
// Make newform 1344.2.h.e in Magma, downloaded from the LMFDB on 18 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1344_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1344_h_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_1344_2_h_e();".
// 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_1344_2_h_e();".
// 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, 0, 0, 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_1344_h();"
function MakeCharacter_1344_h()
    N := 1344;
    order := 2;
    char_gens := [127, 1093, 449, 577];
    v := [1, 2, 1, 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_1344_h_Hecke();"
function MakeCharacter_1344_h_Hecke( : Kf := false)
    N := 1344;
    order := 2;
    char_gens := [127, 1093, 449, 577];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, -1, 2, -1], [0, 0, 1, 0], [2, 0, 0, 0], [-2, 1, 0, -1], [0, 2, -2, 2], [0, -3, 2, -3], [4, 0, 0, 0], [0, -2, 2, -2], [0, 2, 4, 2], [-2, -2, 0, 2], [0, 0, 10, 0], [0, -6, 2, -6], [4, 4, 0, -4], [0, -8, 2, -8], [-2, 3, 0, -3], [-10, 3, 0, -3], [0, 8, 2, 8], [6, 4, 0, -4], [-6, -2, 0, 2], [0, 0, 4, 0], [10, -1, 0, 1], [0, 4, -2, 4], [-6, -8, 0, 8], [0, 3, -14, 3], [0, -12, 0, -12], [6, -6, 0, 6], [-6, -6, 0, 6], [0, -8, -6, -8], [0, 0, -2, 0], [6, -11, 0, 11], [0, 8, 0, 8], [0, -3, -6, -3], [0, -4, 2, -4], [0, 0, -18, 0], [2, 3, 0, -3], [0, 10, 6, 10], [8, -2, 0, 2], [0, 11, 6, 11], [-2, -6, 0, 6], [-2, 13, 0, -13], [-2, -4, 0, 4], [12, -4, 0, 4], [0, 0, -26, 0], [0, 8, 4, 8], [0, -4, -6, -4], [0, 2, 0, 2], [-14, 7, 0, -7], [-6, -5, 0, 5], [0, 12, 8, 12], [0, -8, 0, 8], [-2, -4, 0, 4], [-22, -1, 0, 1], [0, -6, 6, -6], [-10, -8, 0, 8], [0, -9, 6, -9], [0, -6, 4, -6], [-10, -8, 0, 8], [0, -12, -8, -12], [0, 3, -6, 3], [0, 9, 14, 9], [0, -11, -6, -11], [0, -14, 0, 14], [-6, 2, 0, -2], [0, 0, 10, 0], [0, 2, -2, 2], [16, -8, 0, 8], [6, 4, 0, -4], [-22, -5, 0, 5], [0, -18, -6, -18], [4, -16, 0, 16], [0, 2, 16, 2], [6, -12, 0, 12], [0, 6, -26, 6], [0, 4, 0, -4], [0, -6, -2, -6], [-14, 3, 0, -3], [0, -4, 26, -4], [18, -6, 0, 6], [-6, 9, 0, -9], [10, -4, 0, 4], [16, 0, 0, 0], [-2, -12, 0, 12], [0, 20, -12, 20], [26, 2, 0, -2], [0, -4, -24, -4], [-24, 0, 0, 0], [0, 17, 2, 17], [0, -4, -4, -4], [-6, -17, 0, 17], [0, -8, 0, 8], [0, 0, -22, 0], [-10, -2, 0, 2], [0, -16, 14, -16], [0, 18, 0, -18], [0, 19, 6, 19], [0, 4, 18, 4], [0, -7, 30, -7], [10, 16, 0, -16], [0, 10, 6, 10], [0, -8, -18, -8], [14, 15, 0, -15], [0, -8, -10, -8], [0, -10, -6, -10], [18, -8, 0, 8], [18, -9, 0, 9], [0, -6, 14, -6], [-6, -16, 0, 16], [-2, 10, 0, -10], [0, 10, 12, 10], [10, 26, 0, -26], [0, -16, 6, -16], [0, -3, 14, -3], [0, -12, -8, -12], [0, -4, -10, -4], [0, 3, 42, 3], [4, 18, 0, -18], [0, 6, 6, 6], [2, 12, 0, -12], [-2, 27, 0, -27], [2, 0, 0, 0], [0, 19, 22, 19], [38, 6, 0, -6], [0, 17, 2, 17], [0, -10, 14, -10], [-14, -6, 0, 6], [32, 8, 0, -8], [0, -16, -12, -16], [18, 13, 0, -13], [0, -10, -26, -10], [12, 24, 0, -24], [0, 20, -2, 20], [18, -22, 0, 22], [0, -12, -18, -12], [14, 4, 0, -4], [0, -9, -14, -9], [0, 3, 38, 3], [0, -7, -22, -7], [0, 8, 22, 8], [0, 11, -14, 11], [0, 12, -10, 12], [0, 12, -16, 12], [-22, -18, 0, 18], [6, -13, 0, 13], [-4, -2, 0, 2], [-6, -1, 0, 1], [0, 0, -38, 0], [0, -1, -26, -1], [34, 