
// Make newform 3920.2.a.c in Magma, downloaded from the LMFDB on 25 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_c();".
// 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_c();".
// 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], [-3], [-1], [0], [5], [5], [7], [-2], [2], [7], [4], [-6], [12], [2], [1], [0], [-4], [-4], [-8], [0], [-6], [3], [-4], [0], [-13], [18], [13], [-18], [5], [6], [12], [-6], [-8], [18], [-22], [19], [-10], [14], [-3], [7], [4], [-8], [13], [-8], [-8], [-4], [-5], [-19], [9], [-4], [-24], [-9], [22], [6], [-18], [30], [-26], [12], [2], [19], [-25], [-13], [-17], [-34], [1], [6], [12], [34], [-26], [-18], [-5], [0], [-7], [16], [20], [12], [-15], [23], [-15], [14], [-24], [-3], [25], [-14], [-26], [-6], [-9], [-8], [0], [36], [11], [34], [-22], [-9], [1], [3], [14], [-22], [-44], [7], [28], [-16], [4], [-22], [-4], [37], [0], [-27], [-15], [-10], [3], [-30], [6], [26], [-7], [18], [21], [24], [38], [-15], [0], [0], [29], [-12], [44], [35], [-25], [-14], [-40], [33], [-51], [-24], [3], [-40], [-18], [-34], [21], [17], [39], [-39], [-10], [1], [-28], [-6], [20], [-34], [22], [-6], [-4], [-48], [22], [0], [40], [16], [18], [8], [55], [32], [-13], [-28], [-40], [0], [-6], [-16], [26], [3], [-32], [3], [5], [34], [40], [-22], [-35], [-58], [38], [-12], [-55], [17], [28], [44], [-56], [-48], [26], [38], [16], [34], [44], [29], [65], [52], [-14], [66], [7], [40], [20], [-35], [-9], [12], [7], [-43], [54], [10], [58], [-18], [42], [19], [-2], [32], [6], [-28], [52], [-2], [4], [9], [10], [24], [2], [-68], [54], [-38], [50], [34], [-11], [42], [33], [-50], [-29], [1], [-67], [28], [40], [30], [64], [-47], [-39], [-27], [48], [-48], [-73], [42], [-16], [-8], [1], [46], [13], [-22], [16], [2], [-37], [-62], [30], [50], [-24], [-56], [-26], [33], [-58], [-12], [21], [-3], [-22], [-50], [-24], [-47], [10], [-62], [25], [-30], [24], [-50], [40], [-52], [-69], [-44], [-2], [41], [20], [2], [-79], [18], [21], [27], [-16], [50], [-47], [-79], [26], [30], [48], [0], [25], [-69], [-24], [-12], [46], [-72], [38], [-3], [76], [-51], [-33], [-6], [85], [40], [39], [58], [-54], [-40], [-48], [-52], [40], [51], [78], [14], [-14], [-40], [58], [-39], [47], [-59], [-82], [32], [54], [20], [-50], [-48], [23], [-4], [-8], [-60], [-82], [-44], [10], [-65], [68], [93], [-91], [66], [26], [-15], [38], [36], [-46], [85], [-50], [-28], [26], [56], [-30], [15], [4], [12], [-23], [-55], [17], [-58], [6], [-50], [22], [46], [-52], [-28], [-56], [-63], [84], [70], [-57], [-74], [5], [-87], [-28], [76], [-69], [-18], [-46], [-24], [-59], [-11], [-6], [1], [70], [82], [96], [18], [0], [-59], [65], [-51], [84], [31], [1], [75], [-21], [-68], [-23], [-16], [-64], [48], [-83], [66], [0], [24], [98], [-21], [60], [90], [22], [-57], [-68], [-60], [11], [6], [46], [-52], [22], [-24], [-7], [52], [-54], [69], [-51], [38], [-46], [100], [81], [-57], [-36], [42], [80], [-52]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<11,R![-5, 1]>,<13,R![-5, 1]>,<17,R![-7, 1]>],Snew);
    return Vf;
end function;