Newform orbit 91.3.z.a

• Label: 91.3.z.a
• Level: $91$
• Weight: $3$
• Character orbit: 91.z
• Analytic conductor: $2.480$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	91.z (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47957040568\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 2 q^{2} - 4 q^{3} - 2 q^{5} + 8 q^{6} - 2 q^{7} - 24 q^{8} - 64 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
18.1	−3.63762	−	0.974698i	0.474554	−	0.821952i	8.81816	+	5.09116i	−3.31141	−	0.887291i	−2.52740	+	2.52740i	−5.34298	+	4.52245i	−16.4631	−	16.4631i	4.04960	+	7.01411i	11.1808	+	6.45526i
18.2	−2.94467	−	0.789022i	1.92646	−	3.33673i	4.58442	+	2.64681i	5.86383	+	1.57121i	−8.30554	+	8.30554i	6.99826	+	0.156132i	−2.78861	−	2.78861i	−2.92250	−	5.06192i	−16.0273	−	9.25338i
18.3	−2.85496	−	0.764984i	−2.56382	+	4.44067i	4.10149	+	2.36799i	4.69477	+	1.25796i	10.7166	−	10.7166i	−5.07863	+	4.81742i	−1.53820	−	1.53820i	−8.64636	−	14.9759i	−12.4411	−	7.18284i
18.4	−2.57823	−	0.690835i	−1.69296	+	2.93230i	2.70592	+	1.56226i	−7.64871	−	2.04947i	6.39059	−	6.39059i	6.11878	−	3.40009i	1.65237	+	1.65237i	−1.23225	−	2.13433i	18.3043	+	10.5680i
18.5	−2.06162	−	0.552409i	1.51330	−	2.62111i	0.481019	+	0.277716i	−3.15000	−	0.844041i	−4.56776	+	4.56776i	−5.37923	−	4.47927i	5.19858	+	5.19858i	−0.0801289	−	0.138787i	6.02785	+	3.48018i
18.6	−1.32854	−	0.355981i	−0.527841	+	0.914248i	−1.82581	−	1.05413i	0.571033	+	0.153008i	1.02671	−	1.02671i	4.27975	+	5.53929i	5.94064	+	5.94064i	3.94277	+	6.82907i	−0.704172	−	0.406554i
18.7	−1.11174	−	0.297890i	−0.547345	+	0.948029i	−2.31687	−	1.33765i	7.64950	+	2.04968i	0.890915	−	0.890915i	−4.58582	−	5.28869i	5.43270	+	5.43270i	3.90083	+	6.75643i	−7.89370	−	4.55743i
18.8	0.258928	+	0.0693796i	2.74989	−	4.76295i	−3.40187	−	1.96407i	5.02241	+	1.34575i	1.04248	−	1.04248i	−4.06883	+	5.69602i	−1.50277	−	1.50277i	−10.6238	−	18.4009i	1.20708	+	0.696906i
18.9	0.300602	+	0.0805459i	−2.35332	+	4.07607i	−3.38023	−	1.95158i	−1.07352	−	0.287650i	−1.03572	+	1.03572i	−0.908147	−	6.94084i	−1.73913	−	1.73913i	−6.57624	−	11.3904i	−0.299534	−	0.172936i
18.10	0.639350	+	0.171313i	−0.766761	+	1.32807i	−3.08468	−	1.78094i	−7.77723	−	2.08390i	−0.717745	+	0.717745i	−5.14954	+	4.74154i	−3.53924	−	3.53924i	3.32416	+	5.75761i	−4.61537	−	2.66469i
18.11	0.666811	+	0.178671i	1.69002	−	2.92721i	−3.05139	−	1.76172i	−4.03010	−	1.07986i	1.64993	−	1.64993i	4.71763	−	5.17146i	−3.67249	−	3.67249i	−1.21237	−	2.09988i	−2.49437	−	1.44013i
18.12	1.70421	+	0.456642i	−0.0320023	+	0.0554296i	−0.768291	−	0.443573i	5.72352	+	1.53361i	−0.0798501	+	0.0798501i	6.99775	+	0.177470i	−6.09705	−	6.09705i	4.49795	+	7.79068i	9.05376	+	5.22719i
18.13	2.32316	+	0.622489i	−2.28115	+	3.95107i	1.54548	+	0.892283i	2.38613	+	0.639362i	−7.75898	+	7.75898i	−1.21070	+	6.89451i	−3.76773	−	3.76773i	−5.90732	−	10.2318i	5.14537	+	2.97068i
18.14	2.68203	+	0.718648i	1.16854	−	2.02398i	3.21272	+	1.85487i	2.83733	+	0.760261i	4.58860	−	4.58860i	−6.91411	−	1.09323i	−0.569900	−	0.569900i	1.76901	+	3.06401i	7.06345	+	4.07808i
18.15	3.18019	+	0.852129i	1.63982	−	2.84025i	5.92338	+	3.41987i	−6.96800	−	1.86707i	7.63519	−	7.63519i	2.85769	+	6.39012i	6.61107	+	6.61107i	−0.878007	−	1.52075i	−20.5686	−	11.8753i
18.16	3.39607	+	0.909975i	−1.39738	+	2.42033i	7.24117	+	4.18069i	−2.15558	−	0.577586i	−6.94805	+	6.94805i	0.105956	−	6.99920i	10.8428	+	10.8428i	0.594658	+	1.02998i	−6.79492	−	3.92305i
44.1	−0.909975	−	3.39607i	−1.39738	−	2.42033i	−7.24117	+	4.18069i	0.577586	+	2.15558i	−6.94805	+	6.94805i	−6.99920	+	0.105956i	10.8428	+	10.8428i	0.594658	−	1.02998i	6.79492	−	3.92305i
44.2	−0.852129	−	3.18019i	1.63982	+	2.84025i	−5.92338	+	3.41987i	1.86707	+	6.96800i	7.63519	−	7.63519i	6.39012	+	2.85769i	6.61107	+	6.61107i	−0.878007	+	1.52075i	20.5686	−	11.8753i
44.3	−0.718648	−	2.68203i	1.16854	+	2.02398i	−3.21272	+	1.85487i	−0.760261	−	2.83733i	4.58860	−	4.58860i	−1.09323	−	6.91411i	−0.569900	−	0.569900i	1.76901	−	3.06401i	−7.06345	+	4.07808i
44.4	−0.622489	−	2.32316i	−2.28115	−	3.95107i	−1.54548	+	0.892283i	−0.639362	−	2.38613i	−7.75898	+	7.75898i	6.89451	−	1.21070i	−3.76773	−	3.76773i	−5.90732	+	10.2318i	−5.14537	+	2.97068i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
13.d	odd	4	1	inner
91.z	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	91.3.z.a	✓	64
7.c	even	3	1	inner	91.3.z.a	✓	64
13.d	odd	4	1	inner	91.3.z.a	✓	64
91.z	odd	12	1	inner	91.3.z.a	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.3.z.a	✓	64	1.a	even	1	1	trivial
91.3.z.a	✓	64	7.c	even	3	1	inner
91.3.z.a	✓	64	13.d	odd	4	1	inner
91.3.z.a	✓	64	91.z	odd	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(91, [\chi])\).