
// Make newform 1056.3.p.c in Magma, downloaded from the LMFDB on 30 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1056_p();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1056_p_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_1056_3_p_c();".
// 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_1056_3_p_c();".
// 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 := [2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 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_1056_p();"
function MakeCharacter_1056_p()
    N := 1056;
    order := 2;
    char_gens := [991, 133, 353, 673];
    v := [1, 1, 1, 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_1056_p_Hecke();"
function MakeCharacter_1056_p_Hecke( : Kf := false)
    N := 1056;
    order := 2;
    char_gens := [991, 133, 353, 673];
    char_values := [[-1, 0], [-1, 0], [-1, 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 := 3;
    raw_aps := [[0, 0], [-1, -1], [0, 0], [0, 0], [-7, -3], [0, 0], [2, 0], [0, 6], [0, 0], [0, 0], [0, 0], [0, 0], [46, 0], [0, 30], [0, 0], [0, 0], [0, 30], [0, 0], [-62, 0], [0, 0], [0, -12], [0, 0], [158, 0], [0, 36], [94, 0], [0, 0], [0, 0], [178, 0], [0, 0], [0, 72], [0, 0], [62, 0], [0, 48], [0, 66], [0, 0], [0, 0], [0, 0], [-322, 0], [0, 0], [0, 0], [0, -126], [0, 0], [0, 0], [0, 132], [0, 0], [0, 0], [0, -126], [0, 0], [-446, 0], [0, 0], [-434, 0], [0, 0], [0, 156], [0, -66], [0, -120], [0, 0], [0, 0], [0, 0], [0, 0], [-238, 0], [0, 198], [0, 0], [0, 102], [0, 0], [-526, 0], [0, 0], [14, 0], [0, -168], [-658, 0], [0, 0], [0, -240], [0, 0], [0, 0], [0, 0], [686, 0], [0, 0], [0, 0], [0, 0], [0, 84], [0, -264], [0, 234], [0, 0], [0, 0], [578, 0], [0, 0], [0, -42], [0, 84], [0, 312], [0, 0], [0, 0], [0, -330], [0, 0], [0, 0], [-782, 0], [-802, 0], [0, 0], [0, 0], [0, 96], [0, 342], [0, 0], [0, -138], [0, 0], [226, 0], [626, 0], [0, 330], [-2, 0], [0, 102], [-862, 0], [0, 0], [0, 276], [0, 0], [0, 0], [0, -420], [562, 0], [0, 0], [0, 420], [-1214, 0], [0, 0], [0, 0], [994, 0], [0, 0], [0, -180], [0, 0], [0, 462], [-734, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -510], [0, 0], [0, 0], [0, 0], [1394, 0], [0, 396], [0, 0], [0, 450], [0, 0], [1582, 0], [0, -570], [0, 0], [0, 0], [1262, 0], [0, 0], [0, 0], [0, 0], [-1202, 0], [1646, 0], [0, 0], [0, 0], [0, 360], [1762, 0], [0, 0], [1714, 0], [0, 0], [0, 0], [0, 540], [0, -612], [0, 0], [0, -570], [142, 0], [0, 0], [0, -594], [0, -132], [0, 0], [0, 0], [0, 0], [0, -684], [0, 0], [-274, 0], [0, 0], [0, 0], [0, 372], [0, 0], [0, -396], [0, 546], [0, 0], [0, 0], [0, 0], [0, 0], [2174, 0], [0, 0], [-2162, 0], [0, 0], [0, 0], [0, 0], [-1598, 0], [0, 696], [0, 0], [2302, 0], [-562, 0], [-1186, 0], [0, 0], [-1982, 0], [0, -660], [0, -336], [0, 0], [-1922, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 744], [0, 150], [0, 0], [0, 0], [-1666, 0], [1778, 0], [-782, 0], [0, -420], [0, 0], [0, 0], [0, -918], [0, 0], [1966, 0], [0, 0], [-2686, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 924], [0, 0], [-1502, 0], [0, 0], [0, -468], [0, 0], [0, 0], [-14, 0], [0, 0], [0, -54], [0, 0], [1394, 0], [-2866, 0], [0, 0], [386, 0], [0, 0], [0, 966], [0, 0], [0, 78], [0, -666], [0, 0], [0, 0], [2978, 0], [0, 0], [0, 0], [0, 390], [0, 594], [0, 0], [0, 0], [-1154, 0], [0, 0], [-626, 0], [0, 0], [2846, 0], [0, 0], [0, 702], [0, 0], [0, -888], [0, 0], [1022, 0], [0, 0], [0, 0], [0, -1188], [-3326, 0], [0, 0], [0, -780], [0, 1218], [0, 0], [0, 0], [2338, 0], [-3218, 0], [0, 0], [0, 1200], [0, 0], [0, 1242], [0, 0], [0, 60], [-734, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 258], [0, 0], [-1154, 0], [0, 0], [0, 0], [578, 0], [0, 0], [0, -870], [2258, 0], [3826, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-1042, 0], [0, 1242], [0, 1392], [0, 0], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1056_p_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1056_3_p_c();".
// 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_1056_3_p_c(:prec:=2)
    chi := MakeCharacter_1056_p();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_1056_3_p_c();".
// 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_1056_3_p_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1056_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![0, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;