Properties

Label 260.3.l.a
Level $260$
Weight $3$
Character orbit 260.l
Analytic conductor $7.084$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [260,3,Mod(47,260)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 260.l (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: \(7.08448687337\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 4 q^{4} + (3 i - 4) q^{5} - 8 i q^{8} - 9 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 4 q^{4} + (3 i - 4) q^{5} - 8 i q^{8} - 9 i q^{9} + ( - 8 i - 6) q^{10} + ( - 5 i - 12) q^{13} + 16 q^{16} + ( - 7 i + 7) q^{17} + 18 q^{18} + ( - 12 i + 16) q^{20} + ( - 24 i + 7) q^{25} + ( - 24 i + 10) q^{26} - 42 i q^{29} + 32 i q^{32} + (14 i + 14) q^{34} + 36 i q^{36} - 70 q^{37} + (32 i + 24) q^{40} + ( - 31 i - 31) q^{41} + (36 i + 27) q^{45} + 49 q^{49} + (14 i + 48) q^{50} + (20 i + 48) q^{52} + (17 i - 17) q^{53} + 84 q^{58} - 120 q^{61} - 64 q^{64} + ( - 16 i + 63) q^{65} + (28 i - 28) q^{68} - 72 q^{72} + 96 i q^{73} - 140 i q^{74} + (48 i - 64) q^{80} - 81 q^{81} + ( - 62 i + 62) q^{82} + (49 i - 7) q^{85} + ( - 119 i - 119) q^{89} + (54 i - 72) q^{90} + 130 i q^{97} + 98 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 8 q^{5} - 12 q^{10} - 24 q^{13} + 32 q^{16} + 14 q^{17} + 36 q^{18} + 32 q^{20} + 14 q^{25} + 20 q^{26} + 28 q^{34} - 140 q^{37} + 48 q^{40} - 62 q^{41} + 54 q^{45} + 98 q^{49} + 96 q^{50} + 96 q^{52} - 34 q^{53} + 168 q^{58} - 240 q^{61} - 128 q^{64} + 126 q^{65} - 56 q^{68} - 144 q^{72} - 128 q^{80} - 162 q^{81} + 124 q^{82} - 14 q^{85} - 238 q^{89} - 144 q^{90}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.00000i
1.00000i
2.00000i 0 −4.00000 −4.00000 + 3.00000i 0 0 8.00000i 9.00000i −6.00000 8.00000i
83.1 2.00000i 0 −4.00000 −4.00000 3.00000i 0 0 8.00000i 9.00000i −6.00000 + 8.00000i
\(n\): 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.f even 4 1 inner
260.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.3.l.a 2
4.b odd 2 1 CM 260.3.l.a 2
5.c odd 4 1 260.3.s.b yes 2
13.d odd 4 1 260.3.s.b yes 2
20.e even 4 1 260.3.s.b yes 2
52.f even 4 1 260.3.s.b yes 2
65.f even 4 1 inner 260.3.l.a 2
260.l odd 4 1 inner 260.3.l.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.3.l.a 2 1.a even 1 1 trivial
260.3.l.a 2 4.b odd 2 1 CM
260.3.l.a 2 65.f even 4 1 inner
260.3.l.a 2 260.l odd 4 1 inner
260.3.s.b yes 2 5.c odd 4 1
260.3.s.b yes 2 13.d odd 4 1
260.3.s.b yes 2 20.e even 4 1
260.3.s.b yes 2 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(260, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{17}^{2} - 14T_{17} + 98 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 8T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 24T + 169 \) Copy content Toggle raw display
$17$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1764 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 70)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 62T + 1922 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 34T + 578 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 120)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 9216 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 238T + 28322 \) Copy content Toggle raw display
$97$ \( T^{2} + 16900 \) Copy content Toggle raw display
show more
show less