LMFDB - Newform orbit 1792.4.a.v

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,4,Mod(1,1792)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1792.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1792.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$105.731422730$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 58x^{6} + 873x^{4} - 3904x^{2} + 512 $ x^8 - 58x^6 + 873x^4 - 3904*x^2 + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 896)
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 $1,\beta_1,\ldots,\beta_{7}$ 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} - 7 q^{7} + ( - \beta_{3} + 7) q^{9}+O(q^{10}) $ q + b4 * q^3 + b2 * q^5 - 7 * q^7 + (-b3 + 7) * q^9	$ q + \beta_{4} q^{3} + \beta_{2} q^{5} - 7 q^{7} + ( - \beta_{3} + 7) q^{9} + (\beta_{7} + 2 \beta_1) q^{11} + (2 \beta_{7} + 3 \beta_{4} + \cdots + \beta_1) q^{13}+ \cdots + ( - 3 \beta_{7} + 38 \beta_{4} + \cdots + 56 \beta_1) q^{99}+O(q^{100}) $ q + b4 * q^3 + b2 * q^5 - 7 * q^7 + (-b3 + 7) * q^9 + (b7 + 2b1) q^11 + (2b7 + 3b4 - 2b2 + b1) q^13 + (-b6 - b3 + 2) * q^15 + (b5 - 18) * q^17 + (2b7 - 6b4 - b2 - 5b1) q^19 - 7b4 q^21 + (2b6 + b5 - 3b3 + 46) * q^23 + (-b5 + b3 - 39) * q^25 + (11b4 + 13b2 + 5b1) q^27 + (3b7 + 16b4 + 2b2 - b1) q^29 + (2b6 + 2b5 - 2b3 + 28) q^31 + (-7b6 - 3b5 - 48) * q^33 - 7b2 q^35 + (-5b7 + 22b4 - 4b2 + 13b1) * q^37 + (-3b6 - 3b5 - b3 + 50) * q^39 + (7b6 + 2b5 + 2b3 - 46) q^41 + (b7 + 22b4 + 26b2 + 12b1) q^43 + (41b4 - 6b2 + 13b1) q^45 + (-2b6 - 6b5 - 4b3 + 160) q^47 + 49 * q^49 + (-8b7 - 28b4 + 6b2 - 18b1) * q^51 + (5b7 + 34b4 + 10b2 - 13b1) * q^53 + (-4b6 + 2b5 - 2b3 + 28) q^55 + (14b6 + 3b5 + 7b3 - 158) q^57 + (-12b7 + 2b4 + 47b2 + 7b1) * q^59 + (-12b7 + 30b4 + 7b2 - 14b1) * q^61 + (7b3 - 49) q^63 + (-14b6 + 6b5 - 3b3 - 98) q^65 + (5b7 - 10b4 + 44b2 - 42b1) * q^67 + (-8b7 + 113b4 + 29b2 - 19b1) * q^69 + (4b6 - b5 + 20b3 + 32) * q^71 + (21b6 - 3b5 - 16b3 + 378) q^73 + (8b7 - 60b4 - 19b2 + 13b1) * q^75 + (-7b7 - 14b1) * q^77 + (17b6 + 9b5 + 8b3 + 232) q^79 + (-28b6 - 5b5 + 3b3 + 131) q^81 + (-10b7 - 11b4 - 26b2 + 26b1) * q^83 + (24b7 + 22b4 - 76b2 - 2b1) * q^85 + (-2b6 - 2b5 - 18b3 + 516) q^87 + (21b6 - 11b5 - 2b3 - 2) q^89 + (-14b7 - 21b4 + 14b2 - 7b1) * q^91 + (-16b7 + 54b4 + 22b2 - 42b1) * q^93 + (-11b6 + 5b5 + 19b3 - 6) q^95 + (-28b6 + 5b5 + 6b3 + 18) q^97 + (-3b7 + 38b4 + 38b2 + 56b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 56 q^{7} + 56 q^{9}+O(q^{10}) $ 8 * q - 56 * q^7 + 56 * q^9	$ 8 q - 56 q^{7} + 56 q^{9} + 16 q^{15} - 144 q^{17} + 368 q^{23} - 312 q^{25} + 224 q^{31} - 384 q^{33} + 400 q^{39} - 368 q^{41} + 1280 q^{47} + 392 q^{49} + 224 q^{55} - 1264 q^{57} - 392 q^{63} - 784 q^{65} + 256 q^{71} + 3024 q^{73} + 1856 q^{79} + 1048 q^{81} + 4128 q^{87} - 16 q^{89} - 48 q^{95} + 144 q^{97}+O(q^{100}) $ 8 * q - 56 * q^7 + 56 * q^9 + 16 * q^15 - 144 * q^17 + 368 * q^23 - 312 * q^25 + 224 * q^31 - 384 * q^33 + 400 * q^39 - 368 * q^41 + 1280 * q^47 + 392 * q^49 + 224 * q^55 - 1264 * q^57 - 392 * q^63 - 784 * q^65 + 256 * q^71 + 3024 * q^73 + 1856 * q^79 + 1048 * q^81 + 4128 * q^87 - 16 * q^89 - 48 * q^95 + 144 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 58x^{6} + 873x^{4} - 3904x^{2} + 512 $ x^8 - 58*x^6 + 873*x^4 - 3904*x^2 + 512 :

