gp: [N,k,chi] = [1386,2,Mod(881,1386)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 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("1386.881");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [16,0,0,-16,0,0,-8]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 − 4 x 14 + 8 x 12 + 80 x 10 + 1189 x 8 − 2028 x 6 + 1800 x 4 + 1080 x 2 + 324 x^{16} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 x 1 6 − 4 x 1 4 + 8 x 1 2 + 8 0 x 1 0 + 1 1 8 9 x 8 − 2 0 2 8 x 6 + 1 8 0 0 x 4 + 1 0 8 0 x 2 + 3 2 4
x^16 - 4*x^14 + 8*x^12 + 80*x^10 + 1189*x^8 - 2028*x^6 + 1800*x^4 + 1080*x^2 + 324
:
β 1 \beta_{1} β 1 = = =
( − 96575 ν 14 + 438755 ν 12 − 939814 ν 10 − 7435564 ν 8 − 110537567 ν 6 + ⋯ − 51358644 ) / 86730966 ( - 96575 \nu^{14} + 438755 \nu^{12} - 939814 \nu^{10} - 7435564 \nu^{8} - 110537567 \nu^{6} + \cdots - 51358644 ) / 86730966 ( − 9 6 5 7 5 ν 1 4 + 4 3 8 7 5 5 ν 1 2 − 9 3 9 8 1 4 ν 1 0 − 7 4 3 5 5 6 4 ν 8 − 1 1 0 5 3 7 5 6 7 ν 6 + ⋯ − 5 1 3 5 8 6 4 4 ) / 8 6 7 3 0 9 6 6
(-96575*v^14 + 438755*v^12 - 939814*v^10 - 7435564*v^8 - 110537567*v^6 + 264348225*v^4 - 229027878*v^2 - 51358644) / 86730966
β 2 \beta_{2} β 2 = = =
( − 3833 ν 14 + 11990 ν 12 − 16306 ν 10 − 332152 ν 8 − 4851221 ν 6 + ⋯ − 7232976 ) / 1867860 ( - 3833 \nu^{14} + 11990 \nu^{12} - 16306 \nu^{10} - 332152 \nu^{8} - 4851221 \nu^{6} + \cdots - 7232976 ) / 1867860 ( − 3 8 3 3 ν 1 4 + 1 1 9 9 0 ν 1 2 − 1 6 3 0 6 ν 1 0 − 3 3 2 1 5 2 ν 8 − 4 8 5 1 2 2 1 ν 6 + ⋯ − 7 2 3 2 9 7 6 ) / 1 8 6 7 8 6 0
(-3833*v^14 + 11990*v^12 - 16306*v^10 - 332152*v^8 - 4851221*v^6 + 3917118*v^4 + 2396970*v^2 - 7232976) / 1867860
β 3 \beta_{3} β 3 = = =
( − 3079634 ν 15 + 11597979 ν 13 − 22413962 ν 11 − 248772494 ν 9 + ⋯ − 5487054570 ν ) / 2601928980 ( - 3079634 \nu^{15} + 11597979 \nu^{13} - 22413962 \nu^{11} - 248772494 \nu^{9} + \cdots - 5487054570 \nu ) / 2601928980 ( − 3 0 7 9 6 3 4 ν 1 5 + 1 1 5 9 7 9 7 9 ν 1 3 − 2 2 4 1 3 9 6 2 ν 1 1 − 2 4 8 7 7 2 4 9 4 ν 9 + ⋯ − 5 4 8 7 0 5 4 5 7 0 ν ) / 2 6 0 1 9 2 8 9 8 0
(-3079634*v^15 + 11597979*v^13 - 22413962*v^11 - 248772494*v^9 - 3735647454*v^7 + 5354283043*v^5 - 4822618578*v^3 - 5487054570*v) / 2601928980
β 4 \beta_{4} β 4 = = =
( 12591283 ν 15 − 57579817 ν 13 + 129110888 ν 11 + 952830506 ν 9 + ⋯ + 3202119162 ν ) / 7805786940 ( 12591283 \nu^{15} - 57579817 \nu^{13} + 129110888 \nu^{11} + 952830506 \nu^{9} + \cdots + 3202119162 \nu ) / 7805786940 ( 1 2 5 9 1 2 8 3 ν 1 5 − 5 7 5 7 9 8 1 7 ν 1 3 + 1 2 9 1 1 0 8 8 8 ν 1 1 + 9 5 2 8 3 0 5 0 6 ν 9 + ⋯ + 3 2 0 2 1 1 9 1 6 2 ν ) / 7 8 0 5 7 8 6 9 4 0
(12591283*v^15 - 57579817*v^13 + 129110888*v^11 + 952830506*v^9 + 14397572539*v^7 - 34397644005*v^5 + 38419343964*v^3 + 3202119162*v) / 7805786940
