Properties

Label 476.2.q.a
Level 476476
Weight 22
Character orbit 476.q
Analytic conductor 3.8013.801
Analytic rank 00
Dimension 1616
CM discriminant -68
Inner twists 88

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [476,2,Mod(271,476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(476, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("476.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 476=22717 476 = 2^{2} \cdot 7 \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 476.q (of order 66, 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: 3.800879136213.80087913621
Analytic rank: 00
Dimension: 1616
Relative dimension: 88 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x1616x12+175x81296x4+6561 x^{16} - 16x^{12} + 175x^{8} - 1296x^{4} + 6561 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 7 7
Twist minimal: yes
Sato-Tate group: U(1)[D6]\mathrm{U}(1)[D_{6}]

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,,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β8β4)q2+(β14+β12)q3+(2β32)q4+(β13+β10β6)q6+(β7+β6)q7+2β8q8++(2β15+9β14+β2)q99+O(q100) q + ( - \beta_{8} - \beta_{4}) q^{2} + ( - \beta_{14} + \beta_{12}) q^{3} + (2 \beta_{3} - 2) q^{4} + ( - \beta_{13} + \beta_{10} - \beta_{6}) q^{6} + (\beta_{7} + \beta_{6}) q^{7} + 2 \beta_{8} q^{8}+ \cdots + (2 \beta_{15} + 9 \beta_{14} + \cdots - \beta_{2}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q16q4+24q932q168q1816q21+40q25+72q2696q3396q36+40q42+48q53+128q6416q7248q77136q81+64q84+40q93+72q98+O(q100) 16 q - 16 q^{4} + 24 q^{9} - 32 q^{16} - 8 q^{18} - 16 q^{21} + 40 q^{25} + 72 q^{26} - 96 q^{33} - 96 q^{36} + 40 q^{42} + 48 q^{53} + 128 q^{64} - 16 q^{72} - 48 q^{77} - 136 q^{81} + 64 q^{84} + 40 q^{93}+ \cdots - 72 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x1616x12+175x81296x4+6561 x^{16} - 16x^{12} + 175x^{8} - 1296x^{4} + 6561 : Copy content Toggle raw display

β1\beta_{1}== ν2 \nu^{2} Copy content Toggle raw display
β2\beta_{2}== (4ν15+1225ν115425ν7+80784ν3+127575ν)/127575 ( -4\nu^{15} + 1225\nu^{11} - 5425\nu^{7} + 80784\nu^{3} + 127575\nu ) / 127575 Copy content Toggle raw display
β3\beta_{3}== (16ν12+175ν82800ν4+20736)/14175 ( -16\nu^{12} + 175\nu^{8} - 2800\nu^{4} + 20736 ) / 14175 Copy content Toggle raw display
β4\beta_{4}== (31ν141225ν10+5425ν640176ν2)/127575 ( 31\nu^{14} - 1225\nu^{10} + 5425\nu^{6} - 40176\nu^{2} ) / 127575 Copy content Toggle raw display
β5\beta_{5}== (47ν12+1400ν88225ν4+60912)/14175 ( -47\nu^{12} + 1400\nu^{8} - 8225\nu^{4} + 60912 ) / 14175 Copy content Toggle raw display
β6\beta_{6}== (16ν15+175ν11+2870ν7+6561ν3)/76545 ( -16\nu^{15} + 175\nu^{11} + 2870\nu^{7} + 6561\nu^{3} ) / 76545 Copy content Toggle raw display
β7\beta_{7}== (82ν15875ν1114350ν7+106272ν3)/382725 ( -82\nu^{15} - 875\nu^{11} - 14350\nu^{7} + 106272\nu^{3} ) / 382725 Copy content Toggle raw display
β8\beta_{8}== (16ν14+175ν10775ν6+6561ν2)/18225 ( -16\nu^{14} + 175\nu^{10} - 775\nu^{6} + 6561\nu^{2} ) / 18225 Copy content Toggle raw display
β9\beta_{9}== (ν12104)/175 ( -\nu^{12} - 104 ) / 175 Copy content Toggle raw display
