
// Make newform 1300.2.bs.a in Magma, downloaded from the LMFDB on 25 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_a();".
// 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_a();".
// 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, -1, 0, 0], [0, 0, 0, 0], [-1, -2, -1, 0], [1, -3, 2, 2], [-3, 0, 0, 2], [1, -1, 2, 3], [1, 1, -2, 1], [3, 1, -4, -4], [-6, -4, 3, 4], [5, 2, -2, -5], [2, -2, -1, 2], [-3, 1, 4, -4], [5, 5, 4, -9], [2, 0, -4, 4], [3, -4, 4, -3], [9, -9, -6, 3], [1, 0, -1, 0], [0, -2, 9, -2], [1, -1, 2, 3], [6, 8, 0, -4], [2, 0, -4, 8], [-2, 0, 4, 0], [5, 9, 4, -4], [1, 2, -1, -4], [2, 0, -1, 0], [7, -10, -10, 7], [-1, -3, 4, 4], [-7, 12, 12, -7], [-11, 11, 10, -1], [-9, -9, 0, 0], [0, 16, 0, -8], [-3, 6, -3, 0], [-1, -8, -1, 0], [13, 13, -12, -1], [9, -6, -6, 9], [-9, 12, 12, -9], [-5, 2, 5, -4], [10, -2, -5, 2], [-3, -3, -6, 9], [0, 4, 5, 4], [4, 0, -8, -8], [-17, -4, 17, 8], [0, -10, -7, -10], [0, 10, -3, 10], [3, 4, -3, -8], [0, 4, -3, 4], [2, -18, -1, 18], [3, 2, -3, -4], [-11, -16, 16, 11], [11, 0, 0, 11], [-7, 6, 6, -7], [-7, -7, -8, 15], [-9, 12, -9, 0], [-1, -11, -10, 10], [3, -9, -12, 12], [-11, 8, -11, 0], [-15, 9, 6, 6], [-15, 15, 10, -5], [-3, 12, 12, -3], [-5, -11, 16, 16], [-19, 2, 19, -4], [6, 0, -12, 20], [14, 0, -28, -8], [7, 20, -20, -7], [10, 8, 0, -4], [5, -5, 2, 7], [11, 8, -8, -11], [5, 5, 12, -17], [-19, -11, 8, -8], [2, -14, -1, 14], [17, 18, -18, -17], [3, 9, -12, -12], [-15, 15, 2, -13], [17, -11, -6, -6], [3, 6, 3, 0], [10, 16, 0, -8], [13, 14, 13, 0], [9, -1, -8, -8], [-7, -7, 8, -1], [18, -16, -9, 16], [25, 8, -8, -25], [25, 11, -14, 14], [17, 17, -10, -7], [0, -4, 5, -4], [3, 22, -22, -3], [-15, 15, 8, -7], [0, 6, -19, 6], [-15, 15, 8, -7], [16, -16, 0, 8], [-1, 14, -14, 1], [13, 15, 2, -2], [-17, 10, 17, -20], [-14, 4, 7, -4], [1, -6, -6, 1], [-7, 7, -4, -11], [-7, -9, 16, 16], [-10, -16, 0, 8], [-13, 7, 20, -20], [17, 8, 8, 17], [7, -6, -6, 7], [26, 2, -13, -2], [-11, -11, 20, -9], [0, 8, 17, 8], [-22, 0, 44, -4], [22, -8, 0, 4], [0, -10, 9, -10], [14, 8, 0, -4], [-6, 0, 12, 16], [0, -8, -23, -8], [-7, -7, -16, 23], [10, -2, -5, 2], [1, -18, -1, 36], [25, 14, -14, -25], [9, 9, 10, -19], [13, -24, 13, 0], [-25, -2, -25, 0], [-5, -9, -4, 4], [15, 21, 6, -6], [-1, 32, -1, 0], [5, 3, -8, -8], [-13, 15, -2, -2], [3, -20, -20, 3], [27, -2, -27, 4], [5, 7, 2, -2], [-4, 0, 8, 36], [-3, 3, 0, -3], [-9, 12, 9, -24], [11, 18, -18, -11], [-20, 0, 40, -4], [-31, -13, 18, -18], [-6, 30, 3, -30], [10, -28, -5, 28], [-13, 7, 6, 6], [5, 5, 4, -9], [-3, -5, 8, 8], [-7, 30, -7, 0], [15, -18, 15, 0], [17, -17, -14, 3], [26, -12, -13, 12], [13, -14, 14, -13], [-11, -3, 8, -8], [-23, -23, 16, 7], [-2, 0, 4, 28], [0, 4, 9, 4], [-11, 11, -10, -21], [14, -48, 0, 24], [-25, -12, 12, 25], [-22, 0, 44, -8], [-14, 0, 28, 4], [-3, 22, 3, -44], [2, 4, -1, -4], [19, 6, 6, 19], [19, 