Newform orbit 88.2.k.b

• Label: 88.2.k.b
• Level: $88$
• Weight: $2$
• Character orbit: 88.k
• Analytic conductor: $0.703$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.702683537787\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 5 q^{2} - 2 q^{3} - 5 q^{4} + 15 q^{6} - 5 q^{8} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.37526	+	0.329627i	−0.303809	−	0.935028i	1.78269	−	0.906646i	−0.398383	+	0.548327i	0.726027	+	1.18576i	1.40393	−	4.32085i	−2.15281	+	1.83450i	1.64507	−	1.19522i	0.367138	−	0.885411i
19.2	−1.32760	+	0.487314i	0.852681	+	2.62428i	1.52505	−	1.29392i	1.35292	−	1.86213i	−2.41087	−	3.06848i	−1.08979	+	3.35402i	−1.39411	+	2.46098i	−3.73274	+	2.71199i	−0.888693	+	3.13147i
19.3	−1.16173	−	0.806458i	0.385688	+	1.18702i	0.699251	+	1.87378i	−2.03454	+	2.80031i	0.509219	−	1.69005i	−0.442181	+	1.36089i	0.698780	−	2.74075i	1.16678	−	0.847714i	4.62193	−	1.61244i
19.4	−0.557019	+	1.29990i	−0.625543	−	1.92522i	−1.37946	−	1.44813i	1.75152	−	2.41076i	2.85103	+	0.259245i	−0.216882	+	0.667493i	2.65081	−	0.986518i	−0.888128	+	0.645263i	2.15811	+	3.61964i
19.5	−0.407991	−	1.35408i	0.385688	+	1.18702i	−1.66709	+	1.10491i	2.03454	−	2.80031i	1.44997	−	1.00655i	0.442181	−	1.36089i	2.17630	+	1.80658i	1.16678	−	0.847714i	−4.62193	−	1.61244i
19.6	0.738473	−	1.20609i	−0.303809	−	0.935028i	−0.909315	−	1.78133i	0.398383	−	0.548327i	−1.35208	−	0.324071i	−1.40393	+	4.32085i	−2.81996	−	0.218749i	1.64507	−	1.19522i	−0.367138	−	0.885411i
19.7	0.873715	−	1.11204i	0.852681	+	2.62428i	−0.473246	−	1.94320i	−1.35292	+	1.86213i	3.66329	+	1.34466i	1.08979	−	3.35402i	−2.57439	−	1.17154i	−3.73274	+	2.71199i	0.888693	+	3.13147i
19.8	1.40840	−	0.128066i	−0.625543	−	1.92522i	1.96720	−	0.360738i	−1.75152	+	2.41076i	−1.12757	−	2.63138i	0.216882	−	0.667493i	2.72441	−	0.759996i	−0.888128	+	0.645263i	−2.15811	+	3.61964i
35.1	−1.39431	+	0.236423i	−1.63407	+	1.18722i	1.88821	−	0.659294i	−1.62415	+	0.527718i	1.99771	−	2.04168i	−3.70164	−	2.68940i	−2.47688	+	1.36568i	0.333632	−	1.02681i	2.13981	−	1.11979i
35.2	−0.989056	−	1.01083i	−1.63407	+	1.18722i	−0.0435373	+	1.99953i	1.62415	−	0.527718i	2.81625	+	0.477530i	3.70164	+	2.68940i	2.06423	−	1.93363i	0.333632	−	1.02681i	−2.13981	−	1.11979i
35.3	−0.729904	+	1.21130i	1.70387	−	1.23794i	−0.934480	−	1.76826i	1.49538	−	0.485879i	0.255844	+	2.96747i	−1.63043	−	1.18458i	2.82397	+	0.158728i	0.443645	−	1.36540i	−0.502942	+	2.16600i
35.4	−0.253518	+	1.39130i	−0.903665	+	0.656551i	−1.87146	−	0.705442i	−3.74056	+	1.21538i	−0.684367	−	1.42372i	2.25832	+	1.64076i	1.45593	−	2.42492i	−0.541500	+	1.66657i	−0.742666	−	5.51238i
35.5	0.121478	−	1.40899i	1.70387	−	1.23794i	−1.97049	−	0.342321i	−1.49538	+	0.485879i	−1.53725	−	2.55111i	1.63043	+	1.18458i	−0.721696	+	2.73480i	0.443645	−	1.36540i	0.502942	+	2.16600i
35.6	0.612688	−	1.27460i	−0.903665	+	0.656551i	−1.24923	−	1.56187i	3.74056	−	1.21538i	0.283177	+	1.55407i	−2.25832	−	1.64076i	−2.75615	+	0.635331i	−0.541500	+	1.66657i	0.742666	−	5.51238i
35.7	0.668361	+	1.24631i	0.0248408	−	0.0180479i	−1.10659	+	1.66597i	1.78547	−	0.580134i	0.0390960	+	0.0188969i	−0.623146	−	0.452742i	−2.81592	−	0.265679i	−0.926760	+	2.85227i	1.91637	+	1.83751i
