Newform orbit 429.2.bq.a

• Label: 429.2.bq.a
• Level: $429$
• Weight: $2$
• Character orbit: 429.bq
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $416$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bq (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(52\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 416q - 3q^{3} + 42q^{4} - 5q^{6} - 10q^{7} + 3q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−0.276807	+	2.63364i	−1.73182	−	0.0281544i	−4.90316	−	1.04220i	−1.71868	−	2.36556i	0.553529	−	4.55321i	−0.0575585	+	0.270792i	2.46537	−	7.58762i	2.99841	+	0.0975169i	6.70579	−	3.87159i
29.2	−0.274469	+	2.61140i	1.03166	−	1.39129i	−4.78776	−	1.01767i	1.02930	+	1.41670i	3.35004	+	3.07594i	0.916640	−	4.31245i	2.34881	−	7.22888i	−0.871350	−	2.87067i	−3.98208	+	2.29906i
29.3	−0.270097	+	2.56980i	0.0599513	−	1.73101i	−4.57463	−	0.972368i	−0.821874	−	1.13121i	4.43217	+	0.621604i	−0.850348	+	4.00057i	2.13742	−	6.57829i	−2.99281	−	0.207553i	3.12898	−	1.80652i
29.4	−0.260719	+	2.48057i	1.72227	+	0.183844i	−4.12897	−	0.877640i	−2.22258	−	3.05912i	−0.905066	+	4.22428i	0.668489	−	3.14500i	1.71203	−	5.26908i	2.93240	+	0.633258i	8.16784	−	4.71570i
29.5	−0.256618	+	2.44156i	1.38266	+	1.04320i	−3.93907	−	0.837274i	1.76595	+	2.43063i	−2.90184	+	3.10814i	−0.156026	+	0.734044i	1.53781	−	4.73290i	0.823485	+	2.88477i	−6.38769	+	3.68794i
29.6	−0.244335	+	2.32469i	−1.09963	+	1.33821i	−3.38819	−	0.720181i	0.672844	+	0.926090i	−2.84225	−	2.88327i	−0.747086	+	3.51476i	1.05740	−	3.25434i	−0.581631	−	2.94308i	−2.31727	+	1.33788i
29.7	−0.237841	+	2.26290i	−0.837076	−	1.51635i	−3.10787	−	0.660597i	0.619671	+	0.852904i	3.63043	−	1.53357i	0.272540	−	1.28220i	0.827791	−	2.54768i	−1.59861	+	2.53859i	−2.07742	+	1.19940i
29.8	−0.230857	+	2.19646i	−1.65359	−	0.515394i	−2.81485	−	0.598316i	1.87210	+	2.57672i	1.51379	−	3.51307i	−0.269880	+	1.26968i	0.599043	−	1.84366i	2.46874	+	1.70450i	−6.09186	+	3.51714i
29.9	−0.219545	+	2.08883i	1.04164	+	1.38383i	−2.35871	−	0.501359i	−0.372847	−	0.513179i	−3.11927	+	1.87199i	−0.107574	+	0.506098i	0.267016	−	0.821790i	−0.829974	+	2.88291i	1.15380	−	0.666147i
29.10	−0.198720	+	1.89069i	−1.52365	+	0.823714i	−1.57894	−	0.335614i	−0.924919	−	1.27304i	−1.25461	−	3.04444i	0.668009	−	3.14273i	−0.226639	+	0.697522i	1.64299	−	2.51010i	2.59073	−	1.49576i
29.11	−0.188305	+	1.79160i	1.68236	−	0.411909i	−1.21807	−	0.258909i	−1.00123	−	1.37808i	0.421180	+	3.09168i	−0.457516	+	2.15245i	−0.420139	+	1.29305i	2.66066	−	1.38596i	2.65749	−	1.53430i
