Space of modular forms of level 616, weight 2, and trivial character

• Label: 616.2.a
• Level: $616$
• Weight: $2$
• Character orbit: 616.a
• Rep. character: $\chi_{616}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $8$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	616.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(616))\).

	Total	New	Old
Modular forms	104	14	90
Cusp forms	89	14	75
Eisenstein series	15	0	15

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

\(2\)	\(7\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(11\)	\(1\)	\(10\)		\(10\)	\(1\)	\(9\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(13\)	\(4\)	\(9\)		\(11\)	\(4\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(15\)	\(2\)	\(13\)		\(13\)	\(2\)	\(11\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(13\)	\(1\)	\(12\)		\(11\)	\(1\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(13\)	\(0\)	\(13\)		\(11\)	\(0\)	\(11\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(11\)	\(2\)	\(9\)		\(9\)	\(2\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(13\)	\(1\)	\(12\)		\(11\)	\(1\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(15\)	\(3\)	\(12\)		\(13\)	\(3\)	\(10\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(48\)	\(5\)	\(43\)		\(41\)	\(5\)	\(36\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(56\)	\(9\)	\(47\)		\(48\)	\(9\)	\(39\)		\(8\)	\(0\)	\(8\)

Trace form

14 * q + 4 * q^5 + 10 * q^9 + 6 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^17 - 12 * q^23 + 6 * q^25 + 12 * q^27 - 4 * q^29 - 4 * q^31 + 24 * q^37 - 8 * q^39 - 12 * q^41 - 8 * q^43 + 32 * q^45 + 8 * q^47 + 14 * q^49 - 8 * q^51 - 4 * q^53 - 8 * q^59 + 20 * q^61 - 8 * q^63 + 32 * q^65 - 4 * q^67 - 12 * q^69 + 12 * q^71 - 12 * q^73 + 12 * q^75 - 4 * q^77 - 24 * q^79 + 14 * q^81 + 16 * q^83 - 56 * q^87 - 16 * q^89 + 4 * q^91 - 4 * q^93 - 32 * q^95 - 32 * q^97 + 18 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(616))\) 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	7	11
616.2.a.a	$616$	$2$	616.a	1.a	$1$	$1$	$1$	$4.919$	\(\Q\)	None	✓	✓			616.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{5}+q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
616.2.a.b	$616$	$2$	616.a	1.a	$1$	$1$	$1$	$4.919$	\(\Q\)	None	✓	✓	✓	✓	616.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}-2q^{9}+q^{11}+q^{15}+\cdots\)
616.2.a.c	$616$	$2$	616.a	1.a	$1$	$1$	$1$	$4.919$	\(\Q\)	None	✓	✓	✓	✓	616.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}-3q^{9}-q^{11}+2q^{13}+\cdots\)
616.2.a.d	$616$	$2$	616.a	1.a	$1$	$1$	$1$	$4.919$	\(\Q\)	None	✓	✓	✓	✓	616.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}-q^{11}-6q^{13}-2q^{19}+\cdots\)
616.2.a.e	$616$	$2$	616.a	1.a	$1$	$1$	$1$	$4.919$	\(\Q\)	None	✓	✓			616.2.a.e	$1$	$0$	\(0\)	\(2\)	\(2\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}+q^{7}+q^{9}-q^{11}+4q^{15}+\cdots\)
616.2.a.f	$616$	$2$	616.a	1.a	$1$	$2$	$2$	$4.919$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	616.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2+\beta )q^{5}-q^{7}+(1+\beta )q^{9}+\cdots\)
616.2.a.g	$616$	$2$	616.a	1.a	$1$	$3$	$3$	$4.919$	3.3.229.1	None	✓	✓	✓	✓	616.2.a.g	$1$	$0$	\(0\)	\(1\)	\(1\)	\(3\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-\beta _{2}q^{5}+q^{7}+(1-\beta _{1})q^{9}+\cdots\)
616.2.a.h	$616$	$2$	616.a	1.a	$1$	$4$	$4$	$4.919$	4.4.11348.1	None	✓	✓	✓	✓	616.2.a.h	$1$	$0$	\(0\)	\(1\)	\(5\)	\(-4\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(1-\beta _{1})q^{5}-q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(616))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(616)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 2}\)