Newform orbit 429.2.bi.a

• Label: 429.2.bi.a
• Level: $429$
• Weight: $2$
• Character orbit: 429.bi
• 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.bi (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(52\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 416 q - 12 q^{3} - 14 q^{6} - 12 q^{7} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.23921	+	2.43208i	0.596820	−	1.62598i	−3.20380	−	4.40966i	−2.41132	+	1.22863i	3.21492	+	3.46643i	3.29586	+	0.522013i	9.30284	−	1.47343i	−2.28761	−	1.94083i		−	7.38704i
5.2	−1.21529	+	2.38515i	1.68212	−	0.412863i	−3.03642	−	4.17927i	2.86072	−	1.45761i	−1.05954	+	4.51387i	−3.01745	−	0.477918i	8.37041	−	1.32574i	2.65909	−	1.38897i			8.59467i
5.3	−1.17839	+	2.31273i	−1.66274	+	0.485085i	−2.78453	−	3.83258i	−1.47438	+	0.751236i	0.837488	−	4.41708i	0.301846	+	0.0478077i	7.01764	−	1.11149i	2.52938	−	1.61314i		−	4.29510i
5.4	−1.15405	+	2.26495i	−0.969560	+	1.43525i	−2.62261	−	3.60971i	2.77016	−	1.41147i	−2.13186	−	3.85237i	1.58890	+	0.251657i	6.18102	−	0.978978i	−1.11991	−	2.78313i			7.90318i
5.5	−1.12326	+	2.20452i	−0.671689	−	1.59651i	−2.42261	−	3.33444i	0.958973	−	0.488621i	4.27400	+	0.312539i	−2.80706	−	0.444595i	5.18460	−	0.821160i	−2.09767	+	2.14471i			2.66292i
5.6	−1.04506	+	2.05104i	1.56544	+	0.741218i	−1.93906	−	2.66889i	0.172910	−	0.0881019i	−3.15625	+	2.43617i	3.48662	+	0.552227i	2.95325	−	0.467749i	1.90119	+	2.32066i			0.446717i
5.7	−1.02970	+	2.02089i	0.194927	+	1.72105i	−1.84816	−	2.54377i	0.495829	−	0.252638i	−3.67876	−	1.37823i	−3.71395	−	0.588231i	2.56337	−	0.405999i	−2.92401	+	0.670957i			1.26216i
5.8	−0.942586	+	1.84993i	−1.52624	−	0.818902i	−1.35820	−	1.86940i	−2.61854	+	1.33421i	2.95352	−	2.05155i	−3.28014	−	0.519524i	0.637161	−	0.100916i	1.65880	+	2.49968i		−	6.10173i
5.9	−0.881321	+	1.72969i	0.252737	−	1.71351i	−1.03953	−	1.43079i	3.05267	−	1.55541i	2.74110	+	1.94731i	1.80249	+	0.285486i	−0.443769	+	0.0702861i	−2.87225	−	0.866135i			6.65099i
5.10	−0.838827	+	1.64629i	1.70467	+	0.306759i	−0.831074	−	1.14388i	−1.38189	+	0.704110i	−1.93494	+	2.54907i	−0.769893	−	0.121939i	−1.06958	+	0.169404i	2.81180	+	1.04585i		−	2.86563i
5.11	−0.816343	+	1.60216i	−0.787192	−	1.54283i	−0.724943	−	0.997799i	−0.438293	+	0.223321i	3.11449	−	0.00172985i	3.92191	+	0.621170i	−1.36159	+	0.215654i	−1.76066	+	2.42901i		−	0.884524i
5.12	−0.804126	+	1.57819i	1.16294	−	1.28358i	−0.668483	−	0.920087i	−3.07456	+	1.56656i	1.09058	+	2.86749i	−2.02034	−	0.319990i	−1.50925	+	0.239042i	−0.295157	−	2.98545i		−	6.11194i
5.13	−0.698951	+	1.37177i	−1.70131	+	0.324864i	−0.217646	−	0.299564i	−0.540754	+	0.275528i	0.743496	−	2.56087i	4.34543	+	0.688249i	−2.47818	+	0.392505i	2.78893	−	1.10539i		−	0.934371i
5.14	−0.683621	+	1.34168i	−1.71144	+	0.266392i	−0.157203	−	0.216371i	3.31336	−	1.68824i	0.812565	−	2.47832i	−3.03316	−	0.480405i	−2.57676	+	0.408119i	2.85807	−	0.911829i			5.59959i
5.15	−0.667273	+	1.30960i	0.852386	+	1.50779i	−0.0942200	−	0.129683i	3.33356	−	1.69853i	−2.54337	+	0.110172i	0.619167	+	0.0980665i	−2.67070	+	0.422997i	−1.54688	+	2.57044i			5.49900i
5.16	−0.655660	+	1.28681i	−0.748233	+	1.56210i	−0.0504074	−	0.0693798i	−2.04378	+	1.04136i	−1.51953	−	1.98704i	−1.36274	−	0.215838i	−2.73054	+	0.432475i	−1.88029	−	2.33763i		−	3.31273i
5.17	−0.507638	+	0.996296i	1.52061	−	0.829312i	0.440661	+	0.606518i	2.02501	−	1.03179i	0.0543225	+	1.93596i	2.69211	+	0.426388i	−3.03677	+	0.480978i	1.62448	−	2.52211i			2.54129i
5.18	−0.482212	+	0.946394i	−1.19043	+	1.25813i	0.512438	+	0.705310i	−0.107143	+	0.0545921i	−0.616644	−	1.73330i	−1.02889	−	0.162960i	−3.01277	+	0.477177i	−0.165763	−	2.99542i		−	0.127724i
5.19	−0.339713	+	0.666725i	−1.42645	−	0.982471i	0.846454	+	1.16504i	0.476840	−	0.242962i	1.13962	−	0.617289i	−1.13337	−	0.179507i	−2.54245	+	0.402685i	1.06950	+	2.80289i			0.400459i
5.20	−0.319309	+	0.626680i	1.28422	+	1.16223i	0.884801	+	1.21782i	−2.81651	+	1.43508i	−1.13841	+	0.433681i	−1.31181	−	0.207770i	−2.43507	+	0.385677i	0.298429	+	2.98512i		−	2.22328i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.c	even	5	1	inner
13.d	odd	4	1	inner
33.h	odd	10	1	inner
39.f	even	4	1	inner
143.r	odd	20	1	inner
429.bi	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	429.2.bi.a	✓	416
3.b	odd	2	1	inner	429.2.bi.a	✓	416
11.c	even	5	1	inner	429.2.bi.a	✓	416
13.d	odd	4	1	inner	429.2.bi.a	✓	416
33.h	odd	10	1	inner	429.2.bi.a	✓	416
39.f	even	4	1	inner	429.2.bi.a	✓	416
143.r	odd	20	1	inner	429.2.bi.a	✓	416
429.bi	even	20	1	inner	429.2.bi.a	✓	416

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
429.2.bi.a	✓	416	1.a	even	1	1	trivial
429.2.bi.a	✓	416	3.b	odd	2	1	inner
429.2.bi.a	✓	416	11.c	even	5	1	inner
429.2.bi.a	✓	416	13.d	odd	4	1	inner
429.2.bi.a	✓	416	33.h	odd	10	1	inner
429.2.bi.a	✓	416	39.f	even	4	1	inner
429.2.bi.a	✓	416	143.r	odd	20	1	inner
429.2.bi.a	✓	416	429.bi	even	20	1	inner

Hecke kernels

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