Space of modular forms of level 6936, weight 2, and trivial character

• Label: 6936.2.a
• Level: $6936$
• Weight: $2$
• Character orbit: 6936.a
• Rep. character: $\chi_{6936}(1,\cdot)$
• Character field: $\Q$
• Dimension: $135$
• Newform subspaces: $43$
• Sturm bound: $2448$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 6936 = 2^{3} \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6936.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 43 \)
Sturm bound:			\(2448\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6936))\).

	Total	New	Old
Modular forms	1296	135	1161
Cusp forms	1153	135	1018
Eisenstein series	143	0	143

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

\(2\)	\(3\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(18\)
\(+\)	\(+\)	\(-\)	$-$	\(16\)
\(+\)	\(-\)	\(+\)	$-$	\(18\)
\(+\)	\(-\)	\(-\)	$+$	\(16\)
\(-\)	\(+\)	\(+\)	$-$	\(22\)
\(-\)	\(+\)	\(-\)	$+$	\(12\)
\(-\)	\(-\)	\(+\)	$+$	\(13\)
\(-\)	\(-\)	\(-\)	$-$	\(20\)
Plus space			\(+\)	\(59\)
Minus space			\(-\)	\(76\)

Trace form

\( 135 q - q^{3} + 2 q^{5} + 135 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6936))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	17
6936.2.a.a	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-4\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+5q^{7}+q^{9}-q^{13}+4q^{15}+\cdots\)
6936.2.a.b	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
6936.2.a.c	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-4q^{7}+q^{9}-3q^{11}+2q^{13}+\cdots\)
6936.2.a.d	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}+q^{9}+2q^{11}-3q^{13}+\cdots\)
6936.2.a.e	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}+2q^{13}+4q^{19}+\cdots\)
6936.2.a.f	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-4q^{11}-5q^{13}+\cdots\)
6936.2.a.g	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+6q^{11}+5q^{13}+\cdots\)
6936.2.a.h	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}+4q^{7}+q^{9}-q^{11}-5q^{13}+\cdots\)
6936.2.a.i	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+q^{7}+q^{9}+2q^{11}-3q^{13}+\cdots\)
6936.2.a.j	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots\)
6936.2.a.k	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+q^{9}+q^{11}+3q^{13}+\cdots\)
6936.2.a.l	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-6q^{11}+5q^{13}+\cdots\)
6936.2.a.m	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+4q^{11}-5q^{13}+\cdots\)
6936.2.a.n	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots\)
6936.2.a.o	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+4q^{7}+q^{9}+3q^{11}+2q^{13}+\cdots\)
6936.2.a.p	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
6936.2.a.q	$6936$	$2$	6936.a	1.a	$1$	$1$	$1$	$55.384$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(4\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}-5q^{7}+q^{9}-q^{13}+4q^{15}+\cdots\)
6936.2.a.r	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta )q^{5}-2q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
6936.2.a.s	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(2-\beta )q^{7}+q^{9}-3q^{11}+\cdots\)
6936.2.a.t	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta q^{5}+(1-\beta )q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
6936.2.a.u	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{57}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(-8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}-4q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
6936.2.a.v	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+\beta )q^{5}-2q^{7}+q^{9}+4q^{11}+\cdots\)
6936.2.a.w	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}+2q^{7}+q^{9}-4q^{11}+\cdots\)
6936.2.a.x	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(1\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+(-1+\beta )q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
6936.2.a.y	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(1\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+(2-2\beta )q^{7}+q^{9}+(4+\cdots)q^{11}+\cdots\)
