Space of modular forms of level 91, weight 4, and trivial character

• Label: 91.4.a
• Level: $91$
• Weight: $4$
• Character orbit: 91.a
• Rep. character: $\chi_{91}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $4$
• Sturm bound: $37$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	91.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(37\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(91))\).

	Total	New	Old
Modular forms	30	18	12
Cusp forms	26	18	8
Eisenstein series	4	0	4

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

\(7\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(9\)	\(5\)	\(4\)		\(8\)	\(5\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(6\)	\(3\)	\(3\)		\(5\)	\(3\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(7\)	\(4\)	\(3\)		\(6\)	\(4\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(8\)	\(6\)	\(2\)		\(7\)	\(6\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(17\)	\(11\)	\(6\)		\(15\)	\(11\)	\(4\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(13\)	\(7\)	\(6\)		\(11\)	\(7\)	\(4\)		\(2\)	\(0\)	\(2\)

Trace form

18 * q + 6 * q^2 + 4 * q^3 + 86 * q^4 - 16 * q^5 - 64 * q^6 + 14 * q^7 + 30 * q^8 + 158 * q^9 - 48 * q^10 - 32 * q^11 + 104 * q^12 - 70 * q^14 - 32 * q^15 + 214 * q^16 - 164 * q^17 + 490 * q^18 + 268 * q^19 + 52 * q^20 + 84 * q^21 - 148 * q^22 - 360 * q^23 - 1156 * q^24 + 842 * q^25 - 776 * q^27 - 98 * q^28 + 128 * q^29 - 408 * q^30 - 216 * q^31 + 586 * q^32 - 152 * q^33 - 28 * q^35 + 818 * q^36 - 284 * q^37 - 320 * q^38 + 312 * q^39 + 316 * q^40 + 500 * q^41 - 140 * q^42 - 644 * q^43 + 404 * q^44 + 712 * q^45 - 1296 * q^46 - 1072 * q^47 - 1020 * q^48 + 882 * q^49 + 1002 * q^50 - 592 * q^51 - 936 * q^52 + 448 * q^53 - 880 * q^54 + 232 * q^55 - 462 * q^56 - 2088 * q^57 - 992 * q^58 - 20 * q^59 - 3428 * q^60 + 2560 * q^61 + 3944 * q^62 + 742 * q^63 - 842 * q^64 + 312 * q^65 + 1956 * q^66 - 152 * q^67 - 2072 * q^68 + 2504 * q^69 - 728 * q^70 + 3160 * q^71 + 1526 * q^72 + 2220 * q^73 - 3864 * q^74 - 292 * q^75 - 1496 * q^76 - 952 * q^77 + 312 * q^78 - 776 * q^79 + 512 * q^80 + 2 * q^81 - 1564 * q^82 + 1028 * q^83 + 1008 * q^84 + 1688 * q^85 - 628 * q^86 + 4952 * q^87 + 612 * q^88 - 1524 * q^89 + 404 * q^90 + 364 * q^91 + 2828 * q^92 - 2136 * q^93 + 3076 * q^94 + 4004 * q^95 - 316 * q^96 - 1916 * q^97 + 294 * q^98 - 2816 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(91))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	13
91.4.a.a	$91$	$4$	91.a	1.a	$1$	$3$	$3$	$5.369$	3.3.1384.1	None	✓	✓	✓	✓	91.4.a.a	$1$	$1$	\(1\)	\(1\)	\(-22\)	\(-21\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+(-1-\beta _{1}+\cdots)q^{4}+\cdots\)
91.4.a.b	$91$	$4$	91.a	1.a	$1$	$4$	$4$	$5.369$	4.4.5364412.1	None	✓	✓	✓	✓	91.4.a.b	$1$	$1$	\(-4\)	\(-5\)	\(-36\)	\(28\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+\cdots\)
91.4.a.c	$91$	$4$	91.a	1.a	$1$	$5$	$5$	$5.369$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	91.4.a.c	$1$	$0$	\(7\)	\(-5\)	\(16\)	\(-35\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{4})q^{2}+(-1-\beta _{3})q^{3}+(10-\beta _{2}+\cdots)q^{4}+\cdots\)
91.4.a.d	$91$	$4$	91.a	1.a	$1$	$6$	$6$	$5.369$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	91.4.a.d	$1$	$0$	\(2\)	\(13\)	\(26\)	\(42\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2-\beta _{4})q^{3}+(2+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(91))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(91)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)