
// Make newform 2304.2.a.l in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_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_2304_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_2304_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_2304_a();"
function MakeCharacter_2304_a()
    N := 2304;
    order := 1;
    char_gens := [1279, 2053, 1793];
    v := [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_2304_a_Hecke(Kf)
    return MakeCharacter_2304_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 := [[0], [0], [2], [-2], [0], [-4], [2], [-4], [-4], [6], [-2], [-8], [2], [-4], [-12], [6], [-4], [0], [12], [-12], [6], [-10], [16], [-10], [-2], [-10], [-6], [12], [4], [6], [2], [-20], [18], [4], [-6], [-18], [8], [4], [-8], [-6], [4], [20], [-8], [-6], [-2], [10], [-20], [-14], [8], [-4], [14], [0], [2], [0], [-18], [16], [6], [18], [-28], [-18], [-4], [6], [12], [8], [-14], [22], [20], [-2], [-8], [-16], [6], [20], [-18], [-16], [4], [24], [34], [-32], [18], [-10], [24], [20], [12], [14], [-30], [24], [-30], [22], [-30], [26], [8], [20], [38], [20], [36], [36], [6], [42], [36], [20], [-28], [42], [-24], [-30], [-20], [-42], [-28], [6], [20], [-2], [2], [-16], [-2], [-36], [22], [-22], [4], [12], [-6], [-36], [40], [14], [18], [-16], [20], [-50], [-4], [-20], [-42], [-4], [-44], [16], [-2], [12], [-38], [10], [-34], [12], [2], [-30], [20], [-10], [14], [12], [4], [-20], [24], [18], [44], [24], [-32], [-2], [4], [-48], [28], [-48], [30], [50], [-38], [-30], [-12], [-6], [-22], [0], [2], [16], [-2], [-48], [30], [26], [-44], [40], [-32], [26], [-10], [30], [-20], [10], [34], [-56], [-58], [0], [-36], [-2], [24], [-6], [-52], [4], [10], [-48], [-46], [-36], [20], [50], [-52], [-66], [-58], [-4], [-58], [-24], [66], [-62], [-28], [30], [56], [-18], [10], [-24], [30], [-20], [58], [-30], [-46], [12], [-20], [-42], [22], [-22], [-8], [-46], [20], [-30], [10], [26], [-32], [56], [54], [20], [-62], [-60], [-44], [-44], [-22], [-18], [-4], [-12], [-50], [26], [-64], [-52], [56], [20], [54], [24], [6], [-20], [22], [-60], [4], [24], [-12], [-42], [12], [10], [14], [4], [-60], [28], [-42], [22], [26], [-12], [36], [56], [42], [76], [-14], [62], [-44], [26], [80], [-28], [26], [30], [-62], [34], [-8], [-16], [-2], [20], [84], [2], [-28], [0], [-12], [-8], [34], [-22], [30], [70], [10], [68], [-26], [-20], [-24], [-14], [-62], [6], [-24], [-68], [-14], [2], [70], [-4], [-20], [-62], [32], [-36], [60], [-16], [24], [54], [58], [44], [-8], [-70], [-36], [12], [14], [-50], [-60], [2], [70], [46], [-46], [62], [-84], [-84], [48], [26], [-60], [82], [50], [36], [-60], [-8], [24], [46], [38], [-78], [-56], [-22], [-66], [2], [-26], [24], [-20], [68], [12], [-42], [-20], [-58], [30], [26], [-24], [54], [-80], [-60], [42], [56], [-8], [22], [8], [56], [52], [-34], [-18], [-26], [38], [0], [-76], [64], [54], [-58], [28], [-56], [-88], [-46], [-70], [42], [-70], [-6], [-2], [-18], [-44], [56], [78], [52], [-44], [-52], [50], [-54], [36], [92], [-32], [6], [-70], [-70], [100], [90], [64], [-54], [2], [18], [-66], [-58], [68], [18], [-36], [24], [-26], [-22], [64], [-60], [22], [30], [40], [38], [2], [-84], [-54], [-8], [-72], [-64], [-14], [-98], [-104], [-10], [-20], [-20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_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_2304_2_a_l(:prec:=1)
    chi := MakeCharacter_2304_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_2304_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_2304_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![2, 1]>,<11,R![0, 1]>,<13,R![4, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;