
// Make newform 400.2.u.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_400_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_400_u_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_400_2_u_a();".
// 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_400_2_u_a();".
// 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, -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_400_u();"
function MakeCharacter_400_u()
    N := 400;
    order := 5;
    char_gens := [351, 101, 177];
    v := [5, 5, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_400_u_Hecke();"
function MakeCharacter_400_u_Hecke(Kf)
    N := 400;
    order := 5;
    char_gens := [351, 101, 177];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0], [-2, 2, 0, 1], [-2, 1, -2, 0], [-1, 0, 1, -1], [-2, 0, 0, 2], [-1, -4, -1, 0], [0, -2, 4, -2], [0, 1, -7, 1], [-1, -3, 3, 1], [1, -1, 0, -6], [0, -6, 1, -6], [-2, 7, -2, 0], [6, -2, 6, 0], [-6, 0, -1, 1], [-3, 3, 0, 5], [-8, 8, 0, 5], [1, 3, 1, 0], [-3, -7, 7, 3], [0, 6, -10, 6], [-1, 1, 0, -7], [-11, 11, -11, 11], [-7, 7, 0, 6], [0, 4, 7, 4], [-4, 4, -4, 4], [15, -15, 0, -9], [-5, 0, 8, -8], [3, -3, 0, -8], [-7, 0, 6, -6], [8, 2, 8, 0], [-9, 5, -9, 0], [14, -10, 10, -14], [0, -5, -2, -5], [0, -5, 0, 0], [13, -11, 11, -13], [1, 0, 12, -12], [-2, 0, -1, 1], [1, 0, -2, 2], [-10, 9, -10, 0], [0, -13, 11, -13], [-13, 9, -9, 13], [-6, 6, 0, 13], [0, -6, -12, -6], [-6, -4, -6, 0], [2, 0, 6, -6], [-2, 2, 0, 18], [-11, 0, -1, 1], [7, -15, 15, -7], [-4, -2, 2, 4], [10, 12, -12, -10], [-10, 10, 0, -4], [0, 0, 6, 0], [-19, 21, -21, 19], [4, -9, 4, 0], [21, 0, 8, -8], [-2, 0, 1, -1], [3, -12, 3, 0], [0, -6, -18, -6], [8, -8, 0, 0], [-5, 15, -15, 5], [0, 3, 11, 3], [0, -15, -2, -15], [-6, 0, -4, 4], [-14, 0, -2, 2], [-13, 8, -8, 13], [6, -2, 6, 0], [0, 2, -13, 2], [0, 26, -16, 26], [-5, 6, -5, 0], [3, -3, 0, 2], [-4, 0, -18, 18], [9, -9, 0, 15], [-15, 16, -15, 0], [0, 15, -10, 15], [-8, 13, -13, 8], [0, 0, 0, -5], [0, -2, -15, -2], [-7, -5, 5, 7], [-5, 5, 0, -12], [-1, 0, 22, -22], [4, 23, 4, 0], [0, 18, -15, 18], [-8, 8, 0, 8], [0, -6, -7, -6], [0, -15, -7, -15], [-1, -10, 10, 1], [31, 0, 14, -14], [-32, 0, -5, 5], [2, 0, 4, -4], [31, -25, 25, -31], [8, -11, 8, 0], [0, -15, -2, -15], [-1, 1, 0, 24], [6, -23, 6, 0], [2, 5, 2, 0], [-15, 0, 9, -9], [-16, 16, 0, 10], [5, -28, 5, 0], [-12, 12, 0, 11], [-20, 19, -19, 20], [14, 0, 14, 0], [20, -20, 0, -23], [-24, 0, -10, 10], [13, 10, 13, 0], [0, -4, 15, -4], [-28, 28, 0, 13], [10, 6, -6, -10], [4, -29, 4, 0], [5, 0, -9, 9], [-1, 0, -10, 10], [-37, 0, -9, 9], [23, 0, 15, -15], [-13, 2, -13, 0], [0, -22, 17, -22], [0, -6, 37, -6], [0, 24, 7, 24], [-25, 2, -25, 0], [-32, 0, -8, 8], [-14, 14, 0, 0], [-17, 17, 0, -3], [16, 10, 16, 0], [20, -29, 29, -20], [4, -22, 22, -4], [3, -6, 6, -3], [0, -6, 17, -6], [3, -45, 3, 0], [-14, 0, -10, 10], [-22, 23, -22, 0], [0, -28, 21, -28], [0, -7, 7, 0], [0, -3, 9, -3], [0, -7, 7, 0], [-12, 0, -36, 36], [35, 0, 2, -2], [15, 0, 0, 0], [30, -27, 27, -30], [0, 4, 42, 4], [-38, 31, -31, 38], [6, 34, 6, 0], [-6, 6, 0, 15], [-11, -14, -11, 0], [-13, 33, -33, 13], [3, -3, 0, 9], [22, 0, 0, -22], [28, -31, 31, -28], [-9, 9, 0, 19], [8, 3, -3, -8], [-12, 12, 0, 14], [-8, 0, -3, 3], [-22, 17, -22, 0], [17, -3, 17, 0], [-12, 26, -26, 12], [0, 12, -36, 12], [0, 4, -28, 4], [-16, -12, -16, 0], [-17, 0, 10, -10], [-5, -15, 15, 5], [0, -9, -1, -9], [9, -9, 0, -53], [9, -15, 9, 0], [-8, 43, -8, 0], [-34, 34, 0, 0], [23, -23, 0, 5], [0, 4, 2, 4], [-9, 9, 0, 42], [9, -15, 15, -9], [0, 11, 3, 11], [10, -45, 10, 0], [-4, 4, 0, 45]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_400_u_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_400_2_u_a();".
// 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_400_2_u_a(:prec:=4)
    chi := MakeCharacter_400_u();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_400_2_u_a();".
// 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_400_2_u_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_400_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![25, 25, 15, 5, 1]>],Snew);
    return Vf;
end function;