
// Make newform 5184.2.a.ca in Magma, downloaded from the LMFDB on 12 March 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5184_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_5184_2_a_ca();".
// 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_5184_2_a_ca();".
// 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, 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_5184_a();"
function MakeCharacter_5184_a()
    N := 5184;
    order := 1;
    char_gens := [2431, 325, 1217];
    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_5184_a_Hecke(Kf)
    return MakeCharacter_5184_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, 0], [0, 0], [2, 1], [0, 0], [0, 0], [-3, -2], [-4, -1], [0, 0], [0, 0], [-2, -5], [0, 0], [-1, 6], [-8, 0], [0, 0], [0, 0], [4, 0], [0, 0], [-5, -6], [0, 0], [0, 0], [-3, -8], [0, 0], [0, 0], [-8, 5], [-18, 0], [20, 0], [0, 0], [0, 0], [-3, 10], [-8, 7], [0, 0], [0, 0], [-4, -11], [0, 0], [10, 7], [0, 0], [11, 6], [0, 0], [0, 0], [2, -13], [0, 0], [-18, 0], [0, 0], [7, 12], [14, -1], [0, 0], [0, 0], [0, 0], [0, 0], [-15, 2], [8, 13], [0, 0], [-15, -4], [0, 0], [-16, 1], [0, 0], [-10, 13], [0, 0], [-18, 0], [16, -5], [0, 0], [-2, 17], [0, 0], [0, 0], [-13, -12], [14, -11], [0, 0], [-18, 0], [0, 0], [10, 0], [16, 0], [0, 0], [0, 0], [-14, 0], [0, 0], [0, 0], [-20, 0], [19, -6], [20, 1], [-3, -20], [0, 0], [-15, 14], [0, 0], [-17, 12], [0, 0], [0, 0], [40, 0], [21, -4], [-20, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-44, 0], [-40, 0], [0, 0], [21, -10], [0, 0], [-14, 19], [0, 0], [20, -13], [0, 0], [-1, 24], [0, 0], [8, -23], [0, 0], [-5, -24], [0, 0], [34, 0], [16, -19], [0, 0], [0, 0], [-4, 25], [0, 0], [0, 0], [44, 0], [0, 0], [-25, 6], [23, -12], [-52, 0], [0, 0], [0, 0], [-26, 5], [-15, -22], [0, 0], [0, 0], [54, 0], [0, 0], [0, 0], [0, 0], [-18, 0], [-20, -19], [-25, -12], [22, -17], [0, 0], [26, 11], [28, 5], [0, 0], [14, 25], [0, 0], [0, 0], [54, 0], [0, 0], [-46, 0], [4, 29], [0, 0], [0, 0], [-29, -6], [32, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [20, 23], [19, -24], [10, -29], [0, 0], [-28, 13], [0, 0], [0, 0], [-8, 0], [0, 0], [0, 0], [31, -6], [-15, -28], [44, 0], [0, 0], [11, -30], [0, 0], [-3, -32], [0, 0], [64, 0], [0, 0], [10, -31], [0, 0], [-13, -30], [0, 0], [0, 0], [33, -2], [-32, 0], [0, 0], [-22, 25], [21, 26], [0, 0], [54, 0], [0, 0], [33, -8], [0, 0], [0, 0], [34, -5], [0, 0], [-64, 0], [-25, 24], [54, 0], [16, 31], [0, 0], [2, -35], [0, 0], [-18, 0], [-15, 32], [0, 0], [68, 0], [0, 0], [0, 0], [-16, 0], [0, 0], [2, 0], [26, -25], [0, 0], [0, 0], [0, 0], [10, 0], [0, 0], [-20, 31], [0, 0], [2, 37], [-15, -34], [0, 0], [-28, -25], [0, 0], [0, 0], [23, -30], [-16, 0], [0, 0], [0, 0], [0, 0], [-3, -38], [0, 0], [0, 0], [-16, -35], [0, 0], [0, 0], [33, -20], [76, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-70, 0], [-64, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [21, -34], [-40, -1], [0, 0], [-3, 40], [38, -13], [0, 0], [-39, 10], [0, 0], [-26, -31], [-38, 0], [0, 0], [0, 0], [-15, 38], [74, 0], [-8, 0], [0, 0], [-22, 35], [-40, -11], [0, 0], [-38, 17], [-29, 30], [0, 0], [54, 0], [0, 0], [-39, 16], [0, 0], [0, 0], [-5, -42], [35, 24], [0, 0], [0, 0], [0, 0], [0, 0], [31, 30], [0, 0], [0, 0], [33, 28], [28, 0], [0, 0], [40, -17], [-52, 0], [0, 0], [-8, 43], [0, 0], [-13, 42], [10, 43], [0, 0], [38, 23], [0, 0], [0, 0], [43, 12], [-68, 0], [0, 0], [0, 0], [0, 0], [-18, 0], [0, 0], [-90, 0], [0, 0], [-17, -42], [0, 0], [-76, 0], [-20, -41], [0, 0], [0, 0], [-90, 0], [0, 0], [0, 0], [33, -32], [80, 0], [0, 0], [58, 0], [46, 5], [0, 0], [56, 0], [-15, 44], [0, 0], [0, 0], [0, 0], [-4, 0], [-90, 0], [-46, 11], [0, 0], [0, 0], [0, 0], [0, 0], [-37, -30], [8, 47], [-90, 0], [0, 0], [23, 42], [88, 0], [-10, 47], [0, 0], [22, -43], [0, 0], [-15, -46], [0, 0], [0, 0], [-26, 41], [0, 0], [21, 44], [-34, -35], [0, 0], [-25, 42], [32, -37], [0, 0], [0, 0], [8, 0], [0, 0], [-49, 6], [40, 29], [0, 0], [0, 0], [0, 0], [-13, -48], [-92, 0], [0, 0], [-70, 0], [0, 0], [0, 0], [0, 0], [-50, -7], [0, 0], [21, -46], [0, 0], [0, 0], [-17, 48], [-40, 0], [-51, -4], [50, -11], [28, 43], [0, 0], [-16, -49], [0, 0], [0, 0], [0, 0], [-39, 34], [0, 0], [0, 0], [33, 40], [-22, -47], [0, 0], [0, 0], [0, 0], [-3, 52], [0, 0], [52, -5], [0, 0], [-46, -25], [43, 30], [-104, 0], [0, 0], [-44, -29], [-50, -17], [0, 0], [-51, 14], [20, -49], [0, 0], [0, 0], [23, -48], [34, -41], [0, 0], [0, 0], [-51, -16], [-50, 19], [0, 0], [0, 0], [-44, -31], [0, 0], [-20, 0], [2, 0], [0, 0], [0, 0], [-53, 12], [46, -29], [0, 0], [-40, -37], [0, 0], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5184_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5184_2_a_ca();".
// 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_5184_2_a_ca(:prec:=2)
    chi := MakeCharacter_5184_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_5184_2_a_ca();".
// 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_5184_2_a_ca( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5184_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, -4, 1]>,<7,R![0, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;