Newform orbit 261.3.j.a

• Label: 261.3.j.a
• Level: $261$
• Weight: $3$
• Character orbit: 261.j
• Analytic conductor: $7.112$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	261.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.11173489980\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 2 q^{3} + 112 q^{4} - 18 q^{5} - 8 q^{6} + 2 q^{7} + 22 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−3.35970	+	1.93973i	0.736601	+	2.90816i	5.52507	−	9.56970i	−5.48999	−	3.16965i	−8.11580	−	8.34176i	4.27519	+	7.40484i			27.3507i	−7.91484	+	4.28431i	24.5930
59.2	−3.23269	+	1.86640i	2.96726	−	0.442038i	4.96687	−	8.60287i	0.0153376	+	0.00885515i	−8.76721	+	6.96705i	−4.25369	−	7.36760i			22.1494i	8.60920	−	2.62328i	−0.0661089
59.3	−3.17154	+	1.83109i	0.946350	−	2.84683i	4.70577	−	8.15063i	0.902045	+	0.520796i	2.21141	+	10.7617i	4.41170	+	7.64129i			19.8180i	−7.20884	−	5.38819i	−3.81449
59.4	−3.10304	+	1.79154i	−2.63122	+	1.44107i	4.41923	−	7.65433i	−1.41090	−	0.814586i	5.58303	−	9.18564i	−3.77670	−	6.54143i			17.3366i	4.84662	−	7.58356i	5.83745
59.5	−3.03255	+	1.75085i	−2.32085	−	1.90096i	4.13093	−	7.15497i	6.76865	+	3.90788i	10.3664	+	1.70131i	−1.10446	−	1.91298i			14.9237i	1.77270	+	8.82369i	−27.3684
59.6	−2.99235	+	1.72763i	1.16738	+	2.76355i	3.96942	−	6.87524i	6.48871	+	3.74626i	−8.26760	−	6.25271i	−1.65833	−	2.87231i			13.6098i	−6.27446	+	6.45222i	−25.8886
59.7	−2.87020	+	1.65711i	−2.58319	−	1.52549i	3.49202	−	6.04835i	−6.88861	−	3.97714i	9.94217	+	0.0978219i	4.37135	+	7.57140i			9.88975i	4.34577	+	7.88126i	26.3622
59.8	−2.53746	+	1.46500i	−0.365986	−	2.97759i	2.29247	−	3.97067i	−6.37890	−	3.68286i	5.29085	+	7.01935i	−5.68247	−	9.84232i			1.71385i	−8.73211	+	2.17951i	21.5816
59.9	−2.46451	+	1.42289i	0.659048	−	2.92671i	2.04922	−	3.54935i	1.09510	+	0.632255i	2.54015	+	8.15068i	−1.09493	−	1.89648i			0.280134i	−8.13131	−	3.85769i	−3.59851
59.10	−2.44704	+	1.41280i	2.99713	+	0.131208i	1.99201	−	3.45027i	5.81365	+	3.35651i	−7.51947	+	3.91328i	5.56871	+	9.64529i		−	0.0451395i	8.96557	+	0.786492i	−18.9683
59.11	−2.41979	+	1.39707i	−2.97638	−	0.375707i	1.90360	−	3.29713i	0.309250	+	0.178546i	7.72711	−	3.24907i	2.29454	+	3.97426i		−	0.538717i	8.71769	+	2.23650i	−0.997761
59.12	−2.37925	+	1.37366i	−0.740781	+	2.90710i	1.77390	−	3.07248i	3.65019	+	2.10744i	−2.23087	−	7.93431i	1.89594	+	3.28386i		−	1.24237i	−7.90249	−	4.30705i	−11.5796
59.13	−2.33438	+	1.34775i	2.93438	−	0.624030i	1.63288	−	2.82822i	−7.31151	−	4.22130i	−6.00891	+	5.41154i	1.54000	+	2.66736i		−	1.97917i	8.22117	−	3.66228i	22.7571
59.14	−2.24592	+	1.29668i	2.06844	+	2.17291i	1.36278	−	2.36041i	−3.95710	−	2.28463i	−7.46314	−	2.19808i	−2.10178	−	3.64039i		−	3.30508i	−0.443107	+	8.98909i	11.8498
59.15	−2.12284	+	1.22562i	−1.48754	+	2.60523i	1.00430	−	1.73950i	−1.61120	−	0.930228i	−0.0352052	−	7.35365i	1.43764	+	2.49006i		−	4.88140i	−4.57442	−	7.75079i	4.56043
59.16	−1.61566	+	0.932800i	2.29735	+	1.92930i	−0.259769	+	0.449933i	2.48164	+	1.43278i	−5.51137	−	0.974124i	−4.21812	−	7.30599i		−	8.43165i	1.55560	+	8.86454i	−5.34598
59.17	−1.41922	+	0.819389i	2.58189	−	1.52770i	−0.657205	+	1.13831i	1.62817	+	0.940025i	−2.41250	+	4.28371i	−1.60428	−	2.77870i		−	8.70913i	4.33230	−	7.88868i	−3.08098
59.18	−1.35521	+	0.782430i	−2.47592	+	1.69405i	−0.775607	+	1.34339i	7.71604	+	4.45486i	2.02991	−	4.23303i	−6.44947	−	11.1708i		−	8.68687i	3.26037	−	8.38868i	−13.9425
59.19	−1.23566	+	0.713406i	−2.96671	+	0.445711i	−0.982103	+	1.70105i	5.43808	+	3.13968i	3.34785	−	2.66721i	6.04301	+	10.4668i		−	8.50980i	8.60268	−	2.64459i	−8.95946
59.20	−1.23164	+	0.711090i	1.42886	−	2.63787i	−0.988703	+	1.71248i	−3.68773	−	2.12911i	0.115921	+	4.26496i	4.80914	+	8.32968i		−	8.50094i	−4.91674	−	7.53828i	6.05596

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	261.3.j.a	✓	112
9.d	odd	6	1	inner	261.3.j.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
261.3.j.a	✓	112	1.a	even	1	1	trivial
261.3.j.a	✓	112	9.d	odd	6	1	inner

Hecke kernels

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