Space of modular forms of level 1617, weight 4, and trivial character

• Label: 1617.4.a
• Level: $1617$
• Weight: $4$
• Character orbit: 1617.a
• Rep. character: $\chi_{1617}(1,\cdot)$
• Character field: $\Q$
• Dimension: $206$
• Newform subspaces: $32$
• Sturm bound: $896$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(896\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1617))\).

	Total	New	Old
Modular forms	688	206	482
Cusp forms	656	206	450
Eisenstein series	32	0	32

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

\(3\)	\(7\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(26\)
\(+\)	\(+\)	\(-\)	\(-\)	\(24\)
\(+\)	\(-\)	\(+\)	\(-\)	\(25\)
\(+\)	\(-\)	\(-\)	\(+\)	\(28\)
\(-\)	\(+\)	\(+\)	\(-\)	\(26\)
\(-\)	\(+\)	\(-\)	\(+\)	\(28\)
\(-\)	\(-\)	\(+\)	\(+\)	\(26\)
\(-\)	\(-\)	\(-\)	\(-\)	\(23\)
Plus space			\(+\)	\(108\)
Minus space			\(-\)	\(98\)

Trace form

206 * q - 4 * q^2 + 844 * q^4 - 16 * q^5 - 12 * q^6 - 132 * q^8 + 1854 * q^9 - 112 * q^10 + 24 * q^12 - 172 * q^13 + 60 * q^15 + 3300 * q^16 + 152 * q^17 - 36 * q^18 + 120 * q^19 + 180 * q^20 + 44 * q^22 - 108 * q^23 - 324 * q^24 + 4890 * q^25 - 204 * q^26 - 632 * q^29 - 336 * q^30 - 728 * q^31 - 1260 * q^32 - 66 * q^33 - 528 * q^34 + 7596 * q^36 + 1188 * q^37 + 1032 * q^38 + 1176 * q^39 - 1632 * q^40 - 632 * q^41 + 576 * q^43 + 616 * q^44 - 144 * q^45 - 384 * q^46 - 196 * q^47 + 816 * q^48 + 2132 * q^50 - 864 * q^51 - 2984 * q^52 + 1944 * q^53 - 108 * q^54 - 440 * q^55 - 756 * q^57 + 232 * q^58 + 888 * q^59 - 372 * q^60 + 1548 * q^61 + 24 * q^62 + 15900 * q^64 + 832 * q^65 - 264 * q^66 + 1928 * q^67 + 3976 * q^68 + 516 * q^69 - 44 * q^71 - 1188 * q^72 - 52 * q^73 - 848 * q^74 + 216 * q^75 + 1456 * q^76 + 204 * q^78 - 2936 * q^79 + 8444 * q^80 + 16686 * q^81 + 5544 * q^82 + 1352 * q^83 - 5120 * q^85 + 1976 * q^86 - 504 * q^87 + 924 * q^88 - 2356 * q^89 - 1008 * q^90 - 2372 * q^92 - 1128 * q^93 - 824 * q^94 + 3528 * q^95 + 2076 * q^96 - 420 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7	11
1617.4.a.a	$1617$	$4$	1617.a	1.a	$1$	$1$	$1$	$95.406$	\(\Q\)	None	✓				33.4.a.a	$1$	$1$	\(-5\)	\(-3\)	\(14\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-3q^{3}+17q^{4}+14q^{5}+15q^{6}+\cdots\)
1617.4.a.b	$1617$	$4$	1617.a	1.a	$1$	$1$	$1$	$95.406$	\(\Q\)	None	✓				231.4.a.a	$1$	$1$	\(-3\)	\(3\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+3q^{3}+q^{4}+4q^{5}-9q^{6}+\cdots\)
1617.4.a.c	$1617$	$4$	1617.a	1.a	$1$	$1$	$1$	$95.406$	\(\Q\)	None	✓				231.4.a.b	$1$	$1$	\(-2\)	\(-3\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}-4q^{4}-q^{5}+6q^{6}+\cdots\)
1617.4.a.d	$1617$	$4$	1617.a	1.a	$1$	$1$	$1$	$95.406$	\(\Q\)	None	✓				33.4.a.b	$1$	$1$	\(-1\)	\(3\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}-7q^{4}+4q^{5}-3q^{6}+\cdots\)
1617.4.a.e	$1617$	$4$	1617.a	1.a	$1$	$1$	$1$	$95.406$	\(\Q\)	None	✓				231.4.a.c	$1$	$1$	\(2\)	\(3\)	\(-11\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}-4q^{4}-11q^{5}+6q^{6}+\cdots\)
