LMFDB - Newform orbit 1860.2.i.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1860,2,Mod(929,1860)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 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("1860.929"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1860.i (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$14.8521747760$
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} + 13x^{12} + 88x^{8} + 1053x^{4} + 6561 $ x^16 + 13x^12 + 88x^8 + 1053*x^4 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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_{10} q^{3} + \beta_{5} q^{5} + \beta_{8} q^{9} - \beta_{12} q^{13} - \beta_{6} q^{15} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{17} + ( - \beta_{11} + \beta_{9}) q^{19} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_{3}) q^{23}+ \cdots + (3 \beta_{7} - \beta_{4} - \beta_{2}) q^{95}+O(q^{100}) $ q - b10 * q^3 + b5 * q^5 + b8 * q^9 - b12 * q^13 - b6 * q^15 + (-b15 + b13 - b10 - b6 + b1) * q^17 + (-b11 + b9) * q^19 + (-b15 + b14 + b10 + b6 + b3) * q^23 - 5 * q^25 + (-b13 - b12) * q^27 + (-b11 - b8) * q^31 + (-b15 - b13 - b10 - b6 - b1) * q^37 + (b9 + b5 + b2) * q^39 + (b7 - b4 - b2) * q^41 + (2b15 + b14 + b13 + b12 + 2b10 - b3 + b1) * q^43 + (b4 + 1) * q^45 + 7 * q^49 + (b11 + b8 - b7 + 2b5 + b4 - 2) q^51 + (-b15 + b14 - b12 - b6 + 2b3 + b1) q^53 + (2b14 + b13 - b12 + b3 + b1) q^57 + (-b7 - b4 - b2) * q^59 + (2b15 - 2b13 - b12 - 2b10) q^65 + (b11 - b9 - b8 - b7 - 2b5 + b2 + 3) q^69 + (-b11 - b9 + 4b8 + 2b5) * q^71 + (b15 + b14 - b12 - 2b10 - b6 - b1) q^73 + 5b10 q^75 + (b7 + b2 - 3) * q^81 + (2b15 - b14 - b13 - b12 + 2b10 - 2b6 + b3 - b1) q^83 + (b15 + b14 - b12 + 4b10 - b6 - 2b3 + b1) * q^85 + (b14 + 2b13 + b12 - b3 + 2b1) * q^93 + (3b7 - b4 - b2) q^95
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 80 q^{25} - 4 q^{39} + 20 q^{45} + 112 q^{49} - 28 q^{51} + 44 q^{69} - 52 q^{81}+O(q^{100}) $ 16 * q - 80 * q^25 - 4 * q^39 + 20 * q^45 + 112 * q^49 - 28 * q^51 + 44 * q^69 - 52 * q^81

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 13x^{12} + 88x^{8} + 1053x^{4} + 6561 $ x^16 + 13*x^12 + 88*x^8 + 1053*x^4 + 6561 :

