Space of modular forms of level 792, weight 4, and trivial character

• Label: 792.4.a
• Level: $792$
• Weight: $4$
• Character orbit: 792.a
• Rep. character: $\chi_{792}(1,\cdot)$
• Character field: $\Q$
• Dimension: $37$
• Newform subspaces: $17$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$792 = 2^{3} \cdot 3^{2} \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	792.a (trivial)
Character field:			$\Q$
Newform subspaces:			$17$
Sturm bound:			$576$
Trace bound:			$7$
Distinguishing $T_p$:			$5$, $7$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(792))$.

	Total	New	Old
Modular forms	448	37	411
Cusp forms	416	37	379
Eisenstein series	32	0	32

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$2$	$3$	$11$	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$		$58$	$4$	$54$		$54$	$4$	$50$		$4$	$0$	$4$
$+$	$+$	$-$	$-$		$56$	$3$	$53$		$52$	$3$	$49$		$4$	$0$	$4$
$+$	$-$	$+$	$-$		$55$	$6$	$49$		$51$	$6$	$45$		$4$	$0$	$4$
$+$	$-$	$-$	$+$		$57$	$5$	$52$		$53$	$5$	$48$		$4$	$0$	$4$
$-$	$+$	$+$	$-$		$54$	$3$	$51$		$50$	$3$	$47$		$4$	$0$	$4$
$-$	$+$	$-$	$+$		$56$	$4$	$52$		$52$	$4$	$48$		$4$	$0$	$4$
$-$	$-$	$+$	$+$		$57$	$7$	$50$		$53$	$7$	$46$		$4$	$0$	$4$
$-$	$-$	$-$	$-$		$55$	$5$	$50$		$51$	$5$	$46$		$4$	$0$	$4$
Plus space			$+$		$228$	$20$	$208$		$212$	$20$	$192$		$16$	$0$	$16$
Minus space			$-$		$220$	$17$	$203$		$204$	$17$	$187$		$16$	$0$	$16$

Trace form

37 * q + 24 * q^5 + 36 * q^7 - 33 * q^11 - 102 * q^13 + 86 * q^17 + 180 * q^19 - 142 * q^23 + 881 * q^25 - 194 * q^29 + 130 * q^31 - 260 * q^35 + 468 * q^37 - 574 * q^41 + 88 * q^43 + 820 * q^47 + 981 * q^49 - 502 * q^53 - 954 * q^59 + 874 * q^61 - 724 * q^65 + 1274 * q^67 + 374 * q^71 + 1598 * q^73 + 308 * q^77 - 1508 * q^79 + 568 * q^83 - 1080 * q^85 + 1084 * q^89 - 2336 * q^91 + 3048 * q^95 - 724 * q^97

