Properties

Label 10.20.a.b
Level $10$
Weight $20$
Character orbit 10.a
Self dual yes
Analytic conductor $22.882$
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] = [10,20,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(22.8816696556\)
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)
 
\(f(q)\) \(=\) \( q - 512 q^{2} + 38628 q^{3} + 262144 q^{4} + 1953125 q^{5} - 19777536 q^{6} - 144185776 q^{7} - 134217728 q^{8} + 329860917 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 512 q^{2} + 38628 q^{3} + 262144 q^{4} + 1953125 q^{5} - 19777536 q^{6} - 144185776 q^{7} - 134217728 q^{8} + 329860917 q^{9} - 1000000000 q^{10} - 5156500668 q^{11} + 10126098432 q^{12} + 3435515798 q^{13} + 73823117312 q^{14} + 75445312500 q^{15} + 68719476736 q^{16} + 366347849874 q^{17} - 168888789504 q^{18} - 1604379002260 q^{19} + 512000000000 q^{20} - 5569608155328 q^{21} + 2640128342016 q^{22} - 13649676405552 q^{23} - 5184562397184 q^{24} + 3814697265625 q^{25} - 1758984088576 q^{26} - 32153968445400 q^{27} - 37797436063744 q^{28} + 50171441088390 q^{29} - 38628000000000 q^{30} - 135506765375248 q^{31} - 35184372088832 q^{32} - 199185307803504 q^{33} - 187570099135488 q^{34} - 281612843750000 q^{35} + 86471060226048 q^{36} + 91268665304894 q^{37} + 821442049157120 q^{38} + 132707104245144 q^{39} - 262144000000000 q^{40} + 827681875418682 q^{41} + 28\!\cdots\!36 q^{42}+ \cdots - 17\!\cdots\!56 q^{99}+O(q^{100}) \) 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
−512.000 38628.0 262144. 1.95312e6 −1.97775e7 −1.44186e8 −1.34218e8 3.29861e8 −1.00000e9
\(n\): 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 10.20.a.b 1
3.b odd 2 1 90.20.a.c 1
4.b odd 2 1 80.20.a.a 1
5.b even 2 1 50.20.a.c 1
5.c odd 4 2 50.20.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.20.a.b 1 1.a even 1 1 trivial
50.20.a.c 1 5.b even 2 1
50.20.b.b 2 5.c odd 4 2
80.20.a.a 1 4.b odd 2 1
90.20.a.c 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 38628 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 512 \) Copy content Toggle raw display
$3$ \( T - 38628 \) Copy content Toggle raw display
$5$ \( T - 1953125 \) Copy content Toggle raw display
$7$ \( T + 144185776 \) Copy content Toggle raw display
$11$ \( T + 5156500668 \) Copy content Toggle raw display
$13$ \( T - 3435515798 \) Copy content Toggle raw display
$17$ \( T - 366347849874 \) Copy content Toggle raw display
$19$ \( T + 1604379002260 \) Copy content Toggle raw display
$23$ \( T + 13649676405552 \) Copy content Toggle raw display
$29$ \( T - 50171441088390 \) Copy content Toggle raw display
$31$ \( T + 135506765375248 \) Copy content Toggle raw display
$37$ \( T - 91268665304894 \) Copy content Toggle raw display
$41$ \( T - 827681875418682 \) Copy content Toggle raw display
$43$ \( T + 5871395954951812 \) Copy content Toggle raw display
$47$ \( T - 13\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T + 36\!\cdots\!62 \) Copy content Toggle raw display
$59$ \( T + 12\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T + 71\!\cdots\!98 \) Copy content Toggle raw display
$67$ \( T - 31\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T + 24\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T - 37\!\cdots\!58 \) Copy content Toggle raw display
$79$ \( T - 16\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T - 18\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T + 30\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T - 69\!\cdots\!34 \) Copy content Toggle raw display
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