
// Make newform 7920.2.a.bo in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7920_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_7920_2_a_bo();".
// 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_7920_2_a_bo();".
// 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 := [-3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [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_7920_a();"
function MakeCharacter_7920_a()
    N := 7920;
    order := 1;
    char_gens := [991, 5941, 3521, 6337, 6481];
    v := [1, 1, 1, 1, 1];
    // 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;

function MakeCharacter_7920_a_Hecke(Kf)
    return MakeCharacter_7920_a();
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], [-1, 0], [-1, -1], [1, 0], [-1, -1], [-3, -1], [0, 2], [0, 0], [-8, 0], [-2, -2], [4, -2], [-8, 0], [7, -1], [2, -2], [-2, 0], [8, 0], [0, 2], [4, 0], [0, -4], [-3, 1], [4, 2], [-7, -1], [-6, 0], [2, -4], [2, -2], [6, -2], [5, 3], [-10, 0], [8, -2], [5, 1], [-14, 2], [6, 0], [-2, -4], [2, 2], [12, 2], [-4, 2], [-2, 2], [-9, -3], [-11, 3], [12, 0], [16, 2], [4, 4], [-11, -3], [17, -1], [-4, 4], [16, 2], [-2, -2], [-5, -3], [6, -4], [7, 1], [14, -2], [-8, -2], [4, -4], [16, 2], [1, -5], [6, 4], [12, -2], [7, 7], [0, 0], [-1, -1], [-9, -3], [-5, -1], [16, 4], [16, -2], [-4, -2], [14, -2], [-3, -7], [-11, -5], [-16, 2], [26, 0], [26, -2], [-14, 2], [1, 1], [4, 4], [26, -2], [-2, -8], [14, 4], [-6, -4], [-2, 0], [12, 0], [-12, -2], [-16, 4], [-6, 4], [10, 0], [18, 2], [10, -8], [-1, -1], [2, 2], [6, -2], [-4, 0], [-22, 2], [-20, 4], [-20, 4], [4, 0], [15, -3], [14, 4], [-6, 4], [-3, -7], [18, -4], [25, -3], [25, -5], [-9, -3], [6, 6], [2, -4], [36, -2], [-8, 4], [5, -5], [-24, 0], [-16, 2], [-23, 5], [-11, -3], [32, -2], [-2, 6], [-8, -8], [10, -8], [-22, 6], [38, 2], [-12, 2], [-36, 0], [2, 12], [-15, 1], [-9, 1], [-38, 2], [28, -4], [-32, -4], [-16, 6], [-12, -4], [0, 8], [-11, 1], [2, -12], [41, -1], [-8, 0], [-20, 2], [-6, -6], [-36, -2], [12, 10], [-21, -5], [-4, -10], [12, 0], [22, -4], [-16, 8], [-34, -2], [1, -1], [28, -6], [16, 4], [-15, -7], [-7, -5], [-50, -2], [18, 6], [1, -7], [2, -8], [8, -4], [-5, 5], [-2, 10], [4, 4], [26, 0], [2, -8], [39, -5], [-46, -2], [18, -6], [3, 9], [-15, -3], [12, 0], [30, 0], [2, 14], [14, 6], [-3, 5], [-40, 6], [-18, 0], [52, 0], [0, 6], [14, 2], [3, 11], [14, 2], [-42, 4], [6, 8], [-6, 0], [13, 1], [34, -8], [24, 8], [-36, 0], [12, 10], [3, 9], [-22, -2], [34, -4], [-3, 1], [-10, -6], [30, 8], [12, 0], [-8, -14], [1, -9], [20, 0], [-2, 0], [35, -7], [-36, 6], [36, -2], [-46, 4], [-1, -11], [-7, 7], [4, 0], [-22, -4], [-14, 4], [-16, 6], [-20, 8], [-56, 2], [-20, 0], [-57, -3], [-32, 4], [-10, 10], [-1, 3], [30, -8], [0, -8], [0, 12], [-10, 10], [-22, -12], [-1, -13], [2, 14], [52, 4], [12, -6], [-52, 2], [-16, 6], [-22, 0], [2, 18], [-19, 7], [-6, 8], [46, 0], [-56, -4], [33, -7], [6, 2], [-42, 4], [-18, 4], [30, -8], [40, 4], [68, 0], [1, -9], [24, -10], [7, -11], [12, -4], [-12, -12], [-46, -6], [4, 10], [-36, 