Space of modular forms of level 2475, weight 2, and trivial character

• Label: 2475.2.a
• Level: $2475$
• Weight: $2$
• Character orbit: 2475.a
• Rep. character: $\chi_{2475}(1,\cdot)$
• Character field: $\Q$
• Dimension: $78$
• Newform subspaces: $36$
• Sturm bound: $720$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(720\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2475))\).

	Total	New	Old
Modular forms	384	78	306
Cusp forms	337	78	259
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.

\(3\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(42\)	\(8\)	\(34\)		\(37\)	\(8\)	\(29\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(54\)	\(8\)	\(46\)		\(48\)	\(8\)	\(40\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(50\)	\(7\)	\(43\)		\(44\)	\(7\)	\(37\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(46\)	\(7\)	\(39\)		\(40\)	\(7\)	\(33\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(48\)	\(12\)	\(36\)		\(42\)	\(12\)	\(30\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(48\)	\(10\)	\(38\)		\(42\)	\(10\)	\(32\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(48\)	\(11\)	\(37\)		\(42\)	\(11\)	\(31\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(48\)	\(15\)	\(33\)		\(42\)	\(15\)	\(27\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(184\)	\(36\)	\(148\)		\(161\)	\(36\)	\(125\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(200\)	\(42\)	\(158\)		\(176\)	\(42\)	\(134\)		\(24\)	\(0\)	\(24\)

