Space of modular forms of level 1521, weight 4, and trivial character

• Label: 1521.4.a
• Level: $1521$
• Weight: $4$
• Character orbit: 1521.a
• Rep. character: $\chi_{1521}(1,\cdot)$
• Character field: $\Q$
• Dimension: $188$
• Newform subspaces: $40$
• Sturm bound: $728$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 40 \)
Sturm bound:			\(728\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1521))\).

	Total	New	Old
Modular forms	574	199	375
Cusp forms	518	188	330
Eisenstein series	56	11	45

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

\(3\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(41\)
\(+\)	\(-\)	\(-\)	\(36\)
\(-\)	\(+\)	\(-\)	\(54\)
\(-\)	\(-\)	\(+\)	\(57\)
Plus space		\(+\)	\(98\)
Minus space		\(-\)	\(90\)

Trace form

\( 188 q + 718 q^{4} + 6 q^{5} - 22 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\) 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	13
1521.4.a.a	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				13.4.a.a	$1$	$1$	\(-5\)	\(0\)	\(-7\)	\(13\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}-7q^{5}+13q^{7}-45q^{8}+\cdots\)
1521.4.a.b	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				13.4.c.a	$1$	$1$	\(-4\)	\(0\)	\(-17\)	\(20\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+8q^{4}-17q^{5}+20q^{7}+68q^{10}+\cdots\)
1521.4.a.c	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				39.4.e.a	$1$	$1$	\(-3\)	\(0\)	\(9\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+9q^{5}+2q^{7}+21q^{8}+\cdots\)
1521.4.a.d	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				13.4.b.a	$1$	$0$	\(-3\)	\(0\)	\(9\)	\(15\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+9q^{5}+15q^{7}+21q^{8}+\cdots\)
1521.4.a.e	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				39.4.e.b	$1$	$1$	\(-1\)	\(0\)	\(7\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}+7q^{5}+10q^{7}+15q^{8}+\cdots\)
1521.4.a.f	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				39.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-12\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{4}-12q^{5}-2q^{7}-6^{2}q^{11}+2^{6}q^{16}+\cdots\)
1521.4.a.g	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				9.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}-20q^{7}+2^{6}q^{16}-56q^{19}+\cdots\)
1521.4.a.h	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				39.4.e.b	$1$	$1$	\(1\)	\(0\)	\(-7\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}-7q^{5}-10q^{7}-15q^{8}+\cdots\)
1521.4.a.i	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				13.4.b.a	$1$	$0$	\(3\)	\(0\)	\(-9\)	\(-15\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-9q^{5}-15q^{7}-21q^{8}+\cdots\)
1521.4.a.j	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				39.4.e.a	$1$	$1$	\(3\)	\(0\)	\(-9\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-9q^{5}-2q^{7}-21q^{8}+\cdots\)
1521.4.a.k	$1521$	$4$	1521.a	1.a	$1$	$1$	$1$	$89.742$	\(\Q\)	None	✓				13.4.c.a	$1$	$1$	\(4\)	\(0\)	\(17\)	\(-20\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+8q^{4}+17q^{5}-20q^{7}+68q^{10}+\cdots\)
1521.4.a.l	$1521$	$4$	1521.a	1.a	$1$	$2$	$2$	$89.742$	\(\Q(\sqrt{17}) \)	None	✓				13.4.c.b	$1$	$1$	\(-5\)	\(0\)	\(-15\)	\(-15\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+5\beta q^{4}+(-10+5\beta )q^{5}+\cdots\)
1521.4.a.m	$1521$	$4$	1521.a	1.a	$1$	$2$	$2$	$89.742$	\(\Q(\sqrt{3}) \)	None	✓				39.4.j.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{4}-3\beta q^{5}+6\beta q^{7}+30\beta q^{11}+\cdots\)
