Properties

Label 3420.2.bc.d
Level $3420$
Weight $2$
Character orbit 3420.bc
Analytic conductor $27.309$
Analytic rank $0$
Dimension $48$
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] = [3420,2,Mod(449,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.bc (of order \(6\), 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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{19} + 4 q^{25} - 16 q^{49} + 84 q^{55} - 16 q^{61} - 72 q^{79} + 80 q^{85} + 168 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1 0 0 0 −2.23604 + 0.0110334i 0 2.24660i 0 0 0
449.2 0 0 0 −2.20846 + 0.350284i 0 0.578724i 0 0 0
449.3 0 0 0 −2.14073 0.645964i 0 3.56652i 0 0 0
449.4 0 0 0 −2.10104 + 0.765261i 0 4.46771i 0 0 0
449.5 0 0 0 −1.71326 + 1.43692i 0 4.46771i 0 0 0
449.6 0 0 0 −1.40759 + 1.73744i 0 0.578724i 0 0 0
449.7 0 0 0 −1.40503 1.73951i 0 0.948149i 0 0 0
449.8 0 0 0 −1.31712 1.80699i 0 2.24462i 0 0 0
449.9 0 0 0 −1.12758 + 1.93095i 0 2.24660i 0 0 0
449.10 0 0 0 −0.906340 2.04415i 0 2.24462i 0 0 0
449.11 0 0 0 −0.803945 2.08655i 0 0.948149i 0 0 0
449.12 0 0 0 −0.510944 + 2.17691i 0 3.56652i 0 0 0
449.13 0 0 0 0.510944 2.17691i 0 3.56652i 0 0 0
449.14 0 0 0 0.803945 + 2.08655i 0 0.948149i 0 0 0
449.15 0 0 0 0.906340 + 2.04415i 0 2.24462i 0 0 0
449.16 0 0 0 1.12758 1.93095i 0 2.24660i 0 0 0
449.17 0 0 0 1.31712 + 1.80699i 0 2.24462i 0 0 0
449.18 0 0 0 1.40503 + 1.73951i 0 0.948149i 0 0 0
449.19 0 0 0 1.40759 1.73744i 0 0.578724i 0 0 0
449.20 0 0 0 1.71326 1.43692i 0 4.46771i 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.24
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.d odd 6 1 inner
57.f even 6 1 inner
95.h odd 6 1 inner
285.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.2.bc.d 48
3.b odd 2 1 inner 3420.2.bc.d 48
5.b even 2 1 inner 3420.2.bc.d 48
15.d odd 2 1 inner 3420.2.bc.d 48
19.d odd 6 1 inner 3420.2.bc.d 48
57.f even 6 1 inner 3420.2.bc.d 48
95.h odd 6 1 inner 3420.2.bc.d 48
285.q even 6 1 inner 3420.2.bc.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3420.2.bc.d 48 1.a even 1 1 trivial
3420.2.bc.d 48 3.b odd 2 1 inner
3420.2.bc.d 48 5.b even 2 1 inner
3420.2.bc.d 48 15.d odd 2 1 inner
3420.2.bc.d 48 19.d odd 6 1 inner
3420.2.bc.d 48 57.f even 6 1 inner
3420.2.bc.d 48 95.h odd 6 1 inner
3420.2.bc.d 48 285.q even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\):

\( T_{7}^{12} + 44T_{7}^{10} + 662T_{7}^{8} + 4156T_{7}^{6} + 10825T_{7}^{4} + 8988T_{7}^{2} + 1944 \) Copy content Toggle raw display
\( T_{11}^{12} + 98T_{11}^{10} + 3401T_{11}^{8} + 47902T_{11}^{6} + 219868T_{11}^{4} + 229112T_{11}^{2} + 66564 \) Copy content Toggle raw display