gp: [N,k,chi] = [7225,2,Mod(1,7225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7225, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7225.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [6,-2,0,2,0,0,0]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 − 17 x 4 + 73 x 2 − 67 x^{6} - 17x^{4} + 73x^{2} - 67 x 6 − 1 7 x 4 + 7 3 x 2 − 6 7
x^6 - 17*x^4 + 73*x^2 - 67
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( 2 ν 4 − 21 ν 2 + 18 ) / 17 ( 2\nu^{4} - 21\nu^{2} + 18 ) / 17 ( 2 ν 4 − 2 1 ν 2 + 1 8 ) / 1 7
(2*v^4 - 21*v^2 + 18) / 17
β 3 \beta_{3} β 3 = = =
( ν 4 − 19 ν 2 + 60 ) / 17 ( \nu^{4} - 19\nu^{2} + 60 ) / 17 ( ν 4 − 1 9 ν 2 + 6 0 ) / 1 7
(v^4 - 19*v^2 + 60) / 17
β 4 \beta_{4} β 4 = = =
( ν 5 − 19 ν 3 + 77 ν ) / 17 ( \nu^{5} - 19\nu^{3} + 77\nu ) / 17 ( ν 5 − 1 9 ν 3 + 7 7 ν ) / 1 7
(v^5 - 19*v^3 + 77*v) / 17
β 5 \beta_{5} β 5 = = =
( 3 ν 5 − 40 ν 3 + 95 ν ) / 17 ( 3\nu^{5} - 40\nu^{3} + 95\nu ) / 17 ( 3 ν 5 − 4 0 ν 3 + 9 5 ν ) / 1 7
(3*v^5 - 40*v^3 + 95*v) / 17
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
− 2 β 3 + β 2 + 6 -2\beta_{3} + \beta_{2} + 6 − 2 β 3 + β 2 + 6
-2*b3 + b2 + 6
ν 3 \nu^{3} ν 3 = = =
β 5 − 3 β 4 + 8 β 1 \beta_{5} - 3\beta_{4} + 8\beta_1 β 5 − 3 β 4 + 8 β 1
b5 - 3*b4 + 8*b1
ν 4 \nu^{4} ν 4 = = =
− 21 β 3 + 19 β 2 + 54 -21\beta_{3} + 19\beta_{2} + 54 − 2 1 β 3 + 1 9 β 2 + 5 4
-21*b3 + 19*b2 + 54
ν 5 \nu^{5} ν 5 = = =
19 β 5 − 40 β 4 + 75 β 1 19\beta_{5} - 40\beta_{4} + 75\beta_1 1 9 β 5 − 4 0 β 4 + 7 5 β 1
19*b5 - 40*b4 + 75*b1
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
p p p
Sign
5 5 5
+ 1 +1 + 1
17 17 1 7
+ 1 +1 + 1
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 2 n e w ( Γ 0 ( 7225 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(7225)) S 2 n e w ( Γ 0 ( 7 2 2 5 ) ) :
T 2 3 + T 2 2 − 3 T 2 − 1 T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1 T 2 3 + T 2 2 − 3 T 2 − 1
T2^3 + T2^2 - 3*T2 - 1
T 3 6 − 17 T 3 4 + 73 T 3 2 − 67 T_{3}^{6} - 17T_{3}^{4} + 73T_{3}^{2} - 67 T 3 6 − 1 7 T 3 4 + 7 3 T 3 2 − 6 7
T3^6 - 17*T3^4 + 73*T3^2 - 67
T 7 6 − 19 T 7 4 + 83 T 7 2 − 67 T_{7}^{6} - 19T_{7}^{4} + 83T_{7}^{2} - 67 T 7 6 − 1 9 T 7 4 + 8 3 T 7 2 − 6 7
T7^6 - 19*T7^4 + 83*T7^2 - 67
T 11 6 − 66 T 11 4 + 1292 T 11 2 − 6700 T_{11}^{6} - 66T_{11}^{4} + 1292T_{11}^{2} - 6700 T 1 1 6 − 6 6 T 1 1 4 + 1 2 9 2 T 1 1 2 − 6 7 0 0
T11^6 - 66*T11^4 + 1292*T11^2 - 6700
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 3 + T 2 − 3 T − 1 ) 2 (T^{3} + T^{2} - 3 T - 1)^{2} ( T 3 + T 2 − 3 T − 1 ) 2
(T^3 + T^2 - 3*T - 1)^2
3 3 3
T 6 − 17 T 4 + ⋯ − 67 T^{6} - 17 T^{4} + \cdots - 67 T 6 − 1 7 T 4 + ⋯ − 6 7
T^6 - 17*T^4 + 73*T^2 - 67
5 5 5
T 6 T^{6} T 6
T^6
7 7 7
T 6 − 19 T 4 + ⋯ − 67 T^{6} - 19 T^{4} + \cdots - 67 T 6 − 1 9 T 4 + ⋯ − 6 7
