↑

Properties

Label 80.3.k.a
Level $80$
Weight $3$
Character orbit 80.k
Analytic conductor $2.180$
Analytic rank $0$
Dimension $44$
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] = [80,3,Mod(19,80)]
 
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([2, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 80.k (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: \(2.17984211488\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{4} - 2 q^{5} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{4} - 2 q^{5} - 4 q^{6} - 20 q^{10} - 4 q^{11} + 4 q^{14} - 32 q^{16} - 36 q^{19} + 40 q^{20} + 32 q^{21} + 16 q^{24} - 56 q^{26} - 4 q^{29} - 160 q^{30} - 192 q^{34} + 212 q^{36} - 8 q^{39} - 184 q^{40} + 224 q^{44} + 30 q^{45} + 124 q^{46} - 148 q^{49} + 100 q^{50} + 128 q^{51} + 24 q^{54} - 260 q^{55} + 360 q^{56} - 68 q^{59} - 80 q^{60} + 28 q^{61} - 16 q^{64} - 20 q^{65} + 448 q^{66} + 128 q^{69} + 396 q^{70} - 264 q^{71} + 480 q^{74} + 60 q^{75} - 464 q^{76} + 504 q^{80} - 116 q^{81} - 496 q^{84} + 48 q^{85} - 852 q^{86} + 144 q^{90} + 384 q^{91} - 340 q^{94} - 1128 q^{96} + 484 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.94326 + 0.473000i −0.276605 − 0.276605i 3.55254 − 1.83833i −4.95202 − 0.690989i 0.668351 + 0.406683i − 2.26795i −6.03400 + 5.25270i − 8.84698i 9.94992 − 0.999531i
19.2 −1.91882 − 0.564029i 3.19442 + 3.19442i 3.36374 + 2.16454i 0.631330 − 4.95998i −4.32778 − 7.93127i 10.1623i −5.23356 − 6.05061i 11.4087i −4.00898 + 9.16123i
19.3 −1.85180 − 0.755534i −1.37405 − 1.37405i 2.85834 + 2.79820i 4.77405 + 1.48609i 1.50632 + 3.58260i − 5.00956i −3.17893 − 7.34128i − 5.22398i −7.71779 − 6.35891i
19.4 −1.82198 + 0.824856i 2.39041 + 2.39041i 2.63923 − 3.00574i 3.52510 + 3.54593i −6.32702 − 2.38354i − 5.08144i −2.32932 + 7.65338i 2.42812i −9.34755 − 3.55292i
19.5 −1.54678 + 1.26786i −3.99561 − 3.99561i 0.785083 − 3.92220i 0.520186 + 4.97287i 11.2462 + 1.11449i 4.43038i 3.75843 + 7.06217i 22.9299i −7.10949 − 7.03243i
19.6 −1.47984 − 1.34538i −1.33082 − 1.33082i 0.379881 + 3.98192i −3.92249 + 3.10066i 0.178942 + 3.75986i 9.15426i 4.79505 − 6.40371i − 5.45785i 9.97625 + 0.688754i
19.7 −0.990740 + 1.73736i −0.833926 − 0.833926i −2.03687 − 3.44255i 2.16054 − 4.50911i 2.27504 − 0.622628i − 2.27515i 7.99897 − 0.128102i − 7.60914i 5.69344 + 8.22100i
19.8 −0.860120 − 1.80560i 1.09715 + 1.09715i −2.52039 + 3.10607i −2.32699 − 4.42551i 1.03734 − 2.92470i − 12.9438i 7.77615 + 1.87923i − 6.59251i −5.98921 + 8.00808i
19.9 −0.709363 + 1.86997i 2.38521 + 2.38521i −2.99361 − 2.65298i −4.49841 + 2.18272i −6.15226 + 2.76830i 8.36483i 7.08456 − 3.71604i 2.37846i −0.890630 − 9.96026i
19.10 −0.526759 − 1.92938i 2.84589 + 2.84589i −3.44505 + 2.03264i 1.81586 + 4.65861i 3.99172 − 6.98993i 4.05996i 5.73646 + 5.57611i 7.19823i 8.03173 − 5.95746i
