LMFDB - Newform orbit 392.6.i.n

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [392,6,Mod(177,392)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("392.177"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,92] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$62.8704573667$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 110x^{6} + 11956x^{4} + 15840x^{2} + 20736 $ x^8 + 110x^6 + 11956x^4 + 15840*x^2 + 20736
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{3} - \beta_{5} q^{5} + ( - \beta_{7} - \beta_{4} - 23 \beta_{3} + 23) q^{9} + (3 \beta_{7} + 88 \beta_{3}) q^{11} + ( - 12 \beta_{6} + 11 \beta_{5} + \cdots + 11 \beta_1) q^{13} + (\beta_{4} + 28) q^{15}+ \cdots + ( - 157 \beta_{4} + 140312) q^{99}+O(q^{100}) $ q + b2 * q^3 - b5 * q^5 + (-b7 - b4 - 23b3 + 23) q^9 + (3b7 + 88b3) * q^11 + (-12b6 + 11b5 + 12b2 + 11b1) * q^13 + (b4 + 28) * q^15 + (24b2 - 22b1) * q^17 + (-75b6 + 28b5) * q^19 + (9b7 + 9b4 + 492b3 - 492) q^23 + (-15b7 - 231b3) * q^25 + (-58b6 + 12b5 + 58b2 + 12b1) * q^27 + 1010 * q^29 + (438b2 - 88b1) * q^31 + (-536b6 - 36b5) * q^33 + (-18b7 - 18b4 + 2626b3 - 2626) q^37 + (-b7 + 2332b3) q^39 + (-696b6 + 170b5 + 696b2 + 170b1) * q^41 + (-27b4 + 7184) q^43 + (236b2 + 231b1) * q^45 + (-594b6 + 64b5) * q^47 + (-46b7 - 46b4 + 5896b3 - 5896) q^51 + (114b7 + 7074b3) * q^53 + (708b6 - 712b5 - 708b2 - 712b1) * q^55 + (47b4 + 15716) q^57 + (1755b2 + 220b1) * q^59 + (-1128b6 - 365b5) * q^61 + (177b7 + 177b4 + 36580b3 - 36580) q^65 + (-123b7 + 18144b3) * q^67 + (-1380b6 - 108b5 + 1380b2 - 108b1) * q^69 + (126b4 + 44184) q^71 - 4032b2 q^73 + (2889b6 + 180b5) * q^75 + (24b7 + 24b4 + 33992b3 - 33992) q^79 + (-289b7 + 6835b3) * q^81 + (3171b6 + 1432b5 - 3171b2 + 1432b1) * q^83 + (-306b4 + 74504) q^85 + 1010b2 q^87 + (3240b6 + 312b5) * q^89 + (-526b7 - 526b4 + 98824b3 - 98824) q^93 + (495b7 + 91868b3) * q^95 + (-192b6 - 334b5 + 192b2 - 334b1) * q^97 + (-157b4 + 140312) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 92 q^{9} + 352 q^{11} + 224 q^{15} - 1968 q^{23} - 924 q^{25} + 8080 q^{29} - 10504 q^{37} + 9328 q^{39} + 57472 q^{43} - 23584 q^{51} + 28296 q^{53} + 125728 q^{57} - 146320 q^{65} + 72576 q^{67}+ \cdots + 1122496 q^{99}+O(q^{100}) $ 8 * q + 92 * q^9 + 352 * q^11 + 224 * q^15 - 1968 * q^23 - 924 * q^25 + 8080 * q^29 - 10504 * q^37 + 9328 * q^39 + 57472 * q^43 - 23584 * q^51 + 28296 * q^53 + 125728 * q^57 - 146320 * q^65 + 72576 * q^67 + 353472 * q^71 - 135968 * q^79 + 27340 * q^81 + 596032 * q^85 - 395296 * q^93 + 367472 * q^95 + 1122496 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 110x^{6} + 11956x^{4} + 15840x^{2} + 20736 $ x^8 + 110*x^6 + 11956*x^4 + 15840*x^2 + 20736 :