β10\beta_{10}== (ν159ν13+1504ν3+639ν)/4725 ( \nu^{15} - 9\nu^{13} + 1504\nu^{3} + 639\nu ) / 4725 Copy content Toggle raw display
β11\beta_{11}== (ν1416ν10+175ν61296ν2)/729 ( \nu^{14} - 16\nu^{10} + 175\nu^{6} - 1296\nu^{2} ) / 729 Copy content Toggle raw display
β12\beta_{12}== (ν1516ν11+175ν71296ν3+2187ν)/2187 ( \nu^{15} - 16\nu^{11} + 175\nu^{7} - 1296\nu^{3} + 2187\nu ) / 2187 Copy content Toggle raw display
β13\beta_{13}== (175ν15+279ν13+2800ν1111025ν930625ν7+48825ν5+361584ν)/382725 ( - 175 \nu^{15} + 279 \nu^{13} + 2800 \nu^{11} - 11025 \nu^{9} - 30625 \nu^{7} + 48825 \nu^{5} + \cdots - 361584 \nu ) / 382725 Copy content Toggle raw display
β14\beta_{14}== (256ν15432ν132800ν11+4725ν9+30625ν775600ν5++559872ν)/382725 ( 256 \nu^{15} - 432 \nu^{13} - 2800 \nu^{11} + 4725 \nu^{9} + 30625 \nu^{7} - 75600 \nu^{5} + \cdots + 559872 \nu ) / 382725 Copy content Toggle raw display
β15\beta_{15}== (431ν15432ν13+5600ν11+4725ν961250ν775600ν5++559872ν)/382725 ( - 431 \nu^{15} - 432 \nu^{13} + 5600 \nu^{11} + 4725 \nu^{9} - 61250 \nu^{7} - 75600 \nu^{5} + \cdots + 559872 \nu ) / 382725 Copy content Toggle raw display
ν\nu== (β15β14+5β12+2β7+2β2)/7 ( \beta_{15} - \beta_{14} + 5\beta_{12} + 2\beta_{7} + 2\beta_{2} ) / 7 Copy content Toggle raw display
ν2\nu^{2}== β1 \beta_1 Copy content Toggle raw display
ν3\nu^{3}== (4β15+4β146β12+13β7+7β6+6β2)/7 ( -4\beta_{15} + 4\beta_{14} - 6\beta_{12} + 13\beta_{7} + 7\beta_{6} + 6\beta_{2} ) / 7 Copy content Toggle raw display
ν4\nu^{4}== β9+β58β3+8 \beta_{9} + \beta_{5} - 8\beta_{3} + 8 Copy content Toggle raw display
ν5\nu^{5}== (5β1544β1421β13+31β12+21β10+18β7+18β2)/7 ( -5\beta_{15} - 44\beta_{14} - 21\beta_{13} + 31\beta_{12} + 21\beta_{10} + 18\beta_{7} + 18\beta_{2} ) / 7 Copy content Toggle raw display
ν6\nu^{6}== 7β11+9β87β4+7β1 7\beta_{11} + 9\beta_{8} - 7\beta_{4} + 7\beta_1 Copy content Toggle raw display
ν7\nu^{7}== (15β15+15β145β12+5β7+112β6+5β2)/7 ( -15\beta_{15} + 15\beta_{14} - 5\beta_{12} + 5\beta_{7} + 112\beta_{6} + 5\beta_{2} ) / 7 Copy content Toggle raw display
ν8\nu^{8}== 16β547β3 16\beta_{5} - 47\beta_{3} Copy content Toggle raw display
ν9\nu^{9}== (45β15172β14336β13127β12+127β7+127β2)/7 ( -45\beta_{15} - 172\beta_{14} - 336\beta_{13} - 127\beta_{12} + 127\beta_{7} + 127\beta_{2} ) / 7 Copy content Toggle raw display
ν10\nu^{10}== 31β11175β4 31\beta_{11} - 175\beta_{4} Copy content Toggle raw display
ν11\nu^{11}== (82β1582β14164β121061β7+164β2)/7 ( 82\beta_{15} - 82\beta_{14} - 164\beta_{12} - 1061\beta_{7} + 164\beta_{2} ) / 7 Copy content Toggle raw display
ν12\nu^{12}== 175β9104 -175\beta_{9} - 104 Copy content Toggle raw display
ν13\nu^{13}== (979β15+979β141220β123675β10+1717β7+1717β2)/7 ( -979\beta_{15} + 979\beta_{14} - 1220\beta_{12} - 3675\beta_{10} + 1717\beta_{7} + 1717\beta_{2} ) / 7 Copy content Toggle raw display
ν14\nu^{14}== 1575β81575β4+71β1 -1575\beta_{8} - 1575\beta_{4} + 71\beta_1 Copy content Toggle raw display
ν15\nu^{15}== (3434β15+3434β145151β125377β710528β6+5151β2)/7 ( -3434\beta_{15} + 3434\beta_{14} - 5151\beta_{12} - 5377\beta_{7} - 10528\beta_{6} + 5151\beta_{2} ) / 7 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/476Z)×\left(\mathbb{Z}/476\mathbb{Z}\right)^\times.

