
// Make newform 1300.2.bn.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1300_bn();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1300_bn_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1300_2_bn_b();".
// 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_1300_2_bn_b();".
// 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 := [1, 0, -1, 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_1300_bn();"
function MakeCharacter_1300_bn()
    N := 1300;
    order := 12;
    char_gens := [651, 677, 301];
    v := [12, 9, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1300_bn_Hecke();"
function MakeCharacter_1300_bn_Hecke(Kf)
    N := 1300;
    order := 12;
    char_gens := [651, 677, 301];
    char_values := [[1, 0, 0, 0], [0, 0, 0, -1], [0, 1, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0], [1, 0, -1, -1], [0, 0, 0, 0], [-2, -1, 2, 2], [1, -3, 2, 2], [-2, 0, 0, -3], [2, 2, 1, -3], [-1, -1, 2, -1], [3, 4, 1, -1], [6, 4, -3, -4], [5, 2, -2, -5], [0, -1, 2, -1], [-3, 1, 4, -4], [4, -4, 5, 9], [4, -4, 0, 2], [7, -4, -4, 7], [-9, 9, 6, -3], [1, 0, -1, 0], [-4, 9, 2, -9], [1, -1, 2, 3], [-4, 0, 8, 6], [-2, 0, 4, -8], [0, -4, 0, 2], [-5, -9, -4, 4], [-2, -1, -2, 0], [2, 0, -1, 0], [3, 10, -10, -3], [1, 4, 3, -3], [7, -12, -12, 7], [-10, -10, 11, -1], [-9, 0, 9, 9], [0, 16, 0, -8], [6, -3, -6, 6], [1, 8, 1, 0], [-13, -13, 12, 1], [9, -6, -6, 9], [3, 12, -12, -3], [2, -5, 2, 0], [0, -5, 2, -5], [-6, 6, -3, -9], [0, -4, -5, -4], [4, 0, -8, -8], [-17, -4, 17, 8], [20, 7, -10, -7], [20, -3, -10, 3], [-3, -4, 3, 8], [0, 4, -3, 4], [0, 1, -18, 1], [-2, -3, -2, 0], [11, 16, -16, -11], [-11, 0, 0, 11], [7, -6, -6, 7], [-7, -7, -8, 15], [-9, 12, -9, 0], [-1, -10, 11, 11], [-3, 12, -9, -9], [11, -8, 11, 0], [-15, 9, 6, 6], [10, 10, -15, 5], [-3, 12, 12, -3], [-5, -16, -11, 11], [2, -19, 2, 0], [20, -12, 0, 6], [14, 0, -28, -8], [-13, 20, 20, -13], [4, 0, -8, -10], [5, -5, 2, 7], [-3, -8, -8, -3], [-12, 12, -5, -17], [19, 11, -8, 8], [0, 1, -14, 1], [-17, -18, 18, 17], [-3, -12, -9, 9], [-2, -2, 15, -13], [-17, 11, 6, 6], [-6, -3, 6, 6], [-10, -16, 0, 8], [14, 13, -14, -26], [9, -1, -8, -8], [7, 7, -8, 1], [-18, 16, 9, -16], [25, 8, -8, -25], [25, 11, -14, 14], [-10, 10, 17, 7], [0, 4, -5, 4], [-19, 22, 22, -19], [15, -15, -8, 7], [12, -19, -6, 19], [-15, 15, 8, -7], [8, 0, -16, 16], [15, -14, -14, 15], [-13, -15, -2, 2], [-10, 17, -10, 0], [-14, 4, 7, -4], [-1, 6, 6, -1], [4, 4, 7, -11], [7, 9, -16, -16], [-10, -16, 0, 8], [13, -20, 7, 7], [17, 8, 8, 17], [1, -6, 6, -1], [0, -13, -2, -13], [20, -20, -11, 9], [0, -8, -17, -8], [-22, 0, 44, -4], [-4, 0, 8, -22], [-20, 9, 10, -9], [-4, 0, 8, 14], [6, 0, -12, -16], [0, -8, -23, -8], [16, -16, 7, 23], [0, 5, -2, 5], [18, -1, 18, 0], [-25, -14, 14, 25], [9, 9, 10, -19], [13, -24, 13, 0], [2, 25, -2, -50], [-5, -4, 9, 9], [-15, -6, 21, 21], [1, -32, 1, 0], [5, 3, -8, -8], [-13, 2, 15, -15], [-17, -20, 20, 17], [-2, 27, -2, 0], [5, 7, 2, -2], [-4, 0, 8, 36], [3, -3, 0, 3], [9, -12, -9, 24], [7, -18, -18, 7], [4, -40, 0, 20], [31, 13, -18, 18], [0, -3, 30, -3], [10, -28, -5, 28], [13, 6, -7, 7], [5, 5, 4, -9], [3, 5, -8, -8], [-30, 7, 30, -14], [-18, 15, 18, -30], [-14, -14, 17, -3], [-26, 12, 13, -12], [13, -14, 14, -13], [-11, -3, 8, -8], [16, -16, -23, -7], [28, 4, 0, -2], [0, -4, -9, -4], [11, -11, 10, 21], [24, 0, -48, 14], [13, 12, 