Space of modular forms of level 336, weight 8, and trivial character

• Label: 336.8.a
• Level: $336$
• Weight: $8$
• Character orbit: 336.a
• Rep. character: $\chi_{336}(1,\cdot)$
• Character field: $\Q$
• Dimension: $42$
• Newform subspaces: $23$
• Sturm bound: $512$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(512\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(336))\).

	Total	New	Old
Modular forms	460	42	418
Cusp forms	436	42	394
Eisenstein series	24	0	24

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

\(2\)	\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
Plus space			\(+\)	\(23\)
Minus space			\(-\)	\(19\)

Trace form

\( 42 q + 556 q^{5} - 686 q^{7} + 30618 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(336))\) 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	7
336.8.a.a	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				168.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(-470\)	\(343\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-470q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.b	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				21.8.a.a	$1$	$0$	\(0\)	\(-27\)	\(-278\)	\(343\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-278q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.c	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				84.8.a.b	$1$	$1$	\(0\)	\(-27\)	\(-240\)	\(-343\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-240q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.d	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				42.8.a.c	$1$	$0$	\(0\)	\(-27\)	\(-122\)	\(343\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-122q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.e	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				42.8.a.d	$1$	$1$	\(0\)	\(-27\)	\(270\)	\(-343\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+270q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.f	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				42.8.a.f	$1$	$0$	\(0\)	\(-27\)	\(470\)	\(343\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+470q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.g	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				42.8.a.a	$1$	$1$	\(0\)	\(27\)	\(-410\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-410q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.h	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				42.8.a.b	$1$	$0$	\(0\)	\(27\)	\(-18\)	\(-343\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-18q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.i	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				42.8.a.e	$1$	$0$	\(0\)	\(27\)	\(30\)	\(-343\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+30q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.j	$336$	$8$	336.a	1.a	$1$	$1$	$1$	$104.961$	\(\Q\)	None	✓				84.8.a.a	$1$	$1$	\(0\)	\(27\)	\(100\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+10^{2}q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.k	$336$	$8$	336.a	1.a	$1$	$2$	$2$	$104.961$	\(\Q(\sqrt{21961}) \)	None	✓		✓		84.8.a.d	$1$	$0$	\(0\)	\(-54\)	\(96\)	\(686\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(48-\beta )q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.l	$336$	$8$	336.a	1.a	$1$	$2$	$2$	$104.961$	\(\Q(\sqrt{15}) \)	None	✓				168.8.a.c	$1$	$1$	\(0\)	\(-54\)	\(220\)	\(686\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(110+\beta )q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.m	$336$	$8$	336.a	1.a	$1$	$2$	$2$	$104.961$	\(\Q(\sqrt{799}) \)	None	✓				168.8.a.d	$1$	$1$	\(0\)	\(-54\)	\(348\)	\(686\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(174+\beta )q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.n	$336$	$8$	336.a	1.a	$1$	$2$	$2$	$104.961$	\(\Q(\sqrt{1065}) \)	None	✓				21.8.a.b	$1$	$0$	\(0\)	\(54\)	\(-360\)	\(-686\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-180-5\beta )q^{5}-7^{3}q^{7}+\cdots\)
336.8.a.o	$336$	$8$	336.a	1.a	$1$	$2$	$2$	$104.961$	\(\Q(\sqrt{211}) \)	None	✓				168.8.a.b	$1$	$1$	\(0\)	\(54\)	\(-52\)	\(-686\)		$+$	$-$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-26+\beta )q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.p	$336$	$8$	336.a	1.a	$1$	$2$	$2$	$104.961$	\(\Q(\sqrt{67}) \)	None	✓		✓		21.8.a.c	$1$	$1$	\(0\)	\(54\)	\(-24\)	\(686\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-12+\beta )q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.q	$336$	$8$	336.a	1.a	$1$	$2$	$2$	$104.961$	\(\Q(\sqrt{3649}) \)	None	✓				84.8.a.c	$1$	$0$	\(0\)	\(54\)	\(264\)	\(-686\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(132-\beta )q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.r	$336$	$8$	336.a	1.a	$1$	$3$	$3$	$104.961$	3.3.2910828.1	None	✓		✓		21.8.a.d	$1$	$1$	\(0\)	\(-81\)	\(-114\)	\(-1029\)		$-$	$+$	$+$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-38+\beta _{1}-\beta _{2})q^{5}-7^{3}q^{7}+\cdots\)
336.8.a.s	$336$	$8$	336.a	1.a	$1$	$3$	$3$	$104.961$	3.3.2007672.1	None	✓				168.8.a.h	$1$	$0$	\(0\)	\(-81\)	\(-66\)	\(-1029\)		$+$	$+$	$+$	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-22+2\beta _{1}-\beta _{2})q^{5}-7^{3}q^{7}+\cdots\)
336.8.a.t	$336$	$8$	336.a	1.a	$1$	$3$	$3$	$104.961$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				168.8.a.i	$1$	$0$	\(0\)	\(-81\)	\(414\)	\(-1029\)		$+$	$+$	$+$	$+$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(138+\beta _{1})q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.u	$336$	$8$	336.a	1.a	$1$	$3$	$3$	$104.961$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				168.8.a.e	$1$	$0$	\(0\)	\(81\)	\(94\)	\(1029\)		$+$	$-$	$-$	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(31-\beta _{1})q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.v	$336$	$8$	336.a	1.a	$1$	$3$	$3$	$104.961$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		168.8.a.f	$1$	$1$	\(0\)	\(81\)	\(150\)	\(-1029\)		$+$	$-$	$+$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(50+\beta _{1})q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
336.8.a.w	$336$	$8$	336.a	1.a	$1$	$3$	$3$	$104.961$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				168.8.a.g	$1$	$0$	\(0\)	\(81\)	\(254\)	\(1029\)		$+$	$-$	$-$	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(85+\beta _{1})q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(336))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(336)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)