Properties

Label 350.2.h
Level $350$
Weight $2$
Character orbit 350.h
Rep. character $\chi_{350}(71,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $56$
Newform subspaces $4$
Sturm bound $120$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.h (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(350, [\chi])\).

Total New Old
Modular forms 256 56 200
Cusp forms 224 56 168
Eisenstein series 32 0 32

Trace form

\( 56 q + 2 q^{2} + 8 q^{3} - 14 q^{4} - 14 q^{5} + 2 q^{8} - 2 q^{9} + O(q^{10}) \) \( 56 q + 2 q^{2} + 8 q^{3} - 14 q^{4} - 14 q^{5} + 2 q^{8} - 2 q^{9} + 6 q^{10} + 4 q^{11} - 12 q^{12} + 28 q^{13} + 4 q^{14} - 14 q^{16} + 4 q^{17} - 32 q^{18} - 24 q^{19} - 4 q^{20} + 4 q^{21} - 12 q^{22} + 4 q^{23} - 66 q^{25} + 12 q^{26} - 28 q^{27} + 12 q^{29} + 24 q^{31} - 8 q^{32} + 8 q^{33} - 30 q^{34} + 4 q^{35} - 2 q^{36} + 14 q^{37} + 24 q^{38} - 12 q^{39} + 6 q^{40} + 12 q^{41} + 24 q^{43} + 4 q^{44} + 2 q^{45} - 4 q^{46} - 76 q^{47} - 12 q^{48} + 56 q^{49} + 14 q^{50} + 8 q^{51} + 28 q^{52} + 14 q^{53} + 24 q^{54} - 36 q^{55} + 4 q^{56} + 88 q^{57} + 12 q^{58} + 12 q^{59} - 28 q^{61} + 16 q^{62} - 12 q^{63} - 14 q^{64} - 46 q^{65} - 52 q^{67} + 4 q^{68} - 36 q^{69} + 4 q^{70} + 32 q^{71} + 18 q^{72} - 12 q^{73} - 68 q^{74} + 80 q^{75} + 16 q^{76} - 24 q^{77} - 88 q^{78} - 40 q^{79} + 6 q^{80} - 6 q^{81} + 52 q^{82} + 28 q^{83} + 4 q^{84} - 18 q^{85} + 20 q^{86} - 144 q^{87} + 8 q^{88} + 46 q^{89} + 62 q^{90} + 4 q^{91} - 16 q^{92} + 32 q^{93} + 60 q^{95} + 56 q^{97} + 2 q^{98} + 64 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.2.h.a 350.h 25.d $8$ $2.795$ \(\Q(\zeta_{15})\) None \(-2\) \(3\) \(0\) \(8\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{15}^{4}q^{2}+(-\zeta_{15}^{3}-\zeta_{15}^{4}+\zeta_{15}^{6}+\cdots)q^{3}+\cdots\)
350.2.h.b 350.h 25.d $12$ $2.795$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(3\) \(1\) \(-5\) \(-12\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{5}q^{2}+(\beta _{1}-\beta _{3}+\beta _{7})q^{3}-\beta _{3}q^{4}+\cdots\)
350.2.h.c 350.h 25.d $16$ $2.795$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(-4\) \(1\) \(-4\) \(-16\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{6}q^{2}-\beta _{10}q^{3}-\beta _{1}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
350.2.h.d 350.h 25.d $20$ $2.795$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(5\) \(3\) \(-5\) \(20\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{14}q^{2}+\beta _{1}q^{3}+\beta _{6}q^{4}+(\beta _{10}+\beta _{13}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)