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Properties

Label 504.1.bw.a
Level $504$
Weight $1$
Character orbit 504.bw
Analytic conductor $0.252$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,1,Mod(325,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.325");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 504.bw (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: \(0.251528766367\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.9680832.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} + (\zeta_{12}^{3} + \zeta_{12}) q^{5} + \zeta_{12}^{4} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} + (\zeta_{12}^{3} + \zeta_{12}) q^{5} + \zeta_{12}^{4} q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} - 1) q^{10} - \zeta_{12} q^{11} - \zeta_{12}^{3} q^{14} - \zeta_{12}^{2} q^{16} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{20} + q^{22} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{25} + \zeta_{12}^{2} q^{28} - \zeta_{12}^{3} q^{29} + (\zeta_{12}^{2} + 1) q^{31} + \zeta_{12} q^{32} + (\zeta_{12}^{5} - \zeta_{12}) q^{35} + (\zeta_{12}^{4} - 1) q^{40} + \zeta_{12}^{5} q^{44} - \zeta_{12}^{2} q^{49} + ( - \zeta_{12}^{5} - \zeta_{12}^{3} - \zeta_{12}) q^{50} + \zeta_{12} q^{53} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{55} - \zeta_{12} q^{56} + \zeta_{12}^{2} q^{58} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{59} + (\zeta_{12}^{5} - \zeta_{12}) q^{62} - q^{64} + ( - \zeta_{12}^{4} + 1) q^{70} - \zeta_{12}^{5} q^{77} - \zeta_{12}^{2} q^{79} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{80} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{83} - \zeta_{12}^{4} q^{88} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{97} + \zeta_{12} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{7} - 6 q^{10} - 2 q^{16} + 4 q^{22} - 4 q^{25} + 2 q^{28} + 6 q^{31} - 6 q^{40} - 2 q^{49} + 2 q^{58} - 4 q^{64} + 6 q^{70} - 2 q^{79} + 2 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.866025 + 1.50000i 0 −0.500000 + 0.866025i 1.00000i 0 −1.50000 − 0.866025i
325.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.866025 − 1.50000i 0 −0.500000 + 0.866025i − 1.00000i 0 −1.50000 − 0.866025i
397.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.866025 − 1.50000i 0 −0.500000 − 0.866025i − 1.00000i 0 −1.50000 + 0.866025i
397.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.866025 + 1.50000i 0 −0.500000 − 0.866025i 1.00000i 0 −1.50000 + 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
56.j odd 6 1 inner
168.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.1.bw.a ✓ 4
3.b odd 2 1 inner 504.1.bw.a ✓ 4
4.b odd 2 1 2016.1.ce.a 4
7.b odd 2 1 3528.1.bw.c 4
7.c even 3 1 3528.1.l.a 4
7.c even 3 1 3528.1.bw.c 4
7.d odd 6 1 inner 504.1.bw.a ✓ 4
7.d odd 6 1 3528.1.l.a 4
8.b even 2 1 inner 504.1.bw.a ✓ 4
8.d odd 2 1 2016.1.ce.a 4
12.b even 2 1 2016.1.ce.a 4
21.c even 2 1 3528.1.bw.c 4
21.g even 6 1 inner 504.1.bw.a ✓ 4
21.g even 6 1 3528.1.l.a 4
21.h odd 6 1 3528.1.l.a 4
21.h odd 6 1 3528.1.bw.c 4
24.f even 2 1 2016.1.ce.a 4
24.h odd 2 1 CM 504.1.bw.a ✓ 4
28.f even 6 1 2016.1.ce.a 4
56.h odd 2 1 3528.1.bw.c 4
56.j odd 6 1 inner 504.1.bw.a ✓ 4
56.j odd 6 1 3528.1.l.a 4
56.m even 6 1 2016.1.ce.a 4
56.p even 6 1 3528.1.l.a 4
56.p even 6 1 3528.1.bw.c 4
84.j odd 6 1 2016.1.ce.a 4
168.i even 2 1 3528.1.bw.c 4
168.s odd 6 1 3528.1.l.a 4
168.s odd 6 1 3528.1.bw.c 4
168.ba even 6 1 inner 504.1.bw.a ✓ 4
168.ba even 6 1 3528.1.l.a 4
168.be odd 6 1 2016.1.ce.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.bw.a ✓ 4 1.a even 1 1 trivial
504.1.bw.a ✓ 4 3.b odd 2 1 inner
504.1.bw.a ✓ 4 7.d odd 6 1 inner
504.1.bw.a ✓ 4 8.b even 2 1 inner
504.1.bw.a ✓ 4 21.g even 6 1 inner
504.1.bw.a ✓ 4 24.h odd 2 1 CM
504.1.bw.a ✓ 4 56.j odd 6 1 inner
504.1.bw.a ✓ 4 168.ba even 6 1 inner
2016.1.ce.a 4 4.b odd 2 1
2016.1.ce.a 4 8.d odd 2 1
2016.1.ce.a 4 12.b even 2 1
2016.1.ce.a 4 24.f even 2 1
2016.1.ce.a 4 28.f even 6 1
2016.1.ce.a 4 56.m even 6 1
2016.1.ce.a 4 84.j odd 6 1
2016.1.ce.a 4 168.be odd 6 1
3528.1.l.a 4 7.c even 3 1
3528.1.l.a 4 7.d odd 6 1
3528.1.l.a 4 21.g even 6 1
3528.1.l.a 4 21.h odd 6 1
3528.1.l.a 4 56.j odd 6 1
3528.1.l.a 4 56.p even 6 1
3528.1.l.a 4 168.s odd 6 1
3528.1.l.a 4 168.ba even 6 1
3528.1.bw.c 4 7.b odd 2 1
3528.1.bw.c 4 7.c even 3 1
3528.1.bw.c 4 21.c even 2 1
3528.1.bw.c 4 21.h odd 6 1
3528.1.bw.c 4 56.h odd 2 1
3528.1.bw.c 4 56.p even 6 1
3528.1.bw.c 4 168.i even 2 1
3528.1.bw.c 4 168.s odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

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