LMFDB - Newform orbit 1323.2.c.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1322,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("1323.1322");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.c (of order $2$, degree $1$, 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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 212x^{12} + 872x^{10} + 1815x^{8} + 1928x^{6} + 996x^{4} + 200x^{2} + 1 $ x^16 + 24x^14 + 212x^12 + 872x^10 + 1815x^8 + 1928x^6 + 996x^4 + 200*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{5} q^{2} + (\beta_{13} - 1) q^{4} - \beta_{2} q^{5} + ( - \beta_{15} + \beta_{12} + \cdots - \beta_1) q^{8}+O(q^{10}) $ q + b5 * q^2 + (b13 - 1) * q^4 - b2 * q^5 + (-b15 + b12 - b5 - b1) * q^8	$ q + \beta_{5} q^{2} + (\beta_{13} - 1) q^{4} - \beta_{2} q^{5} + ( - \beta_{15} + \beta_{12} + \cdots - \beta_1) q^{8}+ \cdots + (2 \beta_{14} - 6 \beta_{10} + \cdots + 4 \beta_{4}) q^{97}+O(q^{100}) $ q + b5 * q^2 + (b13 - 1) * q^4 - b2 * q^5 + (-b15 + b12 - b5 - b1) * q^8 + (-b14 - b10 - b6 - b4) * q^10 + (-b15 - b12 - b5) * q^11 + (b14 + b10 - b6 + b4) * q^13 + (-b13 + 3b8 + b7 + 3) q^16 + (2b11 + b9) q^17 + (b10 - b6 + b4) * q^19 + (2b11 - 3b9 + b3 + 4b2) q^20 + (-2b13 - b8 + 4) q^22 + (b15 - b12 - 2b1) q^23 + (-b13 + b8 + b7 - 1) * q^25 + (-2b11 - b9 - 3b3 - 4b2) q^26 + (-2b15 + b12 + 2b1) * q^29 + (2b14 - 2b10 + b6 + 3b4) q^31 + (-2b15 - 2b12 + 2b5 + 4b1) * q^32 + (-3b10 - b6 + 2b4) * q^34 + (-b13 + 3b8 + 2b7 + 2) * q^37 + (-b11 - b9 - 2b3 - b2) q^38 + (2b14 + 3b10 + 4b6 + 5b4) * q^40 + (-b11 - b9 + 2b3) q^41 + (4b8 - b7 - 1) q^43 + (-3b12 + 6b5 + b1) * q^44 + (-b13 + 3b8 + 2b7 + 1) * q^46 + (3b11 - 2b9 + b2) * q^47 + (-2b12 + 4b1) * q^50 + (-2b14 + b10 - 2b6 - 7b4) q^52 + (-2b15 + b12 + 5b5 - 2b1) q^53 + (b14 + 3b10 + b6 + 2b4) * q^55 + (-3b8 - 2b7) * q^58 + (-3b11 + 2b9 - 4b3 - 3b2) * q^59 + (b14 - 4b10 - b6 - 5b4) * q^61 + (-5b11 - 4b3 - 8b2) q^62 + (2b13 - 4b8 - 2b7 - 2) q^64 + (2b15 - 2b12 - 3b5 - 2b1) * q^65 + (b13 + b8 - 2b7 + 3) q^67 + (2b11 - 3b9 - 3b3 + 2b2) * q^68 + (-b15 + 5b12 - 3b5 + b1) * q^71 + (-b14 + b10 - 2b6 + b4) q^73 + (-b15 - 4b12 + 2b5 + 8b1) q^74 + (-b14 + 3b10 - 2b4) * q^76 + (3b13 + 3b8 - b7 - 4) * q^79 + (-3b11 + 5b9 - b3 - 10b2) q^80 + (2b10 - b6 + b4) q^82 + 5b3 q^83 + (-b13 - b8 + 2b7 + 4) q^85 + (b15 - 4b12 + 2b1) * q^86 + (3b13 - 7b8 - b7 - 11) * q^88 + (-2b11 - 5b9 - 3b3 + 2b2) * q^89 + (b15 - 6b12 + b5 + 4b1) * q^92 + (b14 + 3b6 + 4b4) * q^94 - b12 * q^95 + (2b14 - 6b10 + 5b6 + 4b4) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 48 q^{16} + 64 q^{22} - 16 q^{25} + 32 q^{37} - 16 q^{43} + 16 q^{46} - 32 q^{64} + 48 q^{67} - 64 q^{79} + 64 q^{85} - 176 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 48 * q^16 + 64 * q^22 - 16 * q^25 + 32 * q^37 - 16 * q^43 + 16 * q^46 - 32 * q^64 + 48 * q^67 - 64 * q^79 + 64 * q^85 - 176 * q^88

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 24x^{14} + 212x^{12} + 872x^{10} + 1815x^{8} + 1928x^{6} + 996x^{4} + 200x^{2} + 1 $ x^16 + 24*x^14 + 212*x^12 + 872*x^10 + 1815*x^8 + 1928*x^6 + 996*x^4 + 200*x^2 + 1 :

