
// Make newform 62.2.g.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_62_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_62_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_62_2_g_a();".
// 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_62_2_g_a();".
// 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, 0, 1, -1, 1, 0, -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_62_g();"
function MakeCharacter_62_g()
    N := 62;
    order := 15;
    char_gens := [3];
    v := [14];
    // 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_62_g_Hecke();"
function MakeCharacter_62_g_Hecke(Kf)
    N := 62;
    order := 15;
    char_gens := [3];
    char_values := [[-1, 1, 0, -1, 1, -1, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 1, 0, 0, 1], [-1, 1, 1, -1, 2, -1, 1, 1], [1, -1, 0, 1, -1, 1, -1, 0], [-4, 1, 2, -1, 0, -2, -2, 2], [2, 2, -4, 2, -2, 0, 2, -4], [2, -3, 2, -1, -5, 3, -1, -4], [-3, -2, 2, -4, 2, -1, -2, 3], [1, -1, 1, 4, 1, -1, 1, 0], [0, 3, -3, -3, 6, 0, 2, 3], [3, 1, -6, 3, -3, -3, 4, -3], [6, -1, -2, 4, -4, 3, 6, -3], [-3, -6, 0, -2, -2, 0, -6, -3], [-6, 6, -2, -2, 1, -2, -1, 3], [-7, 3, 4, -4, 3, -6, -4, 7], [-3, 6, 1, -4, 3, -5, 1, 5], [0, -6, -5, -2, 3, -2, -5, -6], [9, -6, 3, 1, -2, 4, -2, -5], [-1, 1, 1, 2, 0, -4, -1, 4], [2, 2, -5, 1, 4, -1, 2, -6], [-10, 4, 6, -1, 0, -5, -5, 5], [3, -4, 2, -1, 0, 9, 4, 1], [6, -2, 7, -5, -5, 7, -2, 6], [-1, 2, 3, -1, 1, -1, 0, 4], [4, -8, 3, 7, -1, 7, 3, -1], [2, 3, 0, -1, 0, 3, 2, 0], [-4, 1, 4, -1, -1, 4, 1, -4], [1, 6, -14, 7, 1, -7, 6, -6], [1, -8, 3, 1, 0, 0, -11, 11], [-12, -1, 2, -5, -4, -4, -4, 4], [0, -4, 3, 3, -8, 3, 3, -4], [1, -4, -4, 8, 3, -7, 0, -6], [-4, -2, 8, -8, 6, 2, -9, 5], [14, -2, 4, 5, -9, 18, 3, -12], [10, -2, -8, 8, -10, 2, 16, -8], [-2, 4, 3, -3, 6, 2, -1, 5], [-20, 9, 11, -9, 18, -13, -8, 16], [14, 2, -7, 13, -7, 2, 14, 0], [14, -8, 6, 6, -3, 2, 3, -5], [-9, 12, -1, -13, 11, 2, 2, 12], [-1, 3, -2, -1, 3, -5, 5, -4], [6, -6, -6, 3, -7, 9, -1, -9], [-8, 8, 8, -3, 12, -7, 4, 13], [13, 2, -6, 4, -7, 1, 8, -6], [11, 2, -7, 7, 1, 5, 5, -5], [0, 5, -9, 8, -4, -4, 0, -13], [-7, -10, 2, -6, -6, -3, -8, 4], [-17, 6, -1, -13, 12, -11, -12, 6], [1, -10, 11, -5, -5, 11, -10, 1], [-13, 15, 7, -10, 5, -10, -5, 15], [14, -3, 0, 5, -9, 18, 6, -15], [5, -20, 3, 11, -14, 14, -3, -8], [-13, 12, 6, -18, 7, -1, 6, 6], [0, -6, -5, 7, -8, 7, -5, -6], [15, -15, -7, 19, -12, 11, 8, -26], [5, 5, -12, 10, 5, -10, 12, -7], [-4, 3, -10, 5, 5, -10, 3, -4], [8, -3, -10, -2, -10, -3, 8, 0], [1, -3, 2, 3, -6, -3, 3, -7], [1, 0, -1, 0, 0, 10, 8, 9], [1, -2, 3, -4, 13, 15, -5, 2], [9, -1, -16, 8, -4, -2, 5, -12], [0, 11, 3, -10, 6, -10, 3, 11], [11, -17, 0, 8, 1, 8, -9, -16], [0, 9, 0, -11, 13, -11, 0, 9], [7, 7, 0, -7, -7, 7, 7, -4], [-5, -2, 6, -8, -8, 6, -2, -5], [6, -12, 4, -4, -4, 8, -10, -2], [4, 7, 1, 7, 4, 1, 6, -3], [-15, -4, -8, -1, -8, -4, -15, 0], [-10, 9, -8, 10, -5, -12, 5, 7], [-3, 2, -5, 9, -3, -6, -3, -11], [-3, 3, 3, 9, -2, -10, -5, 15], [-14, -8, 0, -3, 14, -3, -11, 6], [-27, 1, 12, -13, 15, -3, -11, 12], [17, -17, 1, -1, -15, 16, 16, -16], [-10, 9, 2, -1, -2, -13, -5, -1], [4, -13, 20, 8, -7, 10, -12, 6], [-10, 7, 3, -7, 14, 0, -13, 17], [-10, 20, 1, 7, 17, -13, 6, -3], [-23, 10, -5, -14, 9, -18, 6, 3], [5, 1, -11, 15, -10, -5, 9, -12], [0, 22, 3, 6, 10, 6, 3, 22], [-17, 0, 13, -21, 16, -8, -8, 29], [18, -18, -18, 