
// Make newform 1734.2.a.d in Magma, downloaded from the LMFDB on 11 March 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1734_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1734_2_a_d();".
// 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_1734_2_a_d();".
// 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_1734_a();"
function MakeCharacter_1734_a()
    N := 1734;
    order := 1;
    char_gens := [1157, 1159];
    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;

function MakeCharacter_1734_a_Hecke(Kf)
    return MakeCharacter_1734_a();
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]];
    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 := 2;
    raw_aps := [[-1], [-1], [2], [2], [0], [-6], [0], [0], [-6], [-6], [-10], [-2], [0], [-4], [8], [6], [0], [-10], [8], [-10], [16], [6], [16], [10], [-12], [-18], [-16], [8], [-14], [4], [-8], [0], [-2], [-16], [10], [8], [-2], [4], [18], [-14], [20], [-10], [-8], [4], [-18], [-6], [0], [16], [12], [-10], [-4], [0], [20], [12], [2], [16], [-6], [32], [-22], [-22], [-16], [-6], [-12], [10], [4], [-18], [-12], [8], [12], [10], [14], [0], [22], [-6], [4], [-24], [10], [22], [0], [-10], [-4], [2], [10], [-34], [-6], [4], [-24], [22], [-2], [24], [-28], [14], [-18], [-12], [-24], [34], [10], [0], [-16], [-10], [8], [38], [-24], [30], [-20], [-2], [12], [46], [-40], [40], [38], [14], [-12], [-4], [8], [-40], [36], [-32], [34], [-40], [-22], [-36], [-42], [-44], [40], [22], [-6], [-26], [48], [-14], [40], [26], [10], [22], [-18], [30], [-14], [-32], [-18], [-24], [-20], [-10], [14], [32], [30], [-34], [-14], [-48], [-20], [-16], [-38], [20], [-16], [-42], [-8], [-50], [0], [16], [2], [30], [8], [-6], [-8], [-32], [22], [46], [10], [-18], [-4], [-26], [0], [42], [30], [34], [0], [-36], [-20], [30], [16], [-10], [-32], [40], [6], [22], [-24], [30], [38], [44], [-4], [-30], [16], [44], [8], [18], [-52], [-44], [-40], [-66], [52], [-16], [-26], [30], [-42], [50], [40], [42], [0], [36], [44], [-28], [28], [-62], [-46], [12], [6], [-20], [8], [-38], [-18], [-26], [22], [14], [-30], [46], [68], [-30], [36], [-54], [32], [-60], [26], [44], [8], [38], [-56], [-48], [36], [46], [24], [48], [44], [-68], [-16], [-50], [44], [-6], [38], [-60], [-20], [16], [-42], [60], [72], [16], [-14], [20], [-50], [-52], [-2], [-38], [6], [48], [-26], [34], [48], [-40], [70], [22], [44], [34], [10], [-52], [-74], [40], [-38], [16], [-48], [10], [2], [8], [-16], [-10], [22], [-22], [8], [72], [-84], [-18], [-40], [30], [-30], [-52], [-54], [-40], [46], [-14], [-8], [-66], [-4], [12], [-6], [-38], [-14], [76], [-20], [-28], [-72], [-74], [0], [-66], [-6], [74], [40], [-24], [48], [50], [60], [10], [56], [-50], [-20], [-12], [42], [-16], [-16], [-2], [24], [24], [-42], [-74], [10], [-58], [-46], [24], [20], [72], [-70], [36], [40], [-48], [26], [62], [-26], [-8], [14], [-84], [30], [-32], [90], [-18], [-92], [-92], [42], [46], [-50], [54], [0], [-60], [8], [16], [-18], [-20], [-72], [36], [-48], [-14], [18], [-56], [20], [-72], [16], [74], [30], [-48], [42], [-36], [-10], [66], [50], [-22], [-70], [-54], [-22], [-52], [-84], [54], [-72], [-98], [-84], [-72], [-16], [14], [-40], [68], [-72], [44], [80], [90], [80], [-58], [94], [-26], [-32], [-48], [50], [70], [-18], [22], [76], [-76], [64], [-78], [-16], [-80], [-2], [-10], [26], [-42], [-48], [24], [30], [-38], [-2], [-20], [-44], [58], [-24], [76], [-88], [6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1734_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1734_2_a_d();".
// 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_1734_2_a_d(:prec:=1)
    chi := MakeCharacter_1734_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1734_2_a_d();".
// 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_1734_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1734_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![-2, 1]>],Snew);
    return Vf;
end function;