Newform orbit 380.2.s.a

• Label: 380.2.s.a
• Level: $380$
• Weight: $2$
• Character orbit: 380.s
• Analytic conductor: $3.034$
• Analytic rank: $0$
• Dimension: $112$
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.03431527681\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(56\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 112 * q - 2 * q^5 - 8 * q^6 + 44 * q^9 + 6 * q^10 - 36 * q^14 - 4 * q^16 + 44 * q^20 - 48 * q^21 + 2 * q^24 - 2 * q^25 - 36 * q^26 - 12 * q^29 - 32 * q^30 - 30 * q^34 + 20 * q^36 - 24 * q^40 - 24 * q^41 - 14 * q^44 + 56 * q^49 + 32 * q^54 - 6 * q^60 - 36 * q^61 - 36 * q^64 + 22 * q^66 + 12 * q^70 + 16 * q^74 - 26 * q^76 + 12 * q^80 - 48 * q^81 - 26 * q^85 - 120 * q^86 - 24 * q^89 + 24 * q^90 - 84 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
179.1	−1.41225	−	0.0745655i	2.83828	+	1.63868i	1.98888	+	0.210610i	−0.699811	−	2.12374i	−3.88617	−	2.52586i	−0.387143			−2.79308	−	0.445735i	3.87057	+	6.70403i	0.829947	+	3.05142i
179.2	−1.40034	+	0.197594i	0.952596	+	0.549982i	1.92191	−	0.553399i	−2.22480	+	0.224228i	−1.44263	−	0.581934i	−1.22694			−2.58199	+	1.15471i	−0.895041	−	1.55026i	3.07117	−	0.753603i
179.3	−1.37216	−	0.342313i	−1.99347	−	1.15093i	1.76564	+	0.939416i	2.01799	−	0.963178i	2.34138	+	2.26165i	2.46740			−2.10117	−	1.89343i	1.14928	+	1.99061i	−3.09871	+	0.630849i
179.4	−1.37100	+	0.346927i	−1.86434	−	1.07638i	1.75928	−	0.951274i	1.15834	+	1.91265i	2.92944	+	0.828924i	−1.21448			−2.08195	+	1.91454i	0.817179	+	1.41540i	−2.25164	−	2.22038i
179.5	−1.36239	−	0.379325i	−1.08198	−	0.624679i	1.71223	+	1.03358i	−1.92835	+	1.13202i	1.23712	+	1.26148i	−0.151300			−1.94066	−	2.05763i	−0.719553	−	1.24630i	3.05657	−	0.810790i
179.6	−1.34533	−	0.435980i	1.43647	+	0.829346i	1.61984	+	1.17308i	1.59262	+	1.56957i	−1.57095	−	1.74202i	4.79169			−1.66779	−	2.28440i	−0.124369	−	0.215414i	−1.45831	−	2.80595i
179.7	−1.33018	+	0.480217i	0.288375	+	0.166493i	1.53878	−	1.27755i	0.656216	−	2.13761i	−0.463545	−	0.0829845i	2.90949			−1.43336	+	2.43833i	−1.44456	−	2.50205i	0.153629	+	3.15854i
179.8	−1.31940	−	0.509093i	0.369609	+	0.213394i	1.48165	+	1.34340i	2.18710	−	0.465374i	−0.379026	−	0.469718i	−3.44511			−1.27098	−	2.52678i	−1.40893	−	2.44033i	−3.12259	−	0.499423i
179.9	−1.31410	+	0.522619i	1.49300	+	0.861986i	1.45374	−	1.37355i	2.15570	+	0.594110i	−2.41245	−	0.352467i	−2.20094			−1.19252	+	2.56474i	−0.0139597	−	0.0241789i	−3.14330	+	0.345886i
179.10	−1.21903	+	0.716913i	−2.56289	−	1.47969i	0.972072	−	1.74788i	−2.23484	+	0.0740448i	4.18505	−	0.0335865i	4.78381			0.0680894	+	2.82761i	2.87894	+	4.98647i	2.67126	−	1.69245i
