
// Make newform 2415.2.a.b 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_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_2415_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_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 := [[-1], [1], [-1], [1], [4], [2], [-2], [0], [1], [6], [-8], [10], [-2], [4], [-8], [6], [-4], [10], [-12], [-12], [-10], [16], [0], [6], [-2], [-6], [8], [4], [14], [10], [20], [12], [10], [20], [-22], [16], [-14], [16], [-16], [14], [16], [10], [0], [10], [6], [-20], [20], [16], [-8], [10], [-6], [-28], [-18], [-20], [-6], [-16], [10], [-16], [22], [-18], [28], [26], [28], [0], [22], [-2], [-28], [-18], [12], [-2], [26], [-32], [-16], [-14], [20], [12], [-30], [18], [22], [10], [12], [-10], [-32], [-34], [16], [-20], [2], [22], [18], [-20], [8], [-32], [-20], [24], [12], [-12], [-30], [22], [44], [-34], [-8], [-2], [-32], [30], [-44], [-26], [20], [-22], [-36], [42], [-24], [34], [34], [0], [16], [-18], [-36], [-16], [-18], [12], [-14], [26], [-14], [20], [-28], [-6], [-18], [0], [8], [-30], [-12], [0], [-40], [-6], [-42], [-2], [18], [36], [-14], [18], [-12], [30], [4], [12], [-18], [-40], [58], [-22], [44], [-24], [22], [-18], [-24], [32], [-12], [40], [-40], [-18], [-26], [18], [20], [34], [-44], [-52], [42], [-20], [40], [42], [-62], [-18], [12], [-10], [24], [14], [-48], [54], [4], [22], [0], [-14], [-4], [28], [-34], [-54], [0], [-22], [38], [-28], [42], [56], [-26], [-12], [12], [42], [36], [38], [-14], [-30], [38], [32], [-62], [40], [18], [-26], [-12], [-18], [20], [52], [-54], [4], [-38], [42], [24], [-48], [48], [22], [12], [-10], [-8], [-34], [-58], [-44], [-6], [-40], [12], [-10], [6], [-20], [-16], [24], [6], [-64], [-24], [18], [-12], [-44], [-2], [-50], [-8], [-56], [-36], [-52], [40], [-2], [18], [-16], [-32], [20], [-4], [8], [50], [-18], [0], [-2], [-2], [24], [-26], [-12], [-18], [-50], [-16], [-60], [30], [-14], [42], [-8], [-30], [6], [36], [54], [-26], [-44], [70], [8], [-26], [0], [28], [46], [-30], [-44], [-16], [-16], [-44], [18], [-60], [12], [38], [38], [-64], [30], [-86], [-72], [-6], [4], [-66], [-62], [28], [-66], [-52], [-44], [22], [-42], [16], [52], [60], [26], [-28], [58], [-48], [46], [48], [-54], [-2], [-16], [-24], [38], [28], [-60], [-34], [-22], [24], [46], [10], [36], [10], [30], [-60], [-28], [0], [-74], [14], [62], [0], [-60], [52], [-36], [-10], [-18], [74], [0], [46], [2], [-46], [0], [86], [40], [70], [-64], [0], [-38], [44], [-30], [30], [56], [14], [66], [72], [60], [-62], [-56], [-6], [-90], [-48], [60], [4], [-70], [-66], [32], [2], [56], [60], [48], [-70], [-52], [22], [20], [72], [14], [30], [38], [-30], [-30], [52], [-30], [56], [24], [32], [-22], [76], [-84], [50], [94], [0], [28], [0], [46], [-60], [38], [68], [-10], [14], [2], [-88], [-50], [-62], [-96], [82], [42], [-44], [-36], [46], [54], [36], [20], [-42], [74], [-4], [-8], [54], [-100], [10], [-6], [72], [-12], [-18], [-42], [-76], [-78], [-20], [-72]];
    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_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_2415_2_a_b(: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_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_2415_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2415_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;