
// Make newform 1254.2.i.p in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1254_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1254_i_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_1254_2_i_p();".
// 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_1254_2_i_p();".
// 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, -8, 18, 0, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-40, 8, -18, 25, -5, 1], [-121, 16, -36, 9, -10, 2], [72, -162, -25, -45, 9, -10], [-26, -454, -65, -117, 7, -26]];
    Rf_basisdens := [1, 1, 82, 41, 82, 82];
    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_1254_i();"
function MakeCharacter_1254_i()
    N := 1254;
    order := 3;
    char_gens := [419, 343, 1123];
    v := [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_1254_i_Hecke();"
function MakeCharacter_1254_i_Hecke( : Kf := false)
    N := 1254;
    order := 3;
    char_gens := [419, 343, 1123];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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 := [[-1, 0, 0, 0, 1, 0], [1, 0, 0, 0, -1, 0], [1, -1, 0, 0, -1, 1], [2, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0], [0, -3, -3, 0, 0, 0], [0, -2, 0, 0, 0, -1], [1, 1, 2, 1, 0, 1], [0, 2, 2, 1, 4, -1], [0, -3, -3, -1, 0, 1], [-3, 0, 1, 1, 0, 0], [-4, 0, 0, 0, 0, 0], [-4, 4, 0, 0, 4, 0], [-2, -1, 0, 0, 2, 3], [0, 2, 2, -2, 3, 2], [0, 2, 2, 0, -8, 0], [0, -2, 0, 0, 0, 0], [0, 0, 0, 5, -1, -5], [0, -3, -3, -3, -3, 3], [3, -2, 0, 0, -3, 0], [0, -6, 0, 0, 0, 2], [2, 0, 0, 0, -2, 4], [-7, 0, -2, -1, 0, 0], [0, 2, 2, -1, 13, 1], [-1, 2, 0, 0, 1, 4], [0, 3, 3, -1, 6, 1], [4, 0, 3, 2, 0, 0], [-4, 0, -3, -2, 0, 0], [13, -2, 0, 0, -13, 1], [-4, 0, 9, 2, 0, 0], [0, 4, 4, -3, 4, 3], [-4, 7, 0, 0, 4, 4], [0, 6, 6, 5, 5, -5], [0, -8, -8, -4, 4, 4], [1, 11, 0, 0, -1, -4], [-14, 0, 0, -4, 0, 0], [12, 2, 0, 0, -12, -2], [2, 0, 4, 4, 0, 0], [0, -5, -5, -3, -1, 3], [3, -3, 0, 0, -3, -6], [1, 0, -1, 2, 0, 0], [0, -1, -1, 0, -13, 0], [15, 0, 2, 0, 0, 0], [4, -2, 0, 0, -4, -2], [-13, 0, -4, -7, 0, 0], [0, 3, 3, -2, -6, 2], [1, -3, 0, 0, -1, 2], [5, 10, 0, 0, -5, -5], [6, 0, 6, -4, 0, 0], [3, 0, -1, 6, 0, 0], [14, -4, 0, 0, -14, 2], [6, 0, 2, 0, 0, 0], [0, -5, -5, -7, 5, 7], [0, 9, 9, 8, 11, -8], [0, 2, 2, -2, -12, 2], [-2, 6, 0, 0, 2, 2], [-9, -7, 0, 0, 9, 3], [-10, 2, 0, 0, 10, 3], [3, 0, 0, -3, 0, 0], [0, 4, 4, -5, 16, 5], [-7, 9, 0, 0, 7, -6], [-5, 0, -2, -5, 0, 0], [-4, 1, 0, 0, 4, 5], [1, 0, -2, -2, 0, 0], [0, 2, 2, 4, 13, -4], [0, 9, 9, -1, 17, 1], [-4, 0, -6, -2, 0, 0], [-5, 7, 0, 0, 5, -5], [-2, 11, 0, 0, 2, -4], [7, 0, 0, -7, 0, 0], [-6, 0, 5, -2, 0, 0], [11, -9, 0, 0, -11, -5], [0, 9, 9, 2, 0, -2], [-8, 0, 6, 4, 0, 0], [-6, 0, -2, 8, 0, 0], [10, -8, 0, 0, -10, 3], [0, -12, -12, -8, 2, 8], [-21, 7, 0, 0, 21, -2], [-2, 0, 0, 0, 2, -2], [0, -2, -2, -8, 0, 8], [-6, 0, -16, -6, 0, 0], [-7, -1, 0, 0, 7, -12], [0, 4, 4, 8, 14, -8], [0, 12, 12, 10, 