Newform orbit 231.2.bc.a

• Label: 231.2.bc.a
• Level: $231$
• Weight: $2$
• Character orbit: 231.bc
• Analytic conductor: $1.845$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.bc (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.84454428669\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 9 q^{3} - 30 q^{4} - 16 q^{7} + 3 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−0.546758	+	2.57230i	−1.67722	+	0.432351i	−4.49067	−	1.99937i	0.108184	+	0.120151i	−0.195099	−	4.55070i	−0.360360	−	2.62110i	4.50682	−	6.20311i	2.62615	−	1.45030i	−0.368214	+	0.212589i
5.2	−0.513957	+	2.41798i	−0.550359	−	1.64229i	−3.75538	−	1.67200i	1.71178	+	1.90112i	4.25388	−	0.486692i	−1.49416	+	2.18346i	3.06697	−	4.22132i	−2.39421	+	1.80769i	−5.47665	+	3.16195i
5.3	−0.506364	+	2.38226i	0.353670	+	1.69556i	−3.59165	−	1.59911i	1.14381	+	1.27033i	−4.21834	−	0.0160368i	1.23243	+	2.34118i	2.76509	−	3.80582i	−2.74983	+	1.19934i	−3.60545	+	2.08161i
5.4	−0.459196	+	2.16035i	1.71801	−	0.220131i	−2.62916	−	1.17058i	−0.345466	−	0.383679i	−0.313341	+	3.81257i	2.35485	+	1.20611i	1.13977	−	1.56876i	2.90308	−	0.756373i	0.987517	−	0.570143i
5.5	−0.447920	+	2.10730i	0.826050	−	1.52238i	−2.41299	−	1.07433i	−2.59584	−	2.88297i	2.83811	+	2.42264i	−2.63172	+	0.272090i	0.812141	−	1.11782i	−1.63528	−	2.51512i	7.23801	−	4.17887i
5.6	−0.396988	+	1.86768i	−1.30026	−	1.14426i	−1.50354	−	0.669417i	−1.28280	−	1.42469i	2.65330	−	1.97420i	2.43431	−	1.03640i	−0.397497	+	0.547108i	0.381326	+	2.97567i	3.17012	−	1.83027i
5.7	−0.380999	+	1.79246i	1.69357	−	0.363054i	−1.24065	−	0.552375i	2.38945	+	2.65375i	0.00550992	+	3.17398i	−1.61866	−	2.09283i	−0.691439	+	0.951684i	2.73638	−	1.22972i	−5.66712	+	3.27191i
5.8	−0.311281	+	1.46446i	−0.472312	+	1.66641i	−0.220656	−	0.0982422i	−0.116479	−	0.129364i	−2.29337	−	1.21040i	−2.61507	−	0.401743i	−1.54748	+	2.12992i	−2.55384	−	1.57413i	0.225706	−	0.130311i
5.9	−0.234283	+	1.10222i	−1.26866	+	1.17920i	0.667100	+	0.297012i	2.02306	+	2.24683i	−1.00250	−	1.67460i	2.45218	−	0.993382i	−1.80834	+	2.48897i	0.218989	−	2.99200i	−2.95046	+	1.70345i
5.10	−0.228179	+	1.07350i	−1.72896	−	0.103384i	0.726756	+	0.323573i	−0.978563	−	1.08680i	0.505496	−	1.83245i	0.116723	+	2.64318i	−1.80335	+	2.48210i	2.97862	+	0.357493i	1.38997	−	0.802500i
5.11	−0.170302	+	0.801208i	1.42625	+	0.982761i	1.21416	+	0.540578i	−0.800599	−	0.889155i	−1.03029	+	0.975354i	−2.04538	+	1.67822i	−1.60281	+	2.20608i	1.06836	+	2.80332i	0.848742	−	0.490022i
