
// Make newform 3648.2.a.bh 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_bh();".
// 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_bh();".
// 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], [3], [-5], [-1], [-2], [-1], [1], [-4], [2], [-6], [0], [0], [1], [-9], [-10], [8], [1], [-8], [-12], [-11], [16], [-12], [-6], [-10], [-2], [-2], [-6], [-4], [2], [-2], [-7], [-9], [13], [21], [0], [18], [0], [10], [-6], [18], [14], [9], [4], [2], [-21], [-12], [12], [-18], [-25], [9], [-3], [20], [-7], [-8], [23], [14], [12], [11], [10], [13], [28], [12], [-21], [-2], [4], [4], [-14], [25], [-9], [-2], [37], [-8], [-16], [-34], [-34], [27], [-25], [36], [-14], [-28], [-26], [-34], [6], [26], [5], [-36], [-29], [-27], [17], [5], [-16], [-16], [0], [-5], [16], [-6], [36], [34], [-3], [26], [41], [12], [-18], [0], [27], [-7], [-6], [0], [0], [26], [-33], [27], [-4], [15], [18], [1], [-39], [-3], [14], [-12], [-24], [-34], [6], [31], [22], [42], [33], [-23], [14], [5], [-8], [24], [17], [15], [11], [-20], [-40], [44], [-55], [38], [45], [43], [-12], [-52], [54], [14], [-8], [-27], [-44], [6], [-37], [-35], [-12], [26], [-6], [8], [-2], [21], [-42], [28], [-32], [-32], [-12], [54], [-44], [-16], [47], [-20], [-19], [12], [34], [16], [6], [-18], [-51], [12], [-51], [-52], [13], [37], [6], [-34], [8], [29], [-11], [-18], [60], [-10], [40], [-14], [-25], [34], [48], [-12], [-26], [-43], [62], [2], [39], [-40], [8], [-32], [22], [-4], [-51], [-19], [-24], [-21], [-42], [30], [18], [-13], [28], [38], [-40], [40], [48], [18], [31], [-48], [-64], [-6], [1], [-8], [37], [36], [-12], [32], [35], [-31], [-38], [-50], [-60], [-1], [45], [21], [-69], [12], [76], [-34], [3], [4], [-64], [38], [3], [-31], [60], [-40], [-69], [38], [-51], [-12], [-64], [43], [28], [-53], [14], [-44], [2], [-24], [-80], [-55], [-14], [57], [22], [26], [17], [-40], [-33], [-34], [-8], [-19], [-69], [-56], [16], [79], [-10], [52], [-13], [-28], [8], [75], [-68], [-3], [33], [51], [-62], [-17], [30], [54], [-17], [-78], [-20], [-40], [34], [4], [-21], [-18], [-81], [-7], [-6], [11], [22], [47], [-74], [-40], [-26], [-15], [-18], [-83], [-6], [-64], [0], [-72], [34], [-39], [-22], [27], [43], [8], [-3], [-58], [-78], [30], [-4], [46], [-44], [-32], [57], [-11], [-22], [-56], [79], [-39], [73], [70], [42], [15], [-40], [40], [63], [-14], [-32], [84], [30], [30], [-74], [18], [-15], [-54], [12], [11], [-54], [24], [-38], [-72], [20], [-73], [-14], [-77], [26], [12], [-60], [11], [74], [6], [-18], [58], [-52], [26], [-41], [8], [35], [47], [-58], [32], [-25], [45], [56], [-10], [33], [49], [5], [-6], [-18], [88], [-10], [79], [-66], [-36], [34], [72], [-69], [32], [14], [0], [-30], [-50], [-30], [32], [9], [50], [-18], [28], [-63], [-1], [-80], [42], [53], [66], [15], [-86], [-29], [-46], [-42], [-54], [-40], [-78], [18], [80], [-78], [83], [90], [104], [-30], [42], [-15], [17], [51]];
    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_bh();".
// 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_bh(: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_bh();".
// 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_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3648_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-3, 1]>,<7,R![5, 1]>,<11,R![1, 1]>,<23,R![4, 1]>,<31,R![6, 1]>],Snew);
    return Vf;
end function;