
// Make newform 3648.2.a.x in Magma, downloaded from the LMFDB on 24 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_x();".
// 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_x();".
// 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_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], [1], [-2], [0], [4], [-2], [-6], [1], [-4], [2], [4], [-10], [10], [-4], [-4], [10], [-12], [-14], [12], [8], [-6], [-4], [-12], [-6], [10], [-2], [-12], [4], [6], [2], [-12], [-12], [-14], [-12], [6], [20], [-22], [-20], [0], [-6], [-12], [14], [4], [-6], [22], [-16], [-12], [-28], [-28], [10], [-6], [12], [10], [28], [2], [-12], [-6], [-8], [26], [10], [-12], [18], [12], [4], [-22], [-6], [4], [-14], [20], [26], [18], [12], [-8], [-26], [36], [16], [-18], [10], [-14], [-14], [12], [-26], [-24], [26], [-4], [20], [34], [26], [-2], [-8], [20], [4], [4], [-20], [20], [36], [-6], [26], [4], [-38], [-4], [-34], [12], [-38], [-20], [2], [-12], [34], [0], [10], [-4], [2], [42], [-4], [-40], [18], [-4], [-4], [-18], [4], [38], [26], [26], [-4], [36], [22], [42], [28], [-8], [-6], [-20], [32], [44], [2], [-30], [-14], [-30], [20], [-6], [-30], [-12], [30], [8], [28], [-42], [-16], [-6], [42], [-52], [-24], [-34], [18], [20], [-32], [-44], [24], [8], [-22], [26], [-22], [-12], [-22], [48], [28], [-46], [-24], [4], [2], [10], [6], [12], [-26], [36], [26], [12], [-6], [52], [54], [28], [-62], [-8], [-4], [-34], [58], [-36], [54], [22], [20], [50], [-60], [-14], [-20], [44], [18], [-52], [-6], [2], [42], [2], [-36], [-30], [28], [38], [2], [-44], [-26], [-64], [36], [-6], [28], [50], [-2], [-48], [-12], [48], [-30], [0], [18], [-32], [6], [-18], [-4], [-46], [-64], [-68], [42], [26], [-72], [-28], [-60], [-6], [52], [60], [10], [4], [-60], [66], [46], [-68], [-24], [-4], [-52], [-56], [-74], [18], [-12], [64], [-20], [-20], [36], [-22], [-46], [68], [26], [-2], [-52], [42], [4], [-54], [-38], [-44], [60], [-30], [6], [42], [-28], [66], [42], [-20], [-18], [6], [-28], [-54], [56], [-46], [-24], [-36], [-10], [-38], [12], [-48], [-32], [-60], [-18], [-28], [-12], [66], [18], [-32], [50], [-66], [-12], [42], [-20], [70], [-26], [44], [-26], [-28], [-36], [58], [34], [16], [-68], [12], [26], [60], [-26], [-60], [-38], [52], [54], [-14], [-60], [-12], [-6], [36], [-32], [2], [-62], [-52], [42], [-78], [-28], [-30], [-54], [-4], [-4], [-32], [22], [74], [18], [-16], [44], [36], [28], [30], [-78], [-70], [-80], [-10], [18], [-54], [-52], [34], [20], [-30], [36], [8], [30], [-4], [2], [46], [36], [-26], [-38], [-52], [60], [-38], [-24], [98], [26], [72], [-20], [-44], [74], [6], [12], [-62], [-12], [-84], [-76], [-22], [80], [82], [52], [-28], [50], [90], [66], [-30], [98], [64], [-30], [84], [32], [-32], [-70], [44], [-56], [-46], [-86], [12], [-4], [72], [-46], [100], [-70], [20], [30], [22], [-86], [-60], [-38], [18], [-88], [-46], [50], [-28], [-92], [-14], [30], [-76], [76], [-86], [-42], [-24], [20], [42], [-32], [-30], [22], [-52], [-100], [34], [90], [-68], [10], [52], [-4]];
    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_x();".
// 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_x(:prec:=1)
    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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3648_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]>,<23,R![4, 1]>,<31,R![-4, 1]>],Snew);
    return Vf;
end function;