LMFDB - Newform orbit 600.2.y.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,2,Mod(121,600)] 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("600.121"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 6])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.y (of order $5$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$3$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 16 x^{10} - 29 x^{9} + 106 x^{8} - 250 x^{7} + 815 x^{6} - 1250 x^{5} + 2650 x^{4} + \cdots + 15625 $ x^12 - 4x^11 + 16x^10 - 29x^9 + 106x^8 - 250x^7 + 815x^6 - 1250x^5 + 2650x^4 - 3625x^3 + 10000x^2 - 12500*x + 15625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{5} q^{3} - \beta_{4} q^{5} + ( - \beta_{10} - \beta_{6} - \beta_{5} + \cdots - 2) q^{7} - \beta_{2} q^{9} + (\beta_{11} + \beta_{10} + \beta_{5} + \cdots + 1) q^{11} + (\beta_{8} + \beta_{6} - 2 \beta_{2}) q^{13}+ \cdots + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) $ q + b5 * q^3 - b4 * q^5 + (-b10 - b6 - b5 - b4 + b1 - 2) * q^7 - b2 * q^9 + (b11 + b10 + b5 + b3 - b2 + 1) * q^11 + (b8 + b6 - 2b2) q^13 + (b11 - b9 + b6 + b4 + 1) * q^15 + (-b11 - b10 + b9 + b7 - b4 - b3 + b1 - 1) * q^17 + (-b10 + b8 - b7 - b6 + 2b5 - 2b4 + 2b3 - 4b2 - b1) * q^19 + (b10 - b9 + b8 + b6 - b5 + b4 - b1) * q^21 + (2b11 + b10 - 2b9 + b8 + b6 - 2b5 + b4 + b3 - b2 + b1 - 1) q^23 + (-b11 - b10 + b8 + b7 + b6 + b5 - b4 + 2b3 - 4b2 - b1) * q^25 + (-b5 - b3 + b2 - 1) * q^27 + (b11 + b10 - b9 + b8 - b7 + 2b6 + b4 - 3b2 - 2b1 + 3) q^29 + (-b11 + b9 + b7 + 2b5 + b3 - b2 + b1 - 1) q^31 + (-b9 - b7 - b5 - b3) * q^33 + (-b11 - b9 + 3b8 - b7 - 3b5 - b1 - 2) * q^35 + (-b11 + b10 - 2b9 + 2b8 - b7 + b6 + b5 + 6b3 - 4b2 - 3b1 + 5) q^37 + (b11 + b10 - b5 - 2b3 + 2b2 - 1) * q^39 + (-b8 + b7 - b6 - b4 - 3b3 + 3b2 - 3) * q^41 + (-2b11 - b10 + b9 - b8 - b7 - 2b6 - 4b4 + b3 - b2 + 3b1 - 1) * q^43 - b6 * q^45 + (2b11 + b10 - b9 - 2b7 - b6 - 2b5 - b2 + b1 + 1) q^47 + (-2b11 + 3b9 + b8 - 3b7 - 3b6 + b5 - 3b4 + b3 - 3b2 - 2) * q^49 + (b11 + b10 + b8 + b6 - b3 - 1) * q^51 + (b10 - b9 + 2b8 + b6 + 2b5 + 2b4 - 2b2 - b1 + 2) * q^53 + (-2b10 + 2b9 + b7 - 2b6 + 4b5 - 2b4 + 4b3 - 3b2 + 2b1 + 1) * q^55 + (b11 - 2b9 - b8 + 2b7 + 2b6 - 3b5 + 2b4 - 3b3 + 2b2) q^57 + (b11 - b10 - 3b9 - b7 + b6 - b5 + 2b4 - 2b3 + b2 - 2b1 - 1) * q^59 + (-2b11 + b10 + b9 + 2b8 - 2b7 - b6 + 6b5 - 3b4 + 6b3 - 8b2 - 3b1 + 3) * q^61 + (b9 - b8 - b6 + 2b2 + b1) q^63 + (-b10 + b9 - 2b8 + 2b7 - b6 - b5 + b4 - 3b3 + 6b2 + 2) * q^65 + (b11 - 2b10 - 2b8 + 2b7 + 2b6 + 3b5 + 2b3 - 3b2 + b1 + 1) q^67 + (2b10 - b9 + b8 - 2b7 - b6 - 2b5 + b4 - b3 + 2b2 - b1) * q^69 + (b11 + 3b10 - 3b9 - b8 - b7 + b5 - b4) * q^71 + (-b10 + b9 - 2b8 + b7 + 5b5 + b4 + 6b3 - 5b2 + b1 + 6) * q^73 + (2b11 - b8 + 2b7 + b6 - 4b5 + b4 - 5b3 + 5b2 + 2b1 - 1) * q^75 + (2b11 - b10 - 3b8 + 2b7 + b6 - 2b5 + 3b4 - b3 + 3b2 + 3b1 + 1) q^77 + (2b8 + 3b6 - 4b5 + 2b4 + 2b2 - 3b1 - 2) * q^79 + b3 * q^81 + (-b11 + b10 + b9 - b8 + 2b7 + b6 - 2b5 + 2b4 - 5b3 + 5b2 + 2b1 - 1) * q^83 + (-b11 - b10 - b9 - b8 - 2b5 + b4 + 2b3 + b2 - 2b1 + 4) q^85 + (b11 - b10 - 2b8 + b7 - b6 - b5 - 2b3 + 4b2 + b1 - 1) q^87 + (-b11 - 2b10 + b9 - 2b8 - b7 + 2b6 + b5 + b4 - b2 + b1 + 2) q^89 + (b9 - 2b8 + b7 - 2b6 - b4 - 2b3 + 6b2 + b1 - 2) * q^91 + (b10 + b9 + b8 - b7 - b5 - b4 - 2b3 - b2 - 1) q^93 + (b11 - b10 - 2b9 + 4b8 + b6 - 2b5 - 3b3 + b2 - 2b1 - 9) q^95 + (-4b11 - 6b10 + 6b9 - 5b8 + 4b7 - 4b6 + 2b5 - 5b4 + 4b2 + 4b1 - 4) * q^97 + (b5 + b4 + b3 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 4 q^{5} - 8 q^{7} - 3 q^{9} - 4 q^{11} - 4 q^{13} - q^{15} + q^{17} - 5 q^{19} - 3 q^{21} - 21 q^{23} - 16 q^{25} - 3 q^{27} + 11 q^{29} - 19 q^{31} + 11 q^{33} + 10 q^{35} + 34 q^{37}+ \cdots - 14 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 4 * q^5 - 8 * q^7 - 3 * q^9 - 4 * q^11 - 4 * q^13 - q^15 + q^17 - 5 * q^19 - 3 * q^21 - 21 * q^23 - 16 * q^25 - 3 * q^27 + 11 * q^29 - 19 * q^31 + 11 * q^33 + 10 * q^35 + 34 * q^37 - 4 * q^39 - 20 * q^41 + 28 * q^43 + 4 * q^45 + 19 * q^47 + 12 * q^49 - 14 * q^51 + 8 * q^53 - q^55 - 10 * q^57 - 18 * q^59 + 10 * q^61 + 7 * q^63 + 34 * q^65 - 40 * q^67 + 24 * q^69 - 7 * q^71 + 8 * q^73 + 4 * q^75 - 11 * q^77 - 26 * q^79 - 3 * q^81 + 10 * q^83 + 41 * q^85 - 4 * q^87 + 9 * q^89 - q^91 + 6 * q^93 - 81 * q^95 - 12 * q^97 - 14 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4 x^{11} + 16 x^{10} - 29 x^{9} + 106 x^{8} - 250 x^{7} + 815 x^{6} - 1250 x^{5} + 2650 x^{4} + \cdots + 15625 $ x^12 - 4*x^11 + 16*x^10 - 29*x^9 + 106*x^8 - 250*x^7 + 815*x^6 - 1250*x^5 + 2650*x^4 - 3625*x^3 + 10000*x^2 - 12500*x + 15625 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 2 \nu^{11} + 67 \nu^{10} + 27 \nu^{9} + 337 \nu^{8} - 118 \nu^{7} + 5145 \nu^{6} - 4600 \nu^{5} + \cdots + 193125 ) / 35625 $ (2v^11 + 67v^10 + 27v^9 + 337v^8 - 118v^7 + 5145v^6 - 4600v^5 + 17375v^4 - 10275v^3 + 93125v^2 - 125125*v + 193125) / 35625
$\beta_{3}$	$=$	$ ( 10 \nu^{11} - 102 \nu^{10} + 173 \nu^{9} - 367 \nu^{8} - 552 \nu^{7} - 3782 \nu^{6} + 5215 \nu^{5} + \cdots - 376250 ) / 35625 $ (10v^11 - 102v^10 + 173v^9 - 367v^8 - 552v^7 - 3782v^6 + 5215v^5 - 21805v^4 - 22400v^3 - 41675v^2 + 48875*v - 376250) / 35625
$\beta_{4}$	$=$	$ ( \nu^{11} - 4 \nu^{10} + 16 \nu^{9} - 29 \nu^{8} + 106 \nu^{7} - 250 \nu^{6} + 815 \nu^{5} + \cdots - 12500 ) / 3125 $ (v^11 - 4v^10 + 16v^9 - 29v^8 + 106v^7 - 250v^6 + 815v^5 - 1250v^4 + 2650v^3 - 3625v^2 + 10000v - 12500) / 3125
$\beta_{5}$	$=$	$ ( 64 \nu^{11} - 516 \nu^{10} + 1814 \nu^{9} - 4891 \nu^{8} + 8574 \nu^{7} - 28685 \nu^{6} + \cdots - 612500 ) / 178125 $ (64v^11 - 516v^10 + 1814v^9 - 4891v^8 + 8574v^7 - 28685v^6 + 58285v^5 - 141775v^4 + 143350v^3 - 290375v^2 + 254375*v - 612500) / 178125
$\beta_{6}$	$=$	$ ( - 309 \nu^{11} + 1286 \nu^{10} - 3269 \nu^{9} + 9636 \nu^{8} - 24329 \nu^{7} + 74300 \nu^{6} + \cdots + 734375 ) / 178125 $ (-309v^11 + 1286v^10 - 3269v^9 + 9636v^8 - 24329v^7 + 74300v^6 - 123210v^5 + 271250v^4 - 384475v^3 + 863250v^2 - 761875*v + 734375) / 178125
$\beta_{7}$	$=$	$ ( - 64 \nu^{11} - 54 \nu^{10} - 104 \nu^{9} - 1949 \nu^{8} - 1164 \nu^{7} - 8080 \nu^{6} + \cdots - 349375 ) / 35625 $ (-64v^11 - 54v^10 - 104v^9 - 1949v^8 - 1164v^7 - 8080v^6 - 4705v^5 - 66275v^4 + 4850v^3 - 144250v^2 - 126125*v - 349375) / 35625
$\beta_{8}$	$=$	$ ( 15 \nu^{11} - \nu^{10} + 79 \nu^{9} - 66 \nu^{8} + 1129 \nu^{7} - 1246 \nu^{6} + 3975 \nu^{5} + \cdots - 6250 ) / 7125 $ (15v^11 - v^10 + 79v^9 - 66v^8 + 1129v^7 - 1246v^6 + 3975v^5 - 3115v^4 + 20075v^3 - 29025v^2 + 43625v - 6250) / 7125
$\beta_{9}$	$=$	$ ( 602 \nu^{11} - 2158 \nu^{10} + 7082 \nu^{9} - 13133 \nu^{8} + 54637 \nu^{7} - 164300 \nu^{6} + \cdots - 6303125 ) / 178125 $ (602v^11 - 2158v^10 + 7082v^9 - 13133v^8 + 54637v^7 - 164300v^6 + 396080v^5 - 622125v^4 + 1050175v^3 - 2742250v^2 + 4978125*v - 6303125) / 178125
$\beta_{10}$	$=$	$ ( - 1009 \nu^{11} + 1396 \nu^{10} - 7209 \nu^{9} + 271 \nu^{8} - 55894 \nu^{7} + 51285 \nu^{6} + \cdots - 1012500 ) / 178125 $ (-1009v^11 + 1396v^10 - 7209v^9 + 271v^8 - 55894v^7 + 51285v^6 - 265960v^5 - 51100v^4 - 730725v^3 + 374750v^2 - 2235625*v - 1012500) / 178125
$\beta_{11}$	$=$	$ ( 210 \nu^{11} - 736 \nu^{10} + 1999 \nu^{9} - 3546 \nu^{8} + 13849 \nu^{7} - 46096 \nu^{6} + \cdots - 1536250 ) / 35625 $ (210v^11 - 736v^10 + 1999v^9 - 3546v^8 + 13849v^7 - 46096v^6 + 97830v^5 - 155140v^4 + 218825v^3 - 678525v^2 + 1135625*v - 1536250) / 35625

