LMFDB - Newform orbit 320.5.b.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,5,Mod(191,320)] 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("320.191"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	320.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-584] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$33.0783881868$
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} + 35x^{6} + 413x^{4} + 2015x^{2} + 3481 $ x^8 + 35x^6 + 413x^4 + 2015*x^2 + 3481
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{20}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 160)
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_1 q^{3} + \beta_{3} q^{5} + ( - \beta_{7} - 4 \beta_{2} - 2 \beta_1) q^{7} + (\beta_{6} + \beta_{5} + 4 \beta_{3} - 73) q^{9} + (3 \beta_{7} - \beta_{4} + \cdots + \beta_1) q^{11} + ( - \beta_{5} - 8 \beta_{3} - 4) q^{13}+ \cdots + ( - 291 \beta_{7} + 253 \beta_{4} + \cdots + 147 \beta_1) q^{99}+O(q^{100}) $ q - b1 * q^3 + b3 * q^5 + (-b7 - 4b2 - 2b1) * q^7 + (b6 + b5 + 4b3 - 73) q^9 + (3b7 - b4 - 20b2 + b1) * q^11 + (-b5 - 8b3 - 4) q^13 + (-3b7 + 2b4 + 15b2 + 2b1) * q^15 + (3b6 + b5 + 10b3 - 70) * q^17 + (5b7 + 5b4 - 30b2 + b1) q^19 + (4b6 + 6b5 - 10b3 - 298) q^21 + (3b7 - 8b4 - 22b2 + 24b1) * q^23 + 125 * q^25 + (12b7 + 30b4 + 144b2 + 62b1) * q^27 + (6b6 + 2b5 - 72b3 + 258) q^29 + (-4b7 - 6b4 + 30b2 - 58b1) * q^31 + (-5b6 + 23b5 + 138b3 + 68) q^33 + (b7 + 11b4 + 80b2 - 14b1) q^35 + (2b6 + 26b5 - 2b3 - 452) q^37 + (24b7 - 22b4 - 252b2 - 18b1) * q^39 + (7b6 - 37b5 - 4b3 - 652) q^41 + (58b7 + 28b4 - 292b2 - 123b1) * q^43 + (-25b5 - 83b3 + 450) * q^45 + (-25b7 + 64b4 + 474b2 - 6b1) * q^47 + (3b6 - 9b5 + 8b3 + 511) q^49 + (42b7 + 74b4 + 138b2 + 264b1) * q^51 + (-22b6 - 59b5 - 68b3 - 636) q^53 + (14b7 + 4b4 - 105b2 + 104b1) * q^55 + (-21b6 + 9b5 + 66b3 + 384) q^57 + (119b7 - 5b4 + 22b2 + 79b1) * q^59 + (-34b6 - 24b5 - 42b3 + 688) q^61 + (45b7 + 80b4 + 126b2 + 376b1) * q^63 + (5b6 + 5b5 - 2b3 - 950) q^65 + (10b7 + 44b4 - 264b2 + 63b1) * q^67 + (-14b6 + 30b5 + 218b3 + 3218) q^69 + (122b7 + 104b4 + 134b2 - 188b1) * q^71 + (-15b6 + 85b5 - 6b3 + 2242) q^73 - 125b1 q^75 + (-14b6 + 93b5 + 334b3 + 1980) q^77 + (42b7 + 170b4 - 622b2 - 10b1) * q^79 + (-65b6 - 245b5 - 620b3 + 5195) q^81 + (-80b7 + 62b4 - 424b2 - 213b1) * q^83 + (10b6 - 65b5 - 96b3 + 1200) q^85 + (360b7 - 36b4 - 1104b2 - 54b1) * q^87 + (10b6 + 62b5 - 428b3 - 6326) q^89 + (52b7 - 76b4 - 794b2 + 50b1) * q^91 + (78b6 + 52b5 + 212b3 - 9228) q^93 + (18b7 + 38b4 - 565b2 + 38b1) * q^95 + (31b6 + 187b5 + 250b3 + 430) q^97 + (-291b7 + 253b4 + 3726b2 + 147b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 584 q^{9} - 32 q^{13} - 560 q^{17} - 2384 q^{21} + 1000 q^{25} + 2064 q^{29} + 544 q^{33} - 3616 q^{37} - 5216 q^{41} + 3600 q^{45} + 4088 q^{49} - 5088 q^{53} + 3072 q^{57} + 5504 q^{61} - 7600 q^{65}+ \cdots + 3440 q^{97}+O(q^{100}) $ 8 * q - 584 * q^9 - 32 * q^13 - 560 * q^17 - 2384 * q^21 + 1000 * q^25 + 2064 * q^29 + 544 * q^33 - 3616 * q^37 - 5216 * q^41 + 3600 * q^45 + 4088 * q^49 - 5088 * q^53 + 3072 * q^57 + 5504 * q^61 - 7600 * q^65 + 25744 * q^69 + 17936 * q^73 + 15840 * q^77 + 41560 * q^81 + 9600 * q^85 - 50608 * q^89 - 73824 * q^93 + 3440 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 35x^{6} + 413x^{4} + 2015x^{2} + 3481 $ x^8 + 35*x^6 + 413*x^4 + 2015*x^2 + 3481 :

