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Properties

Label 1512.1.ef.c
Level $1512$
Weight $1$
Character orbit 1512.ef
Analytic conductor $0.755$
Analytic rank $0$
Dimension $12$
Projective image $D_{18}$
CM discriminant -56
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,1,Mod(13,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 8, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.13");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1512.ef (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: \(0.754586299101\)
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.

\(f(q)\) \(=\) \( q + \zeta_{36}^{10} q^{2} + \zeta_{36}^{3} q^{3} - \zeta_{36}^{2} q^{4} + (\zeta_{36}^{9} - \zeta_{36}) q^{5} + \zeta_{36}^{13} q^{6} - \zeta_{36}^{16} q^{7} - \zeta_{36}^{12} q^{8} + \zeta_{36}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{36}^{10} q^{2} + \zeta_{36}^{3} q^{3} - \zeta_{36}^{2} q^{4} + (\zeta_{36}^{9} - \zeta_{36}) q^{5} + \zeta_{36}^{13} q^{6} - \zeta_{36}^{16} q^{7} - \zeta_{36}^{12} q^{8} + \zeta_{36}^{6} q^{9} + ( - \zeta_{36}^{11} - \zeta_{36}) q^{10} - \zeta_{36}^{5} q^{12} + (\zeta_{36}^{11} + \zeta_{36}^{5}) q^{13} + \zeta_{36}^{8} q^{14} + (\zeta_{36}^{12} - \zeta_{36}^{4}) q^{15} + \zeta_{36}^{4} q^{16} + \zeta_{36}^{16} q^{18} + ( - \zeta_{36}^{5} + \zeta_{36}) q^{19} + ( - \zeta_{36}^{11} + \zeta_{36}^{3}) q^{20} + \zeta_{36} q^{21} + (\zeta_{36}^{16} - \zeta_{36}^{6}) q^{23} - \zeta_{36}^{15} q^{24} + ( - \zeta_{36}^{10} + \zeta_{36}^{2} - 1) q^{25} + (\zeta_{36}^{15} - \zeta_{36}^{3}) q^{26} + \zeta_{36}^{9} q^{27} - q^{28} + ( - \zeta_{36}^{14} - \zeta_{36}^{4}) q^{30} + \zeta_{36}^{14} q^{32} + (\zeta_{36}^{17} + \zeta_{36}^{7}) q^{35} - \zeta_{36}^{8} q^{36} + ( - \zeta_{36}^{15} + \zeta_{36}^{11}) q^{38} + (\zeta_{36}^{14} + \zeta_{36}^{8}) q^{39} + (\zeta_{36}^{13} + \zeta_{36}^{3}) q^{40} + \zeta_{36}^{11} q^{42} + (\zeta_{36}^{15} - \zeta_{36}^{7}) q^{45} + ( - \zeta_{36}^{16} - \zeta_{36}^{8}) q^{46} + \zeta_{36}^{7} q^{48} - \zeta_{36}^{14} q^{49} + (\zeta_{36}^{12} + \cdots + \zeta_{36}^{2}) q^{50} + \cdots + \zeta_{36}^{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{8} + 6 q^{9} - 6 q^{15} - 6 q^{23} - 12 q^{25} - 12 q^{28} - 6 q^{50} + 6 q^{60} - 6 q^{64} + 12 q^{72} - 18 q^{78} + 6 q^{79} - 6 q^{81} + 12 q^{92} + 6 q^{95} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(-\zeta_{36}^{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−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
0.342020 + 0.939693i
−0.342020 − 0.939693i
−0.173648 − 0.984808i −0.866025 + 0.500000i −0.939693 + 0.342020i 0.984808 + 0.826352i 0.642788 + 0.766044i 0.939693 + 0.342020i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.642788 − 1.11334i
13.2 −0.173648 − 0.984808i 0.866025 − 0.500000i −0.939693 + 0.342020i −0.984808 − 0.826352i −0.642788 − 0.766044i 0.939693 + 0.342020i 0.500000 + 0.866025i 0.500000 − 0.866025i −0.642788 + 1.11334i
