
// Make newform 3600.2.a.t in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3600_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_3600_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_3600_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_3600_a();"
function MakeCharacter_3600_a()
    N := 3600;
    order := 1;
    char_gens := [3151, 901, 2801, 577];
    v := [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_3600_a_Hecke(Kf)
    return MakeCharacter_3600_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], [0], [0], [0], [-4], [-6], [-6], [4], [0], [2], [8], [2], [6], [12], [-8], [6], [12], [14], [4], [8], [6], [8], [12], [-10], [-2], [-6], [0], [4], [-18], [-6], [-8], [4], [2], [-4], [10], [-16], [-6], [4], [-16], [14], [-12], [-10], [0], [14], [-10], [0], [-12], [8], [28], [-10], [2], [0], [-14], [28], [10], [0], [18], [24], [2], [22], [-4], [22], [-28], [8], [-10], [14], [28], [-2], [-12], [14], [26], [-24], [-8], [2], [-4], [-24], [-6], [-38], [14], [26], [36], [-10], [-16], [30], [-16], [4], [30], [22], [-14], [8], [28], [-32], [16], [-36], [20], [16], [2], [-42], [-4], [30], [36], [-18], [-4], [6], [12], [-2], [4], [-6], [-40], [26], [40], [2], [2], [28], [-32], [-18], [4], [48], [14], [4], [6], [14], [38], [-28], [20], [-30], [-10], [0], [-32], [26], [4], [16], [-8], [2], [-26], [-30], [-42], [4], [-2], [6], [28], [10], [48], [36], [-2], [40], [-14], [2], [-4], [-8], [-22], [-2], [52], [0], [28], [0], [16], [30], [-42], [50], [60], [-14], [-48], [12], [58], [-32], [-40], [18], [-14], [-42], [-20], [-2], [-24], [-26], [-24], [22], [-4], [26], [-16], [46], [56], [20], [-46], [18], [56], [-54], [-22], [20], [-22], [32], [14], [36], [-12], [2], [-36], [-14], [50], [10], [-22], [48], [-46], [40], [-46], [66], [12], [14], [8], [12], [6], [-52], [-34], [-54], [-16], [-28], [24], [-54], [-40], [46], [-48], [30], [-10], [-16], [14], [-56], [28], [54], [-30], [-48], [64], [12], [58], [-44], [40], [-58], [-4], [-24], [-46], [-10], [-4], [40], [28], [44], [32], [-34], [74], [-24], [-56], [-12], [-52], [-56], [-6], [30], [0], [10], [62], [36], [-10], [44], [-74], [-10], [40], [12], [22], [-54], [-6], [-28], [-78], [6], [44], [6], [-18], [4], [-10], [56], [-34], [-16], [-12], [14], [10], [-12], [-24], [16], [0], [38], [28], [-80], [78], [-42], [0], [-2], [82], [-52], [18], [12], [58], [66], [56], [6], [60], [4], [38], [-18], [24], [-36], [-4], [-34], [-44], [-34], [72], [-14], [72], [74], [-34], [-44], [-32], [-54], [-12], [80], [-82], [-82], [-60], [-42], [2], [8], [18], [-14], [4], [-68], [-8], [-90], [62], [30], [40], [12], [44], [-60], [-34], [-6], [-54], [72], [-62], [34], [-6], [48], [-18], [-76], [-10], [-84], [0], [70], [-12], [22], [-46], [-40], [-26], [-46], [0], [12], [-22], [-16], [-46], [-42], [72], [-68], [-28], [54], [-2], [-80], [-6], [-92], [-20], [88], [-86], [-16], [-38], [36], [48], [-2], [78], [70], [2], [-62], [80], [-38], [4], [-64], [-8], [34], [44], [56], [2], [70], [-4], [4], [-40], [-58], [56], [-90], [-84], [-54], [-2], [-70], [-56], [-78], [-6], [-80], [-22], [-18], [20], [68], [14], [-74], [52], [68], [102], [82], [0], [64], [10], [-64], [2], [-78], [8], [-84], [-58], [-66], [-52], [-58], [28], [56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3600_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3600_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_3600_2_a_t(:prec:=1)
    chi := MakeCharacter_3600_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_3600_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_3600_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<13,R![6, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;