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Properties

Label 350.6.c
Level $350$
Weight $6$
Character orbit 350.c
Rep. character $\chi_{350}(99,\cdot)$
Character field $\Q$
Dimension $44$
Newform subspaces $14$
Sturm bound $360$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(360\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(350, [\chi])\).

Total New Old
Modular forms 312 44 268
Cusp forms 288 44 244
Eisenstein series 24 0 24

Trace form

\( 44 q - 704 q^{4} - 3432 q^{9} + O(q^{10}) \) \( 44 q - 704 q^{4} - 3432 q^{9} - 1244 q^{11} + 784 q^{14} + 11264 q^{16} - 7024 q^{19} - 3724 q^{21} - 9552 q^{26} + 2912 q^{29} + 27416 q^{31} + 880 q^{34} + 54912 q^{36} - 39352 q^{39} + 43148 q^{41} + 19904 q^{44} - 45376 q^{46} - 105644 q^{49} - 83068 q^{51} - 71856 q^{54} - 12544 q^{56} + 172012 q^{59} + 106548 q^{61} - 180224 q^{64} + 110480 q^{66} + 401744 q^{69} - 86952 q^{71} + 121472 q^{74} + 112384 q^{76} + 53320 q^{79} + 222036 q^{81} + 59584 q^{84} + 22240 q^{86} - 113964 q^{89} - 97804 q^{91} - 34560 q^{94} - 835056 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.6.c.a 350.c 5.b $2$ $56.134$ \(\Q(\sqrt{-1}) \) None 70.6.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{2}+17iq^{3}-2^{4}q^{4}-68q^{6}+\cdots\)
350.6.c.b 350.c 5.b $2$ $56.134$ \(\Q(\sqrt{-1}) \) None 70.6.a.f \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{2}+11iq^{3}-2^{4}q^{4}-44q^{6}+\cdots\)
350.6.c.c 350.c 5.b $2$ $56.134$ \(\Q(\sqrt{-1}) \) None 70.6.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{2}+11iq^{3}-2^{4}q^{4}-44q^{6}+\cdots\)
350.6.c.d 350.c 5.b $2$ $56.134$ \(\Q(\sqrt{-1}) \) None 14.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{2}+10iq^{3}-2^{4}q^{4}-40q^{6}+\cdots\)
350.6.c.e 350.c 5.b $2$ $56.134$ \(\Q(\sqrt{-1}) \) None 70.6.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4iq^{2}+3iq^{3}-2^{4}q^{4}+12q^{6}+\cdots\)
350.6.c.f 350.c 5.b $2$ $56.134$ \(\Q(\sqrt{-1}) \) None 14.6.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4iq^{2}+8iq^{3}-2^{4}q^{4}+2^{5}q^{6}+\cdots\)
350.6.c.g 350.c 5.b $2$ $56.134$ \(\Q(\sqrt{-1}) \) None 70.6.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4iq^{2}+9iq^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
350.6.c.h 350.c 5.b $2$ $56.134$ \(\Q(\sqrt{-1}) \) None 70.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4iq^{2}+23iq^{3}-2^{4}q^{4}+92q^{6}+\cdots\)
350.6.c.i 350.c 5.b $4$ $56.134$ \(\Q(i, \sqrt{79})\) None 350.6.a.r \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}+(-10\beta _{1}+\beta _{3})q^{3}-2^{4}q^{4}+\cdots\)
350.6.c.j 350.c 5.b $4$ $56.134$ \(\Q(i, \sqrt{3369})\) None 70.6.a.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{2}q^{2}+(\beta _{1}-\beta _{2})q^{3}-2^{4}q^{4}+(8+\cdots)q^{6}+\cdots\)
350.6.c.k 350.c 5.b $4$ $56.134$ \(\Q(i, \sqrt{1129})\) None 70.6.a.h \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{2}q^{2}+(\beta _{1}-2\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
350.6.c.l 350.c 5.b $4$ $56.134$ \(\Q(i, \sqrt{79})\) None 350.6.a.o \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{1}q^{2}+(-4\beta _{1}-\beta _{3})q^{3}-2^{4}q^{4}+\cdots\)
350.6.c.m 350.c 5.b $6$ $56.134$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 350.6.a.v \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{2}q^{2}-\beta _{4}q^{3}-2^{4}q^{4}-4\beta _{1}q^{6}+\cdots\)
350.6.c.n 350.c 5.b $6$ $56.134$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 350.6.a.u \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{2}q^{2}+(\beta _{1}+3\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(350, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)