nn 239239 309309 409409
χ(n)\chi(n) 1-1 1-1 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.

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}
271.1
−0.643691 + 1.60800i
1.71441 + 0.246547i
−1.71441 0.246547i
0.643691 1.60800i
−0.246547 + 1.71441i
1.60800 + 0.643691i
−1.60800 0.643691i
0.246547 1.71441i
−0.643691 1.60800i
1.71441 0.246547i
−1.71441 + 0.246547i
0.643691 + 1.60800i
−0.246547 1.71441i
1.60800 0.643691i
−1.60800 + 0.643691i
0.246547 + 1.71441i
−0.707107 + 1.22474i −2.78514 + 1.60800i −1.00000 1.73205i 0 4.54811i 2.58495 + 0.563932i 2.82843 3.67132 6.35892i 0
271.2 −0.707107 + 1.22474i −0.427031 + 0.246547i −1.00000 1.73205i 0 0.697339i 2.58495 0.563932i 2.82843 −1.37843 + 2.38751i 0
271.3 −0.707107 + 1.22474i 0.427031 0.246547i −1.00000 1.73205i 0 0.697339i −2.58495 + 0.563932i 2.82843 −1.37843 + 2.38751i 0
271.4 −0.707107 + 1.22474i 2.78514 1.60800i −1.00000 1.73205i 0 4.54811i −2.58495 0.563932i 2.82843 3.67132 6.35892i 0
271.5 0.707107 1.22474i −2.96945 + 1.71441i −1.00000 1.73205i 0 4.84909i −0.563932 2.58495i −2.82843 4.37843 7.58366i 0
271.6 0.707107 1.22474i −1.11491 + 0.643691i −1.00000 1.73205i 0 1.82063i −0.563932 + 2.58495i −2.82843 −0.671323 + 1.16277i 0
271.7 0.707107 1.22474i 1.11491 0.643691i −1.00000 1.73205i 0 1.82063i 0.563932 2.58495i −2.82843 −0.671323 + 1.16277i 0
271.8 0.707107 1.22474i 2.96945 1.71441i −1.00000 1.73205i 0 4.84909i 0.563932 + 2.58495i −2.82843 4.37843 7.58366i 0
339.1 −0.707107 1.22474i −2.78514 1.60800i −1.00000 + 1.73205i 0 4.54811i 2.58495 0.563932i 2.82843 3.67132 + 6.35892i 0
339.2 −0.707107 1.22474i −0.427031 0.246547i −1.00000 + 1.73205i 0 0.697339i 2.58495 + 0.563932i 2.82843 −1.37843 2.38751i 0
339.3 −0.707107 1.22474i 0.427031 + 0.246547i −1.00000 + 1.73205i 0 0.697339i −2.58495 0.563932i 2.82843 −1.37843 2.38751i 0
339.4 −0.707107 1.22474i 2.78514 + 1.60800i −1.00000 + 1.73205i 0 4.54811i −2.58495 + 0.563932i 2.82843 3.67132 + 6.35892i 0
339.5 0.707107 + 1.22474i −2.96945 1.71441i −1.00000 + 1.73205i 0 4.84909i −0.563932 + 2.58495i −2.82843 4.37843 + 7.58366i 0
339.6 0.707107 + 1.22474i −1.11491 0.643691i −1.00000 + 1.73205i 0 1.82063i −0.563932 2.58495i −2.82843 −0.671323 1.16277i 0
339.7 0.707107 + 1.22474i 1.11491 + 0.643691i −1.00000 + 1.73205i 0 1.82063i 0.563932 + 2.58495i −2.82843 −0.671323 1.16277i 0
339.8 0.707107 + 1.22474i 2.96945 + 1.71441i −1.00000 + 1.73205i 0 4.84909i 0.563932 2.58495i −2.82843 4.37843 + 7.58366i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by Q(17)\Q(\sqrt{-17})
4.b odd 2 1 inner
7.d odd 6 1 inner
17.b even 2 1 inner
28.f even 6 1 inner
119.h odd 6 1 inner
