Properties

Label 1280.3.g.f
Level 12801280
Weight 33
Character orbit 1280.g
Analytic conductor 34.87734.877
Analytic rank 00
Dimension 88
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,3,Mod(1151,1280)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 1280=285 1280 = 2^{8} \cdot 5
Weight: k k == 3 3
Character orbit: [χ][\chi] == 1280.g (of order 22, degree 11, 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: 34.877473838134.8774738381
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x83x6+8x43x2+1 x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 216 2^{16}
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β2q3+β4q5+β6q7+(3β1+9)q92β2q11+(6β42β3)q13+(β6+β5)q15+18q17+(2β72β2)q19++(6β736β2)q99+O(q100) q + \beta_{2} q^{3} + \beta_{4} q^{5} + \beta_{6} q^{7} + (3 \beta_1 + 9) q^{9} - 2 \beta_{2} q^{11} + (6 \beta_{4} - 2 \beta_{3}) q^{13} + ( - \beta_{6} + \beta_{5}) q^{15} + 18 q^{17} + (2 \beta_{7} - 2 \beta_{2}) q^{19}+ \cdots + ( - 6 \beta_{7} - 36 \beta_{2}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+72q9+144q1740q25288q3396q41+56q49192q57240q65+304q73+792q81+48q89208q97+O(q100) 8 q + 72 q^{9} + 144 q^{17} - 40 q^{25} - 288 q^{33} - 96 q^{41} + 56 q^{49} - 192 q^{57} - 240 q^{65} + 304 q^{73} + 792 q^{81} + 48 q^{89} - 208 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x83x6+8x43x2+1 x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 : Copy content Toggle raw display

β1\beta_{1}== (ν69)/2 ( -\nu^{6} - 9 ) / 2 Copy content Toggle raw display
β2\beta_{2}== (3ν78ν5+24ν317ν)/4 ( 3\nu^{7} - 8\nu^{5} + 24\nu^{3} - 17\nu ) / 4 Copy content Toggle raw display
β3\beta_{3}== (3ν78ν5+20ν3ν)/2 ( 3\nu^{7} - 8\nu^{5} + 20\nu^{3} - \nu ) / 2 Copy content Toggle raw display
β4\beta_{4}== (3ν78ν5+22ν3ν)/2 ( 3\nu^{7} - 8\nu^{5} + 22\nu^{3} - \nu ) / 2 Copy content Toggle raw display
β5\beta_{5}== (5ν616ν4+32ν27)/2 ( 5\nu^{6} - 16\nu^{4} + 32\nu^{2} - 7 ) / 2 Copy content Toggle raw display
β6\beta_{6}== (17ν648ν4+128ν227)/4 ( 17\nu^{6} - 48\nu^{4} + 128\nu^{2} - 27 ) / 4 Copy content Toggle raw display
β7\beta_{7}== (19ν7+72ν5184ν3+129ν)/4 ( -19\nu^{7} + 72\nu^{5} - 184\nu^{3} + 129\nu ) / 4 Copy content Toggle raw display
ν\nu== (2β4β32β2)/8 ( 2\beta_{4} - \beta_{3} - 2\beta_{2} ) / 8 Copy content Toggle raw display
ν2\nu^{2}== (2β63β5+2β1+12)/16 ( 2\beta_{6} - 3\beta_{5} + 2\beta _1 + 12 ) / 16 Copy content Toggle raw display
ν3\nu^{3}== β4β3 \beta_{4} - \beta_{3} Copy content Toggle raw display
ν4\nu^{4}== (2β64β53β114)/8 ( 2\beta_{6} - 4\beta_{5} - 3\beta _1 - 14 ) / 8 Copy content Toggle raw display
ν5\nu^{5}== (3β7+20β422β3+23β2)/16 ( 3\beta_{7} + 20\beta_{4} - 22\beta_{3} + 23\beta_{2} ) / 16 Copy content Toggle raw display
ν6\nu^{6}== 2β19 -2\beta _1 - 9 Copy content Toggle raw display
ν7\nu^{7}== (4β726β4+29β3+30β2)/8 ( 4\beta_{7} - 26\beta_{4} + 29\beta_{3} + 30\beta_{2} ) / 8 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1280Z)×\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times.

