Space of modular forms of level 9196, weight 2, and trivial character

• Label: 9196.2.a
• Level: $9196$
• Weight: $2$
• Character orbit: 9196.a
• Rep. character: $\chi_{9196}(1,\cdot)$
• Character field: $\Q$
• Dimension: $163$
• Newform subspaces: $24$
• Sturm bound: $2640$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(2640\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9196))\).

	Total	New	Old
Modular forms	1356	163	1193
Cusp forms	1285	163	1122
Eisenstein series	71	0	71

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

\(2\)	\(11\)	\(19\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(41\)
\(-\)	\(+\)	\(-\)	\(+\)	\(37\)
\(-\)	\(-\)	\(+\)	\(+\)	\(40\)
\(-\)	\(-\)	\(-\)	\(-\)	\(45\)
Plus space			\(+\)	\(77\)
Minus space			\(-\)	\(86\)

Trace form

163 * q - 2 * q^3 - 7 * q^5 + 3 * q^7 + 151 * q^9 - 8 * q^13 + 2 * q^15 - q^17 + q^19 - 10 * q^21 - 8 * q^23 + 152 * q^25 - 20 * q^27 - 10 * q^29 + 4 * q^31 - 19 * q^35 - 22 * q^37 - 4 * q^39 - 18 * q^41 - 13 * q^43 - 39 * q^45 - 7 * q^47 + 138 * q^49 - 34 * q^51 - 12 * q^53 + 2 * q^57 + 2 * q^59 - 19 * q^61 + 19 * q^63 + 8 * q^65 - 4 * q^67 + 4 * q^69 + 18 * q^71 + 3 * q^73 + 28 * q^75 + 24 * q^79 + 135 * q^81 + 28 * q^83 - 19 * q^85 + 32 * q^87 - 24 * q^89 - 20 * q^91 + 36 * q^93 - q^95 - 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9196))\) 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	11	19
9196.2.a.a	$9196$	$2$	9196.a	1.a	$1$	$1$	$1$	$73.430$	\(\Q\)	None	✓	✓			9196.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{7}-2q^{9}-6q^{13}+\cdots\)
9196.2.a.b	$9196$	$2$	9196.a	1.a	$1$	$1$	$1$	$73.430$	\(\Q\)	None	✓				836.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{9}+2q^{13}-q^{15}+\cdots\)
9196.2.a.c	$9196$	$2$	9196.a	1.a	$1$	$1$	$1$	$73.430$	\(\Q\)	None	✓	✓			9196.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+2q^{7}-2q^{9}+6q^{13}+\cdots\)
9196.2.a.d	$9196$	$2$	9196.a	1.a	$1$	$1$	$1$	$73.430$	\(\Q\)	None	✓	✓			9196.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-3q^{9}-q^{13}+7q^{17}-q^{19}+\cdots\)
9196.2.a.e	$9196$	$2$	9196.a	1.a	$1$	$1$	$1$	$73.430$	\(\Q\)	None	✓	✓			9196.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-3q^{9}+q^{13}-7q^{17}+q^{19}+\cdots\)
9196.2.a.f	$9196$	$2$	9196.a	1.a	$1$	$1$	$1$	$73.430$	\(\Q\)	None	✓				76.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+3q^{7}+q^{9}+4q^{13}+\cdots\)
9196.2.a.g	$9196$	$2$	9196.a	1.a	$1$	$1$	$1$	$73.430$	\(\Q\)	None	✓				836.2.a.b	$1$	$1$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}+q^{9}-4q^{13}+4q^{15}+\cdots\)
9196.2.a.h	$9196$	$2$	9196.a	1.a	$1$	$2$	$2$	$73.430$	\(\Q(\sqrt{2}) \)	None	✓				836.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-q^{5}+\beta q^{7}-2\beta q^{9}+\cdots\)
9196.2.a.i	$9196$	$2$	9196.a	1.a	$1$	$6$	$6$	$73.430$	6.6.34963625.1	None	✓				836.2.j.a	$1$	$1$	\(0\)	\(-2\)	\(-7\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{5})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
9196.2.a.j	$9196$	$2$	9196.a	1.a	$1$	$6$	$6$	$73.430$	6.6.34963625.1	None	✓				836.2.j.a	$1$	$1$	\(0\)	\(-2\)	\(-7\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{5})q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
9196.2.a.k	$9196$	$2$	9196.a	1.a	$1$	$6$	$6$	$73.430$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				836.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{5}q^{5}-\beta _{4}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
