LMFDB - Newform orbit 176.6.e.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,6,Mod(175,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("176.175");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 176 = 2^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	176.e (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$28.2275522871$
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} - 4 x^{7} - 7362 x^{6} + 22100 x^{5} + 14183534 x^{4} - 28403906 x^{3} + 315403855 x^{2} + \cdots + 1477121929252 $ x^8 - 4x^7 - 7362x^6 + 22100x^5 + 14183534x^4 - 28403906x^3 + 315403855x^2 - 301198218*x + 1477121929252
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
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_1 + 12) q^{5} - \beta_{7} q^{7} + (5 \beta_1 - 219) q^{9} + (\beta_{7} + \beta_{6} + \cdots - 4 \beta_{4}) q^{11} - \beta_{3} q^{13} + ( - 3 \beta_{6} + 14 \beta_{4}) q^{15}+ \cdots + ( - 604 \beta_{7} + \cdots + 2966 \beta_{4}) q^{99}+O(q^{100}) $ q - b4 * q^3 + (b1 + 12) * q^5 - b7 * q^7 + (5b1 - 219) q^9 + (b7 + b6 + b5 - 4b4) q^11 - b3 * q^13 + (-3b6 + 14b4) * q^15 + (-b3 + b2) * q^17 + (5b7 + b6 + 2b5 + b4) * q^19 + (-5b3 - 3b2) * q^21 + (10b6 - 31b4) * q^23 + (25b1 - 539) q^25 + (-15b6 + 106b4) * q^27 + (2b3 + 5b2) * q^29 + (-6b6 + 85b4) * q^31 + (11b3 + 77b1 - 2112) * q^33 + (29b7 + 10b6 + 20b5 + 10b4) * q^35 + (-75b1 + 868) q^37 + (77b7 + 16b6 + 32b5 + 16b4) * q^39 + (13b3 - 7b2) * q^41 + (-34b7 + 17b6 + 34b5 + 17b4) * q^43 + (-154b1 + 9582) q^45 + (-73b6 + 305b4) * q^47 + (-464b1 + 18041) q^49 + (-17b7 + 26b6 + 52b5 + 26b4) * q^51 + (144b1 - 22122) q^53 + (-65b7 + 23b6 + 34b5 + 370b4) * q^55 + (37b3 + 9b2) * q^57 + (28b6 - 613b4) * q^59 + (-48b3 - 9b2) * q^61 + (424b7 + 50b6 + 100b5 + 50b4) * q^63 + (11b3 + 27b2) * q^65 + (-108b6 + 471b4) * q^67 + (1245b1 - 14982) q^69 + (-10b6 + 1609b4) * q^71 + (-53b3 - 45b2) * q^73 + (-75b6 + 1189b4) * q^75 + (-55b3 - 11b2 + 1232b1 - 20064) q^77 + (124b7 + 36b6 + 72b5 + 36b4) * q^79 + (-950b1 - 3255) q^81 + (368b7 - 5b6 - 10b5 - 5b4) * q^83 + (-59b3 + 63b2) * q^85 + (-624b7 + 18b6 + 36b5 + 18b4) * q^87 + (81b1 + 36288) q^89 + (336b6 - 4736b4) * q^91 + (-1079b1 + 39666) q^93 + (-217b7 + 4b6 + 8b5 + 4b4) * q^95 + (-1391b1 - 73688) q^97 + (-604b7 - 164b6 - 109b5 + 2966b4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 100 q^{5} - 1732 q^{9} - 4212 q^{25} - 16588 q^{33} + 6644 q^{37} + 76040 q^{45} + 142472 q^{49} - 176400 q^{53} - 114876 q^{69} - 155584 q^{77} - 29840 q^{81} + 290628 q^{89} + 313012 q^{93} - 595068 q^{97}+O(q^{100}) $ 8 * q + 100 * q^5 - 1732 * q^9 - 4212 * q^25 - 16588 * q^33 + 6644 * q^37 + 76040 * q^45 + 142472 * q^49 - 176400 * q^53 - 114876 * q^69 - 155584 * q^77 - 29840 * q^81 + 290628 * q^89 + 313012 * q^93 - 595068 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 4 x^{7} - 7362 x^{6} + 22100 x^{5} + 14183534 x^{4} - 28403906 x^{3} + 315403855 x^{2} + \cdots + 1477121929252 $ x^8 - 4*x^7 - 7362*x^6 + 22100*x^5 + 14183534*x^4 - 28403906*x^3 + 315403855*x^2 - 301198218*x + 1477121929252 :

