LMFDB - Newform orbit 2142.2.p.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2142,2,Mod(1135,2142)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2142, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2142.1135"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2142.p (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-12,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$17.1039561130$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
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} + 36x^{10} + 450x^{8} + 2400x^{6} + 5385x^{4} + 4428x^{2} + 1156 $ x^12 + 36x^10 + 450x^8 + 2400x^6 + 5385x^4 + 4428*x^2 + 1156
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 714)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{9} q^{2} - q^{4} + ( - \beta_{8} - \beta_{6}) q^{5} + \beta_{5} q^{7} + \beta_{9} q^{8} + ( - \beta_{7} + \beta_{5}) q^{10} + (\beta_{5} - \beta_{2}) q^{11} + (\beta_{8} - \beta_{7} - \beta_{4} - 1) q^{13}+ \cdots + q^{98}+O(q^{100}) $ q - b9 * q^2 - q^4 + (-b8 - b6) * q^5 + b5 * q^7 + b9 * q^8 + (-b7 + b5) * q^10 + (b5 - b2) * q^11 + (b8 - b7 - b4 - 1) * q^13 + b6 * q^14 + q^16 + (b11 + b9 - b7 + b3 - b2) * q^17 + (-b11 + b10 + b7 - b3 + b2) * q^19 + (b8 + b6) * q^20 + (b6 - b3) * q^22 + (b9 + 2b5 + 1) q^23 + (b11 - 2b9 + b8 + b7 - b3 + b2) q^25 + (b10 + b9 + b7) * q^26 - b5 * q^28 + (b11 + b10 + b9 - 2b8 - b4 + b3 + b1 - 1) q^29 + (b10 - b9 + b8 - 2b6 - b4 + 1) q^31 - b9 * q^32 + (b8 - b3 - b2 - b1 + 1) * q^34 + (-b1 - 1) * q^35 + (-b11 + b9 - b6 - b3 - b1 - 1) * q^37 + (-b8 + b7 + b4 + b3 + b2 + b1) * q^38 + (b7 - b5) * q^40 + (b11 + b10 - b8 + 2b7 - 2b5 + b4 - b1) * q^41 + (-b11 - 3b9 - 2b8 - 2b7) q^43 + (-b5 + b2) * q^44 + (-b9 + 2b6 + 1) q^46 + (b8 - b7 - 3b6 - 3b5 - 2b4 + 2) q^47 + b9 * q^49 + (-b8 + b7 + b3 + b2 - b1 - 2) * q^50 + (-b8 + b7 + b4 + 1) * q^52 + (-3b11 + 2b10 - b9 - b8 + b7 - b6 + b5) * q^53 + (-3b6 - 3b5 - 2b4 - b3 - b2 - 3b1 + 1) * q^55 - b6 * q^56 + (b11 + b10 + b9 - b8 - b7 + b4 - b2 - b1 + 1) * q^58 + (3b11 - b10 + 2b8 + b7 - b6 + b5 + b3 - b2) * q^59 + (-2b11 + b10 - b9 - b8 + 2b7 + b4 + 2b2 + 2b1 - 1) * q^61 + (b10 - b9 - b8 + 2b7 + 2b5 + b4 - 1) * q^62 - q^64 + (b10 + 3b9 - b6 - b4 - 3) q^65 + (3b8 - 3b7 - 2b4 + b3 + b2 - b1 - 1) q^67 + (-b11 - b9 + b7 - b3 + b2) * q^68 + (-b11 + b9) * q^70 + (-2b11 + 2b10 - 3b9 - 2b4 - 2b1 + 3) q^71 + (b10 + 3b9 + 2b8 + b6 - b4 + 2b3 - 3) q^73 + (-b11 + b9 + b5 + b2 + b1 + 1) * q^74 + (b11 - b10 - b7 + b3 - b2) * q^76 + (b10 + b9 - b8) * q^77 + (b11 + b10 + 4b9 - b8 - b7 + 5b5 + b4 - 2b2 - b1 + 4) q^79 + (-b8 - b6) * q^80 + (-b11 - b10 - b8 - 2b6 + b4 - b1) q^82 + (-4b11 - b10 - 3b9 + b8 + b6 - b5) * q^83 + (b11 - b10 + 2b9 + 2b7 - b5 - b4 + b3 - b2 - b1 - 4) * q^85 + (2b8 - 2b7 + b1 - 3) * q^86 + (-b6 + b3) * q^88 + (2b8 - 2b7 + 2b6 + 2b5 - 2b4 - b3 - b2 - 5b1 - 1) * q^89 + (-b11 - b5 + b2 + b1) * q^91 + (-b9 - 2b5 - 1) q^92 + (2b10 - 2b9 - b8 + b7 - 3b6 + 3b5) * q^94 + (-b11 + 2b10 + b9 - 2b8 + 2b5 + 2b4 + 2b2 + b1 + 1) q^95 + (2b11 + b10 - 3b9 - b6 - b4 + 2b3 + 2b1 + 3) * q^97 + q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4} - 12 q^{13} + 12 q^{16} + 12 q^{23} - 12 q^{29} + 12 q^{31} + 12 q^{34} - 12 q^{35} - 12 q^{37} + 12 q^{46} + 24 q^{47} - 24 q^{50} + 12 q^{52} + 12 q^{55} + 12 q^{58} - 12 q^{61} - 12 q^{62}+ \cdots + 12 q^{98}+O(q^{100}) $ 12 * q - 12 * q^4 - 12 * q^13 + 12 * q^16 + 12 * q^23 - 12 * q^29 + 12 * q^31 + 12 * q^34 - 12 * q^35 - 12 * q^37 + 12 * q^46 + 24 * q^47 - 24 * q^50 + 12 * q^52 + 12 * q^55 + 12 * q^58 - 12 * q^61 - 12 * q^62 - 12 * q^64 - 36 * q^65 - 12 * q^67 + 36 * q^71 - 36 * q^73 + 12 * q^74 + 48 * q^79 - 48 * q^85 - 36 * q^86 - 12 * q^89 - 12 * q^92 + 12 * q^95 + 36 * q^97 + 12 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 36x^{10} + 450x^{8} + 2400x^{6} + 5385x^{4} + 4428x^{2} + 1156 $ x^12 + 36*x^10 + 450*x^8 + 2400*x^6 + 5385*x^4 + 4428*x^2 + 1156 :

