Space of modular forms of level 3480, weight 2, and trivial character

• Label: 3480.2.a
• Level: $3480$
• Weight: $2$
• Character orbit: 3480.a
• Rep. character: $\chi_{3480}(1,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $32$
• Sturm bound: $1440$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3480.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(1440\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3480))\).

	Total	New	Old
Modular forms	736	56	680
Cusp forms	705	56	649
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(5\)	\(29\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(40\)	\(4\)	\(36\)		\(39\)	\(4\)	\(35\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(50\)	\(3\)	\(47\)		\(48\)	\(3\)	\(45\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(48\)	\(3\)	\(45\)		\(46\)	\(3\)	\(43\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(44\)	\(5\)	\(39\)		\(42\)	\(5\)	\(37\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(47\)	\(3\)	\(44\)		\(45\)	\(3\)	\(42\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(47\)	\(3\)	\(44\)		\(45\)	\(3\)	\(42\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(49\)	\(3\)	\(46\)		\(47\)	\(3\)	\(44\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(43\)	\(4\)	\(39\)		\(41\)	\(4\)	\(37\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(52\)	\(6\)	\(46\)		\(50\)	\(6\)	\(44\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(42\)	\(2\)	\(40\)		\(40\)	\(2\)	\(38\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(44\)	\(2\)	\(42\)		\(42\)	\(2\)	\(40\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(48\)	\(5\)	\(43\)		\(46\)	\(5\)	\(41\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(45\)	\(3\)	\(42\)		\(43\)	\(3\)	\(40\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(45\)	\(4\)	\(41\)		\(43\)	\(4\)	\(39\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(43\)	\(4\)	\(39\)		\(41\)	\(4\)	\(37\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(49\)	\(2\)	\(47\)		\(47\)	\(2\)	\(45\)		\(2\)	\(0\)	\(2\)
Plus space				\(+\)		\(360\)	\(24\)	\(336\)		\(345\)	\(24\)	\(321\)		\(15\)	\(0\)	\(15\)
Minus space				\(-\)		\(376\)	\(32\)	\(344\)		\(360\)	\(32\)	\(328\)		\(16\)	\(0\)	\(16\)

