
// Make newform 2304.3.b.l in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_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_2304_3_b_l();".
// 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_b_l();".
// 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, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 2], [0, 8, 0, -4], [-4, 0, 8, 0]];
    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_2304_b();"
function MakeCharacter_2304_b()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [1, 1, 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_2304_b_Hecke();"
function MakeCharacter_2304_b_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, 1, 0, 0], [0, 0, 0, -1], [0, 0, -1, 0], [0, -1, 0, 0], [-10, 0, 0, 0], [0, 0, -3, 0], [0, 0, 0, 4], [0, -13, 0, 0], [0, 0, 0, -1], [0, 13, 0, 0], [58, 0, 0, 0], [0, 0, -7, 0], [0, 0, 0, 10], [0, 37, 0, 0], [0, 0, 13, 0], [0, -13, 0, 0], [0, 0, -1, 0], [0, 0, 0, 0], [46, 0, 0, 0], [0, 0, 0, -17], [0, 0, 7, 0], [82, 0, 0, 0], [2, 0, 0, 0], [0, 37, 0, 0], [0, 0, 0, -11], [0, 0, -3, 0], [0, 23, 0, 0], [110, 0, 0, 0], [0, 0, 0, 21], [0, 0, 17, 0], [10, 0, 0, 0], [0, 0, 7, 0], [0, 1, 0, 0], [0, 0, 0, -13], [0, 107, 0, 0], [0, 0, -3, 0], [0, 0, 0, -14], [0, 167, 0, 0], [0, 0, -27, 0], [0, 1, 0, 0], [0, 0, 0, -32], [290, 0, 0, 0], [0, 13, 0, 0], [0, 0, 0, 57], [0, 0, 35, 0], [0, 0, 0, 49], [0, 0, -41, 0], [0, -71, 0, 0], [82, 0, 0, 0], [0, 0, 0, 56], [-46, 0, 0, 0], [0, 0, -21, 0], [254, 0, 0, 0], [0, 0, 0, -22], [0, 131, 0, 0], [0, 0, 0, -3], [0, 145, 0, 0], [226, 0, 0, 0], [0, 0, 43, 0], [0, 181, 0, 0], [0, 0, -21, 0], [0, 0, 0, 34], [478, 0, 0, 0], [0, -85, 0, 0], [0, 0, 59, 0], [338, 0, 0, 0], [0, 0, -29, 0], [0, -253, 0, 0], [-178, 0, 0, 0], [0, 0, 0, -24], [0, 0, 0, -29], [0, -155, 0, 0], [0, 0, -63, 0], [0, 0, 0, -88], [0, 289, 0, 0], [0, -13, 0, 0], [-250, 0, 0, 0], [-290, 0, 0, 0], [0, 0, -49, 0], [0, 337, 0, 0], [0, 0, 0, 78], [-334, 0, 0, 0], [0, 0, 0, 17], [0, 0, 11, 0], [-394, 0, 0, 0], [478, 0, 0, 0], [0, 71, 0, 0], [0, 0, 0, -91], [0, 0, 3, 0], [0, 0, 0, 106], [0, 0, 0, 15], [0, 0, 133, 0], [0, 0, 11, 0], [0, 0, 0, 84], [0, -421, 0, 0], [-326, 0, 0, 0], [0, 0, 45, 0], [0, -265, 0, 0], [0, 0, 49, 0], [0, 383, 0, 0], [0, 0, 71, 0], [-422, 0, 0, 0], [0, 0, -41, 0], [-46, 0, 0, 0], [0, 0, -91, 0], [-82, 0, 0, 0], [0, 0, 0, -8], [334, 0, 0, 0], [0, 0, 0, 53], [0, -107, 0, 0], [-1118, 0, 0, 0], [0, 0, -97, 0], [0, 0, 0, -21], [-10, 0, 0, 0], [0, 0, -175, 0], [0, 0, 0, 48], [0, 335, 0, 0], [0, 0, -119, 0], [0, -611, 0, 0], [-334, 0, 0, 0], [0, -503, 0, 0], [0, 0, 27, 0], [0, 0, -143, 0], [0, -517, 0, 0], [0, 265, 0, 0], [0, 0, 0, -102], [0, 0, 0, 35], [0, -97, 0, 0], [0, 0, 195, 0], [0, 0, 0, -98], [0, 0, 0, 95], [0, -503, 0, 0], [-758, 0, 0, 0], [2, 0, 0, 0], [0, -131, 0, 0], [0, 0, 209, 0], [0, -433, 0, 0], [10, 0, 0, 0], [0, 0, -63, 0], [0, -419, 0, 0], [0, 0, 0, 127], [0, 0, 105, 0], [0, -649, 0, 0], [0, 0, 0, -28], [0, 253, 0, 0], [-998, 0, 0, 0], [0, 0, 73, 0], [0, 0, 0, 24], [0, 323, 0, 