Space of modular forms of level 2704, weight 2, and trivial character

• Label: 2704.2.a
• Level: $2704$
• Weight: $2$
• Character orbit: 2704.a
• Rep. character: $\chi_{2704}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $33$
• Sturm bound: $728$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 33 \)
Sturm bound:			\(728\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2704))\).

	Total	New	Old
Modular forms	406	83	323
Cusp forms	323	72	251
Eisenstein series	83	11	72

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

\(2\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(98\)	\(18\)	\(80\)		\(78\)	\(18\)	\(60\)		\(20\)	\(0\)	\(20\)
\(+\)	\(-\)	\(-\)		\(104\)	\(21\)	\(83\)		\(83\)	\(21\)	\(62\)		\(21\)	\(0\)	\(21\)
\(-\)	\(+\)	\(-\)		\(105\)	\(23\)	\(82\)		\(84\)	\(18\)	\(66\)		\(21\)	\(5\)	\(16\)
\(-\)	\(-\)	\(+\)		\(99\)	\(21\)	\(78\)		\(78\)	\(15\)	\(63\)		\(21\)	\(6\)	\(15\)
Plus space		\(+\)		\(197\)	\(39\)	\(158\)		\(156\)	\(33\)	\(123\)		\(41\)	\(6\)	\(35\)
Minus space		\(-\)		\(209\)	\(44\)	\(165\)		\(167\)	\(39\)	\(128\)		\(42\)	\(5\)	\(37\)

