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Properties

Label 350.8.c
Level $350$
Weight $8$
Character orbit 350.c
Rep. character $\chi_{350}(99,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $15$
Sturm bound $480$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(480\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(350, [\chi])\).

Total New Old
Modular forms 432 64 368
Cusp forms 408 64 344
Eisenstein series 24 0 24

Trace form

\( 64 q - 4096 q^{4} - 55908 q^{9} + O(q^{10}) \) \( 64 q - 4096 q^{4} - 55908 q^{9} + 17140 q^{11} - 10976 q^{14} + 262144 q^{16} + 56656 q^{19} - 75460 q^{21} - 142368 q^{26} + 499608 q^{29} - 940904 q^{31} - 605728 q^{34} + 3578112 q^{36} - 190744 q^{39} + 421436 q^{41} - 1096960 q^{44} + 2364928 q^{46} - 7529536 q^{49} + 2858660 q^{51} - 1899360 q^{54} + 702464 q^{56} + 14006444 q^{59} - 1848636 q^{61} - 16777216 q^{64} - 15662368 q^{66} - 7572208 q^{69} + 878456 q^{71} - 25512128 q^{74} - 3625984 q^{76} + 14309288 q^{79} + 40188984 q^{81} + 4829440 q^{84} + 13031104 q^{86} + 23540900 q^{89} + 19963972 q^{91} + 12282624 q^{94} + 13400640 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.8.c.a 350.c 5.b $2$ $109.335$ \(\Q(\sqrt{-1}) \) None 70.8.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8iq^{2}+93iq^{3}-2^{6}q^{4}-744q^{6}+\cdots\)
350.8.c.b 350.c 5.b $2$ $109.335$ \(\Q(\sqrt{-1}) \) None 14.8.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8iq^{2}+66iq^{3}-2^{6}q^{4}-528q^{6}+\cdots\)
350.8.c.c 350.c 5.b $2$ $109.335$ \(\Q(\sqrt{-1}) \) None 70.8.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8iq^{2}+9iq^{3}-2^{6}q^{4}+72q^{6}+\cdots\)
350.8.c.d 350.c 5.b $2$ $109.335$ \(\Q(\sqrt{-1}) \) None 14.8.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8iq^{2}+82iq^{3}-2^{6}q^{4}+656q^{6}+\cdots\)
350.8.c.e 350.c 5.b $4$ $109.335$ \(\Q(i, \sqrt{1401})\) None 70.8.a.f \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta _{2}q^{2}+(\beta _{1}+23\beta _{2})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.f 350.c 5.b $4$ $109.335$ \(\mathbb{Q}[x]/(x^{4} + \cdots)\) None 70.8.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8\beta _{2}q^{2}+(\beta _{1}-15\beta _{2})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.g 350.c 5.b $4$ $109.335$ \(\mathbb{Q}[x]/(x^{4} + \cdots)\) None 70.8.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8\beta _{2}q^{2}+(\beta _{1}-12\beta _{2})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.h 350.c 5.b $4$ $109.335$ \(\Q(i, \sqrt{9241})\) None 70.8.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8\beta _{2}q^{2}+(\beta _{1}-5\beta _{2})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.i 350.c 5.b $4$ $109.335$ \(\Q(i, \sqrt{8761})\) None 70.8.a.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8\beta _{2}q^{2}+(\beta _{1}-2\beta _{2})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.j 350.c 5.b $4$ $109.335$ \(\Q(i, \sqrt{12121})\) None 70.8.a.h \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta _{2}q^{2}+(\beta _{1}-14\beta _{2})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.k 350.c 5.b $4$ $109.335$ \(\Q(i, \sqrt{1969})\) None 14.8.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta _{1}q^{2}+(-35\beta _{1}+\beta _{2})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.l 350.c 5.b $6$ $109.335$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 350.8.a.s \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8\beta _{1}q^{2}+(-18\beta _{1}+\beta _{4})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.m 350.c 5.b $6$ $109.335$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 350.8.a.r \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta _{3}q^{2}+(\beta _{1}-28\beta _{3})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.n 350.c 5.b $8$ $109.335$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 350.8.a.w \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8\beta _{3}q^{2}+(\beta _{1}+3\beta _{3})q^{3}-2^{6}q^{4}+\cdots\)
350.8.c.o 350.c 5.b $8$ $109.335$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 350.8.a.v \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta _{2}q^{2}+(-10\beta _{2}+\beta _{6})q^{3}-2^{6}q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

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