5, -10], [-10, -4, 0, 0, 10, -1], [0, -4, -4, 2, 6, -2], [1, 0, -13, 3, 0, 0], [19, 0, 13, -7, 0, 0], [17, -4, 0, 0, -17, 3], [13, 0, 12, 13, 0, 0], [-15, 0, 1, -2, 0, 0], [0, 5, 5, -1, 11, 1], [-5, 0, 14, 1, 0, 0], [14, 2, 0, 0, -14, 8], [-9, -17, 0, 0, 9, 3], [0, -8, -8, -4, 14, 4], [0, 4, 4, 4, -4, -4], [-2, 0, -3, 12, 0, 0], [0, 3, 3, 13, 10, -13], [0, 2, 2, 12, 16, -12], [0, -4, -4, -1, -17, 1], [0, -7, -7, 5, -20, -5], [3, 0, 15, 9, 0, 0], [8, 0, -12, -1, 0, 0], [6, 0, -3, -7, 0, 0], [-20, 0, -8, 1, 0, 0], [-9, 3, 0, 0, 9, 2], [0, -12, -12, -8, 0, 8], [0, 6, 6, 2, 25, -2], [-35, 0, 1, 1, 0, 0], [6, 0, 6, 17, 0, 0], [18, -11, 0, 0, -18, 6], [0, 8, 8, -9, 13, 9], [-14, 0, 0, -8, 0, 0], [0, -16, -16, -1, -11, 1], [-30, 7, 0, 0, 30, 0], [2, -18, 0, 0, -2, -4], [-3, 0, 6, -12, 0, 0], [-10, 0, -8, -4, 0, 0], [0, 2, 2, 5, 25, -5], [0, -2, -2, -2, 0, 2], [1, 0, 5, -9, 0, 0], [-2, 0, -1, 1, 0, 0], [3, 0, 11, 10, 0, 0], [3, 0, -3, 1, 0, 0], [18, -13, 0, 0, -18, 9], [0, -10, -10, 2, -4, -2], [13, -20, 0, 0, -13, 8], [8, 5, 0, 0, -8, -8], [4, 0, -6, 2, 0, 0], [0, -13, 0, 0, 0, 17], [-3, -13, 0, 0, 3, -5], [0, 9, 9, 0, 2, 0], [-11, -13, 0, 0, 11, 10], [10, 0, -4, 1, 0, 0], [0, -6, -6, -6, -20, 6], [0, 13, 13, 19, 17, -19], [4, 0, 15, 3, 0, 0], [27, 0, -3, 3, 0, 0], [-14, 0, -16, 3, 0, 0], [0, -10, -10, -15, -3, 15], [0, -16, -16, -12, -6, 12], [0, -22, -22, -7, 7, 7], [0, -23, -23, -2, -8, 2], [-25, 0, -3, 4, 0, 0], [-7, -2, 0, 0, 7, 13], [3, 23, 0, 0, -3, -15], [0, -6, 0, 0, 0, 11], [0, 2, 2, 4, -12, -4], [-27, 0, 8, 0, 0, 0], [-7, 9, 0, 0, 7, -19], [7, 0, -7, -11, 0, 0], [0, 7, 7, 9, 33, -9], [0, -3, -3, 3, 23, -3], [28, 6, 0, 0, -28, 0], [4, 0, 14, 5, 0, 0], [-2, 0, 8, -2, 0, 0], [26, 1, 0, 0, -26, 6], [0, -7, -7, -9, -39, 9], [0, -7, -7, -8, 5, 8], [9, 7, 0, 0, -9, 0], [32, -2, 0, 0, -32, 3], [24, -8, 0, 0, -24, 22], [3, 3, 0, 0, -3, 12], [-23, 0, -12, -17, 0, 0], [15, -2, 0, 0, -15, 8], [-1, -4, 0, 0, 1, -7], [0, 6, 6, -4, -4, 4], [5, 11, 0, 0, -5, -1], [0, 16, 16, 22, 18, -22], [-16, 0, -3, -10, 0, 0], [-25, 7, 0, 0, 25, 10], [13, 1, 0, 0, -13, -7], [9, 0, 1, 1, 0, 0], [0, 20, 20, 2, 8, -2], [0, -8, -8, -19, -27, 19], [0, 7, 7, -1, 0, 1], [18, -12, 0, 0, -18, 4], [-4, 0, -18, -5, 0, 0], [-21, 2, 0, 0, 21, 5], [0, 17, 17, -6, 12, 6], [-40, 0, -5, 8, 0, 0], [0, 27, 27, 10, -25, -10], [24, 0, 0, 0, -24, 1], [29, 0, 4, -8, 0, 0], [7, 0, -25, -15, 0, 0], [0, 23, 23, 4, 12, -4], [-44, -2, 0, 0, 44, 2], [-41, 0, 1, -5, 0, 0], [5, 0, -29, -13, 0, 0], [0, 24, 24, -1, 4, 1], [0, -4, -4, 9, 7, -9], [-19, 0, -15, 5, 0, 0], [-11, 1, 0, 0, 11, 17], [0, 15, 15, 13, 35, -13], [0, -14, -14, -1, -15, 1], [0, -24, -24, -10, 26, 10], [-15, -15, 0, 0, 15, -6], [-10, 0, -6, 5, 0, 0], [31, 0, 11, 25, 0, 0], [0, 7, 7, -5, 18, 5], [0, -2, -2, -7, -40, 7], [-2, -26, 0, 0, 2, 0], [0, 10, 0, 0, 0, 0], [-2, 22, 0, 0, 2, -20], [0, -3, -3, 5, -3, -5], [0, -1, -1, -8, 2, 8], [0, 20, 20, 0, 16, 