β 5 \beta_{5} β 5 = = =
( 14545077 ν 15 − 3765719 ν 14 − 55196585 ν 13 + 11685650 ν 12 + ⋯ − 9136826568 ) / 5203857960 ( 14545077 \nu^{15} - 3765719 \nu^{14} - 55196585 \nu^{13} + 11685650 \nu^{12} + \cdots - 9136826568 ) / 5203857960 ( 1 4 5 4 5 0 7 7 ν 1 5 − 3 7 6 5 7 1 9 ν 1 4 − 5 5 1 9 6 5 8 5 ν 1 3 + 1 1 6 8 5 6 5 0 ν 1 2 + ⋯ − 9 1 3 6 8 2 6 5 6 8 ) / 5 2 0 3 8 5 7 9 6 0
(14545077*v^15 - 3765719*v^14 - 55196585*v^13 + 11685650*v^12 + 105758504*v^11 - 14001178*v^10 + 1189121858*v^9 - 333984496*v^8 + 17514628229*v^7 - 4702927103*v^6 - 25836351497*v^5 + 3800759034*v^4 + 22083943080*v^3 + 2325320910*v^2 + 25498762434*v - 9136826568) / 5203857960
β 6 \beta_{6} β 6 = = =
( − 11395 ν 14 + 45868 ν 12 − 96938 ν 10 − 893276 ν 8 − 13578607 ν 6 + ⋯ − 6275124 ) / 2256660 ( - 11395 \nu^{14} + 45868 \nu^{12} - 96938 \nu^{10} - 893276 \nu^{8} - 13578607 \nu^{6} + \cdots - 6275124 ) / 2256660 ( − 1 1 3 9 5 ν 1 4 + 4 5 8 6 8 ν 1 2 − 9 6 9 3 8 ν 1 0 − 8 9 3 2 7 6 ν 8 − 1 3 5 7 8 6 0 7 ν 6 + ⋯ − 6 2 7 5 1 2 4 ) / 2 2 5 6 6 6 0
(-11395*v^14 + 45868*v^12 - 96938*v^10 - 893276*v^8 - 13578607*v^6 + 23130228*v^4 - 27991926*v^2 - 6275124) / 2256660
β 7 \beta_{7} β 7 = = =
( 14545077 ν 15 + 3765719 ν 14 − 55196585 ν 13 − 11685650 ν 12 + ⋯ + 3932968608 ) / 5203857960 ( 14545077 \nu^{15} + 3765719 \nu^{14} - 55196585 \nu^{13} - 11685650 \nu^{12} + \cdots + 3932968608 ) / 5203857960 ( 1 4 5 4 5 0 7 7 ν 1 5 + 3 7 6 5 7 1 9 ν 1 4 − 5 5 1 9 6 5 8 5 ν 1 3 − 1 1 6 8 5 6 5 0 ν 1 2 + ⋯ + 3 9 3 2 9 6 8 6 0 8 ) / 5 2 0 3 8 5 7 9 6 0
(14545077*v^15 + 3765719*v^14 - 55196585*v^13 - 11685650*v^12 + 105758504*v^11 + 14001178*v^10 + 1189121858*v^9 + 333984496*v^8 + 17514628229*v^7 + 4702927103*v^6 - 25836351497*v^5 - 3800759034*v^4 + 22083943080*v^3 - 2325320910*v^2 + 25498762434*v + 3932968608) / 5203857960
β 8 \beta_{8} β 8 = = =
( 18506049 ν 15 − 1441114 ν 14 − 72148880 ν 13 + 4446220 ν 12 + ⋯ − 13615971048 ) / 5203857960 ( 18506049 \nu^{15} - 1441114 \nu^{14} - 72148880 \nu^{13} + 4446220 \nu^{12} + \cdots - 13615971048 ) / 5203857960 ( 1 8 5 0 6 0 4 9 ν 1 5 − 1 4 4 1 1 1 4 ν 1 4 − 7 2 1 4 8 8 8 0 ν 1 3 + 4 4 4 6 2 2 0 ν 1 2 + ⋯ − 1 3 6 1 5 9 7 1 0 4 8 ) / 5 2 0 3 8 5 7 9 6 0
(18506049*v^15 - 1441114*v^14 - 72148880*v^13 + 4446220*v^12 + 136369838*v^11 - 4803548*v^10 + 1512241796*v^9 - 147925256*v^8 + 22114467473*v^7 - 1782429418*v^6 - 35425496144*v^5 + 1441445244*v^4 + 25614327990*v^3 + 881758260*v^2 + 31120049628*v - 13615971048) / 5203857960
β 9 \beta_{9} β 9 = = =
