
// Make newform 6762.2.a.f in Magma, downloaded from the LMFDB on 18 April 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_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_6762_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_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], [0], [0], [-2], [6], [-2], [6], [1], [-6], [0], [0], [-6], [-6], [-8], [-4], [0], [8], [-2], [0], [2], [8], [2], [2], [-14], [14], [4], [2], [-12], [6], [-8], [0], [-6], [0], [4], [-24], [20], [16], [-16], [-6], [-8], [8], [-12], [10], [10], [-16], [12], [-16], [-26], [-24], [26], [24], [2], [2], [2], [-24], [-6], [-8], [-22], [-10], [-30], [24], [-12], [16], [26], [2], [20], [-6], [8], [2], [14], [-28], [-16], [-8], [-34], [0], [0], [14], [2], [-10], [-38], [0], [-36], [34], [16], [-32], [6], [-18], [10], [16], [-22], [24], [-8], [0], [16], [-12], [-14], [30], [-18], [-22], [4], [-24], [6], [-46], [-2], [-6], [-48], [-14], [-40], [34], [-48], [-20], [-42], [-14], [4], [-30], [26], [0], [22], [-6], [8], [-50], [8], [44], [8], [24], [32], [40], [-40], [-20], [8], [-32], [20], [-28], [-42], [34], [-28], [10], [52], [26], [48], [-42], [-8], [42], [34], [0], [-34], [-42], [20], [-56], [18], [-30], [-36], [-48], [-38], [44], [0], [-30], [38], [48], [32], [58], [-56], [6], [10], [36], [-24], [-38], [-10], [-58], [-30], [50], [0], [34], [16], [26], [16], [-18], [28], [-28], [-8], [18], [-14], [-58], [36], [4], [-58], [10], [-10], [48], [-14], [-44], [-38], [6], [18], [42], [50], [-16], [-6], [-24], [16], [32], [-38], [38], [46], [34], [64], [-4], [70], [-44], [14], [-66], [44], [-2], [-64], [18], [48], [46], [8], [-10], [-2], [36], [66], [-28], [-36], [-58], [-46], [-24], [44], [24], [-54], [-14], [-4], [54], [-38], [28], [-38], [-16], [-4], [0], [-58], [12], [-40], [38], [70], [56], [0], [50], [-22], [-56], [-20], [-30], [-48], [-26], [-2], [48], [68], [-2], [-22], [22], [44], [-6], [-14], [16], [30], [26], [80], [-54], [-22], [34], [2], [-38], [46], [76], [-2], [16], [4], [-50], [26], [54], [0], [4], [-52], [40], [-4], [-72], [-34], [40], [8], [-30], [60], [-30], [6], [-18], [26], [-16], [24], [78], [-60], [16], [34], [0], [32], [28], [38], [34], [56], [-40], [24], [14], [-72], [60], [-14], [-76], [20], [14], [20], [-40], [14], [-18], [-30], [-10], [-34], [16], [-6], [2], [10], [-64], [48], [8], [-62], [46], [-24], [0], [-30], [60], [-24], [42], [-74], [-20], [54], [74], [6], [-8], [0], [64], [-66], [-32], [36], [-40], [-4], [-26], [42], [-32], [-28], [6], [56], [-54], [34], [-80], [-44], [74], [-88], [-10], [48], [54], [46], [32], [46], [24], [28], [0], [-72], [-76], [94], [8], [20], [-66], [54], [-82], [-20], [-54], [-56], [14], [30], [40], [72], [98], [-74], [-68], [90], [66], [12], [-16], [-52], [-58], [-28], [46], [34], [-18], [6], [46], [-40], [78], [-66], [-8], [68], [-34], [2], [20], [-62], [-74], [10], [-98], [74], [18], [48], [88], [30], [92], [-72], [-4], [88], [-32], [-42], [-66], [-90], [-22], [-72], [40]];
    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_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_6762_2_a_f(: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_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_6762_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![2, 1]>,<13,R![-6, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;