
// Make newform 6.7.b.a in Magma, downloaded from the LMFDB on 04 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_6_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_6_7_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_6_7_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 := [2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 4]];
    Rf_basisdens := [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_6_b();"
function MakeCharacter_6_b()
    N := 6;
    order := 2;
    char_gens := [5];
    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_6_b_Hecke();"
function MakeCharacter_6_b_Hecke(Kf)
    N := 6;
    order := 2;
    char_gens := [5];
    char_values := [[-1, 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 := 7;
    raw_aps := [[0, 1], [21, 3], [0, -30], [2, 0], [0, -6], [-2950, 0], [0, 792], [5258, 0], [0, -1812], [0, 390], [22898, 0], [34058, 0], [0, -2964], [-6406, 0], [0, 31800], [0, -34038], [0, -57774], [-62566, 0], [438698, 0], [0, -12060], [-730510, 0], [340562, 0], [0, 87726], [0, -68364], [-281086, 0], [0, -167118], [-865726, 0], [0, 260586], [650810, 0], [0, 308328], [-2280526, 0], [0, -189498], [0, -492036], [4573946, 0], [0, 788442], [-2208094, 0], [-1288870, 0], [879914, 0], [0, 1054932], [0, 74046], [0, 53406], [-6476182, 0], [0, -887712], [3500930, 0], [0, 858114], [-9509758, 0], [7064138, 0], [4668914, 0], [0, 3474102], [-4481782, 0], [0, 405324], [0, -467688], [-6995806, 0], [0, -5037966], [0, -33024], [0, -1523004], [0, -1334898], [13944530, 0], [28129322, 0], [0, 3949716], [10141850, 0], [0, -4922670], [-36325366, 0], [0, 6362556], [40109906, 0], [0, -6971658], [27836282, 0], [-23789662, 0], [0, -9441246], [47167706, 0], [0, -3101424], [0, 10929204], [-34046062, 0], [-51578134, 0], [42882842, 0], [0, 2674752], [0, -10877406], [-85581574, 0], [0, 7243224], [61055570, 0], [0, -5990250], [-19615606, 0], [0, 7099128], [-845854, 0], [-74820382, 0], [0, 22140498], [0, -19942560], [157358450, 0], [0, 32369070], [177977810, 0], [0, 16542702], [0, -19032144], [-4144318, 0], [0, -21097686], [117435530, 0], [0, -35311716], [0, -19927146], [0, 20255292], [-149806150, 0], [-157017382, 0], [-279468502, 0], [0, -26568450], [0, -14634210], [0, 45472260], [28403930, 0], [65247554, 0], [0, -11786406], [0, 27104184], [0, 38589324], [108477650, 0], [-343321486, 0], [296325002, 0], [0, -23453940], [-414773062, 0], [303858146, 0], [0, -32107776], [173810666, 0], [0, 42980868], [0, -7911858], [0, -20077170], [-99346390, 0], [-279411838, 0], [0, 7235346], [0, 66160458], [115164362, 0], [0, 10000566], [-128651830, 0], [0, -35541432], [523207586, 0], [-657372454, 0], [350495210, 0], [0, -82495884], [-33699310, 0], [298551530, 0], [0, -70481244], [-517372414, 0], [0, 32392674], [314717834, 0], [0, 139329798], [0, -185164308], [-117929734, 0], [0, -10640886], [16062050, 0], [0, -64817838], [-363153382, 0], [0, 155687268], [-384724438, 0], [0, 60571956], [-387442726, 0], [0, -11429136], [-499281478, 0], [0, 161714664], [7852106, 0], [0, -195756708], [1161927866, 0], [0, 63893928], [1204308002, 0], [0, -112063056], [-247137550, 0], [0, -165702594], [0, -264733026], [0, 7571268], [-1163220478, 0], [0, 246237786], [0, -138231624], [0, 155602380], [-1839149710, 0], [569927306, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6_7_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_6_7_b_a(:prec:=2)
    chi := MakeCharacter_6_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 7));
    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_6_7_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_6_7_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,7,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;