LMFDB - Newform orbit 3675.1.bf.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3675,1,Mod(68,3675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3675, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 10])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3675.68"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3675.bf (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,8,8,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,-16,0,-8,0,0,0,0,0,0,0,0,8,16,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(61)] == traces)

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

Self dual:	no
Analytic conductor:	$1.83406392143$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{48})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.13897288125.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + ( - \zeta_{48}^{23} + \zeta_{48}^{5}) q^{2} - \zeta_{48}^{10} q^{3} + ( - \zeta_{48}^{22} + \cdots + \zeta_{48}^{4}) q^{4} + ( - \zeta_{48}^{15} - \zeta_{48}^{9}) q^{6} + ( - \zeta_{48}^{21} + \cdots + \zeta_{48}^{3}) q^{8} + \cdots + ( - \zeta_{48}^{23} - \zeta_{48}^{17}) q^{96} +O(q^{100}) $ q + (-z^23 + z^5) * q^2 - z^10 * q^3 + (-z^22 + z^10 + z^4) * q^4 + (-z^15 - z^9) * q^6 + (-z^21 + z^15 + z^9 + z^3) * q^8 + z^20 * q^9 + (-z^20 - z^14 - z^8) * q^12 + (z^14 + z^8 + z^2) * q^16 + (-z^16 + z^4) * q^17 + (z^19 - z) * q^18 + (-z^23 + z^17) * q^19 + (-z^23 - z^5) * q^23 + (-z^19 - z^13 - z^7 + z) * q^24 + z^6 * q^27 + (-z^7 - z) * q^31 + (z^13 + z^7) * q^32 + (-z^21 - z^15 + z^9 + z^3) * q^34 + (z^18 - z^6 - 1) * q^36 + (z^16 + z^4) * q^38 + (-z^22 - z^10) * q^46 + (z^20 + z^8) * q^47 + (-z^18 - z^12 + 1) * q^48 + (-z^14 - z^2) * q^51 + (-z^13 + z^7) * q^53 + (z^11 + z^5) * q^54 + (-z^9 + z^3) * q^57 + (z^23 + z^17) * q^61 + (-z^12 - 2z^6 - 1) q^62 + z^12 * q^64 + (-z^20 + z^8 + 2z^2) q^68 + (z^15 - z^9) * q^69 + (z^23 + z^17 - z^11 - z^5) * q^72 + (z^15 + z^9) * q^76 + (z^14 - z^2) * q^79 - z^16 * q^81 + (-z^12 + 1) * q^83 + (-z^21 - z^15) * q^92 + (z^17 + z^11) * q^93 + (z^19 + z^13 + z^7 - z) * q^94 + (-z^23 - z^17) * q^96
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{12} + 8 q^{16} + 8 q^{17} - 16 q^{36} - 8 q^{38} + 8 q^{47} + 16 q^{48} - 16 q^{62} + 8 q^{68} + 8 q^{81} + 16 q^{83}+O(q^{100}) $ 16 * q - 8 * q^12 + 8 * q^16 + 8 * q^17 - 16 * q^36 - 8 * q^38 + 8 * q^47 + 16 * q^48 - 16 * q^62 + 8 * q^68 + 8 * q^81 + 16 * q^83

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times$.

$n$	$1177$	$1226$	$2551$
$\chi(n)$	$\zeta_{48}^{12}$	$-1$	$\zeta_{48}^{8}$

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} $

68.1

−0.991445	+	0.130526i
−0.130526	−	0.991445i
0.130526	+	0.991445i
0.991445	−	0.130526i
−0.608761	−	0.793353i
0.793353	−	0.608761i
−0.793353	+	0.608761i
0.608761	+	0.793353i
−0.608761	+	0.793353i
0.793353	+	0.608761i
−0.793353	−	0.608761i
0.608761	−	0.793353i
−0.991445	−	0.130526i
−0.130526	+	0.991445i
0.130526	−	0.991445i
0.991445	+	0.130526i

