LMFDB - Newform orbit 8112.2.a.cu

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8112,2,Mod(1,8112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("8112.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-6,0,1,0,5,0,6,0,6,0,0,0,-1,0,9] 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:	yes
Analytic conductor:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27700337.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4056)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - q^{3} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{4} - \beta_{2} + 1) q^{7} + q^{9} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{11} + ( - \beta_{3} + \beta_1) q^{15} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{17}+ \cdots + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) $ q - q^3 + (b3 - b1) * q^5 + (-b4 - b2 + 1) * q^7 + q^9 + (-b5 - b4 - b3 - 2b2 + 2) q^11 + (-b3 + b1) * q^15 + (-b3 - b2 + b1 + 2) * q^17 + (-3b3 - b2 + b1) q^19 + (b4 + b2 - 1) * q^21 + (-b4 + 2b2 + b1 - 3) q^23 + (-b5 - 2b3 - 2b2 + 3) * q^25 - q^27 + (-2b4 - b1 + 1) q^29 + (2b2 + b1 + 1) q^31 + (b5 + b4 + b3 + 2b2 - 2) q^33 + (-b4 - 3b3 - 6b2 - b1 + 2) * q^35 + (b5 - b4 - 2b3 + b2 + 1) q^37 + (-2b5 + b4 - b3 - b2 + 2b1 + 3) * q^41 + (-3b5 - b4 - b3 - 2b2 - b1 - 1) * q^43 + (b3 - b1) * q^45 + (2b5 + 2b4 + 5b3 + 5b2 + b1 - 2) * q^47 + (-2b4 + 6b3 - b1) * q^49 + (b3 + b2 - b1 - 2) * q^51 + (-2b3 + b2 + 4) q^53 + (-b4 + b3 - 8b2 - 2b1 + 2) * q^55 + (3b3 + b2 - b1) q^57 + (2b5 + 4b3 - b2 - b1) * q^59 + (3b3 - b2 - 3b1 + 4) * q^61 + (-b4 - b2 + 1) * q^63 + (-b5 + 2b4 + 3b3 + 2b2 - 2) q^67 + (b4 - 2b2 - b1 + 3) q^69 + (-b5 - 6b3 - 3b2 - b1 + 2) * q^71 + (2b5 + b3 - 6b2 - b1 + 4) * q^73 + (b5 + 2b3 + 2b2 - 3) * q^75 + (-b5 - 2b4 - 2b3 - 5b2 - 2b1 + 10) * q^77 + (3b5 + 7b3 + 9b2 - b1 - 8) q^79 + q^81 + (-b5 + b4 - 2b3 + 2b2 - 3b1 + 3) q^83 + (-b4 + 3b3 + b2 - 2b1 - 7) * q^85 + (2b4 + b1 - 1) q^87 + (-b5 - 9b3 - 4b2 + 10) * q^89 + (-2b2 - b1 - 1) q^93 + (2b5 - b4 + b3 + b2 - 9) q^95 + (-3b5 - 2b4 + 2b3 - b1 + 8) q^97 + (-b5 - b4 - b3 - 2b2 + 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + q^{5} + 5 q^{7} + 6 q^{9} + 6 q^{11} - q^{15} + 9 q^{17} - 7 q^{19} - 5 q^{21} - 12 q^{23} + 9 q^{25} - 6 q^{27} + 7 q^{29} + 11 q^{31} - 6 q^{33} - 6 q^{35} + 6 q^{37} + 13 q^{41} - 15 q^{43}+ \cdots + 6 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + q^5 + 5 * q^7 + 6 * q^9 + 6 * q^11 - q^15 + 9 * q^17 - 7 * q^19 - 5 * q^21 - 12 * q^23 + 9 * q^25 - 6 * q^27 + 7 * q^29 + 11 * q^31 - 6 * q^33 - 6 * q^35 + 6 * q^37 + 13 * q^41 - 15 * q^43 + q^45 + 9 * q^47 + 13 * q^49 - 9 * q^51 + 22 * q^53 - 3 * q^55 + 7 * q^57 + 7 * q^59 + 25 * q^61 + 5 * q^63 - 5 * q^67 + 12 * q^69 - 8 * q^71 + 15 * q^73 - 9 * q^75 + 45 * q^77 - 14 * q^79 + 6 * q^81 + 13 * q^83 - 35 * q^85 - 7 * q^87 + 33 * q^89 - 11 * q^93 - 47 * q^95 + 50 * q^97 + 6 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19*x^4 + 17*x^3 + 103*x^2 - 71*x - 127 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -4\nu^{5} + 11\nu^{4} + 15\nu^{3} - 136\nu^{2} + 160\nu + 338 ) / 167 $ (-4v^5 + 11v^4 + 15v^3 - 136v^2 + 160*v + 338) / 167
$\beta_{3}$	$=$	$ ( -9\nu^{5} - 17\nu^{4} + 159\nu^{3} + 195\nu^{2} - 642\nu - 325 ) / 167 $ (-9v^5 - 17v^4 + 159v^3 + 195v^2 - 642*v - 325) / 167
$\beta_{4}$	$=$	$ ( 10\nu^{5} + 56\nu^{4} - 121\nu^{3} - 662\nu^{2} + 268\nu + 1326 ) / 167 $ (10v^5 + 56v^4 - 121v^3 - 662v^2 + 268*v + 1326) / 167
$\beta_{5}$	$=$	$ ( -13\nu^{5} - 6\nu^{4} + 174\nu^{3} + 226\nu^{2} - 482\nu - 1156 ) / 167 $ (-13v^5 - 6v^4 + 174v^3 + 226v^2 - 482*v - 1156) / 167

