↑

Properties

Label 80.6.q.a
Level $80$
Weight $6$
Character orbit 80.q
Analytic conductor $12.831$
Analytic rank $0$
Dimension $116$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [80,6,Mod(29,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.29");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.q (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: \(12.8307055850\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(58\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 4 q^{4} - 2 q^{5} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q - 4 q^{4} - 2 q^{5} - 4 q^{6} + 532 q^{10} - 4 q^{11} + 1108 q^{14} - 4 q^{15} + 1216 q^{16} + 2356 q^{19} + 824 q^{20} - 976 q^{21} - 2864 q^{24} - 12104 q^{26} - 4 q^{29} + 736 q^{30} + 23056 q^{31} - 38304 q^{34} - 3864 q^{35} - 20588 q^{36} + 15144 q^{40} - 25536 q^{44} + 6734 q^{45} - 36052 q^{46} + 220884 q^{49} + 56116 q^{50} - 9472 q^{51} - 104360 q^{54} - 2552 q^{56} + 14476 q^{59} + 1008 q^{60} - 48084 q^{61} + 72944 q^{64} - 27692 q^{65} - 56736 q^{66} - 23296 q^{69} + 135260 q^{70} - 186128 q^{74} - 33244 q^{75} - 35824 q^{76} + 427312 q^{79} - 129128 q^{80} - 551132 q^{81} - 418448 q^{84} + 6248 q^{85} + 177516 q^{86} + 449576 q^{90} - 231952 q^{91} + 607356 q^{94} + 38420 q^{95} - 440264 q^{96} + 406924 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −5.65319 − 0.203680i −12.8690 + 12.8690i 31.9170 + 2.30288i −50.2546 + 24.4842i 75.3721 − 70.1298i −156.045 −179.964 − 19.5195i − 88.2233i 289.086 − 128.178i
29.2 −5.64889 + 0.300072i −16.1243 + 16.1243i 31.8199 − 3.39015i 33.0407 + 45.0922i 86.2458 − 95.9226i 106.654 −178.730 + 28.6988i − 276.984i −200.174 − 244.806i
29.3 −5.61624 − 0.676673i 18.5525 − 18.5525i 31.0842 + 7.60071i 8.85788 − 55.1955i −116.749 + 91.6413i −206.395 −169.433 − 63.7213i − 445.391i −87.0972 + 303.997i
29.4 −5.57469 − 0.960646i −2.26113 + 2.26113i 30.1543 + 10.7106i −30.6126 − 46.7747i 14.7772 − 10.4329i 66.4138 −157.812 − 88.6759i 232.775i 125.722 + 290.162i
29.5 −5.48884 − 1.36846i 6.61300 − 6.61300i 28.2546 + 15.0225i 49.7455 + 25.5026i −45.3473 + 27.2480i 180.633 −134.527 − 121.121i 155.536i −238.146 − 208.054i
29.6 −5.44124 + 1.54691i 9.88218 − 9.88218i 27.2142 − 16.8342i 27.0375 + 48.9282i −38.4845 + 69.0581i −163.839 −122.038 + 133.697i 47.6849i −222.805 − 224.406i
29.7 −5.29707 + 1.98521i 7.43164 − 7.43164i 24.1179 − 21.0316i −55.3405 + 7.90104i −24.6125 + 54.1193i 59.4492 −86.0017 + 159.285i 132.541i 277.457 − 151.715i
29.8 −5.17693 − 2.28021i 18.1244 − 18.1244i 21.6013 + 23.6090i −35.0084 + 43.5822i −135.156 + 52.5013i 75.8099 −57.9951 − 171.478i − 413.986i 280.613 − 145.796i
29.9 −5.15884 + 2.32087i −9.83803 + 9.83803i 21.2272 − 23.9459i 46.4875 − 31.0470i 27.9200 − 73.5856i −45.7194 −53.9321 + 172.799i 49.4262i −167.765 + 268.057i
29.10 −4.87261 + 2.87361i 14.7569 − 14.7569i 15.4847 − 28.0040i 24.3169 − 50.3357i −29.4990 + 114.310i 197.707 5.02143 + 180.950i − 192.530i 26.1584 + 315.144i
29.11 −4.74064 − 3.08648i −1.81560 + 1.81560i 12.9473 + 29.2637i 48.1357 − 28.4245i 14.2109 − 3.00330i −66.7879 28.9433 − 178.690i 236.407i −315.926 − 13.8197i
29.12 −4.71172 − 3.13045i −21.0051 + 21.0051i 12.4006 + 29.4996i 9.85062 − 55.0270i 164.725 − 33.2148i −32.3244 33.9186 − 177.813i − 639.428i −218.672 + 228.435i
29.13 −4.30125 − 3.67413i −2.38770 + 2.38770i 5.00149 + 31.6067i 4.21193 + 55.7428i 19.0429 − 1.49737i −204.311 94.6147 − 154.325i 231.598i 186.690 − 255.239i
29.14 −4.04395 − 3.95557i −11.3559 + 11.3559i 0.706997 + 31.9922i −38.8598 + 40.1860i 90.8416 − 1.00363i 230.224 123.688 − 132.171i − 14.9127i 316.105 − 8.79746i
29.15 −4.03469 + 3.96501i −19.4902 + 19.4902i 0.557416 − 31.9951i −49.1605 − 26.6128i 1.35807 − 155.915i 138.003 124.612 + 131.301i − 516.733i 303.867 − 87.5474i
29.16 −3.76715 + 4.22003i −3.83997 + 3.83997i −3.61723 − 31.7949i −6.34052 + 55.5410i −1.73905 − 30.6705i 92.8374 147.802 + 104.511i 213.509i −210.499 − 235.988i
29.17 −3.62558 + 4.34225i −3.49000 + 3.49000i −5.71035 − 31.4864i −25.3489 − 49.8240i −2.50120 − 27.8078i −221.972 157.425 + 89.3605i 218.640i 308.253 + 70.5698i
29.18 −3.40088 − 4.52040i 8.86425 − 8.86425i −8.86807 + 30.7467i −48.7561 − 27.3468i −70.2162 − 9.92375i −41.3923 169.146 − 64.4784i 85.8500i 42.1948 + 313.400i
29.19 −2.80183 + 4.91424i 12.4641 − 12.4641i −16.2995 − 27.5377i 55.1914 + 8.88296i 26.3294 + 96.1740i −61.0801 180.995 − 2.94408i − 67.7088i −198.290 + 246.336i
29.20 −2.78634 − 4.92304i 16.5974 − 16.5974i −16.4726 + 27.4345i 55.5811 − 5.97797i −127.956 − 35.4637i 50.7548 180.960 + 4.65334i − 307.949i −184.298 − 256.971i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.6.q.a ✓ 116
5.b even 2 1 inner 80.6.q.a ✓ 116
16.e even 4 1 inner 80.6.q.a ✓ 116
80.q even 4 1 inner 80.6.q.a ✓ 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.q.a ✓ 116 1.a even 1 1 trivial
80.6.q.a ✓ 116 5.b even 2 1 inner
80.6.q.a ✓ 116 16.e even 4 1 inner
80.6.q.a ✓ 116 80.q even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(80, [\chi])\).