
// Make newform 24.8.f.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_24_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_24_f_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_24_8_f_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_24_8_f_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 := [64, 0, 10, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 18, 0, 1], [0, -2, 0, -1], [20, 0, 4, 0]];
    Rf_basisdens := [1, 8, 4, 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_24_f();"
function MakeCharacter_24_f()
    N := 24;
    order := 2;
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_24_f_Hecke();"
function MakeCharacter_24_f_Hecke(Kf)
    N := 24;
    order := 2;
    char_gens := [7, 13, 17];
    char_values := [[-1, 0, 0, 0], [-1, 0, 0, 0], [-1, 0, 0, 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 := 8;
    raw_aps := [[0, 2, 1, 0], [-9, -9, 0, 0], [0, 0, -20, 0], [0, 0, 0, -62], [0, -350, 0, 0], [0, 0, 0, -406], [0, 5404, 0, 0], [11570, 0, 0, 0], [0, 0, -11336, 0], [0, 0, 11180, 0], [0, 0, 0, 2870], [0, 0, 0, -11322], [0, -79160, 0, 0], [495062, 0, 0, 0], [0, 0, 231872, 0], [0, 0, 116844, 0], [0, -282506, 0, 0], [0, 0, 0, -71510], [1400126, 0, 0, 0], [0, 0, -728616, 0], [-2223598, 0, 0, 0], [0, 0, 0, -223802], [0, 590962, 0, 0], [0, 1146364, 0, 0], [6867926, 0, 0, 0], [0, 0, -2334436, 0], [0, 0, 0, 41354], [0, -3642386, 0, 0], [0, 0, 0, 489138], [0, 2769352, 0, 0], [0, 0, 0, -430242], [0, -6144410, 0, 0], [0, -1086416, 0, 0], [-13418230, 0, 0, 0], [0, 0, 4273420, 0], [0, 0, 0, 1212850], [0, 0, 0, 3218018], [108124562, 0, 0, 0], [0, 0, -8020568, 0], [0, 0, -26352196, 0], [0, 540574, 0, 0], [0, 0, 0, -4452810], [0, 0, 22425104, 0], [-56029558, 0, 0, 0], [0, 0, -19292388, 0], [0, 0, 0, -3604542], [-94708258, 0, 0, 0], [0, 0, 0, -579146], [0, -6879086, 0, 0], [0, 0, 0, -407362], [0, -27325508, 0, 0], [0, 0, -12500000, 0], [139553882, 0, 0, 0], [0, 6460330, 0, 0], [0, 63133384, 0, 0], [0, 0, 74614904, 0], [0, 0, -78613860, 0], [0, 0, 0, -14707530], [0, 0, 0, -348242], [0, -91887980, 0, 0], [179280362, 0, 0, 0], [0, 0, -15833156, 0], [-134815954, 0, 0, 0], [0, 0, 11710504, 0], [322597862, 0, 0, 0], [0, 0, 131162012, 0], [1110516602, 0, 0, 0], [-354194194, 0, 0, 0], [0, 82404634, 0, 0], [0, 0, 0, -1088702], [0, 174624112, 0, 0], [0, 0, -231654120, 0], [0, 0, 0, 41641558], [0, 0, 0, -1276186], [53412710, 0, 0, 0], [0, 0, 266392144, 0], [0, 0, 146164460, 0], [0, 0, 0, -1664622], [0, 240290500, 0, 0], [1982464250, 0, 0, 0], [0, 392286634, 0, 0], [0, 0, 0, 93053230], [0, 0, 145232544, 0], [995746982, 0, 0, 0], [0, 0, 0, 21673378], [0, -397653758, 0, 0], [0, 463642204, 0, 0], [536393546, 0, 0, 0], [0, 0, 751227404, 0], [0, 0, 0, -47918146], [0, -315585446, 0, 0], [0, 0, -316306240, 0], [0, 0, 0, -104145702], [0, -941311850, 0, 0], [227878430, 0, 0, 0], [0, 0, 351089544, 0], [0, 0, -485783300, 0], [0, -215699360, 0, 0], [1334212742, 0, 0, 0], [0, 0, 0, -110332470], [1816544546, 0, 0, 0], [0, 0, -550619508, 0], [0, 527057482, 0, 0], [0, 318334384, 0, 0], [2258622122, 0, 0, 0], [-2789545954, 0, 0, 0], [0, 302641534, 0, 0], [0, 492694432, 0, 0], [0, 0, -1265056520, 0], [7791368642, 0, 0, 0], [0, 0, 0, 43398878], [0, 0, 0, -155572506], [0, 427750564, 0, 0], [11643319610, 0, 0, 0], [0, 0, 0, 425308410], [0, -1352274380, 0, 0], [-9211462078, 0, 0, 0], [0, 0, -31277688, 0], [0, 0, 1145533324, 0], [0, -372147026, 0, 0], [0, 0, 0, 102502670], [9404902742, 0, 0, 0], [0, 0, -2683548868, 0], [0, 202783882, 0, 0], [-5748973438, 0, 0, 0], [0, 0, 2668159884, 0], [0, 0, 0, -101514122], [0, 0, -2013644880, 0], [0, 0, 0, 250626058], [0, 0, 0, -493203886], [-4486201330, 0, 0, 0], [0, 0, -1209475256, 0], [0, 0, 0, -359584930], [0, 0, 0, 397390758], [0, 320092360, 0, 0], [-20750220790, 0, 0, 0], [0, 0, 1659519324, 0], [-11115494494, 0, 0, 0], [0, 0, 4071620332, 0], [0, 2927536024, 0, 0], [8800231622, 0, 0, 0], [0, 0, 6302512604, 0], [0, 0, 0, -184424566], [0, -670897586, 0, 0], [0, 0, 0, -1052651982], [0, 0, -1152459400, 0], [0, 0, 0, 394556134], [0, 3853783984, 0, 0], [4488375110, 0, 0, 0], [0, 0, 5719575024, 0], [0, 0, 0, -1314923542], [0, -5917935080, 0, 0], [5444396402, 0, 0, 0], [0, 0, -636911288, 0], [-27750367594, 0, 0, 0], [0, 0, 1745710304, 0], [0, 0, 0, 1019042538], [0, 7952767924, 0, 0], [36568894286, 0, 0, 0], [0, 0, -6720638996, 0], [0, -8661011786, 0, 0], [0, 9973269472, 0, 0], [0, 0, 0, -1093703782], [0, -7659842870, 0, 0], [0, -4722947636, 0, 0], [0, 0, -7541449816, 0], [0, 0, 0, -1400159730], [0, 0, 0, -561419402]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_24_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_24_8_f_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_24_8_f_b(:prec:=4)
    chi := MakeCharacter_24_f();
    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_f_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_24_8_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_24_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<5,R![-9600, 0, 1]>],Snew);
    return Vf;
end function;