Trace form

56 * q - 4 * q^3 - 8 * q^7 + 56 * q^9 + 8 * q^11 - 8 * q^13 - 8 * q^21 + 56 * q^25 - 4 * q^27 - 8 * q^31 + 8 * q^35 - 8 * q^39 - 8 * q^41 - 32 * q^43 - 48 * q^47 + 24 * q^49 + 16 * q^53 - 8 * q^57 - 32 * q^59 - 8 * q^61 - 8 * q^63 - 24 * q^65 + 16 * q^71 + 24 * q^73 - 4 * q^75 + 32 * q^77 - 40 * q^79 + 56 * q^81 + 8 * q^83 + 8 * q^85 + 24 * q^89 - 8 * q^93 + 16 * q^95 + 8 * q^97 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3480))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5	29
3480.2.a.a	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{7}+q^{9}+6q^{11}+6q^{13}+\cdots\)
3480.2.a.b	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}+q^{9}-q^{11}-3q^{13}+\cdots\)
3480.2.a.c	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+2q^{7}+q^{9}+2q^{11}+q^{15}+\cdots\)
3480.2.a.d	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-4\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-4q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
3480.2.a.e	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-3q^{7}+q^{9}+3q^{11}-q^{13}+\cdots\)
3480.2.a.f	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}-4q^{13}-q^{15}-2q^{17}+\cdots\)
3480.2.a.g	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(2\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+2q^{7}+q^{9}-5q^{11}+2q^{13}+\cdots\)
3480.2.a.h	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+2q^{7}+q^{9}+6q^{11}+4q^{13}+\cdots\)
3480.2.a.i	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.i	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
3480.2.a.j	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.j	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots\)
3480.2.a.k	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.k	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}+3q^{11}-6q^{13}+\cdots\)
3480.2.a.l	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.l	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}+6q^{11}+6q^{13}+\cdots\)
3480.2.a.m	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.m	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}+q^{9}-3q^{11}+3q^{13}+\cdots\)
3480.2.a.n	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.n	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
3480.2.a.o	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.o	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(3\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+3q^{7}+q^{9}+q^{11}+q^{13}+\cdots\)
3480.2.a.p	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.p	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}+q^{9}+5q^{11}-5q^{13}+\cdots\)
3480.2.a.q	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.q	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
3480.2.a.r	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.r	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
3480.2.a.s	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.s	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}-2q^{13}+q^{15}-2q^{17}+\cdots\)
3480.2.a.t	$3480$	$2$	3480.a	1.a	$1$	$1$	$1$	$27.788$	\(\Q\)	None	✓	✓			3480.2.a.t	$1$	$0$	\(0\)	\(1\)	\(1\)	\(4\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+4q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
3480.2.a.u	$3480$	$2$	3480.a	1.a	$1$	$2$	$2$	$27.788$	\(\Q(\sqrt{17}) \)	None	✓	✓			3480.2.a.u	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-5\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(-2-\beta )q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
3480.2.a.v	$3480$	$2$	3480.a	1.a	$1$	$2$	$2$	$27.788$	\(\Q(\sqrt{33}) \)	None	✓	✓			3480.2.a.v	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(4\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+2q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
3480.2.a.w	$3480$	$2$	3480.a	1.a	$1$	$2$	$2$	$27.788$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓		3480.2.a.w	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(3\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(1+\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3480.2.a.x	$3480$	$2$	3480.a	1.a	$1$	$2$	$2$	$27.788$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓		3480.2.a.x	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-1-\beta )q^{7}+q^{9}+(-3+\cdots)q^{11}+\cdots\)
3480.2.a.y	$3480$	$2$	3480.a	1.a	$1$	$2$	$2$	$27.788$	\(\Q(\sqrt{33}) \)	None	✓	✓	✓		3480.2.a.y	$1$	$1$	\(0\)	\(2\)	\(2\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-\beta q^{7}+q^{9}+(-4+\beta )q^{11}+\cdots\)
3480.2.a.z	$3480$	$2$	3480.a	1.a	$1$	$2$	$2$	$27.788$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓		3480.2.a.z	$1$	$0$	\(0\)	\(2\)	\(2\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+\beta q^{7}+q^{9}+\beta q^{11}+(2+\cdots)q^{13}+\cdots\)
3480.2.a.ba	$3480$	$2$	3480.a	1.a	$1$	$3$	$3$	$27.788$	3.3.229.1	None	✓	✓	✓	✓	3480.2.a.ba	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-\beta _{2}q^{7}+q^{9}-\beta _{2}q^{11}+\cdots\)
3480.2.a.bb	$3480$	$2$	3480.a	1.a	$1$	$3$	$3$	$27.788$	3.3.229.1	None	✓	✓	✓		3480.2.a.bb	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(-1\)		$+$	$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+\beta _{1}q^{7}+q^{9}+\beta _{1}q^{11}+\cdots\)
3480.2.a.bc	$3480$	$2$	3480.a	1.a	$1$	$4$	$4$	$27.788$	4.4.58397.1	None	✓	✓	✓	✓	3480.2.a.bc	$1$	$0$	\(0\)	\(4\)	\(-4\)	\(-3\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-1-\beta _{3})q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
3480.2.a.bd	$3480$	$2$	3480.a	1.a	$1$	$4$	$4$	$27.788$	4.4.69272.1	None	✓	✓	✓	✓	3480.2.a.bd	$1$	$0$	\(0\)	\(4\)	\(4\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-\beta _{1}q^{7}+q^{9}-\beta _{3}q^{11}+\cdots\)
3480.2.a.be	$3480$	$2$	3480.a	1.a	$1$	$5$	$5$	$27.788$	5.5.18396776.1	None	✓	✓	✓		3480.2.a.be	$1$	$0$	\(0\)	\(-5\)	\(-5\)	\(1\)		$-$	$+$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+\beta _{2}q^{7}+q^{9}+(-1+\beta _{3}+\cdots)q^{11}+\cdots\)
3480.2.a.bf	$3480$	$2$	3480.a	1.a	$1$	$5$	$5$	$27.788$	5.5.1025428.1	None	✓	✓	✓	✓	3480.2.a.bf	$1$	$0$	\(0\)	\(-5\)	\(5\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-\beta _{2}q^{7}+q^{9}+\beta _{1}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3480))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3480)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(290))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(348))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(580))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(696))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(870))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1740))\)\(^{\oplus 2}\)