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Properties

Label 3420.2.bb.f
Level $3420$
Weight $2$
Character orbit 3420.bb
Analytic conductor $27.309$
Analytic rank $0$
Dimension $40$
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] = [3420,2,Mod(37,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.37");
 
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.bb (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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 1140)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{5} + 4 q^{7} + 16 q^{11} - 4 q^{17} + 8 q^{23} - 12 q^{25} + 4 q^{35} + 28 q^{43} + 28 q^{47} + 32 q^{55} - 16 q^{61} + 60 q^{73} + 20 q^{77} + 40 q^{83} + 72 q^{85} - 28 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 0 0 0 −2.23518 + 0.0631414i 0 −0.462950 − 0.462950i 0 0 0
37.2 0 0 0 −2.23518 + 0.0631414i 0 −0.462950 − 0.462950i 0 0 0
37.3 0 0 0 −1.76413 + 1.37399i 0 2.45104 + 2.45104i 0 0 0
37.4 0 0 0 −1.76413 + 1.37399i 0 2.45104 + 2.45104i 0 0 0
37.5 0 0 0 −1.66093 − 1.49710i 0 −0.813205 − 0.813205i 0 0 0
37.6 0 0 0 −1.66093 − 1.49710i 0 −0.813205 − 0.813205i 0 0 0
37.7 0 0 0 −0.260865 + 2.22080i 0 −0.450999 − 0.450999i 0 0 0
37.8 0 0 0 −0.260865 + 2.22080i 0 −0.450999 − 0.450999i 0 0 0
37.9 0 0 0 −0.175336 − 2.22918i 0 2.65404 + 2.65404i 0 0 0
37.10 0 0 0 −0.175336 − 2.22918i 0 2.65404 + 2.65404i 0 0 0
37.11 0 0 0 0.105511 − 2.23358i 0 −2.38118 − 2.38118i 0 0 0
37.12 0 0 0 0.105511 − 2.23358i 0 −2.38118 − 2.38118i 0 0 0
37.13 0 0 0 1.45042 + 1.70184i 0 1.40893 + 1.40893i 0 0 0
37.14 0 0 0 1.45042 + 1.70184i 0 1.40893 + 1.40893i 0 0 0
37.15 0 0 0 1.60666 + 1.55520i 0 −3.59745 − 3.59745i 0 0 0
37.16 0 0 0 1.60666 + 1.55520i 0 −3.59745 − 3.59745i 0 0 0
37.17 0 0 0 1.74270 − 1.40107i 0 2.29396 + 2.29396i 0 0 0
37.18 0 0 0 1.74270 − 1.40107i 0 2.29396 + 2.29396i 0 0 0
37.19 0 0 0 2.19115 + 0.445959i 0 −0.102192 − 0.102192i 0 0 0
37.20 0 0 0 2.19115 + 0.445959i 0 −0.102192 − 0.102192i 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.2.bb.f 40
3.b odd 2 1 1140.2.y.a ✓ 40
5.c odd 4 1 inner 3420.2.bb.f 40
15.e even 4 1 1140.2.y.a ✓ 40
19.b odd 2 1 inner 3420.2.bb.f 40
57.d even 2 1 1140.2.y.a ✓ 40
95.g even 4 1 inner 3420.2.bb.f 40
285.j odd 4 1 1140.2.y.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.y.a ✓ 40 3.b odd 2 1
1140.2.y.a ✓ 40 15.e even 4 1
1140.2.y.a ✓ 40 57.d even 2 1
1140.2.y.a ✓ 40 285.j odd 4 1
3420.2.bb.f 40 1.a even 1 1 trivial
3420.2.bb.f 40 5.c odd 4 1 inner
3420.2.bb.f 40 19.b odd 2 1 inner
3420.2.bb.f 40 95.g even 4 1 inner

Hecke kernels

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

\( T_{7}^{20} - 2 T_{7}^{19} + 2 T_{7}^{18} - 18 T_{7}^{17} + 669 T_{7}^{16} - 1928 T_{7}^{15} + \cdots + 10000 \) Copy content Toggle raw display
\( T_{11}^{10} - 4 T_{11}^{9} - 49 T_{11}^{8} + 184 T_{11}^{7} + 744 T_{11}^{6} - 2720 T_{11}^{5} + \cdots - 664 \) Copy content Toggle raw display