Properties

Label 72.21.b.a
Level $72$
Weight $21$
Character orbit 72.b
Self dual yes
Analytic conductor $182.530$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,21,Mod(19,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.19");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 72.b (of order \(2\), degree \(1\), minimal)

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: \(182.529910874\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 1024 q^{2} + 1048576 q^{4} - 1073741824 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 1024 q^{2} + 1048576 q^{4} - 1073741824 q^{8} + 42383023726 q^{11} + 1099511627776 q^{16} + 3353535763774 q^{17} - 1014654432526 q^{19} - 43400216295424 q^{22} + 95367431640625 q^{25} - 11\!\cdots\!24 q^{32}+ \cdots - 81\!\cdots\!24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(1\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
−1024.00 0 1.04858e6 0 0 0 −1.07374e9 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.21.b.a 1
3.b odd 2 1 8.21.d.a 1
8.d odd 2 1 CM 72.21.b.a 1
12.b even 2 1 32.21.d.a 1
24.f even 2 1 8.21.d.a 1
24.h odd 2 1 32.21.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.21.d.a 1 3.b odd 2 1
8.21.d.a 1 24.f even 2 1
32.21.d.a 1 12.b even 2 1
32.21.d.a 1 24.h odd 2 1
72.21.b.a 1 1.a even 1 1 trivial
72.21.b.a 1 8.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{21}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1024 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 42383023726 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 3353535763774 \) Copy content Toggle raw display
$19$ \( T + 1014654432526 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 25\!\cdots\!26 \) Copy content Toggle raw display
$43$ \( T - 2781113986388498 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 17\!\cdots\!98 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 35\!\cdots\!74 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 60\!\cdots\!74 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 31\!\cdots\!74 \) Copy content Toggle raw display
$89$ \( T + 61\!\cdots\!74 \) Copy content Toggle raw display
$97$ \( T + 50\!\cdots\!02 \) Copy content Toggle raw display
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