$\beta_{1}$	$=$	$ ( - 3566 \nu^{15} - 82183 \nu^{13} - 677054 \nu^{11} - 2451006 \nu^{9} - 4029525 \nu^{7} + \cdots + 102636 \nu ) / 7231 $ (-3566v^15 - 82183v^13 - 677054v^11 - 2451006v^9 - 4029525v^7 - 2653934v^5 - 400750v^3 + 102636v) / 7231
$\beta_{2}$	$=$	$ ( 4109 \nu^{14} + 95961 \nu^{12} + 809596 \nu^{10} + 3069810 \nu^{8} + 5537952 \nu^{6} + 4485192 \nu^{4} + \cdots - 9938 ) / 7231 $ (4109v^14 + 95961v^12 + 809596v^10 + 3069810v^8 + 5537952v^6 + 4485192v^4 + 1280801*v^2 - 9938) / 7231
$\beta_{3}$	$=$	$ ( - 4213 \nu^{14} - 98695 \nu^{12} - 836669 \nu^{10} - 3196410 \nu^{8} - 5829935 \nu^{6} - 4816716 \nu^{4} + \cdots - 8680 ) / 7231 $ (-4213v^14 - 98695v^12 - 836669v^10 - 3196410v^8 - 5829935v^6 - 4816716v^4 - 1459060*v^2 - 8680) / 7231
$\beta_{4}$	$=$	$ ( 11880 \nu^{15} + 277026 \nu^{13} + 2330454 \nu^{11} + 8786075 \nu^{9} + 15693672 \nu^{7} + \cdots + 206322 \nu ) / 7231 $ (11880v^15 + 277026v^13 + 2330454v^11 + 8786075v^9 + 15693672v^7 + 12631032v^5 + 3918733v^3 + 206322v) / 7231
$\beta_{5}$	$=$	$ ( 12411 \nu^{15} + 289535 \nu^{13} + 2437027 \nu^{11} + 9191538 \nu^{9} + 16395526 \nu^{7} + \cdots + 89795 \nu ) / 7231 $ (12411v^15 + 289535v^13 + 2437027v^11 + 9191538v^9 + 16395526v^7 + 13061987v^5 + 3821459v^3 + 89795v) / 7231
$\beta_{6}$	$=$	$ ( - 12661 \nu^{15} - 294796 \nu^{13} - 2472050 \nu^{11} - 9248819 \nu^{9} - 16190613 \nu^{7} + \cdots + 184528 \nu ) / 7231 $ (-12661v^15 - 294796v^13 - 2472050v^11 - 9248819v^9 - 16190613v^7 - 12258167v^5 - 2969225v^3 + 184528v) / 7231
$\beta_{7}$	$=$	$ ( - 15611 \nu^{14} - 362454 \nu^{12} - 3026016 \nu^{10} - 11245322 \nu^{8} - 19531408 \nu^{6} + \cdots + 16978 ) / 7231 $ (-15611v^14 - 362454v^12 - 3026016v^10 - 11245322v^8 - 19531408v^6 - 14787914v^4 - 3904397*v^2 + 16978) / 7231
$\beta_{8}$	$=$	$ ( - 2432 \nu^{14} - 56782 \nu^{12} - 478539 \nu^{10} - 1808300 \nu^{8} - 3232355 \nu^{6} - 2573576 \nu^{4} + \cdots - 5170 ) / 1033 $ (-2432v^14 - 56782v^12 - 478539v^10 - 1808300v^8 - 3232355v^6 - 2573576v^4 - 736177*v^2 - 5170) / 1033
$\beta_{9}$	$=$	$ ( 20\nu^{14} + 467\nu^{12} + 3936\nu^{10} + 14872\nu^{8} + 26564\nu^{6} + 21096\nu^{4} + 5980\nu^{2} + 18 ) / 7 $ (20v^14 + 467v^12 + 3936v^10 + 14872v^8 + 26564v^6 + 21096v^4 + 5980*v^2 + 18) / 7
$\beta_{10}$	$=$	$ ( - 20975 \nu^{15} - 489639 \nu^{13} - 4125450 \nu^{11} - 15583888 \nu^{9} - 27854760 \nu^{7} + \cdots - 109968 \nu ) / 7231 $ (-20975v^15 - 489639v^13 - 4125450v^11 - 15583888v^9 - 27854760v^7 - 22235265v^5 - 6487208v^3 - 109968v) / 7231
$\beta_{11}$	$=$	$ ( 24141 \nu^{14} + 562578 \nu^{12} + 4726427 \nu^{10} + 17764742 \nu^{8} + 31483261 \nu^{6} + \cdots + 62386 ) / 7231 $ (24141v^14 + 562578v^12 + 4726427v^10 + 17764742v^8 + 31483261v^6 + 24752870v^4 + 7015238*v^2 + 62386) / 7231
$\beta_{12}$	$=$	$ ( 25023 \nu^{15} + 585089 \nu^{13} + 4943611 \nu^{11} + 18772057 \nu^{9} + 33876484 \nu^{7} + \cdots + 246543 \nu ) / 7231 $ (25023v^15 + 585089v^13 + 4943611v^11 + 18772057v^9 + 33876484v^7 + 27549033v^5 + 8361354v^3 + 246543v) / 7231
$\beta_{13}$	$=$	$ ( - 25833 \nu^{14} - 602052 \nu^{12} - 5058697 \nu^{10} - 19017892 \nu^{8} - 33715265 \nu^{6} + \cdots - 19311 ) / 7231 $ (-25833v^14 - 602052v^12 - 5058697v^10 - 19017892v^8 - 33715265v^6 - 26500974v^4 - 7440426*v^2 - 19311) / 7231
$\beta_{14}$	$=$	$ ( 25926 \nu^{15} + 603881 \nu^{13} + 5068077 \nu^{11} + 18998083 \nu^{9} + 33405881 \nu^{7} + \cdots - 437333 \nu ) / 7231 $ (25926v^15 + 603881v^13 + 5068077v^11 + 18998083v^9 + 33405881v^7 + 25519016v^5 + 6225156v^3 - 437333v) / 7231
$\beta_{15}$	$=$	$ ( 33297 \nu^{15} + 775702 \nu^{13} + 6512858 \nu^{11} + 24444196 \nu^{9} + 43160577 \nu^{7} + \cdots - 173250 \nu ) / 7231 $ (33297v^15 + 775702v^13 + 6512858v^11 + 24444196v^9 + 43160577v^7 + 33528527v^5 + 8990368v^3 - 173250v) / 7231

