gp: [N,k,chi] = [50,26,Mod(49,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 26, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.49");
S:= CuspForms(chi, 26);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,-33554432,0,1334181888]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == 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 β = 2 i \beta = 2i β = 2 i .
We also show the integral q q q -expansion of the trace form .
Character values
We give the values of χ \chi χ on generators for ( Z / 50 Z ) × \left(\mathbb{Z}/50\mathbb{Z}\right)^\times ( Z / 5 0 Z ) × .
n n n
27 27 2 7
χ ( n ) \chi(n) χ ( n )
− 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 2 + 26524682496 T_{3}^{2} + 26524682496 T 3 2 + 2 6 5 2 4 6 8 2 4 9 6
T3^2 + 26524682496
acting on S 26 n e w ( 50 , [ χ ] ) S_{26}^{\mathrm{new}}(50, [\chi]) S 2 6 n e w ( 5 0 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 + 16777216 T^{2} + 16777216 T 2 + 1 6 7 7 7 2 1 6
T^2 + 16777216
3 3 3
T 2 + 26524682496 T^{2} + 26524682496 T 2 + 2 6 5 2 4 6 8 2 4 9 6
T^2 + 26524682496
5 5 5
T 2 T^{2} T 2
T^2
7 7 7
T 2 + 30 ⋯ 64 T^{2} + 30\!\cdots\!64 T 2 + 3 0 ⋯ 6 4
T^2 + 309791451716727954064
11 11 1 1
( T + 11240373835548 ) 2 (T + 11240373835548)^{2} ( T + 1 1 2 4 0 3 7 3 8 3 5 5 4 8 ) 2
(T + 11240373835548)^2
13 13 1 3
T 2 + 17 ⋯ 56 T^{2} + 17\!\cdots\!56 T 2 + 1 7 ⋯ 5 6
T^2 + 17996336649330746219597956
17 17 1 7
T 2 + 12 ⋯ 04 T^{2} + 12\!\cdots\!04 T 2 + 1 2 ⋯ 0 4
T^2 + 1294716159973548316744920377604
19 19 1 9
( T + 29 ⋯ 20 ) 2 (T + 29\!\cdots\!20)^{2} ( T + 2 9 ⋯ 2 0 ) 2
(T + 2989845386727620)^2
23 23 2 3
T 2 + 14 ⋯ 16 T^{2} + 14\!\cdots\!16 T 2 + 1 4 ⋯ 1 6
T^2 + 14958624406981159138219793278889616
29 29 2 9
( T − 21 ⋯ 70 ) 2 (T - 21\!\cdots\!70)^{2} ( T − 2 1 ⋯ 7 0 ) 2
(T - 2120475579683207970)^2
31 31 3 1
( T − 42 ⋯ 52 ) 2 (T - 42\!\cdots\!52)^{2} ( T − 4 2 ⋯ 5 2 ) 2
(T - 4225863091688971352)^2
37 37 3 7
T 2 + 58 ⋯ 84 T^{2} + 58\!\cdots\!84 T 2 + 5 8 ⋯ 8 4
T^2 + 584086276404138800946434161413514488484
41 41 4 1
( T + 16 ⋯ 98 ) 2 (T + 16\!\cdots\!98)^{2} ( T + 1 6 ⋯ 9 8 ) 2
(T + 161564320765298755398)^2
43 43 4 3
T 2 + 97 ⋯ 36 T^{2} + 97\!\cdots\!36 T 2 + 9 7 ⋯ 3 6
T^2 + 97102211623464890411205192355620672195136
47 47 4 7
T 2 + 15 ⋯ 24 T^{2} + 15\!\cdots\!24 T 2 + 1 5 ⋯ 2 4
T^2 + 1502726117337555608367166305668933612839824
53 53 5 3
T 2 + 22 ⋯ 96 T^{2} + 22\!\cdots\!96 T 2 + 2 2 ⋯ 9 6
T^2 + 229360036645829603211392358925938927768996
59 59 5 9
( T + 12 ⋯ 60 ) 2 (T + 12\!\cdots\!60)^{2} ( T + 1 2 ⋯ 6 0 ) 2
(T + 12143384628398397940860)^2
61 61 6 1
( T + 20 ⋯ 98 ) 2 (T + 20\!\cdots\!98)^{2} ( T + 2 0 ⋯ 9 8 ) 2
(T + 20332807034763696982498)^2
67 67 6 7
T 2 + 37 ⋯ 04 T^{2} + 37\!\cdots\!04 T 2 + 3 7 ⋯ 0 4
T^2 + 3790280364287986831312403304882771865003722304
71 71 7 1
( T − 13 ⋯ 52 ) 2 (T - 13\!\cdots\!52)^{2} ( T − 1 3 ⋯ 5 2 ) 2
(T - 134076626495447231398752)^2
73 73 7 3
T 2 + 50 ⋯ 16 T^{2} + 50\!\cdots\!16 T 2 + 5 0 ⋯ 1 6
T^2 + 50076878117600822782762621022406699290015649316
79 79 7 9
( T + 38 ⋯ 80 ) 2 (T + 38\!\cdots\!80)^{2} ( T + 3 8 ⋯ 8 0 ) 2
(T + 382349749262069041023680)^2
83 83 8 3
T 2 + 59 ⋯ 76 T^{2} + 59\!\cdots\!76 T 2 + 5 9 ⋯ 7 6
T^2 + 595703475598599030820255479178076650016616293376
89 89 8 9
( T − 84 ⋯ 10 ) 2 (T - 84\!\cdots\!10)^{2} ( T − 8 4 ⋯ 1 0 ) 2
(T - 840580682623068403216710)^2
97 97 9 7
T 2 + 38 ⋯ 24 T^{2} + 38\!\cdots\!24 T 2 + 3 8 ⋯ 2 4
T^2 + 38336420721249876567536939990482627370088985513924
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