gp: [N,k,chi] = [504,2,Mod(293,504)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(504, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 5, 3]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("504.293");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [16]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 15 1,\beta_1,\ldots,\beta_{15} 1 , β 1 , … , β 1 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 16 + 10 x 12 + 19 x 8 + 810 x 4 + 6561 x^{16} + 10x^{12} + 19x^{8} + 810x^{4} + 6561 x 1 6 + 1 0 x 1 2 + 1 9 x 8 + 8 1 0 x 4 + 6 5 6 1
x^16 + 10*x^12 + 19*x^8 + 810*x^4 + 6561
:
β 1 \beta_{1} β 1 = = =
( 10 ν 13 + 19 ν 9 − 1349 ν 5 + 6561 ν ) / 9234 ( 10\nu^{13} + 19\nu^{9} - 1349\nu^{5} + 6561\nu ) / 9234 ( 1 0 ν 1 3 + 1 9 ν 9 − 1 3 4 9 ν 5 + 6 5 6 1 ν ) / 9 2 3 4
(10*v^13 + 19*v^9 - 1349*v^5 + 6561*v) / 9234
β 2 \beta_{2} β 2 = = =
( 10 ν 14 + 19 ν 10 − 1349 ν 6 + 6561 ν 2 ) / 27702 ( 10\nu^{14} + 19\nu^{10} - 1349\nu^{6} + 6561\nu^{2} ) / 27702 ( 1 0 ν 1 4 + 1 9 ν 1 0 − 1 3 4 9 ν 6 + 6 5 6 1 ν 2 ) / 2 7 7 0 2
(10*v^14 + 19*v^10 - 1349*v^6 + 6561*v^2) / 27702
β 3 \beta_{3} β 3 = = =
( − 11 ν 13 + 133 ν 9 − 209 ν 5 + 324 ν ) / 9234 ( -11\nu^{13} + 133\nu^{9} - 209\nu^{5} + 324\nu ) / 9234 ( − 1 1 ν 1 3 + 1 3 3 ν 9 − 2 0 9 ν 5 + 3 2 4 ν ) / 9 2 3 4
(-11*v^13 + 133*v^9 - 209*v^5 + 324*v) / 9234
β 4 \beta_{4} β 4 = = =
( 10 ν 12 + 19 ν 8 + 190 ν 4 + 8100 ) / 1539 ( 10\nu^{12} + 19\nu^{8} + 190\nu^{4} + 8100 ) / 1539 ( 1 0 ν 1 2 + 1 9 ν 8 + 1 9 0 ν 4 + 8 1 0 0 ) / 1 5 3 9
(10*v^12 + 19*v^8 + 190*v^4 + 8100) / 1539
β 5 \beta_{5} β 5 = = =
( − 31 ν 12 + 95 ν 8 − 589 ν 4 − 25110 ) / 3078 ( -31\nu^{12} + 95\nu^{8} - 589\nu^{4} - 25110 ) / 3078 ( − 3 1 ν 1 2 + 9 5 ν 8 − 5 8 9 ν 4 − 2 5 1 1 0 ) / 3 0 7 8
(-31*v^12 + 95*v^8 - 589*v^4 - 25110) / 3078
β 6 \beta_{6} β 6 = = =
( − ν 15 − 10 ν 11 − 19 ν 7 − 810 ν 3 ) / 2187 ( -\nu^{15} - 10\nu^{11} - 19\nu^{7} - 810\nu^{3} ) / 2187 ( − ν 1 5 − 1 0 ν 1 1 − 1 9 ν 7 − 8 1 0 ν 3 ) / 2 1 8 7
