LMFDB - Newform orbit 1062.3.d.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1062,3,Mod(235,1062)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1062, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1062.235");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1062 = 2 \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1062.d (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:	$28.9374040751$
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} + 124 x^{14} + 5790 x^{12} + 126836 x^{10} + 1312371 x^{8} + 5713540 x^{6} + 8995714 x^{4} + \cdots + 38025 $ x^16 + 124x^14 + 5790x^12 + 126836x^10 + 1312371x^8 + 5713540x^6 + 8995714x^4 + 4218792*x^2 + 38025
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
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_{11} q^{2} - 2 q^{4} + \beta_1 q^{5} + ( - \beta_{5} + 1) q^{7} + 2 \beta_{11} q^{8}+O(q^{10}) $ q - b11 * q^2 - 2 * q^4 + b1 * q^5 + (-b5 + 1) * q^7 + 2b11 q^8	$ q - \beta_{11} q^{2} - 2 q^{4} + \beta_1 q^{5} + ( - \beta_{5} + 1) q^{7} + 2 \beta_{11} q^{8} + \beta_{10} q^{10} + \beta_{12} q^{11} + \beta_{15} q^{13} + ( - \beta_{13} - \beta_{11}) q^{14} + 4 q^{16} + ( - \beta_{6} + \beta_{4} - 2 \beta_1) q^{17} + (\beta_{2} - 13) q^{19} - 2 \beta_1 q^{20} + (\beta_{5} + \beta_{2} + 1) q^{22} + (\beta_{14} + \beta_{12} - 3 \beta_{11}) q^{23} + ( - 2 \beta_{2} + 6) q^{25} + (\beta_{6} - \beta_{3} - \beta_1) q^{26} + (2 \beta_{5} - 2) q^{28} + (\beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_1) q^{29} + ( - 2 \beta_{15} + 2 \beta_{10} + \cdots + \beta_{8}) q^{31}+ \cdots + ( - 2 \beta_{14} + \cdots - 17 \beta_{11}) q^{98}+O(q^{100}) $ q - b11 * q^2 - 2 * q^4 + b1 * q^5 + (-b5 + 1) * q^7 + 2b11 q^8 + b10 * q^10 + b12 * q^11 + b15 * q^13 + (-b13 - b11) * q^14 + 4 * q^16 + (-b6 + b4 - 2b1) q^17 + (b2 - 13) * q^19 - 2b1 q^20 + (b5 + b2 + 1) * q^22 + (b14 + b12 - 3b11) q^23 + (-2b2 + 6) q^25 + (b6 - b3 - b1) * q^26 + (2b5 - 2) q^28 + (b6 + b4 + 2b3 + b1) q^29 + (-2b15 + 2b10 - 3b9 + b8) q^31 - 4b11 q^32 + (b15 - b10 + b9 - 2b8) q^34 + (2b6 - 4b4 + 3b3 + 3b1) * q^35 + (2b15 - 4b10 + b8) * q^37 + (-b13 - 2b12 + 14b11) * q^38 - 2b10 q^40 + (b6 + 3b4 + 2b3 + 3b1) q^41 + (b15 - 3b10) q^43 - 2b12 q^44 + (b7 + b5 + b2 - 5) * q^46 + (b14 - 3b11) q^47 + (b7 - b5 + 17) * q^49 + (2b13 + 4b12 - 8b11) q^50 - 2b15 q^52 + (-4b6 - 6b4 + 4b3 + 5b1) * q^53 + (-b15 + 3b10 - 6b9 + 6b8) q^55 + (2b13 + 2b11) * q^56 + (b15 - 3b9 - 2b8) * q^58 + (-b14 - b12 + 3b11 + 2b6 - 2b4 + 5b3 + 3b1) q^59 + (-3b15 - 4b10 - 6b9 + b8) q^61 + (-5b6 + b4 - b3 - 6b1) * q^62 - 8 * q^64 + (b12 - 15b11) q^65 + (3b15 - 7b10 + 10b8) q^67 + (2b6 - 2b4 + 4b1) q^68 + (b15 - 4b10 - 5b9 + 8b8) q^70 + (9b6 - 4b4 - b3 + 4b1) q^71 + (7b15 - 4b10 - 6b9 + 7b8) * q^73 + (2b6 + b4 - 2b3 + 5b1) q^74 + (-2b2 + 26) q^76 + (b14 + 3b13 + 6b12 - 30b11) q^77 + (2b7 - 3b5 + b2 - 22) * q^79 + 4b1 q^80 + (b15 + 4b10 - 3b9 - 6b8) q^82 + (b14 - 6b12 + 27b11) * q^83 + (b7 + 7b5 + 6b2 - 46) * q^85 + (b6 - b3 + 5b1) q^86 + (-2b5 - 2b2 - 2) * q^88 + (b14 + 3b13 - 4b12 + 18b11) q^89 + (10b15 - 4b10 - 3b9 + b8) q^91 + (-2b14 - 2b12 + 6b11) q^92 + (b7 - 6) * q^94 + (-5b6 + 2b4 - 2b3 - 29b1) * q^95 + (-3b15 + 2b10 - 6b9 - 8b8) * q^97 + (-2b14 - b13 - 17b11) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 24 q^{7}+O(q^{10}) $ 16 * q - 32 * q^4 + 24 * q^7	$ 16 q - 32 q^{4} + 24 q^{7} + 64 q^{16} - 200 q^{19} + 16 q^{22} + 80 q^{25} - 48 q^{28} - 80 q^{46} + 280 q^{49} - 128 q^{64} + 400 q^{76} - 320 q^{79} - 744 q^{85} - 32 q^{88} - 96 q^{94}+O(q^{100}) $ 16 * q - 32 * q^4 + 24 * q^7 + 64 * q^16 - 200 * q^19 + 16 * q^22 + 80 * q^25 - 48 * q^28 - 80 * q^46 + 280 * q^49 - 128 * q^64 + 400 * q^76 - 320 * q^79 - 744 * q^85 - 32 * q^88 - 96 * q^94

