Properties

Label 504.2.p.c
Level $504$
Weight $2$
Character orbit 504.p
Analytic conductor $4.024$
Analytic rank $0$
Dimension $4$
CM discriminant -168
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(307,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.p (of order \(2\), degree \(1\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 2 q^{4} - \beta_{3} q^{7} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 2 q^{4} - \beta_{3} q^{7} - 2 \beta_1 q^{8} - 2 \beta_{3} q^{13} - \beta_{2} q^{14} + 4 q^{16} - 2 \beta_{2} q^{17} + 2 \beta_1 q^{23} - 5 q^{25} - 2 \beta_{2} q^{26} + 2 \beta_{3} q^{28} - 4 \beta_1 q^{29} - 2 \beta_{3} q^{31} + 4 \beta_1 q^{32} + 4 \beta_{3} q^{34} - 2 \beta_{2} q^{41} + 2 q^{43} - 4 q^{46} + 7 q^{49} - 5 \beta_1 q^{50} + 4 \beta_{3} q^{52} + 8 \beta_1 q^{53} + 2 \beta_{2} q^{56} + 8 q^{58} + 4 \beta_{2} q^{59} - 2 \beta_{3} q^{61} - 2 \beta_{2} q^{62} - 8 q^{64} - 10 q^{67} + 4 \beta_{2} q^{68} - 10 \beta_1 q^{71} + 4 \beta_{3} q^{82} + 4 \beta_{2} q^{83} + 2 \beta_1 q^{86} - 2 \beta_{2} q^{89} + 14 q^{91} - 4 \beta_1 q^{92} + 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 8 q^{43} - 16 q^{46} + 28 q^{49} + 32 q^{58} - 32 q^{64} - 40 q^{67} + 56 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 8x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.16372i
2.57794i
1.16372i
2.57794i
1.41421i 0 −2.00000 0 0 −2.64575 2.82843i 0 0
307.2 1.41421i 0 −2.00000 0 0 2.64575 2.82843i 0 0
307.3 1.41421i 0 −2.00000 0 0 −2.64575 2.82843i 0 0
307.4 1.41421i 0 −2.00000 0 0 2.64575 2.82843i 0 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
168.e odd 2 1 CM by \(\Q(\sqrt{-42}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
8.d odd 2 1 inner
21.c even 2 1 inner
24.f even 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.p.c 4
3.b odd 2 1 inner 504.2.p.c 4
4.b odd 2 1 2016.2.p.c 4
7.b odd 2 1 inner 504.2.p.c 4
8.b even 2 1 2016.2.p.c 4
8.d odd 2 1 inner 504.2.p.c 4
12.b even 2 1 2016.2.p.c 4
21.c even 2 1 inner 504.2.p.c 4
24.f even 2 1 inner 504.2.p.c 4
24.h odd 2 1 2016.2.p.c 4
28.d even 2 1 2016.2.p.c 4
56.e even 2 1 inner 504.2.p.c 4
56.h odd 2 1 2016.2.p.c 4
84.h odd 2 1 2016.2.p.c 4
168.e odd 2 1 CM 504.2.p.c 4
168.i even 2 1 2016.2.p.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.p.c 4 1.a even 1 1 trivial
504.2.p.c 4 3.b odd 2 1 inner
504.2.p.c 4 7.b odd 2 1 inner
504.2.p.c 4 8.d odd 2 1 inner
504.2.p.c 4 21.c even 2 1 inner
504.2.p.c 4 24.f even 2 1 inner
504.2.p.c 4 56.e even 2 1 inner
504.2.p.c 4 168.e odd 2 1 CM
2016.2.p.c 4 4.b odd 2 1
2016.2.p.c 4 8.b even 2 1
2016.2.p.c 4 12.b even 2 1
2016.2.p.c 4 24.h odd 2 1
2016.2.p.c 4 28.d even 2 1
2016.2.p.c 4 56.h odd 2 1
2016.2.p.c 4 84.h odd 2 1
2016.2.p.c 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 56)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 56)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 224)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$67$ \( (T + 10)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 200)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 224)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 56)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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