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

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