LMFDB - Newform orbit 216.2.f.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [216,2,Mod(107,216)] 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("216.107"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	216.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$1.72476868366$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.23123460096.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 2x^{6} + 6x^{4} + 8x^{2} + 16 $ x^8 + 2x^6 + 6x^4 + 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

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

$f(q)$	$=$	$ q + \beta_{3} q^{2} + ( - \beta_{4} - \beta_1 - 1) q^{4} + \beta_{2} q^{5} + \beta_{6} q^{7} + (\beta_{7} - \beta_{3} + \beta_{2}) q^{8} + ( - \beta_{6} - \beta_{4}) q^{10} + (\beta_{7} + \beta_{5} + \beta_{2}) q^{11}+ \cdots + (4 \beta_{5} - 2 \beta_{3} + 4 \beta_{2}) q^{98}+O(q^{100}) $ q + b3 * q^2 + (-b4 - b1 - 1) * q^4 + b2 * q^5 + b6 * q^7 + (b7 - b3 + b2) * q^8 + (-b6 - b4) * q^10 + (b7 + b5 + b2) * q^11 + (b6 + b4 + 2b1 + 1) q^13 + (-b5 + b2) * q^14 + (-b6 + b4 - 2) * q^16 + (-2b5 + 2b3 - b2) * q^17 + (b4 + 2) * q^19 + 2b5 q^20 + (-b6 + b4 - 2b1 - 2) q^22 + (b7 - b5 - 2b3 - b2) q^23 + (2b4 + 1) q^25 + (-2b7 + b3) q^26 + (-b6 - 2b4 + b1 - 1) q^28 + (-2b7 + 2b5 + 4b3) q^29 + (2b4 + 4b1 + 2) * q^31 + (-2b3 - 2b2) * q^32 + (b6 - 3b4 - 4) q^34 + (-b7 + 3b5 - 4b3 + b2) * q^35 + (b6 - b4 - 2b1 - 1) q^37 + (-b5 + 2b3 - b2) q^38 + (2b4 - 2b1 + 2) * q^40 + (-2b7 - 2b5 - 2b2) q^41 + 2 * q^43 + (2b7 - 2b5 - 2b3 - 2b2) * q^44 + (b6 + 3b4 + 2b1 - 2) * q^46 + (b7 - b5 - 2b3 - 3b2) * q^47 + (-4b4 - 2) q^49 + (-2b5 + b3 - 2b2) * q^50 + (-3b4 + b1 + 5) q^52 + (2b7 - 2b5 - 4b3 + 2b2) * q^53 + (-2b4 - 4b1 - 2) * q^55 + (-b7 + 4b5 - b3 + b2) q^56 + (-4b4 - 4b1 + 4) * q^58 + (b7 + b5 + b2) * q^59 + (-b6 - b4 - 2b1 - 1) q^61 + (-4b7 + 2b5 + 2b3 - 2b2) * q^62 + (2b6 + 4b4 + 2b1 + 2) q^64 + (-2b7 - 2b3 - b2) * q^65 + (-3b4 - 4) q^67 + (2b5 - 4b3 + 4b2) q^68 + (-b6 + 5b4 + 2b1 + 10) * q^70 + (-2b7 + 2b5 + 4b3 - 2b2) * q^71 + (4b4 - 1) q^73 + (2b7 - 2b5 - b3 + 2b2) q^74 + (b6 - 2b4 - b1 - 3) q^76 + (-2b7 + 2b5 + 4b3 - b2) q^77 + (-3b6 - 2b4 - 4b1 - 2) q^79 + (2b7 - 4b5 + 2b3 - 2b2) * q^80 + (2b6 - 2b4 + 4b1 + 4) q^82 + (2b7 - 2b5 + 4b3) q^83 + (-4b6 + 2b4 + 4b1 + 2) q^85 + 2b3 q^86 + (2b6 + 4b4 + 2b1 - 6) q^88 + (4b7 - 2b5 + 6b3 + b2) q^89 + (b4 - 6) * q^91 + (-2b7 - 2b5 - 2b3 - 2b2) * q^92 + (3b6 + 5b4 + 2b1 - 2) q^94 + (b7 - b5 - 2b3 + 3b2) * q^95 + (-6b4 - 1) q^97 + (4b5 - 2b3 + 4b2) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4} - 16 q^{16} + 16 q^{19} - 8 q^{22} + 8 q^{25} - 12 q^{28} - 32 q^{34} + 24 q^{40} + 16 q^{43} - 24 q^{46} - 16 q^{49} + 36 q^{52} + 48 q^{58} + 8 q^{64} - 32 q^{67} + 72 q^{70} - 8 q^{73}+ \cdots - 8 q^{97}+O(q^{100}) $ 8 * q - 4 * q^4 - 16 * q^16 + 16 * q^19 - 8 * q^22 + 8 * q^25 - 12 * q^28 - 32 * q^34 + 24 * q^40 + 16 * q^43 - 24 * q^46 - 16 * q^49 + 36 * q^52 + 48 * q^58 + 8 * q^64 - 32 * q^67 + 72 * q^70 - 8 * q^73 - 20 * q^76 + 16 * q^82 - 56 * q^88 - 48 * q^91 - 24 * q^94 - 8 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 2x^{6} + 6x^{4} + 8x^{2} + 16 $ x^8 + 2*x^6 + 6*x^4 + 8*x^2 + 16 :

