gp: [N,k,chi] = [336,4,Mod(193,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.193");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [4,0,-6,0,3]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 2 , β 3 1,\beta_1,\beta_2,\beta_3 1 , β 1 , β 2 , β 3 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 4 − x 3 − 4 x 2 − 5 x + 25 x^{4} - x^{3} - 4x^{2} - 5x + 25 x 4 − x 3 − 4 x 2 − 5 x + 2 5
x^4 - x^3 - 4*x^2 - 5*x + 25
:
β 1 \beta_{1} β 1 = = =
( ν 3 + 4 ν 2 + 56 ν − 25 ) / 20 ( \nu^{3} + 4\nu^{2} + 56\nu - 25 ) / 20 ( ν 3 + 4 ν 2 + 5 6 ν − 2 5 ) / 2 0
(v^3 + 4*v^2 + 56*v - 25) / 20
β 2 \beta_{2} β 2 = = =
( ν 3 + 4 ν 2 − 4 ν − 25 ) / 20 ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 ( ν 3 + 4 ν 2 − 4 ν − 2 5 ) / 2 0
(v^3 + 4*v^2 - 4*v - 25) / 20
β 3 \beta_{3} β 3 = = =
( − 7 ν 3 + 2 ν 2 + 28 ν + 35 ) / 10 ( -7\nu^{3} + 2\nu^{2} + 28\nu + 35 ) / 10 ( − 7 ν 3 + 2 ν 2 + 2 8 ν + 3 5 ) / 1 0
(-7*v^3 + 2*v^2 + 28*v + 35) / 10
ν \nu ν = = =
( − β 2 + β 1 ) / 3 ( -\beta_{2} + \beta_1 ) / 3 ( − β 2 + β 1 ) / 3
(-b2 + b1) / 3
ν 2 \nu^{2} ν 2 = = =
( β 3 + 14 β 2 + 14 ) / 3 ( \beta_{3} + 14\beta_{2} + 14 ) / 3 ( β 3 + 1 4 β 2 + 1 4 ) / 3
(b3 + 14*b2 + 14) / 3
ν 3 \nu^{3} ν 3 = = =
( − 4 β 3 + 4 β 1 + 19 ) / 3 ( -4\beta_{3} + 4\beta _1 + 19 ) / 3 ( − 4 β 3 + 4 β 1 + 1 9 ) / 3
(-4*b3 + 4*b1 + 19) / 3
Character values
We give the values of χ \chi χ on generators for ( Z / 336 Z ) × \left(\mathbb{Z}/336\mathbb{Z}\right)^\times ( Z / 3 3 6 Z ) × .
n n n
85 85 8 5
113 113 1 1 3
127 127 1 2 7
241 241 2 4 1
χ ( n ) \chi(n) χ ( n )
1 1 1
1 1 1
1 1 1
− 1 − β 2 -1 - \beta_{2} − 1 − β 2
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 4 − 3 T 5 3 + 135 T 5 2 + 378 T 5 + 15876 T_{5}^{4} - 3T_{5}^{3} + 135T_{5}^{2} + 378T_{5} + 15876 T 5 4 − 3 T 5 3 + 1 3 5 T 5 2 + 3 7 8 T 5 + 1 5 8 7 6
T5^4 - 3*T5^3 + 135*T5^2 + 378*T5 + 15876
acting on S 4 n e w ( 336 , [ χ ] ) S_{4}^{\mathrm{new}}(336, [\chi]) S 4 n e w ( 3 3 6 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 T^{4} T 4
T^4
3 3 3
( T 2 + 3 T + 9 ) 2 (T^{2} + 3 T + 9)^{2} ( T 2 + 3 T + 9 ) 2
(T^2 + 3*T + 9)^2
5 5 5
T 4 − 3 T 3 + ⋯ + 15876 T^{4} - 3 T^{3} + \cdots + 15876 T 4 − 3 T 3 + ⋯ + 1 5 8 7 6
T^4 - 3*T^3 + 135*T^2 + 378*T + 15876
7 7 7
T 4 − 20 T 3 + ⋯ + 117649 T^{4} - 20 T^{3} + \cdots + 117649 T 4 − 2 0 T 3 + ⋯ + 1 1 7 6 4 9
T^4 - 20*T^3 + 273*T^2 - 6860*T + 117649
11 11 1 1
T 4 + 51 T 3 + ⋯ + 272484 T^{4} + 51 T^{3} + \cdots + 272484 T 4 + 5 1 T 3 + ⋯ + 2 7 2 4 8 4
T^4 + 51*T^3 + 2079*T^2 + 26622*T + 272484
13 13 1 3
( T 2 − 61 T − 2276 ) 2 (T^{2} - 61 T - 2276)^{2} ( T 2 − 6 1 T − 2 2 7 6 ) 2
(T^2 - 61*T - 2276)^2
17 17 1 7
T 4 + 24 T 3 + ⋯ + 65028096 T^{4} + 24 T^{3} + \cdots + 65028096 T 4 + 2 4 T 3 + ⋯ + 6 5 0 2 8 0 9 6
