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Properties

Label 684.2.l.a
Level $684$
Weight $2$
Character orbit 684.l
Analytic conductor $5.462$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,2,Mod(121,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.l (of order \(3\), 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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + q^{3} - q^{7} + q^{9} - q^{11} - q^{13} + 10 q^{15} + 5 q^{17} + q^{19} + 6 q^{21} - 4 q^{23} + 40 q^{25} + 7 q^{27} + 18 q^{29} + 2 q^{31} - 7 q^{33} - 6 q^{35} + 2 q^{37} + 3 q^{39} + 50 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0 −1.70674 − 0.295004i 0 2.51503 0 −0.950930 − 1.64706i 0 2.82594 + 1.00699i 0
121.2 0 −1.66388 + 0.481162i 0 −1.49057 0 0.903772 + 1.56538i 0 2.53697 − 1.60119i 0
121.3 0 −1.63863 + 0.561154i 0 2.68159 0 −0.125175 − 0.216810i 0 2.37021 − 1.83905i 0
121.4 0 −1.54198 − 0.788855i 0 −2.83853 0 0.648230 + 1.12277i 0 1.75541 + 2.43280i 0
121.5 0 −1.05089 + 1.37682i 0 −3.06171 0 −2.37338 − 4.11082i 0 −0.791241 − 2.89378i 0
121.6 0 −1.04626 − 1.38034i 0 −2.40565 0 −1.53172 − 2.65301i 0 −0.810664 + 2.88839i 0
121.7 0 −0.898479 − 1.48079i 0 3.00646 0 0.886284 + 1.53509i 0 −1.38547 + 2.66092i 0
121.8 0 −0.692830 + 1.58745i 0 1.46442 0 1.54051 + 2.66824i 0 −2.03997 − 2.19966i 0
121.9 0 −0.528191 − 1.64955i 0 0.290319 0 −0.636745 − 1.10287i 0 −2.44203 + 1.74255i 0
121.10 0 −0.270575 + 1.71079i 0 0.801046 0 −1.41378 − 2.44874i 0 −2.85358 − 0.925792i 0
121.11 0 0.377158 − 1.69049i 0 −3.07739 0 2.34238 + 4.05712i 0 −2.71550 − 1.27516i 0
121.12 0 0.438235 + 1.67569i 0 −4.24712 0 1.38218 + 2.39401i 0 −2.61590 + 1.46870i 0
121.13 0 0.857237 − 1.50504i 0 3.88804 0 −0.0132784 − 0.0229989i 0 −1.53029 − 2.58035i 0
121.14 0 0.915791 − 1.47015i 0 −1.49623 0 −1.75655 − 3.04244i 0 −1.32265 − 2.69269i 0
121.15 0 0.979052 + 1.42880i 0 −0.273500 0 −0.546712 − 0.946933i 0 −1.08291 + 2.79773i 0
121.16 0 1.42321 + 0.987159i 0 0.958709 0 2.11741 + 3.66746i 0 1.05103 + 2.80986i 0
121.17 0 1.43544 − 0.969287i 0 0.965937 0 0.302844 + 0.524541i 0 1.12097 − 2.78270i 0
121.18 0 1.66089 + 0.491360i 0 3.61559 0 −2.29625 − 3.97722i 0 2.51713 + 1.63219i 0
121.19 0 1.72309 + 0.176004i 0 1.40487 0 1.81436 + 3.14256i 0 2.93805 + 0.606538i 0
121.20 0 1.72837 − 0.112905i 0 −2.70132 0 −0.793446 − 1.37429i 0 2.97450 − 0.390282i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.l.a yes 40
3.b odd 2 1 2052.2.l.a 40
9.c even 3 1 684.2.j.a ✓ 40
9.d odd 6 1 2052.2.j.a 40
19.c even 3 1 684.2.j.a ✓ 40
57.h odd 6 1 2052.2.j.a 40
171.g even 3 1 inner 684.2.l.a yes 40
171.n odd 6 1 2052.2.l.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.j.a ✓ 40 9.c even 3 1
684.2.j.a ✓ 40 19.c even 3 1
684.2.l.a yes 40 1.a even 1 1 trivial
684.2.l.a yes 40 171.g even 3 1 inner
2052.2.j.a 40 9.d odd 6 1
2052.2.j.a 40 57.h odd 6 1
2052.2.l.a 40 3.b odd 2 1
2052.2.l.a 40 171.n odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).