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 124 x^{14} + 5790 x^{12} + 126836 x^{10} + 1312371 x^{8} + 5713540 x^{6} + 8995714 x^{4} + \cdots + 38025 $ x^16 + 124*x^14 + 5790*x^12 + 126836*x^10 + 1312371*x^8 + 5713540*x^6 + 8995714*x^4 + 4218792*x^2 + 38025 :

$\beta_{1}$	$=$	$ ( - 1650212 \nu^{14} - 201595493 \nu^{12} - 9236693019 \nu^{10} - 197138316010 \nu^{8} + \cdots + 927039705477 ) / 1145120641212 $ (-1650212v^14 - 201595493v^12 - 9236693019v^10 - 197138316010v^8 - 1946709533187v^6 - 7440073612520v^4 - 6901460510789*v^2 + 927039705477) / 1145120641212
$\beta_{2}$	$=$	$ ( 1650212 \nu^{14} + 201595493 \nu^{12} + 9236693019 \nu^{10} + 197138316010 \nu^{8} + \cdots + 17394890553915 ) / 1145120641212 $ (1650212v^14 + 201595493v^12 + 9236693019v^10 + 197138316010v^8 + 1946709533187v^6 + 7440073612520v^4 + 8046581152001*v^2 + 17394890553915) / 1145120641212
$\beta_{3}$	$=$	$ ( 11145433420 \nu^{14} + 1418856329068 \nu^{12} + 69018841423905 \nu^{10} + \cdots + 64\!\cdots\!94 ) / 38\!\cdots\!94 $ (11145433420v^14 + 1418856329068v^12 + 69018841423905v^10 + 1615561396846958v^8 + 18721754736125583v^6 + 99355361358326548v^4 + 197288435690739163*v^2 + 64588607805765294) / 3806953571709294
$\beta_{4}$	$=$	$ ( 21157248870 \nu^{14} + 2627019039479 \nu^{12} + 122880980236227 \nu^{10} + \cdots + 24\!\cdots\!06 ) / 38\!\cdots\!94 $ (21157248870v^14 + 2627019039479v^12 + 122880980236227v^10 + 2698199508425123v^8 + 27988626430694293v^6 + 120933074225145927v^4 + 165751818025166473*v^2 + 24244376494878006) / 3806953571709294
$\beta_{5}$	$=$	$ ( - 564341077 \nu^{14} - 69595437347 \nu^{12} - 3215003855630 \nu^{10} + \cdots - 683334264045690 ) / 69852359113932 $ (-564341077v^14 - 69595437347v^12 - 3215003855630v^10 - 68894709572105v^8 - 678206195684988v^6 - 2583570737553927v^4 - 2789707505696903*v^2 - 683334264045690) / 69852359113932
$\beta_{6}$	$=$	$ ( 69812658697 \nu^{14} + 8586058539833 \nu^{12} + 395397371559228 \nu^{10} + \cdots - 70\!\cdots\!26 ) / 76\!\cdots\!88 $ (69812658697v^14 + 8586058539833v^12 + 395397371559228v^10 + 8439545747844915v^8 + 82431248553261938v^6 + 303513367341094429v^4 + 236561295032891925*v^2 - 70864932521859726) / 7613907143418588
$\beta_{7}$	$=$	$ ( 4283651303 \nu^{14} + 525431837271 \nu^{12} + 24122641534868 \nu^{10} + \cdots + 25\!\cdots\!60 ) / 69852359113932 $ (4283651303v^14 + 525431837271v^12 + 24122641534868v^10 + 513482019447355v^8 + 5021799368346126v^6 + 19054946267751277v^4 + 20540039053314147*v^2 + 2501250463173960) / 69852359113932
$\beta_{8}$	$=$	$ ( 1808590770098 \nu^{15} + 237816012825707 \nu^{13} + \cdots + 56\!\cdots\!46 \nu ) / 14\!\cdots\!60 $ (1808590770098v^15 + 237816012825707v^13 + 12106750789166970v^11 + 302934613820605753v^9 + 3899131374504764748v^7 + 24730680597289014785v^5 + 67556031873373673162v^3 + 56534837184398702346v) / 1484711892966624660
$\beta_{9}$	$=$	$ ( 495047358121 \nu^{15} + 65159470330918 \nu^{13} + \cdots + 17\!\cdots\!93 \nu ) / 29\!\cdots\!32 $ (495047358121v^15 + 65159470330918v^13 + 3330910246680114v^11 + 84160120525955831v^9 + 1103166151979284464v^7 + 7193748249903108625v^5 + 20187539778804202129v^3 + 17193885911372432793v) / 296942378593324932
$\beta_{10}$	$=$	$ ( - 1118363773 \nu^{15} - 138355316512 \nu^{13} - 6436015124535 \nu^{11} + \cdots - 29\!\cdots\!81 \nu ) / 223298525036340 $ (-1118363773v^15 - 138355316512v^13 - 6436015124535v^11 - 140047632373523v^9 - 1429266211513833v^7 - 6010207792614955v^5 - 8609666295427522v^3 - 2925762288945681v) / 223298525036340
$\beta_{11}$	$=$	$ ( 1118363773 \nu^{15} + 138355316512 \nu^{13} + 6436015124535 \nu^{11} + \cdots + 33\!\cdots\!61 \nu ) / 223298525036340 $ (1118363773v^15 + 138355316512v^13 + 6436015124535v^11 + 140047632373523v^9 + 1429266211513833v^7 + 6010207792614955v^5 + 8609666295427522v^3 + 3372359339018361v) / 223298525036340
$\beta_{12}$	$=$	$ ( 97813935158 \nu^{15} + 12096372313677 \nu^{13} + 562247593028830 \nu^{11} + \cdots + 23\!\cdots\!41 \nu ) / 45\!\cdots\!80 $ (97813935158v^15 + 12096372313677v^13 + 562247593028830v^11 + 12212423594723358v^9 + 124082896577728198v^7 + 514929332898055170v^5 + 702844905967951682v^3 + 231037576985884641v) / 4540403342405580
$\beta_{13}$	$=$	$ ( 97002031952 \nu^{15} + 12011715328883 \nu^{13} + 559638947982435 \nu^{11} + \cdots + 38\!\cdots\!29 \nu ) / 27\!\cdots\!48 $ (97002031952v^15 + 12011715328883v^13 + 559638947982435v^11 + 12213397471855870v^9 + 125462066614774491v^7 + 537577426516587932v^5 + 820774399150743245v^3 + 385969120058095629v) / 2724242005443348
$\beta_{14}$	$=$	$ ( 36616710101 \nu^{15} + 4544715626820 \nu^{13} + 212563666247359 \nu^{11} + \cdots + 22\!\cdots\!56 \nu ) / 908080668481116 $ (36616710101v^15 + 4544715626820v^13 + 212563666247359v^11 + 4671927111150834v^9 + 48721052877899443v^7 + 217236949897126650v^5 + 373382851662416570v^3 + 225277709114517756v) / 908080668481116
$\beta_{15}$	$=$	$ ( - 22716171357238 \nu^{15} + \cdots - 10\!\cdots\!31 \nu ) / 49\!\cdots\!20 $ (-22716171357238v^15 - 2820348884798137v^13 - 131959995236364920v^11 - 2900981651308329328v^9 - 30225377774938880728v^7 - 133654400664780522600v^5 - 216906565992784007652v^3 - 100334382617985933231v) / 494903964322208220

