Space of modular forms of level 4356, weight 2, and trivial character

• Label: 4356.2.a
• Level: $4356$
• Weight: $2$
• Character orbit: 4356.a
• Rep. character: $\chi_{4356}(1,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $26$
• Sturm bound: $1584$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4356.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(1584\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4356))\).

	Total	New	Old
Modular forms	864	46	818
Cusp forms	721	46	675
Eisenstein series	143	0	143

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

\(2\)	\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(108\)	\(0\)	\(108\)		\(85\)	\(0\)	\(85\)		\(23\)	\(0\)	\(23\)
\(+\)	\(+\)	\(-\)	\(-\)		\(114\)	\(0\)	\(114\)		\(90\)	\(0\)	\(90\)		\(24\)	\(0\)	\(24\)
\(+\)	\(-\)	\(+\)	\(-\)		\(114\)	\(0\)	\(114\)		\(90\)	\(0\)	\(90\)		\(24\)	\(0\)	\(24\)
\(+\)	\(-\)	\(-\)	\(+\)		\(108\)	\(0\)	\(108\)		\(84\)	\(0\)	\(84\)		\(24\)	\(0\)	\(24\)
\(-\)	\(+\)	\(+\)	\(-\)		\(108\)	\(12\)	\(96\)		\(96\)	\(12\)	\(84\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(102\)	\(7\)	\(95\)		\(90\)	\(7\)	\(83\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(102\)	\(12\)	\(90\)		\(90\)	\(12\)	\(78\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(108\)	\(15\)	\(93\)		\(96\)	\(15\)	\(81\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(420\)	\(19\)	\(401\)		\(349\)	\(19\)	\(330\)		\(71\)	\(0\)	\(71\)
Minus space			\(-\)		\(444\)	\(27\)	\(417\)		\(372\)	\(27\)	\(345\)		\(72\)	\(0\)	\(72\)

Trace form

46 * q + 2 * q^7 - 2 * q^13 + 6 * q^17 - 8 * q^19 - 14 * q^23 + 56 * q^25 - 8 * q^29 + 8 * q^31 - 6 * q^35 + 14 * q^37 + 8 * q^41 - 6 * q^43 - 6 * q^47 + 56 * q^49 + 10 * q^53 + 4 * q^59 - 6 * q^61 + 20 * q^65 + 14 * q^67 + 16 * q^71 + 2 * q^73 + 2 * q^79 + 38 * q^83 + 18 * q^85 - 42 * q^89 + 44 * q^91 - 40 * q^95 + 38 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4356))\) 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	11
4356.2.a.a	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	None	✓				1452.2.a.g	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-2q^{7}-5q^{13}-3q^{17}+4q^{19}+\cdots\)
4356.2.a.b	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	None	✓				1452.2.a.g	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+2q^{7}+5q^{13}+3q^{17}-4q^{19}+\cdots\)
4356.2.a.c	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	None	✓				132.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{7}-6q^{13}-4q^{17}+2q^{19}+\cdots\)
4356.2.a.d	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	None	✓				132.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{7}+2q^{13}+4q^{17}+6q^{19}+\cdots\)
4356.2.a.e	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			4356.2.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-q^{7}+2q^{13}-q^{19}-5q^{25}-7q^{31}+\cdots\)
4356.2.a.f	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			4356.2.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+q^{7}-2q^{13}+q^{19}-5q^{25}-7q^{31}+\cdots\)
4356.2.a.g	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				36.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{7}-2q^{13}-8q^{19}-5q^{25}-4q^{31}+\cdots\)
4356.2.a.h	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	None	✓				1452.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}+3q^{13}-7q^{17}-4q^{19}+\cdots\)
4356.2.a.i	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	None	✓				1452.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{7}-3q^{13}+7q^{17}+4q^{19}+\cdots\)
4356.2.a.j	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	None	✓				44.2.a.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-2q^{7}+4q^{13}+6q^{17}-8q^{19}+\cdots\)
4356.2.a.k	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	None	✓				1452.2.a.d	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-5q^{7}-2q^{13}-4q^{17}+3q^{19}+\cdots\)
4356.2.a.l	$4356$	$2$	4356.a	1.a	$1$	$1$	$1$	$34.783$	\(\Q\)	None	✓				1452.2.a.d	$1$	$0$	\(0\)	\(0\)	\(4\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+5q^{7}+2q^{13}+4q^{17}-3q^{19}+\cdots\)
4356.2.a.m	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{3}) \)	None	✓				484.2.a.e	$2$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-2\beta q^{7}+3\beta q^{13}-3\beta q^{17}+\cdots\)
4356.2.a.n	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{33}) \)	\(\Q(\sqrt{-11}) \)	✓				484.2.a.d	$2$	$1$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(-1-\beta )q^{5}+(-5+\beta )q^{23}+(4+3\beta )q^{25}+\cdots\)
4356.2.a.o	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{3}) \)	None	✓				1452.2.a.k	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{7}-\beta q^{19}-6q^{23}-5q^{25}-6\beta q^{29}+\cdots\)
4356.2.a.p	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓	✓			4356.2.a.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+\beta q^{7}-2\beta q^{13}+\beta q^{19}-5q^{25}+\cdots\)
4356.2.a.q	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓	✓			4356.2.a.q	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3\beta q^{7}-4\beta q^{13}-5\beta q^{19}-5q^{25}+\cdots\)
4356.2.a.r	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{5}) \)	None	✓				132.2.i.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-3+2\beta )q^{7}+(-1-2\beta )q^{13}+\cdots\)
4356.2.a.s	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{5}) \)	None	✓				132.2.i.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+3\beta )q^{5}+(-3+2\beta )q^{7}+(3+\cdots)q^{13}+\cdots\)
4356.2.a.t	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{5}) \)	None	✓				44.2.e.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(1-3\beta )q^{7}+(4-\beta )q^{13}+(-4+\cdots)q^{17}+\cdots\)
4356.2.a.u	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{5}) \)	None	✓				44.2.e.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-1+3\beta )q^{7}+(-4+\beta )q^{13}+\cdots\)
4356.2.a.v	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{5}) \)	None	✓				132.2.i.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+3\beta )q^{5}+(3-2\beta )q^{7}+(-3+\cdots)q^{13}+\cdots\)
4356.2.a.w	$4356$	$2$	4356.a	1.a	$1$	$2$	$2$	$34.783$	\(\Q(\sqrt{5}) \)	None	✓				132.2.i.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(3-2\beta )q^{7}+(1+2\beta )q^{13}+\cdots\)
4356.2.a.x	$4356$	$2$	4356.a	1.a	$1$	$4$	$4$	$34.783$	4.4.22000.1	None	✓		✓		396.2.j.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-q^{7}+(-3-2\beta _{2})q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
4356.2.a.y	$4356$	$2$	4356.a	1.a	$1$	$4$	$4$	$34.783$	\(\Q(\sqrt{3}, \sqrt{5})\)	None	✓	✓			4356.2.a.y	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}-2\beta _{1}q^{7}-\beta _{1}q^{13}+\beta _{3}q^{17}+\cdots\)
4356.2.a.z	$4356$	$2$	4356.a	1.a	$1$	$4$	$4$	$34.783$	4.4.22000.1	None	✓				396.2.j.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+q^{7}+(3+2\beta _{2})q^{13}+(\beta _{1}+\cdots)q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4356))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4356)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(484))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(726))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1089))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1452))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2178))\)\(^{\oplus 2}\)