gp: [N,k,chi] = [2744,2,Mod(1,2744)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2744, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("2744.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [12,0,0,0,0,0,0,0,22]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == 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 − 29 x 10 + 304 x 8 − 1393 x 6 + 2574 x 4 − 1164 x 2 + 8 x^{12} - 29x^{10} + 304x^{8} - 1393x^{6} + 2574x^{4} - 1164x^{2} + 8 x 1 2 − 2 9 x 1 0 + 3 0 4 x 8 − 1 3 9 3 x 6 + 2 5 7 4 x 4 − 1 1 6 4 x 2 + 8
x^12 - 29*x^10 + 304*x^8 - 1393*x^6 + 2574*x^4 - 1164*x^2 + 8
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( − 9 ν 10 + 680 ν 8 − 13379 ν 6 + 97733 ν 4 − 226709 ν 2 + 1782 ) / 18452 ( -9\nu^{10} + 680\nu^{8} - 13379\nu^{6} + 97733\nu^{4} - 226709\nu^{2} + 1782 ) / 18452 ( − 9 ν 1 0 + 6 8 0 ν 8 − 1 3 3 7 9 ν 6 + 9 7 7 3 3 ν 4 − 2 2 6 7 0 9 ν 2 + 1 7 8 2 ) / 1 8 4 5 2
(-9*v^10 + 680*v^8 - 13379*v^6 + 97733*v^4 - 226709*v^2 + 1782) / 18452
β 3 \beta_{3} β 3 = = =
( 9 ν 10 − 680 ν 8 + 13379 ν 6 − 97733 ν 4 + 245161 ν 2 − 94042 ) / 18452 ( 9\nu^{10} - 680\nu^{8} + 13379\nu^{6} - 97733\nu^{4} + 245161\nu^{2} - 94042 ) / 18452 ( 9 ν 1 0 − 6 8 0 ν 8 + 1 3 3 7 9 ν 6 − 9 7 7 3 3 ν 4 + 2 4 5 1 6 1 ν 2 − 9 4 0 4 2 ) / 1 8 4 5 2
(9*v^10 - 680*v^8 + 13379*v^6 - 97733*v^4 + 245161*v^2 - 94042) / 18452
β 4 \beta_{4} β 4 = = =
( − 37 ν 10 + 2283 ν 8 − 36038 ν 6 + 196769 ν 4 − 310296 ν 2 + 35004 ) / 73808 ( -37\nu^{10} + 2283\nu^{8} - 36038\nu^{6} + 196769\nu^{4} - 310296\nu^{2} + 35004 ) / 73808 ( − 3 7 ν 1 0 + 2 2 8 3 ν 8 − 3 6 0 3 8 ν 6 + 1 9 6 7 6 9 ν 4 − 3 1 0 2 9 6 ν 2 + 3 5 0 0 4 ) / 7 3 8 0 8
(-37*v^10 + 2283*v^8 - 36038*v^6 + 196769*v^4 - 310296*v^2 + 35004) / 73808
β 5 \beta_{5} β 5 = = =
( − 37 ν 11 + 2283 ν 9 − 36038 ν 7 + 196769 ν 5 − 310296 ν 3 + 35004 ν ) / 73808 ( -37\nu^{11} + 2283\nu^{9} - 36038\nu^{7} + 196769\nu^{5} - 310296\nu^{3} + 35004\nu ) / 73808 ( − 3 7 ν 1 1 + 2 2 8 3 ν 9 − 3 6 0 3 8 ν 7 + 1 9 6 7 6 9 ν 5 − 3 1 0 2 9 6 ν 3 + 3 5 0 0 4 ν ) / 7 3 8 0 8