−1.78480

0.478235i

−0.258819

0.965926i

2.09077

−

1.20711i

−

1.84776i

−1.84776

1.84776i

−0.866025

−

0.500000i

68.2

−0.739288

0.198092i

0.258819

−

0.965926i

−0.358719

0.207107i

0.765367i

0.765367

−

0.765367i

−0.866025

−

0.500000i

68.3

0.739288

−

0.198092i

0.258819

−

0.965926i

−0.358719

0.207107i

−

0.765367i

−0.765367

0.765367i

−0.866025

−

0.500000i

68.4

1.78480

−

0.478235i

−0.258819

0.965926i

2.09077

−

1.20711i

1.84776i

1.84776

−

1.84776i

−0.866025

−

0.500000i

668.1

−0.478235

1.78480i

0.965926

−

0.258819i

−2.09077

−

1.20711i

1.84776i

1.84776

−

1.84776i

0.866025

−

0.500000i

668.2

−0.198092

0.739288i

−0.965926

0.258819i

0.358719

0.207107i

−

0.765367i

−0.765367

0.765367i

0.866025

−

0.500000i

668.3

0.198092

−

0.739288i

−0.965926

0.258819i

0.358719

0.207107i

0.765367i

0.765367

−

0.765367i

0.866025

−

0.500000i

668.4

0.478235

−

1.78480i

0.965926

−

0.258819i

−2.09077

−

1.20711i

−

1.84776i

−1.84776

1.84776i

0.866025

−

0.500000i

1832.1

−0.478235

−

1.78480i

0.965926

0.258819i

−2.09077

1.20711i

−

1.84776i

1.84776

1.84776i

0.866025

0.500000i

1832.2

−0.198092

−

0.739288i

−0.965926

−

0.258819i

0.358719

−

0.207107i

0.765367i

−0.765367

−

0.765367i

0.866025

0.500000i

1832.3

0.198092

0.739288i

−0.965926

−

0.258819i

0.358719

−

0.207107i

−

0.765367i

0.765367

0.765367i

0.866025

0.500000i

1832.4

0.478235

1.78480i

0.965926

0.258819i

−2.09077

1.20711i

1.84776i

−1.84776

−

1.84776i

0.866025

0.500000i

2432.1

−1.78480

−

0.478235i

−0.258819

−

0.965926i

2.09077

1.20711i

1.84776i

−1.84776

−

1.84776i

−0.866025

0.500000i

2432.2

−0.739288

−

0.198092i

0.258819

0.965926i

−0.358719

−

0.207107i

−

0.765367i

0.765367

0.765367i

−0.866025

0.500000i

2432.3

0.739288

0.198092i

0.258819

0.965926i

−0.358719

−

0.207107i

0.765367i

−0.765367

−

0.765367i

−0.866025

0.500000i

2432.4

1.78480

0.478235i

−0.258819

−

0.965926i

2.09077

1.20711i

−

1.84776i

1.84776

1.84776i

−0.866025

0.500000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
5.c	odd	4	1	inner
7.c	even	3	1	inner
15.e	even	4	1	inner
21.c	even	2	1	inner
21.g	even	6	1	inner
35.c	odd	2	1	inner
35.f	even	4	1	inner
35.i	odd	6	1	inner
35.k	even	12	1	inner
35.l	odd	12	1	inner
105.k	odd	4	1	inner
105.o	odd	6	1	inner
105.w	odd	12	1	inner
105.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3675.1.bf.c		16
3.b	odd	2	1		3675.1.bf.d		16
5.b	even	2	1		3675.1.bf.d		16
5.c	odd	4	1	inner	3675.1.bf.c		16
5.c	odd	4	1		3675.1.bf.d		16
7.b	odd	2	1		3675.1.bf.d		16
7.c	even	3	1		3675.1.k.c	yes	8
7.c	even	3	1	inner	3675.1.bf.c		16
7.d	odd	6	1		3675.1.k.a	✓	8
7.d	odd	6	1		3675.1.bf.d		16
15.d	odd	2	1	CM	3675.1.bf.c		16
15.e	even	4	1	inner	3675.1.bf.c		16
