LMFDB - Newform orbit 580.2.f.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [580,2,Mod(289,580)] 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("580.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 580 = 2^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	580.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$4.63132331723$
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} + 27x^{14} + 281x^{12} + 1422x^{10} + 3632x^{8} + 4436x^{6} + 2167x^{4} + 165x^{2} + 1 $ x^16 + 27x^14 + 281x^12 + 1422x^10 + 3632x^8 + 4436x^6 + 2167x^4 + 165*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{15} $
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_1 q^{3} + \beta_{9} q^{5} + \beta_{10} q^{7} + (\beta_{5} + 2) q^{9} - \beta_{8} q^{11} + \beta_{4} q^{13} + \beta_{13} q^{15} - \beta_{2} q^{17} + ( - \beta_{11} + \beta_{8}) q^{19} + ( - \beta_{14} - \beta_{13} - \beta_{11}) q^{21}+ \cdots + (\beta_{15} + \beta_{14} + \cdots - \beta_{7}) q^{99}+O(q^{100}) $ q - b1 * q^3 + b9 * q^5 + b10 * q^7 + (b5 + 2) * q^9 - b8 * q^11 + b4 * q^13 + b13 * q^15 - b2 * q^17 + (-b11 + b8) * q^19 + (-b14 - b13 - b11) * q^21 + (-b10 + b9 + b6 - b4) * q^23 + (-b15 + b12 + 1) * q^25 + (b14 - b13 - b3 - b2 - b1) * q^27 + (-b15 + b12 + b9 + b7) * q^29 + (-b15 + b14 + b13 + b11 + b8 + b7) * q^31 + (b15 + 2b9 + b7 + 2b6 + b4) * q^33 + (-b10 + b5 + b4 + 1) * q^35 + (b3 + b1) * q^37 + (b15 - b14 - b13 - 2b8 - b7) q^39 + (-b15 + b14 + b13 - b11 + 2b8 + b7) q^41 + b1 * q^43 + (b12 - 2b10 + 2b9 + b7 + b6 - b5 - 1) * q^45 + (b3 + b2) * q^47 + (b9 - b6 - b5) * q^49 + (b15 - 2b12 - b9 - b7 - b6 + b5 + 1) q^51 + (-b15 - 2b10 - b7 + b4) q^53 + (b15 - 2b14 - b13 - b8 - b7 + b1) q^55 + (-b15 + 2b10 - b9 - b7 - b6 - b4) q^57 + (-b15 + 2b12 + b9 + b7 + b6 - b5 - 1) q^59 + (-b14 - b13 + b11 - 2b8) q^61 + (-b15 + 3b10 - 4b9 - b7 - 4b6) q^63 + (b12 + 2b10 + b9 + b7 + b6 + b4) q^65 + (b15 - b10 + b9 + b7 + b6 - b4) * q^67 + (-b15 + 3b14 + 3b13 + b11 + 2b8 + b7) q^69 + (b15 - 2b12 - b9 - b7 - b6 - b5 - 1) q^71 + (-b14 + b13 + b1) * q^73 + (-b14 - b11 + 2b8 + b2 - b1) q^75 + (b14 - b13 - b3 - 2b2) q^77 + (2b11 + b8) q^79 + (b15 - 2b12 - 3b9 - b7 + b6 + 3b5 + 4) q^81 + (b15 + b10 + 2b9 + b7 + 2b6) * q^83 + (b15 - b11 - b7 + b3 + b2) * q^85 + (-b14 + b13 + 2b10 - b9 - b6 - b4 + b2) q^87 + (b15 - 2b14 - 2b13 - 2b11 - 2b8 - b7) * q^89 + (-2b9 + 2b6) * q^91 + (-2b10 - 3b4) * q^93 + (2b14 + b8 - b3) q^95 + (b14 - b13 - b3 + b2 - 2b1) q^97 + (b15 + b14 + b13 + 2b11 - 3b8 - b7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} + 28 q^{9} + 10 q^{25} - 8 q^{29} + 12 q^{35} - 20 q^{45} + 24 q^{51} - 24 q^{59} - 6 q^{65} + 72 q^{81} + 8 q^{91}+O(q^{100}) $ 16 * q - 2 * q^5 + 28 * q^9 + 10 * q^25 - 8 * q^29 + 12 * q^35 - 20 * q^45 + 24 * q^51 - 24 * q^59 - 6 * q^65 + 72 * q^81 + 8 * q^91

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 27x^{14} + 281x^{12} + 1422x^{10} + 3632x^{8} + 4436x^{6} + 2167x^{4} + 165x^{2} + 1 $ x^16 + 27*x^14 + 281*x^12 + 1422*x^10 + 3632*x^8 + 4436*x^6 + 2167*x^4 + 165*x^2 + 1 :

