Newform orbit 91.10.w.a

• Label: 91.10.w.a
• Level: $91$
• Weight: $10$
• Character orbit: 91.w
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $328$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(328\)
Relative dimension:	\(82\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 328 q - 2 q^{2} - 6 q^{4} - 6 q^{5} + 3072 q^{6} + 1136 q^{7} + 1016 q^{8} - 2047036 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−42.1214	−	11.2864i		−	124.214i	1203.42	+	694.796i	−2496.57	+	668.955i	−1401.93	+	5232.06i	−785.938	+	6303.64i	−27060.6	−	27060.6i	4253.89			112709.
19.2	−42.0539	−	11.2683i			225.274i	1198.15	+	691.751i	−405.575	+	108.674i	2538.45	−	9473.63i	−4540.62	−	4442.57i	−26829.7	−	26829.7i	−31065.3			18280.6
19.3	−41.9569	−	11.2423i		−	46.2785i	1190.59	+	687.385i	222.605	−	59.6469i	−520.278	+	1941.70i	2698.98	−	5750.57i	−26499.6	−	26499.6i	17541.3			−10010.4
19.4	−41.8918	−	11.2249i			83.6686i	1185.52	+	684.459i	2006.85	−	537.734i	939.169	−	3505.03i	−5369.22	+	3394.86i	−26279.0	−	26279.0i	12682.6			−90106.5
19.5	−39.1361	−	10.4865i		−	223.537i	978.259	+	564.798i	836.127	−	224.039i	−2344.12	+	8748.37i	4429.68	+	4553.19i	−17693.9	−	17693.9i	−30286.0			−35072.1
19.6	−37.6161	−	10.0792i			168.334i	869.976	+	502.281i	−1682.51	+	450.828i	1696.67	−	6332.06i	−28.0274	+	6352.39i	−13563.6	−	13563.6i	−8653.25			67833.6
19.7	−37.3346	−	10.0038i		−	207.766i	850.394	+	490.975i	819.107	−	219.479i	−2078.45	+	7756.88i	−5236.89	−	3595.64i	−12844.1	−	12844.1i	−23483.9			−32776.7
19.8	−36.7419	−	9.84496i			65.3690i	809.638	+	467.445i	−963.441	+	258.153i	643.555	−	2401.78i	5922.16	−	2298.17i	−11374.4	−	11374.4i	15409.9			37940.1
19.9	−36.3027	−	9.72728i			195.464i	779.863	+	450.254i	−142.800	+	38.2632i	1901.33	−	7095.88i	5520.27	+	3143.27i	−10324.8	−	10324.8i	−18523.2			5556.24
19.10	−34.8437	−	9.33633i		−	5.93488i	683.508	+	394.624i	2049.00	−	549.028i	−55.4100	+	206.793i	4779.47	+	4184.53i	−7071.85	−	7071.85i	19647.8			−76520.5
19.11	−34.5640	−	9.26141i			229.045i	665.494	+	384.223i	2307.82	−	618.377i	2121.28	−	7916.72i	5294.56	−	3510.16i	−6488.78	−	6488.78i	−32778.6			−85494.5
19.12	−34.1907	−	9.16138i		−	11.3085i	641.671	+	370.469i	453.395	−	121.487i	−103.602	+	386.647i	−4887.48	+	4057.85i	−5730.18	−	5730.18i	19555.1			−16614.9
19.13	−34.0008	−	9.11049i		−	115.784i	629.650	+	363.528i	−1519.89	+	407.252i	−1054.85	+	3936.76i	−5885.54	−	2390.40i	−5352.84	−	5352.84i	6277.01			55387.6
19.14	−31.8341	−	8.52993i			99.4043i	497.248	+	287.086i	−1864.94	+	499.711i	847.912	−	3164.45i	−2436.99	−	5866.40i	−1448.88	−	1448.88i	9801.78			63631.4
19.15	−29.6101	−	7.93400i		−	244.361i	370.403	+	213.852i	−2005.83	+	537.459i	−1938.76	+	7235.53i	5566.24	−	3061.14i	1827.19	+	1827.19i	−40029.1			63656.8
19.16	−28.5420	−	7.64781i		−	148.258i	312.752	+	180.567i	−1354.51	+	362.939i	−1133.85	+	4231.59i	5348.00	−	3428.20i	3152.21	+	3152.21i	−2297.55			41436.0
19.17	−28.2074	−	7.55814i			239.312i	295.125	+	170.391i	52.3963	−	14.0395i	1808.76	−	6750.37i	−6075.99	+	1853.64i	3535.54	+	3535.54i	−37587.4			−1584.07
19.18	−28.0223	−	7.50855i			109.720i	285.466	+	164.814i	1623.73	−	435.078i	823.836	−	3074.60i	−4554.16	−	4428.68i	3741.14	+	3741.14i	7644.59			−48767.6
19.19	−27.3496	−	7.32831i		−	166.796i	250.892	+	144.853i	2135.35	−	572.165i	−1222.34	+	4561.82i	3723.12	−	5147.04i	4450.62	+	4450.62i	−8138.03			−62593.9
19.20	−25.1179	−	6.73031i		−	51.7685i	142.205	+	82.1022i	93.2990	−	24.9994i	−348.418	+	1300.32i	3979.80	+	4951.24i	6395.11	+	6395.11i	17003.0			−2511.73

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.w	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	91.10.w.a	✓	328
7.d	odd	6	1		91.10.ba.a	yes	328
13.f	odd	12	1		91.10.ba.a	yes	328
91.w	even	12	1	inner	91.10.w.a	✓	328

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.10.w.a	✓	328	1.a	even	1	1	trivial
91.10.w.a	✓	328	91.w	even	12	1	inner
91.10.ba.a	yes	328	7.d	odd	6	1
91.10.ba.a	yes	328	13.f	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).