
// Make newform 3520.2.a.bk in Magma, downloaded from the LMFDB on 14 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3520_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_3520_2_a_bk();".
// 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_3520_2_a_bk();".
// 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 := [-4, -1, 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_3520_a();"
function MakeCharacter_3520_a()
    N := 3520;
    order := 1;
    char_gens := [2751, 1541, 2817, 321];
    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_3520_a_Hecke(Kf)
    return MakeCharacter_3520_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, 0], [0, -1], [-1, 0], [0, 1], [1, 0], [-2, 0], [2, -1], [0, 1], [0, -2], [-2, -3], [-4, -1], [-2, 3], [2, 0], [0, 4], [8, 2], [-2, -1], [-4, 2], [-10, 3], [-4, 4], [4, -3], [-2, 0], [0, 6], [0, 2], [-10, -1], [2, 2], [-6, 4], [0, -4], [-16, 0], [2, -8], [-14, -2], [-4, -4], [0, -3], [-6, 10], [12, 0], [-10, 3], [-8, -2], [-2, -5], [16, -5], [8, 1], [6, -4], [4, -4], [-6, 4], [0, -4], [18, -5], [-10, 6], [-12, 5], [16, -3], [-8, -2], [8, -2], [-6, 12], [2, 7], [0, -10], [-6, -4], [-4, -10], [2, -12], [8, -5], [-6, 10], [-8, -8], [-2, -2], [2, -4], [-8, -6], [6, -8], [0, -10], [28, -1], [10, 8], [-26, 5], [12, -8], [10, -3], [-24, 2], [10, -8], [-14, -8], [-16, -4], [-16, 12], [-2, 0], [-20, -2], [16, 6], [-22, -4], [18, 0], [-10, 1], [2, 2], [4, -12], [18, -10], [16, 0], [18, 6], [24, 4], [-4, 0], [18, 0], [26, 1], [-18, -1], [-8, 6], [-8, 7], [-8, 16], [-8, 4], [-16, 9], [-4, 8], [4, 4], [-6, -6], [-22, 12], [-24, 8], [-10, 17], [-16, 0], [-26, -8], [-8, 2], [18, -18], [-8, 15], [-6, 12], [40, -3], [22, -8], [20, -1], [-30, 0], [-8, -5], [14, -12], [-22, 6], [12, -8], [4, 5], [-10, -11], [16, 3], [8, 2], [-2, 5], [32, -5], [-6, -4], [18, -21], [-2, -10], [-8, -7], [-12, 6], [-42, 3], [-22, 8], [-36, 11], [8, -10], [14, 2], [-12, -12], [0, 1], [-12, 11], [26, 0], [-6, -12], [-6, 12], [30, 3], [-24, -12], [34, -4], [-14, 8], [40, -5], [2, 0], [32, 8], [16, -8], [10, -6], [-32, 8], [14, -4], [-6, 15], [-44, 2], [16, -10], [22, 4], [2, 0], [24, -7], [20, 4], [32, -9], [20, -7], [-8, 0], [22, 9], [-42, 0], [22, -7], [-32, -3], [-6, -11], [32, -17], [-4, 4], [-30, 2], [-24, 2], [-32, 0], [22, -14], [-38, -8], [46, -5], [0, 9], [34, -8], [-40, 10], [34, 3], [16, -8], [30, -15], [-20, 8], [-30, 2], [32, -5], [-2, -1], [-32, 2], [40, -3], [-18, 9], [-14, -5], [8, -2], [-22, 4], [14, 14], [-36, 0], [10, -12], [16, 0], [2, -2], [32, -20], [36, 6], [-22, 26], [32, -4], [-22, -8], [34, -2], [-38, -12], [14, -20], [24, -13], [-2, 5], [8, 10], [6, 1], [2, 14], [-4, 16], [14, -5], [-8, -8], [16, -8], [2, 0], [-20, 20], [-14, 25], [2, -22], [-16, 20], [8, -11], [-32, 