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Properties

Label 1728.1.bn.a
Level $1728$
Weight $1$
Character orbit 1728.bn
Analytic conductor $0.862$
Analytic rank $0$
Dimension $12$
Projective image $D_{18}$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,1,Mod(353,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.353");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1728.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: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) 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.

\(f(q)\) \(=\) \( q + \zeta_{36}^{11} q^{3} - \zeta_{36}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{36}^{11} q^{3} - \zeta_{36}^{4} q^{9} + ( - \zeta_{36}^{3} - \zeta_{36}) q^{11} + (\zeta_{36}^{16} - \zeta_{36}^{8}) q^{17} + (\zeta_{36}^{17} + \zeta_{36}^{13}) q^{19} + \zeta_{36}^{14} q^{25} - \zeta_{36}^{15} q^{27} + ( - \zeta_{36}^{14} - \zeta_{36}^{12}) q^{33} + (\zeta_{36}^{6} - \zeta_{36}^{2}) q^{41} + (\zeta_{36}^{15} + \zeta_{36}^{7}) q^{43} - \zeta_{36}^{16} q^{49} + ( - \zeta_{36}^{9} + \zeta_{36}) q^{51} + ( - \zeta_{36}^{10} - \zeta_{36}^{6}) q^{57} + ( - \zeta_{36}^{9} + \zeta_{36}^{5}) q^{59} + ( - \zeta_{36}^{5} + \zeta_{36}^{3}) q^{67} + ( - \zeta_{36}^{10} - \zeta_{36}^{2}) q^{73} - \zeta_{36}^{7} q^{75} + \zeta_{36}^{8} q^{81} + ( - \zeta_{36}^{17} - \zeta_{36}^{11}) q^{83} + (\zeta_{36}^{12} - 1) q^{89} + ( - \zeta_{36}^{12} + \zeta_{36}^{10}) q^{97} + (\zeta_{36}^{7} + \zeta_{36}^{5}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{33} + 6 q^{41} - 6 q^{57} - 18 q^{89} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{36}^{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1
−0.342020 + 0.939693i
0.342020 − 0.939693i
0.984808 + 0.173648i
−0.984808 − 0.173648i
0.984808 − 0.173648i
−0.984808 + 0.173648i
−0.342020 − 0.939693i
0.342020 + 0.939693i
0.642788 + 0.766044i
−0.642788 − 0.766044i
0.642788 − 0.766044i
−0.642788 + 0.766044i
0 −0.642788 + 0.766044i 0 0 0 0 0 −0.173648 − 0.984808i 0
353.2 0 0.642788 − 0.766044i 0 0 0 0 0 −0.173648 − 0.984808i 0
545.1 0 −0.342020 + 0.939693i 0 0 0 0 0 −0.766044 − 0.642788i 0
545.2 0 0.342020 − 0.939693i 0 0 0 0 0 −0.766044 − 0.642788i 0
929.1 0 −0.342020 − 0.939693i 0 0 0 0 0 −0.766044 + 0.642788i 0
929.2 0 0.342020 + 0.939693i 0 0 0 0 0 −0.766044 + 0.642788i 0
1121.1 0 −0.642788 − 0.766044i 0 0 0 0 0 −0.173648 + 0.984808i 0
1121.2 0 0.642788 + 0.766044i 0 0 0 0 0 −0.173648 + 0.984808i 0
1505.1 0 −0.984808 − 0.173648i 0 0 0 0 0 0.939693 + 0.342020i 0
1505.2 0 0.984808 + 0.173648i 0 0 0 0 0 0.939693 + 0.342020i 0
1697.1 0 −0.984808 + 0.173648i 0 0 0 0 0 0.939693 − 0.342020i 0
1697.2 0 0.984808 − 0.173648i 0 0 0 0 0 0.939693 − 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
27.f odd 18 1 inner
108.l even 18 1 inner
216.v 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 1728.1.bn.a ✓ 12
4.b odd 2 1 inner 1728.1.bn.a ✓ 12
8.b even 2 1 inner 1728.1.bn.a ✓ 12
8.d odd 2 1 CM 1728.1.bn.a ✓ 12
27.f odd 18 1 inner 1728.1.bn.a ✓ 12
108.l even 18 1 inner 1728.1.bn.a ✓ 12
216.v even 18 1 inner 1728.1.bn.a ✓ 12
216.x odd 18 1 inner 1728.1.bn.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.1.bn.a ✓ 12 1.a even 1 1 trivial
1728.1.bn.a ✓ 12 4.b odd 2 1 inner
1728.1.bn.a ✓ 12 8.b even 2 1 inner
1728.1.bn.a ✓ 12 8.d odd 2 1 CM
1728.1.bn.a ✓ 12 27.f odd 18 1 inner
1728.1.bn.a ✓ 12 108.l even 18 1 inner
1728.1.bn.a ✓ 12 216.v even 18 1 inner
1728.1.bn.a ✓ 12 216.x odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1728, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 3 T^{4} + 9 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( (T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} + 6 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( (T^{6} + 3 T^{4} + 2 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} + 27T^{6} + 729 \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T + 3)^{6} \) Copy content Toggle raw display
$97$ \( (T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
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