$\beta_{1}$	$=$	$ ( - 76 \nu^{6} + 228 \nu^{5} + 412647 \nu^{4} - 825674 \nu^{3} + 22803321 \nu^{2} + \cdots - 1084025696458 ) / 23022410833 $ (-76v^6 + 228v^5 + 412647v^4 - 825674v^3 + 22803321v^2 - 22390446v - 1084025696458) / 23022410833
$\beta_{2}$	$=$	$ ( - 161252 \nu^{6} + 483756 \nu^{5} + 2087234176 \nu^{4} - 4175274612 \nu^{3} + \cdots + 10271975481128 ) / 2279218672467 $ (-161252v^6 + 483756v^5 + 2087234176v^4 - 4175274612v^3 - 5473360836132v^2 + 5475448554064v + 10271975481128) / 2279218672467
$\beta_{3}$	$=$	$ ( - 461842 \nu^{6} + 1385526 \nu^{5} + 3113454440 \nu^{4} - 6229218090 \nu^{3} + \cdots + 107683947517024 ) / 2279218672467 $ (-461842v^6 + 1385526v^5 + 3113454440v^4 - 6229218090v^3 - 5661257189886v^2 + 5664372029852v + 107683947517024) / 2279218672467
$\beta_{4}$	$=$	$ ( - 115993072 \nu^{7} + 405975752 \nu^{6} + 801680254270 \nu^{5} - 2005215575055 \nu^{4} + \cdots + 23\!\cdots\!98 ) / 44\!\cdots\!95 $ (-115993072v^7 + 405975752v^6 + 801680254270v^5 - 2005215575055v^4 - 1361339016459548v^3 + 2044013943252253v^2 - 467501576610722996*v + 233410052464634198) / 447680332948180695
$\beta_{5}$	$=$	$ ( 560777242 \nu^{7} - 1962720347 \nu^{6} - 2625291397405 \nu^{5} + 6568135294380 \nu^{4} + \cdots + 49\!\cdots\!57 ) / 13\!\cdots\!85 $ (560777242v^7 - 1962720347v^6 - 2625291397405v^5 + 6568135294380v^4 + 2735568744516383v^3 - 4109922233429128v^2 - 9892780351657212439*v + 4947075381852085657) / 1343040998844542085
$\beta_{6}$	$=$	$ ( 402051692 \nu^{7} - 1407180922 \nu^{6} - 3278956814096 \nu^{5} + 8200909987545 \nu^{4} + \cdots - 10\!\cdots\!96 ) / 26\!\cdots\!17 $ (402051692v^7 - 1407180922v^6 - 3278956814096v^5 + 8200909987545v^4 + 6257003191075006v^3 - 9393706400190515v^2 + 2183889458058254482*v - 1090378837898591596) / 268608199768908417
$\beta_{7}$	$=$	$ ( 2314028092 \nu^{7} - 8099098322 \nu^{6} - 18288430196150 \nu^{5} + 45741323236180 \nu^{4} + \cdots + 61\!\cdots\!72 ) / 13\!\cdots\!85 $ (2314028092v^7 - 8099098322v^6 - 18288430196150v^5 + 45741323236180v^4 + 41586321576218478v^3 - 62425227737113058v^2 - 12351272384297305764*v + 6186041921675115272) / 1343040998844542085

