Properties

Label 65.2.e.a
Level 6565
Weight 22
Character orbit 65.e
Analytic conductor 0.5190.519
Analytic rank 00
Dimension 44
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [65,2,Mod(16,65)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(65, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) 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("65.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 65=513 65 = 5 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 65.e (of order 33, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 0.5190276131380.519027613138
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(3,5)\Q(\sqrt{-3}, \sqrt{5})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4x3+2x2+x+1 x^{4} - x^{3} + 2x^{2} + x + 1 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)[C3]\mathrm{SU}(2)[C_{3}]

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) == qβ1q2+(β3+2β11)q3+(β3+β2+β1)q4+q5+(2β3β2β1)q6+(3β32β22β1)q7++(4β2+2)q99+O(q100) q - \beta_1 q^{2} + ( - \beta_{3} + 2 \beta_1 - 1) q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{4} + q^{5} + ( - 2 \beta_{3} - \beta_{2} - \beta_1) q^{6} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{7}+ \cdots + ( - 4 \beta_{2} + 2) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4qq2+q4+4q5+5q64q74q9q104q1110q124q136q14+3q162q17+4q184q19+q20+20q217q2212q23++16q99+O(q100) 4 q - q^{2} + q^{4} + 4 q^{5} + 5 q^{6} - 4 q^{7} - 4 q^{9} - q^{10} - 4 q^{11} - 10 q^{12} - 4 q^{13} - 6 q^{14} + 3 q^{16} - 2 q^{17} + 4 q^{18} - 4 q^{19} + q^{20} + 20 q^{21} - 7 q^{22} - 12 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4x3+2x2+x+1 x^{4} - x^{3} + 2x^{2} + x + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν3+1)/2 ( \nu^{3} + 1 ) / 2 Copy content Toggle raw display
β3\beta_{3}== (ν3+2ν22ν1)/2 ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3+β2+β1 \beta_{3} + \beta_{2} + \beta_1 Copy content Toggle raw display
ν3\nu^{3}== 2β21 2\beta_{2} - 1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/65Z)×\left(\mathbb{Z}/65\mathbb{Z}\right)^\times.

