
// Make newform 3648.2.a.v in Magma, downloaded from the LMFDB on 16 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_v();".
// 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_v();".
// 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], [-2], [-2], [6], [1], [2], [-4], [-8], [2], [-8], [8], [2], [4], [0], [-2], [-12], [-4], [6], [-16], [-6], [0], [-2], [10], [0], [-8], [-18], [-16], [0], [-18], [22], [-12], [18], [-16], [-10], [4], [0], [24], [-24], [-10], [-26], [-6], [22], [20], [12], [8], [-4], [10], [-6], [6], [-26], [-14], [20], [6], [0], [-20], [26], [4], [-24], [-24], [-12], [10], [14], [-24], [28], [22], [-10], [2], [30], [30], [-20], [10], [-12], [-20], [6], [-14], [16], [-26], [6], [-2], [12], [26], [8], [38], [28], [-10], [-14], [4], [-22], [-2], [-32], [22], [20], [-42], [0], [32], [4], [-2], [20], [-22], [36], [-20], [-8], [38], [-6], [-14], [24], [10], [32], [-34], [-42], [-28], [-16], [24], [-40], [2], [18], [-20], [-22], [26], [-28], [-16], [0], [34], [-42], [-38], [52], [30], [4], [20], [8], [50], [30], [-26], [-24], [20], [24], [18], [-12], [-6], [-16], [40], [6], [44], [42], [0], [20], [12], [-22], [30], [44], [16], [4], [24], [-40], [-46], [26], [-40], [-18], [-4], [-24], [-20], [56], [-48], [40], [14], [-2], [-6], [-24], [10], [42], [26], [48], [-30], [-44], [-30], [-8], [-14], [-20], [-28], [-10], [28], [18], [66], [-26], [-4], [-34], [66], [22], [22], [44], [-60], [14], [24], [2], [-66], [-22], [6], [-36], [-32], [14], [38], [-26], [34], [-28], [48], [-18], [52], [-58], [46], [36], [24], [-60], [18], [-12], [0], [-20], [-18], [-18], [-40], [-64], [-64], [28], [6], [20], [-12], [-64], [6], [-6], [28], [24], [-32], [-20], [-54], [-18], [46], [-50], [-12], [-40], [-16], [-20], [-38], [-12], [-18], [-32], [28], [-20], [-54], [26], [-22], [2], [-34], [22], [-58], [54], [-44], [-12], [22], [64], [-36], [6], [66], [66], [-52], [-36], [-54], [-20], [6], [-54], [44], [66], [-40], [-34], [-48], [-30], [74], [58], [-54], [-84], [40], [-18], [42], [-16], [66], [-30], [48], [-8], [-76], [-30], [-30], [-36], [16], [34], [46], [-40], [-74], [44], [12], [22], [4], [28], [-56], [12], [-34], [60], [46], [66], [-38], [-26], [-18], [-8], [60], [18], [-18], [-66], [-32], [14], [-26], [20], [42], [36], [80], [30], [-42], [-76], [68], [-20], [-62], [-22], [-84], [8], [-10], [48], [10], [42], [-72], [-22], [76], [74], [30], [-84], [32], [-20], [-76], [-78], [-60], [-64], [-78], [20], [38], [22], [72], [34], [-32], [-82], [18], [22], [12], [-70], [62], [-60], [88], [28], [-94], [-6], [48], [34], [66], [-60], [74], [68], [20], [-2], [4], [-22], [2], [42], [-66], [84], [26], [4], [66], [-12], [68], [-20], [-70], [-76], [-56], [-58], [-92], [6], [-52], [-36], [-10], [-56], [-76], [-28], [30], [-38], [-14], [96], [52], [-24], [20], [62], [-76], [-4], [-2], [-38], [6], [44], [88], [22], [42], [-48], [8], [6], [52], [-60], [-62], [50], [92], [106], [0], [-80], [-62], [4], [-94]];
    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_v();".
// 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_v(: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_v();".
// 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_v( : 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![2, 1]>,<23,R![-2, 1]>,<31,R![8, 1]>],Snew);
    return Vf;
end function;