gp: [N,k,chi] = [312,2,Mod(155,312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(312, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 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("312.155");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [12]
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 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 − 16 x 9 + 92 x 6 − 68 x 3 + 27 x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 x 1 2 − 1 6 x 9 + 9 2 x 6 − 6 8 x 3 + 2 7
x^12 - 16*x^9 + 92*x^6 - 68*x^3 + 27
:
β 1 \beta_{1} β 1 = = =
( 2 ν 9 − 21 ν 6 + 16 ν 3 + 372 ) / 105 ( 2\nu^{9} - 21\nu^{6} + 16\nu^{3} + 372 ) / 105 ( 2 ν 9 − 2 1 ν 6 + 1 6 ν 3 + 3 7 2 ) / 1 0 5
(2*v^9 - 21*v^6 + 16*v^3 + 372) / 105
β 2 \beta_{2} β 2 = = =
( − ν 10 + 18 ν 7 − 113 ν 4 + 144 ν ) / 30 ( -\nu^{10} + 18\nu^{7} - 113\nu^{4} + 144\nu ) / 30 ( − ν 1 0 + 1 8 ν 7 − 1 1 3 ν 4 + 1 4 4 ν ) / 3 0
(-v^10 + 18*v^7 - 113*v^4 + 144*v) / 30
β 3 \beta_{3} β 3 = = =
( − 3 ν 9 + 49 ν 6 − 269 ν 3 + 107 ) / 70 ( -3\nu^{9} + 49\nu^{6} - 269\nu^{3} + 107 ) / 70 ( − 3 ν 9 + 4 9 ν 6 − 2 6 9 ν 3 + 1 0 7 ) / 7 0
(-3*v^9 + 49*v^6 - 269*v^3 + 107) / 70
β 4 \beta_{4} β 4 = = =
( 6 ν 11 − 98 ν 8 + 573 ν 5 − 529 ν 2 ) / 105 ( 6\nu^{11} - 98\nu^{8} + 573\nu^{5} - 529\nu^{2} ) / 105 ( 6 ν 1 1 − 9 8 ν 8 + 5 7 3 ν 5 − 5 2 9 ν 2 ) / 1 0 5
(6*v^11 - 98*v^8 + 573*v^5 - 529*v^2) / 105
β 5 \beta_{5} β 5 = = =
( 40 ν 11 − 9 ν 10 − 595 ν 8 + 147 ν 7 + 3050 ν 5 − 1017 ν 4 + 475 ν 2 + 2211 ν ) / 630 ( 40\nu^{11} - 9\nu^{10} - 595\nu^{8} + 147\nu^{7} + 3050\nu^{5} - 1017\nu^{4} + 475\nu^{2} + 2211\nu ) / 630 ( 4 0 ν 1 1 − 9 ν 1 0 − 5 9 5 ν 8 + 1 4 7 ν 7 + 3 0 5 0 ν 5 − 1 0 1 7 ν 4 + 4 7 5 ν 2 + 2 2 1 1 ν ) / 6 3 0
(40*v^11 - 9*v^10 - 595*v^8 + 147*v^7 + 3050*v^5 - 1017*v^4 + 475*v^2 + 2211*v) / 630
β 6 \beta_{6} β 6 = = =
( 4 ν 11 − 6 ν 10 − 67 ν 8 + 93 ν 7 + 422 ν 5 − 498 ν 4 − 521 ν 2 + 159 ν ) / 90 ( 4\nu^{11} - 6\nu^{10} - 67\nu^{8} + 93\nu^{7} + 422\nu^{5} - 498\nu^{4} - 521\nu^{2} + 159\nu ) / 90 ( 4 ν 1 1 − 6 ν 1 0 − 6 7 ν 8 + 9 3 ν 7 + 4 2 2 ν 5 − 4 9 8 ν 4 − 5 2 1 ν 2 + 1 5 9 ν ) / 9 0
(4*v^11 - 6*v^10 - 67*v^8 + 93*v^7 + 422*v^5 - 498*v^4 - 521*v^2 + 159*v) / 90
β 7 \beta_{7} β 7 = = =
( 2 ν 11 − 31 ν 8 + 166 ν 5 − 23 ν 2 ) / 30 ( 2\nu^{11} - 31\nu^{8} + 166\nu^{5} - 23\nu^{2} ) / 30 ( 2 ν 1 1 − 3 1 ν 8 + 1 6 6 ν 5 − 2 3 ν 2 ) / 3 0