8], [30, 1], [32, -13], [18, 20], [16, 0], [14, 9], [-14, 0], [40, -2], [-18, -13], [-8, -12], [-56, 8], [6, 27], [-14, 32], [44, 9], [4, 4], [68, 0], [-46, 12], [32, -5], [-56, 2], [-14, 4], [20, -24], [-16, 11], [22, -31], [6, 4], [-12, 0], [-12, 3], [-40, -11], [4, 8], [0, 4], [-14, 16], [-18, 12], [-40, 10], [-64, -4], [-12, 30], [16, 5], [-32, 27], [14, -28], [-70, 2], [40, -6], [-34, 9], [30, 10], [24, -3], [18, -6], [0, 18], [-38, -8], [42, -3], [52, -4], [-32, 12], [62, 7], [-18, 12], [-38, 10], [-4, 28], [-6, -2], [-58, 7], [-16, -16], [6, 8], [-14, -16], [16, 1], [-6, -10], [8, 26], [26, -17], [-16, -14], [36, -12], [-38, -4], [-30, 28], [-24, 15], [40, -15], [-40, -4], [-4, -12], [6, 19], [-8, 4], [28, 15], [-6, 4], [-26, -4], [-68, 9], [-46, 8], [-6, -22], [-12, 24], [-46, 3], [32, -13], [-18, 24], [-14, -20], [8, 16], [-10, 9], [44, 0], [32, 6], [42, 1], [-34, 30], [0, 12], [-20, -8], [-12, 6], [-22, -26], [-12, -24], [10, -26], [-44, 19], [-26, -6], [-52, -8], [26, 12], [-6, 10], [-24, -5], [-24, 1], [2, 12], [28, 2], [24, -8], [10, 12], [-70, 2], [8, 19], [82, -4], [-38, 12], [-8, -2], [10, 13], [22, 1], [28, 10], [-4, 24], [24, -3], [-10, -14], [14, -39], [-14, -28], [-56, 8], [40, -6], [56, 1], [-8, 31], [50, -12], [-6, -17], [-10, 11], [8, 21], [-70, 4], [26, 8], [14, 17], [12, -27], [-10, -9], [-8, -33], [18, 2], [0, 21], [-16, -28], [-26, 29], [8, -27], [58, 12], [42, -30], [40, 3], [-66, 11], [18, 21], [52, -21], [16, -5], [-18, -16], [0, -20], [-42, 12], [2, -26], [16, 20], [-16, 3], [-64, 19], [-54, -10], [22, 0], [-16, 21], [-6, -18], [-36, 12], [76, -8], [20, -4], [22, -5], [64, -14], [66, 0], [-44, 12], [-32, 22], [-38, -1], [26, -26], [26, 17], [26, 6], [-22, -12], [68, 8], [26, 5], [12, -12], [-32, 16], [-40, -24], [62, -7], [-56, -12], [48, -6], [-18, -13], [-42, 21], [-28, 4], [-36, 4], [16, 16], [6, 4], [24, 14], [-18, -11], [-4, -18], [-18, 27], [-26, 25], [-54, 0], [4, 0], [-46, 22], [58, 8], [-8, 22], [22, 21], [34, 4], [0, 25], [20, 6], [-18, 28], [-18, -6], [12, 0], [20, -32], [46, 8], [10, -42], [0, 4], [-16, 16], [2, -20], [-72, 13], [42, 18], [62, -22], [24, -12], [-40, 25], [-22, -4], [14, 13], [-40, 21], [-14, -18], [52, -32], [-24, 26]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3520_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3520_2_a_bk();".
// 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_3520_2_a_bk(:prec:=2)
    chi := MakeCharacter_3520_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_3520_2_a_bk();".
// 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_3520_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, 1, 1]>,<7,R![-4, -1, 1]>,<13,R![2, 1]>,<17,R![-2, -3, 1]>,<19,R![-4, -1, 1]>],Snew);
    return Vf;
end function;