(-v^15 - 10*v^11 - 19*v^7 - 810*v^3) / 2187
β 7 \beta_{7} β 7 = = =
( 50 ν 12 + 95 ν 8 − 589 ν 4 + 32805 ) / 3078 ( 50\nu^{12} + 95\nu^{8} - 589\nu^{4} + 32805 ) / 3078 ( 5 0 ν 1 2 + 9 5 ν 8 − 5 8 9 ν 4 + 3 2 8 0 5 ) / 3 0 7 8
(50*v^12 + 95*v^8 - 589*v^4 + 32805) / 3078
β 8 \beta_{8} β 8 = = =
( − 70 ν 13 − 133 ν 9 + 209 ν 5 − 36693 ν ) / 9234 ( -70\nu^{13} - 133\nu^{9} + 209\nu^{5} - 36693\nu ) / 9234 ( − 7 0 ν 1 3 − 1 3 3 ν 9 + 2 0 9 ν 5 − 3 6 6 9 3 ν ) / 9 2 3 4
(-70*v^13 - 133*v^9 + 209*v^5 - 36693*v) / 9234
β 9 \beta_{9} β 9 = = =
( − ν 15 − 1133 ν 3 + 1026 ν ) / 1026 ( -\nu^{15} - 1133\nu^{3} + 1026\nu ) / 1026 ( − ν 1 5 − 1 1 3 3 ν 3 + 1 0 2 6 ν ) / 1 0 2 6
(-v^15 - 1133*v^3 + 1026*v) / 1026
β 10 \beta_{10} β 1 0 = = =
( ν 13 + 791 ν ) / 114 ( \nu^{13} + 791\nu ) / 114 ( ν 1 3 + 7 9 1 ν ) / 1 1 4
(v^13 + 791*v) / 114
β 11 \beta_{11} β 1 1 = = =
( ν 14 + 791 ν 2 ) / 342 ( \nu^{14} + 791\nu^{2} ) / 342 ( ν 1 4 + 7 9 1 ν 2 ) / 3 4 2
(v^14 + 791*v^2) / 342
β 12 \beta_{12} β 1 2 = = =
( − ν 14 − 449 ν 2 ) / 342 ( -\nu^{14} - 449\nu^{2} ) / 342 ( − ν 1 4 − 4 4 9 ν 2 ) / 3 4 2
(-v^14 - 449*v^2) / 342
β 13 \beta_{13} β 1 3 = = =
( 109 ν 14 + 361 ν 10 + 2071 ν 6 + 88290 ν 2 ) / 27702 ( 109\nu^{14} + 361\nu^{10} + 2071\nu^{6} + 88290\nu^{2} ) / 27702 ( 1 0 9 ν 1 4 + 3 6 1 ν 1 0 + 2 0 7 1 ν 6 + 8 8 2 9 0 ν 2 ) / 2 7 7 0 2
(109*v^14 + 361*v^10 + 2071*v^6 + 88290*v^2) / 27702
β 14 \beta_{14} β 1 4 = = =
( − ν 15 − 620 ν 3 ) / 513 ( -\nu^{15} - 620\nu^{3} ) / 513 ( − ν 1 5 − 6 2 0 ν 3 ) / 5 1 3
(-v^15 - 620*v^3) / 513
β 15 \beta_{15} β 1 5 = = =
( 170 ν 15 + 323 ν 11 + 4769 ν 7 + 111537 ν 3 − 83106 ν ) / 83106 ( 170\nu^{15} + 323\nu^{11} + 4769\nu^{7} + 111537\nu^{3} - 83106\nu ) / 83106 ( 1 7 0 ν 1 5 + 3 2 3 ν 1 1 + 4 7 6 9 ν 7 + 1 1 1 5 3 7 ν 3 − 8 3 1 0 6 ν ) / 8 3 1 0 6
(170*v^15 + 323*v^11 + 4769*v^7 + 111537*v^3 - 83106*v) / 83106
ν \nu ν = = =
( β 10 + β 8 + β 3 ) / 3 ( \beta_{10} + \beta_{8} + \beta_{3} ) / 3 ( β 1 0 + β 8 + β 3 ) / 3
