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Properties

Label 87.6.c.a
Level $87$
Weight $6$
Character orbit 87.c
Analytic conductor $13.953$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [87,6,Mod(28,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.28");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 87.c (of order \(2\), degree \(1\), 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: \(13.9533923237\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 396 q^{4} - 196 q^{5} + 72 q^{6} - 120 q^{7} - 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 396 q^{4} - 196 q^{5} + 72 q^{6} - 120 q^{7} - 1944 q^{9} - 612 q^{13} + 6852 q^{16} + 8192 q^{20} - 12882 q^{22} - 5492 q^{23} - 3402 q^{24} + 504 q^{25} + 19530 q^{28} - 11304 q^{29} - 864 q^{30} + 18144 q^{33} + 23958 q^{34} - 32540 q^{35} + 32076 q^{36} + 18832 q^{38} + 12726 q^{42} + 15876 q^{45} - 61044 q^{49} + 20844 q^{51} - 34182 q^{52} - 29268 q^{53} - 5832 q^{54} - 1332 q^{57} - 65220 q^{58} + 21456 q^{59} + 47288 q^{62} + 9720 q^{63} - 342906 q^{64} - 259004 q^{65} + 88392 q^{67} + 95180 q^{71} + 562664 q^{74} - 177570 q^{78} + 61452 q^{80} + 157464 q^{81} - 92460 q^{82} - 68424 q^{83} + 246372 q^{86} - 28620 q^{87} + 396984 q^{88} + 203400 q^{91} - 187332 q^{92} - 110844 q^{93} + 68358 q^{94} + 278316 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 − 11.0834i 9.00000i −90.8425 −57.5785 99.7509 52.4884 652.177i −81.0000 638.168i
28.2 − 10.5614i − 9.00000i −79.5437 28.4636 −95.0528 −104.679 502.129i −81.0000 − 300.617i
28.3 − 9.14260i 9.00000i −51.5871 71.0014 82.2834 −119.042 179.077i −81.0000 − 649.137i
28.4 − 8.44694i − 9.00000i −39.3508 −96.2810 −76.0225 −13.5906 62.0922i −81.0000 813.280i
28.5 − 7.08327i 9.00000i −18.1726 −22.3209 63.7494 −60.7443 − 97.9428i −81.0000 158.105i
28.6 − 7.06289i − 9.00000i −17.8844 56.5267 −63.5660 33.3187 − 99.6972i −81.0000 − 399.241i
28.7 − 5.85134i 9.00000i −2.23822 −78.2758 52.6621 209.728 − 174.146i −81.0000 458.019i
28.8 − 5.44169i − 9.00000i 2.38796 −52.3335 −48.9752 −17.1231 − 187.129i −81.0000 284.783i
28.9 − 4.10006i 9.00000i 15.1895 −7.65308 36.9005 −130.904 − 193.480i −81.0000 31.3781i
28.10 − 2.59532i − 9.00000i 25.2643 −5.66962 −23.3579 191.201 − 148.619i −81.0000 14.7145i
28.11 − 1.98314i 9.00000i 28.0671 80.3850 17.8483 82.4422 − 119.122i −81.0000 − 159.415i
28.12 − 1.13557i − 9.00000i 30.7105 −14.2642 −10.2201 −183.096 − 71.2120i −81.0000 16.1980i
28.13 1.13557i 9.00000i 30.7105 −14.2642 −10.2201 −183.096 71.2120i −81.0000 − 16.1980i
28.14 1.98314i − 9.00000i 28.0671 80.3850 17.8483 82.4422 119.122i −81.0000 159.415i
28.15 2.59532i 9.00000i 25.2643 −5.66962 −23.3579 191.201 148.619i −81.0000 − 14.7145i
28.16 4.10006i − 9.00000i 15.1895 −7.65308 36.9005 −130.904 193.480i −81.0000 − 31.3781i
28.17 5.44169i 9.00000i 2.38796 −52.3335 −48.9752 −17.1231 187.129i −81.0000 − 284.783i
28.18 5.85134i − 9.00000i −2.23822 −78.2758 52.6621 209.728 174.146i −81.0000 − 458.019i
28.19 7.06289i 9.00000i −17.8844 56.5267 −63.5660 33.3187 99.6972i −81.0000 399.241i
28.20 7.08327i − 9.00000i −18.1726 −22.3209 63.7494 −60.7443 97.9428i −81.0000 − 158.105i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.6.c.a ✓ 24
3.b odd 2 1 261.6.c.d 24
29.b even 2 1 inner 87.6.c.a ✓ 24
87.d odd 2 1 261.6.c.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.6.c.a ✓ 24 1.a even 1 1 trivial
87.6.c.a ✓ 24 29.b even 2 1 inner
261.6.c.d 24 3.b odd 2 1
261.6.c.d 24 87.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(87, [\chi])\).