Space of modular forms of level 325, weight 4, and trivial character

• Label: 325.4.a
• Level: $325$
• Weight: $4$
• Character orbit: 325.a
• Rep. character: $\chi_{325}(1,\cdot)$
• Character field: $\Q$
• Dimension: $57$
• Newform subspaces: $15$
• Sturm bound: $140$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(140\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(325))\).

	Total	New	Old
Modular forms	112	57	55
Cusp forms	100	57	43
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.

\(5\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(30\)	\(15\)	\(15\)		\(27\)	\(15\)	\(12\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(27\)	\(12\)	\(15\)		\(24\)	\(12\)	\(12\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(26\)	\(14\)	\(12\)		\(23\)	\(14\)	\(9\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(29\)	\(16\)	\(13\)		\(26\)	\(16\)	\(10\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(59\)	\(31\)	\(28\)		\(53\)	\(31\)	\(22\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(53\)	\(26\)	\(27\)		\(47\)	\(26\)	\(21\)		\(6\)	\(0\)	\(6\)

Trace form

57 * q - 4 * q^2 + 6 * q^3 + 250 * q^4 + 28 * q^6 - 6 * q^7 + 36 * q^8 + 423 * q^9 + 86 * q^11 - 10 * q^12 - 13 * q^13 - 154 * q^14 + 770 * q^16 + 64 * q^17 + 256 * q^18 - 10 * q^19 - 28 * q^21 + 8 * q^22 + 68 * q^23 + 392 * q^24 + 78 * q^26 + 6 * q^27 + 292 * q^28 - 54 * q^29 - 350 * q^31 + 1384 * q^32 - 348 * q^33 + 400 * q^34 + 1676 * q^36 - 498 * q^37 - 496 * q^38 + 156 * q^39 - 126 * q^41 - 402 * q^42 + 1318 * q^43 - 620 * q^44 - 536 * q^46 - 1430 * q^47 - 1966 * q^48 + 3031 * q^49 - 2362 * q^51 - 624 * q^52 - 142 * q^53 + 1724 * q^54 - 1050 * q^56 - 704 * q^57 - 112 * q^58 - 1338 * q^59 + 654 * q^61 + 852 * q^62 + 914 * q^63 + 2258 * q^64 + 3028 * q^66 - 2238 * q^67 + 2050 * q^68 - 3868 * q^69 - 1398 * q^71 + 5448 * q^72 - 2206 * q^73 + 438 * q^74 - 540 * q^76 + 2116 * q^77 + 754 * q^78 - 2576 * q^79 + 5257 * q^81 + 644 * q^82 - 1942 * q^83 - 2556 * q^84 + 2416 * q^86 - 1088 * q^87 + 716 * q^88 + 4970 * q^89 + 52 * q^91 + 4424 * q^92 + 3656 * q^93 - 9674 * q^94 - 3672 * q^96 + 1950 * q^97 - 3500 * q^98 + 6262 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(325))\) 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	13
325.4.a.a	$325$	$4$	325.a	1.a	$1$	$1$	$1$	$19.176$	\(\Q\)	None	✓				65.4.a.a	$1$	$1$	\(-5\)	\(-2\)	\(0\)	\(12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-2q^{3}+17q^{4}+10q^{6}+12q^{7}+\cdots\)
325.4.a.b	$325$	$4$	325.a	1.a	$1$	$1$	$1$	$19.176$	\(\Q\)	None	✓				65.4.b.a	$1$	$0$	\(-3\)	\(-4\)	\(0\)	\(28\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-4q^{3}+q^{4}+12q^{6}+28q^{7}+\cdots\)
325.4.a.c	$325$	$4$	325.a	1.a	$1$	$1$	$1$	$19.176$	\(\Q\)	None	✓				65.4.b.a	$1$	$1$	\(3\)	\(4\)	\(0\)	\(-28\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+4q^{3}+q^{4}+12q^{6}-28q^{7}+\cdots\)
325.4.a.d	$325$	$4$	325.a	1.a	$1$	$1$	$1$	$19.176$	\(\Q\)	None	✓				13.4.a.a	$1$	$0$	\(5\)	\(7\)	\(0\)	\(13\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+7q^{3}+17q^{4}+35q^{6}+13q^{7}+\cdots\)
325.4.a.e	$325$	$4$	325.a	1.a	$1$	$2$	$2$	$19.176$	\(\Q(\sqrt{3}) \)	None	✓				65.4.a.d	$1$	$0$	\(-4\)	\(10\)	\(0\)	\(36\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{2}+(5-3\beta )q^{3}+(-1-4\beta )q^{4}+\cdots\)
325.4.a.f	$325$	$4$	325.a	1.a	$1$	$2$	$2$	$19.176$	\(\Q(\sqrt{17}) \)	None	✓				13.4.a.b	$1$	$1$	\(-1\)	\(-5\)	\(0\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-4+3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)
325.4.a.g	$325$	$4$	325.a	1.a	$1$	$2$	$2$	$19.176$	\(\Q(\sqrt{17}) \)	None	✓				65.4.a.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-58\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2-4\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)
325.4.a.h	$325$	$4$	325.a	1.a	$1$	$2$	$2$	$19.176$	\(\Q(\sqrt{6}) \)	None	✓				65.4.a.b	$1$	$1$	\(2\)	\(4\)	\(0\)	\(20\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(2-\beta )q^{3}+(-1+2\beta )q^{4}+\cdots\)
325.4.a.i	$325$	$4$	325.a	1.a	$1$	$5$	$5$	$19.176$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		325.4.a.i	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(5+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
325.4.a.j	$325$	$4$	325.a	1.a	$1$	$5$	$5$	$19.176$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				65.4.a.e	$1$	$0$	\(0\)	\(-8\)	\(0\)	\(-38\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-2-\beta _{1}-\beta _{3})q^{3}+(7-\beta _{2}+\cdots)q^{4}+\cdots\)
325.4.a.k	$325$	$4$	325.a	1.a	$1$	$5$	$5$	$19.176$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			325.4.a.i	$1$	$1$	\(1\)	\(1\)	\(0\)	\(26\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+(5+\beta _{2}+\cdots)q^{4}+\cdots\)
325.4.a.l	$325$	$4$	325.a	1.a	$1$	$7$	$7$	$19.176$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		325.4.a.l	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{3})q^{3}+(7+\beta _{2}+\cdots)q^{4}+\cdots\)
325.4.a.m	$325$	$4$	325.a	1.a	$1$	$7$	$7$	$19.176$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			325.4.a.l	$1$	$0$	\(1\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{3})q^{3}+(7+\beta _{2})q^{4}+\cdots\)
325.4.a.n	$325$	$4$	325.a	1.a	$1$	$8$	$8$	$19.176$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		65.4.b.b	$1$	$1$	\(-9\)	\(-16\)	\(0\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-2-\beta _{3})q^{3}+(4+\cdots)q^{4}+\cdots\)
325.4.a.o	$325$	$4$	325.a	1.a	$1$	$8$	$8$	$19.176$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		65.4.b.b	$1$	$0$	\(9\)	\(16\)	\(0\)	\(12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(2+\beta _{3})q^{3}+(4+\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(325))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(325)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)