Space of modular forms of level 192, weight 13, and Character 192.e

• Label: 192.13.e
• Level: $192$
• Weight: $13$
• Character orbit: 192.e
• Rep. character: $\chi_{192}(65,\cdot)$
• Character field: $\Q$
• Dimension: $94$
• Newform subspaces: $12$
• Sturm bound: $416$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	192.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(416\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(192, [\chi])\).

	Total	New	Old
Modular forms	396	98	298
Cusp forms	372	94	278
Eisenstein series	24	4	20

Trace form

94 * q - 2 * q^9 - 2550236 * q^13 + 1062884 * q^21 - 4199218754 * q^25 + 1577980800 * q^33 + 4945580644 * q^37 - 29776816896 * q^45 + 162140792922 * q^49 - 32659370244 * q^57 - 23651826524 * q^61 + 388392039680 * q^69 - 4 * q^73 + 110046484126 * q^81 + 435844437504 * q^85 + 2276987040164 * q^93 - 1277640517444 * q^97

Decomposition of \(S_{13}^{\mathrm{new}}(192, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
192.13.e.a	$192$	$13$	192.e	3.b	$2$	$1$	$1$	$175.487$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				3.13.b.a	$2$	$0$	\(0\)	\(-729\)	\(0\)	\(-153502\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{6}q^{3}-153502q^{7}+3^{12}q^{9}+9397582q^{13}+\cdots\)
192.13.e.b	$192$	$13$	192.e	3.b	$2$	$1$	$1$	$175.487$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				3.13.b.a	$2$	$0$	\(0\)	\(729\)	\(0\)	\(153502\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{6}q^{3}+153502q^{7}+3^{12}q^{9}+9397582q^{13}+\cdots\)
192.13.e.c	$192$	$13$	192.e	3.b	$2$	$2$	$2$	$175.487$	\(\Q(\sqrt{-26}) \)	None					3.13.b.b	$2$	$0$	\(0\)	\(-1350\)	\(0\)	\(-80500\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-675+3\beta )q^{3}+230\beta q^{5}-40250q^{7}+\cdots\)
192.13.e.d	$192$	$13$	192.e	3.b	$2$	$2$	$2$	$175.487$	\(\Q(\sqrt{-26}) \)	None					3.13.b.b	$2$	$0$	\(0\)	\(1350\)	\(0\)	\(80500\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(675-3\beta )q^{3}+230\beta q^{5}+40250q^{7}+\cdots\)
192.13.e.e	$192$	$13$	192.e	3.b	$2$	$4$	$4$	$175.487$	\(\Q(\sqrt{-2}, \sqrt{1009})\)	None					6.13.b.a	$2$	$0$	\(0\)	\(-780\)	\(0\)	\(153080\)	$2^{11}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-195+\beta _{3})q^{3}+(-18\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
192.13.e.f	$192$	$13$	192.e	3.b	$2$	$4$	$4$	$175.487$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					12.13.c.a	$2$	$0$	\(0\)	\(-300\)	\(0\)	\(15800\)	$2^{10}\cdot 3^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-75-\beta _{1})q^{3}+(3\beta _{1}-\beta _{2})q^{5}+(3950+\cdots)q^{7}+\cdots\)
192.13.e.g	$192$	$13$	192.e	3.b	$2$	$4$	$4$	$175.487$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					12.13.c.a	$2$	$0$	\(0\)	\(300\)	\(0\)	\(-15800\)	$2^{10}\cdot 3^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(75+\beta _{1})q^{3}+(3\beta _{1}-\beta _{2})q^{5}+(-3950+\cdots)q^{7}+\cdots\)
192.13.e.h	$192$	$13$	192.e	3.b	$2$	$4$	$4$	$175.487$	\(\Q(\sqrt{-2}, \sqrt{1009})\)	None					6.13.b.a	$2$	$0$	\(0\)	\(780\)	\(0\)	\(-153080\)	$2^{11}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(195-\beta _{3})q^{3}+(-18\beta _{1}+\beta _{2}+10\beta _{3})q^{5}+\cdots\)
192.13.e.i	$192$	$13$	192.e	3.b	$2$	$12$	$12$	$175.487$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					24.13.e.a	$2$	$0$	\(0\)	\(-460\)	\(0\)	\(198840\)	$2^{104}\cdot 3^{25}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-38-\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(16571+\cdots)q^{7}+\cdots\)
192.13.e.j	$192$	$13$	192.e	3.b	$2$	$12$	$12$	$175.487$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					24.13.e.a	$2$	$0$	\(0\)	\(460\)	\(0\)	\(-198840\)	$2^{104}\cdot 3^{25}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(38+\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-16571+\cdots)q^{7}+\cdots\)
192.13.e.k	$192$	$13$	192.e	3.b	$2$	$24$	$24$	$175.487$		None					96.13.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
192.13.e.l	$192$	$13$	192.e	3.b	$2$	$24$	$24$	$175.487$		None					96.13.e.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{13}^{\mathrm{old}}(192, [\chi])\) into lower level spaces

\( S_{13}^{\mathrm{old}}(192, [\chi]) \simeq \) \(S_{13}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)