LMFDB - Newform orbit 42.2.f.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [42,2,Mod(5,42)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("42.5"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	42.f (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.335371688489$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{12}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 1) q^{6} + ( - \zeta_{12}^{2} - 2) q^{7} + \zeta_{12}^{3} q^{8}+ \cdots + 9 \zeta_{12}^{3} q^{99}+O(q^{100}) $ q + z * q^2 + (-z^3 - z) * q^3 + z^2 * q^4 + (2z^3 - z) q^5 + (-2z^2 + 1) q^6 + (-z^2 - 2) * q^7 + z^3 * q^8 + (3z^2 - 3) q^9 + (z^2 - 2) * q^10 + (-3z^3 + 3z) * q^11 + (-2z^3 + z) q^12 + (-4z^2 + 2) q^13 + (-z^3 - 2z) q^14 + 3 * q^15 + (z^2 - 1) * q^16 + (2z^3 + 2z) * q^17 + (3z^3 - 3z) * q^18 + (2z^2 + 2) q^19 + (z^3 - 2z) q^20 + (4z^3 + z) q^21 + 3 * q^22 - 6z q^23 + (-z^2 + 2) * q^24 + 2z^2 q^25 + (-4z^3 + 2z) * q^26 + (-3z^3 + 6z) * q^27 + (-3z^2 + 1) q^28 - 3z^3 q^29 + 3z q^30 + (z^2 - 2) * q^31 + (z^3 - z) * q^32 + (-3z^2 - 3) q^33 + (4z^2 - 2) q^34 + (-5z^3 + 4z) * q^35 - 3 * q^36 + (-2z^2 + 2) q^37 + (2z^3 + 2z) * q^38 + (6z^3 - 6z) * q^39 + (-z^2 - 1) * q^40 + (4z^3 - 8z) * q^41 + (5z^2 - 4) q^42 - 8 * q^43 + 3z q^44 + (-3z^3 - 3z) * q^45 - 6z^2 q^46 + (8z^3 - 4z) * q^47 + (-z^3 + 2z) q^48 + (5z^2 + 3) q^49 + 2z^3 q^50 + (-6z^2 + 6) q^51 + (-2z^2 + 4) q^52 + (-9z^3 + 9z) * q^53 + (3z^2 + 3) q^54 + (6z^2 - 3) q^55 + (-3z^3 + z) q^56 - 6z^3 q^57 + (-3z^2 + 3) q^58 + (z^3 + z) * q^59 + 3z^2 q^60 + (z^3 - 2z) q^62 + (-6z^2 + 9) q^63 - q^64 + 6z q^65 + (-3z^3 - 3z) * q^66 - 2z^2 q^67 + (4z^3 - 2z) * q^68 + (12z^2 - 6) q^69 + (-z^2 + 5) * q^70 + 12z^3 q^71 - 3z q^72 + (-4z^2 + 8) q^73 + (-2z^3 + 2z) * q^74 + (-4z^3 + 2z) * q^75 + (4z^2 - 2) q^76 + (6z^3 - 9z) * q^77 - 6 * q^78 + (-z^2 + 1) * q^79 + (-z^3 - z) * q^80 - 9z^2 q^81 + (-4z^2 - 4) q^82 + (-5z^3 + 10z) * q^83 + (5z^3 - 4z) * q^84 - 6 * q^85 - 8z q^86 + (3z^2 - 6) q^87 + 3z^2 q^88 + (-12z^3 + 6z) * q^89 + (-6z^2 + 3) q^90 + (10z^2 - 8) q^91 - 6z^3 q^92 + 3z q^93 + (4z^2 - 8) q^94 + (6z^3 - 6z) * q^95 + (z^2 + 1) * q^96 + (-6z^2 + 3) q^97 + (5z^3 + 3z) * q^98 + 9z^3 q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 10 q^{7} - 6 q^{9} - 6 q^{10} + 12 q^{15} - 2 q^{16} + 12 q^{19} + 12 q^{22} + 6 q^{24} + 4 q^{25} - 2 q^{28} - 6 q^{31} - 18 q^{33} - 12 q^{36} + 4 q^{37} - 6 q^{40} - 6 q^{42} - 32 q^{43}+ \cdots + 6 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 - 10 * q^7 - 6 * q^9 - 6 * q^10 + 12 * q^15 - 2 * q^16 + 12 * q^19 + 12 * q^22 + 6 * q^24 + 4 * q^25 - 2 * q^28 - 6 * q^31 - 18 * q^33 - 12 * q^36 + 4 * q^37 - 6 * q^40 - 6 * q^42 - 32 * q^43 - 12 * q^46 + 22 * q^49 + 12 * q^51 + 12 * q^52 + 18 * q^54 + 6 * q^58 + 6 * q^60 + 24 * q^63 - 4 * q^64 - 4 * q^67 + 18 * q^70 + 24 * q^73 - 24 * q^78 + 2 * q^79 - 18 * q^81 - 24 * q^82 - 24 * q^85 - 18 * q^87 + 6 * q^88 - 12 * q^91 - 24 * q^94 + 6 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$.