( 18506049 ν 15 + 1441114 ν 14 − 72148880 ν 13 − 4446220 ν 12 + ⋯ + 18819829008 ) / 5203857960 ( 18506049 \nu^{15} + 1441114 \nu^{14} - 72148880 \nu^{13} - 4446220 \nu^{12} + \cdots + 18819829008 ) / 5203857960 ( 1 8 5 0 6 0 4 9 ν 1 5 + 1 4 4 1 1 1 4 ν 1 4 − 7 2 1 4 8 8 8 0 ν 1 3 − 4 4 4 6 2 2 0 ν 1 2 + ⋯ + 1 8 8 1 9 8 2 9 0 0 8 ) / 5 2 0 3 8 5 7 9 6 0
(18506049*v^15 + 1441114*v^14 - 72148880*v^13 - 4446220*v^12 + 136369838*v^11 + 4803548*v^10 + 1512241796*v^9 + 147925256*v^8 + 22114467473*v^7 + 1782429418*v^6 - 35425496144*v^5 - 1441445244*v^4 + 25614327990*v^3 - 881758260*v^2 + 31120049628*v + 18819829008) / 5203857960
β 10 \beta_{10} β 1 0 = = =
( 60462680 ν 15 − 34694355 ν 14 − 274567721 ν 13 + 143608152 ν 12 + ⋯ − 18889526916 ) / 15611573880 ( 60462680 \nu^{15} - 34694355 \nu^{14} - 274567721 \nu^{13} + 143608152 \nu^{12} + \cdots - 18889526916 ) / 15611573880 ( 6 0 4 6 2 6 8 0 ν 1 5 − 3 4 6 9 4 3 5 5 ν 1 4 − 2 7 4 5 6 7 7 2 1 ν 1 3 + 1 4 3 6 0 8 1 5 2 ν 1 2 + ⋯ − 1 8 8 8 9 5 2 6 9 1 6 ) / 1 5 6 1 1 5 7 3 8 8 0
(60462680*v^15 - 34694355*v^14 - 274567721*v^13 + 143608152*v^12 + 617221546*v^11 - 309169302*v^10 + 4576799302*v^9 - 2703731964*v^8 + 69289811864*v^7 - 40803930363*v^6 - 161200544481*v^5 + 75040099632*v^4 + 178659278442*v^3 - 84253473054*v^2 + 14747147118*v - 18889526916) / 15611573880
β 11 \beta_{11} β 1 1 = = =
( − 60462680 ν 15 − 17310855 ν 14 + 274567721 ν 13 + 64632252 ν 12 + ⋯ − 9644970996 ) / 15611573880 ( - 60462680 \nu^{15} - 17310855 \nu^{14} + 274567721 \nu^{13} + 64632252 \nu^{12} + \cdots - 9644970996 ) / 15611573880 ( − 6 0 4 6 2 6 8 0 ν 1 5 − 1 7 3 1 0 8 5 5 ν 1 4 + 2 7 4 5 6 7 7 2 1 ν 1 3 + 6 4 6 3 2 2 5 2 ν 1 2 + ⋯ − 9 6 4 4 9 7 0 9 9 6 ) / 1 5 6 1 1 5 7 3 8 8 0
(-60462680*v^15 - 17310855*v^14 + 274567721*v^13 + 64632252*v^12 - 617221546*v^11 - 140002782*v^10 - 4576799302*v^9 - 1365330444*v^8 - 69289811864*v^7 - 20907168303*v^6 + 161200544481*v^5 + 27457419132*v^4 - 178659278442*v^3 - 43028455014*v^2 - 14747147118*v - 9644970996) / 15611573880
β 12 \beta_{12} β 1 2 = = =
( − 81296765 ν 15 + 47163030 ν 14 + 357230942 ν 13 − 210264972 ν 12 + ⋯ + 25206770376 ) / 15611573880 ( - 81296765 \nu^{15} + 47163030 \nu^{14} + 357230942 \nu^{13} - 210264972 \nu^{12} + \cdots + 25206770376 ) / 15611573880 ( − 8 1 2 9 6 7 6 5 ν 1 5 + 4 7 1 6 3 0 3 0 ν 1 4 + 3 5 7 2 3 0 9 4 2 ν 1 3 − 2 1 0 2 6 4 9 7 2 ν 1 2 + ⋯ + 2 5 2 0 6 7 7 0 3 7 6 ) / 1 5 6 1 1 5 7 3 8 8 0
(-81296765*v^15 + 47163030*v^14 + 357230942*v^13 - 210264972*v^12 - 785602642*v^11 + 450833772*v^10 - 6207754144*v^9 + 3640498104*v^8 - 94072523573*v^7 + 54294184518*v^6 + 202259564982*v^5 - 121172690412*v^4 - 218620670994*v^3 + 112411745244*v^2 - 17473193856*v + 25206770376) / 15611573880
β 13 \beta_{13} β 1 3 = = =
