↑

Properties

Label 3360.2.ba.f
Level $3360$
Weight $2$
Character orbit 3360.ba
Analytic conductor $26.830$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3360,2,Mod(2591,3360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3360.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.ba (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: \(26.8297350792\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{9} + 8 q^{11} + 8 q^{13} + 4 q^{15} - 4 q^{21} - 24 q^{25} + 8 q^{33} + 24 q^{35} + 8 q^{37} + 36 q^{39} - 24 q^{49} - 12 q^{51} + 8 q^{57} + 48 q^{59} + 56 q^{61} - 88 q^{71} - 40 q^{73} + 44 q^{81} + 24 q^{83} + 32 q^{87} - 48 q^{93} - 24 q^{95} - 8 q^{97} - 52 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1 0 −1.66651 − 0.471960i 0 − 1.00000i 0 1.00000i 0 2.55451 + 1.57305i 0
2591.2 0 −1.66651 + 0.471960i 0 1.00000i 0 − 1.00000i 0 2.55451 − 1.57305i 0
2591.3 0 −1.65252 − 0.518832i 0 − 1.00000i 0 1.00000i 0 2.46163 + 1.71476i 0
2591.4 0 −1.65252 + 0.518832i 0 1.00000i 0 − 1.00000i 0 2.46163 − 1.71476i 0
2591.5 0 −1.55507 − 0.762729i 0 1.00000i 0 − 1.00000i 0 1.83649 + 2.37220i 0
2591.6 0 −1.55507 + 0.762729i 0 − 1.00000i 0 1.00000i 0 1.83649 − 2.37220i 0
2591.7 0 −1.20544 − 1.24375i 0 1.00000i 0 − 1.00000i 0 −0.0938457 + 2.99853i 0
2591.8 0 −1.20544 + 1.24375i 0 − 1.00000i 0 1.00000i 0 −0.0938457 − 2.99853i 0
2591.9 0 −0.263398 − 1.71191i 0 − 1.00000i 0 1.00000i 0 −2.86124 + 0.901826i 0
2591.10 0 −0.263398 + 1.71191i 0 1.00000i 0 − 1.00000i 0 −2.86124 − 0.901826i 0
2591.11 0 −0.204503 − 1.71994i 0 1.00000i 0 − 1.00000i 0 −2.91636 + 0.703463i 0
2591.12 0 −0.204503 + 1.71994i 0 − 1.00000i 0 1.00000i 0 −2.91636 − 0.703463i 0
2591.13 0 0.130301 − 1.72714i 0 1.00000i 0 − 1.00000i 0 −2.96604 − 0.450096i 0
2591.14 0 0.130301 + 1.72714i 0 − 1.00000i 0 1.00000i 0 −2.96604 + 0.450096i 0
2591.15 0 0.330652 − 1.70020i 0 − 1.00000i 0 1.00000i 0 −2.78134 − 1.12435i 0
2591.16 0 0.330652 + 1.70020i 0 1.00000i 0 − 1.00000i 0 −2.78134 + 1.12435i 0
2591.17 0 1.26750 − 1.18044i 0 1.00000i 0 − 1.00000i 0 0.213115 − 2.99242i 0
2591.18 0 1.26750 + 1.18044i 0 − 1.00000i 0 1.00000i 0 0.213115 + 2.99242i 0
2591.19 0 1.46430 − 0.925102i 0 − 1.00000i 0 1.00000i 0 1.28837 − 2.70926i 0
2591.20 0 1.46430 + 0.925102i 0 1.00000i 0 − 1.00000i 0 1.28837 + 2.70926i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.2.ba.f yes 24
3.b odd 2 1 3360.2.ba.e ✓ 24
4.b odd 2 1 3360.2.ba.e ✓ 24
12.b even 2 1 inner 3360.2.ba.f yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.ba.e ✓ 24 3.b odd 2 1
3360.2.ba.e ✓ 24 4.b odd 2 1
3360.2.ba.f yes 24 1.a even 1 1 trivial
3360.2.ba.f yes 24 12.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 \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11}^{12} - 4 T_{11}^{11} - 60 T_{11}^{10} + 124 T_{11}^{9} + 1493 T_{11}^{8} - 336 T_{11}^{7} + \cdots - 2944 \) Copy content Toggle raw display
\( T_{23}^{12} - 160 T_{23}^{10} + 112 T_{23}^{9} + 9040 T_{23}^{8} - 10560 T_{23}^{7} - 209344 T_{23}^{6} + \cdots + 729088 \) Copy content Toggle raw display