$\beta_{1}$	$=$	$ ( \nu^{7} - 725432\nu ) / 71736 $ (v^7 - 725432*v) / 71736
$\beta_{2}$	$=$	$ ( -\nu^{7} + 1299320\nu ) / 71736 $ (-v^7 + 1299320*v) / 71736
$\beta_{3}$	$=$	$ ( 55\nu^{6} + 5978\nu^{4} + 657580\nu^{2} + 871200 ) / 860832 $ (55v^6 + 5978v^4 + 657580*v^2 + 871200) / 860832
$\beta_{4}$	$=$	$ ( -\nu^{6} + 641740 ) / 2989 $ (-v^6 + 641740) / 2989
$\beta_{5}$	$=$	$ ( 1669\nu^{7} + 185318\nu^{5} + 19954564\nu^{3} + 26436960\nu ) / 2582496 $ (1669v^7 + 185318v^5 + 19954564v^3 + 26436960v) / 2582496
$\beta_{6}$	$=$	$ ( \nu^{7} + 110\nu^{5} + 11956\nu^{3} + 15840\nu ) / 864 $ (v^7 + 110v^5 + 11956v^3 + 15840*v) / 864
$\beta_{7}$	$=$	$ ( -2953\nu^{6} - 328790\nu^{4} - 35306068\nu^{2} - 46775520 ) / 215208 $ (-2953v^6 - 328790v^4 - 35306068*v^2 - 46775520) / 215208

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{2} + \beta_1 ) / 8 $ (b2 + b1) / 8
$\nu^{2}$	$=$	$ ( \beta_{7} + \beta_{4} + 220\beta_{3} - 220 ) / 4 $ (b7 + b4 + 220*b3 - 220) / 4
$\nu^{3}$	$=$	$ ( 31\beta_{6} - 55\beta_{5} - 31\beta_{2} - 55\beta_1 ) / 4 $ (31b6 - 55b5 - 31b2 - 55b1) / 4
$\nu^{4}$	$=$	$ ( -55\beta_{7} - 11812\beta_{3} ) / 2 $ (-55b7 - 11812b3) / 2
$\nu^{5}$	$=$	$ ( -1669\beta_{6} + 2989\beta_{5} ) / 2 $ (-1669b6 + 2989b5) / 2
$\nu^{6}$	$=$	$ -2989\beta_{4} + 641740 $ -2989*b4 + 641740
$\nu^{7}$	$=$	$ 90679\beta_{2} + 162415\beta_1 $ 90679b2 + 162415b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$-1 + \beta_{3}$

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} $

177.1

−0.575555	−	0.996890i
−5.21236	−	9.02808i
5.21236	+	9.02808i
0.575555	+	0.996890i
−0.575555	+	0.996890i
−5.21236	+	9.02808i
5.21236	−	9.02808i
0.575555	−	0.996890i

