Space of modular forms of level 3775, weight 2, and trivial character

• Label: 3775.2.a
• Level: $3775$
• Weight: $2$
• Character orbit: 3775.a
• Rep. character: $\chi_{3775}(1,\cdot)$
• Character field: $\Q$
• Dimension: $237$
• Newform subspaces: $24$
• Sturm bound: $760$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 3775 = 5^{2} \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3775.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(760\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3775))\).

	Total	New	Old
Modular forms	386	237	149
Cusp forms	375	237	138
Eisenstein series	11	0	11

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

\(5\)	\(151\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(53\)
\(+\)	\(-\)	\(-\)	\(60\)
\(-\)	\(+\)	\(-\)	\(69\)
\(-\)	\(-\)	\(+\)	\(55\)
Plus space		\(+\)	\(108\)
Minus space		\(-\)	\(129\)

Trace form

\( 237 q + 236 q^{4} - 2 q^{6} - 2 q^{7} - 6 q^{8} + 235 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3775))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	151
3775.2.a.a	$3775$	$2$	3775.a	1.a	$1$	$1$	$1$	$30.144$	\(\Q\)	None	✓				755.2.a.f	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+2q^{4}+6q^{6}+q^{7}+\cdots\)
3775.2.a.b	$3775$	$2$	3775.a	1.a	$1$	$1$	$1$	$30.144$	\(\Q\)	None	✓				755.2.a.e	$1$	$1$	\(-2\)	\(-1\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}+2q^{6}-3q^{7}+\cdots\)
3775.2.a.c	$3775$	$2$	3775.a	1.a	$1$	$1$	$1$	$30.144$	\(\Q\)	None	✓				755.2.a.d	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+2q^{4}-6q^{6}+q^{7}+\cdots\)
3775.2.a.d	$3775$	$2$	3775.a	1.a	$1$	$1$	$1$	$30.144$	\(\Q\)	None	✓				755.2.a.c	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}-2q^{9}+\cdots\)
3775.2.a.e	$3775$	$2$	3775.a	1.a	$1$	$1$	$1$	$30.144$	\(\Q\)	None	✓				755.2.a.b	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}-q^{4}-2q^{6}-2q^{7}+3q^{8}+\cdots\)
3775.2.a.f	$3775$	$2$	3775.a	1.a	$1$	$1$	$1$	$30.144$	\(\Q\)	None	✓				755.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{7}-3q^{9}+3q^{11}+3q^{13}+\cdots\)
3775.2.a.g	$3775$	$2$	3775.a	1.a	$1$	$2$	$2$	$30.144$	\(\Q(\sqrt{13}) \)	None	✓	✓			3775.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+(3-\beta )q^{7}-3q^{8}+\cdots\)
3775.2.a.h	$3775$	$2$	3775.a	1.a	$1$	$2$	$2$	$30.144$	\(\Q(\sqrt{13}) \)	None	✓	✓			3775.2.a.g	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}+(-3+\beta )q^{7}+3q^{8}+\cdots\)
3775.2.a.i	$3775$	$2$	3775.a	1.a	$1$	$2$	$2$	$30.144$	\(\Q(\sqrt{2}) \)	None	✓				755.2.a.g	$1$	$0$	\(2\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}-q^{4}+(1+\beta )q^{6}+\cdots\)
3775.2.a.j	$3775$	$2$	3775.a	1.a	$1$	$3$	$3$	$30.144$	3.3.257.1	None	✓				151.2.a.b	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-2q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
3775.2.a.k	$3775$	$2$	3775.a	1.a	$1$	$3$	$3$	$30.144$	\(\Q(\zeta_{14})^+\)	None	✓				151.2.a.a	$1$	$1$	\(2\)	\(1\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{1}q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3775.2.a.l	$3775$	$2$	3775.a	1.a	$1$	$4$	$4$	$30.144$	\(\Q(\zeta_{24})^+\)	None	✓				755.2.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}-q^{6}+(3\beta _{1}+\cdots)q^{7}+\cdots\)
3775.2.a.m	$3775$	$2$	3775.a	1.a	$1$	$4$	$4$	$30.144$	\(\Q(\zeta_{24})^+\)	None	✓				755.2.b.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{3})q^{2}+(\beta _{1}+\beta _{3})q^{3}+2q^{6}+\cdots\)
3775.2.a.n	$3775$	$2$	3775.a	1.a	$1$	$5$	$5$	$30.144$	5.5.220669.1	None	✓				755.2.a.i	$1$	$1$	\(0\)	\(3\)	\(0\)	\(7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
3775.2.a.o	$3775$	$2$	3775.a	1.a	$1$	$5$	$5$	$30.144$	5.5.106069.1	None	✓				755.2.a.h	$1$	$0$	\(1\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{2}+\beta _{1}q^{3}+(\beta _{1}-\beta _{4})q^{4}+(-1+\cdots)q^{6}+\cdots\)
3775.2.a.p	$3775$	$2$	3775.a	1.a	$1$	$6$	$6$	$30.144$	6.6.4838537.1	None	✓				151.2.a.c	$1$	$0$	\(-1\)	\(5\)	\(0\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2}-\beta _{3}+\beta _{4})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
3775.2.a.q	$3775$	$2$	3775.a	1.a	$1$	$15$	$15$	$30.144$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓				755.2.a.j	$1$	$1$	\(2\)	\(-5\)	\(0\)	\(-11\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{6}+\cdots\)
3775.2.a.r	$3775$	$2$	3775.a	1.a	$1$	$18$	$18$	$30.144$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓				755.2.a.k	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(-4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{10}q^{3}+(2+\beta _{2})q^{4}+(1+\cdots)q^{6}+\cdots\)
3775.2.a.s	$3775$	$2$	3775.a	1.a	$1$	$22$	$22$	$30.144$		None	✓				755.2.b.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
3775.2.a.t	$3775$	$2$	3775.a	1.a	$1$	$23$	$23$	$30.144$		None	✓	✓	✓		3775.2.a.t	$1$	$1$	\(-6\)	\(-7\)	\(0\)	\(-3\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
3775.2.a.u	$3775$	$2$	3775.a	1.a	$1$	$23$	$23$	$30.144$		None	✓	✓			3775.2.a.t	$1$	$0$	\(6\)	\(7\)	\(0\)	\(3\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
3775.2.a.v	$3775$	$2$	3775.a	1.a	$1$	$25$	$25$	$30.144$		None	✓	✓	✓		3775.2.a.v	$1$	$1$	\(-7\)	\(-5\)	\(0\)	\(-8\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
3775.2.a.w	$3775$	$2$	3775.a	1.a	$1$	$25$	$25$	$30.144$		None	✓	✓	✓		3775.2.a.v	$1$	$0$	\(7\)	\(5\)	\(0\)	\(8\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
3775.2.a.x	$3775$	$2$	3775.a	1.a	$1$	$44$	$44$	$30.144$		None	✓		✓		755.2.b.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3775))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3775)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(151))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(755))\)\(^{\oplus 2}\)