Newform orbit 308.2.bc.a

• Label: 308.2.bc.a
• Level: $308$
• Weight: $2$
• Character orbit: 308.bc
• Analytic conductor: $2.459$
• Analytic rank: $0$
• Dimension: $352$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	308.bc (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.45939238226\)
Analytic rank:	\(0\)
Dimension:	\(352\)
Relative dimension:	\(44\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 352 q - 5 q^{2} - 5 q^{4} - 6 q^{5} - 20 q^{6} - 20 q^{8} - 42 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
39.1	−1.40477	−	0.163122i	−0.826119	−	1.85549i	1.94678	+	0.458300i	−0.981453	+	0.208614i	0.857839	+	2.74131i	2.45498	−	0.986456i	−2.66003	−	0.961372i	−0.752994	+	0.836284i	1.41275	−	0.132959i
39.2	−1.39645	−	0.223444i	1.08188	+	2.42995i	1.90015	+	0.624057i	−2.65930	+	0.565252i	−0.967839	−	3.63505i	−1.50072	−	2.17896i	−2.51402	−	1.29604i	−2.72680	+	3.02842i	3.83988	−	0.195141i
39.3	−1.37665	+	0.323767i	−0.522302	−	1.17311i	1.79035	−	0.891430i	−0.370031	+	0.0786525i	1.09884	+	1.44586i	−2.62931	−	0.294453i	−2.17608	+	1.80685i	0.904004	−	1.00400i	0.483939	−	0.228081i
39.4	−1.37165	−	0.344345i	0.130027	+	0.292047i	1.76285	+	0.944642i	2.36334	−	0.502342i	−0.0777876	−	0.445360i	2.58418	+	0.567443i	−2.09274	−	1.90275i	1.93901	−	2.15349i	−3.41465	−	0.124764i
39.5	−1.36103	+	0.384185i	0.601631	+	1.35128i	1.70480	−	1.04577i	2.08253	−	0.442656i	−1.33798	−	1.60800i	−1.20777	+	2.35400i	−1.91852	+	2.07829i	0.543382	−	0.603487i	−2.66433	+	1.40255i
39.6	−1.28345	−	0.593927i	−0.282944	−	0.635502i	1.29450	+	1.52455i	−3.26728	+	0.694481i	−0.0142971	+	0.983685i	−1.84726	+	1.89411i	−0.755956	−	2.72553i	1.68359	−	1.86981i	4.60587	+	1.04919i
39.7	−1.25464	−	0.652601i	0.640942	+	1.43958i	1.14822	+	1.63755i	2.74625	−	0.583734i	0.135322	−	2.22443i	−1.29774	−	2.30562i	−0.371933	−	2.80387i	0.345808	−	0.384058i	−3.82649	−	1.05983i
39.8	−1.17621	+	0.785187i	−0.979911	−	2.20092i	0.766963	−	1.84710i	4.22972	−	0.899054i	2.88072	+	1.81934i	0.401053	−	2.61518i	0.548204	+	2.77479i	−1.87641	+	2.08397i	−4.26913	+	4.37860i
39.9	−1.12932	+	0.851250i	−0.134139	−	0.301281i	0.550746	−	1.92267i	−3.48636	+	0.741049i	0.407952	+	0.226058i	2.58951	+	0.542616i	1.01471	+	2.64015i	1.93461	−	2.14861i	3.30641	−	3.80465i
39.10	−1.11362	−	0.871693i	−1.23249	−	2.76823i	0.480301	+	1.94147i	0.0277534	−	0.00589917i	−1.04051	+	4.15711i	−1.81394	−	1.92604i	1.15749	−	2.58074i	−4.13664	+	4.59421i	−0.0360490	−	0.0176230i
39.11	−1.10877	+	0.877855i	1.30069	+	2.92140i	0.458741	−	1.94668i	1.41295	−	0.300331i	−4.00673	−	2.09734i	2.47917	−	0.923964i	1.20026	+	2.56113i	−4.83539	+	5.37025i	−1.30299	+	1.57336i
