LMFDB - Newform orbit 825.2.x.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(131,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 4, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("825.131");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.x (of order $10$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.228765625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 $ x^8 - x^7 - 2x^6 + 5x^5 + x^4 + 15x^3 - 18x^2 - 27*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{5} q^{3} + 2 \beta_{3} q^{4} + (\beta_{6} - 2) q^{5} + (3 \beta_{3} - \beta_1) q^{9}+O(q^{10}) $ q + b5 * q^3 + 2b3 q^4 + (b6 - 2) * q^5 + (3b3 - b1) q^9	$ q + \beta_{5} q^{3} + 2 \beta_{3} q^{4} + (\beta_{6} - 2) q^{5} + (3 \beta_{3} - \beta_1) q^{9} + (2 \beta_{5} + \beta_{4}) q^{11} + ( - 2 \beta_{7} - 2 \beta_{2}) q^{12} + ( - 2 \beta_{5} - 3 \beta_{4}) q^{15} + ( - 4 \beta_{7} + 4 \beta_{4} + \cdots - 4) q^{16}+ \cdots + ( - 8 \beta_{7} - 5 \beta_{2}) q^{99}+O(q^{100}) $ q + b5 * q^3 + 2b3 q^4 + (b6 - 2) * q^5 + (3b3 - b1) q^9 + (2b5 + b4) q^11 + (-2b7 - 2b2) * q^12 + (-2b5 - 3b4) * q^15 + (-4b7 + 4b4 + 4b3 - 4) q^16 + (-2b3 - 2b1) * q^20 + (9b7 - b6 + b5 - 4b4 - 4b3 - b1 + 5) q^23 + (-3b6 + 1) q^25 + (-5b7 - 2b2) * q^27 + (-b7 - 3b6 + 3b5 + 4b4 + 3b3 + 3b2 - 6b1 - 1) * q^31 + (6b3 - b1) q^33 + (-4b7 - 2b6 + 2b5 + 6b4 + 6b3 + 2b2 - 2b1 - 4) q^36 + (7b7 - 7b4 - 7b3 + 7) q^37 + (-2b7 - 4b2) * q^44 + (-6b3 - b1) q^45 + (2b3 - 4b1) * q^47 + (-4b6 + 4) q^48 + 7 * q^49 + (b7 - 4b5 + b4 - 4b2) * q^53 + (-3b5 - 7b4) * q^55 + (7b7 + b6 - b2 - 8) q^59 + (-2b7 + 4b2) * q^60 + 8b4 q^64 + (-3b6 - 3b5 + 5b4 - 5) q^67 + (-7b7 + 9b6 - 4b5 + 3b4 + 3b3 - 4b2 + 4b1 - 9) q^69 + (4b7 + 5b5 + 4b4 + 5b2) * q^71 + (b5 + 9b4) q^75 + (8b7 - 4b6 + 4b5 - 4b4 - 4b3 + 4b2 - 4b1 + 8) q^80 + (-b7 - 5b6 + 5b5 + 6b4 + 6b3 + 5b2 - 5b1 - 1) * q^81 + (9b7 + 5b6 - 5b5 - 7b4 - 7b3 + 5b1 + 2) * q^89 + (-2b6 + 2b5 - 8b4 - 8) q^92 + (-5b7 - b6 + 4b2 + 10) * q^93 + (7b7 + 3b5 - 7b4 - 3b2) * q^97 + (-8b7 - 5b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} + 4 q^{4} - 12 q^{5} + 5 q^{9}+O(q^{10}) $ 8 * q - q^3 + 4 * q^4 - 12 * q^5 + 5 * q^9	$ 8 q - q^{3} + 4 q^{4} - 12 q^{5} + 5 q^{9} + 2 q^{12} - 4 q^{15} - 8 q^{16} - 6 q^{20} - 4 q^{25} + 8 q^{27} - 10 q^{31} + 11 q^{33} - 10 q^{36} + 14 q^{37} - 13 q^{45} + 16 q^{48} + 56 q^{49} - 11 q^{55} - 75 q^{59} + 8 q^{60} + 16 q^{64} - 39 q^{67} - 6 q^{69} + 17 q^{75} + 12 q^{80} - 7 q^{81} - 90 q^{92} + 90 q^{93} - 34 q^{97} + 11 q^{99}+O(q^{100}) $ 8 * q - q^3 + 4 * q^4 - 12 * q^5 + 5 * q^9 + 2 * q^12 - 4 * q^15 - 8 * q^16 - 6 * q^20 - 4 * q^25 + 8 * q^27 - 10 * q^31 + 11 * q^33 - 10 * q^36 + 14 * q^37 - 13 * q^45 + 16 * q^48 + 56 * q^49 - 11 * q^55 - 75 * q^59 + 8 * q^60 + 16 * q^64 - 39 * q^67 - 6 * q^69 + 17 * q^75 + 12 * q^80 - 7 * q^81 - 90 * q^92 + 90 * q^93 - 34 * q^97 + 11 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 $ x^8 - x^7 - 2*x^6 + 5*x^5 + x^4 + 15*x^3 - 18*x^2 - 27*x + 81 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{7} + \nu^{6} + 2\nu^{5} - 5\nu^{4} - \nu^{3} - 15\nu^{2} + 18\nu + 27 ) / 27 $ (-v^7 + v^6 + 2v^5 - 5v^4 - v^3 - 15v^2 + 18v + 27) / 27
$\beta_{3}$	$=$	$ ( \nu^{6} + 16\nu ) / 3 $ (v^6 + 16*v) / 3
$\beta_{4}$	$=$	$ ( -2\nu^{7} - 4\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 36\nu^{2} - 54\nu + 162 ) / 27 $ (-2v^7 - 4v^6 + 10v^5 + 2v^4 - 5v^3 - 36v^2 - 54*v + 162) / 27
$\beta_{5}$	$=$	$ ( -\nu^{7} - 2\nu^{6} + 5\nu^{5} + \nu^{4} + 2\nu^{3} - 18\nu^{2} - 27\nu + 81 ) / 9 $ (-v^7 - 2v^6 + 5v^5 + v^4 + 2v^3 - 18v^2 - 27*v + 81) / 9
$\beta_{6}$	$=$	$ \nu^{5} + 16 $ v^5 + 16
$\beta_{7}$	$=$	$ ( -5\nu^{7} + 5\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 75\nu^{2} + 90\nu + 135 ) / 27 $ (-5v^7 + 5v^6 + 10v^5 + 2v^4 - 5v^3 - 75v^2 + 90*v + 135) / 27

