Newform orbit 88.2.o.a

• Label: 88.2.o.a
• Level: $88$
• Weight: $2$
• Character orbit: 88.o
• Analytic conductor: $0.703$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.702683537787\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 5 q^{2} - q^{4} - 7 q^{6} - 10 q^{7} - 5 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.41090	+	0.0967440i	1.67918	−	2.31119i	1.98128	−	0.272992i	−2.48027	−	0.805887i	−2.14556	+	3.42332i	−0.158860	+	0.115419i	−2.76898	+	0.576842i	−1.59492	−	4.90866i	3.57737	+	0.897076i
5.2	−1.21001	−	0.732035i	−0.188809	+	0.259874i	0.928249	+	1.77154i	1.22432	+	0.397805i	0.418698	−	0.176235i	2.67516	−	1.94362i	0.173638	−	2.82309i	0.895166	+	2.75504i	−1.19023	−	1.37759i
5.3	−1.20883	+	0.733982i	−0.317655	+	0.437215i	0.922541	−	1.77452i	2.75892	+	0.896426i	0.0630835	−	0.761672i	−2.10709	+	1.53089i	0.187268	+	2.82222i	0.836799	+	2.57540i	−3.99302	+	0.941367i
5.4	−0.568603	−	1.29487i	−1.50170	+	2.06692i	−1.35338	+	1.47254i	−2.56349	−	0.832928i	3.53026	+	0.769256i	−3.19487	+	2.32121i	2.67628	+	0.915165i	−1.08998	−	3.35462i	0.379074	+	3.79299i
5.5	−0.301096	−	1.38179i	1.50170	−	2.06692i	−1.81868	+	0.832103i	2.56349	+	0.832928i	−3.30820	−	1.45270i	−3.19487	+	2.32121i	1.69739	+	2.26249i	−1.08998	−	3.35462i	0.379074	−	3.79299i
5.6	0.126804	+	1.40852i	−1.10501	+	1.52092i	−1.96784	+	0.357211i	0.468972	+	0.152378i	−2.28236	−	1.36357i	−0.141392	+	0.102727i	−0.752668	−	2.72644i	−0.165095	−	0.508111i	−0.155160	+	0.679877i
5.7	0.548639	−	1.30345i	0.188809	−	0.259874i	−1.39799	−	1.43025i	−1.22432	−	0.397805i	−0.235146	−	0.388681i	2.67516	−	1.94362i	−2.63126	+	1.03752i	0.895166	+	2.75504i	−1.19023	+	1.37759i
5.8	0.725319	+	1.21405i	1.10501	−	1.52092i	−0.947824	+	1.76114i	−0.468972	−	0.152378i	2.64796	+	0.238387i	−0.141392	+	0.102727i	−2.82559	+	0.126687i	−0.165095	−	0.508111i	−0.155160	−	0.679877i
5.9	1.19831	−	0.751039i	−1.67918	+	2.31119i	0.871881	−	1.79995i	2.48027	+	0.805887i	−0.276378	+	4.03065i	−0.158860	+	0.115419i	−0.307053	−	2.81171i	−1.59492	−	4.90866i	3.57737	−	0.897076i
5.10	1.40939	−	0.116729i	0.317655	−	0.437215i	1.97275	−	0.329033i	−2.75892	−	0.896426i	0.396664	−	0.653285i	−2.10709	+	1.53089i	2.74196	−	0.694011i	0.836799	+	2.57540i	−3.99302	−	0.941367i
37.1	−1.37754	+	0.319983i	−2.62285	−	0.852217i	1.79522	−	0.881578i	1.70795	+	2.35079i	3.88578	+	0.334692i	0.730368	+	2.24784i	−2.19090	+	1.78885i	3.72604	+	2.70713i	−3.10498	−	2.69179i
37.2	−1.26500	+	0.632278i	2.32812	+	0.756451i	1.20045	−	1.59966i	0.117836	+	0.162187i	−3.42336	+	0.515106i	−0.725001	−	2.23132i	−0.507138	+	2.78259i	2.42086	+	1.75886i	−0.251609	−	0.130662i
37.3	−1.25740	−	0.647266i	−0.826127	−	0.268425i	1.16209	+	1.62774i	−2.15963	−	2.97247i	0.865027	+	0.872241i	−0.369362	−	1.13678i	−0.407629	−	2.79890i	−1.81662	−	1.31985i	0.791527	+	5.13543i
37.4	−1.00414	−	0.995839i	0.826127	+	0.268425i	0.0166095	+	1.99993i	2.15963	+	2.97247i	−0.562242	−	1.09223i	−0.369362	−	1.13678i	1.97493	−	2.02476i	−1.81662	−	1.31985i	0.791527	−	5.13543i
