gp: [N,k,chi] = [650,4,Mod(599,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("650.599");
S:= CuspForms(chi, 4);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
sage: traces = [4,0,0,-16,0,-8,0,0,28,0,-100]
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 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 − 9 x 2 + 25 x^{4} - 9x^{2} + 25 x 4 − 9 x 2 + 2 5
x^4 - 9*x^2 + 25
:
β 1 \beta_{1} β 1 = = =
( ν 3 − 4 ν ) / 5 ( \nu^{3} - 4\nu ) / 5 ( ν 3 − 4 ν ) / 5
(v^3 - 4*v) / 5
β 2 \beta_{2} β 2 = = =
( − ν 3 + 14 ν ) / 5 ( -\nu^{3} + 14\nu ) / 5 ( − ν 3 + 1 4 ν ) / 5
(-v^3 + 14*v) / 5
β 3 \beta_{3} β 3 = = =
2 ν 2 − 9 2\nu^{2} - 9 2 ν 2 − 9
2*v^2 - 9
ν \nu ν = = =
( β 2 + β 1 ) / 2 ( \beta_{2} + \beta_1 ) / 2 ( β 2 + β 1 ) / 2
(b2 + b1) / 2
ν 2 \nu^{2} ν 2 = = =
( β 3 + 9 ) / 2 ( \beta_{3} + 9 ) / 2 ( β 3 + 9 ) / 2
(b3 + 9) / 2
ν 3 \nu^{3} ν 3 = = =
2 β 2 + 7 β 1 2\beta_{2} + 7\beta_1 2 β 2 + 7 β 1
2*b2 + 7*b1
Character values
We give the values of χ \chi χ on generators for ( Z / 650 Z ) × \left(\mathbb{Z}/650\mathbb{Z}\right)^\times ( Z / 6 5 0 Z ) × .
n n n
27 27 2 7
301 301 3 0 1
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
1 1 1
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
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 4 n e w ( 650 , [ χ ] ) S_{4}^{\mathrm{new}}(650, [\chi]) S 4 n e w ( 6 5 0 , [ χ ] ) :
T 3 4 + 40 T 3 2 + 324 T_{3}^{4} + 40T_{3}^{2} + 324 T 3 4 + 4 0 T 3 2 + 3 2 4
T3^4 + 40*T3^2 + 324
T 7 2 + 684 T_{7}^{2} + 684 T 7 2 + 6 8 4
T7^2 + 684
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 + 4 ) 2 (T^{2} + 4)^{2} ( T 2 + 4 ) 2
(T^2 + 4)^2
3 3 3
T 4 + 40 T 2 + 324 T^{4} + 40T^{2} + 324 T 4 + 4 0 T 2 + 3 2 4
T^4 + 40*T^2 + 324
5 5 5
T 4 T^{4} T 4
T^4
7 7 7
( T 2 + 684 ) 2 (T^{2} + 684)^{2} ( T 2 + 6 8 4 ) 2
(T^2 + 684)^2
11 11 1 1
( T 2 + 50 T − 306 ) 2 (T^{2} + 50 T - 306)^{2} ( T 2 + 5 0 T − 3 0 6 ) 2
(T^2 + 50*T - 306)^2
13 13 1 3
( T 2 + 169 ) 2 (T^{2} + 169)^{2} ( T 2 + 1 6 9 ) 2
(T^2 + 169)^2
17 17 1 7
T 4 + 11320 T 2 + 3196944 T^{4} + 11320 T^{2} + 3196944 T 4 + 1 1 3 2 0 T 2 + 3 1 9 6 9 4 4
T^4 + 11320*T^2 + 3196944
19 19 1 9
( T 2 − 94 T − 2066 ) 2 (T^{2} - 94 T - 2066)^{2} ( T 2 − 9 4 T − 2 0 6 6 ) 2
