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Properties

Label 468.2.cg.a
Level $468$
Weight $2$
Character orbit 468.cg
Analytic conductor $3.737$
Analytic rank $0$
Dimension $56$
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] = [468,2,Mod(5,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.cg (of order \(12\), degree \(4\), 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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{7} - 12 q^{11} - 12 q^{15} + 4 q^{19} - 12 q^{21} + 36 q^{27} - 4 q^{31} + 12 q^{33} - 4 q^{37} + 24 q^{41} + 6 q^{45} + 66 q^{47} - 48 q^{57} - 30 q^{63} - 78 q^{65} - 14 q^{67} + 28 q^{73} - 24 q^{79} - 78 q^{83} + 36 q^{85} - 8 q^{91} + 6 q^{93} + 26 q^{97} + 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0 −1.69844 + 0.339558i 0 −0.575123 + 0.154104i 0 −0.0243233 + 0.0907757i 0 2.76940 − 1.15344i 0
5.2 0 −1.61220 − 0.633098i 0 0.932593 − 0.249888i 0 0.612271 − 2.28503i 0 2.19837 + 2.04136i 0
5.3 0 −1.39355 + 1.02859i 0 3.50643 − 0.939546i 0 −0.864930 + 3.22796i 0 0.883987 − 2.86680i 0
5.4 0 −1.13940 − 1.30451i 0 −3.55931 + 0.953715i 0 0.0638481 − 0.238284i 0 −0.403516 + 2.97274i 0
5.5 0 −0.792618 + 1.54005i 0 −0.138031 + 0.0369853i 0 0.968186 − 3.61332i 0 −1.74351 − 2.44134i 0
5.6 0 −0.763329 − 1.55478i 0 0.391929 − 0.105017i 0 −1.14793 + 4.28414i 0 −1.83466 + 2.37361i 0
5.7 0 −0.294902 + 1.70676i 0 −1.14537 + 0.306901i 0 0.0347480 − 0.129681i 0 −2.82607 − 1.00666i 0
5.8 0 −0.0927402 − 1.72957i 0 2.84277 − 0.761717i 0 1.16105 − 4.33311i 0 −2.98280 + 0.320801i 0
5.9 0 0.885873 + 1.48836i 0 −3.51413 + 0.941608i 0 −0.158893 + 0.592998i 0 −1.43046 + 2.63700i 0
5.10 0 0.930898 + 1.46063i 0 2.73979 − 0.734125i 0 −0.507313 + 1.89332i 0 −1.26686 + 2.71939i 0
5.11 0 0.954090 − 1.44558i 0 0.981455 − 0.262980i 0 −0.270133 + 1.00815i 0 −1.17943 − 2.75843i 0
5.12 0 1.55690 − 0.758984i 0 −3.57768 + 0.958637i 0 1.00990 − 3.76900i 0 1.84789 − 2.36333i 0
5.13 0 1.72741 − 0.126737i 0 −1.49243 + 0.399894i 0 −1.05046 + 3.92036i 0 2.96788 − 0.437853i 0
5.14 0 1.73202 − 0.0106951i 0 2.60710 − 0.698570i 0 0.539999 − 2.01531i 0 2.99977 − 0.0370482i 0
281.1 0 −1.69844 − 0.339558i 0 −0.575123 − 0.154104i 0 −0.0243233 − 0.0907757i 0 2.76940 + 1.15344i 0
281.2 0 −1.61220 + 0.633098i 0 0.932593 + 0.249888i 0 0.612271 + 2.28503i 0 2.19837 − 2.04136i 0
281.3 0 −1.39355 − 1.02859i 0 3.50643 + 0.939546i 0 −0.864930 − 3.22796i 0 0.883987 + 2.86680i 0
281.4 0 −1.13940 + 1.30451i 0 −3.55931 − 0.953715i 0 0.0638481 + 0.238284i 0 −0.403516 − 2.97274i 0
281.5 0 −0.792618 − 1.54005i 0 −0.138031 − 0.0369853i 0 0.968186 + 3.61332i 0 −1.74351 + 2.44134i 0
281.6 0 −0.763329 + 1.55478i 0 0.391929 + 0.105017i 0 −1.14793 − 4.28414i 0 −1.83466 − 2.37361i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
13.d odd 4 1 inner
117.z even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.cg.a ✓ 56
3.b odd 2 1 1404.2.cj.a 56
9.c even 3 1 1404.2.cj.a 56
9.d odd 6 1 inner 468.2.cg.a ✓ 56
13.d odd 4 1 inner 468.2.cg.a ✓ 56
39.f even 4 1 1404.2.cj.a 56
117.y odd 12 1 1404.2.cj.a 56
117.z even 12 1 inner 468.2.cg.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.cg.a ✓ 56 1.a even 1 1 trivial
468.2.cg.a ✓ 56 9.d odd 6 1 inner
468.2.cg.a ✓ 56 13.d odd 4 1 inner
468.2.cg.a ✓ 56 117.z even 12 1 inner
1404.2.cj.a 56 3.b odd 2 1
1404.2.cj.a 56 9.c even 3 1
1404.2.cj.a 56 39.f even 4 1
1404.2.cj.a 56 117.y odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(468, [\chi])\).