Properties

Label 315.4.a
Level $315$
Weight $4$
Character orbit 315.a
Rep. character $\chi_{315}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $16$
Sturm bound $192$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(192\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(315))\).

Total New Old
Modular forms 152 30 122
Cusp forms 136 30 106
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
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(17\)
Minus space\(-\)\(13\)

Trace form

\( 30 q - 2 q^{2} + 98 q^{4} + 14 q^{7} - 54 q^{8} + O(q^{10}) \) \( 30 q - 2 q^{2} + 98 q^{4} + 14 q^{7} - 54 q^{8} + 60 q^{10} + 116 q^{11} + 16 q^{13} - 70 q^{14} + 466 q^{16} + 260 q^{17} + 196 q^{19} + 36 q^{22} - 400 q^{23} + 750 q^{25} + 468 q^{26} + 70 q^{28} - 16 q^{29} - 432 q^{31} - 54 q^{32} + 608 q^{34} - 140 q^{35} - 116 q^{37} + 748 q^{38} + 600 q^{40} - 356 q^{41} - 872 q^{43} + 1944 q^{44} - 1064 q^{46} + 272 q^{47} + 1470 q^{49} - 50 q^{50} + 648 q^{52} - 1900 q^{53} + 360 q^{55} - 546 q^{56} - 928 q^{58} + 2276 q^{59} - 1024 q^{61} + 1680 q^{62} + 3178 q^{64} - 40 q^{65} - 1456 q^{67} + 604 q^{68} + 140 q^{70} - 40 q^{71} + 1548 q^{73} - 2772 q^{74} + 2228 q^{76} - 1568 q^{77} + 564 q^{79} + 3200 q^{80} - 6516 q^{82} + 484 q^{83} + 940 q^{85} - 1728 q^{86} - 1024 q^{88} + 4236 q^{89} + 1092 q^{91} - 4896 q^{92} - 2356 q^{94} - 500 q^{95} + 124 q^{97} - 98 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\) 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
315.4.a.a 315.a 1.a $1$ $18.586$ \(\Q\) None 105.4.a.b \(-5\) \(0\) \(-5\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-5q^{2}+17q^{4}-5q^{5}+7q^{7}-45q^{8}+\cdots\)
315.4.a.b 315.a 1.a $1$ $18.586$ \(\Q\) None 315.4.a.b \(-3\) \(0\) \(-5\) \(7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}-5q^{5}+7q^{7}+21q^{8}+\cdots\)
315.4.a.c 315.a 1.a $1$ $18.586$ \(\Q\) None 35.4.a.a \(-1\) \(0\) \(5\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}+5q^{5}+7q^{7}+15q^{8}+\cdots\)
315.4.a.d 315.a 1.a $1$ $18.586$ \(\Q\) None 105.4.a.a \(0\) \(0\) \(-5\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{4}-5q^{5}+7q^{7}-42q^{11}+20q^{13}+\cdots\)
315.4.a.e 315.a 1.a $1$ $18.586$ \(\Q\) None 315.4.a.b \(3\) \(0\) \(5\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}+5q^{5}+7q^{7}-21q^{8}+\cdots\)
315.4.a.f 315.a 1.a $2$ $18.586$ \(\Q(\sqrt{2}) \) None 35.4.a.b \(-8\) \(0\) \(10\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-4+\beta )q^{2}+(10-8\beta )q^{4}+5q^{5}+\cdots\)
315.4.a.g 315.a 1.a $2$ $18.586$ \(\Q(\sqrt{41}) \) None 105.4.a.g \(-3\) \(0\) \(10\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(3+3\beta )q^{4}+5q^{5}+\cdots\)
315.4.a.h 315.a 1.a $2$ $18.586$ \(\Q(\sqrt{17}) \) None 315.4.a.h \(-1\) \(0\) \(-10\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-4+\beta )q^{4}-5q^{5}-7q^{7}+\cdots\)
315.4.a.i 315.a 1.a $2$ $18.586$ \(\Q(\sqrt{65}) \) None 105.4.a.f \(-1\) \(0\) \(-10\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(8+\beta )q^{4}-5q^{5}-7q^{7}+(-2^{4}+\cdots)q^{8}+\cdots\)
315.4.a.j 315.a 1.a $2$ $18.586$ \(\Q(\sqrt{17}) \) None 315.4.a.h \(1\) \(0\) \(10\) \(-14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-4+\beta )q^{4}+5q^{5}-7q^{7}+\cdots\)
315.4.a.k 315.a 1.a $2$ $18.586$ \(\Q(\sqrt{2}) \) None 105.4.a.e \(2\) \(0\) \(-10\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-5q^{5}-7q^{7}+\cdots\)
315.4.a.l 315.a 1.a $2$ $18.586$ \(\Q(\sqrt{5}) \) None 105.4.a.d \(4\) \(0\) \(10\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{2}+(1+4\beta )q^{4}+5q^{5}-7q^{7}+\cdots\)
315.4.a.m 315.a 1.a $2$ $18.586$ \(\Q(\sqrt{17}) \) None 105.4.a.c \(7\) \(0\) \(10\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(4-\beta )q^{2}+(12-7\beta )q^{4}+5q^{5}-7q^{7}+\cdots\)
315.4.a.n 315.a 1.a $3$ $18.586$ 3.3.22952.1 None 315.4.a.n \(-2\) \(0\) \(-15\) \(21\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.4.a.o 315.a 1.a $3$ $18.586$ 3.3.22952.1 None 315.4.a.n \(2\) \(0\) \(15\) \(21\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)
315.4.a.p 315.a 1.a $3$ $18.586$ 3.3.14360.1 None 35.4.a.c \(3\) \(0\) \(-15\) \(21\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}-5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces

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