
// Make newform 3600.2.a.g 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_g();".
// 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_g();".
// 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], [-4], [4], [-4], [-6], [4], [4], [-4], [0], [-4], [8], [0], [12], [-2], [-12], [2], [8], [-8], [16], [8], [8], [0], [8], [-12], [-4], [8], [18], [-10], [12], [12], [-2], [20], [12], [-16], [4], [8], [12], [-6], [4], [10], [-24], [-8], [18], [8], [20], [12], [-8], [6], [18], [-8], [2], [-4], [6], [-12], [-12], [-8], [-12], [0], [-24], [2], [16], [24], [-8], [6], [20], [-32], [-32], [2], [-6], [-16], [28], [20], [4], [36], [12], [12], [-8], [-10], [12], [-26], [32], [-24], [-24], [-24], [24], [24], [12], [4], [0], [-8], [-28], [-20], [4], [-4], [-20], [40], [40], [34], [32], [6], [0], [24], [20], [16], [8], [22], [-16], [-42], [4], [-4], [-46], [-28], [-8], [-24], [24], [-28], [-42], [12], [-22], [40], [-18], [-32], [12], [-20], [-26], [-16], [-12], [-12], [-12], [12], [-32], [-28], [-8], [-2], [14], [-8], [-6], [-24], [12], [-36], [4], [48], [34], [0], [-4], [-46], [20], [-36], [4], [16], [-40], [12], [16], [0], [0], [8], [16], [44], [-24], [-18], [44], [12], [-38], [28], [8], [-12], [50], [-30], [-44], [30], [-48], [32], [-8], [16], [-36], [44], [20], [-18], [4], [60], [36], [-18], [36], [-44], [28], [40], [-26], [24], [16], [24], [20], [36], [-48], [46], [-2], [-52], [-26], [28], [36], [-16], [28], [34], [20], [10], [56], [-16], [-64], [68], [-40], [-12], [-20], [-32], [-8], [-26], [-4], [8], [52], [-26], [38], [-40], [72], [-60], [16], [-10], [-66], [16], [-28], [36], [44], [12], [24], [32], [56], [-60], [62], [66], [52], [-72], [-16], [52], [20], [50], [-54], [56], [-68], [-12], [60], [28], [36], [48], [-36], [-26], [38], [12], [-22], [8], [62], [24], [28], [8], [-26], [76], [54], [36], [20], [-64], [-24], [30], [46], [-16], [-64], [56], [0], [76], [-16], [66], [6], [76], [-52], [32], [-44], [-26], [64], [24], [-8], [30], [56], [-32], [-44], [16], [34], [-12], [60], [-84], [16], [18], [-44], [16], [-56], [70], [16], [72], [4], [-72], [-24], [46], [24], [68], [12], [-84], [40], [-16], [-44], [42], [-12], [40], [0], [-64], [44], [8], [-4], [-12], [2], [2], [-52], [-48], [28], [-14], [-50], [-10], [8], [-56], [-84], [88], [-2], [-38], [-6], [60], [4], [82], [-44], [-64], [-42], [84], [70], [0], [-24], [-94], [-28], [40], [-36], [12], [-22], [-94], [-64], [4], [-26], [68], [20], [0], [44], [36], [-88], [32], [-42], [92], [-10], [-12], [-28], [-60], [68], [80], [-28], [100], [48], [-64], [-56], [-88], [-36], [-2], [-52], [58], [76], [36], [0], [-44], [32], [-12], [-18], [-66], [20], [8], [-32], [40], [80], [-72], [-60], [4], [30], [-74], [-84], [-82], [-60], [88], [68], [-40], [-56], [92], [64], [18], [0], [-68], [64], [4], [48], [-4], [-26], [68], [44], [20], [12], [-84], [24], [90], [-56], [80], [-92], [-32]];
    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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![-4, 1]>,<13,R![4, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;