
// Make newform 5175.2.a.be in Magma, downloaded from the LMFDB on 09 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5175_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_5175_2_a_be();".
// 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_5175_2_a_be();".
// 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 := [-1, -1, 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_5175_a();"
function MakeCharacter_5175_a()
    N := 5175;
    order := 1;
    char_gens := [4601, 3727, 2926];
    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_5175_a_Hecke(Kf)
    return MakeCharacter_5175_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], [0, 0], [0, 0], [-2, 2], [4, -2], [-3, 0], [2, 2], [-2, 0], [1, 0], [3, 0], [3, -6], [0, -2], [1, -4], [0, 0], [-1, 2], [-2, -4], [-4, 4], [-2, 8], [4, 2], [-11, 2], [-9, -4], [-6, 8], [-10, -2], [8, -4], [-14, 6], [-2, 4], [-2, -10], [6, -12], [0, 0], [10, 2], [11, 6], [-15, 6], [-12, 16], [-7, 6], [-14, 16], [3, -2], [4, -12], [7, 2], [4, 4], [18, -8], [3, 6], [8, -14], [20, -10], [-5, 8], [1, 4], [-16, -6], [-16, 12], [-4, 0], [-6, 10], [-12, 0], [-9, -4], [-15, -2], [-12, 18], [-6, -6], [-5, 4], [-2, -8], [3, -8], [8, 0], [-13, 4], [10, 2], [-24, 6], [-4, -4], [-12, -4], [-7, 10], [-12, 20], [18, -12], [-11, 14], [-16, 12], [0, 16], [17, -12], [-3, -20], [10, -16], [-2, -10], [-4, 6], [12, -20], [12, 8], [-28, -4], [17, -12], [8, -10], [9, -20], [12, 12], [-14, -6], [20, 4], [-24, -10], [-15, 6], [27, -18], [10, -8], [-6, -18], [-1, -4], [20, 0], [-18, -8], [-18, 22], [11, -6], [-17, -14], [23, 6], [-22, 8], [11, -28], [-24, 12], [-30, 18], [-27, 12], [11, -30], [0, 12], [-28, 8], [16, -10], [-18, -6], [-13, 16], [-21, 6], [2, 8], [-24, 16], [37, -16], [-24, 4], [-2, 6], [-10, -4], [12, -12], [0, -20], [0, -28], [16, -22], [3, -6], [-21, 28], [2, -14], [-18, 8], [-3, 0], [18, 0], [13, -22], [12, 8], [10, 10], [-26, 26], [8, 8], [24, -6], [30, -2], [33, 10], [30, -18], [-12, -20], [12, 22], [-29, 28], [-12, 18], [-8, -4], [-32, -12], [-22, 20], [-22, -16], [-33, -14], [34, -8], [21, -30], [4, 4], [18, -36], [30, -18], [18, 12], [-1, -4], [-13, 6], [-3, 30], [34, -4], [-38, 10], [-4, 0], [7, -26], [18, -36], [-14, 28], [30, -18], [29, 8], [-28, 10], [2, 14], [-17, -10], [18, -4], [-9, 30], [-14, 4], [-32, -14], [-38, 4], [24, 0], [-2, 24], [6, -12], [-29, 8], [0, 10], [-15, 32], [-10, -26], [0, 0], [45, 2], [54, -6], [9, 30], [-27, 48], [30, -24], [26, -18], [-33, 18], [-4, 40], [-21, -20], [-18, 24], [24, -30], [-6, -6], [-14, -28], [34, 12], [-3, 0], [24, -48], [-51, 12], [-8, 20], [16, 18], [2, -16], [60, -2], [32, -28], [32, -38], [-28, 36], [4, 28], [25, 22], [28, 16], [-16, 24], [-25, 12], [0, -6], [24, -12], [-7, 8], [-24, 42], [3, 2], [-23, 52], [-51, 18], [-35, 36], [-47, 4], [-12, 20], [-2, -4], [-4, -16], [8, 