Space of modular forms of level 975, weight 6, and trivial character

• Label: 975.6.a
• Level: $975$
• Weight: $6$
• Character orbit: 975.a
• Rep. character: $\chi_{975}(1,\cdot)$
• Character field: $\Q$
• Dimension: $190$
• Newform subspaces: $25$
• Sturm bound: $840$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(840\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(975))\).

	Total	New	Old
Modular forms	712	190	522
Cusp forms	688	190	498
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.

\(3\)	\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(23\)
\(+\)	\(+\)	\(-\)	\(-\)	\(22\)
\(+\)	\(-\)	\(+\)	\(-\)	\(26\)
\(+\)	\(-\)	\(-\)	\(+\)	\(24\)
\(-\)	\(+\)	\(+\)	\(-\)	\(25\)
\(-\)	\(+\)	\(-\)	\(+\)	\(20\)
\(-\)	\(-\)	\(+\)	\(+\)	\(22\)
\(-\)	\(-\)	\(-\)	\(-\)	\(28\)
Plus space			\(+\)	\(89\)
Minus space			\(-\)	\(101\)

Trace form

190 * q + 4 * q^2 + 3104 * q^4 - 108 * q^6 - 508 * q^7 + 264 * q^8 + 15390 * q^9 + 892 * q^11 - 396 * q^12 - 338 * q^13 - 5080 * q^14 + 49796 * q^16 + 892 * q^17 + 324 * q^18 + 812 * q^19 + 1764 * q^21 - 332 * q^22 - 2648 * q^23 + 2916 * q^24 - 3592 * q^28 + 1668 * q^29 + 9180 * q^31 - 7748 * q^32 - 19620 * q^33 + 8872 * q^34 + 251424 * q^36 + 11284 * q^37 + 54568 * q^38 + 6084 * q^39 - 21548 * q^41 + 29232 * q^42 - 51792 * q^43 - 128876 * q^44 - 13552 * q^46 - 23860 * q^47 - 57456 * q^48 + 417606 * q^49 + 28584 * q^51 + 3380 * q^52 + 56852 * q^53 - 8748 * q^54 - 108368 * q^56 - 3780 * q^57 - 286856 * q^58 - 102364 * q^59 + 131300 * q^61 - 96424 * q^62 - 41148 * q^63 + 1254376 * q^64 + 117252 * q^66 - 124772 * q^67 + 163936 * q^68 - 21528 * q^69 - 186276 * q^71 + 21384 * q^72 + 46340 * q^73 + 309944 * q^74 + 243216 * q^76 + 294072 * q^77 - 12168 * q^78 + 12624 * q^79 + 1246590 * q^81 + 110684 * q^82 + 316964 * q^83 - 179568 * q^84 + 548808 * q^86 + 144216 * q^87 - 477740 * q^88 + 211212 * q^89 + 29068 * q^91 - 202984 * q^92 - 9756 * q^93 - 668380 * q^94 - 263736 * q^96 + 262772 * q^97 + 326084 * q^98 + 72252 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(975))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	13
975.6.a.a	$975$	$6$	975.a	1.a	$1$	$1$	$1$	$156.374$	\(\Q\)	None	✓				39.6.a.a	$1$	$1$	\(-2\)	\(-9\)	\(0\)	\(112\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-9q^{3}-28q^{4}+18q^{6}+112q^{7}+\cdots\)
975.6.a.b	$975$	$6$	975.a	1.a	$1$	$1$	$1$	$156.374$	\(\Q\)	None	✓				195.6.a.a	$1$	$1$	\(-2\)	\(-9\)	\(0\)	\(168\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-9q^{3}-28q^{4}+18q^{6}+168q^{7}+\cdots\)
975.6.a.c	$975$	$6$	975.a	1.a	$1$	$2$	$2$	$156.374$	\(\Q(\sqrt{14}) \)	None	✓				39.6.a.b	$1$	$1$	\(4\)	\(18\)	\(0\)	\(-72\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{2}+9q^{3}+(28+4\beta )q^{4}+(18+\cdots)q^{6}+\cdots\)
975.6.a.d	$975$	$6$	975.a	1.a	$1$	$3$	$3$	$156.374$	3.3.125308.1	None	✓				39.6.a.c	$1$	$0$	\(0\)	\(27\)	\(0\)	\(-84\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+9q^{3}+(5+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.6.a.e	$975$	$6$	975.a	1.a	$1$	$4$	$4$	$156.374$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				39.6.a.d	$1$	$0$	\(-6\)	\(-36\)	\(0\)	\(-72\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}-9q^{3}+(33-3\beta _{1}+\cdots)q^{4}+\cdots\)
975.6.a.f	$975$	$6$	975.a	1.a	$1$	$4$	$4$	$156.374$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				195.6.a.e	$1$	$1$	\(-2\)	\(36\)	\(0\)	\(-87\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+9q^{3}+(14-\beta _{1}+\beta _{3})q^{4}+\cdots\)
975.6.a.g	$975$	$6$	975.a	1.a	$1$	$4$	$4$	$156.374$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				195.6.a.d	$1$	$0$	\(3\)	\(36\)	\(0\)	\(87\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+9q^{3}+(3-3\beta _{1}+2\beta _{3})q^{4}+\cdots\)
975.6.a.h	$975$	$6$	975.a	1.a	$1$	$4$	$4$	$156.374$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				195.6.a.c	$1$	$1$	\(7\)	\(-36\)	\(0\)	\(-73\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}-9q^{3}+(3-2\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
975.6.a.i	$975$	$6$	975.a	1.a	$1$	$4$	$4$	$156.374$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				195.6.a.b	$1$	$0$	\(12\)	\(-36\)	\(0\)	\(73\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}-9q^{3}+(13-3\beta _{1}+2\beta _{3})q^{4}+\cdots\)
