
// Make newform 3450.2.a.i 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_i();".
// 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_i();".
// 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], [-3], [-1], [6], [-1], [-1], [-9], [-4], [-4], [9], [-1], [6], [12], [-6], [-4], [-4], [-6], [-7], [-1], [-15], [-18], [-4], [6], [-7], [-12], [-4], [18], [14], [0], [-12], [-10], [0], [-10], [8], [-16], [24], [15], [-18], [14], [15], [2], [-9], [-13], [-22], [-10], [-24], [-4], [9], [12], [-10], [12], [18], [12], [-9], [8], [-19], [-6], [-4], [-12], [-4], [12], [8], [3], [8], [20], [12], [5], [-9], [-3], [-37], [-10], [20], [-27], [-30], [2], [12], [-7], [-9], [14], [24], [-28], [20], [-24], [-18], [2], [-21], [-4], [-21], [3], [14], [0], [26], [-9], [30], [36], [-31], [29], [26], [30], [21], [-18], [-4], [-31], [42], [21], [24], [-46], [-34], [26], [6], [-28], [17], [36], [-49], [12], [-39], [-45], [-4], [35], [0], [18], [50], [12], [8], [-18], [32], [44], [-4], [9], [-31], [-4], [27], [26], [-36], [17], [18], [51], [38], [45], [2], [27], [29], [3], [-37], [54], [38], [6], [14], [0], [8], [6], [-19], [-27], [-4], [-27], [-16], [-60], [-36], [12], [-16], [-9], [12], [-45], [-40], [53], [8], [42], [15], [17], [-45], [-10], [50], [-36], [26], [-63], [23], [-4], [8], [-21], [65], [27], [15], [-36], [2], [-7], [17], [-12], [-25], [-6], [-7], [-30], [-63], [6], [50], [26], [0], [48], [0], [-64], [-31], [14], [33], [39], [-52], [24], [-15], [-34], [59], [45], [17], [-27], [36], [8], [38], [-66], [9], [-18], [-25], [5], [-66], [29], [-54], [59], [18], [48], [-64], [30], [50], [32], [-37], [-57], [8], [-21], [50], [-24], [0], [30], [21], [2], [32], [2], [-30], [36], [-52], [12], [-31], [51], [32], [-30], [-27], [44], [-3], [-6], [62], [-43], [3], [-22], [8], [-15], [-10], [-34], [42], [-55], [18], [-72], [47], [-21], [-31], [-28], [62], [-49], [41], [14], [12], [-73], [38], [-12], [30], [-16], [-33], [-10], [-22], [-60], [38], [72], [-34], [-81], [-42], [-69], [3], [60], [38], [78], [-73], [81], [18], [8], [14], [-78], [-25], [-84], [-76], [-1], [-78], [-4], [-84], [11], [36], [-6], [-48], [8], [-72], [-4], [-72], [54], [26], [66], [-4], [-82], [-15], [-34], [-6], [2], [-52], [26], [-39], [54], [-46], [-27], [14], [-30], [68], [-48], [-52], [-54], [11], [-16], [26], [12], [81], [-64], [-96], [-90], [2], [38], [15], [-54], [-58], [2], [63], [56], [50], [-57], [-51], [-87], [27], [72], [44], [-9], [96], [75], [56], [-46], [54], [-88], [-40], [-6], [-52], [-42], [30], [-49], [-37], [42], [93], [-16], [6], [83], [78], [6], [98], [-6], [41], [36], [-82], [-82], [-88], [-48], [-28], [15], [-48], [14], [-72], [32], [-43], [42], [-25], [-54], [-61], [-66], [-13], [-90], [6], [20], [-4], [-66], [77], [-30], [-46], [69], [-27], [-37], [-4], [-9], [18], [-16], [0], [-57], [-60], [14], [90], [42], [-91], [-99], [-69], [-75], [26], [54]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : 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![3, 1]>,<13,R![1, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;