
// Make newform 2520.2.bi.l in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2520_bi();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2520_bi_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_2520_2_bi_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_2520_2_bi_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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_2520_bi();"
function MakeCharacter_2520_bi()
    N := 2520;
    order := 3;
    char_gens := [631, 1261, 281, 2017, 1081];
    v := [3, 3, 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_2520_bi_Hecke();"
function MakeCharacter_2520_bi_Hecke(Kf)
    N := 2520;
    order := 3;
    char_gens := [631, 1261, 281, 2017, 1081];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 1, 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 := 2;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1, 0], [0, 1, 1, -1], [0, 1, 0, 0], [-1, 0, 0, 3], [-4, -1, -4, 0], [0, 2, -3, 2], [0, -5, 0, -5], [2, 0, 0, -5], [5, 0, 5, 0], [0, 1, -9, 1], [4, 0, 0, -3], [-1, 0, 0, -7], [0, -4, 2, -4], [-4, 2, -4, 0], [-10, -3, -10, 0], [0, 2, 12, 2], [1, 7, 1, 0], [-4, 0, 0, -5], [3, 3, 3, 0], [0, 2, -9, 2], [-2, 0, 0, -3], [0, 3, 2, 3], [0, 0, 0, -6], [0, 11, 0, 0], [0, -3, -5, -3], [0, -1, 0, -1], [-7, -6, -7, 0], [-2, 0, 0, -8], [1, 0, 0, 5], [0, 6, 2, 6], [8, -5, 8, 0], [-11, 0, 0, -8], [0, 0, -4, 0], [2, -6, 2, 0], [14, 6, 14, 0], [0, -16, 0, -16], [2, 0, 0, -7], [0, 2, -8, 2], [14, 4, 14, 0], [3, 0, 0, 8], [0, 4, -10, 4], [-11, 1, -11, 0], [0, 0, 0, 17], [-2, 6, -2, 0], [-2, 0, 0, 2], [-8, 0, 0, -10], [-22, 5, -22, 0], [0, 12, 1, 12], [0, -6, 14, -6], [-18, 0, 0, 8], [-4, -4, -4, 0], [10, 0, 0, 7], [0, -11, -2, -11], [0, -14, 0, 0], [-2, -6, -2, 0], [0, -6, -6, -6], [-5, 13, -5, 0], [-8, 0, 0, 4], [-11, -9, -11, 0], [-4, 0, 0, 4], [-1, 0, 0, 5], [-16, 3, -16, 0], [0, -5, 3, -5], [0, 1, 12, 1], [0, 2, 5, 2], [-1, 0, 0, 7], [-4, 4, -4, 0], [-2, 0, 0, 8], [-18, -3, -18, 0], [0, 3, -22, 3], [17, -1, 17, 0], [0, 7, -19, 7], [9, 0, 0, -8], [0, -14, 12, -14], [8, -15, 8, 0], [0, 1, 11, 1], [0, 10, -8, 10], [23, 4, 23, 0], [22, 0, 0, -2], [-1, 0, 0, 18], [22, -2, 22, 0], [-19, 0, 0, -1], [0, -4, -18, -4], [0, 6, -24, 6], [-14, 0, 0, -8], [0, -3, -29, -3], [-10, 0, 0, 9], [31, 0, 0, -3], [0, 8, -10, 8], [24, 8, 24, 0], [33, -7, 33, 0], [-24, 0, 0, 10], [0, 4, -9, 4], [-14, 0, 0, 10], [0, -6, 22, -6], [12, -10, 12, 0], [0, -11, 7, -11], [0, 20, 9, 20], [-16, 0, 0, 2], [2, -12, 2, 0], [-2, -12, -2, 0], [0, -3, -24, -3], [23, 6, 23, 0], [23, 1, 23, 0], [20, 0, 0, 3], [0, -3, 2, -3], [0, -14, 0, 0], [5, 0, 0, -22], [0, 21, -1, 21], [4, -18, 4, 0], [34, 0, 0, 6], [-9, -6, -9, 0], [8, 0, 0, 4], [-2, 19, -2, 0], [-7, 0, 0, 1], [-22, 7, -22, 0], [0, 9, -14, 9], [12, 0, 0, 12], [-21, 20, -21, 0], [1, 0, 0, -7], [0, 31, 0, 31], [4, 23, 4, 0], [0, 18, -19, 18], [26, 0, 0, 1], [0, -10, 36, -10], [0, 22, -18, 22], [-35, 0, 0, 3], [0, 3, 47, 3], [27, 4, 27, 0], [-14, 0, 0, -5], [0, 14, 15, 14], [-38, 0, 0, 0], [0, 21, -2, 21], [-3, 0, 0, 24], [38, 9, 38, 0], [44, 2, 44, 0], [-48, 0, 0, -1], [30, 8, 30, 0], [34, 0, 0, -8], [0, 11, -8, 11], [30, 12, 30, 0], [18, 0, 0, -10], [-27, -20, -27, 0], [-14, 0, 0, 13], [-21, 0, 0, -21], [-40, 