
// Make newform 1470.2.a.v in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1470_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_1470_2_a_v();".
// 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_1470_2_a_v();".
// 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 := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_1470_a();"
function MakeCharacter_1470_a()
    N := 1470;
    order := 1;
    char_gens := [491, 1177, 1081];
    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_1470_a_Hecke(Kf)
    return MakeCharacter_1470_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, 0], [1, 0], [1, 0], [0, 0], [2, 1], [0, 0], [0, -1], [0, 2], [2, -2], [4, -3], [-2, -5], [0, 1], [-6, 2], [6, 1], [-2, 5], [2, 8], [-6, -6], [-6, 4], [-2, -7], [8, -2], [2, -4], [0, 8], [-8, -2], [2, 6], [6, -2], [-12, -4], [4, 10], [-4, 0], [-6, -8], [-2, -8], [0, 10], [-14, 2], [-16, 0], [12, -4], [-4, 7], [-2, -14], [6, 4], [2, -9], [-6, -5], [10, -6], [2, 1], [-6, 2], [0, 8], [-2, 14], [-18, 2], [10, 5], [4, 4], [12, -4], [-8, 4], [6, -14], [-10, -12], [12, -8], [-12, 3], [-12, 4], [-8, -7], [-2, 14], [2, -4], [-10, 3], [12, -7], [-22, -6], [22, -6], [-6, 16], [4, 8], [-32, -2], [26, -2], [-18, 0], [4, 16], [-6, -2], [-4, -4], [14, 14], [12, 9], [16, -2], [8, -6], [4, -5], [-16, 6], [6, -7], [20, -5], [-2, -16], [-14, 8], [0, 13], [-12, 4], [-6, -16], [0, -10], [6, -2], [18, -11], [8, -4], [6, 22], [10, 14], [-14, 0], [28, -4], [-12, 6], [8, -6], [16, -6], [2, -7], [-4, -14], [14, -7], [-24, -4], [2, 10], [10, 22], [6, 6], [26, -1], [-10, -16], [-12, 10], [18, -12], [16, 8], [2, 10], [0, -14], [16, -15], [16, 8], [4, -27], [12, 12], [16, 15], [-16, 20], [16, 10], [22, -2], [-30, 0], [6, -30], [26, -5], [30, -2], [30, 7], [-22, 4], [2, -28], [14, 2], [12, -14], [16, 20], [4, 11], [-14, 22], [16, 20], [40, -4], [18, 24], [-12, 22], [0, -4], [34, 6], [28, 3], [-2, 12], [-32, -1], [34, 4], [12, -12], [18, -6], [-6, -22], [-20, -12], [-20, -11], [-12, -20], [20, 22], [-34, 8], [8, 2], [18, 8], [40, 9], [0, -12], [16, 12], [20, -1], [26, -4], [14, 17], [6, 21], [22, -15], [0, -4], [-10, -26], [-6, 4], [-34, -12], [8, -20], [28, -6], [16, 28], [-16, 20], [28, -12], [-6, -8], [10, 19], [18, -2], [-14, -8], [-52, 4], [-2, -30], [10, 17], [-10, 22], [-24, 16], [-30, -14], [22, 15], [-6, -12], [44, -14], [28, 11], [-32, -8], [-22, -28], [-32, 14], [6, 26], [44, -7], [16, -1], [18, -30], [-6, -20], [0, 11], [-14, -22], [-42, 8], [24, -16], [-38, 10], [-40, 0], [16, -30], [46, -4], [-24, -2], [-8, 25], [12, 28], [-24, 23], [16, -3], [-26, -15], [-8, -19], [34, 5], [-8, 0], [-40, 3], [28, 0], [42, -8], [10, -27], [-28, 22], [2, 0], [8, 8], [-14, -20], [-2, 24], [-28, 24], [8, -10], [8, 22], [56, 9], [-40, -6], [38, -2], [-30, -10], [-14, 