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

$\nu$	$=$	$ ( \beta_{11} + \beta_{10} ) / 2 $ (b11 + b10) / 2
$\nu^{2}$	$=$	$ \beta_{2} + \beta _1 - 16 $ b2 + b1 - 16
$\nu^{3}$	$=$	$ ( 3\beta_{15} + 3\beta_{13} + 6\beta_{12} - 50\beta_{11} - 29\beta_{10} + 7\beta_{9} - 4\beta_{8} ) / 2 $ (3b15 + 3b13 + 6b12 - 50b11 - 29b10 + 7b9 - 4*b8) / 2
$\nu^{4}$	$=$	$ -\beta_{7} - 10\beta_{6} - 13\beta_{5} + 4\beta_{4} - 4\beta_{3} - 38\beta_{2} - 64\beta _1 + 487 $ -b7 - 10b6 - 13b5 + 4b4 - 4b3 - 38b2 - 64b1 + 487
$\nu^{5}$	$=$	$ ( - 103 \beta_{15} - 10 \beta_{14} - 115 \beta_{13} - 360 \beta_{12} + 2459 \beta_{11} + \cdots + 298 \beta_{8} ) / 2 $ (-103b15 - 10b14 - 115b13 - 360b12 + 2459b11 + 912b10 - 351b9 + 298b8) / 2
$\nu^{6}$	$=$	$ 77\beta_{7} + 631\beta_{6} + 818\beta_{5} - 427\beta_{4} + 352\beta_{3} + 1463\beta_{2} + 3237\beta _1 - 16979 $ 77b7 + 631b6 + 818b5 - 427b4 + 352b3 + 1463b2 + 3237*b1 - 16979
$\nu^{7}$	$=$	$ ( 3542 \beta_{15} + 1008 \beta_{14} + 3724 \beta_{13} + 17990 \beta_{12} - 112113 \beta_{11} + \cdots - 15868 \beta_{8} ) / 2 $ (3542b15 + 1008b14 + 3724b13 + 17990b12 - 112113b11 - 31209b10 + 15776b9 - 15868b8) / 2
$\nu^{8}$	$=$	$ - 4219 \beta_{7} - 32840 \beta_{6} - 41305 \beta_{5} + 27788 \beta_{4} - 21220 \beta_{3} + \cdots + 636448 $ -4219b7 - 32840b6 - 41305b5 + 27788b4 - 21220b3 - 58554b2 - 151420*b1 + 636448
$\nu^{9}$	$=$	$ ( - 129101 \beta_{15} - 64014 \beta_{14} - 112437 \beta_{13} - 844044 \beta_{12} + \cdots + 763678 \beta_{8} ) / 2 $ (-129101b15 - 64014b14 - 112437b13 - 844044b12 + 4950178b11 + 1142165b10 - 695777b9 + 763678b8) / 2
$\nu^{10}$	$=$	$ 206923 \beta_{7} + 1586863 \beta_{6} + 1946836 \beta_{5} - 1504171 \beta_{4} + 1108086 \beta_{3} + \cdots - 25056201 $ 206923b7 + 1586863b6 + 1946836b5 - 1504171b4 + 1108086b3 + 2415492b2 + 6853510*b1 - 25056201
$\nu^{11}$	$=$	$ ( 4950725 \beta_{15} + 3422188 \beta_{14} + 3247871 \beta_{13} + 38427092 \beta_{12} + \cdots - 35280380 \beta_{8} ) / 2 $ (4950725b15 + 3422188b14 + 3247871b13 + 38427092b12 - 216156313b11 - 43996872b10 + 30579111b9 - 35280380b8) / 2
$\nu^{12}$	$=$	$ - 9675642 \beta_{7} - 73737546 \beta_{6} - 89069520 \beta_{5} + 74509290 \beta_{4} - 53846890 \beta_{3} + \cdots + 1020896597 $ -9675642b7 - 73737546b6 - 89069520b5 + 74509290b4 - 53846890b3 - 101760582b2 - 305371054*b1 + 1020896597
$\nu^{13}$	$=$	$ ( - 197400314 \beta_{15} - 168369864 \beta_{14} - 90117794 \beta_{13} - 1721499052 \beta_{12} + \cdots + 1597588896 \beta_{8} ) / 2 $ (-197400314b15 - 168369864b14 - 90117794b13 - 1721499052b12 + 9411404625b11 + 1760826731b10 - 1343309962b9 + 1597588896b8) / 2
$\nu^{14}$	$=$	$ 441489690 \beta_{7} + 3349775582 \beta_{6} + 4012089360 \beta_{5} - 3516982136 \beta_{4} + \cdots - 42599852104 $ 441489690b7 + 3349775582b6 + 4012089360b5 - 3516982136b4 + 2513636866b3 + 4347525559b2 + 13498243491*b1 - 42599852104
$\nu^{15}$	$=$	$ ( 8100667273 \beta_{15} + 7916943652 \beta_{14} + 2373308457 \beta_{13} + 76421540518 \beta_{12} + \cdots - 71531430848 \beta_{8} ) / 2 $ (8100667273b15 + 7916943652b14 + 2373308457b13 + 76421540518b12 - 409800018932b11 - 72464563321b10 + 59005476301b9 - 71531430848b8) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1062\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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} $

