Space of modular forms of level 828, weight 4, and trivial character

• Label: 828.4.a
• Level: $828$
• Weight: $4$
• Character orbit: 828.a
• Rep. character: $\chi_{828}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $9$
• Sturm bound: $576$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	828.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(828))\).

	Total	New	Old
Modular forms	444	28	416
Cusp forms	420	28	392
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.

\(2\)	\(3\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(62\)	\(0\)	\(62\)		\(58\)	\(0\)	\(58\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(50\)	\(0\)	\(50\)		\(46\)	\(0\)	\(46\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(56\)	\(0\)	\(56\)		\(52\)	\(0\)	\(52\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(56\)	\(0\)	\(56\)		\(52\)	\(0\)	\(52\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(58\)	\(6\)	\(52\)		\(56\)	\(6\)	\(50\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(52\)	\(6\)	\(46\)		\(50\)	\(6\)	\(44\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(55\)	\(8\)	\(47\)		\(53\)	\(8\)	\(45\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(55\)	\(8\)	\(47\)		\(53\)	\(8\)	\(45\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(225\)	\(14\)	\(211\)		\(213\)	\(14\)	\(199\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(219\)	\(14\)	\(205\)		\(207\)	\(14\)	\(193\)		\(12\)	\(0\)	\(12\)

Trace form

28 * q - 10 * q^5 - 20 * q^7 + 58 * q^11 - 20 * q^13 + 56 * q^17 + 274 * q^19 + 616 * q^25 - 584 * q^29 + 248 * q^31 - 368 * q^35 + 170 * q^37 - 672 * q^41 - 490 * q^43 + 336 * q^47 + 1060 * q^49 - 326 * q^53 + 1360 * q^55 - 576 * q^59 - 1030 * q^61 - 376 * q^65 - 1222 * q^67 - 552 * q^71 - 24 * q^73 + 728 * q^77 - 2508 * q^79 + 1198 * q^83 - 1028 * q^85 - 456 * q^89 - 56 * q^91 + 3464 * q^95 - 144 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(828))\) 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	23
828.4.a.a	$828$	$4$	828.a	1.a	$1$	$1$	$1$	$48.854$	\(\Q\)	None	✓				276.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(34\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{5}+34q^{7}-6^{2}q^{11}-62q^{13}+\cdots\)
828.4.a.b	$828$	$4$	828.a	1.a	$1$	$1$	$1$	$48.854$	\(\Q\)	None	✓				276.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-22\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-22q^{7}+14q^{11}-50q^{13}+\cdots\)
828.4.a.c	$828$	$4$	828.a	1.a	$1$	$2$	$2$	$48.854$	\(\Q(\sqrt{13}) \)	None	✓				276.4.a.d	$1$	$1$	\(0\)	\(0\)	\(-14\)	\(-10\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-7-3\beta )q^{5}+(-5+\beta )q^{7}+(2+2\beta )q^{11}+\cdots\)
828.4.a.d	$828$	$4$	828.a	1.a	$1$	$2$	$2$	$48.854$	\(\Q(\sqrt{13}) \)	None	✓				276.4.a.c	$1$	$1$	\(0\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+2q^{7}+(-7-7\beta )q^{11}+\cdots\)
828.4.a.e	$828$	$4$	828.a	1.a	$1$	$3$	$3$	$48.854$	3.3.28669.1	None	✓				92.4.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(42\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(14-\beta _{1})q^{7}+(-3\beta _{1}+6\beta _{2})q^{11}+\cdots\)
828.4.a.f	$828$	$4$	828.a	1.a	$1$	$3$	$3$	$48.854$	3.3.1229.1	None	✓		✓		92.4.a.a	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-46\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1}-3\beta _{2})q^{5}+(-18+5\beta _{1}+\cdots)q^{7}+\cdots\)
828.4.a.g	$828$	$4$	828.a	1.a	$1$	$4$	$4$	$48.854$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		276.4.a.e	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-14\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{5}+(-3+\beta _{1})q^{7}+(7+\beta _{1}-2\beta _{2}+\cdots)q^{11}+\cdots\)
828.4.a.h	$828$	$4$	828.a	1.a	$1$	$6$	$6$	$48.854$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	828.4.a.h	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-4\)		$-$	$+$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{3})q^{5}+(-\beta _{3}+\beta _{5})q^{7}+(-1+\cdots)q^{11}+\cdots\)
828.4.a.i	$828$	$4$	828.a	1.a	$1$	$6$	$6$	$48.854$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	828.4.a.h	$1$	$0$	\(0\)	\(0\)	\(10\)	\(-4\)		$-$	$+$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{3})q^{5}+(-\beta _{3}+\beta _{5})q^{7}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(828))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(828)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(414))\)\(^{\oplus 2}\)