
// Make newform 280.2.bv.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_280_bv();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_280_bv_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_280_2_bv_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_280_2_bv_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, 0, -1, 0, 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_280_bv();"
function MakeCharacter_280_bv()
    N := 280;
    order := 12;
    char_gens := [71, 141, 57, 241];
    v := [12, 6, 3, 10];
    // 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_280_bv_Hecke();"
function MakeCharacter_280_bv_Hecke(Kf)
    N := 280;
    order := 12;
    char_gens := [71, 141, 57, 241];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1], [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 := [[-1, 1, 1, 0], [0, 0, 1, 1], [0, 1, 2, -1], [-3, 0, 1, 0], [-1, 1, -1, 0], [3, 0, 0, -3], [-3, 3, 3, 0], [4, 0, -2, 0], [-3, 0, 3, -3], [-1, -4, 0, 2], [-3, -5, -3, 0], [2, 4, 2, -2], [-2, 0, 4, -3], [1, 1, 1, 1], [4, -1, -5, 5], [0, 0, 6, -6], [4, 2, 4, 0], [0, -4, -3, -4], [4, -4, 1, 5], [12, 4, 0, -2], [-3, -3, -5, 8], [-8, -6, 4, 6], [2, 7, -7, -2], [0, -3, -12, -3], [7, -2, 2, -7], [-2, 7, 2, -14], [3, 5, -8, -8], [3, -5, -8, 8], [6, -1, -6, 2], [4, -4, 4, -4], [-11, 0, 0, -11], [0, -2, -4, -2], [-6, -6, 6, 0], [-1, 0, 2, -3], [0, 5, 14, 5], [-19, 3, 19, -6], [-11, -11, 7, 4], [10, -11, 1, 1], [0, 9, -9, 0], [10, -4, -14, 14], [-5, 11, 5, -22], [0, -26, 0, 13], [0, -7, 9, -7], [-3, 3, 7, 4], [-7, 4, -4, 7], [0, -2, 0, 4], [-2, 0, 4, -24], [-3, 8, 8, -3], [-5, -5, -7, 12], [4, 12, -2, -12], [6, 3, -3, 3], [-1, 0, 2, 3], [-6, -10, -6, 0], [11, 10, 0, -5], [8, 16, 8, -8], [-12, 12, 3, -9], [2, -7, 2, 0], [0, 12, 0, -12], [-11, 11, 13, 2], [0, -28, 0, 14], [1, -1, 7, 8], [-2, -6, 6, 2], [-17, 13, 13, -17], [4, 18, 4, 0], [6, -20, 14, 14], [-6, -1, 5, -5], [2, -9, -1, 9], [-6, -10, -10, -6], [16, -16, -23, -7], [15, 0, -30, 6], [-12, -12, -6, 18], [2, 3, -1, -3], [-10, 10, 7, -3], [8, -4, -4, -4], [-6, -28, 0, 14], [-1, -4, 5, 5], [-22, 4, 22, -8], [-8, -5, 13, 13], [0, 16, 3, 16], [1, 6, -1, -12], [6, 0, -12, 8], [-7, 0, 14, 18], [15, -1, -15, 2], [-6, -4, -4, -6], [0, -10, 12, -10], [-23, 7, 16, 16], [-3, 0, 6, 18], [-6, -2, 8, 8], [26, -16, 0, 8], [-1, 5, -5, 1], [7, -5, -2, -2], [9, -3, -9, 6], [-11, -11, 3, 8], [-3, 0, 6, 7], [0, 13, -9, 13], [6, -21, -21, 6], [22, -8, -11, 8], [-2, 12, -2, 0], [24, 9, -15, 15], [8, -9, -4, 9], [-9, -15, 15, 9], [13, -13, 13, 26], [7, -7, -16, -9], [32, -6, -16, 6], [-4, 24, -4, 0], [23, -23, -15, 8], [-9, 0, 0, -9], [-12, 6, 6, 6], [14, 0, 14, 0], [6, 0, -12, -18], [-5, -15, -10, 10], [1, 1, 7, -8], [10, -8, -8, 10], [5, 9, 5, 0], [-9, 2, 0, -1], [24, 5, -24, -10], [-9, 0, 0, 9], [4, -4, 13, 17], [0, 23, -23, -23], [22, 20, 0, -10], [33, 4, -33, -8], [-8, -6, 6, 8], [10, -17, 7, 7], [-16, -16, 1, 15], [0, -13, -21, -13], [-15, 0, 30, 8], [0, 4, 9, 4], [0, -1, -9, -1], [-14, -9, 9, 14], [-18, -9, 9, -9], [3, 5, -3, -10], [-25, -23, 23, 25], [0, 6, 0, 6], [29, 22, -22, -29], [-48, 6, 24, -6], [-12, 40, 0, -20], [15, -15, 3, 18], [-10, -10, 17, -7], [-10, 8, 8, -10], [-8, -33, -8, 0], [39, -10, 0, 5], [-4, -8, 2, 8], [5, -5, 6, 11], [8, 25, -25, -8], [-2, -6, -2, 0], [9, 6, 0, -3], [22, 20, -20, -22], [-17, 17, 13, -4], [12, 6, -6, -6], [-11, -13, -2, 2], [6, 12, 6, -6], [14, 0, -28, -3], [-23, 28, 28, -23], [13, -7, -20, 20], [-9, 9, 0, -9], [3, 6, 0, -3], [-52, 0, 26, 0], [0, -10, 9, -10], [-4, 12, -12, 4], [-16, -2, 16, 4], [-31, -5, 26, -26], [2, 4, -4, -2], [-31, 15, 15, -31], [0, 0, 20, 0], [11, 11, -1, -10], [2, 2, 17, -19], [1, -21, -1, 42], [6, 6, 0, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_280_bv_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_280_2_bv_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_280_2_bv_b(:prec:=4)
    chi := MakeCharacter_280_bv();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_280_2_bv_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_280_2_bv_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_280_bv();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -4, 5, -2, 1]>],Snew);
    return Vf;
end function;