
// Make newform 2304.3.e.f in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_e_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_2304_3_e_f();".
// 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_2304_3_e_f();".
// 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, 0, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -1, 0, 1], [0, 28, 0, -4], [-16, 0, 8, 0]];
    Rf_basisdens := [1, 3, 3, 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_2304_e();"
function MakeCharacter_2304_e()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [2, 2, 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_2304_e_Hecke();"
function MakeCharacter_2304_e_Hecke(Kf)
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    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, 0], [0, 3, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [-10, 0, 0, 0], [0, -17, 0, 0], [0, 0, 2, 0], [0, 0, 0, -1], [0, 11, 0, 0], [0, 0, 1, 0], [64, 0, 0, 0], [0, 9, 0, 0], [0, 0, 4, 0], [0, 0, 0, -1], [0, -13, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -6, 0], [0, 0, 0, 7], [-96, 0, 0, 0], [0, 0, -5, 0], [0, 0, 0, -7], [0, 39, 0, 0], [64, 0, 0, 0], [0, 29, 0, 0], [0, 0, 11, 0], [0, 0, 0, -8], [-118, 0, 0, 0], [0, -17, 0, 0], [0, 0, 5, 0], [0, 0, 0, -6], [0, 87, 0, 0], [0, 0, -10, 0], [0, 141, 0, 0], [0, 0, -3, 0], [256, 0, 0, 0], [0, 0, -8, 0], [0, 0, 0, 12], [0, 123, 0, 0], [0, 0, 0, 14], [218, 0, 0, 0], [0, 0, 0, -4], [-30, 0, 0, 0], [0, 93, 0, 0], [0, 0, -1, 0], [0, 0, -10, 0], [0, 0, 1, 0], [0, 0, 0, 11], [-58, 0, 0, 0], [0, -247, 0, 0], [0, 0, 0, 18], [160, 0, 0, 0], [0, 0, 0, 3], [0, -127, 0, 0], [0, 0, 0, -14], [0, 261, 0, 0], [0, 0, -27, 0], [-230, 0, 0, 0], [0, -249, 0, 0], [0, 0, -32, 0], [0, 3, 0, 0], [0, 0, -30, 0], [0, 0, 0, -2], [430, 0, 0, 0], [0, -203, 0, 0], [0, 0, -2, 0], [224, 0, 0, 0], [0, 0, 0, -21], [320, 0, 0, 0], [0, 127, 0, 0], [0, 0, 0, -27], [0, 0, 57, 0], [256, 0, 0, 0], [0, 0, -2, 0], [0, 0, 0, -38], [0, 541, 0, 0], [-64, 0, 0, 0], [0, 239, 0, 0], [640, 0, 0, 0], [0, 0, 0, 13], [-518, 0, 0, 0], [0, 0, 0, 39], [290, 0, 0, 0], [0, 0, -3, 0], [0, 0, 0, -39], [0, -511, 0, 0], [-64, 0, 0, 0], [0, -411, 0, 0], [0, 0, -5, 0], [0, 0, 0, -9], [0, 0, 0, 17], [0, 0, -61, 0], [0, 0, 0, -6], [0, 0, 56, 0], [0, 0, 0, -27], [0, 171, 0, 0], [0, -201, 0, 0], [0, 0, -22, 0], [-662, 0, 0, 0], [0, 0, -12, 0], [0, -283, 0, 0], [0, 0, 0, -15], [0, -409, 0, 0], [0, 0, 4, 0], [-30, 0, 0, 0], [0, 0, 0, 16], [0, 177, 0, 0], [0, 0, 0, 27], [-658, 0, 0, 0], [0, 0, 13, 0], [-576, 0, 0, 0], [0, -137, 0, 0], [0, 0, 44, 0], [0, 0, -41, 0], [0, -609, 0, 0], [0, 0, -24, 0], [0, 0, 0, -15], [0, 347, 0, 0], [0, 0, 0, 36], [640, 0, 0, 0], [830, 0, 0, 0], [0, 163, 0, 0], [0, 0, 0, -59], [0, 0, -64, 0], [0, 139, 0, 0], [-698, 0, 0, 0], [0, 0, 0, -69], [0, 0, -37, 0], [-1130, 0, 0, 0], [0, 0, 4, 0], [0, 0, 0, 8], [0, 0, 61, 0], [-90, 0, 0, 0], [0, 761, 0, 0], [-482, 0, 0, 0], [0, -893, 0, 0], [0, 0, -42, 0], [0, 427, 0, 0], [0, -119, 0, 0], [0, 0, -50, 0], [0, -339, 0, 0], [0, 0, -59, 0], [0, 0, 0, -38], [-598, 0, 