Newform orbit 297.2.u.a

• Label: 297.2.u.a
• Level: $297$
• Weight: $2$
• Character orbit: 297.u
• Analytic conductor: $2.372$
• Analytic rank: $0$
• Dimension: $816$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.u (of order \(45\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.37155694003\)
Analytic rank:	\(0\)
Dimension:	\(816\)
Relative dimension:	\(34\) over \(\Q(\zeta_{45})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{45}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 816 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 21 q^{5} - 18 q^{6} - 18 q^{7} - 27 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−1.52226	−	2.25685i	1.71918	+	0.210751i	−2.02687	+	5.01668i	−0.910054	−	1.86589i	−2.14142	−	4.20075i	3.47442	+	2.17106i	9.08179	−	1.93039i	2.91117	+	0.724637i	−2.82568	+	4.89423i
4.2	−1.49575	−	2.21755i	−0.509134	−	1.65553i	−1.93102	+	4.77944i	−1.22867	−	2.51915i	−2.90968	+	3.60530i	−2.95455	−	1.84621i	8.25417	−	1.75448i	−2.48156	+	1.68578i	−3.74855	+	6.49267i
4.3	−1.47131	−	2.18131i	−1.61978	+	0.613455i	−1.84413	+	4.56439i	0.816983	+	1.67506i	3.72132	+	2.63064i	−1.89577	−	1.18461i	7.52234	−	1.59892i	2.24735	−	1.98732i	2.45179	−	4.24662i
4.4	−1.30690	−	1.93756i	1.16918	+	1.27789i	−1.29694	+	3.21003i	1.40384	+	2.87830i	0.947989	−	3.93544i	−2.43944	−	1.52433i	3.34249	−	0.710468i	−0.266023	+	2.98818i	3.74220	−	6.48168i
4.5	−1.29432	−	1.91891i	0.781385	−	1.54578i	−1.25774	+	3.11301i	1.66518	+	3.41413i	−3.97758	+	0.501330i	2.16812	+	1.35479i	3.07342	−	0.653275i	−1.77888	−	2.41570i	4.39613	−	7.61432i
4.6	−1.15511	−	1.71252i	−1.43407	−	0.971305i	−0.849227	+	2.10191i	0.437533	+	0.897075i	−0.00686860	+	3.57783i	2.15510	+	1.34666i	0.539447	−	0.114663i	1.11313	+	2.78585i	1.03086	−	1.78550i
4.7	−1.12919	−	1.67409i	−1.52126	+	0.828104i	−0.778300	+	1.92636i	−1.72560	−	3.53800i	3.10411	+	1.61165i	1.19288	+	0.745391i	0.153364	−	0.0325984i	1.62849	−	2.51953i	−3.97441	+	6.88387i
4.8	−1.10625	−	1.64008i	0.560981	+	1.63869i	−0.716868	+	1.77431i	−0.966959	−	1.98256i	2.06700	−	2.73285i	−1.37805	−	0.861102i	−0.167084	+	0.0355149i	−2.37060	+	1.83855i	−2.18186	+	3.77910i
4.9	−0.950255	−	1.40881i	1.68370	−	0.406384i	−0.332550	+	0.823090i	−0.690850	−	1.41645i	−2.17246	−	1.98585i	−1.36732	−	0.854399i	−1.84881	+	0.392977i	2.66970	−	1.36846i	−1.33903	+	2.31927i
4.10	−0.839527	−	1.24465i	−0.997785	+	1.41578i	−0.0951349	+	0.235467i	0.419820	+	0.860758i	2.59981	+	0.0533092i	−0.462419	−	0.288952i	−2.56408	+	0.545012i	−1.00885	−	2.82528i	0.718893	−	1.24516i
4.11	−0.708837	−	1.05089i	−1.17988	−	1.26803i	0.147285	−	0.364543i	0.962748	+	1.97393i	−0.496220	+	2.13875i	−2.09512	−	1.30917i	−2.96731	+	0.630721i	−0.215780	+	2.99223i	1.39196	−	2.41094i
