
// Make newform 6762.2.a.p in Magma, downloaded from the LMFDB on 25 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_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_6762_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_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], [2], [0], [-6], [-6], [-2], [4], [1], [2], [6], [8], [0], [2], [-6], [4], [8], [-2], [6], [-8], [4], [4], [12], [6], [-10], [6], [-8], [6], [12], [18], [8], [0], [18], [4], [-12], [0], [18], [-4], [6], [10], [-24], [-2], [-16], [14], [-18], [20], [8], [10], [4], [14], [10], [0], [-14], [-12], [-12], [16], [-6], [26], [2], [-18], [-28], [6], [-32], [6], [34], [-18], [-12], [-18], [12], [10], [-16], [0], [20], [-8], [26], [0], [-24], [26], [2], [0], [-20], [-24], [28], [38], [14], [36], [-38], [-34], [-30], [40], [12], [20], [0], [-28], [28], [12], [42], [-14], [-4], [-22], [-28], [-32], [36], [-46], [-10], [28], [44], [-4], [40], [-4], [-42], [-44], [-2], [36], [0], [26], [28], [-30], [-2], [10], [-10], [14], [-38], [-48], [20], [-40], [-8], [42], [-12], [-34], [-28], [-36], [-4], [44], [-36], [-30], [6], [44], [-18], [-26], [40], [18], [40], [-2], [38], [-32], [42], [16], [48], [-24], [-2], [42], [16], [38], [38], [-16], [20], [-16], [46], [-54], [-4], [-42], [-40], [20], [42], [-36], [-32], [46], [22], [-6], [54], [50], [28], [46], [-34], [14], [-64], [54], [-36], [-14], [32], [-52], [54], [-40], [60], [-10], [22], [-4], [46], [10], [-48], [-64], [38], [26], [18], [6], [14], [0], [42], [2], [16], [-38], [54], [-10], [-12], [-66], [24], [8], [6], [-24], [-14], [54], [20], [-36], [42], [-62], [32], [12], [20], [14], [-62], [28], [2], [-24], [28], [-46], [-42], [32], [32], [0], [-66], [4], [40], [26], [4], [-16], [-22], [-64], [44], [-38], [46], [76], [6], [-34], [36], [-66], [-78], [36], [-22], [-60], [36], [30], [-16], [-26], [54], [24], [-20], [-52], [-66], [32], [-60], [26], [30], [-54], [-8], [36], [24], [-10], [-14], [-62], [2], [-18], [-6], [20], [-68], [26], [60], [-26], [18], [-36], [-10], [-52], [-4], [-30], [8], [24], [-26], [-16], [-66], [-64], [60], [4], [66], [60], [-46], [-62], [-24], [26], [40], [40], [70], [56], [-80], [48], [38], [-34], [-40], [-18], [-64], [14], [-30], [20], [18], [64], [44], [-54], [40], [16], [-38], [-78], [-20], [-30], [78], [16], [22], [82], [-22], [4], [-36], [56], [10], [30], [26], [48], [-14], [48], [0], [-18], [-40], [76], [54], [18], [50], [-24], [24], [48], [58], [48], [-52], [22], [68], [-74], [30], [-32], [-12], [72], [32], [60], [30], [8], [-84], [8], [24], [-18], [-88], [22], [18], [16], [30], [-44], [-64], [-32], [-32], [-64], [-10], [84], [4], [-74], [26], [44], [30], [-34], [-48], [30], [-76], [62], [-32], [14], [54], [28], [70], [62], [-60], [-44], [-88], [-22], [20], [18], [-58], [-18], [-58], [-30], [68], [58], [-46], [90], [-28], [18], [4], [56], [-16], [62], [-46], [2], [-34], [-30], [24], [10], [-54], [84], [-24], [6], [-40], [-56], [16], [-66], [-38], [-74], [36], [-62]];
    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_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_6762_2_a_p(: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_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_6762_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![6, 1]>,<13,R![6, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;