
// Make newform 624.2.cn.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_624_cn();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_624_cn_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_2_cn_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_2_cn_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 := [13, -38, 70, -74, 63, -34, 16, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-159, 235, -426, 221, -172, 32, -15, -1], [-70, 298, -390, 256, -146, 22, -8, -3], [-181, 335, -427, 256, -146, 22, -8, -3], [-152, 294, -498, 373, -261, 89, -29, 3], [97, -266, 366, -378, 215, -114, 28, -8], [39, 38, -84, 276, -184, 159, -41, 17]];
    Rf_basisdens := [1, 1, 37, 37, 37, 37, 37, 37];
    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_cn();"
function MakeCharacter_624_cn()
    N := 624;
    order := 12;
    char_gens := [79, 469, 209, 145];
    v := [12, 12, 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_624_cn_Hecke();"
function MakeCharacter_624_cn_Hecke(Kf)
    N := 624;
    order := 12;
    char_gens := [79, 469, 209, 145];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, -2, -1, 0], [0, -1, 0, 1, -1, 1, 0, 0], [0, 0, 0, 0, 0, 1, 1, 0], [1, -2, 0, 0, -1, 0, 1, -2], [2, 0, 0, 0, -3, 0, -2, 0], [1, -2, -1, 0, -1, 0, 2, -3], [2, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-2, 5, 5, -1, 3, -5, -3, 1], [1, 0, 0, 0, -3, -1, -4, 0], [-2, 0, 0, 0, -3, 5, -3, 0], [-1, -1, -3, 3, -2, 4, 1, 1], [-6, 0, 0, 0, 0, -3, 3, 0], [-4, 4, 0, 4, 0, 0, 0, 0], [0, 1, 0, -1, 1, -2, 1, -2], [0, 0, -2, 0, 0, 0, 2, -2], [0, 0, 0, 0, 0, 0, 7, 0], [4, 0, 0, 0, -2, 6, 2, 0], [0, 0, -2, 0, 0, 2, 0, 2], [-5, 0, 0, 0, 2, -5, 3, 0], [-2, 0, 0, 0, 0, 0, 0, 0], [-1, 2, 2, 0, 1, -1, -2, 2], [-1, 4, -2, -2, 3, 0, -1, 4], [6, 0, 0, 0, -6, 7, -7, 0], [1, -5, 5, 3, -4, 3, -4, 3], [4, 0, 0, 0, 0, 0, -8, 0], [2, 4, 4, -8, 6, -4, -6, 8], [-11, 0, 0, 0, -5, 11, 6, 0], [5, 0, 1, -10, 5, -8, 2, -5], [-7, 0, 0, 0, -3, 0, -7, 0], [3, -4, 0, -2, -1, -1, 2, -4], [0, -4, -3, 4, -4, 6, 1, 1], [0, 0, 0, 0, -12, 6, 8, 0], [-2, 2, -3, 2, 0, 4, -1, 5], [-6, 0, 0, 0, 2, -6, 4, 0], [-10, 0, 0, 0, -3, -3, 0, 0], [8, 0, 0, 0, 8, 0, 0, 0], [-2, 6, -2, -2, 4, 0, -2, 6], [-6, 12, 6, 0, 6, -6, -6, 6], [-11, 4, -4, 18, -7, 18, -7, 18], [0, 0, 0, 0, -3, 3, 0, 0], [-2, 0, 8, 4, -2, -1, -5, 2], [-1, 0, 0, 0, 4, -3, 4, 0], [1, 0, 2, -2, 1, -2, -1, 0], [-8, 0, 0, 0, 0, 6, 4, 0], [-7, 0, 0, 0, 3, -6, 7, 0], [-9, 0, 0, 0, 11, -20, -11, 0], [4, -4, 8, -4, 0, -10, 2, -12], [5, 0, 0, 0, 9, 5, -14, 0], [0, -6, -12, 6, -6, 12, 6, 0], [-3, 4, 6, 2, 1, -1, -6, 6], [9, 0, 0, 0, 9, -7, -7, 0], [-4, 8, 4, 0, 4, -9, 1, -6], [-9, 3, -3, 15, -6, 15, -6, 15], [5, -14, -14, 4, -9, 14, 9, -4], [2, 0, 10, -4, 2, -8, -4, -2], [-4, 0, 0, 0, 0, 4, 0, 0], [18, 0, 0, 0, 0, -12, -9, 0], [1, 5, 12, -7, 6, -18, 0, -12], [9, 0, 0, 0, 9, 0, 9, 0], [0, 6, -1, -6, 6, -9, 4, -7], [8, 0, 0, 0, 4, 8, -12, 0], [2, 2, 4, -6, 4, -6, -2, 0], [2, 0, 0, 0, 0, 0, 0, 0], [-5, 7, 10, 3, 2, -2, -10, 10], [6, 0, 0, 0, -6, -16, 16, 0], [2, 0, 0, 0, 15, -15, -4, 0], [10, 0, 2, -20, 10, -16, 4, -10], [15, 0, 0, 0, -1, -14, -1, 0], [-6, 7, -5, 5, 1, -2, 6, -7], [5, 2, 14, -12, 7, -21, 0, -14], [20, 0, 0, 0, 6, -12, -20, 0], [0, 0, 0, 0, 18, -9, 4, 0], [5, 0, 0, 0, 9, -4, -9, 0], [-16, 16, 0, 16, 0, 8, -8, 16], [-1, -7, -14, 9, -8, 15, 7, 0], [-2, 0, 0, 0, -2, 9, 