Space of modular forms of level 3192, weight 2, and trivial character

• Label: 3192.2.a
• Level: $3192$
• Weight: $2$
• Character orbit: 3192.a
• Rep. character: $\chi_{3192}(1,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $28$
• Sturm bound: $1280$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(1280\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3192))\).

	Total	New	Old
Modular forms	656	56	600
Cusp forms	625	56	569
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.

\(2\)	\(3\)	\(7\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(3\)
Plus space				\(+\)	\(24\)
Minus space				\(-\)	\(32\)

Trace form

\( 56 q - 8 q^{5} + 56 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3192))\) 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	19
3192.2.a.a	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}-q^{7}+q^{9}-2q^{13}+4q^{15}+\cdots\)
3192.2.a.b	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
3192.2.a.c	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-q^{7}+q^{9}-2q^{13}+2q^{15}+\cdots\)
3192.2.a.d	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{7}+q^{9}-2q^{13}+2q^{15}+\cdots\)
3192.2.a.e	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-2q^{13}+q^{19}+q^{21}+\cdots\)
3192.2.a.f	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
3192.2.a.g	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}-4q^{13}+\cdots\)
3192.2.a.h	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
3192.2.a.i	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.i	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
3192.2.a.j	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.j	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
3192.2.a.k	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.k	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
3192.2.a.l	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.l	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}+q^{7}+q^{9}+2q^{11}-4q^{13}+\cdots\)
3192.2.a.m	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.m	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
3192.2.a.n	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.n	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+6q^{11}+4q^{13}+\cdots\)
3192.2.a.o	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.o	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-2q^{13}-8q^{17}+\cdots\)
3192.2.a.p	$3192$	$2$	3192.a	1.a	$1$	$1$	$1$	$25.488$	\(\Q\)	None	✓	✓			3192.2.a.p	$1$	$1$	\(0\)	\(1\)	\(2\)	\(-1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
3192.2.a.q	$3192$	$2$	3192.a	1.a	$1$	$2$	$2$	$25.488$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		3192.2.a.q	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{5}+q^{7}+q^{9}+(-2+\cdots)q^{13}+\cdots\)
3192.2.a.r	$3192$	$2$	3192.a	1.a	$1$	$2$	$2$	$25.488$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		3192.2.a.r	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}-q^{7}+q^{9}+2\beta q^{11}+\cdots\)
3192.2.a.s	$3192$	$2$	3192.a	1.a	$1$	$2$	$2$	$25.488$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		3192.2.a.s	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta )q^{5}+q^{7}+q^{9}-4q^{11}+\cdots\)
3192.2.a.t	$3192$	$2$	3192.a	1.a	$1$	$2$	$2$	$25.488$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		3192.2.a.t	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}-q^{7}+q^{9}-2q^{13}+\beta q^{15}+\cdots\)
3192.2.a.u	$3192$	$2$	3192.a	1.a	$1$	$3$	$3$	$25.488$	3.3.568.1	None	✓	✓	✓		3192.2.a.u	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-3\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{5}-q^{7}+q^{9}+(-1-\beta _{1}+\cdots)q^{11}+\cdots\)
3192.2.a.v	$3192$	$2$	3192.a	1.a	$1$	$3$	$3$	$25.488$	3.3.316.1	None	✓	✓	✓		3192.2.a.v	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+(-1+\beta _{1})q^{11}+\cdots\)
3192.2.a.w	$3192$	$2$	3192.a	1.a	$1$	$4$	$4$	$25.488$	4.4.13068.1	None	✓	✓	✓	✓	3192.2.a.w	$1$	$0$	\(0\)	\(-4\)	\(-4\)	\(4\)		$+$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{2})q^{5}+q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
3192.2.a.x	$3192$	$2$	3192.a	1.a	$1$	$4$	$4$	$25.488$	4.4.9248.1	None	✓	✓	✓	✓	3192.2.a.x	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}-q^{7}+q^{9}+(-\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
3192.2.a.y	$3192$	$2$	3192.a	1.a	$1$	$4$	$4$	$25.488$	4.4.7232.1	None	✓	✓	✓	✓	3192.2.a.y	$1$	$0$	\(0\)	\(4\)	\(4\)	\(-4\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{5}-q^{7}+q^{9}-\beta _{2}q^{11}+\cdots\)
3192.2.a.z	$3192$	$2$	3192.a	1.a	$1$	$4$	$4$	$25.488$	4.4.2225.1	None	✓	✓	✓	✓	3192.2.a.z	$1$	$0$	\(0\)	\(4\)	\(4\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{2})q^{5}+q^{7}+q^{9}+(1+\beta _{3})q^{11}+\cdots\)
3192.2.a.ba	$3192$	$2$	3192.a	1.a	$1$	$5$	$5$	$25.488$	5.5.135076.1	None	✓	✓	✓	✓	3192.2.a.ba	$1$	$0$	\(0\)	\(-5\)	\(2\)	\(5\)		$-$	$+$	$-$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}-\beta _{4}q^{11}+\cdots\)
3192.2.a.bb	$3192$	$2$	3192.a	1.a	$1$	$5$	$5$	$25.488$	5.5.401584.1	None	✓	✓	✓	✓	3192.2.a.bb	$1$	$0$	\(0\)	\(5\)	\(2\)	\(5\)		$+$	$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{1}q^{5}+q^{7}+q^{9}+(1-\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3192))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3192)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(266))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(399))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(456))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(532))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(798))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1064))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1596))\)\(^{\oplus 2}\)