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Properties

Label 87.2.k.a
Level $87$
Weight $2$
Character orbit 87.k
Analytic conductor $0.695$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [87,2,Mod(2,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([14, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 87.k (of order \(28\), degree \(12\), 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: \(0.694698497585\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(8\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 12 q^{3} - 28 q^{4} - 14 q^{6} - 20 q^{7} - 14 q^{9} - 4 q^{10} - 8 q^{12} - 28 q^{13} - 16 q^{15} - 48 q^{16} - 28 q^{18} - 20 q^{19} + 26 q^{21} - 28 q^{22} + 58 q^{24} - 20 q^{25} + 36 q^{27}+ \cdots + 82 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.261972 + 2.32506i 1.64121 + 0.553555i −3.38744 − 0.773161i 0.273793 + 0.343325i −1.71700 + 3.67091i −0.670558 − 2.93790i 1.13950 − 3.25651i 2.38715 + 1.81700i −0.869979 + 0.546644i
2.2 −0.255714 + 2.26953i −1.44205 + 0.959421i −3.13550 − 0.715657i −1.22248 − 1.53294i −1.80868 − 3.51811i 0.968527 + 4.24339i 0.917355 − 2.62165i 1.15902 − 2.76707i 3.79166 − 2.38246i
2.3 −0.128998 + 1.14489i −1.41996 − 0.991816i 0.655718 + 0.149663i 1.88898 + 2.36870i 1.31870 − 1.49776i −0.0467107 − 0.204653i −1.01699 + 2.90639i 1.03260 + 2.81669i −2.95558 + 1.85712i
2.4 −0.108076 + 0.959202i 0.843782 − 1.51262i 1.04147 + 0.237708i −1.18935 − 1.49140i 1.35972 + 0.972836i −0.00234208 − 0.0102613i −0.978186 + 2.79549i −1.57606 − 2.55265i 1.55910 − 0.979646i
2.5 0.108076 − 0.959202i −0.996724 + 1.41652i 1.04147 + 0.237708i 1.18935 + 1.49140i 1.25101 + 1.10915i −0.00234208 − 0.0102613i 0.978186 − 2.79549i −1.01308 − 2.82377i 1.55910 − 0.979646i
2.6 0.128998 − 1.14489i −1.50970 − 0.849011i 0.655718 + 0.149663i −1.88898 − 2.36870i −1.16677 + 1.61892i −0.0467107 − 0.204653i 1.01699 − 2.90639i 1.55836 + 2.56350i −2.95558 + 1.85712i
2.7 0.255714 − 2.26953i 0.238726 − 1.71552i −3.13550 − 0.715657i 1.22248 + 1.53294i −3.83237 − 0.980478i 0.968527 + 4.24339i −0.917355 + 2.62165i −2.88602 − 0.819080i 3.79166 − 2.38246i
2.8 0.261972 − 2.32506i 1.21083 + 1.23850i −3.38744 − 0.773161i −0.273793 − 0.343325i 3.19680 − 2.49081i −0.670558 − 2.93790i −1.13950 + 3.25651i −0.0677777 + 2.99923i −0.869979 + 0.546644i
8.1 −0.909159 + 2.59823i 1.42570 + 0.983555i −4.36055 − 3.47743i −1.78469 + 0.859464i −3.85169 + 2.81008i 0.785988 + 0.985597i 8.33803 − 5.23913i 1.06524 + 2.80451i −0.610511 − 5.41843i
8.2 −0.576259 + 1.64685i 0.598445 − 1.62538i −0.816387 − 0.651047i 1.59790 − 0.769508i 2.33190 + 1.92219i 1.54710 + 1.94000i −1.41204 + 0.887242i −2.28373 − 1.94540i 0.346463 + 3.07494i
8.3 −0.454505 + 1.29890i −0.706640 + 1.58135i 0.0830922 + 0.0662638i 0.181368 − 0.0873422i −1.73284 − 1.63659i −0.770934 − 0.966720i −2.45423 + 1.54210i −2.00132 − 2.23489i 0.0310162 + 0.275277i
8.4 −0.0726507 + 0.207624i 1.66092 − 0.491261i 1.52583 + 1.21681i −2.41194 + 1.16153i −0.0186698 + 0.380537i −1.79859 − 2.25536i −0.735996 + 0.462457i 2.51733 − 1.63189i −0.0659319 − 0.585161i
8.5 0.0726507 − 0.207624i −1.50996 − 0.848534i 1.52583 + 1.21681i 2.41194 − 1.16153i −0.285876 + 0.251858i −1.79859 − 2.25536i 0.735996 − 0.462457i 1.55998 + 2.56251i −0.0659319 − 0.585161i
8.6 0.454505 − 1.29890i 0.337040 + 1.69894i 0.0830922 + 0.0662638i −0.181368 + 0.0873422i 2.35995 + 0.334396i −0.770934 − 0.966720i 2.45423 − 1.54210i −2.77281 + 1.14522i 0.0310162 + 0.275277i
8.7 0.576259 − 1.64685i −0.221759 − 1.71780i −0.816387 − 0.651047i −1.59790 + 0.769508i −2.95675 − 0.624690i 1.54710 + 1.94000i 1.41204 − 0.887242i −2.90165 + 0.761874i 0.346463 + 3.07494i
8.8 0.909159 − 2.59823i −1.60882 + 0.641647i −4.36055 − 3.47743i 1.78469 − 0.859464i 0.204476 + 4.76343i 0.785988 + 0.985597i −8.33803 + 5.23913i 2.17658 − 2.06459i −0.610511 − 5.41843i
11.1 −0.909159 − 2.59823i 1.42570 − 0.983555i −4.36055 + 3.47743i −1.78469 − 0.859464i −3.85169 − 2.81008i 0.785988 − 0.985597i 8.33803 + 5.23913i 1.06524 − 2.80451i −0.610511 + 5.41843i
11.2 −0.576259 − 1.64685i 0.598445 + 1.62538i −0.816387 + 0.651047i 1.59790 + 0.769508i 2.33190 − 1.92219i 1.54710 − 1.94000i −1.41204 − 0.887242i −2.28373 + 1.94540i 0.346463 − 3.07494i
11.3 −0.454505 − 1.29890i −0.706640 − 1.58135i 0.0830922 − 0.0662638i 0.181368 + 0.0873422i −1.73284 + 1.63659i −0.770934 + 0.966720i −2.45423 − 1.54210i −2.00132 + 2.23489i 0.0310162 − 0.275277i
11.4 −0.0726507 − 0.207624i 1.66092 + 0.491261i 1.52583 − 1.21681i −2.41194 − 1.16153i −0.0186698 − 0.380537i −1.79859 + 2.25536i −0.735996 − 0.462457i 2.51733 + 1.63189i −0.0659319 + 0.585161i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.f odd 28 1 inner
87.k even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.2.k.a ✓ 96
3.b odd 2 1 inner 87.2.k.a ✓ 96
29.f odd 28 1 inner 87.2.k.a ✓ 96
87.k even 28 1 inner 87.2.k.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.k.a ✓ 96 1.a even 1 1 trivial
87.2.k.a ✓ 96 3.b odd 2 1 inner
87.2.k.a ✓ 96 29.f odd 28 1 inner
87.2.k.a ✓ 96 87.k even 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(87, [\chi])\).