LMFDB - Newform orbit 960.2.bi.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(353,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 3]))

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

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

chi := DirichletCharacter("960.353");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.bi (of order $4$, degree $2$, 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:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{6} - \beta_{3} - 1) q^{3} - \beta_{7} q^{5} + (\beta_{5} + \beta_{4}) q^{7} + (2 \beta_{6} + \beta_{3} - 2 \beta_1) q^{9}+O(q^{10}) $ q + (-b6 - b3 - 1) * q^3 - b7 * q^5 + (b5 + b4) * q^7 + (2b6 + b3 - 2b1) * q^9	\( q + ( - \beta_{6} - \beta_{3} - 1) q^{3} - \beta_{7} q^{5} + (\beta_{5} + \beta_{4}) q^{7} + (2 \beta_{6} + \beta_{3} - 2 \beta_1) q^{9} + (3 \beta_{6} - 3 \beta_1) q^{11} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{13} + (\beta_{7} - \beta_{5} - \beta_{2}) q^{15} - 4 \beta_{6} q^{17} + 2 q^{19} + ( - \beta_{7} - 2 \beta_{5} + \beta_{2}) q^{21} - 2 \beta_{2} q^{23} - 5 \beta_{3} q^{25} + (\beta_{3} + 5 \beta_1 - 1) q^{27} + ( - 3 \beta_{7} - 3 \beta_{2}) q^{29} - 2 \beta_{4} q^{31} + (3 \beta_{3} + 6 \beta_1 - 3) q^{33} + ( - 5 \beta_{6} + 5 \beta_1) q^{35} + (2 \beta_{7} + 4 \beta_{5} - 2 \beta_{2}) q^{39} + (2 \beta_{6} + 2 \beta_1) q^{41} + (4 \beta_{3} + 4) q^{43} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{45} - 2 \beta_{7} q^{47} + 3 \beta_{3} q^{49} + (4 \beta_{6} - 4 \beta_{3} - 4 \beta_1) q^{51} - 4 \beta_{7} q^{53} + (3 \beta_{5} - 3 \beta_{4}) q^{55} + ( - 2 \beta_{6} - 2 \beta_{3} - 2) q^{57} + ( - 5 \beta_{6} - 5 \beta_1) q^{59} + 4 \beta_{5} q^{61} + (\beta_{5} - \beta_{4} - 4 \beta_{2}) q^{63} + (10 \beta_{6} - 10 \beta_1) q^{65} + (6 \beta_{3} - 6) q^{67} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2}) q^{69} + ( - 2 \beta_{7} - 2 \beta_{2}) q^{71} + (3 \beta_{3} - 3) q^{73} + (5 \beta_{3} - 5 \beta_1 - 5) q^{75} - 6 \beta_{2} q^{77} - 2 \beta_{5} q^{79} + ( - 4 \beta_{6} - 4 \beta_1 + 7) q^{81} + 8 \beta_{6} q^{83} - 4 \beta_{5} q^{85} + (6 \beta_{7} - 3 \beta_{5} - 3 \beta_{4}) q^{87} + ( - 6 \beta_{6} + 6 \beta_1) q^{89} - 20 \beta_{3} q^{91} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4}) q^{93} - 2 \beta_{7} q^{95} + (3 \beta_{3} + 3) q^{97} + ( - 3 \beta_{6} - 3 \beta_1 + 12) q^{99}+O(q^{100}) \) q + (-b6 - b3 - 1) * q^3 - b7 * q^5 + (b5 + b4) * q^7 + (2b6 + b3 - 2b1) * q^9 + (3b6 - 3b1) * q^11 + (-2b5 - 2b4) * q^13 + (b7 - b5 - b2) * q^15 - 4b6 q^17 + 2 * q^19 + (-b7 - 2b5 + b2) q^21 - 2b2 q^23 - 5b3 q^25 + (b3 + 5b1 - 1) q^27 + (-3b7 - 3b2) * q^29 - 2b4 q^31 + (3b3 + 6b1 - 3) * q^33 + (-5b6 + 5b1) * q^35 + (2b7 + 4b5 - 2b2) q^39 + (2b6 + 2b1) * q^41 + (4b3 + 4) q^43 + (2b5 - 2b4 + b2) * q^45 - 2b7 q^47 + 3b3 q^49 + (4b6 - 4b3 - 4b1) q^51 - 4b7 q^53 + (3b5 - 3b4) * q^55 + (-2b6 - 2b3 - 2) * q^57 + (-5b6 - 5b1) * q^59 + 4b5 q^61 + (b5 - b4 - 4b2) q^63 + (10b6 - 10b1) * q^65 + (6b3 - 6) q^67 + (2b7 - 2b4 + 2b2) q^69 + (-2b7 - 2b2) * q^71 + (3b3 - 3) q^73 + (5b3 - 5b1 - 5) * q^75 - 6b2 q^77 - 2b5 q^79 + (-4b6 - 4b1 + 7) * q^81 + 8b6 q^83 - 4b5 q^85 + (6b7 - 3b5 - 3b4) q^87 + (-6b6 + 6b1) * q^89 - 20b3 q^91 + (2b7 + 2b5 + 2b4) q^93 - 2b7 q^95 + (3b3 + 3) q^97 + (-3b6 - 3b1 + 12) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3}+O(q^{10}) $ 8 * q - 8 * q^3	$ 8 q - 8 q^{3} + 16 q^{19} - 8 q^{27} - 24 q^{33} + 32 q^{43} - 16 q^{57} - 48 q^{67} - 24 q^{73} - 40 q^{75} + 56 q^{81} + 24 q^{97} + 96 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 16 * q^19 - 8 * q^27 - 24 * q^33 + 32 * q^43 - 16 * q^57 - 48 * q^67 - 24 * q^73 - 40 * q^75 + 56 * q^81 + 24 * q^97 + 96 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1 :