$\beta_{1}$	$=$	$ ( -\nu^{7} - 22\nu^{5} + 3303\nu^{3} - 39680\nu ) / 1696 $ (-v^7 - 22v^5 + 3303v^3 - 39680*v) / 1696
$\beta_{2}$	$=$	$ ( 9\nu^{7} - 650\nu^{5} + 13521\nu^{3} - 75360\nu ) / 3392 $ (9v^7 - 650v^5 + 13521v^3 - 75360v) / 3392
$\beta_{3}$	$=$	$ ( -13\nu^{6} + 562\nu^{4} - 2005\nu^{2} - 19760 ) / 424 $ (-13v^6 + 562v^4 - 2005*v^2 - 19760) / 424
$\beta_{4}$	$=$	$ ( 33\nu^{7} - 1818\nu^{5} + 24137\nu^{3} - 86368\nu ) / 3392 $ (33v^7 - 1818v^5 + 24137v^3 - 86368v) / 3392
$\beta_{5}$	$=$	$ ( 11\nu^{6} - 606\nu^{4} + 6915\nu^{2} - 6176 ) / 212 $ (11v^6 - 606v^4 + 6915*v^2 - 6176) / 212
$\beta_{6}$	$=$	$ ( 3\nu^{6} - 146\nu^{4} + 1327\nu^{2} - 1376 ) / 53 $ (3v^6 - 146v^4 + 1327*v^2 - 1376) / 53
$\beta_{7}$	$=$	$ ( -25\nu^{7} + 1146\nu^{5} - 7313\nu^{3} - 15104\nu ) / 1696 $ (-25v^7 + 1146v^5 - 7313v^3 - 15104v) / 1696

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

$\nu$	$=$	$ ( \beta_{7} + 2\beta_{4} - 2\beta_{2} - \beta_1 ) / 8 $ (b7 + 2b4 - 2b2 - b1) / 8
$\nu^{2}$	$=$	$ ( 2\beta_{6} - \beta_{5} + 2\beta_{3} + 116 ) / 8 $ (2b6 - b5 + 2b3 + 116) / 8
$\nu^{3}$	$=$	$ ( 29\beta_{7} + 66\beta_{4} - 82\beta_{2} - 5\beta_1 ) / 8 $ (29b7 + 66b4 - 82b2 - 5b1) / 8
$\nu^{4}$	$=$	$ ( 80\beta_{6} - 53\beta_{5} + 58\beta_{3} + 3236 ) / 8 $ (80b6 - 53b5 + 58*b3 + 3236) / 8
$\nu^{5}$	$=$	$ ( 969\beta_{7} + 2346\beta_{4} - 3194\beta_{2} + 111\beta_1 ) / 8 $ (969b7 + 2346b4 - 3194b2 + 111b1) / 8
$\nu^{6}$	$=$	$ ( 3150\beta_{6} - 2137\beta_{5} + 1938\beta_{3} + 109844 ) / 8 $ (3150b6 - 2137b5 + 1938*b3 + 109844) / 8
$\nu^{7}$	$=$	$ ( 34789\beta_{7} + 87026\beta_{4} - 121218\beta_{2} + 7155\beta_1 ) / 8 $ (34789b7 + 87026b4 - 121218b2 + 7155b1) / 8