(2*v^11 - 31*v^8 + 166*v^5 - 23*v^2) / 30
β 8 \beta_{8} β 8 = = =
( 40 ν 11 − 42 ν 10 − 595 ν 8 + 651 ν 7 + 3050 ν 5 − 3486 ν 4 + 475 ν 2 − 147 ν ) / 630 ( 40\nu^{11} - 42\nu^{10} - 595\nu^{8} + 651\nu^{7} + 3050\nu^{5} - 3486\nu^{4} + 475\nu^{2} - 147\nu ) / 630 ( 4 0 ν 1 1 − 4 2 ν 1 0 − 5 9 5 ν 8 + 6 5 1 ν 7 + 3 0 5 0 ν 5 − 3 4 8 6 ν 4 + 4 7 5 ν 2 − 1 4 7 ν ) / 6 3 0
(40*v^11 - 42*v^10 - 595*v^8 + 651*v^7 + 3050*v^5 - 3486*v^4 + 475*v^2 - 147*v) / 630
β 9 \beta_{9} β 9 = = =
( 40 ν 11 + 51 ν 10 − 595 ν 8 − 798 ν 7 + 3050 ν 5 + 4503 ν 4 + 475 ν 2 − 2064 ν ) / 630 ( 40\nu^{11} + 51\nu^{10} - 595\nu^{8} - 798\nu^{7} + 3050\nu^{5} + 4503\nu^{4} + 475\nu^{2} - 2064\nu ) / 630 ( 4 0 ν 1 1 + 5 1 ν 1 0 − 5 9 5 ν 8 − 7 9 8 ν 7 + 3 0 5 0 ν 5 + 4 5 0 3 ν 4 + 4 7 5 ν 2 − 2 0 6 4 ν ) / 6 3 0
(40*v^11 + 51*v^10 - 595*v^8 - 798*v^7 + 3050*v^5 + 4503*v^4 + 475*v^2 - 2064*v) / 630
β 10 \beta_{10} β 1 0 = = =
( − 4 ν 11 − 12 ν 10 + 67 ν 8 + 186 ν 7 − 422 ν 5 − 996 ν 4 + 521 ν 2 + 318 ν ) / 90 ( -4\nu^{11} - 12\nu^{10} + 67\nu^{8} + 186\nu^{7} - 422\nu^{5} - 996\nu^{4} + 521\nu^{2} + 318\nu ) / 90 ( − 4 ν 1 1 − 1 2 ν 1 0 + 6 7 ν 8 + 1 8 6 ν 7 − 4 2 2 ν 5 − 9 9 6 ν 4 + 5 2 1 ν 2 + 3 1 8 ν ) / 9 0
(-4*v^11 - 12*v^10 + 67*v^8 + 186*v^7 - 422*v^5 - 996*v^4 + 521*v^2 + 318*v) / 90
β 11 \beta_{11} β 1 1 = = =
( ν 9 − 15 ν 6 + 83 ν 3 − 33 ) / 6 ( \nu^{9} - 15\nu^{6} + 83\nu^{3} - 33 ) / 6 ( ν 9 − 1 5 ν 6 + 8 3 ν 3 − 3 3 ) / 6
(v^9 - 15*v^6 + 83*v^3 - 33) / 6
ν \nu ν = = =
( β 10 + β 9 − 2 β 8 + β 6 + β 5 ) / 6 ( \beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{6} + \beta_{5} ) / 6 ( β 1 0 + β 9 − 2 β 8 + β 6 + β 5 ) / 6
(b10 + b9 - 2*b8 + b6 + b5) / 6
ν 2 \nu^{2} ν 2 = = =
( − β 10 − β 9 − β 8 + 6 β 7 + 2 β 6 − β 5 − 6 β 4 ) / 6 ( -\beta_{10} - \beta_{9} - \beta_{8} + 6\beta_{7} + 2\beta_{6} - \beta_{5} - 6\beta_{4} ) / 6 ( − β 1 0 − β 9 − β 8 + 6 β 7 + 2 β 6 − β 5 − 6 β 4 ) / 6
(-b10 - b9 - b8 + 6*b7 + 2*b6 - b5 - 6*b4) / 6
ν 3 \nu^{3} ν 3 = = =
( β 11 + 3 β 3 − 2 β 1 + 8 ) / 2 ( \beta_{11} + 3\beta_{3} - 2\beta _1 + 8 ) / 2 ( β 1 1 + 3 β 3 − 2 β 1 + 8 ) / 2
(b11 + 3*b3 - 2*b1 + 8) / 2
ν 4 \nu^{4} ν 4 = = =
( 10 β 10 + 16 β 9 − 11 β 8 + 10 β 6 − 5 β 5 + 3 β 2 ) / 6 ( 10\beta_{10} + 16\beta_{9} - 11\beta_{8} + 10\beta_{6} - 5\beta_{5} + 3\beta_{2} ) / 6 ( 1 0 β 1 0 + 1 6 β 9 − 1 1 β 8 + 1 0 β 6 − 5 β 5 + 3 β 2 ) / 6
