
// Make newform 1300.2.bs.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1300_bs();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1300_bs_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_bs_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_bs_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_bs();"
function MakeCharacter_1300_bs()
    N := 1300;
    order := 12;
    char_gens := [651, 677, 301];
    v := [12, 9, 7];
    // 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_bs_Hecke();"
function MakeCharacter_1300_bs_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, 0], [-3, 1, 2, 2], [2, 0, 0, 3], [-2, 2, 1, -1], [-1, -1, -2, 3], [3, -4, 1, 1], [-2, 0, 1, 0], [-3, -6, 6, 3], [4, -5, -2, 5], [5, 5, 0, 0], [4, 4, -3, -1], [-4, 0, 8, 4], [-1, -4, 4, 1], [-5, 5, 6, 1], [9, 0, -9, 0], [0, 1, -14, 1], [-3, 3, 2, -1], [8, 8, 0, -4], [2, 0, -4, 4], [-4, 0, 8, -4], [-5, -9, -4, 4], [-6, -1, 6, 2], [-6, -12, 3, 12], [3, 6, 6, 3], [1, 4, -5, -5], [-9, 4, 4, -9], [-2, 2, -1, -3], [7, 12, 5, -5], [-8, 16, 0, -8], [10, -3, 10, 0], [-3, 0, -3, 0], [-5, -5, 12, -7], [-3, 2, 2, -3], [-1, -4, -4, -1], [-2, 3, 2, -6], [4, 15, -2, -15], [-2, -2, 13, -11], [0, 4, 7, 4], [4, 0, -8, -16], [15, 0, -15, 0], [0, -9, -10, -9], [0, 1, -6, 1], [5, 8, -5, -16], [0, -4, 1, -4], [-4, 5, 2, -5], [2, -3, -2, 6], [3, -8, 8, -3], [9, -16, -16, 9], [3, 10, 10, 3], [5, 5, 4, -9], [-1, -16, -1, 0], [-5, -14, -9, 9], [5, -12, -17, 17], [-1, 20, -1, 0], [5, 5, -10, -10], [2, -2, 5, 7], [-11, -4, -4, -11], [-13, -4, 17, 17], [-2, 13, 2, -26], [0, 0, 0, -4], [-2, 0, 4, 0], [7, -4, 4, -7], [12, 8, 0, -4], [-19, 19, 14, -5], [13, 8, -8, -13], [8, 8, -9, 1], [-13, -1, 12, -12], [-4, -7, 2, 7], [11, 6, -6, -11], [5, 4, -9, -9], [-6, 6, -9, -15], [3, 3, -6, -6], [-2, 29, -2, 0], [-22, 8, 0, -4], [6, 1, 6, 0], [9, -5, -4, -4], [11, 11, -4, -7], [-42, 0, 21, 0], [1, 16, -16, -1], [-11, -13, -2, 2], [-22, -22, 17, 5], [0, -4, 7, -4], [9, 18, -18, -9], [-9, 9, -8, -17], [0, 9, -2, 9], [21, -21, -4, 17], [32, -12, 0, 6], [-13, -10, 10, 13], [-9, 1, 10, -10], [-14, 9, 14, -18], [-6, 12, 3, -12], [23, -2, -2, 23], [-16, 16, 11, -5], [-1, -19, 20, 20], [6, -16, 0, 8], [-11, -4, 7, -7], [-11, 24, 24, -11], [-19, 14, 14, -19], [-12, -5, 6, 5], [12, 12, 5, -17], [0, 20, 3, 20], [6, 0, -12, 32], [0, -24, 0, 12], [0, 17, 18, 17], [24, 8, 0, -4], [-14, 0, 28, 12], [0, 8, 1, 8], [4, 4, -21, 17], [4, 21, -2, -21], [-2, -9, 2, 18], [11, 10, -10, -11], [25, 25, -10, -15], [13, -16, 13, 0], [18, 5, 18, 0], [-5, 0, 5, -5], [-3, -34, -31, 31], [-19, -8, -19, 0], [5, -9, 4, 4], [-9, -14, 23, 23], [-21, 36, 36, -21], [-6, -5, 6, 10], [-7, -25, -18, 18], [4, 0, -8, 28], [-17, 17, -4, -21], [-35, -4, 35, 8], [27, 18, -18, -27], [-12, 0, 24, 6], [-13, -23, -10, 10], [20, 5, -10, -5], [42, -16, -21, 16], [17, -10, -7, -7], [5, 5, -20, 15], [19, -11, -8, -8], [10, -37, 10, 0], [14, 19, 14, 0], [14, -14, -15, -1], [14, -24, -7, 24], [-15, -22, 22, 15], [-3, -19, -16, 16], [-12, -12, -3, 15], [-8, 0, 16, 12], [0, 8, -5, 8], [-9, 9, 18, 9], [40, 16, 0, -8], [9, 28, -28, -9], [6, 0, -12, 8], [-8, 0, 16, -16], [-22, -9, 22, 18], [2, 4, -1, -4], [-9, 6, 6, -9], [-3, 4, -1, -1], [-41, -16, 25, -25], [-16, 