nn 257257 261261 511511
χ(n)\chi(n) 11 1-1 1-1

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
1151.1
1.40126 0.809017i
1.40126 + 0.809017i
0.535233 0.309017i
0.535233 + 0.309017i
−0.535233 0.309017i
−0.535233 + 0.309017i
−1.40126 0.809017i
−1.40126 + 0.809017i
0 −5.60503 0 2.23607i 0 1.32317i 0 22.4164 0
1151.2 0 −5.60503 0 2.23607i 0 1.32317i 0 22.4164 0
1151.3 0 −2.14093 0 2.23607i 0 9.06914i 0 −4.41641 0
1151.4 0 −2.14093 0 2.23607i 0 9.06914i 0 −4.41641 0
1151.5 0 2.14093 0 2.23607i 0 9.06914i 0 −4.41641 0
1151.6 0 2.14093 0 2.23607i 0 9.06914i 0 −4.41641 0
1151.7 0 5.60503 0 2.23607i 0 1.32317i 0 22.4164 0
1151.8 0 5.60503 0 2.23607i 0 1.32317i 0 22.4164 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.g.f 8
4.b odd 2 1 inner 1280.3.g.f 8
8.b even 2 1 inner 1280.3.g.f 8
8.d odd 2 1 inner 1280.3.g.f 8
16.e even 4 1 80.3.b.a 4
16.e even 4 1 320.3.b.a 4
16.f odd 4 1 80.3.b.a 4
16.f odd 4 1 320.3.b.a 4
48.i odd 4 1 720.3.e.c 4
48.i odd 4 1 2880.3.e.b 4
48.k even 4 1 720.3.e.c 4
48.k even 4 1 2880.3.e.b 4
80.i odd 4 1 400.3.h.d 8
80.i odd 4 1 1600.3.h.p 8
80.j even 4 1 400.3.h.d 8
80.j even 4 1 1600.3.h.p 8
80.k odd 4 1 400.3.b.g 4
80.k odd 4 1 1600.3.b.k 4
80.q even 4 1 400.3.b.g 4
80.q even 4 1 1600.3.b.k 4
80.s even 4 1 400.3.h.d 8
80.s even 4 1 1600.3.h.p 8
80.t odd 4 1 400.3.h.d 8
80.t odd 4 1 1600.3.h.p 8
240.t even 4 1 3600.3.e.bb 4
240.z odd 4 1 3600.3.j.k 8
240.bb even 4 1 3600.3.j.k 8
240.bd odd 4 1 3600.3.j.k 8
240.bf even 4 1 3600.3.j.k 8
240.bm odd 4 1 3600.3.e.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.b.a 4 16.e even 4 1
80.3.b.a 4 16.f odd 4 1
320.3.b.a 4 16.e even 4 1
320.3.b.a 4 16.f odd 4 1
400.3.b.g 4 80.k odd 4 1
400.3.b.g 4 80.q even 4 1
400.3.h.d 8 80.i odd 4 1
400.3.h.d 8 80.j even 4 1
400.3.h.d 8 80.s even 4 1
400.3.h.d 8 80.t odd 4 1
720.3.e.c 4 48.i odd 4 1
720.3.e.c 4 48.k even 4 1
1280.3.g.f 8 1.a even 1 1 trivial
1280.3.g.f 8 4.b odd 2 1 inner
1280.3.g.f 8 8.b even 2 1 inner
1280.3.g.f 8 8.d odd 2 1 inner
1600.3.b.k 4 80.k odd 4 1
1600.3.b.k 4 80.q even 4 1
1600.3.h.p 8 80.i odd 4 1
1600.3.h.p 8 80.j even 4 1
1600.3.h.p 8 80.s even 4 1
1600.3.h.p 8 80.t odd 4 1
2880.3.e.b 4 48.i odd 4 1
2880.3.e.b 4 48.k even 4 1
3600.3.e.bb 4 240.t even 4 1
3600.3.e.bb 4 240.bm odd 4 1
3600.3.j.k 8 240.z odd 4 1
3600.3.j.k 8 240.bb even 4 1
3600.3.j.k 8 240.bd odd 4 1
3600.3.j.k 8 240.bf even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3436T32+144 T_{3}^{4} - 36T_{3}^{2} + 144 acting on S3new(1280,[χ])S_{3}^{\mathrm{new}}(1280, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 (T436T2+144)2 (T^{4} - 36 T^{2} + 144)^{2} Copy content Toggle raw display
55 (T2+5)4 (T^{2} + 5)^{4} Copy content Toggle raw display
77 (T4+84T2+144)2 (T^{4} + 84 T^{2} + 144)^{2} Copy content Toggle raw display
1111 (T4144T2+2304)2 (T^{4} - 144 T^{2} + 2304)^{2} Copy content Toggle raw display
1313 (T4+392T2+26896)2 (T^{4} + 392 T^{2} + 26896)^{2} Copy content Toggle raw display
1717 (T18)8 (T - 18)^{8} Copy content Toggle raw display
1919 (T41344T2+36864)2 (T^{4} - 1344 T^{2} + 36864)^{2} Copy content Toggle raw display
2323 (T4+756T2+121104)2 (T^{4} + 756 T^{2} + 121104)^{2} Copy content Toggle raw display
2929 (T4+2088T2+156816)2 (T^{4} + 2088 T^{2} + 156816)^{2} Copy content Toggle raw display
3131 (T4+3024T2+2214144)2 (T^{4} + 3024 T^{2} + 2214144)^{2} Copy content Toggle raw display
3737 (T4+1160T2+48400)2 (T^{4} + 1160 T^{2} + 48400)^{2} Copy content Toggle raw display
4141 (T2+24T1476)4 (T^{2} + 24 T - 1476)^{4} Copy content Toggle raw display
4343 (T41476T2+535824)2 (T^{4} - 1476 T^{2} + 535824)^{2} Copy content Toggle raw display
4747 (T4+8244T2+898704)2 (T^{4} + 8244 T^{2} + 898704)^{2} Copy content Toggle raw display
5353 (T4+3528T2+2178576)2 (T^{4} + 3528 T^{2} + 2178576)^{2} Copy content Toggle raw display
5959 (T42304T2+589824)2 (T^{4} - 2304 T^{2} + 589824)^{2} Copy content Toggle raw display
6161 (T4+8552T2+15335056)2 (T^{4} + 8552 T^{2} + 15335056)^{2} Copy content Toggle raw display
6767 (T42436T2+1468944)2 (T^{4} - 2436 T^{2} + 1468944)^{2} Copy content Toggle raw display
7171 (T4+9936T2+3873024)2 (T^{4} + 9936 T^{2} + 3873024)^{2} Copy content Toggle raw display
7373 (T276T1436)4 (T^{2} - 76 T - 1436)^{4} Copy content Toggle raw display
7979 (T4+14400T2+23040000)2 (T^{4} + 14400 T^{2} + 23040000)^{2} Copy content Toggle raw display
8383 (T48964T2+4210704)2 (T^{4} - 8964 T^{2} + 4210704)^{2} Copy content Toggle raw display
8989 (T212T2844)4 (T^{2} - 12 T - 2844)^{4} Copy content Toggle raw display
9797 (T2+52T17324)4 (T^{2} + 52 T - 17324)^{4} Copy content Toggle raw display
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