
// Make newform 3648.2.a.bt in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3648_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_3648_2_a_bt();".
// 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_3648_2_a_bt();".
// 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 := [-8, -1, 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_3648_a();"
function MakeCharacter_3648_a()
    N := 3648;
    order := 1;
    char_gens := [2623, 2053, 1217, 1921];
    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_3648_a_Hecke(Kf)
    return MakeCharacter_3648_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], [1, 0], [0, 1], [0, -1], [-4, 1], [4, 0], [-2, -1], [-1, 0], [2, -2], [2, 2], [6, 0], [4, 0], [2, 2], [4, 1], [-2, 3], [10, 0], [4, -2], [-6, -1], [0, -2], [8, 2], [6, -1], [2, 0], [0, 0], [6, -4], [14, -2], [8, 2], [10, 0], [-4, 2], [0, -2], [2, -2], [-2, 6], [-8, 3], [-18, -1], [-12, 1], [4, -5], [2, -4], [-2, 0], [12, 0], [-12, -4], [6, 0], [4, 0], [0, 2], [2, -7], [-2, 8], [12, -6], [-16, -1], [4, -6], [14, 2], [-12, 4], [-6, 1], [10, 1], [-10, -3], [-2, -6], [0, -5], [22, -2], [-10, -1], [6, -8], [8, 0], [-22, -3], [-22, 4], [-12, 5], [22, 2], [0, -6], [14, -5], [6, -8], [14, -4], [8, 6], [2, 6], [-12, 3], [-2, 1], [-6, 8], [-14, 9], [-8, 12], [4, -4], [24, 2], [-24, 2], [24, -3], [6, 9], [10, -4], [6, 8], [-12, 0], [12, 2], [-4, 0], [22, -6], [14, -6], [20, 3], [14, -2], [-22, -3], [-28, 3], [-24, 1], [-16, -1], [-22, 6], [10, -12], [8, -4], [-12, 7], [-2, 10], [-2, -2], [6, 0], [-16, 10], [-2, -9], [-16, 4], [-28, -5], [-20, -4], [-26, 2], [-4, -4], [-22, 5], [-12, -9], [2, -4], [12, -6], [-14, -8], [-6, 14], [-26, 1], [2, 3], [12, -8], [4, 3], [-18, 6], [28, -7], [10, 9], [28, -5], [-12, 6], [20, -4], [6, 0], [-6, -10], [4, 4], [12, -1], [-8, 2], [42, 0], [-26, 1], [-4, 13], [-34, 0], [-20, -3], [20, -10], [2, 2], [-2, 11], [-10, -1], [6, -11], [-2, 0], [12, -16], [-2, 4], [38, -3], [-32, 0], [-8, -9], [20, 3], [-20, -2], [-32, 4], [-4, 2], [2, 4], [-26, 8], [4, 9], [12, 12], [36, -6], [-22, -1], [36, 5], [-4, -10], [-32, 2], [12, -2], [12, 4], [-10, 8], [-2, 15], [-2, -8], [-24, -4], [10, 4], [24, 8], [-4, 8], [-6, 8], [-16, 4], [-2, 6], [-2, -3], [6, -12], [-28, -9], [-28, -4], [8, 6], [-10, 6], [-10, 8], [18, -6], [-54, 1], [-28, 8], [-28, 3], [-2, -6], [18, -5], [-8, -7], [-36, 0], [0, -14], [10, -16], [-10, -15], [-20, 7], [16, 10], [20, -6], [6, -14], [42, 2], [-54, 2], [0, -3], [16, 10], [34, -2], [-12, 0], [-6, 6], [-10, 15], [-58, 0], [42, 4], [10, -5], [2, 2], [58, 2], [20, 6], [-30, 2], [12, -8], [24, 11], [28, -3], [20, -16], [-6, -9], [-28, 2], [34, -12], [-48, -2], [16, 11], [52, -4], [12, 0], [38, -2], [52, -4], [10, -6], [-12, -6], [-56, 