1617.4.a.f	$1617$	$4$	1617.a	1.a	$1$	$1$	$1$	$95.406$	\(\Q\)	None	✓				231.4.a.d	$1$	$1$	\(3\)	\(-3\)	\(14\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}-3q^{3}+q^{4}+14q^{5}-9q^{6}+\cdots\)
1617.4.a.g	$1617$	$4$	1617.a	1.a	$1$	$1$	$1$	$95.406$	\(\Q\)	None	✓				231.4.a.e	$1$	$1$	\(5\)	\(-3\)	\(6\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}-3q^{3}+17q^{4}+6q^{5}-15q^{6}+\cdots\)
1617.4.a.h	$1617$	$4$	1617.a	1.a	$1$	$2$	$2$	$95.406$	\(\Q(\sqrt{17}) \)	None	✓				231.4.a.g	$1$	$1$	\(-3\)	\(-6\)	\(-25\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-3q^{3}+(-3+3\beta )q^{4}+\cdots\)
1617.4.a.i	$1617$	$4$	1617.a	1.a	$1$	$2$	$2$	$95.406$	\(\Q(\sqrt{17}) \)	None	✓				231.4.a.f	$1$	$0$	\(-3\)	\(-6\)	\(19\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-3q^{3}+(-3+3\beta )q^{4}+\cdots\)
1617.4.a.j	$1617$	$4$	1617.a	1.a	$1$	$2$	$2$	$95.406$	\(\Q(\sqrt{33}) \)	None	✓				33.4.a.d	$1$	$0$	\(1\)	\(-6\)	\(-16\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-3q^{3}+\beta q^{4}+(-10+4\beta )q^{5}+\cdots\)
1617.4.a.k	$1617$	$4$	1617.a	1.a	$1$	$2$	$2$	$95.406$	\(\Q(\sqrt{97}) \)	None	✓				33.4.a.c	$1$	$0$	\(1\)	\(6\)	\(14\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+3q^{3}+(2^{4}+\beta )q^{4}+(6+2\beta )q^{5}+\cdots\)
1617.4.a.l	$1617$	$4$	1617.a	1.a	$1$	$2$	$2$	$95.406$	\(\Q(\sqrt{37}) \)	None	✓				231.4.a.i	$1$	$1$	\(3\)	\(-6\)	\(-2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-3q^{3}+(2+3\beta )q^{4}+(1+\cdots)q^{5}+\cdots\)
1617.4.a.m	$1617$	$4$	1617.a	1.a	$1$	$2$	$2$	$95.406$	\(\Q(\sqrt{17}) \)	None	✓				231.4.a.h	$1$	$0$	\(3\)	\(6\)	\(-1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3q^{3}+(-3+3\beta )q^{4}+\cdots\)
1617.4.a.n	$1617$	$4$	1617.a	1.a	$1$	$5$	$5$	$95.406$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				231.4.a.k	$1$	$1$	\(-1\)	\(15\)	\(-21\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.o	$1617$	$4$	1617.a	1.a	$1$	$5$	$5$	$95.406$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				231.4.a.j	$1$	$0$	\(-1\)	\(15\)	\(-7\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
1617.4.a.p	$1617$	$4$	1617.a	1.a	$1$	$5$	$5$	$95.406$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				231.4.a.l	$1$	$0$	\(5\)	\(-15\)	\(-7\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1617.4.a.q	$1617$	$4$	1617.a	1.a	$1$	$7$	$7$	$95.406$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			1617.4.a.q	$1$	$0$	\(-4\)	\(-21\)	\(20\)	\(0\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1617.4.a.r	$1617$	$4$	1617.a	1.a	$1$	$7$	$7$	$95.406$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			1617.4.a.q	$1$	$1$	\(-4\)	\(21\)	\(-20\)	\(0\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1617.4.a.s	$1617$	$4$	1617.a	1.a	$1$	$7$	$7$	$95.406$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			1617.4.a.s	$1$	$1$	\(0\)	\(-21\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.4.a.t	$1617$	$4$	1617.a	1.a	$1$	$7$	$7$	$95.406$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			1617.4.a.s	$1$	$0$	\(0\)	\(21\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+(\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