$\beta_{1}$	$=$	$ ( 94\nu^{7} + 2464\nu^{5} + 16402\nu^{3} + 39904\nu ) / 3009 $ (94v^7 + 2464v^5 + 16402v^3 + 39904v) / 3009
$\beta_{2}$	$=$	$ ( 56\nu^{7} + 1724\nu^{5} + 15576\nu^{3} + 42040\nu ) / 1003 $ (56v^7 + 1724v^5 + 15576v^3 + 42040v) / 1003
$\beta_{3}$	$=$	$ ( 50\nu^{6} + 1430\nu^{4} + 11600\nu^{2} + 28295 ) / 51 $ (50v^6 + 1430v^4 + 11600*v^2 + 28295) / 51
$\beta_{4}$	$=$	$ ( -1252\nu^{7} - 37684\nu^{5} - 336772\nu^{3} - 942760\nu ) / 3009 $ (-1252v^7 - 37684v^5 - 336772v^3 - 942760v) / 3009
$\beta_{5}$	$=$	$ ( -36\nu^{6} - 1084\nu^{4} - 9440\nu^{2} - 24466 ) / 17 $ (-36v^6 - 1084v^4 - 9440*v^2 - 24466) / 17
$\beta_{6}$	$=$	$ ( -224\nu^{6} - 6080\nu^{4} - 43808\nu^{2} - 87920 ) / 51 $ (-224v^6 - 6080v^4 - 43808*v^2 - 87920) / 51
$\beta_{7}$	$=$	$ ( 3878\nu^{7} + 110360\nu^{5} + 882050\nu^{3} + 2090816\nu ) / 3009 $ (3878v^7 + 110360v^5 + 882050v^3 + 2090816v) / 3009

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

$\nu$	$=$	$ ( -\beta_{7} - \beta_{4} + 5\beta_{2} + 19\beta_1 ) / 80 $ (-b7 - b4 + 5b2 + 19b1) / 80
$\nu^{2}$	$=$	$ ( 5\beta_{6} + 10\beta_{5} + 44\beta_{3} - 1400 ) / 160 $ (5b6 + 10b5 + 44*b3 - 1400) / 160
$\nu^{3}$	$=$	$ ( 4\beta_{7} - 11\beta_{4} - 115\beta_{2} - 106\beta_1 ) / 40 $ (4b7 - 11b4 - 115b2 - 106b1) / 40
$\nu^{4}$	$=$	$ ( -50\beta_{6} - 125\beta_{5} - 494\beta_{3} + 7980 ) / 80 $ (-50b6 - 125b5 - 494*b3 + 7980) / 80
$\nu^{5}$	$=$	$ ( -11\beta_{7} + 57\beta_{4} + 517\beta_{2} + 289\beta_1 ) / 8 $ (-11b7 + 57b4 + 517b2 + 289b1) / 8
$\nu^{6}$	$=$	$ ( 850\beta_{6} + 2415\beta_{5} + 9106\beta_{3} - 111100 ) / 80 $ (850b6 + 2415b5 + 9106*b3 - 111100) / 80
$\nu^{7}$	$=$	$ ( 956\beta_{7} - 5339\beta_{4} - 48755\beta_{2} - 22134\beta_1 ) / 40 $ (956b7 - 5339b4 - 48755b2 - 22134b1) / 40

