Newform orbit 380.3.j.a

• Label: 380.3.j.a
• Level: $380$
• Weight: $3$
• Character orbit: 380.j
• Analytic conductor: $10.354$
• Analytic rank: $0$
• Dimension: $232$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3542500457\)
Analytic rank:	\(0\)
Dimension:	\(232\)
Relative dimension:	\(116\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 232 * q - 8 * q^5 - 32 * q^6 - 64 * q^16 - 8 * q^17 + 80 * q^20 - 8 * q^25 + 40 * q^26 + 40 * q^28 - 104 * q^30 + 104 * q^36 + 184 * q^38 + 192 * q^42 + 192 * q^45 + 280 * q^58 + 112 * q^61 - 280 * q^62 + 216 * q^66 - 248 * q^68 - 8 * q^73 - 128 * q^76 + 336 * q^77 - 144 * q^80 - 1528 * q^81 + 552 * q^82 + 136 * q^85 + 552 * q^92 - 272 * q^93 + 504 * q^96

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
227.1	−2.00000	+	0.00413025i	−2.70754	+	2.70754i	3.99997	−	0.0165209i	−4.82977	−	1.29359i	5.40389	−	5.42626i	2.48709	−	2.48709i	−7.99985	+	0.0495627i		−	5.66157i	9.66485	+	2.56722i
227.2	−1.99894	+	0.0650204i	3.73319	−	3.73319i	3.99154	−	0.259944i	−1.04361	+	4.88987i	−7.21971	+	7.70517i	−4.85685	+	4.85685i	−7.96197	+	0.779146i		−	18.8735i	1.76818	−	9.84244i
227.3	−1.99610	+	0.124883i	−0.756575	+	0.756575i	3.96881	−	0.498556i	4.99668	−	0.182084i	1.41571	−	1.60468i	−0.187105	+	0.187105i	−7.85987	+	1.49080i			7.85519i	−9.95113	+	0.987456i
227.4	−1.99490	+	0.142778i	1.34489	−	1.34489i	3.95923	−	0.569656i	−4.96914	−	0.554657i	−2.49090	+	2.87494i	4.10715	−	4.10715i	−7.81692	+	1.70170i			5.38253i	9.99212	+	0.396998i
227.5	−1.99311	+	0.165836i	2.36243	−	2.36243i	3.94500	−	0.661058i	2.79906	+	4.14310i	−4.31682	+	5.10037i	8.32823	−	8.32823i	−7.75320	+	1.97178i		−	2.16219i	−6.26591	−	7.79348i
227.6	−1.98524	−	0.242507i	0.169672	−	0.169672i	3.88238	+	0.962870i	0.857699	−	4.92589i	−0.377988	+	0.295694i	1.67200	−	1.67200i	−7.47397	−	2.85303i			8.94242i	−2.89730	+	9.57108i
227.7	−1.94088	+	0.482683i	−2.93828	+	2.93828i	3.53404	−	1.87366i	−2.06367	+	4.55426i	4.28460	−	7.12112i	7.41693	−	7.41693i	−5.95476	+	5.34236i		−	8.26702i	1.80707	−	9.83537i
227.8	−1.93420	−	0.508790i	−3.89344	+	3.89344i	3.48227	+	1.96820i	3.99170	+	3.01103i	9.51163	−	5.54975i	−5.88298	+	5.88298i	−5.73400	−	5.57864i		−	21.3177i	−6.18877	−	7.85488i
227.9	−1.92854	−	0.529843i	−1.41485	+	1.41485i	3.43853	+	2.04365i	−4.97296	+	0.519307i	3.47824	−	1.97894i	−9.66238	+	9.66238i	−5.54854	−	5.76314i			4.99641i	9.86570	+	1.63338i
227.10	−1.92510	+	0.542224i	2.55276	−	2.55276i	3.41199	−	2.08766i	−3.19089	−	3.84944i	−3.53014	+	6.29847i	−3.72891	+	3.72891i	−5.43642	+	5.86901i		−	4.03317i	8.23003	+	5.68036i
227.11	−1.90237	−	0.617236i	1.61266	−	1.61266i	3.23804	+	2.34842i	1.65685	−	4.71750i	−4.06328	+	2.07249i	−5.55382	+	5.55382i	−4.71042	−	6.46621i			3.79864i	−6.06376	+	7.95178i
227.12	−1.89506	+	0.639336i	−0.113263	+	0.113263i	3.18250	−	2.42316i	−1.31466	+	4.82407i	0.142227	−	0.287054i	−6.53716	+	6.53716i	−4.48181	+	6.62672i			8.97434i	−0.592839	−	9.98241i
227.13	−1.88890	−	0.657300i	−0.483404	+	0.483404i	3.13591	+	2.48315i	1.17376	+	4.86028i	1.23085	−	0.595363i	1.64522	−	1.64522i	−4.29126	−	6.75167i			8.53264i	0.977532	−	9.95211i
227.14	−1.88290	+	0.674294i	−2.71125	+	2.71125i	3.09066	−	2.53926i	2.06771	−	4.55243i	3.27685	−	6.93322i	−4.53921	+	4.53921i	−4.10720	+	6.86519i		−	5.70181i	−0.823628	+	9.96602i
227.15	−1.87058	−	0.707774i	3.26349	−	3.26349i	2.99811	+	2.64789i	4.82525	−	1.31032i	−8.41441	+	3.79479i	0.0356335	−	0.0356335i	−3.73409	−	7.07506i		−	12.3007i	−9.95341	−	0.964142i
227.16	−1.83249	−	0.801229i	−3.50213	+	3.50213i	2.71606	+	2.93649i	−0.641092	−	4.95873i	9.22364	−	3.61162i	2.56617	−	2.56617i	−2.62436	−	7.55730i		−	15.5298i	−2.79828	+	9.60050i
227.17	−1.77962	+	0.912666i	3.66846	−	3.66846i	2.33408	−	3.24839i	4.37305	−	2.42414i	−3.18038	+	9.87655i	−2.51789	+	2.51789i	−1.18908	+	7.91114i		−	17.9153i	−5.56992	+	8.30518i
227.18	−1.74987	+	0.968481i	1.12028	−	1.12028i	2.12409	−	3.38943i	3.50045	−	3.57027i	−0.875371	+	3.04530i	8.74065	−	8.74065i	−0.434280	+	7.98820i			6.48997i	−2.66760	+	9.63763i
227.19	−1.74392	−	0.979150i	3.79260	−	3.79260i	2.08253	+	3.41512i	−2.62716	−	4.25418i	−10.3275	+	2.90047i	8.90960	−	8.90960i	−0.287859	−	7.99482i		−	19.7676i	0.416087	+	9.99134i
227.20	−1.74129	−	0.983821i	2.37942	−	2.37942i	2.06419	+	3.42624i	−3.94275	+	3.07485i	−6.48420	+	1.80234i	−6.50795e−5	0	6.50795e-5i	−0.223556	−	7.99688i		−	2.32331i	9.89058	−	1.47524i

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.b	odd	2	1	inner
20.e	even	4	1	inner
76.d	even	2	1	inner
95.g	even	4	1	inner
380.j	odd	4	1	inner

Twists

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

    

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

Hecke kernels

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