Properties

Label 800.3.b.b
Level 800800
Weight 33
Character orbit 800.b
Analytic conductor 21.79821.798
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,3,Mod(351,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.351");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 800=2552 800 = 2^{5} \cdot 5^{2}
Weight: k k == 3 3
Character orbit: [χ][\chi] == 800.b (of order 22, degree 11, 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: 21.798421148821.7984211488
Analytic rank: 00
Dimension: 44
Coefficient field: Q(i,11)\Q(i, \sqrt{11})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x45x2+9 x^{4} - 5x^{2} + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 26 2^{6}
Twist minimal: yes
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,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ2q3+2β2q7+(β16)q9+(β32β2)q114q13+(β11)q17+(2β3+7β2)q19+(2β1+30)q21++(23β3+49β2)q99+O(q100) q - \beta_{2} q^{3} + 2 \beta_{2} q^{7} + ( - \beta_1 - 6) q^{9} + ( - \beta_{3} - 2 \beta_{2}) q^{11} - 4 q^{13} + ( - \beta_1 - 1) q^{17} + ( - 2 \beta_{3} + 7 \beta_{2}) q^{19} + (2 \beta_1 + 30) q^{21}+ \cdots + ( - 23 \beta_{3} + 49 \beta_{2}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q24q916q134q17+120q21+96q29148q3324q37+132q4144q49104q53+364q57+216q61+232q69172q73+296q77+308q81++184q97+O(q100) 4 q - 24 q^{9} - 16 q^{13} - 4 q^{17} + 120 q^{21} + 96 q^{29} - 148 q^{33} - 24 q^{37} + 132 q^{41} - 44 q^{49} - 104 q^{53} + 364 q^{57} + 216 q^{61} + 232 q^{69} - 172 q^{73} + 296 q^{77} + 308 q^{81}+ \cdots + 184 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x45x2+9 x^{4} - 5x^{2} + 9 : Copy content Toggle raw display

β1\beta_{1}== (4ν3+32ν)/3 ( -4\nu^{3} + 32\nu ) / 3 Copy content Toggle raw display
β2\beta_{2}== (2ν3+6ν24ν15)/3 ( 2\nu^{3} + 6\nu^{2} - 4\nu - 15 ) / 3 Copy content Toggle raw display
β3\beta_{3}== (2ν3+6ν2+4ν15)/3 ( -2\nu^{3} + 6\nu^{2} + 4\nu - 15 ) / 3 Copy content Toggle raw display
ν\nu== (β3+β2+β1)/8 ( -\beta_{3} + \beta_{2} + \beta_1 ) / 8 Copy content Toggle raw display
ν2\nu^{2}== (β3+β2+10)/4 ( \beta_{3} + \beta_{2} + 10 ) / 4 Copy content Toggle raw display
ν3\nu^{3}== (4β3+4β2+β1)/4 ( -4\beta_{3} + 4\beta_{2} + \beta_1 ) / 4 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/800Z)×\left(\mathbb{Z}/800\mathbb{Z}\right)^\times.

nn 101101 351351 577577
χ(n)\chi(n) 11 1-1 11

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}
351.1
1.65831 + 0.500000i
−1.65831 0.500000i
−1.65831 + 0.500000i
1.65831 0.500000i
0 5.31662i 0 0 0 10.6332i 0 −19.2665 0
351.2 0 1.31662i 0 0 0 2.63325i 0 7.26650 0
351.3 0 1.31662i 0 0 0 2.63325i 0 7.26650 0
351.4 0 5.31662i 0 0 0 10.6332i 0 −19.2665 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