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2\beta_{5} + 2\beta_{4} - 3\beta_{3} + 5\beta_{2} + \beta _1 - 2 $ 2b11 + b10 - b9 - b8 - 2b5 + 2b4 - 3b3 + 5*b2 + b1 - 2
$\nu^{3}$	$=$	$ \beta_{11} - 2\beta_{9} + \beta_{8} + 2\beta_{6} + 3\beta_{5} + 5\beta_{4} - \beta_{3} - 3\beta_{2} - 3\beta _1 + 1 $ b11 - 2b9 + b8 + 2b6 + 3b5 + 5b4 - b3 - 3b2 - 3b1 + 1
$\nu^{4}$	$=$	$ - 5 \beta_{11} + 2 \beta_{9} + 4 \beta_{8} - 5 \beta_{7} - \beta_{6} + 14 \beta_{5} - 7 \beta_{4} + \cdots - 16 $ -5b11 + 2b9 + 4b8 - 5b7 - b6 + 14b5 - 7b4 + 6b3 - 16b2 - 5*b1 - 16
$\nu^{5}$	$=$	$ - 11 \beta_{11} + 4 \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - 12 \beta_{7} - 14 \beta_{6} + 4 \beta_{5} + \cdots + 7 $ -11b11 + 4b10 + 11b9 + 2b8 - 12b7 - 14b6 + 4b5 - 14b4 + 13b3 - 6b2 - 21*b1 + 7
$\nu^{6}$	$=$	$ - 52 \beta_{11} - 25 \beta_{10} + 35 \beta_{9} + 9 \beta_{8} + 7 \beta_{7} - 22 \beta_{6} - 9 \beta_{5} + \cdots + 45 $ -52b11 - 25b10 + 35b9 + 9b8 + 7b7 - 22b6 - 9b5 - 36b4 + 75b3 - 75b2 + 3*b1 + 45
$\nu^{7}$	$=$	$ 22 \beta_{11} - 24 \beta_{10} + 24 \beta_{9} - 71 \beta_{8} + 75 \beta_{7} - 20 \beta_{6} - 207 \beta_{5} + \cdots - 122 $ 22b11 - 24b10 + 24b9 - 71b8 + 75b7 - 20b6 - 207b5 - 53b4 - 58b3 + 185b2 + 136*b1 - 122
$\nu^{8}$	$=$	$ 276 \beta_{11} - 20 \beta_{10} - 167 \beta_{9} - 154 \beta_{8} + 220 \beta_{7} + 197 \beta_{6} + \cdots + 296 $ 276b11 - 20b10 - 167b9 - 154b8 + 220b7 + 197b6 - 372b5 + 295b4 - 371b3 + 512b2 + 77*b1 + 296
$\nu^{9}$	$=$	$ 220 \beta_{11} - 95 \beta_{10} - 323 \beta_{9} + 129 \beta_{8} + 155 \beta_{7} + 574 \beta_{6} + \cdots + 259 $ 220b11 - 95b10 - 323b9 + 129b8 + 155b7 + 574b6 + 694b5 + 268b4 - 194b3 - 381b2 + 115*b1 + 259
$\nu^{10}$	$=$	$ 514 \beta_{11} + 869 \beta_{10} - 364 \beta_{9} + 482 \beta_{8} - 1017 \beta_{7} + 86 \beta_{6} + \cdots - 668 $ 514b11 + 869b10 - 364b9 + 482b8 - 1017b7 + 86b6 + 1694b5 - 89b4 - 662b3 + 639b2 - 471*b1 - 668
$\nu^{11}$	$=$	$ - 1477 \beta_{11} + 1075 \beta_{10} + 285 \beta_{9} + 2269 \beta_{8} - 2838 \beta_{7} - 1647 \beta_{6} + \cdots + 2845 $ -1477b11 + 1075b10 + 285b9 + 2269b8 - 2838b7 - 1647b6 + 3616b5 + 314b4 + 3275b3 - 3895b2 - 2717*b1 + 2845

