LMFDB - Newform orbit 1352.2.o.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1352,2,Mod(361,1352)] 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("1352.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1352, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1352.o (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,2,0,0,0,0,0,-6,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$10.7957743533$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.1731891456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{7} - \beta_{2}) q^{3} + ( - \beta_{5} + \beta_1) q^{5} + ( - \beta_{6} + \beta_{5}) q^{7} + ( - \beta_{7} + \beta_{4} - 2 \beta_{2} - 1) q^{9} + (\beta_{3} + 2 \beta_1) q^{11} + ( - 2 \beta_{3} + \beta_1) q^{15}+ \cdots + ( - 3 \beta_{6} + 2 \beta_{5} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) $ q + (-b7 - b2) * q^3 + (-b5 + b1) * q^5 + (-b6 + b5) * q^7 + (-b7 + b4 - 2b2 - 1) q^9 + (b3 + 2b1) q^11 + (-2b3 + b1) q^15 + (-b7 + b4 + b2 + 2) * q^17 - b6 * q^19 + (-2b6 + 3b5 + 2b3 - 3b1) * q^21 + (2b7 + 2b2) * q^23 + (b4 + 1) * q^25 + (-b4 - 4) * q^27 + 2b7 q^29 + (-b6 - 2b5 + b3 + 2b1) * q^31 - 4b6 q^33 + (3b7 - 3b4 + 7b2 + 4) q^35 + (4b3 + b1) q^37 - 4b3 q^41 + (b7 - b4 - 7b2 - 8) q^43 + (-2b6 + 2b5) * q^45 + (-3b6 - b5 + 3b3 + b1) * q^47 + (-5b7 - 6b2) * q^49 + (-b4 - 4) * q^51 + (4b4 + 6) q^53 + 8b2 q^55 + (2b5 - 2b1) * q^57 - 5b6 q^59 + (-2b7 + 2b4 - 8b2 - 6) q^61 + (3b3 - 4b1) * q^63 + (-3b3 - 2b1) * q^67 + (2b7 - 2b4 + 10b2 + 8) q^69 + (-b6 - 3b5) q^71 + (-6b5 + 6b1) * q^73 + 4b2 q^75 + (-4b4 + 4) q^77 + (-2b4 - 8) q^79 - 7b2 q^81 + (b6 + 4b5 - b3 - 4b1) * q^83 + (-2b6 - b5) q^85 + (8b2 + 8) q^87 - 2b1 q^89 - 4b3 q^93 + (2b7 - 2b4 + 2b2) q^95 - 4b5 q^97 + (-3b6 + 2b5 + 3b3 - 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} - 6 q^{9} + 6 q^{17} - 4 q^{23} + 4 q^{25} - 28 q^{27} + 4 q^{29} + 22 q^{35} - 30 q^{43} + 14 q^{49} - 28 q^{51} + 32 q^{53} - 32 q^{55} - 28 q^{61} + 36 q^{69} - 16 q^{75} + 48 q^{77} - 56 q^{79}+ \cdots + 4 q^{95}+O(q^{100}) $ 8 * q + 2 * q^3 - 6 * q^9 + 6 * q^17 - 4 * q^23 + 4 * q^25 - 28 * q^27 + 4 * q^29 + 22 * q^35 - 30 * q^43 + 14 * q^49 - 28 * q^51 + 32 * q^53 - 32 * q^55 - 28 * q^61 + 36 * q^69 - 16 * q^75 + 48 * q^77 - 56 * q^79 + 28 * q^81 + 32 * q^87 + 4 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9*x^6 + 65*x^4 - 144*x^2 + 256 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 $ (9v^6 - 65v^4 + 585*v^2 - 1296) / 1040
$\beta_{3}$	$=$	$ ( \nu^{7} + 181\nu ) / 130 $ (v^7 + 181*v) / 130
$\beta_{4}$	$=$	$ ( \nu^{6} + 116 ) / 65 $ (v^6 + 116) / 65
$\beta_{5}$	$=$	$ ( -9\nu^{7} + 65\nu^{5} - 585\nu^{3} + 1296\nu ) / 1040 $ (-9v^7 + 65v^5 - 585v^3 + 1296v) / 1040
$\beta_{6}$	$=$	$ ( -29\nu^{7} + 325\nu^{5} - 1885\nu^{3} + 4176\nu ) / 2080 $ (-29v^7 + 325v^5 - 1885v^3 + 4176v) / 2080
$\beta_{7}$	$=$	$ ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 $ (-29v^6 + 325v^4 - 1885*v^2 + 4176) / 1040

