
// Make newform 475.2.e.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_475_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_475_e_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_475_2_e_f();".
// 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_475_2_e_f();".
// 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, 33, 1107, -552, 1009, -147, 484, -93, 109, -18, 17, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-68499350729, -726241735, -24355803798, -38817354653, -15342668397, -4177401796, -3678052623, -801763109, -621660414, 38785859, -46922925, -3959208], [-6178545, -1547540653, -6717081048, 7520604435, -5508756455, 2957264154, -1701731499, 833713615, -313859676, 76199247, -11701549, 690858], [-263970961763, -1147600790, -38482449021, 12276068039, -19479845384, -11747270177, 1416907499, -3244495858, 251450977, -731712017, 153015320, -54841341], [-148724948799, -4114960306, -138016601091, -204345695921, -102294574896, -7077581935, -27189880401, 1833052570, -7500316929, 3072233475, -998274384, 134211789], [-4335298296, -159117251, -5335902045, -2448832100, -2951210507, -1358444363, -412414770, -345971017, -29949491, -64765518, 9282581, -5051595], [-612082342, -122568267, -4110047595, -725395450, -2049018039, -1288684281, -259108930, -359601729, 35014563, -93333716, 17844777, -6178545], [402309009, -680489253663, 432662257370, -647754164847, 212867920667, -294644252580, 86418671699, -67720087321, 16279393858, -9648336987, 1742271771, -469619124], [-65422139102, -2194621698825, 1145299361381, -2007194188104, 298838370979, -966498199653, 187219927286, -217868445987, 36042149143, -34029515688, 6017448375, -2004496389], [2372026047, -4859631701974, 2534882432535, -4359372249951, 755327449736, -2086126950865, 426177732417, -469415474758, 81775705089, -72904583371, 12953281828, -4220156417], [-3616495515, 7195985653273, -3864903739992, 6642795442545, -1144205547385, 3178214179566, -650339598681, 717279166315, -124776050724, 111043957893, -19688524501, 6424307742]];
    Rf_basisdens := [1, 1, 66494854340, 1955731010, 66494854340, 66494854340, 1955731010, 1955731010, 66494854340, 66494854340, 66494854340, 66494854340];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_475_e();"
function MakeCharacter_475_e()
    N := 475;
    order := 3;
    char_gens := [77, 401];
    v := [3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_475_e_Hecke();"
function MakeCharacter_475_e_Hecke(Kf)
    N := 475;
    order := 3;