$\beta_{1}$	$=$	$ ( \nu^{5} + 5\nu ) / 3 $ (v^5 + 5*v) / 3
$\beta_{2}$	$=$	$ ( \nu^{5} + 11\nu ) / 3 $ (v^5 + 11*v) / 3
$\beta_{3}$	$=$	$ ( \nu^{6} + 8\nu^{2} ) / 3 $ (v^6 + 8*v^2) / 3
$\beta_{4}$	$=$	$ ( 2\nu^{4} + 7 ) / 3 $ (2*v^4 + 7) / 3
$\beta_{5}$	$=$	$ \nu^{6} + 6\nu^{2} $ v^6 + 6*v^2
$\beta_{6}$	$=$	$ ( 2\nu^{7} + 13\nu^{3} ) / 3 $ (2v^7 + 13v^3) / 3
$\beta_{7}$	$=$	$ ( 4\nu^{7} + 29\nu^{3} ) / 3 $ (4v^7 + 29v^3) / 3

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

$\nu$	$=$	$ ( \beta_{2} - \beta_1 ) / 2 $ (b2 - b1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{5} + 3\beta_{3} ) / 2 $ (-b5 + 3*b3) / 2
$\nu^{3}$	$=$	$ \beta_{7} - 2\beta_{6} $ b7 - 2*b6
$\nu^{4}$	$=$	$ ( 3\beta_{4} - 7 ) / 2 $ (3*b4 - 7) / 2
$\nu^{5}$	$=$	$ ( -5\beta_{2} + 11\beta_1 ) / 2 $ (-5b2 + 11b1) / 2
$\nu^{6}$	$=$	$ 4\beta_{5} - 9\beta_{3} $ 4b5 - 9b3
$\nu^{7}$	$=$	$ ( -13\beta_{7} + 29\beta_{6} ) / 2 $ (-13b7 + 29b6) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$\beta_{3}$	$-1$	$-1$

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} $

353.1

−0.437016	+	0.437016i
1.14412	−	1.14412i
−1.14412	+	1.14412i
0.437016	−	0.437016i
−0.437016	−	0.437016i
1.14412	+	1.14412i
−1.14412	−	1.14412i
0.437016	+	0.437016i

