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Properties

Label 780.2.h
Level $780$
Weight $2$
Character orbit 780.h
Rep. character $\chi_{780}(469,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $4$
Sturm bound $336$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 780.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(336\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(780, [\chi])\).

Total New Old
Modular forms 180 12 168
Cusp forms 156 12 144
Eisenstein series 24 0 24

Trace form

\( 12 q - 4 q^{5} - 12 q^{9} + O(q^{10}) \) \( 12 q - 4 q^{5} - 12 q^{9} + 8 q^{11} + 4 q^{15} + 8 q^{19} - 8 q^{21} + 4 q^{25} - 16 q^{31} - 24 q^{35} - 8 q^{41} + 4 q^{45} - 4 q^{51} + 4 q^{55} + 40 q^{59} + 20 q^{61} + 4 q^{65} + 4 q^{69} - 24 q^{71} - 28 q^{79} + 12 q^{81} + 24 q^{85} - 40 q^{89} - 12 q^{91} + 24 q^{95} - 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(780, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
780.2.h.a 780.h 5.b $2$ $6.228$ \(\Q(\sqrt{-1}) \) None 780.2.h.a \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(-1-2i)q^{5}+iq^{7}-q^{9}+\cdots\)
780.2.h.b 780.h 5.b $2$ $6.228$ \(\Q(\sqrt{-1}) \) None 780.2.h.b \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(1+2i)q^{5}+3iq^{7}-q^{9}+\cdots\)
780.2.h.c 780.h 5.b $4$ $6.228$ \(\Q(i, \sqrt{6})\) None 780.2.h.c \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+(-1-\beta _{2}+\beta _{3})q^{5}-2\beta _{2}q^{7}+\cdots\)
780.2.h.d 780.h 5.b $4$ $6.228$ \(\Q(\zeta_{8})\) None 780.2.h.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{3}+(-\zeta_{8}+2\zeta_{8}^{3})q^{5}+(2\zeta_{8}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(780, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(780, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(390, [\chi])\)\(^{\oplus 2}\)