0, 0], [0, 0, 0, 69], [256, 0, 0, 0], [0, -263, 0, 0], [0, 0, 68, 0], [0, 0, 0, 32], [704, 0, 0, 0], [0, 49, 0, 0], [0, 0, -44, 0], [0, 0, 0, -90], [0, 0, -140, 0], [0, 0, 0, 48], [0, 0, -3, 0], [0, 671, 0, 0], [-1810, 0, 0, 0], [0, 389, 0, 0], [0, 0, 0, -12], [0, -57, 0, 0], [0, 0, 27, 0], [0, 0, 0, -67], [0, -303, 0, 0], [0, 0, 0, -82], [0, 0, 77, 0], [-896, 0, 0, 0], [0, 0, 0, 0], [0, 243, 0, 0], [0, 0, 0, 82], [960, 0, 0, 0], [0, 0, 0, 42], [-704, 0, 0, 0], [0, 0, 127, 0], [0, 199, 0, 0], [0, 0, 20, 0], [0, 29, 0, 0], [0, 0, -7, 0], [640, 0, 0, 0], [0, 0, 55, 0], [0, 0, 0, 35], [1350, 0, 0, 0], [0, 1047, 0, 0], [0, 0, 0, -30], [0, 1171, 0, 0], [-810, 0, 0, 0], [0, 0, -68, 0], [800, 0, 0, 0], [0, 0, 0, -114], [-1984, 0, 0, 0], [0, 0, 0, -10], [0, 0, -76, 0], [0, -1419, 0, 0], [0, 0, 0, -42], [0, -183, 0, 0], [578, 0, 0, 0], [470, 0, 0, 0], [0, -993, 0, 0], [0, 0, 0, 20], [0, -1029, 0, 0], [0, 0, 109, 0], [410, 0, 0, 0], [-480, 0, 0, 0], [0, 0, 0, -45], [0, -747, 0, 0], [0, 0, 107, 0], [0, 0, 0, -53], [0, -1559, 0, 0], [0, 0, -76, 0], [-2050, 0, 0, 0], [0, -141, 0, 0], [0, 0, -75, 0], [0, 0, 0, -42], [0, 0, 0, -27], [18, 0, 0, 0], [0, 0, -161, 0], [0, 561, 0, 0], [0, 0, 0, -6], [0, 1387, 0, 0], [442, 0, 0, 0], [0, 0, -15, 0], [0, -1121, 0, 0], [0, 0, -41, 0], [0, 0, 0, -137], [-1600, 0, 0, 0], [0, -423, 0, 0], [0, 0, 0, -41], [0, 0, -7, 0], [0, 0, 0, -76], [970, 0, 0, 0], [0, 0, -32, 0], [0, 0, -17, 0], [0, 489, 0, 0], [0, 0, 184, 0], [0, 0, 0, -37], [-1600, 0, 0, 0], [0, 1037, 0, 0], [0, 0, 0, -125], [0, 0, 0, -123], [0, 0, 0, -59], [0, 0, -66, 0], [0, 0, -153, 0], [1280, 0, 0, 0], [0, 113, 0, 0], [0, 0, 0, 3], [0, 0, 95, 0], [0, 0, 0, 54], [0, 0, 146, 0], [0, 0, 0, -94], [2730, 0, 0, 0], [0, -479, 0, 0], [0, 0, 0, -39], [0, 0, 0, 0], [0, 1893, 0, 0], [0, 0, 0, -62], [858, 0, 0, 0], [0, 0, 80, 0], [0, 3, 0, 0], [-910, 0, 0, 0], [0, 0, -121, 0], [0, 0, 0, 66], [378, 0, 0, 0], [-1664, 0, 0, 0], [0, 513, 0, 0], [0, 0, 178, 0], [0, -549, 0, 0], [0, 1881, 0, 0], [0, 0, -26, 0], [0, -707, 0, 0], [3200, 0, 0, 0], [0, 0, 212, 0], [1024, 0, 0, 0], [0, 0, 21, 0], [1856, 0, 0, 0], [0, 0, 81, 0], [0, 0, 0, 45], [-320, 0, 0, 0], [1138, 0, 0, 0], [0, 0, 0, 120], [0, 0, 0, -33], [0, 0, 217, 0], [0, 0, 0, 201], [960, 0, 0, 0], [0, 0, -270, 0], [0, 0, 0, 76], [2336, 0, 0, 0], [0, 1677, 0, 0], [0, 0, -147, 0], [0, 1791, 0, 0], [0, -901, 0, 0], [0, 0, 0, -88], [0, -313, 0, 0], [0, 0, 0, 78], [-384, 0, 0, 0], [0, 1141, 0, 0], [0, 0, -175, 0], [0, -723, 0, 0], [0, 0, 0, -195], [0, 0, 254, 0], [-1550, 0, 0, 0], [0, -1563, 0, 0], [0, 0, -217, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_e_f();".
// 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_2304_3_e_f(:prec:=4)
    chi := MakeCharacter_2304_e();
    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_2304_3_e_f();".
// 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_2304_3_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![18, 0, 1]>,<7,R![-160, 0, 1]>,<13,R![10, 1]>,<19,R![-640, 0, 1]>,<31,R![-160, 0, 1]>],Snew);
    return Vf;
end function;