LMFDB - Newform orbit 513.2.m.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(107,513)] 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("513.107"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.m (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-12,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$4.09632562369$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.2569273344.7
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 78x^{4} - 220x^{2} + 484 $ x^8 - 10x^6 + 78x^4 - 220*x^2 + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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} q^{2} + (\beta_{6} + 3 \beta_{5} + 2 \beta_{2}) q^{4} + ( - \beta_{4} + \beta_1) q^{5} + ( - \beta_{6} + \beta_{2} + 1) q^{7} + ( - \beta_{4} - \beta_{3} + 2 \beta_1) q^{8} + ( - 5 \beta_{6} - \beta_{5} - 2) q^{10}+ \cdots + (3 \beta_{7} + 2 \beta_{4} + 2 \beta_1) q^{98}+O(q^{100}) $ q - b7 * q^2 + (b6 + 3b5 + 2b2) * q^4 + (-b4 + b1) * q^5 + (-b6 + b2 + 1) * q^7 + (-b4 - b3 + 2b1) q^8 + (-5b6 - b5 - 2) q^10 + (2b7 - b4 + b3) q^11 + (4b6 - b5 - 2) q^13 + (-b7 + b4 + b1) * q^14 + (-8b6 - 2b5 - 4b2 - 2) q^16 + (-3b4 + 3b1) * q^17 + (-2b6 - 5b2) * q^19 + (2b7 + 3b4 + b3) * q^20 + (-4b5 + 2b2 + 4) * q^22 - 2b1 q^23 + (-b6 - 2b2) q^25 + (2b7 - 4b4 + b3) * q^26 + (-2b6 - 4b2) * q^28 + (-4b7 + 2b4 - 4b3 - b1) q^29 + (-2b6 + 10b5 - 2b2 + 5) q^31 + (4b4 - 2b1) * q^32 + (-15b6 - 3b5 - 6) * q^34 + (b7 - b4 + 2b3 + b1) q^35 + (-6b6 - 6b5 - 6b2 - 3) q^37 + (2b4 - 5b1) * q^38 + (-6b5 - 8b2 + 6) * q^40 + (-b7 - b4 - b1) * q^41 + (4b6 + 5b5 + 2b2 + 5) q^43 + (-2b7 + 2b3) * q^44 + (10b6 + 4b5 + 10b2 + 2) q^46 + 2b1 q^47 + (-2b6 + 2b2 - 3) * q^49 + (b4 - 2b1) q^50 + (b5 + 9b2 - 1) q^52 + (2b4 - b1) q^53 + (8b6 - 2b5 + 4b2 - 2) q^55 + 2b3 q^56 + (b6 - b2 - 17) * q^58 + (b7 - 2b4 - 2b1) * q^59 + (b6 + 2b5 + 2b2) * q^61 + (-5b7 + 2b4 - 10b3 - 2b1) * q^62 + (2b6 - 2b2 + 2) * q^64 + (b4 - 4b3 - 2b1) * q^65 + (5b6 - 3b5 - 6) * q^67 + (6b7 + 9b4 + 3b3) q^68 + (-2b6 + 4b5 + 8) * q^70 + (6b6 + 9b5 + 3b2 + 9) q^73 + (3b7 + 6b4 + 6b3 - 6b1) * q^74 + (15b6 + 8b5 + 9b2 + 7) q^76 + (4b7 - 4b4 + 2b3) q^77 + (-5b5 + 2b2 + 5) * q^79 + (-4b7 + 4b3 - 2b1) q^80 + (6b6 + 8b5 + 12b2) q^82 - b4 * q^83 + (-3b6 - 15b5 - 6b2) q^85 + (-5b7 - 4b4 - 5b3 + 2b1) * q^86 + (10b6 + 4b5 + 10b2 + 2) q^88 + (3b7 - 2b4 + 3b3 + b1) q^89 + (7b6 - 5b5 - 10) * q^91 + (-2b7 - 6b4 - 4b3 + 6b1) * q^92 + (-10b6 - 4b5 - 10b2 - 2) q^94 + (-5b7 - 3b3) * q^95 + (-4b5 - 5b2 + 4) * q^97 + (3b7 + 2b4 + 2b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{4} + 8 q^{7} - 12 q^{10} - 12 q^{13} - 8 q^{16} + 48 q^{22} - 36 q^{34} + 72 q^{40} + 20 q^{43} - 24 q^{49} - 12 q^{52} - 8 q^{55} - 136 q^{58} - 8 q^{61} + 16 q^{64} - 36 q^{67} + 48 q^{70}+ \cdots + 48 q^{97}+O(q^{100}) $ 8 * q - 12 * q^4 + 8 * q^7 - 12 * q^10 - 12 * q^13 - 8 * q^16 + 48 * q^22 - 36 * q^34 + 72 * q^40 + 20 * q^43 - 24 * q^49 - 12 * q^52 - 8 * q^55 - 136 * q^58 - 8 * q^61 + 16 * q^64 - 36 * q^67 + 48 * q^70 + 36 * q^73 + 24 * q^76 + 60 * q^79 - 32 * q^82 + 60 * q^85 - 60 * q^91 + 48 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 10x^{6} + 78x^{4} - 220x^{2} + 484 $ x^8 - 10*x^6 + 78*x^4 - 220*x^2 + 484 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{6} + 65\nu^{4} - 364\nu^{2} + 1650 ) / 858 $ (-v^6 + 65v^4 - 364v^2 + 1650) / 858
$\beta_{3}$	$=$	$ ( -\nu^{7} + 65\nu^{5} - 364\nu^{3} + 1650\nu ) / 858 $ (-v^7 + 65v^5 - 364v^3 + 1650*v) / 858
$\beta_{4}$	$=$	$ ( 5\nu^{7} - 39\nu^{5} + 390\nu^{3} - 242\nu ) / 858 $ (5v^7 - 39v^5 + 390v^3 - 242v) / 858
$\beta_{5}$	$=$	$ ( 5\nu^{6} - 39\nu^{4} + 390\nu^{2} - 1100 ) / 858 $ (5v^6 - 39v^4 + 390*v^2 - 1100) / 858
$\beta_{6}$	$=$	$ ( -12\nu^{6} + 65\nu^{4} - 364\nu^{2} - 220 ) / 858 $ (-12v^6 + 65v^4 - 364*v^2 - 220) / 858
$\beta_{7}$	$=$	$ ( -12\nu^{7} + 65\nu^{5} - 364\nu^{3} - 220\nu ) / 858 $ (-12v^7 + 65v^5 - 364v^3 - 220v) / 858

