LMFDB - Newform orbit 432.2.y.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,2,Mod(37,432)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("432.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	432.y (of order $12$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.44953736732$
Analytic rank:	$0$
Dimension:	$4$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$\zeta_{12}^{3}$	$-1 + \zeta_{12}^{2}$

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} $

37.1

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

−0.366025

1.36603i

−1.73205

−

1.00000i

3.73205

−

1.00000i

−0.633975

0.366025i

2.00000

−

2.00000i

5.46410i

181.1

1.36603

−

0.366025i

1.73205

−

1.00000i

0.267949

−

1.00000i

−2.36603

−

1.36603i

2.00000

−

2.00000i

−

1.46410i

253.1

1.36603

0.366025i

1.73205

1.00000i

0.267949

1.00000i

−2.36603

1.36603i

2.00000

2.00000i

1.46410i

397.1

−0.366025

−

1.36603i

−1.73205

1.00000i

3.73205

1.00000i

−0.633975

−

0.366025i

2.00000

2.00000i

−

5.46410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
144.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	432.2.y.c		4
3.b	odd	2	1		144.2.x.b	✓	4
4.b	odd	2	1		1728.2.bc.d		4
9.c	even	3	1		432.2.y.b		4
9.d	odd	6	1		144.2.x.c	yes	4
12.b	even	2	1		576.2.bb.d		4
16.e	even	4	1		432.2.y.b		4
16.f	odd	4	1		1728.2.bc.a		4
36.f	odd	6	1		1728.2.bc.a		4
36.h	even	6	1		576.2.bb.c		4
48.i	odd	4	1		144.2.x.c	yes	4
48.k	even	4	1		576.2.bb.c		4
144.u	even	12	1		576.2.bb.d		4
144.v	odd	12	1		1728.2.bc.d		4
144.w	odd	12	1		144.2.x.b	✓	4
144.x	even	12	1	inner	432.2.y.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
144.2.x.b	✓	4	3.b	odd	2	1
144.2.x.b	✓	4	144.w	odd	12	1
144.2.x.c	yes	4	9.d	odd	6	1
144.2.x.c	yes	4	48.i	odd	4	1
432.2.y.b		4	9.c	even	3	1
432.2.y.b		4	16.e	even	4	1
432.2.y.c		4	1.a	even	1	1	trivial
432.2.y.c		4	144.x	even	12	1	inner
576.2.bb.c		4	36.h	even	6	1
576.2.bb.c		4	48.k	even	4	1
576.2.bb.d		4	12.b	even	2	1
576.2.bb.d		4	144.u	even	12	1
1728.2.bc.a		4	16.f	odd	4	1
1728.2.bc.a		4	36.f	odd	6	1
1728.2.bc.d		4	4.b	odd	2	1
1728.2.bc.d		4	144.v	odd	12	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} - 8T_{5}^{3} + 20T_{5}^{2} - 16T_{5} + 16 $ T5^4 - 8*T5^3 + 20*T5^2 - 16*T5 + 16 acting on $S_{2}^{\mathrm{new}}(432, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 $ T^4 - 2T^3 + 2T^2 - 4*T + 4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} - 8 T^{3} + 20 T^{2} - 16 T + 16 $ T^4 - 8T^3 + 20T^2 - 16*T + 16
$7$	$ T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4 $ T^4 + 6T^3 + 14T^2 + 12*T + 4
$11$	$ T^{4} - 10 T^{3} + 41 T^{2} + \cdots + 169 $ T^4 - 10T^3 + 41T^2 - 104*T + 169
$13$	$ T^{4} + 10 T^{3} + 74 T^{2} + \cdots + 484 $ T^4 + 10T^3 + 74T^2 + 308*T + 484
$17$	$ (T^{2} - 8 T + 13)^{2} $ (T^2 - 8*T + 13)^2
$19$	$ T^{4} + 6 T^{3} + 18 T^{2} + 18 T + 9 $ T^4 + 6T^3 + 18T^2 + 18*T + 9
$23$	$ T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 $ T^4 + 6T^3 + 6T^2 - 36*T + 36
$29$	$ T^{4} - 6 T^{3} + 18 T^{2} - 36 T + 36 $ T^4 - 6T^3 + 18T^2 - 36*T + 36
$31$	$ T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16 $ T^4 + 8T^3 + 60T^2 + 32*T + 16
$37$	$ T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 144 $ T^4 - 12T^3 + 72T^2 - 144*T + 144
$41$	$ T^{4} - 9T^{2} + 81 $ T^4 - 9*T^2 + 81
$43$	$ T^{4} + 16 T^{3} + 65 T^{2} + \cdots + 121 $ T^4 + 16T^3 + 65T^2 - 22*T + 121
$47$	$ T^{4} - 2 T^{3} + 78 T^{2} + \cdots + 5476 $ T^4 - 2T^3 + 78T^2 + 148*T + 5476
$53$	$ T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 64 $ T^4 - 16T^3 + 128T^2 - 128*T + 64
$59$	$ T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 1521 $ T^4 + 6T^3 + 45T^2 + 468*T + 1521
$61$	$ T^{4} + 12 T^{3} + 180 T^{2} + \cdots + 1296 $ T^4 + 12T^3 + 180T^2 + 864*T + 1296
