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Properties

Label 350.4.e
Level $350$
Weight $4$
Character orbit 350.e
Rep. character $\chi_{350}(51,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $76$
Newform subspaces $15$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 15 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(350, [\chi])\).

Total New Old
Modular forms 384 76 308
Cusp forms 336 76 260
Eisenstein series 48 0 48

Trace form

\( 76 q + 6 q^{3} - 152 q^{4} + 16 q^{6} - 52 q^{7} - 388 q^{9} - 22 q^{11} + 24 q^{12} - 8 q^{13} - 64 q^{14} - 608 q^{16} + 242 q^{17} - 64 q^{18} - 358 q^{19} - 318 q^{21} + 256 q^{22} - 106 q^{23} - 32 q^{24}+ \cdots - 6448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{4}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.4.e.a 350.e 7.c $2$ $20.651$ \(\Q(\sqrt{-3}) \) None 70.4.e.c \(-2\) \(-1\) \(0\) \(-17\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
350.4.e.b 350.e 7.c $2$ $20.651$ \(\Q(\sqrt{-3}) \) None 14.4.c.b \(-2\) \(-1\) \(0\) \(20\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
350.4.e.c 350.e 7.c $2$ $20.651$ \(\Q(\sqrt{-3}) \) None 350.4.e.c \(-2\) \(4\) \(0\) \(-35\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(4-4\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
350.4.e.d 350.e 7.c $2$ $20.651$ \(\Q(\sqrt{-3}) \) None 70.4.e.b \(-2\) \(10\) \(0\) \(-28\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(10-10\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
350.4.e.e 350.e 7.c $2$ $20.651$ \(\Q(\sqrt{-3}) \) None 14.4.c.a \(2\) \(-5\) \(0\) \(28\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-5+5\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
350.4.e.f 350.e 7.c $2$ $20.651$ \(\Q(\sqrt{-3}) \) None 350.4.e.c \(2\) \(-4\) \(0\) \(35\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
350.4.e.g 350.e 7.c $2$ $20.651$ \(\Q(\sqrt{-3}) \) None 70.4.e.a \(2\) \(1\) \(0\) \(-35\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
350.4.e.h 350.e 7.c $4$ $20.651$ \(\Q(\sqrt{-3}, \sqrt{46})\) None 70.4.e.d \(-4\) \(-2\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{2}q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+(-4+\cdots)q^{4}+\cdots\)
350.4.e.i 350.e 7.c $6$ $20.651$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 350.4.e.i \(-6\) \(-3\) \(0\) \(56\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{2}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{3}+\cdots\)
350.4.e.j 350.e 7.c $6$ $20.651$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 350.4.e.i \(6\) \(3\) \(0\) \(-56\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{2})q^{2}+(\beta _{1}+\beta _{2})q^{3}-4\beta _{2}q^{4}+\cdots\)
350.4.e.k 350.e 7.c $6$ $20.651$ 6.0.\(\cdots\).2 None 70.4.e.e \(6\) \(4\) \(0\) \(-14\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{3})q^{2}+(\beta _{1}+\beta _{3})q^{3}-4\beta _{3}q^{4}+\cdots\)
350.4.e.l 350.e 7.c $8$ $20.651$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 350.4.e.l \(-8\) \(1\) \(0\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{2}q^{2}+(-\beta _{3}-\beta _{5})q^{3}+(-4-4\beta _{2}+\cdots)q^{4}+\cdots\)
350.4.e.m 350.e 7.c $8$ $20.651$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 350.4.e.l \(8\) \(-1\) \(0\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2+2\beta _{2})q^{2}-\beta _{3}q^{3}+4\beta _{2}q^{4}+2\beta _{5}q^{6}+\cdots\)
350.4.e.n 350.e 7.c $12$ $20.651$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 70.4.i.a \(-12\) \(-7\) \(0\) \(-9\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{2}q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+(-4+\cdots)q^{4}+\cdots\)
350.4.e.o 350.e 7.c $12$ $20.651$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 70.4.i.a \(12\) \(7\) \(0\) \(9\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{2}q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+(-4+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(350, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)