
// Make newform 300.3.g.f in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_300_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_300_g_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_300_3_g_f();".
// 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_300_3_g_f();".
// 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 := [9, -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_300_g();"
function MakeCharacter_300_g()
    N := 300;
    order := 2;
    char_gens := [151, 101, 277];
    v := [2, 1, 2];
    // 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_300_g_Hecke();"
function MakeCharacter_300_g_Hecke(Kf)
    N := 300;
    order := 2;
    char_gens := [151, 101, 277];
    char_values := [[1, 0], [-1, 0], [1, 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], [0, -1], [0, 0], [-8, 0], [-3, 6], [-2, 0], [-3, 6], [11, 0], [6, -12], [-6, 12], [-46, 0], [16, 0], [-9, 18], [-62, 0], [-6, 12], [-6, 12], [12, -24], [-16, 0], [-113, 0], [18, -36], [-101, 0], [68, 0], [-3, 6], [-9, 18], [22, 0], [24, -48], [-26, 0], [-3, 6], [176, 0], [-21, 42], [106, 0], [12, -24], [9, -18], [-127, 0], [-18, 36], [164, 0], [-38, 0], [-47, 0], [-30, 60], [12, -24], [3, -6], [-106, 0], [-42, 84], [193, 0], [-6, 12], [-4, 0], [209, 0], [148, 0], [0, 0], [116, 0], [36, -72], [-30, 60], [59, 0], [-21, 42], [-84, 168], [-42, 84], [60, -120], [-178, 0], [202, 0], [-12, 24], [-197, 0], [-72, 144], [127, 0], [48, -96], [58, 0], [48, -96], [257, 0], [127, 0], [63, -126], [-292, 0], [108, -216], [-48, 96], [-668, 0], [-242, 0], [191, 0], [-48, 96], [102, -204], [772, 0], [-45, 90], [701, 0], [-51, 102], [-148, 0], [-36, 72], [463, 0], [-394, 0], [-111, 222], [-21, 42], [-509, 0], [36, -72], [64, 0], [-108, 216], [-42, 84], [382, 0], [-120, 240], [-94, 0], [132, -264], [54, -108], [-135, 270], [-467, 0], [-976, 0], [-779, 0], [-126, 252], [12, -24], [-165, 330], [-286, 0], [487, 0], [99, -198], [-15, 30], [108, -216], [-271, 0], [-404, 0], [34, 0], [-72, 144], [1046, 0], [-106, 0], [-36, 72], [-566, 0], [-84, 168], [-84, 168], [93, -186], [-298, 0], [274, 0], [168, -336], [-21, 42], [-373, 0], [114, -228], [-184, 0], [108, -216], [-758, 0], [-26, 0], [1298, 0], [-90, 180], [-478, 0], [1072, 0], [93, -186], [-169, 0], [78, -156], [-878, 0], [138, -276], [36, -72], [-106, 0], [-102, 204], [1348, 0], [183, -366], [-1222, 0], [72, -144], [-902, 0], [-141, 282], [-577, 0], [-246, 492], [-488, 0], [-180, 360], [313, 0], [-192, 384], [-374, 0], [-108, 216], [1286, 0], [84, -168], [1597, 0], [-42, 84], [-60, 120], [105, -210], [1576, 0], [-195, 390], [-81, 162], [240, -480], [-676, 0], [-1004, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_300_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_300_3_g_f();".
// 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_300_3_g_f(:prec:=2)
    chi := MakeCharacter_300_g();
    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_300_3_g_f();".
// 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_300_3_g_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_300_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![8, 1]>,<11,R![315, 0, 1]>],Snew);
    return Vf;
end function;