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Properties

Label 216.2.d.c
Level $216$
Weight $2$
Character orbit 216.d
Analytic conductor $1.725$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,2,Mod(109,216)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.d (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: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.629407744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + \beta_{4} q^{5} + ( - \beta_{6} - \beta_{2}) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + \beta_{4} q^{5} + ( - \beta_{6} - \beta_{2}) q^{7} + \beta_{3} q^{8} + ( - \beta_{6} + \beta_{5} + \beta_{2} - 2) q^{10} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{11} + (\beta_{6} - \beta_{5} - \beta_{2} + 1) q^{13} + ( - \beta_{7} - \beta_{3}) q^{14} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{16} + ( - \beta_{4} + 2 \beta_{3}) q^{17} + (2 \beta_{6} - \beta_{5} - 2 \beta_{2} + 2) q^{19} + (2 \beta_{4} - 2 \beta_1) q^{20} + (\beta_{6} - \beta_{5} + \beta_{2} - 2) q^{22} + (\beta_{7} + \beta_{4} + \cdots + 3 \beta_1) q^{23}+ \cdots + (2 \beta_{6} + 2 \beta_{2} - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 8 q^{10} - 16 q^{22} - 8 q^{25} - 28 q^{28} + 16 q^{31} + 8 q^{34} - 24 q^{40} + 32 q^{46} + 20 q^{52} + 16 q^{55} + 32 q^{58} + 40 q^{64} + 8 q^{73} + 52 q^{76} - 48 q^{79} - 80 q^{82} + 8 q^{88} - 48 q^{94} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 2\nu^{3} - 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{4} - 2\nu^{2} - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} - 2\nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} - 2\nu^{3} + 4\nu ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{6} + 2\beta_{5} + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{4} - 2\beta_{3} + 4\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.38255 − 0.297594i
−1.38255 + 0.297594i
−0.767178 − 1.18804i
−0.767178 + 1.18804i
0.767178 − 1.18804i
0.767178 + 1.18804i
1.38255 − 0.297594i
1.38255 + 0.297594i
−1.38255 − 0.297594i 0 1.82288 + 0.822876i − 3.36028i 0 −2.64575 −2.27533 − 1.68014i 0 −1.00000 + 4.64575i
109.2 −1.38255 + 0.297594i 0 1.82288 − 0.822876i 3.36028i 0 −2.64575 −2.27533 + 1.68014i 0 −1.00000 − 4.64575i
109.3 −0.767178 − 1.18804i 0 −0.822876 + 1.82288i − 0.841723i 0 2.64575 2.79694 − 0.420861i 0 −1.00000 + 0.645751i
109.4 −0.767178 + 1.18804i 0 −0.822876 − 1.82288i 0.841723i 0 2.64575 2.79694 + 0.420861i 0 −1.00000 − 0.645751i
109.5 0.767178 − 1.18804i 0 −0.822876 − 1.82288i − 0.841723i 0 2.64575 −2.79694 − 0.420861i 0 −1.00000 − 0.645751i
109.6 0.767178 + 1.18804i 0 −0.822876 + 1.82288i 0.841723i 0 2.64575 −2.79694 + 0.420861i 0 −1.00000 + 0.645751i
109.7 1.38255 − 0.297594i 0 1.82288 − 0.822876i − 3.36028i 0 −2.64575 2.27533 − 1.68014i 0 −1.00000 − 4.64575i
109.8 1.38255 + 0.297594i 0 1.82288 + 0.822876i 3.36028i 0 −2.64575 2.27533 + 1.68014i 0 −1.00000 + 4.64575i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.d.c ✓ 8
3.b odd 2 1 inner 216.2.d.c ✓ 8
4.b odd 2 1 864.2.d.c 8
8.b even 2 1 inner 216.2.d.c ✓ 8
8.d odd 2 1 864.2.d.c 8
9.c even 3 2 648.2.n.q 16
9.d odd 6 2 648.2.n.q 16
12.b even 2 1 864.2.d.c 8
16.e even 4 1 6912.2.a.cc 4
16.e even 4 1 6912.2.a.cj 4
16.f odd 4 1 6912.2.a.cd 4
16.f odd 4 1 6912.2.a.ci 4
24.f even 2 1 864.2.d.c 8
24.h odd 2 1 inner 216.2.d.c ✓ 8
36.f odd 6 2 2592.2.r.q 16
36.h even 6 2 2592.2.r.q 16
48.i odd 4 1 6912.2.a.cc 4
48.i odd 4 1 6912.2.a.cj 4
48.k even 4 1 6912.2.a.cd 4
48.k even 4 1 6912.2.a.ci 4
72.j odd 6 2 648.2.n.q 16
72.l even 6 2 2592.2.r.q 16
72.n even 6 2 648.2.n.q 16
72.p odd 6 2 2592.2.r.q 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.c ✓ 8 1.a even 1 1 trivial
216.2.d.c ✓ 8 3.b odd 2 1 inner
216.2.d.c ✓ 8 8.b even 2 1 inner
216.2.d.c ✓ 8 24.h odd 2 1 inner
648.2.n.q 16 9.c even 3 2
648.2.n.q 16 9.d odd 6 2
648.2.n.q 16 72.j odd 6 2
648.2.n.q 16 72.n even 6 2
864.2.d.c 8 4.b odd 2 1
864.2.d.c 8 8.d odd 2 1
864.2.d.c 8 12.b even 2 1
864.2.d.c 8 24.f even 2 1
2592.2.r.q 16 36.f odd 6 2
2592.2.r.q 16 36.h even 6 2
2592.2.r.q 16 72.l even 6 2
2592.2.r.q 16 72.p odd 6 2
6912.2.a.cc 4 16.e even 4 1
6912.2.a.cc 4 48.i odd 4 1
6912.2.a.cd 4 16.f odd 4 1
6912.2.a.cd 4 48.k even 4 1
6912.2.a.ci 4 16.f odd 4 1
6912.2.a.ci 4 48.k even 4 1
6912.2.a.cj 4 16.e even 4 1
6912.2.a.cj 4 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 12T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 12 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 20 T^{2} + 72)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 22 T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 52 T^{2} + 648)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 58 T^{2} + 729)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 76 T^{2} + 72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 46 T^{2} + 81)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 160 T^{2} + 4608)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 64 T^{2} + 576)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 108 T^{2} + 648)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 180 T^{2} + 6728)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 198 T^{2} + 729)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T - 111)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 29)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 96 T^{2} + 512)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 52 T^{2} + 648)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 6 T - 19)^{4} \) Copy content Toggle raw display
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