Space of modular forms of level 392, weight 6, and trivial character

• Label: 392.6.a
• Level: $392$
• Weight: $6$
• Character orbit: 392.a
• Rep. character: $\chi_{392}(1,\cdot)$
• Character field: $\Q$
• Dimension: $51$
• Newform subspaces: $14$
• Sturm bound: $336$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(336\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(392))\).

	Total	New	Old
Modular forms	296	51	245
Cusp forms	264	51	213
Eisenstein series	32	0	32

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

\(2\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(72\)	\(11\)	\(61\)		\(64\)	\(11\)	\(53\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)		\(76\)	\(14\)	\(62\)		\(68\)	\(14\)	\(54\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(76\)	\(13\)	\(63\)		\(68\)	\(13\)	\(55\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)		\(72\)	\(13\)	\(59\)		\(64\)	\(13\)	\(51\)		\(8\)	\(0\)	\(8\)
Plus space		\(+\)		\(144\)	\(24\)	\(120\)		\(128\)	\(24\)	\(104\)		\(16\)	\(0\)	\(16\)
Minus space		\(-\)		\(152\)	\(27\)	\(125\)		\(136\)	\(27\)	\(109\)		\(16\)	\(0\)	\(16\)

Trace form

51 * q + 2 * q^3 - 24 * q^5 + 3827 * q^9 + 144 * q^11 + 700 * q^13 - 1388 * q^15 - 1126 * q^17 - 3938 * q^19 - 1260 * q^23 + 26957 * q^25 - 2500 * q^27 + 6134 * q^29 + 5252 * q^31 + 11776 * q^33 - 5070 * q^37 + 1600 * q^39 + 8922 * q^41 + 26588 * q^43 + 20168 * q^45 + 4812 * q^47 + 5044 * q^51 - 15642 * q^53 + 65312 * q^55 - 57316 * q^57 - 50714 * q^59 - 49120 * q^61 + 130144 * q^65 - 75480 * q^67 - 49536 * q^69 + 97864 * q^71 + 66158 * q^73 + 117142 * q^75 - 158484 * q^79 + 407611 * q^81 - 141530 * q^83 + 32228 * q^85 - 283628 * q^87 + 203158 * q^89 + 41732 * q^93 + 291388 * q^95 - 113302 * q^97 + 21148 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(392))\) 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	7
392.6.a.a	$392$	$6$	392.a	1.a	$1$	$1$	$1$	$62.870$	\(\Q\)	None	✓				56.6.a.b	$1$	$0$	\(0\)	\(-30\)	\(-32\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-30q^{3}-2^{5}q^{5}+657q^{9}-624q^{11}+\cdots\)
392.6.a.b	$392$	$6$	392.a	1.a	$1$	$1$	$1$	$62.870$	\(\Q\)	None	✓				8.6.a.a	$1$	$1$	\(0\)	\(-20\)	\(74\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-20q^{3}+74q^{5}+157q^{9}+124q^{11}+\cdots\)
392.6.a.c	$392$	$6$	392.a	1.a	$1$	$1$	$1$	$62.870$	\(\Q\)	None	✓				56.6.a.a	$1$	$1$	\(0\)	\(6\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}-4q^{5}-207q^{9}-240q^{11}+\cdots\)
392.6.a.d	$392$	$6$	392.a	1.a	$1$	$2$	$2$	$62.870$	\(\Q(\sqrt{345}) \)	None	✓				56.6.a.e	$1$	$1$	\(0\)	\(6\)	\(-82\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(-41+3\beta )q^{5}+(111+\cdots)q^{9}+\cdots\)
392.6.a.e	$392$	$6$	392.a	1.a	$1$	$2$	$2$	$62.870$	\(\Q(\sqrt{193}) \)	None	✓				56.6.a.d	$1$	$0$	\(0\)	\(14\)	\(-42\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(-21+5\beta )q^{5}+(-1+\cdots)q^{9}+\cdots\)
392.6.a.f	$392$	$6$	392.a	1.a	$1$	$2$	$2$	$62.870$	\(\Q(\sqrt{177}) \)	None	✓				56.6.a.c	$1$	$0$	\(0\)	\(26\)	\(62\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(13-\beta )q^{3}+(31-5\beta )q^{5}+(103-26\beta )q^{9}+\cdots\)
392.6.a.g	$392$	$6$	392.a	1.a	$1$	$4$	$4$	$62.870$	\(\Q(\sqrt{86}, \sqrt{134})\)	None	✓	✓			392.6.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-23+\beta _{3})q^{9}+\cdots\)
392.6.a.h	$392$	$6$	392.a	1.a	$1$	$4$	$4$	$62.870$	4.4.2732674592.1	None	✓	✓			392.6.a.h	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-\beta _{1}-2\beta _{2})q^{5}+(209-\beta _{3})q^{9}+\cdots\)
392.6.a.i	$392$	$6$	392.a	1.a	$1$	$5$	$5$	$62.870$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		56.6.i.a	$1$	$1$	\(0\)	\(-13\)	\(-31\)	\(0\)		$-$	$-$	$+$	$2^{9}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+(-6-\beta _{3})q^{5}+(47+\cdots)q^{9}+\cdots\)
392.6.a.j	$392$	$6$	392.a	1.a	$1$	$5$	$5$	$62.870$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		56.6.i.b	$1$	$0$	\(0\)	\(-5\)	\(81\)	\(0\)		$+$	$-$	$-$	$2^{9}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(2^{4}+\beta _{1}+\beta _{2})q^{5}+\cdots\)
392.6.a.k	$392$	$6$	392.a	1.a	$1$	$5$	$5$	$62.870$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				56.6.i.b	$1$	$1$	\(0\)	\(5\)	\(-81\)	\(0\)		$+$	$+$	$+$	$2^{9}\cdot 7$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-2^{4}-\beta _{1}-\beta _{2})q^{5}+\cdots\)
392.6.a.l	$392$	$6$	392.a	1.a	$1$	$5$	$5$	$62.870$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				56.6.i.a	$1$	$0$	\(0\)	\(13\)	\(31\)	\(0\)		$-$	$+$	$-$	$2^{9}\cdot 7$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{3}+(6+\beta _{3})q^{5}+(47-4\beta _{1}+\cdots)q^{9}+\cdots\)
392.6.a.m	$392$	$6$	392.a	1.a	$1$	$6$	$6$	$62.870$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		392.6.a.m	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(\beta _{1}-\beta _{3})q^{5}+(-20+\beta _{4}+\cdots)q^{9}+\cdots\)
392.6.a.n	$392$	$6$	392.a	1.a	$1$	$8$	$8$	$62.870$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		392.6.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{10}\cdot 7^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-\beta _{1}-\beta _{4})q^{5}+(116+\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(392))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(392)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)