235.1

		6.61957i
		5.40774i
		2.31123i
		1.31834i
		0.0958769i
	−	0.897013i
	−	3.99353i
	−	5.20536i
	−	6.61957i
	−	5.40774i
	−	2.31123i
	−	1.31834i
	−	0.0958769i
		0.897013i
		3.99353i
		5.20536i

−

1.41421i

−2.00000

−8.36148

7.49600

2.82843i

11.8249i

235.2

−

1.41421i

−2.00000

−6.64771

−9.84865

2.82843i

9.40128i

235.3

−

1.41421i

−2.00000

−2.26857

−2.06591

2.82843i

3.20824i

235.4

−

1.41421i

−2.00000

−0.864410

10.4186

2.82843i

1.22246i

235.5

−

1.41421i

−2.00000

0.864410

10.4186

2.82843i

−

1.22246i

235.6

−

1.41421i

−2.00000

2.26857

−2.06591

2.82843i

−

3.20824i

235.7

−

1.41421i

−2.00000

6.64771

−9.84865

2.82843i

−

9.40128i

235.8

−

1.41421i

−2.00000

8.36148

7.49600

2.82843i

−

11.8249i

235.9

1.41421i

−2.00000

−8.36148

7.49600

−

2.82843i

−

11.8249i

235.10

1.41421i

−2.00000

−6.64771

−9.84865

−

2.82843i

−

9.40128i

235.11

1.41421i

−2.00000

−2.26857

−2.06591

−

2.82843i

−

3.20824i

235.12

1.41421i

−2.00000

−0.864410

10.4186

−

2.82843i

−

1.22246i

235.13

1.41421i

−2.00000

0.864410

10.4186

−

2.82843i

1.22246i

235.14

1.41421i

−2.00000

2.26857

−2.06591

−

2.82843i

3.20824i

235.15

1.41421i

−2.00000

6.64771

−9.84865

−

2.82843i

9.40128i

235.16

1.41421i

−2.00000

8.36148

7.49600

−

2.82843i

11.8249i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
59.b	odd	2	1	inner
177.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1062.3.d.e	✓	16
3.b	odd	2	1	inner	1062.3.d.e	✓	16
59.b	odd	2	1	inner	1062.3.d.e	✓	16
177.d	even	2	1	inner	1062.3.d.e	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1062.3.d.e	✓	16	1.a	even	1	1	trivial
1062.3.d.e	✓	16	3.b	odd	2	1	inner
1062.3.d.e	✓	16	59.b	odd	2	1	inner
1062.3.d.e	✓	16	177.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 120T_{5}^{6} + 3766T_{5}^{4} - 18648T_{5}^{2} + 11881 $ T5^8 - 120*T5^6 + 3766*T5^4 - 18648*T5^2 + 11881 acting on $S_{3}^{\mathrm{new}}(1062, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 2)^{8} $ (T^2 + 2)^8
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 120 T^{6} + \cdots + 11881)^{2} $ (T^8 - 120T^6 + 3766T^4 - 18648*T^2 + 11881)^2
$7$	$ (T^{4} - 6 T^{3} + \cdots + 1589)^{4} $ (T^4 - 6T^3 - 115T^2 + 566*T + 1589)^4
$11$	$ (T^{8} + 492 T^{6} + \cdots + 13950225)^{2} $ (T^8 + 492T^6 + 61798T^4 + 1772748*T^2 + 13950225)^2
$13$	$ (T^{8} + 408 T^{6} + \cdots + 893025)^{2} $ (T^8 + 408T^6 + 16942T^4 + 227016*T^2 + 893025)^2
$17$	$ (T^{8} - 942 T^{6} + \cdots + 20043529)^{2} $ (T^8 - 942T^6 + 165151T^4 - 4224594*T^2 + 20043529)^2
$19$	$ (T^{4} + 50 T^{3} + \cdots - 99)^{4} $ (T^4 + 50T^3 + 529T^2 - 1314*T - 99)^4
$23$	$ (T^{8} + 3670 T^{6} + \cdots + 613441400625)^{2} $ (T^8 + 3670T^6 + 4944835T^4 + 2887217982*T^2 + 613441400625)^2
$29$	$ (T^{8} - 2834 T^{6} + \cdots + 235776025)^{2} $ (T^8 - 2834T^6 + 1805207T^4 - 46420526*T^2 + 235776025)^2
$31$	$ (T^{8} + 4450 T^{6} + \cdots + 2267569161)^{2} $ (T^8 + 4450T^6 + 5618119T^4 + 1862610534*T^2 + 2267569161)^2
$37$	$ (T^{8} + 3026 T^{6} + \cdots + 22994992881)^{2} $ (T^8 + 3026T^6 + 2943559T^4 + 954903150*T^2 + 22994992881)^2
$41$	$ (T^{8} - 7778 T^{6} + \cdots + 486582978025)^{2} $ (T^8 - 7778T^6 + 18994199T^4 - 15127646894*T^2 + 486582978025)^2
$43$	$ (T^{8} + 1872 T^{6} + \cdots + 4564758969)^{2} $ (T^8 + 1872T^6 + 1058326T^4 + 178178544*T^2 + 4564758969)^2
$47$	$ (T^{8} + 3350 T^{6} + \cdots + 122954721201)^{2} $ (T^8 + 3350T^6 + 3120427T^4 + 1078591950*T^2 + 122954721201)^2
$53$	$ (T^{8} + \cdots + 12\!\cdots\!69)^{2} $ (T^8 - 24808T^6 + 223783686T^4 - 870881381416*T^2 + 1233861377607769)^2
$59$	$ T^{16} + \cdots + 21\!\cdots\!41 $ T^16 - 13168T^14 + 111796988T^12 - 610715161488T^10 + 2516676686982854T^8 - 7400256079923393168T^6 + 16415200670885023585148T^4 - 23428471603378166815889008*T^2 + 21559177407076402401757871041
$61$	$ (T^{8} + 17358 T^{6} + \cdots + 94171038129)^{2} $ (T^8 + 17358T^6 + 72830839T^4 + 59367017106*T^2 + 94171038129)^2
