Properties

Label 378.3.p.a
Level $378$
Weight $3$
Character orbit 378.p
Analytic conductor $10.300$
Analytic rank $0$
Dimension $32$
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] = [378,3,Mod(73,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.p (of order \(6\), degree \(2\), 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: \(10.2997539928\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 2 q^{7} + 12 q^{11} + 30 q^{13} + 12 q^{14} + 128 q^{16} - 54 q^{17} + 42 q^{23} + 80 q^{25} + 72 q^{26} - 4 q^{28} + 84 q^{29} + 66 q^{35} - 22 q^{37} + 396 q^{41} - 16 q^{43} + 24 q^{44} + 12 q^{46} + 50 q^{49} + 96 q^{50} + 60 q^{52} + 252 q^{53} + 24 q^{56} - 24 q^{58} + 256 q^{64} - 12 q^{65} - 140 q^{67} - 108 q^{68} + 72 q^{70} - 300 q^{71} + 72 q^{74} - 570 q^{77} - 212 q^{79} - 756 q^{83} - 60 q^{85} + 120 q^{86} - 414 q^{89} - 186 q^{91} + 84 q^{92} - 1104 q^{95} - 114 q^{97} + 96 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −1.41421 0 2.00000 −6.25411 3.61081i 0 −6.40955 + 2.81382i −2.82843 0 8.84464 + 5.10646i
73.2 −1.41421 0 2.00000 −3.75454 2.16769i 0 5.02745 4.87080i −2.82843 0 5.30973 + 3.06557i
73.3 −1.41421 0 2.00000 −2.65919 1.53529i 0 −3.27459 + 6.18685i −2.82843 0 3.76067 + 2.17122i
73.4 −1.41421 0 2.00000 −1.56851 0.905579i 0 6.71333 + 1.98274i −2.82843 0 2.21821 + 1.28068i
73.5 −1.41421 0 2.00000 −1.56104 0.901265i 0 −0.490625 6.98279i −2.82843 0 2.20764 + 1.27458i
73.6 −1.41421 0 2.00000 3.25801 + 1.88101i 0 4.45903 + 5.39603i −2.82843 0 −4.60752 2.66015i
73.7 −1.41421 0 2.00000 6.01208 + 3.47108i 0 −6.97949 0.535409i −2.82843 0 −8.50237 4.90885i
73.8 −1.41421 0 2.00000 6.52730 + 3.76854i 0 −1.66686 6.79864i −2.82843 0 −9.23099 5.32952i
73.9 1.41421 0 2.00000 −6.30090 3.63782i 0 −5.27174 + 4.60530i 2.82843 0 −8.91081 5.14466i
73.10 1.41421 0 2.00000 −4.87915 2.81698i 0 5.20700 + 4.67837i 2.82843 0 −6.90016 3.98381i
73.11 1.41421 0 2.00000 −4.82569 2.78612i 0 4.90353 4.99554i 2.82843 0 −6.82456 3.94016i
73.12 1.41421 0 2.00000 −1.59708 0.922074i 0 −6.32875 2.99115i 2.82843 0 −2.25861 1.30401i
73.13 1.41421 0 2.00000 −0.214369 0.123766i 0 −1.01424 + 6.92613i 2.82843 0 −0.303164 0.175032i
73.14 1.41421 0 2.00000 3.45664 + 1.99569i 0 4.98028 4.91903i 2.82843 0 4.88842 + 2.82233i
73.15 1.41421 0 2.00000 6.47195 + 3.73658i 0 −6.15173 3.34009i 2.82843 0 9.15272 + 5.28432i
73.16 1.41421 0 2.00000 7.88860 + 4.55449i 0 5.29697 + 4.57626i 2.82843 0 11.1562 + 6.44102i
145.1 −1.41421 0 2.00000 −6.25411 + 3.61081i 0 −6.40955 2.81382i −2.82843 0 8.84464 5.10646i
145.2 −1.41421 0 2.00000 −3.75454 + 2.16769i 0 5.02745 + 4.87080i −2.82843 0 5.30973 3.06557i
145.3 −1.41421 0 2.00000 −2.65919 + 1.53529i 0 −3.27459 6.18685i −2.82843 0 3.76067 2.17122i
145.4 −1.41421 0 2.00000 −1.56851 + 0.905579i 0 6.71333 1.98274i −2.82843 0 2.21821 1.28068i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.3.p.a 32
3.b odd 2 1 126.3.p.a yes 32
7.d odd 6 1 378.3.j.a 32
9.c even 3 1 378.3.j.a 32
9.d odd 6 1 126.3.j.a 32
21.g even 6 1 126.3.j.a 32
63.i even 6 1 126.3.p.a yes 32
63.t odd 6 1 inner 378.3.p.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.j.a 32 9.d odd 6 1
126.3.j.a 32 21.g even 6 1
126.3.p.a yes 32 3.b odd 2 1
126.3.p.a yes 32 63.i even 6 1
378.3.j.a 32 7.d odd 6 1
378.3.j.a 32 9.c even 3 1
378.3.p.a 32 1.a even 1 1 trivial
378.3.p.a 32 63.t odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(378, [\chi])\).