gp: [N,k,chi] = [841,2,Mod(190,841)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(841, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("841.190");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [12,2,-2,-2,2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 11 1,\beta_1,\ldots,\beta_{11} 1 , β 1 , … , β 1 1 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 12 + 2 x 10 + 4 x 8 + 8 x 6 + 16 x 4 + 32 x 2 + 64 x^{12} + 2x^{10} + 4x^{8} + 8x^{6} + 16x^{4} + 32x^{2} + 64 x 1 2 + 2 x 1 0 + 4 x 8 + 8 x 6 + 1 6 x 4 + 3 2 x 2 + 6 4
x^12 + 2*x^10 + 4*x^8 + 8*x^6 + 16*x^4 + 32*x^2 + 64
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 2 ) / 2 ( \nu^{2} ) / 2 ( ν 2 ) / 2
(v^2) / 2
β 3 \beta_{3} β 3 = = =
( ν 3 ) / 2 ( \nu^{3} ) / 2 ( ν 3 ) / 2
(v^3) / 2
β 4 \beta_{4} β 4 = = =
( ν 4 ) / 4 ( \nu^{4} ) / 4 ( ν 4 ) / 4
(v^4) / 4
β 5 \beta_{5} β 5 = = =
( ν 5 ) / 4 ( \nu^{5} ) / 4 ( ν 5 ) / 4
(v^5) / 4
β 6 \beta_{6} β 6 = = =
( ν 6 ) / 8 ( \nu^{6} ) / 8 ( ν 6 ) / 8
(v^6) / 8
β 7 \beta_{7} β 7 = = =
( ν 7 ) / 8 ( \nu^{7} ) / 8 ( ν 7 ) / 8
(v^7) / 8
β 8 \beta_{8} β 8 = = =
( ν 8 ) / 16 ( \nu^{8} ) / 16 ( ν 8 ) / 1 6
(v^8) / 16
β 9 \beta_{9} β 9 = = =
( ν 9 ) / 16 ( \nu^{9} ) / 16 ( ν 9 ) / 1 6
(v^9) / 16
β 10 \beta_{10} β 1 0 = = =
( ν 10 ) / 32 ( \nu^{10} ) / 32 ( ν 1 0 ) / 3 2
(v^10) / 32
β 11 \beta_{11} β 1 1 = = =
( ν 11 ) / 32 ( \nu^{11} ) / 32 ( ν 1 1 ) / 3 2
(v^11) / 32
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
2 β 2 2\beta_{2} 2 β 2
2*b2
ν 3 \nu^{3} ν 3 = = =
2 β 3 2\beta_{3} 2 β 3
2*b3
ν 4 \nu^{4} ν 4 = = =
4 β 4 4\beta_{4} 4 β 4
4*b4
ν 5 \nu^{5} ν 5 = = =
4 β 5 4\beta_{5} 4 β 5
4*b5
ν 6 \nu^{6} ν 6 = = =
8 β 6 8\beta_{6} 8 β 6
8*b6
ν 7 \nu^{7} ν 7 = = =
8 β 7 8\beta_{7} 8 β 7
8*b7
ν 8 \nu^{8} ν 8 = = =
16 β 8 16\beta_{8} 1 6 β 8
16*b8
ν 9 \nu^{9} ν 9 = = =
16 β 9 16\beta_{9} 1 6 β 9
16*b9
ν 10 \nu^{10} ν 1 0 = = =
32 β 10 32\beta_{10} 3 2 β 1 0
32*b10
ν 11 \nu^{11} ν 1 1 = = =
32 β 11 32\beta_{11} 3 2 β 1 1
32*b11
Character values
We give the values of χ \chi χ on generators for ( Z / 841 Z ) × \left(\mathbb{Z}/841\mathbb{Z}\right)^\times ( Z / 8 4 1 Z ) × .
