LMFDB - Newform orbit 1680.2.ba.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(911,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.911"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.ba (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-2,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.7278137344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + x^{6} + 6x^{5} - 20x^{4} + 18x^{3} + 9x^{2} - 54x + 81 $ x^8 - 2x^7 + x^6 + 6x^5 - 20x^4 + 18x^3 + 9x^2 - 54x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 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_{4} q^{3} + \beta_{2} q^{5} - \beta_{2} q^{7} + ( - \beta_{7} + \beta_{2} + \beta_1) q^{9} + ( - \beta_{7} - \beta_{6} - \beta_{4} - 1) q^{11} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots - 1) q^{13}+ \cdots + (2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots + 4) q^{99}+O(q^{100}) $ q + b4 * q^3 + b2 * q^5 - b2 * q^7 + (-b7 + b2 + b1) * q^9 + (-b7 - b6 - b4 - 1) * q^11 + (b7 + 2b6 + b5 + 2b4 + b1 - 1) * q^13 + b5 * q^15 + (b6 - b4 - 2b2) q^17 + (b6 + b5 - b4 + b3 - b1) * q^19 - b5 * q^21 - 2b7 q^23 - q^25 + (2b6 + 2b5 - b3 - 2b2 + 1) q^27 + (2b6 - b5 - 2b4 + 2b3 - 2b2 + b1) * q^29 + (-b6 + b5 + b4 - b3 - b1) * q^31 + (b7 + 2b6 + b5 - b4 - b3 - b1 - 2) q^33 + q^35 + (2b6 - 2b5 + 2b4 - 2b1 + 2) * q^37 + (-2b7 - b6 - b5 - b4 + 2b3 - 3b2 + 2b1 + 4) * q^39 - 2b3 q^41 + (-2b3 + 2b2) * q^43 + (b6 + b3 - b2 - 1) * q^45 + (b6 + 2b5 + b4 + 2b1 - 6) * q^47 - q^49 + (b7 - 2b5 - b2 - b1 + 3) q^51 + (3b6 + b5 - 3b4 - b3 - b1) * q^53 + (-b5 + b3 - 2b2 + b1) q^55 + (b6 + b5 - b4 + b3 + 2b2 - 3b1 + 2) * q^57 + (-2b6 - 2b5 - 2b4 - 2b1) * q^59 + (-2b7 + 2b6 + 2b4) q^61 + (-b6 - b3 + b2 + 1) * q^63 + (b6 + 2b5 - b4 - b3 - 2b1) * q^65 + (-2b6 - 2b5 + 2b4 + 8b2 + 2b1) q^67 + (4b6 + 2b5 - 2b3 + 2b2 + 2) * q^69 + (b7 + 3b6 + 3b5 + 3b4 + 3b1 - 7) * q^71 + (-b7 + b6 + b5 + b4 + b1 + 3) * q^73 - b4 * q^75 + (b5 - b3 + 2b2 - b1) q^77 + (2b6 + b5 - 2b4 + 2b3 - 2b2 - b1) * q^79 + (b7 + 2b6 - 3b5 + 2b4 + 2b3 - 3b2 + 2b1 + 4) * q^81 + (4b7 + 2b6 + 2b4 - 8) q^83 + (-b5 - b1 + 2) * q^85 + (-b6 - 2b4 - b3 - 2b2 - 6b1 + 7) q^87 + (-2b6 - 4b5 + 2b4 + 2b2 + 4b1) q^89 + (-b6 - 2b5 + b4 + b3 + 2b1) * q^91 + (b6 - b5 + b4 + b3 + 2b2 + 3b1 - 4) * q^93 + (b7 - b6 - b5 - b4 - b1 - 1) * q^95 + (-b7 - b5 - b1 - 7) * q^97 + (2b7 - b6 - 2b5 - b4 + 2b3 - b2 + b1 + 4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{9} - 12 q^{11} + 2 q^{15} - 2 q^{21} - 8 q^{23} - 8 q^{25} + 12 q^{27} - 12 q^{33} + 8 q^{35} + 8 q^{37} + 26 q^{39} - 8 q^{45} - 40 q^{47} - 8 q^{49} + 22 q^{51} + 12 q^{57} - 8 q^{59} - 8 q^{61}+ \cdots + 38 q^{99}+O(q^{100}) $ 8 * q - 2 * q^9 - 12 * q^11 + 2 * q^15 - 2 * q^21 - 8 * q^23 - 8 * q^25 + 12 * q^27 - 12 * q^33 + 8 * q^35 + 8 * q^37 + 26 * q^39 - 8 * q^45 - 40 * q^47 - 8 * q^49 + 22 * q^51 + 12 * q^57 - 8 * q^59 - 8 * q^61 + 8 * q^63 + 20 * q^69 - 40 * q^71 + 24 * q^73 + 34 * q^81 - 48 * q^83 + 12 * q^85 + 44 * q^87 - 28 * q^93 - 8 * q^95 - 64 * q^97 + 38 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{7} + x^{6} + 6x^{5} - 20x^{4} + 18x^{3} + 9x^{2} - 54x + 81 $ x^8 - 2*x^7 + x^6 + 6*x^5 - 20*x^4 + 18*x^3 + 9*x^2 - 54*x + 81 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{7} + \nu^{6} - 5\nu^{5} + 9\nu^{4} - 2\nu^{3} - 15\nu^{2} + 36\nu - 27 ) / 54 $ (v^7 + v^6 - 5v^5 + 9v^4 - 2v^3 - 15v^2 + 36*v - 27) / 54
$\beta_{3}$	$=$	$ ( -\nu^{5} - \nu^{4} + 2\nu^{3} - 3\nu^{2} - \nu + 12 ) / 6 $ (-v^5 - v^4 + 2v^3 - 3v^2 - v + 12) / 6
$\beta_{4}$	$=$	$ ( -\nu^{7} - \nu^{6} - 4\nu^{5} + 9\nu^{4} - 7\nu^{3} - 12\nu^{2} + 36\nu - 54 ) / 54 $ (-v^7 - v^6 - 4v^5 + 9v^4 - 7v^3 - 12v^2 + 36*v - 54) / 54
$\beta_{5}$	$=$	$ ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 6\nu^{4} + 20\nu^{3} - 18\nu^{2} - 9\nu + 54 ) / 27 $ (-v^7 + 2v^6 - v^5 - 6v^4 + 20v^3 - 18v^2 - 9*v + 54) / 27
$\beta_{6}$	$=$	$ ( \nu^{7} - 2\nu^{6} + \nu^{5} + 6\nu^{4} - 11\nu^{3} + 9\nu^{2} + 9\nu - 27 ) / 18 $ (v^7 - 2v^6 + v^5 + 6v^4 - 11v^3 + 9v^2 + 9*v - 27) / 18
$\beta_{7}$	$=$	$ ( -\nu^{7} + \nu^{6} + \nu^{5} - 7\nu^{4} + 14\nu^{3} + 11\nu^{2} - 18\nu + 45 ) / 18 $ (-v^7 + v^6 + v^5 - 7v^4 + 14v^3 + 11v^2 - 18v + 45) / 18

