LMFDB - Newform orbit 390.2.l.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [390,2,Mod(53,390)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(390, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("390.53"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	390.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$3.11416567883$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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 $\zeta_{8}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8} + 1) q^{3} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + ( - \zeta_{8}^{2} + \zeta_{8} + 1) q^{6} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{7}+ \cdots + (8 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 8 \zeta_{8}) q^{99}+O(q^{100}) $ q + z * q^2 + (-z^3 - z + 1) * q^3 + z^2 * q^4 + (-z^3 - 2z) q^5 + (-z^2 + z + 1) * q^6 + (2z^3 - 2z^2 + 2) * q^7 + z^3 * q^8 + (-2z^3 - 2z - 1) * q^9 + (-2z^2 + 1) q^10 - 4z^2 q^11 + (-z^3 + z^2 + z) * q^12 - z * q^13 + (-2z^3 + 2z - 2) * q^14 + (-z^3 + z^2 - 2z - 3) q^15 - q^16 + (3z^2 + 3) q^17 + (-2z^2 - z + 2) q^18 + (2z^3 + 4z^2 + 2z) q^19 + (-2z^3 + z) q^20 + (2z^3 - 4z + 4) * q^21 - 4z^3 q^22 - 4z^3 q^23 + (z^3 + z^2 + 1) * q^24 + (3z^2 - 4) q^25 - z^2 * q^26 + (-z^3 - z - 5) * q^27 + (2z^2 - 2z + 2) * q^28 + 6 * q^29 + (z^3 - 2z^2 - 3z + 1) * q^30 + (-z^3 + z + 2) * q^31 - z * q^32 + (4z^3 - 4z^2 - 4z) q^33 + (3z^3 + 3z) * q^34 + (2z^3 + 2z^2 - 6z + 4) q^35 + (-2z^3 - z^2 + 2z) * q^36 + (-2z^3 + 4z^2 - 4) * q^37 + (4z^3 + 2z^2 - 2) * q^38 + (z^2 - z - 1) * q^39 + (z^2 + 2) * q^40 - 2z^2 q^41 + (-4z^2 + 4z - 2) * q^42 + (-3z^2 + 6z - 3) * q^43 + 4 * q^44 + (z^3 + 2z^2 + 2z - 6) * q^45 + 4 * q^46 + (-2z^2 + 6z - 2) * q^47 + (z^3 + z - 1) * q^48 + (8z^3 - 5z^2 + 8z) q^49 + (3z^3 - 4z) * q^50 + (-6z^3 + 3z^2 + 3) * q^51 - z^3 * q^52 + (-2z^3 + 2z^2 - 2) * q^53 + (-z^2 - 5z + 1) q^54 + (8z^3 - 4z) * q^55 + (2z^3 - 2z^2 + 2z) q^56 + (-2z^3 + 4z^2 + 6z + 4) q^57 + 6z q^58 + (6z^3 - 6z + 4) * q^59 + (-2z^3 - 3z^2 + z - 1) * q^60 + (-2z^3 + 2z + 2) * q^61 + (z^2 + 2z + 1) q^62 + (-2z^3 + 6z^2 - 8z + 2) q^63 - z^2 * q^64 + (2z^2 - 1) q^65 + (-4z^3 - 4z^2 - 4) * q^66 + (4z^3 - 2z^2 + 2) * q^67 + (3z^2 - 3) q^68 + (-4z^3 - 4z^2 - 4) * q^69 + (2z^3 - 6z^2 + 4z - 2) q^70 + (3z^3 + 10z^2 + 3z) q^71 + (-z^3 + 2z^2 + 2) q^72 + (-4z^2 - 10z - 4) * q^73 + (4z^3 - 4z + 2) * q^74 + (z^3 + 3z^2 + 7z - 4) * q^75 + (2z^3 - 2z - 4) * q^76 + (-8z^2 + 8z - 8) * q^77 + (z^3 - z^2 - z) * q^78 + (6z^3 + 4z^2 + 6z) q^79 + (z^3 + 2z) q^80 + (4z^3 + 4z - 7) * q^81 - 2z^3 q^82 + (-4z^3 - 4z^2 + 4) * q^83 + (-4z^3 + 4z^2 - 2z) q^84 + (-9z^3 - 3z) * q^85 + (-3z^3 + 6z^2 - 3z) q^86 + (-6z^3 - 6z + 6) * q^87 + 4z q^88 + (-2z^3 + 2z - 14) * q^89 + (2z^3 + 2z^2 - 6z - 1) q^90 + (2z^3 - 2z + 2) * q^91 + 4z q^92 + (-3z^3 - 2z^2 - z + 2) * q^93 + (-2z^3 + 6z^2 - 2z) q^94 + (-8z^3 - 2z^2 + 4z + 6) q^95 + (z^2 - z - 1) * q^96 + (-10z^3 - 2z^2 + 2) * q^97 + (-5z^3 + 8z^2 - 8) * q^98 + (8z^3 + 4z^2 - 8z) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{6} + 8 q^{7} - 4 q^{9} + 4 q^{10} - 8 q^{14} - 12 q^{15} - 4 q^{16} + 12 q^{17} + 8 q^{18} + 16 q^{21} + 4 q^{24} - 16 q^{25} - 20 q^{27} + 8 q^{28} + 24 q^{29} + 4 q^{30} + 8 q^{31}+ \cdots - 32 q^{98}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^6 + 8 * q^7 - 4 * q^9 + 4 * q^10 - 8 * q^14 - 12 * q^15 - 4 * q^16 + 12 * q^17 + 8 * q^18 + 16 * q^21 + 4 * q^24 - 16 * q^25 - 20 * q^27 + 8 * q^28 + 24 * q^29 + 4 * q^30 + 8 * q^31 + 16 * q^35 - 16 * q^37 - 8 * q^38 - 4 * q^39 + 8 * q^40 - 8 * q^42 - 12 * q^43 + 16 * q^44 - 24 * q^45 + 16 * q^46 - 8 * q^47 - 4 * q^48 + 12 * q^51 - 8 * q^53 + 4 * q^54 + 16 * q^57 + 16 * q^59 - 4 * q^60 + 8 * q^61 + 4 * q^62 + 8 * q^63 - 4 * q^65 - 16 * q^66 + 8 * q^67 - 12 * q^68 - 16 * q^69 - 8 * q^70 + 8 * q^72 - 16 * q^73 + 8 * q^74 - 16 * q^75 - 16 * q^76 - 32 * q^77 - 28 * q^81 + 16 * q^83 + 24 * q^87 - 56 * q^89 - 4 * q^90 + 8 * q^91 + 8 * q^93 + 24 * q^95 - 4 * q^96 + 8 * q^97 - 32 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$.

