Properties

Label 1.20.a.a
Level $1$
Weight $20$
Character orbit 1.a
Self dual yes
Analytic conductor $2.288$
Analytic rank $0$
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] = [1,20,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(2.28816696556\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 456 q^{2} + 50652 q^{3} - 316352 q^{4} - 2377410 q^{5} + 23097312 q^{6} - 16917544 q^{7} - 383331840 q^{8} + 1403363637 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 456 q^{2} + 50652 q^{3} - 316352 q^{4} - 2377410 q^{5} + 23097312 q^{6} - 16917544 q^{7} - 383331840 q^{8} + 1403363637 q^{9} - 1084098960 q^{10} - 16212108 q^{11} - 16023861504 q^{12} + 50421615062 q^{13} - 7714400064 q^{14} - 120420571320 q^{15} - 8939761664 q^{16} + 225070099506 q^{17} + 639933818472 q^{18} - 1710278572660 q^{19} + 752098408320 q^{20} - 856907438688 q^{21} - 7392721248 q^{22} + 14036534788872 q^{23} - 19416524359680 q^{24} - 13421408020025 q^{25} + 22992256468272 q^{26} + 12212307114840 q^{27} + 5351898879488 q^{28} + 1137835269510 q^{29} - 54911780521920 q^{30} - 104626880141728 q^{31} + 196899752411136 q^{32} - 821175694416 q^{33} + 102631965374736 q^{34} + 40219938281040 q^{35} - 443956893292224 q^{36} - 169392327370594 q^{37} - 779887029132960 q^{38} + 25\!\cdots\!24 q^{39}+ \cdots - 22\!\cdots\!96 q^{99}+O(q^{100}) \) 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
456.000 50652.0 −316352. −2.37741e6 2.30973e7 −1.69175e7 −3.83332e8 1.40336e9 −1.08410e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.20.a.a 1
3.b odd 2 1 9.20.a.a 1
4.b odd 2 1 16.20.a.a 1
5.b even 2 1 25.20.a.a 1
5.c odd 4 2 25.20.b.a 2
7.b odd 2 1 49.20.a.b 1
8.b even 2 1 64.20.a.b 1
8.d odd 2 1 64.20.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.20.a.a 1 1.a even 1 1 trivial
9.20.a.a 1 3.b odd 2 1
16.20.a.a 1 4.b odd 2 1
25.20.a.a 1 5.b even 2 1
25.20.b.a 2 5.c odd 4 2
49.20.a.b 1 7.b odd 2 1
64.20.a.b 1 8.b even 2 1
64.20.a.h 1 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 456 \) Copy content Toggle raw display
$3$ \( T - 50652 \) Copy content Toggle raw display
$5$ \( T + 2377410 \) Copy content Toggle raw display
$7$ \( T + 16917544 \) Copy content Toggle raw display
$11$ \( T + 16212108 \) Copy content Toggle raw display
$13$ \( T - 50421615062 \) Copy content Toggle raw display
$17$ \( T - 225070099506 \) Copy content Toggle raw display
$19$ \( T + 1710278572660 \) Copy content Toggle raw display
$23$ \( T - 14036534788872 \) Copy content Toggle raw display
$29$ \( T - 1137835269510 \) Copy content Toggle raw display
$31$ \( T + 104626880141728 \) Copy content Toggle raw display
$37$ \( T + 169392327370594 \) Copy content Toggle raw display
$41$ \( T + 3309984750560838 \) Copy content Toggle raw display
$43$ \( T - 1127913532193492 \) Copy content Toggle raw display
$47$ \( T - 3498693987674256 \) Copy content Toggle raw display
$53$ \( T - 29\!\cdots\!02 \) Copy content Toggle raw display
$59$ \( T - 58\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T - 23\!\cdots\!42 \) Copy content Toggle raw display
$67$ \( T + 20\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T + 17\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T - 29\!\cdots\!22 \) Copy content Toggle raw display
$79$ \( T + 92\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T - 12\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T - 43\!\cdots\!30 \) Copy content Toggle raw display
$97$ \( T + 63\!\cdots\!94 \) Copy content Toggle raw display
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