Space of modular forms of level 832, weight 4, and trivial character

• Label: 832.4.a
• Level: $832$
• Weight: $4$
• Character orbit: 832.a
• Rep. character: $\chi_{832}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $35$
• Sturm bound: $448$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$832 = 2^{6} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	832.a (trivial)
Character field:			$\Q$
Newform subspaces:			$35$
Sturm bound:			$448$
Trace bound:			$7$
Distinguishing $T_p$:			$3$, $5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(832))$.

	Total	New	Old
Modular forms	348	72	276
Cusp forms	324	72	252
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.

$2$	$13$	Fricke	Dim
$+$	$+$	$+$	$19$
$+$	$-$	$-$	$17$
$-$	$+$	$-$	$17$
$-$	$-$	$+$	$19$
Plus space		$+$	$38$
Minus space		$-$	$34$

Trace form

72 * q + 648 * q^9 + 1800 * q^25 + 400 * q^29 + 464 * q^33 + 16 * q^37 + 80 * q^41 - 1968 * q^45 + 3528 * q^49 - 1568 * q^53 - 688 * q^57 + 1824 * q^61 + 2544 * q^69 - 5408 * q^77 + 6456 * q^81 - 2832 * q^85 + 1696 * q^89 + 4944 * q^93 + 2976 * q^97