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{7} - \beta_{4} + 5\beta_{2} + 4 $ b7 - b4 + 5*b2 + 4
$\nu^{3}$	$=$	$ 2\beta_{6} - 5\beta_{5} - 2\beta_{3} + 5\beta_1 $ 2b6 - 5b5 - 2b3 + 5b1
$\nu^{4}$	$=$	$ 9\beta_{7} + 29\beta_{2} $ 9b7 + 29b2
$\nu^{5}$	$=$	$ 18\beta_{6} - 29\beta_{5} $ 18b6 - 29b5
$\nu^{6}$	$=$	$ 65\beta_{4} - 116 $ 65*b4 - 116
$\nu^{7}$	$=$	$ 130\beta_{3} - 181\beta_1 $ 130b3 - 181b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1185$
$\chi(n)$	$1$	$1$	$1 + \beta_{2}$

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} $

361.1

1.35234	−	0.780776i
−1.35234	+	0.780776i
2.21837	−	1.28078i
−2.21837	+	1.28078i
−1.35234	−	0.780776i
1.35234	+	0.780776i
−2.21837	−	1.28078i
2.21837	+	1.28078i

−0.780776

−

1.35234i

−

1.56155i

−0.379706

−

0.219224i

0.280776

−

0.486319i

361.2

−0.780776

−

1.35234i

1.56155i

0.379706

0.219224i

0.280776

−

0.486319i

361.3

1.28078

2.21837i

−

2.56155i

3.95042

2.28078i

−1.78078

3.08440i

361.4

1.28078

2.21837i

2.56155i

−3.95042

−

2.28078i

−1.78078

3.08440i

1161.1

−0.780776

1.35234i

−

1.56155i

0.379706

−

0.219224i

0.280776

0.486319i

1161.2

−0.780776

1.35234i

1.56155i

−0.379706

0.219224i

0.280776

0.486319i

1161.3

1.28078

−

2.21837i

−

2.56155i

−3.95042

2.28078i

−1.78078

−

3.08440i

1161.4

1.28078

−

2.21837i

2.56155i

3.95042

−

2.28078i

−1.78078

−

3.08440i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner
13.c	even	3	1	inner
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1352.2.o.e		8
13.b	even	2	1	inner	1352.2.o.e		8
13.c	even	3	1		104.2.f.a	✓	4
13.c	even	3	1	inner	1352.2.o.e		8
13.d	odd	4	1		1352.2.i.g		4
13.d	odd	4	1		1352.2.i.h		4
13.e	even	6	1		104.2.f.a	✓	4
13.e	even	6	1	inner	1352.2.o.e		8
13.f	odd	12	1		1352.2.a.d		2
13.f	odd	12	1		1352.2.a.e		2
13.f	odd	12	1		1352.2.i.g		4
13.f	odd	12	1		1352.2.i.h		4
39.h	odd	6	1		936.2.c.d		4
39.i	odd	6	1		936.2.c.d		4
52.i	odd	6	1		208.2.f.b		4
52.j	odd	6	1		208.2.f.b		4
52.l	even	12	1		2704.2.a.s		2
52.l	even	12	1		2704.2.a.t		2
65.l	even	6	1		2600.2.k.a		4
65.n	even	6	1		2600.2.k.a		4
65.q	odd	12	1		2600.2.f.a		4
65.q	odd	12	1		2600.2.f.b		4
65.r	odd	12	1		2600.2.f.a		4
65.r	odd	12	1		2600.2.f.b		4
104.n	odd	6	1		832.2.f.f		4
104.p	odd	6	1		832.2.f.f		4
104.r	even	6	1		832.2.f.i		4
104.s	even	6	1		832.2.f.i		4
156.p	even	6	1		1872.2.c.j		4
156.r	even	6	1		1872.2.c.j		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.f.a	✓	4	13.c	even	3	1
104.2.f.a	✓	4	13.e	even	6	1
208.2.f.b		4	52.i	odd	6	1
208.2.f.b		4	52.j	odd	6	1
832.2.f.f		4	104.n	odd	6	1
832.2.f.f		4	104.p	odd	6	1
832.2.f.i		4	104.r	even	6	1
832.2.f.i		4	104.s	even	6	1
936.2.c.d		4	39.h	odd	6	1
936.2.c.d		4	39.i	odd	6	1
1352.2.a.d		2	13.f	odd	12	1
1352.2.a.e		2	13.f	odd	12	1
1352.2.i.g		4	13.d	odd	4	1
1352.2.i.g		4	13.f	odd	12	1
1352.2.i.h		4	13.d	odd	4	1
1352.2.i.h		4	13.f	odd	12	1
1352.2.o.e		8	1.a	even	1	1	trivial
1352.2.o.e		8	13.b	even	2	1	inner
1352.2.o.e		8	13.c	even	3	1	inner
1352.2.o.e		8	13.e	even	6	1	inner
1872.2.c.j		4	156.p	even	6	1
1872.2.c.j		4	156.r	even	6	1
2600.2.f.a		4	65.q	odd	12	1
2600.2.f.a		4	65.r	odd	12	1
2600.2.f.b		4	65.q	odd	12	1
2600.2.f.b		4	65.r	odd	12	1
2600.2.k.a		4	65.l	even	6	1
2600.2.k.a		4	65.n	even	6	1
2704.2.a.s		2	52.l	even	12	1
2704.2.a.t		2	52.l	even	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1352, [\chi])$:

