gp: [N,k,chi] = [3969,2,Mod(1,3969)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3969, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3969.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [6,-2,0,6,0,0,0,-12,0,0,-8]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 − 12 x 4 + 21 x 2 − 9 x^{6} - 12x^{4} + 21x^{2} - 9 x 6 − 1 2 x 4 + 2 1 x 2 − 9
x^6 - 12*x^4 + 21*x^2 - 9
:
β 1 \beta_{1} β 1 = = =
( ν 5 − 12 ν 3 + 21 ν ) / 3 ( \nu^{5} - 12\nu^{3} + 21\nu ) / 3 ( ν 5 − 1 2 ν 3 + 2 1 ν ) / 3
(v^5 - 12*v^3 + 21*v) / 3
β 2 \beta_{2} β 2 = = =
( ν 4 − 12 ν 2 + 15 ) / 3 ( \nu^{4} - 12\nu^{2} + 15 ) / 3 ( ν 4 − 1 2 ν 2 + 1 5 ) / 3
(v^4 - 12*v^2 + 15) / 3
β 3 \beta_{3} β 3 = = =
( 2 ν 5 − 21 ν 3 + 15 ν ) / 3 ( 2\nu^{5} - 21\nu^{3} + 15\nu ) / 3 ( 2 ν 5 − 2 1 ν 3 + 1 5 ν ) / 3
(2*v^5 - 21*v^3 + 15*v) / 3
β 4 \beta_{4} β 4 = = =
( 2 ν 4 − 21 ν 2 + 15 ) / 3 ( 2\nu^{4} - 21\nu^{2} + 15 ) / 3 ( 2 ν 4 − 2 1 ν 2 + 1 5 ) / 3
(2*v^4 - 21*v^2 + 15) / 3
β 5 \beta_{5} β 5 = = =
ν 5 − 11 ν 3 + 9 ν \nu^{5} - 11\nu^{3} + 9\nu ν 5 − 1 1 ν 3 + 9 ν
v^5 - 11*v^3 + 9*v
ν \nu ν = = =
( − β 5 + β 3 + β 1 ) / 3 ( -\beta_{5} + \beta_{3} + \beta_1 ) / 3 ( − β 5 + β 3 + β 1 ) / 3
(-b5 + b3 + b1) / 3
ν 2 \nu^{2} ν 2 = = =
β 4 − 2 β 2 + 5 \beta_{4} - 2\beta_{2} + 5 β 4 − 2 β 2 + 5
b4 - 2*b2 + 5
ν 3 \nu^{3} ν 3 = = =
− 3 β 5 + 4 β 3 + β 1 -3\beta_{5} + 4\beta_{3} + \beta_1 − 3 β 5 + 4 β 3 + β 1
-3*b5 + 4*b3 + b1
ν 4 \nu^{4} ν 4 = = =
12 β 4 − 21 β 2 + 45 12\beta_{4} - 21\beta_{2} + 45 1 2 β 4 − 2 1 β 2 + 4 5
12*b4 - 21*b2 + 45
ν 5 \nu^{5} ν 5 = = =
− 29 β 5 + 41 β 3 + 8 β 1 -29\beta_{5} + 41\beta_{3} + 8\beta_1 − 2 9 β 5 + 4 1 β 3 + 8 β 1
-29*b5 + 41*b3 + 8*b1
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
3 3 3
− 1 -1 − 1
7 7 7
− 1 -1 − 1
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 2 n e w ( Γ 0 ( 3969 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(3969)) S 2 n e w ( Γ 0 ( 3 9 6 9 ) ) :
T 2 3 + T 2 2 − 4 T 2 − 1 T_{2}^{3} + T_{2}^{2} - 4T_{2} - 1 T 2 3 + T 2 2 − 4 T 2 − 1
T2^3 + T2^2 - 4*T2 - 1
T 5 6 − 21 T 5 4 + 108 T 5 2 − 81 T_{5}^{6} - 21T_{5}^{4} + 108T_{5}^{2} - 81 T 5 6 − 2 1 T 5 4 + 1 0 8 T 5 2 − 8 1
T5^6 - 21*T5^4 + 108*T5^2 - 81
T 11 3 + 4 T 11 2 − T 11 − 1 T_{11}^{3} + 4T_{11}^{2} - T_{11} - 1 T 1 1 3 + 4 T 1 1 2 − T 1 1 − 1
T11^3 + 4*T11^2 - T11 - 1
T 13 6 − 39 T 13 4 + 351 T 13 2 − 81 T_{13}^{6} - 39T_{13}^{4} + 351T_{13}^{2} - 81 T 1 3 6 − 3 9 T 1 3 4 + 3 5 1 T 1 3 2 − 8 1
T13^6 - 39*T13^4 + 351*T13^2 - 81
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 3 + T 2 − 4 T − 1 ) 2 (T^{3} + T^{2} - 4 T - 1)^{2} ( T 3 + T 2 − 4 T − 1 ) 2
(T^3 + T^2 - 4*T - 1)^2
3 3 3
T 6 T^{6} T 6
T^6
5 5 5
T 6 − 21 T 4 + ⋯ − 81 T^{6} - 21 T^{4} + \cdots - 81 T 6 − 2 1 T 4 + ⋯ − 8 1
T^6 - 21*T^4 + 108*T^2 - 81
7 7 7
T 6 T^{6} T 6
T^6
11 11 1 1
