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Properties

Label 87.6.f.a
Level $87$
Weight $6$
Character orbit 87.f
Analytic conductor $13.953$
Analytic rank $0$
Dimension $96$
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] = [87,6,Mod(17,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 87.f (of order \(4\), degree \(2\), 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: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2 q^{3} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2 q^{3} - 8 q^{7} + 396 q^{10} + 1206 q^{12} - 502 q^{15} - 14824 q^{16} + 4202 q^{18} + 1752 q^{19} - 7236 q^{21} - 10884 q^{24} + 49992 q^{25} + 8632 q^{27} + 25220 q^{30} + 8004 q^{31} + 2040 q^{36} - 20672 q^{37} - 13506 q^{39} - 3560 q^{40} - 1020 q^{43} - 20096 q^{45} - 48868 q^{46} - 86310 q^{48} + 71112 q^{49} - 58788 q^{52} + 37228 q^{54} + 93508 q^{55} - 44304 q^{58} + 91050 q^{60} - 24984 q^{61} + 19974 q^{66} - 114360 q^{69} - 371524 q^{70} - 248874 q^{72} + 41504 q^{73} + 195660 q^{75} - 37488 q^{76} + 343788 q^{78} + 151044 q^{79} + 53876 q^{81} + 445688 q^{82} - 536908 q^{84} - 146104 q^{85} - 109022 q^{87} - 723184 q^{88} + 488954 q^{90} + 1236172 q^{94} - 126808 q^{97} + 414762 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −7.54346 − 7.54346i −8.28733 + 13.2030i 81.8075i 28.4617 162.112 − 37.0815i 18.8079 375.721 − 375.721i −105.640 − 218.836i −214.700 − 214.700i
17.2 −7.50124 − 7.50124i 6.07333 − 14.3567i 80.5373i −80.2027 −153.251 + 62.1355i −223.279 364.090 − 364.090i −169.229 − 174.386i 601.620 + 601.620i
17.3 −7.22345 − 7.22345i −5.95651 − 14.4056i 72.3564i 104.302 −61.0312 + 147.084i −0.288990 291.512 − 291.512i −172.040 + 171.614i −753.423 − 753.423i
17.4 −7.21503 − 7.21503i 12.7925 + 8.90801i 72.1132i −51.1287 −28.0265 − 156.570i 118.009 289.418 − 289.418i 84.2948 + 227.911i 368.895 + 368.895i
17.5 −6.86244 − 6.86244i −14.7854 − 4.93882i 62.1861i −44.4847 67.5716 + 135.356i 124.910 207.150 − 207.150i 194.216 + 146.045i 305.273 + 305.273i
17.6 −6.60206 − 6.60206i 15.5882 + 0.0968455i 55.1743i 62.9805 −102.275 − 103.553i −104.563 152.998 − 152.998i 242.981 + 3.01929i −415.801 − 415.801i
17.7 −5.78124 − 5.78124i −15.4378 − 2.16180i 34.8456i −8.19725 76.7520 + 101.748i −234.643 16.4510 − 16.4510i 233.653 + 66.7469i 47.3903 + 47.3903i
17.8 −5.73763 − 5.73763i 1.15626 − 15.5455i 33.8407i −25.4721 −95.8286 + 82.5602i 145.628 10.5615 − 10.5615i −240.326 − 35.9495i 146.149 + 146.149i
17.9 −5.54240 − 5.54240i 4.44335 + 14.9418i 29.4364i −6.79092 58.1865 − 107.440i −102.912 −14.2087 + 14.2087i −203.513 + 132.783i 37.6380 + 37.6380i
17.10 −5.32892 − 5.32892i 12.3037 − 9.57183i 24.7947i 15.8370 −116.573 − 14.5577i 106.461 −38.3963 + 38.3963i 59.7600 − 235.537i −84.3939 − 84.3939i
17.11 −4.90659 − 4.90659i −4.23449 + 15.0023i 16.1493i −89.6228 94.3871 − 52.8332i −48.1183 −77.7730 + 77.7730i −207.138 − 127.054i 439.742 + 439.742i
17.12 −4.58762 − 4.58762i −12.9552 + 8.66964i 10.0926i 62.2636 99.2067 + 19.6606i −46.1441 −100.503 + 100.503i 92.6747 − 224.634i −285.642 − 285.642i
17.13 −4.55998 − 4.55998i 5.81595 + 14.4629i 9.58686i 99.7623 39.4298 − 92.4711i 194.901 −102.204 + 102.204i −175.349 + 168.231i −454.914 − 454.914i
17.14 −3.69285 − 3.69285i 15.4989 − 1.66889i − 4.72568i −93.6900 −63.3980 − 51.0721i −43.9968 −135.623 + 135.623i 237.430 − 51.7317i 345.984 + 345.984i
17.15 −3.57180 − 3.57180i −7.72994 − 13.5369i − 6.48445i 14.2350 −20.7414 + 75.9610i 7.80338 −137.459 + 137.459i −123.496 + 209.279i −50.8445 − 50.8445i
17.16 −3.48326 − 3.48326i −13.2662 + 8.18578i − 7.73383i −34.8184 74.7229 + 17.6965i 234.508 −138.403 + 138.403i 108.986 − 217.189i 121.282 + 121.282i
17.17 −2.86141 − 2.86141i −9.55239 − 12.3188i − 15.6247i −90.8762 −7.91573 + 62.5823i −42.6783 −136.274 + 136.274i −60.5038 + 235.347i 260.034 + 260.034i
17.18 −2.58414 − 2.58414i 5.27054 − 14.6704i − 18.6444i 41.2035 −51.5303 + 24.2907i −180.666 −130.872 + 130.872i −187.443 − 154.642i −106.476 − 106.476i
17.19 −2.48747 − 2.48747i 12.9445 + 8.68562i − 19.6250i 15.3432 −10.5938 − 53.8042i −183.292 −128.415 + 128.415i 92.1199 + 224.862i −38.1656 − 38.1656i
17.20 −2.43001 − 2.43001i −14.2102 − 6.40859i − 20.1901i 86.2353 18.9580 + 50.1038i 58.1718 −126.822 + 126.822i 160.860 + 182.135i −209.552 − 209.552i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.c odd 4 1 inner
87.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.6.f.a ✓ 96
3.b odd 2 1 inner 87.6.f.a ✓ 96
29.c odd 4 1 inner 87.6.f.a ✓ 96
87.f even 4 1 inner 87.6.f.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.6.f.a ✓ 96 1.a even 1 1 trivial
87.6.f.a ✓ 96 3.b odd 2 1 inner
87.6.f.a ✓ 96 29.c odd 4 1 inner
87.6.f.a ✓ 96 87.f even 4 1 inner

Hecke kernels

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