Properties

Label 585.2.j.d
Level 585585
Weight 22
Character orbit 585.j
Analytic conductor 4.6714.671
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] = [585,2,Mod(406,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.406");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 585=32513 585 = 3^{2} \cdot 5 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 585.j (of order 33, degree 22, 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: 4.671248518244.67124851824
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(3,13)\Q(\sqrt{-3}, \sqrt{13})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x3+4x2+3x+9 x^{4} - x^{3} + 4x^{2} + 3x + 9 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 65)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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) == qβ1q2+(β3+β2+β11)q4+q5β2q7+3q8β1q10+(3β2+2β13)q11+(2β31)q13+(β31)q14++(6β36β1+6)q98+O(q100) q - \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + q^{5} - \beta_{2} q^{7} + 3 q^{8} - \beta_1 q^{10} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{11} + (2 \beta_{3} - 1) q^{13} + (\beta_{3} - 1) q^{14}+ \cdots + ( - 6 \beta_{3} - 6 \beta_1 + 6) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4qq23q4+4q5+2q7+12q8q104q112q14+3q16+8q17+4q193q20+11q226q23+4q25+13q26+3q28+2q2916q31++6q98+O(q100) 4 q - q^{2} - 3 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8} - q^{10} - 4 q^{11} - 2 q^{14} + 3 q^{16} + 8 q^{17} + 4 q^{19} - 3 q^{20} + 11 q^{22} - 6 q^{23} + 4 q^{25} + 13 q^{26} + 3 q^{28} + 2 q^{29} - 16 q^{31}+ \cdots + 6 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4x3+4x2+3x+9 x^{4} - x^{3} + 4x^{2} + 3x + 9 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν3+4ν24ν3)/12 ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 Copy content Toggle raw display
β3\beta_{3}== (ν3+7)/4 ( \nu^{3} + 7 ) / 4 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3+3β2+β11 \beta_{3} + 3\beta_{2} + \beta _1 - 1 Copy content Toggle raw display
ν3\nu^{3}== 4β37 4\beta_{3} - 7 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/585Z)×\left(\mathbb{Z}/585\mathbb{Z}\right)^\times.

nn 326326 352352 496496
χ(n)\chi(n) 11 11 β2\beta_{2}

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}
406.1
1.15139 + 1.99426i
−0.651388 1.12824i
1.15139 1.99426i
−0.651388 + 1.12824i
−1.15139 1.99426i 0 −1.65139 + 2.86029i 1.00000 0 0.500000 0.866025i 3.00000 0 −1.15139 1.99426i
406.2 0.651388 + 1.12824i 0 0.151388 0.262211i 1.00000 0 0.500000 0.866025i 3.00000 0 0.651388 + 1.12824i
451.1 −1.15139 + 1.99426i 0 −1.65139 2.86029i 1.00000 0 0.500000 + 0.866025i 3.00000 0 −1.15139 + 1.99426i
451.2 0.651388 1.12824i 0 0.151388 + 0.262211i 1.00000 0 0.500000 + 0.866025i 3.00000 0 0.651388 1.12824i
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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.d 4
3.b odd 2 1 65.2.e.b 4
12.b even 2 1 1040.2.q.o 4
13.c even 3 1 inner 585.2.j.d 4
13.c even 3 1 7605.2.a.bg 2
13.e even 6 1 7605.2.a.bb 2
15.d odd 2 1 325.2.e.a 4
15.e even 4 2 325.2.o.b 8
39.d odd 2 1 845.2.e.d 4
39.f even 4 2 845.2.m.d 8
39.h odd 6 1 845.2.a.f 2
39.h odd 6 1 845.2.e.d 4
39.i odd 6 1 65.2.e.b 4
39.i odd 6 1 845.2.a.c 2
39.k even 12 2 845.2.c.d 4
39.k even 12 2 845.2.m.d 8
156.p even 6 1 1040.2.q.o 4
195.x odd 6 1 325.2.e.a 4
195.x odd 6 1 4225.2.a.x 2
195.y odd 6 1 4225.2.a.t 2
195.bl even 12 2 325.2.o.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.b 4 3.b odd 2 1
65.2.e.b 4 39.i odd 6 1
325.2.e.a 4 15.d odd 2 1
325.2.e.a 4 195.x odd 6 1
325.2.o.b 8 15.e even 4 2
325.2.o.b 8 195.bl even 12 2
585.2.j.d 4 1.a even 1 1 trivial
585.2.j.d 4 13.c even 3 1 inner
845.2.a.c 2 39.i odd 6 1
845.2.a.f 2 39.h odd 6 1
845.2.c.d 4 39.k even 12 2
845.2.e.d 4 39.d odd 2 1
845.2.e.d 4 39.h odd 6 1
845.2.m.d 8 39.f even 4 2
845.2.m.d 8 39.k even 12 2
1040.2.q.o 4 12.b even 2 1
1040.2.q.o 4 156.p even 6 1
4225.2.a.t 2 195.y odd 6 1
4225.2.a.x 2 195.x odd 6 1
7605.2.a.bb 2 13.e even 6 1
7605.2.a.bg 2 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T24+T23+4T223T2+9 T_{2}^{4} + T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 9 acting on S2new(585,[χ])S_{2}^{\mathrm{new}}(585, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4+T3+4T2++9 T^{4} + T^{3} + 4 T^{2} + \cdots + 9 Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 (T1)4 (T - 1)^{4} Copy content Toggle raw display
77 (T2T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
1111 T4+4T3++81 T^{4} + 4 T^{3} + \cdots + 81 Copy content Toggle raw display
1313 (T213)2 (T^{2} - 13)^{2} Copy content Toggle raw display
1717 T48T3++9 T^{4} - 8 T^{3} + \cdots + 9 Copy content Toggle raw display
1919 T44T3++81 T^{4} - 4 T^{3} + \cdots + 81 Copy content Toggle raw display
2323 (T2+3T+9)2 (T^{2} + 3 T + 9)^{2} Copy content Toggle raw display
2929 T42T3++2601 T^{4} - 2 T^{3} + \cdots + 2601 Copy content Toggle raw display
3131 (T+4)4 (T + 4)^{4} Copy content Toggle raw display
3737 T4+13T2+169 T^{4} + 13T^{2} + 169 Copy content Toggle raw display
4141 (T23T+9)2 (T^{2} - 3 T + 9)^{2} Copy content Toggle raw display
4343 T46T3++1849 T^{4} - 6 T^{3} + \cdots + 1849 Copy content Toggle raw display
4747 (T2+4T48)2 (T^{2} + 4 T - 48)^{2} Copy content Toggle raw display
5353 (T2+8T36)2 (T^{2} + 8 T - 36)^{2} Copy content Toggle raw display
5959 T4+117T2+13689 T^{4} + 117 T^{2} + 13689 Copy content Toggle raw display
6161 (T2T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
6767 (T27T+49)2 (T^{2} - 7 T + 49)^{2} Copy content Toggle raw display
7171 T4+12T3++6561 T^{4} + 12 T^{3} + \cdots + 6561 Copy content Toggle raw display
7373 (T2+16T+12)2 (T^{2} + 16 T + 12)^{2} Copy content Toggle raw display
7979 (T2+4T48)2 (T^{2} + 4 T - 48)^{2} Copy content Toggle raw display
8383 (T24T48)2 (T^{2} - 4 T - 48)^{2} Copy content Toggle raw display
8989 T42T3++2601 T^{4} - 2 T^{3} + \cdots + 2601 Copy content Toggle raw display
9797 T424T3++17161 T^{4} - 24 T^{3} + \cdots + 17161 Copy content Toggle raw display
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