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

$\nu$	$=$	$ ( \beta_{10} - \beta_{6} + \beta_{4} + \beta_1 ) / 2 $ (b10 - b6 + b4 + b1) / 2
$\nu^{2}$	$=$	$ ( \beta_{11} + \beta_{7} + 3\beta_{3} + \beta_{2} - 6 ) / 2 $ (b11 + b7 + 3*b3 + b2 - 6) / 2
$\nu^{3}$	$=$	$ ( -\beta_{15} - 5\beta_{10} + 6\beta_{6} + 6\beta_{5} - 8\beta_{4} - 7\beta_1 ) / 2 $ (-b15 - 5b10 + 6b6 + 6b5 - 8b4 - 7*b1) / 2
$\nu^{4}$	$=$	$ ( 2\beta_{13} - 9\beta_{11} - 3\beta_{9} - 6\beta_{8} - 10\beta_{7} - 27\beta_{3} - 10\beta_{2} + 38 ) / 2 $ (2b13 - 9b11 - 3b9 - 6b8 - 10b7 - 27b3 - 10*b2 + 38) / 2
$\nu^{5}$	$=$	$ ( 5\beta_{15} + 3\beta_{14} - \beta_{12} + 33\beta_{10} - 47\beta_{6} - 62\beta_{5} + 71\beta_{4} + 55\beta_1 ) / 2 $ (5b15 + 3b14 - b12 + 33b10 - 47b6 - 62b5 + 71b4 + 55*b1) / 2
$\nu^{6}$	$=$	$ ( -18\beta_{13} + 86\beta_{11} + 42\beta_{9} + 83\beta_{8} + 90\beta_{7} + 229\beta_{3} + 91\beta_{2} - 294 ) / 2 $ (-18b13 + 86b11 + 42b9 + 83b8 + 90b7 + 229b3 + 91*b2 - 294) / 2
$\nu^{7}$	$=$	$ ( - 11 \beta_{15} - 51 \beta_{14} + 34 \beta_{12} - 219 \beta_{10} + 385 \beta_{6} + 559 \beta_{5} + \cdots - 467 \beta_1 ) / 2 $ (-11b15 - 51b14 + 34b12 - 219b10 + 385b6 + 559b5 - 630b4 - 467b1) / 2
$\nu^{8}$	$=$	$ ( 113 \beta_{13} - 820 \beta_{11} - 476 \beta_{9} - 909 \beta_{8} - 787 \beta_{7} - 1948 \beta_{3} + \cdots + 2417 ) / 2 $ (113b13 - 820b11 - 476b9 - 909b8 - 787b7 - 1948b3 - 832*b2 + 2417) / 2
$\nu^{9}$	$=$	$ ( - 145 \beta_{15} + 630 \beta_{14} - 552 \beta_{12} + 1387 \beta_{10} - 3204 \beta_{6} + \cdots + 4085 \beta_1 ) / 2 $ (-145b15 + 630b14 - 552b12 + 1387b10 - 3204b6 - 4908b5 + 5582b4 + 4085b1) / 2
$\nu^{10}$	$=$	$ ( - 514 \beta_{13} + 7743 \beta_{11} + 4970 \beta_{9} + 9227 \beta_{8} + 6847 \beta_{7} + 16682 \beta_{3} + \cdots - 20336 ) / 2 $ (-514b13 + 7743b11 + 4970b9 + 9227b8 + 6847b7 + 16682b3 + 7634*b2 - 20336) / 2
$\nu^{11}$	$=$	$ ( 3077 \beta_{15} - 6896 \beta_{14} + 6970 \beta_{12} - 8084 \beta_{10} + 26945 \beta_{6} + \cdots - 36116 \beta_1 ) / 2 $ (3077b15 - 6896b14 + 6970b12 - 8084b10 + 26945b6 + 42906b5 - 49477b4 - 36116b1) / 2
$\nu^{12}$	$=$	$ 252 \beta_{13} - 36212 \beta_{11} - 24831 \beta_{9} - 45170 \beta_{8} - 29832 \beta_{7} - 71870 \beta_{3} + \cdots + 86566 $ 252b13 - 36212b11 - 24831b9 - 45170b8 - 29832b7 - 71870b3 - 34986*b2 + 86566
$\nu^{13}$	$=$	$ ( - 40723 \beta_{15} + 70975 \beta_{14} - 78066 \beta_{12} + 39791 \beta_{10} - 228507 \beta_{6} + \cdots + 320677 \beta_1 ) / 2 $ (-40723b15 + 70975b14 - 78066b12 + 39791b10 - 228507b6 - 375611b5 + 439060b4 + 320677b1) / 2
$\nu^{14}$	$=$	$ ( 27158 \beta_{13} + 671999 \beta_{11} + 482846 \beta_{9} + 865586 \beta_{8} + 521589 \beta_{7} + \cdots - 1485588 ) / 2 $ (27158b13 + 671999b11 + 482846b9 + 865586b8 + 521589b7 + 1245269b3 + 639533*b2 - 1485588) / 2
$\nu^{15}$	$=$	$ ( 462786 \beta_{15} - 704028 \beta_{14} + 817509 \beta_{12} - 112000 \beta_{10} + 1951782 \beta_{6} + \cdots - 2853438 \beta_1 ) / 2 $ (462786b15 - 704028b14 + 817509b12 - 112000b10 + 1951782b6 + 3297345b5 - 3901487b4 - 2853438b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

