
// Make newform 950.2.e.o in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_950_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_950_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_950_2_e_o();".
// 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_950_2_e_o();".
// 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 := [100, -150, 285, -110, 186, -70, 85, -12, 10, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [154050, -58200, 106700, -163092, 61110, -22892, 5464, -1940, 582, -47], [-740030, 88800, -162800, -61562, -93240, 34928, -13433, 2960, -888, 535], [-919600, 496200, -909700, 855387, -521010, 195172, -51681, 16540, -4962, 864], [-103640, -664685, 12726, -450846, 169886, -250957, 33092, -29646, -94, -3081], [-866100, 1605370, -846025, 959337, -359938, 250962, -41134, 23553, -2572, 1752], [1304860, -780900, 1431650, -530181, 819945, -307154, 128908, -26030, 7809, -3642], [202580, 957675, 19353, 800162, -185422, 444094, -47287, 54392, 1658, 5580], [318760, 1125350, 288666, 785664, 323839, 427658, 27158, 67471, 11216, 6198]];
    Rf_basisdens := [1, 1, 224695, 224695, 224695, 449390, 224695, 224695, 224695, 224695];
    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_950_e();"
function MakeCharacter_950_e()
    N := 950;
    order := 3;
    char_gens := [77, 401];
    v := [3, 2];
    // 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_950_e_Hecke();"
function MakeCharacter_950_e_Hecke(Kf)
    N := 950;
    order := 3;
    char_gens := [77, 401];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, -1, 0, 0, 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, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 1, -1, -1], [0, 0, 0, 0, -1, 0, 1, 0, 0, -1], [2, 0, -1, 1, 0, 1, 0, -1, 1, 0], [-3, -1, 1, -1, 0, -3, 0, 1, -1, -1], [0, 0, 0, -2, 0, 0, 0, 1, -2, -1], [0, 0, 0, 1, -1, 0, 0, -1, 0, 0], [-1, 0, -1, 0, -1, 0, 0, 1, 0, 0], [0, 0, 0, 0, -1, -1, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0], [2, -2, 2, -1, 0, 2, -1, 1, -1, -1], [-1, 1, -1, -1, 0, -1, -1, 1, -1, -1], [0, -1, 0, 0, 0, -4, 0, 0, 1, 1], [0, 4, -4, 0, 0, 0, 0, -1, 0, 1], [-2, -1, 1, 1, 0, -2, 1, 2, 1, -2], [0, -2, 0, 0, 0, 4, 0, 0, -1, -1], [0, -1, 0, 0, -1, 4, 1, 0, 0, -2], [0, 4, 0, 0, 0, 0, 0, 0, -1, 1], [-2, 0, -1, 0, 0, 0, 0, 2, 0, 0], [-1, -1, 1, 1, 0, -1, 0, 2, 1, -2], [0, 5, 0, 0, 0, 2, 0, 0, 0, 1], [4, -2, 2, 1, 0, 4, 0, 0, 1, 0], [-1, 0, 3, -1, 1, 0, 0, 1, 0, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 4, 0, 0, 2, 2], [0, 0, -3, 2, 0, 0, 0, -3, 0, 0], [-3, -1, 1, 3, 0, -3, -1, -3, 3, 3], [0, 2, 0, 0, 0, 5, 0, 0, 3, 2], [-6, 1, -1, 2, 0, -6, 0, -1, 2, 1], [4, -3, 3, -1, 0, 4, 2, -1, -1, 1], [0, -4, 