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

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

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

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

107.1

1.08766	−	0.903873i
1.08766	+	0.903873i
0.563016	−	1.29731i
0.563016	+	1.29731i
−0.563016	−	1.29731i
−0.563016	+	1.29731i
−1.08766	−	0.903873i
−1.08766	+	0.903873i

−1.08766

−

0.903873i

0.366025

1.96622i

−1.59245

−

1.43937i

1.37910

−

2.46943i

1.73205

1.43937i

107.2

−1.08766

0.903873i

0.366025

−

1.96622i

−1.59245

1.43937i

1.37910

2.46943i

1.73205

−

1.43937i

107.3

−0.563016

−

1.29731i

−1.36603

1.46081i

3.07638

3.99102i

2.66422

0.949697i

−1.73205

−

3.99102i

107.4

−0.563016

1.29731i

−1.36603

−

1.46081i

3.07638

−

3.99102i

2.66422

−

0.949697i

−1.73205

3.99102i

107.5

0.563016

−

1.29731i

−1.36603

−

1.46081i

−3.07638

−

3.99102i

−2.66422

0.949697i

−1.73205

3.99102i

107.6

0.563016

1.29731i

−1.36603

1.46081i

−3.07638

3.99102i

−2.66422

−

0.949697i

−1.73205

−

3.99102i

107.7

1.08766

−

0.903873i

0.366025

−

1.96622i

1.59245

1.43937i

−1.37910

−

2.46943i

1.73205

−

1.43937i

107.8

1.08766

0.903873i

0.366025

1.96622i

1.59245

−

1.43937i

−1.37910

2.46943i

1.73205

1.43937i

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
24.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	216.2.f.a	✓	8
3.b	odd	2	1	inner	216.2.f.a	✓	8
4.b	odd	2	1		864.2.f.a		8
8.b	even	2	1		864.2.f.a		8
8.d	odd	2	1	inner	216.2.f.a	✓	8
9.c	even	3	2		648.2.l.f		16
9.d	odd	6	2		648.2.l.f		16
12.b	even	2	1		864.2.f.a		8
24.f	even	2	1	inner	216.2.f.a	✓	8
24.h	odd	2	1		864.2.f.a		8
36.f	odd	6	2		2592.2.p.f		16
36.h	even	6	2		2592.2.p.f		16
72.j	odd	6	2		2592.2.p.f		16
72.l	even	6	2		648.2.l.f		16
72.n	even	6	2		2592.2.p.f		16
72.p	odd	6	2		648.2.l.f		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
216.2.f.a	✓	8	1.a	even	1	1	trivial
216.2.f.a	✓	8	3.b	odd	2	1	inner
216.2.f.a	✓	8	8.d	odd	2	1	inner
216.2.f.a	✓	8	24.f	even	2	1	inner
648.2.l.f		16	9.c	even	3	2
648.2.l.f		16	9.d	odd	6	2
648.2.l.f		16	72.l	even	6	2
648.2.l.f		16	72.p	odd	6	2
864.2.f.a		8	4.b	odd	2	1
864.2.f.a		8	8.b	even	2	1
