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Properties

Label 273.2.g.a
Level $273$
Weight $2$
Character orbit 273.g
Analytic conductor $2.180$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,2,Mod(272,273)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.272");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.g (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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} - 16 q^{16} - 16 q^{25} + 16 q^{30} - 32 q^{36} - 48 q^{42} + 48 q^{43} - 32 q^{49} - 16 q^{51} - 80 q^{64} + 32 q^{78} + 80 q^{79} - 48 q^{81} - 96 q^{88} + 32 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
272.1 −2.35085 −0.813690 − 1.52902i 3.52651 − 0.514868i 1.91287 + 3.59451i −1.49028 + 2.18611i −3.58861 −1.67582 + 2.48830i 1.21038i
272.2 −2.35085 −0.813690 + 1.52902i 3.52651 0.514868i 1.91287 − 3.59451i −1.49028 − 2.18611i −3.58861 −1.67582 − 2.48830i − 1.21038i
272.3 −2.35085 0.813690 − 1.52902i 3.52651 − 0.514868i −1.91287 + 3.59451i 1.49028 − 2.18611i −3.58861 −1.67582 − 2.48830i 1.21038i
272.4 −2.35085 0.813690 + 1.52902i 3.52651 0.514868i −1.91287 − 3.59451i 1.49028 + 2.18611i −3.58861 −1.67582 + 2.48830i − 1.21038i
272.5 −1.77583 −1.65032 − 0.525794i 1.15356 − 3.85624i 2.93067 + 0.933719i −1.56982 − 2.12971i 1.50313 2.44708 + 1.73545i 6.84801i
272.6 −1.77583 −1.65032 + 0.525794i 1.15356 3.85624i 2.93067 − 0.933719i −1.56982 + 2.12971i 1.50313 2.44708 − 1.73545i − 6.84801i
272.7 −1.77583 1.65032 − 0.525794i 1.15356 − 3.85624i −2.93067 + 0.933719i 1.56982 + 2.12971i 1.50313 2.44708 − 1.73545i 6.84801i
272.8 −1.77583 1.65032 + 0.525794i 1.15356 3.85624i −2.93067 − 0.933719i 1.56982 − 2.12971i 1.50313 2.44708 + 1.73545i − 6.84801i
272.9 −1.10741 −0.669305 − 1.59751i −0.773652 1.62666i 0.741192 + 1.76909i 2.53337 + 0.762910i 3.07156 −2.10406 + 2.13844i − 1.80138i
272.10 −1.10741 −0.669305 + 1.59751i −0.773652 − 1.62666i 0.741192 − 1.76909i 2.53337 − 0.762910i 3.07156 −2.10406 − 2.13844i 1.80138i
272.11 −1.10741 0.669305 − 1.59751i −0.773652 1.62666i −0.741192 + 1.76909i −2.53337 − 0.762910i 3.07156 −2.10406 − 2.13844i − 1.80138i
272.12 −1.10741 0.669305 + 1.59751i −0.773652 − 1.62666i −0.741192 − 1.76909i −2.53337 + 0.762910i 3.07156 −2.10406 + 2.13844i 1.80138i
272.13 −0.305902 −1.47187 − 0.913018i −1.90642 2.05385i 0.450247 + 0.279294i −0.946976 − 2.47047i 1.19498 1.33280 + 2.68769i − 0.628276i
272.14 −0.305902 −1.47187 + 0.913018i −1.90642 − 2.05385i 0.450247 − 0.279294i −0.946976 + 2.47047i 1.19498 1.33280 − 2.68769i 0.628276i
272.15 −0.305902 1.47187 − 0.913018i −1.90642 2.05385i −0.450247 + 0.279294i 0.946976 + 2.47047i 1.19498 1.33280 − 2.68769i − 0.628276i
272.16 −0.305902 1.47187 + 0.913018i −1.90642 − 2.05385i −0.450247 − 0.279294i 0.946976 − 2.47047i 1.19498 1.33280 + 2.68769i 0.628276i
272.17 0.305902 −1.47187 − 0.913018i −1.90642 − 2.05385i −0.450247 − 0.279294i 0.946976 + 2.47047i −1.19498 1.33280 + 2.68769i − 0.628276i
272.18 0.305902 −1.47187 + 0.913018i −1.90642 2.05385i −0.450247 + 0.279294i 0.946976 − 2.47047i −1.19498 1.33280 − 2.68769i 0.628276i
272.19 0.305902 1.47187 − 0.913018i −1.90642 − 2.05385i 0.450247 − 0.279294i −0.946976 − 2.47047i −1.19498 1.33280 − 2.68769i − 0.628276i
272.20 0.305902 1.47187 + 0.913018i −1.90642 2.05385i 0.450247 + 0.279294i −0.946976 + 2.47047i −1.19498 1.33280 + 2.68769i 0.628276i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 272.32
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
13.b even 2 1 inner
21.c even 2 1 inner
39.d odd 2 1 inner
91.b odd 2 1 inner
273.g 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.g.a ✓ 32
3.b odd 2 1 inner 273.2.g.a ✓ 32
7.b odd 2 1 inner 273.2.g.a ✓ 32
13.b even 2 1 inner 273.2.g.a ✓ 32
21.c even 2 1 inner 273.2.g.a ✓ 32
39.d odd 2 1 inner 273.2.g.a ✓ 32
91.b odd 2 1 inner 273.2.g.a ✓ 32
273.g even 2 1 inner 273.2.g.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.g.a ✓ 32 1.a even 1 1 trivial
273.2.g.a ✓ 32 3.b odd 2 1 inner
273.2.g.a ✓ 32 7.b odd 2 1 inner
273.2.g.a ✓ 32 13.b even 2 1 inner
273.2.g.a ✓ 32 21.c even 2 1 inner
273.2.g.a ✓ 32 39.d odd 2 1 inner
273.2.g.a ✓ 32 91.b odd 2 1 inner
273.2.g.a ✓ 32 273.g even 2 1 inner

Hecke kernels

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