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Properties

Label 1280.2.c.f
Level $1280$
Weight $2$
Character orbit 1280.c
Analytic conductor $10.221$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{3} q^{5} + 3 \beta_1 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{3} q^{5} + 3 \beta_1 q^{7} - 7 q^{9} + 5 \beta_1 q^{15} - 6 \beta_{3} q^{21} + \beta_1 q^{23} + 5 q^{25} - 4 \beta_{2} q^{27} - 4 \beta_{3} q^{29} + 3 \beta_{2} q^{35} + 12 q^{41} - \beta_{2} q^{43} - 7 \beta_{3} q^{45} + 7 \beta_1 q^{47} - 11 q^{49} + 6 \beta_{3} q^{61} - 21 \beta_1 q^{63} - 5 \beta_{2} q^{67} - 2 \beta_{3} q^{69} + 5 \beta_{2} q^{75} + 19 q^{81} + 3 \beta_{2} q^{83} - 20 \beta_1 q^{87} + 6 q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{9} + 20 q^{25} + 48 q^{41} - 44 q^{49} + 76 q^{81} + 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 6x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{2} + 4\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
− 2.28825i
− 0.874032i
2.28825i
0.874032i
0 − 3.16228i 0 −2.23607 0 4.24264i 0 −7.00000 0
769.2 0 − 3.16228i 0 2.23607 0 − 4.24264i 0 −7.00000 0
769.3 0 3.16228i 0 −2.23607 0 − 4.24264i 0 −7.00000 0
769.4 0 3.16228i 0 2.23607 0 4.24264i 0 −7.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 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.f 4
4.b odd 2 1 inner 1280.2.c.f 4
5.b even 2 1 inner 1280.2.c.f 4
5.c odd 4 2 6400.2.a.cv 4
8.b even 2 1 inner 1280.2.c.f 4
8.d odd 2 1 inner 1280.2.c.f 4
16.e even 4 2 640.2.f.f ✓ 4
16.f odd 4 2 640.2.f.f ✓ 4
20.d odd 2 1 CM 1280.2.c.f 4
20.e even 4 2 6400.2.a.cv 4
40.e odd 2 1 inner 1280.2.c.f 4
40.f even 2 1 inner 1280.2.c.f 4
40.i odd 4 2 6400.2.a.cv 4
40.k even 4 2 6400.2.a.cv 4
80.i odd 4 2 3200.2.d.i 4
80.j even 4 2 3200.2.d.i 4
80.k odd 4 2 640.2.f.f ✓ 4
80.q even 4 2 640.2.f.f ✓ 4
80.s even 4 2 3200.2.d.i 4
80.t odd 4 2 3200.2.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.f ✓ 4 16.e even 4 2
640.2.f.f ✓ 4 16.f odd 4 2
640.2.f.f ✓ 4 80.k odd 4 2
640.2.f.f ✓ 4 80.q even 4 2
1280.2.c.f 4 1.a even 1 1 trivial
1280.2.c.f 4 4.b odd 2 1 inner
1280.2.c.f 4 5.b even 2 1 inner
1280.2.c.f 4 8.b even 2 1 inner
1280.2.c.f 4 8.d odd 2 1 inner
1280.2.c.f 4 20.d odd 2 1 CM
1280.2.c.f 4 40.e odd 2 1 inner
1280.2.c.f 4 40.f even 2 1 inner
3200.2.d.i 4 80.i odd 4 2
3200.2.d.i 4 80.j even 4 2
3200.2.d.i 4 80.s even 4 2
3200.2.d.i 4 80.t odd 4 2
6400.2.a.cv 4 5.c odd 4 2
6400.2.a.cv 4 20.e even 4 2
6400.2.a.cv 4 40.i odd 4 2
6400.2.a.cv 4 40.k even 4 2

Hecke kernels

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

\( T_{3}^{2} + 10 \) Copy content Toggle raw display
\( T_{7}^{2} + 18 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{29}^{2} - 80 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T - 12)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 250)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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