
// Make newform 5415.2.a.s in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5415_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_5415_2_a_s();".
// 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_5415_2_a_s();".
// 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 := [-7, 0, 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_5415_a();"
function MakeCharacter_5415_a()
    N := 5415;
    order := 1;
    char_gens := [3611, 2167, 5056];
    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_5415_a_Hecke(Kf)
    return MakeCharacter_5415_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, 1], [1, 0], [1, 0], [-1, 1], [3, -1], [3, -1], [-4, 0], [0, 0], [4, 2], [-1, -3], [-6, 0], [-1, -1], [7, 1], [3, 1], [4, -2], [-6, -2], [-6, -2], [-6, -2], [-4, 4], [-2, -2], [10, 0], [4, -4], [6, 0], [9, 3], [-5, 3], [4, 2], [0, 4], [0, 0], [0, -2], [4, 0], [-4, 0], [-1, -5], [6, 0], [2, 6], [-10, -4], [-2, 4], [-8, -2], [-5, 1], [16, 2], [0, 0], [2, 2], [-12, -2], [-7, 5], [9, 1], [-4, -8], [2, 2], [8, -4], [8, -4], [16, -2], [14, -2], [-18, 0], [13, 1], [-6, -4], [-19, 1], [16, -4], [-6, 4], [9, -5], [-10, 2], [-10, 4], [19, 5], [7, -7], [-16, 4], [-2, 10], [15, 3], [12, 2], [6, -10], [-22, 4], [-7, -3], [26, 0], [2, 8], [-10, 4], [-7, -11], [-1, 1], [-13, 3], [-10, 0], [8, 4], [-6, 0], [-2, 0], [-3, 3], [12, -2], [31, 3], [-18, -4], [2, -2], [25, 1], [12, -4], [32, -2], [23, 5], [16, -6], [-6, -12], [-17, -7], [4, 6], [7, -1], [-2, -6], [-19, 5], [6, -14], [-2, 0], [5, 11], [7, -11], [18, 2], [30, 0], [14, -6], [-22, 4], [-24, 6], [15, -3], [22, -2], [-8, -2], [-28, -2], [14, -4], [-20, 4], [-28, 2], [6, 14], [-4, -10], [12, 0], [18, 2], [-4, -4], [9, 3], [-11, -1], [-14, 4], [20, -4], [-36, 4], [4, -2], [21, -3], [-14, -10], [12, -12], [-6, -14], [6, 4], [-10, 2], [9, 9], [-15, 11], [18, -4], [-20, 0], [0, 4], [34, 4], [-8, -6], [-6, 12], [14, -12], [14, -2], [-22, -6], [-40, 0], [-30, 4], [4, -16], [0, 18], [-9, 5], [-4, -10], [-8, 2], [-14, 10], [20, -2], [6, -14], [8, -16], [20, 0], [33, -3], [-16, 6], [-13, 9], [24, 0], [10, 14], [-16, -4], [36, -8], [4, -6], [36, -6], [25, -5], [22, -4], [0, 4], [-39, -1], [0, 0], [14, 14], [-8, -10], [4, 12], [32, -2], [-30, -8], [-26, 0], [18, 6], [-18, -4], [-31, 1], [30, 12], [4, 4], [-12, -6], [-6, -6], [-24, -6], [-14, -6], [18, -8], [-27, 11], [28, 4], [29, -3], [-12, -4], [4, 14], [14, 0], [3, 15], [22, -2], [-32, 6], [15, -13], [-1, -9], [-12, 14], [20, -8], [-7, -1], [-12, -18], [-4, 0], [6, -14], [2, 16], [-8, 8], [-4, 2], [-11, 11], [20, 4], [65, 1], [-10, 0], [11, 7], [0, 12], [20, -12], [8, 2], [48, -6], [-4, 8], [16, -14], [32, -10], [7, -7], [4, -6], [2, 10], [-8, 6], [19, -3], [27, 9], [16, 