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} $

1322.1

	−	1.54124i
	−	1.70654i
	−	0.829521i
		2.99366i
	−	0.964652i
	−	0.0716223i
		2.73296i
	−	0.810833i
	−	2.73296i
		0.810833i
		0.964652i
		0.0716223i
		0.829521i
	−	2.99366i
		1.54124i
		1.70654i

−

2.67093i

−5.13386

−3.20318

8.37032i

8.55547i

1322.2

−

2.67093i

−5.13386

3.20318

8.37032i

−

8.55547i

1322.3

−

2.02428i

−2.09771

−0.182734

0.197790i

0.369905i

1322.4

−

2.02428i

−2.09771

0.182734

0.197790i

−

0.369905i

1322.5

−

0.698626i

1.51192

−2.26676

−

2.45352i

1.58361i

1322.6

−

0.698626i

1.51192

2.26676

−

2.45352i

−

1.58361i

1322.7

−

0.529484i

1.71965

−0.753692

−

1.96949i

0.399068i

1322.8

−

0.529484i

1.71965

0.753692

−

1.96949i

−

0.399068i

1322.9

0.529484i

1.71965

−0.753692

1.96949i

−

0.399068i

1322.10

0.529484i

1.71965

0.753692

1.96949i

0.399068i

1322.11

0.698626i

1.51192

−2.26676

2.45352i

−

1.58361i

1322.12

0.698626i

1.51192

2.26676

2.45352i

1.58361i

1322.13

2.02428i

−2.09771

−0.182734

−

0.197790i

−

0.369905i

1322.14

2.02428i

−2.09771

0.182734

−

0.197790i

0.369905i

1322.15

2.67093i

−5.13386

−3.20318

−

8.37032i

−

8.55547i

1322.16

2.67093i

−5.13386

3.20318

−

8.37032i

8.55547i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1323.2.c.f	✓	16
3.b	odd	2	1	inner	1323.2.c.f	✓	16
7.b	odd	2	1	inner	1323.2.c.f	✓	16
21.c	even	2	1	inner	1323.2.c.f	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1323.2.c.f	✓	16	1.a	even	1	1	trivial
1323.2.c.f	✓	16	3.b	odd	2	1	inner
1323.2.c.f	✓	16	7.b	odd	2	1	inner
1323.2.c.f	✓	16	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 12T_{2}^{6} + 38T_{2}^{4} + 24T_{2}^{2} + 4 $ T2^8 + 12*T2^6 + 38*T2^4 + 24*T2^2 + 4 acting on $S_{2}^{\mathrm{new}}(1323, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 12 T^{6} + 38 T^{4} + \cdots + 4)^{2} $ (T^8 + 12T^6 + 38T^4 + 24*T^2 + 4)^2
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 16 T^{6} + 62 T^{4} + \cdots + 1)^{2} $ (T^8 - 16T^6 + 62T^4 - 32*T^2 + 1)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} + 60 T^{6} + \cdots + 2116)^{2} $ (T^8 + 60T^6 + 1142T^4 + 7128*T^2 + 2116)^2
$13$	$ (T^{8} + 76 T^{6} + \cdots + 20164)^{2} $ (T^8 + 76T^6 + 1790T^4 + 12920*T^2 + 20164)^2
$17$	$ (T^{8} - 92 T^{6} + \cdots + 134689)^{2} $ (T^8 - 92T^6 + 2918T^4 - 36124*T^2 + 134689)^2
$19$	$ (T^{8} + 28 T^{6} + 206 T^{4} + \cdots + 4)^{2} $ (T^8 + 28T^6 + 206T^4 + 152*T^2 + 4)^2
$23$	$ (T^{8} + 88 T^{6} + \cdots + 148996)^{2} $ (T^8 + 88T^6 + 2676T^4 + 33712*T^2 + 148996)^2
$29$	$ (T^{8} + 108 T^{6} + \cdots + 4)^{2} $ (T^8 + 108T^6 + 2774T^4 + 10392*T^2 + 4)^2
$31$	$ (T^{8} + 172 T^{6} + \cdots + 9604)^{2} $ (T^8 + 172T^6 + 9662T^4 + 176792*T^2 + 9604)^2
$37$	$ (T^{4} - 8 T^{3} + \cdots + 391)^{4} $ (T^4 - 8T^3 - 88T^2 + 476*T + 391)^4
$41$	$ (T^{8} - 112 T^{6} + \cdots + 529)^{2} $ (T^8 - 112T^6 + 878T^4 - 1664*T^2 + 529)^2
$43$	$ (T^{4} + 4 T^{3} + \cdots + 943)^{4} $ (T^4 + 4T^3 - 96T^2 + 16*T + 943)^4
$47$	$ (T^{8} - 196 T^{6} + \cdots + 18769)^{2} $ (T^8 - 196T^6 + 6806T^4 - 56036*T^2 + 18769)^2
$53$	$ (T^{8} + 344 T^{6} + \cdots + 6791236)^{2} $ (T^8 + 344T^6 + 37044T^4 + 1315376*T^2 + 6791236)^2
$59$	$ (T^{8} - 292 T^{6} + \cdots + 24990001)^{2} $ (T^8 - 292T^6 + 31286T^4 - 1458692*T^2 + 24990001)^2