    char_gens := [77, 401];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0], [1, -1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 1, 1, -1, 0, 0, 0, 0], [-1, 1, 0, 1, 0, 0, 0, 0, -1, -1, -1, 0], [-1, 1, -1, 1, 0, 1, -1, 0, 0, -1, 0, 0], [-1, 0, 0, 0, -1, -1, 0, -1, 1, 1, -1, 1], [1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 3, 0], [0, 1, 0, 1, 0, 0, 0, 0, -1, 0, -1, -2], [1, 0, 1, 0, 1, -1, 1, 2, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0], [-2, 1, -1, 0, -1, 0, -2, 2, 3, 0, -2, 1], [2, -1, 1, 0, 1, 0, 0, 1, 0, 3, 0, -1], [2, 0, 0, 1, 0, 0, 0, 0, -1, 2, -2, -2], [-1, -4, 0, -1, 0, 0, 0, 0, 1, -1, 2, -1], [4, -2, 2, -1, 2, -1, 3, -3, -5, 4, 2, -2], [-5, 1, 0, -2, 0, 0, 0, 0, 0, -5, 1, 1], [1, -2, 0, 0, 0, 0, 0, 0, -3, 1, 5, 0], [-5, 4, -4, 2, -1, 2, -1, -2, -1, 1, 1, 1], [-1, 1, -1, -1, 0, -1, 3, -4, -4, 1, 2, 0], [-2, 1, -1, 0, -1, 0, 3, 4, 5, -2, 3, 1], [-5, 0, -1, 0, 0, -3, 4, -2, 0, 0, 0, 0], [-4, 2, 0, 1, 0, 0, 0, 0, 4, -4, 3, -1], [-2, 3, -3, -2, 1, -2, -4, 2, 1, -1, -6, -1], [5, 2, 0, -2, 0, 0, 0, 0, -3, 5, 1, 0], [8, 0, 0, 0, -1, -1, -2, 1, 0, 0, 0, 0], [1, 0, -1, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, -1, -8, 1, 0], [2, 0, -3, 0, -1, -2, 2, -4, 0, 0, 0, 0], [-6, 1, 0, 2, 0, 0, 0, 0, 5, -6, -1, -1], [2, -3, 3, 1, -1, 1, 0, -3, -2, 4, 1, 1], [3, -2, 0, -2, 0, 0, 0, 0, -8, 3, 2, -2], [-2, -5, 0, -2, 0, 0, 0, 0, 1, -2, -1, 1], [2, 0, 0, 2, 2, 2, -1, 5, 3, 3, 1, -2], [-4, 0, 2, 0, 0, -1, 7, 1, 0, 0, 0, 0], [0, -3, 3, 0, -3, 0, 1, -1, 2, 10, 1, 3], [2, 0, 4, 0, 2, 3, 3, -3, 0, 0, 0, 0], [-9, 1, 0, 0, 0, 0, 0, 0, 2, -9, 2, 2], [3, 0, 0, -1, 3, -1, 3, 2, -1, 9, 2, -3], [3, 0, -6, 0, -1, -1, -3, -4, 0, 0, 0, 0], [-1, -3, 0, 0, 0, 0, 0, 0, 1, -1, 6, -1], [-1, 0, -3, 0, 4, 2, -7, -1, 0, 0, 0, 0], [-1, 0, 0, -1, -1, -1, -6, -5, -4, 1, -7, 1], [-2, 0, 5, 0, -2, 1, 2, 1, 0, 0, 0, 0], [-6, 4, 0, -3, 0, 0, 0, 0, -7, -6, 2, 1], [3, -5, 5, 0, -2, 0, -4, 0, 2, -5, -4, 2], [-4, 3, -3, 5, -1, 5, -5, 2, 3, -9, 0, 1], [7, 0, 6, 0, 2, -3, 4, 6, 0, 0, 0, 0], [0, 0, 2, 0, 1, -3, -1, -2, 0, 0, 0, 0], [-2, 0, 0, 2, -2, 2, 1, -5, -3, 7, 3, 2], [-2, 0, 0, 0, -3, 0, -1, 0, 0, 0, 0, 0], [-2, 5, 0, 4, 0, 0, 0, 0, -2, -2, 5, 1], [-2, -1, 0, 1, 0, 0, 0, 0, 6, -2, -7, 3], [-3, -1, 0, -3, 0, 0, 0, 0, 0, -3, -4, 2], [2, 2, -2, 1, 4, 1, -4, -4, -8, 9, -3, -4], [-4, 5, -5, 2, 1, 2, 0, 6, 5, -9, 2, -1], [7, -3, 3, 2, 4, 2, 5, 1, -3, 0, 7, -4], [-5, 0, -4, 0, 1, 5, -7, 0, 0, 0, 0, 0], [4, -4, 0, -4, 0, 0, 0, 0, 3, 4, -3, -2], [-3, -1, 1, -4, -4, -4, 6, 2, 6, -11, 2, 4], [20, 0, 5, 0, 0, 1, 2, 3, 0, 0, 0, 0], [-1, 4, -4, 4, 