LMFDB - Newform orbit 144.14.c.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,14,Mod(143,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 14, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.143"); S:= CuspForms(chi, 14); N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	144.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$154.412537691$
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} - 2 x^{7} + 4365087 x^{6} + 1640648910 x^{5} + 5227007496627 x^{4} + \cdots + 54\!\cdots\!04 $ x^8 - 2x^7 + 4365087x^6 + 1640648910x^5 + 5227007496627x^4 + 3589835445735744x^3 + 1691776997434821656x^2 + 382610538805924667192*x + 54094475398144125773004
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{30} $
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 + ( - 5 \beta_{4} + 40 \beta_{2}) q^{5} + \beta_{3} q^{7} - \beta_{6} q^{11} + (61 \beta_1 - 10119200) q^{13} + (11282 \beta_{4} + 114823 \beta_{2}) q^{17} + (\beta_{5} - 342 \beta_{3}) q^{19} + (7 \beta_{7} - 50 \beta_{6}) q^{23}+ \cdots + ( - 24046122 \beta_1 - 2969204811200) q^{97}+O(q^{100}) $ q + (-5b4 + 40b2) * q^5 + b3 * q^7 - b6 * q^11 + (61b1 - 10119200) q^13 + (11282b4 + 114823b2) * q^17 + (b5 - 342b3) q^19 + (7b7 - 50b6) * q^23 + (-3650b1 - 56132725) q^25 + (14675b4 - 5247534b2) * q^29 + (-14b5 - 14987b3) * q^31 + (110b7 + 1105b6) * q^35 + (-70252b1 - 9837670750) q^37 + (7217250b4 - 1556087b2) * q^41 + (50b5 + 22394b3) * q^43 + (-637b7 - 8750b6) * q^47 + (-385800b1 - 38120117753) q^49 + (-7894221b4 - 290828194b2) * q^53 + (260b5 + 466390b3) * q^55 + (-1050b7 + 33164b6) * q^59 + (346700b1 + 115600498550) q^61 + (18133630b4 + 1059141110b2) * q^65 + (-2975b5 - 1124368b3) * q^67 + (5775b7 - 49988b6) * q^71 + (6771724b1 + 224925373600) q^73 + (-292746960b4 - 117773040b2) * q^77 + (10766b5 - 2259397b3) * q^79 + (21154b7 - 28625b6) * q^83 + (-992695b1 + 1898820389070) q^85 + (26499100b4 - 4173942061b2) * q^89 + (-13481b5 + 890202b3) * q^91 + (-90090b7 - 7670b6) * q^95 + (-24046122b1 - 2969204811200) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 80953600 q^{13} - 449061800 q^{25} - 78701366000 q^{37} - 304960942024 q^{49} + 924803988400 q^{61} + 1799402988800 q^{73} + 15190563112560 q^{85} - 23753638489600 q^{97}+O(q^{100}) $ 8 * q - 80953600 * q^13 - 449061800 * q^25 - 78701366000 * q^37 - 304960942024 * q^49 + 924803988400 * q^61 + 1799402988800 * q^73 + 15190563112560 * q^85 - 23753638489600 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2 x^{7} + 4365087 x^{6} + 1640648910 x^{5} + 5227007496627 x^{4} + \cdots + 54\!\cdots\!04 $ x^8 - 2*x^7 + 4365087*x^6 + 1640648910*x^5 + 5227007496627*x^4 + 3589835445735744*x^3 + 1691776997434821656*x^2 + 382610538805924667192*x + 54094475398144125773004 :