$61$	$ (T^{8} + 436 T^{6} + \cdots + 5827396)^{2} $ (T^8 + 436T^6 + 63062T^4 + 3061832*T^2 + 5827396)^2
$67$	$ (T^{4} - 12 T^{3} + \cdots + 612)^{4} $ (T^4 - 12T^3 - 42T^2 + 504*T + 612)^4
$71$	$ (T^{8} + 432 T^{6} + \cdots + 68029504)^{2} $ (T^8 + 432T^6 + 63824T^4 + 3723648*T^2 + 68029504)^2
$73$	$ (T^{8} + 104 T^{6} + \cdots + 103684)^{2} $ (T^8 + 104T^6 + 3572T^4 + 43216*T^2 + 103684)^2
$79$	$ (T^{4} + 16 T^{3} + \cdots - 6167)^{4} $ (T^4 + 16T^3 - 58T^2 - 1744*T - 6167)^4
$83$	$ (T^{8} - 400 T^{6} + \cdots + 390625)^{2} $ (T^8 - 400T^6 + 38750T^4 - 500000*T^2 + 390625)^2
$89$	$ (T^{8} - 652 T^{6} + \cdots + 2155024)^{2} $ (T^8 - 652T^6 + 109580T^4 - 1096496*T^2 + 2155024)^2
$97$	$ (T^{8} + 596 T^{6} + \cdots + 36409156)^{2} $ (T^8 + 596T^6 + 113798T^4 + 7276072*T^2 + 36409156)^2
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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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{2} + (\beta_{13} - 1) q^{4} - \beta_{2} q^{5} + ( - \beta_{15} + \beta_{12} + \cdots - \beta_1) q^{8}+O(q^{10}) \) q + b5 * q^2 + (b13 - 1) * q^4 - b2 * q^5 + (-b15 + b12 - b5 - b1) * q^8	\( q + \beta_{5} q^{2} + (\beta_{13} - 1) q^{4} - \beta_{2} q^{5} + ( - \beta_{15} + \beta_{12} + \cdots - \beta_1) q^{8}+ \cdots + (2 \beta_{14} - 6 \beta_{10} + \cdots + 4 \beta_{4}) q^{97}+O(q^{100}) \) q + b5 * q^2 + (b13 - 1) * q^4 - b2 * q^5 + (-b15 + b12 - b5 - b1) * q^8 + (-b14 - b10 - b6 - b4) * q^10 + (-b15 - b12 - b5) * q^11 + (b14 + b10 - b6 + b4) * q^13 + (-b13 + 3b8 + b7 + 3) q^16 + (2b11 + b9) q^17 + (b10 - b6 + b4) * q^19 + (2b11 - 3b9 + b3 + 4b2) q^20 + (-2b13 - b8 + 4) q^22 + (b15 - b12 - 2b1) q^23 + (-b13 + b8 + b7 - 1) * q^25 + (-2b11 - b9 - 3b3 - 4b2) q^26 + (-2b15 + b12 + 2b1) * q^29 + (2b14 - 2b10 + b6 + 3b4) q^31 + (-2b15 - 2b12 + 2b5 + 4b1) * q^32 + (-3b10 - b6 + 2b4) * q^34 + (-b13 + 3b8 + 2b7 + 2) * q^37 + (-b11 - b9 - 2b3 - b2) q^38 + (2b14 + 3b10 + 4b6 + 5b4) * q^40 + (-b11 - b9 + 2b3) q^41 + (4b8 - b7 - 1) q^43 + (-3b12 + 6b5 + b1) * q^44 + (-b13 + 3b8 + 2b7 + 1) * q^46 + (3b11 - 2b9 + b2) * q^47 + (-2b12 + 4b1) * q^50 + (-2b14 + b10 - 2b6 - 7b4) q^52 + (-2b15 + b12 + 5b5 - 2b1) q^53 + (b14 + 3b10 + b6 + 2b4) * q^55 + (-3b8 - 2b7) * q^58 + (-3b11 + 2b9 - 4b3 - 3b2) * q^59 + (b14 - 4b10 - b6 - 5b4) * q^61 + (-5b11 - 4b3 - 8b2) q^62 + (2b13 - 4b8 - 2b7 - 2) q^64 + (2b15 - 2b12 - 3b5 - 2b1) * q^65 + (b13 + b8 - 2b7 + 3) q^67 + (2b11 - 3b9 - 3b3 + 2b2) * q^68 + (-b15 + 5b12 - 3b5 + b1) * q^71 + (-b14 + b10 - 2b6 + b4) q^73 + (-b15 - 4b12 + 2b5 + 8b1) q^74 + (-b14 + 3b10 - 2b4) * q^76 + (3b13 + 3b8 - b7 - 4) * q^79 + (-3b11 + 5b9 - b3 - 10b2) q^80 + (2b10 - b6 + b4) q^82 + 5b3 q^83 + (-b13 - b8 + 2b7 + 4) q^85 + (b15 - 4b12 + 2b1) * q^86 + (3b13 - 7b8 - b7 - 11) * q^88 + (-2b11 - 5b9 - 3b3 + 2b2) * q^89 + (b15 - 6b12 + b5 + 4b1) * q^92 + (b14 + 3b6 + 4b4) * q^94 - b12 * q^95 + (2b14 - 6b10 + 5b6 + 4b4) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 48 q^{16} + 64 q^{22} - 16 q^{25} + 32 q^{37} - 16 q^{43} + 16 q^{46} - 32 q^{64} + 48 q^{67} - 64 q^{79} + 64 q^{85} - 176 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 48 * q^16 + 64 * q^22 - 16 * q^25 + 32 * q^37 - 16 * q^43 + 16 * q^46 - 32 * q^64 + 48 * q^67 - 64 * q^79 + 64 * q^85 - 176 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	1323.2.c.f