$\beta_{1}$	$=$	$ ( - 9443 \nu^{14} - 296198 \nu^{12} - 3739017 \nu^{10} - 24091149 \nu^{8} - 82107943 \nu^{6} + \cdots - 4761763 ) / 1257924 $ (-9443v^14 - 296198v^12 - 3739017v^10 - 24091149v^8 - 82107943v^6 - 136810433v^4 - 86508832*v^2 - 4761763) / 1257924
$\beta_{2}$	$=$	$ ( - 7435 \nu^{14} - 183636 \nu^{12} - 1657949 \nu^{10} - 6510065 \nu^{8} - 9409491 \nu^{6} + \cdots - 709185 ) / 209654 $ (-7435v^14 - 183636v^12 - 1657949v^10 - 6510065v^8 - 9409491v^6 + 1079539v^4 + 6319564*v^2 - 709185) / 209654
$\beta_{3}$	$=$	$ ( 27681 \nu^{14} + 736166 \nu^{12} + 7500059 \nu^{10} + 36758839 \nu^{8} + 88923701 \nu^{6} + \cdots - 279163 ) / 419308 $ (27681v^14 + 736166v^12 + 7500059v^10 + 36758839v^8 + 88923701v^6 + 97173219v^4 + 35363496*v^2 - 279163) / 419308
$\beta_{4}$	$=$	$ ( 36061 \nu^{15} + 949910 \nu^{13} + 9511983 \nu^{11} + 45198331 \nu^{9} + 103719977 \nu^{7} + \cdots + 7221193 \nu ) / 419308 $ (36061v^15 + 949910v^13 + 9511983v^11 + 45198331v^9 + 103719977v^7 + 106050879v^5 + 43039712v^3 + 7221193v) / 419308
$\beta_{5}$	$=$	$ ( 10869 \nu^{14} + 287387 \nu^{12} + 2904866 \nu^{10} + 14092173 \nu^{8} + 33778082 \nu^{6} + \cdots + 745589 ) / 104827 $ (10869v^14 + 287387v^12 + 2904866v^10 + 14092173v^8 + 33778082v^6 + 37492055v^4 + 15987961*v^2 + 745589) / 104827
$\beta_{6}$	$=$	$ ( 442855 \nu^{15} - 100356 \nu^{14} + 11989666 \nu^{13} - 2602356 \nu^{12} + 125241357 \nu^{11} + \cdots + 3109176 ) / 2515848 $ (442855v^15 - 100356v^14 + 11989666v^13 - 2602356v^12 + 125241357v^11 - 25457712v^10 + 636963225v^9 - 116354484v^8 + 1637663819v^7 - 247055832v^6 + 2013804781v^5 - 206991756v^4 + 979478360v^3 - 40553388v^2 + 61735451*v + 3109176) / 2515848
$\beta_{7}$	$=$	$ ( - 228275 \nu^{15} - 6098186 \nu^{13} - 62493369 \nu^{11} - 308997693 \nu^{9} + \cdots - 15821143 \nu ) / 1257924 $ (-228275v^15 - 6098186v^13 - 62493369v^11 - 308997693v^9 - 761373967v^7 - 879997745v^5 - 395046688v^3 - 15821143v) / 1257924
$\beta_{8}$	$=$	$ ( - 41081 \nu^{15} - 1126488 \nu^{13} - 11989151 \nu^{11} - 62734637 \nu^{9} - 168688829 \nu^{7} + \cdots - 12006461 \nu ) / 209654 $ (-41081v^15 - 1126488v^13 - 11989151v^11 - 62734637v^9 - 168688829v^7 - 222672257v^5 - 121015012v^3 - 12006461v) / 209654
$\beta_{9}$	$=$	$ ( 442855 \nu^{15} + 100356 \nu^{14} + 11989666 \nu^{13} + 2602356 \nu^{12} + 125241357 \nu^{11} + \cdots - 3109176 ) / 2515848 $ (442855v^15 + 100356v^14 + 11989666v^13 + 2602356v^12 + 125241357v^11 + 25457712v^10 + 636963225v^9 + 116354484v^8 + 1637663819v^7 + 247055832v^6 + 2013804781v^5 + 206991756v^4 + 979478360v^3 + 40553388v^2 + 61735451*v - 3109176) / 2515848
$\beta_{10}$	$=$	$ ( - 57807 \nu^{15} - 1560214 \nu^{13} - 16232103 \nu^{11} - 82127051 \nu^{9} - 209864801 \nu^{7} + \cdots - 11068957 \nu ) / 209654 $ (-57807v^15 - 1560214v^13 - 16232103v^11 - 82127051v^9 - 209864801v^7 - 257170883v^5 - 127773910v^3 - 11068957v) / 209654
$\beta_{11}$	$=$	$ ( 69357 \nu^{15} + 1858316 \nu^{13} + 19114181 \nu^{11} + 94923403 \nu^{9} + 234890639 \nu^{7} + \cdots + 4153887 \nu ) / 209654 $ (69357v^15 + 1858316v^13 + 19114181v^11 + 94923403v^9 + 234890639v^7 + 271503839v^5 + 119649158v^3 + 4153887v) / 209654
$\beta_{12}$	$=$	$ ( 916409 \nu^{15} - 76452 \nu^{14} + 24544094 \nu^{13} - 2104332 \nu^{12} + 252457563 \nu^{11} + \cdots - 7333800 ) / 2515848 $ (916409v^15 - 76452v^14 + 24544094v^13 - 2104332v^12 + 252457563v^11 - 22334208v^10 + 1254983967v^9 - 115246452v^8 + 3115533709v^7 - 300802248v^6 + 3631292171v^5 - 379573548v^4 + 1635683488v^3 - 192811260v^2 + 74693533*v - 7333800) / 2515848
$\beta_{13}$	$=$	$ ( 427154 \nu^{15} - 60039 \nu^{14} + 11522040 \nu^{13} - 1619346 \nu^{12} + 119763906 \nu^{11} + \cdots - 4372579 ) / 838616 $ (427154v^15 - 60039v^14 + 11522040v^13 - 1619346v^12 + 119763906v^11 - 16758669v^10 + 605052134v^9 - 83531377v^8 + 1541827174v^7 - 205965307v^6 + 1876784422v^5 - 232622165v^4 + 911123444v^3 - 96287128v^2 + 67452198*v - 4372579) / 838616
$\beta_{14}$	$=$	$ ( 427154 \nu^{15} + 60039 \nu^{14} + 11522040 \nu^{13} + 1619346 \nu^{12} + 119763906 \nu^{11} + \cdots + 4372579 ) / 838616 $ (427154v^15 + 60039v^14 + 11522040v^13 + 1619346v^12 + 119763906v^11 + 16758669v^10 + 605052134v^9 + 83531377v^8 + 1541827174v^7 + 205965307v^6 + 1876784422v^5 + 232622165v^4 + 911123444v^3 + 96287128v^2 + 67452198*v + 4372579) / 838616
$\beta_{15}$	$=$	$ ( 1130989 \nu^{15} + 30435574 \nu^{13} + 315205551 \nu^{11} + 1582949499 \nu^{9} + \cdots + 120607841 \nu ) / 1257924 $ (1130989v^15 + 30435574v^13 + 315205551v^11 + 1582949499v^9 + 3991823561v^7 + 4765099207v^5 + 2220115160v^3 + 120607841v) / 1257924

