
// Make newform 2450.2.a.l in Magma, downloaded from the LMFDB on 09 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2450_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_2450_2_a_l();".
// 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_2450_2_a_l();".
// 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_2450_a();"
function MakeCharacter_2450_a()
    N := 2450;
    order := 1;
    char_gens := [1177, 101];
    v := [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_2450_a_Hecke(Kf)
    return MakeCharacter_2450_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], [0], [0], [0], [4], [-6], [2], [0], [0], [6], [-8], [10], [-2], [-4], [8], [2], [8], [14], [12], [-16], [2], [-8], [8], [-10], [2], [6], [16], [-12], [6], [-2], [8], [16], [6], [-16], [6], [8], [10], [4], [0], [-22], [-12], [14], [24], [-2], [-14], [16], [4], [16], [8], [14], [6], [16], [-10], [0], [-22], [-8], [6], [0], [18], [26], [32], [10], [-8], [24], [-22], [-22], [4], [14], [-4], [-10], [-6], [8], [16], [-14], [-12], [24], [14], [10], [-14], [30], [-24], [-10], [0], [2], [8], [20], [-30], [-10], [-10], [16], [-40], [-24], [-32], [12], [-12], [-24], [-10], [-2], [16], [-18], [-12], [-22], [-16], [-6], [-12], [10], [8], [-6], [0], [-42], [-32], [-6], [22], [-8], [-16], [2], [-32], [16], [-14], [28], [38], [14], [18], [4], [32], [22], [-2], [-24], [0], [-14], [28], [-48], [-16], [-22], [-18], [-10], [42], [-8], [-6], [-6], [-8], [6], [8], [-44], [46], [16], [-46], [-54], [48], [32], [-14], [-2], [12], [-16], [52], [0], [-16], [-58], [50], [-2], [-44], [54], [-16], [24], [30], [-16], [40], [-22], [34], [-14], [-52], [-26], [48], [6], [40], [38], [-20], [30], [-40], [-34], [40], [24], [58], [50], [-24], [-18], [-22], [8], [58], [24], [-6], [20], [52], [-26], [12], [18], [34], [-46], [42], [56], [-50], [40], [-6], [-18], [24], [-46], [-56], [36], [-6], [-32], [-18], [54], [-64], [32], [24], [46], [-16], [-42], [56], [-30], [-34], [40], [-30], [24], [-40], [62], [-62], [-16], [-8], [20], [50], [-24], [-8], [-54], [64], [48], [-10], [34], [-4], [-40], [-28], [-8], [72], [46], [-54], [0], [32], [0], [20], [16], [26], [-18], [-24], [-42], [-30], [4], [62], [-8], [-6], [42], [72], [-28], [38], [-46], [18], [-8], [-18], [-26], [68], [-54], [6], [-12], [-46], [48], [-70], [56], [-68], [14], [-6], [-64], [16], [32], [48], [-42], [-16], [48], [-34], [-30], [16], [-18], [-18], [-40], [-58], [40], [-6], [-42], [-64], [34], [48], [32], [42], [-62], [-8], [-60], [-28], [78], [76], [-18], [24], [-14], [-48], [-18], [-14], [-44], [72], [70], [-72], [-72], [42], [50], [-40], [-10], [-50], [0], [54], [22], [44], [48], [8], [-54], [62], [-30], [56], [-48], [-4], [-48], [22], [-54], [-10], [-88], [2], [-58], [30], [80], [18], [28], [30], [-52], [0], [-6], [-56], [6], [78], [-40], [-90], [18], [24], [8], [30], [96], [34], [22], [88], [28], [40], [86], [-86], [-64], [10], [-60], [-16], [80], [-90], [-8], [-70], [16], [0], [-70], [6], [42], [-10], [86], [-72], [-66], [-72], [-24], [-72], [-22], [4], [88], [-46], [58], [76], [48], [40], [-10], [-88], [-82], [12], [30], [78], [-18], [80], [-6], [14], [-16], [58], [-30], [0], [-72], [26], [90], [-36], [4], [-26], [-10], [-24], [96], [-30], [8], [-74], [74], [48], [-80], [-6], [-14], [52], [90], [80], [64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2450_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2450_2_a_l();".
// 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_2450_2_a_l(:prec:=1)
    chi := MakeCharacter_2450_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_2450_2_a_l();".
// 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_2450_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<11,R![-4, 1]>,<13,R![6, 1]>,<17,R![-2, 1]>,<19,R![0, 1]>,<23,R![0, 1]>,<37,R![-10, 1]>],Snew);
    return Vf;
end function;