gp: [N,k,chi] = [294,4,Mod(67,294)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(294, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("294.67");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [4,-4,6,-8,12,-24,0,32,-18,24,-4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 + 2 x 2 + 4 x^{4} + 2x^{2} + 4 x 4 + 2 x 2 + 4
x^4 + 2*x^2 + 4
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 2 ) / 2 ( \nu^{2} ) / 2 ( ν 2 ) / 2
(v^2) / 2
β 3 \beta_{3} β 3 = = =
( ν 3 ) / 2 ( \nu^{3} ) / 2 ( ν 3 ) / 2
(v^3) / 2
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
2 β 2 2\beta_{2} 2 β 2
2*b2
ν 3 \nu^{3} ν 3 = = =
2 β 3 2\beta_{3} 2 β 3
2*b3
Character values
We give the values of χ \chi χ on generators for ( Z / 294 Z ) × \left(\mathbb{Z}/294\mathbb{Z}\right)^\times ( Z / 2 9 4 Z ) × .
n n n
197 197 1 9 7
199 199 1 9 9
χ ( n ) \chi(n) χ ( n )
1 1 1
β 2 \beta_{2} 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 intersection of the kernels of the following linear operators acting on S 4 n e w ( 294 , [ χ ] ) S_{4}^{\mathrm{new}}(294, [\chi]) S 4 n e w ( 2 9 4 , [ χ ] ) :
T 5 4 − 12 T 5 3 + 110 T 5 2 − 408 T 5 + 1156 T_{5}^{4} - 12T_{5}^{3} + 110T_{5}^{2} - 408T_{5} + 1156 T 5 4 − 1 2 T 5 3 + 1 1 0 T 5 2 − 4 0 8 T 5 + 1 1 5 6
T5^4 - 12*T5^3 + 110*T5^2 - 408*T5 + 1156
T 11 4 + 4 T 11 3 + 84 T 11 2 − 272 T 11 + 4624 T_{11}^{4} + 4T_{11}^{3} + 84T_{11}^{2} - 272T_{11} + 4624 T 1 1 4 + 4 T 1 1 3 + 8 4 T 1 1 2 − 2 7 2 T 1 1 + 4 6 2 4
T11^4 + 4*T11^3 + 84*T11^2 - 272*T11 + 4624
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 + 2 T + 4 ) 2 (T^{2} + 2 T + 4)^{2} ( T 2 + 2 T + 4 ) 2
(T^2 + 2*T + 4)^2
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 − 12 T 3 + ⋯ + 1156 T^{4} - 12 T^{3} + \cdots + 1156 T 4 − 1 2 T 3 + ⋯ + 1 1 5 6
T^4 - 12*T^3 + 110*T^2 - 408*T + 1156
7 7 7
T 4 T^{4} T 4
T^4
11 11 1 1
T 4 + 4 T 3 + ⋯ + 4624 T^{4} + 4 T^{3} + \cdots + 4624 T 4 + 4 T 3 + ⋯ + 4 6 2 4
T^4 + 4*T^3 + 84*T^2 - 272*T + 4624
13 13 1 3
( T 2 + 48 T + 126 ) 2 (T^{2} + 48 T + 126)^{2} ( T 2 + 4 8 T + 1 2 6 ) 2
(T^2 + 48*T + 126)^2
17 17 1 7
T 4 − 132 T 3 + ⋯ + 16924996 T^{4} - 132 T^{3} + \cdots + 16924996 T 4 − 1 3 2 T 3 + ⋯ + 1 6 9 2 4 9 9 6
T^4 - 132*T^3 + 13310*T^2 - 543048*T + 16924996
19 19 1 9
T 4 − 120 T 3 + ⋯ + 399424 T^{4} - 120 T^{3} + \cdots + 399424 T 4 − 1 2 0 T 3 + ⋯ + 3 9 9 4 2 4
T^4 - 120*T^3 + 15032*T^2 + 75840*T + 399424
23 23 2 3
