gp: [N,k,chi] = [600,6,Mod(1,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
sage: traces = [4,0,-36,0,0,0,-6]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 2 , β 3 1,\beta_1,\beta_2,\beta_3 1 , β 1 , β 2 , β 3 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 4 − 71 x 2 − 30 x + 360 x^{4} - 71x^{2} - 30x + 360 x 4 − 7 1 x 2 − 3 0 x + 3 6 0
x^4 - 71*x^2 - 30*x + 360
:
β 1 \beta_{1} β 1 = = =
( − 5 ν 3 + 265 ν + 111 ) / 3 ( -5\nu^{3} + 265\nu + 111 ) / 3 ( − 5 ν 3 + 2 6 5 ν + 1 1 1 ) / 3
(-5*v^3 + 265*v + 111) / 3
β 2 \beta_{2} β 2 = = =
− ν 3 − ν 2 + 80 ν + 58 -\nu^{3} - \nu^{2} + 80\nu + 58 − ν 3 − ν 2 + 8 0 ν + 5 8
-v^3 - v^2 + 80*v + 58
β 3 \beta_{3} β 3 = = =
− ν 3 + 4 ν 2 + 65 ν − 120 -\nu^{3} + 4\nu^{2} + 65\nu - 120 − ν 3 + 4 ν 2 + 6 5 ν − 1 2 0
-v^3 + 4*v^2 + 65*v - 120
ν \nu ν = = =
( β 3 + 4 β 2 − 3 β 1 − 1 ) / 120 ( \beta_{3} + 4\beta_{2} - 3\beta _1 - 1 ) / 120 ( β 3 + 4 β 2 − 3 β 1 − 1 ) / 1 2 0
(b3 + 4*b2 - 3*b1 - 1) / 120
ν 2 \nu^{2} ν 2 = = =
( 9 β 3 − 4 β 2 − 3 β 1 + 1423 ) / 40 ( 9\beta_{3} - 4\beta_{2} - 3\beta _1 + 1423 ) / 40 ( 9 β 3 − 4 β 2 − 3 β 1 + 1 4 2 3 ) / 4 0
(9*b3 - 4*b2 - 3*b1 + 1423) / 40
ν 3 \nu^{3} ν 3 = = =
( 53 β 3 + 212 β 2 − 231 β 1 + 2611 ) / 120 ( 53\beta_{3} + 212\beta_{2} - 231\beta _1 + 2611 ) / 120 ( 5 3 β 3 + 2 1 2 β 2 − 2 3 1 β 1 + 2 6 1 1 ) / 1 2 0
(53*b3 + 212*b2 - 231*b1 + 2611) / 120
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
2 2 2
+ 1 +1 + 1
3 3 3
+ 1 +1 + 1
5 5 5
− 1 -1 − 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 7 4 + 6 T 7 3 − 42188 T 7 2 − 4078392 T 7 − 80996000 T_{7}^{4} + 6T_{7}^{3} - 42188T_{7}^{2} - 4078392T_{7} - 80996000 T 7 4 + 6 T 7 3 − 4 2 1 8 8 T 7 2 − 4 0 7 8 3 9 2 T 7 − 8 0 9 9 6 0 0 0
T7^4 + 6*T7^3 - 42188*T7^2 - 4078392*T7 - 80996000
acting on S 6 n e w ( Γ 0 ( 600 ) ) S_{6}^{\mathrm{new}}(\Gamma_0(600)) S 6 n e w ( Γ 0 ( 6 0 0 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 T^{4} T 4
T^4
3 3 3
( T + 9 ) 4 (T + 9)^{4} ( T + 9 ) 4
(T + 9)^4
5 5 5
T 4 T^{4} T 4
T^4
7 7 7
T 4 + 6 T 3 + ⋯ − 80996000 T^{4} + 6 T^{3} + \cdots - 80996000 T 4 + 6 T 3 + ⋯ − 8 0 9 9 6 0 0 0
T^4 + 6*T^3 - 42188*T^2 - 4078392*T - 80996000
11 11 1 1
T 4 + ⋯ − 7954368992 T^{4} + \cdots - 7954368992 T 4 + ⋯ − 7 9 5 4 3 6 8 9 9 2
T^4 - 342*T^3 - 200492*T^2 + 88434840*T - 7954368992
13 13 1 3
T 4 + ⋯ + 531625211008 T^{4} + \cdots + 531625211008 T 4 + ⋯ + 5 3 1 6 2 5 2 1 1 0 0 8
T^4 - 222*T^3 - 1499972*T^2 + 126187320*T + 531625211008
17 17 1 7
T 4 + ⋯ + 1356291692064 T^{4} + \cdots + 1356291692064 T 4 + ⋯ + 1 3 5 6 2 9 1 6 9 2 0 6 4
T^4 + 618*T^3 - 3708300*T^2 + 467899992*T + 1356291692064
19 19 1 9
T 4 + ⋯ − 1452392480768 T^{4} + \cdots - 1452392480768 T 4 + ⋯ − 1 4 5 2 3 9 2 4 8 0 7 6 8
