
// Make newform 810.3.j.e in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_810_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_810_j_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_810_3_j_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_810_3_j_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_810_j();"
function MakeCharacter_810_j()
    N := 810;
    order := 6;
    char_gens := [731, 487];
    v := [1, 3];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_810_j_Hecke();"
function MakeCharacter_810_j_Hecke(Kf)
    N := 810;
    order := 6;
    char_gens := [731, 487];
    char_values := [[1, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 3;
    raw_aps := [[0, -1, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5, 0, 0, 0, 5], [0, 0, 5, 0, 0, 0, -5, 0], [0, -1, 0, 0, 0, 0, 0, 1], [0, 0, -9, 0, 0, 0, 0, 0], [0, -8, 0, -8, 0, 8, 0, 0], [21, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -1, 0, -1], [0, -27, 0, 0, 0, 0, 0, 27], [-40, 0, 0, 0, 40, 0, 0, 0], [0, 0, 0, 0, 0, 0, -25, 0], [0, 0, 0, 37, 0, 37, 0, -37], [0, 0, 64, 0, 0, 0, -64, 0], [0, 16, 0, 0, 0, 0, 0, 16], [0, -51, 0, -51, 0, 51, 0, 0], [0, 0, 0, 64, 0, 64, 0, -64], [0, 0, 0, 0, 97, 0, 0, 0], [0, 0, 131, 0, 0, 0, 0, 0], [0, -63, 0, 63, 0, 63, 0, 0], [0, 0, 0, 0, 0, 0, -17, 0], [0, 0, 0, 0, 117, 0, 0, 0], [0, -41, 0, 0, 0, 0, 0, -41], [0, 104, 0, -104, 0, -104, 0, 0], [0, 0, -41, 0, 0, 0, 41, 0], [0, 64, 0, 0, 0, 0, 0, -64], [0, 0, 13, 0, 0, 0, 0, 0], [0, 87, 0, 87, 0, -87, 0, 0], [8, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -27, 0, 27, 0, 27], [0, 0, 0, 0, 0, 0, 8, 0], [0, 0, 0, -96, 0, -96, 0, 96], [0, 189, 0, 0, 0, 0, 0, 189], [37, 0, 0, 0, -37, 0, 0, 0], [0, 0, 0, 184, 0, 184, 0, -184], [0, 0, 0, 0, -109, 0, 0, 0], [0, 0, -118, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 203, 0], [0, 0, 0, -72, 0, 72, 0, 72], [0, 8, 0, 0, 0, 0, 0, 8], [0, -89, 0, 89, 0, 89, 0, 0], [-127, 0, 0, 0, 0, 0, 0, 0], [0, 72, 0, 0, 0, 0, 0, -72], [0, 0, -271, 0, 0, 0, 0, 0], [0, 224, 0, 224, 0, -224, 0, 0], [147, 0, 0, 0, 0, 0, 0, 0], [-141, 0, 0, 0, 141, 0, 0, 0], [0, 0, -8, 0, 0, 0, 8, 0], [0, -49, 0, 0, 0, 0, 0, -49], [8, 0, 0, 0, -8, 0, 0, 0], [0, -224, 0, -224, 0, 224, 0, 0], [0, 0, 0, -145, 0, -145, 0, 145], [0, 0, 0, 0, -79, 0, 0, 0], [0, -33, 0, 33, 0, 33, 0, 0], [0, 0, 0, 120, 0, -120, 0, -120], [0, -200, 0, 0, 0, 0, 0, -200], [0, -72, 0, 72, 0, 72, 0, 0], [-221, 0, 0, 0, 0, 0, 0, 0], [0, 0, -88, 0, 0, 0, 88, 0], [0, -144, 0, 0, 0, 0, 0, 144], [0, 0, -32, 0, 0, 0, 0, 0], [0, 0, 0, -123, 0, 123, 0, 123], [0, 0, 0, 0, 0, 0, -486, 0], [0, 0, 0, 48, 0, 48, 0, -48], [0, 0, -281, 0, 0, 0, 281, 0], [0, 368, 0, 0, 0, 0, 0, 368], [0, 0, 0, 0, 59, 0, 0, 0], [0, 0, 55, 0, 0, 0, 0, 0], [0, 0, 0, 423, 0, -423, 0, -423], [0, 0, 0, 0, 439, 0, 0, 0], [0, 368, 0, 0, 0, 0, 0, 368], [0, -39, 0, 39, 0, 39, 0, 0], [0, 0, -589, 0, 0, 0, 589, 0], [0, 0, -9, 0, 0, 0, 0, 0], [157, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 200, 0, -200, 0, -200], [0, 424, 0, 0, 0, 0, 0, -424], [0, 0, 0, 0, 0, 0, 296, 0], [0, 0, 0, 275, 0, 275, 0, -275], [-145, 0, 0, 0, 145, 0, 0, 0], [0, 0, 0, -496, 0, -496, 0, 496], [0, 0, 0, 0, -505, 0, 0, 0], [0, 31, 0, -31, 0, -31, 0, 0], [0, 0, 