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{6} + 5\beta_{5} + \beta_{2} + 5 $ 2b6 + 5b5 + b2 + 5
$\nu^{3}$	$=$	$ 2\beta_{7} + 5\beta_{4} + \beta_{3} $ 2b7 + 5b4 + b3
$\nu^{4}$	$=$	$ 10\beta_{6} + 28\beta_{5} + 20\beta_{2} $ 10b6 + 28b5 + 20*b2
$\nu^{5}$	$=$	$ 10\beta_{7} + 28\beta_{4} + 20\beta_{3} - 28\beta_1 $ 10b7 + 28b4 + 20b3 - 28b1
$\nu^{6}$	$=$	$ -78\beta_{6} + 78\beta_{2} - 170 $ -78b6 + 78b2 - 170
$\nu^{7}$	$=$	$ -78\beta_{7} + 78\beta_{3} - 170\beta_1 $ -78b7 + 78b3 - 170*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/513\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$-1$	$-\beta_{5}$

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

2.24701	+	1.29731i
−1.56555	−	0.903873i
1.56555	+	0.903873i
−2.24701	−	1.29731i
2.24701	−	1.29731i
−1.56555	+	0.903873i
1.56555	−	0.903873i
−2.24701	+	1.29731i

−1.29731

−

2.24701i

−2.36603

4.09808i

2.24701

−

1.29731i

−0.732051

7.08863

−5.83013

−

3.36603i

107.2

−0.903873

−

1.56555i

−0.633975

1.09808i

−1.56555

0.903873i

2.73205

−1.32336

2.83013

1.63397i

107.3

0.903873

1.56555i

−0.633975

1.09808i

1.56555

−

0.903873i

2.73205

1.32336

2.83013

1.63397i

107.4

1.29731

2.24701i

−2.36603

4.09808i

−2.24701

1.29731i

−0.732051

−7.08863

−5.83013

−

3.36603i

350.1

−1.29731

2.24701i

−2.36603

−

4.09808i

2.24701

1.29731i

−0.732051

7.08863

−5.83013

3.36603i

350.2

−0.903873

1.56555i

−0.633975

−

1.09808i

−1.56555

−

0.903873i

2.73205

−1.32336

2.83013

−

1.63397i

350.3

0.903873

−

1.56555i

−0.633975

−

1.09808i

1.56555

0.903873i

2.73205

1.32336

2.83013

−

1.63397i

350.4

1.29731

−

2.24701i

−2.36603

−

4.09808i

−2.24701

−

1.29731i

−0.732051

−7.08863

−5.83013

3.36603i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
19.d	odd	6	1	inner
57.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	513.2.m.e	✓	8
3.b	odd	2	1	inner	513.2.m.e	✓	8
19.d	odd	6	1	inner	513.2.m.e	✓	8
57.f	even	6	1	inner	513.2.m.e	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
513.2.m.e	✓	8	1.a	even	1	1	trivial
513.2.m.e	✓	8	3.b	odd	2	1	inner
513.2.m.e	✓	8	19.d	odd	6	1	inner
513.2.m.e	✓	8	57.f	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(513, [\chi])$:

