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Properties

Label 273.2.e.a
Level $273$
Weight $2$
Character orbit 273.e
Analytic conductor $2.180$
Analytic rank $0$
Dimension $32$
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] = [273,2,Mod(209,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.e (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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} + 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} + 4 q^{7} - 8 q^{9} - 12 q^{15} + 16 q^{16} - 20 q^{18} - 4 q^{21} - 16 q^{22} - 28 q^{28} + 16 q^{30} + 24 q^{36} + 24 q^{37} + 32 q^{43} - 24 q^{46} - 24 q^{49} - 8 q^{51} + 32 q^{57} + 24 q^{58} - 28 q^{60} + 8 q^{63} + 48 q^{64} - 32 q^{67} - 8 q^{70} + 64 q^{72} + 20 q^{78} - 32 q^{79} + 32 q^{81} - 48 q^{84} - 16 q^{85} + 64 q^{88} + 4 q^{91} - 52 q^{93} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 − 2.62996i −0.518189 + 1.65272i −4.91669 2.48925 4.34658 + 1.36282i 1.33971 − 2.28149i 7.67077i −2.46296 − 1.71284i − 6.54663i
209.2 − 2.62996i 0.518189 − 1.65272i −4.91669 −2.48925 −4.34658 − 1.36282i 1.33971 + 2.28149i 7.67077i −2.46296 − 1.71284i 6.54663i
209.3 − 2.37766i −1.58497 + 0.698482i −3.65324 −1.30604 1.66075 + 3.76851i −0.0678621 + 2.64488i 3.93085i 2.02425 − 2.21414i 3.10530i
209.4 − 2.37766i 1.58497 − 0.698482i −3.65324 1.30604 −1.66075 − 3.76851i −0.0678621 − 2.64488i 3.93085i 2.02425 − 2.21414i − 3.10530i
209.5 − 1.95979i −1.02881 − 1.39340i −1.84076 −2.34217 −2.73076 + 2.01624i 0.446056 − 2.60788i − 0.312071i −0.883116 + 2.86707i 4.59015i
209.6 − 1.95979i 1.02881 + 1.39340i −1.84076 2.34217 2.73076 − 2.01624i 0.446056 + 2.60788i − 0.312071i −0.883116 + 2.86707i − 4.59015i
209.7 − 1.86253i −0.150306 + 1.72552i −1.46902 −2.96439 3.21383 + 0.279950i −2.41066 − 1.09028i − 0.988960i −2.95482 − 0.518713i 5.52126i
209.8 − 1.86253i 0.150306 − 1.72552i −1.46902 2.96439 −3.21383 − 0.279950i −2.41066 + 1.09028i − 0.988960i −2.95482 − 0.518713i − 5.52126i
209.9 − 1.71574i −1.66803 − 0.466569i −0.943779 2.59497 −0.800512 + 2.86191i 2.59496 + 0.515930i − 1.81221i 2.56463 + 1.55650i − 4.45230i
209.10 − 1.71574i 1.66803 + 0.466569i −0.943779 −2.59497 0.800512 − 2.86191i 2.59496 − 0.515930i − 1.81221i 2.56463 + 1.55650i 4.45230i
209.11 − 0.875536i −1.71925 + 0.210227i 1.23344 −0.171607 0.184061 + 1.50526i −2.26296 − 1.37077i − 2.83099i 2.91161 − 0.722864i 0.150248i
209.12 − 0.875536i 1.71925 − 0.210227i 1.23344 0.171607 −0.184061 − 1.50526i −2.26296 + 1.37077i − 2.83099i 2.91161 − 0.722864i − 0.150248i
209.13 − 0.633614i −0.915380 + 1.47040i 1.59853 0.119728 0.931666 + 0.579998i 2.16525 − 1.52042i − 2.28008i −1.32416 − 2.69195i − 0.0758615i
209.14 − 0.633614i 0.915380 − 1.47040i 1.59853 −0.119728 −0.931666 − 0.579998i 2.16525 + 1.52042i − 2.28008i −1.32416 − 2.69195i 0.0758615i
209.15 − 0.0920595i −0.749855 + 1.56132i 1.99153 3.32369 0.143734 + 0.0690313i −0.804490 + 2.52048i − 0.367458i −1.87543 − 2.34153i − 0.305977i
209.16 − 0.0920595i 0.749855 − 1.56132i 1.99153 −3.32369 −0.143734 − 0.0690313i −0.804490 − 2.52048i − 0.367458i −1.87543 − 2.34153i 0.305977i
209.17 0.0920595i −0.749855 − 1.56132i 1.99153 3.32369 0.143734 − 0.0690313i −0.804490 − 2.52048i 0.367458i −1.87543 + 2.34153i 0.305977i
209.18 0.0920595i 0.749855 + 1.56132i 1.99153 −3.32369 −0.143734 + 0.0690313i −0.804490 + 2.52048i 0.367458i −1.87543 + 2.34153i − 0.305977i
209.19 0.633614i −0.915380 − 1.47040i 1.59853 0.119728 0.931666 − 0.579998i 2.16525 + 1.52042i 2.28008i −1.32416 + 2.69195i 0.0758615i
209.20 0.633614i 0.915380 + 1.47040i 1.59853 −0.119728 −0.931666 + 0.579998i 2.16525 − 1.52042i 2.28008i −1.32416 + 2.69195i − 0.0758615i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.e.a ✓ 32
3.b odd 2 1 inner 273.2.e.a ✓ 32
7.b odd 2 1 inner 273.2.e.a ✓ 32
21.c even 2 1 inner 273.2.e.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.e.a ✓ 32 1.a even 1 1 trivial
273.2.e.a ✓ 32 3.b odd 2 1 inner
273.2.e.a ✓ 32 7.b odd 2 1 inner
273.2.e.a ✓ 32 21.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(273, [\chi])\).