
// Make newform 8085.2.a.ba in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8085_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_8085_2_a_ba();".
// 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_8085_2_a_ba();".
// 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 := [-2, 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_8085_a();"
function MakeCharacter_8085_a()
    N := 8085;
    order := 1;
    char_gens := [2696, 4852, 1816, 3676];
    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_8085_a_Hecke(Kf)
    return MakeCharacter_8085_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 := [[-1, 1], [1, 0], [1, 0], [0, 0], [-1, 0], [0, -4], [4, 2], [4, -2], [-4, 0], [-2, 2], [0, 0], [6, -4], [-2, 2], [-6, 2], [4, 0], [-2, -8], [4, 0], [6, -4], [0, 4], [8, 4], [0, 8], [0, -6], [10, 0], [2, 4], [-6, 4], [-2, -2], [-8, -8], [6, -8], [6, -8], [-2, 12], [-6, 6], [8, 8], [-2, 8], [8, 6], [10, 6], [8, -6], [-18, 0], [4, 8], [2, -8], [0, 2], [-12, 4], [14, 0], [0, -4], [8, -4], [0, 6], [16, -4], [4, 2], [12, 4], [14, 0], [2, 0], [-16, 2], [12, -4], [18, 4], [-4, 12], [22, 4], [18, 0], [2, 16], [-8, -14], [-8, -4], [-14, -2], [6, -2], [4, -2], [6, 2], [-8, -8], [-10, 4], [2, 20], [12, 4], [-8, -8], [-18, 8], [6, 12], [-26, 0], [12, 0], [-4, 4], [-12, -16], [-12, 8], [8, 0], [-18, 4], [-2, -12], [-18, -8], [14, -4], [20, -12], [-6, 0], [16, -4], [10, -4], [12, -6], [12, 0], [14, -12], [-12, 8], [14, 10], [-12, -12], [0, -16], [-20, 12], [4, 12], [20, 4], [-28, -4], [-6, 8], [-30, -8], [10, -16], [18, 6], [6, -8], [6, 10], [8, 2], [-26, 4], [-10, 18], [-8, 6], [-14, 0], [8, 16], [20, -2], [16, -8], [-6, -8], [-10, -6], [-12, 24], [6, 4], [20, 4], [40, -4], [-10, -12], [-4, 24], [-16, -8], [-10, -12], [-4, 8], [2, 8], [24, 4], [-4, 18], [24, 0], [-20, 0], [22, -10], [18, 8], [8, -28], [0, 24], [12, 4], [16, 22], [42, 4], [-8, -20], [-14, 16], [26, -2], [-22, -8], [2, 4], [6, -10], [-18, 12], [-14, -30], [8, 6], [-10, 2], [16, 0], [30, -8], [38, 0], [0, -16], [20, 8], [20, -6], [36, -12], [-32, -8], [-20, -16], [30, -12], [-28, -16], [10, -4], [24, -12], [-32, 12], [20, 6], [6, -28], [-28, -12], [-2, -26], [-16, -16], [8, 14], [-14, -18], [12, -16], [14, 8], [8, -16], [-8, 16], [28, -8], [-26, 12], [-38, -16], [4, -16], [22, 16], [-28, 0], [12, 8], [-24, -12], [-10, 4], [16, 14], [22, -8], [-26, -14], [-14, 12], [-8, 24], [4, -40], [2, -16], [-24, -14], [-28, -24], [-18, 20], [36, -12], [0, -20], [2, -4], [-24, 28], [-22, 8], [-2, 16], [12, -28], [2, 12], [-38, -12], [22, -4], [46, 16], [30, 0], [-24, 2], [22, -20], [30, 14], [28, 6], [18, 24], [-22, 12], [20, -28], [-10, 20], [32, -24], [-6, -28], [10, 10], [-12, 24], [-36, -20], [-34, -12], [-32, 24], [48, 0], [-52, 0], [-34, 