9, 0], [-10, 9, 11, 11, -1, 0, -10, 9], [17, 0, 0, 0, -17, 2, -2, 0], [-1, -2, -2, 4, -3, 2, 3, -4], [-6, 0, 0, 0, -1, 6, 5, 0], [-7, -6, -20, 20, -13, 26, 7, 6], [-22, 0, 0, 0, 0, 12, 11, 0], [1, 0, 0, 0, 3, 0, 1, 0], [-6, 14, 0, -2, 8, -10, 2, -4], [0, 10, 10, -10, 10, -15, -5, 0], [-15, 0, 0, 0, -2, -13, 2, 0], [14, -14, -5, -14, 0, -2, 7, -9], [7, 0, 0, 0, 15, 7, -22, 0], [-9, 0, 0, 18, -9, 9, 0, 0], [-5, 10, 0, 0, 5, 0, -5, 10], [8, 0, 0, 0, -8, 15, -15, 0], [9, -12, 12, -6, -3, -6, -3, -6], [1, 0, 0, 0, -3, -1, -4, 0], [4, 0, -10, -8, 4, -1, 7, -4], [-1, -7, -9, 9, -8, 16, 1, 7], [3, -11, 0, 5, -8, 13, -5, 10], [-13, 0, 0, 0, -15, 30, 13, 0], [16, 0, 0, 0, 5, 16, -21, 0], [-2, 0, 0, 0, 0, 0, 0, 0], [17, -17, -17, -17, 0, 0, 17, -17], [-2, 4, 2, 0, 2, 0, -4, 6], [-2, -2, 2, 6, -4, 6, -4, 6], [-11, 0, 0, 0, 21, -21, 22, 0], [-17, 0, 0, 0, -2, 17, 15, 0], [6, -4, 8, -8, 2, -4, -6, 4], [-8, 7, -2, 9, -1, 3, 0, 2], [9, -16, 0, -2, -7, 5, 2, -4], [-8, 0, 0, 0, -9, 18, 8, 0], [0, 0, 0, 0, 6, -3, 5, 0], [16, 0, 0, 0, 27, -11, -27, 0], [-2, 2, 15, 2, 0, -14, -1, -13], [18, 0, 0, 0, -28, 18, 10, 0], [-10, 0, 0, 0, -10, 12, 12, 0], [-9, 18, 9, 0, 9, 0, -18, 27], [-14, 0, 0, 0, 14, -14, 14, 0], [7, -2, 2, -12, 5, -12, 5, -12], [10, -16, -16, -4, -6, 16, 6, 4], [-2, 0, -22, 4, -2, 14, 10, 2], [8, 0, 0, 0, -11, 3, -11, 0], [-16, 0, 0, 0, 15, 0, -16, 0], [-6, 16, 0, -4, 10, -14, 4, -8], [0, 14, 26, -14, 14, -21, -19, 12], [27, 0, 0, 0, 17, 10, -17, 0], [-6, 0, 0, 12, -6, 6, 0, 0], [-9, 0, 0, 0, -9, 5, 5, 0], [-4, 8, 4, 0, 4, -6, -2, 0], [-13, 0, 0, 0, 3, -3, 26, 0], [-13, 0, 0, 0, -30, 13, -17, 0], [14, 0, 0, 0, -30, 16, -30, 0], [-8, 16, 0, 0, 8, -16, 8, -16], [-3, 0, 0, 0, -39, 0, -3, 0], [4, 0, 0, 0, -12, 24, -4, 0], [0, -6, -10, 6, -6, 9, 7, -4], [5, 0, 0, 0, 23, -18, -23, 0], [-18, 18, -4, 18, 0, 13, -9, 22], [-2, 0, 0, 0, -2, 11, 11, 0], [8, -16, -8, 0, -8, 0, 16, -24], [-2, -7, -7, 11, -9, 7, 9, -11], [-19, 0, 0, 0, -19, 19, 0, 0], [9, 4, 22, -22, 13, -26, -9, -4], [-4, 0, 0, 0, 0, -12, 2, 0], [-16, 8, -16, 24, -8, 24, 0, 16], [7, 0, 0, 0, -36, 0, 7, 0], [0, -2, -6, 2, -2, 3, 5, -4], [-21, 0, 0, 0, 22, -21, -1, 0], [8, -7, -14, -9, 1, 6, 7, 0], [-8, 0, 0, 0, 18, 18, 0, 0], [7, -12, -14, -2, -5, 5, 14, -14], [17, 0, 0, 0, -17, 2, -2, 0], [-7, 11, -11, 3, 4, 3, 4, 3], [21, 0, 0, 0, -3, 3, -42, 0], [-10, 4, 4, 16, -6, -4, 6, -16], [10, 0, 0, 0, 0, 0, 10, 0], [0, 2, 0, -2, 2, -4, 2, -4], [0, 0, 0, 0, 18, -9, 29, 0], [-4, 4, 19, 4, 0, -17, -2, -15], [-37, 0, 0, 0, 0, 0, 0, 0], [-9, 2, 18, 16, -7, 7, -18, 18], [-5, -2, 12, 12, -7, 0, -5, -2], [-8, 16, 8, 0, 8, -18, 2, -12], [22, 0, 0, 0, 6, -22, -16, 0], [4, 0, 38, -8, 4, -25, -17, -4], [-1, -13, -15, 15, -14, 28, 1, 13], [-13, 4, -18, 22, -9, 27, 0, 18], [-16, 0, 0, 0, -24, 48, 16, 0], [0, 0, 0, 0, 0, 0, 7, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_624_cn_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_624_2_cn_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_2_cn_c(:prec:=8)
    chi := MakeCharacter_624_cn();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_2_cn_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_2_cn_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_624_cn();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![169, 0, 0, 0, 38, 0, 0, 0, 1]>,<7,R![4, -4, 2, -2, 1]>],Snew);
    return Vf;
end function;