
// Make newform 490.3.f.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_490_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_490_f_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_490_3_f_b();".
// 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_490_3_f_b();".
// 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
        poly := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_490_f();"
function MakeCharacter_490_f()
    N := 490;
    order := 4;
    char_gens := [197, 101];
    v := [1, 4];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_490_f_Hecke();"
function MakeCharacter_490_f_Hecke(Kf)
    N := 490;
    order := 4;
    char_gens := [197, 101];
    char_values := [[0, 1], [1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 3;
    raw_aps := [[-1, -1], [2, -2], [0, 5], [0, 0], [-8, 0], [-3, 3], [-7, -7], [0, 20], [-2, 2], [0, 40], [-52, 0], [-3, -3], [8, 0], [-42, 42], [18, 18], [53, -53], [0, 20], [48, 0], [62, 62], [-28, 0], [47, -47], [0, 0], [-18, 18], [0, -80], [63, 63], [-62, 0], [-118, 118], [142, 142], [0, -10], [23, -23], [-118, -118], [128, 0], [-63, -63], [0, -140], [0, -150], [52, 0], [-27, -27], [-82, 82], [-62, -62], [107, -107], [0, -220], [-2, 0], [212, 0], [-57, 57], [-3, -3], [0, 120], [-328, 0], [-138, 138], [-2, -2], [0, -120], [183, -183], [0, 120], [-232, 0], [48, 0], [313, 313], [-262, 262], [0, -10], [-252, 0], [267, 267], [312, 0], [262, -262], [-243, 243], [18, 18], [388, 0], [-183, 183], [-213, -213], [232, 0], [417, 417], [202, 202], [0, 440], [447, -447], [0, -400], [118, 118], [-107, 107], [0, 340], [342, -342], [0, 390], [323, 323], [642, 0], [0, 150], [0, -300], [-208, 0], [-788, 0], [367, -367], [0, -560], [378, -378], [0, -410], [-393, -393], [-622, 0], [278, -278], [38, 38], [0, 440], [522, 522], [-328, 0], [0, 380], [42, -42], [0, 440], [258, 0], [-258, 258], [-338, 0], [-558, -558], [-3, -3], [42, -42], [0, 950], [392, 0], [473, 473], [198, 198], [47, -47], [0, 520], [328, 0], [-462, -462], [723, -723], [327, 327], [0, 660], [-548, 0], [-568, 0], [342, -342], [118, 118], [453, -453], [0, 140], [-512, 0], [193, -193], [-157, -157], [438, -438], [-1032, 0], [-128, 0], [0, 760], [0, 1160], [558, 558], [827, -827], [0, -700], [-382, 382], [-588, 0], [987, 987], [158, 0], [0, -80], [-243, 243], [-262, -262], [-267, -267], [0, 560], [208, 0], [-1568, 0], [-562, 562], [762, 762], [0, -170], [0, 280], [-1123, 1123], [-417, -417], [0, 1300], [-242, 242], [-453, -453], [-712, 0], [118, -118], [1158, 1158], [142, 142], [1172, 0], [0, 920], [0, 1190], [233, 233], [78, 0], [62, 62], [-1017, 1017], [502, 502], [-992, 0], [-783, -783], [-1058, 1058], [-68, 0], [773, 773]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_490_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_490_3_f_b();".
// 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_490_3_f_b(:prec:=2)
    chi := MakeCharacter_490_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_490_3_f_b();".
// 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_490_3_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_490_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![8, -4, 1]>,<11,R![8, 1]>],Snew);
    return Vf;
end function;