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{5} - \beta_{3} - \beta_{2} + 7 $ b5 - b3 - b2 + 7
$\nu^{3}$	$=$	$ \beta_{4} + 2\beta_{3} - 2\beta_{2} + 8\beta_1 $ b4 + 2b3 - 2b2 + 8*b1
$\nu^{4}$	$=$	$ 12\beta_{5} + 3\beta_{4} - 10\beta_{3} - 9\beta_{2} + 58 $ 12b5 + 3b4 - 10b3 - 9b2 + 58
$\nu^{5}$	$=$	$ -\beta_{5} + 12\beta_{4} + 14\beta_{3} - 40\beta_{2} + 70\beta _1 + 6 $ -b5 + 12b4 + 14b3 - 40b2 + 70b1 + 6

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

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} $

1.1

1.88227
3.16419
2.72245
−3.12925
−0.920510
−2.71914

−1.00000

−3.12925

1.48962

1.00000

1.2

−1.00000

−2.71914

−0.735542

1.00000

1.3

−1.00000

−0.920510

4.87031

1.00000

1.4

−1.00000

1.88227

−1.29156

1.00000

1.5

−1.00000

2.72245

−3.31535

1.00000

1.6

−1.00000

3.16419

3.98252

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$13$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8112.2.a.cu		6
4.b	odd	2	1		4056.2.a.bi	yes	6
13.b	even	2	1		8112.2.a.ct		6
52.b	odd	2	1		4056.2.a.bh	✓	6
52.f	even	4	2		4056.2.c.r		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4056.2.a.bh	✓	6	52.b	odd	2	1
4056.2.a.bi	yes	6	4.b	odd	2	1
4056.2.c.r		12	52.f	even	4	2
8112.2.a.ct		6	13.b	even	2	1
8112.2.a.cu		6	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8112))$:

$ T_{5}^{6} - T_{5}^{5} - 19T_{5}^{4} + 17T_{5}^{3} + 103T_{5}^{2} - 71T_{5} - 127 $

T5^6 - T5^5 - 19*T5^4 + 17*T5^3 + 103*T5^2 - 71*T5 - 127

$ T_{7}^{6} - 5T_{7}^{5} - 15T_{7}^{4} + 69T_{7}^{3} + 63T_{7}^{2} - 119T_{7} - 91 $

T7^6 - 5*T7^5 - 15*T7^4 + 69*T7^3 + 63*T7^2 - 119*T7 - 91

$ T_{11}^{6} - 6T_{11}^{5} - 23T_{11}^{4} + 118T_{11}^{3} + 126T_{11}^{2} - 112T_{11} - 91 $

T11^6 - 6*T11^5 - 23*T11^4 + 118*T11^3 + 126*T11^2 - 112*T11 - 91

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ (T + 1)^{6} $ (T + 1)^6
$5$	$ T^{6} - T^{5} + \cdots - 127 $ T^6 - T^5 - 19T^4 + 17T^3 + 103T^2 - 71T - 127
$7$	$ T^{6} - 5 T^{5} + \cdots - 91 $ T^6 - 5T^5 - 15T^4 + 69T^3 + 63T^2 - 119*T - 91
$11$	$ T^{6} - 6 T^{5} + \cdots - 91 $ T^6 - 6T^5 - 23T^4 + 118T^3 + 126T^2 - 112*T - 91
$13$	$ T^{6} $ T^6
$17$	$ T^{6} - 9 T^{5} + \cdots + 8 $ T^6 - 9T^5 + 12T^4 + 49T^3 - 60T^2 - 92*T + 8
$19$	$ T^{6} + 7 T^{5} + \cdots + 832 $ T^6 + 7T^5 - 20T^4 - 161T^3 + 12T^2 + 896*T + 832
$23$	$ T^{6} + 12 T^{5} + \cdots + 16856 $ T^6 + 12T^5 - 15T^4 - 573T^3 - 924T^2 + 6860*T + 16856
$29$	$ T^{6} - 7 T^{5} + \cdots + 27397 $ T^6 - 7T^5 - 87T^4 + 707T^3 + 1011T^2 - 15799*T + 27397
$31$	$ T^{6} - 11 T^{5} + \cdots - 301 $ T^6 - 11T^5 + 17T^4 + 155T^3 - 595T^2 + 735*T - 301
$37$	$ T^{6} - 6 T^{5} + \cdots + 664 $ T^6 - 6T^5 - 89T^4 + 683T^3 - 408T^2 - 3436*T + 664
$41$	$ T^{6} - 13 T^{5} + \cdots - 78728 $ T^6 - 13T^5 - 116T^4 + 1673T^3 + 2350T^2 - 37000*T - 78728
$43$	$ T^{6} + 15 T^{5} + \cdots - 249992 $ T^6 + 15T^5 - 117T^4 - 1795T^3 + 7292T^2 + 60460*T - 249992
$47$	$ T^{6} - 9 T^{5} + \cdots + 287552 $ T^6 - 9T^5 - 188T^4 + 2229T^3 + 2852T^2 - 101056*T + 287552
$53$	$ (T^{3} - 11 T^{2} + \cdots - 13)^{2} $ (T^3 - 11T^2 + 24T - 13)^2
$59$	$ T^{6} - 7 T^{5} + \cdots + 1112 $ T^6 - 7T^5 - 150T^4 + 1827T^3 - 6738T^2 + 7784*T + 1112
$61$	$ T^{6} - 25 T^{5} + \cdots + 22504 $ T^6 - 25T^5 + 74T^4 + 1725T^3 - 7134T^2 - 23816*T + 22504
$67$	$ T^{6} + 5 T^{5} + \cdots + 7112 $ T^6 + 5T^5 - 176T^4 - 2029T^3 - 7378T^2 - 6776*T + 7112
$71$	$ T^{6} + 8 T^{5} + \cdots - 23192 $ T^6 + 8T^5 - 145T^4 - 171T^3 + 4114T^2 - 1616*T - 23192
$73$	$ T^{6} - 15 T^{5} + \cdots + 109067 $ T^6 - 15T^5 - 219T^4 + 4019T^3 - 6755T^2 - 61061*T + 109067
$79$	$ T^{6} + 14 T^{5} + \cdots + 784147 $ T^6 + 14T^5 - 329T^4 - 4914T^3 + 22148T^2 + 427966*T + 784147
$83$	$ T^{6} - 13 T^{5} + \cdots + 598949 $ T^6 - 13T^5 - 277T^4 + 3871T^3 + 9259T^2 - 227939*T + 598949
$89$	$ T^{6} - 33 T^{5} + \cdots + 786344 $ T^6 - 33T^5 + 176T^4 + 3861T^3 - 39806T^2 - 12904*T + 786344
$97$	$ T^{6} - 50 T^{5} + \cdots + 711061 $ T^6 - 50T^5 + 766T^4 - 1365T^3 - 50164T^2 + 198287*T + 711061