−10.4247

−

18.0562i

−5.82029

10.0810i

−95.8499

166.017i

177.2

−1.15111

−

1.99378i

40.5478

−

70.2309i

118.850

−

205.854i

177.3

1.15111

1.99378i

−40.5478

70.2309i

118.850

−

205.854i

177.4

10.4247

18.0562i

5.82029

−

10.0810i

−95.8499

166.017i

361.1

−10.4247

18.0562i

−5.82029

−

10.0810i

−95.8499

−

166.017i

361.2

−1.15111

1.99378i

40.5478

70.2309i

118.850

205.854i

361.3

1.15111

−

1.99378i

−40.5478

−

70.2309i

118.850

205.854i

361.4

10.4247

−

18.0562i

5.82029

10.0810i

−95.8499

−

166.017i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.6.i.n		8
7.b	odd	2	1	inner	392.6.i.n		8
7.c	even	3	1		392.6.a.g	✓	4
7.c	even	3	1	inner	392.6.i.n		8
7.d	odd	6	1		392.6.a.g	✓	4
7.d	odd	6	1	inner	392.6.i.n		8
28.f	even	6	1		784.6.a.be		4
28.g	odd	6	1		784.6.a.be		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
392.6.a.g	✓	4	7.c	even	3	1
392.6.a.g	✓	4	7.d	odd	6	1
392.6.i.n		8	1.a	even	1	1	trivial
392.6.i.n		8	7.b	odd	2	1	inner
392.6.i.n		8	7.c	even	3	1	inner
392.6.i.n		8	7.d	odd	6	1	inner
784.6.a.be		4	28.f	even	6	1
784.6.a.be		4	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 440T_{3}^{6} + 191296T_{3}^{4} + 1013760T_{3}^{2} + 5308416 $ T3^8 + 440*T3^6 + 191296*T3^4 + 1013760*T3^2 + 5308416 acting on $S_{6}^{\mathrm{new}}(392, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + 440 T^{6} + \cdots + 5308416 $ T^8 + 440T^6 + 191296T^4 + 1013760*T^2 + 5308416
$5$	$ T^{8} + \cdots + 794123370496 $ T^8 + 6712T^6 + 44159808T^4 + 5981304832*T^2 + 794123370496
$7$	$ T^{8} $ T^8
$11$	$ (T^{4} - 176 T^{3} + \cdots + 165746694400)^{2} $ (T^4 - 176T^3 + 438096T^2 + 71653120*T + 165746694400)^2
$13$	$ (T^{4} - 860728 T^{2} + 12619376896)^{2} $ (T^4 - 860728*T^2 + 12619376896)^2
$17$	$ T^{8} + \cdots + 29\!\cdots\!00 $ T^8 + 3561184T^6 + 10971586016256T^4 + 6091211024967270400*T^2 + 2925623690791600783360000
$19$	$ T^{8} + \cdots + 83\!\cdots\!56 $ T^8 + 7502008T^6 + 47133709347648T^4 + 68616476133806307328*T^2 + 83656901579300636873261056
$23$	$ (T^{4} + \cdots + 12192052690944)^{2} $ (T^4 + 984T^3 + 4459968T^2 - 3435844608*T + 12192052690944)^2
$29$	$ (T - 1010)^{8} $ (T - 1010)^8
$31$	$ T^{8} + \cdots + 15\!\cdots\!00 $ T^8 + 140706016T^6 + 15924484838470656T^4 + 545052626854879451545600*T^2 + 15005536970885693377934786560000
$37$	$ (T^{4} + \cdots + 64629186835984)^{2} $ (T^4 + 5252T^3 + 35622732T^2 - 42222025456*T + 64629186835984)^2
$41$	$ (T^{4} + \cdots + 37\!\cdots\!04)^{2} $ (T^4 - 393868000*T^2 + 37192992397103104)^2
$43$	$ (T^{2} - 14368 T + 18005872)^{4} $ (T^2 - 14368*T + 18005872)^4
$47$	$ T^{8} + \cdots + 33\!\cdots\!56 $ T^8 + 178482400T^6 + 26029983985775616T^4 + 1039835450328230418841600*T^2 + 33942079360950842271071475859456
$53$	$ (T^{4} + \cdots + 30\!\cdots\!00)^{2} $ (T^4 - 14148T^3 + 749188044T^2 + 7767565236720*T + 301425310210179600)^2
$59$	$ T^{8} + \cdots + 30\!\cdots\!00 $ T^8 + 1636828600T^6 + 2124745942893960000T^4 + 907559132987525682400000000*T^2 + 307428023928222762877456000000000000
$61$	$ T^{8} + \cdots + 31\!\cdots\!16 $ T^8 + 1500167800T^6 + 1688315312912805696T^4 + 843376508031789404956211200*T^2 + 316055476921639595781746661528764416