(-37*v^11 + 2283*v^9 - 36038*v^7 + 196769*v^5 - 310296*v^3 + 35004*v) / 73808
β 6 \beta_{6} β 6 = = =
( 30 ν 10 − 729 ν 8 + 6155 ν 6 − 24394 ν 4 + 52983 ν 2 − 33618 ) / 18452 ( 30\nu^{10} - 729\nu^{8} + 6155\nu^{6} - 24394\nu^{4} + 52983\nu^{2} - 33618 ) / 18452 ( 3 0 ν 1 0 − 7 2 9 ν 8 + 6 1 5 5 ν 6 − 2 4 3 9 4 ν 4 + 5 2 9 8 3 ν 2 − 3 3 6 1 8 ) / 1 8 4 5 2
(30*v^10 - 729*v^8 + 6155*v^6 - 24394*v^4 + 52983*v^2 - 33618) / 18452
β 7 \beta_{7} β 7 = = =
( 1395 ν 10 − 36205 ν 8 + 325418 ν 6 − 1198903 ν 4 + 1543416 ν 2 − 174724 ) / 73808 ( 1395\nu^{10} - 36205\nu^{8} + 325418\nu^{6} - 1198903\nu^{4} + 1543416\nu^{2} - 174724 ) / 73808 ( 1 3 9 5 ν 1 0 − 3 6 2 0 5 ν 8 + 3 2 5 4 1 8 ν 6 − 1 1 9 8 9 0 3 ν 4 + 1 5 4 3 4 1 6 ν 2 − 1 7 4 7 2 4 ) / 7 3 8 0 8
(1395*v^10 - 36205*v^8 + 325418*v^6 - 1198903*v^4 + 1543416*v^2 - 174724) / 73808
β 8 \beta_{8} β 8 = = =
( − 2299 ν 11 + 65553 ν 9 − 670806 ν 7 + 2962367 ν 5 − 5110644 ν 3 + 1866780 ν ) / 73808 ( -2299\nu^{11} + 65553\nu^{9} - 670806\nu^{7} + 2962367\nu^{5} - 5110644\nu^{3} + 1866780\nu ) / 73808 ( − 2 2 9 9 ν 1 1 + 6 5 5 5 3 ν 9 − 6 7 0 8 0 6 ν 7 + 2 9 6 2 3 6 7 ν 5 − 5 1 1 0 6 4 4 ν 3 + 1 8 6 6 7 8 0 ν ) / 7 3 8 0 8
(-2299*v^11 + 65553*v^9 - 670806*v^7 + 2962367*v^5 - 5110644*v^3 + 1866780*v) / 73808
β 9 \beta_{9} β 9 = = =
( 2419 ν 11 − 68469 ν 9 + 695426 ν 7 − 3059943 ν 5 + 5322576 ν 3 − 1927444 ν ) / 73808 ( 2419\nu^{11} - 68469\nu^{9} + 695426\nu^{7} - 3059943\nu^{5} + 5322576\nu^{3} - 1927444\nu ) / 73808 ( 2 4 1 9 ν 1 1 − 6 8 4 6 9 ν 9 + 6 9 5 4 2 6 ν 7 − 3 0 5 9 9 4 3 ν 5 + 5 3 2 2 5 7 6 ν 3 − 1 9 2 7 4 4 4 ν ) / 7 3 8 0 8
(2419*v^11 - 68469*v^9 + 695426*v^7 - 3059943*v^5 + 5322576*v^3 - 1927444*v) / 73808
β 10 \beta_{10} β 1 0 = = =
( 3009 ν 11 − 87419 ν 9 + 919498 ν 7 − 4234717 ν 5 + 7845348 ν 3 − 3428164 ν ) / 73808 ( 3009\nu^{11} - 87419\nu^{9} + 919498\nu^{7} - 4234717\nu^{5} + 7845348\nu^{3} - 3428164\nu ) / 73808 ( 3 0 0 9 ν 1 1 − 8 7 4 1 9 ν 9 + 9 1 9 4 9 8 ν 7 − 4 2 3 4 7 1 7 ν 5 + 7 8 4 5 3 4 8 ν 3 − 3 4 2 8 1 6 4 ν ) / 7 3 8 0 8
