
// Make newform 3525.2.a.v in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3525_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_3525_2_a_v();".
// 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_3525_2_a_v();".
// 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, 23, 0, -10, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-4, 0, 1, 0, 0], [4, -5, -1, 1, 0], [-2, 10, -5, -2, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_3525_a();"
function MakeCharacter_3525_a()
    N := 3525;
    order := 1;
    char_gens := [2351, 1552, 2026];
    v := [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_3525_a_Hecke(Kf)
    return MakeCharacter_3525_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, 0, 0, 0], [-1, 0, 0, 0, 0], [0, 0, 0, 0, 0], [-2, 0, 0, -1, 0], [0, 0, 0, 0, 1], [-1, -1, -1, 0, 1], [2, -1, 0, 0, 0], [0, 1, -1, -1, -1], [-1, 0, -1, -1, -1], [0, 0, -1, 0, 1], [-2, 0, -1, -1, -2], [-2, 0, 2, 2, -1], [0, -4, -1, 2, 0], [-3, 0, 1, 2, -1], [-1, 0, 0, 0, 0], [2, -1, 3, 3, 0], [-3, 3, 2, -1, -2], [3, 0, 0, 2, 2], [-4, -2, 3, 1, 2], [-3, -1, 2, -1, -2], [-3, 0, -4, 1, 1], [-1, 2, -1, 0, 0], [0, -2, -2, 2, 0], [-1, 2, 0, -1, -1], [2, -2, 2, 0, -2], [-5, 0, 2, -1, -1], [0, 2, 0, 1, -1], [-9, 0, 1, 0, -1], [0, 2, -1, -3, 2], [-1, -2, -3, -2, 0], [-4, 0, -2, 0, 2], [1, -4, 0, -5, -1], [1, 2, -1, -4, -1], [-4, -1, -2, 4, 1], [-1, 0, 5, 4, -3], [0, -3, -5, -1, 3], [6, 0, 4, 4, -2], [3, 8, -1, -2, 1], [-1, 2, -3, -1, -4], [6, -1, 0, 0, 2], [0, 4, 4, -2, 0], [6, -6, 5, 3, 0], [-11, -4, -2, 1, -1], [2, 4, -1, -7, 2], [4, 3, -2, 0, 2], [8, -3, 5, 5, 3], [4, 0, 3, 1, 6], [-11, 4, 2, 5, 1], [4, 0, -2, 1, 4], [-8, 0, 1, -3, -4], [2, -4, 8, 2, -2], [8, 0, -1, 1, 2], [-5, 0, 2, -6, -6], [-2, -3, -2, 6, 1], [5, 6, -1, -2, 4], [-2, -6, -7, 1, 4], [-7, 6, -3, -4, 1], [4, 2, 2, -2, 0], [-4, -4, -4, 4, -5], [8, -4, -3, 2, 1], [-20, 0, -2, 0, -2], [1, -6, 1, 6, 5], [-10, 6, -4, -7, 1], [6, 2, 0, 4, 1], [-6, 5, 1, -7, -6], [5, 10, -3, -2, -3], [-7, 0, 1, 4, -1], [4, -4, -4, 0, -1], [4, -2, 4, 2, 2], [2, 6, -1, -7, -2], [0, 12, 5, -3, -6], [-2, 6, 0, 2, 0], [8, -2, 4, 2, 0], [-9, 8, 2, 1, -1], [-6, 4, 2, -2, -5], [-10, 2, 1, -5, 4], [4, 0, -3, -4, -1], [12, 6, 2, -2, 2], [-3, -2, -3, 2, -1], [0, 6, -5, -5, 0], [-22, 2, -2, -4, -2], [-10, 6, -2, 0, 0], [14, 1, 1, -5, -5], [-2, -9, 0, 0, 2], [3, -6, 7, 6, 6], [-18, 6, 0, -1, 1], [-10, 4, 5, -4, -1], [8, 8, 4, -8, 0], [0, -8, -2, -2, 0], [3, 2, -6, -5, 3], [3, -4, 1, 5, -1], [3, -5, -5, 4, 6], [0, 2, 4, -9, 1], [-2, 7, 1, -9, 