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

$\nu$	$=$	$ ( -\beta_{15} + \beta_{14} + \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} ) / 4 $ (-b15 + b14 + b13 + b10 + b9 + b8 + b6) / 4
$\nu^{2}$	$=$	$ ( - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 2 \beta_{9} + 2 \beta_{7} + 6 \beta_{6} + \cdots - 26 ) / 8 $ (-2b15 + 2b14 - 2b13 + 4b12 - 2b9 + 2b7 + 6b6 + 2b5 - b3 - b2 + 3*b1 - 26) / 8
$\nu^{3}$	$=$	$ ( 5 \beta_{15} - 4 \beta_{14} - 4 \beta_{13} + 2 \beta_{11} - 4 \beta_{10} - 8 \beta_{9} + \cdots - 8 \beta_{6} ) / 4 $ (5b15 - 4b14 - 4b13 + 2b11 - 4b10 - 8b9 - 6b8 + 3b7 - 8*b6) / 4
$\nu^{4}$	$=$	$ ( 18 \beta_{15} - 16 \beta_{14} + 16 \beta_{13} - 36 \beta_{12} + 12 \beta_{9} - 18 \beta_{7} - 48 \beta_{6} + \cdots + 156 ) / 8 $ (18b15 - 16b14 + 16b13 - 36b12 + 12b9 - 18b7 - 48b6 - 20b5 + 9b3 - b2 - 33b1 + 156) / 8
$\nu^{5}$	$=$	$ ( - 28 \beta_{15} + 21 \beta_{14} + 21 \beta_{13} - 30 \beta_{11} + 25 \beta_{10} + 65 \beta_{9} + \cdots + 6 \beta_{4} ) / 4 $ (-28b15 + 21b14 + 21b13 - 30b11 + 25b10 + 65b9 + 43b8 - 31b7 + 65b6 + 6b4) / 4
$\nu^{6}$	$=$	$ ( - 38 \beta_{15} + 32 \beta_{14} - 32 \beta_{13} + 76 \beta_{12} - 17 \beta_{9} + 38 \beta_{7} + \cdots - 271 ) / 2 $ (-38b15 + 32b14 - 32b13 + 76b12 - 17b9 + 38b7 + 93b6 + 45b5 - 24b3 + 12b2 + 72*b1 - 271) / 2
$\nu^{7}$	$=$	$ ( 157 \beta_{15} - 120 \beta_{14} - 120 \beta_{13} + 333 \beta_{11} - 203 \beta_{10} - 532 \beta_{9} + \cdots - 93 \beta_{4} ) / 4 $ (157b15 - 120b14 - 120b13 + 333b11 - 203b10 - 532b9 - 319b8 + 288b7 - 532b6 - 93b4) / 4
$\nu^{8}$	$=$	$ ( 1300 \beta_{15} - 1056 \beta_{14} + 1056 \beta_{13} - 2600 \beta_{12} + 328 \beta_{9} - 1300 \beta_{7} + \cdots + 8056 ) / 8 $ (1300b15 - 1056b14 + 1056b13 - 2600b12 + 328b9 - 1300b7 - 2928b6 - 1564b5 + 979b3 - 615b2 - 2403*b1 + 8056) / 8
$\nu^{9}$	$=$	$ ( - 851 \beta_{15} + 698 \beta_{14} + 698 \beta_{13} - 3320 \beta_{11} + 1792 \beta_{10} + \cdots + 1076 \beta_{4} ) / 4 $ (-851b15 + 698b14 + 698b13 - 3320b11 + 1792b10 + 4400b9 + 2406b8 - 2625b7 + 4400b6 + 1076b4) / 4
$\nu^{10}$	$=$	$ ( - 11270 \beta_{15} + 8890 \beta_{14} - 8890 \beta_{13} + 22540 \beta_{12} - 1030 \beta_{9} + \cdots - 62414 ) / 8 $ (-11270b15 + 8890b14 - 8890b13 + 22540b12 - 1030b9 + 11270b7 + 23570b6 + 13502b5 - 9503b3 + 6477b2 + 19929*b1 - 62414) / 8
$\nu^{11}$	$=$	$ ( 4268 \beta_{15} - 4004 \beta_{14} - 4004 \beta_{13} + 31465 \beta_{11} - 16141 \beta_{10} + \cdots - 11139 \beta_{4} ) / 4 $ (4268b15 - 4004b14 - 4004b13 + 31465b11 - 16141b10 - 36874b9 - 18491b8 + 23705b7 - 36874b6 - 11139b4) / 4
$\nu^{12}$	$=$	$ ( 24629 \beta_{15} - 18936 \beta_{14} + 18936 \beta_{13} - 49258 \beta_{12} - 818 \beta_{9} + \cdots + 124640 ) / 2 $ (24629b15 - 18936b14 + 18936b13 - 49258b12 - 818b9 - 24629b7 - 48440b6 - 29220b5 + 22300b3 - 15804b2 - 41556*b1 + 124640) / 2
$\nu^{13}$	$=$	$ ( - 17805 \beta_{15} + 22031 \beta_{14} + 22031 \beta_{13} - 290092 \beta_{11} + 145603 \beta_{10} + \cdots + 108944 \beta_{4} ) / 4 $ (-17805b15 + 22031b14 + 22031b13 - 290092b11 + 145603b10 + 312901b9 + 145079b8 - 212596b7 + 312901b6 + 108944b4) / 4
$\nu^{14}$	$=$	$ ( - 865104 \beta_{15} + 650402 \beta_{14} - 650402 \beta_{13} + 1730208 \beta_{12} + 109402 \beta_{9} + \cdots - 4078158 ) / 8 $ (-865104b15 + 650402b14 - 650402b13 + 1730208b12 + 109402b9 + 865104b7 + 1620806b6 + 1016386b5 - 819299b3 + 593565b2 + 1398861*b1 - 4078158) / 8
$\nu^{15}$	$=$	$ ( 34863 \beta_{15} - 111080 \beta_{14} - 111080 \beta_{13} + 2631150 \beta_{11} - 1308988 \beta_{10} + \cdots - 1030340 \beta_{4} ) / 4 $ (34863b15 - 111080b14 - 111080b13 + 2631150b11 - 1308988b10 - 2682476b9 - 1161798b8 + 1896737b7 - 2682476b6 - 1030340b4) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/580\mathbb{Z}\right)^\times$.

