Properties

Label 72.22.a.d
Level $72$
Weight $22$
Character orbit 72.a
Self dual yes
Analytic conductor $201.224$
Analytic rank $1$
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: \(1\)
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: \( x^{3} - x^{2} - 2295485x - 828958533 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 693342) q^{5} + ( - \beta_{2} - 401760688) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 693342) q^{5} + ( - \beta_{2} - 401760688) q^{7} + ( - 160 \beta_{2} + 1186 \beta_1 - 4613082500) q^{11} + (161 \beta_{2} - 7159 \beta_1 + 239618517230) q^{13} + (7450 \beta_{2} + 194532 \beta_1 - 711729947682) q^{17} + ( - 1702 \beta_{2} - 693676 \beta_1 - 13374108228996) q^{19} + (256122 \beta_{2} - 2178680 \beta_1 - 92808139227544) q^{23} + (709430 \beta_{2} + 15078686 \beta_1 + 449347753517967) q^{25} + (1261598 \beta_{2} - 33914689 \beta_1 - 147569389330598) q^{29} + ( - 256009 \beta_{2} - 110688876 \beta_1 + 26\!\cdots\!84) q^{31}+ \cdots + (432471338728 \beta_{2} - 12488719178940 \beta_1 - 53\!\cdots\!74) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2080026 q^{5} - 1205282064 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2080026 q^{5} - 1205282064 q^{7} - 13839247500 q^{11} + 718855551690 q^{13} - 2135189843046 q^{17} - 40122324686988 q^{19} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots - 15\!\cdots\!22 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 2295485x - 828958533 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 384\nu^{2} + 177408\nu - 587703424 ) / 19 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13440\nu^{2} + 13402368\nu + 20563082624 ) / 19 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 35\beta _1 + 344064 ) / 1032192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -11\beta_{2} + 831\beta _1 + 37609234432 ) / 24576 \) 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
1671.27
−1283.97
−386.305
0 0 0 −4.18179e7 0 −6.87132e8 0 0 0
1.2 0 0 0 8.90852e6 0 5.87822e8 0 0 0
1.3 0 0 0 3.08294e7 0 −1.10597e9 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.d 3
3.b odd 2 1 24.22.a.c 3
12.b even 2 1 48.22.a.l 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.a.c 3 3.b odd 2 1
48.22.a.l 3 12.b even 2 1
72.22.a.d 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 2080026T_{5}^{2} - 1387114113501300T_{5} + 11485065000261414395000 \) 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
$5$ \( T^{3} + 2080026 T^{2} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + 1205282064 T^{2} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{3} + 13839247500 T^{2} + \cdots - 32\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{3} - 718855551690 T^{2} + \cdots + 76\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{3} + 2135189843046 T^{2} + \cdots - 14\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{3} + 40122324686988 T^{2} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{3} + 278424417682632 T^{2} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{3} + 442708167991794 T^{2} + \cdots - 22\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 17\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 64\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 63\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 90\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 55\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 43\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 74\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 62\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 55\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
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