
// Make newform 2450.2.a.y in Magma, downloaded from the LMFDB on 23 April 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_y();".
// 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_y();".
// 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], [-2], [0], [-7], [0], [3], [6], [-7], [-4], [-7], [-8], [-7], [4], [-14], [-14], [12], [-1], [14], [-11], [14], [7], [-7], [0], [7], [-6], [12], [-1], [-8], [14], [-3], [-14], [-18], [-16], [-14], [8], [0], [14], [18], [-14], [15], [5], [20], [-7], [16], [7], [14], [-14], [-18], [-5], [-14], [0], [14], [5], [0], [-21], [18], [-1], [14], [-14], [28], [21], [-7], [-32], [10], [-5], [-2], [28], [-21], [-4], [28], [-4], [6], [21], [-16], [-14], [10], [21], [0], [20], [-3], [-7], [7], [-20], [-15], [10], [28], [23], [14], [21], [-37], [-36], [-6], [0], [-14], [-7], [-14], [24], [-36], [-2], [-28], [45], [12], [-14], [14], [-21], [33], [-42], [7], [-30], [-1], [14], [-43], [5], [-14], [28], [8], [-26], [-14], [37], [-42], [26], [-14], [16], [22], [21], [21], [-14], [-44], [9], [8], [-8], [21], [-14], [0], [28], [-42], [-30], [-28], [-48], [-44], [-22], [14], [-49], [-28], [-42], [-42], [-11], [-34], [35], [6], [56], [-10], [27], [-25], [-14], [14], [-28], [8], [6], [37], [42], [-51], [56], [-23], [-28], [13], [28], [26], [-28], [24], [-45], [-49], [7], [34], [18], [-28], [-14], [-4], [42], [-22], [14], [-51], [-42], [-2], [14], [-11], [0], [-42], [-8], [10], [14], [-30], [42], [-2], [10], [-42], [56], [4], [35], [42], [-21], [-42], [-28], [35], [-6], [27], [14], [-39], [0], [-29], [-28], [-21], [-7], [61], [42], [-32], [48], [-46], [-49], [-39], [3], [-70], [-40], [7], [5], [-35], [-4], [40], [-42], [-47], [75], [-56], [63], [-14], [38], [-64], [-35], [-44], [14], [0], [10], [-21], [21], [-7], [-42], [32], [65], [-68], [-42], [-3], [-42], [0], [-20], [-16], [-56], [42], [21], [-24], [34], [70], [-28], [-63], [14], [6], [14], [-50], [-30], [-42], [18], [-49], [3], [14], [77], [38], [-10], [3], [-14], [49], [11], [63], [-42], [-28], [-39], [61], [-78], [56], [63], [-72], [56], [31], [-28], [48], [-42], [-56], [28], [84], [-28], [21], [50], [-65], [-78], [80], [9], [26], [0], [24], [-34], [63], [-66], [10], [-10], [27], [14], [42], [-45], [-21], [-1], [70], [-62], [14], [48], [33], [14], [80], [-84], [-53], [-78], [8], [-30], [49], [0], [74], [42], [-50], [63], [70], [77], [46], [13], [0], [-64], [-18], [-8], [-42], [10], [0], [84], [14], [33], [30], [-7], [-18], [-21], [-21], [28], [57], [72], [62], [14], [-73], [52], [-56], [22], [-14], [19], [34], [-30], [-56], [-53], [-96], [35], [94], [56], [48], [-49], [21], [21], [-56], [34], [48], [-30], [0], [63], [75], [14], [-46], [7], [-1], [28], [40], [-42], [31], [-23], [28], [-14], [-12], [12], [42], [-18], [-53], [-42], [28], [7], [-58], [9], [-84], [0], [98], [-102], [-96], [-74], [-85], [70], [-48], [-21], [21], [56], [-38], [56], [-12], [-28], [105], [70], [80], [30], [28], [-7]];
    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_y();".
// 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_y(: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_y();".
// 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_y( : 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![2, 1]>,<13,R![0, 1]>,<17,R![7, 1]>,<19,R![0, 1]>,<23,R![-3, 1]>,<37,R![4, 1]>],Snew);
    return Vf;
end function;