$\beta_{1}$	$=$	$ ( 47\nu^{10} + 1528\nu^{8} + 15305\nu^{6} + 44682\nu^{4} - 40198\nu^{2} - 136816 ) / 32164 $ (47v^10 + 1528v^8 + 15305v^6 + 44682v^4 - 40198*v^2 - 136816) / 32164
$\beta_{2}$	$=$	$ ( - 57 \nu^{11} + 335 \nu^{10} - 2467 \nu^{9} + 12773 \nu^{8} - 39313 \nu^{7} + 173759 \nu^{6} + \cdots + 1199656 ) / 128656 $ (-57v^11 + 335v^10 - 2467v^9 + 12773v^8 - 39313v^7 + 173759v^6 - 282205v^5 + 1025915v^4 - 865206v^3 + 2382410v^2 - 664944*v + 1199656) / 128656
$\beta_{3}$	$=$	$ ( 57 \nu^{11} + 335 \nu^{10} + 2467 \nu^{9} + 12773 \nu^{8} + 39313 \nu^{7} + 173759 \nu^{6} + \cdots + 1199656 ) / 128656 $ (57v^11 + 335v^10 + 2467v^9 + 12773v^8 + 39313v^7 + 173759v^6 + 282205v^5 + 1025915v^4 + 865206v^3 + 2382410v^2 + 664944*v + 1199656) / 128656
$\beta_{4}$	$=$	$ ( -463\nu^{10} - 14197\nu^{8} - 131609\nu^{6} - 361295\nu^{4} + 116440\nu^{2} + 432820 ) / 128656 $ (-463v^10 - 14197v^8 - 131609v^6 - 361295v^4 + 116440*v^2 + 432820) / 128656
$\beta_{5}$	$=$	$ ( 99 \nu^{11} - 867 \nu^{10} + 3289 \nu^{9} - 30753 \nu^{8} + 36465 \nu^{7} - 371977 \nu^{6} + \cdots - 1485732 ) / 257312 $ (99v^11 - 867v^10 + 3289v^9 - 30753v^8 + 36465v^7 - 371977v^6 + 167211v^5 - 1832787v^4 + 350548v^3 - 3340908v^2 + 394020*v - 1485732) / 257312
$\beta_{6}$	$=$	$ ( - 99 \nu^{11} - 867 \nu^{10} - 3289 \nu^{9} - 30753 \nu^{8} - 36465 \nu^{7} - 371977 \nu^{6} + \cdots - 1485732 ) / 257312 $ (-99v^11 - 867v^10 - 3289v^9 - 30753v^8 - 36465v^7 - 371977v^6 - 167211v^5 - 1832787v^4 - 350548v^3 - 3340908v^2 - 394020*v - 1485732) / 257312
$\beta_{7}$	$=$	$ ( 57 \nu^{11} - 1023 \nu^{10} + 2467 \nu^{9} - 36509 \nu^{8} + 39313 \nu^{7} - 447755 \nu^{6} + \cdots - 2094400 ) / 128656 $ (57v^11 - 1023v^10 + 2467v^9 - 36509v^8 + 39313v^7 - 447755v^6 + 282205v^5 - 2282203v^4 + 865206v^3 - 4492334v^2 + 793600*v - 2094400) / 128656
$\beta_{8}$	$=$	$ ( 57 \nu^{11} + 1023 \nu^{10} + 2467 \nu^{9} + 36509 \nu^{8} + 39313 \nu^{7} + 447755 \nu^{6} + \cdots + 2094400 ) / 128656 $ (57v^11 + 1023v^10 + 2467v^9 + 36509v^8 + 39313v^7 + 447755v^6 + 282205v^5 + 2282203v^4 + 865206v^3 + 4492334v^2 + 793600*v + 2094400) / 128656
$\beta_{9}$	$=$	$ ( -9\nu^{11} - 316\nu^{9} - 3774\nu^{7} - 18414\nu^{5} - 33857\nu^{3} - 15318\nu ) / 1496 $ (-9v^11 - 316v^9 - 3774v^7 - 18414v^5 - 33857v^3 - 15318v) / 1496
$\beta_{10}$	$=$	$ ( - 92 \nu^{11} + 93 \nu^{10} - 3238 \nu^{9} + 3319 \nu^{8} - 38932 \nu^{7} + 40705 \nu^{6} + \cdots + 190400 ) / 11696 $ (-92v^11 + 93v^10 - 3238v^9 + 3319v^8 - 38932v^7 + 40705v^6 - 192970v^5 + 207473v^4 - 362608v^3 + 408394v^2 - 141872*v + 190400) / 11696
$\beta_{11}$	$=$	$ ( -88\nu^{11} - 3129\nu^{9} - 38161\nu^{7} - 192621\nu^{5} - 372491\nu^{3} - 174542\nu ) / 5848 $ (-88v^11 - 3129v^9 - 38161v^7 - 192621v^5 - 372491v^3 - 174542v) / 5848

