
// Make newform 32.6.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_32_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_32_6_a_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_32_6_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_32_a();"
function MakeCharacter_32_a()
    N := 32;
    order := 1;
    char_gens := [31, 5];
    v := [1, 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;

function MakeCharacter_32_a_Hecke(Kf)
    return MakeCharacter_32_a();
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 := 6;
    raw_aps := [[0], [8], [14], [208], [536], [694], [-1278], [-1112], [-3216], [2918], [2624], [-9458], [170], [19928], [-32], [-22178], [-41480], [15462], [20744], [-28592], [-53670], [69152], [37800], [-126806], [62290], [6414], [108432], [-103976], [2486], [15794], [-1024], [22664], [-53238], [-19816], [452190], [263280], [-353530], [100936], [284944], [484374], [-406680], [570302], [-138624], [34482], [643598], [-1107376], [-229976], [1089472], [687048], [-699730], [937722], [-643488], [157282], [1586040], [-654334], [330192], [1569558], [-957792], [565438], [-1341270], [734264], [-1133202], [2913768], [1438128], [-1376006], [-1234938], [1489304], [838226], [350008], [-383642], [4093090], [-3144304], [1476192], [-3739810], [-1899656], [-1743104], [-2691474], [5373526], [156418], [-306086], [6708680], [4023470], [7043040], [-1251422], [1254064], [-5180808], [-5730638], [-5241526], [-173994], [-4012768], [-774616], [-2335296], [-1039472], [-7850920], [-2716440], [4620336], [-4461978], [3743754], [-9284328], [862150], [-1754424], [-10029226], [5274600], [8369402], [-4027016], [2375682], [3440280], [-3229422], [9297136], [-11278166], [-7001152], [-6194322], [-2624070], [2837208], [-12965584], [-11679758], [-7027320], [9728208], [-9814250], [-14665176], [-14183650], [5499410], [17737486], [9396696], [13476712], [21559398], [-6381650], [16356640], [21313008], [12157094], [19233736], [-1665648], [-9812896], [-19275330], [-11786310], [12294130], [2570862], [-5545944], [20961030], [27029674], [2130520], [15805982], [22884784], [25533560], [8317862], [-36626096], [17480190], [9310618], [34952504], [20234944], [-19834250], [-38102286], [24156008], [-20236752], [-19797608], [21324192], [-34994064], [-28810734], [12885434], [-52761482], [-5539608], [-228102], [-7367088], [19116056], [55704002], [15546864], [-23689024], [3717198]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_32_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_32_6_a_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_32_6_a_c(:prec:=1)
    chi := MakeCharacter_32_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_32_6_a_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_32_6_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_32_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![-8, 1]>],Snew);
    return Vf;
end function;