Properties

Label 1280.4.d.y
Level $1280$
Weight $4$
Character orbit 1280.d
Analytic conductor $75.522$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{51})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 25x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 640)
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 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.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{3} + 5 \beta_1 q^{5} + ( - \beta_{2} + 4) q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{3} + 5 \beta_1 q^{5} + ( - \beta_{2} + 4) q^{7} + 23 q^{9} + ( - \beta_{3} + 18 \beta_1) q^{11} + ( - 2 \beta_{3} + 2 \beta_1) q^{13} - 10 q^{15} + (2 \beta_{2} - 2) q^{17} + (\beta_{3} + 34 \beta_1) q^{19} + ( - 2 \beta_{3} + 8 \beta_1) q^{21} + (\beta_{2} - 132) q^{23} - 25 q^{25} + 100 \beta_1 q^{27} + ( - 2 \beta_{3} + 170 \beta_1) q^{29} + ( - 2 \beta_{2} + 160) q^{31} + (2 \beta_{2} - 36) q^{33} + ( - 5 \beta_{3} + 20 \beta_1) q^{35} + ( - 8 \beta_{3} - 106 \beta_1) q^{37} + (4 \beta_{2} - 4) q^{39} + (12 \beta_{2} - 90) q^{41} + (10 \beta_{3} + 138 \beta_1) q^{43} + 115 \beta_1 q^{45} + (11 \beta_{2} + 24) q^{47} + ( - 8 \beta_{2} + 489) q^{49} + (4 \beta_{3} - 4 \beta_1) q^{51} + (14 \beta_{3} - 106 \beta_1) q^{53} + (5 \beta_{2} - 90) q^{55} + ( - 2 \beta_{2} - 68) q^{57} + ( - 11 \beta_{3} + 162 \beta_1) q^{59} + (24 \beta_{3} - 78 \beta_1) q^{61} + ( - 23 \beta_{2} + 92) q^{63} + (10 \beta_{2} - 10) q^{65} + (10 \beta_{3} - 118 \beta_1) q^{67} + (2 \beta_{3} - 264 \beta_1) q^{69} + ( - 24 \beta_{2} - 364) q^{71} + ( - 26 \beta_{2} - 94) q^{73} - 50 \beta_1 q^{75} + ( - 22 \beta_{3} + 888 \beta_1) q^{77} + (20 \beta_{2} + 24) q^{79} + 421 q^{81} + ( - 4 \beta_{3} - 582 \beta_1) q^{83} + (10 \beta_{3} - 10 \beta_1) q^{85} + (4 \beta_{2} - 340) q^{87} + ( - 24 \beta_{2} - 702) q^{89} + ( - 10 \beta_{3} + 1640 \beta_1) q^{91} + ( - 4 \beta_{3} + 320 \beta_1) q^{93} + ( - 5 \beta_{2} - 170) q^{95} + ( - 38 \beta_{2} + 574) q^{97} + ( - 23 \beta_{3} + 414 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{7} + 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{7} + 92 q^{9} - 40 q^{15} - 8 q^{17} - 528 q^{23} - 100 q^{25} + 640 q^{31} - 144 q^{33} - 16 q^{39} - 360 q^{41} + 96 q^{47} + 1956 q^{49} - 360 q^{55} - 272 q^{57} + 368 q^{63} - 40 q^{65} - 1456 q^{71} - 376 q^{73} + 96 q^{79} + 1684 q^{81} - 1360 q^{87} - 2808 q^{89} - 680 q^{95} + 2296 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 25x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 12\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{3} + 152\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} - 100 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 4\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 100 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{2} + 38\beta_1 ) / 2 \) 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
3.57071 0.500000i
−3.57071 0.500000i
3.57071 + 0.500000i
−3.57071 + 0.500000i
0 2.00000i 0 5.00000i 0 −24.5657 0 23.0000 0
641.2 0 2.00000i 0 5.00000i 0 32.5657 0 23.0000 0
641.3 0 2.00000i 0 5.00000i 0 −24.5657 0 23.0000 0
641.4 0 2.00000i 0 5.00000i 0 32.5657 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.y 4
4.b odd 2 1 1280.4.d.r 4
8.b even 2 1 inner 1280.4.d.y 4
8.d odd 2 1 1280.4.d.r 4
16.e even 4 1 640.4.a.f yes 2
16.e even 4 1 640.4.a.k yes 2
16.f odd 4 1 640.4.a.e 2
16.f odd 4 1 640.4.a.l yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.a.e 2 16.f odd 4 1
640.4.a.f yes 2 16.e even 4 1
640.4.a.k yes 2 16.e even 4 1
640.4.a.l yes 2 16.f odd 4 1
1280.4.d.r 4 4.b odd 2 1
1280.4.d.r 4 8.d odd 2 1
1280.4.d.y 4 1.a even 1 1 trivial
1280.4.d.y 4 8.b even 2 1 inner

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
\( T_{7}^{2} - 8T_{7} - 800 \) Copy content Toggle raw display
\( T_{11}^{4} + 2280T_{11}^{2} + 242064 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 8 T - 800)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 2280 T^{2} + 242064 \) Copy content Toggle raw display
$13$ \( T^{4} + 6536 T^{2} + \cdots + 10627600 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 3260)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 3944 T^{2} + 115600 \) Copy content Toggle raw display
$23$ \( (T^{2} + 264 T + 16608)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 64328 T^{2} + \cdots + 657204496 \) Copy content Toggle raw display
$31$ \( (T^{2} - 320 T + 22336)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 126920 T^{2} + \cdots + 1680016144 \) Copy content Toggle raw display
$41$ \( (T^{2} + 180 T - 109404)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 201288 T^{2} + \cdots + 3913253136 \) Copy content Toggle raw display
$47$ \( (T^{2} - 48 T - 98160)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 342344 T^{2} + \cdots + 22111690000 \) Copy content Toggle raw display
$59$ \( T^{4} + 249960 T^{2} + \cdots + 5255090064 \) Copy content Toggle raw display
$61$ \( T^{4} + 952200 T^{2} + \cdots + 215232900624 \) Copy content Toggle raw display
$67$ \( T^{4} + 191048 T^{2} + \cdots + 4580040976 \) Copy content Toggle raw display
$71$ \( (T^{2} + 728 T - 337520)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 188 T - 542780)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 48 T - 325824)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 703560 T^{2} + \cdots + 106059646224 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1404 T + 22788)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 1148 T - 848828)^{2} \) Copy content Toggle raw display
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