LMFDB - Newform orbit 2370.4.a.j

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2370,4,Mod(1,2370)] 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("2370.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2370, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2370 = 2 \cdot 3 \cdot 5 \cdot 79 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2370.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,-20,-30,40,-50,60,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$139.834526714$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 1411 x^{8} - 10836 x^{7} + 510263 x^{6} + 7885642 x^{5} - 20087358 x^{4} + \cdots - 28223592960 $ x^10 - 4x^9 - 1411x^8 - 10836x^7 + 510263x^6 + 7885642x^5 - 20087358x^4 - 1125633336x^3 - 8611887056x^2 - 26186766720*x - 28223592960
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{9}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} + 6 q^{6} + (\beta_{2} + 2) q^{7} - 8 q^{8} + 9 q^{9} + 10 q^{10} + ( - \beta_{8} + \beta_{6} - \beta_{5} + \cdots - 7) q^{11} - 12 q^{12} + (\beta_{9} + \beta_{8} - \beta_{6} + \cdots + 8) q^{13}+ \cdots + ( - 9 \beta_{8} + 9 \beta_{6} + \cdots - 63) q^{99}+O(q^{100}) $ q - 2 * q^2 - 3 * q^3 + 4 * q^4 - 5 * q^5 + 6 * q^6 + (b2 + 2) * q^7 - 8 * q^8 + 9 * q^9 + 10 * q^10 + (-b8 + b6 - b5 - b3 + b2 - 7) * q^11 - 12 * q^12 + (b9 + b8 - b6 - b3 + 8) * q^13 + (-2b2 - 4) q^14 + 15 * q^15 + 16 * q^16 + (-b8 - b7 + 2b6 - 2b1 - 2) * q^17 - 18 * q^18 + (b9 + b8 + b7 - b6 + b5 - b4 + 2b3 + b2 + 9) q^19 - 20 * q^20 + (-3b2 - 6) q^21 + (2b8 - 2b6 + 2b5 + 2b3 - 2b2 + 14) q^22 + (-3b9 - b8 + 2b7 - b6 + b3 + 2b1 - 22) q^23 + 24 * q^24 + 25 * q^25 + (-2b9 - 2b8 + 2b6 + 2b3 - 16) * q^26 - 27 * q^27 + (4b2 + 8) q^28 + (-b9 - b8 - 3b7 - b6 - 2b5 + 5b4 - 2b3 - 2b2 - 45) q^29 - 30 * q^30 + (5b8 - 3b7 + b6 + 2b5 - 3b4 + 2b2 + 3b1 - 11) * q^31 - 32 * q^32 + (3b8 - 3b6 + 3b5 + 3b3 - 3b2 + 21) q^33 + (2b8 + 2b7 - 4b6 + 4b1 + 4) * q^34 + (-5b2 - 10) q^35 + 36 * q^36 + (3b9 - 2b8 - 2b6 - b5 + 4b4 - b3 - 4b2 + 7b1 + 50) * q^37 + (-2b9 - 2b8 - 2b7 + 2b6 - 2b5 + 2b4 - 4b3 - 2b2 - 18) * q^38 + (-3b9 - 3b8 + 3b6 + 3b3 - 24) * q^39 + 40 * q^40 + (-6b9 - 2b8 + 7b7 - 6b6 + 6b5 + 3b3 - 5b2 - 3b1 - 66) * q^41 + (6b2 + 12) q^42 + (-4b9 - b8 + 9b7 + 3b6 + 3b5 - 4b4 + 7b3 - 3b2 + 2b1 + 56) * q^43 + (-4b8 + 4b6 - 4b5 - 4b3 + 4b2 - 28) q^44 - 45 * q^45 + (6b9 + 2b8 - 4b7 + 2b6 - 2b3 - 4b1 + 44) * q^46 + (-b9 + b8 - 3b7 - 4b6 - 3b5 - b4 + 5b3 - 8b2 - 3b1 + 8) * q^47 - 48 * q^48 + (4b9 + 4b8 - 7b7 - 5b6 + 2b5 - 2b4 + 4b3 - 5b2 - 11b1 + 3) q^49 - 50 * q^50 + (3b8 + 3b7 - 6b6 + 6b1 + 6) * q^51 + (4b9 + 4b8 - 4b6 - 4b3 + 32) * q^52 + (5b9 + 6b8 - 12b7 + 15b6 - 2b5 - 3b4 + 2b3 - 4b2 - 3b1 + 4) q^53 + 54 * q^54 + (5b8 - 5b6 + 5b5 + 5b3 - 5b2 + 35) q^55 + (-8b2 - 16) q^56 + (-3b9 - 3b8 - 3b7 + 3b6 - 3b5 + 3b4 - 6b3 - 3b2 - 27) * q^57 + (2b9 + 2b8 + 6b7 + 2b6 + 4b5 - 10b4 + 4b3 + 4b2 + 90) * q^58 + (-3b8 + 5b7 + 6b6 + 11b5 - b4 - 2b3 - 4b2 + 6b1 - 128) q^59 + 60 * q^60 + (-5b9 + 6b8 + 4b7 - 8b6 + 4b5 + b4 + 9b3 + b2 - 8b1 - 15) q^61 + (-10b8 + 6b7 - 2b6 - 4b5 + 6b4 - 4b2 - 6b1 + 22) q^62 + (9b2 + 18) q^63 + 64 * q^64 + (-5b9 - 5b8 + 5b6 + 5b3 - 40) * q^65 + (-6b8 + 6b6 - 6b5 - 6b3 + 6b2 - 42) q^66 + (-2b8 - 2b7 - 3b6 + 5b5 + 9b4 - 4b3 - 11b2 - b1 + 215) q^67 + (-4b8 - 4b7 + 8b6 - 8b1 - 8) * q^68 + (9b9 + 