Properties

Label 8.20.a.b
Level $8$
Weight $20$
Character orbit 8.a
Self dual yes
Analytic conductor $18.305$
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] = [8,20,Mod(1,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(18.3053357245\)
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: \( x^{3} - x^{2} - 2519x + 43659 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 7911) q^{3} + (\beta_{2} - 70 \beta_1 + 713429) q^{5} + ( - 20 \beta_{2} - 2722 \beta_1 + 18618154) q^{7} + (150 \beta_{2} - 21956 \beta_1 + 215591919) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 7911) q^{3} + (\beta_{2} - 70 \beta_1 + 713429) q^{5} + ( - 20 \beta_{2} - 2722 \beta_1 + 18618154) q^{7} + (150 \beta_{2} - 21956 \beta_1 + 215591919) q^{9} + ( - 360 \beta_{2} + 63333 \beta_1 - 99151979) q^{11} + ( - 1615 \beta_{2} + 1369882 \beta_1 - 4954589763) q^{13} + (12276 \beta_{2} - 5829910 \beta_1 + 97547157054) q^{15} + ( - 15370 \beta_{2} + \cdots + 267776864140) q^{17}+ \cdots + (28907244960 \beta_{2} + \cdots - 38\!\cdots\!71) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9} - 297392964 q^{11} - 14862401022 q^{13} + 292635653528 q^{15} + 803332464534 q^{17} + 3212269666884 q^{19} + 11192319829728 q^{21} + 24948509305560 q^{23} + 72340360289109 q^{25} + 64092343553864 q^{27} + 77667139511058 q^{29} - 248431735193568 q^{31} - 252071696774128 q^{33} - 13\!\cdots\!08 q^{35}+ \cdots - 11\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 2519x + 43659 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 64\nu^{2} + 1664\nu - 108053 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -896\nu^{2} + 124160\nu + 1463595 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 14\beta _1 + 49147 ) / 147456 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -13\beta_{2} + 970\beta _1 + 123838145 ) / 73728 \) 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
37.1696
−56.8359
20.6663
0 −34307.5 0 2.59881e6 0 −1.93114e8 0 1.47441e7 0
1.2 0 3798.34 0 −8.06198e6 0 1.77174e8 0 −1.14783e9 0
1.3 0 54241.2 0 7.60339e6 0 7.17920e7 0 1.77984e9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)

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 8.20.a.b 3
3.b odd 2 1 72.20.a.f 3
4.b odd 2 1 16.20.a.f 3
8.b even 2 1 64.20.a.l 3
8.d odd 2 1 64.20.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.20.a.b 3 1.a even 1 1 trivial
16.20.a.f 3 4.b odd 2 1
64.20.a.l 3 8.b even 2 1
64.20.a.m 3 8.d odd 2 1
72.20.a.f 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 23732T_{3}^{2} - 1785165264T_{3} + 7068250797120 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + \cdots + 7068250797120 \) Copy content Toggle raw display
$5$ \( T^{3} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots + 24\!\cdots\!88 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 59\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 44\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 42\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 68\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 52\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 41\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 10\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 59\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 21\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 93\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 32\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 27\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 34\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 28\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 55\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 60\!\cdots\!48 \) Copy content Toggle raw display
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