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

$\nu$	$=$	$ ( \beta_{8} + \beta_{7} - \beta_{3} + \beta_{2} ) / 2 $ (b8 + b7 - b3 + b2) / 2
$\nu^{2}$	$=$	$ ( -\beta_{8} + \beta_{7} - 4\beta_{6} - 4\beta_{5} - \beta_{3} - \beta_{2} - 4\beta _1 - 12 ) / 2 $ (-b8 + b7 - 4b6 - 4b5 - b3 - b2 - 4*b1 - 12) / 2
$\nu^{3}$	$=$	$ ( - 6 \beta_{11} + 6 \beta_{10} + 8 \beta_{9} - 15 \beta_{8} - 9 \beta_{7} - 10 \beta_{6} + \cdots - 9 \beta_{2} ) / 2 $ (-6b11 + 6b10 + 8b9 - 15b8 - 9b7 - 10b6 + 10b5 + 9b3 - 9*b2) / 2
$\nu^{4}$	$=$	$ ( 25\beta_{8} - 25\beta_{7} + 72\beta_{6} + 72\beta_{5} + 4\beta_{4} + 5\beta_{3} + 5\beta_{2} + 52\beta _1 + 132 ) / 2 $ (25b8 - 25b7 + 72b6 + 72b5 + 4b4 + 5b3 + 5b2 + 52b1 + 132) / 2
$\nu^{5}$	$=$	$ ( 134 \beta_{11} - 98 \beta_{10} - 220 \beta_{9} + 221 \beta_{8} + 123 \beta_{7} + 190 \beta_{6} + \cdots + 95 \beta_{2} ) / 2 $ (134b11 - 98b10 - 220b9 + 221b8 + 123b7 + 190b6 - 190b5 - 95b3 + 95*b2) / 2
$\nu^{6}$	$=$	$ ( - 447 \beta_{8} + 447 \beta_{7} - 1124 \beta_{6} - 1124 \beta_{5} - 120 \beta_{4} + 33 \beta_{3} + \cdots - 1752 ) / 2 $ (-447b8 + 447b7 - 1124b6 - 1124b5 - 120b4 + 33b3 + 33b2 - 732b1 - 1752) / 2
$\nu^{7}$	$=$	$ ( - 2402 \beta_{11} + 1418 \beta_{10} + 4368 \beta_{9} - 3287 \beta_{8} - 1869 \beta_{7} + \cdots - 1149 \beta_{2} ) / 2 $ (-2402b11 + 1418b10 + 4368b9 - 3287b8 - 1869b7 - 3318b6 + 3318b5 + 1149b3 - 1149*b2) / 2
$\nu^{8}$	$=$	$ ( 7381 \beta_{8} - 7381 \beta_{7} + 17232 \beta_{6} + 17232 \beta_{5} + 2620 \beta_{4} - 1527 \beta_{3} + \cdots + 25292 ) / 2 $ (7381b8 - 7381b7 + 17232b6 + 17232b5 + 2620b4 - 1527b3 - 1527b2 + 11036b1 + 25292) / 2
$\nu^{9}$	$=$	$ ( 40230 \beta_{11} - 20466 \beta_{10} - 77468 \beta_{9} + 49941 \beta_{8} + 29475 \beta_{7} + \cdots + 15327 \beta_{2} ) / 2 $ (40230b11 - 20466b10 - 77468b9 + 49941b8 + 29475b7 + 56222b6 - 56222b5 - 15327b3 + 15327*b2) / 2
$\nu^{10}$	$=$	$ ( - 119023 \beta_{8} + 119023 \beta_{7} - 266076 \beta_{6} - 266076 \beta_{5} - 49904 \beta_{4} + \cdots - 381672 ) / 2 $ (-119023b8 + 119023b7 - 266076b6 - 266076b5 - 49904b4 + 33289b3 + 33289b2 - 171908b1 - 381672) / 2
$\nu^{11}$	$=$	$ ( - 656874 \beta_{11} + 301906 \beta_{10} + 1308032 \beta_{9} - 772575 \beta_{8} + \cdots - 218549 \beta_{2} ) / 2 $ (-656874b11 + 301906b10 + 1308032b9 - 772575b8 - 470669b7 - 933790b6 + 933790b5 + 218549b3 - 218549*b2) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2142\mathbb{Z}\right)^\times$.

$n$	$1261$	$1667$	$1837$
$\chi(n)$	$-\beta_{9}$	$1$	$1$

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} $

1135.1

		4.00120i
		0.725963i
	−	3.11211i
	−	0.889088i
	−	2.45154i
		1.72558i
	−	4.00120i
	−	0.725963i
		3.11211i
		0.889088i
		2.45154i
	−	1.72558i

