
// Make newform 7600.2.a.bz in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7600_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_7600_2_a_bz();".
// 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_7600_2_a_bz();".
// 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, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 1]];
    Rf_basisdens := [1, 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_7600_a();"
function MakeCharacter_7600_a()
    N := 7600;
    order := 1;
    char_gens := [4751, 5701, 5777, 401];
    v := [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_7600_a_Hecke(Kf)
    return MakeCharacter_7600_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], [1, -1, 0], [0, 0, 0], [1, 0, -1], [0, -2, 0], [2, -1, 1], [-4, 1, -1], [-1, 0, 0], [-1, -1, 2], [2, 1, -1], [-2, -2, 0], [1, -2, 0], [0, 0, 0], [-4, -2, 0], [1, 2, -2], [5, -4, -1], [-1, 1, -4], [0, 0, -2], [1, 2, -1], [4, -4, -2], [5, -1, -2], [-8, 4, 0], [-2, 4, -2], [6, -4, 0], [8, 6, -2], [-2, 0, -2], [0, -2, -2], [-6, 5, 3], [1, 0, -1], [6, 0, 0], [0, -6, 4], [6, 2, 0], [5, -2, -1], [0, 2, -4], [2, -6, 2], [-20, 0, 0], [4, 2, 0], [12, 4, -2], [12, -2, 0], [-1, 6, -4], [5, 0, 2], [-1, -6, 0], [5, 1, 2], [-16, 0, -2], [12, 4, 0], [11, 4, 1], [-17, 1, 0], [-10, 2, 0], [-7, 6, -1], [-10, 4, 0], [-3, -4, 0], [-11, 1, -2], [4, 2, -4], [12, 6, 0], [-12, 2, 0], [0, 8, 0], [11, -6, -4], [-15, -3, 2], [20, -6, -2], [-2, -4, 4], [4, -6, 8], [2, 1, -1], [-5, 0, 2], [11, 0, 5], [4, -9, -3], [11, -5, 8], [-18, 7, 5], [2, -8, -2], [6, -12, 0], [-6, 8, 4], [-13, -6, 5], [-12, -7, 3], [4, -8, -4], [-13, 3, 4], [-13, 10, 1], [6, 8, 0], [-4, 6, 2], [12, 4, -4], [0, 0, -6], [10, 6, 2], [-8, 0, -8], [7, 15, -4], [-10, -8, 2], [-22, 2, 4], [-4, -4, 4], [6, 6, -6], [18, 2, 0], [-4, 3, 1], [2, -2, 2], [-1, -10, 6], [6, -2, 6], [-3, 6, 6], [10, -6, -4], [8, 4, -4], [10, 2, 6], [-16, 1, -1], [-5, 10, -8], [-18, 6, 0], [-9, -6, 1], [-14, -12, 2], [0, 8, -8], [14, 10, -4], [-19, -12, 2], [-4, 10, 2], [-24, -6, 2], [7, -3, -6], [12, 0, -6], [13, -4, 4], [2, 6, -10], [8, -8, 6], [6, -2, -2], [14, -6, 10], [-17, -12, 4], [0, 0, 8], [17, 2, -2], [10, -2, -8], [8, 12, -6], [5, 4, -1], [20, 2, -4], [-3, -6, 3], [6, 3, -11], [0, 2, 2], [27, 0, -3], [17, -4, 2], [-18, -6, 2], [-10, -14, 2], [2, -6, -6], [5, 13, -10], [3, -13, -2], [-30, 6, -4], [16, 0, 6], [8, 4, -10], [4, -2, -6], [-12, 12, -4], [8, 3, 5], [-12, 1, -5], [19, 11, -8], [30, 3, 1], [22, 7, 1], [5, 2, -1], [17, 6, -5], [-20, 10, 4], [-19, 9, -6], [-26, -1, 1], [-11, 1, 8], [6, -14, 12], [2, 16, -2], [4, 12, -2], [-12, 4, 2], [6, -4, 0], [3, 15, -4], [-19, 8, -4], [-24, -6, -2], [2, 0, 2], [-25, 6, -3], [-4, -2, 8], [-20, 1, 3], [-21, -3, 6], [16, -9, 9], [-37, 4, 5], [-16, -8, -4], [24, -12, 6], [0, 4, 4], [-24, -4, 0], [-4, 6, 8], [-2, -4, -8], [-12, 6, 8], [-22, 0, -2], [50, 0, 0], [42, -8, 0], [-19, -5, -4], [-1, -17, 0], [-20, 0, -8], [-5, -3, 6], [-4, 16, -14], [-41, 8, 4], [0, 16, 2], [12, -6, -6], [-2, -2, 2], [-12, -10, 10], [13, -3, 2], [21, 6, -9], [21, -14, -4], [-2, 0, -2], [-12, 13, 3], [-18, 6, 0], [-1, 15, -8], [-36, 4, 4], [22, 0, -4], [-5, -16, 1], [10, -12, 6], [-6, -14, 12], [-1, 0, 10], [42, 7, -3], [16, 8, -2], [14, -16, -4], [-12, 9, -5], [12, -4, -10], [-19, -14, -1], [-5, 16, -11], [15, -10, 0], [-18, 0, -4], [1, -8, -9], [-4, 8, -12], [-8, -8, -2], [-16, 14, 8], [-7, -15, -2], [13, -9, -8], [-28, 5, -1], [-18, 5, -1], [3, -22, 5], [26, 10, -4], [-17, -10, 2], [3, 5, 0], [-14, 2, 4], [-12, -2, 10], [8, -3, 15], [0, -20, 6], [0, 