$67$	$ (T^{8} + \cdots + 29356576985241)^{2} $ (T^8 + 22080T^6 + 137550646T^4 + 208972908288*T^2 + 29356576985241)^2
$71$	$ (T^{8} + \cdots + 12\!\cdots\!29)^{2} $ (T^8 - 29312T^6 + 275942798T^4 - 1017426486896*T^2 + 1287168241774129)^2
$73$	$ (T^{8} + \cdots + 73563848301969)^{2} $ (T^8 + 37006T^6 + 414961687T^4 + 1349976468402*T^2 + 73563848301969)^2
$79$	$ (T^{4} + 80 T^{3} + \cdots + 14624821)^{4} $ (T^4 + 80T^3 - 11578T^2 - 439720*T + 14624821)^4
$83$	$ (T^{8} + \cdots + 226971067982481)^{2} $ (T^8 + 26414T^6 + 175741027T^4 + 369096038694*T^2 + 226971067982481)^2
$89$	$ (T^{8} + \cdots + 12424757865129)^{2} $ (T^8 + 23418T^6 + 165763915T^4 + 370014530058*T^2 + 12424757865129)^2
$97$	$ (T^{8} + \cdots + 1921619750625)^{2} $ (T^8 + 33336T^6 + 71611822T^4 + 41430343464*T^2 + 1921619750625)^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:	\( N \)	\(=\)	\( 1062 = 2 \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1062.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{11} q^{2} - 2 q^{4} + \beta_1 q^{5} + ( - \beta_{5} + 1) q^{7} + 2 \beta_{11} q^{8}+O(q^{10}) \) q - b11 * q^2 - 2 * q^4 + b1 * q^5 + (-b5 + 1) * q^7 + 2b11 q^8	\( q - \beta_{11} q^{2} - 2 q^{4} + \beta_1 q^{5} + ( - \beta_{5} + 1) q^{7} + 2 \beta_{11} q^{8} + \beta_{10} q^{10} + \beta_{12} q^{11} + \beta_{15} q^{13} + ( - \beta_{13} - \beta_{11}) q^{14} + 4 q^{16} + ( - \beta_{6} + \beta_{4} - 2 \beta_1) q^{17} + (\beta_{2} - 13) q^{19} - 2 \beta_1 q^{20} + (\beta_{5} + \beta_{2} + 1) q^{22} + (\beta_{14} + \beta_{12} - 3 \beta_{11}) q^{23} + ( - 2 \beta_{2} + 6) q^{25} + (\beta_{6} - \beta_{3} - \beta_1) q^{26} + (2 \beta_{5} - 2) q^{28} + (\beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_1) q^{29} + ( - 2 \beta_{15} + 2 \beta_{10} + \cdots + \beta_{8}) q^{31}+ \cdots + ( - 2 \beta_{14} + \cdots - 17 \beta_{11}) q^{98}+O(q^{100}) \) q - b11 * q^2 - 2 * q^4 + b1 * q^5 + (-b5 + 1) * q^7 + 2b11 q^8 + b10 * q^10 + b12 * q^11 + b15 * q^13 + (-b13 - b11) * q^14 + 4 * q^16 + (-b6 + b4 - 2b1) q^17 + (b2 - 13) * q^19 - 2b1 q^20 + (b5 + b2 + 1) * q^22 + (b14 + b12 - 3b11) q^23 + (-2b2 + 6) q^25 + (b6 - b3 - b1) * q^26 + (2b5 - 2) q^28 + (b6 + b4 + 2b3 + b1) q^29 + (-2b15 + 2b10 - 3b9 + b8) q^31 - 4b11 q^32 + (b15 - b10 + b9 - 2b8) q^34 + (2b6 - 4b4 + 3b3 + 3b1) * q^35 + (2b15 - 4b10 + b8) * q^37 + (-b13 - 2b12 + 14b11) * q^38 - 2b10 q^40 + (b6 + 3b4 + 2b3 + 3b1) q^41 + (b15 - 3b10) q^43 - 2b12 q^44 + (b7 + b5 + b2 - 5) * q^46 + (b14 - 3b11) q^47 + (b7 - b5 + 17) * q^49 + (2b13 + 4b12 - 8b11) q^50 - 2b15 q^52 + (-4b6 - 6b4 + 4b3 + 5b1) * q^53 + (-b15 + 3b10 - 6b9 + 6b8) q^55 + (2b13 + 2b11) * q^56 + (b15 - 3b9 - 2b8) * q^58 + (-b14 - b12 + 3b11 + 2b6 - 2b4 + 5b3 + 3b1) q^59 + (-3b15 - 4b10 - 6b9 + b8) q^61 + (-5b6 + b4 - b3 - 6b1) * q^62 - 8 * q^64 + (b12 - 15b11) q^65 + (3b15 - 7b10 + 10b8) q^67 + (2b6 - 2b4 + 4b1) q^68 + (b15 - 4b10 - 5b9 + 8b8) q^70 + (9b6 - 4b4 - b3 + 4b1) q^71 + (7b15 - 4b10 - 6b9 + 7b8) * q^73 + (2b6 + b4 - 2b3 + 5b1) q^74 + (-2b2 + 26) q^76 + (b14 + 3b13 + 6b12 - 30b11) q^77 + (2b7 - 3b5 + b2 - 22) * q^79 + 4b1 q^80 + (b15 + 4b10 - 3b9 - 6b8) q^82 + (b14 - 6b12 + 27b11) * q^83 + (b7 + 7b5 + 6b2 - 46) * q^85 + (b6 - b3 + 5b1) q^86 + (-2b5 - 2b2 - 2) * q^88 + (b14 + 3b13 - 4b12 + 18b11) q^89 + (10b15 - 4b10 - 3b9 + b8) q^91 + (-2b14 - 2b12 + 6b11) q^92 + (b7 - 6) * q^94 + (-5b6 + 2b4 - 2b3 - 29b1) * q^95 + (-3b15 + 2b10 - 6b9 - 8b8) * q^97 + (-2b14 - b13 - 17b11) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 24 q^{7}+O(q^{10}) \) 16 * q - 32 * q^4 + 24 * q^7	\( 16 q - 32 q^{4} + 24 q^{7} + 64 q^{16} - 200 q^{19} + 16 q^{22} + 80 q^{25} - 48 q^{28} - 80 q^{46} + 280 q^{49} - 128 q^{64} + 400 q^{76} - 320 q^{79} - 744 q^{85} - 32 q^{88} - 96 q^{94}+O(q^{100}) \) 16 * q - 32 * q^4 + 24 * q^7 + 64 * q^16 - 200 * q^19 + 16 * q^22 + 80 * q^25 - 48 * q^28 - 80 * q^46 + 280 * q^49 - 128 * q^64 + 400 * q^76 - 320 * q^79 - 744 * q^85 - 32 * q^88 - 96 * q^94

