
// Make newform 3456.2.i.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3456_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3456_i_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_3456_2_i_b();".
// 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_3456_2_i_b();".
// 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];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_3456_i();"
function MakeCharacter_3456_i()
    N := 3456;
    order := 3;
    char_gens := [2431, 2053, 2945];
    v := [3, 3, 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_3456_i_Hecke();"
function MakeCharacter_3456_i_Hecke(Kf)
    N := 3456;
    order := 3;
    char_gens := [2431, 2053, 2945];
    char_values := [[1, 0], [1, 0], [0, -1]];
    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 := 2;
    raw_aps := [[0, 0], [0, 0], [0, -2], [2, -2], [-5, 5], [0, -4], [-1, 0], [5, 0], [0, -4], [-6, 6], [0, 0], [-10, 0], [0, -3], [-9, 9], [8, -8], [12, 0], [0, -7], [-4, 4], [0, -7], [-6, 0], [-13, 0], [-2, 2], [-12, 12], [-10, 0], [-13, 13], [-6, 6], [0, 8], [11, 0], [-12, 0], [0, 14], [20, 0], [0, 12], [1, -1], [0, 1], [0, 0], [16, -16], [0, 24], [-4, 0], [0, 2], [-18, 18], [-12, 0], [-20, 0], [-2, 2], [0, -1], [-20, 0], [-2, 0], [0, -20], [-24, 24], [3, -3], [0, -14], [9, 0], [0, -12], [-25, 25], [-3, 0], [0, 3], [18, -18], [4, 0], [-8, 0], [0, 0], [30, -30], [0, -24], [0, -10], [3, 0], [0, 10], [-25, 25], [-12, 12], [-20, 20], [0, -19], [0, 3], [16, -16], [-9, 9], [-34, 0], [22, -22], [0, -32], [25, 0], [0, 6], [38, -38], [-18, 0], [0, -3], [0, -25], [0, 20], [-2, 2], [10, 0], [33, 0], [-24, 24], [-15, 15], [-17, 0], [3, -3], [20, -20], [0, -2], [29, 0], [-18, 18], [-10, 0], [0, -25], [0, -5], [-4, 0], [0, -28], [7, 0], [44, 0], [2, 0], [-1, 1], [26, 0], [0, -33], [-19, 19], [0, 19], [23, 0], [5, -5], [-6, 0], [0, 42], [9, -9], [0, 8], [26, 0], [0, -17], [35, -35], [-22, 0], [33, -33], [0, -21], [-42, 0], [0, 18], [24, -24], [0, -2], [-10, 10], [28, -28], [15, 0], [-12, 12], [-16, 0], [20, -20], [-30, 0], [48, -48], [0, 18], [21, 0], [0, -38], [0, 2], [-14, 0], [0, 10], [0, -2], [-24, 0], [0, 28], [0, -30], [3, 0], [-9, 0], [14, -14], [0, -34], [44, 0], [2, 0], [-48, 48], [8, -8], [18, -18], [0, -13], [22, 0], [0, -16], [-14, 0], [7, 0], [0, 30], [-11, 11], [-12, 12], [36, 0], [-6, 6], [-26, 0], [0, -28], [-23, 23], [-1, 0], [0, -14], [28, 0], [0, 57], [46, -46], [-40, 0], [0, 0], [-22, 0], [0, -28], [3, -3], [0, -30], [0, 8], [13, -13], [0, -38], [0, 6], [-33, 33], [-58, 0], [-14, 0], [-10, 10], [-38, 38], [-3, 3], [0, -6], [-19, 0], [0, 6], [-24, 24], [16, 0], [28, -28], [0, 47], [10, 0], [-41, 0], [3, -3], [-11, 0], [-20, 20], [-19, 0], [0, -1], [0, -11], [16, -16], [-37, 37], [-60, 0], [0, -38], [-16, 16], [0, 40], [65, -65], [-27, 0], [48, 0], [14, 0], [0, -41], [-13, 13], [0, 5], [-31, 0], [0, -28], [22, -22], [-20, 20], [0, 48], [-25, 25], [0, -12], [-59, 59], [-60, 0], [0, 0], [0, -52], [0, -6], [0, -15], [30, 0], [0, 61], [-58, 58], [-34, 34], [14, 0], [-26, 26], [52, -52], [0, -22], [-16, 0], [0, 50], [0, 67], [40, -40], [2, -2], [0, -14], [-22, 0], [0, -75], [-48, 0], [45, -45], [8, 0], [0, 70], [-52, 0], [0, 19], [44, -44], [28, 0], [0, 27], [0, -4], [24, 0], [0, -66], [17, 0], [0, -64], [50, -50], [-16, 16], [-35, 0], [28, 0], [-60, 60], [18, 0], [33, 0], [44, -44], [-25, 25], [0, -28], [-26, 0], [0, -30], [45, -45], [-68, 0], [-45, 45], [0, 41], [0, 34], [0, 44], [44, 0], [-55, 55], [0, 62], [0, 9], [4, 0], [0, 68], [-30, 30], [-45, 0], [15, -15], [0, -2], [0, 32], [42, -42], [-44, 44], [0, 35], [52, 0], [-38, 0], [0, 54], [28, -28], [-55, 0], [74, -74], [-57, 0], [0, -57], [0, 33], [82, -82], [0, -60], [4, -4], [-10, 10], [-24, 0], [-76, 76], [0, -29], [-24, 0], [-18, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3456_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3456_2_i_b();".
// 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_3456_2_i_b(:prec:=2)
    chi := MakeCharacter_3456_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_3456_2_i_b();".
// 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_3456_2_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3456_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 2, 1]>,<7,R![4, -2, 1]>],Snew);
    return Vf;
end function;