
// Make newform 615.2.j.b in Magma, downloaded from the LMFDB on 19 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_615_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_615_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_615_2_j_b();".
// 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_615_2_j_b();".
// 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, 0, 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_615_j();"
function MakeCharacter_615_j()
    N := 615;
    order := 4;
    char_gens := [206, 247, 211];
    v := [4, 4, 3];
    // 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_615_j_Hecke();"
function MakeCharacter_615_j_Hecke( : Kf := false)
    N := 615;
    order := 4;
    char_gens := [206, 247, 211];
    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 := 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, 1, 0, 1], [0, 0, 0, 1], [0, 0, -1, 0], [2, 0, -2, 0], [0, 0, 0, 3], [-1, 0, 1, -4], [3, -1, 3, 0], [5, 0, 5, 0], [0, 4, 0, -4], [0, 0, 0, 5], [1, 4, 0, -4], [-7, -1, 0, 1], [4, 0, 4, -3], [0, 3, -3, 3], [-7, -1, -7, 0], [2, 0, -2, -2], [-2, -5, 0, 5], [0, -4, 9, -4], [2, -10, 2, 0], [2, 0, -2, 1], [0, -9, 1, -9], [10, 0, -10, -2], [-2, 3, 0, -3], [-2, 0, 2, 8], [2, -2, 2, 0], [4, 5, 4, 0], [0, 3, -11, 3], [10, 6, 0, -6], [-4, 12, -4, 0], [2, -3, 0, 3], [0, -6, 0, 6], [0, 0, 4, 0], [-7, -5, -7, 0], [-10, -8, 0, 8], [-2, -12, -2, 0], [-6, 0, 6, -12], [9, 0, -9, 2], [5, 5, 0, -5], [-2, 0, 2, -8], [0, -6, -2, -6], [-10, -7, -10, 0], [-9, -6, -9, 0], [8, -14, 8, 0], [1, 0, -1, 8], [0, 7, -14, 7], [10, -10, 10, 0], [5, -6, 5, 0], [12, 8, 0, -8], [-15, -7, -15, 0], [0, -6, 0, 0], [-8, 0, 8, 2], [2, 0, -2, -26], [0, 14, 3, 14], [0, 10, -10, 10], [-11, 0, 11, -5], [7, -5, 7, 0], [-16, 4, 0, -4], [-7, -16, 0, 16], [-5, -7, 0, 7], [-8, 11, -8, 0], [-11, -5, 0, 5], [1, 15, 1, 0], [0, 7, 3, 7], [-20, -4, -20, 0], [0, 8, 0, 0], [-7, 0, 7, -9], [0, 0, 0, -34], [0, 13, -9, 13], [-9, 11, -9, 0], [0, 10, 11, 10], [-34, -1, 0, 1], [12, -12, 0, 12], [0, 1, -5, 1], [21, 5, 0, -5], [-10, 4, 0, -4], [3, -3, 3, 0], [0, -4, -6, -4], [-16, 0, 16, 12], [0, -3, -6, -3], [15, 2, 0, -2], [0, 4, 16, 4], [0, 0, 0, -12], [0, 4, -32, 4], [-15, 7, 0, -7], [0, 0, 0, 14], [0, 4, 20, 4], [0, -3, 6, -3], [1, -4, 1, 0], [0, 11, 0, -11], [22, 0, -22, -4], [-2, -9, 0, 9], [2, 0, -2, 7], [0, -7, 19, -7], [8, 13, 0, -13], [13, 0, -13, -10], [7, 0, -7, -21], [-6, 13, -6, 0], [18, 0, -18, -13], [10, 12, 0, -12], [0, 10, 16, 10], [-10, -16, -10, 0], [-23, 7, -23, 0], [-1, 0, 1, -43], [0, 7, 18, 7], [8, 0, -8, 12], [6, 0, -6, 8], [-9, 0, 9, -11], [-9, -23, -9, 0], [10, -13, 0, 13], [20, 6, 20, 0], [0, -16, 2, -16], [0, -10, 30, -10], [0, -3, -10, -3], [-31, 0, 0, 0], [-19, 4, 0, -4], [0, -27, 0, 0], [-11, 0, 11, 0], [0, -5, 20, -5], [-15, 0, 15, 9], [-8, 0, 8, 2], [0, -6, -12, -6], [-18, 14, -18, 0], [0, 6, -10, 6], [-18, 8, -18, 0], [19, -2, 19, 0], [0, -11, 0, 11], [-18, 0, 18, -2], [-2, 29, -2, 0], [-6, 0, 6, -2], [0, 27, 13, 27], [-5, 4, 0, -4], [0, -9, -2, -9], [0, 0, 0, 34], [9, 6, 9, 0], [12, 6, 0, -6], [29, -4, 0, 4], [9, -3, 9, 0], [0, 11, -11, 11], [14, -14, 0, 14], [-2, 0, 2, -20], [0, 32, 11, 32], [-24, 4, 0, -4], [25, 0, -25, -18], [9, 0, -9, -1], [0, -22, -19, -22], [28, 18, 28, 0], [0, -6, -4, -6], [-4, 19, 0, -19], [0, 8, -3, 8], [0, -14, 6, -14], [-15, -7, 0, 7], [0, 10, 20, 10], [10, -30, 10, 0], [21, 9, 21, 0], [0, 36, 6, 36], [0, 11, 34, 11], [-13, 30, -13, 0], [8, 25, 8, 0], [9, -4, 9, 0], [0, 13, -20, 13], [34, -15, 0, 15], [44, 0, 0, 0], [-14, 14, -14, 0], [26, 0, -26, -23], [33, 0, -33, 15], [-6, -1, 0, 1], [20, 0, -20, 28], [-16, 0, 16, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_615_j_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_615_2_j_b();".
// 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_615_2_j_b(:prec:=4)
    chi := MakeCharacter_615_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_615_2_j_b();".
// 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_615_2_j_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_615_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 0, 1]>],Snew);
    return Vf;
end function;