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$\beta_{5}$

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} $

121.1

−1.51022	+	1.64901i
1.06989	+	1.96350i
1.44033	−	1.71039i
−1.47248	+	1.68280i
0.865280	−	2.06187i
1.60720	+	1.55464i
−1.47248	−	1.68280i
0.865280	+	2.06187i
1.60720	−	1.55464i
−1.51022	−	1.64901i
1.06989	−	1.96350i
1.44033	+	1.71039i

−0.809017

−

0.587785i

−1.51022

−

1.64901i

−2.81722

0.309017

0.951057i

121.2

−0.809017

−

0.587785i

1.06989

−

1.96350i

4.53581

0.309017

0.951057i

121.3

−0.809017

−

0.587785i

1.44033

1.71039i

−1.48253

0.309017

0.951057i

241.1

0.309017

0.951057i

−1.47248

−

1.68280i

−0.584192

−0.809017

0.587785i

241.2

0.309017

0.951057i

0.865280

2.06187i

−4.09336

−0.809017

0.587785i

241.3

0.309017

0.951057i

1.60720

−

1.55464i

0.441485

−0.809017

0.587785i

361.1

0.309017

−

0.951057i

−1.47248

1.68280i

−0.584192

−0.809017

−

0.587785i

361.2

0.309017

−

0.951057i

0.865280

−

2.06187i

−4.09336

−0.809017

−

0.587785i

361.3

0.309017

−

0.951057i

1.60720

1.55464i

0.441485

−0.809017

−

0.587785i

481.1

−0.809017

0.587785i

−1.51022

1.64901i

−2.81722

0.309017

−

0.951057i

481.2

−0.809017

0.587785i

1.06989

1.96350i

4.53581

0.309017

−

0.951057i

481.3

−0.809017

0.587785i

1.44033

−

1.71039i

−1.48253

0.309017

−

0.951057i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.d	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	600.2.y.c	✓	12
25.d	even	5	1	inner	600.2.y.c	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
600.2.y.c	✓	12	1.a	even	1	1	trivial
600.2.y.c	✓	12	25.d	even	5	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{6} + 4T_{7}^{5} - 16T_{7}^{4} - 85T_{7}^{3} - 85T_{7}^{2} + 10T_{7} + 20 $ T7^6 + 4*T7^5 - 16*T7^4 - 85*T7^3 - 85*T7^2 + 10*T7 + 20 acting on $S_{2}^{\mathrm{new}}(600, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} $ (T^4 + T^3 + T^2 + T + 1)^3
$5$	$ T^{12} - 4 T^{11} + \cdots + 15625 $ T^12 - 4T^11 + 16T^10 - 29T^9 + 106T^8 - 250T^7 + 815T^6 - 1250T^5 + 2650T^4 - 3625T^3 + 10000T^2 - 12500*T + 15625
$7$	$ (T^{6} + 4 T^{5} - 16 T^{4} + \cdots + 20)^{2} $ (T^6 + 4T^5 - 16T^4 - 85T^3 - 85T^2 + 10*T + 20)^2
$11$	$ T^{12} + 4 T^{11} + \cdots + 59536 $ T^12 + 4T^11 + 31T^10 + 21T^9 + 122T^8 - 409T^7 + 1921T^6 - 487T^5 + 21109T^4 - 54092T^3 + 103688T^2 - 104920*T + 59536
$13$	$ T^{12} + 4 T^{11} + \cdots + 1 $ T^12 + 4T^11 + 21T^10 + 106T^9 + 517T^8 - 484T^7 + 341T^6 - 42T^5 + 624T^4 + 338T^3 + 88T^2 + 10*T + 1
$17$	$ T^{12} - T^{11} + \cdots + 361 $ T^12 - T^11 + 47T^10 - 252T^9 + 858T^8 - 1537T^7 + 1303T^6 + 803T^5 + 1178T^4 - 8422T^3 + 11867T^2 - 1501T + 361
$19$	$ T^{12} + 5 T^{11} + \cdots + 234256 $ T^12 + 5T^11 + 43T^10 - 167T^9 + 654T^8 + 14273T^7 + 208311T^6 - 29469T^5 + 883447T^4 + 816926T^3 + 500456T^2 + 191664*T + 234256
$23$	$ T^{12} + \cdots + 206784400 $ T^12 + 21T^11 + 216T^10 + 1201T^9 + 4831T^8 + 21445T^7 + 213300T^6 + 1760105T^5 + 10814135T^4 + 40508950T^3 + 117565600T^2 + 158180000*T + 206784400
$29$	$ T^{12} - 11 T^{11} + \cdots + 43681 $ T^12 - 11T^11 + 84T^10 - 370T^9 + 2345T^8 - 3336T^7 + 30541T^6 + 63771T^5 + 45435T^4 - 142385T^3 + 129699T^2 + 13376*T + 43681
$31$	$ T^{12} + 19 T^{11} + \cdots + 250000 $ T^12 + 19T^11 + 252T^10 + 2067T^9 + 11755T^8 + 50603T^7 + 191412T^6 + 588351T^5 + 1290311T^4 + 1679250T^3 + 1531000T^2 + 850000*T + 250000
$37$	$ T^{12} + \cdots + 5586815025 $ T^12 - 34T^11 + 672T^10 - 9392T^9 + 105410T^8 - 966758T^7 + 7501107T^6 - 47994436T^5 + 248662261T^4 - 999951990T^3 + 2956420215T^2 - 5686599600*T + 5586815025
$41$	$ T^{12} + 20 T^{11} + \cdots + 229441 $ T^12 + 20T^11 + 313T^10 + 2807T^9 + 18014T^8 + 82157T^7 + 304566T^6 + 839949T^5 + 2245312T^4 + 3970209T^3 + 4017861T^2 + 1474841*T + 229441
$43$	$ (T^{6} - 14 T^{5} + \cdots + 100804)^{2} $ (T^6 - 14T^5 - 129T^4 + 2392T^3 - 2941T^2 - 50774*T + 100804)^2
$47$	$ T^{12} - 19 T^{11} + \cdots + 144400 $ T^12 - 19T^11 + 262T^10 - 2887T^9 + 31315T^8 - 192223T^7 + 1025592T^6 - 2556291T^5 + 3437091T^4 - 2422370T^3 + 871940T^2 - 193800*T + 144400
$53$	$ T^{12} + \cdots + 260661025 $ T^12 - 8T^11 + 108T^10 - 256T^9 + 2410T^8 + 6934T^7 + 87533T^6 + 423692T^5 + 3512281T^4 + 18277690T^3 + 85001415T^2 + 209723550*T + 260661025
$59$	$ T^{12} + \cdots + 1353062656 $ T^12 + 18T^11 + 159T^10 + 358T^9 + 10687T^8 + 235402T^7 + 4205769T^6 + 37189744T^5 + 234561049T^4 + 864258676T^3 + 1746758832T^2 - 400945600*T + 1353062656
$61$	$ T^{12} - 10 T^{11} + \cdots + 4330561 $ T^12 - 10T^11 + 183T^10 - 2063T^9 + 23864T^8 - 44933T^7 + 525646T^6 + 3784469T^5 + 11166512T^4 + 18102989T^3 + 19461741T^2 + 12696181*T + 4330561
$67$	$ T^{12} + \cdots + 321269776 $ T^12 + 40T^11 + 893T^10 + 12962T^9 + 132134T^8 + 944382T^7 + 5129116T^6 + 23184964T^5 + 95131617T^4 + 250967054T^3 + 541785236T^2 + 606799096*T + 321269776
$71$	$ T^{12} + \cdots + 98668861456 $ T^12 + 7T^11 + 89T^10 + 47T^9 + 30922T^8 + 338383T^7 + 8149639T^6 + 41158216T^5 + 682639109T^4 + 3822539094T^3 + 12078330352T^2 + 7394290640*T + 98668861456
