Properties

Label 1470.2.d.f
Level 14701470
Weight 22
Character orbit 1470.d
Analytic conductor 11.73811.738
Analytic rank 00
Dimension 88
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1470,2,Mod(1469,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1470.1469");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1470=23572 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1470.d (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: 11.738009097111.7380090971
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x84x6+7x436x2+81 x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 24 2^{4}
Twist minimal: no (minimal twist has level 210)
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,,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+q2+(β5+β2)q3+q4+(β6β2)q5+(β5+β2)q6+q8+(β72)q9+(β6β2)q10+(β7+β4)q11++(2β7β4+2β15)q99+O(q100) q + q^{2} + (\beta_{5} + \beta_{2}) q^{3} + q^{4} + ( - \beta_{6} - \beta_{2}) q^{5} + (\beta_{5} + \beta_{2}) q^{6} + q^{8} + ( - \beta_{7} - 2) q^{9} + ( - \beta_{6} - \beta_{2}) q^{10} + ( - \beta_{7} + \beta_{4}) q^{11}+ \cdots + (2 \beta_{7} - \beta_{4} + 2 \beta_1 - 5) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+8q2+8q4+8q816q98q15+8q1616q188q238q258q30+8q3216q36+16q398q468q5040q51+40q5340q57+40q99+O(q100) 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} - 16 q^{9} - 8 q^{15} + 8 q^{16} - 16 q^{18} - 8 q^{23} - 8 q^{25} - 8 q^{30} + 8 q^{32} - 16 q^{36} + 16 q^{39} - 8 q^{46} - 8 q^{50} - 40 q^{51} + 40 q^{53} - 40 q^{57}+ \cdots - 40 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x84x6+7x436x2+81 x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 : Copy content Toggle raw display

β1\beta_{1}== (2ν7+28ν549ν3+180ν)/189 ( 2\nu^{7} + 28\nu^{5} - 49\nu^{3} + 180\nu ) / 189 Copy content Toggle raw display
β2\beta_{2}== (4ν7+7ν5+35ν3+81ν)/189 ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 Copy content Toggle raw display
β3\beta_{3}== (ν6+4ν4+2ν2+18)/9 ( -\nu^{6} + 4\nu^{4} + 2\nu^{2} + 18 ) / 9 Copy content Toggle raw display
β4\beta_{4}== (2ν7ν55ν3+63ν)/27 ( -2\nu^{7} - \nu^{5} - 5\nu^{3} + 63\nu ) / 27 Copy content Toggle raw display
β5\beta_{5}== (16ν728ν5+49ν3324ν)/189 ( 16\nu^{7} - 28\nu^{5} + 49\nu^{3} - 324\nu ) / 189 Copy content Toggle raw display
β6\beta_{6}== (8ν6+14ν456ν2+225)/63 ( -8\nu^{6} + 14\nu^{4} - 56\nu^{2} + 225 ) / 63 Copy content Toggle raw display
β7\beta_{7}== (ν6+22)/7 ( -\nu^{6} + 22 ) / 7 Copy content Toggle raw display
ν\nu== (β5+β4+β2+β1)/2 ( \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β72β6+β3+2)/2 ( \beta_{7} - 2\beta_{6} + \beta_{3} + 2 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== β5+4β2 \beta_{5} + 4\beta_{2} Copy content Toggle raw display
ν4\nu^{4}== (4β7+β6+4β3+1)/2 ( -4\beta_{7} + \beta_{6} + 4\beta_{3} + 1 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (5β57β4+7β2+5β1)/2 ( -5\beta_{5} - 7\beta_{4} + 7\beta_{2} + 5\beta_1 ) / 2 Copy content Toggle raw display
ν6\nu^{6}== 7β7+22 -7\beta_{7} + 22 Copy content Toggle raw display
ν7\nu^{7}== (29β5+8β4+8β2+29β1)/2 ( 29\beta_{5} + 8\beta_{4} + 8\beta_{2} + 29\beta_1 ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1470Z)×\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times.

