Properties

Label 1280.2.c.a
Level $1280$
Weight $2$
Character orbit 1280.c
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
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] = [1280,2,Mod(769,1280)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.c (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: \(10.2208514587\)
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: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

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

\(f(q)\) \(=\) \( q + (i - 2) q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (i - 2) q^{5} + 3 q^{9} - 6 i q^{13} + 8 i q^{17} + ( - 4 i + 3) q^{25} + 4 q^{29} + 2 i q^{37} + 10 q^{41} + (3 i - 6) q^{45} + 7 q^{49} + 14 i q^{53} + 12 q^{61} + (12 i + 6) q^{65} + 16 i q^{73} + 9 q^{81} + ( - 16 i - 8) q^{85} + 10 q^{89} - 8 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} + 6 q^{9} + 6 q^{25} + 8 q^{29} + 20 q^{41} - 12 q^{45} + 14 q^{49} + 24 q^{61} + 12 q^{65} + 18 q^{81} - 16 q^{85} + 20 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
1.00000i
1.00000i
0 0 0 −2.00000 1.00000i 0 0 0 3.00000 0
769.2 0 0 0 −2.00000 + 1.00000i 0 0 0 3.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d 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.c.a 2
4.b odd 2 1 CM 1280.2.c.a 2
5.b even 2 1 inner 1280.2.c.a 2
5.c odd 4 1 6400.2.a.j 1
5.c odd 4 1 6400.2.a.o 1
8.b even 2 1 1280.2.c.d 2
8.d odd 2 1 1280.2.c.d 2
16.e even 4 1 640.2.f.a 2
16.e even 4 1 640.2.f.b yes 2
16.f odd 4 1 640.2.f.a 2
16.f odd 4 1 640.2.f.b yes 2
20.d odd 2 1 inner 1280.2.c.a 2
20.e even 4 1 6400.2.a.j 1
20.e even 4 1 6400.2.a.o 1
40.e odd 2 1 1280.2.c.d 2
40.f even 2 1 1280.2.c.d 2
40.i odd 4 1 6400.2.a.k 1
40.i odd 4 1 6400.2.a.n 1
40.k even 4 1 6400.2.a.k 1
40.k even 4 1 6400.2.a.n 1
80.i odd 4 1 3200.2.d.d 2
80.i odd 4 1 3200.2.d.f 2
80.j even 4 1 3200.2.d.d 2
80.j even 4 1 3200.2.d.f 2
80.k odd 4 1 640.2.f.a 2
80.k odd 4 1 640.2.f.b yes 2
80.q even 4 1 640.2.f.a 2
80.q even 4 1 640.2.f.b yes 2
80.s even 4 1 3200.2.d.d 2
80.s even 4 1 3200.2.d.f 2
80.t odd 4 1 3200.2.d.d 2
80.t odd 4 1 3200.2.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.a 2 16.e even 4 1
640.2.f.a 2 16.f odd 4 1
640.2.f.a 2 80.k odd 4 1
640.2.f.a 2 80.q even 4 1
640.2.f.b yes 2 16.e even 4 1
640.2.f.b yes 2 16.f odd 4 1
640.2.f.b yes 2 80.k odd 4 1
640.2.f.b yes 2 80.q even 4 1
1280.2.c.a 2 1.a even 1 1 trivial
1280.2.c.a 2 4.b odd 2 1 CM
1280.2.c.a 2 5.b even 2 1 inner
1280.2.c.a 2 20.d odd 2 1 inner
1280.2.c.d 2 8.b even 2 1
1280.2.c.d 2 8.d odd 2 1
1280.2.c.d 2 40.e odd 2 1
1280.2.c.d 2 40.f even 2 1
3200.2.d.d 2 80.i odd 4 1
3200.2.d.d 2 80.j even 4 1
3200.2.d.d 2 80.s even 4 1
3200.2.d.d 2 80.t odd 4 1
3200.2.d.f 2 80.i odd 4 1
3200.2.d.f 2 80.j even 4 1
3200.2.d.f 2 80.s even 4 1
3200.2.d.f 2 80.t odd 4 1
6400.2.a.j 1 5.c odd 4 1
6400.2.a.j 1 20.e even 4 1
6400.2.a.k 1 40.i odd 4 1
6400.2.a.k 1 40.k even 4 1
6400.2.a.n 1 40.i odd 4 1
6400.2.a.n 1 40.k even 4 1
6400.2.a.o 1 5.c odd 4 1
6400.2.a.o 1 20.e even 4 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{29} - 4 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 64 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 196 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 256 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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