
// Make newform 3630.2.a.z in Magma, downloaded from the LMFDB on 19 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_z();".
// 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_z();".
// 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], [2], [0], [0], [2], [2], [0], [6], [0], [-6], [-6], [2], [-4], [2], [-8], [0], [4], [12], [16], [-2], [8], [-6], [10], [6], [-16], [-16], [4], [-18], [10], [0], [-2], [18], [-14], [-18], [-22], [-4], [24], [-22], [4], [14], [12], [12], [2], [0], [-6], [0], [-24], [-6], [6], [-20], [28], [-20], [10], [-16], [2], [2], [24], [-14], [-6], [6], [14], [24], [-10], [6], [28], [-8], [0], [24], [-6], [24], [-8], [4], [4], [-16], [6], [-18], [-18], [-28], [-20], [-30], [12], [2], [-14], [-24], [-6], [32], [-38], [16], [0], [-28], [0], [12], [-28], [8], [-22], [14], [-6], [-28], [34], [-42], [-12], [-10], [14], [30], [8], [-34], [0], [-40], [18], [-4], [-26], [36], [8], [2], [-12], [-40], [-6], [-40], [26], [-4], [18], [36], [-4], [18], [26], [-20], [8], [16], [-50], [-36], [24], [42], [6], [-48], [-46], [50], [-10], [-18], [2], [6], [24], [16], [-2], [24], [-4], [-6], [36], [48], [44], [30], [20], [-36], [-20], [20], [-38], [-42], [28], [14], [36], [-6], [-22], [24], [46], [8], [-24], [-44], [-20], [-22], [-24], [46], [8], [32], [0], [-42], [-42], [-38], [2], [-8], [8], [32], [10], [10], [-44], [18], [16], [52], [28], [12], [46], [24], [44], [30], [60], [-46], [-16], [-34], [-30], [-20], [-46], [62], [38], [28], [-36], [62], [64], [12], [-50], [28], [-20], [-30], [8], [-40], [0], [-50], [46], [42], [60], [18], [-44], [22], [-58], [64], [-12], [0], [6], [-16], [70], [28], [14], [58], [-50], [30], [36], [-12], [-50], [-18], [-12], [72], [-8], [-26], [48], [-46], [6], [-48], [-32], [12], [54], [20], [24], [42], [68], [38], [6], [-36], [-42], [-2], [6], [76], [62], [-4], [-28], [-20], [66], [-4], [42], [-82], [46], [34], [-50], [-36], [38], [10], [40], [48], [-68], [0], [-48], [72], [16], [-32], [44], [64], [-14], [24], [-54], [42], [-8], [50], [54], [48], [46], [64], [28], [-6], [32], [26], [44], [-50], [20], [-66], [-26], [-24], [12], [-74], [-60], [-6], [-48], [76], [48], [-10], [50], [28], [32], [-20], [8], [24], [-18], [-34], [-46], [54], [-6], [24], [-10], [-78], [76], [68], [24], [34], [76], [54], [70], [32], [-82], [64], [-18], [30], [-70], [-78], [-10], [-30], [74], [16], [34], [36], [-34], [-44], [48], [50], [-70], [-22], [-54], [-94], [32], [34], [-36], [-36], [18], [-84], [28], [42], [72], [-20], [28], [-58], [-26], [34], [-72], [-16], [-20], [-76], [78], [74], [34], [60], [52], [-20], [-26], [-48], [-66], [-30], [90], [-54], [46], [60], [80], [54], [-38], [68], [66], [66], [-52], [68], [40], [-72], [-26], [-70], [-20], [-42], [-64], [78], [-54], [18], [82], [-26], [-82], [-14], [-4], [-20], [60], [-10], [-12], [46], [-40], [70], [52], [-104], [-30], [64], [-46], [-92], [12], [100], [26], [42], [76], [90], [28], [-76]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3630_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;