Level	$1323$
Weight	$2$
Character orbit	1323.c
Analytic conductor	$10.564$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(10.5642081874\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 24x^{14} + 212x^{12} + 872x^{10} + 1815x^{8} + 1928x^{6} + 996x^{4} + 200x^{2} + 1 \) x^16 + 24x^14 + 212x^12 + 872x^10 + 1815x^8 + 1928x^6 + 996x^4 + 200*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 3566 \nu^{15} - 82183 \nu^{13} - 677054 \nu^{11} - 2451006 \nu^{9} - 4029525 \nu^{7} + \cdots + 102636 \nu ) / 7231 \) (-3566v^15 - 82183v^13 - 677054v^11 - 2451006v^9 - 4029525v^7 - 2653934v^5 - 400750v^3 + 102636v) / 7231
\(\beta_{2}\)	\(=\)	\( ( 4109 \nu^{14} + 95961 \nu^{12} + 809596 \nu^{10} + 3069810 \nu^{8} + 5537952 \nu^{6} + 4485192 \nu^{4} + \cdots - 9938 ) / 7231 \) (4109v^14 + 95961v^12 + 809596v^10 + 3069810v^8 + 5537952v^6 + 4485192v^4 + 1280801*v^2 - 9938) / 7231
\(\beta_{3}\)	\(=\)	\( ( - 4213 \nu^{14} - 98695 \nu^{12} - 836669 \nu^{10} - 3196410 \nu^{8} - 5829935 \nu^{6} - 4816716 \nu^{4} + \cdots - 8680 ) / 7231 \) (-4213v^14 - 98695v^12 - 836669v^10 - 3196410v^8 - 5829935v^6 - 4816716v^4 - 1459060*v^2 - 8680) / 7231
\(\beta_{4}\)	\(=\)	\( ( 11880 \nu^{15} + 277026 \nu^{13} + 2330454 \nu^{11} + 8786075 \nu^{9} + 15693672 \nu^{7} + \cdots + 206322 \nu ) / 7231 \) (11880v^15 + 277026v^13 + 2330454v^11 + 8786075v^9 + 15693672v^7 + 12631032v^5 + 3918733v^3 + 206322v) / 7231
\(\beta_{5}\)	\(=\)	\( ( 12411 \nu^{15} + 289535 \nu^{13} + 2437027 \nu^{11} + 9191538 \nu^{9} + 16395526 \nu^{7} + \cdots + 89795 \nu ) / 7231 \) (12411v^15 + 289535v^13 + 2437027v^11 + 9191538v^9 + 16395526v^7 + 13061987v^5 + 3821459v^3 + 89795v) / 7231
\(\beta_{6}\)	\(=\)	\( ( - 12661 \nu^{15} - 294796 \nu^{13} - 2472050 \nu^{11} - 9248819 \nu^{9} - 16190613 \nu^{7} + \cdots + 184528 \nu ) / 7231 \) (-12661v^15 - 294796v^13 - 2472050v^11 - 9248819v^9 - 16190613v^7 - 12258167v^5 - 2969225v^3 + 184528v) / 7231
\(\beta_{7}\)	\(=\)	\( ( - 15611 \nu^{14} - 362454 \nu^{12} - 3026016 \nu^{10} - 11245322 \nu^{8} - 19531408 \nu^{6} + \cdots + 16978 ) / 7231 \) (-15611v^14 - 362454v^12 - 3026016v^10 - 11245322v^8 - 19531408v^6 - 14787914v^4 - 3904397*v^2 + 16978) / 7231
\(\beta_{8}\)	\(=\)	\( ( - 2432 \nu^{14} - 56782 \nu^{12} - 478539 \nu^{10} - 1808300 \nu^{8} - 3232355 \nu^{6} - 2573576 \nu^{4} + \cdots - 5170 ) / 1033 \) (-2432v^14 - 56782v^12 - 478539v^10 - 1808300v^8 - 3232355v^6 - 2573576v^4 - 736177*v^2 - 5170) / 1033
\(\beta_{9}\)	\(=\)	\( ( 20\nu^{14} + 467\nu^{12} + 3936\nu^{10} + 14872\nu^{8} + 26564\nu^{6} + 21096\nu^{4} + 5980\nu^{2} + 18 ) / 7 \) (20v^14 + 467v^12 + 3936v^10 + 14872v^8 + 26564v^6 + 21096v^4 + 5980*v^2 + 18) / 7
\(\beta_{10}\)	\(=\)	\( ( - 20975 \nu^{15} - 489639 \nu^{13} - 4125450 \nu^{11} - 15583888 \nu^{9} - 27854760 \nu^{7} + \cdots - 109968 \nu ) / 7231 \) (-20975v^15 - 489639v^13 - 4125450v^11 - 15583888v^9 - 27854760v^7 - 22235265v^5 - 6487208v^3 - 109968v) / 7231
\(\beta_{11}\)	\(=\)	\( ( 24141 \nu^{14} + 562578 \nu^{12} + 4726427 \nu^{10} + 17764742 \nu^{8} + 31483261 \nu^{6} + \cdots + 62386 ) / 7231 \) (24141v^14 + 562578v^12 + 4726427v^10 + 17764742v^8 + 31483261v^6 + 24752870v^4 + 7015238*v^2 + 62386) / 7231
\(\beta_{12}\)	\(=\)	\( ( 25023 \nu^{15} + 585089 \nu^{13} + 4943611 \nu^{11} + 18772057 \nu^{9} + 33876484 \nu^{7} + \cdots + 246543 \nu ) / 7231 \) (25023v^15 + 585089v^13 + 4943611v^11 + 18772057v^9 + 33876484v^7 + 27549033v^5 + 8361354v^3 + 246543v) / 7231
\(\beta_{13}\)	\(=\)	\( ( - 25833 \nu^{14} - 602052 \nu^{12} - 5058697 \nu^{10} - 19017892 \nu^{8} - 33715265 \nu^{6} + \cdots - 19311 ) / 7231 \) (-25833v^14 - 602052v^12 - 5058697v^10 - 19017892v^8 - 33715265v^6 - 26500974v^4 - 7440426*v^2 - 19311) / 7231
\(\beta_{14}\)	\(=\)	\( ( 25926 \nu^{15} + 603881 \nu^{13} + 5068077 \nu^{11} + 18998083 \nu^{9} + 33405881 \nu^{7} + \cdots - 437333 \nu ) / 7231 \) (25926v^15 + 603881v^13 + 5068077v^11 + 18998083v^9 + 33405881v^7 + 25519016v^5 + 6225156v^3 - 437333v) / 7231
\(\beta_{15}\)	\(=\)	\( ( 33297 \nu^{15} + 775702 \nu^{13} + 6512858 \nu^{11} + 24444196 \nu^{9} + 43160577 \nu^{7} + \cdots - 173250 \nu ) / 7231 \) (33297v^15 + 775702v^13 + 6512858v^11 + 24444196v^9 + 43160577v^7 + 33528527v^5 + 8990368v^3 - 173250v) / 7231