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 2 $ 2b7 + b6 - b5 - 3b4 - 3*b3 - b2 + b1 + 2
$\nu^{3}$	$=$	$ 2\beta_{5} - 3\beta_{4} $ 2b5 - 3b4
$\nu^{4}$	$=$	$ \beta_{7} - 5\beta_{2} $ b7 - 5*b2
$\nu^{5}$	$=$	$ \beta_{6} - 16 $ b6 - 16
$\nu^{6}$	$=$	$ 3\beta_{3} - 16\beta_1 $ 3b3 - 16b1
$\nu^{7}$	$=$	$ -35\beta_{7} - 13\beta_{6} + 13\beta_{5} + 48\beta_{4} + 48\beta_{3} + 13\beta_{2} - 13\beta _1 - 35 $ -35b7 - 13b6 + 13b5 + 48b4 + 48b3 + 13b2 - 13*b1 - 35

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$-1$	$-1$	$-\beta_{3}$

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} $

131.1

1.37924	+	1.04771i
−0.570223	−	1.63550i
1.37924	−	1.04771i
−0.570223	+	1.63550i
−1.73166	−	0.0369185i
1.42264	+	0.987975i
−1.73166	+	0.0369185i
1.42264	−	0.987975i

−1.42264

0.987975i

1.61803

−

1.17557i

−1.50000

−

1.65831i

1.04781

−

2.81107i

131.2

1.73166

−

0.0369185i

1.61803

−

1.17557i

−1.50000

1.65831i

2.99727

−

0.127860i

296.1

−1.42264

−

0.987975i

1.61803

1.17557i

−1.50000

1.65831i

1.04781

2.81107i

296.2

1.73166

0.0369185i

1.61803

1.17557i

−1.50000

−

1.65831i

2.99727

0.127860i

461.1

−1.37924

−

1.04771i

−0.618034

1.90211i

−1.50000

−

1.65831i

0.804606

2.89009i

461.2

0.570223

1.63550i

−0.618034

1.90211i

−1.50000

1.65831i

−2.34969

1.86519i

791.1

−1.37924

1.04771i

−0.618034

−

1.90211i

−1.50000

1.65831i

0.804606

−

2.89009i

791.2

0.570223

−

1.63550i

−0.618034

−

1.90211i

−1.50000

−

1.65831i

−2.34969

−

1.86519i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
75.j	odd	10	1	inner
825.x	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.x.a	✓	8
3.b	odd	2	1		825.2.x.b	yes	8
11.b	odd	2	1	CM	825.2.x.a	✓	8
25.d	even	5	1		825.2.x.b	yes	8
33.d	even	2	1		825.2.x.b	yes	8
75.j	odd	10	1	inner	825.2.x.a	✓	8
275.v	odd	10	1		825.2.x.b	yes	8
825.x	even	10	1	inner	825.2.x.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.x.a	✓	8	1.a	even	1	1	trivial
825.2.x.a	✓	8	11.b	odd	2	1	CM
825.2.x.a	✓	8	75.j	odd	10	1	inner
825.2.x.a	✓	8	825.x	even	10	1	inner
825.2.x.b	yes	8	3.b	odd	2	1
825.2.x.b	yes	8	25.d	even	5	1
825.2.x.b	yes	8	33.d	even	2	1
825.2.x.b	yes	8	275.v	odd	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(825, [\chi])$:

$ T_{2} $

$ T_{23}^{8} - 11T_{23}^{6} + 1035T_{23}^{5} + 4146T_{23}^{4} - 11385T_{23}^{3} + 258569T_{23}^{2} + 1553535T_{23} + 2253001 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + T^{7} + \cdots + 81 $ T^8 + T^7 - 2T^6 - 5T^5 + T^4 - 15T^3 - 18T^2 + 27*T + 81
$5$	$ (T^{2} + 3 T + 5)^{4} $ (T^2 + 3*T + 5)^4
$7$	$ T^{8} $ T^8
$11$	$ T^{8} - 11 T^{6} + \cdots + 14641 $ T^8 - 11T^6 + 121T^4 - 1331*T^2 + 14641
$13$	$ T^{8} $ T^8
$17$	$ T^{8} $ T^8
$19$	$ T^{8} $ T^8
$23$	$ T^{8} - 11 T^{6} + \cdots + 2253001 $ T^8 - 11T^6 + 1035T^5 + 4146T^4 - 11385T^3 + 258569T^2 + 1553535T + 2253001
$29$	$ T^{8} $ T^8
$31$	$ T^{8} + 10 T^{7} + \cdots + 2418025 $ T^8 + 10T^7 + 75T^6 - 275T^5 + 1110T^4 + 2425T^3 + 741275T^2 - 816375*T + 2418025
$37$	$ (T^{4} - 7 T^{3} + \cdots + 2401)^{2} $ (T^4 - 7T^3 + 49T^2 - 343*T + 2401)^2
$41$	$ T^{8} $ T^8
$43$	$ T^{8} $ T^8
$47$	$ T^{8} - 44 T^{6} + \cdots + 3748096 $ T^8 - 44T^6 + 1936T^4 - 85184*T^2 + 3748096
$53$	$ T^{8} - 176 T^{6} + \cdots + 2621161 $ T^8 - 176T^6 - 1590T^5 + 12426T^4 + 279840T^3 + 1534949T^2 + 2574210T + 2621161
$59$	$ T^{8} + 75 T^{7} + \cdots + 203946961 $ T^8 + 75T^7 + 2534T^6 + 50625T^5 + 658311T^4 + 5726625T^3 + 32705284T^2 + 114605025*T + 203946961
$61$	$ T^{8} $ T^8
$67$	$ T^{8} + 39 T^{7} + \cdots + 31460881 $ T^8 + 39T^7 + 842T^6 + 10907T^5 + 101505T^4 + 510653T^3 + 2761122T^2 + 11885471*T + 31460881
$71$	$ T^{8} - 275 T^{6} + \cdots + 10272025 $ T^8 - 275T^6 - 1065T^5 + 28410T^4 + 292875T^3 + 1135025T^2 - 3413325T + 10272025
$73$	$ T^{8} $ T^8
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ T^{8} - 275 T^{6} + \cdots + 51051025 $ T^8 - 275T^6 + 4005T^5 + 32460T^4 - 1101375T^3 + 8697725T^2 - 28615725T + 51051025
$97$	$ T^{8} + 34 T^{7} + \cdots + 92140801 $ T^8 + 34T^7 + 867T^6 + 11407T^5 + 91200T^4 + 130543T^3 + 898487T^2 - 15176019*T + 92140801
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Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.x (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{3} + 2 \beta_{3} q^{4} + (\beta_{6} - 2) q^{5} + (3 \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) q + b5 * q^3 + 2b3 q^4 + (b6 - 2) * q^5 + (3b3 - b1) q^9	\( q + \beta_{5} q^{3} + 2 \beta_{3} q^{4} + (\beta_{6} - 2) q^{5} + (3 \beta_{3} - \beta_1) q^{9} + (2 \beta_{5} + \beta_{4}) q^{11} + ( - 2 \beta_{7} - 2 \beta_{2}) q^{12} + ( - 2 \beta_{5} - 3 \beta_{4}) q^{15} + ( - 4 \beta_{7} + 4 \beta_{4} + \cdots - 4) q^{16}+ \cdots + ( - 8 \beta_{7} - 5 \beta_{2}) q^{99}+O(q^{100}) \) q + b5 * q^3 + 2b3 q^4 + (b6 - 2) * q^5 + (3b3 - b1) q^9 + (2b5 + b4) q^11 + (-2b7 - 2b2) * q^12 + (-2b5 - 3b4) * q^15 + (-4b7 + 4b4 + 4b3 - 4) q^16 + (-2b3 - 2b1) * q^20 + (9b7 - b6 + b5 - 4b4 - 4b3 - b1 + 5) q^23 + (-3b6 + 1) q^25 + (-5b7 - 2b2) * q^27 + (-b7 - 3b6 + 3b5 + 4b4 + 3b3 + 3b2 - 6b1 - 1) * q^31 + (6b3 - b1) q^33 + (-4b7 - 2b6 + 2b5 + 6b4 + 6b3 + 2b2 - 2b1 - 4) q^36 + (7b7 - 7b4 - 7b3 + 7) q^37 + (-2b7 - 4b2) * q^44 + (-6b3 - b1) q^45 + (2b3 - 4b1) * q^47 + (-4b6 + 4) q^48 + 7 * q^49 + (b7 - 4b5 + b4 - 4b2) * q^53 + (-3b5 - 7b4) * q^55 + (7b7 + b6 - b2 - 8) q^59 + (-2b7 + 4b2) * q^60 + 8b4 q^64 + (-3b6 - 3b5 + 5b4 - 5) q^67 + (-7b7 + 9b6 - 4b5 + 3b4 + 3b3 - 4b2 + 4b1 - 9) q^69 + (4b7 + 5b5 + 4b4 + 5b2) * q^71 + (b5 + 9b4) q^75 + (8b7 - 4b6 + 4b5 - 4b4 - 4b3 + 4b2 - 4b1 + 8) q^80 + (-b7 - 5b6 + 5b5 + 6b4 + 6b3 + 5b2 - 5b1 - 1) * q^81 + (9b7 + 5b6 - 5b5 - 7b4 - 7b3 + 5b1 + 2) * q^89 + (-2b6 + 2b5 - 8b4 - 8) q^92 + (-5b7 - b6 + 4b2 + 10) * q^93 + (7b7 + 3b5 - 7b4 - 3b2) * q^97 + (-8b7 - 5b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} + 4 q^{4} - 12 q^{5} + 5 q^{9}+O(q^{10}) \) 8 * q - q^3 + 4 * q^4 - 12 * q^5 + 5 * q^9	\( 8 q - q^{3} + 4 q^{4} - 12 q^{5} + 5 q^{9} + 2 q^{12} - 4 q^{15} - 8 q^{16} - 6 q^{20} - 4 q^{25} + 8 q^{27} - 10 q^{31} + 11 q^{33} - 10 q^{36} + 14 q^{37} - 13 q^{45} + 16 q^{48} + 56 q^{49} - 11 q^{55} - 75 q^{59} + 8 q^{60} + 16 q^{64} - 39 q^{67} - 6 q^{69} + 17 q^{75} + 12 q^{80} - 7 q^{81} - 90 q^{92} + 90 q^{93} - 34 q^{97} + 11 q^{99}+O(q^{100}) \) 8 * q - q^3 + 4 * q^4 - 12 * q^5 + 5 * q^9 + 2 * q^12 - 4 * q^15 - 8 * q^16 - 6 * q^20 - 4 * q^25 + 8 * q^27 - 10 * q^31 + 11 * q^33 - 10 * q^36 + 14 * q^37 - 13 * q^45 + 16 * q^48 + 56 * q^49 - 11 * q^55 - 75 * q^59 + 8 * q^60 + 16 * q^64 - 39 * q^67 - 6 * q^69 + 17 * q^75 + 12 * q^80 - 7 * q^81 - 90 * q^92 + 90 * q^93 - 34 * q^97 + 11 * q^99

