
// Make newform 1400.2.q.h in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1400_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1400_q_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_1400_2_q_h();".
// 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_1400_2_q_h();".
// 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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_1400_q();"
function MakeCharacter_1400_q()
    N := 1400;
    order := 3;
    char_gens := [351, 701, 1177, 801];
    v := [3, 3, 3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1400_q_Hecke();"
function MakeCharacter_1400_q_Hecke(Kf)
    N := 1400;
    order := 3;
    char_gens := [351, 701, 1177, 801];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0]];
    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], [-1, 1, -1, 0], [0, 0, 0, 0], [1, 1, 1, 2], [2, -2, 2, 0], [-2, 0, 0, 0], [-2, -4, -2, 0], [0, -4, 0, -4], [0, -1, 7, -1], [-5, 0, 0, 2], [2, -2, 2, 0], [0, -4, 0, -4], [3, 0, 0, -2], [-3, 0, 0, -7], [0, -4, -6, -4], [0, -4, 0, 0], [4, 0, 4, 0], [0, 4, 1, 4], [-3, -7, -3, 0], [-12, 0, 0, 0], [-2, 4, -2, 0], [0, 0, -4, 0], [9, 0, 0, 3], [0, 4, -11, 4], [-6, 0, 0, 0], [3, 6, 3, 0], [0, 1, -9, 1], [0, -1, 7, -1], [-7, -8, -7, 0], [0, 0, 0, 8], [10, 0, 0, 4], [0, 4, 8, 4], [-4, 0, -4, 0], [-6, 0, 0, -6], [0, 4, -1, 4], [14, -2, 14, 0], [-10, 8, -10, 0], [0, 4, 10, 4], [5, 0, 0, 5], [0, 4, -16, 4], [-2, 6, -2, 0], [-7, 0, 0, -2], [0, 6, -14, 6], [6, -8, 6, 0], [-6, 0, 0, -4], [12, 8, 12, 0], [4, 0, 0, -16], [-2, 0, 0, -12], [14, 0, 14, 0], [0, 0, -14, 0], [0, -4, -6, -4], [-22, 0, 0, 6], [10, 0, 10, 0], [-2, 0, 0, 18], [0, -8, -4, -8], [-3, -9, -3, 0], [1, 4, 1, 0], [0, -8, 8, -8], [6, 16, 6, 0], [14, 0, 0, 8], [18, 0, 18, 0], [0, 0, 0, 12], [-9, 0, 0, -3], [-16, -4, -16, 0], [0, -12, -4, -12], [0, 0, 22, 0], [0, 6, 18, 6], [-8, 0, 0, 12], [-7, 13, -7, 0], [7, 0, 0, -2], [28, 4, 28, 0], [0, -6, 2, -6], [23, -1, 23, 0], [0, 0, -12, 0], [16, 0, 0, 8], [0, 7, -9, 7], [6, -8, 6, 0], [0, 16, 2, 16], [0, -8, -21, -8], [11, 10, 11, 0], [-4, 0, 0, 8], [33, 0, 0, 4], [-2, 2, -2, 0], [8, 0, 0, -8], [0, 4, -24, 4], [0, -1, 17, -1], [1, 0, 0, 6], [0, -12, -12, -12], [-10, 0, 0, -8], [-5, 0, 0, 7], [0, 19, 11, 19], [10, -6, 10, 0], [-10, -20, -10, 0], [10, 0, 0, 6], [0, 10, 30, 10], [11, 0, 0, 5], [0, -6, -25, -6], [-14, -16, -14, 0], [0, -24, 6, -24], [0, -6, -15, -6], [15, 0, 0, 9], [-6, -8, -6, 0], [3, -5, 3, 0], [0, -16, -14, -16], [4, 12, 4, 0], [6, -20, 6, 0], [34, 0, 0, 8], [0, -24, 4, -24], [-20, -16, -20, 0], [-34, 0, 0, 0], [0, -3, -23, -3], [22, 12, 22, 0], [24, 0, 0, 8], [-14, -18, -14, 0], [-10, 0, 0, -6], [17, -22, 17, 0], [-26, 0, 0, 0], [-25, 3, -25, 0], [0, 4, 10, 4], [24, 0, 0, 8], [39, 8, 39, 0], [0, 0, 0, -4], [0, -12, -6, -12], [-9, 5, -9, 0], [0, -26, -6, -26], [-11, 0, 0, 0], [0, -30, 7, -30], [0, -14, -6, -14], [-31, 0, 0, 5], [0, -4, -2, -4], [-14, -2, -14, 0], [15, 0, 0, 3], [0, 0, -12, 0], [-6, 0, 0, 0], [0, -24, -10, -24], [-2, 0, 0, -32], [30, -4, 30, 0], [-3, 1, -3, 0], [24, 0, 0, -8], [19, 12, 19, 0], [38, 0, 0, -10], [0, -16, -10, -16], [-11, 29, -11, 0], [19, 0, 0, 23], [6, -8, 6, 0], [6, 0, 0, -10], [-24, 0, 