↑

Properties

Label 1716.2.d.a
Level $1716$
Weight $2$
Character orbit 1716.d
Analytic conductor $13.702$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1716,2,Mod(989,1716)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.989");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1716.d (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: \(13.7023289869\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{3} - 2 q^{9} + 6 q^{15} - 28 q^{25} + 20 q^{27} + 4 q^{31} - 6 q^{33} - 28 q^{37} + 38 q^{45} - 104 q^{49} + 36 q^{55} - 52 q^{67} - 50 q^{69} - 32 q^{75} + 6 q^{81} - 8 q^{91} + 18 q^{93} + 52 q^{97} - 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
989.1 0 −1.65794 − 0.501243i 0 − 1.91814i 0 − 3.25087i 0 2.49751 + 1.66206i 0
989.2 0 −1.65794 − 0.501243i 0 − 1.91814i 0 3.25087i 0 2.49751 + 1.66206i 0
989.3 0 −1.65794 + 0.501243i 0 1.91814i 0 − 3.25087i 0 2.49751 − 1.66206i 0
989.4 0 −1.65794 + 0.501243i 0 1.91814i 0 3.25087i 0 2.49751 − 1.66206i 0
989.5 0 −1.62112 − 0.609899i 0 1.04276i 0 − 1.76345i 0 2.25605 + 1.97744i 0
989.6 0 −1.62112 − 0.609899i 0 1.04276i 0 1.76345i 0 2.25605 + 1.97744i 0
989.7 0 −1.62112 + 0.609899i 0 − 1.04276i 0 − 1.76345i 0 2.25605 − 1.97744i 0
989.8 0 −1.62112 + 0.609899i 0 − 1.04276i 0 1.76345i 0 2.25605 − 1.97744i 0
989.9 0 −1.23992 − 1.20938i 0 − 1.91395i 0 − 3.14813i 0 0.0748174 + 2.99907i 0
989.10 0 −1.23992 − 1.20938i 0 − 1.91395i 0 3.14813i 0 0.0748174 + 2.99907i 0
989.11 0 −1.23992 + 1.20938i 0 1.91395i 0 − 3.14813i 0 0.0748174 − 2.99907i 0
989.12 0 −1.23992 + 1.20938i 0 1.91395i 0 3.14813i 0 0.0748174 − 2.99907i 0
989.13 0 −0.937481 − 1.45641i 0 3.28485i 0 − 3.65568i 0 −1.24226 + 2.73071i 0
989.14 0 −0.937481 − 1.45641i 0 3.28485i 0 3.65568i 0 −1.24226 + 2.73071i 0
989.15 0 −0.937481 + 1.45641i 0 − 3.28485i 0 − 3.65568i 0 −1.24226 − 2.73071i 0
989.16 0 −0.937481 + 1.45641i 0 − 3.28485i 0 3.65568i 0 −1.24226 − 2.73071i 0
989.17 0 −0.669559 − 1.59740i 0 − 3.11598i 0 − 3.35439i 0 −2.10338 + 2.13911i 0
989.18 0 −0.669559 − 1.59740i 0 − 3.11598i 0 3.35439i 0 −2.10338 + 2.13911i 0
989.19 0 −0.669559 + 1.59740i 0 3.11598i 0 − 3.35439i 0 −2.10338 − 2.13911i 0
989.20 0 −0.669559 + 1.59740i 0 3.11598i 0 3.35439i 0 −2.10338 − 2.13911i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 989.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.2.d.a ✓ 48
3.b odd 2 1 inner 1716.2.d.a ✓ 48
11.b odd 2 1 inner 1716.2.d.a ✓ 48
33.d even 2 1 inner 1716.2.d.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.d.a ✓ 48 1.a even 1 1 trivial
1716.2.d.a ✓ 48 3.b odd 2 1 inner
1716.2.d.a ✓ 48 11.b odd 2 1 inner
1716.2.d.a ✓ 48 33.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1716, [\chi])\).