Properties

Label 3549.1.bk.e
Level $3549$
Weight $1$
Character orbit 3549.bk
Analytic conductor $1.771$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3549,1,Mod(170,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3549.170");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3549.bk (of order \(6\), degree \(2\), 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: \(1.77118172983\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.24069811311.4

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{3} - \zeta_{12}^{2} q^{4} - \zeta_{12}^{3} q^{7} + \zeta_{12}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{2} q^{3} - \zeta_{12}^{2} q^{4} - \zeta_{12}^{3} q^{7} + \zeta_{12}^{4} q^{9} - \zeta_{12}^{4} q^{12} + \zeta_{12}^{4} q^{16} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{19} - \zeta_{12}^{5} q^{21} - \zeta_{12}^{2} q^{25} - q^{27} + \zeta_{12}^{5} q^{28} + q^{36} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{37} + q^{43} - q^{48} - q^{49} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{57} + \zeta_{12}^{4} q^{61} + \zeta_{12} q^{63} + q^{64} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{73} - \zeta_{12}^{4} q^{75} + (\zeta_{12}^{5} - \zeta_{12}) q^{76} - 2 \zeta_{12}^{4} q^{79} - \zeta_{12}^{2} q^{81} - \zeta_{12} q^{84} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{4} - 2 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{25} - 4 q^{27} + 4 q^{36} + 4 q^{43} - 4 q^{48} - 4 q^{49} - 2 q^{61} + 4 q^{64} + 2 q^{75} + 4 q^{79} - 2 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3549\mathbb{Z}\right)^\times\).