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

$\nu$	$=$	$ ( -\beta_{6} - 2\beta_{5} - 9\beta_{4} + 8 ) / 16 $ (-b6 - 2b5 - 9b4 + 8) / 16
$\nu^{2}$	$=$	$ ( -\beta_{6} - 2\beta_{5} - 9\beta_{4} - 12\beta_{3} + 6\beta_{2} + 608\beta _1 + 29176 ) / 16 $ (-b6 - 2b5 - 9b4 - 12b3 + 6b2 + 608*b1 + 29176) / 16
$\nu^{3}$	$=$	$ ( 2916\beta_{7} - 5970\beta_{6} - 6588\beta_{5} - 25714\beta_{4} - 18\beta_{3} + 9\beta_{2} + 912\beta _1 + 43760 ) / 16 $ (2916b7 - 5970b6 - 6588b5 - 25714b4 - 18b3 + 9b2 + 912*b1 + 43760) / 16
$\nu^{4}$	$=$	$ ( 5832 \beta_{7} - 11939 \beta_{6} - 13174 \beta_{5} - 51419 \beta_{4} - 54720 \beta_{3} + \cdots + 102499816 ) / 16 $ (5832b7 - 11939b6 - 13174b5 - 51419b4 - 54720b3 + 57456b2 + 2127472*b1 + 102499816) / 16
$\nu^{5}$	$=$	$ ( 8982828 \beta_{7} - 28104686 \beta_{6} - 22538212 \beta_{5} - 138945318 \beta_{4} - 136770 \beta_{3} + \cdots + 256176608 ) / 16 $ (8982828b7 - 28104686b6 - 22538212b5 - 138945318b4 - 136770b3 + 143625b2 + 5317160*b1 + 256176608) / 16
$\nu^{6}$	$=$	$ ( 26933904 \beta_{7} - 84284211 \beta_{6} - 67581702 \beta_{5} - 416707411 \beta_{4} + \cdots + 337358391736 ) / 16 $ (26933904b7 - 84284211b6 - 67581702b5 - 416707411b4 - 300921120b3 + 314094492b2 + 6892895816*b1 + 337358391736) / 16
$\nu^{7}$	$=$	$ ( 27854520120 \beta_{7} - 120253711402 \beta_{6} - 70435610012 \beta_{5} - 684731244834 \beta_{4} + \cdots + 1179857804000 ) / 16 $ (27854520120b7 - 120253711402b6 - 70435610012b5 - 684731244834b4 - 1052745246b3 + 1098828045b2 + 24106526360*b1 + 1179857804000) / 16

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$.

$n$	$111$	$133$	$145$
$\chi(n)$	$-1$	$1$	$-1$

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} $

175.1

−11.3732	−	13.2909i
12.3732	−	13.2909i
61.9270	−	7.28705i
−60.9270	−	7.28705i
61.9270	+	7.28705i
−60.9270	+	7.28705i
−11.3732	+	13.2909i
12.3732	+	13.2909i