0], [-898, 0, 0, 0], [0, 0, 105, 0], [0, 0, 0, -122], [0, 0, 197, 0], [0, 0, 0, -56], [0, 0, 0, 87], [-1594, 0, 0, 0], [-674, 0, 0, 0], [0, 215, 0, 0], [0, 0, -11, 0], [730, 0, 0, 0], [0, 0, 0, 133], [0, 0, -213, 0], [-346, 0, 0, 0], [0, 0, 0, -106], [0, 0, 0, -141], [0, 229, 0, 0], [-1342, 0, 0, 0], [0, 169, 0, 0], [0, 0, -127, 0], [0, -757, 0, 0], [0, 0, 0, 214], [1054, 0, 0, 0], [0, 0, 0, 251], [1570, 0, 0, 0], [0, 0, -133, 0], [0, 625, 0, 0], [0, 0, 0, -147], [0, 491, 0, 0], [0, 0, 0, 67], [0, 0, 259, 0], [0, 505, 0, 0], [-254, 0, 0, 0], [0, 0, 0, -76], [0, 121, 0, 0], [0, -481, 0, 0], [0, 0, 223, 0], [814, 0, 0, 0], [0, 0, 0, 288], [-478, 0, 0, 0], [0, 0, -7, 0], [0, 0, -45, 0], [0, -337, 0, 0], [0, 0, -279, 0], [-158, 0, 0, 0], [-718, 0, 0, 0], [0, 575, 0, 0], [1790, 0, 0, 0], [0, 0, 0, -66], [0, 215, 0, 0], [0, 0, 0, -203], [0, 97, 0, 0], [1970, 0, 0, 0], [0, 0, 147, 0], [0, 971, 0, 0], [0, 0, 0, -3], [0, 0, 59, 0], [1906, 0, 0, 0], [0, 0, -251, 0], [866, 0, 0, 0], [0, -71, 0, 0], [0, 0, 0, 161], [0, 0, -115, 0], [0, 0, 0, 76], [142, 0, 0, 0], [0, 0, 0, -1], [2006, 0, 0, 0], [0, 0, 0, -66], [0, 71, 0, 0], [0, 433, 0, 0], [0, 0, 0, -119], [1766, 0, 0, 0], [0, 0, 0, -3], [0, 0, 119, 0], [0, 589, 0, 0], [754, 0, 0, 0], [0, 0, 0, -342], [0, 0, 0, 17], [0, 0, 389, 0], [0, -649, 0, 0], [0, 0, -267, 0], [0, 0, 0, -175], [1186, 0, 0, 0], [0, 0, 245, 0], [0, 0, 0, 22], [-1150, 0, 0, 0], [0, 1, 0, 0], [0, 0, -161, 0], [0, 0, 0, 168], [0, 0, -41, 0], [0, 0, -81, 0], [0, 0, 0, -41], [0, -181, 0, 0], [446, 0, 0, 0], [0, 0, 0, 368], [0, 0, 0, 231], [0, 0, 197, 0], [0, 0, 211, 0], [0, 0, 0, 168], [0, -505, 0, 0], [1934, 0, 0, 0], [0, 0, 0, -344], [1342, 0, 0, 0], [0, 1139, 0, 0], [0, 0, -39, 0], [0, -527, 0, 0], [0, 0, -97, 0], [0, 13, 0, 0], [-1874, 0, 0, 0], [0, 0, 0, 181], [0, 0, -287, 0], [0, 1153, 0, 0], [0, -757, 0, 0], [1670, 0, 0, 0], [0, 0, 67, 0], [0, -769, 0, 0], [1234, 0, 0, 0], [0, 0, -203, 0], [0, 1297, 0, 0], [0, -685, 0, 0], [0, 0, 279, 0], [1726, 0, 0, 0], [0, 0, 0, 179], [-2014, 0, 0, 0], [0, 0, 0, 465], [0, 0, -353, 0], [0, 1283, 0, 0], [-530, 0, 0, 0], [0, 0, -263, 0], [0, 0, 0, -118], [0, 0, 0, 253], [0, 0, 0, -280], [0, 1093, 0, 0], [0, 0, 409, 0], [0, 0, 0, 360], [-2062, 0, 0, 0], [0, 601, 0, 0], [0, 0, 0, -181], [-3250, 0, 0, 0], [0, -1297, 0, 0], [0, 0, -195, 0], [754, 0, 0, 0], [0, 0, 217, 0], [0, 923, 0, 0], [0, 215, 0, 0], [0, 0, 0, 109], [0, 1525, 0, 0], [0, 0, 39, 0], [0, 0, 133, 0], [3166, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 531]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_b_l();".
// 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_b_l(:prec:=4)
    chi := MakeCharacter_2304_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(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_b_l();".
// 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_b_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![4, 0, 1]>,<7,R![48, 0, 1]>,<11,R![-48, 0, 1]>,<17,R![10, 1]>,<19,R![-432, 0, 1]>],Snew);
    return Vf;
end function;