$n$	$117$	$291$	$321$
$\chi(n)$	$-1$	$1$	$-1$

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} $

289.1

		1.78049i
	−	1.78049i
		1.08596i
	−	1.08596i
		0.290190i
	−	0.290190i
	−	2.43830i
		2.43830i
	−	2.96708i
		2.96708i
	−	1.19824i
		1.19824i
	−	2.52488i
		2.52488i
	−	0.0814257i
		0.0814257i

−3.33421

−1.33912

−

1.79074i

3.97432i

8.11698

289.2

−3.33421

−1.33912

1.79074i

−

3.97432i

8.11698

289.3

−2.24181

2.14815

−

0.620854i

1.65209i

2.02571

289.4

−2.24181

2.14815

0.620854i

−

1.65209i

2.02571

289.5

−1.27260

−2.06615

−

0.854997i

−

2.78421i

−1.38049

289.6

−1.27260

−2.06615

0.854997i

2.78421i

−1.38049

289.7

−1.11256

0.757123

−

2.10399i

−

1.31284i

−1.76220

289.8

−1.11256

0.757123

2.10399i

1.31284i

−1.76220

289.9

1.11256

0.757123

−

2.10399i

−

1.31284i

−1.76220

289.10

1.11256

0.757123

2.10399i

1.31284i

−1.76220

289.11

1.27260

−2.06615

−

0.854997i

−

2.78421i

−1.38049

289.12

1.27260

−2.06615

0.854997i

2.78421i

−1.38049

289.13

2.24181

2.14815

−

0.620854i

1.65209i

2.02571

289.14

2.24181

2.14815

0.620854i

−

1.65209i

2.02571

289.15

3.33421

−1.33912

−

1.79074i

3.97432i

8.11698

289.16

3.33421

−1.33912

1.79074i

−

3.97432i

8.11698

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
29.b	even	2	1	inner
145.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	580.2.f.a	✓	16
3.b	odd	2	1		5220.2.b.e		16
4.b	odd	2	1		2320.2.j.f		16
5.b	even	2	1	inner	580.2.f.a	✓	16
5.c	odd	4	2		2900.2.d.g		16
15.d	odd	2	1		5220.2.b.e		16
20.d	odd	2	1		2320.2.j.f		16
29.b	even	2	1	inner	580.2.f.a	✓	16
87.d	odd	2	1		5220.2.b.e		16
116.d	odd	2	1		2320.2.j.f		16
145.d	even	2	1	inner	580.2.f.a	✓	16
145.h	odd	4	2		2900.2.d.g		16
435.b	odd	2	1		5220.2.b.e		16
580.e	odd	2	1		2320.2.j.f		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
580.2.f.a	✓	16	1.a	even	1	1	trivial
580.2.f.a	✓	16	5.b	even	2	1	inner
580.2.f.a	✓	16	29.b	even	2	1	inner
580.2.f.a	✓	16	145.d	even	2	1	inner
2320.2.j.f		16	4.b	odd	2	1
2320.2.j.f		16	20.d	odd	2	1
2320.2.j.f		16	116.d	odd	2	1
2320.2.j.f		16	580.e	odd	2	1
2900.2.d.g		16	5.c	odd	4	2
2900.2.d.g		16	145.h	odd	4	2
5220.2.b.e		16	3.b	odd	2	1
5220.2.b.e		16	15.d	odd	2	1
5220.2.b.e		16	87.d	odd	2	1
5220.2.b.e		16	435.b	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(580, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 19 T^{6} + \cdots + 112)^{2} $ (T^8 - 19T^6 + 104T^4 - 192*T^2 + 112)^2
$5$	$ (T^{8} + T^{7} - 2 T^{6} + \cdots + 625)^{2} $ (T^8 + T^7 - 2T^6 - 5T^5 + 2T^4 - 25T^3 - 50T^2 + 125T + 625)^2
$7$	$ (T^{8} + 28 T^{6} + \cdots + 576)^{2} $ (T^8 + 28T^6 + 232T^4 + 656*T^2 + 576)^2
$11$	$ (T^{8} + 57 T^{6} + \cdots + 112)^{2} $ (T^8 + 57T^6 + 1080T^4 + 6808*T^2 + 112)^2
$13$	$ (T^{8} + 43 T^{6} + \cdots + 2304)^{2} $ (T^8 + 43T^6 + 504T^4 + 2048*T^2 + 2304)^2
$17$	$ (T^{8} - 92 T^{6} + \cdots + 100800)^{2} $ (T^8 - 92T^6 + 2784T^4 - 30544*T^2 + 100800)^2
$19$	$ (T^{8} + 80 T^{6} + \cdots + 16128)^{2} $ (T^8 + 80T^6 + 1976T^4 + 14576*T^2 + 16128)^2
$23$	$ (T^{8} + 108 T^{6} + \cdots + 3136)^{2} $ (T^8 + 108T^6 + 2936T^4 + 10064*T^2 + 3136)^2
$29$	$ (T^{8} + 4 T^{7} + \cdots + 707281)^{2} $ (T^8 + 4T^7 + 4T^6 - 212T^5 - 826T^4 - 6148T^3 + 3364T^2 + 97556*T + 707281)^2
$31$	$ (T^{8} + 153 T^{6} + \cdots + 49392)^{2} $ (T^8 + 153T^6 + 6440T^4 + 44552*T^2 + 49392)^2
$37$	$ (T^{8} - 240 T^{6} + \cdots + 4845568)^{2} $ (T^8 - 240T^6 + 18816T^4 - 529616*T^2 + 4845568)^2
$41$	$ (T^{8} + 252 T^{6} + \cdots + 4480000)^{2} $ (T^8 + 252T^6 + 20384T^4 + 571904*T^2 + 4480000)^2
