Properties

Label 40.11.g.a
Level $40$
Weight $11$
Character orbit 40.g
Analytic conductor $25.414$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [40,11,Mod(11,40)] 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("40.11"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 40.g (of order \(2\), degree \(1\), 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: \(25.4142901069\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 22 q^{2} - 1864 q^{4} - 19644 q^{6} - 3448 q^{8} + 787320 q^{9} - 31250 q^{10} - 91808 q^{11} - 111340 q^{12} + 371356 q^{14} + 2799880 q^{16} - 974678 q^{18} - 5107040 q^{19} - 1187500 q^{20} - 21770780 q^{22}+ \cdots + 19036908320 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −31.9776 − 1.19585i −112.570 1021.14 + 76.4809i 1397.54i 3599.74 + 134.617i − 15404.2i −32562.2 − 3666.81i −46376.9 1671.25 − 44690.1i
11.2 −31.9776 + 1.19585i −112.570 1021.14 − 76.4809i − 1397.54i 3599.74 − 134.617i 15404.2i −32562.2 + 3666.81i −46376.9 1671.25 + 44690.1i
11.3 −30.9125 − 8.27149i 193.599 887.165 + 511.385i 1397.54i −5984.62 − 1601.35i 10909.2i −23194.6 − 23146.4i −21568.5 11559.8 − 43201.5i
11.4 −30.9125 + 8.27149i 193.599 887.165 − 511.385i − 1397.54i −5984.62 + 1601.35i − 10909.2i −23194.6 + 23146.4i −21568.5 11559.8 + 43201.5i
11.5 −29.7310 − 11.8351i 457.207 743.862 + 703.736i − 1397.54i −13593.2 − 5411.07i 22956.3i −13787.0 − 29726.4i 149989. −16540.0 + 41550.3i
11.6 −29.7310 + 11.8351i 457.207 743.862 − 703.736i 1397.54i −13593.2 + 5411.07i − 22956.3i −13787.0 + 29726.4i 149989. −16540.0 − 41550.3i
11.7 −27.0186 − 17.1463i −202.048 436.007 + 926.539i − 1397.54i 5459.05 + 3464.38i − 4689.91i 4106.46 − 32509.7i −18225.7 −23962.7 + 37759.6i
11.8 −27.0186 + 17.1463i −202.048 436.007 − 926.539i 1397.54i 5459.05 − 3464.38i 4689.91i 4106.46 + 32509.7i −18225.7 −23962.7 − 37759.6i
11.9 −24.4987 − 20.5867i −266.761 176.373 + 1008.70i 1397.54i 6535.29 + 5491.73i 30081.9i 16444.9 − 28342.7i 12112.2 28770.8 − 34238.0i
11.10 −24.4987 + 20.5867i −266.761 176.373 − 1008.70i − 1397.54i 6535.29 − 5491.73i − 30081.9i 16444.9 + 28342.7i 12112.2 28770.8 + 34238.0i
11.11 −18.4957 − 26.1134i −309.151 −339.818 + 965.971i 1397.54i 5717.96 + 8072.98i − 27916.6i 31509.9 − 8992.52i 36525.3 36494.6 − 25848.5i
11.12 −18.4957 + 26.1134i −309.151 −339.818 − 965.971i − 1397.54i 5717.96 − 8072.98i 27916.6i 31509.9 + 8992.52i 36525.3 36494.6 + 25848.5i
11.13 −16.1343 − 27.6348i 268.045 −503.366 + 891.739i 1397.54i −4324.73 − 7407.38i 425.075i 32764.5 − 477.182i 12799.3 38620.8 − 22548.4i
11.14 −16.1343 + 27.6348i 268.045 −503.366 − 891.739i − 1397.54i −4324.73 + 7407.38i − 425.075i 32764.5 + 477.182i 12799.3 38620.8 + 22548.4i
11.15 −14.9888 − 28.2725i 94.5150 −574.672 + 847.543i − 1397.54i −1416.67 − 2672.18i 9969.20i 32575.8 + 3543.77i −50115.9 −39512.1 + 20947.5i
11.16 −14.9888 + 28.2725i 94.5150 −574.672 − 847.543i 1397.54i −1416.67 + 2672.18i − 9969.20i 32575.8 − 3543.77i −50115.9 −39512.1 − 20947.5i
11.17 −5.88008 − 31.4551i −406.713 −954.849 + 369.917i − 1397.54i 2391.50 + 12793.2i − 1327.68i 17250.4 + 27859.8i 106366. −43959.9 + 8217.67i
11.18 −5.88008 + 31.4551i −406.713 −954.849 − 369.917i 1397.54i 2391.50 − 12793.2i 1327.68i 17250.4 − 27859.8i 106366. −43959.9 − 8217.67i
11.19 −3.78229 − 31.7757i 431.966 −995.389 + 240.370i − 1397.54i −1633.82 − 13726.0i − 17773.3i 11402.8 + 30720.0i 127545. −44407.9 + 5285.91i
11.20 −3.78229 + 31.7757i 431.966 −995.389 − 240.370i 1397.54i −1633.82 + 13726.0i 17773.3i 11402.8 − 30720.0i 127545. −44407.9 − 5285.91i
See all 40 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.40
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.11.g.a ✓ 40
4.b odd 2 1 160.11.g.a 40
8.b even 2 1 160.11.g.a 40
8.d odd 2 1 inner 40.11.g.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.11.g.a ✓ 40 1.a even 1 1 trivial
40.11.g.a ✓ 40 8.d odd 2 1 inner
160.11.g.a 40 4.b odd 2 1
160.11.g.a 40 8.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(40, [\chi])\).