Newform orbit 59.8.c.a

• Label: 59.8.c.a
• Level: $59$
• Weight: $8$
• Character orbit: 59.c
• Analytic conductor: $18.431$
• Analytic rank: $0$
• Dimension: $952$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	59.c (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.4307165036\)
Analytic rank:	\(0\)
Dimension:	\(952\)
Relative dimension:	\(34\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 952 q - 35 q^{2} + 25 q^{3} - 2139 q^{4} - 169 q^{5} - 1065 q^{6} + 1321 q^{7} + 289 q^{8} - 22159 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−19.4252	+	8.98705i	8.18271	+	2.75708i	213.705	−	251.593i	−166.561	−	100.216i	−183.729	+	19.9817i	−75.6425	+	1395.14i	−1157.25	+	4168.06i	−1681.70	−	1278.40i	4136.13	+	449.831i
3.2	−19.0356	+	8.80680i	64.6687	+	21.7894i	201.928	−	237.729i	400.319	+	240.864i	−1422.90	+	154.750i	−0.501802	+	9.25519i	−1031.97	+	3716.82i	1966.21	+	1494.67i	−9741.54	−	1059.46i
3.3	−17.7431	+	8.20883i	−53.4543	−	18.0109i	164.567	−	193.744i	320.264	+	192.696i	1096.29	−	119.229i	18.2856	−	337.258i	−660.063	+	2377.33i	791.917	+	602.000i	−7264.28	−	790.038i
3.4	−17.4659	+	8.08061i	−4.41894	−	1.48891i	156.898	−	184.714i	−189.353	−	113.930i	89.2123	−	9.70242i	86.0214	−	1586.57i	−588.758	+	2120.51i	−1723.75	−	1310.36i	4227.85	+	459.806i
3.5	−15.6583	+	7.24431i	−67.4051	−	22.7114i	109.837	−	129.311i	−231.713	−	139.417i	1219.98	−	132.681i	−31.6596	+	583.928i	−192.301	+	692.604i	2286.58	+	1738.21i	4638.21	+	504.436i
3.6	−14.5859	+	6.74816i	77.5304	+	26.1230i	84.3458	−	99.2995i	−302.724	−	182.143i	−1307.13	+	142.159i	−4.47353	+	82.5095i	−9.83137	+	35.4094i	3587.50	+	2727.15i	5644.65	+	613.892i
3.7	−14.3515	+	6.63969i	42.3455	+	14.2679i	79.0134	−	93.0218i	109.188	+	65.6961i	−702.455	+	76.3966i	18.6008	−	343.071i	25.1713	−	90.6590i	−151.483	−	115.154i	−2003.21	−	217.862i
3.8	−11.5946	+	5.36423i	−24.8998	−	8.38971i	22.7941	−	26.8353i	241.772	+	145.470i	333.707	−	36.2928i	−56.0378	+	1033.56i	317.136	−	1142.22i	−1191.44	−	905.712i	−3583.58	−	389.738i
3.9	−10.9467	+	5.06448i	30.6438	+	10.3251i	11.3156	−	13.3217i	46.6054	+	28.0415i	−387.740	+	42.1693i	−41.6923	+	768.969i	356.628	−	1284.46i	−908.618	−	690.714i	−652.190	−	70.9300i
3.10	−9.68811	+	4.48220i	−64.1128	−	21.6021i	−9.09610	+	10.7088i	−167.470	−	100.763i	717.956	−	78.0824i	23.3397	−	430.475i	405.666	−	1461.08i	1902.74	+	1446.42i	2074.10	+	225.572i
3.11	−8.39027	+	3.88175i	3.42256	+	1.15319i	−27.5368	+	32.4189i	−407.113	−	244.952i	−33.1926	+	3.60991i	−1.18975	+	21.9436i	421.772	−	1519.08i	−1730.67	−	1315.62i	4366.63	+	474.899i
3.12	−7.76192	+	3.59105i	−10.6380	−	3.58437i	−35.5137	+	41.8100i	292.460	+	175.967i	95.4431	−	10.3801i	40.3910	−	744.968i	418.377	−	1506.86i	−1640.74	−	1247.25i	−2901.95	−	315.606i
3.13	−5.65798	+	2.61766i	59.4622	+	20.0351i	−57.7048	+	67.9354i	167.896	+	101.020i	−388.881	+	42.2934i	92.6654	−	1709.11i	362.142	−	1304.32i	1393.29	+	1059.15i	−1214.39	−	132.073i
3.14	−3.44005	+	1.59154i	−82.9941	−	27.9640i	−73.5645	+	86.6068i	257.444	+	154.899i	330.010	−	35.8907i	43.3224	−	799.034i	245.024	−	882.496i	4364.98	+	3318.17i	−1132.15	−	123.128i
3.15	−2.70915	+	1.25338i	70.5802	+	23.7812i	−77.0969	+	90.7655i	359.444	+	216.270i	−221.019	+	24.0373i	−62.5382	+	1153.45i	197.321	−	710.687i	2674.96	+	2033.45i	−1244.86	−	135.386i
3.16	−2.20401	+	1.01968i	−29.6212	−	9.98055i	−79.0476	+	93.0620i	−175.745	−	105.743i	75.4623	−	8.20702i	58.1267	−	1072.08i	162.487	−	585.224i	−963.250	−	732.244i	495.168	+	53.8527i
3.17	−1.46713	+	0.678768i	37.1036	+	12.5017i	−81.1737	+	95.5651i	−234.710	−	141.220i	−62.9217	+	6.84315i	−56.4400	+	1040.97i	109.582	−	394.680i	−520.669	−	395.802i	440.206	+	47.8753i
3.18	−1.03327	+	0.478041i	−52.1031	−	17.5556i	−82.0263	+	96.5689i	−53.4152	−	32.1388i	62.2287	−	6.76778i	−91.4124	+	1686.00i	77.5775	−	279.409i	665.475	+	505.881i	70.5559	+	7.67341i
3.19	0.217282	−	0.100525i	74.1466	+	24.9829i	−82.8283	+	97.5131i	−172.078	−	103.536i	18.6221	−	2.02528i	26.3057	−	485.180i	−16.3928	+	59.0416i	3132.52	+	2381.28i	−47.7973	−	5.19827i
3.20	3.78522	−	1.75123i	−18.7327	−	6.31177i	−71.6044	+	84.2992i	383.893	+	230.981i	−81.9606	+	8.91376i	−13.2493	+	244.370i	−266.231	+	958.877i	−1429.98	−	1087.04i	1857.62	+	202.028i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
59.c	even	29	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	59.8.c.a	✓	952
59.c	even	29	1	inner	59.8.c.a	✓	952

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
59.8.c.a	✓	952	1.a	even	1	1	trivial
59.8.c.a	✓	952	59.c	even	29	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(59, [\chi])\).