Newform orbit 380.3.bi.a

• Label: 380.3.bi.a
• Level: $380$
• Weight: $3$
• Character orbit: 380.bi
• Analytic conductor: $10.354$
• Analytic rank: $0$
• Dimension: $1392$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.bi (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3542500457\)
Analytic rank:	\(0\)
Dimension:	\(1392\)
Relative dimension:	\(116\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1392 q - 12 q^{2} - 24 q^{5} + 12 q^{6} - 18 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−1.99877	+	0.0702414i	−4.43630	−	2.06868i	3.99013	−	0.280792i	−4.95221	+	0.689650i	9.01243	+	3.82320i	2.76071	−	10.3031i	−7.95562	+	0.841511i	9.61623	+	11.4602i	9.84987	−	1.72630i
3.2	−1.99648	+	0.118606i	−4.00938	−	1.86961i	3.97187	−	0.473589i	0.0730664	+	4.99947i	8.22640	+	3.25710i	−2.59927	+	9.70060i	−7.87358	+	1.41660i	6.79465	+	8.09754i	−0.738843	−	9.97267i
3.3	−1.99574	+	0.130475i	0.731432	+	0.341073i	3.96595	−	0.520788i	−4.51630	−	2.14547i	−1.50425	−	0.585258i	−0.448042	+	1.67211i	−7.84706	+	1.55681i	−5.36643	−	6.39546i	9.29329	+	3.69253i
3.4	−1.99213	+	0.177244i	−1.23599	−	0.576352i	3.93717	−	0.706188i	3.24120	−	3.80718i	2.56441	+	0.929097i	2.60015	−	9.70391i	−7.71819	+	2.10466i	−4.58960	−	5.46967i	−5.78210	+	8.15888i
3.5	−1.99054	−	0.194255i	5.23571	+	2.44145i	3.92453	+	0.773346i	1.95101	+	4.60365i	−9.94764	−	5.87687i	−0.265927	+	0.992454i	−7.66172	−	2.30174i	15.6669	+	18.6710i	−2.98929	−	9.54275i
3.6	−1.98698	−	0.227818i	3.63842	+	1.69662i	3.89620	+	0.905342i	3.18076	−	3.85782i	−6.84295	−	4.20006i	−0.278829	+	1.04061i	−7.53542	−	2.68652i	4.57448	+	5.45166i	−7.19899	+	6.94078i
3.7	−1.96707	−	0.361460i	−3.22252	−	1.50269i	3.73869	+	1.42203i	−0.435029	−	4.98104i	5.79575	+	4.12069i	−0.844431	+	3.15146i	−6.84025	−	4.14861i	2.34149	+	2.79048i	−0.944714	+	9.95528i
3.8	−1.96134	+	0.391347i	−1.27648	−	0.595235i	3.69370	−	1.53513i	3.81080	+	3.23694i	2.73656	+	0.667908i	1.44442	−	5.39066i	−6.64382	+	4.45642i	−4.50998	−	5.37478i	−8.74104	−	4.85740i
3.9	−1.94897	+	0.448909i	3.93815	+	1.83639i	3.59696	−	1.74982i	−3.81592	−	3.23090i	−8.49970	−	1.81120i	2.91932	−	10.8951i	−6.22486	+	5.02505i	6.35161	+	7.56955i	8.88750	+	4.58392i
3.10	−1.93940	−	0.488598i	0.490435	+	0.228693i	3.52254	+	1.89517i	3.21139	+	3.83236i	−0.839410	−	0.683154i	−0.551370	+	2.05774i	−5.90564	−	5.39661i	−5.59686	−	6.67008i	−4.35568	−	9.00156i
3.11	−1.93450	+	0.507652i	1.30140	+	0.606854i	3.48458	−	1.96411i	4.96614	−	0.580911i	−2.82563	−	0.513299i	−3.57093	+	13.3269i	−5.74383	+	5.56852i	−4.45971	−	5.31488i	−9.31209	+	3.64484i
