
// Make newform 1470.2.a.p in Magma, downloaded from the LMFDB on 25 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_p();".
// 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_p();".
// 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_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], [1], [-1], [0], [2], [-2], [4], [0], [8], [0], [2], [8], [2], [-2], [-10], [-2], [-4], [10], [2], [-12], [-10], [16], [-16], [-14], [-6], [14], [20], [12], [-2], [-14], [-12], [-12], [-2], [4], [-16], [0], [10], [-10], [-18], [6], [-2], [22], [0], [-18], [18], [-10], [-4], [-16], [12], [-10], [-14], [-8], [20], [20], [-12], [0], [10], [14], [-28], [14], [-4], [-30], [20], [20], [26], [6], [-4], [-2], [-4], [-10], [-24], [-20], [28], [-36], [8], [-14], [-24], [-14], [-14], [32], [-36], [38], [12], [-38], [26], [28], [6], [-42], [18], [16], [-8], [-4], [28], [6], [40], [6], [-18], [-30], [-20], [2], [-14], [30], [-24], [-38], [12], [2], [16], [20], [32], [-4], [-8], [-8], [6], [24], [-8], [42], [36], [2], [2], [-34], [26], [10], [-26], [24], [28], [16], [42], [48], [-32], [-30], [48], [-48], [-8], [-20], [30], [-16], [-30], [12], [-2], [-10], [28], [24], [-16], [8], [10], [-12], [-38], [12], [4], [24], [12], [46], [34], [2], [10], [16], [56], [30], [-42], [-50], [0], [-58], [32], [20], [-30], [46], [-24], [-6], [-2], [14], [18], [-50], [0], [30], [-58], [30], [-16], [-40], [40], [26], [4], [12], [52], [60], [-40], [18], [-40], [44], [42], [0], [-54], [12], [-24], [42], [-8], [12], [-50], [-32], [-4], [-6], [-36], [-2], [-26], [56], [12], [-30], [-34], [48], [-38], [-52], [-38], [-54], [16], [-32], [4], [8], [-4], [10], [-72], [2], [18], [-50], [-18], [-20], [-64], [10], [-12], [-48], [64], [-66], [-48], [-56], [-40], [50], [52], [2], [-64], [42], [30], [-20], [-12], [-64], [48], [50], [-40], [-56], [24], [-44], [4], [-24], [12], [-6], [-32], [20], [26], [22], [30], [12], [26], [-30], [-24], [-44], [10], [30], [-16], [60], [52], [-30], [30], [-30], [22], [-62], [82], [-16], [2], [-12], [-52], [-14], [22], [44], [30], [24], [26], [-62], [4], [28], [70], [-50], [-70], [-26], [28], [12], [34], [-76], [-32], [-18], [78], [-42], [-4], [76], [14], [74], [8], [-24], [68], [2], [12], [66], [60], [68], [-34], [16], [74], [-30], [72], [-8], [-36], [20], [-14], [-50], [68], [2], [-18], [56], [-26], [-52], [-4], [-28], [-8], [-2], [14], [50], [34], [60], [-40], [-60], [18], [48], [32], [20], [28], [-90], [38], [48], [50], [66], [-26], [62], [-36], [-46], [-36], [-2], [96], [8], [58], [24], [40], [60], [-6], [16], [60], [-90], [48], [-10], [4], [62], [-18], [52], [-14], [-6], [84], [-32], [36], [-10], [-12], [-52], [-100], [-70], [-70], [-26], [34], [34], [-28], [50], [28], [18], [-40], [-22], [-94], [66], [-50], [50], [86], [20], [-60], [66], [-70], [62], [80], [68], [42], [-66], [68], [16], [38], [-78], [-12], [-10], [-20], [12], [22], [-58], [12], [-80], [-58], [14], [8], [12], [-8], [-38], [48], [-82], [8], [-36], [-30], [-62], [56], [86], [56], [60]];
    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_p();".
// 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_p(:prec:=1)
    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_p();".
// 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_p( : 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, 1]>,<13,R![2, 1]>,<17,R![-4, 1]>,<19,R![0, 1]>,<31,R![-2, 1]>],Snew);
    return Vf;
end function;