1], [-10, -5, -2, -6, 1], [-24, 8, 2, -1, 0], [2, 0, -5, 6, 5], [1, 4, 3, 8, -3], [-2, -4, -2, 1, -2], [-3, -4, 0, 6, 5], [-4, -8, -8, 4, 0], [11, 2, 1, -8, -5], [5, -4, 1, 0, 1], [2, -8, 8, 0, -4], [3, -6, -7, -6, 2], [9, -9, 2, 9, 1], [-3, 12, -13, -6, 1], [8, -6, -2, 0, -1], [24, 2, -8, -4, 4], [-9, 0, 2, 10, 1], [-13, -12, 7, 2, 1], [8, 0, -2, 0, -5], [14, 2, 5, -5, 0], [5, 4, 3, -4, 7], [0, 4, 3, 13, -2], [-18, 0, -4, 6, 0], [-6, -8, 4, -3, 6], [-4, 4, -4, -10, 0], [6, -3, -2, -10, -2], [-1, 2, -10, -5, -1], [-13, 4, 4, 2, -1], [-1, -1, -9, 0, 3], [19, -14, 1, 8, 3], [0, -4, 8, 4, -2], [-8, -1, -5, 11, 1], [4, 8, -1, 2, 4], [7, 4, -1, -8, -5], [-2, 7, -1, -7, -7], [13, -10, -9, 0, -3], [0, -6, 8, 8, -1], [-21, 6, -1, 2, 3], [-6, -6, 4, -1, 1], [18, -9, -5, 1, 3], [-14, 8, 5, -5, -4], [-7, 0, -5, 0, 1], [-3, -4, -9, -6, 5], [-2, -12, 1, -5, -2], [-17, -2, -3, 2, -7], [8, -6, 0, 10, 3], [0, 20, 4, -8, 2], [4, -4, -4, -10, 4], [-22, 0, 3, 0, 2], [-8, -8, 2, 8, -2], [-8, -4, 7, 1, 2], [16, -4, 2, 2, -4], [-8, -4, 4, -2, 0], [-28, 10, 0, -6, -2], [5, -8, -3, 4, 2], [-14, 3, 7, 13, -1], [-6, 0, -6, 4, -2], [14, 1, -6, -6, 10], [-8, -12, 1, -6, -1], [14, -10, 2, -1, -1], [3, 12, 5, -6, -9], [6, -2, 8, -1, 1], [8, 4, 9, -5, 8], [-4, -11, 1, -7, -7], [1, 0, -1, -4, 7], [9, 4, 4, -7, -9], [-18, 10, 6, -4, -10], [-2, 6, 5, -5, -6], [9, -6, -1, 4, 1], [-18, 4, -4, 4, 2], [-8, 6, 0, -6, -2], [12, 5, 4, 6, 8], [-3, -2, -1, -3, 4], [-5, 4, -9, -2, -7], [-18, -3, 6, 2, 8], [-10, -2, 0, 8, 0], [9, 10, 11, -2, -2], [14, 7, -11, -5, -7], [0, -16, 11, -2, -1], [34, 2, -6, 0, -2], [-18, -1, 12, 4, 2], [-12, -15, 6, 4, -3], [18, -4, 3, 12, 3], [-19, -2, 1, 6, 1], [-22, -8, -10, 8, 8], [17, 6, -11, -2, 9], [20, 14, -4, 2, -8], [-30, 0, 4, -6, -14], [-14, 2, -6, 10, 2], [-2, -10, 0, 6, 7], [-6, 19, -17, -5, 0], [-21, 6, 7, -3, -2], [34, 12, -7, -1, 6], [14, 2, -2, -10, 4], [-2, 0, -16, -9, 2], [-9, -8, 2, -8, 6], [2, 0, -10, 4, 6], [-24, 2, -4, -6, 3], [3, 4, -11, -9, 1], [16, 3, 12, 2, 1], [1, -2, -1, -8, -5], [-12, 12, 9, -5, -10], [14, 8, 9, -1, -8], [-20, 0, 1, 3, 6], [-29, -3, 5, 6, -3], [-16, 15, -12, -10, -4], [6, -14, 13, 3, 4], [-6, -20, -1, 9, 0], [13, 2, -1, 2, -2], [-11, -1, 2, 17, 7], [-13, -16, 6, 0, 1], [-2, -21, 11, 5, -3], [14, -4, 15, -7, -6], [-24, 11, -6, -8, 1], [4, -10, 8, 14, 14], [10, 8, 2, 4, 10], [2, 2, 11, 7, 6], [-2, 12, -10, -4, -3], [-6, -2, -5, -13, 4], [8, 12, 2, 8, 0], [-39, 2, 1, 7, 2], [15, -9, 10, 5, 4], [12, 2, 10, -4, 4], [-12, -10, 1, -7, 4], [2, 16, -11, 6, 2], [-26, 