$\beta_{1}$	$=$	$ ( \nu^{15} - 176\nu^{11} + 1276\nu^{7} + 16497\nu^{3} + 64152\nu ) / 32076 $ (v^15 - 176v^11 + 1276v^7 + 16497v^3 + 64152v) / 32076
$\beta_{2}$	$=$	$ ( -5\nu^{12} - 308\nu^{8} - 440\nu^{4} - 5265 ) / 3564 $ (-5v^12 - 308v^8 - 440*v^4 - 5265) / 3564
$\beta_{3}$	$=$	$ ( \nu^{15} - 176\nu^{11} + 1276\nu^{7} + 16497\nu^{3} - 32076\nu ) / 32076 $ (v^15 - 176v^11 + 1276v^7 + 16497v^3 - 32076v) / 32076
$\beta_{4}$	$=$	$ ( 13\nu^{12} + 88\nu^{8} - 638\nu^{4} + 4779 ) / 1782 $ (13v^12 + 88v^8 - 638*v^4 + 4779) / 1782
$\beta_{5}$	$=$	$ ( 13\nu^{14} + 88\nu^{10} - 638\nu^{6} + 6561\nu^{2} ) / 16038 $ (13v^14 + 88v^10 - 638v^6 + 6561v^2) / 16038
$\beta_{6}$	$=$	$ ( 13\nu^{13} + 88\nu^{9} - 638\nu^{5} + 6561\nu ) / 5346 $ (13v^13 + 88v^9 - 638v^5 + 6561v) / 5346
$\beta_{7}$	$=$	$ ( -13\nu^{12} - 88\nu^{8} - 1144\nu^{4} - 10125 ) / 1188 $ (-13v^12 - 88v^8 - 1144*v^4 - 10125) / 1188
$\beta_{8}$	$=$	$ ( -\nu^{14} - 13\nu^{10} - 88\nu^{6} - 1053\nu^{2} ) / 729 $ (-v^14 - 13v^10 - 88v^6 - 1053*v^2) / 729
$\beta_{9}$	$=$	$ ( 5\nu^{14} - 16\nu^{10} + 116\nu^{6} + 6885\nu^{2} ) / 2916 $ (5v^14 - 16v^10 + 116v^6 + 6885v^2) / 2916
$\beta_{10}$	$=$	$ ( -\nu^{15} - 13\nu^{11} - 88\nu^{7} - 1053\nu^{3} ) / 2187 $ (-v^15 - 13v^11 - 88v^7 - 1053*v^3) / 2187
$\beta_{11}$	$=$	$ ( -\nu^{14} - 4\nu^{10} - 52\nu^{6} - 261\nu^{2} ) / 324 $ (-v^14 - 4v^10 - 52v^6 - 261*v^2) / 324
$\beta_{12}$	$=$	$ ( 9\nu^{15} - 5\nu^{13} - 308\nu^{9} - 440\nu^{5} + 9873\nu^{3} - 5265\nu ) / 10692 $ (9v^15 - 5v^13 - 308v^9 - 440v^5 + 9873v^3 - 5265v) / 10692
$\beta_{13}$	$=$	$ ( -3\nu^{15} - 13\nu^{13} - 88\nu^{9} - 1144\nu^{5} - 3291\nu^{3} - 13689\nu ) / 3564 $ (-3v^15 - 13v^13 - 88v^9 - 1144v^5 - 3291v^3 - 13689v) / 3564
$\beta_{14}$	$=$	$ ( 13\nu^{15} - 90\nu^{13} + 88\nu^{11} - 198\nu^{9} - 638\nu^{7} - 2574\nu^{5} + 6561\nu^{3} - 57348\nu ) / 16038 $ (13v^15 - 90v^13 + 88v^11 - 198v^9 - 638v^7 - 2574v^5 + 6561v^3 - 57348v) / 16038
$\beta_{15}$	$=$	$ ( -125\nu^{15} - 572\nu^{11} - 3872\nu^{7} - 38961\nu^{3} ) / 96228 $ (-125v^15 - 572v^11 - 3872v^7 - 38961v^3) / 96228

