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Properties

Label 80.20.c.d
Level $80$
Weight $20$
Character orbit 80.c
Analytic conductor $183.053$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [80,20,Mod(49,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.49");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 80.c (of order \(2\), degree \(1\), not 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: \(183.053357245\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 40)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 3935164 q^{5} - 10352560732 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 3935164 q^{5} - 10352560732 q^{9} - 18570995888 q^{11} - 79041415120 q^{15} + 1685869130672 q^{19} + 140419722832 q^{21} - 29558822439924 q^{25} + 160448197635496 q^{29} - 64251929934720 q^{31} + 88899634227664 q^{35} - 45\!\cdots\!04 q^{39}+ \cdots + 24\!\cdots\!92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 − 62769.7i 0 −4.29791e6 + 775547.i 0 1.11877e8i 0 −2.77777e9 0
49.2 0 − 61700.0i 0 560272. + 4.33123e6i 0 4.95314e7i 0 −2.64463e9 0
49.3 0 − 52989.4i 0 2.69002e6 − 3.44054e6i 0 − 1.42817e8i 0 −1.64561e9 0
49.4 0 − 51907.0i 0 3.18724e6 − 2.98580e6i 0 9.49555e7i 0 −1.53207e9 0
49.5 0 − 46696.1i 0 3.00829e6 + 3.16602e6i 0 − 1.36132e8i 0 −1.01826e9 0
49.6 0 − 41425.1i 0 −4.36383e6 − 174471.i 0 − 4.72501e7i 0 −5.53775e8 0
49.7 0 − 40783.0i 0 −1.74028e6 − 4.00561e6i 0 − 1.19363e8i 0 −5.00995e8 0
49.8 0 − 35066.5i 0 −1.65053e6 − 4.04342e6i 0 1.45627e8i 0 −6.73972e7 0
49.9 0 − 22508.3i 0 4.34396e6 + 451090.i 0 7.05814e7i 0 6.55637e8 0
49.10 0 − 20594.2i 0 −4.00905e6 + 1.73235e6i 0 − 2.36626e7i 0 7.38139e8 0
49.11 0 − 15095.8i 0 913879. + 4.27063e6i 0 1.11343e8i 0 9.34378e8 0
49.12 0 − 14155.3i 0 −2.17015e6 + 3.78998e6i 0 − 1.89772e8i 0 9.61890e8 0
49.13 0 − 7086.75i 0 3.46294e6 − 2.66112e6i 0 8.07683e7i 0 1.11204e9 0
49.14 0 − 332.013i 0 −1.90243e6 + 3.93119e6i 0 1.13879e8i 0 1.16215e9 0
49.15 0 332.013i 0 −1.90243e6 − 3.93119e6i 0 − 1.13879e8i 0 1.16215e9 0
49.16 0 7086.75i 0 3.46294e6 + 2.66112e6i 0 − 8.07683e7i 0 1.11204e9 0
49.17 0 14155.3i 0 −2.17015e6 − 3.78998e6i 0 1.89772e8i 0 9.61890e8 0
49.18 0 15095.8i 0 913879. − 4.27063e6i 0 − 1.11343e8i 0 9.34378e8 0
49.19 0 20594.2i 0 −4.00905e6 − 1.73235e6i 0 2.36626e7i 0 7.38139e8 0
49.20 0 22508.3i 0 4.34396e6 − 451090.i 0 − 7.05814e7i 0 6.55637e8 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.20.c.d 28
4.b odd 2 1 40.20.c.a ✓ 28
5.b even 2 1 inner 80.20.c.d 28
20.d odd 2 1 40.20.c.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.20.c.a ✓ 28 4.b odd 2 1
40.20.c.a ✓ 28 20.d odd 2 1
80.20.c.d 28 1.a even 1 1 trivial
80.20.c.d 28 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} + 21447940904 T_{3}^{26} + \cdots + 47\!\cdots\!00 \) acting on \(S_{20}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display