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 800.3.b.b 4
4.b odd 2 1 inner 800.3.b.b 4
5.b even 2 1 800.3.b.c yes 4
5.c odd 4 1 800.3.h.d 4
5.c odd 4 1 800.3.h.i 4
8.b even 2 1 1600.3.b.m 4
8.d odd 2 1 1600.3.b.m 4
20.d odd 2 1 800.3.b.c yes 4
20.e even 4 1 800.3.h.d 4
20.e even 4 1 800.3.h.i 4
40.e odd 2 1 1600.3.b.l 4
40.f even 2 1 1600.3.b.l 4
40.i odd 4 1 1600.3.h.e 4
40.i odd 4 1 1600.3.h.l 4
40.k even 4 1 1600.3.h.e 4
40.k even 4 1 1600.3.h.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.3.b.b 4 1.a even 1 1 trivial
800.3.b.b 4 4.b odd 2 1 inner
800.3.b.c yes 4 5.b even 2 1
800.3.b.c yes 4 20.d odd 2 1
800.3.h.d 4 5.c odd 4 1
800.3.h.d 4 20.e even 4 1
800.3.h.i 4 5.c odd 4 1
800.3.h.i 4 20.e even 4 1
1600.3.b.l 4 40.e odd 2 1
1600.3.b.l 4 40.f even 2 1
1600.3.b.m 4 8.b even 2 1
1600.3.b.m 4 8.d odd 2 1
1600.3.h.e 4 40.i odd 4 1
1600.3.h.e 4 40.k even 4 1
1600.3.h.l 4 40.i odd 4 1
1600.3.h.l 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S3new(800,[χ])S_{3}^{\mathrm{new}}(800, [\chi]):

T34+30T32+49 T_{3}^{4} + 30T_{3}^{2} + 49 Copy content Toggle raw display
T13+4 T_{13} + 4 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 T4+30T2+49 T^{4} + 30T^{2} + 49 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 T4+120T2+784 T^{4} + 120T^{2} + 784 Copy content Toggle raw display
1111 T4+206T2+9025 T^{4} + 206T^{2} + 9025 Copy content Toggle raw display
1313 (T+4)4 (T + 4)^{4} Copy content Toggle raw display
1717 (T2+2T175)2 (T^{2} + 2 T - 175)^{2} Copy content Toggle raw display
1919 T4+1198T2+2401 T^{4} + 1198 T^{2} + 2401 Copy content Toggle raw display
2323 T4+824T2+144400 T^{4} + 824 T^{2} + 144400 Copy content Toggle raw display
2929 (T248T128)2 (T^{2} - 48 T - 128)^{2} Copy content Toggle raw display
3131 T4+2680T2+1567504 T^{4} + 2680 T^{2} + 1567504 Copy content Toggle raw display
3737 (T2+12T2780)2 (T^{2} + 12 T - 2780)^{2} Copy content Toggle raw display
4141 (T266T+385)2 (T^{2} - 66 T + 385)^{2} Copy content Toggle raw display
4343 T4+1440T2+473344 T^{4} + 1440 T^{2} + 473344 Copy content Toggle raw display
4747 (T2+176)2 (T^{2} + 176)^{2} Copy content Toggle raw display
5353 (T2+52T28)2 (T^{2} + 52 T - 28)^{2} Copy content Toggle raw display
5959 T4+10368T2+13075456 T^{4} + 10368 T^{2} + 13075456 Copy content Toggle raw display
6161 (T2108T+2212)2 (T^{2} - 108 T + 2212)^{2} Copy content Toggle raw display
6767 T4+21886T2+97318225 T^{4} + 21886 T^{2} + 97318225 Copy content Toggle raw display
7171 T4+25440T2+152967424 T^{4} + 25440 T^{2} + 152967424 Copy content Toggle raw display
7373 (T2+86T+1673)2 (T^{2} + 86 T + 1673)^{2} Copy content Toggle raw display
7979 T4+6200T2+5326864 T^{4} + 6200 T^{2} + 5326864 Copy content Toggle raw display
8383 T4+4014T2+429025 T^{4} + 4014 T^{2} + 429025 Copy content Toggle raw display
8989 (T2110T1375)2 (T^{2} - 110 T - 1375)^{2} Copy content Toggle raw display
9797 (T46)4 (T - 46)^{4} Copy content Toggle raw display
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