$ T_{2}^{8} + 10T_{2}^{6} + 78T_{2}^{4} + 220T_{2}^{2} + 484 $

T2^8 + 10*T2^6 + 78*T2^4 + 220*T2^2 + 484

$ T_{5}^{8} - 10T_{5}^{6} + 78T_{5}^{4} - 220T_{5}^{2} + 484 $

T5^8 - 10*T5^6 + 78*T5^4 - 220*T5^2 + 484

$ T_{7}^{2} - 2T_{7} - 2 $

T7^2 - 2*T7 - 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 10 T^{6} + \cdots + 484 $ T^8 + 10T^6 + 78T^4 + 220*T^2 + 484
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - 10 T^{6} + \cdots + 484 $ T^8 - 10T^6 + 78T^4 - 220*T^2 + 484
$7$	$ (T^{2} - 2 T - 2)^{4} $ (T^2 - 2*T - 2)^4
$11$	$ (T^{4} + 28 T^{2} + 88)^{2} $ (T^4 + 28*T^2 + 88)^2
$13$	$ (T^{4} + 6 T^{3} + \cdots + 169)^{2} $ (T^4 + 6T^3 - T^2 - 78T + 169)^2
$17$	$ T^{8} - 90 T^{6} + \cdots + 3175524 $ T^8 - 90T^6 + 6318T^4 - 160380*T^2 + 3175524
$19$	$ (T^{4} + 11 T^{2} + 361)^{2} $ (T^4 + 11*T^2 + 361)^2
$23$	$ T^{8} - 40 T^{6} + \cdots + 123904 $ T^8 - 40T^6 + 1248T^4 - 14080*T^2 + 123904
$29$	$ T^{8} + 142 T^{6} + \cdots + 13823524 $ T^8 + 142T^6 + 16446T^4 + 527956*T^2 + 13823524
$31$	$ (T^{4} + 158 T^{2} + 5041)^{2} $ (T^4 + 158*T^2 + 5041)^2
$37$	$ (T^{4} + 126 T^{2} + 81)^{2} $ (T^4 + 126*T^2 + 81)^2
$41$	$ T^{8} + 52 T^{6} + \cdots + 7744 $ T^8 + 52T^6 + 2616T^4 + 4576*T^2 + 7744
$43$	$ (T^{4} - 10 T^{3} + \cdots + 169)^{2} $ (T^4 - 10T^3 + 87T^2 - 130*T + 169)^2
$47$	$ T^{8} - 40 T^{6} + \cdots + 123904 $ T^8 - 40T^6 + 1248T^4 - 14080*T^2 + 123904
$53$	$ T^{8} + 30 T^{6} + \cdots + 39204 $ T^8 + 30T^6 + 702T^4 + 5940*T^2 + 39204
$59$	$ T^{8} + 106 T^{6} + \cdots + 7086244 $ T^8 + 106T^6 + 8574T^4 + 282172*T^2 + 7086244
$61$	$ (T^{4} + 4 T^{3} + 15 T^{2} + \cdots + 1)^{2} $ (T^4 + 4T^3 + 15T^2 + 4*T + 1)^2
$67$	$ (T^{4} + 18 T^{3} + 110 T^{2} + \cdots + 4)^{2} $ (T^4 + 18T^3 + 110T^2 + 36*T + 4)^2
$71$	$ T^{8} $ T^8
$73$	$ (T^{4} - 18 T^{3} + \cdots + 2916)^{2} $ (T^4 - 18T^3 + 270T^2 - 972*T + 2916)^2
$79$	$ (T^{4} - 30 T^{3} + \cdots + 5041)^{2} $ (T^4 - 30T^3 + 371T^2 - 2130*T + 5041)^2
$83$	$ (T^{4} + 10 T^{2} + 22)^{2} $ (T^4 + 10*T^2 + 22)^2
$89$	$ T^{8} + 84 T^{6} + \cdots + 627264 $ T^8 + 84T^6 + 6264T^4 + 66528*T^2 + 627264
$97$	$ (T^{4} - 24 T^{3} + \cdots + 529)^{2} $ (T^4 - 24T^3 + 215T^2 - 552*T + 529)^2
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Elliptic curves over $\Q$ over number fields
