gp: [N,k,chi] = [135,2,Mod(109,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, 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("135.109");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [4,0,0,0]
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 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
β 1 \beta_{1} β 1 = = =
3 ζ 8 2 3\zeta_{8}^{2} 3 ζ 8 2
3*v^2
β 2 \beta_{2} β 2 = = =
ζ 8 3 + ζ 8 \zeta_{8}^{3} + \zeta_{8} ζ 8 3 + ζ 8
v^3 + v
β 3 \beta_{3} β 3 = = =
− ζ 8 3 + 2 ζ 8 -\zeta_{8}^{3} + 2\zeta_{8} − ζ 8 3 + 2 ζ 8
-v^3 + 2*v
ζ 8 \zeta_{8} ζ 8 = = =
( β 3 + β 2 ) / 3 ( \beta_{3} + \beta_{2} ) / 3 ( β 3 + β 2 ) / 3
(b3 + b2) / 3
ζ 8 2 \zeta_{8}^{2} ζ 8 2 = = =
( β 1 ) / 3 ( \beta_1 ) / 3 ( β 1 ) / 3
(b1) / 3
ζ 8 3 \zeta_{8}^{3} ζ 8 3 = = =
( − β 3 + 2 β 2 ) / 3 ( -\beta_{3} + 2\beta_{2} ) / 3 ( − β 3 + 2 β 2 ) / 3
(-b3 + 2*b2) / 3
Character values
We give the values of χ \chi χ on generators for ( Z / 135 Z ) × \left(\mathbb{Z}/135\mathbb{Z}\right)^\times ( Z / 1 3 5 Z ) × .
n n n
56 56 5 6
82 82 8 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 2 + 2 T_{2}^{2} + 2 T 2 2 + 2
T2^2 + 2
acting on S 2 n e w ( 135 , [ χ ] ) S_{2}^{\mathrm{new}}(135, [\chi]) S 2 n e w ( 1 3 5 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 + 2 ) 2 (T^{2} + 2)^{2} ( T 2 + 2 ) 2
(T^2 + 2)^2
3 3 3
T 4 T^{4} T 4
T^4
5 5 5
T 4 − 8 T 2 + 25 T^{4} - 8T^{2} + 25 T 4 − 8 T 2 + 2 5
T^4 - 8*T^2 + 25
7 7 7
( T 2 + 9 ) 2 (T^{2} + 9)^{2} ( T 2 + 9 ) 2
(T^2 + 9)^2
11 11 1 1
( T 2 − 18 ) 2 (T^{2} - 18)^{2} ( T 2 − 1 8 ) 2
(T^2 - 18)^2
13 13 1 3
( T 2 + 9 ) 2 (T^{2} + 9)^{2} ( T 2 + 9 ) 2
(T^2 + 9)^2
17 17 1 7
( T 2 + 8 ) 2 (T^{2} + 8)^{2} ( T 2 + 8 ) 2
(T^2 + 8)^2
19 19 1 9
( T + 1 ) 4 (T + 1)^{4} ( T + 1 ) 4
(T + 1)^4
23 23 2 3
( T 2 + 50 ) 2 (T^{2} + 50)^{2} ( T 2 + 5 0 ) 2
(T^2 + 50)^2
29 29 2 9
( T 2 − 18 ) 2 (T^{2} - 18)^{2} ( T 2 − 1 8 ) 2
(T^2 - 18)^2
31 31 3 1
( T − 2 ) 4 (T - 2)^{4} ( T − 2 ) 4
(T - 2)^4
37 37 3 7
( T 2 + 81 ) 2 (T^{2} + 81)^{2} ( T 2 + 8 1 ) 2
(T^2 + 81)^2
41 41 4 1
( T 2 − 18 ) 2 (T^{2} - 18)^{2} ( T 2 − 1 8 ) 2
(T^2 - 18)^2
43 43 4 3
( T 2 + 36 ) 2 (T^{2} + 36)^{2} ( T 2 + 3 6 ) 2
(T^2 + 36)^2
47 47 4 7
( T 2 + 8 ) 2 (T^{2} + 8)^{2} ( T 2 + 8 ) 2
(T^2 + 8)^2
53 53 5 3
( T 2 + 98 ) 2 (T^{2} + 98)^{2} ( T 2 + 9 8 ) 2
(T^2 + 98)^2
59 59 5 9
( T 2 − 72 ) 2 (T^{2} - 72)^{2} ( T 2 − 7 2 ) 2
(T^2 - 72)^2
61 61 6 1
( T + 13 ) 4 (T + 13)^{4} ( T + 1 3 ) 4
(T + 13)^4
67 67 6 7
( T 2 + 9 ) 2 (T^{2} + 9)^{2} ( T 2 + 9 ) 2
(T^2 + 9)^2
71 71 7 1
( T 2 − 162 ) 2 (T^{2} - 162)^{2} ( T 2 − 1 6 2 ) 2
(T^2 - 162)^2
73 73 7 3
( T 2 + 81 ) 2 (T^{2} + 81)^{2} ( T 2 + 8 1 ) 2
(T^2 + 81)^2
79 79 7 9
( T − 5 ) 4 (T - 5)^{4} ( T − 5 ) 4
(T - 5)^4
83 83 8 3
( T 2 + 2 ) 2 (T^{2} + 2)^{2} ( T 2 + 2 ) 2
(T^2 + 2)^2
89 89 8 9
T 4 T^{4} T 4
T^4
97 97 9 7
( T 2 + 9 ) 2 (T^{2} + 9)^{2} ( T 2 + 9 ) 2
(T^2 + 9)^2
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