476.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 476.2.q.a 16
4.b odd 2 1 inner 476.2.q.a 16
7.d odd 6 1 inner 476.2.q.a 16
17.b even 2 1 inner 476.2.q.a 16
28.f even 6 1 inner 476.2.q.a 16
68.d odd 2 1 CM 476.2.q.a 16
119.h odd 6 1 inner 476.2.q.a 16
476.q even 6 1 inner 476.2.q.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.q.a 16 1.a even 1 1 trivial
476.2.q.a 16 4.b odd 2 1 inner
476.2.q.a 16 7.d odd 6 1 inner
476.2.q.a 16 17.b even 2 1 inner
476.2.q.a 16 28.f even 6 1 inner
476.2.q.a 16 68.d odd 2 1 CM
476.2.q.a 16 119.h odd 6 1 inner
476.2.q.a 16 476.q even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T31624T314+412T3123456T310+21087T3837008T36+49564T3411760T32+2401 T_{3}^{16} - 24T_{3}^{14} + 412T_{3}^{12} - 3456T_{3}^{10} + 21087T_{3}^{8} - 37008T_{3}^{6} + 49564T_{3}^{4} - 11760T_{3}^{2} + 2401 acting on S2new(476,[χ])S_{2}^{\mathrm{new}}(476, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4+2T2+4)4 (T^{4} + 2 T^{2} + 4)^{4} Copy content Toggle raw display
33 T1624T14++2401 T^{16} - 24 T^{14} + \cdots + 2401 Copy content Toggle raw display
55 T16 T^{16} Copy content Toggle raw display
77 (T864T4+2401)2 (T^{8} - 64 T^{4} + 2401)^{2} Copy content Toggle raw display
1111 T16++11716114081 T^{16} + \cdots + 11716114081 Copy content Toggle raw display
1313 (T4+44T2+25)4 (T^{4} + 44 T^{2} + 25)^{4} Copy content Toggle raw display
1717 (T417T2+289)4 (T^{4} - 17 T^{2} + 289)^{4} Copy content Toggle raw display
1919 T16 T^{16} Copy content Toggle raw display
2323 (T8+92T6++2775556)2 (T^{8} + 92 T^{6} + \cdots + 2775556)^{2} Copy content Toggle raw display
2929 T16 T^{16} Copy content Toggle raw display
3131 (T8124T6++2775556)2 (T^{8} - 124 T^{6} + \cdots + 2775556)^{2} Copy content Toggle raw display
3737 T16 T^{16} Copy content Toggle raw display
4141 T16 T^{16} Copy content Toggle raw display
4343 T16 T^{16} Copy content Toggle raw display
4747 T16 T^{16} Copy content Toggle raw display
5353 (T412T3++225)4 (T^{4} - 12 T^{3} + \cdots + 225)^{4} Copy content Toggle raw display
5959 T16 T^{16} Copy content Toggle raw display
6161 T16 T^{16} Copy content Toggle raw display
6767 T16 T^{16} Copy content Toggle raw display
7171 (T8568T6++8196769)2 (T^{8} - 568 T^{6} + \cdots + 8196769)^{2} Copy content Toggle raw display
7373 T16 T^{16} Copy content Toggle raw display
7979 T16++62 ⁣ ⁣81 T^{16} + \cdots + 62\!\cdots\!81 Copy content Toggle raw display
8383 T16 T^{16} Copy content Toggle raw display
8989 (T8228T6++2313441)2 (T^{8} - 228 T^{6} + \cdots + 2313441)^{2} Copy content Toggle raw display
9797 T16 T^{16} Copy content Toggle raw display
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