
// Make newform 3630.2.a.t 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_t();".
// 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_t();".
// 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], [-4], [-6], [2], [4], [-2], [-8], [10], [2], [-6], [-8], [-10], [0], [-12], [-4], [-8], [-8], [-6], [8], [-6], [10], [-10], [8], [16], [-16], [6], [14], [0], [-2], [2], [-14], [-6], [18], [12], [-8], [18], [12], [6], [16], [12], [18], [-16], [-14], [-16], [-24], [-22], [-26], [20], [-12], [20], [2], [-16], [-10], [-10], [28], [18], [-14], [14], [-2], [20], [-10], [18], [-12], [-32], [32], [20], [18], [24], [-8], [-16], [12], [36], [-22], [-34], [6], [12], [4], [26], [-28], [34], [-34], [-32], [18], [32], [2], [24], [-32], [-36], [-32], [12], [-4], [-24], [-2], [6], [-30], [16], [-14], [14], [-12], [-42], [46], [-18], [16], [22], [-20], [8], [38], [-32], [30], [-12], [16], [42], [-36], [12], [-10], [40], [10], [-36], [18], [-36], [-4], [-6], [-38], [-24], [-8], [-4], [-26], [4], [-32], [-54], [22], [-8], [-42], [42], [42], [6], [10], [22], [-40], [48], [-2], [-4], [-16], [10], [-20], [28], [-8], [46], [-20], [-44], [-12], [16], [46], [14], [52], [46], [-28], [10], [-34], [-32], [-18], [-36], [-16], [48], [-12], [6], [24], [-10], [-24], [8], [-64], [6], [-58], [-26], [46], [-44], [-48], [0], [26], [26], [-56], [-66], [52], [-36], [20], [36], [14], [24], [20], [-30], [28], [42], [-32], [30], [-30], [20], [-30], [-46], [54], [-4], [-12], [26], [8], [-4], [-2], [-36], [-36], [-2], [16], [-8], [-32], [62], [18], [42], [0], [-50], [-8], [-62], [14], [-64], [-12], [-4], [-34], [36], [10], [-68], [22], [-46], [18], [30], [20], [28], [30], [6], [60], [-36], [24], [14], [-24], [-46], [-18], [-16], [-32], [44], [-26], [-4], [12], [-30], [64], [38], [-74], [-36], [-42], [-74], [2], [-20], [-38], [-20], [64], [-56], [-78], [28], [-26], [-2], [-34], [-22], [-58], [-44], [38], [-10], [16], [0], [28], [68], [-32], [-8], [-32], [32], [-12], [-68], [50], [-76], [-22], [-14], [-56], [50], [26], [-72], [6], [-48], [40], [42], [32], [-66], [44], [54], [4], [6], [42], [72], [76], [54], [52], [-30], [-60], [-72], [-16], [-62], [58], [-20], [-48], [12], [24], [-72], [-2], [-58], [-14], [86], [-22], [8], [-18], [-46], [-4], [4], [-24], [74], [32], [-6], [42], [-48], [-74], [48], [54], [54], [90], [54], [-10], [-62], [-30], [-16], [-10], [-60], [38], [-60], [0], [54], [-14], [-6], [-18], [22], [68], [-30], [24], [60], [-30], [-72], [64], [-22], [-84], [60], [-60], [-26], [-90], [70], [16], [-24], [-28], [-36], [-34], [86], [-62], [-52], [76], [20], [62], [-80], [-78], [34], [-2], [-30], [-58], [-56], [40], [-2], [66], [0], [34], [46], [-28], [-12], [60], [40], [74], [50], [4], [14], [20], [-66], [22], [98], [-38], [-86], [-10], [-54], [52], [12], [4], [-10], [-76], [-98], [-32], [2], [76], [-64], [82], [-48], [14], [24], [48], [-12], [26], [-18], [12], [50], [-92], [60]];
    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_t();".
// 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_t(: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_t();".
// 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_t( : 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![4, 1]>],Snew);
    return Vf;
end function;