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Properties

Label 71.8.a.b
Level $71$
Weight $8$
Character orbit 71.a
Self dual yes
Analytic conductor $22.179$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $1$

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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] = [71,8,Mod(1,71)] 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("71.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(71, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 71.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.1793368094\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 40 q^{3} + 1818 q^{4} + 570 q^{5} + 1254 q^{6} + 2400 q^{7} + 23522 q^{9} + 6430 q^{10} + 5760 q^{11} + 7680 q^{12} + 26660 q^{13} + 6528 q^{14} - 3888 q^{15} + 144258 q^{16} + 33360 q^{17} - 123445 q^{18}+ \cdots + 26843490 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −21.9409 −76.6441 353.402 −313.547 1681.64 1023.90 −4945.53 3687.31 6879.50
1.2 −20.9330 79.0778 310.189 −254.815 −1655.33 1300.86 −3813.75 4066.29 5334.04
1.3 −17.2388 −83.4843 169.178 544.588 1439.17 −632.084 −709.858 4782.62 −9388.07
1.4 −16.8369 24.4501 155.482 −504.377 −411.664 −891.927 −462.716 −1589.19 8492.16
1.5 −16.0075 −28.5065 128.239 47.2760 456.317 377.866 −3.83297 −1374.38 −756.769
1.6 −15.2942 −47.3129 105.914 306.266 723.615 1524.48 337.790 51.5112 −4684.11
1.7 −14.6492 85.9397 86.5986 −6.36437 −1258.95 −1755.01 606.497 5198.63 93.2328
1.8 −13.7705 49.0513 61.6262 278.251 −675.460 969.459 913.999 219.031 −3831.66
1.9 −8.26701 −30.4778 −59.6566 76.1158 251.960 −1080.34 1551.36 −1258.10 −629.250
1.10 −1.64138 16.7176 −125.306 −237.156 −27.4399 85.2710 415.772 −1907.52 389.264
1.11 −1.09868 26.9839 −126.793 −305.803 −29.6468 −1258.12 279.937 −1458.87 335.981
1.12 −0.997940 7.31862 −127.004 470.148 −7.30354 1407.18 254.479 −2133.44 −469.180
1.13 0.158322 82.9163 −127.975 384.519 13.1275 89.0017 −40.5265 4688.12 60.8779
1.14 4.09330 −38.1209 −111.245 −396.273 −156.041 530.103 −979.302 −733.795 −1622.07
1.15 5.99496 −66.4393 −92.0605 516.447 −398.301 −1311.13 −1319.25 2227.18 3096.08
1.16 6.00907 −55.9464 −91.8911 −300.280 −336.186 −1399.84 −1321.34 943.000 −1804.40
1.17 9.98759 60.1350 −28.2481 −180.167 600.603 1547.20 −1560.54 1429.21 −1799.44
1.18 10.8199 −41.1330 −10.9306 139.218 −445.054 18.3242 −1503.21 −495.074 1506.32
1.19 15.4481 69.2612 110.645 272.315 1069.96 −453.610 −268.110 2610.12 4206.76
1.20 15.4949 −79.5869 112.092 −381.837 −1233.19 247.838 −246.492 4147.08 −5916.52
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.24
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(71\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.8.a.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.8.a.b ✓ 24 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 2445 T_{2}^{22} + 2581812 T_{2}^{20} - 239600 T_{2}^{19} - 1543388120 T_{2}^{18} + \cdots + 96\!\cdots\!24 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(71))\). Copy content Toggle raw display