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Properties

Label 8649.2.a.bf
Level 86498649
Weight 22
Character orbit 8649.a
Self dual yes
Analytic conductor 69.06369.063
Analytic rank 00
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] = [8649,2,Mod(1,8649)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8649, 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("8649.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 8649=32⋅312 8649 = 3^{2} \cdot 31^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 8649.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: 69.062612708269.0626127082
Analytic rank: 00
Dimension: 88
Coefficient field: 8.8.2051578125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8−2x7−9x6+19x5+14x4−28x3−11x2+6x+1 x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 31)
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+(β5−β3−1)q2+(−β6+β1+1)q4+(β6−β5+β3+⋯+β1)q5+(−β7−β5+⋯+β1)q7+(β7+β6+⋯+β2)q8+⋯+(−2β7+β6−3β5+⋯−1)q98+O(q100) q + (\beta_{5} - \beta_{3} - 1) q^{2} + ( - \beta_{6} + \beta_1 + 1) q^{4} + (\beta_{6} - \beta_{5} + \beta_{3} + \cdots + \beta_1) q^{5} + ( - \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{7} + (\beta_{7} + \beta_{6} + \cdots + \beta_{2}) q^{8}+ \cdots + ( - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} + \cdots - 1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−2q2+8q4−3q5−2q7+9q8−13q10+18q11−8q13+9q14+4q16+14q17−6q19+7q20−4q22+22q23+13q25+9q26−5q28+⋯−10q98+O(q100) 8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8} - 13 q^{10} + 18 q^{11} - 8 q^{13} + 9 q^{14} + 4 q^{16} + 14 q^{17} - 6 q^{19} + 7 q^{20} - 4 q^{22} + 22 q^{23} + 13 q^{25} + 9 q^{26} - 5 q^{28}+ \cdots - 10 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−2x7−9x6+19x5+14x4−28x3−11x2+6x+1 x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν6−7ν4+2ν3+4ν2+5ν+1)/3 ( \nu^{6} - 7\nu^{4} + 2\nu^{3} + 4\nu^{2} + 5\nu + 1 ) / 3 Copy content Toggle raw display
β3\beta_{3}== (−ν7+ν6+7ν5−9ν4+ν3−ν2−11ν+4)/3 ( -\nu^{7} + \nu^{6} + 7\nu^{5} - 9\nu^{4} + \nu^{3} - \nu^{2} - 11\nu + 4 ) / 3 Copy content Toggle raw display
β4\beta_{4}== (−2ν6+17ν4−4ν3−26ν2−ν+1)/3 ( -2\nu^{6} + 17\nu^{4} - 4\nu^{3} - 26\nu^{2} - \nu + 1 ) / 3 Copy content Toggle raw display
β5\beta_{5}== (ν7−10ν5+2ν4+25ν3−4ν2−14ν+3)/3 ( \nu^{7} - 10\nu^{5} + 2\nu^{4} + 25\nu^{3} - 4\nu^{2} - 14\nu + 3 ) / 3 Copy content Toggle raw display
β6\beta_{6}== (ν7−3ν6−7ν5+26ν4−5ν3−28ν2+10ν+6)/3 ( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 26\nu^{4} - 5\nu^{3} - 28\nu^{2} + 10\nu + 6 ) / 3 Copy content Toggle raw display
β7\beta_{7}== (ν7−4ν6−7ν5+36ν4−4ν3−47ν2−ν+2)/3 ( \nu^{7} - 4\nu^{6} - 7\nu^{5} + 36\nu^{4} - 4\nu^{3} - 47\nu^{2} - \nu + 2 ) / 3 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}== −β6+β4−β3+3 -\beta_{6} + \beta_{4} - \beta_{3} + 3 Copy content Toggle raw display