(3009*v^11 - 87419*v^9 + 919498*v^7 - 4234717*v^5 + 7845348*v^3 - 3428164*v) / 73808
β 11 \beta_{11} β 1 1 = = =
( 1643 ν 11 − 47767 ν 9 + 502388 ν 7 − 2313319 ν 5 + 4326658 ν 3 − 2087480 ν ) / 36904 ( 1643\nu^{11} - 47767\nu^{9} + 502388\nu^{7} - 2313319\nu^{5} + 4326658\nu^{3} - 2087480\nu ) / 36904 ( 1 6 4 3 ν 1 1 − 4 7 7 6 7 ν 9 + 5 0 2 3 8 8 ν 7 − 2 3 1 3 3 1 9 ν 5 + 4 3 2 6 6 5 8 ν 3 − 2 0 8 7 4 8 0 ν ) / 3 6 9 0 4
(1643*v^11 - 47767*v^9 + 502388*v^7 - 2313319*v^5 + 4326658*v^3 - 2087480*v) / 36904
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 3 + β 2 + 5 \beta_{3} + \beta_{2} + 5 β 3 + β 2 + 5
b3 + b2 + 5
ν 3 \nu^{3} ν 3 = = =
β 11 − β 10 − 2 β 9 − 2 β 8 + β 5 + 8 β 1 \beta_{11} - \beta_{10} - 2\beta_{9} - 2\beta_{8} + \beta_{5} + 8\beta_1 β 1 1 − β 1 0 − 2 β 9 − 2 β 8 + β 5 + 8 β 1
b11 - b10 - 2*b9 - 2*b8 + b5 + 8*b1
ν 4 \nu^{4} ν 4 = = =
β 7 − 11 β 6 + 3 β 4 + 11 β 3 + 10 β 2 + 36 \beta_{7} - 11\beta_{6} + 3\beta_{4} + 11\beta_{3} + 10\beta_{2} + 36 β 7 − 1 1 β 6 + 3 β 4 + 1 1 β 3 + 1 0 β 2 + 3 6
b7 - 11*b6 + 3*b4 + 11*b3 + 10*b2 + 36
ν 5 \nu^{5} ν 5 = = =
9 β 11 − 9 β 10 − 35 β 9 − 36 β 8 + 16 β 5 + 80 β 1 9\beta_{11} - 9\beta_{10} - 35\beta_{9} - 36\beta_{8} + 16\beta_{5} + 80\beta_1 9 β 1 1 − 9 β 1 0 − 3 5 β 9 − 3 6 β 8 + 1 6 β 5 + 8 0 β 1
9*b11 - 9*b10 - 35*b9 - 36*b8 + 16*b5 + 80*b1
ν 6 \nu^{6} ν 6 = = =
17 β 7 − 186 β 6 + 63 β 4 + 124 β 3 + 98 β 2 + 294 17\beta_{7} - 186\beta_{6} + 63\beta_{4} + 124\beta_{3} + 98\beta_{2} + 294 1 7 β 7 − 1 8 6 β 6 + 6 3 β 4 + 1 2 4 β 3 + 9 8 β 2 + 2 9 4
17*b7 - 186*b6 + 63*b4 + 124*b3 + 98*b2 + 294
ν 7 \nu^{7} ν 7 = = =
72 β 11 − 63 β 10 − 477 β 9 − 485 β 8 + 221 β 5 + 852 β 1 72\beta_{11} - 63\beta_{10} - 477\beta_{9} - 485\beta_{8} + 221\beta_{5} + 852\beta_1 7 2 β 1 1 − 6 3 β 1 0 − 4 7 7 β 9 − 4 8 5 β 8 + 2 2 1 β 5 + 8 5 2 β 1
72*b11 - 63*b10 - 477*b9 - 485*b8 + 221*b5 + 852*b1
ν 8 \nu^{8} ν 8 = = =
229 β 7 − 2482 β 6 + 987 β 4 + 1401 β 3 + 987 β 2 + 2597 229\beta_{7} - 2482\beta_{6} + 987\beta_{4} + 1401\beta_{3} + 987\beta_{2} + 2597 2 2 9 β 7 − 2 4 8 2 β 6 + 9 8 7 β 4 + 1 4 0 1 β 3 + 9 8 7 β 2 + 2 5 9 7
