Newform orbit 79.8.g.a

• Label: 79.8.g.a
• Level: $79$
• Weight: $8$
• Character orbit: 79.g
• Analytic conductor: $24.678$
• Analytic rank: $0$
• Dimension: $1104$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 79 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	79.g (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.6784170132\)
Analytic rank:	\(0\)
Dimension:	\(1104\)
Relative dimension:	\(46\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1104 q - 18 q^{2} + 2 q^{3} + 2776 q^{4} - 25 q^{5} + 43 q^{6} + 2818 q^{7} + 8430 q^{8} + 32206 q^{9}+O(q^{10}) \)

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	−17.2929	−	12.9948i	78.7756	+	12.8023i	94.5691	+	326.490i	−31.9275	−	20.1897i	−1195.90	−	1245.06i	−914.013	+	1119.46i	1625.47	−	4286.01i	3967.24	+	1324.46i	289.758	+	764.030i
2.2	−17.1276	−	12.8705i	−69.2272	−	11.2505i	92.0911	+	317.935i	−12.8122	−	8.10193i	1040.89	+	1083.68i	231.391	−	283.403i	1542.26	−	4066.60i	2591.38	+	865.131i	115.165	+	303.666i
2.3	−16.3776	−	12.3070i	−3.09806	−	0.503484i	81.1527	+	280.172i	−48.2828	−	30.5322i	44.5425	+	46.3737i	37.9617	−	46.4946i	1189.12	−	3135.44i	−2065.10	−	689.433i	414.998	+	1094.26i
2.4	−16.1281	−	12.1195i	18.6077	+	3.02404i	77.6222	+	267.983i	390.867	+	247.169i	−263.457	−	274.288i	364.938	−	446.968i	1080.22	−	2848.31i	−1737.35	−	580.012i	−3308.38	−	8723.49i
2.5	−15.3311	−	11.5206i	−36.3720	−	5.91102i	66.7074	+	230.301i	−341.240	−	215.787i	489.525	+	509.649i	−109.944	+	134.657i	760.056	−	2004.10i	−786.469	−	262.562i	2745.60	+	7239.54i
2.6	−14.2829	−	10.7329i	55.6423	+	9.04275i	53.1946	+	183.649i	−182.316	−	115.290i	−697.680	−	726.361i	775.566	−	949.896i	400.384	−	1055.73i	939.843	+	313.766i	1366.61	+	3603.46i
2.7	−13.1499	−	9.88152i	1.48766	+	0.241769i	39.6640	+	136.936i	160.133	+	101.262i	−17.1736	−	17.8796i	−951.628	+	1165.53i	84.9533	−	224.004i	−2072.29	−	691.834i	−1105.11	−	2913.94i
2.8	−12.8031	−	9.62091i	−70.5987	−	11.4734i	35.7458	+	123.409i	371.348	+	234.826i	793.498	+	826.119i	412.173	−	504.820i	2.73475	−	7.21095i	2778.09	+	927.463i	−2495.16	−	6579.21i
2.9	−11.9834	−	9.00491i	−67.4865	−	10.9676i	26.9007	+	92.8720i	59.2752	+	37.4834i	709.953	+	739.140i	−763.174	+	934.719i	−166.428	+	438.835i	2359.70	+	787.782i	−372.782	−	982.945i
2.10	−11.7252	−	8.81095i	27.9134	+	4.53637i	24.2367	+	83.6747i	−436.057	−	275.746i	−287.321	−	299.133i	−587.084	+	719.047i	−212.644	+	560.695i	−1315.87	−	439.302i	2683.30	+	7075.27i
2.11	−10.5512	−	7.92869i	80.9477	+	13.1553i	12.8512	+	44.3675i	294.886	+	186.474i	−749.788	−	780.612i	47.8376	−	58.5904i	−382.876	+	1009.56i	4305.02	+	1437.23i	−1632.89	−	4305.58i
2.12	−9.57639	−	7.19619i	−31.3126	−	5.08880i	4.31023	+	14.8806i	−29.1343	−	18.4234i	263.242	+	274.064i	888.156	−	1087.79i	−477.905	+	1260.13i	−1119.86	−	373.866i	146.423	+	386.085i
2.13	−9.45313	−	7.10356i	49.5027	+	8.04498i	3.28914	+	11.3554i	150.841	+	95.3863i	−410.808	−	427.696i	46.9588	−	57.5140i	−487.142	+	1284.49i	311.351	+	103.944i	−748.340	−	1973.21i
2.14	−8.88450	−	6.67627i	−29.0420	−	4.71979i	−1.25004	−	4.31562i	−113.672	−	71.8818i	226.513	+	235.825i	743.231	−	910.292i	−522.136	+	1376.76i	−1253.29	−	418.409i	530.017	+	1397.54i
2.15	−7.94443	−	5.96985i	67.3201	+	10.9406i	−8.13698	−	28.0921i	−174.942	−	110.627i	−469.506	−	488.808i	−450.530	+	551.799i	−554.118	+	1461.09i	2337.85	+	780.489i	729.390	+	1923.24i
2.16	−7.83161	−	5.88507i	−83.6131	−	13.5884i	−8.91181	−	30.7671i	−418.558	−	264.680i	574.856	+	598.488i	59.9798	−	73.4619i	−555.923	+	1465.85i	4732.05	+	1579.79i	1720.32	+	4536.11i
2.17	−7.14595	−	5.36984i	4.90622	+	0.797338i	−13.3823	−	46.2011i	288.967	+	182.732i	−30.7780	−	32.0433i	−188.732	+	231.154i	−558.184	+	1471.81i	−2051.01	−	684.729i	−1083.71	−	2857.50i
2.18	−5.04577	−	3.79165i	2.84922	+	0.463043i	−24.5287	−	84.6828i	−257.393	−	162.766i	−12.6208	−	13.1396i	−238.595	+	292.226i	−483.801	+	1275.68i	−2066.55	−	689.914i	681.596	+	1797.22i
2.19	−2.57132	−	1.93222i	−46.7908	−	7.60424i	−32.7336	−	113.010i	409.010	+	258.643i	105.621	+	109.963i	17.4610	−	21.3859i	−280.181	+	738.777i	57.1007	+	19.0630i	−551.943	−	1455.35i
2.20	−2.35162	−	1.76713i	−49.1503	−	7.98770i	−33.2045	−	114.635i	−33.3451	−	21.0861i	101.467	+	105.639i	−667.087	+	817.033i	−258.007	+	680.308i	277.498	+	92.6424i	41.1530	+	108.512i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
79.g	even	39	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	79.8.g.a	✓	1104
79.g	even	39	1	inner	79.8.g.a	✓	1104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
79.8.g.a	✓	1104	1.a	even	1	1	trivial
79.8.g.a	✓	1104	79.g	even	39	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(79, [\chi])\).