3, 4, -1, 6, 3, -3, 3, -3], [15, 0, 6, 0, 0, 1, -2, 8, 0, 0, 0, 0], [13, -6, 0, -5, 0, 0, 0, 0, -5, 13, 0, -3], [-12, -2, 0, -3, 0, 0, 0, 0, 1, -12, 0, 5], [-5, 0, 0, 0, -3, -5, 3, -12, 0, 0, 0, 0], [-6, 4, -4, 0, -2, 0, -3, -9, -7, -2, -3, 2], [1, 3, -3, -2, 4, -2, 2, 2, -2, 5, 0, -4], [-8, 0, 0, 0, 1, 7, 3, 0, 0, 0, 0, 0], [-3, 0, -2, 0, -2, -2, -12, 10, 0, 0, 0, 0], [4, -1, 1, -2, 3, -2, 0, 5, 2, -7, -2, -3], [-7, 1, 0, 4, 0, 0, 0, 0, -2, -7, -4, -2], [11, 0, 7, 0, 2, 1, 1, 14, 0, 0, 0, 0], [10, 0, 10, 0, -3, 3, 0, -1, 0, 0, 0, 0], [5, -8, 8, 0, -3, 0, 14, -9, -6, 14, 14, 3], [-4, 5, 0, 3, 0, 0, 0, 0, 2, -4, -8, 2], [1, 2, -2, -3, 3, -3, -3, 0, -3, -1, -6, -3], [-5, 2, -2, -3, -3, -3, 8, -5, -2, -9, 5, 3], [6, 2, 0, 3, 0, 0, 0, 0, -3, 6, 0, -5], [9, 0, 2, 0, 4, 0, -3, 9, 0, 0, 0, 0], [2, -2, 2, 0, 0, 0, 3, -13, -13, 3, 3, 0], [5, -5, 0, -3, 0, 0, 0, 0, 1, 5, 6, -6], [-10, 0, 0, 4, 0, 0, 0, 0, -5, -10, 5, 2], [1, 2, -2, 4, 3, 4, -12, 1, -2, 18, -8, -3], [-12, 2, 0, -1, 0, 0, 0, 0, 9, -12, -10, 1], [11, 0, 0, 0, 1, 3, 4, 1, 0, 0, 0, 0], [-3, 0, 4, 0, -5, 1, 5, -4, 0, 0, 0, 0], [-13, 9, -9, -5, -4, -5, -6, -3, 1, -2, -11, 4], [-16, 0, -3, 0, 1, -1, 0, 3, 0, 0, 0, 0], [-4, 0, -6, 0, -2, 4, -18, 0, 0, 0, 0, 0], [6, 2, 0, 0, 0, 0, 0, 0, -4, 6, 10, 1], [16, 0, 8, 0, 6, -3, 7, -3, 0, 0, 0, 0], [9, -6, 6, -2, 3, -2, 6, -5, -8, 0, 4, -3], [-5, 2, -2, -2, -3, -2, -7, 6, 9, -5, -9, 3], [5, -2, 0, -1, 0, 0, 0, 0, 0, 5, -3, 4], [3, -2, 0, 3, 0, 0, 0, 0, 0, 3, 13, 3], [0, 0, -6, 0, 8, 0, -6, 8, 0, 0, 0, 0], [2, 0, 0, 4, 0, 0, 0, 0, 2, 2, 6, 1], [4, -7, 0, -3, 0, 0, 0, 0, -15, 4, 5, -6], [5, -2, 0, -1, 0, 0, 0, 0, 6, 5, -7, -5], [8, -3, 0, 2, 0, 0, 0, 0, -2, 8, -3, 1], [4, 0, -9, 0, -4, 0, 2, -2, 0, 0, 0, 0], [2, 0, 6, 0, 5, 7, 4, -11, 0, 0, 0, 0], [5, 0, 4, 0, -1, -1, 9, -10, 0, 0, 0, 0], [7, 0, 0, 0, 4, -4, 5, -7, 0, 0, 0, 0], [1, -6, 6, 5, -5, 5, -7, 0, 5, -8, -2, 5], [4, 4, 0, -2, 0, 0, 0, 0, -15, 4, 7, -2], [16, -5, 0, 1, 0, 0, 0, 0, -8, 16, 9, 1], [8, 0, 6, 0, 0, -2, 6, 12, 0, 0, 0, 0], [-10, 0, -1, 0, 1, 2, -11, -6, 0, 0, 0, 0], [0, 4, -4, 2, 4, 2, 1, -1, -5, 6, 3, -4], [10, 4, 0, -5, 0, 0, 0, 0, 8, 10, -3, -3], [-7, 0, 1, 0, -7, 3, 7, -8, 0, 0, 0, 0], [-8, -1, 0, 6, 0, 0, 0, 0, 7, -8, 1, 1], [0, 1, -1, 6, 1, 6, -1, -5, -6, 15, 5, -1], [6, -8, 8, 2, -2, 2, 5, -3, -1, 3, 7, 2], [-4, 0, -7, 0, 1, 7, -2, -1, 0, 0, 0, 0], [18, 0, 2, 0, 9, 7, 1, 10, 0, 0, 0, 0], [11, 2, 0, 7, 0, 0, 0, 0, -10, 11, -5, -8], [11, 1, 0, -1, 0, 0, 0, 0, -9, 11, 3, 6], [9, 0, 7, 0, -3, -2, -4, -8, 0, 0, 0, 0], [-9, 0, 6, 