
// Make newform 6240.2.a.n in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6240_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_6240_2_a_n();".
// 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_6240_2_a_n();".
// 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_6240_a();"
function MakeCharacter_6240_a()
    N := 6240;
    order := 1;
    char_gens := [1951, 2341, 2081, 2497, 5761];
    v := [1, 1, 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_6240_a_Hecke(Kf)
    return MakeCharacter_6240_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 := [[0], [-1], [1], [2], [4], [1], [0], [-6], [-6], [-8], [0], [10], [-10], [-12], [-12], [-6], [12], [-10], [-4], [-8], [-4], [-8], [-4], [10], [-4], [12], [4], [-4], [4], [-4], [4], [-10], [14], [-12], [18], [-16], [18], [16], [0], [-6], [-6], [10], [0], [24], [18], [-16], [-16], [14], [4], [20], [0], [-16], [26], [18], [-28], [-26], [-28], [-16], [10], [-30], [12], [-6], [28], [8], [6], [-18], [10], [-18], [16], [-36], [-2], [-24], [28], [10], [6], [0], [-4], [26], [30], [22], [30], [-8], [-32], [-34], [8], [-8], [34], [20], [-10], [-2], [-20], [16], [18], [-2], [14], [-42], [-42], [-10], [20], [16], [20], [-18], [0], [-18], [-28], [32], [12], [2], [8], [-10], [-20], [-22], [6], [30], [-36], [42], [28], [38], [-18], [38], [24], [-14], [-42], [44], [42], [12], [-20], [-36], [-32], [42], [-10], [16], [-16], [-2], [-42], [18], [-6], [-32], [-30], [-22], [-46], [-22], [36], [-36], [-26], [40], [42], [-44], [-40], [-24], [-2], [-50], [-4], [10], [-12], [-44], [-24], [10], [34], [10], [52], [8], [2], [18], [18], [32], [16], [34], [-30], [34], [8], [36], [-60], [48], [0], [30], [42], [-54], [40], [18], [10], [38], [-50], [34], [-32], [-24], [-18], [28], [-26], [-32], [34], [-36], [16], [-6], [0], [4], [-2], [-62], [2], [-2], [-2], [16], [62], [-2], [4], [-26], [-40], [24], [-42], [-40], [22], [-40], [-16], [4], [-56], [-14], [-8], [46], [72], [14], [10], [4], [-42], [-38], [-56], [-70], [48], [36], [40], [-36], [18], [32], [64], [-18], [-76], [24], [70], [-2], [18], [60], [52], [-40], [40], [-24], [66], [-16], [14], [-24], [34], [6], [34], [-46], [-12], [-70], [-14], [40], [-2], [-8], [2], [-44], [-24], [-12], [-52], [-2], [42], [20], [-10], [-54], [-68], [-58], [-62], [-16], [-64], [32], [-6], [34], [12], [-8], [38], [10], [14], [4], [-26], [60], [44], [-36], [-74], [-54], [-20], [-14], [32], [-60], [-14], [-48], [-58], [-20], [0], [30], [-18], [8], [22], [26], [-8], [56], [40], [16], [-4], [-14], [48], [6], [-62], [-82], [-18], [44], [-68], [-38], [60], [32], [-44], [34], [-36], [44], [44], [-34], [-30], [-18], [-82], [-36], [6], [2], [-32], [6], [-16], [-12], [-22], [-44], [8], [-18], [-10], [68], [-90], [-20], [10], [-48], [-46], [-50], [14], [-92], [-48], [-10], [-18], [24], [90], [16], [34], [-24], [-96], [28], [48], [-36], [-18], [50], [-42], [40], [-28], [-34], [18], [-26], [90], [-58], [60], [24], [16], [-40], [-42], [-16], [16], [-64], [-10], [-26], [-30], [-26], [14], [-78], [22], [-60], [-60], [102], [-76], [78], [2], [-42], [-68], [-76], [72], [-2], [32], [18], [-64], [-10], [-80], [92], [62], [-78], [78], [104], [-62], [34], [-24], [88], [-74], [-62], [0], [-28], [-6], [56], [48], [92], [34], [-66], [56], [30], [36], [34], [44], [-50], [-12], [90], [-58], [-16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6240_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6240_2_a_n();".
// 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_6240_2_a_n(:prec:=1)
    chi := MakeCharacter_6240_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_6240_2_a_n();".
// 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_6240_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![-4, 1]>,<17,R![0, 1]>,<19,R![6, 1]>],Snew);
    return Vf;
end function;