Newform orbit 336.3.v.a

• Label: 336.3.v.a
• Level: $336$
• Weight: $3$
• Character orbit: 336.v
• Analytic conductor: $9.155$
• Analytic rank: $0$
• Dimension: $248$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.15533688251\)
Analytic rank:	\(0\)
Dimension:	\(248\)
Relative dimension:	\(124\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 248 q - 8 q^{4} - 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
83.1	−1.99997	−	0.0109030i	−2.26001	+	1.97290i	3.99976	+	0.0436113i	2.64420	−	2.64420i	4.54147	−	3.92110i	−0.333254	+	6.99206i	−7.99893	−	0.130831i	1.21533	−	8.91757i	−5.31715	+	5.25949i
83.2	−1.99997	−	0.0109030i	2.26001	−	1.97290i	3.99976	+	0.0436113i	−2.64420	+	2.64420i	−4.54147	+	3.92110i	−0.333254	−	6.99206i	−7.99893	−	0.130831i	1.21533	−	8.91757i	5.31715	−	5.25949i
83.3	−1.99225	−	0.175893i	−2.28007	−	1.94969i	3.93812	+	0.700844i	2.66511	−	2.66511i	4.19952	+	4.28532i	2.25515	+	6.62679i	−7.72245	−	2.08894i	1.39739	+	8.89085i	−5.77834	+	4.84080i
83.4	−1.99225	−	0.175893i	2.28007	+	1.94969i	3.93812	+	0.700844i	−2.66511	+	2.66511i	−4.19952	−	4.28532i	2.25515	−	6.62679i	−7.72245	−	2.08894i	1.39739	+	8.89085i	5.77834	−	4.84080i
83.5	−1.98114	+	0.274017i	−2.69090	+	1.32629i	3.84983	−	1.08573i	−3.06068	+	3.06068i	4.96762	−	3.36492i	−5.88126	−	3.79615i	−7.32954	+	3.20591i	5.48190	−	7.13784i	5.22495	−	6.90230i
83.6	−1.98114	+	0.274017i	2.69090	−	1.32629i	3.84983	−	1.08573i	3.06068	−	3.06068i	−4.96762	+	3.36492i	−5.88126	+	3.79615i	−7.32954	+	3.20591i	5.48190	−	7.13784i	−5.22495	+	6.90230i
83.7	−1.97661	+	0.304950i	−0.249596	+	2.98960i	3.81401	−	1.20554i	−4.01887	+	4.01887i	−0.418325	−	5.98540i	0.723024	+	6.96256i	−7.17120	+	3.54597i	−8.87540	−	1.49238i	6.71820	−	9.16931i
83.8	−1.97661	+	0.304950i	0.249596	−	2.98960i	3.81401	−	1.20554i	4.01887	−	4.01887i	0.418325	+	5.98540i	0.723024	−	6.96256i	−7.17120	+	3.54597i	−8.87540	−	1.49238i	−6.71820	+	9.16931i
83.9	−1.96115	−	0.392291i	−2.12005	−	2.12259i	3.69222	+	1.53868i	−3.99269	+	3.99269i	3.32507	+	4.99438i	−6.99574	+	0.244101i	−6.63738	−	4.46600i	−0.0107348	+	8.99999i	9.39655	−	6.26397i
83.10	−1.96115	−	0.392291i	2.12005	+	2.12259i	3.69222	+	1.53868i	3.99269	−	3.99269i	−3.32507	−	4.99438i	−6.99574	−	0.244101i	−6.63738	−	4.46600i	−0.0107348	+	8.99999i	−9.39655	+	6.26397i
83.11	−1.90536	+	0.607935i	−1.43686	−	2.63352i	3.26083	−	2.31668i	−6.39629	+	6.39629i	4.33875	+	4.14431i	6.65514	−	2.17006i	−4.80468	+	6.39649i	−4.87089	+	7.56799i	8.29874	−	16.0758i