\(\nu\)	\(=\)	\( ( \beta_{10} - \beta_{6} + \beta_{4} + \beta_1 ) / 2 \) (b10 - b6 + b4 + b1) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} + \beta_{7} + 3\beta_{3} + \beta_{2} - 6 ) / 2 \) (b11 + b7 + 3*b3 + b2 - 6) / 2
\(\nu^{3}\)	\(=\)	\( ( -\beta_{15} - 5\beta_{10} + 6\beta_{6} + 6\beta_{5} - 8\beta_{4} - 7\beta_1 ) / 2 \) (-b15 - 5b10 + 6b6 + 6b5 - 8b4 - 7*b1) / 2
\(\nu^{4}\)	\(=\)	\( ( 2\beta_{13} - 9\beta_{11} - 3\beta_{9} - 6\beta_{8} - 10\beta_{7} - 27\beta_{3} - 10\beta_{2} + 38 ) / 2 \) (2b13 - 9b11 - 3b9 - 6b8 - 10b7 - 27b3 - 10*b2 + 38) / 2
\(\nu^{5}\)	\(=\)	\( ( 5\beta_{15} + 3\beta_{14} - \beta_{12} + 33\beta_{10} - 47\beta_{6} - 62\beta_{5} + 71\beta_{4} + 55\beta_1 ) / 2 \) (5b15 + 3b14 - b12 + 33b10 - 47b6 - 62b5 + 71b4 + 55*b1) / 2
\(\nu^{6}\)	\(=\)	\( ( -18\beta_{13} + 86\beta_{11} + 42\beta_{9} + 83\beta_{8} + 90\beta_{7} + 229\beta_{3} + 91\beta_{2} - 294 ) / 2 \) (-18b13 + 86b11 + 42b9 + 83b8 + 90b7 + 229b3 + 91*b2 - 294) / 2
\(\nu^{7}\)	\(=\)	\( ( - 11 \beta_{15} - 51 \beta_{14} + 34 \beta_{12} - 219 \beta_{10} + 385 \beta_{6} + 559 \beta_{5} + \cdots - 467 \beta_1 ) / 2 \) (-11b15 - 51b14 + 34b12 - 219b10 + 385b6 + 559b5 - 630b4 - 467b1) / 2
\(\nu^{8}\)	\(=\)	\( ( 113 \beta_{13} - 820 \beta_{11} - 476 \beta_{9} - 909 \beta_{8} - 787 \beta_{7} - 1948 \beta_{3} + \cdots + 2417 ) / 2 \) (113b13 - 820b11 - 476b9 - 909b8 - 787b7 - 1948b3 - 832*b2 + 2417) / 2
\(\nu^{9}\)	\(=\)	\( ( - 145 \beta_{15} + 630 \beta_{14} - 552 \beta_{12} + 1387 \beta_{10} - 3204 \beta_{6} + \cdots + 4085 \beta_1 ) / 2 \) (-145b15 + 630b14 - 552b12 + 1387b10 - 3204b6 - 4908b5 + 5582b4 + 4085b1) / 2
\(\nu^{10}\)	\(=\)	\( ( - 514 \beta_{13} + 7743 \beta_{11} + 4970 \beta_{9} + 9227 \beta_{8} + 6847 \beta_{7} + 16682 \beta_{3} + \cdots - 20336 ) / 2 \) (-514b13 + 7743b11 + 4970b9 + 9227b8 + 6847b7 + 16682b3 + 7634*b2 - 20336) / 2
\(\nu^{11}\)	\(=\)	\( ( 3077 \beta_{15} - 6896 \beta_{14} + 6970 \beta_{12} - 8084 \beta_{10} + 26945 \beta_{6} + \cdots - 36116 \beta_1 ) / 2 \) (3077b15 - 6896b14 + 6970b12 - 8084b10 + 26945b6 + 42906b5 - 49477b4 - 36116b1) / 2
\(\nu^{12}\)	\(=\)	\( 252 \beta_{13} - 36212 \beta_{11} - 24831 \beta_{9} - 45170 \beta_{8} - 29832 \beta_{7} - 71870 \beta_{3} + \cdots + 86566 \) 252b13 - 36212b11 - 24831b9 - 45170b8 - 29832b7 - 71870b3 - 34986*b2 + 86566
\(\nu^{13}\)	\(=\)	\( ( - 40723 \beta_{15} + 70975 \beta_{14} - 78066 \beta_{12} + 39791 \beta_{10} - 228507 \beta_{6} + \cdots + 320677 \beta_1 ) / 2 \) (-40723b15 + 70975b14 - 78066b12 + 39791b10 - 228507b6 - 375611b5 + 439060b4 + 320677b1) / 2
\(\nu^{14}\)	\(=\)	\( ( 27158 \beta_{13} + 671999 \beta_{11} + 482846 \beta_{9} + 865586 \beta_{8} + 521589 \beta_{7} + \cdots - 1485588 ) / 2 \) (27158b13 + 671999b11 + 482846b9 + 865586b8 + 521589b7 + 1245269b3 + 639533*b2 - 1485588) / 2
