
// Make newform 4290.2.a.y in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4290_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_4290_2_a_y();".
// 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_4290_2_a_y();".
// 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_4290_a();"
function MakeCharacter_4290_a()
    N := 4290;
    order := 1;
    char_gens := [2861, 1717, 3511, 2641];
    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_4290_a_Hecke(Kf)
    return MakeCharacter_4290_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], [0], [-1], [1], [6], [4], [0], [-6], [4], [-2], [2], [-8], [8], [6], [12], [6], [-8], [-8], [-6], [4], [12], [-2], [10], [18], [16], [-12], [14], [2], [-4], [20], [14], [8], [-10], [-8], [-2], [-16], [0], [-6], [12], [-10], [-16], [2], [6], [0], [-8], [-20], [4], [6], [14], [-16], [-30], [-4], [-6], [-32], [6], [-32], [-10], [-14], [32], [-26], [-12], [16], [-22], [-14], [8], [10], [28], [-18], [14], [-8], [0], [14], [-16], [-32], [14], [22], [14], [-22], [12], [-10], [-40], [-14], [4], [-4], [-10], [10], [-42], [-36], [-4], [16], [-20], [12], [-24], [0], [-6], [-6], [-16], [30], [40], [14], [20], [-10], [32], [2], [12], [18], [40], [-6], [-4], [-42], [30], [0], [-28], [18], [-48], [-8], [-26], [-20], [14], [2], [42], [-4], [-16], [-22], [22], [16], [48], [30], [28], [16], [-32], [-42], [26], [18], [-6], [28], [-26], [6], [-4], [30], [16], [-20], [-34], [-32], [6], [6], [-28], [-32], [14], [50], [12], [0], [44], [0], [20], [38], [-22], [6], [12], [54], [16], [-60], [-18], [0], [-40], [14], [34], [46], [-60], [-10], [-32], [-54], [8], [-14], [-28], [-38], [-36], [-58], [-44], [-44], [-58], [-14], [-56], [22], [46], [-8], [26], [40], [-46], [20], [-20], [42], [36], [50], [-62], [30], [42], [0], [54], [-68], [38], [42], [20], [6], [-4], [68], [18], [-44], [-30], [6], [-8], [-60], [-24], [42], [-52], [-18], [24], [2], [-10], [32], [-18], [-20], [28], [62], [2], [-48], [28], [-60], [-18], [-48], [0], [-2], [20], [24], [-22], [6], [-12], [-48], [4], [-32], [-8], [-10], [50], [-64], [52], [-36], [4], [72], [62], [-6], [-56], [58], [2], [-12], [-10], [28], [30], [42], [36], [-36], [-58], [-26], [-42], [36], [58], [14], [12], [82], [-50], [48], [-14], [28], [42], [36], [28], [-18], [10], [44], [-32], [-4], [48], [38], [4], [0], [-62], [-82], [36], [-42], [-58], [76], [26], [-68], [22], [58], [-8], [-10], [-20], [4], [-22], [54], [-12], [28], [-60], [-54], [-28], [46], [-48], [30], [24], [58], [-26], [-44], [-48], [50], [-20], [-16], [26], [54], [88], [58], [-54], [-36], [-54], [-62], [72], [-16], [8], [-46], [-50], [30], [4], [4], [-60], [-36], [-10], [42], [2], [52], [-18], [34], [70], [-16], [-54], [-20], [38], [8], [-80], [6], [60], [18], [66], [-4], [46], [70], [-56], [60], [-2], [-64], [38], [54], [48], [-84], [-28], [26], [-42], [56], [74], [-12], [-44], [48], [-38], [68], [-34], [-100], [-88], [98], [6], [98], [10], [54], [48], [-38], [-44], [24], [-100], [-10], [20], [-72], [-70], [74], [12], [4], [-8], [82], [32], [-54], [100], [-42], [-34], [-70], [88], [78], [-98], [-68], [-18], [50], [16], [100], [50], [-62], [44], [-32], [18], [54], [-16], [-48], [-42], [72], [-90], [-42], [-48], [84], [42], [18], [84], [90], [-56], [-72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4290_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4290_2_a_y();".
// 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_4290_2_a_y(:prec:=1)
    chi := MakeCharacter_4290_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_4290_2_a_y();".
// 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_4290_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4290_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<17,R![-6, 1]>,<19,R![-4, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;