0], [-2, 26, 0, 0, 2, -13], [5, 0, 7, -19, 0, 0], [18, 2, 0, 0, -18, 8], [0, -7, -7, 3, 29, -3], [34, 0, 1, -6, 0, 0], [0, -9, -9, 4, -47, -4], [26, 0, -6, -6, 0, 0], [0, 12, 12, -3, 25, 3], [20, 0, 0, 0, -20, 11], [-14, 0, -24, 3, 0, 0], [16, 0, -6, -15, 0, 0], [19, -15, 0, 0, -19, -17], [0, 3, 3, 14, -12, -14], [-10, 0, -10, -5, 0, 0], [10, -7, 0, 0, -10, -10], [-11, 7, 0, 0, 11, -15], [-9, 20, 0, 0, 9, -1], [0, 0, 0, -4, 46, 4], [-15, 0, 7, -3, 0, 0], [-50, -2, 0, 0, 50, 4], [-16, -6, 0, 0, 16, -2], [22, 0, -9, -10, 0, 0], [0, -1, -1, 2, 5, -2], [0, -17, -17, -11, 32, 11], [-18, 0, -30, -8, 0, 0], [-30, 0, -16, -3, 0, 0], [15, 0, -19, 1, 0, 0], [-22, 6, 0, 0, 22, -14], [3, 0, 4, -10, 0, 0], [-4, 0, 15, 23, 0, 0], [33, -3, 0, 0, -33, -4], [0, -14, -14, -6, 5, 6], [-3, 3, 0, 0, 3, -2], [-37, 0, -10, 7, 0, 0], [0, 0, 0, -14, -12, 14], [0, 4, 4, -6, 28, 6], [-20, 14, 0, 0, 20, 11], [-19, 0, 19, 3, 0, 0], [0, 0, 0, -13, -29, 13], [0, -17, -17, 6, 11, -6], [-21, 23, 0, 0, 21, -14], [0, 2, 2, 0, -70, 0], [-29, 0, -12, -5, 0, 0], [-22, -6, 0, 0, 22, -3], [-23, 0, 10, 3, 0, 0], [0, -18, -18, -17, 5, 17], [-7, 27, 0, 0, 7, -10], [0, 2, 2, 10, -6, -10], [0, -9, -9, -6, -31, 6], [-26, 0, -26, 10, 0, 0], [20, 8, 0, 0, -20, 4], [0, -19, -19, -5, 39, 5], [0, 14, 14, -9, 2, 9], [9, -4, 0, 0, -9, 19], [-6, -3, 0, 0, 6, -8], [-21, 11, 0, 0, 21, 7], [0, -13, -13, -16, -20, 16], [-5, 0, -7, -15, 0, 0], [-17, 0, -15, -19, 0, 0], [15, 0, -12, -9, 0, 0], [0, -16, -16, -1, 7, 1], [0, 0, 0, 5, 19, -5], [-16, 0, -6, 22, 0, 0], [31, 0, -7, 9, 0, 0], [-9, 24, 0, 0, 9, -26], [10, 0, -12, -11, 0, 0], [0, -9, -9, 11, 15, -11], [13, -4, 0, 0, -13, 1], [0, 0, -20, -20, 0, 0], [-3, -15, 0, 0, 3, 9], [0, -31, -31, -17, -15, 17], [0, -9, -9, -7, -49, 7], [18, 0, 34, 12, 0, 0], [-26, 0, -3, -26, 0, 0], [0, -23, -23, 1, -21, -1], [55, 0, 12, -7, 0, 0], [-16, -19, 0, 0, 16, 19], [0, -4, -4, 4, -1, -4], [43, 0, 8, 16, 0, 0], [0, 5, 5, 10, 33, -10], [8, -8, 0, 0, -8, 20], [-2, 0, 28, 14, 0, 0], [-24, 0, 8, 20, 0, 0], [14, 0, -22, -14, 0, 0], [0, 16, 16, -9, 12, 9], [26, 0, -1, 0, 0, 0], [15, -14, 0, 0, -15, 11], [-17, 0, 31, 14, 0, 0], [0, 3, 3, 4, -6, -4], [-24, 16, 0, 0, 24, 12], [-20, 3, 0, 0, 20, -28], [27, 0, 8, -23, 0, 0], [-1, 7, 0, 0, 1, 13], [-39, 8, 0, 0, 39, -9], [0, -16, -16, 10, 22, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1254_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1254_2_i_p();".
// 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_1254_2_i_p(:prec:=6)
    chi := MakeCharacter_1254_i();
    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_1254_2_i_p();".
// 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_1254_2_i_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1254_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![64, -48, 52, -4, 10, -2, 1]>,<7,R![8, 5, -6, 1]>,<13,R![2916, -1944, 1458, 0, 45, -3, 1]>],Snew);
    return Vf;
end function;