Newform orbit 336.3.t.a

• Label: 336.3.t.a
• Level: $336$
• Weight: $3$
• Character orbit: 336.t
• Analytic conductor: $9.155$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.15533688251\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(96\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 12 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−2.00000	+	0.00312796i	1.78471	−	2.41139i	3.99998	−	0.0125118i	2.26094	+	2.26094i	−3.56188	+	4.82836i			2.64575i	−7.99991	+	0.0375353i	−2.62960	−	8.60728i	−4.52895	−	4.51481i
29.2	−1.99663	−	0.116080i	2.85963	−	0.906930i	3.97305	+	0.463536i	6.89997	+	6.89997i	−5.81489	+	1.47886i		−	2.64575i	−7.87890	−	1.38670i	7.35496	−	5.18697i	−12.9757	−	14.5776i
29.3	−1.99534	+	0.136487i	−1.54841	−	2.56952i	3.96274	−	0.544675i	−3.45764	−	3.45764i	3.44031	+	4.91572i			2.64575i	−7.83267	+	1.62767i	−4.20485	+	7.95734i	7.37107	+	6.42723i
29.4	−1.99499	−	0.141418i	2.99222	−	0.215927i	3.96000	+	0.564255i	−5.69938	−	5.69938i	−6.00000	+	0.00761992i			2.64575i	−7.82038	−	1.68570i	8.90675	−	1.29220i	10.5642	+	12.1762i
29.5	−1.99097	+	0.189870i	1.19591	+	2.75133i	3.92790	−	0.756052i	−4.68892	−	4.68892i	−2.90341	−	5.25073i		−	2.64575i	−7.67676	+	2.25107i	−6.13960	+	6.58068i	10.2258	+	8.44519i
29.6	−1.97640	−	0.306366i	−0.271501	+	2.98769i	3.81228	+	1.21100i	3.10851	+	3.10851i	1.45192	−	5.82168i		−	2.64575i	−7.16356	−	3.56137i	−8.85257	−	1.62232i	−5.19131	−	7.09599i
29.7	−1.94829	+	0.451834i	−2.66782	+	1.37213i	3.59169	−	1.76061i	−4.56547	−	4.56547i	4.57772	−	3.87872i			2.64575i	−6.20217	+	5.05303i	5.23454	−	7.32118i	10.9577	+	6.83204i
29.8	−1.94130	+	0.480993i	−1.06131	+	2.80600i	3.53729	−	1.86750i	4.09211	+	4.09211i	0.710650	−	5.95777i			2.64575i	−5.96869	+	5.32679i	−6.74726	−	5.95605i	−9.91228	−	5.97574i
29.9	−1.93826	+	0.493099i	−0.0439263	−	2.99968i	3.51371	−	1.91151i	0.736310	+	0.736310i	1.56428	+	5.79250i		−	2.64575i	−5.86791	+	5.43761i	−8.99614	+	0.263529i	−1.79023	−	1.06409i
29.10	−1.92491	+	0.542872i	2.70917	+	1.28857i	3.41058	−	2.08996i	0.0579151	+	0.0579151i	−5.91444	−	1.00965i		−	2.64575i	−5.43049	+	5.87450i	5.67918	+	6.98190i	−0.142922	−	0.0800410i
29.11	−1.90710	−	0.602477i	2.17141	+	2.07002i	3.27404	+	2.29796i	1.46144	+	1.46144i	−2.89395	−	5.25595i			2.64575i	−4.85945	−	6.35498i	0.430048	+	8.98972i	−1.90662	−	3.66759i
29.12	−1.89862	−	0.628676i	−2.43165	+	1.75700i	3.20953	+	2.38724i	−2.93392	−	2.93392i	5.72138	−	1.80716i		−	2.64575i	−4.59289	−	6.55022i	2.82587	−	8.54485i	3.72592	+	7.41489i
29.13	−1.84284	+	0.777147i	−2.92540	−	0.664856i	2.79208	−	2.86431i	−0.526176	−	0.526176i	5.90772	−	1.04825i		−	2.64575i	−2.91936	+	7.44831i	8.11593	+	3.88994i	1.37857	+	0.560740i
29.14	−1.83112	−	0.804365i	−2.41255	−	1.78315i	2.70599	+	2.94578i	−5.05435	−	5.05435i	2.98336	+	5.20572i		−	2.64575i	−2.58552	−	7.57067i	2.64079	+	8.60385i	5.18958	+	13.3207i
29.15	−1.70626	+	1.04340i	−2.61808	+	1.46480i	1.82265	−	3.56061i	3.61830	+	3.61830i	2.93877	−	5.23102i		−	2.64575i	0.605206	+	7.97708i	4.70874	−	7.66993i	−9.94909	−	2.39845i
29.16	−1.68609	+	1.07569i	2.18089	−	2.06002i	1.68577	−	3.62742i	−6.39300	−	6.39300i	−1.46122	+	5.81935i		−	2.64575i	1.05964	+	7.92951i	0.512600	−	8.98539i	17.6561	+	3.90224i
29.17	−1.66368	−	1.11003i	−0.699683	−	2.91727i	1.53568	+	3.69347i	5.11623	+	5.11623i	−2.07419	+	5.63007i		−	2.64575i	1.54496	−	7.84940i	−8.02089	+	4.08232i	−2.83263	−	14.1909i
29.18	−1.66016	−	1.11528i	2.98465	−	0.303046i	1.51229	+	3.70310i	−1.55904	−	1.55904i	−5.29300	−	2.82562i		−	2.64575i	1.61934	−	7.83439i	8.81633	−	1.80898i	0.849497	+	4.32705i
29.19	−1.64280	+	1.14070i	−1.84901	−	2.36245i	1.39759	−	3.74790i	6.40343	+	6.40343i	5.73241	+	1.77186i			2.64575i	1.97928	+	7.75129i	−2.16233	+	8.73638i	−17.8240	−	3.21514i
29.20	−1.64232	−	1.14140i	−1.05966	−	2.80662i	1.39443	+	3.74907i	2.24404	+	2.24404i	−1.46316	+	5.81886i			2.64575i	1.98908	−	7.74878i	−6.75422	+	5.94815i	−1.12409	−	6.24677i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
16.e	even	4	1	inner
48.i	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.t.a	✓	192
3.b	odd	2	1	inner	336.3.t.a	✓	192
16.e	even	4	1	inner	336.3.t.a	✓	192
48.i	odd	4	1	inner	336.3.t.a	✓	192

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.3.t.a	✓	192	1.a	even	1	1	trivial
336.3.t.a	✓	192	3.b	odd	2	1	inner
336.3.t.a	✓	192	16.e	even	4	1	inner
336.3.t.a	✓	192	48.i	odd	4	1	inner

Hecke kernels

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