(b10 + b8 + b3) / 3
ν 2 \nu^{2} ν 2 = = =
β 12 + β 11 \beta_{12} + \beta_{11} β 1 2 + β 1 1
b12 + b11
ν 3 \nu^{3} ν 3 = = =
( 3 β 14 + 2 β 10 − 6 β 9 + 2 β 8 + 2 β 3 ) / 3 ( 3\beta_{14} + 2\beta_{10} - 6\beta_{9} + 2\beta_{8} + 2\beta_{3} ) / 3 ( 3 β 1 4 + 2 β 1 0 − 6 β 9 + 2 β 8 + 2 β 3 ) / 3
(3*b14 + 2*b10 - 6*b9 + 2*b8 + 2*b3) / 3
ν 4 \nu^{4} ν 4 = = =
− 2 β 7 + 5 β 4 − 5 -2\beta_{7} + 5\beta_{4} - 5 − 2 β 7 + 5 β 4 − 5
-2*b7 + 5*b4 - 5
ν 5 \nu^{5} ν 5 = = =
( β 10 − 2 β 8 + β 3 − 21 β 1 ) / 3 ( \beta_{10} - 2\beta_{8} + \beta_{3} - 21\beta_1 ) / 3 ( β 1 0 − 2 β 8 + β 3 − 2 1 β 1 ) / 3
(b10 - 2*b8 + b3 - 21*b1) / 3
ν 6 \nu^{6} ν 6 = = =
β 13 − β 12 − 19 β 2 \beta_{13} - \beta_{12} - 19\beta_{2} β 1 3 − β 1 2 − 1 9 β 2
b13 - b12 - 19*b2
ν 7 \nu^{7} ν 7 = = =
( 60 β 15 + 51 β 14 + 20 β 10 + 20 β 8 + 51 β 6 + 20 β 3 ) / 3 ( 60\beta_{15} + 51\beta_{14} + 20\beta_{10} + 20\beta_{8} + 51\beta_{6} + 20\beta_{3} ) / 3 ( 6 0 β 1 5 + 5 1 β 1 4 + 2 0 β 1 0 + 2 0 β 8 + 5 1 β 6 + 2 0 β 3 ) / 3
(60*b15 + 51*b14 + 20*b10 + 20*b8 + 51*b6 + 20*b3) / 3
ν 8 \nu^{8} ν 8 = = =
20 β 5 + 31 β 4 20\beta_{5} + 31\beta_{4} 2 0 β 5 + 3 1 β 4
20*b5 + 31*b4
ν 9 \nu^{9} ν 9 = = =
( − 38 β 10 − 71 β 8 + 142 β 3 − 33 β 1 ) / 3 ( -38\beta_{10} - 71\beta_{8} + 142\beta_{3} - 33\beta_1 ) / 3 ( − 3 8 β 1 0 − 7 1 β 8 + 1 4 2 β 3 − 3 3 β 1 ) / 3
(-38*b10 - 71*b8 + 142*b3 - 33*b1) / 3
ν 10 \nu^{10} ν 1 0 = = =
71 β 13 − 109 β 11 + 109 β 2 71\beta_{13} - 109\beta_{11} + 109\beta_{2} 7 1 β 1 3 − 1 0 9 β 1 1 + 1 0 9 β 2
71*b13 - 109*b11 + 109*b2
ν 11 \nu^{11} ν 1 1 = = =
( − 114 β 15 − 76 β 10 + 114 β 9 − 76 β 8 − 753 β 6 − 76 β 3 ) / 3 ( -114\beta_{15} - 76\beta_{10} + 114\beta_{9} - 76\beta_{8} - 753\beta_{6} - 76\beta_{3} ) / 3 ( − 1 1 4 β 1 5 − 7 6 β 1 0 + 1 1 4 β 9 − 7 6 β 8 − 7 5 3 β 6 − 7 6 β 3 ) / 3
(-114*b15 - 76*b10 + 114*b9 - 76*b8 - 753*b6 - 76*b3) / 3
ν 12 \nu^{12} ν 1 2 = = =
38 β 7 − 38 β 5 − 715 38\beta_{7} - 38\beta_{5} - 715 3 8 β 7 − 3 8 β 5 − 7 1 5
38*b7 - 38*b5 - 715
