Newform orbit 392.3.s.a

• Label: 392.3.s.a
• Level: $392$
• Weight: $3$
• Character orbit: 392.s
• Analytic conductor: $10.681$
• Analytic rank: $0$
• Dimension: $660$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	392.s (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.6812263629\)
Analytic rank:	\(0\)
Dimension:	\(660\)
Relative dimension:	\(110\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 660 q - 5 q^{2} - 10 q^{3} - 5 q^{4} + 3 q^{6} + q^{8} - 328 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−2.00000	+	0.00291544i	−3.04788	−	1.46778i	3.99998	−	0.0116617i	−3.44393	+	7.15140i	6.10003	+	2.92667i	0.257911	+	6.99525i	−7.99992	+	0.0349852i	1.52377	+	1.91074i	6.86701	−	14.3128i
43.2	−1.99972	−	0.0332616i	−0.272192	−	0.131081i	3.99779	+	0.133028i	−1.42253	+	2.95391i	0.539948	+	0.271178i	−6.80945	−	1.62216i	−7.99004	−	0.398992i	−5.55450	−	6.96513i	2.94291	−	5.85969i
43.3	−1.99554	+	0.133431i	4.89938	+	2.35942i	3.96439	−	0.532536i	−2.82394	+	5.86398i	−10.0917	−	4.05459i	−5.15897	+	4.73128i	−7.84006	+	1.59167i	12.8257	+	16.0829i	4.85287	−	12.0786i
43.4	−1.98691	−	0.228458i	−4.26928	−	2.05598i	3.89561	+	0.907850i	2.16573	−	4.49718i	8.01297	+	5.06040i	6.91158	+	1.10909i	−7.53282	−	2.69380i	8.38833	+	10.5186i	−5.33053	+	8.44072i
43.5	−1.98671	−	0.230157i	2.20511	+	1.06193i	3.89406	+	0.914512i	−1.02486	+	2.12815i	−4.13651	−	2.61726i	1.14437	−	6.90582i	−7.52589	−	2.71312i	−1.87658	−	2.35315i	2.52592	−	3.99214i
43.6	−1.98433	+	0.249894i	2.29999	+	1.10762i	3.87511	−	0.991743i	1.65823	−	3.44334i	−4.84073	−	1.62312i	6.14271	+	3.35666i	−7.44165	+	2.93631i	−1.54825	−	1.94145i	−2.42999	+	7.24709i
43.7	−1.96499	−	0.372558i	0.852005	+	0.410304i	3.72240	+	1.46415i	−2.37865	+	4.93931i	−1.52132	−	1.12367i	6.74304	+	1.87921i	−6.76901	−	4.26386i	−5.05384	−	6.33732i	6.51421	−	8.81953i
43.8	−1.96494	+	0.372863i	4.01107	+	1.93163i	3.72195	−	1.46530i	3.56253	−	7.39767i	−8.60173	−	2.29995i	−3.22991	−	6.21029i	−6.76703	+	4.26700i	6.74608	+	8.45932i	−4.24182	+	15.8643i
43.9	−1.92621	+	0.538269i	−2.60061	−	1.25239i	3.42053	−	2.07363i	−0.866215	+	1.79871i	5.68342	+	1.01253i	3.10421	−	6.27406i	−5.47248	+	5.83541i	−0.416723	−	0.522554i	0.700316	−	3.93095i
43.10	−1.92541	+	0.541090i	−1.11848	−	0.538632i	3.41444	−	2.08365i	3.12417	−	6.48740i	2.44499	+	0.431891i	−4.20630	+	5.59527i	−5.44678	+	5.85940i	−4.65053	−	5.83158i	−2.50505	+	14.1814i
43.11	−1.89792	+	0.630779i	−4.84433	−	2.33291i	3.20424	−	2.39434i	1.75545	−	3.64522i	10.6657	+	1.37198i	−6.14562	−	3.35132i	−4.57110	+	6.56544i	12.4137	+	15.5663i	−1.03238	+	8.02565i
43.12	−1.87765	−	0.688792i	−1.71029	−	0.823632i	3.05113	+	2.58662i	3.47847	−	7.22313i	2.64401	+	2.72452i	−0.773838	−	6.95710i	−3.94731	−	6.95836i	−3.36469	−	4.21919i	−11.5066	+	11.1666i
43.13	−1.86806	−	0.714394i	−0.106419	−	0.0512487i	2.97928	+	2.66906i	2.35910	−	4.89872i	0.162185	+	0.171761i	0.798328	+	6.95433i	−3.65871	−	7.11434i	−5.60271	−	7.02558i	−7.90655	+	7.46577i
43.14	−1.85439	−	0.749165i	−2.86809	−	1.38120i	2.87750	+	2.77848i	−0.404243	+	0.839419i	4.28380	+	4.70995i	−6.90547	+	1.14654i	−3.25446	−	7.30811i	0.706809	+	0.886311i	1.37849	−	1.25376i
43.15	−1.85037	+	0.759027i	2.24638	+	1.08180i	2.84776	−	2.80896i	−0.901861	+	1.87273i	−4.97775	−	0.296669i	−3.77269	+	5.89634i	−3.13733	+	7.35915i	−1.73548	−	2.17622i	0.247324	−	4.14979i
43.16	−1.84079	−	0.781991i	3.51697	+	1.69368i	2.77698	+	2.87895i	1.74935	−	3.63256i	−5.14954	−	5.86794i	−5.12370	+	4.76946i	−2.86051	−	7.47111i	3.88909	+	4.87676i	−6.06081	+	5.31879i
43.17	−1.81214	−	0.846248i	−4.94506	−	2.38141i	2.56773	+	3.06704i	−3.69065	+	7.66371i	6.94589	+	8.50020i	1.15614	−	6.90386i	−2.05762	−	7.73086i	13.1710	+	16.5160i	13.1734	−	10.7645i
43.18	−1.79944	−	0.872932i	5.00768	+	2.41157i	2.47598	+	3.14158i	−0.601541	+	1.24911i	−6.90590	−	8.71085i	5.93541	−	3.71092i	−1.71300	−	7.81445i	13.6498	+	17.1163i	2.17283	−	1.72260i
43.19	−1.73043	+	1.00280i	2.57338	+	1.23927i	1.98878	−	3.47055i	−4.22478	+	8.77284i	−5.69579	+	0.436108i	6.75636	−	1.83073i	0.0388276	+	7.99991i	−0.524944	−	0.658260i	−1.48673	−	19.4174i
43.20	−1.66971	+	1.10095i	−2.76541	−	1.33175i	1.57583	−	3.67651i	−1.29618	+	2.69154i	6.08362	−	0.820937i	5.39101	+	4.46509i	1.41646	+	7.87360i	0.262539	+	0.329213i	−0.799007	−	5.92110i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
49.e	even	7	1	inner
392.s	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.3.s.a	✓	660
8.d	odd	2	1	inner	392.3.s.a	✓	660
49.e	even	7	1	inner	392.3.s.a	✓	660
392.s	odd	14	1	inner	392.3.s.a	✓	660

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
392.3.s.a	✓	660	1.a	even	1	1	trivial
392.3.s.a	✓	660	8.d	odd	2	1	inner
392.3.s.a	✓	660	49.e	even	7	1	inner
392.3.s.a	✓	660	392.s	odd	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(392, [\chi])\).