864.2.f.a		8	12.b	even	2	1
864.2.f.a		8	24.h	odd	2	1
2592.2.p.f		16	36.f	odd	6	2
2592.2.p.f		16	36.h	even	6	2
2592.2.p.f		16	72.j	odd	6	2
2592.2.p.f		16	72.n	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} - 12T_{5}^{2} + 24 $ T5^4 - 12*T5^2 + 24 acting on $S_{2}^{\mathrm{new}}(216, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 2 T^{6} + \cdots + 16 $ T^8 + 2T^6 + 6T^4 + 8*T^2 + 16
$3$	$ T^{8} $ T^8
$5$	$ (T^{4} - 12 T^{2} + 24)^{2} $ (T^4 - 12*T^2 + 24)^2
$7$	$ (T^{4} + 18 T^{2} + 33)^{2} $ (T^4 + 18*T^2 + 33)^2
$11$	$ (T^{4} + 28 T^{2} + 88)^{2} $ (T^4 + 28*T^2 + 88)^2
$13$	$ (T^{4} + 30 T^{2} + 33)^{2} $ (T^4 + 30*T^2 + 33)^2
$17$	$ (T^{4} + 52 T^{2} + 88)^{2} $ (T^4 + 52*T^2 + 88)^2
$19$	$ (T^{2} - 4 T + 1)^{4} $ (T^2 - 4*T + 1)^4
$23$	$ (T^{4} - 36 T^{2} + 24)^{2} $ (T^4 - 36*T^2 + 24)^2
$29$	$ (T^{4} - 96 T^{2} + 1536)^{2} $ (T^4 - 96*T^2 + 1536)^2
$31$	$ (T^{4} + 96 T^{2} + 2112)^{2} $ (T^4 + 96*T^2 + 2112)^2
$37$	$ (T^{4} + 54 T^{2} + 297)^{2} $ (T^4 + 54*T^2 + 297)^2
$41$	$ (T^{4} + 112 T^{2} + 1408)^{2} $ (T^4 + 112*T^2 + 1408)^2
$43$	$ (T - 2)^{8} $ (T - 2)^8
$47$	$ (T^{4} - 132 T^{2} + 4056)^{2} $ (T^4 - 132*T^2 + 4056)^2
$53$	$ (T^{4} - 144 T^{2} + 3456)^{2} $ (T^4 - 144*T^2 + 3456)^2
$59$	$ (T^{4} + 28 T^{2} + 88)^{2} $ (T^4 + 28*T^2 + 88)^2
$61$	$ (T^{4} + 30 T^{2} + 33)^{2} $ (T^4 + 30*T^2 + 33)^2
$67$	$ (T^{2} + 8 T - 11)^{4} $ (T^2 + 8*T - 11)^4
$71$	$ (T^{4} - 144 T^{2} + 3456)^{2} $ (T^4 - 144*T^2 + 3456)^2
$73$	$ (T^{2} + 2 T - 47)^{4} $ (T^2 + 2*T - 47)^4
$79$	$ (T^{4} + 186 T^{2} + 5577)^{2} $ (T^4 + 186*T^2 + 5577)^2
$83$	$ (T^{4} + 160 T^{2} + 5632)^{2} $ (T^4 + 160*T^2 + 5632)^2
$89$	$ (T^{4} + 436 T^{2} + 46552)^{2} $ (T^4 + 436*T^2 + 46552)^2
$97$	$ (T^{2} + 2 T - 107)^{4} $ (T^2 + 2*T - 107)^4
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Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	216.