4.12	−0.648656	−	0.961673i	0.277433	−	1.70969i	0.245154	−	0.606777i	−0.349371	−	0.716317i	−1.82412	+	0.842200i	−1.97338	−	1.23310i	−3.01182	+	0.640182i	−2.84606	−	0.948648i	−0.462241	+	0.800624i
4.13	−0.646233	−	0.958080i	0.502071	+	1.65769i	0.248913	−	0.616081i	0.477156	+	0.978315i	1.26374	−	1.55228i	4.03264	+	2.51987i	−3.01191	+	0.640201i	−2.49585	+	1.66455i	0.628950	−	1.08937i
4.14	−0.550495	−	0.816142i	0.913366	−	1.47165i	0.386170	−	0.955803i	−1.20860	−	2.47799i	−1.70388	+	0.0647011i	4.06473	+	2.53993i	−2.91852	+	0.620351i	−1.33153	−	2.68832i	−1.35707	+	2.35051i
4.15	−0.404130	−	0.599148i	1.70916	−	0.280688i	0.553556	−	1.37010i	1.50295	+	3.08151i	−0.858895	−	0.910603i	−0.0855010	−	0.0534269i	−2.45842	+	0.522554i	2.84243	−	0.959479i	1.23889	−	2.14583i
4.16	−0.279995	−	0.415110i	−1.72620	+	0.142227i	0.655294	−	1.62191i	0.222884	+	0.456980i	0.542368	+	0.676741i	2.31953	+	1.44940i	−1.83629	+	0.390316i	2.95954	−	0.491023i	0.127290	−	0.220473i
4.17	−0.145184	−	0.215244i	−0.559375	+	1.63924i	0.723962	−	1.79187i	−0.336481	−	0.689889i	0.434048	−	0.117589i	−3.97018	−	2.48084i	−0.998710	+	0.212282i	−2.37420	−	1.83390i	−0.0996426	+	0.172586i
4.18	0.0281155	+	0.0416829i	1.63951	+	0.558582i	0.748266	−	1.85202i	−1.17408	−	2.40721i	0.0228122	+	0.0840443i	−2.07448	−	1.29628i	0.196596	−	0.0417877i	2.37597	+	1.83160i	0.0673300	−	0.116619i
4.19	0.0343340	+	0.0509022i	1.36049	+	1.07194i	0.747801	−	1.85087i	0.192374	+	0.394426i	−0.00785293	+	0.106056i	0.816264	+	0.510058i	0.240003	−	0.0510143i	0.701890	+	2.91674i	−0.0134722	+	0.0233345i
4.20	0.109922	+	0.162966i	−1.04504	−	1.38127i	0.734738	−	1.81854i	−1.33182	−	2.73063i	0.110227	−	0.322137i	0.804741	+	0.502858i	0.761679	−	0.161900i	−0.815795	+	2.88695i	0.298604	−	0.517198i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner
27.e	even	9	1	inner
297.u	even	45	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	297.2.u.a	✓	816
3.b	odd	2	1		891.2.v.a		816
11.c	even	5	1	inner	297.2.u.a	✓	816
27.e	even	9	1	inner	297.2.u.a	✓	816
27.f	odd	18	1		891.2.v.a		816
33.h	odd	10	1		891.2.v.a		816
297.u	even	45	1	inner	297.2.u.a	✓	816
297.v	odd	90	1		891.2.v.a		816

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
297.2.u.a	✓	816	1.a	even	1	1	trivial
297.2.u.a	✓	816	11.c	even	5	1	inner
297.2.u.a	✓	816	27.e	even	9	1	inner
297.2.u.a	✓	816	297.u	even	45	1	inner
891.2.v.a		816	3.b	odd	2	1
891.2.v.a		816	27.f	odd	18	1
891.2.v.a		816	33.h	odd	10	1
891.2.v.a		816	297.v	odd	90	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(297, [\chi])\).