
// Make newform 62.3.b.a in Magma, downloaded from the LMFDB on 06 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_62_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_62_b_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_3_b_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_3_b_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 := [47, 0, 14, 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, 7, 0, 1], [7, 0, 1, 0]];
    Rf_basisdens := [1, 1, 1, 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_62_b();"
function MakeCharacter_62_b()
    N := 62;
    order := 2;
    char_gens := [3];
    v := [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_62_b_Hecke();"
function MakeCharacter_62_b_Hecke( : Kf := false)
    N := 62;
    order := 2;
    char_gens := [3];
    char_values := [[-1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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 := [[0, 0, 0, 1], [0, 0, 1, 0], [-3, 0, 0, 2], [-3, 0, 0, 1], [0, -2, 2, 0], [0, -1, -3, 0], [0, -3, -1, 0], [9, 0, 0, -15], [0, 4, 1, 0], [0, 1, 3, 0], [23, -4, -1, -1], [0, -6, 6, 0], [11, 0, 0, -4], [0, 12, 1, 0], [-14, 0, 0, 38], [0, 14, 8, 0], [35, 0, 0, -15], [0, -7, -11, 0], [-44, 0, 0, -46], [7, 0, 0, -3], [0, -10, -8, 0], [0, -14, -11, 0], [0, -2, 36, 0], [0, 7, -25, 0], [-43, 0, 0, -50], [127, 0, 0, 32], [91, 0, 0, -67], [43, 0, 0, -27], [-7, 0, 0, 62], [91, 0, 0, 60], [0, 4, 26, 0], [14, 0, 0, -130], [0, 11, -15, 0], [0, -8, 4, 0], [20, 0, 0, -58], [0, -18, 17, 0], [123, 0, 0, 40], [53, 0, 0, 149], [0, -18, -12, 0], [64, 0, 0, 50], [0, -28, -55, 0], [0, 35, 49, 0], [-229, 0, 0, -15], [-11, 0, 0, -42], [0, 44, 34, 0], [0, 34, -16, 0], [-113, 0, 0, 41], [0, -8, 74, 0], [-182, 0, 0, 6], [0, -6, 20, 0], [-45, 0, 0, 128], [0, -38, -65, 0], [0, -28, -6, 0], [0, 18, 19, 0], [-183, 0, 0, -142], [0, 32, 13, 0], [0, 24, -88, 0], [0, -4, -9, 0], [0, -23, -75, 0], [11, 0, 0, 136], [-130, 0, 0, -154], [212, 0, 0, -98], [407, 0, 0, -47], [65, 0, 0, -271], [0, -9, -101, 0], [79, 0, 0, 216], [0, 54, 46, 0], [0, 5, -41, 0], [0, -48, 25, 0], [-80, 0, 0, 94], [0, 80, 66, 0], [211, 0, 0, 113], [0, -54, -70, 0], [-497, 0, 0, 52], [138, 0, 0, 50], [0, -54, -138, 0], [0, -70, 78, 0], [-117, 0, 0, -260], [0, 7, -137, 0], [0, -55, -115, 0], [-399, 0, 0, -87], [-273, 0, 0, -220], [-404, 0, 0, 2], [0, 73, -89, 0], [7, 0, 0, 129], [125, 0, 0, -191], [0, 54, 148, 0], [0, 54, 130, 0], [0, -48, 10, 0], [0, 4, -58, 0], [-525, 0, 0, -179], [239, 0, 0, 305], [0, -4, -165, 0], [0, -50, -52, 0], [0, -36, 112, 0], [401, 0, 0, -7], [0, 3, 61, 0], [-204, 0, 0, 190], [0, 70, 185, 0], [-15, 0, 0, 474], [89, 0, 0, -531], [0, -24, 48, 0], [-471, 0, 0, -11], [0, 78, 20, 0], [0, 46, -71, 0], [72, 0, 0, 86], [0, -30, 106, 0], [869, 0, 0, 106], [-11, 0, 0, 133], [0, -1, 193, 0], [-534, 0, 0, -238], [0, -152, -28, 0], [-180, 0, 0, -350], [0, 16, 21, 0], [0, 164, -26, 0], [0, 102, -50, 0], [0, -192, -58, 0], [0, -56, 92, 0], [-612, 0, 0, -334], [649, 0, 0, -419], [37, 0, 0, 394], [0, 19, 99, 0], [0, 106, 118, 0], [625, 0, 0, -75], [195, 0, 0, 341], [23, 0, 0, -252], [0, -72, -206, 0], [0, 8, -29, 0], [123, 0, 0, 353], [-303, 0, 0, -490], [0, 116, -99, 0], [0, 212, 13, 0], [-353, 0, 0, -187], [0, 35, 161, 0], [0, 150, -2, 0], [43, 0, 0, -516], [0, -69, -157, 0], [0, 14, 71, 0], [0, -116, -54, 0], [0, 36, 158, 0], [-270, 0, 0, 754], [0, -151, -107, 0], [0, -106, -51, 0], [0, 46, 243, 0], [0, 6, -198, 0], [-42, 0, 0, -318], [-284, 0, 0, 670], [708, 0, 0, 102], [0, -112, 108, 0], [0, 140, 73, 0], [261, 0, 0, -778], [0, -36, -290, 0], [0, 112, -56, 0], [165, 0, 0, 681], [39, 0, 0, -387], [0, -14, 245, 0], [-274, 0, 0, -450], [0, -96, -152, 0], [144, 0, 0, -466], [0, -28, 260, 0], [0, -102, 40, 0], [0, -137, 9, 0], [0, -2, -314, 0], [-1506, 0, 0, 278], [1381, 0, 0, 338], [0, 232, -52, 0], [0, -282, -61, 0], [1267, 0, 0, 48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_62_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_62_3_b_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_3_b_a(:prec:=4)
    chi := MakeCharacter_62_b();
    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_62_3_b_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_3_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_62_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;