Space of modular forms of level 289, weight 4, and trivial character

• Label: 289.4.a
• Level: $289$
• Weight: $4$
• Character orbit: 289.a
• Rep. character: $\chi_{289}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $9$
• Sturm bound: $102$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	289.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(102\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(289))\).

	Total	New	Old
Modular forms	86	75	11
Cusp forms	68	60	8
Eisenstein series	18	15	3

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

\(17\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(45\)	\(39\)	\(6\)		\(36\)	\(32\)	\(4\)		\(9\)	\(7\)	\(2\)
\(-\)		\(41\)	\(36\)	\(5\)		\(32\)	\(28\)	\(4\)		\(9\)	\(8\)	\(1\)

Trace form

60 * q + 2 * q^2 + 4 * q^3 + 198 * q^4 + 2 * q^5 + 50 * q^6 + 6 * q^7 + 18 * q^8 + 336 * q^9 + 74 * q^10 + 52 * q^11 - 14 * q^12 + 4 * q^13 - 176 * q^14 - 72 * q^15 + 574 * q^16 + 142 * q^18 - 184 * q^19 + 162 * q^20 - 52 * q^21 - 358 * q^22 - 82 * q^23 + 834 * q^24 + 1160 * q^25 - 160 * q^26 + 100 * q^27 - 448 * q^28 + 426 * q^29 - 834 * q^30 - 58 * q^31 - 204 * q^32 + 360 * q^33 + 360 * q^35 - 596 * q^36 - 298 * q^37 - 592 * q^38 - 732 * q^39 + 298 * q^40 + 636 * q^41 + 2102 * q^42 - 280 * q^43 + 1146 * q^44 + 162 * q^45 + 524 * q^46 - 1020 * q^47 - 1342 * q^48 + 532 * q^49 - 448 * q^50 + 432 * q^52 - 312 * q^53 + 860 * q^54 - 16 * q^55 - 96 * q^56 + 1648 * q^57 + 1394 * q^58 - 744 * q^59 - 1294 * q^60 - 26 * q^61 - 1024 * q^62 - 86 * q^63 + 242 * q^64 - 60 * q^65 - 2056 * q^66 - 520 * q^67 - 1540 * q^69 + 756 * q^70 + 1110 * q^71 + 320 * q^72 - 1200 * q^73 - 1010 * q^74 + 836 * q^75 + 660 * q^76 - 572 * q^77 + 84 * q^78 + 1078 * q^79 + 466 * q^80 + 1244 * q^81 - 1384 * q^82 + 1796 * q^83 + 516 * q^84 + 1866 * q^86 - 980 * q^87 - 1334 * q^88 + 812 * q^89 + 2674 * q^90 - 608 * q^91 - 4836 * q^92 - 2100 * q^93 + 2972 * q^94 - 224 * q^95 - 318 * q^96 + 652 * q^97 - 1356 * q^98 + 3808 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(289))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		17
289.4.a.a	$289$	$4$	289.a	1.a	$1$	$1$	$1$	$17.052$	\(\Q\)	None	✓				17.4.a.a	$1$	$0$	\(-3\)	\(8\)	\(-6\)	\(28\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+8q^{3}+q^{4}-6q^{5}-24q^{6}+\cdots\)
289.4.a.b	$289$	$4$	289.a	1.a	$1$	$3$	$3$	$17.052$	3.3.2636.1	None	✓				17.4.a.b	$1$	$0$	\(1\)	\(-4\)	\(8\)	\(-22\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{2})q^{2}+(-2+\beta _{1}-2\beta _{2})q^{3}+\cdots\)
289.4.a.c	$289$	$4$	289.a	1.a	$1$	$4$	$4$	$17.052$	4.4.2555057.1	None	✓	✓			289.4.a.c	$1$	$0$	\(1\)	\(-2\)	\(14\)	\(36\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(3+\beta _{1}+\cdots)q^{4}+\cdots\)
289.4.a.d	$289$	$4$	289.a	1.a	$1$	$4$	$4$	$17.052$	4.4.2555057.1	None	✓	✓			289.4.a.c	$1$	$1$	\(1\)	\(2\)	\(-14\)	\(-36\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+(3+\beta _{1}+\cdots)q^{4}+\cdots\)
289.4.a.e	$289$	$4$	289.a	1.a	$1$	$4$	$4$	$17.052$	4.4.4669632.2	None	✓				17.4.b.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+(1+\beta _{2})q^{4}+\cdots\)
289.4.a.f	$289$	$4$	289.a	1.a	$1$	$8$	$8$	$17.052$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				17.4.c.a	$2$	$0$	\(8\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{3}q^{3}+(5-\beta _{1}+\beta _{6}+\cdots)q^{4}+\cdots\)
289.4.a.g	$289$	$4$	289.a	1.a	$1$	$12$	$12$	$17.052$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				17.4.d.a	$2$	$1$	\(-8\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{5})q^{2}+\beta _{7}q^{3}+(2+2\beta _{1}+\cdots)q^{4}+\cdots\)
289.4.a.h	$289$	$4$	289.a	1.a	$1$	$12$	$12$	$17.052$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			289.4.a.h	$1$	$1$	\(0\)	\(-18\)	\(-30\)	\(-24\)		$-$	$-$	$2^{3}\cdot 17^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2+\beta _{5})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
289.4.a.i	$289$	$4$	289.a	1.a	$1$	$12$	$12$	$17.052$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		289.4.a.h	$1$	$0$	\(0\)	\(18\)	\(30\)	\(24\)		$+$	$+$	$2^{3}\cdot 17^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2-\beta _{5})q^{3}+(4+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(289))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(289)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)