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(792))$ 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	11
792.4.a.a	$792$	$4$	792.a	1.a	$1$	$1$	$1$	$46.730$	$\Q$	None	✓				264.4.a.b	$1$	$1$	$0$	$0$	$-12$	$22$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-12q^{5}+22q^{7}-11q^{11}-48q^{13}+\cdots$
792.4.a.b	$792$	$4$	792.a	1.a	$1$	$1$	$1$	$46.730$	$\Q$	None	✓				88.4.a.b	$1$	$1$	$0$	$0$	$-9$	$2$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-9q^{5}+2q^{7}+11q^{11}+38q^{17}+\cdots$
792.4.a.c	$792$	$4$	792.a	1.a	$1$	$1$	$1$	$46.730$	$\Q$	None	✓				264.4.a.c	$1$	$1$	$0$	$0$	$6$	$-14$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+6q^{5}-14q^{7}-11q^{11}+6q^{13}+\cdots$
792.4.a.d	$792$	$4$	792.a	1.a	$1$	$1$	$1$	$46.730$	$\Q$	None	✓				264.4.a.d	$1$	$1$	$0$	$0$	$6$	$-8$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+6q^{5}-8q^{7}+11q^{11}-30q^{13}+\cdots$
792.4.a.e	$792$	$4$	792.a	1.a	$1$	$1$	$1$	$46.730$	$\Q$	None	✓				88.4.a.a	$1$	$0$	$0$	$0$	$7$	$-6$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+7q^{5}-6q^{7}+11q^{11}-40q^{13}+\cdots$
792.4.a.f	$792$	$4$	792.a	1.a	$1$	$1$	$1$	$46.730$	$\Q$	None	✓				264.4.a.a	$1$	$1$	$0$	$0$	$18$	$-28$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+18q^{5}-28q^{7}-11q^{11}-18q^{13}+\cdots$
792.4.a.g	$792$	$4$	792.a	1.a	$1$	$2$	$2$	$46.730$	$\Q(\sqrt{5})$	None	✓				88.4.a.c	$1$	$0$	$0$	$0$	$6$	$-56$		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+(3+4\beta )q^{5}+(-28+\beta )q^{7}-11q^{11}+\cdots$
792.4.a.h	$792$	$4$	792.a	1.a	$1$	$2$	$2$	$46.730$	$\Q(\sqrt{17})$	None	✓				264.4.a.e	$1$	$0$	$0$	$0$	$6$	$10$		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q+(3+\beta )q^{5}+(5+\beta )q^{7}-11q^{11}+(-1+\cdots)q^{13}+\cdots$
792.4.a.i	$792$	$4$	792.a	1.a	$1$	$2$	$2$	$46.730$	$\Q(\sqrt{137})$	None	✓				264.4.a.f	$1$	$0$	$0$	$0$	$6$	$16$		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+(3+\beta )q^{5}+(8+2\beta )q^{7}+11q^{11}+\cdots$
792.4.a.j	$792$	$4$	792.a	1.a	$1$	$2$	$2$	$46.730$	$\Q(\sqrt{185})$	None	✓				264.4.a.g	$1$	$0$	$0$	$0$	$6$	$22$		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+(3+\beta )q^{5}+(11-\beta )q^{7}+11q^{11}+\cdots$
792.4.a.k	$792$	$4$	792.a	1.a	$1$	$3$	$3$	$46.730$	3.3.4364.1	None	✓	✓	✓	✓	792.4.a.k	$1$	$1$	$0$	$0$	$-8$	$10$		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q+(-3-\beta _{1})q^{5}+(3-\beta _{2})q^{7}+11q^{11}+\cdots$
792.4.a.l	$792$	$4$	792.a	1.a	$1$	$3$	$3$	$46.730$	3.3.11109.1	None	✓		✓		88.4.a.d	$1$	$1$	$0$	$0$	$-8$	$24$		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q+(-3-\beta _{2})q^{5}+(8-\beta _{1}-\beta _{2})q^{7}+\cdots$
792.4.a.m	$792$	$4$	792.a	1.a	$1$	$3$	$3$	$46.730$	3.3.142161.1	None	✓		✓		264.4.a.h	$1$	$1$	$0$	$0$	$-4$	$-12$		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{5}+(-4+\beta _{2})q^{7}+11q^{11}+\cdots$
792.4.a.n	$792$	$4$	792.a	1.a	$1$	$3$	$3$	$46.730$	3.3.123209.1	None	✓		✓		264.4.a.i	$1$	$0$	$0$	$0$	$-4$	$28$		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{5}+(10+\beta _{1}+\beta _{2})q^{7}+\cdots$
792.4.a.o	$792$	$4$	792.a	1.a	$1$	$3$	$3$	$46.730$	3.3.4364.1	None	✓	✓	✓	✓	792.4.a.k	$1$	$1$	$0$	$0$	$8$	$10$		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q+(3+\beta _{1})q^{5}+(3-\beta _{2})q^{7}-11q^{11}+\cdots$
792.4.a.p	$792$	$4$	792.a	1.a	$1$	$4$	$4$	$46.730$	4.4.8611212.1	None	✓	✓	✓	✓	792.4.a.p	$1$	$0$	$0$	$0$	$-8$	$8$		$+$	$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	$q+(-2-\beta _{1})q^{5}+(2+\beta _{2})q^{7}-11q^{11}+\cdots$
792.4.a.q	$792$	$4$	792.a	1.a	$1$	$4$	$4$	$46.730$	4.4.8611212.1	None	✓	✓	✓	✓	792.4.a.p	$1$	$0$	$0$	$0$	$8$	$8$		$-$	$+$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	$q+(2+\beta _{1})q^{5}+(2+\beta _{2})q^{7}+11q^{11}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(792))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(792)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))$$^{\oplus 12}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(9))$$^{\oplus 8}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(11))$$^{\oplus 12}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(12))$$^{\oplus 8}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(18))$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(22))$$^{\oplus 9}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(24))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(33))$$^{\oplus 8}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(36))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(44))$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(66))$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(72))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(88))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(99))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(132))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(198))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(264))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(396))$$^{\oplus 2}$