Space of modular forms of level 300, weight 9, and Character 300.g

• Label: 300.9.g
• Level: $300$
• Weight: $9$
• Character orbit: 300.g
• Rep. character: $\chi_{300}(101,\cdot)$
• Character field: $\Q$
• Dimension: $51$
• Newform subspaces: $8$
• Sturm bound: $540$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	300.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(540\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(300, [\chi])\).

	Total	New	Old
Modular forms	498	51	447
Cusp forms	462	51	411
Eisenstein series	36	0	36

Trace form

51 * q - 91 * q^3 - 1994 * q^7 - 1939 * q^9 + 54466 * q^13 - 323276 * q^19 - 274396 * q^21 + 93869 * q^27 + 1220744 * q^31 - 1440060 * q^33 - 882062 * q^37 + 3798264 * q^39 + 6461026 * q^43 + 44800155 * q^49 + 13072850 * q^51 - 1576694 * q^57 - 40919376 * q^61 + 15229606 * q^63 - 12796814 * q^67 + 13142600 * q^69 + 13092238 * q^73 + 121145662 * q^79 - 28903279 * q^81 + 40636620 * q^87 + 157439246 * q^91 + 62977666 * q^93 - 31762634 * q^97 + 191259250 * q^99

Decomposition of \(S_{9}^{\mathrm{new}}(300, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
300.9.g.a	$300$	$9$	300.g	3.b	$2$	$1$	$1$	$122.214$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				12.9.c.a	$2$	$0$	\(0\)	\(-81\)	\(0\)	\(-4034\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{4}q^{3}-4034q^{7}+3^{8}q^{9}+35806q^{13}+\cdots\)
300.9.g.b	$300$	$9$	300.g	3.b	$2$	$1$	$1$	$122.214$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			300.9.g.b	$2$	$0$	\(0\)	\(-81\)	\(0\)	\(-239\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{4}q^{3}-239q^{7}+3^{8}q^{9}+20641q^{13}+\cdots\)
300.9.g.c	$300$	$9$	300.g	3.b	$2$	$1$	$1$	$122.214$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			300.9.g.b	$2$	$0$	\(0\)	\(81\)	\(0\)	\(239\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{4}q^{3}+239q^{7}+3^{8}q^{9}-20641q^{13}+\cdots\)
300.9.g.d	$300$	$9$	300.g	3.b	$2$	$2$	$2$	$122.214$	\(\Q(\sqrt{-110}) \)	None					12.9.c.b	$2$	$0$	\(0\)	\(102\)	\(0\)	\(6188\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(51+\beta )q^{3}+3094q^{7}+(-1359+\cdots)q^{9}+\cdots\)
300.9.g.e	$300$	$9$	300.g	3.b	$2$	$10$	$10$	$122.214$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		✓			300.9.g.e	$2$	$0$	\(0\)	\(-137\)	\(0\)	\(-1048\)	$2^{15}\cdot 3^{17}\cdot 5^{7}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-14-\beta _{1})q^{3}+(-104+2\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
300.9.g.f	$300$	$9$	300.g	3.b	$2$	$10$	$10$	$122.214$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					60.9.g.a	$2$	$0$	\(0\)	\(-112\)	\(0\)	\(-4148\)	$2^{13}\cdot 3^{17}\cdot 5^{13}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-11+\beta _{1})q^{3}+(-417-11\beta _{1}+\cdots)q^{7}+\cdots\)
300.9.g.g	$300$	$9$	300.g	3.b	$2$	$10$	$10$	$122.214$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		✓			300.9.g.e	$2$	$0$	\(0\)	\(137\)	\(0\)	\(1048\)	$2^{15}\cdot 3^{17}\cdot 5^{7}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(14+\beta _{1})q^{3}+(104-2\beta _{1}+\beta _{3})q^{7}+\cdots\)
300.9.g.h	$300$	$9$	300.g	3.b	$2$	$16$	$16$	$122.214$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None			✓		60.9.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{38}\cdot 3^{20}\cdot 5^{18}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{7}q^{7}+(922+\beta _{2})q^{9}-\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(300, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)