12, 13], [22, 0, -44, 8], [-4, -28, 0, 14], [-22, 3, -22, 0], [2, 4, -1, -4], [-25, -6, 6, 25], [-19, -28, -9, 9], [15, -4, -11, -11], [-36, -8, 0, 4], [9, -28, 9, 0], [-9, -9, 4, 5], [-9, 24, -24, 9], [-19, 0, 0, 19], [0, -13, -26, -13], [14, -14, -15, -1], [-20, 0, 40, 24], [-33, 0, 33, 0], [28, 25, -14, -25], [4, 0, -8, 36], [0, -4, -43, -4], [6, -6, -5, 1], [-21, -12, 12, 21], [-3, -4, 4, 3], [11, -22, -22, 11], [21, 21, -36, 15], [-21, 20, -21, 0], [-10, -9, 10, 18], [28, 24, 0, -12], [7, 16, 7, 0], [-23, 29, -6, -6], [-3, -28, -28, -3], [7, 20, 13, -13], [6, -24, -3, 24], [-40, -12, 0, 6], [-2, 0, 4, 0], [-25, 32, 32, -25], [8, 0, -16, -18], [4, -17, -2, 17], [-25, 0, 25, 0], [-11, -8, -8, -11], [8, -4, 0, 2], [-13, -17, -4, 4], [29, 29, -2, -27], [-2, -2, -17, 19], [-46, 9, 46, -18], [8, -24, 0, 12], [17, -41, 24, 24], [-20, -20, -19, 39], [-21, -22, -1, 1], [-3, -4, 4, 3], [-14, 14, -27, -41], [12, -32, 0, 16], [-7, -6, -6, -7], [3, -3, 12, 15], [7, -8, -7, 16], [-4, 5, 2, -5], [-24, 0, 48, -16], [11, -19, -30, 30], [-7, -10, -3, 3], [19, -14, -14, 19], [-24, -24, 15, 9], [-13, 21, -8, -8], [-13, -44, -13, 0], [-17, -10, 27, 27], [-50, -8, 0, 4], [-27, 24, 3, 3], [18, -1, -18, 2], [15, 15, -18, 3], [21, 4, 4, 21], [-15, 10, -10, 15], [29, 8, -29, -16], [28, -3, -14, 3], [0, 0, 0, -18], [0, 8, -31, 8], [-1, -18, 18, 1], [43, -20, -20, 43], [6, -3, -6, 6], [-49, 24, 25, 25], [46, 0, 0, 0], [-27, 14, 13, 13], [-19, -4, -19, 0], [36, 36, -35, -1], [13, -6, -6, 13], [35, 10, -25, 25], [-18, 28, 9, -28], [9, -9, -18, 18], [8, 0, -16, -12], [17, -34, -34, 17], [-4, 0, 8, -28], [3, -3, -8, -5], [20, 27, -10, -27], [-3, 24, 3, -48], [0, -16, 9, -16], [0, 5, -18, 5], [-22, 32, 11, -32], [-36, -36, 47, -11], [-17, 21, -4, -4], [10, -29, -10, 58], [36, -24, 0, 12], [-6, 7, 6, -14], [25, 17, -42, -42], [-33, -33, 10, 23], [35, 16, -19, 19], [0, -17, 18, -17], [13, 17, 4, -4], [12, -4, 0, 2], [0, 0, 35, 0], [35, -12, -12, 35], [-21, 21, -34, -55], [-27, 24, 27, -48], [-12, -19, 6, 19], [-43, -4, -4, -43], [-2, 31, -2, 0], [-13, 10, -10, 13], [-11, 28, 39, -39], [-9, 24, 24, -9], [9, -10, 1, 1], [66, -3, -66, 6], [-7, 4, -7, 0], [-9, -9, 4, 5], [5, -4, 4, -5], [30, -1, 30, 0], [26, -26, 17, 43], [-24, 0, 48, -8], [-28, 0, 56, 16], [36, 35, -18, -35], [9, 24, -9, -48], [26, -26, -25, 1], [0, 1, 22, 1], [18, -15, 18, 0], [-9, 16, -16, 9], [9, 9, -12, 3], [-1, 36, -1, 0], [-47, 36, 11, 11], [21, -47, 26, 26], [-19, 22, -22, 19], [13, 25, 12, -12], [24, -4, 0, 2], [-10, 0, 20, -4], [19, -16, -16, 19], [-4, 0, 8, 22], [-29, 29, 26, -3], [27, 24, -27, -48], [0, 0, 81, 0], [-4, 4, 23, 19], [0, -7, -10, -7], [37, 37, -26, -11], [18, 18, -45, 27], [-14, 24, 0, -12], [12, -64, 0, 32], [-5, 18, 23, -23], [-34, 0, 17, 0], [0, -33, -2, -33], [2, -2, 5, 7], [0, 0, 0, 0], [0, 4, -73, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1300_bn_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1300_2_bn_b();".
// 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_1300_2_bn_b(:prec:=4)
    chi := MakeCharacter_1300_bn();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1300_2_bn_b();".
// 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_1300_2_bn_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1300_bn();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -4, 5, -2, 1]>],Snew);
    return Vf;
end function;