Properties

Label 3456.1.bn.b
Level $3456$
Weight $1$
Character orbit 3456.bn
Analytic conductor $1.725$
Analytic rank $0$
Dimension $6$
Projective image $D_{18}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3456,1,Mod(65,3456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3456, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 13]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3456.65");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3456 = 2^{7} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3456.bn (of order \(18\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} + \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{18}^{4} q^{3} + \zeta_{18}^{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{18}^{4} q^{3} + \zeta_{18}^{8} q^{9} + ( - \zeta_{18}^{6} - \zeta_{18}^{2}) q^{11} + ( - \zeta_{18}^{7} + \zeta_{18}^{5}) q^{17} + ( - \zeta_{18}^{8} - \zeta_{18}^{7}) q^{19} + \zeta_{18} q^{25} - \zeta_{18}^{3} q^{27} + ( - \zeta_{18}^{6} + \zeta_{18}) q^{33} + (\zeta_{18}^{4} + \zeta_{18}^{3}) q^{41} + ( - \zeta_{18}^{5} + \zeta_{18}^{3}) q^{43} + \zeta_{18}^{5} q^{49} + (\zeta_{18}^{2} - 1) q^{51} + (\zeta_{18}^{3} + \zeta_{18}^{2}) q^{57} + ( - \zeta_{18} + 1) q^{59} + (\zeta_{18}^{6} + \zeta_{18}) q^{67} + ( - \zeta_{18}^{4} - \zeta_{18}^{2}) q^{73} + \zeta_{18}^{5} q^{75} - \zeta_{18}^{7} q^{81} - \zeta_{18} q^{83} + ( - \zeta_{18}^{6} + 1) q^{89} + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{97} + (\zeta_{18}^{5} + \zeta_{18}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{11} - 3 q^{27} + 3 q^{33} + 3 q^{41} + 3 q^{43} - 6 q^{51} + 3 q^{57} + 6 q^{59} - 3 q^{67} + 9 q^{89} - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(2431\) \(2945\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{18}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 + 0.984808i
−0.173648 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
0 −0.939693 + 0.342020i 0 0 0 0 0 0.766044 0.642788i 0
833.1 0 0.173648 + 0.984808i 0 0 0 0 0 −0.939693 + 0.342020i 0
1217.1 0 0.766044 + 0.642788i 0 0 0 0 0 0.173648 + 0.984808i 0
1985.1 0 0.766044 0.642788i 0 0 0 0 0 0.173648 0.984808i 0
2369.1 0 0.173648 0.984808i 0 0 0 0 0 −0.939693 0.342020i 0
3137.1 0 −0.939693 0.342020i 0 0 0 0 0 0.766044 + 0.642788i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
108.l even 18 1 inner
216.x odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3456.1.bn.b yes 6
4.b odd 2 1 3456.1.bn.a 6
8.b even 2 1 3456.1.bn.a 6
8.d odd 2 1 CM 3456.1.bn.b yes 6
27.f odd 18 1 3456.1.bn.a 6
108.l even 18 1 inner 3456.1.bn.b yes 6
216.v even 18 1 3456.1.bn.a 6
216.x odd 18 1 inner 3456.1.bn.b yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3456.1.bn.a 6 4.b odd 2 1
3456.1.bn.a 6 8.b even 2 1
3456.1.bn.a 6 27.f odd 18 1
3456.1.bn.a 6 216.v even 18 1
3456.1.bn.b yes 6 1.a even 1 1 trivial
3456.1.bn.b yes 6 8.d odd 2 1 CM
3456.1.bn.b yes 6 108.l even 18 1 inner
3456.1.bn.b yes 6 216.x odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{6} - 3T_{11}^{5} + 6T_{11}^{4} - 8T_{11}^{3} + 12T_{11}^{2} - 6T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3456, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{4} + 9 T^{2} - 9 T + 3 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{4} + 9 T^{2} + 9 T + 3 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} + 6 T^{4} - 6 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + 6 T^{4} - 6 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} + 3 T^{5} + 6 T^{4} + 6 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T + 3)^{3} \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
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