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	1062.3.d.e

Level	$1062$
Weight	$3$
Character orbit	1062.d
Analytic conductor	$28.937$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(28.9374040751\)
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:	\( x^{16} + 124 x^{14} + 5790 x^{12} + 126836 x^{10} + 1312371 x^{8} + 5713540 x^{6} + 8995714 x^{4} + \cdots + 38025 \) x^16 + 124x^14 + 5790x^12 + 126836x^10 + 1312371x^8 + 5713540x^6 + 8995714x^4 + 4218792*x^2 + 38025
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{10} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 1650212 \nu^{14} - 201595493 \nu^{12} - 9236693019 \nu^{10} - 197138316010 \nu^{8} + \cdots + 927039705477 ) / 1145120641212 \) (-1650212v^14 - 201595493v^12 - 9236693019v^10 - 197138316010v^8 - 1946709533187v^6 - 7440073612520v^4 - 6901460510789*v^2 + 927039705477) / 1145120641212
\(\beta_{2}\)	\(=\)	\( ( 1650212 \nu^{14} + 201595493 \nu^{12} + 9236693019 \nu^{10} + 197138316010 \nu^{8} + \cdots + 17394890553915 ) / 1145120641212 \) (1650212v^14 + 201595493v^12 + 9236693019v^10 + 197138316010v^8 + 1946709533187v^6 + 7440073612520v^4 + 8046581152001*v^2 + 17394890553915) / 1145120641212
\(\beta_{3}\)	\(=\)	\( ( 11145433420 \nu^{14} + 1418856329068 \nu^{12} + 69018841423905 \nu^{10} + \cdots + 64\!\cdots\!94 ) / 38\!\cdots\!94 \) (11145433420v^14 + 1418856329068v^12 + 69018841423905v^10 + 1615561396846958v^8 + 18721754736125583v^6 + 99355361358326548v^4 + 197288435690739163*v^2 + 64588607805765294) / 3806953571709294
\(\beta_{4}\)	\(=\)	\( ( 21157248870 \nu^{14} + 2627019039479 \nu^{12} + 122880980236227 \nu^{10} + \cdots + 24\!\cdots\!06 ) / 38\!\cdots\!94 \) (21157248870v^14 + 2627019039479v^12 + 122880980236227v^10 + 2698199508425123v^8 + 27988626430694293v^6 + 120933074225145927v^4 + 165751818025166473*v^2 + 24244376494878006) / 3806953571709294
\(\beta_{5}\)	\(=\)	\( ( - 564341077 \nu^{14} - 69595437347 \nu^{12} - 3215003855630 \nu^{10} + \cdots - 683334264045690 ) / 69852359113932 \) (-564341077v^14 - 69595437347v^12 - 3215003855630v^10 - 68894709572105v^8 - 678206195684988v^6 - 2583570737553927v^4 - 2789707505696903*v^2 - 683334264045690) / 69852359113932
\(\beta_{6}\)	\(=\)	\( ( 69812658697 \nu^{14} + 8586058539833 \nu^{12} + 395397371559228 \nu^{10} + \cdots - 70\!\cdots\!26 ) / 76\!\cdots\!88 \) (69812658697v^14 + 8586058539833v^12 + 395397371559228v^10 + 8439545747844915v^8 + 82431248553261938v^6 + 303513367341094429v^4 + 236561295032891925*v^2 - 70864932521859726) / 7613907143418588
\(\beta_{7}\)	\(=\)	\( ( 4283651303 \nu^{14} + 525431837271 \nu^{12} + 24122641534868 \nu^{10} + \cdots + 25\!\cdots\!60 ) / 69852359113932 \) (4283651303v^14 + 525431837271v^12 + 24122641534868v^10 + 513482019447355v^8 + 5021799368346126v^6 + 19054946267751277v^4 + 20540039053314147*v^2 + 2501250463173960) / 69852359113932
\(\beta_{8}\)	\(=\)	\( ( 1808590770098 \nu^{15} + 237816012825707 \nu^{13} + \cdots + 56\!\cdots\!46 \nu ) / 14\!\cdots\!60 \) (1808590770098v^15 + 237816012825707v^13 + 12106750789166970v^11 + 302934613820605753v^9 + 3899131374504764748v^7 + 24730680597289014785v^5 + 67556031873373673162v^3 + 56534837184398702346v) / 1484711892966624660
\(\beta_{9}\)	\(=\)	\( ( 495047358121 \nu^{15} + 65159470330918 \nu^{13} + \cdots + 17\!\cdots\!93 \nu ) / 29\!\cdots\!32 \) (495047358121v^15 + 65159470330918v^13 + 3330910246680114v^11 + 84160120525955831v^9 + 1103166151979284464v^7 + 7193748249903108625v^5 + 20187539778804202129v^3 + 17193885911372432793v) / 296942378593324932
\(\beta_{10}\)	\(=\)	\( ( - 1118363773 \nu^{15} - 138355316512 \nu^{13} - 6436015124535 \nu^{11} + \cdots - 29\!\cdots\!81 \nu ) / 223298525036340 \) (-1118363773v^15 - 138355316512v^13 - 6436015124535v^11 - 140047632373523v^9 - 1429266211513833v^7 - 6010207792614955v^5 - 8609666295427522v^3 - 2925762288945681v) / 223298525036340
\(\beta_{11}\)	\(=\)	\( ( 1118363773 \nu^{15} + 138355316512 \nu^{13} + 6436015124535 \nu^{11} + \cdots + 33\!\cdots\!61 \nu ) / 223298525036340 \) (1118363773v^15 + 138355316512v^13 + 6436015124535v^11 + 140047632373523v^9 + 1429266211513833v^7 + 6010207792614955v^5 + 8609666295427522v^3 + 3372359339018361v) / 223298525036340
\(\beta_{12}\)	\(=\)	\( ( 97813935158 \nu^{15} + 12096372313677 \nu^{13} + 562247593028830 \nu^{11} + \cdots + 23\!\cdots\!41 \nu ) / 45\!\cdots\!80 \) (97813935158v^15 + 12096372313677v^13 + 562247593028830v^11 + 12212423594723358v^9 + 124082896577728198v^7 + 514929332898055170v^5 + 702844905967951682v^3 + 231037576985884641v) / 4540403342405580
\(\beta_{13}\)	\(=\)	\( ( 97002031952 \nu^{15} + 12011715328883 \nu^{13} + 559638947982435 \nu^{11} + \cdots + 38\!\cdots\!29 \nu ) / 27\!\cdots\!48 \) (97002031952v^15 + 12011715328883v^13 + 559638947982435v^11 + 12213397471855870v^9 + 125462066614774491v^7 + 537577426516587932v^5 + 820774399150743245v^3 + 385969120058095629v) / 2724242005443348
\(\beta_{14}\)	\(=\)	\( ( 36616710101 \nu^{15} + 4544715626820 \nu^{13} + 212563666247359 \nu^{11} + \cdots + 22\!\cdots\!56 \nu ) / 908080668481116 \) (36616710101v^15 + 4544715626820v^13 + 212563666247359v^11 + 4671927111150834v^9 + 48721052877899443v^7 + 217236949897126650v^5 + 373382851662416570v^3 + 225277709114517756v) / 908080668481116
\(\beta_{15}\)	\(=\)	\( ( - 22716171357238 \nu^{15} + \cdots - 10\!\cdots\!31 \nu ) / 49\!\cdots\!20 \) (-22716171357238v^15 - 2820348884798137v^13 - 131959995236364920v^11 - 2900981651308329328v^9 - 30225377774938880728v^7 - 133654400664780522600v^5 - 216906565992784007652v^3 - 100334382617985933231v) / 494903964322208220