Introduction

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Modular forms

Varieties

Fields

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Properties

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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	825.2.x.a

Level	$825$
Weight	$2$
Character orbit	825.x
Analytic conductor	$6.588$
Analytic rank	$0$
Dimension	$8$
CM discriminant	-11
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	8.0.228765625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 \) x^8 - x^7 - 2x^6 + 5x^5 + x^4 + 15x^3 - 18x^2 - 27*x + 81
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + 2\nu^{5} - 5\nu^{4} - \nu^{3} - 15\nu^{2} + 18\nu + 27 ) / 27 \) (-v^7 + v^6 + 2v^5 - 5v^4 - v^3 - 15v^2 + 18v + 27) / 27
\(\beta_{3}\)	\(=\)	\( ( \nu^{6} + 16\nu ) / 3 \) (v^6 + 16*v) / 3
\(\beta_{4}\)	\(=\)	\( ( -2\nu^{7} - 4\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 36\nu^{2} - 54\nu + 162 ) / 27 \) (-2v^7 - 4v^6 + 10v^5 + 2v^4 - 5v^3 - 36v^2 - 54*v + 162) / 27
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} - 2\nu^{6} + 5\nu^{5} + \nu^{4} + 2\nu^{3} - 18\nu^{2} - 27\nu + 81 ) / 9 \) (-v^7 - 2v^6 + 5v^5 + v^4 + 2v^3 - 18v^2 - 27*v + 81) / 9
\(\beta_{6}\)	\(=\)	\( \nu^{5} + 16 \) v^5 + 16
\(\beta_{7}\)	\(=\)	\( ( -5\nu^{7} + 5\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 75\nu^{2} + 90\nu + 135 ) / 27 \) (-5v^7 + 5v^6 + 10v^5 + 2v^4 - 5v^3 - 75v^2 + 90*v + 135) / 27