1521.4.a.n	$1521$	$4$	1521.a	1.a	$1$	$2$	$2$	$89.742$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓				117.4.b.b	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2\cdot 3^{2}$	$N(\mathrm{U}(1))$	\(q-8q^{4}+\beta q^{7}+2^{6}q^{16}-5\beta q^{19}+\cdots\)
1521.4.a.o	$1521$	$4$	1521.a	1.a	$1$	$2$	$2$	$89.742$	\(\Q(\sqrt{3}) \)	None	✓				13.4.e.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-5q^{4}+\beta q^{5}+8\beta q^{7}-13\beta q^{8}+\cdots\)
1521.4.a.p	$1521$	$4$	1521.a	1.a	$1$	$2$	$2$	$89.742$	\(\Q(\sqrt{7}) \)	None	✓				117.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(44\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{4}-4\beta q^{5}+22q^{7}-9\beta q^{8}+\cdots\)
1521.4.a.q	$1521$	$4$	1521.a	1.a	$1$	$2$	$2$	$89.742$	\(\Q(\sqrt{3}) \)	None	✓				13.4.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{2}+4q^{4}+8\beta q^{5}+13\beta q^{7}+\cdots\)
1521.4.a.r	$1521$	$4$	1521.a	1.a	$1$	$2$	$2$	$89.742$	\(\Q(\sqrt{17}) \)	None	✓				13.4.a.b	$1$	$1$	\(1\)	\(0\)	\(-3\)	\(9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-4+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
1521.4.a.s	$1521$	$4$	1521.a	1.a	$1$	$2$	$2$	$89.742$	\(\Q(\sqrt{14}) \)	None	✓				39.4.a.b	$1$	$1$	\(2\)	\(0\)	\(24\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(7+2\beta )q^{4}+(12-2\beta )q^{5}+\cdots\)
1521.4.a.t	$1521$	$4$	1521.a	1.a	$1$	$2$	$2$	$89.742$	\(\Q(\sqrt{17}) \)	None	✓				13.4.c.b	$1$	$1$	\(5\)	\(0\)	\(15\)	\(15\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(5-5\beta )q^{4}+(5+5\beta )q^{5}+\cdots\)
1521.4.a.u	$1521$	$4$	1521.a	1.a	$1$	$3$	$3$	$89.742$	3.3.3144.1	None	✓				39.4.a.c	$1$	$1$	\(2\)	\(0\)	\(4\)	\(-30\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(2-2\beta _{2})q^{5}+\cdots\)
1521.4.a.v	$1521$	$4$	1521.a	1.a	$1$	$4$	$4$	$89.742$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				39.4.e.c	$1$	$1$	\(-2\)	\(0\)	\(6\)	\(-14\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(5+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
1521.4.a.w	$1521$	$4$	1521.a	1.a	$1$	$4$	$4$	$89.742$	4.4.1362828.1	None	✓				39.4.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{3})q^{4}+(2\beta _{1}+\beta _{2})q^{5}+\cdots\)
1521.4.a.x	$1521$	$4$	1521.a	1.a	$1$	$4$	$4$	$89.742$	4.4.5054412.1	None	✓				39.4.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(7+\beta _{3})q^{4}-\beta _{2}q^{5}+6\beta _{1}q^{7}+\cdots\)
1521.4.a.y	$1521$	$4$	1521.a	1.a	$1$	$4$	$4$	$89.742$	4.4.8112.1	\(\Q(\sqrt{-39}) \)	✓				117.4.b.c	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}\cdot 3^{2}$	$N(\mathrm{U}(1))$	\(q-\beta _{1}q^{2}+(8-\beta _{2})q^{4}+(2\beta _{1}-\beta _{3})q^{5}+\cdots\)
1521.4.a.z	$1521$	$4$	1521.a	1.a	$1$	$4$	$4$	$89.742$	\(\Q(\sqrt{3}, \sqrt{17})\)	None	✓				39.4.j.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+9q^{4}+(3\beta _{1}-2\beta _{3})q^{5}+(11\beta _{1}+\cdots)q^{7}+\cdots\)
1521.4.a.ba	$1521$	$4$	1521.a	1.a	$1$	$4$	$4$	$89.742$	4.4.1520092.1	None	✓				117.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-36\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(14+\beta _{2})q^{4}-\beta _{3}q^{5}+(-8+\cdots)q^{7}+\cdots\)
1521.4.a.bb	$1521$	$4$	1521.a	1.a	$1$	$4$	$4$	$89.742$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				39.4.e.c	$1$	$1$	\(2\)	\(0\)	\(-6\)	\(14\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1521.4.a.bc	$1521$	$4$	1521.a	1.a	$1$	$8$	$8$	$89.742$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				117.4.g.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-22\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+\beta _{3}q^{5}+(-3+\cdots)q^{7}+\cdots\)