$n$	$29$	$31$
$\chi(n)$	$-1$	$1 - \zeta_{12}^{2}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

5.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

−0.866025

−

0.500000i

0.866025

1.50000i

0.500000

0.866025i

0.866025

−

1.50000i

−

1.73205i

−2.50000

−

0.866025i

−

1.00000i

−1.50000

2.59808i

−1.50000

0.866025i

5.2

0.866025

0.500000i

−0.866025

−

1.50000i

0.500000

0.866025i

−0.866025

1.50000i

−

1.73205i

−2.50000

−

0.866025i

1.00000i

−1.50000

2.59808i

−1.50000

0.866025i

17.1

−0.866025

0.500000i

0.866025

−

1.50000i

0.500000

−

0.866025i

0.866025

1.50000i

1.73205i

−2.50000

0.866025i

1.00000i

−1.50000

−

2.59808i

−1.50000

−

0.866025i

17.2

0.866025

−

0.500000i

−0.866025

1.50000i

0.500000

−

0.866025i

−0.866025

−

1.50000i

1.73205i

−2.50000

0.866025i

−

1.00000i

−1.50000

−

2.59808i

−1.50000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	42.2.f.a	✓	4
3.b	odd	2	1	inner	42.2.f.a	✓	4
4.b	odd	2	1		336.2.bc.e		4
5.b	even	2	1		1050.2.s.b		4
5.c	odd	4	1		1050.2.u.a		4
5.c	odd	4	1		1050.2.u.d		4
7.b	odd	2	1		294.2.f.a		4
7.c	even	3	1		294.2.d.a		4
7.c	even	3	1		294.2.f.a		4
7.d	odd	6	1	inner	42.2.f.a	✓	4
7.d	odd	6	1		294.2.d.a		4
9.c	even	3	1		1134.2.l.c		4
9.c	even	3	1		1134.2.t.d		4
9.d	odd	6	1		1134.2.l.c		4
9.d	odd	6	1		1134.2.t.d		4
12.b	even	2	1		336.2.bc.e		4
15.d	odd	2	1		1050.2.s.b		4
15.e	even	4	1		1050.2.u.a		4
15.e	even	4	1		1050.2.u.d		4
21.c	even	2	1		294.2.f.a		4
21.g	even	6	1	inner	42.2.f.a	✓	4
21.g	even	6	1		294.2.d.a		4
21.h	odd	6	1		294.2.d.a		4
21.h	odd	6	1		294.2.f.a		4
28.f	even	6	1		336.2.bc.e		4
28.f	even	6	1		2352.2.k.e		4
28.g	odd	6	1		2352.2.k.e		4
35.i	odd	6	1		1050.2.s.b		4
35.k	even	12	1		1050.2.u.a		4
35.k	even	12	1		1050.2.u.d		4
63.i	even	6	1		1134.2.t.d		4
63.k	odd	6	1		1134.2.l.c		4
63.s	even	6	1		1134.2.l.c		4
63.t	odd	6	1		1134.2.t.d		4
84.j	odd	6	1		336.2.bc.e		4
84.j	odd	6	1		2352.2.k.e		4
84.n	even	6	1		2352.2.k.e		4
105.p	even	6	1		1050.2.s.b		4
105.w	odd	12	1		1050.2.u.a		4
105.w	odd	12	1		1050.2.u.d		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.2.f.a	✓	4	1.a	even	1	1	trivial
42.2.f.a	✓	4	3.b	odd	2	1	inner
42.2.f.a	✓	4	7.d	odd	6	1	inner
42.2.f.a	✓	4	21.g	even	6	1	inner
294.2.d.a		4	7.c	even	3	1
294.2.d.a		4	7.d	odd	6	1
294.2.d.a		4	21.g	even	6	1
294.2.d.a		4	21.h	odd	6	1
294.2.f.a		4	7.b	odd	2	1
