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Properties

Label 71.10.g.a
Level $71$
Weight $10$
Character orbit 71.g
Analytic conductor $36.568$
Analytic rank $0$
Dimension $1272$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [71,10,Mod(2,71)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("71.2"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(71, base_ring=CyclotomicField(70)) chi = DirichletCharacter(H, H._module([6])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 71.g (of order \(35\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1272 q - 76 q^{2} - 395 q^{3} + 13542 q^{4} - 1610 q^{5} - 4334 q^{6} - 4827 q^{7} - 18627 q^{8} + 286810 q^{9} - 53448 q^{10} - 170685 q^{11} + 138323 q^{12} - 93291 q^{13} + 985964 q^{14} - 966194 q^{15}+ \cdots - 4074592265 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
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
See next 80 embeddings (of 1272 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2.53
Significant digits:
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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 \(S_{10}^{\mathrm{new}}(71, [\chi])\).