
// Make newform 3024.2.a.i in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3024_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_3024_2_a_i();".
// 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_3024_2_a_i();".
// 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_3024_a();"
function MakeCharacter_3024_a()
    N := 3024;
    order := 1;
    char_gens := [1135, 757, 785, 2593];
    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_3024_a_Hecke(Kf)
    return MakeCharacter_3024_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], [-1], [1], [-2], [-4], [3], [-2], [6], [6], [4], [-11], [5], [7], [5], [-6], [9], [6], [4], [8], [-2], [11], [-5], [6], [-6], [14], [10], [-6], [-1], [-16], [5], [4], [12], [2], [-6], [-15], [18], [9], [17], [6], [-8], [-8], [10], [19], [18], [-24], [12], [24], [-4], [16], [8], [16], [-8], [27], [-22], [-12], [-17], [-30], [13], [18], [-10], [21], [-10], [13], [28], [6], [7], [11], [-24], [24], [-5], [-18], [-10], [-29], [-35], [21], [-26], [-2], [-24], [28], [-29], [-2], [20], [-20], [0], [-42], [16], [-38], [21], [1], [-40], [15], [-8], [0], [37], [41], [19], [39], [-4], [-7], [39], [-28], [-32], [-46], [23], [4], [12], [-17], [-30], [-40], [-20], [30], [-2], [-22], [21], [-30], [14], [32], [4], [12], [-28], [-14], [-42], [-36], [32], [24], [3], [13], [2], [-52], [-4], [-14], [20], [-29], [-3], [-4], [-31], [-14], [54], [-46], [56], [20], [9], [-32], [-10], [33], [-16], [21], [-14], [-10], [-3], [-18], [-31], [-47], [1], [30], [25], [13], [-14], [15], [44], [-6], [40], [-21], [32], [33], [19], [10], [-19], [-51], [-12], [22], [36], [-61], [2], [63], [20], [-20], [-12], [62], [-13], [-33], [-22], [63], [-38], [9], [30], [56], [58], [7], [-16], [-16], [-43], [-63], [24], [39], [2], [-57], [-22], [3], [40], [-8], [-6], [-18], [61], [61], [-42], [-44], [42], [24], [53], [66], [-53], [39], [-21], [14], [27], [18], [-48], [64], [-25], [20], [-68], [27], [-1], [21], [33], [-14], [62], [32], [-34], [0], [-53], [40], [-22], [-8], [20], [62], [-6], [61], [-32], [-8], [-56], [74], [30], [9], [-62], [23], [-67], [-42], [-35], [30], [38], [50], [-51], [30], [-5], [-38], [26], [6], [20], [54], [-8], [-22], [-2], [44], [32], [14], [45], [-70], [-56], [48], [-52], [-61], [6], [70], [64], [-17], [67], [3], [-21], [23], [-43], [8], [6], [4], [-43], [-50], [-38], [-83], [54], [75], [-62], [75], [7], [-23], [74], [-9], [55], [10], [-48], [76], [-56], [60], [-23], [22], [68], [-38], [-36], [38], [-36], [-44], [-10], [-31], [-72], [-38], [-7], [-78], [-2], [44], [10], [-59], [-11], [-64], [64], [-74], [-5], [28], [-42], [2], [38], [-26], [38], [31], [65], [29], [6], [-75], [64], [52], [-73], [-54], [-9], [80], [22], [-34], [28], [-81], [-5], [-50], [0], [33], [-14], [18], [-25], [-53], [20], [-75], [-72], [2], [51], [65], [20], [4], [68], [21], [51], [95], [-93], [24], [-20], [38], [-4], [52], [62], [36], [-12], [60], [-18], [-68], [-39], [-30], [24], [6], [-90], [-72], [-64], [-96], [-19], [-40], [47], [-46], [30], [88], [14], [30], [-74], [19], [20], [38], [4], [-84], [-51], [1], [-94], [-94], [21], [-78], [-4], [12], [-64], [-48], [-12], [21], [23], [50], [80], [14], [39], [-36], [0], [-50], [-108], [-79], [-84], [-7], [-8], [-18], [42], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3024_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3024_2_a_i();".
// 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_3024_2_a_i(:prec:=1)
    chi := MakeCharacter_3024_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_3024_2_a_i();".
// 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_3024_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3024_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<11,R![2, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;