Space of modular forms of level 336, weight 6, and trivial character

• Label: 336.6.a
• Level: $336$
• Weight: $6$
• Character orbit: 336.a
• Rep. character: $\chi_{336}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $24$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	336.a (trivial)
Character field:			$\Q$
Newform subspaces:			$24$
Sturm bound:			$384$
Trace bound:			$7$
Distinguishing $T_p$:			$5$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{6}(\Gamma_0(336))$.

	Total	New	Old
Modular forms	332	30	302
Cusp forms	308	30	278
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
$+$	$+$	$+$	$+$	$3$
$+$	$+$	$-$	$-$	$4$
$+$	$-$	$+$	$-$	$4$
$+$	$-$	$-$	$+$	$3$
$-$	$+$	$+$	$-$	$4$
$-$	$+$	$-$	$+$	$4$
$-$	$-$	$+$	$+$	$3$
$-$	$-$	$-$	$-$	$5$
Plus space			$+$	$13$
Minus space			$-$	$17$

Trace form

30 * q - 76 * q^5 + 98 * q^7 + 2430 * q^9 - 604 * q^11 + 244 * q^13 + 900 * q^15 - 404 * q^17 - 2360 * q^19 + 836 * q^23 + 21866 * q^25 + 4492 * q^29 - 7160 * q^31 + 14412 * q^37 + 12168 * q^39 + 2476 * q^41 - 5496 * q^43 - 6156 * q^45 - 5016 * q^47 + 72030 * q^49 - 20772 * q^51 + 64972 * q^53 - 70408 * q^55 + 90024 * q^59 - 66396 * q^61 + 7938 * q^63 - 86904 * q^65 + 54944 * q^67 + 22320 * q^69 + 18860 * q^71 + 25180 * q^73 - 96624 * q^75 + 7448 * q^77 + 365960 * q^79 + 196830 * q^81 + 245264 * q^83 + 283136 * q^85 + 222696 * q^87 + 285484 * q^89 - 99372 * q^91 - 792 * q^93 - 318064 * q^95 - 76740 * q^97 - 48924 * q^99