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{7} - \beta_{5} + \beta_{2} $ b7 - b5 + b2
$\nu^{3}$	$=$	$ \beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{2} - 3 $ b7 + 2b6 + 2b5 + b2 - 3
$\nu^{4}$	$=$	$ \beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 2\beta_{3} + 3\beta_{2} - 3\beta _1 + 4 $ b7 + 2b6 + 2b5 + 2b4 - 2b3 + 3b2 - 3b1 + 4
$\nu^{5}$	$=$	$ -2\beta_{7} + 2\beta_{6} + 5\beta_{5} - 2\beta_{4} - 4\beta_{3} - 4\beta_{2} + 2\beta _1 + 2 $ -2b7 + 2b6 + 5b5 - 2b4 - 4b3 - 4b2 + 2*b1 + 2
$\nu^{6}$	$=$	$ -10\beta_{6} - 6\beta_{5} - 6\beta_{4} - 6\beta_{3} + 12\beta_{2} - 2\beta _1 + 9 $ -10b6 - 6b5 - 6b4 - 6b3 + 12b2 - 2b1 + 9
$\nu^{7}$	$=$	$ -2\beta_{7} + 6\beta_{6} + 2\beta_{5} - 22\beta_{4} + 4\beta_{3} + 12\beta_{2} + 3\beta _1 - 14 $ -2b7 + 6b6 + 2b5 - 22b4 + 4b3 + 12b2 + 3*b1 - 14

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$-1$	$-1$

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} $

911.1

0.828750	+	1.52091i
0.828750	−	1.52091i
1.67936	+	0.423958i
1.67936	−	0.423958i
−1.71731	+	0.225499i
−1.71731	−	0.225499i
0.209196	−	1.71937i
0.209196	+	1.71937i

