Properties

Label 9.22.a.c
Level $9$
Weight $22$
Character orbit 9.a
Self dual yes
Analytic conductor $25.153$
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] = [9,22,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(25.1529609858\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
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)
 
\(f(q)\) \(=\) \( q + 288 q^{2} - 2014208 q^{4} - 21640950 q^{5} - 768078808 q^{7} - 1184071680 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 288 q^{2} - 2014208 q^{4} - 21640950 q^{5} - 768078808 q^{7} - 1184071680 q^{8} - 6232593600 q^{10} + 94724929188 q^{11} - 80621789794 q^{13} - 221206696704 q^{14} + 3883087691776 q^{16} - 3052282930002 q^{17} - 7920788351740 q^{19} + 43589374617600 q^{20} + 27280779606144 q^{22} + 73845437470344 q^{23} - 8506441300625 q^{25} - 23219075460672 q^{26} + 15\!\cdots\!64 q^{28}+ \cdots + 90\!\cdots\!16 q^{98}+O(q^{100}) \) 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
288.000 0 −2.01421e6 −2.16410e7 0 −7.68079e8 −1.18407e9 0 −6.23259e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

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 9.22.a.c 1
3.b odd 2 1 1.22.a.a 1
12.b even 2 1 16.22.a.c 1
15.d odd 2 1 25.22.a.a 1
15.e even 4 2 25.22.b.a 2
21.c even 2 1 49.22.a.a 1
24.f even 2 1 64.22.a.a 1
24.h odd 2 1 64.22.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.22.a.a 1 3.b odd 2 1
9.22.a.c 1 1.a even 1 1 trivial
16.22.a.c 1 12.b even 2 1
25.22.a.a 1 15.d odd 2 1
25.22.b.a 2 15.e even 4 2
49.22.a.a 1 21.c even 2 1
64.22.a.a 1 24.f even 2 1
64.22.a.g 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 288 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 288 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 21640950 \) Copy content Toggle raw display
$7$ \( T + 768078808 \) Copy content Toggle raw display
$11$ \( T - 94724929188 \) Copy content Toggle raw display
$13$ \( T + 80621789794 \) Copy content Toggle raw display
$17$ \( T + 3052282930002 \) Copy content Toggle raw display
$19$ \( T + 7920788351740 \) Copy content Toggle raw display
$23$ \( T - 73845437470344 \) Copy content Toggle raw display
$29$ \( T - 4253031736469010 \) Copy content Toggle raw display
$31$ \( T - 1900541176310432 \) Copy content Toggle raw display
$37$ \( T - 22\!\cdots\!22 \) Copy content Toggle raw display
$41$ \( T - 20\!\cdots\!58 \) Copy content Toggle raw display
$43$ \( T + 19\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T + 14\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T + 20\!\cdots\!06 \) Copy content Toggle raw display
$59$ \( T - 59\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T - 61\!\cdots\!62 \) Copy content Toggle raw display
$67$ \( T - 16\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T - 56\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T + 43\!\cdots\!94 \) Copy content Toggle raw display
$79$ \( T + 51\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T + 48\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T - 50\!\cdots\!30 \) Copy content Toggle raw display
$97$ \( T - 80\!\cdots\!82 \) Copy content Toggle raw display
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