Properties

Label 20.12.a.a
Level $20$
Weight $12$
Character orbit 20.a
Self dual yes
Analytic conductor $15.367$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,12,Mod(1,20)]
 
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([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 20.a (trivial)

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: \(15.3668636112\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 306 q^{3} - 3125 q^{5} - 32074 q^{7} - 83511 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 306 q^{3} - 3125 q^{5} - 32074 q^{7} - 83511 q^{9} - 5280 q^{11} + 227834 q^{13} - 956250 q^{15} - 5097318 q^{17} - 16279036 q^{19} - 9814644 q^{21} - 33055038 q^{23} + 9765625 q^{25} - 79761348 q^{27} - 2112786 q^{29} + 91337396 q^{31} - 1615680 q^{33} + 100231250 q^{35} - 109132054 q^{37} + 69717204 q^{39} + 1202079126 q^{41} + 1112512490 q^{43} + 260971875 q^{45} + 507908142 q^{47} - 948585267 q^{49} - 1559779308 q^{51} - 1900361502 q^{53} + 16500000 q^{55} - 4981385016 q^{57} + 2802066708 q^{59} - 9660996838 q^{61} + 2678531814 q^{63} - 711981250 q^{65} + 8370234446 q^{67} - 10114841628 q^{69} - 12173973252 q^{71} + 18053518034 q^{73} + 2988281250 q^{75} + 169350720 q^{77} + 22759013912 q^{79} - 9613249371 q^{81} + 65367228042 q^{83} + 15929118750 q^{85} - 646512516 q^{87} - 13526251734 q^{89} - 7307547716 q^{91} + 27949243176 q^{93} + 50871987500 q^{95} - 155553294334 q^{97} + 440938080 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 306.000 0 −3125.00 0 −32074.0 0 −83511.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.12.a.a 1
3.b odd 2 1 180.12.a.a 1
4.b odd 2 1 80.12.a.c 1
5.b even 2 1 100.12.a.a 1
5.c odd 4 2 100.12.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.12.a.a 1 1.a even 1 1 trivial
80.12.a.c 1 4.b odd 2 1
100.12.a.a 1 5.b even 2 1
100.12.c.b 2 5.c odd 4 2
180.12.a.a 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 306 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(20))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 306 \) Copy content Toggle raw display
$5$ \( T + 3125 \) Copy content Toggle raw display
$7$ \( T + 32074 \) Copy content Toggle raw display
$11$ \( T + 5280 \) Copy content Toggle raw display
$13$ \( T - 227834 \) Copy content Toggle raw display
$17$ \( T + 5097318 \) Copy content Toggle raw display
$19$ \( T + 16279036 \) Copy content Toggle raw display
$23$ \( T + 33055038 \) Copy content Toggle raw display
$29$ \( T + 2112786 \) Copy content Toggle raw display
$31$ \( T - 91337396 \) Copy content Toggle raw display
$37$ \( T + 109132054 \) Copy content Toggle raw display
$41$ \( T - 1202079126 \) Copy content Toggle raw display
$43$ \( T - 1112512490 \) Copy content Toggle raw display
$47$ \( T - 507908142 \) Copy content Toggle raw display
$53$ \( T + 1900361502 \) Copy content Toggle raw display
$59$ \( T - 2802066708 \) Copy content Toggle raw display
$61$ \( T + 9660996838 \) Copy content Toggle raw display
$67$ \( T - 8370234446 \) Copy content Toggle raw display
$71$ \( T + 12173973252 \) Copy content Toggle raw display
$73$ \( T - 18053518034 \) Copy content Toggle raw display
$79$ \( T - 22759013912 \) Copy content Toggle raw display
$83$ \( T - 65367228042 \) Copy content Toggle raw display
$89$ \( T + 13526251734 \) Copy content Toggle raw display
$97$ \( T + 155553294334 \) Copy content Toggle raw display
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