Space of modular forms of level 1275, weight 4, and trivial character

• Label: 1275.4.a
• Level: $1275$
• Weight: $4$
• Character orbit: 1275.a
• Rep. character: $\chi_{1275}(1,\cdot)$
• Character field: $\Q$
• Dimension: $152$
• Newform subspaces: $33$
• Sturm bound: $720$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1275.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 33 \)
Sturm bound:			\(720\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1275))\).

	Total	New	Old
Modular forms	552	152	400
Cusp forms	528	152	376
Eisenstein series	24	0	24

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

\(3\)	\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(18\)
\(+\)	\(+\)	\(-\)	\(-\)	\(16\)
\(+\)	\(-\)	\(+\)	\(-\)	\(19\)
\(+\)	\(-\)	\(-\)	\(+\)	\(23\)
\(-\)	\(+\)	\(+\)	\(-\)	\(18\)
\(-\)	\(+\)	\(-\)	\(+\)	\(20\)
\(-\)	\(-\)	\(+\)	\(+\)	\(21\)
\(-\)	\(-\)	\(-\)	\(-\)	\(17\)
Plus space			\(+\)	\(82\)
Minus space			\(-\)	\(70\)

Trace form

152 * q + 4 * q^2 + 588 * q^4 - 12 * q^6 - 16 * q^7 - 120 * q^8 + 1368 * q^9 - 96 * q^11 - 48 * q^12 + 4 * q^13 + 8 * q^14 + 2404 * q^16 + 36 * q^18 + 164 * q^19 - 36 * q^21 - 36 * q^22 - 352 * q^23 - 144 * q^24 + 364 * q^26 + 264 * q^28 + 368 * q^29 - 312 * q^31 - 976 * q^32 + 132 * q^33 - 136 * q^34 + 5292 * q^36 - 480 * q^37 + 884 * q^38 - 816 * q^39 - 720 * q^41 - 252 * q^42 - 756 * q^43 + 744 * q^44 + 1700 * q^46 - 408 * q^47 - 480 * q^48 + 7864 * q^49 - 204 * q^51 - 1040 * q^52 - 792 * q^53 - 108 * q^54 - 624 * q^56 + 312 * q^57 - 2672 * q^58 + 1032 * q^59 - 2664 * q^61 + 376 * q^62 - 144 * q^63 + 8860 * q^64 + 828 * q^66 + 1960 * q^67 + 552 * q^69 - 1552 * q^71 - 1080 * q^72 + 2096 * q^73 - 2984 * q^74 + 4200 * q^76 + 312 * q^77 + 1764 * q^78 + 2152 * q^79 + 12312 * q^81 + 6908 * q^82 - 128 * q^83 + 108 * q^84 - 4220 * q^86 + 1836 * q^87 + 3336 * q^88 + 4360 * q^89 + 1680 * q^91 + 7440 * q^92 - 1836 * q^93 + 1168 * q^94 + 1536 * q^96 - 1904 * q^97 + 7916 * q^98 - 864 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1275))\) 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	17
1275.4.a.a	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓	✓			1275.4.a.a	$1$	$1$	\(-5\)	\(3\)	\(0\)	\(-13\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+3q^{3}+17q^{4}-15q^{6}-13q^{7}+\cdots\)
1275.4.a.b	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓	✓			1275.4.a.b	$1$	$1$	\(-3\)	\(3\)	\(0\)	\(-13\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+3q^{3}+q^{4}-9q^{6}-13q^{7}+\cdots\)
1275.4.a.c	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓	✓			1275.4.a.c	$1$	$1$	\(-3\)	\(3\)	\(0\)	\(20\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+3q^{3}+q^{4}-9q^{6}+20q^{7}+\cdots\)
1275.4.a.d	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓				51.4.a.c	$1$	$1$	\(-1\)	\(3\)	\(0\)	\(8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}-7q^{4}-3q^{6}+8q^{7}+\cdots\)
1275.4.a.e	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓				51.4.a.b	$1$	$1$	\(1\)	\(-3\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-7q^{4}-3q^{6}+2q^{7}+\cdots\)
1275.4.a.f	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓				51.4.a.a	$1$	$0$	\(1\)	\(3\)	\(0\)	\(-34\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-7q^{4}+3q^{6}-34q^{7}+\cdots\)
1275.4.a.g	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓				255.4.a.b	$1$	$0$	\(2\)	\(3\)	\(0\)	\(17\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}-4q^{4}+6q^{6}+17q^{7}+\cdots\)
1275.4.a.h	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓	✓			1275.4.a.c	$1$	$0$	\(3\)	\(-3\)	\(0\)	\(-20\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}-3q^{3}+q^{4}-9q^{6}-20q^{7}+\cdots\)
1275.4.a.i	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓	✓			1275.4.a.b	$1$	$0$	\(3\)	\(-3\)	\(0\)	\(13\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}-3q^{3}+q^{4}-9q^{6}+13q^{7}+\cdots\)
1275.4.a.j	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓				255.4.a.a	$1$	$1$	\(4\)	\(3\)	\(0\)	\(8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+3q^{3}+8q^{4}+12q^{6}+8q^{7}+\cdots\)
1275.4.a.k	$1275$	$4$	1275.a	1.a	$1$	$1$	$1$	$75.227$	\(\Q\)	None	✓	✓			1275.4.a.a	$1$	$0$	\(5\)	\(-3\)	\(0\)	\(13\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}-3q^{3}+17q^{4}-15q^{6}+13q^{7}+\cdots\)
1275.4.a.l	$1275$	$4$	1275.a	1.a	$1$	$2$	$2$	$75.227$	\(\Q(\sqrt{33}) \)	None	✓				255.4.a.f	$1$	$1$	\(-3\)	\(6\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3q^{3}+(1+3\beta )q^{4}+\cdots\)
1275.4.a.m	$1275$	$4$	1275.a	1.a	$1$	$2$	$2$	$75.227$	\(\Q(\sqrt{2}) \)	None	✓				51.4.a.d	$1$	$0$	\(0\)	\(6\)	\(0\)	\(8\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+3q^{3}+10q^{4}+3\beta q^{6}+(4+\cdots)q^{7}+\cdots\)
