
// Make newform 1152.3.b.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1152_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1152_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_1152_3_b_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_1152_3_b_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, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 2, 0], [0, 12, 0, 4], [0, 24, 0, -8]];
    Rf_basisdens := [1, 3, 3, 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_1152_b();"
function MakeCharacter_1152_b()
    N := 1152;
    order := 2;
    char_gens := [127, 901, 641];
    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_1152_b_Hecke();"
function MakeCharacter_1152_b_Hecke(Kf)
    N := 1152;
    order := 2;
    char_gens := [127, 901, 641];
    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, 1, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 25, 0, 0], [0, 0, -5, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 47, 0, 0], [0, 0, 0, -6], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [50, 0, 0, 0], [0, 0, 15, 0], [0, 0, 0, 5], [0, 0, 0, 0], [-190, 0, 0, 0], [0, 95, 0, 0], [0, 0, -21, 0], [0, 0, 0, 10], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -11, 0], [0, 0, 0, -4], [0, 0, 0, 0], [0, 0, 0, 0], [0, 145, 0, 0], [0, 0, 5, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 23, 0, 0], [0, 0, 0, -14], [0, 0, 0, 0], [0, 0, 0, 0], [-290, 0, 0, 0], [0, -97, 0, 0], [0, 0, 35, 0], [0, 0, 0, 0], [0, 0, -39, 0], [0, 0, 0, 15], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-382, 0, 0, 0], [0, 0, 0, 9], [0, 0, 0, 0], [0, 0, 0, 0], [0, -215, 0, 0], [0, 0, -55, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -193, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [530, 0, 0, 0], [0, -167, 0, 0], [0, 0, 0, 0], [190, 0, 0, 0], [0, 0, 0, 35], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 19, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -95, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [718, 0, 0, 0], [0, 0, 0, -21], [0, 0, 0, 0], [0, 0, 0, 0], [670, 0, 0, 0], [0, 0, -85, 0], [0, 0, 0, -45], [0, 0, 0, 0], [-530, 0, 0, 0], [0, -265, 0, 0], [0, 0, -91, 0], [0, 0, 0, 35], [0, 0, 0, 0], [0, 0, 9, 0], [0, 0, 0, 44], [0, 0, 0, 0], [0, 0, 0, 0], [0, 505, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -457, 0, 0], [0, 0, 0, 55], [0, 0, 0, 0], [0, 0, 0, 0], [290, 0, 0, 0], [0, 0, 0, 40], [0, 0, 0, 0], [0, 0, 0, 0], [1198, 0, 0, 0], [0, 0, -99, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 25, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 503, 0, 0], [0, 0, 0, -56], [0, 0, 0, 0], [190, 0, 0, 0], [0, 673, 0, 0], [0, 0, 0, -15], [0, 0, 0, 0], [0, 25, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -11, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 55, 0], [0, 0, 0, 0], [0, 0, 0, 0], [862, 0, 0, 0], [0, -577, 0, 0], [0, 0, 0, 0], [0, 647, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 335, 0, 0], [0, 0, 161, 0], [0, 0, 0, 30], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -175, 0], [0, 0, 0, 0], [-1490, 0, 0, 0], [0, 215, 0, 0], [0, 0, 0, 70], [0, 0, 0, 0], [0, 0, 31, 0], [0, 0, 0, -11], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -145, 0], [0, 0, 0, 0], [482, 0, 0, 0], [0, 337, 0, 0], [0, 0, 0, 104], [0, 0, 0, 0], [0, 0, 0, 0], [-1390, 0, 0, 0], [0, 0, -65, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1055, 0, 0], [0, 0, 91, 0], [0, 0, 0, 0], [0, 0, -209, 0], [0, 0, 0, -19], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 625, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [142, 0, 0, 0], [0, 0, 0, 0], [1150, 0, 0, 0], [0, 0, 0, 110], [0, 0, 0, 0], [0, -985, 0, 0], [0, 0, 0, -120], [0, 0, 0, 0], [-2302, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 215, 0, 0], [0, 0, -35, 0], [0, 0, 0, 0], [-2402, 0, 0, 0], [0, 0, 0, 119], [0, -1223, 0, 0], [0, 0, 105, 0], [0, 0, 0, 95], [0, 0, 0, 0], [0, 0, 0, 0], [2110, 0, 0, 0], [0, -815, 0, 0], [0, 0, -221, 0], [0, 0, 0, -130], [0, 0, 0, 0], [2258, 0, 0, 0], [0, 0, -261, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1273, 0, 0], [0, 0, 0, 0], [0, 0, -105, 0], [0, 0, 0, 0], [0, 0, -111, 0], [0, 0, 0, 65], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 279, 0], [0, 0, 0, -84], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 165, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-478, 0, 0, 0], [0, -1007, 0, 0], [0, 0, 0, -91], [0, 0, 0, 0], [0, 0, 0, -105], [0, 0, 0, 0], [0, 0, -299, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 319, 0], [0, 0, 0, 14], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1682, 0, 0, 0], [0, -887, 0, 0], [0, 0, 0, -46], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1537, 0, 0], [-2830, 0, 0, 0], [0, 0, -259, 0], [0, 0, 0, -170], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1655, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -383, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1970, 0, 0, 0], [0, 0, -85, 0], [-3170, 0, 0, 0], [0, 0, 119, 0], [0, 0, 0, 115], [0, 0, 0, 0], [1102, 0, 0, 0], [0, 0, 0, 126], [0, 0, 0, 0], [0, 0, -205, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [3650, 0, 0, 0], [0, 1873, 0, 0], [0, 0, 345, 0], [0, 0, 0, 0], [0, -745, 0, 0], [0, 0, 0, 50], [0, 0, 0, 0], [0, 0, 0, -154], [0, 0, 0, 0], [0, 1895, 0, 0], [0, 0, 385, 0], [0, -1777, 0, 0], [0, 0, 0, -161], [0, 0, 0, 0], [-3790, 0, 0, 0], [0, 1847, 0, 0], [0, 0, 215, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1152_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1152_3_b_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_1152_3_b_f(:prec:=4)
    chi := MakeCharacter_1152_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_1152_3_b_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_1152_3_b_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1152_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![4, 0, 1]>,<7,R![96, 0, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;