Space of modular forms of level 3392, weight 2, and trivial character

• Label: 3392.2.a
• Level: $3392$
• Weight: $2$
• Character orbit: 3392.a
• Rep. character: $\chi_{3392}(1,\cdot)$
• Character field: $\Q$
• Dimension: $104$
• Newform subspaces: $41$
• Sturm bound: $864$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 3392 = 2^{6} \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3392.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 41 \)
Sturm bound:			\(864\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3392))\).

	Total	New	Old
Modular forms	444	104	340
Cusp forms	421	104	317
Eisenstein series	23	0	23

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

\(2\)	\(53\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(104\)	\(23\)	\(81\)		\(99\)	\(23\)	\(76\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(116\)	\(29\)	\(87\)		\(110\)	\(29\)	\(81\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(118\)	\(29\)	\(89\)		\(112\)	\(29\)	\(83\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(106\)	\(23\)	\(83\)		\(100\)	\(23\)	\(77\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(210\)	\(46\)	\(164\)		\(199\)	\(46\)	\(153\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(234\)	\(58\)	\(176\)		\(222\)	\(58\)	\(164\)		\(12\)	\(0\)	\(12\)

Trace form

\( 104 q + 104 q^{9} + 16 q^{13} + 16 q^{21} + 104 q^{25} - 16 q^{33} + 16 q^{37} - 16 q^{41} + 104 q^{49} - 16 q^{57} + 16 q^{61} + 48 q^{77} + 56 q^{81} + 48 q^{85} - 32 q^{89} - 32 q^{93}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3392))\) 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	53
3392.2.a.a	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				53.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{7}+6q^{9}+3q^{13}-3q^{17}+\cdots\)
3392.2.a.b	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				1696.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{5}+2q^{7}+6q^{9}-6q^{11}+\cdots\)
3392.2.a.c	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				106.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-3q^{5}-2q^{7}+q^{9}-3q^{11}+\cdots\)
3392.2.a.d	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				212.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{5}+q^{9}+4q^{11}+2q^{13}+\cdots\)
3392.2.a.e	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				106.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}-2q^{7}+q^{9}-5q^{11}+\cdots\)
3392.2.a.f	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				1696.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+2q^{7}+q^{9}+3q^{11}+\cdots\)
3392.2.a.g	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				106.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}-5q^{13}-3q^{17}+\cdots\)
3392.2.a.h	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				212.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+2q^{7}-2q^{9}+2q^{11}+\cdots\)
3392.2.a.i	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				106.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}-2q^{9}-4q^{11}-q^{13}+\cdots\)
3392.2.a.j	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				1696.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+4q^{7}-2q^{9}+3q^{13}+\cdots\)
3392.2.a.k	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				106.2.a.d	$1$	$1$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}-5q^{13}-3q^{17}+\cdots\)
3392.2.a.l	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				212.2.a.a	$1$	$1$	\(0\)	\(1\)	\(2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-2q^{7}-2q^{9}-2q^{11}+\cdots\)
3392.2.a.m	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				1696.2.a.c	$1$	$0$	\(0\)	\(1\)	\(4\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}-4q^{7}-2q^{9}+3q^{13}+\cdots\)
3392.2.a.n	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				106.2.a.a	$1$	$0$	\(0\)	\(1\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}-2q^{9}+4q^{11}-q^{13}+\cdots\)
3392.2.a.o	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				106.2.a.c	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{5}+2q^{7}+q^{9}+3q^{11}+\cdots\)
3392.2.a.p	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				212.2.a.b	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{5}+q^{9}-4q^{11}+2q^{13}+\cdots\)
3392.2.a.q	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				1696.2.a.b	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}-2q^{7}+q^{9}-3q^{11}+\cdots\)
3392.2.a.r	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				106.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+2q^{7}+q^{9}+5q^{11}+\cdots\)
3392.2.a.s	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				53.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{7}+6q^{9}+3q^{13}-3q^{17}+\cdots\)
3392.2.a.t	$3392$	$2$	3392.a	1.a	$1$	$1$	$1$	$27.085$	\(\Q\)	None	✓				1696.2.a.a	$1$	$0$	\(0\)	\(3\)	\(2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}-2q^{7}+6q^{9}+6q^{11}+\cdots\)