35.8	1.27328	−	0.615434i	0.0248408	−	0.0180479i	1.24248	−	1.56724i	−1.78547	+	0.580134i	0.0205220	−	0.0382680i	0.623146	+	0.452742i	0.617492	−	2.76020i	−0.926760	+	2.85227i	−1.91637	+	1.83751i
51.1	−1.37526	−	0.329627i	−0.303809	+	0.935028i	1.78269	+	0.906646i	−0.398383	−	0.548327i	0.726027	−	1.18576i	1.40393	+	4.32085i	−2.15281	−	1.83450i	1.64507	+	1.19522i	0.367138	+	0.885411i
51.2	−1.32760	−	0.487314i	0.852681	−	2.62428i	1.52505	+	1.29392i	1.35292	+	1.86213i	−2.41087	+	3.06848i	−1.08979	−	3.35402i	−1.39411	−	2.46098i	−3.73274	−	2.71199i	−0.888693	−	3.13147i
51.3	−1.16173	+	0.806458i	0.385688	−	1.18702i	0.699251	−	1.87378i	−2.03454	−	2.80031i	0.509219	+	1.69005i	−0.442181	−	1.36089i	0.698780	+	2.74075i	1.16678	+	0.847714i	4.62193	+	1.61244i
51.4	−0.557019	−	1.29990i	−0.625543	+	1.92522i	−1.37946	+	1.44813i	1.75152	+	2.41076i	2.85103	−	0.259245i	−0.216882	−	0.667493i	2.65081	+	0.986518i	−0.888128	−	0.645263i	2.15811	−	3.61964i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	88.2.k.b	✓	32
3.b	odd	2	1		792.2.bp.b		32
4.b	odd	2	1		352.2.s.b		32
8.b	even	2	1		352.2.s.b		32
8.d	odd	2	1	inner	88.2.k.b	✓	32
11.b	odd	2	1		968.2.k.h		32
11.c	even	5	1		968.2.g.e		32
11.c	even	5	1		968.2.k.e		32
11.c	even	5	1		968.2.k.h		32
11.c	even	5	1		968.2.k.i		32
11.d	odd	10	1	inner	88.2.k.b	✓	32
11.d	odd	10	1		968.2.g.e		32
11.d	odd	10	1		968.2.k.e		32
11.d	odd	10	1		968.2.k.i		32
24.f	even	2	1		792.2.bp.b		32
33.f	even	10	1		792.2.bp.b		32
44.g	even	10	1		352.2.s.b		32
44.g	even	10	1		3872.2.g.d		32
44.h	odd	10	1		3872.2.g.d		32
88.g	even	2	1		968.2.k.h		32
88.k	even	10	1	inner	88.2.k.b	✓	32
88.k	even	10	1		968.2.g.e		32
88.k	even	10	1		968.2.k.e		32
88.k	even	10	1		968.2.k.i		32
88.l	odd	10	1		968.2.g.e		32
88.l	odd	10	1		968.2.k.e		32
88.l	odd	10	1		968.2.k.h		32
88.l	odd	10	1		968.2.k.i		32
88.o	even	10	1		3872.2.g.d		32
88.p	odd	10	1		352.2.s.b		32
88.p	odd	10	1		3872.2.g.d		32
264.r	odd	10	1		792.2.bp.b		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.2.k.b	✓	32	1.a	even	1	1	trivial
88.2.k.b	✓	32	8.d	odd	2	1	inner
88.2.k.b	✓	32	11.d	odd	10	1	inner
88.2.k.b	✓	32	88.k	even	10	1	inner
352.2.s.b		32	4.b	odd	2	1
352.2.s.b		32	8.b	even	2	1
352.2.s.b		32	44.g	even	10	1
352.2.s.b		32	88.p	odd	10	1
792.2.bp.b		32	3.b	odd	2	1
792.2.bp.b		32	24.f	even	2	1
792.2.bp.b		32	33.f	even	10	1
792.2.bp.b		32	264.r	odd	10	1
968.2.g.e		32	11.c	even	5	1
968.2.g.e		32	11.d	odd	10	1
968.2.g.e		32	88.k	even	10	1
968.2.g.e		32	88.l	odd	10	1
968.2.k.e		32	11.c	even	5	1
968.2.k.e		32	11.d	odd	10	1
968.2.k.e		32	88.k	even	10	1
968.2.k.e		32	88.l	odd	10	1
968.2.k.h		32	11.b	odd	2	1
968.2.k.h		32	11.c	even	5	1
968.2.k.h		32	88.g	even	2	1
968.2.k.h		32	88.l	odd	10	1
968.2.k.i		32	11.c	even	5	1
968.2.k.i		32	11.d	odd	10	1
968.2.k.i		32	88.k	even	10	1
968.2.k.i		32	88.l	odd	10	1
3872.2.g.d		32	44.g	even	10	1
3872.2.g.d		32	44.h	odd	10	1
3872.2.g.d		32	88.o	even	10	1
3872.2.g.d		32	88.p	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^16 + T3^15 + 9*T3^14 + 16*T3^13 + 51*T3^12 + 58*T3^11 + 181*T3^10 + 288*T3^9 + 1012*T3^8 + 1432*T3^7 + 2281*T3^6 + 2517*T3^5 + 2741*T3^4 + 1659*T3^3 + 974*T3^2 - 51*T3 + 1