37.5	−0.321268	+	1.37724i	0.582243	+	0.189182i	−1.79357	−	0.884925i	0.858340	+	1.18140i	−0.447605	+	0.741110i	1.36837	+	4.21140i	1.79497	−	2.18588i	−2.12383	−	1.54306i	−1.90283	+	0.802592i
37.6	−0.121361	−	1.40900i	2.62285	+	0.852217i	−1.97054	+	0.341993i	−1.70795	−	2.35079i	0.882460	−	3.79902i	0.730368	+	2.24784i	0.721013	+	2.73498i	3.72604	+	2.70713i	−3.10498	+	2.69179i
37.7	0.210425	−	1.39847i	−2.32812	−	0.756451i	−1.91144	−	0.588548i	−0.117836	−	0.162187i	−1.54777	+	3.09663i	−0.725001	−	2.23132i	−1.22528	+	2.54925i	2.42086	+	1.75886i	−0.251609	+	0.130662i
37.8	0.742578	+	1.20357i	1.22337	+	0.397498i	−0.897156	+	1.78749i	−0.929834	−	1.27981i	0.430033	+	1.76759i	−0.577320	−	1.77681i	−2.81757	+	0.247559i	−1.08841	−	0.790780i	0.849862	−	2.06948i
37.9	1.21055	−	0.731134i	−0.582243	−	0.189182i	0.930886	−	1.77016i	−0.858340	−	1.18140i	−0.843155	+	0.196682i	1.36837	+	4.21140i	−0.167332	−	2.82347i	−2.12383	−	1.54306i	−1.90283	−	0.802592i
37.10	1.37413	+	0.334310i	−1.22337	−	0.397498i	1.77647	+	0.918772i	0.929834	+	1.27981i	−1.54819	−	0.955200i	−0.577320	−	1.77681i	2.13395	+	1.85641i	−1.08841	−	0.790780i	0.849862	+	2.06948i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
11.c	even	5	1	inner
88.o	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.o.a	✓	40
3.b	odd	2	1		792.2.br.b		40
4.b	odd	2	1		352.2.w.a		40
8.b	even	2	1	inner	88.2.o.a	✓	40
8.d	odd	2	1		352.2.w.a		40
11.b	odd	2	1		968.2.o.i		40
11.c	even	5	1	inner	88.2.o.a	✓	40
11.c	even	5	1		968.2.c.h		20
11.c	even	5	2		968.2.o.j		40
11.d	odd	10	1		968.2.c.i		20
11.d	odd	10	2		968.2.o.d		40
11.d	odd	10	1		968.2.o.i		40
24.h	odd	2	1		792.2.br.b		40
33.h	odd	10	1		792.2.br.b		40
44.g	even	10	1		3872.2.c.i		20
44.h	odd	10	1		352.2.w.a		40
44.h	odd	10	1		3872.2.c.h		20
88.b	odd	2	1		968.2.o.i		40
88.k	even	10	1		3872.2.c.i		20
88.l	odd	10	1		352.2.w.a		40
88.l	odd	10	1		3872.2.c.h		20
88.o	even	10	1	inner	88.2.o.a	✓	40
88.o	even	10	1		968.2.c.h		20
88.o	even	10	2		968.2.o.j		40
88.p	odd	10	1		968.2.c.i		20
88.p	odd	10	2		968.2.o.d		40
88.p	odd	10	1		968.2.o.i		40
264.t	odd	10	1		792.2.br.b		40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.2.o.a	✓	40	1.a	even	1	1	trivial
88.2.o.a	✓	40	8.b	even	2	1	inner
88.2.o.a	✓	40	11.c	even	5	1	inner
88.2.o.a	✓	40	88.o	even	10	1	inner
352.2.w.a		40	4.b	odd	2	1
352.2.w.a		40	8.d	odd	2	1
352.2.w.a		40	44.h	odd	10	1
352.2.w.a		40	88.l	odd	10	1
792.2.br.b		40	3.b	odd	2	1
792.2.br.b		40	24.h	odd	2	1
792.2.br.b		40	33.h	odd	10	1
792.2.br.b		40	264.t	odd	10	1
968.2.c.h		20	11.c	even	5	1
968.2.c.h		20	88.o	even	10	1
968.2.c.i		20	11.d	odd	10	1
968.2.c.i		20	88.p	odd	10	1
968.2.o.d		40	11.d	odd	10	2
968.2.o.d		40	88.p	odd	10	2
968.2.o.i		40	11.b	odd	2	1
968.2.o.i		40	11.d	odd	10	1
968.2.o.i		40	88.b	odd	2	1
968.2.o.i		40	88.p	odd	10	1
968.2.o.j		40	11.c	even	5	2
968.2.o.j		40	88.o	even	10	2
3872.2.c.h		20	44.h	odd	10	1
3872.2.c.h		20	88.l	odd	10	1
3872.2.c.i		20	44.g	even	10	1
3872.2.c.i		20	88.k	even	10	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(88, [\chi])\).