
// Make newform 5610.2.a.ba in Magma, downloaded from the LMFDB on 17 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_ba();".
// 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_ba();".
// 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], [4], [-1], [-6], [-1], [2], [6], [-6], [4], [-2], [8], [-4], [12], [8], [0], [-2], [6], [-4], [-6], [10], [16], [6], [-16], [-8], [4], [12], [10], [-12], [8], [0], [-14], [16], [12], [-24], [-24], [20], [20], [6], [16], [-10], [0], [2], [-6], [-4], [-12], [24], [12], [-14], [26], [20], [-4], [12], [-6], [16], [-10], [-12], [10], [-2], [4], [-18], [-32], [16], [8], [22], [8], [18], [-36], [16], [2], [16], [-26], [22], [-20], [36], [-26], [-10], [10], [-10], [4], [22], [-14], [-2], [34], [-18], [38], [38], [0], [-32], [-6], [-38], [42], [2], [20], [-4], [-34], [18], [-12], [38], [28], [26], [-24], [-6], [8], [-34], [6], [-18], [-24], [-28], [28], [-10], [-36], [-20], [-32], [-30], [-48], [-48], [10], [18], [-34], [26], [18], [4], [-20], [8], [14], [-24], [-36], [-34], [-50], [-16], [4], [-12], [-54], [-30], [4], [12], [-12], [-44], [-44], [-18], [2], [12], [-46], [16], [-26], [54], [24], [0], [14], [-26], [22], [8], [16], [12], [-48], [50], [-14], [-10], [4], [-22], [-8], [36], [18], [18], [40], [50], [-24], [6], [62], [-30], [-50], [18], [-16], [-50], [0], [-58], [16], [16], [-36], [-52], [22], [10], [36], [54], [30], [42], [-20], [66], [56], [44], [-4], [-38], [36], [12], [-12], [22], [18], [-24], [-58], [-42], [4], [-62], [12], [-68], [-24], [-8], [24], [60], [-10], [38], [-34], [-6], [-30], [-18], [-48], [-18], [46], [44], [-12], [26], [30], [-26], [28], [44], [-36], [-52], [56], [-12], [-4], [36], [36], [42], [-30], [-56], [-18], [54], [-36], [-24], [-24], [-34], [-12], [-50], [-54], [74], [-34], [-12], [-26], [0], [18], [48], [0], [-46], [10], [-18], [-38], [12], [-10], [34], [48], [52], [-10], [34], [32], [24], [6], [14], [76], [-18], [50], [34], [-58], [-16], [-2], [-24], [-34], [-4], [-58], [-42], [24], [24], [-28], [-4], [44], [32], [32], [-78], [56], [-46], [-14], [16], [-34], [-84], [34], [-46], [40], [60], [0], [28], [-6], [18], [46], [-8], [-20], [-56], [82], [-50], [-48], [-34], [0], [14], [48], [-54], [8], [78], [80], [-70], [-8], [50], [-80], [-24], [-36], [-60], [26], [-34], [44], [-20], [52], [54], [-46], [78], [-10], [20], [68], [0], [-22], [-30], [14], [-32], [-28], [-22], [-10], [24], [16], [-44], [42], [42], [-90], [54], [-58], [-52], [38], [40], [-40], [66], [56], [-8], [-38], [-84], [10], [64], [40], [-8], [2], [22], [-42], [-56], [-20], [-60], [92], [72], [-40], [-20], [-6], [-60], [-26], [-2], [90], [-14], [2], [74], [-4], [-82], [68], [18], [0], [-28], [-4], [68], [-2], [54], [24], [-30], [-16], [30], [4], [78], [28], [0], [14], [-62], [64], [-20], [56], [26], [-76], [42], [-90], [-60], [70], [-2], [-26], [8], [-6], [-82], [54], [26], [12], [0], [-54], [-98], [66], [6], [32], [-32], [60], [-64], [28], [14]];
    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_ba();".
// 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_ba(: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_ba();".
// 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_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<13,R![6, 1]>,<19,R![-2, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;