
// Make newform 3630.2.a.y in Magma, downloaded from the LMFDB on 23 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_y();".
// 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_y();".
// 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], [-3], [0], [-5], [-7], [-7], [0], [7], [6], [-5], [-10], [6], [-10], [-12], [0], [12], [-2], [-9], [-6], [-10], [13], [4], [2], [-3], [-3], [0], [20], [10], [-4], [16], [3], [-5], [6], [20], [-11], [4], [-16], [-4], [-16], [-8], [-3], [-24], [12], [-16], [-7], [-1], [0], [16], [-2], [-3], [25], [-12], [17], [22], [-19], [-22], [-26], [-4], [-24], [16], [-34], [-8], [-2], [12], [-17], [6], [3], [14], [6], [-24], [-27], [29], [1], [6], [34], [35], [-34], [6], [-14], [-6], [-1], [-28], [-26], [15], [22], [4], [-33], [-8], [-7], [-3], [29], [30], [33], [16], [-2], [8], [12], [-26], [-38], [-42], [15], [-2], [-24], [42], [7], [14], [8], [-10], [-33], [3], [3], [25], [4], [-26], [-44], [-32], [30], [42], [-30], [16], [10], [25], [-31], [-33], [30], [4], [7], [46], [27], [-16], [-46], [-26], [36], [-9], [-12], [28], [40], [18], [17], [-6], [11], [47], [0], [35], [7], [-51], [-28], [-30], [-47], [26], [-14], [-16], [6], [-57], [48], [-16], [-18], [-33], [3], [-18], [16], [48], [42], [-14], [-34], [-23], [-35], [-34], [20], [-10], [-17], [-14], [-58], [0], [7], [19], [-20], [-32], [-9], [24], [14], [13], [-24], [5], [-41], [-24], [6], [47], [14], [27], [-7], [-27], [-57], [45], [17], [-41], [49], [-68], [6], [-8], [-54], [-1], [32], [-20], [34], [-55], [-62], [-67], [12], [30], [4], [-28], [-69], [55], [16], [-44], [-58], [50], [-44], [4], [66], [24], [-7], [-4], [41], [-39], [-61], [24], [43], [36], [-2], [58], [34], [0], [15], [18], [24], [35], [23], [15], [0], [10], [-53], [5], [-72], [58], [65], [18], [-62], [28], [22], [-70], [-44], [18], [-40], [-70], [-6], [80], [-57], [-53], [60], [43], [47], [-64], [17], [72], [-2], [-46], [28], [-10], [24], [12], [40], [-64], [-12], [-16], [53], [34], [-2], [-66], [-42], [44], [-82], [-20], [4], [-6], [0], [-70], [30], [-37], [-69], [-48], [-25], [74], [-10], [-2], [-48], [-80], [22], [-44], [-64], [-15], [-63], [22], [69], [24], [-1], [-27], [24], [3], [-10], [-30], [68], [-37], [16], [-11], [66], [32], [75], [32], [-45], [-41], [21], [1], [-41], [28], [-10], [0], [52], [-38], [-2], [31], [23], [-20], [22], [75], [-5], [-5], [-26], [-15], [-50], [26], [-54], [-46], [30], [38], [25], [56], [-4], [-12], [51], [-11], [64], [-89], [-64], [-60], [82], [80], [13], [-60], [0], [34], [88], [26], [-44], [-13], [31], [-12], [29], [36], [-19], [88], [-14], [36], [-3], [18], [-48], [-54], [-89], [-33], [31], [50], [-48], [-2], [-32], [-3], [52], [10], [-11], [-30], [-102], [-48], [-68], [-46], [36], [-24], [79], [-18], [-52], [-37], [-41], [38], [-6], [20], [17], [-30], [-76], [-28], [46], [12], [83], [28], [22], [81], [48], [28], [-14], [84], [43], [-11], [-18], [-42], [24], [-60], [60], [-26], [-105], [-39]];
    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_y();".
// 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_y(: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_y();".
// 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_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3630_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3, 1]>,<13,R![5, 1]>],Snew);
    return Vf;
end function;