ν 13 \nu^{13} ν 1 3 = = =
( − 449 β 10 − 791 β 8 − 791 β 3 ) / 3 ( -449\beta_{10} - 791\beta_{8} - 791\beta_{3} ) / 3 ( − 4 4 9 β 1 0 − 7 9 1 β 8 − 7 9 1 β 3 ) / 3
(-449*b10 - 791*b8 - 791*b3) / 3
ν 14 \nu^{14} ν 1 4 = = =
− 791 β 12 − 449 β 11 -791\beta_{12} - 449\beta_{11} − 7 9 1 β 1 2 − 4 4 9 β 1 1
-791*b12 - 449*b11
ν 15 \nu^{15} ν 1 5 = = =
( − 3399 β 14 − 1240 β 10 + 3720 β 9 − 1240 β 8 − 1240 β 3 ) / 3 ( -3399\beta_{14} - 1240\beta_{10} + 3720\beta_{9} - 1240\beta_{8} - 1240\beta_{3} ) / 3 ( − 3 3 9 9 β 1 4 − 1 2 4 0 β 1 0 + 3 7 2 0 β 9 − 1 2 4 0 β 8 − 1 2 4 0 β 3 ) / 3
(-3399*b14 - 1240*b10 + 3720*b9 - 1240*b8 - 1240*b3) / 3
Character values
We give the values of χ \chi χ on generators for ( Z / 504 Z ) × \left(\mathbb{Z}/504\mathbb{Z}\right)^\times ( Z / 5 0 4 Z ) × .
n n n
73 73 7 3
127 127 1 2 7
253 253 2 5 3
281 281 2 8 1
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
1 1 1
− 1 -1 − 1
1 − β 4 1 - \beta_{4} 1 − β 4
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 16 − 40 T 5 14 + 1122 T 5 12 − 16000 T 5 10 + 166075 T 5 8 − 744960 T 5 6 + ⋯ + 81 T_{5}^{16} - 40 T_{5}^{14} + 1122 T_{5}^{12} - 16000 T_{5}^{10} + 166075 T_{5}^{8} - 744960 T_{5}^{6} + \cdots + 81 T 5 1 6 − 4 0 T 5 1 4 + 1 1 2 2 T 5 1 2 − 1 6 0 0 0 T 5 1 0 + 1 6 6 0 7 5 T 5 8 − 7 4 4 9 6 0 T 5 6 + ⋯ + 8 1
T5^16 - 40*T5^14 + 1122*T5^12 - 16000*T5^10 + 166075*T5^8 - 744960*T5^6 + 2429298*T5^4 - 14040*T5^2 + 81
acting on S 2 n e w ( 504 , [ χ ] ) S_{2}^{\mathrm{new}}(504, [\chi]) S 2 n e w ( 5 0 4 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 4 − 2 T 2 + 4 ) 4 (T^{4} - 2 T^{2} + 4)^{4} ( T 4 − 2 T 2 + 4 ) 4
(T^4 - 2*T^2 + 4)^4
3 3 3
( T 8 − 10 T 4 + 81 ) 2 (T^{8} - 10 T^{4} + 81)^{2} ( T 8 − 1 0 T 4 + 8 1 ) 2
(T^8 - 10*T^4 + 81)^2
5 5 5
T 16 − 40 T 14 + ⋯ + 81 T^{16} - 40 T^{14} + \cdots + 81 T 1 6 − 4 0 T 1 4 + ⋯ + 8 1
T^16 - 40*T^14 + 1122*T^12 - 16000*T^10 + 166075*T^8 - 744960*T^6 + 2429298*T^4 - 14040*T^2 + 81
7 7 7
