gp: [N,k,chi] = [10,16,Mod(1,10)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 16, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
Newform invariants
sage: traces = [2,256,-1844]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 = 10 239569 \beta = 10\sqrt{239569} β = 1 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)
p p p
Sign
2 2 2
− 1 -1 − 1
5 5 5
− 1 -1 − 1
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 + 1844 T 3 − 23106816 T_{3}^{2} + 1844T_{3} - 23106816 T 3 2 + 1 8 4 4 T 3 − 2 3 1 0 6 8 1 6 S 16 n e w ( Γ 0 ( 10 ) ) S_{16}^{\mathrm{new}}(\Gamma_0(10)) S 1 6 n e w ( Γ 0 ( 1 0 ) )
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
( T − 128 ) 2 (T - 128)^{2} ( T − 1 2 8 ) 2
3 3 3
T 2 + 1844 T − 23106816 T^{2} + 1844 T - 23106816 T 2 + 1 8 4 4 T − 2 3 1 0 6 8 1 6
5 5 5
( T − 78125 ) 2 (T - 78125)^{2} ( T − 7 8 1 2 5 ) 2
7 7 7
T 2 + ⋯ − 4044061398944 T^{2} + \cdots - 4044061398944 T 2 + ⋯ − 4 0 4 4 0 6 1 3 9 8 9 4 4
11 11 1 1
T 2 + ⋯ − 342274048299216 T^{2} + \cdots - 342274048299216 T 2 + ⋯ − 3 4 2 2 7 4 0 4 8 2 9 9 2 1 6
13 13 1 3
T 2 + ⋯ + 14 ⋯ 04 T^{2} + \cdots + 14\!\cdots\!04 T 2 + ⋯ + 1 4 ⋯ 0 4
17 17 1 7
T 2 + ⋯ + 50 ⋯ 76 T^{2} + \cdots + 50\!\cdots\!76 T 2 + ⋯ + 5 0 ⋯ 7 6
19 19 1 9
T 2 + ⋯ + 90 ⋯ 00 T^{2} + \cdots + 90\!\cdots\!00 T 2 + ⋯ + 9 0 ⋯ 0 0
23 23 2 3
T 2 + ⋯ − 64 ⋯ 36 T^{2} + \cdots - 64\!\cdots\!36 T 2 + ⋯ − 6 4 ⋯ 3 6
29 29 2 9
T 2 + ⋯ − 28 ⋯ 00 T^{2} + \cdots - 28\!\cdots\!00 T 2 + ⋯ − 2 8 ⋯ 0 0
31 31 3 1
T 2 + ⋯ − 79 ⋯ 36 T^{2} + \cdots - 79\!\cdots\!36 T 2 + ⋯ − 7 9 ⋯ 3 6
37 37 3 7
T 2 + ⋯ + 71 ⋯ 96 T^{2} + \cdots + 71\!\cdots\!96 T 2 + ⋯ + 7 1 ⋯ 9 6
41 41 4 1
T 2 + ⋯ − 11 ⋯ 56 T^{2} + \cdots - 11\!\cdots\!56 T 2 + ⋯ − 1 1 ⋯ 5 6
43 43 4 3
T 2 + ⋯ + 21 ⋯ 04 T^{2} + \cdots + 21\!\cdots\!04 T 2 + ⋯ + 2 1 ⋯ 0 4
47 47 4 7
T 2 + ⋯ − 22 ⋯ 04 T^{2} + \cdots - 22\!\cdots\!04 T 2 + ⋯ − 2 2 ⋯ 0 4
53 53 5 3
T 2 + ⋯ − 15 ⋯ 36 T^{2} + \cdots - 15\!\cdots\!36 T 2 + ⋯ − 1 5 ⋯ 3 6
59 59 5 9
T 2 + ⋯ − 27 ⋯ 00 T^{2} + \cdots - 27\!\cdots\!00 T 2 + ⋯ − 2 7 ⋯ 0 0
61 61 6 1
T 2 + ⋯ − 23 ⋯ 76 T^{2} + \cdots - 23\!\cdots\!76 T 2 + ⋯ − 2 3 ⋯ 7 6
67 67 6 7
T 2 + ⋯ + 14 ⋯ 56 T^{2} + \cdots + 14\!\cdots\!56 T 2 + ⋯ + 1 4 ⋯ 5 6
71 71 7 1
T 2 + ⋯ − 89 ⋯ 76 T^{2} + \cdots - 89\!\cdots\!76 T 2 + ⋯ − 8 9 ⋯ 7 6
73 73 7 3
T 2 + ⋯ − 56 ⋯ 56 T^{2} + \cdots - 56\!\cdots\!56 T 2 + ⋯ − 5 6 ⋯ 5 6
79 79 7 9
T 2 + ⋯ + 43 ⋯ 00 T^{2} + \cdots + 43\!\cdots\!00 T 2 + ⋯ + 4 3 ⋯ 0 0
83 83 8 3
T 2 + ⋯ + 12 ⋯ 24 T^{2} + \cdots + 12\!\cdots\!24 T 2 + ⋯ + 1 2 ⋯ 2 4
89 89 8 9
T 2 + ⋯ − 99 ⋯ 00 T^{2} + \cdots - 99\!\cdots\!00 T 2 + ⋯ − 9 9 ⋯ 0 0
97 97 9 7
T 2 + ⋯ − 85 ⋯ 24 T^{2} + \cdots - 85\!\cdots\!24 T 2 + ⋯ − 8 5 ⋯ 2 4
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