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Properties

Label 87.3.j.a
Level $87$
Weight $3$
Character orbit 87.j
Analytic conductor $2.371$
Analytic rank $0$
Dimension $108$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [87,3,Mod(20,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 12]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.20");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 87.j (of order \(14\), degree \(6\), 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: \(2.37057829993\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 5 q^{3} + 22 q^{4} + 13 q^{6} - 2 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 5 q^{3} + 22 q^{4} + 13 q^{6} - 2 q^{7} + 15 q^{9} - 26 q^{10} - 32 q^{12} - 32 q^{13} - 37 q^{15} - 90 q^{16} - 6 q^{18} + 10 q^{19} - 25 q^{21} + 80 q^{22} - 185 q^{24} + 16 q^{25} - 107 q^{27} - 332 q^{28} + 158 q^{30} - 38 q^{31} - 29 q^{33} + 252 q^{34} + 161 q^{36} + 26 q^{37} + 445 q^{39} + 162 q^{40} + 34 q^{42} - 132 q^{43} - 39 q^{45} - 1032 q^{46} - 486 q^{48} - 92 q^{49} - 148 q^{51} + 92 q^{52} + 109 q^{54} - 118 q^{55} + 266 q^{57} + 70 q^{58} + 550 q^{60} - 34 q^{61} - 87 q^{63} + 558 q^{64} - 185 q^{66} + 130 q^{67} - 235 q^{69} + 1624 q^{70} + 1070 q^{72} + 684 q^{73} + 512 q^{75} + 496 q^{76} + 95 q^{78} - 716 q^{79} - 93 q^{81} + 310 q^{82} - 421 q^{84} + 894 q^{85} + 315 q^{87} - 700 q^{88} - 219 q^{90} + 520 q^{91} + 5 q^{93} - 1240 q^{94} - 434 q^{96} - 1052 q^{97} - 2518 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −1.64742 − 3.42091i −2.99809 + 0.106938i −6.49464 + 8.14402i −1.72071 − 3.57310i 5.30495 + 10.0800i 2.26711 + 2.84287i 23.7524 + 5.42134i 8.97713 − 0.641223i −9.38849 + 11.7728i
20.2 −1.43528 − 2.98039i 2.38549 + 1.81919i −4.32874 + 5.42807i 3.26791 + 6.78589i 1.99806 − 9.72073i 4.70347 + 5.89797i 9.49056 + 2.16616i 2.38109 + 8.67931i 15.5342 − 19.4793i
20.3 −1.37671 − 2.85877i 2.15394 − 2.08820i −3.78326 + 4.74406i −1.15319 − 2.39463i −8.93501 − 3.28276i −2.01278 − 2.52395i 6.39686 + 1.46004i 0.278883 − 8.99568i −5.25808 + 6.59342i
20.4 −1.19567 − 2.48283i −0.0701062 + 2.99918i −2.24085 + 2.80994i −1.24077 − 2.57649i 7.53027 − 3.41196i −8.06493 − 10.1131i −1.09066 − 0.248936i −8.99017 − 0.420522i −4.91342 + 6.16123i
20.5 −0.875687 − 1.81838i −2.46982 + 1.70293i −0.0457274 + 0.0573404i 2.53628 + 5.26663i 5.25937 + 2.99985i 4.70651 + 5.90178i −7.72629 − 1.76348i 3.20006 − 8.41187i 7.35577 − 9.22384i
20.6 −0.746868 − 1.55089i −0.806531 − 2.88955i 0.646519 − 0.810709i −2.18117 − 4.52925i −3.87900 + 3.40895i 6.07633 + 7.61947i −8.45297 − 1.92934i −7.69901 + 4.66103i −5.39531 + 6.76550i
20.7 −0.633893 − 1.31629i −2.69110 − 1.32589i 1.16315 − 1.45855i 0.826032 + 1.71527i −0.0393844 + 4.38275i −6.06047 − 7.59959i −8.35457 − 1.90688i 5.48405 + 7.13619i 1.73419 − 2.17460i