26, -12], [-10, -6, 2], [-2, 12, -4], [28, 4, 0], [12, -2, 6], [-28, -3, -9], [-1, 22, -7], [22, -10, 2], [-32, 4, -2], [-48, 0, 0], [10, 0, -4], [26, -6, 2], [14, 0, 10], [23, 9, -8], [-2, 6, -12], [22, -6, -14], [-12, 18, -2], [-7, 13, -10], [3, -8, -8], [0, -4, 18], [-20, -4, 10], [36, -2, 6], [45, -4, 6], [14, -24, -2], [0, 3, -11], [19, -16, 5], [-6, 6, 6], [3, 6, 6], [-8, 12, -4], [-3, 4, -6], [-29, 4, 6], [11, -17, 14], [2, 0, -14], [27, 2, -3], [8, -23, -1], [-8, 0, -10], [32, -2, -4], [-50, 10, 4], [-2, 12, 2], [-32, 0, -4], [-13, -2, -4], [13, 0, 0], [-22, 0, 0], [3, -4, -6], [-12, 22, -8], [-7, 0, -5], [23, 4, -4], [-13, 18, -11], [9, -12, 15], [-29, 6, 1], [-5, 9, -16], [14, -4, -10], [-1, 3, -12], [18, -15, 7], [-57, 6, 5], [-5, 15, 6], [-42, 8, -4], [-20, 3, 5], [26, -16, 8], [-1, -8, 9], [8, 16, 0], [-6, -2, -6], [22, 12, -8], [12, 15, -7], [11, -9, 2], [-29, 8, 5], [-10, -18, 0], [-35, 18, -2], [-8, 7, -15], [27, -14, 12], [36, -11, -1], [-22, 12, 8], [52, 2, -2], [16, -6, 4], [-8, 2, -2], [31, -8, -14], [10, -28, 0], [-30, 4, -12], [4, 0, -6], [-8, -8, 16], [-22, -13, -7], [-2, 4, 14], [-30, 12, -4], [-25, 7, 8], [-68, -1, -3], [25, -17, -8], [2, -4, -8], [30, -2, 2], [18, 3, -3], [-51, -3, -8], [-28, 16, -4], [-18, -16, 2], [-12, -8, 0], [56, -8, 14], [2, 4, 14], [43, -12, 2], [12, -4, 0], [-10, 12, -6], [60, 2, 0], [34, 2, -2], [21, -14, 11], [-23, 8, 16], [61, -2, 3], [-5, -11, 18], [38, -11, -13], [-4, 32, -12], [-12, -11, 3], [-14, -2, -4], [46, -7, 3], [-1, -23, 0], [10, -2, -2], [8, -20, 14], [22, 4, -10], [2, -3, -1], [-4, 15, 1], [8, 16, -4], [-44, 0, 0], [-18, 18, -18], [50, -9, -3], [-30, 2, 6], [-45, -2, 1], [20, -21, -3], [7, 27, 0], [13, 30, -5], [30, -15, 15], [-20, -30, 6], [22, 23, 1], [-16, 16, -4], [34, -6, 20], [19, -6, 5], [6, -10, 6], [-44, -4, 10], [-39, 12, -10], [-6, 14, 2], [-36, -18, 12], [-40, -24, 6], [-25, -6, 12], [20, -4, 14], [-20, -13, 1], [2, -20, 8], [-48, 5, -9], [30, -8, 12], [-20, -6, 6], [49, -12, 4], [10, 4, -2], [47, -11, 8], [18, -6, 4], [44, 0, -8], [50, 4, 8], [34, -14, 20], [-56, -8, -4], [-2, -34, 16], [1, 30, 3], [-15, -8, -9], [-49, 15, -4], [37, -14, -6], [24, 30, -4], [-31, -2, -1], [31, 26, -14], [62, -12, -8], [9, 14, 3], [50, -18, -14], [-14, -32, 16], [-18, 24, -12], [45, -15, 10], [22, -4, -8], [18, -19, -13], [24, -16, 18], [1, -4, 15], [6, 0, -4], [12, -24, -2], [-12, 24, -6], [24, 2, 16], [31, -12, -11], [14, -20, 26], [-42, 16, 10], [14, -10, 14], [50, 12, 4], [-40, 8, 10], [10, 4, -14], [0, 16, -16], [30, -6, 12], [-34, 24, 12], [35, 28, -4], [8, -12, 6], [-2, -30, 4], [-2, 4, 4], [-25, 12, 1], [24, 12, -4], [20, -8, 2], [-22, -3, 15], [-22, 8, -16], [24, -14, 14], [32, -4, 8], [-47, 2, 7], [8, -10, -20], [-5, 6, -15], [22, -18, -8], [-44, -26, 4], [2, 18, -6], [-13, -3, 2], [26, 28, -10], [-37, -18, -4], [10, 29, -9], [24, -17, 9], [-6, 15, 9], [26, -12, -2], [59, -21, -4], [-15, 2, 3], [44, 7, 5], [0, -26, 8], [56, -5, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7600_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7600_2_a_bz();".
// 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_7600_2_a_bz(:prec:=3)
    chi := MakeCharacter_7600_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(2999) - 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_7600_2_a_bz();".
// 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_7600_2_a_bz( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, -5, -2, 1]>,<7,R![-1, -13, -2, 1]>,<11,R![-24, -24, 2, 1]>,<13,R![35, -3, -6, 1]>],Snew);
    return Vf;
end function;