15.e	even	4	1		3675.1.bf.d		16
21.c	even	2	1	inner	3675.1.bf.c		16
21.g	even	6	1		3675.1.k.c	yes	8
21.g	even	6	1	inner	3675.1.bf.c		16
21.h	odd	6	1		3675.1.k.a	✓	8
21.h	odd	6	1		3675.1.bf.d		16
35.c	odd	2	1	inner	3675.1.bf.c		16
35.f	even	4	1	inner	3675.1.bf.c		16
35.f	even	4	1		3675.1.bf.d		16
35.i	odd	6	1		3675.1.k.c	yes	8
35.i	odd	6	1	inner	3675.1.bf.c		16
35.j	even	6	1		3675.1.k.a	✓	8
35.j	even	6	1		3675.1.bf.d		16
35.k	even	12	1		3675.1.k.a	✓	8
35.k	even	12	1		3675.1.k.c	yes	8
35.k	even	12	1	inner	3675.1.bf.c		16
35.k	even	12	1		3675.1.bf.d		16
35.l	odd	12	1		3675.1.k.a	✓	8
35.l	odd	12	1		3675.1.k.c	yes	8
35.l	odd	12	1	inner	3675.1.bf.c		16
35.l	odd	12	1		3675.1.bf.d		16
105.g	even	2	1		3675.1.bf.d		16
105.k	odd	4	1	inner	3675.1.bf.c		16
105.k	odd	4	1		3675.1.bf.d		16
105.o	odd	6	1		3675.1.k.c	yes	8
105.o	odd	6	1	inner	3675.1.bf.c		16
105.p	even	6	1		3675.1.k.a	✓	8
105.p	even	6	1		3675.1.bf.d		16
105.w	odd	12	1		3675.1.k.a	✓	8
105.w	odd	12	1		3675.1.k.c	yes	8
105.w	odd	12	1	inner	3675.1.bf.c		16
105.w	odd	12	1		3675.1.bf.d		16
105.x	even	12	1		3675.1.k.a	✓	8
105.x	even	12	1		3675.1.k.c	yes	8
105.x	even	12	1	inner	3675.1.bf.c		16
105.x	even	12	1		3675.1.bf.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3675.1.k.a	✓	8	7.d	odd	6	1
3675.1.k.a	✓	8	21.h	odd	6	1
3675.1.k.a	✓	8	35.j	even	6	1
3675.1.k.a	✓	8	35.k	even	12	1
3675.1.k.a	✓	8	35.l	odd	12	1
3675.1.k.a	✓	8	105.p	even	6	1
3675.1.k.a	✓	8	105.w	odd	12	1
3675.1.k.a	✓	8	105.x	even	12	1
3675.1.k.c	yes	8	7.c	even	3	1
3675.1.k.c	yes	8	21.g	even	6	1
3675.1.k.c	yes	8	35.i	odd	6	1
3675.1.k.c	yes	8	35.k	even	12	1
3675.1.k.c	yes	8	35.l	odd	12	1
3675.1.k.c	yes	8	105.o	odd	6	1
3675.1.k.c	yes	8	105.w	odd	12	1
3675.1.k.c	yes	8	105.x	even	12	1
3675.1.bf.c		16	1.a	even	1	1	trivial
3675.1.bf.c		16	5.c	odd	4	1	inner
3675.1.bf.c		16	7.c	even	3	1	inner
3675.1.bf.c		16	15.d	odd	2	1	CM
3675.1.bf.c		16	15.e	even	4	1	inner
3675.1.bf.c		16	21.c	even	2	1	inner
3675.1.bf.c		16	21.g	even	6	1	inner
3675.1.bf.c		16	35.c	odd	2	1	inner
3675.1.bf.c		16	35.f	even	4	1	inner
3675.1.bf.c		16	35.i	odd	6	1	inner
3675.1.bf.c		16	35.k	even	12	1	inner
3675.1.bf.c		16	35.l	odd	12	1	inner
3675.1.bf.c		16	105.k	odd	4	1	inner
3675.1.bf.c		16	105.o	odd	6	1	inner
3675.1.bf.c		16	105.w	odd	12	1	inner
3675.1.bf.c		16	105.x	even	12	1	inner
3675.1.bf.d		16	3.b	odd	2	1
3675.1.bf.d		16	5.b	even	2	1
3675.1.bf.d		16	5.c	odd	4	1
3675.1.bf.d		16	7.b	odd	2	1
3675.1.bf.d		16	7.d	odd	6	1
3675.1.bf.d		16	15.e	even	4	1
3675.1.bf.d		16	21.h	odd	6	1
3675.1.bf.d		16	35.f	even	4	1
3675.1.bf.d		16	35.j	even	6	1
3675.1.bf.d		16	35.k	even	12	1
3675.1.bf.d		16	35.l	odd	12	1
3675.1.bf.d		16	105.g	even	2	1
3675.1.bf.d		16	105.k	odd	4	1
3675.1.bf.d		16	105.p	even	6	1
3675.1.bf.d		16	105.w	odd	12	1
3675.1.bf.d		16	105.x	even	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3675, [\chi])$:

$ T_{2}^{16} - 12T_{2}^{12} + 140T_{2}^{8} - 48T_{2}^{4} + 16 $

T2^16 - 12*T2^12 + 140*T2^8 - 48*T2^4 + 16

$ T_{13} $

T13

$ T_{17}^{4} - 2T_{17}^{3} + 2T_{17}^{2} - 4T_{17} + 4 $

T17^4 - 2*T17^3 + 2*T17^2 - 4*T17 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - 12 T^{12} + \cdots + 16 $ T^16 - 12T^12 + 140T^8 - 48*T^4 + 16
$3$	$ (T^{8} - T^{4} + 1)^{2} $ (T^8 - T^4 + 1)^2
$5$	$ T^{16} $ T^16
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} $ T^16
$17$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} $ (T^4 - 2T^3 + 2T^2 - 4*T + 4)^4
$19$	$ (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$23$	$ T^{16} - 12 T^{12} + \cdots + 16 $ T^16 - 12T^12 + 140T^8 - 48*T^4 + 16
$29$	$ T^{16} $ T^16
$31$	$ (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$37$	$ T^{16} $ T^16
$41$	$ T^{16} $ T^16
$43$	$ T^{16} $ T^16
$47$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} $ (T^4 - 2T^3 + 2T^2 - 4*T + 4)^4
$53$	$ T^{16} - 12 T^{12} + \cdots + 16 $ T^16 - 12T^12 + 140T^8 - 48*T^4 + 16
$59$	$ T^{16} $ T^16
$61$	$ (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} $ T^16
$79$	$ (T^{4} - 2 T^{2} + 4)^{4} $ (T^4 - 2*T^2 + 4)^4
$83$	$ (T^{2} - 2 T + 2)^{8} $ (T^2 - 2*T + 2)^8
$89$	$ T^{16} $ T^16
$97$	$ T^{16} $ T^16
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3675.bf (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{48}^{23} + \zeta_{48}^{5}) q^{2} - \zeta_{48}^{10} q^{3} + ( - \zeta_{48}^{22} + \cdots + \zeta_{48}^{4}) q^{4} + ( - \zeta_{48}^{15} - \zeta_{48}^{9}) q^{6} + ( - \zeta_{48}^{21} + \cdots + \zeta_{48}^{3}) q^{8} + \cdots + ( - \zeta_{48}^{23} - \zeta_{48}^{17}) q^{96} +O(q^{100}) \) q + (-z^23 + z^5) * q^2 - z^10 * q^3 + (-z^22 + z^10 + z^4) * q^4 + (-z^15 - z^9) * q^6 + (-z^21 + z^15 + z^9 + z^3) * q^8 + z^20 * q^9 + (-z^20 - z^14 - z^8) * q^12 + (z^14 + z^8 + z^2) * q^16 + (-z^16 + z^4) * q^17 + (z^19 - z) * q^18 + (-z^23 + z^17) * q^19 + (-z^23 - z^5) * q^23 + (-z^19 - z^13 - z^7 + z) * q^24 + z^6 * q^27 + (-z^7 - z) * q^31 + (z^13 + z^7) * q^32 + (-z^21 - z^15 + z^9 + z^3) * q^34 + (z^18 - z^6 - 1) * q^36 + (z^16 + z^4) * q^38 + (-z^22 - z^10) * q^46 + (z^20 + z^8) * q^47 + (-z^18 - z^12 + 1) * q^48 + (-z^14 - z^2) * q^51 + (-z^13 + z^7) * q^53 + (z^11 + z^5) * q^54 + (-z^9 + z^3) * q^57 + (z^23 + z^17) * q^61 + (-z^12 - 2z^6 - 1) q^62 + z^12 * q^64 + (-z^20 + z^8 + 2z^2) q^68 + (z^15 - z^9) * q^69 + (z^23 + z^17 - z^11 - z^5) * q^72 + (z^15 + z^9) * q^76 + (z^14 - z^2) * q^79 - z^16 * q^81 + (-z^12 + 1) * q^83 + (-z^21 - z^15) * q^92 + (z^17 + z^11) * q^93 + (z^19 + z^13 + z^7 - z) * q^94 + (-z^23 - z^17) * q^96
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{12} + 8 q^{16} + 8 q^{17} - 16 q^{36} - 8 q^{38} + 8 q^{47} + 16 q^{48} - 16 q^{62} + 8 q^{68} + 8 q^{81} + 16 q^{83}+O(q^{100}) \) 16 * q - 8 * q^12 + 8 * q^16 + 8 * q^17 - 16 * q^36 - 8 * q^38 + 8 * q^47 + 16 * q^48 - 16 * q^62 + 8 * q^68 + 8 * q^81 + 16 * q^83

