Space of modular forms of level 575, weight 4, and trivial character

• Label: 575.4.a
• Level: $575$
• Weight: $4$
• Character orbit: 575.a
• Rep. character: $\chi_{575}(1,\cdot)$
• Character field: $\Q$
• Dimension: $105$
• Newform subspaces: $18$
• Sturm bound: $240$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(240\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(575))\).

	Total	New	Old
Modular forms	186	105	81
Cusp forms	174	105	69
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.

\(5\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(29\)
\(+\)	\(-\)	\(-\)	\(20\)
\(-\)	\(+\)	\(-\)	\(25\)
\(-\)	\(-\)	\(+\)	\(31\)
Plus space		\(+\)	\(60\)
Minus space		\(-\)	\(45\)

Trace form

105 * q - 4 * q^2 + 2 * q^3 + 424 * q^4 + 17 * q^6 + 32 * q^7 - 3 * q^8 + 965 * q^9 + 58 * q^11 + q^12 - 114 * q^13 - 52 * q^14 + 1712 * q^16 + 202 * q^17 + 349 * q^18 - 46 * q^19 - 100 * q^21 - 616 * q^22 - 69 * q^23 + 688 * q^24 + 5 * q^26 + 278 * q^27 - 34 * q^28 - 246 * q^29 - 110 * q^31 + 552 * q^32 + 388 * q^33 - 566 * q^34 + 4531 * q^36 + 984 * q^37 - 228 * q^38 + 1146 * q^39 + 190 * q^41 + 152 * q^42 + 954 * q^43 - 934 * q^44 + 92 * q^46 - 362 * q^47 - 1029 * q^48 + 5169 * q^49 - 2520 * q^51 - 1045 * q^52 - 772 * q^53 + 479 * q^54 - 934 * q^56 + 1032 * q^57 + 127 * q^58 + 1128 * q^59 + 2244 * q^61 + 1433 * q^62 + 884 * q^63 + 9647 * q^64 - 862 * q^66 + 782 * q^67 + 4036 * q^68 + 276 * q^69 - 854 * q^71 + 4383 * q^72 - 2246 * q^73 + 1178 * q^74 - 348 * q^76 - 1248 * q^77 + 1071 * q^78 - 1280 * q^79 + 8753 * q^81 - 799 * q^82 + 634 * q^83 + 5866 * q^84 - 162 * q^86 - 3066 * q^87 - 2784 * q^88 + 2438 * q^89 - 1188 * q^91 - 552 * q^92 - 1992 * q^93 + 4069 * q^94 + 3071 * q^96 + 2098 * q^97 + 3736 * q^98 + 3726 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(575))\) 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	23
575.4.a.a	$575$	$4$	575.a	1.a	$1$	$1$	$1$	$33.926$	\(\Q\)	None	✓				115.4.a.b	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}-4q^{4}-6q^{6}+2q^{7}+\cdots\)
575.4.a.b	$575$	$4$	575.a	1.a	$1$	$1$	$1$	$33.926$	\(\Q\)	None	✓				115.4.a.a	$1$	$0$	\(-1\)	\(-4\)	\(0\)	\(32\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-4q^{3}-7q^{4}+4q^{6}+2^{5}q^{7}+\cdots\)
575.4.a.c	$575$	$4$	575.a	1.a	$1$	$1$	$1$	$33.926$	\(\Q\)	None	✓	✓			575.4.a.c	$1$	$1$	\(-1\)	\(6\)	\(0\)	\(7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+6q^{3}-7q^{4}-6q^{6}+7q^{7}+\cdots\)
575.4.a.d	$575$	$4$	575.a	1.a	$1$	$1$	$1$	$33.926$	\(\Q\)	None	✓	✓			575.4.a.d	$1$	$0$	\(-1\)	\(10\)	\(0\)	\(-23\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+10q^{3}-7q^{4}-10q^{6}-23q^{7}+\cdots\)
575.4.a.e	$575$	$4$	575.a	1.a	$1$	$1$	$1$	$33.926$	\(\Q\)	None	✓	✓			575.4.a.d	$1$	$0$	\(1\)	\(-10\)	\(0\)	\(23\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-10q^{3}-7q^{4}-10q^{6}+23q^{7}+\cdots\)
575.4.a.f	$575$	$4$	575.a	1.a	$1$	$1$	$1$	$33.926$	\(\Q\)	None	✓	✓			575.4.a.c	$1$	$1$	\(1\)	\(-6\)	\(0\)	\(-7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-6q^{3}-7q^{4}-6q^{6}-7q^{7}+\cdots\)