Trace form

72 * q + 2 * q^7 + 64 * q^9 - 2 * q^11 - 8 * q^15 + 4 * q^17 + 2 * q^19 + 8 * q^21 - 8 * q^23 + 56 * q^25 + 14 * q^31 + 8 * q^33 - 8 * q^37 - 4 * q^41 + 4 * q^43 + 8 * q^45 + 10 * q^47 + 44 * q^49 + 28 * q^51 - 20 * q^53 - 12 * q^55 - 18 * q^59 - 16 * q^61 - 6 * q^63 + 26 * q^67 - 8 * q^69 + 6 * q^71 + 20 * q^73 - 20 * q^75 - 8 * q^77 + 40 * q^79 + 24 * q^81 - 22 * q^83 + 28 * q^87 - 4 * q^89 + 16 * q^93 - 24 * q^95 + 4 * q^97 - 34 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2704))\) 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	13
2704.2.a.a	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				52.2.e.a	$1$	$0$	\(0\)	\(-3\)	\(-2\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{5}+q^{7}+6q^{9}-5q^{11}+\cdots\)
2704.2.a.b	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				52.2.e.a	$1$	$0$	\(0\)	\(-3\)	\(2\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{5}-q^{7}+6q^{9}+5q^{11}+\cdots\)
2704.2.a.c	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				104.2.i.a	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-q^{7}-2q^{9}+q^{11}+2q^{15}+\cdots\)
2704.2.a.d	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				104.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+5q^{7}-2q^{9}-2q^{11}+\cdots\)
2704.2.a.e	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				104.2.i.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{7}-2q^{9}-q^{11}-2q^{15}+\cdots\)
2704.2.a.f	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				26.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-q^{7}-2q^{9}+6q^{11}+\cdots\)
2704.2.a.g	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				52.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{7}-3q^{9}-2q^{11}+6q^{17}+\cdots\)
2704.2.a.h	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				26.2.c.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{7}-3q^{9}-4q^{11}+3q^{17}+\cdots\)
2704.2.a.i	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				26.2.c.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{7}-3q^{9}+4q^{11}+3q^{17}+\cdots\)
2704.2.a.j	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				26.2.b.a	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+3q^{7}-2q^{9}-3q^{15}+\cdots\)
2704.2.a.k	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				26.2.b.a	$1$	$1$	\(0\)	\(1\)	\(3\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}-3q^{7}-2q^{9}+3q^{15}+\cdots\)
2704.2.a.l	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				52.2.e.b	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{5}+4q^{7}+q^{9}-6q^{15}+\cdots\)
2704.2.a.m	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				52.2.e.b	$1$	$0$	\(0\)	\(2\)	\(3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{5}-4q^{7}+q^{9}+6q^{15}+\cdots\)
2704.2.a.n	$2704$	$2$	2704.a	1.a	$1$	$1$	$1$	$21.592$	\(\Q\)	None	✓				26.2.a.b	$1$	$0$	\(0\)	\(3\)	\(1\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}+q^{7}+6q^{9}-2q^{11}+\cdots\)
2704.2.a.o	$2704$	$2$	2704.a	1.a	$1$	$2$	$2$	$21.592$	\(\Q(\sqrt{3}) \)	None	✓				13.2.e.a	$2$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-\beta q^{5}+q^{9}+2\beta q^{15}+3q^{17}+\cdots\)
2704.2.a.p	$2704$	$2$	2704.a	1.a	$1$	$2$	$2$	$21.592$	\(\Q(\sqrt{17}) \)	None	✓				104.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2+\beta )q^{5}-\beta q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.q	$2704$	$2$	2704.a	1.a	$1$	$2$	$2$	$21.592$	\(\Q(\sqrt{17}) \)	None	✓				104.2.i.b	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1-\beta )q^{5}+\beta q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.r	$2704$	$2$	2704.a	1.a	$1$	$2$	$2$	$21.592$	\(\Q(\sqrt{17}) \)	None	✓				104.2.i.b	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1+\beta )q^{5}-\beta q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.s	$2704$	$2$	2704.a	1.a	$1$	$2$	$2$	$21.592$	\(\Q(\sqrt{17}) \)	None	✓				104.2.f.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{5}+(-2-\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.t	$2704$	$2$	2704.a	1.a	$1$	$2$	$2$	$21.592$	\(\Q(\sqrt{17}) \)	None	✓				104.2.f.a	$1$	$0$	\(0\)	\(1\)	\(1\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{5}+(2+\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.u	$2704$	$2$	2704.a	1.a	$1$	$2$	$2$	$21.592$	\(\Q(\sqrt{3}) \)	None	✓				52.2.h.a	$2$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta q^{7}-2q^{9}+3\beta q^{11}-3q^{17}+\cdots\)
2704.2.a.v	$2704$	$2$	2704.a	1.a	$1$	$3$	$3$	$21.592$	\(\Q(\zeta_{14})^+\)	None	✓				338.2.a.g	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{3}-2\beta _{1}q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
2704.2.a.w	$2704$	$2$	2704.a	1.a	$1$	$3$	$3$	$21.592$	\(\Q(\zeta_{14})^+\)	None	✓				338.2.a.g	$1$	$0$	\(0\)	\(-3\)	\(2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{3}+2\beta _{1}q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
2704.2.a.x	$2704$	$2$	2704.a	1.a	$1$	$3$	$3$	$21.592$	\(\Q(\zeta_{14})^+\)	None	✓				676.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta _{1}+\beta _{2})q^{3}+(-3-\beta _{2})q^{5}+\cdots\)
2704.2.a.y	$2704$	$2$	2704.a	1.a	$1$	$3$	$3$	$21.592$	\(\Q(\zeta_{14})^+\)	None	✓				676.2.a.g	$1$	$0$	\(0\)	\(0\)	\(8\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta _{1}+\beta _{2})q^{3}+(3+\beta _{2})q^{5}+(2+\cdots)q^{7}+\cdots\)
2704.2.a.z	$2704$	$2$	2704.a	1.a	$1$	$3$	$3$	$21.592$	\(\Q(\zeta_{14})^+\)	None	✓				169.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1-\beta _{1})q^{5}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
2704.2.a.ba	$2704$	$2$	2704.a	1.a	$1$	$3$	$3$	$21.592$	\(\Q(\zeta_{14})^+\)	None	✓				169.2.a.b	$1$	$1$	\(0\)	\(2\)	\(4\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{1})q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
2704.2.a.bb	$2704$	$2$	2704.a	1.a	$1$	$3$	$3$	$21.592$	\(\Q(\zeta_{14})^+\)	None	✓				1352.2.a.i	$1$	$1$	\(0\)	\(4\)	\(-4\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-1-\beta _{1})q^{5}+\beta _{1}q^{7}+\cdots\)
2704.2.a.bc	$2704$	$2$	2704.a	1.a	$1$	$3$	$3$	$21.592$	\(\Q(\zeta_{14})^+\)	None	✓				1352.2.a.i	$1$	$0$	\(0\)	\(4\)	\(4\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(1+\beta _{1})q^{5}-\beta _{1}q^{7}+\cdots\)
2704.2.a.bd	$2704$	$2$	2704.a	1.a	$1$	$4$	$4$	$21.592$	\(\Q(\sqrt{17 +8 \sqrt{3}})\)	None	✓				104.2.o.a	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
2704.2.a.be	$2704$	$2$	2704.a	1.a	$1$	$4$	$4$	$21.592$	\(\Q(\sqrt{17 +8 \sqrt{3}})\)	None	✓				104.2.o.a	$1$	$0$	\(0\)	\(2\)	\(4\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{5}+(-1+\cdots)q^{7}+\cdots\)
2704.2.a.bf	$2704$	$2$	2704.a	1.a	$1$	$6$	$6$	$21.592$	6.6.3728753.1	None	✓		✓		1352.2.a.m	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}-\beta _{2}-\beta _{5})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
2704.2.a.bg	$2704$	$2$	2704.a	1.a	$1$	$6$	$6$	$21.592$	6.6.3728753.1	None	✓		✓		1352.2.a.m	$1$	$0$	\(0\)	\(-3\)	\(2\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}-\beta _{2}-\beta _{5})q^{3}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2704))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2704)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1352))\)\(^{\oplus 2}\)