
// Make newform 62.2.g.b in Magma, downloaded from the LMFDB on 19 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_b();".
// 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_b();".
// 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, 0, 1, 1, -1], [-3, -1, 2, -1, 1, 1, -3, 2], [0, -1, 2, -1, 0, 0, 0, 0], [-2, -2, 4, -2, 2, 0, -2, 4], [2, 1, -6, 3, -1, -1, 3, -4], [-3, 4, 2, -4, 2, -1, -2, 3], [3, -1, -3, 0, -3, -1, 3, 0], [2, -1, -1, 1, -2, -2, 0, -5], [-7, 5, 0, -5, 3, -5, -2, 3], [2, 3, 0, 2, 2, 1, 2, -1], [1, -6, 0, 2, 2, 0, -6, 1], [2, -8, 0, 2, -1, 0, 1, -1], [3, 3, 2, 2, -1, 2, 2, -3], [-5, -4, -1, 2, -1, -3, 5, -3], [0, 4, 3, 4, 1, 4, 3, 4], [-7, 0, 9, -9, 0, 0, 0, 9], [1, -1, -1, 2, 2, 4, 3, 0], [6, 0, -1, 7, -6, -1, 8, -8], [-2, 4, -2, 3, 0, -1, -1, 1], [-1, 6, -10, 5, 2, -9, 2, -3], [0, -2, -1, -1, -1, -1, -2, 0], [-1, 2, 3, -9, 5, 3, -8, 4], [4, -8, -5, 3, -9, -1, -1, -9], [2, 3, -2, -1, -2, 3, 2, 0], [4, 7, -2, 5, 5, -2, 7, 4], [-3, 2, 2, -1, 3, -5, -4, 4], [11, -4, -7, 5, -6, 12, 9, -15], [-8, 3, -6, -1, 0, -4, -4, 4], [0, -8, 1, -1, -10, -1, 1, -8], [-13, 0, 4, -4, 9, -5, -4, 4], [-4, 6, 0, -8, 6, 2, -1, 5], [-2, 12, -2, 5, 7, -14, 7, 0], [10, -10, -4, 4, -14, 6, 8, -4], [-6, 12, 9, -3, 8, -4, 3, 15], [-10, 3, 7, -3, 6, -5, -8, 8], [-2, -4, -1, -1, -1, -4, -2, 0], [4, -2, -4, 10, -5, -2, 5, -3], [-9, 8, -3, -1, 5, -4, -2, 0], [7, -3, 4, -1, -11, 15, -7, -6], [4, -4, -4, -1, -7, 7, -3, -9], [-2, 2, 2, -5, 2, -7, 0, -1], [1, 6, -6, 0, 5, -11, 12, 6], [9, 2, -7, 7, -1, 5, 5, -5], [-8, -11, 17, -4, 6, -2, -16, 7], [7, -4, -2, -10, 0, 11, -8, 4], [13, 0, -7, -3, -2, 11, 4, -2], [-11, -10, 5, -7, -7, 5, -10, -11], [5, -5, -3, 6, -3, 2, 3, -5], [10, -7, -4, 5, -5, 10, 2, -7], [-11, -6, 5, -9, 4, 4, -13, 14], [-1, 4, -2, -2, 3, -5, 6, 2], [0, 0, 5, 7, -4, 7, 5, 0], [3, -7, 5, -1, -4, 11, -12, 6], [7, 9, 10, 4, -3, 6, 2, -5], [-6, 1, -14, 3, 3, -14, 1, -6], [-12, -1, 8, -14, 8, -1, -12, 0], [-27, 17, 10, -17, 34, -21, -13, 23], [-11, 4, 7, -4, 8, -14, 4, 1], [-1, 2, 23, -10, 5, 3, -9, 24], [21, -5, -32, 16, -10, 4, 15, -26], [0, -7, -13, -8, 10, -8, -13, -7], [7, 7, -22, 20, -3, -2, 5, -18], [0, 13, 0, 1, 1, 1, 0, 13], [19, -15, -4, 19, -15, 11, -11, -16], [5, 6, 8, -6, -6, 8, 6, 5], [-6, 12, -8, 16, 8, -4, 22, -2], [6, -13, 9, 11, -14, 5, 2, -1], [5, -4, 10, 1, 10, -4, 5, 0], [-14, 21, 12, -22, 11, -4, -11, 15], [-1, 4, -3, -1, 1, 0, 17, 15], [-11, 11, 11, -7, 10, -12, -1, 15], [-4, 16, -2, -1, -10, -3, 13, 4], [-11, -5, 2, 3, 9, -7, -7, 10], [-9, -7, 15, -9, -5, -2, -2, 2], [-4, -3, 14, -7, -10, 7, -3, -3], [4, 13, -10, -2, 9, 0, 8, -4], [6, -1, -5, 1, -2, 20, 11, 13], [-6, 0, -23, -1, -11, -17, 22, -11], [-15, 16, 3, -2, 13, -26, 0, 13], [19, -3, 1, -15, -2, 17, -3, -4], [0, -6, -3, -4, 14, -4, -3, -6], [15, -12, -3, 19, -20, 16, 4, -35], [2, -2, -2, 13, -8, -4, -6, 9], [-12, 7, -14, -11, 25, -8, -3, -3], [-32, 1, 25, -25, 