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Properties

Label 87.4.g.a
Level $87$
Weight $4$
Character orbit 87.g
Analytic conductor $5.133$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [87,4,Mod(7,87)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("87.7"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 87.g (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.13316617050\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 2 q^{2} + 21 q^{3} - 34 q^{4} - 47 q^{5} + 6 q^{6} + 20 q^{7} - 81 q^{8} - 63 q^{9} - 108 q^{10} + 85 q^{11} - 402 q^{12} - 96 q^{13} - 197 q^{14} + 141 q^{15} + 150 q^{16} + 488 q^{17} - 18 q^{18}+ \cdots + 1206 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.23219 − 5.39858i 2.70291 + 1.30165i −20.4186 + 9.83308i −3.60054 − 15.7750i 3.69657 − 16.1957i −12.3000 − 5.92337i 50.6241 + 63.4806i 5.61141 + 7.03648i −80.7259 + 38.8756i
7.2 −1.00927 − 4.42191i 2.70291 + 1.30165i −11.3269 + 5.45473i 3.81294 + 16.7056i 3.02781 − 13.2657i 25.7918 + 12.4207i 12.9289 + 16.2123i 5.61141 + 7.03648i 70.0221 − 33.7209i
7.3 −0.578310 − 2.53374i 2.70291 + 1.30165i 1.12235 − 0.540496i −1.84710 − 8.09269i 1.73493 − 7.60122i 13.1067 + 6.31183i −14.9816 − 18.7864i 5.61141 + 7.03648i −19.4366 + 9.36017i
7.4 −0.0397057 − 0.173962i 2.70291 + 1.30165i 7.17906 − 3.45726i −3.77759 − 16.5507i 0.119117 − 0.521887i −18.2541 − 8.79069i −1.77651 − 2.22767i 5.61141 + 7.03648i −2.72921 + 1.31432i
7.5 −0.0329050 − 0.144166i 2.70291 + 1.30165i 7.18805 − 3.46158i 3.38147 + 14.8152i 0.0987151 − 0.432499i −3.00984 − 1.44946i −1.47315 − 1.84727i 5.61141 + 7.03648i 2.02459 − 0.974990i
7.6 0.562107 + 2.46275i 2.70291 + 1.30165i 1.45858 − 0.702415i −0.920546 − 4.03317i −1.68632 + 7.38825i 14.5553 + 7.00946i 15.1496 + 18.9971i 5.61141 + 7.03648i 9.41525 − 4.53415i
7.7 1.03922 + 4.55313i 2.70291 + 1.30165i −12.4433 + 5.99237i 1.95017 + 8.54425i −3.11767 + 13.6594i −8.47913 − 4.08333i −16.9207 − 21.2179i 5.61141 + 7.03648i −36.8765 + 17.7588i
16.1 −4.75225 − 2.28856i −1.87047 + 2.34549i 12.3585 + 15.4970i 0.923827 + 0.444892i 14.2568 − 6.86569i 1.75640 − 2.20246i −13.8749 − 60.7901i −2.00269 − 8.77435i −3.37210 − 4.22848i
16.2 −2.78928 − 1.34325i −1.87047 + 2.34549i 0.987842 + 1.23871i −13.3238 − 6.41638i 8.36783 − 4.02974i 12.4583 − 15.6222i 4.41969 + 19.3639i −2.00269 − 8.77435i 28.5449 + 35.7941i
16.3 −2.60781 − 1.25586i −1.87047 + 2.34549i 0.235602 + 0.295435i 4.97984 + 2.39816i 7.82344 − 3.76757i −4.10088 + 5.14234i 4.90924 + 21.5088i −2.00269 − 8.77435i −9.97474 − 12.5079i
16.4 −0.181202 − 0.0872622i −1.87047 + 2.34549i −4.96270 − 6.22303i 10.5620 + 5.08637i 0.543606 − 0.261787i −11.8767 + 14.8930i 0.714240 + 3.12929i −2.00269 − 8.77435i −1.47000 − 1.84332i
16.5 1.12228 + 0.540460i −1.87047 + 2.34549i −4.02051 − 5.04156i −2.86104 − 1.37780i −3.36683 + 1.62138i 14.3709 − 18.0205i −4.00480 − 17.5462i −2.00269 − 8.77435i −2.46623 − 3.09255i
16.6 2.35299 + 1.13314i −1.87047 + 2.34549i −0.735357 − 0.922108i −16.2366 − 7.81914i −7.05898 + 3.39942i −15.9094 + 19.9498i −5.33454 − 23.3721i −2.00269 − 8.77435i −29.3444 − 36.7967i
16.7 4.37489 + 2.10684i −1.87047 + 2.34549i 9.71301 + 12.1797i 3.94824 + 1.90137i −13.1247 + 6.32051i −2.05691 + 2.57929i 8.18860 + 35.8766i −2.00269 − 8.77435i 13.2673 + 16.6366i
25.1 −1.23219 + 5.39858i 2.70291 − 1.30165i −20.4186 − 9.83308i −3.60054 + 15.7750i 3.69657 + 16.1957i −12.3000 + 5.92337i 50.6241 − 63.4806i 5.61141 − 7.03648i −80.7259 − 38.8756i
25.2 −1.00927 + 4.42191i 2.70291 − 1.30165i −11.3269 − 5.45473i 3.81294 − 16.7056i 3.02781 + 13.2657i 25.7918 − 12.4207i 12.9289 − 16.2123i 5.61141 − 7.03648i 70.0221 + 33.7209i
25.3 −0.578310 + 2.53374i 2.70291 − 1.30165i 1.12235 + 0.540496i −1.84710 + 8.09269i 1.73493 + 7.60122i 13.1067 − 6.31183i −14.9816 + 18.7864i 5.61141 − 7.03648i −19.4366 − 9.36017i
25.4 −0.0397057 + 0.173962i 2.70291 − 1.30165i 7.17906 + 3.45726i −3.77759 + 16.5507i 0.119117 + 0.521887i −18.2541 + 8.79069i −1.77651 + 2.22767i 5.61141 − 7.03648i −2.72921 − 1.31432i
25.5 −0.0329050 + 0.144166i 2.70291 − 1.30165i 7.18805 + 3.46158i 3.38147 − 14.8152i 0.0987151 + 0.432499i −3.00984 + 1.44946i −1.47315 + 1.84727i 5.61141 − 7.03648i 2.02459 + 0.974990i
25.6 0.562107 − 2.46275i 2.70291 − 1.30165i 1.45858 + 0.702415i −0.920546 + 4.03317i −1.68632 − 7.38825i 14.5553 − 7.00946i 15.1496 − 18.9971i 5.61141 − 7.03648i 9.41525 + 4.53415i
See all 42 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.4.g.a ✓ 42
29.d even 7 1 inner 87.4.g.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.4.g.a ✓ 42 1.a even 1 1 trivial
87.4.g.a ✓ 42 29.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 2 T_{2}^{41} + 47 T_{2}^{40} + 129 T_{2}^{39} + 1359 T_{2}^{38} + 5982 T_{2}^{37} + \cdots + 1388098686976 \) acting on \(S_{4}^{\mathrm{new}}(87, [\chi])\). Copy content Toggle raw display