6936.2.a.z	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(2\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(-2-\beta )q^{7}+q^{9}+3q^{11}+\cdots\)
6936.2.a.ba	$6936$	$2$	6936.a	1.a	$1$	$2$	$2$	$55.384$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(3\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{5}+2q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
6936.2.a.bb	$6936$	$2$	6936.a	1.a	$1$	$3$	$3$	$55.384$	3.3.316.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta _{1})q^{5}+(1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
6936.2.a.bc	$6936$	$2$	6936.a	1.a	$1$	$3$	$3$	$55.384$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{1})q^{5}+(1+\beta _{1}-\beta _{2})q^{7}+\cdots\)
6936.2.a.bd	$6936$	$2$	6936.a	1.a	$1$	$3$	$3$	$55.384$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{1})q^{5}+(-1-\beta _{1}+\beta _{2})q^{7}+\cdots\)
6936.2.a.be	$6936$	$2$	6936.a	1.a	$1$	$3$	$3$	$55.384$	3.3.316.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(-2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta _{1})q^{5}+(-1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
6936.2.a.bf	$6936$	$2$	6936.a	1.a	$1$	$6$	$6$	$55.384$	6.6.19952001.1	None	✓	$1$	$0$	\(0\)	\(-6\)	\(-3\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{5}+(-\beta _{2}-\beta _{3}+\beta _{5})q^{7}+\cdots\)
6936.2.a.bg	$6936$	$2$	6936.a	1.a	$1$	$6$	$6$	$55.384$	6.6.50874368.1	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{5}q^{5}+(-1+\beta _{3})q^{7}+q^{9}+\cdots\)
6936.2.a.bh	$6936$	$2$	6936.a	1.a	$1$	$6$	$6$	$55.384$	6.6.14178321.1	None	✓	$1$	$1$	\(0\)	\(-6\)	\(3\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{3})q^{5}+(-1-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
6936.2.a.bi	$6936$	$2$	6936.a	1.a	$1$	$6$	$6$	$55.384$	6.6.14178321.1	None	✓	$1$	$1$	\(0\)	\(6\)	\(-3\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{3})q^{5}+(1+\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
6936.2.a.bj	$6936$	$2$	6936.a	1.a	$1$	$6$	$6$	$55.384$	6.6.50874368.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{5}q^{5}+(1-\beta _{3})q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
6936.2.a.bk	$6936$	$2$	6936.a	1.a	$1$	$6$	$6$	$55.384$	6.6.19952001.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(3\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{5}+\beta _{5}q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
6936.2.a.bl	$6936$	$2$	6936.a	1.a	$1$	$8$	$8$	$55.384$	8.8.\(\cdots\).1	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3})q^{5}-\beta _{7}q^{7}+\cdots\)
6936.2.a.bm	$6936$	$2$	6936.a	1.a	$1$	$8$	$8$	$55.384$	8.8.\(\cdots\).1	None	✓	$1$	$0$	\(0\)	\(-8\)	\(4\)	\(8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(\beta _{2}+\beta _{3}-\beta _{7})q^{5}+(\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
6936.2.a.bn	$6936$	$2$	6936.a	1.a	$1$	$8$	$8$	$55.384$	8.8.\(\cdots\).1	None	✓	$1$	$1$	\(0\)	\(8\)	\(-4\)	\(-8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-\beta _{2}-\beta _{3}+\beta _{7})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
6936.2.a.bo	$6936$	$2$	6936.a	1.a	$1$	$8$	$8$	$55.384$	8.8.\(\cdots\).1	None	✓	$1$	$0$	\(0\)	\(8\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(\beta _{1}+\beta _{2}+\beta _{3})q^{5}+\beta _{7}q^{7}+\cdots\)
6936.2.a.bp	$6936$	$2$	6936.a	1.a	$1$	$9$	$9$	$55.384$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-9\)	\(-3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{5}-\beta _{7}q^{7}+q^{9}+(-\beta _{3}+\cdots)q^{11}+\cdots\)
6936.2.a.bq	$6936$	$2$	6936.a	1.a	$1$	$9$	$9$	$55.384$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}+\beta _{7}q^{7}+q^{9}+(\beta _{3}+\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6936))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6936)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(408))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(578))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(867))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1734))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2312))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3468))\)\(^{\oplus 2}\)