349.1 −0.173648 + 0.984808i −0.866025 − 0.500000i −0.939693 − 0.342020i 0.984808 − 0.826352i 0.642788 − 0.766044i 0.939693 − 0.342020i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.642788 + 1.11334i
349.2 −0.173648 + 0.984808i 0.866025 + 0.500000i −0.939693 − 0.342020i −0.984808 + 0.826352i −0.642788 + 0.766044i 0.939693 − 0.342020i 0.500000 − 0.866025i 0.500000 + 0.866025i −0.642788 − 1.11334i
517.1 0.939693 + 0.342020i −0.866025 + 0.500000i 0.766044 + 0.642788i −0.342020 + 1.93969i −0.984808 + 0.173648i −0.766044 + 0.642788i 0.500000 + 0.866025i 0.500000 − 0.866025i −0.984808 + 1.70574i
517.2 0.939693 + 0.342020i 0.866025 − 0.500000i 0.766044 + 0.642788i 0.342020 − 1.93969i 0.984808 − 0.173648i −0.766044 + 0.642788i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.984808 − 1.70574i
853.1 −0.766044 − 0.642788i −0.866025 − 0.500000i 0.173648 + 0.984808i −0.642788 − 0.233956i 0.342020 + 0.939693i −0.173648 + 0.984808i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.342020 + 0.592396i
853.2 −0.766044 − 0.642788i 0.866025 + 0.500000i 0.173648 + 0.984808i 0.642788 + 0.233956i −0.342020 − 0.939693i −0.173648 + 0.984808i 0.500000 − 0.866025i 0.500000 + 0.866025i −0.342020 − 0.592396i
1021.1 −0.766044 + 0.642788i −0.866025 + 0.500000i 0.173648 − 0.984808i −0.642788 + 0.233956i 0.342020 − 0.939693i −0.173648 − 0.984808i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.342020 − 0.592396i
1021.2 −0.766044 + 0.642788i 0.866025 − 0.500000i 0.173648 − 0.984808i 0.642788 − 0.233956i −0.342020 + 0.939693i −0.173648 − 0.984808i 0.500000 + 0.866025i 0.500000 − 0.866025i −0.342020 + 0.592396i
1357.1 0.939693 − 0.342020i −0.866025 − 0.500000i 0.766044 − 0.642788i −0.342020 − 1.93969i −0.984808 − 0.173648i −0.766044 − 0.642788i 0.500000 − 0.866025i 0.500000 + 0.866025i −0.984808 − 1.70574i
1357.2 0.939693 − 0.342020i 0.866025 + 0.500000i 0.766044 − 0.642788i 0.342020 + 1.93969i 0.984808 + 0.173648i −0.766044 − 0.642788i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.984808 + 1.70574i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner
27.e even 9 1 inner
189.y odd 18 1 inner
216.t even 18 1 inner
1512.ef odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.1.ef.c ✓ 12
7.b odd 2 1 inner 1512.1.ef.c ✓ 12
8.b even 2 1 inner 1512.1.ef.c ✓ 12
27.e even 9 1 inner 1512.1.ef.c ✓ 12
56.h odd 2 1 CM 1512.1.ef.c ✓ 12
189.y odd 18 1 inner 1512.1.ef.c ✓ 12
216.t even 18 1 inner 1512.1.ef.c ✓ 12
1512.ef odd 18 1 inner 1512.1.ef.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.1.ef.c ✓ 12 1.a even 1 1 trivial
1512.1.ef.c ✓ 12 7.b odd 2 1 inner
1512.1.ef.c ✓ 12 8.b even 2 1 inner
1512.1.ef.c ✓ 12 27.e even 9 1 inner
1512.1.ef.c ✓ 12 56.h odd 2 1 CM
1512.1.ef.c ✓ 12 189.y odd 18 1 inner
1512.1.ef.c ✓ 12 216.t even 18 1 inner
1512.1.ef.c ✓ 12 1512.ef odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 6T_{5}^{10} + 9T_{5}^{8} + 3T_{5}^{6} + 36T_{5}^{4} - 27T_{5}^{2} + 9 \) acting on \(S_{1}^{\mathrm{new}}(1512, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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