$67$	$ (T^{4} + \cdots + 13\!\cdots\!04)^{2} $ (T^4 - 36288T^3 + 1685000592T^2 + 13360575642624*T + 135557725923995904)^2
$71$	$ (T^{2} - 88368 T + 1220405760)^{4} $ (T^2 - 88368*T + 1220405760)^4
$73$	$ T^{8} + \cdots + 37\!\cdots\!16 $ T^8 + 7153090560T^6 + 50557778488769642496T^4 + 4355703328716363682535178240*T^2 + 370790959689539678380989068466978816
$79$	$ (T^{4} + \cdots + 12\!\cdots\!24)^{2} $ (T^4 + 67984T^3 + 3492919488T^2 + 76747461747712*T + 1274425975213133824)^2
$83$	$ (T^{4} + \cdots + 81\!\cdots\!16)^{2} $ (T^4 - 18696671992*T^2 + 81164756622979062016)^2
$89$	$ T^{8} + \cdots + 26\!\cdots\!96 $ T^8 + 5385535488T^6 + 27390807349665988608T^4 + 8687865835486760212069613568*T^2 + 2602366305084258888332598231271735296
$97$	$ (T^{4} + \cdots + 44\!\cdots\!56)^{2} $ (T^4 - 772166368*T^2 + 44204382974611456)^2
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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} - \beta_{5} q^{5} + ( - \beta_{7} - \beta_{4} - 23 \beta_{3} + 23) q^{9} + (3 \beta_{7} + 88 \beta_{3}) q^{11} + ( - 12 \beta_{6} + 11 \beta_{5} + \cdots + 11 \beta_1) q^{13} + (\beta_{4} + 28) q^{15}+ \cdots + ( - 157 \beta_{4} + 140312) q^{99}+O(q^{100}) \) q + b2 * q^3 - b5 * q^5 + (-b7 - b4 - 23b3 + 23) q^9 + (3b7 + 88b3) * q^11 + (-12b6 + 11b5 + 12b2 + 11b1) * q^13 + (b4 + 28) * q^15 + (24b2 - 22b1) * q^17 + (-75b6 + 28b5) * q^19 + (9b7 + 9b4 + 492b3 - 492) q^23 + (-15b7 - 231b3) * q^25 + (-58b6 + 12b5 + 58b2 + 12b1) * q^27 + 1010 * q^29 + (438b2 - 88b1) * q^31 + (-536b6 - 36b5) * q^33 + (-18b7 - 18b4 + 2626b3 - 2626) q^37 + (-b7 + 2332b3) q^39 + (-696b6 + 170b5 + 696b2 + 170b1) * q^41 + (-27b4 + 7184) q^43 + (236b2 + 231b1) * q^45 + (-594b6 + 64b5) * q^47 + (-46b7 - 46b4 + 5896b3 - 5896) q^51 + (114b7 + 7074b3) * q^53 + (708b6 - 712b5 - 708b2 - 712b1) * q^55 + (47b4 + 15716) q^57 + (1755b2 + 220b1) * q^59 + (-1128b6 - 365b5) * q^61 + (177b7 + 177b4 + 36580b3 - 36580) q^65 + (-123b7 + 18144b3) * q^67 + (-1380b6 - 108b5 + 1380b2 - 108b1) * q^69 + (126b4 + 44184) q^71 - 4032b2 q^73 + (2889b6 + 180b5) * q^75 + (24b7 + 24b4 + 33992b3 - 33992) q^79 + (-289b7 + 6835b3) * q^81 + (3171b6 + 1432b5 - 3171b2 + 1432b1) * q^83 + (-306b4 + 74504) q^85 + 1010b2 q^87 + (3240b6 + 312b5) * q^89 + (-526b7 - 526b4 + 98824b3 - 98824) q^93 + (495b7 + 91868b3) * q^95 + (-192b6 - 334b5 + 192b2 - 334b1) * q^97 + (-157b4 + 140312) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 92 q^{9} + 352 q^{11} + 224 q^{15} - 1968 q^{23} - 924 q^{25} + 8080 q^{29} - 10504 q^{37} + 9328 q^{39} + 57472 q^{43} - 23584 q^{51} + 28296 q^{53} + 125728 q^{57} - 146320 q^{65} + 72576 q^{67}+ \cdots + 1122496 q^{99}+O(q^{100}) \) 8 * q + 92 * q^9 + 352 * q^11 + 224 * q^15 - 1968 * q^23 - 924 * q^25 + 8080 * q^29 - 10504 * q^37 + 9328 * q^39 + 57472 * q^43 - 23584 * q^51 + 28296 * q^53 + 125728 * q^57 - 146320 * q^65 + 72576 * q^67 + 353472 * q^71 - 135968 * q^79 + 27340 * q^81 + 596032 * q^85 - 395296 * q^93 + 367472 * q^95 + 1122496 * 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	392.6.i.n

