
// Make newform 1062.2.a.h in Magma, downloaded from the LMFDB on 03 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1062_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_1062_2_a_h();".
// 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_1062_2_a_h();".
// 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_1062_a();"
function MakeCharacter_1062_a()
    N := 1062;
    order := 1;
    char_gens := [119, 415];
    v := [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_1062_a_Hecke(Kf)
    return MakeCharacter_1062_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], [-2], [0], [-4], [-6], [-2], [4], [-8], [-2], [8], [2], [-2], [0], [-8], [6], [-1], [10], [-8], [12], [-14], [-16], [-4], [-6], [2], [6], [8], [12], [10], [2], [-8], [12], [-18], [-4], [22], [8], [18], [-20], [-12], [-2], [-4], [-10], [0], [-14], [6], [16], [-8], [0], [-4], [-22], [18], [-28], [2], [4], [6], [-12], [14], [-16], [-18], [-10], [-16], [22], [28], [-20], [18], [-2], [-12], [10], [20], [10], [34], [-36], [-8], [-26], [-28], [-4], [-34], [34], [-30], [10], [-12], [34], [-24], [-14], [8], [-36], [14], [-22], [6], [-24], [-20], [4], [40], [-20], [-12], [-24], [30], [-2], [-20], [-22], [4], [6], [4], [18], [-32], [2], [-36], [-26], [12], [10], [24], [-46], [-34], [-28], [-16], [30], [-44], [20], [-42], [20], [-10], [42], [-18], [4], [40], [30], [22], [16], [16], [-10], [-40], [36], [-48], [-10], [38], [-6], [6], [12], [-34], [18], [48], [22], [40], [36], [-50], [48], [30], [-22], [-40], [16], [-10], [26], [20], [0], [-4], [44], [-16], [-22], [-22], [-2], [28], [-26], [24], [-12], [18], [-24], [0], [14], [34], [6], [-28], [50], [-36], [-30], [56], [30], [20], [38], [8], [30], [32], [60], [50], [54], [-12], [-42], [42], [-48], [10], [-24], [10], [-44], [8], [6], [12], [-54], [2], [-46], [42], [-32], [54], [-32], [-42], [42], [44], [-34], [-48], [-68], [10], [40], [-54], [-66], [24], [-12], [-28], [-62], [40], [30], [0], [54], [18], [48], [26], [32], [36], [-46], [-42], [-40], [-24], [-12], [58], [24], [40], [-30], [48], [60], [34], [38], [12], [-52], [-12], [8], [56], [54], [54], [-20], [-40], [52], [36], [-12], [30], [-54], [-48], [26], [-50], [12], [-58], [-16], [-26], [-70], [8], [12], [46], [-26], [46], [-8], [6], [-30], [60], [-18], [26], [12], [-46], [-80], [-30], [8], [-12], [-2], [74], [12], [4], [-80], [-72], [-70], [-48], [24], [18], [14], [56], [-66], [-50], [-60], [-18], [-60], [-26], [78], [24], [-58], [-12], [-24], [-54], [-82], [-16], [4], [-52], [-46], [-12], [58], [72], [26], [-44], [-42], [-82], [-16], [84], [2], [68], [-12], [18], [6], [-20], [-30], [30], [56], [22], [50], [40], [-44], [88], [-26], [-54], [30], [-56], [-4], [12], [4], [70], [-78], [66], [-64], [-26], [-62], [70], [-24], [30], [-20], [58], [-52], [-16], [-42], [32], [74], [-18], [8], [-50], [42], [8], [-36], [46], [-12], [-14], [22], [-20], [-20], [60], [58], [78], [16], [18], [12], [-56], [-8], [54], [8], [54], [84], [8], [50], [-54], [-86], [-26], [66], [56], [-70], [76], [48], [-16], [6], [-44], [-8], [10], [54], [-28], [80], [8], [26], [-24], [-18], [92], [46], [70], [-54], [-16], [22], [-26], [-64], [34], [-10], [40], [60], [-94], [-82], [-84], [-4], [58], [-50], [-56], [-72], [-6], [-76], [-2], [-74], [-36], [-52], [-86], [-18], [44], [70], [-68], [-52]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1062_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1062_2_a_h();".
// 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_1062_2_a_h(:prec:=1)
    chi := MakeCharacter_1062_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_1062_2_a_h();".
// 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_1062_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1062_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![0, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;