Properties

Label 1280.4.d.g
Level $1280$
Weight $4$
Character orbit 1280.d
Analytic conductor $75.522$
Analytic rank $0$
Dimension $2$
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 10)
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.

\(f(q)\) \(=\) \( q + 8 i q^{3} - 5 i q^{5} - 4 q^{7} - 37 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 8 i q^{3} - 5 i q^{5} - 4 q^{7} - 37 q^{9} + 12 i q^{11} - 58 i q^{13} + 40 q^{15} + 66 q^{17} + 100 i q^{19} - 32 i q^{21} + 132 q^{23} - 25 q^{25} - 80 i q^{27} - 90 i q^{29} - 152 q^{31} - 96 q^{33} + 20 i q^{35} + 34 i q^{37} + 464 q^{39} + 438 q^{41} + 32 i q^{43} + 185 i q^{45} + 204 q^{47} - 327 q^{49} + 528 i q^{51} - 222 i q^{53} + 60 q^{55} - 800 q^{57} + 420 i q^{59} + 902 i q^{61} + 148 q^{63} - 290 q^{65} + 1024 i q^{67} + 1056 i q^{69} + 432 q^{71} - 362 q^{73} - 200 i q^{75} - 48 i q^{77} + 160 q^{79} - 359 q^{81} - 72 i q^{83} - 330 i q^{85} + 720 q^{87} - 810 q^{89} + 232 i q^{91} - 1216 i q^{93} + 500 q^{95} + 1106 q^{97} - 444 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} - 74 q^{9} + 80 q^{15} + 132 q^{17} + 264 q^{23} - 50 q^{25} - 304 q^{31} - 192 q^{33} + 928 q^{39} + 876 q^{41} + 408 q^{47} - 654 q^{49} + 120 q^{55} - 1600 q^{57} + 296 q^{63} - 580 q^{65} + 864 q^{71} - 724 q^{73} + 320 q^{79} - 718 q^{81} + 1440 q^{87} - 1620 q^{89} + 1000 q^{95} + 2212 q^{97}+O(q^{100}) \) 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 8.00000i 0 5.00000i 0 −4.00000 0 −37.0000 0
641.2 0 8.00000i 0 5.00000i 0 −4.00000 0 −37.0000 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.g 2
4.b odd 2 1 1280.4.d.j 2
8.b even 2 1 inner 1280.4.d.g 2
8.d odd 2 1 1280.4.d.j 2
16.e even 4 1 80.4.a.f 1
16.e even 4 1 320.4.a.b 1
16.f odd 4 1 10.4.a.a 1
16.f odd 4 1 320.4.a.m 1
48.i odd 4 1 720.4.a.j 1
48.k even 4 1 90.4.a.a 1
80.i odd 4 1 400.4.c.c 2
80.j even 4 1 50.4.b.a 2
80.k odd 4 1 50.4.a.c 1
80.k odd 4 1 1600.4.a.d 1
80.q even 4 1 400.4.a.b 1
80.q even 4 1 1600.4.a.bx 1
80.s even 4 1 50.4.b.a 2
80.t odd 4 1 400.4.c.c 2
112.j even 4 1 490.4.a.o 1
112.u odd 12 2 490.4.e.i 2
112.v even 12 2 490.4.e.a 2
144.u even 12 2 810.4.e.w 2
144.v odd 12 2 810.4.e.c 2
176.i even 4 1 1210.4.a.b 1
208.o odd 4 1 1690.4.a.a 1
240.t even 4 1 450.4.a.q 1
240.z odd 4 1 450.4.c.d 2
240.bd odd 4 1 450.4.c.d 2
560.be even 4 1 2450.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 16.f odd 4 1
50.4.a.c 1 80.k odd 4 1
50.4.b.a 2 80.j even 4 1
50.4.b.a 2 80.s even 4 1
80.4.a.f 1 16.e even 4 1
90.4.a.a 1 48.k even 4 1
320.4.a.b 1 16.e even 4 1
320.4.a.m 1 16.f odd 4 1
400.4.a.b 1 80.q even 4 1
400.4.c.c 2 80.i odd 4 1
400.4.c.c 2 80.t odd 4 1
450.4.a.q 1 240.t even 4 1
450.4.c.d 2 240.z odd 4 1
450.4.c.d 2 240.bd odd 4 1
490.4.a.o 1 112.j even 4 1
490.4.e.a 2 112.v even 12 2
490.4.e.i 2 112.u odd 12 2
720.4.a.j 1 48.i odd 4 1
810.4.e.c 2 144.v odd 12 2
810.4.e.w 2 144.u even 12 2
1210.4.a.b 1 176.i even 4 1
1280.4.d.g 2 1.a even 1 1 trivial
1280.4.d.g 2 8.b even 2 1 inner
1280.4.d.j 2 4.b odd 2 1
1280.4.d.j 2 8.d odd 2 1
1600.4.a.d 1 80.k odd 4 1
1600.4.a.bx 1 80.q even 4 1
1690.4.a.a 1 208.o odd 4 1
2450.4.a.b 1 560.be even 4 1

Hecke kernels

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

\( T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 144 \) Copy content Toggle raw display
$13$ \( T^{2} + 3364 \) Copy content Toggle raw display
$17$ \( (T - 66)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 10000 \) Copy content Toggle raw display
$23$ \( (T - 132)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 8100 \) Copy content Toggle raw display
$31$ \( (T + 152)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1156 \) Copy content Toggle raw display
$41$ \( (T - 438)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1024 \) Copy content Toggle raw display
$47$ \( (T - 204)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 49284 \) Copy content Toggle raw display
$59$ \( T^{2} + 176400 \) Copy content Toggle raw display
$61$ \( T^{2} + 813604 \) Copy content Toggle raw display
$67$ \( T^{2} + 1048576 \) Copy content Toggle raw display
$71$ \( (T - 432)^{2} \) Copy content Toggle raw display
$73$ \( (T + 362)^{2} \) Copy content Toggle raw display
$79$ \( (T - 160)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5184 \) Copy content Toggle raw display
$89$ \( (T + 810)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1106)^{2} \) Copy content Toggle raw display
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