1.00000i

−1.00000

−3.02305

−

3.02305i

0.707107

−

0.707107i

−

1.00000i

3.02305

−

3.02305i

1135.2

1.00000i

−1.00000

−1.60883

−

1.60883i

−0.707107

0.707107i

−

1.00000i

1.60883

−

1.60883i

1135.3

1.00000i

−1.00000

−0.240016

−

0.240016i

0.707107

−

0.707107i

−

1.00000i

0.240016

−

0.240016i

1135.4

1.00000i

−1.00000

1.14174

1.14174i

0.707107

−

0.707107i

−

1.00000i

−1.14174

1.14174i

1135.5

1.00000i

−1.00000

1.17420

1.17420i

−0.707107

0.707107i

−

1.00000i

−1.17420

1.17420i

1135.6

1.00000i

−1.00000

2.55596

2.55596i

−0.707107

0.707107i

−

1.00000i

−2.55596

2.55596i

1891.1

−

1.00000i

−1.00000

−3.02305

3.02305i

0.707107

0.707107i

1.00000i

3.02305

3.02305i

1891.2

−

1.00000i

−1.00000

−1.60883

1.60883i

−0.707107

−

0.707107i

1.00000i

1.60883

1.60883i

1891.3

−

1.00000i

−1.00000

−0.240016

0.240016i

0.707107

0.707107i

1.00000i

0.240016

0.240016i

1891.4

−

1.00000i

−1.00000

1.14174

−

1.14174i

0.707107

0.707107i

1.00000i

−1.14174

−

1.14174i

1891.5

−

1.00000i

−1.00000

1.17420

−

1.17420i

−0.707107

−

0.707107i

1.00000i

−1.17420

−

1.17420i

1891.6

−

1.00000i

−1.00000

2.55596

−

2.55596i

−0.707107

−

0.707107i

1.00000i

−2.55596

−

2.55596i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.c	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2142.2.p.f		12
3.b	odd	2	1		714.2.m.e	✓	12
17.c	even	4	1	inner	2142.2.p.f		12
51.f	odd	4	1		714.2.m.e	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
714.2.m.e	✓	12	3.b	odd	2	1
714.2.m.e	✓	12	51.f	odd	4	1
2142.2.p.f		12	1.a	even	1	1	trivial
2142.2.p.f		12	17.c	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} - 16 T_{5}^{9} + 273 T_{5}^{8} - 240 T_{5}^{7} + 128 T_{5}^{6} - 672 T_{5}^{5} + 5712 T_{5}^{4} + \cdots + 1024 $ T5^12 - 16*T5^9 + 273*T5^8 - 240*T5^7 + 128*T5^6 - 672*T5^5 + 5712*T5^4 - 7552*T5^3 + 4608*T5^2 + 3072*T5 + 1024 acting on $S_{2}^{\mathrm{new}}(2142, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$3$	$ T^{12} $ T^12
$5$	$ T^{12} - 16 T^{9} + \cdots + 1024 $ T^12 - 16T^9 + 273T^8 - 240T^7 + 128T^6 - 672T^5 + 5712T^4 - 7552T^3 + 4608T^2 + 3072*T + 1024
$7$	$ (T^{4} + 1)^{3} $ (T^4 + 1)^3
$11$	$ T^{12} + 4 T^{9} + \cdots + 1201216 $ T^12 + 4T^9 + 405T^8 + 84T^7 + 8T^6 + 216T^5 + 44628T^4 + 12032T^3 + 1152T^2 - 52608*T + 1201216
$13$	$ (T^{6} + 6 T^{5} - 15 T^{4} + \cdots + 32)^{2} $ (T^6 + 6T^5 - 15T^4 - 56T^3 + 174T^2 - 144*T + 32)^2
$17$	$ (T^{6} + 33 T^{4} + \cdots + 4913)^{2} $ (T^6 + 33T^4 - 16T^3 + 561*T^2 + 4913)^2
$19$	$ T^{12} + 120 T^{10} + \cdots + 73984 $ T^12 + 120T^10 + 4104T^8 + 45184T^6 + 199824T^4 + 323712*T^2 + 73984
$23$	$ (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 4)^{3} $ (T^4 - 4T^3 + 8T^2 + 8*T + 4)^3
$29$	$ T^{12} + \cdots + 300814336 $ T^12 + 12T^11 + 72T^10 - 56T^9 + 6036T^8 + 63936T^7 + 334208T^6 + 198144T^5 + 8308032T^4 + 83576320T^3 + 420268032T^2 + 502837248*T + 300814336
$31$	$ T^{12} - 12 T^{11} + \cdots + 21233664 $ T^12 - 12T^11 + 72T^10 - 264T^9 + 3924T^8 - 42912T^7 + 267264T^6 - 870912T^5 + 1553472T^4 - 1050624T^3 + 2654208T^2 - 10616832*T + 21233664
$37$	$ T^{12} + 12 T^{11} + \cdots + 246016 $ T^12 + 12T^11 + 72T^10 + 152T^9 + 21T^8 - 276T^7 + 6728T^6 + 9816T^5 - 2940T^4 - 98720T^3 + 415872T^2 - 452352*T + 246016
$41$	$ T^{12} + \cdots + 1073741824 $ T^12 - 64T^9 + 8976T^8 - 8448T^7 + 2048T^6 + 270336T^5 + 9191424T^4 + 2097152*T^3 + 1073741824
$43$	$ T^{12} + \cdots + 4507242496 $ T^12 + 378T^10 + 50481T^8 + 3031560T^6 + 85187088T^4 + 1057282560*T^2 + 4507242496
$47$	$ (T^{6} - 12 T^{5} + \cdots - 2176)^{2} $ (T^6 - 12T^5 - 78T^4 + 944T^3 - 192T^2 - 7296*T - 2176)^2
$53$	$ T^{12} + 366 T^{10} + \cdots + 40551424 $ T^12 + 366T^10 + 46593T^8 + 2435592T^6 + 45310416T^4 + 83519232*T^2 + 40551424
$59$	$ T^{12} + \cdots + 2102772736 $ T^12 + 468T^10 + 77604T^8 + 5444544T^6 + 150401664T^4 + 1143601152*T^2 + 2102772736
$61$	$ T^{12} + 12 T^{11} + \cdots + 4096 $ T^12 + 12T^11 + 72T^10 + 424T^9 + 15828T^8 + 200064T^7 + 1351040T^6 + 4527360T^5 + 7368000T^4 - 1456640T^3 + 165888T^2 + 36864*T + 4096