−

26.5819i

−36.9191

−239.888

−463.596

175.2

−

26.5819i

−36.9191

239.888

−463.596

175.3

−

14.5741i

61.9191

−108.100

30.5956

175.4

−

14.5741i

61.9191

108.100

30.5956

175.5

14.5741i

61.9191

−108.100

30.5956

175.6

14.5741i

61.9191

108.100

30.5956

175.7

26.5819i

−36.9191

−239.888

−463.596

175.8

26.5819i

−36.9191

239.888

−463.596

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	176.6.e.c	✓	8
4.b	odd	2	1	inner	176.6.e.c	✓	8
11.b	odd	2	1	inner	176.6.e.c	✓	8
44.c	even	2	1	inner	176.6.e.c	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
176.6.e.c	✓	8	1.a	even	1	1	trivial
176.6.e.c	✓	8	4.b	odd	2	1	inner
176.6.e.c	✓	8	11.b	odd	2	1	inner
176.6.e.c	✓	8	44.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} + 919T_{3}^{2} + 150084 $ T3^4 + 919*T3^2 + 150084 acting on $S_{6}^{\mathrm{new}}(176, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} + 919 T^{2} + 150084)^{2} $ (T^4 + 919*T^2 + 150084)^2
$5$	$ (T^{2} - 25 T - 2286)^{4} $ (T^2 - 25*T - 2286)^4
$7$	$ (T^{4} - 69232 T^{2} + 672460800)^{2} $ (T^4 - 69232*T^2 + 672460800)^2
$11$	$ T^{8} + \cdots + 67\!\cdots\!01 $ T^8 - 392084T^6 + 85465642262T^4 - 10169649187258484*T^2 + 672749994932560009201
$13$	$ (T^{4} + 912784 T^{2} + 168904424448)^{2} $ (T^4 + 912784*T^2 + 168904424448)^2
$17$	$ (T^{4} + 3533392 T^{2} + 3707092992)^{2} $ (T^4 + 3533392*T^2 + 3707092992)^2
$19$	$ (T^{4} - 2126128 T^{2} + 377714506752)^{2} $ (T^4 - 2126128*T^2 + 377714506752)^2
$23$	$ (T^{4} + \cdots + 85363997233284)^{2} $ (T^4 + 18624119*T^2 + 85363997233284)^2
$29$	$ (T^{4} + \cdots + 10\!\cdots\!28)^{2} $ (T^4 + 64410816*T^2 + 1033478212681728)^2
$31$	$ (T^{4} + \cdots + 11477677652100)^{2} $ (T^4 + 13044847*T^2 + 11477677652100)^2
$37$	$ (T^{2} - 1661 T - 13047926)^{4} $ (T^2 - 1661*T - 13047926)^4
$41$	$ (T^{4} + \cdots + 67\!\cdots\!00)^{2} $ (T^4 + 288376912*T^2 + 6781223024947200)^2
$43$	$ (T^{4} + \cdots + 10\!\cdots\!08)^{2} $ (T^4 - 439760896*T^2 + 10084915302494208)^2
$47$	$ (T^{4} + \cdots + 26\!\cdots\!00)^{2} $ (T^4 + 1031170008*T^2 + 265721844350771600)^2
$53$	$ (T^{2} + 44100 T + 435560004)^{4} $ (T^2 + 44100*T + 435560004)^4
$59$	$ (T^{4} + \cdots + 937504392372900)^{2} $ (T^4 + 485098583*T^2 + 937504392372900)^2
$61$	$ (T^{4} + \cdots + 30\!\cdots\!72)^{2} $ (T^4 + 2245621248*T^2 + 304190787871567872)^2
$67$	$ (T^{4} + \cdots + 12\!\cdots\!76)^{2} $ (T^4 + 2273854095*T^2 + 1290098666758227876)^2
$71$	$ (T^{4} + \cdots + 78\!\cdots\!04)^{2} $ (T^4 + 2397648479*T^2 + 783821937818343204)^2
