↑

Properties

Label 975.4.a
Level $975$
Weight $4$
Character orbit 975.a
Rep. character $\chi_{975}(1,\cdot)$
Character field $\Q$
Dimension $114$
Newform subspaces $29$
Sturm bound $560$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 29 \)
Sturm bound: \(560\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(975))\).

Total New Old
Modular forms 432 114 318
Cusp forms 408 114 294
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.

\(3\)\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)$+$\(13\)
\(+\)\(+\)\(-\)$-$\(14\)
\(+\)\(-\)\(+\)$-$\(14\)
\(+\)\(-\)\(-\)$+$\(16\)
\(-\)\(+\)\(+\)$-$\(11\)
\(-\)\(+\)\(-\)$+$\(16\)
\(-\)\(-\)\(+\)$+$\(18\)
\(-\)\(-\)\(-\)$-$\(12\)
Plus space\(+\)\(63\)
Minus space\(-\)\(51\)

Trace form

\( 114 q + 4 q^{2} + 424 q^{4} + 24 q^{7} - 120 q^{8} + 1026 q^{9} + O(q^{10}) \) \( 114 q + 4 q^{2} + 424 q^{4} + 24 q^{7} - 120 q^{8} + 1026 q^{9} + 16 q^{11} - 12 q^{12} + 26 q^{13} + 32 q^{14} + 1796 q^{16} + 196 q^{17} + 36 q^{18} - 136 q^{19} - 84 q^{21} - 172 q^{22} - 536 q^{23} + 180 q^{24} + 328 q^{28} - 60 q^{29} + 112 q^{31} - 1316 q^{32} + 264 q^{33} + 40 q^{34} + 3816 q^{36} + 420 q^{37} - 368 q^{38} - 156 q^{39} + 424 q^{41} + 456 q^{42} - 1040 q^{43} + 1780 q^{44} + 2272 q^{46} + 368 q^{47} + 432 q^{48} + 5986 q^{49} + 456 q^{51} + 780 q^{52} + 260 q^{53} + 2320 q^{56} + 540 q^{57} + 2072 q^{58} + 1232 q^{59} - 252 q^{61} + 1328 q^{62} + 216 q^{63} + 7496 q^{64} - 492 q^{66} + 3048 q^{67} + 2704 q^{68} - 840 q^{69} - 624 q^{71} - 1080 q^{72} + 380 q^{73} - 4888 q^{74} - 1248 q^{76} - 3432 q^{77} + 156 q^{78} + 2512 q^{79} + 9234 q^{81} + 404 q^{82} + 4112 q^{83} + 2976 q^{84} - 7560 q^{86} + 624 q^{87} + 1076 q^{88} - 2496 q^{89} - 416 q^{91} - 6280 q^{92} - 1500 q^{93} + 9988 q^{94} + 6120 q^{96} - 3348 q^{97} + 7748 q^{98} + 144 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(975))\) 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 13
975.4.a.a 975.a 1.a $1$ $57.527$ \(\Q\) None \(-4\) \(-3\) \(0\) \(-18\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-3q^{3}+8q^{4}+12q^{6}-18q^{7}+\cdots\)
975.4.a.b 975.a 1.a $1$ $57.527$ \(\Q\) None \(-3\) \(3\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+3q^{3}+q^{4}-9q^{6}+4q^{7}+\cdots\)
975.4.a.c 975.a 1.a $1$ $57.527$ \(\Q\) None \(0\) \(-3\) \(0\) \(17\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{4}+17q^{7}+9q^{9}+39q^{11}+\cdots\)
975.4.a.d 975.a 1.a $1$ $57.527$ \(\Q\) None \(0\) \(3\) \(0\) \(-17\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-8q^{4}-17q^{7}+9q^{9}+39q^{11}+\cdots\)
975.4.a.e 975.a 1.a $1$ $57.527$ \(\Q\) None \(0\) \(3\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-8q^{4}-2q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
975.4.a.f 975.a 1.a $1$ $57.527$ \(\Q\) None \(3\) \(-3\) \(0\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}-3q^{3}+q^{4}-9q^{6}-4q^{7}+\cdots\)
975.4.a.g 975.a 1.a $1$ $57.527$ \(\Q\) None \(3\) \(-3\) \(0\) \(16\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}-3q^{3}+q^{4}-9q^{6}+2^{4}q^{7}+\cdots\)
975.4.a.h 975.a 1.a $1$ $57.527$ \(\Q\) None \(3\) \(3\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}+9q^{6}-2q^{7}+\cdots\)
975.4.a.i 975.a 1.a $1$ $57.527$ \(\Q\) None \(5\) \(-3\) \(0\) \(-8\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{2}-3q^{3}+17q^{4}-15q^{6}-8q^{7}+\cdots\)
975.4.a.j 975.a 1.a $2$ $57.527$ \(\Q(\sqrt{14}) \) None \(-2\) \(6\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+3q^{3}+(7-2\beta )q^{4}+\cdots\)
