
// Make newform 3570.2.a.t in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3570_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_3570_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_3570_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_3570_a();"
function MakeCharacter_3570_a()
    N := 3570;
    order := 1;
    char_gens := [2381, 2857, 2551, 1261];
    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_3570_a_Hecke(Kf)
    return MakeCharacter_3570_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], [-1], [4], [6], [1], [4], [0], [-2], [-8], [-10], [2], [4], [0], [6], [-12], [-2], [12], [-8], [2], [8], [4], [10], [10], [-2], [-8], [4], [14], [18], [16], [12], [10], [-12], [-2], [-8], [-10], [-20], [-16], [-18], [-4], [-10], [0], [26], [6], [0], [-20], [0], [-4], [-10], [-6], [0], [2], [4], [2], [-24], [-18], [-16], [22], [-22], [4], [-10], [-4], [-8], [2], [-2], [-20], [26], [36], [30], [-30], [24], [-16], [14], [20], [16], [14], [-2], [-22], [26], [28], [6], [-32], [34], [0], [-36], [-6], [26], [-10], [16], [-12], [16], [-8], [-12], [-20], [-32], [-42], [18], [4], [-2], [44], [-2], [4], [-6], [4], [-14], [-4], [-14], [24], [26], [-32], [30], [-38], [-44], [40], [-38], [12], [-24], [-18], [-20], [-10], [26], [-26], [-44], [-36], [22], [6], [0], [8], [38], [-28], [48], [-40], [-34], [-6], [2], [6], [-4], [-18], [-46], [52], [-26], [-40], [-44], [-2], [-8], [22], [-6], [-4], [-32], [46], [-38], [-20], [-48], [20], [16], [8], [26], [-38], [30], [12], [58], [40], [20], [18], [32], [-40], [-42], [-14], [-42], [-60], [-50], [24], [26], [-16], [-14], [-28], [-10], [56], [62], [-32], [-20], [22], [-22], [0], [-18], [-18], [-4], [-38], [-48], [10], [20], [-12], [38], [44], [-6], [34], [14], [-14], [-24], [30], [-8], [62], [-30], [-12], [-18], [32], [36], [34], [-4], [-38], [-34], [-8], [-52], [24], [10], [-16], [34], [64], [-18], [70], [48], [-30], [16], [-12], [22], [-54], [0], [-24], [68], [-10], [44], [-16], [-54], [20], [48], [-14], [54], [-12], [-24], [-4], [-20], [-56], [-34], [-14], [-72], [32], [12], [-52], [32], [-26], [-6], [-56], [-54], [14], [60], [-42], [20], [-58], [-22], [-64], [52], [-26], [-66], [18], [-60], [54], [-38], [4], [-10], [-18], [-52], [10], [0], [66], [8], [-52], [78], [42], [-68], [0], [-48], [0], [54], [52], [-48], [10], [22], [-40], [34], [-50], [28], [-6], [-28], [-66], [30], [0], [70], [-60], [44], [-22], [-66], [-24], [60], [68], [10], [12], [46], [40], [-82], [72], [-74], [42], [44], [-88], [10], [76], [16], [42], [50], [76], [-62], [-10], [32], [-70], [-14], [-4], [84], [-72], [38], [14], [30], [-8], [-44], [52], [36], [-34], [66], [-38], [64], [78], [58], [-10], [-72], [-18], [-4], [-90], [-60], [16], [-10], [4], [66], [54], [64], [70], [26], [64], [-60], [18], [-72], [86], [-30], [-16], [-44], [-68], [-38], [-50], [40], [-22], [12], [-12], [56], [94], [24], [-2], [12], [-16], [-94], [50], [74], [-66], [-54], [8], [82], [44], [32], [16], [14], [-60], [96], [18], [-26], [36], [12], [88], [-94], [-48], [42], [-12], [-66], [-2], [-62], [-48], [42], [14], [80], [54], [-14], [44], [-52], [-6], [54], [-4], [-36], [-86], [30], [-80], [40], [2], [-24], [54], [38], [104], [-76], [18], [62], [60], [-62], [-52], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3570_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3570_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_3570_2_a_t(:prec:=1)
    chi := MakeCharacter_3570_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_3570_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_3570_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3570_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, 1]>,<13,R![-6, 1]>,<19,R![-4, 1]>,<37,R![10, 1]>],Snew);
    return Vf;
end function;