gp: [N,k,chi] = [1216,2,Mod(255,1216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1216.255");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [6,0,1]
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 -expansion are expressed in terms of a basis 1 , β 1 , … , β 5 1,\beta_1,\ldots,\beta_{5} 1 , β 1 , … , β 5 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 6 − x 5 + 10 x 4 + 3 x 3 + 84 x 2 − 27 x + 9 x^{6} - x^{5} + 10x^{4} + 3x^{3} + 84x^{2} - 27x + 9 x 6 − x 5 + 1 0 x 4 + 3 x 3 + 8 4 x 2 − 2 7 x + 9
x^6 - x^5 + 10*x^4 + 3*x^3 + 84*x^2 - 27*x + 9
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( − ν 5 + 10 ν 4 − 100 ν 3 + 84 ν 2 − 27 ν + 270 ) / 813 ( -\nu^{5} + 10\nu^{4} - 100\nu^{3} + 84\nu^{2} - 27\nu + 270 ) / 813 ( − ν 5 + 1 0 ν 4 − 1 0 0 ν 3 + 8 4 ν 2 − 2 7 ν + 2 7 0 ) / 8 1 3
(-v^5 + 10*v^4 - 100*v^3 + 84*v^2 - 27*v + 270) / 813
β 3 \beta_{3} β 3 = = =
( 30 ν 5 − 29 ν 4 + 290 ν 3 + 190 ν 2 + 2436 ν − 783 ) / 813 ( 30\nu^{5} - 29\nu^{4} + 290\nu^{3} + 190\nu^{2} + 2436\nu - 783 ) / 813 ( 3 0 ν 5 − 2 9 ν 4 + 2 9 0 ν 3 + 1 9 0 ν 2 + 2 4 3 6 ν − 7 8 3 ) / 8 1 3
(30*v^5 - 29*v^4 + 290*v^3 + 190*v^2 + 2436*v - 783) / 813
β 4 \beta_{4} β 4 = = =
( − 161 ν 5 − 16 ν 4 − 1466 ν 3 − 1923 ν 2 − 14916 ν − 5310 ) / 2439 ( -161\nu^{5} - 16\nu^{4} - 1466\nu^{3} - 1923\nu^{2} - 14916\nu - 5310 ) / 2439 ( − 1 6 1 ν 5 − 1 6 ν 4 − 1 4 6 6 ν 3 − 1 9 2 3 ν 2 − 1 4 9 1 6 ν − 5 3 1 0 ) / 2 4 3 9
(-161*v^5 - 16*v^4 - 1466*v^3 - 1923*v^2 - 14916*v - 5310) / 2439
β 5 \beta_{5} β 5 = = =
( 191 ν 5 − 284 ν 4 + 2027 ν 3 − 597 ν 2 + 15726 ν − 10107 ) / 2439 ( 191\nu^{5} - 284\nu^{4} + 2027\nu^{3} - 597\nu^{2} + 15726\nu - 10107 ) / 2439 ( 1 9 1 ν 5 − 2 8 4 ν 4 + 2 0 2 7 ν 3 − 5 9 7 ν 2 + 1 5 7 2 6 ν − 1 0 1 0 7 ) / 2 4 3 9
(191*v^5 - 284*v^4 + 2027*v^3 - 597*v^2 + 15726*v - 10107) / 2439
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
− 2 β 5 + β 4 + 6 β 3 − β 2 + β 1 -2\beta_{5} + \beta_{4} + 6\beta_{3} - \beta_{2} + \beta_1 − 2 β 5 + β 4 + 6 β 3 − β 2 + β 1
-2*b5 + b4 + 6*b3 - b2 + b1
ν 3 \nu^{3} ν 3 = = =
− β 5 − β 4 − 10 β 2 − 3 -\beta_{5} - \beta_{4} - 10\beta_{2} - 3 − β 5 − β 4 − 1 0 β 2 − 3
-b5 - b4 - 10*b2 - 3
ν 4 \nu^{4} ν 4 = = =
10 β 5 − 20 β 4 − 57 β 3 − 16 β 1 − 57 10\beta_{5} - 20\beta_{4} - 57\beta_{3} - 16\beta _1 - 57 1 0 β 5 − 2 0 β 4 − 5 7 β 3 − 1 6 β 1 − 5 7
10*b5 - 20*b4 - 57*b3 - 16*b1 - 57
ν 5 \nu^{5} ν 5 = = =
32 β 5 − 16 β 4 − 66 β 3 + 103 β 2 − 103 β 1 32\beta_{5} - 16\beta_{4} - 66\beta_{3} + 103\beta_{2} - 103\beta_1 3 2 β 5 − 1 6 β 4 − 6 6 β 3 + 1 0 3 β 2 − 1 0 3 β 1
32*b5 - 16*b4 - 66*b3 + 103*b2 - 103*b1
Character values
We give the values of χ \chi χ on generators for ( Z / 1216 Z ) × \left(\mathbb{Z}/1216\mathbb{Z}\right)^\times ( Z / 1 2 1 6 Z ) × .