(10*b10 + 16*b9 - 11*b8 + 10*b6 - 5*b5 + 3*b2) / 6
ν 5 \nu^{5} ν 5 = = =
( − 13 β 10 − β 9 − β 8 + 18 β 7 + 26 β 6 − β 5 − 48 β 4 ) / 6 ( -13\beta_{10} - \beta_{9} - \beta_{8} + 18\beta_{7} + 26\beta_{6} - \beta_{5} - 48\beta_{4} ) / 6 ( − 1 3 β 1 0 − β 9 − β 8 + 1 8 β 7 + 2 6 β 6 − β 5 − 4 8 β 4 ) / 6
(-13*b10 - b9 - b8 + 18*b7 + 26*b6 - b5 - 48*b4) / 6
ν 6 \nu^{6} ν 6 = = =
7 β 11 + 25 β 3 − 5 β 1 + 18 7\beta_{11} + 25\beta_{3} - 5\beta _1 + 18 7 β 1 1 + 2 5 β 3 − 5 β 1 + 1 8
7*b11 + 25*b3 - 5*b1 + 18
ν 7 \nu^{7} ν 7 = = =
( 61 β 10 + 145 β 9 − 38 β 8 + 61 β 6 − 107 β 5 + 108 β 2 ) / 6 ( 61\beta_{10} + 145\beta_{9} - 38\beta_{8} + 61\beta_{6} - 107\beta_{5} + 108\beta_{2} ) / 6 ( 6 1 β 1 0 + 1 4 5 β 9 − 3 8 β 8 + 6 1 β 6 − 1 0 7 β 5 + 1 0 8 β 2 ) / 6
(61*b10 + 145*b9 - 38*b8 + 61*b6 - 107*b5 + 108*b2) / 6
ν 8 \nu^{8} ν 8 = = =
( − 103 β 10 + 77 β 9 + 77 β 8 − 174 β 7 + 206 β 6 + 77 β 5 − 294 β 4 ) / 6 ( -103\beta_{10} + 77\beta_{9} + 77\beta_{8} - 174\beta_{7} + 206\beta_{6} + 77\beta_{5} - 294\beta_{4} ) / 6 ( − 1 0 3 β 1 0 + 7 7 β 9 + 7 7 β 8 − 1 7 4 β 7 + 2 0 6 β 6 + 7 7 β 5 − 2 9 4 β 4 ) / 6
(-103*b10 + 77*b9 + 77*b8 - 174*b7 + 206*b6 + 77*b5 - 294*b4) / 6
ν 9 \nu^{9} ν 9 = = =
( 139 β 11 + 501 β 3 + 16 β 1 − 58 ) / 2 ( 139\beta_{11} + 501\beta_{3} + 16\beta _1 - 58 ) / 2 ( 1 3 9 β 1 1 + 5 0 1 β 3 + 1 6 β 1 − 5 8 ) / 2
(139*b11 + 501*b3 + 16*b1 - 58) / 2
ν 10 \nu^{10} ν 1 0 = = =
( 112 β 10 + 946 β 9 + 271 β 8 + 112 β 6 − 1217 β 5 + 1425 β 2 ) / 6 ( 112\beta_{10} + 946\beta_{9} + 271\beta_{8} + 112\beta_{6} - 1217\beta_{5} + 1425\beta_{2} ) / 6 ( 1 1 2 β 1 0 + 9 4 6 β 9 + 2 7 1 β 8 + 1 1 2 β 6 − 1 2 1 7 β 5 + 1 4 2 5 β 2 ) / 6
(112*b10 + 946*b9 + 271*b8 + 112*b6 - 1217*b5 + 1425*b2) / 6
ν 11 \nu^{11} ν 1 1 = = =
( − 529 β 10 + 1265 β 9 + 1265 β 8 − 4032 β 7 + 1058 β 6 + 1265 β 5 − 642 β 4 ) / 6 ( -529\beta_{10} + 1265\beta_{9} + 1265\beta_{8} - 4032\beta_{7} + 1058\beta_{6} + 1265\beta_{5} - 642\beta_{4} ) / 6 ( − 5 2 9 β 1 0 + 1 2 6 5 β 9 + 1 2 6 5 β 8 − 4 0 3 2 β 7 + 1 0 5 8 β 6 + 1 2 6 5 β 5 − 6 4 2 β 4 ) / 6
(-529*b10 + 1265*b9 + 1265*b8 - 4032*b7 + 1058*b6 + 1265*b5 - 642*b4) / 6
Character values
We give the values of χ \chi χ on generators for ( Z / 312 Z ) × \left(\mathbb{Z}/312\mathbb{Z}\right)^\times ( Z / 3 1 2 Z ) × .
