gp: [N,k,chi] = [324,3,Mod(163,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.163");
S:= CuspForms(chi, 3);
N := Newforms(S);
Newform invariants
sage: traces = [6,-1]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 + 8 x 4 + 12 x 2 + 3 x^{6} + 8x^{4} + 12x^{2} + 3 x 6 + 8 x 4 + 1 2 x 2 + 3
x^6 + 8*x^4 + 12*x^2 + 3
:
β 1 \beta_{1} β 1 = = =
( ν 5 + ν 4 + 9 ν 3 + 5 ν 2 + 17 ν + 1 ) / 2 ( \nu^{5} + \nu^{4} + 9\nu^{3} + 5\nu^{2} + 17\nu + 1 ) / 2 ( ν 5 + ν 4 + 9 ν 3 + 5 ν 2 + 1 7 ν + 1 ) / 2
(v^5 + v^4 + 9*v^3 + 5*v^2 + 17*v + 1) / 2
β 2 \beta_{2} β 2 = = =
ν 5 + 8 ν 3 + ν 2 + 12 ν + 3 \nu^{5} + 8\nu^{3} + \nu^{2} + 12\nu + 3 ν 5 + 8 ν 3 + ν 2 + 1 2 ν + 3
v^5 + 8*v^3 + v^2 + 12*v + 3
β 3 \beta_{3} β 3 = = =
( ν 5 − ν 4 + 7 ν 3 − 7 ν 2 + 7 ν − 5 ) / 2 ( \nu^{5} - \nu^{4} + 7\nu^{3} - 7\nu^{2} + 7\nu - 5 ) / 2 ( ν 5 − ν 4 + 7 ν 3 − 7 ν 2 + 7 ν − 5 ) / 2
(v^5 - v^4 + 7*v^3 - 7*v^2 + 7*v - 5) / 2
β 4 \beta_{4} β 4 = = =
( 5 ν 5 + ν 4 + 37 ν 3 + 5 ν 2 + 33 ν + 1 ) / 2 ( 5\nu^{5} + \nu^{4} + 37\nu^{3} + 5\nu^{2} + 33\nu + 1 ) / 2 ( 5 ν 5 + ν 4 + 3 7 ν 3 + 5 ν 2 + 3 3 ν + 1 ) / 2
(5*v^5 + v^4 + 37*v^3 + 5*v^2 + 33*v + 1) / 2
β 5 \beta_{5} β 5 = = =
− ν 5 − 2 ν 4 − 6 ν 3 − 13 ν 2 − 2 ν − 8 -\nu^{5} - 2\nu^{4} - 6\nu^{3} - 13\nu^{2} - 2\nu - 8 − ν 5 − 2 ν 4 − 6 ν 3 − 1 3 ν 2 − 2 ν − 8
-v^5 - 2*v^4 - 6*v^3 - 13*v^2 - 2*v - 8
ν \nu ν = = =
( − β 5 − 2 β 4 + 4 β 3 + β 2 + 2 β 1 − 1 ) / 12 ( -\beta_{5} - 2\beta_{4} + 4\beta_{3} + \beta_{2} + 2\beta _1 - 1 ) / 12 ( − β 5 − 2 β 4 + 4 β 3 + β 2 + 2 β 1 − 1 ) / 1 2
(-b5 - 2*b4 + 4*b3 + b2 + 2*b1 - 1) / 12
ν 2 \nu^{2} ν 2 = = =
( − β 3 + β 2 − β 1 − 5 ) / 2 ( -\beta_{3} + \beta_{2} - \beta _1 - 5 ) / 2 ( − β 3 + β 2 − β 1 − 5 ) / 2
(-b3 + b2 - b1 - 5) / 2
ν 3 \nu^{3} ν 3 = = =
( 4 β 5 + 5 β 4 − 13 β 3 − β 2 − 2 β 1 + 1 ) / 6 ( 4\beta_{5} + 5\beta_{4} - 13\beta_{3} - \beta_{2} - 2\beta _1 + 1 ) / 6 ( 4 β 5 + 5 β 4 − 1 3 β 3 − β 2 − 2 β 1 + 1 ) / 6
(4*b5 + 5*b4 - 13*b3 - b2 - 2*b1 + 1) / 6
ν 4 \nu^{4} ν 4 = = =
( − β 5 + 10 β 3 − 13 β 2 + 14 β 1 + 49 ) / 4 ( -\beta_{5} + 10\beta_{3} - 13\beta_{2} + 14\beta _1 + 49 ) / 4 ( − β 5 + 1 0 β 3 − 1 3 β 2 + 1 4 β 1 + 4 9 ) / 4
(-b5 + 10*b3 - 13*b2 + 14*b1 + 49) / 4
ν 5 \nu^{5} ν 5 = = =
( − 26 β 5 − 28 β 4 + 83 β 3 + 5 β 2 + 7 β 1 − 5 ) / 6 ( -26\beta_{5} - 28\beta_{4} + 83\beta_{3} + 5\beta_{2} + 7\beta _1 - 5 ) / 6 ( − 2 6 β 5 − 2 8 β 4 + 8 3 β 3 + 5 β 2 + 7 β 1 − 5 ) / 6
(-26*b5 - 28*b4 + 83*b3 + 5*b2 + 7*b1 - 5) / 6
Character values
We give the values of χ \chi χ on generators for ( Z / 324 Z ) × \left(\mathbb{Z}/324\mathbb{Z}\right)^\times ( Z / 3 2 4 Z ) × .
