gp: [N,k,chi] = [605,4,Mod(1,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [6,3,12,27]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
gp: f = lf[1] \\ Warning: the index may be different
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the q q q -expansion are expressed in terms of a basis 1 , β 1 , … , β 5 1,\beta_1,\ldots,\beta_{5} 1 , β 1 , … , β 5 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 6 − 3 x 5 − 33 x 4 + 67 x 3 + 256 x 2 − 236 x + 44 x^{6} - 3x^{5} - 33x^{4} + 67x^{3} + 256x^{2} - 236x + 44 x 6 − 3 x 5 − 3 3 x 4 + 6 7 x 3 + 2 5 6 x 2 − 2 3 6 x + 4 4
x^6 - 3*x^5 - 33*x^4 + 67*x^3 + 256*x^2 - 236*x + 44
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
ν 2 − ν − 12 \nu^{2} - \nu - 12 ν 2 − ν − 1 2
v^2 - v - 12
β 3 \beta_{3} β 3 = = =
ν 3 − 2 ν 2 − 19 ν + 14 \nu^{3} - 2\nu^{2} - 19\nu + 14 ν 3 − 2 ν 2 − 1 9 ν + 1 4
v^3 - 2*v^2 - 19*v + 14
β 4 \beta_{4} β 4 = = =
( ν 5 − 2 ν 4 − 35 ν 3 + 40 ν 2 + 288 ν − 108 ) / 8 ( \nu^{5} - 2\nu^{4} - 35\nu^{3} + 40\nu^{2} + 288\nu - 108 ) / 8 ( ν 5 − 2 ν 4 − 3 5 ν 3 + 4 0 ν 2 + 2 8 8 ν − 1 0 8 ) / 8
(v^5 - 2*v^4 - 35*v^3 + 40*v^2 + 288*v - 108) / 8
β 5 \beta_{5} β 5 = = =
( − 5 ν 5 + 14 ν 4 + 163 ν 3 − 280 ν 2 − 1264 ν + 596 ) / 8 ( -5\nu^{5} + 14\nu^{4} + 163\nu^{3} - 280\nu^{2} - 1264\nu + 596 ) / 8 ( − 5 ν 5 + 1 4 ν 4 + 1 6 3 ν 3 − 2 8 0 ν 2 − 1 2 6 4 ν + 5 9 6 ) / 8
(-5*v^5 + 14*v^4 + 163*v^3 - 280*v^2 - 1264*v + 596) / 8
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 2 + β 1 + 12 \beta_{2} + \beta _1 + 12 β 2 + β 1 + 1 2
b2 + b1 + 12
ν 3 \nu^{3} ν 3 = = =
β 3 + 2 β 2 + 21 β 1 + 10 \beta_{3} + 2\beta_{2} + 21\beta _1 + 10 β 3 + 2 β 2 + 2 1 β 1 + 1 0
b3 + 2*b2 + 21*b1 + 10
ν 4 \nu^{4} ν 4 = = =
2 β 5 + 10 β 4 + 3 β 3 + 26 β 2 + 39 β 1 + 256 2\beta_{5} + 10\beta_{4} + 3\beta_{3} + 26\beta_{2} + 39\beta _1 + 256 2 β 5 + 1 0 β 4 + 3 β 3 + 2 6 β 2 + 3 9 β 1 + 2 5 6
2*b5 + 10*b4 + 3*b3 + 26*b2 + 39*b1 + 256
ν 5 \nu^{5} ν 5 = = =
4 β 5 + 28 β 4 + 41 β 3 + 82 β 2 + 485 β 1 + 490 4\beta_{5} + 28\beta_{4} + 41\beta_{3} + 82\beta_{2} + 485\beta _1 + 490 4 β 5 + 2 8 β 4 + 4 1 β 3 + 8 2 β 2 + 4 8 5 β 1 + 4 9 0
4*b5 + 28*b4 + 41*b3 + 82*b2 + 485*b1 + 490
For each embedding ι m \iota_m ι m of the coefficient field, the values ι m ( a n ) \iota_m(a_n) ι m ( a n ) are shown below.