$67$	$ (T^{6} + 6 T^{5} + \cdots - 214784)^{2} $ (T^6 + 6T^5 - 327T^4 - 1872T^3 + 24984T^2 + 112896*T - 214784)^2
$71$	$ T^{12} + \cdots + 13184550976 $ T^12 - 36T^11 + 648T^10 - 4968T^9 + 16524T^8 - 23328T^7 + 2472768T^6 - 13292352T^5 + 33893424T^4 + 49581504T^3 + 1700611200T^2 - 6696535680*T + 13184550976
$73$	$ T^{12} + \cdots + 9491435776 $ T^12 + 36T^11 + 648T^10 + 4212T^9 + 8097T^8 + 20208T^7 + 4351104T^6 + 11274048T^5 + 14310432T^4 + 21719808T^3 + 3968335872T^2 + 8679309312*T + 9491435776
$79$	$ T^{12} + \cdots + 702038016 $ T^12 - 48T^11 + 1152T^10 - 13704T^9 + 77841T^8 + 104904T^7 - 808416T^6 - 665280T^5 + 54575424T^4 + 289778688T^3 + 767066112T^2 + 1037795328*T + 702038016
$83$	$ T^{12} + \cdots + 19606720576 $ T^12 + 798T^10 + 222909T^8 + 24902868T^6 + 897275892T^4 + 8923036128*T^2 + 19606720576
$89$	$ (T^{6} + 6 T^{5} + \cdots - 451996)^{2} $ (T^6 + 6T^5 - 339T^4 - 848T^3 + 23280T^2 + 25992*T - 451996)^2
$97$	$ T^{12} + \cdots + 1473331456 $ T^12 - 36T^11 + 648T^10 - 6692T^9 + 61041T^8 - 790560T^7 + 11297024T^6 - 113617920T^5 + 748235808T^4 - 3008800256T^3 + 6702041088T^2 - 4443945984*T + 1473331456
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2142.p (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{9} q^{2} - q^{4} + ( - \beta_{8} - \beta_{6}) q^{5} + \beta_{5} q^{7} + \beta_{9} q^{8} + ( - \beta_{7} + \beta_{5}) q^{10} + (\beta_{5} - \beta_{2}) q^{11} + (\beta_{8} - \beta_{7} - \beta_{4} - 1) q^{13}+ \cdots + q^{98}+O(q^{100}) \) q - b9 * q^2 - q^4 + (-b8 - b6) * q^5 + b5 * q^7 + b9 * q^8 + (-b7 + b5) * q^10 + (b5 - b2) * q^11 + (b8 - b7 - b4 - 1) * q^13 + b6 * q^14 + q^16 + (b11 + b9 - b7 + b3 - b2) * q^17 + (-b11 + b10 + b7 - b3 + b2) * q^19 + (b8 + b6) * q^20 + (b6 - b3) * q^22 + (b9 + 2b5 + 1) q^23 + (b11 - 2b9 + b8 + b7 - b3 + b2) q^25 + (b10 + b9 + b7) * q^26 - b5 * q^28 + (b11 + b10 + b9 - 2b8 - b4 + b3 + b1 - 1) q^29 + (b10 - b9 + b8 - 2b6 - b4 + 1) q^31 - b9 * q^32 + (b8 - b3 - b2 - b1 + 1) * q^34 + (-b1 - 1) * q^35 + (-b11 + b9 - b6 - b3 - b1 - 1) * q^37 + (-b8 + b7 + b4 + b3 + b2 + b1) * q^38 + (b7 - b5) * q^40 + (b11 + b10 - b8 + 2b7 - 2b5 + b4 - b1) * q^41 + (-b11 - 3b9 - 2b8 - 2b7) q^43 + (-b5 + b2) * q^44 + (-b9 + 2b6 + 1) q^46 + (b8 - b7 - 3b6 - 3b5 - 2b4 + 2) q^47 + b9 * q^49 + (-b8 + b7 + b3 + b2 - b1 - 2) * q^50 + (-b8 + b7 + b4 + 1) * q^52 + (-3b11 + 2b10 - b9 - b8 + b7 - b6 + b5) * q^53 + (-3b6 - 3b5 - 2b4 - b3 - b2 - 3b1 + 1) * q^55 - b6 * q^56 + (b11 + b10 + b9 - b8 - b7 + b4 - b2 - b1 + 1) * q^58 + (3b11 - b10 + 2b8 + b7 - b6 + b5 + b3 - b2) * q^59 + (-2b11 + b10 - b9 - b8 + 2b7 + b4 + 2b2 + 2b1 - 1) * q^61 + (b10 - b9 - b8 + 2b7 + 2b5 + b4 - 1) * q^62 - q^64 + (b10 + 3b9 - b6 - b4 - 3) q^65 + (3b8 - 3b7 - 2b4 + b3 + b2 - b1 - 1) q^67 + (-b11 - b9 + b7 - b3 + b2) * q^68 + (-b11 + b9) * q^70 + (-2b11 + 2b10 - 3b9 - 2b4 - 2b1 + 3) q^71 + (b10 + 3b9 + 2b8 + b6 - b4 + 2b3 - 3) q^73 + (-b11 + b9 + b5 + b2 + b1 + 1) * q^74 + (b11 - b10 - b7 + b3 - b2) * q^76 + (b10 + b9 - b8) * q^77 + (b11 + b10 + 4b9 - b8 - b7 + 5b5 + b4 - 2b2 - b1 + 4) q^79 + (-b8 - b6) * q^80 + (-b11 - b10 - b8 - 2b6 + b4 - b1) q^82 + (-4b11 - b10 - 3b9 + b8 + b6 - b5) * q^83 + (b11 - b10 + 2b9 + 2b7 - b5 - b4 + b3 - b2 - b1 - 4) * q^85 + (2b8 - 2b7 + b1 - 3) * q^86 + (-b6 + b3) * q^88 + (2b8 - 2b7 + 2b6 + 2b5 - 2b4 - b3 - b2 - 5b1 - 1) * q^89 + (-b11 - b5 + b2 + b1) * q^91 + (-b9 - 2b5 - 1) q^92 + (2b10 - 2b9 - b8 + b7 - 3b6 + 3b5) * q^94 + (-b11 + 2b10 + b9 - 2b8 + 2b5 + 2b4 + 2b2 + b1 + 1) q^95 + (2b11 + b10 - 3b9 - b6 - b4 + 2b3 + 2b1 + 3) * q^97 + q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4} - 12 q^{13} + 12 q^{16} + 12 q^{23} - 12 q^{29} + 12 q^{31} + 12 q^{34} - 12 q^{35} - 12 q^{37} + 12 q^{46} + 24 q^{47} - 24 q^{50} + 12 q^{52} + 12 q^{55} + 12 q^{58} - 12 q^{61} - 12 q^{62}+ \cdots + 12 q^{98}+O(q^{100}) \) 12 * q - 12 * q^4 - 12 * q^13 + 12 * q^16 + 12 * q^23 - 12 * q^29 + 12 * q^31 + 12 * q^34 - 12 * q^35 - 12 * q^37 + 12 * q^46 + 24 * q^47 - 24 * q^50 + 12 * q^52 + 12 * q^55 + 12 * q^58 - 12 * q^61 - 12 * q^62 - 12 * q^64 - 36 * q^65 - 12 * q^67 + 36 * q^71 - 36 * q^73 + 12 * q^74 + 48 * q^79 - 48 * q^85 - 36 * q^86 - 12 * q^89 - 12 * q^92 + 12 * q^95 + 36 * q^97 + 12 * q^98

