Properties

Label 612.2.k.d
Level 612612
Weight 22
Character orbit 612.k
Analytic conductor 4.8874.887
Analytic rank 00
Dimension 44
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [612,2,Mod(217,612)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(612, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("612.217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 612=223217 612 = 2^{2} \cdot 3^{2} \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 612.k (of order 44, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 4.886844603704.88684460370
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(i)\Q(i)
Coefficient field: Q(i,34)\Q(i, \sqrt{34})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4+289 x^{4} + 289 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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+β1q5+(3β2+3)q7β3q115q13β3q17+β2q19+β3q23+12β2q25+(4β24)q31+(3β3+3β1)q35++(6β2+6)q97+O(q100) q + \beta_1 q^{5} + ( - 3 \beta_{2} + 3) q^{7} - \beta_{3} q^{11} - 5 q^{13} - \beta_{3} q^{17} + \beta_{2} q^{19} + \beta_{3} q^{23} + 12 \beta_{2} q^{25} + ( - 4 \beta_{2} - 4) q^{31} + ( - 3 \beta_{3} + 3 \beta_1) q^{35}+ \cdots + (6 \beta_{2} + 6) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+12q720q1316q3112q37+68q55+12q6124q6716q73+20q79+68q8560q91+24q97+O(q100) 4 q + 12 q^{7} - 20 q^{13} - 16 q^{31} - 12 q^{37} + 68 q^{55} + 12 q^{61} - 24 q^{67} - 16 q^{73} + 20 q^{79} + 68 q^{85} - 60 q^{91} + 24 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4+289 x^{4} + 289 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν2)/17 ( \nu^{2} ) / 17 Copy content Toggle raw display
β3\beta_{3}== (ν3)/17 ( \nu^{3} ) / 17 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== 17β2 17\beta_{2} Copy content Toggle raw display
ν3\nu^{3}== 17β3 17\beta_{3} Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/612Z)×\left(\mathbb{Z}/612\mathbb{Z}\right)^\times.

nn 3737 137137 307307
χ(n)\chi(n) β2\beta_{2} 11 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
217.1
−2.91548 2.91548i
2.91548 + 2.91548i
−2.91548 + 2.91548i
2.91548 2.91548i
0 0 0 −2.91548 2.91548i 0 3.00000 3.00000i 0 0 0
217.2 0 0 0 2.91548 + 2.91548i 0 3.00000 3.00000i 0 0 0
361.1 0 0 0 −2.91548 + 2.91548i 0 3.00000 + 3.00000i 0 0 0
361.2 0 0 0 2.91548 2.91548i 0 3.00000 + 3.00000i 0 0 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
3.b odd 2 1 inner
17.c even 4 1 inner
51.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 612.2.k.d 4
3.b odd 2 1 inner 612.2.k.d 4
4.b odd 2 1 2448.2.be.q 4
12.b even 2 1 2448.2.be.q 4
17.c even 4 1 inner 612.2.k.d 4
51.f odd 4 1 inner 612.2.k.d 4
68.f odd 4 1 2448.2.be.q 4
204.l even 4 1 2448.2.be.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
612.2.k.d 4 1.a even 1 1 trivial
612.2.k.d 4 3.b odd 2 1 inner
612.2.k.d 4 17.c even 4 1 inner
612.2.k.d 4 51.f odd 4 1 inner
2448.2.be.q 4 4.b odd 2 1
2448.2.be.q 4 12.b even 2 1
2448.2.be.q 4 68.f odd 4 1
2448.2.be.q 4 204.l even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(612,[χ])S_{2}^{\mathrm{new}}(612, [\chi]):

T54+289 T_{5}^{4} + 289 Copy content Toggle raw display
T726T7+18 T_{7}^{2} - 6T_{7} + 18 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 T^{4} Copy content Toggle raw display
55 T4+289 T^{4} + 289 Copy content Toggle raw display
77 (T26T+18)2 (T^{2} - 6 T + 18)^{2} Copy content Toggle raw display
1111 T4+289 T^{4} + 289 Copy content Toggle raw display
1313 (T+5)4 (T + 5)^{4} Copy content Toggle raw display
1717 T4+289 T^{4} + 289 Copy content Toggle raw display
1919 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
2323 T4+289 T^{4} + 289 Copy content Toggle raw display
2929 T4 T^{4} Copy content Toggle raw display
3131 (T2+8T+32)2 (T^{2} + 8 T + 32)^{2} Copy content Toggle raw display
3737 (T2+6T+18)2 (T^{2} + 6 T + 18)^{2} Copy content Toggle raw display
4141 T4+289 T^{4} + 289 Copy content Toggle raw display
4343 (T2+25)2 (T^{2} + 25)^{2} Copy content Toggle raw display
4747 (T2136)2 (T^{2} - 136)^{2} Copy content Toggle raw display
5353 (T2+136)2 (T^{2} + 136)^{2} Copy content Toggle raw display
5959 (T2+34)2 (T^{2} + 34)^{2} Copy content Toggle raw display
6161 (T26T+18)2 (T^{2} - 6 T + 18)^{2} Copy content Toggle raw display
6767 (T+6)4 (T + 6)^{4} Copy content Toggle raw display
7171 T4+4624 T^{4} + 4624 Copy content Toggle raw display
7373 (T2+8T+32)2 (T^{2} + 8 T + 32)^{2} Copy content Toggle raw display
7979 (T210T+50)2 (T^{2} - 10 T + 50)^{2} Copy content Toggle raw display
8383 (T2+136)2 (T^{2} + 136)^{2} Copy content Toggle raw display
8989 (T2306)2 (T^{2} - 306)^{2} Copy content Toggle raw display
9797 (T212T+72)2 (T^{2} - 12 T + 72)^{2} Copy content Toggle raw display
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