Space of modular forms of level 912, weight 6, and trivial character

• Label: 912.6.a
• Level: $912$
• Weight: $6$
• Character orbit: 912.a
• Rep. character: $\chi_{912}(1,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $27$
• Sturm bound: $960$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$912 = 2^{4} \cdot 3 \cdot 19$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	912.a (trivial)
Character field:			$\Q$
Newform subspaces:			$27$
Sturm bound:			$960$
Trace bound:			$5$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{6}(\Gamma_0(912))$.

	Total	New	Old
Modular forms	812	90	722
Cusp forms	788	90	698
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.

$2$	$3$	$19$	Fricke	Dim
$+$	$+$	$+$	$+$	$12$
$+$	$+$	$-$	$-$	$10$
$+$	$-$	$+$	$-$	$13$
$+$	$-$	$-$	$+$	$9$
$-$	$+$	$+$	$-$	$12$
$-$	$+$	$-$	$+$	$11$
$-$	$-$	$+$	$+$	$11$
$-$	$-$	$-$	$-$	$12$
Plus space			$+$	$43$
Minus space			$-$	$47$

Trace form

90 * q - 76 * q^5 + 196 * q^7 + 7290 * q^9 - 1208 * q^11 + 244 * q^13 - 404 * q^17 - 2166 * q^19 + 5500 * q^23 + 62094 * q^25 + 420 * q^29 - 9432 * q^33 + 16788 * q^35 + 25060 * q^37 + 12168 * q^39 + 2476 * q^41 + 10540 * q^43 - 6156 * q^45 - 5016 * q^47 + 197698 * q^49 + 89700 * q^53 - 6636 * q^55 + 118984 * q^59 - 18316 * q^61 + 15876 * q^63 - 86904 * q^65 + 12232 * q^67 - 15448 * q^71 + 31748 * q^73 - 96624 * q^75 + 14896 * q^77 + 132976 * q^79 + 590490 * q^81 - 104172 * q^83 + 100968 * q^85 + 90828 * q^87 + 175580 * q^89 - 229384 * q^91 - 146492 * q^97 - 97848 * q^99

