
// Make newform 1805.2.a.p in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1805_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_1805_2_a_p();".
// 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_1805_2_a_p();".
// 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 := [3, 4, -4, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -1, 1, 0], [2, -3, -2, 1]];
    Rf_basisdens := [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_1805_a();"
function MakeCharacter_1805_a()
    N := 1805;
    order := 1;
    char_gens := [362, 1446];
    v := [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_1805_a_Hecke(Kf)
    return MakeCharacter_1805_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, 0, -1, 0], [0, 0, 0, 1], [-1, 0, 0, 0], [2, -2, 0, 0], [0, 2, 0, 0], [0, -2, 0, -1], [2, 0, 0, 2], [0, 0, 0, 0], [-2, -2, 0, -2], [-2, 0, 0, -2], [-4, 2, -2, -2], [0, 0, -2, -1], [-6, 2, -2, 0], [2, -2, 0, 0], [-6, 2, -4, 0], [4, 0, 2, 1], [0, -2, -2, 0], [2, 4, -2, 0], [4, 2, -2, 3], [4, 2, 2, -2], [6, 0, 0, -2], [4, -2, -2, 0], [2, -2, 0, 2], [2, 0, 4, 2], [-4, -4, 2, 1], [2, -4, -2, -4], [0, -2, 2, -1], [0, -2, 6, -3], [2, -6, 2, -2], [4, -2, 0, 1], [4, 0, 0, -3], [4, 0, 0, 4], [2, 0, 4, 2], [4, -6, 4, -4], [2, 0, 2, -4], [0, 4, 4, 2], [10, 0, 4, 2], [2, 6, -4, 0], [8, -4, 0, -1], [4, 2, 0, 1], [-8, 6, -2, 4], [2, 4, 4, 4], [-4, 4, 0, -4], [12, 0, 6, 5], [-10, 0, 0, -4], [0, 8, 0, 4], [4, -4, 0, -6], [4, 0, 0, 3], [-8, 2, -6, 1], [2, -4, 2, 0], [-2, 0, 0, 4], [12, -4, 8, 0], [-6, 6, 2, 8], [0, -4, 8, 0], [0, -4, -2, 3], [-2, -2, 4, -2], [-6, 2, -6, -6], [4, -2, -4, -4], [2, 8, -4, 6], [2, 6, 6, -4], [2, 6, 4, 6], [4, -8, -6, 1], [16, -2, 6, -3], [4, -6, 0, -8], [14, -4, -4, 0], [24, -6, 4, 3], [4, 0, -4, -6], [-12, -2, -8, 5], [18, -6, 4, 6], [-14, -4, -4, -4], [-6, -4, 4, -6], [-8, -2, -8, 4], [14, 2, -8, 0], [12, -12, -2, 5], [4, -4, -4, 0], [20, 2, 2, 5], [6, -12, -4, 0], [-2, -12, 0, 2], [-10, -8, 4, -4], [14, 2, 6, -2], [24, 0, 4, 0], [-2, 2, -2, 0], [8, -16, 4, 2], [-20, 10, -4, -3], [20, -6, 6, 4], [-2, -10, 8, -2], [-2, 6, 2, 10], [2, 4, -4, 0], [-2, 0, 0, -8], [14, -2, 0, 6], [2, 2, -4, 2], [-16, 2, -4, -4], [12, -2, -10, -1], [-8, 8, -12, 4], [-4, 6, 12, 0], [6, 2, 12, 6], [-10, 4, -4, 2], [-10, 8, 0, 8], [4, -4, 8, -9], [-14, 4, 2, -4], [4, 8, 4, -3], [-14, -4, 8, -2], [-4, 12, 0, -1], [-18, 18, -6, -6], [0, -10, -4, -8], [22, -8, 0, -4], [-10, 10, -8, 2], [-14, 12, 4, 4], [-20, 2, -2, 4], [-22, -10, 2, -8], [4, 2, 2, 3], [-2, 0, -8, 2], [6, 4, 4, -6], [4, 2, -4, -4], [-20, 2, 0, -4], [-14, 10, -10, 4], [14, -2, 4, 6], [-10, 2, 4, -10], [14, -20, 4, 2], [8, -2, 10, 8], [22, -18, -6, 0], [0, 10, -8, 11], [-28, 4, -2, 5], [12, -8, -4, -9], [0, -6, 0, 4], [26, 0, 6, 8], [18, -12, 12, 4], [0, -14, 0, 0], [-34, 10, 0, 0], [14, -4, 0, 12], [-4, -8, 0, 0], [16, -8, 20, 1], [-16, -8, -4, -2], [-2, 0, 0, -16], [-18, 0, -6, 