ν3\nu^{3}== β7−2β4+β3−β2+5β1−1 \beta_{7} - 2\beta_{4} + \beta_{3} - \beta_{2} + 5\beta _1 - 1 Copy content Toggle raw display
ν4\nu^{4}== −6β6+7β4−6β3+2β2−3β1+17 -6\beta_{6} + 7\beta_{4} - 6\beta_{3} + 2\beta_{2} - 3\beta _1 + 17 Copy content Toggle raw display
ν5\nu^{5}== 8β7+3β6−β5−19β4+10β3−7β2+30β1−15 8\beta_{7} + 3\beta_{6} - \beta_{5} - 19\beta_{4} + 10\beta_{3} - 7\beta_{2} + 30\beta _1 - 15 Copy content Toggle raw display
ν6\nu^{6}== −2β7−38β6+49β4−40β3+19β2−36β1+108 -2\beta_{7} - 38\beta_{6} + 49\beta_{4} - 40\beta_{3} + 19\beta_{2} - 36\beta _1 + 108 Copy content Toggle raw display
ν7\nu^{7}== 55β7+38β6−7β5−150β4+83β3−49β2+195β1−150 55\beta_{7} + 38\beta_{6} - 7\beta_{5} - 150\beta_{4} + 83\beta_{3} - 49\beta_{2} + 195\beta _1 - 150 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
2.11562
−0.662608
0.431370
−0.143490
−1.04940
−2.73366
1.76152
2.28064
−2.30753 0 3.32468 2.49846 0 1.60188 −3.05673 0 −5.76526
1.2 −2.07212 0 2.29369 −2.34791 0 −3.67454 −0.608557 0 4.86516
1.3 −1.26660 0 −0.395721 3.80032 0 2.18899 3.03442 0 −4.81349
1.4 −1.23217 0 −0.481752 1.54562 0 −3.80376 3.05795 0 −1.90447
1.5 −0.351432 0 −1.87650 −2.97323 0 −1.08213 1.36233 0 1.04489
1.6 0.689493 0 −1.52460 −3.70752 0 0.763394 −2.43019 0 −2.55631
1.7 1.85021 0 1.42326 −1.20736 0 3.73304 −1.06708 0 −2.23387
1.8 2.69016 0 5.23694 −0.608384 0 −1.72688 8.70786 0 −1.63665
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 8649.2.a.bf 8
3.b odd 2 1 961.2.a.i 8
31.b odd 2 1 8649.2.a.be 8
31.g even 15 2 279.2.y.c 16
93.c even 2 1 961.2.a.j 8
93.g even 6 2 961.2.c.i 16
93.h odd 6 2 961.2.c.j 16
93.k even 10 2 961.2.d.n 16
93.k even 10 2 961.2.d.q 16
93.l odd 10 2 961.2.d.o 16
93.l odd 10 2 961.2.d.p 16
93.o odd 30 2 31.2.g.a ✓ 16
93.o odd 30 2 961.2.g.k 16
93.o odd 30 2 961.2.g.s 16
93.o odd 30 2 961.2.g.t 16
93.p even 30 2 961.2.g.j 16
93.p even 30 2 961.2.g.l 16
93.p even 30 2 961.2.g.m 16
93.p even 30 2 961.2.g.n 16
372.bd even 30 2 496.2.bg.c 16
465.bl odd 30 2 775.2.bl.a 16
465.bt even 60 4 775.2.ck.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.g.a ✓ 16 93.o odd 30 2
279.2.y.c 16 31.g even 15 2
496.2.bg.c 16 372.bd even 30 2
775.2.bl.a 16 465.bl odd 30 2
775.2.ck.a 32 465.bt even 60 4
961.2.a.i 8 3.b odd 2 1
961.2.a.j 8 93.c even 2 1
961.2.c.i 16 93.g even 6 2
961.2.c.j 16 93.h odd 6 2
961.2.d.n 16 93.k even 10 2
961.2.d.o 16 93.l odd 10 2
961.2.d.p 16 93.l odd 10 2
961.2.d.q 16 93.k even 10 2
961.2.g.j 16 93.p even 30 2
961.2.g.k 16 93.o odd 30 2
961.2.g.l 16 93.p even 30 2
961.2.g.m 16 93.p even 30 2
961.2.g.n 16 93.p even 30 2
961.2.g.s 16 93.o odd 30 2
961.2.g.t 16 93.o odd 30 2
8649.2.a.be 8 31.b odd 2 1
8649.2.a.bf 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(8649))S_{2}^{\mathrm{new}}(\Gamma_0(8649)):

T28+2T27−10T26−23T25+19T24+63T23+15T22−27T2−9 T_{2}^{8} + 2T_{2}^{7} - 10T_{2}^{6} - 23T_{2}^{5} + 19T_{2}^{4} + 63T_{2}^{3} + 15T_{2}^{2} - 27T_{2} - 9 Copy content Toggle raw display