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

$\nu$	$=$	$ ( -\beta_{3} + \beta_1 ) / 3 $ (-b3 + b1) / 3
$\nu^{2}$	$=$	$ ( \beta_{11} + \beta_{9} - \beta_{8} ) / 3 $ (b11 + b9 - b8) / 3
$\nu^{3}$	$=$	$ ( 3\beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - 3\beta_{10} + \beta_{6} + \beta_{3} ) / 3 $ (3b15 + b14 - b13 + b12 - 3b10 + b6 + b3) / 3
$\nu^{4}$	$=$	$ ( -2\beta_{7} - 3\beta_{4} - 9 ) / 3 $ (-2b7 - 3b4 - 9) / 3
$\nu^{5}$	$=$	$ ( -2\beta_{14} - 4\beta_{13} - 2\beta_{12} - 11\beta_{6} + 2\beta_{3} - 4\beta_1 ) / 3 $ (-2b14 - 4b13 - 2b12 - 11b6 + 2b3 - 4b1) / 3
$\nu^{6}$	$=$	$ -2\beta_{11} - 2\beta_{8} - 11\beta_{5} $ -2b11 - 2b8 - 11*b5
$\nu^{7}$	$=$	$ ( -18\beta_{15} - 13\beta_{14} + 13\beta_{13} - 13\beta_{12} - 18\beta_{10} - 13\beta_{6} + 13\beta_{3} + 13\beta_1 ) / 3 $ (-18b15 - 13b14 + 13b13 - 13b12 - 18b10 - 13b6 + 13b3 + 13b1) / 3
$\nu^{8}$	$=$	$ ( 5\beta_{7} - 39\beta_{2} - 15 ) / 3 $ (5b7 - 39b2 - 15) / 3
$\nu^{9}$	$=$	$ ( 44\beta_{14} - 29\beta_{13} - 73\beta_{12} + 44\beta_{6} + 44\beta_{3} ) / 3 $ (44b14 - 29b13 - 73b12 + 44b6 + 44*b3) / 3
$\nu^{10}$	$=$	$ ( 44\beta_{11} - 88\beta_{9} - 131\beta_{8} + 132\beta_{5} ) / 3 $ (44b11 - 88b9 - 131b8 + 132b5) / 3
$\nu^{11}$	$=$	$ ( 132\beta_{15} - 393\beta_{10} - 176\beta_{3} - 88\beta_1 ) / 3 $ (132b15 - 393b10 - 176b3 - 88b1) / 3
$\nu^{12}$	$=$	$ -44\beta_{7} + 88\beta_{4} + 88\beta_{2} - 481 $ -44b7 + 88b4 + 88*b2 - 481
$\nu^{13}$	$=$	$ ( -396\beta_{14} + 396\beta_{12} + 396\beta_{6} + 305\beta_{3} - 701\beta_1 ) / 3 $ (-396b14 + 396b12 + 396b6 + 305b3 - 701*b1) / 3
$\nu^{14}$	$=$	$ ( -1097\beta_{11} + 91\beta_{9} + 1097\beta_{8} + 1188\beta_{5} ) / 3 $ (-1097b11 + 91b9 + 1097b8 + 1188b5) / 3
$\nu^{15}$	$=$	$ ( -3291\beta_{15} + 91\beta_{14} - 91\beta_{13} + 91\beta_{12} + 3291\beta_{10} + 91\beta_{6} + 91\beta_{3} ) / 3 $ (-3291b15 + 91b14 - 91b13 + 91b12 + 3291b10 + 91b6 + 91*b3) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times$.

$n$	$931$	$1117$	$1241$	$1801$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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} $

929.1

1.63710	−	0.565603i
1.63710	+	0.565603i
1.30838	−	1.13497i
1.30838	+	1.13497i
1.13497	−	1.30838i
1.13497	+	1.30838i
0.565603	−	1.63710i
0.565603	+	1.63710i
−0.565603	−	1.63710i
−0.565603	+	1.63710i
−1.13497	−	1.30838i
−1.13497	+	1.30838i
−1.30838	−	1.13497i
−1.30838	+	1.13497i
−1.63710	−	0.565603i
−1.63710	+	0.565603i

