Newform orbit 336.3.bt.a

• Label: 336.3.bt.a
• Level: $336$
• Weight: $3$
• Character orbit: 336.bt
• Analytic conductor: $9.155$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	336.bt (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.15533688251\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(124\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 496 q - 2 q^{3} - 4 q^{4} - 8 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−1.99973	+	0.0330430i	1.25560	−	2.72460i	3.99782	−	0.132154i	−0.642918	−	0.172269i	−2.42083	+	5.48995i	2.42792	+	6.56545i	−7.99017	+	0.396372i	−5.84694	−	6.84202i	1.29135	+	0.323248i
53.2	−1.99744	+	0.101093i	−2.95539	−	0.515460i	3.97956	−	0.403853i	7.96178	+	2.13335i	5.95532	+	0.730835i	5.54600	+	4.27106i	−7.90812	+	1.20898i	8.46860	+	3.04677i	−16.1189	−	3.45637i
53.3	−1.99638	+	0.120311i	−2.39492	+	1.80675i	3.97105	−	0.480373i	0.199225	+	0.0533823i	4.56380	−	3.89509i	2.75032	−	6.43706i	−7.86992	+	1.43677i	2.47130	−	8.65405i	−0.404152	−	0.0826022i
53.4	−1.99436	+	0.150055i	−0.0965741	+	2.99845i	3.95497	−	0.598529i	−7.88140	−	2.11182i	−0.257329	−	5.99448i	−6.84280	−	1.47517i	−7.79783	+	1.78715i	−8.98135	−	0.579144i	16.0353	+	3.02908i
53.5	−1.99258	−	0.172085i	−1.73337	−	2.44856i	3.94077	+	0.685787i	−5.59379	−	1.49885i	3.03252	+	5.17724i	6.99380	+	0.294464i	−7.73430	−	2.04464i	−2.99087	+	8.48850i	10.8882	+	3.94919i
53.6	−1.97921	+	0.287655i	−1.18773	−	2.75487i	3.83451	−	1.13866i	6.09740	+	1.63379i	3.14321	+	5.11080i	−5.79598	−	3.92513i	−7.26174	+	3.35665i	−6.17861	+	6.54407i	−12.5380	−	1.47967i
53.7	−1.97078	+	0.340606i	2.75929	+	1.17741i	3.76797	−	1.34252i	3.07194	+	0.823123i	−5.83901	−	1.38059i	−1.65102	−	6.80251i	−6.96859	+	3.92922i	6.22741	+	6.49764i	−6.33448	−	0.575876i
53.8	−1.96771	+	0.357931i	2.98510	+	0.298667i	3.74377	−	1.40861i	−5.14621	−	1.37892i	−5.98071	+	0.480768i	−3.41403	+	6.11100i	−6.86247	+	4.11175i	8.82160	+	1.78310i	10.6198	+	0.871332i
53.9	−1.94659	−	0.459134i	−2.13606	+	2.10647i	3.57839	+	1.78749i	1.79413	+	0.480737i	5.12518	−	3.11969i	−5.96381	+	3.66510i	−6.14495	−	5.12246i	0.125537	−	8.99912i	−3.27171	−	1.75954i
53.10	−1.93667	+	0.499290i	0.901194	+	2.86144i	3.50142	−	1.93392i	9.12344	+	2.44462i	−3.17401	−	5.09173i	−3.15696	+	6.24769i	−5.81552	+	5.49361i	−7.37570	+	5.15743i	−18.8897	−	0.179189i
53.11	−1.93499	−	0.505766i	1.25182	+	2.72634i	3.48840	+	1.95731i	−1.43566	−	0.384683i	−1.04337	−	5.90859i	5.73373	+	4.01551i	−5.76010	−	5.55169i	−5.86590	+	6.82577i	2.58342	+	1.47046i
