Properties

Label 72.22.a.h
Level $72$
Weight $22$
Character orbit 72.a
Self dual yes
Analytic conductor $201.224$
Analytic rank $0$
Dimension $5$
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] = [72,22,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, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
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: \(201.223687887\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 10642080274x^{3} + 257327290824634x^{2} - 1056913897140193890x - 892208868339633807375 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{13}\cdot 5\cdot 7 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2064899) q^{5} + (\beta_{2} + 6 \beta_1 - 74076006) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2064899) q^{5} + (\beta_{2} + 6 \beta_1 - 74076006) q^{7} + (\beta_{3} - 15 \beta_{2} - 158 \beta_1 + 17280501613) q^{11} + ( - \beta_{4} + \beta_{3} + 74 \beta_{2} + 2030 \beta_1 - 54179632965) q^{13} + ( - 16 \beta_{4} + 4 \beta_{3} + 1812 \beta_{2} + \cdots - 1107669848430) q^{17}+ \cdots + ( - 576968284 \beta_{4} + 4128325212 \beta_{3} + \cdots + 95\!\cdots\!46) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10324496 q^{5} - 370380036 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 10324496 q^{5} - 370380036 q^{7} + 86402508224 q^{11} - 270898166854 q^{13} - 5538349279328 q^{17} + 23316304328776 q^{19} - 99884715182464 q^{23} + 538732738730551 q^{25} + 858399022266864 q^{29} - 32\!\cdots\!72 q^{31}+ \cdots + 47\!\cdots\!98 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 10642080274x^{3} + 257327290824634x^{2} - 1056913897140193890x - 892208868339633807375 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 22287213162208 \nu^{4} + \cdots - 53\!\cdots\!97 ) / 19\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 361945148223296 \nu^{4} + \cdots - 17\!\cdots\!08 ) / 19\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33\!\cdots\!72 \nu^{4} + \cdots + 78\!\cdots\!01 ) / 19\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28\!\cdots\!40 \nu^{4} + \cdots + 25\!\cdots\!96 ) / 27\!\cdots\!71 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 5\beta_{3} - 1002\beta_{2} + 28914\beta _1 + 3721675 ) / 18579456 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 221605\beta_{4} - 1139889\beta_{3} + 320146870\beta_{2} - 2883820566\beta _1 + 237268874073712671 ) / 55738368 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4602054689 \beta_{4} - 24646945741 \beta_{3} - 6373617258450 \beta_{2} + 136719475778946 \beta _1 - 14\!\cdots\!65 ) / 9289728 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 223598521944013 \beta_{4} + \cdots + 36\!\cdots\!07 ) / 7962624 \) 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
19603.8
88760.9
6287.62
−113936.
−715.745
0 0 0 −3.27849e7 0 1.92379e8 0 0 0
1.2 0 0 0 −1.28149e7 0 −3.84305e7 0 0 0
1.3 0 0 0 −1.08925e6 0 −4.86930e8 0 0 0
1.4 0 0 0 2.31503e7 0 1.20078e9 0 0 0
1.5 0 0 0 3.38633e7 0 −1.23818e9 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 72.22.a.h yes 5
3.b odd 2 1 72.22.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.22.a.g 5 3.b odd 2 1
72.22.a.h yes 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 10324496 T_{5}^{4} + \cdots + 35\!\cdots\!00 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(72))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 10324496 T^{4} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{5} + 370380036 T^{4} + \cdots + 53\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{5} - 86402508224 T^{4} + \cdots - 71\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{5} + 270898166854 T^{4} + \cdots - 14\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{5} + 5538349279328 T^{4} + \cdots - 59\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{5} - 23316304328776 T^{4} + \cdots - 83\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{5} + 99884715182464 T^{4} + \cdots - 85\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{5} - 858399022266864 T^{4} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 91\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 11\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 51\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 24\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots - 80\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 53\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 78\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 53\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 18\!\cdots\!52 \) Copy content Toggle raw display
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