( − 81296765 ν 15 − 47163030 ν 14 + 357230942 ν 13 + 210264972 ν 12 + ⋯ − 25206770376 ) / 15611573880 ( - 81296765 \nu^{15} - 47163030 \nu^{14} + 357230942 \nu^{13} + 210264972 \nu^{12} + \cdots - 25206770376 ) / 15611573880 ( − 8 1 2 9 6 7 6 5 ν 1 5 − 4 7 1 6 3 0 3 0 ν 1 4 + 3 5 7 2 3 0 9 4 2 ν 1 3 + 2 1 0 2 6 4 9 7 2 ν 1 2 + ⋯ − 2 5 2 0 6 7 7 0 3 7 6 ) / 1 5 6 1 1 5 7 3 8 8 0
(-81296765*v^15 - 47163030*v^14 + 357230942*v^13 + 210264972*v^12 - 785602642*v^11 - 450833772*v^10 - 6207754144*v^9 - 3640498104*v^8 - 94072523573*v^7 - 54294184518*v^6 + 202259564982*v^5 + 121172690412*v^4 - 218620670994*v^3 - 112411745244*v^2 - 17473193856*v - 25206770376) / 15611573880
β 14 \beta_{14} β 1 4 = = =
( 23298266 ν 15 − 93544963 ν 13 + 176425470 ν 11 + 1897812810 ν 9 + ⋯ + 37162486386 ν ) / 2601928980 ( 23298266 \nu^{15} - 93544963 \nu^{13} + 176425470 \nu^{11} + 1897812810 \nu^{9} + \cdots + 37162486386 \nu ) / 2601928980 ( 2 3 2 9 8 2 6 6 ν 1 5 − 9 3 5 4 4 9 6 3 ν 1 3 + 1 7 6 4 2 5 4 7 0 ν 1 1 + 1 8 9 7 8 1 2 8 1 0 ν 9 + ⋯ + 3 7 1 6 2 4 8 6 3 8 6 ν ) / 2 6 0 1 9 2 8 9 8 0
(23298266*v^15 - 93544963*v^13 + 176425470*v^11 + 1897812810*v^9 + 27613656994*v^7 - 48610532639*v^5 + 28420002066*v^3 + 37162486386*v) / 2601928980
β 15 \beta_{15} β 1 5 = = =
( − 15573353 ν 15 + 65944679 ν 13 − 141046900 ν 11 − 1206672370 ν 9 + ⋯ − 2860280478 ν ) / 1115112420 ( - 15573353 \nu^{15} + 65944679 \nu^{13} - 141046900 \nu^{11} - 1206672370 \nu^{9} + \cdots - 2860280478 \nu ) / 1115112420 ( − 1 5 5 7 3 3 5 3 ν 1 5 + 6 5 9 4 4 6 7 9 ν 1 3 − 1 4 1 0 4 6 9 0 0 ν 1 1 − 1 2 0 6 6 7 2 3 7 0 ν 9 + ⋯ − 2 8 6 0 2 8 0 4 7 8 ν ) / 1 1 1 5 1 1 2 4 2 0
(-15573353*v^15 + 65944679*v^13 - 141046900*v^11 - 1206672370*v^9 - 18241907237*v^7 + 35778301767*v^5 - 37432034028*v^3 - 2860280478*v) / 1115112420
ν \nu ν = = =
( β 15 − β 13 − β 12 − 3 β 3 ) / 6 ( \beta_{15} - \beta_{13} - \beta_{12} - 3\beta_{3} ) / 6 ( β 1 5 − β 1 3 − β 1 2 − 3 β 3 ) / 6
(b15 - b13 - b12 - 3*b3) / 6
ν 2 \nu^{2} ν 2 = = =
( β 11 + β 10 + β 7 − β 6 − β 5 + β 2 + β 1 + 1 ) / 2 ( \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 + 1 ) / 2 ( β 1 1 + β 1 0 + β 7 − β 6 − β 5 + β 2 + β 1 + 1 ) / 2
(b11 + b10 + b7 - b6 - b5 + b2 + b1 + 1) / 2
ν 3 \nu^{3} ν 3 = = =
( − β 14 + β 11 − β 10 + 2 β 9 + 2 β 8 − β 7 − β 5 + 5 β 4 + β 1 − 3 ) / 2 ( -\beta_{14} + \beta_{11} - \beta_{10} + 2\beta_{9} + 2\beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + \beta _1 - 3 ) / 2 ( − β 1 4 + β 1 1 − β 1 0 + 2 β 9 + 2 β 8 − β 7 − β 5 + 5 β 4 + β 1 − 3 ) / 2
(-b14 + b11 - b10 + 2*b9 + 2*b8 - b7 - b5 + 5*b4 + b1 - 3) / 2
ν 4 \nu^{4} ν 4 = = =