11, -40, 0], [0, 12, -22, 12], [0, 2, -42, 2], [0, 12, 30, 12], [-20, 0, 0, -20], [-47, 0, 0, -1], [0, 29, 10, 29], [27, 15, 27, 0], [-10, 0, 0, 15], [0, -18, -11, -18], [0, -17, 16, -17], [-13, 0, 0, -13], [-10, 27, -10, 0], [0, 19, -30, 19], [-16, 0, 0, 10], [-31, 0, 0, -9], [0, -20, 0, -20], [-6, 11, -6, 0], [20, -13, 20, 0], [39, 10, 39, 0], [-3, -37, -3, 0], [21, 0, 0, 2], [0, -5, 8, -5], [0, 10, 0, 0], [12, 0, 0, 24], [0, -3, -2, -3], [-15, 7, -15, 0], [-27, 22, -27, 0], [12, 0, 0, 12], [-3, 0, 0, -16], [-14, -22, -14, 0], [12, 0, 0, 12], [0, -16, 7, -16], [0, 15, -33, 15], [-22, 0, 0, 13], [-5, 0, 0, 7], [0, -26, -20, -26], [22, -8, 22, 0], [20, -22, 20, 0], [10, -18, 10, 0], [11, -13, 11, 0], [0, -6, 7, -6], [-28, -9, -28, 0], [0, -29, -21, -29], [-18, 0, 0, -13], [0, 28, 24, 28], [0, -8, 8, -8], [-40, -3, -40, 0], [-12, 3, -12, 0], [-7, 32, -7, 0], [0, -14, 4, -14], [-18, 0, 0, -18], [0, 1, -16, 1], [-14, 30, -14, 0], [9, 0, 0, -16], [0, 6, 10, 6], [-4, 14, -4, 0], [8, 0, 0, -27], [-2, -30, -2, 0], [0, 4, -16, 4], [0, 8, -38, 8], [-20, 0, 0, 26], [-22, 28, -22, 0], [0, -1, -31, -1], [-22, 0, 0, 1], [0, 0, 0, -14], [0, -16, -48, -16], [50, 9, 50, 0], [0, -18, 37, -18], [-24, -4, -24, 0], [-8, 12, -8, 0], [0, 34, -4, 34], [-2, 0, 0, -20], [0, 0, 15, 0], [-7, 0, 0, -2], [0, 7, -16, 7], [0, 20, -8, 20], [-42, 0, 0, -10], [16, 0, 0, -30], [0, -16, 36, -16], [-2, 38, -2, 0], [0, -19, 13, -19], [0, -17, -24, -17], [-21, 9, -21, 0], [-28, -16, -28, 0], [-21, 0, 0, 32], [16, 7, 16, 0], [27, 0, 0, 5], [-14, 43, -14, 0], [0, -2, 21, -2], [0, 5, -22, 5], [-12, 0, 0, -8], [12, 0, 0, -27], [26, 9, 26, 0], [0, 6, -23, 6], [-25, -23, -25, 0], [0, -8, 32, -8], [16, 0, 0, 22], [0, -2, -48, -2], [29, 0, 0, -27], [-8, -7, -8, 0], [10, 44, 10, 0], [-4, 0, 0, 24], [63, 0, 0, 3], [0, -4, 20, -4], [-22, -28, -22, 0], [-17, 0, 0, -42], [32, -6, 32, 0], [0, 11, 2, 11], [37, 0, 37, 0], [1, 51, 1, 0], [-10, 0, 0, 25], [0, 12, 2, 12], [-13, 19, -13, 0], [0, 0, 0, 26], [-21, -32, -21, 0], [-3, 0, 0, 15], [10, 30, 10, 0], [0, -38, -5, -38], [-56, 0, 0, 3], [66, 0, 0, 0], [10, 0, 0, 48], [-22, 21, -22, 0], [0, -26, 9, -26], [-46, 0, -46, 0], [-29, -1, -29, 0], [0, -8, -55, -8], [-50, 0, 0, 14], [0, 22, -24, 22], [0, -27, 10, -27], [-11, 24, -11, 0], [0, -16, 25, -16], [0, 24, 44, 24], [-16, -30, -16, 0], [-7, -48, -7, 0], [10, 0, 0, 48], [41, 0, 0, -14], [0, -10, 24, -10], [0, 25, -4, 25], [7, -3, 7, 0], [18, 0, 0, 19], [3, 0, 3, 0], [24, 0, 0, 1], [-24, 30, -24, 0], [12, -26, 12, 0], [0, 29, 34, 29], [-14, 0, 0, -5], [15, 0, 0, 13], [42, -20, 42, 0], [0, -8, -11, -8], [12, 0, 0, 29], [0, 19, -32, 19], [13, 0, 0, -17], [0, 4, 22, 4], [0, -26, 18, -26], [70, 8, 70, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2520_bi_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2520_2_bi_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_2520_2_bi_l(:prec:=4)
    chi := MakeCharacter_2520_bi();
    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_2520_2_bi_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_2520_2_bi_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2520_bi();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 0, 2, 0, 1]>,<13,R![-17, 2, 1]>],Snew);
    return Vf;
end function;