39.12	−0.983323	−	1.01640i	1.23249	+	2.76823i	−0.0661511	+	1.99891i	0.0277534	−	0.00589917i	1.60169	−	3.97477i	1.81394	+	1.92604i	2.09674	−	1.89833i	−4.13664	+	4.59421i	−0.0332865	−	0.0224079i
39.13	−0.943212	+	1.05373i	−0.697435	−	1.56646i	−0.220702	−	1.98779i	0.755878	−	0.160667i	2.30846	+	0.742598i	0.409649	+	2.61385i	2.30276	+	1.64234i	0.0399981	−	0.0444223i	−0.543653	+	0.948035i
39.14	−0.780171	−	1.17955i	−0.640942	−	1.43958i	−0.782665	+	1.84050i	2.74625	−	0.583734i	−1.19801	+	1.87914i	1.29774	+	2.30562i	2.78157	−	0.512713i	0.345808	−	0.384058i	−2.83109	−	2.78392i
39.15	−0.724831	−	1.21434i	0.282944	+	0.635502i	−0.949240	+	1.76038i	−3.26728	+	0.694481i	0.566629	−	0.804221i	1.84726	−	1.89411i	2.82574	−	0.123279i	1.68359	−	1.86981i	3.21156	+	3.46420i
39.16	−0.721241	+	1.21647i	0.993877	+	2.23229i	−0.959623	−	1.75474i	−2.87637	+	0.611391i	−3.43234	−	0.400989i	−2.00256	+	1.72909i	2.82672	+	0.0982363i	−1.98791	+	2.20780i	1.33081	−	3.93999i
39.17	−0.607133	+	1.27726i	0.00672284	+	0.0150997i	−1.26278	−	1.55093i	−1.02532	+	0.217938i	−0.0233679		0.000580741i	−1.66603	−	2.05532i	2.74762	−	0.671276i	2.00721	−	2.22923i	0.344141	−	1.44191i
39.18	−0.485835	−	1.32814i	−0.130027	−	0.292047i	−1.52793	+	1.29052i	2.36334	−	0.502342i	−0.324708	+	0.314582i	−2.58418	−	0.567443i	2.45631	+	1.40233i	1.93901	−	2.15349i	−1.81537	−	2.89479i
39.19	−0.368189	−	1.36544i	−1.08188	−	2.42995i	−1.72887	+	1.00548i	−2.65930	+	0.565252i	−2.91963	+	2.37193i	1.50072	+	2.17896i	2.00948	+	1.99047i	−2.72680	+	3.02842i	1.75094	+	3.42301i
39.20	−0.341056	+	1.37247i	0.287646	+	0.646064i	−1.76736	−	0.936181i	2.22123	−	0.472137i	−0.984808	+	0.174442i	2.53597	−	0.754235i	1.88765	−	2.10636i	1.67273	−	1.85776i	−0.109569	+	3.20960i

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
11.d	odd	10	1	inner
28.g	odd	6	1	inner
44.g	even	10	1	inner
77.o	odd	30	1	inner
308.bc	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	308.2.bc.a	✓	352
4.b	odd	2	1	inner	308.2.bc.a	✓	352
7.c	even	3	1	inner	308.2.bc.a	✓	352
11.d	odd	10	1	inner	308.2.bc.a	✓	352
28.g	odd	6	1	inner	308.2.bc.a	✓	352
44.g	even	10	1	inner	308.2.bc.a	✓	352
77.o	odd	30	1	inner	308.2.bc.a	✓	352
308.bc	even	30	1	inner	308.2.bc.a	✓	352

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
308.2.bc.a	✓	352	1.a	even	1	1	trivial
308.2.bc.a	✓	352	4.b	odd	2	1	inner
308.2.bc.a	✓	352	7.c	even	3	1	inner
308.2.bc.a	✓	352	11.d	odd	10	1	inner
308.2.bc.a	✓	352	28.g	odd	6	1	inner
308.2.bc.a	✓	352	44.g	even	10	1	inner
308.2.bc.a	✓	352	77.o	odd	30	1	inner
308.2.bc.a	✓	352	308.bc	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(308, [\chi])\).