gp: [N,k,chi] = [3276,2,Mod(2521,3276)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3276, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3276.2521");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,-4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == 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 + 16 x 6 + 80 x 4 + 132 x 2 + 64 x^{8} + 16x^{6} + 80x^{4} + 132x^{2} + 64 x 8 + 1 6 x 6 + 8 0 x 4 + 1 3 2 x 2 + 6 4
x^8 + 16*x^6 + 80*x^4 + 132*x^2 + 64
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 4 + 8 ν 2 + 8 ) / 2 ( \nu^{4} + 8\nu^{2} + 8 ) / 2 ( ν 4 + 8 ν 2 + 8 ) / 2
(v^4 + 8*v^2 + 8) / 2
β 3 \beta_{3} β 3 = = =
( − ν 7 − 16 ν 5 − 72 ν 3 − 68 ν ) / 16 ( -\nu^{7} - 16\nu^{5} - 72\nu^{3} - 68\nu ) / 16 ( − ν 7 − 1 6 ν 5 − 7 2 ν 3 − 6 8 ν ) / 1 6
(-v^7 - 16*v^5 - 72*v^3 - 68*v) / 16
β 4 \beta_{4} β 4 = = =
( ν 7 + 16 ν 5 + 80 ν 3 + 116 ν ) / 16 ( \nu^{7} + 16\nu^{5} + 80\nu^{3} + 116\nu ) / 16 ( ν 7 + 1 6 ν 5 + 8 0 ν 3 + 1 1 6 ν ) / 1 6
(v^7 + 16*v^5 + 80*v^3 + 116*v) / 16
β 5 \beta_{5} β 5 = = =
( − ν 6 − 14 ν 4 − 52 ν 2 − 36 ) / 4 ( -\nu^{6} - 14\nu^{4} - 52\nu^{2} - 36 ) / 4 ( − ν 6 − 1 4 ν 4 − 5 2 ν 2 − 3 6 ) / 4
(-v^6 - 14*v^4 - 52*v^2 - 36) / 4
β 6 \beta_{6} β 6 = = =
( 3 ν 7 + 40 ν 5 + 144 ν 3 + 92 ν ) / 16 ( 3\nu^{7} + 40\nu^{5} + 144\nu^{3} + 92\nu ) / 16 ( 3 ν 7 + 4 0 ν 5 + 1 4 4 ν 3 + 9 2 ν ) / 1 6
(3*v^7 + 40*v^5 + 144*v^3 + 92*v) / 16
β 7 \beta_{7} β 7 = = =
( ν 6 + 14 ν 4 + 56 ν 2 + 52 ) / 4 ( \nu^{6} + 14\nu^{4} + 56\nu^{2} + 52 ) / 4 ( ν 6 + 1 4 ν 4 + 5 6 ν 2 + 5 2 ) / 4
(v^6 + 14*v^4 + 56*v^2 + 52) / 4
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 7 + β 5 − 4 \beta_{7} + \beta_{5} - 4 β 7 + β 5 − 4
b7 + b5 - 4
ν 3 \nu^{3} ν 3 = = =
2 β 4 + 2 β 3 − 6 β 1 2\beta_{4} + 2\beta_{3} - 6\beta_1 2 β 4 + 2 β 3 − 6 β 1
2*b4 + 2*b3 - 6*b1
ν 4 \nu^{4} ν 4 = = =
− 8 β 7 − 8 β 5 + 2 β 2 + 24 -8\beta_{7} - 8\beta_{5} + 2\beta_{2} + 24 − 8 β 7 − 8 β 5 + 2 β 2 + 2 4
-8*b7 - 8*b5 + 2*b2 + 24
ν 5 \nu^{5} ν 5 = = =
− 2 β 6 − 18 β 4 − 24 β 3 + 40 β 1 -2\beta_{6} - 18\beta_{4} - 24\beta_{3} + 40\beta_1 − 2 β 6 − 1 8 β 4 − 2 4 β 3 + 4 0 β 1
-2*b6 - 18*b4 - 24*b3 + 40*b1
ν 6 \nu^{6} ν 6 = = =
60 β 7 + 56 β 5 − 28 β 2 − 164 60\beta_{7} + 56\beta_{5} - 28\beta_{2} - 164 6 0 β 7 + 5 6 β 5 − 2 8 β 2 − 1 6 4
60*b7 + 56*b5 - 28*b2 - 164
ν 7 \nu^{7} ν 7 = = =
32 β 6 + 144 β 4 + 224 β 3 − 276 β 1 32\beta_{6} + 144\beta_{4} + 224\beta_{3} - 276\beta_1 3 2 β 6 + 1 4 4 β 4 + 2 2 4 β 3 − 2 7 6 β 1
32*b6 + 144*b4 + 224*b3 - 276*b1
Character values
We give the values of χ \chi χ on generators for ( Z / 3276 Z ) × \left(\mathbb{Z}/3276\mathbb{Z}\right)^\times ( Z / 3 2 7 6 Z ) × .
