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Properties

Label 40.8.f.a
Level $40$
Weight $8$
Character orbit 40.f
Analytic conductor $12.495$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [40,8,Mod(29,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.29");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 40.f (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.4954010194\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 28 q^{4} - 204 q^{6} + 26240 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 28 q^{4} - 204 q^{6} + 26240 q^{9} + 1312 q^{10} - 15108 q^{14} - 4376 q^{15} + 23000 q^{16} + 70044 q^{20} + 117008 q^{24} - 8064 q^{25} + 55464 q^{26} + 347052 q^{30} - 625520 q^{31} - 24232 q^{34} + 1063116 q^{36} - 957856 q^{39} + 869792 q^{40} - 441288 q^{41} - 314856 q^{44} - 1959564 q^{46} - 3294176 q^{49} - 912648 q^{50} + 2702384 q^{54} + 2251752 q^{55} - 4150176 q^{56} - 5626808 q^{60} + 1967264 q^{64} - 2463240 q^{65} + 12743048 q^{66} + 11782012 q^{70} + 3847152 q^{71} - 12318816 q^{74} + 26732232 q^{76} + 9729840 q^{79} + 15525480 q^{80} + 9702688 q^{81} - 16965112 q^{84} - 43598652 q^{86} + 15759840 q^{89} - 35763560 q^{90} + 1478820 q^{94} - 2170824 q^{95} - 57790288 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −11.2434 − 1.25952i 70.5054 124.827 + 28.3224i −35.7394 − 277.214i −792.719 − 88.8027i − 1255.84i −1367.81 − 475.662i 2784.01 52.6756 + 3161.84i
29.2 −11.2434 + 1.25952i 70.5054 124.827 − 28.3224i −35.7394 + 277.214i −792.719 + 88.8027i 1255.84i −1367.81 + 475.662i 2784.01 52.6756 − 3161.84i
29.3 −11.0855 − 2.26111i −35.0518 117.775 + 50.1309i −246.912 − 130.994i 388.566 + 79.2560i 1440.38i −1192.24 − 822.025i −958.369 2440.94 + 2010.42i
29.4 −11.0855 + 2.26111i −35.0518 117.775 − 50.1309i −246.912 + 130.994i 388.566 − 79.2560i − 1440.38i −1192.24 + 822.025i −958.369 2440.94 − 2010.42i
29.5 −10.4160 − 4.41671i 3.15519 88.9853 + 92.0088i 235.904 + 149.914i −32.8644 − 13.9356i − 123.632i −520.493 − 1351.38i −2177.04 −1795.05 − 2603.42i
29.6 −10.4160 + 4.41671i 3.15519 88.9853 − 92.0088i 235.904 − 149.914i −32.8644 + 13.9356i 123.632i −520.493 + 1351.38i −2177.04 −1795.05 + 2603.42i
29.7 −10.0446 − 5.20635i −85.1361 73.7879 + 104.591i 163.129 − 226.967i 855.158 + 443.248i − 1132.18i −196.632 − 1434.74i 5061.16 −2820.23 + 1430.49i
29.8 −10.0446 + 5.20635i −85.1361 73.7879 − 104.591i 163.129 + 226.967i 855.158 − 443.248i 1132.18i −196.632 + 1434.74i 5061.16 −2820.23 − 1430.49i
29.9 −8.11437 − 7.88397i 50.9148 3.68600 + 127.947i −244.226 + 135.936i −413.142 − 401.411i − 534.587i 978.820 − 1067.27i 405.322 3053.46 + 822.431i
29.10 −8.11437 + 7.88397i 50.9148 3.68600 − 127.947i −244.226 − 135.936i −413.142 + 401.411i 534.587i 978.820 + 1067.27i 405.322 3053.46 − 822.431i
29.11 −7.07891 − 8.82547i 18.1977 −27.7779 + 124.950i −5.66487 − 279.451i −128.820 − 160.603i 646.347i 1299.38 − 639.354i −1855.84 −2426.19 + 2028.21i
29.12 −7.07891 + 8.82547i 18.1977 −27.7779 − 124.950i −5.66487 + 279.451i −128.820 + 160.603i − 646.347i 1299.38 + 639.354i −1855.84 −2426.19 − 2028.21i
29.13 −6.15121 − 9.49540i −54.6083 −52.3254 + 116.816i −96.2712 + 262.406i 335.907 + 518.528i 75.2701i 1431.08 − 221.711i 795.063 3083.83 − 699.978i
29.14 −6.15121 + 9.49540i −54.6083 −52.3254 − 116.816i −96.2712 − 262.406i 335.907 − 518.528i − 75.2701i 1431.08 + 221.711i 795.063 3083.83 + 699.978i
29.15 −4.44800 − 10.4027i 80.4449 −88.4307 + 92.5420i 276.446 + 41.2614i −357.819 − 836.841i 281.883i 1356.02 + 508.288i 4284.38 −800.404 − 3059.31i
29.16 −4.44800 + 10.4027i 80.4449 −88.4307 − 92.5420i 276.446 − 41.2614i −357.819 + 836.841i − 281.883i 1356.02 − 508.288i 4284.38 −800.404 + 3059.31i
29.17 −1.58390 − 11.2023i −21.4733 −122.983 + 35.4866i −162.054 − 227.735i 34.0116 + 240.551i − 1461.90i 592.323 + 1321.48i −1725.90 −2294.48 + 2176.09i
29.18 −1.58390 + 11.2023i −21.4733 −122.983 − 35.4866i −162.054 + 227.735i 34.0116 − 240.551i 1461.90i 592.323 − 1321.48i −1725.90 −2294.48 − 2176.09i
29.19 −1.31439 − 11.2371i −46.1976 −124.545 + 29.5398i 271.075 − 68.1421i 60.7216 + 519.126i 1088.21i 495.642 + 1360.69i −52.7861 −1122.02 − 2956.53i
29.20 −1.31439 + 11.2371i −46.1976 −124.545 − 29.5398i 271.075 + 68.1421i 60.7216 − 519.126i − 1088.21i 495.642 − 1360.69i −52.7861 −1122.02 + 2956.53i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.8.f.a ✓ 40
4.b odd 2 1 160.8.f.a 40
5.b even 2 1 inner 40.8.f.a ✓ 40
8.b even 2 1 inner 40.8.f.a ✓ 40
8.d odd 2 1 160.8.f.a 40
20.d odd 2 1 160.8.f.a 40
40.e odd 2 1 160.8.f.a 40
40.f even 2 1 inner 40.8.f.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.8.f.a ✓ 40 1.a even 1 1 trivial
40.8.f.a ✓ 40 5.b even 2 1 inner
40.8.f.a ✓ 40 8.b even 2 1 inner
40.8.f.a ✓ 40 40.f even 2 1 inner
160.8.f.a 40 4.b odd 2 1
160.8.f.a 40 8.d odd 2 1
160.8.f.a 40 20.d odd 2 1
160.8.f.a 40 40.e odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(40, [\chi])\).