$43$	$ (T^{8} - 19 T^{6} + \cdots + 112)^{2} $ (T^8 - 19T^6 + 104T^4 - 192*T^2 + 112)^2
$47$	$ (T^{8} - 219 T^{6} + \cdots + 2041200)^{2} $ (T^8 - 219T^6 + 12360T^4 - 269344*T^2 + 2041200)^2
$53$	$ (T^{8} + 331 T^{6} + \cdots + 4804864)^{2} $ (T^8 + 331T^6 + 30312T^4 + 734976*T^2 + 4804864)^2
$59$	$ (T^{4} + 6 T^{3} + \cdots + 3360)^{4} $ (T^4 + 6T^3 - 120T^2 - 304*T + 3360)^4
$61$	$ (T^{8} + 220 T^{6} + \cdots + 64512)^{2} $ (T^8 + 220T^6 + 12640T^4 + 61184*T^2 + 64512)^2
$67$	$ (T^{8} + 304 T^{6} + \cdots + 1806336)^{2} $ (T^8 + 304T^6 + 28696T^4 + 848048*T^2 + 1806336)^2
$71$	$ (T^{4} - 112 T^{2} + \cdots - 576)^{4} $ (T^4 - 112T^2 - 496T - 576)^4
$73$	$ (T^{8} - 248 T^{6} + \cdots + 114688)^{2} $ (T^8 - 248T^6 + 17040T^4 - 355920*T^2 + 114688)^2
$79$	$ (T^{8} + 449 T^{6} + \cdots + 4258800)^{2} $ (T^8 + 449T^6 + 55496T^4 + 1576856*T^2 + 4258800)^2
$83$	$ (T^{8} + 252 T^{6} + \cdots + 270400)^{2} $ (T^8 + 252T^6 + 18696T^4 + 386704*T^2 + 270400)^2
$89$	$ (T^{8} + 500 T^{6} + \cdots + 83607552)^{2} $ (T^8 + 500T^6 + 78912T^4 + 4727808*T^2 + 83607552)^2
$97$	$ (T^{8} - 532 T^{6} + \cdots + 64351168)^{2} $ (T^8 - 532T^6 + 85776T^4 - 4444816*T^2 + 64351168)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	580.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} + \beta_{9} q^{5} + \beta_{10} q^{7} + (\beta_{5} + 2) q^{9} - \beta_{8} q^{11} + \beta_{4} q^{13} + \beta_{13} q^{15} - \beta_{2} q^{17} + ( - \beta_{11} + \beta_{8}) q^{19} + ( - \beta_{14} - \beta_{13} - \beta_{11}) q^{21}+ \cdots + (\beta_{15} + \beta_{14} + \cdots - \beta_{7}) q^{99}+O(q^{100}) \) q - b1 * q^3 + b9 * q^5 + b10 * q^7 + (b5 + 2) * q^9 - b8 * q^11 + b4 * q^13 + b13 * q^15 - b2 * q^17 + (-b11 + b8) * q^19 + (-b14 - b13 - b11) * q^21 + (-b10 + b9 + b6 - b4) * q^23 + (-b15 + b12 + 1) * q^25 + (b14 - b13 - b3 - b2 - b1) * q^27 + (-b15 + b12 + b9 + b7) * q^29 + (-b15 + b14 + b13 + b11 + b8 + b7) * q^31 + (b15 + 2b9 + b7 + 2b6 + b4) * q^33 + (-b10 + b5 + b4 + 1) * q^35 + (b3 + b1) * q^37 + (b15 - b14 - b13 - 2b8 - b7) q^39 + (-b15 + b14 + b13 - b11 + 2b8 + b7) q^41 + b1 * q^43 + (b12 - 2b10 + 2b9 + b7 + b6 - b5 - 1) * q^45 + (b3 + b2) * q^47 + (b9 - b6 - b5) * q^49 + (b15 - 2b12 - b9 - b7 - b6 + b5 + 1) q^51 + (-b15 - 2b10 - b7 + b4) q^53 + (b15 - 2b14 - b13 - b8 - b7 + b1) q^55 + (-b15 + 2b10 - b9 - b7 - b6 - b4) q^57 + (-b15 + 2b12 + b9 + b7 + b6 - b5 - 1) q^59 + (-b14 - b13 + b11 - 2b8) q^61 + (-b15 + 3b10 - 4b9 - b7 - 4b6) q^63 + (b12 + 2b10 + b9 + b7 + b6 + b4) q^65 + (b15 - b10 + b9 + b7 + b6 - b4) * q^67 + (-b15 + 3b14 + 3b13 + b11 + 2b8 + b7) q^69 + (b15 - 2b12 - b9 - b7 - b6 - b5 - 1) q^71 + (-b14 + b13 + b1) * q^73 + (-b14 - b11 + 2b8 + b2 - b1) q^75 + (b14 - b13 - b3 - 2b2) q^77 + (2b11 + b8) q^79 + (b15 - 2b12 - 3b9 - b7 + b6 + 3b5 + 4) q^81 + (b15 + b10 + 2b9 + b7 + 2b6) * q^83 + (b15 - b11 - b7 + b3 + b2) * q^85 + (-b14 + b13 + 2b10 - b9 - b6 - b4 + b2) q^87 + (b15 - 2b14 - 2b13 - 2b11 - 2b8 - b7) * q^89 + (-2b9 + 2b6) * q^91 + (-2b10 - 3b4) * q^93 + (2b14 + b8 - b3) q^95 + (b14 - b13 - b3 + b2 - 2b1) q^97 + (b15 + b14 + b13 + 2b11 - 3b8 - b7) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} + 28 q^{9} + 10 q^{25} - 8 q^{29} + 12 q^{35} - 20 q^{45} + 24 q^{51} - 24 q^{59} - 6 q^{65} + 72 q^{81} + 8 q^{91}+O(q^{100}) \) 16 * q - 2 * q^5 + 28 * q^9 + 10 * q^25 - 8 * q^29 + 12 * q^35 - 20 * q^45 + 24 * q^51 - 24 * q^59 - 6 * q^65 + 72 * q^81 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	580.2.f.a

