
// Make newform 6080.2.a.c in Magma, downloaded from the LMFDB on 09 May 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_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_6080_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_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], [-2], [-1], [-4], [4], [4], [-2], [-1], [0], [-2], [-8], [-4], [6], [0], [-12], [8], [0], [-2], [-14], [-8], [6], [4], [4], [14], [-12], [-10], [-14], [-18], [2], [-12], [6], [12], [14], [12], [-10], [12], [14], [-24], [2], [16], [24], [26], [8], [4], [10], [8], [8], [-22], [22], [6], [-6], [24], [-10], [-20], [-24], [-16], [10], [8], [-2], [-10], [20], [-24], [-26], [32], [-14], [4], [-8], [28], [20], [26], [30], [32], [12], [24], [4], [-2], [-6], [22], [10], [34], [-12], [22], [-28], [-12], [12], [12], [10], [-22], [-14], [-40], [36], [0], [2], [-20], [-36], [16], [-34], [18], [26], [-26], [22], [-42], [-14], [-6], [28], [-30], [20], [10], [20], [6], [42], [38], [-2], [-20], [24], [-10], [20], [0], [14], [-8], [-26], [-8], [24], [-14], [36], [-2], [34], [48], [-28], [42], [28], [22], [-44], [2], [-42], [-50], [36], [14], [0], [-6], [28], [18], [-44], [-6], [-38], [12], [58], [32], [-4], [-26], [8], [30], [-8], [-10], [46], [-12], [-32], [-30], [14], [42], [-36], [32], [48], [20], [-32], [22], [-4], [-34], [30], [-50], [12], [-6], [-40], [58], [-16], [-26], [4], [-42], [30], [-58], [16], [-4], [-4], [24], [-8], [-30], [-32], [10], [18], [24], [56], [-20], [52], [30], [48], [12], [10], [18], [-38], [-12], [-10], [28], [-28], [62], [-52], [-18], [24], [-10], [-30], [12], [-6], [22], [16], [42], [0], [-42], [52], [-30], [30], [-6], [-2], [56], [-2], [-64], [-42], [10], [-24], [24], [22], [4], [2], [44], [28], [-42], [44], [-76], [30], [18], [-12], [-68], [-54], [68], [68], [-10], [60], [24], [-12], [0], [-44], [12], [-22], [-62], [24], [34], [42], [-4], [26], [10], [-32], [-34], [74], [-6], [6], [-32], [-62], [-8], [22], [-2], [46], [-22], [58], [50], [-18], [40], [-68], [52], [32], [-18], [-74], [76], [-18], [-32], [-72], [2], [20], [40], [26], [-52], [8], [-6], [58], [-72], [12], [-72], [80], [-42], [48], [-46], [4], [0], [-26], [48], [-48], [-74], [76], [20], [-30], [46], [-8], [-30], [4], [-10], [66], [26], [-48], [62], [-4], [-12], [-14], [66], [20], [-26], [6], [-2], [86], [30], [-24], [66], [-2], [78], [-66], [0], [-16], [32], [-52], [24], [82], [0], [42], [-8], [-24], [6], [-14], [48], [32], [16], [10], [-86], [16], [38], [-44], [48], [30], [62], [-10], [48], [8], [12], [-42], [30], [30], [-90], [-30], [28], [-28], [36], [6], [-74], [70], [-28], [-40], [-12], [66], [-40], [38], [-12], [-16], [50], [-14], [-28], [42], [-14], [-28], [-30], [0], [58], [96], [38], [16], [42], [-94], [-56], [-76], [12], [12], [0], [-52], [22], [4], [10], [-62], [-38], [82], [48], [90], [-8], [-98], [2], [98], [-4], [-64], [-94], [46], [12], [86], [18], [60], [102], [86], [22], [-70], [96], [-76], [92], [76], [-64], [-78], [78], [-44], [-40]];
    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_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_6080_2_a_c(: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_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_6080_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6080_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![4, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;