3b8 - 6b7 + 3b6 - 3b3 - 6b1 + 66) * q^69 + (10b2 + 20) q^70 + (12b9 + 6b8 + 16b7 + 4b6 - 9b5 + 4b4 + 3b3 + 5b2 + 14b1 - 118) q^71 - 72 * q^72 + (-6b9 - 13b8 + b7 + 6b6 - 12b5 + b4 - 19b3 - 22b2 - 4b1 + 164) q^73 + (-6b9 + 4b8 + 4b6 + 2b5 - 8b4 + 2b3 + 8b2 - 14b1 - 100) * q^74 - 75 * q^75 + (4b9 + 4b8 + 4b7 - 4b6 + 4b5 - 4b4 + 8b3 + 4b2 + 36) * q^76 + (-3b9 + 2b8 + 4b7 - 9b6 + b5 - 18b4 + 9b3 - 23b2 - 8b1 + 173) * q^77 + (6b9 + 6b8 - 6b6 - 6b3 + 48) * q^78 + 79 * q^79 - 80 * q^80 + 81 * q^81 + (12b9 + 4b8 - 14b7 + 12b6 - 12b5 - 6b3 + 10b2 + 6b1 + 132) * q^82 + (-13b8 - 20b7 - 10b6 - 4b5 - 9b4 - 22b3 - 23b2 + 2b1 + 143) * q^83 + (-12b2 - 24) q^84 + (5b8 + 5b7 - 10b6 + 10b1 + 10) * q^85 + (8b9 + 2b8 - 18b7 - 6b6 - 6b5 + 8b4 - 14b3 + 6b2 - 4b1 - 112) q^86 + (3b9 + 3b8 + 9b7 + 3b6 + 6b5 - 15b4 + 6b3 + 6b2 + 135) * q^87 + (8b8 - 8b6 + 8b5 + 8b3 - 8b2 + 56) q^88 + (26b9 - 2b8 - 2b7 - 2b6 - 6b5 + 8b4 - 7b3 - 9b2 + 7b1 - 29) q^89 + 90 * q^90 + (2b9 + 3b8 - 11b7 - 21b6 - 15b5 + 12b4 - 10b3 + 4b2 + 11b1 + 135) q^91 + (-12b9 - 4b8 + 8b7 - 4b6 + 4b3 + 8b1 - 88) * q^92 + (-15b8 + 9b7 - 3b6 - 6b5 + 9b4 - 6b2 - 9b1 + 33) q^93 + (2b9 - 2b8 + 6b7 + 8b6 + 6b5 + 2b4 - 10b3 + 16b2 + 6b1 - 16) q^94 + (-5b9 - 5b8 - 5b7 + 5b6 - 5b5 + 5b4 - 10b3 - 5b2 - 45) * q^95 + 96 * q^96 + (5b9 + 12b8 - 8b7 + 12b6 + 7b5 + 13b4 + 4b3 - 9b2 - 2b1 + 367) q^97 + (-8b9 - 8b8 + 14b7 + 10b6 - 4b5 + 4b4 - 8b3 + 10b2 + 22b1 - 6) q^98 + (-9b8 + 9b6 - 9b5 - 9b3 + 9b2 - 63) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 20 q^{2} - 30 q^{3} + 40 q^{4} - 50 q^{5} + 60 q^{6} + 21 q^{7} - 80 q^{8} + 90 q^{9} + 100 q^{10} - 74 q^{11} - 120 q^{12} + 76 q^{13} - 42 q^{14} + 150 q^{15} + 160 q^{16} - 16 q^{17} - 180 q^{18}+ \cdots - 666 q^{99}+O(q^{100}) $ 10 * q - 20 * q^2 - 30 * q^3 + 40 * q^4 - 50 * q^5 + 60 * q^6 + 21 * q^7 - 80 * q^8 + 90 * q^9 + 100 * q^10 - 74 * q^11 - 120 * q^12 + 76 * q^13 - 42 * q^14 + 150 * q^15 + 160 * q^16 - 16 * q^17 - 180 * q^18 + 94 * q^19 - 200 * q^20 - 63 * q^21 + 148 * q^22 - 222 * q^23 + 240 * q^24 + 250 * q^25 - 152 * q^26 - 270 * q^27 + 84 * q^28 - 439 * q^29 - 300 * q^30 - 116 * q^31 - 320 * q^32 + 222 * q^33 + 32 * q^34 - 105 * q^35 + 360 * q^36 + 512 * q^37 - 188 * q^38 - 228 * q^39 + 400 * q^40 - 647 * q^41 + 126 * q^42 + 547 * q^43 - 296 * q^44 - 450 * q^45 + 444 * q^46 + 81 * q^47 - 480 * q^48 + 57 * q^49 - 500 * q^50 + 48 * q^51 + 304 * q^52 + 33 * q^53 + 540 * q^54 + 370 * q^55 - 168 * q^56 - 282 * q^57 + 878 * q^58 - 1261 * q^59 + 600 * q^60 - 130 * q^61 + 232 * q^62 + 189 * q^63 + 640 * q^64 - 380 * q^65 - 444 * q^66 + 2189 * q^67 - 64 * q^68 + 666 * q^69 + 210 * q^70 - 1248 * q^71 - 720 * q^72 + 1540 * q^73 - 1024 * q^74 - 750 * q^75 + 376 * q^76 + 1672 * q^77 + 456 * q^78 + 790 * q^79 - 800 * q^80 + 810 * q^81 + 1294 * q^82 + 1407 * q^83 - 252 * q^84 + 80 * q^85 - 1094 * q^86 + 1317 * q^87 + 592 * q^88 - 280 * q^89 + 900 * q^90 + 1347 * q^91 - 888 * q^92 + 348 * q^93 - 162 * q^94 - 470 * q^95 + 960 * q^96 + 3721 * q^97 - 114 * q^98 - 666 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{10} - 4 x^{9} - 1411 x^{8} - 10836 x^{7} + 510263 x^{6} + 7885642 x^{5} - 20087358 x^{4} + \cdots - 28223592960 $ x^10 - 4*x^9 - 1411*x^8 - 10836*x^7 + 510263*x^6 + 7885642*x^5 - 20087358*x^4 - 1125633336*x^3 - 8611887056*x^2 - 26186766720*x - 28223592960 :