nn 2727 4141
χ(n)\chi(n) 11 β3\beta_{3}

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}
16.1
0.809017 + 1.40126i
−0.309017 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i
−0.809017 1.40126i 1.11803 + 1.93649i −0.309017 + 0.535233i 1.00000 1.80902 3.13331i 0.118034 0.204441i −2.23607 −1.00000 + 1.73205i −0.809017 1.40126i
16.2 0.309017 + 0.535233i −1.11803 1.93649i 0.809017 1.40126i 1.00000 0.690983 1.19682i −2.11803 + 3.66854i 2.23607 −1.00000 + 1.73205i 0.309017 + 0.535233i
61.1 −0.809017 + 1.40126i 1.11803 1.93649i −0.309017 0.535233i 1.00000 1.80902 + 3.13331i 0.118034 + 0.204441i −2.23607 −1.00000 1.73205i −0.809017 + 1.40126i
61.2 0.309017 0.535233i −1.11803 + 1.93649i 0.809017 + 1.40126i 1.00000 0.690983 + 1.19682i −2.11803 3.66854i 2.23607 −1.00000 1.73205i 0.309017 0.535233i
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 65.2.e.a 4
3.b odd 2 1 585.2.j.e 4
4.b odd 2 1 1040.2.q.n 4
5.b even 2 1 325.2.e.b 4
5.c odd 4 2 325.2.o.a 8
13.b even 2 1 845.2.e.g 4
13.c even 3 1 inner 65.2.e.a 4
13.c even 3 1 845.2.a.e 2
13.d odd 4 2 845.2.m.e 8
13.e even 6 1 845.2.a.b 2
13.e even 6 1 845.2.e.g 4
13.f odd 12 2 845.2.c.c 4
13.f odd 12 2 845.2.m.e 8
39.h odd 6 1 7605.2.a.bf 2
39.i odd 6 1 585.2.j.e 4
39.i odd 6 1 7605.2.a.ba 2
52.j odd 6 1 1040.2.q.n 4
65.l even 6 1 4225.2.a.y 2
65.n even 6 1 325.2.e.b 4
65.n even 6 1 4225.2.a.u 2
65.q odd 12 2 325.2.o.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 1.a even 1 1 trivial
65.2.e.a 4 13.c even 3 1 inner
325.2.e.b 4 5.b even 2 1
325.2.e.b 4 65.n even 6 1
325.2.o.a 8 5.c odd 4 2
325.2.o.a 8 65.q odd 12 2
585.2.j.e 4 3.b odd 2 1
585.2.j.e 4 39.i odd 6 1
845.2.a.b 2 13.e even 6 1
845.2.a.e 2 13.c even 3 1
845.2.c.c 4 13.f odd 12 2
845.2.e.g 4 13.b even 2 1
845.2.e.g 4 13.e even 6 1
845.2.m.e 8 13.d odd 4 2
845.2.m.e 8 13.f odd 12 2
1040.2.q.n 4 4.b odd 2 1
1040.2.q.n 4 52.j odd 6 1
4225.2.a.u 2 65.n even 6 1
4225.2.a.y 2 65.l even 6 1
7605.2.a.ba 2 39.i odd 6 1
7605.2.a.bf 2 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T24+T23+2T22T2+1 T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - T_{2} + 1 acting on S2new(65,[χ])S_{2}^{\mathrm{new}}(65, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4+T3+2T2++1 T^{4} + T^{3} + 2 T^{2} + \cdots + 1 Copy content Toggle raw display
33 T4+5T2+25 T^{4} + 5T^{2} + 25 Copy content Toggle raw display
55 (T1)4 (T - 1)^{4} Copy content Toggle raw display
77 T4+4T3++1 T^{4} + 4 T^{3} + \cdots + 1 Copy content Toggle raw display
1111 T4+4T3++1 T^{4} + 4 T^{3} + \cdots + 1 Copy content Toggle raw display
1313 (T2+2T+13)2 (T^{2} + 2 T + 13)^{2} Copy content Toggle raw display
1717 T4+2T3++361 T^{4} + 2 T^{3} + \cdots + 361 Copy content Toggle raw display
1919 T4+4T3++1 T^{4} + 4 T^{3} + \cdots + 1 Copy content Toggle raw display
2323 T4+12T3++961 T^{4} + 12 T^{3} + \cdots + 961 Copy content Toggle raw display
2929 T46T3++121 T^{4} - 6 T^{3} + \cdots + 121 Copy content Toggle raw display
3131 T4 T^{4} Copy content Toggle raw display
3737 (T2+3T+9)2 (T^{2} + 3 T + 9)^{2} Copy content Toggle raw display
4141 T4+6T3++5041 T^{4} + 6 T^{3} + \cdots + 5041 Copy content Toggle raw display
4343 T48T3++121 T^{4} - 8 T^{3} + \cdots + 121 Copy content Toggle raw display
4747 (T28T64)2 (T^{2} - 8 T - 64)^{2} Copy content Toggle raw display
5353 (T6)4 (T - 6)^{4} Copy content Toggle raw display
5959 T4+12T3++81 T^{4} + 12 T^{3} + \cdots + 81 Copy content Toggle raw display
6161 T4+2T3++32041 T^{4} + 2 T^{3} + \cdots + 32041 Copy content Toggle raw display
6767 T4+8T3++841 T^{4} + 8 T^{3} + \cdots + 841 Copy content Toggle raw display
7171 T48T3++121 T^{4} - 8 T^{3} + \cdots + 121 Copy content Toggle raw display
7373 (T+6)4 (T + 6)^{4} Copy content Toggle raw display
7979 T4 T^{4} Copy content Toggle raw display
8383 (T280)2 (T^{2} - 80)^{2} Copy content Toggle raw display
8989 (T29T+81)2 (T^{2} - 9 T + 81)^{2} Copy content Toggle raw display
9797 T4+2T3++361 T^{4} + 2 T^{3} + \cdots + 361 Copy content Toggle raw display
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