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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$		$85$	$23$	$62$		$82$	$23$	$59$		$3$	$0$	$3$
$+$	$+$	$-$	$-$		$91$	$22$	$69$		$88$	$22$	$66$		$3$	$0$	$3$
$+$	$-$	$+$	$-$		$91$	$26$	$65$		$88$	$26$	$62$		$3$	$0$	$3$
$+$	$-$	$-$	$+$		$89$	$24$	$65$		$86$	$24$	$62$		$3$	$0$	$3$
$-$	$+$	$+$	$-$		$93$	$25$	$68$		$90$	$25$	$65$		$3$	$0$	$3$
$-$	$+$	$-$	$+$		$87$	$20$	$67$		$84$	$20$	$64$		$3$	$0$	$3$
$-$	$-$	$+$	$+$		$87$	$22$	$65$		$84$	$22$	$62$		$3$	$0$	$3$
$-$	$-$	$-$	$-$		$89$	$28$	$61$		$86$	$28$	$58$		$3$	$0$	$3$
Plus space			$+$		$348$	$89$	$259$		$336$	$89$	$247$		$12$	$0$	$12$
Minus space			$-$		$364$	$101$	$263$		$352$	$101$	$251$		$12$	$0$	$12$

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}$