
// Make newform 5225.2.a.f in Magma, downloaded from the LMFDB on 27 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5225_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_5225_2_a_f();".
// 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_5225_2_a_f();".
// 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
        poly := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_5225_a();"
function MakeCharacter_5225_a()
    N := 5225;
    order := 1;
    char_gens := [2927, 2851, 4676];
    v := [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_5225_a_Hecke(Kf)
    return MakeCharacter_5225_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], [1, -1], [0, 0], [2, -1], [-1, 0], [2, 3], [-2, 1], [-1, 0], [3, 0], [-2, 3], [-5, 1], [-3, 5], [4, 4], [-6, -4], [-6, 2], [-4, -6], [-3, -1], [-4, 5], [9, -1], [-11, 1], [-4, -6], [-16, -1], [-2, 1], [-5, -7], [-1, 1], [2, -3], [-8, 4], [-10, 4], [-6, -4], [5, 1], [18, -1], [6, -6], [-3, -4], [2, -13], [10, -8], [-2, 6], [-5, 0], [6, -10], [6, -9], [8, 7], [9, -5], [-1, 13], [-3, -8], [-4, -2], [-6, 7], [-2, -10], [-14, -1], [15, -1], [-12, -5], [-3, 8], [-4, -3], [6, 0], [0, 12], [-17, -4], [22, 2], [18, 6], [-2, 0], [-8, -10], [-6, -6], [18, -4], [4, -9], [2, -21], [-6, 1], [-6, 4], [-3, -12], [27, -1], [13, -15], [-4, 9], [-4, 21], [2, -15], [13, -8], [8, -6], [7, -2], [14, 8], [-1, 21], [-1, 23], [-17, 8], [-26, 0], [-18, 12], [-12, -9], [-2, 18], [-14, 10], [22, -6], [7, 13], [-10, 3], [-5, -12], [5, 13], [-24, -5], [-4, -12], [13, 18], [-21, -8], [-8, 23], [3, 11], [32, 3], [-10, 16], [14, -10], [13, 15], [-15, -11], [-8, -4], [-12, -15], [2, -9], [-6, -24], [2, -15], [-22, 3], [4, -21], [-13, 12], [-2, 4], [-20, 12], [2, -8], [-4, -11], [12, 7], [12, -1], [-20, 2], [15, -6], [-25, -12], [-9, -7], [-19, 4], [-3, -24], [17, 6], [2, -14], [-23, 19], [18, 10], [-10, 8], [8, 22], [9, -12], [10, 8], [3, 20], [27, 12], [-13, -22], [-12, 1], [-26, 1], [14, 5], [19, 11], [-8, 24], [20, 10], [16, 10], [-14, 4], [-36, 0], [9, -3], [44, 6], [48, 3], [-6, 23], [23, 0], [-14, 7], [-13, 17], [25, 11], [14, 0], [-18, -21], [-1, -20], [-8, -28], [24, -10], [-15, 26], [-42, 6], [6, 32], [36, 10], [-4, -30], [-38, 13], [-28, -20], [-8, -11], [24, -3], [11, -12], [-16, -7], [12, 11], [13, 9], [1, 31], [-23, 7], [-20, 8], [22, 11], [-34, -14], [-1, 6], [2, -26], [-6, 20], [-16, -16], [-12, -6], [-11, 19], [21, -4], [-26, 0], [-5, 28], [-18, 7], [0, 43], [16, -4], [26, 6], [1, 15], [-14, -18], [9, 26], [-20, 2], [-2, 5], [-8, -6], [48, -1], [-26, -4], [-19, 21], [10, 21], [-41, 3], [14, -38], [-12, 10], [47, 5], [22, -22], [25, -4], [-2, 3], [-6, -14], [-8, 12], [54, 0], [-12, 18], [12, -22], [33, 0], [49, 4], [-9, -28], [36, -22], [-16, -19], [6, -20], [-12, 23], [61, -6], [17, 12], [28, -20], [-28, 