−1.52091

−

0.828750i

1.00000i

−

1.00000i

1.62635

2.52091i

911.2

−1.52091

0.828750i

−

1.00000i

1.62635

−

2.52091i

911.3

−0.423958

−

1.67936i

1.00000i

−

1.00000i

−2.64052

1.42396i

911.4

−0.423958

1.67936i

−

1.00000i

−2.64052

−

1.42396i

911.5

0.225499

−

1.71731i

−

1.00000i

−2.89830

−

0.774501i

911.6

0.225499

1.71731i

1.00000i

−

1.00000i

−2.89830

0.774501i

911.7

1.71937

−

0.209196i

1.00000i

−

1.00000i

2.91247

−

0.719371i

911.8

1.71937

0.209196i

−

1.00000i

2.91247

0.719371i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1680.2.ba.a	✓	8
3.b	odd	2	1		1680.2.ba.b	yes	8
4.b	odd	2	1		1680.2.ba.b	yes	8
12.b	even	2	1	inner	1680.2.ba.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1680.2.ba.a	✓	8	1.a	even	1	1	trivial
1680.2.ba.a	✓	8	12.b	even	2	1	inner
1680.2.ba.b	yes	8	3.b	odd	2	1
1680.2.ba.b	yes	8	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{4} + 6T_{11}^{3} - T_{11}^{2} - 48T_{11} - 58 $ T11^4 + 6*T11^3 - T11^2 - 48*T11 - 58 acting on $S_{2}^{\mathrm{new}}(1680, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + T^{6} + \cdots + 81 $ T^8 + T^6 - 4T^5 - 8T^4 - 12T^3 + 9T^2 + 81
$5$	$ (T^{2} + 1)^{4} $ (T^2 + 1)^4
$7$	$ (T^{2} + 1)^{4} $ (T^2 + 1)^4
$11$	$ (T^{4} + 6 T^{3} - T^{2} + \cdots - 58)^{2} $ (T^4 + 6T^3 - T^2 - 48T - 58)^2
$13$	$ (T^{4} - 41 T^{2} + \cdots + 262)^{2} $ (T^4 - 41T^2 + 28T + 262)^2
$17$	$ T^{8} + 34 T^{6} + \cdots + 16 $ T^8 + 34T^6 + 137T^4 + 152*T^2 + 16
$19$	$ T^{8} + 68 T^{6} + \cdots + 64 $ T^8 + 68T^6 + 628T^4 + 480*T^2 + 64
$23$	$ (T^{4} + 4 T^{3} + \cdots + 176)^{2} $ (T^4 + 4T^3 - 48T^2 - 48*T + 176)^2
$29$	$ T^{8} + 166 T^{6} + \cdots + 53824 $ T^8 + 166T^6 + 7353T^4 + 54896*T^2 + 53824
$31$	$ T^{8} + 60 T^{6} + \cdots + 256 $ T^8 + 60T^6 + 772T^4 + 896*T^2 + 256
$37$	$ (T^{4} - 4 T^{3} + \cdots + 3728)^{2} $ (T^4 - 4T^3 - 120T^2 + 240*T + 3728)^2
$41$	$ T^{8} + 112 T^{6} + \cdots + 4096 $ T^8 + 112T^6 + 3712T^4 + 31744*T^2 + 4096
$43$	$ T^{8} + 112 T^{6} + \cdots + 30976 $ T^8 + 112T^6 + 3040T^4 + 19200*T^2 + 30976
$47$	$ (T^{4} + 20 T^{3} + \cdots + 16)^{2} $ (T^4 + 20T^3 + 105T^2 + 136*T + 16)^2
$53$	$ T^{8} + 252 T^{6} + \cdots + 256 $ T^8 + 252T^6 + 14980T^4 + 14720*T^2 + 256
$59$	$ (T^{4} + 4 T^{3} + \cdots + 640)^{2} $ (T^4 + 4T^3 - 56T^2 - 128*T + 640)^2
$61$	$ (T^{4} + 4 T^{3} + \cdots - 832)^{2} $ (T^4 + 4T^3 - 132T^2 - 768*T - 832)^2
$67$	$ T^{8} + 320 T^{6} + \cdots + 16384 $ T^8 + 320T^6 + 25408T^4 + 202752*T^2 + 16384
$71$	$ (T^{4} + 20 T^{3} + \cdots + 1600)^{2} $ (T^4 + 20T^3 + 6T^2 - 944*T + 1600)^2
$73$	$ (T^{4} - 12 T^{3} + \cdots - 32)^{2} $ (T^4 - 12T^3 + 22T^2 + 16*T - 32)^2
$79$	$ T^{8} + 182 T^{6} + \cdots + 193600 $ T^8 + 182T^6 + 9161T^4 + 85264*T^2 + 193600
$83$	$ (T^{4} + 24 T^{3} + \cdots - 3712)^{2} $ (T^4 + 24T^3 + 36T^2 - 1600*T - 3712)^2