9196.2.a.l	$9196$	$2$	9196.a	1.a	$1$	$6$	$6$	$73.430$	6.6.114134848.1	None	✓	✓			9196.2.a.l	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{5})q^{5}+(-\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots\)
9196.2.a.m	$9196$	$2$	9196.a	1.a	$1$	$6$	$6$	$73.430$	6.6.114134848.1	None	✓	✓			9196.2.a.l	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{5})q^{5}+(\beta _{1}+\beta _{4}+\cdots)q^{7}+\cdots\)
9196.2.a.n	$9196$	$2$	9196.a	1.a	$1$	$6$	$6$	$73.430$	6.6.744786576.1	None	✓				836.2.a.e	$1$	$0$	\(0\)	\(1\)	\(5\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{5})q^{5}+(1-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
9196.2.a.o	$9196$	$2$	9196.a	1.a	$1$	$7$	$7$	$73.430$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			9196.2.a.o	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{4}q^{5}-\beta _{2}q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
9196.2.a.p	$9196$	$2$	9196.a	1.a	$1$	$7$	$7$	$73.430$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			9196.2.a.o	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{4}q^{5}+\beta _{2}q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
9196.2.a.q	$9196$	$2$	9196.a	1.a	$1$	$8$	$8$	$73.430$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			9196.2.a.q	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{4}q^{5}+(-\beta _{2}+\beta _{3}+\beta _{6}+\cdots)q^{7}+\cdots\)
9196.2.a.r	$9196$	$2$	9196.a	1.a	$1$	$8$	$8$	$73.430$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			9196.2.a.q	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{4}q^{5}+(\beta _{2}-\beta _{3}-\beta _{6})q^{7}+\cdots\)
9196.2.a.s	$9196$	$2$	9196.a	1.a	$1$	$10$	$10$	$73.430$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				836.2.j.b	$1$	$1$	\(0\)	\(0\)	\(8\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{7})q^{5}+(-\beta _{5}-\beta _{8}+\cdots)q^{7}+\cdots\)
9196.2.a.t	$9196$	$2$	9196.a	1.a	$1$	$10$	$10$	$73.430$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		836.2.j.b	$1$	$1$	\(0\)	\(0\)	\(8\)	\(5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{7})q^{5}+(\beta _{5}+\beta _{8})q^{7}+\cdots\)
9196.2.a.u	$9196$	$2$	9196.a	1.a	$1$	$14$	$14$	$73.430$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	✓	✓		9196.2.a.u	$1$	$1$	\(0\)	\(-2\)	\(-6\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{6}q^{5}+\beta _{9}q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
9196.2.a.v	$9196$	$2$	9196.a	1.a	$1$	$14$	$14$	$73.430$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	✓			9196.2.a.u	$1$	$0$	\(0\)	\(-2\)	\(-6\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{6}q^{5}-\beta _{9}q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
9196.2.a.w	$9196$	$2$	9196.a	1.a	$1$	$20$	$20$	$73.430$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		836.2.j.c	$1$	$0$	\(0\)	\(6\)	\(2\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{14}q^{5}+\beta _{10}q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
9196.2.a.x	$9196$	$2$	9196.a	1.a	$1$	$20$	$20$	$73.430$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		836.2.j.c	$1$	$0$	\(0\)	\(6\)	\(2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{14}q^{5}-\beta _{10}q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9196))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9196)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(418))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(484))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(836))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\)\(^{\oplus 2}\)