Space of modular forms of level 1089, weight 4, and trivial character

• Label: 1089.4.a
• Level: $1089$
• Weight: $4$
• Character orbit: 1089.a
• Rep. character: $\chi_{1089}(1,\cdot)$
• Character field: $\Q$
• Dimension: $132$
• Newform subspaces: $40$
• Sturm bound: $528$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 40 \)
Sturm bound:			\(528\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1089))\).

	Total	New	Old
Modular forms	420	141	279
Cusp forms	372	132	240
Eisenstein series	48	9	39

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

\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(108\)	\(30\)	\(78\)		\(96\)	\(30\)	\(66\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(102\)	\(25\)	\(77\)		\(90\)	\(25\)	\(65\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(102\)	\(41\)	\(61\)		\(90\)	\(37\)	\(53\)		\(12\)	\(4\)	\(8\)
\(-\)	\(-\)	\(+\)		\(108\)	\(45\)	\(63\)		\(96\)	\(40\)	\(56\)		\(12\)	\(5\)	\(7\)
Plus space		\(+\)		\(216\)	\(75\)	\(141\)		\(192\)	\(70\)	\(122\)		\(24\)	\(5\)	\(19\)
Minus space		\(-\)		\(204\)	\(66\)	\(138\)		\(180\)	\(62\)	\(118\)		\(24\)	\(4\)	\(20\)