−1.70711

0.292893i

−1.58114

−

1.58114i

2.23607

−

2.23607i

2.82843

−

1.00000i

353.2

−1.70711

0.292893i

1.58114

1.58114i

−2.23607

2.23607i

2.82843

−

1.00000i

353.3

−0.292893

1.70711i

−1.58114

−

1.58114i

−2.23607

2.23607i

−2.82843

−

1.00000i

353.4

−0.292893

1.70711i

1.58114

1.58114i

2.23607

−

2.23607i

−2.82843

−

1.00000i

737.1

−1.70711

−

0.292893i

−1.58114

1.58114i

2.23607

2.23607i

2.82843

1.00000i

737.2

−1.70711

−

0.292893i

1.58114

−

1.58114i

−2.23607

−

2.23607i

2.82843

1.00000i

737.3

−0.292893

−

1.70711i

−1.58114

1.58114i

−2.23607

−

2.23607i

−2.82843

1.00000i

737.4

−0.292893

−

1.70711i

1.58114

−

1.58114i

2.23607

2.23607i

−2.82843

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
20.e	even	4	1	inner
24.f	even	2	1	inner
40.i	odd	4	1	inner
60.l	odd	4	1	inner
120.w	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	960.2.bi.e	✓	8
3.b	odd	2	1	inner	960.2.bi.e	✓	8
4.b	odd	2	1		960.2.bi.f	yes	8
5.c	odd	4	1		960.2.bi.f	yes	8
8.b	even	2	1		960.2.bi.f	yes	8
8.d	odd	2	1	inner	960.2.bi.e	✓	8
12.b	even	2	1		960.2.bi.f	yes	8
15.e	even	4	1		960.2.bi.f	yes	8
20.e	even	4	1	inner	960.2.bi.e	✓	8
24.f	even	2	1	inner	960.2.bi.e	✓	8
24.h	odd	2	1		960.2.bi.f	yes	8
40.i	odd	4	1	inner	960.2.bi.e	✓	8
40.k	even	4	1		960.2.bi.f	yes	8
60.l	odd	4	1	inner	960.2.bi.e	✓	8
120.q	odd	4	1		960.2.bi.f	yes	8
120.w	even	4	1	inner	960.2.bi.e	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
960.2.bi.e	✓	8	1.a	even	1	1	trivial
960.2.bi.e	✓	8	3.b	odd	2	1	inner
960.2.bi.e	✓	8	8.d	odd	2	1	inner
960.2.bi.e	✓	8	20.e	even	4	1	inner
960.2.bi.e	✓	8	24.f	even	2	1	inner
960.2.bi.e	✓	8	40.i	odd	4	1	inner
960.2.bi.e	✓	8	60.l	odd	4	1	inner
960.2.bi.e	✓	8	120.w	even	4	1	inner
960.2.bi.f	yes	8	4.b	odd	2	1
960.2.bi.f	yes	8	5.c	odd	4	1
960.2.bi.f	yes	8	8.b	even	2	1
960.2.bi.f	yes	8	12.b	even	2	1
960.2.bi.f	yes	8	15.e	even	4	1
960.2.bi.f	yes	8	24.h	odd	2	1
960.2.bi.f	yes	8	40.k	even	4	1
960.2.bi.f	yes	8	120.q	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(960, [\chi])$:

