
// Make newform 845.2.a.n in Magma, downloaded from the LMFDB on 25 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_845_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_845_2_a_n();".
// 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_845_2_a_n();".
// 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, 1, -23, -44, 32, 59, -9, -17, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0], [446, 904, 4655, -926, -2939, 310, 554, -10, -27], [-424, 6746, 10697, -14450, -16339, 4476, 5028, -110, -297], [458, 14913, 18698, -17211, -26004, 4568, 7572, -27, -446], [-2435, -1814, 22645, 2252, -28560, -648, 8013, 296, -480], [1524, -9605, -28206, 16901, 36890, -5026, -10550, -27, 620], [621, -8842, -57984, 15248, 80123, -4308, -22977, -480, 1369], [6725, -7897, -91090, 12477, 109615, -3096, -30295, -777, 1793]];
    Rf_basisdens := [1, 1, 1066, 1066, 1066, 1066, 1066, 1066, 1066];
    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_845_a();"
function MakeCharacter_845_a()
    N := 845;
    order := 1;
    char_gens := [677, 171];
    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_845_a_Hecke( : Kf := Kf)
    return MakeCharacter_845_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]];
    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, 0, -1, 0, 0], [1, 0, 0, 0, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 0, 0, -1], [1, -1, -1, -1, 0, -1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 1, 0, 0, -1], [0, 2, 1, 0, -1, 0, 1, -1, 0], [2, 0, -2, 0, 1, 1, 0, -1, 0], [3, 1, -1, 1, 0, 2, 0, 0, 1], [2, 1, 0, 2, 1, 1, 0, 0, 1], [0, 1, -1, 2, -2, -1, 3, -1, -1], [-2, -1, 1, -1, 0, -6, 3, 0, 1], [4, -1, 0, 0, 0, -2, 0, -1, 1], [-6, 0, 2, 0, -1, -3, 2, -1, 0], [-3, -2, -2, -1, 2, -2, 3, -1, 0], [2, -1, 1, 0, 0, -1, -1, 3, -1], [0, 0, -1, 1, -1, -1, 1, 2, 0], [-1, 1, 2, 1, 0, 2, 3, 0, -1], [3, 1, 0, 1, -1, 3, -5, 1, -1], [2, 2, 0, 2, 0, 4, -2, 2, -2], [3, 1, -1, -1, 0, -3, 2, 0, 1], [-2, 2, 2, -1, 1, 0, 3, -1, 2], [3, 0, -2, -3, 0, 2, -2, 1, 0], [2, 2, 1, -2, 1, 2, 3, -3, 0], [-2, -2, -1, -4, -1, -2, -2, -1, 4], [3, -2, 1, 2, 0, 0, 2, 2, 3], [1, 0, 0, 1, 2, -2, -1, 1, -4], [-3, -1, -2, -2, 1, -4, 3, 2, -1], [0, 2, 1, -2, 1, -2, -7, 1, -2], [8, 1, 3, 1, 3, 5, -9, 1, -2], [1, -2, 1, -3, 1, -2, -6, 0, -2], [7, -2, -2, -3, 0, 2, -9, -1, 2], [7, 2, -3, -1, 1, 0, 4, -2, 2], [5, -1, 0, 3, 3, 8, -3, 1, -3], [-2, -2, 1, 0, -3, -8, 5, 1, 0], [2, -1, 3, 2, 2, 1, 1, -1, -1], [4, -4, 1, -1, 0, 2, -8, 2, -2], [-6, 0, 0, -1, 