$\beta_{1}$	$=$	$ ( - 18\!\cdots\!16 \nu^{7} + \cdots + 65\!\cdots\!28 ) / 26\!\cdots\!29 $ (-1834812271496016v^7 - 910228937774946552v^6 - 5754666480600957648048v^5 - 5627750978602773382653360v^4 - 3039973392032158842150624000v^3 - 943766828102861033270053506336v^2 - 173707176665495623487033377768896*v + 6559964777737067006210188690059331428) / 26168252310704088116635187333929
$\beta_{2}$	$=$	$ ( 12\!\cdots\!29 \nu^{7} + \cdots + 91\!\cdots\!52 ) / 35\!\cdots\!28 $ (1232975624007247771329v^7 - 338241506804136399427362v^6 + 5309771372177355730664994861v^5 + 670169963729031359231171514510v^4 + 5723974235342304011139648177077145v^3 + 2935609559564352379523604489736423764v^2 + 933256891932125662884713909745769177914*v + 91690730291616152562883565663617747909152) / 3538633585904579961638023052964921428
$\beta_{3}$	$=$	$ ( - 87\!\cdots\!77 \nu^{7} + \cdots - 20\!\cdots\!92 ) / 35\!\cdots\!66 $ (-87376926125821298770811790777v^7 + 11087426476623166917894052157898v^6 - 390184322320818145796051300720640885v^5 - 89650761518959644199541223937660390026v^4 - 477443613098461724307983187655816645109169v^3 - 245636429762074347170325309733042955492569860v^2 - 143254977053979475758032928852095090695866297414*v - 20638776123530748871373718852024163176868218655392) / 35781723313450054906103874216460723293296866
$\beta_{4}$	$=$	$ ( - 19\!\cdots\!51 \nu^{7} + \cdots - 15\!\cdots\!64 ) / 30\!\cdots\!28 $ (-199641211972374444199280315851v^7 + 41572356187328488857331757200182v^6 - 878902811091898827281535494843203455v^5 - 141071239103201820461457657775760211018v^4 - 979498750075920071202207124247720184564867v^3 - 502409884421738539207521620851685899743219996v^2 - 158677413776218899799107026390521563844175331038*v - 15725260513028251510371864850158708809383720911264) / 30670048554385761348089035042680619965683028
$\beta_{5}$	$=$	$ ( 49\!\cdots\!49 \nu^{7} + \cdots + 11\!\cdots\!72 ) / 61\!\cdots\!77 $ (494392190147453895213230566449v^7 - 105065183525328782791945266988602v^6 + 2180753641670583698445903109188440589v^5 + 368708302626091040808040462837474831290v^4 + 2533526099520067464947541200426262923245545v^3 + 1304750650728510849596440530948451785711937764v^2 + 779135866913745145759256234637069060527539324566*v + 112270131600069047649347684035501173306040469088672) / 616926264025000946656963348559667642987877
$\beta_{6}$	$=$	$ ( 17\!\cdots\!66 \nu^{7} + \cdots + 11\!\cdots\!46 ) / 18\!\cdots\!03 $ (177477484684464232866v^7 + 212368060708160516545455v^6 + 608127176029787255198758788v^5 + 1083491863324549830304347353283v^4 + 697935948694694998732155492504360v^3 + 1265382442109525769614782388565812259v^2 + 441138292535842277959465058150053292376*v + 110210215631137660029290942317793914070946) / 183177766174928616816446311337503
$\beta_{7}$	$=$	$ ( 50\!\cdots\!26 \nu^{7} + \cdots + 45\!\cdots\!46 ) / 18\!\cdots\!03 $ (5050448958007318385826v^7 + 6434333742581389882696959v^6 + 19446431773284329295109281732v^5 + 37946189094792922254485769096627v^4 + 24850225297078200228293167007664936v^3 + 51528595344082565460317811941920636851v^2 + 18202174849684974058617016834076831544792*v + 4592743747033659334348869313104727296559746) / 183177766174928616816446311337503

