Space of modular forms of level 3825, weight 2, and trivial character

• Label: 3825.2.a
• Level: $3825$
• Weight: $2$
• Character orbit: 3825.a
• Rep. character: $\chi_{3825}(1,\cdot)$
• Character field: $\Q$
• Dimension: $126$
• Newform subspaces: $46$
• Sturm bound: $1080$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3825.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 46 \)
Sturm bound:			\(1080\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3825))\).

	Total	New	Old
Modular forms	564	126	438
Cusp forms	517	126	391
Eisenstein series	47	0	47

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

\(3\)	\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(12\)
\(+\)	\(+\)	\(-\)	$-$	\(12\)
\(+\)	\(-\)	\(+\)	$-$	\(13\)
\(+\)	\(-\)	\(-\)	$+$	\(13\)
\(-\)	\(+\)	\(+\)	$-$	\(21\)
\(-\)	\(+\)	\(-\)	$+$	\(15\)
\(-\)	\(-\)	\(+\)	$+$	\(18\)
\(-\)	\(-\)	\(-\)	$-$	\(22\)
Plus space			\(+\)	\(58\)
Minus space			\(-\)	\(68\)

Trace form

\( 126 q + 2 q^{2} + 126 q^{4} + 4 q^{7} - 6 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3825))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	17
3825.2.a.a	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{7}-2q^{11}+7q^{13}+\cdots\)
3825.2.a.b	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+2q^{7}+3q^{11}+5q^{13}+\cdots\)
3825.2.a.c	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-4q^{7}+3q^{8}-6q^{11}+\cdots\)
3825.2.a.d	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-4q^{7}+3q^{8}+2q^{13}+\cdots\)
3825.2.a.e	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-4q^{7}+3q^{8}+2q^{11}+\cdots\)
3825.2.a.f	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{7}+3q^{8}+4q^{11}-q^{13}+\cdots\)
3825.2.a.g	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{7}-2q^{11}+q^{13}+4q^{16}+\cdots\)
3825.2.a.h	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{7}-2q^{11}-q^{13}+4q^{16}+\cdots\)
3825.2.a.i	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+4q^{7}+3q^{11}+q^{13}+4q^{16}+\cdots\)
3825.2.a.j	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-4q^{7}-3q^{8}-2q^{11}+\cdots\)
3825.2.a.k	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{7}-3q^{8}+4q^{11}+q^{13}+\cdots\)
3825.2.a.l	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{7}-3q^{8}-2q^{11}+\cdots\)
3825.2.a.m	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+4q^{7}-3q^{8}-6q^{11}+\cdots\)
3825.2.a.n	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{7}-2q^{11}-7q^{13}+\cdots\)
3825.2.a.o	$3825$	$2$	3825.a	1.a	$1$	$1$	$1$	$30.543$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+2q^{7}-3q^{11}+5q^{13}+\cdots\)
3825.2.a.p	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(2-\beta )q^{7}+\cdots\)
3825.2.a.q	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+q^{7}+(-1+2\beta )q^{8}+\cdots\)
3825.2.a.r	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(1+2\beta )q^{7}+\cdots\)
3825.2.a.s	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{4}+(-4-\beta )q^{8}+(1+\cdots)q^{11}+\cdots\)
3825.2.a.t	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-3-\beta )q^{7}-2\beta q^{8}+(2-2\beta )q^{11}+\cdots\)
3825.2.a.u	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(3-\beta )q^{7}-2\beta q^{8}+(2+2\beta )q^{11}+\cdots\)
3825.2.a.v	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+(1-\beta )q^{7}-\beta q^{8}+(-3+\cdots)q^{11}+\cdots\)
3825.2.a.w	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}-q^{7}+(1-2\beta )q^{8}+\cdots\)
3825.2.a.x	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(1+2\beta )q^{7}+\cdots\)
