Properties

Label 216.6.a.e
Level $216$
Weight $6$
Character orbit 216.a
Self dual yes
Analytic conductor $34.643$
Analytic rank $1$
Dimension $2$
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] = [216,6,Mod(1,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 216.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: \(34.6429050796\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{209}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(\beta = 3\sqrt{209}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 10) q^{5} + (\beta + 6) q^{7} + (12 \beta + 119) q^{11} + ( - 16 \beta - 236) q^{13} + (16 \beta + 184) q^{17} + (38 \beta - 28) q^{19} + ( - 32 \beta - 514) q^{23} + (20 \beta - 1144) q^{25}+ \cdots + ( - 856 \beta - 39913) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 20 q^{5} + 12 q^{7} + 238 q^{11} - 472 q^{13} + 368 q^{17} - 56 q^{19} - 1028 q^{23} - 2288 q^{25} - 144 q^{29} - 8228 q^{31} - 3882 q^{35} - 14076 q^{37} + 6696 q^{41} - 15112 q^{43} - 156 q^{47}+ \cdots - 79826 q^{97}+O(q^{100}) \) 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
7.72842
−6.72842
0 0 0 −53.3705 0 49.3705 0 0 0
1.2 0 0 0 33.3705 0 −37.3705 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 216.6.a.e 2
3.b odd 2 1 216.6.a.f yes 2
4.b odd 2 1 432.6.a.n 2
12.b even 2 1 432.6.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.6.a.e 2 1.a even 1 1 trivial
216.6.a.f yes 2 3.b odd 2 1
432.6.a.n 2 4.b odd 2 1
432.6.a.s 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 20T_{5} - 1781 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(216))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 20T - 1781 \) Copy content Toggle raw display
$7$ \( T^{2} - 12T - 1845 \) Copy content Toggle raw display
$11$ \( T^{2} - 238T - 256703 \) Copy content Toggle raw display
$13$ \( T^{2} + 472T - 425840 \) Copy content Toggle raw display
$17$ \( T^{2} - 368T - 447680 \) Copy content Toggle raw display
$19$ \( T^{2} + 56T - 2715380 \) Copy content Toggle raw display
$23$ \( T^{2} + 1028 T - 1661948 \) Copy content Toggle raw display
$29$ \( T^{2} + 144 T - 13906692 \) Copy content Toggle raw display
$31$ \( T^{2} + 8228 T + 2025595 \) Copy content Toggle raw display
$37$ \( T^{2} + 14076 T + 34966980 \) Copy content Toggle raw display
$41$ \( T^{2} - 6696 T - 16787700 \) Copy content Toggle raw display
$43$ \( T^{2} + 15112 T - 60469364 \) Copy content Toggle raw display
$47$ \( T^{2} + 156 T - 20338812 \) Copy content Toggle raw display
$53$ \( T^{2} - 3004 T - 7226117 \) Copy content Toggle raw display
$59$ \( T^{2} + 27704 T + 122536720 \) Copy content Toggle raw display
$61$ \( T^{2} + 57856 T + 836708800 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1312916812 \) Copy content Toggle raw display
$71$ \( T^{2} + 58768 T + 131003200 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 2258407367 \) Copy content Toggle raw display
$79$ \( T^{2} + 82984 T + 529754368 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 1732621871 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 11547040500 \) Copy content Toggle raw display
$97$ \( T^{2} + 79826 T + 214771153 \) Copy content Toggle raw display
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