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

0.367693
−2.91484
−6.12162
3.44880
−3.44880
6.12162
2.91484
−0.367693

−9.01216

−7.97216

−7.00000

54.2189

1.2

−6.62600

1.61841

−7.00000

16.9039

1.3

−3.15500

14.1596

−7.00000

−17.0460

1.4

−0.960829

8.79386

−7.00000

−26.0768

1.5

0.960829

−8.79386

−7.00000

−26.0768

1.6

3.15500

−14.1596

−7.00000

−17.0460

1.7

6.62600

−1.61841

−7.00000

16.9039

1.8

9.01216

7.97216

−7.00000

54.2189

Atkin-Lehner signs

$ p $	Sign
$2$	$1$
$7$	$1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1792.4.a.v		8
4.b	odd	2	1		1792.4.a.x		8
8.b	even	2	1	inner	1792.4.a.v		8
8.d	odd	2	1		1792.4.a.x		8
16.e	even	4	2		896.4.b.c	yes	8
16.f	odd	4	2		896.4.b.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
896.4.b.a	✓	8	16.f	odd	4	2
896.4.b.c	yes	8	16.e	even	4	2
1792.4.a.v		8	1.a	even	1	1	trivial
1792.4.a.v		8	8.b	even	2	1	inner
1792.4.a.x		8	4.b	odd	2	1
1792.4.a.x		8	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1792))$:

