Properties

Label 8.20.a.a
Level $8$
Weight $20$
Character orbit 8.a
Self dual yes
Analytic conductor $18.305$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1453}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 363 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5 \)
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 \(\beta = 960\sqrt{1453}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 13956) q^{3} + (44 \beta + 613310) q^{5} + (3190 \beta + 44255256) q^{7} + (27912 \beta + 371593269) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 13956) q^{3} + (44 \beta + 613310) q^{5} + (3190 \beta + 44255256) q^{7} + (27912 \beta + 371593269) q^{9} + ( - 223467 \beta - 3581893804) q^{11} + (71660 \beta - 5063461802) q^{13} + ( - 1227374 \beta - 67479085560) q^{15} + (17338504 \beta - 36022539470) q^{17} + ( - 26137813 \beta - 1560240236116) q^{19} + ( - 88774896 \beta - 4889306864736) q^{21} + (184291330 \beta - 7379603545144) q^{23} + (53971280 \beta - 16104868999225) q^{25} + (401128326 \beta - 26341969566312) q^{27} + ( - 893092484 \beta - 15124769622522) q^{29} + ( - 3864337064 \beta - 61694781388960) q^{31} + (6700599256 \beta + 349230172930224) q^{33} + (3903690164 \beta + 215096133585360) q^{35} + ( - 3072429508 \beta + 10\!\cdots\!62) q^{37}+ \cdots + ( - 183016652900871 \beta - 96\!\cdots\!76) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9} - 7163787608 q^{11} - 10126923604 q^{13} - 134958171120 q^{15} - 72045078940 q^{17} - 3120480472232 q^{19} - 9778613729472 q^{21} - 14759207090288 q^{23} - 32209737998450 q^{25} - 52683939132624 q^{27} - 30249539245044 q^{29} - 123389562777920 q^{31} + 698460345860448 q^{33} + 430192267170720 q^{35} + 20\!\cdots\!24 q^{37}+ \cdots - 19\!\cdots\!52 q^{99}+O(q^{100}) \) 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
19.5591
−18.5591
0 −50549.5 0 2.22342e6 0 1.60989e8 0 1.39299e9 0
1.2 0 22637.5 0 −996804. 0 −7.24780e7 0 −6.49805e8 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.a 2
3.b odd 2 1 72.20.a.a 2
4.b odd 2 1 16.20.a.e 2
8.b even 2 1 64.20.a.k 2
8.d odd 2 1 64.20.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.20.a.a 2 1.a even 1 1 trivial
16.20.a.e 2 4.b odd 2 1
64.20.a.j 2 8.d odd 2 1
64.20.a.k 2 8.b even 2 1
72.20.a.a 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 27912T_{3} - 1144314864 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + \cdots - 1144314864 \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 2216319016700 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 11\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 54\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 89\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 83\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 85\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 66\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 54\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 21\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 91\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 92\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 45\!\cdots\!56 \) Copy content Toggle raw display
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