$89$	$ T^{8} + 328 T^{6} + \cdots + 1638400 $ T^8 + 328T^6 + 33040T^4 + 989184*T^2 + 1638400
$97$	$ (T^{4} + 32 T^{3} + \cdots + 2582)^{2} $ (T^4 + 32T^3 + 351T^2 + 1596*T + 2582)^2
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Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.ba (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{3} + \beta_{2} q^{5} - \beta_{2} q^{7} + ( - \beta_{7} + \beta_{2} + \beta_1) q^{9} + ( - \beta_{7} - \beta_{6} - \beta_{4} - 1) q^{11} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots - 1) q^{13}+ \cdots + (2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots + 4) q^{99}+O(q^{100}) \) q + b4 * q^3 + b2 * q^5 - b2 * q^7 + (-b7 + b2 + b1) * q^9 + (-b7 - b6 - b4 - 1) * q^11 + (b7 + 2b6 + b5 + 2b4 + b1 - 1) * q^13 + b5 * q^15 + (b6 - b4 - 2b2) q^17 + (b6 + b5 - b4 + b3 - b1) * q^19 - b5 * q^21 - 2b7 q^23 - q^25 + (2b6 + 2b5 - b3 - 2b2 + 1) q^27 + (2b6 - b5 - 2b4 + 2b3 - 2b2 + b1) * q^29 + (-b6 + b5 + b4 - b3 - b1) * q^31 + (b7 + 2b6 + b5 - b4 - b3 - b1 - 2) q^33 + q^35 + (2b6 - 2b5 + 2b4 - 2b1 + 2) * q^37 + (-2b7 - b6 - b5 - b4 + 2b3 - 3b2 + 2b1 + 4) * q^39 - 2b3 q^41 + (-2b3 + 2b2) * q^43 + (b6 + b3 - b2 - 1) * q^45 + (b6 + 2b5 + b4 + 2b1 - 6) * q^47 - q^49 + (b7 - 2b5 - b2 - b1 + 3) q^51 + (3b6 + b5 - 3b4 - b3 - b1) * q^53 + (-b5 + b3 - 2b2 + b1) q^55 + (b6 + b5 - b4 + b3 + 2b2 - 3b1 + 2) * q^57 + (-2b6 - 2b5 - 2b4 - 2b1) * q^59 + (-2b7 + 2b6 + 2b4) q^61 + (-b6 - b3 + b2 + 1) * q^63 + (b6 + 2b5 - b4 - b3 - 2b1) * q^65 + (-2b6 - 2b5 + 2b4 + 8b2 + 2b1) q^67 + (4b6 + 2b5 - 2b3 + 2b2 + 2) * q^69 + (b7 + 3b6 + 3b5 + 3b4 + 3b1 - 7) * q^71 + (-b7 + b6 + b5 + b4 + b1 + 3) * q^73 - b4 * q^75 + (b5 - b3 + 2b2 - b1) q^77 + (2b6 + b5 - 2b4 + 2b3 - 2b2 - b1) * q^79 + (b7 + 2b6 - 3b5 + 2b4 + 2b3 - 3b2 + 2b1 + 4) * q^81 + (4b7 + 2b6 + 2b4 - 8) q^83 + (-b5 - b1 + 2) * q^85 + (-b6 - 2b4 - b3 - 2b2 - 6b1 + 7) q^87 + (-2b6 - 4b5 + 2b4 + 2b2 + 4b1) q^89 + (-b6 - 2b5 + b4 + b3 + 2b1) * q^91 + (b6 - b5 + b4 + b3 + 2b2 + 3b1 - 4) * q^93 + (b7 - b6 - b5 - b4 - b1 - 1) * q^95 + (-b7 - b5 - b1 - 7) * q^97 + (2b7 - b6 - 2b5 - b4 + 2b3 - b2 + b1 + 4) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{9} - 12 q^{11} + 2 q^{15} - 2 q^{21} - 8 q^{23} - 8 q^{25} + 12 q^{27} - 12 q^{33} + 8 q^{35} + 8 q^{37} + 26 q^{39} - 8 q^{45} - 40 q^{47} - 8 q^{49} + 22 q^{51} + 12 q^{57} - 8 q^{59} - 8 q^{61}+ \cdots + 38 q^{99}+O(q^{100}) \) 8 * q - 2 * q^9 - 12 * q^11 + 2 * q^15 - 2 * q^21 - 8 * q^23 - 8 * q^25 + 12 * q^27 - 12 * q^33 + 8 * q^35 + 8 * q^37 + 26 * q^39 - 8 * q^45 - 40 * q^47 - 8 * q^49 + 22 * q^51 + 12 * q^57 - 8 * q^59 - 8 * q^61 + 8 * q^63 + 20 * q^69 - 40 * q^71 + 24 * q^73 + 34 * q^81 - 48 * q^83 + 12 * q^85 + 44 * q^87 - 28 * q^93 - 8 * q^95 - 64 * q^97 + 38 * q^99

