
// Make newform 5610.2.a.v 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_v();".
// 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_v();".
// 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], [2], [-1], [4], [-8], [2], [0], [-2], [6], [4], [4], [-10], [4], [2], [8], [-4], [10], [0], [4], [10], [2], [-10], [-8], [0], [-14], [10], [8], [-4], [-2], [4], [6], [-20], [10], [4], [-8], [-18], [12], [18], [24], [-6], [-2], [8], [-12], [-8], [0], [-18], [6], [-24], [14], [4], [-26], [-24], [14], [-20], [-22], [6], [4], [26], [-4], [12], [26], [-34], [-28], [18], [-16], [10], [-18], [0], [-8], [10], [32], [12], [6], [-2], [30], [26], [20], [14], [0], [2], [16], [-36], [-18], [-10], [6], [8], [-12], [16], [16], [-16], [-16], [8], [-34], [22], [28], [-14], [-44], [-30], [12], [6], [-20], [-6], [12], [-18], [0], [-2], [28], [-6], [-30], [24], [-16], [-2], [4], [12], [30], [40], [-18], [-38], [6], [-36], [-8], [6], [34], [36], [-8], [2], [-12], [-32], [-8], [2], [6], [26], [-10], [-28], [-18], [6], [44], [18], [16], [32], [6], [28], [10], [-58], [20], [20], [26], [-18], [32], [24], [36], [-36], [20], [22], [-2], [-38], [-44], [54], [-24], [12], [-2], [24], [-56], [-46], [-10], [-26], [40], [14], [-8], [14], [56], [-18], [-28], [-34], [-24], [-22], [8], [-4], [-26], [-42], [-12], [54], [-30], [48], [-42], [-56], [-30], [-64], [-60], [30], [0], [2], [-58], [-18], [6], [0], [-54], [-16], [10], [18], [-60], [-58], [8], [44], [38], [-4], [-46], [-26], [16], [12], [32], [-30], [-16], [54], [0], [-26], [42], [-64], [-54], [16], [-52], [-62], [-30], [-20], [-24], [36], [-54], [4], [-44], [-66], [32], [32], [26], [-34], [-20], [72], [-60], [28], [-40], [-34], [-66], [48], [-16], [12], [-44], [-32], [34], [-34], [52], [50], [26], [-24], [2], [-28], [38], [-58], [20], [12], [-6], [10], [-6], [-20], [-18], [-30], [52], [-54], [26], [16], [-54], [-68], [22], [0], [-28], [2], [10], [8], [-40], [32], [-56], [-30], [12], [32], [-62], [-42], [-24], [30], [30], [12], [6], [12], [-86], [-6], [-16], [70], [-4], [4], [-2], [42], [16], [36], [32], [50], [60], [34], [88], [-54], [72], [14], [62], [-16], [16], [66], [44], [-40], [-46], [70], [-60], [-38], [30], [72], [-2], [2], [64], [4], [16], [62], [82], [-2], [-64], [52], [84], [12], [14], [62], [82], [-40], [-22], [-58], [-14], [-64], [-2], [4], [90], [-8], [48], [-10], [4], [-70], [78], [-36], [26], [-10], [-24], [36], [-34], [-12], [-6], [54], [-12], [84], [-32], [-46], [86], [-64], [-90], [68], [88], [8], [-42], [-20], [6], [60], [-8], [-66], [-34], [-58], [-18], [-34], [92], [-90], [-20], [96], [-40], [-62], [44], [68], [98], [-26], [-20], [48], [0], [-86], [-52], [58], [-88], [-10], [66], [54], [-48], [-38], [-18], [40], [58], [-42], [-16], [-100], [-54], [42], [68], [52], [-2], [-58], [-80], [32], [50], [96], [78], [-38], [72], [-96], [-54], [-2], [36], [70], [-28], [0]];
    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_v();".
// 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_v(: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_v();".
// 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_v( : 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![-2, 1]>,<19,R![-4, 1]>,<23,R![8, 1]>],Snew);
    return Vf;
end function;