gp: [N,k,chi] = [4608,2,Mod(1153,4608)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4608, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 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("4608.1153");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,0,0,-8,0,0,0,0,0,0,0,-8,0,0,0,0,0,16]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == 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
β 1 \beta_{1} β 1 = = =
2 ζ 16 2 2\zeta_{16}^{2} 2 ζ 1 6 2
2*v^2
β 2 \beta_{2} β 2 = = =
2 ζ 16 3 2\zeta_{16}^{3} 2 ζ 1 6 3
2*v^3
β 3 \beta_{3} β 3 = = =
ζ 16 4 \zeta_{16}^{4} ζ 1 6 4
v^4
β 4 \beta_{4} β 4 = = =
2 ζ 16 6 2\zeta_{16}^{6} 2 ζ 1 6 6
2*v^6
β 5 \beta_{5} β 5 = = =
ζ 16 7 + ζ 16 5 + ζ 16 3 + ζ 16 \zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16} ζ 1 6 7 + ζ 1 6 5 + ζ 1 6 3 + ζ 1 6
v^7 + v^5 + v^3 + v
β 6 \beta_{6} β 6 = = =
− ζ 16 7 − ζ 16 5 − ζ 16 3 + ζ 16 -\zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16} − ζ 1 6 7 − ζ 1 6 5 − ζ 1 6 3 + ζ 1 6
-v^7 - v^5 - v^3 + v
β 7 \beta_{7} β 7 = = =
− ζ 16 7 + ζ 16 5 − ζ 16 3 − ζ 16 -\zeta_{16}^{7} + \zeta_{16}^{5} - \zeta_{16}^{3} - \zeta_{16} − ζ 1 6 7 + ζ 1 6 5 − ζ 1 6 3 − ζ 1 6
-v^7 + v^5 - v^3 - v
ζ 16 \zeta_{16} ζ 1 6 = = =
( β 6 + β 5 ) / 2 ( \beta_{6} + \beta_{5} ) / 2 ( β 6 + β 5 ) / 2
(b6 + b5) / 2
ζ 16 2 \zeta_{16}^{2} ζ 1 6 2 = = =
( β 1 ) / 2 ( \beta_1 ) / 2 ( β 1 ) / 2
(b1) / 2
ζ 16 3 \zeta_{16}^{3} ζ 1 6 3 = = =
( β 2 ) / 2 ( \beta_{2} ) / 2 ( β 2 ) / 2
(b2) / 2
ζ 16 4 \zeta_{16}^{4} ζ 1 6 4 = = =
β 3 \beta_{3} β 3
b3
ζ 16 5 \zeta_{16}^{5} ζ 1 6 5 = = =
( β 7 + β 5 ) / 2 ( \beta_{7} + \beta_{5} ) / 2 ( β 7 + β 5 ) / 2
(b7 + b5) / 2
ζ 16 6 \zeta_{16}^{6} ζ 1 6 6 = = =
( β 4 ) / 2 ( \beta_{4} ) / 2 ( β 4 ) / 2
(b4) / 2
ζ 16 7 \zeta_{16}^{7} ζ 1 6 7 = = =
( − β 7 − β 6 − β 2 ) / 2 ( -\beta_{7} - \beta_{6} - \beta_{2} ) / 2 ( − β 7 − β 6 − β 2 ) / 2
(-b7 - b6 - b2) / 2
Character values
We give the values of χ \chi χ on generators for ( Z / 4608 Z ) × \left(\mathbb{Z}/4608\mathbb{Z}\right)^\times ( Z / 4 6 0 8 Z ) × .
