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Properties

Label 780.4.a
Level $780$
Weight $4$
Character orbit 780.a
Rep. character $\chi_{780}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $10$
Sturm bound $672$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 780.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(672\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(780))\).

Total New Old
Modular forms 516 24 492
Cusp forms 492 24 468
Eisenstein series 24 0 24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)\(13\)FrickeDim
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(4\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(3\)
\(-\)\(-\)\(+\)\(+\)$+$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(3\)
\(-\)\(-\)\(-\)\(+\)$-$\(2\)
\(-\)\(-\)\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(14\)
Minus space\(-\)\(10\)

Trace form

\( 24 q + 216 q^{9} + O(q^{10}) \) \( 24 q + 216 q^{9} + 52 q^{13} - 120 q^{17} - 80 q^{19} + 16 q^{23} + 600 q^{25} - 344 q^{29} - 416 q^{31} + 72 q^{33} - 80 q^{35} + 888 q^{37} + 144 q^{41} - 224 q^{43} - 320 q^{47} - 400 q^{49} + 72 q^{51} - 264 q^{53} - 120 q^{55} + 456 q^{57} + 2096 q^{59} + 1512 q^{61} + 728 q^{67} - 1632 q^{69} + 3136 q^{71} + 1776 q^{73} - 784 q^{77} - 464 q^{79} + 1944 q^{81} - 672 q^{83} + 1800 q^{85} + 1464 q^{87} + 3888 q^{89} + 312 q^{91} - 72 q^{93} - 80 q^{95} + 1136 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(780))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 13
780.4.a.a 780.a 1.a $1$ $46.021$ \(\Q\) None \(0\) \(-3\) \(-5\) \(-4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}-4q^{7}+9q^{9}+8q^{11}+\cdots\)
780.4.a.b 780.a 1.a $1$ $46.021$ \(\Q\) None \(0\) \(-3\) \(-5\) \(7\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+7q^{7}+9q^{9}-3q^{11}+\cdots\)
780.4.a.c 780.a 1.a $1$ $46.021$ \(\Q\) None \(0\) \(3\) \(5\) \(24\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+24q^{7}+9q^{9}+60q^{11}+\cdots\)
780.4.a.d 780.a 1.a $2$ $46.021$ \(\Q(\sqrt{409}) \) None \(0\) \(6\) \(10\) \(-15\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+(-7-\beta )q^{7}+9q^{9}+\cdots\)
780.4.a.e 780.a 1.a $3$ $46.021$ 3.3.99465.1 None \(0\) \(-9\) \(15\) \(-17\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+(-6+\beta _{1})q^{7}+9q^{9}+\cdots\)
780.4.a.f 780.a 1.a $3$ $46.021$ 3.3.96488.1 None \(0\) \(-9\) \(15\) \(5\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+(2-\beta _{1})q^{7}+9q^{9}+\cdots\)
780.4.a.g 780.a 1.a $3$ $46.021$ 3.3.3302457.1 None \(0\) \(9\) \(-15\) \(-5\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}+(-2-\beta _{2})q^{7}+9q^{9}+\cdots\)
780.4.a.h 780.a 1.a $3$ $46.021$ 3.3.38472.1 None \(0\) \(9\) \(-15\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}+(1-\beta _{1}-\beta _{2})q^{7}+9q^{9}+\cdots\)
780.4.a.i 780.a 1.a $3$ $46.021$ 3.3.233772.1 None \(0\) \(9\) \(15\) \(-5\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+(-2-\beta _{2})q^{7}+9q^{9}+\cdots\)
780.4.a.j 780.a 1.a $4$ $46.021$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(-20\) \(9\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+(2-\beta _{2})q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(780))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(780)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 2}\)