Properties

Label 528.4.b.a
Level 528528
Weight 44
Character orbit 528.b
Analytic conductor 31.15331.153
Analytic rank 00
Dimension 22
CM discriminant -11
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [528,4,Mod(65,528)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(528, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("528.65"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 528=24311 528 = 2^{4} \cdot 3 \cdot 11
Weight: k k == 4 4
Character orbit: [χ][\chi] == 528.b (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,8,0,0,0,0,0,10,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 31.153008483031.1530084830
Analytic rank: 00
Dimension: 22
Coefficient field: Q(11)\Q(\sqrt{-11})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+3 x^{2} - x + 3 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{2}]

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 β=11\beta = \sqrt{-11}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β+4)q34βq5+(8β+5)q9+11βq11+(16β+44)q15+58βq2351q25+(37β68)q27+340q31+(44β121)q33+434q37++(55β968)q99+O(q100) q + (\beta + 4) q^{3} - 4 \beta q^{5} + (8 \beta + 5) q^{9} + 11 \beta q^{11} + ( - 16 \beta + 44) q^{15} + 58 \beta q^{23} - 51 q^{25} + (37 \beta - 68) q^{27} + 340 q^{31} + (44 \beta - 121) q^{33} + 434 q^{37} + \cdots + (55 \beta - 968) q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+8q3+10q9+88q15102q25136q27+680q31242q33+868q37+704q45+686q49+968q55832q671276q69408q751358q81+2720q93+1936q99+O(q100) 2 q + 8 q^{3} + 10 q^{9} + 88 q^{15} - 102 q^{25} - 136 q^{27} + 680 q^{31} - 242 q^{33} + 868 q^{37} + 704 q^{45} + 686 q^{49} + 968 q^{55} - 832 q^{67} - 1276 q^{69} - 408 q^{75} - 1358 q^{81} + 2720 q^{93}+ \cdots - 1936 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/528Z)×\left(\mathbb{Z}/528\mathbb{Z}\right)^\times.

nn 133133 145145 353353 463463
χ(n)\chi(n) 11 1-1 1-1 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}
65.1
0.500000 1.65831i
0.500000 + 1.65831i
0 4.00000 3.31662i 0 13.2665i 0 0 0 5.00000 26.5330i 0
65.2 0 4.00000 + 3.31662i 0 13.2665i 0 0 0 5.00000 + 26.5330i 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
11.b odd 2 1 CM by Q(11)\Q(\sqrt{-11})
3.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.4.b.a 2
3.b odd 2 1 inner 528.4.b.a 2
4.b odd 2 1 33.4.d.a 2
11.b odd 2 1 CM 528.4.b.a 2
12.b even 2 1 33.4.d.a 2
33.d even 2 1 inner 528.4.b.a 2
44.c even 2 1 33.4.d.a 2
132.d odd 2 1 33.4.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.d.a 2 4.b odd 2 1
33.4.d.a 2 12.b even 2 1
33.4.d.a 2 44.c even 2 1
33.4.d.a 2 132.d odd 2 1
528.4.b.a 2 1.a even 1 1 trivial
528.4.b.a 2 3.b odd 2 1 inner
528.4.b.a 2 11.b odd 2 1 CM
528.4.b.a 2 33.d 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 S4new(528,[χ])S_{4}^{\mathrm{new}}(528, [\chi]):

T52+176 T_{5}^{2} + 176 Copy content Toggle raw display
T17 T_{17} Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T28T+27 T^{2} - 8T + 27 Copy content Toggle raw display
55 T2+176 T^{2} + 176 Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2+1331 T^{2} + 1331 Copy content Toggle raw display
1313 T2 T^{2} Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 T2 T^{2} Copy content Toggle raw display
2323 T2+37004 T^{2} + 37004 Copy content Toggle raw display
2929 T2 T^{2} Copy content Toggle raw display
3131 (T340)2 (T - 340)^{2} Copy content Toggle raw display
3737 (T434)2 (T - 434)^{2} Copy content Toggle raw display
4141 T2 T^{2} Copy content Toggle raw display
4343 T2 T^{2} Copy content Toggle raw display
4747 T2+413996 T^{2} + 413996 Copy content Toggle raw display
5353 T2+50864 T^{2} + 50864 Copy content Toggle raw display
5959 T2+303116 T^{2} + 303116 Copy content Toggle raw display
6161 T2 T^{2} Copy content Toggle raw display
6767 (T+416)2 (T + 416)^{2} Copy content Toggle raw display
7171 T2+1057100 T^{2} + 1057100 Copy content Toggle raw display
7373 T2 T^{2} Copy content Toggle raw display
7979 T2 T^{2} Copy content Toggle raw display
8383 T2 T^{2} Copy content Toggle raw display
8989 T2+17600 T^{2} + 17600 Copy content Toggle raw display
9797 (T+34)2 (T + 34)^{2} Copy content Toggle raw display
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