
// Make newform 1248.2.q.g in Magma, downloaded from the LMFDB on 08 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1248_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1248_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_1248_2_q_g();".
// 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_1248_2_q_g();".
// 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, 2, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 1];
    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_1248_q();"
function MakeCharacter_1248_q()
    N := 1248;
    order := 3;
    char_gens := [703, 1093, 833, 769];
    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_1248_q_Hecke();"
function MakeCharacter_1248_q_Hecke( : Kf := false)
    N := 1248;
    order := 3;
    char_gens := [703, 1093, 833, 769];
    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 := 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, 0, -1, 0], [-1, 0, 0, 1], [0, 1, 0, 1], [0, 1, 0, 0], [3, 0, -1, 0], [0, 1, 3, 1], [0, -1, -4, -1], [-4, -1, -4, 0], [1, 1, 1, 0], [-4, 0, 0, 2], [-3, -2, -3, 0], [1, -1, 1, 0], [0, -1, -8, -1], [-4, 0, 0, -1], [-1, 0, 0, -1], [0, 2, -8, 2], [0, 2, -1, 2], [-4, -3, -4, 0], [0, -3, -4, -3], [3, 0, 0, -2], [0, 0, 0, -4], [0, 0, 0, 1], [6, 0, 6, 0], [0, 4, -6, 4], [-7, 3, -7, 0], [4, 0, 0, -3], [-4, 1, -4, 0], [6, 0, 0, 4], [0, -5, -1, -5], [4, 0, 4, 0], [-4, 0, 0, 4], [0, -1, -1, -1], [0, 2, -16, 2], [0, -1, 7, -1], [-12, 0, 0, -1], [19, 0, 0, 2], [0, -2, 0, -2], [8, -2, 8, 0], [0, -8, -2, -8], [8, -5, 8, 0], [3, 0, 0, 0], [0, -6, 4, -6], [-3, 0, -3, 0], [-14, -4, -14, 0], [0, 1, 12, 1], [12, -4, 12, 0], [4, -8, 4, 0], [0, -3, 12, -3], [6, 0, 0, 0], [-6, 0, 0, 4], [0, 0, 0, -1], [0, 6, 3, 6], [0, -6, 4, -6], [9, 3, 9, 0], [-8, 3, -8, 0], [0, 0, -2, 0], [24, -2, 24, 0], [0, 4, 7, 4], [-5, 0, 0, 9], [8, -5, 8, 0], [0, 7, 3, 7], [-12, 0, 0, -3], [4, 0, 0, 5], [2, 0, 0, 4], [-13, 0, 0, -1], [0, -2, 16, -2], [-17, 0, 0, 2], [0, 3, -20, 3], [-6, 4, -6, 0], [1, -3, 1, 0], [-16, 0, 0, -3], [-12, 1, -12, 0], [0, -2, -5, -2], [12, 4, 12, 0], [0, 0, 8, 0], [-1, 0, 0, 1], [0, -4, -10, -4], [21, -3, 21, 0], [0, -2, -9, -2], [4, 4, 4, 0], [-5, 0, 0, -4], [16, -1, 16, 0], [0, 0, 27, 0], [0, 3, 0, 0], [-8, 0, 0, 2], [0, 4, -30, 4], [1, 10, 1, 0], [0, -7, -1, -7], [-8, 0, 0, -5], [8, 0, 0, 5], [12, -6, 12, 0], [0, -1, -32, -1], [8, -1, 8, 0], [-12, 0, 0, -10], [0, -1, -32, -1], [21, -1, 21, 0], [-9, 0, 0, -1], [-16, -1, -16, 0], [-1, 0, 0, 4], [-20, 0, 0, 7], [-31, -1, -31, 0], [0, -2, 16, -2], [14, 0, 14, 0], [8, 0, 0, 9], [-17, 0, 0, 0], [-16, 8, -16, 0], [23, 0, 0, 5], [12, 0, 0, 0], [-3, -8, -3, 0], [0, 2, -28, 2], [-11, -10, -11, 0], [0, -11, -5, -11], [-24, 0, 0, -2], [0, -6, -8, -6], [0, -5, -21, -5], [0, 2, 8, 2], [-20, 0, -20, 0], [2, -4, 2, 0], [0, 0, -40, 0], [-15, -2, -15, 0], [13, 4, 13, 0], [6, 0, 0, -12], [0, 0, -44, 0], [8, -3, 8, 0], [14, 0, 0, -8], [0, -4, 35, -4], [0, 10, 16, 10], [24, 0, 0, 5], [-17, 0, 0, -4], [-8, -8, -8, 0], [0, 10, 0, 0], [24, 7, 24, 0], [2, 12, 2, 0], [0, -4, 10, -4], [6, 4, 6, 0], [0, -4, -42, -4], [0, 2, -4, 2], [0, -4, 6, -4], [-39, -3, -39, 0], [-36, 0, 0, -2], [26, 0, 26, 0], [0, -10, -8, -10], [0, 0, 0, 6], [9, -16, 9, 