LMFDB - Newform orbit 102.2.h.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [102,2,Mod(19,102)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(102, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 7]))

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

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

chi := DirichletCharacter("102.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 102 = 2 \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	102.h (of order $8$, degree $4$, 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:	$0.814474100617$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/102\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$1$	$\zeta_{16}^{6}$

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} $

19.1

0.382683	−	0.923880i
−0.382683	+	0.923880i
−0.923880	+	0.382683i
0.923880	−	0.382683i
0.382683	+	0.923880i
−0.382683	−	0.923880i
−0.923880	−	0.382683i
0.923880	+	0.382683i

0.707107

0.707107i

−0.923880

0.382683i

1.00000i

0.834089

2.01367i

−0.923880

−

0.382683i

0.224171

−

0.541196i

−0.707107

0.707107i

0.707107

−

0.707107i

−0.834089

2.01367i

19.2

0.707107

0.707107i

0.923880

−

0.382683i

1.00000i

−0.248303

−

0.599456i

0.923880

0.382683i

−0.224171

0.541196i

−0.707107

0.707107i

0.707107

−

0.707107i

0.248303

−

0.599456i

25.1

−0.707107

0.707107i

−0.382683

0.923880i

−

1.00000i

3.01367

1.24830i

−0.382683

−

0.923880i

−3.15432

1.30656i

0.707107

0.707107i

−0.707107

−

0.707107i

−3.01367

1.24830i

25.2

−0.707107

0.707107i

0.382683

−

0.923880i

−

1.00000i

0.400544

0.165911i

0.382683

0.923880i

3.15432

−

1.30656i

0.707107

0.707107i

−0.707107

−

0.707107i

−0.400544

0.165911i

43.1

0.707107

−

0.707107i

−0.923880

−

0.382683i

−

1.00000i

0.834089

−

2.01367i

−0.923880

0.382683i

0.224171

0.541196i

−0.707107

−

0.707107i

0.707107

0.707107i

−0.834089

−

2.01367i

43.2

0.707107

−

0.707107i

0.923880

0.382683i

−

1.00000i

−0.248303

0.599456i

0.923880

−

0.382683i

−0.224171

−

0.541196i

−0.707107

−

0.707107i

0.707107

0.707107i

0.248303

0.599456i

49.1

−0.707107

−

0.707107i

−0.382683

−

0.923880i

1.00000i

3.01367

−

1.24830i

−0.382683

0.923880i

−3.15432

−

1.30656i

0.707107

−

0.707107i

−0.707107

0.707107i

−3.01367

−

1.24830i

49.2

−0.707107

−

0.707107i

0.382683

0.923880i

1.00000i

0.400544

−

0.165911i

0.382683

−

0.923880i

3.15432

1.30656i

0.707107

−

0.707107i

−0.707107

0.707107i

−0.400544

−

0.165911i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.d	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	102.2.h.a	✓	8
3.b	odd	2	1		306.2.l.d		8
4.b	odd	2	1		816.2.bq.b		8
17.d	even	8	1	inner	102.2.h.a	✓	8
17.e	odd	16	1		1734.2.a.v		4
17.e	odd	16	1		1734.2.a.w		4
17.e	odd	16	2		1734.2.b.k		8
17.e	odd	16	2		1734.2.f.j		8
17.e	odd	16	2		1734.2.f.m		8
51.g	odd	8	1		306.2.l.d		8
51.i	even	16	1		5202.2.a.br		4
51.i	even	16	1		5202.2.a.bt		4
68.g	odd	8	1		816.2.bq.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
102.2.h.a	✓	8	1.a	even	1	1	trivial
102.2.h.a	✓	8	17.d	even	8	1	inner
306.2.l.d		8	3.b	odd	2	1
306.2.l.d		8	51.g	odd	8	1
816.2.bq.b		8	4.b	odd	2	1
816.2.bq.b		8	68.g	odd	8	1
1734.2.a.v		4	17.e	odd	16	1
1734.2.a.w		4	17.e	odd	16	1
1734.2.b.k		8	17.e	odd	16	2