3.12	−1.91228	−	0.585814i	−1.02895	−	0.479806i	3.31364	+	2.24048i	−4.81034	+	1.36407i	1.68656	+	1.52030i	−1.28335	+	4.78951i	−5.02411	−	6.22562i	−4.95657	−	5.90701i	9.99781	+	0.209486i
3.13	−1.90877	−	0.597161i	1.55536	+	0.725276i	3.28680	+	2.27969i	−2.54463	+	4.30405i	−2.53572	−	2.31319i	1.81697	−	6.78101i	−4.91240	−	6.31414i	−3.89197	−	4.63827i	7.42733	−	6.69588i
3.14	−1.87073	+	0.707375i	2.11578	+	0.986605i	2.99924	−	2.64661i	−3.76680	+	3.28804i	−4.65595	−	0.349017i	−1.60089	+	5.97461i	−3.73861	+	7.07268i	−2.28195	−	2.71952i	4.72078	−	8.81557i
3.15	−1.80846	−	0.854083i	−4.52403	−	2.10959i	2.54108	+	3.08916i	4.89841	+	1.00277i	6.37978	+	7.67902i	0.683144	−	2.54953i	−1.95706	−	7.75693i	10.2314	+	12.1933i	−8.00215	−	5.99713i
3.16	−1.80117	+	0.869363i	−1.06056	−	0.494548i	2.48842	−	3.13174i	−1.57637	−	4.74500i	2.34019	−	0.0312491i	−1.90276	+	7.10119i	−1.75944	+	7.80413i	−4.90488	−	5.84540i	6.96444	+	7.17611i
3.17	−1.74244	+	0.981777i	3.58787	+	1.67305i	2.07223	−	3.42138i	−2.56275	+	4.29328i	−7.89422	+	0.607287i	−0.539643	+	2.01397i	−0.251709	+	7.99604i	4.28860	+	5.11096i	0.250410	−	9.99686i
3.18	−1.73108	+	1.00167i	−4.16925	−	1.94415i	1.99331	−	3.46796i	3.67769	−	3.38741i	9.16473	−	0.810729i	0.594198	−	2.21758i	0.0231768	+	7.99997i	7.81785	+	9.31695i	−2.97332	+	9.54774i
3.19	−1.72414	+	1.01359i	3.22312	+	1.50296i	1.94529	−	3.49512i	4.59861	+	1.96285i	−7.08048	+	0.675589i	2.51871	−	9.39994i	0.188654	+	7.99778i	2.34450	+	2.79407i	−9.91815	+	1.27687i
3.20	−1.69008	−	1.06941i	2.72030	+	1.26850i	1.71273	+	3.61477i	−1.93087	−	4.61213i	−3.24098	−	5.05297i	−1.49238	+	5.56964i	0.971019	−	7.94085i	0.00585713	+	0.00698026i	−1.66892	+	9.85975i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
19.f	odd	18	1	inner
20.e	even	4	1	inner
76.k	even	18	1	inner
95.r	even	36	1	inner
380.bi	odd	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.3.bi.a	✓	1392
4.b	odd	2	1	inner	380.3.bi.a	✓	1392
5.c	odd	4	1	inner	380.3.bi.a	✓	1392
19.f	odd	18	1	inner	380.3.bi.a	✓	1392
20.e	even	4	1	inner	380.3.bi.a	✓	1392
76.k	even	18	1	inner	380.3.bi.a	✓	1392
95.r	even	36	1	inner	380.3.bi.a	✓	1392
380.bi	odd	36	1	inner	380.3.bi.a	✓	1392

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.3.bi.a	✓	1392	1.a	even	1	1	trivial
380.3.bi.a	✓	1392	4.b	odd	2	1	inner
380.3.bi.a	✓	1392	5.c	odd	4	1	inner
380.3.bi.a	✓	1392	19.f	odd	18	1	inner
380.3.bi.a	✓	1392	20.e	even	4	1	inner
380.3.bi.a	✓	1392	76.k	even	18	1	inner
380.3.bi.a	✓	1392	95.r	even	36	1	inner
380.3.bi.a	✓	1392	380.bi	odd	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).