Space of modular forms of level 7644, weight 2, and trivial character

• Label: 7644.2.a
• Level: $7644$
• Weight: $2$
• Character orbit: 7644.a
• Rep. character: $\chi_{7644}(1,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $32$
• Sturm bound: $3136$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7644.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(3136\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7644))\).

	Total	New	Old
Modular forms	1616	82	1534
Cusp forms	1521	82	1439
Eisenstein series	95	0	95

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

\(2\)	\(3\)	\(7\)	\(13\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(11\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(12\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(9\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(11\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(9\)
Plus space				\(+\)	\(41\)
Minus space				\(-\)	\(41\)

Trace form

\( 82 q + 4 q^{5} + 82 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7644))\) 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	7	13
7644.2.a.a	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-q^{13}+6q^{17}-2q^{19}+\cdots\)
7644.2.a.b	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}+2q^{11}-q^{13}-q^{15}+\cdots\)
7644.2.a.c	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}+q^{13}-2q^{15}+\cdots\)
7644.2.a.d	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}+q^{9}+2q^{11}+q^{13}+\cdots\)
7644.2.a.e	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+q^{9}+2q^{11}-q^{13}+\cdots\)
7644.2.a.f	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}-6q^{11}+q^{13}-q^{15}+\cdots\)
7644.2.a.g	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}+2q^{11}+q^{13}-q^{15}+\cdots\)
7644.2.a.h	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-2q^{11}+q^{13}-4q^{17}+\cdots\)
7644.2.a.i	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}+2q^{11}-q^{13}+q^{15}+\cdots\)
7644.2.a.j	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+q^{13}+2q^{15}+\cdots\)
7644.2.a.k	$7644$	$2$	7644.a	1.a	$1$	$1$	$1$	$61.038$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(4\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+q^{9}-4q^{11}-q^{13}+\cdots\)
7644.2.a.l	$7644$	$2$	7644.a	1.a	$1$	$2$	$2$	$61.038$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta )q^{5}+q^{9}+(-3+\beta )q^{11}+\cdots\)
7644.2.a.m	$7644$	$2$	7644.a	1.a	$1$	$2$	$2$	$61.038$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta )q^{5}+q^{9}+(1-3\beta )q^{11}+\cdots\)
7644.2.a.n	$7644$	$2$	7644.a	1.a	$1$	$2$	$2$	$61.038$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta q^{5}+q^{9}+2q^{11}+q^{13}+\cdots\)
7644.2.a.o	$7644$	$2$	7644.a	1.a	$1$	$2$	$2$	$61.038$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2\beta q^{5}+q^{9}+(-1-\beta )q^{11}+\cdots\)
7644.2.a.p	$7644$	$2$	7644.a	1.a	$1$	$2$	$2$	$61.038$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta )q^{5}+q^{9}+(1-\beta )q^{11}+\cdots\)
7644.2.a.q	$7644$	$2$	7644.a	1.a	$1$	$2$	$2$	$61.038$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2\beta q^{5}+q^{9}+(-1+\beta )q^{11}+\cdots\)
7644.2.a.r	$7644$	$2$	7644.a	1.a	$1$	$2$	$2$	$61.038$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{5}+q^{9}+(-3-\beta )q^{11}+\cdots\)
7644.2.a.s	$7644$	$2$	7644.a	1.a	$1$	$2$	$2$	$61.038$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{5}+q^{9}+(1+3\beta )q^{11}+\cdots\)
7644.2.a.t	$7644$	$2$	7644.a	1.a	$1$	$3$	$3$	$61.038$	3.3.1373.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{2}q^{5}+q^{9}+\beta _{1}q^{11}-q^{13}+\cdots\)
7644.2.a.u	$7644$	$2$	7644.a	1.a	$1$	$3$	$3$	$61.038$	3.3.321.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{2}q^{5}+q^{9}+(1+\beta _{1}-\beta _{2})q^{11}+\cdots\)
7644.2.a.v	$7644$	$2$	7644.a	1.a	$1$	$3$	$3$	$61.038$	3.3.321.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}+q^{9}+(2\beta _{1}+\beta _{2})q^{11}+\cdots\)
7644.2.a.w	$7644$	$2$	7644.a	1.a	$1$	$3$	$3$	$61.038$	3.3.321.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2-\beta _{1})q^{5}+q^{9}-\beta _{2}q^{11}+\cdots\)
7644.2.a.x	$7644$	$2$	7644.a	1.a	$1$	$3$	$3$	$61.038$	3.3.321.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta _{1})q^{5}+q^{9}-\beta _{2}q^{11}+\cdots\)
7644.2.a.y	$7644$	$2$	7644.a	1.a	$1$	$3$	$3$	$61.038$	3.3.321.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(-1\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{1}q^{5}+q^{9}+(2\beta _{1}+\beta _{2})q^{11}+\cdots\)
7644.2.a.z	$7644$	$2$	7644.a	1.a	$1$	$3$	$3$	$61.038$	3.3.321.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{2}q^{5}+q^{9}+(1+\beta _{1}-\beta _{2})q^{11}+\cdots\)
7644.2.a.ba	$7644$	$2$	7644.a	1.a	$1$	$5$	$5$	$61.038$	5.5.20094696.1	None	✓	$1$	$1$	\(0\)	\(-5\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{5}+q^{9}+(\beta _{1}-\beta _{4})q^{11}+\cdots\)
7644.2.a.bb	$7644$	$2$	7644.a	1.a	$1$	$5$	$5$	$61.038$	5.5.20094696.1	None	✓	$1$	$0$	\(0\)	\(5\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}+q^{9}+(\beta _{1}-\beta _{4})q^{11}+\cdots\)
7644.2.a.bc	$7644$	$2$	7644.a	1.a	$1$	$6$	$6$	$61.038$	6.6.566222848.1	None	✓	$1$	$0$	\(0\)	\(-6\)	\(-2\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{5}+q^{9}+(2-\beta _{1}+\beta _{3}+\cdots)q^{11}+\cdots\)
7644.2.a.bd	$7644$	$2$	7644.a	1.a	$1$	$6$	$6$	$61.038$	6.6.71950336.1	None	✓	$1$	$1$	\(0\)	\(-6\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{3}q^{5}+q^{9}+(-2-\beta _{2}-\beta _{3}+\cdots)q^{11}+\cdots\)
7644.2.a.be	$7644$	$2$	7644.a	1.a	$1$	$6$	$6$	$61.038$	6.6.71950336.1	None	✓	$1$	$1$	\(0\)	\(6\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{3}q^{5}+q^{9}+(-2-\beta _{2}-\beta _{3}+\cdots)q^{11}+\cdots\)
7644.2.a.bf	$7644$	$2$	7644.a	1.a	$1$	$6$	$6$	$61.038$	6.6.566222848.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(2\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}+q^{9}+(2-\beta _{1}+\beta _{3}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7644))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7644)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(273))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(364))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(546))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(637))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1092))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1274))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1911))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2548))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3822))\)\(^{\oplus 2}\)