
// Make newform 2520.2.a.w in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2520_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_2520_2_a_w();".
// 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_2520_2_a_w();".
// 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
        poly := [-4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2520_a();"
function MakeCharacter_2520_a()
    N := 2520;
    order := 1;
    char_gens := [631, 1261, 281, 2017, 1081];
    v := [1, 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_2520_a_Hecke(Kf)
    return MakeCharacter_2520_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], [0, 0], [-1, 0], [1, 0], [0, 1], [2, -3], [-6, 1], [-4, 2], [0, 2], [-2, -1], [0, -4], [-2, 4], [-2, -2], [-4, 2], [-4, -1], [10, -2], [4, 0], [-10, -2], [-4, -4], [0, 0], [2, 4], [-4, -3], [-12, 0], [-2, 2], [6, 3], [-6, 0], [-4, 5], [-12, 2], [-6, -1], [-2, 4], [0, 4], [-12, 2], [-2, -6], [-12, 6], [-6, -4], [-12, 5], [-2, -4], [4, -10], [-12, -5], [6, -11], [4, 0], [-2, 2], [-4, -3], [10, -2], [10, -2], [-8, 4], [-16, 1], [20, -3], [16, 3], [-2, 6], [14, 2], [4, -9], [2, 0], [-4, 6], [6, -8], [0, 6], [2, 4], [0, 4], [-2, 4], [18, -1], [0, 3], [-18, 9], [-24, 3], [16, 2], [22, -3], [10, 0], [-12, 0], [2, 0], [-12, 2], [-18, -4], [10, 5], [-32, 0], [4, -7], [6, -6], [20, -8], [16, 4], [-2, 13], [-6, 11], [2, -7], [-6, 8], [12, -4], [2, -1], [12, -7], [26, 4], [16, -2], [4, -10], [26, 3], [18, 6], [-22, -2], [0, 4], [-24, 1], [-8, 6], [0, -2], [8, 3], [-8, -5], [12, -11], [-14, 12], [6, -4], [-12, 0], [18, -15], [12, 0], [2, -10], [20, 8], [6, 16], [20, 0], [-2, 1], [-4, 4], [-14, 15], [36, -7], [-22, 0], [-4, 3], [-2, -12], [22, 4], [20, 6], [-4, 15], [-2, -16], [-16, 9], [-8, 0], [26, -8], [8, 5], [-2, 10], [26, -10], [6, 3], [-20, 16], [4, -8], [14, 3], [10, 9], [8, -10], [24, -16], [18, 1], [-8, -1], [24, 0], [-36, -3], [6, -6], [-26, 0], [18, -16], [22, -5], [16, -11], [-10, 5], [10, 9], [-12, 10], [-34, 13], [-24, -4], [-28, -2], [-10, -10], [-16, 2], [22, 4], [-34, 12], [12, -16], [-16, 8], [-26, -8], [-26, 18], [28, 4], [24, -8], [-20, 18], [-8, 0], [4, 1], [6, 2], [30, -5], [-38, -6], [4, -20], [30, -14], [16, -18], [4, -12], [14, 16], [12, -11], [-8, 8], [-46, 7], [14, 11], [-38, -8], [-36, -4], [-34, 8], [36, -11], [-22, -4], [-24, -2], [-18, 14], [16, -21], [30, -19], [8, -12], [6, -18], [-32, 0], [-36, 12], [-2, 4], [14, -8], [0, -8], [-22, 0], [-34, 6], [64, 1], [-10, -1], [-16, 4], [-22, -12], [20, -2], [16, 5], [-6, -2], [-12, -16], [-6, -1], [-10, 13], [-18, -18], [26, 1], [-28, 19], [-30, 8], [-8, 10], [-42, 0], [18, 8], [-52, -6], [-10, 17], [8, -2], [-44, 4], [6, -28], [12, 0], [26, -22], [-6, -12], [24, -4], [-8, 