Newform orbit 968.2.o.i

• Label: 968.2.o.i
• Level: $968$
• Weight: $2$
• Character orbit: 968.o
• Analytic conductor: $7.730$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.o (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.72951891566\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{10})\)
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
245.1	−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
245.2	−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
245.3	−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
245.4	−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
245.5	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
245.6	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
245.7	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
245.8	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
245.9	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
245.10	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
269.1	−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
269.2	−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
269.3	−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
269.4	−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
269.5	−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
269.6	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
269.7	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
269.8	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
269.9	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
269.10	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

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	968.2.o.i		40
8.b	even	2	1	inner	968.2.o.i		40
11.b	odd	2	1		88.2.o.a	✓	40
11.c	even	5	1		968.2.c.i		20
11.c	even	5	2		968.2.o.d		40
11.c	even	5	1	inner	968.2.o.i		40
11.d	odd	10	1		88.2.o.a	✓	40
11.d	odd	10	1		968.2.c.h		20
11.d	odd	10	2		968.2.o.j		40
33.d	even	2	1		792.2.br.b		40
33.f	even	10	1		792.2.br.b		40
44.c	even	2	1		352.2.w.a		40
44.g	even	10	1		352.2.w.a		40
44.g	even	10	1		3872.2.c.h		20
44.h	odd	10	1		3872.2.c.i		20
88.b	odd	2	1		88.2.o.a	✓	40
88.g	even	2	1		352.2.w.a		40
88.k	even	10	1		352.2.w.a		40
88.k	even	10	1		3872.2.c.h		20
88.l	odd	10	1		3872.2.c.i		20
88.o	even	10	1		968.2.c.i		20
88.o	even	10	2		968.2.o.d		40
88.o	even	10	1	inner	968.2.o.i		40
88.p	odd	10	1		88.2.o.a	✓	40
88.p	odd	10	1		968.2.c.h		20
88.p	odd	10	2		968.2.o.j		40
264.m	even	2	1		792.2.br.b		40
264.u	even	10	1		792.2.br.b		40

    

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

T3^40 - 15*T3^38 + 181*T3^36 - 1921*T3^34 + 18015*T3^32 - 120568*T3^30 + 754980*T3^28 - 4134743*T3^26 + 18406849*T3^24 - 57215545*T3^22 + 187206324*T3^20 - 455243831*T3^18 + 702932013*T3^16 - 614433944*T3^14 + 315579280*T3^12 - 96090729*T3^10 + 26350107*T3^8 - 6799423*T3^6 + 1212825*T3^4 + 7744*T3^2 + 14641

T5^40 - 23*T5^38 + 425*T5^36 - 7117*T5^34 + 97237*T5^32 - 998258*T5^30 + 8607287*T5^28 - 64894840*T5^26 + 403742319*T5^24 - 1827891519*T5^22 + 6010541286*T5^20 - 16312036420*T5^18 + 46558002353*T5^16 - 83735970028*T5^14 + 144334767520*T5^12 - 217660353792*T5^10 + 244640825600*T5^8 - 73379909632*T5^6 + 8822587392*T5^4 + 130023424*T5^2 + 16777216

T7^20 - 5*T7^19 + 31*T7^18 - 147*T7^17 + 691*T7^16 - 1836*T7^15 + 6915*T7^14 - 21030*T7^13 + 82651*T7^12 - 246377*T7^11 + 748072*T7^10 - 1717404*T7^9 + 3898041*T7^8 - 6281620*T7^7 + 9393848*T7^6 - 8896824*T7^5 + 7190256*T7^4 - 2916032*T7^3 + 649984*T7^2 - 76800*T7 + 4096