Properties

Label 1280.4.d.f
Level $1280$
Weight $4$
Character orbit 1280.d
Analytic conductor $75.522$
Analytic rank $0$
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] = [1280,4,Mod(641,1280)]
 
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, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.641");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.d (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: \(75.5224448073\)
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 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{3} - 5 i q^{5} - 6 q^{7} + 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{3} - 5 i q^{5} - 6 q^{7} + 23 q^{9} + 60 i q^{11} - 50 i q^{13} + 10 q^{15} - 30 q^{17} - 40 i q^{19} - 12 i q^{21} - 178 q^{23} - 25 q^{25} + 100 i q^{27} - 166 i q^{29} + 20 q^{31} - 120 q^{33} + 30 i q^{35} + 10 i q^{37} + 100 q^{39} + 250 q^{41} + 142 i q^{43} - 115 i q^{45} + 214 q^{47} - 307 q^{49} - 60 i q^{51} + 490 i q^{53} + 300 q^{55} + 80 q^{57} - 800 i q^{59} - 250 i q^{61} - 138 q^{63} - 250 q^{65} + 774 i q^{67} - 356 i q^{69} - 100 q^{71} + 230 q^{73} - 50 i q^{75} - 360 i q^{77} - 1320 q^{79} + 421 q^{81} - 982 i q^{83} + 150 i q^{85} + 332 q^{87} - 874 q^{89} + 300 i q^{91} + 40 i q^{93} - 200 q^{95} - 310 q^{97} + 1380 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{7} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{7} + 46 q^{9} + 20 q^{15} - 60 q^{17} - 356 q^{23} - 50 q^{25} + 40 q^{31} - 240 q^{33} + 200 q^{39} + 500 q^{41} + 428 q^{47} - 614 q^{49} + 600 q^{55} + 160 q^{57} - 276 q^{63} - 500 q^{65} - 200 q^{71} + 460 q^{73} - 2640 q^{79} + 842 q^{81} + 664 q^{87} - 1748 q^{89} - 400 q^{95} - 620 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
1.00000i
1.00000i
0 2.00000i 0 5.00000i 0 −6.00000 0 23.0000 0
641.2 0 2.00000i 0 5.00000i 0 −6.00000 0 23.0000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.4.d.f 2
4.b odd 2 1 1280.4.d.k 2
8.b even 2 1 inner 1280.4.d.f 2
8.d odd 2 1 1280.4.d.k 2
16.e even 4 1 160.4.a.a 1
16.e even 4 1 320.4.a.i 1
16.f odd 4 1 160.4.a.b yes 1
16.f odd 4 1 320.4.a.f 1
48.i odd 4 1 1440.4.a.o 1
48.k even 4 1 1440.4.a.n 1
80.i odd 4 1 800.4.c.f 2
80.j even 4 1 800.4.c.e 2
80.k odd 4 1 800.4.a.d 1
80.k odd 4 1 1600.4.a.bj 1
80.q even 4 1 800.4.a.h 1
80.q even 4 1 1600.4.a.r 1
80.s even 4 1 800.4.c.e 2
80.t odd 4 1 800.4.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 16.e even 4 1
160.4.a.b yes 1 16.f odd 4 1
320.4.a.f 1 16.f odd 4 1
320.4.a.i 1 16.e even 4 1
800.4.a.d 1 80.k odd 4 1
800.4.a.h 1 80.q even 4 1
800.4.c.e 2 80.j even 4 1
800.4.c.e 2 80.s even 4 1
800.4.c.f 2 80.i odd 4 1
800.4.c.f 2 80.t odd 4 1
1280.4.d.f 2 1.a even 1 1 trivial
1280.4.d.f 2 8.b even 2 1 inner
1280.4.d.k 2 4.b odd 2 1
1280.4.d.k 2 8.d odd 2 1
1440.4.a.n 1 48.k even 4 1
1440.4.a.o 1 48.i odd 4 1
1600.4.a.r 1 80.q even 4 1
1600.4.a.bj 1 80.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7} + 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 3600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3600 \) Copy content Toggle raw display
$13$ \( T^{2} + 2500 \) Copy content Toggle raw display
$17$ \( (T + 30)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1600 \) Copy content Toggle raw display
$23$ \( (T + 178)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 27556 \) Copy content Toggle raw display
$31$ \( (T - 20)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T - 250)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 20164 \) Copy content Toggle raw display
$47$ \( (T - 214)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 240100 \) Copy content Toggle raw display
$59$ \( T^{2} + 640000 \) Copy content Toggle raw display
$61$ \( T^{2} + 62500 \) Copy content Toggle raw display
$67$ \( T^{2} + 599076 \) Copy content Toggle raw display
$71$ \( (T + 100)^{2} \) Copy content Toggle raw display
$73$ \( (T - 230)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1320)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 964324 \) Copy content Toggle raw display
$89$ \( (T + 874)^{2} \) Copy content Toggle raw display
$97$ \( (T + 310)^{2} \) Copy content Toggle raw display
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