Newform orbit 356.3.j.a

• Label: 356.3.j.a
• Level: $356$
• Weight: $3$
• Character orbit: 356.j
• Analytic conductor: $9.700$
• Analytic rank: $0$
• Dimension: $880$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 356 = 2^{2} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	356.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.70029741123\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(88\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 880 q - 9 q^{2} - 9 q^{4} - 22 q^{5} - 11 q^{6} - 9 q^{8} - 262 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.99998	−	0.00947902i	−0.0295839	−	0.205760i	3.99982	+	0.0379157i	−5.99811	+	3.85475i	0.0572166	+	0.411796i	3.05344	−	1.96232i	−7.99919	−	0.113745i	8.59397	−	2.52342i	12.0326	−	7.65256i
11.2	−1.99672	+	0.114514i	−0.518994	−	3.60968i	3.97377	−	0.457305i	6.53760	−	4.20146i	1.44964	+	7.14808i	−6.70539	+	4.30930i	−7.88214	+	1.36816i	−4.12500	+	1.21121i	−12.5726	+	9.13779i
11.3	−1.98630	+	0.233683i	0.603978	+	4.20076i	3.89078	−	0.928331i	−4.90413	+	3.15169i	−2.18133	−	8.20283i	−7.11018	+	4.56943i	−7.51133	+	2.75316i	−8.64616	+	2.53874i	9.00459	−	7.40623i
11.4	−1.98442	+	0.249176i	0.0111439	+	0.0775072i	3.87582	−	0.988940i	2.63766	−	1.69512i	−0.0414270	−	0.151030i	9.83619	−	6.32133i	−7.44483	+	2.92823i	8.62955	−	2.53387i	−4.81183	+	4.02107i
11.5	−1.98101	−	0.274982i	−0.601487	−	4.18344i	3.84877	+	1.08948i	3.88722	−	2.49817i	0.0411805	+	8.45281i	4.10548	−	2.63843i	−7.32485	−	3.21661i	−8.50391	+	2.49697i	−8.38756	+	3.87996i
11.6	−1.95913	−	0.402253i	0.174367	+	1.21275i	3.67638	+	1.57613i	2.73089	−	1.75503i	0.146224	−	2.44607i	−7.01877	+	4.51069i	−6.56851	−	4.56669i	7.19509	−	2.11267i	−6.05613	+	2.33983i
11.7	−1.95757	−	0.409777i	0.699124	+	4.86252i	3.66417	+	1.60434i	−0.741814	+	0.476735i	0.623963	−	9.80521i	4.64062	−	2.98234i	−6.51544	−	4.64209i	−14.5199	+	4.26342i	1.64751	−	0.629263i
11.8	−1.93555	+	0.503616i	0.320897	+	2.23189i	3.49274	−	1.94955i	4.02050	−	2.58382i	−1.74513	−	4.15833i	0.387548	−	0.249062i	−5.77857	+	5.53247i	3.75710	−	1.10318i	−6.48064	+	7.02591i
11.9	−1.93138	−	0.519412i	−0.836330	−	5.81681i	3.46042	+	2.00636i	−7.34803	+	4.72229i	−1.40605	+	11.6688i	−6.44973	+	4.14499i	−5.64125	−	5.67242i	−24.5003	+	7.19395i	16.6446	−	5.30387i
11.10	−1.89078	+	0.651878i	0.781672	+	5.43665i	3.15011	−	2.46512i	8.12377	−	5.22083i	−5.02200	−	9.76996i	1.34076	−	0.861656i	−4.34921	+	6.71449i	−20.3107	+	5.96376i	−11.9569	+	15.1672i
11.11	−1.84498	+	0.772040i	−0.454453	−	3.16079i	2.80791	−	2.84880i	−0.327283	+	0.210332i	3.27871	+	5.48074i	−4.77497	+	3.06869i	−2.98115	+	7.42380i	−1.14863	+	0.337267i	0.441447	−	0.640735i
11.12	−1.82512	+	0.817884i	−0.757454	−	5.26821i	2.66213	−	2.98547i	−1.59211	+	1.02318i	5.69123	+	8.99560i	6.14615	−	3.94989i	−2.41694	+	7.62617i	−18.5448	+	5.44525i	2.06894	−	3.16959i
11.13	−1.81631	+	0.837264i	−0.335743	−	2.33514i	2.59798	−	3.04147i	−4.67876	+	3.00685i	2.56495	+	3.96025i	−1.10951	+	0.713039i	−2.17223	+	7.69944i	3.29526	−	0.967575i	5.98055	−	9.37874i
11.14	−1.81437	−	0.841467i	−0.269316	−	1.87314i	2.58387	+	3.05346i	−2.86791	+	1.84309i	−1.08754	+	3.62518i	2.19736	−	1.41216i	−2.11870	−	7.71434i	5.19933	−	1.52666i	6.75434	−	0.930800i
11.15	−1.65560	−	1.12204i	0.489963	+	3.40777i	1.48205	+	3.71531i	−3.22923	+	2.07530i	3.01247	−	6.19168i	6.05144	−	3.88902i	1.71505	−	7.81400i	−2.73740	+	0.803773i	7.67489	+	0.187454i
11.16	−1.64593	−	1.13619i	−0.184538	−	1.28349i	1.41814	+	3.74017i	−1.39534	+	0.896731i	−1.15455	+	2.32220i	−5.14543	+	3.30676i	1.91540	−	7.76732i	7.02215	−	2.06189i	3.31548	+	0.109422i
11.17	−1.63680	+	1.14929i	0.302253	+	2.10221i	1.35826	−	3.76233i	1.33903	−	0.860540i	−2.91079	−	3.09354i	−7.40484	+	4.75880i	2.10081	+	7.71924i	4.30749	−	1.26479i	−1.20271	+	2.94747i
11.18	−1.62718	−	1.16287i	0.130765	+	0.909488i	1.29546	+	3.78441i	6.78179	−	4.35839i	0.844840	−	1.63197i	7.60040	−	4.88448i	2.29285	−	7.66439i	7.82537	−	2.29774i	−16.1035	−	0.794451i
11.19	−1.61325	−	1.18213i	−0.603183	−	4.19523i	1.20512	+	3.81414i	2.21346	−	1.42250i	−3.98624	+	7.48098i	−1.22890	+	0.789764i	2.56466	−	7.57777i	−8.60069	+	2.52539i	−5.25244	−	0.321753i
11.20	−1.60992	+	1.18666i	0.444297	+	3.09016i	1.18367	−	3.82085i	−4.36561	+	2.80561i	−4.38225	−	4.44767i	7.89058	−	5.07097i	2.62845	+	7.55588i	−0.716226	+	0.210303i	3.69897	−	9.69729i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
89.f	even	22	1	inner
356.j	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	356.3.j.a	✓	880
4.b	odd	2	1	inner	356.3.j.a	✓	880
89.f	even	22	1	inner	356.3.j.a	✓	880
356.j	odd	22	1	inner	356.3.j.a	✓	880

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
356.3.j.a	✓	880	1.a	even	1	1	trivial
356.3.j.a	✓	880	4.b	odd	2	1	inner
356.3.j.a	✓	880	89.f	even	22	1	inner
356.3.j.a	✓	880	356.j	odd	22	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(356, [\chi])\).