
// Make newform 2880.2.a.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2880_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_2880_2_a_b();".
// 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_2880_2_a_b();".
// 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_2880_a();"
function MakeCharacter_2880_a()
    N := 2880;
    order := 1;
    char_gens := [2431, 901, 641, 577];
    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_2880_a_Hecke(Kf)
    return MakeCharacter_2880_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], [-4], [0], [6], [2], [4], [-8], [-6], [0], [6], [-10], [-4], [8], [10], [0], [-6], [-4], [0], [-14], [-16], [-12], [-2], [2], [-14], [-4], [-4], [10], [-6], [4], [-16], [18], [-12], [-6], [0], [-10], [-4], [-8], [18], [-8], [-14], [8], [-6], [10], [0], [12], [-12], [12], [-6], [10], [-24], [2], [8], [-6], [-24], [-6], [-8], [-10], [-2], [4], [10], [4], [16], [-14], [2], [28], [10], [36], [26], [18], [8], [-28], [-2], [4], [8], [-6], [-2], [6], [-38], [16], [-6], [-32], [10], [32], [-12], [-26], [-38], [2], [-4], [20], [-24], [36], [40], [20], [0], [-6], [14], [36], [10], [4], [-14], [-36], [14], [-20], [34], [-36], [-22], [8], [-22], [12], [-26], [26], [-28], [-40], [-18], [-36], [48], [18], [16], [2], [-22], [-6], [-36], [4], [-38], [-22], [48], [4], [46], [4], [0], [48], [-42], [-50], [-30], [-38], [12], [34], [6], [-12], [50], [4], [20], [-38], [-40], [-10], [-38], [36], [16], [-10], [-26], [28], [24], [28], [-32], [-40], [6], [26], [-14], [44], [18], [-20], [56], [50], [0], [-56], [30], [2], [-54], [-16], [-14], [56], [34], [-32], [30], [-12], [-54], [36], [-14], [-52], [16], [38], [-30], [-24], [26], [-26], [-4], [-38], [-24], [58], [-60], [28], [-30], [4], [34], [-14], [-10], [42], [64], [50], [16], [62], [50], [16], [18], [16], [-4], [-18], [-20], [26], [-22], [-52], [4], [32], [58], [20], [38], [-32], [-30], [58], [16], [-18], [-20], [-28], [26], [-30], [48], [68], [0], [-26], [-52], [-8], [-34], [-12], [0], [-14], [42], [32], [16], [12], [-52], [-12], [10], [-46], [0], [-28], [-24], [12], [-64], [14], [30], [-8], [26], [-14], [-48], [2], [52], [-6], [-46], [-76], [-68], [-22], [46], [10], [4], [2], [-10], [4], [-70], [-62], [-60], [34], [32], [42], [12], [-12], [-54], [42], [-48], [48], [32], [16], [-22], [-20], [24], [-78], [-22], [80], [-66], [34], [36], [-6], [80], [14], [-46], [-64], [-6], [48], [68], [34], [-46], [-40], [36], [12], [-46], [-20], [66], [-32], [-50], [-64], [66], [22], [52], [32], [58], [-40], [72], [-54], [-66], [-76], [-78], [42], [12], [74], [66], [76], [44], [-24], [26], [50], [2], [32], [-44], [-44], [28], [-30], [-30], [-86], [36], [14], [-6], [-14], [-32], [34], [8], [10], [-68], [40], [-6], [92], [58], [-54], [4], [34], [26], [48], [-32], [26], [88], [94], [-42], [-72], [16], [28], [-22], [-78], [28], [-70], [72], [20], [-8], [-14], [16], [-82], [72], [96], [-78], [-42], [26], [-78], [-46], [-36], [66], [92], [-40], [64], [-42], [76], [-64], [66], [-38], [80], [28], [88], [42], [56], [70], [92], [10], [50], [-6], [-12], [-62], [-30], [8], [14], [6], [-68], [0], [66], [42], [-52], [-92], [50], [-38], [-96], [28], [-54], [24], [50], [62], [-24], [40], [82], [66], [20], [-90], [-68], [72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2880_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2880_2_a_b();".
// 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_2880_2_a_b(:prec:=1)
    chi := MakeCharacter_2880_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_2880_2_a_b();".
// 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_2880_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2880_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![0, 1]>,<13,R![-6, 1]>,<17,R![-2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;