Properties

Label 200.2.f.d
Level 200200
Weight 22
Character orbit 200.f
Analytic conductor 1.5971.597
Analytic rank 00
Dimension 44
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,2,Mod(149,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 200=2352 200 = 2^{3} \cdot 5^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 200.f (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: 1.597008040431.59700804043
Analytic rank: 00
Dimension: 44
Coefficient field: Q(i,7)\Q(i, \sqrt{7})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x43x2+4 x^{4} - 3x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
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) == q+β1q2+(β32β1)q3+(β2+1)q4+(β23)q6+4β3q7+(2β3+β1)q8+4q9+(2β21)q11+(2β33β1)q12++(8β24)q99+O(q100) q + \beta_1 q^{2} + (\beta_{3} - 2 \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} - 3) q^{6} + 4 \beta_{3} q^{7} + (2 \beta_{3} + \beta_1) q^{8} + 4 q^{9} + (2 \beta_{2} - 1) q^{11} + ( - 2 \beta_{3} - 3 \beta_1) q^{12}+ \cdots + (8 \beta_{2} - 4) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+6q414q6+16q98q14+2q1614q24+16q31+6q34+24q3620q4114q44+8q4636q4914q5440q5618q64+14q66+32q71++14q96+O(q100) 4 q + 6 q^{4} - 14 q^{6} + 16 q^{9} - 8 q^{14} + 2 q^{16} - 14 q^{24} + 16 q^{31} + 6 q^{34} + 24 q^{36} - 20 q^{41} - 14 q^{44} + 8 q^{46} - 36 q^{49} - 14 q^{54} - 40 q^{56} - 18 q^{64} + 14 q^{66} + 32 q^{71}+ \cdots + 14 q^{96}+O(q^{100}) Copy content Toggle raw display

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

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν21 \nu^{2} - 1 Copy content Toggle raw display
β3\beta_{3}== (ν3ν)/2 ( \nu^{3} - \nu ) / 2 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+1 \beta_{2} + 1 Copy content Toggle raw display
ν3\nu^{3}== 2β3+β1 2\beta_{3} + \beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/200Z)×\left(\mathbb{Z}/200\mathbb{Z}\right)^\times.