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	2142.2.p.f

Level	$2142$
Weight	$2$
Character orbit	2142.p
Analytic conductor	$17.104$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(17.1039561130\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} + 36x^{10} + 450x^{8} + 2400x^{6} + 5385x^{4} + 4428x^{2} + 1156 \) x^12 + 36x^10 + 450x^8 + 2400x^6 + 5385x^4 + 4428*x^2 + 1156
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 714)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 47\nu^{10} + 1528\nu^{8} + 15305\nu^{6} + 44682\nu^{4} - 40198\nu^{2} - 136816 ) / 32164 \) (47v^10 + 1528v^8 + 15305v^6 + 44682v^4 - 40198*v^2 - 136816) / 32164
\(\beta_{2}\)	\(=\)	\( ( - 57 \nu^{11} + 335 \nu^{10} - 2467 \nu^{9} + 12773 \nu^{8} - 39313 \nu^{7} + 173759 \nu^{6} + \cdots + 1199656 ) / 128656 \) (-57v^11 + 335v^10 - 2467v^9 + 12773v^8 - 39313v^7 + 173759v^6 - 282205v^5 + 1025915v^4 - 865206v^3 + 2382410v^2 - 664944*v + 1199656) / 128656
\(\beta_{3}\)	\(=\)	\( ( 57 \nu^{11} + 335 \nu^{10} + 2467 \nu^{9} + 12773 \nu^{8} + 39313 \nu^{7} + 173759 \nu^{6} + \cdots + 1199656 ) / 128656 \) (57v^11 + 335v^10 + 2467v^9 + 12773v^8 + 39313v^7 + 173759v^6 + 282205v^5 + 1025915v^4 + 865206v^3 + 2382410v^2 + 664944*v + 1199656) / 128656
\(\beta_{4}\)	\(=\)	\( ( -463\nu^{10} - 14197\nu^{8} - 131609\nu^{6} - 361295\nu^{4} + 116440\nu^{2} + 432820 ) / 128656 \) (-463v^10 - 14197v^8 - 131609v^6 - 361295v^4 + 116440*v^2 + 432820) / 128656
\(\beta_{5}\)	\(=\)	\( ( 99 \nu^{11} - 867 \nu^{10} + 3289 \nu^{9} - 30753 \nu^{8} + 36465 \nu^{7} - 371977 \nu^{6} + \cdots - 1485732 ) / 257312 \) (99v^11 - 867v^10 + 3289v^9 - 30753v^8 + 36465v^7 - 371977v^6 + 167211v^5 - 1832787v^4 + 350548v^3 - 3340908v^2 + 394020*v - 1485732) / 257312
\(\beta_{6}\)	\(=\)	\( ( - 99 \nu^{11} - 867 \nu^{10} - 3289 \nu^{9} - 30753 \nu^{8} - 36465 \nu^{7} - 371977 \nu^{6} + \cdots - 1485732 ) / 257312 \) (-99v^11 - 867v^10 - 3289v^9 - 30753v^8 - 36465v^7 - 371977v^6 - 167211v^5 - 1832787v^4 - 350548v^3 - 3340908v^2 - 394020*v - 1485732) / 257312
\(\beta_{7}\)	\(=\)	\( ( 57 \nu^{11} - 1023 \nu^{10} + 2467 \nu^{9} - 36509 \nu^{8} + 39313 \nu^{7} - 447755 \nu^{6} + \cdots - 2094400 ) / 128656 \) (57v^11 - 1023v^10 + 2467v^9 - 36509v^8 + 39313v^7 - 447755v^6 + 282205v^5 - 2282203v^4 + 865206v^3 - 4492334v^2 + 793600*v - 2094400) / 128656
\(\beta_{8}\)	\(=\)	\( ( 57 \nu^{11} + 1023 \nu^{10} + 2467 \nu^{9} + 36509 \nu^{8} + 39313 \nu^{7} + 447755 \nu^{6} + \cdots + 2094400 ) / 128656 \) (57v^11 + 1023v^10 + 2467v^9 + 36509v^8 + 39313v^7 + 447755v^6 + 282205v^5 + 2282203v^4 + 865206v^3 + 4492334v^2 + 793600*v + 2094400) / 128656
\(\beta_{9}\)	\(=\)	\( ( -9\nu^{11} - 316\nu^{9} - 3774\nu^{7} - 18414\nu^{5} - 33857\nu^{3} - 15318\nu ) / 1496 \) (-9v^11 - 316v^9 - 3774v^7 - 18414v^5 - 33857v^3 - 15318v) / 1496
\(\beta_{10}\)	\(=\)	\( ( - 92 \nu^{11} + 93 \nu^{10} - 3238 \nu^{9} + 3319 \nu^{8} - 38932 \nu^{7} + 40705 \nu^{6} + \cdots + 190400 ) / 11696 \) (-92v^11 + 93v^10 - 3238v^9 + 3319v^8 - 38932v^7 + 40705v^6 - 192970v^5 + 207473v^4 - 362608v^3 + 408394v^2 - 141872*v + 190400) / 11696
\(\beta_{11}\)	\(=\)	\( ( -88\nu^{11} - 3129\nu^{9} - 38161\nu^{7} - 192621\nu^{5} - 372491\nu^{3} - 174542\nu ) / 5848 \) (-88v^11 - 3129v^9 - 38161v^7 - 192621v^5 - 372491v^3 - 174542v) / 5848