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$1$	$1$

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} $

191.1

		4.10910i
		2.72964i
		2.49107i
		2.11161i
	−	2.11161i
	−	2.49107i
	−	2.72964i
	−	4.10910i

−

17.6725i

−11.1803

−

41.7166i

−231.317

191.2

−

14.1546i

11.1803

−

49.8485i

−119.354

191.3

−

8.72821i

−11.1803

47.9490i

4.81829

191.4

−

5.21036i

11.1803

−

32.1829i

53.8521

191.5

5.21036i

11.1803

32.1829i

53.8521

191.6

8.72821i

−11.1803

−

47.9490i

4.81829

191.7

14.1546i

11.1803

49.8485i

−119.354

191.8

17.6725i

−11.1803

41.7166i

−231.317

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	320.5.b.c		8
4.b	odd	2	1	inner	320.5.b.c		8
8.b	even	2	1		160.5.b.a	✓	8
8.d	odd	2	1		160.5.b.a	✓	8
24.f	even	2	1		1440.5.e.b		8
24.h	odd	2	1		1440.5.e.b		8
40.e	odd	2	1		800.5.b.g		8
40.f	even	2	1		800.5.b.g		8
40.i	odd	4	1		800.5.h.k		8
40.i	odd	4	1		800.5.h.l		8
40.k	even	4	1		800.5.h.k		8
40.k	even	4	1		800.5.h.l		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.5.b.a	✓	8	8.b	even	2	1
160.5.b.a	✓	8	8.d	odd	2	1
320.5.b.c		8	1.a	even	1	1	trivial
320.5.b.c		8	4.b	odd	2	1	inner
800.5.b.g		8	40.e	odd	2	1
800.5.b.g		8	40.f	even	2	1
800.5.h.k		8	40.i	odd	4	1
800.5.h.k		8	40.k	even	4	1
800.5.h.l		8	40.i	odd	4	1
800.5.h.l		8	40.k	even	4	1
1440.5.e.b		8	24.f	even	2	1
1440.5.e.b		8	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 616T_{3}^{6} + 117616T_{3}^{4} + 7526016T_{3}^{2} + 129413376 $ T3^8 + 616*T3^6 + 117616*T3^4 + 7526016*T3^2 + 129413376 acting on $S_{5}^{\mathrm{new}}(320, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + 616 T^{6} + \cdots + 129413376 $ T^8 + 616T^6 + 117616T^4 + 7526016*T^2 + 129413376
$5$	$ (T^{2} - 125)^{4} $ (T^2 - 125)^4
$7$	$ T^{8} + \cdots + 10297526968576 $ T^8 + 7560T^6 + 20795888T^4 + 24482378880*T^2 + 10297526968576
$11$	$ T^{8} + \cdots + 7518651744256 $ T^8 + 65120T^6 + 938179328T^4 + 2300121866240*T^2 + 7518651744256
$13$	$ (T^{4} + 16 T^{3} + \cdots + 24176144)^{2} $ (T^4 + 16T^3 - 18632T^2 - 368448*T + 24176144)^2
$17$	$ (T^{4} + 280 T^{3} + \cdots + 3410634256)^{2} $ (T^4 + 280T^3 - 169192T^2 - 37582240*T + 3410634256)^2
$19$	$ T^{8} + \cdots + 22\!\cdots\!36 $ T^8 + 235456T^6 + 15322410496T^4 + 226231969038336*T^2 + 227436143299854336
$23$	$ T^{8} + \cdots + 56\!\cdots\!00 $ T^8 + 638408T^6 + 108881252336T^4 + 2949649420496000*T^2 + 560758350244000000