$\beta_{1}$	$=$	$ ( 52\!\cdots\!84 \nu^{9} + \cdots + 54\!\cdots\!40 ) / 65\!\cdots\!20 $ (52673262242122489684v^9 - 7249567385006809086157v^8 + 51525944050085515225824v^7 + 7785326420953314203900935v^6 - 15639003380722476554915988v^5 - 2800392627370805345245497015v^4 - 11016361185581660427343939822v^3 + 305683227540181653106372162854v^2 + 2697408234425127043458342618360*v + 5470899035689027104808707923240) / 6581884742775441832319264920
$\beta_{2}$	$=$	$ ( 91\!\cdots\!43 \nu^{9} + \cdots - 11\!\cdots\!80 ) / 52\!\cdots\!60 $ (912715035205246271643v^9 - 4852791171636913706896v^8 - 1299000634192230000646513v^7 - 7718516085447619542085808v^6 + 491917060346576092665505021v^5 + 6325270955478490374551901666v^4 - 31526773634154529684649154418v^3 - 980939310186016569925580070320v^2 - 6113986151225373725187734804880*v - 11301765413797659968311564433280) / 52655077942203534658554119360
$\beta_{3}$	$=$	$ ( 75\!\cdots\!83 \nu^{9} + \cdots + 20\!\cdots\!20 ) / 52\!\cdots\!60 $ (7592351900021275439383v^9 - 162227809534674363198416v^8 - 8581295581066119753538453v^7 + 79267007089157322712058592v^6 + 3309107754122370964210266081v^5 - 2148632464691253621604381894v^4 - 425904792018002102722119079978v^3 - 1949011445991183546768236430320v^2 + 3782751806620884388017320442000*v + 20404704751862914588025412827520) / 52655077942203534658554119360
$\beta_{4}$	$=$	$ ( - 49\!\cdots\!15 \nu^{9} + \cdots + 10\!\cdots\!40 ) / 26\!\cdots\!80 $ (-4929007902923026423315v^9 + 66231119248052767491246v^8 + 6376406979413789569137313v^7 - 7568718353496918554561666v^6 - 2499856360988579427907389821v^5 - 15017936110272512637426358688v^4 + 262477513556466249576678021782v^3 + 3144087981919701916855006601164v^2 + 10313003273494343312660785054800*v + 10322240804292173276651280178240) / 26327538971101767329277059680
$\beta_{5}$	$=$	$ ( - 12\!\cdots\!95 \nu^{9} + \cdots + 41\!\cdots\!80 ) / 65\!\cdots\!20 $ (-1276233122455915919695v^9 + 15094547393005111231429v^8 + 1689202897856400784359417v^7 + 411709698531672126407801v^6 - 659677638806826374946403569v^5 - 4805625885171338337217203227v^4 + 64885042030948605632579150328v^3 + 922119557062721010235804434426v^2 + 3597241134771925785888900135240*v + 4183324647908152391647649840080) / 6581884742775441832319264920
$\beta_{6}$	$=$	$ ( - 11\!\cdots\!99 \nu^{9} + \cdots + 31\!\cdots\!60 ) / 52\!\cdots\!60 $ (-11162658891261166940699v^9 + 164813828687695098225796v^8 + 13968519938958356246147313v^7 - 29307113911554243827174364v^6 - 5368760086616069059512198221v^5 - 30255717849686768333125625382v^4 + 544285224694795290304364148170v^3 + 6675241397840186105148218752312v^2 + 25121759945873929322650478389200*v + 31640038453882658495562518856960) / 52655077942203534658554119360
$\beta_{7}$	$=$	$ ( 20\!\cdots\!62 \nu^{9} + \cdots - 12\!\cdots\!00 ) / 65\!\cdots\!20 $ (2002701688005076523262v^9 - 20961523599004946249047v^8 - 2656090721404321450295166v^7 - 5197089260656381321980259v^6 + 1016728834681071147127627552v^5 + 9480644506951203442312219713v^4 - 86980121647592672283526265228v^3 - 1664249477684670898380701955102v^2 - 8137380338581405985910337131060*v - 12346329722758702442887704194200) / 6581884742775441832319264920
$\beta_{8}$	$=$	$ ( - 21\!\cdots\!55 \nu^{9} + \cdots + 52\!\cdots\!00 ) / 52\!\cdots\!60 $ (-21412602747317087609755v^9 + 324774866203753282744696v^8 + 26638039243724482491648113v^7 - 66332743908556107196434536v^6 - 10245603112885562026358891421v^5 - 54186164743895046291699349098v^4 + 1057043675755436050924079141922v^3 + 12369543485494355640370857434304v^2 + 43971568506695874316137559615440*v + 52030966571909860557472027400000) / 52655077942203534658554119360
$\beta_{9}$	$=$	$ ( 28\!\cdots\!67 \nu^{9} + \cdots - 69\!\cdots\!80 ) / 52\!\cdots\!60 $ (28033772602746220517067v^9 - 427541615297420793787864v^8 - 34779891529195066980739777v^7 + 88334611735822626970298648v^6 + 13348564889220241611470199709v^5 + 70474408369895069857505063834v^4 - 1371150548099608850632379859522v^3 - 16117647622990303799294161970400v^2 - 58069778320552098946604021782800*v - 69696602340670227052149527142080) / 52655077942203534658554119360

