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Properties

Label 1638.2.x.e
Level $1638$
Weight $2$
Character orbit 1638.x
Analytic conductor $13.079$
Analytic rank $0$
Dimension $24$
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] = [1638,2,Mod(307,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.x (of order \(4\), degree \(2\), 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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{7} - 4 q^{14} - 24 q^{16} + 8 q^{22} + 8 q^{28} + 16 q^{29} - 20 q^{35} - 8 q^{37} + 24 q^{50} - 16 q^{53} + 8 q^{58} - 88 q^{65} - 96 q^{67} - 44 q^{70} + 64 q^{71} + 16 q^{74} - 8 q^{79} - 56 q^{85} - 56 q^{86} + 44 q^{91} + 8 q^{92} - 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −0.707107 − 0.707107i 0 1.00000i −0.345286 + 0.345286i 0 1.89132 − 1.85011i 0.707107 − 0.707107i 0 0.488308
307.2 −0.707107 − 0.707107i 0 1.00000i 0.345286 − 0.345286i 0 1.85011 − 1.89132i 0.707107 − 0.707107i 0 −0.488308
307.3 −0.707107 − 0.707107i 0 1.00000i −1.29628 + 1.29628i 0 −0.536277 + 2.59083i 0.707107 − 0.707107i 0 1.83321
307.4 −0.707107 − 0.707107i 0 1.00000i 1.29628 − 1.29628i 0 −2.59083 + 0.536277i 0.707107 − 0.707107i 0 −1.83321
307.5 −0.707107 − 0.707107i 0 1.00000i −2.56498 + 2.56498i 0 −0.504434 − 2.59722i 0.707107 − 0.707107i 0 3.62743
307.6 −0.707107 − 0.707107i 0 1.00000i 2.56498 − 2.56498i 0 2.59722 + 0.504434i 0.707107 − 0.707107i 0 −3.62743
307.7 0.707107 + 0.707107i 0 1.00000i −2.98401 + 2.98401i 0 −0.000706405 2.64575i −0.707107 + 0.707107i 0 −4.22003
307.8 0.707107 + 0.707107i 0 1.00000i 2.98401 − 2.98401i 0 −2.64575 0.000706405i −0.707107 + 0.707107i 0 4.22003
307.9 0.707107 + 0.707107i 0 1.00000i −1.86233 + 1.86233i 0 2.61979 − 0.369730i −0.707107 + 0.707107i 0 −2.63373
307.10 0.707107 + 0.707107i 0 1.00000i 1.86233 − 1.86233i 0 0.369730 − 2.61979i −0.707107 + 0.707107i 0 2.63373
307.11 0.707107 + 0.707107i 0 1.00000i −0.498747 + 0.498747i 0 −1.33463 − 2.28446i −0.707107 + 0.707107i 0 −0.705335
307.12 0.707107 + 0.707107i 0 1.00000i 0.498747 − 0.498747i 0 2.28446 + 1.33463i −0.707107 + 0.707107i 0 0.705335
811.1 −0.707107 + 0.707107i 0 − 1.00000i −0.345286 − 0.345286i 0 1.89132 + 1.85011i 0.707107 + 0.707107i 0 0.488308
811.2 −0.707107 + 0.707107i 0 − 1.00000i 0.345286 + 0.345286i 0 1.85011 + 1.89132i 0.707107 + 0.707107i 0 −0.488308
811.3 −0.707107 + 0.707107i 0 − 1.00000i −1.29628 − 1.29628i 0 −0.536277 − 2.59083i 0.707107 + 0.707107i 0 1.83321
811.4 −0.707107 + 0.707107i 0 − 1.00000i 1.29628 + 1.29628i 0 −2.59083 − 0.536277i 0.707107 + 0.707107i 0 −1.83321
811.5 −0.707107 + 0.707107i 0 − 1.00000i −2.56498 − 2.56498i 0 −0.504434 + 2.59722i 0.707107 + 0.707107i 0 3.62743
811.6 −0.707107 + 0.707107i 0 − 1.00000i 2.56498 + 2.56498i 0 2.59722 − 0.504434i 0.707107 + 0.707107i 0 −3.62743
811.7 0.707107 − 0.707107i 0 − 1.00000i −2.98401 − 2.98401i 0 −0.000706405 − 2.64575i −0.707107 − 0.707107i 0 −4.22003
811.8 0.707107 − 0.707107i 0 − 1.00000i 2.98401 + 2.98401i 0 −2.64575 0.000706405i −0.707107 − 0.707107i 0 4.22003
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.d odd 4 1 inner
91.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.x.e 24
3.b odd 2 1 182.2.i.a ✓ 24
7.b odd 2 1 inner 1638.2.x.e 24
13.d odd 4 1 inner 1638.2.x.e 24
21.c even 2 1 182.2.i.a ✓ 24
39.f even 4 1 182.2.i.a ✓ 24
91.i even 4 1 inner 1638.2.x.e 24
273.o odd 4 1 182.2.i.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.i.a ✓ 24 3.b odd 2 1
182.2.i.a ✓ 24 21.c even 2 1
182.2.i.a ✓ 24 39.f even 4 1
182.2.i.a ✓ 24 273.o odd 4 1
1638.2.x.e 24 1.a even 1 1 trivial
1638.2.x.e 24 7.b odd 2 1 inner
1638.2.x.e 24 13.d odd 4 1 inner
1638.2.x.e 24 91.i even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 550T_{5}^{20} + 84749T_{5}^{16} + 3554396T_{5}^{12} + 30914740T_{5}^{8} + 9131616T_{5}^{4} + 419904 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display