$29$	$ (T^{4} - 1032 T^{3} + \cdots - 131615691120)^{2} $ (T^4 - 1032T^3 - 1627784T^2 + 1315582560*T - 131615691120)^2
$31$	$ T^{8} + \cdots + 35\!\cdots\!00 $ T^8 + 2461728T^6 + 1713528123136T^4 + 240325717471272960*T^2 + 3531885571712681574400
$37$	$ (T^{4} + 1808 T^{3} + \cdots + 482453699600)^{2} $ (T^4 + 1808T^3 - 1644744T^2 - 2693677120*T + 482453699600)^2
$41$	$ (T^{4} + \cdots + 8307294475280)^{2} $ (T^4 + 2608T^3 - 5056584T^2 - 9559110080*T + 8307294475280)^2
$43$	$ T^{8} + \cdots + 12\!\cdots\!00 $ T^8 + 27765672T^6 + 272442447852656T^4 + 1071511320772593951360*T^2 + 1255206865991528449383174400
$47$	$ T^{8} + \cdots + 69\!\cdots\!36 $ T^8 + 34838856T^6 + 352804168090096T^4 + 1041153008639654468736*T^2 + 697383257008381340531527936
$53$	$ (T^{4} + \cdots - 21920933051120)^{2} $ (T^4 + 2544T^3 - 19001736T^2 - 46570872000*T - 21920933051120)^2
$59$	$ T^{8} + \cdots + 43\!\cdots\!16 $ T^8 + 60413760T^6 + 972237942814208T^4 + 1835980136714791895040*T^2 + 43531593640268699853193216
$61$	$ (T^{4} + \cdots + 121504764169616)^{2} $ (T^4 - 2752T^3 - 20677416T^2 + 30294324992*T + 121504764169616)^2
$67$	$ T^{8} + \cdots + 85\!\cdots\!36 $ T^8 + 17156520T^6 + 88033061654128T^4 + 162591386517098547840*T^2 + 85644342617096071638036736
$71$	$ T^{8} + \cdots + 21\!\cdots\!16 $ T^8 + 105454880T^6 + 2324825293553408T^4 + 1034605170901861867520*T^2 + 21177417085869407396233216
$73$	$ (T^{4} + \cdots + 119575623950096)^{2} $ (T^4 - 8968T^3 - 9304616T^2 + 144790350048*T + 119575623950096)^2
$79$	$ T^{8} + \cdots + 13\!\cdots\!76 $ T^8 + 156089984T^6 + 2520880561647616T^4 + 10728468983831956291584*T^2 + 1361616791894498500543512576
$83$	$ T^{8} + \cdots + 74\!\cdots\!00 $ T^8 + 86118440T^6 + 2337768512620400T^4 + 23691159858187120400000*T^2 + 74601700634578874832100000000
$89$	$ (T^{4} + \cdots - 17\!\cdots\!00)^{2} $ (T^4 + 25304T^3 + 172118744T^2 + 9655693920*T - 1774158379599600)^2
$97$	$ (T^{4} + \cdots + 774583434485136)^{2} $ (T^4 - 1720T^3 - 159376328T^2 + 648046365600*T + 774583434485136)^2
