Properties

Label 1521.2.b.h
Level 15211521
Weight 22
Character orbit 1521.b
Analytic conductor 12.14512.145
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] = [1521,2,Mod(1351,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1521.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1521=32132 1521 = 3^{2} \cdot 13^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1521.b (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: 12.145246147412.1452461474
Analytic rank: 00
Dimension: 44
Coefficient field: Q(i,17)\Q(i, \sqrt{17})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+9x2+16 x^{4} + 9x^{2} + 16 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 39)
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+(β33)q4+(2β2+β1)q5+(β2β1)q7+(4β2β1)q8+(β33)q102β2q11+(2β3+6)q14++(12β2β1)q98+O(q100) q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + (2 \beta_{2} + \beta_1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8} + ( - \beta_{3} - 3) q^{10} - 2 \beta_{2} q^{11} + ( - 2 \beta_{3} + 6) q^{14}+ \cdots + (12 \beta_{2} - \beta_1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q10q414q10+20q14+6q16+2q174q22+8q236q252q29+8q3528q3810q4010q43+2q4922q53+12q5544q56+32q61+52q95+O(q100) 4 q - 10 q^{4} - 14 q^{10} + 20 q^{14} + 6 q^{16} + 2 q^{17} - 4 q^{22} + 8 q^{23} - 6 q^{25} - 2 q^{29} + 8 q^{35} - 28 q^{38} - 10 q^{40} - 10 q^{43} + 2 q^{49} - 22 q^{53} + 12 q^{55} - 44 q^{56} + 32 q^{61}+ \cdots - 52 q^{95}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4+9x2+16 x^{4} + 9x^{2} + 16 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν3+5ν)/4 ( \nu^{3} + 5\nu ) / 4 Copy content Toggle raw display
β3\beta_{3}== ν2+5 \nu^{2} + 5 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β35 \beta_{3} - 5 Copy content Toggle raw display
ν3\nu^{3}== 4β25β1 4\beta_{2} - 5\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1521Z)×\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times.

nn 677677 847847
χ(n)\chi(n) 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}
1351.1
2.56155i
1.56155i
1.56155i
2.56155i
2.56155i 0 −4.56155 0.561553i 0 3.56155i 6.56155i 0 −1.43845
1351.2 1.56155i 0 −0.438447 3.56155i 0 0.561553i 2.43845i 0 −5.56155
1351.3 1.56155i 0 −0.438447 3.56155i 0 0.561553i 2.43845i 0 −5.56155
1351.4 2.56155i 0 −4.56155 0.561553i 0 3.56155i 6.56155i 0 −1.43845
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.b.h 4
3.b odd 2 1 507.2.b.d 4
13.b even 2 1 inner 1521.2.b.h 4
13.d odd 4 1 1521.2.a.g 2
13.d odd 4 1 1521.2.a.m 2
13.f odd 12 2 117.2.g.c 4
39.d odd 2 1 507.2.b.d 4
39.f even 4 1 507.2.a.d 2
39.f even 4 1 507.2.a.g 2
39.h odd 6 2 507.2.j.g 8
39.i odd 6 2 507.2.j.g 8
39.k even 12 2 39.2.e.b 4
39.k even 12 2 507.2.e.g 4
52.l even 12 2 1872.2.t.r 4
156.l odd 4 1 8112.2.a.bk 2
156.l odd 4 1 8112.2.a.bo 2
156.v odd 12 2 624.2.q.h 4
195.bc odd 12 2 975.2.bb.i 8
195.bh even 12 2 975.2.i.k 4
195.bn odd 12 2 975.2.bb.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 39.k even 12 2
117.2.g.c 4 13.f odd 12 2
507.2.a.d 2 39.f even 4 1
507.2.a.g 2 39.f even 4 1
507.2.b.d 4 3.b odd 2 1
507.2.b.d 4 39.d odd 2 1
507.2.e.g 4 39.k even 12 2
507.2.j.g 8 39.h odd 6 2
507.2.j.g 8 39.i odd 6 2
624.2.q.h 4 156.v odd 12 2
975.2.i.k 4 195.bh even 12 2
975.2.bb.i 8 195.bc odd 12 2
975.2.bb.i 8 195.bn odd 12 2
1521.2.a.g 2 13.d odd 4 1
1521.2.a.m 2 13.d odd 4 1
1521.2.b.h 4 1.a even 1 1 trivial
1521.2.b.h 4 13.b even 2 1 inner
1872.2.t.r 4 52.l even 12 2
8112.2.a.bk 2 156.l odd 4 1
8112.2.a.bo 2 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1521,[χ])S_{2}^{\mathrm{new}}(1521, [\chi]):

T24+9T22+16 T_{2}^{4} + 9T_{2}^{2} + 16 Copy content Toggle raw display
T54+13T52+4 T_{5}^{4} + 13T_{5}^{2} + 4 Copy content Toggle raw display
T74+13T72+4 T_{7}^{4} + 13T_{7}^{2} + 4 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4+9T2+16 T^{4} + 9T^{2} + 16 Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 T4+13T2+4 T^{4} + 13T^{2} + 4 Copy content Toggle raw display
77 T4+13T2+4 T^{4} + 13T^{2} + 4 Copy content Toggle raw display
1111 (T2+4)2 (T^{2} + 4)^{2} Copy content Toggle raw display
1313 T4 T^{4} Copy content Toggle raw display
1717 (T2T4)2 (T^{2} - T - 4)^{2} Copy content Toggle raw display
1919 T4+52T2+64 T^{4} + 52T^{2} + 64 Copy content Toggle raw display
2323 (T2)4 (T - 2)^{4} Copy content Toggle raw display
2929 (T2+T38)2 (T^{2} + T - 38)^{2} Copy content Toggle raw display
3131 T4+9T2+16 T^{4} + 9T^{2} + 16 Copy content Toggle raw display
3737 T4+69T2+676 T^{4} + 69T^{2} + 676 Copy content Toggle raw display
4141 T4+9T2+16 T^{4} + 9T^{2} + 16 Copy content Toggle raw display
4343 (T2+5T+2)2 (T^{2} + 5 T + 2)^{2} Copy content Toggle raw display
4747 (T2+68)2 (T^{2} + 68)^{2} Copy content Toggle raw display
5353 (T2+11T8)2 (T^{2} + 11 T - 8)^{2} Copy content Toggle raw display
5959 T4+132T2+1024 T^{4} + 132T^{2} + 1024 Copy content Toggle raw display
6161 (T216T+47)2 (T^{2} - 16 T + 47)^{2} Copy content Toggle raw display
6767 T4+21T2+4 T^{4} + 21T^{2} + 4 Copy content Toggle raw display
7171 (T2+196)2 (T^{2} + 196)^{2} Copy content Toggle raw display
7373 T4+106T2+361 T^{4} + 106T^{2} + 361 Copy content Toggle raw display
7979 (T215T+52)2 (T^{2} - 15 T + 52)^{2} Copy content Toggle raw display
8383 T4+84T2+64 T^{4} + 84T^{2} + 64 Copy content Toggle raw display
8989 T4+196T2+4096 T^{4} + 196T^{2} + 4096 Copy content Toggle raw display
9797 T4+93T2+1444 T^{4} + 93T^{2} + 1444 Copy content Toggle raw display
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