
// Make newform 1232.2.a.h in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1232_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_1232_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_1232_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_1232_a();"
function MakeCharacter_1232_a()
    N := 1232;
    order := 1;
    char_gens := [463, 309, 353, 673];
    v := [1, 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_1232_a_Hecke(Kf)
    return MakeCharacter_1232_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 := [[0], [0], [2], [1], [1], [2], [2], [0], [8], [-2], [8], [-2], [10], [-4], [-8], [6], [0], [10], [12], [-16], [-14], [0], [0], [-6], [10], [18], [0], [4], [-2], [2], [-8], [0], [-6], [16], [14], [0], [-14], [20], [-16], [-14], [-4], [-22], [-8], [2], [-18], [-16], [-4], [-16], [-16], [-22], [-6], [8], [18], [-24], [-30], [24], [18], [-16], [30], [-22], [24], [-30], [16], [-8], [-14], [-18], [-4], [18], [28], [2], [18], [-32], [-16], [14], [-4], [-8], [-18], [18], [18], [10], [-16], [22], [8], [26], [-40], [-12], [-30], [-22], [-30], [8], [-24], [-8], [40], [-12], [28], [24], [2], [34], [-8], [38], [36], [-10], [-8], [-6], [44], [2], [-24], [42], [0], [-46], [16], [22], [-6], [16], [0], [-30], [32], [0], [-26], [-28], [2], [-14], [26], [-12], [-24], [30], [30], [8], [-32], [-22], [-12], [0], [32], [14], [-6], [34], [18], [-32], [2], [42], [40], [30], [-32], [52], [-38], [0], [26], [-22], [56], [-40], [-2], [-30], [-4], [32], [4], [48], [-24], [-6], [18], [-38], [52], [-22], [-32], [16], [2], [0], [8], [-30], [-14], [26], [-60], [-46], [-40], [58], [-8], [-38], [-44], [14], [24], [-38], [0], [-8], [-2], [-14], [-64], [42], [22], [-24], [26], [8], [34], [-28], [-20], [-14], [-4], [-6], [50], [38], [42], [24], [-26], [56], [34], [-22], [0], [-6], [-56], [36], [-6], [-8], [-46], [-14], [40], [-32], [24], [-14], [-32], [18], [48], [-10], [-42], [-40], [-30], [-16], [0], [-26], [10], [0], [-40], [-20], [-10], [-56], [32], [42], [0], [0], [42], [-18], [-60], [40], [36], [8], [-24], [14], [42], [0], [32], [56], [-36], [-48], [54], [26], [-16], [10], [26], [-36], [-34], [32], [2], [26], [56], [20], [18], [-22], [-6], [0], [30], [10], [68], [-10], [82], [68], [-6], [40], [58], [-8], [12], [-10], [58], [0], [48], [-32], [-48], [-62], [8], [48], [-14], [46], [-48], [42], [-18], [-8], [-6], [40], [-74], [2], [0], [-86], [64], [-72], [26], [46], [-80], [36], [-20], [50], [44], [58], [-72], [-18], [0], [-34], [50], [84], [-72], [18], [80], [-48], [-78], [-30], [-56], [-22], [10], [-40], [42], [-54], [-44], [48], [24], [6], [-42], [-58], [-8], [-56], [20], [0], [22], [-54], [90], [-72], [38], [-6], [26], [32], [54], [84], [-70], [92], [-48], [-30], [24], [-22], [-2], [-8], [-18], [66], [8], [-40], [-62], [24], [-82], [34], [-16], [-60], [40], [-6], [-14], [32], [26], [-36], [-56], [32], [-82], [-56], [30], [-72], [-56], [26], [-46], [90], [-94], [58], [40], [2], [-88], [-88], [-8], [-78], [-76], [-8], [-14], [34], [-28], [64], [24], [-54], [40], [-46], [4], [-58], [-54], [66], [-48], [-78], [42], [-16], [-2], [18], [-32], [32], [-6], [-90], [12], [-52], [58], [66], [16], [-80], [10], [-56], [70], [-30], [8], [-16], [98], [-6], [36], [-54], [-8], [80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1232_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1232_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_1232_2_a_h(:prec:=1)
    chi := MakeCharacter_1232_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_1232_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_1232_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1232_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![-2, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;