229*b7 - 2482*b6 + 987*b4 + 1401*b3 + 987*b2 + 2597
ν 9 \nu^{9} ν 9 = = =
573 β 11 − 388 β 10 − 5927 β 9 − 5971 β 8 + 2846 β 5 + 9282 β 1 573\beta_{11} - 388\beta_{10} - 5927\beta_{9} - 5971\beta_{8} + 2846\beta_{5} + 9282\beta_1 5 7 3 β 1 1 − 3 8 8 β 1 0 − 5 9 2 7 β 9 − 5 9 7 1 β 8 + 2 8 4 6 β 5 + 9 2 8 2 β 1
573*b11 - 388*b10 - 5927*b9 - 5971*b8 + 2846*b5 + 9282*b1
ν 10 \nu^{10} ν 1 0 = = =
2890 β 7 − 30481 β 6 + 13498 β 4 + 15782 β 3 + 10243 β 2 + 24351 2890\beta_{7} - 30481\beta_{6} + 13498\beta_{4} + 15782\beta_{3} + 10243\beta_{2} + 24351 2 8 9 0 β 7 − 3 0 4 8 1 β 6 + 1 3 4 9 8 β 4 + 1 5 7 8 2 β 3 + 1 0 2 4 3 β 2 + 2 4 3 5 1
2890*b7 - 30481*b6 + 13498*b4 + 15782*b3 + 10243*b2 + 24351
ν 11 \nu^{11} ν 1 1 = = =
4704 β 11 − 2055 β 10 − 70474 β 9 − 70715 β 8 + 35060 β 5 + 102178 β 1 4704\beta_{11} - 2055\beta_{10} - 70474\beta_{9} - 70715\beta_{8} + 35060\beta_{5} + 102178\beta_1 4 7 0 4 β 1 1 − 2 0 5 5 β 1 0 − 7 0 4 7 4 β 9 − 7 0 7 1 5 β 8 + 3 5 0 6 0 β 5 + 1 0 2 1 7 8 β 1
4704*b11 - 2055*b10 - 70474*b9 - 70715*b8 + 35060*b5 + 102178*b1
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
p p p
Sign
2 2 2
+ 1 +1 + 1
7 7 7
− 1 -1 − 1
This newform subspace can be constructed as the kernel of the linear operator
T 3 12 − 29 T 3 10 + 304 T 3 8 − 1393 T 3 6 + 2574 T 3 4 − 1164 T 3 2 + 8 T_{3}^{12} - 29T_{3}^{10} + 304T_{3}^{8} - 1393T_{3}^{6} + 2574T_{3}^{4} - 1164T_{3}^{2} + 8 T 3 1 2 − 2 9 T 3 1 0 + 3 0 4 T 3 8 − 1 3 9 3 T 3 6 + 2 5 7 4 T 3 4 − 1 1 6 4 T 3 2 + 8
T3^12 - 29*T3^10 + 304*T3^8 - 1393*T3^6 + 2574*T3^4 - 1164*T3^2 + 8
acting on S 2 n e w ( Γ 0 ( 2744 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(2744)) S 2 n e w ( Γ 0 ( 2 7 4 4 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 12 T^{12} T 1 2
T^12
3 3 3
T 12 − 29 T 10 + ⋯ + 8 T^{12} - 29 T^{10} + \cdots + 8 T 1 2 − 2 9 T 1 0 + ⋯ + 8
T^12 - 29*T^10 + 304*T^8 - 1393*T^6 + 2574*T^4 - 1164*T^2 + 8
5 5 5
