
// Make newform 74.3.d.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_74_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_74_d_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_74_3_d_b();".
// 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_74_3_d_b();".
// 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, 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_74_d();"
function MakeCharacter_74_d()
    N := 74;
    order := 4;
    char_gens := [39];
    v := [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_74_d_Hecke();"
function MakeCharacter_74_d_Hecke(Kf)
    N := 74;
    order := 4;
    char_gens := [39];
    char_values := [[0, 1]];
    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 := [[1, 1], [0, 4], [3, -3], [-4, 0], [0, 4], [3, -3], [23, -23], [10, -10], [-10, 10], [-19, -19], [-18, -18], [0, -37], [0, 74], [42, -42], [-44, 0], [-80, 0], [-54, 54], [-3, -3], [0, -12], [124, 0], [0, -10], [14, -14], [-64, 0], [17, 17], [129, -129], [0, 118], [-42, -42], [-24, 0], [-91, 91], [7, 7], [76, 0], [70, 70], [216, 0], [0, 180], [-34, 0], [0, -72], [-14, 0], [-66, -66], [-230, 230], [0, -88], [-166, -166], [304, 0], [186, -186], [-9, 9], [34, 0], [46, 46], [-208, 0], [-124, 0], [-90, 90], [-2, 0], [0, 42], [22, -22], [-175, 175], [-198, -198], [-159, 159], [0, -88], [178, 0], [404, 0], [-147, -147], [135, -135], [-258, -258], [0, 0], [0, 364], [-238, -238], [87, -87], [0, 264], [-326, 326], [0, -230], [362, 362], [-48, 0], [343, 343], [-404, 0], [492, 0], [0, -488], [-440, 0], [-450, 450], [21, -21], [0, -646], [273, 273], [39, 39], [464, 0], [-469, -469], [450, 450], [200, 0], [246, 246], [0, -668], [199, -199], [-255, 255], [-555, 555], [554, -554], [158, -158], [290, -290], [266, -266], [312, 0], [-210, -210], [298, -298], [0, 808], [0, -170], [654, -654], [147, -147], [-294, -294], [-179, -179], [282, -282], [375, 375], [-184, 0], [-623, 623], [-382, -382], [734, 0], [-1004, 0], [306, 0], [298, 298], [0, 470], [0, 304], [0, 956], [-362, -362], [-398, 0], [218, 218], [-358, -358], [555, 555], [0, 788], [-661, -661], [50, 0], [0, -296], [6, -6], [0, -708], [291, -291], [-77, 77], [-412, 0], [-262, -262], [0, 682], [0, 100], [0, -800], [0, 536], [-157, 157], [0, 272], [-761, -761], [-1282, 0], [-376, 0], [-69, -69], [631, 631], [1184, 0], [784, 0], [1116, 0], [382, -382], [603, 603], [0, -48], [-163, -163], [-815, 815], [962, -962], [156, 0], [210, 0], [0, -720], [-294, -294], [0, 904], [618, -618], [-882, 882], [-1218, -1218], [0, -666], [-424, 0], [-912, 0], [-134, 134], [0, -1264], [1006, -1006], [0, 0], [-921, -921], [0, 1480], [1038, 1038], [355, -355]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_74_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_74_3_d_b();".
// 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_74_3_d_b(:prec:=2)
    chi := MakeCharacter_74_d();
    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_74_3_d_b();".
// 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_74_3_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_74_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![16, 0, 1]>,<5,R![18, -6, 1]>],Snew);
    return Vf;
end function;