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Properties

Label 627.2.k.a
Level $627$
Weight $2$
Character orbit 627.k
Analytic conductor $5.007$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [627,2,Mod(274,627)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(627, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("627.274");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 627 = 3 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 627.k (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: \(5.00662020673\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 40 q^{4} + 4 q^{5} + 40 q^{9} + 4 q^{11} - 24 q^{14} + 12 q^{15} - 40 q^{16} + 48 q^{20} - 24 q^{23} - 56 q^{25} + 80 q^{26} + 6 q^{33} - 48 q^{34} + 40 q^{36} + 60 q^{38} - 4 q^{42} + 14 q^{44}+ \cdots + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1 −1.35430 + 2.34572i 0.866025 + 0.500000i −2.66827 − 4.62157i −1.05875 + 1.83380i −2.34572 + 1.35430i 3.93679i 9.03734 0.500000 + 0.866025i −2.86773 − 4.96705i
274.2 −1.34279 + 2.32577i −0.866025 − 0.500000i −2.60614 − 4.51397i −1.81012 + 3.13522i 2.32577 − 1.34279i − 0.952391i 8.62682 0.500000 + 0.866025i −4.86121 − 8.41986i
274.3 −1.29260 + 2.23886i 0.866025 + 0.500000i −2.34165 − 4.05585i 1.87956 − 3.25549i −2.23886 + 1.29260i 1.35502i 6.93688 0.500000 + 0.866025i 4.85905 + 8.41611i
274.4 −1.24629 + 2.15863i −0.866025 − 0.500000i −2.10647 − 3.64851i 1.28883 − 2.23232i 2.15863 − 1.24629i − 0.861110i 5.51591 0.500000 + 0.866025i 3.21251 + 5.56422i
274.5 −1.10110 + 1.90715i 0.866025 + 0.500000i −1.42482 − 2.46787i 0.631882 − 1.09445i −1.90715 + 1.10110i 0.523096i 1.87109 0.500000 + 0.866025i 1.39153 + 2.41019i
274.6 −1.10029 + 1.90576i 0.866025 + 0.500000i −1.42128 − 2.46172i −1.35732 + 2.35095i −1.90576 + 1.10029i − 3.75702i 1.85410 0.500000 + 0.866025i −2.98690 − 5.17346i
274.7 −1.01165 + 1.75223i −0.866025 − 0.500000i −1.04687 − 1.81324i −0.897161 + 1.55393i 1.75223 − 1.01165i − 4.05167i 0.189683 0.500000 + 0.866025i −1.81523 − 3.14407i
274.8 −1.01051 + 1.75025i −0.866025 − 0.500000i −1.04226 − 1.80524i 1.27988 − 2.21683i 1.75025 − 1.01051i 4.75094i 0.170810 0.500000 + 0.866025i 2.58667 + 4.48025i
274.9 −0.890907 + 1.54310i −0.866025 − 0.500000i −0.587432 − 1.01746i −1.80116 + 3.11970i 1.54310 − 0.890907i 3.47801i −1.47024 0.500000 + 0.866025i −3.20933 − 5.55873i
274.10 −0.794908 + 1.37682i 0.866025 + 0.500000i −0.263759 − 0.456844i 0.635326 − 1.10042i −1.37682 + 0.794908i − 3.77458i −2.34098 0.500000 + 0.866025i 1.01005 + 1.74946i
274.11 −0.706504 + 1.22370i −0.866025 − 0.500000i 0.00170454 + 0.00295235i −0.192549 + 0.333505i 1.22370 − 0.706504i − 0.136702i −2.83083 0.500000 + 0.866025i −0.272074 − 0.471245i
274.12 −0.702805 + 1.21729i 0.866025 + 0.500000i 0.0121308 + 0.0210111i 0.600102 − 1.03941i −1.21729 + 0.702805i 2.32368i −2.84532 0.500000 + 0.866025i 0.843508 + 1.46100i
274.13 −0.697148 + 1.20750i −0.866025 − 0.500000i 0.0279699 + 0.0484453i 1.69812 − 2.94122i 1.20750 − 0.697148i − 2.09693i −2.86659 0.500000 + 0.866025i 2.36768 + 4.10094i
274.14 −0.515776 + 0.893349i −0.866025 − 0.500000i 0.467951 + 0.810515i −1.08337 + 1.87645i 0.893349 − 0.515776i 1.35171i −3.02853 0.500000 + 0.866025i −1.11755 − 1.93565i
274.15 −0.460857 + 0.798227i 0.866025 + 0.500000i 0.575222 + 0.996315i −0.869775 + 1.50649i −0.798227 + 0.460857i 3.56103i −2.90381 0.500000 + 0.866025i −0.801683 − 1.38856i
274.16 −0.390401 + 0.676194i 0.866025 + 0.500000i 0.695174 + 1.20408i 0.326365 − 0.565281i −0.676194 + 0.390401i − 4.16312i −2.64719 0.500000 + 0.866025i 0.254827 + 0.441373i
274.17 −0.266436 + 0.461482i 0.866025 + 0.500000i 0.858023 + 1.48614i −1.50574 + 2.60802i −0.461482 + 0.266436i − 0.319406i −1.98018 0.500000 + 0.866025i −0.802370 − 1.38975i
274.18 −0.213122 + 0.369138i −0.866025 − 0.500000i 0.909158 + 1.57471i 1.43711 − 2.48915i 0.369138 − 0.213122i 3.80160i −1.62753 0.500000 + 0.866025i 0.612560 + 1.06099i
274.19 −0.101927 + 0.176542i 0.866025 + 0.500000i 0.979222 + 1.69606i 2.08438 − 3.61025i −0.176542 + 0.101927i 1.40693i −0.806942 0.500000 + 0.866025i 0.424908 + 0.735963i
274.20 −0.0938248 + 0.162509i −0.866025 − 0.500000i 0.982394 + 1.70156i −0.285605 + 0.494682i 0.162509 − 0.0938248i − 1.76300i −0.743991 0.500000 + 0.866025i −0.0535936 − 0.0928268i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.d odd 6 1 inner
209.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 627.2.k.a ✓ 80
11.b odd 2 1 inner 627.2.k.a ✓ 80
19.d odd 6 1 inner 627.2.k.a ✓ 80
209.g even 6 1 inner 627.2.k.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.k.a ✓ 80 1.a even 1 1 trivial
627.2.k.a ✓ 80 11.b odd 2 1 inner
627.2.k.a ✓ 80 19.d odd 6 1 inner
627.2.k.a ✓ 80 209.g even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(627, [\chi])\).