
// Make newform 7200.2.a.bk in Magma, downloaded from the LMFDB on 09 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7200_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_7200_2_a_bk();".
// 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_7200_2_a_bk();".
// 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_7200_a();"
function MakeCharacter_7200_a()
    N := 7200;
    order := 1;
    char_gens := [6751, 901, 6401, 577];
    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_7200_a_Hecke(Kf)
    return MakeCharacter_7200_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], [0], [1], [4], [-3], [4], [-1], [0], [8], [1], [2], [-2], [-11], [2], [10], [6], [11], [9], [-6], [-14], [16], [-2], [0], [-11], [-14], [8], [-6], [-11], [12], [0], [14], [-2], [20], [2], [23], [-5], [-9], [22], [24], [18], [-15], [-20], [-13], [-2], [11], [-5], [-19], [18], [5], [-26], [-10], [21], [8], [-8], [-26], [12], [28], [-7], [16], [-17], [-6], [23], [26], [15], [28], [-8], [13], [-30], [34], [-24], [26], [3], [19], [33], [8], [10], [23], [30], [5], [30], [-2], [-16], [-1], [-7], [12], [-40], [-6], [6], [-20], [12], [6], [11], [8], [-13], [-10], [-18], [0], [-17], [-7], [-8], [-2], [-12], [-24], [31], [7], [26], [-30], [24], [-5], [40], [38], [-18], [23], [-15], [0], [12], [12], [30], [-14], [-26], [-2], [-44], [-8], [12], [12], [17], [-32], [-23], [6], [-20], [36], [16], [-11], [20], [-9], [2], [-53], [24], [-6], [43], [-46], [-39], [10], [-34], [12], [-17], [-24], [-4], [28], [-17], [52], [25], [-34], [20], [-50], [-49], [36], [-53], [-38], [32], [8], [-32], [-18], [18], [-44], [-47], [18], [29], [34], [-48], [19], [42], [-55], [-45], [12], [-13], [-40], [25], [-41], [57], [-32], [-47], [-42], [42], [26], [33], [39], [-47], [60], [1], [10], [-31], [48], [48], [2], [38], [-50], [42], [20], [6], [40], [-50], [-22], [-44], [8], [-53], [44], [-8], [9], [-57], [-10], [-19], [20], [26], [22], [-31], [8], [18], [36], [-47], [-49], [-74], [-12], [-42], [-21], [12], [-34], [-33], [-72], [25], [-48], [-41], [-46], [16], [48], [-38], [18], [-30], [-54], [-52], [-7], [59], [51], [26], [-34], [-13], [48], [-11], [-66], [-30], [36], [-32], [-3], [0], [-4], [49], [-72], [-18], [-18], [-11], [-40], [-38], [23], [14], [-60], [-30], [-28], [-4], [-54], [55], [-68], [-26], [-3], [29], [41], [6], [-17], [-77], [-28], [54], [24], [16], [-33], [3], [-70], [9], [-18], [-64], [-26], [18], [-24], [-24], [84], [59], [-66], [-23], [-34], [-24], [28], [62], [84], [53], [78], [-44], [-59], [24], [29], [22], [-14], [-72], [16], [76], [13], [36], [65], [-64], [68], [49], [20], [-28], [-9], [58], [13], [-54], [-62], [-1], [-1], [72], [-82], [1], [-46], [-3], [-24], [-53], [-84], [-77], [46], [-55], [0], [11], [-54], [-72], [7], [-76], [34], [53], [-53], [-78], [-64], [-53], [-26], [72], [79], [87], [54], [62], [20], [56], [-24], [-53], [-34], [-36], [84], [40], [23], [48], [-32], [-57], [12], [25], [-12], [-72], [56], [17], [6], [18], [39], [-12], [-55], [-42], [60], [8], [30], [4], [-90], [85], [10], [-32], [-100], [1], [-18], [62], [93], [-6], [13], [-75], [50], [-55], [-24], [-30], [78], [-73], [-22], [-2], [96], [1], [22], [-21], [-24], [31], [12], [-78], [-9], [17], [44], [-6], [79], [74], [66], [-68], [59], [78], [-10], [-90], [54], [-74], [82], [-43], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7200_2_a_bk();".
// 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_7200_2_a_bk(:prec:=1)
    chi := MakeCharacter_7200_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_7200_2_a_bk();".
// 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_7200_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-1, 1]>,<11,R![-4, 1]>,<13,R![3, 1]>,<17,R![-4, 1]>,<19,R![1, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;