Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,4,Mod(1,1280)]
 
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, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.a (trivial)

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: yes
Analytic conductor: \(75.5224448073\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{29})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 17x^{2} + 18x + 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + 5 q^{5} + ( - \beta_{2} + 3 \beta_1) q^{7} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + 5 q^{5} + ( - \beta_{2} + 3 \beta_1) q^{7} + 31 q^{9} + ( - 2 \beta_{2} + \beta_1) q^{11} + (\beta_{3} + 34) q^{13} - 5 \beta_{2} q^{15} + (\beta_{3} - 6) q^{17} + 15 \beta_1 q^{19} + ( - 3 \beta_{3} + 58) q^{21} + ( - 9 \beta_{2} - 13 \beta_1) q^{23} + 25 q^{25} - 4 \beta_{2} q^{27} + (5 \beta_{3} + 40) q^{29} + ( - 14 \beta_{2} + 32 \beta_1) q^{31} + ( - \beta_{3} + 116) q^{33} + ( - 5 \beta_{2} + 15 \beta_1) q^{35} + ( - 6 \beta_{3} + 146) q^{37} + ( - 34 \beta_{2} - 58 \beta_1) q^{39} + (4 \beta_{3} - 136) q^{41} + (13 \beta_{2} + 58 \beta_1) q^{43} + 155 q^{45} + ( - 41 \beta_{2} + 13 \beta_1) q^{47} + ( - 6 \beta_{3} + 3) q^{49} + (6 \beta_{2} - 58 \beta_1) q^{51} + (\beta_{3} - 366) q^{53} + ( - 10 \beta_{2} + 5 \beta_1) q^{55} - 15 \beta_{3} q^{57} + ( - 8 \beta_{2} - 91 \beta_1) q^{59} + (2 \beta_{3} - 302) q^{61} + ( - 31 \beta_{2} + 93 \beta_1) q^{63} + (5 \beta_{3} + 170) q^{65} + ( - 17 \beta_{2} + 2 \beta_1) q^{67} + (13 \beta_{3} + 522) q^{69} + ( - 94 \beta_{2} - 28 \beta_1) q^{71} + (7 \beta_{3} + 658) q^{73} - 25 \beta_{2} q^{75} + ( - 7 \beta_{3} + 212) q^{77} + ( - 80 \beta_{2} + 90 \beta_1) q^{79} - 605 q^{81} + (81 \beta_{2} - 36 \beta_1) q^{83} + (5 \beta_{3} - 30) q^{85} + ( - 40 \beta_{2} - 290 \beta_1) q^{87} + (26 \beta_{3} - 490) q^{89} + (62 \beta_{2} + 44 \beta_1) q^{91} + ( - 32 \beta_{3} + 812) q^{93} + 75 \beta_1 q^{95} + (35 \beta_{3} - 250) q^{97} + ( - 62 \beta_{2} + 31 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{5} + 124 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 20 q^{5} + 124 q^{9} + 136 q^{13} - 24 q^{17} + 232 q^{21} + 100 q^{25} + 160 q^{29} + 464 q^{33} + 584 q^{37} - 544 q^{41} + 620 q^{45} + 12 q^{49} - 1464 q^{53} - 1208 q^{61} + 680 q^{65} + 2088 q^{69} + 2632 q^{73} + 848 q^{77} - 2420 q^{81} - 120 q^{85} - 1960 q^{89} + 3248 q^{93} - 1000 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 17x^{2} + 18x + 23 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8\nu^{3} - 12\nu^{2} - 100\nu + 52 ) / 21 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -32\nu^{3} + 48\nu^{2} + 736\nu - 376 ) / 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 4\beta _1 + 8 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 16\beta_{2} + 4\beta _1 + 152 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{3} + 12\beta_{2} + 49\beta _1 + 112 ) / 8 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.60680
4.60680
1.77837
−0.778369
0 −7.61577 0 5.00000 0 −24.5863 0 31.0000 0
1.2 0 −7.61577 0 5.00000 0 9.35479 0 31.0000 0
1.3 0 7.61577 0 5.00000 0 −9.35479 0 31.0000 0
1.4 0 7.61577 0 5.00000 0 24.5863 0 31.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.4.a.y 4
4.b odd 2 1 inner 1280.4.a.y 4
8.b even 2 1 1280.4.a.t 4
8.d odd 2 1 1280.4.a.t 4
16.e even 4 2 640.4.d.g 8
16.f odd 4 2 640.4.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.d.g 8 16.e even 4 2
640.4.d.g 8 16.f odd 4 2
1280.4.a.t 4 8.b even 2 1
1280.4.a.t 4 8.d odd 2 1
1280.4.a.y 4 1.a even 1 1 trivial
1280.4.a.y 4 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1280))\):

\( T_{3}^{2} - 58 \) Copy content Toggle raw display
\( T_{7}^{4} - 692T_{7}^{2} + 52900 \) Copy content Toggle raw display
\( T_{13}^{2} - 68T_{13} - 700 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 58)^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 692 T^{2} + 52900 \) Copy content Toggle raw display
$11$ \( T^{4} - 528 T^{2} + 40000 \) Copy content Toggle raw display
$13$ \( (T^{2} - 68 T - 700)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12 T - 1820)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 7200)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 20212 T^{2} + \cdots + 504100 \) Copy content Toggle raw display
$29$ \( (T^{2} - 80 T - 44800)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 88272 T^{2} + \cdots + 457960000 \) Copy content Toggle raw display
$37$ \( (T^{2} - 292 T - 45500)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 272 T - 11200)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 234900 T^{2} + \cdots + 9573839716 \) Copy content Toggle raw display
$47$ \( T^{4} - 205812 T^{2} + \cdots + 8480568100 \) Copy content Toggle raw display
$53$ \( (T^{2} + 732 T + 132100)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 537408 T^{2} + \cdots + 68267238400 \) Copy content Toggle raw display
$61$ \( (T^{2} + 604 T + 83780)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 33780 T^{2} + \cdots + 276689956 \) Copy content Toggle raw display
$71$ \( T^{4} - 1075152 T^{2} + \cdots + 237558760000 \) Copy content Toggle raw display
$73$ \( (T^{2} - 1316 T + 342020)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 1260800 T^{2} + \cdots + 12544000000 \) Copy content Toggle raw display
$83$ \( T^{4} - 844020 T^{2} + \cdots + 114965752356 \) Copy content Toggle raw display
$89$ \( (T^{2} + 980 T - 1014556)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 500 T - 2211100)^{2} \) Copy content Toggle raw display
show more
show less