gp: [N,k,chi] = [270,10,Mod(1,270)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(270, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("270.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
Newform invariants
sage: traces = [2,-32,0,512,-1250,0,-6461]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 = 1 2 ( − 1 + 3 9569 ) \beta = \frac{1}{2}(-1 + 3\sqrt{9569}) β = 2 1 ( − 1 + 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
3 3 3
+ 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 intersection of the kernels of the following linear operators acting on S 10 n e w ( Γ 0 ( 270 ) ) S_{10}^{\mathrm{new}}(\Gamma_0(270)) S 1 0 n e w ( Γ 0 ( 2 7 0 ) )
T 7 2 + 6461 T 7 + 6797518 T_{7}^{2} + 6461T_{7} + 6797518 T 7 2 + 6 4 6 1 T 7 + 6 7 9 7 5 1 8
T 11 2 + 28533 T 11 − 683707050 T_{11}^{2} + 28533T_{11} - 683707050 T 1 1 2 + 2 8 5 3 3 T 1 1 − 6 8 3 7 0 7 0 5 0
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
( T + 16 ) 2 (T + 16)^{2} ( T + 1 6 ) 2
3 3 3
T 2 T^{2} T 2
5 5 5
( T + 625 ) 2 (T + 625)^{2} ( T + 6 2 5 ) 2
7 7 7
T 2 + 6461 T + 6797518 T^{2} + 6461 T + 6797518 T 2 + 6 4 6 1 T + 6 7 9 7 5 1 8
11 11 1 1
T 2 + 28533 T − 683707050 T^{2} + 28533 T - 683707050 T 2 + 2 8 5 3 3 T − 6 8 3 7 0 7 0 5 0
13 13 1 3
T 2 + ⋯ − 5833626347 T^{2} + \cdots - 5833626347 T 2 + ⋯ − 5 8 3 3 6 2 6 3 4 7
17 17 1 7
T 2 + ⋯ − 28210449888 T^{2} + \cdots - 28210449888 T 2 + ⋯ − 2 8 2 1 0 4 4 9 8 8 8
19 19 1 9
T 2 + ⋯ − 723888544328 T^{2} + \cdots - 723888544328 T 2 + ⋯ − 7 2 3 8 8 8 5 4 4 3 2 8
23 23 2 3
T 2 + ⋯ − 31338087822 T^{2} + \cdots - 31338087822 T 2 + ⋯ − 3 1 3 3 8 0 8 7 8 2 2
29 29 2 9
T 2 + ⋯ + 11463639031674 T^{2} + \cdots + 11463639031674 T 2 + ⋯ + 1 1 4 6 3 6 3 9 0 3 1 6 7 4
31 31 3 1
T 2 + ⋯ − 24058563518240 T^{2} + \cdots - 24058563518240 T 2 + ⋯ − 2 4 0 5 8 5 6 3 5 1 8 2 4 0
37 37 3 7
T 2 + ⋯ − 73593367962134 T^{2} + \cdots - 73593367962134 T 2 + ⋯ − 7 3 5 9 3 3 6 7 9 6 2 1 3 4
41 41 4 1
T 2 + ⋯ − 130557815243784 T^{2} + \cdots - 130557815243784 T 2 + ⋯ − 1 3 0 5 5 7 8 1 5 2 4 3 7 8 4
43 43 4 3
T 2 + ⋯ − 8660758086452 T^{2} + \cdots - 8660758086452 T 2 + ⋯ − 8 6 6 0 7 5 8 0 8 6 4 5 2
47 47 4 7
T 2 + ⋯ − 22 ⋯ 56 T^{2} + \cdots - 22\!\cdots\!56 T 2 + ⋯ − 2 2 ⋯ 5 6
53 53 5 3
T 2 + ⋯ + 669926813760360 T^{2} + \cdots + 669926813760360 T 2 + ⋯ + 6 6 9 9 2 6 8 1 3 7 6 0 3 6 0
59 59 5 9
T 2 + ⋯ + 14 ⋯ 88 T^{2} + \cdots + 14\!\cdots\!88 T 2 + ⋯ + 1 4 ⋯ 8 8
61 61 6 1
T 2 + ⋯ − 32 ⋯ 70 T^{2} + \cdots - 32\!\cdots\!70 T 2 + ⋯ − 3 2 ⋯ 7 0
67 67 6 7
T 2 + ⋯ + 67 ⋯ 36 T^{2} + \cdots + 67\!\cdots\!36 T 2 + ⋯ + 6 7 ⋯ 3 6
71 71 7 1
T 2 + ⋯ − 54 ⋯ 60 T^{2} + \cdots - 54\!\cdots\!60 T 2 + ⋯ − 5 4 ⋯ 6 0
73 73 7 3
T 2 + ⋯ − 10 ⋯ 34 T^{2} + \cdots - 10\!\cdots\!34 T 2 + ⋯ − 1 0 ⋯ 3 4
79 79 7 9
T 2 + ⋯ − 97 ⋯ 39 T^{2} + \cdots - 97\!\cdots\!39 T 2 + ⋯ − 9 7 ⋯ 3 9
83 83 8 3
T 2 + ⋯ + 22 ⋯ 92 T^{2} + \cdots + 22\!\cdots\!92 T 2 + ⋯ + 2 2 ⋯ 9 2
89 89 8 9
T 2 + ⋯ − 33 ⋯ 40 T^{2} + \cdots - 33\!\cdots\!40 T 2 + ⋯ − 3 3 ⋯ 4 0
97 97 9 7
T 2 + ⋯ + 12 ⋯ 10 T^{2} + \cdots + 12\!\cdots\!10 T 2 + ⋯ + 1 2 ⋯ 1 0
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