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Properties

Label 2883.2.a.r
Level 28832883
Weight 22
Character orbit 2883.a
Self dual yes
Analytic conductor 23.02123.021
Analytic rank 11
Dimension 88
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2883,2,Mod(1,2883)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2883, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2883.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 2883=3⋅312 2883 = 3 \cdot 31^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2883.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: 23.020870902723.0208709027
Analytic rank: 11
Dimension: 88
Coefficient field: 8.8.1413480448.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8−4x7−2x6+16x5−x4−16x3+2x2+4x−1 x^{8} - 4x^{7} - 2x^{6} + 16x^{5} - x^{4} - 16x^{3} + 2x^{2} + 4x - 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(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+(−β7+β1)q2+q3+(−β5−β3−β2−1)q4+(2β7+β4−β1+1)q5+(−β7+β1)q6+(−β7+β6−β5+⋯−1)q7+⋯+(β5+β4−β2+⋯−1)q99+O(q100) q + ( - \beta_{7} + \beta_1) q^{2} + q^{3} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{4} + (2 \beta_{7} + \beta_{4} - \beta_1 + 1) q^{5} + ( - \beta_{7} + \beta_1) q^{6} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 1) q^{7}+ \cdots + (\beta_{5} + \beta_{4} - \beta_{2} + \cdots - 1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q+4q2+8q3+4q4+4q6−8q7+8q9−20q10−8q11+4q12−16q13−12q14+4q16+4q18−16q19−12q20−8q21−16q23+8q25+⋯−8q99+O(q100) 8 q + 4 q^{2} + 8 q^{3} + 4 q^{4} + 4 q^{6} - 8 q^{7} + 8 q^{9} - 20 q^{10} - 8 q^{11} + 4 q^{12} - 16 q^{13} - 12 q^{14} + 4 q^{16} + 4 q^{18} - 16 q^{19} - 12 q^{20} - 8 q^{21} - 16 q^{23} + 8 q^{25}+ \cdots - 8 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−4x7−2x6+16x5−x4−16x3+2x2+4x−1 x^{8} - 4x^{7} - 2x^{6} + 16x^{5} - x^{4} - 16x^{3} + 2x^{2} + 4x - 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν7−4ν6−2ν5+16ν4−ν3−16ν2+2ν+3 \nu^{7} - 4\nu^{6} - 2\nu^{5} + 16\nu^{4} - \nu^{3} - 16\nu^{2} + 2\nu + 3 Copy content Toggle raw display
β3\beta_{3}== −2ν7+8ν6+3ν5−28ν4+3ν3+20ν2−2ν−3 -2\nu^{7} + 8\nu^{6} + 3\nu^{5} - 28\nu^{4} + 3\nu^{3} + 20\nu^{2} - 2\nu - 3 Copy content Toggle raw display
β4\beta_{4}== −2ν7+9ν6−ν5−29ν4+16ν3+15ν2−8ν+1 -2\nu^{7} + 9\nu^{6} - \nu^{5} - 29\nu^{4} + 16\nu^{3} + 15\nu^{2} - 8\nu + 1 Copy content Toggle raw display
β5\beta_{5}== −3ν7+12ν6+5ν5−44ν4+4ν3+35ν2−2ν−5 -3\nu^{7} + 12\nu^{6} + 5\nu^{5} - 44\nu^{4} + 4\nu^{3} + 35\nu^{2} - 2\nu - 5 Copy content Toggle raw display
β6\beta_{6}== 2ν7−7ν6−8ν5+30ν4+14ν3−33ν2−11ν+8 2\nu^{7} - 7\nu^{6} - 8\nu^{5} + 30\nu^{4} + 14\nu^{3} - 33\nu^{2} - 11\nu + 8 Copy content Toggle raw display
