Properties

Label 315.8.a
Level $315$
Weight $8$
Character orbit 315.a
Rep. character $\chi_{315}(1,\cdot)$
Character field $\Q$
Dimension $70$
Newform subspaces $17$
Sturm bound $384$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(384\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(315))\).

Total New Old
Modular forms 344 70 274
Cusp forms 328 70 258
Eisenstein series 16 0 16

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

\(3\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(10\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(12\)
\(-\)\(-\)\(-\)\(-\)\(9\)
Plus space\(+\)\(37\)
Minus space\(-\)\(33\)

Trace form

\( 70 q - 30 q^{2} + 4714 q^{4} + 686 q^{7} + 906 q^{8} + O(q^{10}) \) \( 70 q - 30 q^{2} + 4714 q^{4} + 686 q^{7} + 906 q^{8} - 6500 q^{10} + 5110 q^{11} - 9572 q^{13} - 8918 q^{14} + 308898 q^{16} - 62612 q^{17} + 33720 q^{19} - 184700 q^{22} - 47624 q^{23} + 1093750 q^{25} - 125404 q^{26} + 107702 q^{28} + 131186 q^{29} - 530052 q^{31} + 306778 q^{32} + 1001784 q^{34} - 171500 q^{35} + 495264 q^{37} + 100804 q^{38} - 1365000 q^{40} + 2349068 q^{41} + 2095844 q^{43} - 1647960 q^{44} - 2073176 q^{46} - 3620 q^{47} + 8235430 q^{49} - 468750 q^{50} + 5834120 q^{52} - 2337772 q^{53} - 811000 q^{55} - 1524978 q^{56} + 2851352 q^{58} - 6506252 q^{59} - 3382656 q^{61} + 22740672 q^{62} + 13580954 q^{64} - 2150250 q^{65} + 1684700 q^{67} - 17493108 q^{68} + 686000 q^{70} - 672024 q^{71} + 17383592 q^{73} + 872308 q^{74} - 959548 q^{76} + 2304960 q^{77} - 6066738 q^{79} - 15120000 q^{80} + 1520820 q^{82} - 15755040 q^{83} + 8672750 q^{85} + 47422128 q^{86} + 662336 q^{88} + 26929096 q^{89} - 11956294 q^{91} + 1016960 q^{92} + 5191388 q^{94} - 27587500 q^{95} - 33125748 q^{97} - 3529470 q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(315))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 7
315.8.a.a 315.a 1.a $1$ $98.401$ \(\Q\) None 105.8.a.b \(-18\) \(0\) \(125\) \(343\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-18q^{2}+14^{2}q^{4}+5^{3}q^{5}+7^{3}q^{7}+\cdots\)
315.8.a.b 315.a 1.a $1$ $98.401$ \(\Q\) None 105.8.a.a \(2\) \(0\) \(125\) \(343\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-124q^{4}+5^{3}q^{5}+7^{3}q^{7}+\cdots\)
315.8.a.c 315.a 1.a $2$ $98.401$ \(\Q(\sqrt{11}) \) None 35.8.a.a \(-16\) \(0\) \(-250\) \(-686\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-8+\beta )q^{2}+(-20-2^{4}\beta )q^{4}-5^{3}q^{5}+\cdots\)
315.8.a.d 315.a 1.a $2$ $98.401$ \(\Q(\sqrt{2}) \) None 105.8.a.c \(12\) \(0\) \(-250\) \(686\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(6+2\beta )q^{2}+(6^{2}+24\beta )q^{4}-5^{3}q^{5}+\cdots\)
315.8.a.e 315.a 1.a $3$ $98.401$ 3.3.2268428.1 None 35.8.a.b \(23\) \(0\) \(375\) \(1029\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(8-\beta _{1})q^{2}+(80-8\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.f 315.a 1.a $4$ $98.401$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 105.8.a.i \(-25\) \(0\) \(500\) \(1372\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-6-\beta _{1})q^{2}+(2^{5}+12\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
315.8.a.g 315.a 1.a $4$ $98.401$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 105.8.a.h \(-11\) \(0\) \(-500\) \(1372\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{2}+(6^{2}+5\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.h 315.a 1.a $4$ $98.401$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 105.8.a.g \(-4\) \(0\) \(500\) \(-1372\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(26+\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.i 315.a 1.a $4$ $98.401$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 105.8.a.f \(1\) \(0\) \(-500\) \(-1372\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(105+\beta _{2})q^{4}-5^{3}q^{5}-7^{3}q^{7}+\cdots\)
315.8.a.j 315.a 1.a $4$ $98.401$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 35.8.a.c \(2\) \(0\) \(500\) \(-1372\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(69-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.k 315.a 1.a $4$ $98.401$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 105.8.a.e \(5\) \(0\) \(500\) \(-1372\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(105+2\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.8.a.l 315.a 1.a $4$ $98.401$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 105.8.a.d \(10\) \(0\) \(-500\) \(-1372\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{2}+(29-4\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.m 315.a 1.a $5$ $98.401$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 35.8.a.d \(-11\) \(0\) \(-625\) \(1715\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{2}+(111-\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.8.a.n 315.a 1.a $6$ $98.401$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 315.8.a.n \(-21\) \(0\) \(750\) \(-2058\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-4+\beta _{1})q^{2}+(77-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.o 315.a 1.a $6$ $98.401$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 315.8.a.n \(21\) \(0\) \(-750\) \(-2058\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(4-\beta _{1})q^{2}+(77-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.p 315.a 1.a $8$ $98.401$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 315.8.a.p \(-5\) \(0\) \(1000\) \(2744\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(3^{4}+\beta _{2})q^{4}+5^{3}q^{5}+\cdots\)
315.8.a.q 315.a 1.a $8$ $98.401$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 315.8.a.p \(5\) \(0\) \(-1000\) \(2744\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(3^{4}+\beta _{2})q^{4}-5^{3}q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(315)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)