Space of modular forms of level 961, weight 4, and trivial character

• Label: 961.4.a
• Level: $961$
• Weight: $4$
• Character orbit: 961.a
• Rep. character: $\chi_{961}(1,\cdot)$
• Character field: $\Q$
• Dimension: $218$
• Newform subspaces: $14$
• Sturm bound: $330$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(330\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(961))\).

	Total	New	Old
Modular forms	264	247	17
Cusp forms	232	218	14
Eisenstein series	32	29	3

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

\(31\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(136\)	\(127\)	\(9\)		\(120\)	\(113\)	\(7\)		\(16\)	\(14\)	\(2\)
\(-\)		\(128\)	\(120\)	\(8\)		\(112\)	\(105\)	\(7\)		\(16\)	\(15\)	\(1\)

Trace form

218 * q + 2 * q^2 - 2 * q^3 + 806 * q^4 + 10 * q^5 - 12 * q^6 + 10 * q^7 - 18 * q^8 + 1744 * q^9 + 96 * q^10 - 62 * q^11 + 110 * q^12 - 36 * q^13 - 60 * q^14 + 156 * q^15 + 2814 * q^16 - 54 * q^17 + 146 * q^18 + 94 * q^19 + 298 * q^20 - 104 * q^21 - 76 * q^22 - 216 * q^23 + 140 * q^24 + 3520 * q^25 + 406 * q^26 - 176 * q^27 + 496 * q^28 - 160 * q^29 - 810 * q^30 - 772 * q^32 + 312 * q^33 - 316 * q^34 - 400 * q^35 + 5898 * q^36 + 476 * q^37 - 116 * q^38 + 364 * q^39 + 852 * q^40 - 18 * q^41 + 710 * q^42 - 138 * q^43 - 984 * q^44 - 422 * q^45 + 670 * q^46 + 234 * q^47 + 1124 * q^48 + 7684 * q^49 + 1256 * q^50 + 1132 * q^51 - 1328 * q^52 - 320 * q^53 - 1948 * q^54 + 320 * q^55 - 1692 * q^56 - 388 * q^57 - 1768 * q^58 + 1238 * q^59 + 1708 * q^60 - 812 * q^61 - 180 * q^63 + 8614 * q^64 - 292 * q^65 - 3500 * q^66 + 1922 * q^67 - 3510 * q^68 - 3200 * q^69 + 820 * q^70 - 454 * q^71 + 1178 * q^72 + 1538 * q^73 - 2162 * q^74 + 1282 * q^75 + 1960 * q^76 + 652 * q^77 - 804 * q^78 + 2376 * q^79 + 5478 * q^80 + 9658 * q^81 - 3094 * q^82 - 3002 * q^83 - 360 * q^84 - 1648 * q^85 + 3754 * q^86 - 1044 * q^87 - 514 * q^88 - 3474 * q^89 + 3984 * q^90 + 3500 * q^91 + 132 * q^92 + 630 * q^94 + 300 * q^95 - 350 * q^96 - 2270 * q^97 + 4572 * q^98 - 4426 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(961))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		31
961.4.a.a	$961$	$4$	961.a	1.a	$1$	$1$	$1$	$56.701$	\(\Q\)	\(\Q(\sqrt{-31}) \)	✓	✓			961.4.a.a	$2$	$1$	\(1\)	\(0\)	\(2\)	\(16\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+q^{2}-7q^{4}+2q^{5}+2^{4}q^{7}-15q^{8}+\cdots\)
961.4.a.b	$961$	$4$	961.a	1.a	$1$	$2$	$2$	$56.701$	\(\Q(\sqrt{17}) \)	None	✓				31.4.a.a	$1$	$1$	\(-5\)	\(2\)	\(-25\)	\(-19\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+(2-2\beta )q^{3}+5\beta q^{4}+\cdots\)
961.4.a.c	$961$	$4$	961.a	1.a	$1$	$2$	$2$	$56.701$	\(\Q(\sqrt{93}) \)	\(\Q(\sqrt{-31}) \)	✓	✓			961.4.a.c	$2$	$1$	\(-1\)	\(0\)	\(-2\)	\(-16\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-\beta q^{2}+(15+\beta )q^{4}+(1-4\beta )q^{5}+(-11+\cdots)q^{7}+\cdots\)
961.4.a.d	$961$	$4$	961.a	1.a	$1$	$4$	$4$	$56.701$	4.4.27702880.2	None	✓	✓			961.4.a.d	$2$	$1$	\(2\)	\(0\)	\(-26\)	\(-18\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+\beta _{1}q^{3}+(3+\beta _{2})q^{4}+\cdots\)
961.4.a.e	$961$	$4$	961.a	1.a	$1$	$5$	$5$	$56.701$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				31.4.a.b	$1$	$1$	\(3\)	\(-4\)	\(15\)	\(9\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(\beta _{2}-\beta _{3}+\beta _{4})q^{3}+(7+\cdots)q^{4}+\cdots\)
961.4.a.f	$961$	$4$	961.a	1.a	$1$	$7$	$7$	$56.701$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				31.4.c.a	$1$	$1$	\(-2\)	\(-11\)	\(5\)	\(-31\)		$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2+\beta _{4})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
961.4.a.g	$961$	$4$	961.a	1.a	$1$	$7$	$7$	$56.701$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				31.4.c.a	$1$	$0$	\(-2\)	\(11\)	\(5\)	\(-31\)		$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2-\beta _{4})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
961.4.a.h	$961$	$4$	961.a	1.a	$1$	$14$	$14$	$56.701$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	✓			961.4.a.h	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(-28\)		$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+\beta _{6}q^{3}+(2-\beta _{4})q^{4}-\beta _{8}q^{5}+\cdots\)
961.4.a.i	$961$	$4$	961.a	1.a	$1$	$14$	$14$	$56.701$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓				31.4.d.a	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(5\)		$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{8}q^{3}+(4+\beta _{3}+\beta _{4})q^{4}+\cdots\)
961.4.a.j	$961$	$4$	961.a	1.a	$1$	$14$	$14$	$56.701$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓				31.4.d.a	$1$	$1$	\(1\)	\(1\)	\(0\)	\(5\)		$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{8}q^{3}+(4+\beta _{3}+\beta _{4})q^{4}+\cdots\)
961.4.a.k	$961$	$4$	961.a	1.a	$1$	$28$	$28$	$56.701$		None	✓	✓			961.4.a.k	$2$	$1$	\(-16\)	\(0\)	\(-40\)	\(-56\)		$-$	$-$		$\mathrm{SU}(2)$
961.4.a.l	$961$	$4$	961.a	1.a	$1$	$28$	$28$	$56.701$		None	✓				31.4.g.a	$1$	$1$	\(2\)	\(-19\)	\(0\)	\(31\)		$-$	$-$		$\mathrm{SU}(2)$
961.4.a.m	$961$	$4$	961.a	1.a	$1$	$28$	$28$	$56.701$		None	✓				31.4.g.a	$1$	$0$	\(2\)	\(19\)	\(0\)	\(31\)		$+$	$+$		$\mathrm{SU}(2)$
961.4.a.n	$961$	$4$	961.a	1.a	$1$	$64$	$64$	$56.701$		None	✓	✓	✓		961.4.a.n	$2$	$0$	\(16\)	\(0\)	\(80\)	\(112\)		$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(961))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(961)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 2}\)