Space of modular forms of level 2775, weight 2, and trivial character

• Label: 2775.2.a
• Level: $2775$
• Weight: $2$
• Character orbit: 2775.a
• Rep. character: $\chi_{2775}(1,\cdot)$
• Character field: $\Q$
• Dimension: $114$
• Newform subspaces: $34$
• Sturm bound: $760$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2775.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(760\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2775))\).

	Total	New	Old
Modular forms	392	114	278
Cusp forms	369	114	255
Eisenstein series	23	0	23

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

\(3\)	\(5\)	\(37\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(42\)	\(13\)	\(29\)		\(40\)	\(13\)	\(27\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(54\)	\(14\)	\(40\)		\(51\)	\(14\)	\(37\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(52\)	\(18\)	\(34\)		\(49\)	\(18\)	\(31\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(48\)	\(12\)	\(36\)		\(45\)	\(12\)	\(33\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(56\)	\(17\)	\(39\)		\(53\)	\(17\)	\(36\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(44\)	\(10\)	\(34\)		\(41\)	\(10\)	\(31\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(46\)	\(10\)	\(36\)		\(43\)	\(10\)	\(33\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(50\)	\(20\)	\(30\)		\(47\)	\(20\)	\(27\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(180\)	\(45\)	\(135\)		\(169\)	\(45\)	\(124\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(212\)	\(69\)	\(143\)		\(200\)	\(69\)	\(131\)		\(12\)	\(0\)	\(12\)

