
// Make newform 5520.2.a.bq in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5520_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_5520_2_a_bq();".
// 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_5520_2_a_bq();".
// 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 := [-8, -1, 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_5520_a();"
function MakeCharacter_5520_a()
    N := 5520;
    order := 1;
    char_gens := [4831, 1381, 1841, 4417, 1201];
    v := [1, 1, 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_5520_a_Hecke(Kf)
    return MakeCharacter_5520_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], [1, 0], [0, -1], [-4, 0], [-2, 2], [2, 1], [-4, 0], [-1, 0], [-2, -1], [0, -1], [-2, -1], [2, -3], [4, 0], [0, 2], [-2, 1], [-12, 1], [-2, 0], [-12, -1], [0, -1], [-6, -2], [0, 4], [-4, 5], [2, -2], [2, 4], [-10, -1], [8, 0], [4, 1], [-2, 2], [2, -3], [-8, 2], [4, 0], [2, -4], [-12, 1], [-10, 4], [-16, 0], [6, -7], [-12, 2], [-8, 2], [6, -6], [4, -4], [6, -6], [0, -6], [-14, -2], [-2, 4], [-8, 2], [-4, -5], [-24, 2], [-12, 0], [22, 2], [-6, -6], [8, 1], [-14, 2], [12, 2], [2, -4], [16, -5], [-2, 5], [16, -3], [22, 0], [2, 6], [-4, 1], [-2, -5], [4, 0], [-8, 0], [-14, 5], [22, -4], [-12, 3], [-14, -4], [-4, 0], [6, -3], [-14, 0], [-8, 10], [8, 5], [-10, 0], [4, -4], [8, 3], [-10, 4], [14, 6], [10, 4], [18, -1], [-12, 0], [-10, 2], [32, 0], [-6, 11], [0, -8], [28, -2], [10, -5], [-14, -7], [6, -12], [8, 0], [12, 9], [0, 10], [0, -10], [20, -1], [-20, -7], [32, -1], [6, -8], [2, 6], [-12, -8], [14, 0], [4, 0], [6, 9], [-4, -1], [-14, 4], [20, -6], [-14, 2], [-4, -6], [-14, 8], [24, -4], [2, 5], [40, -2], [-26, -4], [10, -9], [-12, -8], [24, -2], [10, 10], [-44, -1], [8, 4], [-10, 6], [4, -8], [22, 0], [-30, 4], [30, 3], [12, 6], [-12, 4], [14, -16], [6, -8], [8, -11], [0, 1], [-10, 3], [-36, -1], [-8, -8], [16, 0], [-2, -7], [2, 13], [34, -4], [22, -8], [4, -1], [6, 3], [34, -1], [-28, 3], [-18, 8], [-16, 8], [20, 7], [6, 11], [-8, -2], [6, 8], [10, 12], [4, 7], [0, 2], [-34, 4], [-22, 0], [4, 14], [-8, 10], [12, -1], [16, 12], [-24, -4], [10, 7], [-6, 12], [14, -8], [-28, -8], [2, 4], [-32, 6], [12, -4], [34, -1], [32, 5], [32, 3], [-26, -6], [-14, -6], [14, 8], [-20, 16], [-10, 3], [-24, -6], [-30, 7], [-8, 8], [2, -4], [-4, 0], [6, -9], [-16, 13], [-2, 2], [-8, 0], [-12, -6], [6, 4], [10, -6], [-24, 3], [22, 10], [-18, 18], [-44, 1], [34, -9], [24, -13], [18, -14], [-4, -18], [-20, 4], [14, 5], [-44, -4], [-6, 7], [34, -10], [-2, -20], [26, 8], [-40, 2], [-2, -8], [-24, 8], [6, -6], [34, -10], [12, 2], [22, 8], [0, -10], [4, 12], [-14, -3], [-20, -4], [-14, -8], [-2, -12], [40, 0], [-52, 8], [16, 1], [-22, 16], [-24, 