
// Make newform 3450.2.a.bs in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3450_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_3450_2_a_bs();".
// 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_3450_2_a_bs();".
// 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 := [11, -15, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-11, 0, 1]];
    Rf_basisdens := [1, 1, 2];
    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_3450_a();"
function MakeCharacter_3450_a()
    N := 3450;
    order := 1;
    char_gens := [1151, 277, 1201];
    v := [1, 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_3450_a_Hecke(Kf)
    return MakeCharacter_3450_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 := [[1, 0, 0], [-1, 0, 0], [0, 0, 0], [0, -1, 0], [1, -1, -1], [1, -1, 1], [1, 0, 1], [-1, 1, -1], [1, 0, 0], [-1, 0, 0], [6, 0, 0], [-1, 0, -1], [7, -1, 1], [-5, 1, -1], [2, 1, -1], [-4, 0, -2], [0, 2, 2], [6, 2, 0], [0, 2, -2], [4, 1, 1], [-1, 2, 0], [-1, 1, 1], [-4, -3, 0], [3, 0, 1], [-4, 2, 0], [2, 3, 1], [-4, -1, -2], [4, 2, -2], [3, -2, -3], [-1, 0, 1], [8, 0, 4], [4, -4, -2], [17, 0, -1], [4, -1, -3], [6, -2, 2], [-4, 0, -2], [-4, 4, -2], [-2, -2, 2], [8, -5, -1], [-3, -1, -1], [0, 0, -2], [3, 2, 5], [1, 3, -1], [2, 1, 3], [13, -4, -2], [8, -3, 2], [-4, -5, 3], [0, 6, -4], [-1, 4, -1], [0, 4, 2], [7, 3, -3], [-6, 5, 3], [-6, 2, -6], [-5, 2, 1], [10, 0, 4], [-12, 2, -6], [-7, 3, -5], [2, -2, -2], [1, -1, 3], [15, -2, 3], [4, 0, 4], [-4, 2, 2], [-24, 3, 1], [-6, 1, 5], [-2, -6, 4], [11, 0, 0], [-4, 1, 1], [12, 0, 2], [-18, 4, 2], [-9, -3, 3], [-9, 5, 1], [11, -1, 3], [-25, 1, -3], [3, -4, -5], [-2, -6, 0], [-1, 3, 7], [14, -4, 4], [-14, 2, -6], [14, -2, -4], [3, 4, 6], [-26, -1, 2], [-4, 4, 2], [-6, 4, -2], [-6, -4, 2], [-2, -2, -4], [4, 4, -2], [-14, -4, 0], [-10, -2, 2], [-7, 6, 0], [12, 0, 0], [18, -3, 0], [5, -5, -5], [-12, 8, 2], [18, 0, 6], [6, 1, 3], [-13, -1, 1], [4, 9, -3], [27, 0, -1], [-13, 3, -1], [13, -7, -3], [-20, -5, 3], [18, 8, -2], [-7, 3, 3], [-26, -2, 0], [2, 4, 2], [-21, 1, -3], [22, 2, 0], [-7, -3, 1], [-2, -2, -6], [-8, -5, -1], [6, -8, 8], [-11, -6, -5], [2, 2, 4], [18, 6, 0], [-8, 7, 2], [17, -10, -5], [-7, -1, -1], [-28, -1, 3], [23, -2, 0], [10, 5, 4], [-29, -2, 3], [23, 6, -2], [-8, 8, 0], [4, -8, -6], [-6, -11, 5], [-8, -2, 6], [-5, -6, 3], [-30, 1, 1], [-28, 2, 2], [3, -8, 1], [24, -1, 11], [13, 3, -3], [14, 7, 0], [27, 6, 1], [-19, -1, -7], [-34, 4, 2], [-2, -8, 6], [7, 7, -1], [22, -4, 6], [7, -3, 5], [8, -4, -8], [35, 1, 1], [2, 6, 0], [6, 1, 12], [-13, -3, -1], [11, 7, -3], [-5, 7, -1], [24, -6, 0], [-8, 4, -12], [8, 5, -3], [-14, 0, 0], [18, 2, 0], [-6, 5, -7], [0, -1, -7], [-31, 3, 1], [13, -3, -1], [-25, 0, -1], [13, -5, 3], [-2, -4, 4], [-30, 0, -4], [-10, -2, -2], [21, 0, 5], [-8, 2, 2], [-2, 11, 6], [1, 10, -3], [9, 5, 5], [14, 2, -2], [-1, -3, -1], [-20, -10, 2], [2, 13, -1], [30, -3, 4], [9, 3, 11], [1, -13, 7], [-6, 8, 0], [18, 0, -4], [15, 0, 3], [4, 1, 5], [-5, 1, 5], [2, 1, -4], [-1, 0, -5], [0, 10, -4], [12, 7, -8], [27, -9, -7], [15, -5, 7], [17, 1, 1], [2, 8, 6], [14, 2, 14], [-29, -5, -5], [11, -4, 14], [-24, 7, 1], [5, -9, -9], [-12, 8, 4], [9, -1, -7], [24, 1, -3], [-23, 3, 3], [17, -8, -5], [-8, -10, -2], [-25, -8, -3], [-5, -4, -5], [14, -11, -1], [-6, -4, -12], [8, 8, 2], [43, -3, 3], [-36, 2, -2], [5, 7, 3], [9, 7, -5], [43, -2, 5], [12, -16, -4], [31, 3, 1], [-8, 7, -7], [-31, 11, 3], [-41, 5, 5], [34, -1, 8], [-28, 3, 6], [34, 