T 12 − 37 T 10 + ⋯ + 1352 T^{12} - 37 T^{10} + \cdots + 1352 T 1 2 − 3 7 T 1 0 + ⋯ + 1 3 5 2
T^12 - 37*T^10 + 474*T^8 - 2513*T^6 + 5694*T^4 - 4880*T^2 + 1352
7 7 7
T 12 T^{12} T 1 2
T^12
11 11 1 1
( T 6 − 9 T 5 + ⋯ + 728 ) 2 (T^{6} - 9 T^{5} + \cdots + 728)^{2} ( T 6 − 9 T 5 + ⋯ + 7 2 8 ) 2
(T^6 - 9*T^5 - 8*T^4 + 225*T^3 - 336*T^2 - 588*T + 728)^2
13 13 1 3
T 12 − 129 T 10 + ⋯ + 724808 T^{12} - 129 T^{10} + \cdots + 724808 T 1 2 − 1 2 9 T 1 0 + ⋯ + 7 2 4 8 0 8
T^12 - 129*T^10 + 6492*T^8 - 158957*T^6 + 1891400*T^4 - 8786092*T^2 + 724808
17 17 1 7
T 12 − 108 T 10 + ⋯ + 86528 T^{12} - 108 T^{10} + \cdots + 86528 T 1 2 − 1 0 8 T 1 0 + ⋯ + 8 6 5 2 8
T^12 - 108*T^10 + 3572*T^8 - 39192*T^6 + 161312*T^4 - 248064*T^2 + 86528
19 19 1 9
T 12 − 162 T 10 + ⋯ + 73544192 T^{12} - 162 T^{10} + \cdots + 73544192 T 1 2 − 1 6 2 T 1 0 + ⋯ + 7 3 5 4 4 1 9 2
T^12 - 162*T^10 + 10048*T^8 - 301224*T^6 + 4500480*T^4 - 30677376*T^2 + 73544192
23 23 2 3
( T 6 − 10 T 5 + ⋯ + 2639 ) 2 (T^{6} - 10 T^{5} + \cdots + 2639)^{2} ( T 6 − 1 0 T 5 + ⋯ + 2 6 3 9 ) 2
(T^6 - 10*T^5 - 11*T^4 + 300*T^3 - 532*T^2 - 1120*T + 2639)^2
29 29 2 9
( T 6 + 9 T 5 + ⋯ + 9304 ) 2 (T^{6} + 9 T^{5} + \cdots + 9304)^{2} ( T 6 + 9 T 5 + ⋯ + 9 3 0 4 ) 2
(T^6 + 9*T^5 - 94*T^4 - 895*T^3 + 1766*T^2 + 20472*T + 9304)^2
31 31 3 1
T 12 + ⋯ + 5776835072 T^{12} + \cdots + 5776835072 T 1 2 + ⋯ + 5 7 7 6 8 3 5 0 7 2
T^12 - 354*T^10 + 47032*T^8 - 2928744*T^6 + 88202848*T^4 - 1189597056*T^2 + 5776835072
37 37 3 7
( T 6 + 9 T 5 + ⋯ − 28904 ) 2 (T^{6} + 9 T^{5} + \cdots - 28904)^{2} ( T 6 + 9 T 5 + ⋯ − 2 8 9 0 4 ) 2
(T^6 + 9*T^5 - 146*T^4 - 1221*T^3 + 5694*T^2 + 41024*T - 28904)^2
41 41 4 1
T 12 − 276 T 10 + ⋯ + 512 T^{12} - 276 T^{10} + \cdots + 512 T 1 2 − 2 7 6 T 1 0 + ⋯ + 5 1 2
T^12 - 276*T^10 + 20708*T^8 - 232056*T^6 + 712352*T^4 - 41088*T^2 + 512
43 43 4 3
( T 6 − 5 T 5 + ⋯ − 5384 ) 2 (T^{6} - 5 T^{5} + \cdots - 5384)^{2} ( T 6 − 5 T 5 + ⋯ − 5 3 8 4 ) 2
(T^6 - 5*T^5 - 90*T^4 + 235*T^3 + 2306*T^2 + 480*T - 5384)^2
47 47 4 7
T 12 − 74 T 10 + ⋯ + 86528 T^{12} - 74 T^{10} + \cdots + 86528 T 1 2 − 7 4 T 1 0 + ⋯ + 8 6 5 2 8