T^4 + 24*T^3 + 8640*T^2 - 193536*T + 65028096
19 19 1 9
T 4 − 169 T 3 + ⋯ + 49168144 T^{4} - 169 T^{3} + \cdots + 49168144 T 4 − 1 6 9 T 3 + ⋯ + 4 9 1 6 8 1 4 4
T^4 - 169*T^3 + 21549*T^2 - 1185028*T + 49168144
23 23 2 3
( T 2 + 96 T + 9216 ) 2 (T^{2} + 96 T + 9216)^{2} ( T 2 + 9 6 T + 9 2 1 6 ) 2
(T^2 + 96*T + 9216)^2
29 29 2 9
( T 2 + 39 T − 36684 ) 2 (T^{2} + 39 T - 36684)^{2} ( T 2 + 3 9 T − 3 6 6 8 4 ) 2
(T^2 + 39*T - 36684)^2
31 31 3 1
T 4 + 92 T 3 + ⋯ + 114682681 T^{4} + 92 T^{3} + \cdots + 114682681 T 4 + 9 2 T 3 + ⋯ + 1 1 4 6 8 2 6 8 1
T^4 + 92*T^3 + 19173*T^2 - 985228*T + 114682681
37 37 3 7
T 4 + ⋯ + 1506681856 T^{4} + \cdots + 1506681856 T 4 + ⋯ + 1 5 0 6 6 8 1 8 5 6
T^4 - 173*T^3 + 68745*T^2 + 6715168*T + 1506681856
41 41 4 1
( T 2 − 174 T − 140688 ) 2 (T^{2} - 174 T - 140688)^{2} ( T 2 − 1 7 4 T − 1 4 0 6 8 8 ) 2
(T^2 - 174*T - 140688)^2
43 43 4 3
( T 2 − 497 T + 15454 ) 2 (T^{2} - 497 T + 15454)^{2} ( T 2 − 4 9 7 T + 1 5 4 5 4 ) 2
(T^2 - 497*T + 15454)^2
47 47 4 7
T 4 − 180 T 3 + ⋯ + 611671824 T^{4} - 180 T^{3} + \cdots + 611671824 T 4 − 1 8 0 T 3 + ⋯ + 6 1 1 6 7 1 8 2 4
T^4 - 180*T^3 + 57132*T^2 + 4451760*T + 611671824
53 53 5 3
T 4 + ⋯ + 5356483344 T^{4} + \cdots + 5356483344 T 4 + ⋯ + 5 3 5 6 4 8 3 3 4 4
T^4 + 285*T^3 + 154413*T^2 - 20858580*T + 5356483344
59 59 5 9
T 4 + ⋯ + 161150862096 T^{4} + \cdots + 161150862096 T 4 + ⋯ + 1 6 1 1 5 0 8 6 2 0 9 6
T^4 + 1269*T^3 + 1208925*T^2 + 509422284*T + 161150862096
61 61 6 1
T 4 + ⋯ + 19408947856 T^{4} + \cdots + 19408947856 T 4 + ⋯ + 1 9 4 0 8 9 4 7 8 5 6
T^4 + 328*T^3 + 246900*T^2 - 45695648*T + 19408947856
67 67 6 7
T 4 + ⋯ + 32767516324 T^{4} + \cdots + 32767516324 T 4 + ⋯ + 3 2 7 6 7 5 1 6 3 2 4
T^4 + 875*T^3 + 584607*T^2 + 158390750*T + 32767516324
71 71 7 1
( T 2 − 1404 T + 361476 ) 2 (T^{2} - 1404 T + 361476)^{2} ( T 2 − 1 4 0 4 T + 3 6 1 4 7 6 ) 2
(T^2 - 1404*T + 361476)^2
73 73 7 3
T 4 + ⋯ + 165260136484 T^{4} + \cdots + 165260136484 T 4 + ⋯ + 1 6 5 2 6 0 1 3 6 4 8 4
T^4 - 1361*T^3 + 1445799*T^2 - 553276442*T + 165260136484
79 79 7 9
T 4 + 182 T 3 + ⋯ + 38800441 T^{4} + 182 T^{3} + \cdots + 38800441 T 4 + 1 8 2 T 3 + ⋯ + 3 8 8 0 0 4 4 1
T^4 + 182*T^3 + 26895*T^2 + 1133678*T + 38800441
83 83 8 3
( T 2 − 399 T − 197334 ) 2 (T^{2} - 399 T - 197334)^{2} ( T 2 − 3 9 9 T − 1 9 7 3 3 4 ) 2
(T^2 - 399*T - 197334)^2
89 89 8 9
T 4 + ⋯ + 11416495104 T^{4} + \cdots + 11416495104 T 4 + ⋯ + 1 1 4 1 6 4 9 5 1 0 4
T^4 + 822*T^3 + 568836*T^2 + 87829056*T + 11416495104
97 97 9 7
( T 2 − 841 T + 120262 ) 2 (T^{2} - 841 T + 120262)^{2} ( T 2 − 8 4 1 T + 1 2 0 2 6 2 ) 2
(T^2 - 841*T + 120262)^2
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