2, -2], [-20, -2, 0], [40, 0, -4], [-28, 12, 6], [-49, -1, 3], [10, -8, 4], [-3, -9, 1], [37, -1, 7], [20, -4, -6], [-39, 7, 3], [2, -10, 8], [17, 1, -5], [12, 12, -6], [24, 0, 0], [8, 18, -6], [8, -4, 4], [4, 14, -4], [10, -4, -2], [-33, 9, 5], [-19, 7, -11], [-30, -10, 0], [37, -7, 1], [2, 0, -10], [-36, 2, 2], [22, 8, -6], [14, -9, -11], [-52, 1, -4], [8, 11, 9], [28, -2, 0], [0, 8, 6], [32, -2, 4], [12, 7, -7], [-22, -4, -8], [4, -14, 6], [15, -9, 7], [9, -3, 1], [0, 4, -2], [16, 12, 2], [15, 1, -11], [20, -6, -2], [-3, -4, 8], [-2, 8, -8], [-24, 0, -14], [-37, 3, 7], [17, -16, -2], [-2, -4, -8], [3, -6, -7], [4, 9, 4], [-10, 0, -4], [3, 14, -5], [12, 0, -10], [-3, -3, -1], [18, -2, 4], [25, -8, -5], [-21, 13, 3], [21, 3, 3], [-19, 5, 5], [-66, 0, 2], [4, 8, 0], [52, -3, 6], [59, -3, 1], [-2, -16, 2], [-14, 14, 4], [-7, 5, 7], [22, -6, 8], [-17, -2, -11], [-8, 0, 0], [8, 10, 10], [-7, 11, 7], [3, 2, 7], [4, 0, 16], [-30, 6, -18], [-4, 10, 10], [-32, 4, -6], [2, 0, -6], [35, -1, -1], [2, 0, -8], [20, 5, 2], [-3, 9, -1], [-41, 0, 3], [-10, 14, 2], [4, -4, 12], [3, 9, 13], [-1, 4, 2], [2, 2, 6], [22, 9, 7], [42, -12, -2], [-50, -2, -4], [44, -9, -4], [-12, -12, -2], [-12, -12, 4], [45, 11, -1], [-40, -6, -8], [-43, 0, 7], [-8, 0, -8], [1, -9, 13], [-30, 2, -2], [24, -6, -6], [-7, 4, -9], [24, -5, -7], [-28, 12, -8], [-10, -6, -18], [26, 2, 0], [-32, -16, 4], [-2, -10, 2], [-44, -10, 0], [42, 8, -6], [-2, -4, 4], [-23, -12, -4], [38, 12, -4], [-17, -14, 3], [-20, 10, 4], [32, 2, -6], [18, -11, -5], [-13, -17, -1], [28, 2, 16], [-8, -2, -12], [35, 5, -3], [34, -8, 16], [6, 8, 8], [0, 2, 18], [-22, 16, 4], [12, 8, -10], [-19, 10, -5], [-3, 10, 4], [-3, -12, -11], [2, 12, -4], [14, -2, -8], [15, -15, 9], [-11, -8, 13], [20, 4, 0], [-70, 0, -2], [34, -6, 2], [-30, -3, 7], [67, -5, -3], [22, 6, -8], [42, -5, -9], [0, -17, 11], [-23, 0, -22], [41, -6, -9], [7, -12, 1], [29, -3, -9], [-17, 9, 5], [-44, 3, 2], [-19, -3, 3], [-14, -6, -18], [-5, -2, -17], [7, -19, 1], [-8, 8, 4], [-24, -3, -10], [34, -10, 6], [22, 0, -12], [6, -8, 8], [-1, 0, 15], [18, 2, 4], [-26, 0, 4], [-50, 7, -7], [0, -3, -3], [20, 6, -8], [-28, -13, -4], [17, -7, -1], [40, -8, 14], [-29, -7, 9], [-32, -16, 2], [9, 10, 9], [-15, 8, -14], [-42, 14, 6], [59, -10, -3], [-24, 4, 10], [30, 0, 16], [5, -19, 3], [56, 1, 11], [12, 6, 0], [20, 6, -12], [-4, 0, 8], [-14, 8, -6], [4, 14, 6], [-27, 7, 11], [-6, -8, -2], [62, -3, 5], [30, 8, 14], [42, -6, 24], [-19, 11, -1], [24, 0, 6], [-43, -1, -9], [-38, 16, 12], [-3, -15, 11], [-32, 10, -4], [-24, 9, 6], [-44, 4, 6], [-68, -5, -1], [34, 10, 8], [-33, -10, 5], [36, 6, 4], [43, 3, 11], [-30, 16, 8], [-14, 4, -8], [-27, -10, 2], [-3, 11, 7], [71, -7, -5], [68, 2, 4], [-41, -12, 0], [56, 1, 3], [14, 16, -6], [2, -10, 4], [-3, 7, -1], [38, 0, 6], [-68, 4, -2], [16, 17, 7], [-46, 0, 2], [-71, 2, -6], [-61, 12, 6], [-56, 9, 6], [-29, 5, -5], [52, -11, 1], [-40, 3, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3450_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3450_2_a_bs();".
// 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_3450_2_a_bs(:prec:=3)
    chi := MakeCharacter_3450_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_3450_2_a_bs();".
// 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_3450_2_a_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-11, -15, 1, 1]>,<11,R![100, -32, -3, 1]>,<13,R![-12, -32, -1, 1]>,<17,R![40, -17, -2, 1]>],Snew);
    return Vf;
end function;