( − 7 β 13 + 7 β 12 + 2 β 11 + 2 β 10 + 32 β 1 ) / 2 ( -7\beta_{13} + 7\beta_{12} + 2\beta_{11} + 2\beta_{10} + 32\beta_1 ) / 2 ( − 7 β 1 3 + 7 β 1 2 + 2 β 1 1 + 2 β 1 0 + 3 2 β 1 ) / 2
(-7*b13 + 7*b12 + 2*b11 + 2*b10 + 32*b1) / 2
ν 5 \nu^{5} ν 5 = = =
( − 5 β 14 + β 13 + β 12 − 7 β 11 + 7 β 10 + 13 β 9 + 13 β 8 + ⋯ − 22 ) / 2 ( - 5 \beta_{14} + \beta_{13} + \beta_{12} - 7 \beta_{11} + 7 \beta_{10} + 13 \beta_{9} + 13 \beta_{8} + \cdots - 22 ) / 2 ( − 5 β 1 4 + β 1 3 + β 1 2 − 7 β 1 1 + 7 β 1 0 + 1 3 β 9 + 1 3 β 8 + ⋯ − 2 2 ) / 2
(-5*b14 + b13 + b12 - 7*b11 + 7*b10 + 13*b9 + 13*b8 - 9*b7 - 9*b5 - 27*b4 - 2*b3 - 7*b1 - 22) / 2
ν 6 \nu^{6} ν 6 = = =
( − 11 β 13 + 11 β 12 + 45 β 11 + 45 β 10 + 11 β 9 − 11 β 8 + ⋯ − 68 ) / 2 ( - 11 \beta_{13} + 11 \beta_{12} + 45 \beta_{11} + 45 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} + \cdots - 68 ) / 2 ( − 1 1 β 1 3 + 1 1 β 1 2 + 4 5 β 1 1 + 4 5 β 1 0 + 1 1 β 9 − 1 1 β 8 + ⋯ − 6 8 ) / 2
(-11*b13 + 11*b12 + 45*b11 + 45*b10 + 11*b9 - 11*b8 - 45*b7 - 29*b6 + 45*b5 - 29*b2 + 57*b1 - 68) / 2
ν 7 \nu^{7} ν 7 = = =
( − 31 β 15 − 2 β 14 + 87 β 13 + 87 β 12 − 67 β 11 + 67 β 10 + ⋯ − 58 ) / 2 ( - 31 \beta_{15} - 2 \beta_{14} + 87 \beta_{13} + 87 \beta_{12} - 67 \beta_{11} + 67 \beta_{10} + \cdots - 58 ) / 2 ( − 3 1 β 1 5 − 2 β 1 4 + 8 7 β 1 3 + 8 7 β 1 2 − 6 7 β 1 1 + 6 7 β 1 0 + ⋯ − 5 8 ) / 2
(-31*b15 - 2*b14 + 87*b13 + 87*b12 - 67*b11 + 67*b10 + 13*b9 + 13*b8 - 45*b7 - 45*b5 - 28*b4 - 149*b3 - 67*b1 - 58) / 2
ν 8 \nu^{8} ν 8 = = =
( 289 β 9 − 289 β 8 − 190 β 7 + 190 β 5 − 56 β 2 − 1541 ) / 2 ( 289\beta_{9} - 289\beta_{8} - 190\beta_{7} + 190\beta_{5} - 56\beta_{2} - 1541 ) / 2 ( 2 8 9 β 9 − 2 8 9 β 8 − 1 9 0 β 7 + 1 9 0 β 5 − 5 6 β 2 − 1 5 4 1 ) / 2
(289*b9 - 289*b8 - 190*b7 + 190*b5 - 56*b2 - 1541) / 2
ν 9 \nu^{9} ν 9 = = =
( − 205 β 15 + 28 β 14 + 589 β 13 + 589 β 12 − 479 β 11 + 479 β 10 + ⋯ + 412 ) / 2 ( - 205 \beta_{15} + 28 \beta_{14} + 589 \beta_{13} + 589 \beta_{12} - 479 \beta_{11} + 479 \beta_{10} + \cdots + 412 ) / 2 ( − 2 0 5 β 1 5 + 2 8 β 1 4 + 5 8 9 β 1 3 + 5 8 9 β 1 2 − 4 7 9 β 1 1 + 4 7 9 β 1 0 + ⋯ + 4 1 2 ) / 2
(-205*b15 + 28*b14 + 589*b13 + 589*b12 - 479*b11 + 479*b10 - 123*b9 - 123*b8 + 289*b7 + 289*b5 - 274*b4 + 835*b3 - 479*b1 + 412) / 2
ν 10 \nu^{10} ν 1 0 = = =
( 753 β 13 − 753 β 12 − 1877 β 11 − 1877 β 10 + 753 β 9 − 753 β 8 + ⋯ − 4274 ) / 2 ( 753 \beta_{13} - 753 \beta_{12} - 1877 \beta_{11} - 1877 \beta_{10} + 753 \beta_{9} - 753 \beta_{8} + \cdots - 4274 ) / 2 ( 7 5 3 β 1 3 − 7 5 3 β 1 2 − 1 8 7 7 β 1 1 − 1 8 7 7 β 1 0 + 7 5 3 β 9 − 7 5 3 β 8 + ⋯ − 4 2 7 4 ) / 2
(753*b13 - 753*b12 - 1877*b11 - 1877*b10 + 753*b9 - 753*b8 - 1877*b7 + 1109*b6 + 1877*b5 - 1109*b2 - 3521*b1 - 4274) / 2