0, 0, -2, -6, 2, 0, 2, -1], [2, 0, -2, 5, 1, 0, 0, -1, 0, 0], [0, 3, 0, 0, 1, 7, -1, 0, 1, 3], [6, 0, 1, 3, -1, 0, 0, 0, 0, 0], [3, 3, -3, -3, 0, 3, -1, -1, -3, 1], [0, 3, 0, 0, -1, -3, 1, 0, 1, -1], [-1, 0, 4, 1, 0, 0, 0, 0, 0, 0], [-8, -2, 2, 2, 0, -8, 0, -1, 2, 1], [0, 0, 0, -4, 0, 0, 0, 2, 0, 0], [0, 6, 0, 0, 1, 10, -1, 0, 0, 3], [3, 0, 1, 1, 1, 0, 0, -4, 0, 0], [4, -6, 6, -2, 0, 4, 0, 0, -2, 0], [0, -1, 0, 0, 0, -11, 0, 0, -2, -1], [0, -1, 0, 0, 0, -7, 0, 0, 0, 0], [-2, 0, -3, 0, 2, 0, 0, 2, 0, 0], [0, 0, 0, -2, -2, 0, 0, -2, 0, 0], [0, -3, 0, 0, -1, 0, 1, 0, -2, 2], [-4, 0, -8, 0, 0, 0, 0, 0, 0, 0], [6, -3, 3, -3, 0, 6, 2, 2, -3, -2], [-6, 5, -5, 0, 0, -6, -1, 1, 0, -1], [-10, -3, 3, 0, 0, -10, -1, 2, 0, -2], [0, 1, 0, 0, 4, 13, -4, 0, -1, -1], [0, 8, 0, 0, -2, 0, 2, 0, 2, -2], [0, 0, 0, 0, -3, -2, 3, 0, 0, 0], [10, 0, 4, 0, -2, 0, 0, -1, 0, 0], [3, 2, -2, -2, 0, 3, -1, 3, -2, -3], [0, -3, 0, 0, 0, 12, 0, 0, -2, 2], [3, 0, 3, 1, -3, 0, 0, 0, 0, 0], [0, 5, 0, 0, -1, 4, 1, 0, 3, 2], [8, 0, 4, -2, -3, 0, 0, -2, 0, 0], [0, 3, -3, 0, 0, 0, 3, 0, 0, 0], [-7, 5, -5, -2, 0, -7, 1, 2, -2, -2], [-13, 0, 4, -1, 0, 0, 0, -2, 0, 0], [0, 5, 0, 0, 2, -8, -2, 0, 2, 5], [0, -11, 0, 0, -1, -2, 1, 0, -3, -6], [10, 0, 0, -1, 2, 0, 0, 5, 0, 0], [0, 0, 5, -6, 2, 0, 0, 1, 0, 0], [0, 2, 0, 0, -1, 10, 1, 0, 4, -2], [12, 2, -2, 4, 0, 12, -1, 0, 4, 0], [-3, 0, 1, -4, -1, 0, 0, 0, 0, 0], [-6, 0, 8, 3, -1, 0, 0, -2, 0, 0], [0, 6, 0, 0, -3, 2, 3, 0, 5, -3], [8, -6, 6, 1, 0, 8, -2, 3, 1, -3], [0, -3, 0, 0, 3, 21, -3, 0, 1, 0], [0, 3, 0, 0, -1, -10, 1, 0, -1, -3], [-17, -4, 4, 4, 0, -17, 3, 1, 4, -1], [3, 0, 3, 3, 3, 0, 0, 1, 0, 0], [0, 0, 0, 0, -4, -2, 4, 0, 2, 5], [-4, -10, 10, 1, 0, -4, 0, -1, 1, 1], [0, -2, 2, 2, 0, 0, 1, -1, 2, 1], [0, 8, 0, 0, 1, 8, -1, 0, 3, 3], [6, -1, 1, 2, 0, 6, 2, 6, 2, -6], [-3, 0, 2, -8, 1, 0, 0, -1, 0, 0], [0, 0, 5, -4, 1, 0, 0, 6, 0, 0], [0, -4, 0, 0, 4, 0, -4, 0, 5, 5], [11, 0, -1, -7, -1, 0, 0, -1, 0, 0], [8, 0, -7, -5, -1, 0, 0, 4, 0, 0], [-10, 2, -2, 2, 0, -10, 0, -4, 2, 4], [11, 0, 11, -3, 2, 0, 0, 3, 0, 0], [0, -5, 0, 0, -1, -15, 1, 0, 5, -1], [0, 2, 0, 0, 0, 5, 0, 0, -1, 2], [5, 3, -3, -6, 0, 5, -2, 4, -6, -4], [-2, 0, 0, -1, 0, -2, 2, 5, -1, -5], [-2, 0, 1, -1, 1, 0, 0, 1, 0, 0], [0, 2, -2, -4, 0, 0, -4, 6, -4, -6], [8, -2, 2, -1, 0, 8, 0, 5, -1, -5], [6, -4, 4, -4, 0, 6, 2, 2, -4, -2], [-3, 5, -5, -3, 0, -3, 1, -1, -3, 1], [10, 0, -5, -2, -2, 0, 0, -4, 0, 0], [-19, 0, -1, 1, -4, 0, 0, -4, 0, 0], [6, 0, -3, 10, -1, 0, 0, -1, 0, 0], [24, 0, 7, 4, -4, 0, 0, 1, 0, 0], [0, -4, 0, 0, 0, 4, 0, 0, 6, 8], [-2, -3, 3, 0, 0, -2, -5, -2, 0, 2], [-8, -4, 4, -6, 0, -8, 1, 2, -6, -2], [-11, 0, -4, -2, -1, 0, 0, -5, 0, 0], [-5, 0, 1, 