
// Make newform 810.3.j.c 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_c();".
// 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_c();".
// 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 := [6561, 0, -1296, 0, 175, 0, -16, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [20736, 0, -2800, 0, 175, 0, -16, 0], [0, -40176, 0, 5425, 0, -1225, 0, 31], [60912, 0, -8225, 0, 1400, 0, -47, 0], [0, 3079, 0, 0, 0, 0, 0, 1], [-104, 0, 0, 0, 0, 0, -1, 0], [0, 6561, 0, -775, 0, 175, 0, -16], [0, 413424, 0, -55825, 0, 4375, 0, -319]];
    Rf_basisdens := [1, 14175, 127575, 14175, 1575, 175, 18225, 127575];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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, -1, 0, 0, 0, 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, 0, -1, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 1, 0, 1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0], [12, 0, 0, 0, 0, 0, 0, 0], [0, 0, 17, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [32, -32, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, -4], [0, 0, 0, 14, 0, 14, 0, 0], [0, 0, 0, 0, 7, 0, 0, 0], [0, 0, 25, 0, 0, 0, 25, 0], [0, 0, 0, 0, 0, 0, 48, 0], [0, 0, 0, -4, 0, -4, 0, 0], [0, 16, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 20, 0, 0, -20], [0, 72, 0, 0, 0, 0, 0, 0], [0, 0, 31, 0, 0, 0, 31, 0], [0, 0, 0, 0, 0, -16, 0, 0], [0, 0, 0, 0, 28, 0, 0, 0], [0, 0, 0, 32, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -17], [0, 0, 0, 0, 0, 0, -39, 0], [80, 0, 0, 0, 0, 0, 0, 0], [0, 0, -108, 0, 0, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, -7], [0, 0, 0, -12, 0, -12, 0, 0], [0, 0, 36, 0, 0, 0, 36, 0], [-44, 44, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 2, 0, 0], [0, -136, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 20], [0, 0, 0, 0, -17, 0, 0, 17], [0, 0, 207, 0, 0, 0, 0, 0], [0, 0, 116, 0, 0, 0, 116, 0], [0, 0, 0, 0, 0, 4, 0, 0], [-82, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -72, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 20], [0, 0, 0, 0, 0, 0, -136, 0], [-312, 0, 0, 0, 0, 0, 0, 0], [12, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, 0], [0, 0, 113, 0, 0, 0, 113, 0], [-82, 82, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 136, 0], [0, 0, 0, 112, 0, 112, 0, 0], [0, -304, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 84, 0, 0], [0, 0, 276, 0, 0, 0, 0, 0], [0, 0, -209, 0, 0, 0, -209, 0], [0, 0, 0, 0, 0, 18, 0, 0], [40, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -76, 0, 0, 0], [0, 0, 0, 126, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 55], [0, 0, 60, 0, 0, 0, 0, 0], [0, 0, 0, 0, -63, 0, 0, 63], [0, 0, 0, 24, 0, 24, 0, 0], [0, 0, 0, 0, -32, 0, 0, 0], [0, 0, 368, 0, 0, 0, 368, 0], [0, -292, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -56], [0, 0, 279, 0, 0, 0, 0, 0], [0, -254, 0, 0, 0, 0, 0, 0], [0, 0, -244, 0, 0, 0, -244, 0], [0, 0, 0, 0, 0, 96, 0, 0], [0, 0, 0, 0, 71, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -108], [-572, 0, 0, 0, 0, 0, 0, 0], [0, 0, -137, 0, 0, 0, 0, 0], [0, 0, 0, -94, 0, 0, 0, 0], [0, 0, 0, 0, 88, 0, 0, -88], [0, 0, 0, 16, 0, 16, 0, 0], [-640, 640, 0, 0, 0, 0, 0, 0], [0, 0, 0, -140, 0, -140, 0, 0], [0, 656, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 88, 0, 0], [0, 0, 0, 0, -28, 0, 0, 28], [0, -432, 0, 0, 0, 0, 0, 0], [0, 0, -87, 0, 0, 0, -87, 0], [0, 0, 0, 0, 0, -210, 0, 0], [0, 0, 0, 0, 80, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 105], [0, 0, 0, 0, 0, 0, -543, 0], [0, 0, 0, -136, 0, 0, 0, 0], [0, 0, 0, 0, 111, 0, 0, -111], [0, 0, 0, -84, 0, -84, 0, 0], [660, -660, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -129, 0], [0, 0, 0, -96, 0, -96, 0, 0], [0, 0, 0, 0, 0, 32, 0, 0], [0, 0, 0, 0, 25, 0, 0, -25], [418, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -49, 0, 0, 0], [0, 0, 0, 0, 0, 0, 300, 0], [0, 0, -575, 0, 0, 0, 0, 0], [0, 0, 0, 110, 0, 0, 0, 0], [-220, 220, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -8, 0, 0, 8], [0, 0, -39, 0, 0, 0, -39, 0], [0, 0, 0, 0, 0, 0, 276, 0], [0, 0, 0, 24, 0, 24, 0, 0], [0, 880, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -73], [0, 0, 0, 0, 104, 0, 0, -104], [0, 0, -80, 0, 0, 0, 0, 0], [0, -52, 0, 0, 0, 0, 0, 0], [-544, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -206, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -63], [0, 0, 0, 0, 0, 0, -687, 0], [0, 0, 248, 0, 0, 0, 0, 0], [0, 0, 0, -140, 0, 0, 0, 0], [-80, 80, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -84, 0, 0, 0], [0, 0, -136, 0, 0, 0, -136, 0], [0, 0, 0, 0, 0, 0, 167, 0], [0, 548, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -14, 0, 0], [0, -1230, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 152, 0, 0], [0, 0, 0, 0, -63, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 172], [-340, 0, 0, 0, 0, 0, 0, 0], [0, 0, 879, 0, 0, 0, 0, 0], [520, -520, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -140, 0, 0, 140], [0, 0, 0, -96, 0, -96, 0, 0], [306, -306, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 216, 0], [0, 0, 0, 0, 0, 0, 0, 97], [0, 0, -752, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -192, 0, 0], [436, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 194, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -225], [0, 0, 0, 0, 0, 0, 697, 0], [-1488, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 112, 0, 0, 0, 0], [0, 0, 0, 0, 56, 0, 0, 0], [0, 0, 204, 0, 0, 0, 204, 0], [-796, 796, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -249, 0], [0, 0, 0, 0, 0, 0, 0, 116], [0, 0, 0, 0, 0, -126, 0, 0], [0, 0, 0, 0, -273, 0, 0, 273], [0, 0, -543, 0, 0, 0, 0, 0], [0, 0, 0, 0, 57, 0, 0, 0], [0, 0, 0, 168, 0, 0, 0, 0], [-912, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 306, 0, 0, 0, 0], [0, 0, 0, 0, 140, 0, 0, -140], [0, 0, 0, -256, 0, -256, 0, 0], [0, 0, 81, 0, 0, 0, 81, 0], [0, 0, 0, 0, 0, 0, 172, 0], [0, 0, 0, 0, 0, 0, 0, 161], [0, 0, 0, 0, 0, -324, 0, 0], [0, 0, 216, 0, 0, 0, 0, 0], [0, 0, 785, 0, 0, 0, 785, 0], [640, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -48, 0, 0, 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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_810_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![1156, 0, -34, 0, 1]>,<17,R![-128, 0, 1]>],Snew);
    return Vf;
end function;