Properties

Label 2523.1.b.b
Level $2523$
Weight $1$
Character orbit 2523.b
Analytic conductor $1.259$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -87
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2523,1,Mod(842,2523)]
 
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, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2523.842");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2523 = 3 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2523.b (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: \(1.25914102687\)
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: no (minimal twist has level 87)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.87.1
Artin image: $C_4\times S_3$
Artin field: Galois closure of 12.0.1175078824045389.1

$q$-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.

\(f(q)\) \(=\) \( q - i q^{2} + i q^{3} + q^{6} - q^{7} - i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} + i q^{3} + q^{6} - q^{7} - i q^{8} - q^{9} - i q^{11} + q^{13} + i q^{14} - q^{16} - i q^{17} + i q^{18} - i q^{21} - q^{22} + q^{24} + q^{25} - i q^{26} - i q^{27} + q^{33} - q^{34} + i q^{39} - 2 i q^{41} - q^{42} + i q^{47} - i q^{48} - i q^{50} + q^{51} - q^{54} + i q^{56} + q^{63} - q^{64} - i q^{66} + q^{67} + i q^{72} + i q^{75} + i q^{77} + q^{78} + q^{81} - 2 q^{82} - q^{88} - i q^{89} - q^{91} + q^{94} + i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{6} - 2 q^{7} - 2 q^{9} + 2 q^{13} - 2 q^{16} - 2 q^{22} + 2 q^{24} + 2 q^{25} + 2 q^{33} - 2 q^{34} - 2 q^{42} + 2 q^{51} - 2 q^{54} + 2 q^{63} - 2 q^{64} + 2 q^{67} + 2 q^{78} + 2 q^{81} - 4 q^{82} - 2 q^{88} - 2 q^{91} + 2 q^{94}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
842.1
1.00000i
1.00000i
1.00000i 1.00000i 0 0 1.00000 −1.00000 1.00000i −1.00000 0
842.2 1.00000i 1.00000i 0 0 1.00000 −1.00000 1.00000i −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
87.d odd 2 1 CM by \(\Q(\sqrt{-87}) \)
3.b odd 2 1 inner
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2523.1.b.b 2
3.b odd 2 1 inner 2523.1.b.b 2
29.b even 2 1 inner 2523.1.b.b 2
29.c odd 4 1 87.1.d.a 1
29.c odd 4 1 87.1.d.b yes 1
29.d even 7 6 2523.1.j.b 12
29.e even 14 6 2523.1.j.b 12
29.f odd 28 6 2523.1.h.a 6
29.f odd 28 6 2523.1.h.b 6
87.d odd 2 1 CM 2523.1.b.b 2
87.f even 4 1 87.1.d.a 1
87.f even 4 1 87.1.d.b yes 1
87.h odd 14 6 2523.1.j.b 12
87.j odd 14 6 2523.1.j.b 12
87.k even 28 6 2523.1.h.a 6
87.k even 28 6 2523.1.h.b 6
116.e even 4 1 1392.1.i.a 1
116.e even 4 1 1392.1.i.b 1
145.e even 4 1 2175.1.b.a 2
145.e even 4 1 2175.1.b.b 2
145.f odd 4 1 2175.1.h.a 1
145.f odd 4 1 2175.1.h.b 1
145.j even 4 1 2175.1.b.a 2
145.j even 4 1 2175.1.b.b 2
261.l even 12 2 2349.1.h.a 2
261.l even 12 2 2349.1.h.b 2
261.m odd 12 2 2349.1.h.a 2
261.m odd 12 2 2349.1.h.b 2
348.k odd 4 1 1392.1.i.a 1
348.k odd 4 1 1392.1.i.b 1
435.i odd 4 1 2175.1.b.a 2
435.i odd 4 1 2175.1.b.b 2
435.l even 4 1 2175.1.h.a 1
435.l even 4 1 2175.1.h.b 1
435.t odd 4 1 2175.1.b.a 2
435.t odd 4 1 2175.1.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 29.c odd 4 1
87.1.d.a 1 87.f even 4 1
87.1.d.b yes 1 29.c odd 4 1
87.1.d.b yes 1 87.f even 4 1
1392.1.i.a 1 116.e even 4 1
1392.1.i.a 1 348.k odd 4 1
1392.1.i.b 1 116.e even 4 1
1392.1.i.b 1 348.k odd 4 1
2175.1.b.a 2 145.e even 4 1
2175.1.b.a 2 145.j even 4 1
2175.1.b.a 2 435.i odd 4 1
2175.1.b.a 2 435.t odd 4 1
2175.1.b.b 2 145.e even 4 1
2175.1.b.b 2 145.j even 4 1
2175.1.b.b 2 435.i odd 4 1
2175.1.b.b 2 435.t odd 4 1
2175.1.h.a 1 145.f odd 4 1
2175.1.h.a 1 435.l even 4 1
2175.1.h.b 1 145.f odd 4 1
2175.1.h.b 1 435.l even 4 1
2349.1.h.a 2 261.l even 12 2
2349.1.h.a 2 261.m odd 12 2
2349.1.h.b 2 261.l even 12 2
2349.1.h.b 2 261.m odd 12 2
2523.1.b.b 2 1.a even 1 1 trivial
2523.1.b.b 2 3.b odd 2 1 inner
2523.1.b.b 2 29.b even 2 1 inner
2523.1.b.b 2 87.d odd 2 1 CM
2523.1.h.a 6 29.f odd 28 6
2523.1.h.a 6 87.k even 28 6
2523.1.h.b 6 29.f odd 28 6
2523.1.h.b 6 87.k even 28 6
2523.1.j.b 12 29.d even 7 6
2523.1.j.b 12 29.e even 14 6
2523.1.j.b 12 87.h odd 14 6
2523.1.j.b 12 87.j odd 14 6

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2523, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

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