Decomposition of $S_{6}^{\mathrm{new}}(\Gamma_0(912))$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	19
912.6.a.a	$912$	$6$	912.a	1.a	$1$	$1$	$1$	$146.270$	$\Q$	None	✓				57.6.a.a	$1$	$1$	$0$	$-9$	$-98$	$-240$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}-98q^{5}-240q^{7}+3^{4}q^{9}+\cdots$
912.6.a.b	$912$	$6$	912.a	1.a	$1$	$1$	$1$	$146.270$	$\Q$	None	✓				114.6.a.d	$1$	$1$	$0$	$-9$	$-91$	$33$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}-91q^{5}+33q^{7}+3^{4}q^{9}+91q^{11}+\cdots$
912.6.a.c	$912$	$6$	912.a	1.a	$1$	$1$	$1$	$146.270$	$\Q$	None	✓				114.6.a.a	$1$	$0$	$0$	$-9$	$-54$	$-104$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}-54q^{5}-104q^{7}+3^{4}q^{9}+\cdots$
912.6.a.d	$912$	$6$	912.a	1.a	$1$	$1$	$1$	$146.270$	$\Q$	None	✓				57.6.a.b	$1$	$1$	$0$	$-9$	$6$	$176$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}+6q^{5}+176q^{7}+3^{4}q^{9}+496q^{11}+\cdots$
912.6.a.e	$912$	$6$	912.a	1.a	$1$	$1$	$1$	$146.270$	$\Q$	None	✓				114.6.a.b	$1$	$0$	$0$	$-9$	$81$	$247$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}+3^{4}q^{5}+247q^{7}+3^{4}q^{9}+\cdots$
912.6.a.f	$912$	$6$	912.a	1.a	$1$	$1$	$1$	$146.270$	$\Q$	None	✓				114.6.a.c	$1$	$1$	$0$	$9$	$21$	$143$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}+21q^{5}+143q^{7}+3^{4}q^{9}+\cdots$
912.6.a.g	$912$	$6$	912.a	1.a	$1$	$2$	$2$	$146.270$	$\Q(\sqrt{17})$	None	✓				57.6.a.c	$1$	$0$	$0$	$-18$	$-87$	$251$		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	$q-9q^{3}+(-41-5\beta )q^{5}+(11^{2}+9\beta )q^{7}+\cdots$
912.6.a.h	$912$	$6$	912.a	1.a	$1$	$2$	$2$	$146.270$	$\Q(\sqrt{4089})$	None	✓				114.6.a.f	$1$	$1$	$0$	$-18$	$-49$	$105$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}+(-24-\beta )q^{5}+(50+5\beta )q^{7}+\cdots$
912.6.a.i	$912$	$6$	912.a	1.a	$1$	$2$	$2$	$146.270$	$\Q(\sqrt{201})$	None	✓				114.6.a.e	$1$	$0$	$0$	$18$	$-13$	$33$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}+(-4-5\beta )q^{5}+(22-11\beta )q^{7}+\cdots$
912.6.a.j	$912$	$6$	912.a	1.a	$1$	$2$	$2$	$146.270$	$\Q(\sqrt{2441})$	None	✓				114.6.a.g	$1$	$0$	$0$	$18$	$-5$	$105$		$-$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	$q+9q^{3}+(-3-\beta )q^{5}+(53+\beta )q^{7}+\cdots$
912.6.a.k	$912$	$6$	912.a	1.a	$1$	$3$	$3$	$146.270$	3.3.286833.1	None	✓				228.6.a.a	$1$	$1$	$0$	$-27$	$-5$	$33$		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-9q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+(13+\cdots)q^{7}+\cdots$
912.6.a.l	$912$	$6$	912.a	1.a	$1$	$3$	$3$	$146.270$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓				114.6.a.i	$1$	$0$	$0$	$-27$	$135$	$-125$		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	$q-9q^{3}+(45-\beta _{1})q^{5}+(-42-\beta _{2})q^{7}+\cdots$
912.6.a.m	$912$	$6$	912.a	1.a	$1$	$3$	$3$	$146.270$	3.3.616092.1	None	✓				57.6.a.d	$1$	$1$	$0$	$-27$	$206$	$-186$		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q-9q^{3}+(69+\beta _{1})q^{5}+(-61+\beta _{1}+\cdots)q^{7}+\cdots$
912.6.a.n	$912$	$6$	912.a	1.a	$1$	$3$	$3$	$146.270$	3.3.9153.1	None	✓				57.6.a.e	$1$	$1$	$0$	$27$	$-9$	$-141$		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	$q+9q^{3}+(-3-3\beta _{1}-2\beta _{2})q^{5}+(-47+\cdots)q^{7}+\cdots$
912.6.a.o	$912$	$6$	912.a	1.a	$1$	$3$	$3$	$146.270$	3.3.2922585.1	None	✓				114.6.a.h	$1$	$1$	$0$	$27$	$63$	$-125$		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	$q+9q^{3}+(21+\beta _{1}-\beta _{2})q^{5}+(-40+\cdots)q^{7}+\cdots$
912.6.a.p	$912$	$6$	912.a	1.a	$1$	$4$	$4$	$146.270$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓		✓		228.6.a.b	$1$	$1$	$0$	$36$	$-84$	$-54$		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q+9q^{3}+(-21+\beta _{2})q^{5}+(-13-\beta _{1}+\cdots)q^{7}+\cdots$
912.6.a.q	$912$	$6$	912.a	1.a	$1$	$4$	$4$	$146.270$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓				456.6.a.a	$1$	$1$	$0$	$36$	$-20$	$-70$		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q+9q^{3}+(-5+\beta _{2})q^{5}+(-19+4\beta _{1}+\cdots)q^{7}+\cdots$
