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Properties

Label 3960.1.eg.b
Level 39603960
Weight 11
Character orbit 3960.eg
Analytic conductor 1.9761.976
Analytic rank 00
Dimension 1616
Projective image D20D_{20}
CM discriminant -40
Inner twists 88

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3960,1,Mod(899,3960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3960, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 5, 5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3960.899");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 3960=23⋅32⋅5⋅11 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11
Weight: k k == 1 1
Character orbit: [χ][\chi] == 3960.eg (of order 1010, degree 44, 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: 1.976297450031.97629745003
Analytic rank: 00
Dimension: 1616
Relative dimension: 44 over Q(ζ10)\Q(\zeta_{10})
Coefficient field: Q(ζ40)\Q(\zeta_{40})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x16−x12+x8−x4+1 x^{16} - x^{12} + x^{8} - x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D20D_{20}
Projective field: Galois closure of Q[x]/(x20−⋯ )\mathbb{Q}[x]/(x^{20} - \cdots)

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The qq-expansion and trace form are shown below.

f(q)f(q) == q−ζ4016q2−ζ4012q4−ζ4014q5+(−ζ4015+ζ409)q7−ζ408q8−ζ4010q10−ζ4017q11+(−ζ4017+ζ4015)q13+⋯+(ζ4014−ζ406+1)q98+O(q100) q - \zeta_{40}^{16} q^{2} - \zeta_{40}^{12} q^{4} - \zeta_{40}^{14} q^{5} + ( - \zeta_{40}^{15} + \zeta_{40}^{9}) q^{7} - \zeta_{40}^{8} q^{8} - \zeta_{40}^{10} q^{10} - \zeta_{40}^{17} q^{11} + ( - \zeta_{40}^{17} + \zeta_{40}^{15}) q^{13} + \cdots + (\zeta_{40}^{14} - \zeta_{40}^{6} + 1) q^{98} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q+4q2−4q4+4q8−4q16+4q25−16q32+4q49−4q50−4q64−4q77−8q91+20q94−8q95+16q98+O(q100) 16 q + 4 q^{2} - 4 q^{4} + 4 q^{8} - 4 q^{16} + 4 q^{25} - 16 q^{32} + 4 q^{49} - 4 q^{50} - 4 q^{64} - 4 q^{77} - 8 q^{91} + 20 q^{94} - 8 q^{95} + 16 q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3960Z)×\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times.