$ T_{3}^{4} - T_{3}^{3} + 5T_{3}^{2} + 4T_{3} + 16 $

T3^4 - T3^3 + 5*T3^2 + 4*T3 + 16

$ T_{5}^{4} + 9T_{5}^{2} + 16 $

T5^4 + 9*T5^2 + 16

$ T_{11}^{8} - 36T_{11}^{6} + 1040T_{11}^{4} - 9216T_{11}^{2} + 65536 $

T11^8 - 36*T11^6 + 1040*T11^4 - 9216*T11^2 + 65536

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} - T^{3} + 5 T^{2} + \cdots + 16)^{2} $ (T^4 - T^3 + 5T^2 + 4T + 16)^2
$5$	$ (T^{4} + 9 T^{2} + 16)^{2} $ (T^4 + 9*T^2 + 16)^2
$7$	$ T^{8} - 21 T^{6} + \cdots + 16 $ T^8 - 21T^6 + 437T^4 - 84*T^2 + 16
$11$	$ T^{8} - 36 T^{6} + \cdots + 65536 $ T^8 - 36T^6 + 1040T^4 - 9216*T^2 + 65536
$13$	$ T^{8} $ T^8
$17$	$ (T^{4} - 3 T^{3} + 11 T^{2} + \cdots + 4)^{2} $ (T^4 - 3T^3 + 11T^2 + 6*T + 4)^2
$19$	$ (T^{4} - 4 T^{2} + 16)^{2} $ (T^4 - 4*T^2 + 16)^2
$23$	$ (T^{4} + 2 T^{3} + \cdots + 256)^{2} $ (T^4 + 2T^3 + 20T^2 - 32*T + 256)^2
$29$	$ (T^{4} - 2 T^{3} + \cdots + 256)^{2} $ (T^4 - 2T^3 + 20T^2 + 32*T + 256)^2
$31$	$ (T^{4} + 36 T^{2} + 256)^{2} $ (T^4 + 36*T^2 + 256)^2
$37$	$ T^{8} - 121 T^{6} + \cdots + 7311616 $ T^8 - 121T^6 + 11937T^4 - 327184*T^2 + 7311616
$41$	$ (T^{4} - 64 T^{2} + 4096)^{2} $ (T^4 - 64*T^2 + 4096)^2
$43$	$ (T^{4} + 15 T^{3} + \cdots + 2704)^{2} $ (T^4 + 15T^3 + 173T^2 + 780*T + 2704)^2
$47$	$ (T^{4} + 69 T^{2} + 676)^{2} $ (T^4 + 69*T^2 + 676)^2
$53$	$ (T^{2} - 8 T - 52)^{4} $ (T^2 - 8*T - 52)^4
$59$	$ (T^{4} - 100 T^{2} + 10000)^{2} $ (T^4 - 100*T^2 + 10000)^2
$61$	$ (T^{4} + 14 T^{3} + \cdots + 1024)^{2} $ (T^4 + 14T^3 + 164T^2 + 448*T + 1024)^2
$67$	$ T^{8} - 84 T^{6} + \cdots + 4096 $ T^8 - 84T^6 + 6992T^4 - 5376*T^2 + 4096
$71$	$ T^{8} - 77 T^{6} + \cdots + 2085136 $ T^8 - 77T^6 + 4485T^4 - 111188*T^2 + 2085136
$73$	$ (T^{4} + 324 T^{2} + 20736)^{2} $ (T^4 + 324*T^2 + 20736)^2
$79$	$ (T^{2} + 14 T + 32)^{4} $ (T^2 + 14*T + 32)^4
$83$	$ (T^{2} + 68)^{4} $ (T^2 + 68)^4
$89$	$ T^{8} - 36 T^{6} + \cdots + 65536 $ T^8 - 36T^6 + 1040T^4 - 9216*T^2 + 65536
$97$	$ T^{8} - 144 T^{6} + \cdots + 16777216 $ T^8 - 144T^6 + 16640T^4 - 589824*T^2 + 16777216
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{7} - \beta_{2}) q^{3} + ( - \beta_{5} + \beta_1) q^{5} + ( - \beta_{6} + \beta_{5}) q^{7} + ( - \beta_{7} + \beta_{4} - 2 \beta_{2} - 1) q^{9} + (\beta_{3} + 2 \beta_1) q^{11} + ( - 2 \beta_{3} + \beta_1) q^{15}+ \cdots + ( - 3 \beta_{6} + 2 \beta_{5} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) q + (-b7 - b2) * q^3 + (-b5 + b1) * q^5 + (-b6 + b5) * q^7 + (-b7 + b4 - 2b2 - 1) q^9 + (b3 + 2b1) q^11 + (-2b3 + b1) q^15 + (-b7 + b4 + b2 + 2) * q^17 - b6 * q^19 + (-2b6 + 3b5 + 2b3 - 3b1) * q^21 + (2b7 + 2b2) * q^23 + (b4 + 1) * q^25 + (-b4 - 4) * q^27 + 2b7 q^29 + (-b6 - 2b5 + b3 + 2b1) * q^31 - 4b6 q^33 + (3b7 - 3b4 + 7b2 + 4) q^35 + (4b3 + b1) q^37 - 4b3 q^41 + (b7 - b4 - 7b2 - 8) q^43 + (-2b6 + 2b5) * q^45 + (-3b6 - b5 + 3b3 + b1) * q^47 + (-5b7 - 6b2) * q^49 + (-b4 - 4) * q^51 + (4b4 + 6) q^53 + 8b2 q^55 + (2b5 - 2b1) * q^57 - 5b6 q^59 + (-2b7 + 2b4 - 8b2 - 6) q^61 + (3b3 - 4b1) * q^63 + (-3b3 - 2b1) * q^67 + (2b7 - 2b4 + 10b2 + 8) q^69 + (-b6 - 3b5) q^71 + (-6b5 + 6b1) * q^73 + 4b2 q^75 + (-4b4 + 4) q^77 + (-2b4 - 8) q^79 - 7b2 q^81 + (b6 + 4b5 - b3 - 4b1) * q^83 + (-2b6 - b5) q^85 + (8b2 + 8) q^87 - 2b1 q^89 - 4b3 q^93 + (2b7 - 2b4 + 2b2) q^95 - 4b5 q^97 + (-3b6 + 2b5 + 3b3 - 2b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 6 q^{9} + 6 q^{17} - 4 q^{23} + 4 q^{25} - 28 q^{27} + 4 q^{29} + 22 q^{35} - 30 q^{43} + 14 q^{49} - 28 q^{51} + 32 q^{53} - 32 q^{55} - 28 q^{61} + 36 q^{69} - 16 q^{75} + 48 q^{77} - 56 q^{79}+ \cdots + 4 q^{95}+O(q^{100}) \) 8 * q + 2 * q^3 - 6 * q^9 + 6 * q^17 - 4 * q^23 + 4 * q^25 - 28 * q^27 + 4 * q^29 + 22 * q^35 - 30 * q^43 + 14 * q^49 - 28 * q^51 + 32 * q^53 - 32 * q^55 - 28 * q^61 + 36 * q^69 - 16 * q^75 + 48 * q^77 - 56 * q^79 + 28 * q^81 + 32 * q^87 + 4 * q^95

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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	1352.2.o.e

Level	$1352$
Weight	$2$
Character orbit	1352.o
Analytic conductor	$10.796$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(10.7957743533\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.1731891456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 \) (9v^6 - 65v^4 + 585*v^2 - 1296) / 1040
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} + 181\nu ) / 130 \) (v^7 + 181*v) / 130
\(\beta_{4}\)	\(=\)	\( ( \nu^{6} + 116 ) / 65 \) (v^6 + 116) / 65
\(\beta_{5}\)	\(=\)	\( ( -9\nu^{7} + 65\nu^{5} - 585\nu^{3} + 1296\nu ) / 1040 \) (-9v^7 + 65v^5 - 585v^3 + 1296v) / 1040
\(\beta_{6}\)	\(=\)	\( ( -29\nu^{7} + 325\nu^{5} - 1885\nu^{3} + 4176\nu ) / 2080 \) (-29v^7 + 325v^5 - 1885v^3 + 4176v) / 2080
\(\beta_{7}\)	\(=\)	\( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \) (-29v^6 + 325v^4 - 1885*v^2 + 4176) / 1040