7, -14, -31, 38], [2, 5, 4, 9, 4, 5, 2, 0], [0, 0, -13, 7, -11, -11, 7, -13], [-6, 6, 0, -6, 12, -10, -15, 2], [15, -4, -11, 4, -8, 3, 2, -16], [-6, 4, 4, -2, 5, -6, -4, 7], [21, -13, 2, 20, -22, 14, 6, -18], [12, -12, -12, 12, -16, 9, -4, -12], [0, -3, -9, 12, 20, 12, -9, -3], [-9, 7, 4, -11, 5, -1, 3, -3], [8, -6, -2, 8, -8, 0, -9, -19], [39, -20, 5, 25, -30, 29, 20, -10], [-8, 16, -13, 18, -4, -20, 26, -5], [2, -9, 14, -7, 0, 10, -6, 7], [6, 0, -6, 15, -24, 15, 15, -15], [3, -12, 9, -4, 2, 5, -7, -1], [26, -26, -26, 14, -15, -6, 11, -38], [-10, 11, 12, 10, -8, -1, -1, 1], [-6, -9, 12, 0, 3, -3, -3, 12], [22, -24, -3, 12, -6, 13, 6, -19], [11, 3, -14, -3, 6, 6, 22, -2], [2, -4, 17, -6, 1, 5, -8, 15], [4, 15, 2, -1, -1, 2, 15, 4], [8, -6, -32, 24, -12, 16, 12, -28], [14, -11, -2, 12, -13, 1, 11, -4], [-21, -10, 2, 8, 19, -17, -12, 20], [0, 8, 5, -8, -8, -8, 5, 8], [23, -17, -5, 13, 1, 8, -9, -21], [-4, 11, -2, 4, 10, -7, -7, 7], [11, -10, -4, -5, -5, -4, -10, 11], [-28, 1, 3, -26, 3, 1, -28, 0], [4, -8, 16, -12, -20, -12, -16, 12], [-1, -4, 5, 4, -8, 3, 1, 0], [5, 2, -5, 13, -5, 2, 5, 0], [0, -4, -1, 7, 7, -1, -4, 0], [31, -16, -3, 25, -22, 22, 3, -28], [20, -20, -20, 11, 1, -3, 21, -29], [34, -11, 3, 8, -11, 11, 0, -19], [-17, 17, 18, -22, 4, -12, -10, 40], [18, -4, -23, 26, -13, 15, 13, -28], [-5, 10, -9, -7, 14, 4, -2, -4], [-32, 5, -5, -21, 13, -24, -16, 8], [6, -7, 2, -1, -12, 21, 13, -11], [0, -8, -5, -14, 12, -6, -6, 6], [-18, 18, 18, -18, 21, 2, 3, 18], [-34, 16, -12, -2, 12, -14, 2, 16], [-20, 14, 17, -31, 3, 14, -3, 39], [-3, -13, 20, -11, 7, 4, -4, 23], [-5, -3, 16, -8, 4, 8, -3, 12], [1, 2, 16, 18, 1, 0, 2, -1], [-15, -1, 16, 1, -2, -13, 20, 1], [9, 3, 14, 0, 0, -5, 0, 5], [24, 11, 8, 17, -25, -4, 21, -9], [0, -16, -12, 4, 8, 4, -12, -16], [-2, 2, 2, 7, -5, 6, -7, 11], [9, -5, -11, 7, -2, 4, 8, -10], [15, 1, -32, 16, 13, 1, 18, -3], [-21, 9, -3, 5, 5, -3, 9, -21], [-11, -1, -3, 0, -3, -1, -11, 0], [-14, 13, 1, -13, 26, 15, -23, 42], [3, -15, 23, -14, -17, 34, -21, 4], [-5, -4, -4, -22, 31, -18, -18, 18], [17, 19, -2, -3, 5, 1, -4, 1], [10, -10, -10, 32, -9, 7, 1, 12], [-16, 16, -3, -2, 5, -30, 28, -1], [-56, 15, 23, -20, 38, -18, -24, 37], [33, -15, -9, 16, -17, 34, 11, -28], [22, 4, 6, -4, 2, 20, -2, -18], [1, -11, 0, -6, 0, -11, 1, 0], [36, -36, -7, 24, -12, 19, 12, -31], [4, -11, -6, -15, 12, -4, -4, 4], [-27, 8, 15, -22, 6, -7, 1, 29], [-5, 1, 6, -7, -1, 7, -5, 0], [1, 1, 27, -10, 23, -11, -11, 11], [-20, 3, 34, -17, 29, -13, -27, 46], [13, 18, -12, -4, -12, 18, 13, 0], [-23, 13, 2, -20, 24, -12, -22, 11], [6, 20, -7, -3, -3, -7, 20, 6], [12, 2, 10, 4, -8, 16, 0, -8], [-6, -22, 7, 9, -16, 7, 2, -2], [-24, 30, -6, -12, 6, -18, 12, 30], [5, -5, -5, -9, 4, 17, 9, -19]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : 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, 4, 10, 26, 39, -14, -5, -1, 1]>],Snew);
    return Vf;
end function;