$n$	$131$	$157$	$301$
$\chi(n)$	$-1$	$-\zeta_{8}^{2}$	$1$

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} $

53.1

−0.707107	−	0.707107i
0.707107	+	0.707107i
−0.707107	+	0.707107i
0.707107	−	0.707107i

−0.707107

−

0.707107i

1.00000

1.41421i

1.00000i

0.707107

2.12132i

0.292893

−

1.70711i

3.41421

−

3.41421i

0.707107

−

0.707107i

−1.00000

2.82843i

1.00000

−

2.00000i

53.2

0.707107

0.707107i

1.00000

−

1.41421i

1.00000i

−0.707107

−

2.12132i

1.70711

−

0.292893i

0.585786

−

0.585786i

−0.707107

0.707107i

−1.00000

−

2.82843i

1.00000

−

2.00000i

287.1

−0.707107

0.707107i

1.00000

−

1.41421i

−

1.00000i

0.707107

−

2.12132i

0.292893

1.70711i

3.41421

3.41421i

0.707107

0.707107i

−1.00000

−

2.82843i

1.00000

2.00000i

287.2

0.707107

−

0.707107i

1.00000

1.41421i

−

1.00000i

−0.707107

2.12132i

1.70711

0.292893i

0.585786

0.585786i

−0.707107

−

0.707107i

−1.00000

2.82843i

1.00000

2.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	390.2.l.b	yes	4
3.b	odd	2	1		390.2.l.a	✓	4
5.c	odd	4	1		390.2.l.a	✓	4
15.e	even	4	1	inner	390.2.l.b	yes	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.2.l.a	✓	4	3.b	odd	2	1
390.2.l.a	✓	4	5.c	odd	4	1
390.2.l.b	yes	4	1.a	even	1	1	trivial
390.2.l.b	yes	4	15.e	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(390, [\chi])$:

