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Properties

Label 216.2.d.b
Level $216$
Weight $2$
Character orbit 216.d
Analytic conductor $1.725$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{3} q^{5} + q^{7} + (2 \beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{3} q^{5} + q^{7} + (2 \beta_{3} + \beta_1) q^{8} + ( - \beta_{2} + 1) q^{10} - 3 \beta_{3} q^{11} + (4 \beta_{2} - 2) q^{13} + \beta_1 q^{14} + (3 \beta_{2} - 1) q^{16} + (2 \beta_{3} - 4 \beta_1) q^{17} + ( - 4 \beta_{2} + 2) q^{19} + ( - 2 \beta_{3} + \beta_1) q^{20} + ( - 3 \beta_{2} + 3) q^{22} + (2 \beta_{3} - 4 \beta_1) q^{23} + 4 q^{25} + (8 \beta_{3} - 2 \beta_1) q^{26} + (\beta_{2} + 1) q^{28} - 6 \beta_{3} q^{29} - 7 q^{31} + (6 \beta_{3} - \beta_1) q^{32} + ( - 2 \beta_{2} - 6) q^{34} - \beta_{3} q^{35} + (4 \beta_{2} - 2) q^{37} + ( - 8 \beta_{3} + 2 \beta_1) q^{38} + ( - \beta_{2} + 3) q^{40} + ( - 2 \beta_{3} + 4 \beta_1) q^{41} + ( - 8 \beta_{2} + 4) q^{43} + ( - 6 \beta_{3} + 3 \beta_1) q^{44} + ( - 2 \beta_{2} - 6) q^{46} - 6 q^{49} + 4 \beta_1 q^{50} + (6 \beta_{2} - 10) q^{52} + 9 \beta_{3} q^{53} - 3 q^{55} + (2 \beta_{3} + \beta_1) q^{56} + ( - 6 \beta_{2} + 6) q^{58} + 4 \beta_{3} q^{59} - 7 \beta_1 q^{62} + (5 \beta_{2} - 7) q^{64} + ( - 2 \beta_{3} + 4 \beta_1) q^{65} + ( - 4 \beta_{3} - 6 \beta_1) q^{68} + ( - \beta_{2} + 1) q^{70} + ( - 6 \beta_{3} + 12 \beta_1) q^{71} + 3 q^{73} + (8 \beta_{3} - 2 \beta_1) q^{74} + ( - 6 \beta_{2} + 10) q^{76} - 3 \beta_{3} q^{77} + 4 q^{79} + ( - 2 \beta_{3} + 3 \beta_1) q^{80} + (2 \beta_{2} + 6) q^{82} + 7 \beta_{3} q^{83} + (4 \beta_{2} - 2) q^{85} + ( - 16 \beta_{3} + 4 \beta_1) q^{86} + ( - 3 \beta_{2} + 9) q^{88} + (4 \beta_{3} - 8 \beta_1) q^{89} + (4 \beta_{2} - 2) q^{91} + ( - 4 \beta_{3} - 6 \beta_1) q^{92} + (2 \beta_{3} - 4 \beta_1) q^{95} + 7 q^{97} - 6 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} + 4 q^{7} + 2 q^{10} + 2 q^{16} + 6 q^{22} + 16 q^{25} + 6 q^{28} - 28 q^{31} - 28 q^{34} + 10 q^{40} - 28 q^{46} - 24 q^{49} - 28 q^{52} - 12 q^{55} + 12 q^{58} - 18 q^{64} + 2 q^{70} + 12 q^{73} + 28 q^{76} + 16 q^{79} + 28 q^{82} + 30 q^{88} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 3x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + \beta_1 \) 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
−1.32288 − 0.500000i
−1.32288 + 0.500000i
1.32288 − 0.500000i
1.32288 + 0.500000i
−1.32288 − 0.500000i 0 1.50000 + 1.32288i 1.00000i 0 1.00000 −1.32288 − 2.50000i 0 0.500000 − 1.32288i
109.2 −1.32288 + 0.500000i 0 1.50000 − 1.32288i − 1.00000i 0 1.00000 −1.32288 + 2.50000i 0 0.500000 + 1.32288i
109.3 1.32288 − 0.500000i 0 1.50000 − 1.32288i 1.00000i 0 1.00000 1.32288 − 2.50000i 0 0.500000 + 1.32288i
109.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i − 1.00000i 0 1.00000 1.32288 + 2.50000i 0 0.500000 − 1.32288i
\(n\): e.g. 2-40 or 990-1000
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.b ✓ 4
3.b odd 2 1 inner 216.2.d.b ✓ 4
4.b odd 2 1 864.2.d.a 4
8.b even 2 1 inner 216.2.d.b ✓ 4
8.d odd 2 1 864.2.d.a 4
9.c even 3 2 648.2.n.n 8
9.d odd 6 2 648.2.n.n 8
12.b even 2 1 864.2.d.a 4
16.e even 4 1 6912.2.a.bc 2
16.e even 4 1 6912.2.a.bu 2
16.f odd 4 1 6912.2.a.bd 2
16.f odd 4 1 6912.2.a.bv 2
24.f even 2 1 864.2.d.a 4
24.h odd 2 1 inner 216.2.d.b ✓ 4
36.f odd 6 2 2592.2.r.p 8
36.h even 6 2 2592.2.r.p 8
48.i odd 4 1 6912.2.a.bc 2
48.i odd 4 1 6912.2.a.bu 2
48.k even 4 1 6912.2.a.bd 2
48.k even 4 1 6912.2.a.bv 2
72.j odd 6 2 648.2.n.n 8
72.l even 6 2 2592.2.r.p 8
72.n even 6 2 648.2.n.n 8
72.p odd 6 2 2592.2.r.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b ✓ 4 1.a even 1 1 trivial
216.2.d.b ✓ 4 3.b odd 2 1 inner
216.2.d.b ✓ 4 8.b even 2 1 inner
216.2.d.b ✓ 4 24.h odd 2 1 inner
648.2.n.n 8 9.c even 3 2
648.2.n.n 8 9.d odd 6 2
648.2.n.n 8 72.j odd 6 2
648.2.n.n 8 72.n even 6 2
864.2.d.a 4 4.b odd 2 1
864.2.d.a 4 8.d odd 2 1
864.2.d.a 4 12.b even 2 1
864.2.d.a 4 24.f even 2 1
2592.2.r.p 8 36.f odd 6 2
2592.2.r.p 8 36.h even 6 2
2592.2.r.p 8 72.l even 6 2
2592.2.r.p 8 72.p odd 6 2
6912.2.a.bc 2 16.e even 4 1
6912.2.a.bc 2 48.i odd 4 1
6912.2.a.bd 2 16.f odd 4 1
6912.2.a.bd 2 48.k even 4 1
6912.2.a.bu 2 16.e even 4 1
6912.2.a.bu 2 48.i odd 4 1
6912.2.a.bv 2 16.f odd 4 1
6912.2.a.bv 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 252)^{2} \) Copy content Toggle raw display
$73$ \( (T - 3)^{4} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$97$ \( (T - 7)^{4} \) Copy content Toggle raw display
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