
// Make newform 2450.2.a.s in Magma, downloaded from the LMFDB on 25 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_s();".
// 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_s();".
// 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], [-3], [0], [0], [0], [-2], [-2], [-2], [-1], [-1], [10], [-8], [-3], [5], [8], [-6], [2], [-9], [-7], [6], [-10], [-10], [-9], [-7], [0], [-15], [-11], [-7], [5], [-10], [-8], [20], [16], [-8], [-15], [6], [-12], [12], [-9], [12], [-26], [5], [-20], [20], [-8], [12], [-18], [-8], [12], [10], [-14], [-10], [18], [-10], [12], [21], [5], [6], [-26], [18], [8], [24], [-19], [6], [8], [22], [14], [-2], [21], [-9], [-6], [-14], [-23], [8], [-2], [-9], [-18], [-32], [3], [-17], [-40], [31], [32], [-14], [8], [-3], [23], [32], [-14], [25], [1], [18], [4], [-18], [16], [-5], [35], [-18], [4], [5], [37], [-4], [11], [-18], [-10], [-16], [28], [42], [36], [-22], [5], [18], [20], [-20], [-24], [-35], [-20], [21], [42], [30], [-41], [20], [-2], [25], [-20], [-1], [-9], [20], [-29], [-16], [-32], [-3], [4], [-6], [-50], [34], [-18], [21], [-8], [25], [6], [-42], [-45], [-37], [34], [32], [-10], [-4], [0], [-11], [-14], [-3], [4], [9], [-25], [8], [26], [27], [-56], [14], [-3], [36], [17], [0], [36], [25], [-26], [34], [37], [-14], [-12], [21], [10], [22], [10], [-51], [62], [-38], [-33], [-55], [7], [38], [40], [0], [56], [-45], [-60], [-44], [-29], [24], [8], [-24], [-20], [-9], [-4], [-68], [53], [-12], [62], [68], [7], [-2], [-54], [-51], [54], [-34], [-54], [-44], [69], [-4], [58], [-41], [-63], [-21], [64], [-10], [3], [51], [-60], [-8], [-29], [-4], [45], [0], [11], [-51], [-48], [-28], [-17], [30], [26], [-6], [8], [-3], [61], [-3], [-29], [6], [-18], [48], [-36], [-8], [-40], [75], [36], [-24], [1], [-42], [60], [60], [44], [63], [-59], [-1], [42], [46], [53], [-13], [48], [2], [-59], [-39], [34], [-34], [-32], [52], [-30], [10], [-27], [-24], [1], [-12], [-38], [-4], [48], [24], [-36], [-29], [-55], [30], [-21], [-8], [-32], [-55], [23], [36], [2], [-72], [-70], [-30], [15], [-27], [30], [56], [-10], [49], [-62], [-2], [48], [-52], [14], [26], [60], [-39], [58], [-50], [16], [35], [-42], [58], [13], [-6], [75], [-3], [15], [-19], [84], [22], [-56], [-45], [-50], [-2], [-25], [-49], [-62], [41], [22], [-65], [24], [-22], [-38], [24], [-2], [9], [-68], [-19], [78], [-16], [49], [56], [16], [-44], [21], [58], [60], [10], [-37], [-39], [-30], [56], [-60], [-2], [-13], [57], [47], [28], [58], [72], [-78], [-45], [-6], [6], [19], [-16], [7], [-78], [-14], [-88], [9], [54], [-10], [81], [30], [16], [22], [74], [12], [-90], [-26], [-2], [21], [-50], [-21], [48], [50], [93], [-68], [-76], [77], [-81], [-58], [36], [-14], [-44], [-48], [-6], [70], [61], [-16], [35], [45], [-76], [69], [-60], [81], [62], [48], [-69], [77], [18], [-6], [84], [56], [36], [32], [-66], [-36], [8], [-12], [-87], [-75], [-28], [17], [2], [0], [-60], [-80], [51], [-96], [76]];
    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_s();".
// 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_s(: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_s();".
// 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_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<11,R![0, 1]>,<13,R![2, 1]>,<17,R![2, 1]>,<19,R![2, 1]>,<23,R![1, 1]>,<37,R![8, 1]>],Snew);
    return Vf;
end function;