16, 0, -16], [38, -14, 0, 14], [0, 10, 22, 10], [0, 8, 2, 8], [0, -2, 0, 2], [0, -28, -2, -28], [0, -12, 0, 12], [0, -4, 24, -4], [0, -14, 14, -14], [30, 6, 0, -6], [0, -21, -14, -21], [-46, 8, 0, -8], [0, -8, -12, -8], [0, 4, -22, 4], [14, -21, 0, 21], [0, 0, 44, 0], [8, 18, 0, -18], [0, 24, -12, 24], [-34, -17, 0, 17], [-4, 20, 0, -20], [0, 17, -22, 17], [-6, 38, 0, -38], [6, -11, 0, 11], [-46, -12, 0, 12], [-30, -4, 0, 4], [0, -6, 28, -6], [0, -8, -26, -8], [0, 8, -42, 8], [0, -6, 26, -6], [0, -24, -4, -24], [-18, -13, 0, 13], [0, 8, 34, 8], [-30, 17, 0, -17], [18, 6, 0, -6], [0, 20, -26, 20], [40, -4, 0, 4], [0, 21, 18, 21], [30, 0, 0, 0], [0, -3, 50, -3], [-10, 12, 0, -12], [-60, 0, 0, 0], [-2, -4, 0, 4], [18, -2, 0, 2], [0, 24, -26, 24], [0, 9, -30, 9], [-14, -30, 0, 30], [0, 0, 30, 0], [16, -8, 0, 8], [10, -2, 0, 2], [0, 6, -10, 6], [36, -18, 0, 18], [0, 12, 26, 12], [0, 14, -40, 14], [10, 29, 0, -29], [-22, 12, 0, -12], [-26, -25, 0, 25], [0, -23, 30, -23], [0, -10, 0, -10], [22, -12, 0, 12], [0, 8, 46, 8], [0, 39, 14, 39], [-8, 4, 0, -4], [0, 7, 34, 7], [0, -36, 2, -36], [22, -21, 0, 21], [28, -14, 0, 14], [6, -14, 0, 14], [0, -16, 28, -16], [0, -10, 30, -10], [-22, 8, 0, -8], [0, 22, -22, 22], [22, 2, 0, -2], [0, 12, 8, 12], [0, -20, 16, -20], [0, -28, 6, -28], [22, -23, 0, 23], [-50, 10, 0, -10], [0, -32, -6, -32], [24, -4, 0, 4], [0, -8, -32, -8], [-34, 12, 0, -12], [42, 14, 0, -14], [0, 1, 14, 1], [0, 8, -52, 8], [0, 4, -52, 4], [0, 13, -10, 13], [36, 4, 0, -4], [-2, -32, 0, 32], [0, -4, 18, -4], [30, -10, 0, 10], [20, 6, 0, -6], [-30, 0, 0, 0], [0, -17, 10, -17], [0, 8, 52, 8], [-14, 4, 0, -4], [0, 6, -26, 6], [-28, -14, 0, 14], [0, -26, -16, -26], [42, 7, 0, -7], [0, 14, 38, 14], [-34, 12, 0, -12], [14, -16, 0, 16], [0, -10, -2, -10], [-46, 8, 0, -8], [-6, -26, 0, 26], [0, 25, 38, 25], [10, 10, 0, -10], [-2, -42, 0, 42], [0, 7, -18, 7], [0, 1, -10, 1], [34, 14, 0, -14], [0, 20, -6, 20], [46, 12, 0, -12], [-6, -39, 0, 39], [-6, -19, 0, 19], [0, 10, -14, 10], [0, -29, 18, -29], [0, 24, 6, 24], [0, 0, -30, 0], [0, -12, -22, -12], [0, 30, -26, 30], [-22, -17, 0, 17], [0, 14, -34, 14], [-38, -26, 0, 26], [0, -24, 38, -24], [-6, 8, 0, -8], [0, -16, 28, -16], [-30, 16, 0, -16], [-70, 0, 0, 0], [8, -36, 0, 36], [-22, 23, 0, -23], [4, -40, 0, 40], [0, -20, 0, -20], [20, -6, 0, 6], [-14, 9, 0, -9], [0, 19, -6, 19], [-24, -24, 0, 24], [34, 16, 0, -16], [0, 28, -14, 28], [0, 20, 20, 20], [0, -22, -14, -22], [0, -36, 22, -36], [-2, 9, 0, -9], [0, 16, -30, 16], [2, -3, 0, 3], [-66, -4, 0, 4], [0, 9, 18, 9], [0, -26, 0, -26], [0, -19, -46, -19], [42, 27, 0, -27], [0, -41, 10, -41], [22, 30, 0, -30], [0, -2, 22, -2], [0, -24, -4, -24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1344_h_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1344_2_h_e();".
// 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_1344_2_h_e(:prec:=4)
    chi := MakeCharacter_1344_h();
    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_1344_2_h_e();".
// 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_1344_2_h_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1344_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 0, 12, 0, 1]>,<11,R![-2, 1]>,<13,R![2, 4, 1]>],Snew);
    return Vf;
end function;