
// Make newform 363.3.h.d in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_363_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_363_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_363_3_h_d();".
// 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_363_3_h_d();".
// 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 := [81, -27, -18, 15, 1, 5, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, -1, 0], [0, -16, 0, 0, 0, 0, -1, 0], [162, -54, -36, -5, 2, 10, -4, -2], [135, 90, -75, -5, 2, 10, 5, -5], [-189, -126, 105, 7, 8, -14, -7, 7], [31, 0, 0, 0, 0, 2, 0, 0], [-648, 216, 144, -7, -8, -40, 16, 8]];
    Rf_basisdens := [1, 3, 3, 27, 27, 27, 1, 27];
    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_363_h();"
function MakeCharacter_363_h()
    N := 363;
    order := 10;
    char_gens := [122, 244];
    v := [5, 8];
    // 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_363_h_Hecke();"
function MakeCharacter_363_h_Hecke(Kf)
    N := 363;
    order := 10;
    char_gens := [122, 244];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 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 := 3;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 3, 0, 0, 0], [0, -2, 0, 0, 0, -2, -2, -2], [0, 0, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 4, 0, 0, 0, 0], [0, -4, 0, 0, 0, -4, -4, -4], [0, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0], [0, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 26, 0, 0, 0, 0], [0, 0, 30, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 0, 0], [-42, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -26, 0, 0], [0, 0, 0, 0, 0, 0, 0, 18], [0, -20, 0, 0, 0, 0, 0, 0], [12, 0, 12, -12, 12, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0], [0, -18, 0, 0, 0, -18, -18, -18], [0, 0, 74, 0, 0, 0, 0, 0], [0, 0, 0, -40, 0, 0, 0, 0], [0, 12, 0, 0, 0, 12, 12, 12], [0, 0, 0, 0, 0, 0, 36, 0], [0, 0, 0, -62, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 32], [0, 0, 74, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -12, 0, 0], [200, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 12, 0, 0], [188, 0, 188, -188, 188, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0], [0, 32, 0, 0, 0, 32, 32, 32], [0, 0, 74, 0, 0, 0, 0, 0], [0, 28, 0, 0, 0, 28, 28, 28], [0, 0, 0, 0, 160, 0, 0, 0], [0, 0, 0, 0, 182, 0, 0, 0], [0, 0, 0, 290, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -72], [0, 0, 0, 0, 0, 60, 0, 0], [0, 0, 0, 0, 0, -60, 0, 0], [-10, 0, -10, 10, -10, 0, 0, 0], [0, 34, 0, 0, 0, 0, 0, 0], [298, 0, 298, -298, 298, 0, 0, 0], [0, 0, 0, 0, 0, 0, 40, 0], [-42, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 246, 0, 0, 0, 0], [0, 0, 0, 0, -302, 0, 0, 0], [0, -60, 0, 0, 0, 0, 0, 0], [0, 0, 0, -150, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -132], [0, 0, 0, 0, 0, 0, 0, 0], [-130, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -88], [0, -148, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 68, 0], [0, 22, 0, 0, 0, 22, 22, 22], [0, 0, 448, 0, 0, 0, 0, 0], [0, 0, 0, -260, 0, 0, 0, 0], [0, 60, 0, 0, 0, 60, 60, 60], [0, 0, 0, 0, 50, 0, 0, 0], [0, 144, 0, 0, 0, 0, 0, 0], [-86, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, 0, 0], [-98, 0, -98, 98, -98, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -94], [-218, 0, 0, 0, 0, 0, 0, 0], [0, 0, -278, 0, 0, 0, 0, 0], [0, -100, 0, 0, 0, -100, -100, -100], [0, 0, 0, 0, -324, 0, 0, 0], [0, 0, 0, 0, 0, 0, -120, 0], [0, -92, 0, 0, 0, 0, 0, 0], [0, 0, -278, 0, 0, 0, 0, 0], [68, 0, 0, 0, 0, 0, 0, 0], [-670, 0, -670, 670, -670, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10], [0, 126, 0, 0, 0, 0, 0, 