Properties

Label 40.11.e.c
Level $40$
Weight $11$
Character orbit 40.e
Analytic conductor $25.414$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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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] = [40,11,Mod(19,40)] 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("40.19"); 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, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 40.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.4142901069\)
Analytic rank: \(0\)
Dimension: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2660 q^{4} + 4476 q^{6} - 1180984 q^{9} - 34040 q^{10} - 642208 q^{11} + 157684 q^{14} - 3599144 q^{16} - 2529440 q^{19} + 7209620 q^{20} - 16455824 q^{24} - 18792760 q^{25} - 9893384 q^{26} + 49545460 q^{30}+ \cdots - 23796212768 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −31.8732 − 2.84572i − 227.354i 1007.80 + 181.405i −1323.91 + 2830.70i −646.985 + 7246.50i −8327.09 −31605.7 − 8649.87i 7359.27 50252.6 − 86456.2i
19.2 −31.8732 + 2.84572i 227.354i 1007.80 − 181.405i −1323.91 − 2830.70i −646.985 − 7246.50i −8327.09 −31605.7 + 8649.87i 7359.27 50252.6 + 86456.2i
19.3 −31.3026 − 6.64441i − 415.330i 935.704 + 415.975i −1829.08 − 2533.79i −2759.63 + 13000.9i 30519.9 −26526.0 − 19238.3i −113450. 40419.4 + 91467.3i
19.4 −31.3026 + 6.64441i 415.330i 935.704 − 415.975i −1829.08 + 2533.79i −2759.63 − 13000.9i 30519.9 −26526.0 + 19238.3i −113450. 40419.4 − 91467.3i
19.5 −30.6229 − 9.28657i 302.298i 851.519 + 568.763i 2172.74 + 2246.07i 2807.31 − 9257.23i −24260.0 −20794.1 − 25324.8i −32335.1 −45677.2 − 88958.4i
19.6 −30.6229 + 9.28657i − 302.298i 851.519 − 568.763i 2172.74 − 2246.07i 2807.31 + 9257.23i −24260.0 −20794.1 + 25324.8i −32335.1 −45677.2 + 88958.4i
19.7 −27.8541 − 15.7527i 14.5140i 527.703 + 877.557i 2258.09 − 2160.25i 228.636 − 404.275i 4652.12 −874.765 − 32756.3i 58838.3 −96926.8 + 24600.7i
19.8 −27.8541 + 15.7527i − 14.5140i 527.703 − 877.557i 2258.09 + 2160.25i 228.636 + 404.275i 4652.12 −874.765 + 32756.3i 58838.3 −96926.8 − 24600.7i
19.9 −26.5665 − 17.8388i 135.709i 387.554 + 947.828i −3017.77 + 811.587i 2420.88 − 3605.30i 7097.43 6612.16 − 32093.9i 40632.1 94649.3 + 32272.4i
19.10 −26.5665 + 17.8388i − 135.709i 387.554 − 947.828i −3017.77 − 811.587i 2420.88 + 3605.30i 7097.43 6612.16 + 32093.9i 40632.1 94649.3 − 32272.4i
19.11 −24.3986 − 20.7053i − 398.950i 166.579 + 1010.36i 3066.76 + 600.514i −8260.38 + 9733.80i −3548.75 16855.5 − 28100.4i −100112. −62390.6 − 78149.9i
19.12 −24.3986 + 20.7053i 398.950i 166.579 − 1010.36i 3066.76 − 600.514i −8260.38 − 9733.80i −3548.75 16855.5 + 28100.4i −100112. −62390.6 + 78149.9i
19.13 −23.4375 − 21.7873i − 223.782i 74.6308 + 1021.28i −2093.20 − 2320.37i −4875.59 + 5244.88i −28042.9 20501.7 − 25562.2i 8970.71 −1495.27 + 99988.8i
19.14 −23.4375 + 21.7873i 223.782i 74.6308 − 1021.28i −2093.20 + 2320.37i −4875.59 − 5244.88i −28042.9 20501.7 + 25562.2i 8970.71 −1495.27 − 99988.8i
19.15 −22.6252 − 22.6296i 462.958i −0.196147 + 1024.00i 916.532 − 2987.57i 10476.5 − 10474.5i 13440.8 23177.1 − 23163.8i −155281. −88344.3 + 46853.8i
19.16 −22.6252 + 22.6296i − 462.958i −0.196147 − 1024.00i 916.532 + 2987.57i 10476.5 + 10474.5i 13440.8 23177.1 + 23163.8i −155281. −88344.3 − 46853.8i
19.17 −17.9281 − 26.5063i − 149.954i −381.167 + 950.415i 1130.63 + 2913.30i −3974.73 + 2688.39i 13082.9 32025.6 − 6935.81i 36562.8 56950.7 − 82198.7i
19.18 −17.9281 + 26.5063i 149.954i −381.167 − 950.415i 1130.63 − 2913.30i −3974.73 − 2688.39i 13082.9 32025.6 + 6935.81i 36562.8 56950.7 + 82198.7i
19.19 −14.5220 − 28.5151i 292.137i −602.221 + 828.194i 668.290 + 3052.71i 8330.33 − 4242.43i 4269.43 32361.5 + 5145.35i −26295.3 77343.3 − 63387.8i
19.20 −14.5220 + 28.5151i − 292.137i −602.221 − 828.194i 668.290 − 3052.71i 8330.33 + 4242.43i 4269.43 32361.5 − 5145.35i −26295.3 77343.3 + 63387.8i
See all 56 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 19.56
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
40.e 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.e.c ✓ 56
4.b odd 2 1 160.11.e.c 56
5.b even 2 1 inner 40.11.e.c ✓ 56
8.b even 2 1 160.11.e.c 56
8.d odd 2 1 inner 40.11.e.c ✓ 56
20.d odd 2 1 160.11.e.c 56
40.e odd 2 1 inner 40.11.e.c ✓ 56
40.f even 2 1 160.11.e.c 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.11.e.c ✓ 56 1.a even 1 1 trivial
40.11.e.c ✓ 56 5.b even 2 1 inner
40.11.e.c ✓ 56 8.d odd 2 1 inner
40.11.e.c ✓ 56 40.e odd 2 1 inner
160.11.e.c 56 4.b odd 2 1
160.11.e.c 56 8.b even 2 1
160.11.e.c 56 20.d odd 2 1
160.11.e.c 56 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(40, [\chi])\):

\( T_{3}^{28} + 1121932 T_{3}^{26} + 552887698916 T_{3}^{24} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{28} - 4198529092 T_{7}^{26} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display