19.11 −0.456673 − 1.94716i −3.41958 − 3.41958i −3.58290 + 1.77844i 2.09356 − 4.54060i −5.09686 + 8.22013i 7.35293i 5.09912 + 6.16433i 14.3871i −9.79736 − 2.00293i
19.12 0.456673 + 1.94716i 3.41958 + 3.41958i −3.58290 + 1.77844i 4.54060 − 2.09356i −5.09686 + 8.22013i − 7.35293i −5.09912 − 6.16433i 14.3871i 6.15007 + 7.88522i
19.13 0.526759 + 1.92938i −2.84589 − 2.84589i −3.44505 + 2.03264i −4.65861 − 1.81586i 3.99172 − 6.98993i − 4.05996i −5.73646 − 5.57611i 7.19823i 1.04953 − 9.94477i
19.14 0.709363 − 1.86997i −2.38521 − 2.38521i −2.99361 − 2.65298i −2.18272 + 4.49841i −6.15226 + 2.76830i − 8.36483i −7.08456 + 3.71604i 2.37846i 6.86357 + 7.27265i
19.15 0.860120 + 1.80560i −1.09715 − 1.09715i −2.52039 + 3.10607i 4.42551 + 2.32699i 1.03734 − 2.92470i 12.9438i −7.77615 − 1.87923i − 6.59251i −0.395152 + 9.99219i
19.16 0.990740 − 1.73736i 0.833926 + 0.833926i −2.03687 − 3.44255i 4.50911 − 2.16054i 2.27504 − 0.622628i 2.27515i −7.99897 + 0.128102i − 7.60914i 0.713722 − 9.97450i
19.17 1.47984 + 1.34538i 1.33082 + 1.33082i 0.379881 + 3.98192i −3.10066 + 3.92249i 0.178942 + 3.75986i − 9.15426i −4.79505 + 6.40371i − 5.45785i −9.86575 + 1.63309i
19.18 1.54678 − 1.26786i 3.99561 + 3.99561i 0.785083 − 3.92220i −4.97287 − 0.520186i 11.2462 + 1.11449i − 4.43038i −3.75843 − 7.06217i 22.9299i −8.35147 + 5.50026i
19.19 1.82198 − 0.824856i −2.39041 − 2.39041i 2.63923 − 3.00574i −3.54593 − 3.52510i −6.32702 − 2.38354i 5.08144i 2.32932 − 7.65338i 2.42812i −9.36832 − 3.49778i
19.20 1.85180 + 0.755534i 1.37405 + 1.37405i 2.85834 + 2.79820i −1.48609 − 4.77405i 1.50632 + 3.58260i 5.00956i 3.17893 + 7.34128i − 5.22398i 0.855008 − 9.96338i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.f odd 4 1 inner
80.k 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.k.a ✓ 44
4.b odd 2 1 320.3.k.a 44
5.b even 2 1 inner 80.3.k.a ✓ 44
5.c odd 4 2 400.3.r.g 44
8.b even 2 1 640.3.k.b 44
8.d odd 2 1 640.3.k.a 44
16.e even 4 1 320.3.k.a 44
16.e even 4 1 640.3.k.a 44
16.f odd 4 1 inner 80.3.k.a ✓ 44
16.f odd 4 1 640.3.k.b 44
20.d odd 2 1 320.3.k.a 44
40.e odd 2 1 640.3.k.a 44
40.f even 2 1 640.3.k.b 44
80.j even 4 1 400.3.r.g 44
80.k odd 4 1 inner 80.3.k.a ✓ 44
80.k odd 4 1 640.3.k.b 44
80.q even 4 1 320.3.k.a 44
80.q even 4 1 640.3.k.a 44
80.s even 4 1 400.3.r.g 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.k.a ✓ 44 1.a even 1 1 trivial
80.3.k.a ✓ 44 5.b even 2 1 inner
80.3.k.a ✓ 44 16.f odd 4 1 inner
80.3.k.a ✓ 44 80.k odd 4 1 inner
320.3.k.a 44 4.b odd 2 1
320.3.k.a 44 16.e even 4 1
320.3.k.a 44 20.d odd 2 1
320.3.k.a 44 80.q even 4 1
400.3.r.g 44 5.c odd 4 2
400.3.r.g 44 80.j even 4 1
400.3.r.g 44 80.s even 4 1
640.3.k.a 44 8.d odd 2 1
640.3.k.a 44 16.e even 4 1
640.3.k.a 44 40.e odd 2 1
640.3.k.a 44 80.q even 4 1
640.3.k.b 44 8.b even 2 1
640.3.k.b 44 16.f odd 4 1
640.3.k.b 44 40.f even 2 1
640.3.k.b 44 80.k odd 4 1

Hecke kernels

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