
// Make newform 624.4.c.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_624_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_624_c_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_624_4_c_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_624_4_c_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 := [48, 0, 23, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [0, 19, 0, 1], [46, 0, 4, 0]];
    Rf_basisdens := [1, 1, 2, 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_624_c();"
function MakeCharacter_624_c()
    N := 624;
    order := 2;
    char_gens := [79, 469, 209, 145];
    v := [2, 2, 2, 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_624_c_Hecke();"
function MakeCharacter_624_c_Hecke(Kf)
    N := 624;
    order := 2;
    char_gens := [79, 469, 209, 145];
    char_values := [[1, 0, 0, 0], [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 := 4;
    raw_aps := [[0, 0, 0, 0], [-3, 0, 0, 0], [0, -1, 1, 0], [0, -1, 2, 0], [0, -4, -3, 0], [-3, -4, -1, -1], [24, 0, 0, -3], [0, -5, -8, 0], [72, 0, 0, 0], [-96, 0, 0, 3], [0, -19, 20, 0], [0, 34, 4, 0], [0, -33, -13, 0], [322, 0, 0, -3], [0, 18, 1, 0], [246, 0, 0, 12], [0, -26, -1, 0], [72, 0, 0, 7], [0, -27, -54, 0], [0, 84, -41, 0], [0, -78, -42, 0], [-1080, 0, 0, -4], [0, -94, 63, 0], [0, -69, -25, 0], [0, 34, -14, 0], [-1050, 0, 0, 12], [298, 0, 0, -9], [624, 0, 0, 30], [0, 76, 82, 0], [564, 0, 0, 15], [660, 0, 0, 14], [564, 0, 0, 24], [0, 29, -47, 0], [-14, 0, 0, 21], [0, -111, -229, 0], [0, 261, 0, 0], [-616, 0, 0, 15], [0, 175, -188, 0], [0, -46, -267, 0], [-126, 0, 0, 90], [1860, 0, 0, 24], [-16, 0, 0, -105], [708, 0, 0, 6], [0, 492, 60, 0], [0, 203, -227, 0], [1428, 0, 0, -46], [-2400, 0, 0, -94], [0, 653, 116, 0], [0, -252, -359, 0], [0, -124, -346, 0], [1950, 0, 0, 78], [0, -248, -73, 0], [0, 410, -46, 0], [600, 0, 0, 42], [-1170, 0, 0, -54], [2388, 0, 0, -6], [-4020, 0, 0, 21], [0, 219, -438, 0], [-1138, 0, 0, -48], [0, 421, -447, 0], [-192, 0, 0, -10], [0, -569, 681, 0], [0, -1071, -450, 0], [-8604, 0, 0, 18], [1182, 0, 0, -170], [0, -525, -559, 0], [0, 1129, -134, 0], [-1302, 0, 0, -44], [-1116, 0, 0, -108], [0, 518, -460, 0], [0, 375, 167, 0], [0, 14, -351, 0], [-1454, 0, 0, 87], [1470, 0, 0, -8], [0, 407, 464, 0], [0, -942, 73, 0], [4572, 0, 0, 81], [0, 370, -56, 0], [0, -377, 15, 0], [0, -1598, 46, 0], [3120, 0, 0, 186], [0, -68, 46, 0], [0, 362, 105, 0], [-8388, 0, 0, -77], [6646, 0, 0, -231], [-6984, 0, 0, -126], [0, -293, 1179, 0], [0, -730, 362, 0], [0, 223, -555, 0], [0, -713, 166, 0], [-4992, 0, 0, -78], [0, -20, -1021, 0], [0, -811, 200, 0], [7728, 0, 0, -282], [0, 873, -810, 0], [-1848, 0, 0, 168], [0, -2389, -407, 0], [12060, 0, 0, 63], [17938, 0, 0, -27], [0, -1160, 214, 0], [238, 0, 0, 483], [0, -1389, -199, 0], [-7896, 0, 0, -138], [-7986, 0, 0, 222], [-6312, 0, 0, 314], [0, -468, 1116, 0], [0, -2114, 383, 0], [0, 1027, -549, 0], [-3684, 0, 0, -66], [7950, 0, 0, 46], [-7526, 0, 0, -117], [0, -1242, 936, 0], [0, 2823, 1499, 0], [0, 45, 2286, 0], [0, 1193, -460, 0], [13740, 0, 0, 123], [0, 405, -198, 0], [3396, 0, 0, 354], [21936, 0, 0, -141], [12036, 0, 0, 480], [0, 6, 816, 0], [7664, 0, 0, 321], [8130, 0, 0, -402], [0, 1852, -643, 0], [0, -3143, 454, 0], [-21612, 0, 0, -327], [0, -3136, -1342, 0], [-20064, 0, 0, 96], [1488, 0, 0, -388], [0, 1112, -1882, 0], [0, -3647, 1462, 0], [0, 252, 1423, 0], [9898, 0, 0, -837], [1572, 0, 0, -611], [0, 2817, -1027, 0], [0, -1472, -1520, 0], [0, -1211, -2025, 0], [0, 3441, -2706, 0], [15684, 0, 0, -399], [2352, 0, 0, 741], [0, -2645, -1748, 0], [0, 4293, 2063, 0], [-10238, 0, 0, -429], [0, 1920, 1081, 0], [13836, 0, 0, -395], [0, 3804, 1951, 0], [0, -402, -3048, 0], [-1806, 0, 0, -420], [34200, 0, 0, -22], [0, -1922, 1829, 0], [0, 4738, -8, 0], [17694, 0, 0, -630], [14724, 0, 0, 176], [26760, 0, 0, -384], [-14592, 0, 0, 494], [-3576, 0, 0, 204], [-28800, 0, 0, 596], [0, -4213, 715, 0], [23700, 0, 0, 667], [0, 3365, 1291, 0], [0, 1006, 2219, 0], [4818, 0, 0, -420], [0, -471, 492, 0], [-4080, 0, 0, -1434], [0, 2797, -5019, 0], [0, 2272, -2341, 0], [20386, 0, 0, 711], [11726, 0, 0, 360]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_624_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_624_4_c_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_624_4_c_c(:prec:=4)
    chi := MakeCharacter_624_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_624_4_c_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_624_4_c_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_624_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![15552, 0, 260, 0, 1]>],Snew);
    return Vf;
end function;