Properties

Label 350.2.e
Level $350$
Weight $2$
Character orbit 350.e
Rep. character $\chi_{350}(51,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $24$
Newform subspaces $12$
Sturm bound $120$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(120\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\), \(17\)

Dimensions

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

Total New Old
Modular forms 144 24 120
Cusp forms 96 24 72
Eisenstein series 48 0 48

Trace form

\( 24 q - 12 q^{4} + 8 q^{7} - 8 q^{9} + O(q^{10}) \) \( 24 q - 12 q^{4} + 8 q^{7} - 8 q^{9} - 4 q^{14} - 12 q^{16} - 4 q^{17} + 8 q^{18} + 4 q^{19} + 4 q^{21} - 8 q^{22} + 12 q^{23} + 8 q^{26} - 24 q^{27} - 4 q^{28} - 8 q^{29} - 20 q^{31} - 24 q^{33} - 16 q^{34} + 16 q^{36} - 4 q^{37} - 8 q^{38} + 8 q^{39} - 24 q^{41} - 4 q^{42} + 40 q^{43} - 8 q^{46} + 20 q^{47} - 12 q^{51} + 4 q^{53} + 12 q^{54} - 4 q^{56} + 24 q^{57} + 4 q^{59} + 32 q^{61} + 48 q^{62} - 36 q^{63} + 24 q^{64} - 4 q^{68} + 48 q^{69} - 24 q^{71} + 8 q^{72} - 28 q^{73} + 20 q^{74} - 8 q^{76} + 8 q^{77} + 16 q^{78} + 12 q^{79} - 20 q^{81} - 16 q^{82} + 32 q^{83} + 28 q^{84} - 8 q^{86} + 24 q^{87} + 4 q^{88} - 8 q^{89} + 28 q^{91} - 24 q^{92} - 12 q^{93} + 4 q^{94} - 64 q^{97} - 24 q^{98} + O(q^{100}) \)

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.e.a 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(-1\) \(-2\) \(0\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.b 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(-1\) \(-2\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.c 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
350.2.e.d 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(2-3\zeta_{6})q^{7}+\cdots\)
350.2.e.e 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.f 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(-1\) \(3\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.g 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(0\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.h 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(1\) \(-2\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.i 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-2+3\zeta_{6})q^{7}+\cdots\)
350.2.e.j 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(3-2\zeta_{6})q^{7}+\cdots\)
350.2.e.k 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(1\) \(2\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.l 350.e 7.c $2$ $2.795$ \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)