
// Make newform 6048.2.a.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6048_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_6048_2_a_h();".
// 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_6048_2_a_h();".
// 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_6048_a();"
function MakeCharacter_6048_a()
    N := 6048;
    order := 1;
    char_gens := [4159, 3781, 3809, 2593];
    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_6048_a_Hecke(Kf)
    return MakeCharacter_6048_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], [1], [-2], [4], [3], [-4], [0], [8], [-8], [-11], [-7], [-11], [11], [-2], [11], [-6], [4], [-4], [12], [1], [-3], [10], [4], [-18], [-12], [-4], [11], [14], [7], [-20], [0], [-14], [-18], [-9], [8], [-25], [-1], [-18], [18], [2], [-18], [15], [-16], [-26], [8], [-16], [-12], [2], [24], [-6], [-8], [-3], [14], [-12], [-9], [-6], [13], [-12], [-2], [9], [0], [15], [-2], [-12], [1], [23], [-28], [-32], [-29], [-14], [6], [-1], [-25], [23], [-30], [-8], [-2], [4], [13], [38], [24], [10], [26], [24], [40], [-10], [-39], [-5], [4], [9], [8], [-20], [15], [-21], [-21], [3], [6], [25], [-27], [32], [12], [20], [5], [30], [12], [-21], [-46], [-32], [-38], [-6], [12], [-36], [-29], [10], [18], [0], [10], [-6], [14], [26], [-26], [12], [-34], [42], [-29], [-1], [12], [2], [20], [36], [16], [27], [21], [-50], [21], [12], [18], [46], [2], [-16], [-37], [22], [-4], [11], [-30], [37], [-14], [-10], [-47], [30], [43], [27], [47], [-42], [3], [-3], [-52], [-41], [-48], [36], [24], [-11], [2], [-33], [25], [24], [61], [1], [-24], [12], [-2], [43], [-16], [43], [52], [6], [26], [-50], [-11], [49], [-14], [23], [-42], [-59], [6], [-20], [-10], [33], [-44], [46], [-17], [13], [-58], [39], [10], [7], [-46], [17], [30], [70], [-46], [-24], [-17], [21], [50], [56], [-32], [10], [-67], [-42], [69], [9], [65], [-60], [-35], [-50], [-2], [12], [11], [-50], [42], [17], [-15], [41], [33], [0], [-10], [16], [-14], [16], [-55], [-54], [-26], [-8], [-10], [-6], [-48], [-41], [-6], [-6], [-54], [-18], [34], [19], [20], [45], [71], [-16], [69], [50], [78], [60], [29], [-24], [-69], [32], [-18], [-38], [-4], [66], [50], [-26], [-42], [44], [-8], [-58], [43], [-18], [-64], [-16], [32], [69], [-26], [-36], [82], [11], [-5], [5], [33], [37], [-49], [36], [-40], [-60], [-23], [32], [62], [-35], [66], [45], [-18], [-7], [51], [45], [-4], [39], [57], [-50], [10], [32], [16], [-74], [59], [-2], [0], [30], [-18], [26], [36], [16], [74], [-61], [78], [-50], [39], [16], [-86], [-48], [54], [-31], [17], [20], [-42], [-26], [81], [-18], [-62], [-86], [-26], [-22], [-80], [-91], [83], [67], [30], [-71], [-94], [-42], [59], [-2], [-41], [32], [38], [36], [-24], [-23], [5], [-10], [16], [29], [40], [18], [-45], [51], [72], [-69], [-76], [-26], [-17], [93], [-32], [94], [-40], [-59], [27], [-31], [7], [36], [20], [12], [24], [-22], [38], [60], [-50], [44], [-78], [-48], [81], [44], [-52], [-6], [-2], [88], [-24], [14], [-53], [-68], [59], [-38], [24], [-74], [-56], [-90], [-80], [15], [92], [44], [-8], [-66], [-65], [33], [-38], [50], [-7], [-58], [0], [-76], [-32], [80], [-90], [39], [67], [-14], [-50], [-52], [-77], [44], [-82], [-4], [2], [-9], [-70], [29], [24], [14], [-74], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6048_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6048_2_a_h();".
// 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_6048_2_a_h(:prec:=1)
    chi := MakeCharacter_6048_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_6048_2_a_h();".
// 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_6048_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6048_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<11,R![2, 1]>,<13,R![-4, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;