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Properties

Label 504.1.l.b
Level $504$
Weight $1$
Character orbit 504.l
Analytic conductor $0.252$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -7, -24, 168
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,1,Mod(181,504)]
 
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([0, 1, 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("504.181");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 504.l (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: \(0.251528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-6}, \sqrt{-7})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.112021056.1

$q$-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.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + q^{7} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + q^{7} + i q^{8} - 2 i q^{11} - i q^{14} + q^{16} - 2 q^{22} - q^{25} - q^{28} + 2 i q^{29} - i q^{32} + 2 i q^{44} + q^{49} + i q^{50} + 2 i q^{53} + i q^{56} + 2 q^{58} - q^{64} - 2 i q^{77} - 2 q^{79} + 2 q^{88} - i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{7} + 2 q^{16} - 4 q^{22} - 2 q^{25} - 2 q^{28} + 2 q^{49} + 4 q^{58} - 2 q^{64} - 4 q^{79} + 4 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.00000i
− 1.00000i
− 1.00000i 0 −1.00000 0 0 1.00000 1.00000i 0 0
181.2 1.00000i 0 −1.00000 0 0 1.00000 − 1.00000i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
168.i even 2 1 RM by \(\Q(\sqrt{42}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
56.h odd 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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