\(\nu^{15}\)	\(=\)	\( ( 462786 \beta_{15} - 704028 \beta_{14} + 817509 \beta_{12} - 112000 \beta_{10} + 1951782 \beta_{6} + \cdots - 2853438 \beta_1 ) / 2 \) (462786b15 - 704028b14 + 817509b12 - 112000b10 + 1951782b6 + 3297345b5 - 3901487b4 - 2853438b1) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1322.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 12 T^{6} + 38 T^{4} + \cdots + 4)^{2} \) (T^8 + 12T^6 + 38T^4 + 24*T^2 + 4)^2
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 16 T^{6} + 62 T^{4} + \cdots + 1)^{2} \) (T^8 - 16T^6 + 62T^4 - 32*T^2 + 1)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} + 60 T^{6} + \cdots + 2116)^{2} \) (T^8 + 60T^6 + 1142T^4 + 7128*T^2 + 2116)^2
$13$	\( (T^{8} + 76 T^{6} + \cdots + 20164)^{2} \) (T^8 + 76T^6 + 1790T^4 + 12920*T^2 + 20164)^2
$17$	\( (T^{8} - 92 T^{6} + \cdots + 134689)^{2} \) (T^8 - 92T^6 + 2918T^4 - 36124*T^2 + 134689)^2
$19$	\( (T^{8} + 28 T^{6} + 206 T^{4} + \cdots + 4)^{2} \) (T^8 + 28T^6 + 206T^4 + 152*T^2 + 4)^2
$23$	\( (T^{8} + 88 T^{6} + \cdots + 148996)^{2} \) (T^8 + 88T^6 + 2676T^4 + 33712*T^2 + 148996)^2
$29$	\( (T^{8} + 108 T^{6} + \cdots + 4)^{2} \) (T^8 + 108T^6 + 2774T^4 + 10392*T^2 + 4)^2
$31$	\( (T^{8} + 172 T^{6} + \cdots + 9604)^{2} \) (T^8 + 172T^6 + 9662T^4 + 176792*T^2 + 9604)^2
$37$	\( (T^{4} - 8 T^{3} + \cdots + 391)^{4} \) (T^4 - 8T^3 - 88T^2 + 476*T + 391)^4
$41$	\( (T^{8} - 112 T^{6} + \cdots + 529)^{2} \) (T^8 - 112T^6 + 878T^4 - 1664*T^2 + 529)^2
$43$	\( (T^{4} + 4 T^{3} + \cdots + 943)^{4} \) (T^4 + 4T^3 - 96T^2 + 16*T + 943)^4
$47$	\( (T^{8} - 196 T^{6} + \cdots + 18769)^{2} \) (T^8 - 196T^6 + 6806T^4 - 56036*T^2 + 18769)^2
$53$	\( (T^{8} + 344 T^{6} + \cdots + 6791236)^{2} \) (T^8 + 344T^6 + 37044T^4 + 1315376*T^2 + 6791236)^2
$59$	\( (T^{8} - 292 T^{6} + \cdots + 24990001)^{2} \) (T^8 - 292T^6 + 31286T^4 - 1458692*T^2 + 24990001)^2
$61$	\( (T^{8} + 436 T^{6} + \cdots + 5827396)^{2} \) (T^8 + 436T^6 + 63062T^4 + 3061832*T^2 + 5827396)^2
$67$	\( (T^{4} - 12 T^{3} + \cdots + 612)^{4} \) (T^4 - 12T^3 - 42T^2 + 504*T + 612)^4
$71$	\( (T^{8} + 432 T^{6} + \cdots + 68029504)^{2} \) (T^8 + 432T^6 + 63824T^4 + 3723648*T^2 + 68029504)^2
$73$	\( (T^{8} + 104 T^{6} + \cdots + 103684)^{2} \) (T^8 + 104T^6 + 3572T^4 + 43216*T^2 + 103684)^2
$79$	\( (T^{4} + 16 T^{3} + \cdots - 6167)^{4} \) (T^4 + 16T^3 - 58T^2 - 1744*T - 6167)^4
$83$	\( (T^{8} - 400 T^{6} + \cdots + 390625)^{2} \) (T^8 - 400T^6 + 38750T^4 - 500000*T^2 + 390625)^2
$89$	\( (T^{8} - 652 T^{6} + \cdots + 2155024)^{2} \) (T^8 - 652T^6 + 109580T^4 - 1096496*T^2 + 2155024)^2
$97$	\( (T^{8} + 596 T^{6} + \cdots + 36409156)^{2} \) (T^8 + 596T^6 + 113798T^4 + 7276072*T^2 + 36409156)^2
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\(n\)	\(785\)	\(1081\)
\(\chi(n)\)	\(-1\)	\(-1\)