
// Make newform 8.15.d.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8_d_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_8_15_d_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_8_15_d_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_8_d();"
function MakeCharacter_8_d()
    N := 8;
    order := 2;
    char_gens := [7, 5];
    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_8_d_Hecke();"
function MakeCharacter_8_d_Hecke(Kf)
    N := 8;
    order := 2;
    char_gens := [7, 5];
    char_values := [[-1], [-1]];
    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 := 15;
    raw_aps := [[-128], [3022], [0], [0], [38712254], [0], [-328636222], [1778973806], [0], [0], [0], [0], [-333393570766], [-495012562114], [0], [0], [-3914494552162], [0], [2711103884558], [0], [1579402558802], [0], [-31146255762898], [-38433671549134], [-62815034524126], [0], [0], [179453810669438], [0], [-933669648478], [0], [-1317302898312178], [1600852645891058], [856226373493886], [0], [0], [0], [-2785985847869938], [0], [0], [7274030110759406], [0], [0], [-19441823973739678], [0], [0], [-27087016674035026], [0], [15653950970849294], [0], [-64243922279126254], [0], [-22626994974655486], [111662818803793694], [48893872373757314], [0], [0], [0], [0], [22262556281081522], [247410912417815582], [0], [-493679953839131602], [0], [379697224754709266], [0], [-128421807285940546], [-683482215240466462], [774680466926475998], [0], [-1269681729500112574], [0], [0], [0], [-2241614561356653154], [0], [0], [0], [1698696244851398594], [750795173364327506], [-4516057752779467954], [0], [0], [5250395658012370562], [0], [3942082054352205662], [-2199296058580372894], [8017305152556028658], [0], [0], [2442653996794815854], [0], [0], [-2263871735287660738], [3813916435610978798], [0], [0], [5650894149820304114], [-8518146844054122178], [0], [-24386132406615005362], [0], [35421587586352395566], [31089771482832065426], [-36236178441292128418], [-516621503302458622], [7992477322200439358], [-28585123628626668958], [0], [13227136170443426354], [0], [0], [64004986946238695954], [-10876676801710200322], [0], [-38155182870822635038], [-64091357962953491314], [0], [0], [-32152157181656332114], [0], [113871852962417030306], [0], [-121775042240225888578], [105570425149073837294], [0], [0], [0], [0], [0], [240601480308873970958], [0], [0], [0], [-286405589627086616686], [-267301206933125461534], [0], [356201212153856178734], [0], [-41402490136057523662], [321182665919107337342], [0], [0], [109065380786194228958], [0], [0], [0], [506914710209234634578], [-308239628757158784034], [0], [0], [320409846733074556322], [-745827773261702280658], [0], [698907504513868815614], [0], [0], [1063940103607060634018], [481304852197603299218], [0], [-906816949080074857426], [711936767361432531602], [0], [831377721371598299774], [-379186294538396328382], [0], [0], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8_15_d_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_8_15_d_a(:prec:=1)
    chi := MakeCharacter_8_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 15));
    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_8_15_d_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_8_15_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,15,sign)));
    Vf := Kernel([<3,R![-3022, 1]>],Snew);
    return Vf;
end function;