Properties

Label 45.22.a.a
Level $45$
Weight $22$
Character orbit 45.a
Self dual yes
Analytic conductor $125.765$
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] = [45,22,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(125.764804929\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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)
 
\(f(q)\) \(=\) \( q - 544 q^{2} - 1801216 q^{4} + 9765625 q^{5} + 1277698380 q^{7} + 2120712192 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 544 q^{2} - 1801216 q^{4} + 9765625 q^{5} + 1277698380 q^{7} + 2120712192 q^{8} - 5312500000 q^{10} + 77585921744 q^{11} - 434110898702 q^{13} - 695067918720 q^{14} + 2623756304384 q^{16} - 12848917115782 q^{17} - 28605256159796 q^{19} - 17590000000000 q^{20} - 42206741428736 q^{22} - 224022192208080 q^{23} + 95367431640625 q^{25} + 236156328893888 q^{26} - 23\!\cdots\!80 q^{28}+ \cdots - 58\!\cdots\!92 q^{98}+O(q^{100}) \) 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
−544.000 0 −1.80122e6 9.76562e6 0 1.27770e9 2.12071e9 0 −5.31250e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

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 45.22.a.a 1
3.b odd 2 1 15.22.a.a 1
15.d odd 2 1 75.22.a.b 1
15.e even 4 2 75.22.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.a 1 3.b odd 2 1
45.22.a.a 1 1.a even 1 1 trivial
75.22.a.b 1 15.d odd 2 1
75.22.b.c 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 544 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 544 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 9765625 \) Copy content Toggle raw display
$7$ \( T - 1277698380 \) Copy content Toggle raw display
$11$ \( T - 77585921744 \) Copy content Toggle raw display
$13$ \( T + 434110898702 \) Copy content Toggle raw display
$17$ \( T + 12848917115782 \) Copy content Toggle raw display
$19$ \( T + 28605256159796 \) Copy content Toggle raw display
$23$ \( T + 224022192208080 \) Copy content Toggle raw display
$29$ \( T - 51676030833142 \) Copy content Toggle raw display
$31$ \( T - 8921108838285000 \) Copy content Toggle raw display
$37$ \( T - 43\!\cdots\!90 \) Copy content Toggle raw display
$41$ \( T + 58\!\cdots\!70 \) Copy content Toggle raw display
$43$ \( T + 16\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T - 16\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T + 22\!\cdots\!70 \) Copy content Toggle raw display
$59$ \( T + 51\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T - 12\!\cdots\!98 \) Copy content Toggle raw display
$67$ \( T + 54\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T - 11\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T + 37\!\cdots\!50 \) Copy content Toggle raw display
$79$ \( T - 63\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T - 14\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T + 13\!\cdots\!14 \) Copy content Toggle raw display
$97$ \( T + 32\!\cdots\!18 \) Copy content Toggle raw display
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