Newform orbit 380.3.p.a

• Label: 380.3.p.a
• Level: $380$
• Weight: $3$
• Character orbit: 380.p
• Analytic conductor: $10.354$
• Analytic rank: $0$
• Dimension: $232$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3542500457\)
Analytic rank:	\(0\)
Dimension:	\(232\)
Relative dimension:	\(116\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
159.1	−1.99940	+	0.0491004i	2.13447	+	3.69701i	3.99518	−	0.196342i	3.83161	−	3.21229i	−4.44918	−	7.28700i	−2.09290			−7.97831	+	0.588731i	−4.61194	+	7.98812i	−7.50318	+	6.61077i
159.2	−1.99929	−	0.0533207i	−2.05947	−	3.56711i	3.99431	+	0.213207i	4.96777	+	0.566763i	3.92728	+	7.24149i	−2.21466			−7.97442	−	0.639242i	−3.98284	+	6.89847i	−9.90180	−	1.39801i
159.3	−1.99798	−	0.0898151i	0.192588	+	0.333573i	3.98387	+	0.358898i	0.758288	+	4.94217i	−0.354828	−	0.683770i	5.98368			−7.92746	−	1.07488i	4.42582	−	7.66574i	−1.07117	−	9.94246i
159.4	−1.99491	+	0.142649i	1.65352	+	2.86398i	3.95930	−	0.569141i	−4.35243	−	2.46097i	−3.70716	−	5.47750i	−11.5548			−7.81725	+	1.70017i	−0.968248	+	1.67706i	9.03374	+	4.28854i
159.5	−1.99219	−	0.176550i	1.30777	+	2.26512i	3.93766	+	0.703443i	−3.13801	+	3.89267i	−2.20541	−	4.74343i	−0.966370			−7.72038	−	2.09659i	1.07950	−	1.86975i	6.93877	−	7.20093i
159.6	−1.97327	+	0.325875i	−0.501671	−	0.868919i	3.78761	−	1.28608i	−2.04342	−	4.56338i	1.27309	+	1.55113i	6.90502			−7.05489	+	3.77207i	3.99665	−	6.92241i	5.51932	+	8.33889i
159.7	−1.97102	−	0.339262i	−1.97599	−	3.42251i	3.76980	+	1.33738i	−0.693065	−	4.95173i	2.73357	+	7.41619i	−2.56913			−6.97662	−	3.91494i	−3.30904	+	5.73143i	−0.313891	+	9.99507i
159.8	−1.97064	−	0.341417i	−2.91038	−	5.04092i	3.76687	+	1.34562i	−4.21638	+	2.68741i	4.01426	+	10.9275i	4.09062			−6.96373	−	3.93782i	−12.4406	+	21.5477i	9.22650	−	3.85638i
159.9	−1.95148	−	0.437846i	2.80787	+	4.86337i	3.61658	+	1.70890i	4.08358	+	2.88520i	−3.35010	−	10.7202i	7.39194			−6.30947	−	4.91840i	−11.2682	+	19.5171i	−6.70576	−	7.41841i
159.10	−1.94086	+	0.482776i	0.480142	+	0.831631i	3.53385	−	1.87400i	4.92739	+	0.849034i	−1.33338	−	1.38227i	−5.58137			−5.95398	+	5.34323i	4.03893	−	6.99563i	−9.97325	+	0.730972i
159.11	−1.92793	+	0.532049i	−1.00965	−	1.74877i	3.43385	−	2.05151i	−4.76464	+	1.51598i	2.87698	+	2.83433i	3.07442			−5.52873	+	5.78214i	2.46120	−	4.26292i	8.37933	−	5.45773i
159.12	−1.87977	−	0.682971i	1.12368	+	1.94628i	3.06710	+	2.56766i	2.49657	−	4.33210i	−0.783020	−	4.42601i	−2.72652			−4.01181	−	6.92137i	1.97467	−	3.42023i	−7.65169	+	6.43829i
159.13	−1.85215	+	0.754679i	2.72599	+	4.72155i	2.86092	−	2.79556i	−4.08217	−	2.88720i	−8.61220	−	6.68778i	7.14895			−3.18910	+	7.33687i	−10.3621	+	17.9476i	9.73969	+	2.26680i
159.14	−1.83289	−	0.800316i	−0.529478	−	0.917082i	2.71899	+	2.93379i	4.84073	−	1.25194i	0.236520	+	2.10466i	13.3882			−2.63566	−	7.55336i	3.93931	−	6.82308i	−9.87448	−	1.57944i
159.15	−1.82919	−	0.808734i	−0.866507	−	1.50083i	2.69190	+	2.95866i	−4.28911	−	2.56974i	0.371233	+	3.44609i	−10.8203			−2.53123	−	7.58900i	2.99833	−	5.19326i	5.76738	+	8.16929i
159.16	−1.82898	−	0.809227i	−0.0580338	−	0.100518i	2.69030	+	2.96011i	2.00334	+	4.58111i	0.0248009	+	0.230807i	−7.81754			−2.52509	−	7.59104i	4.49326	−	7.78256i	0.0430974	−	9.99991i
159.17	−1.76297	+	0.944431i	−1.63664	−	2.83475i	2.21610	−	3.33000i	−0.368392	+	4.98641i	5.56256	+	3.45187i	−8.62575			−0.761952	+	7.96363i	−0.857187	+	1.48469i	−4.05986	−	9.13879i
159.18	−1.74433	+	0.978432i	2.25815	+	3.91123i	2.08534	−	3.41341i	2.67107	+	4.22675i	−7.76582	−	4.61301i	−11.4308			−0.297731	+	7.99446i	−5.69848	+	9.87006i	−8.79480	−	4.75936i
159.19	−1.73403	+	0.996556i	1.05335	+	1.82446i	2.01375	−	3.45613i	2.78233	−	4.15435i	−3.64473	−	2.11396i	8.55585			−0.0476908	+	7.99986i	2.28089	−	3.95061i	−0.684605	+	9.97654i
159.20	−1.73006	+	1.00344i	−1.05335	−	1.82446i	1.98622	−	3.47202i	2.78233	−	4.15435i	3.65311	+	2.09945i	−8.55585			0.0476908	+	7.99986i	2.28089	−	3.95061i	−0.644950	+	9.97918i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
19.c	even	3	1	inner
20.d	odd	2	1	inner
76.g	odd	6	1	inner
95.i	even	6	1	inner
380.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.3.p.a	✓	232
4.b	odd	2	1	inner	380.3.p.a	✓	232
5.b	even	2	1	inner	380.3.p.a	✓	232
19.c	even	3	1	inner	380.3.p.a	✓	232
20.d	odd	2	1	inner	380.3.p.a	✓	232
76.g	odd	6	1	inner	380.3.p.a	✓	232
95.i	even	6	1	inner	380.3.p.a	✓	232
380.p	odd	6	1	inner	380.3.p.a	✓	232

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.3.p.a	✓	232	1.a	even	1	1	trivial
380.3.p.a	✓	232	4.b	odd	2	1	inner
380.3.p.a	✓	232	5.b	even	2	1	inner
380.3.p.a	✓	232	19.c	even	3	1	inner
380.3.p.a	✓	232	20.d	odd	2	1	inner
380.3.p.a	✓	232	76.g	odd	6	1	inner
380.3.p.a	✓	232	95.i	even	6	1	inner
380.3.p.a	✓	232	380.p	odd	6	1	inner

Hecke kernels

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