
// Make newform 6762.2.a.bl in Magma, downloaded from the LMFDB on 17 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_bl();".
// 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_bl();".
// 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], [-4], [2], [6], [-4], [-1], [-2], [8], [6], [6], [-4], [8], [6], [-4], [10], [4], [-8], [6], [0], [12], [-2], [-10], [-6], [-8], [12], [14], [18], [-16], [-12], [-6], [12], [6], [8], [-22], [4], [0], [2], [-12], [2], [-16], [2], [-26], [24], [4], [-24], [12], [-14], [10], [-16], [22], [-12], [-18], [-24], [2], [-24], [22], [10], [20], [2], [-12], [16], [14], [-2], [-4], [-30], [-36], [2], [30], [-24], [0], [6], [-36], [-16], [6], [2], [34], [-10], [12], [-10], [0], [-26], [16], [-20], [2], [-22], [-30], [0], [-4], [32], [8], [-4], [-28], [-24], [-14], [30], [4], [-34], [20], [30], [-4], [-22], [-20], [14], [12], [-34], [24], [22], [24], [6], [26], [-12], [40], [18], [44], [0], [14], [20], [18], [-30], [-14], [-36], [52], [-34], [-26], [-8], [8], [26], [4], [8], [32], [22], [38], [38], [-46], [-20], [-54], [42], [-52], [-42], [56], [-36], [34], [40], [-38], [-42], [28], [-16], [-18], [-42], [4], [-48], [-4], [48], [-24], [-18], [-18], [-38], [52], [-6], [40], [-44], [-62], [-40], [-32], [58], [-14], [-54], [-4], [34], [24], [58], [40], [-2], [60], [-26], [24], [10], [-16], [12], [-26], [22], [-64], [50], [-2], [-4], [10], [-8], [-34], [-36], [4], [-30], [-12], [14], [-30], [30], [6], [48], [-2], [-8], [-54], [-10], [-28], [18], [-48], [36], [-54], [44], [-46], [42], [40], [-44], [0], [62], [-32], [14], [40], [-34], [6], [-24], [-62], [16], [-28], [22], [30], [0], [8], [-36], [14], [-36], [-48], [-54], [-28], [-32], [22], [6], [-36], [-64], [-44], [28], [16], [14], [-34], [0], [-8], [-4], [-36], [-64], [14], [-10], [8], [46], [-14], [20], [22], [-44], [-54], [-42], [16], [-60], [-6], [-22], [46], [76], [-2], [-34], [28], [6], [50], [-76], [46], [80], [62], [64], [12], [-34], [74], [28], [-24], [-24], [-8], [-62], [-4], [-48], [50], [70], [-64], [30], [-34], [28], [58], [20], [-82], [-6], [64], [10], [-36], [-60], [46], [46], [-16], [20], [-52], [-30], [-68], [58], [-24], [70], [72], [22], [-14], [-28], [-24], [-34], [-60], [48], [70], [50], [44], [-22], [-62], [48], [10], [-10], [-76], [12], [32], [-58], [46], [-2], [-24], [4], [-52], [44], [94], [54], [22], [-32], [86], [10], [-6], [72], [30], [84], [42], [28], [16], [-14], [68], [74], [78], [32], [-90], [-58], [48], [4], [-14], [-24], [-42], [-10], [-32], [-52], [52], [-54], [18], [24], [-6], [52], [-52], [48], [6], [-40], [-66], [-12], [-48], [-74], [54], [-90], [-54], [-22], [-40], [-94], [44], [64], [48], [10], [-4], [-48], [50], [-54], [12], [-92], [88], [74], [-80], [30], [60], [-42], [-14], [66], [64], [78], [-38], [-48], [62], [-78], [44], [20], [-82], [-42], [60], [-12], [58], [-30], [32], [48], [54], [8], [30], [-78], [-48], [-20], [-58], [-14], [68], [58], [60], [0]];
    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_bl();".
// 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_bl(: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_bl();".
// 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_bl( : 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![4, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;