294.2.f.a		4	7.c	even	3	1
294.2.f.a		4	21.c	even	2	1
294.2.f.a		4	21.h	odd	6	1
336.2.bc.e		4	4.b	odd	2	1
336.2.bc.e		4	12.b	even	2	1
336.2.bc.e		4	28.f	even	6	1
336.2.bc.e		4	84.j	odd	6	1
1050.2.s.b		4	5.b	even	2	1
1050.2.s.b		4	15.d	odd	2	1
1050.2.s.b		4	35.i	odd	6	1
1050.2.s.b		4	105.p	even	6	1
1050.2.u.a		4	5.c	odd	4	1
1050.2.u.a		4	15.e	even	4	1
1050.2.u.a		4	35.k	even	12	1
1050.2.u.a		4	105.w	odd	12	1
1050.2.u.d		4	5.c	odd	4	1
1050.2.u.d		4	15.e	even	4	1
1050.2.u.d		4	35.k	even	12	1
1050.2.u.d		4	105.w	odd	12	1
1134.2.l.c		4	9.c	even	3	1
1134.2.l.c		4	9.d	odd	6	1
1134.2.l.c		4	63.k	odd	6	1
1134.2.l.c		4	63.s	even	6	1
1134.2.t.d		4	9.c	even	3	1
1134.2.t.d		4	9.d	odd	6	1
1134.2.t.d		4	63.i	even	6	1
1134.2.t.d		4	63.t	odd	6	1
2352.2.k.e		4	28.f	even	6	1
2352.2.k.e		4	28.g	odd	6	1
2352.2.k.e		4	84.j	odd	6	1
2352.2.k.e		4	84.n	even	6	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(42, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} - T^{2} + 1 $ T^4 - T^2 + 1
$3$	$ T^{4} + 3T^{2} + 9 $ T^4 + 3*T^2 + 9
$5$	$ T^{4} + 3T^{2} + 9 $ T^4 + 3*T^2 + 9
$7$	$ (T^{2} + 5 T + 7)^{2} $ (T^2 + 5*T + 7)^2
$11$	$ T^{4} - 9T^{2} + 81 $ T^4 - 9*T^2 + 81
$13$	$ (T^{2} + 12)^{2} $ (T^2 + 12)^2
$17$	$ T^{4} + 12T^{2} + 144 $ T^4 + 12*T^2 + 144
$19$	$ (T^{2} - 6 T + 12)^{2} $ (T^2 - 6*T + 12)^2
$23$	$ T^{4} - 36T^{2} + 1296 $ T^4 - 36*T^2 + 1296
$29$	$ (T^{2} + 9)^{2} $ (T^2 + 9)^2
$31$	$ (T^{2} + 3 T + 3)^{2} $ (T^2 + 3*T + 3)^2
$37$	$ (T^{2} - 2 T + 4)^{2} $ (T^2 - 2*T + 4)^2
$41$	$ (T^{2} - 48)^{2} $ (T^2 - 48)^2
$43$	$ (T + 8)^{4} $ (T + 8)^4
$47$	$ T^{4} + 48T^{2} + 2304 $ T^4 + 48*T^2 + 2304
$53$	$ T^{4} - 81T^{2} + 6561 $ T^4 - 81*T^2 + 6561
$59$	$ T^{4} + 3T^{2} + 9 $ T^4 + 3*T^2 + 9
$61$	$ T^{4} $ T^4
$67$	$ (T^{2} + 2 T + 4)^{2} $ (T^2 + 2*T + 4)^2
$71$	$ (T^{2} + 144)^{2} $ (T^2 + 144)^2
$73$	$ (T^{2} - 12 T + 48)^{2} $ (T^2 - 12*T + 48)^2
$79$	$ (T^{2} - T + 1)^{2} $ (T^2 - T + 1)^2
$83$	$ (T^{2} - 75)^{2} $ (T^2 - 75)^2
$89$	$ T^{4} + 108 T^{2} + 11664 $ T^4 + 108*T^2 + 11664
$97$	$ (T^{2} + 27)^{2} $ (T^2 + 27)^2