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Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.m (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{2} + (\beta_{6} + 3 \beta_{5} + 2 \beta_{2}) q^{4} + ( - \beta_{4} + \beta_1) q^{5} + ( - \beta_{6} + \beta_{2} + 1) q^{7} + ( - \beta_{4} - \beta_{3} + 2 \beta_1) q^{8} + ( - 5 \beta_{6} - \beta_{5} - 2) q^{10}+ \cdots + (3 \beta_{7} + 2 \beta_{4} + 2 \beta_1) q^{98}+O(q^{100}) \) q - b7 * q^2 + (b6 + 3b5 + 2b2) * q^4 + (-b4 + b1) * q^5 + (-b6 + b2 + 1) * q^7 + (-b4 - b3 + 2b1) q^8 + (-5b6 - b5 - 2) q^10 + (2b7 - b4 + b3) q^11 + (4b6 - b5 - 2) q^13 + (-b7 + b4 + b1) * q^14 + (-8b6 - 2b5 - 4b2 - 2) q^16 + (-3b4 + 3b1) * q^17 + (-2b6 - 5b2) * q^19 + (2b7 + 3b4 + b3) * q^20 + (-4b5 + 2b2 + 4) * q^22 - 2b1 q^23 + (-b6 - 2b2) q^25 + (2b7 - 4b4 + b3) * q^26 + (-2b6 - 4b2) * q^28 + (-4b7 + 2b4 - 4b3 - b1) q^29 + (-2b6 + 10b5 - 2b2 + 5) q^31 + (4b4 - 2b1) * q^32 + (-15b6 - 3b5 - 6) * q^34 + (b7 - b4 + 2b3 + b1) q^35 + (-6b6 - 6b5 - 6b2 - 3) q^37 + (2b4 - 5b1) * q^38 + (-6b5 - 8b2 + 6) * q^40 + (-b7 - b4 - b1) * q^41 + (4b6 + 5b5 + 2b2 + 5) q^43 + (-2b7 + 2b3) * q^44 + (10b6 + 4b5 + 10b2 + 2) q^46 + 2b1 q^47 + (-2b6 + 2b2 - 3) * q^49 + (b4 - 2b1) q^50 + (b5 + 9b2 - 1) q^52 + (2b4 - b1) q^53 + (8b6 - 2b5 + 4b2 - 2) q^55 + 2b3 q^56 + (b6 - b2 - 17) * q^58 + (b7 - 2b4 - 2b1) * q^59 + (b6 + 2b5 + 2b2) * q^61 + (-5b7 + 2b4 - 10b3 - 2b1) * q^62 + (2b6 - 2b2 + 2) * q^64 + (b4 - 4b3 - 2b1) * q^65 + (5b6 - 3b5 - 6) * q^67 + (6b7 + 9b4 + 3b3) q^68 + (-2b6 + 4b5 + 8) * q^70 + (6b6 + 9b5 + 3b2 + 9) q^73 + (3b7 + 6b4 + 6b3 - 6b1) * q^74 + (15b6 + 8b5 + 9b2 + 7) q^76 + (4b7 - 4b4 + 2b3) q^77 + (-5b5 + 2b2 + 5) * q^79 + (-4b7 + 4b3 - 2b1) q^80 + (6b6 + 8b5 + 12b2) q^82 - b4 * q^83 + (-3b6 - 15b5 - 6b2) q^85 + (-5b7 - 4b4 - 5b3 + 2b1) * q^86 + (10b6 + 4b5 + 10b2 + 2) q^88 + (3b7 - 2b4 + 3b3 + b1) q^89 + (7b6 - 5b5 - 10) * q^91 + (-2b7 - 6b4 - 4b3 + 6b1) * q^92 + (-10b6 - 4b5 - 10b2 - 2) q^94 + (-5b7 - 3b3) * q^95 + (-4b5 - 5b2 + 4) * q^97 + (3b7 + 2b4 + 2b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{4} + 8 q^{7} - 12 q^{10} - 12 q^{13} - 8 q^{16} + 48 q^{22} - 36 q^{34} + 72 q^{40} + 20 q^{43} - 24 q^{49} - 12 q^{52} - 8 q^{55} - 136 q^{58} - 8 q^{61} + 16 q^{64} - 36 q^{67} + 48 q^{70}+ \cdots + 48 q^{97}+O(q^{100}) \) 8 * q - 12 * q^4 + 8 * q^7 - 12 * q^10 - 12 * q^13 - 8 * q^16 + 48 * q^22 - 36 * q^34 + 72 * q^40 + 20 * q^43 - 24 * q^49 - 12 * q^52 - 8 * q^55 - 136 * q^58 - 8 * q^61 + 16 * q^64 - 36 * q^67 + 48 * q^70 + 36 * q^73 + 24 * q^76 + 60 * q^79 - 32 * q^82 + 60 * q^85 - 60 * q^91 + 48 * q^97

