Space of modular forms of level 261, weight 4, and trivial character

• Label: 261.4.a
• Level: $261$
• Weight: $4$
• Character orbit: 261.a
• Rep. character: $\chi_{261}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $8$
• Sturm bound: $120$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(120\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(261))\).

	Total	New	Old
Modular forms	94	35	59
Cusp forms	86	35	51
Eisenstein series	8	0	8

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

\(3\)	\(29\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(27\)	\(7\)	\(20\)		\(25\)	\(7\)	\(18\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(21\)	\(7\)	\(14\)		\(19\)	\(7\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(23\)	\(9\)	\(14\)		\(21\)	\(9\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(23\)	\(12\)	\(11\)		\(21\)	\(12\)	\(9\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(50\)	\(19\)	\(31\)		\(46\)	\(19\)	\(27\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(44\)	\(16\)	\(28\)		\(40\)	\(16\)	\(24\)		\(4\)	\(0\)	\(4\)

Trace form

35 * q + 2 * q^2 + 152 * q^4 - 16 * q^5 - 40 * q^7 - 18 * q^8 - 14 * q^10 - 26 * q^11 + 44 * q^13 + 76 * q^14 + 396 * q^16 + 34 * q^17 - 70 * q^19 - 232 * q^20 - 112 * q^22 - 224 * q^23 + 915 * q^25 + 674 * q^26 - 160 * q^28 + 87 * q^29 - 610 * q^31 - 282 * q^32 - 396 * q^34 + 684 * q^35 - 358 * q^37 + 612 * q^38 + 230 * q^40 + 190 * q^41 + 562 * q^43 - 294 * q^44 - 312 * q^46 - 494 * q^47 + 2379 * q^49 + 1420 * q^50 - 728 * q^52 - 536 * q^53 + 324 * q^55 + 2584 * q^56 - 58 * q^58 - 832 * q^59 - 1042 * q^61 - 676 * q^62 + 1060 * q^64 + 70 * q^65 + 2044 * q^67 - 1332 * q^68 - 1312 * q^70 - 192 * q^71 - 910 * q^73 + 880 * q^74 + 1028 * q^76 - 1832 * q^77 - 1742 * q^79 - 2208 * q^80 - 932 * q^82 - 1624 * q^83 - 968 * q^85 - 1872 * q^86 + 872 * q^88 - 1618 * q^89 + 2844 * q^91 - 1780 * q^92 - 2332 * q^94 + 1152 * q^95 + 346 * q^97 - 1474 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(261))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	29
261.4.a.a	$261$	$4$	261.a	1.a	$1$	$2$	$2$	$15.399$	\(\Q(\sqrt{41}) \)	None	✓				87.4.a.b	$1$	$1$	\(1\)	\(0\)	\(1\)	\(-24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{4}+(2-3\beta )q^{5}+(-13+\cdots)q^{7}+\cdots\)
261.4.a.b	$261$	$4$	261.a	1.a	$1$	$2$	$2$	$15.399$	\(\Q(\sqrt{2}) \)	None	✓				29.4.a.a	$1$	$1$	\(2\)	\(0\)	\(10\)	\(-16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-5+2\beta )q^{4}+(5+4\beta )q^{5}+\cdots\)
261.4.a.c	$261$	$4$	261.a	1.a	$1$	$2$	$2$	$15.399$	\(\Q(\sqrt{17}) \)	None	✓				87.4.a.a	$1$	$0$	\(5\)	\(0\)	\(11\)	\(-24\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(5-5\beta )q^{4}+(7-3\beta )q^{5}+\cdots\)
261.4.a.d	$261$	$4$	261.a	1.a	$1$	$5$	$5$	$15.399$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		87.4.a.d	$1$	$1$	\(-3\)	\(0\)	\(-29\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(6+\beta _{2}+\beta _{3})q^{4}+\cdots\)
261.4.a.e	$261$	$4$	261.a	1.a	$1$	$5$	$5$	$15.399$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				87.4.a.c	$1$	$0$	\(-3\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(6-2\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
261.4.a.f	$261$	$4$	261.a	1.a	$1$	$5$	$5$	$15.399$	5.5.13458092.1	None	✓				29.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(40\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+2\beta _{1}+2\beta _{3}+\beta _{4})q^{4}+\cdots\)
261.4.a.g	$261$	$4$	261.a	1.a	$1$	$7$	$7$	$15.399$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	261.4.a.g	$1$	$1$	\(-2\)	\(0\)	\(-20\)	\(-12\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(5+\beta _{1}+\beta _{3})q^{4}+(-3+\beta _{1}+\cdots)q^{5}+\cdots\)
261.4.a.h	$261$	$4$	261.a	1.a	$1$	$7$	$7$	$15.399$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	261.4.a.g	$1$	$0$	\(2\)	\(0\)	\(20\)	\(-12\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{1}+\beta _{3})q^{4}+(3-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(261))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(261)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 2}\)