Trace form

114 * q + 118 * q^4 + 6 * q^6 + 8 * q^7 + 12 * q^8 + 114 * q^9 + 8 * q^12 - 4 * q^13 + 20 * q^14 + 150 * q^16 - 8 * q^17 + 20 * q^19 + 8 * q^22 + 12 * q^23 + 6 * q^24 + 8 * q^26 + 16 * q^28 + 12 * q^31 + 24 * q^32 + 16 * q^33 + 20 * q^34 + 118 * q^36 - 2 * q^37 - 28 * q^38 + 8 * q^39 - 12 * q^41 - 12 * q^42 + 12 * q^43 + 16 * q^44 - 40 * q^46 - 8 * q^47 + 8 * q^48 + 114 * q^49 + 8 * q^51 + 4 * q^52 - 36 * q^53 + 6 * q^54 + 64 * q^56 - 4 * q^57 - 28 * q^58 + 16 * q^59 + 4 * q^61 + 68 * q^62 + 8 * q^63 + 190 * q^64 - 8 * q^66 + 40 * q^67 - 44 * q^68 + 24 * q^69 + 24 * q^71 + 12 * q^72 + 12 * q^73 + 76 * q^76 + 24 * q^77 - 4 * q^78 + 36 * q^79 + 114 * q^81 + 16 * q^82 + 32 * q^83 - 16 * q^84 + 52 * q^86 + 16 * q^87 - 24 * q^88 + 4 * q^89 + 32 * q^91 + 128 * q^92 - 4 * q^93 - 64 * q^94 - 42 * q^96 + 12 * q^97 - 8 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2775))\) 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	37
2775.2.a.a	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓				555.2.c.a	$1$	$1$	\(-2\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+q^{9}-2q^{11}+\cdots\)
2775.2.a.b	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓	✓			2775.2.a.b	$1$	$1$	\(-2\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+q^{7}+q^{9}+\cdots\)
2775.2.a.c	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓	✓			2775.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-3q^{7}+q^{9}+4q^{11}+\cdots\)
2775.2.a.d	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓				555.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-2q^{7}+q^{9}+2q^{12}+\cdots\)
2775.2.a.e	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓	✓			2775.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-q^{7}+q^{9}-2q^{11}+2q^{12}+\cdots\)
2775.2.a.f	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓				555.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+2q^{7}+q^{9}+4q^{11}+\cdots\)
2775.2.a.g	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓	✓			2775.2.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+q^{7}+q^{9}-2q^{11}-2q^{12}+\cdots\)
2775.2.a.h	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓	✓			2775.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+3q^{7}+q^{9}+4q^{11}+\cdots\)
2775.2.a.i	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓	✓			2775.2.a.b	$1$	$0$	\(2\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-q^{7}+q^{9}+\cdots\)
2775.2.a.j	$2775$	$2$	2775.a	1.a	$1$	$1$	$1$	$22.158$	\(\Q\)	None	✓				555.2.c.a	$1$	$1$	\(2\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+q^{9}-2q^{11}+\cdots\)
2775.2.a.k	$2775$	$2$	2775.a	1.a	$1$	$2$	$2$	$22.158$	\(\Q(\sqrt{2}) \)	None	✓	✓			2775.2.a.k	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(1+\cdots)q^{6}+\cdots\)
2775.2.a.l	$2775$	$2$	2775.a	1.a	$1$	$2$	$2$	$22.158$	\(\Q(\sqrt{13}) \)	None	✓				555.2.a.g	$1$	$0$	\(-1\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+\beta q^{6}-q^{7}+\cdots\)
2775.2.a.m	$2775$	$2$	2775.a	1.a	$1$	$2$	$2$	$22.158$	\(\Q(\sqrt{5}) \)	None	✓				555.2.a.f	$1$	$1$	\(-1\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
2775.2.a.n	$2775$	$2$	2775.a	1.a	$1$	$2$	$2$	$22.158$	\(\Q(\sqrt{5}) \)	None	✓				555.2.a.e	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
2775.2.a.o	$2775$	$2$	2775.a	1.a	$1$	$2$	$2$	$22.158$	\(\Q(\sqrt{13}) \)	None	✓				555.2.a.d	$1$	$0$	\(1\)	\(2\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(1+\beta )q^{4}+\beta q^{6}+3q^{7}+\cdots\)
2775.2.a.p	$2775$	$2$	2775.a	1.a	$1$	$2$	$2$	$22.158$	\(\Q(\sqrt{2}) \)	None	✓	✓			2775.2.a.k	$1$	$0$	\(2\)	\(2\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
2775.2.a.q	$2775$	$2$	2775.a	1.a	$1$	$2$	$2$	$22.158$	\(\Q(\sqrt{5}) \)	None	✓				555.2.a.c	$1$	$0$	\(3\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+3\beta q^{4}+(-1-\beta )q^{6}+\cdots\)
2775.2.a.r	$2775$	$2$	2775.a	1.a	$1$	$3$	$3$	$22.158$	3.3.148.1	None	✓				111.2.a.a	$1$	$0$	\(-3\)	\(3\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2775.2.a.s	$2775$	$2$	2775.a	1.a	$1$	$3$	$3$	$22.158$	3.3.229.1	None	✓				555.2.a.i	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}-q^{3}+(2+\beta _{1})q^{4}+\cdots\)
2775.2.a.t	$2775$	$2$	2775.a	1.a	$1$	$3$	$3$	$22.158$	3.3.469.1	None	✓				555.2.a.h	$1$	$1$	\(-1\)	\(3\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
2775.2.a.u	$2775$	$2$	2775.a	1.a	$1$	$3$	$3$	$22.158$	3.3.257.1	None	✓	✓			2775.2.a.u	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\beta _{2}q^{6}+\cdots\)
2775.2.a.v	$2775$	$2$	2775.a	1.a	$1$	$3$	$3$	$22.158$	3.3.257.1	None	✓	✓			2775.2.a.u	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\beta _{2}q^{6}+\cdots\)
2775.2.a.w	$2775$	$2$	2775.a	1.a	$1$	$4$	$4$	$22.158$	\(\Q(\sqrt{14 +2 \sqrt{5}})\)	None	✓				555.2.c.b	$1$	$1$	\(-2\)	\(-4\)	\(0\)	\(8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}-q^{3}+\beta _{2}q^{4}+(1+\beta _{2}+\cdots)q^{6}+\cdots\)
2775.2.a.x	$2775$	$2$	2775.a	1.a	$1$	$4$	$4$	$22.158$	4.4.6224.1	None	✓		✓		111.2.a.b	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
2775.2.a.y	$2775$	$2$	2775.a	1.a	$1$	$4$	$4$	$22.158$	\(\Q(\sqrt{14 +2 \sqrt{5}})\)	None	✓		✓		555.2.c.b	$1$	$1$	\(2\)	\(4\)	\(0\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+q^{3}+\beta _{2}q^{4}+(1+\beta _{2}+\cdots)q^{6}+\cdots\)
2775.2.a.z	$2775$	$2$	2775.a	1.a	$1$	$5$	$5$	$22.158$	5.5.528933.1	None	✓	✓	✓		2775.2.a.z	$1$	$1$	\(-2\)	\(-5\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}-q^{3}+(2-\beta _{2}+\beta _{4})q^{4}+\beta _{3}q^{6}+\cdots\)
2775.2.a.ba	$2775$	$2$	2775.a	1.a	$1$	$5$	$5$	$22.158$	5.5.176684.1	None	✓	✓	✓		2775.2.a.ba	$1$	$1$	\(-2\)	\(5\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+q^{3}+(\beta _{2}-\beta _{4})q^{4}-\beta _{2}q^{6}+\cdots\)
2775.2.a.bb	$2775$	$2$	2775.a	1.a	$1$	$5$	$5$	$22.158$	5.5.176684.1	None	✓	✓			2775.2.a.ba	$1$	$0$	\(2\)	\(-5\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-q^{3}+(\beta _{2}-\beta _{4})q^{4}-\beta _{2}q^{6}+\cdots\)
2775.2.a.bc	$2775$	$2$	2775.a	1.a	$1$	$5$	$5$	$22.158$	5.5.528933.1	None	✓	✓			2775.2.a.z	$1$	$0$	\(2\)	\(5\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+q^{3}+(2-\beta _{2}+\beta _{4})q^{4}+\beta _{3}q^{6}+\cdots\)
2775.2.a.bd	$2775$	$2$	2775.a	1.a	$1$	$5$	$5$	$22.158$	5.5.600268.1	None	✓				555.2.a.j	$1$	$0$	\(3\)	\(5\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{2}+q^{3}+(2-\beta _{3}+\beta _{4})q^{4}+\cdots\)
2775.2.a.be	$2775$	$2$	2775.a	1.a	$1$	$6$	$6$	$22.158$	6.6.297869800.1	None	✓	✓	✓		2775.2.a.be	$1$	$1$	\(-4\)	\(-6\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-q^{3}+(2+\beta _{2})q^{4}+\cdots\)
2775.2.a.bf	$2775$	$2$	2775.a	1.a	$1$	$6$	$6$	$22.158$	6.6.297869800.1	None	✓	✓	✓		2775.2.a.be	$1$	$0$	\(4\)	\(6\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1+\cdots)q^{6}+\cdots\)
2775.2.a.bg	$2775$	$2$	2775.a	1.a	$1$	$13$	$13$	$22.158$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		555.2.c.c	$1$	$0$	\(0\)	\(-13\)	\(0\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
2775.2.a.bh	$2775$	$2$	2775.a	1.a	$1$	$13$	$13$	$22.158$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		555.2.c.c	$1$	$0$	\(0\)	\(13\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2775))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2775)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(555))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(925))\)\(^{\oplus 2}\)