T^6 - 19*T^4 + 83*T^2 - 67
11 11 1 1
T 6 − 66 T 4 + ⋯ − 6700 T^{6} - 66 T^{4} + \cdots - 6700 T 6 − 6 6 T 4 + ⋯ − 6 7 0 0
T^6 - 66*T^4 + 1292*T^2 - 6700
13 13 1 3
( T 3 + T 2 − 13 T − 23 ) 2 (T^{3} + T^{2} - 13 T - 23)^{2} ( T 3 + T 2 − 1 3 T − 2 3 ) 2
(T^3 + T^2 - 13*T - 23)^2
17 17 1 7
T 6 T^{6} T 6
T^6
19 19 1 9
( T 3 + 6 T 2 + ⋯ − 100 ) 2 (T^{3} + 6 T^{2} + \cdots - 100)^{2} ( T 3 + 6 T 2 + ⋯ − 1 0 0 ) 2
(T^3 + 6*T^2 - 16*T - 100)^2
23 23 2 3
T 6 − 54 T 4 + ⋯ − 268 T^{6} - 54 T^{4} + \cdots - 268 T 6 − 5 4 T 4 + ⋯ − 2 6 8
T^6 - 54*T^4 + 236*T^2 - 268
29 29 2 9
T 6 − 116 T 4 + ⋯ − 1072 T^{6} - 116 T^{4} + \cdots - 1072 T 6 − 1 1 6 T 4 + ⋯ − 1 0 7 2
T^6 - 116*T^4 + 2632*T^2 - 1072
31 31 3 1
T 6 − 89 T 4 + ⋯ − 1675 T^{6} - 89 T^{4} + \cdots - 1675 T 6 − 8 9 T 4 + ⋯ − 1 6 7 5
T^6 - 89*T^4 + 1377*T^2 - 1675
37 37 3 7
T 6 − 144 T 4 + ⋯ − 1072 T^{6} - 144 T^{4} + \cdots - 1072 T 6 − 1 4 4 T 4 + ⋯ − 1 0 7 2
T^6 - 144*T^4 + 4088*T^2 - 1072
41 41 4 1
T 6 − 100 T 4 + ⋯ − 26800 T^{6} - 100 T^{4} + \cdots - 26800 T 6 − 1 0 0 T 4 + ⋯ − 2 6 8 0 0
T^6 - 100*T^4 + 3032*T^2 - 26800
43 43 4 3
( T 3 + 4 T 2 + ⋯ − 452 ) 2 (T^{3} + 4 T^{2} + \cdots - 452)^{2} ( T 3 + 4 T 2 + ⋯ − 4 5 2 ) 2
(T^3 + 4*T^2 - 88*T - 452)^2
47 47 4 7
( T 3 + 20 T 2 + ⋯ + 208 ) 2 (T^{3} + 20 T^{2} + \cdots + 208)^{2} ( T 3 + 2 0 T 2 + ⋯ + 2 0 8 ) 2
(T^3 + 20*T^2 + 120*T + 208)^2
53 53 5 3
( T 3 + 5 T 2 − 69 T + 43 ) 2 (T^{3} + 5 T^{2} - 69 T + 43)^{2} ( T 3 + 5 T 2 − 6 9 T + 4 3 ) 2
(T^3 + 5*T^2 - 69*T + 43)^2
59 59 5 9
( T 3 − 2 T 2 − 44 T + 20 ) 2 (T^{3} - 2 T^{2} - 44 T + 20)^{2} ( T 3 − 2 T 2 − 4 4 T + 2 0 ) 2
(T^3 - 2*T^2 - 44*T + 20)^2
61 61 6 1
T 6 − 228 T 4 + ⋯ − 107200 T^{6} - 228 T^{4} + \cdots - 107200 T 6 − 2 2 8 T 4 + ⋯ − 1 0 7 2 0 0
T^6 - 228*T^4 + 13008*T^2 - 107200
67 67 6 7
( T 3 + 12 T 2 + ⋯ − 16 ) 2 (T^{3} + 12 T^{2} + \cdots - 16)^{2} ( T 3 + 1 2 T 2 + ⋯ − 1 6 ) 2
(T^3 + 12*T^2 + 32*T - 16)^2
71 71 7 1
T 6 − 91 T 4 + ⋯ − 1675 T^{6} - 91 T^{4} + \cdots - 1675 T 6 − 9 1 T 4 + ⋯ − 1 6 7 5
T^6 - 91*T^4 + 1067*T^2 - 1675
73 73 7 3
T 6 − 32 T 4 + ⋯ − 1072 T^{6} - 32 T^{4} + \cdots - 1072 T 6 − 3 2 T 4 + ⋯ − 1 0 7 2
T^6 - 32*T^4 + 328*T^2 - 1072
79 79 7 9
T 6 − 135 T 4 + ⋯ − 67 T^{6} - 135 T^{4} + \cdots - 67 T 6 − 1 3 5 T 4 + ⋯ − 6 7
T^6 - 135*T^4 + 491*T^2 - 67
83 83 8 3
( T 3 + 10 T 2 + ⋯ + 184 ) 2 (T^{3} + 10 T^{2} + \cdots + 184)^{2} ( T 3 + 1 0 T 2 + ⋯ + 1 8 4 ) 2
(T^3 + 10*T^2 - 100*T + 184)^2
89 89 8 9
( T 3 + 4 T 2 − 4 T − 20 ) 2 (T^{3} + 4 T^{2} - 4 T - 20)^{2} ( T 3 + 4 T 2 − 4 T − 2 0 ) 2
(T^3 + 4*T^2 - 4*T - 20)^2
97 97 9 7
T 6 − 332 T 4 + ⋯ − 1239232 T^{6} - 332 T^{4} + \cdots - 1239232 T 6 − 3 3 2 T 4 + ⋯ − 1 2 3 9 2 3 2
T^6 - 332*T^4 + 35760*T^2 - 1239232
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