Properties

Label 3.22.a.b
Level $3$
Weight $22$
Character orbit 3.a
Self dual yes
Analytic conductor $8.384$
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] = [3,22,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, 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("3.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(8.38432032861\)
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)
 
\(f(q)\) \(=\) \( q + 1728 q^{2} - 59049 q^{3} + 888832 q^{4} - 41512770 q^{5} - 102036672 q^{6} + 538429808 q^{7} - 2087976960 q^{8} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 1728 q^{2} - 59049 q^{3} + 888832 q^{4} - 41512770 q^{5} - 102036672 q^{6} + 538429808 q^{7} - 2087976960 q^{8} + 3486784401 q^{9} - 71734066560 q^{10} - 64113040188 q^{11} - 52484640768 q^{12} - 130980107986 q^{13} + 930406708224 q^{14} + 2451287555730 q^{15} - 5472039993344 q^{16} + 8242029723618 q^{17} + 6025163444928 q^{18} + 13492101753020 q^{19} - 36897878384640 q^{20} - 31793741732592 q^{21} - 110787333444864 q^{22} - 233184825844776 q^{23} + 123292951511040 q^{24} + 12\!\cdots\!75 q^{25}+ \cdots - 22\!\cdots\!88 q^{99}+O(q^{100}) \) 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
1728.00 −59049.0 888832. −4.15128e7 −1.02037e8 5.38430e8 −2.08798e9 3.48678e9 −7.17341e10
\(n\): 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 3.22.a.b 1
3.b odd 2 1 9.22.a.a 1
4.b odd 2 1 48.22.a.d 1
5.b even 2 1 75.22.a.a 1
5.c odd 4 2 75.22.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.b 1 1.a even 1 1 trivial
9.22.a.a 1 3.b odd 2 1
48.22.a.d 1 4.b odd 2 1
75.22.a.a 1 5.b even 2 1
75.22.b.b 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1728 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1728 \) Copy content Toggle raw display
$3$ \( T + 59049 \) Copy content Toggle raw display
$5$ \( T + 41512770 \) Copy content Toggle raw display
$7$ \( T - 538429808 \) Copy content Toggle raw display
$11$ \( T + 64113040188 \) Copy content Toggle raw display
$13$ \( T + 130980107986 \) Copy content Toggle raw display
$17$ \( T - 8242029723618 \) Copy content Toggle raw display
$19$ \( T - 13492101753020 \) Copy content Toggle raw display
$23$ \( T + 233184825844776 \) Copy content Toggle raw display
$29$ \( T + 2024562031123770 \) Copy content Toggle raw display
$31$ \( T + 6869194988701768 \) Copy content Toggle raw display
$37$ \( T - 3443998107027638 \) Copy content Toggle raw display
$41$ \( T + 21\!\cdots\!58 \) Copy content Toggle raw display
$43$ \( T + 71\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T - 28\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T + 21\!\cdots\!46 \) Copy content Toggle raw display
$59$ \( T - 15\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T - 43\!\cdots\!62 \) Copy content Toggle raw display
$67$ \( T - 92\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T + 20\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T - 16\!\cdots\!74 \) Copy content Toggle raw display
$79$ \( T - 67\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T - 39\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T - 41\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T - 57\!\cdots\!98 \) Copy content Toggle raw display
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