
// Make newform 684.3.y.c in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_684_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_684_y_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_684_3_y_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_684_3_y_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 := [1, -1, 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_684_y();"
function MakeCharacter_684_y()
    N := 684;
    order := 6;
    char_gens := [343, 533, 325];
    v := [6, 6, 5];
    // 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_684_y_Hecke();"
function MakeCharacter_684_y_Hecke(Kf)
    N := 684;
    order := 6;
    char_gens := [343, 533, 325];
    char_values := [[1, 0], [1, 0], [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 := [[0, 0], [0, 0], [0, 0], [-11, 0], [0, 0], [30, -15], [0, 0], [-21, 16], [0, 0], [0, 0], [-35, 70], [-33, 66], [0, 0], [-83, 83], [0, 0], [0, 0], [0, 0], [0, -121], [154, -77], [0, 0], [-143, 143], [51, 51], [0, 0], [0, 0], [112, 112], [0, 0], [-77, 154], [0, 0], [24, 24], [0, 0], [240, -120], [0, 0], [0, 0], [0, -229], [0, 0], [-56, 112], [-193, 193], [-299, 0], [0, 0], [0, 0], [0, 0], [208, -104], [0, 0], [-175, -175], [0, 0], [0, 109], [29, 29], [-85, -85], [0, 0], [409, 0], [0, 0], [0, 0], [-62, 31], [0, 0], [0, 0], [0, 0], [0, 0], [-242, 242], [-122, 0], [0, 0], [-458, 458], [0, 0], [-288, -288], [0, 0], [0, -142], [0, 0], [21, -42], [-377, -377], [0, 0], [671, 0], [0, 0], [0, 0], [0, -491], [-152, 304], [-259, 518], [0, 0], [0, 0], [-431, 431], [0, 0], [-608, 304], [0, 0], [440, 440], [0, 0], [-814, 407], [-299, -299], [0, 0], [0, 0], [767, 0], [0, 0], [-397, 0], [0, 0], [0, 0], [88, -176], [0, 0], [851, -851], [0, 0], [0, 0], [0, 0], [774, -387], [0, 241], [54, -27], [0, 0], [0, 0], [0, 0], [-181, 0], [-962, 0], [0, 0], [0, 0], [0, 0], [-49, 98], [-667, 1334], [-1126, 1126], [0, 0], [949, 0], [0, 587], [0, 0], [1237, -1237], [0, 0], [0, 0], [0, 0], [1520, -760], [-377, 754], [0, 0], [0, 0], [-1318, 0], [0, 0], [0, 457], [0, 0], [1429, -1429], [1034, 0], [-109, 109], [0, 0], [1722, -861], [-673, 673], [0, 0], [0, 863], [0, 0], [837, -1674], [0, 0], [0, 0], [672, -336], [0, 0], [0, 1058], [0, 0], [689, -1378], [0, 0], [1177, -1177], [0, 0], [0, -1549], [0, 0], [177, 177], [0, 0], [0, 443], [0, 0], [1040, 1040], [0, 0], [1837, 0], [0, 0], [0, -1847], [0, 0], [0, 0], [0, 0], [1787, -1787], [0, 0], [0, 0], [0, 0], [-549, -549], [0, 1487]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_684_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_684_3_y_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_684_3_y_c(:prec:=2)
    chi := MakeCharacter_684_y();
    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_684_3_y_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_684_3_y_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_684_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![11, 1]>],Snew);
    return Vf;
end function;