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 2 \) 2b7 + b6 - b5 - 3b4 - 3*b3 - b2 + b1 + 2
\(\nu^{3}\)	\(=\)	\( 2\beta_{5} - 3\beta_{4} \) 2b5 - 3b4
\(\nu^{4}\)	\(=\)	\( \beta_{7} - 5\beta_{2} \) b7 - 5*b2
\(\nu^{5}\)	\(=\)	\( \beta_{6} - 16 \) b6 - 16
\(\nu^{6}\)	\(=\)	\( 3\beta_{3} - 16\beta_1 \) 3b3 - 16b1
\(\nu^{7}\)	\(=\)	\( -35\beta_{7} - 13\beta_{6} + 13\beta_{5} + 48\beta_{4} + 48\beta_{3} + 13\beta_{2} - 13\beta _1 - 35 \) -35b7 - 13b6 + 13b5 + 48b4 + 48b3 + 13b2 - 13*b1 - 35

\(n\)	\(376\)	\(551\)	\(727\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\beta_{3}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 131.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} + T^{7} + \cdots + 81 \) T^8 + T^7 - 2T^6 - 5T^5 + T^4 - 15T^3 - 18T^2 + 27*T + 81
$5$	\( (T^{2} + 3 T + 5)^{4} \) (T^2 + 3*T + 5)^4
$7$	\( T^{8} \) T^8
$11$	\( T^{8} - 11 T^{6} + \cdots + 14641 \) T^8 - 11T^6 + 121T^4 - 1331*T^2 + 14641
$13$	\( T^{8} \) T^8
$17$	\( T^{8} \) T^8
$19$	\( T^{8} \) T^8
$23$	\( T^{8} - 11 T^{6} + \cdots + 2253001 \) T^8 - 11T^6 + 1035T^5 + 4146T^4 - 11385T^3 + 258569T^2 + 1553535T + 2253001
$29$	\( T^{8} \) T^8
$31$	\( T^{8} + 10 T^{7} + \cdots + 2418025 \) T^8 + 10T^7 + 75T^6 - 275T^5 + 1110T^4 + 2425T^3 + 741275T^2 - 816375*T + 2418025
$37$	\( (T^{4} - 7 T^{3} + \cdots + 2401)^{2} \) (T^4 - 7T^3 + 49T^2 - 343*T + 2401)^2
$41$	\( T^{8} \) T^8
$43$	\( T^{8} \) T^8
$47$	\( T^{8} - 44 T^{6} + \cdots + 3748096 \) T^8 - 44T^6 + 1936T^4 - 85184*T^2 + 3748096
$53$	\( T^{8} - 176 T^{6} + \cdots + 2621161 \) T^8 - 176T^6 - 1590T^5 + 12426T^4 + 279840T^3 + 1534949T^2 + 2574210T + 2621161
$59$	\( T^{8} + 75 T^{7} + \cdots + 203946961 \) T^8 + 75T^7 + 2534T^6 + 50625T^5 + 658311T^4 + 5726625T^3 + 32705284T^2 + 114605025*T + 203946961
$61$	\( T^{8} \) T^8
$67$	\( T^{8} + 39 T^{7} + \cdots + 31460881 \) T^8 + 39T^7 + 842T^6 + 10907T^5 + 101505T^4 + 510653T^3 + 2761122T^2 + 11885471*T + 31460881
$71$	\( T^{8} - 275 T^{6} + \cdots + 10272025 \) T^8 - 275T^6 - 1065T^5 + 28410T^4 + 292875T^3 + 1135025T^2 - 3413325T + 10272025
$73$	\( T^{8} \) T^8
$79$	\( T^{8} \) T^8
$83$	\( T^{8} \) T^8
$89$	\( T^{8} - 275 T^{6} + \cdots + 51051025 \) T^8 - 275T^6 + 4005T^5 + 32460T^4 - 1101375T^3 + 8697725T^2 - 28615725T + 51051025
$97$	\( T^{8} + 34 T^{7} + \cdots + 92140801 \) T^8 + 34T^7 + 867T^6 + 11407T^5 + 91200T^4 + 130543T^3 + 898487T^2 - 15176019*T + 92140801
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