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} + \beta_{3} q^{5} + ( - \beta_{7} - 4 \beta_{2} - 2 \beta_1) q^{7} + (\beta_{6} + \beta_{5} + 4 \beta_{3} - 73) q^{9} + (3 \beta_{7} - \beta_{4} + \cdots + \beta_1) q^{11} + ( - \beta_{5} - 8 \beta_{3} - 4) q^{13}+ \cdots + ( - 291 \beta_{7} + 253 \beta_{4} + \cdots + 147 \beta_1) q^{99}+O(q^{100}) \) q - b1 * q^3 + b3 * q^5 + (-b7 - 4b2 - 2b1) * q^7 + (b6 + b5 + 4b3 - 73) q^9 + (3b7 - b4 - 20b2 + b1) * q^11 + (-b5 - 8b3 - 4) q^13 + (-3b7 + 2b4 + 15b2 + 2b1) * q^15 + (3b6 + b5 + 10b3 - 70) * q^17 + (5b7 + 5b4 - 30b2 + b1) q^19 + (4b6 + 6b5 - 10b3 - 298) q^21 + (3b7 - 8b4 - 22b2 + 24b1) * q^23 + 125 * q^25 + (12b7 + 30b4 + 144b2 + 62b1) * q^27 + (6b6 + 2b5 - 72b3 + 258) q^29 + (-4b7 - 6b4 + 30b2 - 58b1) * q^31 + (-5b6 + 23b5 + 138b3 + 68) q^33 + (b7 + 11b4 + 80b2 - 14b1) q^35 + (2b6 + 26b5 - 2b3 - 452) q^37 + (24b7 - 22b4 - 252b2 - 18b1) * q^39 + (7b6 - 37b5 - 4b3 - 652) q^41 + (58b7 + 28b4 - 292b2 - 123b1) * q^43 + (-25b5 - 83b3 + 450) * q^45 + (-25b7 + 64b4 + 474b2 - 6b1) * q^47 + (3b6 - 9b5 + 8b3 + 511) q^49 + (42b7 + 74b4 + 138b2 + 264b1) * q^51 + (-22b6 - 59b5 - 68b3 - 636) q^53 + (14b7 + 4b4 - 105b2 + 104b1) * q^55 + (-21b6 + 9b5 + 66b3 + 384) q^57 + (119b7 - 5b4 + 22b2 + 79b1) * q^59 + (-34b6 - 24b5 - 42b3 + 688) q^61 + (45b7 + 80b4 + 126b2 + 376b1) * q^63 + (5b6 + 5b5 - 2b3 - 950) q^65 + (10b7 + 44b4 - 264b2 + 63b1) * q^67 + (-14b6 + 30b5 + 218b3 + 3218) q^69 + (122b7 + 104b4 + 134b2 - 188b1) * q^71 + (-15b6 + 85b5 - 6b3 + 2242) q^73 - 125b1 q^75 + (-14b6 + 93b5 + 334b3 + 1980) q^77 + (42b7 + 170b4 - 622b2 - 10b1) * q^79 + (-65b6 - 245b5 - 620b3 + 5195) q^81 + (-80b7 + 62b4 - 424b2 - 213b1) * q^83 + (10b6 - 65b5 - 96b3 + 1200) q^85 + (360b7 - 36b4 - 1104b2 - 54b1) * q^87 + (10b6 + 62b5 - 428b3 - 6326) q^89 + (52b7 - 76b4 - 794b2 + 50b1) * q^91 + (78b6 + 52b5 + 212b3 - 9228) q^93 + (18b7 + 38b4 - 565b2 + 38b1) * q^95 + (31b6 + 187b5 + 250b3 + 430) q^97 + (-291b7 + 253b4 + 3726b2 + 147b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 584 q^{9} - 32 q^{13} - 560 q^{17} - 2384 q^{21} + 1000 q^{25} + 2064 q^{29} + 544 q^{33} - 3616 q^{37} - 5216 q^{41} + 3600 q^{45} + 4088 q^{49} - 5088 q^{53} + 3072 q^{57} + 5504 q^{61} - 7600 q^{65}+ \cdots + 3440 q^{97}+O(q^{100}) \) 8 * q - 584 * q^9 - 32 * q^13 - 560 * q^17 - 2384 * q^21 + 1000 * q^25 + 2064 * q^29 + 544 * q^33 - 3616 * q^37 - 5216 * q^41 + 3600 * q^45 + 4088 * q^49 - 5088 * q^53 + 3072 * q^57 + 5504 * q^61 - 7600 * q^65 + 25744 * q^69 + 17936 * q^73 + 15840 * q^77 + 41560 * q^81 + 9600 * q^85 - 50608 * q^89 - 73824 * q^93 + 3440 * q^97