n n n
2053 2053 2 0 5 3
3583 3583 3 5 8 3
4097 4097 4 0 9 7
χ ( n ) \chi(n) χ ( n )
β 3 \beta_{3} β 3
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 2 n e w ( 4608 , [ χ ] ) S_{2}^{\mathrm{new}}(4608, [\chi]) S 2 n e w ( 4 6 0 8 , [ χ ] ) :
T 5 8 + 8 T 5 7 + 32 T 5 6 + 48 T 5 5 + 24 T 5 4 − 32 T 5 3 + 128 T 5 2 + 64 T 5 + 16 T_{5}^{8} + 8T_{5}^{7} + 32T_{5}^{6} + 48T_{5}^{5} + 24T_{5}^{4} - 32T_{5}^{3} + 128T_{5}^{2} + 64T_{5} + 16 T 5 8 + 8 T 5 7 + 3 2 T 5 6 + 4 8 T 5 5 + 2 4 T 5 4 − 3 2 T 5 3 + 1 2 8 T 5 2 + 6 4 T 5 + 1 6
T5^8 + 8*T5^7 + 32*T5^6 + 48*T5^5 + 24*T5^4 - 32*T5^3 + 128*T5^2 + 64*T5 + 16
T 7 8 + 32 T 7 6 + 240 T 7 4 + 256 T 7 2 + 64 T_{7}^{8} + 32T_{7}^{6} + 240T_{7}^{4} + 256T_{7}^{2} + 64 T 7 8 + 3 2 T 7 6 + 2 4 0 T 7 4 + 2 5 6 T 7 2 + 6 4
T7^8 + 32*T7^6 + 240*T7^4 + 256*T7^2 + 64
T 11 8 + 192 T 11 4 + 1024 T_{11}^{8} + 192T_{11}^{4} + 1024 T 1 1 8 + 1 9 2 T 1 1 4 + 1 0 2 4
T11^8 + 192*T11^4 + 1024
T 13 8 + 8 T 13 7 + 32 T 13 6 + 48 T 13 5 + 1160 T 13 4 + 9120 T 13 3 + 36992 T 13 2 + 51136 T 13 + 35344 T_{13}^{8} + 8T_{13}^{7} + 32T_{13}^{6} + 48T_{13}^{5} + 1160T_{13}^{4} + 9120T_{13}^{3} + 36992T_{13}^{2} + 51136T_{13} + 35344 T 1 3 8 + 8 T 1 3 7 + 3 2 T 1 3 6 + 4 8 T 1 3 5 + 1 1 6 0 T 1 3 4 + 9 1 2 0 T 1 3 3 + 3 6 9 9 2 T 1 3 2 + 5 1 1 3 6 T 1 3 + 3 5 3 4 4
T13^8 + 8*T13^7 + 32*T13^6 + 48*T13^5 + 1160*T13^4 + 9120*T13^3 + 36992*T13^2 + 51136*T13 + 35344
T 19 8 − 16 T 19 7 + 128 T 19 6 − 512 T 19 5 + 1088 T 19 4 − 512 T 19 3 + 1024 T_{19}^{8} - 16T_{19}^{7} + 128T_{19}^{6} - 512T_{19}^{5} + 1088T_{19}^{4} - 512T_{19}^{3} + 1024 T 1 9 8 − 1 6 T 1 9 7 + 1 2 8 T 1 9 6 − 5 1 2 T 1 9 5 + 1 0 8 8 T 1 9 4 − 5 1 2 T 1 9 3 + 1 0 2 4
T19^8 - 16*T19^7 + 128*T19^6 - 512*T19^5 + 1088*T19^4 - 512*T19^3 + 1024
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 + 8 T 7 + ⋯ + 16 T^{8} + 8 T^{7} + \cdots + 16 T 8 + 8 T 7 + ⋯ + 1 6
T^8 + 8*T^7 + 32*T^6 + 48*T^5 + 24*T^4 - 32*T^3 + 128*T^2 + 64*T + 16
7 7 7
T 8 + 32 T 6 + ⋯ + 64 T^{8} + 32 T^{6} + \cdots + 64 T 8 + 3 2 T 6 + ⋯ + 6 4
T^8 + 32*T^6 + 240*T^4 + 256*T^2 + 64
11 11 1 1
T 8 + 192 T 4 + 1024 T^{8} + 192T^{4} + 1024 T 8 + 1 9 2 T 4 + 1 0 2 4
T^8 + 192*T^4 + 1024
13 13 1 3
T 8 + 8 T 7 + ⋯ + 35344 T^{8} + 8 T^{7} + \cdots + 35344 T 8 + 8 T 7 + ⋯ + 3 5 3 4 4
T^8 + 8*T^7 + 32*T^6 + 48*T^5 + 1160*T^4 + 9120*T^3 + 36992*T^2 + 51136*T + 35344
17 17 1 7
( T 4 − 32 T 2 + ⋯ − 32 ) 2 (T^{4} - 32 T^{2} + \cdots - 32)^{2} ( T 4 − 3 2 T 2 + ⋯ − 3 2 ) 2
(T^4 - 32*T^2 - 64*T - 32)^2
19 19 1 9
T 8 − 16 T 7 + ⋯ + 1024 T^{8} - 16 T^{7} + \cdots + 1024 T 8 − 1 6 T 7 + ⋯ + 1 0 2 4
T^8 - 16*T^7 + 128*T^6 - 512*T^5 + 1088*T^4 - 512*T^3 + 1024
23 23 2 3
( T 2 + 16 ) 4 (T^{2} + 16)^{4} ( T 2 + 1 6 ) 4
(T^2 + 16)^4
