
// Make newform 60.3.f.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_60_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_60_f_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_60_3_f_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_60_3_f_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 := [256, 0, 80, 0, 24, 0, 5, 0, 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], [0, 4, 0, 1, 0, 1, 0, 0], [16, 0, 4, 0, 1, 0, 1, 0], [1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 12, 0, -1, 0, -1], [128, 80, 16, 24, 16, 5, 0, 1], [128, -80, 16, -24, 16, -5, 0, -1]];
    Rf_basisdens := [1, 1, 16, 16, 1, 32, 64, 64];
    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_60_f();"
function MakeCharacter_60_f()
    N := 60;
    order := 2;
    char_gens := [31, 41, 37];
    v := [1, 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_60_f_Hecke();"
function MakeCharacter_60_f_Hecke(Kf)
    N := 60;
    order := 2;
    char_gens := [31, 41, 37];
    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]];
    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 := 3;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [1, 1, 0, -1, 0, 0, 0, -1], [0, 2, 4, 0, 0, 0, -1, 1], [0, 0, 0, -2, -4, 0, -1, -1], [0, -2, 0, 0, 0, 0, -1, 1], [0, -2, 0, 0, 0, -4, 1, -1], [0, 0, 0, -4, 0, 0, 2, 2], [0, 4, -4, 0, 0, 0, -2, 2], [-26, 0, 0, 6, 0, 0, 3, 3], [0, 0, 0, 8, 8, 0, 0, 0], [0, 6, 0, 0, 0, 0, 3, -3], [-26, 0, 0, -12, 0, 0, -6, -6], [0, -16, -12, 0, 0, 0, 8, -8], [0, 16, -4, 0, 0, 0, -8, 8], [0, 6, 0, 0, 0, 0, 3, -3], [0, 0, 0, -14, 4, 0, 9, 9], [38, 0, 0, 0, 0, 0, 0, 0], [0, -28, 4, 0, 0, 0, 14, -14], [0, 0, 0, -4, -24, 0, -10, -10], [0, -4, 0, 0, 0, -8, 2, -2], [0, 0, 0, -8, -8, 0, 0, 0], [0, -8, 20, 0, 0, 0, 4, -4], [66, 0, 0, 8, 0, 0, 4, 4], [0, 48, 0, 0, 0, 8, 20, -20], [38, 0, 0, -2, 0, 0, -1, -1], [0, 62, 36, 0, 0, 0, -31, 31], [0, -32, 4, 0, 0, 0, 16, -16], [-66, 0, 0, -20, 0, 0, -10, -10], [0, -2, 0, 0, 0, 4, -3, 3], [0, -46, -76, 0, 0, 0, 23, -23], [0, 0, 0, -6, 4, 0, 5, 5], [0, -22, 0, 0, 0, 20, -21, 21], [0, 0, 0, 8, 8, 0, 0, 0], [-14, 0, 0, 6, 0, 0, 3, 3], [0, 0, 0, 24, 40, 0, 8, 8], [0, 26, 0, 0, 0, -16, 21, -21], [0, 44, -44, 0, 0, 0, -22, 22], [0, 76, -4, 0, 0, 0, -38, 38], [0, 18, 0, 0, 0, -8, 13, -13], [0, 0, 0, 34, -12, 0, -23, -23], [-30, 0, 0, -44, 0, 0, -22, -22], [0, 0, 0, -20, 8, 0, 14, 14], [0, 36, 0, 0, 0, 16, 10, -10], [0, -14, 0, 0, 0, -40, 13, -13], [0, 0, 0, 24, -24, 0, -24, -24], [0, 0, 0, -8, -40, 0, -16, -16], [0, -58, 100, 0, 0, 0, 29, -29], [0, 136, 132, 0, 0, 0, -68, 68], [146, 0, 0, -36, 0, 0, -18, -18], [0, -14, 0, 0, 0, -4, -5, 5], [0, 0, 0, -52, -56, 0, -2, -2], [158, 0, 0, 72, 0, 0, 36, 36], [0, 0, 0, 26, 36, 0, 5, 5], [0, -70, 0, 0, 0, -36, -17, 17], [0, -44, -84, 0, 0, 0, 22, -22], [-150, 0, 0, -22, 0, 0, -11, -11], [0, 0, 0, -80, -72, 0, 4, 4], [0, -26, 0, 0, 0, 64, -45, 45], [-146, 0, 0, 24, 0, 0, 12, 12], [0, -52, -140, 0, 0, 0, 26, -26], [0, 78, 0, 0, 0, 8, 35, -35], [0, 104, 116, 0, 0, 0, -52, 52], [0, 0, 0, 76, 88, 0, 6, 6], [0, 4, 0, 0, 0, 16, -6, 6], [0, 62, 0, 0, 0, 56, 3, -3], [0, 0, 0, 28, 64, 0, 18, 18], [0, 60, 0, 0, 0, 40, 10, -10], [0, -104, 84, 0, 0, 0, 52, -52], [122, 0, 0, -24, 0, 0, -12, -12], [0, -54, 0, 0, 0, -20, -17, 17], [0, 0, 0, -48, 0, 0, 24, 24], [0, -94, -204, 0, 0, 0, 47, -47], [0, -82, 0, 0, 0, -64, -9, 9], [0, 0, 0, -112, -120, 0, -4, -4], [0, -104, 204, 0, 0, 0, 52, -52], [-62, 0, 0, 126, 0, 0, 63, 63], [0, -90, 0, 0, 0, -48, -21, 21], [122, 0, 0, -68, 