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

$\nu$	$=$	$ ( -\beta_{8} + 2\beta_{6} + \beta_{2} + 1 ) / 3 $ (-b8 + 2*b6 + b2 + 1) / 3
$\nu^{2}$	$=$	$ ( - 5 \beta_{9} - 14 \beta_{8} + \beta_{7} + 27 \beta_{6} + 18 \beta_{5} - 19 \beta_{4} + 15 \beta_{3} + \cdots + 843 ) / 3 $ (-5b9 - 14b8 + b7 + 27b6 + 18b5 - 19b4 + 15b3 + 17b2 - 17b1 + 843) / 3
$\nu^{3}$	$=$	$ ( - 233 \beta_{9} - 515 \beta_{8} + 199 \beta_{7} + 1359 \beta_{6} + 549 \beta_{5} - 619 \beta_{4} + \cdots + 14514 ) / 3 $ (-233b9 - 515b8 + 199b7 + 1359b6 + 549b5 - 619b4 + 789b3 + 1310b2 - 467*b1 + 14514) / 3
$\nu^{4}$	$=$	$ ( - 9617 \beta_{9} - 12797 \beta_{8} + 7771 \beta_{7} + 39411 \beta_{6} + 25824 \beta_{5} - 28963 \beta_{4} + \cdots + 648825 ) / 3 $ (-9617b9 - 12797b8 + 7771b7 + 39411b6 + 25824b5 - 28963b4 + 34665b3 + 39476b2 - 23888*b1 + 648825) / 3
$\nu^{5}$	$=$	$ ( - 350234 \beta_{9} - 428738 \beta_{8} + 332974 \beta_{7} + 1522134 \beta_{6} + 926541 \beta_{5} + \cdots + 19921317 ) / 3 $ (-350234b9 - 428738b8 + 332974b7 + 1522134b6 + 926541b5 - 1057006b4 + 1335294b3 + 1687940b2 - 817511*b1 + 19921317) / 3
$\nu^{6}$	$=$	$ ( - 12981281 \beta_{9} - 13665023 \beta_{8} + 12404227 \beta_{7} + 53203143 \beta_{6} + 35669361 \beta_{5} + \cdots + 750888216 ) / 3 $ (-12981281b9 - 13665023b8 + 12404227b7 + 53203143b6 + 35669361b5 - 40721287b4 + 51209421b3 + 59253026b2 - 32439197*b1 + 750888216) / 3
$\nu^{7}$	$=$	$ ( - 471509249 \beta_{9} - 478066619 \beta_{8} + 471747415 \beta_{7} + 1969732359 \beta_{6} + \cdots + 26473222281 ) / 3 $ (-471509249b9 - 478066619b8 + 471747415b7 + 1969732359b6 + 1307443344b5 - 1498770943b4 + 1899713229b3 + 2251321934b2 - 1170817490*b1 + 26473222281) / 3
$\nu^{8}$	$=$	$ ( - 17260129388 \beta_{9} - 16779767924 \beta_{8} + 17367360592 \beta_{7} + 71550629472 \beta_{6} + \cdots + 974456368095 ) / 3 $ (-17260129388b9 - 16779767924b8 + 17367360592b7 + 71550629472b6 + 48482867337b5 - 55631255944b4 + 70514227260b3 + 81837541604b2 - 43815598715*b1 + 974456368095) / 3
$\nu^{9}$	$=$	$ ( - 630625162637 \beta_{9} - 604949656055 \beta_{8} + 643276448635 \beta_{7} + 2629585743315 \beta_{6} + \cdots + 35426401897182 ) / 3 $ (-630625162637b9 - 604949656055b8 + 643276448635b7 + 2629585743315b6 + 1780184856963b5 - 2044640356219b4 + 2595168759501b3 + 3027428672906b2 - 1601876063351*b1 + 35426401897182) / 3

