gp: [N,k,chi] = [100,5,Mod(51,100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("100.51");
S:= CuspForms(chi, 5);
N := Newforms(S);
Newform invariants
sage: traces = [8,0]
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 7 1,\beta_1,\ldots,\beta_{7} 1 , β 1 , … , β 7 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 8 − 6 x 6 + 31 x 4 − 96 x 2 + 256 x^{8} - 6x^{6} + 31x^{4} - 96x^{2} + 256 x 8 − 6 x 6 + 3 1 x 4 − 9 6 x 2 + 2 5 6
x^8 - 6*x^6 + 31*x^4 - 96*x^2 + 256
:
β 1 \beta_{1} β 1 = = =
( ν 7 + 10 ν 5 − ν 3 + 80 ν ) / 32 ( \nu^{7} + 10\nu^{5} - \nu^{3} + 80\nu ) / 32 ( ν 7 + 1 0 ν 5 − ν 3 + 8 0 ν ) / 3 2
(v^7 + 10*v^5 - v^3 + 80*v) / 32
β 2 \beta_{2} β 2 = = =
( ν 7 − 6 ν 5 + 31 ν 3 − 96 ν ) / 32 ( \nu^{7} - 6\nu^{5} + 31\nu^{3} - 96\nu ) / 32 ( ν 7 − 6 ν 5 + 3 1 ν 3 − 9 6 ν ) / 3 2
(v^7 - 6*v^5 + 31*v^3 - 96*v) / 32
β 3 \beta_{3} β 3 = = =
( ν 7 − 6 ν 5 + 31 ν 3 + 160 ν ) / 32 ( \nu^{7} - 6\nu^{5} + 31\nu^{3} + 160\nu ) / 32 ( ν 7 − 6 ν 5 + 3 1 ν 3 + 1 6 0 ν ) / 3 2
(v^7 - 6*v^5 + 31*v^3 + 160*v) / 32
β 4 \beta_{4} β 4 = = =
( − ν 6 + 6 ν 4 + ν 2 + 24 ) / 4 ( -\nu^{6} + 6\nu^{4} + \nu^{2} + 24 ) / 4 ( − ν 6 + 6 ν 4 + ν 2 + 2 4 ) / 4
(-v^6 + 6*v^4 + v^2 + 24) / 4
β 5 \beta_{5} β 5 = = =
( ν 6 + 2 ν 4 − ν 2 + 28 ) / 2 ( \nu^{6} + 2\nu^{4} - \nu^{2} + 28 ) / 2 ( ν 6 + 2 ν 4 − ν 2 + 2 8 ) / 2
(v^6 + 2*v^4 - v^2 + 28) / 2
β 6 \beta_{6} β 6 = = =
( − ν 6 + 6 ν 4 − 31 ν 2 + 72 ) / 4 ( -\nu^{6} + 6\nu^{4} - 31\nu^{2} + 72 ) / 4 ( − ν 6 + 6 ν 4 − 3 1 ν 2 + 7 2 ) / 4
(-v^6 + 6*v^4 - 31*v^2 + 72) / 4
β 7 \beta_{7} β 7 = = =
( − 3 ν 7 + 10 ν 5 − 13 ν 3 + 72 ν ) / 8 ( -3\nu^{7} + 10\nu^{5} - 13\nu^{3} + 72\nu ) / 8 ( − 3 ν 7 + 1 0 ν 5 − 1 3 ν 3 + 7 2 ν ) / 8
(-3*v^7 + 10*v^5 - 13*v^3 + 72*v) / 8
ν \nu ν = = =
( β 3 − β 2 ) / 8 ( \beta_{3} - \beta_{2} ) / 8 ( β 3 − β 2 ) / 8
(b3 - b2) / 8
ν 2 \nu^{2} ν 2 = = =
( − β 6 + β 4 + 12 ) / 8 ( -\beta_{6} + \beta_{4} + 12 ) / 8 ( − β 6 + β 4 + 1 2 ) / 8
(-b6 + b4 + 12) / 8
ν 3 \nu^{3} ν 3 = = =
( β 7 + 2 β 3 + 8 β 2 + 2 β 1 ) / 8 ( \beta_{7} + 2\beta_{3} + 8\beta_{2} + 2\beta_1 ) / 8 ( β 7 + 2 β 3 + 8 β 2 + 2 β 1 ) / 8
(b7 + 2*b3 + 8*b2 + 2*b1) / 8
ν 4 \nu^{4} ν 4 = = =
( β 5 + 2 β 4 − 26 ) / 4 ( \beta_{5} + 2\beta_{4} - 26 ) / 4 ( β 5 + 2 β 4 − 2 6 ) / 4
(b5 + 2*b4 - 26) / 4
ν 5 \nu^{5} ν 5 = = =
( 2 β 7 − 7 β 3 + 11 β 2 + 20 β 1 ) / 8 ( 2\beta_{7} - 7\beta_{3} + 11\beta_{2} + 20\beta_1 ) / 8 ( 2 β 7 − 7 β 3 + 1 1 β 2 + 2 0 β 1 ) / 8
(2*b7 - 7*b3 + 11*b2 + 20*b1) / 8
ν 6 \nu^{6} ν 6 = = =
( − β 6 + 12 β 5 − 7 β 4 − 108 ) / 8 ( -\beta_{6} + 12\beta_{5} - 7\beta_{4} - 108 ) / 8 ( − β 6 + 1 2 β 5 − 7 β 4 − 1 0 8 ) / 8
(-b6 + 12*b5 - 7*b4 - 108) / 8
ν 7 \nu^{7} ν 7 = = =
( − 19 β 7 − 8 β 3 − 22 β 2 + 58 β 1 ) / 8 ( -19\beta_{7} - 8\beta_{3} - 22\beta_{2} + 58\beta_1 ) / 8 ( − 1 9 β 7 − 8 β 3 − 2 2 β 2 + 5 8 β 1 ) / 8
(-19*b7 - 8*b3 - 22*b2 + 58*b1) / 8
Character values
We give the values of χ \chi χ on generators for ( Z / 100 Z ) × \left(\mathbb{Z}/100\mathbb{Z}\right)^\times ( Z / 1 0 0 Z ) × .
