
// Make newform 3696.2.a.bb in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3696_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_3696_2_a_bb();".
// 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_3696_2_a_bb();".
// 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_3696_a();"
function MakeCharacter_3696_a()
    N := 3696;
    order := 1;
    char_gens := [463, 2773, 2465, 1585, 673];
    v := [1, 1, 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_3696_a_Hecke(Kf)
    return MakeCharacter_3696_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 := [[0], [1], [2], [1], [-1], [2], [6], [8], [-4], [2], [-8], [6], [6], [-8], [-4], [10], [-4], [-14], [4], [4], [-14], [8], [-4], [-14], [18], [-6], [16], [12], [10], [10], [-8], [4], [-6], [16], [10], [-16], [-2], [-12], [8], [-14], [12], [-10], [4], [-6], [-6], [8], [-8], [-16], [4], [-10], [-18], [8], [2], [20], [26], [-24], [2], [-8], [-6], [30], [0], [-6], [8], [-12], [-6], [18], [-4], [-22], [36], [-22], [26], [-24], [0], [-22], [-20], [28], [-30], [-18], [34], [-22], [12], [-26], [-32], [2], [40], [-4], [-6], [10], [2], [24], [-20], [-24], [24], [-20], [28], [24], [2], [-30], [-16], [-38], [-40], [18], [-36], [38], [-32], [18], [12], [30], [12], [42], [-24], [2], [-46], [-4], [16], [-6], [-28], [-12], [10], [28], [-26], [-22], [-6], [36], [-12], [-30], [-34], [-12], [-40], [-14], [16], [-40], [-8], [6], [30], [-38], [-22], [-8], [-22], [-10], [-56], [10], [-8], [-28], [-26], [-36], [10], [-18], [4], [44], [2], [18], [-44], [-32], [-12], [-20], [-16], [-22], [-22], [18], [36], [22], [48], [12], [2], [4], [40], [18], [2], [18], [-4], [14], [-24], [-30], [32], [-22], [0], [-62], [56], [26], [8], [4], [-26], [-18], [12], [-46], [-62], [4], [10], [8], [50], [-60], [44], [42], [-36], [66], [-14], [6], [38], [40], [50], [48], [38], [26], [-44], [-46], [40], [52], [6], [-60], [42], [-30], [-32], [36], [0], [58], [-64], [-66], [4], [42], [10], [-64], [66], [16], [12], [10], [-54], [-36], [-8], [60], [-50], [32], [16], [70], [44], [-48], [34], [-22], [-28], [44], [44], [0], [-64], [38], [-2], [48], [64], [4], [0], [16], [-14], [-50], [44], [-38], [-14], [-52], [22], [40], [-70], [-62], [-24], [-12], [10], [-6], [-14], [60], [74], [42], [8], [-22], [-18], [28], [10], [64], [10], [-24], [12], [-70], [-22], [28], [24], [-40], [0], [-62], [-72], [36], [34], [26], [-40], [-18], [26], [52], [-18], [-12], [34], [-46], [80], [-54], [4], [-32], [-14], [18], [64], [-4], [-4], [-62], [12], [86], [-76], [26], [0], [42], [-42], [44], [-72], [18], [44], [24], [66], [78], [40], [26], [34], [-64], [46], [50], [-44], [60], [24], [-54], [82], [42], [16], [-44], [8], [84], [6], [14], [58], [-88], [-50], [66], [90], [88], [-94], [-36], [-58], [-52], [-24], [-6], [16], [26], [34], [72], [50], [30], [52], [44], [-18], [-84], [34], [86], [-12], [-36], [-84], [74], [18], [-16], [34], [4], [84], [-8], [-54], [24], [-34], [-60], [72], [18], [70], [10], [-70], [34], [48], [-66], [8], [20], [-8], [14], [-24], [-84], [82], [2], [52], [-28], [-76], [-38], [80], [-54], [-20], [-70], [66], [-78], [-32], [42], [-38], [-40], [-2], [-42], [-20], [-60], [-30], [-22], [76], [-24], [2], [2], [104], [-16], [-78], [40], [74], [90], [76], [-20], [10], [50], [-76], [-66], [52], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3696_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3696_2_a_bb();".
// 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_3696_2_a_bb(:prec:=1)
    chi := MakeCharacter_3696_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_3696_2_a_bb();".
// 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_3696_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3696_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;