Newform orbit 71.10.g.a

• Label: 71.10.g.a
• Level: $71$
• Weight: $10$
• Character orbit: 71.g
• Analytic conductor: $36.568$
• Analytic rank: $0$
• Dimension: $1272$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	71.g (of order \(35\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(36.5675443676\)
Analytic rank:	\(0\)
Dimension:	\(1272\)
Relative dimension:	\(53\) over \(\Q(\zeta_{35})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1272 q - 76 q^{2} - 395 q^{3} + 13542 q^{4} - 1610 q^{5} - 4334 q^{6} - 4827 q^{7} - 18627 q^{8} + 286810 q^{9}+O(q^{10}) \)

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	−1.98523	+	44.2047i	177.763	+	155.306i	−1440.17	−	129.618i	−755.978	−	2326.66i	−7218.17	+	7549.61i	5364.21	+	1480.43i	5547.65	−	40954.4i	4837.31	+	35710.5i	104350.	−	28798.8i
2.2	−1.95578	+	43.5489i	−130.482	−	113.998i	−1382.74	−	124.449i	729.287	+	2244.51i	5219.69	−	5459.36i	−9550.39	−	2635.74i	5127.92	−	37855.8i	1387.71	+	10244.5i	−99172.4	+	27369.8i
2.3	−1.94237	+	43.2502i	−65.2666	−	57.0217i	−1356.87	−	122.121i	57.5622	+	177.158i	2592.98	−	2712.04i	10262.9	+	2832.38i	4941.81	−	36481.9i	−1633.86	−	12061.6i	−7773.94	+	2145.47i
2.4	−1.76410	+	39.2807i	−78.6078	−	68.6776i	−1029.92	−	92.6948i	−479.392	−	1475.42i	2836.38	−	2966.62i	−2440.90	−	673.645i	2755.62	−	20342.8i	−1179.54	−	8707.68i	58801.2	−	16228.1i
2.5	−1.75596	+	39.0994i	134.490	+	117.500i	−1015.74	−	91.4186i	629.359	+	1936.97i	−4830.34	+	5052.14i	4014.04	+	1107.80i	2668.11	−	19696.8i	1639.08	+	12100.2i	−76839.5	+	21206.3i
2.6	−1.71897	+	38.2758i	57.7916	+	50.4910i	−952.140	−	85.6942i	−148.076	−	455.730i	−2031.92	+	2125.23i	−6864.78	−	1894.56i	2283.46	−	16857.2i	−1851.58	−	13668.9i	17697.9	−	4884.32i
2.7	−1.68601	+	37.5420i	77.6854	+	67.8717i	−896.623	−	80.6975i	226.028	+	695.642i	−2679.02	+	2802.03i	−2823.28	−	779.175i	1958.50	−	14458.2i	−1213.66	−	8959.59i	−26496.9	+	7312.68i
2.8	−1.60524	+	35.7435i	−201.017	−	175.623i	−765.079	−	68.8584i	−311.108	−	957.493i	6600.05	−	6903.12i	3843.59	+	1060.76i	1230.34	−	9082.74i	6922.16	+	51101.5i	34723.5	−	9583.08i
2.9	−1.34663	+	29.9850i	−129.450	−	113.097i	−387.347	−	34.8619i	459.484	+	1414.15i	3565.52	−	3729.25i	3763.22	+	1038.58i	−495.923	+	3661.05i	1324.22	+	9775.79i	−43021.9	+	11873.3i
2.10	−1.33405	+	29.7049i	−5.04413	−	4.40692i	−370.660	−	33.3600i	−108.717	−	334.598i	137.636	−	143.956i	8583.36	+	2368.86i	−558.163	+	4120.53i	−2636.09	−	19460.4i	10084.2	−	2783.07i
2.11	−1.25358	+	27.9132i	194.677	+	170.084i	−267.637	−	24.0878i	−39.4486	−	121.410i	−4991.64	+	5220.85i	−6331.96	−	1747.51i	−912.463	+	6736.08i	6328.41	+	46718.2i	3438.41	−	948.939i
2.12	−1.22566	+	27.2915i	89.6817	+	78.3526i	−233.387	−	21.0052i	−738.642	−	2273.31i	−2248.28	+	2351.52i	2604.45	+	718.783i	−1018.25	+	7517.03i	−738.425	−	5451.27i	62947.3	−	17372.4i
2.13	−1.21652	+	27.0880i	−157.432	−	137.544i	−222.342	−	20.0111i	0.550643	+	1.69470i	3917.33	−	4097.20i	−7924.62	−	2187.06i	−1051.02	+	7758.94i	3224.33	+	23803.0i	−46.5760	+	12.8542i
2.14	−1.21648	+	27.0870i	−47.8440	−	41.8000i	−222.290	−	20.0064i	598.391	+	1841.66i	1190.44	−	1245.10i	1606.04	+	443.240i	−1051.17	+	7760.08i	−2100.31	−	15505.1i	−50613.0	+	13968.3i
2.15	−1.07546	+	23.9470i	111.649	+	97.5446i	−62.3628	−	5.61275i	−241.002	−	741.727i	−2455.97	+	2568.75i	8147.66	+	2248.61i	−1446.00	+	10674.8i	308.388	+	2276.61i	18021.3	−	4973.57i
2.16	−1.05362	+	23.4606i	−62.0217	−	54.1867i	−39.3519	−	3.54173i	−781.496	−	2405.20i	1336.60	−	1397.98i	−5181.59	−	1430.03i	−1489.46	+	10995.6i	−1731.62	−	12783.4i	57250.8	−	15800.2i
2.17	−0.814302	+	18.1318i	44.5835	+	38.9515i	181.838	+	16.3657i	601.072	+	1849.91i	−742.566	+	776.664i	−8764.18	−	2418.76i	−1692.22	+	12492.5i	−2171.64	−	16031.7i	−34031.7	+	9392.15i
2.18	−0.686555	+	15.2873i	30.1497	+	26.3410i	276.707	+	24.9041i	−137.078	−	421.881i	−423.383	+	442.824i	−9439.10	−	2605.03i	−1622.41	+	11977.1i	−2426.96	−	17916.5i	6543.55	−	1805.91i
2.19	−0.665801	+	14.8252i	166.472	+	145.442i	290.595	+	26.1541i	466.454	+	1435.60i	−2267.05	+	2371.15i	6112.18	+	1686.86i	−1601.14	+	11820.1i	3917.40	+	28919.4i	−21593.6	+	5959.46i
2.20	−0.552600	+	12.3046i	−87.2823	−	76.2563i	358.841	+	32.2963i	−62.7331	−	193.073i	986.535	−	1031.83i	−1072.51	−	295.994i	−1442.20	+	10646.8i	−838.927	−	6193.21i	2410.35	−	665.213i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
71.g	even	35	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	71.10.g.a	✓	1272
71.g	even	35	1	inner	71.10.g.a	✓	1272

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
71.10.g.a	✓	1272	1.a	even	1	1	trivial
71.10.g.a	✓	1272	71.g	even	35	1	inner

Hecke kernels

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