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Properties

Label 87.7.b.a
Level $87$
Weight $7$
Character orbit 87.b
Analytic conductor $20.015$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [87,7,Mod(59,87)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("87.59"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 87.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.0147052749\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 52 q^{3} - 1924 q^{4} - 160 q^{6} + 160 q^{7} - 1060 q^{9} - 3588 q^{10} - 2166 q^{12} - 1400 q^{13} - 6240 q^{15} + 56588 q^{16} - 5978 q^{18} + 25000 q^{19} + 7520 q^{21} + 20970 q^{22} + 1238 q^{24}+ \cdots + 4793544 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 − 15.4133i −7.71721 − 25.8736i −173.570 − 221.187i −398.798 + 118.948i 394.709 1688.84i −609.889 + 399.345i −3409.22
59.2 − 15.1008i 21.0756 + 16.8766i −164.033 − 118.473i 254.850 − 318.257i −348.513 1510.57i 159.358 + 711.369i −1789.03
59.3 − 14.9948i 26.9046 − 2.26729i −160.844 110.651i −33.9976 − 403.430i 552.617 1452.16i 718.719 − 122.001i 1659.19
59.4 − 14.7970i −25.2055 + 9.67893i −154.951 − 25.9593i 143.219 + 372.966i −511.916 1345.81i 541.637 − 487.925i −384.120
59.5 − 14.2052i 4.68709 − 26.5901i −137.789 131.841i −377.718 − 66.5813i −416.698 1048.19i −685.062 − 249.260i 1872.83
59.6 − 13.5783i 0.291464 + 26.9984i −120.371 197.511i 366.594 − 3.95760i −18.6779 765.430i −728.830 + 15.7381i 2681.87
59.7 − 13.2552i −25.3475 − 9.30085i −111.699 106.724i −123.284 + 335.985i 284.634 632.265i 555.988 + 471.506i 1414.65
59.8 − 12.4411i 1.67176 + 26.9482i −90.7821 − 69.4567i 335.266 − 20.7986i 181.456 333.200i −723.410 + 90.1019i −864.121
59.9 − 12.3060i −17.8037 + 20.2985i −87.4378 − 148.813i 249.793 + 219.093i 382.073 288.426i −95.0556 − 722.776i −1831.30
59.10 − 11.6762i 20.6297 − 17.4188i −72.3347 17.6911i −203.386 − 240.878i 117.179 97.3182i 122.172 − 718.690i 206.566
59.11 − 11.3220i 21.7587 − 15.9862i −64.1869 − 190.317i −180.996 − 246.351i −248.040 2.11620i 217.881 − 695.679i −2154.77
59.12 − 9.96396i 24.5636 + 11.2085i −35.2806 − 27.8839i 111.681 − 244.750i 60.7681 − 286.159i 477.737 + 550.643i −277.834
59.13 − 9.80663i −13.8860 − 23.1556i −32.1699 − 31.8600i −227.078 + 136.175i −247.832 − 312.146i −343.359 + 643.075i −312.439
59.14 − 9.12250i 26.3711 + 5.79335i −19.2200 194.806i 52.8499 − 240.571i −587.861 − 408.506i 661.874 + 305.555i 1777.12
59.15 − 8.99068i −26.6506 − 4.32950i −16.8323 − 174.149i −38.9251 + 239.607i −298.105 − 424.069i 691.511 + 230.768i −1565.72
59.16 − 8.21957i 1.43004 − 26.9621i −3.56134 27.4201i −221.617 − 11.7543i 526.273 − 496.780i −724.910 − 77.1139i 225.382
59.17 − 7.77900i −23.4760 + 13.3371i 3.48715 148.308i 103.749 + 182.620i 172.143 − 524.983i 373.244 − 626.203i 1153.69
59.18 − 7.42129i −14.8480 + 22.5508i 8.92448 51.0538i 167.356 + 110.191i −600.212 − 541.194i −288.074 − 669.667i 378.885
59.19 − 5.77572i 6.84673 + 26.1175i 30.6411 − 163.398i 150.847 − 39.5448i −327.324 − 546.620i −635.245 + 357.638i −943.742
59.20 − 5.40547i 14.0726 + 23.0426i 34.7809 110.739i 124.556 − 76.0688i 411.141 − 533.957i −332.926 + 648.538i 598.599
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.56
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.7.b.a ✓ 56
3.b odd 2 1 inner 87.7.b.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.7.b.a ✓ 56 1.a even 1 1 trivial
87.7.b.a ✓ 56 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(87, [\chi])\).