
// Make newform 2415.2.a.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2415_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_2415_2_a_e();".
// 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_2415_2_a_e();".
// 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_2415_a();"
function MakeCharacter_2415_a()
    N := 2415;
    order := 1;
    char_gens := [806, 967, 346, 1891];
    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_2415_a_Hecke(Kf)
    return MakeCharacter_2415_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], [1], [-3], [0], [-2], [7], [1], [-2], [-4], [2], [3], [10], [-1], [-3], [-5], [3], [-14], [0], [2], [-6], [-8], [-12], [-6], [11], [-5], [-4], [-2], [6], [-1], [9], [-7], [-16], [-11], [-11], [-11], [7], [-13], [-14], [-8], [-10], [-1], [11], [-20], [19], [-3], [16], [12], [11], [-26], [-6], [25], [8], [-15], [-9], [14], [14], [-15], [-2], [-8], [4], [-4], [23], [1], [22], [17], [-36], [-18], [16], [-30], [24], [-23], [26], [30], [-8], [-18], [-20], [-15], [4], [-12], [2], [-19], [-13], [36], [22], [6], [-12], [6], [-13], [10], [0], [8], [-20], [-16], [2], [19], [-12], [5], [11], [-12], [-34], [16], [-7], [-24], [28], [31], [-34], [-40], [10], [22], [-28], [-38], [36], [4], [5], [11], [-44], [44], [20], [-13], [27], [0], [-44], [-8], [11], [-40], [40], [-37], [-26], [32], [3], [-12], [-16], [-35], [-26], [-24], [35], [-20], [-20], [-8], [18], [-19], [-1], [-22], [48], [-40], [-15], [-14], [-36], [25], [40], [20], [-44], [16], [-4], [-44], [-34], [-13], [40], [-10], [-15], [-32], [52], [9], [54], [1], [-2], [-28], [-51], [-51], [-30], [-15], [-26], [-10], [58], [-5], [54], [53], [38], [-23], [-18], [19], [21], [-21], [42], [-1], [32], [-31], [48], [-18], [40], [-50], [15], [-5], [14], [36], [54], [-2], [-33], [-33], [6], [0], [-61], [-6], [-17], [-28], [10], [28], [2], [-37], [25], [-8], [18], [-32], [62], [-35], [-10], [12], [36], [-17], [23], [-16], [32], [35], [65], [-14], [-26], [16], [-44], [-17], [28], [-50], [44], [31], [2], [-63], [42], [30], [56], [49], [6], [14], [-57], [-47], [-41], [-4], [8], [4], [-3], [10], [-48], [69], [-17], [-14], [66], [-62], [67], [-17], [-2], [60], [27], [46], [-47], [-37], [-1], [63], [-36], [6], [-82], [-30], [-62], [26], [-4], [42], [28], [2], [19], [-40], [-22], [-17], [-4], [38], [-70], [82], [54], [38], [-63], [22], [-86], [45], [-16], [16], [8], [46], [0], [56], [-29], [35], [30], [-70], [58], [-20], [2], [38], [-43], [-44], [37], [-7], [-66], [-3], [74], [7], [56], [15], [45], [-49], [-62], [-37], [34], [-7], [-8], [11], [25], [9], [5], [70], [64], [-33], [59], [-50], [36], [8], [-81], [50], [-83], [10], [-14], [64], [-32], [-34], [83], [37], [32], [57], [66], [-52], [29], [-54], [18], [32], [-51], [72], [39], [50], [57], [58], [-62], [74], [64], [-44], [85], [-22], [-33], [28], [-7], [19], [-20], [34], [58], [-36], [18], [49], [-39], [62], [-72], [-21], [-9], [86], [-12], [-24], [58], [73], [-46], [100], [-39], [-89], [12], [-70], [90], [-50], [-3], [60], [-26], [23], [-70], [55], [-84], [-42], [-12], [6], [-24], [76], [42], [-63], [-20], [46], [-78], [4], [-36], [16], [72], [-33], [-58], [12], [-6], [60], [-38], [76], [-56], [38], [63], [-18], [48], [-18], [-79], [37], [-54], [-76], [-13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2415_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2415_2_a_e();".
// 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_2415_2_a_e(:prec:=1)
    chi := MakeCharacter_2415_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_2415_2_a_e();".
// 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_2415_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2415_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<11,R![3, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;