Newform orbit 261.3.w.a

• Label: 261.3.w.a
• Level: $261$
• Weight: $3$
• Character orbit: 261.w
• Analytic conductor: $7.112$
• Analytic rank: $0$
• Dimension: $1392$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	261.w (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.11173489980\)
Analytic rank:	\(0\)
Dimension:	\(1392\)
Relative dimension:	\(58\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1392 q - 12 q^{2} - 24 q^{3} - 14 q^{4} - 14 q^{5} - 28 q^{6} - 10 q^{7} - 68 q^{8} - 28 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1	−1.56108	−	3.57803i	−1.74847	−	2.43780i	−7.64466	+	8.23899i	0.312977	+	2.07647i	−5.99301	+	10.0617i	8.04531	−	7.46496i	26.6745	+	9.33383i	−2.88570	+	8.52483i	6.94109	−	4.36138i
31.2	−1.53431	−	3.51667i	1.67752	−	2.48715i	−7.29215	+	7.85907i	−1.14237	−	7.57916i	−11.3203	−	2.08321i	−3.75526	+	3.48437i	24.3402	+	8.51699i	−3.37187	−	8.34449i	−24.9006	+	15.6461i
31.3	−1.48859	−	3.41188i	0.414307	+	2.97125i	−6.70435	+	7.22557i	1.12508	+	7.46444i	9.52083	−	5.83654i	1.45436	−	1.34945i	20.5785	+	7.20071i	−8.65670	+	2.46202i	23.7930	−	14.9501i
31.4	−1.43255	−	3.28344i	−0.918134	+	2.85605i	−6.00806	+	6.47515i	−0.755238	−	5.01068i	10.6929	−	1.07680i	−2.80786	+	2.60531i	16.3423	+	5.71844i	−7.31406	−	5.24447i	−15.3703	+	9.65781i
31.5	−1.41410	−	3.24114i	−2.99195	+	0.219643i	−5.78466	+	6.23438i	0.481468	+	3.19433i	4.94280	+	9.38674i	−6.11382	+	5.67279i	15.0356	+	5.26117i	8.90351	−	1.31432i	9.67245	−	6.07760i
31.6	−1.34412	−	3.08077i	2.96843	+	0.434062i	−4.96376	+	5.34966i	0.289924	+	1.92352i	−2.65270	−	9.72848i	4.23914	−	3.93334i	10.4626	+	3.66101i	8.62318	+	2.57697i	5.53622	−	3.47864i
31.7	−1.32719	−	3.04195i	2.76364	+	1.16717i	−4.77136	+	5.14230i	−0.180576	−	1.19805i	−0.117394	−	9.95593i	−8.80397	+	8.16889i	9.44464	+	3.30482i	6.27542	+	6.45129i	−3.40474	+	2.13934i
31.8	−1.29935	−	2.97814i	−2.84231	+	0.959839i	−4.46030	+	4.80706i	−1.02232	−	6.78267i	6.55168	+	7.21761i	8.85390	−	8.21522i	7.84392	+	2.74471i	7.15742	−	5.45632i	−18.8714	+	11.8577i
31.9	−1.24196	−	2.84660i	2.23707	−	1.99888i	−3.84000	+	4.13854i	0.331363	+	2.19845i	−8.46838	−	3.88550i	3.75402	−	3.48322i	4.82408	+	1.68802i	1.00892	−	8.94327i	5.84657	−	3.67365i
31.10	−1.19316	−	2.73476i	0.648455	−	2.92908i	−3.33457	+	3.59382i	1.11912	+	7.42488i	−8.78404	+	1.72150i	−4.76061	+	4.41720i	2.54180	+	0.889415i	−8.15901	−	3.79875i	18.9700	−	11.9196i
31.11	−1.12031	−	2.56777i	−2.05976	−	2.18115i	−2.61767	+	2.82118i	−0.469195	−	3.11291i	−3.29313	+	7.73254i	−1.32324	+	1.22779i	−0.400509	−	0.140144i	−0.514810	+	8.98526i	−7.46759	+	4.69220i
31.12	−1.02524	−	2.34987i	−1.77513	+	2.41845i	−1.75008	+	1.88614i	0.0658336	+	0.436777i	7.50297	+	1.69184i	1.35764	−	1.25970i	−3.45323	−	1.20834i	−2.69781	−	8.58614i	0.958874	−	0.602500i
31.13	−1.02162	−	2.34158i	−0.677718	−	2.92245i	−1.71860	+	1.85221i	−1.16761	−	7.74658i	−6.15077	+	4.57256i	−1.91261	+	1.77465i	−3.55267	−	1.24313i	−8.08140	+	3.96119i	−16.9464	+	10.6481i
31.14	−0.885481	−	2.02954i	1.32550	+	2.69129i	−0.614278	+	0.662034i	−0.896954	−	5.95090i	4.28838	−	5.07325i	−1.09052	+	1.01186i	−6.47260	−	2.26486i	−5.48609	+	7.13462i	−11.2834	+	7.08981i
31.15	−0.872188	−	1.99908i	0.850162	+	2.87702i	−0.514904	+	0.554935i	0.509498	+	3.38030i	5.00988	−	4.20884i	7.63190	−	7.08137i	−6.67621	−	2.33611i	−7.55445	+	4.89186i	6.31309	−	3.96678i
31.16	−0.850509	−	1.94939i	2.97461	−	0.389511i	−0.356053	+	0.383734i	−1.24438	−	8.25593i	−3.28924	−	5.46738i	6.34642	−	5.88862i	−6.97911	−	2.44209i	8.69656	−	2.31728i	−15.0356	+	9.44752i
31.17	−0.822012	−	1.88407i	−2.91781	−	0.697420i	−0.153334	+	0.165255i	0.404187	+	2.68161i	1.08449	+	6.07065i	−2.46459	+	2.28680i	−7.32354	−	2.56262i	8.02721	+	4.06987i	4.72010	−	2.96583i
31.18	−0.788432	−	1.80710i	2.53797	+	1.59960i	0.0766886	−	0.0826507i	1.13728	+	7.54539i	0.889638	−	5.84755i	−4.86607	+	4.51506i	−7.65371	−	2.67815i	3.88254	+	8.11948i	12.7386	−	8.00422i
31.19	−0.743209	−	1.70345i	−2.06848	−	2.17287i	0.371300	−	0.400166i	1.36851	+	9.07947i	−2.16408	+	5.13846i	8.35295	−	7.75040i	−7.97453	−	2.79041i	−0.442771	+	8.98910i	14.4494	−	9.07913i
31.20	−0.611217	−	1.40092i	−1.50155	+	2.59718i	1.13169	−	1.21967i	1.02064	+	6.77151i	4.55623	+	0.516125i	−6.73495	+	6.24912i	−8.17111	−	2.85919i	−4.49067	−	7.79961i	8.86254	−	5.56870i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
29.f	odd	28	1	inner
261.w	odd	84	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	261.3.w.a	✓	1392
9.c	even	3	1	inner	261.3.w.a	✓	1392
29.f	odd	28	1	inner	261.3.w.a	✓	1392
261.w	odd	84	1	inner	261.3.w.a	✓	1392

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
261.3.w.a	✓	1392	1.a	even	1	1	trivial
261.3.w.a	✓	1392	9.c	even	3	1	inner
261.3.w.a	✓	1392	29.f	odd	28	1	inner
261.3.w.a	✓	1392	261.w	odd	84	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(261, [\chi])\).