Space of modular forms of level 3364, weight 2, and trivial character

• Label: 3364.2.a
• Level: $3364$
• Weight: $2$
• Character orbit: 3364.a
• Rep. character: $\chi_{3364}(1,\cdot)$
• Character field: $\Q$
• Dimension: $67$
• Newform subspaces: $18$
• Sturm bound: $870$
• Trace bound: $45$

Defining parameters

Level:	$N$	$=$	$3364 = 2^{2} \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3364.a (trivial)
Character field:			$\Q$
Newform subspaces:			$18$
Sturm bound:			$870$
Trace bound:			$45$
Distinguishing $T_p$:			$3$, $5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(3364))$.

	Total	New	Old
Modular forms	480	67	413
Cusp forms	391	67	324
Eisenstein series	89	0	89

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

$2$	$29$	Fricke	Dim
$-$	$+$	$-$	$37$
$-$	$-$	$+$	$30$
Plus space		$+$	$30$
Minus space		$-$	$37$

Trace form

67 * q - 4 * q^5 - 4 * q^7 + 65 * q^9 + 4 * q^11 - 4 * q^13 + 10 * q^15 + 2 * q^17 + 6 * q^19 + 8 * q^21 + 8 * q^23 + 63 * q^25 + 18 * q^27 - 8 * q^31 + 8 * q^35 - 2 * q^37 - 18 * q^39 + 6 * q^41 - 4 * q^43 - 16 * q^45 + 12 * q^47 + 43 * q^49 + 12 * q^51 - 4 * q^53 - 18 * q^55 + 32 * q^57 + 2 * q^59 - 22 * q^61 - 32 * q^63 - 8 * q^65 - 6 * q^67 - 20 * q^69 - 8 * q^71 + 6 * q^73 + 10 * q^75 + 40 * q^77 - 4 * q^79 + 75 * q^81 - 30 * q^83 + 16 * q^85 - 2 * q^89 + 24 * q^91 + 32 * q^93 - 12 * q^95 - 18 * q^97 + 18 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(3364))$ 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	29
3364.2.a.a	$3364$	$2$	3364.a	1.a	$1$	$1$	$1$	$26.862$	$\Q$	None	✓				116.2.a.c	$1$	$0$	$0$	$-2$	$-2$	$4$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}-2q^{5}+4q^{7}+q^{9}+6q^{11}+\cdots$
3364.2.a.b	$3364$	$2$	3364.a	1.a	$1$	$1$	$1$	$26.862$	$\Q$	None	✓				116.2.a.b	$1$	$0$	$0$	$-1$	$3$	$-4$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{3}+3q^{5}-4q^{7}-2q^{9}-3q^{11}+\cdots$
3364.2.a.c	$3364$	$2$	3364.a	1.a	$1$	$1$	$1$	$26.862$	$\Q$	None	✓				116.2.a.a	$1$	$0$	$0$	$3$	$3$	$4$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+3q^{3}+3q^{5}+4q^{7}+6q^{9}+q^{11}+\cdots$
3364.2.a.d	$3364$	$2$	3364.a	1.a	$1$	$2$	$2$	$26.862$	$\Q(\sqrt{5})$	None	✓	✓			3364.2.a.d	$1$	$0$	$0$	$-2$	$1$	$-3$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2\beta q^{3}+(2-3\beta )q^{5}+(-2+\beta )q^{7}+\cdots$
3364.2.a.e	$3364$	$2$	3364.a	1.a	$1$	$2$	$2$	$26.862$	$\Q(\sqrt{13})$	None	✓	✓			3364.2.a.e	$1$	$1$	$0$	$-1$	$1$	$2$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{3}+\beta q^{5}+q^{7}+\beta q^{9}+(2+\beta )q^{11}+\cdots$
3364.2.a.f	$3364$	$2$	3364.a	1.a	$1$	$2$	$2$	$26.862$	$\Q(\sqrt{7})$	None	✓				116.2.c.a	$2$	$1$	$0$	$0$	$-2$	$-4$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-q^{5}-2q^{7}+4q^{9}+\beta q^{11}+\cdots$
3364.2.a.g	$3364$	$2$	3364.a	1.a	$1$	$2$	$2$	$26.862$	$\Q(\sqrt{13})$	None	✓	✓			3364.2.a.e	$1$	$0$	$0$	$1$	$1$	$2$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+\beta q^{5}+q^{7}+\beta q^{9}+(-2-\beta )q^{11}+\cdots$
