LMFDB - Newform orbit 850.2.o.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [850,2,Mod(49,850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(850, base_ring=CyclotomicField(8))

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

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

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

chi := DirichletCharacter("850.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.o (of order $8$, 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:	$6.78728417181$
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:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/850\mathbb{Z}\right)^\times$.

$n$	$477$	$751$
$\chi(n)$	$-1$	$-\zeta_{16}^{6}$

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} $

49.1

−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

−2.08979

0.865619i

−

1.00000i

−0.865619

2.08979i

−0.834089

2.01367i

−0.707107

−

0.707107i

1.49661

−

1.49661i

49.2

0.707107

−

0.707107i

−1.32442

0.548594i

−

1.00000i

−0.548594

1.32442i

0.248303

−

0.599456i

−0.707107

−

0.707107i

−0.668179

0.668179i

349.1

−0.707107

−

0.707107i

−1.21677

2.93755i

1.00000i

2.93755

−

1.21677i

−0.400544

0.165911i

0.707107

−

0.707107i

−5.02734

−

5.02734i

349.2

−0.707107

−

0.707107i

0.630986

−

1.52334i

1.00000i

−1.52334

0.630986i

−3.01367

1.24830i

0.707107

−

0.707107i

0.198912

0.198912i

399.1

0.707107

0.707107i

−2.08979

−

0.865619i

1.00000i

−0.865619

−

2.08979i

−0.834089

−

2.01367i

−0.707107

0.707107i

1.49661

1.49661i

399.2

0.707107

0.707107i

−1.32442

−

0.548594i

1.00000i

−0.548594

−

1.32442i

0.248303

0.599456i

−0.707107

0.707107i

−0.668179

−

0.668179i

699.1

−0.707107

0.707107i

−1.21677

−

2.93755i

−

1.00000i

2.93755

1.21677i

−0.400544

−

0.165911i

0.707107

0.707107i

−5.02734

5.02734i

699.2

−0.707107

0.707107i

0.630986

1.52334i

−

1.00000i

−1.52334

−

0.630986i

−3.01367

−

1.24830i

0.707107

0.707107i

0.198912

−

0.198912i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
85.m	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	850.2.o.c		8
5.b	even	2	1		850.2.o.f		8
5.c	odd	4	1		170.2.k.a	✓	8
5.c	odd	4	1		850.2.l.d		8
17.d	even	8	1		850.2.o.f		8
85.k	odd	8	1		850.2.l.d		8
85.m	even	8	1	inner	850.2.o.c		8
85.n	odd	8	1		170.2.k.a	✓	8
85.r	even	16	1		2890.2.a.bc		4
85.r	even	16	1		2890.2.a.bf		4
85.r	even	16	2		2890.2.b.p		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
170.2.k.a	✓	8	5.c	odd	4	1
170.2.k.a	✓	8	85.n	odd	8	1
850.2.l.d		8	5.c	odd	4	1
850.2.l.d		8	85.k	odd	8	1
850.2.o.c		8	1.a	even	1	1	trivial
850.2.o.c		8	85.m	even	8	1	inner
850.2.o.f		8	5.b	even	2	1
850.2.o.f		8	17.d	even	8	1
2890.2.a.bc		4	85.r	even	16	1
2890.2.a.bf		4	85.r	even	16	1
