
// Make newform 8450.2.a.m in Magma, downloaded from the LMFDB on 13 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8450_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_8450_2_a_m();".
// 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_8450_2_a_m();".
// 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_8450_a();"
function MakeCharacter_8450_a()
    N := 8450;
    order := 1;
    char_gens := [677, 3551];
    v := [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_8450_a_Hecke( : Kf := Kf)
    return MakeCharacter_8450_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 := [[1], [-3], [0], [0], [3], [0], [7], [-1], [4], [4], [10], [12], [5], [-12], [-4], [-6], [4], [4], [-5], [0], [-11], [4], [15], [11], [-2], [12], [-14], [9], [8], [-19], [-10], [4], [-9], [-5], [0], [2], [-12], [-1], [-14], [18], [-5], [16], [-18], [15], [16], [-6], [-11], [-14], [-4], [16], [-6], [-20], [-23], [-19], [10], [-8], [24], [4], [2], [-2], [11], [30], [25], [12], [-6], [10], [-25], [-27], [25], [10], [-6], [-4], [0], [-24], [21], [26], [-12], [18], [33], [-23], [-9], [-16], [24], [23], [30], [39], [9], [31], [10], [14], [-20], [10], [-32], [-4], [-20], [16], [-24], [-7], [5], [26], [13], [14], [0], [13], [32], [-19], [-5], [29], [-10], [-25], [42], [2], [-22], [-36], [0], [18], [-44], [46], [2], [-15], [-22], [-10], [44], [-35], [31], [32], [10], [-50], [10], [14], [24], [-8], [-22], [-28], [-27], [5], [-38], [-4], [-8], [-22], [-12], [-36], [4], [-21], [-10], [-10], [-20], [-21], [-13], [-50], [-40], [14], [-55], [-44], [-44], [-6], [8], [-22], [13], [-18], [32], [7], [-36], [-13], [-9], [-24], [14], [56], [45], [-36], [5], [26], [30], [17], [-16], [-9], [-55], [-26], [22], [-6], [-32], [-45], [-8], [-15], [54], [28], [-14], [29], [41], [-34], [9], [27], [-51], [-6], [25], [2], [43], [-4], [41], [24], [66], [20], [-16], [-1], [36], [34], [-46], [-36], [10], [4], [38], [-18], [24], [-11], [14], [69], [-10], [-27], [-50], [26], [-22], [-2], [-59], [-50], [25], [66], [21], [24], [70], [-23], [22], [37], [8], [-21], [-4], [22], [49], [16], [13], [2], [45], [13], [-52], [50], [57], [-46], [52], [0], [13], [54], [-52], [-45], [32], [-21], [-42], [-27], [2], [-9], [-60], [3], [-40], [-9], [8], [-16], [27], [11], [-82], [6], [-23], [48], [36], [23], [1], [-52], [-38], [-64], [45], [-56], [-42], [23], [4], [60], [-48], [28], [79], [40], [-26], [52], [64], [-18], [66], [-36], [81], [-59], [4], [42], [26], [16], [-48], [43], [-49], [-52], [-22], [1], [-35], [7], [-36], [16], [-60], [-16], [-18], [-50], [21], [-28], [36], [-26], [-29], [72], [33], [15], [76], [-53], [42], [74], [-10], [-14], [-43], [4], [10], [0], [-14], [-48], [-92], [81], [15], [-36], [-38], [-34], [47], [-32], [-34], [-45], [-90], [-40], [-24], [-44], [60], [-5], [90], [12], [-71], [38], [-22], [-16], [-56], [81], [-42], [23], [54], [-90], [70], [-45], [30], [-36], [-17], [-51], [82], [14], [-59], [-72], [11], [46], [-96], [-6], [88], [63], [-2], [21], [-39], [46], [-28], [27], [-18], [-39], [20], [-30], [-86], [62], [4], [-66], [79], [-38], [21], [23], [40], [-25], [76], [61], [55], [-26], [-64], [43], [54], [-25], [10], [46], [-24], [42], [-32], [77], [1], [-4], [-13], [76], [-55], [18], [-66], [54], [-77], [-90], [28], [38], [24], [-27], [91], [-84], [-12], [-65], [67], [-44]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8450_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8450_2_a_m();".
// 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_8450_2_a_m(:prec:=1)
    chi := MakeCharacter_8450_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_8450_2_a_m();".
// 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_8450_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<7,R![0, 1]>,<11,R![-3, 1]>,<17,R![-7, 1]>,<31,R![-10, 1]>],Snew);
    return Vf;
end function;