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Properties

Label 209.2.f.b
Level $209$
Weight $2$
Character orbit 209.f
Analytic conductor $1.669$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [209,2,Mod(20,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.20");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.f (of order \(5\), 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: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} + q^{3} - 6 q^{4} + 7 q^{5} - 14 q^{6} - 12 q^{7} + 4 q^{8} + 6 q^{9} - 12 q^{10} + 2 q^{11} - 16 q^{12} + 5 q^{13} + 23 q^{14} + 15 q^{15} + 30 q^{16} - 15 q^{17} - 38 q^{18} + 7 q^{19}+ \cdots - 64 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −0.816838 + 2.51397i 0.225457 − 0.163804i −4.03478 − 2.93144i −1.00025 − 3.07845i 0.227636 + 0.700593i −1.23421 − 0.896705i 6.38829 − 4.64136i −0.903052 + 2.77931i 8.55617
20.2 −0.577060 + 1.77601i −0.224936 + 0.163425i −1.20317 − 0.874155i 0.450929 + 1.38782i −0.160443 − 0.493794i −3.13571 − 2.27823i −0.774717 + 0.562865i −0.903163 + 2.77965i −2.72498
20.3 −0.262601 + 0.808204i 1.60321 − 1.16480i 1.03380 + 0.751099i 0.485568 + 1.49442i 0.520392 + 1.60160i 0.244920 + 0.177945i −2.25352 + 1.63728i 0.286477 − 0.881686i −1.33531
20.4 −0.0477727 + 0.147029i −1.42347 + 1.03421i 1.59870 + 1.16152i −0.143849 − 0.442722i −0.0840567 − 0.258700i 1.63386 + 1.18707i −0.497293 + 0.361305i 0.0296284 − 0.0911869i 0.0719653
20.5 0.113205 − 0.348410i 1.51853 − 1.10328i 1.50946 + 1.09669i −0.692692 − 2.13189i −0.212488 − 0.653969i −3.58854 − 2.60723i 1.14573 − 0.832420i 0.161666 − 0.497556i −0.821188
20.6 0.323754 − 0.996414i −2.33617 + 1.69733i 0.730011 + 0.530384i −1.21350 − 3.73478i 0.934895 + 2.87731i −0.157180 − 0.114198i 2.46003 − 1.78731i 1.64972 − 5.07733i −4.11427
20.7 0.767312 − 2.36154i 1.44640 − 1.05087i −3.37009 − 2.44851i −0.0493208 − 0.151794i −1.37183 − 4.22207i 1.00080 + 0.727123i −4.35048 + 3.16081i 0.0606861 − 0.186773i −0.396313
58.1 −1.63827 + 1.19028i −0.0193086 − 0.0594258i 0.649153 − 1.99789i 2.71423 + 1.97200i 0.102366 + 0.0743732i 1.46905 − 4.52128i 0.0630158 + 0.193943i 2.42389 − 1.76106i −6.79389
58.2 −1.56377 + 1.13615i 0.625891 + 1.92630i 0.536518 − 1.65123i 1.70533 + 1.23900i −3.16730 − 2.30118i −1.01637 + 3.12806i −0.157565 − 0.484936i −0.891823 + 0.647947i −4.07443
58.3 −0.732184 + 0.531963i −0.173387 − 0.533629i −0.364925 + 1.12312i −2.22723 − 1.61818i 0.410822 + 0.298480i 0.170311 − 0.524163i −0.889607 − 2.73793i 2.17235 − 1.57831i 2.49156
58.4 −0.161825 + 0.117573i 0.378923 + 1.16621i −0.605670 + 1.86406i −0.0660344 − 0.0479768i −0.198433 − 0.144170i −0.276802 + 0.851908i −0.244773 − 0.753335i 1.21060 − 0.879550i 0.0163268
58.5 0.507210 − 0.368509i −0.519183 − 1.59788i −0.496571 + 1.52829i 3.26196 + 2.36995i −0.852170 − 0.619138i −1.09252 + 3.36244i 0.698797 + 2.15068i 0.143374 − 0.104168i 2.52785
58.6 1.50708 − 1.09496i 0.282042 + 0.868036i 0.454323 − 1.39826i 0.593812 + 0.431430i 1.37552 + 0.999375i 0.0915185 − 0.281665i 0.304970 + 0.938602i 1.75311 − 1.27371i 1.36732
58.7 1.58177 − 1.14922i −0.883995 − 2.72066i 0.563242 − 1.73348i −0.318947 − 0.231728i −4.52491 − 3.28754i −0.109120 + 0.335838i 0.107128 + 0.329706i −4.19347 + 3.04674i −0.770806
115.1 −0.816838 − 2.51397i 0.225457 + 0.163804i −4.03478 + 2.93144i −1.00025 + 3.07845i 0.227636 − 0.700593i −1.23421 + 0.896705i 6.38829 + 4.64136i −0.903052 − 2.77931i 8.55617
115.2 −0.577060 − 1.77601i −0.224936 − 0.163425i −1.20317 + 0.874155i 0.450929 − 1.38782i −0.160443 + 0.493794i −3.13571 + 2.27823i −0.774717 − 0.562865i −0.903163 − 2.77965i −2.72498
115.3 −0.262601 − 0.808204i 1.60321 + 1.16480i 1.03380 − 0.751099i 0.485568 − 1.49442i 0.520392 − 1.60160i 0.244920 − 0.177945i −2.25352 − 1.63728i 0.286477 + 0.881686i −1.33531
115.4 −0.0477727 − 0.147029i −1.42347 − 1.03421i 1.59870 − 1.16152i −0.143849 + 0.442722i −0.0840567 + 0.258700i 1.63386 − 1.18707i −0.497293 − 0.361305i 0.0296284 + 0.0911869i 0.0719653
115.5 0.113205 + 0.348410i 1.51853 + 1.10328i 1.50946 − 1.09669i −0.692692 + 2.13189i −0.212488 + 0.653969i −3.58854 + 2.60723i 1.14573 + 0.832420i 0.161666 + 0.497556i −0.821188
115.6 0.323754 + 0.996414i −2.33617 − 1.69733i 0.730011 − 0.530384i −1.21350 + 3.73478i 0.934895 − 2.87731i −0.157180 + 0.114198i 2.46003 + 1.78731i 1.64972 + 5.07733i −4.11427
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.f.b ✓ 28
11.c even 5 1 inner 209.2.f.b ✓ 28
11.c even 5 1 2299.2.a.t 14
11.d odd 10 1 2299.2.a.u 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.f.b ✓ 28 1.a even 1 1 trivial
209.2.f.b ✓ 28 11.c even 5 1 inner
2299.2.a.t 14 11.c even 5 1
2299.2.a.u 14 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 2 T_{2}^{27} + 12 T_{2}^{26} + 16 T_{2}^{25} + 68 T_{2}^{24} + 84 T_{2}^{23} + 302 T_{2}^{22} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display