\(\nu\)	\(=\)	\( ( \beta_{11} + \beta_{10} ) / 2 \) (b11 + b10) / 2
\(\nu^{2}\)	\(=\)	\( \beta_{2} + \beta _1 - 16 \) b2 + b1 - 16
\(\nu^{3}\)	\(=\)	\( ( 3\beta_{15} + 3\beta_{13} + 6\beta_{12} - 50\beta_{11} - 29\beta_{10} + 7\beta_{9} - 4\beta_{8} ) / 2 \) (3b15 + 3b13 + 6b12 - 50b11 - 29b10 + 7b9 - 4*b8) / 2
\(\nu^{4}\)	\(=\)	\( -\beta_{7} - 10\beta_{6} - 13\beta_{5} + 4\beta_{4} - 4\beta_{3} - 38\beta_{2} - 64\beta _1 + 487 \) -b7 - 10b6 - 13b5 + 4b4 - 4b3 - 38b2 - 64b1 + 487
\(\nu^{5}\)	\(=\)	\( ( - 103 \beta_{15} - 10 \beta_{14} - 115 \beta_{13} - 360 \beta_{12} + 2459 \beta_{11} + \cdots + 298 \beta_{8} ) / 2 \) (-103b15 - 10b14 - 115b13 - 360b12 + 2459b11 + 912b10 - 351b9 + 298b8) / 2
\(\nu^{6}\)	\(=\)	\( 77\beta_{7} + 631\beta_{6} + 818\beta_{5} - 427\beta_{4} + 352\beta_{3} + 1463\beta_{2} + 3237\beta _1 - 16979 \) 77b7 + 631b6 + 818b5 - 427b4 + 352b3 + 1463b2 + 3237*b1 - 16979
\(\nu^{7}\)	\(=\)	\( ( 3542 \beta_{15} + 1008 \beta_{14} + 3724 \beta_{13} + 17990 \beta_{12} - 112113 \beta_{11} + \cdots - 15868 \beta_{8} ) / 2 \) (3542b15 + 1008b14 + 3724b13 + 17990b12 - 112113b11 - 31209b10 + 15776b9 - 15868b8) / 2
\(\nu^{8}\)	\(=\)	\( - 4219 \beta_{7} - 32840 \beta_{6} - 41305 \beta_{5} + 27788 \beta_{4} - 21220 \beta_{3} + \cdots + 636448 \) -4219b7 - 32840b6 - 41305b5 + 27788b4 - 21220b3 - 58554b2 - 151420*b1 + 636448
\(\nu^{9}\)	\(=\)	\( ( - 129101 \beta_{15} - 64014 \beta_{14} - 112437 \beta_{13} - 844044 \beta_{12} + \cdots + 763678 \beta_{8} ) / 2 \) (-129101b15 - 64014b14 - 112437b13 - 844044b12 + 4950178b11 + 1142165b10 - 695777b9 + 763678b8) / 2
\(\nu^{10}\)	\(=\)	\( 206923 \beta_{7} + 1586863 \beta_{6} + 1946836 \beta_{5} - 1504171 \beta_{4} + 1108086 \beta_{3} + \cdots - 25056201 \) 206923b7 + 1586863b6 + 1946836b5 - 1504171b4 + 1108086b3 + 2415492b2 + 6853510*b1 - 25056201
\(\nu^{11}\)	\(=\)	\( ( 4950725 \beta_{15} + 3422188 \beta_{14} + 3247871 \beta_{13} + 38427092 \beta_{12} + \cdots - 35280380 \beta_{8} ) / 2 \) (4950725b15 + 3422188b14 + 3247871b13 + 38427092b12 - 216156313b11 - 43996872b10 + 30579111b9 - 35280380b8) / 2
\(\nu^{12}\)	\(=\)	\( - 9675642 \beta_{7} - 73737546 \beta_{6} - 89069520 \beta_{5} + 74509290 \beta_{4} - 53846890 \beta_{3} + \cdots + 1020896597 \) -9675642b7 - 73737546b6 - 89069520b5 + 74509290b4 - 53846890b3 - 101760582b2 - 305371054*b1 + 1020896597
\(\nu^{13}\)	\(=\)	\( ( - 197400314 \beta_{15} - 168369864 \beta_{14} - 90117794 \beta_{13} - 1721499052 \beta_{12} + \cdots + 1597588896 \beta_{8} ) / 2 \) (-197400314b15 - 168369864b14 - 90117794b13 - 1721499052b12 + 9411404625b11 + 1760826731b10 - 1343309962b9 + 1597588896b8) / 2
\(\nu^{14}\)	\(=\)	\( 441489690 \beta_{7} + 3349775582 \beta_{6} + 4012089360 \beta_{5} - 3516982136 \beta_{4} + \cdots - 42599852104 \) 441489690b7 + 3349775582b6 + 4012089360b5 - 3516982136b4 + 2513636866b3 + 4347525559b2 + 13498243491*b1 - 42599852104
