gp: [N,k,chi] = [11,3,Mod(2,11)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.2");
S:= CuspForms(chi, 3);
N := Newforms(S);
Newform invariants
sage: traces = []
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == 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 a primitive root of unity ζ 10 \zeta_{10} ζ 1 0 .
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 / 11 Z ) × \left(\mathbb{Z}/11\mathbb{Z}\right)^\times ( Z / 1 1 Z ) × .
n n n
2 2 2
χ ( n ) \chi(n) χ ( n )
ζ 10 \zeta_{10} ζ 1 0
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 is the entire newspace S 3 n e w ( 11 , [ χ ] ) S_{3}^{\mathrm{new}}(11, [\chi]) S 3 n e w ( 1 1 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 + 5 T 3 + ⋯ + 5 T^{4} + 5 T^{3} + \cdots + 5 T 4 + 5 T 3 + ⋯ + 5
T^4 + 5*T^3 + 15*T^2 + 15*T + 5
3 3 3
T 4 + 10 T 2 + ⋯ + 25 T^{4} + 10 T^{2} + \cdots + 25 T 4 + 1 0 T 2 + ⋯ + 2 5
T^4 + 10*T^2 + 25*T + 25
5 5 5
T 4 + 4 T 3 + ⋯ + 256 T^{4} + 4 T^{3} + \cdots + 256 T 4 + 4 T 3 + ⋯ + 2 5 6
T^4 + 4*T^3 + 16*T^2 + 64*T + 256
7 7 7
T 4 − 10 T 3 + ⋯ + 80 T^{4} - 10 T^{3} + \cdots + 80 T 4 − 1 0 T 3 + ⋯ + 8 0
T^4 - 10*T^3 + 80*T^2 + 160*T + 80
11 11 1 1
T 4 − T 3 + ⋯ + 14641 T^{4} - T^{3} + \cdots + 14641 T 4 − T 3 + ⋯ + 1 4 6 4 1
T^4 - T^3 - 209*T^2 - 121*T + 14641
13 13 1 3
T 4 + 20 T 3 + ⋯ + 2000 T^{4} + 20 T^{3} + \cdots + 2000 T 4 + 2 0 T 3 + ⋯ + 2 0 0 0
T^4 + 20*T^3 + 200*T^2 + 1000*T + 2000
17 17 1 7
T 4 − 10985 T + 142805 T^{4} - 10985 T + 142805 T 4 − 1 0 9 8 5 T + 1 4 2 8 0 5
T^4 - 10985*T + 142805
19 19 1 9
T 4 − 25 T 3 + ⋯ + 605 T^{4} - 25 T^{3} + \cdots + 605 T 4 − 2 5 T 3 + ⋯ + 6 0 5
T^4 - 25*T^3 + 200*T^2 - 440*T + 605
23 23 2 3
( T 2 + 10 T + 20 ) 2 (T^{2} + 10 T + 20)^{2} ( T 2 + 1 0 T + 2 0 ) 2
(T^2 + 10*T + 20)^2
29 29 2 9
T 4 + 40 T 3 + ⋯ + 9680 T^{4} + 40 T^{3} + \cdots + 9680 T 4 + 4 0 T 3 + ⋯ + 9 6 8 0
T^4 + 40*T^3 + 1040*T^2 + 5720*T + 9680
31 31 3 1
T 4 + 58 T 3 + ⋯ + 55696 T^{4} + 58 T^{3} + \cdots + 55696 T 4 + 5 8 T 3 + ⋯ + 5 5 6 9 6
T^4 + 58*T^3 + 1384*T^2 + 7552*T + 55696
37 37 3 7