9], [19, -19], [24, -15], [14, 15], [-20, -6], [-47, -8], [16, 14], [-60, 2], [21, 19], [-37, 20], [-8, -4], [6, 16], [62, 4], [-8, 8], [32, -14], [32, 7], [21, -10], [-20, -34], [-18, 22], [2, 21], [-9, -44], [30, 26], [27, 32], [-40, -16], [-57, 6], [-7, -43], [51, -7], [-16, -6], [6, -8], [-13, -13], [0, -26], [16, 34], [-68, 4], [-52, -10], [-20, -1], [62, 7], [-48, -4], [-14, -2], [17, -30], [-26, 26], [-6, 14], [12, 5], [12, 6], [4, -15], [-7, -1], [-38, -15], [4, 6], [18, -21], [46, -3], [-22, 0], [4, 2], [0, -22], [-33, 1], [-11, -46], [-8, -39], [-26, -30], [15, 39], [29, -27], [-8, 14], [-32, -14], [-34, 17], [53, 6], [-7, -20], [-10, 10], [-8, 12], [-16, -15], [-36, -6], [-13, -6], [24, -16], [18, -20], [8, -39], [27, 18], [-2, 16], [22, 18], [13, -8], [-8, -13], [-25, -28], [66, 10], [-44, 16], [-18, 20], [18, 14], [20, -8], [21, 19], [11, 10], [-56, -9], [46, 16], [-6, -34], [34, 3], [16, 39], [-28, 36], [-23, -34], [-13, 7], [27, 21], [-25, 9], [-2, 20], [-28, -23], [-44, 19], [32, -38], [-50, 4], [-3, -41], [8, 14], [-2, -4], [59, -6], [-40, 13], [-46, 0], [-12, 34], [-42, -27], [41, -8], [-44, 1], [-35, -7], [30, 27], [23, 19], [-27, 3], [-31, 23], [14, 3], [60, -6], [-18, -27], [-36, 22], [-66, 16], [32, 13], [-24, 23], [39, 22], [-25, -17], [-4, -18], [-9, -16], [6, 36], [39, 3], [-57, 12], [36, 28], [-35, 37], [47, 33], [-64, -3], [67, -2], [7, 9], [12, 6], [11, -26], [-14, -39], [-35, -21], [-12, -52], [-56, 8], [0, 21], [44, 31], [41, 2], [-22, -20], [-42, 16], [-11, -29], [-66, -20], [-2, -3], [-11, -29], [-34, -4], [49, 0], [-37, -31], [-4, 15], [40, 25], [12, -20], [-29, -2], [-56, -14], [-30, -25], [4, 26], [-18, 43], [-11, -18], [88, 0], [6, 4], [54, 29], [2, -23], [-10, 40], [-68, -18], [39, 12], [-54, -13], [-18, 9], [16, -38], [-11, 7], [-6, -46], [9, 14], [-16, 45], [-35, -7], [-41, -37], [-31, -3], [-77, 6], [3, 6], [23, 21], [32, -4], [-14, 0], [2, 18], [-32, 12], [42, 3], [38, -26], [19, -2], [0, 54], [8, -18], [-22, -19], [10, 22], [-4, 14], [14, -1], [26, -40], [11, 26], [78, 16], [-34, -13], [-44, 0], [18, -44], [-10, 25], [-19, -42], [-10, 59], [63, -7], [48, 14], [-40, -17], [-5, 33], [8, -28], [-78, -12], [-34, 20], [20, 26], [61, 15], [-77, 9], [50, 10], [-19, -40], [80, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5225_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5225_2_a_f();".
// 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_5225_2_a_f(:prec:=2)
    chi := MakeCharacter_5225_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_5225_2_a_f();".
// 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_5225_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5225_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 0, 1]>,<7,R![2, -4, 1]>],Snew);
    return Vf;
end function;