T^4 - 1520*T^3 - 1886784*T^2 + 3787611136*T - 1452392480768
23 23 2 3
T 4 + ⋯ + 9590662585600 T^{4} + \cdots + 9590662585600 T 4 + ⋯ + 9 5 9 0 6 6 2 5 8 5 6 0 0
T^4 + 752*T^3 - 6731232*T^2 - 3674058496*T + 9590662585600
29 29 2 9
T 4 + ⋯ − 318720439844864 T^{4} + \cdots - 318720439844864 T 4 + ⋯ − 3 1 8 7 2 0 4 3 9 8 4 4 8 6 4
T^4 - 4002*T^3 - 40798736*T^2 + 244204826112*T - 318720439844864
31 31 3 1
T 4 + ⋯ − 51134602752000 T^{4} + \cdots - 51134602752000 T 4 + ⋯ − 5 1 1 3 4 6 0 2 7 5 2 0 0 0
T^4 - 3012*T^3 - 50140800*T^2 + 153716486400*T - 51134602752000
37 37 3 7
T 4 + ⋯ − 11 ⋯ 00 T^{4} + \cdots - 11\!\cdots\!00 T 4 + ⋯ − 1 1 ⋯ 0 0
T^4 + 14142*T^3 - 9577940*T^2 - 616129409400*T - 1132098494168000
41 41 4 1
T 4 + ⋯ + 39 ⋯ 92 T^{4} + \cdots + 39\!\cdots\!92 T 4 + ⋯ + 3 9 ⋯ 9 2
T^4 + 18900*T^3 - 45850064*T^2 - 1227533332176*T + 3978546127865392
43 43 4 3
T 4 + ⋯ + 47 ⋯ 64 T^{4} + \cdots + 47\!\cdots\!64 T 4 + ⋯ + 4 7 ⋯ 6 4
T^4 + 11424*T^3 - 132415200*T^2 - 882271097856*T + 4754460581086464
47 47 4 7
T 4 + ⋯ + 64 ⋯ 00 T^{4} + \cdots + 64\!\cdots\!00 T 4 + ⋯ + 6 4 ⋯ 0 0
T^4 + 7572*T^3 - 414530400*T^2 - 725403110400*T + 6439566901248000
53 53 5 3
T 4 + ⋯ − 50 ⋯ 00 T^{4} + \cdots - 50\!\cdots\!00 T 4 + ⋯ − 5 0 ⋯ 0 0
T^4 + 47374*T^3 - 358645020*T^2 - 43213821044600*T - 504218693049812000
59 59 5 9
T 4 + ⋯ − 69 ⋯ 32 T^{4} + \cdots - 69\!\cdots\!32 T 4 + ⋯ − 6 9 ⋯ 3 2
T^4 - 43014*T^3 + 279226212*T^2 + 6744017047320*T - 69521982305008032
61 61 6 1
T 4 + ⋯ − 15 ⋯ 28 T^{4} + \cdots - 15\!\cdots\!28 T 4 + ⋯ − 1 5 ⋯ 2 8
T^4 + 19480*T^3 - 1559402664*T^2 - 38180267839136*T - 157467631166578928
67 67 6 7
T 4 + ⋯ + 58 ⋯ 32 T^{4} + \cdots + 58\!\cdots\!32 T 4 + ⋯ + 5 8 ⋯ 3 2
T^4 + 25620*T^3 - 4727473184*T^2 - 60879451950336*T + 5898428346168850432
71 71 7 1
T 4 + ⋯ − 12 ⋯ 32 T^{4} + \cdots - 12\!\cdots\!32 T 4 + ⋯ − 1 2 ⋯ 3 2
T^4 - 81036*T^3 + 911625712*T^2 + 59365340863680*T - 1218719162661343232
73 73 7 3
T 4 + ⋯ + 18 ⋯ 96 T^{4} + \cdots + 18\!\cdots\!96 T 4 + ⋯ + 1 8 ⋯ 9 6
T^4 + 111324*T^3 + 3871156816*T^2 + 47732211929664*T + 188830040180087296
79 79 7 9
T 4 + ⋯ + 12 ⋯ 56 T^{4} + \cdots + 12\!\cdots\!56 T 4 + ⋯ + 1 2 ⋯ 5 6
T^4 - 84988*T^3 - 6401793696*T^2 + 374203814532608*T + 12635467423946899456
83 83 8 3
T 4 + ⋯ − 36 ⋯ 88 T^{4} + \cdots - 36\!\cdots\!88 T 4 + ⋯ − 3 6 ⋯ 8 8
T^4 + 83200*T^3 - 1678533216*T^2 - 222916935254528*T - 3697613923871007488
89 89 8 9
T 4 + ⋯ − 17 ⋯ 40 T^{4} + \cdots - 17\!\cdots\!40 T 4 + ⋯ − 1 7 ⋯ 4 0
T^4 - 226488*T^3 - 3507369192*T^2 + 3436316564346144*T - 177773946658700793840
97 97 9 7
T 4 + ⋯ + 13 ⋯ 64 T^{4} + \cdots + 13\!\cdots\!64 T 4 + ⋯ + 1 3 ⋯ 6 4
T^4 - 32016*T^3 - 4724207360*T^2 - 39959882505216*T + 1342778387682340864
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