Trace form

78 * q - q^2 + 79 * q^4 - 2 * q^7 - 15 * q^8 + 2 * q^11 - 10 * q^13 - 12 * q^14 + 77 * q^16 + q^22 + 3 * q^23 + 22 * q^26 + 16 * q^28 + 18 * q^29 - 7 * q^31 - 11 * q^32 + 14 * q^34 - 17 * q^37 + 22 * q^41 - 18 * q^43 + q^44 + 10 * q^46 - 12 * q^47 + 30 * q^49 - 2 * q^52 - 12 * q^53 - 24 * q^56 + 42 * q^58 - 7 * q^59 + 6 * q^61 + 10 * q^62 + 101 * q^64 - 9 * q^67 + 66 * q^68 - 15 * q^71 + 2 * q^73 - 28 * q^74 - 24 * q^76 - 2 * q^77 - 50 * q^79 + 30 * q^82 - 6 * q^83 + 52 * q^86 + 15 * q^88 + 5 * q^89 - 36 * q^91 - 22 * q^92 - 40 * q^94 - 35 * q^97 + 63 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\) 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	5	11
2475.2.a.a	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓				11.2.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+2q^{7}-q^{11}-4q^{13}+\cdots\)
2475.2.a.b	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓	✓			2475.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-3q^{7}+3q^{8}+q^{11}+2q^{13}+\cdots\)
2475.2.a.c	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓				99.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{7}+3q^{8}+q^{11}+2q^{13}+\cdots\)
2475.2.a.d	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓	✓			2475.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{7}+3q^{8}-q^{11}-2q^{13}+\cdots\)
2475.2.a.e	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓				825.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{7}+q^{11}-q^{13}+4q^{16}+\cdots\)
2475.2.a.f	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓				825.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{7}+q^{11}+q^{13}+4q^{16}+\cdots\)
2475.2.a.g	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓				33.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-4q^{7}-3q^{8}-q^{11}+2q^{13}+\cdots\)
2475.2.a.h	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓	✓			2475.2.a.b	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{7}-3q^{8}-q^{11}+2q^{13}+\cdots\)
2475.2.a.i	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓				55.2.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{8}+q^{11}-2q^{13}+\cdots\)
2475.2.a.j	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓				99.2.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{7}-3q^{8}-q^{11}+2q^{13}+\cdots\)
2475.2.a.k	$2475$	$2$	2475.a	1.a	$1$	$1$	$1$	$19.763$	\(\Q\)	None	✓	✓			2475.2.a.b	$1$	$0$	\(1\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+3q^{7}-3q^{8}+q^{11}-2q^{13}+\cdots\)
2475.2.a.l	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{2}) \)	None	✓				825.2.a.d	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(-1-\beta )q^{7}+\cdots\)
2475.2.a.m	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{2}) \)	None	✓				165.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(2+2\beta )q^{7}+\cdots\)
2475.2.a.n	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{5}) \)	None	✓				275.2.a.d	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(2-3\beta )q^{7}+\cdots\)
2475.2.a.o	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{13}) \)	None	✓				275.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+(2+\beta )q^{7}-3q^{8}+\cdots\)
2475.2.a.p	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{17}) \)	None	✓	✓			2475.2.a.p	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{4}-q^{7}+(-4-\beta )q^{8}+\cdots\)
2475.2.a.q	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{17}) \)	None	✓	✓			2475.2.a.p	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{4}+q^{7}+(-4-\beta )q^{8}+\cdots\)
2475.2.a.r	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{3}) \)	None	✓				165.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}-2q^{7}-\beta q^{8}+q^{11}+\cdots\)
2475.2.a.s	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{5}) \)	None	✓				275.2.a.d	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(-2+3\beta )q^{7}+\cdots\)
2475.2.a.t	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{13}) \)	None	✓				275.2.a.e	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}+(-2-\beta )q^{7}+3q^{8}+\cdots\)
2475.2.a.u	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{17}) \)	None	✓	✓			2475.2.a.p	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{4}-q^{7}+(4+\beta )q^{8}+\cdots\)
2475.2.a.v	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{17}) \)	None	✓	✓			2475.2.a.p	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{4}+q^{7}+(4+\beta )q^{8}+\cdots\)
2475.2.a.w	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{2}) \)	None	✓				825.2.a.d	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(1-\beta )q^{7}+\cdots\)
2475.2.a.x	$2475$	$2$	2475.a	1.a	$1$	$2$	$2$	$19.763$	\(\Q(\sqrt{2}) \)	None	✓				55.2.a.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+2q^{7}+(3+\cdots)q^{8}+\cdots\)
2475.2.a.y	$2475$	$2$	2475.a	1.a	$1$	$3$	$3$	$19.763$	3.3.148.1	None	✓				165.2.c.a	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2475.2.a.z	$2475$	$2$	2475.a	1.a	$1$	$3$	$3$	$19.763$	3.3.568.1	None	✓				825.2.a.i	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
2475.2.a.ba	$2475$	$2$	2475.a	1.a	$1$	$3$	$3$	$19.763$	3.3.148.1	None	✓				165.2.c.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
2475.2.a.bb	$2475$	$2$	2475.a	1.a	$1$	$3$	$3$	$19.763$	3.3.148.1	None	✓				165.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
2475.2.a.bc	$2475$	$2$	2475.a	1.a	$1$	$3$	$3$	$19.763$	3.3.148.1	None	✓				165.2.c.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
2475.2.a.bd	$2475$	$2$	2475.a	1.a	$1$	$3$	$3$	$19.763$	3.3.568.1	None	✓				825.2.a.i	$1$	$0$	\(2\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
2475.2.a.be	$2475$	$2$	2475.a	1.a	$1$	$3$	$3$	$19.763$	3.3.148.1	None	✓				165.2.c.a	$1$	$0$	\(3\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(-3+\cdots)q^{7}+\cdots\)
2475.2.a.bf	$2475$	$2$	2475.a	1.a	$1$	$4$	$4$	$19.763$	4.4.48704.2	None	✓		✓		495.2.a.f	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
2475.2.a.bg	$2475$	$2$	2475.a	1.a	$1$	$4$	$4$	$19.763$	\(\Q(\zeta_{24})^+\)	None	✓		✓		495.2.c.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}-\beta _{3})q^{7}+\beta _{3}q^{8}+\cdots\)
2475.2.a.bh	$2475$	$2$	2475.a	1.a	$1$	$4$	$4$	$19.763$	\(\Q(\zeta_{24})^+\)	None	✓		✓		495.2.c.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
2475.2.a.bi	$2475$	$2$	2475.a	1.a	$1$	$4$	$4$	$19.763$	\(\Q(\sqrt{3}, \sqrt{11})\)	None	✓		✓		55.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{3})q^{4}+2\beta _{2}q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
2475.2.a.bj	$2475$	$2$	2475.a	1.a	$1$	$4$	$4$	$19.763$	4.4.48704.2	None	✓		✓		495.2.a.f	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2475))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2475)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(495))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(825))\)\(^{\oplus 2}\)