$67$	$ T^{4} - 16 T^{3} + 113 T^{2} + \cdots + 1369 $ T^4 - 16T^3 + 113T^2 - 518*T + 1369
$71$	$ T^{4} + 128T^{2} + 1024 $ T^4 + 128*T^2 + 1024
$73$	$ T^{4} + 134T^{2} + 3721 $ T^4 + 134*T^2 + 3721
$79$	$ (T^{2} + 12 T + 144)^{2} $ (T^2 + 12*T + 144)^2
$83$	$ T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 $ T^4 - 2T^3 + 2T^2 - 4*T + 4
$89$	$ (T^{2} + 4)^{2} $ (T^2 + 4)^2
$97$	$ T^{4} + 20 T^{3} + 303 T^{2} + \cdots + 9409 $ T^4 + 20T^3 + 303T^2 + 1940*T + 9409
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.y (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12} + 2) q^{5} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 2) q^{7} + ( - 2 \zeta_{12}^{3} + 2) q^{8} +O(q^{10}) \) q + (z^3 + z^2 - z) * q^2 - 2z q^4 + (-2z^3 + 2z + 2) * q^5 + (-z^3 + z^2 + z - 2) * q^7 + (-2z^3 + 2) q^8	\( q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12} + 2) q^{5} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 2) q^{7} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + (2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{10} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 3) q^{11} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} - 1) q^{13} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12} - 2) q^{14} + 4 \zeta_{12}^{2} q^{16} + (\zeta_{12}^{3} - 2 \zeta_{12} + 4) q^{17} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{19} + ( - 4 \zeta_{12} - 4) q^{20} + (3 \zeta_{12}^{2} - \zeta_{12} + 3) q^{22} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{23} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 8) q^{25} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 2) q^{26} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 2) q^{28} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{29} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{31} + (4 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{32} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 5 \zeta_{12} + 2) q^{34} + (2 \zeta_{12}^{3} - 2) q^{35} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{37} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{38} + ( - 8 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{40} + 3 \zeta_{12} q^{41} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 5 \zeta_{12} - 3) q^{43} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} - 2) q^{44} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{46} + (5 \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12}) q^{47} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{49} + (8 \zeta_{12}^{3} + 7 \zeta_{12}^{2} - \zeta_{12} + 1) q^{50} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 2 \zeta_{12} - 8) q^{52} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 6) q^{53} + ( - 8 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{55} + (4 \zeta_{12} - 4) q^{56} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{58} + ( - \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{59} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6) q^{61} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} - 2) q^{62} - 8 \zeta_{12}^{3} q^{64} + ( - 6 \zeta_{12}^{3} - 14 \zeta_{12}^{2} - 6 \zeta_{12}) q^{65} + (5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{67} + (2 \zeta_{12}^{2} - 8 \zeta_{12} + 2) q^{68} - 4 \zeta_{12}^{2} q^{70} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{71} + (8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{73} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 6 \zeta_{12} + 4) q^{74} + (2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{76} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 7 \zeta_{12} - 7) q^{77} - 12 \zeta_{12}^{2} q^{79} + (8 \zeta_{12}^{2} + 8 \zeta_{12}) q^{80} + (3 \zeta_{12}^{3} - 3) q^{82} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{83} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{85} + ( - 3 \zeta_{12}^{2} + 7 \zeta_{12} - 3) q^{86} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 6 \zeta_{12}) q^{88} - 2 \zeta_{12}^{3} q^{89} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{91} + (2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12}) q^{92} + (10 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{94} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{95} + (\zeta_{12}^{3} - 10 \zeta_{12}^{2} + \zeta_{12}) q^{97} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} - 1) q^{98} +O(q^{100}) \) q + (z^3 + z^2 - z) * q^2 - 2z q^4 + (-2z^3 + 2z + 2) * q^5 + (-z^3 + z^2 + z - 2) * q^7 + (-2z^3 + 2) q^8 + (2z^3 + 4z^2 - 2) * q^10 + (-z^3 - z^2 - 2z + 3) q^11 + (-4z^3 - 3z^2 + z - 1) * q^13 + (-2z^3 + 2z - 2) * q^14 + 4z^2 q^16 + (z^3 - 2z + 4) q^17 + (z^3 - z^2 + z - 1) * q^19 + (-4z - 4) q^20 + (3z^2 - z + 3) q^22 + (-z^2 + 3z - 1) q^23 + (-3z^3 - 4z^2 + 3z + 8) q^25 + (-4z^3 + 8z + 2) * q^26 + (-2z^3 + 4z - 2) * q^28 + (z^3 - z^2 + z + 2) * q^29 + (4z^3 + 4z^2 - 2z - 4) q^31 + (4z^2 - 4z - 4) * q^32 + (3z^3 + 3z^2 - 5z + 2) q^34 + (2z^3 - 2) q^35 + (4z^3 - 2z^2 - 2z + 4) q^37 + (z^3 - 3z^2 + z) q^38 + (-8z^3 - 4z^2 + 4z + 4) q^40 + 3z q^41 + (-2z^3 - 2z^2 + 5z - 3) q^43 + (2z^3 + 6z^2 - 6z - 2) q^44 + (2z^3 - 2z^2 + 2z - 2) q^46 + (5z^3 + z^2 + 5z) * q^47 + (4z^3 + 3z^2 - 2z - 3) q^49 + (8z^3 + 7z^2 - z + 1) * q^50 + (6z^3 + 6z^2 + 2z - 8) q^52 + (6z^3 - 4z^2 - 4z + 6) q^53 + (-8z^3 - 4z^2 + 2) * q^55 + (4z - 4) q^56 + (4z^3 - 2z) * q^58 + (-z^3 + 5z^2 - 4z - 4) * q^59 + (-6z^3 + 6z^2 - 6) * q^61 + (-2z^3 - 4z^2 - 4z - 2) q^62 - 8z^3 q^64 + (-6z^3 - 14z^2 - 6z) q^65 + (5z^3 + 2z^2 - 3z + 3) q^67 + (2z^2 - 8z + 2) * q^68 - 4z^2 q^70 + (-4z^3 - 8z^2 + 4) * q^71 + (8z^3 + 2z^2 - 1) * q^73 + (6z^3 - 2z^2 - 6z + 4) q^74 + (2z^3 - 4z^2 + 2z + 2) q^76 + (-4z^3 + 3z^2 + 7z - 7) q^77 - 12z^2 q^79 + (8z^2 + 8z) * q^80 + (3z^3 - 3) q^82 + (-z^3 + z^2 + z) * q^83 + (-6z^3 + 2z^2 + 4z + 4) q^85 + (-3z^2 + 7z - 3) * q^86 + (-6z^3 + 2z^2 - 6z) q^88 - 2z^3 q^89 + (6z^3 - 2z^2 - 2z + 6) q^91 + (2z^3 - 6z^2 + 2z) q^92 + (10z^3 - 4z^2 - 6z - 6) q^94 + (4z^3 - 2z) * q^95 + (z^3 - 10z^2 + z) q^97 + (-z^3 - 4z^2 - 4z - 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{5} - 6 q^{7} + 8 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^5 - 6 * q^7 + 8 * q^8	\( 4 q + 2 q^{2} + 8 q^{5} - 6 q^{7} + 8 q^{8} + 10 q^{11} - 10 q^{13} - 8 q^{14} + 8 q^{16} + 16 q^{17} - 6 q^{19} - 16 q^{20} + 18 q^{22} - 6 q^{23} + 24 q^{25} + 8 q^{26} - 8 q^{28} + 6 q^{29} - 8 q^{31} - 8 q^{32} + 14 q^{34} - 8 q^{35} + 12 q^{37} - 6 q^{38} + 8 q^{40} - 16 q^{43} + 4 q^{44} - 12 q^{46} + 2 q^{47} - 6 q^{49} + 18 q^{50} - 20 q^{52} + 16 q^{53} - 16 q^{56} - 6 q^{59} - 12 q^{61} - 16 q^{62} - 28 q^{65} + 16 q^{67} + 12 q^{68} - 8 q^{70} + 12 q^{74} - 22 q^{77} - 24 q^{79} + 16 q^{80} - 12 q^{82} + 2 q^{83} + 20 q^{85} - 18 q^{86} + 4 q^{88} + 20 q^{91} - 12 q^{92} - 32 q^{94} - 20 q^{97} - 12 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^5 - 6 * q^7 + 8 * q^8 + 10 * q^11 - 10 * q^13 - 8 * q^14 + 8 * q^16 + 16 * q^17 - 6 * q^19 - 16 * q^20 + 18 * q^22 - 6 * q^23 + 24 * q^25 + 8 * q^26 - 8 * q^28 + 6 * q^29 - 8 * q^31 - 8 * q^32 + 14 * q^34 - 8 * q^35 + 12 * q^37 - 6 * q^38 + 8 * q^40 - 16 * q^43 + 4 * q^44 - 12 * q^46 + 2 * q^47 - 6 * q^49 + 18 * q^50 - 20 * q^52 + 16 * q^53 - 16 * q^56 - 6 * q^59 - 12 * q^61 - 16 * q^62 - 28 * q^65 + 16 * q^67 + 12 * q^68 - 8 * q^70 + 12 * q^74 - 22 * q^77 - 24 * q^79 + 16 * q^80 - 12 * q^82 + 2 * q^83 + 20 * q^85 - 18 * q^86 + 4 * q^88 + 20 * q^91 - 12 * q^92 - 32 * q^94 - 20 * q^97 - 12 * q^98