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{2} + ( - \beta_{4} - \beta_1 - 1) q^{4} + \beta_{2} q^{5} + \beta_{6} q^{7} + (\beta_{7} - \beta_{3} + \beta_{2}) q^{8} + ( - \beta_{6} - \beta_{4}) q^{10} + (\beta_{7} + \beta_{5} + \beta_{2}) q^{11}+ \cdots + (4 \beta_{5} - 2 \beta_{3} + 4 \beta_{2}) q^{98}+O(q^{100}) \) q + b3 * q^2 + (-b4 - b1 - 1) * q^4 + b2 * q^5 + b6 * q^7 + (b7 - b3 + b2) * q^8 + (-b6 - b4) * q^10 + (b7 + b5 + b2) * q^11 + (b6 + b4 + 2b1 + 1) q^13 + (-b5 + b2) * q^14 + (-b6 + b4 - 2) * q^16 + (-2b5 + 2b3 - b2) * q^17 + (b4 + 2) * q^19 + 2b5 q^20 + (-b6 + b4 - 2b1 - 2) q^22 + (b7 - b5 - 2b3 - b2) q^23 + (2b4 + 1) q^25 + (-2b7 + b3) q^26 + (-b6 - 2b4 + b1 - 1) q^28 + (-2b7 + 2b5 + 4b3) q^29 + (2b4 + 4b1 + 2) * q^31 + (-2b3 - 2b2) * q^32 + (b6 - 3b4 - 4) q^34 + (-b7 + 3b5 - 4b3 + b2) * q^35 + (b6 - b4 - 2b1 - 1) q^37 + (-b5 + 2b3 - b2) q^38 + (2b4 - 2b1 + 2) * q^40 + (-2b7 - 2b5 - 2b2) q^41 + 2 * q^43 + (2b7 - 2b5 - 2b3 - 2b2) * q^44 + (b6 + 3b4 + 2b1 - 2) * q^46 + (b7 - b5 - 2b3 - 3b2) * q^47 + (-4b4 - 2) q^49 + (-2b5 + b3 - 2b2) * q^50 + (-3b4 + b1 + 5) q^52 + (2b7 - 2b5 - 4b3 + 2b2) * q^53 + (-2b4 - 4b1 - 2) * q^55 + (-b7 + 4b5 - b3 + b2) q^56 + (-4b4 - 4b1 + 4) * q^58 + (b7 + b5 + b2) * q^59 + (-b6 - b4 - 2b1 - 1) q^61 + (-4b7 + 2b5 + 2b3 - 2b2) * q^62 + (2b6 + 4b4 + 2b1 + 2) q^64 + (-2b7 - 2b3 - b2) * q^65 + (-3b4 - 4) q^67 + (2b5 - 4b3 + 4b2) q^68 + (-b6 + 5b4 + 2b1 + 10) * q^70 + (-2b7 + 2b5 + 4b3 - 2b2) * q^71 + (4b4 - 1) q^73 + (2b7 - 2b5 - b3 + 2b2) q^74 + (b6 - 2b4 - b1 - 3) q^76 + (-2b7 + 2b5 + 4b3 - b2) q^77 + (-3b6 - 2b4 - 4b1 - 2) q^79 + (2b7 - 4b5 + 2b3 - 2b2) * q^80 + (2b6 - 2b4 + 4b1 + 4) q^82 + (2b7 - 2b5 + 4b3) q^83 + (-4b6 + 2b4 + 4b1 + 2) q^85 + 2b3 q^86 + (2b6 + 4b4 + 2b1 - 6) q^88 + (4b7 - 2b5 + 6b3 + b2) q^89 + (b4 - 6) * q^91 + (-2b7 - 2b5 - 2b3 - 2b2) * q^92 + (3b6 + 5b4 + 2b1 - 2) q^94 + (b7 - b5 - 2b3 + 3b2) * q^95 + (-6b4 - 1) q^97 + (4b5 - 2b3 + 4b2) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} - 16 q^{16} + 16 q^{19} - 8 q^{22} + 8 q^{25} - 12 q^{28} - 32 q^{34} + 24 q^{40} + 16 q^{43} - 24 q^{46} - 16 q^{49} + 36 q^{52} + 48 q^{58} + 8 q^{64} - 32 q^{67} + 72 q^{70} - 8 q^{73}+ \cdots - 8 q^{97}+O(q^{100}) \) 8 * q - 4 * q^4 - 16 * q^16 + 16 * q^19 - 8 * q^22 + 8 * q^25 - 12 * q^28 - 32 * q^34 + 24 * q^40 + 16 * q^43 - 24 * q^46 - 16 * q^49 + 36 * q^52 + 48 * q^58 + 8 * q^64 - 32 * q^67 + 72 * q^70 - 8 * q^73 - 20 * q^76 + 16 * q^82 - 56 * q^88 - 48 * q^91 - 24 * q^94 - 8 * q^97