$ T_{3}^{8} - 136T_{3}^{6} + 4936T_{3}^{4} - 39936T_{3}^{2} + 32768 $

$ T_{5}^{8} - 344T_{5}^{6} + 34056T_{5}^{4} - 1072256T_{5}^{2} + 2580992 $

$ T_{23}^{4} - 184T_{23}^{3} - 13048T_{23}^{2} + 1666176T_{23} + 92570112 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} - 136 T^{6} + \cdots + 32768 $ T^8 - 136T^6 + 4936T^4 - 39936*T^2 + 32768
$5$	$ T^{8} - 344 T^{6} + \cdots + 2580992 $ T^8 - 344T^6 + 34056T^4 - 1072256*T^2 + 2580992
$7$	$ (T + 7)^{8} $ (T + 7)^8
$11$	$ T^{8} + \cdots + 176294469632 $ T^8 - 5232T^6 + 4871360T^4 - 1616883712*T^2 + 176294469632
$13$	$ T^{8} + \cdots + 7851321287168 $ T^8 - 8952T^6 + 23568392T^4 - 23650716800*T^2 + 7851321287168
$17$	$ (T^{4} + 72 T^{3} + \cdots + 6110608)^{2} $ (T^4 + 72T^3 - 6024T^2 - 168288*T + 6110608)^2
$19$	$ T^{8} + \cdots + 98200806760448 $ T^8 - 27112T^6 + 205923528T^4 - 376064531968*T^2 + 98200806760448
$23$	$ (T^{4} - 184 T^{3} + \cdots + 92570112)^{2} $ (T^4 - 184T^3 - 13048T^2 + 1666176*T + 92570112)^2
$29$	$ T^{8} + \cdots + 10\!\cdots\!72 $ T^8 - 51808T^6 + 856606848T^4 - 5293159505920*T^2 + 10871711465603072
$31$	$ (T^{4} - 112 T^{3} + \cdots - 83968)^{2} $ (T^4 - 112T^3 - 35040T^2 + 1035776*T - 83968)^2
$37$	$ T^{8} + \cdots + 24\!\cdots\!72 $ T^8 - 219104T^6 + 11691925632T^4 - 102744441380864*T^2 + 248241087106383872
$41$	$ (T^{4} + 184 T^{3} + \cdots - 93253104)^{2} $ (T^4 + 184T^3 - 74152T^2 - 6150432*T - 93253104)^2
$43$	$ T^{8} + \cdots + 93\!\cdots\!28 $ T^8 - 380528T^6 + 35222504640T^4 - 1095707436756992*T^2 + 9374434481286422528
$47$	$ (T^{4} - 640 T^{3} + \cdots + 15493566464)^{2} $ (T^4 - 640T^3 - 156800T^2 + 85204992*T + 15493566464)^2
$53$	$ T^{8} + \cdots + 97\!\cdots\!68 $ T^8 - 414720T^6 + 54791149568T^4 - 2291273759457280*T^2 + 976070864606855168
$59$	$ T^{8} + \cdots + 82\!\cdots\!12 $ T^8 - 887848T^6 + 128912854088T^4 - 1305795210015744*T^2 + 825734795408474112
$61$	$ T^{8} + \cdots + 50\!\cdots\!12 $ T^8 - 669848T^6 + 83745945096T^4 - 3661894724293760*T^2 + 50636186573965517312
$67$	$ T^{8} + \cdots + 17\!\cdots\!08 $ T^8 - 2088208T^6 + 1358417428544T^4 - 282065843390754816*T^2 + 1715214900569587089408
$71$	$ (T^{4} - 128 T^{3} + \cdots + 56776476672)^{2} $ (T^4 - 128T^3 - 879520T^2 + 256003584*T + 56776476672)^2
$73$	$ (T^{4} - 1512 T^{3} + \cdots - 322329535344)^{2} $ (T^4 - 1512T^3 + 5560T^2 + 854213088*T - 322329535344)^2
$79$	$ (T^{4} - 928 T^{3} + \cdots + 2198996992)^{2} $ (T^4 - 928T^3 - 690592T^2 + 436976128*T + 2198996992)^2
$83$	$ T^{8} + \cdots + 12\!\cdots\!28 $ T^8 - 1029768T^6 + 231706523336T^4 - 11553688699637248*T^2 + 121490583442969468928
$89$	$ (T^{4} + 8 T^{3} + \cdots - 49467394928)^{2} $ (T^4 + 8T^3 - 1566472T^2 - 620219488*T - 49467394928)^2
$97$	$ (T^{4} - 72 T^{3} + \cdots + 73503910544)^{2} $ (T^4 - 72T^3 - 1061512T^2 - 51654304*T + 73503910544)^2
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1792.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{3} + \beta_{2} q^{5} - 7 q^{7} + ( - \beta_{3} + 7) q^{9}+O(q^{10}) \) q + b4 * q^3 + b2 * q^5 - 7 * q^7 + (-b3 + 7) * q^9	\( q + \beta_{4} q^{3} + \beta_{2} q^{5} - 7 q^{7} + ( - \beta_{3} + 7) q^{9} + (\beta_{7} + 2 \beta_1) q^{11} + (2 \beta_{7} + 3 \beta_{4} + \cdots + \beta_1) q^{13}+ \cdots + ( - 3 \beta_{7} + 38 \beta_{4} + \cdots + 56 \beta_1) q^{99}+O(q^{100}) \) q + b4 * q^3 + b2 * q^5 - 7 * q^7 + (-b3 + 7) * q^9 + (b7 + 2b1) q^11 + (2b7 + 3b4 - 2b2 + b1) q^13 + (-b6 - b3 + 2) * q^15 + (b5 - 18) * q^17 + (2b7 - 6b4 - b2 - 5b1) q^19 - 7b4 q^21 + (2b6 + b5 - 3b3 + 46) * q^23 + (-b5 + b3 - 39) * q^25 + (11b4 + 13b2 + 5b1) q^27 + (3b7 + 16b4 + 2b2 - b1) q^29 + (2b6 + 2b5 - 2b3 + 28) q^31 + (-7b6 - 3b5 - 48) * q^33 - 7b2 q^35 + (-5b7 + 22b4 - 4b2 + 13b1) * q^37 + (-3b6 - 3b5 - b3 + 50) * q^39 + (7b6 + 2b5 + 2b3 - 46) q^41 + (b7 + 22b4 + 26b2 + 12b1) q^43 + (41b4 - 6b2 + 13b1) q^45 + (-2b6 - 6b5 - 4b3 + 160) q^47 + 49 * q^49 + (-8b7 - 28b4 + 6b2 - 18b1) * q^51 + (5b7 + 34b4 + 10b2 - 13b1) * q^53 + (-4b6 + 2b5 - 2b3 + 28) q^55 + (14b6 + 3b5 + 7b3 - 158) q^57 + (-12b7 + 2b4 + 47b2 + 7b1) * q^59 + (-12b7 + 30b4 + 7b2 - 14b1) * q^61 + (7b3 - 49) q^63 + (-14b6 + 6b5 - 3b3 - 98) q^65 + (5b7 - 10b4 + 44b2 - 42b1) * q^67 + (-8b7 + 113b4 + 29b2 - 19b1) * q^69 + (4b6 - b5 + 20b3 + 32) * q^71 + (21b6 - 3b5 - 16b3 + 378) q^73 + (8b7 - 60b4 - 19b2 + 13b1) * q^75 + (-7b7 - 14b1) * q^77 + (17b6 + 9b5 + 8b3 + 232) q^79 + (-28b6 - 5b5 + 3b3 + 131) q^81 + (-10b7 - 11b4 - 26b2 + 26b1) * q^83 + (24b7 + 22b4 - 76b2 - 2b1) * q^85 + (-2b6 - 2b5 - 18b3 + 516) q^87 + (21b6 - 11b5 - 2b3 - 2) q^89 + (-14b7 - 21b4 + 14b2 - 7b1) * q^91 + (-16b7 + 54b4 + 22b2 - 42b1) * q^93 + (-11b6 + 5b5 + 19b3 - 6) q^95 + (-28b6 + 5b5 + 6b3 + 18) q^97 + (-3b7 + 38b4 + 38b2 + 56b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 56 q^{7} + 56 q^{9}+O(q^{10}) \) 8 * q - 56 * q^7 + 56 * q^9	\( 8 q - 56 q^{7} + 56 q^{9} + 16 q^{15} - 144 q^{17} + 368 q^{23} - 312 q^{25} + 224 q^{31} - 384 q^{33} + 400 q^{39} - 368 q^{41} + 1280 q^{47} + 392 q^{49} + 224 q^{55} - 1264 q^{57} - 392 q^{63} - 784 q^{65} + 256 q^{71} + 3024 q^{73} + 1856 q^{79} + 1048 q^{81} + 4128 q^{87} - 16 q^{89} - 48 q^{95} + 144 q^{97}+O(q^{100}) \) 8 * q - 56 * q^7 + 56 * q^9 + 16 * q^15 - 144 * q^17 + 368 * q^23 - 312 * q^25 + 224 * q^31 - 384 * q^33 + 400 * q^39 - 368 * q^41 + 1280 * q^47 + 392 * q^49 + 224 * q^55 - 1264 * q^57 - 392 * q^63 - 784 * q^65 + 256 * q^71 + 3024 * q^73 + 1856 * q^79 + 1048 * q^81 + 4128 * q^87 - 16 * q^89 - 48 * q^95 + 144 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1792.4.a.v