27], [-48, -2], [-14, -10], [-24, 22], [-26, 22], [0, -26], [-6, 28], [6, 12], [0, 26], [-30, 21], [24, -10], [0, -5], [22, 8], [-46, 5], [44, 1], [44, 5], [-4, 16], [-18, 10], [12, -6], [-8, -8], [-22, 1], [-40, 5], [28, -5], [10, -26], [-22, 30], [56, 3], [16, -10], [-28, -4], [-20, -20], [-20, 9], [-2, 16], [18, -9], [0, -10], [32, 0], [36, 2], [8, -7], [-24, -6], [54, 0], [-18, 4], [-24, -16], [2, 14], [2, 4], [-8, -11], [22, 0], [-28, 16], [22, -5], [6, -15], [8, -2], [12, 10], [-18, 18], [-14, 1], [-10, -12], [-36, 22], [6, -3], [-10, -4], [-44, 4], [2, -24], [-10, -14], [-4, 0], [-18, 3], [-40, 4], [-22, 8], [12, -15], [-36, -8], [30, -12], [22, 15], [-44, 2], [12, 11], [36, -3], [-8, -8], [6, -8], [40, 13], [-4, -31], [-30, 0], [2, -8], [-24, -24], [38, 18], [6, 13], [-8, 1], [30, -22], [36, -16], [38, 0], [-38, -2], [-24, 22], [54, 3], [20, -16], [32, -15], [30, -1], [-22, 12], [-12, -5], [-12, 4], [-24, 29], [34, 16], [-12, 6], [-10, 10], [-16, 16], [70, -2], [-16, -8], [-42, -5], [-18, 20], [4, -10], [-48, 18], [18, 6], [-12, -22], [-4, 25], [-26, 23], [50, 5], [60, -6], [-46, 14], [50, 4], [-48, 4], [-10, 12], [-38, 14], [-48, 21], [36, -24], [32, 16], [-38, -14], [30, -16], [-14, -6], [-40, -6], [-56, -3], [4, -24], [24, -25], [10, 31], [6, 12], [10, 4], [52, -15], [-34, 8], [30, 6], [-38, 0], [12, -29], [58, 0], [52, -16], [6, 16], [4, -20], [56, -10], [14, 5], [-12, -16], [-30, 26], [6, -19], [-60, -7], [-30, 3], [46, -24], [-8, 10], [28, 22], [-50, 20], [-16, 6], [70, -10], [-18, -2], [-64, 0], [-16, -3], [-20, 8], [10, -28], [38, 5], [-48, 26], [22, -13], [40, -19], [-68, 16], [-16, -20], [-58, 3], [-16, 38], [6, 28], [-52, 12], [-52, 5], [-34, 1], [62, -20], [22, 13], [18, -12], [-58, 24], [-56, -8], [62, 0], [68, -12], [20, 3], [4, -1], [26, -19], [-60, 24], [4, 33], [46, 7], [14, -15], [-8, 39], [44, -16], [4, 1], [18, -10], [-32, 40], [-50, 2], [-32, -9], [-22, 36], [-26, 26], [-42, -22], [-56, 22], [2, 13], [34, 18], [-32, -14], [6, 16], [-22, -13], [-4, 0], [12, 16], [-58, 27], [34, 8], [-28, -6], [12, -24], [42, 4], [10, -26], [-20, 9], [8, 12], [-26, 4], [4, -29], [-10, -7], [22, 20], [8, -30], [-60, 8], [-58, -19], [-74, 5], [4, -24], [38, -32], [76, -4], [-24, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2520_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2520_2_a_w();".
// 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_2520_2_a_w(:prec:=2)
    chi := MakeCharacter_2520_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_2520_2_a_w();".
// 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_2520_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, -1, 1]>,<13,R![-38, -1, 1]>,<17,R![26, 11, 1]>,<19,R![-8, 6, 1]>],Snew);
    return Vf;
end function;