T^12 - 74*T^10 + 1896*T^8 - 20104*T^6 + 91104*T^4 - 156160*T^2 + 86528
53 53 5 3
( T 6 + 19 T 5 + ⋯ + 24408 ) 2 (T^{6} + 19 T^{5} + \cdots + 24408)^{2} ( T 6 + 1 9 T 5 + ⋯ + 2 4 4 0 8 ) 2
(T^6 + 19*T^5 + 50*T^4 - 725*T^3 - 2874*T^2 + 6912*T + 24408)^2
59 59 5 9
T 12 + ⋯ + 42838986632 T^{12} + \cdots + 42838986632 T 1 2 + ⋯ + 4 2 8 3 8 9 8 6 6 3 2
T^12 - 503*T^10 + 92814*T^8 - 7870057*T^6 + 325717974*T^4 - 6258273160*T^2 + 42838986632
61 61 6 1
T 12 − 193 T 10 + ⋯ + 3623432 T^{12} - 193 T^{10} + \cdots + 3623432 T 1 2 − 1 9 3 T 1 0 + ⋯ + 3 6 2 3 4 3 2
T^12 - 193*T^10 + 11136*T^8 - 226709*T^6 + 1805964*T^4 - 5162588*T^2 + 3623432
67 67 6 7
( T 6 − 21 T 5 + ⋯ + 5144 ) 2 (T^{6} - 21 T^{5} + \cdots + 5144)^{2} ( T 6 − 2 1 T 5 + ⋯ + 5 1 4 4 ) 2
(T^6 - 21*T^5 + 114*T^4 + 161*T^3 - 2290*T^2 + 2128*T + 5144)^2
71 71 7 1
( T 6 − 28 T 5 + ⋯ + 323743 ) 2 (T^{6} - 28 T^{5} + \cdots + 323743)^{2} ( T 6 − 2 8 T 5 + ⋯ + 3 2 3 7 4 3 ) 2
(T^6 - 28*T^5 + 77*T^4 + 3038*T^3 - 15974*T^2 - 72226*T + 323743)^2
73 73 7 3
T 12 + ⋯ + 97067704832 T^{12} + \cdots + 97067704832 T 1 2 + ⋯ + 9 7 0 6 7 7 0 4 8 3 2
T^12 - 448*T^10 + 80556*T^8 - 7439768*T^6 + 371658336*T^4 - 9503196416*T^2 + 97067704832
79 79 7 9
( T 6 − 28 T 5 + ⋯ + 41957 ) 2 (T^{6} - 28 T^{5} + \cdots + 41957)^{2} ( T 6 − 2 8 T 5 + ⋯ + 4 1 9 5 7 ) 2
(T^6 - 28*T^5 + 123*T^4 + 2254*T^3 - 20136*T^2 + 28840*T + 41957)^2
83 83 8 3
T 12 + ⋯ + 5051733128 T^{12} + \cdots + 5051733128 T 1 2 + ⋯ + 5 0 5 1 7 3 3 1 2 8
T^12 - 379*T^10 + 53756*T^8 - 3539445*T^6 + 108894392*T^4 - 1365893356*T^2 + 5051733128
89 89 8 9
T 12 + ⋯ + 45858455552 T^{12} + \cdots + 45858455552 T 1 2 + ⋯ + 4 5 8 5 8 4 5 5 5 5 2
T^12 - 804*T^10 + 230172*T^8 - 28046600*T^6 + 1330130816*T^4 - 19367735296*T^2 + 45858455552
97 97 9 7
T 12 + ⋯ + 796164608 T^{12} + \cdots + 796164608 T 1 2 + ⋯ + 7 9 6 1 6 4 6 0 8
T^12 - 556*T^10 + 116020*T^8 - 11269208*T^6 + 510743328*T^4 - 8731291392*T^2 + 796164608
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