$73$	$ (T^{4} + \cdots + 73\!\cdots\!48)^{2} $ (T^4 + 7271545936*T^2 + 7341135863972917248)^2
$79$	$ (T^{4} + \cdots + 64\!\cdots\!68)^{2} $ (T^4 - 1821485824*T^2 + 646501688789434368)^2
$83$	$ (T^{4} + \cdots + 15\!\cdots\!00)^{2} $ (T^4 - 9623998528*T^2 + 15234643978562764800)^2
$89$	$ (T^{2} - 72657 T + 1303736310)^{4} $ (T^2 - 72657*T + 1303736310)^4
$97$	$ (T^{2} + 148767 T + 807441950)^{4} $ (T^2 + 148767*T + 807441950)^4
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	176.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{3} + (\beta_1 + 12) q^{5} - \beta_{7} q^{7} + (5 \beta_1 - 219) q^{9} + (\beta_{7} + \beta_{6} + \cdots - 4 \beta_{4}) q^{11} - \beta_{3} q^{13} + ( - 3 \beta_{6} + 14 \beta_{4}) q^{15}+ \cdots + ( - 604 \beta_{7} + \cdots + 2966 \beta_{4}) q^{99}+O(q^{100}) \) q - b4 * q^3 + (b1 + 12) * q^5 - b7 * q^7 + (5b1 - 219) q^9 + (b7 + b6 + b5 - 4b4) q^11 - b3 * q^13 + (-3b6 + 14b4) * q^15 + (-b3 + b2) * q^17 + (5b7 + b6 + 2b5 + b4) * q^19 + (-5b3 - 3b2) * q^21 + (10b6 - 31b4) * q^23 + (25b1 - 539) q^25 + (-15b6 + 106b4) * q^27 + (2b3 + 5b2) * q^29 + (-6b6 + 85b4) * q^31 + (11b3 + 77b1 - 2112) * q^33 + (29b7 + 10b6 + 20b5 + 10b4) * q^35 + (-75b1 + 868) q^37 + (77b7 + 16b6 + 32b5 + 16b4) * q^39 + (13b3 - 7b2) * q^41 + (-34b7 + 17b6 + 34b5 + 17b4) * q^43 + (-154b1 + 9582) q^45 + (-73b6 + 305b4) * q^47 + (-464b1 + 18041) q^49 + (-17b7 + 26b6 + 52b5 + 26b4) * q^51 + (144b1 - 22122) q^53 + (-65b7 + 23b6 + 34b5 + 370b4) * q^55 + (37b3 + 9b2) * q^57 + (28b6 - 613b4) * q^59 + (-48b3 - 9b2) * q^61 + (424b7 + 50b6 + 100b5 + 50b4) * q^63 + (11b3 + 27b2) * q^65 + (-108b6 + 471b4) * q^67 + (1245b1 - 14982) q^69 + (-10b6 + 1609b4) * q^71 + (-53b3 - 45b2) * q^73 + (-75b6 + 1189b4) * q^75 + (-55b3 - 11b2 + 1232b1 - 20064) q^77 + (124b7 + 36b6 + 72b5 + 36b4) * q^79 + (-950b1 - 3255) q^81 + (368b7 - 5b6 - 10b5 - 5b4) * q^83 + (-59b3 + 63b2) * q^85 + (-624b7 + 18b6 + 36b5 + 18b4) * q^87 + (81b1 + 36288) q^89 + (336b6 - 4736b4) * q^91 + (-1079b1 + 39666) q^93 + (-217b7 + 4b6 + 8b5 + 4b4) * q^95 + (-1391b1 - 73688) q^97 + (-604b7 - 164b6 - 109b5 + 2966b4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 100 q^{5} - 1732 q^{9} - 4212 q^{25} - 16588 q^{33} + 6644 q^{37} + 76040 q^{45} + 142472 q^{49} - 176400 q^{53} - 114876 q^{69} - 155584 q^{77} - 29840 q^{81} + 290628 q^{89} + 313012 q^{93} - 595068 q^{97}+O(q^{100}) \) 8 * q + 100 * q^5 - 1732 * q^9 - 4212 * q^25 - 16588 * q^33 + 6644 * q^37 + 76040 * q^45 + 142472 * q^49 - 176400 * q^53 - 114876 * q^69 - 155584 * q^77 - 29840 * q^81 + 290628 * q^89 + 313012 * q^93 - 595068 * q^97