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

$\nu$	$=$	$ ( \beta_{5} + 170\beta_{3} - 1152\beta_{2} - 216\beta _1 + 69984 ) / 279936 $ (b5 + 170b3 - 1152b2 - 216*b1 + 69984) / 279936
$\nu^{2}$	$=$	$ ( - 4 \beta_{7} + 324 \beta_{6} + 1011 \beta_{5} - 69984 \beta_{4} + 498462 \beta_{3} + \cdots - 916458325920 ) / 839808 $ (-4b7 + 324b6 + 1011b5 - 69984b4 + 498462b3 - 9504b2 + 3549096*b1 - 916458325920) / 839808
$\nu^{3}$	$=$	$ ( - 1009 \beta_{7} + 27297 \beta_{6} - 968716 \beta_{5} + 95825592 \beta_{4} - 313716536 \beta_{3} + \cdots - 86572609221024 ) / 139968 $ (-1009b7 + 27297b6 - 968716b5 + 95825592b4 - 313716536b3 + 1896361704b2 + 362135016*b1 - 86572609221024) / 139968
$\nu^{4}$	$=$	$ ( 30218912 \beta_{7} - 1255453344 \beta_{6} - 4679892339 \beta_{5} + 469326420864 \beta_{4} + \cdots + 18\!\cdots\!24 ) / 839808 $ (30218912b7 - 1255453344b6 - 4679892339b5 + 469326420864b4 - 1509279602718b3 + 8602528523904b2 - 7210925681688*b1 + 1804195639820886624) / 839808
$\nu^{5}$	$=$	$ ( 10094900830 \beta_{7} - 421189582590 \beta_{6} + 3717218087125 \beta_{5} - 648980756002800 \beta_{4} + \cdots + 62\!\cdots\!84 ) / 279936 $ (10094900830b7 - 421189582590b6 + 3717218087125b5 - 648980756002800b4 + 1192702332716210b3 - 12446514378685008b2 - 2505976909357896*b1 + 629748773908545770784) / 279936
$\nu^{6}$	$=$	$ ( - 43072460330174 \beta_{7} + \cdots - 16\!\cdots\!64 ) / 419904 $ (-43072460330174b7 + 1806555435814494b6 + 7385405137728819b5 - 1217900084684848560b4 + 2364969771013177758b3 - 23459265391099712400b2 + 6559061003259805008*b1 - 1647737972083574540691264) / 419904
$\nu^{7}$	$=$	$ ( - 44\!\cdots\!00 \beta_{7} + \cdots - 18\!\cdots\!36 ) / 279936 $ (-44282860578536300b7 + 1861110301403426220b6 - 6374429760364989365b5 + 1652865201374612738688b4 - 2041838955958314963058b3 + 31827107138989753689984b2 + 7282386151217710913304*b1 - 1829949377544417361677450336) / 279936

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$-1$

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} $

143.1

−193.632	+	260.013i
−193.632	−	257.184i
194.132	−	1481.56i
194.132	+	1478.74i
194.132	−	1478.74i
194.132	+	1481.56i
−193.632	+	257.184i
−193.632	−	260.013i

−

46839.9i

−

481611.i

143.2

−

46839.9i

481611.i

143.3

−

18965.7i

−

195113.i

143.4

−

18965.7i

195113.i

143.5

18965.7i

−

195113.i

143.6

18965.7i

195113.i

143.7

46839.9i

−

481611.i

143.8

46839.9i

481611.i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
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	144.14.c.b	✓	8
3.b	odd	2	1	inner	144.14.c.b	✓	8
4.b	odd	2	1	inner	144.14.c.b	✓	8
12.b	even	2	1	inner	144.14.c.b	✓	8