29 29 2 9
T 8 + 8 T 7 + ⋯ + 4624 T^{8} + 8 T^{7} + \cdots + 4624 T 8 + 8 T 7 + ⋯ + 4 6 2 4
T^8 + 8*T^7 + 32*T^6 - 80*T^5 + 920*T^4 + 4576*T^3 + 10368*T^2 + 9792*T + 4624
31 31 3 1
( T 4 − 8 T 3 + ⋯ + 248 ) 2 (T^{4} - 8 T^{3} + \cdots + 248)^{2} ( T 4 − 8 T 3 + ⋯ + 2 4 8 ) 2
(T^4 - 8*T^3 - 16*T^2 + 128*T + 248)^2
37 37 3 7
T 8 + 8 T 7 + ⋯ + 150544 T^{8} + 8 T^{7} + \cdots + 150544 T 8 + 8 T 7 + ⋯ + 1 5 0 5 4 4
T^8 + 8*T^7 + 32*T^6 - 464*T^5 + 2824*T^4 - 2144*T^3 + 128*T^2 - 6208*T + 150544
41 41 4 1
T 8 + 192 T 6 + ⋯ + 984064 T^{8} + 192 T^{6} + \cdots + 984064 T 8 + 1 9 2 T 6 + ⋯ + 9 8 4 0 6 4
T^8 + 192*T^6 + 11200*T^4 + 194560*T^2 + 984064
43 43 4 3
T 8 − 16 T 7 + ⋯ + 984064 T^{8} - 16 T^{7} + \cdots + 984064 T 8 − 1 6 T 7 + ⋯ + 9 8 4 0 6 4
T^8 - 16*T^7 + 128*T^6 - 960*T^4 - 512*T^3 + 131072*T^2 + 507904*T + 984064
47 47 4 7
( T 2 − 8 T − 16 ) 4 (T^{2} - 8 T - 16)^{4} ( T 2 − 8 T − 1 6 ) 4
(T^2 - 8*T - 16)^4
53 53 5 3
T 8 − 8 T 7 + ⋯ + 35344 T^{8} - 8 T^{7} + \cdots + 35344 T 8 − 8 T 7 + ⋯ + 3 5 3 4 4
T^8 - 8*T^7 + 32*T^6 + 336*T^5 + 8088*T^4 - 35296*T^3 + 80000*T^2 + 75200*T + 35344
59 59 5 9
T 8 − 32 T 7 + ⋯ + 18939904 T^{8} - 32 T^{7} + \cdots + 18939904 T 8 − 3 2 T 7 + ⋯ + 1 8 9 3 9 9 0 4
T^8 - 32*T^7 + 512*T^6 - 4096*T^5 + 17920*T^4 - 40960*T^3 + 524288*T^2 - 4456448*T + 18939904
61 61 6 1
T 8 + 8 T 7 + ⋯ + 16 T^{8} + 8 T^{7} + \cdots + 16 T 8 + 8 T 7 + ⋯ + 1 6
T^8 + 8*T^7 + 32*T^6 - 1232*T^5 + 15368*T^4 - 29792*T^3 + 28800*T^2 + 960*T + 16
67 67 6 7
T 8 + 32 T 7 + ⋯ + 62980096 T^{8} + 32 T^{7} + \cdots + 62980096 T 8 + 3 2 T 7 + ⋯ + 6 2 9 8 0 0 9 6
T^8 + 32*T^7 + 512*T^6 + 3584*T^5 + 16896*T^4 + 172032*T^3 + 3276800*T^2 + 20316160*T + 62980096
71 71 7 1
T 8 + 256 T 6 + ⋯ + 4734976 T^{8} + 256 T^{6} + \cdots + 4734976 T 8 + 2 5 6 T 6 + ⋯ + 4 7 3 4 9 7 6
T^8 + 256*T^6 + 20224*T^4 + 589824*T^2 + 4734976
73 73 7 3
T 8 + 304 T 6 + ⋯ + 2408704 T^{8} + 304 T^{6} + \cdots + 2408704 T 8 + 3 0 4 T 6 + ⋯ + 2 4 0 8 7 0 4
T^8 + 304*T^6 + 18016*T^4 + 373504*T^2 + 2408704
79 79 7 9
( T 4 + 24 T 3 + ⋯ − 7688 ) 2 (T^{4} + 24 T^{3} + \cdots - 7688)^{2} ( T 4 + 2 4 T 3 + ⋯ − 7 6 8 8 ) 2
(T^4 + 24*T^3 + 80*T^2 - 1344*T - 7688)^2
83 83 8 3
T 8 + 32 T 7 + ⋯ + 295936 T^{8} + 32 T^{7} + \cdots + 295936 T 8 + 3 2 T 7 + ⋯ + 2 9 5 9 3 6
T^8 + 32*T^7 + 512*T^6 + 4864*T^5 + 29888*T^4 + 117760*T^3 + 294912*T^2 + 417792*T + 295936
89 89 8 9
T 8 + 400 T 6 + ⋯ + 73984 T^{8} + 400 T^{6} + \cdots + 73984 T 8 + 4 0 0 T 6 + ⋯ + 7 3 9 8 4
T^8 + 400*T^6 + 24160*T^4 + 174336*T^2 + 73984
97 97 9 7
( T 4 + 16 T 3 + ⋯ − 256 ) 2 (T^{4} + 16 T^{3} + \cdots - 256)^{2} ( T 4 + 1 6 T 3 + ⋯ − 2 5 6 ) 2
(T^4 + 16*T^3 - 32*T^2 - 256*T - 256)^2
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