Space of modular forms of level 3249, weight 2, and trivial character

• Label: 3249.2.a
• Level: $3249$
• Weight: $2$
• Character orbit: 3249.a
• Rep. character: $\chi_{3249}(1,\cdot)$
• Character field: $\Q$
• Dimension: $133$
• Newform subspaces: $40$
• Sturm bound: $760$
• Trace bound: $91$

Defining parameters

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 40 \)
Sturm bound:			\(760\)
Trace bound:			\(91\)
Distinguishing \(T_p\):			\(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3249))\).

	Total	New	Old
Modular forms	420	150	270
Cusp forms	341	133	208
Eisenstein series	79	17	62

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

\(3\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(23\)
\(+\)	\(-\)	\(-\)	\(33\)
\(-\)	\(+\)	\(-\)	\(41\)
\(-\)	\(-\)	\(+\)	\(36\)
Plus space		\(+\)	\(59\)
Minus space		\(-\)	\(74\)

Trace form

133 * q - 3 * q^2 + 121 * q^4 + 2 * q^7 - 3 * q^8 + 10 * q^10 + 2 * q^11 - 6 * q^13 + 4 * q^14 + 97 * q^16 - 6 * q^17 + 14 * q^20 + 20 * q^22 + 8 * q^23 + 93 * q^25 + 6 * q^26 + 34 * q^28 - 4 * q^29 + 4 * q^31 + 21 * q^32 + 22 * q^34 + 20 * q^35 - 20 * q^37 + 66 * q^40 - 16 * q^41 + 18 * q^43 - 14 * q^44 + 8 * q^46 - 2 * q^47 + 75 * q^49 - q^50 - 14 * q^52 + 10 * q^53 + 8 * q^55 - 36 * q^58 - 26 * q^59 - 20 * q^61 + 22 * q^62 + 31 * q^64 - 36 * q^65 + 8 * q^67 + 26 * q^68 - 24 * q^70 + 6 * q^71 + 2 * q^73 - 26 * q^74 + 12 * q^77 - 8 * q^79 + 60 * q^80 + 24 * q^82 + 44 * q^83 - 28 * q^85 + 12 * q^88 + 14 * q^89 + 20 * q^91 + 84 * q^92 - 24 * q^94 + 10 * q^97 - 47 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	19
3249.2.a.a	$3249$	$2$	3249.a	1.a	$1$	$1$	$1$	$25.943$	\(\Q\)	None	✓				57.2.a.b	$1$	$1$	\(-2\)	\(0\)	\(-1\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{5}+3q^{7}+2q^{10}+\cdots\)
3249.2.a.b	$3249$	$2$	3249.a	1.a	$1$	$1$	$1$	$25.943$	\(\Q\)	None	✓				57.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(3\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+3q^{5}-5q^{7}-6q^{10}+\cdots\)
3249.2.a.c	$3249$	$2$	3249.a	1.a	$1$	$1$	$1$	$25.943$	\(\Q\)	None	✓				57.2.e.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{7}+3q^{8}+2q^{11}+5q^{13}+\cdots\)
3249.2.a.d	$3249$	$2$	3249.a	1.a	$1$	$1$	$1$	$25.943$	\(\Q\)	None	✓				19.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}-q^{7}-3q^{11}+4q^{13}+\cdots\)
3249.2.a.e	$3249$	$2$	3249.a	1.a	$1$	$1$	$1$	$25.943$	\(\Q\)	\(\Q(\sqrt{-19}) \)	✓				361.2.a.a	$2$	$0$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+q^{5}+3q^{7}+5q^{11}+4q^{16}+\cdots\)
3249.2.a.f	$3249$	$2$	3249.a	1.a	$1$	$1$	$1$	$25.943$	\(\Q\)	None	✓				57.2.e.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{7}-3q^{8}+2q^{11}-5q^{13}+\cdots\)
3249.2.a.g	$3249$	$2$	3249.a	1.a	$1$	$1$	$1$	$25.943$	\(\Q\)	None	✓				57.2.a.c	$1$	$1$	\(1\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{5}-3q^{8}+2q^{10}+\cdots\)
3249.2.a.h	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{5}) \)	None	✓				1083.2.a.f	$1$	$1$	\(-3\)	\(0\)	\(-1\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3\beta q^{4}-\beta q^{5}+(-2+\cdots)q^{7}+\cdots\)
3249.2.a.i	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{5}) \)	None	✓				361.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}-2\beta q^{5}+3q^{7}+\cdots\)
3249.2.a.j	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{5}) \)	None	✓				1083.2.a.h	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(-1+3\beta )q^{5}+\cdots\)
3249.2.a.k	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{17}) \)	None	✓				1083.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(3\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{4}+(1+\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.l	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{19}) \)	\(\Q(\sqrt{-19}) \)	✓	✓			3249.2.a.l	$4$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+\beta q^{5}-3q^{7}-\beta q^{11}+4q^{16}+\cdots\)
3249.2.a.m	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{5}) \)	None	✓				361.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}+3q^{4}+(-1+\beta )q^{5}+\cdots\)
3249.2.a.n	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{5}) \)	None	✓				361.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}+3q^{4}-\beta q^{5}+(-2+2\beta )q^{7}+\cdots\)
3249.2.a.o	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{5}) \)	None	✓				361.2.a.c	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}-2\beta q^{5}+3q^{7}+\cdots\)
3249.2.a.p	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{5}) \)	None	✓				1083.2.a.h	$1$	$1$	\(1\)	\(0\)	\(1\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(-1+3\beta )q^{5}+\cdots\)
