
// Make newform 8712.2.a.bb in Magma, downloaded from the LMFDB on 13 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8712_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_8712_2_a_bb();".
// 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_8712_2_a_bb();".
// 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 := [-4, -1, 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_8712_a();"
function MakeCharacter_8712_a()
    N := 8712;
    order := 1;
    char_gens := [6535, 4357, 1937, 5689];
    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_8712_a_Hecke( : Kf := Kf)
    return MakeCharacter_8712_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, -1], [2, -2], [0, 0], [0, 2], [2, 0], [4, 0], [-5, 1], [0, -2], [-3, -1], [-5, -1], [4, -2], [2, 2], [-8, 0], [-2, -4], [5, -5], [4, -2], [7, 1], [1, 3], [0, -2], [6, 2], [6, -2], [5, -3], [13, 1], [-2, 0], [0, 0], [2, -6], [-2, -4], [3, -1], [4, -4], [-2, 6], [11, -1], [-10, -2], [-6, 4], [-6, 6], [-7, 1], [-4, 0], [8, 0], [2, 4], [9, -1], [-9, 3], [13, -1], [4, 2], [6, -8], [8, -8], [12, -8], [1, -5], [18, 2], [7, 3], [8, -6], [-10, 2], [-24, -2], [1, -1], [-22, 4], [14, 2], [-10, 12], [8, 8], [18, 0], [6, 4], [-20, 0], [12, -6], [12, 8], [-20, 4], [3, -5], [27, -5], [17, 7], [-14, 12], [-8, 12], [22, 4], [1, 9], [-16, 8], [5, 7], [-2, 4], [13, -5], [-21, 9], [-1, -1], [-10, 8], [-10, -8], [-2, -8], [12, 8], [-18, 8], [22, 2], [-15, -3], [16, -8], [-19, 3], [-7, 17], [12, -14], [-4, 2], [23, -3], [5, -13], [-16, 0], [-3, 7], [24, -4], [12, -16], [-6, -10], [-15, 13], [-9, 3], [12, 0], [-12, 14], [32, 4], [-14, 4], [4, 8], [-32, 2], [4, -8], [13, 1], [4, 8], [-12, 6], [16, 0], [-14, 4], [10, 14], [24, -14], [-18, 8], [1, 7], [-19, -9], [13, 13], [19, -11], [-11, -1], [-7, -11], [14, -10], [29, 5], [18, -20], [10, -12], [24, -12], [-33, 9], [30, 8], [15, -5], [3, -15], [-31, 3], [4, -2], [18, -6], [-20, 12], [3, 1], [26, -4], [0, 2], [0, -10], [-10, -12], [40, -4], [27, -5], [-20, -10], [2, 2], [-2, -16], [-11, -17], [8, 12], [33, -7], [7, 5], [26, 8], [-16, 18], [7, 9], [16, 8], [-40, -6], [-1, -13], [-12, 8], [6, 10], [-4, 8], [-4, 12], [-22, 6], [-26, -8], [-4, 18], [-22, -4], [5, 3], [-10, 4], [-8, 0], [-9, 17], [-41, -5], [-13, -7], [16, -8], [-42, -4], [-14, 28], [-39, 13], [26, 2], [-26, 8], [4, 20], [36, 10], [-25, 13], [-37, -1], [-10, 14], [37, 1], [-4, -4], [12, -10], [-32, 8], [10, -6], [-19, 5], [30, -4], [41, -13], [54, 4], [-44, -2], [-16, -4], [-14, 4], [-42, 2], [-27, 9], [2, 10], [-5, -3], [18, 0], [8, 20], [-31, 1], [-22, -12], [31, 3], [46, 4], [54, -6], [-24, 6], [46, -6], [-30, 4], [-8, 14], [31, 9], [-17, 23], [11, 9], [12, 0], [-16, 18], [-32, -12], [-8, -18], [21, 1], [-27, 15], [20, -8], [-22, -10], [11, 3], [0, 16], [28, -18], [8, 24], [-31, -3], [16, 10], [16, -8], [35, -17], [47, -11], [-8, 4], [6, -4], [-6, -4], [-16, -8], [8, -16], [-8, 