
// Make newform 3920.2.a.ca 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_ca();".
// 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_ca();".
// 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
        poly := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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, 0], [2, 1], [-1, 0], [0, 0], [-2, 2], [2, 2], [-4, 1], [2, -1], [4, -2], [-2, -2], [0, -2], [-2, -4], [-4, 5], [6, 2], [8, -2], [-2, 6], [10, -1], [2, -8], [4, -4], [-4, 6], [8, 1], [4, 2], [2, -3], [0, -9], [12, -3], [-2, -8], [-12, -2], [-4, -4], [-6, 6], [-4, -8], [0, -2], [-2, -3], [-16, 0], [10, 7], [-6, 0], [0, 8], [2, -6], [6, 10], [-4, 14], [6, 2], [-4, 0], [-6, 6], [4, -10], [0, 4], [-6, -14], [8, 6], [4, -16], [4, -8], [14, 3], [2, 10], [12, 8], [4, -4], [12, 1], [2, -1], [0, 7], [-28, 0], [-10, -6], [12, 0], [6, -10], [8, 0], [2, -3], [14, -4], [22, -5], [4, 0], [4, -13], [2, 6], [-14, -14], [6, 0], [6, -2], [6, -6], [-16, 13], [0, -20], [8, 12], [-22, -6], [-26, -6], [-24, -6], [14, 10], [-14, -14], [-6, 0], [12, -1], [6, -19], [6, 8], [28, -4], [-12, 1], [-8, -12], [-20, 12], [6, -16], [16, 4], [10, 2], [0, 12], [-6, -7], [-20, 2], [-12, -14], [-4, -20], [-36, 4], [12, 4], [14, -14], [-12, -5], [14, 7], [2, -12], [2, 6], [18, -2], [-38, 1], [-18, -8], [22, -14], [-16, 7], [10, 5], [-16, 9], [8, 2], [-8, 1], [-12, 20], [-18, 4], [-22, -8], [22, 5], [4, 6], [-2, -16], [26, -15], [8, -22], [10, -4], [30, -2], [10, 20], [-48, 0], [-14, -18], [-28, -4], [6, -3], [-6, 6], [14, 22], [16, 18], [-12, 6], [-34, 0], [-6, 2], [-32, -14], [28, 8], [-2, 4], [-4, -13], [-4, -11], [2, 0], [-14, 9], [22, 14], [20, -12], [34, -1], [6, 8], [8, -20], [28, -16], [22, -12], [-28, -10], [46, -2], [12, -5], [14, -15], [-12, 12], [-2, -20], [-16, 13], [36, 4], [-24, -22], [28, -20], [-8, -20], [-24, 6], [-4, 15], [-16, -3], [50, 0], [2, 2], [6, -4], [12, 18], [2, -25], [-24, 8], [-32, 14], [-8, -26], [-10, 20], [4, -4], [-18, -2], [20, 24], [10, 26], [-4, 14], [-44, 0], [24, 18], [32, -11], [4, 28], [-2, -14], [12, -8], [14, 24], [12, -30], [-26, -15], [18, 16], [32, -11], [-24, 18], [-26, 2], [14, 22], [18, 9], [12, -4], [-4, 42], [-8, -21], [-20, -8], [12, 24], [2, 18], [12, -32], [-16, 13], [6, -4], [-10, 16], [-24, 7], [32, 0], [30, 12], [-4, -18], [10, 4], [40, 5], [-18, -3], [-38, 20], [8, -38], [-10, -6], [-32, 20], [18, -5], [30, -16], [-26, 28], [0, 46], [18, -27], [-16, 8], [36, 23], [-16, -16], [-4, -9], [-8, 0], [2, 32], [18, 22], [24, 24], [-18, -8], [36, -2], [-14, 1], [58, -10], [-32, 13], [20, 8], [-52, 0], [38, 14], [-14, -36], [50, -13], [24, 14], [34, 0], [2, -23], [20, 16], [44, 7], [50, 6], [20, -12], [52, 4], [26, 6], [14, -7], [-24, 12], [-18, -40], [-16, -27], [24, -6], [-12, -4], [-14, -21], [-6, 22], [8, 16], [-6, 26], [24, 27], [-20, 8], [56, 3], [-38, 0], [12, 36], [-6, -6], [18, -19], [-34, 16], [36, 11], [-4, 22], [-2, 18], [-18, -24], [-6, 24], [-12, -17], [14, -7], [26, -36], [-8, -29], [-36, -20], [-22, 24], [10, -14], [10, 26], [0, -7], [0, -8], [24, 1], [24, -4], [38, -10], [50, -12], [0, -12], [-10, 13], [-12, 4], [16, -14], [-32, -10], [78, -4], [42, -3], [56, 8], [-50, -4], [26, -30], [-40, 10], [-24, -3], [-10, 28], [-10, 27], [50, -16], [-34, 1], [-42, -2], [22, -22], [-32, 8], [-42, 10], [30, 41], [-18, -21], [12, -7], [42, 8], [16, 2], [54, 22], [26, 6], [32, 0], [-50, 14], [-2, 28], [68, -12], [38, -10], [40, 16], [6, 46], [12, -12], [6, 30], [-36, -12], [24, 9], [26, 31], [-16, -38], [16, 25], [-24, -16], [18, 7], [-8, -16], [62, 12], [28, 16], [56, -20], [-60, 9], [30, -34], [-18, 27], [-16, 22], [50, 0], [10, 28], [10, 12], [-4, -42], [6, -49], [-52, -20], [22, -21], [-30, 42], [-8, -1], [-8, -19], [56, -2], [10, -18], [-14, -8], [18, 22], [-36, -6], [10, 28], [38, -6], [-38, 2], [34, -18], [24, -16], [-18, -32], [50, 15], [22, -20], [6, -32], [-36, -6], [-6, -4], [8, -57], [16, 6], [-34, 25], [-30, 4], [20, -10], [58, 10], [-32, -13], [-40, 30], [-20, -24], [-22, 33], [30, 32], [-30, 14], [-4, -38], [-12, 52], [4, 48], [-6, 31], [0, 40], [18, -12], [-12, -36], [-22, -14], [74, -5], [52, 32], [-32, 37], [36, 45], [-12, -43], [-50, -30], [16, -28], [-40, -4], [26, 0], [-42, 1], [12, -52], [4, -58], [-14, -4], [36, 0], [16, -30], [6, -28], [-54, -12], [-68, 8], [-14, -31], [-16, -40], [16, -24], [-36, 34], [-28, -17], [12, 40], [-10, 30], [-30, -12], [14, 4], [-16, 24], [60, -7], [-10, 42], [-44, -6], [-42, -18], [20, 36], [62, -17], [18, 43], [80, -9], [-78, -4], [50, -2], [-60, -20], [-20, 0], [-46, 18], [-52, 22], [-48, -14], [68, -1], [-44, 28], [-54, -22], [26, 40], [-72, 0], [14, 17], [-60, -19], [-18, -28], [-6, 2], [48, 8], [-26, -7], [32, -10]];
    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_ca();".
// 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_ca(:prec:=2)
    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_ca();".
// 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_ca( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, -4, 1]>,<11,R![-4, 4, 1]>,<13,R![-4, -4, 1]>,<17,R![14, 8, 1]>],Snew);
    return Vf;
end function;