n n n
191 191 1 9 1
705 705 7 0 5
837 837 8 3 7
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
− β 3 -\beta_{3} − β 3
1 1 1
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 can be constructed as the kernel of the linear operator
T 3 6 − T 3 5 + 10 T 3 4 + 3 T 3 3 + 84 T 3 2 − 27 T 3 + 9 T_{3}^{6} - T_{3}^{5} + 10T_{3}^{4} + 3T_{3}^{3} + 84T_{3}^{2} - 27T_{3} + 9 T 3 6 − T 3 5 + 1 0 T 3 4 + 3 T 3 3 + 8 4 T 3 2 − 2 7 T 3 + 9
T3^6 - T3^5 + 10*T3^4 + 3*T3^3 + 84*T3^2 - 27*T3 + 9
acting on S 2 n e w ( 1216 , [ χ ] ) S_{2}^{\mathrm{new}}(1216, [\chi]) S 2 n e w ( 1 2 1 6 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 6 T^{6} T 6
T^6
3 3 3
T 6 − T 5 + 10 T 4 + ⋯ + 9 T^{6} - T^{5} + 10 T^{4} + \cdots + 9 T 6 − T 5 + 1 0 T 4 + ⋯ + 9
T^6 - T^5 + 10*T^4 + 3*T^3 + 84*T^2 - 27*T + 9
5 5 5
T 6 + 2 T 5 + ⋯ + 36 T^{6} + 2 T^{5} + \cdots + 36 T 6 + 2 T 5 + ⋯ + 3 6
T^6 + 2*T^5 + 12*T^4 - 4*T^3 + 76*T^2 + 48*T + 36
7 7 7
T 6 + 20 T 4 + ⋯ + 108 T^{6} + 20 T^{4} + \cdots + 108 T 6 + 2 0 T 4 + ⋯ + 1 0 8
T^6 + 20*T^4 + 100*T^2 + 108
11 11 1 1
T 6 + 29 T 4 + ⋯ + 507 T^{6} + 29 T^{4} + \cdots + 507 T 6 + 2 9 T 4 + ⋯ + 5 0 7
T^6 + 29*T^4 + 235*T^2 + 507
13 13 1 3
( T 2 + 6 T + 12 ) 3 (T^{2} + 6 T + 12)^{3} ( T 2 + 6 T + 1 2 ) 3
(T^2 + 6*T + 12)^3
17 17 1 7
T 6 + 2 T 5 + ⋯ + 144 T^{6} + 2 T^{5} + \cdots + 144 T 6 + 2 T 5 + ⋯ + 1 4 4
T^6 + 2*T^5 + 48*T^4 - 112*T^3 + 1912*T^2 - 528*T + 144
19 19 1 9
T 6 − 17 T 5 + ⋯ + 6859 T^{6} - 17 T^{5} + \cdots + 6859 T 6 − 1 7 T 5 + ⋯ + 6 8 5 9
T^6 - 17*T^5 + 144*T^4 - 775*T^3 + 2736*T^2 - 6137*T + 6859
23 23 2 3
T 6 − 10 T 4 + ⋯ + 108 T^{6} - 10 T^{4} + \cdots + 108 T 6 − 1 0 T 4 + ⋯ + 1 0 8
T^6 - 10*T^4 + 100*T^2 + 180*T + 108
29 29 2 9
T 6 − 12 T 5 + ⋯ + 2700 T^{6} - 12 T^{5} + \cdots + 2700 T 6 − 1 2 T 5 + ⋯ + 2 7 0 0
T^6 - 12*T^5 + 36*T^4 + 144*T^3 - 216*T^2 - 1080*T + 2700
31 31 3 1
( T 3 + 2 T 2 − 54 T + 66 ) 2 (T^{3} + 2 T^{2} - 54 T + 66)^{2} ( T 3 + 2 T 2 − 5 4 T + 6 6 ) 2
(T^3 + 2*T^2 - 54*T + 66)^2