912.6.a.r	$912$	$6$	912.a	1.a	$1$	$4$	$4$	$146.270$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓				57.6.a.f	$1$	$0$	$0$	$36$	$-8$	$142$		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	$q+9q^{3}+(-3-\beta _{1}-3\beta _{3})q^{5}+(33+\cdots)q^{7}+\cdots$
912.6.a.s	$912$	$6$	912.a	1.a	$1$	$4$	$4$	$146.270$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓				228.6.a.c	$1$	$0$	$0$	$36$	$117$	$33$		$-$	$-$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	$q+9q^{3}+(29-\beta _{2})q^{5}+(8-\beta _{1})q^{7}+\cdots$
912.6.a.t	$912$	$6$	912.a	1.a	$1$	$5$	$5$	$146.270$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				456.6.a.c	$1$	$1$	$0$	$-45$	$-59$	$149$		$+$	$+$	$+$	$+$	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)$	$q-9q^{3}+(-12-\beta _{1})q^{5}+(30-\beta _{1}+\cdots)q^{7}+\cdots$
912.6.a.u	$912$	$6$	912.a	1.a	$1$	$5$	$5$	$146.270$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				456.6.a.d	$1$	$0$	$0$	$-45$	$-54$	$-70$		$+$	$+$	$-$	$-$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	$q-9q^{3}+(-11+\beta _{2})q^{5}+(-14-\beta _{4})q^{7}+\cdots$
912.6.a.v	$912$	$6$	912.a	1.a	$1$	$5$	$5$	$146.270$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓		✓		228.6.a.d	$1$	$0$	$0$	$-45$	$-6$	$-54$		$-$	$+$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	$q-9q^{3}+(-1-\beta _{1})q^{5}+(-11+\beta _{1}+\cdots)q^{7}+\cdots$
912.6.a.w	$912$	$6$	912.a	1.a	$1$	$5$	$5$	$146.270$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				456.6.a.e	$1$	$0$	$0$	$-45$	$66$	$2$		$+$	$+$	$-$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	$q-9q^{3}+(13+\beta _{1})q^{5}-\beta _{2}q^{7}+3^{4}q^{9}+\cdots$
912.6.a.x	$912$	$6$	912.a	1.a	$1$	$5$	$5$	$146.270$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓		✓		456.6.a.b	$1$	$1$	$0$	$45$	$-90$	$2$		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	$q+9q^{3}+(-18+\beta _{2}-\beta _{3})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots$
912.6.a.y	$912$	$6$	912.a	1.a	$1$	$6$	$6$	$146.270$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓				456.6.a.f	$1$	$0$	$0$	$54$	$-65$	$149$		$+$	$-$	$+$	$-$	$2^{9}$	$\mathrm{SU}(2)$	$q+9q^{3}+(-11-\beta _{2})q^{5}+(5^{2}+\beta _{4}+\cdots)q^{7}+\cdots$
912.6.a.z	$912$	$6$	912.a	1.a	$1$	$7$	$7$	$146.270$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓		✓		456.6.a.h	$1$	$1$	$0$	$-63$	$-29$	$-119$		$+$	$+$	$+$	$+$	$2^{10}$	$\mathrm{SU}(2)$	$q-9q^{3}+(-4-\beta _{1})q^{5}+(-17+\beta _{3}+\cdots)q^{7}+\cdots$
912.6.a.ba	$912$	$6$	912.a	1.a	$1$	$7$	$7$	$146.270$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓		✓		456.6.a.g	$1$	$0$	$0$	$63$	$55$	$-119$		$+$	$-$	$+$	$-$	$2^{12}$	$\mathrm{SU}(2)$	$q+9q^{3}+(8-\beta _{1})q^{5}+(-17-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots$

Decomposition of $S_{6}^{\mathrm{old}}(\Gamma_0(912))$ into lower level spaces

$S_{6}^{\mathrm{old}}(\Gamma_0(912)) \simeq$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))$$^{\oplus 10}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(4))$$^{\oplus 12}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(6))$$^{\oplus 8}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(8))$$^{\oplus 8}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(16))$$^{\oplus 4}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(19))$$^{\oplus 10}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(24))$$^{\oplus 4}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(38))$$^{\oplus 8}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(48))$$^{\oplus 2}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(57))$$^{\oplus 5}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(76))$$^{\oplus 6}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(114))$$^{\oplus 4}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(152))$$^{\oplus 4}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(228))$$^{\oplus 3}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(304))$$^{\oplus 2}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(456))$$^{\oplus 2}$