gp: [N,k,chi] = [1875,2,Mod(1249,1875)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1875.1249");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,0,-2]
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 7 1,\beta_1,\ldots,\beta_{7} 1 , β 1 , … , β 7 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 8 + 9 x 6 + 26 x 4 + 24 x 2 + 1 x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 x 8 + 9 x 6 + 2 6 x 4 + 2 4 x 2 + 1
x^8 + 9*x^6 + 26*x^4 + 24*x^2 + 1
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
ν 2 + 2 \nu^{2} + 2 ν 2 + 2
v^2 + 2
β 3 \beta_{3} β 3 = = =
ν 3 + 3 ν \nu^{3} + 3\nu ν 3 + 3 ν
v^3 + 3*v
β 4 \beta_{4} β 4 = = =
ν 4 + 4 ν 2 + 2 \nu^{4} + 4\nu^{2} + 2 ν 4 + 4 ν 2 + 2
v^4 + 4*v^2 + 2
β 5 \beta_{5} β 5 = = =
ν 5 + 5 ν 3 + 5 ν \nu^{5} + 5\nu^{3} + 5\nu ν 5 + 5 ν 3 + 5 ν
v^5 + 5*v^3 + 5*v
β 6 \beta_{6} β 6 = = =
ν 6 + 6 ν 4 + 9 ν 2 + 2 \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 ν 6 + 6 ν 4 + 9 ν 2 + 2
v^6 + 6*v^4 + 9*v^2 + 2
β 7 \beta_{7} β 7 = = =
ν 7 + 7 ν 5 + 14 ν 3 + 7 ν \nu^{7} + 7\nu^{5} + 14\nu^{3} + 7\nu ν 7 + 7 ν 5 + 1 4 ν 3 + 7 ν
v^7 + 7*v^5 + 14*v^3 + 7*v
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 2 − 2 \beta_{2} - 2 β 2 − 2
b2 - 2
ν 3 \nu^{3} ν 3 = = =
β 3 − 3 β 1 \beta_{3} - 3\beta_1 β 3 − 3 β 1
b3 - 3*b1
ν 4 \nu^{4} ν 4 = = =
β 4 − 4 β 2 + 6 \beta_{4} - 4\beta_{2} + 6 β 4 − 4 β 2 + 6
b4 - 4*b2 + 6
ν 5 \nu^{5} ν 5 = = =
β 5 − 5 β 3 + 10 β 1 \beta_{5} - 5\beta_{3} + 10\beta_1 β 5 − 5 β 3 + 1 0 β 1
b5 - 5*b3 + 10*b1
ν 6 \nu^{6} ν 6 = = =
β 6 − 6 β 4 + 15 β 2 − 20 \beta_{6} - 6\beta_{4} + 15\beta_{2} - 20 β 6 − 6 β 4 + 1 5 β 2 − 2 0
b6 - 6*b4 + 15*b2 - 20
ν 7 \nu^{7} ν 7 = = =
β 7 − 7 β 5 + 21 β 3 − 35 β 1 \beta_{7} - 7\beta_{5} + 21\beta_{3} - 35\beta_1 β 7 − 7 β 5 + 2 1 β 3 − 3 5 β 1
b7 - 7*b5 + 21*b3 - 35*b1
Character values
We give the values of χ \chi χ on generators for ( Z / 1875 Z ) × \left(\mathbb{Z}/1875\mathbb{Z}\right)^\times ( Z / 1 8 7 5 Z ) × .
