Properties

Label 1280.2.a.i
Level $1280$
Weight $2$
Character orbit 1280.a
Self dual yes
Analytic conductor $10.221$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,2,Mod(1,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.a (trivial)

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: yes
Analytic conductor: \(10.2208514587\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} - 3 \beta q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{5} - 3 \beta q^{7} - q^{9} + 2 \beta q^{11} - 6 q^{13} + \beta q^{15} - 6 q^{17} - 6 q^{21} - 3 \beta q^{23} + q^{25} - 4 \beta q^{27} + 6 \beta q^{31} + 4 q^{33} - 3 \beta q^{35} - 6 q^{37} - 6 \beta q^{39} + 3 \beta q^{43} - q^{45} - 3 \beta q^{47} + 11 q^{49} - 6 \beta q^{51} - 6 q^{53} + 2 \beta q^{55} - 8 \beta q^{59} - 6 q^{61} + 3 \beta q^{63} - 6 q^{65} + 9 \beta q^{67} - 6 q^{69} + 6 \beta q^{71} + 2 q^{73} + \beta q^{75} - 12 q^{77} - 5 q^{81} + 7 \beta q^{83} - 6 q^{85} + 6 q^{89} + 18 \beta q^{91} + 12 q^{93} - 10 q^{97} - 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{9} - 12 q^{13} - 12 q^{17} - 12 q^{21} + 2 q^{25} + 8 q^{33} - 12 q^{37} - 2 q^{45} + 22 q^{49} - 12 q^{53} - 12 q^{61} - 12 q^{65} - 12 q^{69} + 4 q^{73} - 24 q^{77} - 10 q^{81} - 12 q^{85} + 12 q^{89} + 24 q^{93} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −1.41421 0 1.00000 0 4.24264 0 −1.00000 0
1.2 0 1.41421 0 1.00000 0 −4.24264 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.a.i 2
4.b odd 2 1 inner 1280.2.a.i 2
5.b even 2 1 6400.2.a.bp 2
8.b even 2 1 1280.2.a.g 2
8.d odd 2 1 1280.2.a.g 2
16.e even 4 2 640.2.d.b 4
16.f odd 4 2 640.2.d.b 4
20.d odd 2 1 6400.2.a.bp 2
40.e odd 2 1 6400.2.a.bk 2
40.f even 2 1 6400.2.a.bk 2
48.i odd 4 2 5760.2.k.v 4
48.k even 4 2 5760.2.k.v 4
80.i odd 4 2 3200.2.f.l 4
80.j even 4 2 3200.2.f.g 4
80.k odd 4 2 3200.2.d.v 4
80.q even 4 2 3200.2.d.v 4
80.s even 4 2 3200.2.f.l 4
80.t odd 4 2 3200.2.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.b 4 16.e even 4 2
640.2.d.b 4 16.f odd 4 2
1280.2.a.g 2 8.b even 2 1
1280.2.a.g 2 8.d odd 2 1
1280.2.a.i 2 1.a even 1 1 trivial
1280.2.a.i 2 4.b odd 2 1 inner
3200.2.d.v 4 80.k odd 4 2
3200.2.d.v 4 80.q even 4 2
3200.2.f.g 4 80.j even 4 2
3200.2.f.g 4 80.t odd 4 2
3200.2.f.l 4 80.i odd 4 2
3200.2.f.l 4 80.s even 4 2
5760.2.k.v 4 48.i odd 4 2
5760.2.k.v 4 48.k even 4 2
6400.2.a.bk 2 40.e odd 2 1
6400.2.a.bk 2 40.f even 2 1
6400.2.a.bp 2 5.b even 2 1
6400.2.a.bp 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1280))\):

\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 18 \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 18 \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 18 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 72 \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 18 \) Copy content Toggle raw display
$47$ \( T^{2} - 18 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 128 \) Copy content Toggle raw display
$61$ \( (T + 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 162 \) Copy content Toggle raw display
$71$ \( T^{2} - 72 \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 98 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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