Newform orbit 71.6.c.a

• Label: 71.6.c.a
• Level: $71$
• Weight: $6$
• Character orbit: 71.c
• Analytic conductor: $11.387$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	71.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.3872512067\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(29\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 116 q - q^{3} - 418 q^{4} - 61 q^{5} - 110 q^{6} + 95 q^{7} - 1065 q^{8} - 2458 q^{9}+O(q^{10}) \)

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	−3.33100	−	10.2518i	−7.26205	−	22.3503i	−68.1147	+	49.4882i	−19.9276	−	14.4782i	−204.940	+	148.898i	−64.3591	−	198.077i	455.170	+	330.701i	−250.207	+	181.786i	−82.0487	+	252.520i
5.2	−3.20618	−	9.86762i	6.01209	+	18.5033i	−61.2018	+	44.4657i	−11.5116	−	8.36365i	163.308	−	118.650i	18.5953	+	57.2306i	366.390	+	266.198i	−109.637	+	79.6556i	−45.6211	+	140.407i
5.3	−2.99295	−	9.21136i	−2.42281	−	7.45665i	−50.0029	+	36.3292i	20.2283	+	14.6968i	−61.4346	+	44.6348i	49.1630	+	151.308i	233.557	+	169.689i	146.859	−	106.700i	74.8346	−	230.317i
5.4	−2.66477	−	8.20132i	3.29458	+	10.1397i	−34.2722	+	24.9002i	−47.8417	−	34.7590i	74.3794	−	54.0398i	−47.7925	−	147.090i	72.2951	+	52.5254i	104.632	−	76.0199i	−157.583	+	484.990i
5.5	−2.40876	−	7.41339i	−1.36768	−	4.20929i	−23.2676	+	16.9049i	72.3702	+	52.5800i	−27.9107	+	20.2783i	−28.1890	−	86.7568i	−20.4296	−	14.8430i	180.744	−	131.318i	215.474	−	663.161i
5.6	−2.27398	−	6.99860i	8.43856	+	25.9712i	−17.9208	+	13.0202i	63.2579	+	45.9596i	162.573	−	118.116i	−20.2955	−	62.4631i	−58.6327	−	42.5992i	−406.704	+	295.487i	177.805	−	547.228i
5.7	−2.17245	−	6.68612i	−3.87537	−	11.9272i	−14.0961	+	10.2415i	−88.4058	−	64.2306i	−71.3274	+	51.8224i	30.8226	+	94.8624i	−82.9030	−	60.2325i	69.3524	−	50.3875i	−237.396	+	730.630i
5.8	−1.94507	−	5.98630i	−9.56859	−	29.4491i	−6.16396	+	4.47838i	21.7735	+	15.8194i	−157.680	+	114.561i	46.0600	+	141.758i	−124.154	−	90.2030i	−579.100	+	420.741i	52.3487	−	161.113i
5.9	−1.44413	−	4.44459i	3.29320	+	10.1354i	8.21970	−	5.97196i	−19.4412	−	14.1249i	40.2920	−	29.2739i	12.5400	+	38.5942i	−159.399	−	115.810i	104.709	−	76.0757i	−34.7035	+	106.806i
5.10	−1.40376	−	4.32034i	−5.82706	−	17.9338i	9.19374	−	6.67964i	0.0676466	+	0.0491482i	−69.3005	+	50.3498i	−41.7929	−	128.625i	−159.367	−	115.787i	−91.0770	+	66.1713i	0.117377	−	0.361249i
5.11	−1.35119	−	4.15853i	5.70847	+	17.5689i	10.4209	−	7.57124i	15.7827	+	11.4668i	65.3474	−	47.4776i	63.5118	+	195.469i	−158.764	−	115.349i	−79.4874	+	57.7510i	26.3596	−	81.1266i
5.12	−0.660124	−	2.03165i	9.55316	+	29.4016i	22.1967	−	16.1268i	−76.5190	−	55.5943i	53.4276	−	38.8174i	−35.5872	−	109.526i	−102.720	−	74.6305i	−576.600	+	418.924i	−62.4363	+	192.159i
5.13	−0.630690	−	1.94106i	−1.89821	−	5.84208i	22.5186	−	16.3607i	−19.5333	−	14.1918i	−10.1427	+	7.36908i	−3.61343	−	11.1210i	−98.7967	−	71.7800i	166.064	−	120.653i	−15.2277	+	46.8660i
5.14	−0.187375	−	0.576681i	−4.06995	−	12.5260i	25.5911	−	18.5930i	64.3536	+	46.7556i	−6.46090	+	4.69412i	30.8651	+	94.9931i	−31.2151	−	22.6791i	56.2548	−	40.8715i	14.9048	−	45.8723i
5.15	−0.167057	−	0.514147i	3.29325	+	10.1356i	25.6521	−	18.6373i	25.4151	+	18.4651i	4.66102	−	3.38643i	−75.2779	−	231.681i	−27.8632	−	20.2438i	104.707	−	76.0739i	5.24804	−	16.1518i
5.16	0.370897	+	1.14150i	4.98292	+	15.3358i	24.7231	−	17.9624i	75.9402	+	55.1738i	−15.6578	+	11.3760i	5.96485	+	18.3579i	60.7464	+	44.1349i	−13.7675	+	10.0026i	−34.8150	+	107.150i
5.17	0.546101	+	1.68073i	−6.90607	−	21.2547i	23.3619	−	16.9734i	−62.3192	−	45.2776i	31.9519	−	23.2144i	−30.5984	−	94.1721i	87.0365	+	63.2357i	−207.477	+	150.741i	42.0666	−	129.468i
5.18	0.725500	+	2.23286i	3.62874	+	11.1681i	21.4292	−	15.5692i	−57.1245	−	41.5034i	−22.3042	+	16.2049i	46.1923	+	142.165i	111.091	+	80.7124i	85.0320	−	61.7793i	51.2274	−	157.662i
5.19	0.736167	+	2.26569i	−4.14679	−	12.7625i	21.2971	−	15.4733i	−8.08116	−	5.87131i	25.8632	−	18.7907i	64.8422	+	199.564i	112.410	+	81.6705i	50.9052	−	36.9848i	7.35347	−	22.6317i
5.20	1.45383	+	4.47443i	−8.36952	−	25.7587i	7.98162	−	5.79899i	52.3512	+	38.0353i	103.088	−	74.8977i	−41.3536	−	127.273i	159.349	+	115.774i	−396.873	+	288.345i	−94.0768	+	289.539i

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	71.6.c.a	✓	116
71.c	even	5	1	inner	71.6.c.a	✓	116

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
71.6.c.a	✓	116	1.a	even	1	1	trivial
71.6.c.a	✓	116	71.c	even	5	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(71, [\chi])\).