n n n
2 2 2
χ ( n ) \chi(n) χ ( n )
β 4 \beta_{4} 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 2 12 − 2 T 2 11 + 5 T 2 10 − 12 T 2 9 + 29 T 2 8 − 70 T 2 7 + 169 T 2 6 + ⋯ + 1 T_{2}^{12} - 2 T_{2}^{11} + 5 T_{2}^{10} - 12 T_{2}^{9} + 29 T_{2}^{8} - 70 T_{2}^{7} + 169 T_{2}^{6} + \cdots + 1 T 2 1 2 − 2 T 2 1 1 + 5 T 2 1 0 − 1 2 T 2 9 + 2 9 T 2 8 − 7 0 T 2 7 + 1 6 9 T 2 6 + ⋯ + 1
T2^12 - 2*T2^11 + 5*T2^10 - 12*T2^9 + 29*T2^8 - 70*T2^7 + 169*T2^6 + 70*T2^5 + 29*T2^4 + 12*T2^3 + 5*T2^2 + 2*T2 + 1
acting on S 2 n e w ( 841 , [ χ ] ) S_{2}^{\mathrm{new}}(841, [\chi]) S 2 n e w ( 8 4 1 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 12 − 2 T 11 + ⋯ + 1 T^{12} - 2 T^{11} + \cdots + 1 T 1 2 − 2 T 1 1 + ⋯ + 1
T^12 - 2*T^11 + 5*T^10 - 12*T^9 + 29*T^8 - 70*T^7 + 169*T^6 + 70*T^5 + 29*T^4 + 12*T^3 + 5*T^2 + 2*T + 1
3 3 3
T 12 + 2 T 11 + ⋯ + 1 T^{12} + 2 T^{11} + \cdots + 1 T 1 2 + 2 T 1 1 + ⋯ + 1
T^12 + 2*T^11 + 5*T^10 + 12*T^9 + 29*T^8 + 70*T^7 + 169*T^6 - 70*T^5 + 29*T^4 - 12*T^3 + 5*T^2 - 2*T + 1
5 5 5
( T 6 − T 5 + T 4 + ⋯ + 1 ) 2 (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} ( T 6 − T 5 + T 4 + ⋯ + 1 ) 2
(T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
7 7 7
T 12 + 8 T 10 + ⋯ + 262144 T^{12} + 8 T^{10} + \cdots + 262144 T 1 2 + 8 T 1 0 + ⋯ + 2 6 2 1 4 4
T^12 + 8*T^10 + 64*T^8 + 512*T^6 + 4096*T^4 + 32768*T^2 + 262144
11 11 1 1
T 12 + 2 T 11 + ⋯ + 1 T^{12} + 2 T^{11} + \cdots + 1 T 1 2 + 2 T 1 1 + ⋯ + 1
T^12 + 2*T^11 + 5*T^10 + 12*T^9 + 29*T^8 + 70*T^7 + 169*T^6 - 70*T^5 + 29*T^4 - 12*T^3 + 5*T^2 - 2*T + 1
13 13 1 3
T 12 − 2 T 11 + ⋯ + 117649 T^{12} - 2 T^{11} + \cdots + 117649 T 1 2 − 2 T 1 1 + ⋯ + 1 1 7 6 4 9
T^12 - 2*T^11 + 11*T^10 - 36*T^9 + 149*T^8 - 550*T^7 + 2143*T^6 + 3850*T^5 + 7301*T^4 + 12348*T^3 + 26411*T^2 + 33614*T + 117649
17 17 1 7
( T 2 + 4 T − 4 ) 6 (T^{2} + 4 T - 4)^{6} ( T 2 + 4 T − 4 ) 6
(T^2 + 4*T - 4)^6
19 19 1 9
( T 6 + 6 T 5 + ⋯ + 46656 ) 2 (T^{6} + 6 T^{5} + \cdots + 46656)^{2} ( T 6 + 6 T 5 + ⋯ + 4 6 6 5 6 ) 2
(T^6 + 6*T^5 + 36*T^4 + 216*T^3 + 1296*T^2 + 7776*T + 46656)^2
23 23 2 3
T 12 + ⋯ + 481890304 T^{12} + \cdots + 481890304 T 1 2 + ⋯ + 4 8 1 8 9 0 3 0 4
T^12 - 4*T^11 + 44*T^10 - 288*T^9 + 2384*T^8 - 17600*T^7 + 137152*T^6 + 492800*T^5 + 1869056*T^4 + 6322176*T^3 + 27044864*T^2 + 68841472*T + 481890304
29 29 2 9
T 12 T^{12} T 1 2
T^12
31 31 3 1
T 12 + ⋯ + 4750104241 T^{12} + \cdots + 4750104241 T 1 2 + ⋯ + 4 7 5 0 1 0 4 2 4 1
T^12 + 6*T^11 + 77*T^10 + 708*T^9 + 7405*T^8 + 73458*T^7 + 744353*T^6 - 3011778*T^5 + 12447805*T^4 - 48796068*T^3 + 217583597*T^2 - 695137206*T + 4750104241
37 37 3 7
( T 6 − 4 T 5 + ⋯ + 4096 ) 2 (T^{6} - 4 T^{5} + \cdots + 4096)^{2} ( T 6 − 4 T 5 + ⋯ + 4 0 9 6 ) 2
(T^6 - 4*T^5 + 16*T^4 - 64*T^3 + 256*T^2 - 1024*T + 4096)^2
41 41 4 1