n n n
626 626 6 2 6
1252 1252 1 2 5 2
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 kernel of the linear operator
T 2 8 + 9 T 2 6 + 26 T 2 4 + 24 T 2 2 + 1 T_{2}^{8} + 9T_{2}^{6} + 26T_{2}^{4} + 24T_{2}^{2} + 1 T 2 8 + 9 T 2 6 + 2 6 T 2 4 + 2 4 T 2 2 + 1
T2^8 + 9*T2^6 + 26*T2^4 + 24*T2^2 + 1
acting on S 2 n e w ( 1875 , [ χ ] ) S_{2}^{\mathrm{new}}(1875, [\chi]) S 2 n e w ( 1 8 7 5 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 8 + 9 T 6 + ⋯ + 1 T^{8} + 9 T^{6} + \cdots + 1 T 8 + 9 T 6 + ⋯ + 1
T^8 + 9*T^6 + 26*T^4 + 24*T^2 + 1
3 3 3
( T 2 + 1 ) 4 (T^{2} + 1)^{4} ( T 2 + 1 ) 4
(T^2 + 1)^4
5 5 5
T 8 T^{8} T 8
T^8
7 7 7
T 8 + 25 T 6 + ⋯ + 25 T^{8} + 25 T^{6} + \cdots + 25 T 8 + 2 5 T 6 + ⋯ + 2 5
T^8 + 25*T^6 + 90*T^4 + 100*T^2 + 25
11 11 1 1
( T 4 + 6 T 3 + 6 T 2 + ⋯ − 9 ) 2 (T^{4} + 6 T^{3} + 6 T^{2} + \cdots - 9)^{2} ( T 4 + 6 T 3 + 6 T 2 + ⋯ − 9 ) 2
(T^4 + 6*T^3 + 6*T^2 - 9*T - 9)^2
13 13 1 3
T 8 + 61 T 6 + ⋯ + 7921 T^{8} + 61 T^{6} + \cdots + 7921 T 8 + 6 1 T 6 + ⋯ + 7 9 2 1
T^8 + 61*T^6 + 1146*T^4 + 7396*T^2 + 7921
17 17 1 7
T 8 + 81 T 6 + ⋯ + 128881 T^{8} + 81 T^{6} + \cdots + 128881 T 8 + 8 1 T 6 + ⋯ + 1 2 8 8 8 1
T^8 + 81*T^6 + 2366*T^4 + 29316*T^2 + 128881
19 19 1 9
( T 4 − 9 T 3 + 6 T 2 + ⋯ − 9 ) 2 (T^{4} - 9 T^{3} + 6 T^{2} + \cdots - 9)^{2} ( T 4 − 9 T 3 + 6 T 2 + ⋯ − 9 ) 2
(T^4 - 9*T^3 + 6*T^2 + 36*T - 9)^2
23 23 2 3
T 8 + 60 T 6 + ⋯ + 25 T^{8} + 60 T^{6} + \cdots + 25 T 8 + 6 0 T 6 + ⋯ + 2 5
T^8 + 60*T^6 + 590*T^4 + 300*T^2 + 25
29 29 2 9
( T 4 − 28 T 3 + ⋯ + 1801 ) 2 (T^{4} - 28 T^{3} + \cdots + 1801)^{2} ( T 4 − 2 8 T 3 + ⋯ + 1 8 0 1 ) 2
(T^4 - 28*T^3 + 284*T^2 - 1217*T + 1801)^2
31 31 3 1
( T 4 + 10 T 3 + ⋯ − 125 ) 2 (T^{4} + 10 T^{3} + \cdots - 125)^{2} ( T 4 + 1 0 T 3 + ⋯ − 1 2 5 ) 2
(T^4 + 10*T^3 - 125*T - 125)^2
37 37 3 7
T 8 + 280 T 6 + ⋯ + 2175625 T^{8} + 280 T^{6} + \cdots + 2175625 T 8 + 2 8 0 T 6 + ⋯ + 2 1 7 5 6 2 5
T^8 + 280*T^6 + 25650*T^4 + 785125*T^2 + 2175625
41 41 4 1
( T 4 − 70 T 2 + ⋯ + 145 ) 2 (T^{4} - 70 T^{2} + \cdots + 145)^{2} ( T 4 − 7 0 T 2 + ⋯ + 1 4 5 ) 2