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} + 2553671700T_{5}^{2} + 789168831728062500 $ T5^4 + 2553671700*T5^2 + 789168831728062500 acting on $S_{14}^{\mathrm{new}}(144, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ (T^{4} + \cdots + 78\!\cdots\!00)^{2} $ (T^4 + 2553671700*T^2 + 789168831728062500)^2
$7$	$ (T^{4} + \cdots + 88\!\cdots\!00)^{2} $ (T^4 + 270018256320*T^2 + 8830069349338382745600)^2
$11$	$ (T^{4} + \cdots + 15\!\cdots\!00)^{2} $ (T^4 - 115989736199040*T^2 + 1559034166831341363988070400)^2
$13$	$ (T^{2} + \cdots - 132534086177936)^{4} $ (T^2 + 20238400*T - 132534086177936)^4
$17$	$ (T^{4} + \cdots + 15\!\cdots\!04)^{2} $ (T^4 + 14070168741526596*T^2 + 15909120960274682670053347836804)^2
$19$	$ (T^{4} + \cdots + 10\!\cdots\!00)^{2} $ (T^4 + 200624241908920320*T^2 + 10060466712677001105036820453785600)^2
$23$	$ (T^{4} + \cdots + 87\!\cdots\!00)^{2} $ (T^4 - 1866610795023008640*T^2 + 870984300730373940966594089715302400)^2
$29$	$ (T^{4} + \cdots + 10\!\cdots\!44)^{2} $ (T^4 + 6526731877332722676*T^2 + 10528172352073439053506482834620790244)^2
$31$	$ (T^{4} + \cdots + 17\!\cdots\!00)^{2} $ (T^4 + 93823700951927862720*T^2 + 1745902817890024286681763534828339609600)^2
$37$	$ (T^{2} + \cdots - 21\!\cdots\!64)^{4} $ (T^2 + 19675341500*T - 214822377485089815164)^4
$41$	$ (T^{4} + \cdots + 50\!\cdots\!44)^{2} $ (T^4 + 4512585375079457407524*T^2 + 5084932071690921839094030126294465036952644)^2
$43$	$ (T^{4} + \cdots + 51\!\cdots\!00)^{2} $ (T^4 + 558320615252133707520*T^2 + 5179769440299749272308869192502745497600)^2
$47$	$ (T^{4} + \cdots + 39\!\cdots\!00)^{2} $ (T^4 - 21595424786696067027840*T^2 + 39622129687944477114564489727603096815206400)^2
$53$	$ (T^{4} + \cdots + 52\!\cdots\!24)^{2} $ (T^4 + 25254648708495453617364*T^2 + 52273696929605466655330258867747124438537124)^2
$59$	$ (T^{4} + \cdots + 30\!\cdots\!00)^{2} $ (T^4 - 164130881456845582179840*T^2 + 302057555851899835630928057857165368518246400)^2
$61$	$ (T^{2} + \cdots + 57\!\cdots\!00)^{4} $ (T^2 - 231200997100*T + 5774367249190681862500)^4
$67$	$ (T^{4} + \cdots + 12\!\cdots\!00)^{2} $ (T^4 + 1838465590881344627623680*T^2 + 12487867775647709818999298835501794051496345600)^2
$71$	$ (T^{4} + \cdots + 45\!\cdots\!00)^{2} $ (T^4 - 1364965425773169988901760*T^2 + 454427490896224585775504292577873438219699814400)^2
$73$	$ (T^{2} + \cdots - 28\!\cdots\!16)^{4} $ (T^2 - 449850747200*T - 2844628433494923659945216)^4
$79$	$ (T^{4} + \cdots + 10\!\cdots\!00)^{2} $ (T^4 + 20973858927340406888041920*T^2 + 101390124961932851926179180532354923233978089881600)^2
$83$	$ (T^{4} + \cdots + 33\!\cdots\!00)^{2} $ (T^4 - 14389316836758768319781760*T^2 + 33388892283413610052689562772309063652923213414400)^2
$89$	$ (T^{4} + \cdots + 41\!\cdots\!64)^{2} $ (T^4 + 4181581169445673072198116*T^2 + 4120796861000654313193173143223371249793792647364)^2
$97$	$ (T^{2} + \cdots - 27\!\cdots\!44)^{4} $ (T^2 + 5938409622400*T - 27690570395360219435015744)^4
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	144.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + ( - 5 \beta_{4} + 40 \beta_{2}) q^{5} + \beta_{3} q^{7} - \beta_{6} q^{11} + (61 \beta_1 - 10119200) q^{13} + (11282 \beta_{4} + 114823 \beta_{2}) q^{17} + (\beta_{5} - 342 \beta_{3}) q^{19} + (7 \beta_{7} - 50 \beta_{6}) q^{23}+ \cdots + ( - 24046122 \beta_1 - 2969204811200) q^{97}+O(q^{100}) \) q + (-5b4 + 40b2) * q^5 + b3 * q^7 - b6 * q^11 + (61b1 - 10119200) q^13 + (11282b4 + 114823b2) * q^17 + (b5 - 342b3) q^19 + (7b7 - 50b6) * q^23 + (-3650b1 - 56132725) q^25 + (14675b4 - 5247534b2) * q^29 + (-14b5 - 14987b3) * q^31 + (110b7 + 1105b6) * q^35 + (-70252b1 - 9837670750) q^37 + (7217250b4 - 1556087b2) * q^41 + (50b5 + 22394b3) * q^43 + (-637b7 - 8750b6) * q^47 + (-385800b1 - 38120117753) q^49 + (-7894221b4 - 290828194b2) * q^53 + (260b5 + 466390b3) * q^55 + (-1050b7 + 33164b6) * q^59 + (346700b1 + 115600498550) q^61 + (18133630b4 + 1059141110b2) * q^65 + (-2975b5 - 1124368b3) * q^67 + (5775b7 - 49988b6) * q^71 + (6771724b1 + 224925373600) q^73 + (-292746960b4 - 117773040b2) * q^77 + (10766b5 - 2259397b3) * q^79 + (21154b7 - 28625b6) * q^83 + (-992695b1 + 1898820389070) q^85 + (26499100b4 - 4173942061b2) * q^89 + (-13481b5 + 890202b3) * q^91 + (-90090b7 - 7670b6) * q^95 + (-24046122b1 - 2969204811200) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 80953600 q^{13} - 449061800 q^{25} - 78701366000 q^{37} - 304960942024 q^{49} + 924803988400 q^{61} + 1799402988800 q^{73} + 15190563112560 q^{85} - 23753638489600 q^{97}+O(q^{100}) \) 8 * q - 80953600 * q^13 - 449061800 * q^25 - 78701366000 * q^37 - 304960942024 * q^49 + 924803988400 * q^61 + 1799402988800 * q^73 + 15190563112560 * q^85 - 23753638489600 * q^97

