Properties

Label 315.6.a
Level $315$
Weight $6$
Character orbit 315.a
Rep. character $\chi_{315}(1,\cdot)$
Character field $\Q$
Dimension $50$
Newform subspaces $16$
Sturm bound $288$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(288\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(315))\).

Total New Old
Modular forms 248 50 198
Cusp forms 232 50 182
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
\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(4\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(7\)
\(-\)\(-\)\(+\)$+$\(6\)
\(-\)\(-\)\(-\)$-$\(9\)
Plus space\(+\)\(23\)
Minus space\(-\)\(27\)

Trace form

\( 50 q + 10 q^{2} + 842 q^{4} - 98 q^{7} - 582 q^{8} + O(q^{10}) \) \( 50 q + 10 q^{2} + 842 q^{4} - 98 q^{7} - 582 q^{8} - 100 q^{10} + 974 q^{11} + 404 q^{13} - 1078 q^{14} + 14946 q^{16} - 2524 q^{17} - 3864 q^{19} - 460 q^{22} + 7448 q^{23} + 31250 q^{25} + 14436 q^{26} - 1274 q^{28} + 2390 q^{29} + 13404 q^{31} - 43254 q^{32} + 12552 q^{34} + 4900 q^{35} + 3168 q^{37} + 44692 q^{38} + 3000 q^{40} - 47132 q^{41} - 27068 q^{43} + 15864 q^{44} + 101480 q^{46} - 16420 q^{47} + 120050 q^{49} + 6250 q^{50} - 16760 q^{52} + 88604 q^{53} - 27800 q^{55} - 29106 q^{56} - 102344 q^{58} - 55276 q^{59} + 44160 q^{61} + 135744 q^{62} + 365242 q^{64} + 55850 q^{65} - 122660 q^{67} + 64396 q^{68} - 9800 q^{70} - 172936 q^{71} - 238424 q^{73} + 129876 q^{74} - 84380 q^{76} + 54880 q^{77} + 262926 q^{79} + 32000 q^{80} + 321060 q^{82} + 314320 q^{83} - 170150 q^{85} - 408000 q^{86} - 382832 q^{88} - 20568 q^{89} - 38318 q^{91} + 293760 q^{92} + 819772 q^{94} + 146500 q^{95} + 319524 q^{97} + 24010 q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(315))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 7
315.6.a.a 315.a 1.a $1$ $50.521$ \(\Q\) None \(8\) \(0\) \(-25\) \(49\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{5}q^{4}-5^{2}q^{5}+7^{2}q^{7}-200q^{10}+\cdots\)
315.6.a.b 315.a 1.a $2$ $50.521$ \(\Q(\sqrt{65}) \) None \(-3\) \(0\) \(-50\) \(98\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(-15+3\beta )q^{4}-5^{2}q^{5}+\cdots\)
315.6.a.c 315.a 1.a $2$ $50.521$ \(\Q(\sqrt{65}) \) None \(-1\) \(0\) \(50\) \(-98\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-2^{4}+\beta )q^{4}+5^{2}q^{5}-7^{2}q^{7}+\cdots\)
315.6.a.d 315.a 1.a $2$ $50.521$ \(\Q(\sqrt{233}) \) None \(-1\) \(0\) \(50\) \(-98\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(26+\beta )q^{4}+5^{2}q^{5}-7^{2}q^{7}+\cdots\)
315.6.a.e 315.a 1.a $2$ $50.521$ \(\Q(\sqrt{73}) \) None \(1\) \(0\) \(50\) \(98\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-14+\beta )q^{4}+5^{2}q^{5}+7^{2}q^{7}+\cdots\)
315.6.a.f 315.a 1.a $2$ $50.521$ \(\Q(\sqrt{2}) \) None \(4\) \(0\) \(-50\) \(-98\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{2}+(4+4\beta )q^{4}-5^{2}q^{5}-7^{2}q^{7}+\cdots\)
315.6.a.g 315.a 1.a $2$ $50.521$ \(\Q(\sqrt{5}) \) None \(8\) \(0\) \(50\) \(-98\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(4+\beta )q^{2}+(4+8\beta )q^{4}+5^{2}q^{5}-7^{2}q^{7}+\cdots\)
315.6.a.h 315.a 1.a $2$ $50.521$ \(\Q(\sqrt{65}) \) None \(13\) \(0\) \(-50\) \(-98\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(7-\beta )q^{2}+(33-13\beta )q^{4}-5^{2}q^{5}+\cdots\)
315.6.a.i 315.a 1.a $3$ $50.521$ 3.3.577880.1 None \(6\) \(0\) \(75\) \(147\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+(38+2\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)
315.6.a.j 315.a 1.a $4$ $50.521$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-10\) \(0\) \(-100\) \(196\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{2}+(30-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.6.a.k 315.a 1.a $4$ $50.521$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(0\) \(100\) \(196\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+(29-\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.6.a.l 315.a 1.a $4$ $50.521$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-7\) \(0\) \(-100\) \(-196\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+(13-4\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.6.a.m 315.a 1.a $4$ $50.521$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-5\) \(0\) \(100\) \(196\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(11+2\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.6.a.n 315.a 1.a $4$ $50.521$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(5\) \(0\) \(-100\) \(196\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(11+2\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.6.a.o 315.a 1.a $6$ $50.521$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-3\) \(0\) \(-150\) \(-294\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(5^{2}+\beta _{2})q^{4}-5^{2}q^{5}+\cdots\)
315.6.a.p 315.a 1.a $6$ $50.521$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(3\) \(0\) \(150\) \(-294\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(5^{2}+\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces

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