n n n
79 79 7 9
145 145 1 4 5
157 157 1 5 7
209 209 2 0 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 6 + 30 T 5 4 + 225 T 5 2 + 32 T_{5}^{6} + 30T_{5}^{4} + 225T_{5}^{2} + 32 T 5 6 + 3 0 T 5 4 + 2 2 5 T 5 2 + 3 2
T5^6 + 30*T5^4 + 225*T5^2 + 32
acting on S 2 n e w ( 312 , [ χ ] ) S_{2}^{\mathrm{new}}(312, [\chi]) S 2 n e w ( 3 1 2 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 + 2 ) 6 (T^{2} + 2)^{6} ( T 2 + 2 ) 6
(T^2 + 2)^6
3 3 3
( T 6 + 2 T 3 + 27 ) 2 (T^{6} + 2 T^{3} + 27)^{2} ( T 6 + 2 T 3 + 2 7 ) 2
(T^6 + 2*T^3 + 27)^2
5 5 5
( T 6 + 30 T 4 + ⋯ + 32 ) 2 (T^{6} + 30 T^{4} + \cdots + 32)^{2} ( T 6 + 3 0 T 4 + ⋯ + 3 2 ) 2
(T^6 + 30*T^4 + 225*T^2 + 32)^2
7 7 7
( T 6 − 42 T 4 + ⋯ − 1300 ) 2 (T^{6} - 42 T^{4} + \cdots - 1300)^{2} ( T 6 − 4 2 T 4 + ⋯ − 1 3 0 0 ) 2
(T^6 - 42*T^4 + 441*T^2 - 1300)^2
11 11 1 1
T 12 T^{12} T 1 2
T^12
13 13 1 3
( T 2 − 13 ) 6 (T^{2} - 13)^{6} ( T 2 − 1 3 ) 6
(T^2 - 13)^6
17 17 1 7
( T 6 + 102 T 4 + ⋯ + 6656 ) 2 (T^{6} + 102 T^{4} + \cdots + 6656)^{2} ( T 6 + 1 0 2 T 4 + ⋯ + 6 6 5 6 ) 2
(T^6 + 102*T^4 + 2601*T^2 + 6656)^2
19 19 1 9
T 12 T^{12} T 1 2
T^12
23 23 2 3
T 12 T^{12} T 1 2
T^12
29 29 2 9
T 12 T^{12} T 1 2
T^12
31 31 3 1
( T 2 − 52 ) 6 (T^{2} - 52)^{6} ( T 2 − 5 2 ) 6
(T^2 - 52)^6
37 37 3 7
( T 6 − 222 T 4 + ⋯ − 87412 ) 2 (T^{6} - 222 T^{4} + \cdots - 87412)^{2} ( T 6 − 2 2 2 T 4 + ⋯ − 8 7 4 1 2 ) 2
(T^6 - 222*T^4 + 12321*T^2 - 87412)^2
41 41 4 1
T 12 T^{12} T 1 2
T^12
43 43 4 3
( T 3 − 129 T − 218 ) 4 (T^{3} - 129 T - 218)^{4} ( T 3 − 1 2 9 T − 2 1 8 ) 4
(T^3 - 129*T - 218)^4
47 47 4 7
( T 6 + 282 T 4 + ⋯ + 336200 ) 2 (T^{6} + 282 T^{4} + \cdots + 336200)^{2} ( T 6 + 2 8 2 T 4 + ⋯ + 3 3 6 2 0 0 ) 2
(T^6 + 282*T^4 + 19881*T^2 + 336200)^2
53 53 5 3
T 12 T^{12} T 1 2
T^12
59 59 5 9
T 12 T^{12} T 1 2
T^12
61 61 6 1
T 12 T^{12} T 1 2
T^12
67 67 6 7
T 12 T^{12} T 1 2
T^12
71 71 7 1
( T 6 + 426 T 4 + ⋯ + 397832 ) 2 (T^{6} + 426 T^{4} + \cdots + 397832)^{2} ( T 6 + 4 2 6 T 4 + ⋯ + 3 9 7 8 3 2 ) 2
(T^6 + 426*T^4 + 45369*T^2 + 397832)^2
73 73 7 3
T 12 T^{12} T 1 2
T^12
79 79 7 9
T 12 T^{12} T 1 2
T^12
83 83 8 3
T 12 T^{12} T 1 2
T^12
89 89 8 9
T 12 T^{12} T 1 2
T^12
97 97 9 7
T 12 T^{12} T 1 2
T^12
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