Space of modular forms of level 325, weight 8, and trivial character

• Label: 325.8.a
• Level: $325$
• Weight: $8$
• Character orbit: 325.a
• Rep. character: $\chi_{325}(1,\cdot)$
• Character field: $\Q$
• Dimension: $133$
• Newform subspaces: $13$
• Sturm bound: $280$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(280\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(325))\).

	Total	New	Old
Modular forms	252	133	119
Cusp forms	240	133	107
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
\(+\)	\(+\)	\(+\)		\(65\)	\(33\)	\(32\)		\(62\)	\(33\)	\(29\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(62\)	\(30\)	\(32\)		\(59\)	\(30\)	\(29\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(61\)	\(34\)	\(27\)		\(58\)	\(34\)	\(24\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(64\)	\(36\)	\(28\)		\(61\)	\(36\)	\(25\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(129\)	\(69\)	\(60\)		\(123\)	\(69\)	\(54\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(123\)	\(64\)	\(59\)		\(117\)	\(64\)	\(53\)		\(6\)	\(0\)	\(6\)

Trace form

133 * q + 6 * q^2 + 8278 * q^4 - 1292 * q^6 - 2428 * q^7 + 684 * q^8 + 103977 * q^9 + 10752 * q^11 - 18562 * q^12 - 2197 * q^13 + 66418 * q^14 + 501762 * q^16 - 46606 * q^17 + 134818 * q^18 - 19716 * q^19 - 9676 * q^21 + 341288 * q^22 - 134640 * q^23 - 616408 * q^24 + 52728 * q^26 + 602196 * q^27 - 100604 * q^28 - 301002 * q^29 + 123056 * q^31 - 406072 * q^32 - 1043220 * q^33 - 639084 * q^34 + 5325104 * q^36 + 710590 * q^37 + 1684248 * q^38 + 237276 * q^39 + 591358 * q^41 - 1922022 * q^42 - 541276 * q^43 - 128572 * q^44 - 1816016 * q^46 - 1483764 * q^47 + 3880514 * q^48 + 16192421 * q^49 + 567104 * q^51 - 430612 * q^52 - 3118490 * q^53 + 3984812 * q^54 + 13766758 * q^56 + 194068 * q^57 - 997996 * q^58 + 6932564 * q^59 - 2884114 * q^61 + 5619236 * q^62 - 2761108 * q^63 + 39327650 * q^64 - 1195988 * q^66 + 3360352 * q^67 + 6853970 * q^68 - 15745204 * q^69 + 11651556 * q^71 + 16161648 * q^72 + 4616562 * q^73 - 7706050 * q^74 - 9852540 * q^76 + 3907228 * q^77 - 7122674 * q^78 + 2212820 * q^79 + 91341349 * q^81 - 25667736 * q^82 + 13381960 * q^83 - 23385708 * q^84 - 20029712 * q^86 - 23526284 * q^87 + 47750348 * q^88 + 36911318 * q^89 + 5114616 * q^91 - 13208360 * q^92 + 1396100 * q^93 - 30150398 * q^94 - 132063480 * q^96 + 36213134 * q^97 - 78176562 * q^98 + 45948784 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(325))\) 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
325.8.a.a	$325$	$8$	325.a	1.a	$1$	$1$	$1$	$101.525$	\(\Q\)	None	✓				13.8.a.a	$1$	$0$	\(-10\)	\(73\)	\(0\)	\(-1373\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}+73q^{3}-28q^{4}-730q^{6}+\cdots\)
325.8.a.b	$325$	$8$	325.a	1.a	$1$	$2$	$2$	$101.525$	\(\Q(\sqrt{337}) \)	None	✓				13.8.a.b	$1$	$0$	\(19\)	\(-45\)	\(0\)	\(2009\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{2}+(-24+3\beta )q^{3}+(56+\cdots)q^{4}+\cdots\)
325.8.a.c	$325$	$8$	325.a	1.a	$1$	$4$	$4$	$101.525$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				13.8.a.c	$1$	$1$	\(-15\)	\(-80\)	\(0\)	\(-1692\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{2}+(-21+2\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)
325.8.a.d	$325$	$8$	325.a	1.a	$1$	$6$	$6$	$101.525$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				65.8.a.b	$1$	$1$	\(-6\)	\(40\)	\(0\)	\(1988\)		$+$	$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(7-2\beta _{1}+\beta _{2})q^{3}+\cdots\)
325.8.a.e	$325$	$8$	325.a	1.a	$1$	$6$	$6$	$101.525$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				65.8.a.a	$1$	$0$	\(20\)	\(94\)	\(0\)	\(756\)		$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{2}+(15+\beta _{1}-\beta _{3})q^{3}+(55+\cdots)q^{4}+\cdots\)
325.8.a.f	$325$	$8$	325.a	1.a	$1$	$7$	$7$	$101.525$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				65.8.a.c	$1$	$1$	\(12\)	\(-14\)	\(0\)	\(-2196\)		$+$	$-$	$-$	$2^{8}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(-2+\beta _{1}+\beta _{4})q^{3}+\cdots\)
325.8.a.g	$325$	$8$	325.a	1.a	$1$	$9$	$9$	$101.525$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				65.8.a.d	$1$	$0$	\(-14\)	\(-68\)	\(0\)	\(-1920\)		$+$	$+$	$+$	$2^{8}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(-8+\beta _{1}-\beta _{2})q^{3}+\cdots\)
325.8.a.h	$325$	$8$	325.a	1.a	$1$	$13$	$13$	$101.525$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓		325.8.a.h	$1$	$1$	\(-15\)	\(-13\)	\(0\)	\(-208\)		$+$	$-$	$-$	$2^{8}\cdot 3\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-1+\beta _{3})q^{3}+(45+\cdots)q^{4}+\cdots\)
325.8.a.i	$325$	$8$	325.a	1.a	$1$	$13$	$13$	$101.525$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓			325.8.a.h	$1$	$1$	\(15\)	\(13\)	\(0\)	\(208\)		$-$	$+$	$-$	$2^{8}\cdot 3\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(1-\beta _{3})q^{3}+(45+2\beta _{1}+\cdots)q^{4}+\cdots\)
325.8.a.j	$325$	$8$	325.a	1.a	$1$	$15$	$15$	$101.525$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓	✓		325.8.a.j	$1$	$0$	\(-15\)	\(-13\)	\(0\)	\(-1164\)		$+$	$+$	$+$	$2^{14}\cdot 3^{2}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{1}-\beta _{4})q^{3}+\cdots\)
325.8.a.k	$325$	$8$	325.a	1.a	$1$	$15$	$15$	$101.525$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓			325.8.a.j	$1$	$0$	\(15\)	\(13\)	\(0\)	\(1164\)		$-$	$-$	$+$	$2^{14}\cdot 3^{2}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{1}+\beta _{4})q^{3}+(65+\cdots)q^{4}+\cdots\)
325.8.a.l	$325$	$8$	325.a	1.a	$1$	$21$	$21$	$101.525$		None	✓		✓		65.8.b.a	$1$	$1$	\(-24\)	\(-108\)	\(0\)	\(-894\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
325.8.a.m	$325$	$8$	325.a	1.a	$1$	$21$	$21$	$101.525$		None	✓		✓		65.8.b.a	$1$	$0$	\(24\)	\(108\)	\(0\)	\(894\)		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(325))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(325)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)