Properties

Label 414.2.i
Level $414$
Weight $2$
Character orbit 414.i
Rep. character $\chi_{414}(55,\cdot)$
Character field $\Q(\zeta_{11})$
Dimension $100$
Newform subspaces $8$
Sturm bound $144$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.i (of order \(11\) and degree \(10\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 23 \)
Character field: \(\Q(\zeta_{11})\)
Newform subspaces: \( 8 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(414, [\chi])\).

Total New Old
Modular forms 800 100 700
Cusp forms 640 100 540
Eisenstein series 160 0 160

Trace form

\( 100 q - 10 q^{4} + 2 q^{5} - 4 q^{7} + O(q^{10}) \) \( 100 q - 10 q^{4} + 2 q^{5} - 4 q^{7} - 2 q^{10} - 10 q^{11} + 4 q^{13} + 4 q^{14} - 10 q^{16} + 14 q^{17} + 20 q^{19} + 24 q^{20} + 28 q^{22} + 44 q^{23} + 26 q^{25} + 22 q^{26} + 18 q^{28} + 22 q^{29} + 6 q^{31} + 12 q^{34} - 10 q^{35} - 46 q^{37} - 10 q^{38} - 2 q^{40} + 38 q^{41} - 50 q^{43} - 10 q^{44} + 2 q^{46} + 52 q^{47} + 56 q^{49} - 4 q^{50} + 4 q^{52} + 40 q^{53} - 72 q^{55} - 18 q^{56} - 24 q^{58} - 58 q^{59} - 94 q^{61} - 74 q^{62} - 10 q^{64} - 142 q^{65} - 22 q^{67} - 52 q^{68} - 88 q^{70} - 110 q^{71} - 28 q^{73} - 54 q^{74} - 2 q^{76} - 86 q^{77} - 38 q^{79} - 20 q^{80} - 16 q^{82} - 10 q^{83} + 38 q^{85} - 28 q^{86} + 6 q^{88} + 34 q^{89} + 68 q^{91} + 74 q^{95} + 80 q^{97} + 60 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(414, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
414.2.i.a 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(-1\) \(0\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{11}]$ \(q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(-1-\zeta_{22}^{2}+\cdots)q^{5}+\cdots\)
414.2.i.b 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(-1\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{11}]$ \(q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(1-2\zeta_{22}+\cdots)q^{5}+\cdots\)
414.2.i.c 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(-1\) \(0\) \(4\) \(-7\) $\mathrm{SU}(2)[C_{11}]$ \(q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(\zeta_{22}+\zeta_{22}^{3}+\cdots)q^{5}+\cdots\)
414.2.i.d 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(1\) \(0\) \(-8\) \(8\) $\mathrm{SU}(2)[C_{11}]$ \(q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(-1+\zeta_{22}^{2}+\cdots)q^{5}+\cdots\)
414.2.i.e 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(1\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{11}]$ \(q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(1-2\zeta_{22}+\cdots)q^{5}+\cdots\)
414.2.i.f 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(1\) \(0\) \(6\) \(3\) $\mathrm{SU}(2)[C_{11}]$ \(q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(\zeta_{22}-2\zeta_{22}^{2}+\cdots)q^{5}+\cdots\)
414.2.i.g 414.i 23.c $20$ $3.306$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(-2\) \(0\) \(-2\) \(-4\) $\mathrm{SU}(2)[C_{11}]$ \(q+\beta _{3}q^{2}+\beta _{7}q^{4}+(\beta _{2}-\beta _{5}+\beta _{13}+\cdots)q^{5}+\cdots\)
414.2.i.h 414.i 23.c $20$ $3.306$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(2\) \(0\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{11}]$ \(q-\beta _{3}q^{2}+\beta _{7}q^{4}+(-\beta _{2}+\beta _{5}-\beta _{13}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(414, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(414, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(23, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(46, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(69, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(138, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(207, [\chi])\)\(^{\oplus 2}\)