3249.2.a.q	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{17}) \)	None	✓				1083.2.a.g	$1$	$0$	\(1\)	\(0\)	\(3\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{4}+(1+\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.r	$3249$	$2$	3249.a	1.a	$1$	$2$	$2$	$25.943$	\(\Q(\sqrt{5}) \)	None	✓				1083.2.a.f	$1$	$1$	\(3\)	\(0\)	\(-1\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3\beta q^{4}-\beta q^{5}+(-2-\beta )q^{7}+\cdots\)
3249.2.a.s	$3249$	$2$	3249.a	1.a	$1$	$3$	$3$	$25.943$	\(\Q(\zeta_{18})^+\)	None	✓				19.2.e.a	$1$	$1$	\(-3\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3249.2.a.t	$3249$	$2$	3249.a	1.a	$1$	$3$	$3$	$25.943$	3.3.564.1	None	✓				57.2.e.b	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
3249.2.a.u	$3249$	$2$	3249.a	1.a	$1$	$3$	$3$	$25.943$	\(\Q(\zeta_{18})^+\)	\(\Q(\sqrt{-3}) \)	✓				171.2.u.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+(-3\beta _{1}+\beta _{2})q^{7}+(4\beta _{1}-\beta _{2})q^{13}+\cdots\)
3249.2.a.v	$3249$	$2$	3249.a	1.a	$1$	$3$	$3$	$25.943$	\(\Q(\zeta_{18})^+\)	\(\Q(\sqrt{-3}) \)	✓				171.2.u.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+(-3\beta _{1}+\beta _{2})q^{7}+(-4\beta _{1}+\cdots)q^{13}+\cdots\)
3249.2.a.w	$3249$	$2$	3249.a	1.a	$1$	$3$	$3$	$25.943$	\(\Q(\zeta_{18})^+\)	None	✓				57.2.i.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}+(1-\beta _{1})q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.x	$3249$	$2$	3249.a	1.a	$1$	$3$	$3$	$25.943$	\(\Q(\zeta_{18})^+\)	None	✓				57.2.i.a	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(1-\beta _{1})q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.y	$3249$	$2$	3249.a	1.a	$1$	$3$	$3$	$25.943$	3.3.564.1	None	✓				57.2.e.b	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
3249.2.a.z	$3249$	$2$	3249.a	1.a	$1$	$3$	$3$	$25.943$	\(\Q(\zeta_{18})^+\)	None	✓				19.2.e.a	$1$	$0$	\(3\)	\(0\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
3249.2.a.ba	$3249$	$2$	3249.a	1.a	$1$	$4$	$4$	$25.943$	\(\Q(\zeta_{20})^+\)	None	✓	✓			3249.2.a.ba	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
3249.2.a.bb	$3249$	$2$	3249.a	1.a	$1$	$4$	$4$	$25.943$	\(\Q(\zeta_{20})^+\)	None	✓	✓			3249.2.a.ba	$2$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{2})q^{5}+\cdots\)
3249.2.a.bc	$3249$	$2$	3249.a	1.a	$1$	$4$	$4$	$25.943$	\(\Q(\zeta_{20})^+\)	None	✓				361.2.a.i	$2$	$0$	\(0\)	\(0\)	\(4\)	\(-8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(2+2\beta _{2})q^{5}+\cdots\)
3249.2.a.bd	$3249$	$2$	3249.a	1.a	$1$	$4$	$4$	$25.943$	4.4.27648.1	None	✓				171.2.f.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.be	$3249$	$2$	3249.a	1.a	$1$	$4$	$4$	$25.943$	4.4.27648.1	None	✓				171.2.f.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.bf	$3249$	$2$	3249.a	1.a	$1$	$4$	$4$	$25.943$	4.4.13068.1	None	✓				171.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{3})q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
3249.2.a.bg	$3249$	$2$	3249.a	1.a	$1$	$6$	$6$	$25.943$	6.6.6357609.1	None	✓		✓		57.2.i.b	$1$	$1$	\(0\)	\(0\)	\(-9\)	\(9\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{3}+\beta _{4}+\beta _{5})q^{4}+(-2+\cdots)q^{5}+\cdots\)
3249.2.a.bh	$3249$	$2$	3249.a	1.a	$1$	$6$	$6$	$25.943$	6.6.6357609.1	None	✓				57.2.i.b	$1$	$0$	\(0\)	\(0\)	\(-9\)	\(9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{3}+\beta _{4}+\beta _{5})q^{4}+(-2+\cdots)q^{5}+\cdots\)
3249.2.a.bi	$3249$	$2$	3249.a	1.a	$1$	$6$	$6$	$25.943$	6.6.21415104.1	None	✓		✓		171.2.u.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
3249.2.a.bj	$3249$	$2$	3249.a	1.a	$1$	$6$	$6$	$25.943$	6.6.21415104.1	None	✓				171.2.u.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
3249.2.a.bk	$3249$	$2$	3249.a	1.a	$1$	$8$	$8$	$25.943$	8.8.9764000000.1	None	✓				1083.2.a.r	$1$	$0$	\(-4\)	\(0\)	\(2\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{2}+\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots\)
3249.2.a.bl	$3249$	$2$	3249.a	1.a	$1$	$8$	$8$	$25.943$	8.8.\(\cdots\).1	None	✓	✓			3249.2.a.bl	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{7})q^{5}+\cdots\)
3249.2.a.bm	$3249$	$2$	3249.a	1.a	$1$	$8$	$8$	$25.943$	8.8.\(\cdots\).1	None	✓	✓			3249.2.a.bl	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}-\beta _{7})q^{5}+\cdots\)
3249.2.a.bn	$3249$	$2$	3249.a	1.a	$1$	$8$	$8$	$25.943$	8.8.9764000000.1	None	✓				1083.2.a.r	$1$	$0$	\(4\)	\(0\)	\(2\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{2}+\beta _{4}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3249))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3249)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1083))\)\(^{\oplus 2}\)