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

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	1680.2.ba.a

Level	$1680$
Weight	$2$
Character orbit	1680.ba
Analytic conductor	$13.415$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(13.4148675396\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.7278137344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - 2x^{7} + x^{6} + 6x^{5} - 20x^{4} + 18x^{3} + 9x^{2} - 54x + 81 \) x^8 - 2x^7 + x^6 + 6x^5 - 20x^4 + 18x^3 + 9x^2 - 54x + 81
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} + \nu^{6} - 5\nu^{5} + 9\nu^{4} - 2\nu^{3} - 15\nu^{2} + 36\nu - 27 ) / 54 \) (v^7 + v^6 - 5v^5 + 9v^4 - 2v^3 - 15v^2 + 36*v - 27) / 54
\(\beta_{3}\)	\(=\)	\( ( -\nu^{5} - \nu^{4} + 2\nu^{3} - 3\nu^{2} - \nu + 12 ) / 6 \) (-v^5 - v^4 + 2v^3 - 3v^2 - v + 12) / 6
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} - \nu^{6} - 4\nu^{5} + 9\nu^{4} - 7\nu^{3} - 12\nu^{2} + 36\nu - 54 ) / 54 \) (-v^7 - v^6 - 4v^5 + 9v^4 - 7v^3 - 12v^2 + 36*v - 54) / 54
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 6\nu^{4} + 20\nu^{3} - 18\nu^{2} - 9\nu + 54 ) / 27 \) (-v^7 + 2v^6 - v^5 - 6v^4 + 20v^3 - 18v^2 - 9*v + 54) / 27
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 6\nu^{4} - 11\nu^{3} + 9\nu^{2} + 9\nu - 27 ) / 18 \) (v^7 - 2v^6 + v^5 + 6v^4 - 11v^3 + 9v^2 + 9*v - 27) / 18
\(\beta_{7}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + \nu^{5} - 7\nu^{4} + 14\nu^{3} + 11\nu^{2} - 18\nu + 45 ) / 18 \) (-v^7 + v^6 + v^5 - 7v^4 + 14v^3 + 11v^2 - 18v + 45) / 18

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{7} - \beta_{5} + \beta_{2} \) b7 - b5 + b2
\(\nu^{3}\)	\(=\)	\( \beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{2} - 3 \) b7 + 2b6 + 2b5 + b2 - 3
\(\nu^{4}\)	\(=\)	\( \beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 2\beta_{3} + 3\beta_{2} - 3\beta _1 + 4 \) b7 + 2b6 + 2b5 + 2b4 - 2b3 + 3b2 - 3b1 + 4
\(\nu^{5}\)	\(=\)	\( -2\beta_{7} + 2\beta_{6} + 5\beta_{5} - 2\beta_{4} - 4\beta_{3} - 4\beta_{2} + 2\beta _1 + 2 \) -2b7 + 2b6 + 5b5 - 2b4 - 4b3 - 4b2 + 2*b1 + 2
\(\nu^{6}\)	\(=\)	\( -10\beta_{6} - 6\beta_{5} - 6\beta_{4} - 6\beta_{3} + 12\beta_{2} - 2\beta _1 + 9 \) -10b6 - 6b5 - 6b4 - 6b3 + 12b2 - 2b1 + 9
\(\nu^{7}\)	\(=\)	\( -2\beta_{7} + 6\beta_{6} + 2\beta_{5} - 22\beta_{4} + 4\beta_{3} + 12\beta_{2} + 3\beta _1 - 14 \) -2b7 + 6b6 + 2b5 - 22b4 + 4b3 + 12b2 + 3*b1 - 14