3, 4, 0, 2, 1], [0, -1, 0, 4, 1, 1, 0, 0, -1], [-4, 2, -1, -2, -1, 0, 1, -1, -2], [-2, -5, 2, 0, 1, -7, 3, 0, 1], [0, -2, 1, 6, 3, 0, 1, 1, 2], [1, -1, -1, -5, 0, 1, -4, -2, 3], [-2, 2, 0, 4, -2, -6, 4, 2, -2], [-4, -1, 1, 2, 4, -5, 1, 3, -3], [8, -2, 2, 0, 2, 0, 2, 0, 4], [-2, -1, -2, -5, -4, -1, -5, 1, 1], [-5, 3, 0, 0, 4, -1, -4, 2, -1], [-3, -1, -2, 1, 3, 0, -3, -5, -3], [-6, 5, -3, 0, -2, -1, 5, -1, -1], [-4, 0, 3, 2, -1, -8, 7, -3, -4], [-3, 4, 7, 5, -5, 4, 3, 0, 0], [-3, 1, 2, 3, 1, 3, -1, -1, -3], [-2, -4, -2, -4, 0, 0, -4, 6, 0], [-4, 2, 1, -3, 0, 4, 7, 3, 3], [0, -2, 3, -4, 1, 3, -3, -3, 4], [7, -6, -1, 1, -1, 2, 0, -2, 4], [5, 3, 6, 3, 3, -1, 1, -1, -1], [1, 1, -3, -7, 4, -2, -2, 0, 5], [6, 1, 1, 1, 5, 3, 2, -2, -1], [-7, 3, 0, 3, -1, -3, 7, -7, 1], [-6, 1, -1, 7, 5, 7, 2, -2, -1], [0, -3, -1, -6, 0, 3, -1, 5, 3], [0, 5, -2, 6, 1, 3, 8, -2, -1], [-1, -7, 4, -5, -1, -5, -3, 1, 1], [3, -6, -3, -3, -1, 2, -10, 0, -4], [3, -2, 1, -1, -5, -10, -2, 0, 2], [3, -3, 3, -2, -1, -7, 7, 3, 3], [1, 3, 0, 7, -3, 3, 5, 7, 1], [-2, -2, 2, 4, 4, 0, 0, 4, 2], [6, -2, 0, 0, 4, 4, 0, 4, 4], [9, -1, -6, 0, 4, 11, -6, 0, -5], [1, 8, -2, 5, 0, 4, 7, -1, 0], [7, -3, -2, 1, 5, 5, -11, -1, -1], [-14, 5, -6, 3, -2, 1, -1, 1, -5], [0, 8, 3, 2, 5, -2, -2, -1, -4], [-7, -4, 1, 1, -1, -4, -12, -2, 0], [5, -3, 2, 0, 7, 2, 3, -4, -1], [12, 1, 1, -1, 2, 6, -11, 4, -5], [1, -2, 0, -3, 2, -2, 9, 1, 0], [-15, 0, -4, -3, 4, -7, -1, 1, -4], [6, 1, 4, -4, -5, -3, -6, 0, -1], [-4, 6, 3, 2, -3, 6, 5, -7, 4], [-9, -6, 3, 1, -3, -14, 16, -4, 0], [5, 4, 1, 4, -2, 6, -5, -7, 0], [1, -2, 2, 1, -4, 3, 9, 3, 6], [-12, -5, 1, 0, 0, -1, -5, -7, -1], [1, 3, -1, -1, -2, 3, 3, 2, 11], [-5, 1, 3, -1, 0, 1, 14, 0, 5], [-2, 3, 3, 8, -5, 12, 5, 0, -1], [5, 3, 2, -3, -3, 13, 1, -1, -1], [1, -6, 3, 3, -2, -5, 8, 3, 4], [7, -5, -3, -1, 0, 1, -12, 6, 1], [2, 0, -3, -2, 7, 2, -3, 1, 2], [-5, -1, -1, -4, 1, 1, -2, -4, -2], [-1, 4, -1, -9, -1, 7, -2, 0, -2], [5, -5, -2, 5, -1, 3, -8, 7, 1], [15, 5, 0, 5, 4, -4, 1, 0, -5], [4, -7, 3, 0, 2, 2, -3, -7, 1], [27, 5, -4, -2, 0, 9, -1, -5, 6], [-7, 1, -2, -5, -3, -15, 5, 3, -3], [16, 1, 1, -3, -7, 9, -11, 3, 2], [-3, 0, 1, 4, 3, 5, 4, 5, -4], [17, 3, 1, -1, -4, 7, -8, 2, -5], [-9, 2, 4, 1, 0, -2, 7, -11, 2], [-9, 4, -3, 2, 4, 0, 6, 2, -3], [13, 2, -5, 5, 9, 12, -6, 6, 0], [3, 1, 3, 5, -2, 3, 0, 0, 5], [-9, 0, 3, -4, 1, -9, -4, 3, 0], [7, -4, 0, -1, -1, -3, -1, 4, -4], [18, -5, 2, 2, -1, 15, -8, 6, -5], [9, -1, -3, -3, 0, 13, -4, 6, 3], [19, -5, 4, 1, -3, 