Properties

Label 1512.1.bn.a
Level $1512$
Weight $1$
Character orbit 1512.bn
Analytic conductor $0.755$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -56
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,1,Mod(181,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.181");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1512.bn (of order \(6\), degree \(2\), not 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.4536.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{5} - \zeta_{6} q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{5} - \zeta_{6} q^{7} - q^{8} - q^{10} + \zeta_{6}^{2} q^{13} - \zeta_{6}^{2} q^{14} - \zeta_{6} q^{16} - q^{19} - \zeta_{6} q^{20} + \zeta_{6}^{2} q^{23} - 2 q^{26} + q^{28} - \zeta_{6}^{2} q^{32} + q^{35} - \zeta_{6} q^{38} - \zeta_{6}^{2} q^{40} - q^{46} + \zeta_{6}^{2} q^{49} - 2 \zeta_{6} q^{52} + \zeta_{6} q^{56} - \zeta_{6}^{2} q^{59} + \zeta_{6} q^{61} + q^{64} - 2 \zeta_{6} q^{65} + \zeta_{6} q^{70} + q^{71} - \zeta_{6}^{2} q^{76} + \zeta_{6} q^{79} + q^{80} + \zeta_{6} q^{83} + 2 q^{91} - \zeta_{6} q^{92} - \zeta_{6}^{2} q^{95} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - q^{5} - q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - q^{5} - q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{13} + q^{14} - q^{16} - 2 q^{19} - q^{20} - q^{23} - 4 q^{26} + 2 q^{28} + q^{32} + 2 q^{35} - q^{38} + q^{40} - 2 q^{46} - q^{49} - 2 q^{52} + q^{56} + 2 q^{59} + q^{61} + 2 q^{64} - 2 q^{65} + q^{70} + 2 q^{71} + q^{76} + q^{79} + 2 q^{80} + 2 q^{83} + 4 q^{91} - q^{92} + q^{95} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 −1.00000
685.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
9.c even 3 1 inner
504.bn odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.1.bn.a 2
3.b odd 2 1 504.1.bn.b yes 2
7.b odd 2 1 1512.1.bn.b 2
8.b even 2 1 1512.1.bn.b 2
9.c even 3 1 inner 1512.1.bn.a 2
9.d odd 6 1 504.1.bn.b yes 2
12.b even 2 1 2016.1.bv.a 2
21.c even 2 1 504.1.bn.a 2
21.g even 6 1 3528.1.bp.b 2
21.g even 6 1 3528.1.cw.b 2
21.h odd 6 1 3528.1.bp.a 2
21.h odd 6 1 3528.1.cw.a 2
24.f even 2 1 2016.1.bv.b 2
24.h odd 2 1 504.1.bn.a 2
36.h even 6 1 2016.1.bv.a 2
56.h odd 2 1 CM 1512.1.bn.a 2
63.i even 6 1 3528.1.cw.b 2
63.j odd 6 1 3528.1.cw.a 2
63.l odd 6 1 1512.1.bn.b 2
63.n odd 6 1 3528.1.bp.a 2
63.o even 6 1 504.1.bn.a 2
63.s even 6 1 3528.1.bp.b 2
72.j odd 6 1 504.1.bn.a 2
72.l even 6 1 2016.1.bv.b 2
72.n even 6 1 1512.1.bn.b 2
84.h odd 2 1 2016.1.bv.b 2
168.e odd 2 1 2016.1.bv.a 2
168.i even 2 1 504.1.bn.b yes 2
168.s odd 6 1 3528.1.bp.b 2
168.s odd 6 1 3528.1.cw.b 2
168.ba even 6 1 3528.1.bp.a 2
168.ba even 6 1 3528.1.cw.a 2
252.s odd 6 1 2016.1.bv.b 2
504.y even 6 1 3528.1.bp.a 2
504.bi odd 6 1 3528.1.cw.b 2
504.bn odd 6 1 inner 1512.1.bn.a 2
504.ca even 6 1 3528.1.cw.a 2
504.cc even 6 1 504.1.bn.b yes 2
504.co odd 6 1 2016.1.bv.a 2
504.db odd 6 1 3528.1.bp.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.bn.a 2 21.c even 2 1
504.1.bn.a 2 24.h odd 2 1
504.1.bn.a 2 63.o even 6 1
504.1.bn.a 2 72.j odd 6 1
504.1.bn.b yes 2 3.b odd 2 1
504.1.bn.b yes 2 9.d odd 6 1
504.1.bn.b yes 2 168.i even 2 1
504.1.bn.b yes 2 504.cc even 6 1
1512.1.bn.a 2 1.a even 1 1 trivial
1512.1.bn.a 2 9.c even 3 1 inner
1512.1.bn.a 2 56.h odd 2 1 CM
1512.1.bn.a 2 504.bn odd 6 1 inner
1512.1.bn.b 2 7.b odd 2 1
1512.1.bn.b 2 8.b even 2 1
1512.1.bn.b 2 63.l odd 6 1
1512.1.bn.b 2 72.n even 6 1
2016.1.bv.a 2 12.b even 2 1
2016.1.bv.a 2 36.h even 6 1
2016.1.bv.a 2 168.e odd 2 1
2016.1.bv.a 2 504.co odd 6 1
2016.1.bv.b 2 24.f even 2 1
2016.1.bv.b 2 72.l even 6 1
2016.1.bv.b 2 84.h odd 2 1
2016.1.bv.b 2 252.s odd 6 1
3528.1.bp.a 2 21.h odd 6 1
3528.1.bp.a 2 63.n odd 6 1
3528.1.bp.a 2 168.ba even 6 1
3528.1.bp.a 2 504.y even 6 1
3528.1.bp.b 2 21.g even 6 1
3528.1.bp.b 2 63.s even 6 1
3528.1.bp.b 2 168.s odd 6 1
3528.1.bp.b 2 504.db odd 6 1
3528.1.cw.a 2 21.h odd 6 1
3528.1.cw.a 2 63.j odd 6 1
3528.1.cw.a 2 168.ba even 6 1
3528.1.cw.a 2 504.ca even 6 1
3528.1.cw.b 2 21.g even 6 1
3528.1.cw.b 2 63.i even 6 1
3528.1.cw.b 2 168.s odd 6 1
3528.1.cw.b 2 504.bi odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1512, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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