nn 491491 10811081 11771177
χ(n)\chi(n) 1-1 1-1 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}
1469.1
1.01575 1.40294i
−1.72286 0.178197i
−1.72286 + 0.178197i
1.01575 + 1.40294i
1.72286 0.178197i
−1.01575 1.40294i
−1.01575 + 1.40294i
1.72286 + 0.178197i
1.00000 −0.707107 1.58114i 1.00000 1.41421 1.73205i −0.707107 1.58114i 0 1.00000 −2.00000 + 2.23607i 1.41421 1.73205i
1469.2 1.00000 −0.707107 1.58114i 1.00000 1.41421 + 1.73205i −0.707107 1.58114i 0 1.00000 −2.00000 + 2.23607i 1.41421 + 1.73205i
1469.3 1.00000 −0.707107 + 1.58114i 1.00000 1.41421 1.73205i −0.707107 + 1.58114i 0 1.00000 −2.00000 2.23607i 1.41421 1.73205i
1469.4 1.00000 −0.707107 + 1.58114i 1.00000 1.41421 + 1.73205i −0.707107 + 1.58114i 0 1.00000 −2.00000 2.23607i 1.41421 + 1.73205i
1469.5 1.00000 0.707107 1.58114i 1.00000 −1.41421 1.73205i 0.707107 1.58114i 0 1.00000 −2.00000 2.23607i −1.41421 1.73205i
1469.6 1.00000 0.707107 1.58114i 1.00000 −1.41421 + 1.73205i 0.707107 1.58114i 0 1.00000 −2.00000 2.23607i −1.41421 + 1.73205i
1469.7 1.00000 0.707107 + 1.58114i 1.00000 −1.41421 1.73205i 0.707107 + 1.58114i 0 1.00000 −2.00000 + 2.23607i −1.41421 1.73205i
1469.8 1.00000 0.707107 + 1.58114i 1.00000 −1.41421 + 1.73205i 0.707107 + 1.58114i 0 1.00000 −2.00000 + 2.23607i −1.41421 + 1.73205i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1469.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.d.f 8
3.b odd 2 1 1470.2.d.e 8
5.b even 2 1 1470.2.d.e 8
7.b odd 2 1 inner 1470.2.d.f 8
7.c even 3 1 210.2.t.e 8
7.d odd 6 1 210.2.t.e 8
15.d odd 2 1 inner 1470.2.d.f 8
21.c even 2 1 1470.2.d.e 8
21.g even 6 1 210.2.t.f yes 8
21.h odd 6 1 210.2.t.f yes 8
35.c odd 2 1 1470.2.d.e 8
35.i odd 6 1 210.2.t.f yes 8
35.j even 6 1 210.2.t.f yes 8
35.k even 12 2 1050.2.s.i 16
35.l odd 12 2 1050.2.s.i 16
105.g even 2 1 inner 1470.2.d.f 8
105.o odd 6 1 210.2.t.e 8
105.p even 6 1 210.2.t.e 8
105.w odd 12 2 1050.2.s.i 16
105.x even 12 2 1050.2.s.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.e 8 7.c even 3 1
210.2.t.e 8 7.d odd 6 1
210.2.t.e 8 105.o odd 6 1
210.2.t.e 8 105.p even 6 1
210.2.t.f yes 8 21.g even 6 1
210.2.t.f yes 8 21.h odd 6 1
210.2.t.f yes 8 35.i odd 6 1
210.2.t.f yes 8 35.j even 6 1
1050.2.s.i 16 35.k even 12 2
1050.2.s.i 16 35.l odd 12 2
1050.2.s.i 16 105.w odd 12 2
1050.2.s.i 16 105.x even 12 2
1470.2.d.e 8 3.b odd 2 1
1470.2.d.e 8 5.b even 2 1
1470.2.d.e 8 21.c even 2 1
1470.2.d.e 8 35.c odd 2 1
1470.2.d.f 8 1.a even 1 1 trivial
1470.2.d.f 8 7.b odd 2 1 inner
1470.2.d.f 8 15.d odd 2 1 inner
1470.2.d.f 8 105.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1470,[χ])S_{2}^{\mathrm{new}}(1470, [\chi]):

T114+22T112+1 T_{11}^{4} + 22T_{11}^{2} + 1 Copy content Toggle raw display
T13446T132+49 T_{13}^{4} - 46T_{13}^{2} + 49 Copy content Toggle raw display
T232+2T2329 T_{23}^{2} + 2T_{23} - 29 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T1)8 (T - 1)^{8} Copy content Toggle raw display
33 (T4+4T2+9)2 (T^{4} + 4 T^{2} + 9)^{2} Copy content Toggle raw display
55 (T4+2T2+25)2 (T^{4} + 2 T^{2} + 25)^{2} Copy content Toggle raw display
77 T8 T^{8} Copy content Toggle raw display
1111 (T4+22T2+1)2 (T^{4} + 22 T^{2} + 1)^{2} Copy content Toggle raw display
1313 (T446T2+49)2 (T^{4} - 46 T^{2} + 49)^{2} Copy content Toggle raw display
1717 (T2+10)4 (T^{2} + 10)^{4} Copy content Toggle raw display
1919 (T4+26T2+49)2 (T^{4} + 26 T^{2} + 49)^{2} Copy content Toggle raw display
2323 (T2+2T29)4 (T^{2} + 2 T - 29)^{4} Copy content Toggle raw display
2929 (T4+52T2+196)2 (T^{4} + 52 T^{2} + 196)^{2} Copy content Toggle raw display
3131 (T4+44T2+4)2 (T^{4} + 44 T^{2} + 4)^{2} Copy content Toggle raw display
3737 (T4+58T2+361)2 (T^{4} + 58 T^{2} + 361)^{2} Copy content Toggle raw display
4141 (T446T2+49)2 (T^{4} - 46 T^{2} + 49)^{2} Copy content Toggle raw display
4343 (T4+52T2+196)2 (T^{4} + 52 T^{2} + 196)^{2} Copy content Toggle raw display
4747 (T4+186T2+7569)2 (T^{4} + 186 T^{2} + 7569)^{2} Copy content Toggle raw display
5353 (T5)8 (T - 5)^{8} Copy content Toggle raw display
5959 (T4136T2+2704)2 (T^{4} - 136 T^{2} + 2704)^{2} Copy content Toggle raw display
6161 (T2+12)4 (T^{2} + 12)^{4} Copy content Toggle raw display
6767 (T4+232T2+5776)2 (T^{4} + 232 T^{2} + 5776)^{2} Copy content Toggle raw display
7171 (T4+52T2+196)2 (T^{4} + 52 T^{2} + 196)^{2} Copy content Toggle raw display
7373 (T4156T2+1764)2 (T^{4} - 156 T^{2} + 1764)^{2} Copy content Toggle raw display
7979 (T212T+6)4 (T^{2} - 12 T + 6)^{4} Copy content Toggle raw display
8383 (T4+296T2+4624)2 (T^{4} + 296 T^{2} + 4624)^{2} Copy content Toggle raw display
8989 (T4136T2+2704)2 (T^{4} - 136 T^{2} + 2704)^{2} Copy content Toggle raw display
9797 (T4156T2+1764)2 (T^{4} - 156 T^{2} + 1764)^{2} Copy content Toggle raw display
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