Properties

Label 72.16.a.b
Level $72$
Weight $16$
Character orbit 72.a
Self dual yes
Analytic conductor $102.739$
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] = [72,16,Mod(1,72)]
 
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([0, 0, 0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(102.739323672\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
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)
 
\(f(q)\) \(=\) \( q + 42010 q^{5} + 385728 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 42010 q^{5} + 385728 q^{7} - 20444300 q^{11} + 75221302 q^{13} - 639707746 q^{17} + 267006356 q^{19} - 571925576 q^{23} - 28752738025 q^{25} + 62437913154 q^{29} - 1243357000 q^{31} + 16204433280 q^{35} - 408247831602 q^{37} + 1376815491990 q^{41} - 1923521494772 q^{43} + 4743411679104 q^{47} - 4598775419959 q^{49} + 3260491936570 q^{53} - 858865043000 q^{55} - 6734040423500 q^{59} - 10498750519274 q^{61} + 3160046897020 q^{65} + 42186307399892 q^{67} - 81526680652600 q^{71} - 37550447155142 q^{73} - 7885938950400 q^{77} + 169301409686344 q^{79} - 357833604131476 q^{83} - 26874122409460 q^{85} + 157672146871542 q^{89} + 29014962377856 q^{91} + 11216937015560 q^{95} - 12\!\cdots\!26 q^{97}+O(q^{100}) \) 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
0 0 0 42010.0 0 385728. 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

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 72.16.a.b 1
3.b odd 2 1 24.16.a.a 1
4.b odd 2 1 144.16.a.i 1
12.b even 2 1 48.16.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.16.a.a 1 3.b odd 2 1
48.16.a.e 1 12.b even 2 1
72.16.a.b 1 1.a even 1 1 trivial
144.16.a.i 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 42010 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(72))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 42010 \) Copy content Toggle raw display
$7$ \( T - 385728 \) Copy content Toggle raw display
$11$ \( T + 20444300 \) Copy content Toggle raw display
$13$ \( T - 75221302 \) Copy content Toggle raw display
$17$ \( T + 639707746 \) Copy content Toggle raw display
$19$ \( T - 267006356 \) Copy content Toggle raw display
$23$ \( T + 571925576 \) Copy content Toggle raw display
$29$ \( T - 62437913154 \) Copy content Toggle raw display
$31$ \( T + 1243357000 \) Copy content Toggle raw display
$37$ \( T + 408247831602 \) Copy content Toggle raw display
$41$ \( T - 1376815491990 \) Copy content Toggle raw display
$43$ \( T + 1923521494772 \) Copy content Toggle raw display
$47$ \( T - 4743411679104 \) Copy content Toggle raw display
$53$ \( T - 3260491936570 \) Copy content Toggle raw display
$59$ \( T + 6734040423500 \) Copy content Toggle raw display
$61$ \( T + 10498750519274 \) Copy content Toggle raw display
$67$ \( T - 42186307399892 \) Copy content Toggle raw display
$71$ \( T + 81526680652600 \) Copy content Toggle raw display
$73$ \( T + 37550447155142 \) Copy content Toggle raw display
$79$ \( T - 169301409686344 \) Copy content Toggle raw display
$83$ \( T + 357833604131476 \) Copy content Toggle raw display
$89$ \( T - 157672146871542 \) Copy content Toggle raw display
$97$ \( T + 1220442820436926 \) Copy content Toggle raw display
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