
// Make newform 30.6.c.a in Magma, downloaded from the LMFDB on 30 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_30_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_30_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_30_6_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_30_6_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 := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_30_c();"
function MakeCharacter_30_c()
    N := 30;
    order := 2;
    char_gens := [11, 7];
    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_30_c_Hecke();"
function MakeCharacter_30_c_Hecke(Kf)
    N := 30;
    order := 2;
    char_gens := [11, 7];
    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 := 6;
    raw_aps := [[0, 4], [0, 9], [-55, 10], [0, -4], [-500, 0], [0, 288], [0, 1516], [1344, 0], [0, 4100], [2646, 0], [-5612, 0], [0, -7288], [-18986, 0], [0, 2404], [0, 8900], [0, -39804], [28300, 0], [18290, 0], [0, 65956], [-28800, 0], [0, 30808], [-60228, 0], [0, 2468], [-22678, 0], [0, -36968], [167918, 0], [0, 154364], [0, 136788], [53810, 0], [0, -82692], [0, -211780], [169500, 0], [0, 252036], [-192016, 0], [-235694, 0], [-371492, 0], [0, 264952], [0, 403124], [0, -261900], [0, -326228], [109516, 0], [-53146, 0], [232056, 0], [0, 1030672], [0, -522796], [215292, 0], [-1030080, 0], [0, -456020], [0, -434252], [722710, 0], [0, 565348], [-324904, 0], [915262, 0], [1367076, 0], [0, 892932], [0, 1866500], [1372266, 0], [458644, 0], [0, -985408], [165798, 0], [0, 1664708], [0, -2551044], [0, 736020], [1716600, 0], [0, -2838512], [0, -1276052], [443992, 0], [0, 2713256], [0, -1310508], [298910, 0], [0, -737996], [2340744, 0], [0, 127292], [0, -4038696], [-1012136, 0], [0, 2376100], [-1424974, 0], [0, 1693448], [-2845006, 0], [1890694, 0], [-4609300, 0], [-6041510, 0], [-3800, 0], [0, -250736], [3588732, 0], [0, 1414788], [829806, 0], [0, 4681976], [-141930, 0], [0, -727476], [0, -4476396], [1327184, 0], [0, -4116468], [-6123156, 0], [7904904, 0], [0, -3976284], [-781914, 0], [5826938, 0], [0, 9789380], [4760590, 0], [0, -1165948], [0, -1612932], [0, -3406028], [14400902, 0], [4747720, 0], [0, -10909408], [0, 8532228], [0, 4631820], [-6275976, 0], [7719878, 0], [0, -6061604], [0, 3664888], [0, -9325220], [7401616, 0], [160052, 0], [-16956530, 0], [0, -11012796], [0, 3338484], [0, 4761812], [-798188, 0], [-15404806, 0], [0, 976704], [0, 19388524], [0, 5255732], [-5450336, 0], [-4439606, 0], [-4559178, 0], [20663000, 0], [0, 5481612], [0, 8555792], [5291192, 0], [0, 2364324], [-8796940, 0], [0, 29580752], [-12629622, 0], [23218578, 0], [0, 17320068], [0, 556676], [0, 3005620], [-2231526, 0], [22486216, 0], [-16592062, 0], [0, -14754420], [0, -3394748], [509442, 0], [-40060904, 0], [0, -9675064], [0, 32753532], [21742024, 0], [0, -20874444], [0, 39680392], [26074206, 0], [0, -41048556], [0, 13755300], [0, -5869356], [46398192, 0], [-22785924, 0], [-27035154, 0], [0, -28614944], [-36789242, 0], [0, 7968276], [0, 48220204], [0, 48350956], [-40551516, 0], [0, -43492932], [0, 3348964], [-55572596, 0], [0, -12734184]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_30_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_30_6_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_30_6_c_a(:prec:=2)
    chi := MakeCharacter_30_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_30_6_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_30_6_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_30_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<7,R![16, 0, 1]>],Snew);
    return Vf;
end function;