Newform orbit 348.3.v.a

• Label: 348.3.v.a
• Level: $348$
• Weight: $3$
• Character orbit: 348.v
• Analytic conductor: $9.482$
• Analytic rank: $0$
• Dimension: $1392$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	348.v (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.48231319974\)
Analytic rank:	\(0\)
Dimension:	\(1392\)
Relative dimension:	\(116\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1392 q - 28 q^{4} - 14 q^{6} - 28 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.99998	−	0.00816944i	−1.36972	+	2.66906i	3.99987	+	0.0326775i	−1.58516	−	0.763371i	2.76122	−	5.32688i	6.46637	+	5.15676i	−7.99940	−	0.0980311i	−5.24775	−	7.31171i	3.16405	+	1.53968i
11.2	−1.99996	−	0.0119881i	1.19688	−	2.75090i	3.99971	+	0.0479516i	0.596982	+	0.287491i	−2.42670	+	5.48736i	3.41280	+	2.72161i	−7.99871	−	0.143851i	−6.13494	−	6.58502i	−1.19050	−	0.582129i
11.3	−1.99993	−	0.0162047i	2.79341	+	1.09402i	3.99947	+	0.0648168i	5.41792	+	2.60913i	−5.56890	−	2.23323i	−5.69744	−	4.54356i	−7.99764	−	0.194440i	6.60625	+	6.11208i	−10.7932	−	5.30589i
11.4	−1.99177	−	0.181301i	2.75467	−	1.18820i	3.93426	+	0.722217i	−1.03892	−	0.500318i	−5.70207	+	1.86718i	−1.00735	−	0.803338i	−7.70519	−	2.15177i	6.17639	−	6.54616i	1.97858	+	1.18487i
11.5	−1.99160	+	0.183101i	−2.97363	−	0.396887i	3.93295	−	0.729327i	7.22883	+	3.48122i	5.99496	+	0.245967i	7.47601	+	5.96192i	−7.69932	+	2.17265i	8.68496	+	2.36039i	−15.0344	−	5.60960i
11.6	−1.96981	+	0.346217i	−2.38019	−	1.82612i	3.76027	−	1.36396i	0.175987	+	0.0847507i	5.32074	+	2.77304i	−0.380227	−	0.303221i	−6.93477	+	3.98860i	2.33057	+	8.69301i	−0.376002	−	0.106013i
11.7	−1.95045	+	0.442425i	−2.70372	+	1.29995i	3.60852	−	1.72586i	−2.57114	−	1.23820i	4.69835	−	3.73169i	−9.99302	−	7.96916i	−6.27468	+	4.96269i	5.62024	−	7.02943i	5.56270	+	1.27751i
11.8	−1.94931	+	0.447423i	1.05396	+	2.80877i	3.59962	−	1.74433i	−5.66454	−	2.72790i	−3.31120	−	5.00359i	−1.27592	−	1.01751i	−6.23633	+	5.01081i	−6.77834	+	5.92065i	12.2625	+	2.78307i
11.9	−1.92888	+	0.528587i	0.0596975	−	2.99941i	3.44119	−	2.03917i	7.68944	+	3.70304i	1.47030	+	5.81706i	−7.69057	−	6.13302i	−5.55978	+	5.75229i	−8.99287	−	0.358114i	−16.7894	−	3.07819i
11.10	−1.92447	−	0.544429i	−2.98211	−	0.327168i	3.40719	+	2.09548i	−5.81585	−	2.80077i	5.56087	+	2.25317i	3.11357	+	2.48299i	−5.41622	−	5.88766i	8.78592	+	1.95130i	9.66763	+	8.55632i
11.11	−1.92037	−	0.558718i	0.209671	−	2.99266i	3.37567	+	2.14590i	−7.42336	−	3.57490i	−2.07470	+	5.62989i	−9.85035	−	7.85539i	−5.28359	−	6.00697i	−8.91208	−	1.25495i	12.2583	+	11.0127i
