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Properties

Label 504.1.ce.a
Level $504$
Weight $1$
Character orbit 504.ce
Analytic conductor $0.252$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,1,Mod(403,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.403"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 504.ce (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.251528766367\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.254016.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{12}^{3} q^{2} - q^{3} - q^{4} + \zeta_{12} q^{5} + \zeta_{12}^{3} q^{6} + \zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} + q^{9} - \zeta_{12}^{4} q^{10} + \zeta_{12}^{2} q^{11} + q^{12} + \zeta_{12}^{5} q^{13} + \cdots + \zeta_{12}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 2 q^{10} + 2 q^{11} + 4 q^{12} + 4 q^{14} + 4 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{26} - 4 q^{27} - 2 q^{30} - 2 q^{33} - 2 q^{35} - 4 q^{36} - 2 q^{40} + 2 q^{41} - 4 q^{42}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(-1\) \(-\zeta_{12}^{2}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
403.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
− 1.00000i −1.00000 −1.00000 0.866025 + 0.500000i 1.00000i 1.00000i 1.00000i 1.00000 0.500000 − 0.866025i
403.2 1.00000i −1.00000 −1.00000 −0.866025 − 0.500000i − 1.00000i − 1.00000i − 1.00000i 1.00000 0.500000 − 0.866025i
499.1 − 1.00000i −1.00000 −1.00000 −0.866025 + 0.500000i 1.00000i 1.00000i 1.00000i 1.00000 0.500000 + 0.866025i
499.2 1.00000i −1.00000 −1.00000 0.866025 − 0.500000i − 1.00000i − 1.00000i − 1.00000i 1.00000 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
63.h even 3 1 inner
504.ce odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.1.ce.a yes 4
3.b odd 2 1 1512.1.ce.a 4
4.b odd 2 1 2016.1.cm.a 4
7.b odd 2 1 3528.1.ce.c 4
7.c even 3 1 504.1.ba.a ✓ 4
7.c even 3 1 3528.1.cg.d 4
7.d odd 6 1 3528.1.ba.d 4
7.d odd 6 1 3528.1.cg.c 4
8.b even 2 1 2016.1.cm.a 4
8.d odd 2 1 inner 504.1.ce.a yes 4
9.c even 3 1 504.1.ba.a ✓ 4
9.d odd 6 1 1512.1.ba.a 4
21.h odd 6 1 1512.1.ba.a 4
24.f even 2 1 1512.1.ce.a 4
28.g odd 6 1 2016.1.bi.a 4
36.f odd 6 1 2016.1.bi.a 4
56.e even 2 1 3528.1.ce.c 4
56.k odd 6 1 504.1.ba.a ✓ 4
56.k odd 6 1 3528.1.cg.d 4
56.m even 6 1 3528.1.ba.d 4
56.m even 6 1 3528.1.cg.c 4
56.p even 6 1 2016.1.bi.a 4
63.g even 3 1 3528.1.cg.d 4
63.h even 3 1 inner 504.1.ce.a yes 4
63.j odd 6 1 1512.1.ce.a 4
63.k odd 6 1 3528.1.cg.c 4
63.l odd 6 1 3528.1.ba.d 4
63.t odd 6 1 3528.1.ce.c 4
72.l even 6 1 1512.1.ba.a 4
72.n even 6 1 2016.1.bi.a 4
72.p odd 6 1 504.1.ba.a ✓ 4
168.v even 6 1 1512.1.ba.a 4
252.u odd 6 1 2016.1.cm.a 4
504.ba odd 6 1 3528.1.cg.d 4
504.be even 6 1 3528.1.ba.d 4
504.bf even 6 1 3528.1.ce.c 4
504.bt even 6 1 1512.1.ce.a 4
504.ce odd 6 1 inner 504.1.ce.a yes 4
504.cq even 6 1 2016.1.cm.a 4
504.cz even 6 1 3528.1.cg.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.ba.a ✓ 4 7.c even 3 1
504.1.ba.a ✓ 4 9.c even 3 1
504.1.ba.a ✓ 4 56.k odd 6 1
504.1.ba.a ✓ 4 72.p odd 6 1
504.1.ce.a yes 4 1.a even 1 1 trivial
504.1.ce.a yes 4 8.d odd 2 1 inner
504.1.ce.a yes 4 63.h even 3 1 inner
504.1.ce.a yes 4 504.ce odd 6 1 inner
1512.1.ba.a 4 9.d odd 6 1
1512.1.ba.a 4 21.h odd 6 1
1512.1.ba.a 4 72.l even 6 1
1512.1.ba.a 4 168.v even 6 1
1512.1.ce.a 4 3.b odd 2 1
1512.1.ce.a 4 24.f even 2 1
1512.1.ce.a 4 63.j odd 6 1
1512.1.ce.a 4 504.bt even 6 1
2016.1.bi.a 4 28.g odd 6 1
2016.1.bi.a 4 36.f odd 6 1
2016.1.bi.a 4 56.p even 6 1
2016.1.bi.a 4 72.n even 6 1
2016.1.cm.a 4 4.b odd 2 1
2016.1.cm.a 4 8.b even 2 1
2016.1.cm.a 4 252.u odd 6 1
2016.1.cm.a 4 504.cq even 6 1
3528.1.ba.d 4 7.d odd 6 1
3528.1.ba.d 4 56.m even 6 1
3528.1.ba.d 4 63.l odd 6 1
3528.1.ba.d 4 504.be even 6 1
3528.1.ce.c 4 7.b odd 2 1
3528.1.ce.c 4 56.e even 2 1
3528.1.ce.c 4 63.t odd 6 1
3528.1.ce.c 4 504.bf even 6 1
3528.1.cg.c 4 7.d odd 6 1
3528.1.cg.c 4 56.m even 6 1
3528.1.cg.c 4 63.k odd 6 1
3528.1.cg.c 4 504.cz even 6 1
3528.1.cg.d 4 7.c even 3 1
3528.1.cg.d 4 56.k odd 6 1
3528.1.cg.d 4 63.g even 3 1
3528.1.cg.d 4 504.ba odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(504, [\chi])\).

Hecke characteristic polynomials

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