15, -6, 0, 12, -21], [8, 11, -8, 1, 7, -4, 5, -9], [4, -3, 9, 3, -1, 2, -11, 10], [18, 1, -8, 17, -8, 1, 18, 0], [-4, 8, -1, -15, 11, 3, -11, 3], [10, -8, -2, 8, -16, -6, 5, -24], [27, -10, -17, 10, -20, 15, 8, -22], [24, -20, -8, 4, -9, 22, 6, -13], [-11, -21, 4, -4, 0, 14, -18, 8], [-10, 10, 10, -10, 6, -9, -4, 10], [0, 11, 15, -2, -4, -2, 15, 11], [-15, 1, 12, -13, 3, 9, -11, 1], [-34, 22, -4, -2, 18, -16, 5, 19], [3, -4, 1, 5, -6, 1, 4, -2], [8, -16, -11, 4, -12, 4, -4, -19], [-10, 5, 10, -5, -2, -8, -8, 3], [-18, 10, 6, 1, 0, -9, -9, 9], [17, 18, -11, 12, -20, 1, 19, -13], [8, -8, -8, 6, -11, -10, -3, -10], [-6, -1, 8, -6, 4, -5, -5, 5], [40, -19, -4, 6, -23, 17, 7, -20], [22, -4, -3, 4, -2, 21, 2, -23], [-21, 13, 8, -13, 26, -12, -8, 22], [-2, 4, -23, 14, -11, -15, 16, -21], [12, 1, 8, 5, 5, 8, 1, 12], [-8, 14, 16, -24, 12, 0, -12, 12], [-14, -5, 12, 0, 7, -7, 5, 24], [-11, 14, -8, -6, 19, -27, 22, 22], [0, 8, 5, -12, 24, -12, 5, 8], [17, -21, 1, 9, 1, 10, -11, -19], [14, -3, -8, 6, -4, 9, 9, -9], [11, -22, 16, 7, 7, 16, -22, 11], [2, -1, 15, 18, 15, -1, 2, 0], [0, 0, 4, 0, -8, -8, 0, 4], [5, 4, -9, -4, 8, 7, -21, 6], [-5, 0, 1, -31, 1, 0, -5, 0], [-2, 36, -15, 7, 7, -15, 36, -2], [11, 16, 1, -3, 2, -2, -1, 4], [14, -14, -14, -5, -7, 3, 7, -33], [6, 15, -13, 20, -29, 7, 22, -27], [-7, 11, 8, -16, 8, -2, -14, 24], [2, -4, -27, 30, -15, -1, 15, -14], [15, -30, -5, -13, -14, 16, -28, -20], [-8, -9, 19, -5, 3, 4, -24, 12], [-26, 21, 10, -5, 12, -17, -1, 17], [-2, -2, 17, -12, 18, -10, -10, 10], [36, -36, -36, 38, -35, 26, 1, -34], [-2, 4, 8, -14, 8, -6, -2, 20], [12, 8, -11, 3, -1, -10, 19, -29], [-1, 13, -10, -5, 3, 2, 6, -1], [-33, 13, 40, -20, 12, 0, -7, 32], [-15, -10, -4, -10, -7, -12, -6, 3], [-1, 1, 0, -1, 2, -3, 30, -1], [-13, 31, 6, -16, 8, -3, -8, 11], [18, 23, -6, 7, -1, -12, 19, -13], [0, -4, 4, 0, 28, 0, 4, -4], [-16, 16, 16, -17, -7, -12, -23, 15], [1, 5, 21, 5, 4, -8, -20, 24], [51, -37, -28, 14, -27, 25, 2, -41], [19, -5, 9, -13, -13, 9, -5, 19], [31, 21, -3, 2, -3, 21, 31, 0], [16, -15, -1, 15, -30, 17, 25, -14], [13, -21, -9, 6, -7, 14, -5, -2], [11, 8, 4, 14, -1, 6, 6, -6], [-27, 21, 6, -31, 25, -17, -14, 37], [-32, 32, 32, -28, 37, -19, 5, 36], [-30, -8, -1, -26, 27, 4, -30, 25], [28, -13, -9, 16, -22, 6, 4, -27], [-1, 29, 9, 4, 5, -10, 15, -10], [-6, -4, -14, 12, -6, -4, 6, -2], [7, -11, 2, -6, 2, -11, 7, 0], [16, -26, -27, 40, -20, 3, 20, -23], [-4, 19, 8, 3, 28, -16, -16, 16], [1, -8, -13, 18, -2, 5, -3, -23], [19, -15, -10, 25, -9, -1, -5, -24], [-9, 9, -7, -4, 17, -13, -13, 13], [4, -19, 30, -15, -21, 27, -7, -6], [-35, -14, 0, 0, 0, -14, -35, 0], [23, -25, 26, 12, -18, 30, -14, 7], [-8, -8, 13, -19, -19, 13, -8, -8], [-28, -2, 30, -32, -4, 8, -28, 24], [18, 8, -9, 3, 6, 7, -4, -12], [4, 2, -30, 28, 2, -2, 0, -26], [-17, 17, 17, -23, 22, -17, 5, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_62_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_62_2_g_a();".
// 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_62_2_g_a(:prec:=8)
    chi := MakeCharacter_62_g();
    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_62_2_g_a();".
// 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_62_2_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_62_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -2, 0, -2, 9, -8, 5, -3, 1]>],Snew);
    return Vf;
end function;