Trace form

132 * q - 2 * q^2 + 512 * q^4 - 18 * q^5 + 24 * q^7 + 24 * q^8 + 46 * q^10 - 54 * q^13 - 84 * q^14 + 1908 * q^16 + 28 * q^17 - 64 * q^19 - 552 * q^20 + 60 * q^23 + 2948 * q^25 + 540 * q^26 + 160 * q^28 + 24 * q^29 + 454 * q^31 + 260 * q^32 + 518 * q^34 + 748 * q^35 + 8 * q^37 + 78 * q^38 - 228 * q^40 - 224 * q^41 + 68 * q^43 + 2 * q^46 - 1290 * q^47 + 4970 * q^49 + 380 * q^50 - 8 * q^52 - 1074 * q^53 + 144 * q^56 + 2512 * q^58 + 144 * q^59 + 350 * q^61 + 3166 * q^62 + 7028 * q^64 - 1760 * q^65 + 1402 * q^67 + 1640 * q^68 + 1012 * q^70 + 582 * q^71 - 2370 * q^73 + 2262 * q^74 + 1208 * q^76 - 1904 * q^79 - 7296 * q^80 + 390 * q^82 + 732 * q^83 - 2180 * q^85 + 498 * q^86 - 2322 * q^89 - 528 * q^91 + 1164 * q^92 - 3064 * q^94 + 2784 * q^95 - 1838 * q^97 - 9450 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\) 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	11
1089.4.a.a	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	None	✓				33.4.a.a	$1$	$0$	\(-5\)	\(0\)	\(14\)	\(32\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}+14q^{5}+2^{5}q^{7}-45q^{8}+\cdots\)
1089.4.a.b	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	None	✓				363.4.a.a	$1$	$0$	\(-4\)	\(0\)	\(13\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+8q^{4}+13q^{5}+26q^{7}-52q^{10}+\cdots\)
1089.4.a.c	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	None	✓				363.4.a.b	$1$	$1$	\(-3\)	\(0\)	\(12\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+12q^{5}+12q^{7}+21q^{8}+\cdots\)
1089.4.a.d	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	None	✓				363.4.a.c	$1$	$0$	\(-1\)	\(0\)	\(-7\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}-7q^{5}+4q^{7}+15q^{8}+\cdots\)
1089.4.a.e	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	None	✓				33.4.a.b	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}+4q^{5}+26q^{7}+15q^{8}+\cdots\)
1089.4.a.f	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	\(\Q(\sqrt{-11}) \)	✓				121.4.a.a	$2$	$1$	\(0\)	\(0\)	\(-18\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}-18q^{5}+2^{6}q^{16}+12^{2}q^{20}+\cdots\)
1089.4.a.g	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				9.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-20\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}-20q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
1089.4.a.h	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	None	✓				363.4.a.c	$1$	$0$	\(1\)	\(0\)	\(-7\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}-7q^{5}-4q^{7}-15q^{8}+\cdots\)
1089.4.a.i	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	None	✓				363.4.a.b	$1$	$1$	\(3\)	\(0\)	\(12\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+12q^{5}-12q^{7}-21q^{8}+\cdots\)
1089.4.a.j	$1089$	$4$	1089.a	1.a	$1$	$1$	$1$	$64.253$	\(\Q\)	None	✓				363.4.a.a	$1$	$0$	\(4\)	\(0\)	\(13\)	\(-26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+8q^{4}+13q^{5}-26q^{7}+52q^{10}+\cdots\)
1089.4.a.k	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{3}) \)	None	✓				121.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(10\)	\(8\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(5-2\beta )q^{4}+(5-\beta )q^{5}+\cdots\)
1089.4.a.l	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{3}) \)	None	✓				99.4.a.d	$1$	$1$	\(-2\)	\(0\)	\(20\)	\(16\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(5-2\beta )q^{4}+(10+\beta )q^{5}+\cdots\)
1089.4.a.m	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓	✓			1089.4.a.m	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2\cdot 3^{2}$	$N(\mathrm{U}(1))$	\(q-8q^{4}-\beta q^{7}-2\beta q^{13}+2^{6}q^{16}+\cdots\)
1089.4.a.n	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{3}) \)	None	✓				363.4.a.m	$2$	$1$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-5q^{4}+3q^{5}-2\beta q^{7}-13\beta q^{8}+\cdots\)
1089.4.a.o	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{5}) \)	None	✓				363.4.a.k	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-3q^{4}+2q^{5}-10\beta q^{7}+11\beta q^{8}+\cdots\)
1089.4.a.p	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{5}) \)	None	✓				33.4.e.a	$1$	$1$	\(0\)	\(0\)	\(13\)	\(-44\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2-4\beta )q^{2}+12q^{4}+(1+11\beta )q^{5}+\cdots\)
1089.4.a.q	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{5}) \)	None	✓				33.4.e.a	$1$	$0$	\(0\)	\(0\)	\(13\)	\(44\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2-4\beta )q^{2}+12q^{4}+(12-11\beta )q^{5}+\cdots\)
1089.4.a.r	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{26}) \)	None	✓				121.4.a.d	$2$	$1$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+18q^{4}-5q^{5}-4\beta q^{7}+10\beta q^{8}+\cdots\)
1089.4.a.s	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{3}) \)	None	✓				363.4.a.l	$2$	$1$	\(0\)	\(0\)	\(-18\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3\beta q^{2}+19q^{4}-9q^{5}-14\beta q^{7}+\cdots\)
1089.4.a.t	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{33}) \)	None	✓				33.4.a.d	$1$	$0$	\(1\)	\(0\)	\(-16\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+\beta q^{4}+(-10+4\beta )q^{5}+(-2+\cdots)q^{7}+\cdots\)