3364.2.a.h	$3364$	$2$	3364.a	1.a	$1$	$2$	$2$	$26.862$	$\Q(\sqrt{5})$	None	✓	✓			3364.2.a.d	$1$	$1$	$0$	$2$	$1$	$-3$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2\beta q^{3}+(2-3\beta )q^{5}+(-2+\beta )q^{7}+\cdots$
3364.2.a.i	$3364$	$2$	3364.a	1.a	$1$	$3$	$3$	$26.862$	$\Q(\zeta_{14})^+$	None	✓				116.2.g.a	$1$	$2$	$0$	$-5$	$-5$	$-9$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-2-\beta _{2})q^{3}+(-2+\beta _{1})q^{5}+(-2+\cdots)q^{7}+\cdots$
3364.2.a.j	$3364$	$2$	3364.a	1.a	$1$	$3$	$3$	$26.862$	$\Q(\zeta_{14})^+$	None	✓				116.2.g.a	$1$	$0$	$0$	$5$	$-5$	$-9$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(2-\beta _{1})q^{3}+(-1-\beta _{1}+\beta _{2})q^{5}+\cdots$
3364.2.a.k	$3364$	$2$	3364.a	1.a	$1$	$4$	$4$	$26.862$	$\Q(\zeta_{15})^+$	None	✓	✓			3364.2.a.k	$1$	$1$	$0$	$-2$	$-3$	$1$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1}+\beta _{2})q^{3}+(-2+\beta _{1}-2\beta _{3})q^{5}+\cdots$
3364.2.a.l	$3364$	$2$	3364.a	1.a	$1$	$4$	$4$	$26.862$	$\Q(\zeta_{15})^+$	None	✓	✓			3364.2.a.k	$1$	$0$	$0$	$2$	$-3$	$1$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1-\beta _{1}-\beta _{2})q^{3}+(-2+\beta _{1}-2\beta _{3})q^{5}+\cdots$
3364.2.a.m	$3364$	$2$	3364.a	1.a	$1$	$6$	$6$	$26.862$	6.6.6456289.1	None	✓				116.2.g.b	$1$	$0$	$0$	$-5$	$10$	$7$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{3}+(1-\beta _{2}-\beta _{5})q^{5}+\cdots$
3364.2.a.n	$3364$	$2$	3364.a	1.a	$1$	$6$	$6$	$26.862$	$\Q(\zeta_{28})^+$	None	✓				116.2.i.a	$2$	$1$	$0$	$0$	$-6$	$2$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(-1-\beta _{2}+\beta _{4})q^{5}+(2-2\beta _{2}+\cdots)q^{7}+\cdots$
3364.2.a.o	$3364$	$2$	3364.a	1.a	$1$	$6$	$6$	$26.862$	$\Q(\zeta_{28})^+$	None	✓				116.2.i.b	$2$	$1$	$0$	$0$	$-6$	$2$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(-1+\beta _{2}-\beta _{4})q^{5}+\beta _{4}q^{7}+\cdots$
3364.2.a.p	$3364$	$2$	3364.a	1.a	$1$	$6$	$6$	$26.862$	6.6.6456289.1	None	✓				116.2.g.b	$1$	$0$	$0$	$5$	$10$	$7$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{3}+(1-\beta _{2}-\beta _{5})q^{5}+(1+\cdots)q^{7}+\cdots$
3364.2.a.q	$3364$	$2$	3364.a	1.a	$1$	$8$	$8$	$26.862$	8.8.3266578125.2	None	✓	✓	✓		3364.2.a.q	$1$	$1$	$0$	$-4$	$-1$	$-2$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{3}-\beta _{4}-\beta _{5}-\beta _{6})q^{3}+(-1+\cdots)q^{5}+\cdots$
3364.2.a.r	$3364$	$2$	3364.a	1.a	$1$	$8$	$8$	$26.862$	8.8.3266578125.2	None	✓	✓	✓		3364.2.a.q	$1$	$0$	$0$	$4$	$-1$	$-2$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta _{3}+\beta _{4}+\beta _{5}+\beta _{6})q^{3}+(-1+\cdots)q^{5}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(3364))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(3364)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(58))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(116))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(841))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1682))$$^{\oplus 2}$