ν 11 \nu^{11} ν 1 1 = = =
( − 274 β 15 + 1383 β 14 + 1027 β 13 + 1027 β 12 − 1877 β 11 + 1877 β 10 + ⋯ + 7396 ) / 2 ( - 274 \beta_{15} + 1383 \beta_{14} + 1027 \beta_{13} + 1027 \beta_{12} - 1877 \beta_{11} + 1877 \beta_{10} + \cdots + 7396 ) / 2 ( − 2 7 4 β 1 5 + 1 3 8 3 β 1 4 + 1 0 2 7 β 1 3 + 1 0 2 7 β 1 2 − 1 8 7 7 β 1 1 + 1 8 7 7 β 1 0 + ⋯ + 7 3 9 6 ) / 2
(-274*b15 + 1383*b14 + 1027*b13 + 1027*b12 - 1877*b11 + 1877*b10 - 4013*b9 - 4013*b8 + 3383*b7 + 3383*b5 - 4753*b4 + 2328*b3 - 1877*b1 + 7396) / 2
ν 12 \nu^{12} ν 1 2 = = =
( 12341 β 13 − 12341 β 12 − 11422 β 11 − 11422 β 10 + 4656 β 6 − 53908 β 1 ) / 2 ( 12341\beta_{13} - 12341\beta_{12} - 11422\beta_{11} - 11422\beta_{10} + 4656\beta_{6} - 53908\beta_1 ) / 2 ( 1 2 3 4 1 β 1 3 − 1 2 3 4 1 β 1 2 − 1 1 4 2 2 β 1 1 − 1 1 4 2 2 β 1 0 + 4 6 5 6 β 6 − 5 3 9 0 8 β 1 ) / 2
(12341*b13 - 12341*b12 - 11422*b11 - 11422*b10 + 4656*b6 - 53908*b1) / 2
ν 13 \nu^{13} ν 1 3 = = =
( 2328 β 15 + 9409 β 14 − 8039 β 13 − 8039 β 12 + 12341 β 11 − 12341 β 10 + ⋯ + 51224 ) / 2 ( 2328 \beta_{15} + 9409 \beta_{14} - 8039 \beta_{13} - 8039 \beta_{12} + 12341 \beta_{11} - 12341 \beta_{10} + \cdots + 51224 ) / 2 ( 2 3 2 8 β 1 5 + 9 4 0 9 β 1 4 − 8 0 3 9 β 1 3 − 8 0 3 9 β 1 2 + 1 2 3 4 1 β 1 1 − 1 2 3 4 1 β 1 0 + ⋯ + 5 1 2 2 4 ) / 2
(2328*b15 + 9409*b14 - 8039*b13 - 8039*b12 + 12341*b11 - 12341*b10 - 27461*b9 - 27461*b8 + 23763*b7 + 23763*b5 + 27519*b4 + 18406*b3 + 12341*b1 + 51224) / 2
ν 14 \nu^{14} ν 1 4 = = =
( 42169 β 13 − 42169 β 12 − 82029 β 11 − 82029 β 10 − 42169 β 9 + ⋯ + 234634 ) / 2 ( 42169 \beta_{13} - 42169 \beta_{12} - 82029 \beta_{11} - 82029 \beta_{10} - 42169 \beta_{9} + \cdots + 234634 ) / 2 ( 4 2 1 6 9 β 1 3 − 4 2 1 6 9 β 1 2 − 8 2 0 2 9 β 1 1 − 8 2 0 2 9 β 1 0 − 4 2 1 6 9 β 9 + ⋯ + 2 3 4 6 3 4 ) / 2
(42169*b13 - 42169*b12 - 82029*b11 - 82029*b10 - 42169*b9 + 42169*b8 + 82029*b7 + 45925*b6 - 82029*b5 + 45925*b2 - 192465*b1 + 234634) / 2
ν 15 \nu^{15} ν 1 5 = = =
( 64331 β 15 + 18406 β 14 − 188529 β 13 − 188529 β 12 + 166367 β 11 + ⋯ + 142604 ) / 2 ( 64331 \beta_{15} + 18406 \beta_{14} - 188529 \beta_{13} - 188529 \beta_{12} + 166367 \beta_{11} + \cdots + 142604 ) / 2 ( 6 4 3 3 1 β 1 5 + 1 8 4 0 6 β 1 4 − 1 8 8 5 2 9 β 1 3 − 1 8 8 5 2 9 β 1 2 + 1 6 6 3 6 7 β 1 1 + ⋯ + 1 4 2 6 0 4 ) / 2
(64331*b15 + 18406*b14 - 188529*b13 - 188529*b12 + 166367*b11 - 166367*b10 - 60575*b9 - 60575*b8 + 82029*b7 + 82029*b5 + 139556*b4 + 162277*b3 + 166367*b1 + 142604) / 2
Character values
We give the values of χ \chi χ on generators for ( Z / 1386 Z ) × \left(\mathbb{Z}/1386\mathbb{Z}\right)^\times ( Z / 1 3 8 6 Z ) × .
