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Properties

Label 400.3.bg
Level $400$
Weight $3$
Character orbit 400.bg
Rep. character $\chi_{400}(17,\cdot)$
Character field $\Q(\zeta_{20})$
Dimension $232$
Newform subspaces $6$
Sturm bound $180$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.bg (of order \(20\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{20})\)
Newform subspaces: \( 6 \)
Sturm bound: \(180\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(400, [\chi])\).

Total New Old
Modular forms 1008 248 760
Cusp forms 912 232 680
Eisenstein series 96 16 80

Trace form

\( 232 q + 8 q^{3} - 8 q^{5} + 8 q^{7} - 10 q^{9} + O(q^{10}) \) \( 232 q + 8 q^{3} - 8 q^{5} + 8 q^{7} - 10 q^{9} + 6 q^{11} + 4 q^{13} - 40 q^{15} - 12 q^{17} + 10 q^{19} - 6 q^{21} + 56 q^{23} + 4 q^{25} + 122 q^{27} - 10 q^{29} + 6 q^{31} + 58 q^{33} + 56 q^{35} + 12 q^{37} + 10 q^{39} - 6 q^{41} - 56 q^{43} - 118 q^{45} - 232 q^{47} + 16 q^{51} - 36 q^{53} - 138 q^{55} + 6 q^{57} + 10 q^{59} - 6 q^{61} + 232 q^{63} - 132 q^{65} + 344 q^{67} - 10 q^{69} + 6 q^{71} - 180 q^{73} + 312 q^{75} + 90 q^{77} + 10 q^{79} + 372 q^{81} + 696 q^{83} - 568 q^{85} + 954 q^{87} + 490 q^{89} + 6 q^{91} - 374 q^{93} + 250 q^{95} + 92 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(400, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.3.bg.a 400.bg 25.f $16$ $10.899$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(-2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{20}]$ \(q+(-1-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{8}-\beta _{9}+\cdots)q^{3}+\cdots\)
400.3.bg.b 400.bg 25.f $24$ $10.899$ None \(0\) \(-2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{20}]$
400.3.bg.c 400.bg 25.f $32$ $10.899$ None \(0\) \(10\) \(-10\) \(10\) $\mathrm{SU}(2)[C_{20}]$
400.3.bg.d 400.bg 25.f $40$ $10.899$ None \(0\) \(2\) \(6\) \(-14\) $\mathrm{SU}(2)[C_{20}]$
400.3.bg.e 400.bg 25.f $56$ $10.899$ None \(0\) \(0\) \(-10\) \(4\) $\mathrm{SU}(2)[C_{20}]$
400.3.bg.f 400.bg 25.f $64$ $10.899$ None \(0\) \(0\) \(6\) \(4\) $\mathrm{SU}(2)[C_{20}]$

Decomposition of \(S_{3}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(400, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)