$ T_{7}^{4} - 8T_{7}^{3} + 32T_{7}^{2} - 32T_{7} + 16 $

T7^4 - 8*T7^3 + 32*T7^2 - 32*T7 + 16

$ T_{17}^{2} - 6T_{17} + 18 $

T17^2 - 6*T17 + 18

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} + 1 $ T^4 + 1
$3$	$ (T^{2} - 2 T + 3)^{2} $ (T^2 - 2*T + 3)^2
$5$	$ T^{4} + 8T^{2} + 25 $ T^4 + 8*T^2 + 25
$7$	$ T^{4} - 8 T^{3} + \cdots + 16 $ T^4 - 8T^3 + 32T^2 - 32*T + 16
$11$	$ (T^{2} + 16)^{2} $ (T^2 + 16)^2
$13$	$ T^{4} + 1 $ T^4 + 1
$17$	$ (T^{2} - 6 T + 18)^{2} $ (T^2 - 6*T + 18)^2
$19$	$ T^{4} + 48T^{2} + 64 $ T^4 + 48*T^2 + 64
$23$	$ T^{4} + 256 $ T^4 + 256
$29$	$ (T - 6)^{4} $ (T - 6)^4
$31$	$ (T^{2} - 4 T + 2)^{2} $ (T^2 - 4*T + 2)^2
$37$	$ T^{4} + 16 T^{3} + \cdots + 784 $ T^4 + 16T^3 + 128T^2 + 448*T + 784
$41$	$ (T^{2} + 4)^{2} $ (T^2 + 4)^2
$43$	$ T^{4} + 12 T^{3} + \cdots + 324 $ T^4 + 12T^3 + 72T^2 - 216*T + 324
$47$	$ T^{4} + 8 T^{3} + \cdots + 784 $ T^4 + 8T^3 + 32T^2 - 224*T + 784
$53$	$ T^{4} + 8 T^{3} + \cdots + 16 $ T^4 + 8T^3 + 32T^2 + 32*T + 16
$59$	$ (T^{2} - 8 T - 56)^{2} $ (T^2 - 8*T - 56)^2
$61$	$ (T^{2} - 4 T - 4)^{2} $ (T^2 - 4*T - 4)^2
$67$	$ T^{4} - 8 T^{3} + \cdots + 64 $ T^4 - 8T^3 + 32T^2 + 64*T + 64
$71$	$ T^{4} + 236T^{2} + 6724 $ T^4 + 236*T^2 + 6724
$73$	$ T^{4} + 16 T^{3} + \cdots + 4624 $ T^4 + 16T^3 + 128T^2 - 1088*T + 4624
$79$	$ T^{4} + 176T^{2} + 3136 $ T^4 + 176*T^2 + 3136
$83$	$ T^{4} - 16 T^{3} + \cdots + 256 $ T^4 - 16T^3 + 128T^2 - 256*T + 256
$89$	$ (T^{2} + 28 T + 188)^{2} $ (T^2 + 28*T + 188)^2
$97$	$ T^{4} - 8 T^{3} + \cdots + 8464 $ T^4 - 8T^3 + 32T^2 + 736*T + 8464
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Belyi maps

