gp: [N,k,chi] = [441,6,Mod(1,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
sage: traces = [2,3,0,37,-72,0,0,249,0,-108,-480,0,1296]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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 + 193 ) \beta = \frac{1}{2}(1 + \sqrt{193}) β = 2 1 ( 1 + 1 9 3 ) 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
3 3 3
− 1 -1 − 1
7 7 7
− 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 6 n e w ( Γ 0 ( 441 ) ) S_{6}^{\mathrm{new}}(\Gamma_0(441)) S 6 n e w ( Γ 0 ( 4 4 1 ) )
T 2 2 − 3 T 2 − 46 T_{2}^{2} - 3T_{2} - 46 T 2 2 − 3 T 2 − 4 6
T 5 + 36 T_{5} + 36 T 5 + 3 6
T 13 2 − 1296 T 13 + 169776 T_{13}^{2} - 1296T_{13} + 169776 T 1 3 2 − 1 2 9 6 T 1 3 + 1 6 9 7 7 6
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 − 3 T − 46 T^{2} - 3T - 46 T 2 − 3 T − 4 6
3 3 3
T 2 T^{2} T 2
5 5 5
( T + 36 ) 2 (T + 36)^{2} ( T + 3 6 ) 2
7 7 7
T 2 T^{2} T 2
11 11 1 1
T 2 + 480 T + 54512 T^{2} + 480T + 54512 T 2 + 4 8 0 T + 5 4 5 1 2
13 13 1 3
T 2 − 1296 T + 169776 T^{2} - 1296 T + 169776 T 2 − 1 2 9 6 T + 1 6 9 7 7 6
17 17 1 7
T 2 + 936 T − 2032128 T^{2} + 936 T - 2032128 T 2 + 9 3 6 T − 2 0 3 2 1 2 8
19 19 1 9
T 2 + 216 T − 3990384 T^{2} + 216 T - 3990384 T 2 + 2 1 6 T − 3 9 9 0 3 8 4
23 23 2 3
T 2 − 504 T − 87808 T^{2} - 504T - 87808 T 2 − 5 0 4 T − 8 7 8 0 8
29 29 2 9
T 2 + 6372 T − 9612604 T^{2} + 6372 T - 9612604 T 2 + 6 3 7 2 T − 9 6 1 2 6 0 4
31 31 3 1
T 2 − 9936 T + 8672832 T^{2} - 9936 T + 8672832 T 2 − 9 9 3 6 T + 8 6 7 2 8 3 2
37 37 3 7
T 2 − 11124 T − 22869468 T^{2} - 11124 T - 22869468 T 2 − 1 1 1 2 4 T − 2 2 8 6 9 4 6 8
41 41 4 1
T 2 + 20952 T + 107495424 T^{2} + 20952 T + 107495424 T 2 + 2 0 9 5 2 T + 1 0 7 4 9 5 4 2 4
43 43 4 3
T 2 + 6264 T − 268110576 T^{2} + 6264 T - 268110576 T 2 + 6 2 6 4 T − 2 6 8 1 1 0 5 7 6
47 47 4 7
T 2 + 7920 T − 560613312 T^{2} + 7920 T - 560613312 T 2 + 7 9 2 0 T − 5 6 0 6 1 3 3 1 2
53 53 5 3
T 2 + 2220 T − 66407452 T^{2} + 2220 T - 66407452 T 2 + 2 2 2 0 T − 6 6 4 0 7 4 5 2
59 59 5 9
T 2 + 29736 T + 185038992 T^{2} + 29736 T + 185038992 T 2 + 2 9 7 3 6 T + 1 8 5 0 3 8 9 9 2
61 61 6 1
T 2 + 17280 T − 982141200 T^{2} + 17280 T - 982141200 T 2 + 1 7 2 8 0 T − 9 8 2 1 4 1 2 0 0
67 67 6 7
T 2 + ⋯ − 1919121200 T^{2} + \cdots - 1919121200 T 2 + ⋯ − 1 9 1 9 1 2 1 2 0 0
71 71 7 1
T 2 + ⋯ + 1814661632 T^{2} + \cdots + 1814661632 T 2 + ⋯ + 1 8 1 4 6 6 1 6 3 2
73 73 7 3
T 2 − 56592 T + 631577088 T^{2} - 56592 T + 631577088 T 2 − 5 6 5 9 2 T + 6 3 1 5 7 7 0 8 8
79 79 7 9
T 2 + ⋯ − 1239346496 T^{2} + \cdots - 1239346496 T 2 + ⋯ − 1 2 3 9 3 4 6 4 9 6
83 83 8 3
T 2 + ⋯ − 3085453296 T^{2} + \cdots - 3085453296 T 2 + ⋯ − 3 0 8 5 4 5 3 2 9 6
89 89 8 9
T 2 + ⋯ + 3143484288 T^{2} + \cdots + 3143484288 T 2 + ⋯ + 3 1 4 3 4 8 4 2 8 8
97 97 9 7
T 2 + ⋯ − 18184056768 T^{2} + \cdots - 18184056768 T 2 + ⋯ − 1 8 1 8 4 0 5 6 7 6 8
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