n n n
51 51 5 1
77 77 7 7
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 intersection of the kernels of the following linear operators acting on S 5 n e w ( 100 , [ χ ] ) S_{5}^{\mathrm{new}}(100, [\chi]) S 5 n e w ( 1 0 0 , [ χ ] ) :
T 3 4 + 240 T 3 2 + 8640 T_{3}^{4} + 240T_{3}^{2} + 8640 T 3 4 + 2 4 0 T 3 2 + 8 6 4 0
T3^4 + 240*T3^2 + 8640
T 13 4 − 20928 T 13 2 + 82861056 T_{13}^{4} - 20928T_{13}^{2} + 82861056 T 1 3 4 − 2 0 9 2 8 T 1 3 2 + 8 2 8 6 1 0 5 6
T13^4 - 20928*T13^2 + 82861056
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 8 − 24 T 6 + ⋯ + 65536 T^{8} - 24 T^{6} + \cdots + 65536 T 8 − 2 4 T 6 + ⋯ + 6 5 5 3 6
T^8 - 24*T^6 + 496*T^4 - 6144*T^2 + 65536
3 3 3
( T 4 + 240 T 2 + 8640 ) 2 (T^{4} + 240 T^{2} + 8640)^{2} ( T 4 + 2 4 0 T 2 + 8 6 4 0 ) 2
(T^4 + 240*T^2 + 8640)^2
5 5 5
T 8 T^{8} T 8
T^8
7 7 7
( T 4 + 2800 T 2 + 1944000 ) 2 (T^{4} + 2800 T^{2} + 1944000)^{2} ( T 4 + 2 8 0 0 T 2 + 1 9 4 4 0 0 0 ) 2
(T^4 + 2800*T^2 + 1944000)^2
11 11 1 1
( T 4 + 48000 T 2 + 552407040 ) 2 (T^{4} + 48000 T^{2} + 552407040)^{2} ( T 4 + 4 8 0 0 0 T 2 + 5 5 2 4 0 7 0 4 0 ) 2
(T^4 + 48000*T^2 + 552407040)^2
13 13 1 3
( T 4 − 20928 T 2 + 82861056 ) 2 (T^{4} - 20928 T^{2} + 82861056)^{2} ( T 4 − 2 0 9 2 8 T 2 + 8 2 8 6 1 0 5 6 ) 2
(T^4 - 20928*T^2 + 82861056)^2
17 17 1 7
( T 4 − 26368 T 2 + 16367616 ) 2 (T^{4} - 26368 T^{2} + 16367616)^{2} ( T 4 − 2 6 3 6 8 T 2 + 1 6 3 6 7 6 1 6 ) 2
(T^4 - 26368*T^2 + 16367616)^2
19 19 1 9
( T 4 + 447360 T 2 + 10372976640 ) 2 (T^{4} + 447360 T^{2} + 10372976640)^{2} ( T 4 + 4 4 7 3 6 0 T 2 + 1 0 3 7 2 9 7 6 6 4 0 ) 2
(T^4 + 447360*T^2 + 10372976640)^2
23 23 2 3
( T 4 + 350320 T 2 + 30678152640 ) 2 (T^{4} + 350320 T^{2} + 30678152640)^{2} ( T 4 + 3 5 0 3 2 0 T 2 + 3 0 6 7 8 1 5 2 6 4 0 ) 2
(T^4 + 350320*T^2 + 30678152640)^2
29 29 2 9
( T 2 + 676 T + 104004 ) 4 (T^{2} + 676 T + 104004)^{4} ( T 2 + 6 7 6 T + 1 0 4 0 0 4 ) 4
(T^2 + 676*T + 104004)^4
31 31 3 1
( T 4 + 1013760 T 2 + 1745879040 ) 2 (T^{4} + 1013760 T^{2} + 1745879040)^{2} ( T 4 + 1 0 1 3 7 6 0 T 2 + 1 7 4 5 8 7 9 0 4 0 ) 2
(T^4 + 1013760*T^2 + 1745879040)^2
37 37 3 7
( T 4 − 3008448 T 2 + 126031666176 ) 2 (T^{4} - 3008448 T^{2} + 126031666176)^{2} ( T 4 − 3 0 0 8 4 4 8 T 2 + 1 2 6 0 3 1 6 6 6 1 7 6 ) 2
