Newform orbit 88.3.p.a

• Label: 88.3.p.a
• Level: $88$
• Weight: $3$
• Character orbit: 88.p
• Analytic conductor: $2.398$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	88.p (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.39782632637\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 5 q^{2} - 9 q^{4} - 5 q^{6} - 10 q^{7} - 5 q^{8} + 48 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−1.98759	+	0.222477i	3.22728	+	4.44197i	3.90101	−	0.884384i	4.95028	−	1.60844i	−7.40274	−	8.11081i	3.83911	−	5.28408i	−7.55684	+	2.62568i	−6.53461	+	20.1115i	−9.48128	+	4.29825i
13.2	−1.93575	−	0.502853i	0.938146	+	1.29125i	3.49428	+	1.94680i	−5.62313	+	1.82707i	−1.16671	−	2.97128i	−3.68080	+	5.06619i	−5.78511	−	5.52562i	1.99395	−	6.13675i	11.8037	−	0.709142i
13.3	−1.88083	+	0.680054i	−0.213163	−	0.293394i	3.07505	−	2.55813i	−2.92700	+	0.951041i	0.600447	+	0.406862i	4.88796	−	6.72770i	−4.04399	+	6.90262i	2.74051	−	8.43443i	4.85844	−	3.77927i
13.4	−1.86163	−	0.730991i	−0.938146	−	1.29125i	2.93131	+	2.72166i	5.62313	−	1.82707i	0.802587	+	3.08959i	−3.68080	+	5.06619i	−3.46748	−	7.20948i	1.99395	−	6.13675i	−11.8037	−	0.709142i
13.5	−1.75361	+	0.961697i	−2.88037	−	3.96449i	2.15028	−	3.37288i	−0.108335	+	0.0352003i	8.86368	+	4.18211i	−5.01905	+	6.90814i	−0.527058	+	7.98262i	−4.63949	+	14.2789i	0.156126	−	0.165913i
13.6	−1.47722	−	1.34826i	−3.22728	−	4.44197i	0.364378	+	3.98337i	−4.95028	+	1.60844i	−1.22153	+	10.9130i	3.83911	−	5.28408i	4.83236	−	6.37560i	−6.53461	+	20.1115i	9.48128	+	4.29825i
13.7	−1.12980	+	1.65032i	0.382792	+	0.526868i	−1.44712	−	3.72905i	8.93287	−	2.90247i	−1.30198	+	0.0364769i	−2.38295	+	3.27985i	7.78909	+	1.82485i	2.65009	−	8.15615i	−5.30232	+	18.0213i
13.8	−1.12190	−	1.65570i	0.213163	+	0.293394i	−1.48269	+	3.71506i	2.92700	−	0.951041i	0.246624	−	0.682092i	4.88796	−	6.72770i	7.81444	−	1.71304i	2.74051	−	8.43443i	−4.85844	−	3.77927i
13.9	−0.853427	−	1.80877i	2.88037	+	3.96449i	−2.54332	+	3.08731i	0.108335	−	0.0352003i	4.71268	−	8.59334i	−5.01905	+	6.90814i	7.75479	+	1.96550i	−4.63949	+	14.2789i	−0.156126	−	0.165913i
13.10	−0.741063	+	1.85764i	2.56174	+	3.52593i	−2.90165	−	2.75326i	−6.04544	+	1.96428i	−8.44833	+	2.14585i	−1.57329	+	2.16545i	7.26486	−	3.34989i	−3.08854	+	9.50556i	0.831122	−	12.6859i
13.11	−0.593419	+	1.90994i	−1.35631	−	1.86679i	−3.29571	−	2.26678i	−2.41906	+	0.786001i	4.37031	−	1.48266i	5.38917	−	7.41755i	6.28515	−	4.94943i	1.13580	−	3.49562i	−0.0656931	−	5.08668i
13.12	0.0560111	−	1.99922i	−0.382792	−	0.526868i	−3.99373	−	0.223956i	−8.93287	+	2.90247i	−1.07476	+	0.735773i	−2.38295	+	3.27985i	−0.671430	+	7.97177i	2.65009	−	8.15615i	5.30232	+	18.0213i
13.13	0.492361	−	1.93845i	−2.56174	−	3.52593i	−3.51516	−	1.90883i	6.04544	−	1.96428i	−8.09614	+	3.22977i	−1.57329	+	2.16545i	−5.43090	+	5.87412i	−3.08854	+	9.50556i	−0.831122	−	12.6859i
13.14	0.534057	+	1.92738i	−1.67055	−	2.29932i	−3.42957	+	2.05866i	−4.39401	+	1.42770i	3.53949	−	4.44775i	−7.65935	+	10.5422i	−5.79939	−	5.51063i	0.285032	−	0.877238i	−5.09837	−	7.70645i
13.15	0.642546	−	1.89397i	1.35631	+	1.86679i	−3.17427	−	2.43393i	2.41906	−	0.786001i	4.40715	−	1.36931i	5.38917	−	7.41755i	−6.64941	+	4.44807i	1.13580	−	3.49562i	0.0656931	−	5.08668i
13.16	0.695031	+	1.87535i	2.05231	+	2.82477i	−3.03386	+	2.60685i	3.89171	−	1.26449i	−3.87100	+	5.81211i	1.36274	−	1.87565i	−6.99738	−	3.87771i	−0.986170	+	3.03512i	5.07622	+	6.41945i
13.17	1.17658	+	1.61730i	−2.74432	−	3.77723i	−1.23130	+	3.80577i	7.17582	−	2.33157i	2.87998	−	8.88260i	4.79703	−	6.60254i	−7.60379	+	2.48643i	−3.95502	+	12.1723i	12.2138	+	8.86215i
13.18	1.56494	−	1.24537i	1.67055	+	2.29932i	0.898105	−	3.89787i	4.39401	−	1.42770i	5.47783	+	1.51785i	−7.65935	+	10.5422i	−3.44881	−	7.21843i	0.285032	−	0.877238i	5.09837	−	7.70645i
13.19	1.66459	−	1.10866i	−2.05231	−	2.82477i	1.54175	−	3.69094i	−3.89171	+	1.26449i	−6.54798	−	2.42678i	1.36274	−	1.87565i	−1.52561	−	7.85319i	−0.986170	+	3.03512i	−5.07622	+	6.41945i
13.20	1.68397	+	1.07900i	1.01262	+	1.39375i	1.67152	+	3.63401i	−2.46770	+	0.801803i	0.201366	+	3.43965i	−0.651539	+	0.896767i	−1.10629	+	7.92314i	1.86401	−	5.73684i	−5.02067	−	1.31243i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
11.d	odd	10	1	inner
88.p	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	88.3.p.a	✓	88
4.b	odd	2	1		352.3.x.a		88
8.b	even	2	1	inner	88.3.p.a	✓	88
8.d	odd	2	1		352.3.x.a		88
11.d	odd	10	1	inner	88.3.p.a	✓	88
44.g	even	10	1		352.3.x.a		88
88.k	even	10	1		352.3.x.a		88
88.p	odd	10	1	inner	88.3.p.a	✓	88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.3.p.a	✓	88	1.a	even	1	1	trivial
88.3.p.a	✓	88	8.b	even	2	1	inner
88.3.p.a	✓	88	11.d	odd	10	1	inner
88.3.p.a	✓	88	88.p	odd	10	1	inner
352.3.x.a		88	4.b	odd	2	1
352.3.x.a		88	8.d	odd	2	1
352.3.x.a		88	44.g	even	10	1
352.3.x.a		88	88.k	even	10	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(88, [\chi])\).