T 4 − 90 T 3 + ⋯ + 2624400 T^{4} - 90 T^{3} + \cdots + 2624400 T 4 − 9 0 T 3 + ⋯ + 2 6 2 4 4 0 0
T^4 - 90*T^3 + 4860*T^2 - 145800*T + 2624400
41 41 4 1
T 4 + 80 T 3 + ⋯ + 8405 T^{4} + 80 T^{3} + \cdots + 8405 T 4 + 8 0 T 3 + ⋯ + 8 4 0 5
T^4 + 80*T^3 + 4720*T^2 - 12095*T + 8405
43 43 4 3
T 4 + 1625 T 2 + 581405 T^{4} + 1625 T^{2} + 581405 T 4 + 1 6 2 5 T 2 + 5 8 1 4 0 5
T^4 + 1625*T^2 + 581405
47 47 4 7
T 4 + 30 T 3 + ⋯ + 384400 T^{4} + 30 T^{3} + \cdots + 384400 T 4 + 3 0 T 3 + ⋯ + 3 8 4 4 0 0
T^4 + 30*T^3 + 640*T^2 + 12400*T + 384400
53 53 5 3
T 4 − 120 T 3 + ⋯ + 810000 T^{4} - 120 T^{3} + \cdots + 810000 T 4 − 1 2 0 T 3 + ⋯ + 8 1 0 0 0 0
T^4 - 120*T^3 + 5400*T^2 + 27000*T + 810000
59 59 5 9
T 4 − 23 T 3 + ⋯ + 5041 T^{4} - 23 T^{3} + \cdots + 5041 T 4 − 2 3 T 3 + ⋯ + 5 0 4 1
T^4 - 23*T^3 + 1054*T^2 - 3692*T + 5041
61 61 6 1
T 4 − 10 T 3 + ⋯ + 403280 T^{4} - 10 T^{3} + \cdots + 403280 T 4 − 1 0 T 3 + ⋯ + 4 0 3 2 8 0
T^4 - 10*T^3 - 120*T^2 - 17040*T + 403280
67 67 6 7
( T 2 + 115 T + 2945 ) 2 (T^{2} + 115 T + 2945)^{2} ( T 2 + 1 1 5 T + 2 9 4 5 ) 2
(T^2 + 115*T + 2945)^2
71 71 7 1
T 4 − 148 T 3 + ⋯ + 22619536 T^{4} - 148 T^{3} + \cdots + 22619536 T 4 − 1 4 8 T 3 + ⋯ + 2 2 6 1 9 5 3 6
T^4 - 148*T^3 + 14464*T^2 - 770472*T + 22619536
73 73 7 3
T 4 − 300 T 3 + ⋯ + 93787805 T^{4} - 300 T^{3} + \cdots + 93787805 T 4 − 3 0 0 T 3 + ⋯ + 9 3 7 8 7 8 0 5
T^4 - 300*T^3 + 41100*T^2 - 2966735*T + 93787805
79 79 7 9
T 4 − 70 T 3 + ⋯ + 67280 T^{4} - 70 T^{3} + \cdots + 67280 T 4 − 7 0 T 3 + ⋯ + 6 7 2 8 0
T^4 - 70*T^3 + 3780*T^2 + 31320*T + 67280
83 83 8 3
T 4 − 225 T 3 + ⋯ + 22281605 T^{4} - 225 T^{3} + \cdots + 22281605 T 4 − 2 2 5 T 3 + ⋯ + 2 2 2 8 1 6 0 5
T^4 - 225*T^3 + 20700*T^2 - 971060*T + 22281605
89 89 8 9
( T 2 − 61 T − 7681 ) 2 (T^{2} - 61 T - 7681)^{2} ( T 2 − 6 1 T − 7 6 8 1 ) 2
(T^2 - 61*T - 7681)^2
97 97 9 7
T 4 + 165 T 3 + ⋯ + 31416025 T^{4} + 165 T^{3} + \cdots + 31416025 T 4 + 1 6 5 T 3 + ⋯ + 3 1 4 1 6 0 2 5
T^4 + 165*T^3 + 16840*T^2 + 952850*T + 31416025
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