5, -2, 0, 0, -3, 0, 0], [0, -5, 0, 0, 1, -1, -1, 0, -2, -4], [20, 3, -3, -4, 0, 20, 0, -3, -4, 3], [13, 0, -9, 7, -1, 0, 0, -7, 0, 0], [-12, 2, -2, 3, 0, -12, -3, 1, 3, -1], [0, 1, 0, 0, 0, -2, 0, 0, 5, -6], [0, 1, 0, 0, -2, 14, 2, 0, 4, 8], [-9, 0, -5, 3, 3, 0, 0, -5, 0, 0], [3, 0, 9, 1, 3, 0, 0, -3, 0, 0], [-15, 1, -1, -5, 0, -15, 1, 5, -5, -5], [0, 2, -2, 7, 0, 0, -2, -4, 7, 4], [0, 0, 0, 4, 3, 0, 0, -3, 0, 0], [23, 0, -1, -7, 1, 0, 0, 3, 0, 0], [4, 0, 0, 0, 2, 0, 0, 2, 0, 0], [3, 0, -1, 3, -3, 0, 0, -1, 0, 0], [0, 8, 0, 0, -2, 6, 2, 0, -5, 0], [8, 6, -6, 0, 0, 8, -4, -1, 0, 1], [0, -4, 0, 0, 2, -26, -2, 0, -8, -6], [0, -8, 0, 0, -2, 20, 2, 0, -6, 2], [23, 0, 3, -2, 1, 0, 0, 1, 0, 0], [0, 2, 0, 0, -1, -21, 1, 0, 6, -2], [0, -5, 0, 0, -2, 5, 2, 0, -4, 2], [2, -4, 4, -8, 0, 2, -4, 0, -8, 0], [0, -1, 0, 0, -3, 23, 3, 0, -2, 0], [-7, 0, 2, 0, 2, 0, 0, -2, 0, 0], [6, -2, 2, 0, 0, 6, 3, -7, 0, 7], [-3, 9, -9, -2, 0, -3, 3, -3, -2, 3], [2, 0, -5, 8, 6, 0, 0, 4, 0, 0], [9, 0, -5, 3, -7, 0, 0, -6, 0, 0], [-10, 0, 11, -9, -4, 0, 0, 2, 0, 0], [-21, 1, -1, -1, 0, -21, 5, 1, -1, -1], [10, 4, -4, 2, 0, 10, 0, -6, 2, 6], [-5, -3, 3, -7, 0, -5, -1, -1, -7, 1], [-2, 13, -13, 0, 0, -2, 6, 2, 0, -2], [-22, 0, 0, 1, 4, 0, 0, 1, 0, 0], [0, -10, 0, 0, 1, 0, -1, 0, 4, 2], [0, -4, 0, 0, 0, 20, 0, 0, 1, -1], [0, 11, 0, 0, 0, 14, 0, 0, 8, 5], [17, 0, 0, 5, 0, 17, 0, -6, 5, 6], [-1, 0, 3, 11, -2, 0, 0, -5, 0, 0], [0, 1, 0, 0, 1, -1, -1, 0, 7, -2], [9, 0, -4, 6, 1, 0, 0, -1, 0, 0], [8, 1, -1, 6, 0, 8, 2, -2, 6, 2], [-10, -6, 6, 3, 0, -10, -2, 1, 3, -1], [0, 3, 0, 0, -1, -16, 1, 0, 9, 2], [2, 0, -10, 0, -1, 0, 0, -12, 0, 0], [16, 0, -10, -5, 0, 0, 0, 5, 0, 0], [0, 12, 0, 0, 2, 5, -2, 0, 4, 6], [4, 5, -5, 4, 0, 4, 1, -4, 4, 4], [20, -2, 2, 8, 0, 20, 2, -6, 8, 6], [0, -8, 0, 0, -4, 12, 4, 0, -2, -6], [0, 1, 0, 0, 1, -10, -1, 0, -6, -4], [0, 2, 0, 0, -6, 8, 6, 0, -2, -2], [0, -1, 0, 0, 2, -8, -2, 0, -5, -3], [-18, 0, -3, -8, 2, 0, 0, -1, 0, 0], [0, 9, 0, 0, 1, 5, -1, 0, 6, 10], [0, -8, 0, 0, 2, 0, -2, 0, 2, 6], [15, -5, 5, -12, 0, 15, -5, -3, -12, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_950_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_950_2_e_o();".
// 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_950_2_e_o(:prec:=10)
    chi := MakeCharacter_950_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_950_2_e_o();".
// 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_950_2_e_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_950_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![100, 150, 285, 110, 186, 70, 85, 12, 10, 0, 1]>,<7,R![-2, 2, 14, 0, -5, 1]>,<11,R![1555, 389, -176, -48, 3, 1]>],Snew);
    return Vf;
end function;