Properties

Label 1280.4.d.q
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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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 160)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 4 \beta_1) q^{3} - 5 \beta_1 q^{5} + ( - 5 \beta_{3} - 4) q^{7} + (8 \beta_{3} - 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 4 \beta_1) q^{3} - 5 \beta_1 q^{5} + ( - 5 \beta_{3} - 4) q^{7} + (8 \beta_{3} - 13) q^{9} + (6 \beta_{2} - 32 \beta_1) q^{11} + ( - 8 \beta_{2} + 6 \beta_1) q^{13} + ( - 5 \beta_{3} + 20) q^{15} + (24 \beta_{3} + 2) q^{17} + (8 \beta_{2} + 104 \beta_1) q^{19} + ( - 16 \beta_{2} + 104 \beta_1) q^{21} + ( - 11 \beta_{3} - 60) q^{23} - 25 q^{25} + (18 \beta_{2} - 136 \beta_1) q^{27} + (16 \beta_{2} - 146 \beta_1) q^{29} + (6 \beta_{3} + 88) q^{31} + ( - 56 \beta_{3} + 272) q^{33} + (25 \beta_{2} + 20 \beta_1) q^{35} + (16 \beta_{2} + 178 \beta_1) q^{37} + (38 \beta_{3} - 216) q^{39} + ( - 40 \beta_{3} - 50) q^{41} + (37 \beta_{2} + 188 \beta_1) q^{43} + ( - 40 \beta_{2} + 65 \beta_1) q^{45} + (13 \beta_{3} - 140) q^{47} + (40 \beta_{3} + 273) q^{49} + (94 \beta_{2} - 568 \beta_1) q^{51} + (24 \beta_{2} - 158 \beta_1) q^{53} + (30 \beta_{3} - 160) q^{55} + (72 \beta_{3} - 224) q^{57} + ( - 20 \beta_{2} - 360 \beta_1) q^{59} + (32 \beta_{2} - 634 \beta_1) q^{61} + (33 \beta_{3} - 908) q^{63} + ( - 40 \beta_{3} + 30) q^{65} + ( - 47 \beta_{2} - 372 \beta_1) q^{67} + (16 \beta_{2} + 24 \beta_1) q^{69} + (218 \beta_{3} + 24) q^{71} + ( - 120 \beta_{3} + 470) q^{73} + (25 \beta_{2} - 100 \beta_1) q^{75} + (136 \beta_{2} - 592 \beta_1) q^{77} + (172 \beta_{3} + 16) q^{79} + (8 \beta_{3} + 625) q^{81} + (47 \beta_{2} - 796 \beta_1) q^{83} + ( - 120 \beta_{2} - 10 \beta_1) q^{85} + ( - 210 \beta_{3} + 968) q^{87} + (48 \beta_{3} + 390) q^{89} + (2 \beta_{2} + 936 \beta_1) q^{91} + ( - 64 \beta_{2} + 208 \beta_1) q^{93} + (40 \beta_{3} + 520) q^{95} + (40 \beta_{3} + 610) q^{97} + ( - 334 \beta_{2} + 1568 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{7} - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{7} - 52 q^{9} + 80 q^{15} + 8 q^{17} - 240 q^{23} - 100 q^{25} + 352 q^{31} + 1088 q^{33} - 864 q^{39} - 200 q^{41} - 560 q^{47} + 1092 q^{49} - 640 q^{55} - 896 q^{57} - 3632 q^{63} + 120 q^{65} + 96 q^{71} + 1880 q^{73} + 64 q^{79} + 2500 q^{81} + 3872 q^{87} + 1560 q^{89} + 2080 q^{95} + 2440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 6\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 6\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 4 \) 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
−1.22474 + 1.22474i
1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
0 8.89898i 0 5.00000i 0 20.4949 0 −52.1918 0
641.2 0 0.898979i 0 5.00000i 0 −28.4949 0 26.1918 0
641.3 0 0.898979i 0 5.00000i 0 −28.4949 0 26.1918 0
641.4 0 8.89898i 0 5.00000i 0 20.4949 0 −52.1918 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.q 4
4.b odd 2 1 1280.4.d.x 4
8.b even 2 1 inner 1280.4.d.q 4
8.d odd 2 1 1280.4.d.x 4
16.e even 4 1 160.4.a.g yes 2
16.e even 4 1 320.4.a.o 2
16.f odd 4 1 160.4.a.c 2
16.f odd 4 1 320.4.a.s 2
48.i odd 4 1 1440.4.a.x 2
48.k even 4 1 1440.4.a.t 2
80.i odd 4 1 800.4.c.k 4
80.j even 4 1 800.4.c.i 4
80.k odd 4 1 800.4.a.s 2
80.k odd 4 1 1600.4.a.cd 2
80.q even 4 1 800.4.a.m 2
80.q even 4 1 1600.4.a.cn 2
80.s even 4 1 800.4.c.i 4
80.t odd 4 1 800.4.c.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.c 2 16.f odd 4 1
160.4.a.g yes 2 16.e even 4 1
320.4.a.o 2 16.e even 4 1
320.4.a.s 2 16.f odd 4 1
800.4.a.m 2 80.q even 4 1
800.4.a.s 2 80.k odd 4 1
800.4.c.i 4 80.j even 4 1
800.4.c.i 4 80.s even 4 1
800.4.c.k 4 80.i odd 4 1
800.4.c.k 4 80.t odd 4 1
1280.4.d.q 4 1.a even 1 1 trivial
1280.4.d.q 4 8.b even 2 1 inner
1280.4.d.x 4 4.b odd 2 1
1280.4.d.x 4 8.d odd 2 1
1440.4.a.t 2 48.k even 4 1
1440.4.a.x 2 48.i odd 4 1
1600.4.a.cd 2 80.k odd 4 1
1600.4.a.cn 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 \(S_{4}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{4} + 80T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{2} + 8T_{7} - 584 \) Copy content Toggle raw display
\( T_{11}^{4} + 3776T_{11}^{2} + 25600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8 T - 584)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 3776 T^{2} + 25600 \) Copy content Toggle raw display
$13$ \( T^{4} + 3144 T^{2} + \cdots + 2250000 \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 13820)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 24704 T^{2} + \cdots + 86118400 \) Copy content Toggle raw display
$23$ \( (T^{2} + 120 T + 696)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 54920 T^{2} + \cdots + 230189584 \) Copy content Toggle raw display
$31$ \( (T^{2} - 176 T + 6880)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 75656 T^{2} + \cdots + 652291600 \) Copy content Toggle raw display
$41$ \( (T^{2} + 100 T - 35900)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 136400 T^{2} + \cdots + 6190144 \) Copy content Toggle raw display
$47$ \( (T^{2} + 280 T + 15544)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 77576 T^{2} + \cdots + 124099600 \) Copy content Toggle raw display
$59$ \( T^{4} + 278400 T^{2} + \cdots + 14400000000 \) Copy content Toggle raw display
$61$ \( T^{4} + 853064 T^{2} + \cdots + 142415664400 \) Copy content Toggle raw display
$67$ \( T^{4} + 382800 T^{2} + \cdots + 7287695424 \) Copy content Toggle raw display
$71$ \( (T^{2} - 48 T - 1140000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 940 T - 124700)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 32 T - 709760)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1373264 T^{2} + \cdots + 337096360000 \) Copy content Toggle raw display
$89$ \( (T^{2} - 780 T + 96804)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 1220 T + 333700)^{2} \) Copy content Toggle raw display
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