0], [-86, 0, 0, 0, 0, 0, 0, 0], [0, -76, 0, 0, 0, -76, -76, -76], [0, 0, 0, 510, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 160, 0], [0, 0, 0, 0, -170, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 84], [0, 0, -542, 0, 0, 0, 0, 0], [-328, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -40, 0, 0], [0, 0, 0, 0, 0, 0, 0, 136], [342, 0, 342, -342, 342, 0, 0, 0], [0, 0, 0, 0, 0, 0, -24, 0], [-86, 0, 0, 0, 0, 0, 0, 0], [0, -180, 0, 0, 0, -180, -180, -180], [0, -68, 0, 0, 0, -68, -68, -68], [0, 0, 0, 0, 446, 0, 0, 0], [0, -212, 0, 0, 0, 0, 0, 0], [0, 0, -58, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -152, 0, 0], [0, 0, 0, 0, 0, -222, 0, 0], [0, -84, 0, 0, 0, 0, 0, 0], [-142, 0, -142, 142, -142, 0, 0, 0], [0, 0, 0, 400, 0, 0, 0, 0], [0, 0, 0, 0, -170, 0, 0, 0], [0, 256, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -212], [0, 0, 0, 0, 0, -36, 0, 0], [706, 0, 0, 0, 0, 0, 0, 0], [738, 0, 738, -738, 738, 0, 0, 0], [0, 284, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 164, 0], [0, -14, 0, 0, 0, -14, -14, -14], [0, 0, -542, 0, 0, 0, 0, 0], [0, 0, 0, -700, 0, 0, 0, 0], [0, 0, 0, 0, -764, 0, 0, 0], [0, 0, 0, 0, 0, 0, -12, 0], [0, 0, 0, 0, -742, 0, 0, 0], [0, 0, -410, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 228, 0, 0], [-890, 0, -890, 890, -890, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -146], [0, -118, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -116, 0], [-746, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -634, 0, 0, 0, 0], [0, -240, 0, 0, 0, -240, -240, -240], [0, 0, 0, 0, 0, 0, 136, 0], [0, 0, 0, -458, 0, 0, 0, 0], [0, 0, 0, 0, 0, 152, 0, 0], [562, 0, 562, -562, 562, 0, 0, 0], [0, -102, 0, 0, 0, 0, 0, 0], [-42, 0, 0, 0, 0, 0, 0, 0], [0, 0, 624, 0, 0, 0, 0, 0], [0, 0, 0, 686, 0, 0, 0, 0], [0, 260, 0, 0, 0, 260, 260, 260], [0, 0, 0, 0, 94, 0, 0, 0], [0, 0, 0, -1118, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -348], [-1274, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 282, 0, 0], [298, 0, 298, -298, 298, 0, 0, 0], [0, 158, 0, 0, 0, 158, 158, 158], [0, 276, 0, 0, 0, 276, 276, 276], [0, 0, 0, 0, 182, 0, 0, 0], [0, -248, 0, 0, 0, 0, 0, 0], [0, 0, 0, 246, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 164], [0, 0, 250, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -142, 0, 0], [56, 0, 56, -56, 56, 0, 0, 0], [0, 0, 0, 0, 0, 0, -76, 0], [1278, 0, 0, 0, 0, 0, 0, 0], [0, -342, 0, 0, 0, -342, -342, -342], [0, 0, 0, 0, -456, 0, 0, 0], [0, 0, 0, 0, 0, 0, 184, 0], [0, 0, 0, 0, -1094, 0, 0, 0], [0, 44, 0, 0, 0, 0, 0, 0], [-186, 0, -186, 186, -186, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -258], [-428, 0, -428, 428, -428, 0, 0, 0], [0, 192, 0, 0, 0, 192, 192, 192], [0, 0, 0, 290, 0, 0, 0, 0], [0, -264, 0, 0, 0, -264, -264, -264], [0, 0, 0, 0, 0, 0, 24, 0], [0, -236, 0, 0, 0, 0, 0, 0], [-460, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 352, 0, 0], [0, 0, 0, 0, 0, 0, 0, -396], [0, -126, 0, 0, 0, 0, 0, 0], [838, 0, 0, 0, 0, 0, 0, 0], [0, 0, 52, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_363_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_363_3_h_d();".
// 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_363_3_h_d(:prec:=8)
    chi := MakeCharacter_363_h();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_363_3_h_d();".
// 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_363_3_h_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_363_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![14641, 0, -1331, 0, 121, 0, -11, 0, 1]>,<5,R![3748096, 0, -85184, 0, 1936, 0, -44, 0, 1]>,<7,R![4096, 512, 64, 8, 1]>],Snew);
    return Vf;
end function;