n n n
163 163 1 6 3
245 245 2 4 5
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
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 5 3 + T 5 2 − 37 T 5 + 59 T_{5}^{3} + T_{5}^{2} - 37T_{5} + 59 T 5 3 + T 5 2 − 3 7 T 5 + 5 9
T5^3 + T5^2 - 37*T5 + 59
acting on S 3 n e w ( 324 , [ χ ] ) S_{3}^{\mathrm{new}}(324, [\chi]) S 3 n e w ( 3 2 4 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 6 + T 5 + ⋯ + 64 T^{6} + T^{5} + \cdots + 64 T 6 + T 5 + ⋯ + 6 4
T^6 + T^5 + 2*T^4 + 4*T^3 + 8*T^2 + 16*T + 64
3 3 3
T 6 T^{6} T 6
T^6
5 5 5
( T 3 + T 2 − 37 T + 59 ) 2 (T^{3} + T^{2} - 37 T + 59)^{2} ( T 3 + T 2 − 3 7 T + 5 9 ) 2
(T^3 + T^2 - 37*T + 59)^2
7 7 7
T 6 + 180 T 4 + ⋯ + 27648 T^{6} + 180 T^{4} + \cdots + 27648 T 6 + 1 8 0 T 4 + ⋯ + 2 7 6 4 8
T^6 + 180*T^4 + 6864*T^2 + 27648
11 11 1 1
T 6 + 516 T 4 + ⋯ + 1769472 T^{6} + 516 T^{4} + \cdots + 1769472 T 6 + 5 1 6 T 4 + ⋯ + 1 7 6 9 4 7 2
T^6 + 516*T^4 + 67920*T^2 + 1769472
13 13 1 3
( T 3 − 3 T 2 + ⋯ + 991 ) 2 (T^{3} - 3 T^{2} + \cdots + 991)^{2} ( T 3 − 3 T 2 + ⋯ + 9 9 1 ) 2
(T^3 - 3*T^2 - 165*T + 991)^2
17 17 1 7
( T 3 − 5 T 2 + ⋯ + 233 ) 2 (T^{3} - 5 T^{2} + \cdots + 233)^{2} ( T 3 − 5 T 2 + ⋯ + 2 3 3 ) 2
(T^3 - 5*T^2 - 157*T + 233)^2
19 19 1 9
T 6 + 1476 T 4 + ⋯ + 20155392 T^{6} + 1476 T^{4} + \cdots + 20155392 T 6 + 1 4 7 6 T 4 + ⋯ + 2 0 1 5 5 3 9 2
T^6 + 1476*T^4 + 352080*T^2 + 20155392
23 23 2 3
T 6 + 2868 T 4 + ⋯ + 425115648 T^{6} + 2868 T^{4} + \cdots + 425115648 T 6 + 2 8 6 8 T 4 + ⋯ + 4 2 5 1 1 5 6 4 8
T^6 + 2868*T^4 + 2259024*T^2 + 425115648
29 29 2 9
( T 3 − 11 T 2 + ⋯ + 32519 ) 2 (T^{3} - 11 T^{2} + \cdots + 32519)^{2} ( T 3 − 1 1 T 2 + ⋯ + 3 2 5 1 9 ) 2
(T^3 - 11*T^2 - 1861*T + 32519)^2
31 31 3 1
T 6 + 5376 T 4 + ⋯ + 7077888 T^{6} + 5376 T^{4} + \cdots + 7077888 T 6 + 5 3 7 6 T 4 + ⋯ + 7 0 7 7 8 8 8
T^6 + 5376*T^4 + 7176192*T^2 + 7077888
37 37 3 7
( T 3 − 27 T 2 + ⋯ + 144607 ) 2 (T^{3} - 27 T^{2} + \cdots + 144607)^{2} ( T 3 − 2 7 T 2 + ⋯ + 1 4 4 6 0 7 ) 2
(T^3 - 27*T^2 - 4029*T + 144607)^2
41 41 4 1
