Newform orbit 229.3.j.a

• Label: 229.3.j.a
• Level: $229$
• Weight: $3$
• Character orbit: 229.j
• Analytic conductor: $6.240$
• Analytic rank: $0$
• Dimension: $1368$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 229 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	229.j (of order \(76\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.23979805385\)
Analytic rank:	\(0\)
Dimension:	\(1368\)
Relative dimension:	\(38\) over \(\Q(\zeta_{76})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{76}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1368 q - 38 q^{2} - 34 q^{3} - 38 q^{4} - 38 q^{5} - 32 q^{6} - 20 q^{7} - 32 q^{8} - 242 q^{9}+O(q^{10}) \)

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	−3.80098	+	0.796982i	−4.17334	−	2.25850i	10.1492	−	4.45184i	8.06314	+	0.668131i	17.6628	+	5.25844i	7.10784	−	3.47481i	−22.3860	+	15.9833i	7.39342	+	11.3165i	−31.1803	+	3.88662i
2.2	−3.61504	+	0.757994i	−1.32705	−	0.718166i	8.83087	−	3.87358i	−5.61066	−	0.464912i	5.34172	+	1.59030i	0.486753	−	0.237959i	−16.9635	+	12.1117i	−3.67722	−	5.62841i	20.6351	−	2.57217i
2.3	−3.55893	+	0.746229i	4.18703	+	2.26591i	8.44604	−	3.70478i	−1.04723	−	0.0867756i	−16.5923	−	4.93973i	6.67354	−	3.26250i	−15.4566	+	11.0358i	7.47438	+	11.4404i	3.79176	−	0.472642i
2.4	−3.53314	+	0.740823i	2.71340	+	1.46842i	8.27120	−	3.62808i	7.69168	+	0.637351i	−10.6747	−	3.17798i	−11.5224	+	5.63294i	−14.7837	+	10.5553i	0.283741	+	0.434297i	−27.6480	+	3.44632i
2.5	−3.29926	+	0.691781i	0.124122	+	0.0671715i	6.74343	−	2.95794i	−0.362856	−	0.0300671i	−0.455978	−	0.135751i	−0.903136	+	0.441516i	−9.22812	+	6.58875i	−4.91164	−	7.51782i	1.21795	−	0.151818i
2.6	−2.99165	+	0.627283i	−4.16513	−	2.25406i	4.89338	−	2.14644i	−1.25098	−	0.103659i	13.8745	+	4.13063i	−9.36336	+	4.57747i	−3.34208	+	2.38620i	7.34501	+	11.2424i	3.80751	−	0.474605i
2.7	−2.55131	+	0.534954i	0.264748	+	0.143274i	2.55992	−	1.12288i	6.99679	+	0.579770i	−0.752098	−	0.223909i	7.17138	−	3.50587i	2.55568	−	1.82472i	−4.87297	−	7.45864i	−18.1611	+	2.26378i
2.8	−2.36818	+	0.496555i	2.36169	+	1.27808i	1.69862	−	0.745083i	−5.20631	−	0.431407i	−6.22755	−	1.85402i	−9.80223	+	4.79202i	4.22435	−	3.01613i	−0.978451	−	1.49763i	12.5437	−	1.56357i
2.9	−2.34414	+	0.491514i	−2.77460	−	1.50154i	1.59031	−	0.697575i	1.79372	+	0.148632i	7.24209	+	2.15606i	−2.52727	+	1.23551i	4.41200	−	3.15011i	0.521268	+	0.797860i	−4.27779	+	0.533226i
2.10	−2.33517	+	0.489633i	4.12933	+	2.23468i	1.55017	−	0.679969i	−4.82452	−	0.399771i	−10.7368	−	3.19650i	0.995453	−	0.486647i	4.48022	−	3.19882i	7.13502	+	10.9210i	11.4618	−	1.42871i
2.11	−2.16437	+	0.453820i	−0.420617	−	0.227626i	0.815431	−	0.357681i	−8.60573	−	0.713091i	1.01367	+	0.301782i	7.80875	−	3.81746i	5.59652	−	3.99584i	−4.79743	−	7.34301i	18.9496	−	2.36206i
2.12	−2.06087	+	0.432119i	−4.39642	−	2.37922i	0.397366	−	0.174301i	−1.98839	−	0.164762i	10.0886	+	3.00350i	8.91982	−	4.36063i	6.11124	−	4.36334i	8.74529	+	13.3857i	4.16900	−	0.519666i
2.13	−1.88662	+	0.395583i	4.10205	+	2.21992i	−0.260232	+	0.114148i	6.90878	+	0.572478i	−8.61719	−	2.56545i	−1.88661	+	0.922309i	6.72107	−	4.79875i	6.97625	+	10.6779i	−13.2607	+	1.65295i
2.14	−1.38122	+	0.289612i	−1.57523	−	0.852473i	−1.83919	+	0.806742i	4.85842	+	0.402580i	2.42263	+	0.721250i	−3.61313	+	1.76635i	6.90090	−	4.92715i	−3.16789	−	4.84882i	−6.82716	+	0.851004i
2.15	−0.844161	+	0.177002i	3.34040	+	1.80774i	−2.98182	+	1.30795i	1.78615	+	0.148004i	−3.13981	−	0.934762i	9.80847	−	4.79507i	5.09346	−	3.63666i	2.96785	+	4.54264i	−1.53399	+	0.191212i
2.16	−0.723194	+	0.151638i	1.12055	+	0.606409i	−3.16308	+	1.38745i	1.80041	+	0.149186i	−0.902326	−	0.268634i	−4.91157	+	2.40112i	4.48261	−	3.20052i	−4.03464	−	6.17548i	−1.32467	+	0.165120i
2.17	−0.643155	+	0.134856i	−2.93887	−	1.59044i	−3.26763	+	1.43332i	−7.64386	−	0.633388i	2.10463	+	0.626576i	−9.28248	+	4.53793i	4.04756	−	2.88990i	1.18495	+	1.81371i	5.00161	−	0.623450i
2.18	−0.219716	+	0.0460696i	1.17662	+	0.636756i	−3.61694	+	1.58654i	−4.48868	−	0.371943i	−0.287858	−	0.0856990i	5.65778	−	2.76592i	1.45243	−	1.03701i	−3.94355	−	6.03606i	1.00337	−	0.125070i
2.19	−0.181862	+	0.0381325i	−4.58731	−	2.48253i	−3.63147	+	1.59291i	9.27215	+	0.768312i	0.928924	+	0.276553i	−4.35005	+	2.12661i	1.20459	−	0.860064i	9.95793	+	15.2417i	−1.71555	+	0.213843i
2.20	0.110668	−	0.0232047i	−3.79642	−	2.05452i	−3.65138	+	1.60165i	−5.20436	−	0.431246i	−0.467817	−	0.139275i	1.12513	−	0.550045i	−0.735029	+	0.524801i	5.26921	+	8.06512i	−0.585964	+	0.0730403i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
229.j	odd	76	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	229.3.j.a	✓	1368
229.j	odd	76	1	inner	229.3.j.a	✓	1368

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
229.3.j.a	✓	1368	1.a	even	1	1	trivial
229.3.j.a	✓	1368	229.j	odd	76	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(229, [\chi])\).