Properties

Label 216.3.h.a
Level $216$
Weight $3$
Character orbit 216.h
Self dual yes
Analytic conductor $5.886$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(53,216)]
 
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, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.h (of order \(2\), degree \(1\), minimal)

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: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 6\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + (\beta - 1) q^{5} + (\beta + 5) q^{7} - 8 q^{8} + ( - 2 \beta + 2) q^{10} + ( - 2 \beta + 5) q^{11} + ( - 2 \beta - 10) q^{14} + 16 q^{16} + (4 \beta - 4) q^{20} + (4 \beta - 10) q^{22}+ \cdots + ( - 20 \beta - 96) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} - 2 q^{5} + 10 q^{7} - 16 q^{8} + 4 q^{10} + 10 q^{11} - 20 q^{14} + 32 q^{16} - 8 q^{20} - 20 q^{22} + 96 q^{25} + 40 q^{28} + 100 q^{29} - 38 q^{31} - 64 q^{32} + 134 q^{35} + 16 q^{40}+ \cdots - 192 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.41421
1.41421
−2.00000 0 4.00000 −9.48528 0 −3.48528 −8.00000 0 18.9706
53.2 −2.00000 0 4.00000 7.48528 0 13.4853 −8.00000 0 −14.9706
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.h.a 2
3.b odd 2 1 216.3.h.b yes 2
4.b odd 2 1 864.3.h.a 2
8.b even 2 1 216.3.h.b yes 2
8.d odd 2 1 864.3.h.b 2
12.b even 2 1 864.3.h.b 2
24.f even 2 1 864.3.h.a 2
24.h odd 2 1 CM 216.3.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.h.a 2 1.a even 1 1 trivial
216.3.h.a 2 24.h odd 2 1 CM
216.3.h.b yes 2 3.b odd 2 1
216.3.h.b yes 2 8.b even 2 1
864.3.h.a 2 4.b odd 2 1
864.3.h.a 2 24.f even 2 1
864.3.h.b 2 8.d odd 2 1
864.3.h.b 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2T_{5} - 71 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 71 \) Copy content Toggle raw display
$7$ \( T^{2} - 10T - 47 \) Copy content Toggle raw display
$11$ \( T^{2} - 10T - 263 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 50)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 38T - 1439 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 94T + 409 \) Copy content Toggle raw display
$59$ \( (T + 10)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 50T - 13487 \) Copy content Toggle raw display
$79$ \( (T + 58)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 134T - 2711 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 190T + 7873 \) Copy content Toggle raw display
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