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Properties

Label 350.8.a
Level $350$
Weight $8$
Character orbit 350.a
Rep. character $\chi_{350}(1,\cdot)$
Character field $\Q$
Dimension $66$
Newform subspaces $27$
Sturm bound $480$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 27 \)
Sturm bound: \(480\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(350))\).

Total New Old
Modular forms 432 66 366
Cusp forms 408 66 342
Eisenstein series 24 0 24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(8\)
\(+\)\(+\)\(-\)\(-\)\(7\)
\(+\)\(-\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(-\)\(+\)\(10\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(9\)
\(-\)\(-\)\(+\)\(+\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(36\)
Minus space\(-\)\(30\)

Trace form

\( 66 q + 26 q^{3} + 4224 q^{4} + 688 q^{6} + 40074 q^{9} + O(q^{10}) \) \( 66 q + 26 q^{3} + 4224 q^{4} + 688 q^{6} + 40074 q^{9} + 5520 q^{11} + 1664 q^{12} - 15410 q^{13} - 5488 q^{14} + 270336 q^{16} + 42064 q^{17} - 10528 q^{18} + 3558 q^{19} - 74774 q^{21} - 52128 q^{22} + 71520 q^{23} + 44032 q^{24} + 66000 q^{26} - 15508 q^{27} - 590776 q^{29} + 616388 q^{31} - 126000 q^{33} - 344000 q^{34} + 2564736 q^{36} + 334176 q^{37} + 370256 q^{38} - 1126088 q^{39} + 222612 q^{41} + 148176 q^{42} - 652556 q^{43} + 353280 q^{44} - 1255360 q^{46} + 262468 q^{47} + 106496 q^{48} + 7764834 q^{49} + 5522240 q^{51} - 986240 q^{52} - 1106692 q^{53} - 182528 q^{54} - 351232 q^{56} + 3297948 q^{57} + 2873824 q^{58} + 796682 q^{59} + 7993390 q^{61} - 1095264 q^{62} - 2841412 q^{63} + 17301504 q^{64} - 19030944 q^{66} + 4167872 q^{67} + 2692096 q^{68} + 19739040 q^{69} + 10063536 q^{71} - 673792 q^{72} + 5821708 q^{73} + 9733920 q^{74} + 227712 q^{76} + 5664988 q^{77} - 6097856 q^{78} - 5836864 q^{79} - 1543122 q^{81} - 6424352 q^{82} + 6363814 q^{83} - 4785536 q^{84} + 142816 q^{86} - 14454764 q^{87} - 3336192 q^{88} - 20703952 q^{89} + 10827138 q^{91} + 4577280 q^{92} - 13065544 q^{93} - 8504352 q^{94} + 2818048 q^{96} + 5287456 q^{97} + 26161484 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(350))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 7
350.8.a.a 350.a 1.a $1$ $109.335$ \(\Q\) None 70.8.a.b \(-8\) \(-9\) \(0\) \(343\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}-9q^{3}+2^{6}q^{4}+72q^{6}+7^{3}q^{7}+\cdots\)
350.8.a.b 350.a 1.a $1$ $109.335$ \(\Q\) None 70.8.c.b \(-8\) \(3\) \(0\) \(-343\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+3q^{3}+2^{6}q^{4}-24q^{6}-7^{3}q^{7}+\cdots\)
350.8.a.c 350.a 1.a $1$ $109.335$ \(\Q\) None 70.8.c.a \(-8\) \(63\) \(0\) \(-343\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+63q^{3}+2^{6}q^{4}-504q^{6}+\cdots\)
350.8.a.d 350.a 1.a $1$ $109.335$ \(\Q\) None 14.8.a.b \(-8\) \(66\) \(0\) \(343\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+66q^{3}+2^{6}q^{4}-528q^{6}+\cdots\)
350.8.a.e 350.a 1.a $1$ $109.335$ \(\Q\) None 70.8.a.a \(-8\) \(93\) \(0\) \(-343\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+93q^{3}+2^{6}q^{4}-744q^{6}+\cdots\)
350.8.a.f 350.a 1.a $1$ $109.335$ \(\Q\) None 70.8.c.a \(8\) \(-63\) \(0\) \(343\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-63q^{3}+2^{6}q^{4}-504q^{6}+\cdots\)
350.8.a.g 350.a 1.a $1$ $109.335$ \(\Q\) None 70.8.c.b \(8\) \(-3\) \(0\) \(343\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-3q^{3}+2^{6}q^{4}-24q^{6}+7^{3}q^{7}+\cdots\)
350.8.a.h 350.a 1.a $1$ $109.335$ \(\Q\) None 14.8.a.a \(8\) \(82\) \(0\) \(343\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+82q^{3}+2^{6}q^{4}+656q^{6}+\cdots\)
350.8.a.i 350.a 1.a $2$ $109.335$ \(\Q(\sqrt{214}) \) None 70.8.c.c \(-16\) \(-80\) \(0\) \(-686\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+(-40+\beta )q^{3}+2^{6}q^{4}+(320+\cdots)q^{6}+\cdots\)
350.8.a.j 350.a 1.a $2$ $109.335$ \(\Q(\sqrt{1969}) \) None 14.8.a.c \(-16\) \(-70\) \(0\) \(-686\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+(-35-\beta )q^{3}+2^{6}q^{4}+(280+\cdots)q^{6}+\cdots\)
