
// Make newform 6080.2.a.n in Magma, downloaded from the LMFDB on 23 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_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_6080_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_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], [2], [-4], [4], [6], [1], [2], [6], [8], [-4], [6], [-6], [-6], [-8], [-12], [-6], [0], [0], [-10], [8], [14], [14], [16], [14], [20], [-4], [2], [0], [-12], [-8], [6], [-8], [10], [16], [18], [-14], [-16], [8], [4], [10], [-12], [16], [22], [24], [4], [-4], [-4], [6], [-14], [-16], [6], [0], [-20], [-14], [-14], [24], [26], [30], [22], [24], [4], [8], [-22], [12], [20], [24], [-6], [-14], [-2], [-4], [10], [8], [12], [-32], [-26], [10], [-30], [10], [-4], [-10], [-24], [-16], [-8], [30], [2], [10], [18], [-22], [-14], [12], [32], [-24], [24], [-6], [-26], [-30], [-36], [-34], [20], [18], [-4], [-30], [36], [2], [-6], [42], [-16], [30], [-16], [2], [-10], [-48], [40], [14], [-34], [6], [10], [-36], [-42], [28], [0], [-44], [-36], [18], [10], [12], [-14], [-42], [44], [-24], [-40], [-42], [-10], [-2], [-36], [12], [-16], [10], [28], [22], [-50], [24], [-14], [40], [-18], [-52], [-28], [40], [24], [10], [-6], [4], [44], [-24], [32], [-42], [38], [18], [18], [60], [14], [4], [4], [0], [48], [-22], [-10], [10], [-60], [42], [-8], [-46], [24], [-42], [-16], [-18], [40], [34], [14], [36], [-44], [20], [26], [-6], [32], [52], [-14], [36], [-52], [-2], [-20], [-34], [30], [-64], [-66], [14], [-6], [-26], [22], [-48], [-28], [-2], [40], [58], [-4], [64], [34], [-20], [34], [-30], [-6], [-16], [0], [-50], [58], [-38], [-68], [-34], [62], [-64], [-50], [-38], [12], [6], [-44], [0], [-28], [-8], [14], [-44], [-32], [22], [-66], [-30], [-14], [-58], [-48], [-56], [-52], [64], [30], [6], [24], [48], [26], [68], [52], [-66], [62], [50], [6], [-38], [-18], [4], [74], [-56], [72], [-10], [4], [8], [38], [56], [-78], [28], [-42], [-50], [-8], [-50], [-38], [-72], [-42], [-20], [40], [-10], [70], [-42], [-66], [-52], [32], [24], [70], [-30], [-70], [-8], [-38], [76], [64], [-22], [-18], [-74], [0], [-44], [-24], [-46], [64], [-58], [-52], [-34], [78], [0], [36], [8], [56], [-16], [72], [30], [0], [14], [38], [54], [-30], [28], [22], [-34], [64], [0], [-46], [-18], [28], [22], [-42], [56], [62], [62], [-84], [-60], [-68], [-30], [10], [32], [20], [66], [44], [54], [-14], [-44], [22], [-2], [-8], [-58], [-54], [40], [24], [36], [38], [-8], [-16], [-30], [36], [76], [-6], [88], [-50], [-4], [0], [0], [-26], [-88], [-38], [70], [-32], [-60], [66], [40], [-54], [-80], [38], [-12], [-28], [38], [10], [68], [-30], [-84], [-36], [10], [-10], [-8], [10], [-6], [-30], [42], [28], [-44], [72], [42], [66], [-48], [-70], [40], [-36], [58], [-8], [28], [-8], [-34], [-4], [-42], [-62], [-30], [-8], [-52], [-14], [-52], [-6], [-54], [-4], [-28], [68], [46], [-96], [-52], [-34], [-46], [8], [-68], [-2], [-24], [10], [-24], [2], [-12], [16], [-64], [-100], [-102], [-16], [-12]];
    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_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_6080_2_a_n(: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_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_6080_2_a_n( : 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![-2, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;