−1.63710

−

0.565603i

2.23607i

2.36019

1.85190i

929.2

−1.63710

0.565603i

−

2.23607i

2.36019

−

1.85190i

929.3

−1.30838

−

1.13497i

−

2.23607i

0.423695

2.96993i

929.4

−1.30838

1.13497i

2.23607i

0.423695

−

2.96993i

929.5

−1.13497

−

1.30838i

−

2.23607i

−0.423695

2.96993i

929.6

−1.13497

1.30838i

2.23607i

−0.423695

−

2.96993i

929.7

−0.565603

−

1.63710i

2.23607i

−2.36019

1.85190i

929.8

−0.565603

1.63710i

−

2.23607i

−2.36019

−

1.85190i

929.9

0.565603

−

1.63710i

−

2.23607i

−2.36019

−

1.85190i

929.10

0.565603

1.63710i

2.23607i

−2.36019

1.85190i

929.11

1.13497

−

1.30838i

2.23607i

−0.423695

−

2.96993i

929.12

1.13497

1.30838i

−

2.23607i

−0.423695

2.96993i

929.13

1.30838

−

1.13497i

2.23607i

0.423695

−

2.96993i

929.14

1.30838

1.13497i

−

2.23607i

0.423695

2.96993i

929.15

1.63710

−

0.565603i

−

2.23607i

2.36019

−

1.85190i

929.16

1.63710

0.565603i

2.23607i

2.36019

1.85190i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
155.c	odd	2	1	CM by $\Q(\sqrt{-155}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner
31.b	odd	2	1	inner
93.c	even	2	1	inner
465.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1860.2.i.a	✓	16
3.b	odd	2	1	inner	1860.2.i.a	✓	16
5.b	even	2	1	inner	1860.2.i.a	✓	16
15.d	odd	2	1	inner	1860.2.i.a	✓	16
31.b	odd	2	1	inner	1860.2.i.a	✓	16
93.c	even	2	1	inner	1860.2.i.a	✓	16
155.c	odd	2	1	CM	1860.2.i.a	✓	16
465.g	even	2	1	inner	1860.2.i.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1860.2.i.a	✓	16	1.a	even	1	1	trivial
1860.2.i.a	✓	16	3.b	odd	2	1	inner
1860.2.i.a	✓	16	5.b	even	2	1	inner
1860.2.i.a	✓	16	15.d	odd	2	1	inner
1860.2.i.a	✓	16	31.b	odd	2	1	inner
1860.2.i.a	✓	16	93.c	even	2	1	inner
1860.2.i.a	✓	16	155.c	odd	2	1	CM
1860.2.i.a	✓	16	465.g	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} $ T7 acting on $S_{2}^{\mathrm{new}}(1860, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 13 T^{12} + \cdots + 6561 $ T^16 + 13T^12 + 88T^8 + 1053*T^4 + 6561
$5$	$ (T^{2} + 5)^{8} $ (T^2 + 5)^8
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ (T^{4} - 52 T^{2} + 180)^{4} $ (T^4 - 52*T^2 + 180)^4
$17$	$ (T^{8} + 136 T^{6} + \cdots + 66564)^{2} $ (T^8 + 136T^6 + 5233T^4 + 41412*T^2 + 66564)^2
$19$	$ (T^{4} - 83 T^{2} + 676)^{4} $ (T^4 - 83*T^2 + 676)^4
$23$	$ (T^{4} + 92 T^{2} + 1620)^{4} $ (T^4 + 92*T^2 + 1620)^4
$29$	$ T^{16} $ T^16
$31$	$ (T^{2} - 31)^{8} $ (T^2 - 31)^8
$37$	$ (T^{8} - 296 T^{6} + \cdots + 1790244)^{2} $ (T^8 - 296T^6 + 24673T^4 - 409812*T^2 + 1790244)^2
$41$	$ (T^{4} + 91 T^{2} + 1024)^{4} $ (T^4 + 91*T^2 + 1024)^4
$43$	$ (T^{8} - 344 T^{6} + \cdots + 3617604)^{2} $ (T^8 - 344T^6 + 37033T^4 - 1281228*T^2 + 3617604)^2
$47$	$ T^{16} $ T^16
$53$	$ (T^{8} + 424 T^{6} + \cdots + 70257924)^{2} $ (T^8 + 424T^6 + 61753T^4 + 3563508*T^2 + 70257924)^2
$59$	$ (T^{4} + 199 T^{2} + 484)^{4} $ (T^4 + 199*T^2 + 484)^4
$61$	$ T^{16} $ T^16
$67$	$ T^{16} $ T^16
$71$	$ (T^{4} + 421 T^{2} + 43264)^{4} $ (T^4 + 421*T^2 + 43264)^4
$73$	$ (T^{8} - 584 T^{6} + \cdots + 34363044)^{2} $ (T^8 - 584T^6 + 107113T^4 - 6379908*T^2 + 34363044)^2
$79$	$ T^{16} $ T^16
$83$	$ (T^{8} + 664 T^{6} + \cdots + 9847044)^{2} $ (T^8 + 664T^6 + 127753T^4 + 5819628*T^2 + 9847044)^2
$89$	$ T^{16} $ T^16
$97$	$ T^{16} $ T^16
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1860.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{10} q^{3} + \beta_{5} q^{5} + \beta_{8} q^{9} - \beta_{12} q^{13} - \beta_{6} q^{15} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{17} + ( - \beta_{11} + \beta_{9}) q^{19} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_{3}) q^{23}+ \cdots + (3 \beta_{7} - \beta_{4} - \beta_{2}) q^{95}+O(q^{100}) \) q - b10 * q^3 + b5 * q^5 + b8 * q^9 - b12 * q^13 - b6 * q^15 + (-b15 + b13 - b10 - b6 + b1) * q^17 + (-b11 + b9) * q^19 + (-b15 + b14 + b10 + b6 + b3) * q^23 - 5 * q^25 + (-b13 - b12) * q^27 + (-b11 - b8) * q^31 + (-b15 - b13 - b10 - b6 - b1) * q^37 + (b9 + b5 + b2) * q^39 + (b7 - b4 - b2) * q^41 + (2b15 + b14 + b13 + b12 + 2b10 - b3 + b1) * q^43 + (b4 + 1) * q^45 + 7 * q^49 + (b11 + b8 - b7 + 2b5 + b4 - 2) q^51 + (-b15 + b14 - b12 - b6 + 2b3 + b1) q^53 + (2b14 + b13 - b12 + b3 + b1) q^57 + (-b7 - b4 - b2) * q^59 + (2b15 - 2b13 - b12 - 2b10) q^65 + (b11 - b9 - b8 - b7 - 2b5 + b2 + 3) q^69 + (-b11 - b9 + 4b8 + 2b5) * q^71 + (b15 + b14 - b12 - 2b10 - b6 - b1) q^73 + 5b10 q^75 + (b7 + b2 - 3) * q^81 + (2b15 - b14 - b13 - b12 + 2b10 - 2b6 + b3 - b1) q^83 + (b15 + b14 - b12 + 4b10 - b6 - 2b3 + b1) * q^85 + (b14 + 2b13 + b12 - b3 + 2b1) * q^93 + (3b7 - b4 - b2) q^95
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 80 q^{25} - 4 q^{39} + 20 q^{45} + 112 q^{49} - 28 q^{51} + 44 q^{69} - 52 q^{81}+O(q^{100}) \) 16 * q - 80 * q^25 - 4 * q^39 + 20 * q^45 + 112 * q^49 - 28 * q^51 + 44 * q^69 - 52 * q^81