1521.4.a.bd	$1521$	$4$	1521.a	1.a	$1$	$8$	$8$	$89.742$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				117.4.g.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(22\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+\beta _{3}q^{5}+(3-\beta _{5}+\cdots)q^{7}+\cdots\)
1521.4.a.be	$1521$	$4$	1521.a	1.a	$1$	$9$	$9$	$89.742$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				507.4.a.n	$1$	$1$	\(-8\)	\(0\)	\(-41\)	\(1\)		$-$	$+$	$-$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(3-2\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
1521.4.a.bf	$1521$	$4$	1521.a	1.a	$1$	$9$	$9$	$89.742$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				507.4.a.o	$1$	$0$	\(-6\)	\(0\)	\(-33\)	\(83\)		$-$	$-$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{2}+(5+\beta _{3}+\beta _{5})q^{4}+\cdots\)
1521.4.a.bg	$1521$	$4$	1521.a	1.a	$1$	$9$	$9$	$89.742$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				169.4.a.k	$1$	$1$	\(-5\)	\(0\)	\(-30\)	\(38\)		$-$	$+$	$-$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(5+\beta _{5}+\beta _{6}+\beta _{8})q^{4}+\cdots\)
1521.4.a.bh	$1521$	$4$	1521.a	1.a	$1$	$9$	$9$	$89.742$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				169.4.a.k	$1$	$0$	\(5\)	\(0\)	\(30\)	\(-38\)		$-$	$-$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(5+\beta _{5}+\beta _{6}+\beta _{8})q^{4}+\cdots\)
1521.4.a.bi	$1521$	$4$	1521.a	1.a	$1$	$9$	$9$	$89.742$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				507.4.a.o	$1$	$1$	\(6\)	\(0\)	\(33\)	\(-83\)		$-$	$+$	$-$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{3})q^{2}+(5+\beta _{3}+\beta _{5})q^{4}+(3+\cdots)q^{5}+\cdots\)
1521.4.a.bj	$1521$	$4$	1521.a	1.a	$1$	$9$	$9$	$89.742$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				507.4.a.n	$1$	$0$	\(8\)	\(0\)	\(41\)	\(-1\)		$-$	$-$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(3-2\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
1521.4.a.bk	$1521$	$4$	1521.a	1.a	$1$	$10$	$10$	$89.742$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		39.4.j.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{4})q^{4}+(\beta _{1}+\beta _{2}-\beta _{8}+\cdots)q^{5}+\cdots\)
1521.4.a.bl	$1521$	$4$	1521.a	1.a	$1$	$12$	$12$	$89.742$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				117.4.q.f	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{9}\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{6}q^{2}+(3+\beta _{2})q^{4}+(-\beta _{6}+\beta _{8}+\cdots)q^{5}+\cdots\)
1521.4.a.bm	$1521$	$4$	1521.a	1.a	$1$	$18$	$18$	$89.742$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓	✓		1521.4.a.bm	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-94\)		$+$	$-$	$-$	$2^{3}\cdot 7\cdot 13^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(\beta _{1}-\beta _{14})q^{5}+\cdots\)
1521.4.a.bn	$1521$	$4$	1521.a	1.a	$1$	$18$	$18$	$89.742$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓	✓		1521.4.a.bm	$2$	$0$	\(0\)	\(0\)	\(0\)	\(94\)		$+$	$+$	$+$	$2^{3}\cdot 7\cdot 13^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(\beta _{1}-\beta _{14})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1521))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1521)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 2}\)