975.6.a.j	$975$	$6$	975.a	1.a	$1$	$5$	$5$	$156.374$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				195.6.a.g	$1$	$1$	\(-5\)	\(45\)	\(0\)	\(-164\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+9q^{3}+(8+4\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.6.a.k	$975$	$6$	975.a	1.a	$1$	$5$	$5$	$156.374$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				195.6.a.f	$1$	$0$	\(-1\)	\(-45\)	\(0\)	\(-128\)		$+$	$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-9q^{3}+(7+4\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.6.a.l	$975$	$6$	975.a	1.a	$1$	$6$	$6$	$156.374$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				195.6.a.h	$1$	$1$	\(-2\)	\(-54\)	\(0\)	\(-236\)		$+$	$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-9q^{3}+(35+\beta _{3})q^{4}+9\beta _{1}q^{6}+\cdots\)
975.6.a.m	$975$	$6$	975.a	1.a	$1$	$7$	$7$	$156.374$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				195.6.a.i	$1$	$0$	\(-2\)	\(63\)	\(0\)	\(-32\)		$-$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+9q^{3}+(26-\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.6.a.n	$975$	$6$	975.a	1.a	$1$	$9$	$9$	$156.374$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		975.6.a.n	$1$	$1$	\(-10\)	\(81\)	\(0\)	\(-251\)		$-$	$+$	$-$	$+$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+9q^{3}+(14+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.6.a.o	$975$	$6$	975.a	1.a	$1$	$9$	$9$	$156.374$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			975.6.a.o	$1$	$1$	\(-8\)	\(81\)	\(0\)	\(55\)		$-$	$-$	$+$	$+$	$2^{6}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+9q^{3}+(14-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.6.a.p	$975$	$6$	975.a	1.a	$1$	$9$	$9$	$156.374$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		975.6.a.o	$1$	$0$	\(8\)	\(-81\)	\(0\)	\(-55\)		$+$	$+$	$-$	$-$	$2^{6}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-9q^{3}+(14-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.6.a.q	$975$	$6$	975.a	1.a	$1$	$9$	$9$	$156.374$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			975.6.a.n	$1$	$0$	\(10\)	\(-81\)	\(0\)	\(251\)		$+$	$-$	$+$	$-$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}-9q^{3}+(14+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.6.a.r	$975$	$6$	975.a	1.a	$1$	$11$	$11$	$156.374$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓		975.6.a.r	$1$	$0$	\(-2\)	\(99\)	\(0\)	\(55\)		$-$	$+$	$+$	$-$	$2^{5}\cdot 3\cdot 5^{3}\cdot 11$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+9q^{3}+(20+\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.6.a.s	$975$	$6$	975.a	1.a	$1$	$11$	$11$	$156.374$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓		975.6.a.s	$1$	$1$	\(0\)	\(-99\)	\(0\)	\(-141\)		$+$	$+$	$+$	$+$	$2^{5}\cdot 3^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-9q^{3}+(20+\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.6.a.t	$975$	$6$	975.a	1.a	$1$	$11$	$11$	$156.374$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓			975.6.a.s	$1$	$0$	\(0\)	\(99\)	\(0\)	\(141\)		$-$	$-$	$-$	$-$	$2^{5}\cdot 3^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+9q^{3}+(20+\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.6.a.u	$975$	$6$	975.a	1.a	$1$	$11$	$11$	$156.374$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓			975.6.a.r	$1$	$1$	\(2\)	\(-99\)	\(0\)	\(-55\)		$+$	$-$	$-$	$+$	$2^{5}\cdot 3\cdot 5^{3}\cdot 11$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-9q^{3}+(20+\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.6.a.v	$975$	$6$	975.a	1.a	$1$	$13$	$13$	$156.374$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		195.6.c.a	$1$	$1$	\(-9\)	\(-117\)	\(0\)	\(153\)		$+$	$-$	$-$	$+$	$2^{10}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-9q^{3}+(13-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.6.a.w	$975$	$6$	975.a	1.a	$1$	$13$	$13$	$156.374$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		195.6.c.a	$1$	$1$	\(9\)	\(117\)	\(0\)	\(-153\)		$-$	$-$	$+$	$+$	$2^{10}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+9q^{3}+(13-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
975.6.a.x	$975$	$6$	975.a	1.a	$1$	$17$	$17$	$156.374$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓		✓		195.6.c.b	$1$	$0$	\(-9\)	\(-153\)	\(0\)	\(-349\)		$+$	$-$	$+$	$-$	$2^{12}\cdot 5^{9}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-9q^{3}+(18+\beta _{2})q^{4}+\cdots\)
975.6.a.y	$975$	$6$	975.a	1.a	$1$	$17$	$17$	$156.374$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓		✓		195.6.c.b	$1$	$0$	\(9\)	\(153\)	\(0\)	\(349\)		$-$	$-$	$-$	$-$	$2^{12}\cdot 5^{9}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+9q^{3}+(18+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(975))\) into lower level spaces

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