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Properties

Label 87.5.d.c
Level $87$
Weight $5$
Character orbit 87.d
Analytic conductor $8.993$
Analytic rank $0$
Dimension $32$
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] = [87,5,Mod(86,87)]
 
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([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.86");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 87.d (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: \(8.99318678829\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 188 q^{4} - 36 q^{6} - 84 q^{7} - 452 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 188 q^{4} - 36 q^{6} - 84 q^{7} - 452 q^{9} - 224 q^{13} - 52 q^{16} - 4216 q^{22} - 832 q^{24} - 7684 q^{25} - 396 q^{28} + 3384 q^{30} - 3308 q^{33} + 9124 q^{34} - 3680 q^{36} + 19764 q^{42} + 44 q^{45} + 4796 q^{49} - 7052 q^{51} + 34048 q^{52} + 3008 q^{54} + 7976 q^{57} + 5936 q^{58} - 22848 q^{63} - 14668 q^{64} - 1872 q^{67} + 15192 q^{78} - 41088 q^{81} - 27028 q^{82} - 35572 q^{87} - 60032 q^{88} - 34272 q^{91} + 38000 q^{93} + 59636 q^{94} + 13884 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1 −7.23588 2.40914 − 8.67157i 36.3580 19.6231i −17.4322 + 62.7464i −0.405357 −147.308 −69.3921 − 41.7820i − 141.990i
86.2 −7.23588 2.40914 + 8.67157i 36.3580 − 19.6231i −17.4322 − 62.7464i −0.405357 −147.308 −69.3921 + 41.7820i 141.990i
86.3 −6.51985 −5.97966 − 6.72634i 26.5085 − 31.1356i 38.9865 + 43.8547i 44.9223 −68.5136 −9.48733 + 80.4425i 202.999i
86.4 −6.51985 −5.97966 + 6.72634i 26.5085 31.1356i 38.9865 − 43.8547i 44.9223 −68.5136 −9.48733 − 80.4425i − 202.999i
86.5 −5.73762 7.98970 − 4.14303i 16.9203 − 47.2179i −45.8419 + 23.7712i −77.2621 −5.28056 46.6705 − 66.2032i 270.918i
86.6 −5.73762 7.98970 + 4.14303i 16.9203 47.2179i −45.8419 − 23.7712i −77.2621 −5.28056 46.6705 + 66.2032i − 270.918i
86.7 −4.29093 −7.78044 − 4.52380i 2.41206 28.6540i 33.3853 + 19.4113i 23.9440 58.3049 40.0705 + 70.3943i − 122.952i
86.8 −4.29093 −7.78044 + 4.52380i 2.41206 − 28.6540i 33.3853 − 19.4113i 23.9440 58.3049 40.0705 − 70.3943i 122.952i
86.9 −3.96406 6.34485 − 6.38301i −0.286250 7.48339i −25.1514 + 25.3026i 18.5615 64.5596 −0.485670 − 80.9985i − 29.6646i
86.10 −3.96406 6.34485 + 6.38301i −0.286250 − 7.48339i −25.1514 − 25.3026i 18.5615 64.5596 −0.485670 + 80.9985i 29.6646i
86.11 −3.29168 −1.45168 − 8.88215i −5.16484 6.84103i 4.77846 + 29.2372i −58.5203 69.6679 −76.7853 + 25.7881i − 22.5185i
86.12 −3.29168 −1.45168 + 8.88215i −5.16484 − 6.84103i 4.77846 − 29.2372i −58.5203 69.6679 −76.7853 − 25.7881i 22.5185i
86.13 −1.36205 1.79151 − 8.81989i −14.1448 − 35.6756i −2.44013 + 12.0132i 75.8945 41.0588 −74.5810 − 31.6018i 48.5920i
86.14 −1.36205 1.79151 + 8.81989i −14.1448 35.6756i −2.44013 − 12.0132i 75.8945 41.0588 −74.5810 + 31.6018i − 48.5920i
86.15 −0.630141 −7.48299 − 5.00048i −15.6029 − 33.7696i 4.71534 + 3.15101i −48.1347 19.9143 30.9904 + 74.8371i 21.2796i
86.16 −0.630141 −7.48299 + 5.00048i −15.6029 33.7696i 4.71534 − 3.15101i −48.1347 19.9143 30.9904 − 74.8371i − 21.2796i
86.17 0.630141 7.48299 − 5.00048i −15.6029 33.7696i 4.71534 − 3.15101i −48.1347 −19.9143 30.9904 − 74.8371i 21.2796i
86.18 0.630141 7.48299 + 5.00048i −15.6029 − 33.7696i 4.71534 + 3.15101i −48.1347 −19.9143 30.9904 + 74.8371i − 21.2796i
86.19 1.36205 −1.79151 − 8.81989i −14.1448 35.6756i −2.44013 − 12.0132i 75.8945 −41.0588 −74.5810 + 31.6018i 48.5920i
86.20 1.36205 −1.79151 + 8.81989i −14.1448 − 35.6756i −2.44013 + 12.0132i 75.8945 −41.0588 −74.5810 − 31.6018i − 48.5920i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.32
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.5.d.c ✓ 32
3.b odd 2 1 inner 87.5.d.c ✓ 32
29.b even 2 1 inner 87.5.d.c ✓ 32
87.d odd 2 1 inner 87.5.d.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.5.d.c ✓ 32 1.a even 1 1 trivial
87.5.d.c ✓ 32 3.b odd 2 1 inner
87.5.d.c ✓ 32 29.b even 2 1 inner
87.5.d.c ✓ 32 87.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 175 T_{2}^{14} + 12143 T_{2}^{12} - 427707 T_{2}^{10} + 8132138 T_{2}^{8} + \cdots + 169201312 \) acting on \(S_{5}^{\mathrm{new}}(87, [\chi])\). Copy content Toggle raw display