8], [30, -6], [18, -22], [-32, -8], [14, -28], [-30, -16], [-32, -10], [2, -4], [-24, -8], [-14, -12], [54, -4], [-54, 0], [-24, -16], [62, 6], [-8, -20], [2, 36], [8, 30], [28, 2], [2, -30], [0, -4], [-26, 20], [-14, -16], [-60, 10], [4, -28], [-24, 0], [-4, -24], [36, 26], [24, -24], [18, 16], [24, -2], [-12, 24], [-12, -28], [12, -4], [-8, 6], [-38, -4], [-48, 4], [18, -6], [-8, 32], [-2, 16], [24, 30], [-12, 12], [-30, 16], [-46, -14], [22, 36], [-4, -12], [-10, -30], [-46, 20], [54, -8], [-24, 0], [6, -48], [4, 44], [-6, -4], [14, 28], [42, -10], [36, -26], [26, 0], [32, -12], [-6, -8], [28, 14], [-20, -32], [-4, 52], [24, 0], [-30, 4], [-2, -16], [8, 16], [-18, -16], [-24, 32], [38, 12], [-14, 36], [-6, 14], [-8, 32], [14, 12], [-44, -2], [32, 32], [-14, 2], [30, 12], [-12, 24], [24, -30], [0, 32], [-40, 16], [6, -6], [-8, 36], [-6, -20], [4, 12], [-66, 2], [-68, 4], [8, 10], [44, 2], [24, -8], [60, 0], [-22, -12], [-16, -40], [-6, -16], [8, -24], [-4, -40], [66, 0], [-58, 12], [-50, -10], [-16, 12], [-2, 48], [-38, -4], [-60, 8], [-4, 56], [10, 16], [-26, 18], [-12, 54], [-26, 0], [-22, -38], [-20, 12], [0, -30], [50, -24], [4, -8], [-4, 8], [-6, -8], [0, 22], [10, -24], [-6, 0], [-24, 6], [-82, -8], [-20, 46], [-28, 32], [-10, -16], [-24, -18], [-6, 56], [18, 18], [26, -44], [-30, -40], [-34, 2], [40, -12], [-42, 20], [0, 36], [-6, 16], [-36, -16], [56, 0], [50, -4], [-16, -26], [26, 8], [74, 4], [-10, 10], [-30, 4], [92, 2], [40, 12], [32, 20], [-16, -2], [0, -32], [0, 16], [-14, -30], [28, 24], [-24, 36], [-52, 0], [-2, 40], [-24, -26], [-50, 30], [46, 32], [20, 0], [4, 4], [10, -4], [2, 18], [4, -22], [42, 8], [52, 28], [12, -36], [28, 4], [-22, 38], [20, 0], [-54, -20], [-6, 60], [-30, -26], [44, -34], [-36, 2], [8, -56], [-64, -20], [22, -44], [-2, 2], [48, 8], [6, 8], [-62, 0], [36, 44], [24, -20], [-8, -24], [36, 32], [12, 14], [6, -48], [52, -8], [46, 18], [42, -20], [14, -60], [6, 6], [-2, -36], [38, 26], [16, -26], [-30, 12], [14, -18], [-64, -4], [-52, 20], [12, -12], [-4, 66], [20, -32], [-64, 6], [-24, -8], [14, 0], [-24, 12], [-8, 48], [-18, 8], [-38, -32], [34, 20], [-8, 16], [32, 24], [-32, 20], [30, -48], [18, 36], [-20, -32], [-42, 14], [60, 16], [-4, -36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8085_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8085_2_a_ba();".
// 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_8085_2_a_ba(:prec:=2)
    chi := MakeCharacter_8085_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_8085_2_a_ba();".
// 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_8085_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8085_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 2, 1]>,<13,R![-32, 0, 1]>,<17,R![8, -8, 1]>],Snew);
    return Vf;
end function;