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

−17.5304
15.7999
−2.86229
19.9050
−14.3165
−3.53822
−11.8705
36.6916
−10.7085
−7.57012

−2.00000

−3.00000

4.00000

−5.00000

6.00000

−33.5805

−8.00000

9.00000

10.0000

1.2

−2.00000

−3.00000

4.00000

−5.00000

6.00000

−13.2820

−8.00000

9.00000

10.0000

1.3

−2.00000

−3.00000

4.00000

−5.00000

6.00000

−12.6463

−8.00000

9.00000

10.0000

1.4

−2.00000

−3.00000

4.00000

−5.00000

6.00000

−5.81125

−8.00000

9.00000

10.0000

1.5

−2.00000

−3.00000

4.00000

−5.00000

6.00000

−5.66286

−8.00000

9.00000

10.0000

1.6

−2.00000

−3.00000

4.00000

−5.00000

6.00000

5.02310

−8.00000

9.00000

10.0000

1.7

−2.00000

−3.00000

4.00000

−5.00000

6.00000

17.1172

−8.00000

9.00000

10.0000

1.8

−2.00000

−3.00000

4.00000

−5.00000

6.00000

20.8260

−8.00000

9.00000

10.0000

1.9

−2.00000

−3.00000

4.00000

−5.00000

6.00000

23.1123

−8.00000

9.00000

10.0000

1.10

−2.00000

−3.00000

4.00000

−5.00000

6.00000

25.9044

−8.00000

9.00000

10.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$5$	$ +1 $
$79$	$ -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	2370.4.a.j	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2370.4.a.j	✓	10	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{10} - 21 T_{7}^{9} - 1523 T_{7}^{8} + 34639 T_{7}^{7} + 567433 T_{7}^{6} - 12355582 T_{7}^{5} + \cdots - 198998150736 $ T7^10 - 21*T7^9 - 1523*T7^8 + 34639*T7^7 + 567433*T7^6 - 12355582*T7^5 - 103299467*T7^4 + 1454871430*T7^3 + 10066211710*T7^2 - 28939171448*T7 - 198998150736 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2370))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T + 2)^{10} $ (T + 2)^10
$3$	$ (T + 3)^{10} $ (T + 3)^10
$5$	$ (T + 5)^{10} $ (T + 5)^10
$7$	$ T^{10} + \cdots - 198998150736 $ T^10 - 21T^9 - 1523T^8 + 34639T^7 + 567433T^6 - 12355582T^5 - 103299467T^4 + 1454871430T^3 + 10066211710T^2 - 28939171448*T - 198998150736
$11$	$ T^{10} + \cdots - 92230570090800 $ T^10 + 74T^9 - 4994T^8 - 344142T^7 + 11827352T^6 + 528395576T^5 - 16712708279T^4 - 229669157730T^3 + 10257635048970T^2 - 70606883485800*T - 92230570090800
$13$	$ T^{10} + \cdots - 10\!\cdots\!00 $ T^10 - 76T^9 - 10793T^8 + 1021997T^7 + 25073181T^6 - 4389103359T^5 + 62851735381T^4 + 5370308101640T^3 - 213688406618020T^2 + 2695784355870000*T - 10934225416248800
$17$	$ T^{10} + \cdots - 13\!\cdots\!00 $ T^10 + 16T^9 - 23072T^8 - 293698T^7 + 176226532T^6 + 1418191556T^5 - 545636176317T^4 - 2850432073230T^3 + 608016696278310T^2 + 3998506690994400*T - 131319323659296000
$19$	$ T^{10} + \cdots + 71\!\cdots\!04 $ T^10 - 94T^9 - 21710T^8 + 1596227T^7 + 103696613T^6 - 5253436828T^5 - 96339817734T^4 + 5990388156192T^3 - 43717894462880T^2 - 671458093186816*T + 7161821928059904
$23$	$ T^{10} + \cdots - 37\!\cdots\!40 $ T^10 + 222T^9 - 23081T^8 - 6014869T^7 + 191756583T^6 + 58760750337T^5 - 820497108157T^4 - 247214105186160T^3 + 2352695008232004T^2 + 377232837308070480*T - 3749225301479233440
$29$	$ T^{10} + \cdots - 49\!\cdots\!00 $ T^10 + 439T^9 - 42996T^8 - 41227950T^7 - 3531714644T^6 + 693643540499T^5 + 126458918185089T^4 + 6556213014066420T^3 + 77420214963145080T^2 - 2450821942192819500*T - 49003591330540417500
$31$	$ T^{10} + \cdots + 18\!\cdots\!00 $ T^10 + 116T^9 - 191850T^8 - 28222298T^7 + 11127705651T^6 + 2020610060270T^5 - 163557902819970T^4 - 44852072273218744T^3 - 1850353450979095336T^2 + 4254357160378222560*T + 185508705261936750400
