
// Make newform 8464.2.a.f in Magma, downloaded from the LMFDB on 27 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8464_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_8464_2_a_f();".
// 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_8464_2_a_f();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8464_a();"
function MakeCharacter_8464_a()
    N := 8464;
    order := 1;
    char_gens := [7407, 2117, 6353];
    v := [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_8464_a_Hecke( : Kf := Kf)
    return MakeCharacter_8464_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], [-1], [0], [2], [0], [-1], [6], [2], [0], [-3], [-5], [-8], [3], [8], [-9], [-6], [12], [-14], [8], [15], [-7], [-10], [6], [0], [10], [6], [2], [-6], [16], [-18], [-11], [-21], [12], [-11], [-6], [7], [4], [1], [-12], [6], [9], [16], [0], [-7], [-15], [-4], [16], [-8], [18], [4], [-9], [-21], [4], [-18], [27], [-6], [-15], [-8], [-19], [-30], [-4], [-12], [-20], [15], [4], [-18], [25], [-8], [12], [29], [-27], [-30], [26], [-20], [-28], [-36], [36], [11], [-24], [17], [36], [-2], [12], [16], [-35], [-21], [6], [10], [-27], [16], [-30], [42], [-23], [-3], [-5], [6], [-3], [24], [38], [-7], [1], [-24], [-12], [-24], [-34], [-7], [-21], [-42], [-24], [-19], [-8], [-26], [18], [20], [-40], [-24], [-4], [-33], [27], [6], [-14], [-13], [-18], [-39], [28], [-30], [-38], [0], [-28], [22], [25], [6], [-4], [-20], [-15], [28], [-24], [-40], [30], [-30], [7], [-6], [-41], [-36], [38], [18], [2], [-33], [19], [9], [50], [-30], [52], [51], [50], [18], [38], [15], [28], [-30], [3], [6], [-47], [18], [-48], [-18], [16], [62], [10], [15], [36], [-31], [42], [-8], [25], [6], [-35], [-33], [14], [46], [49], [12], [5], [-6], [-36], [18], [2], [26], [29], [-48], [23], [48], [44], [-6], [48], [-24], [16], [-20], [-36], [-15], [60], [-8], [17], [-32], [24], [-3], [-4], [51], [51], [-35], [-25], [-9], [44], [-54], [48], [-8], [31], [-3], [36], [63], [23], [32], [-15], [-34], [63], [-73], [36], [45], [14], [-36], [-46], [2], [-40], [-6], [-28], [-66], [-20], [-24], [39], [9], [48], [-53], [-53], [-13], [6], [-21], [-5], [-54], [14], [36], [22], [48], [-42], [-50], [18], [-15], [28], [80], [45], [-43], [56], [36], [26], [22], [-42], [-10], [-72], [36], [-22], [-39], [-7], [-58], [64], [-22], [41], [43], [48], [-46], [52], [36], [21], [-40], [-12], [70], [-68], [12], [46], [66], [52], [-30], [30], [-6], [-30], [-42], [-55], [-78], [38], [39], [45], [37], [58], [30], [-34], [-39], [32], [11], [-12], [-80], [12], [26], [75], [-54], [42], [-11], [-66], [-74], [-63], [-9], [64], [39], [-4], [46], [27], [-20], [60], [-20], [56], [1], [12], [-18], [5], [-33], [61], [81], [-34], [-15], [4], [-54], [-13], [56], [47], [-48], [-30], [2], [84], [-24], [-25], [28], [-30], [-48], [-77], [41], [78], [-46], [-26], [-6], [30], [48], [-39], [-24], [-20], [-51], [-63], [24], [49], [53], [39], [20], [-50], [57], [-20], [-69], [42], [92], [47], [-39], [84], [-26], [36], [-1], [-36], [6], [76], [39], [38], [9], [37], [38], [-4], [-42], [-26], [6], [27], [-20], [-78], [64], [-64], [-102], [-58], [-78], [-85], [9], [-94], [6], [-81], [7], [46], [57], [74], [36], [-1], [-9], [36], [50], [-14], [-39], [72], [-32], [60], [-24], [78], [16], [-24], [-12], [101], [75], [78], [-6], [43], [27]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8464_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8464_2_a_f();".
// 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_8464_2_a_f(:prec:=1)
    chi := MakeCharacter_8464_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_8464_2_a_f();".
// 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_8464_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8464_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<5,R![0, 1]>,<7,R![-2, 1]>,<13,R![1, 1]>],Snew);
    return Vf;
end function;