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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(59\)	\(6\)	\(53\)		\(56\)	\(6\)	\(50\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(57\)	\(5\)	\(52\)		\(54\)	\(5\)	\(49\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(56\)	\(5\)	\(51\)		\(53\)	\(5\)	\(48\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(58\)	\(6\)	\(52\)		\(55\)	\(6\)	\(49\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(56\)	\(5\)	\(51\)		\(53\)	\(5\)	\(48\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(58\)	\(5\)	\(53\)		\(55\)	\(5\)	\(50\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(59\)	\(6\)	\(53\)		\(56\)	\(6\)	\(50\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(57\)	\(4\)	\(53\)		\(54\)	\(4\)	\(50\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(234\)	\(23\)	\(211\)		\(222\)	\(23\)	\(199\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(226\)	\(19\)	\(207\)		\(214\)	\(19\)	\(195\)		\(12\)	\(0\)	\(12\)

Trace form

42 * q + 556 * q^5 - 686 * q^7 + 30618 * q^9 + 1204 * q^11 - 13108 * q^13 - 13500 * q^15 - 2908 * q^17 + 60584 * q^19 - 71708 * q^23 + 640126 * q^25 + 172532 * q^29 - 268024 * q^31 - 498348 * q^37 - 474552 * q^39 + 441284 * q^41 + 1890024 * q^43 + 405324 * q^45 - 1252248 * q^47 + 4941258 * q^49 + 95148 * q^51 + 2923028 * q^53 + 710200 * q^55 - 4443576 * q^59 + 2102364 * q^61 - 500094 * q^63 + 4613976 * q^65 - 3708704 * q^67 - 4790448 * q^69 + 2646220 * q^71 - 6713260 * q^73 - 5630256 * q^75 + 5957224 * q^77 - 467576 * q^79 + 22320522 * q^81 - 15764912 * q^83 - 3458048 * q^85 + 3673512 * q^87 - 20303356 * q^89 + 9042852 * q^91 + 4062744 * q^93 + 39594256 * q^95 + 6695124 * q^97 + 877716 * q^99

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

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