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Properties

Label 119.2.k.a
Level $119$
Weight $2$
Character orbit 119.k
Analytic conductor $0.950$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [119,2,Mod(8,119)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(119, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("119.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 119 = 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 119.k (of order \(8\), degree \(4\), 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: \(0.950219784053\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{6} - 24 q^{9} - 16 q^{10} - 8 q^{11} + 24 q^{12} - 8 q^{15} - 8 q^{16} + 8 q^{17} - 24 q^{18} - 24 q^{19} + 16 q^{20} + 8 q^{23} + 40 q^{24} - 8 q^{25} - 16 q^{26} + 24 q^{27} + 16 q^{31} + 40 q^{32}+ \cdots - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −1.18907 + 1.18907i −0.875473 + 2.11358i − 0.827752i 3.95772 + 1.63934i −1.47219 − 3.55418i −0.923880 + 0.382683i −1.39388 − 1.39388i −1.57944 − 1.57944i −6.65527 + 2.75670i
8.2 −0.906953 + 0.906953i 0.388298 − 0.937435i 0.354874i 1.29163 + 0.535011i 0.498041 + 1.20238i 0.923880 − 0.382683i −2.13576 − 2.13576i 1.39331 + 1.39331i −1.65668 + 0.686219i
8.3 −0.686718 + 0.686718i −0.839618 + 2.02702i 1.05684i −2.58346 − 1.07010i −0.815409 − 1.96857i 0.923880 − 0.382683i −2.09919 − 2.09919i −1.28252 − 1.28252i 2.50897 − 1.03925i
8.4 −0.0228885 + 0.0228885i 1.02310 − 2.46998i 1.99895i 1.90867 + 0.790598i 0.0331169 + 0.0799512i −0.923880 + 0.382683i −0.0915300 − 0.0915300i −2.93273 − 2.93273i −0.0617823 + 0.0255910i
8.5 0.315329 − 0.315329i −0.558583 + 1.34854i 1.80113i −0.410302 − 0.169953i 0.249096 + 0.601371i −0.923880 + 0.382683i 1.19861 + 1.19861i 0.614779 + 0.614779i −0.182971 + 0.0757892i
8.6 0.881269 − 0.881269i 0.818760 − 1.97666i 0.446731i −2.87188 − 1.18957i −1.02042 − 2.46352i 0.923880 − 0.382683i 2.15623 + 2.15623i −1.11550 − 1.11550i −3.57923 + 1.48257i
8.7 1.41951 − 1.41951i −1.29872 + 3.13538i − 2.03001i 1.17530 + 0.486825i 2.60716 + 6.29424i 0.923880 − 0.382683i −0.0426034 − 0.0426034i −6.02263 − 6.02263i 2.35940 − 0.977296i
8.8 1.60373 − 1.60373i −0.0719773 + 0.173769i − 3.14391i −1.05347 − 0.436361i 0.163246 + 0.394110i −0.923880 + 0.382683i −1.83452 − 1.83452i 2.09631 + 2.09631i −2.38928 + 0.989674i
15.1 −1.18907 − 1.18907i −0.875473 − 2.11358i 0.827752i 3.95772 − 1.63934i −1.47219 + 3.55418i −0.923880 − 0.382683i −1.39388 + 1.39388i −1.57944 + 1.57944i −6.65527 − 2.75670i
15.2 −0.906953 − 0.906953i 0.388298 + 0.937435i − 0.354874i 1.29163 − 0.535011i 0.498041 − 1.20238i 0.923880 + 0.382683i −2.13576 + 2.13576i 1.39331 − 1.39331i −1.65668 − 0.686219i
15.3 −0.686718 − 0.686718i −0.839618 − 2.02702i − 1.05684i −2.58346 + 1.07010i −0.815409 + 1.96857i 0.923880 + 0.382683i −2.09919 + 2.09919i −1.28252 + 1.28252i 2.50897 + 1.03925i
15.4 −0.0228885 − 0.0228885i 1.02310 + 2.46998i − 1.99895i 1.90867 − 0.790598i 0.0331169 − 0.0799512i −0.923880 − 0.382683i −0.0915300 + 0.0915300i −2.93273 + 2.93273i −0.0617823 − 0.0255910i
15.5 0.315329 + 0.315329i −0.558583 − 1.34854i − 1.80113i −0.410302 + 0.169953i 0.249096 − 0.601371i −0.923880 − 0.382683i 1.19861 − 1.19861i 0.614779 − 0.614779i −0.182971 − 0.0757892i
15.6 0.881269 + 0.881269i 0.818760 + 1.97666i − 0.446731i −2.87188 + 1.18957i −1.02042 + 2.46352i 0.923880 + 0.382683i 2.15623 − 2.15623i −1.11550 + 1.11550i −3.57923 − 1.48257i
15.7 1.41951 + 1.41951i −1.29872 − 3.13538i 2.03001i 1.17530 − 0.486825i 2.60716 − 6.29424i 0.923880 + 0.382683i −0.0426034 + 0.0426034i −6.02263 + 6.02263i 2.35940 + 0.977296i
15.8 1.60373 + 1.60373i −0.0719773 − 0.173769i 3.14391i −1.05347 + 0.436361i 0.163246 − 0.394110i −0.923880 − 0.382683i −1.83452 + 1.83452i 2.09631 − 2.09631i −2.38928 − 0.989674i
36.1 −1.92629 − 1.92629i 2.29131 − 0.949092i 5.42117i −0.942466 − 2.27531i −6.24195 − 2.58550i 0.382683 − 0.923880i 6.59016 − 6.59016i 2.22801 − 2.22801i −2.56745 + 6.19837i
36.2 −1.61205 − 1.61205i −0.183371 + 0.0759548i 3.19742i 1.49140 + 3.60057i 0.418046 + 0.173160i −0.382683 + 0.923880i 1.93030 − 1.93030i −2.09346 + 2.09346i 3.40008 − 8.20852i
36.3 −1.26441 − 1.26441i −1.09343 + 0.452913i 1.19747i −0.942787 − 2.27609i 1.95521 + 0.809875i −0.382683 + 0.923880i −1.01472 + 1.01472i −1.13087 + 1.13087i −1.68584 + 4.06999i
36.4 −0.570914 − 0.570914i 1.90914 − 0.790793i − 1.34811i 0.917613 + 2.21531i −1.54143 − 0.638482i 0.382683 − 0.923880i −1.91149 + 1.91149i 0.898153 − 0.898153i 0.740876 − 1.78863i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 119.2.k.a ✓ 32
7.b odd 2 1 833.2.l.c 32
7.c even 3 2 833.2.v.c 64
7.d odd 6 2 833.2.v.e 64
17.d even 8 1 inner 119.2.k.a ✓ 32
17.e odd 16 1 2023.2.a.q 16
17.e odd 16 1 2023.2.a.r 16
119.l odd 8 1 833.2.l.c 32
119.q even 24 2 833.2.v.c 64
119.r odd 24 2 833.2.v.e 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.2.k.a ✓ 32 1.a even 1 1 trivial
119.2.k.a ✓ 32 17.d even 8 1 inner
833.2.l.c 32 7.b odd 2 1
833.2.l.c 32 119.l odd 8 1
833.2.v.c 64 7.c even 3 2
833.2.v.c 64 119.q even 24 2
833.2.v.e 64 7.d odd 6 2
833.2.v.e 64 119.r odd 24 2
2023.2.a.q 16 17.e odd 16 1
2023.2.a.r 16 17.e odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(119, [\chi])\).