n n n
155 155 1 5 5
199 199 1 9 9
1135 1135 1 1 3 5
χ ( n ) \chi(n) χ ( n )
− 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 8 − 24 T 5 6 + 118 T 5 4 − 168 T 5 2 + 72 T_{5}^{8} - 24T_{5}^{6} + 118T_{5}^{4} - 168T_{5}^{2} + 72 T 5 8 − 2 4 T 5 6 + 1 1 8 T 5 4 − 1 6 8 T 5 2 + 7 2
T5^8 - 24*T5^6 + 118*T5^4 - 168*T5^2 + 72
acting on S 2 n e w ( 1386 , [ χ ] ) S_{2}^{\mathrm{new}}(1386, [\chi]) S 2 n e w ( 1 3 8 6 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 + 1 ) 8 (T^{2} + 1)^{8} ( T 2 + 1 ) 8
(T^2 + 1)^8
3 3 3
T 16 T^{16} T 1 6
T^16
5 5 5
( T 8 − 24 T 6 + ⋯ + 72 ) 2 (T^{8} - 24 T^{6} + \cdots + 72)^{2} ( T 8 − 2 4 T 6 + ⋯ + 7 2 ) 2
(T^8 - 24*T^6 + 118*T^4 - 168*T^2 + 72)^2
7 7 7
( T 8 + 4 T 7 + ⋯ + 2401 ) 2 (T^{8} + 4 T^{7} + \cdots + 2401)^{2} ( T 8 + 4 T 7 + ⋯ + 2 4 0 1 ) 2
(T^8 + 4*T^7 + 6*T^6 + 28*T^5 + 122*T^4 + 196*T^3 + 294*T^2 + 1372*T + 2401)^2
11 11 1 1
( T 2 + 1 ) 8 (T^{2} + 1)^{8} ( T 2 + 1 ) 8
(T^2 + 1)^8
13 13 1 3
T 16 T^{16} T 1 6
T^16
17 17 1 7
( T 8 − 84 T 6 + ⋯ + 83232 ) 2 (T^{8} - 84 T^{6} + \cdots + 83232)^{2} ( T 8 − 8 4 T 6 + ⋯ + 8 3 2 3 2 ) 2
(T^8 - 84*T^6 + 2386*T^4 - 25872*T^2 + 83232)^2
19 19 1 9
( T 8 + 104 T 6 + ⋯ + 72 ) 2 (T^{8} + 104 T^{6} + \cdots + 72)^{2} ( T 8 + 1 0 4 T 6 + ⋯ + 7 2 ) 2
(T^8 + 104*T^6 + 2710*T^4 + 1320*T^2 + 72)^2
23 23 2 3
( T 8 + 120 T 6 + ⋯ + 331776 ) 2 (T^{8} + 120 T^{6} + \cdots + 331776)^{2} ( T 8 + 1 2 0 T 6 + ⋯ + 3 3 1 7 7 6 ) 2
(T^8 + 120*T^6 + 4624*T^4 + 69120*T^2 + 331776)^2
29 29 2 9
( T 8 + 108 T 6 + ⋯ + 5184 ) 2 (T^{8} + 108 T^{6} + \cdots + 5184)^{2} ( T 8 + 1 0 8 T 6 + ⋯ + 5 1 8 4 ) 2
(T^8 + 108*T^6 + 2932*T^4 + 7776*T^2 + 5184)^2
31 31 3 1
( T 8 + 268 T 6 + ⋯ + 6139008 ) 2 (T^{8} + 268 T^{6} + \cdots + 6139008)^{2} ( T 8 + 2 6 8 T 6 + ⋯ + 6 1 3 9 0 0 8 ) 2
(T^8 + 268*T^6 + 23458*T^4 + 727008*T^2 + 6139008)^2
37 37 3 7
( T 4 + 4 T 3 + ⋯ + 544 ) 4 (T^{4} + 4 T^{3} + \cdots + 544)^{4} ( T 4 + 4 T 3 + ⋯ + 5 4 4 ) 4
(T^4 + 4*T^3 - 52*T^2 - 128*T + 544)^4
41 41 4 1
( T 8 − 132 T 6 + ⋯ + 23328 ) 2 (T^{8} - 132 T^{6} + \cdots + 23328)^{2} ( T 8 − 1 3 2 T 6 + ⋯ + 2 3 3 2 8 ) 2
