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Properties

Label 1710.2.c.d
Level $1710$
Weight $2$
Character orbit 1710.c
Analytic conductor $13.654$
Analytic rank $0$
Dimension $24$
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] = [1710,2,Mod(1709,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1710.1709");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.c (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: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 24 q^{16} - 56 q^{19} - 8 q^{25} - 104 q^{49} - 128 q^{55} + 48 q^{61} - 24 q^{64} + 56 q^{76} - 48 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1709.1 − 1.00000i 0 −1.00000 1.33763 − 1.79186i 0 4.18206i 1.00000i 0 −1.79186 − 1.33763i
1709.2 − 1.00000i 0 −1.00000 1.33763 + 1.79186i 0 − 4.18206i 1.00000i 0 1.79186 − 1.33763i
1709.3 1.00000i 0 −1.00000 1.33763 − 1.79186i 0 4.18206i − 1.00000i 0 1.79186 + 1.33763i
1709.4 1.00000i 0 −1.00000 1.33763 + 1.79186i 0 − 4.18206i − 1.00000i 0 −1.79186 + 1.33763i
1709.5 − 1.00000i 0 −1.00000 −1.33763 − 1.79186i 0 − 4.18206i 1.00000i 0 −1.79186 + 1.33763i
1709.6 − 1.00000i 0 −1.00000 −1.33763 + 1.79186i 0 4.18206i 1.00000i 0 1.79186 + 1.33763i
1709.7 1.00000i 0 −1.00000 −1.33763 − 1.79186i 0 − 4.18206i − 1.00000i 0 1.79186 − 1.33763i
1709.8 1.00000i 0 −1.00000 −1.33763 + 1.79186i 0 4.18206i − 1.00000i 0 −1.79186 − 1.33763i
1709.9 − 1.00000i 0 −1.00000 0.586990 − 2.15765i 0 − 3.03449i 1.00000i 0 −2.15765 − 0.586990i
1709.10 − 1.00000i 0 −1.00000 0.586990 + 2.15765i 0 3.03449i 1.00000i 0 2.15765 − 0.586990i
1709.11 1.00000i 0 −1.00000 0.586990 − 2.15765i 0 − 3.03449i − 1.00000i 0 2.15765 + 0.586990i
1709.12 1.00000i 0 −1.00000 0.586990 + 2.15765i 0 3.03449i − 1.00000i 0 −2.15765 + 0.586990i
1709.13 − 1.00000i 0 −1.00000 2.20595 − 0.365789i 0 − 2.70226i 1.00000i 0 −0.365789 − 2.20595i
1709.14 − 1.00000i 0 −1.00000 2.20595 + 0.365789i 0 2.70226i 1.00000i 0 0.365789 − 2.20595i
1709.15 1.00000i 0 −1.00000 2.20595 − 0.365789i 0 − 2.70226i − 1.00000i 0 0.365789 + 2.20595i
1709.16 1.00000i 0 −1.00000 2.20595 + 0.365789i 0 2.70226i − 1.00000i 0 −0.365789 + 2.20595i
1709.17 − 1.00000i 0 −1.00000 −0.586990 − 2.15765i 0 3.03449i 1.00000i 0 −2.15765 + 0.586990i
1709.18 − 1.00000i 0 −1.00000 −0.586990 + 2.15765i 0 − 3.03449i 1.00000i 0 2.15765 + 0.586990i
1709.19 1.00000i 0 −1.00000 −0.586990 − 2.15765i 0 3.03449i − 1.00000i 0 2.15765 − 0.586990i
1709.20 1.00000i 0 −1.00000 −0.586990 + 2.15765i 0 − 3.03449i − 1.00000i 0 −2.15765 − 0.586990i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1709.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner
95.d odd 2 1 inner
285.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.c.d ✓ 24
3.b odd 2 1 inner 1710.2.c.d ✓ 24
5.b even 2 1 inner 1710.2.c.d ✓ 24
15.d odd 2 1 inner 1710.2.c.d ✓ 24
19.b odd 2 1 inner 1710.2.c.d ✓ 24
57.d even 2 1 inner 1710.2.c.d ✓ 24
95.d odd 2 1 inner 1710.2.c.d ✓ 24
285.b even 2 1 inner 1710.2.c.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.c.d ✓ 24 1.a even 1 1 trivial
1710.2.c.d ✓ 24 3.b odd 2 1 inner
1710.2.c.d ✓ 24 5.b even 2 1 inner
1710.2.c.d ✓ 24 15.d odd 2 1 inner
1710.2.c.d ✓ 24 19.b odd 2 1 inner
1710.2.c.d ✓ 24 57.d even 2 1 inner
1710.2.c.d ✓ 24 95.d odd 2 1 inner
1710.2.c.d ✓ 24 285.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{6} + 34T_{7}^{4} + 356T_{7}^{2} + 1176 \) Copy content Toggle raw display
\( T_{11}^{6} + 38T_{11}^{4} + 364T_{11}^{2} + 968 \) Copy content Toggle raw display
\( T_{29}^{6} - 40T_{29}^{4} + 416T_{29}^{2} - 384 \) Copy content Toggle raw display