12], [-2, -8, 2, -4], [8, -6, 4, -1], [12, -8, -4, -7], [12, 4, 6, -3], [-18, 12, 4, 0], [24, -2, 2, 2], [30, 8, 0, 0], [-6, 2, -4, 4], [0, 24, -8, 3], [-14, -6, -6, 6], [20, 6, -2, 8], [-6, 4, 0, 4], [-16, 0, -6, -1], [0, 8, 4, -8], [0, -8, 12, -3], [4, 16, -6, -3], [-26, 4, -14, -8], [-2, 2, 16, -4], [-32, 14, 10, 1], [16, 2, 2, 9], [16, -16, 12, -2], [28, 0, 4, -4], [6, 8, -16, 0], [-2, 12, 4, 2], [-6, 4, -4, 0], [-14, 6, -12, -2], [-16, 8, -6, -1], [22, 6, 12, 14], [-8, 10, 2, -2], [8, 10, 0, -1], [4, -6, -6, -11], [8, -16, 8, -2], [-22, 16, 0, -6], [2, 16, 4, -14], [6, 4, -8, -6], [28, -8, -4, 4], [-30, 16, -8, -4], [4, 6, 8, 4], [-22, 24, -4, -12], [-8, -4, -20, 0], [-10, 0, 10, -4], [20, 0, 0, 12], [-30, 4, -2, -12], [-16, 0, 4, -11], [2, 4, 10, 0], [6, 6, 8, 10], [-8, -14, 2, -14], [16, -10, 0, 15], [24, -18, 12, 3], [-2, 14, 4, -2], [30, 8, 0, 12], [28, -10, 12, -3], [-4, 0, -12, 7], [22, -2, 6, 6], [12, -8, 20, 0], [-40, -4, -2, -5], [-14, -6, 4, 10], [16, 10, 2, 6], [-26, -6, 2, 4], [26, -10, -8, -4], [-20, 10, 4, -11], [22, -4, 4, 8], [30, 4, 12, 4], [-14, -20, 16, -8], [-26, 10, -12, -8], [-18, -16, 8, -6], [-24, 0, -8, -10], [8, 18, -12, 7], [-6, 16, 8, 14], [-4, 2, 0, 8], [-26, 16, 4, -2], [16, -6, 20, -16], [-12, -14, 6, -15], [-18, -4, -12, 12], [-16, 22, -10, -2], [2, -4, -4, 12], [-6, 4, 6, 0], [-10, 10, -4, 6], [-8, -20, 4, 7], [-16, -4, 8, 8], [-6, 18, -14, 8], [2, 2, 0, 0], [30, -16, 12, 0], [28, -20, 4, 1], [-10, -16, 4, -4], [-2, 2, 2, -12], [24, -8, 24, -4], [-14, 4, -8, -2], [-30, 6, -12, -2], [-24, -18, 6, -3], [18, -8, -16, -8], [-8, 14, 4, 19], [32, -8, -8, -4], [16, 10, 6, 21], [8, -8, -4, 8], [-14, 8, -16, 8], [-4, 12, 8, -8], [0, 12, -16, 2], [26, 2, -6, -4], [26, 14, 4, -6], [14, -22, 4, -4], [14, -4, -2, 12], [-34, 12, 4, 8], [-4, 6, 4, 8], [0, 0, -4, -6], [32, 10, 2, -1], [4, -12, -4, 8], [-2, -6, -12, 8], [30, -8, 20, -10], [4, -20, -6, -11], [0, -4, 12, 0], [-2, 14, 4, -4], [-60, 0, -4, -6], [4, 8, 8, 12], [-38, 10, 0, 4], [-10, 12, -8, 12], [-30, 24, 8, 0], [18, 2, 0, 6], [-26, -10, 2, -6], [18, 20, 4, 12], [4, -8, 0, -8], [-22, 4, -20, 0], [4, 22, 2, -9], [-4, -8, 10, 5], [38, -4, 4, -6], [0, 14, -14, -1], [-8, 8, -24, 7], [-22, -4, -2, 12], [4, -12, -18, -3], [-10, 8, -12, 8], [-8, -6, -6, 0], [-34, -12, 0, -10], [-26, 8, 6, 4], [12, 6, 2, 5], [-14, 0, -12, -20], [-6, 8, -8, 8], [0, 12, 20, -1], [-10, 16, -20, 6], [-32, 18, -16, -4], [-52, 0, 6, 1], [10, 6, 20, -4], [26, 6, 12, 8], [-2, -4, 0, -18], [2, 10, 22, 4], [-24, 10, -4, 12], [-60, 14, -18, -15], [44, 2, 12, -12], [-6, 6, 12, 6], [26, 8, -12, 4], [-22, 6, -12, -18], [4, -6, 4, 4], [30, -20, -12, -8], [0, 22, -8, 15], [-12, 20, 8, 4], [22, 6, -2, -2], [14, -16, 16, -16], [-18, 2, -20, 12], [12, -16, 22, -3], [28, -12, 8, 10], [-28, 18, -16, 1], [10, -28, -6, -8], [-4, -30, 6, -2], [26, -12, 12, -4], [52, -8, 4, 4], [6, 10, 16, 16], [-14, 4, 4, 