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Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\( q - q^{3} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{4} - \beta_{2} + 1) q^{7} + q^{9} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{11} + ( - \beta_{3} + \beta_1) q^{15} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{17}+ \cdots + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) \) q - q^3 + (b3 - b1) * q^5 + (-b4 - b2 + 1) * q^7 + q^9 + (-b5 - b4 - b3 - 2b2 + 2) q^11 + (-b3 + b1) * q^15 + (-b3 - b2 + b1 + 2) * q^17 + (-3b3 - b2 + b1) q^19 + (b4 + b2 - 1) * q^21 + (-b4 + 2b2 + b1 - 3) q^23 + (-b5 - 2b3 - 2b2 + 3) * q^25 - q^27 + (-2b4 - b1 + 1) q^29 + (2b2 + b1 + 1) q^31 + (b5 + b4 + b3 + 2b2 - 2) q^33 + (-b4 - 3b3 - 6b2 - b1 + 2) * q^35 + (b5 - b4 - 2b3 + b2 + 1) q^37 + (-2b5 + b4 - b3 - b2 + 2b1 + 3) * q^41 + (-3b5 - b4 - b3 - 2b2 - b1 - 1) * q^43 + (b3 - b1) * q^45 + (2b5 + 2b4 + 5b3 + 5b2 + b1 - 2) * q^47 + (-2b4 + 6b3 - b1) * q^49 + (b3 + b2 - b1 - 2) * q^51 + (-2b3 + b2 + 4) q^53 + (-b4 + b3 - 8b2 - 2b1 + 2) * q^55 + (3b3 + b2 - b1) q^57 + (2b5 + 4b3 - b2 - b1) * q^59 + (3b3 - b2 - 3b1 + 4) * q^61 + (-b4 - b2 + 1) * q^63 + (-b5 + 2b4 + 3b3 + 2b2 - 2) q^67 + (b4 - 2b2 - b1 + 3) q^69 + (-b5 - 6b3 - 3b2 - b1 + 2) * q^71 + (2b5 + b3 - 6b2 - b1 + 4) * q^73 + (b5 + 2b3 + 2b2 - 3) * q^75 + (-b5 - 2b4 - 2b3 - 5b2 - 2b1 + 10) * q^77 + (3b5 + 7b3 + 9b2 - b1 - 8) q^79 + q^81 + (-b5 + b4 - 2b3 + 2b2 - 3b1 + 3) q^83 + (-b4 + 3b3 + b2 - 2b1 - 7) * q^85 + (2b4 + b1 - 1) q^87 + (-b5 - 9b3 - 4b2 + 10) * q^89 + (-2b2 - b1 - 1) q^93 + (2b5 - b4 + b3 + b2 - 9) q^95 + (-3b5 - 2b4 + 2b3 - b1 + 8) q^97 + (-b5 - b4 - b3 - 2b2 + 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + q^{5} + 5 q^{7} + 6 q^{9} + 6 q^{11} - q^{15} + 9 q^{17} - 7 q^{19} - 5 q^{21} - 12 q^{23} + 9 q^{25} - 6 q^{27} + 7 q^{29} + 11 q^{31} - 6 q^{33} - 6 q^{35} + 6 q^{37} + 13 q^{41} - 15 q^{43}+ \cdots + 6 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + q^5 + 5 * q^7 + 6 * q^9 + 6 * q^11 - q^15 + 9 * q^17 - 7 * q^19 - 5 * q^21 - 12 * q^23 + 9 * q^25 - 6 * q^27 + 7 * q^29 + 11 * q^31 - 6 * q^33 - 6 * q^35 + 6 * q^37 + 13 * q^41 - 15 * q^43 + q^45 + 9 * q^47 + 13 * q^49 - 9 * q^51 + 22 * q^53 - 3 * q^55 + 7 * q^57 + 7 * q^59 + 25 * q^61 + 5 * q^63 - 5 * q^67 + 12 * q^69 - 8 * q^71 + 15 * q^73 - 9 * q^75 + 45 * q^77 - 14 * q^79 + 6 * q^81 + 13 * q^83 - 35 * q^85 - 7 * q^87 + 33 * q^89 - 11 * q^93 - 47 * q^95 + 50 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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	8112.2.a.cu