\(\nu\)	\(=\)	\( ( \beta_{8} + \beta_{7} - \beta_{3} + \beta_{2} ) / 2 \) (b8 + b7 - b3 + b2) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{8} + \beta_{7} - 4\beta_{6} - 4\beta_{5} - \beta_{3} - \beta_{2} - 4\beta _1 - 12 ) / 2 \) (-b8 + b7 - 4b6 - 4b5 - b3 - b2 - 4*b1 - 12) / 2
\(\nu^{3}\)	\(=\)	\( ( - 6 \beta_{11} + 6 \beta_{10} + 8 \beta_{9} - 15 \beta_{8} - 9 \beta_{7} - 10 \beta_{6} + \cdots - 9 \beta_{2} ) / 2 \) (-6b11 + 6b10 + 8b9 - 15b8 - 9b7 - 10b6 + 10b5 + 9b3 - 9*b2) / 2
\(\nu^{4}\)	\(=\)	\( ( 25\beta_{8} - 25\beta_{7} + 72\beta_{6} + 72\beta_{5} + 4\beta_{4} + 5\beta_{3} + 5\beta_{2} + 52\beta _1 + 132 ) / 2 \) (25b8 - 25b7 + 72b6 + 72b5 + 4b4 + 5b3 + 5b2 + 52b1 + 132) / 2
\(\nu^{5}\)	\(=\)	\( ( 134 \beta_{11} - 98 \beta_{10} - 220 \beta_{9} + 221 \beta_{8} + 123 \beta_{7} + 190 \beta_{6} + \cdots + 95 \beta_{2} ) / 2 \) (134b11 - 98b10 - 220b9 + 221b8 + 123b7 + 190b6 - 190b5 - 95b3 + 95*b2) / 2
\(\nu^{6}\)	\(=\)	\( ( - 447 \beta_{8} + 447 \beta_{7} - 1124 \beta_{6} - 1124 \beta_{5} - 120 \beta_{4} + 33 \beta_{3} + \cdots - 1752 ) / 2 \) (-447b8 + 447b7 - 1124b6 - 1124b5 - 120b4 + 33b3 + 33b2 - 732b1 - 1752) / 2
\(\nu^{7}\)	\(=\)	\( ( - 2402 \beta_{11} + 1418 \beta_{10} + 4368 \beta_{9} - 3287 \beta_{8} - 1869 \beta_{7} + \cdots - 1149 \beta_{2} ) / 2 \) (-2402b11 + 1418b10 + 4368b9 - 3287b8 - 1869b7 - 3318b6 + 3318b5 + 1149b3 - 1149*b2) / 2
\(\nu^{8}\)	\(=\)	\( ( 7381 \beta_{8} - 7381 \beta_{7} + 17232 \beta_{6} + 17232 \beta_{5} + 2620 \beta_{4} - 1527 \beta_{3} + \cdots + 25292 ) / 2 \) (7381b8 - 7381b7 + 17232b6 + 17232b5 + 2620b4 - 1527b3 - 1527b2 + 11036b1 + 25292) / 2
\(\nu^{9}\)	\(=\)	\( ( 40230 \beta_{11} - 20466 \beta_{10} - 77468 \beta_{9} + 49941 \beta_{8} + 29475 \beta_{7} + \cdots + 15327 \beta_{2} ) / 2 \) (40230b11 - 20466b10 - 77468b9 + 49941b8 + 29475b7 + 56222b6 - 56222b5 - 15327b3 + 15327*b2) / 2
\(\nu^{10}\)	\(=\)	\( ( - 119023 \beta_{8} + 119023 \beta_{7} - 266076 \beta_{6} - 266076 \beta_{5} - 49904 \beta_{4} + \cdots - 381672 ) / 2 \) (-119023b8 + 119023b7 - 266076b6 - 266076b5 - 49904b4 + 33289b3 + 33289b2 - 171908b1 - 381672) / 2
\(\nu^{11}\)	\(=\)	\( ( - 656874 \beta_{11} + 301906 \beta_{10} + 1308032 \beta_{9} - 772575 \beta_{8} + \cdots - 218549 \beta_{2} ) / 2 \) (-656874b11 + 301906b10 + 1308032b9 - 772575b8 - 470669b7 - 933790b6 + 933790b5 + 218549b3 - 218549*b2) / 2