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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	320.5.b.c

Level	$320$
Weight	$5$
Character orbit	320.b
Analytic conductor	$33.078$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(33.0783881868\)
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:	\( x^{8} + 35x^{6} + 413x^{4} + 2015x^{2} + 3481 \) x^8 + 35x^6 + 413x^4 + 2015*x^2 + 3481
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{20}\cdot 5^{2} \)
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 94\nu^{7} + 2464\nu^{5} + 16402\nu^{3} + 39904\nu ) / 3009 \) (94v^7 + 2464v^5 + 16402v^3 + 39904v) / 3009
\(\beta_{2}\)	\(=\)	\( ( 56\nu^{7} + 1724\nu^{5} + 15576\nu^{3} + 42040\nu ) / 1003 \) (56v^7 + 1724v^5 + 15576v^3 + 42040v) / 1003
\(\beta_{3}\)	\(=\)	\( ( 50\nu^{6} + 1430\nu^{4} + 11600\nu^{2} + 28295 ) / 51 \) (50v^6 + 1430v^4 + 11600*v^2 + 28295) / 51
\(\beta_{4}\)	\(=\)	\( ( -1252\nu^{7} - 37684\nu^{5} - 336772\nu^{3} - 942760\nu ) / 3009 \) (-1252v^7 - 37684v^5 - 336772v^3 - 942760v) / 3009
\(\beta_{5}\)	\(=\)	\( ( -36\nu^{6} - 1084\nu^{4} - 9440\nu^{2} - 24466 ) / 17 \) (-36v^6 - 1084v^4 - 9440*v^2 - 24466) / 17
\(\beta_{6}\)	\(=\)	\( ( -224\nu^{6} - 6080\nu^{4} - 43808\nu^{2} - 87920 ) / 51 \) (-224v^6 - 6080v^4 - 43808*v^2 - 87920) / 51
\(\beta_{7}\)	\(=\)	\( ( 3878\nu^{7} + 110360\nu^{5} + 882050\nu^{3} + 2090816\nu ) / 3009 \) (3878v^7 + 110360v^5 + 882050v^3 + 2090816v) / 3009

\(\nu\)	\(=\)	\( ( -\beta_{7} - \beta_{4} + 5\beta_{2} + 19\beta_1 ) / 80 \) (-b7 - b4 + 5b2 + 19b1) / 80
\(\nu^{2}\)	\(=\)	\( ( 5\beta_{6} + 10\beta_{5} + 44\beta_{3} - 1400 ) / 160 \) (5b6 + 10b5 + 44*b3 - 1400) / 160
\(\nu^{3}\)	\(=\)	\( ( 4\beta_{7} - 11\beta_{4} - 115\beta_{2} - 106\beta_1 ) / 40 \) (4b7 - 11b4 - 115b2 - 106b1) / 40
\(\nu^{4}\)	\(=\)	\( ( -50\beta_{6} - 125\beta_{5} - 494\beta_{3} + 7980 ) / 80 \) (-50b6 - 125b5 - 494*b3 + 7980) / 80
\(\nu^{5}\)	\(=\)	\( ( -11\beta_{7} + 57\beta_{4} + 517\beta_{2} + 289\beta_1 ) / 8 \) (-11b7 + 57b4 + 517b2 + 289b1) / 8
\(\nu^{6}\)	\(=\)	\( ( 850\beta_{6} + 2415\beta_{5} + 9106\beta_{3} - 111100 ) / 80 \) (850b6 + 2415b5 + 9106*b3 - 111100) / 80
\(\nu^{7}\)	\(=\)	\( ( 956\beta_{7} - 5339\beta_{4} - 48755\beta_{2} - 22134\beta_1 ) / 40 \) (956b7 - 5339b4 - 48755b2 - 22134b1) / 40