Level	$392$
Weight	$6$
Character orbit	392.i
Analytic conductor	$62.870$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(62.8704573667\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 110x^{6} + 11956x^{4} + 15840x^{2} + 20736 \) x^8 + 110x^6 + 11956x^4 + 15840*x^2 + 20736
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{14} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} - 725432\nu ) / 71736 \) (v^7 - 725432*v) / 71736
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} + 1299320\nu ) / 71736 \) (-v^7 + 1299320*v) / 71736
\(\beta_{3}\)	\(=\)	\( ( 55\nu^{6} + 5978\nu^{4} + 657580\nu^{2} + 871200 ) / 860832 \) (55v^6 + 5978v^4 + 657580*v^2 + 871200) / 860832
\(\beta_{4}\)	\(=\)	\( ( -\nu^{6} + 641740 ) / 2989 \) (-v^6 + 641740) / 2989
\(\beta_{5}\)	\(=\)	\( ( 1669\nu^{7} + 185318\nu^{5} + 19954564\nu^{3} + 26436960\nu ) / 2582496 \) (1669v^7 + 185318v^5 + 19954564v^3 + 26436960v) / 2582496
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} + 110\nu^{5} + 11956\nu^{3} + 15840\nu ) / 864 \) (v^7 + 110v^5 + 11956v^3 + 15840*v) / 864
\(\beta_{7}\)	\(=\)	\( ( -2953\nu^{6} - 328790\nu^{4} - 35306068\nu^{2} - 46775520 ) / 215208 \) (-2953v^6 - 328790v^4 - 35306068*v^2 - 46775520) / 215208

\(\nu\)	\(=\)	\( ( \beta_{2} + \beta_1 ) / 8 \) (b2 + b1) / 8
\(\nu^{2}\)	\(=\)	\( ( \beta_{7} + \beta_{4} + 220\beta_{3} - 220 ) / 4 \) (b7 + b4 + 220*b3 - 220) / 4
\(\nu^{3}\)	\(=\)	\( ( 31\beta_{6} - 55\beta_{5} - 31\beta_{2} - 55\beta_1 ) / 4 \) (31b6 - 55b5 - 31b2 - 55b1) / 4
\(\nu^{4}\)	\(=\)	\( ( -55\beta_{7} - 11812\beta_{3} ) / 2 \) (-55b7 - 11812b3) / 2
\(\nu^{5}\)	\(=\)	\( ( -1669\beta_{6} + 2989\beta_{5} ) / 2 \) (-1669b6 + 2989b5) / 2
\(\nu^{6}\)	\(=\)	\( -2989\beta_{4} + 641740 \) -2989*b4 + 641740
\(\nu^{7}\)	\(=\)	\( 90679\beta_{2} + 162415\beta_1 \) 90679b2 + 162415b1

