Properties

Label 3549.1.x.b
Level $3549$
Weight $1$
Character orbit 3549.x
Analytic conductor $1.771$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3549,1,Mod(485,3549)]
 
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, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3549.485");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3549.x (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: \(1.77118172983\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.74529.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{2} q^{2} - \zeta_{12}^{3} q^{3} + \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{6} + \zeta_{12}^{3} q^{7} - q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{2} q^{2} - \zeta_{12}^{3} q^{3} + \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{6} + \zeta_{12}^{3} q^{7} - q^{8} - q^{9} + q^{10} - \zeta_{12}^{5} q^{14} + \zeta_{12} q^{15} + \zeta_{12}^{2} q^{16} - \zeta_{12} q^{17} + \zeta_{12}^{2} q^{18} + q^{21} - \zeta_{12}^{5} q^{23} + \zeta_{12}^{3} q^{24} + \zeta_{12}^{3} q^{27} + \zeta_{12} q^{29} - \zeta_{12}^{3} q^{30} - \zeta_{12}^{5} q^{31} + \zeta_{12}^{3} q^{34} - \zeta_{12} q^{35} - \zeta_{12}^{5} q^{37} - \zeta_{12}^{4} q^{40} + \zeta_{12}^{4} q^{41} - \zeta_{12}^{2} q^{42} + \zeta_{12}^{2} q^{43} - \zeta_{12}^{4} q^{45} - \zeta_{12} q^{46} - \zeta_{12}^{4} q^{47} - \zeta_{12}^{5} q^{48} - q^{49} + \zeta_{12}^{4} q^{51} - \zeta_{12}^{5} q^{53} - \zeta_{12}^{5} q^{54} - \zeta_{12}^{3} q^{56} - \zeta_{12}^{3} q^{58} - \zeta_{12}^{4} q^{59} - \zeta_{12} q^{62} - \zeta_{12}^{3} q^{63} + q^{64} - \zeta_{12}^{2} q^{69} + \zeta_{12}^{3} q^{70} + \zeta_{12}^{2} q^{71} + q^{72} - \zeta_{12}^{5} q^{73} - \zeta_{12} q^{74} + \zeta_{12}^{4} q^{79} - q^{80} + q^{81} + q^{82} - \zeta_{12}^{5} q^{85} - \zeta_{12}^{4} q^{86} - \zeta_{12}^{4} q^{87} + \zeta_{12}^{2} q^{89} - q^{90} - \zeta_{12}^{2} q^{93} - q^{94} - \zeta_{12}^{5} q^{97} + \zeta_{12}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{5} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{5} - 4 q^{8} - 4 q^{9} + 4 q^{10} + 2 q^{16} + 2 q^{18} + 4 q^{21} + 2 q^{40} - 2 q^{41} - 2 q^{42} + 2 q^{43} + 2 q^{45} + 2 q^{47} - 4 q^{49} - 2 q^{51} + 2 q^{59} + 4 q^{64} - 2 q^{69} + 2 q^{71} + 4 q^{72} - 2 q^{79} - 4 q^{80} + 4 q^{81} + 4 q^{82} + 2 q^{86} + 2 q^{87} + 2 q^{89} - 4 q^{90} - 2 q^{93} - 4 q^{94} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1184\) \(1522\) \(3382\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.500000 0.866025i 1.00000i 0 −0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −1.00000 −1.00000 1.00000
485.2 −0.500000 0.866025i 1.00000i 0 −0.500000 + 0.866025i 0.866025 0.500000i 1.00000i −1.00000 −1.00000 1.00000
1544.1 −0.500000 + 0.866025i 1.00000i 0 −0.500000 0.866025i 0.866025 + 0.500000i 1.00000i −1.00000 −1.00000 1.00000
1544.2 −0.500000 + 0.866025i 1.00000i 0 −0.500000 0.866025i −0.866025 0.500000i 1.00000i −1.00000 −1.00000 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
39.d odd 2 1 inner
91.g even 3 1 inner
273.x 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.x.b 4
3.b odd 2 1 3549.1.x.d 4
7.c even 3 1 3549.1.bp.d 4
13.b even 2 1 3549.1.x.d 4
13.c even 3 1 3549.1.w.c 4
13.c even 3 1 3549.1.bp.d 4
13.d odd 4 1 273.1.bm.b yes 4
13.d odd 4 1 3549.1.bm.c 4
13.e even 6 1 3549.1.w.e 4
13.e even 6 1 3549.1.bp.b 4
13.f odd 12 1 273.1.s.b 4
13.f odd 12 1 3549.1.s.b 4
13.f odd 12 1 3549.1.bk.c 4
13.f odd 12 1 3549.1.bk.d 4