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8} + 1) q^{3} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + ( - \zeta_{8}^{2} + \zeta_{8} + 1) q^{6} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{7}+ \cdots + (8 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 8 \zeta_{8}) q^{99}+O(q^{100}) \) q + z * q^2 + (-z^3 - z + 1) * q^3 + z^2 * q^4 + (-z^3 - 2z) q^5 + (-z^2 + z + 1) * q^6 + (2z^3 - 2z^2 + 2) * q^7 + z^3 * q^8 + (-2z^3 - 2z - 1) * q^9 + (-2z^2 + 1) q^10 - 4z^2 q^11 + (-z^3 + z^2 + z) * q^12 - z * q^13 + (-2z^3 + 2z - 2) * q^14 + (-z^3 + z^2 - 2z - 3) q^15 - q^16 + (3z^2 + 3) q^17 + (-2z^2 - z + 2) q^18 + (2z^3 + 4z^2 + 2z) q^19 + (-2z^3 + z) q^20 + (2z^3 - 4z + 4) * q^21 - 4z^3 q^22 - 4z^3 q^23 + (z^3 + z^2 + 1) * q^24 + (3z^2 - 4) q^25 - z^2 * q^26 + (-z^3 - z - 5) * q^27 + (2z^2 - 2z + 2) * q^28 + 6 * q^29 + (z^3 - 2z^2 - 3z + 1) * q^30 + (-z^3 + z + 2) * q^31 - z * q^32 + (4z^3 - 4z^2 - 4z) q^33 + (3z^3 + 3z) * q^34 + (2z^3 + 2z^2 - 6z + 4) q^35 + (-2z^3 - z^2 + 2z) * q^36 + (-2z^3 + 4z^2 - 4) * q^37 + (4z^3 + 2z^2 - 2) * q^38 + (z^2 - z - 1) * q^39 + (z^2 + 2) * q^40 - 2z^2 q^41 + (-4z^2 + 4z - 2) * q^42 + (-3z^2 + 6z - 3) * q^43 + 4 * q^44 + (z^3 + 2z^2 + 2z - 6) * q^45 + 4 * q^46 + (-2z^2 + 6z - 2) * q^47 + (z^3 + z - 1) * q^48 + (8z^3 - 5z^2 + 8z) q^49 + (3z^3 - 4z) * q^50 + (-6z^3 + 3z^2 + 3) * q^51 - z^3 * q^52 + (-2z^3 + 2z^2 - 2) * q^53 + (-z^2 - 5z + 1) q^54 + (8z^3 - 4z) * q^55 + (2z^3 - 2z^2 + 2z) q^56 + (-2z^3 + 4z^2 + 6z + 4) q^57 + 6z q^58 + (6z^3 - 6z + 4) * q^59 + (-2z^3 - 3z^2 + z - 1) * q^60 + (-2z^3 + 2z + 2) * q^61 + (z^2 + 2z + 1) q^62 + (-2z^3 + 6z^2 - 8z + 2) q^63 - z^2 * q^64 + (2z^2 - 1) q^65 + (-4z^3 - 4z^2 - 4) * q^66 + (4z^3 - 2z^2 + 2) * q^67 + (3z^2 - 3) q^68 + (-4z^3 - 4z^2 - 4) * q^69 + (2z^3 - 6z^2 + 4z - 2) q^70 + (3z^3 + 10z^2 + 3z) q^71 + (-z^3 + 2z^2 + 2) q^72 + (-4z^2 - 10z - 4) * q^73 + (4z^3 - 4z + 2) * q^74 + (z^3 + 3z^2 + 7z - 4) * q^75 + (2z^3 - 2z - 4) * q^76 + (-8z^2 + 8z - 8) * q^77 + (z^3 - z^2 - z) * q^78 + (6z^3 + 4z^2 + 6z) q^79 + (z^3 + 2z) q^80 + (4z^3 + 4z - 7) * q^81 - 2z^3 q^82 + (-4z^3 - 4z^2 + 4) * q^83 + (-4z^3 + 4z^2 - 2z) q^84 + (-9z^3 - 3z) * q^85 + (-3z^3 + 6z^2 - 3z) q^86 + (-6z^3 - 6z + 6) * q^87 + 4z q^88 + (-2z^3 + 2z - 14) * q^89 + (2z^3 + 2z^2 - 6z - 1) q^90 + (2z^3 - 2z + 2) * q^91 + 4z q^92 + (-3z^3 - 2z^2 - z + 2) * q^93 + (-2z^3 + 6z^2 - 2z) q^94 + (-8z^3 - 2z^2 + 4z + 6) q^95 + (z^2 - z - 1) * q^96 + (-10z^3 - 2z^2 + 2) * q^97 + (-5z^3 + 8z^2 - 8) * q^98 + (8z^3 + 4z^2 - 8z) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{6} + 8 q^{7} - 4 q^{9} + 4 q^{10} - 8 q^{14} - 12 q^{15} - 4 q^{16} + 12 q^{17} + 8 q^{18} + 16 q^{21} + 4 q^{24} - 16 q^{25} - 20 q^{27} + 8 q^{28} + 24 q^{29} + 4 q^{30} + 8 q^{31}+ \cdots - 32 q^{98}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^6 + 8 * q^7 - 4 * q^9 + 4 * q^10 - 8 * q^14 - 12 * q^15 - 4 * q^16 + 12 * q^17 + 8 * q^18 + 16 * q^21 + 4 * q^24 - 16 * q^25 - 20 * q^27 + 8 * q^28 + 24 * q^29 + 4 * q^30 + 8 * q^31 + 16 * q^35 - 16 * q^37 - 8 * q^38 - 4 * q^39 + 8 * q^40 - 8 * q^42 - 12 * q^43 + 16 * q^44 - 24 * q^45 + 16 * q^46 - 8 * q^47 - 4 * q^48 + 12 * q^51 - 8 * q^53 + 4 * q^54 + 16 * q^57 + 16 * q^59 - 4 * q^60 + 8 * q^61 + 4 * q^62 + 8 * q^63 - 4 * q^65 - 16 * q^66 + 8 * q^67 - 12 * q^68 - 16 * q^69 - 8 * q^70 + 8 * q^72 - 16 * q^73 + 8 * q^74 - 16 * q^75 - 16 * q^76 - 32 * q^77 - 28 * q^81 + 16 * q^83 + 24 * q^87 - 56 * q^89 - 4 * q^90 + 8 * q^91 + 8 * q^93 + 24 * q^95 - 4 * q^96 + 8 * q^97 - 32 * q^98

