Space of modular forms of level 275, weight 4, and trivial character

• Label: 275.4.a
• Level: $275$
• Weight: $4$
• Character orbit: 275.a
• Rep. character: $\chi_{275}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $11$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(275))\).

	Total	New	Old
Modular forms	96	48	48
Cusp forms	84	48	36
Eisenstein series	12	0	12

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

\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(12\)
\(+\)	\(-\)	\(-\)	\(10\)
\(-\)	\(+\)	\(-\)	\(11\)
\(-\)	\(-\)	\(+\)	\(15\)
Plus space		\(+\)	\(27\)
Minus space		\(-\)	\(21\)

Trace form

48 * q - 2 * q^2 + 2 * q^3 + 208 * q^4 - 34 * q^6 + 20 * q^7 + 24 * q^8 + 406 * q^9 + 22 * q^11 + 24 * q^12 - 88 * q^13 - 96 * q^14 + 808 * q^16 + 60 * q^17 + 420 * q^18 + 48 * q^19 + 44 * q^21 - 22 * q^22 - 246 * q^23 - 232 * q^24 - 280 * q^26 - 82 * q^27 + 100 * q^28 - 324 * q^29 + 94 * q^31 + 1040 * q^32 - 154 * q^33 - 148 * q^34 + 2200 * q^36 + 758 * q^37 - 1416 * q^38 + 200 * q^39 - 436 * q^41 + 372 * q^42 - 252 * q^43 - 88 * q^44 + 454 * q^46 - 696 * q^47 + 760 * q^48 + 3080 * q^49 - 2276 * q^51 - 540 * q^52 - 888 * q^53 - 170 * q^54 - 660 * q^56 - 152 * q^57 - 1824 * q^58 + 1366 * q^59 + 3100 * q^61 - 1366 * q^62 + 2112 * q^63 + 1116 * q^64 + 814 * q^66 - 698 * q^67 + 1932 * q^68 - 2322 * q^69 - 582 * q^71 + 1188 * q^72 + 1856 * q^73 - 206 * q^74 + 188 * q^76 - 220 * q^77 + 1600 * q^78 - 3156 * q^79 + 3024 * q^81 + 2928 * q^82 - 300 * q^83 + 3356 * q^84 + 776 * q^86 + 3616 * q^87 - 660 * q^88 + 6442 * q^89 + 5000 * q^91 - 3488 * q^92 - 130 * q^93 - 8328 * q^94 - 9128 * q^96 + 1194 * q^97 - 4050 * q^98 + 924 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(275))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	11
275.4.a.a	$275$	$4$	275.a	1.a	$1$	$1$	$1$	$16.226$	\(\Q\)	None	✓				55.4.a.a	$1$	$1$	\(-1\)	\(3\)	\(0\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}-7q^{4}-3q^{6}+9q^{7}+\cdots\)
275.4.a.b	$275$	$4$	275.a	1.a	$1$	$2$	$2$	$16.226$	\(\Q(\sqrt{3}) \)	None	✓				11.4.a.a	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(-20\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-4\beta )q^{3}+(-4-2\beta )q^{4}+\cdots\)
275.4.a.c	$275$	$4$	275.a	1.a	$1$	$2$	$2$	$16.226$	\(\Q(\sqrt{17}) \)	None	✓				55.4.a.b	$1$	$0$	\(7\)	\(3\)	\(0\)	\(25\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(4-\beta )q^{2}+(2-\beta )q^{3}+(12-7\beta )q^{4}+\cdots\)
275.4.a.d	$275$	$4$	275.a	1.a	$1$	$3$	$3$	$16.226$	3.3.568.1	None	✓				55.4.a.c	$1$	$0$	\(-5\)	\(3\)	\(0\)	\(15\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}+\beta _{2})q^{2}+(2+3\beta _{2})q^{3}+\cdots\)
275.4.a.e	$275$	$4$	275.a	1.a	$1$	$4$	$4$	$16.226$	4.4.1539480.1	None	✓				55.4.a.d	$1$	$1$	\(-1\)	\(-9\)	\(0\)	\(-9\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2+\beta _{2})q^{3}+(5+\beta _{1}+\beta _{3})q^{4}+\cdots\)
275.4.a.f	$275$	$4$	275.a	1.a	$1$	$5$	$5$	$16.226$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		275.4.a.f	$1$	$1$	\(-2\)	\(-12\)	\(0\)	\(-24\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2-\beta _{3})q^{3}+(2+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
275.4.a.g	$275$	$4$	275.a	1.a	$1$	$5$	$5$	$16.226$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			275.4.a.g	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-40\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{4})q^{3}+(8+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
275.4.a.h	$275$	$4$	275.a	1.a	$1$	$5$	$5$	$16.226$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		275.4.a.g	$1$	$0$	\(2\)	\(0\)	\(0\)	\(40\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{4})q^{3}+(8+\beta _{2}+\cdots)q^{4}+\cdots\)
275.4.a.i	$275$	$4$	275.a	1.a	$1$	$5$	$5$	$16.226$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			275.4.a.f	$1$	$0$	\(2\)	\(12\)	\(0\)	\(24\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{3})q^{3}+(2+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
275.4.a.j	$275$	$4$	275.a	1.a	$1$	$6$	$6$	$16.226$	6.6.2301792529.1	None	✓		✓		55.4.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{5})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)
275.4.a.k	$275$	$4$	275.a	1.a	$1$	$10$	$10$	$16.226$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		55.4.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(6+\beta _{2})q^{4}+(3-\beta _{3}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(275))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(275)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)