
// Make newform 324.3.f.m in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_324_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_324_f_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_324_3_f_m();".
// 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_324_3_f_m();".
// 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, 3, 4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -4, 4, -1], [7, 0, 0, 1]];
    Rf_basisdens := [1, 1, 12, 4];
    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_324_f();"
function MakeCharacter_324_f()
    N := 324;
    order := 6;
    char_gens := [163, 245];
    v := [3, 4];
    // 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_324_f_Hecke();"
function MakeCharacter_324_f_Hecke(Kf)
    N := 324;
    order := 6;
    char_gens := [163, 245];
    char_values := [[-1, 0, 0, 0], [0, 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 := [[1, -1, 0, 0], [0, 0, 0, 0], [-2, 2, -1, 2], [-3, 2, -1, 4], [7, 0, -7, 0], [0, 0, -16, 0], [4, 0, 0, -8], [18, -24, 12, -12], [20, 0, 10, 0], [-14, 28, -14, 0], [0, -2, 1, 2], [26, 0, 0, 0], [4, -4, 2, -4], [6, -4, 2, -8], [-2, 0, 2, 0], [19, 0, 0, -38], [-88, 0, -44, 0], [-8, 0, -8, 0], [0, -20, 10, 20], [-36, 0, -72, 0], [-19, 0, 0, 0], [24, -16, 8, -32], [67, 0, -67, 0], [22, 0, 0, -44], [-119, 0, -119, 0], [-17, 34, -17, 0], [0, 40, -20, -40], [75, 0, 150, 0], [-46, 0, 0, 0], [52, -52, 26, -52], [9, -12, 6, -6], [50, 0, 25, 0], [22, -44, 22, 0], [0, 40, -20, -40], [-2, 2, -1, 2], [-93, 62, -31, 124], [0, 0, 92, 0], [72, -96, 48, -48], [-196, 0, -98, 0], [49, -98, 49, 0], [153, 0, 306, 0], [272, 0, 0, 0], [-152, 0, 152, 0], [0, 0, -187, 0], [-5, 0, 0, 10], [45, -60, 30, -30], [0, -20, 10, 20], [96, -64, 32, -128], [-44, 0, 44, 0], [0, 0, -214, 0], [-86, 0, 0, 172], [476, 0, 238, 0], [-206, 0, -206, 0], [48, 0, 96, 0], [16, -16, 8, -16], [-26, 0, 26, 0], [-14, 0, 0, 28], [-9, 12, -6, 6], [160, 0, 160, 0], [-140, 280, -140, 0], [0, 4, -2, -4], [-92, 92, -46, 92], [-252, 336, -168, 168], [-28, 0, -14, 0], [-233, 0, -233, 0], [139, -278, 139, 0], [-66, 44, -22, 88], [0, 0, 50, 0], [-226, 0, -113, 0], [538, 0, 538, 0], [-62, 124, -62, 0], [258, 0, 516, 0], [-129, 86, -43, 172], [0, 0, 212, 0], [108, -144, 72, -72], [-280, 0, -140, 0], [-143, 286, -143, 0], [80, 0, 0, 0], [-344, 344, -172, 344], [0, 0, -349, 0], [344, 0, 172, 0], [208, 0, 208, 0], [-318, 0, -636, 0], [125, 0, 0, 0], [-273, 182, -91, 364], [220, 0, -220, 0], [-134, 0, 0, 268], [-827, 0, -827, 0], [-179, 358, -179, 0], [0, 34, -17, -34], [-429, 0, -858, 0], [-194, 0, 194, 0], [-288, 384, -192, 192], [566, 0, 283, 0], [0, -212, 106, 212], [186, 0, 372, 0], [-302, 302, -151, 302], [64, 0, 0, -128], [324, -432, 216, -216], [260, 0, 0, 0], [420, -280, 140, -560], [-131, 0, 0, 262], [806, 0, 403, 0], [76, -152, 76, 0], [0, -80, 40, 80], [494, 0, 0, 0], [-137, 0, 137, 0], [130, 0, 0, -260], [-652, 0, -326, 0], [-647, 0, -647, 0], [0, -128, 64, 128], [866, 0, 0, 0], [124, -124, 62, -124], [-480, 320, -160, 640], [-45, 60, -30, 30], [238, -476, 238, 0], [0, -8, 4, 8], [312, 0, 624, 0], [238, -238, 119, -238], [-563, 0, 563, 0], [0, 0, -970, 0], [595, 0, 595, 0], [118, -236, 118, 0], [36, 0, 72, 0], [-516, 344, -172, 688], [-257, 0, 0, 514], [28, 0, 28, 0], [-156, 0, -312, 0], [-3, 2, -1, 4], [0, 0, 338, 0], [396, -528, 264, -264], [-688, 0, -344, 0], [0, -170, 85, 170], [-250, 0, 0, 0], [-272, 272, -136, 272], [0, 0, 65, 0], [-230, 0, 0, 460], [0, 256, -128, -256], [430, -430, 215, -430], [58, 0, 0, -116], [-702, 936, -468, 468], [-14, 28, -14, 0], [0, 430, -215, -430], [336, 0, 672, 0], [-1186, 0, 0, 0], [484, 0, -484, 0], [1294, 0, 1294, 0], [364, -728, 364, 0], [0, -128, 64, 128], [492, 0, 984, 0], [0, 0, 1118, 0], [208, 0, 0, -416], [-630, 840, -420, 420], [1280, 0, 640, 0], [294, -196, 98, -392], [454, 0, -454, 0], [-423, 564, -282, 282], [-68, 136, -68, 0], [-649, 0, 0, 0], [-866, 866, -433, 866], [-59, 0, 59, 0], [-140, 0, 0, 280], [0, -302, 151, 302], [303, 0, 606, 0], [724, -724, 362, -724], [-986, 0, 986, 0], [-153, 204, -102, 102], [-1304, 0, -1304, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_324_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_324_3_f_m();".
// 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_324_3_f_m(:prec:=4)
    chi := MakeCharacter_324_f();
    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_324_3_f_m();".
// 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_324_3_f_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![169, 0, 13, 0, 1]>,<7,R![1521, 0, -39, 0, 1]>,<11,R![147, -21, 1]>],Snew);
    return Vf;
end function;