0, 0, -34, -34], [-122, 0, 0, 116, 0, 0, 58, 58], [0, 0, 0, -10, 28, 0, 19, 19], [190, 0, 0, -28, 0, 0, -14, -14], [0, 0, 0, -32, 48, 0, 40, 40], [0, 116, 0, 0, 0, -32, 74, -74], [0, 0, 0, -136, -152, 0, -8, -8], [0, -176, -220, 0, 0, 0, 88, -88], [22, 0, 0, 36, 0, 0, 18, 18], [0, -136, 0, 0, 0, -24, -56, 56], [334, 0, 0, -6, 0, 0, -3, -3], [0, -118, 36, 0, 0, 0, 59, -59], [0, -8, 356, 0, 0, 0, 4, -4], [0, 0, 0, 28, 56, 0, 14, 14], [0, 158, -236, 0, 0, 0, -79, 79], [0, 0, 0, -70, -140, 0, -35, -35], [0, 0, 0, 124, 176, 0, 26, 26], [0, 232, 76, 0, 0, 0, -116, 116], [70, 0, 0, -42, 0, 0, -21, -21], [-726, 0, 0, -4, 0, 0, -2, -2], [0, -52, -172, 0, 0, 0, 26, -26], [-462, 0, 0, 148, 0, 0, 74, 74], [0, 128, 196, 0, 0, 0, -64, 64], [0, -14, 0, 0, 0, -136, 61, -61], [0, -296, -316, 0, 0, 0, 148, -148], [-10, 0, 0, -20, 0, 0, -10, -10], [0, 0, 0, -132, 32, 0, 82, 82], [0, -112, 0, 0, 0, 16, -64, 64], [0, -80, 116, 0, 0, 0, 40, -40], [0, -226, 0, 0, 0, 4, -115, 115], [0, 0, 0, 144, -32, 0, -88, -88], [502, 0, 0, -52, 0, 0, -26, -26], [0, 374, 148, 0, 0, 0, -187, 187], [0, 10, 0, 0, 0, 80, -35, 35], [0, 78, 0, 0, 0, -76, 77, -77], [0, 0, 0, -80, -200, 0, -60, -60], [0, 0, 0, 24, 120, 0, 48, 48], [-650, 0, 0, -108, 0, 0, -54, -54], [0, -28, 292, 0, 0, 0, 14, -14], [0, 16, -68, 0, 0, 0, -8, 8], [0, 34, 0, 0, 0, -128, 81, -81], [0, 0, 0, -114, -196, 0, -41, -41], [70, 0, 0, -88, 0, 0, -44, -44], [0, -36, 0, 0, 0, -112, 38, -38], [0, 94, 0, 0, 0, 48, 23, -23], [0, 208, -364, 0, 0, 0, -104, 104], [0, 0, 0, -92, 16, 0, 54, 54], [234, 0, 0, -262, 0, 0, -131, -131], [-94, 0, 0, -156, 0, 0, -78, -78], [0, 0, 0, 88, 0, 0, -44, -44], [0, 62, -236, 0, 0, 0, -31, 31], [0, 146, 0, 0, 0, -64, 105, -105], [0, 0, 0, -28, 0, 0, 14, 14], [0, -8, -100, 0, 0, 0, 4, -4], [0, 0, 0, 48, -40, 0, -44, -44], [0, 38, 0, 0, 0, 112, -37, 37], [-518, 0, 0, 180, 0, 0, 90, 90], [122, 0, 0, 108, 0, 0, 54, 54], [0, -286, 0, 0, 0, 40, -163, 163], [0, -292, -540, 0, 0, 0, 146, -146], [0, -194, 0, 0, 0, 88, -141, 141], [-38, 0, 0, 84, 0, 0, 42, 42], [0, 0, 0, 136, -72, 0, -104, -104], [682, 0, 0, 114, 0, 0, 57, 57], [0, -262, -524, 0, 0, 0, 131, -131], [0, -368, -236, 0, 0, 0, 184, -184], [366, 0, 0, 148, 0, 0, 74, 74], [0, 0, 0, -72, 176, 0, 124, 124], [0, -94, 0, 0, 0, -192, 49, -49], [0, -98, 0, 0, 0, -84, -7, 7], [0, 0, 0, 96, 296, 0, 100, 100], [0, 340, 556, 0, 0, 0, -170, 170], [0, 362, 0, 0, 0, 80, 141, -141], [-162, 0, 0, 296, 0, 0, 148, 148], [0, -124, 404, 0, 0, 0, 62, -62], [0, 28, 492, 0, 0, 0, -14, 14], [0, 632, 484, 0, 0, 0, -316, 316], [0, 0, 0, 256, 224, 0, -16, -16], [0, 0, 0, -176, -72, 0, 52, 52], [1334, 0, 0, -44, 0, 0, -22, -22], [0, -232, 0, 0, 0, 64, -148, 148], [1246, 0, 0, -126, 0, 0, -63, -63], [0, 64, 116, 0, 0, 0, -32, 32], [0, -138, 0, 0, 0, -132, -3, 3], [0, -310, 436, 0, 0, 0, 155, -155], [0, 0, 0, 138, 132, 0, -3, -3], [0, -250, 0, 0, 0, -36, -107, 107], [0, -416, -676, 0, 0, 0, 208, -208], [0, 0, 0, -32, 168, 0, 100, 100], [0, -78, 0, 0, 0, -304, 113, -113]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_60_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_60_3_f_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_60_3_f_b(:prec:=8)
    chi := MakeCharacter_60_f();
    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_60_3_f_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_60_3_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_60_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![64, 0, -92, 0, 1]>],Snew);
    return Vf;
end function;