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Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	42.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 1) q^{6} + ( - \zeta_{12}^{2} - 2) q^{7} + \zeta_{12}^{3} q^{8}+ \cdots + 9 \zeta_{12}^{3} q^{99}+O(q^{100}) \) q + z * q^2 + (-z^3 - z) * q^3 + z^2 * q^4 + (2z^3 - z) q^5 + (-2z^2 + 1) q^6 + (-z^2 - 2) * q^7 + z^3 * q^8 + (3z^2 - 3) q^9 + (z^2 - 2) * q^10 + (-3z^3 + 3z) * q^11 + (-2z^3 + z) q^12 + (-4z^2 + 2) q^13 + (-z^3 - 2z) q^14 + 3 * q^15 + (z^2 - 1) * q^16 + (2z^3 + 2z) * q^17 + (3z^3 - 3z) * q^18 + (2z^2 + 2) q^19 + (z^3 - 2z) q^20 + (4z^3 + z) q^21 + 3 * q^22 - 6z q^23 + (-z^2 + 2) * q^24 + 2z^2 q^25 + (-4z^3 + 2z) * q^26 + (-3z^3 + 6z) * q^27 + (-3z^2 + 1) q^28 - 3z^3 q^29 + 3z q^30 + (z^2 - 2) * q^31 + (z^3 - z) * q^32 + (-3z^2 - 3) q^33 + (4z^2 - 2) q^34 + (-5z^3 + 4z) * q^35 - 3 * q^36 + (-2z^2 + 2) q^37 + (2z^3 + 2z) * q^38 + (6z^3 - 6z) * q^39 + (-z^2 - 1) * q^40 + (4z^3 - 8z) * q^41 + (5z^2 - 4) q^42 - 8 * q^43 + 3z q^44 + (-3z^3 - 3z) * q^45 - 6z^2 q^46 + (8z^3 - 4z) * q^47 + (-z^3 + 2z) q^48 + (5z^2 + 3) q^49 + 2z^3 q^50 + (-6z^2 + 6) q^51 + (-2z^2 + 4) q^52 + (-9z^3 + 9z) * q^53 + (3z^2 + 3) q^54 + (6z^2 - 3) q^55 + (-3z^3 + z) q^56 - 6z^3 q^57 + (-3z^2 + 3) q^58 + (z^3 + z) * q^59 + 3z^2 q^60 + (z^3 - 2z) q^62 + (-6z^2 + 9) q^63 - q^64 + 6z q^65 + (-3z^3 - 3z) * q^66 - 2z^2 q^67 + (4z^3 - 2z) * q^68 + (12z^2 - 6) q^69 + (-z^2 + 5) * q^70 + 12z^3 q^71 - 3z q^72 + (-4z^2 + 8) q^73 + (-2z^3 + 2z) * q^74 + (-4z^3 + 2z) * q^75 + (4z^2 - 2) q^76 + (6z^3 - 9z) * q^77 - 6 * q^78 + (-z^2 + 1) * q^79 + (-z^3 - z) * q^80 - 9z^2 q^81 + (-4z^2 - 4) q^82 + (-5z^3 + 10z) * q^83 + (5z^3 - 4z) * q^84 - 6 * q^85 - 8z q^86 + (3z^2 - 6) q^87 + 3z^2 q^88 + (-12z^3 + 6z) * q^89 + (-6z^2 + 3) q^90 + (10z^2 - 8) q^91 - 6z^3 q^92 + 3z q^93 + (4z^2 - 8) q^94 + (6z^3 - 6z) * q^95 + (z^2 + 1) * q^96 + (-6z^2 + 3) q^97 + (5z^3 + 3z) * q^98 + 9z^3 q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 10 q^{7} - 6 q^{9} - 6 q^{10} + 12 q^{15} - 2 q^{16} + 12 q^{19} + 12 q^{22} + 6 q^{24} + 4 q^{25} - 2 q^{28} - 6 q^{31} - 18 q^{33} - 12 q^{36} + 4 q^{37} - 6 q^{40} - 6 q^{42} - 32 q^{43}+ \cdots + 6 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 - 10 * q^7 - 6 * q^9 - 6 * q^10 + 12 * q^15 - 2 * q^16 + 12 * q^19 + 12 * q^22 + 6 * q^24 + 4 * q^25 - 2 * q^28 - 6 * q^31 - 18 * q^33 - 12 * q^36 + 4 * q^37 - 6 * q^40 - 6 * q^42 - 32 * q^43 - 12 * q^46 + 22 * q^49 + 12 * q^51 + 12 * q^52 + 18 * q^54 + 6 * q^58 + 6 * q^60 + 24 * q^63 - 4 * q^64 - 4 * q^67 + 18 * q^70 + 24 * q^73 - 24 * q^78 + 2 * q^79 - 18 * q^81 - 24 * q^82 - 24 * q^85 - 18 * q^87 + 6 * q^88 - 12 * q^91 - 24 * q^94 + 6 * q^96

