
// Make newform 95.9.d.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_95_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_95_d_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_95_9_d_c();".
// 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_95_9_d_c();".
// 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 := [5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 3]];
    Rf_basisdens := [1, 1];
    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_95_d();"
function MakeCharacter_95_d()
    N := 95;
    order := 2;
    char_gens := [77, 21];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_95_d_Hecke();"
function MakeCharacter_95_d_Hecke(Kf)
    N := 95;
    order := 2;
    char_gens := [77, 21];
    char_values := [[-1, 0], [-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 := 9;
    raw_aps := [[0, 0], [0, 0], [-191, 93], [-365, 730], [-25007, 0], [0, 0], [-12355, 24710], [130321, 0], [-12640, 25280], [0, 0], [0, 0], [0, 0], [0, 0], [299795, -599590], [-392275, 784550], [0, 0], [0, 0], [17661793, 0], [0, 0], [0, 0], [-2759075, 5518150], [0, 0], [-5450880, 10901760], [0, 0], [0, 0], [108160798, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-466963633, 0], [-53737085, 107474170], [746574193, 0], [814865473, 0], [0, 0], [-11807520, 23615040], [-37448000, 74896000], [0, 0], [0, 0], [0, 0], [0, 0], [1474435247, 0], [0, 0], [183527520, -367055040], [-2370683473, 0], [0, 0], [0, 0], [0, 0], [-5439578687, 0], [-116796995, 233593990], [1749377807, 0], [0, 0], [7755072527, 0], [0, 0], [-267016365, 534032730], [0, 0], [-5270290562, 0], [-781333795, 1562667590], [0, 0], [50853485, -101706970], [0, 0], [0, 0], [-10246904207, 0], [1436409280, -2872818560], [0, 0], [0, 0], [0, 0], [1788268275, -3576536550], [-28506315073, 0], [-207193280, 414386560], [-6266232047, 0], [747258400, -1494516800], [0, 0], [0, 0], [0, 0], [32174978207, 0], [3751780515, -7503561030], [0, 0], [0, 0], [-8962980958, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-1188990285, 2377980570], [0, 0], [-862951875, 1725903750], [-39600446593, 0], [-4373539085, 8747078170], [-5259731405, 10519462810], [-78310479362, 0], [0, 0], [13764858722, 0], [73614868273, 0], [-9790300960, 19580601920], [0, 0], [0, 0], [0, 0], [29562296353, 0], [0, 0], [9924630685, -19849261370], [0, 0], [0, 0], [16960254238, 0], [8125878365, -16251756730], [-16646062125, 33292124250], [1910522560, -3821045120], [0, 0], [0, 0], [0, 0], [17838851485, -35677702970], [18927577955, -37855155910], [185989529758, 0], [239197135633, 0], [0, 0], [22746025645, -45492051290], [-10825494515, 21650989030], [-25549923875, 51099847750], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-409506067247, 0], [433309154398, 0], [30099249122, 0], [523143809233, 0], [-9905174355, 19810348710], [38285460640, -76570921280], [32405045807, 0], [0, 0], [0, 0], [41179080195, -82358160390], [-540925723007, 0], [696632785633, 0], [0, 0], [0, 0], [0, 0], [652980985247, 0], [0, 0], [-908471906113, 0], [-66806896845, 133613793690], [0, 0], [0, 0], [0, 0], [42239344160, -84478688320], [0, 0], [-521495435953, 0], [0, 0], [0, 0], [-392725390367, 0], [-85667832845, 171335665690], [0, 0], [0, 0], [0, 0], [577135967042, 0], [236700975362, 0], [77640783555, -155281567110], [0, 0], [-106256866240, 212513732480], [0, 0], [133712341280, -267424682560], [0, 0], [0, 0], [0, 0], [0, 0], [-79325375325, 158650750650]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_95_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_95_9_d_c();".
// 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_95_9_d_c(:prec:=2)
    chi := MakeCharacter_95_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 9));
    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_95_9_d_c();".
// 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_95_9_d_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_95_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,9,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;