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Properties

Label 820.2.bi.a
Level $820$
Weight $2$
Character orbit 820.bi
Analytic conductor $6.548$
Analytic rank $0$
Dimension $80$
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] = [820,2,Mod(189,820)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(820, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("820.189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.bi (of order \(10\), degree \(4\), 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: \(6.54773296574\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 68 q^{9} + 10 q^{15} - 26 q^{21} + 10 q^{25} - 20 q^{29} + 4 q^{31} + 15 q^{35} - 8 q^{39} + 4 q^{41} - 4 q^{45} + 18 q^{49} + 52 q^{51} - 36 q^{59} - 42 q^{61} - 15 q^{65} + 30 q^{69} - 20 q^{75} + 32 q^{81} + 10 q^{89} - 132 q^{91} + 50 q^{95} + 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1 0 −3.03349 0 1.66279 + 1.49503i 0 −0.0669129 − 0.205937i 0 6.20209 0
189.2 0 −2.68584 0 0.992463 − 2.00375i 0 −0.267597 − 0.823579i 0 4.21375 0
189.3 0 −2.59331 0 −2.08605 − 0.805227i 0 0.600140 + 1.84704i 0 3.72526 0
189.4 0 −2.47904 0 −2.21434 + 0.310997i 0 −1.40286 − 4.31755i 0 3.14566 0
189.5 0 −2.03336 0 −0.602525 + 2.15336i 0 1.21876 + 3.75095i 0 1.13456 0
189.6 0 −1.56760 0 2.00558 + 0.988762i 0 −0.633766 − 1.95053i 0 −0.542630 0
189.7 0 −0.900352 0 −1.04421 − 1.97727i 0 0.798803 + 2.45846i 0 −2.18937 0
189.8 0 −0.660390 0 −2.12425 + 0.698249i 0 0.328706 + 1.01165i 0 −2.56389 0
189.9 0 −0.389760 0 2.10341 − 0.758728i 0 0.300283 + 0.924177i 0 −2.84809 0
189.10 0 −0.323438 0 −0.368890 + 2.20543i 0 −1.10691 − 3.40672i 0 −2.89539 0
189.11 0 0.323438 0 1.98349 − 1.03235i 0 1.10691 + 3.40672i 0 −2.89539 0
189.12 0 0.389760 0 −0.0716037 + 2.23492i 0 −0.300283 − 0.924177i 0 −2.84809 0
189.13 0 0.660390 0 0.00764381 − 2.23605i 0 −0.328706 − 1.01165i 0 −2.56389 0
189.14 0 0.900352 0 −2.20318 − 0.382096i 0 −0.798803 − 2.45846i 0 −2.18937 0
189.15 0 1.56760 0 1.56013 + 1.60188i 0 0.633766 + 1.95053i 0 −0.542630 0
189.16 0 2.03336 0 1.86178 − 1.23846i 0 −1.21876 − 3.75095i 0 1.13456 0
189.17 0 2.47904 0 −0.388491 − 2.20206i 0 1.40286 + 4.31755i 0 3.14566 0
189.18 0 2.59331 0 −1.41044 − 1.73512i 0 −0.600140 − 1.84704i 0 3.72526 0
189.19 0 2.68584 0 −1.59899 + 1.56308i 0 0.267597 + 0.823579i 0 4.21375 0
189.20 0 3.03349 0 1.93569 + 1.11942i 0 0.0669129 + 0.205937i 0 6.20209 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 189.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
41.f even 10 1 inner
205.r even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.2.bi.a ✓ 80
5.b even 2 1 inner 820.2.bi.a ✓ 80
41.f even 10 1 inner 820.2.bi.a ✓ 80
205.r even 10 1 inner 820.2.bi.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.bi.a ✓ 80 1.a even 1 1 trivial
820.2.bi.a ✓ 80 5.b even 2 1 inner
820.2.bi.a ✓ 80 41.f even 10 1 inner
820.2.bi.a ✓ 80 205.r even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(820, [\chi])\).