
// Make newform 1632.2.a.c in Magma, downloaded from the LMFDB on 02 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1632_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_1632_2_a_c();".
// 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_1632_2_a_c();".
// 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_1632_a();"
function MakeCharacter_1632_a()
    N := 1632;
    order := 1;
    char_gens := [511, 613, 545, 1057];
    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_1632_a_Hecke( : Kf := Kf)
    return MakeCharacter_1632_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]];
    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], [-1], [-1], [2], [1], [-1], [1], [-1], [5], [2], [2], [-8], [5], [5], [12], [0], [2], [2], [4], [8], [0], [6], [4], [6], [-2], [12], [13], [-3], [12], [15], [7], [-13], [-8], [8], [4], [4], [11], [-6], [-3], [3], [-2], [4], [-4], [2], [-9], [-6], [-22], [3], [-17], [-6], [9], [6], [-26], [8], [6], [0], [1], [23], [-16], [-10], [26], [12], [-24], [-24], [-14], [-14], [11], [-12], [4], [21], [30], [-24], [-16], [30], [-34], [34], [8], [-2], [-11], [-23], [4], [-1], [-12], [7], [-32], [-32], [30], [-27], [-18], [-16], [12], [17], [-2], [12], [-20], [-17], [0], [13], [36], [16], [22], [-30], [18], [6], [22], [17], [44], [-32], [-44], [-14], [8], [-5], [18], [-10], [-17], [-17], [22], [0], [-33], [-12], [-19], [16], [1], [51], [32], [-16], [-20], [-21], [-8], [14], [43], [-8], [46], [49], [22], [-39], [18], [-34], [-12], [-35], [20], [-17], [-52], [47], [22], [51], [-26], [-42], [28], [30], [24], [-54], [39], [15], [-44], [-19], [-21], [-45], [34], [-42], [0], [20], [21], [42], [2], [-21], [-6], [-26], [38], [-34], [0], [53], [25], [-49], [-41], [-9], [-2], [9], [57], [-37], [37], [12], [28], [-32], [-18], [42], [8], [-51], [-8], [32], [34], [8], [-25], [22], [-57], [42], [26], [-28], [-22], [42], [5], [-6], [25], [1], [-70], [-52], [-8], [64], [1], [-52], [40], [-24], [6], [30], [-33], [-52], [17], [-22], [-53], [-50], [-7], [-24], [-4], [-52], [-28], [-58], [-45], [-63], [-7], [-72], [-29], [-38], [56], [30], [-4], [-30], [66], [21], [5], [38], [21], [-40], [31], [-33], [7], [-15], [56], [-45], [-47], [-54], [-11], [-6], [78], [32], [72], [-66], [-16], [2], [27], [-50], [-54], [-28], [-76], [76], [-3], [-57], [-12], [14], [22], [-38], [36], [-31], [-41], [-32], [78], [67], [-42], [-55], [5], [18], [4], [4], [32], [-50], [24], [38], [18], [-58], [31], [4], [59], [-3], [12], [3], [-60], [19], [60], [50], [9], [28], [-39], [-30], [-64], [61], [-86], [74], [30], [-8], [18], [-26], [40], [42], [39], [39], [76], [11], [42], [-21], [-16], [84], [2], [-48], [-60], [35], [-2], [-21], [30], [64], [-28], [85], [46], [-19], [-36], [46], [44], [-51], [13], [39], [-8], [45], [-75], [48], [43], [71], [28], [57], [-24], [-41], [9], [18], [77], [64], [76], [-4], [-5], [26], [68], [-44], [-25], [-18], [-82], [-27], [20], [44], [-15], [-75], [29], [-56], [38], [-12], [-70], [-69], [30], [32], [68], [3], [40], [11], [60], [-43], [95], [74], [-2], [-69], [8], [41], [-11], [-58], [66], [48], [-40], [-55], [-12], [-9], [4], [20], [-64], [-4], [28], [-36], [-58], [-32], [45], [-26], [64], [64], [2], [-12], [77], [69], [-60], [62], [-2], [-80], [13], [-18], [-55], [44], [-77], [-76], [94], [26], [-29], [-60], [-28], [-30], [72], [21], [-35], [-57]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1632_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1632_2_a_c();".
// 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_1632_2_a_c(:prec:=1)
    chi := MakeCharacter_1632_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_1632_2_a_c();".
// 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_1632_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1632_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<7,R![-2, 1]>,<11,R![-1, 1]>],Snew);
    return Vf;
end function;