gp: [N,k,chi] = [308,2,Mod(9,308)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(308, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 10, 18]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("308.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [48]
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
The algebraic q q q -expansion of this newform has not been computed, but we have computed the trace expansion .
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 3 48 + T 3 47 − 15 T 3 46 − 8 T 3 45 + 73 T 3 44 − 24 T 3 43 + ⋯ + 1982119441 T_{3}^{48} + T_{3}^{47} - 15 T_{3}^{46} - 8 T_{3}^{45} + 73 T_{3}^{44} - 24 T_{3}^{43} + \cdots + 1982119441 T 3 4 8 + T 3 4 7 − 1 5 T 3 4 6 − 8 T 3 4 5 + 7 3 T 3 4 4 − 2 4 T 3 4 3 + ⋯ + 1 9 8 2 1 1 9 4 4 1
T3^48 + T3^47 - 15*T3^46 - 8*T3^45 + 73*T3^44 - 24*T3^43 + 512*T3^42 + 666*T3^41 - 12540*T3^40 - 9337*T3^39 + 164846*T3^38 + 160821*T3^37 - 614311*T3^36 - 570448*T3^35 - 7854285*T3^34 - 4631021*T3^33 + 130103066*T3^32 + 50073069*T3^31 - 398203464*T3^30 + 5250800*T3^29 - 1292228727*T3^28 + 1277263065*T3^27 + 16998015553*T3^26 - 9386485399*T3^25 - 49288631377*T3^24 + 81694550142*T3^23 + 143645305897*T3^22 - 87879223463*T3^21 + 133925300089*T3^20 + 122004775487*T3^19 - 1673417787878*T3^18 - 965699930991*T3^17 + 4317488610092*T3^16 + 1223714261507*T3^15 - 7434804638624*T3^14 + 2114512477893*T3^13 + 12641712041123*T3^12 + 1504215757906*T3^11 - 2866166390735*T3^10 + 4911933268738*T3^9 + 4392670719529*T3^8 + 1214752153232*T3^7 + 810263588074*T3^6 + 154740786960*T3^5 - 81903775828*T3^4 - 11350451710*T3^3 + 4762366849*T3^2 - 150302896*T3 + 1982119441
acting on S 2 n e w ( 308 , [ χ ] ) S_{2}^{\mathrm{new}}(308, [\chi]) S 2 n e w ( 3 0 8 , [ χ ] ) .