
// Make newform 1290.2.a.j in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1290_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_1290_2_a_j();".
// 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_1290_2_a_j();".
// 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_1290_a();"
function MakeCharacter_1290_a()
    N := 1290;
    order := 1;
    char_gens := [431, 517, 691];
    v := [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_1290_a_Hecke(Kf)
    return MakeCharacter_1290_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], [-4], [-4], [4], [4], [4], [8], [6], [-4], [2], [10], [-1], [4], [-2], [-12], [0], [8], [-12], [-14], [8], [6], [-4], [2], [6], [6], [2], [-10], [18], [18], [18], [22], [-4], [-14], [-22], [18], [-20], [8], [18], [18], [-6], [16], [-26], [-2], [10], [-8], [16], [8], [-2], [-22], [0], [-14], [0], [18], [-32], [-6], [16], [-18], [22], [28], [-10], [-28], [-24], [-14], [-2], [-32], [-2], [4], [-12], [24], [8], [-2], [14], [-12], [32], [18], [-4], [-18], [10], [-18], [8], [24], [14], [28], [-18], [8], [-10], [-10], [4], [-12], [-8], [18], [-26], [-4], [16], [42], [0], [-44], [10], [0], [-30], [-14], [-6], [-12], [22], [4], [-14], [-48], [-6], [-40], [16], [-28], [20], [-2], [0], [-12], [32], [6], [-40], [26], [-34], [6], [2], [8], [2], [34], [0], [44], [30], [0], [24], [-2], [2], [-20], [30], [-26], [16], [-14], [10], [-20], [-42], [-22], [-34], [-28], [56], [-44], [48], [-40], [-16], [52], [18], [-16], [-48], [-4], [-36], [-20], [-28], [38], [50], [18], [-6], [54], [12], [8], [-16], [2], [-38], [2], [-22], [6], [-20], [52], [-34], [14], [6], [20], [2], [-6], [-60], [52], [-32], [-50], [38], [-16], [-30], [-50], [-4], [46], [24], [6], [52], [-20], [-14], [-52], [-66], [-10], [-40], [28], [64], [18], [-2], [-62], [-10], [10], [-14], [-30], [-50], [-60], [20], [-2], [-18], [-26], [-46], [-16], [-26], [-40], [-40], [-48], [-6], [-44], [32], [56], [50], [72], [50], [-20], [-28], [36], [42], [-30], [52], [-40], [-16], [-60], [12], [26], [-38], [-22], [0], [-72], [32], [62], [-42], [-74], [40], [56], [-60], [20], [4], [28], [70], [8], [38], [14], [18], [52], [-48], [-30], [30], [-16], [-16], [2], [48], [26], [-12], [-62], [-30], [76], [-78], [2], [52], [-26], [82], [14], [-8], [54], [52], [-2], [38], [8], [-20], [-12], [44], [28], [-12], [10], [58], [46], [-78], [38], [-82], [40], [6], [44], [2], [-28], [-26], [28], [-88], [-18], [30], [-44], [6], [16], [-82], [-38], [-52], [80], [-58], [0], [70], [-78], [76], [44], [-10], [-36], [32], [-34], [84], [20], [2], [46], [82], [42], [-58], [-56], [56], [72], [38], [-40], [26], [-2], [-32], [-20], [-78], [-44], [-26], [86], [56], [80], [-14], [-58], [-58], [-50], [-68], [-52], [8], [-56], [78], [-28], [10], [18], [-56], [42], [-34], [-44], [-10], [36], [16], [58], [-72], [-32], [90], [-20], [26], [-30], [-26], [-58], [-26], [-8], [32], [78], [12], [14], [-74], [8], [98], [64], [-26], [46], [-24], [-18], [-22], [20], [-52], [82], [92], [64], [20], [-50], [78], [30], [-80], [32], [-54], [-84], [0], [24], [-62], [34], [-60], [74], [-16], [6], [-42], [-62], [18], [20], [-72], [-22], [-54], [-24], [-100], [-22], [-14], [16], [58], [-24], [40], [66], [-80], [48], [84], [-26], [-50], [64], [-36], [12], [40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1290_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1290_2_a_j();".
// 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_1290_2_a_j(:prec:=1)
    chi := MakeCharacter_1290_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_1290_2_a_j();".
// 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_1290_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1290_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;