5.12	−0.148555	+	0.698897i	0.250717	−	1.71381i	1.36070	+	0.605824i	1.08864	+	1.20906i	1.16053	+	0.429821i	1.71199	+	2.01719i	−1.46551	+	2.01709i	−2.87428	−	0.859364i	−1.00673	+	0.581238i
5.13	−0.0876680	+	0.412446i	1.04792	−	1.37908i	1.66467	+	0.741157i	−0.908673	−	1.00918i	0.476926	+	0.553112i	0.442873	−	2.60842i	−0.947316	+	1.30387i	−0.803721	−	2.89033i	0.495895	−	0.286305i
5.14	−0.0516949	+	0.243205i	−1.39546	−	1.02601i	1.77061	+	0.788328i	2.20131	+	2.44480i	0.321668	−	0.286345i	−1.85363	−	1.88787i	−0.575550	+	0.792176i	0.894626	+	2.86350i	−0.708384	+	0.408986i
5.15	0.0516949	−	0.243205i	−1.57829	+	0.713452i	1.77061	+	0.788328i	−2.20131	−	2.44480i	0.0919260	+	0.420730i	−1.85363	−	1.88787i	0.575550	−	0.792176i	1.98197	−	2.25206i	−0.708384	+	0.408986i
5.16	0.0876680	−	0.412446i	0.738295	+	1.56682i	1.66467	+	0.741157i	0.908673	+	1.00918i	0.710952	−	0.167147i	0.442873	−	2.60842i	0.947316	−	1.30387i	−1.90984	+	2.31355i	0.495895	−	0.286305i
5.17	0.148555	−	0.698897i	−0.111082	+	1.72849i	1.36070	+	0.605824i	−1.08864	−	1.20906i	1.19153	+	0.334410i	1.71199	+	2.01719i	1.46551	−	2.01709i	−2.97532	−	0.384008i	−1.00673	+	0.581238i
5.18	0.170302	−	0.801208i	1.59941	−	0.664752i	1.21416	+	0.540578i	0.800599	+	0.889155i	−0.260223	−	1.39467i	−2.04538	+	1.67822i	1.60281	−	2.20608i	2.11621	−	2.12642i	0.848742	−	0.490022i
5.19	0.228179	−	1.07350i	−1.71268	−	0.258347i	0.726756	+	0.323573i	0.978563	+	1.08680i	−0.668132	+	1.77961i	0.116723	+	2.64318i	1.80335	−	2.48210i	2.86651	+	0.884929i	1.38997	−	0.802500i
5.20	0.234283	−	1.10222i	−0.995767	−	1.41720i	0.667100	+	0.297012i	−2.02306	−	2.24683i	−1.79535	+	0.765524i	2.45218	−	0.993382i	1.80834	−	2.48897i	−1.01690	+	2.82240i	−2.95046	+	1.70345i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
11.c	even	5	1	inner
21.g	even	6	1	inner
33.h	odd	10	1	inner
77.p	odd	30	1	inner
231.bc	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	231.2.bc.a	✓	224
3.b	odd	2	1	inner	231.2.bc.a	✓	224
7.d	odd	6	1	inner	231.2.bc.a	✓	224
11.c	even	5	1	inner	231.2.bc.a	✓	224
21.g	even	6	1	inner	231.2.bc.a	✓	224
33.h	odd	10	1	inner	231.2.bc.a	✓	224
77.p	odd	30	1	inner	231.2.bc.a	✓	224
231.bc	even	30	1	inner	231.2.bc.a	✓	224

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
231.2.bc.a	✓	224	1.a	even	1	1	trivial
231.2.bc.a	✓	224	3.b	odd	2	1	inner
231.2.bc.a	✓	224	7.d	odd	6	1	inner
231.2.bc.a	✓	224	11.c	even	5	1	inner
231.2.bc.a	✓	224	21.g	even	6	1	inner
231.2.bc.a	✓	224	33.h	odd	10	1	inner
231.2.bc.a	✓	224	77.p	odd	30	1	inner
231.2.bc.a	✓	224	231.bc	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(231, [\chi])\).