
// Make newform 1334.2.a.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1334_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_1334_2_a_c();".
// 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_1334_2_a_c();".
// 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_1334_a();"
function MakeCharacter_1334_a()
    N := 1334;
    order := 1;
    char_gens := [465, 553];
    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_1334_a_Hecke(Kf)
    return MakeCharacter_1334_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], [-3], [1], [-2], [-8], [-1], [-1], [9], [-10], [0], [-11], [9], [5], [8], [-6], [2], [6], [6], [1], [-4], [0], [6], [-2], [14], [-18], [5], [-10], [-8], [-4], [-6], [-10], [3], [6], [0], [-21], [14], [18], [-20], [11], [24], [-26], [-18], [-14], [27], [-10], [24], [-12], [19], [6], [-17], [-13], [21], [-1], [6], [-25], [-22], [-3], [22], [-12], [15], [32], [-17], [10], [19], [16], [22], [7], [-10], [13], [32], [27], [-16], [20], [-28], [25], [-7], [4], [12], [-8], [-24], [-22], [-6], [0], [-4], [-14], [8], [-24], [-7], [-27], [-24], [-31], [-24], [29], [-33], [21], [28], [-22], [40], [-10], [33], [30], [-40], [-2], [-32], [-17], [-9], [-42], [-13], [-31], [-12], [9], [22], [-24], [-20], [14], [-6], [-21], [38], [21], [-40], [36], [-10], [-25], [-29], [46], [8], [-2], [-27], [-24], [-8], [28], [-38], [-16], [36], [40], [-40], [-26], [-22], [-15], [32], [-27], [16], [35], [-14], [-23], [-31], [40], [-17], [40], [30], [-3], [4], [-55], [-32], [42], [-54], [29], [-33], [-51], [13], [56], [-37], [-37], [16], [6], [31], [36], [40], [63], [-24], [-6], [12], [-14], [18], [-38], [17], [-46], [-17], [-15], [-58], [9], [10], [15], [-46], [52], [-24], [-18], [49], [33], [-43], [16], [-60], [-18], [-66], [14], [-1], [40], [24], [-20], [4], [-18], [-36], [57], [-35], [32], [-21], [-11], [-32], [-10], [9], [-43], [-40], [-5], [58], [-28], [70], [-36], [50], [-30], [-2], [33], [-28], [-8], [48], [-72], [-13], [-36], [-4], [26], [-8], [10], [-50], [-32], [-54], [14], [-66], [-11], [19], [-42], [42], [-34], [6], [40], [-76], [-58], [42], [33], [16], [45], [17], [20], [-28], [-80], [10], [-41], [-30], [73], [24], [-51], [-22], [50], [74], [45], [-66], [-30], [-20], [-21], [-13], [-34], [11], [-36], [-22], [37], [12], [61], [52], [70], [-16], [-16], [84], [6], [-80], [52], [10], [-24], [-8], [37], [-57], [-72], [-39], [7], [40], [-1], [-17], [-21], [38], [28], [26], [18], [-16], [-33], [-32], [89], [63], [43], [20], [37], [-20], [-2], [6], [-60], [-30], [-47], [60], [-88], [19], [54], [-5], [-37], [-18], [-1], [-75], [32], [-36], [-46], [-83], [-51], [80], [81], [-14], [-24], [-25], [-24], [82], [-24], [14], [38], [56], [-66], [-42], [-38], [58], [-41], [-16], [-76], [9], [34], [20], [10], [-36], [82], [54], [46], [-81], [0], [-48], [36], [-1], [-7], [35], [-6], [-61], [8], [44], [-80], [0], [43], [-36], [44], [-4], [42], [-77], [92], [77], [38], [-31], [-39], [-94], [-41], [21], [64], [-26], [62], [40], [21], [-15], [16], [26], [-89], [45], [29], [91], [79], [52], [-24], [76], [-76], [-66], [-14], [44], [-1], [45], [-10], [17], [50], [-47], [74], [-22], [23], [66], [20], [-76], [-12], [-13], [78], [44], [33], [30], [88], [23], [-75], [-53], [-5], [-6], [4], [-30], [11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1334_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1334_2_a_c();".
// 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_1334_2_a_c(:prec:=1)
    chi := MakeCharacter_1334_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_1334_2_a_c();".
// 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_1334_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1334_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<5,R![-1, 1]>],Snew);
    return Vf;
end function;