
// Make newform 6080.2.a.k in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6080_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_6080_2_a_k();".
// 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_6080_2_a_k();".
// 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_6080_a();"
function MakeCharacter_6080_a()
    N := 6080;
    order := 1;
    char_gens := [191, 5701, 1217, 1921];
    v := [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_6080_a_Hecke(Kf)
    return MakeCharacter_6080_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], [0], [-1], [0], [4], [6], [-6], [1], [8], [2], [0], [-2], [2], [-4], [-8], [6], [4], [2], [-8], [8], [2], [-8], [-4], [-14], [14], [10], [4], [8], [10], [14], [20], [-12], [-14], [20], [-6], [-16], [10], [-12], [-12], [14], [12], [2], [8], [14], [10], [16], [-4], [12], [0], [26], [-6], [-24], [26], [28], [22], [-24], [-6], [16], [2], [18], [-28], [6], [16], [-24], [26], [-26], [20], [-2], [20], [34], [-22], [-24], [8], [-10], [-20], [20], [26], [-6], [-30], [10], [-12], [10], [-8], [-34], [32], [4], [18], [-22], [-30], [-16], [12], [0], [-28], [28], [-20], [-40], [18], [-22], [-16], [2], [32], [42], [24], [42], [-28], [-14], [-44], [-30], [0], [2], [28], [2], [18], [4], [24], [42], [-4], [8], [-6], [-36], [42], [-26], [6], [-16], [12], [34], [-6], [-16], [48], [18], [-28], [12], [40], [10], [10], [-14], [-18], [16], [-2], [42], [28], [10], [-32], [-48], [10], [48], [26], [-34], [-4], [36], [-50], [-14], [4], [-36], [24], [-24], [0], [-30], [-46], [-22], [-36], [30], [56], [44], [-58], [-36], [-56], [2], [10], [-14], [-20], [58], [24], [-38], [32], [-54], [-36], [10], [-36], [18], [-16], [-12], [30], [-18], [-32], [10], [-50], [24], [10], [-24], [-10], [36], [4], [-46], [-12], [6], [66], [-46], [-30], [-16], [-46], [24], [-50], [26], [52], [-6], [32], [-48], [26], [-36], [2], [10], [8], [32], [-72], [-14], [-8], [50], [28], [-14], [58], [-72], [-6], [-16], [-32], [10], [62], [48], [-4], [-36], [-14], [4], [-8], [-30], [-44], [56], [2], [-54], [-60], [16], [56], [76], [32], [-14], [30], [16], [-40], [-52], [60], [-24], [-14], [2], [72], [10], [2], [-12], [-70], [-16], [-26], [2], [-36], [-72], [26], [30], [-62], [-68], [-30], [-22], [40], [-54], [-54], [-24], [18], [-16], [-2], [0], [12], [50], [50], [-84], [4], [72], [-40], [34], [20], [48], [50], [-2], [32], [18], [18], [-12], [-74], [-60], [-50], [-30], [16], [-6], [-36], [-28], [18], [62], [-64], [-16], [76], [-18], [0], [34], [-48], [58], [88], [42], [-30], [-24], [-8], [-46], [76], [-72], [66], [18], [-28], [-14], [82], [-12], [34], [-70], [-4], [-32], [-12], [2], [18], [-82], [80], [-44], [68], [12], [90], [-58], [10], [-16], [22], [18], [66], [-8], [-66], [-4], [58], [-80], [16], [50], [-76], [14], [18], [-44], [58], [-18], [-40], [12], [10], [-68], [98], [26], [-52], [-36], [-84], [-10], [26], [-52], [34], [60], [-12], [8], [-78], [64], [-22], [84], [-72], [-46], [50], [-10], [42], [42], [48], [50], [-100], [4], [-16], [-78], [-36], [60], [18], [6], [12], [-36], [32], [94], [24], [-30], [12], [42], [58], [34], [-68], [-34], [66], [16], [10], [-30], [104], [-92], [54], [-54], [56], [28], [34], [66], [-72], [-4], [-6], [-36], [-86], [22], [-8], [76], [86], [94], [-8], [10], [-52], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6080_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6080_2_a_k();".
// 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_6080_2_a_k(:prec:=1)
    chi := MakeCharacter_6080_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_6080_2_a_k();".
// 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_6080_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6080_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![0, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;