n n n
1639 1639 1 6 3 9
2017 2017 2 0 1 7
2341 2341 2 3 4 1
2549 2549 2 5 4 9
χ ( n ) \chi(n) χ ( n )
1 1 1
− 1 -1 − 1
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 kernel of the linear operator
T 5 8 + 19 T 5 6 + 119 T 5 4 + 249 T 5 2 + 16 T_{5}^{8} + 19T_{5}^{6} + 119T_{5}^{4} + 249T_{5}^{2} + 16 T 5 8 + 1 9 T 5 6 + 1 1 9 T 5 4 + 2 4 9 T 5 2 + 1 6
T5^8 + 19*T5^6 + 119*T5^4 + 249*T5^2 + 16
acting on S 2 n e w ( 3276 , [ χ ] ) S_{2}^{\mathrm{new}}(3276, [\chi]) S 2 n e w ( 3 2 7 6 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 8 T^{8} T 8
T^8
3 3 3
T 8 T^{8} T 8
T^8
5 5 5
T 8 + 19 T 6 + ⋯ + 16 T^{8} + 19 T^{6} + \cdots + 16 T 8 + 1 9 T 6 + ⋯ + 1 6
T^8 + 19*T^6 + 119*T^4 + 249*T^2 + 16
7 7 7
( T 2 + 1 ) 4 (T^{2} + 1)^{4} ( T 2 + 1 ) 4
(T^2 + 1)^4
11 11 1 1
T 8 + 36 T 6 + ⋯ + 256 T^{8} + 36 T^{6} + \cdots + 256 T 8 + 3 6 T 6 + ⋯ + 2 5 6
T^8 + 36*T^6 + 356*T^4 + 580*T^2 + 256
13 13 1 3
T 8 − 2 T 7 + ⋯ + 28561 T^{8} - 2 T^{7} + \cdots + 28561 T 8 − 2 T 7 + ⋯ + 2 8 5 6 1
T^8 - 2*T^7 + 20*T^6 - 94*T^5 + 278*T^4 - 1222*T^3 + 3380*T^2 - 4394*T + 28561
17 17 1 7
( T 4 + 2 T 3 − 52 T 2 + ⋯ + 44 ) 2 (T^{4} + 2 T^{3} - 52 T^{2} + \cdots + 44)^{2} ( T 4 + 2 T 3 − 5 2 T 2 + ⋯ + 4 4 ) 2
(T^4 + 2*T^3 - 52*T^2 - 78*T + 44)^2
19 19 1 9
T 8 + 71 T 6 + ⋯ + 30976 T^{8} + 71 T^{6} + \cdots + 30976 T 8 + 7 1 T 6 + ⋯ + 3 0 9 7 6
T^8 + 71*T^6 + 1515*T^4 + 11889*T^2 + 30976
23 23 2 3
( T 4 − 3 T 3 − 13 T 2 + ⋯ − 36 ) 2 (T^{4} - 3 T^{3} - 13 T^{2} + \cdots - 36)^{2} ( T 4 − 3 T 3 − 1 3 T 2 + ⋯ − 3 6 ) 2
(T^4 - 3*T^3 - 13*T^2 + 47*T - 36)^2
29 29 2 9
( T 4 − T 3 − 83 T 2 + ⋯ + 1182 ) 2 (T^{4} - T^{3} - 83 T^{2} + \cdots + 1182)^{2} ( T 4 − T 3 − 8 3 T 2 + ⋯ + 1 1 8 2 ) 2
(T^4 - T^3 - 83*T^2 + 85*T + 1182)^2
31 31 3 1
T 8 + 123 T 6 + ⋯ + 322624 T^{8} + 123 T^{6} + \cdots + 322624 T 8 + 1 2 3 T 6 + ⋯ + 3 2 2 6 2 4
T^8 + 123*T^6 + 4463*T^4 + 64153*T^2 + 322624
37 37 3 7
T 8 + 212 T 6 + ⋯ + 20736 T^{8} + 212 T^{6} + \cdots + 20736 T 8 + 2 1 2 T 6 + ⋯ + 2 0 7 3 6
T^8 + 212*T^6 + 10948*T^4 + 164836*T^2 + 20736
41 41 4 1
