Newform orbit 349.2.e.a

• Label: 349.2.e.a
• Level: $349$
• Weight: $2$
• Character orbit: 349.e
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $58$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(58\)
Relative dimension:	\(29\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 58 q - 3 q^{2} + 27 q^{4} + 2 q^{5} - 29 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
123.1	−2.39209	+	1.38107i	−0.0982202	+	0.170122i	2.81473	−	4.87525i	1.45632	−	2.52242i		−	0.542597i	−2.31421	+	1.33611i			10.0251i	1.48071	+	2.56466i			8.04515i
123.2	−2.28298	+	1.31808i	−1.40419	+	2.43213i	2.47467	−	4.28626i	0.108819	−	0.188480i		−	7.40335i	3.00446	−	1.73462i			7.77496i	−2.44351	−	4.23228i			0.573727i
123.3	−2.10033	+	1.21263i	0.180949	−	0.313412i	1.94093	−	3.36178i	−1.51394	+	2.62221i			0.877693i	0.636711	−	0.367605i			4.56397i	1.43452	+	2.48465i		−	7.34336i
123.4	−2.00748	+	1.15902i	1.04619	−	1.81206i	1.68665	−	2.92136i	−0.198863	+	0.344441i			4.85022i	−2.47763	+	1.43046i			3.18335i	−0.689035	−	1.19344i		−	0.921944i
123.5	−1.85370	+	1.07024i	1.18706	−	2.05605i	1.29081	−	2.23575i	1.34675	−	2.33263i			5.08174i	2.87980	−	1.66266i			1.24494i	−1.31823	−	2.28323i			5.76535i
123.6	−1.71754	+	0.991624i	−1.29117	+	2.23637i	0.966636	−	1.67426i	−0.796466	+	1.37952i		−	5.12141i	−1.81826	+	1.04977i		−	0.132337i	−1.83422	−	3.17696i		−	3.15918i
123.7	−1.34862	+	0.778626i	1.56432	−	2.70948i	0.212518	−	0.368092i	−1.71886	+	2.97716i			4.87208i	0.0378395	−	0.0218466i		−	2.45262i	−3.39418	−	5.87890i		−	5.35340i
123.8	−1.33595	+	0.771310i	−0.677977	+	1.17429i	0.189837	−	0.328808i	1.91694	−	3.32023i		−	2.09172i	−1.98739	+	1.14742i		−	2.49955i	0.580694	+	1.00579i			5.91420i
123.9	−1.06897	+	0.617172i	0.625735	−	1.08381i	−0.238198	+	0.412572i	0.733047	−	1.26968i			1.54474i	−0.833145	+	0.481016i		−	3.05672i	0.716910	+	1.24172i			1.80966i
123.10	−1.03662	+	0.598491i	0.00513702	−	0.00889758i	−0.283617	+	0.491238i	−0.193021	+	0.334323i			0.0122978i	3.23720	−	1.86900i		−	3.07293i	1.49995	+	2.59798i		−	0.462086i
123.11	−0.941972	+	0.543848i	−0.443096	+	0.767465i	−0.408460	+	0.707473i	−1.75571	+	3.04098i		−	0.963907i	2.73929	−	1.58153i		−	3.06395i	1.10733	+	1.91796i		−	3.81936i
123.12	−0.788972	+	0.455513i	−1.46240	+	2.53295i	−0.585016	+	1.01328i	1.45267	−	2.51610i		−	2.66457i	3.29342	−	1.90146i		−	2.88798i	−2.77724	−	4.81032i			2.64684i
123.13	−0.529073	+	0.305460i	0.484492	−	0.839165i	−0.813388	+	1.40883i	−0.0329412	+	0.0570558i			0.591972i	−3.18284	+	1.83761i		−	2.21567i	1.03054	+	1.78494i		−	0.0402489i
123.14	−0.327556	+	0.189114i	−1.06274	+	1.84073i	−0.928472	+	1.60816i	−1.07891	+	1.86873i		−	0.803920i	−2.19494	+	1.26725i		−	1.45881i	−0.758847	−	1.31436i		−	0.816152i
123.15	−0.0377450	+	0.0217921i	1.71622	−	2.97259i	−0.999050	+	1.73041i	1.73771	−	3.00980i			0.149601i	−2.79927	+	1.61616i		−	0.174254i	−4.39085	−	7.60517i			0.151473i
123.16	0.228167	−	0.131732i	0.150552	−	0.260763i	−0.965293	+	1.67194i	0.807178	−	1.39807i		−	0.0793300i	1.23534	−	0.713223i			1.03557i	1.45467	+	2.51956i		−	0.425325i
123.17	0.499056	−	0.288130i	−0.844268	+	1.46231i	−0.833962	+	1.44447i	0.707762	−	1.22588i			0.973035i	1.68926	−	0.975293i			2.11368i	0.0744233	+	0.128905i		−	0.815710i
123.18	0.531309	−	0.306751i	0.869046	−	1.50523i	−0.811807	+	1.40609i	−1.66390	+	2.88196i		−	1.06632i	−1.62203	+	0.936479i			2.22310i	−0.0104823	−	0.0181558i			2.04162i
123.19	0.544820	−	0.314552i	1.24137	−	2.15012i	−0.802114	+	1.38930i	−0.0260007	+	0.0450346i		−	1.56191i	3.86429	−	2.23105i			2.26743i	−1.58202	−	2.74014i			0.0327144i
123.20	0.831968	−	0.480337i	−1.62465	+	2.81397i	−0.538552	+	0.932800i	−1.69786	+	2.94079i			3.12152i	3.20576	−	1.85085i			2.95610i	−3.77897	−	6.54537i			3.26219i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
349.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	349.2.e.a	✓	58
349.e	even	6	1	inner	349.2.e.a	✓	58

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
349.2.e.a	✓	58	1.a	even	1	1	trivial
349.2.e.a	✓	58	349.e	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(349, [\chi])\).