9, 1, 3, 1], [-5, 3, 6, 1, -3, -23, 17, -5, -1], [-2, -5, 0, -3, 1, -2, 2, -7, 11], [-5, 3, 10, 1, 3, -17, 5, -3, 1], [0, -3, -4, -4, 1, 3, -12, -8, 3], [-11, -1, -2, -1, 3, 3, 1, 5, 3], [11, 3, 0, -7, -7, 3, 3, 1, -1], [-3, 0, -5, -3, 3, 0, -11, -4, -8], [-2, 5, 3, -6, 2, -7, 3, 3, 1], [2, 11, 9, 6, 6, 19, -5, -3, -7], [-3, 7, 4, -1, 2, 4, 5, -8, 1], [13, 7, -5, -7, 2, 13, -4, 2, 3], [-19, 2, -4, 2, 6, -5, 7, -6, -8], [-12, -5, -5, -13, -2, -10, -1, 6, 3], [-9, -5, 2, 3, -7, -17, 9, 1, 3], [-9, -3, -1, -1, 1, -6, -14, 3, 5], [-7, 7, -6, 9, -3, -5, 23, -3, 5], [2, -1, 3, -2, 2, -1, -9, -5, 3], [-5, 10, 0, 8, 1, 16, 5, -5, -3], [-3, 12, -1, 3, -9, -8, 20, 4, 2], [-13, 6, -2, 3, 0, -10, 7, -3, 6], [3, 6, 4, -3, 0, 1, -13, -1, -2], [15, 2, 3, 4, -5, 3, 0, -5, 4], [4, 7, 1, 2, -6, 7, 7, 3, -3], [-3, -3, 3, -7, 4, -7, -4, -2, 5], [-23, -4, -5, -11, 3, 2, -2, 2, -4], [-4, 3, 2, 7, 3, 12, 6, 1, 1], [15, -6, 3, 1, 3, 8, -26, 6, 6], [-9, 3, -2, -5, 3, -15, -4, -1, -1], [9, 5, 2, 11, 1, 13, -1, 3, -13], [0, 4, -4, -7, 7, 8, -5, 3, 2], [-17, -6, 2, -5, 4, 5, -1, 1, -10], [15, 5, -5, 5, 8, 3, 12, -6, 3], [19, -3, -6, -5, 5, 9, -15, -3, 7], [-12, 12, -4, 0, -8, 0, 16, -6, 0], [-13, -1, 11, -3, -2, -11, 6, -4, 5], [-9, 7, -2, 10, -2, 1, 20, 6, 1], [4, 1, 7, 6, -8, -17, 15, -5, -3], [-3, -5, 7, 1, -2, 3, -8, 0, -9], [12, 9, 9, 2, -4, -5, 13, 3, 5], [9, 3, 6, -2, 6, 11, -21, -5, -2], [34, 0, -2, 1, -1, 8, -2, 0, 3], [-10, -8, 4, -2, 0, 0, -4, -4, -8], [6, -4, 4, -6, -6, 0, 2, 10, 4], [6, 8, -4, 6, -8, 6, 15, 4, 2], [1, -18, 0, -3, 2, -6, -7, 11, 0], [-12, 7, 10, -1, -1, -20, 10, 1, 1], [-7, 5, 1, -2, 9, -9, 6, -4, 4], [-11, -1, 0, -1, 1, -9, -7, -7, 5], [7, 6, 0, -5, -8, 4, 5, -1, 2], [-1, -7, -2, -1, 11, 11, -1, -5, -7], [-16, 7, -2, -8, -7, -15, 18, 2, 9], [-27, -4, -5, -2, 2, -18, 23, 1, 4], [20, 2, 7, 2, -7, -4, -7, -5, 12], [-25, 2, 10, 1, 4, -14, 3, -11, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_845_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_845_2_a_n();".
// 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_845_2_a_n(:prec:=9)
    chi := MakeCharacter_845_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(997) - 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_845_2_a_n();".
// 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_845_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_845_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 38, -152, -51, 156, 49, -40, -13, 3, 1]>],Snew);
    return Vf;
end function;