T 8 + 156 T 6 + ⋯ + 123904 T^{8} + 156 T^{6} + \cdots + 123904 T 8 + 1 5 6 T 6 + ⋯ + 1 2 3 9 0 4
T^8 + 156*T^6 + 7920*T^4 + 133952*T^2 + 123904
43 43 4 3
( T 4 + 3 T 3 + ⋯ + 232 ) 2 (T^{4} + 3 T^{3} + \cdots + 232)^{2} ( T 4 + 3 T 3 + ⋯ + 2 3 2 ) 2
(T^4 + 3*T^3 - 85*T^2 - 367*T + 232)^2
47 47 4 7
T 8 + 239 T 6 + ⋯ + 4032064 T^{8} + 239 T^{6} + \cdots + 4032064 T 8 + 2 3 9 T 6 + ⋯ + 4 0 3 2 0 6 4
T^8 + 239*T^6 + 19227*T^4 + 571401*T^2 + 4032064
53 53 5 3
( T 4 + 11 T 3 + ⋯ − 54 ) 2 (T^{4} + 11 T^{3} + \cdots - 54)^{2} ( T 4 + 1 1 T 3 + ⋯ − 5 4 ) 2
(T^4 + 11*T^3 - 3*T^2 - 63*T - 54)^2
59 59 5 9
T 8 + 180 T 6 + ⋯ + 147456 T^{8} + 180 T^{6} + \cdots + 147456 T 8 + 1 8 0 T 6 + ⋯ + 1 4 7 4 5 6
T^8 + 180*T^6 + 5744*T^4 + 53008*T^2 + 147456
61 61 6 1
( T 4 − 4 T 3 − 112 T 2 + ⋯ − 16 ) 2 (T^{4} - 4 T^{3} - 112 T^{2} + \cdots - 16)^{2} ( T 4 − 4 T 3 − 1 1 2 T 2 + ⋯ − 1 6 ) 2
(T^4 - 4*T^3 - 112*T^2 - 336*T - 16)^2
67 67 6 7
T 8 + 368 T 6 + ⋯ + 11943936 T^{8} + 368 T^{6} + \cdots + 11943936 T 8 + 3 6 8 T 6 + ⋯ + 1 1 9 4 3 9 3 6
T^8 + 368*T^6 + 40576*T^4 + 1345792*T^2 + 11943936
71 71 7 1
T 8 + 408 T 6 + ⋯ + 2214144 T^{8} + 408 T^{6} + \cdots + 2214144 T 8 + 4 0 8 T 6 + ⋯ + 2 2 1 4 1 4 4
T^8 + 408*T^6 + 43580*T^4 + 701284*T^2 + 2214144
73 73 7 3
T 8 + 279 T 6 + ⋯ + 2421136 T^{8} + 279 T^{6} + \cdots + 2421136 T 8 + 2 7 9 T 6 + ⋯ + 2 4 2 1 1 3 6
T^8 + 279*T^6 + 18539*T^4 + 398689*T^2 + 2421136
79 79 7 9
( T 4 + 13 T 3 + ⋯ − 36 ) 2 (T^{4} + 13 T^{3} + \cdots - 36)^{2} ( T 4 + 1 3 T 3 + ⋯ − 3 6 ) 2
(T^4 + 13*T^3 - 97*T^2 - 377*T - 36)^2
83 83 8 3
T 8 + 547 T 6 + ⋯ + 75481344 T^{8} + 547 T^{6} + \cdots + 75481344 T 8 + 5 4 7 T 6 + ⋯ + 7 5 4 8 1 3 4 4
T^8 + 547*T^6 + 94823*T^4 + 5809657*T^2 + 75481344
89 89 8 9
T 8 + 439 T 6 + ⋯ + 57335184 T^{8} + 439 T^{6} + \cdots + 57335184 T 8 + 4 3 9 T 6 + ⋯ + 5 7 3 3 5 1 8 4
T^8 + 439*T^6 + 67043*T^4 + 3935161*T^2 + 57335184
97 97 9 7
T 8 + 447 T 6 + ⋯ + 75134224 T^{8} + 447 T^{6} + \cdots + 75134224 T 8 + 4 4 7 T 6 + ⋯ + 7 5 1 3 4 2 2 4
T^8 + 447*T^6 + 66099*T^4 + 3864161*T^2 + 75134224
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