1275.4.a.n	$1275$	$4$	1275.a	1.a	$1$	$2$	$2$	$75.227$	\(\Q(\sqrt{41}) \)	None	✓				255.4.a.e	$1$	$1$	\(1\)	\(-6\)	\(0\)	\(25\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-3q^{3}+(2+\beta )q^{4}-3\beta q^{6}+\cdots\)
1275.4.a.o	$1275$	$4$	1275.a	1.a	$1$	$2$	$2$	$75.227$	\(\Q(\sqrt{57}) \)	None	✓				255.4.a.d	$1$	$0$	\(3\)	\(6\)	\(0\)	\(-10\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3q^{3}+(7+3\beta )q^{4}+(3+\cdots)q^{6}+\cdots\)
1275.4.a.p	$1275$	$4$	1275.a	1.a	$1$	$2$	$2$	$75.227$	\(\Q(\sqrt{17}) \)	None	✓				255.4.a.c	$1$	$0$	\(5\)	\(-6\)	\(0\)	\(7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}-3q^{3}+(5-5\beta )q^{4}+(-9+\cdots)q^{6}+\cdots\)
1275.4.a.q	$1275$	$4$	1275.a	1.a	$1$	$3$	$3$	$75.227$	3.3.5912.1	None	✓				51.4.a.e	$1$	$0$	\(-5\)	\(-9\)	\(0\)	\(8\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}-3q^{3}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
1275.4.a.r	$1275$	$4$	1275.a	1.a	$1$	$5$	$5$	$75.227$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			1275.4.a.r	$1$	$0$	\(-5\)	\(-15\)	\(0\)	\(57\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1275.4.a.s	$1275$	$4$	1275.a	1.a	$1$	$5$	$5$	$75.227$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				255.4.a.h	$1$	$1$	\(-1\)	\(15\)	\(0\)	\(-27\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1275.4.a.t	$1275$	$4$	1275.a	1.a	$1$	$5$	$5$	$75.227$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				255.4.a.g	$1$	$0$	\(-1\)	\(15\)	\(0\)	\(47\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1275.4.a.u	$1275$	$4$	1275.a	1.a	$1$	$5$	$5$	$75.227$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			1275.4.a.r	$1$	$1$	\(5\)	\(15\)	\(0\)	\(-57\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1275.4.a.v	$1275$	$4$	1275.a	1.a	$1$	$6$	$6$	$75.227$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				255.4.a.j	$1$	$1$	\(-1\)	\(-18\)	\(0\)	\(-43\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(7+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1275.4.a.w	$1275$	$4$	1275.a	1.a	$1$	$6$	$6$	$75.227$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		255.4.a.i	$1$	$0$	\(-1\)	\(-18\)	\(0\)	\(-37\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(7+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1275.4.a.x	$1275$	$4$	1275.a	1.a	$1$	$7$	$7$	$75.227$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		1275.4.a.x	$1$	$1$	\(-3\)	\(-21\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1275.4.a.y	$1275$	$4$	1275.a	1.a	$1$	$7$	$7$	$75.227$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			1275.4.a.x	$1$	$0$	\(3\)	\(21\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1275.4.a.z	$1275$	$4$	1275.a	1.a	$1$	$8$	$8$	$75.227$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			1275.4.a.z	$1$	$0$	\(-2\)	\(-24\)	\(0\)	\(-9\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(3+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1275.4.a.ba	$1275$	$4$	1275.a	1.a	$1$	$8$	$8$	$75.227$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		1275.4.a.z	$1$	$1$	\(2\)	\(24\)	\(0\)	\(9\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(3+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1275.4.a.bb	$1275$	$4$	1275.a	1.a	$1$	$9$	$9$	$75.227$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			1275.4.a.bb	$1$	$1$	\(-1\)	\(-27\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1275.4.a.bc	$1275$	$4$	1275.a	1.a	$1$	$9$	$9$	$75.227$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		1275.4.a.bb	$1$	$0$	\(1\)	\(27\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1275.4.a.bd	$1275$	$4$	1275.a	1.a	$1$	$10$	$10$	$75.227$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		255.4.b.a	$1$	$1$	\(-6\)	\(30\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1275.4.a.be	$1275$	$4$	1275.a	1.a	$1$	$10$	$10$	$75.227$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		255.4.b.a	$1$	$1$	\(6\)	\(-30\)	\(0\)	\(-8\)		$+$	$-$	$+$	$-$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1275.4.a.bf	$1275$	$4$	1275.a	1.a	$1$	$14$	$14$	$75.227$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓		✓		255.4.b.b	$1$	$0$	\(-2\)	\(-42\)	\(0\)	\(-8\)		$+$	$-$	$-$	$+$	$2^{8}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1275.4.a.bg	$1275$	$4$	1275.a	1.a	$1$	$14$	$14$	$75.227$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓		✓		255.4.b.b	$1$	$0$	\(2\)	\(42\)	\(0\)	\(8\)		$-$	$-$	$+$	$+$	$2^{8}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(5+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1275))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1275)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(425))\)\(^{\oplus 2}\)