Introduction

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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	513.2.m.e

Level	$513$
Weight	$2$
Character orbit	513.m
Analytic conductor	$4.096$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.09632562369\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.2569273344.7
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - 10x^{6} + 78x^{4} - 220x^{2} + 484 \) x^8 - 10x^6 + 78x^4 - 220*x^2 + 484
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -\nu^{6} + 65\nu^{4} - 364\nu^{2} + 1650 ) / 858 \) (-v^6 + 65v^4 - 364v^2 + 1650) / 858
\(\beta_{3}\)	\(=\)	\( ( -\nu^{7} + 65\nu^{5} - 364\nu^{3} + 1650\nu ) / 858 \) (-v^7 + 65v^5 - 364v^3 + 1650*v) / 858
\(\beta_{4}\)	\(=\)	\( ( 5\nu^{7} - 39\nu^{5} + 390\nu^{3} - 242\nu ) / 858 \) (5v^7 - 39v^5 + 390v^3 - 242v) / 858
\(\beta_{5}\)	\(=\)	\( ( 5\nu^{6} - 39\nu^{4} + 390\nu^{2} - 1100 ) / 858 \) (5v^6 - 39v^4 + 390*v^2 - 1100) / 858
\(\beta_{6}\)	\(=\)	\( ( -12\nu^{6} + 65\nu^{4} - 364\nu^{2} - 220 ) / 858 \) (-12v^6 + 65v^4 - 364*v^2 - 220) / 858
\(\beta_{7}\)	\(=\)	\( ( -12\nu^{7} + 65\nu^{5} - 364\nu^{3} - 220\nu ) / 858 \) (-12v^7 + 65v^5 - 364v^3 - 220v) / 858

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{6} + 5\beta_{5} + \beta_{2} + 5 \) 2b6 + 5b5 + b2 + 5
\(\nu^{3}\)	\(=\)	\( 2\beta_{7} + 5\beta_{4} + \beta_{3} \) 2b7 + 5b4 + b3
\(\nu^{4}\)	\(=\)	\( 10\beta_{6} + 28\beta_{5} + 20\beta_{2} \) 10b6 + 28b5 + 20*b2
\(\nu^{5}\)	\(=\)	\( 10\beta_{7} + 28\beta_{4} + 20\beta_{3} - 28\beta_1 \) 10b7 + 28b4 + 20b3 - 28b1
\(\nu^{6}\)	\(=\)	\( -78\beta_{6} + 78\beta_{2} - 170 \) -78b6 + 78b2 - 170
\(\nu^{7}\)	\(=\)	\( -78\beta_{7} + 78\beta_{3} - 170\beta_1 \) -78b7 + 78b3 - 170*b1

