Space of modular forms of level 1156, weight 2, and trivial character

• Label: 1156.2.a
• Level: $1156$
• Weight: $2$
• Character orbit: 1156.a
• Rep. character: $\chi_{1156}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $8$
• Sturm bound: $306$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1156 = 2^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1156.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(306\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1156))\).

	Total	New	Old
Modular forms	180	22	158
Cusp forms	127	22	105
Eisenstein series	53	0	53

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

\(2\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(45\)	\(0\)	\(45\)		\(28\)	\(0\)	\(28\)		\(17\)	\(0\)	\(17\)
\(+\)	\(-\)	\(-\)		\(49\)	\(0\)	\(49\)		\(31\)	\(0\)	\(31\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(-\)		\(45\)	\(13\)	\(32\)		\(36\)	\(13\)	\(23\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)		\(41\)	\(9\)	\(32\)		\(32\)	\(9\)	\(23\)		\(9\)	\(0\)	\(9\)
Plus space		\(+\)		\(86\)	\(9\)	\(77\)		\(60\)	\(9\)	\(51\)		\(26\)	\(0\)	\(26\)
Minus space		\(-\)		\(94\)	\(13\)	\(81\)		\(67\)	\(13\)	\(54\)		\(27\)	\(0\)	\(27\)

Trace form

22 * q - 2 * q^3 + 2 * q^7 + 22 * q^9 + 6 * q^11 - 2 * q^13 + 14 * q^15 - 2 * q^19 + 4 * q^21 + 6 * q^23 + 12 * q^25 - 8 * q^27 + 2 * q^31 - 2 * q^33 - 16 * q^35 - 16 * q^37 - 16 * q^39 + 12 * q^41 - 6 * q^43 + 24 * q^45 + 2 * q^47 + 32 * q^49 - 14 * q^53 + 10 * q^55 + 8 * q^57 - 14 * q^59 + 8 * q^61 + 14 * q^63 + 24 * q^65 - 18 * q^67 - 2 * q^69 + 6 * q^71 - 4 * q^73 - 14 * q^75 - 2 * q^77 + 14 * q^79 + 22 * q^81 + 12 * q^83 - 12 * q^87 - 14 * q^89 + 16 * q^91 - 16 * q^93 - 24 * q^95 - 4 * q^97 - 6 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1156))\) 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	17
1156.2.a.a	$1156$	$2$	1156.a	1.a	$1$	$2$	$2$	$9.231$	\(\Q(\sqrt{3}) \)	None	✓				68.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2\beta q^{5}+(1-\beta )q^{7}+\cdots\)
1156.2.a.b	$1156$	$2$	1156.a	1.a	$1$	$2$	$2$	$9.231$	\(\Q(\sqrt{21}) \)	None	✓	✓			1156.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1-\beta )q^{5}+(-3+\beta )q^{7}+\cdots\)
1156.2.a.c	$1156$	$2$	1156.a	1.a	$1$	$2$	$2$	$9.231$	\(\Q(\sqrt{2}) \)	None	✓				68.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}+3\beta q^{7}-q^{9}+\beta q^{11}+\cdots\)
1156.2.a.d	$1156$	$2$	1156.a	1.a	$1$	$2$	$2$	$9.231$	\(\Q(\sqrt{21}) \)	None	✓	✓			1156.2.a.b	$1$	$0$	\(0\)	\(1\)	\(3\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+\beta )q^{5}+(3-\beta )q^{7}+(2+\beta )q^{9}+\cdots\)
1156.2.a.e	$1156$	$2$	1156.a	1.a	$1$	$3$	$3$	$9.231$	\(\Q(\zeta_{18})^+\)	None	✓	✓			1156.2.a.e	$1$	$1$	\(0\)	\(-3\)	\(-6\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-2+\beta _{1}-\beta _{2})q^{5}+\cdots\)
1156.2.a.f	$1156$	$2$	1156.a	1.a	$1$	$3$	$3$	$9.231$	\(\Q(\zeta_{18})^+\)	None	✓	✓			1156.2.a.e	$1$	$0$	\(0\)	\(3\)	\(6\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
1156.2.a.g	$1156$	$2$	1156.a	1.a	$1$	$4$	$4$	$9.231$	\(\Q(\zeta_{16})^+\)	None	✓		✓		68.2.h.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{1}q^{5}+\beta _{3}q^{7}+(-1+\beta _{2}+\cdots)q^{9}+\cdots\)
1156.2.a.h	$1156$	$2$	1156.a	1.a	$1$	$4$	$4$	$9.231$	\(\Q(\sqrt{2}, \sqrt{13})\)	None	✓		✓		68.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{2})q^{3}-\beta _{2}q^{5}+(\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1156))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1156)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(578))\)\(^{\oplus 2}\)