Properties

Label 20.9.d.a
Level $20$
Weight $9$
Character orbit 20.d
Self dual yes
Analytic conductor $8.148$
Analytic rank $0$
Dimension $1$
CM discriminant -20
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,9,Mod(19,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 20.d (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: yes
Analytic conductor: \(8.14757220122\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 16 q^{2} + 158 q^{3} + 256 q^{4} + 625 q^{5} - 2528 q^{6} - 1922 q^{7} - 4096 q^{8} + 18403 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{2} + 158 q^{3} + 256 q^{4} + 625 q^{5} - 2528 q^{6} - 1922 q^{7} - 4096 q^{8} + 18403 q^{9} - 10000 q^{10} + 40448 q^{12} + 30752 q^{14} + 98750 q^{15} + 65536 q^{16} - 294448 q^{18} + 160000 q^{20} - 303676 q^{21} - 211202 q^{23} - 647168 q^{24} + 390625 q^{25} + 1871036 q^{27} - 492032 q^{28} + 20642 q^{29} - 1580000 q^{30} - 1048576 q^{32} - 1201250 q^{35} + 4711168 q^{36} - 2560000 q^{40} - 5419198 q^{41} + 4858816 q^{42} + 2519518 q^{43} + 11501875 q^{45} + 3379232 q^{46} - 9618242 q^{47} + 10354688 q^{48} - 2070717 q^{49} - 6250000 q^{50} - 29936576 q^{54} + 7872512 q^{56} - 330272 q^{58} + 25280000 q^{60} - 11061598 q^{61} - 35370566 q^{63} + 16777216 q^{64} + 20249758 q^{67} - 33369916 q^{69} + 19220000 q^{70} - 75378688 q^{72} + 61718750 q^{75} + 40960000 q^{80} + 174881605 q^{81} + 86707168 q^{82} + 30884638 q^{83} - 77741056 q^{84} - 40312288 q^{86} + 3261436 q^{87} - 106804798 q^{89} - 184030000 q^{90} - 54067712 q^{92} + 153891872 q^{94} - 165675008 q^{96} + 33131472 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
−16.0000 158.000 256.000 625.000 −2528.00 −1922.00 −4096.00 18403.0 −10000.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.9.d.a 1
4.b odd 2 1 20.9.d.b yes 1
5.b even 2 1 20.9.d.b yes 1
5.c odd 4 2 100.9.b.b 2
8.b even 2 1 320.9.h.a 1
8.d odd 2 1 320.9.h.b 1
20.d odd 2 1 CM 20.9.d.a 1
20.e even 4 2 100.9.b.b 2
40.e odd 2 1 320.9.h.a 1
40.f even 2 1 320.9.h.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.d.a 1 1.a even 1 1 trivial
20.9.d.a 1 20.d odd 2 1 CM
20.9.d.b yes 1 4.b odd 2 1
20.9.d.b yes 1 5.b even 2 1
100.9.b.b 2 5.c odd 4 2
100.9.b.b 2 20.e even 4 2
320.9.h.a 1 8.b even 2 1
320.9.h.a 1 40.e odd 2 1
320.9.h.b 1 8.d odd 2 1
320.9.h.b 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 158 \) acting on \(S_{9}^{\mathrm{new}}(20, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 16 \) Copy content Toggle raw display
$3$ \( T - 158 \) Copy content Toggle raw display
$5$ \( T - 625 \) Copy content Toggle raw display
$7$ \( T + 1922 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 211202 \) Copy content Toggle raw display
$29$ \( T - 20642 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T + 5419198 \) Copy content Toggle raw display
$43$ \( T - 2519518 \) Copy content Toggle raw display
$47$ \( T + 9618242 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 11061598 \) Copy content Toggle raw display
$67$ \( T - 20249758 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 30884638 \) Copy content Toggle raw display
$89$ \( T + 106804798 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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