
// Make newform 7872.2.a.bf in Magma, downloaded from the LMFDB on 02 August 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_7872_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_7872_2_a_bf();".
// 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_7872_2_a_bf();".
// 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_7872_a();"
function MakeCharacter_7872_a()
    N := 7872;
    order := 1;
    char_gens := [3199, 1477, 5249, 5377];
    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_7872_a_Hecke( : Kf := Kf)
    return MakeCharacter_7872_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], [1], [2], [0], [-1], [-4], [-7], [-2], [4], [9], [-7], [3], [1], [-1], [-9], [2], [8], [-1], [-2], [5], [-11], [10], [-2], [-18], [2], [3], [13], [-18], [-4], [-18], [-20], [16], [-3], [-12], [-2], [-4], [-10], [-9], [0], [-18], [-3], [0], [-12], [8], [-6], [-14], [20], [16], [9], [-10], [10], [8], [15], [10], [-7], [25], [16], [-1], [-31], [5], [1], [9], [33], [-12], [0], [-21], [-14], [-3], [27], [-3], [-10], [0], [13], [25], [16], [9], [-34], [-8], [0], [-33], [-24], [28], [20], [-29], [30], [-8], [40], [-4], [38], [20], [-6], [-19], [-23], [-6], [24], [-21], [21], [5], [32], [-6], [40], [-37], [-31], [28], [0], [-4], [-43], [-5], [-12], [10], [32], [-2], [-18], [35], [-47], [25], [12], [-28], [3], [-24], [22], [46], [26], [-28], [-20], [6], [22], [25], [18], [-11], [25], [34], [18], [34], [-28], [-49], [-39], [-49], [2], [-10], [-7], [-36], [50], [-41], [23], [-36], [-14], [0], [3], [10], [-3], [20], [-26], [41], [-16], [-8], [16], [-7], [-12], [6], [-18], [36], [26], [-3], [-3], [-42], [-20], [-28], [17], [-46], [-48], [-30], [-16], [-53], [20], [-1], [-14], [30], [58], [58], [53], [34], [36], [-66], [-18], [-42], [26], [36], [28], [21], [-53], [36], [20], [26], [60], [-28], [-38], [-28], [62], [-45], [4], [-25], [34], [34], [39], [27], [8], [-3], [68], [41], [-24], [-27], [64], [-8], [11], [-67], [-52], [28], [-57], [6], [10], [-39], [-15], [14], [-6], [22], [-54], [-66], [-12], [20], [6], [44], [-67], [-32], [-54], [8], [68], [14], [-8], [-41], [12], [-32], [-62], [35], [12], [26], [40], [39], [-13], [-46], [-3], [-48], [-34], [-29], [18], [-78], [58], [14], [57], [58], [61], [33], [74], [48], [-38], [-53], [49], [50], [-20], [37], [-76], [61], [-5], [-17], [-14], [56], [-75], [-44], [82], [71], [19], [-68], [-74], [10], [-68], [55], [2], [42], [-40], [5], [11], [-6], [55], [-36], [-42], [35], [32], [6], [36], [52], [-70], [22], [77], [-7], [51], [-46], [0], [-53], [8], [10], [-39], [30], [-52], [53], [82], [50], [-78], [-48], [26], [-33], [-52], [-62], [-38], [76], [-68], [-6], [48], [-84], [-53], [-66], [-52], [6], [-72], [-12], [-27], [-69], [-52], [-6], [-4], [-9], [-40], [60], [-18], [-42], [92], [68], [-42], [-67], [-24], [-32], [48], [-13], [29], [-81], [52], [50], [82], [-80], [-50], [48], [85], [-18], [-63], [28], [26], [24], [42], [13], [-71], [-68], [-60], [8], [-43], [-80], [58], [-48], [-18], [-50], [-45], [-34], [-75], [44], [-27], [72], [-34], [42], [36], [4], [-55], [-48], [-14], [-85], [-72], [60], [36], [4], [66], [-18], [13], [-10], [25], [-42], [-53], [-46], [-4], [22], [23], [-21], [34], [24], [-13], [68], [76], [32], [6], [32], [-54], [86], [-87], [-34], [-56], [82], [62], [-95], [-23], [88], [41], [39], [18], [-95]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7872_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7872_2_a_bf();".
// 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_7872_2_a_bf(:prec:=1)
    chi := MakeCharacter_7872_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_7872_2_a_bf();".
// 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_7872_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7872_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![0, 1]>,<11,R![1, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;