
// Make newform 7920.2.a.bn in Magma, downloaded from the LMFDB on 01 October 2025.
// 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_bn();".
// 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_bn();".
// 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 := 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]];
    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], [3, -1], [-1, 1], [-2, 0], [0, 0], [-2, 2], [2, 2], [0, 2], [2, -2], [-1, -1], [-2, 2], [6, 0], [0, 0], [4, -2], [0, 4], [-12, 0], [1, 1], [-14, 0], [-13, 1], [-2, -4], [-10, 0], [0, 0], [2, 2], [-9, -3], [14, 0], [4, 2], [1, -3], [-10, -2], [6, 0], [0, -2], [-12, 0], [10, 0], [-12, 2], [18, -2], [1, -1], [11, 1], [-8, -4], [0, 2], [0, 0], [-11, 1], [-9, -3], [-8, 0], [-2, 0], [-2, 6], [9, 3], [6, -4], [-11, -1], [6, 6], [12, 2], [-24, 0], [-4, -2], [-1, 1], [-10, 4], [-2, 0], [3, -1], [-22, -2], [23, -1], [1, -1], [7, 3], [8, 4], [12, 2], [0, -6], [-18, -2], [-23, 1], [15, -3], [-4, 6], [-30, 0], [6, -6], [-22, 2], [17, -3], [-12, 4], [6, -6], [10, -4], [-22, 0], [2, 4], [10, 4], [-16, 4], [4, -2], [4, 8], [-10, 0], [-24, -2], [6, 6], [22, -4], [-21, -1], [-12, 0], [-18, -2], [4, 8], [18, -6], [16, 0], [0, 0], [-16, -4], [17, 7], [22, -4], [-22, 4], [17, 5], [-6, -4], [21, 1], [27, -3], [9, -9], [4, 8], [8, 2], [-20, -2], [8, 4], [-13, 1], [-8, 8], [-16, -6], [-19, -7], [-35, 1], [-12, -6], [6, -2], [-12, 4], [-26, -4], [6, 10], [2, -2], [24, 6], [-12, 0], [10, 4], [33, 5], [-11, -1], [14, -2], [4, -12], [-6, -6], [40, -2], [-16, -8], [-24, -8], [9, -7], [-16, 2], [-5, -7], [4, 12], [-12, 2], [0, -12], [4, -2], [-24, 6], [3, -5], [16, 2], [-6, -6], [16, 6], [6, -6], [-14, -6], [-29, -7], [-20, 10], [-32, -4], [-15, 5], [-21, -3], [22, 6], [6, -6], [33, 5], [-10, 4], [24, 4], [1, -13], [-34, 2], [0, 0], [-16, -10], [-10, 4], [7, 7], [-48, 0], [6, 6], [-27, 3], [21, 1], [-12, 0], [-18, 0], [14, 10], [-14, -6], [21, -7], [-12, 2], [14, -8], [4, 8], [32, 6], [26, -2], [-9, -13], [-26, -6], [-6, 0], [12, -2], [34, 8], [-15, -11], [6, -4], [-16, 8], [-12, 0], [20, -6], [-11, -13], [-18, -6], [-10, -8], [-15, -7], [-10, 2], [-14, 4], [40, -4], [8, -6], [3, -15], [-8, -12], [-6, -12], [-27, 3], [0, -6], [-28, 6], [42, -4], [1, -1], [11, 1], [-42, 6], [12, -2], [34, 4], [20, -6], [16, -4], [4, -10], [36, -8], [1, -1], [-2, 14], [-14, 6], [-41, 7], [18, -12], [24, -8], [16, -4], [-30, 6], [18, -4], [3, 7], [4, -4], [4, -4], [32, -2], [-36, 2], [30, 4], [30, 12], [-10, 14], [19, 5], [14, 12], [-10, -8], [24, -12], [1, 9], [-42, -6], [-6, 8], [8, 14], [24, -14], [-26, 14], [48, 4], [-21, 9], [-32, -2], [-15, -9], [-48, 0], [-44, -4], [-2, -10], [-10, -4], [-4, -4], [48, 2], [37, -1], [70, 2], [-28, -4], [44, 4], [-20, 6], [-33, -3], [15, -13], [4, 20], [-28, 