Introduction

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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	144.14.c.b

Level	$144$
Weight	$14$
Character orbit	144.c
Analytic conductor	$154.413$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(154.412537691\)
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:	\( x^{8} - 2 x^{7} + 4365087 x^{6} + 1640648910 x^{5} + 5227007496627 x^{4} + \cdots + 54\!\cdots\!04 \) x^8 - 2x^7 + 4365087x^6 + 1640648910x^5 + 5227007496627x^4 + 3589835445735744x^3 + 1691776997434821656x^2 + 382610538805924667192*x + 54094475398144125773004
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{28}\cdot 3^{30} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 18\!\cdots\!16 \nu^{7} + \cdots + 65\!\cdots\!28 ) / 26\!\cdots\!29 \) (-1834812271496016v^7 - 910228937774946552v^6 - 5754666480600957648048v^5 - 5627750978602773382653360v^4 - 3039973392032158842150624000v^3 - 943766828102861033270053506336v^2 - 173707176665495623487033377768896*v + 6559964777737067006210188690059331428) / 26168252310704088116635187333929
\(\beta_{2}\)	\(=\)	\( ( 12\!\cdots\!29 \nu^{7} + \cdots + 91\!\cdots\!52 ) / 35\!\cdots\!28 \) (1232975624007247771329v^7 - 338241506804136399427362v^6 + 5309771372177355730664994861v^5 + 670169963729031359231171514510v^4 + 5723974235342304011139648177077145v^3 + 2935609559564352379523604489736423764v^2 + 933256891932125662884713909745769177914*v + 91690730291616152562883565663617747909152) / 3538633585904579961638023052964921428
\(\beta_{3}\)	\(=\)	\( ( - 87\!\cdots\!77 \nu^{7} + \cdots - 20\!\cdots\!92 ) / 35\!\cdots\!66 \) (-87376926125821298770811790777v^7 + 11087426476623166917894052157898v^6 - 390184322320818145796051300720640885v^5 - 89650761518959644199541223937660390026v^4 - 477443613098461724307983187655816645109169v^3 - 245636429762074347170325309733042955492569860v^2 - 143254977053979475758032928852095090695866297414*v - 20638776123530748871373718852024163176868218655392) / 35781723313450054906103874216460723293296866
\(\beta_{4}\)	\(=\)	\( ( - 19\!\cdots\!51 \nu^{7} + \cdots - 15\!\cdots\!64 ) / 30\!\cdots\!28 \) (-199641211972374444199280315851v^7 + 41572356187328488857331757200182v^6 - 878902811091898827281535494843203455v^5 - 141071239103201820461457657775760211018v^4 - 979498750075920071202207124247720184564867v^3 - 502409884421738539207521620851685899743219996v^2 - 158677413776218899799107026390521563844175331038*v - 15725260513028251510371864850158708809383720911264) / 30670048554385761348089035042680619965683028
\(\beta_{5}\)	\(=\)	\( ( 49\!\cdots\!49 \nu^{7} + \cdots + 11\!\cdots\!72 ) / 61\!\cdots\!77 \) (494392190147453895213230566449v^7 - 105065183525328782791945266988602v^6 + 2180753641670583698445903109188440589v^5 + 368708302626091040808040462837474831290v^4 + 2533526099520067464947541200426262923245545v^3 + 1304750650728510849596440530948451785711937764v^2 + 779135866913745145759256234637069060527539324566*v + 112270131600069047649347684035501173306040469088672) / 616926264025000946656963348559667642987877
\(\beta_{6}\)	\(=\)	\( ( 17\!\cdots\!66 \nu^{7} + \cdots + 11\!\cdots\!46 ) / 18\!\cdots\!03 \) (177477484684464232866v^7 + 212368060708160516545455v^6 + 608127176029787255198758788v^5 + 1083491863324549830304347353283v^4 + 697935948694694998732155492504360v^3 + 1265382442109525769614782388565812259v^2 + 441138292535842277959465058150053292376*v + 110210215631137660029290942317793914070946) / 183177766174928616816446311337503
\(\beta_{7}\)	\(=\)	\( ( 50\!\cdots\!26 \nu^{7} + \cdots + 45\!\cdots\!46 ) / 18\!\cdots\!03 \) (5050448958007318385826v^7 + 6434333742581389882696959v^6 + 19446431773284329295109281732v^5 + 37946189094792922254485769096627v^4 + 24850225297078200228293167007664936v^3 + 51528595344082565460317811941920636851v^2 + 18202174849684974058617016834076831544792*v + 4592743747033659334348869313104727296559746) / 183177766174928616816446311337503