9, -28, -28], [15, 11, -4, 4], [-36, -8, 0, 4], [-9, 28, -9, 0], [9, 9, -4, -5], [-33, 24, 24, -33], [-19, 0, 0, 19], [26, 26, -13, -26], [-15, -15, 14, 1], [-24, -40, 0, 20], [-33, 0, 33, 0], [0, 14, 25, 14], [36, -8, 0, 4], [0, -4, -43, -4], [5, 5, -6, 1], [21, 12, -12, -21], [-1, 4, 4, -1], [-11, 22, 22, -11], [21, 21, -36, 15], [-21, 20, -21, 0], [9, 10, 9, 0], [-28, -24, 0, 12], [-7, -16, -7, 0], [-23, 29, -6, -6], [-3, -28, -28, -3], [7, 13, -20, -20], [-6, 24, 3, -24], [6, 0, -12, -40], [-2, 0, 4, 0], [7, 32, -32, -7], [18, 16, 0, -8], [0, -2, 17, -2], [25, 0, -25, 0], [19, 8, -8, -19], [-2, 0, 4, -8], [13, 17, 4, -4], [29, 29, -2, -27], [17, -17, 2, 19], [-9, 46, -9, 0], [8, -24, 0, 12], [17, -41, 24, 24], [-19, 19, -20, -39], [-21, -1, 22, 22], [-3, -4, 4, 3], [-27, -27, -14, 41], [16, 0, -32, 12], [-13, -6, 6, 13], [-3, 3, -12, -15], [7, -8, -7, 16], [0, -2, 5, -2], [24, 0, -48, 16], [-11, 19, 30, -30], [7, 3, -10, -10], [-19, 14, 14, -19], [-15, 15, 24, 9], [13, -21, 8, 8], [-13, -44, -13, 0], [-17, -27, -10, 10], [-50, -8, 0, 4], [27, 3, -24, 24], [-1, 18, -1, 0], [-15, -15, 18, -3], [21, 4, 4, 21], [-25, 10, 10, -25], [29, 8, -29, -16], [0, 14, -3, 14], [-18, 0, 0, 0], [0, 8, -31, 8], [1, 18, -18, -1], [-43, 20, 20, -43], [3, -6, 3, 0], [-49, -25, 24, -24], [-46, 0, 0, 0], [27, 13, -14, 14], [19, 4, 19, 0], [-35, 35, 36, 1], [13, -6, -6, 13], [35, -25, -10, -10], [18, -28, -9, 28], [9, -9, -18, 18], [8, 0, -16, -12], [-17, -34, 34, 17], [28, -8, 0, 4], [-3, 3, 8, 5], [0, -10, -27, -10], [3, -24, -3, 48], [0, -16, 9, -16], [10, -18, -5, 18], [-22, 32, 11, -32], [-47, 47, 36, -11], [17, -21, 4, 4], [29, -10, 29, 0], [-36, 24, 0, -12], [7, -6, 7, 0], [25, 17, -42, -42], [33, 33, -10, -23], [35, -19, -16, -16], [34, -18, -17, 18], [13, 17, 4, -4], [2, 0, -4, 12], [0, 0, -35, 0], [23, -12, 12, -23], [21, -21, 34, 55], [-27, 24, 27, -48], [0, -6, -19, -6], [47, 4, -4, -47], [-31, 2, 31, -4], [23, -10, -10, 23], [11, -39, 28, 28], [9, -24, -24, 9], [-9, 1, 10, -10], [-3, 66, -3, 0], [7, -4, 7, 0], [9, 9, -4, -5], [5, -4, 4, -5], [-1, 30, 1, -60], [17, 17, 26, -43], [-24, 0, 48, -8], [-16, -56, 0, 28], [0, -18, -35, -18], [-9, -24, 9, 48], [25, 25, -26, 1], [2, 22, -1, -22], [15, -18, -15, 36], [9, -16, 16, -9], [9, 9, -12, 3], [-1, 36, -1, 0], [47, 11, -36, 36], [21, -47, 26, 26], [-41, 22, 22, -41], [13, 25, 12, -12], [2, 0, -4, 24], [-10, 0, 20, -4], [3, -16, 16, -3], [-22, -8, 0, 4], [29, -29, -26, 3], [-27, -24, 27, 48], [0, 0, 81, 0], [-23, -23, 4, 19], [-14, -10, 7, 10], [37, 37, -26, -11], [45, -45, -18, 27], [14, -24, 0, 12], [12, -64, 0, 32], [-5, 23, -18, -18], [34, 0, -17, 0], [66, 2, -33, -2], [5, 5, 2, -7], [0, 0, 0, 0], [0, -4, 73, -4]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : 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, 2, 5, 4, 1]>],Snew);
    return Vf;
end function;