
// Make newform 6240.2.a.c in Magma, downloaded from the LMFDB on 19 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_c();".
// 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_c();".
// 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], [0], [-4], [1], [-2], [4], [8], [-6], [0], [-10], [6], [-4], [0], [6], [12], [-10], [-12], [-12], [-10], [8], [16], [14], [-2], [10], [4], [12], [2], [-18], [4], [-4], [22], [-16], [-6], [16], [-2], [-4], [24], [22], [-12], [22], [8], [14], [-22], [8], [8], [24], [-8], [-6], [-18], [20], [-22], [-12], [30], [16], [-14], [-8], [6], [-10], [4], [2], [20], [0], [-14], [18], [36], [26], [28], [10], [14], [-20], [12], [-10], [28], [8], [18], [30], [22], [-22], [-12], [-14], [-28], [-22], [8], [28], [22], [-10], [-14], [32], [-36], [28], [40], [-36], [4], [16], [34], [42], [4], [42], [-4], [-6], [-12], [10], [-32], [14], [0], [30], [8], [10], [28], [22], [-42], [4], [8], [-46], [-36], [16], [-2], [-20], [-38], [18], [-18], [0], [-28], [10], [-38], [-32], [-4], [22], [20], [-48], [-32], [22], [-42], [-14], [34], [28], [-2], [50], [12], [-6], [-28], [-24], [6], [12], [-42], [38], [-16], [56], [22], [-30], [-20], [-40], [12], [32], [-16], [-18], [-54], [34], [-32], [30], [56], [44], [30], [-56], [-40], [54], [10], [-18], [4], [10], [-24], [-18], [16], [-30], [-12], [42], [28], [62], [32], [-28], [-26], [-42], [48], [10], [30], [4], [-38], [-36], [-46], [-8], [0], [2], [28], [-26], [-6], [30], [-18], [-56], [-14], [-48], [-42], [34], [-36], [6], [-40], [-4], [14], [-40], [10], [42], [-28], [24], [-20], [58], [44], [-6], [24], [-22], [-10], [-8], [6], [32], [-28], [-10], [-10], [0], [-4], [-60], [-18], [56], [-8], [-46], [44], [56], [18], [34], [20], [-48], [16], [56], [-28], [-30], [14], [-8], [-8], [-20], [-12], [-32], [-42], [14], [72], [26], [6], [60], [62], [4], [-42], [-2], [-44], [-12], [26], [-34], [-18], [8], [-46], [54], [44], [70], [-18], [-68], [-82], [24], [-78], [-56], [8], [-62], [18], [-60], [-24], [-32], [48], [66], [36], [-16], [58], [2], [-64], [10], [66], [4], [6], [-20], [14], [26], [40], [14], [60], [12], [-30], [-62], [16], [36], [-16], [22], [68], [-2], [-36], [-42], [-56], [42], [-62], [-52], [16], [-38], [44], [12], [46], [34], [-40], [-34], [42], [24], [22], [50], [-4], [44], [0], [78], [10], [-2], [-80], [88], [-52], [80], [82], [30], [-22], [-20], [-50], [6], [-6], [-72], [-78], [-20], [14], [44], [-12], [-66], [60], [-2], [-62], [68], [6], [-18], [-76], [36], [-42], [16], [46], [10], [-8], [4], [-60], [66], [18], [24], [10], [12], [-16], [64], [26], [48], [-2], [60], [88], [-50], [-38], [82], [-46], [54], [-40], [-34], [60], [48], [-80], [70], [36], [72], [-54], [18], [100], [4], [20], [-14], [-64], [10], [40], [-86], [2], [-42], [40], [70], [26], [-8], [-10], [-42], [-60], [-68], [-6], [22], [-12], [-8], [-6], [-14], [-12], [-28], [78], [-80], [-22], [62], [-32], [60], [-50], [-102], [-60], [-66], [4], [-48]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<17,R![2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;