48, 0, -24], [1, -40, 1, 0], [15, 15, -12, -3], [19, -16, -16, 19], [1, 16, -16, -1], [-28, 19, 14, -19], [14, 14, -27, 13], [-48, -12, 0, 6], [15, 12, -15, -24], [0, 9, 2, 9], [-12, 12, 0, -6], [0, -20, -7, -20], [-2, -2, -9, 11], [-17, 4, -4, 17], [9, -12, -12, 9], [-25, -6, -6, -25], [-27, -27, 36, -9], [3, 24, 3, 0], [-10, 19, -10, 0], [-28, 8, 0, -4], [3, 52, 3, 0], [29, -15, -14, -14], [-11, -4, -4, -11], [-1, -8, 9, 9], [30, -28, -15, 28], [0, 0, 0, 48], [-2, 0, 4, 24], [27, 0, 0, -27], [16, 8, 0, -4], [0, -13, 2, -13], [23, 0, -23, 0], [25, 32, -32, -25], [20, 0, -40, -28], [-21, -5, 16, -16], [13, 13, 2, -15], [-30, 30, 15, -15], [-6, -3, -6, 0], [12, 0, 0, 0], [-7, 11, -4, -4], [8, -8, 25, 33], [-9, 22, -13, -13], [9, 12, -12, -9], [6, 6, -27, 21], [12, 0, -24, -10], [25, 46, -46, -25], [11, -11, 12, 23], [11, 8, -11, -16], [0, -15, 14, -15], [-8, 0, 16, -48], [-1, -27, -26, 26], [37, -30, -7, -7], [-5, 18, 18, -5], [20, -20, 19, 39], [-5, 33, -28, -28], [-21, -8, -21, 0], [35, 26, -9, 9], [-6, 16, 0, -8], [13, -16, -29, 29], [10, 27, 10, 0], [39, 39, -46, 7], [13, -12, -12, 13], [17, 22, 22, 17], [-15, 4, 15, -8], [0, -15, -10, -15], [-4, 8, 0, -4], [0, -4, 37, -4], [-13, -2, 2, 13], [23, -28, -28, 23], [30, 9, 30, 0], [-1, -20, -19, 19], [22, 0, 0, 0], [9, 2, -7, 7], [13, -24, 13, 0], [-8, 8, 1, -7], [21, -22, -22, 21], [-9, -18, 27, 27], [14, 32, -7, -32], [-3, 11, 14, -14], [28, 0, -56, -8], [-15, -6, 6, 15], [16, 28, 0, -14], [-21, 21, -16, -37], [0, 7, -22, 7], [-11, 4, 11, -8], [0, 4, 9, 4], [-12, -3, 6, 3], [-30, -24, 15, 24], [-16, 16, -29, -45], [23, -19, -4, -4], [-18, 11, -18, 0], [20, -24, 0, 12], [-2, -49, -2, 0], [5, -11, 6, 6], [-5, -5, 2, 3], [43, -32, -11, -11], [20, 11, -10, -11], [-3, 25, 28, -28], [8, 0, -16, 8], [0, 24, 3, 24], [31, 28, -28, -31], [-29, 29, 26, -3], [49, -4, -49, 8], [0, 17, -30, 17], [-35, 4, -4, 35], [-26, 3, 26, -6], [19, -2, -2, 19], [21, -8, -13, -13], [-17, 16, 16, -17], [-27, -6, 21, -21], [-6, -23, -6, 0], [17, 40, 17, 0], [-5, -5, 8, -3], [13, -4, 4, -13], [10, -9, -10, 18], [-6, -6, -3, 9], [-12, 0, 24, -28], [24, -28, 0, 14], [0, 3, 14, 3], [33, -12, -33, 24], [14, 14, -25, 11], [4, -55, -2, 55], [54, 1, -54, -2], [-41, -32, 32, 41], [-35, -35, -8, 43], [7, -16, 7, 0], [49, 36, -13, 13], [-31, 13, 18, 18], [41, -14, -14, 41], [37, 41, 4, -4], [-12, 0, 24, -16], [-10, 0, 20, 60], [-9, -24, 24, 9], [8, 48, 0, -24], [-17, 17, -14, -31], [-1, 44, 1, -88], [0, -12, -35, -12], [-24, -24, -5, 29], [4, -31, -2, 31], [5, 5, 10, -15], [-10, 10, 19, 9], [-38, 8, 0, -4], [-12, 32, 0, -16], [31, 18, -49, -49], [-18, -36, 9, 36], [4, 31, -2, -31], [-22, -22, 17, 5], [-24, 0, 48, -26], [0, -8, -49, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1300_bs_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1300_2_bs_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_bs_b(:prec:=4)
    chi := MakeCharacter_1300_bs();
    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_bs_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_bs_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1300_bs();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -4, 5, -2, 1]>],Snew);
    return Vf;
end function;