
// Make newform 1575.2.a.u in Magma, downloaded from the LMFDB on 16 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_u();".
// 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_u();".
// 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_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], [-2, 2], [2, 2], [2, -4], [0, 2], [2, 4], [-8, 0], [0, -6], [6, 0], [2, -4], [4, 4], [-4, 0], [8, 2], [4, 0], [6, 0], [4, -8], [-2, 6], [-2, -10], [0, -4], [-8, 0], [-6, -8], [6, 2], [-10, -4], [4, -8], [-2, 0], [10, 0], [4, -6], [-8, -4], [8, -4], [8, -10], [4, 2], [-4, 4], [0, 4], [6, -2], [12, 4], [-12, -8], [10, -4], [-14, -6], [-6, 4], [-2, 2], [2, 12], [0, -2], [20, 2], [-12, 0], [0, 4], [-20, 0], [-6, 8], [-8, 10], [10, -10], [-2, 4], [4, -16], [6, 8], [-14, -8], [2, 12], [-16, 6], [10, -4], [20, -4], [4, 0], [2, 0], [-12, 12], [28, 4], [18, 2], [0, 10], [-12, 0], [-6, 20], [-6, 4], [-18, -8], [-6, 16], [-2, -6], [-8, -12], [10, 0], [12, -4], [16, 8], [-24, -4], [-18, 10], [-4, 16], [14, -8], [-12, 16], [14, 0], [-10, -2], [-6, -14], [-12, 18], [18, -12], [8, -8], [10, -8], [6, 16], [16, -4], [20, -8], [36, 4], [8, 4], [2, 10], [20, 4], [8, -16], [-22, 4], [-10, -12], [24, -4], [22, 8], [-12, 8], [24, 2], [-4, 24], [28, 8], [-12, -8], [34, -6], [12, 16], [-6, -8], [6, 10], [2, -20], [-32, 12], [2, -12], [-24, 2], [-20, -6], [0, -4], [4, -12], [-28, 4], [-4, 16], [20, -10], [-18, -10], [38, 8], [18, -16], [10, 8], [18, -12], [4, -10], [8, -8], [-6, 16], [-32, 8], [12, -4], [10, -6], [4, -16], [-18, -8], [24, 16], [-18, -12], [2, 16], [-14, -4], [42, -4], [-8, 4], [34, 12], [8, -4], [-28, 2], [4, -24], [-16, 4], [46, 0], [-6, -4], [-12, -4], [-34, -6], [6, -4], [-24, -10], [26, 20], [26, -4], [22, -8], [4, -8], [-20, -8], [-4, 0], [-18, 10], [0, -16], [18, 24], [-6, 14], [18, -16], [-2, -12], [-20, -22], [16, 16], [0, 12], [-8, 10], [-8, -24], [-40, -8], [46, -6], [-14, -16], [26, 24], [-10, -38], [14, 20], [-34, -10], [6, 24], [-12, -2], [22, -12], [20, 4], [8, 36], [-20, -12], [-10, 20], [-48, 0], [8, 20], [-22, 16], [-14, 4], [-14, 16], [-18, -12], [14, 24], [-16, 4], [18, 24], [-24, 0], [22, 18], [54, -8], [-36, -20], [-2, -40], [-18, -4], [18, 24], [6, 24], [-14, -32], [-42, 0], [-4, -8], [-20, -16], [-28, 2], [2, 14], [14, 4], [-24, -4], [10, -12], [-4, -42], [14, 16], [0, 12], [-8, -18], [6, -4], [10, 36], [-8, 16], [-8, -8], [28, -20], [18, 28], [8, -20], [-10, 32], [42, 16], [-20, 14], [42, -16], [8, -30], [8, -4], [8, -24], [-24, 16], [-30, -32], [14, 24], [-6, 26], [4, -24], [-38, -22], [-58, 4], [-16, 10], [-16, 28], [8, -4], [-36, 12], [16, -8], [14, -8], [-32, 14], [-14, 18], [-32, 8], [-6, -40], [4, 42], [0, -40], [-30, -16], [46, 4], [-4, 12], [20, 28], [4, 0], [-4, 20], [18, -12], [-2, -40], [-2, -8], [-42, 4], [14, -24], [58, -4], [-2, -6], [-18, -16], [-12, 24], [-6, -12], [10, -14], [16, -28], [34, -12], [-30, -24], [-58, 14], [2, -16], [36, 6], [-4, -24], [6, -44], [4, 8], [-36, 14], [46, -20], [4, 28], [-26, 10], [-24, 24], [38, -22], [-12, 20], [14, 0], [-38, 16], [-18, 24], [36, 16], [-24, 32], [-16, 4], [56, 0], [-18, -4], [40, 4], [-50, 18], [18, -8], [-32, 10], [-40, 18], [2, 32], [-52, 8], [0, 0], [24, 2], [12, 8], [34, 16], [6, 0], [-20, -26], [46, -16], [-12, -32], [16, 4], [-14, -10], [-28, 6], [0, 24], [-10, -4], [20, -40], [-6, -12], [-14, -8], [-70, 0], [6, 38], [-10, 44], [-8, 24], [40, -16], [4, 56], [-20, 8], [-30, -28], [38, 28], [40, -4], [66, 2], [46, -6], [4, -40], [-24, 30], [10, 8], [-34, 0], [-40, -16], [28, -10], [14, -28], [4, 24], [-4, 0], [-18, -12], [-4, -6], [10, 8], [36, -2], [-8, -18], [4, 0], [12, -12], [4, -64], [-18, 24], [-2, 24], [6, -12], [-36, -36], [-6, -48], [0, 34], [-14, 12], [48, -28], [-4, -26], [-26, -18], [2, 32], [-12, -32], [-56, 0], [2, 0], [40, 18], [6, 44], [4, 32], [0, 40], [-34, -8], [-22, -24], [-32, 24], [48, -12], [12, 2], [-50, 0], [-6, 24], [-6, 4], [22, -16], [-10, -42], [-32, -28], [42, 8], [-42, 24], [-8, 8], [-2, 8], [42, 10], [16, 14], [14, 16], [28, 28], [56, 14], [-46, -4], [40, 4], [42, 18], [-14, 34], [46, -12], [22, -26], [-2, -4], [52, -14], [24, -28], [-12, -2], [68, -14], [-48, 0], [8, -56], [66, 2], [4, 0], [20, 16], [74, 8], [6, -20], [38, 22], [-20, -28], [46, 6], [2, -28], [-12, 6], [-2, 16], [28, 16], [56, 32], [-26, -20], [-68, 18], [0, 56], [54, -28], [-30, 8], [-36, -26], [-46, -20], [-32, 8], [56, 0], [84, 0], [50, 38], [52, 2], [42, 12], [-52, 16], [-10, 48], [66, 4], [-38, -18], [-20, 20], [-38, 12], [12, 8], [-48, 28], [-10, -22], [30, 48], [24, 44], [10, 18], [-6, -40], [-38, -8], [-12, 0], [4, 30], [28, -20]];
    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_u();".
// 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_u(: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_u();".
// 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_u( : 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, -2, 1]>,<11,R![-4, 4, 1]>,<13,R![-4, -4, 1]>],Snew);
    return Vf;
end function;