975.4.a.k 975.a 1.a $2$ $57.527$ \(\Q(\sqrt{17}) \) None \(-1\) \(-6\) \(0\) \(20\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-3q^{3}+(-4+\beta )q^{4}+3\beta q^{6}+\cdots\)
975.4.a.l 975.a 1.a $3$ $57.527$ 3.3.3144.1 None \(-2\) \(-9\) \(0\) \(-30\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(3+\beta _{2})q^{4}+\cdots\)
975.4.a.m 975.a 1.a $3$ $57.527$ 3.3.48812.1 None \(-1\) \(-9\) \(0\) \(12\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(8+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.4.a.n 975.a 1.a $3$ $57.527$ 3.3.1016.1 None \(2\) \(9\) \(0\) \(30\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(3-\beta _{1}+2\beta _{2})q^{4}+\cdots\)
975.4.a.o 975.a 1.a $4$ $57.527$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-2\) \(12\) \(0\) \(11\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.4.a.p 975.a 1.a $4$ $57.527$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-2\) \(12\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.4.a.q 975.a 1.a $4$ $57.527$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(0\) \(6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.4.a.r 975.a 1.a $4$ $57.527$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(2\) \(-12\) \(0\) \(-11\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.4.a.s 975.a 1.a $4$ $57.527$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(3\) \(12\) \(0\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(8+\beta _{3})q^{4}+(3+\cdots)q^{6}+\cdots\)
975.4.a.t 975.a 1.a $5$ $57.527$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-15\) \(0\) \(22\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(1+\beta _{1}+\beta _{3}-\beta _{4})q^{4}+\cdots\)
975.4.a.u 975.a 1.a $5$ $57.527$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(15\) \(0\) \(-22\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(1+\beta _{1}+\beta _{3}-\beta _{4})q^{4}+\cdots\)
975.4.a.v 975.a 1.a $6$ $57.527$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-18\) \(0\) \(52\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}-3q^{3}+(4-\beta _{2}-\beta _{3})q^{4}+\cdots\)
975.4.a.w 975.a 1.a $6$ $57.527$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(18\) \(0\) \(-52\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+3q^{3}+(4-\beta _{2}-\beta _{3})q^{4}+\cdots\)
975.4.a.x 975.a 1.a $7$ $57.527$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-4\) \(-21\) \(0\) \(6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(5-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.4.a.y 975.a 1.a $7$ $57.527$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-2\) \(-21\) \(0\) \(-34\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
975.4.a.z 975.a 1.a $7$ $57.527$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(2\) \(21\) \(0\) \(34\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
975.4.a.ba 975.a 1.a $7$ $57.527$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(4\) \(21\) \(0\) \(-6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(5-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.4.a.bb 975.a 1.a $11$ $57.527$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-3\) \(33\) \(0\) \(20\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(5+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
975.4.a.bc 975.a 1.a $11$ $57.527$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(3\) \(-33\) \(0\) \(-20\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(975))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(975)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 2}\)