\(n\)	\(1261\)	\(1667\)	\(1837\)
\(\chi(n)\)	\(-\beta_{9}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1135.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$3$	\( T^{12} \) T^12
$5$	\( T^{12} - 16 T^{9} + \cdots + 1024 \) T^12 - 16T^9 + 273T^8 - 240T^7 + 128T^6 - 672T^5 + 5712T^4 - 7552T^3 + 4608T^2 + 3072*T + 1024
$7$	\( (T^{4} + 1)^{3} \) (T^4 + 1)^3
$11$	\( T^{12} + 4 T^{9} + \cdots + 1201216 \) T^12 + 4T^9 + 405T^8 + 84T^7 + 8T^6 + 216T^5 + 44628T^4 + 12032T^3 + 1152T^2 - 52608*T + 1201216
$13$	\( (T^{6} + 6 T^{5} - 15 T^{4} + \cdots + 32)^{2} \) (T^6 + 6T^5 - 15T^4 - 56T^3 + 174T^2 - 144*T + 32)^2
$17$	\( (T^{6} + 33 T^{4} + \cdots + 4913)^{2} \) (T^6 + 33T^4 - 16T^3 + 561*T^2 + 4913)^2
$19$	\( T^{12} + 120 T^{10} + \cdots + 73984 \) T^12 + 120T^10 + 4104T^8 + 45184T^6 + 199824T^4 + 323712*T^2 + 73984
$23$	\( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 4)^{3} \) (T^4 - 4T^3 + 8T^2 + 8*T + 4)^3
$29$	\( T^{12} + \cdots + 300814336 \) T^12 + 12T^11 + 72T^10 - 56T^9 + 6036T^8 + 63936T^7 + 334208T^6 + 198144T^5 + 8308032T^4 + 83576320T^3 + 420268032T^2 + 502837248*T + 300814336
$31$	\( T^{12} - 12 T^{11} + \cdots + 21233664 \) T^12 - 12T^11 + 72T^10 - 264T^9 + 3924T^8 - 42912T^7 + 267264T^6 - 870912T^5 + 1553472T^4 - 1050624T^3 + 2654208T^2 - 10616832*T + 21233664
$37$	\( T^{12} + 12 T^{11} + \cdots + 246016 \) T^12 + 12T^11 + 72T^10 + 152T^9 + 21T^8 - 276T^7 + 6728T^6 + 9816T^5 - 2940T^4 - 98720T^3 + 415872T^2 - 452352*T + 246016
$41$	\( T^{12} + \cdots + 1073741824 \) T^12 - 64T^9 + 8976T^8 - 8448T^7 + 2048T^6 + 270336T^5 + 9191424T^4 + 2097152*T^3 + 1073741824
$43$	\( T^{12} + \cdots + 4507242496 \) T^12 + 378T^10 + 50481T^8 + 3031560T^6 + 85187088T^4 + 1057282560*T^2 + 4507242496
$47$	\( (T^{6} - 12 T^{5} + \cdots - 2176)^{2} \) (T^6 - 12T^5 - 78T^4 + 944T^3 - 192T^2 - 7296*T - 2176)^2
$53$	\( T^{12} + 366 T^{10} + \cdots + 40551424 \) T^12 + 366T^10 + 46593T^8 + 2435592T^6 + 45310416T^4 + 83519232*T^2 + 40551424
$59$	\( T^{12} + \cdots + 2102772736 \) T^12 + 468T^10 + 77604T^8 + 5444544T^6 + 150401664T^4 + 1143601152*T^2 + 2102772736
$61$	\( T^{12} + 12 T^{11} + \cdots + 4096 \) T^12 + 12T^11 + 72T^10 + 424T^9 + 15828T^8 + 200064T^7 + 1351040T^6 + 4527360T^5 + 7368000T^4 - 1456640T^3 + 165888T^2 + 36864*T + 4096
$67$	\( (T^{6} + 6 T^{5} + \cdots - 214784)^{2} \) (T^6 + 6T^5 - 327T^4 - 1872T^3 + 24984T^2 + 112896*T - 214784)^2
$71$	\( T^{12} + \cdots + 13184550976 \) T^12 - 36T^11 + 648T^10 - 4968T^9 + 16524T^8 - 23328T^7 + 2472768T^6 - 13292352T^5 + 33893424T^4 + 49581504T^3 + 1700611200T^2 - 6696535680*T + 13184550976
$73$	\( T^{12} + \cdots + 9491435776 \) T^12 + 36T^11 + 648T^10 + 4212T^9 + 8097T^8 + 20208T^7 + 4351104T^6 + 11274048T^5 + 14310432T^4 + 21719808T^3 + 3968335872T^2 + 8679309312*T + 9491435776
$79$	\( T^{12} + \cdots + 702038016 \) T^12 - 48T^11 + 1152T^10 - 13704T^9 + 77841T^8 + 104904T^7 - 808416T^6 - 665280T^5 + 54575424T^4 + 289778688T^3 + 767066112T^2 + 1037795328*T + 702038016
$83$	\( T^{12} + \cdots + 19606720576 \) T^12 + 798T^10 + 222909T^8 + 24902868T^6 + 897275892T^4 + 8923036128*T^2 + 19606720576
$89$	\( (T^{6} + 6 T^{5} + \cdots - 451996)^{2} \) (T^6 + 6T^5 - 339T^4 - 848T^3 + 23280T^2 + 25992*T - 451996)^2
$97$	\( T^{12} + \cdots + 1473331456 \) T^12 - 36T^11 + 648T^10 - 6692T^9 + 61041T^8 - 790560T^7 + 11297024T^6 - 113617920T^5 + 748235808T^4 - 3008800256T^3 + 6702041088T^2 - 4443945984*T + 1473331456
show more
show less