(T^8 - 132*T^6 + 3154*T^4 - 18576*T^2 + 23328)^2
43 43 4 3
( T 4 − 12 T 3 + ⋯ − 16 ) 4 (T^{4} - 12 T^{3} + \cdots - 16)^{4} ( T 4 − 1 2 T 3 + ⋯ − 1 6 ) 4
(T^4 - 12*T^3 + 26*T^2 + 64*T - 16)^4
47 47 4 7
( T 8 − 196 T 6 + ⋯ + 288 ) 2 (T^{8} - 196 T^{6} + \cdots + 288)^{2} ( T 8 − 1 9 6 T 6 + ⋯ + 2 8 8 ) 2
(T^8 - 196*T^6 + 7474*T^4 - 3120*T^2 + 288)^2
53 53 5 3
( T 8 + 260 T 6 + ⋯ + 8714304 ) 2 (T^{8} + 260 T^{6} + \cdots + 8714304)^{2} ( T 8 + 2 6 0 T 6 + ⋯ + 8 7 1 4 3 0 4 ) 2
(T^8 + 260*T^6 + 22516*T^4 + 776736*T^2 + 8714304)^2
59 59 5 9
( T 8 − 96 T 6 + ⋯ + 18432 ) 2 (T^{8} - 96 T^{6} + \cdots + 18432)^{2} ( T 8 − 9 6 T 6 + ⋯ + 1 8 4 3 2 ) 2
(T^8 - 96*T^6 + 1888*T^4 - 10752*T^2 + 18432)^2
61 61 6 1
( T 8 + 528 T 6 + ⋯ + 294912 ) 2 (T^{8} + 528 T^{6} + \cdots + 294912)^{2} ( T 8 + 5 2 8 T 6 + ⋯ + 2 9 4 9 1 2 ) 2
(T^8 + 528*T^6 + 89152*T^4 + 4706304*T^2 + 294912)^2
67 67 6 7
( T 4 − 4 T 3 − 70 T 2 + ⋯ + 16 ) 4 (T^{4} - 4 T^{3} - 70 T^{2} + \cdots + 16)^{4} ( T 4 − 4 T 3 − 7 0 T 2 + ⋯ + 1 6 ) 4
(T^4 - 4*T^3 - 70*T^2 + 176*T + 16)^4
71 71 7 1
( T 8 + 312 T 6 + ⋯ + 82944 ) 2 (T^{8} + 312 T^{6} + \cdots + 82944)^{2} ( T 8 + 3 1 2 T 6 + ⋯ + 8 2 9 4 4 ) 2
(T^8 + 312*T^6 + 25552*T^4 + 232704*T^2 + 82944)^2
73 73 7 3
( T 8 + 196 T 6 + ⋯ + 288 ) 2 (T^{8} + 196 T^{6} + \cdots + 288)^{2} ( T 8 + 1 9 6 T 6 + ⋯ + 2 8 8 ) 2
(T^8 + 196*T^6 + 7474*T^4 + 3120*T^2 + 288)^2
79 79 7 9
( T 4 + 4 T 3 + ⋯ − 292 ) 4 (T^{4} + 4 T^{3} + \cdots - 292)^{4} ( T 4 + 4 T 3 + ⋯ − 2 9 2 ) 4
(T^4 + 4*T^3 - 176*T^2 - 840*T - 292)^4
83 83 8 3
( T 8 − 348 T 6 + ⋯ + 1492992 ) 2 (T^{8} - 348 T^{6} + \cdots + 1492992)^{2} ( T 8 − 3 4 8 T 6 + ⋯ + 1 4 9 2 9 9 2 ) 2
(T^8 - 348*T^6 + 33282*T^4 - 549504*T^2 + 1492992)^2
89 89 8 9
( T 8 − 420 T 6 + ⋯ + 1936512 ) 2 (T^{8} - 420 T^{6} + \cdots + 1936512)^{2} ( T 8 − 4 2 0 T 6 + ⋯ + 1 9 3 6 5 1 2 ) 2
(T^8 - 420*T^6 + 40708*T^4 - 889152*T^2 + 1936512)^2
97 97 9 7
( T 8 + 480 T 6 + ⋯ + 18874368 ) 2 (T^{8} + 480 T^{6} + \cdots + 18874368)^{2} ( T 8 + 4 8 0 T 6 + ⋯ + 1 8 8 7 4 3 6 8 ) 2
(T^8 + 480*T^6 + 69760*T^4 + 3047424*T^2 + 18874368)^2
show more
show less