
// Make newform 21.22.c.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_21_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_21_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_21_22_c_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_21_22_c_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 := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-9, 18]];
    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_21_c();"
function MakeCharacter_21_c()
    N := 21;
    order := 2;
    char_gens := [8, 10];
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_21_c_Hecke();"
function MakeCharacter_21_c_Hecke(Kf)
    N := 21;
    order := 2;
    char_gens := [8, 10];
    char_values := [[-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 := 22;
    raw_aps := [[0, 0], [0, 6561], [0, 0], [-561991510, 31603993], [0, 0], [0, -59188659092], [0, 0], [0, -2560945382530], [0, 0], [0, 0], [0, 80612910215030], [57776323439003290, 0], [0, 0], [-265258444413820520, 0], [0, 0], [0, 0], [0, 0], [0, -159819254191212780], [6944332370266921520, 0], [0, 0], [0, -3987850521047987688], [168068289047083465196, 0], [0, 0], [0, 0], [0, -58385106087771462296], [0, 0], [0, 77684456387134540878], [0, 0], [-4464490110295350699506, 0], [0, 0], [-19751213115221011867420, 0], [0, 0], [0, 0], [0, -4068313822834600324310], [0, 0], [-43540286684050669327148, 0], [0, -14525549413224922802236], [134412476509758772807400, 0], [0, 0], [0, 0], [0, 0], [0, 47377848737160965903660], [0, 0], [-1989088012190431606203650, 0], [0, 0], [0, 93132819803455869943090], [-1589030837992603331339312, 0], [0, -538768047289524141138038], [0, 0], [0, -144673834008950636075780], [0, 0], [0, 0], [0, -1213205164936356899002160], [0, 0], [0, 0], [0, 0], [0, 0], [0, 3624927045310965424070310], [-2196176843605863273086230, 0], [0, 0], [0, 2177102722811170662823718], [0, 0], [0, 14403823681224119194821486], [0, 0], [0, -952396557483069504595408], [0, 0], [313447617509409161536886368, 0], [-186904955140296883584842270, 0], [0, 0], [0, 13493903648553321093086620], [0, 0], [0, 0], [0, 63797306357141637605542074], [-1608394608067600292513943850, 0], [2222654448182155576664390296, 0], [0, 0], [0, 0], [0, -214878386478316110979259204], [0, 0], [0, -296436401635688809047997160], [0, 0], [4536622701202252855670163578, 0], [0, 0], [0, 329181108056134639249119032], [0, -673540290420407666701164030], [0, 0], [0, 0], [-16480060634764858265173002710, 0], [0, 0], [-12924477976387041445195243420, 0], [0, 0], [0, 0], [2617736265465959525758080100, 0], [0, 0], [37284622445785187987116576864, 0], [0, 0], [0, 0], [0, 0], [0, -3828939248502171394135227462], [99907966081872095883883741442, 0], [-48689480703923618112935361160, 0], [0, 0], [0, 0], [0, 0], [-136363722291235548002011614928, 0], [0, 1746280106523322689880382616], [0, 0], [0, 0], [0, 0], [0, 13067811467050204214935561400], [0, -17115131108912279984292950614], [335321716776379799624306436710, 0], [0, 0], [0, 13372515737087770846907227530], [-204088505898430665332988356068, 0], [0, 0], [0, -38235561692127851072536716498], [0, 0], [0, 0], [0, 0], [0, 9659654922529399465238041380], [-812893964861900582902896806450, 0], [0, 0], [0, 0], [0, 83519681817470568712432616930], [0, 0], [791583928314597930879670988806, 0], [0, 0], [0, -52803495504185814650315016834], [0, 142065546932384434023189263932], [27302015506378880847554510384, 0], [0, 0], [2790249849777085127200026882052, 0], [2684015304503838485848370744470, 0], [0, 0], [0, -186958715454508375918141543840], [0, 0], [0, 315880347097663485602717106306], [0, 0], [0, 0], [0, 293719952354047622357545082650], [0, 0], [7901318280153990761223245865020, 0], [0, 0], [0, 472913640932255751087065729380], [0, 0], [0, -459380001902989314474785954772], [0, 0], [0, -13176477469664704629810307690], [0, 0], [-4312289842603101751290104568850, 0], [0, 0], [-5161311414856187785384690978280, 0], [0, 0], [9997349288611245089007675249160, 0], [0, 0], [8844530600847718444858023444676, 0], [0, 0], [0, 2002148250162290342628733646944], [0, 0], [0, 0], [0, 0], [24816574923533403717605808508580, 0], [0, 0], [0, 0], [0, 0], [29655034771287320882428302086308, 0], [0, 3854517857257302963210976257804]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_21_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_21_22_c_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_21_22_c_a(:prec:=2)
    chi := MakeCharacter_21_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 22));
    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_22_c_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_21_22_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_21_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,22,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;