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

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	390.2.l.b

Level	$390$
Weight	$2$
Character orbit	390.l
Analytic conductor	$3.114$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.11416567883\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{8})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} + 1 \) x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(n\)	\(131\)	\(157\)	\(301\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{8}^{2}\)	\(1\)

$p$	$F_p(T)$
$2$	\( T^{4} + 1 \) T^4 + 1
$3$	\( (T^{2} - 2 T + 3)^{2} \) (T^2 - 2*T + 3)^2
$5$	\( T^{4} + 8T^{2} + 25 \) T^4 + 8*T^2 + 25
$7$	\( T^{4} - 8 T^{3} + \cdots + 16 \) T^4 - 8T^3 + 32T^2 - 32*T + 16
$11$	\( (T^{2} + 16)^{2} \) (T^2 + 16)^2
$13$	\( T^{4} + 1 \) T^4 + 1
$17$	\( (T^{2} - 6 T + 18)^{2} \) (T^2 - 6*T + 18)^2
$19$	\( T^{4} + 48T^{2} + 64 \) T^4 + 48*T^2 + 64
$23$	\( T^{4} + 256 \) T^4 + 256
$29$	\( (T - 6)^{4} \) (T - 6)^4
$31$	\( (T^{2} - 4 T + 2)^{2} \) (T^2 - 4*T + 2)^2
$37$	\( T^{4} + 16 T^{3} + \cdots + 784 \) T^4 + 16T^3 + 128T^2 + 448*T + 784
$41$	\( (T^{2} + 4)^{2} \) (T^2 + 4)^2
$43$	\( T^{4} + 12 T^{3} + \cdots + 324 \) T^4 + 12T^3 + 72T^2 - 216*T + 324
$47$	\( T^{4} + 8 T^{3} + \cdots + 784 \) T^4 + 8T^3 + 32T^2 - 224*T + 784
$53$	\( T^{4} + 8 T^{3} + \cdots + 16 \) T^4 + 8T^3 + 32T^2 + 32*T + 16
$59$	\( (T^{2} - 8 T - 56)^{2} \) (T^2 - 8*T - 56)^2
$61$	\( (T^{2} - 4 T - 4)^{2} \) (T^2 - 4*T - 4)^2
$67$	\( T^{4} - 8 T^{3} + \cdots + 64 \) T^4 - 8T^3 + 32T^2 + 64*T + 64
$71$	\( T^{4} + 236T^{2} + 6724 \) T^4 + 236*T^2 + 6724
$73$	\( T^{4} + 16 T^{3} + \cdots + 4624 \) T^4 + 16T^3 + 128T^2 - 1088*T + 4624
$79$	\( T^{4} + 176T^{2} + 3136 \) T^4 + 176*T^2 + 3136
$83$	\( T^{4} - 16 T^{3} + \cdots + 256 \) T^4 - 16T^3 + 128T^2 - 256*T + 256
$89$	\( (T^{2} + 28 T + 188)^{2} \) (T^2 + 28*T + 188)^2
$97$	\( T^{4} - 8 T^{3} + \cdots + 8464 \) T^4 - 8T^3 + 32T^2 + 736*T + 8464
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\(n\):		e.g. 2-40 or 990-1000
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