10], [-6, -11], [-16, 11], [38, -2], [46, 3], [56, 2], [-6, 15], [24, -15], [4, -12], [-34, -7], [-30, 4], [56, -7], [-8, 12], [4, -5], [-2, -6], [12, 4], [0, -14], [18, -21], [12, -5], [0, 4], [2, 0], [6, -8], [-12, 1], [-48, 3], [-20, -7], [36, 11], [16, 6], [46, 8], [34, -2], [24, 0], [8, 6], [4, -8], [-28, 6], [-8, 1], [-2, -20], [-22, 4], [-64, 3], [26, -20], [-58, 2], [28, -11], [-10, 0], [-44, -4], [-18, 6], [-6, 16], [-24, -11], [44, -9], [-2, 1], [-26, 9], [-30, 18], [-4, 2], [-2, -4], [34, 8], [12, -23], [-34, 10], [38, 3], [52, 3], [-14, 19], [-16, 2], [-46, 12], [-16, 16], [12, 6], [-42, 7], [10, 8], [-60, 10], [-48, 14], [-40, -2], [24, -20], [-10, -2], [-20, -8], [-24, -1], [-22, -17], [6, -16], [48, 0], [-14, 4], [46, 4], [28, -13], [-38, 10], [-52, -6], [-2, 24], [-18, 14], [16, -16], [-50, 10], [4, 15], [36, -14], [-30, -11], [38, -7], [48, -12], [20, 2], [-28, 4], [2, -12], [-4, 24], [46, -12], [-56, 2], [-10, -12], [0, -20], [-26, -10], [74, 0], [-12, 24], [32, 3], [26, 0], [28, 13], [-8, -23], [-6, -21], [-14, 16], [28, -18], [50, -3], [-42, 12], [-72, -2], [-38, -11], [34, -12], [28, -14], [-52, 8], [0, 0], [-50, -5], [6, -19], [6, -26], [48, 1], [-60, 0], [-12, 28], [-4, -12], [-34, 16], [42, 8], [42, -8], [72, -1], [6, -28], [34, -4], [-66, 0], [40, 2], [6, -1], [76, 3], [14, -21], [-36, 20], [-16, 14], [-50, 11], [-4, 1], [-22, 20], [-18, 15], [-40, -3], [-42, 4], [-22, -12], [16, 4], [12, -14], [66, 2], [-8, -2], [-2, 3], [-30, -7], [32, 2], [-68, 4], [52, 10], [42, 4], [-42, 12], [16, 11], [10, -22], [28, 3], [4, 15], [0, -18], [-74, 2], [40, 8], [-18, -14], [-20, -7], [-48, 14], [-22, 15], [10, -30], [26, 4], [46, 4], [-30, 12], [-48, -12], [18, 6], [12, 8], [24, -6], [-16, -1], [-74, 0], [-20, 7], [-40, -15], [2, 18], [14, -8], [52, -1], [4, 6], [-8, 4], [-46, 7], [0, -14], [2, -10], [-12, 16], [-26, 13], [-26, 21], [34, 16], [-8, 21], [-6, -5], [-74, -1], [72, -3], [54, -7], [-14, 12], [-4, 4], [4, 16], [18, 16], [-34, -18], [-4, -16], [-68, -4], [-14, 33], [-42, 4], [64, -12], [-16, 14], [-46, -11], [-8, -8], [-34, -10], [6, -20], [-48, 16], [28, -8], [-54, 8], [6, 14], [4, 12], [-78, -1], [-68, 8], [32, 19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5520_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5520_2_a_bq();".
// 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_5520_2_a_bq(:prec:=2)
    chi := MakeCharacter_5520_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_5520_2_a_bq();".
// 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_5520_2_a_bq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-8, 1, 1]>,<11,R![4, 1]>,<13,R![-32, 2, 1]>,<17,R![-2, -5, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;