( T 4 + 7 T 2 + 49 ) 4 (T^{4} + 7 T^{2} + 49)^{4} ( T 4 + 7 T 2 + 4 9 ) 4
(T^4 + 7*T^2 + 49)^4
11 11 1 1
T 16 T^{16} T 1 6
T^16
13 13 1 3
( T 8 + 52 T 6 + ⋯ + 419904 ) 2 (T^{8} + 52 T^{6} + \cdots + 419904)^{2} ( T 8 + 5 2 T 6 + ⋯ + 4 1 9 9 0 4 ) 2
(T^8 + 52*T^6 + 2056*T^4 + 33696*T^2 + 419904)^2
17 17 1 7
T 16 T^{16} T 1 6
T^16
19 19 1 9
( T 8 − 152 T 6 + ⋯ + 1022121 ) 2 (T^{8} - 152 T^{6} + \cdots + 1022121)^{2} ( T 8 − 1 5 2 T 6 + ⋯ + 1 0 2 2 1 2 1 ) 2
(T^8 - 152*T^6 + 7870*T^4 - 159144*T^2 + 1022121)^2
23 23 2 3
( T 4 − 18 T 3 + ⋯ + 169 ) 4 (T^{4} - 18 T^{3} + \cdots + 169)^{4} ( T 4 − 1 8 T 3 + ⋯ + 1 6 9 ) 4
(T^4 - 18*T^3 + 121*T^2 - 234*T + 169)^4
29 29 2 9
T 16 T^{16} T 1 6
T^16
31 31 3 1
T 16 T^{16} T 1 6
T^16
37 37 3 7
T 16 T^{16} T 1 6
T^16
41 41 4 1
T 16 T^{16} T 1 6
T^16
43 43 4 3
T 16 T^{16} T 1 6
T^16
47 47 4 7
T 16 T^{16} T 1 6
T^16
53 53 5 3
T 16 T^{16} T 1 6
T^16
59 59 5 9
( T 8 − 236 T 6 + ⋯ + 34012224 ) 2 (T^{8} - 236 T^{6} + \cdots + 34012224)^{2} ( T 8 − 2 3 6 T 6 + ⋯ + 3 4 0 1 2 2 2 4 ) 2
(T^8 - 236*T^6 + 49864*T^4 - 1376352*T^2 + 34012224)^2
61 61 6 1
T 16 + ⋯ + 15 ⋯ 61 T^{16} + \cdots + 15\!\cdots\!61 T 1 6 + ⋯ + 1 5 ⋯ 6 1
T^16 + 488*T^14 + 156354*T^12 + 29053568*T^10 + 3916765531*T^8 + 324059702784*T^6 + 19423628477586*T^4 + 667942873583256*T^2 + 15131529231698961
67 67 6 7
T 16 T^{16} T 1 6
T^16
71 71 7 1
( T 4 + 394 T 2 + 32761 ) 4 (T^{4} + 394 T^{2} + 32761)^{4} ( T 4 + 3 9 4 T 2 + 3 2 7 6 1 ) 4
(T^4 + 394*T^2 + 32761)^4
73 73 7 3
T 16 T^{16} T 1 6
T^16
79 79 7 9
( T 8 + 446 T 6 + ⋯ + 1908029761 ) 2 (T^{8} + 446 T^{6} + \cdots + 1908029761)^{2} ( T 8 + 4 4 6 T 6 + ⋯ + 1 9 0 8 0 2 9 7 6 1 ) 2
(T^8 + 446*T^6 + 155235*T^4 + 19481726*T^2 + 1908029761)^2
83 83 8 3
( T 8 − 332 T 6 + ⋯ + 419904 ) 2 (T^{8} - 332 T^{6} + \cdots + 419904)^{2} ( T 8 − 3 3 2 T 6 + ⋯ + 4 1 9 9 0 4 ) 2
(T^8 - 332*T^6 + 109576*T^4 - 215136*T^2 + 419904)^2
89 89 8 9
T 16 T^{16} T 1 6
T^16
97 97 9 7
T 16 T^{16} T 1 6
T^16
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