
// Make newform 3920.2.a.k in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3920_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_3920_2_a_k();".
// 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_3920_2_a_k();".
// 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_3920_a();"
function MakeCharacter_3920_a()
    N := 3920;
    order := 1;
    char_gens := [1471, 981, 3137, 3041];
    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_3920_a_Hecke(Kf)
    return MakeCharacter_3920_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], [-1], [-1], [0], [-6], [2], [-6], [-8], [-3], [3], [-2], [8], [-3], [-5], [0], [12], [0], [-1], [7], [0], [-10], [4], [-3], [-3], [-10], [-3], [7], [3], [17], [-12], [-8], [-12], [-12], [-2], [-15], [10], [14], [16], [21], [0], [6], [17], [18], [2], [-18], [-20], [4], [-8], [12], [2], [6], [-6], [26], [18], [-24], [3], [-15], [16], [-22], [-6], [-20], [-12], [19], [0], [-4], [18], [22], [20], [27], [-1], [-12], [-6], [-5], [-16], [-8], [-3], [6], [26], [3], [11], [-24], [23], [-30], [-28], [28], [-9], [-9], [-4], [30], [-29], [-33], [30], [-8], [-6], [-2], [9], [39], [30], [-44], [17], [19], [30], [9], [-18], [16], [-10], [-12], [-12], [0], [-22], [43], [2], [-12], [-14], [-38], [-39], [-20], [-21], [-42], [-12], [-25], [32], [-18], [15], [-14], [27], [-1], [30], [-23], [2], [46], [39], [-20], [2], [30], [-34], [-42], [31], [-12], [-51], [-2], [18], [49], [45], [-10], [42], [32], [42], [4], [-33], [-40], [9], [28], [-57], [-23], [-30], [-8], [-45], [44], [54], [-9], [36], [7], [12], [-54], [39], [-26], [32], [-19], [-60], [12], [-31], [42], [32], [34], [-3], [40], [-42], [-11], [-31], [13], [-30], [-34], [42], [0], [51], [20], [-32], [-25], [-30], [8], [-48], [34], [63], [-48], [-54], [29], [-10], [12], [48], [27], [58], [-4], [-19], [54], [-24], [-20], [-12], [21], [40], [44], [15], [19], [9], [-18], [50], [-23], [-57], [48], [-12], [11], [-38], [-15], [52], [21], [53], [0], [-30], [-23], [18], [2], [16], [34], [-27], [-5], [-33], [-25], [6], [12], [54], [-60], [28], [16], [47], [-12], [54], [55], [-54], [-32], [0], [56], [51], [27], [-13], [6], [-24], [-19], [13], [54], [20], [-41], [-21], [-70], [-10], [72], [-50], [-18], [30], [31], [-36], [29], [52], [-76], [-20], [50], [16], [-12], [23], [-19], [0], [-27], [16], [24], [-31], [49], [48], [-28], [60], [58], [54], [51], [-21], [-6], [66], [68], [45], [40], [84], [-6], [4], [50], [24], [70], [3], [-38], [-82], [12], [47], [78], [26], [-9], [-42], [-9], [31], [-27], [17], [-48], [72], [-22], [-21], [22], [38], [63], [37], [78], [-55], [82], [-59], [-72], [-36], [50], [-72], [-68], [39], [10], [-45], [2], [-42], [-7], [-44], [68], [48], [33], [40], [54], [-72], [35], [-17], [-30], [-48], [-50], [38], [51], [19], [-25], [72], [-30], [-30], [-6], [69], [8], [-18], [-39], [0], [-35], [-46], [-6], [-44], [65], [-60], [46], [-21], [-42], [52], [26], [36], [90], [38], [-30], [-4], [-3], [-66], [-83], [-12], [-14], [87], [-50], [-4], [19], [93], [-94], [-72], [6], [4], [-54], [32], [52], [21], [-20], [-45], [77], [78], [67], [48], [-15], [-8], [44], [75], [-29], [-54], [-64], [18], [-36], [-8], [2], [6], [30], [-104], [-54], [-57], [-3], [8], [15], [-42], [38], [54], [-36], [99], [-92], [-42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3920_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3920_2_a_k();".
// 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_3920_2_a_k(:prec:=1)
    chi := MakeCharacter_3920_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_3920_2_a_k();".
// 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_3920_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<11,R![6, 1]>,<13,R![-2, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;