
// Make newform 5610.2.a.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5610_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_5610_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_5610_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_5610_a();"
function MakeCharacter_5610_a()
    N := 5610;
    order := 1;
    char_gens := [1871, 3367, 1531, 3301];
    v := [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_5610_a_Hecke(Kf)
    return MakeCharacter_5610_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], [1], [-1], [0], [1], [-2], [-1], [0], [8], [-6], [-8], [-10], [-6], [12], [8], [-10], [-4], [14], [-4], [0], [-14], [8], [12], [10], [-18], [-6], [-16], [0], [-2], [-2], [4], [12], [-18], [-12], [10], [16], [-18], [-12], [16], [14], [-12], [10], [20], [-22], [-18], [-8], [-28], [-16], [-24], [14], [-18], [-8], [-14], [4], [14], [0], [18], [0], [6], [-26], [-4], [-6], [4], [8], [6], [-30], [-12], [10], [0], [18], [22], [8], [-28], [14], [0], [8], [-30], [-18], [-10], [10], [-28], [-34], [28], [-6], [8], [16], [30], [-18], [-30], [24], [-8], [12], [-4], [4], [32], [16], [-6], [14], [4], [6], [-20], [-6], [12], [30], [-4], [-30], [-16], [-34], [36], [10], [0], [-42], [22], [48], [-8], [-26], [-44], [-32], [-6], [-20], [6], [-6], [-42], [-12], [-40], [-46], [-38], [48], [32], [22], [32], [-16], [-48], [-42], [14], [-14], [46], [28], [6], [-30], [44], [18], [36], [-48], [14], [0], [22], [54], [20], [-48], [46], [6], [52], [-8], [-44], [48], [-48], [-18], [14], [10], [-12], [6], [-36], [-52], [-18], [-16], [24], [-10], [-14], [26], [44], [14], [36], [14], [0], [-42], [52], [58], [4], [-14], [8], [-12], [14], [38], [48], [42], [30], [28], [26], [36], [46], [-24], [-4], [-30], [8], [-42], [-62], [-34], [62], [0], [-14], [-8], [22], [66], [-12], [54], [56], [36], [-6], [-4], [-6], [-54], [44], [48], [-12], [-6], [-12], [38], [32], [6], [-62], [56], [50], [-36], [36], [34], [-26], [-24], [-68], [20], [14], [20], [24], [-42], [36], [56], [18], [6], [-36], [36], [-36], [56], [-56], [-10], [54], [68], [52], [12], [40], [-8], [-26], [-46], [32], [50], [-14], [-36], [-62], [-28], [42], [-34], [8], [-52], [-18], [-66], [-66], [-76], [-62], [2], [76], [66], [18], [44], [-14], [0], [54], [48], [-32], [26], [34], [36], [0], [-8], [-16], [-46], [-4], [12], [62], [-42], [-48], [-26], [-14], [-36], [-26], [60], [46], [-22], [8], [-34], [28], [-36], [6], [-78], [8], [12], [-80], [-18], [0], [50], [12], [62], [-72], [26], [34], [-12], [32], [58], [60], [-12], [46], [30], [-12], [-42], [34], [-80], [6], [-14], [-40], [4], [72], [78], [-10], [-38], [24], [-36], [52], [-44], [-34], [-42], [74], [28], [-42], [22], [10], [-32], [30], [-84], [-78], [44], [-36], [66], [48], [-82], [-6], [24], [58], [-90], [36], [-84], [-74], [-56], [70], [-70], [72], [-4], [-20], [90], [46], [-44], [10], [-12], [-16], [-16], [-54], [56], [-26], [52], [4], [-2], [-58], [14], [34], [-34], [-64], [54], [-100], [-32], [64], [86], [-100], [8], [74], [-6], [68], [4], [36], [2], [-64], [34], [-32], [10], [-34], [6], [4], [38], [-38], [-56], [46], [-82], [-76], [-20], [-70], [-102], [-64], [-76], [-34], [18], [12], [-36], [-18], [64], [-54], [22], [64], [-68], [-26], [-2], [-60], [-54], [-4], [-60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5610_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5610_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_5610_2_a_n(:prec:=1)
    chi := MakeCharacter_5610_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_5610_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_5610_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<13,R![2, 1]>,<19,R![0, 1]>,<23,R![-8, 1]>],Snew);
    return Vf;
end function;