Level	$8112$
Weight	$2$
Character orbit	8112.a
Self dual	yes
Analytic conductor	$64.775$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(64.7746461197\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.6.27700337.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 \) x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 4056)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -4\nu^{5} + 11\nu^{4} + 15\nu^{3} - 136\nu^{2} + 160\nu + 338 ) / 167 \) (-4v^5 + 11v^4 + 15v^3 - 136v^2 + 160*v + 338) / 167
\(\beta_{3}\)	\(=\)	\( ( -9\nu^{5} - 17\nu^{4} + 159\nu^{3} + 195\nu^{2} - 642\nu - 325 ) / 167 \) (-9v^5 - 17v^4 + 159v^3 + 195v^2 - 642*v - 325) / 167
\(\beta_{4}\)	\(=\)	\( ( 10\nu^{5} + 56\nu^{4} - 121\nu^{3} - 662\nu^{2} + 268\nu + 1326 ) / 167 \) (10v^5 + 56v^4 - 121v^3 - 662v^2 + 268*v + 1326) / 167
\(\beta_{5}\)	\(=\)	\( ( -13\nu^{5} - 6\nu^{4} + 174\nu^{3} + 226\nu^{2} - 482\nu - 1156 ) / 167 \) (-13v^5 - 6v^4 + 174v^3 + 226v^2 - 482*v - 1156) / 167

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{5} - \beta_{3} - \beta_{2} + 7 \) b5 - b3 - b2 + 7
\(\nu^{3}\)	\(=\)	\( \beta_{4} + 2\beta_{3} - 2\beta_{2} + 8\beta_1 \) b4 + 2b3 - 2b2 + 8*b1
\(\nu^{4}\)	\(=\)	\( 12\beta_{5} + 3\beta_{4} - 10\beta_{3} - 9\beta_{2} + 58 \) 12b5 + 3b4 - 10b3 - 9b2 + 58
\(\nu^{5}\)	\(=\)	\( -\beta_{5} + 12\beta_{4} + 14\beta_{3} - 40\beta_{2} + 70\beta _1 + 6 \) -b5 + 12b4 + 14b3 - 40b2 + 70b1 + 6

