
// Make newform 10.5.c.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_10_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_10_c_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_10_5_c_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_10_5_c_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, 0, 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_10_c();"
function MakeCharacter_10_c()
    N := 10;
    order := 4;
    char_gens := [7];
    v := [3];
    // 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_10_c_Hecke();"
function MakeCharacter_10_c_Hecke(Kf)
    N := 10;
    order := 4;
    char_gens := [7];
    char_values := [[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 := 5;
    raw_aps := [[2, 2], [1, -1], [-15, -20], [-19, -19], [202, 0], [-99, 99], [-239, -239], [0, 40], [541, -541], [0, -200], [-758, 0], [141, 141], [1042, 0], [-759, 759], [-459, -459], [-1819, 1819], [0, 4600], [2082, 0], [5081, 5081], [-3478, 0], [-3479, 3479], [0, -7680], [6081, -6081], [0, -5680], [561, 561], [1682, 0], [7021, -7021], [-2159, -2159], [0, -280], [-8479, 8479], [821, 821], [-2198, 0], [-9399, -9399], [0, -13960], [0, -9000], [-23798, 0], [29781, 29781], [12641, -12641], [29981, 29981], [-4739, 4739], [0, -32920], [-40558, 0], [33002, 0], [-23199, 23199], [-16899, -16899], [0, 14160], [48842, 0], [-35019, 35019], [-68599, -68599], [0, 98760], [53721, -53721], [0, -45600], [-57038, 0], [39402, 0], [31121, 31121], [-60739, 60739], [0, -63800], [-113238, 0], [-14739, -14739], [-7278, 0], [58601, -58601], [-95499, 95499], [38601, 38601], [29162, 0], [1881, -1881], [83781, 83781], [106282, 0], [-142479, -142479], [-6479, -6479], [0, 32920], [-53919, 53919], [0, -171760], [152261, 152261], [-71339, 71339], [0, 172600], [158421, -158421], [0, -146760], [-83579, -83579], [-42078, 0], [0, -300960], [0, 208680], [86882, 0], [-125078, 0], [5921, -5921], [0, 55280], [63561, -63561], [0, 204880], [-10599, -10599], [224242, 0], [-243499, 243499], [-226919, -226919], [0, 334240], [278541, 278541], [-84118, 0], [0, -166840], [190461, -190461], [0, 223960], [-297918, 0], [200601, -200601], [-288398, 0], [123081, 123081], [162261, 162261], [264081, -264081], [0, -8320], [283082, 0], [260401, 260401], [-281439, -281439], [419761, -419761], [0, 136240], [234962, 0], [-406779, -406779], [135621, -135621], [-151959, -151959], [0, -22440], [-199958, 0], [448562, 0], [73041, -73041], [-90259, -90259], [-56019, 56019], [0, 438920], [593762, 0], [424561, -424561], [229021, 229021], [-450999, 450999], [-432438, 0], [-895838, 0], [0, 64360], [0, -239840], [-438339, -438339], [145261, -145261], [0, -738040], [579101, -579101], [-495318, 0], [-536979, -536979], [-908798, 0], [0, 1027040], [161061, -161061], [772201, 772201], [299781, 299781], [0, -897040], [-115798, 0], [1241602, 0], [-13219, 13219], [394641, 394641], [0, 694760], [0, 124400], [-432299, 432299], [-669159, -669159], [0, 370040], [490981, -490981], [206181, 206181], [-118478, 0], [-204719, 204719], [-562179, -562179], [492241, 492241], [1152842, 0], [0, 337520], [0, 760240], [75721, 75721], [-1165518, 0], [-331639, -331639], [-573639, 573639], [-49859, -49859], [-1411318, 0], [-708639, -708639], [234221, -234221], [898762, 0], [-223379, -223379]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_10_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_10_5_c_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_10_5_c_b(:prec:=2)
    chi := MakeCharacter_10_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_10_5_c_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_10_5_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_10_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<3,R![2, -2, 1]>],Snew);
    return Vf;
end function;