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1860.2.i.a

Level	$1860$
Weight	$2$
Character orbit	1860.i
Analytic conductor	$14.852$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-155
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(14.8521747760\)
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:	\( x^{16} + 13x^{12} + 88x^{8} + 1053x^{4} + 6561 \) x^16 + 13x^12 + 88x^8 + 1053*x^4 + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{15} - 176\nu^{11} + 1276\nu^{7} + 16497\nu^{3} + 64152\nu ) / 32076 \) (v^15 - 176v^11 + 1276v^7 + 16497v^3 + 64152v) / 32076
\(\beta_{2}\)	\(=\)	\( ( -5\nu^{12} - 308\nu^{8} - 440\nu^{4} - 5265 ) / 3564 \) (-5v^12 - 308v^8 - 440*v^4 - 5265) / 3564
\(\beta_{3}\)	\(=\)	\( ( \nu^{15} - 176\nu^{11} + 1276\nu^{7} + 16497\nu^{3} - 32076\nu ) / 32076 \) (v^15 - 176v^11 + 1276v^7 + 16497v^3 - 32076v) / 32076
\(\beta_{4}\)	\(=\)	\( ( 13\nu^{12} + 88\nu^{8} - 638\nu^{4} + 4779 ) / 1782 \) (13v^12 + 88v^8 - 638*v^4 + 4779) / 1782
\(\beta_{5}\)	\(=\)	\( ( 13\nu^{14} + 88\nu^{10} - 638\nu^{6} + 6561\nu^{2} ) / 16038 \) (13v^14 + 88v^10 - 638v^6 + 6561v^2) / 16038
\(\beta_{6}\)	\(=\)	\( ( 13\nu^{13} + 88\nu^{9} - 638\nu^{5} + 6561\nu ) / 5346 \) (13v^13 + 88v^9 - 638v^5 + 6561v) / 5346
\(\beta_{7}\)	\(=\)	\( ( -13\nu^{12} - 88\nu^{8} - 1144\nu^{4} - 10125 ) / 1188 \) (-13v^12 - 88v^8 - 1144*v^4 - 10125) / 1188
\(\beta_{8}\)	\(=\)	\( ( -\nu^{14} - 13\nu^{10} - 88\nu^{6} - 1053\nu^{2} ) / 729 \) (-v^14 - 13v^10 - 88v^6 - 1053*v^2) / 729
\(\beta_{9}\)	\(=\)	\( ( 5\nu^{14} - 16\nu^{10} + 116\nu^{6} + 6885\nu^{2} ) / 2916 \) (5v^14 - 16v^10 + 116v^6 + 6885v^2) / 2916
\(\beta_{10}\)	\(=\)	\( ( -\nu^{15} - 13\nu^{11} - 88\nu^{7} - 1053\nu^{3} ) / 2187 \) (-v^15 - 13v^11 - 88v^7 - 1053*v^3) / 2187
\(\beta_{11}\)	\(=\)	\( ( -\nu^{14} - 4\nu^{10} - 52\nu^{6} - 261\nu^{2} ) / 324 \) (-v^14 - 4v^10 - 52v^6 - 261*v^2) / 324
\(\beta_{12}\)	\(=\)	\( ( 9\nu^{15} - 5\nu^{13} - 308\nu^{9} - 440\nu^{5} + 9873\nu^{3} - 5265\nu ) / 10692 \) (9v^15 - 5v^13 - 308v^9 - 440v^5 + 9873v^3 - 5265v) / 10692
\(\beta_{13}\)	\(=\)	\( ( -3\nu^{15} - 13\nu^{13} - 88\nu^{9} - 1144\nu^{5} - 3291\nu^{3} - 13689\nu ) / 3564 \) (-3v^15 - 13v^13 - 88v^9 - 1144v^5 - 3291v^3 - 13689v) / 3564
\(\beta_{14}\)	\(=\)	\( ( 13\nu^{15} - 90\nu^{13} + 88\nu^{11} - 198\nu^{9} - 638\nu^{7} - 2574\nu^{5} + 6561\nu^{3} - 57348\nu ) / 16038 \) (13v^15 - 90v^13 + 88v^11 - 198v^9 - 638v^7 - 2574v^5 + 6561v^3 - 57348v) / 16038
\(\beta_{15}\)	\(=\)	\( ( -125\nu^{15} - 572\nu^{11} - 3872\nu^{7} - 38961\nu^{3} ) / 96228 \) (-125v^15 - 572v^11 - 3872v^7 - 38961v^3) / 96228

