
// Make newform 2450.2.a.k in Magma, downloaded from the LMFDB on 24 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_k();".
// 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_k();".
// 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], [3], [5], [2], [-5], [7], [-4], [-2], [-1], [3], [-2], [7], [-9], [-4], [6], [-2], [-6], [16], [14], [6], [2], [12], [0], [8], [16], [2], [-14], [-7], [-1], [8], [16], [-18], [-6], [9], [12], [-15], [-9], [13], [26], [-20], [10], [5], [18], [-9], [8], [6], [16], [8], [20], [-9], [5], [6], [0], [-10], [24], [2], [9], [-14], [9], [22], [6], [22], [2], [5], [-10], [12], [-12], [-24], [16], [-13], [-26], [-29], [-21], [-6], [14], [-15], [-14], [35], [-20], [2], [-28], [-28], [-30], [5], [10], [-32], [17], [-34], [36], [32], [24], [4], [-40], [-34], [-27], [-16], [-16], [26], [-23], [-2], [-15], [32], [4], [-34], [6], [-12], [-22], [13], [15], [-14], [19], [-18], [-5], [14], [27], [-3], [-36], [16], [-32], [17], [44], [-44], [26], [12], [26], [29], [-41], [-29], [21], [28], [-42], [1], [-29], [-45], [-18], [2], [-5], [-33], [-18], [24], [22], [-26], [-34], [43], [-8], [-12], [11], [-31], [-15], [-16], [-36], [40], [2], [20], [21], [26], [-28], [12], [24], [-32], [-33], [-54], [29], [-8], [38], [43], [47], [-24], [-18], [-26], [20], [46], [42], [-1], [-32], [33], [-4], [59], [-13], [-13], [-24], [-29], [18], [57], [-4], [-11], [30], [52], [-12], [25], [24], [-50], [38], [-67], [30], [22], [-56], [-56], [10], [-42], [-51], [-12], [13], [30], [26], [-3], [-31], [-46], [-50], [-36], [-12], [64], [23], [9], [-3], [-3], [17], [4], [56], [-9], [57], [-20], [0], [-42], [-10], [35], [36], [25], [-57], [68], [45], [26], [-48], [-14], [27], [21], [30], [54], [-11], [40], [-60], [66], [6], [17], [3], [12], [-45], [-2], [-57], [23], [-37], [30], [25], [-16], [76], [3], [34], [-31], [36], [-50], [-62], [8], [-71], [6], [19], [-30], [-15], [-32], [-48], [-16], [-52], [66], [3], [-18], [-20], [53], [-39], [-39], [16], [-13], [68], [-32], [-84], [4], [33], [-4], [-57], [18], [-66], [-36], [47], [-53], [-32], [64], [-58], [9], [-2], [4], [85], [0], [78], [-20], [-34], [44], [50], [54], [-31], [-13], [24], [90], [0], [-87], [29], [-33], [-80], [-4], [54], [-20], [2], [-16], [67], [-88], [-61], [55], [44], [-72], [-62], [-2], [-30], [-56], [-60], [-11], [-32], [-60], [-38], [85], [-47], [74], [92], [30], [76], [18], [52], [38], [-90], [90], [61], [9], [-88], [-10], [-93], [-28], [26], [64], [-12], [18], [-72], [-27], [54], [-52], [-73], [-64], [-42], [-61], [31], [39], [75], [-91], [-72], [24], [70], [26], [41], [18], [-6], [46], [-76], [-6], [-4], [87], [30], [5], [-3], [40], [31], [76], [-12], [14], [42], [-20], [-88], [66], [-12], [-2], [91], [-37], [-88], [-18], [-32], [-27], [72], [48], [32], [63], [-6], [94], [15], [-18], [42], [-14], [-39], [-20], [90], [-18], [61], [24], [69], [-78], [-11], [-8], [-28], [60], [45], [-70], [-45], [-87], [-62]];
    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_k();".
// 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_k(: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_k();".
// 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_k( : 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![-3, 1]>,<13,R![-5, 1]>,<17,R![-2, 1]>,<19,R![5, 1]>,<23,R![-7, 1]>,<37,R![1, 1]>],Snew);
    return Vf;
end function;