nn 991991 19811981 23772377 25212521 35213521
χ(n)\chi(n) −1-1 −1-1 −1-1 ζ4012\zeta_{40}^{12} −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}
899.1
−0.156434 + 0.987688i
0.156434 − 0.987688i
−0.987688 − 0.156434i
0.987688 + 0.156434i
0.891007 − 0.453990i
−0.891007 + 0.453990i
0.453990 + 0.891007i
−0.453990 − 0.891007i
−0.156434 − 0.987688i
0.156434 + 0.987688i
−0.987688 + 0.156434i
0.987688 − 0.156434i
0.891007 + 0.453990i
−0.891007 − 0.453990i
0.453990 − 0.891007i
−0.453990 + 0.891007i
0.809017 − 0.587785i 0 0.309017 − 0.951057i −0.587785 + 0.809017i 0 −1.69480 − 0.550672i −0.309017 − 0.951057i 0 1.00000i
899.2 0.809017 − 0.587785i 0 0.309017 − 0.951057i −0.587785 + 0.809017i 0 1.69480 + 0.550672i −0.309017 − 0.951057i 0 1.00000i
899.3 0.809017 − 0.587785i 0 0.309017 − 0.951057i 0.587785 − 0.809017i 0 −0.863541 − 0.280582i −0.309017 − 0.951057i 0 − 1.00000i
899.4 0.809017 − 0.587785i 0 0.309017 − 0.951057i 0.587785 − 0.809017i 0 0.863541 + 0.280582i −0.309017 − 0.951057i 0 − 1.00000i
1619.1 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −0.951057 + 0.309017i 0 −1.16110 + 1.59811i 0.809017 − 0.587785i 0 − 1.00000i
1619.2 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −0.951057 + 0.309017i 0 1.16110 − 1.59811i 0.809017 − 0.587785i 0 − 1.00000i
1619.3 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i 0.951057 − 0.309017i 0 −0.183900 + 0.253116i 0.809017 − 0.587785i 0 1.00000i
1619.4 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i 0.951057 − 0.309017i 0 0.183900 − 0.253116i 0.809017 − 0.587785i 0 1.00000i
2339.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.587785 − 0.809017i 0 −1.69480 + 0.550672i −0.309017 + 0.951057i 0 − 1.00000i
2339.2 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.587785 − 0.809017i 0 1.69480 − 0.550672i −0.309017 + 0.951057i 0 − 1.00000i
2339.3 0.809017 + 0.587785i 0 0.309017 + 0.951057i 0.587785 + 0.809017i 0 −0.863541 + 0.280582i −0.309017 + 0.951057i 0 1.00000i
2339.4 0.809017 + 0.587785i 0 0.309017 + 0.951057i 0.587785 + 0.809017i 0 0.863541 − 0.280582i −0.309017 + 0.951057i 0 1.00000i
3779.1 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i −0.951057 − 0.309017i 0 −1.16110 − 1.59811i 0.809017 + 0.587785i 0 1.00000i
3779.2 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i −0.951057 − 0.309017i 0 1.16110 + 1.59811i 0.809017 + 0.587785i 0 1.00000i
3779.3 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i 0.951057 + 0.309017i 0 −0.183900 − 0.253116i 0.809017 + 0.587785i 0 − 1.00000i
3779.4 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i 0.951057 + 0.309017i 0 0.183900 + 0.253116i 0.809017 + 0.587785i 0 − 1.00000i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 899.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by Q(−10)\Q(\sqrt{-10})
15.d odd 2 1 inner
24.f even 2 1 inner
33.f even 10 1 inner
55.h odd 10 1 inner
88.k even 10 1 inner
1320.da odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3960.1.eg.b yes 16
3.b odd 2 1 3960.1.eg.a ✓ 16
5.b even 2 1 3960.1.eg.a ✓ 16
8.d odd 2 1 3960.1.eg.a ✓ 16
11.d odd 10 1 3960.1.eg.a ✓ 16
15.d odd 2 1 inner 3960.1.eg.b yes 16
24.f even 2 1 inner 3960.1.eg.b yes 16
33.f even 10 1 inner 3960.1.eg.b yes 16
40.e odd 2 1 CM 3960.1.eg.b yes 16
55.h odd 10 1 inner 3960.1.eg.b yes 16
88.k even 10 1 inner 3960.1.eg.b yes 16
120.m even 2 1 3960.1.eg.a ✓ 16
165.r even 10 1 3960.1.eg.a ✓ 16
264.r odd 10 1 3960.1.eg.a ✓ 16
440.bm even 10 1 3960.1.eg.a ✓ 16
1320.da odd 10 1 inner 3960.1.eg.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3960.1.eg.a ✓ 16 3.b odd 2 1
3960.1.eg.a ✓ 16 5.b even 2 1
3960.1.eg.a ✓ 16 8.d odd 2 1
3960.1.eg.a ✓ 16 11.d odd 10 1
3960.1.eg.a ✓ 16 120.m even 2 1
3960.1.eg.a ✓ 16 165.r even 10 1
3960.1.eg.a ✓ 16 264.r odd 10 1
3960.1.eg.a ✓ 16 440.bm even 10 1
3960.1.eg.b yes 16 1.a even 1 1 trivial
3960.1.eg.b yes 16 15.d odd 2 1 inner
3960.1.eg.b yes 16 24.f even 2 1 inner
3960.1.eg.b yes 16 33.f even 10 1 inner
3960.1.eg.b yes 16 40.e odd 2 1 CM
3960.1.eg.b yes 16 55.h odd 10 1 inner
3960.1.eg.b yes 16 88.k even 10 1 inner
3960.1.eg.b yes 16 1320.da odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T474−5T47+5 T_{47}^{4} - 5T_{47} + 5 acting on S1new(3960,[χ])S_{1}^{\mathrm{new}}(3960, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4−T3+T2+⋯+1)4 (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} Copy content Toggle raw display
33 T16 T^{16} Copy content Toggle raw display
55 (T8−T6+T4+⋯+1)2 (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} Copy content Toggle raw display
77 T16−4T14+⋯+1 T^{16} - 4 T^{14} + \cdots + 1 Copy content Toggle raw display
1111 T16−T12+⋯+1 T^{16} - T^{12} + \cdots + 1 Copy content Toggle raw display
1313 T16−4T14+⋯+1 T^{16} - 4 T^{14} + \cdots + 1 Copy content Toggle raw display
1717 T16 T^{16} Copy content Toggle raw display
1919 (T8−4T6+6T4+⋯+1)2 (T^{8} - 4 T^{6} + 6 T^{4} + \cdots + 1)^{2} Copy content Toggle raw display
2323 (T4+3T2+1)4 (T^{4} + 3 T^{2} + 1)^{4} Copy content Toggle raw display
2929 T16 T^{16} Copy content Toggle raw display
3131 T16 T^{16} Copy content Toggle raw display
3737 T16+4T14+⋯+1 T^{16} + 4 T^{14} + \cdots + 1 Copy content Toggle raw display
4141 T16+4T14+⋯+1 T^{16} + 4 T^{14} + \cdots + 1 Copy content Toggle raw display
4343 T16 T^{16} Copy content Toggle raw display
4747 (T4−5T+5)4 (T^{4} - 5 T + 5)^{4} Copy content Toggle raw display
5353 (T4+5T+5)4 (T^{4} + 5 T + 5)^{4} Copy content Toggle raw display
5959 T16−4T14+⋯+1 T^{16} - 4 T^{14} + \cdots + 1 Copy content Toggle raw display
6161 T16 T^{16} Copy content Toggle raw display
6767 T16 T^{16} Copy content Toggle raw display
7171 T16 T^{16} Copy content Toggle raw display
7373 T16 T^{16} Copy content Toggle raw display
7979 T16 T^{16} Copy content Toggle raw display
8383 T16 T^{16} Copy content Toggle raw display
8989 (T8+8T6+19T4+⋯+1)2 (T^{8} + 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} Copy content Toggle raw display
9797 T16 T^{16} Copy content Toggle raw display
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