
// Make newform 1575.2.a.v in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1575_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_1575_2_a_v();".
// 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_1575_2_a_v();".
// 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_1575_a();"
function MakeCharacter_1575_a()
    N := 1575;
    order := 1;
    char_gens := [1226, 127, 451];
    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_1575_a_Hecke(Kf)
    return MakeCharacter_1575_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 := [[1, 1], [0, 0], [0, 0], [1, 0], [-1, 4], [-4, 2], [2, -6], [0, -2], [5, 0], [5, -6], [-2, 4], [1, -4], [8, 0], [-5, -2], [4, 2], [-6, 4], [4, -2], [4, -12], [-7, 6], [7, -4], [2, -2], [5, -2], [6, 4], [-4, 6], [-6, -4], [14, -6], [0, -6], [8, -8], [13, -12], [-13, -2], [9, 2], [14, -6], [6, -4], [-6, 6], [-5, -2], [11, -6], [-4, -8], [-12, -4], [-16, 6], [-20, 10], [8, -8], [-18, 14], [-8, 12], [19, -12], [-9, 2], [-16, 0], [0, 8], [-2, 2], [-8, 2], [-10, 20], [17, -6], [-4, 20], [16, 2], [-6, 2], [-2, -4], [15, -8], [22, 4], [20, 2], [-10, 16], [9, -14], [-6, 12], [-4, -14], [-2, -14], [-10, -2], [-4, 6], [-5, 14], [17, -2], [22, 4], [-13, 0], [4, -2], [14, -8], [-13, -8], [-4, 8], [15, 0], [-19, 22], [-4, 18], [5, -2], [18, 0], [17, 6], [8, -10], [-22, 8], [19, -12], [-8, -16], [8, -28], [-8, -2], [32, -12], [-13, 6], [23, 4], [12, -14], [4, -20], [-2, 16], [20, 2], [23, -10], [7, -4], [20, -32], [-2, 12], [8, 14], [-14, 32], [-16, 32], [-5, -8], [-7, 26], [15, -14], [-10, 28], [9, 2], [-31, -6], [-8, -2], [22, -18], [-20, 28], [33, 8], [4, -20], [-12, 6], [-23, 8], [1, -14], [-4, 18], [-11, -18], [25, -6], [-20, 40], [-14, 8], [6, 12], [-28, -8], [-8, 10], [6, 0], [-2, 28], [31, -16], [20, -20], [-14, -12], [18, 16], [16, -36], [-30, 30], [-12, 0], [7, -2], [28, -36], [8, 20], [1, 8], [36, -28], [16, 12], [-22, -8], [-42, 2], [2, -26], [23, -10], [-6, 0], [-2, 8], [-11, 34], [19, 8], [30, -2], [2, -18], [-20, -20], [20, 12], [6, -4], [15, -8], [-6, 0], [6, 20], [-31, -2], [4, -12], [-16, 20], [-27, 12], [-19, 34], [14, 10], [28, -20], [-6, -24], [20, 8], [-13, 18], [28, -16], [24, -22], [13, -6], [-22, 12], [-27, 22], [-4, -30], [39, 4], [-26, -6], [-3, 0], [28, -6], [-11, -28], [-14, -8], [-22, -2], [26, -6], [-51, 18], [-18, -12], [-16, -10], [-22, -24], [7, -18], [-20, 20], [21, -8], [-6, 0], [13, 12], [-10, 8], [15, 4], [30, -28], [9, -28], [26, 2], [-2, 32], [25, 20], [27, 10], [-18, 4], [-5, 16], [12, 12], [13, -36], [-19, -12], [-56, 10], [-14, -4], [2, -36], [-6, 16], [18, -8], [12, 8], [-16, 44], [52, -20], [2, 0], [-8, -16], [-25, -14], [-64, 4], [-3, 40], [-46, 38], [-9, 10], [60, -4], [-22, 34], [46, -2], [11, 6], [-8, -18], [-47, 0], [-11, 2], [-7, 0], [-46, 26], [11, 34], [9, 38], [-36, 22], [-38, 16], [50, -38], [-11, -4], [22, -30], [43, -12], [-47, -8], [6, 16], [-64, -4], [-37, -6], [8, 26], [-22, -20], [-20, -4], [3, -10], [27, -56], [-28, -20], [9, -8], [16, -34], [-18, 0], [-29, 0], [36, -30], [-42, 34], [10, 24], [38, -16], [0, -20], [16, 16], [-34, -4], [8, 40], [-20, -32], [-30, -14], [4, 26], [-48, 8], [29, 12], [30, -54], [16, 2], [4, 10], [-4, 24], [19, -12], [18, 24], [-50, -12], [-34, 54], [-48, 0], [-13, -2], [-6, -34], [-5, -26], [-31, 2], [20, 0], [-44, 32], [-40, 0], [-19, 6], [-14, 2], [34, -50], [39, -4], [42, -44], [23, 4], [-66, 6], [-42, 6], [36, 0], [-52, 14], [-62, -6], [-18, 38], [-29, 56], [27, -28], [19, -6], [-20, -8], [2, 8], [-17, 18], [-26, 16], [-19, 50], [2, -34], [-9, -12], [14, 34], [28, 24], [-56, 10], [-44, 12], [-14, 12], [50, -22], [26, -8], [-19, -22], [51, -12], [9, -18], [42, -56], [-39, 24], [-56, 42], [-27, 44], [6, 0], [-62, -2], [-50, 40], [57, -10], [20, 24], [-11, 0], [38, -14], [10, -36], [-28, 24], [-60, 48], [21, 6], [30, 14], [-66, 8], [40, -62], [45, 2], [-17, -2], [6, -44], [25, -22], [-2, -24], [29, 24], [-23, -22], [-2, 4], [-19, 18], [-14, -8], [-26, 58], [24, -28], [0, 10], [41, 0], [28, 2], [-4, -42], [58, -38], [-18, 24], [65, -14], [16, -12], [-31, 6], [25, -18], [15, 4], [26, -50], [-45, 54], [-80, 14], [6, -50], [2, 10], [-69, -4], [-38, -20], [-16, 50], [-5, -16], [52, -10], [42, -24], [2, -18], [-34, 76], [44, -4], [-25, 68], [2, 40], [32, -12], [63, -44], [-64, 14], [-14, 20], [-18, 40], [-37, 34], [71, 8], [-17, -20], [70, 8], [33, -72], [-10, -20], [-62, 36], [19, 36], [-60, 36], [39, -28], [62, -18], [16, -22], [-26, -10], [-90, 14], [-41, 14], [1, 58], [-29, -38], [-2, -6], [-40, 36], [-53, 70], [-70, 40], [-21, -30], [-16, -24], [-15, 68], [46, 16], [43, 32], [-22, 0], [43, 4], [33, 8], [48, -4], [12, 50], [12, -60], [33, -62], [40, -78], [11, -10], [-71, 42], [0, 14], [32, 26], [0, 54], [15, 36], [26, 28], [-32, 16], [-22, -24], [-64, -22], [41, -30], [76, 12], [-31, -34], [-3, -56], [32, 34], [-76, 0], [66, -26], [-4, 8], [-2, -52], [-66, -16], [42, -14], [-64, 24], [32, -64], [26, 20], [62, 2], [-49, 44], [-30, -12], [-18, -6], [30, -32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1575_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1575_2_a_v();".
// 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_1575_2_a_v(:prec:=2)
    chi := MakeCharacter_1575_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_1575_2_a_v();".
// 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_1575_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1575_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -3, 1]>,<11,R![-19, -2, 1]>,<13,R![4, 6, 1]>],Snew);
    return Vf;
end function;