Space of modular forms of level 1176, weight 2, and trivial character

• Label: 1176.2.a
• Level: $1176$
• Weight: $2$
• Character orbit: 1176.a
• Rep. character: $\chi_{1176}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $15$
• Sturm bound: $448$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(448\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1176))\).

	Total	New	Old
Modular forms	256	21	235
Cusp forms	193	21	172
Eisenstein series	63	0	63

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

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(28\)	\(3\)	\(25\)		\(21\)	\(3\)	\(18\)		\(7\)	\(0\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)		\(35\)	\(2\)	\(33\)		\(27\)	\(2\)	\(25\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(32\)	\(3\)	\(29\)		\(24\)	\(3\)	\(21\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(33\)	\(2\)	\(31\)		\(25\)	\(2\)	\(23\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(36\)	\(4\)	\(32\)		\(28\)	\(4\)	\(24\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)		\(29\)	\(1\)	\(28\)		\(21\)	\(1\)	\(20\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)		\(32\)	\(2\)	\(30\)		\(24\)	\(2\)	\(22\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)		\(31\)	\(4\)	\(27\)		\(23\)	\(4\)	\(19\)		\(8\)	\(0\)	\(8\)
Plus space			\(+\)		\(122\)	\(8\)	\(114\)		\(91\)	\(8\)	\(83\)		\(31\)	\(0\)	\(31\)
Minus space			\(-\)		\(134\)	\(13\)	\(121\)		\(102\)	\(13\)	\(89\)		\(32\)	\(0\)	\(32\)

Trace form

21 * q + q^3 - 2 * q^5 + 21 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 - 6 * q^17 + 4 * q^19 + 16 * q^23 + 23 * q^25 + q^27 - 2 * q^29 + 8 * q^31 + 4 * q^33 - 6 * q^37 + 6 * q^39 + 18 * q^41 - 16 * q^43 - 2 * q^45 - 10 * q^51 + 46 * q^53 + 8 * q^55 - 20 * q^59 + 14 * q^61 + 28 * q^65 - 24 * q^67 - 8 * q^69 - 8 * q^71 + 2 * q^73 - q^75 + 21 * q^81 - 12 * q^83 + 44 * q^85 - 10 * q^87 + 2 * q^89 + 20 * q^93 + 8 * q^95 - 22 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\) 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	7
1176.2.a.a	$1176$	$2$	1176.a	1.a	$1$	$1$	$1$	$9.390$	\(\Q\)	None	✓				168.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}+2q^{13}+2q^{15}+\cdots\)
1176.2.a.b	$1176$	$2$	1176.a	1.a	$1$	$1$	$1$	$9.390$	\(\Q\)	None	✓	✓	✓	✓	1176.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-4q^{13}-4q^{17}+4q^{19}+\cdots\)
1176.2.a.c	$1176$	$2$	1176.a	1.a	$1$	$1$	$1$	$9.390$	\(\Q\)	None	✓				168.2.q.a	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}+3q^{11}-4q^{13}+\cdots\)
1176.2.a.d	$1176$	$2$	1176.a	1.a	$1$	$1$	$1$	$9.390$	\(\Q\)	None	✓				168.2.q.b	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}-6q^{11}-3q^{13}+\cdots\)
1176.2.a.e	$1176$	$2$	1176.a	1.a	$1$	$1$	$1$	$9.390$	\(\Q\)	None	✓				168.2.q.b	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}-6q^{11}+3q^{13}+\cdots\)
1176.2.a.f	$1176$	$2$	1176.a	1.a	$1$	$1$	$1$	$9.390$	\(\Q\)	None	✓				168.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}-6q^{13}-2q^{15}+\cdots\)
1176.2.a.g	$1176$	$2$	1176.a	1.a	$1$	$1$	$1$	$9.390$	\(\Q\)	None	✓				168.2.q.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}+3q^{11}+4q^{13}+\cdots\)
1176.2.a.h	$1176$	$2$	1176.a	1.a	$1$	$1$	$1$	$9.390$	\(\Q\)	None	✓	✓			1176.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+4q^{13}+4q^{17}-4q^{19}+\cdots\)
1176.2.a.i	$1176$	$2$	1176.a	1.a	$1$	$1$	$1$	$9.390$	\(\Q\)	None	✓				24.2.a.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+4q^{11}+2q^{13}+\cdots\)
1176.2.a.j	$1176$	$2$	1176.a	1.a	$1$	$2$	$2$	$9.390$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		1176.2.a.j	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2+\beta )q^{5}+q^{9}+(2-2\beta )q^{11}+\cdots\)
1176.2.a.k	$1176$	$2$	1176.a	1.a	$1$	$2$	$2$	$9.390$	\(\Q(\sqrt{57}) \)	None	✓				168.2.q.c	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta q^{5}+q^{9}-\beta q^{11}+(3-\beta )q^{13}+\cdots\)
1176.2.a.l	$1176$	$2$	1176.a	1.a	$1$	$2$	$2$	$9.390$	\(\Q(\sqrt{2}) \)	None	✓	✓			1176.2.a.l	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+\beta )q^{5}+q^{9}+(-2+2\beta )q^{11}+\cdots\)
1176.2.a.m	$1176$	$2$	1176.a	1.a	$1$	$2$	$2$	$9.390$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	1176.2.a.l	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}+q^{9}+(-2-2\beta )q^{11}+\cdots\)
1176.2.a.n	$1176$	$2$	1176.a	1.a	$1$	$2$	$2$	$9.390$	\(\Q(\sqrt{57}) \)	None	✓		✓		168.2.q.c	$1$	$0$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+q^{9}-\beta q^{11}+(-3+\beta )q^{13}+\cdots\)
1176.2.a.o	$1176$	$2$	1176.a	1.a	$1$	$2$	$2$	$9.390$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		1176.2.a.j	$1$	$0$	\(0\)	\(2\)	\(4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(2+\beta )q^{5}+q^{9}+(2+2\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1176))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1176)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 2}\)