\(n\) \(1184\) \(1522\) \(3382\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0.500000 + 0.866025i −0.500000 0.866025i 0 0 1.00000i 0 −0.500000 + 0.866025i 0
170.2 0 0.500000 + 0.866025i −0.500000 0.866025i 0 0 1.00000i 0 −0.500000 + 0.866025i 0
1691.1 0 0.500000 0.866025i −0.500000 + 0.866025i 0 0 1.00000i 0 −0.500000 0.866025i 0
1691.2 0 0.500000 0.866025i −0.500000 + 0.866025i 0 0 1.00000i 0 −0.500000 0.866025i 0
\(n\): 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 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
13.b even 2 1 inner
21.h odd 6 1 inner
39.d odd 2 1 inner
91.r even 6 1 inner
273.w odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3549.1.bk.e 4
3.b odd 2 1 CM 3549.1.bk.e 4
7.c even 3 1 inner 3549.1.bk.e 4
13.b even 2 1 inner 3549.1.bk.e 4
13.c even 3 1 3549.1.s.c 4
13.c even 3 1 3549.1.bm.b 4
13.d odd 4 1 3549.1.w.a 2
13.d odd 4 1 3549.1.w.b 2
13.e even 6 1 3549.1.s.c 4
13.e even 6 1 3549.1.bm.b 4
13.f odd 12 1 273.1.x.a 2
13.f odd 12 1 273.1.bp.a yes 2
13.f odd 12 1 3549.1.x.a 2
13.f odd 12 1 3549.1.bp.a 2
21.h odd 6 1 inner 3549.1.bk.e 4
39.d odd 2 1 inner 3549.1.bk.e 4
39.f even 4 1 3549.1.w.a 2
39.f even 4 1 3549.1.w.b 2
39.h odd 6 1 3549.1.s.c 4
39.h odd 6 1 3549.1.bm.b 4
39.i odd 6 1 3549.1.s.c 4
39.i odd 6 1 3549.1.bm.b 4
39.k even 12 1 273.1.x.a 2
39.k even 12 1 273.1.bp.a yes 2
39.k even 12 1 3549.1.x.a 2
39.k even 12 1 3549.1.bp.a 2
91.g even 3 1 3549.1.s.c 4
91.h even 3 1 3549.1.bm.b 4
91.k even 6 1 3549.1.bm.b 4
91.r even 6 1 inner 3549.1.bk.e 4
91.u even 6 1 3549.1.s.c 4
91.w even 12 1 1911.1.bc.a 2
91.w even 12 1 1911.1.bp.a 2
91.x odd 12 1 273.1.x.a 2
91.x odd 12 1 1911.1.bc.b 2
91.x odd 12 1 3549.1.x.a 2
91.z odd 12 1 3549.1.w.a 2
91.z odd 12 1 3549.1.w.b 2
91.ba even 12 1 1911.1.x.a 2
91.ba even 12 1 1911.1.bc.a 2
91.bc even 12 1 1911.1.x.a 2
91.bc even 12 1 1911.1.bp.a 2
91.bd odd 12 1 273.1.bp.a yes 2
91.bd odd 12 1 1911.1.bc.b 2
91.bd odd 12 1 3549.1.bp.a 2
273.s odd 6 1 3549.1.bm.b 4
273.w odd 6 1 inner 3549.1.bk.e 4
273.x odd 6 1 3549.1.s.c 4
273.bm odd 6 1 3549.1.s.c 4
273.bp odd 6 1 3549.1.bm.b 4
273.bs odd 12 1 1911.1.x.a 2
273.bs odd 12 1 1911.1.bc.a 2
273.bv even 12 1 273.1.x.a 2
273.bv even 12 1 1911.1.bc.b 2
273.bv even 12 1 3549.1.x.a 2
273.bw even 12 1 273.1.bp.a yes 2
273.bw even 12 1 1911.1.bc.b 2
273.bw even 12 1 3549.1.bp.a 2
273.ca odd 12 1 1911.1.x.a 2
273.ca odd 12 1 1911.1.bp.a 2
273.cd even 12 1 3549.1.w.a 2
273.cd even 12 1 3549.1.w.b 2
273.ch odd 12 1 1911.1.bc.a 2
273.ch odd 12 1 1911.1.bp.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.x.a 2 13.f odd 12 1
273.1.x.a 2 39.k even 12 1
273.1.x.a 2 91.x odd 12 1
273.1.x.a 2 273.bv even 12 1
273.1.bp.a yes 2 13.f odd 12 1
273.1.bp.a yes 2 39.k even 12 1
273.1.bp.a yes 2 91.bd odd 12 1
273.1.bp.a yes 2 273.bw even 12 1
1911.1.x.a 2 91.ba even 12 1
1911.1.x.a 2 91.bc even 12 1
1911.1.x.a 2 273.bs odd 12 1
1911.1.x.a 2 273.ca odd 12 1
1911.1.bc.a 2 91.w even 12 1
1911.1.bc.a 2 91.ba even 12 1
1911.1.bc.a 2 273.bs odd 12 1
1911.1.bc.a 2 273.ch odd 12 1
1911.1.bc.b 2 91.x odd 12 1
1911.1.bc.b 2 91.bd odd 12 1
1911.1.bc.b 2 273.bv even 12 1
1911.1.bc.b 2 273.bw even 12 1
1911.1.bp.a 2 91.w even 12 1
1911.1.bp.a 2 91.bc even 12 1
1911.1.bp.a 2 273.ca odd 12 1
1911.1.bp.a 2 273.ch odd 12 1
3549.1.s.c 4 13.c even 3 1
3549.1.s.c 4 13.e even 6 1
3549.1.s.c 4 39.h odd 6 1
3549.1.s.c 4 39.i odd 6 1
3549.1.s.c 4 91.g even 3 1
3549.1.s.c 4 91.u even 6 1
3549.1.s.c 4 273.x odd 6 1
3549.1.s.c 4 273.bm odd 6 1
3549.1.w.a 2 13.d odd 4 1
3549.1.w.a 2 39.f even 4 1
3549.1.w.a 2 91.z odd 12 1
3549.1.w.a 2 273.cd even 12 1
3549.1.w.b 2 13.d odd 4 1
3549.1.w.b 2 39.f even 4 1
3549.1.w.b 2 91.z odd 12 1
3549.1.w.b 2 273.cd even 12 1
3549.1.x.a 2 13.f odd 12 1
3549.1.x.a 2 39.k even 12 1
3549.1.x.a 2 91.x odd 12 1
3549.1.x.a 2 273.bv even 12 1
3549.1.bk.e 4 1.a even 1 1 trivial
3549.1.bk.e 4 3.b odd 2 1 CM
3549.1.bk.e 4 7.c even 3 1 inner
3549.1.bk.e 4 13.b even 2 1 inner
3549.1.bk.e 4 21.h odd 6 1 inner
3549.1.bk.e 4 39.d odd 2 1 inner
3549.1.bk.e 4 91.r even 6 1 inner
3549.1.bk.e 4 273.w odd 6 1 inner
3549.1.bm.b 4 13.c even 3 1
3549.1.bm.b 4 13.e even 6 1
3549.1.bm.b 4 39.h odd 6 1
3549.1.bm.b 4 39.i odd 6 1
3549.1.bm.b 4 91.h even 3 1
3549.1.bm.b 4 91.k even 6 1
3549.1.bm.b 4 273.s odd 6 1
3549.1.bm.b 4 273.bp odd 6 1
3549.1.bp.a 2 13.f odd 12 1
3549.1.bp.a 2 39.k even 12 1
3549.1.bp.a 2 91.bd odd 12 1
3549.1.bp.a 2 273.bw even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3549, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{19}^{4} + 3T_{19}^{2} + 9 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

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