
// Make newform 676.6.d.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_676_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_676_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_676_6_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_676_6_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
        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_676_d();"
function MakeCharacter_676_d()
    N := 676;
    order := 2;
    char_gens := [339, 509];
    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_676_d_Hecke();"
function MakeCharacter_676_d_Hecke(Kf)
    N := 676;
    order := 2;
    char_gens := [339, 509];
    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, 0], [-12, 0], [0, 27], [0, 44], [0, -270], [0, 0], [-594, 0], [0, 418], [4104, 0], [-594, 0], [0, 2128], [0, 149], [0, 8613], [12100, 0], [0, 648], [19494, 0], [0, 3834], [-34738, 0], [0, 10906], [0, -23436], [0, -33781], [-76912, 0], [0, 33858], [0, -14877], [0, -61199], [-11286, 0], [27256, 0], [122364, 0], [0, 49951], [-29646, 0], [-336512, 0], [100980, 0], [0, 158571], [-148324, 0], [0, 98307], [0, -37180], [120878, 0], [0, 55670], [0, 245916], [-707454, 0], [-493668, 0], [559450, 0], [-724032, 0], [0, -3553], [0, -265221], [-56168, 0], [-339196, 0], [0, 389680], [0, -372438], [0, 136373], [153846, 0], [0, 577368], [0, -328537], [-1341900, 0], [-132354, 0], [943272, 0], [967518, 0], [0, 259160], [-2222726, 0], [0, 98307], [1552276, 0], [0, 536085], [0, -792946], [730728, 0], [584858, 0], [0, -1241433], [0, 188974], [-639122, 0], [-2904660, 0], [0, 1995785], [0, 714609], [0, -580932], [-1089232, 0], [3505766, 0], [0, 2021926], [0, 2593728], [950346, 0], [0, 260369], [0, -382185], [0, 1320253], [-4980204, 0], [0, -118997], [0, -1941192], [66958, 0], [6501352, 0], [-4607604, 0], [0, -1888353], [0, -1590347], [0, 3342735], [0, 2175608], [-7079940, 0], [0, -1610928], [0, 1148548], [-2821500, 0], [0, -2068142], [8332632, 0], [0, 2170503], [-6741846, 0], [-7721956, 0], [0, 341033], [2157716, 0], [0, 1337985], [3553308, 0], [12922470, 0], [6083572, 0], [0, -7912031], [0, 2301102], [0, -4305609], [-7982280, 0], [10174010, 0], [-9958432, 0], [0, 2097931], [0, 4562757], [0, -3228670], [0, 7025684], [-8471682, 0], [0, 244282], [-2481192, 0], [-5291298, 0], [4724676, 0], [0, 3087101], [9409246, 0], [15008598, 0], [0, 6485346], [0, 11327794], [-19016910, 0], [0, -7565563], [15032304, 0], [7412296, 0], [0, -1389641], [0, 6052310], [0, 2234628], [-28846256, 0], [9608678, 0], [0, -2272941], [0, -10796159], [0, -7420005], [0, 12439262], [-31607982, 0], [-3100086, 0], [0, 935902], [0, -10009197], [-15311768, 0], [0, -4796550], [-25220942, 0], [0, 8881164], [0, 243485], [1926342, 0], [22353836, 0], [0, 9291888], [0, -14552417], [3146958, 0], [-15999524, 0], [-34587432, 0], [-17439644, 0], [-2595888, 0], [-17641096, 0], [0, 19839249], [39341258, 0], [0, 23125311], [0, 18951246], [26646246, 0], [0, 20489524], [-27203364, 0], [0, 12677769], [0, -5956956], [29193056, 0], [-17300074, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_676_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_676_6_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_676_6_d_a(:prec:=2)
    chi := MakeCharacter_676_d();
    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_676_6_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_676_6_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_676_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![12, 1]>],Snew);
    return Vf;
end function;