24], [-3, 42], [-9, 0], [24, -30], [-13, -4], [-1, 12], [-12, 6], [31, -44], [-6, -26], [-21, -10], [3, 12], [-28, 26], [-1, 14], [6, 8], [8, 32], [-2, 0], [66, -10], [12, -54], [-6, 8], [36, 8], [46, -26], [-4, -36], [-24, 22], [13, 2], [31, -10], [-24, 56], [-9, 18], [41, -18], [43, 0], [-6, 8], [-39, -10], [-23, 42], [-6, -28], [18, 0], [32, -26], [6, 14], [48, -22], [-46, 28], [-6, -18], [18, -48], [-25, 22], [28, -18], [12, -18], [-35, -4], [-33, -12], [8, -44], [-36, 48], [-38, 32], [-42, 30], [-38, 8], [42, 12], [0, -2], [-4, 8], [2, 40], [-47, 4], [41, 16], [-2, -12], [-32, 24], [2, 30], [-13, -24], [57, -6], [40, 4], [-18, -16], [4, -42], [-28, 52], [17, -34], [-52, 10], [32, -70], [-30, 44], [-20, 44], [12, 12], [38, -8], [22, -32], [-20, 12], [2, -44], [14, 8], [74, -10], [-62, 16], [-2, 20], [-21, -4], [58, -20], [-2, -6], [27, 0], [63, -34], [15, 26], [-62, 6], [-46, 8], [62, -30], [13, 22], [40, -12], [41, -36], [44, -16], [16, -4], [0, -46], [-10, -8], [-33, -6], [-30, -24], [-6, -34], [-37, -18], [18, 36], [2, -14], [-13, -34], [33, -22], [-40, 24], [-47, 16], [36, 20], [46, 2], [57, -4], [-28, 60], [-36, 18], [36, -12], [28, -6], [-33, -6], [-48, 18], [-46, 46], [-3, -12], [-69, 12], [-11, 38], [-15, 6], [-30, -20], [1, -26], [-12, -42], [-18, 6], [-9, -12], [24, -36], [25, -52], [8, -8], [-38, -20], [30, 24], [-60, -6], [8, 20], [-33, 48], [-20, -16], [42, -16], [-56, 2], [-53, 2], [43, -84], [-26, 8], [-38, 12], [-46, 46], [78, -24], [-6, 6], [-44, 34], [-7, -20], [0, 28], [-44, 6], [51, -16], [57, 6], [-12, 2], [47, -38], [-21, 44], [75, -20], [-8, 18], [42, 24], [23, 34], [-12, 0], [-69, 42], [34, -70], [-44, 44], [-39, 0], [-9, 10], [56, 10], [6, -28], [-12, 8], [-39, 64], [20, -14], [34, 28], [24, -48], [39, -48], [-2, -22], [45, 18], [-7, -2], [-10, 24], [64, -30], [-18, 22], [2, -38], [54, 0], [-15, -18], [-12, 48], [78, 6], [32, -38], [32, -60], [50, 10], [-2, 42], [-14, 44], [51, -60], [45, -12], [-66, 6], [74, -14], [17, 52], [31, -22], [-34, 12], [43, 28], [-50, -6], [48, 32], [-35, -24], [7, -16], [-8, 10], [6, -32], [26, -36], [19, -4], [-12, 32], [-8, 36], [12, -60], [48, -28], [-90, -6], [-80, 0], [12, 40], [20, -40], [-1, -24], [-29, 64], [-30, -8], [-6, 20], [-33, 14], [-75, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5175_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5175_2_a_be();".
// 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_5175_2_a_be(:prec:=2)
    chi := MakeCharacter_5175_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_5175_2_a_be();".
// 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_5175_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5175_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1, 1]>,<7,R![-4, 2, 1]>,<11,R![4, -6, 1]>],Snew);
    return Vf;
end function;