
// Make newform 12.15.c.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_12_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_12_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_12_15_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_12_15_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_12_c();"
function MakeCharacter_12_c()
    N := 12;
    order := 2;
    char_gens := [7, 5];
    v := [2, 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_12_c_Hecke();"
function MakeCharacter_12_c_Hecke(Kf)
    N := 12;
    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 := [[0], [-2187], [0], [-1389022], [0], [88071962], [0], [1512529226], [0], [0], [-23014293166], [-111626070166], [0], [528051507962], [0], [0], [0], [6214599846074], [-11973142765462], [0], [17278443497906], [38383122173618], [0], [0], [-11651694897022], [0], [-181554565596574], [0], [-82560107906854], [0], [970569860053394], [0], [0], [1055778395752058], [0], [2355113278757954], [2023382846839418], [1518633172942442], [0], [0], [0], [-11144273272619446], [0], [-10633733625505918], [0], [-3754744686332254], [11416039276365194], [14266400147685842], [0], [-25574941734778774], [0], [0], [-91122277589703646], [0], [0], [0], [0], [21073128773336882], [-250224002298217462], [0], [-90410572611632998], [0], [-482000066333093302], [0], [588344327485203602], [0], [686498758122537722], [640149744052518818], [0], [-777937365832208902], [0], [0], [-190714462270323982], [-1638820860758420278], [2179712822418264986], [0], [0], [532786231128047258], [0], [696209258588407826], [0], [316126818784379882], [0], [-4462802223279132958], [4287863724226598786], [0], [0], [8215115618081499314], [0], [-8076168725889072334], [0], [0], [5894524061592226658], [0], [-2794559535367563574], [0], [0], [0], [16641959661091882682], [-13527543936829561606], [6517486797482829674], [0], [0], [0], [-2648253596886622246], [-24613447602987418942], [0], [0], [0], [-1638646448950274926], [-55146890768915418286], [62245750687098751466], [0], [65306513088765945146], [57270498816431628674], [0], [-89786700651705312214], [0], [0], [0], [-66459828376904442358], [-100073141273645623294], [0], [0], [80912452487477671562], [0], [-170916357885542484694], [0], [-150414255843136949182], [163598227829503780346], [121801903847712530858], [0], [-56434228167125389198], [-234099107758614585142], [0], [-278367189653616861694], [0], [242602909269807984458], [0], [0], [-387857339816251882246], [0], [503723684507178662402], [0], [-498089301656077206982], [0], [-61448187999353354998], [0], [-338702276706248733862], [0], [517897041519948447578], [0], [262906392966611034122], [0], [743690235101930717114], [0], [319457337854078119106], [0], [-1255403399107188319822], [0], [0], [0], [203182527012407192354], [0], [0], [0], [303685677320738993234], [1193734679462045450474]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_12_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_12_15_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_12_15_c_a(:prec:=1)
    chi := MakeCharacter_12_c();
    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_12_15_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_12_15_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_12_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,15,sign)));
    Vf := Kernel([<5,R![0, 1]>],Snew);
    return Vf;
end function;