Newform orbit 71.10.c.a

• Label: 71.10.c.a
• Level: $71$
• Weight: $10$
• Character orbit: 71.c
• Analytic conductor: $36.568$
• Analytic rank: $0$
• Dimension: $212$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	71.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(36.5675443676\)
Analytic rank:	\(0\)
Dimension:	\(212\)
Relative dimension:	\(53\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 212 q + 48 q^{2} - 151 q^{3} - 13570 q^{4} + 1589 q^{5} + 4306 q^{6} + 4799 q^{7} + 18599 q^{8} - 357286 q^{9}+O(q^{10}) \)

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	−13.6411	−	41.9831i	62.7093	+	192.999i	−1162.28	+	844.448i	−46.1963	−	33.5636i	7247.28	−	5265.46i	2279.58	+	7015.83i	33022.4	+	23992.1i	−17392.4	+	12636.3i	−778.932	+	2397.31i
5.2	−13.2245	−	40.7010i	−12.2824	−	37.8014i	−1067.46	+	775.558i	353.473	+	256.813i	−1376.13	+	999.814i	273.849	+	842.821i	27956.1	+	20311.3i	14645.8	−	10640.8i	5778.02	−	17782.9i
5.3	−12.4123	−	38.2013i	−39.6632	−	122.071i	−891.053	+	647.388i	−1354.24	−	983.917i	−4170.95	+	3030.37i	−2314.86	−	7124.40i	19153.2	+	13915.6i	2595.75	−	1885.92i	−20777.5	+	63946.6i
5.4	−12.2160	−	37.5970i	−37.3129	−	114.837i	−850.087	+	617.624i	1577.69	+	1146.26i	−3861.73	+	2805.71i	−2.12635	−	6.54422i	17230.7	+	12518.9i	4128.51	−	2999.54i	23822.9	−	73319.3i
5.5	−12.1675	−	37.4478i	−71.7797	−	220.915i	−840.070	+	610.347i	−812.295	−	590.167i	−7399.40	+	5375.98i	2331.82	+	7176.62i	16768.0	+	12182.6i	−27727.3	+	20145.1i	−12216.8	+	37599.5i
5.6	−12.0705	−	37.1493i	43.9867	+	135.377i	−820.158	+	595.879i	674.925	+	490.362i	4498.23	−	3268.15i	−2195.54	−	6757.18i	15856.5	+	11520.4i	−468.254	+	340.207i	10069.9	−	30991.9i
5.7	−11.6833	−	35.9575i	51.3695	+	158.099i	−742.228	+	539.260i	−2256.74	−	1639.62i	5084.68	−	3694.24i	−1736.56	−	5344.60i	12401.5	+	9010.18i	−6432.58	+	4673.55i	−32590.4	+	100303.i
5.8	−10.4491	−	32.1589i	9.95813	+	30.6480i	−510.795	+	371.114i	−558.038	−	405.438i	881.552	−	640.485i	3797.16	+	11686.4i	3265.67	+	2372.65i	15083.7	−	10959.0i	−7207.47	+	22182.3i
5.9	−9.68063	−	29.7939i	14.5810	+	44.8757i	−379.746	+	275.901i	−1191.92	−	865.980i	1195.87	−	868.850i	748.075	+	2302.34i	−1079.88	−	784.575i	14122.7	−	10260.7i	−14262.4	+	43895.2i
5.10	−9.65631	−	29.7191i	−81.5060	−	250.850i	−375.761	+	273.006i	1070.74	+	777.937i	−6667.97	+	4844.56i	−3710.65	−	11420.2i	−1201.68	−	873.071i	−40358.5	+	29322.2i	12780.2	−	39333.3i
5.11	−9.56884	−	29.4499i	40.7548	+	125.431i	−361.514	+	262.656i	2110.69	+	1533.50i	3303.93	−	2400.45i	812.899	+	2501.85i	−1631.95	−	1185.68i	1852.02	−	1345.57i	24964.6	−	76833.2i
5.12	−8.71019	−	26.8072i	83.7987	+	257.906i	−228.543	+	166.046i	27.1116	+	19.6978i	6183.83	−	4492.82i	1254.77	+	3861.79i	−5233.55	−	3802.40i	−43569.3	+	31655.0i	291.895	−	898.359i
5.13	−8.03302	−	24.7231i	−25.6263	−	78.8695i	−132.485	+	96.2563i	1426.71	+	1036.56i	−1744.04	+	1267.12i	1149.82	+	3538.79i	−7323.72	−	5320.99i	10360.2	−	7527.11i	14166.3	−	43599.3i
5.14	−7.93660	−	24.4263i	−24.4702	−	75.3116i	−119.440	+	86.7784i	−238.159	−	173.033i	−1645.38	+	1195.44i	−1517.90	−	4671.62i	−7570.86	−	5500.55i	10850.8	−	7883.59i	−2336.38	+	7190.65i
5.15	−7.84134	−	24.1332i	−70.4680	−	216.878i	−106.707	+	77.5269i	750.463	+	545.243i	−4681.40	+	3401.23i	2561.80	+	7884.42i	−7803.10	−	5669.29i	−26146.5	+	18996.6i	7273.81	−	22386.5i
5.16	−7.68354	−	23.6475i	56.1556	+	172.829i	−85.9513	+	62.4473i	342.030	+	248.499i	3655.51	−	2655.88i	−2956.31	−	9098.60i	−8162.14	−	5930.14i	−10792.6	+	7841.31i	3248.39	−	9997.50i
5.17	−5.95018	−	18.3128i	−56.7828	−	174.759i	114.264	−	83.0175i	−1670.78	−	1213.90i	−2862.46	+	2079.70i	188.376	+	579.761i	−10176.0	−	7393.29i	−11392.7	+	8277.26i	−12288.3	+	37819.6i
5.18	−5.38827	−	16.5834i	1.44096	+	4.43483i	168.242	−	122.235i	440.752	+	320.225i	65.7802	−	47.7921i	−2558.34	−	7873.75i	−10156.2	−	7378.92i	15906.3	−	11556.6i	2935.53	−	9034.62i
5.19	−4.48780	−	13.8120i	59.6413	+	183.557i	243.585	−	176.975i	−1556.51	−	1130.87i	2267.64	−	1647.53i	604.218	+	1859.59i	−9553.14	−	6940.76i	−14212.2	+	10325.8i	−8634.30	+	26573.7i
5.20	−3.16926	−	9.75399i	−48.0894	−	148.004i	329.121	−	239.120i	1034.28	+	751.449i	−1291.22	+	938.128i	1527.41	+	4700.90i	−7623.63	−	5538.89i	−3668.72	+	2665.48i	4051.72	−	12469.9i

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	71.10.c.a	✓	212
71.c	even	5	1	inner	71.10.c.a	✓	212

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
71.10.c.a	✓	212	1.a	even	1	1	trivial
71.10.c.a	✓	212	71.c	even	5	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(71, [\chi])\).