4, -2, -2, 6], [21, 2, -5, 2, 8], [26, 12, 8, -4, -8], [-13, 0, -5, 4, -1], [24, 4, -8, -6, 4], [-17, 0, -1, -12, -15], [8, 14, -4, -2, -10], [0, -6, 2, 10, 16], [17, -12, -17, -4, 4], [20, 20, 10, -8, -11], [-8, 8, 12, -4, -6], [10, -2, -2, 8, -10], [-23, -1, -12, 7, -3], [-33, -4, 3, -8, -8], [15, 20, -7, 2, -2], [7, -16, 6, -3, 1], [5, -6, 2, 13, 7], [-23, 0, 13, 12, -7], [-3, -20, -11, 4, 7], [10, -5, 6, 10, 2], [9, -3, -5, -6, 4], [1, 9, 5, -12, -4], [19, -2, -13, -13, -6], [-6, 2, 10, 6, -5], [-1, -8, 12, 3, -1], [6, -2, -2, -18, 0], [6, 8, -7, -17, 8], [-21, 4, -14, 3, 1], [-8, 14, -20, -1, 5], [-34, 12, -6, -10, -9], [-17, -4, 1, -12, -8], [-8, 16, -7, -19, 2], [-34, 3, 3, -5, 2], [-3, 6, -2, -1, -11], [-22, 8, -2, 2, 0], [-22, -10, 2, 2, -14], [19, -8, 5, 22, 6], [-5, 9, -3, -2, 0], [-8, 20, -1, -5, 0], [-21, 6, 0, -9, -11], [19, 26, -7, -2, -5], [14, 6, -18, -8, 4], [2, 12, -8, -7, -14], [6, -10, 4, 21, 3], [4, 0, 11, 2, 1], [-20, -4, 10, 8, -3], [-4, -4, 8, -10, 9], [-33, -12, 5, 6, 6], [3, 12, 0, 5, -1], [14, 8, 5, -16, -5], [11, 10, 4, -13, -5], [-13, 10, 13, 0, -16], [1, 4, -8, -18, 7], [26, 12, 8, -9, 2], [-16, -8, 2, -4, 0], [18, -18, 9, 21, -2], [-7, -5, 2, 9, 3], [11, 0, -10, 3, -7], [8, -14, -7, -1, -6], [33, 0, -4, 0, 12], [-10, -18, 1, 13, -8], [-5, 1, -2, 13, 8], [-18, 2, -4, -8, -14], [36, 7, 1, 5, -1], [-36, -8, 0, 0, 8], [-3, 8, 9, 0, -15], [-4, 22, 12, -9, 3], [54, 4, -2, -2, 1], [-25, -13, 6, 3, 15], [18, -10, 16, 10, 8], [6, -13, 2, 10, 11], [9, 2, -6, -19, -9], [18, 2, -12, -16, -2], [-4, -4, -3, -9, -14], [-17, 4, -5, 12, 0], [-25, -11, 11, 0, 12], [-18, 2, -4, -12, -2], [18, 4, -13, -10, -2], [13, 2, -3, -10, -8], [-11, -12, -3, -6, -9], [-2, -16, -2, 12, -5], [-3, -4, 22, 11, 1], [-2, 13, 2, 6, 4], [3, -16, -1, 4, 4], [-6, 16, 6, -2, 1], [-5, 12, 11, -4, -9], [5, -8, -1, 10, 15], [-9, -7, -9, 8, 5], [4, -12, 9, 11, 12], [-30, 4, 15, 14, 8], [-19, -17, 3, 2, 6], [16, 0, -6, 6, -12], [24, -8, 5, 15, 8], [19, -8, 3, 18, 13], [36, 4, -7, 4, 21], [7, 0, -6, -11, -11], [-3, -22, -5, 9, 10], [-23, 4, 6, -6, -16], [-14, -2, 0, 0, -16], [4, 0, 4, -10, -5], [-11, 5, 3, -6, -17], [-9, 14, 10, 5, -7], [22, 18, -2, 4, -1], [1, -13, 7, -10, -5], [0, 4, 2, 16, 2], [20, 8, -6, -13, -16], [-3, -6, -1, -18, 5], [60, -4, 3, 5, -4], [-44, 8, -4, -20, -8], [19, -12, 5, 4, 7], [-11, -16, 3, 0, 11], [4, 16, -3, -15, 10], [-3, 8, -8, -10, -17], [-22, -11, 3, -9, -4], [14, 5, 7, -7, 11], [18, -10, 4, 24, 10], [-43, -4, -1, -8, 3], [-19, -10, -7, 21, -2], [-4, 6, -18, -10, 