Newform orbit 308.3.o.a

• Label: 308.3.o.a
• Level: $308$
• Weight: $3$
• Character orbit: 308.o
• Analytic conductor: $8.392$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	308.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.39239214230\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 12 q^{8} + 240 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−1.99972	−	0.0332033i	−2.23781	+	1.29200i	3.99780	+	0.132795i	−4.36162	+	7.55454i	4.51789	−	2.50934i	−6.43401	+	2.75744i	−7.99008	−	0.398293i	−1.16148	+	2.01175i	8.97287	−	14.9622i
23.2	−1.99911	+	0.0595885i	2.47614	−	1.42960i	3.99290	−	0.238248i	−2.93379	+	5.08147i	−4.86488	+	3.00547i	−3.59358	−	6.00718i	−7.96805	+	0.714216i	−0.412503	+	0.714476i	5.56217	−	10.3332i
23.3	−1.99799	+	0.0896554i	−3.89515	+	2.24887i	3.98392	−	0.358261i	1.29592	−	2.24460i	7.58085	−	4.84243i	−2.35317	+	6.59262i	−7.92772	+	1.07298i	5.61480	−	9.72512i	−2.38800	+	4.60087i
23.4	−1.98642	+	0.232712i	−2.25207	+	1.30024i	3.89169	−	0.924525i	−1.03060	+	1.78505i	4.17097	−	3.10689i	4.12660	−	5.65431i	−7.51537	+	2.74213i	−1.11878	+	1.93778i	1.63180	−	3.78569i
23.5	−1.98215	−	0.266585i	3.93824	−	2.27374i	3.85786	+	1.05682i	−2.83512	+	4.91057i	−8.41234	+	3.45703i	5.95269	+	3.68313i	−7.36515	−	3.12324i	5.83983	−	10.1149i	6.92873	−	8.97771i
23.6	−1.93491	+	0.506069i	−0.213669	+	0.123362i	3.48779	−	1.95840i	0.772593	−	1.33817i	0.351002	−	0.346826i	6.55123	+	2.46604i	−5.75749	+	5.55440i	−4.46956	+	7.74151i	−0.817695	+	2.98023i
23.7	−1.92942	−	0.526616i	−0.593204	+	0.342487i	3.44535	+	2.03213i	1.63928	−	2.83932i	1.32490	−	0.348411i	−3.61614	−	5.99363i	−5.57739	−	5.73522i	−4.26541	+	7.38790i	−4.65810	+	4.61498i
23.8	−1.88458	+	0.669600i	4.68405	−	2.70434i	3.10327	−	2.52383i	2.72844	−	4.72579i	−7.01664	+	8.23298i	−6.76751	−	1.78909i	−4.15841	+	6.83430i	10.1269	−	17.5403i	−1.97756	+	10.7331i
23.9	−1.85602	+	0.745113i	2.70091	−	1.55937i	2.88961	−	2.76589i	−2.45279	+	4.24835i	−3.85104	+	4.90671i	−3.10970	+	6.27134i	−3.30228	+	7.28663i	0.363280	−	0.629220i	1.38692	−	9.71263i
23.10	−1.84810	−	0.764539i	3.60280	−	2.08008i	2.83096	+	2.82589i	2.21215	−	3.83155i	−8.24864	+	1.08972i	4.38538	−	5.45605i	−3.07140	−	7.38691i	4.15345	−	7.19398i	−7.01765	+	5.38983i
23.11	−1.83336	+	0.799243i	−4.63007	+	2.67317i	2.72242	−	2.93060i	4.28212	−	7.41686i	6.35207	−	8.60144i	−0.417310	−	6.98755i	−2.64891	+	7.54873i	9.79169	−	16.9597i	−1.92280	+	17.0202i
23.12	−1.79261	−	0.886871i	−4.85645	+	2.80387i	2.42692	+	3.17963i	−3.20442	+	5.55022i	11.1924	−	0.719215i	4.02748	−	5.72533i	−1.53061	−	7.85221i	11.2234	−	19.4396i	10.6666	−	7.10748i
23.13	−1.74575	−	0.975894i	−0.562763	+	0.324911i	2.09526	+	3.40733i	−2.78677	+	4.82683i	1.29952	−	0.0180157i	5.68205	+	4.08831i	−0.332602	−	7.99308i	−4.28887	+	7.42853i	9.57547	−	5.70682i
23.14	−1.72108	+	1.01876i	1.48202	−	0.855645i	1.92425	−	3.50674i	4.30074	−	7.44910i	−1.67898	+	2.98246i	6.79118	−	1.69701i	0.260731	+	7.99575i	−3.03574	+	5.25806i	0.186919	+	17.2019i
23.15	−1.66011	−	1.11536i	−2.83119	+	1.63459i	1.51196	+	3.70324i	4.02586	−	6.97299i	6.52325	+	0.444182i	5.94725	+	3.69191i	1.62040	−	7.83417i	0.843768	−	1.46145i	−14.4608	+	7.08570i
23.16	−1.65343	+	1.12524i	−1.82724	+	1.05496i	1.46769	−	3.72101i	−0.157290	+	0.272434i	1.83414	−	3.80037i	−6.96745	+	0.674311i	1.76028	+	7.80394i	−2.27414	+	3.93892i	−0.0464839	−	0.627440i
23.17	−1.65052	−	1.12951i	−3.74516	+	2.16227i	1.44840	+	3.72856i	0.922288	−	1.59745i	8.62375	+	0.661347i	−6.96343	−	0.714619i	1.82084	−	7.79003i	4.85081	−	8.40186i	−3.32659	+	1.59488i
23.18	−1.57413	+	1.23374i	0.645781	−	0.372842i	0.955776	−	3.88413i	0.748699	−	1.29679i	−0.556554	+	1.38363i	−6.14136	−	3.35911i	3.28749	+	7.29331i	−4.22198	+	7.31268i	0.421344	+	2.96501i
23.19	−1.55017	−	1.26372i	3.62472	−	2.09273i	0.806033	+	3.91795i	−3.08113	+	5.33668i	−8.26354	−	1.33654i	−6.54030	−	2.49490i	3.70170	−	7.09207i	4.25906	−	7.37691i	11.5203	−	4.37906i
23.20	−1.45541	−	1.37178i	1.50570	−	0.869314i	0.236435	+	3.99301i	0.463768	−	0.803271i	−3.38391	−	0.800277i	−6.06428	+	3.49635i	5.13342	−	6.13580i	−2.98859	+	5.17638i	−1.77688	+	0.532899i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.c	even	3	1	inner
28.g	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	308.3.o.a	✓	160
4.b	odd	2	1	inner	308.3.o.a	✓	160
7.c	even	3	1	inner	308.3.o.a	✓	160
28.g	odd	6	1	inner	308.3.o.a	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
308.3.o.a	✓	160	1.a	even	1	1	trivial
308.3.o.a	✓	160	4.b	odd	2	1	inner
308.3.o.a	✓	160	7.c	even	3	1	inner
308.3.o.a	✓	160	28.g	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(308, [\chi])\).