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
Refresh table
p p p
Sign
5 5 5
− 1 -1 − 1
11 11 1 1
− 1 -1 − 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 4 n e w ( Γ 0 ( 605 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(605)) S 4 n e w ( Γ 0 ( 6 0 5 ) ) :
T 2 6 − 3 T 2 5 − 33 T 2 4 + 75 T 2 3 + 244 T 2 2 − 336 T 2 + 96 T_{2}^{6} - 3T_{2}^{5} - 33T_{2}^{4} + 75T_{2}^{3} + 244T_{2}^{2} - 336T_{2} + 96 T 2 6 − 3 T 2 5 − 3 3 T 2 4 + 7 5 T 2 3 + 2 4 4 T 2 2 − 3 3 6 T 2 + 9 6
T2^6 - 3*T2^5 - 33*T2^4 + 75*T2^3 + 244*T2^2 - 336*T2 + 96
T 3 6 − 12 T 3 5 − 61 T 3 4 + 978 T 3 3 − 835 T 3 2 − 7374 T 3 − 4577 T_{3}^{6} - 12T_{3}^{5} - 61T_{3}^{4} + 978T_{3}^{3} - 835T_{3}^{2} - 7374T_{3} - 4577 T 3 6 − 1 2 T 3 5 − 6 1 T 3 4 + 9 7 8 T 3 3 − 8 3 5 T 3 2 − 7 3 7 4 T 3 − 4 5 7 7
T3^6 - 12*T3^5 - 61*T3^4 + 978*T3^3 - 835*T3^2 - 7374*T3 - 4577
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 6 − 3 T 5 + ⋯ + 96 T^{6} - 3 T^{5} + \cdots + 96 T 6 − 3 T 5 + ⋯ + 9 6
T^6 - 3*T^5 - 33*T^4 + 75*T^3 + 244*T^2 - 336*T + 96
3 3 3
T 6 − 12 T 5 + ⋯ − 4577 T^{6} - 12 T^{5} + \cdots - 4577 T 6 − 1 2 T 5 + ⋯ − 4 5 7 7
T^6 - 12*T^5 - 61*T^4 + 978*T^3 - 835*T^2 - 7374*T - 4577
5 5 5
( T − 5 ) 6 (T - 5)^{6} ( T − 5 ) 6
(T - 5)^6
7 7 7
T 6 + 10 T 5 + ⋯ + 31877147 T^{6} + 10 T^{5} + \cdots + 31877147 T 6 + 1 0 T 5 + ⋯ + 3 1 8 7 7 1 4 7
T^6 + 10*T^5 - 1485*T^4 - 18194*T^3 + 480993*T^2 + 8321008*T + 31877147
11 11 1 1
T 6 T^{6} T 6
T^6
13 13 1 3
T 6 − 80 T 5 + ⋯ − 9114688 T^{6} - 80 T^{5} + \cdots - 9114688 T 6 − 8 0 T 5 + ⋯ − 9 1 1 4 6 8 8
T^6 - 80*T^5 - 100*T^4 + 40924*T^3 - 72240*T^2 - 4243008*T - 9114688
17 17 1 7
T 6 + ⋯ − 3820861440 T^{6} + \cdots - 3820861440 T 6 + ⋯ − 3 8 2 0 8 6 1 4 4 0
T^6 + 12*T^5 - 8856*T^4 - 7152*T^3 + 14861440*T^2 - 94047744*T - 3820861440
19 19 1 9
T 6 + ⋯ − 111603969216 T^{6} + \cdots - 111603969216 T 6 + ⋯ − 1 1 1 6 0 3 9 6 9 2 1 6
T^6 - 86*T^5 - 22268*T^4 + 1451100*T^3 + 90838400*T^2 - 2539541952*T - 111603969216
23 23 2 3
T 6 + ⋯ + 4087676116032 T^{6} + \cdots + 4087676116032 T 6 + ⋯ + 4 0 8 7 6 7 6 1 1 6 0 3 2
T^6 - 406*T^5 + 21380*T^4 + 8354972*T^3 - 773507792*T^2 - 41324967072*T + 4087676116032
29 29 2 9
T 6 + ⋯ − 11701424064 T^{6} + \cdots - 11701424064 T 6 + ⋯ − 1 1 7 0 1 4 2 4 0 6 4
T^6 + 372*T^5 + 7956*T^4 - 4541616*T^3 + 69857296*T^2 + 4734449280*T - 11701424064
31 31 3 1
T 6 + ⋯ − 15269134498272 T^{6} + \cdots - 15269134498272 T 6 + ⋯ − 1 5 2 6 9 1 3 4 4 9 8 2 7 2
T^6 + 122*T^5 - 100292*T^4 - 11950692*T^3 + 2627480088*T^2 + 248300869392*T - 15269134498272
37 37 3 7
T 6 + ⋯ − 5305026789376 T^{6} + \cdots - 5305026789376 T 6 + ⋯ − 5 3 0 5 0 2 6 7 8 9 3 7 6