( T 3 + 4 T 2 − T − 1 ) 2 (T^{3} + 4 T^{2} - T - 1)^{2} ( T 3 + 4 T 2 − T − 1 ) 2
(T^3 + 4*T^2 - T - 1)^2
13 13 1 3
T 6 − 39 T 4 + ⋯ − 81 T^{6} - 39 T^{4} + \cdots - 81 T 6 − 3 9 T 4 + ⋯ − 8 1
T^6 - 39*T^4 + 351*T^2 - 81
17 17 1 7
T 6 − 84 T 4 + ⋯ − 3969 T^{6} - 84 T^{4} + \cdots - 3969 T 6 − 8 4 T 4 + ⋯ − 3 9 6 9
T^6 - 84*T^4 + 1593*T^2 - 3969
19 19 1 9
T 6 − 75 T 4 + ⋯ − 3969 T^{6} - 75 T^{4} + \cdots - 3969 T 6 − 7 5 T 4 + ⋯ − 3 9 6 9
T^6 - 75*T^4 + 1296*T^2 - 3969
23 23 2 3
( T 3 + 2 T 2 − 25 T − 59 ) 2 (T^{3} + 2 T^{2} - 25 T - 59)^{2} ( T 3 + 2 T 2 − 2 5 T − 5 9 ) 2
(T^3 + 2*T^2 - 25*T - 59)^2
29 29 2 9
( T 3 + 11 T 2 + ⋯ − 89 ) 2 (T^{3} + 11 T^{2} + \cdots - 89)^{2} ( T 3 + 1 1 T 2 + ⋯ − 8 9 ) 2
(T^3 + 11*T^2 + 14*T - 89)^2
31 31 3 1
T 6 − 129 T 4 + ⋯ − 77841 T^{6} - 129 T^{4} + \cdots - 77841 T 6 − 1 2 9 T 4 + ⋯ − 7 7 8 4 1
T^6 - 129*T^4 + 5508*T^2 - 77841
37 37 3 7
( T 3 + 3 T 2 − 24 T + 27 ) 2 (T^{3} + 3 T^{2} - 24 T + 27)^{2} ( T 3 + 3 T 2 − 2 4 T + 2 7 ) 2
(T^3 + 3*T^2 - 24*T + 27)^2
41 41 4 1
T 6 − 162 T 4 + ⋯ − 6561 T^{6} - 162 T^{4} + \cdots - 6561 T 6 − 1 6 2 T 4 + ⋯ − 6 5 6 1
T^6 - 162*T^4 + 6075*T^2 - 6561
43 43 4 3
( T 3 − 3 T 2 − 24 T − 27 ) 2 (T^{3} - 3 T^{2} - 24 T - 27)^{2} ( T 3 − 3 T 2 − 2 4 T − 2 7 ) 2
(T^3 - 3*T^2 - 24*T - 27)^2
47 47 4 7
T 6 − 183 T 4 + ⋯ − 194481 T^{6} - 183 T^{4} + \cdots - 194481 T 6 − 1 8 3 T 4 + ⋯ − 1 9 4 4 8 1
T^6 - 183*T^4 + 10584*T^2 - 194481
53 53 5 3
( T 3 + 14 T 2 + ⋯ − 263 ) 2 (T^{3} + 14 T^{2} + \cdots - 263)^{2} ( T 3 + 1 4 T 2 + ⋯ − 2 6 3 ) 2
(T^3 + 14*T^2 + 11*T - 263)^2
59 59 5 9
T 6 − 183 T 4 + ⋯ − 149769 T^{6} - 183 T^{4} + \cdots - 149769 T 6 − 1 8 3 T 4 + ⋯ − 1 4 9 7 6 9
T^6 - 183*T^4 + 9450*T^2 - 149769
61 61 6 1
T 6 − 264 T 4 + ⋯ − 558009 T^{6} - 264 T^{4} + \cdots - 558009 T 6 − 2 6 4 T 4 + ⋯ − 5 5 8 0 0 9
T^6 - 264*T^4 + 21519*T^2 - 558009
67 67 6 7
( T 3 − 111 T + 353 ) 2 (T^{3} - 111 T + 353)^{2} ( T 3 − 1 1 1 T + 3 5 3 ) 2
(T^3 - 111*T + 353)^2
71 71 7 1
( T 3 + 19 T 2 + ⋯ + 227 ) 2 (T^{3} + 19 T^{2} + \cdots + 227)^{2} ( T 3 + 1 9 T 2 + ⋯ + 2 2 7 ) 2
(T^3 + 19*T^2 + 116*T + 227)^2
73 73 7 3
T 6 − 75 T 4 + ⋯ − 3969 T^{6} - 75 T^{4} + \cdots - 3969 T 6 − 7 5 T 4 + ⋯ − 3 9 6 9
T^6 - 75*T^4 + 1296*T^2 - 3969
79 79 7 9
( T 3 + 3 T 2 + ⋯ − 107 ) 2 (T^{3} + 3 T^{2} + \cdots - 107)^{2} ( T 3 + 3 T 2 + ⋯ − 1 0 7 ) 2
(T^3 + 3*T^2 - 78*T - 107)^2
83 83 8 3
T 6 − 228 T 4 + ⋯ − 227529 T^{6} - 228 T^{4} + \cdots - 227529 T 6 − 2 2 8 T 4 + ⋯ − 2 2 7 5 2 9
T^6 - 228*T^4 + 13365*T^2 - 227529
89 89 8 9
T 6 − 246 T 4 + ⋯ − 3969 T^{6} - 246 T^{4} + \cdots - 3969 T 6 − 2 4 6 T 4 + ⋯ − 3 9 6 9
T^6 - 246*T^4 + 5751*T^2 - 3969
97 97 9 7
T 6 − 111 T 4 + ⋯ − 9801 T^{6} - 111 T^{4} + \cdots - 9801 T 6 − 1 1 1 T 4 + ⋯ − 9 8 0 1
T^6 - 111*T^4 + 1998*T^2 - 9801
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