1089.4.a.u	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{97}) \)	None	✓				33.4.a.c	$1$	$0$	\(1\)	\(0\)	\(14\)	\(-24\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2^{4}+\beta )q^{4}+(6+2\beta )q^{5}+(-14+\cdots)q^{7}+\cdots\)
1089.4.a.v	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{3}) \)	None	✓				11.4.a.a	$1$	$0$	\(2\)	\(0\)	\(-2\)	\(-20\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-4+2\beta )q^{4}+(-1-8\beta )q^{5}+\cdots\)
1089.4.a.w	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{3}) \)	None	✓				99.4.a.d	$1$	$1$	\(2\)	\(0\)	\(-20\)	\(16\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(5+2\beta )q^{4}+(-10+\beta )q^{5}+\cdots\)
1089.4.a.x	$1089$	$4$	1089.a	1.a	$1$	$2$	$2$	$64.253$	\(\Q(\sqrt{3}) \)	None	✓				121.4.a.b	$1$	$0$	\(2\)	\(0\)	\(10\)	\(-8\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(5+2\beta )q^{4}+(5+\beta )q^{5}+\cdots\)
1089.4.a.y	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\Q(\sqrt{5}, \sqrt{37})\)	None	✓				11.4.c.a	$1$	$1$	\(-4\)	\(0\)	\(11\)	\(25\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{2}+(3-3\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
1089.4.a.z	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\Q(\sqrt{62 +10 \sqrt{5}})\)	None	✓				33.4.e.b	$1$	$0$	\(-3\)	\(0\)	\(-12\)	\(11\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}-\beta _{2})q^{2}+(-2+3\beta _{1}+\cdots)q^{4}+\cdots\)
1089.4.a.ba	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				363.4.a.q	$1$	$0$	\(-1\)	\(0\)	\(-14\)	\(-20\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(6+\beta _{1}+\beta _{3})q^{4}+(-3+\beta _{2}+\cdots)q^{5}+\cdots\)
1089.4.a.bb	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\Q(\sqrt{3}, \sqrt{5})\)	None	✓				363.4.a.r	$2$	$1$	\(0\)	\(0\)	\(-16\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}+\beta _{3})q^{2}+2\beta _{2}q^{4}+(-4+4\beta _{2}+\cdots)q^{5}+\cdots\)
1089.4.a.bc	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\Q(\sqrt{34 +2 \sqrt{201}})\)	None	✓	✓			1089.4.a.bc	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+\beta _{2}q^{7}+\cdots\)
1089.4.a.bd	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\Q(\sqrt{34 +2 \sqrt{201}})\)	None	✓	✓			1089.4.a.bc	$2$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}-\beta _{2}q^{7}+\cdots\)
1089.4.a.be	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\Q(\sqrt{5}, \sqrt{17})\)	None	✓	✓			1089.4.a.be	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+9q^{4}-\beta _{3}q^{5}+\beta _{2}q^{7}+\beta _{1}q^{8}+\cdots\)
1089.4.a.bf	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				363.4.a.q	$1$	$0$	\(1\)	\(0\)	\(-14\)	\(20\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{1}+\beta _{3})q^{4}+(-3+\beta _{2}+\cdots)q^{5}+\cdots\)
1089.4.a.bg	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\Q(\sqrt{62 +10 \sqrt{5}})\)	None	✓				33.4.e.b	$1$	$1$	\(3\)	\(0\)	\(-12\)	\(-11\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+(-2+3\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
1089.4.a.bh	$1089$	$4$	1089.a	1.a	$1$	$4$	$4$	$64.253$	\(\Q(\sqrt{5}, \sqrt{37})\)	None	✓				11.4.c.a	$1$	$0$	\(4\)	\(0\)	\(11\)	\(-25\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+(3+\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots\)
1089.4.a.bi	$1089$	$4$	1089.a	1.a	$1$	$6$	$6$	$64.253$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				33.4.e.c	$1$	$1$	\(-5\)	\(0\)	\(-9\)	\(1\)		$-$	$+$	$-$	$11$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1089.4.a.bj	$1089$	$4$	1089.a	1.a	$1$	$6$	$6$	$64.253$	6.6.\(\cdots\).1	None	✓				121.4.a.h	$2$	$1$	\(0\)	\(0\)	\(-14\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{3}+\beta _{4})q^{4}+(-2-2\beta _{3}+\cdots)q^{5}+\cdots\)
1089.4.a.bk	$1089$	$4$	1089.a	1.a	$1$	$6$	$6$	$64.253$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		33.4.e.c	$1$	$0$	\(5\)	\(0\)	\(-9\)	\(-1\)		$-$	$-$	$+$	$11$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1089.4.a.bl	$1089$	$4$	1089.a	1.a	$1$	$12$	$12$	$64.253$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		99.4.f.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-66\)		$+$	$-$	$-$	$2^{2}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-\beta _{1}-\beta _{5}+\cdots)q^{5}+\cdots\)
1089.4.a.bm	$1089$	$4$	1089.a	1.a	$1$	$12$	$12$	$64.253$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				99.4.f.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(66\)		$+$	$+$	$+$	$2^{2}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(\beta _{1}+\beta _{5})q^{5}+\cdots\)
1089.4.a.bn	$1089$	$4$	1089.a	1.a	$1$	$12$	$12$	$64.253$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			1089.4.a.bn	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{7})q^{4}+\beta _{4}q^{5}+(-2\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1089))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1089)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 2}\)