
// Make newform 3450.2.a.x in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3450_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_3450_2_a_x();".
// 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_3450_2_a_x();".
// 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_3450_a();"
function MakeCharacter_3450_a()
    N := 3450;
    order := 1;
    char_gens := [1151, 277, 1201];
    v := [1, 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_3450_a_Hecke(Kf)
    return MakeCharacter_3450_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+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 := 2;
    raw_aps := [[1], [1], [0], [-1], [0], [2], [3], [2], [-1], [9], [-10], [11], [12], [-10], [-9], [0], [-6], [14], [14], [3], [11], [8], [-9], [3], [-10], [-9], [-7], [12], [11], [-9], [8], [12], [9], [5], [0], [8], [14], [-16], [-3], [-6], [0], [-1], [-18], [-25], [-15], [-1], [23], [-22], [-21], [2], [-24], [27], [8], [3], [-18], [-18], [6], [-28], [26], [3], [-16], [18], [5], [-15], [8], [-27], [5], [-28], [-30], [-10], [-6], [36], [8], [-13], [20], [-36], [-30], [8], [-18], [17], [15], [26], [30], [2], [20], [0], [-6], [32], [3], [8], [-33], [-18], [-28], [-30], [-19], [36], [-27], [-39], [-4], [44], [-1], [-42], [24], [-42], [26], [-22], [-12], [36], [24], [-19], [14], [-19], [30], [44], [5], [-21], [14], [21], [9], [15], [-31], [-7], [12], [-12], [11], [-6], [-19], [-15], [32], [29], [-1], [-30], [-13], [-13], [-42], [-34], [-42], [-28], [-30], [-6], [-52], [18], [8], [-33], [2], [0], [44], [-6], [-4], [27], [38], [30], [11], [39], [38], [-24], [-31], [-42], [50], [42], [-12], [15], [2], [-21], [21], [36], [-40], [26], [62], [-27], [51], [-10], [-18], [-22], [14], [-51], [-19], [6], [-19], [-37], [-46], [21], [-16], [-48], [36], [-18], [-16], [-4], [-31], [15], [2], [-24], [50], [-15], [0], [-39], [-10], [41], [45], [-33], [30], [-28], [2], [14], [-36], [6], [65], [0], [12], [17], [26], [30], [-1], [3], [-24], [-46], [-4], [-12], [-24], [-42], [50], [-40], [-60], [-16], [0], [-16], [6], [0], [8], [36], [14], [62], [32], [-18], [-28], [-12], [-10], [66], [18], [-39], [-21], [29], [-22], [56], [-42], [75], [38], [60], [20], [72], [-22], [-18], [78], [62], [-9], [66], [-10], [32], [9], [-10], [-19], [3], [-10], [-7], [48], [-64], [48], [33], [26], [-30], [8], [2], [-28], [59], [-82], [50], [12], [-4], [-40], [27], [48], [-28], [-54], [-19], [68], [36], [-34], [72], [2], [-36], [-78], [-3], [24], [39], [20], [0], [8], [3], [-12], [53], [-58], [48], [11], [60], [-40], [38], [6], [-13], [36], [50], [0], [-78], [69], [5], [48], [-88], [0], [0], [44], [54], [-22], [38], [3], [86], [-33], [38], [2], [-19], [24], [42], [62], [-18], [2], [-42], [-58], [-66], [74], [27], [77], [47], [-10], [-42], [-30], [-37], [12], [18], [-52], [-19], [36], [12], [-7], [-55], [33], [11], [-67], [72], [6], [-21], [-12], [-24], [-55], [-30], [48], [-87], [-28], [-94], [-78], [-79], [-64], [-6], [-73], [-21], [-90], [-1], [20], [-84], [78], [20], [-27], [-25], [-84], [69], [56], [-18], [-28], [-39], [-82], [50], [80], [42], [50], [6], [18], [-97], [78], [-40], [-52], [-78], [14], [6], [44], [-42], [-37], [30], [87], [-16], [-1], [-18], [-40], [6], [14], [33], [-12], [98], [2], [-51], [-15], [-10], [-18], [6], [-6], [38], [-63], [-30], [23], [39], [-3], [18], [17], [-21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3450_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3450_2_a_x();".
// 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_3450_2_a_x(:prec:=1)
    chi := MakeCharacter_3450_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_3450_2_a_x();".
// 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_3450_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;