1617.4.a.u	$1617$	$4$	1617.a	1.a	$1$	$8$	$8$	$95.406$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				231.4.i.a	$1$	$1$	\(-4\)	\(-24\)	\(20\)	\(0\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(3+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.v	$1617$	$4$	1617.a	1.a	$1$	$8$	$8$	$95.406$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		231.4.i.a	$1$	$1$	\(-4\)	\(24\)	\(-20\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(3+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.w	$1617$	$4$	1617.a	1.a	$1$	$10$	$10$	$95.406$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		231.4.i.c	$1$	$1$	\(-4\)	\(-30\)	\(20\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.x	$1617$	$4$	1617.a	1.a	$1$	$10$	$10$	$95.406$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				231.4.i.c	$1$	$1$	\(-4\)	\(30\)	\(-20\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.y	$1617$	$4$	1617.a	1.a	$1$	$10$	$10$	$95.406$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				231.4.i.b	$1$	$0$	\(4\)	\(-30\)	\(-20\)	\(0\)		$+$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.z	$1617$	$4$	1617.a	1.a	$1$	$10$	$10$	$95.406$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		231.4.i.b	$1$	$0$	\(4\)	\(30\)	\(20\)	\(0\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.ba	$1617$	$4$	1617.a	1.a	$1$	$12$	$12$	$95.406$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		231.4.i.d	$1$	$0$	\(4\)	\(-36\)	\(-20\)	\(0\)		$+$	$-$	$-$	$+$	$2^{3}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(7+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
1617.4.a.bb	$1617$	$4$	1617.a	1.a	$1$	$12$	$12$	$95.406$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				231.4.i.d	$1$	$0$	\(4\)	\(36\)	\(20\)	\(0\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(7+\beta _{2})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.4.a.bc	$1617$	$4$	1617.a	1.a	$1$	$16$	$16$	$95.406$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓	✓		1617.4.a.bc	$1$	$0$	\(-4\)	\(-48\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.4.a.bd	$1617$	$4$	1617.a	1.a	$1$	$16$	$16$	$95.406$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓	✓		1617.4.a.bc	$1$	$1$	\(-4\)	\(48\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(5+\beta _{2})q^{4}+(\beta _{1}-\beta _{5}+\cdots)q^{5}+\cdots\)
1617.4.a.be	$1617$	$4$	1617.a	1.a	$1$	$16$	$16$	$95.406$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓	✓		1617.4.a.be	$1$	$1$	\(4\)	\(-48\)	\(-40\)	\(0\)		$+$	$+$	$-$	$-$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
1617.4.a.bf	$1617$	$4$	1617.a	1.a	$1$	$16$	$16$	$95.406$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓	✓		1617.4.a.be	$1$	$0$	\(4\)	\(48\)	\(40\)	\(0\)		$-$	$+$	$-$	$+$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(5+\beta _{2})q^{4}+(2+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1617))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1617)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(539))\)\(^{\oplus 2}\)