
// Make newform 3630.2.a.p in Magma, downloaded from the LMFDB on 17 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_p();".
// 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_p();".
// 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], [1], [-7], [-5], [-8], [-5], [-10], [7], [6], [10], [-10], [-4], [4], [0], [-6], [9], [-2], [6], [-7], [0], [-14], [-15], [13], [-8], [-12], [14], [4], [-12], [5], [-7], [-10], [-4], [-23], [8], [-16], [0], [0], [16], [3], [4], [8], [-12], [3], [23], [24], [-4], [6], [-19], [15], [28], [23], [-30], [-1], [18], [2], [-20], [-28], [24], [6], [-8], [18], [-12], [3], [2], [-25], [-18], [26], [8], [-11], [31], [-27], [-6], [14], [31], [18], [-14], [-14], [-34], [15], [8], [-26], [-19], [34], [-28], [-21], [-16], [-21], [-3], [29], [-18], [13], [36], [2], [16], [-36], [22], [-26], [10], [-13], [34], [-32], [42], [21], [-34], [-24], [-38], [41], [17], [-3], [-11], [0], [-30], [-20], [-32], [30], [-22], [-10], [36], [-34], [-13], [-43], [-5], [-26], [-36], [47], [2], [9], [-40], [50], [38], [12], [-7], [28], [8], [-36], [30], [35], [-22], [3], [27], [-44], [21], [37], [21], [4], [34], [43], [-14], [34], [8], [46], [49], [-32], [-40], [-54], [-5], [-23], [-26], [32], [12], [-2], [-2], [22], [43], [-5], [30], [-36], [22], [31], [10], [-50], [52], [-59], [9], [-28], [-20], [7], [0], [-26], [21], [0], [15], [53], [32], [18], [-17], [38], [-17], [-35], [-57], [-5], [11], [7], [-45], [41], [20], [-26], [44], [50], [-7], [-44], [-20], [-2], [-3], [-50], [49], [44], [-30], [52], [28], [-21], [-17], [-40], [-16], [-38], [-18], [4], [-56], [38], [-24], [-51], [44], [47], [-9], [-51], [28], [55], [20], [34], [18], [-10], [52], [-41], [-38], [36], [-19], [-51], [13], [24], [-70], [27], [13], [-24], [-50], [-13], [6], [-34], [56], [-6], [-30], [-52], [22], [4], [34], [-54], [-64], [17], [7], [-4], [-71], [81], [24], [35], [72], [-22], [14], [56], [42], [36], [8], [-52], [-32], [-60], [-60], [-29], [10], [-54], [-42], [2], [-52], [-78], [-44], [-48], [-62], [16], [34], [18], [-7], [-21], [-40], [29], [-30], [14], [30], [48], [0], [-14], [64], [-4], [-29], [-67], [-6], [23], [0], [-71], [31], [-20], [57], [18], [54], [40], [-51], [-8], [-3], [-14], [72], [-7], [4], [-33], [-41], [21], [-55], [-77], [-36], [-2], [12], [48], [-66], [-14], [59], [-19], [12], [-46], [-13], [3], [-91], [-18], [-49], [-90], [6], [14], [-50], [-58], [-54], [-23], [60], [-28], [-80], [-95], [59], [-84], [39], [-24], [-12], [18], [-32], [-25], [60], [-36], [46], [-76], [-2], [-48], [-43], [97], [-40], [-55], [-4], [57], [-4], [-14], [36], [-59], [-50], [96], [-78], [5], [25], [-7], [-78], [-40], [66], [-68], [-7], [-52], [82], [61], [-42], [-66], [8], [-12], [-58], [8], [36], [-5], [46], [-32], [45], [9], [-78], [-38], [-4], [-59], [18], [8], [-56], [-82], [-84], [73], [60], [14], [-21], [88], [-52], [30], [-84], [65], [15], [-54], [102], [-12], [-4], [-36], [6], [91], [-55]];
    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_p();".
// 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_p(: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_p();".
// 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_p( : 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![-1, 1]>],Snew);
    return Vf;
end function;