Space of modular forms of level 2156, weight 2, and trivial character

• Label: 2156.2.a
• Level: $2156$
• Weight: $2$
• Character orbit: 2156.a
• Rep. character: $\chi_{2156}(1,\cdot)$
• Character field: $\Q$
• Dimension: $33$
• Newform subspaces: $13$
• Sturm bound: $672$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(672\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2156))\).

	Total	New	Old
Modular forms	360	33	327
Cusp forms	313	33	280
Eisenstein series	47	0	47

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

\(2\)	\(7\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(40\)	\(0\)	\(40\)		\(33\)	\(0\)	\(33\)		\(7\)	\(0\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)		\(52\)	\(0\)	\(52\)		\(44\)	\(0\)	\(44\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(52\)	\(0\)	\(52\)		\(44\)	\(0\)	\(44\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(40\)	\(0\)	\(40\)		\(32\)	\(0\)	\(32\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(44\)	\(9\)	\(35\)		\(40\)	\(9\)	\(31\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(44\)	\(5\)	\(39\)		\(40\)	\(5\)	\(35\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(44\)	\(8\)	\(36\)		\(40\)	\(8\)	\(32\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(44\)	\(11\)	\(33\)		\(40\)	\(11\)	\(29\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(168\)	\(13\)	\(155\)		\(145\)	\(13\)	\(132\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(192\)	\(20\)	\(172\)		\(168\)	\(20\)	\(148\)		\(24\)	\(0\)	\(24\)

Trace form

33 * q + q^3 + q^5 + 34 * q^9 - q^11 - 8 * q^13 + q^15 - 6 * q^17 - 4 * q^19 + 5 * q^23 + 20 * q^25 + 7 * q^27 - 8 * q^29 - 3 * q^31 + 3 * q^33 + 23 * q^37 + 48 * q^39 + 42 * q^43 + 14 * q^45 + 20 * q^47 + 30 * q^51 - 58 * q^53 + 3 * q^55 + 16 * q^57 + 19 * q^59 - 4 * q^61 - 32 * q^65 + 31 * q^67 + 9 * q^69 - 5 * q^71 + 12 * q^73 + 40 * q^75 + 42 * q^79 + 65 * q^81 + 10 * q^83 - 42 * q^85 - 12 * q^87 + 3 * q^89 + 45 * q^93 + 24 * q^95 - 19 * q^97 + 2 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2156))\) 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	7	11
2156.2.a.a	$2156$	$2$	2156.a	1.a	$1$	$1$	$1$	$17.216$	\(\Q\)	None	✓				44.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-2q^{9}-q^{11}+4q^{13}+\cdots\)
2156.2.a.b	$2156$	$2$	2156.a	1.a	$1$	$1$	$1$	$17.216$	\(\Q\)	None	✓				308.2.a.a	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}+q^{11}+4q^{13}+\cdots\)
2156.2.a.c	$2156$	$2$	2156.a	1.a	$1$	$2$	$2$	$17.216$	\(\Q(\sqrt{6}) \)	None	✓				308.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2q^{5}+3q^{9}-q^{11}+(-2+\cdots)q^{13}+\cdots\)
2156.2.a.d	$2156$	$2$	2156.a	1.a	$1$	$2$	$2$	$17.216$	\(\Q(\sqrt{2}) \)	None	✓	✓			2156.2.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-3q^{9}+q^{11}-3\beta q^{13}-\beta q^{17}+\cdots\)
2156.2.a.e	$2156$	$2$	2156.a	1.a	$1$	$2$	$2$	$17.216$	\(\Q(\sqrt{2}) \)	None	✓	✓			2156.2.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{9}-q^{11}-\beta q^{13}-5\beta q^{17}+\cdots\)
2156.2.a.f	$2156$	$2$	2156.a	1.a	$1$	$2$	$2$	$17.216$	\(\Q(\sqrt{2}) \)	None	✓	✓			2156.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+\beta q^{5}+5q^{9}-q^{11}-\beta q^{13}+\cdots\)
2156.2.a.g	$2156$	$2$	2156.a	1.a	$1$	$3$	$3$	$17.216$	3.3.257.1	None	✓		✓		308.2.i.b	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-1+\beta _{1})q^{5}+(1+\cdots)q^{9}+\cdots\)
2156.2.a.h	$2156$	$2$	2156.a	1.a	$1$	$3$	$3$	$17.216$	3.3.321.1	None	✓		✓		308.2.i.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(1-\beta _{1}-2\beta _{2})q^{9}+\cdots\)
2156.2.a.i	$2156$	$2$	2156.a	1.a	$1$	$3$	$3$	$17.216$	3.3.321.1	None	✓				308.2.i.a	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-\beta _{1}+\beta _{2})q^{5}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
2156.2.a.j	$2156$	$2$	2156.a	1.a	$1$	$3$	$3$	$17.216$	3.3.1016.1	None	✓				308.2.a.c	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
2156.2.a.k	$2156$	$2$	2156.a	1.a	$1$	$3$	$3$	$17.216$	3.3.257.1	None	✓				308.2.i.b	$1$	$0$	\(0\)	\(3\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
2156.2.a.l	$2156$	$2$	2156.a	1.a	$1$	$4$	$4$	$17.216$	\(\Q(\zeta_{24})^+\)	None	✓	✓	✓		2156.2.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2\beta _{1}+\beta _{3})q^{5}+(-1+\beta _{2}+\cdots)q^{9}+\cdots\)
2156.2.a.m	$2156$	$2$	2156.a	1.a	$1$	$4$	$4$	$17.216$	4.4.301088.1	None	✓	✓	✓		2156.2.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(4+\beta _{3})q^{9}+q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2156))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2156)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(539))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\)\(^{\oplus 2}\)