Space of modular forms of level 300, weight 8, and trivial character

• Label: 300.8.a
• Level: $300$
• Weight: $8$
• Character orbit: 300.a
• Rep. character: $\chi_{300}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $14$
• Sturm bound: $480$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	300.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(480\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(300))\).

	Total	New	Old
Modular forms	438	22	416
Cusp forms	402	22	380
Eisenstein series	36	0	36

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(57\)	\(0\)	\(57\)		\(51\)	\(0\)	\(51\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(54\)	\(0\)	\(54\)		\(48\)	\(0\)	\(48\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(54\)	\(0\)	\(54\)		\(48\)	\(0\)	\(48\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(57\)	\(0\)	\(57\)		\(51\)	\(0\)	\(51\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(54\)	\(5\)	\(49\)		\(51\)	\(5\)	\(46\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(54\)	\(6\)	\(48\)		\(51\)	\(6\)	\(45\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(54\)	\(6\)	\(48\)		\(51\)	\(6\)	\(45\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(54\)	\(5\)	\(49\)		\(51\)	\(5\)	\(46\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(222\)	\(12\)	\(210\)		\(204\)	\(12\)	\(192\)		\(18\)	\(0\)	\(18\)
Minus space			\(-\)		\(216\)	\(10\)	\(206\)		\(198\)	\(10\)	\(188\)		\(18\)	\(0\)	\(18\)

Trace form

22 * q + 840 * q^7 + 16038 * q^9 + 5572 * q^11 + 11700 * q^13 + 8220 * q^17 - 58510 * q^19 + 28998 * q^21 + 86160 * q^23 - 231108 * q^29 - 292042 * q^31 + 272160 * q^33 + 5700 * q^37 - 155790 * q^39 - 1301600 * q^41 + 2095920 * q^43 - 1736400 * q^47 + 2468112 * q^49 - 818532 * q^51 + 3320700 * q^53 - 320760 * q^57 + 686252 * q^59 - 1936066 * q^61 + 612360 * q^63 - 2282880 * q^67 - 1040148 * q^69 - 2087760 * q^71 - 304620 * q^73 - 2270400 * q^77 + 1837424 * q^79 + 11691702 * q^81 - 1929840 * q^83 + 3314520 * q^87 + 10635760 * q^89 + 13246510 * q^91 - 5336280 * q^93 + 14536500 * q^97 + 4061988 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(300))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5
300.8.a.a	$300$	$8$	300.a	1.a	$1$	$1$	$1$	$93.716$	\(\Q\)	None	✓				12.8.a.b	$1$	$1$	\(0\)	\(-27\)	\(0\)	\(-1112\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-1112q^{7}+3^{6}q^{9}-5724q^{11}+\cdots\)
300.8.a.b	$300$	$8$	300.a	1.a	$1$	$1$	$1$	$93.716$	\(\Q\)	None	✓				60.8.d.a	$1$	$0$	\(0\)	\(-27\)	\(0\)	\(-722\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-722q^{7}+3^{6}q^{9}-3994q^{11}+\cdots\)
300.8.a.c	$300$	$8$	300.a	1.a	$1$	$1$	$1$	$93.716$	\(\Q\)	None	✓				60.8.a.c	$1$	$1$	\(0\)	\(-27\)	\(0\)	\(-92\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-92q^{7}+3^{6}q^{9}+3456q^{11}+\cdots\)
300.8.a.d	$300$	$8$	300.a	1.a	$1$	$1$	$1$	$93.716$	\(\Q\)	None	✓				60.8.a.d	$1$	$1$	\(0\)	\(-27\)	\(0\)	\(1408\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+1408q^{7}+3^{6}q^{9}-4044q^{11}+\cdots\)
300.8.a.e	$300$	$8$	300.a	1.a	$1$	$1$	$1$	$93.716$	\(\Q\)	None	✓				60.8.a.a	$1$	$0$	\(0\)	\(27\)	\(0\)	\(-1028\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-1028q^{7}+3^{6}q^{9}+3096q^{11}+\cdots\)
300.8.a.f	$300$	$8$	300.a	1.a	$1$	$1$	$1$	$93.716$	\(\Q\)	None	✓				60.8.d.a	$1$	$1$	\(0\)	\(27\)	\(0\)	\(722\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+722q^{7}+3^{6}q^{9}-3994q^{11}+\cdots\)
300.8.a.g	$300$	$8$	300.a	1.a	$1$	$1$	$1$	$93.716$	\(\Q\)	None	✓				12.8.a.a	$1$	$0$	\(0\)	\(27\)	\(0\)	\(832\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+832q^{7}+3^{6}q^{9}-2484q^{11}+\cdots\)
300.8.a.h	$300$	$8$	300.a	1.a	$1$	$1$	$1$	$93.716$	\(\Q\)	None	✓				60.8.a.b	$1$	$0$	\(0\)	\(27\)	\(0\)	\(832\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+832q^{7}+3^{6}q^{9}+3156q^{11}+\cdots\)
300.8.a.i	$300$	$8$	300.a	1.a	$1$	$2$	$2$	$93.716$	\(\Q(\sqrt{319}) \)	None	✓	✓	✓		300.8.a.i	$1$	$1$	\(0\)	\(-54\)	\(0\)	\(-154\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-77+\beta )q^{7}+3^{6}q^{9}+(-714+\cdots)q^{11}+\cdots\)
300.8.a.j	$300$	$8$	300.a	1.a	$1$	$2$	$2$	$93.716$	\(\Q(\sqrt{1129}) \)	None	✓				60.8.d.b	$1$	$0$	\(0\)	\(-54\)	\(0\)	\(906\)		$-$	$+$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(453-\beta )q^{7}+3^{6}q^{9}+(3171+\cdots)q^{11}+\cdots\)
300.8.a.k	$300$	$8$	300.a	1.a	$1$	$2$	$2$	$93.716$	\(\Q(\sqrt{1129}) \)	None	✓				60.8.d.b	$1$	$1$	\(0\)	\(54\)	\(0\)	\(-906\)		$-$	$-$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-453-\beta )q^{7}+3^{6}q^{9}+\cdots\)
300.8.a.l	$300$	$8$	300.a	1.a	$1$	$2$	$2$	$93.716$	\(\Q(\sqrt{319}) \)	None	✓	✓			300.8.a.i	$1$	$1$	\(0\)	\(54\)	\(0\)	\(154\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(77+\beta )q^{7}+3^{6}q^{9}+(-714+\cdots)q^{11}+\cdots\)
300.8.a.m	$300$	$8$	300.a	1.a	$1$	$3$	$3$	$93.716$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓		300.8.a.m	$1$	$0$	\(0\)	\(-81\)	\(0\)	\(-351\)		$-$	$+$	$-$	$+$	$2^{5}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-117-\beta _{1})q^{7}+3^{6}q^{9}+\cdots\)
300.8.a.n	$300$	$8$	300.a	1.a	$1$	$3$	$3$	$93.716$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓		300.8.a.m	$1$	$0$	\(0\)	\(81\)	\(0\)	\(351\)		$-$	$-$	$+$	$+$	$2^{5}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(117+\beta _{1})q^{7}+3^{6}q^{9}+(1046+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(300))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(300)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)