
// Make newform 320.3.b.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_320_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_320_b_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_320_3_b_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_320_3_b_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 := [1, -1, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 4, 4], [1, 0, 2, -2], [-2, 4, -2, 2]];
    Rf_basisdens := [1, 1, 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_320_b();"
function MakeCharacter_320_b()
    N := 320;
    order := 2;
    char_gens := [191, 261, 257];
    v := [1, 2, 2];
    // 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_320_b_Hecke();"
function MakeCharacter_320_b_Hecke(Kf)
    N := 320;
    order := 2;
    char_gens := [191, 261, 257];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [1, 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, 1, 0], [0, -1, 0, 1], [0, -2, 0, -2], [4, 0, 2, 0], [-6, 0, 8, 0], [0, 2, 0, 0], [0, -1, 0, 3], [2, 0, 4, 0], [0, 2, 0, -10], [-4, 0, 10, 0], [-28, 0, 6, 0], [0, -2, 0, -3], [0, 3, 0, 13], [44, 0, 10, 0], [0, -6, 0, 12], [-32, 0, -26, 0], [0, -10, 0, 11], [0, -10, 0, -2], [66, 0, -32, 0], [0, -8, 0, 20], [0, 6, 0, 13], [-22, 0, -40, 0], [-66, 0, -12, 0], [-82, 0, 8, 0], [0, 11, 0, 23], [0, 12, 0, -17], [32, 0, -74, 0], [126, 0, 20, 0], [0, -19, 0, -23], [0, -14, 0, 42], [-26, 0, 12, 0], [0, 30, 0, 4], [-48, 0, 38, 0], [0, 18, 0, 62], [-84, 0, -54, 0], [0, -28, 0, 45], [0, 9, 0, -15], [204, 0, 10, 0], [0, -6, 0, -76], [-122, 0, 16, 0], [0, -22, 0, 58], [146, 0, -16, 0], [16, 0, 70, 0], [0, -16, 0, -64], [0, 14, 0, 42], [0, 35, 0, -13], [0, 8, 0, -1], [-118, 0, -20, 0], [6, 0, 140, 0], [0, 20, 0, 36], [-68, 0, 30, 0], [0, 26, 0, -26], [-266, 0, -4, 0], [0, 29, 0, 79], [192, 0, 70, 0], [0, -6, 0, -86], [256, 0, -90, 0], [-28, 0, -10, 0], [0, 24, 0, -15], [-76, 0, -46, 0], [0, 34, 0, 11], [0, 34, 0, 10], [-174, 0, -176, 0], [-264, 0, 46, 0], [0, -22, 0, -70], [-126, 0, -64, 0], [0, 38, 0, 91], [242, 0, -108, 0], [-14, 0, -144, 0], [0, 8, 0, 16], [0, 17, 0, 157], [-256, 0, -154, 0], [0, -30, 0, -32], [0, -57, 0, 43], [-48, 0, -154, 0], [-124, 0, -94, 0], [282, 0, -72, 0], [-272, 0, -150, 0], [0, 70, 0, -28], [328, 0, -114, 0], [0, -22, 0, -126], [-234, 0, 60, 0], [0, 64, 0, -48], [0, 44, 0, -19], [528, 0, -102, 0], [554, 0, 120, 0], [278, 0, 80, 0], [0, 1, 0, 167], [0, -116, 0, -13], [0, -16, 0, -172], [0, 79, 0, -67], [0, 26, 0, 38], [0, -114, 0, 24], [0, 107, 0, -73], [602, 0, 140, 0], [202, 0, 360, 0], [0, 52, 0, 161], [438, 0, -80, 0], [0, 20, 0, -1], [-704, 0, 86, 0], [0, -92, 0, 89], [-112, 0, -54, 0], [0, -10, 0, 122], [-186, 0, 236, 0], [0, -34, 0, -153], [-94, 0, 96, 0], [0, -108, 0, -12], [172, 0, -82, 0], [0, -69, 0, -251], [-656, 0, 70, 0], [54, 0, 220, 0], [0, 2, 0, -200], [0, -26, 0, 198], [212, 0, -282, 0], [0, 26, 0, -271], [0, 19, 0, 117], [464, 0, -42, 0], [0, 14, 0, 92], [48, 0, 342, 0], [-114, 0, 260, 0], [136, 0, -306, 0], [0, -22, 0, 61], [0, -2, 0, 86], [568, 0, 62, 0], [-518, 0, 316, 0], [0, 44, 0, -76], [0, -77, 0, -55], [-236, 0, -446, 0], [0, 10, 0, -36], [0, -67, 0, -213], [0, -58, 0, -282], [576, 0, 134, 0], [122, 0, -344, 0], [-102, 0, 328, 0], [104, 0, 142, 0], [0, -58, 0, -169], [-744, 0, -210, 0], [218, 0, -200, 0], [0, -22, 0, -122], [-232, 0, -402, 0], [0, 67, 0, 115], [0, -110, 0, 267], [352, 0, 70, 0], [0, -104, 0, 220], [264, 0, 590, 0], [-426, 0, -260, 0], [0, -6, 0, 52], [0, -79, 0, 147], [-284, 0, -142, 0], [-348, 0, 262, 0], [0, 32, 0, -103], [0, -9, 0, -199], [0, -52, 0, 27], [0, -54, 0, -442], [0, 44, 0, 204], [968, 0, -254, 0], [294, 0, -20, 0], [-1082, 0, 160, 0], [0, 70, 0, 259], [866, 0, 192, 0], [0, -91, 0, 125], [0, 198, 0, -246], [-1226, 0, -68, 0], [0, 61, 0, -65], [0, 122, 0, 14], [316, 0, 346, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_320_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_320_3_b_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_320_3_b_c(:prec:=4)
    chi := MakeCharacter_320_b();
    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_320_3_b_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_320_3_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_320_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![80, 0, 20, 0, 1]>],Snew);
    return Vf;
end function;