Properties

Label 72.22.a.e
Level $72$
Weight $22$
Character orbit 72.a
Self dual yes
Analytic conductor $201.224$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 53560x - 70812 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{4}\cdot 5\cdot 7 \)
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)
 

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

\(f(q)\) \(=\) \( q + (\beta_1 + 1611042) q^{5} + ( - 11 \beta_{2} + 25 \beta_1 + 90477008) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1611042) q^{5} + ( - 11 \beta_{2} + 25 \beta_1 + 90477008) q^{7} + (416 \beta_{2} - 3842 \beta_1 - 20398219396) q^{11} + ( - 6677 \beta_{2} - 32706 \beta_1 - 198162067474) q^{13} + ( - 96866 \beta_{2} + 258706 \beta_1 - 474939839778) q^{17} + ( - 107426 \beta_{2} - 1737246 \beta_1 - 11698275227268) q^{19} + (1820478 \beta_{2} - 15793522 \beta_1 + 31637014357864) q^{23} + ( - 2291630 \beta_{2} + 7718556 \beta_1 - 204593341765233) q^{25} + ( - 20187574 \beta_{2} + 102873171 \beta_1 + 10\!\cdots\!78) q^{29}+ \cdots + (3277635789560 \beta_{2} + \cdots + 66\!\cdots\!86) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4833126 q^{5} + 271431024 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4833126 q^{5} + 271431024 q^{7} - 61194658188 q^{11} - 594486202422 q^{13} - 1424819519334 q^{17} - 35094825681804 q^{19} + 94911043073592 q^{23} - 613780025295699 q^{25} + 32\!\cdots\!34 q^{29}+ \cdots + 19\!\cdots\!58 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 53560x - 70812 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 768\nu^{2} + 948864\nu - 27422720 ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -3072\nu^{2} + 75264\nu + 109690880 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 44\beta_1 ) / 3870720 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -353\beta_{2} + 308\beta _1 + 39488716800 ) / 1105920 \) 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
−230.766
−1.32215
232.089
0 0 0 −1.70699e7 0 4.07437e8 0 0 0
1.2 0 0 0 −995860. 0 −1.18014e9 0 0 0
1.3 0 0 0 2.28989e7 0 1.04414e9 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 72.22.a.e 3
3.b odd 2 1 24.22.a.b 3
12.b even 2 1 48.22.a.k 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.a.b 3 3.b odd 2 1
48.22.a.k 3 12.b even 2 1
72.22.a.e 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 4833126T_{5}^{2} - 396686171190900T_{5} - 389262902874527045000 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(72))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 4833126 T^{2} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} - 271431024 T^{2} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{3} + 61194658188 T^{2} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{3} + 594486202422 T^{2} + \cdots - 47\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{3} + 1424819519334 T^{2} + \cdots + 16\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{3} + 35094825681804 T^{2} + \cdots - 23\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{3} - 94911043073592 T^{2} + \cdots + 20\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 74\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 51\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 61\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 57\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 32\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 19\!\cdots\!32 \) Copy content Toggle raw display
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