83.12	−1.90536	+	0.607935i	1.43686	+	2.63352i	3.26083	−	2.31668i	6.39629	−	6.39629i	−4.33875	−	4.14431i	6.65514	+	2.17006i	−4.80468	+	6.39649i	−4.87089	+	7.56799i	−8.29874	+	16.0758i
83.13	−1.88235	−	0.675832i	−0.947745	+	2.84636i	3.08650	+	2.54431i	1.77344	−	1.77344i	3.70765	−	4.71734i	3.81994	−	5.86584i	−4.09036	−	6.87524i	−7.20356	−	5.39525i	−4.53678	+	2.13969i
83.14	−1.88235	−	0.675832i	0.947745	−	2.84636i	3.08650	+	2.54431i	−1.77344	+	1.77344i	−3.70765	+	4.71734i	3.81994	+	5.86584i	−4.09036	−	6.87524i	−7.20356	−	5.39525i	4.53678	−	2.13969i
83.15	−1.85057	−	0.758553i	−2.95969	−	0.490165i	2.84919	+	2.80751i	5.98359	−	5.98359i	5.10528	+	3.15216i	−3.84316	−	5.85065i	−3.14298	−	7.35674i	8.51948	+	2.90147i	−15.6119	+	6.53417i
83.16	−1.85057	−	0.758553i	2.95969	+	0.490165i	2.84919	+	2.80751i	−5.98359	+	5.98359i	−5.10528	−	3.15216i	−3.84316	+	5.85065i	−3.14298	−	7.35674i	8.51948	+	2.90147i	15.6119	−	6.53417i
83.17	−1.84203	+	0.779062i	−2.98965	−	0.249035i	2.78613	−	2.87011i	1.75425	−	1.75425i	5.70102	−	1.87039i	6.21074	−	3.22904i	−2.89613	+	7.45737i	8.87596	+	1.48905i	−1.86470	+	4.59803i
83.18	−1.84203	+	0.779062i	2.98965	+	0.249035i	2.78613	−	2.87011i	−1.75425	+	1.75425i	−5.70102	+	1.87039i	6.21074	+	3.22904i	−2.89613	+	7.45737i	8.87596	+	1.48905i	1.86470	−	4.59803i
83.19	−1.76319	−	0.944009i	−2.90906	+	0.733040i	2.21769	+	3.32894i	−4.17594	+	4.17594i	5.82124	+	1.45369i	6.96301	+	0.718664i	−0.767673	−	7.96308i	7.92530	−	4.26492i	11.3051	−	3.42086i
83.20	−1.76319	−	0.944009i	2.90906	−	0.733040i	2.21769	+	3.32894i	4.17594	−	4.17594i	−5.82124	−	1.45369i	6.96301	−	0.718664i	−0.767673	−	7.96308i	7.92530	−	4.26492i	−11.3051	+	3.42086i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
16.f	odd	4	1	inner
21.c	even	2	1	inner
48.k	even	4	1	inner
112.j	even	4	1	inner
336.v	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.3.v.a	✓	248
3.b	odd	2	1	inner	336.3.v.a	✓	248
7.b	odd	2	1	inner	336.3.v.a	✓	248
16.f	odd	4	1	inner	336.3.v.a	✓	248
21.c	even	2	1	inner	336.3.v.a	✓	248
48.k	even	4	1	inner	336.3.v.a	✓	248
112.j	even	4	1	inner	336.3.v.a	✓	248
336.v	odd	4	1	inner	336.3.v.a	✓	248

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.3.v.a	✓	248	1.a	even	1	1	trivial
336.3.v.a	✓	248	3.b	odd	2	1	inner
336.3.v.a	✓	248	7.b	odd	2	1	inner
336.3.v.a	✓	248	16.f	odd	4	1	inner
336.3.v.a	✓	248	21.c	even	2	1	inner
336.3.v.a	✓	248	48.k	even	4	1	inner
336.3.v.a	✓	248	112.j	even	4	1	inner
336.3.v.a	✓	248	336.v	odd	4	1	inner

Hecke kernels

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