8], [-14, -23, -1, -1, 10], [15, -8, -6, 6, 12], [-11, 22, -8, -3, -5], [30, -4, -10, -4, 7], [40, -4, 0, 4, 13], [-14, 8, 23, 9, -12], [10, -14, -4, 10, 1], [-25, 0, -3, 14, 5], [-17, 1, 16, 23, 6], [-34, -8, 5, 1, -4], [25, 14, -9, -8, 9], [-38, -3, 7, 13, 5], [-10, -3, -12, 4, 4], [-20, -26, 10, 2, -4], [16, -14, 6, 20, 3], [-50, 12, 10, 4, 4], [0, 4, -3, -3, 14], [-4, -28, -10, 6, 2], [-14, -12, -8, 14, 2], [-21, -4, 0, -15, 7], [7, 7, 5, -14, 2], [-23, -8, 11, 6, -6], [12, -12, -4, 17, -2], [-7, -15, 6, 13, 7], [48, 4, 1, 4, -7], [16, -12, 2, 18, 8], [26, 14, -6, -8, 2], [40, -12, 7, 15, -2], [-3, 14, -14, 3, 7], [29, 14, 1, -2, 6], [-8, 20, 2, 3, -6], [22, -4, -3, 3, 2], [-50, -8, 8, -2, -23], [-26, -8, -2, 20, 11], [-21, 12, -3, -4, 3], [-8, 0, -8, 12, -8], [42, -15, -10, 6, 13], [-10, -21, -1, 11, -8], [-18, -8, -6, 24, -2], [27, -15, 14, 9, 8], [6, 6, 4, 18, 8], [19, -14, 5, 22, 23], [30, 14, 10, 0, 1], [-14, -8, 25, 13, -12], [-32, 1, -5, -15, 0], [-7, 0, -16, 1, -5], [0, 7, -4, 6, -16], [-13, 0, 1, -14, 7], [1, 4, -3, 7, -1], [26, -13, 10, 6, 5], [11, 7, -4, 13, -1], [-4, 18, 8, 7, -11], [-6, -18, -16, 2, 0], [-4, 22, 1, -1, 2], [22, -1, 11, -13, -10], [22, -12, 4, 20, 13], [20, 4, 6, 17, 8], [29, -33, 9, 20, 8], [10, 20, 6, -8, -2], [-6, -1, 13, 19, 5], [13, 2, 10, 7, -5], [-18, 10, -11, -3, 2], [-8, -4, 11, 14, 0], [-4, 4, -12, -12, 0], [-14, -25, -6, 14, 12], [-19, 14, -3, -28, -11], [-4, 21, 1, 15, 0], [-26, 14, 5, 1, -4], [2, -30, -10, 18, 17], [-2, -20, 8, 8, 0], [19, 30, 7, -12, 1], [36, -16, -9, 1, 6], [-38, 18, -2, -16, -13], [-18, 15, 12, 0, -4], [26, 9, -3, -19, 12], [12, -16, 4, 4, -8], [26, 16, 5, -11, -12], [24, 4, -24, -10, 2], [-2, -4, 4, -4, 8], [5, -7, -1, 2, -10], [-7, -22, 5, 20, 3], [-9, 4, 1, -12, -8], [24, 4, -16, -10, 14], [19, 0, -5, -10, -5], [-44, 20, -4, -24, -10], [-16, -4, -2, -11, 8], [8, 16, -15, -17, 12], [-21, -5, -2, -9, 5], [8, -12, -20, 2, -9], [36, 4, 11, 3, -4], [27, 2, 11, 4, 13], [-30, -3, 14, 14, -3], [38, -18, 8, 8, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3525_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3525_2_a_v();".
// 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_3525_2_a_v(:prec:=5)
    chi := MakeCharacter_3525_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_3525_2_a_v();".
// 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_3525_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-4, 23, 0, -10, 0, 1]>,<7,R![18, -43, -4, 26, 10, 1]>,<11,R![8, 20, -8, -20, 2, 1]>],Snew);
    return Vf;
end function;