
// Make newform 2310.2.a.u in Magma, downloaded from the LMFDB on 16 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2310_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_2310_2_a_u();".
// 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_2310_2_a_u();".
// 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_2310_a();"
function MakeCharacter_2310_a()
    N := 2310;
    order := 1;
    char_gens := [1541, 1387, 661, 211];
    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_2310_a_Hecke(Kf)
    return MakeCharacter_2310_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], [1], [-1], [2], [6], [-4], [0], [6], [8], [-10], [-6], [8], [0], [6], [12], [2], [-4], [12], [-10], [-4], [-12], [6], [-10], [-6], [-4], [12], [2], [6], [8], [0], [6], [-4], [-18], [-4], [-10], [-4], [-12], [6], [-12], [2], [-12], [14], [6], [-16], [-4], [-4], [-12], [-22], [18], [0], [14], [12], [-18], [24], [-30], [-16], [-10], [-6], [-4], [-18], [-28], [24], [-10], [-18], [20], [14], [-12], [2], [30], [0], [-4], [14], [-4], [24], [-30], [-34], [-30], [14], [12], [-10], [0], [-34], [8], [24], [-6], [-10], [-6], [32], [12], [0], [-40], [12], [-4], [12], [-30], [-42], [20], [2], [8], [30], [12], [42], [-4], [38], [12], [6], [12], [14], [-40], [14], [-42], [8], [8], [42], [-28], [24], [30], [-36], [2], [-34], [6], [48], [8], [-18], [-10], [-24], [-4], [-22], [-4], [-48], [-40], [38], [42], [-34], [-6], [-52], [-54], [-6], [20], [54], [32], [12], [-46], [-24], [2], [54], [8], [-24], [-34], [-18], [20], [36], [-52], [-36], [20], [6], [38], [-30], [48], [18], [32], [-12], [54], [0], [56], [50], [2], [42], [36], [-22], [0], [62], [-16], [-42], [-28], [-6], [56], [-46], [8], [48], [-10], [-18], [24], [-54], [38], [44], [26], [0], [-34], [-12], [44], [-30], [-36], [6], [50], [62], [30], [36], [6], [56], [14], [38], [36], [42], [-16], [-36], [42], [8], [38], [-6], [8], [36], [0], [26], [8], [-54], [24], [-66], [-22], [56], [-30], [-64], [12], [26], [54], [12], [8], [12], [14], [44], [68], [-54], [20], [60], [-70], [-42], [12], [0], [0], [-28], [-52], [38], [6], [0], [-76], [-12], [-28], [0], [-34], [-78], [48], [-46], [-66], [12], [-34], [-28], [-30], [-58], [32], [-36], [-22], [-22], [30], [56], [-54], [54], [8], [6], [-22], [68], [38], [-4], [14], [20], [-48], [50], [2], [48], [-60], [-64], [-12], [26], [20], [-36], [2], [-42], [80], [-30], [18], [-12], [-78], [48], [14], [-54], [-64], [-78], [48], [68], [86], [-18], [-76], [-24], [-76], [2], [-72], [74], [-36], [-82], [36], [18], [-30], [-4], [-48], [-34], [36], [-72], [-82], [-78], [-76], [74], [42], [-88], [-6], [-22], [-4], [20], [-72], [-42], [-46], [-18], [80], [12], [68], [36], [-34], [54], [-22], [8], [38], [-42], [-30], [-64], [6], [36], [74], [-28], [0], [-54], [-28], [2], [-78], [-64], [-22], [54], [24], [0], [42], [72], [62], [-6], [-48], [-60], [68], [74], [30], [-16], [74], [-60], [32], [72], [54], [32], [-34], [-84], [0], [-34], [-78], [-58], [18], [-66], [8], [-6], [68], [-24], [-40], [14], [-16], [24], [98], [-78], [12], [-28], [36], [38], [-88], [-18], [-4], [-18], [-94], [30], [8], [6], [66], [-16], [-82], [66], [68], [-12], [-34], [-42], [-24], [68], [38], [18], [96], [-52], [30], [84], [18], [-70], [72], [0], [-10], [18], [-48], [-102], [-16], [96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2310_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2310_2_a_u();".
// 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_2310_2_a_u(:prec:=1)
    chi := MakeCharacter_2310_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_2310_2_a_u();".
// 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_2310_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2310_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![-2, 1]>,<17,R![-6, 1]>,<19,R![4, 1]>,<23,R![0, 1]>,<29,R![-6, 1]>,<31,R![-8, 1]>],Snew);
    return Vf;
end function;