\(n\)	\(241\)	\(337\)	\(421\)	\(1121\)	\(1471\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 911.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} + T^{6} + \cdots + 81 \) T^8 + T^6 - 4T^5 - 8T^4 - 12T^3 + 9T^2 + 81
$5$	\( (T^{2} + 1)^{4} \) (T^2 + 1)^4
$7$	\( (T^{2} + 1)^{4} \) (T^2 + 1)^4
$11$	\( (T^{4} + 6 T^{3} - T^{2} + \cdots - 58)^{2} \) (T^4 + 6T^3 - T^2 - 48T - 58)^2
$13$	\( (T^{4} - 41 T^{2} + \cdots + 262)^{2} \) (T^4 - 41T^2 + 28T + 262)^2
$17$	\( T^{8} + 34 T^{6} + \cdots + 16 \) T^8 + 34T^6 + 137T^4 + 152*T^2 + 16
$19$	\( T^{8} + 68 T^{6} + \cdots + 64 \) T^8 + 68T^6 + 628T^4 + 480*T^2 + 64
$23$	\( (T^{4} + 4 T^{3} + \cdots + 176)^{2} \) (T^4 + 4T^3 - 48T^2 - 48*T + 176)^2
$29$	\( T^{8} + 166 T^{6} + \cdots + 53824 \) T^8 + 166T^6 + 7353T^4 + 54896*T^2 + 53824
$31$	\( T^{8} + 60 T^{6} + \cdots + 256 \) T^8 + 60T^6 + 772T^4 + 896*T^2 + 256
$37$	\( (T^{4} - 4 T^{3} + \cdots + 3728)^{2} \) (T^4 - 4T^3 - 120T^2 + 240*T + 3728)^2
$41$	\( T^{8} + 112 T^{6} + \cdots + 4096 \) T^8 + 112T^6 + 3712T^4 + 31744*T^2 + 4096
$43$	\( T^{8} + 112 T^{6} + \cdots + 30976 \) T^8 + 112T^6 + 3040T^4 + 19200*T^2 + 30976
$47$	\( (T^{4} + 20 T^{3} + \cdots + 16)^{2} \) (T^4 + 20T^3 + 105T^2 + 136*T + 16)^2
$53$	\( T^{8} + 252 T^{6} + \cdots + 256 \) T^8 + 252T^6 + 14980T^4 + 14720*T^2 + 256
$59$	\( (T^{4} + 4 T^{3} + \cdots + 640)^{2} \) (T^4 + 4T^3 - 56T^2 - 128*T + 640)^2
$61$	\( (T^{4} + 4 T^{3} + \cdots - 832)^{2} \) (T^4 + 4T^3 - 132T^2 - 768*T - 832)^2
$67$	\( T^{8} + 320 T^{6} + \cdots + 16384 \) T^8 + 320T^6 + 25408T^4 + 202752*T^2 + 16384
$71$	\( (T^{4} + 20 T^{3} + \cdots + 1600)^{2} \) (T^4 + 20T^3 + 6T^2 - 944*T + 1600)^2
$73$	\( (T^{4} - 12 T^{3} + \cdots - 32)^{2} \) (T^4 - 12T^3 + 22T^2 + 16*T - 32)^2
$79$	\( T^{8} + 182 T^{6} + \cdots + 193600 \) T^8 + 182T^6 + 9161T^4 + 85264*T^2 + 193600
$83$	\( (T^{4} + 24 T^{3} + \cdots - 3712)^{2} \) (T^4 + 24T^3 + 36T^2 - 1600*T - 3712)^2
$89$	\( T^{8} + 328 T^{6} + \cdots + 1638400 \) T^8 + 328T^6 + 33040T^4 + 989184*T^2 + 1638400
$97$	\( (T^{4} + 32 T^{3} + \cdots + 2582)^{2} \) (T^4 + 32T^3 + 351T^2 + 1596*T + 2582)^2
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