350.8.a.k 350.a 1.a $2$ $109.335$ \(\Q(\sqrt{12121}) \) None 70.8.a.h \(-16\) \(-29\) \(0\) \(686\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+(-14-\beta )q^{3}+2^{6}q^{4}+(112+\cdots)q^{6}+\cdots\)
350.8.a.l 350.a 1.a $2$ $109.335$ \(\Q(\sqrt{8761}) \) None 70.8.a.g \(-16\) \(5\) \(0\) \(-686\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+(3-\beta )q^{3}+2^{6}q^{4}+(-24+\cdots)q^{6}+\cdots\)
350.8.a.m 350.a 1.a $2$ $109.335$ \(\Q(\sqrt{1401}) \) None 70.8.a.f \(16\) \(-45\) \(0\) \(686\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+(-22-\beta )q^{3}+2^{6}q^{4}+(-176+\cdots)q^{6}+\cdots\)
350.8.a.n 350.a 1.a $2$ $109.335$ \(\Q(\sqrt{18481}) \) None 70.8.a.e \(16\) \(-31\) \(0\) \(-686\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+(-15-\beta )q^{3}+2^{6}q^{4}+(-120+\cdots)q^{6}+\cdots\)
350.8.a.o 350.a 1.a $2$ $109.335$ \(\Q(\sqrt{11761}) \) None 70.8.a.d \(16\) \(-25\) \(0\) \(686\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+(-12-\beta )q^{3}+2^{6}q^{4}+(-96+\cdots)q^{6}+\cdots\)
350.8.a.p 350.a 1.a $2$ $109.335$ \(\Q(\sqrt{9241}) \) None 70.8.a.c \(16\) \(-11\) \(0\) \(-686\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+(-5-\beta )q^{3}+2^{6}q^{4}+(-40+\cdots)q^{6}+\cdots\)
350.8.a.q 350.a 1.a $2$ $109.335$ \(\Q(\sqrt{214}) \) None 70.8.c.c \(16\) \(80\) \(0\) \(686\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+(40+\beta )q^{3}+2^{6}q^{4}+(320+\cdots)q^{6}+\cdots\)
350.8.a.r 350.a 1.a $3$ $109.335$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 350.8.a.r \(-24\) \(-83\) \(0\) \(1029\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+(-28+\beta _{1})q^{3}+2^{6}q^{4}+(224+\cdots)q^{6}+\cdots\)
350.8.a.s 350.a 1.a $3$ $109.335$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 350.8.a.s \(-24\) \(53\) \(0\) \(-1029\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+(18+\beta _{1})q^{3}+2^{6}q^{4}+(-12^{2}+\cdots)q^{6}+\cdots\)
350.8.a.t 350.a 1.a $3$ $109.335$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 350.8.a.s \(24\) \(-53\) \(0\) \(1029\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+(-18-\beta _{1})q^{3}+2^{6}q^{4}+(-12^{2}+\cdots)q^{6}+\cdots\)
350.8.a.u 350.a 1.a $3$ $109.335$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 350.8.a.r \(24\) \(83\) \(0\) \(-1029\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+(28-\beta _{1})q^{3}+2^{6}q^{4}+(224+\cdots)q^{6}+\cdots\)
350.8.a.v 350.a 1.a $4$ $109.335$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 350.8.a.v \(-32\) \(-42\) \(0\) \(1372\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+(-10+\beta _{1})q^{3}+2^{6}q^{4}+(80+\cdots)q^{6}+\cdots\)
350.8.a.w 350.a 1.a $4$ $109.335$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 350.8.a.w \(-32\) \(-14\) \(0\) \(-1372\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+(-4+\beta _{1})q^{3}+2^{6}q^{4}+(2^{5}+\cdots)q^{6}+\cdots\)
350.8.a.x 350.a 1.a $4$ $109.335$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 350.8.a.w \(32\) \(14\) \(0\) \(1372\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+(4-\beta _{1})q^{3}+2^{6}q^{4}+(2^{5}-8\beta _{1}+\cdots)q^{6}+\cdots\)
350.8.a.y 350.a 1.a $4$ $109.335$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 350.8.a.v \(32\) \(42\) \(0\) \(-1372\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+(10-\beta _{1})q^{3}+2^{6}q^{4}+(80+\cdots)q^{6}+\cdots\)
350.8.a.z 350.a 1.a $6$ $109.335$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 70.8.c.d \(-48\) \(14\) \(0\) \(2058\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+(2+\beta _{1})q^{3}+2^{6}q^{4}+(-2^{4}+\cdots)q^{6}+\cdots\)
350.8.a.ba 350.a 1.a $6$ $109.335$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 70.8.c.d \(48\) \(-14\) \(0\) \(-2058\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+(-2-\beta _{1})q^{3}+2^{6}q^{4}+(-2^{4}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(350))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(350)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)