Level	$580$
Weight	$2$
Character orbit	580.f
Analytic conductor	$4.631$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.63132331723\)
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:	\( x^{16} + 27x^{14} + 281x^{12} + 1422x^{10} + 3632x^{8} + 4436x^{6} + 2167x^{4} + 165x^{2} + 1 \) x^16 + 27x^14 + 281x^12 + 1422x^10 + 3632x^8 + 4436x^6 + 2167x^4 + 165*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index:	\( 2^{15} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 9443 \nu^{14} - 296198 \nu^{12} - 3739017 \nu^{10} - 24091149 \nu^{8} - 82107943 \nu^{6} + \cdots - 4761763 ) / 1257924 \) (-9443v^14 - 296198v^12 - 3739017v^10 - 24091149v^8 - 82107943v^6 - 136810433v^4 - 86508832*v^2 - 4761763) / 1257924
\(\beta_{2}\)	\(=\)	\( ( - 7435 \nu^{14} - 183636 \nu^{12} - 1657949 \nu^{10} - 6510065 \nu^{8} - 9409491 \nu^{6} + \cdots - 709185 ) / 209654 \) (-7435v^14 - 183636v^12 - 1657949v^10 - 6510065v^8 - 9409491v^6 + 1079539v^4 + 6319564*v^2 - 709185) / 209654
\(\beta_{3}\)	\(=\)	\( ( 27681 \nu^{14} + 736166 \nu^{12} + 7500059 \nu^{10} + 36758839 \nu^{8} + 88923701 \nu^{6} + \cdots - 279163 ) / 419308 \) (27681v^14 + 736166v^12 + 7500059v^10 + 36758839v^8 + 88923701v^6 + 97173219v^4 + 35363496*v^2 - 279163) / 419308
\(\beta_{4}\)	\(=\)	\( ( 36061 \nu^{15} + 949910 \nu^{13} + 9511983 \nu^{11} + 45198331 \nu^{9} + 103719977 \nu^{7} + \cdots + 7221193 \nu ) / 419308 \) (36061v^15 + 949910v^13 + 9511983v^11 + 45198331v^9 + 103719977v^7 + 106050879v^5 + 43039712v^3 + 7221193v) / 419308
\(\beta_{5}\)	\(=\)	\( ( 10869 \nu^{14} + 287387 \nu^{12} + 2904866 \nu^{10} + 14092173 \nu^{8} + 33778082 \nu^{6} + \cdots + 745589 ) / 104827 \) (10869v^14 + 287387v^12 + 2904866v^10 + 14092173v^8 + 33778082v^6 + 37492055v^4 + 15987961*v^2 + 745589) / 104827
\(\beta_{6}\)	\(=\)	\( ( 442855 \nu^{15} - 100356 \nu^{14} + 11989666 \nu^{13} - 2602356 \nu^{12} + 125241357 \nu^{11} + \cdots + 3109176 ) / 2515848 \) (442855v^15 - 100356v^14 + 11989666v^13 - 2602356v^12 + 125241357v^11 - 25457712v^10 + 636963225v^9 - 116354484v^8 + 1637663819v^7 - 247055832v^6 + 2013804781v^5 - 206991756v^4 + 979478360v^3 - 40553388v^2 + 61735451*v + 3109176) / 2515848
\(\beta_{7}\)	\(=\)	\( ( - 228275 \nu^{15} - 6098186 \nu^{13} - 62493369 \nu^{11} - 308997693 \nu^{9} + \cdots - 15821143 \nu ) / 1257924 \) (-228275v^15 - 6098186v^13 - 62493369v^11 - 308997693v^9 - 761373967v^7 - 879997745v^5 - 395046688v^3 - 15821143v) / 1257924
\(\beta_{8}\)	\(=\)	\( ( - 41081 \nu^{15} - 1126488 \nu^{13} - 11989151 \nu^{11} - 62734637 \nu^{9} - 168688829 \nu^{7} + \cdots - 12006461 \nu ) / 209654 \) (-41081v^15 - 1126488v^13 - 11989151v^11 - 62734637v^9 - 168688829v^7 - 222672257v^5 - 121015012v^3 - 12006461v) / 209654
\(\beta_{9}\)	\(=\)	\( ( 442855 \nu^{15} + 100356 \nu^{14} + 11989666 \nu^{13} + 2602356 \nu^{12} + 125241357 \nu^{11} + \cdots - 3109176 ) / 2515848 \) (442855v^15 + 100356v^14 + 11989666v^13 + 2602356v^12 + 125241357v^11 + 25457712v^10 + 636963225v^9 + 116354484v^8 + 1637663819v^7 + 247055832v^6 + 2013804781v^5 + 206991756v^4 + 979478360v^3 + 40553388v^2 + 61735451*v - 3109176) / 2515848
\(\beta_{10}\)	\(=\)	\( ( - 57807 \nu^{15} - 1560214 \nu^{13} - 16232103 \nu^{11} - 82127051 \nu^{9} - 209864801 \nu^{7} + \cdots - 11068957 \nu ) / 209654 \) (-57807v^15 - 1560214v^13 - 16232103v^11 - 82127051v^9 - 209864801v^7 - 257170883v^5 - 127773910v^3 - 11068957v) / 209654
\(\beta_{11}\)	\(=\)	\( ( 69357 \nu^{15} + 1858316 \nu^{13} + 19114181 \nu^{11} + 94923403 \nu^{9} + 234890639 \nu^{7} + \cdots + 4153887 \nu ) / 209654 \) (69357v^15 + 1858316v^13 + 19114181v^11 + 94923403v^9 + 234890639v^7 + 271503839v^5 + 119649158v^3 + 4153887v) / 209654
\(\beta_{12}\)	\(=\)	\( ( 916409 \nu^{15} - 76452 \nu^{14} + 24544094 \nu^{13} - 2104332 \nu^{12} + 252457563 \nu^{11} + \cdots - 7333800 ) / 2515848 \) (916409v^15 - 76452v^14 + 24544094v^13 - 2104332v^12 + 252457563v^11 - 22334208v^10 + 1254983967v^9 - 115246452v^8 + 3115533709v^7 - 300802248v^6 + 3631292171v^5 - 379573548v^4 + 1635683488v^3 - 192811260v^2 + 74693533*v - 7333800) / 2515848
\(\beta_{13}\)	\(=\)	\( ( 427154 \nu^{15} - 60039 \nu^{14} + 11522040 \nu^{13} - 1619346 \nu^{12} + 119763906 \nu^{11} + \cdots - 4372579 ) / 838616 \) (427154v^15 - 60039v^14 + 11522040v^13 - 1619346v^12 + 119763906v^11 - 16758669v^10 + 605052134v^9 - 83531377v^8 + 1541827174v^7 - 205965307v^6 + 1876784422v^5 - 232622165v^4 + 911123444v^3 - 96287128v^2 + 67452198*v - 4372579) / 838616
\(\beta_{14}\)	\(=\)	\( ( 427154 \nu^{15} + 60039 \nu^{14} + 11522040 \nu^{13} + 1619346 \nu^{12} + 119763906 \nu^{11} + \cdots + 4372579 ) / 838616 \) (427154v^15 + 60039v^14 + 11522040v^13 + 1619346v^12 + 119763906v^11 + 16758669v^10 + 605052134v^9 + 83531377v^8 + 1541827174v^7 + 205965307v^6 + 1876784422v^5 + 232622165v^4 + 911123444v^3 + 96287128v^2 + 67452198*v + 4372579) / 838616
\(\beta_{15}\)	\(=\)	\( ( 1130989 \nu^{15} + 30435574 \nu^{13} + 315205551 \nu^{11} + 1582949499 \nu^{9} + \cdots + 120607841 \nu ) / 1257924 \) (1130989v^15 + 30435574v^13 + 315205551v^11 + 1582949499v^9 + 3991823561v^7 + 4765099207v^5 + 2220115160v^3 + 120607841v) / 1257924

