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Properties

Label 1638.2.x.f
Level $1638$
Weight $2$
Character orbit 1638.x
Analytic conductor $13.079$
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] = [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: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{7} - 32 q^{16} + 16 q^{22} - 4 q^{28} + 72 q^{37} - 24 q^{46} + 40 q^{58} + 16 q^{67} + 8 q^{70} - 16 q^{79} + 80 q^{85} - 52 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −0.707107 − 0.707107i 0 1.00000i 2.17132 − 2.17132i 0 −1.71625 − 2.01358i 0.707107 − 0.707107i 0 −3.07071
307.2 −0.707107 − 0.707107i 0 1.00000i −0.345779 + 0.345779i 0 0.851822 − 2.50488i 0.707107 − 0.707107i 0 0.489006
307.3 −0.707107 − 0.707107i 0 1.00000i 0.345779 − 0.345779i 0 2.50488 − 0.851822i 0.707107 − 0.707107i 0 −0.489006
307.4 −0.707107 − 0.707107i 0 1.00000i −1.09262 + 1.09262i 0 −2.35215 − 1.21136i 0.707107 − 0.707107i 0 1.54520
307.5 −0.707107 − 0.707107i 0 1.00000i −1.72394 + 1.72394i 0 −2.40028 + 1.11295i 0.707107 − 0.707107i 0 2.43802
307.6 −0.707107 − 0.707107i 0 1.00000i 1.09262 − 1.09262i 0 1.21136 + 2.35215i 0.707107 − 0.707107i 0 −1.54520
307.7 −0.707107 − 0.707107i 0 1.00000i −2.17132 + 2.17132i 0 2.01358 + 1.71625i 0.707107 − 0.707107i 0 3.07071
307.8 −0.707107 − 0.707107i 0 1.00000i 1.72394 − 1.72394i 0 −1.11295 + 2.40028i 0.707107 − 0.707107i 0 −2.43802
307.9 0.707107 + 0.707107i 0 1.00000i −1.72394 + 1.72394i 0 −1.11295 + 2.40028i −0.707107 + 0.707107i 0 −2.43802
307.10 0.707107 + 0.707107i 0 1.00000i 2.17132 − 2.17132i 0 2.01358 + 1.71625i −0.707107 + 0.707107i 0 3.07071
307.11 0.707107 + 0.707107i 0 1.00000i −1.09262 + 1.09262i 0 1.21136 + 2.35215i −0.707107 + 0.707107i 0 −1.54520
307.12 0.707107 + 0.707107i 0 1.00000i 1.72394 − 1.72394i 0 −2.40028 + 1.11295i −0.707107 + 0.707107i 0 2.43802
307.13 0.707107 + 0.707107i 0 1.00000i 1.09262 − 1.09262i 0 −2.35215 − 1.21136i −0.707107 + 0.707107i 0 1.54520
307.14 0.707107 + 0.707107i 0 1.00000i −0.345779 + 0.345779i 0 2.50488 − 0.851822i −0.707107 + 0.707107i 0 −0.489006
307.15 0.707107 + 0.707107i 0 1.00000i 0.345779 − 0.345779i 0 0.851822 − 2.50488i −0.707107 + 0.707107i 0 0.489006
307.16 0.707107 + 0.707107i 0 1.00000i −2.17132 + 2.17132i 0 −1.71625 − 2.01358i −0.707107 + 0.707107i 0 −3.07071
811.1 −0.707107 + 0.707107i 0 − 1.00000i 2.17132 + 2.17132i 0 −1.71625 + 2.01358i 0.707107 + 0.707107i 0 −3.07071
811.2 −0.707107 + 0.707107i 0 − 1.00000i −0.345779 − 0.345779i 0 0.851822 + 2.50488i 0.707107 + 0.707107i 0 0.489006
811.3 −0.707107 + 0.707107i 0 − 1.00000i 0.345779 + 0.345779i 0 2.50488 + 0.851822i 0.707107 + 0.707107i 0 −0.489006
811.4 −0.707107 + 0.707107i 0 − 1.00000i −1.09262 − 1.09262i 0 −2.35215 + 1.21136i 0.707107 + 0.707107i 0 1.54520
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.16
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.d odd 4 1 inner
21.c even 2 1 inner
39.f even 4 1 inner
91.i even 4 1 inner
273.o odd 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.f ✓ 32
3.b odd 2 1 inner 1638.2.x.f ✓ 32
7.b odd 2 1 inner 1638.2.x.f ✓ 32
13.d odd 4 1 inner 1638.2.x.f ✓ 32
21.c even 2 1 inner 1638.2.x.f ✓ 32
39.f even 4 1 inner 1638.2.x.f ✓ 32
91.i even 4 1 inner 1638.2.x.f ✓ 32
273.o odd 4 1 inner 1638.2.x.f ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.x.f ✓ 32 1.a even 1 1 trivial
1638.2.x.f ✓ 32 3.b odd 2 1 inner
1638.2.x.f ✓ 32 7.b odd 2 1 inner
1638.2.x.f ✓ 32 13.d odd 4 1 inner
1638.2.x.f ✓ 32 21.c even 2 1 inner
1638.2.x.f ✓ 32 39.f even 4 1 inner
1638.2.x.f ✓ 32 91.i even 4 1 inner
1638.2.x.f ✓ 32 273.o odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 130T_{5}^{12} + 3857T_{5}^{8} + 18128T_{5}^{4} + 1024 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display