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Properties

Label 403.2.k
Level $403$
Weight $2$
Character orbit 403.k
Rep. character $\chi_{403}(66,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $128$
Newform subspaces $5$
Sturm bound $74$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.k (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 31 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 5 \)
Sturm bound: \(74\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(403, [\chi])\).

Total New Old
Modular forms 152 128 24
Cusp forms 136 128 8
Eisenstein series 16 0 16

Trace form

\( 128 q - 4 q^{3} - 40 q^{4} - 4 q^{5} + 8 q^{6} + 12 q^{7} - 6 q^{8} - 36 q^{9} + O(q^{10}) \) \( 128 q - 4 q^{3} - 40 q^{4} - 4 q^{5} + 8 q^{6} + 12 q^{7} - 6 q^{8} - 36 q^{9} - 22 q^{10} - 8 q^{11} - 16 q^{12} - 2 q^{13} - 26 q^{15} - 16 q^{16} + 6 q^{17} + 24 q^{19} - 22 q^{20} - 20 q^{21} - 8 q^{22} - 10 q^{23} + 4 q^{24} + 72 q^{25} + 24 q^{26} + 38 q^{27} - 30 q^{28} + 4 q^{29} - 68 q^{30} + 24 q^{31} + 8 q^{32} + 2 q^{33} - 48 q^{34} - 32 q^{35} + 40 q^{36} + 40 q^{37} - 42 q^{38} - 12 q^{40} - 8 q^{41} + 26 q^{42} - 28 q^{43} + 18 q^{44} + 44 q^{45} + 8 q^{46} + 36 q^{47} + 16 q^{48} + 22 q^{49} - 66 q^{50} + 26 q^{51} - 6 q^{52} - 10 q^{53} + 42 q^{54} - 28 q^{55} - 48 q^{56} + 2 q^{58} + 24 q^{59} - 50 q^{60} - 28 q^{61} + 4 q^{62} - 60 q^{63} - 20 q^{64} - 2 q^{65} + 2 q^{66} + 68 q^{67} + 16 q^{68} - 28 q^{69} + 4 q^{70} + 186 q^{72} - 22 q^{74} + 12 q^{75} - 110 q^{76} - 4 q^{77} - 68 q^{79} + 60 q^{80} - 42 q^{81} + 52 q^{82} - 36 q^{83} + 40 q^{84} - 40 q^{85} - 12 q^{86} - 40 q^{87} + 64 q^{88} - 58 q^{89} - 110 q^{90} - 8 q^{91} - 32 q^{92} + 76 q^{93} + 112 q^{94} + 18 q^{95} - 68 q^{96} - 46 q^{97} - 36 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(403, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
403.2.k.a 403.k 31.d $4$ $3.218$ \(\Q(\zeta_{10})\) None \(-2\) \(-1\) \(-6\) \(-7\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2})q^{2}+(1-2\zeta_{10}+\cdots)q^{3}+\cdots\)
403.2.k.b 403.k 31.d $4$ $3.218$ \(\Q(\zeta_{10})\) None \(-1\) \(-2\) \(-4\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q-\zeta_{10}q^{2}+(-2+2\zeta_{10}-2\zeta_{10}^{2}+\cdots)q^{3}+\cdots\)
403.2.k.c 403.k 31.d $4$ $3.218$ \(\Q(\zeta_{10})\) None \(-1\) \(3\) \(6\) \(-9\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+2\zeta_{10}-\zeta_{10}^{2})q^{2}+(1-\zeta_{10}^{3})q^{3}+\cdots\)
403.2.k.d 403.k 31.d $48$ $3.218$ None \(7\) \(-2\) \(-12\) \(25\) $\mathrm{SU}(2)[C_{5}]$
403.2.k.e 403.k 31.d $68$ $3.218$ None \(-3\) \(-2\) \(12\) \(2\) $\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{2}^{\mathrm{old}}(403, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(403, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(31, [\chi])\)\(^{\oplus 2}\)