11.12	−1.91821	−	0.566110i	2.49988	+	1.65850i	3.35904	+	2.17183i	−4.98910	−	2.40262i	−3.85639	−	4.59655i	0.973874	+	0.776638i	−5.21384	−	6.06761i	3.49877	+	8.29208i	8.20997	+	7.43310i
11.13	−1.89796	−	0.630686i	−2.51067	+	1.64211i	3.20447	+	2.39403i	6.25873	+	3.01405i	5.80080	−	1.53321i	−2.93986	−	2.34446i	−4.57206	−	6.56477i	3.60694	−	8.24560i	−9.97788	−	9.66782i
11.14	−1.86539	−	0.721345i	−0.384883	+	2.97521i	2.95932	+	2.69117i	2.46531	+	1.18723i	2.86411	−	5.27228i	−5.14870	−	4.10595i	−3.57902	−	7.15476i	−8.70373	−	2.29021i	−3.74236	−	3.99299i
11.15	−1.83763	+	0.789374i	−0.725631	−	2.91092i	2.75378	−	2.90116i	−7.68944	−	3.70304i	3.63125	+	4.77641i	7.69057	+	6.13302i	−2.77034	+	7.50501i	−7.94692	+	4.22451i	17.0534	+	0.734981i
11.16	−1.80300	+	0.865566i	−0.402524	+	2.97287i	2.50159	−	3.12122i	5.66454	+	2.72790i	−1.84747	−	5.70849i	1.27592	+	1.01751i	−1.80874	+	7.79285i	−8.67595	−	2.39331i	−12.5743	−	0.0153599i
11.17	−1.80077	+	0.870185i	2.92520	+	0.665726i	2.48556	−	3.13401i	2.57114	+	1.23820i	−5.84693	+	1.34665i	9.99302	+	7.96916i	−1.74875	+	7.80653i	8.11362	+	3.89477i	−5.70750	+	0.00766004i
11.18	−1.75592	+	0.957470i	1.91416	−	2.30998i	2.16650	−	3.36248i	−0.175987	−	0.0847507i	−1.14938	+	5.88888i	0.380227	+	0.303221i	−0.584724	+	7.97860i	−1.67198	−	8.84333i	0.390164	−	0.0196867i
11.19	−1.75527	−	0.958668i	−0.599245	−	2.93954i	2.16191	+	3.36543i	0.300429	+	0.144679i	−1.76621	+	5.73415i	4.89702	+	3.90524i	−0.568392	−	7.97978i	−8.28181	+	3.52301i	−0.388633	−	0.541962i
11.20	−1.74887	−	0.970288i	2.46087	−	1.71584i	2.11708	+	3.39381i	8.15481	+	3.92715i	−5.96860	+	0.613030i	6.08043	+	4.84898i	−0.409528	−	7.98951i	3.11177	−	8.44493i	−10.4512	−	14.7806i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
29.f	odd	28	1	inner
87.k	even	28	1	inner
116.l	even	28	1	inner
348.v	odd	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	348.3.v.a	✓	1392
3.b	odd	2	1	inner	348.3.v.a	✓	1392
4.b	odd	2	1	inner	348.3.v.a	✓	1392
12.b	even	2	1	inner	348.3.v.a	✓	1392
29.f	odd	28	1	inner	348.3.v.a	✓	1392
87.k	even	28	1	inner	348.3.v.a	✓	1392
116.l	even	28	1	inner	348.3.v.a	✓	1392
348.v	odd	28	1	inner	348.3.v.a	✓	1392

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
348.3.v.a	✓	1392	1.a	even	1	1	trivial
348.3.v.a	✓	1392	3.b	odd	2	1	inner
348.3.v.a	✓	1392	4.b	odd	2	1	inner
348.3.v.a	✓	1392	12.b	even	2	1	inner
348.3.v.a	✓	1392	29.f	odd	28	1	inner
348.3.v.a	✓	1392	87.k	even	28	1	inner
348.3.v.a	✓	1392	116.l	even	28	1	inner
348.3.v.a	✓	1392	348.v	odd	28	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(348, [\chi])\).