nn 101101 151151 177177
χ(n)\chi(n) 1-1 11 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}
149.1
−1.32288 0.500000i
−1.32288 + 0.500000i
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i 2.64575 1.50000 + 1.32288i 0 −3.50000 1.32288i 4.00000i −1.32288 2.50000i 4.00000 0
149.2 −1.32288 + 0.500000i 2.64575 1.50000 1.32288i 0 −3.50000 + 1.32288i 4.00000i −1.32288 + 2.50000i 4.00000 0
149.3 1.32288 0.500000i −2.64575 1.50000 1.32288i 0 −3.50000 + 1.32288i 4.00000i 1.32288 2.50000i 4.00000 0
149.4 1.32288 + 0.500000i −2.64575 1.50000 + 1.32288i 0 −3.50000 1.32288i 4.00000i 1.32288 + 2.50000i 4.00000 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.2.f.d 4
3.b odd 2 1 1800.2.d.m 4
4.b odd 2 1 800.2.f.d 4
5.b even 2 1 inner 200.2.f.d 4
5.c odd 4 1 200.2.d.b 2
5.c odd 4 1 200.2.d.c yes 2
8.b even 2 1 inner 200.2.f.d 4
8.d odd 2 1 800.2.f.d 4
12.b even 2 1 7200.2.d.m 4
15.d odd 2 1 1800.2.d.m 4
15.e even 4 1 1800.2.k.d 2
15.e even 4 1 1800.2.k.f 2
20.d odd 2 1 800.2.f.d 4
20.e even 4 1 800.2.d.a 2
20.e even 4 1 800.2.d.d 2
24.f even 2 1 7200.2.d.m 4
24.h odd 2 1 1800.2.d.m 4
40.e odd 2 1 800.2.f.d 4
40.f even 2 1 inner 200.2.f.d 4
40.i odd 4 1 200.2.d.b 2
40.i odd 4 1 200.2.d.c yes 2
40.k even 4 1 800.2.d.a 2
40.k even 4 1 800.2.d.d 2
60.h even 2 1 7200.2.d.m 4
60.l odd 4 1 7200.2.k.b 2
60.l odd 4 1 7200.2.k.i 2
80.i odd 4 1 6400.2.a.bg 2
80.i odd 4 1 6400.2.a.cc 2
80.j even 4 1 6400.2.a.bh 2
80.j even 4 1 6400.2.a.cb 2
80.s even 4 1 6400.2.a.bh 2
80.s even 4 1 6400.2.a.cb 2
80.t odd 4 1 6400.2.a.bg 2
80.t odd 4 1 6400.2.a.cc 2
120.i odd 2 1 1800.2.d.m 4
120.m even 2 1 7200.2.d.m 4
120.q odd 4 1 7200.2.k.b 2
120.q odd 4 1 7200.2.k.i 2
120.w even 4 1 1800.2.k.d 2
120.w even 4 1 1800.2.k.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.b 2 5.c odd 4 1
200.2.d.b 2 40.i odd 4 1
200.2.d.c yes 2 5.c odd 4 1
200.2.d.c yes 2 40.i odd 4 1
200.2.f.d 4 1.a even 1 1 trivial
200.2.f.d 4 5.b even 2 1 inner
200.2.f.d 4 8.b even 2 1 inner
200.2.f.d 4 40.f even 2 1 inner
800.2.d.a 2 20.e even 4 1
800.2.d.a 2 40.k even 4 1
800.2.d.d 2 20.e even 4 1
800.2.d.d 2 40.k even 4 1
800.2.f.d 4 4.b odd 2 1
800.2.f.d 4 8.d odd 2 1
800.2.f.d 4 20.d odd 2 1
800.2.f.d 4 40.e odd 2 1
1800.2.d.m 4 3.b odd 2 1
1800.2.d.m 4 15.d odd 2 1
1800.2.d.m 4 24.h odd 2 1
1800.2.d.m 4 120.i odd 2 1
1800.2.k.d 2 15.e even 4 1
1800.2.k.d 2 120.w even 4 1
1800.2.k.f 2 15.e even 4 1
1800.2.k.f 2 120.w even 4 1
6400.2.a.bg 2 80.i odd 4 1
6400.2.a.bg 2 80.t odd 4 1
6400.2.a.bh 2 80.j even 4 1
6400.2.a.bh 2 80.s even 4 1
6400.2.a.cb 2 80.j even 4 1
6400.2.a.cb 2 80.s even 4 1
6400.2.a.cc 2 80.i odd 4 1
6400.2.a.cc 2 80.t odd 4 1
7200.2.d.m 4 12.b even 2 1
7200.2.d.m 4 24.f even 2 1
7200.2.d.m 4 60.h even 2 1
7200.2.d.m 4 120.m even 2 1
7200.2.k.b 2 60.l odd 4 1
7200.2.k.b 2 120.q odd 4 1
7200.2.k.i 2 60.l odd 4 1
7200.2.k.i 2 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T327 T_{3}^{2} - 7 acting on S2new(200,[χ])S_{2}^{\mathrm{new}}(200, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T43T2+4 T^{4} - 3T^{2} + 4 Copy content Toggle raw display
33 (T27)2 (T^{2} - 7)^{2} Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 (T2+16)2 (T^{2} + 16)^{2} Copy content Toggle raw display
1111 (T2+7)2 (T^{2} + 7)^{2} Copy content Toggle raw display
1313 T4 T^{4} Copy content Toggle raw display
1717 (T2+9)2 (T^{2} + 9)^{2} Copy content Toggle raw display
1919 (T2+7)2 (T^{2} + 7)^{2} Copy content Toggle raw display
2323 (T2+16)2 (T^{2} + 16)^{2} Copy content Toggle raw display
2929 T4 T^{4} Copy content Toggle raw display
3131 (T4)4 (T - 4)^{4} Copy content Toggle raw display
3737 (T2112)2 (T^{2} - 112)^{2} Copy content Toggle raw display
4141 (T+5)4 (T + 5)^{4} Copy content Toggle raw display
4343 (T228)2 (T^{2} - 28)^{2} Copy content Toggle raw display
4747 (T2+64)2 (T^{2} + 64)^{2} Copy content Toggle raw display
5353 (T2112)2 (T^{2} - 112)^{2} Copy content Toggle raw display
5959 (T2+28)2 (T^{2} + 28)^{2} Copy content Toggle raw display
6161 (T2+112)2 (T^{2} + 112)^{2} Copy content Toggle raw display
6767 (T263)2 (T^{2} - 63)^{2} Copy content Toggle raw display
7171 (T8)4 (T - 8)^{4} Copy content Toggle raw display
7373 (T2+49)2 (T^{2} + 49)^{2} Copy content Toggle raw display
7979 (T+4)4 (T + 4)^{4} Copy content Toggle raw display
8383 (T263)2 (T^{2} - 63)^{2} Copy content Toggle raw display
8989 (T1)4 (T - 1)^{4} Copy content Toggle raw display
9797 (T2+4)2 (T^{2} + 4)^{2} Copy content Toggle raw display
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