3392.2.a.u	$3392$	$2$	3392.a	1.a	$1$	$2$	$2$	$27.085$	\(\Q(\sqrt{2}) \)	None	✓				424.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(4\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+2q^{5}+(-2+2\beta )q^{7}+\cdots\)
3392.2.a.v	$3392$	$2$	3392.a	1.a	$1$	$2$	$2$	$27.085$	\(\Q(\sqrt{5}) \)	None	✓				1696.2.a.g	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2q^{5}-2\beta q^{7}+2q^{9}-2\beta q^{11}+\cdots\)
3392.2.a.w	$3392$	$2$	3392.a	1.a	$1$	$2$	$2$	$27.085$	\(\Q(\sqrt{2}) \)	None	✓				424.2.a.a	$1$	$0$	\(0\)	\(2\)	\(4\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2q^{5}+(2+2\beta )q^{7}+2\beta q^{9}+\cdots\)
3392.2.a.x	$3392$	$2$	3392.a	1.a	$1$	$3$	$3$	$27.085$	3.3.756.1	None	✓				212.2.a.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+\beta _{2}q^{5}+(-2-\beta _{2})q^{7}+\cdots\)
3392.2.a.y	$3392$	$2$	3392.a	1.a	$1$	$3$	$3$	$27.085$	3.3.148.1	None	✓				53.2.a.b	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}+(1-\beta _{1}-\beta _{2})q^{5}+\cdots\)
3392.2.a.z	$3392$	$2$	3392.a	1.a	$1$	$3$	$3$	$27.085$	3.3.316.1	None	✓				424.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(-3\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{2})q^{5}+2q^{7}+\cdots\)
3392.2.a.ba	$3392$	$2$	3392.a	1.a	$1$	$3$	$3$	$27.085$	3.3.148.1	None	✓				424.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(3-\beta _{1}+\cdots)q^{7}+\cdots\)
3392.2.a.bb	$3392$	$2$	3392.a	1.a	$1$	$3$	$3$	$27.085$	3.3.148.1	None	✓				424.2.a.b	$1$	$1$	\(0\)	\(1\)	\(2\)	\(-8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(-3+\beta _{1}+\cdots)q^{7}+\cdots\)
3392.2.a.bc	$3392$	$2$	3392.a	1.a	$1$	$3$	$3$	$27.085$	3.3.316.1	None	✓				424.2.a.c	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{2})q^{5}-2q^{7}+\cdots\)
3392.2.a.bd	$3392$	$2$	3392.a	1.a	$1$	$3$	$3$	$27.085$	3.3.756.1	None	✓				212.2.a.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+\beta _{2}q^{5}+(2+\beta _{2})q^{7}+\cdots\)
3392.2.a.be	$3392$	$2$	3392.a	1.a	$1$	$3$	$3$	$27.085$	3.3.148.1	None	✓				53.2.a.b	$1$	$0$	\(0\)	\(3\)	\(2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{3}+(1-\beta _{1}-\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots\)
3392.2.a.bf	$3392$	$2$	3392.a	1.a	$1$	$4$	$4$	$27.085$	4.4.6224.1	None	✓				1696.2.a.j	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
3392.2.a.bg	$3392$	$2$	3392.a	1.a	$1$	$4$	$4$	$27.085$	4.4.6224.1	None	✓				1696.2.a.j	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
3392.2.a.bh	$3392$	$2$	3392.a	1.a	$1$	$4$	$4$	$27.085$	\(\Q(\zeta_{24})^+\)	None	✓				1696.2.a.h	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{5}+(-\beta _{1}-\beta _{3})q^{7}+\cdots\)
3392.2.a.bi	$3392$	$2$	3392.a	1.a	$1$	$4$	$4$	$27.085$	\(\Q(\sqrt{2}, \sqrt{7})\)	None	✓				1696.2.a.i	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{3})q^{5}-\beta _{2}q^{7}+(1+\beta _{3})q^{9}+\cdots\)
3392.2.a.bj	$3392$	$2$	3392.a	1.a	$1$	$5$	$5$	$27.085$	5.5.4321108.1	None	✓		✓		424.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-5\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{3})q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
3392.2.a.bk	$3392$	$2$	3392.a	1.a	$1$	$5$	$5$	$27.085$	5.5.4321108.1	None	✓				424.2.a.d	$1$	$0$	\(0\)	\(1\)	\(-5\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{3})q^{5}+(1+\beta _{2}+\beta _{4})q^{7}+\cdots\)
3392.2.a.bl	$3392$	$2$	3392.a	1.a	$1$	$6$	$6$	$27.085$	6.6.27160832.1	None	✓		✓		1696.2.a.l	$2$	$1$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(-2+\beta _{2})q^{5}+\beta _{4}q^{7}+(2+\cdots)q^{9}+\cdots\)
3392.2.a.bm	$3392$	$2$	3392.a	1.a	$1$	$7$	$7$	$27.085$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1696.2.a.m	$1$	$0$	\(0\)	\(-5\)	\(-1\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+\beta _{3}q^{5}+(\beta _{1}+\beta _{5}+\cdots)q^{7}+\cdots\)
3392.2.a.bn	$3392$	$2$	3392.a	1.a	$1$	$7$	$7$	$27.085$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1696.2.a.m	$1$	$0$	\(0\)	\(5\)	\(-1\)	\(-4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+\beta _{3}q^{5}+(-\beta _{1}-\beta _{5}+\cdots)q^{7}+\cdots\)
3392.2.a.bo	$3392$	$2$	3392.a	1.a	$1$	$8$	$8$	$27.085$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		1696.2.a.o	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{3})q^{5}+\beta _{4}q^{7}+(1+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3392))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3392)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(53))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(106))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(212))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(424))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(848))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1696))\)\(^{\oplus 2}\)