T58+3T57−22T56−69T55+115T54+411T53−57T52−612T5−279 T_{5}^{8} + 3T_{5}^{7} - 22T_{5}^{6} - 69T_{5}^{5} + 115T_{5}^{4} + 411T_{5}^{3} - 57T_{5}^{2} - 612T_{5} - 279 Copy content Toggle raw display
T78+2T77−25T76−38T75+184T74+153T73−435T72−162T7+261 T_{7}^{8} + 2T_{7}^{7} - 25T_{7}^{6} - 38T_{7}^{5} + 184T_{7}^{4} + 153T_{7}^{3} - 435T_{7}^{2} - 162T_{7} + 261 Copy content Toggle raw display
T118−18T117+131T116−489T115+964T114−852T113−51T112+594T11−279 T_{11}^{8} - 18T_{11}^{7} + 131T_{11}^{6} - 489T_{11}^{5} + 964T_{11}^{4} - 852T_{11}^{3} - 51T_{11}^{2} + 594T_{11} - 279 Copy content Toggle raw display
T138+8T137−4T136−161T135−281T134+627T133+2064T132+1296T13−279 T_{13}^{8} + 8T_{13}^{7} - 4T_{13}^{6} - 161T_{13}^{5} - 281T_{13}^{4} + 627T_{13}^{3} + 2064T_{13}^{2} + 1296T_{13} - 279 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+2T7+⋯−9 T^{8} + 2 T^{7} + \cdots - 9 Copy content Toggle raw display
33 T8 T^{8} Copy content Toggle raw display
55 T8+3T7+⋯−279 T^{8} + 3 T^{7} + \cdots - 279 Copy content Toggle raw display
77 T8+2T7+⋯+261 T^{8} + 2 T^{7} + \cdots + 261 Copy content Toggle raw display
1111 T8−18T7+⋯−279 T^{8} - 18 T^{7} + \cdots - 279 Copy content Toggle raw display
1313 T8+8T7+⋯−279 T^{8} + 8 T^{7} + \cdots - 279 Copy content Toggle raw display
1717 T8−14T7+⋯−8649 T^{8} - 14 T^{7} + \cdots - 8649 Copy content Toggle raw display
1919 T8+6T7+⋯+601 T^{8} + 6 T^{7} + \cdots + 601 Copy content Toggle raw display
2323 T8−22T7+⋯−279 T^{8} - 22 T^{7} + \cdots - 279 Copy content Toggle raw display
2929 T8−12T7+⋯−279 T^{8} - 12 T^{7} + \cdots - 279 Copy content Toggle raw display
3131 T8 T^{8} Copy content Toggle raw display
3737 T8−8T7+⋯−18569 T^{8} - 8 T^{7} + \cdots - 18569 Copy content Toggle raw display
4141 T8−22T7+⋯−9 T^{8} - 22 T^{7} + \cdots - 9 Copy content Toggle raw display
4343 T8−2T7+⋯−2759 T^{8} - 2 T^{7} + \cdots - 2759 Copy content Toggle raw display
4747 T8−18T7+⋯+57501 T^{8} - 18 T^{7} + \cdots + 57501 Copy content Toggle raw display
5353 T8−6T7+⋯+605151 T^{8} - 6 T^{7} + \cdots + 605151 Copy content Toggle raw display
5959 T8−4T7+⋯+12951 T^{8} - 4 T^{7} + \cdots + 12951 Copy content Toggle raw display
6161 T8+30T7+⋯+38161 T^{8} + 30 T^{7} + \cdots + 38161 Copy content Toggle raw display
6767 T8+13T7+⋯+86521 T^{8} + 13 T^{7} + \cdots + 86521 Copy content Toggle raw display
7171 T8−T7+⋯+14661 T^{8} - T^{7} + \cdots + 14661 Copy content Toggle raw display
7373 T8+2T7+⋯+4176351 T^{8} + 2 T^{7} + \cdots + 4176351 Copy content Toggle raw display
7979 T8+8T7+⋯+9198351 T^{8} + 8 T^{7} + \cdots + 9198351 Copy content Toggle raw display
8383 T8−39T7+⋯−1202769 T^{8} - 39 T^{7} + \cdots - 1202769 Copy content Toggle raw display
8989 T8−27T7+⋯−343449 T^{8} - 27 T^{7} + \cdots - 343449 Copy content Toggle raw display
9797 T8−34T7+⋯−2670579 T^{8} - 34 T^{7} + \cdots - 2670579 Copy content Toggle raw display
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