Newform orbit 304.2.v.a

• Label: 304.2.v.a
• Level: $304$
• Weight: $2$
• Character orbit: 304.v
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $152$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(152\)
Relative dimension:	\(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 152 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 20 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
45.1	−1.41027	−	0.105599i	0.189454	−	0.707053i	1.97770	+	0.297844i	2.25138	+	0.603257i	−0.341845	+	0.977126i			3.60246i	−2.75763	−	0.628882i	2.13405	+	1.23209i	−3.11135	−	1.08850i
45.2	−1.40583	+	0.153775i	−0.748343	+	2.79285i	1.95271	−	0.432363i	−0.781039	−	0.209279i	0.622570	−	4.04135i		−	3.41596i	−2.67868	+	0.908105i	−4.64194	−	2.68002i	1.13019	+	0.174106i
45.3	−1.39918	−	0.205658i	0.641478	−	2.39403i	1.91541	+	0.575504i	0.952796	+	0.255301i	−1.38989	+	3.21775i		−	3.66664i	−2.56165	−	1.19915i	−2.72180	−	1.57143i	−1.28063	−	0.553161i
45.4	−1.37219	−	0.342188i	−0.221425	+	0.826371i	1.76582	+	0.939094i	−3.30093	−	0.884483i	0.586612	−	1.05817i		−	0.780889i	−2.10169	−	1.89286i	1.96422	+	1.13404i	4.22685	+	2.34322i
45.5	−1.31504	+	0.520268i	0.290706	−	1.08493i	1.45864	−	1.36834i	−1.30410	−	0.349433i	0.182165	+	1.57797i			0.561402i	−1.20626	+	2.55831i	1.50552	+	0.869210i	1.89674	−	0.218965i
45.6	−1.25568	+	0.650588i	−0.493160	+	1.84050i	1.15347	−	1.63386i	2.82071	+	0.755807i	−0.578154	−	2.63192i			0.864823i	−0.385418	+	2.80204i	−0.546145	−	0.315317i	−4.03363	+	0.886069i
45.7	−1.20113	−	0.746517i	−0.441592	+	1.64804i	0.885425	+	1.79333i	3.49316	+	0.935990i	1.76070	−	1.64986i		−	2.31632i	0.275239	−	2.81500i	0.0770322	+	0.0444745i	−3.49701	−	3.73195i
45.8	−1.15573	−	0.815040i	0.692850	−	2.58575i	0.671421	+	1.88393i	−3.58957	−	0.961823i	−2.90824	+	2.42373i			4.00771i	0.759498	−	2.72455i	−3.60800	−	2.08308i	3.36465	+	4.03725i
45.9	−1.09092	+	0.899945i	−0.137950	+	0.514836i	0.380199	−	1.96353i	−1.58882	−	0.425723i	−0.312832	−	0.685790i		−	0.191074i	1.35230	+	2.48421i	2.35205	+	1.35796i	2.11640	−	0.965421i
45.10	−0.950827	−	1.04687i	−0.709349	+	2.64733i	−0.191855	+	1.99078i	−1.07967	−	0.289296i	3.44587	−	1.77456i			3.11727i	2.26650	−	1.69204i	−3.90709	−	2.25576i	0.723722	+	1.40534i
45.11	−0.939521	+	1.05702i	0.545442	−	2.03562i	−0.234600	−	1.98619i	4.05417	+	1.08631i	1.63924	+	2.48905i		−	1.60823i	2.31987	+	1.61809i	−1.24815	−	0.720620i	−4.95724	+	3.26474i
45.12	−0.840675	+	1.13722i	0.755564	−	2.81980i	−0.586532	−	1.91206i	−2.08871	−	0.559668i	2.57155	+	3.22978i		−	1.48841i	2.66751	+	0.940408i	−4.78233	−	2.76108i	2.39239	−	1.90482i
45.13	−0.786643	−	1.17524i	0.237599	−	0.886733i	−0.762384	+	1.84899i	−1.46925	−	0.393685i	−1.22903	+	0.418306i		−	4.81431i	2.77274	−	0.558512i	1.86823	+	1.07863i	0.693104	+	2.03642i
45.14	−0.716301	−	1.21939i	0.0733751	−	0.273840i	−0.973825	+	1.74690i	0.515567	+	0.138146i	−0.386476	+	0.106679i			0.825212i	2.82771	−	0.0638340i	2.52847	+	1.45981i	−0.200848	−	0.727631i
45.15	−0.433173	+	1.34624i	0.363878	−	1.35801i	−1.62472	−	1.16631i	−1.31521	−	0.352409i	1.67059	+	1.07812i			5.06974i	2.27392	−	1.68205i	0.886290	+	0.511700i	1.04414	−	1.61793i
45.16	−0.404912	+	1.35501i	−0.425492	+	1.58796i	−1.67209	−	1.09732i	2.31475	+	0.620236i	−1.97941	−	1.21953i			2.08781i	2.16393	−	1.82138i	0.257509	+	0.148673i	−1.77769	+	2.88536i
45.17	−0.0870758	−	1.41153i	0.0982442	−	0.366652i	−1.98484	+	0.245820i	2.63873	+	0.707046i	−0.526096	−	0.106748i			3.93198i	0.519814	+	2.78025i	2.47329	+	1.42796i	0.768247	−	3.78621i
45.18	−0.0760279	−	1.41217i	0.836115	−	3.12042i	−1.98844	+	0.214728i	0.873351	+	0.234014i	−4.47013	−	0.943496i		−	0.731308i	0.454410	+	2.79169i	−6.43988	−	3.71807i	0.264068	−	1.25111i
45.19	−0.0246786	−	1.41400i	−0.652360	+	2.43464i	−1.99878	+	0.0697910i	−1.43027	−	0.383241i	3.45868	+	0.862353i		−	4.08887i	0.148012	+	2.82455i	−2.90383	−	1.67653i	−0.506604	+	2.03186i
45.20	0.0220522	+	1.41404i	−0.291788	+	1.08897i	−1.99903	+	0.0623654i	−3.94741	−	1.05771i	−1.54628	−	0.388587i		−	1.27067i	−0.132270	−	2.82533i	1.49736	+	0.864503i	1.40859	−	5.60513i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner
19.c	even	3	1	inner
304.v	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	304.2.v.a	✓	152
16.e	even	4	1	inner	304.2.v.a	✓	152
19.c	even	3	1	inner	304.2.v.a	✓	152
304.v	even	12	1	inner	304.2.v.a	✓	152

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
304.2.v.a	✓	152	1.a	even	1	1	trivial
304.2.v.a	✓	152	16.e	even	4	1	inner
304.2.v.a	✓	152	19.c	even	3	1	inner
304.2.v.a	✓	152	304.v	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(304, [\chi])\).