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Properties

Label 1980.2.f.a
Level $1980$
Weight $2$
Character orbit 1980.f
Analytic conductor $15.810$
Analytic rank $0$
Dimension $40$
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] = [1980,2,Mod(1871,1980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1980, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1980.1871");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1980.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: \(15.8103796002\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} - 4 q^{10} - 40 q^{11} - 16 q^{13} - 8 q^{14} + 4 q^{16} + 16 q^{23} - 40 q^{25} - 32 q^{26} + 8 q^{34} + 16 q^{37} + 4 q^{40} + 4 q^{44} + 8 q^{49} + 64 q^{56} - 80 q^{59} - 16 q^{61} - 56 q^{62} - 4 q^{64} + 16 q^{71} - 48 q^{73} - 40 q^{74} - 8 q^{80} - 16 q^{83} + 32 q^{85} + 48 q^{92} - 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1871.1 −1.41066 − 0.100255i 0 1.97990 + 0.282851i − 1.00000i 0 − 3.62050i −2.76460 − 0.597501i 0 −0.100255 + 1.41066i
1871.2 −1.41066 + 0.100255i 0 1.97990 − 0.282851i 1.00000i 0 3.62050i −2.76460 + 0.597501i 0 −0.100255 − 1.41066i
1871.3 −1.36355 − 0.375149i 0 1.71853 + 1.02307i 1.00000i 0 − 0.474085i −1.95949 − 2.03970i 0 0.375149 − 1.36355i
1871.4 −1.36355 + 0.375149i 0 1.71853 − 1.02307i − 1.00000i 0 0.474085i −1.95949 + 2.03970i 0 0.375149 + 1.36355i
1871.5 −1.28810 − 0.583782i 0 1.31840 + 1.50394i 1.00000i 0 2.56623i −0.820256 − 2.70688i 0 0.583782 − 1.28810i
1871.6 −1.28810 + 0.583782i 0 1.31840 − 1.50394i − 1.00000i 0 − 2.56623i −0.820256 + 2.70688i 0 0.583782 + 1.28810i
1871.7 −1.23500 − 0.689048i 0 1.05043 + 1.70194i − 1.00000i 0 2.59637i −0.124553 − 2.82568i 0 −0.689048 + 1.23500i
1871.8 −1.23500 + 0.689048i 0 1.05043 − 1.70194i 1.00000i 0 − 2.59637i −0.124553 + 2.82568i 0 −0.689048 − 1.23500i
1871.9 −0.918589 − 1.07526i 0 −0.312387 + 1.97545i − 1.00000i 0 − 2.59874i 2.41109 − 1.47873i 0 −1.07526 + 0.918589i
1871.10 −0.918589 + 1.07526i 0 −0.312387 − 1.97545i 1.00000i 0 2.59874i 2.41109 + 1.47873i 0 −1.07526 − 0.918589i
1871.11 −0.788911 − 1.17372i 0 −0.755239 + 1.85192i − 1.00000i 0 4.08701i 2.76945 − 0.574561i 0 −1.17372 + 0.788911i
1871.12 −0.788911 + 1.17372i 0 −0.755239 − 1.85192i 1.00000i 0 − 4.08701i 2.76945 + 0.574561i 0 −1.17372 − 0.788911i
1871.13 −0.783111 − 1.17760i 0 −0.773475 + 1.84438i 1.00000i 0 − 2.31892i 2.77765 − 0.533512i 0 1.17760 − 0.783111i
1871.14 −0.783111 + 1.17760i 0 −0.773475 − 1.84438i − 1.00000i 0 2.31892i 2.77765 + 0.533512i 0 1.17760 + 0.783111i
1871.15 −0.603826 − 1.27883i 0 −1.27079 + 1.54438i 1.00000i 0 − 0.397913i 2.74232 + 0.692583i 0 1.27883 − 0.603826i
1871.16 −0.603826 + 1.27883i 0 −1.27079 − 1.54438i − 1.00000i 0 0.397913i 2.74232 − 0.692583i 0 1.27883 + 0.603826i
1871.17 −0.172088 − 1.40370i 0 −1.94077 + 0.483121i − 1.00000i 0 0.822967i 1.01214 + 2.64113i 0 −1.40370 + 0.172088i
1871.18 −0.172088 + 1.40370i 0 −1.94077 − 0.483121i 1.00000i 0 − 0.822967i 1.01214 − 2.64113i 0 −1.40370 − 0.172088i
1871.19 −0.134258 − 1.40783i 0 −1.96395 + 0.378023i − 1.00000i 0 − 3.38883i 0.795866 + 2.71415i 0 −1.40783 + 0.134258i
1871.20 −0.134258 + 1.40783i 0 −1.96395 − 0.378023i 1.00000i 0 3.38883i 0.795866 − 2.71415i 0 −1.40783 − 0.134258i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1871.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1980.2.f.a ✓ 40
3.b odd 2 1 1980.2.f.b yes 40
4.b odd 2 1 1980.2.f.b yes 40
12.b even 2 1 inner 1980.2.f.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1980.2.f.a ✓ 40 1.a even 1 1 trivial
1980.2.f.a ✓ 40 12.b even 2 1 inner
1980.2.f.b yes 40 3.b odd 2 1
1980.2.f.b yes 40 4.b odd 2 1