
// Make newform 8.11.d.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8_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_8_11_d_a();".
// 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_8_11_d_a();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8_d();"
function MakeCharacter_8_d()
    N := 8;
    order := 2;
    char_gens := [7, 5];
    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_8_d_Hecke();"
function MakeCharacter_8_d_Hecke(Kf)
    N := 8;
    order := 2;
    char_gens := [7, 5];
    char_values := [[-1], [-1]];
    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 := 11;
    raw_aps := [[-32], [-482], [0], [0], [-97426], [0], [823682], [3353726], [0], [0], [0], [0], [-37778926], [214485614], [0], [0], [921043598], [0], [1813708382], [0], [-1605781582], [0], [96051518], [-11116019374], [-9872978014], [0], [0], [27496855982], [0], [28842011618], [0], [71761374302], [82309120082], [-89035358674], [0], [0], [0], [-162941557282], [0], [0], [-168275103874], [0], [0], [513616024418], [0], [0], [-259805532226], [0], [710946942686], [0], [-396603990286], [0], [1426636951874], [653736226574], [-2004507263614], [0], [0], [0], [0], [-2860618609198], [-2412756487282], [0], [-4185951895618], [0], [5780277876914], [0], [838750251374], [6237407621282], [470042088782], [0], [10787284888514], [0], [0], [0], [-9165042703474], [0], [0], [0], [-1388492780926], [-19723838860174], [4114425956126], [0], [0], [-14919700555582], [0], [26695081533518], [8391037859426], [-38591189004718], [0], [0], [-8042864611714], [0], [0], [-56751493118098], [61812802959998], [0], [0], [-19192094090926], [72651699220718], [0], [-24546509563618], [0], [-95825796555586], [27106987397426], [-8653631781298], [1108403873282], [-45085125318418], [117278475453218], [0], [-144880026641326], [0], [0], [-175258017843214], [-128542706766802], [0], [202826966258402], [-24185774660386], [0], [0], [224853971048126], [0], [99515866113314], [0], [295985891726318], [105673823042174], [0], [0], [0], [0], [0], [-392476480109602], [0], [0], [0], [-242238255653326], [318401239705826], [0], [-1958843582914], [0], [-340806757073902], [-366643026594898], [0], [0], [-720561357844018], [0], [0], [0], [-626409076267918], [110234334606926], [0], [0], [1059505510456802], [-1013335312642882], [0], [119075838259886], [0], [0], [155409062731298], [-1333212818501518], [0], [-563746269096514], [-572694362936782], [0], [849794318864174], [-1019214885765118], [0], [0], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8_11_d_a();".
// 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_8_11_d_a(:prec:=1)
    chi := MakeCharacter_8_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 11));
    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_8_11_d_a();".
// 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_8_11_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,11,sign)));
    Vf := Kernel([<3,R![482, 1]>],Snew);
    return Vf;
end function;