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    Choi, SoYoung 2013. CONGRUENCE PROPERTIES OF COEFFICIENTS OF MODULAR FORMS FOR Γ+0(5). Journal of the Chungcheng Mathematical Society, Vol. 26, Issue. 4, p. 923.

    Choi, D. and Choie, Y. 2010. p-Adic limit of the Fourier coefficients of weakly holomorphic modular forms of half integral weight. Israel Journal of Mathematics, Vol. 175, Issue. 1, p. 61.

    Choi, SoYoung 2008. Linear relations and congruences for the coefficients of Drinfeld modular forms. Israel Journal of Mathematics, Vol. 165, Issue. 1, p. 93.

    KAZALICKI, MATIJA 2008. LINEAR RELATIONS FOR COEFFICIENTS OF DRINFELD MODULAR FORMS. International Journal of Number Theory, Vol. 04, Issue. 02, p. 171.

    KOHNEN, WINFRIED LAU, YUK-KAM and SHPARLINSKI, IGOR E. 2008. ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS. Journal of the Australian Mathematical Society, Vol. 85, Issue. 01, p. 87.

    Lachterman, Samuel Schayer, Rhiannon and Younger, Brendan 2008. A new proof of the Ramanujan congruences for the partition function. The Ramanujan Journal, Vol. 15, Issue. 2, p. 197.

    Choi, D. and Choie, Y. 2007. Weight-dependent congruence properties of modular forms. Journal of Number Theory, Vol. 122, Issue. 2, p. 301.

    Choi, D. and Choie, Y. 2007. Linear relations among the Fourier coefficients of modular forms on groups <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="" xmlns:xs="" xmlns:xsi="" xmlns="" xmlns:ja="" xmlns:mml="" xmlns:tb="" xmlns:sb="" xmlns:ce="" xmlns:xlink="" xmlns:cals=""><mml:msub><mml:mi>Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> of genus zero and their applications. Journal of Mathematical Analysis and Applications, Vol. 326, Issue. 1, p. 655.

    EL-GUINDY, AHMAD 2007. LINEAR CONGRUENCES AND RELATIONS ON SPACES OF CUSP FORMS. International Journal of Number Theory, Vol. 03, Issue. 04, p. 529.

    Kohnen, Winfried and Sengupta, Jyoti 2006. On the first sign change of Hecke eigenvalues of newforms. Mathematische Zeitschrift, Vol. 254, Issue. 1, p. 173.

  • Bulletin of the London Mathematical Society, Volume 37, Issue 3
  • June 2005, pp. 335-341


  • DOI:
  • Published online: 01 June 2005

Here, a classical observation of Siegel is generalized by determining all the linear relations among the initial Fourier coefficients of a modular form on $\SL_2(\ZZ)$. As a consequence, spaces $M_k$ are identified, in which there are universal $p$-divisibility properties for certain $p$-power coefficients. As a corollary, let $f(z)=\sum_{n=1}^{\infty}a_f(n)q^n \in S_k\cap O_{L}[[q]]$ be a normalized Hecke eigenform (note that $q:=e^{2\pi i z}$), and let $k\equiv \delta(k)\pmod{12}$, where $\delta(k)\in \{4, 6, 8, 10, 14\}$. Reproducing earlier results of Hatada and Hida, if $p$ is a prime for which $k\equiv \delta(k)\pmod{p-1}$, and $\mathfrak{p}\subset O_L$ is a prime ideal above $p$, a proof is given to show that $a_f(p)\equiv 0\pmod{\mathfrak{p}}$.

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