$37$	$ T^{10} + \cdots + 24\!\cdots\!00 $ T^10 - 512T^9 - 73165T^8 + 61146854T^7 - 874646029T^6 - 1783523042064T^5 + 26214121829756T^4 + 13643263389087960T^3 - 15083989028865500T^2 - 24236712958461530000*T + 24101474474851240400
$41$	$ T^{10} + \cdots + 34\!\cdots\!60 $ T^10 + 647T^9 - 172158T^8 - 161049694T^7 - 136926613T^6 + 11160795856018T^5 + 633418845378530T^4 - 236617103062699560T^3 - 15946366664553828024T^2 + 227992596709009318560*T + 3427893826402732061760
$43$	$ T^{10} + \cdots + 45\!\cdots\!04 $ T^10 - 547T^9 - 197526T^8 + 152995534T^7 - 3360274422T^6 - 9920967297387T^5 + 963618829933885T^4 + 190237073635495412T^3 - 22434154655147273064T^2 - 164294777960489936384*T + 45305654782775730469504
$47$	$ T^{10} + \cdots + 35\!\cdots\!72 $ T^10 - 81T^9 - 431584T^8 - 31270180T^7 + 52834156587T^6 + 10218195992000T^5 - 609723713571192T^4 - 239551333910406480T^3 - 13779982952783865036T^2 - 109533202637352247296*T + 3528075950150166361872
$53$	$ T^{10} + \cdots - 23\!\cdots\!84 $ T^10 - 33T^9 - 937880T^8 - 30713770T^7 + 279885366569T^6 + 17557709110084T^5 - 26461695879731772T^4 - 195317573198858304T^3 + 720370004702113125504T^2 - 18694608502043040491520*T - 2338752072776418852240384
$59$	$ T^{10} + \cdots + 13\!\cdots\!84 $ T^10 + 1261T^9 - 402135T^8 - 1075868673T^7 - 208914255439T^6 + 248359026077392T^5 + 97364048061838910T^4 - 11978278696576502208T^3 - 9091368052174939278648T^2 - 468727854449755359613824*T + 133391815742881160537261184
$61$	$ T^{10} + \cdots + 25\!\cdots\!64 $ T^10 + 130T^9 - 643900T^8 - 107488105T^7 + 123798265887T^6 + 24130503458308T^5 - 7132215381148938T^4 - 1591114914549783424T^3 - 9323373792238726072T^2 + 7654758858589062437888*T + 2527149549210891099264
$67$	$ T^{10} + \cdots - 14\!\cdots\!52 $ T^10 - 2189T^9 + 1412528T^8 + 73852360T^7 - 387081152431T^6 + 99988797348422T^5 + 13149342704476514T^4 - 5419577952532486464T^3 - 169006961709060068256T^2 + 53940369313861833218720*T - 1458455940149489296157152
$71$	$ T^{10} + \cdots + 11\!\cdots\!48 $ T^10 + 1248T^9 - 1749004T^8 - 2539300105T^7 + 906810179181T^6 + 1847989267388360T^5 - 34519338056618814T^4 - 556351995448400659992T^3 - 76677754002708405254256T^2 + 55446364437785343489259008*T + 11568245604456327412564021248
$73$	$ T^{10} + \cdots + 57\!\cdots\!40 $ T^10 - 1540T^9 - 1721418T^8 + 3300409440T^7 + 630749775514T^6 - 2243179580167760T^5 + 116245186147282057T^4 + 533538344888753041820T^3 - 37347648963682279468424T^2 - 39830829389220383244821200*T + 578606709815229877271004240
$79$	$ (T - 79)^{10} $ (T - 79)^10
$83$	$ T^{10} + \cdots - 16\!\cdots\!12 $ T^10 - 1407T^9 - 3558034T^8 + 5257644750T^7 + 3523484449086T^6 - 5870899729532363T^5 - 823976447371224671T^4 + 1904347391092869793524T^3 + 133974547851473105386440T^2 - 142625983796640885878789184*T - 16924908540211400129285897712
$89$	$ T^{10} + \cdots - 10\!\cdots\!32 $ T^10 + 280T^9 - 2837204T^8 - 1614113931T^7 + 1165700915297T^6 + 1076347664527060T^5 + 216230582921468002T^4 - 11365059086173748352T^3 - 7024705003906735967424T^2 - 561453775597785149851392*T - 10561655842218896896782432
$97$	$ T^{10} + \cdots + 42\!\cdots\!00 $ T^10 - 3721T^9 + 2837173T^8 + 4742479301T^7 - 9313800447663T^6 + 6043206316958646T^5 - 1688770584821227727T^4 + 108719493968961300344T^3 + 41449112404205429949416T^2 - 8179886140456683611277120*T + 422507295947051597317915600
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2370 = 2 \cdot 3 \cdot 5 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2370.a (trivial)

