Newform orbit 71.8.c.a

• Label: 71.8.c.a
• Level: $71$
• Weight: $8$
• Character orbit: 71.c
• Analytic conductor: $22.179$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	71.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1793368094\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(41\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 164 q - 24 q^{2} + 23 q^{3} - 2290 q^{4} - 145 q^{5} + 514 q^{6} - 689 q^{7} + 7007 q^{8} - 28870 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−6.93300	−	21.3376i	0.364373	+	1.12143i	−303.672	+	220.631i	−352.606	−	256.184i	21.4023	−	15.5497i	177.303	+	545.682i	4489.78	+	3262.02i	1768.20	−	1284.67i	−3021.72	+	9299.89i
5.2	−6.51967	−	20.0655i	−20.2475	−	62.3154i	−256.564	+	186.404i	304.298	+	221.085i	−1118.38	+	812.552i	193.121	+	594.364i	3228.21	+	2345.43i	−1703.92	+	1237.97i	2452.27	−	7547.30i
5.3	−6.23970	−	19.2038i	1.99957	+	6.15405i	−226.299	+	164.416i	191.290	+	138.980i	105.705	−	76.7989i	−476.532	−	1466.61i	2478.48	+	1800.72i	1735.45	−	1260.88i	1475.36	−	4540.69i
5.4	−5.91496	−	18.2044i	25.0402	+	77.0657i	−192.859	+	140.120i	−160.635	−	116.708i	1254.82	−	911.681i	−295.568	−	909.664i	1709.39	+	1241.95i	−3542.79	+	2573.99i	−1174.45	+	3614.58i
5.5	−5.89857	−	18.1539i	19.4037	+	59.7186i	−191.218	+	138.928i	292.252	+	212.334i	969.672	−	704.508i	292.607	+	900.553i	1673.33	+	1215.75i	−1420.48	+	1032.04i	2130.82	−	6557.98i
5.6	−5.40238	−	16.6268i	−22.1625	−	68.2091i	−143.711	+	104.412i	−238.233	−	173.086i	−1014.37	+	736.982i	9.27602	+	28.5486i	702.037	+	510.059i	−2391.98	+	1737.88i	−1590.85	+	4896.13i
5.7	−5.39670	−	16.6093i	1.75916	+	5.41413i	−143.191	+	104.035i	−4.10341	−	2.98130i	80.4314	−	58.4369i	221.109	+	680.503i	692.227	+	502.932i	1743.10	−	1266.44i	−27.3726	+	84.2441i
5.8	−4.57245	−	14.0725i	−4.38759	−	13.5036i	−73.5750	+	53.4553i	−182.376	−	132.504i	−169.968	+	123.489i	−262.544	−	808.027i	−443.595	−	322.291i	1606.22	−	1166.99i	−1030.76	+	3172.37i
5.9	−4.56651	−	14.0543i	−18.4491	−	56.7806i	−73.1157	+	53.1217i	105.290	+	76.4975i	−713.763	+	518.579i	−236.022	−	726.402i	−449.806	−	326.803i	−1114.35	+	809.622i	594.310	−	1829.10i
5.10	−4.22295	−	12.9969i	15.7844	+	48.5794i	−47.5322	+	34.5342i	−303.187	−	220.279i	564.726	−	410.297i	282.918	+	870.732i	−765.582	−	556.228i	−341.492	+	248.108i	−1582.60	+	4870.73i
5.11	−3.54545	−	10.9118i	−12.7834	−	39.3433i	−2.94253	+	2.13788i	221.645	+	161.035i	−383.983	+	278.980i	503.884	+	1550.79i	−1154.35	−	838.684i	384.839	−	279.602i	971.343	−	2989.49i
5.12	−3.32987	−	10.2483i	6.41874	+	19.7549i	9.61486	−	6.98561i	292.318	+	212.381i	181.080	−	131.562i	−6.62913	−	20.4024i	−1219.47	−	886.000i	1420.27	−	1031.88i	1203.16	−	3702.95i
5.13	−3.06892	−	9.44516i	19.1042	+	58.7966i	23.7614	−	17.2637i	−132.769	−	96.4624i	496.714	−	360.884i	−13.8081	−	42.4970i	−1264.40	−	918.640i	−1322.75	+	961.038i	−503.645	+	1550.06i
5.14	−2.33944	−	7.20005i	20.0680	+	61.7630i	57.1864	−	41.5484i	246.833	+	179.335i	397.749	−	288.981i	−420.227	−	1293.33i	−1216.90	−	884.129i	−1642.62	+	1193.43i	713.767	−	2196.75i
5.15	−2.27884	−	7.01354i	−19.8378	−	61.0546i	59.5576	−	43.2711i	357.700	+	259.884i	−383.001	+	278.267i	−219.765	−	676.366i	−1202.86	−	873.931i	−1564.80	+	1136.90i	1007.57	−	3100.98i
5.16	−2.23677	−	6.88408i	−13.7481	−	42.3123i	61.1668	−	44.4403i	−245.479	−	178.351i	−260.529	+	189.286i	538.846	+	1658.40i	−1192.31	−	866.262i	168.003	−	122.061i	−678.700	+	2088.82i
5.17	−2.14382	−	6.59799i	−1.13673	−	3.49849i	64.6167	−	46.9468i	−431.608	−	313.582i	−20.6460	+	15.0002i	−548.780	−	1688.97i	−1166.69	−	847.651i	1758.37	−	1277.53i	−1143.72	+	3520.01i
5.18	−1.01407	−	3.12098i	−4.38685	−	13.5013i	94.8420	−	68.9067i	−50.9781	−	37.0377i	−37.6888	+	27.3825i	11.7105	+	36.0411i	−651.056	−	473.020i	1606.28	−	1167.03i	−63.8988	+	196.660i
5.19	−0.992398	−	3.05429i	−25.7357	−	79.2065i	95.2104	−	69.1744i	−175.334	−	127.387i	−216.379	+	157.209i	−95.8503	−	294.997i	−638.326	−	463.771i	−3842.02	+	2791.39i	−215.077	+	661.939i
5.20	−0.749526	−	2.30681i	27.7977	+	85.5524i	98.7946	−	71.7785i	193.778	+	140.788i	176.518	−	128.248i	531.212	+	1634.90i	−490.801	−	356.588i	−4777.19	+	3470.83i	179.529	−	552.532i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
71.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	71.8.c.a	✓	164
71.c	even	5	1	inner	71.8.c.a	✓	164

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
71.8.c.a	✓	164	1.a	even	1	1	trivial
71.8.c.a	✓	164	71.c	even	5	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(71, [\chi])\).