$ T_{7}^{4} + 100 $

$ T_{11}^{2} - 18 $

$ T_{19} - 2 $

$ T_{23}^{4} + 400 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 9)^{2} $ (T^4 + 4T^3 + 8T^2 + 12*T + 9)^2
$5$	$ (T^{4} + 25)^{2} $ (T^4 + 25)^2
$7$	$ (T^{4} + 100)^{2} $ (T^4 + 100)^2
$11$	$ (T^{2} - 18)^{4} $ (T^2 - 18)^4
$13$	$ (T^{4} + 1600)^{2} $ (T^4 + 1600)^2
$17$	$ (T^{4} + 256)^{2} $ (T^4 + 256)^2
$19$	$ (T - 2)^{8} $ (T - 2)^8
$23$	$ (T^{4} + 400)^{2} $ (T^4 + 400)^2
$29$	$ (T^{2} + 90)^{4} $ (T^2 + 90)^4
$31$	$ (T^{2} - 20)^{4} $ (T^2 - 20)^4
$37$	$ T^{8} $ T^8
$41$	$ (T^{2} + 8)^{4} $ (T^2 + 8)^4
$43$	$ (T^{2} - 8 T + 32)^{4} $ (T^2 - 8*T + 32)^4
$47$	$ (T^{4} + 400)^{2} $ (T^4 + 400)^2
$53$	$ (T^{4} + 6400)^{2} $ (T^4 + 6400)^2
$59$	$ (T^{2} + 50)^{4} $ (T^2 + 50)^4
$61$	$ (T^{2} + 80)^{4} $ (T^2 + 80)^4
$67$	$ (T^{2} + 12 T + 72)^{4} $ (T^2 + 12*T + 72)^4
$71$	$ (T^{2} + 40)^{4} $ (T^2 + 40)^4
$73$	$ (T^{2} + 6 T + 18)^{4} $ (T^2 + 6*T + 18)^4
$79$	$ (T^{2} + 20)^{4} $ (T^2 + 20)^4
$83$	$ (T^{4} + 4096)^{2} $ (T^4 + 4096)^2
$89$	$ (T^{2} - 72)^{4} $ (T^2 - 72)^4
$97$	$ (T^{2} - 6 T + 18)^{4} $ (T^2 - 6*T + 18)^4
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.bi (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{6} - \beta_{3} - 1) q^{3} - \beta_{7} q^{5} + (\beta_{5} + \beta_{4}) q^{7} + (2 \beta_{6} + \beta_{3} - 2 \beta_1) q^{9}+O(q^{10}) \) q + (-b6 - b3 - 1) * q^3 - b7 * q^5 + (b5 + b4) * q^7 + (2b6 + b3 - 2b1) * q^9	\( q + ( - \beta_{6} - \beta_{3} - 1) q^{3} - \beta_{7} q^{5} + (\beta_{5} + \beta_{4}) q^{7} + (2 \beta_{6} + \beta_{3} - 2 \beta_1) q^{9} + (3 \beta_{6} - 3 \beta_1) q^{11} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{13} + (\beta_{7} - \beta_{5} - \beta_{2}) q^{15} - 4 \beta_{6} q^{17} + 2 q^{19} + ( - \beta_{7} - 2 \beta_{5} + \beta_{2}) q^{21} - 2 \beta_{2} q^{23} - 5 \beta_{3} q^{25} + (\beta_{3} + 5 \beta_1 - 1) q^{27} + ( - 3 \beta_{7} - 3 \beta_{2}) q^{29} - 2 \beta_{4} q^{31} + (3 \beta_{3} + 6 \beta_1 - 3) q^{33} + ( - 5 \beta_{6} + 5 \beta_1) q^{35} + (2 \beta_{7} + 4 \beta_{5} - 2 \beta_{2}) q^{39} + (2 \beta_{6} + 2 \beta_1) q^{41} + (4 \beta_{3} + 4) q^{43} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{45} - 2 \beta_{7} q^{47} + 3 \beta_{3} q^{49} + (4 \beta_{6} - 4 \beta_{3} - 4 \beta_1) q^{51} - 4 \beta_{7} q^{53} + (3 \beta_{5} - 3 \beta_{4}) q^{55} + ( - 2 \beta_{6} - 2 \beta_{3} - 2) q^{57} + ( - 5 \beta_{6} - 5 \beta_1) q^{59} + 4 \beta_{5} q^{61} + (\beta_{5} - \beta_{4} - 4 \beta_{2}) q^{63} + (10 \beta_{6} - 10 \beta_1) q^{65} + (6 \beta_{3} - 6) q^{67} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2}) q^{69} + ( - 2 \beta_{7} - 2 \beta_{2}) q^{71} + (3 \beta_{3} - 3) q^{73} + (5 \beta_{3} - 5 \beta_1 - 5) q^{75} - 6 \beta_{2} q^{77} - 2 \beta_{5} q^{79} + ( - 4 \beta_{6} - 4 \beta_1 + 7) q^{81} + 8 \beta_{6} q^{83} - 4 \beta_{5} q^{85} + (6 \beta_{7} - 3 \beta_{5} - 3 \beta_{4}) q^{87} + ( - 6 \beta_{6} + 6 \beta_1) q^{89} - 20 \beta_{3} q^{91} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4}) q^{93} - 2 \beta_{7} q^{95} + (3 \beta_{3} + 3) q^{97} + ( - 3 \beta_{6} - 3 \beta_1 + 12) q^{99}+O(q^{100}) \) q + (-b6 - b3 - 1) * q^3 - b7 * q^5 + (b5 + b4) * q^7 + (2b6 + b3 - 2b1) * q^9 + (3b6 - 3b1) * q^11 + (-2b5 - 2b4) * q^13 + (b7 - b5 - b2) * q^15 - 4b6 q^17 + 2 * q^19 + (-b7 - 2b5 + b2) q^21 - 2b2 q^23 - 5b3 q^25 + (b3 + 5b1 - 1) q^27 + (-3b7 - 3b2) * q^29 - 2b4 q^31 + (3b3 + 6b1 - 3) * q^33 + (-5b6 + 5b1) * q^35 + (2b7 + 4b5 - 2b2) q^39 + (2b6 + 2b1) * q^41 + (4b3 + 4) q^43 + (2b5 - 2b4 + b2) * q^45 - 2b7 q^47 + 3b3 q^49 + (4b6 - 4b3 - 4b1) q^51 - 4b7 q^53 + (3b5 - 3b4) * q^55 + (-2b6 - 2b3 - 2) * q^57 + (-5b6 - 5b1) * q^59 + 4b5 q^61 + (b5 - b4 - 4b2) q^63 + (10b6 - 10b1) * q^65 + (6b3 - 6) q^67 + (2b7 - 2b4 + 2b2) q^69 + (-2b7 - 2b2) * q^71 + (3b3 - 3) q^73 + (5b3 - 5b1 - 5) * q^75 - 6b2 q^77 - 2b5 q^79 + (-4b6 - 4b1 + 7) * q^81 + 8b6 q^83 - 4b5 q^85 + (6b7 - 3b5 - 3b4) q^87 + (-6b6 + 6b1) * q^89 - 20b3 q^91 + (2b7 + 2b5 + 2b4) q^93 - 2b7 q^95 + (3b3 + 3) q^97 + (-3b6 - 3b1 + 12) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3}+O(q^{10}) \) 8 * q - 8 * q^3	\( 8 q - 8 q^{3} + 16 q^{19} - 8 q^{27} - 24 q^{33} + 32 q^{43} - 16 q^{57} - 48 q^{67} - 24 q^{73} - 40 q^{75} + 56 q^{81} + 24 q^{97} + 96 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 16 * q^19 - 8 * q^27 - 24 * q^33 + 32 * q^43 - 16 * q^57 - 48 * q^67 - 24 * q^73 - 40 * q^75 + 56 * q^81 + 24 * q^97 + 96 * q^99