Level	$1792$
Weight	$4$
Character orbit	1792.a
Self dual	yes
Analytic conductor	$105.731$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	yes
Analytic conductor:	\(105.731422730\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - 58x^{6} + 873x^{4} - 3904x^{2} + 512 \) x^8 - 58x^6 + 873x^4 - 3904*x^2 + 512
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	no (minimal twist has level 896)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{7} - 22\nu^{5} + 3303\nu^{3} - 39680\nu ) / 1696 \) (-v^7 - 22v^5 + 3303v^3 - 39680*v) / 1696
\(\beta_{2}\)	\(=\)	\( ( 9\nu^{7} - 650\nu^{5} + 13521\nu^{3} - 75360\nu ) / 3392 \) (9v^7 - 650v^5 + 13521v^3 - 75360v) / 3392
\(\beta_{3}\)	\(=\)	\( ( -13\nu^{6} + 562\nu^{4} - 2005\nu^{2} - 19760 ) / 424 \) (-13v^6 + 562v^4 - 2005*v^2 - 19760) / 424
\(\beta_{4}\)	\(=\)	\( ( 33\nu^{7} - 1818\nu^{5} + 24137\nu^{3} - 86368\nu ) / 3392 \) (33v^7 - 1818v^5 + 24137v^3 - 86368v) / 3392
\(\beta_{5}\)	\(=\)	\( ( 11\nu^{6} - 606\nu^{4} + 6915\nu^{2} - 6176 ) / 212 \) (11v^6 - 606v^4 + 6915*v^2 - 6176) / 212
\(\beta_{6}\)	\(=\)	\( ( 3\nu^{6} - 146\nu^{4} + 1327\nu^{2} - 1376 ) / 53 \) (3v^6 - 146v^4 + 1327*v^2 - 1376) / 53
\(\beta_{7}\)	\(=\)	\( ( -25\nu^{7} + 1146\nu^{5} - 7313\nu^{3} - 15104\nu ) / 1696 \) (-25v^7 + 1146v^5 - 7313v^3 - 15104v) / 1696