2890.2.b.p		8	85.r	even	16	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 8T_{3}^{7} + 36T_{3}^{6} + 104T_{3}^{5} + 200T_{3}^{4} + 304T_{3}^{3} + 468T_{3}^{2} + 544T_{3} + 289 $ T3^8 + 8*T3^7 + 36*T3^6 + 104*T3^5 + 200*T3^4 + 304*T3^3 + 468*T3^2 + 544*T3 + 289 acting on $S_{2}^{\mathrm{new}}(850, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$3$	$ T^{8} + 8 T^{7} + \cdots + 289 $ T^8 + 8T^7 + 36T^6 + 104T^5 + 200T^4 + 304T^3 + 468T^2 + 544*T + 289
$5$	$ T^{8} $ T^8
$7$	$ T^{8} + 8 T^{7} + \cdots + 4 $ T^8 + 8T^7 + 28T^6 + 56T^5 + 72T^4 + 32T^3 + 24T^2 + 16*T + 4
$11$	$ T^{8} - 16 T^{7} + \cdots + 51076 $ T^8 - 16T^7 + 116T^6 - 552T^5 + 2248T^4 - 8016T^3 + 22536T^2 - 45200*T + 51076
$13$	$ (T^{4} - 12 T^{3} + \cdots + 17)^{2} $ (T^4 - 12T^3 + 46T^2 - 60*T + 17)^2
$17$	$ T^{8} + 48 T^{6} + \cdots + 83521 $ T^8 + 48T^6 + 1152T^4 + 13872*T^2 + 83521
$19$	$ T^{8} - 16 T^{7} + \cdots + 452929 $ T^8 - 16T^7 + 128T^6 - 504T^5 + 1446T^4 - 7328T^3 + 59168T^2 - 231512*T + 452929
$23$	$ T^{8} + 8 T^{7} + \cdots + 3182656 $ T^8 + 8T^7 + 24T^6 - 352T^5 - 3040T^4 + 2368T^3 + 158784T^2 + 913408*T + 3182656
$29$	$ T^{8} + 8 T^{7} + \cdots + 289 $ T^8 + 8T^7 - 72T^5 + 448T^4 - 1032T^3 + 800T^2 + 136T + 289
$31$	$ T^{8} + 16 T^{7} + \cdots + 277729 $ T^8 + 16T^7 + 68T^6 + 96T^5 + 2824T^4 - 4584T^3 + 35220T^2 - 181288*T + 277729
$37$	$ (T^{4} + 16 T^{3} + \cdots + 98)^{2} $ (T^4 + 16T^3 + 82T^2 + 84*T + 98)^2
$41$	$ T^{8} + 68 T^{6} + \cdots + 1156 $ T^8 + 68T^6 - 328T^5 + 2312T^4 - 5264T^3 + 2072T^2 + 4080T + 1156
$43$	$ T^{8} + 8 T^{7} + \cdots + 4 $ T^8 + 8T^7 + 32T^6 - 144T^5 + 260T^4 - 240T^3 + 128T^2 - 32*T + 4
$47$	$ (T^{4} - 4 T^{3} - 34 T^{2} + \cdots - 17)^{2} $ (T^4 - 4T^3 - 34T^2 + 148*T - 17)^2
$53$	$ T^{8} - 8 T^{7} + \cdots + 1 $ T^8 - 8T^7 + 32T^6 + 200T^5 + 898T^4 - 1192T^3 + 800T^2 + 40*T + 1
$59$	$ T^{8} - 40 T^{7} + \cdots + 113569 $ T^8 - 40T^7 + 800T^6 - 9408T^5 + 69318T^4 - 294344T^3 + 574592T^2 + 361264*T + 113569
$61$	$ T^{8} + 24 T^{7} + \cdots + 6723649 $ T^8 + 24T^7 + 184T^6 + 1128T^5 + 20960T^4 + 63192T^3 + 398632T^2 + 2841928*T + 6723649
$67$	$ T^{8} + 112 T^{6} + \cdots + 61504 $ T^8 + 112T^6 + 2640T^4 + 22400*T^2 + 61504
$71$	$ T^{8} - 24 T^{7} + \cdots + 8025889 $ T^8 - 24T^7 + 372T^6 - 2904T^5 + 7560T^4 + 35216T^3 - 228956T^2 + 634592*T + 8025889
$73$	$ T^{8} - 16 T^{7} + \cdots + 57121 $ T^8 - 16T^7 + 112T^6 - 656T^5 + 4480T^4 - 19952T^3 + 48400T^2 - 65008*T + 57121
$79$	$ T^{8} - 8 T^{7} + \cdots + 35344 $ T^8 - 8T^7 - 40T^6 - 656T^5 + 9632T^4 + 126496T^3 + 383200T^2 - 81216*T + 35344
$83$	$ T^{8} + 8 T^{7} + \cdots + 9604 $ T^8 + 8T^7 + 32T^6 - 96T^5 + 1796T^4 + 9744T^3 + 25088T^2 + 21952*T + 9604
$89$	$ T^{8} + 260 T^{6} + \cdots + 1324801 $ T^8 + 260T^6 + 18694T^4 + 302852*T^2 + 1324801
$97$	$ T^{8} - 16 T^{7} + \cdots + 16394401 $ T^8 - 16T^7 + 176T^6 + 2784T^5 - 57728T^4 - 265376T^3 + 8124688T^2 + 8616272*T + 16394401
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.o (of order \(8\), degree \(4\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	850.2.o.c

