Properties

Label 1280.4.d.s
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: \(\Q(i, \sqrt{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{3} + 5 \beta_1 q^{5} + ( - 3 \beta_{3} - 3) q^{7} + (2 \beta_{3} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{3} + 5 \beta_1 q^{5} + ( - 3 \beta_{3} - 3) q^{7} + (2 \beta_{3} + 5) q^{9} - 14 \beta_1 q^{11} + ( - 10 \beta_{2} + 16 \beta_1) q^{13} + ( - 5 \beta_{3} + 5) q^{15} + ( - 14 \beta_{3} - 48) q^{17} - 2 \beta_{2} q^{19} - 60 \beta_1 q^{21} + (3 \beta_{3} + 75) q^{23} - 25 q^{25} + (30 \beta_{2} + 10 \beta_1) q^{27} + ( - 32 \beta_{2} + 86 \beta_1) q^{29} + (42 \beta_{3} + 74) q^{31} + (14 \beta_{3} - 14) q^{33} + ( - 15 \beta_{2} - 15 \beta_1) q^{35} + ( - 20 \beta_{2} - 22 \beta_1) q^{37} + ( - 26 \beta_{3} + 226) q^{39} + (62 \beta_{3} + 120) q^{41} + ( - 5 \beta_{2} + 137 \beta_1) q^{43} + (10 \beta_{2} + 25 \beta_1) q^{45} + (11 \beta_{3} - 201) q^{47} + (18 \beta_{3} - 145) q^{49} + ( - 34 \beta_{2} - 246 \beta_1) q^{51} + ( - 58 \beta_{2} - 188 \beta_1) q^{53} + 70 q^{55} + ( - 2 \beta_{3} + 42) q^{57} + ( - 46 \beta_{2} + 516 \beta_1) q^{59} + ( - 96 \beta_{2} + 306 \beta_1) q^{61} + ( - 21 \beta_{3} - 141) q^{63} + (50 \beta_{3} - 80) q^{65} + ( - 45 \beta_{2} - 47 \beta_1) q^{67} + (72 \beta_{2} - 12 \beta_1) q^{69} + (58 \beta_{3} + 518) q^{71} + ( - 2 \beta_{3} + 400) q^{73} + ( - 25 \beta_{2} + 25 \beta_1) q^{75} + (42 \beta_{2} + 42 \beta_1) q^{77} + ( - 172 \beta_{3} + 52) q^{79} + (74 \beta_{3} - 485) q^{81} + (253 \beta_{2} - 29 \beta_1) q^{83} + ( - 70 \beta_{2} - 240 \beta_1) q^{85} + ( - 118 \beta_{3} + 758) q^{87} + (132 \beta_{3} + 214) q^{89} + ( - 18 \beta_{2} + 582 \beta_1) q^{91} + (32 \beta_{2} + 808 \beta_1) q^{93} + 10 \beta_{3} q^{95} + (206 \beta_{3} - 732) q^{97} + ( - 28 \beta_{2} - 70 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{7} + 20 q^{9} + 20 q^{15} - 192 q^{17} + 300 q^{23} - 100 q^{25} + 296 q^{31} - 56 q^{33} + 904 q^{39} + 480 q^{41} - 804 q^{47} - 580 q^{49} + 280 q^{55} + 168 q^{57} - 564 q^{63} - 320 q^{65} + 2072 q^{71} + 1600 q^{73} + 208 q^{79} - 1940 q^{81} + 3032 q^{87} + 856 q^{89} - 2928 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 6\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 16\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{2} + 8\beta_1 \) 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
2.79129i
1.79129i
1.79129i
2.79129i
0 5.58258i 0 5.00000i 0 10.7477 0 −4.16515 0
641.2 0 3.58258i 0 5.00000i 0 −16.7477 0 14.1652 0
641.3 0 3.58258i 0 5.00000i 0 −16.7477 0 14.1652 0
641.4 0 5.58258i 0 5.00000i 0 10.7477 0 −4.16515 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.s 4
4.b odd 2 1 1280.4.d.w 4
8.b even 2 1 inner 1280.4.d.s 4
8.d odd 2 1 1280.4.d.w 4
16.e even 4 1 640.4.a.g 2
16.e even 4 1 640.4.a.j yes 2
16.f odd 4 1 640.4.a.h yes 2
16.f odd 4 1 640.4.a.i yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.a.g 2 16.e even 4 1
640.4.a.h yes 2 16.f odd 4 1
640.4.a.i yes 2 16.f odd 4 1
640.4.a.j yes 2 16.e even 4 1
1280.4.d.s 4 1.a even 1 1 trivial
1280.4.d.s 4 8.b even 2 1 inner
1280.4.d.w 4 4.b odd 2 1
1280.4.d.w 4 8.d odd 2 1

Hecke kernels

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

\( T_{3}^{4} + 44T_{3}^{2} + 400 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} - 180 \) Copy content Toggle raw display
\( T_{11}^{2} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 44T^{2} + 400 \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6 T - 180)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 4712 T^{2} + \cdots + 3400336 \) Copy content Toggle raw display
$17$ \( (T^{2} + 96 T - 1812)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 84)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 150 T + 5436)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 57800 T^{2} + \cdots + 199035664 \) Copy content Toggle raw display
$31$ \( (T^{2} - 148 T - 31568)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 17768 T^{2} + \cdots + 62663056 \) Copy content Toggle raw display
$41$ \( (T^{2} - 240 T - 66324)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 38588 T^{2} + \cdots + 332843536 \) Copy content Toggle raw display
$47$ \( (T^{2} + 402 T + 37860)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 211976 T^{2} + \cdots + 1246090000 \) Copy content Toggle raw display
$59$ \( T^{4} + 621384 T^{2} + \cdots + 49204112400 \) Copy content Toggle raw display
$61$ \( T^{4} + 574344 T^{2} + \cdots + 9980010000 \) Copy content Toggle raw display
$67$ \( T^{4} + 89468 T^{2} + \cdots + 1625379856 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1036 T + 197680)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 800 T + 159916)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 104 T - 618560)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 2690060 T^{2} + \cdots + 1804583849104 \) Copy content Toggle raw display
$89$ \( (T^{2} - 428 T - 320108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1464 T - 355332)^{2} \) Copy content Toggle raw display
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