Properties

Label 1280.2.o.f
Level $1280$
Weight $2$
Character orbit 1280.o
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,2,Mod(127,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.o (of order \(4\), degree \(2\), 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (i - 1) q^{3} + ( - i + 2) q^{5} + ( - 3 i + 3) q^{7} + i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (i - 1) q^{3} + ( - i + 2) q^{5} + ( - 3 i + 3) q^{7} + i q^{9} - 2 q^{11} + (3 i + 3) q^{13} + (3 i - 1) q^{15} + (i + 1) q^{17} + 4 i q^{19} + 6 i q^{21} + (i + 1) q^{23} + ( - 4 i + 3) q^{25} + ( - 4 i - 4) q^{27} - 10 i q^{31} + ( - 2 i + 2) q^{33} + ( - 9 i + 3) q^{35} + (i - 1) q^{37} - 6 q^{39} + 10 q^{41} + ( - 5 i + 5) q^{43} + (2 i + 1) q^{45} + (3 i - 3) q^{47} - 11 i q^{49} - 2 q^{51} + (5 i + 5) q^{53} + (2 i - 4) q^{55} + ( - 4 i - 4) q^{57} + 12 i q^{59} + 2 i q^{61} + (3 i + 3) q^{63} + (3 i + 9) q^{65} + ( - i - 1) q^{67} - 2 q^{69} + 2 i q^{71} + (i - 1) q^{73} + (7 i + 1) q^{75} + (6 i - 6) q^{77} + 8 q^{79} + 5 q^{81} + (5 i - 5) q^{83} + (i + 3) q^{85} - 16 i q^{89} + 18 q^{91} + (10 i + 10) q^{93} + (8 i + 4) q^{95} + ( - 3 i - 3) q^{97} - 2 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 4 q^{5} + 6 q^{7} - 4 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 2 q^{23} + 6 q^{25} - 8 q^{27} + 4 q^{33} + 6 q^{35} - 2 q^{37} - 12 q^{39} + 20 q^{41} + 10 q^{43} + 2 q^{45} - 6 q^{47} - 4 q^{51} + 10 q^{53} - 8 q^{55} - 8 q^{57} + 6 q^{63} + 18 q^{65} - 2 q^{67} - 4 q^{69} - 2 q^{73} + 2 q^{75} - 12 q^{77} + 16 q^{79} + 10 q^{81} - 10 q^{83} + 6 q^{85} + 36 q^{91} + 20 q^{93} + 8 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 −1.00000 + 1.00000i 0 2.00000 1.00000i 0 3.00000 3.00000i 0 1.00000i 0
383.1 0 −1.00000 1.00000i 0 2.00000 + 1.00000i 0 3.00000 + 3.00000i 0 1.00000i 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
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.o.f 2
4.b odd 2 1 1280.2.o.m 2
5.c odd 4 1 1280.2.o.c 2
8.b even 2 1 1280.2.o.l 2
8.d odd 2 1 1280.2.o.c 2
16.e even 4 1 160.2.n.d yes 2
16.e even 4 1 320.2.n.b 2
16.f odd 4 1 160.2.n.c 2
16.f odd 4 1 320.2.n.g 2
20.e even 4 1 1280.2.o.l 2
40.i odd 4 1 1280.2.o.m 2
40.k even 4 1 inner 1280.2.o.f 2
48.i odd 4 1 1440.2.x.a 2
48.k even 4 1 1440.2.x.f 2
80.i odd 4 1 320.2.n.g 2
80.i odd 4 1 800.2.n.f 2
80.j even 4 1 160.2.n.d yes 2
80.j even 4 1 1600.2.n.m 2
80.k odd 4 1 800.2.n.f 2
80.k odd 4 1 1600.2.n.c 2
80.q even 4 1 800.2.n.e 2
80.q even 4 1 1600.2.n.m 2
80.s even 4 1 320.2.n.b 2
80.s even 4 1 800.2.n.e 2
80.t odd 4 1 160.2.n.c 2
80.t odd 4 1 1600.2.n.c 2
240.bd odd 4 1 1440.2.x.a 2
240.bf even 4 1 1440.2.x.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.c 2 16.f odd 4 1
160.2.n.c 2 80.t odd 4 1
160.2.n.d yes 2 16.e even 4 1
160.2.n.d yes 2 80.j even 4 1
320.2.n.b 2 16.e even 4 1
320.2.n.b 2 80.s even 4 1
320.2.n.g 2 16.f odd 4 1
320.2.n.g 2 80.i odd 4 1
800.2.n.e 2 80.q even 4 1
800.2.n.e 2 80.s even 4 1
800.2.n.f 2 80.i odd 4 1
800.2.n.f 2 80.k odd 4 1
1280.2.o.c 2 5.c odd 4 1
1280.2.o.c 2 8.d odd 2 1
1280.2.o.f 2 1.a even 1 1 trivial
1280.2.o.f 2 40.k even 4 1 inner
1280.2.o.l 2 8.b even 2 1
1280.2.o.l 2 20.e even 4 1
1280.2.o.m 2 4.b odd 2 1
1280.2.o.m 2 40.i odd 4 1
1440.2.x.a 2 48.i odd 4 1
1440.2.x.a 2 240.bd odd 4 1
1440.2.x.f 2 48.k even 4 1
1440.2.x.f 2 240.bf even 4 1
1600.2.n.c 2 80.k odd 4 1
1600.2.n.c 2 80.t odd 4 1
1600.2.n.m 2 80.j even 4 1
1600.2.n.m 2 80.q even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 100 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$71$ \( T^{2} + 4 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$89$ \( T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
show more
show less