Properties

Label 2523.1.d.a
Level $2523$
Weight $1$
Character orbit 2523.d
Analytic conductor $1.259$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2523,1,Mod(2522,2523)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2523, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2523.2522");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2523 = 3 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2523.d (of order \(2\), degree \(1\), 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: \(1.25914102687\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.6365529.1

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - q^{4} + \beta_{2} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - q^{4} + \beta_{2} q^{7} - q^{9} - \beta_{3} q^{12} + (\beta_{2} + 1) q^{13} + q^{16} + ( - \beta_{3} - \beta_1) q^{19} + \beta_1 q^{21} + q^{25} - \beta_{3} q^{27} - \beta_{2} q^{28} + \beta_1 q^{31} + q^{36} + (\beta_{3} + \beta_1) q^{37} + (\beta_{3} + \beta_1) q^{39} + \beta_1 q^{43} + \beta_{3} q^{48} - \beta_{2} q^{49} + ( - \beta_{2} - 1) q^{52} + (\beta_{2} + 1) q^{57} + \beta_1 q^{61} - \beta_{2} q^{63} - q^{64} + (\beta_{2} + 1) q^{67} + (\beta_{3} + \beta_1) q^{73} + \beta_{3} q^{75} + (\beta_{3} + \beta_1) q^{76} + ( - \beta_{3} - \beta_1) q^{79} + q^{81} - \beta_1 q^{84} + q^{91} - \beta_{2} q^{93} - \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 2 q^{7} - 4 q^{9} + 2 q^{13} + 4 q^{16} + 4 q^{25} + 2 q^{28} + 4 q^{36} + 2 q^{49} - 2 q^{52} + 2 q^{57} + 2 q^{63} - 4 q^{64} + 2 q^{67} + 4 q^{81} + 4 q^{91} + 2 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2523\mathbb{Z}\right)^\times\).

\(n\) \(842\) \(1684\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2522.1
1.61803i
0.618034i
1.61803i
0.618034i
0 1.00000i −1.00000 0 0 −1.61803 0 −1.00000 0
2522.2 0 1.00000i −1.00000 0 0 0.618034 0 −1.00000 0
2522.3 0 1.00000i −1.00000 0 0 −1.61803 0 −1.00000 0
2522.4 0 1.00000i −1.00000 0 0 0.618034 0 −1.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
29.b even 2 1 inner
87.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2523.1.d.a 4
3.b odd 2 1 CM 2523.1.d.a 4
29.b even 2 1 inner 2523.1.d.a 4
29.c odd 4 1 2523.1.b.a 2
29.c odd 4 1 2523.1.b.c yes 2
29.d even 7 6 2523.1.h.c 24
29.e even 14 6 2523.1.h.c 24
29.f odd 28 6 2523.1.j.a 12
29.f odd 28 6 2523.1.j.c 12
87.d odd 2 1 inner 2523.1.d.a 4
87.f even 4 1 2523.1.b.a 2
87.f even 4 1 2523.1.b.c yes 2
87.h odd 14 6 2523.1.h.c 24
87.j odd 14 6 2523.1.h.c 24
87.k even 28 6 2523.1.j.a 12
87.k even 28 6 2523.1.j.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2523.1.b.a 2 29.c odd 4 1
2523.1.b.a 2 87.f even 4 1
2523.1.b.c yes 2 29.c odd 4 1
2523.1.b.c yes 2 87.f even 4 1
2523.1.d.a 4 1.a even 1 1 trivial
2523.1.d.a 4 3.b odd 2 1 CM
2523.1.d.a 4 29.b even 2 1 inner
2523.1.d.a 4 87.d odd 2 1 inner
2523.1.h.c 24 29.d even 7 6
2523.1.h.c 24 29.e even 14 6
2523.1.h.c 24 87.h odd 14 6
2523.1.h.c 24 87.j odd 14 6
2523.1.j.a 12 29.f odd 28 6
2523.1.j.a 12 87.k even 28 6
2523.1.j.c 12 29.f odd 28 6
2523.1.j.c 12 87.k even 28 6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2523, [\chi])\).

Hecke characteristic polynomials

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