\(f(q)\)	\(=\)	\( q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} + 6 q^{6} + (\beta_{2} + 2) q^{7} - 8 q^{8} + 9 q^{9} + 10 q^{10} + ( - \beta_{8} + \beta_{6} - \beta_{5} + \cdots - 7) q^{11} - 12 q^{12} + (\beta_{9} + \beta_{8} - \beta_{6} + \cdots + 8) q^{13}+ \cdots + ( - 9 \beta_{8} + 9 \beta_{6} + \cdots - 63) q^{99}+O(q^{100}) \) q - 2 * q^2 - 3 * q^3 + 4 * q^4 - 5 * q^5 + 6 * q^6 + (b2 + 2) * q^7 - 8 * q^8 + 9 * q^9 + 10 * q^10 + (-b8 + b6 - b5 - b3 + b2 - 7) * q^11 - 12 * q^12 + (b9 + b8 - b6 - b3 + 8) * q^13 + (-2b2 - 4) q^14 + 15 * q^15 + 16 * q^16 + (-b8 - b7 + 2b6 - 2b1 - 2) * q^17 - 18 * q^18 + (b9 + b8 + b7 - b6 + b5 - b4 + 2b3 + b2 + 9) q^19 - 20 * q^20 + (-3b2 - 6) q^21 + (2b8 - 2b6 + 2b5 + 2b3 - 2b2 + 14) q^22 + (-3b9 - b8 + 2b7 - b6 + b3 + 2b1 - 22) q^23 + 24 * q^24 + 25 * q^25 + (-2b9 - 2b8 + 2b6 + 2b3 - 16) * q^26 - 27 * q^27 + (4b2 + 8) q^28 + (-b9 - b8 - 3b7 - b6 - 2b5 + 5b4 - 2b3 - 2b2 - 45) q^29 - 30 * q^30 + (5b8 - 3b7 + b6 + 2b5 - 3b4 + 2b2 + 3b1 - 11) * q^31 - 32 * q^32 + (3b8 - 3b6 + 3b5 + 3b3 - 3b2 + 21) q^33 + (2b8 + 2b7 - 4b6 + 4b1 + 4) * q^34 + (-5b2 - 10) q^35 + 36 * q^36 + (3b9 - 2b8 - 2b6 - b5 + 4b4 - b3 - 4b2 + 7b1 + 50) * q^37 + (-2b9 - 2b8 - 2b7 + 2b6 - 2b5 + 2b4 - 4b3 - 2b2 - 18) * q^38 + (-3b9 - 3b8 + 3b6 + 3b3 - 24) * q^39 + 40 * q^40 + (-6b9 - 2b8 + 7b7 - 6b6 + 6b5 + 3b3 - 5b2 - 3b1 - 66) * q^41 + (6b2 + 12) q^42 + (-4b9 - b8 + 9b7 + 3b6 + 3b5 - 4b4 + 7b3 - 3b2 + 2b1 + 56) * q^43 + (-4b8 + 4b6 - 4b5 - 4b3 + 4b2 - 28) q^44 - 45 * q^45 + (6b9 + 2b8 - 4b7 + 2b6 - 2b3 - 4b1 + 44) * q^46 + (-b9 + b8 - 3b7 - 4b6 - 3b5 - b4 + 5b3 - 8b2 - 3b1 + 8) * q^47 - 48 * q^48 + (4b9 + 4b8 - 7b7 - 5b6 + 2b5 - 2b4 + 4b3 - 5b2 - 11b1 + 3) q^49 - 50 * q^50 + (3b8 + 3b7 - 6b6 + 6b1 + 6) * q^51 + (4b9 + 4b8 - 4b6 - 4b3 + 32) * q^52 + (5b9 + 6b8 - 12b7 + 15b6 - 2b5 - 3b4 + 2b3 - 4b2 - 3b1 + 4) q^53 + 54 * q^54 + (5b8 - 5b6 + 5b5 + 5b3 - 5b2 + 35) q^55 + (-8b2 - 16) q^56 + (-3b9 - 3b8 - 3b7 + 3b6 - 3b5 + 3b4 - 6b3 - 3b2 - 27) * q^57 + (2b9 + 2b8 + 6b7 + 2b6 + 4b5 - 10b4 + 4b3 + 4b2 + 90) * q^58 + (-3b8 + 5b7 + 6b6 + 11b5 - b4 - 2b3 - 4b2 + 6b1 - 128) q^59 + 60 * q^60 + (-5b9 + 6b8 + 4b7 - 8b6 + 4b5 + b4 + 9b3 + b2 - 8b1 - 15) q^61 + (-10b8 + 6b7 - 2b6 - 4b5 + 6b4 - 4b2 - 6b1 + 22) q^62 + (9b2 + 18) q^63 + 64 * q^64 + (-5b9 - 5b8 + 5b6 + 5b3 - 40) * q^65 + (-6b8 + 6b6 - 6b5 - 6b3 + 6b2 - 42) q^66 + (-2b8 - 2b7 - 3b6 + 5b5 + 9b4 - 4b3 - 11b2 - b1 + 215) q^67 + (-4b8 - 4b7 + 8b6 - 8b1 - 8) * q^68 + (9b9 + 3b8 - 6b7 + 3b6 - 3b3 - 6b1 + 66) * q^69 + (10b2 + 20) q^70 + (12b9 + 6b8 + 16b7 + 4b6 - 9b5 + 4b4 + 3b3 + 5b2 + 14b1 - 118) q^71 - 72 * q^72 + (-6b9 - 13b8 + b7 + 6b6 - 12b5 + b4 - 19b3 - 22b2 - 4b1 + 164) q^73 + (-6b9 + 4b8 + 4b6 + 2b5 - 8b4 + 2b3 + 8b2 - 14b1 - 100) * q^74 - 75 * q^75 + (4b9 + 4b8 + 4b7 - 4b6 + 4b5 - 4b4 + 8b3 + 4b2 + 36) * q^76 + (-3b9 + 2b8 + 4b7 - 9b6 + b5 - 18b4 + 9b3 - 23b2 - 8b1 + 173) * q^77 + (6b9 + 6b8 - 6b6 - 6b3 + 48) * q^78 + 79 * q^79 - 