\(\nu\)	\(=\)	\( ( \beta_{7} + 2\beta_{4} - 2\beta_{2} - \beta_1 ) / 8 \) (b7 + 2b4 - 2b2 - b1) / 8
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{6} - \beta_{5} + 2\beta_{3} + 116 ) / 8 \) (2b6 - b5 + 2b3 + 116) / 8
\(\nu^{3}\)	\(=\)	\( ( 29\beta_{7} + 66\beta_{4} - 82\beta_{2} - 5\beta_1 ) / 8 \) (29b7 + 66b4 - 82b2 - 5b1) / 8
\(\nu^{4}\)	\(=\)	\( ( 80\beta_{6} - 53\beta_{5} + 58\beta_{3} + 3236 ) / 8 \) (80b6 - 53b5 + 58*b3 + 3236) / 8
\(\nu^{5}\)	\(=\)	\( ( 969\beta_{7} + 2346\beta_{4} - 3194\beta_{2} + 111\beta_1 ) / 8 \) (969b7 + 2346b4 - 3194b2 + 111b1) / 8
\(\nu^{6}\)	\(=\)	\( ( 3150\beta_{6} - 2137\beta_{5} + 1938\beta_{3} + 109844 ) / 8 \) (3150b6 - 2137b5 + 1938*b3 + 109844) / 8
\(\nu^{7}\)	\(=\)	\( ( 34789\beta_{7} + 87026\beta_{4} - 121218\beta_{2} + 7155\beta_1 ) / 8 \) (34789b7 + 87026b4 - 121218b2 + 7155b1) / 8

