
// Make newform 315.2.j.c in Magma, downloaded from the LMFDB on 15 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_315_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_315_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_315_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_315_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, 0, -1, 0, 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_315_j();"
function MakeCharacter_315_j()
    N := 315;
    order := 3;
    char_gens := [281, 127, 136];
    v := [3, 3, 2];
    // 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_315_j_Hecke();"
function MakeCharacter_315_j_Hecke(Kf)
    N := 315;
    order := 3;
    char_gens := [281, 127, 136];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 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]];
    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, 1, -1, 1], [0, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 3], [-1, -1, 1, 2], [4, 2, 0, -1], [5, 1, -5, -2], [0, 2, -1, 2], [0, 1, -3, 1], [-1, -6, 0, 3], [-3, -2, 3, 4], [0, -3, -2, -3], [-1, -2, 0, 1], [-2, -6, 0, 3], [0, 0, 2, 0], [2, -6, -2, 12], [5, 3, -5, -6], [0, 0, -4, 0], [6, -5, -6, 10], [-1, 6, 0, -3], [-4, 5, 4, -10], [0, -6, -3, -6], [-3, 14, 0, -7], [0, -7, 3, -7], [8, -8, 0, 4], [-9, -1, 9, 2], [0, -3, 4, -3], [0, -3, -3, -3], [-11, 0, 11, 0], [-2, 8, 0, -4], [10, 6, 0, -3], [0, -2, 12, -2], [-3, -3, 3, 6], [-1, -8, 0, 4], [0, -8, -8, -8], [2, -4, -2, 8], [4, 6, -4, -12], [0, 8, -8, 8], [9, -10, 0, 5], [0, 2, 18, 2], [-10, 0, 10, 0], [-7, 20, 0, -10], [0, 4, -2, 4], [4, 3, -4, -6], [9, 10, 0, -5], [-22, 0, 22, 0], [14, 8, 0, -4], [-10, 12, 0, -6], [7, 5, -7, -10], [0, 0, 3, 0], [0, -10, 0, -10], [-14, -8, 0, 4], [-10, 2, 10, -4], [9, -18, 0, 9], [0, 5, 3, 5], [-2, -6, 2, 12], [-16, -2, 16, 4], [0, -4, -10, -4], [6, -5, -6, 10], [0, 16, 0, -8], [12, -7, -12, 14], [-12, 8, 0, -4], [-20, 14, 0, -7], [-3, -7, 3, 14], [0, -5, 4, -5], [0, 17, 1, 17], [0, -4, -15, -4], [-8, -30, 0, 15], [-28, -4, 28, 8], [22, 0, 0, 0], [9, 7, -9, -14], [0, -1, -3, -1], [-6, 3, 6, -6], [0, 13, -4, 13], [-11, -20, 0, 10], [0, 10, -6, 10], [21, 9, -21, -18], [0, -3, 26, -3], [0, 6, -6, 6], [-17, -8, 17, 16], [32, -4, 0, 2], [7, 8, 0, -4], [0, -10, 0, 20], [10, -6, 0, 3], [0, -2, -4, -2], [0, -2, 6, -2], [22, -16, 0, 8], [0, -5, 8, -5], [9, 30, 0, -15], [24, -2, 0, 1], [0, -8, -14, -8], [-12, -12, 12, 24], [16, 9, -16, -18], [14, -28, 0, 14], [0, 2, 17, 2], [-4, 12, 0, -6], [0, 2, 8, 2], [-2, 2, 2, -4], [0, -1, 26, -1], [0, 12, 15, 12], [-2, 28, 0, -14], [-6, -12, 6, 24], [-18, 0, 18, 0], [0, -17, 3, -17], [-29, -6, 29, 12], [-20, 5, 20, -10], [-39, 2, 0, -1], [0, -19, -5, -19], [-14, 14, 14, -28], [-7, -8, 0, 4], [0, 3, 2, 3], [20, -4, -20, 8], [12, -28, 0, 14], [37, 4, -37, -8], [28, -24, 0, 12], [9, 3, -9, -6], [-18, -26, 0, 13], [5, 19, -5, -38], [0, 9, -29, 9], [4, 8, 0, -4], [-7, 2, 7, -4], [-36, -10, 0, 5], [0, 7, -21, 7], [-9, 11, 9, -22], [0, 8, 5, 8], [-7, -18, 0, 9], [0, 4, 8, 4], [0, -2, -24, -2], [-22, -10, 0, 5], [0, 5, 10, 5], [-7, 12, 7, -24], [-17, -38, 0, 19], [0, -4, -25, -4], [-14, 16, 0, -8], [0, -1, 45, -1], [35, -20, 0, 10], [21, -13, -21, 26], [-10, 2, 10, -4], [23, 22, 0, -11], [-2, -16, 2, 32], [0, 4, 0, -2], [0, 13, -3, 13], [-8, 18, 8, -36], [28, 28, 0, -14], [15, 6, -15, -12], [11, 34, 0, -17], [-6, 30, 0, -15], [17, -7, -17, 14], [0, 2, -4, 2], [0, -6, 4, -6], [0, 8, 18, 8], [4, 16, 0, -8], [38, -18, 0, 9], [0, -1, -27, -1], [-18, 9, 18, -18], [13, -2, 0, 1], [0, -8, 17, -8], [0, 17, -23, 17], [30, -2, 0, 1], [-11, 11, 11, -22], [0, 5, 3, 5], [-2, -44, 0, 22], [2, 14, 0, -7], [0, 0, -24, 0], [-1, 9, 1, -18], [15, 3, -15, -6], [-17, -22, 17, 44], [30, -15, -30, 30]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_315_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_315_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_315_2_j_c(:prec:=4)
    chi := MakeCharacter_315_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_315_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_315_2_j_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_315_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, -4, 6, 2, 1]>],Snew);
    return Vf;
end function;