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 191.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} + 616 T^{6} + \cdots + 129413376 \) T^8 + 616T^6 + 117616T^4 + 7526016*T^2 + 129413376
$5$	\( (T^{2} - 125)^{4} \) (T^2 - 125)^4
$7$	\( T^{8} + \cdots + 10297526968576 \) T^8 + 7560T^6 + 20795888T^4 + 24482378880*T^2 + 10297526968576
$11$	\( T^{8} + \cdots + 7518651744256 \) T^8 + 65120T^6 + 938179328T^4 + 2300121866240*T^2 + 7518651744256
$13$	\( (T^{4} + 16 T^{3} + \cdots + 24176144)^{2} \) (T^4 + 16T^3 - 18632T^2 - 368448*T + 24176144)^2
$17$	\( (T^{4} + 280 T^{3} + \cdots + 3410634256)^{2} \) (T^4 + 280T^3 - 169192T^2 - 37582240*T + 3410634256)^2
$19$	\( T^{8} + \cdots + 22\!\cdots\!36 \) T^8 + 235456T^6 + 15322410496T^4 + 226231969038336*T^2 + 227436143299854336
$23$	\( T^{8} + \cdots + 56\!\cdots\!00 \) T^8 + 638408T^6 + 108881252336T^4 + 2949649420496000*T^2 + 560758350244000000
$29$	\( (T^{4} - 1032 T^{3} + \cdots - 131615691120)^{2} \) (T^4 - 1032T^3 - 1627784T^2 + 1315582560*T - 131615691120)^2
$31$	\( T^{8} + \cdots + 35\!\cdots\!00 \) T^8 + 2461728T^6 + 1713528123136T^4 + 240325717471272960*T^2 + 3531885571712681574400
$37$	\( (T^{4} + 1808 T^{3} + \cdots + 482453699600)^{2} \) (T^4 + 1808T^3 - 1644744T^2 - 2693677120*T + 482453699600)^2
$41$	\( (T^{4} + \cdots + 8307294475280)^{2} \) (T^4 + 2608T^3 - 5056584T^2 - 9559110080*T + 8307294475280)^2
$43$	\( T^{8} + \cdots + 12\!\cdots\!00 \) T^8 + 27765672T^6 + 272442447852656T^4 + 1071511320772593951360*T^2 + 1255206865991528449383174400
$47$	\( T^{8} + \cdots + 69\!\cdots\!36 \) T^8 + 34838856T^6 + 352804168090096T^4 + 1041153008639654468736*T^2 + 697383257008381340531527936
$53$	\( (T^{4} + \cdots - 21920933051120)^{2} \) (T^4 + 2544T^3 - 19001736T^2 - 46570872000*T - 21920933051120)^2
$59$	\( T^{8} + \cdots + 43\!\cdots\!16 \) T^8 + 60413760T^6 + 972237942814208T^4 + 1835980136714791895040*T^2 + 43531593640268699853193216
$61$	\( (T^{4} + \cdots + 121504764169616)^{2} \) (T^4 - 2752T^3 - 20677416T^2 + 30294324992*T + 121504764169616)^2
$67$	\( T^{8} + \cdots + 85\!\cdots\!36 \) T^8 + 17156520T^6 + 88033061654128T^4 + 162591386517098547840*T^2 + 85644342617096071638036736
$71$	\( T^{8} + \cdots + 21\!\cdots\!16 \) T^8 + 105454880T^6 + 2324825293553408T^4 + 1034605170901861867520*T^2 + 21177417085869407396233216
$73$	\( (T^{4} + \cdots + 119575623950096)^{2} \) (T^4 - 8968T^3 - 9304616T^2 + 144790350048*T + 119575623950096)^2
$79$	\( T^{8} + \cdots + 13\!\cdots\!76 \) T^8 + 156089984T^6 + 2520880561647616T^4 + 10728468983831956291584*T^2 + 1361616791894498500543512576
$83$	\( T^{8} + \cdots + 74\!\cdots\!00 \) T^8 + 86118440T^6 + 2337768512620400T^4 + 23691159858187120400000*T^2 + 74601700634578874832100000000
$89$	\( (T^{4} + \cdots - 17\!\cdots\!00)^{2} \) (T^4 + 25304T^3 + 172118744T^2 + 9655693920*T - 1774158379599600)^2
$97$	\( (T^{4} + \cdots + 774583434485136)^{2} \) (T^4 - 1720T^3 - 159376328T^2 + 648046365600*T + 774583434485136)^2
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\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)