
// Make newform 25.8.b.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_25_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_25_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_25_8_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_25_8_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 := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    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_25_b();"
function MakeCharacter_25_b()
    N := 25;
    order := 2;
    char_gens := [2];
    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_25_b_Hecke();"
function MakeCharacter_25_b_Hecke(Kf)
    N := 25;
    order := 2;
    char_gens := [2];
    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 := 8;
    raw_aps := [[0, 7], [0, -24], [0, 0], [0, 822], [172, 0], [0, 1931], [0, 6127], [25940, 0], [0, 6486], [81610, 0], [-156888, 0], [0, -55063], [467882, 0], [0, -249604], [0, 198442], [0, -640249], [1337420, 0], [-923978, 0], [0, 398652], [5103392, 0], [0, -2133739], [960, 0], [0, 3070416], [-2010570, 0], [0, 2440967], [9726702, 0], [0, 8157526], [0, 2044872], [26831770, 0], [0, -8740519], [0, 6250882], [-7756188, 0], [0, -18086013], [-10909220, 0], [36458050, 0], [7189552, 0], [0, -43996603], [0, -27443744], [0, -4307298], [0, -25626189], [50162660, 0], [69081662, 0], [154745432, 0], [0, -7970279], [0, 84094017], [-177772600, 0], [-161996428, 0], [0, 8796586], [0, 101517932], [-159558990, 0], [0, -9749259], [160220480, 0], [-378778718, 0], [161304252, 0], [0, -113693553], [0, 228863806], [-467286310, 0], [-44593208, 0], [0, -158328543], [-225817638, 0], [0, 10410516], [0, -89412729], [0, 42757972], [-484705728, 0], [0, 285410581], [0, 275095277], [-939838788, 0], [0, -266815913], [0, -539669408], [427217450, 0], [0, -744829599], [-841275480, 0], [0, 375230802], [0, 85536911], [-46623940, 0], [0, -221132934], [464032530, 0], [0, 158896117], [-1195629198, 0], [-2213047130, 0], [802298740, 0], [34471342, 0], [1726923912, 0], [0, 244153841], [2886397880, 0], [0, 463291596], [-1355352850, 0], [0, -23171453], [-152117178, 0], [0, 817251906], [0, 556901852], [-1277448240, 0], [0, -489836338], [-4921252468, 0], [3657864500, 0], [0, -1941790274], [-3900717430, 0], [2883986042, 0], [0, 438594236], [-653484618, 0], [0, 2296493692], [0, -3415820403], [0, 1712547456], [7509302790, 0], [-135841108, 0], [0, -818875393], [0, 2986025712], [0, 4156732921], [-9783675800, 0], [5401594202, 0], [0, 1422197122], [0, -3510528769], [0, -1675827573], [3923624140, 0], [-681544688, 0], [9651992082, 0], [0, -2539130304], [0, 1041651742], [0, 3424405051], [-6999128380, 0], [-1995943778, 0], [0, -5896952439], [0, -3901109343], [0, 2255764716], [-10633131068, 0], [-4385141098, 0], [5988049970, 0], [11776791840, 0], [0, -4205252818], [0, 7204196491], [-8217083020, 0], [0, -8626533754], [15849821752, 0], [0, -3569282303], [-2599929078, 0], [4964772190, 0], [0, -7484057889], [0, 4899950812], [0, -1965845683], [-12632426330, 0], [11665325772, 0], [-8174997858, 0], [0, -8760563114], [0, -6111272768], [10663419210, 0], [2313998680, 0], [0, -2111883849], [0, 1760518747], [-24493004980, 0], [0, -2702864894], [0, 14457749757], [780642562, 0], [0, -6845347184], [0, -1792017138], [0, 15055402272], [-24879995528, 0], [-10842048760, 0], [-10704503090, 0], [0, -17139344613], [-3736949818, 0], [0, -16607496108], [0, 23494773701], [0, 7114688102], [545473692, 0], [0, 9895789807], [0, -23575136634], [38796825032, 0], [0, 28319700417]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_25_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_25_8_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_25_8_b_a(:prec:=2)
    chi := MakeCharacter_25_b();
    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_25_8_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_25_8_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_25_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![196, 0, 1]>],Snew);
    return Vf;
end function;