Introduction

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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	176.6.e.c

Level	$176$
Weight	$6$
Character orbit	176.e
Analytic conductor	$28.228$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(28.2275522871\)
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:	\( x^{8} - 4 x^{7} - 7362 x^{6} + 22100 x^{5} + 14183534 x^{4} - 28403906 x^{3} + 315403855 x^{2} + \cdots + 1477121929252 \) x^8 - 4x^7 - 7362x^6 + 22100x^5 + 14183534x^4 - 28403906x^3 + 315403855x^2 - 301198218*x + 1477121929252
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 76 \nu^{6} + 228 \nu^{5} + 412647 \nu^{4} - 825674 \nu^{3} + 22803321 \nu^{2} + \cdots - 1084025696458 ) / 23022410833 \) (-76v^6 + 228v^5 + 412647v^4 - 825674v^3 + 22803321v^2 - 22390446v - 1084025696458) / 23022410833
\(\beta_{2}\)	\(=\)	\( ( - 161252 \nu^{6} + 483756 \nu^{5} + 2087234176 \nu^{4} - 4175274612 \nu^{3} + \cdots + 10271975481128 ) / 2279218672467 \) (-161252v^6 + 483756v^5 + 2087234176v^4 - 4175274612v^3 - 5473360836132v^2 + 5475448554064v + 10271975481128) / 2279218672467
\(\beta_{3}\)	\(=\)	\( ( - 461842 \nu^{6} + 1385526 \nu^{5} + 3113454440 \nu^{4} - 6229218090 \nu^{3} + \cdots + 107683947517024 ) / 2279218672467 \) (-461842v^6 + 1385526v^5 + 3113454440v^4 - 6229218090v^3 - 5661257189886v^2 + 5664372029852v + 107683947517024) / 2279218672467
\(\beta_{4}\)	\(=\)	\( ( - 115993072 \nu^{7} + 405975752 \nu^{6} + 801680254270 \nu^{5} - 2005215575055 \nu^{4} + \cdots + 23\!\cdots\!98 ) / 44\!\cdots\!95 \) (-115993072v^7 + 405975752v^6 + 801680254270v^5 - 2005215575055v^4 - 1361339016459548v^3 + 2044013943252253v^2 - 467501576610722996*v + 233410052464634198) / 447680332948180695
\(\beta_{5}\)	\(=\)	\( ( 560777242 \nu^{7} - 1962720347 \nu^{6} - 2625291397405 \nu^{5} + 6568135294380 \nu^{4} + \cdots + 49\!\cdots\!57 ) / 13\!\cdots\!85 \) (560777242v^7 - 1962720347v^6 - 2625291397405v^5 + 6568135294380v^4 + 2735568744516383v^3 - 4109922233429128v^2 - 9892780351657212439*v + 4947075381852085657) / 1343040998844542085
\(\beta_{6}\)	\(=\)	\( ( 402051692 \nu^{7} - 1407180922 \nu^{6} - 3278956814096 \nu^{5} + 8200909987545 \nu^{4} + \cdots - 10\!\cdots\!96 ) / 26\!\cdots\!17 \) (402051692v^7 - 1407180922v^6 - 3278956814096v^5 + 8200909987545v^4 + 6257003191075006v^3 - 9393706400190515v^2 + 2183889458058254482*v - 1090378837898591596) / 268608199768908417
\(\beta_{7}\)	\(=\)	\( ( 2314028092 \nu^{7} - 8099098322 \nu^{6} - 18288430196150 \nu^{5} + 45741323236180 \nu^{4} + \cdots + 61\!\cdots\!72 ) / 13\!\cdots\!85 \) (2314028092v^7 - 8099098322v^6 - 18288430196150v^5 + 45741323236180v^4 + 41586321576218478v^3 - 62425227737113058v^2 - 12351272384297305764*v + 6186041921675115272) / 1343040998844542085

\(\nu\)	\(=\)	\( ( -\beta_{6} - 2\beta_{5} - 9\beta_{4} + 8 ) / 16 \) (-b6 - 2b5 - 9b4 + 8) / 16
\(\nu^{2}\)	\(=\)	\( ( -\beta_{6} - 2\beta_{5} - 9\beta_{4} - 12\beta_{3} + 6\beta_{2} + 608\beta _1 + 29176 ) / 16 \) (-b6 - 2b5 - 9b4 - 12b3 + 6b2 + 608*b1 + 29176) / 16
\(\nu^{3}\)	\(=\)	\( ( 2916\beta_{7} - 5970\beta_{6} - 6588\beta_{5} - 25714\beta_{4} - 18\beta_{3} + 9\beta_{2} + 912\beta _1 + 43760 ) / 16 \) (2916b7 - 5970b6 - 6588b5 - 25714b4 - 18b3 + 9b2 + 912*b1 + 43760) / 16
\(\nu^{4}\)	\(=\)	\( ( 5832 \beta_{7} - 11939 \beta_{6} - 13174 \beta_{5} - 51419 \beta_{4} - 54720 \beta_{3} + \cdots + 102499816 ) / 16 \) (5832b7 - 11939b6 - 13174b5 - 51419b4 - 54720b3 + 57456b2 + 2127472*b1 + 102499816) / 16
\(\nu^{5}\)	\(=\)	\( ( 8982828 \beta_{7} - 28104686 \beta_{6} - 22538212 \beta_{5} - 138945318 \beta_{4} - 136770 \beta_{3} + \cdots + 256176608 ) / 16 \) (8982828b7 - 28104686b6 - 22538212b5 - 138945318b4 - 136770b3 + 143625b2 + 5317160*b1 + 256176608) / 16
\(\nu^{6}\)	\(=\)	\( ( 26933904 \beta_{7} - 84284211 \beta_{6} - 67581702 \beta_{5} - 416707411 \beta_{4} + \cdots + 337358391736 ) / 16 \) (26933904b7 - 84284211b6 - 67581702b5 - 416707411b4 - 300921120b3 + 314094492b2 + 6892895816*b1 + 337358391736) / 16
\(\nu^{7}\)	\(=\)	\( ( 27854520120 \beta_{7} - 120253711402 \beta_{6} - 70435610012 \beta_{5} - 684731244834 \beta_{4} + \cdots + 1179857804000 ) / 16 \) (27854520120b7 - 120253711402b6 - 70435610012b5 - 684731244834b4 - 1052745246b3 + 1098828045b2 + 24106526360*b1 + 1179857804000) / 16

