
// Make newform 3630.2.a.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3630_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_3630_2_a_g();".
// 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_3630_2_a_g();".
// 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_3630_a();"
function MakeCharacter_3630_a()
    N := 3630;
    order := 1;
    char_gens := [1211, 727, 3511];
    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_3630_a_Hecke(Kf)
    return MakeCharacter_3630_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], [0], [-1], [-3], [-1], [6], [3], [-4], [-1], [0], [-4], [6], [6], [-6], [-10], [2], [-15], [-4], [-10], [3], [12], [8], [-3], [-1], [-12], [-16], [-6], [-16], [-6], [-21], [5], [18], [2], [-7], [14], [12], [-6], [-18], [-22], [-9], [-4], [12], [20], [-13], [-19], [12], [8], [6], [-21], [-19], [0], [-3], [24], [9], [2], [-10], [30], [-4], [-24], [32], [-24], [8], [18], [-19], [32], [-3], [14], [6], [24], [-13], [-31], [11], [-18], [18], [-25], [-24], [-22], [-6], [8], [21], [-28], [-10], [-27], [-6], [-10], [-9], [-4], [-9], [15], [23], [12], [-13], [-42], [-6], [-18], [-34], [20], [26], [30], [9], [-24], [-4], [2], [33], [-6], [36], [-22], [41], [23], [-45], [-25], [38], [-18], [20], [24], [36], [-42], [8], [44], [12], [15], [-37], [-9], [32], [24], [17], [14], [5], [18], [20], [38], [36], [-49], [18], [-46], [-42], [24], [11], [6], [-31], [21], [14], [-27], [23], [21], [20], [30], [53], [-12], [20], [36], [38], [9], [2], [-6], [38], [27], [-15], [-6], [32], [-48], [-54], [24], [14], [-31], [41], [18], [0], [-46], [45], [-28], [44], [6], [-19], [27], [-40], [-22], [53], [-18], [-22], [9], [-12], [9], [47], [62], [-34], [45], [-52], [21], [23], [-51], [-3], [9], [5], [-37], [-51], [-42], [54], [-34], [-46], [11], [60], [-36], [44], [-9], [-42], [71], [56], [6], [8], [-12], [-15], [35], [-16], [24], [42], [-48], [-22], [-4], [-24], [-4], [-21], [-70], [9], [51], [17], [48], [-13], [-4], [-70], [-54], [-4], [-6], [-1], [12], [30], [-39], [21], [-55], [32], [74], [-45], [-57], [8], [6], [71], [66], [-34], [-60], [-12], [-58], [-6], [-60], [50], [44], [-42], [20], [-31], [9], [-10], [23], [15], [20], [57], [-6], [-22], [6], [50], [26], [-34], [38], [-4], [-16], [-60], [-34], [-79], [66], [-30], [-28], [-48], [-40], [2], [-12], [-34], [-72], [-16], [-54], [18], [-51], [51], [-18], [-1], [-78], [62], [12], [12], [-46], [26], [-6], [44], [3], [-13], [-34], [39], [-40], [-27], [41], [-36], [15], [66], [-16], [-6], [47], [60], [-21], [62], [-72], [-43], [-76], [-69], [29], [33], [-55], [-43], [-88], [-54], [-42], [-40], [-78], [56], [-51], [65], [48], [56], [-57], [-13], [53], [-22], [-39], [-18], [62], [-78], [48], [56], [44], [-45], [-12], [-76], [-22], [27], [-1], [92], [-69], [0], [0], [30], [6], [5], [24], [54], [-48], [-22], [8], [-42], [-7], [47], [-42], [23], [72], [57], [-4], [2], [12], [-57], [38], [96], [-100], [63], [3], [-55], [-42], [44], [-18], [-100], [-31], [92], [24], [-19], [42], [12], [62], [48], [-28], [-34], [30], [5], [30], [-4], [3], [17], [-42], [-18], [38], [-103], [-48], [56], [48], [-52], [-96], [-15], [-40], [50], [9], [-60], [104], [66], [54], [51], [-43], [-12], [-6], [-58], [-60], [36], [54], [-43], [51]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3630_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3630_2_a_g();".
// 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_3630_2_a_g(:prec:=1)
    chi := MakeCharacter_3630_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_3630_2_a_g();".
// 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_3630_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3630_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<13,R![1, 1]>],Snew);
    return Vf;
end function;