2], [36, 2, 24, 9], [-40, 14, -20, -12], [-24, -4, -4, -15], [20, -12, 12, -12], [-28, -2, 16, 1], [-16, 8, -24, -1], [-26, 24, -8, 6], [40, -12, 12, 16], [-10, -4, 20, 0], [-50, 6, -8, 4], [34, -4, 14, -20], [30, -12, 24, 0], [52, -16, 12, 15], [-26, -6, -8, -2], [26, 0, 8, 22], [-20, -6, 8, 4], [-8, 16, 4, 22], [22, 0, 12, -16], [-10, 28, 4, -4], [8, 18, -6, 10], [22, -28, -20, -10], [-18, 6, -10, -24], [4, -10, 14, 9], [26, -20, 8, 14], [-14, 2, 26, 0], [-8, 26, 2, 24], [12, 0, 12, -1], [-12, -8, 0, -15], [22, 0, 4, -14], [-54, 4, -10, -8], [28, -18, 4, -11], [24, 22, -8, 16], [38, 14, 0, 8], [32, 14, 12, 24], [-18, -2, -12, -24], [26, 14, 6, -2], [-16, -4, 6, -1], [6, 8, -8, 16], [14, 2, 20, 6], [-4, 30, 4, -11], [2, 12, 8, -10], [2, 14, 2, 2], [-20, -6, -6, 6], [-12, -8, 10, 9], [-16, 18, -6, -4], [26, -8, 8, -24], [16, 16, -16, 3], [-12, 42, -14, 6], [-18, -4, -20, -6], [16, -10, 2, 6], [0, -14, 8, -13], [10, 4, -2, -8], [44, 6, -18, 1], [22, -16, 0, -6], [-48, -10, -4, -9], [12, -36, 0, 0], [-44, 28, -4, 4], [-30, -12, -4, 6], [32, -4, 0, -7], [18, 0, 16, -14], [-2, 0, -30, 4], [-36, -18, -2, 3], [20, -12, 8, 20], [-34, -2, -20, 6], [-20, 26, -4, 9], [-34, 16, -36, 2], [-28, 22, -10, 1], [-14, -10, -26, 0], [48, -14, 0, 12], [0, 30, -14, 8], [46, -22, -8, 4], [-62, 2, 6, -2], [-20, 16, -12, -4], [10, 8, 8, 26], [4, -36, 16, -8], [12, -16, 28, -12], [-2, -40, 16, -4], [42, -20, 4, 0], [-20, 12, -30, -3], [-30, 0, 24, 0], [-14, 12, 0, -8], [22, -6, -12, 8], [-10, -12, 4, -4], [-24, -6, -6, -4], [-24, -10, 18, 3], [68, -22, -4, 0], [-38, 4, -8, -22], [14, 18, 8, 12], [4, 18, 2, -5], [-50, -6, -10, -6], [-44, 14, -4, -11], [16, 0, 4, 28], [-50, -10, -8, -22], [32, 0, 24, -10], [-16, 6, -20, -17], [-12, 14, 6, 8], [-14, -8, -12, -2], [-56, 6, 2, -6], [18, -16, 8, 0], [-14, 30, 2, 18], [18, -32, -12, 0], [0, 20, 0, -13], [40, 18, -8, 19], [-14, -6, 14, 10], [-24, 8, -24, 16], [-34, -4, -16, 6], [14, -32, 0, 20], [-4, 0, 20, -5], [-20, -20, 4, 4], [-64, 12, -18, -17], [30, 8, 8, 0], [4, 2, -10, 1], [-64, 26, -24, -24], [-22, 28, -32, 6], [2, 12, 12, 8], [-4, 14, -26, 8], [4, -4, -4, -3], [70, -4, 4, 10], [-4, 26, -6, 5], [2, 10, -14, 14], [4, 0, -6, -3], [-22, 6, -12, 8], [-28, -12, 28, -4], [-20, -34, 16, -15], [-12, 14, -12, -3], [-20, 20, -8, 7], [-10, 0, -26, 8], [36, 16, 8, -20], [-16, 30, 8, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1805_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1805_2_a_p();".
// 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_1805_2_a_p(:prec:=4)
    chi := MakeCharacter_1805_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_1805_2_a_p();".
// 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_1805_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1805_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 8, -6, -2, 1]>,<3,R![-4, -16, -8, 2, 1]>],Snew);
    return Vf;
end function;