\(\nu\)	\(=\)	\( ( -\beta_{15} + \beta_{14} + \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} ) / 4 \) (-b15 + b14 + b13 + b10 + b9 + b8 + b6) / 4
\(\nu^{2}\)	\(=\)	\( ( - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 2 \beta_{9} + 2 \beta_{7} + 6 \beta_{6} + \cdots - 26 ) / 8 \) (-2b15 + 2b14 - 2b13 + 4b12 - 2b9 + 2b7 + 6b6 + 2b5 - b3 - b2 + 3*b1 - 26) / 8
\(\nu^{3}\)	\(=\)	\( ( 5 \beta_{15} - 4 \beta_{14} - 4 \beta_{13} + 2 \beta_{11} - 4 \beta_{10} - 8 \beta_{9} + \cdots - 8 \beta_{6} ) / 4 \) (5b15 - 4b14 - 4b13 + 2b11 - 4b10 - 8b9 - 6b8 + 3b7 - 8*b6) / 4
\(\nu^{4}\)	\(=\)	\( ( 18 \beta_{15} - 16 \beta_{14} + 16 \beta_{13} - 36 \beta_{12} + 12 \beta_{9} - 18 \beta_{7} - 48 \beta_{6} + \cdots + 156 ) / 8 \) (18b15 - 16b14 + 16b13 - 36b12 + 12b9 - 18b7 - 48b6 - 20b5 + 9b3 - b2 - 33b1 + 156) / 8
\(\nu^{5}\)	\(=\)	\( ( - 28 \beta_{15} + 21 \beta_{14} + 21 \beta_{13} - 30 \beta_{11} + 25 \beta_{10} + 65 \beta_{9} + \cdots + 6 \beta_{4} ) / 4 \) (-28b15 + 21b14 + 21b13 - 30b11 + 25b10 + 65b9 + 43b8 - 31b7 + 65b6 + 6b4) / 4
\(\nu^{6}\)	\(=\)	\( ( - 38 \beta_{15} + 32 \beta_{14} - 32 \beta_{13} + 76 \beta_{12} - 17 \beta_{9} + 38 \beta_{7} + \cdots - 271 ) / 2 \) (-38b15 + 32b14 - 32b13 + 76b12 - 17b9 + 38b7 + 93b6 + 45b5 - 24b3 + 12b2 + 72*b1 - 271) / 2
\(\nu^{7}\)	\(=\)	\( ( 157 \beta_{15} - 120 \beta_{14} - 120 \beta_{13} + 333 \beta_{11} - 203 \beta_{10} - 532 \beta_{9} + \cdots - 93 \beta_{4} ) / 4 \) (157b15 - 120b14 - 120b13 + 333b11 - 203b10 - 532b9 - 319b8 + 288b7 - 532b6 - 93b4) / 4
\(\nu^{8}\)	\(=\)	\( ( 1300 \beta_{15} - 1056 \beta_{14} + 1056 \beta_{13} - 2600 \beta_{12} + 328 \beta_{9} - 1300 \beta_{7} + \cdots + 8056 ) / 8 \) (1300b15 - 1056b14 + 1056b13 - 2600b12 + 328b9 - 1300b7 - 2928b6 - 1564b5 + 979b3 - 615b2 - 2403*b1 + 8056) / 8
\(\nu^{9}\)	\(=\)	\( ( - 851 \beta_{15} + 698 \beta_{14} + 698 \beta_{13} - 3320 \beta_{11} + 1792 \beta_{10} + \cdots + 1076 \beta_{4} ) / 4 \) (-851b15 + 698b14 + 698b13 - 3320b11 + 1792b10 + 4400b9 + 2406b8 - 2625b7 + 4400b6 + 1076b4) / 4
\(\nu^{10}\)	\(=\)	\( ( - 11270 \beta_{15} + 8890 \beta_{14} - 8890 \beta_{13} + 22540 \beta_{12} - 1030 \beta_{9} + \cdots - 62414 ) / 8 \) (-11270b15 + 8890b14 - 8890b13 + 22540b12 - 1030b9 + 11270b7 + 23570b6 + 13502b5 - 9503b3 + 6477b2 + 19929*b1 - 62414) / 8
\(\nu^{11}\)	\(=\)	\( ( 4268 \beta_{15} - 4004 \beta_{14} - 4004 \beta_{13} + 31465 \beta_{11} - 16141 \beta_{10} + \cdots - 11139 \beta_{4} ) / 4 \) (4268b15 - 4004b14 - 4004b13 + 31465b11 - 16141b10 - 36874b9 - 18491b8 + 23705b7 - 36874b6 - 11139b4) / 4
\(\nu^{12}\)	\(=\)	\( ( 24629 \beta_{15} - 18936 \beta_{14} + 18936 \beta_{13} - 49258 \beta_{12} - 818 \beta_{9} + \cdots + 124640 ) / 2 \) (24629b15 - 18936b14 + 18936b13 - 49258b12 - 818b9 - 24629b7 - 48440b6 - 29220b5 + 22300b3 - 15804b2 - 41556*b1 + 124640) / 2
\(\nu^{13}\)	\(=\)	\( ( - 17805 \beta_{15} + 22031 \beta_{14} + 22031 \beta_{13} - 290092 \beta_{11} + 145603 \beta_{10} + \cdots + 108944 \beta_{4} ) / 4 \) (-17805b15 + 22031b14 + 22031b13 - 290092b11 + 145603b10 + 312901b9 + 145079b8 - 212596b7 + 312901b6 + 108944b4) / 4
\(\nu^{14}\)	\(=\)	\( ( - 865104 \beta_{15} + 650402 \beta_{14} - 650402 \beta_{13} + 1730208 \beta_{12} + 109402 \beta_{9} + \cdots - 4078158 ) / 8 \) (-865104b15 + 650402b14 - 650402b13 + 1730208b12 + 109402b9 + 865104b7 + 1620806b6 + 1016386b5 - 819299b3 + 593565b2 + 1398861*b1 - 4078158) / 8
\(\nu^{15}\)	\(=\)	\( ( 34863 \beta_{15} - 111080 \beta_{14} - 111080 \beta_{13} + 2631150 \beta_{11} - 1308988 \beta_{10} + \cdots - 1030340 \beta_{4} ) / 4 \) (34863b15 - 111080b14 - 111080b13 + 2631150b11 - 1308988b10 - 2682476b9 - 1161798b8 + 1896737b7 - 2682476b6 - 1030340b4) / 4

