Newform orbit 819.2.ft.a

• Label: 819.2.ft.a
• Level: $819$
• Weight: $2$
• Character orbit: 819.ft
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.ft (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(108\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 432 q + 2 q^{2} - 6 q^{3} + 6 q^{4} - 20 q^{8} + 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
409.1	−2.68758	−	0.720134i	0.377990	+	1.69030i	4.97242	+	2.87083i	−1.10221	−	1.10221i	0.201366	−	4.81502i	−2.64200	+	0.140928i	−7.36150	−	7.36150i	−2.71425	+	1.27784i	2.16854	+	3.75602i
409.2	−2.65200	−	0.710602i	−1.38791	−	1.03619i	4.79611	+	2.76903i	−0.669009	−	0.669009i	2.94443	+	3.73424i	2.55561	−	0.684747i	−6.86880	−	6.86880i	0.852604	+	2.87629i	1.29881	+	2.24961i
409.3	−2.65126	−	0.710402i	1.70950	+	0.278615i	4.79245	+	2.76692i	−0.766298	−	0.766298i	−4.33438	−	1.95311i	1.64019	+	2.07600i	−6.85867	−	6.85867i	2.84475	+	0.952581i	1.48727	+	2.57603i
409.4	−2.53732	−	0.679873i	−1.13348	−	1.30966i	4.24372	+	2.45011i	2.27276	+	2.27276i	1.98560	+	4.09366i	−2.57859	+	0.592327i	−5.38702	−	5.38702i	−0.430436	+	2.96896i	−4.22154	−	7.31192i
409.5	−2.53368	−	0.678897i	0.623805	−	1.61582i	4.22658	+	2.44021i	−1.79609	−	1.79609i	−2.67750	+	3.67046i	−1.73501	−	1.99743i	−5.34257	−	5.34257i	−2.22173	−	2.01591i	3.33136	+	5.77009i
409.6	−2.47854	−	0.664122i	−1.10862	+	1.33078i	3.97005	+	2.29211i	2.12715	+	2.12715i	3.63155	−	2.56212i	1.17764	+	2.36921i	−4.68884	−	4.68884i	−0.541934	−	2.95065i	−3.85953	−	6.68490i
409.7	−2.47098	−	0.662098i	0.936207	+	1.45723i	3.93533	+	2.27207i	2.93021	+	2.93021i	−1.34852	−	4.22065i	−0.368421	−	2.61997i	−4.60204	−	4.60204i	−1.24703	+	2.72854i	−5.30042	−	9.18060i
409.8	−2.40108	−	0.643367i	1.66065	−	0.492188i	3.61920	+	2.08955i	0.882920	+	0.882920i	−4.30400	+	0.113376i	1.82986	−	1.91092i	−3.83021	−	3.83021i	2.51550	−	1.63470i	−1.55192	−	2.68800i
409.9	−2.34933	−	0.629501i	0.0798414	−	1.73021i	3.39103	+	1.95781i	−1.78884	−	1.78884i	−1.27674	+	4.01457i	0.676928	+	2.55769i	−3.29454	−	3.29454i	−2.98725	−	0.276285i	3.07650	+	5.32866i
409.10	−2.30642	−	0.618002i	−1.02485	+	1.39631i	3.20558	+	1.85074i	−3.10454	−	3.10454i	3.22665	−	2.58711i	2.49441	−	0.881985i	−2.87280	−	2.87280i	−0.899368	−	2.86202i	5.24175	+	9.07898i
409.11	−2.28735	−	0.612892i	−1.69707	+	0.346346i	3.12426	+	1.80379i	−2.11536	−	2.11536i	4.09406	+	0.247908i	−0.987900	+	2.45439i	−2.69183	−	2.69183i	2.76009	−	1.17555i	3.54207	+	6.13504i
409.12	−2.28634	−	0.612624i	−1.72350	+	0.171897i	3.12001	+	1.80134i	0.501081	+	0.501081i	4.04582	+	0.662843i	−2.51346	−	0.826157i	−2.68244	−	2.68244i	2.94090	−	0.592527i	−0.838669	−	1.45262i
409.13	−2.24825	−	0.602416i	1.57233	−	0.726491i	2.95966	+	1.70876i	1.01775	+	1.01775i	−3.97263	+	0.686138i	−2.43960	−	1.02389i	−2.33301	−	2.33301i	1.94442	−	2.28456i	−1.67505	−	2.90127i
409.14	−2.17610	−	0.583084i	0.263826	+	1.71184i	2.66337	+	1.53769i	−0.149378	−	0.149378i	0.424034	−	3.87896i	2.62472	−	0.332924i	−1.71311	−	1.71311i	−2.86079	+	0.903257i	0.237961	+	0.412160i
409.15	−2.13943	−	0.573259i	−0.437657	−	1.67585i	2.51649	+	1.45290i	2.14173	+	2.14173i	−0.0243570	+	3.83625i	2.55096	−	0.701842i	−1.41862	−	1.41862i	−2.61691	+	1.46689i	−3.35432	−	5.80986i
409.16	−2.09970	−	0.562612i	1.34541	+	1.09081i	2.36015	+	1.36263i	−1.98908	−	1.98908i	−2.21126	−	3.04731i	0.382611	−	2.61794i	−1.11479	−	1.11479i	0.620277	+	2.93518i	3.05738	+	5.29555i
409.17	−2.06444	−	0.553166i	0.818898	−	1.52624i	2.22389	+	1.28396i	1.14350	+	1.14350i	−2.53483	+	2.69785i	2.16144	+	1.52584i	−0.858290	−	0.858290i	−1.65881	−	2.49967i	−1.72815	−	2.99325i
409.18	−1.98969	−	0.533136i	−0.410988	+	1.68258i	1.94258	+	1.12155i	0.903047	+	0.903047i	1.71478	−	3.12871i	−0.495152	+	2.59900i	−0.354087	−	0.354087i	−2.66218	−	1.38304i	−1.31534	−	2.27823i
409.19	−1.97207	−	0.528414i	1.69562	−	0.353394i	1.87778	+	1.08414i	−2.48637	−	2.48637i	−3.53061	−	0.199070i	−2.14333	+	1.55118i	−0.242926	−	0.242926i	2.75023	−	1.19844i	3.58945	+	6.21712i
409.20	−1.94984	−	0.522457i	−1.54793	−	0.777125i	1.79685	+	1.03741i	−1.60152	−	1.60152i	2.61219	+	2.32399i	−0.285696	−	2.63028i	−0.106795	−	0.106795i	1.79215	+	2.40586i	2.28598	+	3.95944i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
819.ft	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	819.2.ft.a	yes	432
7.d	odd	6	1		819.2.gi.a	yes	432
9.c	even	3	1		819.2.fs.a	yes	432
13.f	odd	12	1		819.2.fa.a	✓	432
63.k	odd	6	1		819.2.fa.a	✓	432
91.ba	even	12	1		819.2.fs.a	yes	432
117.w	odd	12	1		819.2.gi.a	yes	432
819.ft	even	12	1	inner	819.2.ft.a	yes	432

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
819.2.fa.a	✓	432	13.f	odd	12	1
819.2.fa.a	✓	432	63.k	odd	6	1
819.2.fs.a	yes	432	9.c	even	3	1
819.2.fs.a	yes	432	91.ba	even	12	1
819.2.ft.a	yes	432	1.a	even	1	1	trivial
819.2.ft.a	yes	432	819.ft	even	12	1	inner
819.2.gi.a	yes	432	7.d	odd	6	1
819.2.gi.a	yes	432	117.w	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).