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Newspace parameters

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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	960.2.bi.e

Level	$960$
Weight	$2$
Character orbit	960.bi
Analytic conductor	$7.666$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(7.66563859404\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 7x^{4} + 1 \) x^8 + 7*x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{5} + 5\nu ) / 3 \) (v^5 + 5*v) / 3
\(\beta_{2}\)	\(=\)	\( ( \nu^{5} + 11\nu ) / 3 \) (v^5 + 11*v) / 3
\(\beta_{3}\)	\(=\)	\( ( \nu^{6} + 8\nu^{2} ) / 3 \) (v^6 + 8*v^2) / 3
\(\beta_{4}\)	\(=\)	\( ( 2\nu^{4} + 7 ) / 3 \) (2*v^4 + 7) / 3
\(\beta_{5}\)	\(=\)	\( \nu^{6} + 6\nu^{2} \) v^6 + 6*v^2
\(\beta_{6}\)	\(=\)	\( ( 2\nu^{7} + 13\nu^{3} ) / 3 \) (2v^7 + 13v^3) / 3
\(\beta_{7}\)	\(=\)	\( ( 4\nu^{7} + 29\nu^{3} ) / 3 \) (4v^7 + 29v^3) / 3

\(\nu\)	\(=\)	\( ( \beta_{2} - \beta_1 ) / 2 \) (b2 - b1) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{5} + 3\beta_{3} ) / 2 \) (-b5 + 3*b3) / 2
\(\nu^{3}\)	\(=\)	\( \beta_{7} - 2\beta_{6} \) b7 - 2*b6
\(\nu^{4}\)	\(=\)	\( ( 3\beta_{4} - 7 ) / 2 \) (3*b4 - 7) / 2
\(\nu^{5}\)	\(=\)	\( ( -5\beta_{2} + 11\beta_1 ) / 2 \) (-5b2 + 11b1) / 2
\(\nu^{6}\)	\(=\)	\( 4\beta_{5} - 9\beta_{3} \) 4b5 - 9b3
\(\nu^{7}\)	\(=\)	\( ( -13\beta_{7} + 29\beta_{6} ) / 2 \) (-13b7 + 29b6) / 2

\(n\)	\(511\)	\(577\)	\(641\)	\(901\)
\(\chi(n)\)	\(1\)	\(\beta_{3}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 353.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 9)^{2} \) (T^4 + 4T^3 + 8T^2 + 12*T + 9)^2
$5$	\( (T^{4} + 25)^{2} \) (T^4 + 25)^2
$7$	\( (T^{4} + 100)^{2} \) (T^4 + 100)^2
$11$	\( (T^{2} - 18)^{4} \) (T^2 - 18)^4
$13$	\( (T^{4} + 1600)^{2} \) (T^4 + 1600)^2
$17$	\( (T^{4} + 256)^{2} \) (T^4 + 256)^2
$19$	\( (T - 2)^{8} \) (T - 2)^8
$23$	\( (T^{4} + 400)^{2} \) (T^4 + 400)^2
$29$	\( (T^{2} + 90)^{4} \) (T^2 + 90)^4
$31$	\( (T^{2} - 20)^{4} \) (T^2 - 20)^4
$37$	\( T^{8} \) T^8
$41$	\( (T^{2} + 8)^{4} \) (T^2 + 8)^4
$43$	\( (T^{2} - 8 T + 32)^{4} \) (T^2 - 8*T + 32)^4
$47$	\( (T^{4} + 400)^{2} \) (T^4 + 400)^2
$53$	\( (T^{4} + 6400)^{2} \) (T^4 + 6400)^2
$59$	\( (T^{2} + 50)^{4} \) (T^2 + 50)^4
$61$	\( (T^{2} + 80)^{4} \) (T^2 + 80)^4
$67$	\( (T^{2} + 12 T + 72)^{4} \) (T^2 + 12*T + 72)^4
$71$	\( (T^{2} + 40)^{4} \) (T^2 + 40)^4
$73$	\( (T^{2} + 6 T + 18)^{4} \) (T^2 + 6*T + 18)^4
$79$	\( (T^{2} + 20)^{4} \) (T^2 + 20)^4
$83$	\( (T^{4} + 4096)^{2} \) (T^4 + 4096)^2
$89$	\( (T^{2} - 72)^{4} \) (T^2 - 72)^4
$97$	\( (T^{2} - 6 T + 18)^{4} \) (T^2 - 6*T + 18)^4
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