
// Make newform 256.6.b.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_256_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_256_b_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_256_6_b_h();".
// 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_256_6_b_h();".
// 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;
    Rf_num := [[1, 0], [0, 2]];
    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_256_b();"
function MakeCharacter_256_b()
    N := 256;
    order := 2;
    char_gens := [255, 5];
    v := [2, 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_256_b_Hecke();"
function MakeCharacter_256_b_Hecke(Kf)
    N := 256;
    order := 2;
    char_gens := [255, 5];
    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 := 6;
    raw_aps := [[0, 0], [0, 4], [0, 7], [208, 0], [0, -268], [0, -347], [-1278, 0], [0, -556], [-3216, 0], [0, -1459], [-2624, 0], [0, -4729], [-170, 0], [0, -9964], [32, 0], [0, -11089], [0, 20740], [0, -7731], [0, 10372], [-28592, 0], [53670, 0], [-69152, 0], [0, 18900], [126806, 0], [62290, 0], [0, 3207], [108432, 0], [0, 51988], [0, -1243], [15794, 0], [1024, 0], [0, 11332], [53238, 0], [0, 9908], [0, 226095], [263280, 0], [0, 176765], [0, 50468], [284944, 0], [0, -242187], [0, -203340], [0, 285151], [138624, 0], [34482, 0], [0, 321799], [-1107376, 0], [0, -114988], [-1089472, 0], [0, 343524], [0, -349865], [-937722, 0], [643488, 0], [157282, 0], [0, -793020], [-654334, 0], [330192, 0], [0, -784779], [957792, 0], [0, 282719], [1341270, 0], [0, -367132], [0, -566601], [0, 1456884], [1438128, 0], [1376006, 0], [0, 617469], [0, -744652], [838226, 0], [0, -175004], [0, 191821], [4093090, 0], [-3144304, 0], [-1476192, 0], [0, -1869905], [0, 949828], [1743104, 0], [0, -1345737], [0, -2686763], [156418, 0], [306086, 0], [0, 3354340], [0, 2011735], [-7043040, 0], [-1251422, 0], [1254064, 0], [0, 2590404], [-5730638, 0], [5241526, 0], [0, 86997], [4012768, 0], [0, -387308], [2335296, 0], [-1039472, 0], [0, 3925460], [0, -1358220], [4620336, 0], [0, 2230989], [-3743754, 0], [0, 4642164], [0, -431075], [0, -877212], [0, 5014613], [0, 2637300], [-8369402, 0], [0, 2013508], [2375682, 0], [0, -1720140], [-3229422, 0], [9297136, 0], [11278166, 0], [7001152, 0], [0, -3097161], [2624070, 0], [0, -1418604], [-12965584, 0], [-11679758, 0], [0, -3513660], [9728208, 0], [0, 4907125], [0, -7332588], [0, -7091825], [5499410, 0], [0, 8868743], [0, -4698348], [0, 6738356], [0, -10779699], [0, -3190825], [-16356640, 0], [21313008, 0], [0, -6078547], [0, 9616868], [-1665648, 0], [9812896, 0], [0, -9637665], [11786310, 0], [12294130, 0], [0, 1285431], [0, -2772972], [0, -10480515], [-27029674, 0], [0, -1065260], [0, 7902991], [22884784, 0], [0, -12766780], [0, -4158931], [-36626096, 0], [0, 8740095], [-9310618, 0], [0, -17476252], [-20234944, 0], [0, 9917125], [-38102286, 0], [0, 12078004], [-20236752, 0], [0, 9898804], [-21324192, 0], [-34994064, 0], [-28810734, 0], [-12885434, 0], [0, 26380741], [0, -2769804], [228102, 0], [-7367088, 0], [0, -9558028], [55704002, 0], [15546864, 0], [23689024, 0], [0, 1858599]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_256_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_256_6_b_h();".
// 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_256_6_b_h(:prec:=2)
    chi := MakeCharacter_256_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_256_6_b_h();".
// 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_256_6_b_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_256_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![64, 0, 1]>,<7,R![-208, 1]>],Snew);
    return Vf;
end function;