
// Make newform 392.3.k.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_392_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_392_k_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_392_3_k_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_392_3_k_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 := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_392_k();"
function MakeCharacter_392_k()
    N := 392;
    order := 6;
    char_gens := [295, 197, 297];
    v := [3, 3, 4];
    // 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_392_k_Hecke();"
function MakeCharacter_392_k_Hecke(Kf)
    N := 392;
    order := 6;
    char_gens := [295, 197, 297];
    char_values := [[-1, 0], [-1, 0], [-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 := 3;
    raw_aps := [[0, 2], [0, 1], [-3, -3], [0, 0], [0, -17], [-8, 16], [0, -25], [-7, 7], [-3, -3], [-8, 16], [38, -19], [-5, -5], [-26, 0], [14, 0], [-29, -29], [-106, 53], [0, -55], [13, 13], [0, -17], [0, 0], [0, 119], [-43, -43], [-110, 0], [71, -71], [22, 0], [90, -45], [-93, -93], [-65, 65], [-10, 5], [122, 0], [96, -192], [17, -17], [0, 145], [82, 0], [3, 3], [42, -21], [-358, 179], [-17, 17], [-8, 16], [61, 61], [0, -89], [144, -288], [125, 125], [0, 73], [-120, 240], [-74, 37], [302, 0], [80, -160], [0, -55], [189, 189], [385, -385], [248, -496], [0, -145], [58, 0], [119, -119], [378, -189], [154, -77], [251, 251], [-234, 117], [74, 0], [0, -463], [64, -128], [274, 0], [-58, 29], [-409, 409], [-109, -109], [295, -295], [26, 0], [0, -377], [56, -112], [0, 503], [-107, -107], [342, -171], [-69, -69], [-634, 0], [-141, -141], [678, -339], [-139, -139], [-119, 119], [0, -145], [-302, 0], [232, -464], [-934, 467], [-410, 0], [283, 283], [-401, 401], [-310, 0], [-167, 167], [8, -16], [-352, 704], [785, -785], [-714, 357], [-454, 227], [422, 0], [-65, 65], [-144, 288], [-315, -315], [0, -25], [593, -593], [-437, -437], [662, 0], [-682, 341], [0, 737], [121, -121], [0, -737], [0, 47], [-446, 0], [215, -215], [-326, 163], [-266, 0], [-381, -381], [806, -403], [-118, 0], [0, -919], [96, -192], [0, 1], [514, 0], [70, -35], [-189, -189], [542, 0], [-1366, 683], [218, 0], [-371, -371], [0, 367], [497, -497], [192, -384], [-229, -229], [-37, -37], [32, -64], [477, 477], [0, -713], [368, -736], [-675, -675], [600, -1200], [863, -863], [-410, 0], [922, -461], [0, -31], [-344, 688], [0, 313], [1138, 0], [707, 707], [-134, 67], [-754, 0], [-1046, 523], [608, -1216], [488, -976], [0, 887], [-1663, 1663], [325, 325], [-69, -69], [574, 0], [1166, 0], [315, 315], [0, -521], [688, -1376], [805, 805], [-961, 961], [142, 0], [-1414, 707], [175, -175], [-454, 0], [416, -832], [-1639, 1639], [0, 793], [1782, -891], [1034, -517], [-918, 459]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_392_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_392_3_k_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_392_3_k_c(:prec:=2)
    chi := MakeCharacter_392_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_392_3_k_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_392_3_k_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_392_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![1, -1, 1]>,<5,R![27, 9, 1]>],Snew);
    return Vf;
end function;