
// Make newform 5610.2.a.l in Magma, downloaded from the LMFDB on 06 May 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_l();".
// 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_l();".
// 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], [-3], [1], [1], [-1], [-3], [-1], [6], [1], [5], [6], [0], [-10], [2], [-4], [5], [-7], [-6], [4], [-10], [3], [-14], [-15], [18], [-7], [18], [7], [-2], [-20], [3], [-21], [6], [7], [-17], [0], [-24], [-8], [-7], [9], [-20], [8], [-4], [21], [1], [-10], [23], [-18], [-13], [-6], [-26], [-17], [-11], [-10], [15], [15], [12], [-24], [10], [-4], [-12], [-32], [14], [15], [-18], [12], [4], [-6], [18], [13], [14], [-28], [-34], [-15], [2], [12], [18], [5], [10], [32], [35], [-8], [-6], [-22], [-26], [-33], [15], [-30], [3], [4], [-15], [-16], [22], [-28], [40], [18], [-19], [-8], [6], [-41], [-12], [33], [-15], [26], [6], [14], [-10], [-21], [-41], [-27], [42], [10], [-9], [-26], [-26], [-8], [22], [-48], [34], [-3], [-36], [6], [3], [-13], [26], [-14], [18], [8], [19], [11], [-52], [48], [6], [-31], [40], [1], [37], [3], [-18], [-46], [6], [6], [18], [35], [12], [4], [33], [44], [12], [4], [18], [-20], [-32], [16], [-24], [3], [3], [-49], [28], [36], [-36], [6], [11], [33], [20], [3], [-46], [10], [-4], [-58], [23], [-36], [5], [48], [-27], [-20], [-26], [4], [52], [-43], [21], [-7], [-34], [24], [0], [6], [-8], [-10], [-21], [-14], [6], [-22], [30], [26], [-30], [55], [-19], [-7], [-21], [52], [-8], [-50], [60], [51], [30], [20], [33], [-18], [2], [30], [-42], [-34], [0], [15], [30], [18], [50], [44], [18], [52], [50], [44], [-24], [0], [16], [-56], [-60], [-2], [-55], [20], [38], [33], [-51], [-12], [8], [-30], [6], [18], [45], [-12], [-55], [67], [47], [-42], [-16], [-62], [48], [76], [4], [7], [26], [80], [56], [52], [-66], [-50], [-13], [-6], [-22], [-37], [-52], [63], [0], [-12], [20], [-32], [-16], [-59], [54], [42], [35], [70], [39], [57], [33], [-32], [-22], [-32], [24], [0], [19], [62], [-28], [-49], [-72], [-31], [33], [-48], [25], [61], [-72], [-8], [-21], [-20], [14], [8], [-79], [-41], [24], [63], [72], [-16], [15], [61], [-63], [-18], [44], [0], [83], [60], [35], [-8], [33], [-16], [-44], [-12], [33], [22], [-9], [-24], [-45], [-35], [-77], [39], [1], [20], [-32], [-66], [51], [-1], [88], [18], [-60], [34], [73], [59], [39], [44], [-68], [-12], [-17], [-20], [10], [-21], [36], [-54], [-4], [48], [-48], [-57], [71], [-90], [24], [70], [-36], [-36], [-36], [19], [40], [-68], [62], [30], [-85], [43], [-30], [-17], [64], [-26], [63], [-52], [26], [15], [-91], [67], [-92], [79], [-83], [-7], [-37], [-53], [53], [8], [-6], [20], [-8], [-62], [-22], [-4], [-46], [-16], [36], [-97], [67], [-63], [44], [-88], [-20], [-11], [1], [26], [18], [40], [-16], [43], [34], [-26], [62], [-7], [16], [62], [-12], [2], [-58], [53], [-21], [-24], [-54], [-24], [1], [-30], [-8], [-71], [-20], [103], [-53], [-60], [-90], [20], [-36]];
    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_l();".
// 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_l(: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_l();".
// 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_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3, 1]>,<13,R![-1, 1]>,<19,R![3, 1]>,<23,R![1, 1]>],Snew);
    return Vf;
end function;