β7\beta_{7}== −2ν7+6ν6+12ν5−29ν4−26ν3+35ν2+16ν−9 -2\nu^{7} + 6\nu^{6} + 12\nu^{5} - 29\nu^{4} - 26\nu^{3} + 35\nu^{2} + 16\nu - 9 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== −β5+β3−β2+2β1+1 -\beta_{5} + \beta_{3} - \beta_{2} + 2\beta _1 + 1 Copy content Toggle raw display
ν3\nu^{3}== β7+β6−3β5+β4+2β3−3β2+7β1 \beta_{7} + \beta_{6} - 3\beta_{5} + \beta_{4} + 2\beta_{3} - 3\beta_{2} + 7\beta_1 Copy content Toggle raw display
ν4\nu^{4}== 3β7+2β6−13β5+4β4+9β3−11β2+20β1+2 3\beta_{7} + 2\beta_{6} - 13\beta_{5} + 4\beta_{4} + 9\beta_{3} - 11\beta_{2} + 20\beta _1 + 2 Copy content Toggle raw display
ν5\nu^{5}== 13β7+9β6−43β5+17β4+25β3−37β2+65β1−1 13\beta_{7} + 9\beta_{6} - 43\beta_{5} + 17\beta_{4} + 25\beta_{3} - 37\beta_{2} + 65\beta _1 - 1 Copy content Toggle raw display
ν6\nu^{6}== 42β7+25β6−151β5+60β4+87β3−125β2+205β1−1 42\beta_{7} + 25\beta_{6} - 151\beta_{5} + 60\beta_{4} + 87\beta_{3} - 125\beta_{2} + 205\beta _1 - 1 Copy content Toggle raw display
ν7\nu^{7}== 147β7+87β6−501β5+211β4+272β3−416β2+667β1−25 147\beta_{7} + 87\beta_{6} - 501\beta_{5} + 211\beta_{4} + 272\beta_{3} - 416\beta_{2} + 667\beta _1 - 25 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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}
1.1
−0.867710
0.426500
−1.55182
1.98771
−0.733914
0.319701
3.28192
1.13761
−2.28192 1.00000 3.20718 2.11277 −2.28192 1.56903 −2.75468 1.00000 −4.82118
1.2 −0.987714 1.00000 −1.02442 4.03512 −0.987714 −3.85422 2.98726 1.00000 −3.98554
1.3 −0.137609 1.00000 −1.98106 −0.677863 −0.137609 2.14378 0.547830 1.00000 0.0932799
1.4 0.573500 1.00000 −1.67110 −1.97206 0.573500 0.440009 −2.10538 1.00000 −1.13098
1.5 0.680299 1.00000 −1.53719 0.966483 0.680299 −4.84476 −2.40635 1.00000 0.657497
1.6 1.73391 1.00000 1.00646 −1.94715 1.73391 4.25897 −1.72271 1.00000 −3.37619
1.7 1.86771 1.00000 1.48834 1.48102 1.86771 −4.98324 −0.955629 1.00000 2.76613
1.8 2.55182 1.00000 4.51180 −3.99832 2.55182 −2.72957 6.40966 1.00000 −10.2030
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 −1 -1
3131 −1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2883.2.a.r yes 8
3.b odd 2 1 8649.2.a.bd 8
31.b odd 2 1 2883.2.a.q ✓ 8
93.c even 2 1 8649.2.a.bc 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2883.2.a.q ✓ 8 31.b odd 2 1
2883.2.a.r yes 8 1.a even 1 1 trivial
8649.2.a.bc 8 93.c even 2 1
8649.2.a.bd 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(2883))S_{2}^{\mathrm{new}}(\Gamma_0(2883)):

T28−4T27−2T26+24T25−21T24−16T23+22T22−4T2−1 T_{2}^{8} - 4T_{2}^{7} - 2T_{2}^{6} + 24T_{2}^{5} - 21T_{2}^{4} - 16T_{2}^{3} + 22T_{2}^{2} - 4T_{2} - 1 Copy content Toggle raw display