Introduction

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Varieties

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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	216.2.f.a

Level	$216$
Weight	$2$
Character orbit	216.f
Analytic conductor	$1.725$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.72476868366\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.23123460096.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 2x^{6} + 6x^{4} + 8x^{2} + 16 \) x^8 + 2x^6 + 6x^4 + 8*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

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

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 107.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 2 T^{6} + \cdots + 16 \) T^8 + 2T^6 + 6T^4 + 8*T^2 + 16
$3$	\( T^{8} \) T^8
$5$	\( (T^{4} - 12 T^{2} + 24)^{2} \) (T^4 - 12*T^2 + 24)^2
$7$	\( (T^{4} + 18 T^{2} + 33)^{2} \) (T^4 + 18*T^2 + 33)^2
$11$	\( (T^{4} + 28 T^{2} + 88)^{2} \) (T^4 + 28*T^2 + 88)^2
$13$	\( (T^{4} + 30 T^{2} + 33)^{2} \) (T^4 + 30*T^2 + 33)^2
$17$	\( (T^{4} + 52 T^{2} + 88)^{2} \) (T^4 + 52*T^2 + 88)^2
$19$	\( (T^{2} - 4 T + 1)^{4} \) (T^2 - 4*T + 1)^4
$23$	\( (T^{4} - 36 T^{2} + 24)^{2} \) (T^4 - 36*T^2 + 24)^2
$29$	\( (T^{4} - 96 T^{2} + 1536)^{2} \) (T^4 - 96*T^2 + 1536)^2
$31$	\( (T^{4} + 96 T^{2} + 2112)^{2} \) (T^4 + 96*T^2 + 2112)^2
$37$	\( (T^{4} + 54 T^{2} + 297)^{2} \) (T^4 + 54*T^2 + 297)^2
$41$	\( (T^{4} + 112 T^{2} + 1408)^{2} \) (T^4 + 112*T^2 + 1408)^2
$43$	\( (T - 2)^{8} \) (T - 2)^8
$47$	\( (T^{4} - 132 T^{2} + 4056)^{2} \) (T^4 - 132*T^2 + 4056)^2
$53$	\( (T^{4} - 144 T^{2} + 3456)^{2} \) (T^4 - 144*T^2 + 3456)^2
$59$	\( (T^{4} + 28 T^{2} + 88)^{2} \) (T^4 + 28*T^2 + 88)^2
$61$	\( (T^{4} + 30 T^{2} + 33)^{2} \) (T^4 + 30*T^2 + 33)^2
$67$	\( (T^{2} + 8 T - 11)^{4} \) (T^2 + 8*T - 11)^4
$71$	\( (T^{4} - 144 T^{2} + 3456)^{2} \) (T^4 - 144*T^2 + 3456)^2
$73$	\( (T^{2} + 2 T - 47)^{4} \) (T^2 + 2*T - 47)^4
$79$	\( (T^{4} + 186 T^{2} + 5577)^{2} \) (T^4 + 186*T^2 + 5577)^2
$83$	\( (T^{4} + 160 T^{2} + 5632)^{2} \) (T^4 + 160*T^2 + 5632)^2
$89$	\( (T^{4} + 436 T^{2} + 46552)^{2} \) (T^4 + 436*T^2 + 46552)^2
$97$	\( (T^{2} + 2 T - 107)^{4} \) (T^2 + 2*T - 107)^4
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\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)