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} \) T^6
$3$	\( (T + 1)^{6} \) (T + 1)^6
$5$	\( T^{6} - T^{5} + \cdots - 127 \) T^6 - T^5 - 19T^4 + 17T^3 + 103T^2 - 71T - 127
$7$	\( T^{6} - 5 T^{5} + \cdots - 91 \) T^6 - 5T^5 - 15T^4 + 69T^3 + 63T^2 - 119*T - 91
$11$	\( T^{6} - 6 T^{5} + \cdots - 91 \) T^6 - 6T^5 - 23T^4 + 118T^3 + 126T^2 - 112*T - 91
$13$	\( T^{6} \) T^6
$17$	\( T^{6} - 9 T^{5} + \cdots + 8 \) T^6 - 9T^5 + 12T^4 + 49T^3 - 60T^2 - 92*T + 8
$19$	\( T^{6} + 7 T^{5} + \cdots + 832 \) T^6 + 7T^5 - 20T^4 - 161T^3 + 12T^2 + 896*T + 832
$23$	\( T^{6} + 12 T^{5} + \cdots + 16856 \) T^6 + 12T^5 - 15T^4 - 573T^3 - 924T^2 + 6860*T + 16856
$29$	\( T^{6} - 7 T^{5} + \cdots + 27397 \) T^6 - 7T^5 - 87T^4 + 707T^3 + 1011T^2 - 15799*T + 27397
$31$	\( T^{6} - 11 T^{5} + \cdots - 301 \) T^6 - 11T^5 + 17T^4 + 155T^3 - 595T^2 + 735*T - 301
$37$	\( T^{6} - 6 T^{5} + \cdots + 664 \) T^6 - 6T^5 - 89T^4 + 683T^3 - 408T^2 - 3436*T + 664
$41$	\( T^{6} - 13 T^{5} + \cdots - 78728 \) T^6 - 13T^5 - 116T^4 + 1673T^3 + 2350T^2 - 37000*T - 78728
$43$	\( T^{6} + 15 T^{5} + \cdots - 249992 \) T^6 + 15T^5 - 117T^4 - 1795T^3 + 7292T^2 + 60460*T - 249992
$47$	\( T^{6} - 9 T^{5} + \cdots + 287552 \) T^6 - 9T^5 - 188T^4 + 2229T^3 + 2852T^2 - 101056*T + 287552
$53$	\( (T^{3} - 11 T^{2} + \cdots - 13)^{2} \) (T^3 - 11T^2 + 24T - 13)^2
$59$	\( T^{6} - 7 T^{5} + \cdots + 1112 \) T^6 - 7T^5 - 150T^4 + 1827T^3 - 6738T^2 + 7784*T + 1112
$61$	\( T^{6} - 25 T^{5} + \cdots + 22504 \) T^6 - 25T^5 + 74T^4 + 1725T^3 - 7134T^2 - 23816*T + 22504
$67$	\( T^{6} + 5 T^{5} + \cdots + 7112 \) T^6 + 5T^5 - 176T^4 - 2029T^3 - 7378T^2 - 6776*T + 7112
$71$	\( T^{6} + 8 T^{5} + \cdots - 23192 \) T^6 + 8T^5 - 145T^4 - 171T^3 + 4114T^2 - 1616*T - 23192
$73$	\( T^{6} - 15 T^{5} + \cdots + 109067 \) T^6 - 15T^5 - 219T^4 + 4019T^3 - 6755T^2 - 61061*T + 109067
$79$	\( T^{6} + 14 T^{5} + \cdots + 784147 \) T^6 + 14T^5 - 329T^4 - 4914T^3 + 22148T^2 + 427966*T + 784147
$83$	\( T^{6} - 13 T^{5} + \cdots + 598949 \) T^6 - 13T^5 - 277T^4 + 3871T^3 + 9259T^2 - 227939*T + 598949
$89$	\( T^{6} - 33 T^{5} + \cdots + 786344 \) T^6 - 33T^5 + 176T^4 + 3861T^3 - 39806T^2 - 12904*T + 786344
$97$	\( T^{6} - 50 T^{5} + \cdots + 711061 \) T^6 - 50T^5 + 766T^4 - 1365T^3 - 50164T^2 + 198287*T + 711061
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\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( +1 \)
\(13\)	\( -1 \)