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(832))$ 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	13
832.4.a.a	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				13.4.a.a	$1$	$1$	$0$	$-7$	$7$	$13$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-7q^{3}+7q^{5}+13q^{7}+22q^{9}-26q^{11}+\cdots$
832.4.a.b	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				104.4.a.b	$1$	$1$	$0$	$-5$	$-19$	$-3$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-5q^{3}-19q^{5}-3q^{7}-2q^{9}+2q^{11}+\cdots$
832.4.a.c	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				416.4.a.a	$1$	$1$	$0$	$-5$	$3$	$-5$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-5q^{3}+3q^{5}-5q^{7}-2q^{9}+30q^{11}+\cdots$
832.4.a.d	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				26.4.a.c	$1$	$0$	$0$	$-4$	$18$	$20$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-4q^{3}+18q^{5}+20q^{7}-11q^{9}+48q^{11}+\cdots$
832.4.a.e	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				26.4.a.a	$1$	$1$	$0$	$-3$	$-11$	$19$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-3q^{3}-11q^{5}+19q^{7}-18q^{9}+38q^{11}+\cdots$
832.4.a.f	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				52.4.a.a	$1$	$0$	$0$	$-3$	$13$	$11$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-3q^{3}+13q^{5}+11q^{7}-18q^{9}-2q^{11}+\cdots$
832.4.a.g	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				26.4.a.b	$1$	$1$	$0$	$-1$	$-17$	$35$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{3}-17q^{5}+35q^{7}-26q^{9}+2q^{11}+\cdots$
832.4.a.h	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				416.4.a.b	$1$	$1$	$0$	$-1$	$1$	$5$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{3}+q^{5}+5q^{7}-26q^{9}-10q^{11}+\cdots$
832.4.a.i	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				104.4.a.a	$1$	$0$	$0$	$-1$	$7$	$-21$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{3}+7q^{5}-21q^{7}-26q^{9}-6q^{11}+\cdots$
832.4.a.j	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				26.4.a.b	$1$	$2$	$0$	$1$	$-17$	$-35$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-17q^{5}-35q^{7}-26q^{9}-2q^{11}+\cdots$
832.4.a.k	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				416.4.a.b	$1$	$1$	$0$	$1$	$1$	$-5$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}+q^{5}-5q^{7}-26q^{9}+10q^{11}+\cdots$
832.4.a.l	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				104.4.a.a	$1$	$1$	$0$	$1$	$7$	$21$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}+7q^{5}+21q^{7}-26q^{9}+6q^{11}+\cdots$
832.4.a.m	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				26.4.a.a	$1$	$0$	$0$	$3$	$-11$	$-19$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+3q^{3}-11q^{5}-19q^{7}-18q^{9}-38q^{11}+\cdots$
832.4.a.n	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				52.4.a.a	$1$	$1$	$0$	$3$	$13$	$-11$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+3q^{3}+13q^{5}-11q^{7}-18q^{9}+2q^{11}+\cdots$
832.4.a.o	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				26.4.a.c	$1$	$1$	$0$	$4$	$18$	$-20$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+4q^{3}+18q^{5}-20q^{7}-11q^{9}-48q^{11}+\cdots$
832.4.a.p	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				104.4.a.b	$1$	$0$	$0$	$5$	$-19$	$3$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+5q^{3}-19q^{5}+3q^{7}-2q^{9}-2q^{11}+\cdots$
832.4.a.q	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				416.4.a.a	$1$	$1$	$0$	$5$	$3$	$5$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+5q^{3}+3q^{5}+5q^{7}-2q^{9}-30q^{11}+\cdots$
832.4.a.r	$832$	$4$	832.a	1.a	$1$	$1$	$1$	$49.090$	$\Q$	None	✓				13.4.a.a	$1$	$0$	$0$	$7$	$7$	$-13$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+7q^{3}+7q^{5}-13q^{7}+22q^{9}+26q^{11}+\cdots$
832.4.a.s	$832$	$4$	832.a	1.a	$1$	$2$	$2$	$49.090$	$\Q(\sqrt{17})$	None	✓				13.4.a.b	$1$	$1$	$0$	$-5$	$3$	$-9$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1-3\beta )q^{3}+(1+\beta )q^{5}+(1-11\beta )q^{7}+\cdots$
832.4.a.t	$832$	$4$	832.a	1.a	$1$	$2$	$2$	$49.090$	$\Q(\sqrt{217})$	None	✓				52.4.a.b	$1$	$1$	$0$	$-3$	$-23$	$-27$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta )q^{3}+(-11-\beta )q^{5}+(-13+\cdots)q^{7}+\cdots$
832.4.a.u	$832$	$4$	832.a	1.a	$1$	$2$	$2$	$49.090$	$\Q(\sqrt{73})$	None	✓				104.4.a.c	$1$	$0$	$0$	$-3$	$3$	$25$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta )q^{3}+(3-3\beta )q^{5}+(13-\beta )q^{7}+\cdots$
832.4.a.v	$832$	$4$	832.a	1.a	$1$	$2$	$2$	$49.090$	$\Q(\sqrt{321})$	None	✓				104.4.a.d	$1$	$0$	$0$	$-1$	$11$	$-1$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{3}+(6-\beta )q^{5}+(-2+3\beta )q^{7}+\cdots$
832.4.a.w	$832$	$4$	832.a	1.a	$1$	$2$	$2$	$49.090$	$\Q(\sqrt{321})$	None	✓				104.4.a.d	$1$	$1$	$0$	$1$	$11$	$1$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+(6-\beta )q^{5}+(2-3\beta )q^{7}+(53+\cdots)q^{9}+\cdots$
832.4.a.x	$832$	$4$	832.a	1.a	$1$	$2$	$2$	$49.090$	$\Q(\sqrt{217})$	None	✓				52.4.a.b	$1$	$0$	$0$	$3$	$-23$	$27$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(-11-\beta )q^{5}+(13+\beta )q^{7}+\cdots$
832.4.a.y	$832$	$4$	832.a	1.a	$1$	$2$	$2$	$49.090$	$\Q(\sqrt{73})$	None	✓				104.4.a.c	$1$	$1$	$0$	$3$	$3$	$-25$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(3-3\beta )q^{5}+(-13+\beta )q^{7}+\cdots$
832.4.a.z	$832$	$4$	832.a	1.a	$1$	$2$	$2$	$49.090$	$\Q(\sqrt{17})$	None	✓				13.4.a.b	$1$	$0$	$0$	$5$	$3$	$9$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1+3\beta )q^{3}+(1+\beta )q^{5}+(-1+11\beta )q^{7}+\cdots$
832.4.a.ba	$832$	$4$	832.a	1.a	$1$	$3$	$3$	$49.090$	3.3.24965.1	None	✓				416.4.a.e	$1$	$1$	$0$	$-4$	$-16$	$-26$		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{3}+(-6-\beta _{1}+\beta _{2})q^{5}+\cdots$
832.4.a.bb	$832$	$4$	832.a	1.a	$1$	$3$	$3$	$49.090$	3.3.18257.1	None	✓				104.4.a.e	$1$	$1$	$0$	$0$	$8$	$-36$		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{3}+(3+2\beta _{1}+\beta _{2})q^{5}+(-12+\cdots)q^{7}+\cdots$
832.4.a.bc	$832$	$4$	832.a	1.a	$1$	$3$	$3$	$49.090$	3.3.18257.1	None	✓				104.4.a.e	$1$	$0$	$0$	$0$	$8$	$36$		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(3+2\beta _{1}+\beta _{2})q^{5}+(12+\beta _{1}+\cdots)q^{7}+\cdots$
832.4.a.bd	$832$	$4$	832.a	1.a	$1$	$3$	$3$	$49.090$	3.3.24965.1	None	✓				416.4.a.e	$1$	$1$	$0$	$4$	$-16$	$26$		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+(1+\beta _{2})q^{3}+(-6-\beta _{1}+\beta _{2})q^{5}+\cdots$
832.4.a.be	$832$	$4$	832.a	1.a	$1$	$4$	$4$	$49.090$	4.4.1847677.1	None	✓				416.4.a.g	$2$	$0$	$0$	$0$	$14$	$0$		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q-\beta _{3}q^{3}+(3-\beta _{2})q^{5}+(-\beta _{1}-\beta _{3})q^{7}+\cdots$
832.4.a.bf	$832$	$4$	832.a	1.a	$1$	$5$	$5$	$49.090$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				416.4.a.h	$1$	$0$	$0$	$-11$	$5$	$21$		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	$q+(-2+\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(4+\beta _{1}+\cdots)q^{7}+\cdots$
832.4.a.bg	$832$	$4$	832.a	1.a	$1$	$5$	$5$	$49.090$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				416.4.a.h	$1$	$0$	$0$	$11$	$5$	$-21$		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	$q+(2-\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(-4-\beta _{1}+\cdots)q^{7}+\cdots$
832.4.a.bh	$832$	$4$	832.a	1.a	$1$	$6$	$6$	$49.090$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓		✓		416.4.a.k	$2$	$1$	$0$	$0$	$-16$	$0$		$-$	$+$	$-$	$2^{7}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(-3+\beta _{5})q^{5}+(-2\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots$
832.4.a.bi	$832$	$4$	832.a	1.a	$1$	$6$	$6$	$49.090$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓		✓		416.4.a.j	$2$	$0$	$0$	$0$	$16$	$0$		$-$	$-$	$+$	$2^{8}$	$\mathrm{SU}(2)$	$q-\beta _{3}q^{3}+(3+\beta _{1})q^{5}-\beta _{4}q^{7}+(20+\cdots)q^{9}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(832))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(832)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))$$^{\oplus 8}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(13))$$^{\oplus 7}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(26))$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(32))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(52))$$^{\oplus 5}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(64))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(104))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(208))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(416))$$^{\oplus 2}$