Properties

Label 260.1.s.a
Level $260$
Weight $1$
Character orbit 260.s
Analytic conductor $0.130$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [260,1,Mod(187,260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(260, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.187");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 260.s (of order \(4\), 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: \(0.129756903285\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.1098500.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.0.70304000.3

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + i q^{5} - q^{8} + i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + i q^{5} - q^{8} + i q^{9} - i q^{10} - i q^{13} + q^{16} + (i + 1) q^{17} - i q^{18} + i q^{20} - q^{25} + i q^{26} - i q^{29} - q^{32} + ( - i - 1) q^{34} + i q^{36} - i q^{40} + ( - i - 1) q^{41} - q^{45} - q^{49} + q^{50} - i q^{52} + ( - i - 1) q^{53} + 2 i q^{58} + q^{64} + q^{65} + (i + 1) q^{68} - i q^{72} + q^{73} + i q^{80} - q^{81} + (i + 1) q^{82} + (i - 1) q^{85} + (i + 1) q^{89} + q^{90} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{16} + 2 q^{17} - 2 q^{25} - 2 q^{32} - 2 q^{34} - 2 q^{41} - 2 q^{45} - 2 q^{49} + 2 q^{50} - 2 q^{53} + 2 q^{64} + 2 q^{65} + 2 q^{68} + 4 q^{73} - 2 q^{81} + 2 q^{82} - 2 q^{85} + 2 q^{89} + 2 q^{90} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(131\) \(157\)
\(\chi(n)\) \(i\) \(-1\) \(-i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
187.1
1.00000i
1.00000i
−1.00000 0 1.00000 1.00000i 0 0 −1.00000 1.00000i 1.00000i
203.1 −1.00000 0 1.00000 1.00000i 0 0 −1.00000 1.00000i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
65.k even 4 1 inner
260.s odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.1.s.a yes 2
3.b odd 2 1 2340.1.bo.c 2
4.b odd 2 1 CM 260.1.s.a yes 2
5.b even 2 1 1300.1.s.a 2
5.c odd 4 1 260.1.l.a 2
5.c odd 4 1 1300.1.l.a 2
12.b even 2 1 2340.1.bo.c 2
13.b even 2 1 3380.1.s.a 2
13.c even 3 2 3380.1.be.d 4
13.d odd 4 1 260.1.l.a 2
13.d odd 4 1 3380.1.l.a 2
13.e even 6 2 3380.1.be.a 4
13.f odd 12 2 3380.1.bl.a 4
13.f odd 12 2 3380.1.bl.e 4
15.e even 4 1 2340.1.v.a 2
20.d odd 2 1 1300.1.s.a 2
20.e even 4 1 260.1.l.a 2
20.e even 4 1 1300.1.l.a 2
39.f even 4 1 2340.1.v.a 2
52.b odd 2 1 3380.1.s.a 2
52.f even 4 1 260.1.l.a 2
52.f even 4 1 3380.1.l.a 2
52.i odd 6 2 3380.1.be.a 4
52.j odd 6 2 3380.1.be.d 4
52.l even 12 2 3380.1.bl.a 4
52.l even 12 2 3380.1.bl.e 4
60.l odd 4 1 2340.1.v.a 2
65.f even 4 1 1300.1.s.a 2
65.f even 4 1 3380.1.s.a 2
65.g odd 4 1 1300.1.l.a 2
65.h odd 4 1 3380.1.l.a 2
65.k even 4 1 inner 260.1.s.a yes 2
65.o even 12 2 3380.1.be.d 4
65.q odd 12 2 3380.1.bl.e 4
65.r odd 12 2 3380.1.bl.a 4
65.t even 12 2 3380.1.be.a 4
156.l odd 4 1 2340.1.v.a 2
195.j odd 4 1 2340.1.bo.c 2
260.l odd 4 1 1300.1.s.a 2
260.l odd 4 1 3380.1.s.a 2
260.p even 4 1 3380.1.l.a 2
260.s odd 4 1 inner 260.1.s.a yes 2
260.u even 4 1 1300.1.l.a 2
260.be odd 12 2 3380.1.be.d 4
260.bg even 12 2 3380.1.bl.a 4
260.bj even 12 2 3380.1.bl.e 4
260.bl odd 12 2 3380.1.be.a 4
780.bn even 4 1 2340.1.bo.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.1.l.a 2 5.c odd 4 1
260.1.l.a 2 13.d odd 4 1
260.1.l.a 2 20.e even 4 1
260.1.l.a 2 52.f even 4 1
260.1.s.a yes 2 1.a even 1 1 trivial
260.1.s.a yes 2 4.b odd 2 1 CM
260.1.s.a yes 2 65.k even 4 1 inner
260.1.s.a yes 2 260.s odd 4 1 inner
1300.1.l.a 2 5.c odd 4 1
1300.1.l.a 2 20.e even 4 1
1300.1.l.a 2 65.g odd 4 1
1300.1.l.a 2 260.u even 4 1
1300.1.s.a 2 5.b even 2 1
1300.1.s.a 2 20.d odd 2 1
1300.1.s.a 2 65.f even 4 1
1300.1.s.a 2 260.l odd 4 1
2340.1.v.a 2 15.e even 4 1
2340.1.v.a 2 39.f even 4 1
2340.1.v.a 2 60.l odd 4 1
2340.1.v.a 2 156.l odd 4 1
2340.1.bo.c 2 3.b odd 2 1
2340.1.bo.c 2 12.b even 2 1
2340.1.bo.c 2 195.j odd 4 1
2340.1.bo.c 2 780.bn even 4 1
3380.1.l.a 2 13.d odd 4 1
3380.1.l.a 2 52.f even 4 1
3380.1.l.a 2 65.h odd 4 1
3380.1.l.a 2 260.p even 4 1
3380.1.s.a 2 13.b even 2 1
3380.1.s.a 2 52.b odd 2 1
3380.1.s.a 2 65.f even 4 1
3380.1.s.a 2 260.l odd 4 1
3380.1.be.a 4 13.e even 6 2
3380.1.be.a 4 52.i odd 6 2
3380.1.be.a 4 65.t even 12 2
3380.1.be.a 4 260.bl odd 12 2
3380.1.be.d 4 13.c even 3 2
3380.1.be.d 4 52.j odd 6 2
3380.1.be.d 4 65.o even 12 2
3380.1.be.d 4 260.be odd 12 2
3380.1.bl.a 4 13.f odd 12 2
3380.1.bl.a 4 52.l even 12 2
3380.1.bl.a 4 65.r odd 12 2
3380.1.bl.a 4 260.bg even 12 2
3380.1.bl.e 4 13.f odd 12 2
3380.1.bl.e 4 52.l even 12 2
3380.1.bl.e 4 65.q odd 12 2
3380.1.bl.e 4 260.bj even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(260, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 2 \) 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
$41$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 2 \) 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
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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