$73$	$ T^{12} + \cdots + 635090401 $ T^12 - 8T^11 + 231T^10 - 3435T^9 + 35215T^8 - 262443T^7 + 2183009T^6 - 12315018T^5 + 50207215T^4 - 158304810T^3 + 609204456T^2 - 939568883*T + 635090401
$79$	$ T^{12} + \cdots + 41609472256 $ T^12 + 26T^11 + 409T^10 + 3395T^9 + 17870T^8 + 179261T^7 + 4907031T^6 + 66624479T^5 + 540623385T^4 + 2730726960T^3 + 10562872224T^2 + 23106491584*T + 41609472256
$83$	$ T^{12} + \cdots + 4710900496 $ T^12 - 10T^11 + 227T^10 - 1764T^9 + 25689T^8 - 70756T^7 + 1160899T^6 + 10234142T^5 + 116212577T^4 - 289301452T^3 + 676172384T^2 - 1470320392*T + 4710900496
$89$	$ T^{12} + \cdots + 18164300625 $ T^12 - 9T^11 + 311T^10 - 1254T^9 + 32596T^8 - 115035T^7 + 2010395T^6 + 19338525T^5 + 272975650T^4 + 617746500T^3 + 3763137375T^2 - 13939104375*T + 18164300625
$97$	$ T^{12} + \cdots + 6878877721 $ T^12 + 12T^11 + 443T^10 + 4461T^9 + 101108T^8 + 330649T^7 + 10656822T^6 - 109874689T^5 + 1490128078T^4 - 8335334921T^3 + 32255256253T^2 + 24616875773*T + 6878877721
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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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.y (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{3} - \beta_{4} q^{5} + ( - \beta_{10} - \beta_{6} - \beta_{5} + \cdots - 2) q^{7} - \beta_{2} q^{9} + (\beta_{11} + \beta_{10} + \beta_{5} + \cdots + 1) q^{11} + (\beta_{8} + \beta_{6} - 2 \beta_{2}) q^{13}+ \cdots + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) q + b5 * q^3 - b4 * q^5 + (-b10 - b6 - b5 - b4 + b1 - 2) * q^7 - b2 * q^9 + (b11 + b10 + b5 + b3 - b2 + 1) * q^11 + (b8 + b6 - 2b2) q^13 + (b11 - b9 + b6 + b4 + 1) * q^15 + (-b11 - b10 + b9 + b7 - b4 - b3 + b1 - 1) * q^17 + (-b10 + b8 - b7 - b6 + 2b5 - 2b4 + 2b3 - 4b2 - b1) * q^19 + (b10 - b9 + b8 + b6 - b5 + b4 - b1) * q^21 + (2b11 + b10 - 2b9 + b8 + b6 - 2b5 + b4 + b3 - b2 + b1 - 1) q^23 + (-b11 - b10 + b8 + b7 + b6 + b5 - b4 + 2b3 - 4b2 - b1) * q^25 + (-b5 - b3 + b2 - 1) * q^27 + (b11 + b10 - b9 + b8 - b7 + 2b6 + b4 - 3b2 - 2b1 + 3) q^29 + (-b11 + b9 + b7 + 2b5 + b3 - b2 + b1 - 1) q^31 + (-b9 - b7 - b5 - b3) * q^33 + (-b11 - b9 + 3b8 - b7 - 3b5 - b1 - 2) * q^35 + (-b11 + b10 - 2b9 + 2b8 - b7 + b6 + b5 + 6b3 - 4b2 - 3b1 + 5) q^37 + (b11 + b10 - b5 - 2b3 + 2b2 - 1) * q^39 + (-b8 + b7 - b6 - b4 - 3b3 + 3b2 - 3) * q^41 + (-2b11 - b10 + b9 - b8 - b7 - 2b6 - 4b4 + b3 - b2 + 3b1 - 1) * q^43 - b6 * q^45 + (2b11 + b10 - b9 - 2b7 - b6 - 2b5 - b2 + b1 + 1) q^47 + (-2b11 + 3b9 + b8 - 3b7 - 3b6 + b5 - 3b4 + b3 - 3b2 - 2) * q^49 + (b11 + b10 + b8 + b6 - b3 - 1) * q^51 + (b10 - b9 + 2b8 + b6 + 2b5 + 2b4 - 2b2 - b1 + 2) * q^53 + (-2b10 + 2b9 + b7 - 2b6 + 4b5 - 2b4 + 4b3 - 3b2 + 2b1 + 1) * q^55 + (b11 - 2b9 - b8 + 2b7 + 2b6 - 3b5 + 2b4 - 3b3 + 2b2) q^57 + (b11 - b10 - 3b9 - b7 + b6 - b5 + 2b4 - 2b3 + b2 - 2b1 - 1) * q^59 + (-2b11 + b10 + b9 + 2b8 - 2b7 - b6 + 6b5 - 3b4 + 6b3 - 8b2 - 3b1 + 3) * q^61 + (b9 - b8 - b6 + 2b2 + b1) q^63 + (-b10 + b9 - 2b8 + 2b7 - b6 - b5 + b4 - 3b3 + 6b2 + 2) * q^65 + (b11 - 2b10 - 2b8 + 2b7 + 2b6 + 3b5 + 2b3 - 3b2 + b1 + 1) q^67 + (2b10 - b9 + b8 - 2b7 - b6 - 2b5 + b4 - b3 + 2b2 - b1) * q^69 + (b11 + 3b10 - 3b9 - b8 - b7 + b5 - b4) * q^71 + (-b10 + b9 - 2b8 + b7 + 5b5 + b4 + 6b3 - 5b2 + b1 + 6) * q^73 + (2b11 - b8 + 2b7 + b6 - 4b5 + b4 - 5b3 + 5b2 + 2b1 - 1) * q^75 + (2b11 - b10 - 3b8 + 2b7 + b6 - 2b5 + 3b4 - b3 + 3b2 + 3b1 + 1) q^77 + (2b8 + 3b6 - 4b5 + 2b4 + 2b2 - 3b1 - 2) * q^79 + b3 * q^81 + (-b11 + b10 + b9 - b8 + 2b7 + b6 - 2b5 + 2b4 - 5b3 + 5b2 + 2b1 - 1) * q^83 + (-b11 - b10 - b9 - b8 - 2b5 + b4 + 2b3 + b2 - 2b1 + 4) q^85 + (b11 - b10 - 2b8 + b7 - b6 - b5 - 2b3 + 4b2 + b1 - 1) q^87 + (-b11 - 2b10 + b9 - 2b8 - b7 + 2b6 + b5 + b4 - b2 + b1 + 2) q^89 + (b9 - 2b8 + b7 - 2b6 - b4 - 2b3 + 6b2 + b1 - 2) * q^91 + (b10 + b9 + b8 - b7 - b5 - b4 - 2b3 - b2 - 1) q^93 + (b11 - b10 - 2b9 + 4b8 + b6 - 2b5 - 3b3 + b2 - 2b1 - 9) q^95 + (-4b11 - 6b10 + 6b9 - 5b8 + 4b7 - 4b6 + 2b5 - 5b4 + 4b2 + 4b1 - 4) * q^97 + (b5 + b4 + b3 - b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 4 q^{5} - 8 q^{7} - 3 q^{9} - 4 q^{11} - 4 q^{13} - q^{15} + q^{17} - 5 q^{19} - 3 q^{21} - 21 q^{23} - 16 q^{25} - 3 q^{27} + 11 q^{29} - 19 q^{31} + 11 q^{33} + 10 q^{35} + 34 q^{37}+ \cdots - 14 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 4 * q^5 - 8 * q^7 - 3 * q^9 - 4 * q^11 - 4 * q^13 - q^15 + q^17 - 5 * q^19 - 3 * q^21 - 21 * q^23 - 16 * q^25 - 3 * q^27 + 11 * q^29 - 19 * q^31 + 11 * q^33 + 10 * q^35 + 34 * q^37 - 4 * q^39 - 20 * q^41 + 28 * q^43 + 4 * q^45 + 19 * q^47 + 12 * q^49 - 14 * q^51 + 8 * q^53 - q^55 - 10 * q^57 - 18 * q^59 + 10 * q^61 + 7 * q^63 + 34 * q^65 - 40 * q^67 + 24 * q^69 - 7 * q^71 + 8 * q^73 + 4 * q^75 - 11 * q^77 - 26 * q^79 - 3 * q^81 + 10 * q^83 + 41 * q^85 - 4 * q^87 + 9 * q^89 - q^91 + 6 * q^93 - 81 * q^95 - 12 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	600.2.y.c