\(\nu^{15}\)	\(=\)	\( ( 8100667273 \beta_{15} + 7916943652 \beta_{14} + 2373308457 \beta_{13} + 76421540518 \beta_{12} + \cdots - 71531430848 \beta_{8} ) / 2 \) (8100667273b15 + 7916943652b14 + 2373308457b13 + 76421540518b12 - 409800018932b11 - 72464563321b10 + 59005476301b9 - 71531430848b8) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 235.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + 2)^{8} \) (T^2 + 2)^8
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 120 T^{6} + \cdots + 11881)^{2} \) (T^8 - 120T^6 + 3766T^4 - 18648*T^2 + 11881)^2
$7$	\( (T^{4} - 6 T^{3} + \cdots + 1589)^{4} \) (T^4 - 6T^3 - 115T^2 + 566*T + 1589)^4
$11$	\( (T^{8} + 492 T^{6} + \cdots + 13950225)^{2} \) (T^8 + 492T^6 + 61798T^4 + 1772748*T^2 + 13950225)^2
$13$	\( (T^{8} + 408 T^{6} + \cdots + 893025)^{2} \) (T^8 + 408T^6 + 16942T^4 + 227016*T^2 + 893025)^2
$17$	\( (T^{8} - 942 T^{6} + \cdots + 20043529)^{2} \) (T^8 - 942T^6 + 165151T^4 - 4224594*T^2 + 20043529)^2
$19$	\( (T^{4} + 50 T^{3} + \cdots - 99)^{4} \) (T^4 + 50T^3 + 529T^2 - 1314*T - 99)^4
$23$	\( (T^{8} + 3670 T^{6} + \cdots + 613441400625)^{2} \) (T^8 + 3670T^6 + 4944835T^4 + 2887217982*T^2 + 613441400625)^2
$29$	\( (T^{8} - 2834 T^{6} + \cdots + 235776025)^{2} \) (T^8 - 2834T^6 + 1805207T^4 - 46420526*T^2 + 235776025)^2
$31$	\( (T^{8} + 4450 T^{6} + \cdots + 2267569161)^{2} \) (T^8 + 4450T^6 + 5618119T^4 + 1862610534*T^2 + 2267569161)^2
$37$	\( (T^{8} + 3026 T^{6} + \cdots + 22994992881)^{2} \) (T^8 + 3026T^6 + 2943559T^4 + 954903150*T^2 + 22994992881)^2
$41$	\( (T^{8} - 7778 T^{6} + \cdots + 486582978025)^{2} \) (T^8 - 7778T^6 + 18994199T^4 - 15127646894*T^2 + 486582978025)^2
$43$	\( (T^{8} + 1872 T^{6} + \cdots + 4564758969)^{2} \) (T^8 + 1872T^6 + 1058326T^4 + 178178544*T^2 + 4564758969)^2
$47$	\( (T^{8} + 3350 T^{6} + \cdots + 122954721201)^{2} \) (T^8 + 3350T^6 + 3120427T^4 + 1078591950*T^2 + 122954721201)^2
$53$	\( (T^{8} + \cdots + 12\!\cdots\!69)^{2} \) (T^8 - 24808T^6 + 223783686T^4 - 870881381416*T^2 + 1233861377607769)^2
$59$	\( T^{16} + \cdots + 21\!\cdots\!41 \) T^16 - 13168T^14 + 111796988T^12 - 610715161488T^10 + 2516676686982854T^8 - 7400256079923393168T^6 + 16415200670885023585148T^4 - 23428471603378166815889008*T^2 + 21559177407076402401757871041
$61$	\( (T^{8} + 17358 T^{6} + \cdots + 94171038129)^{2} \) (T^8 + 17358T^6 + 72830839T^4 + 59367017106*T^2 + 94171038129)^2
$67$	\( (T^{8} + \cdots + 29356576985241)^{2} \) (T^8 + 22080T^6 + 137550646T^4 + 208972908288*T^2 + 29356576985241)^2
$71$	\( (T^{8} + \cdots + 12\!\cdots\!29)^{2} \) (T^8 - 29312T^6 + 275942798T^4 - 1017426486896*T^2 + 1287168241774129)^2
$73$	\( (T^{8} + \cdots + 73563848301969)^{2} \) (T^8 + 37006T^6 + 414961687T^4 + 1349976468402*T^2 + 73563848301969)^2
$79$	\( (T^{4} + 80 T^{3} + \cdots + 14624821)^{4} \) (T^4 + 80T^3 - 11578T^2 - 439720*T + 14624821)^4
$83$	\( (T^{8} + \cdots + 226971067982481)^{2} \) (T^8 + 26414T^6 + 175741027T^4 + 369096038694*T^2 + 226971067982481)^2
$89$	\( (T^{8} + \cdots + 12424757865129)^{2} \) (T^8 + 23418T^6 + 165763915T^4 + 370014530058*T^2 + 12424757865129)^2
$97$	\( (T^{8} + \cdots + 1921619750625)^{2} \) (T^8 + 33336T^6 + 71611822T^4 + 41430343464*T^2 + 1921619750625)^2
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\(n\)	\(119\)	\(415\)
\(\chi(n)\)	\(1\)	\(-1\)