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Newspace parameters

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Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	42.2.f.a

Level	$42$
Weight	$2$
Character orbit	42.f
Analytic conductor	$0.335$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.335371688489\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - x^{2} + 1 \) x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$p$	$F_p(T)$
$2$	\( T^{4} - T^{2} + 1 \) T^4 - T^2 + 1
$3$	\( T^{4} + 3T^{2} + 9 \) T^4 + 3*T^2 + 9
$5$	\( T^{4} + 3T^{2} + 9 \) T^4 + 3*T^2 + 9
$7$	\( (T^{2} + 5 T + 7)^{2} \) (T^2 + 5*T + 7)^2
$11$	\( T^{4} - 9T^{2} + 81 \) T^4 - 9*T^2 + 81
$13$	\( (T^{2} + 12)^{2} \) (T^2 + 12)^2
$17$	\( T^{4} + 12T^{2} + 144 \) T^4 + 12*T^2 + 144
$19$	\( (T^{2} - 6 T + 12)^{2} \) (T^2 - 6*T + 12)^2
$23$	\( T^{4} - 36T^{2} + 1296 \) T^4 - 36*T^2 + 1296
$29$	\( (T^{2} + 9)^{2} \) (T^2 + 9)^2
$31$	\( (T^{2} + 3 T + 3)^{2} \) (T^2 + 3*T + 3)^2
$37$	\( (T^{2} - 2 T + 4)^{2} \) (T^2 - 2*T + 4)^2
$41$	\( (T^{2} - 48)^{2} \) (T^2 - 48)^2
$43$	\( (T + 8)^{4} \) (T + 8)^4
$47$	\( T^{4} + 48T^{2} + 2304 \) T^4 + 48*T^2 + 2304
$53$	\( T^{4} - 81T^{2} + 6561 \) T^4 - 81*T^2 + 6561
$59$	\( T^{4} + 3T^{2} + 9 \) T^4 + 3*T^2 + 9
$61$	\( T^{4} \) T^4
$67$	\( (T^{2} + 2 T + 4)^{2} \) (T^2 + 2*T + 4)^2
$71$	\( (T^{2} + 144)^{2} \) (T^2 + 144)^2
$73$	\( (T^{2} - 12 T + 48)^{2} \) (T^2 - 12*T + 48)^2
$79$	\( (T^{2} - T + 1)^{2} \) (T^2 - T + 1)^2
$83$	\( (T^{2} - 75)^{2} \) (T^2 - 75)^2
$89$	\( T^{4} + 108 T^{2} + 11664 \) T^4 + 108*T^2 + 11664
$97$	\( (T^{2} + 27)^{2} \) (T^2 + 27)^2
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\(n\)	\(29\)	\(31\)
\(\chi(n)\)	\(-1\)	\(1 - \zeta_{12}^{2}\)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
Format: