
// Make newform 6080.2.a.o in Magma, downloaded from the LMFDB on 18 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_o();".
// 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_o();".
// 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], [-6], [-6], [8], [4], [6], [-2], [10], [0], [-4], [2], [-16], [8], [-2], [16], [-6], [-10], [-12], [-10], [-4], [4], [6], [-4], [12], [8], [-2], [16], [-22], [-8], [10], [-2], [8], [0], [-12], [-18], [0], [4], [-2], [20], [28], [20], [-12], [14], [26], [12], [-10], [8], [0], [26], [-10], [-12], [-30], [6], [-30], [8], [12], [4], [-6], [-12], [-28], [20], [6], [34], [-18], [0], [26], [24], [20], [-8], [6], [2], [-22], [-30], [-28], [10], [-8], [4], [-32], [18], [-6], [-38], [2], [34], [6], [32], [24], [8], [8], [-6], [2], [26], [20], [-34], [-20], [26], [-28], [42], [44], [-38], [-2], [34], [-16], [-26], [-8], [18], [46], [-16], [-28], [-50], [-14], [6], [10], [-20], [-30], [40], [-16], [12], [12], [-30], [-38], [-24], [26], [6], [44], [32], [-16], [46], [22], [46], [4], [-28], [-56], [-54], [12], [30], [14], [24], [-10], [-24], [14], [-40], [-44], [-16], [8], [-6], [-34], [-52], [-12], [-48], [-20], [6], [22], [-18], [-50], [-56], [-2], [-20], [16], [-56], [32], [-46], [6], [-54], [-20], [38], [4], [26], [-16], [22], [40], [-58], [16], [-38], [-10], [36], [44], [24], [-46], [10], [-16], [-52], [-22], [-32], [32], [18], [-44], [-22], [42], [-36], [-18], [14], [42], [14], [-46], [48], [52], [-10], [-24], [18], [24], [0], [18], [-28], [-22], [-22], [-46], [-64], [-48], [-50], [50], [-30], [36], [-58], [10], [40], [30], [-70], [-12], [-10], [56], [-32], [28], [-48], [-34], [52], [48], [-66], [-46], [74], [-46], [30], [-24], [-48], [-60], [0], [46], [-70], [-44], [-12], [-30], [-28], [44], [-26], [-10], [66], [78], [-6], [22], [-12], [-70], [56], [-48], [-26], [-68], [8], [38], [16], [50], [28], [-30], [78], [8], [-18], [22], [56], [54], [24], [-28], [14], [-62], [-22], [-26], [-52], [-24], [4], [-18], [62], [22], [12], [34], [12], [-44], [-70], [38], [-54], [44], [44], [-32], [74], [80], [-34], [-76], [50], [78], [-16], [-56], [-8], [-16], [-44], [56], [26], [-60], [62], [14], [-26], [-86], [-44], [-90], [-42], [-16], [40], [10], [46], [4], [38], [42], [16], [54], [14], [76], [76], [-28], [-6], [2], [48], [-48], [-42], [-20], [18], [-26], [8], [-74], [-26], [24], [6], [-26], [0], [-16], [12], [6], [-24], [0], [42], [20], [-8], [18], [0], [-6], [-32], [-20], [-48], [46], [-48], [-54], [6], [40], [20], [78], [-12], [10], [88], [22], [84], [-20], [-66], [-10], [80], [34], [-36], [-8], [66], [6], [-20], [-66], [58], [-30], [74], [52], [-4], [68], [-54], [30], [-8], [-22], [-56], [100], [22], [-72], [72], [-64], [-66], [-4], [54], [-98], [-6], [0], [-88], [70], [64], [50], [-30], [-60], [-92], [-8], [-66], [48], [20], [-50], [-14], [48], [-12], [14], [0], [78], [-8], [-38], [-44], [-44], [32], [68], [74], [-80], [104]];
    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_o();".
// 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_o(: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_o();".
// 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_o( : 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;