37 37 3 7
T 6 + 72 T 4 + ⋯ + 2700 T^{6} + 72 T^{4} + \cdots + 2700 T 6 + 7 2 T 4 + ⋯ + 2 7 0 0
T^6 + 72*T^4 + 864*T^2 + 2700
41 41 4 1
T 6 − 3 T 5 + ⋯ + 54675 T^{6} - 3 T^{5} + \cdots + 54675 T 6 − 3 T 5 + ⋯ + 5 4 6 7 5
T^6 - 3*T^5 - 96*T^4 + 297*T^3 + 9396*T^2 - 40095*T + 54675
43 43 4 3
( T 2 − 6 T + 12 ) 3 (T^{2} - 6 T + 12)^{3} ( T 2 − 6 T + 1 2 ) 3
(T^2 - 6*T + 12)^3
47 47 4 7
T 6 − 18 T 5 + ⋯ + 12 T^{6} - 18 T^{5} + \cdots + 12 T 6 − 1 8 T 5 + ⋯ + 1 2
T^6 - 18*T^5 + 134*T^4 - 468*T^3 + 640*T^2 + 156*T + 12
53 53 5 3
( T 2 − 12 T + 48 ) 3 (T^{2} - 12 T + 48)^{3} ( T 2 − 1 2 T + 4 8 ) 3
(T^2 - 12*T + 48)^3
59 59 5 9
T 6 − 27 T 5 + ⋯ + 263169 T^{6} - 27 T^{5} + \cdots + 263169 T 6 − 2 7 T 5 + ⋯ + 2 6 3 1 6 9
T^6 - 27*T^5 + 516*T^4 - 4725*T^3 + 31518*T^2 - 109269*T + 263169
61 61 6 1
T 6 − 10 T 5 + ⋯ + 36 T^{6} - 10 T^{5} + \cdots + 36 T 6 − 1 0 T 5 + ⋯ + 3 6
T^6 - 10*T^5 + 76*T^4 - 228*T^3 + 516*T^2 - 144*T + 36
67 67 6 7
T 6 + 11 T 5 + ⋯ + 245025 T^{6} + 11 T^{5} + \cdots + 245025 T 6 + 1 1 T 5 + ⋯ + 2 4 5 0 2 5
T^6 + 11*T^5 + 160*T^4 + 561*T^3 + 6966*T^2 + 19305*T + 245025
71 71 7 1
( T 2 + 6 T + 36 ) 3 (T^{2} + 6 T + 36)^{3} ( T 2 + 6 T + 3 6 ) 3
(T^2 + 6*T + 36)^3
73 73 7 3
T 6 + 5 T 5 + ⋯ + 289 T^{6} + 5 T^{5} + \cdots + 289 T 6 + 5 T 5 + ⋯ + 2 8 9
T^6 + 5*T^5 + 50*T^4 - 91*T^3 + 710*T^2 + 425*T + 289
79 79 7 9
T 6 − 16 T 5 + ⋯ + 2304 T^{6} - 16 T^{5} + \cdots + 2304 T 6 − 1 6 T 5 + ⋯ + 2 3 0 4
T^6 - 16*T^5 + 208*T^4 - 864*T^3 + 3072*T^2 + 2304*T + 2304
83 83 8 3
T 6 + 113 T 4 + ⋯ + 3 T^{6} + 113 T^{4} + \cdots + 3 T 6 + 1 1 3 T 4 + ⋯ + 3
T^6 + 113*T^4 + 3031*T^2 + 3
89 89 8 9
T 6 + 24 T 5 + ⋯ + 97200 T^{6} + 24 T^{5} + \cdots + 97200 T 6 + 2 4 T 5 + ⋯ + 9 7 2 0 0
T^6 + 24*T^5 + 120*T^4 - 1728*T^3 + 864*T^2 + 38880*T + 97200
97 97 9 7
T 6 − 21 T 5 + ⋯ + 151875 T^{6} - 21 T^{5} + \cdots + 151875 T 6 − 2 1 T 5 + ⋯ + 1 5 1 8 7 5
T^6 - 21*T^5 + 84*T^4 + 1323*T^3 - 756*T^2 - 42525*T + 151875
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