
// Make newform 6762.2.a.k in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6762_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_6762_2_a_k();".
// 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_6762_2_a_k();".
// 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_6762_a();"
function MakeCharacter_6762_a()
    N := 6762;
    order := 1;
    char_gens := [2255, 3727, 3823];
    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_6762_a_Hecke(Kf)
    return MakeCharacter_6762_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], [-3], [0], [2], [7], [-4], [6], [1], [7], [8], [-5], [-5], [-5], [-3], [6], [8], [-12], [8], [-12], [14], [-6], [-12], [8], [3], [4], [13], [-12], [-1], [1], [13], [0], [-3], [-7], [-14], [-19], [22], [-8], [8], [6], [9], [-22], [20], [-21], [-3], [-1], [-2], [26], [3], [-16], [28], [20], [5], [11], [14], [-1], [12], [-18], [6], [27], [28], [-14], [11], [-12], [14], [17], [-20], [14], [29], [10], [-15], [-19], [-1], [-22], [25], [30], [-30], [22], [-2], [30], [-20], [15], [-25], [9], [-4], [-37], [14], [-16], [12], [-17], [9], [-34], [-17], [-12], [-38], [0], [26], [-10], [10], [32], [14], [24], [15], [35], [-28], [-22], [6], [27], [26], [-4], [22], [9], [-22], [-20], [18], [35], [26], [-28], [-21], [-34], [10], [-1], [-26], [-4], [39], [-12], [46], [-47], [12], [46], [-44], [-16], [2], [-46], [6], [-49], [33], [22], [-35], [46], [33], [-38], [-3], [-42], [-54], [40], [25], [-5], [45], [16], [32], [54], [36], [-20], [-7], [33], [-32], [-33], [11], [-39], [-21], [6], [40], [-4], [1], [-24], [-16], [-50], [-6], [-32], [12], [-53], [39], [2], [32], [-6], [24], [-14], [-29], [8], [8], [41], [36], [1], [-51], [29], [-40], [56], [35], [1], [-58], [3], [-44], [28], [48], [-24], [-44], [-26], [24], [11], [6], [12], [-49], [-6], [-23], [48], [7], [-4], [24], [0], [23], [-30], [34], [48], [39], [-34], [-24], [-53], [3], [51], [-66], [4], [36], [-20], [-40], [30], [34], [-54], [-43], [52], [12], [-20], [32], [46], [34], [12], [53], [-36], [15], [-68], [-16], [-20], [-34], [-4], [1], [-17], [2], [-33], [71], [39], [57], [20], [3], [-27], [-66], [13], [-33], [-34], [-6], [-14], [-50], [48], [-53], [16], [42], [34], [-8], [52], [51], [58], [-41], [-60], [7], [-26], [-26], [-28], [21], [-26], [-4], [63], [-15], [-54], [-58], [-28], [17], [24], [-50], [-8], [46], [-65], [-60], [1], [40], [-7], [-70], [-22], [-65], [50], [-26], [-15], [-15], [-38], [-14], [43], [59], [-39], [13], [-22], [25], [-80], [51], [-36], [-34], [26], [-32], [70], [-30], [-8], [-21], [42], [42], [34], [28], [53], [-43], [2], [-52], [-11], [-51], [-68], [-84], [-93], [-60], [-40], [35], [-26], [-73], [-86], [86], [17], [-10], [-25], [-30], [36], [60], [-51], [-25], [34], [-88], [-57], [-64], [-35], [-59], [-64], [43], [-79], [80], [60], [-42], [66], [-10], [-89], [12], [54], [65], [83], [76], [54], [-26], [96], [-61], [84], [66], [-64], [6], [32], [-59], [-17], [34], [-2], [-18], [-22], [-48], [-18], [-50], [-1], [-65], [59], [53], [-56], [-38], [66], [17], [-15], [72], [-2], [-11], [-34], [-33], [99], [-13], [44], [82], [-48], [22], [-22], [73], [-62], [50], [32], [26], [-73], [96], [100], [4], [-74], [26], [52], [84], [-66], [54], [82], [54], [2], [96], [-26], [-50], [44], [87], [9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6762_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6762_2_a_k();".
// 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_6762_2_a_k(:prec:=1)
    chi := MakeCharacter_6762_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_6762_2_a_k();".
// 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_6762_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, 1]>,<11,R![-2, 1]>,<13,R![-7, 1]>,<17,R![4, 1]>],Snew);
    return Vf;
end function;