\(\nu\)	\(=\)	\( ( \beta_{5} + 170\beta_{3} - 1152\beta_{2} - 216\beta _1 + 69984 ) / 279936 \) (b5 + 170b3 - 1152b2 - 216*b1 + 69984) / 279936
\(\nu^{2}\)	\(=\)	\( ( - 4 \beta_{7} + 324 \beta_{6} + 1011 \beta_{5} - 69984 \beta_{4} + 498462 \beta_{3} + \cdots - 916458325920 ) / 839808 \) (-4b7 + 324b6 + 1011b5 - 69984b4 + 498462b3 - 9504b2 + 3549096*b1 - 916458325920) / 839808
\(\nu^{3}\)	\(=\)	\( ( - 1009 \beta_{7} + 27297 \beta_{6} - 968716 \beta_{5} + 95825592 \beta_{4} - 313716536 \beta_{3} + \cdots - 86572609221024 ) / 139968 \) (-1009b7 + 27297b6 - 968716b5 + 95825592b4 - 313716536b3 + 1896361704b2 + 362135016*b1 - 86572609221024) / 139968
\(\nu^{4}\)	\(=\)	\( ( 30218912 \beta_{7} - 1255453344 \beta_{6} - 4679892339 \beta_{5} + 469326420864 \beta_{4} + \cdots + 18\!\cdots\!24 ) / 839808 \) (30218912b7 - 1255453344b6 - 4679892339b5 + 469326420864b4 - 1509279602718b3 + 8602528523904b2 - 7210925681688*b1 + 1804195639820886624) / 839808
\(\nu^{5}\)	\(=\)	\( ( 10094900830 \beta_{7} - 421189582590 \beta_{6} + 3717218087125 \beta_{5} - 648980756002800 \beta_{4} + \cdots + 62\!\cdots\!84 ) / 279936 \) (10094900830b7 - 421189582590b6 + 3717218087125b5 - 648980756002800b4 + 1192702332716210b3 - 12446514378685008b2 - 2505976909357896*b1 + 629748773908545770784) / 279936
\(\nu^{6}\)	\(=\)	\( ( - 43072460330174 \beta_{7} + \cdots - 16\!\cdots\!64 ) / 419904 \) (-43072460330174b7 + 1806555435814494b6 + 7385405137728819b5 - 1217900084684848560b4 + 2364969771013177758b3 - 23459265391099712400b2 + 6559061003259805008*b1 - 1647737972083574540691264) / 419904
\(\nu^{7}\)	\(=\)	\( ( - 44\!\cdots\!00 \beta_{7} + \cdots - 18\!\cdots\!36 ) / 279936 \) (-44282860578536300b7 + 1861110301403426220b6 - 6374429760364989365b5 + 1652865201374612738688b4 - 2041838955958314963058b3 + 31827107138989753689984b2 + 7282386151217710913304*b1 - 1829949377544417361677450336) / 279936

