
// Make newform 5610.2.a.f 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_f();".
// 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_f();".
// 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], [-4], [-1], [4], [-1], [-3], [-5], [-6], [8], [13], [-12], [-4], [-6], [-10], [8], [14], [-6], [4], [-8], [-4], [-1], [-2], [-11], [5], [8], [10], [-6], [12], [17], [1], [16], [12], [-18], [21], [18], [-6], [-6], [-5], [-9], [-6], [-12], [8], [15], [-19], [21], [-24], [23], [6], [-9], [0], [-17], [11], [26], [-25], [-2], [3], [-2], [-3], [20], [0], [17], [-31], [-11], [18], [12], [26], [25], [14], [22], [22], [-8], [8], [22], [-18], [3], [8], [25], [8], [35], [12], [-34], [13], [41], [-22], [-26], [-24], [12], [29], [2], [8], [4], [28], [30], [-18], [-11], [-28], [2], [-5], [-12], [34], [44], [38], [27], [-6], [1], [-46], [40], [10], [-6], [14], [-22], [-27], [11], [22], [-39], [-23], [-16], [-8], [18], [12], [12], [46], [50], [-6], [5], [44], [-10], [8], [45], [-23], [-3], [2], [48], [-12], [-52], [-54], [8], [-33], [10], [-37], [34], [18], [-51], [-7], [35], [32], [13], [19], [-24], [24], [-37], [-32], [1], [-7], [-38], [46], [18], [-38], [-42], [-12], [2], [-41], [-47], [7], [-15], [-39], [47], [12], [-4], [27], [36], [50], [-28], [-10], [52], [-53], [21], [-12], [-46], [-10], [0], [26], [61], [-24], [-13], [-8], [19], [-28], [-65], [-45], [-31], [4], [10], [-34], [-33], [-19], [46], [14], [-11], [-40], [30], [60], [-34], [-24], [-48], [-23], [52], [-27], [4], [-23], [24], [-8], [-6], [-43], [48], [30], [-13], [44], [-28], [-66], [0], [-11], [6], [-38], [50], [-10], [49], [23], [-21], [-13], [-24], [25], [-8], [64], [15], [-16], [-64], [-66], [-55], [62], [65], [29], [-58], [19], [-14], [-61], [64], [0], [-60], [-50], [-17], [-49], [10], [42], [21], [42], [60], [10], [46], [19], [-28], [-35], [43], [0], [78], [23], [-7], [2], [-74], [-49], [-21], [52], [-12], [13], [38], [-60], [51], [8], [-48], [37], [-38], [32], [14], [-10], [-34], [63], [62], [-16], [-36], [-54], [25], [-18], [-18], [0], [-18], [-77], [31], [-11], [26], [84], [-60], [62], [77], [14], [59], [-6], [-74], [-2], [-20], [60], [-13], [-56], [-80], [-40], [22], [5], [-35], [-23], [-36], [31], [-58], [-41], [80], [61], [36], [-36], [30], [25], [6], [24], [12], [68], [-56], [37], [12], [34], [-10], [-54], [39], [62], [-76], [-6], [-10], [-52], [-27], [-35], [8], [-6], [-13], [12], [-6], [18], [-21], [84], [-29], [-2], [-63], [18], [-76], [62], [4], [-16], [-30], [-10], [-70], [2], [-4], [6], [94], [-35], [-28], [4], [45], [-65], [-14], [-38], [56], [39], [-8], [27], [-1], [91], [70], [53], [50], [96], [-28], [-9], [94], [102], [64], [-24], [4], [30], [50], [-36], [-76], [74], [48], [-72], [30], [-2], [82], [-58], [-26], [-75], [-26], [66], [12], [1], [63], [-46], [24], [8], [-36], [-40], [102], [-54], [48], [68], [26], [-42], [48], [-70], [77], [-13]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : 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![4, 1]>,<19,R![-4, 1]>,<23,R![1, 1]>],Snew);
    return Vf;
end function;