
// Make newform 315.2.cg.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_315_cg();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_315_cg_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_cg_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_315_2_cg_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
        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_cg();"
function MakeCharacter_315_cg()
    N := 315;
    order := 12;
    char_gens := [281, 127, 136];
    v := [4, 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_cg_Hecke();"
function MakeCharacter_315_cg_Hecke(Kf)
    N := 315;
    order := 12;
    char_gens := [281, 127, 136];
    char_values := [[-1, 0, 1, 0], [0, 0, 0, 1], [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+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 := 2;
    raw_aps := [[0, 0, -1, 1], [-1, 0, 2, 0], [-2, 0, 0, 1], [0, -1, 0, -2], [-2, 4, 0, -2], [0, 0, -2, 2], [-2, 2, 2, 0], [0, -1, -5, -1], [2, -1, 1, -2], [-8, 0, 4, 0], [-6, -5, 3, 5], [0, 3, -3, -3], [-2, -6, -2, 0], [-3, -3, 0, 0], [6, -6, 3, 9], [3, -3, 1, 4], [0, 2, 0, 2], [-6, 1, -6, 0], [3, 2, -5, -5], [-6, 0, 0, 0], [1, 1, -5, 4], [2, 8, 2, 0], [0, -1, 1, 1], [0, 3, -4, 3], [-2, 3, 5, -5], [-9, 0, 18, -2], [4, 3, 3, 4], [2, -1, -1, -1], [0, -2, 0, 0], [-3, 3, -3, -6], [-11, 13, 13, -11], [4, 0, -8, -12], [0, -4, -4, 0], [0, -4, 0, 8], [5, 0, -10, -12], [3, -2, 0, 1], [0, -1, -1, 1], [4, -1, -5, 5], [-11, 11, 14, 3], [-10, -10, 10, 0], [-9, -3, -9, 0], [2, 0, -4, 8], [9, -5, -9, 10], [-4, 9, 13, -13], [-16, 4, 4, -16], [-6, -8, 6, 16], [-2, 4, 2, -8], [1, 4, -5, -5], [3, 10, -10, -3], [4, 10, 0, -5], [10, 5, -5, 5], [0, 22, 0, 0], [2, 0, -4, -13], [-8, 0, 16, 8], [-17, -6, 6, 17], [-16, 3, -3, 16], [-3, 4, 3, -8], [-4, 8, 2, -8], [-12, 6, 6, -12], [0, -1, 12, -1], [13, -11, -2, -2], [-7, -7, -9, 16], [23, 4, -4, -23], [0, 16, 0, -16], [-3, -3, -3, 6], [-3, -3, 7, -4], [-27, -5, 27, 10], [6, 6, -4, -2], [5, -13, 8, 8], [0, 7, -22, 7], [2, -16, -16, 2], [-2, 19, 1, -19], [2, 5, -5, -2], [3, 0, 0, -3], [2, 0, -4, -16], [9, -7, -7, 9], [-3, 0, 6, 4], [0, -24, -24, 24], [13, -12, 0, 6], [0, -6, -5, -6], [8, 0, -8, 0], [0, 11, -4, 11], [19, -3, -19, 6], [-12, -6, -6, -12], [8, 8, -8, -16], [-3, 3, 13, 10], [-9, 0, 18, 14], [21, 21, -11, -10], [-6, -4, 3, 4], [21, -21, -8, 13], [15, 11, -4, 4], [7, 2, 0, -1], [-21, -21, 23, -2], [10, -4, -10, 8], [4, 0, -8, -14], [-13, -8, -8, -13], [-18, 0, 0, 0], [17, -14, 17, 0], [1, 4, -5, -5], [0, -4, 3, -4], [-22, 1, 21, 21], [-3, -3, 9, -6], [-5, -17, 22, 22], [-12, -2, -12, 0], [0, -14, -4, -14], [8, -8, -20, -12], [7, -14, -21, 21], [0, 2, -2, -2], [-30, 5, 15, -5], [-16, 10, 8, -10], [3, 0, 0, -3], [23, -23, -3, 20], [-7, -7, 3, 4], [-1, -30, 0, 15], [20, -20, 0, 10], [-2, 34, 0, -17], [21, 21, -6, -15], [5, -5, 19, 24], [15, -2, 2, -15], [0, -26, 0, 26], [0, 11, 0, -11], [-8, 8, 16, -16], [-22, 22, -16, -38], [4, -4, 15, 19], [9, 5, 9, 0], [-44, 4, 0, -2], [16, 21, 16, 0], [0, 7, -19, 7], [-13, -14, -1, 1], [-12, 10, 10, -12], [6, -14, 6, 0], [5, -5, -1, 4], [-35, 14, 0, -7], [-9, -2, -2, -9], [9, 0, -18, 4], [-10, -15, 10, 30], [27, 27, -11, -16], [9, 1, -8, 8], [29, -29, -33, -4], [-26, -3, -26, 0], [8, 0, -16, -12], [16, -5, -16, 10], [8, 25, 17, -17], [17, -12, -12, 17], [-27, 10, 27, -20], [0, -11, -19, -11], [-10, 26, -16, -16], [-9, -22, 22, 9], [17, -30, 0, 15], [-27, -19, 8, -8], [7, -4, -4, 7], [12, 0, -24, -3], [26, 37, -37, -26], [2, 11, -11, -2], [9, -16, -16, 9], [0, -7, -27, -7], [12, -42, -6, 42], [-19, 10, 19, -20], [4, -4, 4, -4], [-30, 12, 15, -12], [5, 5, 12, -17], [21, 0, 0, -21], [22, 22, -5, -17], [-8, -4, 4, 4], [36, -20, -16, -16], [7, 12, 12, 7], [14, -2, -14, 4], [9, 30, -30, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_315_cg_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_315_2_cg_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_315_2_cg_a(:prec:=4)
    chi := MakeCharacter_315_cg();
    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_cg_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_315_2_cg_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_315_cg();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 4, 5, 2, 1]>,<11,R![-8, 4, 1]>],Snew);
    return Vf;
end function;