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Newspace parameters

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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	432.2.y.c

Level	$432$
Weight	$2$
Character orbit	432.y
Analytic conductor	$3.450$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.44953736732\)
Analytic rank:	\(0\)
Dimension:	\(4\)
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:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(\zeta_{12}^{3}\)	\(-1 + \zeta_{12}^{2}\)

$p$	$F_p(T)$
$2$	\( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) T^4 - 2T^3 + 2T^2 - 4*T + 4
$3$	\( T^{4} \) T^4
$5$	\( T^{4} - 8 T^{3} + 20 T^{2} - 16 T + 16 \) T^4 - 8T^3 + 20T^2 - 16*T + 16
$7$	\( T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4 \) T^4 + 6T^3 + 14T^2 + 12*T + 4
$11$	\( T^{4} - 10 T^{3} + 41 T^{2} + \cdots + 169 \) T^4 - 10T^3 + 41T^2 - 104*T + 169
$13$	\( T^{4} + 10 T^{3} + 74 T^{2} + \cdots + 484 \) T^4 + 10T^3 + 74T^2 + 308*T + 484
$17$	\( (T^{2} - 8 T + 13)^{2} \) (T^2 - 8*T + 13)^2
$19$	\( T^{4} + 6 T^{3} + 18 T^{2} + 18 T + 9 \) T^4 + 6T^3 + 18T^2 + 18*T + 9
$23$	\( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) T^4 + 6T^3 + 6T^2 - 36*T + 36
$29$	\( T^{4} - 6 T^{3} + 18 T^{2} - 36 T + 36 \) T^4 - 6T^3 + 18T^2 - 36*T + 36
$31$	\( T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16 \) T^4 + 8T^3 + 60T^2 + 32*T + 16
$37$	\( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 144 \) T^4 - 12T^3 + 72T^2 - 144*T + 144
$41$	\( T^{4} - 9T^{2} + 81 \) T^4 - 9*T^2 + 81
$43$	\( T^{4} + 16 T^{3} + 65 T^{2} + \cdots + 121 \) T^4 + 16T^3 + 65T^2 - 22*T + 121
$47$	\( T^{4} - 2 T^{3} + 78 T^{2} + \cdots + 5476 \) T^4 - 2T^3 + 78T^2 + 148*T + 5476
$53$	\( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 64 \) T^4 - 16T^3 + 128T^2 - 128*T + 64
$59$	\( T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 1521 \) T^4 + 6T^3 + 45T^2 + 468*T + 1521
$61$	\( T^{4} + 12 T^{3} + 180 T^{2} + \cdots + 1296 \) T^4 + 12T^3 + 180T^2 + 864*T + 1296
$67$	\( T^{4} - 16 T^{3} + 113 T^{2} + \cdots + 1369 \) T^4 - 16T^3 + 113T^2 - 518*T + 1369
$71$	\( T^{4} + 128T^{2} + 1024 \) T^4 + 128*T^2 + 1024
$73$	\( T^{4} + 134T^{2} + 3721 \) T^4 + 134*T^2 + 3721
$79$	\( (T^{2} + 12 T + 144)^{2} \) (T^2 + 12*T + 144)^2
$83$	\( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) T^4 - 2T^3 + 2T^2 - 4*T + 4
$89$	\( (T^{2} + 4)^{2} \) (T^2 + 4)^2
$97$	\( T^{4} + 20 T^{3} + 303 T^{2} + \cdots + 9409 \) T^4 + 20T^3 + 303T^2 + 1940*T + 9409
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\(n\):		e.g. 2-40 or 990-1000
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