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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	3675.1.bf.c

Level	$3675$
Weight	$1$
Character orbit	3675.bf
Analytic conductor	$1.834$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{8}$
CM discriminant	-15
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(1.83406392143\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\Q(\zeta_{48})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{8} + 1 \) x^16 - x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Projective image:	\(D_{8}\)
Projective field:	Galois closure of 8.0.13897288125.2

\(n\)	\(1177\)	\(1226\)	\(2551\)
\(\chi(n)\)	\(\zeta_{48}^{12}\)	\(-1\)	\(\zeta_{48}^{8}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 68.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - 12 T^{12} + \cdots + 16 \) T^16 - 12T^12 + 140T^8 - 48*T^4 + 16
$3$	\( (T^{8} - T^{4} + 1)^{2} \) (T^8 - T^4 + 1)^2
$5$	\( T^{16} \) T^16
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} \) T^16
$17$	\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} \) (T^4 - 2T^3 + 2T^2 - 4*T + 4)^4
$19$	\( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$23$	\( T^{16} - 12 T^{12} + \cdots + 16 \) T^16 - 12T^12 + 140T^8 - 48*T^4 + 16
$29$	\( T^{16} \) T^16
$31$	\( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$37$	\( T^{16} \) T^16
$41$	\( T^{16} \) T^16
$43$	\( T^{16} \) T^16
$47$	\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} \) (T^4 - 2T^3 + 2T^2 - 4*T + 4)^4
$53$	\( T^{16} - 12 T^{12} + \cdots + 16 \) T^16 - 12T^12 + 140T^8 - 48*T^4 + 16
$59$	\( T^{16} \) T^16
$61$	\( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} \) T^16
$79$	\( (T^{4} - 2 T^{2} + 4)^{4} \) (T^4 - 2*T^2 + 4)^4
$83$	\( (T^{2} - 2 T + 2)^{8} \) (T^2 - 2*T + 2)^8
$89$	\( T^{16} \) T^16
$97$	\( T^{16} \) T^16
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