\(\nu\)	\(=\)	\( ( -\beta_{3} + \beta_1 ) / 3 \) (-b3 + b1) / 3
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} + \beta_{9} - \beta_{8} ) / 3 \) (b11 + b9 - b8) / 3
\(\nu^{3}\)	\(=\)	\( ( 3\beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - 3\beta_{10} + \beta_{6} + \beta_{3} ) / 3 \) (3b15 + b14 - b13 + b12 - 3b10 + b6 + b3) / 3
\(\nu^{4}\)	\(=\)	\( ( -2\beta_{7} - 3\beta_{4} - 9 ) / 3 \) (-2b7 - 3b4 - 9) / 3
\(\nu^{5}\)	\(=\)	\( ( -2\beta_{14} - 4\beta_{13} - 2\beta_{12} - 11\beta_{6} + 2\beta_{3} - 4\beta_1 ) / 3 \) (-2b14 - 4b13 - 2b12 - 11b6 + 2b3 - 4b1) / 3
\(\nu^{6}\)	\(=\)	\( -2\beta_{11} - 2\beta_{8} - 11\beta_{5} \) -2b11 - 2b8 - 11*b5
\(\nu^{7}\)	\(=\)	\( ( -18\beta_{15} - 13\beta_{14} + 13\beta_{13} - 13\beta_{12} - 18\beta_{10} - 13\beta_{6} + 13\beta_{3} + 13\beta_1 ) / 3 \) (-18b15 - 13b14 + 13b13 - 13b12 - 18b10 - 13b6 + 13b3 + 13b1) / 3
\(\nu^{8}\)	\(=\)	\( ( 5\beta_{7} - 39\beta_{2} - 15 ) / 3 \) (5b7 - 39b2 - 15) / 3
\(\nu^{9}\)	\(=\)	\( ( 44\beta_{14} - 29\beta_{13} - 73\beta_{12} + 44\beta_{6} + 44\beta_{3} ) / 3 \) (44b14 - 29b13 - 73b12 + 44b6 + 44*b3) / 3
\(\nu^{10}\)	\(=\)	\( ( 44\beta_{11} - 88\beta_{9} - 131\beta_{8} + 132\beta_{5} ) / 3 \) (44b11 - 88b9 - 131b8 + 132b5) / 3
\(\nu^{11}\)	\(=\)	\( ( 132\beta_{15} - 393\beta_{10} - 176\beta_{3} - 88\beta_1 ) / 3 \) (132b15 - 393b10 - 176b3 - 88b1) / 3
\(\nu^{12}\)	\(=\)	\( -44\beta_{7} + 88\beta_{4} + 88\beta_{2} - 481 \) -44b7 + 88b4 + 88*b2 - 481
\(\nu^{13}\)	\(=\)	\( ( -396\beta_{14} + 396\beta_{12} + 396\beta_{6} + 305\beta_{3} - 701\beta_1 ) / 3 \) (-396b14 + 396b12 + 396b6 + 305b3 - 701*b1) / 3
\(\nu^{14}\)	\(=\)	\( ( -1097\beta_{11} + 91\beta_{9} + 1097\beta_{8} + 1188\beta_{5} ) / 3 \) (-1097b11 + 91b9 + 1097b8 + 1188b5) / 3
\(\nu^{15}\)	\(=\)	\( ( -3291\beta_{15} + 91\beta_{14} - 91\beta_{13} + 91\beta_{12} + 3291\beta_{10} + 91\beta_{6} + 91\beta_{3} ) / 3 \) (-3291b15 + 91b14 - 91b13 + 91b12 + 3291b10 + 91b6 + 91*b3) / 3

