
// Make newform 8450.2.a.p in Magma, downloaded from the LMFDB on 05 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_p();".
// 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_p();".
// 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], [-2], [0], [1], [-3], [0], [3], [4], [-6], [-3], [1], [-2], [0], [-10], [3], [3], [15], [-13], [13], [0], [10], [-4], [-15], [-6], [4], [9], [8], [-6], [-14], [18], [-16], [-6], [-18], [2], [-12], [-11], [5], [16], [0], [-15], [-24], [-25], [0], [4], [24], [20], [-22], [4], [-9], [-20], [6], [21], [-8], [0], [-15], [18], [21], [-23], [-22], [-24], [26], [12], [4], [18], [-19], [12], [4], [-13], [-6], [-32], [-6], [-9], [-10], [-31], [-11], [12], [-30], [-2], [-24], [-14], [6], [-20], [12], [-22], [8], [-6], [6], [10], [36], [-29], [30], [-27], [-11], [-30], [7], [24], [36], [-6], [-4], [28], [-22], [-12], [-36], [21], [8], [-32], [-33], [-42], [-42], [5], [14], [-44], [6], [-44], [-20], [3], [4], [18], [21], [-12], [10], [23], [6], [-39], [-17], [3], [-8], [18], [44], [22], [43], [3], [14], [-13], [24], [40], [-24], [7], [-33], [6], [43], [0], [-4], [-15], [17], [24], [-26], [-6], [-10], [39], [46], [-39], [-28], [-12], [44], [36], [44], [6], [-13], [-30], [-3], [-3], [13], [-30], [30], [9], [50], [35], [-2], [57], [36], [-56], [48], [40], [-22], [27], [-29], [42], [2], [-55], [64], [-12], [26], [48], [-9], [-54], [47], [-17], [34], [-51], [2], [-27], [-16], [18], [-12], [-63], [46], [47], [42], [36], [-6], [2], [-44], [5], [-39], [42], [55], [0], [24], [-58], [-43], [42], [26], [-21], [-9], [-62], [44], [3], [-3], [54], [-58], [64], [0], [-29], [24], [17], [6], [48], [-10], [63], [2], [74], [-47], [54], [2], [24], [10], [-36], [-18], [-30], [60], [56], [44], [-2], [54], [48], [-20], [36], [49], [-66], [4], [12], [24], [71], [6], [15], [-58], [-35], [-45], [34], [-16], [-42], [52], [-70], [-24], [-10], [-60], [-72], [13], [33], [-58], [-17], [28], [-4], [-19], [13], [-12], [40], [-2], [24], [-72], [40], [30], [-44], [49], [-30], [-61], [6], [31], [-15], [27], [24], [54], [21], [-79], [-45], [-34], [-42], [-42], [49], [-19], [-66], [-10], [-84], [44], [-20], [36], [26], [-12], [-25], [12], [-60], [69], [-16], [-87], [-22], [-3], [-24], [-44], [51], [38], [-32], [-30], [49], [-66], [-1], [-11], [-77], [12], [21], [-80], [66], [62], [-9], [61], [-39], [-62], [0], [70], [-52], [16], [-6], [48], [68], [-12], [6], [-79], [85], [9], [-21], [25], [-92], [-42], [38], [86], [-27], [3], [21], [-63], [57], [34], [6], [84], [-36], [-28], [-10], [-54], [-35], [-61], [24], [8], [72], [15], [-94], [23], [87], [-84], [-2], [3], [-10], [72], [-24], [43], [72], [-44], [-48], [-59], [-7], [4], [78], [-86], [-42], [-9], [-64], [99], [-97], [25], [99], [38], [-30], [-86], [51], [67], [12], [-42], [-52], [82], [48], [-59], [12], [-19], [63], [48], [-28], [-70], [99], [48], [-28], [-18], [-66], [-27], [70], [-3], [-84], [-38], [6], [30], [78], [-23], [-24]];
    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_p();".
// 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_p(: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_p();".
// 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_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![-1, 1]>,<11,R![3, 1]>,<17,R![-3, 1]>,<31,R![-1, 1]>],Snew);
    return Vf;
end function;