(T^4 - 3008448*T^2 + 126031666176)^2
41 41 4 1
( T 2 − 364 T − 3961116 ) 4 (T^{2} - 364 T - 3961116)^{4} ( T 2 − 3 6 4 T − 3 9 6 1 1 1 6 ) 4
(T^2 - 364*T - 3961116)^4
43 43 4 3
( T 4 + 10034800 T 2 + 325194264000 ) 2 (T^{4} + 10034800 T^{2} + 325194264000)^{2} ( T 4 + 1 0 0 3 4 8 0 0 T 2 + 3 2 5 1 9 4 2 6 4 0 0 0 ) 2
(T^4 + 10034800*T^2 + 325194264000)^2
47 47 4 7
( T 4 + 751600 T 2 + 49724936640 ) 2 (T^{4} + 751600 T^{2} + 49724936640)^{2} ( T 4 + 7 5 1 6 0 0 T 2 + 4 9 7 2 4 9 3 6 6 4 0 ) 2
(T^4 + 751600*T^2 + 49724936640)^2
53 53 5 3
( T 4 + ⋯ + 20248378776576 ) 2 (T^{4} + \cdots + 20248378776576)^{2} ( T 4 + ⋯ + 2 0 2 4 8 3 7 8 7 7 6 5 7 6 ) 2
(T^4 - 24538048*T^2 + 20248378776576)^2
59 59 5 9
( T 4 + ⋯ + 225559394979840 ) 2 (T^{4} + \cdots + 225559394979840)^{2} ( T 4 + ⋯ + 2 2 5 5 5 9 3 9 4 9 7 9 8 4 0 ) 2
(T^4 + 30276480*T^2 + 225559394979840)^2
61 61 6 1
( T 2 − 1644 T − 10307356 ) 4 (T^{2} - 1644 T - 10307356)^{4} ( T 2 − 1 6 4 4 T − 1 0 3 0 7 3 5 6 ) 4
(T^2 - 1644*T - 10307356)^4
67 67 6 7
( T 4 + ⋯ + 7632038946240 ) 2 (T^{4} + \cdots + 7632038946240)^{2} ( T 4 + ⋯ + 7 6 3 2 0 3 8 9 4 6 2 4 0 ) 2
(T^4 + 11037040*T^2 + 7632038946240)^2
71 71 7 1
( T 4 + ⋯ + 13443377725440 ) 2 (T^{4} + \cdots + 13443377725440)^{2} ( T 4 + ⋯ + 1 3 4 4 3 3 7 7 7 2 5 4 4 0 ) 2
(T^4 + 64765440*T^2 + 13443377725440)^2
73 73 7 3
( T 4 + ⋯ + 11 ⋯ 16 ) 2 (T^{4} + \cdots + 11\!\cdots\!16)^{2} ( T 4 + ⋯ + 1 1 ⋯ 1 6 ) 2
(T^4 - 68179968*T^2 + 1161507835478016)^2
79 79 7 9
( T 4 + ⋯ + 152966937968640 ) 2 (T^{4} + \cdots + 152966937968640)^{2} ( T 4 + ⋯ + 1 5 2 9 6 6 9 3 7 9 6 8 6 4 0 ) 2
(T^4 + 27279360*T^2 + 152966937968640)^2
83 83 8 3
( T 4 + ⋯ + 10 ⋯ 40 ) 2 (T^{4} + \cdots + 10\!\cdots\!40)^{2} ( T 4 + ⋯ + 1 0 ⋯ 4 0 ) 2
(T^4 + 118219120*T^2 + 1012072567842240)^2
89 89 8 9
( T 2 + 6916 T − 62026236 ) 4 (T^{2} + 6916 T - 62026236)^{4} ( T 2 + 6 9 1 6 T − 6 2 0 2 6 2 3 6 ) 4
(T^2 + 6916*T - 62026236)^4
97 97 9 7
( T 4 + ⋯ + 13 ⋯ 76 ) 2 (T^{4} + \cdots + 13\!\cdots\!76)^{2} ( T 4 + ⋯ + 1 3 ⋯ 7 6 ) 2
(T^4 - 86769408*T^2 + 1349748268056576)^2
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