
// Make newform 2667.2.a.c in Magma, downloaded from the LMFDB on 11 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2667_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_2667_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_2667_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_2667_a();"
function MakeCharacter_2667_a()
    N := 2667;
    order := 1;
    char_gens := [890, 1144, 2416];
    v := [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_2667_a_Hecke( : Kf := Kf)
    return MakeCharacter_2667_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]];
    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 := [[-1], [1], [0], [1], [0], [2], [-2], [-6], [-8], [6], [-2], [-6], [-6], [4], [2], [2], [4], [10], [4], [0], [-10], [-12], [8], [-12], [-16], [4], [-6], [0], [2], [-2], [-1], [-6], [-22], [-12], [10], [-8], [-6], [8], [-8], [0], [8], [-12], [16], [-14], [-6], [-2], [16], [4], [14], [4], [10], [0], [-12], [-14], [16], [12], [18], [-22], [-26], [18], [4], [24], [8], [0], [0], [-30], [-28], [22], [28], [4], [22], [8], [10], [14], [-28], [-18], [-6], [6], [-6], [16], [6], [-2], [-24], [-26], [-28], [-36], [-6], [-22], [-32], [0], [16], [14], [-24], [20], [36], [30], [-18], [-42], [-2], [30], [20], [-18], [-20], [-10], [-44], [-42], [-26], [12], [0], [40], [-2], [10], [18], [0], [8], [10], [14], [16], [42], [4], [26], [-2], [-18], [20], [48], [6], [38], [-2], [28], [18], [32], [48], [32], [-2], [0], [8], [18], [-26], [-18], [50], [18], [-30], [4], [-4], [20], [40], [28], [0], [12], [-32], [-46], [20], [-24], [28], [-32], [20], [-32], [-24], [-56], [-42], [12], [-6], [-8], [-50], [30], [18], [-16], [-20], [-2], [-34], [44], [-28], [-8], [34], [8], [-24], [40], [-10], [-54], [-8], [28], [20], [50], [-54], [40], [-12], [22], [30], [10], [-30], [32], [-4], [44], [-50], [48], [-30], [22], [-18], [-42], [-20], [18], [-10], [-2], [28], [-36], [-16], [10], [-36], [58], [-10], [-10], [-22], [16], [-18], [-66], [16], [8], [-20], [24], [-34], [42], [30], [38], [-28], [-22], [-22], [26], [24], [50], [36], [-38], [46], [68], [6], [-28], [12], [0], [18], [44], [16], [4], [64], [-30], [-2], [36], [-74], [-60], [-38], [20], [-72], [-14], [0], [64], [16], [-16], [52], [-54], [-50], [-22], [-16], [32], [-40], [42], [-46], [-24], [20], [-6], [22], [-76], [-38], [64], [20], [-28], [-48], [-32], [-8], [76], [-18], [50], [-28], [64], [24], [-54], [12], [-40], [0], [-46], [-22], [-36], [12], [10], [-42], [-6], [-42], [46], [50], [-28], [30], [30], [-10], [62], [38], [48], [76], [20], [-42], [52], [70], [80], [26], [-82], [-74], [22], [68], [80], [-52], [0], [-12], [2], [26], [82], [22], [-12], [8], [22], [-30], [20], [58], [-88], [-2], [-70], [-90], [-12], [-78], [12], [36], [-70], [68], [-18], [70], [30], [-18], [28], [-80], [-18], [84], [-48], [-4], [60], [70], [-80], [-70], [-54], [-52], [42], [30], [-2], [-72], [-2], [-24], [-82], [-84], [-60], [60], [-8], [-26], [-26], [-56], [58], [84], [64], [-40], [-86], [-98], [46], [100], [0], [-76], [-26], [-36], [30], [-50], [-72], [-2], [-92], [-76], [-32], [-68], [52], [20], [-22], [-66], [32], [-4], [60], [-34], [-90], [-70], [-64], [82], [-90], [30], [28], [-60], [-18], [82], [62], [-34], [38], [98], [0], [-10], [12], [-4], [102], [0], [-24], [24], [-74], [-52], [10], [-8], [8], [-6], [-14], [38], [36], [-6], [22], [-72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2667_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2667_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_2667_2_a_c(:prec:=1)
    chi := MakeCharacter_2667_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_2667_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_2667_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2667_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<5,R![0, 1]>],Snew);
    return Vf;
end function;