
// Make newform 7938.2.a.o in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7938_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_7938_2_a_o();".
// 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_7938_2_a_o();".
// 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_7938_a();"
function MakeCharacter_7938_a()
    N := 7938;
    order := 1;
    char_gens := [6077, 3727];
    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_7938_a_Hecke(Kf)
    return MakeCharacter_7938_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], [0], [3], [0], [3], [5], [-3], [5], [3], [3], [-4], [-7], [9], [11], [0], [3], [-12], [2], [-4], [0], [11], [8], [-3], [-15], [-1], [3], [5], [15], [-7], [-15], [-16], [-3], [-3], [5], [3], [11], [14], [17], [-3], [-6], [-3], [-10], [-12], [14], [6], [-7], [5], [17], [-9], [17], [-27], [-27], [23], [-12], [-15], [9], [-21], [-13], [-7], [-3], [8], [27], [-28], [24], [14], [-30], [20], [-25], [12], [5], [9], [-15], [-1], [17], [-16], [15], [-9], [29], [-27], [-22], [3], [-31], [3], [14], [8], [0], [-30], [-34], [-9], [35], [3], [9], [-31], [39], [11], [12], [27], [-3], [-7], [17], [11], [3], [12], [-30], [20], [11], [33], [21], [0], [-1], [-43], [-31], [-3], [-19], [8], [45], [29], [3], [-9], [-39], [14], [11], [-6], [33], [20], [-6], [-10], [-39], [5], [41], [47], [-3], [29], [14], [-3], [-1], [-21], [44], [27], [-39], [20], [54], [-40], [24], [41], [39], [17], [33], [11], [-15], [-43], [-6], [-4], [39], [17], [-9], [-1], [18], [-34], [-54], [12], [6], [-49], [27], [-6], [21], [29], [41], [-34], [39], [51], [-31], [3], [23], [-40], [-3], [-1], [51], [-16], [-34], [-40], [45], [29], [33], [12], [-9], [41], [5], [-13], [3], [-13], [-51], [17], [-6], [57], [-54], [-10], [-10], [-39], [-15], [-54], [17], [5], [-25], [-24], [-45], [47], [-39], [21], [20], [-25], [-57], [-43], [-36], [24], [-10], [-40], [-63], [57], [-69], [-67], [-25], [9], [-61], [63], [53], [42], [-45], [32], [0], [2], [-25], [-37], [42], [-1], [-27], [-1], [-9], [-21], [-12], [-45], [-43], [-40], [-67], [-51], [24], [-16], [60], [68], [12], [65], [-39], [-48], [-37], [-33], [-33], [65], [17], [-78], [-25], [11], [33], [14], [2], [18], [8], [-6], [-63], [41], [-6], [29], [-1], [-73], [-31], [23], [59], [45], [-19], [59], [-60], [-24], [32], [-9], [-19], [-37], [21], [-25], [15], [35], [66], [51], [-27], [21], [-81], [-31], [-54], [-64], [75], [-21], [-73], [23], [-69], [-43], [27], [-13], [38], [39], [41], [15], [41], [-24], [-81], [42], [-40], [36], [-13], [69], [24], [47], [45], [-55], [-1], [-18], [-76], [81], [-25], [-55], [32], [-24], [-42], [-10], [18], [-31], [-57], [-37], [36], [14], [-75], [-1], [71], [-19], [-87], [-9], [-67], [-66], [-84], [53], [8], [-57], [63], [-25], [47], [-33], [-73], [77], [-54], [-69], [-3], [45], [-57], [-91], [-42], [87], [48], [11], [-58], [87], [-43], [-70], [3], [29], [63], [63], [8], [-7], [36], [24], [47], [-75], [-37], [-33], [57], [8], [-39], [-67], [-81], [-13], [38], [5], [69], [-61], [66], [-3], [-28], [60], [62], [53], [-27], [-67], [-54], [41], [-3], [-31], [69], [-45], [-49], [-19], [-51], [68], [-84], [-82], [-78], [24], [20], [23], [75], [-39], [-97], [-102], [3], [-45], [77], [51], [63], [74], [-78], [36], [-102], [11], [105], [-85], [-45], [-49], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7938_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7938_2_a_o();".
// 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_7938_2_a_o(:prec:=1)
    chi := MakeCharacter_7938_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(3023) - 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_7938_2_a_o();".
// 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_7938_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7938_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-3, 1]>,<11,R![-3, 1]>,<13,R![-5, 1]>,<17,R![3, 1]>,<23,R![-3, 1]>],Snew);
    return Vf;
end function;