(T^2 - 94*T - 2066)^2
23 23 2 3
T 4 + 7784 T 2 + 12602500 T^{4} + 7784 T^{2} + 12602500 T 4 + 7 7 8 4 T 2 + 1 2 6 0 2 5 0 0
T^4 + 7784*T^2 + 12602500
29 29 2 9
( T 2 − 260 T + 7704 ) 2 (T^{2} - 260 T + 7704)^{2} ( T 2 − 2 6 0 T + 7 7 0 4 ) 2
(T^2 - 260*T + 7704)^2
31 31 3 1
( T 2 + 402 T + 40382 ) 2 (T^{2} + 402 T + 40382)^{2} ( T 2 + 4 0 2 T + 4 0 3 8 2 ) 2
(T^2 + 402*T + 40382)^2
37 37 3 7
T 4 + 90112 T 2 + 37748736 T^{4} + 90112 T^{2} + 37748736 T 4 + 9 0 1 1 2 T 2 + 3 7 7 4 8 7 3 6
T^4 + 90112*T^2 + 37748736
41 41 4 1
( T 2 + 536 T + 49860 ) 2 (T^{2} + 536 T + 49860)^{2} ( T 2 + 5 3 6 T + 4 9 8 6 0 ) 2
(T^2 + 536*T + 49860)^2
43 43 4 3
T 4 + ⋯ + 22361613444 T^{4} + \cdots + 22361613444 T 4 + ⋯ + 2 2 3 6 1 6 1 3 4 4 4
T^4 + 302920*T^2 + 22361613444
47 47 4 7
T 4 + ⋯ + 19453554576 T^{4} + \cdots + 19453554576 T 4 + ⋯ + 1 9 4 5 3 5 5 4 5 7 6
T^4 + 286552*T^2 + 19453554576
53 53 5 3
T 4 + ⋯ + 71513456400 T^{4} + \cdots + 71513456400 T 4 + ⋯ + 7 1 5 1 3 4 5 6 4 0 0
T^4 + 596344*T^2 + 71513456400
59 59 5 9
( T 2 − 330 T + 24014 ) 2 (T^{2} - 330 T + 24014)^{2} ( T 2 − 3 3 0 T + 2 4 0 1 4 ) 2
(T^2 - 330*T + 24014)^2
61 61 6 1
( T 2 − 620 T + 13336 ) 2 (T^{2} - 620 T + 13336)^{2} ( T 2 − 6 2 0 T + 1 3 3 3 6 ) 2
(T^2 - 620*T + 13336)^2
67 67 6 7
T 4 + ⋯ + 57237691536 T^{4} + \cdots + 57237691536 T 4 + ⋯ + 5 7 2 3 7 6 9 1 5 3 6
T^4 + 690088*T^2 + 57237691536
71 71 7 1
( T 2 + 734 T + 131478 ) 2 (T^{2} + 734 T + 131478)^{2} ( T 2 + 7 3 4 T + 1 3 1 4 7 8 ) 2
(T^2 + 734*T + 131478)^2
73 73 7 3
T 4 + ⋯ + 210717721600 T^{4} + \cdots + 210717721600 T 4 + ⋯ + 2 1 0 7 1 7 7 2 1 6 0 0
T^4 + 1126016*T^2 + 210717721600
79 79 7 9
( T 2 − 924 T − 51112 ) 2 (T^{2} - 924 T - 51112)^{2} ( T 2 − 9 2 4 T − 5 1 1 1 2 ) 2
(T^2 - 924*T - 51112)^2
83 83 8 3
T 4 + ⋯ + 1759390016400 T^{4} + \cdots + 1759390016400 T 4 + ⋯ + 1 7 5 9 3 9 0 0 1 6 4 0 0
T^4 + 2655576*T^2 + 1759390016400
89 89 8 9
( T 2 − 748 T + 71476 ) 2 (T^{2} - 748 T + 71476)^{2} ( T 2 − 7 4 8 T + 7 1 4 7 6 ) 2
(T^2 - 748*T + 71476)^2
97 97 9 7
T 4 + ⋯ + 79625552400 T^{4} + \cdots + 79625552400 T 4 + ⋯ + 7 9 6 2 5 5 5 2 4 0 0
T^4 + 3137416*T^2 + 79625552400
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