Level	$600$
Weight	$2$
Character orbit	600.y
Analytic conductor	$4.791$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(4.79102412128\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4 x^{11} + 16 x^{10} - 29 x^{9} + 106 x^{8} - 250 x^{7} + 815 x^{6} - 1250 x^{5} + 2650 x^{4} + \cdots + 15625 \) x^12 - 4x^11 + 16x^10 - 29x^9 + 106x^8 - 250x^7 + 815x^6 - 1250x^5 + 2650x^4 - 3625x^3 + 10000x^2 - 12500*x + 15625
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 2 \nu^{11} + 67 \nu^{10} + 27 \nu^{9} + 337 \nu^{8} - 118 \nu^{7} + 5145 \nu^{6} - 4600 \nu^{5} + \cdots + 193125 ) / 35625 \) (2v^11 + 67v^10 + 27v^9 + 337v^8 - 118v^7 + 5145v^6 - 4600v^5 + 17375v^4 - 10275v^3 + 93125v^2 - 125125*v + 193125) / 35625
\(\beta_{3}\)	\(=\)	\( ( 10 \nu^{11} - 102 \nu^{10} + 173 \nu^{9} - 367 \nu^{8} - 552 \nu^{7} - 3782 \nu^{6} + 5215 \nu^{5} + \cdots - 376250 ) / 35625 \) (10v^11 - 102v^10 + 173v^9 - 367v^8 - 552v^7 - 3782v^6 + 5215v^5 - 21805v^4 - 22400v^3 - 41675v^2 + 48875*v - 376250) / 35625
\(\beta_{4}\)	\(=\)	\( ( \nu^{11} - 4 \nu^{10} + 16 \nu^{9} - 29 \nu^{8} + 106 \nu^{7} - 250 \nu^{6} + 815 \nu^{5} + \cdots - 12500 ) / 3125 \) (v^11 - 4v^10 + 16v^9 - 29v^8 + 106v^7 - 250v^6 + 815v^5 - 1250v^4 + 2650v^3 - 3625v^2 + 10000v - 12500) / 3125
\(\beta_{5}\)	\(=\)	\( ( 64 \nu^{11} - 516 \nu^{10} + 1814 \nu^{9} - 4891 \nu^{8} + 8574 \nu^{7} - 28685 \nu^{6} + \cdots - 612500 ) / 178125 \) (64v^11 - 516v^10 + 1814v^9 - 4891v^8 + 8574v^7 - 28685v^6 + 58285v^5 - 141775v^4 + 143350v^3 - 290375v^2 + 254375*v - 612500) / 178125
\(\beta_{6}\)	\(=\)	\( ( - 309 \nu^{11} + 1286 \nu^{10} - 3269 \nu^{9} + 9636 \nu^{8} - 24329 \nu^{7} + 74300 \nu^{6} + \cdots + 734375 ) / 178125 \) (-309v^11 + 1286v^10 - 3269v^9 + 9636v^8 - 24329v^7 + 74300v^6 - 123210v^5 + 271250v^4 - 384475v^3 + 863250v^2 - 761875*v + 734375) / 178125
\(\beta_{7}\)	\(=\)	\( ( - 64 \nu^{11} - 54 \nu^{10} - 104 \nu^{9} - 1949 \nu^{8} - 1164 \nu^{7} - 8080 \nu^{6} + \cdots - 349375 ) / 35625 \) (-64v^11 - 54v^10 - 104v^9 - 1949v^8 - 1164v^7 - 8080v^6 - 4705v^5 - 66275v^4 + 4850v^3 - 144250v^2 - 126125*v - 349375) / 35625
\(\beta_{8}\)	\(=\)	\( ( 15 \nu^{11} - \nu^{10} + 79 \nu^{9} - 66 \nu^{8} + 1129 \nu^{7} - 1246 \nu^{6} + 3975 \nu^{5} + \cdots - 6250 ) / 7125 \) (15v^11 - v^10 + 79v^9 - 66v^8 + 1129v^7 - 1246v^6 + 3975v^5 - 3115v^4 + 20075v^3 - 29025v^2 + 43625v - 6250) / 7125
\(\beta_{9}\)	\(=\)	\( ( 602 \nu^{11} - 2158 \nu^{10} + 7082 \nu^{9} - 13133 \nu^{8} + 54637 \nu^{7} - 164300 \nu^{6} + \cdots - 6303125 ) / 178125 \) (602v^11 - 2158v^10 + 7082v^9 - 13133v^8 + 54637v^7 - 164300v^6 + 396080v^5 - 622125v^4 + 1050175v^3 - 2742250v^2 + 4978125*v - 6303125) / 178125
\(\beta_{10}\)	\(=\)	\( ( - 1009 \nu^{11} + 1396 \nu^{10} - 7209 \nu^{9} + 271 \nu^{8} - 55894 \nu^{7} + 51285 \nu^{6} + \cdots - 1012500 ) / 178125 \) (-1009v^11 + 1396v^10 - 7209v^9 + 271v^8 - 55894v^7 + 51285v^6 - 265960v^5 - 51100v^4 - 730725v^3 + 374750v^2 - 2235625*v - 1012500) / 178125
\(\beta_{11}\)	\(=\)	\( ( 210 \nu^{11} - 736 \nu^{10} + 1999 \nu^{9} - 3546 \nu^{8} + 13849 \nu^{7} - 46096 \nu^{6} + \cdots - 1536250 ) / 35625 \) (210v^11 - 736v^10 + 1999v^9 - 3546v^8 + 13849v^7 - 46096v^6 + 97830v^5 - 155140v^4 + 218825v^3 - 678525v^2 + 1135625*v - 1536250) / 35625

