Properties

Label 220.2.g.b
Level $220$
Weight $2$
Character orbit 220.g
Analytic conductor $1.757$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,2,Mod(219,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.219");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 220.g (of order \(2\), degree \(1\), 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: \(1.75670884447\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{4} - 4 q^{5} + 40 q^{9} + 12 q^{14} - 12 q^{16} - 20 q^{20} - 36 q^{25} - 48 q^{26} + 28 q^{34} - 48 q^{36} + 56 q^{44} - 48 q^{45} + 48 q^{49} + 20 q^{56} - 8 q^{60} + 20 q^{64} - 52 q^{66}+ \cdots - 16 q^{89}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
219.1 −1.33078 0.478573i −1.30496 1.54194 + 1.27375i −0.393401 + 2.20119i 1.73662 + 0.624520i 1.57789i −1.44239 2.43300i −1.29707 1.57696 2.74102i
219.2 −1.33078 0.478573i 1.30496 1.54194 + 1.27375i −0.393401 2.20119i −1.73662 0.624520i 1.57789i −1.44239 2.43300i −1.29707 −0.529900 + 3.11756i
219.3 −1.33078 + 0.478573i −1.30496 1.54194 1.27375i −0.393401 2.20119i 1.73662 0.624520i 1.57789i −1.44239 + 2.43300i −1.29707 1.57696 + 2.74102i
219.4 −1.33078 + 0.478573i 1.30496 1.54194 1.27375i −0.393401 + 2.20119i −1.73662 + 0.624520i 1.57789i −1.44239 + 2.43300i −1.29707 −0.529900 3.11756i
219.5 −0.896270 1.09394i −2.84322 −0.393401 + 1.96093i −1.64854 1.51074i 2.54829 + 3.11030i 3.37357i 2.49773 1.32716i 5.08387 −0.175122 + 3.15742i
219.6 −0.896270 1.09394i 2.84322 −0.393401 + 1.96093i −1.64854 + 1.51074i −2.54829 3.11030i 3.37357i 2.49773 1.32716i 5.08387 3.13019 + 0.449366i
219.7 −0.896270 + 1.09394i −2.84322 −0.393401 1.96093i −1.64854 + 1.51074i 2.54829 3.11030i 3.37357i 2.49773 + 1.32716i 5.08387 −0.175122 3.15742i
219.8 −0.896270 + 1.09394i 2.84322 −0.393401 1.96093i −1.64854 1.51074i −2.54829 + 3.11030i 3.37357i 2.49773 + 1.32716i 5.08387 3.13019 0.449366i
219.9 −0.419204 1.35065i −2.05261 −1.64854 + 1.13240i 1.54194 + 1.61939i 0.860462 + 2.77236i 1.06270i 2.22056 + 1.75189i 1.21320 1.54085 2.76148i
219.10 −0.419204 1.35065i 2.05261 −1.64854 + 1.13240i 1.54194 1.61939i −0.860462 2.77236i 1.06270i 2.22056 + 1.75189i 1.21320 −2.83363 1.40377i
219.11 −0.419204 + 1.35065i −2.05261 −1.64854 1.13240i 1.54194 1.61939i 0.860462 2.77236i 1.06270i 2.22056 1.75189i 1.21320 1.54085 + 2.76148i
219.12 −0.419204 + 1.35065i 2.05261 −1.64854 1.13240i 1.54194 + 1.61939i −0.860462 + 2.77236i 1.06270i 2.22056 1.75189i 1.21320 −2.83363 + 1.40377i
219.13 0.419204 1.35065i −2.05261 −1.64854 1.13240i 1.54194 1.61939i −0.860462 + 2.77236i 1.06270i −2.22056 + 1.75189i 1.21320 −1.54085 2.76148i
219.14 0.419204 1.35065i 2.05261 −1.64854 1.13240i 1.54194 + 1.61939i 0.860462 2.77236i 1.06270i −2.22056 + 1.75189i 1.21320 2.83363 1.40377i
219.15 0.419204 + 1.35065i −2.05261 −1.64854 + 1.13240i 1.54194 + 1.61939i −0.860462 2.77236i 1.06270i −2.22056 1.75189i 1.21320 −1.54085 + 2.76148i
219.16 0.419204 + 1.35065i 2.05261 −1.64854 + 1.13240i 1.54194 1.61939i 0.860462 + 2.77236i 1.06270i −2.22056 1.75189i 1.21320 2.83363 + 1.40377i
219.17 0.896270 1.09394i −2.84322 −0.393401 1.96093i −1.64854 + 1.51074i −2.54829 + 3.11030i 3.37357i −2.49773 1.32716i 5.08387 0.175122 + 3.15742i
219.18 0.896270 1.09394i 2.84322 −0.393401 1.96093i −1.64854 1.51074i 2.54829 3.11030i 3.37357i −2.49773 1.32716i 5.08387 −3.13019 + 0.449366i
219.19 0.896270 + 1.09394i −2.84322 −0.393401 + 1.96093i −1.64854 1.51074i −2.54829 3.11030i 3.37357i −2.49773 + 1.32716i 5.08387 0.175122 3.15742i
219.20 0.896270 + 1.09394i 2.84322 −0.393401 + 1.96093i −1.64854 + 1.51074i 2.54829 + 3.11030i 3.37357i −2.49773 + 1.32716i 5.08387 −3.13019 0.449366i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 219.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
20.d odd 2 1 inner
44.c even 2 1 inner
55.d odd 2 1 inner
220.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.2.g.b 24
4.b odd 2 1 inner 220.2.g.b 24
5.b even 2 1 inner 220.2.g.b 24
11.b odd 2 1 inner 220.2.g.b 24
20.d odd 2 1 inner 220.2.g.b 24
44.c even 2 1 inner 220.2.g.b 24
55.d odd 2 1 inner 220.2.g.b 24
220.g even 2 1 inner 220.2.g.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.g.b 24 1.a even 1 1 trivial
220.2.g.b 24 4.b odd 2 1 inner
220.2.g.b 24 5.b even 2 1 inner
220.2.g.b 24 11.b odd 2 1 inner
220.2.g.b 24 20.d odd 2 1 inner
220.2.g.b 24 44.c even 2 1 inner
220.2.g.b 24 55.d odd 2 1 inner
220.2.g.b 24 220.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 14T_{3}^{4} + 55T_{3}^{2} - 58 \) acting on \(S_{2}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display