T^6 - 576*T^5 + 75216*T^4 + 8010608*T^3 - 2584774400*T^2 + 203634059264*T - 5305026789376
41 41 4 1
T 6 + ⋯ − 255688255776375 T^{6} + \cdots - 255688255776375 T 6 + ⋯ − 2 5 5 6 8 8 2 5 5 7 7 6 3 7 5
T^6 - 46*T^5 - 265213*T^4 + 20024616*T^3 + 16970478915*T^2 - 1137137934474*T - 255688255776375
43 43 4 3
T 6 + ⋯ − 588984378067305 T^{6} + \cdots - 588984378067305 T 6 + ⋯ − 5 8 8 9 8 4 3 7 8 0 6 7 3 0 5
T^6 - 10*T^5 - 390517*T^4 - 7575958*T^3 + 32807860189*T^2 + 566649523224*T - 588984378067305
47 47 4 7
T 6 + ⋯ + 38794898412339 T^{6} + \cdots + 38794898412339 T 6 + ⋯ + 3 8 7 9 4 8 9 8 4 1 2 3 3 9
T^6 - 568*T^5 - 5593*T^4 + 46781762*T^3 - 6230493179*T^2 - 143975057346*T + 38794898412339
53 53 5 3
T 6 + ⋯ − 20 ⋯ 92 T^{6} + \cdots - 20\!\cdots\!92 T 6 + ⋯ − 2 0 ⋯ 9 2
T^6 + 184*T^5 - 448116*T^4 - 64488860*T^3 + 55990257680*T^2 + 4763263456128*T - 2073577902441792
59 59 5 9
T 6 + ⋯ − 160905290704224 T^{6} + \cdots - 160905290704224 T 6 + ⋯ − 1 6 0 9 0 5 2 9 0 7 0 4 2 2 4
T^6 - 1598*T^5 + 769456*T^4 - 68179164*T^3 - 32104234552*T^2 + 6037200024240*T - 160905290704224
61 61 6 1
T 6 + ⋯ − 19 ⋯ 87 T^{6} + \cdots - 19\!\cdots\!87 T 6 + ⋯ − 1 9 ⋯ 8 7
T^6 - 1598*T^5 + 163403*T^4 + 1069814656*T^3 - 784759070589*T^2 + 212589254798574*T - 19734292541201887
67 67 6 7
T 6 + ⋯ − 17 ⋯ 89 T^{6} + \cdots - 17\!\cdots\!89 T 6 + ⋯ − 1 7 ⋯ 8 9
T^6 - 528*T^5 - 1313089*T^4 + 659569962*T^3 + 409624066889*T^2 - 180007454341842*T - 17410974449174489
71 71 7 1
T 6 + ⋯ + 30 ⋯ 12 T^{6} + \cdots + 30\!\cdots\!12 T 6 + ⋯ + 3 0 ⋯ 1 2
T^6 - 142*T^5 - 1756560*T^4 + 66499844*T^3 + 705733678712*T^2 + 119589294793584*T + 3098871683558112
73 73 7 3
T 6 + ⋯ − 64 ⋯ 36 T^{6} + \cdots - 64\!\cdots\!36 T 6 + ⋯ − 6 4 ⋯ 3 6
T^6 + 268*T^5 - 1460176*T^4 - 232599968*T^3 + 581862287616*T^2 + 43001976453120*T - 64927663145537536
79 79 7 9
T 6 + ⋯ − 21 ⋯ 60 T^{6} + \cdots - 21\!\cdots\!60 T 6 + ⋯ − 2 1 ⋯ 6 0
T^6 + 624*T^5 - 666236*T^4 - 788359344*T^3 - 292622791856*T^2 - 44632694484288*T - 2139297427316160
83 83 8 3
T 6 + ⋯ + 39 ⋯ 12 T^{6} + \cdots + 39\!\cdots\!12 T 6 + ⋯ + 3 9 ⋯ 1 2
T^6 + 2786*T^5 + 1742140*T^4 - 866617692*T^3 - 892382211840*T^2 - 101549823611040*T + 3914541053963712
89 89 8 9
T 6 + ⋯ + 88 ⋯ 97 T^{6} + \cdots + 88\!\cdots\!97 T 6 + ⋯ + 8 8 ⋯ 9 7
T^6 - 634*T^5 - 1765981*T^4 + 1499579764*T^3 + 268190700295*T^2 - 453396021884058*T + 88352276398846197
97 97 9 7
T 6 + ⋯ − 46 ⋯ 36 T^{6} + \cdots - 46\!\cdots\!36 T 6 + ⋯ − 4 6 ⋯ 3 6
T^6 - 648*T^5 - 1177404*T^4 + 336305828*T^3 + 216721548352*T^2 - 22353188444992*T - 4608877431228736
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