53.12	−1.92918	−	0.527509i	2.36831	−	1.84150i	3.44347	+	2.03532i	7.86569	+	2.10760i	−5.54030	+	2.30328i	5.42076	−	4.42892i	−5.56942	−	5.74296i	2.21775	−	8.72248i	−14.0625	−	8.21517i
53.13	−1.91689	−	0.570540i	1.50400	−	2.59576i	3.34897	+	2.18733i	1.94438	+	0.520994i	−4.36400	+	4.11771i	−6.97036	+	0.643458i	−5.17166	−	6.10360i	−4.47595	−	7.80806i	−3.42992	−	2.10804i
53.14	−1.86745	−	0.715987i	−2.91595	−	0.705131i	2.97472	+	2.67414i	−2.28226	−	0.611530i	4.94053	+	3.40458i	−3.20004	−	6.22573i	−3.64049	−	7.12368i	8.00558	+	4.11226i	3.82416	+	2.77607i
53.15	−1.86351	+	0.726181i	−2.36295	−	1.84837i	2.94532	−	2.70649i	−4.87656	−	1.30667i	5.74562	+	1.72853i	−6.82723	+	1.54563i	−3.52323	+	7.18240i	2.16705	+	8.73521i	10.0364	−	1.10627i
53.16	−1.83070	−	0.805328i	2.91419	−	0.712399i	2.70289	+	2.94862i	−7.15007	−	1.91585i	−5.90870	−	1.04269i	5.33882	−	4.52737i	−2.57357	−	7.57474i	7.98498	−	4.15213i	11.5467	+	9.26550i
53.17	−1.82635	+	0.815144i	−0.00566031	+	2.99999i	2.67108	−	2.97747i	1.79821	+	0.481829i	−2.43509	−	5.48364i	5.32471	−	4.54394i	−2.45125	+	7.61521i	−8.99994	−	0.0339618i	−3.67691	+	0.585814i
53.18	−1.79723	+	0.877478i	−2.96627	+	0.448570i	2.46007	−	3.15406i	−2.29000	−	0.613604i	4.93747	−	3.40902i	0.702247	+	6.96469i	−1.65368	+	7.82722i	8.59757	−	2.66116i	4.65408	−	0.906639i
53.19	−1.79207	+	0.887970i	1.19622	−	2.75119i	2.42302	−	3.18261i	−4.99673	−	1.33887i	0.299267	+	5.99253i	0.706769	−	6.96423i	−1.51616	+	7.85502i	−6.13811	−	6.58206i	10.1434	−	2.03760i
53.20	−1.76874	−	0.933573i	2.94473	+	0.573211i	2.25688	+	3.30250i	5.74497	+	1.53936i	−4.67333	−	3.76298i	−5.41596	+	4.43479i	−0.908716	−	7.94822i	8.34286	+	3.37590i	−8.72426	−	8.08608i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
16.e	even	4	1	inner
21.h	odd	6	1	inner
48.i	odd	4	1	inner
112.w	even	12	1	inner
336.bt	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.3.bt.a	✓	496
3.b	odd	2	1	inner	336.3.bt.a	✓	496
7.c	even	3	1	inner	336.3.bt.a	✓	496
16.e	even	4	1	inner	336.3.bt.a	✓	496
21.h	odd	6	1	inner	336.3.bt.a	✓	496
48.i	odd	4	1	inner	336.3.bt.a	✓	496
112.w	even	12	1	inner	336.3.bt.a	✓	496
336.bt	odd	12	1	inner	336.3.bt.a	✓	496

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.3.bt.a	✓	496	1.a	even	1	1	trivial
336.3.bt.a	✓	496	3.b	odd	2	1	inner
336.3.bt.a	✓	496	7.c	even	3	1	inner
336.3.bt.a	✓	496	16.e	even	4	1	inner
336.3.bt.a	✓	496	21.h	odd	6	1	inner
336.3.bt.a	✓	496	48.i	odd	4	1	inner
336.3.bt.a	✓	496	112.w	even	12	1	inner
336.3.bt.a	✓	496	336.bt	odd	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(336, [\chi])\).