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{7} - \beta_{4} + 5\beta_{2} + 4 \) b7 - b4 + 5*b2 + 4
\(\nu^{3}\)	\(=\)	\( 2\beta_{6} - 5\beta_{5} - 2\beta_{3} + 5\beta_1 \) 2b6 - 5b5 - 2b3 + 5b1
\(\nu^{4}\)	\(=\)	\( 9\beta_{7} + 29\beta_{2} \) 9b7 + 29b2
\(\nu^{5}\)	\(=\)	\( 18\beta_{6} - 29\beta_{5} \) 18b6 - 29b5
\(\nu^{6}\)	\(=\)	\( 65\beta_{4} - 116 \) 65*b4 - 116
\(\nu^{7}\)	\(=\)	\( 130\beta_{3} - 181\beta_1 \) 130b3 - 181b1

\(n\)	\(677\)	\(1015\)	\(1185\)
\(\chi(n)\)	\(1\)	\(1\)	\(1 + \beta_{2}\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 361.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( (T^{4} - T^{3} + 5 T^{2} + \cdots + 16)^{2} \) (T^4 - T^3 + 5T^2 + 4T + 16)^2
$5$	\( (T^{4} + 9 T^{2} + 16)^{2} \) (T^4 + 9*T^2 + 16)^2
$7$	\( T^{8} - 21 T^{6} + \cdots + 16 \) T^8 - 21T^6 + 437T^4 - 84*T^2 + 16
$11$	\( T^{8} - 36 T^{6} + \cdots + 65536 \) T^8 - 36T^6 + 1040T^4 - 9216*T^2 + 65536
$13$	\( T^{8} \) T^8
$17$	\( (T^{4} - 3 T^{3} + 11 T^{2} + \cdots + 4)^{2} \) (T^4 - 3T^3 + 11T^2 + 6*T + 4)^2
$19$	\( (T^{4} - 4 T^{2} + 16)^{2} \) (T^4 - 4*T^2 + 16)^2
$23$	\( (T^{4} + 2 T^{3} + \cdots + 256)^{2} \) (T^4 + 2T^3 + 20T^2 - 32*T + 256)^2
$29$	\( (T^{4} - 2 T^{3} + \cdots + 256)^{2} \) (T^4 - 2T^3 + 20T^2 + 32*T + 256)^2
$31$	\( (T^{4} + 36 T^{2} + 256)^{2} \) (T^4 + 36*T^2 + 256)^2
$37$	\( T^{8} - 121 T^{6} + \cdots + 7311616 \) T^8 - 121T^6 + 11937T^4 - 327184*T^2 + 7311616
$41$	\( (T^{4} - 64 T^{2} + 4096)^{2} \) (T^4 - 64*T^2 + 4096)^2
$43$	\( (T^{4} + 15 T^{3} + \cdots + 2704)^{2} \) (T^4 + 15T^3 + 173T^2 + 780*T + 2704)^2
$47$	\( (T^{4} + 69 T^{2} + 676)^{2} \) (T^4 + 69*T^2 + 676)^2
$53$	\( (T^{2} - 8 T - 52)^{4} \) (T^2 - 8*T - 52)^4
$59$	\( (T^{4} - 100 T^{2} + 10000)^{2} \) (T^4 - 100*T^2 + 10000)^2
$61$	\( (T^{4} + 14 T^{3} + \cdots + 1024)^{2} \) (T^4 + 14T^3 + 164T^2 + 448*T + 1024)^2
$67$	\( T^{8} - 84 T^{6} + \cdots + 4096 \) T^8 - 84T^6 + 6992T^4 - 5376*T^2 + 4096
$71$	\( T^{8} - 77 T^{6} + \cdots + 2085136 \) T^8 - 77T^6 + 4485T^4 - 111188*T^2 + 2085136
$73$	\( (T^{4} + 324 T^{2} + 20736)^{2} \) (T^4 + 324*T^2 + 20736)^2
$79$	\( (T^{2} + 14 T + 32)^{4} \) (T^2 + 14*T + 32)^4
$83$	\( (T^{2} + 68)^{4} \) (T^2 + 68)^4
$89$	\( T^{8} - 36 T^{6} + \cdots + 65536 \) T^8 - 36T^6 + 1040T^4 - 9216*T^2 + 65536
$97$	\( T^{8} - 144 T^{6} + \cdots + 16777216 \) T^8 - 144T^6 + 16640T^4 - 589824*T^2 + 16777216
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