10], [40, 8], [22, 16], [-38, 12], [9, -21], [25, 3], [0, 14], [-46, 0], [12, 12], [10, 22], [-24, -16], [11, -17], [8, 22], [-30, 8], [-60, 4], [-7, -13], [9, 11], [-36, 6], [-42, 6], [62, -4], [27, 15], [-2, -10], [4, -4], [0, 12], [-17, -3], [28, -10], [14, 10], [7, 7], [-19, -9], [44, 8], [-54, -8], [58, 4], [-20, 6], [-2, 4], [-6, 24], [8, 22], [-12, 4], [37, 5], [-6, 10], [-60, 4], [26, 10], [-10, 24], [-44, -8], [0, -4], [18, 8], [-43, 1], [-24, 0], [18, 8], [45, -5], [-24, -2], [-24, 4], [52, 4], [18, -8], [-36, -4], [-28, -6], [-54, -6], [-11, -7], [-27, -13], [34, -4], [4, 26], [-20, -10], [5, 1], [-24, -4], [12, 16], [-28, 2], [14, 4], [21, 3], [-21, 15], [-22, -16], [-46, -2], [-12, -4], [15, 21], [-16, 2], [6, -8], [18, 18], [38, 10], [-33, 7], [-32, 14], [-36, 4], [-12, 0], [-24, 0], [13, -17], [58, 8], [-10, 26], [0, 4], [32, 10], [-26, -6], [9, -11], [-8, -4], [-38, -12], [15, 3], [-48, -6], [-28, 10], [-30, -12], [-7, 7], [-46, -10], [-34, -16], [-10, 0], [-21, 15], [-52, 4], [-24, -6], [-58, -8], [18, 16], [-20, -26], [-3, -5], [-62, -2], [12, 0], [42, -4], [4, 0], [-18, 14], [4, 2], [4, -8], [34, 8], [-6, 22], [-42, 2], [18, 16], [-2, 10], [18, -20], [-26, 24], [70, -2], [-2, 10], [-5, 17], [57, 5], [26, 8], [17, -5], [-74, -4], [-22, 14], [12, 0], [-2, -14], [-48, -4], [-60, 12], [-82, -4], [18, -12], [-3, 9], [4, -6], [34, -14], [-24, 18], [-2, 14], [47, -1], [11, 19], [64, 8], [36, 16], [-78, 4], [62, -12], [48, 12], [-34, -10], [3, -7], [3, -1], [30, 0], [66, 6], [-12, 10], [-37, -5], [40, 20], [-6, 20], [61, -1], [4, 32], [-56, -6], [0, -32], [31, 11], [38, -12], [0, 6], [-3, 17], [-55, 3], [-24, -24], [1, -21], [-42, -8], [54, -12], [-4, -34], [66, 2], [-42, -4], [-11, -13], [12, -18], [-24, 6], [-38, 6], [31, 19], [1, 31], [22, -30], [31, -9], [14, -4], [15, -15], [-66, 8], [18, -24], [26, 12], [24, 0], [-42, 14], [40, 4], [39, -7], [68, 8], [-48, 14], [-37, 9], [42, 6], [-1, -5], [-21, -9], [48, -12], [16, -12], [-62, 2], [-34, 12], [-38, 16], [4, 16], [-24, -16], [42, 0], [-8, -20], [-1, 9], [13, -23], [-8, 14], [20, 20], [-23, 9], [32, 8], [-8, -8], [-30, 0], [-78, 2], [31, 19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5415_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5415_2_a_s();".
// 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_5415_2_a_s(:prec:=2)
    chi := MakeCharacter_5415_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_5415_2_a_s();".
// 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_5415_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5415_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-7, 0, 1]>,<7,R![-6, 2, 1]>,<11,R![2, -6, 1]>,<13,R![2, -6, 1]>],Snew);
    return Vf;
end function;