Level	$850$
Weight	$2$
Character orbit	850.o
Analytic conductor	$6.787$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(6.78728417181\)
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:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 1)^{2} \) (T^4 + 1)^2
$3$	\( T^{8} + 8 T^{7} + \cdots + 289 \) T^8 + 8T^7 + 36T^6 + 104T^5 + 200T^4 + 304T^3 + 468T^2 + 544*T + 289
$5$	\( T^{8} \) T^8
$7$	\( T^{8} + 8 T^{7} + \cdots + 4 \) T^8 + 8T^7 + 28T^6 + 56T^5 + 72T^4 + 32T^3 + 24T^2 + 16*T + 4
$11$	\( T^{8} - 16 T^{7} + \cdots + 51076 \) T^8 - 16T^7 + 116T^6 - 552T^5 + 2248T^4 - 8016T^3 + 22536T^2 - 45200*T + 51076
$13$	\( (T^{4} - 12 T^{3} + \cdots + 17)^{2} \) (T^4 - 12T^3 + 46T^2 - 60*T + 17)^2
$17$	\( T^{8} + 48 T^{6} + \cdots + 83521 \) T^8 + 48T^6 + 1152T^4 + 13872*T^2 + 83521
$19$	\( T^{8} - 16 T^{7} + \cdots + 452929 \) T^8 - 16T^7 + 128T^6 - 504T^5 + 1446T^4 - 7328T^3 + 59168T^2 - 231512*T + 452929
$23$	\( T^{8} + 8 T^{7} + \cdots + 3182656 \) T^8 + 8T^7 + 24T^6 - 352T^5 - 3040T^4 + 2368T^3 + 158784T^2 + 913408*T + 3182656
$29$	\( T^{8} + 8 T^{7} + \cdots + 289 \) T^8 + 8T^7 - 72T^5 + 448T^4 - 1032T^3 + 800T^2 + 136T + 289
$31$	\( T^{8} + 16 T^{7} + \cdots + 277729 \) T^8 + 16T^7 + 68T^6 + 96T^5 + 2824T^4 - 4584T^3 + 35220T^2 - 181288*T + 277729
$37$	\( (T^{4} + 16 T^{3} + \cdots + 98)^{2} \) (T^4 + 16T^3 + 82T^2 + 84*T + 98)^2
$41$	\( T^{8} + 68 T^{6} + \cdots + 1156 \) T^8 + 68T^6 - 328T^5 + 2312T^4 - 5264T^3 + 2072T^2 + 4080T + 1156
$43$	\( T^{8} + 8 T^{7} + \cdots + 4 \) T^8 + 8T^7 + 32T^6 - 144T^5 + 260T^4 - 240T^3 + 128T^2 - 32*T + 4
$47$	\( (T^{4} - 4 T^{3} - 34 T^{2} + \cdots - 17)^{2} \) (T^4 - 4T^3 - 34T^2 + 148*T - 17)^2
$53$	\( T^{8} - 8 T^{7} + \cdots + 1 \) T^8 - 8T^7 + 32T^6 + 200T^5 + 898T^4 - 1192T^3 + 800T^2 + 40*T + 1
$59$	\( T^{8} - 40 T^{7} + \cdots + 113569 \) T^8 - 40T^7 + 800T^6 - 9408T^5 + 69318T^4 - 294344T^3 + 574592T^2 + 361264*T + 113569
$61$	\( T^{8} + 24 T^{7} + \cdots + 6723649 \) T^8 + 24T^7 + 184T^6 + 1128T^5 + 20960T^4 + 63192T^3 + 398632T^2 + 2841928*T + 6723649
$67$	\( T^{8} + 112 T^{6} + \cdots + 61504 \) T^8 + 112T^6 + 2640T^4 + 22400*T^2 + 61504
$71$	\( T^{8} - 24 T^{7} + \cdots + 8025889 \) T^8 - 24T^7 + 372T^6 - 2904T^5 + 7560T^4 + 35216T^3 - 228956T^2 + 634592*T + 8025889
$73$	\( T^{8} - 16 T^{7} + \cdots + 57121 \) T^8 - 16T^7 + 112T^6 - 656T^5 + 4480T^4 - 19952T^3 + 48400T^2 - 65008*T + 57121
$79$	\( T^{8} - 8 T^{7} + \cdots + 35344 \) T^8 - 8T^7 - 40T^6 - 656T^5 + 9632T^4 + 126496T^3 + 383200T^2 - 81216*T + 35344
$83$	\( T^{8} + 8 T^{7} + \cdots + 9604 \) T^8 + 8T^7 + 32T^6 - 96T^5 + 1796T^4 + 9744T^3 + 25088T^2 + 21952*T + 9604
$89$	\( T^{8} + 260 T^{6} + \cdots + 1324801 \) T^8 + 260T^6 + 18694T^4 + 302852*T^2 + 1324801
$97$	\( T^{8} - 16 T^{7} + \cdots + 16394401 \) T^8 - 16T^7 + 176T^6 + 2784T^5 - 57728T^4 - 265376T^3 + 8124688T^2 + 8616272*T + 16394401
show more
show less

\(n\)	\(477\)	\(751\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{16}^{6}\)