Newform orbit 197.3.i.a

• Label: 197.3.i.a
• Level: $197$
• Weight: $3$
• Character orbit: 197.i
• Analytic conductor: $5.368$
• Analytic rank: $0$
• Dimension: $2688$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 197 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	197.i (of order \(196\), degree \(84\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.36786120790\)
Analytic rank:	\(0\)
Dimension:	\(2688\)
Relative dimension:	\(32\) over \(\Q(\zeta_{196})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{196}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 2688 q - 84 q^{2} - 84 q^{3} - 84 q^{4} - 84 q^{5} - 98 q^{6} - 84 q^{7} - 140 q^{8} - 84 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−3.77189	−	0.0604630i	−3.32606	+	0.815453i	10.2255	+	0.327913i	0.114711	+	0.789665i	12.5948	−	2.87469i	1.23384	−	1.27404i	−23.4777	−	1.12981i	2.41835	−	1.26165i	−0.384933	−	2.98546i
2.2	−3.55486	−	0.0569840i	1.99334	−	0.488708i	8.63581	+	0.276934i	−0.673805	−	4.63842i	−7.11389	+	1.62370i	−5.67297	+	5.85781i	−16.4785	−	0.792989i	−4.24482	+	2.21452i	2.13096	+	16.5273i
2.3	−3.34701	−	0.0536522i	5.54436	−	1.35931i	7.20163	+	0.230942i	0.0803709	+	0.553268i	−18.6299	+	4.25216i	3.57428	−	3.69074i	−10.7172	−	0.515743i	20.9128	−	10.9102i	−0.239318	−	1.85610i
2.4	−3.03519	−	0.0486538i	1.10928	−	0.271963i	5.21208	+	0.167141i	0.311397	+	2.14364i	−3.38012	+	0.771489i	6.85710	−	7.08052i	−3.68323	−	0.177247i	−6.82285	+	3.55948i	−0.840854	−	6.52149i
2.5	−2.90911	−	0.0466327i	−3.04003	+	0.745325i	4.46279	+	0.143113i	1.10267	+	7.59067i	8.87853	−	2.02647i	−1.50854	+	1.55769i	−1.35160	−	0.0650426i	0.706873	−	0.368775i	−2.85380	−	22.1335i
2.6	−2.90099	−	0.0465026i	−0.438834	+	0.107589i	4.41565	+	0.141601i	−1.33348	−	9.17958i	1.27806	−	0.291708i	6.15258	−	6.35304i	−1.21114	−	0.0582831i	−7.79839	+	4.06842i	3.44154	+	26.6919i
2.7	−2.51939	−	0.0403857i	2.86079	−	0.701380i	2.34777	+	0.0752885i	1.17768	+	8.10706i	−7.23578	+	1.65152i	−4.47328	+	4.61903i	4.15530	+	0.199964i	−0.287225	+	0.149845i	−2.63963	−	20.4724i
2.8	−2.39696	−	0.0384230i	−2.46500	+	0.604344i	1.74599	+	0.0559904i	−0.118182	−	0.813558i	5.93172	−	1.35388i	−8.04223	+	8.30426i	5.39507	+	0.259626i	−2.26841	+	1.18343i	0.252018	+	1.95460i
2.9	−2.29979	−	0.0368654i	−5.44672	+	1.33537i	1.28973	+	0.0413592i	−0.353836	−	2.43578i	12.5755	−	2.87028i	3.36359	−	3.47318i	6.22512	+	0.299570i	19.9041	−	10.3840i	0.723952	+	5.61483i
2.10	−1.56922	−	0.0251544i	3.06562	−	0.751600i	−1.53613	−	0.0492607i	−0.0487173	−	0.335366i	−4.82954	+	1.10231i	−4.37223	+	4.51469i	8.67971	+	0.417691i	0.853757	−	0.445404i	0.0680121	+	0.527489i
2.11	−1.39730	−	0.0223985i	4.32840	−	1.06119i	−2.04601	−	0.0656116i	−0.976931	−	6.72513i	−6.07182	+	1.38585i	−0.551726	+	0.569703i	8.44085	+	0.406197i	9.62948	−	5.02369i	1.21443	+	9.41887i
2.12	−1.20152	−	0.0192602i	−1.83823	+	0.450679i	−2.55467	−	0.0819232i	−1.02118	−	7.02975i	2.21735	−	0.506095i	0.840357	−	0.867738i	7.86904	+	0.378680i	−4.80343	+	2.50594i	1.09158	+	8.46605i
2.13	−1.16597	−	0.0186905i	0.0922996	−	0.0226291i	−2.63880	−	0.0846211i	0.771412	+	5.31035i	−0.108042	+	0.0246599i	5.57777	−	5.75951i	7.73430	+	0.372195i	−7.97139	+	4.15867i	−0.800194	−	6.20615i
2.14	−0.800946	−	0.0128391i	−1.49515	+	0.366567i	−3.35660	−	0.107639i	0.284989	+	1.96185i	1.20224	−	0.274404i	3.19566	−	3.29978i	5.88756	+	0.283325i	−5.87829	+	3.06670i	−0.203073	−	1.57499i
2.15	−0.215672	−	0.00345720i	4.98112	−	1.22122i	−3.95144	−	0.126715i	0.981634	+	6.75750i	−1.07851	+	0.246163i	3.12516	−	3.22698i	1.71358	+	0.0824619i	15.3408	−	8.00328i	−0.188349	−	1.46079i
2.16	−0.192762	−	0.00308995i	−4.80460	+	1.17795i	−3.96080	−	0.127015i	1.32718	+	9.13619i	0.929784	−	0.212217i	−3.47871	+	3.59205i	1.53335	+	0.0737891i	13.7173	−	7.15629i	−0.227598	−	1.76521i
2.17	0.0441796		0.000708195i	−4.54280	+	1.11376i	−3.99599	−	0.128144i	−0.843380	−	5.80577i	−0.201488	+	0.0459883i	−5.49223	+	5.67118i	−0.352988	−	0.0169867i	11.4172	−	5.95634i	−0.0331486	−	0.257094i
2.18	0.648427	+	0.0103942i	0.725334	−	0.177830i	−3.57760	−	0.114727i	−0.578637	−	3.98330i	0.472175	−	0.107771i	−3.67258	+	3.79224i	−4.90966	−	0.236266i	−7.48491	+	3.90487i	−0.333801	−	2.58889i
2.19	0.701592	+	0.0112464i	1.42484	−	0.349329i	−3.50584	−	0.112425i	0.386593	+	2.66128i	1.00358	−	0.229062i	−6.67457	+	6.89204i	−5.26189	−	0.253216i	−6.07125	+	3.16737i	0.241300	+	1.87148i
2.20	0.830612	+	0.0133146i	3.37180	−	0.826664i	−3.30821	−	0.106088i	−0.509948	−	3.51044i	2.81166	−	0.641743i	7.89308	−	8.15026i	−6.06546	−	0.291886i	2.70624	−	1.41184i	−0.376829	−	2.92261i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
197.i	odd	196	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	197.3.i.a	✓	2688
197.i	odd	196	1	inner	197.3.i.a	✓	2688

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
197.3.i.a	✓	2688	1.a	even	1	1	trivial
197.3.i.a	✓	2688	197.i	odd	196	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(197, [\chi])\).