0, -2, 2, 8, 0, 0, 0, 0, 0], [-3, 0, -4, 0, -8, 1, 2, -12, 0, 0, 0, 0], [13, 0, 15, 0, 0, -1, 10, 6, 0, 0, 0, 0], [4, -1, 1, -8, 3, -8, 3, -5, -8, -11, -5, -3], [2, -6, 0, 1, 0, 0, 0, 0, 7, 2, 10, -1], [-2, 0, 0, -3, -2, -3, -5, -7, -5, 0, -8, 2], [-7, 10, -10, 2, 3, 2, -6, 5, 2, 8, -4, -3], [0, 0, -2, 0, -5, 3, 6, -17, 0, 0, 0, 0], [0, -2, 2, -3, -2, -3, 11, -9, -7, 4, 8, 2], [-1, -6, 6, -2, -7, -2, 9, -2, 5, 15, 7, 7], [-9, 2, 0, -5, 0, 0, 0, 0, 6, -9, 1, 3], [14, -10, 10, 4, 4, 4, 5, 1, -3, -8, 9, -4], [3, 0, 8, 0, -4, 0, 11, 7, 0, 0, 0, 0], [1, -6, 0, 6, 0, 0, 0, 0, -9, 1, 17, 2], [-1, 2, 0, 6, 0, 0, 0, 0, 14, -1, -2, 4], [8, 0, 0, 0, -7, -4, 7, 16, 0, 0, 0, 0], [4, 0, 1, 0, 7, 0, 0, 10, 0, 0, 0, 0], [-1, 0, -5, 0, 3, -2, -7, 11, 0, 0, 0, 0], [14, -11, 0, 0, 0, 0, 0, 0, -1, 14, 3, 2], [-4, 6, 0, 2, 0, 0, 0, 0, 6, -4, 0, -10], [-2, 2, 0, 3, 0, 0, 0, 0, -7, -2, -2, -7], [-9, -2, 0, -3, 0, 0, 0, 0, -8, -9, 3, -4], [-12, 0, -5, 0, -2, -1, -5, 4, 0, 0, 0, 0], [-4, 3, -3, -5, -1, -5, 9, -10, -9, -11, 4, 1], [5, 3, -3, 1, 8, 1, 8, 11, 3, 5, 9, -8], [4, -6, 6, -4, -2, -4, -5, 1, 3, 4, -9, 2], [-17, -3, 0, -3, 0, 0, 0, 0, 21, -17, -2, 1], [10, 0, 10, 0, 2, -6, 8, 4, 0, 0, 0, 0], [-12, 14, -14, 0, 2, 0, -8, 0, -2, -9, -8, -2], [-9, 0, -9, 0, -1, 6, -24, -2, 0, 0, 0, 0], [4, -2, 0, 3, 0, 0, 0, 0, -15, 4, -2, -1], [-30, 5, 0, -4, 0, 0, 0, 0, -5, -30, -9, -1], [0, 1, -1, -7, 1, -7, 17, 0, -1, 3, 10, -1], [-6, 0, 0, 0, -3, 1, -18, -5, 0, 0, 0, 0], [9, 0, 1, 0, 8, -8, -1, 7, 0, 0, 0, 0], [7, -1, 1, -1, 6, -1, 3, -2, -8, 25, 2, -6], [4, -17, 0, 5, 0, 0, 0, 0, -6, 4, 16, -4], [-14, -6, 0, 1, 0, 0, 0, 0, -8, -14, 11, -1], [-6, -4, 4, -3, -10, -3, 11, -17, -7, 10, 8, 10], [-9, 9, -9, 5, 0, 5, 7, -6, -6, -4, 12, 0], [5, -10, 10, -3, -5, -3, -1, -18, -13, 12, -4, 5], [9, -6, 6, -2, 3, -2, -10, 13, 10, 10, -12, -3], [2, 0, -9, 0, -4, 3, 1, -12, 0, 0, 0, 0], [0, 0, 0, -4, 0, -4, -11, 9, 9, 25, -15, 0], [-7, 9, -9, 1, 2, 1, -11, -13, -15, -9, -10, -2], [5, 0, 0, 2, 0, 0, 0, 0, 14, 5, 2, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_475_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_475_2_e_f();".
// 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_475_2_e_f(:prec:=12)
    chi := MakeCharacter_475_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_475_2_e_f();".
// 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_475_2_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_475_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 3, 10, -1, 4, 1, 1]>],Snew);
    return Vf;
end function;