10], [-18, -14], [-10, 15], [-10, 30], [-28, -6], [-24, -6], [-30, -22], [0, -5], [-32, 24], [8, 4], [-38, -19], [40, -7], [36, -6], [-16, 20], [-58, 0], [20, 12], [-30, -33], [-16, -9], [14, -6], [-50, -13], [-20, 2], [-52, -16], [12, -2], [-52, 0], [-18, -14], [-20, -33], [56, -12], [-8, 12], [-22, -38], [-36, 16], [-16, 20], [4, -31], [22, 18], [0, 4], [52, -1], [-6, 8], [-54, 5], [-6, 48], [38, 26], [-6, 44], [-30, 26], [8, -36], [-4, -12], [-26, -4], [-56, 0], [-52, 21], [60, -4], [40, -1], [30, -10], [18, 39], [-6, -22], [30, -22], [10, 31], [30, 10], [0, 8], [-34, -26], [-56, 2], [16, 40], [42, -10], [20, -36], [-34, 18], [-58, 9], [0, -24], [-34, 9], [-34, -12], [14, -26], [0, -6], [-22, 30], [38, -12], [54, -5], [2, -52], [-16, 35], [-28, -36], [22, -4], [-2, 10], [72, 7], [-16, -12], [-2, 19], [14, -14], [20, 0], [12, -4], [-30, 40], [-46, 14], [32, 12], [16, 14], [-12, -44], [22, 12], [60, 12], [18, 8], [52, 14], [28, -41], [-18, 1], [-20, -15], [-18, 28], [18, -9], [24, -40], [0, -1], [-28, 28], [-56, -10], [14, 14], [34, -6], [48, 4], [2, -16], [8, -12], [8, 40], [20, 16], [4, 31], [-32, -28], [12, -44], [-16, 4], [26, -36], [42, -20], [74, -2], [34, 5], [-40, 24], [0, 18], [32, 36], [34, 22], [4, 45], [32, -27], [-52, -18], [12, 35], [28, -32], [28, 0], [24, 44], [42, -6], [-30, -43], [30, -18], [66, 7], [-12, -10], [-6, -24], [0, -12], [38, 10], [12, 13], [20, -24], [38, 14], [-52, -27], [-20, -24], [-28, 48], [50, -4], [26, -22], [-20, -37], [66, 8], [2, 22], [-30, 25], [20, 32], [-54, -2], [38, 34], [-8, -26], [46, 4], [-18, -25], [28, 16], [18, 30], [56, -11], [-2, 51], [36, 3], [-42, 34], [-12, -14], [-30, 30], [-14, 58], [2, -52], [46, 0], [-10, 4], [-28, 10], [-54, 8], [0, 0], [-10, 21], [8, -36], [2, 24], [-66, -19], [-46, -1], [60, 4], [18, -4], [-38, 33], [36, -24], [44, -30], [-10, 40], [-62, 1], [6, -16], [-72, 6], [40, -21], [-74, -20], [64, -4], [-80, 2], [-12, -57], [-36, -24], [-2, -9], [-12, -13], [-42, -18], [18, -14], [-34, -38], [46, -32], [-14, 36], [-12, -28], [-20, 10], [22, 26], [-68, -8], [-32, 0], [-40, 38], [-84, 7], [10, -41], [12, -21], [-12, -8], [-16, 20], [-28, 0], [14, -14], [-38, 28], [-16, -10], [-62, 26], [28, 6], [32, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1470_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1470_2_a_v();".
// 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_1470_2_a_v(:prec:=2)
    chi := MakeCharacter_1470_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_1470_2_a_v();".
// 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_1470_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1470_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![2, -4, 1]>,<13,R![0, 1]>,<17,R![-2, 0, 1]>,<19,R![-8, 0, 1]>,<31,R![-46, 4, 1]>],Snew);
    return Vf;
end function;