1], [-8, -2], [14, 6], [-2, -14], [-20, -7], [-36, -10], [50, -1], [-2, -8], [-20, -6], [-30, -4], [28, -11], [-18, 23], [-20, 2], [26, 4], [-42, 0], [20, -5], [-54, -3], [22, 3], [-32, -11], [0, -12], [-8, 10], [36, 6], [-4, -13], [-36, -4], [24, 2], [-26, -2], [70, -1], [-12, 5], [-20, -10], [52, 4], [10, -5], [14, -12], [-22, 9], [46, -10], [-22, -4], [36, -7], [-12, -16], [-70, 1], [32, -12], [42, 6], [54, -4], [-34, 2], [20, 14], [6, -1], [-8, -10], [22, -15], [12, 2], [-14, 0], [42, -7], [20, -6], [40, -11], [0, 14], [-12, 20], [54, -5], [8, -13], [2, 0], [-12, 4], [56, -7], [-16, 22], [-14, -16], [52, 1], [-32, -6], [40, 8], [22, -17], [8, -2], [-28, 17], [-14, -7], [2, -7], [30, 10], [-16, 7], [34, -2], [28, -2], [0, 5], [-38, 6], [4, 18], [48, 2], [-16, -2], [2, -2], [20, -11], [20, -14], [4, 7], [-22, -9], [-38, 8], [20, 11], [20, -14], [-12, -5], [-10, 4], [-12, -8], [48, -16], [54, 1], [10, 4], [-38, 1], [-12, -6], [-66, 6], [-20, 24], [6, -6], [18, -16], [-24, -17], [40, 10], [10, 13], [2, 23], [-24, 20], [42, 3], [58, -8], [-2, -8], [30, 12], [-2, 20], [0, 8], [-20, -4], [40, 4], [4, 23], [-26, 11], [-30, -12], [-68, -4], [-20, 1], [-28, 5], [8, -13], [-4, -18], [2, 2], [14, -15], [-52, 0], [-20, -10], [-18, 3], [-6, -26], [-2, 24], [18, -8], [-36, 24], [-34, 8], [-8, 0], [-4, 4], [8, -1], [-56, 12], [62, 6], [-20, 9], [46, -8], [-44, 10], [-26, 22], [22, -22], [0, 16], [-18, 15], [-56, 4], [-26, 17], [-30, -12], [-28, 4], [-12, 30], [-60, -1], [2, 22], [40, -14], [14, 20], [30, 8], [8, -4], [52, -16], [-38, 7], [42, -14], [-16, -21], [50, -11], [28, 16], [66, 6], [-50, -3], [78, -3], [46, -10], [26, -4], [14, -3], [-16, -15], [-6, 17], [-12, -12], [28, 4], [68, 0], [-38, 0], [28, 15], [-12, 20], [18, -12], [2, 0], [-80, 4], [-28, -13], [24, 22], [46, -24], [22, 0], [26, 6], [-24, 8], [-32, -10], [64, -10], [6, -15], [-74, -6], [22, 4], [14, 8], [12, -3], [-6, 25], [-34, -10], [-68, -8], [24, 15], [34, 4], [-40, 13], [28, -14], [-4, -21], [58, -16], [16, -14], [28, 0], [10, -28], [-42, 4], [4, 14], [2, 0], [-44, 0], [-74, 3], [-60, 4], [-26, 6], [22, -10], [12, 8], [42, -27], [68, -15], [18, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3648_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3648_2_a_bt();".
// 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_3648_2_a_bt(:prec:=2)
    chi := MakeCharacter_3648_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_3648_2_a_bt();".
// 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_3648_2_a_bt( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3648_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, -1, 1]>,<7,R![-8, 1, 1]>,<11,R![4, 7, 1]>,<23,R![-32, -2, 1]>,<31,R![-6, 1]>],Snew);
    return Vf;
end function;