575.4.a.g	$575$	$4$	575.a	1.a	$1$	$1$	$1$	$33.926$	\(\Q\)	None	✓				23.4.a.a	$1$	$0$	\(2\)	\(5\)	\(0\)	\(8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+5q^{3}-4q^{4}+10q^{6}+8q^{7}+\cdots\)
575.4.a.h	$575$	$4$	575.a	1.a	$1$	$2$	$2$	$33.926$	\(\Q(\sqrt{109}) \)	None	✓				115.4.a.c	$1$	$1$	\(6\)	\(3\)	\(0\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+(2-\beta )q^{3}+q^{4}+(6-3\beta )q^{6}+\cdots\)
575.4.a.i	$575$	$4$	575.a	1.a	$1$	$4$	$4$	$33.926$	4.4.334189.1	None	✓				23.4.a.b	$1$	$1$	\(-2\)	\(-7\)	\(0\)	\(-16\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+\cdots\)
575.4.a.j	$575$	$4$	575.a	1.a	$1$	$5$	$5$	$33.926$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				115.4.a.e	$1$	$1$	\(-6\)	\(-4\)	\(0\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-1-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{3}+\cdots\)
575.4.a.k	$575$	$4$	575.a	1.a	$1$	$5$	$5$	$33.926$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				115.4.a.d	$1$	$0$	\(5\)	\(6\)	\(0\)	\(15\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{2}+\beta _{3}+\beta _{4})q^{3}+\cdots\)
575.4.a.l	$575$	$4$	575.a	1.a	$1$	$7$	$7$	$33.926$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		575.4.a.l	$1$	$1$	\(-3\)	\(-1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{5})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
575.4.a.m	$575$	$4$	575.a	1.a	$1$	$7$	$7$	$33.926$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			575.4.a.l	$1$	$1$	\(3\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{1}-\beta _{5})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
575.4.a.n	$575$	$4$	575.a	1.a	$1$	$8$	$8$	$33.926$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				115.4.a.f	$1$	$0$	\(-6\)	\(0\)	\(0\)	\(-11\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-\beta _{6}q^{3}+(5+\beta _{2})q^{4}+\cdots\)
575.4.a.o	$575$	$4$	575.a	1.a	$1$	$13$	$13$	$33.926$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓		575.4.a.o	$1$	$0$	\(-3\)	\(3\)	\(0\)	\(1\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(6+\beta _{2})q^{4}+(5-\beta _{1}+\cdots)q^{6}+\cdots\)
575.4.a.p	$575$	$4$	575.a	1.a	$1$	$13$	$13$	$33.926$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓			575.4.a.o	$1$	$0$	\(3\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(6+\beta _{2})q^{4}+(5-\beta _{1}+\cdots)q^{6}+\cdots\)
575.4.a.q	$575$	$4$	575.a	1.a	$1$	$17$	$17$	$33.926$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓		✓		115.4.b.a	$1$	$1$	\(-4\)	\(-12\)	\(0\)	\(-72\)		$-$	$+$	$-$	$2^{3}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{4})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
575.4.a.r	$575$	$4$	575.a	1.a	$1$	$17$	$17$	$33.926$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓		✓		115.4.b.a	$1$	$0$	\(4\)	\(12\)	\(0\)	\(72\)		$-$	$-$	$+$	$2^{3}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{4})q^{3}+(4+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(575))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(575)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)