179.11	−1.10059	−	0.888090i	0.369609	+	0.213394i	0.422592	+	1.95484i	−0.690526	−	2.12678i	−0.217274	−	0.563105i	−3.44511			1.27098	−	2.52678i	−1.40893	−	2.44033i	−1.12878	+	2.95395i
179.12	−1.08964	+	0.901488i	2.51144	+	1.44998i	0.374639	−	1.96460i	−0.944748	+	2.02668i	−4.04370	+	0.684072i	2.35031			1.36284	+	2.47844i	2.70487	+	4.68497i	−0.797594	−	3.06004i
179.13	−1.05024	−	0.947103i	1.43647	+	0.829346i	0.205992	+	1.98936i	−2.15560	−	0.594467i	−0.723156	−	2.23149i	4.79169			1.66779	−	2.28440i	−0.124369	−	0.215414i	1.70087	+	2.66591i
179.14	−1.00970	−	0.990204i	−1.08198	−	0.624679i	0.0389922	+	1.99962i	−0.0161872	+	2.23601i	0.473912	+	1.70211i	−0.151300			1.94066	−	2.05763i	−0.719553	−	1.24630i	2.23045	−	2.24167i
179.15	−0.993958	+	1.00601i	−0.753647	−	0.435118i	−0.0240957	−	1.99985i	−1.67009	−	1.48688i	1.18682	−	0.325684i	−1.85484			2.03582	+	1.96353i	−1.12134	−	1.94223i	3.15580	−	0.202225i
179.16	−0.982532	−	1.01717i	−1.99347	−	1.15093i	−0.0692634	+	1.99880i	−0.174859	−	2.22922i	0.787956	+	3.15852i	2.46740			2.10117	−	1.89343i	1.14928	+	1.99061i	−2.09569	+	2.36814i
179.17	−0.948703	+	1.04879i	0.350497	+	0.202360i	−0.199926	−	1.98998i	−0.789605	+	2.09201i	−0.544751	+	0.175619i	−3.85760			2.27675	+	1.67822i	−1.41810	−	2.45622i	−1.44499	−	2.81283i
179.18	−0.784760	+	1.17650i	−2.02176	−	1.16726i	−0.768302	−	1.84654i	1.88679	−	1.20001i	2.95988	−	1.46258i	−0.943490			2.77539	+	0.545184i	1.22501	+	2.12179i	−0.0688652	+	3.16153i
179.19	−0.770699	−	1.18576i	2.83828	+	1.63868i	−0.812047	+	1.82773i	2.18912	−	0.455815i	−0.244379	−	4.62845i	−0.387143			2.79308	−	0.445735i	3.87057	+	6.70403i	−2.22764	−	2.24447i
179.20	−0.626498	+	1.26787i	2.02176	+	1.16726i	−1.21500	−	1.58864i	1.88679	−	1.20001i	−2.74657	+	1.83205i	0.943490			2.77539	−	0.545184i	1.22501	+	2.12179i	0.339390	+	3.14401i

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.d	odd	6	1	inner
20.d	odd	2	1	inner
76.f	even	6	1	inner
95.h	odd	6	1	inner
380.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.2.s.a	✓	112
4.b	odd	2	1	inner	380.2.s.a	✓	112
5.b	even	2	1	inner	380.2.s.a	✓	112
19.d	odd	6	1	inner	380.2.s.a	✓	112
20.d	odd	2	1	inner	380.2.s.a	✓	112
76.f	even	6	1	inner	380.2.s.a	✓	112
95.h	odd	6	1	inner	380.2.s.a	✓	112
380.s	even	6	1	inner	380.2.s.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.2.s.a	✓	112	1.a	even	1	1	trivial
380.2.s.a	✓	112	4.b	odd	2	1	inner
380.2.s.a	✓	112	5.b	even	2	1	inner
380.2.s.a	✓	112	19.d	odd	6	1	inner
380.2.s.a	✓	112	20.d	odd	2	1	inner
380.2.s.a	✓	112	76.f	even	6	1	inner
380.2.s.a	✓	112	95.h	odd	6	1	inner
380.2.s.a	✓	112	380.s	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(380, [\chi])\).