29.12	−0.179632	+	1.70909i	1.33049	−	1.10896i	−0.932422	−	0.198192i	1.96134	+	2.69955i	1.65631	+	2.47313i	−0.393158	+	1.84966i	−0.555871	+	1.71080i	0.540418	−	2.95092i	−4.96609	+	2.86717i
29.13	−0.170529	+	1.62248i	−0.280130	+	1.70925i	−0.647056	−	0.137536i	−2.20891	−	3.04030i	−2.72544	−	0.745981i	−0.757508	+	3.56379i	−0.674778	+	2.07675i	−2.84305	−	0.957623i	5.30950	−	3.06544i
29.14	−0.159737	+	1.51979i	−1.11400	−	1.32627i	−0.327964	−	0.0697108i	−0.0308880	−	0.0425136i	2.19361	−	1.48120i	0.593356	−	2.79152i	−0.786124	+	2.41944i	−0.517989	+	2.95494i	0.0695459	−	0.0401523i
29.15	−0.144098	+	1.37101i	0.143318	+	1.72611i	0.0974043	+	0.0207039i	0.756574	+	1.04133i	−2.38716	−	0.0522407i	0.619507	−	2.91455i	−0.894416	+	2.75273i	−2.95892	+	0.494764i	−1.53670	+	0.887212i
29.16	−0.127461	+	1.21271i	0.478976	−	1.66451i	0.501877	+	0.106677i	−1.76681	−	2.43180i	1.95751	+	0.793018i	0.706704	−	3.32478i	−0.946962	+	2.91445i	−2.54116	−	1.59452i	3.17427	−	1.83266i
29.17	−0.123261	+	1.17275i	−1.50937	−	0.849596i	0.596149	+	0.126715i	−2.37562	−	3.26976i	1.18241	−	1.66539i	−0.489968	+	2.30512i	−0.950878	+	2.92650i	1.55637	+	2.56470i	4.12743	−	2.38297i
29.18	−0.115554	+	1.09942i	1.72615	−	0.142897i	0.760914	+	0.161737i	1.07486	+	1.47941i	−0.0423587	+	1.91428i	0.576184	−	2.71073i	−0.948969	+	2.92063i	2.95916	−	0.493323i	−1.75071	+	1.01077i
29.19	−0.0983746	+	0.935971i	−0.0424277	+	1.73153i	1.08993	+	0.231672i	1.83190	+	2.52139i	−1.61649	−	0.210050i	−0.116038	+	0.545914i	−0.905708	+	2.78748i	−2.99640	−	0.146930i	−2.54016	+	1.46656i
29.20	−0.0790544	+	0.752152i	−1.44283	+	0.958255i	1.39681	+	0.296902i	0.221314	+	0.304612i	−0.606692	−	1.16098i	0.0924831	−	0.435099i	−0.801155	+	2.46570i	1.16349	−	2.76519i	−0.246611	+	0.142381i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.d	odd	10	1	inner
13.c	even	3	1	inner
33.f	even	10	1	inner
39.i	odd	6	1	inner
143.t	odd	30	1	inner
429.bq	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	429.2.bq.a	✓	416
3.b	odd	2	1	inner	429.2.bq.a	✓	416
11.d	odd	10	1	inner	429.2.bq.a	✓	416
13.c	even	3	1	inner	429.2.bq.a	✓	416
33.f	even	10	1	inner	429.2.bq.a	✓	416
39.i	odd	6	1	inner	429.2.bq.a	✓	416
143.t	odd	30	1	inner	429.2.bq.a	✓	416
429.bq	even	30	1	inner	429.2.bq.a	✓	416

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
429.2.bq.a	✓	416	1.a	even	1	1	trivial
429.2.bq.a	✓	416	3.b	odd	2	1	inner
429.2.bq.a	✓	416	11.d	odd	10	1	inner
429.2.bq.a	✓	416	13.c	even	3	1	inner
429.2.bq.a	✓	416	33.f	even	10	1	inner
429.2.bq.a	✓	416	39.i	odd	6	1	inner
429.2.bq.a	✓	416	143.t	odd	30	1	inner
429.2.bq.a	✓	416	429.bq	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(429, [\chi])\).