
// Make newform 405.3.h.c in Magma, downloaded from the LMFDB on 25 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_405_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_405_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_405_3_h_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_405_3_h_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 := [9, -3, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-9, 16, 2, 1], [-9, -2, 2, 1], [12, 8, 1, -4]];
    Rf_basisdens := [1, 6, 6, 3];
    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_405_h();"
function MakeCharacter_405_h()
    N := 405;
    order := 6;
    char_gens := [326, 82];
    v := [5, 3];
    // 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_405_h_Hecke();"
function MakeCharacter_405_h_Hecke( : Kf := false)
    N := 405;
    order := 6;
    char_gens := [326, 82];
    char_values := [[1, 0, 1, 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 := 3;
    raw_aps := [[0, 0, 1, 0], [0, 0, 0, 0], [-1, 0, -1, -1], [0, 2, 1, 0], [0, -2, -1, 0], [-2, 0, -2, -4], [22, 0, 0, 0], [-4, 0, 0, 0], [20, 0, 20, 0], [0, -8, -4, 0], [-29, 0, -29, 0], [0, 0, 0, 0], [-4, 0, -4, -8], [0, -4, -2, 0], [0, 0, 58, 0], [31, 0, 0, 0], [-4, 0, -4, -8], [0, 0, 44, 0], [-2, 0, -2, -4], [-6, 12, 0, -12], [9, -18, 0, 18], [0, 0, -10, 0], [0, 0, 19, 0], [-6, 12, 0, -12], [0, 26, 13, 0], [0, 34, 17, 0], [4, 0, 4, 8], [-29, 0, 0, 0], [104, 0, 0, 0], [80, 0, 80, 0], [-15, 30, 0, -30], [17, 0, 17, 34], [0, 0, -98, 0], [64, 0, 64, 0], [-1, 0, -1, -2], [0, 0, 257, 0], [16, 0, 16, 32], [-30, 60, 0, -60], [254, 0, 254, 0], [0, 0, 235, 0], [-15, 30, 0, -30], [-292, 0, 0, 0], [0, -20, -10, 0], [-17, 0, -17, -34], [-203, 0, 0, 0], [155, 0, 0, 0], [-308, 0, -308, 0], [0, 8, 4, 0], [0, 0, -242, 0], [106, 0, 106, 0], [142, 0, 0, 0], [20, 0, 20, 40], [0, 0, 170, 0], [36, -72, 0, 72], [290, 0, 290, 0], [0, 0, -104, 0], [-12, 24, 0, -24], [-121, 0, 0, 0], [0, 32, 16, 0], [0, -44, -22, 0], [28, 0, 28, 56], [-286, 0, -286, 0], [-6, 12, 0, -12], [20, 0, 20, 40], [0, 14, 7, 0], [0, 0, 541, 0], [0, 0, -274, 0], [-44, 0, -44, -88], [179, 0, 179, 0], [0, 0, 620, 0], [0, 0, -32, 0], [54, -108, 0, 108], [0, -70, -35, 0], [34, 0, 34, 68], [248, 0, 0, 0], [-394, 0, -394, 0], [0, -2, -1, 0], [-6, 12, 0, -12], [20, 0, 20, 40], [331, 0, 331, 0], [68, 0, 68, 136], [0, 0, 8, 0], [30, -60, 0, 60], [69, -138, 0, 138], [0, 0, 215, 0], [0, 0, 190, 0], [-12, 24, 0, -24], [0, 26, 13, 0], [0, -38, -19, 0], [-83, 0, -83, -166], [193, 0, 0, 0], [0, 148, 74, 0], [48, -96, 0, 96], [29, 0, 29, 58], [-290, 0, -290, 0], [220, 0, 0, 0], [-1, 0, -1, -2], [-30, 60, 0, -60], [-54, 108, 0, -108], [704, 0, 0, 0], [0, 20, 10, 0], [-101, 0, 0, 0], [935, 0, 935, 0], [0, -152, -76, 0], [-506, 0, -506, 0], [-108, 216, 0, -216], [0, 0, -125, 0], [-50, 0, 0, 0], [14, 0, 14, 28], [0, 0, 53, 0], [40, 0, 40, 80], [-90, 180, 0, -180], [-1132, 0, -1132, 0], [0, 0, 302, 0], [1103, 0, 0, 0], [0, -80, -40, 0], [28, 0, 28, 56], [-1256, 0, 0, 0], [-7, 0, -7, 0], [0, -194, -97, 0], [640, 0, 640, 0], [0, 62, 31, 0], [0, 0, 22, 0], [166, 0, 0, 0], [0, 0, -460, 0], [27, -54, 0, 54], [0, 0, -544, 0], [36, -72, 0, 72], [0, 278, 139, 0], [64, 0, 64, 128], [-958, 0, 0, 0], [602, 0, 602, 0], [-533, 0, -533, 0], [-90, 180, 0, -180], [-100, 0, -100, -200], [-605, 0, -605, 0], [886, 0, 0, 0], [52, 0, 52, 104], [335, 0, 335, 0], [-42, 84, 0, -84], [-700, 0, 0, 0], [0, -224, -112, 0], [-95, 0, -95, -190], [-230, 0, 0, 0], [686, 0, 0, 0], [0, 244, 122, 0], [0, -148, -74, 0], [0, 0, 154, 0], [1132, 0, 1132, 0], [-1040, 0, 0, 0], [166, 0, 166, 332], [42, -84, 0, 84], [24, -48, 0, 48], [614, 0, 614, 0], [0, 80, 40, 0], [0, 292, 146, 0], [-187, 0, 0, 0], [0, 136, 68, 0], [99, -198, 0, 198], [-31, 0, -31, -62], [0, 0, -1385, 0], [-1496, 0, 0, 0], [7, 0, 7, 14], [-75, 150, 0, -150], [-1054, 0, -1054, 0], [0, 0, -1394, 0], [-265, 0, 0, 0], [0, 92, 46, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_405_h_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_405_3_h_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_405_3_h_c(:prec:=4)
    chi := MakeCharacter_405_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_405_3_h_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_405_3_h_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_405_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![1, 1, 1]>,<7,R![9801, 0, -99, 0, 1]>],Snew);
    return Vf;
end function;