
// Make newform 3450.2.a.m in Magma, downloaded from the LMFDB on 20 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_m();".
// 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_m();".
// 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], [5], [0], [2], [3], [2], [1], [3], [2], [-7], [0], [2], [-3], [-12], [6], [2], [2], [-15], [11], [8], [-9], [3], [-10], [-3], [-13], [12], [17], [-9], [-16], [-12], [-15], [5], [12], [-4], [2], [-16], [15], [6], [12], [5], [-6], [-1], [3], [-19], [23], [26], [3], [14], [12], [-15], [8], [3], [-18], [-18], [18], [-16], [2], [27], [-4], [-30], [29], [27], [8], [-9], [-19], [20], [-18], [26], [-6], [0], [-16], [17], [-16], [24], [18], [20], [6], [-7], [15], [-10], [6], [2], [-28], [-24], [30], [-28], [-15], [-40], [15], [30], [8], [-6], [5], [12], [15], [-39], [20], [20], [-1], [18], [24], [30], [26], [2], [-12], [-24], [-24], [-43], [14], [-25], [-18], [-40], [-1], [3], [14], [-9], [-45], [-33], [-13], [17], [48], [24], [35], [-42], [-49], [3], [-40], [-49], [-25], [-6], [41], [-7], [-18], [-34], [-42], [32], [-42], [-18], [-4], [30], [-28], [-9], [2], [24], [-4], [54], [-4], [33], [-22], [-42], [11], [-27], [26], [0], [11], [-6], [-58], [18], [-24], [-33], [-58], [27], [-51], [-48], [-16], [26], [-22], [-9], [3], [-22], [-6], [62], [50], [-27], [29], [-54], [-1], [-55], [26], [45], [-52], [-24], [-60], [-42], [-40], [-52], [17], [45], [26], [12], [-58], [51], [48], [9], [50], [-37], [-27], [-51], [18], [44], [14], [26], [36], [18], [35], [36], [36], [-7], [-22], [66], [-43], [27], [-48], [-10], [-4], [60], [24], [-54], [-22], [-64], [-36], [56], [48], [20], [30], [24], [8], [72], [-34], [2], [-16], [66], [44], [72], [14], [18], [30], [75], [-69], [29], [-34], [8], [54], [-15], [-46], [12], [-40], [24], [-58], [-66], [54], [26], [-3], [30], [-22], [-28], [27], [38], [23], [3], [14], [-37], [12], [44], [36], [-15], [50], [-18], [-64], [2], [-28], [17], [-34], [-34], [72], [8], [80], [-69], [-48], [-76], [18], [23], [44], [36], [38], [-36], [-10], [72], [30], [-27], [24], [-57], [-64], [-72], [-40], [33], [-36], [53], [-22], [-48], [41], [-48], [68], [-34], [-54], [-7], [-12], [74], [-72], [18], [21], [-19], [12], [8], [-12], [72], [-16], [30], [-10], [-34], [33], [62], [-81], [-70], [86], [29], [0], [-6], [26], [66], [50], [-78], [38], [-30], [-82], [3], [-67], [29], [86], [54], [54], [41], [84], [-18], [-4], [-67], [-24], [48], [41], [65], [-21], [-55], [-25], [24], [-54], [-93], [-12], [24], [-13], [-30], [72], [-15], [20], [-46], [30], [-61], [-28], [-54], [47], [81], [-54], [5], [56], [-96], [-78], [20], [69], [71], [-24], [21], [-16], [18], [-4], [51], [74], [-94], [92], [-54], [-10], [90], [66], [95], [42], [-52], [-52], [18], [-22], [-6], [-76], [18], [-79], [-66], [-75], [-88], [53], [6], [32], [90], [-10], [3], [-60], [-10], [-58], [-33], [27], [74], [-18], [-18], [-54], [26], [-93], [-18], [47], [-27], [45], [-6], [-55], [105]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-5, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;