Newform orbit 89.8.e.a

• Label: 89.8.e.a
• Level: $89$
• Weight: $8$
• Character orbit: 89.e
• Analytic conductor: $27.802$
• Analytic rank: $0$
• Dimension: $520$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	89.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(27.8022672681\)
Analytic rank:	\(0\)
Dimension:	\(520\)
Relative dimension:	\(52\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 520 q - 25 q^{2} - 100 q^{3} - 3337 q^{4} - 13 q^{5} + 487 q^{6} - 9 q^{7} - 2825 q^{8} - 26360 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−21.2001	+	6.22490i	−33.4095	−	21.4710i	303.013	−	194.734i	263.036	−	303.560i	841.939	+	247.216i	−0.516662	+	0.596260i	−3359.63	+	3877.21i	−253.319	−	554.692i	−3686.75	+	8072.86i
2.2	−20.5565	+	6.03594i	11.9643	+	7.68901i	278.457	−	178.954i	−48.2949	+	55.7353i	−292.355	−	85.8433i	928.914	−	1072.02i	−2848.13	+	3286.91i	−824.488	−	1805.38i	656.360	−	1437.23i
2.3	−20.5363	+	6.03001i	−70.3873	−	45.2351i	277.699	−	178.467i	−294.237	+	339.568i	1718.26	+	504.528i	−185.636	+	214.236i	−2832.70	+	3269.11i	1999.64	+	4378.59i	3994.95	−	8747.72i
2.4	−20.1719	+	5.92300i	47.9334	+	30.8049i	264.142	−	169.754i	−212.353	+	245.068i	−1149.36	−	337.484i	−863.515	+	996.550i	−2560.56	+	2955.04i	440.157	+	963.809i	2832.01	−	6201.24i
2.5	−18.2426	+	5.35651i	38.0824	+	24.4741i	196.420	−	126.232i	40.0015	−	46.1642i	−825.818	−	242.482i	69.3538	−	80.0385i	−1313.36	+	1515.70i	−57.2230	−	125.301i	−482.453	+	1056.42i
2.6	−18.0262	+	5.29297i	67.4808	+	43.3672i	189.247	−	121.622i	344.724	−	397.833i	−1445.96	−	424.573i	152.883	−	176.436i	−1192.89	+	1376.66i	1764.42	+	3863.54i	−4108.35	+	8996.02i
2.7	−17.3159	+	5.08442i	−17.8251	−	11.4555i	166.310	−	106.881i	−102.105	+	117.835i	366.903	+	107.733i	−550.699	+	635.540i	−823.654	+	950.548i	−722.007	−	1580.97i	1168.92	−	2559.58i
2.8	−16.8471	+	4.94675i	−5.05763	−	3.25034i	151.673	−	97.4745i	264.971	−	305.792i	101.285	+	29.7399i	−756.962	+	873.581i	−601.294	+	693.931i	−893.498	−	1956.49i	−2951.30	+	6462.45i
2.9	−16.0704	+	4.71869i	−37.9848	−	24.4114i	128.310	−	82.4600i	−78.3375	+	90.4063i	725.620	+	213.061i	850.478	−	981.503i	−268.968	+	310.406i	−61.5796	−	134.841i	832.314	−	1822.51i
2.10	−15.9834	+	4.69316i	−61.8700	−	39.7614i	125.764	−	80.8235i	133.709	−	154.308i	1175.50	+	345.158i	−363.499	+	419.500i	−234.492	+	270.619i	1338.41	+	2930.71i	−1412.93	+	3093.89i
2.11	−14.9883	+	4.40096i	69.5461	+	44.6946i	97.5996	−	62.7235i	−310.556	+	358.400i	−1239.08	−	363.825i	1121.77	−	1294.59i	122.583	−	141.468i	1930.55	+	4227.31i	3077.39	−	6738.54i
2.12	−13.5119	+	3.96747i	36.9806	+	23.7660i	59.1515	−	38.0144i	77.9143	−	89.9179i	−593.971	−	174.406i	275.511	−	317.957i	531.984	−	613.943i	−105.768	−	231.601i	−696.028	+	1524.09i
2.13	−12.8781	+	3.78134i	−4.79284	−	3.08017i	43.8653	−	28.1905i	−332.998	+	384.300i	73.3696	+	21.5432i	−1.23373	+	1.42380i	666.737	−	769.455i	−895.029	−	1959.84i	2835.20	−	6208.22i
2.14	−12.5833	+	3.69480i	−64.2506	−	41.2914i	37.0083	−	23.7838i	194.842	−	224.859i	961.051	+	282.190i	538.997	−	622.035i	721.480	−	832.632i	1514.65	+	3316.63i	−1620.95	+	3549.38i
2.15	−11.2460	+	3.30213i	73.6479	+	47.3307i	7.88855	−	5.06967i	−40.5184	+	46.7607i	−984.539	−	289.087i	−886.851	+	1023.48i	910.489	−	1050.76i	2275.32	+	4982.25i	301.261	−	659.669i
2.16	−9.60772	+	2.82108i	−4.29476	−	2.76008i	−23.3306	+	14.9937i	257.829	−	297.550i	49.0493	+	14.4022i	682.330	−	787.451i	1021.20	−	1178.52i	−897.686	−	1965.66i	−1637.73	+	3586.14i
2.17	−9.38620	+	2.75604i	40.4729	+	26.0104i	−27.1754	+	17.4646i	25.6024	−	29.5467i	−451.573	−	132.594i	−106.327	+	122.708i	1026.93	−	1185.14i	53.0067	+	116.068i	−158.877	+	347.893i
2.18	−8.44733	+	2.48036i	−56.0085	−	35.9945i	−42.4752	+	27.2972i	−115.185	+	132.931i	562.402	+	165.136i	−800.208	+	923.489i	1029.06	−	1187.60i	932.838	+	2042.63i	643.292	−	1408.61i
2.19	−8.20805	+	2.41010i	20.3231	+	13.0608i	−46.1169	+	29.6375i	−196.806	+	227.126i	−198.291	−	58.2234i	−232.155	+	267.922i	1024.16	−	1181.95i	−666.071	−	1458.49i	1067.99	−	2338.58i
2.20	−8.19033	+	2.40490i	2.16744	+	1.39293i	−46.3825	+	29.8082i	228.112	−	263.255i	−21.1019	−	6.19607i	−1145.57	+	1322.06i	1023.72	−	1181.43i	−905.755	−	1983.33i	−1235.21	+	2704.73i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
89.e	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	89.8.e.a	✓	520
89.e	even	11	1	inner	89.8.e.a	✓	520

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
89.8.e.a	✓	520	1.a	even	1	1	trivial
89.8.e.a	✓	520	89.e	even	11	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(89, [\chi])\).