4], [38, 0], [-29, 5], [36, 12], [-28, -6], [13, -3], [-12, 6], [35, 3], [23, 11], [-19, -5], [16, -2], [-21, 11], [-58, 4], [-44, 0], [-34, -8], [30, -12], [27, 7], [41, -5], [18, -16], [34, 6], [4, -2], [-28, 2], [-21, -1], [16, 0], [42, -6], [-52, 6], [-28, 6], [62, -2], [47, 1], [26, 2], [-33, -3], [-46, 0], [-35, 9], [8, 16], [18, -4], [57, 3], [26, 2], [-28, 4], [-6, -12], [-2, -10], [33, 3], [-18, 6], [-13, 3], [-64, 4], [64, 0], [-8, 14], [12, -12], [-33, -5], [15, -1], [55, -7], [16, -6], [-36, 0], [-22, 14], [-36, 2], [54, 6], [58, 4], [16, -16], [-23, 13], [9, 15], [30, 0], [-4, 4], [32, -4], [-47, -1], [20, -6], [44, -8], [50, -2], [28, -2], [-28, -8], [-34, 8], [20, 6], [54, 6], [0, 16], [45, -9], [-48, -10], [6, -14], [48, 4], [-17, -7], [15, -3], [-12, 2], [70, -4], [6, 4], [5, -17], [-50, 12], [-22, -14], [-30, -16], [-15, 3], [24, -10], [-33, 7], [-18, -4], [-18, -12], [-8, 20], [-26, 18], [-12, -6], [-2, -22], [-26, 16], [-62, 6], [-4, -20], [-28, -14], [16, 6], [22, 4], [-42, 0], [21, 1], [0, -22], [7, 17], [-80, -4], [18, 6], [49, -1], [-32, 8], [3, 11], [24, 0], [-48, 0], [-10, 10], [-24, 4], [20, 6], [-17, -7], [-5, 3], [-12, -10], [48, 0], [-2, -18], [-21, 9], [32, -8], [-72, -2], [0, -10], [48, -12], [-42, 6], [-5, -5], [-24, -12], [7, -5], [10, 20], [12, 6], [21, 1], [71, 1], [-22, -16], [56, -8], [-24, -8], [22, 4], [-27, 1], [44, 4], [-48, 2], [18, 12], [0, -24], [32, -4], [-28, -8], [-1, -21], [44, -10], [-6, -12], [-24, -8], [18, 6], [38, 12], [20, 10], [-9, 7], [-64, 10], [0, 12], [-38, 0], [-34, -12], [-48, 0], [76, 0], [-32, 8], [15, -1], [-39, 15], [16, 20], [-60, -2], [21, -19], [30, 12], [0, 12], [-6, 10], [34, -16], [-27, 3], [-26, 8], [39, -1], [36, -12], [-30, 6], [46, 4], [36, 18], [48, -12], [16, 8], [-76, -4], [6, 6], [-4, 6], [44, 4], [60, -8], [58, -10], [36, 2], [-30, -12], [12, 2], [20, 18], [58, 6], [-36, -2], [-66, 6], [90, 0], [-26, 4], [8, -8], [-10, 0], [-3, -9], [-71, 9], [17, -17], [24, 2], [-24, 14], [-33, -5], [-16, 4], [81, 3], [0, 12], [-52, -6], [38, 4], [-26, -8], [-68, -4], [75, -1], [-10, 16], [-62, 6], [-40, -4], [6, 6], [-20, 22], [-33, -5], [-19, -3], [64, 6], [0, 12], [-98, 2], [18, 10], [-47, 9], [-30, 18], [-4, 4], [48, -10], [0, 0], [55, -5], [54, 0], [-18, 10], [15, -3], [12, 6], [20, -6], [28, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7920_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7920_2_a_bn();".
// 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_bn(: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_bn();".
// 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_bn( : 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![-4, -6, 1]>,<17,R![-12, 2, 1]>,<19,R![2, 1]>,<23,R![0, 1]>,<29,R![-48, 4, 1]>],Snew);
    return Vf;
end function;