0], [0, -8, 4, -8], [-25, 0, 0, 6], [-9, 0, 0, -1], [20, 0, 0, -5], [24, 0, 0, -11], [0, 2, 7, 2], [-15, 13, -15, 0], [44, 0, 0, -2], [-32, -6, -32, 0], [0, 2, 0, 0], [-16, 0, 0, 10], [0, -2, 12, -2], [0, -1, -21, -1], [-13, 0, 0, 8], [14, 0, 0, -4], [-4, -14, -4, 0], [0, 0, -30, 0], [12, 0, 0, 3], [0, 12, 4, 12], [21, 5, 21, 0], [-52, 0, 0, -2], [12, 9, 12, 0], [0, -2, 23, -2], [-6, 0, 0, -8], [23, 0, 0, 9], [-4, 0, 0, 19], [0, -14, 7, -14], [0, -6, 36, -6], [0, 2, 27, 2], [-36, 0, 0, 6], [0, -1, 35, -1], [44, 5, 44, 0], [19, 0, 0, -5], [0, -10, 0, 0], [-6, 8, -6, 0], [-28, 0, 0, -11], [24, 0, 0, 8], [-5, 0, 0, 4], [-30, 0, 0, 8], [40, 9, 40, 0], [0, 3, -49, 3], [22, 0, 0, -8], [-36, 0, 0, 6], [-23, -6, -23, 0], [0, -3, 48, -3], [0, -2, 23, -2], [0, -1, 4, -1], [0, 0, 0, 15], [-6, -12, -6, 0], [0, 17, 8, 17], [6, -4, 6, 0], [23, 0, 0, -4], [0, -12, 27, -12], [23, 0, 0, -1], [-12, 0, 0, 0], [0, 1, 47, 1], [0, 0, 44, 0], [-3, 8, -3, 0], [15, 0, 0, 18], [-36, 3, -36, 0], [1, 13, 1, 0], [8, 0, 0, -13], [0, -4, -48, -4], [-19, 15, -19, 0], [0, 2, 24, 2], [37, 8, 37, 0], [-10, 0, 0, -4], [-28, 1, -28, 0], [0, 13, 4, 13], [0, -2, 12, -2], [55, 0, 0, 0], [4, 0, 0, 22], [0, -9, 35, -9], [-16, -6, -16, 0], [-2, 0, 0, -4], [-23, 16, -23, 0], [-12, 0, 0, 10], [3, 0, 0, -1], [0, -8, 0, -8], [-24, -3, -24, 0], [-13, 0, 0, -10], [-11, 3, -11, 0], [0, 0, -16, 0], [0, -2, -4, -2], [8, 0, 0, -12], [25, 4, 25, 0], [-8, -10, -8, 0], [4, 4, 4, 0], [-13, 0, 0, -13], [-20, 0, 0, -8], [20, 0, 0, 4], [0, -8, 10, -8], [-3, -1, -3, 0], [0, 6, -20, 6], [16, -5, 16, 0], [-36, -13, -36, 0], [44, -9, 44, 0], [0, -11, 24, -11], [-31, 10, -31, 0], [0, -3, 23, -3], [-24, 0, 0, -12], [0, 8, -36, 8], [-32, 0, -32, 0], [0, 1, -24, 1], [-12, -15, -12, 0], [-38, 0, -38, 0], [-58, 0, -58, 0], [32, 0, 0, -6], [13, -10, 13, 0], [-42, 0, 0, -4], [0, 19, 8, 19], [0, -4, -50, -4], [0, -7, 0, 0], [6, 0, 0, -8], [0, -12, -9, -12], [-16, 0, 0, -8], [8, 16, 8, 0], [-9, 0, 0, -4], [-7, -2, -7, 0], [0, 1, -37, 1], [0, 5, -52, 5], [0, 4, -10, 4], [-21, 0, 0, -21], [0, 15, -28, 15], [0, -9, -25, -9], [-1, 0, 0, 26], [-48, 0, 0, 4], [-27, 10, -27, 0], [0, 2, -20, 2], [0, 12, -22, 12], [-4, 19, -4, 0], [0, -6, 8, -6], [-9, 0, 0, 14], [0, 20, 19, 20], [0, 1, -12, 1], [-24, 13, -24, 0], [44, 4, 44, 0], [64, 0, 0, 2], [-47, -6, -47, 0], [-48, 0, 0, 11], [-36, 0, 0, 5], [35, 0, 0, -18], [-9, 0, 0, 9], [0, -1, 56, -1], [0, 17, -25, 17], [50, -4, 50, 0], [0, 9, 8, 9], [-11, -13, -11, 0], [0, 18, 20, 18], [0, 0, 22, 0], [15, 0, 0, -9], [20, 0, 0, -12], [25, 19, 25, 0], [48, -8, 48, 0], [-60, 5, -60, 0], [0, -24, -1, -24], [-33, 0, 0, 11], [-24, -5, -24, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1248_q_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1248_2_q_g();".
// 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_1248_2_q_g(:prec:=4)
    chi := MakeCharacter_1248_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_1248_2_q_g();".
// 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_1248_2_q_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1248_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-7, 2, 1]>,<7,R![64, 0, 8, 0, 1]>,<19,R![64, -64, 56, -8, 1]>],Snew);
    return Vf;
end function;