
// Make newform 24.8.a.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_24_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_24_8_a_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_24_8_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_24_a();"
function MakeCharacter_24_a()
    N := 24;
    order := 1;
    char_gens := [7, 13, 17];
    v := [1, 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_24_a_Hecke(Kf)
    return MakeCharacter_24_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 := 8;
    raw_aps := [[0], [-27], [-26], [1056], [6412], [5206], [-6238], [41492], [-29432], [-210498], [185240], [507630], [360042], [620044], [-847680], [1423750], [-2548724], [-706058], [-2418796], [265976], [-5791238], [2955688], [3462932], [-2211126], [-15594814], [12956310], [-6820016], [-13778532], [9677206], [-21918414], [-661016], [1203092], [12004650], [22291356], [-39235498], [11301296], [-22230218], [-32381052], [51509544], [-100283874], [69317892], [-74671794], [170909952], [11506562], [79303638], [23861968], [109739044], [-271084952], [-22102908], [-151778306], [-89441222], [-269284720], [198807090], [-338049412], [-164488702], [421663816], [213467726], [223317464], [172178542], [684835562], [-78000340], [-650814186], [367913220], [442350552], [-1061715318], [430645374], [-777479044], [238365426], [-917421156], [396693462], [-227148862], [1591745304], [290258728], [651930686], [858738044], [-1244408832], [435596518], [-1079399578], [-2222493438], [370640970], [-1059168636], [2156327774], [-1940088544], [-484392686], [368469216], [-1757268868], [2937039538], [-411304870], [1929673918], [307870424], [1502977332], [-39220912], [2802622528], [-1714586724], [-2468725500], [-1680069912], [-2511527682], [4174523594], [283171852], [-6669233370], [5122013540], [-1454906114], [7502270452], [-4236610598], [-1483012884], [-236380798], [1929175228], [-4582118318], [4355641704], [1600730410], [-3168556024], [5293001102], [-219315750], [-4665522500], [3939629200], [-6745480142], [-7701749628], [-9656419336], [2954273118], [8901702660], [12657927422], [7301121250], [-6157172570], [10720883980], [-14040413932], [3684953982], [-15553217506], [7471080160], [-1777860240], [13314012614], [-3060643180], [10350602440], [-1797199992], [-16124386834], [7651722938], [-19064526334], [6372598614], [-2688725516], [-5103586610], [23720649482], [14815129612], [7388847270], [12307028528], [6190404908], [6547411654], [-7985597544], [4322657150], [22592503482], [-4203385188], [10976575264], [-14347350138], [-16511928622], [8585085716], [3555406904], [-2506807796], [18205706608], [2415714064], [16113914546], [-36469903782], [-21999926754], [15273429252], [-8792558694], [-43165646992], [28477560300], [-2499084414], [40867969368], [-8314930952], [41131641934]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_24_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_24_8_a_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_24_8_a_a(:prec:=1)
    chi := MakeCharacter_24_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_24_8_a_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_24_8_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_24_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<5,R![26, 1]>],Snew);
    return Vf;
end function;