1734.2.f.j		8	17.e	odd	16	2
1734.2.f.m		8	17.e	odd	16	2
5202.2.a.br		4	51.i	even	16	1
5202.2.a.bt		4	51.i	even	16	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 8T_{5}^{7} + 28T_{5}^{6} - 56T_{5}^{5} + 72T_{5}^{4} - 32T_{5}^{3} + 24T_{5}^{2} - 16T_{5} + 4 $ T5^8 - 8*T5^7 + 28*T5^6 - 56*T5^5 + 72*T5^4 - 32*T5^3 + 24*T5^2 - 16*T5 + 4 acting on $S_{2}^{\mathrm{new}}(102, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$3$	$ T^{8} + 1 $ T^8 + 1
$5$	$ T^{8} - 8 T^{7} + \cdots + 4 $ T^8 - 8T^7 + 28T^6 - 56T^5 + 72T^4 - 32T^3 + 24T^2 - 16*T + 4
$7$	$ T^{8} - 16 T^{6} + \cdots + 16 $ T^8 - 16T^6 + 128T^4 + 64*T^2 + 16
$11$	$ T^{8} + 16 T^{6} + \cdots + 256 $ T^8 + 16T^6 + 128T^5 + 128T^4 - 256T^3 + 1280T^2 - 1024T + 256
$13$	$ T^{8} + 24 T^{6} + \cdots + 16 $ T^8 + 24T^6 + 136T^4 + 160*T^2 + 16
$17$	$ T^{8} + 8 T^{7} + \cdots + 83521 $ T^8 + 8T^7 + 12T^6 - 136T^5 - 922T^4 - 2312T^3 + 3468T^2 + 39304*T + 83521
$19$	$ T^{8} - 32 T^{5} + \cdots + 61504 $ T^8 - 32T^5 + 1808T^4 - 1536T^3 + 512T^2 + 7936*T + 61504
$23$	$ T^{8} + 16 T^{7} + \cdots + 15376 $ T^8 + 16T^7 + 128T^6 + 640T^5 + 2048T^4 + 4096T^3 + 6400T^2 + 11904*T + 15376
$29$	$ T^{8} - 12 T^{6} + \cdots + 749956 $ T^8 - 12T^6 + 456T^5 + 72T^4 - 22992T^3 + 143112T^2 - 436464T + 749956
$31$	$ T^{8} - 32 T^{6} + \cdots + 38416 $ T^8 - 32T^6 + 512T^4 + 6272*T^2 + 38416
$37$	$ T^{8} - 44 T^{6} + \cdots + 148996 $ T^8 - 44T^6 - 1032T^5 + 968T^4 + 122064T^3 + 712520T^2 + 268656T + 148996
$41$	$ T^{8} - 16 T^{7} + \cdots + 264196 $ T^8 - 16T^7 + 36T^6 + 328T^5 + 2568T^4 + 9904T^3 + 29016T^2 + 102800*T + 264196
$43$	$ T^{8} + 16 T^{7} + \cdots + 13075456 $ T^8 + 16T^7 + 128T^6 + 640T^5 + 11328T^4 + 164352T^3 + 1384448T^2 + 6017024*T + 13075456
$47$	$ T^{8} + 112 T^{6} + \cdots + 61504 $ T^8 + 112T^6 + 2640T^4 + 22400*T^2 + 61504
$53$	$ T^{8} - 8 T^{7} + \cdots + 15376 $ T^8 - 8T^7 + 32T^6 + 464T^5 + 15128T^4 - 64480T^3 + 139392T^2 + 65472*T + 15376
$59$	$ T^{8} + 224 T^{5} + \cdots + 3655744 $ T^8 + 224T^5 + 8720T^4 + 25088T^3 + 25088T^2 - 428288*T + 3655744
$61$	$ T^{8} + 32 T^{7} + \cdots + 857476 $ T^8 + 32T^7 + 484T^6 + 4552T^5 + 29320T^4 + 130992T^3 + 399464T^2 + 792656*T + 857476
$67$	$ (T^{4} - 24 T^{3} + \cdots - 2312)^{2} $ (T^4 - 24T^3 + 64T^2 + 1184*T - 2312)^2
$71$	$ T^{8} + 16 T^{7} + \cdots + 4624 $ T^8 + 16T^7 - 32T^6 - 1280T^5 + 4608T^4 + 6656T^3 + 88960T^2 - 15232*T + 4624
$73$	$ T^{8} + 24 T^{7} + \cdots + 62948356 $ T^8 + 24T^7 + 284T^6 + 1352T^5 - 7864T^4 - 141440T^3 - 26584T^2 + 3617904*T + 62948356
$79$	$ T^{8} + 208 T^{6} + \cdots + 99856 $ T^8 + 208T^6 - 4736T^5 + 21632T^4 - 297728T^3 + 6901952T^2 + 1537024T + 99856
$83$	$ T^{8} + 32 T^{7} + \cdots + 141376 $ T^8 + 32T^7 + 512T^6 + 4896T^5 + 30224T^4 + 117504T^3 + 270848T^2 + 276736*T + 141376
$89$	$ T^{8} + 360 T^{6} + \cdots + 4624 $ T^8 + 360T^6 + 32536T^4 + 50080*T^2 + 4624
$97$	$ T^{8} - 16 T^{7} + \cdots + 219573124 $ T^8 - 16T^7 + 228T^6 - 7960T^5 + 111368T^4 - 591248T^3 + 16891608T^2 - 108941936*T + 219573124
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 102 = 2 \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	102.h (of order \(8\), degree \(4\), minimal)

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

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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	102.2.h.a