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 107.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 10 T^{6} + \cdots + 484 \) T^8 + 10T^6 + 78T^4 + 220*T^2 + 484
$3$	\( T^{8} \) T^8
$5$	\( T^{8} - 10 T^{6} + \cdots + 484 \) T^8 - 10T^6 + 78T^4 - 220*T^2 + 484
$7$	\( (T^{2} - 2 T - 2)^{4} \) (T^2 - 2*T - 2)^4
$11$	\( (T^{4} + 28 T^{2} + 88)^{2} \) (T^4 + 28*T^2 + 88)^2
$13$	\( (T^{4} + 6 T^{3} + \cdots + 169)^{2} \) (T^4 + 6T^3 - T^2 - 78T + 169)^2
$17$	\( T^{8} - 90 T^{6} + \cdots + 3175524 \) T^8 - 90T^6 + 6318T^4 - 160380*T^2 + 3175524
$19$	\( (T^{4} + 11 T^{2} + 361)^{2} \) (T^4 + 11*T^2 + 361)^2
$23$	\( T^{8} - 40 T^{6} + \cdots + 123904 \) T^8 - 40T^6 + 1248T^4 - 14080*T^2 + 123904
$29$	\( T^{8} + 142 T^{6} + \cdots + 13823524 \) T^8 + 142T^6 + 16446T^4 + 527956*T^2 + 13823524
$31$	\( (T^{4} + 158 T^{2} + 5041)^{2} \) (T^4 + 158*T^2 + 5041)^2
$37$	\( (T^{4} + 126 T^{2} + 81)^{2} \) (T^4 + 126*T^2 + 81)^2
$41$	\( T^{8} + 52 T^{6} + \cdots + 7744 \) T^8 + 52T^6 + 2616T^4 + 4576*T^2 + 7744
$43$	\( (T^{4} - 10 T^{3} + \cdots + 169)^{2} \) (T^4 - 10T^3 + 87T^2 - 130*T + 169)^2
$47$	\( T^{8} - 40 T^{6} + \cdots + 123904 \) T^8 - 40T^6 + 1248T^4 - 14080*T^2 + 123904
$53$	\( T^{8} + 30 T^{6} + \cdots + 39204 \) T^8 + 30T^6 + 702T^4 + 5940*T^2 + 39204
$59$	\( T^{8} + 106 T^{6} + \cdots + 7086244 \) T^8 + 106T^6 + 8574T^4 + 282172*T^2 + 7086244
$61$	\( (T^{4} + 4 T^{3} + 15 T^{2} + \cdots + 1)^{2} \) (T^4 + 4T^3 + 15T^2 + 4*T + 1)^2
$67$	\( (T^{4} + 18 T^{3} + 110 T^{2} + \cdots + 4)^{2} \) (T^4 + 18T^3 + 110T^2 + 36*T + 4)^2
$71$	\( T^{8} \) T^8
$73$	\( (T^{4} - 18 T^{3} + \cdots + 2916)^{2} \) (T^4 - 18T^3 + 270T^2 - 972*T + 2916)^2
$79$	\( (T^{4} - 30 T^{3} + \cdots + 5041)^{2} \) (T^4 - 30T^3 + 371T^2 - 2130*T + 5041)^2
$83$	\( (T^{4} + 10 T^{2} + 22)^{2} \) (T^4 + 10*T^2 + 22)^2
$89$	\( T^{8} + 84 T^{6} + \cdots + 627264 \) T^8 + 84T^6 + 6264T^4 + 66528*T^2 + 627264
$97$	\( (T^{4} - 24 T^{3} + \cdots + 529)^{2} \) (T^4 - 24T^3 + 215T^2 - 552*T + 529)^2
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\(n\)	\(191\)	\(325\)
\(\chi(n)\)	\(-1\)	\(-\beta_{5}\)