Space of modular forms of level 1656, weight 2, and trivial character

• Label: 1656.2.a
• Level: $1656$
• Weight: $2$
• Character orbit: 1656.a
• Rep. character: $\chi_{1656}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $16$
• Sturm bound: $576$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1656.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(576\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1656))\).

	Total	New	Old
Modular forms	304	28	276
Cusp forms	273	28	245
Eisenstein series	31	0	31

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
\(+\)	\(+\)	\(+\)	\(+\)		\(30\)	\(2\)	\(28\)		\(27\)	\(2\)	\(25\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(46\)	\(4\)	\(42\)		\(42\)	\(4\)	\(38\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(40\)	\(4\)	\(36\)		\(36\)	\(4\)	\(32\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(36\)	\(3\)	\(33\)		\(32\)	\(3\)	\(29\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(34\)	\(4\)	\(30\)		\(30\)	\(4\)	\(26\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(42\)	\(2\)	\(40\)		\(38\)	\(2\)	\(36\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(36\)	\(4\)	\(32\)		\(32\)	\(4\)	\(28\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(40\)	\(5\)	\(35\)		\(36\)	\(5\)	\(31\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(144\)	\(11\)	\(133\)		\(129\)	\(11\)	\(118\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(160\)	\(17\)	\(143\)		\(144\)	\(17\)	\(127\)		\(16\)	\(0\)	\(16\)

Trace form

28 * q - 2 * q^5 - 2 * q^11 + 4 * q^13 + 8 * q^17 + 10 * q^19 + 40 * q^25 - 12 * q^31 + 2 * q^37 - 2 * q^43 + 4 * q^47 + 52 * q^49 - 6 * q^53 + 8 * q^55 + 8 * q^59 - 6 * q^61 - 8 * q^65 + 14 * q^67 + 20 * q^71 - 8 * q^73 - 24 * q^77 + 36 * q^79 + 10 * q^83 + 12 * q^85 + 24 * q^89 + 28 * q^91 + 64 * q^95

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1656))\) 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
1656.2.a.a	$1656$	$2$	1656.a	1.a	$1$	$1$	$1$	$13.223$	\(\Q\)	None	✓				552.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+2q^{7}+2q^{13}+4q^{17}-6q^{19}+\cdots\)
1656.2.a.b	$1656$	$2$	1656.a	1.a	$1$	$1$	$1$	$13.223$	\(\Q\)	None	✓				552.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{7}+4q^{11}-2q^{13}+2q^{17}+\cdots\)
1656.2.a.c	$1656$	$2$	1656.a	1.a	$1$	$1$	$1$	$13.223$	\(\Q\)	None	✓				184.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-5q^{13}+6q^{17}+6q^{19}-q^{23}+\cdots\)
1656.2.a.d	$1656$	$2$	1656.a	1.a	$1$	$1$	$1$	$13.223$	\(\Q\)	None	✓				552.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+2q^{13}-8q^{17}+6q^{19}-q^{23}+\cdots\)
1656.2.a.e	$1656$	$2$	1656.a	1.a	$1$	$1$	$1$	$13.223$	\(\Q\)	None	✓				184.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-6q^{11}-2q^{13}-6q^{17}-6q^{19}+\cdots\)
1656.2.a.f	$1656$	$2$	1656.a	1.a	$1$	$1$	$1$	$13.223$	\(\Q\)	None	✓				552.2.a.e	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-4q^{7}-2q^{13}+2q^{17}-4q^{19}+\cdots\)
1656.2.a.g	$1656$	$2$	1656.a	1.a	$1$	$1$	$1$	$13.223$	\(\Q\)	None	✓				184.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-4q^{7}+2q^{11}+7q^{13}+4q^{17}+\cdots\)
1656.2.a.h	$1656$	$2$	1656.a	1.a	$1$	$1$	$1$	$13.223$	\(\Q\)	None	✓				552.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{7}+2q^{11}-2q^{13}+4q^{17}+\cdots\)
1656.2.a.i	$1656$	$2$	1656.a	1.a	$1$	$1$	$1$	$13.223$	\(\Q\)	None	✓				184.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+2q^{7}+4q^{11}-5q^{13}+2q^{17}+\cdots\)
1656.2.a.j	$1656$	$2$	1656.a	1.a	$1$	$2$	$2$	$13.223$	\(\Q(\sqrt{17}) \)	None	✓		✓		184.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+(-2+2\beta )q^{11}+(3-\beta )q^{13}+\cdots\)
1656.2.a.k	$1656$	$2$	1656.a	1.a	$1$	$2$	$2$	$13.223$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	1656.2.a.k	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(-1+\beta )q^{7}+(2-2\beta )q^{11}+\cdots\)
1656.2.a.l	$1656$	$2$	1656.a	1.a	$1$	$2$	$2$	$13.223$	\(\Q(\sqrt{5}) \)	None	✓		✓		552.2.a.f	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+2q^{7}+(-1-\beta )q^{11}+\cdots\)
1656.2.a.m	$1656$	$2$	1656.a	1.a	$1$	$2$	$2$	$13.223$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	1656.2.a.k	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-1-\beta )q^{7}+(-2-2\beta )q^{11}+\cdots\)
1656.2.a.n	$1656$	$2$	1656.a	1.a	$1$	$3$	$3$	$13.223$	3.3.148.1	None	✓		✓		552.2.a.g	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}+(1+\beta _{1})q^{7}+(-1+\beta _{1}+\beta _{2})q^{11}+\cdots\)
1656.2.a.o	$1656$	$2$	1656.a	1.a	$1$	$4$	$4$	$13.223$	\(\Q(\sqrt{13 +4 \sqrt{7}})\)	None	✓	✓	✓	✓	1656.2.a.o	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{5}-\beta _{1}q^{7}+(-\beta _{2}+\beta _{3})q^{11}+\cdots\)
1656.2.a.p	$1656$	$2$	1656.a	1.a	$1$	$4$	$4$	$13.223$	\(\Q(\sqrt{13 +4 \sqrt{7}})\)	None	✓	✓	✓	✓	1656.2.a.o	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{5}-\beta _{1}q^{7}+(\beta _{2}-\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1656))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1656)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(414))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(552))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(828))\)\(^{\oplus 2}\)