Level	$102$
Weight	$2$
Character orbit	102.h
Analytic conductor	$0.814$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.814474100617\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{8})\)
Coefficient field:	\(\Q(\zeta_{16})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 1 \) x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 1)^{2} \) (T^4 + 1)^2
$3$	\( T^{8} + 1 \) T^8 + 1
$5$	\( T^{8} - 8 T^{7} + \cdots + 4 \) T^8 - 8T^7 + 28T^6 - 56T^5 + 72T^4 - 32T^3 + 24T^2 - 16*T + 4
$7$	\( T^{8} - 16 T^{6} + \cdots + 16 \) T^8 - 16T^6 + 128T^4 + 64*T^2 + 16
$11$	\( T^{8} + 16 T^{6} + \cdots + 256 \) T^8 + 16T^6 + 128T^5 + 128T^4 - 256T^3 + 1280T^2 - 1024T + 256
$13$	\( T^{8} + 24 T^{6} + \cdots + 16 \) T^8 + 24T^6 + 136T^4 + 160*T^2 + 16
$17$	\( T^{8} + 8 T^{7} + \cdots + 83521 \) T^8 + 8T^7 + 12T^6 - 136T^5 - 922T^4 - 2312T^3 + 3468T^2 + 39304*T + 83521
$19$	\( T^{8} - 32 T^{5} + \cdots + 61504 \) T^8 - 32T^5 + 1808T^4 - 1536T^3 + 512T^2 + 7936*T + 61504
$23$	\( T^{8} + 16 T^{7} + \cdots + 15376 \) T^8 + 16T^7 + 128T^6 + 640T^5 + 2048T^4 + 4096T^3 + 6400T^2 + 11904*T + 15376
$29$	\( T^{8} - 12 T^{6} + \cdots + 749956 \) T^8 - 12T^6 + 456T^5 + 72T^4 - 22992T^3 + 143112T^2 - 436464T + 749956
$31$	\( T^{8} - 32 T^{6} + \cdots + 38416 \) T^8 - 32T^6 + 512T^4 + 6272*T^2 + 38416
$37$	\( T^{8} - 44 T^{6} + \cdots + 148996 \) T^8 - 44T^6 - 1032T^5 + 968T^4 + 122064T^3 + 712520T^2 + 268656T + 148996
$41$	\( T^{8} - 16 T^{7} + \cdots + 264196 \) T^8 - 16T^7 + 36T^6 + 328T^5 + 2568T^4 + 9904T^3 + 29016T^2 + 102800*T + 264196
$43$	\( T^{8} + 16 T^{7} + \cdots + 13075456 \) T^8 + 16T^7 + 128T^6 + 640T^5 + 11328T^4 + 164352T^3 + 1384448T^2 + 6017024*T + 13075456
$47$	\( T^{8} + 112 T^{6} + \cdots + 61504 \) T^8 + 112T^6 + 2640T^4 + 22400*T^2 + 61504
$53$	\( T^{8} - 8 T^{7} + \cdots + 15376 \) T^8 - 8T^7 + 32T^6 + 464T^5 + 15128T^4 - 64480T^3 + 139392T^2 + 65472*T + 15376
$59$	\( T^{8} + 224 T^{5} + \cdots + 3655744 \) T^8 + 224T^5 + 8720T^4 + 25088T^3 + 25088T^2 - 428288*T + 3655744
$61$	\( T^{8} + 32 T^{7} + \cdots + 857476 \) T^8 + 32T^7 + 484T^6 + 4552T^5 + 29320T^4 + 130992T^3 + 399464T^2 + 792656*T + 857476
$67$	\( (T^{4} - 24 T^{3} + \cdots - 2312)^{2} \) (T^4 - 24T^3 + 64T^2 + 1184*T - 2312)^2
$71$	\( T^{8} + 16 T^{7} + \cdots + 4624 \) T^8 + 16T^7 - 32T^6 - 1280T^5 + 4608T^4 + 6656T^3 + 88960T^2 - 15232*T + 4624
$73$	\( T^{8} + 24 T^{7} + \cdots + 62948356 \) T^8 + 24T^7 + 284T^6 + 1352T^5 - 7864T^4 - 141440T^3 - 26584T^2 + 3617904*T + 62948356
$79$	\( T^{8} + 208 T^{6} + \cdots + 99856 \) T^8 + 208T^6 - 4736T^5 + 21632T^4 - 297728T^3 + 6901952T^2 + 1537024T + 99856
$83$	\( T^{8} + 32 T^{7} + \cdots + 141376 \) T^8 + 32T^7 + 512T^6 + 4896T^5 + 30224T^4 + 117504T^3 + 270848T^2 + 276736*T + 141376
$89$	\( T^{8} + 360 T^{6} + \cdots + 4624 \) T^8 + 360T^6 + 32536T^4 + 50080*T^2 + 4624
$97$	\( T^{8} - 16 T^{7} + \cdots + 219573124 \) T^8 - 16T^7 + 228T^6 - 7960T^5 + 111368T^4 - 591248T^3 + 16891608T^2 - 108941936*T + 219573124
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\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(1\)	\(\zeta_{16}^{6}\)