Newform orbit 189.2.u.a

• Label: 189.2.u.a
• Level: $189$
• Weight: $2$
• Character orbit: 189.u
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} - 15 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−0.467294	−	2.65016i	1.66541	+	0.475808i	−4.92558	+	1.79277i	2.75387	−	1.00233i	0.482727	−	4.63595i	−0.609446	−	2.57460i	4.36176	+	7.55480i	2.54721	+	1.58484i	−3.94319	−	6.82980i
4.2	−0.460894	−	2.61386i	−1.36647	+	1.06432i	−4.74046	+	1.72539i	−0.528985	+	0.192535i	3.41177	+	3.08122i	0.317729	+	2.62660i	4.04059	+	6.99852i	0.734460	−	2.90871i	0.747066	+	1.29396i
4.3	−0.422628	−	2.39684i	−0.118761	−	1.72797i	−3.68685	+	1.34190i	−2.38847	+	0.869333i	−4.09149	+	1.01494i	2.35689	−	1.20212i	2.34068	+	4.05418i	−2.97179	+	0.410433i	3.09309	+	5.35738i
4.4	−0.324131	−	1.83824i	0.743955	−	1.56414i	−1.39467	+	0.507617i	1.37479	−	0.500384i	−3.11640	−	0.860580i	−2.42382	+	1.06071i	−0.481419	−	0.833843i	−1.89306	−	2.32730i	−1.36544	−	2.36501i
4.5	−0.310937	−	1.76341i	0.625951	+	1.61499i	−1.13356	+	0.412582i	−0.00172074		0.000626297i	2.65326	−	1.60597i	2.59651	+	0.508047i	−0.710598	−	1.23079i	−2.21637	+	2.02181i	0.00163946	+	0.00283963i
4.6	−0.279530	−	1.58529i	−1.44281	+	0.958286i	−0.555633	+	0.202234i	0.230811	−	0.0840085i	1.92247	+	2.01940i	−1.31413	−	2.29631i	−1.13383	−	1.96386i	1.16338	−	2.76524i	−0.197697	−	0.342421i
4.7	−0.266701	−	1.51254i	−1.44858	−	0.949540i	−0.337258	+	0.122752i	3.71318	−	1.35149i	−1.04988	+	2.44427i	2.56740	+	0.639091i	−1.26026	−	2.18283i	1.19675	+	2.75096i	−3.03449	−	5.25589i
4.8	−0.203963	−	1.15673i	−1.51129	−	0.846167i	0.582964	−	0.212181i	−3.41115	+	1.24156i	−0.670539	+	1.92074i	−1.99948	+	1.73265i	−1.53891	−	2.66548i	1.56800	+	2.55761i	2.13189	+	3.69255i
4.9	−0.165988	−	0.941364i	1.70872	−	0.283332i	1.02077	−	0.371530i	−2.75749	+	1.00364i	−0.550345	−	1.56150i	0.0940141	−	2.64408i	−1.47507	−	2.55489i	2.83945	−	0.968269i	1.40250	+	2.42921i
4.10	−0.134854	−	0.764795i	1.54799	+	0.776991i	1.31266	−	0.477769i	1.10529	−	0.402293i	0.385486	−	1.28868i	−0.975813	+	2.45923i	−1.31901	−	2.28459i	1.79257	+	2.40555i	−0.456725	−	0.791071i
4.11	−0.0305699	−	0.173370i	−0.226706	+	1.71715i	1.85026	−	0.673440i	2.52551	−	0.919210i	0.304634	−	0.0131890i	−1.78991	−	1.94839i	−0.349362	−	0.605113i	−2.89721	−	0.778577i	−0.236568	−	0.409748i
4.12	0.0290601	+	0.164808i	0.733837	−	1.56891i	1.85307	−	0.674462i	−0.999212	+	0.363683i	0.279894	+	0.0753494i	1.97858	+	1.75648i	0.332357	+	0.575660i	−1.92297	−	2.30265i	−0.0889751	−	0.154109i
4.13	0.0570934	+	0.323793i	−0.691412	−	1.58806i	1.77780	−	0.647067i	1.85642	−	0.675680i	0.474729	−	0.314543i	−2.33644	−	1.24140i	0.639805	+	1.10817i	−2.04390	+	2.19601i	0.324770	+	0.562517i
4.14	0.0954712	+	0.541444i	−1.72856	−	0.109886i	1.59534	−	0.580656i	−0.969531	+	0.352880i	−0.105531	−	0.946411i	2.17686	−	1.50376i	1.01650	+	1.76063i	2.97585	+	0.379890i	−0.283627	−	0.491257i
4.15	0.101938	+	0.578121i	−0.121715	+	1.72777i	1.55555	−	0.566175i	−3.06658	+	1.11614i	−1.01127	+	0.105760i	−1.48906	+	2.18694i	1.07293	+	1.85836i	−2.97037	−	0.420592i	−0.957867	−	1.65908i
4.16	0.241593	+	1.37014i	1.22928	+	1.22020i	0.0604707	−	0.0220095i	−1.40760	+	0.512324i	−1.37486	+	1.97907i	1.17416	−	2.37094i	1.43604	+	2.48730i	0.0222396	+	2.99992i	−1.04202	−	1.80483i
4.17	0.246187	+	1.39619i	−1.04527	+	1.38109i	−0.00936568	+	0.00340883i	3.04103	−	1.10685i	−2.18560	−	1.11939i	1.78778	+	1.95034i	1.41067	+	2.44335i	−0.814825	−	2.88722i	2.29403	+	3.97338i
4.18	0.340870	+	1.93317i	−0.913186	−	1.47176i	−1.74157	+	0.633879i	1.35943	−	0.494793i	2.53389	−	2.26702i	−0.147009	+	2.64166i	0.143948	+	0.249325i	−1.33218	+	2.68799i	1.41991	+	2.45935i
4.19	0.371090	+	2.10456i	0.871315	−	1.49693i	−2.41207	+	0.877922i	2.18602	−	0.795646i	3.47372	+	1.27824i	1.27150	−	2.32019i	−0.605711	−	1.04912i	−1.48162	−	2.60860i	2.48569	+	4.30535i
4.20	0.373934	+	2.12069i	−1.62126	+	0.609530i	−2.47810	+	0.901956i	−1.18205	+	0.430229i	−1.89887	−	3.21026i	−2.62839	−	0.302569i	−0.686013	−	1.18821i	2.25695	−	1.97641i	−1.35439	−	2.34587i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
189.u	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	189.2.u.a	✓	132
3.b	odd	2	1		567.2.u.a		132
7.c	even	3	1		189.2.w.a	yes	132
21.h	odd	6	1		567.2.w.a		132
27.e	even	9	1		189.2.w.a	yes	132
27.f	odd	18	1		567.2.w.a		132
189.u	even	9	1	inner	189.2.u.a	✓	132
189.bc	odd	18	1		567.2.u.a		132

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.2.u.a	✓	132	1.a	even	1	1	trivial
189.2.u.a	✓	132	189.u	even	9	1	inner
189.2.w.a	yes	132	7.c	even	3	1
189.2.w.a	yes	132	27.e	even	9	1
567.2.u.a		132	3.b	odd	2	1
567.2.u.a		132	189.bc	odd	18	1
567.2.w.a		132	21.h	odd	6	1
567.2.w.a		132	27.f	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(189, [\chi])\).