
// Make newform 1850.2.a.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1850_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_1850_2_a_c();".
// 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_1850_2_a_c();".
// 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_1850_a();"
function MakeCharacter_1850_a()
    N := 1850;
    order := 1;
    char_gens := [1777, 1001];
    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_1850_a_Hecke(Kf)
    return MakeCharacter_1850_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], [-1], [0], [4], [3], [6], [3], [-3], [2], [0], [0], [-1], [-3], [-4], [4], [-2], [-12], [12], [9], [-2], [-9], [-2], [-7], [-3], [2], [-10], [14], [-1], [18], [5], [8], [12], [-7], [3], [-18], [22], [24], [-17], [-18], [24], [21], [-10], [6], [11], [12], [8], [9], [8], [28], [0], [-26], [-12], [-21], [-9], [-22], [26], [-28], [-28], [28], [-6], [7], [12], [-29], [28], [-10], [-18], [-3], [-9], [-31], [-16], [-14], [34], [-2], [28], [1], [24], [-12], [-30], [-5], [15], [27], [-32], [2], [5], [16], [5], [-37], [-33], [20], [6], [-28], [16], [28], [-20], [-16], [-12], [0], [-11], [37], [-2], [5], [32], [4], [19], [-4], [-15], [-13], [-9], [-36], [-33], [-26], [-46], [-42], [-16], [22], [-6], [20], [42], [36], [9], [-16], [-34], [-8], [17], [-1], [-10], [32], [-36], [-44], [40], [-12], [6], [20], [16], [45], [23], [46], [32], [16], [-38], [20], [-4], [42], [7], [6], [-42], [40], [3], [9], [6], [26], [6], [-7], [-36], [36], [18], [-34], [-54], [7], [0], [-24], [33], [-48], [3], [23], [18], [-12], [-18], [-19], [-6], [51], [-50], [-12], [-33], [-22], [-51], [23], [24], [-46], [40], [26], [45], [-16], [33], [-18], [56], [14], [-11], [15], [-34], [57], [-7], [-61], [-22], [-57], [-2], [-27], [-58], [-9], [-20], [4], [-18], [-2], [11], [-28], [-42], [-2], [16], [-58], [-48], [22], [24], [-14], [51], [-14], [-27], [36], [-25], [-10], [-54], [-12], [12], [-31], [-36], [39], [20], [39], [72], [-68], [35], [-16], [37], [-58], [-57], [-20], [-32], [61], [-60], [15], [60], [-75], [11], [-4], [-66], [-21], [-50], [-32], [28], [-59], [-16], [-38], [-45], [18], [49], [50], [45], [10], [33], [-74], [-5], [-38], [-21], [-46], [-10], [-37], [-65], [48], [-6], [-61], [4], [-8], [71], [53], [-22], [-14], [6], [-53], [82], [38], [57], [-74], [12], [-44], [44], [-41], [-42], [-46], [-28], [28], [30], [-30], [4], [-25], [1], [-24], [16], [-20], [42], [84], [53], [31], [-6], [26], [-15], [-33], [-21], [-60], [-60], [70], [-34], [6], [82], [-39], [-60], [84], [34], [-9], [-58], [19], [69], [-44], [1], [74], [46], [-26], [14], [-11], [0], [-60], [70], [-42], [8], [-32], [-19], [33], [4], [16], [22], [77], [48], [-6], [-75], [66], [-28], [-36], [-44], [-8], [53], [82], [26], [-11], [-70], [-80], [-16], [-28], [39], [76], [37], [78], [36], [-56], [63], [56], [92], [57], [65], [-18], [18], [-17], [92], [-59], [54], [60], [-14], [8], [-21], [44], [47], [-81], [26], [-42], [-57], [8], [-57], [-28], [58], [-14], [38], [-56], [6], [-5], [82], [11], [71], [-30], [-31], [80], [85], [27], [-28], [-84], [11], [-42], [-17], [30], [-68], [-52], [-50], [-68], [-99], [7], [24], [-33], [0], [57], [6], [-32], [-32], [5], [-78], [-8], [20], [26], [-27], [9], [-24], [-36], [-77], [55], [-98]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1850_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1850_2_a_c();".
// 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_1850_2_a_c(:prec:=1)
    chi := MakeCharacter_1850_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_1850_2_a_c();".
// 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_1850_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1850_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<7,R![-4, 1]>],Snew);
    return Vf;
end function;