\(n\)	\(931\)	\(1117\)	\(1241\)	\(1801\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 929.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 13 T^{12} + \cdots + 6561 \) T^16 + 13T^12 + 88T^8 + 1053*T^4 + 6561
$5$	\( (T^{2} + 5)^{8} \) (T^2 + 5)^8
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( (T^{4} - 52 T^{2} + 180)^{4} \) (T^4 - 52*T^2 + 180)^4
$17$	\( (T^{8} + 136 T^{6} + \cdots + 66564)^{2} \) (T^8 + 136T^6 + 5233T^4 + 41412*T^2 + 66564)^2
$19$	\( (T^{4} - 83 T^{2} + 676)^{4} \) (T^4 - 83*T^2 + 676)^4
$23$	\( (T^{4} + 92 T^{2} + 1620)^{4} \) (T^4 + 92*T^2 + 1620)^4
$29$	\( T^{16} \) T^16
$31$	\( (T^{2} - 31)^{8} \) (T^2 - 31)^8
$37$	\( (T^{8} - 296 T^{6} + \cdots + 1790244)^{2} \) (T^8 - 296T^6 + 24673T^4 - 409812*T^2 + 1790244)^2
$41$	\( (T^{4} + 91 T^{2} + 1024)^{4} \) (T^4 + 91*T^2 + 1024)^4
$43$	\( (T^{8} - 344 T^{6} + \cdots + 3617604)^{2} \) (T^8 - 344T^6 + 37033T^4 - 1281228*T^2 + 3617604)^2
$47$	\( T^{16} \) T^16
$53$	\( (T^{8} + 424 T^{6} + \cdots + 70257924)^{2} \) (T^8 + 424T^6 + 61753T^4 + 3563508*T^2 + 70257924)^2
$59$	\( (T^{4} + 199 T^{2} + 484)^{4} \) (T^4 + 199*T^2 + 484)^4
$61$	\( T^{16} \) T^16
$67$	\( T^{16} \) T^16
$71$	\( (T^{4} + 421 T^{2} + 43264)^{4} \) (T^4 + 421*T^2 + 43264)^4
$73$	\( (T^{8} - 584 T^{6} + \cdots + 34363044)^{2} \) (T^8 - 584T^6 + 107113T^4 - 6379908*T^2 + 34363044)^2
$79$	\( T^{16} \) T^16
$83$	\( (T^{8} + 664 T^{6} + \cdots + 9847044)^{2} \) (T^8 + 664T^6 + 127753T^4 + 5819628*T^2 + 9847044)^2
$89$	\( T^{16} \) T^16
$97$	\( T^{16} \) T^16
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