Newform orbit 91.8.g.a

• Label: 91.8.g.a
• Level: $91$
• Weight: $8$
• Character orbit: 91.g
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(63\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 126 q + q^{2} + 160 q^{3} - 3839 q^{4} - 2 q^{5} - 258 q^{6} - 1816 q^{7} - 264 q^{8} + 83414 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1	−11.1216	−	19.2632i	2.12093			−183.381	+	317.626i	144.888	−	250.953i	−23.5882	−	40.8560i	−600.660	−	680.257i	5310.86			−2182.50			−6445.54
9.2	−10.2852	−	17.8144i	80.9349			−147.570	+	255.598i	−183.635	+	318.065i	−832.430	−	1441.81i	−822.766	−	382.883i	3438.12			4363.47			7554.87
9.3	−10.2674	−	17.7837i	16.7911			−146.839	+	254.333i	−140.214	+	242.858i	−172.401	−	298.607i	−111.419	+	900.627i	3402.16			−1905.06			5758.53
9.4	−10.1524	−	17.5845i	−84.0693			−142.143	+	246.199i	−87.2987	+	151.206i	853.506	+	1478.32i	−907.393	+	13.4657i	3173.35			4880.65			3545.17
9.5	−10.0509	−	17.4086i	39.1708			−138.040	+	239.092i	68.9494	−	119.424i	−393.700	−	681.909i	800.714	+	427.083i	2976.65			−652.649			−2772.00
9.6	−9.91903	−	17.1803i	−59.8380			−132.774	+	229.972i	52.4514	−	90.8484i	593.535	+	1028.03i	835.329	+	354.640i	2728.70			1393.59			−2081.07
9.7	−9.75137	−	16.8899i	−33.0220			−126.179	+	218.548i	−238.580	+	413.232i	322.010	+	557.738i	286.802	−	860.980i	2425.31			−1096.55			9305.93
9.8	−9.30897	−	16.1236i	82.6777			−109.314	+	189.337i	79.5295	−	137.749i	−769.645	−	1333.06i	833.829	−	358.152i	1687.30			4648.61			−2961.35
9.9	−8.91844	−	15.4472i	−33.8094			−95.0771	+	164.678i	226.811	−	392.849i	301.527	+	522.260i	−538.169	+	730.696i	1108.64			−1043.92			−8091.22
9.10	−8.16987	−	14.1506i	33.7329			−69.4935	+	120.366i	140.577	−	243.487i	−275.594	−	477.342i	−71.4247	−	904.678i	179.525			−1049.09			−4593.99
9.11	−7.97222	−	13.8083i	10.3622			−63.1125	+	109.314i	−123.748	+	214.338i	−82.6094	−	143.084i	696.052	−	582.284i	−28.3024			−2079.63			3946.19
9.12	−7.47686	−	12.9503i	50.8547			−47.8068	+	82.8038i	−18.5461	+	32.1228i	−380.234	−	658.584i	−548.996	+	722.597i	−484.298			399.204			554.666
9.13	−7.37896	−	12.7807i	70.9222			−44.8980	+	77.7656i	233.511	−	404.454i	−523.332	−	906.437i	−826.030	+	375.788i	−563.812			2842.96			−6892.28
9.14	−7.29630	−	12.6376i	−62.9903			−42.4721	+	73.5638i	159.912	−	276.976i	459.596	+	796.044i	409.569	−	809.812i	−628.297			1780.78			−4667.07
9.15	−7.23497	−	12.5313i	−23.7301			−40.6896	+	70.4765i	−54.6650	+	94.6825i	171.686	+	297.370i	−874.352	−	243.006i	−674.600			−1623.88			1582.00
9.16	−6.31833	−	10.9437i	−31.4852			−15.8425	+	27.4401i	−91.3850	+	158.284i	198.934	+	344.563i	−243.304	+	874.269i	−1217.10			−1195.68			2309.60
9.17	−5.77207	−	9.99752i	63.1946			−2.63359	+	4.56152i	−256.044	+	443.482i	−364.764	−	631.789i	884.311	+	203.805i	−1416.84			1806.56			5911.62
9.18	−5.71921	−	9.90597i	−71.9046			−1.41879	+	2.45741i	−197.077	+	341.347i	411.238	+	712.284i	529.566	+	736.955i	−1431.66			2983.27			4508.49
9.19	−4.96126	−	8.59316i	9.59815			14.7717	−	25.5853i	209.709	−	363.226i	−47.6190	−	82.4785i	704.944	+	571.487i	−1563.23			−2094.88			−4161.68
9.20	−4.62761	−	8.01526i	47.9981			21.1704	−	36.6682i	−120.307	+	208.377i	−222.117	−	384.718i	−628.292	−	654.823i	−1576.54			116.821			2226.93

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	91.8.g.a	✓	126
7.c	even	3	1		91.8.h.a	yes	126
13.c	even	3	1		91.8.h.a	yes	126
91.g	even	3	1	inner	91.8.g.a	✓	126

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.8.g.a	✓	126	1.a	even	1	1	trivial
91.8.g.a	✓	126	91.g	even	3	1	inner
91.8.h.a	yes	126	7.c	even	3	1
91.8.h.a	yes	126	13.c	even	3	1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(91, [\chi])\).