21.h odd 6 1 3549.1.bp.b 4
39.d odd 2 1 inner 3549.1.x.b 4
39.f even 4 1 273.1.bm.b yes 4
39.f even 4 1 3549.1.bm.c 4
39.h odd 6 1 3549.1.w.c 4
39.h odd 6 1 3549.1.bp.d 4
39.i odd 6 1 3549.1.w.e 4
39.i odd 6 1 3549.1.bp.b 4
39.k even 12 1 273.1.s.b 4
39.k even 12 1 3549.1.s.b 4
39.k even 12 1 3549.1.bk.c 4
39.k even 12 1 3549.1.bk.d 4
91.g even 3 1 inner 3549.1.x.b 4
91.h even 3 1 3549.1.w.c 4
91.i even 4 1 1911.1.bm.b 4
91.k even 6 1 3549.1.w.e 4
91.r even 6 1 3549.1.bp.b 4
91.u even 6 1 3549.1.x.d 4
91.w even 12 1 1911.1.bm.b 4
91.x odd 12 1 1911.1.be.c 4
91.x odd 12 1 3549.1.bk.c 4
91.x odd 12 1 3549.1.bk.d 4
91.z odd 12 1 273.1.s.b 4
91.z odd 12 1 1911.1.be.c 4
91.z odd 12 1 3549.1.s.b 4
91.ba even 12 1 1911.1.be.d 4
91.bb even 12 1 1911.1.s.b 4
91.bb even 12 1 1911.1.be.d 4
91.bc even 12 1 1911.1.s.b 4
91.bd odd 12 1 273.1.bm.b yes 4
91.bd odd 12 1 3549.1.bm.c 4
273.o odd 4 1 1911.1.bm.b 4
273.s odd 6 1 3549.1.w.e 4
273.w odd 6 1 3549.1.bp.d 4
273.x odd 6 1 inner 3549.1.x.b 4
273.bm odd 6 1 3549.1.x.d 4
273.bp odd 6 1 3549.1.w.c 4
273.bs odd 12 1 1911.1.be.d 4
273.bv even 12 1 1911.1.be.c 4
273.bv even 12 1 3549.1.bk.c 4
273.bv even 12 1 3549.1.bk.d 4
273.bw even 12 1 273.1.bm.b yes 4
273.bw even 12 1 3549.1.bm.c 4
273.ca odd 12 1 1911.1.s.b 4
273.cb odd 12 1 1911.1.s.b 4
273.cb odd 12 1 1911.1.be.d 4
273.cd even 12 1 273.1.s.b 4
273.cd even 12 1 1911.1.be.c 4
273.cd even 12 1 3549.1.s.b 4
273.ch odd 12 1 1911.1.bm.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.s.b 4 13.f odd 12 1
273.1.s.b 4 39.k even 12 1
273.1.s.b 4 91.z odd 12 1
273.1.s.b 4 273.cd even 12 1
273.1.bm.b yes 4 13.d odd 4 1
273.1.bm.b yes 4 39.f even 4 1
273.1.bm.b yes 4 91.bd odd 12 1
273.1.bm.b yes 4 273.bw even 12 1
1911.1.s.b 4 91.bb even 12 1
1911.1.s.b 4 91.bc even 12 1
1911.1.s.b 4 273.ca odd 12 1
1911.1.s.b 4 273.cb odd 12 1
1911.1.be.c 4 91.x odd 12 1
1911.1.be.c 4 91.z odd 12 1
1911.1.be.c 4 273.bv even 12 1
1911.1.be.c 4 273.cd even 12 1
1911.1.be.d 4 91.ba even 12 1
1911.1.be.d 4 91.bb even 12 1
1911.1.be.d 4 273.bs odd 12 1
1911.1.be.d 4 273.cb odd 12 1
1911.1.bm.b 4 91.i even 4 1
1911.1.bm.b 4 91.w even 12 1
1911.1.bm.b 4 273.o odd 4 1
1911.1.bm.b 4 273.ch odd 12 1
3549.1.s.b 4 13.f odd 12 1
3549.1.s.b 4 39.k even 12 1
3549.1.s.b 4 91.z odd 12 1
3549.1.s.b 4 273.cd even 12 1
3549.1.w.c 4 13.c even 3 1
3549.1.w.c 4 39.h odd 6 1
3549.1.w.c 4 91.h even 3 1
3549.1.w.c 4 273.bp odd 6 1
3549.1.w.e 4 13.e even 6 1
3549.1.w.e 4 39.i odd 6 1
3549.1.w.e 4 91.k even 6 1
3549.1.w.e 4 273.s odd 6 1
3549.1.x.b 4 1.a even 1 1 trivial
3549.1.x.b 4 39.d odd 2 1 inner
3549.1.x.b 4 91.g even 3 1 inner
3549.1.x.b 4 273.x odd 6 1 inner
3549.1.x.d 4 3.b odd 2 1
3549.1.x.d 4 13.b even 2 1
3549.1.x.d 4 91.u even 6 1
3549.1.x.d 4 273.bm odd 6 1
3549.1.bk.c 4 13.f odd 12 1
3549.1.bk.c 4 39.k even 12 1
3549.1.bk.c 4 91.x odd 12 1
3549.1.bk.c 4 273.bv even 12 1
3549.1.bk.d 4 13.f odd 12 1
3549.1.bk.d 4 39.k even 12 1
3549.1.bk.d 4 91.x odd 12 1
3549.1.bk.d 4 273.bv even 12 1
3549.1.bm.c 4 13.d odd 4 1
3549.1.bm.c 4 39.f even 4 1
3549.1.bm.c 4 91.bd odd 12 1
3549.1.bm.c 4 273.bw even 12 1
3549.1.bp.b 4 13.e even 6 1
3549.1.bp.b 4 21.h odd 6 1
3549.1.bp.b 4 39.i odd 6 1
3549.1.bp.b 4 91.r even 6 1
3549.1.bp.d 4 7.c even 3 1
3549.1.bp.d 4 13.c even 3 1
3549.1.bp.d 4 39.h odd 6 1
3549.1.bp.d 4 273.w odd 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

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