
// Make newform 21.3.h.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_21_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_21_h_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_21_3_h_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_21_3_h_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 := [25, 0, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 5, 5];
    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_21_h();"
function MakeCharacter_21_h()
    N := 21;
    order := 6;
    char_gens := [8, 10];
    v := [3, 2];
    // 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_21_h_Hecke();"
function MakeCharacter_21_h_Hecke(Kf)
    N := 21;
    order := 6;
    char_gens := [8, 10];
    char_values := [[-1, 0, 0, 0], [-1, 0, 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 := 3;
    raw_aps := [[0, 1, 0, 0], [0, -1, -2, 1], [0, -1, 0, 0], [0, 0, 7, 0], [0, -5, 0, 5], [-2, 0, 0, 0], [0, 12, 0, -12], [-16, 0, 16, 0], [0, -6, 0, 0], [0, 0, 0, 7], [0, 0, 3, 0], [-12, 0, 12, 0], [0, 0, 0, 14], [44, 0, 0, 0], [0, 6, 0, 0], [0, 9, 0, -9], [0, -9, 0, 9], [26, 0, -26, 0], [0, 0, -52, 0], [0, 0, 0, 42], [0, 0, -18, 0], [79, 0, -79, 0], [0, 0, 0, -63], [0, -22, 0, 0], [-93, 0, 0, 0], [0, 26, 0, -26], [82, 0, -82, 0], [0, 29, 0, 0], [0, 0, 144, 0], [0, 0, 0, -14], [177, 0, 0, 0], [0, -85, 0, 0], [0, -96, 0, 96], [-114, 0, 0, 0], [0, -6, 0, 0], [0, 0, -59, 0], [0, 0, 248, 0], [2, 0, -2, 0], [0, 0, 0, 112], [0, 118, 0, 0], [0, 86, 0, -86], [82, 0, 0, 0], [0, -76, 0, 0], [0, 0, -59, 0], [0, 0, 0, -98], [0, 0, 234, 0], [-26, 0, 0, 0], [-107, 0, 0, 0], [0, -135, 0, 135], [334, 0, -334, 0], [0, 120, 0, 0], [0, 0, 0, -126], [0, 0, -179, 0], [0, 0, 0, -77], [0, 118, 0, 0], [0, 16, 0, -16], [0, -177, 0, 177], [-457, 0, 457, 0], [0, 0, -52, 0], [0, 0, 0, 56], [0, 0, -88, 0], [0, 0, 0, -7], [166, 0, 0, 0], [0, 138, 0, -138], [271, 0, -271, 0], [0, -195, 0, 0], [16, 0, -16, 0], [-509, 0, 0, 0], [0, 170, 0, -170], [628, 0, 0, 0], [0, 208, 0, -208], [0, -104, 0, 0], [0, 0, -501, 0], [-698, 0, 698, 0], [366, 0, 0, 0], [0, 6, 0, 0], [0, 30, 0, -30], [264, 0, -264, 0], [0, -48, 0, 0], [0, 0, 199, 0], [0, 0, 0, -42], [254, 0, 0, 0], [0, -54, 0, 54], [-506, 0, 0, 0], [-471, 0, 471, 0], [0, -27, 0, 0], [0, 0, 0, -126], [23, 0, -23, 0], [0, 0, 0, 266], [58, 0, 0, 0], [0, 258, 0, 0], [0, 152, 0, -152], [0, 0, 809, 0], [0, 0, 0, 203], [254, 0, -254, 0], [0, 0, 0, -420], [0, 181, 0, 0], [0, -156, 0, 156], [-394, 0, 394, 0], [-404, 0, 404, 0], [16, 0, 0, 0], [0, 331, 0, -331], [0, -107, 0, 107], [0, 428, 0, 0], [0, 0, 46, 0], [0, 0, -991, 0], [0, 0, 0, 343], [0, -78, 0, 0], [0, -432, 0, 432], [-471, 0, 0, 0], [943, 0, -943, 0], [0, 0, -934, 0], [0, 0, 0, 42], [0, 0, 122, 0], [-61, 0, 0, 0], [0, 366, 0, -366], [908, 0, 0, 0], [0, -142, 0, 142], [0, -447, 0, 0], [0, 0, 0, -294], [0, 0, 836, 0], [-677, 0, 0, 0], [0, -477, 0, 0], [0, -285, 0, 285], [-674, 0, 674, 0], [0, 0, 0, 483], [114, 0, -114, 0], [0, 174, 0, 0], [1307, 0, 0, 0], [-408, 0, 408, 0], [0, 0, 354, 0], [0, 0, 0, -336], [1031, 0, -1031, 0], [-1258, 0, 0, 0], [0, -414, 0, 0], [-877, 0, 0, 0], [0, -226, 0, 226], [0, 0, -1222, 0], [0, 0, 0, -245], [0, -96, 0, 96], [-436, 0, 0, 0], [0, -209, 0, 0], [0, 0, 522, 0], [0, 0, 0, -441], [0, 0, 304, 0], [0, 0, 0, 448], [824, 0, 0, 0], [0, -2, 0, 2], [-128, 0, 128, 0], [0, 666, 0, 0], [-1552, 0, 1552, 0], [0, 0, 0, -756], [-502, 0, 0, 0], [0, 622, 0, 0], [0, 0, 298, 0], [0, 0, 0, 294], [-278, 0, 278, 0], [0, 496, 0, 0], [-219, 0, 0, 0], [0, -457, 0, 457], [0, -398, 0, 0], [0, 0, 0, 224], [1087, 0, 0, 0], [0, -267, 0, 0], [0, -264, 0, 264], [0, -450, 0, 450], [0, 0, -17, 0], [0, 0, -592, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_21_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_21_3_h_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_21_3_h_b(:prec:=4)
    chi := MakeCharacter_21_h();
    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_21_3_h_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_21_3_h_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_21_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![25, 0, -5, 0, 1]>],Snew);
    return Vf;
end function;