T118+8T117−4T116−184T115−506T114−112T113+1328T112+1872T11+764 T_{11}^{8} + 8T_{11}^{7} - 4T_{11}^{6} - 184T_{11}^{5} - 506T_{11}^{4} - 112T_{11}^{3} + 1328T_{11}^{2} + 1872T_{11} + 764 Copy content Toggle raw display
T138+16T137+68T136−176T135−2226T134−6240T133−5112T132+4000T13+5956 T_{13}^{8} + 16T_{13}^{7} + 68T_{13}^{6} - 176T_{13}^{5} - 2226T_{13}^{4} - 6240T_{13}^{3} - 5112T_{13}^{2} + 4000T_{13} + 5956 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8−4T7+⋯−1 T^{8} - 4 T^{7} + \cdots - 1 Copy content Toggle raw display
33 (T−1)8 (T - 1)^{8} Copy content Toggle raw display
55 T8−24T6+⋯+127 T^{8} - 24 T^{6} + \cdots + 127 Copy content Toggle raw display
77 T8+8T7+⋯+1601 T^{8} + 8 T^{7} + \cdots + 1601 Copy content Toggle raw display
1111 T8+8T7+⋯+764 T^{8} + 8 T^{7} + \cdots + 764 Copy content Toggle raw display
1313 T8+16T7+⋯+5956 T^{8} + 16 T^{7} + \cdots + 5956 Copy content Toggle raw display
1717 T8−56T6+⋯−64 T^{8} - 56 T^{6} + \cdots - 64 Copy content Toggle raw display
1919 T8+16T7+⋯+10369 T^{8} + 16 T^{7} + \cdots + 10369 Copy content Toggle raw display
2323 T8+16T7+⋯−772 T^{8} + 16 T^{7} + \cdots - 772 Copy content Toggle raw display
2929 T8+16T7+⋯+7996 T^{8} + 16 T^{7} + \cdots + 7996 Copy content Toggle raw display
3131 T8 T^{8} Copy content Toggle raw display
3737 T8+16T7+⋯+4 T^{8} + 16 T^{7} + \cdots + 4 Copy content Toggle raw display
4141 T8−244T6+⋯+2938879 T^{8} - 244 T^{6} + \cdots + 2938879 Copy content Toggle raw display
4343 T8+32T7+⋯+4 T^{8} + 32 T^{7} + \cdots + 4 Copy content Toggle raw display
4747 T8+8T7+⋯−69376 T^{8} + 8 T^{7} + \cdots - 69376 Copy content Toggle raw display
5353 T8+16T7+⋯−392192 T^{8} + 16 T^{7} + \cdots - 392192 Copy content Toggle raw display
5959 T8−24T7+⋯+813343 T^{8} - 24 T^{7} + \cdots + 813343 Copy content Toggle raw display
6161 T8+48T7+⋯+43072 T^{8} + 48 T^{7} + \cdots + 43072 Copy content Toggle raw display
6767 T8+40T7+⋯+10211584 T^{8} + 40 T^{7} + \cdots + 10211584 Copy content Toggle raw display
7171 T8−16T7+⋯−92993 T^{8} - 16 T^{7} + \cdots - 92993 Copy content Toggle raw display
7373 T8+48T7+⋯+1604 T^{8} + 48 T^{7} + \cdots + 1604 Copy content Toggle raw display
7979 T8−296T6+⋯+12356 T^{8} - 296 T^{6} + \cdots + 12356 Copy content Toggle raw display
8383 T8+24T7+⋯−31610948 T^{8} + 24 T^{7} + \cdots - 31610948 Copy content Toggle raw display
8989 T8+24T7+⋯−23670788 T^{8} + 24 T^{7} + \cdots - 23670788 Copy content Toggle raw display
9797 T8−252T6+⋯+3697 T^{8} - 252 T^{6} + \cdots + 3697 Copy content Toggle raw display
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