(T^4 - 70*T^2 - 135*T + 145)^2
43 43 4 3
T 8 + 159 T 6 + ⋯ + 175561 T^{8} + 159 T^{6} + \cdots + 175561 T 8 + 1 5 9 T 6 + ⋯ + 1 7 5 5 6 1
T^8 + 159*T^6 + 4721*T^4 + 50079*T^2 + 175561
47 47 4 7
T 8 + 351 T 6 + ⋯ + 37075921 T^{8} + 351 T^{6} + \cdots + 37075921 T 8 + 3 5 1 T 6 + ⋯ + 3 7 0 7 5 9 2 1
T^8 + 351*T^6 + 42341*T^4 + 2110011*T^2 + 37075921
53 53 5 3
T 8 + 290 T 6 + ⋯ + 8970025 T^{8} + 290 T^{6} + \cdots + 8970025 T 8 + 2 9 0 T 6 + ⋯ + 8 9 7 0 0 2 5
T^8 + 290*T^6 + 27015*T^4 + 941450*T^2 + 8970025
59 59 5 9
( T 4 + 4 T 3 + ⋯ + 1531 ) 2 (T^{4} + 4 T^{3} + \cdots + 1531)^{2} ( T 4 + 4 T 3 + ⋯ + 1 5 3 1 ) 2
(T^4 + 4*T^3 - 154*T^2 - 421*T + 1531)^2
61 61 6 1
( T 4 + 43 T 3 + ⋯ + 10261 ) 2 (T^{4} + 43 T^{3} + \cdots + 10261)^{2} ( T 4 + 4 3 T 3 + ⋯ + 1 0 2 6 1 ) 2
(T^4 + 43*T^3 + 669*T^2 + 4417*T + 10261)^2
67 67 6 7
T 8 + 186 T 6 + ⋯ + 22801 T^{8} + 186 T^{6} + \cdots + 22801 T 8 + 1 8 6 T 6 + ⋯ + 2 2 8 0 1
T^8 + 186*T^6 + 11351*T^4 + 228186*T^2 + 22801
71 71 7 1
( T 4 + 27 T 3 + ⋯ + 271 ) 2 (T^{4} + 27 T^{3} + \cdots + 271)^{2} ( T 4 + 2 7 T 3 + ⋯ + 2 7 1 ) 2
(T^4 + 27*T^3 + 134*T^2 - 672*T + 271)^2
73 73 7 3
T 8 + 345 T 6 + ⋯ + 7535025 T^{8} + 345 T^{6} + \cdots + 7535025 T 8 + 3 4 5 T 6 + ⋯ + 7 5 3 5 0 2 5
T^8 + 345*T^6 + 36090*T^4 + 1139400*T^2 + 7535025
79 79 7 9
( T 4 + 10 T 3 + ⋯ − 3155 ) 2 (T^{4} + 10 T^{3} + \cdots - 3155)^{2} ( T 4 + 1 0 T 3 + ⋯ − 3 1 5 5 ) 2
(T^4 + 10*T^3 - 105*T^2 - 1370*T - 3155)^2
83 83 8 3
T 8 + 501 T 6 + ⋯ + 182007081 T^{8} + 501 T^{6} + \cdots + 182007081 T 8 + 5 0 1 T 6 + ⋯ + 1 8 2 0 0 7 0 8 1
T^8 + 501*T^6 + 88686*T^4 + 6676776*T^2 + 182007081
89 89 8 9
( T 4 − 9 T 3 − 4 T 2 + ⋯ + 61 ) 2 (T^{4} - 9 T^{3} - 4 T^{2} + \cdots + 61)^{2} ( T 4 − 9 T 3 − 4 T 2 + ⋯ + 6 1 ) 2
(T^4 - 9*T^3 - 4*T^2 + 96*T + 61)^2
97 97 9 7
T 8 + 601 T 6 + ⋯ + 216119401 T^{8} + 601 T^{6} + \cdots + 216119401 T 8 + 6 0 1 T 6 + ⋯ + 2 1 6 1 1 9 4 0 1
T^8 + 601*T^6 + 113706*T^4 + 8447536*T^2 + 216119401
show more
show less