0, 0, 0, 0, -32, 0], [0, 0, 0, 0, 504, 0, 0, 0], [0, -168, 0, 0, 0, 0, 0, -168], [0, -48, 0, 48, 0, 48, 0, 0], [0, 0, 752, 0, 0, 0, -752, 0], [0, 432, 0, 0, 0, 0, 0, -432], [0, 0, -597, 0, 0, 0, 0, 0], [0, 600, 0, 600, 0, -600, 0, 0], [0, -119, 0, 0, 0, 0, 0, 119], [0, 0, 0, 0, 0, 0, -507, 0], [0, 0, 0, -303, 0, -303, 0, 303], [-870, 0, 0, 0, 870, 0, 0, 0], [0, -327, 0, -327, 0, 327, 0, 0], [0, 0, 0, -579, 0, -579, 0, 579], [0, 611, 0, -611, 0, -611, 0, 0], [0, 0, 0, 0, 0, 0, -163, 0], [697, 0, 0, 0, 0, 0, 0, 0], [0, 0, 389, 0, 0, 0, -389, 0], [0, 768, 0, 768, 0, -768, 0, 0], [0, 0, 0, 440, 0, -440, 0, -440], [0, 328, 0, 0, 0, 0, 0, -328], [923, 0, 0, 0, -923, 0, 0, 0], [0, 0, 0, 0, 0, 0, -247, 0], [0, 321, 0, 0, 0, 0, 0, 321], [0, -3, 0, -3, 0, 3, 0, 0], [0, 0, 0, -159, 0, -159, 0, 159], [0, 0, 0, 0, 736, 0, 0, 0], [0, 0, 437, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -335, 0], [0, 0, 0, -181, 0, 181, 0, 181], [0, 0, 0, 0, 965, 0, 0, 0], [275, 0, 0, 0, 0, 0, 0, 0], [0, 344, 0, 0, 0, 0, 0, -344], [0, 0, 1152, 0, 0, 0, 0, 0], [0, -489, 0, -489, 0, 489, 0, 0], [0, 0, 0, -248, 0, 248, 0, 248], [0, -352, 0, 0, 0, 0, 0, 352], [577, 0, 0, 0, -577, 0, 0, 0], [0, 0, -489, 0, 0, 0, 489, 0], [0, -424, 0, 0, 0, 0, 0, -424], [0, 167, 0, 167, 0, -167, 0, 0], [0, 0, 0, 0, -640, 0, 0, 0], [0, -773, 0, 773, 0, 773, 0, 0], [0, 0, 0, 0, 489, 0, 0, 0], [0, -439, 0, 439, 0, 439, 0, 0], [0, 0, -1080, 0, 0, 0, 1080, 0], [0, 0, -248, 0, 0, 0, 0, 0], [848, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -24, 0, 24, 0, 24], [133, 0, 0, 0, -133, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1271, 0], [0, 0, 0, -381, 0, -381, 0, 381], [81, 0, 0, 0, -81, 0, 0, 0], [0, 315, 0, 315, 0, -315, 0, 0], [0, 0, -395, 0, 0, 0, 0, 0], [0, 0, 0, -760, 0, 760, 0, 760], [0, 888, 0, -888, 0, -888, 0, 0], [-752, 0, 0, 0, 0, 0, 0, 0], [0, 352, 0, 0, 0, 0, 0, -352], [0, 0, 531, 0, 0, 0, 0, 0], [0, -248, 0, -248, 0, 248, 0, 0], [-705, 0, 0, 0, 0, 0, 0, 0], [0, -1144, 0, 0, 0, 0, 0, 1144], [0, 0, -1639, 0, 0, 0, 1639, 0], [0, 699, 0, 0, 0, 0, 0, 699], [-355, 0, 0, 0, 355, 0, 0, 0], [0, -1032, 0, -1032, 0, 1032, 0, 0], [0, 0, -1129, 0, 0, 0, 0, 0], [0, -117, 0, 117, 0, 117, 0, 0], [0, 0, 0, 0, 0, 0, 1227, 0], [0, 0, 0, -168, 0, 168, 0, 168], [0, 0, -1005, 0, 0, 0, 1005, 0], [0, 552, 0, 0, 0, 0, 0, -552], [600, 0, 0, 0, 0, 0, 0, 0], [0, -891, 0, 0, 0, 0, 0, 891], [0, 0, 0, 0, 0, 0, 1465, 0], [0, 0, 0, 181, 0, 181, 0, -181], [0, -288, 0, 0, 0, 0, 0, -288], [0, 352, 0, 352, 0, -352, 0, 0], [0, 0, 491, 0, 0, 0, 0, 0], [0, 360, 0, -360, 0, -360, 0, 0], [0, 0, 0, 333, 0, -333, 0, -333], [0, 425, 0, 0, 0, 0, 0, 425], [-755, 0, 0, 0, 0, 0, 0, 0], [0, 0, 24, 0, 0, 0, -24, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_810_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_810_3_j_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_810_3_j_e(:prec:=8)
    chi := MakeCharacter_810_j();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_810_3_j_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_810_3_j_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_810_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![625, 0, -25, 0, 1]>,<17,R![-128, 0, 1]>],Snew);
    return Vf;
end function;