gp: [N,k,chi] = [150,8,Mod(49,150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 8, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("150.49");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,-128,0,-432,0,0,-1458,0,-16080]
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 i = − 1 i = \sqrt{-1} i = − 1 .
We also show the integral q q q -expansion of the trace form .
Character values
We give the values of χ \chi χ on generators for ( Z / 150 Z ) × \left(\mathbb{Z}/150\mathbb{Z}\right)^\times ( Z / 1 5 0 Z ) × .
n n n
101 101 1 0 1
127 127 1 2 7
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 7 2 + 976144 T_{7}^{2} + 976144 T 7 2 + 9 7 6 1 4 4
T7^2 + 976144
acting on S 8 n e w ( 150 , [ χ ] ) S_{8}^{\mathrm{new}}(150, [\chi]) S 8 n e w ( 1 5 0 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 + 64 T^{2} + 64 T 2 + 6 4
T^2 + 64
3 3 3
T 2 + 729 T^{2} + 729 T 2 + 7 2 9
T^2 + 729
5 5 5
T 2 T^{2} T 2
T^2
7 7 7
T 2 + 976144 T^{2} + 976144 T 2 + 9 7 6 1 4 4
T^2 + 976144
11 11 1 1
( T + 8040 ) 2 (T + 8040)^{2} ( T + 8 0 4 0 ) 2
(T + 8040)^2
13 13 1 3
T 2 + 11115556 T^{2} + 11115556 T 2 + 1 1 1 1 5 5 5 6
T^2 + 11115556
17 17 1 7
T 2 + 43322724 T^{2} + 43322724 T 2 + 4 3 3 2 2 7 2 4
T^2 + 43322724
19 19 1 9
( T − 27436 ) 2 (T - 27436)^{2} ( T − 2 7 4 3 6 ) 2
(T - 27436)^2
23 23 2 3
T 2 + 2361960000 T^{2} + 2361960000 T 2 + 2 3 6 1 9 6 0 0 0 0
T^2 + 2361960000
29 29 2 9
( T − 132414 ) 2 (T - 132414)^{2} ( T − 1 3 2 4 1 4 ) 2
(T - 132414)^2
31 31 3 1
( T − 254408 ) 2 (T - 254408)^{2} ( T − 2 5 4 4 0 8 ) 2
(T - 254408)^2
37 37 3 7
T 2 + 269811680356 T^{2} + 269811680356 T 2 + 2 6 9 8 1 1 6 8 0 3 5 6
T^2 + 269811680356
41 41 4 1
( T − 92394 ) 2 (T - 92394)^{2} ( T − 9 2 3 9 4 ) 2
(T - 92394)^2
43 43 4 3
T 2 + 55005259024 T^{2} + 55005259024 T 2 + 5 5 0 0 5 2 5 9 0 2 4
T^2 + 55005259024
47 47 4 7
T 2 + 1632363969600 T^{2} + 1632363969600 T 2 + 1 6 3 2 3 6 3 9 6 9 6 0 0
T^2 + 1632363969600
53 53 5 3
T 2 + 697689337284 T^{2} + 697689337284 T 2 + 6 9 7 6 8 9 3 3 7 2 8 4
T^2 + 697689337284
59 59 5 9
( T − 3068760 ) 2 (T - 3068760)^{2} ( T − 3 0 6 8 7 6 0 ) 2
(T - 3068760)^2
61 61 6 1
( T + 1009330 ) 2 (T + 1009330)^{2} ( T + 1 0 0 9 3 3 0 ) 2
(T + 1009330)^2
67 67 6 7
T 2 + 9499784237584 T^{2} + 9499784237584 T 2 + 9 4 9 9 7 8 4 2 3 7 5 8 4
T^2 + 9499784237584
71 71 7 1
( T + 3666720 ) 2 (T + 3666720)^{2} ( T + 3 6 6 6 7 2 0 ) 2
(T + 3666720)^2
73 73 7 3
T 2 + 1260828053956 T^{2} + 1260828053956 T 2 + 1 2 6 0 8 2 8 0 5 3 9 5 6
T^2 + 1260828053956
79 79 7 9
( T − 4128808 ) 2 (T - 4128808)^{2} ( T − 4 1 2 8 8 0 8 ) 2
(T - 4128808)^2
83 83 8 3
T 2 + 21036495941136 T^{2} + 21036495941136 T 2 + 2 1 0 3 6 4 9 5 9 4 1 1 3 6
T^2 + 21036495941136
89 89 8 9
( T − 5763678 ) 2 (T - 5763678)^{2} ( T − 5 7 6 3 6 7 8 ) 2
(T - 5763678)^2
97 97 9 7
T 2 + 45529484982916 T^{2} + 45529484982916 T 2 + 4 5 5 2 9 4 8 4 9 8 2 9 1 6
T^2 + 45529484982916
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