28], [-7, -15], [8, -4], [-42, 2], [16, 18], [23, 9], [-2, -6], [-11, -23], [-8, -2], [-25, -15], [-7, -5], [7, -7], [-48, 12], [36, 12], [-7, 17], [0, 10], [20, -28], [52, -12], [-12, 32], [-62, 2], [-50, 10], [-24, 26], [-2, 20], [29, 7], [-2, 28], [-2, 24], [-52, 0], [22, -16], [42, -6], [-17, 15], [22, 0], [-52, 4], [44, -16], [-6, 24], [10, 0], [-14, -4], [-16, -4], [-19, 17], [-1, 11], [10, -6], [-22, -12], [-15, 25], [41, -9], [42, -16], [36, 12], [-12, -14], [-7, -5], [15, 25], [38, 4], [-40, 6], [48, -4], [4, 28], [-17, -19], [48, -8], [-62, 0], [68, -8], [-23, 19], [-6, 24], [-26, 24], [-41, 5], [-14, 16], [11, -5], [-28, -16], [68, -2], [54, -10], [46, -4], [-20, 2], [35, -15], [-7, -19], [38, -10], [18, 10], [-2, 8], [-28, -6], [-44, -4], [-12, -16], [-35, 27], [3, 3], [15, 9], [-27, -11], [-36, -12], [-24, 18], [0, 0], [-70, 0], [-18, -12], [-25, 1], [-48, 8], [-10, 0], [-5, 5], [6, 2], [2, 0], [24, -14], [-16, -12], [5, 1], [-14, 4], [-25, 13], [-18, -4], [-1, 7], [-43, -5], [-23, 15], [44, -12], [76, -6], [-78, 8], [10, 8], [-74, 2], [32, -36], [40, 4], [-27, 27], [-31, -7], [-20, -2], [51, -5], [50, -10], [13, -11], [29, -27], [8, -18], [-23, 11], [37, 9], [6, -18], [27, -1], [13, -5], [-20, 20], [-39, 13], [16, -28], [39, -9], [-14, 32], [12, 4], [-2, -12], [-50, 28], [-23, -29], [34, -6], [18, 16], [13, 23], [10, 8], [38, -12], [-17, -27], [-18, -2], [-3, -29], [1, -27], [-36, 18], [-34, 2], [36, 10], [41, -33], [-8, -20], [2, 22], [-16, 38], [22, -6], [21, -11], [44, 8], [54, -14], [-30, 12], [-22, -8], [42, -4], [-30, -16], [61, -11], [-46, 14], [10, -8], [40, 12], [-45, -7], [-8, 32], [41, 9], [4, -24], [59, -7], [-15, 45], [-55, 13], [-31, -1], [-7, 7], [-69, 1], [-68, 18], [22, 2], [-22, -4], [28, -8], [30, -32], [50, 8], [35, -17], [32, 16], [-14, 36], [-84, -6], [14, 2], [22, -24], [48, -22], [4, 24], [-49, 1], [-18, 0], [6, -24], [52, 8], [-2, -34], [-42, 0], [-17, 7], [16, 0], [29, 7], [26, 4], [10, -26], [-71, 5], [8, -30], [8, -16], [-26, 38], [42, -32], [-3, 1], [55, -23], [-30, -8], [-35, 11], [-48, 8], [15, -33], [-46, 2], [-13, -19], [11, 25], [77, -3], [78, -16], [4, 10], [-27, -3], [-29, -3], [-16, 8], [-43, 27], [27, -25], [12, -34], [62, -22], [14, -32], [8, -14], [-14, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8712_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8712_2_a_bb();".
// 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_8712_2_a_bb(:prec:=2)
    chi := MakeCharacter_8712_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(3163) - 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_8712_2_a_bb();".
// 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_8712_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8712_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 3, 1]>,<7,R![-16, -2, 1]>,<13,R![-16, -2, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;