
// Make newform 3630.2.a.n in Magma, downloaded from the LMFDB on 24 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_n();".
// 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_n();".
// 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], [0], [0], [-2], [2], [-8], [4], [-2], [8], [-2], [-6], [-8], [-4], [2], [4], [6], [-12], [-12], [-2], [0], [-4], [-6], [-14], [6], [16], [-20], [6], [2], [16], [-12], [2], [-16], [-10], [8], [-2], [20], [-16], [-18], [-12], [-2], [-12], [22], [-10], [-24], [0], [-16], [-12], [14], [18], [-16], [-18], [-20], [26], [24], [-22], [0], [14], [10], [8], [6], [0], [-20], [-6], [-6], [-12], [-10], [20], [30], [26], [32], [16], [-2], [36], [12], [-22], [-2], [18], [-26], [4], [38], [24], [2], [16], [-28], [10], [38], [-18], [8], [36], [-24], [8], [12], [4], [0], [-22], [-38], [-24], [-26], [-8], [46], [-4], [-14], [40], [18], [-12], [26], [-36], [22], [-16], [38], [42], [-20], [0], [42], [-20], [-44], [-30], [-4], [14], [6], [38], [-4], [20], [46], [46], [-36], [-40], [14], [-24], [8], [-40], [-34], [-30], [-10], [34], [32], [-30], [-30], [8], [-10], [-24], [-12], [-2], [-12], [-26], [42], [4], [4], [22], [58], [4], [-40], [52], [-20], [-56], [50], [54], [-50], [28], [-38], [8], [36], [-38], [4], [40], [-2], [46], [-6], [-36], [22], [-32], [-2], [16], [-14], [40], [-6], [-64], [46], [-56], [-60], [-2], [-6], [-12], [-6], [-10], [44], [6], [-56], [-46], [4], [-44], [-30], [-20], [-46], [-50], [-18], [50], [-16], [46], [56], [38], [38], [-20], [-38], [-56], [-12], [34], [52], [-58], [-54], [64], [-20], [0], [10], [8], [2], [28], [-30], [22], [40], [-46], [48], [12], [-58], [26], [12], [0], [28], [22], [-16], [-8], [-30], [-28], [-56], [34], [22], [-36], [68], [-60], [-32], [0], [-10], [-6], [64], [48], [-68], [-40], [-48], [-10], [50], [-12], [-54], [14], [-20], [22], [-32], [66], [-34], [-64], [52], [-34], [-26], [18], [-36], [2], [-30], [8], [38], [-10], [60], [42], [8], [-58], [-8], [-12], [54], [22], [-68], [-48], [-24], [-40], [-58], [48], [52], [-30], [22], [-56], [34], [-30], [44], [18], [-28], [-10], [30], [80], [66], [-60], [24], [46], [-50], [8], [-44], [20], [-30], [-76], [30], [52], [46], [-8], [-30], [18], [-84], [40], [-18], [52], [-48], [34], [-6], [-40], [10], [-18], [64], [66], [-46], [-68], [36], [-56], [6], [-66], [18], [88], [20], [-16], [-36], [-10], [-22], [-54], [32], [6], [-22], [-42], [72], [26], [28], [62], [-84], [40], [18], [-16], [58], [10], [-64], [-66], [-6], [-36], [60], [-30], [52], [-58], [-6], [-36], [-20], [-28], [-86], [78], [-40], [-2], [-4], [20], [56], [-42], [32], [-26], [60], [16], [-50], [42], [6], [-46], [10], [-88], [-30], [-40], [36], [40], [-34], [8], [-68], [82], [-54], [-36], [-52], [-12], [-10], [-40], [66], [4], [22], [-26], [42], [72], [-6], [-74], [-64], [70], [-30], [-28], [76], [62], [54], [4], [48], [-34], [-102], [-80], [32], [18], [48], [2], [54], [36], [12], [-54], [74], [-36], [-6], [4], [80]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3630_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;