0, -8], [-2, -12, -2, 0], [0, -16, -8, -16], [0, 29, 15, 29], [0, 20, 0, 20], [-1, 0, 0, 18], [-18, 0, 0, -8], [0, 1, -19, 1], [25, 15, 25, 0], [-10, 0, 0, 6], [0, -32, 12, -32], [0, -14, 29, -14], [4, 0, 0, 16], [10, 0, 10, 0], [0, 7, -21, 7], [32, 0, 0, -16], [29, 0, 0, 3], [0, 0, 8, 0], [2, 20, 2, 0], [-23, -13, -23, 0], [-2, -2, -2, 0], [-20, 4, -20, 0], [21, 0, 0, 4], [0, -8, -24, -8], [28, -4, 28, 0], [-15, 0, 0, -2], [0, 38, -2, 38], [-12, 8, -12, 0], [-2, 38, -2, 0], [-19, 0, 0, 8], [4, 0, 0, 24], [10, 8, 10, 0], [31, 0, 0, -15], [0, 24, -9, 24], [0, -21, 5, -21], [-6, 0, 0, -6], [18, 0, 0, 4], [0, 4, 22, 4], [10, -4, 10, 0], [-19, 6, -19, 0], [8, 12, 8, 0], [6, -20, 6, 0], [0, -22, 9, -22], [-14, 22, -14, 0], [0, 0, -20, 0], [-14, 0, 0, -4], [0, -22, 14, -22], [0, 8, -23, 8], [-22, 4, -22, 0], [-10, -8, -10, 0], [-19, 2, -19, 0], [0, 0, -30, 0], [-8, 0, 0, -16], [0, -20, -26, -20], [13, -10, 13, 0], [-6, 0, 0, 14], [0, 12, 12, 12], [-13, 4, -13, 0], [-2, 0, 0, -30], [0, -8, 0, 0], [0, 16, -8, 16], [0, -8, 30, -8], [5, 0, 0, 8], [24, 0, 24, 0], [0, 28, -24, 28], [-15, 0, 0, -12], [11, 0, 0, -25], [0, 21, 7, 21], [2, 22, 2, 0], [0, 8, -34, 8], [-49, 1, -49, 0], [9, -8, 9, 0], [0, 28, 26, 28], [8, 0, 0, -16], [0, 24, -27, 24], [22, 0, 0, -14], [0, -24, 33, -24], [0, -8, 26, -8], [11, 0, 0, -25], [11, 0, 0, -8], [0, 8, 24, 8], [-26, 22, -26, 0], [0, -17, 9, -17], [0, 2, 38, 2], [-10, -8, -10, 0], [-72, 0, -72, 0], [6, 0, 0, 2], [37, -14, 37, 0], [21, 0, 0, -7], [3, 11, 3, 0], [0, -2, 33, -2], [0, 0, -46, 0], [16, 0, 0, -36], [-26, 0, 0, -10], [42, -8, 42, 0], [0, 40, -12, 40], [10, 12, 10, 0], [0, 22, -9, 22], [24, 0, 0, -12], [0, -26, -10, -26], [-35, 0, 0, -21], [-38, -10, -38, 0], [60, 4, 60, 0], [6, 0, 0, -12], [24, 0, 0, 36], [0, 20, 19, 20], [-1, -5, -1, 0], [-3, 0, 0, 10], [-22, -16, -22, 0], [0, -4, -36, -4], [-37, -18, -37, 0], [-11, -17, -11, 0], [6, 0, 0, -4], [0, -12, 0, -12], [-23, -5, -23, 0], [-1, 0, 0, -7], [6, 0, 6, 0], [30, 0, 0, 28], [16, -24, 16, 0], [0, 34, 2, 34], [6, 0, 0, 24], [30, 0, 0, 8], [33, 0, 0, 1], [28, -24, 28, 0], [0, -50, -11, -50], [14, 24, 14, 0], [48, -12, 48, 0], [0, 28, -8, 28], [30, 0, 0, -28], [0, 12, 30, 12], [0, -32, 2, -32], [9, 6, 9, 0], [0, 18, 43, 18], [0, 16, 28, 16], [-7, -11, -7, 0], [12, 0, 12, 0], [42, 0, 0, -28], [-7, 0, 0, 50], [0, 5, 39, 5], [0, -32, 24, -32], [60, -12, 60, 0], [-32, 0, 0, 32], [-6, -42, -6, 0], [-14, 0, 0, 0], [11, -44, 11, 0], [-39, 29, -39, 0], [0, -4, -26, -4], [-46, 0, 0, -22], [-32, 0, 0, -36], [-27, -20, -27, 0], [0, -8, -52, -8], [-36, 0, 0, 0], [0, -2, 42, -2], [-14, 0, 0, 24], [0, 4, -10, 4], [0, 12, 8, 12], [10, 2, 10, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1400_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1400_2_q_h();".
// 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_1400_2_q_h(:prec:=4)
    chi := MakeCharacter_1400_q();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1400_2_q_h();".
// 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_1400_2_q_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1400_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -2, 5, 2, 1]>,<11,R![16, 16, 20, -4, 1]>],Snew);
    return Vf;
end function;