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} - 136 T^{6} + \cdots + 32768 \) T^8 - 136T^6 + 4936T^4 - 39936*T^2 + 32768
$5$	\( T^{8} - 344 T^{6} + \cdots + 2580992 \) T^8 - 344T^6 + 34056T^4 - 1072256*T^2 + 2580992
$7$	\( (T + 7)^{8} \) (T + 7)^8
$11$	\( T^{8} + \cdots + 176294469632 \) T^8 - 5232T^6 + 4871360T^4 - 1616883712*T^2 + 176294469632
$13$	\( T^{8} + \cdots + 7851321287168 \) T^8 - 8952T^6 + 23568392T^4 - 23650716800*T^2 + 7851321287168
$17$	\( (T^{4} + 72 T^{3} + \cdots + 6110608)^{2} \) (T^4 + 72T^3 - 6024T^2 - 168288*T + 6110608)^2
$19$	\( T^{8} + \cdots + 98200806760448 \) T^8 - 27112T^6 + 205923528T^4 - 376064531968*T^2 + 98200806760448
$23$	\( (T^{4} - 184 T^{3} + \cdots + 92570112)^{2} \) (T^4 - 184T^3 - 13048T^2 + 1666176*T + 92570112)^2
$29$	\( T^{8} + \cdots + 10\!\cdots\!72 \) T^8 - 51808T^6 + 856606848T^4 - 5293159505920*T^2 + 10871711465603072
$31$	\( (T^{4} - 112 T^{3} + \cdots - 83968)^{2} \) (T^4 - 112T^3 - 35040T^2 + 1035776*T - 83968)^2
$37$	\( T^{8} + \cdots + 24\!\cdots\!72 \) T^8 - 219104T^6 + 11691925632T^4 - 102744441380864*T^2 + 248241087106383872
$41$	\( (T^{4} + 184 T^{3} + \cdots - 93253104)^{2} \) (T^4 + 184T^3 - 74152T^2 - 6150432*T - 93253104)^2
$43$	\( T^{8} + \cdots + 93\!\cdots\!28 \) T^8 - 380528T^6 + 35222504640T^4 - 1095707436756992*T^2 + 9374434481286422528
$47$	\( (T^{4} - 640 T^{3} + \cdots + 15493566464)^{2} \) (T^4 - 640T^3 - 156800T^2 + 85204992*T + 15493566464)^2
$53$	\( T^{8} + \cdots + 97\!\cdots\!68 \) T^8 - 414720T^6 + 54791149568T^4 - 2291273759457280*T^2 + 976070864606855168
$59$	\( T^{8} + \cdots + 82\!\cdots\!12 \) T^8 - 887848T^6 + 128912854088T^4 - 1305795210015744*T^2 + 825734795408474112
$61$	\( T^{8} + \cdots + 50\!\cdots\!12 \) T^8 - 669848T^6 + 83745945096T^4 - 3661894724293760*T^2 + 50636186573965517312
$67$	\( T^{8} + \cdots + 17\!\cdots\!08 \) T^8 - 2088208T^6 + 1358417428544T^4 - 282065843390754816*T^2 + 1715214900569587089408
$71$	\( (T^{4} - 128 T^{3} + \cdots + 56776476672)^{2} \) (T^4 - 128T^3 - 879520T^2 + 256003584*T + 56776476672)^2
$73$	\( (T^{4} - 1512 T^{3} + \cdots - 322329535344)^{2} \) (T^4 - 1512T^3 + 5560T^2 + 854213088*T - 322329535344)^2
$79$	\( (T^{4} - 928 T^{3} + \cdots + 2198996992)^{2} \) (T^4 - 928T^3 - 690592T^2 + 436976128*T + 2198996992)^2
$83$	\( T^{8} + \cdots + 12\!\cdots\!28 \) T^8 - 1029768T^6 + 231706523336T^4 - 11553688699637248*T^2 + 121490583442969468928
$89$	\( (T^{4} + 8 T^{3} + \cdots - 49467394928)^{2} \) (T^4 + 8T^3 - 1566472T^2 - 620219488*T - 49467394928)^2
$97$	\( (T^{4} - 72 T^{3} + \cdots + 73503910544)^{2} \) (T^4 - 72T^3 - 1061512T^2 - 51654304*T + 73503910544)^2
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