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1 + \beta_{3}\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 177.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} + 440 T^{6} + \cdots + 5308416 \) T^8 + 440T^6 + 191296T^4 + 1013760*T^2 + 5308416
$5$	\( T^{8} + \cdots + 794123370496 \) T^8 + 6712T^6 + 44159808T^4 + 5981304832*T^2 + 794123370496
$7$	\( T^{8} \) T^8
$11$	\( (T^{4} - 176 T^{3} + \cdots + 165746694400)^{2} \) (T^4 - 176T^3 + 438096T^2 + 71653120*T + 165746694400)^2
$13$	\( (T^{4} - 860728 T^{2} + 12619376896)^{2} \) (T^4 - 860728*T^2 + 12619376896)^2
$17$	\( T^{8} + \cdots + 29\!\cdots\!00 \) T^8 + 3561184T^6 + 10971586016256T^4 + 6091211024967270400*T^2 + 2925623690791600783360000
$19$	\( T^{8} + \cdots + 83\!\cdots\!56 \) T^8 + 7502008T^6 + 47133709347648T^4 + 68616476133806307328*T^2 + 83656901579300636873261056
$23$	\( (T^{4} + \cdots + 12192052690944)^{2} \) (T^4 + 984T^3 + 4459968T^2 - 3435844608*T + 12192052690944)^2
$29$	\( (T - 1010)^{8} \) (T - 1010)^8
$31$	\( T^{8} + \cdots + 15\!\cdots\!00 \) T^8 + 140706016T^6 + 15924484838470656T^4 + 545052626854879451545600*T^2 + 15005536970885693377934786560000
$37$	\( (T^{4} + \cdots + 64629186835984)^{2} \) (T^4 + 5252T^3 + 35622732T^2 - 42222025456*T + 64629186835984)^2
$41$	\( (T^{4} + \cdots + 37\!\cdots\!04)^{2} \) (T^4 - 393868000*T^2 + 37192992397103104)^2
$43$	\( (T^{2} - 14368 T + 18005872)^{4} \) (T^2 - 14368*T + 18005872)^4
$47$	\( T^{8} + \cdots + 33\!\cdots\!56 \) T^8 + 178482400T^6 + 26029983985775616T^4 + 1039835450328230418841600*T^2 + 33942079360950842271071475859456
$53$	\( (T^{4} + \cdots + 30\!\cdots\!00)^{2} \) (T^4 - 14148T^3 + 749188044T^2 + 7767565236720*T + 301425310210179600)^2
$59$	\( T^{8} + \cdots + 30\!\cdots\!00 \) T^8 + 1636828600T^6 + 2124745942893960000T^4 + 907559132987525682400000000*T^2 + 307428023928222762877456000000000000
$61$	\( T^{8} + \cdots + 31\!\cdots\!16 \) T^8 + 1500167800T^6 + 1688315312912805696T^4 + 843376508031789404956211200*T^2 + 316055476921639595781746661528764416
$67$	\( (T^{4} + \cdots + 13\!\cdots\!04)^{2} \) (T^4 - 36288T^3 + 1685000592T^2 + 13360575642624*T + 135557725923995904)^2
$71$	\( (T^{2} - 88368 T + 1220405760)^{4} \) (T^2 - 88368*T + 1220405760)^4
$73$	\( T^{8} + \cdots + 37\!\cdots\!16 \) T^8 + 7153090560T^6 + 50557778488769642496T^4 + 4355703328716363682535178240*T^2 + 370790959689539678380989068466978816
$79$	\( (T^{4} + \cdots + 12\!\cdots\!24)^{2} \) (T^4 + 67984T^3 + 3492919488T^2 + 76747461747712*T + 1274425975213133824)^2
$83$	\( (T^{4} + \cdots + 81\!\cdots\!16)^{2} \) (T^4 - 18696671992*T^2 + 81164756622979062016)^2
$89$	\( T^{8} + \cdots + 26\!\cdots\!96 \) T^8 + 5385535488T^6 + 27390807349665988608T^4 + 8687865835486760212069613568*T^2 + 2602366305084258888332598231271735296
$97$	\( (T^{4} + \cdots + 44\!\cdots\!56)^{2} \) (T^4 - 772166368*T^2 + 44204382974611456)^2
show more
show less