
// Make newform 24.3.h.c in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_24_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_24_h_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_24_3_h_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_24_3_h_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 := [16, 0, 6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 2, 0, 1], [3, 0, 1, 0]];
    Rf_basisdens := [1, 1, 4, 1];
    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_24_h();"
function MakeCharacter_24_h()
    N := 24;
    order := 2;
    char_gens := [7, 13, 17];
    v := [2, 1, 1];
    // 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_24_h_Hecke();"
function MakeCharacter_24_h_Hecke(Kf)
    N := 24;
    order := 2;
    char_gens := [7, 13, 17];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [-1, 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, -1, -1], [0, 0, 4, 0], [4, 0, 0, 0], [0, 0, -6, 0], [0, 0, 0, 4], [0, 8, 4, 0], [0, 0, 0, 2], [0, -16, -8, 0], [0, 0, -12, 0], [-4, 0, 0, 0], [0, 0, 0, -20], [0, 16, 8, 0], [0, 0, 0, -2], [0, 0, 0, 0], [0, 0, 36, 0], [0, 0, -34, 0], [0, 0, 0, 36], [0, 0, 0, -18], [0, 48, 24, 0], [-6, 0, 0, 0], [124, 0, 0, 0], [0, 0, 2, 0], [0, -56, -28, 0], [118, 0, 0, 0], [0, 0, -44, 0], [-108, 0, 0, 0], [0, 0, 102, 0], [0, 0, 0, 20], [0, 48, 24, 0], [76, 0, 0, 0], [0, 0, -10, 0], [0, 0, 0, 0], [0, 0, 0, -46], [0, 0, 132, 0], [-60, 0, 0, 0], [0, 0, 0, -44], [0, 0, 0, -110], [0, -176, -88, 0], [0, 0, 132, 0], [0, 0, -226, 0], [0, 0, 0, 44], [0, 32, 16, 0], [-102, 0, 0, 0], [0, 0, -172, 0], [-188, 0, 0, 0], [0, 0, 0, 94], [188, 0, 0, 0], [0, 0, 274, 0], [0, 0, 0, 92], [0, -56, -28, 0], [0, -192, -96, 0], [122, 0, 0, 0], [0, 0, 114, 0], [0, 240, 120, 0], [0, 112, 56, 0], [0, 0, 36, 0], [348, 0, 0, 0], [0, 0, 0, 92], [0, -168, -84, 0], [0, 0, 0, 34], [0, 0, -12, 0], [0, 0, 0, 174], [0, 80, 40, 0], [-562, 0, 0, 0], [0, 0, 4, 0], [0, 0, 0, -158], [-50, 0, 0, 0], [0, 0, -254, 0], [0, 0, 0, -44], [0, -96, -48, 0], [0, 176, 88, 0], [-228, 0, 0, 0], [0, 0, 0, -52], [0, 0, 0, 62], [0, -96, -48, 0], [0, 0, 68, 0], [0, 0, 0, 20], [0, -72, -36, 0], [-158, 0, 0, 0], [0, 0, 234, 0], [0, 0, 0, -196], [0, 192, 96, 0], [86, 0, 0, 0], [-156, 0, 0, 0], [0, 0, -150, 0], [0, 264, 132, 0], [194, 0, 0, 0], [0, 0, -396, 0], [-404, 0, 0, 0], [0, 0, -470, 0], [0, -128, -64, 0], [500, 0, 0, 0], [0, 0, -82, 0], [0, 0, 0, 126], [0, -240, -120, 0], [0, 0, 228, 0], [0, -96, -48, 0], [0, 0, 0, -210], [0, 0, 0, 292], [0, 0, 0, -190], [0, 0, 372, 0], [0, 0, -6, 0], [0, 192, 96, 0], [0, 0, 0, 2], [-18, 0, 0, 0], [0, 0, 86, 0], [0, 384, 192, 0], [0, 368, 184, 0], [-358, 0, 0, 0], [-884, 0, 0, 0], [0, 0, 0, -148], [0, 56, 28, 0], [0, 0, 0, 258], [-428, 0, 0, 0], [0, -424, -212, 0], [0, 0, 0, 322], [0, -240, -120, 0], [0, 0, -780, 0], [0, 0, 654, 0], [0, 0, 0, -164], [566, 0, 0, 0], [0, 0, 564, 0], [0, 0, -286, 0], [0, 0, 0, -46], [0, 0, -236, 0], [0, 0, 0, 140], [0, -544, -272, 0], [1140, 0, 0, 0], [0, 0, 0, -140], [0, 0, 0, -2], [0, 400, 200, 0], [-1108, 0, 0, 0], [0, 0, 0, 396], [0, -688, -344, 0], [538, 0, 0, 0], [0, 0, -44, 0], [0, 0, 0, -558], [0, 0, 884, 0], [0, 240, 120, 0], [0, 0, 0, 526], [0, 0, -332, 0], [1268, 0, 0, 0], [0, 0, -810, 0], [0, 0, 0, -460], [0, -656, -328, 0], [0, 0, 0, 396], [0, 192, 96, 0], [0, 0, 0, -194], [0, -288, -144, 0], [0, 0, 0, -508], [0, 400, 200, 0], [0, 0, 0, -158], [0, -624, -312, 0], [0, 0, 0, -226], [0, -192, -96, 0], [-780, 0, 0, 0], [0, 728, 364, 0], [566, 0, 0, 0], [0, 0, -300, 0], [0, 0, -330, 0], [0, 480, 240, 0], [692, 0, 0, 0], [0, 0, -318, 0], [0, -472, -236, 0], [0, -176, -88, 0], [-340, 0, 0, 0], [0, 0, 0, 588]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_24_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_24_3_h_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_24_3_h_c(:prec:=4)
    chi := MakeCharacter_24_h();
    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_24_3_h_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_24_3_h_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_24_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-32, 0, 1]>],Snew);
    return Vf;
end function;