
// Make newform 396.2.j.c in Magma, downloaded from the LMFDB on 15 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_396_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_396_j_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_396_2_j_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_396_2_j_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, -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_396_j();"
function MakeCharacter_396_j()
    N := 396;
    order := 5;
    char_gens := [199, 353, 145];
    v := [5, 5, 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_396_j_Hecke();"
function MakeCharacter_396_j_Hecke( : Kf := false)
    N := 396;
    order := 5;
    char_gens := [199, 353, 145];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 0, 0, 0], [3, -2, 3, 0], [2, -2, 0, -3], [-2, -1, 0, -2], [3, -5, 5, -3], [1, 3, 1, 0], [0, 5, -2, 5], [-3, 0, -2, 2], [4, -4, 0, -2], [8, -7, 7, -8], [-2, 2, 0, 7], [0, 4, -7, 4], [3, 0, -4, 4], [0, 1, -3, 1], [-8, -1, 1, 8], [-3, 3, 0, 7], [-1, 2, -1, 0], [-5, 0, -3, 3], [-1, -2, -1, 0], [-2, 2, 0, 0], [-11, 13, -13, 11], [-4, 13, -4, 0], [-1, 0, 0, 0], [-3, -2, 2, 3], [7, 1, -1, -7], [-4, 4, 0, -6], [0, -4, -9, -4], [-8, 0, 0, 0], [0, 2, -11, 2], [6, -12, 6, 0], [-6, 0, -9, 9], [0, -21, 0, 0], [-11, 11, 0, 14], [16, -7, 16, 0], [0, -4, 2, -4], [0, 2, -16, 2], [0, -3, 3, 0], [1, 0, 0, -1], [0, 9, 0, 9], [0, -8, 13, -8], [8, -1, 8, 0], [-14, 14, 0, 1], [7, -11, 7, 0], [14, 0, 5, -5], [9, 0, 12, -12], [12, 7, -7, -12], [0, 4, -5, 4], [6, -6, 0, 3], [2, 6, -6, -2], [19, -8, 8, -19], [0, 17, -7, 17], [-14, 0, 6, -6], [16, -21, 21, -16], [-11, 11, 0, 15], [-19, 0, 3, -3], [-4, 26, -4, 0], [-9, 9, 0, 17], [12, -3, 3, -12], [-18, 6, -18, 0], [0, -2, 22, -2], [0, 0, 0, -5], [-12, 0, -7, 7], [0, -10, 23, -10], [0, 1, 0, 0], [15, -11, 11, -15], [-1, 0, -8, 8], [22, -22, 0, -18], [0, -8, 0, 0], [0, 2, -27, 2], [4, 0, -4, 4], [-2, 2, 0, -20], [-1, 1, 0, 7], [-1, 0, 8, -8], [-6, 9, -6, 0], [7, -9, 9, -7], [5, -5, 0, 8], [11, 0, -10, 10], [3, -1, 3, 0], [16, 0, 0, -16], [-13, 0, 3, -3], [0, 15, 0, 15], [2, -9, 9, -2], [8, -8, 0, 2], [13, 0, 20, -20], [0, 10, 6, 10], [-14, 10, -10, 14], [-3, -3, -3, 0], [-5, 0, 9, -9], [-16, 0, -23, 23], [-24, 5, -24, 0], [-7, 9, -7, 0], [0, -24, 21, -24], [3, -3, 0, -10], [-15, 15, 0, 9], [0, 2, -16, 2], [0, 19, -1, 19], [24, -24, 0, -24], [-11, 15, -11, 0], [-5, -2, 2, 5], [0, -7, 9, -7], [5, -5, 0, -3], [31, -37, 37, -31], [0, -18, -16, -18], [13, 0, -7, 7], [2, -28, 2, 0], [-2, 2, 0, 29], [21, 0, 25, -25], [-14, 32, -14, 0], [-8, 8, 0, -7], [-20, 13, -13, 20], [0, -18, 6, -18], [7, 0, 0, 0], [0, 4, -5, 4], [15, -15, 0, -32], [0, 21, -12, 21], [-31, 21, -31, 0], [-8, -19, 19, 8], [3, -3, 0, -26], [0, 0, 6, -6], [-2, 0, 7, -7], [-29, 33, -33, 29], [-4, 0, -4, 0], [15, 0, -14, 14], [16, 14, -14, -16], [0, -17, 18, -17], [19, -5, 19, 0], [26, -26, 0, -23], [-4, 0, 3, -3], [-30, 30, 0, 4], [29, -19, 19, -29], [-4, 9, -4, 0], [0, 35, -20, 35], [-9, -3, 3, 9], [29, 7, -7, -29], [-19, 0, 9, -9], [0, 20, -17, 20], [-2, -46, -2, 0], [34, -28, 34, 0], [-15, 26, -15, 0], [0, -11, -9, -11], [20, -20, 0, -5], [-7, 17, -17, 7], [-12, 30, -30, 12], [-19, 19, 0, 26], [0, 24, -23, 24], [-20, -23, -20, 0], [-21, 0, -44, 44], [25, 0, 10, -10], [20, -2, 20, 0], [0, -16, -9, -16], [-26, 0, 9, -9], [0, -8, 10, -8], [6, -6, 0, -43], [-37, 20, -37, 0], [-23, -7, 7, 23], [-24, 1, -24, 0], [12, 27, 12, 0], [-23, 21, -21, 23], [9, 3, 9, 0], [-15, 0, -11, 11], [-2, 2, 0, 38], [39, 0, 15, -15], [0, 20, 10, 20], [-7, 23, -23, 7], [-24, 24, 0, 2], [19, 0, -15, 15], [-15, 15, 0, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_396_j_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_396_2_j_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_396_2_j_c(:prec:=4)
    chi := MakeCharacter_396_j();
    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_396_2_j_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_396_2_j_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_396_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![121, -88, 34, -7, 1]>,<7,R![1, -7, 19, -3, 1]>],Snew);
    return Vf;
end function;