( T 3 + 46 T 2 + ⋯ − 21976 ) 2 (T^{3} + 46 T^{2} + \cdots - 21976)^{2} ( T 3 + 4 6 T 2 + ⋯ − 2 1 9 7 6 ) 2
(T^3 + 46*T^2 - 1012*T - 21976)^2
43 43 4 3
T 6 + 6372 T 4 + ⋯ + 204484608 T^{6} + 6372 T^{4} + \cdots + 204484608 T 6 + 6 3 7 2 T 4 + ⋯ + 2 0 4 4 8 4 6 0 8
T^6 + 6372*T^4 + 8273616*T^2 + 204484608
47 47 4 7
T 6 + 8208 T 4 + ⋯ + 743620608 T^{6} + 8208 T^{4} + \cdots + 743620608 T 6 + 8 2 0 8 T 4 + ⋯ + 7 4 3 6 2 0 6 0 8
T^6 + 8208*T^4 + 9076992*T^2 + 743620608
53 53 5 3
( T 3 + 58 T 2 + ⋯ − 5128 ) 2 (T^{3} + 58 T^{2} + \cdots - 5128)^{2} ( T 3 + 5 8 T 2 + ⋯ − 5 1 2 8 ) 2
(T^3 + 58*T^2 + 140*T - 5128)^2
59 59 5 9
T 6 + ⋯ + 15635054592 T^{6} + \cdots + 15635054592 T 6 + ⋯ + 1 5 6 3 5 0 5 4 5 9 2
T^6 + 13392*T^4 + 37600512*T^2 + 15635054592
61 61 6 1
( T 3 − 27 T 2 + ⋯ + 101191 ) 2 (T^{3} - 27 T^{2} + \cdots + 101191)^{2} ( T 3 − 2 7 T 2 + ⋯ + 1 0 1 1 9 1 ) 2
(T^3 - 27*T^2 - 4821*T + 101191)^2
67 67 6 7
T 6 + ⋯ + 12897487872 T^{6} + \cdots + 12897487872 T 6 + ⋯ + 1 2 8 9 7 4 8 7 8 7 2
T^6 + 14148*T^4 + 47514192*T^2 + 12897487872
71 71 7 1
T 6 + 8532 T 4 + ⋯ + 808455168 T^{6} + 8532 T^{4} + \cdots + 808455168 T 6 + 8 5 3 2 T 4 + ⋯ + 8 0 8 4 5 5 1 6 8
T^6 + 8532*T^4 + 16256592*T^2 + 808455168
73 73 7 3
( T 3 + 39 T 2 + ⋯ − 447851 ) 2 (T^{3} + 39 T^{2} + \cdots - 447851)^{2} ( T 3 + 3 9 T 2 + ⋯ − 4 4 7 8 5 1 ) 2
(T^3 + 39*T^2 - 8901*T - 447851)^2
79 79 7 9
T 6 + 7764 T 4 + ⋯ + 113246208 T^{6} + 7764 T^{4} + \cdots + 113246208 T 6 + 7 7 6 4 T 4 + ⋯ + 1 1 3 2 4 6 2 0 8
T^6 + 7764*T^4 + 12273360*T^2 + 113246208
83 83 8 3
T 6 + ⋯ + 900806787072 T^{6} + \cdots + 900806787072 T 6 + ⋯ + 9 0 0 8 0 6 7 8 7 0 7 2
T^6 + 34896*T^4 + 341295360*T^2 + 900806787072
89 89 8 9
( T 3 − 125 T 2 + ⋯ + 107777 ) 2 (T^{3} - 125 T^{2} + \cdots + 107777)^{2} ( T 3 − 1 2 5 T 2 + ⋯ + 1 0 7 7 7 7 ) 2
(T^3 - 125*T^2 - 4477*T + 107777)^2
97 97 9 7
( T 3 + 102 T 2 + ⋯ − 369272 ) 2 (T^{3} + 102 T^{2} + \cdots - 369272)^{2} ( T 3 + 1 0 2 T 2 + ⋯ − 3 6 9 2 7 2 ) 2
(T^3 + 102*T^2 - 4500*T - 369272)^2
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