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 289.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 19 T^{6} + \cdots + 112)^{2} \) (T^8 - 19T^6 + 104T^4 - 192*T^2 + 112)^2
$5$	\( (T^{8} + T^{7} - 2 T^{6} + \cdots + 625)^{2} \) (T^8 + T^7 - 2T^6 - 5T^5 + 2T^4 - 25T^3 - 50T^2 + 125T + 625)^2
$7$	\( (T^{8} + 28 T^{6} + \cdots + 576)^{2} \) (T^8 + 28T^6 + 232T^4 + 656*T^2 + 576)^2
$11$	\( (T^{8} + 57 T^{6} + \cdots + 112)^{2} \) (T^8 + 57T^6 + 1080T^4 + 6808*T^2 + 112)^2
$13$	\( (T^{8} + 43 T^{6} + \cdots + 2304)^{2} \) (T^8 + 43T^6 + 504T^4 + 2048*T^2 + 2304)^2
$17$	\( (T^{8} - 92 T^{6} + \cdots + 100800)^{2} \) (T^8 - 92T^6 + 2784T^4 - 30544*T^2 + 100800)^2
$19$	\( (T^{8} + 80 T^{6} + \cdots + 16128)^{2} \) (T^8 + 80T^6 + 1976T^4 + 14576*T^2 + 16128)^2
$23$	\( (T^{8} + 108 T^{6} + \cdots + 3136)^{2} \) (T^8 + 108T^6 + 2936T^4 + 10064*T^2 + 3136)^2
$29$	\( (T^{8} + 4 T^{7} + \cdots + 707281)^{2} \) (T^8 + 4T^7 + 4T^6 - 212T^5 - 826T^4 - 6148T^3 + 3364T^2 + 97556*T + 707281)^2
$31$	\( (T^{8} + 153 T^{6} + \cdots + 49392)^{2} \) (T^8 + 153T^6 + 6440T^4 + 44552*T^2 + 49392)^2
$37$	\( (T^{8} - 240 T^{6} + \cdots + 4845568)^{2} \) (T^8 - 240T^6 + 18816T^4 - 529616*T^2 + 4845568)^2
$41$	\( (T^{8} + 252 T^{6} + \cdots + 4480000)^{2} \) (T^8 + 252T^6 + 20384T^4 + 571904*T^2 + 4480000)^2
$43$	\( (T^{8} - 19 T^{6} + \cdots + 112)^{2} \) (T^8 - 19T^6 + 104T^4 - 192*T^2 + 112)^2
$47$	\( (T^{8} - 219 T^{6} + \cdots + 2041200)^{2} \) (T^8 - 219T^6 + 12360T^4 - 269344*T^2 + 2041200)^2
$53$	\( (T^{8} + 331 T^{6} + \cdots + 4804864)^{2} \) (T^8 + 331T^6 + 30312T^4 + 734976*T^2 + 4804864)^2
$59$	\( (T^{4} + 6 T^{3} + \cdots + 3360)^{4} \) (T^4 + 6T^3 - 120T^2 - 304*T + 3360)^4
$61$	\( (T^{8} + 220 T^{6} + \cdots + 64512)^{2} \) (T^8 + 220T^6 + 12640T^4 + 61184*T^2 + 64512)^2
$67$	\( (T^{8} + 304 T^{6} + \cdots + 1806336)^{2} \) (T^8 + 304T^6 + 28696T^4 + 848048*T^2 + 1806336)^2
$71$	\( (T^{4} - 112 T^{2} + \cdots - 576)^{4} \) (T^4 - 112T^2 - 496T - 576)^4
$73$	\( (T^{8} - 248 T^{6} + \cdots + 114688)^{2} \) (T^8 - 248T^6 + 17040T^4 - 355920*T^2 + 114688)^2
$79$	\( (T^{8} + 449 T^{6} + \cdots + 4258800)^{2} \) (T^8 + 449T^6 + 55496T^4 + 1576856*T^2 + 4258800)^2
$83$	\( (T^{8} + 252 T^{6} + \cdots + 270400)^{2} \) (T^8 + 252T^6 + 18696T^4 + 386704*T^2 + 270400)^2
$89$	\( (T^{8} + 500 T^{6} + \cdots + 83607552)^{2} \) (T^8 + 500T^6 + 78912T^4 + 4727808*T^2 + 83607552)^2
$97$	\( (T^{8} - 532 T^{6} + \cdots + 64351168)^{2} \) (T^8 - 532T^6 + 85776T^4 - 4444816*T^2 + 64351168)^2
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\(n\)	\(117\)	\(291\)	\(321\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)