
// Make newform 1872.2.a.n in Magma, downloaded from the LMFDB on 17 March 2026.
// 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_n();".
// 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_n();".
// 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 := 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]];
    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], [-4], [0], [1], [-2], [-8], [8], [2], [-4], [-10], [-2], [4], [-12], [-6], [0], [-2], [-8], [-12], [10], [8], [0], [14], [2], [-6], [-8], [4], [-18], [-18], [0], [12], [-2], [-4], [18], [12], [-2], [16], [12], [18], [20], [-10], [-24], [-14], [-14], [-8], [4], [-20], [0], [-10], [-26], [-20], [-14], [28], [14], [-16], [18], [20], [-10], [30], [4], [2], [32], [24], [10], [10], [-16], [-30], [20], [30], [6], [-20], [-8], [22], [16], [12], [-22], [-18], [22], [10], [-4], [22], [12], [18], [-8], [4], [6], [-6], [10], [4], [-12], [4], [-28], [12], [24], [16], [10], [22], [-20], [30], [20], [-38], [-44], [22], [4], [-30], [8], [-26], [0], [10], [-8], [22], [-18], [-40], [12], [-18], [-40], [-40], [-46], [-20], [-10], [-14], [42], [32], [-8], [18], [-42], [16], [40], [14], [-16], [4], [-16], [-42], [46], [-14], [-14], [-16], [-14], [22], [-48], [-30], [-56], [-16], [-34], [12], [22], [-42], [-20], [-36], [14], [-2], [-20], [-16], [52], [8], [0], [-42], [-22], [-22], [40], [-42], [-12], [20], [-10], [-36], [-24], [38], [-46], [42], [8], [-18], [-8], [42], [-32], [6], [8], [-14], [-24], [14], [-44], [-12], [-58], [-2], [-12], [-6], [-2], [56], [58], [-60], [18], [-40], [20], [42], [20], [-58], [18], [30], [22], [-16], [26], [8], [22], [50], [0], [18], [-4], [-20], [30], [52], [50], [-6], [-16], [-64], [-44], [10], [24], [14], [-44], [42], [6], [-20], [6], [-20], [36], [22], [6], [-40], [8], [-32], [-18], [-60], [-20], [-26], [28], [12], [50], [-30], [12], [48], [-16], [20], [-48], [46], [-26], [8], [-12], [24], [0], [0], [14], [-26], [-52], [58], [18], [-40], [-42], [-32], [42], [-38], [-16], [60], [-26], [-34], [-26], [-12], [10], [30], [-8], [-6], [-50], [-24], [26], [32], [-62], [20], [-72], [14], [-6], [20], [72], [-20], [48], [22], [-24], [0], [18], [-14], [-4], [-50], [34], [12], [14], [0], [-50], [18], [-40], [58], [36], [16], [-54], [-6], [-80], [44], [28], [2], [12], [-18], [60], [38], [88], [34], [-2], [-44], [60], [-6], [16], [-68], [-78], [-66], [-4], [58], [-46], [52], [-18], [-78], [-64], [-48], [0], [-38], [30], [-30], [8], [-72], [-16], [24], [62], [6], [74], [-80], [-10], [-90], [66], [8], [10], [-44], [70], [8], [20], [-70], [40], [-38], [-70], [-16], [54], [6], [52], [-24], [94], [-76], [-26], [-90], [-88], [-56], [-4], [26], [42], [-4], [74], [-20], [-12], [-12], [-38], [32], [-34], [-48], [48], [-94], [78], [-22], [42], [-66], [-4], [-42], [0], [76], [-84], [-42], [48], [-88], [-46], [66], [0], [52], [28], [26], [44], [70], [28], [2], [78], [-18], [36], [94], [66], [48], [-66], [22], [64], [-24], [18], [-70], [-12], [60], [-22], [-30], [12], [-48], [-42], [-24], [66], [86], [92], [20], [42], [-6], [-44], [14], [-40], [48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1872_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1872_2_a_n();".
// 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_n(: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_n();".
// 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_n( : 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![4, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;