
// Make newform 2450.2.a.p in Magma, downloaded from the LMFDB on 10 June 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_p();".
// 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_p();".
// 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]];
    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], [2], [0], [0], [3], [-5], [-6], [-1], [-3], [-6], [-4], [-11], [3], [10], [-3], [-3], [0], [-4], [4], [12], [4], [-10], [12], [6], [-14], [-12], [4], [12], [-4], [-12], [19], [3], [-12], [-4], [18], [14], [-5], [4], [-9], [-3], [-3], [2], [12], [4], [3], [-4], [-1], [-8], [24], [-28], [6], [6], [-25], [-15], [12], [0], [-12], [-16], [-2], [-3], [-26], [27], [-2], [12], [-8], [-18], [-7], [-14], [-24], [-10], [-12], [6], [1], [34], [-25], [-15], [-24], [-2], [21], [-22], [15], [-34], [0], [16], [-10], [24], [-3], [22], [6], [19], [18], [24], [16], [-12], [-28], [-24], [6], [33], [-20], [8], [28], [27], [18], [-3], [20], [-20], [-12], [36], [42], [2], [19], [-47], [6], [-1], [32], [-45], [-38], [21], [-21], [24], [44], [34], [3], [12], [32], [18], [14], [-36], [-29], [-47], [-37], [-9], [26], [-26], [-51], [-49], [-39], [34], [30], [39], [47], [-18], [-44], [54], [14], [-6], [1], [18], [32], [-39], [7], [33], [-8], [-24], [10], [-30], [38], [33], [-2], [24], [-30], [12], [-32], [27], [30], [-57], [20], [-14], [47], [-63], [-36], [-40], [-36], [-14], [-4], [-6], [-1], [-30], [-41], [8], [-35], [-45], [13], [54], [33], [30], [-5], [-26], [41], [-48], [64], [-6], [-55], [0], [18], [54], [17], [10], [-12], [-48], [-24], [26], [-14], [-55], [0], [27], [-64], [-60], [-51], [-19], [-56], [-24], [52], [66], [-18], [35], [31], [-27], [27], [-57], [32], [44], [-9], [-29], [-48], [-52], [6], [6], [1], [36], [-65], [11], [-22], [33], [-62], [-60], [-46], [9], [57], [-30], [-6], [-7], [16], [14], [-30], [-6], [43], [-45], [-76], [-15], [34], [-45], [-63], [-1], [30], [-15], [2], [76], [-3], [-74], [31], [0], [50], [-2], [6], [5], [0], [-21], [-50], [3], [20], [-38], [-14], [14], [-14], [37], [60], [-10], [5], [57], [39], [-16], [45], [44], [10], [30], [52], [63], [26], [-57], [18], [30], [-18], [15], [-11], [-36], [62], [-6], [69], [28], [-32], [-33], [-52], [-36], [-4], [-32], [-18], [2], [-36], [-41], [33], [24], [-54], [-38], [3], [-43], [-45], [54], [52], [18], [92], [22], [48], [49], [-72], [-25], [-25], [52], [72], [30], [44], [-78], [-10], [12], [-43], [12], [8], [-84], [5], [-53], [-26], [-42], [54], [-22], [18], [-12], [-40], [28], [66], [63], [-79], [-92], [-96], [73], [-40], [-90], [36], [60], [-84], [-72], [-77], [6], [-72], [-33], [-74], [58], [21], [25], [-13], [-21], [65], [-24], [6], [62], [-2], [-27], [30], [4], [-30], [22], [-48], [-66], [-47], [12], [5], [15], [20], [49], [-50], [-96], [38], [-54], [-60], [64], [6], [-38], [86], [-21], [-37], [-6], [-40], [84], [67], [18], [-24], [8], [-23], [-78], [22], [-33], [10], [78], [-78], [-7], [-68], [96], [-78], [31], [-18], [51], [-6], [7], [72], [-84], [88], [-45], [84], [87], [-7], [-36]];
    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_p();".
// 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_p(: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_p();".
// 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_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<11,R![-3, 1]>,<13,R![5, 1]>,<17,R![6, 1]>,<19,R![1, 1]>,<23,R![3, 1]>,<37,R![11, 1]>],Snew);
    return Vf;
end function;