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2\beta_{5} + 2\beta_{4} - 3\beta_{3} + 5\beta_{2} + \beta _1 - 2 \) 2b11 + b10 - b9 - b8 - 2b5 + 2b4 - 3b3 + 5*b2 + b1 - 2
\(\nu^{3}\)	\(=\)	\( \beta_{11} - 2\beta_{9} + \beta_{8} + 2\beta_{6} + 3\beta_{5} + 5\beta_{4} - \beta_{3} - 3\beta_{2} - 3\beta _1 + 1 \) b11 - 2b9 + b8 + 2b6 + 3b5 + 5b4 - b3 - 3b2 - 3b1 + 1
\(\nu^{4}\)	\(=\)	\( - 5 \beta_{11} + 2 \beta_{9} + 4 \beta_{8} - 5 \beta_{7} - \beta_{6} + 14 \beta_{5} - 7 \beta_{4} + \cdots - 16 \) -5b11 + 2b9 + 4b8 - 5b7 - b6 + 14b5 - 7b4 + 6b3 - 16b2 - 5*b1 - 16
\(\nu^{5}\)	\(=\)	\( - 11 \beta_{11} + 4 \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - 12 \beta_{7} - 14 \beta_{6} + 4 \beta_{5} + \cdots + 7 \) -11b11 + 4b10 + 11b9 + 2b8 - 12b7 - 14b6 + 4b5 - 14b4 + 13b3 - 6b2 - 21*b1 + 7
\(\nu^{6}\)	\(=\)	\( - 52 \beta_{11} - 25 \beta_{10} + 35 \beta_{9} + 9 \beta_{8} + 7 \beta_{7} - 22 \beta_{6} - 9 \beta_{5} + \cdots + 45 \) -52b11 - 25b10 + 35b9 + 9b8 + 7b7 - 22b6 - 9b5 - 36b4 + 75b3 - 75b2 + 3*b1 + 45
\(\nu^{7}\)	\(=\)	\( 22 \beta_{11} - 24 \beta_{10} + 24 \beta_{9} - 71 \beta_{8} + 75 \beta_{7} - 20 \beta_{6} - 207 \beta_{5} + \cdots - 122 \) 22b11 - 24b10 + 24b9 - 71b8 + 75b7 - 20b6 - 207b5 - 53b4 - 58b3 + 185b2 + 136*b1 - 122
\(\nu^{8}\)	\(=\)	\( 276 \beta_{11} - 20 \beta_{10} - 167 \beta_{9} - 154 \beta_{8} + 220 \beta_{7} + 197 \beta_{6} + \cdots + 296 \) 276b11 - 20b10 - 167b9 - 154b8 + 220b7 + 197b6 - 372b5 + 295b4 - 371b3 + 512b2 + 77*b1 + 296
\(\nu^{9}\)	\(=\)	\( 220 \beta_{11} - 95 \beta_{10} - 323 \beta_{9} + 129 \beta_{8} + 155 \beta_{7} + 574 \beta_{6} + \cdots + 259 \) 220b11 - 95b10 - 323b9 + 129b8 + 155b7 + 574b6 + 694b5 + 268b4 - 194b3 - 381b2 + 115*b1 + 259
\(\nu^{10}\)	\(=\)	\( 514 \beta_{11} + 869 \beta_{10} - 364 \beta_{9} + 482 \beta_{8} - 1017 \beta_{7} + 86 \beta_{6} + \cdots - 668 \) 514b11 + 869b10 - 364b9 + 482b8 - 1017b7 + 86b6 + 1694b5 - 89b4 - 662b3 + 639b2 - 471*b1 - 668
\(\nu^{11}\)	\(=\)	\( - 1477 \beta_{11} + 1075 \beta_{10} + 285 \beta_{9} + 2269 \beta_{8} - 2838 \beta_{7} - 1647 \beta_{6} + \cdots + 2845 \) -1477b11 + 1075b10 + 285b9 + 2269b8 - 2838b7 - 1647b6 + 3616b5 + 314b4 + 3275b3 - 3895b2 - 2717*b1 + 2845

