Properties

Label 418.2.f
Level $418$
Weight $2$
Character orbit 418.f
Rep. character $\chi_{418}(115,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $72$
Newform subspaces $8$
Sturm bound $120$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.f (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 8 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(418, [\chi])\).

Total New Old
Modular forms 256 72 184
Cusp forms 224 72 152
Eisenstein series 32 0 32

Trace form

\( 72 q + 4 q^{3} - 18 q^{4} + 8 q^{5} + 4 q^{6} + 2 q^{7} - 38 q^{9} + O(q^{10}) \) \( 72 q + 4 q^{3} - 18 q^{4} + 8 q^{5} + 4 q^{6} + 2 q^{7} - 38 q^{9} + 8 q^{10} - 6 q^{11} - 16 q^{12} - 12 q^{13} - 28 q^{15} - 18 q^{16} + 10 q^{17} + 16 q^{18} + 2 q^{19} + 8 q^{20} + 56 q^{21} + 4 q^{22} + 20 q^{23} + 4 q^{24} + 6 q^{25} - 24 q^{26} - 20 q^{27} - 8 q^{28} + 16 q^{29} + 8 q^{30} - 48 q^{33} + 16 q^{34} - 30 q^{35} + 2 q^{36} + 12 q^{37} + 48 q^{39} - 12 q^{40} - 4 q^{41} - 36 q^{42} + 20 q^{43} + 4 q^{44} - 20 q^{45} - 24 q^{46} - 32 q^{47} + 4 q^{48} - 16 q^{51} + 8 q^{52} + 36 q^{53} - 64 q^{54} + 18 q^{55} + 4 q^{57} - 40 q^{58} + 20 q^{59} + 32 q^{60} - 38 q^{61} + 40 q^{62} - 42 q^{63} - 18 q^{64} - 56 q^{65} - 4 q^{66} - 136 q^{67} + 10 q^{68} + 72 q^{69} + 12 q^{71} - 24 q^{72} - 16 q^{73} - 4 q^{74} + 124 q^{75} - 8 q^{76} + 68 q^{77} + 8 q^{78} + 96 q^{79} - 2 q^{80} + 30 q^{81} + 28 q^{83} - 24 q^{84} + 30 q^{85} - 4 q^{86} + 120 q^{87} + 4 q^{88} - 28 q^{90} - 36 q^{91} - 20 q^{92} - 48 q^{93} + 32 q^{94} - 6 q^{95} + 4 q^{96} - 76 q^{97} + 64 q^{98} + 22 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(418, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
418.2.f.a 418.f 11.c $4$ $3.338$ \(\Q(\zeta_{10})\) None \(-1\) \(-4\) \(-8\) \(-7\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
418.2.f.b 418.f 11.c $4$ $3.338$ \(\Q(\zeta_{10})\) None \(-1\) \(0\) \(8\) \(3\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
418.2.f.c 418.f 11.c $4$ $3.338$ \(\Q(\zeta_{10})\) None \(-1\) \(1\) \(-1\) \(-3\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
418.2.f.d 418.f 11.c $4$ $3.338$ \(\Q(\zeta_{10})\) None \(-1\) \(2\) \(2\) \(9\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
418.2.f.e 418.f 11.c $4$ $3.338$ \(\Q(\zeta_{10})\) None \(-1\) \(5\) \(3\) \(-7\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
418.2.f.f 418.f 11.c $16$ $3.338$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-4\) \(0\) \(4\) \(6\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{4}q^{2}+(-1+\beta _{1}+\beta _{2}-\beta _{4}+\beta _{6}+\cdots)q^{3}+\cdots\)
418.2.f.g 418.f 11.c $16$ $3.338$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(4\) \(1\) \(1\) \(-12\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1+\beta _{6}-\beta _{8}+\beta _{10})q^{2}+\beta _{1}q^{3}+\cdots\)
418.2.f.h 418.f 11.c $20$ $3.338$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(5\) \(-1\) \(-1\) \(13\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{9}q^{2}-\beta _{4}q^{3}+(-1+\beta _{8}+\beta _{9}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(418, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(418, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(22, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(209, [\chi])\)\(^{\oplus 2}\)