T 4 − 76 T 3 + ⋯ + 374964496 T^{4} - 76 T^{3} + \cdots + 374964496 T 4 − 7 6 T 3 + ⋯ + 3 7 4 9 6 4 4 9 6
T^4 - 76*T^3 + 25140*T^2 + 1471664*T + 374964496
29 29 2 9
( T 2 + 112 T − 41864 ) 2 (T^{2} + 112 T - 41864)^{2} ( T 2 + 1 1 2 T − 4 1 8 6 4 ) 2
(T^2 + 112*T - 41864)^2
31 31 3 1
T 4 + ⋯ + 1666598976 T^{4} + \cdots + 1666598976 T 4 + ⋯ + 1 6 6 6 5 9 8 9 7 6
T^4 - 432*T^3 + 145800*T^2 - 17635968*T + 1666598976
37 37 3 7
T 4 + ⋯ + 2043040000 T^{4} + \cdots + 2043040000 T 4 + ⋯ + 2 0 4 3 0 4 0 0 0 0
T^4 - 280*T^3 + 123600*T^2 + 12656000*T + 2043040000
41 41 4 1
( T 2 + 36 T − 31934 ) 2 (T^{2} + 36 T - 31934)^{2} ( T 2 + 3 6 T − 3 1 9 3 4 ) 2
(T^2 + 36*T - 31934)^2
43 43 4 3
( T 2 + 128 T − 161792 ) 2 (T^{2} + 128 T - 161792)^{2} ( T 2 + 1 2 8 T − 1 6 1 7 9 2 ) 2
(T^2 + 128*T - 161792)^2
47 47 4 7
T 4 + ⋯ + 44548012096 T^{4} + \cdots + 44548012096 T 4 + ⋯ + 4 4 5 4 8 0 1 2 0 9 6
T^4 + 264*T^3 + 280760*T^2 - 55720896*T + 44548012096
53 53 5 3
T 4 + ⋯ + 3110515984 T^{4} + \cdots + 3110515984 T 4 + ⋯ + 3 1 1 0 5 1 5 9 8 4
T^4 + 268*T^3 + 127596*T^2 - 14946896*T + 3110515984
59 59 5 9
T 4 + ⋯ + 22315579456 T^{4} + \cdots + 22315579456 T 4 + ⋯ + 2 2 3 1 5 5 7 9 4 5 6
T^4 - 336*T^3 + 262280*T^2 + 50193024*T + 22315579456
61 61 6 1
T 4 + ⋯ + 34489689796 T^{4} + \cdots + 34489689796 T 4 + ⋯ + 3 4 4 8 9 6 8 9 7 9 6
T^4 + 504*T^3 + 439730*T^2 - 93599856*T + 34489689796
67 67 6 7
T 4 − 384 T 3 + ⋯ + 21233664 T^{4} - 384 T^{3} + \cdots + 21233664 T 4 − 3 8 4 T 3 + ⋯ + 2 1 2 3 3 6 6 4
T^4 - 384*T^3 + 152064*T^2 + 1769472*T + 21233664
71 71 7 1
( T 2 + 396 T − 194724 ) 2 (T^{2} + 396 T - 194724)^{2} ( T 2 + 3 9 6 T − 1 9 4 7 2 4 ) 2
(T^2 + 396*T - 194724)^2
73 73 7 3
T 4 + ⋯ + 467464833796 T^{4} + \cdots + 467464833796 T 4 + ⋯ + 4 6 7 4 6 4 8 3 3 7 9 6
T^4 + 312*T^3 + 781058*T^2 - 213318768*T + 467464833796
79 79 7 9
T 4 − 848 T 3 + ⋯ + 50176 T^{4} - 848 T^{3} + \cdots + 50176 T 4 − 8 4 8 T 3 + ⋯ + 5 0 1 7 6
T^4 - 848*T^3 + 719328*T^2 + 189952*T + 50176
83 83 8 3
( T 2 − 648 T + 92176 ) 2 (T^{2} - 648 T + 92176)^{2} ( T 2 − 6 4 8 T + 9 2 1 7 6 ) 2
(T^2 - 648*T + 92176)^2
89 89 8 9
T 4 + ⋯ + 1048852996 T^{4} + \cdots + 1048852996 T 4 + ⋯ + 1 0 4 8 8 5 2 9 9 6
T^4 + 612*T^3 + 342158*T^2 + 19820232*T + 1048852996
97 97 9 7
( T 2 − 2184 T + 1157086 ) 2 (T^{2} - 2184 T + 1157086)^{2} ( T 2 − 2 1 8 4 T + 1 1 5 7 0 8 6 ) 2
(T^2 - 2184*T + 1157086)^2
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