gp: [N,k,chi] = [2,42,Mod(1,2)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 42, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 42);
N := Newforms(S);
Newform invariants
sage: traces = [2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 β = 2880 4559670239569 \beta = 2880\sqrt{4559670239569} β = 2 8 8 0 4 5 5 9 6 7 0 2 3 9 5 6 9 q q q trace form .
For each embedding ι m \iota_m ι m ι m ( a n ) \iota_m(a_n) ι m ( a n )
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 3 2 − 8863347528 T 3 − 18179996484555185904 T_{3}^{2} - 8863347528T_{3} - 18179996484555185904 T 3 2 − 8 8 6 3 3 4 7 5 2 8 T 3 − 1 8 1 7 9 9 9 6 4 8 4 5 5 5 1 8 5 9 0 4 S 42 n e w ( Γ 0 ( 2 ) ) S_{42}^{\mathrm{new}}(\Gamma_0(2)) S 4 2 n e w ( Γ 0 ( 2 ) )
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
( T − 1048576 ) 2 (T - 1048576)^{2} ( T − 1 0 4 8 5 7 6 ) 2
3 3 3
T 2 + ⋯ − 18 ⋯ 04 T^{2} + \cdots - 18\!\cdots\!04 T 2 + ⋯ − 1 8 ⋯ 0 4
5 5 5
T 2 + ⋯ − 89 ⋯ 00 T^{2} + \cdots - 89\!\cdots\!00 T 2 + ⋯ − 8 9 ⋯ 0 0
7 7 7
T 2 + ⋯ − 23 ⋯ 16 T^{2} + \cdots - 23\!\cdots\!16 T 2 + ⋯ − 2 3 ⋯ 1 6
11 11 1 1
T 2 + ⋯ − 13 ⋯ 36 T^{2} + \cdots - 13\!\cdots\!36 T 2 + ⋯ − 1 3 ⋯ 3 6
13 13 1 3
T 2 + ⋯ + 74 ⋯ 56 T^{2} + \cdots + 74\!\cdots\!56 T 2 + ⋯ + 7 4 ⋯ 5 6
17 17 1 7
T 2 + ⋯ + 26 ⋯ 04 T^{2} + \cdots + 26\!\cdots\!04 T 2 + ⋯ + 2 6 ⋯ 0 4
19 19 1 9
T 2 + ⋯ − 84 ⋯ 00 T^{2} + \cdots - 84\!\cdots\!00 T 2 + ⋯ − 8 4 ⋯ 0 0
23 23 2 3
T 2 + ⋯ + 55 ⋯ 56 T^{2} + \cdots + 55\!\cdots\!56 T 2 + ⋯ + 5 5 ⋯ 5 6
29 29 2 9
T 2 + ⋯ − 42 ⋯ 00 T^{2} + \cdots - 42\!\cdots\!00 T 2 + ⋯ − 4 2 ⋯ 0 0
31 31 3 1
T 2 + ⋯ − 17 ⋯ 36 T^{2} + \cdots - 17\!\cdots\!36 T 2 + ⋯ − 1 7 ⋯ 3 6
37 37 3 7
T 2 + ⋯ + 34 ⋯ 24 T^{2} + \cdots + 34\!\cdots\!24 T 2 + ⋯ + 3 4 ⋯ 2 4
41 41 4 1
T 2 + ⋯ − 19 ⋯ 76 T^{2} + \cdots - 19\!\cdots\!76 T 2 + ⋯ − 1 9 ⋯ 7 6
43 43 4 3
T 2 + ⋯ + 20 ⋯ 76 T^{2} + \cdots + 20\!\cdots\!76 T 2 + ⋯ + 2 0 ⋯ 7 6
47 47 4 7
T 2 + ⋯ + 12 ⋯ 24 T^{2} + \cdots + 12\!\cdots\!24 T 2 + ⋯ + 1 2 ⋯ 2 4
53 53 5 3
T 2 + ⋯ + 20 ⋯ 76 T^{2} + \cdots + 20\!\cdots\!76 T 2 + ⋯ + 2 0 ⋯ 7 6
59 59 5 9
T 2 + ⋯ − 16 ⋯ 00 T^{2} + \cdots - 16\!\cdots\!00 T 2 + ⋯ − 1 6 ⋯ 0 0
61 61 6 1
T 2 + ⋯ − 13 ⋯ 76 T^{2} + \cdots - 13\!\cdots\!76 T 2 + ⋯ − 1 3 ⋯ 7 6
67 67 6 7
T 2 + ⋯ + 11 ⋯ 84 T^{2} + \cdots + 11\!\cdots\!84 T 2 + ⋯ + 1 1 ⋯ 8 4
71 71 7 1
T 2 + ⋯ − 58 ⋯ 36 T^{2} + \cdots - 58\!\cdots\!36 T 2 + ⋯ − 5 8 ⋯ 3 6
73 73 7 3
T 2 + ⋯ + 57 ⋯ 36 T^{2} + \cdots + 57\!\cdots\!36 T 2 + ⋯ + 5 7 ⋯ 3 6
79 79 7 9
T 2 + ⋯ + 16 ⋯ 00 T^{2} + \cdots + 16\!\cdots\!00 T 2 + ⋯ + 1 6 ⋯ 0 0
83 83 8 3
T 2 + ⋯ − 47 ⋯ 44 T^{2} + \cdots - 47\!\cdots\!44 T 2 + ⋯ − 4 7 ⋯ 4 4
89 89 8 9
T 2 + ⋯ + 11 ⋯ 00 T^{2} + \cdots + 11\!\cdots\!00 T 2 + ⋯ + 1 1 ⋯ 0 0
97 97 9 7
T 2 + ⋯ + 19 ⋯ 04 T^{2} + \cdots + 19\!\cdots\!04 T 2 + ⋯ + 1 9 ⋯ 0 4
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