3825.2.a.y	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}+(1-2\beta )q^{7}+3q^{8}+\cdots\)
3825.2.a.z	$3825$	$2$	3825.a	1.a	$1$	$2$	$2$	$30.543$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3\beta q^{4}+(-1+2\beta )q^{7}+\cdots\)
3825.2.a.ba	$3825$	$2$	3825.a	1.a	$1$	$3$	$3$	$30.543$	3.3.148.1	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
3825.2.a.bb	$3825$	$2$	3825.a	1.a	$1$	$3$	$3$	$30.543$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{1}+\beta _{2})q^{7}+\cdots\)
3825.2.a.bc	$3825$	$2$	3825.a	1.a	$1$	$3$	$3$	$30.543$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
3825.2.a.bd	$3825$	$2$	3825.a	1.a	$1$	$3$	$3$	$30.543$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
3825.2.a.be	$3825$	$2$	3825.a	1.a	$1$	$3$	$3$	$30.543$	3.3.621.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}-\beta _{2}q^{7}+\cdots\)
3825.2.a.bf	$3825$	$2$	3825.a	1.a	$1$	$3$	$3$	$30.543$	3.3.621.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}-\beta _{2}q^{7}+\cdots\)
3825.2.a.bg	$3825$	$2$	3825.a	1.a	$1$	$3$	$3$	$30.543$	3.3.148.1	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
3825.2.a.bh	$3825$	$2$	3825.a	1.a	$1$	$4$	$4$	$30.543$	4.4.6224.1	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(10\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+\cdots\)
3825.2.a.bi	$3825$	$2$	3825.a	1.a	$1$	$4$	$4$	$30.543$	4.4.13768.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+(2-\beta _{1})q^{4}+(-1+\beta _{2}-\beta _{3})q^{7}+\cdots\)
3825.2.a.bj	$3825$	$2$	3825.a	1.a	$1$	$4$	$4$	$30.543$	4.4.6224.1	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-10\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+(-2+\cdots)q^{7}+\cdots\)
3825.2.a.bk	$3825$	$2$	3825.a	1.a	$1$	$5$	$5$	$30.543$	5.5.3717884.1	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}-\beta _{2})q^{7}+\cdots\)
3825.2.a.bl	$3825$	$2$	3825.a	1.a	$1$	$5$	$5$	$30.543$	5.5.1893456.1	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(2+\beta _{1}+\beta _{2}+\beta _{4})q^{4}+(\beta _{3}+\cdots)q^{7}+\cdots\)
3825.2.a.bm	$3825$	$2$	3825.a	1.a	$1$	$5$	$5$	$30.543$	5.5.2716368.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{1})q^{7}+\cdots\)
3825.2.a.bn	$3825$	$2$	3825.a	1.a	$1$	$5$	$5$	$30.543$	5.5.2716368.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{1})q^{7}+\cdots\)
3825.2.a.bo	$3825$	$2$	3825.a	1.a	$1$	$5$	$5$	$30.543$	5.5.2716368.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{1})q^{7}+\cdots\)
3825.2.a.bp	$3825$	$2$	3825.a	1.a	$1$	$5$	$5$	$30.543$	5.5.2716368.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{1})q^{7}+\cdots\)
3825.2.a.bq	$3825$	$2$	3825.a	1.a	$1$	$5$	$5$	$30.543$	5.5.1893456.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(2+\beta _{1}+\beta _{2}+\beta _{4})q^{4}+(-\beta _{3}+\cdots)q^{7}+\cdots\)
3825.2.a.br	$3825$	$2$	3825.a	1.a	$1$	$5$	$5$	$30.543$	5.5.3717884.1	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
3825.2.a.bs	$3825$	$2$	3825.a	1.a	$1$	$8$	$8$	$30.543$	8.8.\(\cdots\).1	None	✓	$2$	$1$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{6})q^{2}+(1-\beta _{2}-\beta _{6})q^{4}+\cdots\)
3825.2.a.bt	$3825$	$2$	3825.a	1.a	$1$	$8$	$8$	$30.543$	8.8.\(\cdots\).1	None	✓	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{6})q^{2}+(1-\beta _{2}-\beta _{6})q^{4}-\beta _{4}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3825))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3825)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(425))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(765))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1275))\)\(^{\oplus 2}\)