Properties

Label 780.2.a
Level $780$
Weight $2$
Character orbit 780.a
Rep. character $\chi_{780}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $6$
Sturm bound $336$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 780.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(336\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(780))\).

Total New Old
Modular forms 180 8 172
Cusp forms 157 8 149
Eisenstein series 23 0 23

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
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)\(+\)$+$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(1\)
\(-\)\(-\)\(+\)\(+\)$+$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(1\)
\(-\)\(-\)\(-\)\(+\)$-$\(2\)
Plus space\(+\)\(2\)
Minus space\(-\)\(6\)

Trace form

\( 8 q + 8 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{9} - 4 q^{13} + 8 q^{19} + 8 q^{23} + 8 q^{25} + 8 q^{29} + 16 q^{31} + 8 q^{35} - 8 q^{37} + 16 q^{43} + 32 q^{47} + 12 q^{49} + 12 q^{51} + 4 q^{55} - 8 q^{57} + 16 q^{59} - 4 q^{61} - 4 q^{69} - 16 q^{71} - 24 q^{73} - 8 q^{77} + 12 q^{79} + 8 q^{81} + 8 q^{87} + 4 q^{91} - 24 q^{93} - 16 q^{95} - 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(780))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 13
780.2.a.a 780.a 1.a $1$ $6.228$ \(\Q\) None 780.2.a.a \(0\) \(-1\) \(1\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
780.2.a.b 780.a 1.a $1$ $6.228$ \(\Q\) None 780.2.a.b \(0\) \(-1\) \(1\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+3q^{7}+q^{9}+q^{11}+q^{13}+\cdots\)
780.2.a.c 780.a 1.a $1$ $6.228$ \(\Q\) None 780.2.a.c \(0\) \(1\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}-6q^{11}-q^{13}+\cdots\)
780.2.a.d 780.a 1.a $1$ $6.228$ \(\Q\) None 780.2.a.d \(0\) \(1\) \(-1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}+3q^{11}+q^{13}+\cdots\)
780.2.a.e 780.a 1.a $2$ $6.228$ \(\Q(\sqrt{73}) \) None 780.2.a.e \(0\) \(-2\) \(-2\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-\beta q^{7}+q^{9}+\beta q^{11}-q^{13}+\cdots\)
780.2.a.f 780.a 1.a $2$ $6.228$ \(\Q(\sqrt{17}) \) None 780.2.a.f \(0\) \(2\) \(2\) \(3\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+(1+\beta )q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(780))\) into lower level spaces

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