( T 2 − 8 T − 56 ) 6 (T^{2} - 8 T - 56)^{6} ( T 2 − 8 T − 5 6 ) 6
(T^2 - 8*T - 56)^6
43 43 4 3
T 12 + ⋯ + 148035889 T^{12} + \cdots + 148035889 T 1 2 + ⋯ + 1 4 8 0 3 5 8 8 9
T^12 + 10*T^11 + 77*T^10 + 540*T^9 + 3629*T^8 + 23870*T^7 + 155233*T^6 + 549010*T^5 + 1919741*T^4 + 6570180*T^3 + 21547757*T^2 + 64363430*T + 148035889
47 47 4 7
T 12 + 2 T 11 + ⋯ + 24137569 T^{12} + 2 T^{11} + \cdots + 24137569 T 1 2 + 2 T 1 1 + ⋯ + 2 4 1 3 7 5 6 9
T^12 + 2*T^11 + 21*T^10 + 76*T^9 + 509*T^8 + 2310*T^7 + 13273*T^6 - 39270*T^5 + 147101*T^4 - 373388*T^3 + 1753941*T^2 - 2839714*T + 24137569
53 53 5 3
T 12 + ⋯ + 128100283921 T^{12} + \cdots + 128100283921 T 1 2 + ⋯ + 1 2 8 1 0 0 2 8 3 9 2 1
T^12 + 2*T^11 + 75*T^10 + 292*T^9 + 5909*T^8 + 32550*T^7 + 484639*T^6 - 2311050*T^5 + 29787269*T^4 - 104510012*T^3 + 1905876075*T^2 - 3608458702*T + 128100283921
59 59 5 9
( T 2 − 4 T − 28 ) 6 (T^{2} - 4 T - 28)^{6} ( T 2 − 4 T − 2 8 ) 6
(T^2 - 4*T - 28)^6
61 61 6 1
T 12 − 4 T 11 + ⋯ + 4096 T^{12} - 4 T^{11} + \cdots + 4096 T 1 2 − 4 T 1 1 + ⋯ + 4 0 9 6
T^12 - 4*T^11 + 20*T^10 - 96*T^9 + 464*T^8 - 2240*T^7 + 10816*T^6 + 8960*T^5 + 7424*T^4 + 6144*T^3 + 5120*T^2 + 4096*T + 4096
67 67 6 7
T 12 + ⋯ + 1073741824 T^{12} + \cdots + 1073741824 T 1 2 + ⋯ + 1 0 7 3 7 4 1 8 2 4
T^12 + 32*T^10 + 1024*T^8 + 32768*T^6 + 1048576*T^4 + 33554432*T^2 + 1073741824
71 71 7 1
T 12 + ⋯ + 481890304 T^{12} + \cdots + 481890304 T 1 2 + ⋯ + 4 8 1 8 9 0 3 0 4
T^12 - 12*T^11 + 116*T^10 - 1056*T^9 + 9424*T^8 - 83520*T^7 + 738368*T^6 - 2338560*T^5 + 7388416*T^4 - 23181312*T^3 + 71300096*T^2 - 206524416*T + 481890304
73 73 7 3
( T 6 + 4 T 5 + ⋯ + 4096 ) 2 (T^{6} + 4 T^{5} + \cdots + 4096)^{2} ( T 6 + 4 T 5 + ⋯ + 4 0 9 6 ) 2
(T^6 + 4*T^5 + 16*T^4 + 64*T^3 + 256*T^2 + 1024*T + 4096)^2
79 79 7 9
T 12 − 2 T 11 + ⋯ + 1 T^{12} - 2 T^{11} + \cdots + 1 T 1 2 − 2 T 1 1 + ⋯ + 1
T^12 - 2*T^11 + 5*T^10 - 12*T^9 + 29*T^8 - 70*T^7 + 169*T^6 + 70*T^5 + 29*T^4 + 12*T^3 + 5*T^2 + 2*T + 1
83 83 8 3
T 12 + ⋯ + 481890304 T^{12} + \cdots + 481890304 T 1 2 + ⋯ + 4 8 1 8 9 0 3 0 4
T^12 + 4*T^11 + 44*T^10 + 288*T^9 + 2384*T^8 + 17600*T^7 + 137152*T^6 - 492800*T^5 + 1869056*T^4 - 6322176*T^3 + 27044864*T^2 - 68841472*T + 481890304
89 89 8 9
T 12 + ⋯ + 30840979456 T^{12} + \cdots + 30840979456 T 1 2 + ⋯ + 3 0 8 4 0 9 7 9 4 5 6
T^12 - 8*T^11 + 120*T^10 - 1408*T^9 + 17984*T^8 - 222720*T^7 + 2788864*T^6 + 12472320*T^5 + 56397824*T^4 + 247267328*T^3 + 1180139520*T^2 + 4405854208*T + 30840979456
97 97 9 7
T 12 + ⋯ + 30840979456 T^{12} + \cdots + 30840979456 T 1 2 + ⋯ + 3 0 8 4 0 9 7 9 4 5 6
T^12 - 8*T^11 + 120*T^10 - 1408*T^9 + 17984*T^8 - 222720*T^7 + 2788864*T^6 + 12472320*T^5 + 56397824*T^4 + 247267328*T^3 + 1180139520*T^2 + 4405854208*T + 30840979456
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