20.8 −0.373428 − 0.775431i 2.82018 + 1.02304i 2.03211 − 2.54819i −1.44697 − 3.00467i −0.259839 − 2.56888i 0.543291 + 0.681265i −6.09113 − 1.39026i 6.90680 + 5.77028i −1.78957 + 2.24405i
20.9 −0.243171 − 0.504951i 1.89872 − 2.32268i 2.29812 − 2.88175i 3.65926 + 7.59853i −1.63456 − 0.393950i −0.689021 − 0.864005i −4.19958 − 0.958527i −1.78972 − 8.82026i 2.94706 − 3.69549i
20.10 0.243171 + 0.504951i −2.68695 + 1.33427i 2.29812 − 2.88175i −3.65926 − 7.59853i −1.32713 − 1.03232i −0.689021 − 0.864005i 4.19958 + 0.958527i 5.43945 − 7.17025i 2.94706 − 3.69549i
20.11 0.373428 + 0.775431i 0.369837 + 2.97712i 2.03211 − 2.54819i 1.44697 + 3.00467i −2.17044 + 1.39852i 0.543291 + 0.681265i 6.09113 + 1.39026i −8.72644 + 2.20210i −1.78957 + 2.24405i
20.12 0.633893 + 1.31629i −0.693818 − 2.91867i 1.16315 − 1.45855i −0.826032 − 1.71527i 3.40201 − 2.76339i −6.06047 − 7.59959i 8.35457 + 1.90688i −8.03723 + 4.05005i 1.73419 − 2.17460i
20.13 0.746868 + 1.55089i −2.63763 − 1.42930i 0.646519 − 0.810709i 2.18117 + 4.52925i 0.246712 − 5.15817i 6.07633 + 7.61947i 8.45297 + 1.92934i 4.91423 + 7.53992i −5.39531 + 6.76550i
20.14 0.875687 + 1.81838i 2.20982 − 2.02896i −0.0457274 + 0.0573404i −2.53628 − 5.26663i 5.62454 + 2.24156i 4.70651 + 5.90178i 7.72629 + 1.76348i 0.766618 − 8.96729i 7.35577 − 9.22384i
20.15 1.19567 + 2.48283i 2.93959 + 0.599032i −2.24085 + 2.80994i 1.24077 + 2.57649i 2.02747 + 8.01472i −8.06493 − 10.1131i 1.09066 + 0.248936i 8.28232 + 3.52181i −4.91342 + 6.16123i
20.16 1.37671 + 2.85877i −2.51514 + 1.63527i −3.78326 + 4.74406i 1.15319 + 2.39463i −8.13746 − 4.93890i −2.01278 − 2.52395i −6.39686 − 1.46004i 3.65181 − 8.22583i −5.25808 + 6.59342i
20.17 1.43528 + 2.98039i 1.24276 + 2.73049i −4.32874 + 5.42807i −3.26791 − 6.78589i −6.35421 + 7.62292i 4.70347 + 5.89797i −9.49056 − 2.16616i −5.91110 + 6.78667i 15.5342 − 19.4793i
20.18 1.64742 + 3.42091i 0.771396 − 2.89913i −6.49464 + 8.14402i 1.72071 + 3.57310i 11.1885 − 2.13721i 2.26711 + 2.84287i −23.7524 − 5.42134i −7.80990 − 4.47275i −9.38849 + 11.7728i
23.1 −2.91004 − 2.32068i −1.67807 − 2.48678i 2.19270 + 9.60683i 6.77615 + 5.40380i −0.887771 + 11.1309i −0.840468 + 3.68233i 9.45374 − 19.6309i −3.36816 + 8.34599i −7.17838 − 31.4505i
23.2 −2.50185 − 1.99516i 2.04266 − 2.19717i 1.38851 + 6.08347i −4.59381 − 3.66344i −9.49413 + 1.42155i 0.756169 − 3.31299i 3.10996 − 6.45789i −0.655079 − 8.97613i 4.18388 + 18.3308i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.d even 7 1 inner
87.j odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.3.j.a ✓ 108
3.b odd 2 1 inner 87.3.j.a ✓ 108
29.d even 7 1 inner 87.3.j.a ✓ 108
87.j odd 14 1 inner 87.3.j.a ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.3.j.a ✓ 108 1.a even 1 1 trivial
87.3.j.a ✓ 108 3.b odd 2 1 inner
87.3.j.a ✓ 108 29.d even 7 1 inner
87.3.j.a ✓ 108 87.j odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(87, [\chi])\).