
// Make newform 12.7.c.a in Magma, downloaded from the LMFDB on 19 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_7_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_7_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
        poly := [5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 12]];
    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_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, 0], [-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 := 7;
    raw_aps := [[0, 0], [-3, 1], [0, 6], [242, 0], [0, -66], [2618, 0], [0, 264], [5786, 0], [0, -348], [0, -462], [-20446, 0], [-46774, 0], [0, 132], [68618, 0], [0, -792], [0, -6402], [0, 5574], [24794, 0], [-84358, 0], [0, -12084], [-113806, 0], [-159742, 0], [0, -19206], [0, 46812], [899522, 0], [0, -44682], [-1597006, 0], [0, -7986], [2218106, 0], [0, -74952], [709346, 0], [0, 29634], [0, 54708], [-5074102, 0], [0, 124014], [-3931246, 0], [-3611878, 0], [3763898, 0], [0, 17820], [0, 186714], [0, 358026], [3855914, 0], [0, -50784], [-4531582, 0], [0, -226842], [-2917774, 0], [11207834, 0], [-14873182, 0], [0, 101970], [-8961334, 0], [0, 396132], [0, -891000], [-10021726, 0], [0, -962202], [0, 519552], [0, 1072236], [0, -759990], [-9575038, 0], [32866922, 0], [0, 884796], [39304298, 0], [0, -865194], [9051482, 0], [0, 80916], [10851218, 0], [0, -1189374], [-56859958, 0], [45057122, 0], [0, 787446], [-14110822, 0], [0, 599856], [0, -416196], [-5396158, 0], [-21215062, 0], [-6902614, 0], [0, -2283456], [0, 2793414], [93935162, 0], [0, 1062600], [-113436334, 0], [0, 4318578], [65833802, 0], [0, -4298712], [-91723102, 0], [17064146, 0], [0, -3804282], [0, -2577312], [49805426, 0], [0, 1866282], [-4159486, 0], [0, 3731514], [0, 7633296], [145736882, 0], [0, -3782130], [-94205734, 0], [0, -2439756], [0, 327138], [0, 5200500], [166441418, 0], [-82628326, 0], [-44235334, 0], [0, -8711142], [0, 3109194], [0, 1873740], [-344756566, 0], [-146691838, 0], [0, 13811358], [0, -8214360], [0, -11041404], [378140114, 0], [271119266, 0], [-89558326, 0], [0, 3114276], [-301493494, 0], [178991954, 0], [0, 9509568], [420260474, 0], [0, 4475052], [0, 15030282], [0, -12430374], [490583402, 0], [78096194, 0], [0, -4888554], [0, -51282], [-5596198, 0], [0, 16528578], [212263946, 0], [0, 15727896], [-459540718, 0], [-7631206, 0], [-162220102, 0], [0, 22209660], [47204162, 0], [-650504278, 0], [0, -24218964], [241672706, 0], [0, -9801594], [-326576998, 0], [0, 3554226], [0, 10539012], [-215000566, 0], [0, 29717886], [495961298, 0], [0, -857274], [503474138, 0], [0, -11414196], [-2745622, 0], [0, -11494692], [1040430698, 0], [0, -28775472], [-317817478, 0], [0, -17119560], [777966938, 0], [0, -22586124], [-752746294, 0], [0, -32949576], [-1083340654, 0], [0, 53550288], [1127142962, 0], [0, -45884454], [0, 258570], [0, 1863180], [-1775511694, 0], [0, 16319070], [0, 37343208], [0, -42209340], [1346043554, 0], [262968266, 0]];
    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_7_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_7_c_a(:prec:=2)
    chi := MakeCharacter_12_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 7));
    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_7_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_7_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_12_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,7,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;