
// Make newform 315.2.cg.d in Magma, downloaded from the LMFDB on 19 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_d();".
// 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_d();".
// 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 := [[1, 1, 0, -1], [1, 0, 1, 0], [-1, 0, 0, 2], [2, 0, -3, 0], [-2, 0, 0, 0], [3, 3, -1, -2], [0, 0, -4, -4], [0, -1, -1, -1], [-1, -4, 4, 1], [-8, 3, 4, -3], [-4, 2, 2, -2], [4, -5, 1, 1], [0, 3, 0, 0], [1, 2, 1, -1], [4, -4, -5, -1], [0, 0, -2, -2], [0, 7, 3, 7], [4, -4, 4, 0], [-4, 1, 3, 3], [3, 2, 0, -1], [4, 4, -8, 4], [-2, 8, -2, 0], [-9, 4, 5, 5], [0, -4, -8, -4], [10, 7, -3, 3], [-1, 0, 2, 2], [-5, -1, -1, -5], [-1, -7, 8, 8], [1, -8, 1, 0], [4, -4, 4, 8], [11, -1, -1, 11], [4, 0, -8, 12], [1, 0, 0, 1], [-3, 9, 3, -18], [2, 0, -4, -12], [-9, -10, 0, 5], [-6, -5, 1, -1], [-6, 9, 15, -15], [-3, 3, -1, -4], [16, 16, -16, 0], [-1, 3, -1, 0], [7, 0, -14, -8], [15, -3, -15, 6], [-6, 2, 8, -8], [4, 0, 0, 4], [-6, 6, 6, -12], [-20, 4, 20, -8], [1, 4, -5, -5], [-7, -18, 18, 7], [14, -10, 0, 5], [14, 4, -10, 10], [0, -2, 0, 0], [-2, 0, 4, 19], [-5, 0, 10, -5], [2, -6, 6, -2], [5, -3, 3, -5], [18, 6, -18, -12], [-18, -17, 9, 17], [0, -6, -6, 0], [0, 1, -24, 1], [-1, 11, -10, -10], [-2, -2, -6, 8], [-5, -1, 1, 5], [-14, 5, 7, -5], [-6, -6, 18, -12], [-7, -7, 3, 4], [6, 0, -6, 0], [-1, -1, -11, 12], [-1, 8, -7, -7], [0, 0, 10, 0], [-7, -4, -4, -7], [8, 4, -4, -4], [5, -3, 3, -5], [-7, 4, -4, 7], [13, 0, -26, 5], [10, -13, -13, 10], [11, 0, -22, 10], [4, 20, 16, -16], [1, -12, 0, 6], [0, 10, 5, 10], [-11, 5, 11, -10], [0, 3, -10, 3], [-5, -9, 5, 18], [19, -14, -14, 19], [22, 8, -22, -16], [-10, 10, 9, -1], [-5, 0, 10, -16], [15, 15, 7, -22], [30, -2, -15, 2], [10, -10, 5, 15], [-5, 0, 5, -5], [-19, 2, 0, -1], [-13, -13, -11, 24], [1, 9, -1, -18], [6, 0, -12, 28], [3, 1, 1, 3], [0, -38, 0, 19], [5, -16, 5, 0], [11, 2, -13, -13], [0, 4, 0, 4], [-23, 3, 20, 20], [20, 20, 8, -28], [0, -3, 3, 3], [4, 4, 4, 0], [0, -3, -1, -3], [6, -6, 16, 22], [-25, -10, 15, -15], [6, 13, -19, -19], [-16, 32, 8, -32], [16, 20, -8, -20], [6, -9, 9, -6], [-3, 3, 13, 10], [12, 12, -20, 8], [-20, -24, 0, 12], [2, 16, 0, -8], [28, 12, 0, -6], [-25, -25, 20, 5], [-3, 3, -14, -17], [12, -2, 2, -12], [10, 31, -5, -31], [12, 13, -6, -13], [-8, -7, 1, -1], [-1, 1, -25, -26], [-23, 23, 0, -23], [-2, 16, -2, 0], [-8, -22, 0, 11], [20, -6, 20, 0], [0, -4, 16, -4], [4, -13, -17, 17], [3, 20, 20, 3], [-12, -20, -12, 0], [-23, 23, 16, -7], [37, 14, 0, -7], [-10, 12, 12, -10], [-3, 0, 6, -4], [-20, 3, 20, -6], [-23, -23, -3, 26], [5, 36, 31, -31], [27, -27, -31, -4], [10, -27, 10, 0], [9, 0, -18, -9], [-26, 9, 26, -18], [-7, -17, -10, 10], [16, 5, 5, 16], [-27, 10, 27, -20], [0, -29, -5, -29], [-10, 14, -4, -4], [2, 8, -8, -2], [19, -10, 0, 5], [4, -29, -33, 33], [-1, -6, -6, -1], [-26, 0, 52, 6], [22, -7, 7, -22], [-1, -15, 15, 1], [8, 1, 1, 8], [0, 1, -3, 1], [18, 3, -9, -3], [-11, -12, 11, 24], [7, -4, 4, -7], [-22, 12, 11, -12], [-5, -5, 5, 0], [14, 22, -22, -14], [-33, -33, 21, 12], [2, -17, -1, 17], [-6, -20, 26, 26], [36, -11, -11, 36], [8, 0, -8, 0], [33, 6, -6, -33]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : 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, -2, 5, -4, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;