\(n\)	\(151\)	\(301\)	\(401\)	\(577\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(\beta_{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 121.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) (T^4 + T^3 + T^2 + T + 1)^3
$5$	\( T^{12} - 4 T^{11} + \cdots + 15625 \) T^12 - 4T^11 + 16T^10 - 29T^9 + 106T^8 - 250T^7 + 815T^6 - 1250T^5 + 2650T^4 - 3625T^3 + 10000T^2 - 12500*T + 15625
$7$	\( (T^{6} + 4 T^{5} - 16 T^{4} + \cdots + 20)^{2} \) (T^6 + 4T^5 - 16T^4 - 85T^3 - 85T^2 + 10*T + 20)^2
$11$	\( T^{12} + 4 T^{11} + \cdots + 59536 \) T^12 + 4T^11 + 31T^10 + 21T^9 + 122T^8 - 409T^7 + 1921T^6 - 487T^5 + 21109T^4 - 54092T^3 + 103688T^2 - 104920*T + 59536
$13$	\( T^{12} + 4 T^{11} + \cdots + 1 \) T^12 + 4T^11 + 21T^10 + 106T^9 + 517T^8 - 484T^7 + 341T^6 - 42T^5 + 624T^4 + 338T^3 + 88T^2 + 10*T + 1
$17$	\( T^{12} - T^{11} + \cdots + 361 \) T^12 - T^11 + 47T^10 - 252T^9 + 858T^8 - 1537T^7 + 1303T^6 + 803T^5 + 1178T^4 - 8422T^3 + 11867T^2 - 1501T + 361
$19$	\( T^{12} + 5 T^{11} + \cdots + 234256 \) T^12 + 5T^11 + 43T^10 - 167T^9 + 654T^8 + 14273T^7 + 208311T^6 - 29469T^5 + 883447T^4 + 816926T^3 + 500456T^2 + 191664*T + 234256
$23$	\( T^{12} + \cdots + 206784400 \) T^12 + 21T^11 + 216T^10 + 1201T^9 + 4831T^8 + 21445T^7 + 213300T^6 + 1760105T^5 + 10814135T^4 + 40508950T^3 + 117565600T^2 + 158180000*T + 206784400
$29$	\( T^{12} - 11 T^{11} + \cdots + 43681 \) T^12 - 11T^11 + 84T^10 - 370T^9 + 2345T^8 - 3336T^7 + 30541T^6 + 63771T^5 + 45435T^4 - 142385T^3 + 129699T^2 + 13376*T + 43681
$31$	\( T^{12} + 19 T^{11} + \cdots + 250000 \) T^12 + 19T^11 + 252T^10 + 2067T^9 + 11755T^8 + 50603T^7 + 191412T^6 + 588351T^5 + 1290311T^4 + 1679250T^3 + 1531000T^2 + 850000*T + 250000
$37$	\( T^{12} + \cdots + 5586815025 \) T^12 - 34T^11 + 672T^10 - 9392T^9 + 105410T^8 - 966758T^7 + 7501107T^6 - 47994436T^5 + 248662261T^4 - 999951990T^3 + 2956420215T^2 - 5686599600*T + 5586815025
$41$	\( T^{12} + 20 T^{11} + \cdots + 229441 \) T^12 + 20T^11 + 313T^10 + 2807T^9 + 18014T^8 + 82157T^7 + 304566T^6 + 839949T^5 + 2245312T^4 + 3970209T^3 + 4017861T^2 + 1474841*T + 229441
$43$	\( (T^{6} - 14 T^{5} + \cdots + 100804)^{2} \) (T^6 - 14T^5 - 129T^4 + 2392T^3 - 2941T^2 - 50774*T + 100804)^2
$47$	\( T^{12} - 19 T^{11} + \cdots + 144400 \) T^12 - 19T^11 + 262T^10 - 2887T^9 + 31315T^8 - 192223T^7 + 1025592T^6 - 2556291T^5 + 3437091T^4 - 2422370T^3 + 871940T^2 - 193800*T + 144400
$53$	\( T^{12} + \cdots + 260661025 \) T^12 - 8T^11 + 108T^10 - 256T^9 + 2410T^8 + 6934T^7 + 87533T^6 + 423692T^5 + 3512281T^4 + 18277690T^3 + 85001415T^2 + 209723550*T + 260661025
$59$	\( T^{12} + \cdots + 1353062656 \) T^12 + 18T^11 + 159T^10 + 358T^9 + 10687T^8 + 235402T^7 + 4205769T^6 + 37189744T^5 + 234561049T^4 + 864258676T^3 + 1746758832T^2 - 400945600*T + 1353062656
$61$	\( T^{12} - 10 T^{11} + \cdots + 4330561 \) T^12 - 10T^11 + 183T^10 - 2063T^9 + 23864T^8 - 44933T^7 + 525646T^6 + 3784469T^5 + 11166512T^4 + 18102989T^3 + 19461741T^2 + 12696181*T + 4330561
$67$	\( T^{12} + \cdots + 321269776 \) T^12 + 40T^11 + 893T^10 + 12962T^9 + 132134T^8 + 944382T^7 + 5129116T^6 + 23184964T^5 + 95131617T^4 + 250967054T^3 + 541785236T^2 + 606799096*T + 321269776
$71$	\( T^{12} + \cdots + 98668861456 \) T^12 + 7T^11 + 89T^10 + 47T^9 + 30922T^8 + 338383T^7 + 8149639T^6 + 41158216T^5 + 682639109T^4 + 3822539094T^3 + 12078330352T^2 + 7394290640*T + 98668861456
$73$	\( T^{12} + \cdots + 635090401 \) T^12 - 8T^11 + 231T^10 - 3435T^9 + 35215T^8 - 262443T^7 + 2183009T^6 - 12315018T^5 + 50207215T^4 - 158304810T^3 + 609204456T^2 - 939568883*T + 635090401
$79$	\( T^{12} + \cdots + 41609472256 \) T^12 + 26T^11 + 409T^10 + 3395T^9 + 17870T^8 + 179261T^7 + 4907031T^6 + 66624479T^5 + 540623385T^4 + 2730726960T^3 + 10562872224T^2 + 23106491584*T + 41609472256
$83$	\( T^{12} + \cdots + 4710900496 \) T^12 - 10T^11 + 227T^10 - 1764T^9 + 25689T^8 - 70756T^7 + 1160899T^6 + 10234142T^5 + 116212577T^4 - 289301452T^3 + 676172384T^2 - 1470320392*T + 4710900496
$89$	\( T^{12} + \cdots + 18164300625 \) T^12 - 9T^11 + 311T^10 - 1254T^9 + 32596T^8 - 115035T^7 + 2010395T^6 + 19338525T^5 + 272975650T^4 + 617746500T^3 + 3763137375T^2 - 13939104375*T + 18164300625
$97$	\( T^{12} + \cdots + 6878877721 \) T^12 + 12T^11 + 443T^10 + 4461T^9 + 101108T^8 + 330649T^7 + 10656822T^6 - 109874689T^5 + 1490128078T^4 - 8335334921T^3 + 32255256253T^2 + 24616875773*T + 6878877721
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