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Properties

Label 80.3.r.a
Level $80$
Weight $3$
Character orbit 80.r
Analytic conductor $2.180$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,3,Mod(11,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.11"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 80.r (of order \(4\), degree \(2\), 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: \(2.17984211488\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{4} + 12 q^{6} - 20 q^{10} + 32 q^{11} - 60 q^{12} - 36 q^{14} + 48 q^{16} + 160 q^{18} - 32 q^{19} + 40 q^{20} - 12 q^{22} - 128 q^{23} - 120 q^{24} - 48 q^{26} - 96 q^{27} - 180 q^{28} + 32 q^{29}+ \cdots + 608 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.99970 + 0.0345663i −0.374900 − 0.374900i 3.99761 − 0.138245i −1.58114 − 1.58114i 0.762647 + 0.736729i 2.42442 −7.98925 + 0.414630i − 8.71890i 3.21646 + 3.10715i
11.2 −1.96150 − 0.390534i 3.81615 + 3.81615i 3.69497 + 1.53206i 1.58114 + 1.58114i −5.99504 − 8.97572i −7.06228 −6.64935 − 4.44816i 20.1260i −2.48392 − 3.71889i
11.3 −1.96125 + 0.391780i −2.56741 − 2.56741i 3.69302 − 1.53676i 1.58114 + 1.58114i 6.04119 + 4.02947i −8.10325 −6.64087 + 4.46082i 4.18314i −3.72047 − 2.48155i
11.4 −1.58308 − 1.22223i −0.313290 − 0.313290i 1.01229 + 3.86979i 1.58114 + 1.58114i 0.113050 + 0.878878i 10.1627 3.12725 − 7.36344i − 8.80370i −0.570549 − 4.43559i
11.5 −1.35625 − 1.46989i −3.96156 − 3.96156i −0.321158 + 3.98709i −1.58114 − 1.58114i −0.450185 + 11.1959i 0.171519 6.29615 − 4.93543i 22.3880i −0.179678 + 4.46853i
11.6 −0.945244 + 1.76253i −2.05924 − 2.05924i −2.21303 − 3.33204i 1.58114 + 1.58114i 5.57595 − 1.68299i 10.3931 7.96468 − 0.750937i − 0.519079i −4.28137 + 1.29224i
11.7 −0.522340 − 1.93059i 3.58499 + 3.58499i −3.45432 + 2.01684i −1.58114 − 1.58114i 5.04854 − 8.79371i 10.0090 5.69802 + 5.61539i 16.7043i −2.22663 + 3.87842i
11.8 −0.403471 + 1.95888i −1.45771 − 1.45771i −3.67442 − 1.58070i −1.58114 − 1.58114i 3.44363 − 2.26734i −11.3889 4.57893 − 6.55999i − 4.75014i 3.73520 − 2.45932i
11.9 0.0515452 + 1.99934i 2.58103 + 2.58103i −3.99469 + 0.206112i 1.58114 + 1.58114i −5.02730 + 5.29338i 0.523915 −0.617995 − 7.97609i 4.32341i −3.07973 + 3.24273i
11.10 0.724117 − 1.86431i −1.40411 − 1.40411i −2.95131 − 2.69996i −1.58114 − 1.58114i −3.63445 + 1.60096i 0.552025 −7.17066 + 3.54707i − 5.05693i −4.09266 + 1.80280i
11.11 1.21377 − 1.58958i 1.92589 + 1.92589i −1.05352 − 3.85877i 1.58114 + 1.58114i 5.39895 − 0.723763i 4.10918 −7.41254 − 3.00902i − 1.58188i 4.43249 − 0.594202i
11.12 1.25514 + 1.55712i 1.91324 + 1.91324i −0.849242 + 3.90881i −1.58114 − 1.58114i −0.577758 + 5.38054i 1.82022 −7.15240 + 3.58374i − 1.67900i 0.477470 − 4.44657i
11.13 1.71579 − 1.02766i −3.48227 − 3.48227i 1.88784 − 3.52648i 1.58114 + 1.58114i −9.55341 − 2.39625i −6.27123 −0.384885 − 7.99074i 15.2524i 4.33776 + 1.08803i
11.14 1.88884 + 0.657496i 0.0991349 + 0.0991349i 3.13540 + 2.48380i 1.58114 + 1.58114i 0.122069 + 0.252430i −3.75219 4.28916 + 6.75301i − 8.98034i 1.94692 + 4.02610i
11.15 1.89747 − 0.632136i 3.09850 + 3.09850i 3.20081 − 2.39892i −1.58114 − 1.58114i 7.83800 + 3.92065i −13.0357 4.55701 − 6.57523i 10.2014i −3.99966 − 2.00067i
11.16 1.98617 − 0.234782i −1.39844 − 1.39844i 3.88975 − 0.932635i −1.58114 − 1.58114i −3.10588 − 2.44922i 9.44746 7.50675 − 2.76562i − 5.08871i −3.51164 − 2.76919i
51.1 −1.99970 − 0.0345663i −0.374900 + 0.374900i 3.99761 + 0.138245i −1.58114 + 1.58114i 0.762647 − 0.736729i 2.42442 −7.98925 − 0.414630i 8.71890i 3.21646 − 3.10715i
51.2 −1.96150 + 0.390534i 3.81615 − 3.81615i 3.69497 − 1.53206i 1.58114 − 1.58114i −5.99504 + 8.97572i −7.06228 −6.64935 + 4.44816i − 20.1260i −2.48392 + 3.71889i
51.3 −1.96125 − 0.391780i −2.56741 + 2.56741i 3.69302 + 1.53676i 1.58114 − 1.58114i 6.04119 − 4.02947i −8.10325 −6.64087 − 4.46082i − 4.18314i −3.72047 + 2.48155i
51.4 −1.58308 + 1.22223i −0.313290 + 0.313290i 1.01229 − 3.86979i 1.58114 − 1.58114i 0.113050 − 0.878878i 10.1627 3.12725 + 7.36344i 8.80370i −0.570549 + 4.43559i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.3.r.a ✓ 32
4.b odd 2 1 320.3.r.a 32
5.b even 2 1 400.3.r.f 32
5.c odd 4 1 400.3.k.g 32
5.c odd 4 1 400.3.k.h 32
8.b even 2 1 640.3.r.a 32
8.d odd 2 1 640.3.r.b 32
16.e even 4 1 320.3.r.a 32
16.e even 4 1 640.3.r.b 32
16.f odd 4 1 inner 80.3.r.a ✓ 32
16.f odd 4 1 640.3.r.a 32
80.j even 4 1 400.3.k.g 32
80.k odd 4 1 400.3.r.f 32
80.s even 4 1 400.3.k.h 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.r.a ✓ 32 1.a even 1 1 trivial
80.3.r.a ✓ 32 16.f odd 4 1 inner
320.3.r.a 32 4.b odd 2 1
320.3.r.a 32 16.e even 4 1
400.3.k.g 32 5.c odd 4 1
400.3.k.g 32 80.j even 4 1
400.3.k.h 32 5.c odd 4 1
400.3.k.h 32 80.s even 4 1
400.3.r.f 32 5.b even 2 1
400.3.r.f 32 80.k odd 4 1
640.3.r.a 32 8.b even 2 1
640.3.r.a 32 16.f odd 4 1
640.3.r.b 32 8.d odd 2 1
640.3.r.b 32 16.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(80, [\chi])\).