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 143.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} \) T^8
$5$	\( (T^{4} + \cdots + 78\!\cdots\!00)^{2} \) (T^4 + 2553671700*T^2 + 789168831728062500)^2
$7$	\( (T^{4} + \cdots + 88\!\cdots\!00)^{2} \) (T^4 + 270018256320*T^2 + 8830069349338382745600)^2
$11$	\( (T^{4} + \cdots + 15\!\cdots\!00)^{2} \) (T^4 - 115989736199040*T^2 + 1559034166831341363988070400)^2
$13$	\( (T^{2} + \cdots - 132534086177936)^{4} \) (T^2 + 20238400*T - 132534086177936)^4
$17$	\( (T^{4} + \cdots + 15\!\cdots\!04)^{2} \) (T^4 + 14070168741526596*T^2 + 15909120960274682670053347836804)^2
$19$	\( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \) (T^4 + 200624241908920320*T^2 + 10060466712677001105036820453785600)^2
$23$	\( (T^{4} + \cdots + 87\!\cdots\!00)^{2} \) (T^4 - 1866610795023008640*T^2 + 870984300730373940966594089715302400)^2
$29$	\( (T^{4} + \cdots + 10\!\cdots\!44)^{2} \) (T^4 + 6526731877332722676*T^2 + 10528172352073439053506482834620790244)^2
$31$	\( (T^{4} + \cdots + 17\!\cdots\!00)^{2} \) (T^4 + 93823700951927862720*T^2 + 1745902817890024286681763534828339609600)^2
$37$	\( (T^{2} + \cdots - 21\!\cdots\!64)^{4} \) (T^2 + 19675341500*T - 214822377485089815164)^4
$41$	\( (T^{4} + \cdots + 50\!\cdots\!44)^{2} \) (T^4 + 4512585375079457407524*T^2 + 5084932071690921839094030126294465036952644)^2
$43$	\( (T^{4} + \cdots + 51\!\cdots\!00)^{2} \) (T^4 + 558320615252133707520*T^2 + 5179769440299749272308869192502745497600)^2
$47$	\( (T^{4} + \cdots + 39\!\cdots\!00)^{2} \) (T^4 - 21595424786696067027840*T^2 + 39622129687944477114564489727603096815206400)^2
$53$	\( (T^{4} + \cdots + 52\!\cdots\!24)^{2} \) (T^4 + 25254648708495453617364*T^2 + 52273696929605466655330258867747124438537124)^2
$59$	\( (T^{4} + \cdots + 30\!\cdots\!00)^{2} \) (T^4 - 164130881456845582179840*T^2 + 302057555851899835630928057857165368518246400)^2
$61$	\( (T^{2} + \cdots + 57\!\cdots\!00)^{4} \) (T^2 - 231200997100*T + 5774367249190681862500)^4
$67$	\( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) (T^4 + 1838465590881344627623680*T^2 + 12487867775647709818999298835501794051496345600)^2
$71$	\( (T^{4} + \cdots + 45\!\cdots\!00)^{2} \) (T^4 - 1364965425773169988901760*T^2 + 454427490896224585775504292577873438219699814400)^2
$73$	\( (T^{2} + \cdots - 28\!\cdots\!16)^{4} \) (T^2 - 449850747200*T - 2844628433494923659945216)^4
$79$	\( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \) (T^4 + 20973858927340406888041920*T^2 + 101390124961932851926179180532354923233978089881600)^2
$83$	\( (T^{4} + \cdots + 33\!\cdots\!00)^{2} \) (T^4 - 14389316836758768319781760*T^2 + 33388892283413610052689562772309063652923213414400)^2
$89$	\( (T^{4} + \cdots + 41\!\cdots\!64)^{2} \) (T^4 + 4181581169445673072198116*T^2 + 4120796861000654313193173143223371249793792647364)^2
$97$	\( (T^{2} + \cdots - 27\!\cdots\!44)^{4} \) (T^2 + 5938409622400*T - 27690570395360219435015744)^4
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\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)