4], [0, 2], [-1, -7], [-30, -2], [8, -8], [-16, 8], [-26, 0], [-43, -5], [-45, -5], [-14, -10], [-52, 4], [6, -16], [-15, 3], [24, 8], [-28, -6], [-19, 5], [-8, 2], [35, 7], [-13, 7], [-21, 1], [0, -2], [11, 7], [-2, 4], [-8, -12], [-54, -4], [18, 0], [23, -5], [-13, 5], [-14, 0], [26, -10], [-12, -10], [-50, -4], [31, -1], [28, 4], [30, -10], [40, -6], [20, -2], [-6, 18], [33, -5], [-42, -2], [53, -5], [6, -12], [33, 5], [32, 0], [-18, 8], [3, 13], [-42, -2], [18, 6], [22, 8], [-14, 10], [-21, 13], [10, -6], [-5, -9], [-34, -10], [-4, -4], [-60, -6], [8, 8], [11, 15], [23, 11], [-35, -1], [38, 0], [-36, 0], [6, 10], [24, -10], [58, 2], [-6, 4], [-16, 8], [-47, 9], [15, 13], [-6, 20], [-14, 14], [-20, -16], [-25, 13], [-8, -18], [-20, -16], [14, 10], [-44, -2], [6, -10], [-24, 2], [-4, 6], [-20, 8], [-4, 4], [-9, -15], [-16, 6], [-30, 14], [-56, 4], [-15, 3], [25, -17], [-12, -14], [22, 12], [40, 6], [7, 9], [44, -6], [30, 6], [66, 0], [11, 13], [-40, 6], [-17, -9], [-22, -8], [14, 12], [62, -2], [-10, 2], [-48, 6], [54, 10], [22, 0], [54, 2], [-20, -4], [-80, -2], [-18, 4], [14, 12], [10, -4], [9, 5], [-44, 6], [-35, 7], [-12, 16], [2, -18], [-33, 5], [32, 0], [-21, -1], [-18, -6], [-24, 8], [-18, 10], [56, -4], [36, 14], [-3, -9], [-5, 3], [-4, 22], [24, 16], [22, -10], [33, -9], [18, 22], [-30, 8], [-16, 14], [20, 8], [46, -2], [67, -9], [2, 26], [-45, -13], [-18, 0], [-48, 10], [65, -3], [21, -1], [16, 22], [-48, 8], [-44, -12], [10, 8], [9, -19], [28, -4], [16, 2], [-14, 4], [-28, -4], [36, 8], [4, 0], [-45, -5], [26, -12], [-22, 20], [-32, 0], [-12, 12], [-30, 8], [16, -2], [39, 7], [-20, 22], [46, -2], [52, 10], [18, -8], [-34, -14], [-24, -12], [-28, -4], [39, 11], [-41, 5], [48, 12], [-18, -8], [65, -3], [-50, -4], [44, 8], [-26, -18], [18, 8], [-13, -7], [-6, -12], [-33, -13], [-12, -20], [42, 6], [-2, -12], [-24, 6], [64, -4], [38, -14], [12, 4], [22, -10], [36, -2], [48, -16], [8, -12], [14, 2], [-4, 18], [-62, 4], [-48, -10], [12, -22], [2, 6], [10, 12], [-70, 2], [-10, 4], [10, 24], [-8, 8], [10, -20], [39, -15], [17, 9], [-9, -3], [16, 26], [48, 6], [-61, 7], [8, 4], [-13, -19], [2, 10], [16, 6], [-58, 4], [6, -16], [0, -8], [15, 7], [38, 16], [30, 2], [32, 4], [18, -6], [-12, -2], [-37, -1], [-63, -11], [42, 0], [0, -12], [-12, 4], [50, -6], [-63, -7], [62, 14], [52, 4], [52, 10], [80, -8], [51, -5], [58, 12], [-42, 2], [-59, -5], [40, 10], [-32, 14], [22, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7920_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7920_2_a_bo();".
// 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_7920_2_a_bo(:prec:=2)
    chi := MakeCharacter_7920_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(3449) - 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_7920_2_a_bo();".
// 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_7920_2_a_bo( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-12, 2, 1]>,<13,R![-12, 2, 1]>,<17,R![-4, 6, 1]>,<19,R![-52, 0, 1]>,<23,R![0, 1]>,<29,R![8, 1]>],Snew);
    return Vf;
end function;