Decomposition of $S_{6}^{\mathrm{new}}(\Gamma_0(336))$ 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
336.6.a.a	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				21.6.a.d	$1$	$1$	$0$	$-9$	$-106$	$49$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}-106q^{5}+7^{2}q^{7}+3^{4}q^{9}+\cdots$
336.6.a.b	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				42.6.a.c	$1$	$0$	$0$	$-9$	$-72$	$-49$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}-72q^{5}-7^{2}q^{7}+3^{4}q^{9}+414q^{11}+\cdots$
336.6.a.c	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				168.6.a.d	$1$	$1$	$0$	$-9$	$-64$	$-49$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}-2^{6}q^{5}-7^{2}q^{7}+3^{4}q^{9}+54q^{11}+\cdots$
336.6.a.d	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				168.6.a.e	$1$	$0$	$0$	$-9$	$-38$	$49$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}-38q^{5}+7^{2}q^{7}+3^{4}q^{9}-600q^{11}+\cdots$
336.6.a.e	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				84.6.a.b	$1$	$1$	$0$	$-9$	$-34$	$49$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}-34q^{5}+7^{2}q^{7}+3^{4}q^{9}+332q^{11}+\cdots$
336.6.a.f	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				168.6.a.f	$1$	$0$	$0$	$-9$	$14$	$49$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}+14q^{5}+7^{2}q^{7}+3^{4}q^{9}+700q^{11}+\cdots$
336.6.a.g	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				42.6.a.f	$1$	$0$	$0$	$-9$	$24$	$-49$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}+24q^{5}-7^{2}q^{7}+3^{4}q^{9}-66q^{11}+\cdots$
336.6.a.h	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				42.6.a.d	$1$	$1$	$0$	$-9$	$26$	$49$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}+26q^{5}+7^{2}q^{7}+3^{4}q^{9}-664q^{11}+\cdots$
336.6.a.i	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				21.6.a.c	$1$	$1$	$0$	$-9$	$94$	$49$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-9q^{3}+94q^{5}+7^{2}q^{7}+3^{4}q^{9}-52q^{11}+\cdots$
336.6.a.j	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				42.6.a.a	$1$	$1$	$0$	$9$	$-54$	$-49$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}-54q^{5}-7^{2}q^{7}+3^{4}q^{9}-6^{3}q^{11}+\cdots$
336.6.a.k	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				168.6.a.a	$1$	$0$	$0$	$9$	$-34$	$-49$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}-34q^{5}-7^{2}q^{7}+3^{4}q^{9}+756q^{11}+\cdots$
336.6.a.l	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				21.6.a.b	$1$	$0$	$0$	$9$	$-34$	$49$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}-34q^{5}+7^{2}q^{7}+3^{4}q^{9}+340q^{11}+\cdots$
336.6.a.m	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				168.6.a.b	$1$	$1$	$0$	$9$	$4$	$49$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}+4q^{5}+7^{2}q^{7}+3^{4}q^{9}-370q^{11}+\cdots$
336.6.a.n	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				84.6.a.a	$1$	$1$	$0$	$9$	$6$	$-49$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}+6q^{5}-7^{2}q^{7}+3^{4}q^{9}+108q^{11}+\cdots$
336.6.a.o	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				42.6.a.b	$1$	$0$	$0$	$9$	$44$	$49$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}+44q^{5}+7^{2}q^{7}+3^{4}q^{9}+470q^{11}+\cdots$
336.6.a.p	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				168.6.a.c	$1$	$0$	$0$	$9$	$74$	$-49$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}+74q^{5}-7^{2}q^{7}+3^{4}q^{9}-6^{3}q^{11}+\cdots$
336.6.a.q	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				42.6.a.e	$1$	$0$	$0$	$9$	$76$	$49$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}+76q^{5}+7^{2}q^{7}+3^{4}q^{9}-650q^{11}+\cdots$
336.6.a.r	$336$	$6$	336.a	1.a	$1$	$1$	$1$	$53.889$	$\Q$	None	✓				21.6.a.a	$1$	$1$	$0$	$9$	$78$	$-49$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+9q^{3}+78q^{5}-7^{2}q^{7}+3^{4}q^{9}-444q^{11}+\cdots$
336.6.a.s	$336$	$6$	336.a	1.a	$1$	$2$	$2$	$53.889$	$\Q(\sqrt{4281})$	None	✓		✓		168.6.a.i	$1$	$1$	$0$	$-18$	$-10$	$-98$		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-9q^{3}+(-5-\beta )q^{5}-7^{2}q^{7}+3^{4}q^{9}+\cdots$
336.6.a.t	$336$	$6$	336.a	1.a	$1$	$2$	$2$	$53.889$	$\Q(\sqrt{1129})$	None	✓		✓		168.6.a.j	$1$	$0$	$0$	$-18$	$0$	$98$		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q-9q^{3}-\beta q^{5}+7^{2}q^{7}+3^{4}q^{9}+(-50+\cdots)q^{11}+\cdots$
336.6.a.u	$336$	$6$	336.a	1.a	$1$	$2$	$2$	$53.889$	$\Q(\sqrt{505})$	None	✓		✓		84.6.a.d	$1$	$0$	$0$	$-18$	$78$	$-98$		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	$q-9q^{3}+(39-\beta )q^{5}-7^{2}q^{7}+3^{4}q^{9}+\cdots$
336.6.a.v	$336$	$6$	336.a	1.a	$1$	$2$	$2$	$53.889$	$\Q(\sqrt{193})$	None	✓		✓		168.6.a.g	$1$	$1$	$0$	$18$	$-78$	$98$		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+9q^{3}+(-39-5\beta )q^{5}+7^{2}q^{7}+3^{4}q^{9}+\cdots$
336.6.a.w	$336$	$6$	336.a	1.a	$1$	$2$	$2$	$53.889$	$\Q(\sqrt{249})$	None	✓		✓		168.6.a.h	$1$	$0$	$0$	$18$	$-64$	$-98$		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q+9q^{3}+(-2^{5}-\beta )q^{5}-7^{2}q^{7}+3^{4}q^{9}+\cdots$
336.6.a.x	$336$	$6$	336.a	1.a	$1$	$2$	$2$	$53.889$	$\Q(\sqrt{5569})$	None	✓		✓		84.6.a.c	$1$	$0$	$0$	$18$	$-6$	$98$		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+9q^{3}+(-3-\beta )q^{5}+7^{2}q^{7}+3^{4}q^{9}+\cdots$

Decomposition of $S_{6}^{\mathrm{old}}(\Gamma_0(336))$ into lower level spaces

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