
// Make newform 1872.2.a.f in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1872_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_1872_2_a_f();".
// 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_1872_2_a_f();".
// 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_1872_a();"
function MakeCharacter_1872_a()
    N := 1872;
    order := 1;
    char_gens := [703, 469, 209, 145];
    v := [1, 1, 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_1872_a_Hecke(Kf)
    return MakeCharacter_1872_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 := [[0], [0], [-2], [2], [-2], [-1], [-6], [6], [8], [-2], [-10], [-6], [6], [-4], [-2], [-6], [-10], [-2], [-10], [10], [2], [4], [-6], [6], [2], [2], [8], [-16], [-14], [-14], [8], [-16], [-18], [-16], [-18], [-6], [2], [10], [6], [10], [12], [-6], [4], [2], [6], [16], [-8], [6], [18], [18], [-10], [18], [-6], [12], [-18], [-12], [-18], [2], [30], [14], [-4], [14], [34], [0], [-10], [-18], [22], [-10], [-24], [10], [14], [30], [4], [-22], [-10], [18], [6], [2], [6], [-14], [24], [10], [-30], [-14], [16], [-24], [38], [34], [6], [-22], [20], [-18], [-2], [-20], [14], [28], [14], [-6], [-24], [26], [32], [-42], [-4], [-10], [12], [-6], [26], [-26], [4], [-26], [12], [26], [-34], [2], [-22], [26], [-2], [-32], [-18], [0], [2], [-26], [-6], [-2], [-34], [-18], [-22], [-40], [32], [26], [-6], [-14], [0], [-42], [-18], [-54], [46], [-38], [-18], [-6], [22], [-2], [24], [6], [22], [-10], [-38], [38], [-8], [34], [18], [-18], [-28], [4], [-36], [-36], [12], [54], [42], [-10], [-42], [-26], [-50], [-48], [-18], [14], [-12], [-58], [50], [34], [42], [-30], [-8], [-14], [24], [-10], [10], [38], [-64], [50], [18], [-44], [22], [-2], [22], [-10], [34], [54], [-46], [6], [-2], [-6], [56], [22], [52], [6], [2], [-14], [22], [-68], [30], [-20], [-46], [-2], [46], [-30], [14], [0], [-18], [60], [-46], [-54], [4], [-26], [18], [-62], [-52], [-50], [58], [-66], [-26], [10], [30], [-58], [12], [62], [-26], [36], [8], [54], [-62], [32], [-18], [50], [16], [-70], [34], [-34], [-36], [44], [-6], [20], [-36], [-22], [70], [24], [34], [46], [-2], [0], [-46], [46], [-26], [14], [14], [22], [42], [-26], [54], [34], [-64], [48], [74], [46], [30], [-40], [14], [22], [-54], [-30], [-26], [-58], [-30], [-32], [62], [-18], [-22], [-30], [-62], [36], [-76], [-6], [20], [74], [-54], [40], [2], [-2], [82], [46], [18], [-64], [-42], [38], [-14], [30], [12], [-26], [64], [-14], [26], [-26], [0], [32], [8], [-22], [-44], [78], [-26], [-6], [12], [-2], [-30], [8], [-34], [-58], [-30], [-70], [2], [10], [4], [82], [-78], [34], [46], [-62], [26], [-42], [-32], [-26], [-62], [-2], [-68], [30], [-2], [26], [-22], [-26], [42], [-32], [-54], [-54], [62], [80], [-26], [-44], [-38], [10], [38], [54], [-10], [34], [6], [-8], [30], [-6], [78], [42], [-18], [-78], [10], [18], [68], [-46], [76], [-26], [30], [-46], [30], [0], [20], [-58], [90], [-84], [78], [-30], [24], [-70], [-50], [-2], [6], [22], [18], [38], [-46], [6], [78], [14], [94], [92], [90], [-82], [-66], [-80], [54], [10], [-42], [-62], [-72], [78], [10], [-38], [2], [-58], [-66], [28], [58], [-66], [26], [-42], [-62], [42], [-40], [44], [26], [18], [90], [92], [30], [-72], [-86], [-14], [-54], [96], [98], [14], [-28], [-18], [-46], [4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1872_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1872_2_a_f();".
// 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_1872_2_a_f(:prec:=1)
    chi := MakeCharacter_1872_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_1872_2_a_f();".
// 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_1872_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1872_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![-2, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;