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 175.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( (T^{4} + 919 T^{2} + 150084)^{2} \) (T^4 + 919*T^2 + 150084)^2
$5$	\( (T^{2} - 25 T - 2286)^{4} \) (T^2 - 25*T - 2286)^4
$7$	\( (T^{4} - 69232 T^{2} + 672460800)^{2} \) (T^4 - 69232*T^2 + 672460800)^2
$11$	\( T^{8} + \cdots + 67\!\cdots\!01 \) T^8 - 392084T^6 + 85465642262T^4 - 10169649187258484*T^2 + 672749994932560009201
$13$	\( (T^{4} + 912784 T^{2} + 168904424448)^{2} \) (T^4 + 912784*T^2 + 168904424448)^2
$17$	\( (T^{4} + 3533392 T^{2} + 3707092992)^{2} \) (T^4 + 3533392*T^2 + 3707092992)^2
$19$	\( (T^{4} - 2126128 T^{2} + 377714506752)^{2} \) (T^4 - 2126128*T^2 + 377714506752)^2
$23$	\( (T^{4} + \cdots + 85363997233284)^{2} \) (T^4 + 18624119*T^2 + 85363997233284)^2
$29$	\( (T^{4} + \cdots + 10\!\cdots\!28)^{2} \) (T^4 + 64410816*T^2 + 1033478212681728)^2
$31$	\( (T^{4} + \cdots + 11477677652100)^{2} \) (T^4 + 13044847*T^2 + 11477677652100)^2
$37$	\( (T^{2} - 1661 T - 13047926)^{4} \) (T^2 - 1661*T - 13047926)^4
$41$	\( (T^{4} + \cdots + 67\!\cdots\!00)^{2} \) (T^4 + 288376912*T^2 + 6781223024947200)^2
$43$	\( (T^{4} + \cdots + 10\!\cdots\!08)^{2} \) (T^4 - 439760896*T^2 + 10084915302494208)^2
$47$	\( (T^{4} + \cdots + 26\!\cdots\!00)^{2} \) (T^4 + 1031170008*T^2 + 265721844350771600)^2
$53$	\( (T^{2} + 44100 T + 435560004)^{4} \) (T^2 + 44100*T + 435560004)^4
$59$	\( (T^{4} + \cdots + 937504392372900)^{2} \) (T^4 + 485098583*T^2 + 937504392372900)^2
$61$	\( (T^{4} + \cdots + 30\!\cdots\!72)^{2} \) (T^4 + 2245621248*T^2 + 304190787871567872)^2
$67$	\( (T^{4} + \cdots + 12\!\cdots\!76)^{2} \) (T^4 + 2273854095*T^2 + 1290098666758227876)^2
$71$	\( (T^{4} + \cdots + 78\!\cdots\!04)^{2} \) (T^4 + 2397648479*T^2 + 783821937818343204)^2
$73$	\( (T^{4} + \cdots + 73\!\cdots\!48)^{2} \) (T^4 + 7271545936*T^2 + 7341135863972917248)^2
$79$	\( (T^{4} + \cdots + 64\!\cdots\!68)^{2} \) (T^4 - 1821485824*T^2 + 646501688789434368)^2
$83$	\( (T^{4} + \cdots + 15\!\cdots\!00)^{2} \) (T^4 - 9623998528*T^2 + 15234643978562764800)^2
$89$	\( (T^{2} - 72657 T + 1303736310)^{4} \) (T^2 - 72657*T + 1303736310)^4
$97$	\( (T^{2} + 148767 T + 807441950)^{4} \) (T^2 + 148767*T + 807441950)^4
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\(n\)	\(111\)	\(133\)	\(145\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)