80 * q^80 + 81 * q^81 + (12b9 + 4b8 - 14b7 + 12b6 - 12b5 - 6b3 + 10b2 + 6b1 + 132) * q^82 + (-13b8 - 20b7 - 10b6 - 4b5 - 9b4 - 22b3 - 23b2 + 2b1 + 143) * q^83 + (-12b2 - 24) q^84 + (5b8 + 5b7 - 10b6 + 10b1 + 10) * q^85 + (8b9 + 2b8 - 18b7 - 6b6 - 6b5 + 8b4 - 14b3 + 6b2 - 4b1 - 112) q^86 + (3b9 + 3b8 + 9b7 + 3b6 + 6b5 - 15b4 + 6b3 + 6b2 + 135) * q^87 + (8b8 - 8b6 + 8b5 + 8b3 - 8b2 + 56) q^88 + (26b9 - 2b8 - 2b7 - 2b6 - 6b5 + 8b4 - 7b3 - 9b2 + 7b1 - 29) q^89 + 90 * q^90 + (2b9 + 3b8 - 11b7 - 21b6 - 15b5 + 12b4 - 10b3 + 4b2 + 11b1 + 135) q^91 + (-12b9 - 4b8 + 8b7 - 4b6 + 4b3 + 8b1 - 88) * q^92 + (-15b8 + 9b7 - 3b6 - 6b5 + 9b4 - 6b2 - 9b1 + 33) q^93 + (2b9 - 2b8 + 6b7 + 8b6 + 6b5 + 2b4 - 10b3 + 16b2 + 6b1 - 16) q^94 + (-5b9 - 5b8 - 5b7 + 5b6 - 5b5 + 5b4 - 10b3 - 5b2 - 45) * q^95 + 96 * q^96 + (5b9 + 12b8 - 8b7 + 12b6 + 7b5 + 13b4 + 4b3 - 9b2 - 2b1 + 367) q^97 + (-8b9 - 8b8 + 14b7 + 10b6 - 4b5 + 4b4 - 8b3 + 10b2 + 22b1 - 6) q^98 + (-9b8 + 9b6 - 9b5 - 9b3 + 9b2 - 63) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 20 q^{2} - 30 q^{3} + 40 q^{4} - 50 q^{5} + 60 q^{6} + 21 q^{7} - 80 q^{8} + 90 q^{9} + 100 q^{10} - 74 q^{11} - 120 q^{12} + 76 q^{13} - 42 q^{14} + 150 q^{15} + 160 q^{16} - 16 q^{17} - 180 q^{18}+ \cdots - 666 q^{99}+O(q^{100}) \) 10 * q - 20 * q^2 - 30 * q^3 + 40 * q^4 - 50 * q^5 + 60 * q^6 + 21 * q^7 - 80 * q^8 + 90 * q^9 + 100 * q^10 - 74 * q^11 - 120 * q^12 + 76 * q^13 - 42 * q^14 + 150 * q^15 + 160 * q^16 - 16 * q^17 - 180 * q^18 + 94 * q^19 - 200 * q^20 - 63 * q^21 + 148 * q^22 - 222 * q^23 + 240 * q^24 + 250 * q^25 - 152 * q^26 - 270 * q^27 + 84 * q^28 - 439 * q^29 - 300 * q^30 - 116 * q^31 - 320 * q^32 + 222 * q^33 + 32 * q^34 - 105 * q^35 + 360 * q^36 + 512 * q^37 - 188 * q^38 - 228 * q^39 + 400 * q^40 - 647 * q^41 + 126 * q^42 + 547 * q^43 - 296 * q^44 - 450 * q^45 + 444 * q^46 + 81 * q^47 - 480 * q^48 + 57 * q^49 - 500 * q^50 + 48 * q^51 + 304 * q^52 + 33 * q^53 + 540 * q^54 + 370 * q^55 - 168 * q^56 - 282 * q^57 + 878 * q^58 - 1261 * q^59 + 600 * q^60 - 130 * q^61 + 232 * q^62 + 189 * q^63 + 640 * q^64 - 380 * q^65 - 444 * q^66 + 2189 * q^67 - 64 * q^68 + 666 * q^69 + 210 * q^70 - 1248 * q^71 - 720 * q^72 + 1540 * q^73 - 1024 * q^74 - 750 * q^75 + 376 * q^76 + 1672 * q^77 + 456 * q^78 + 790 * q^79 - 800 * q^80 + 810 * q^81 + 1294 * q^82 + 1407 * q^83 - 252 * q^84 + 80 * q^85 - 1094 * q^86 + 1317 * q^87 + 592 * q^88 - 280 * q^89 + 900 * q^90 + 1347 * q^91 - 888 * q^92 + 348 * q^93 - 162 * q^94 - 470 * q^95 + 960 * q^96 + 3721 * q^97 - 114 * q^98 - 666 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	2370.4.a.j

Level	$2370$
Weight	$4$
Character orbit	2370.a
Self dual	yes
Analytic conductor	$139.835$
Analytic rank	$1$
Dimension	$10$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(139.834526714\)
Analytic rank:	\(1\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - 4 x^{9} - 1411 x^{8} - 10836 x^{7} + 510263 x^{6} + 7885642 x^{5} - 20087358 x^{4} + \cdots - 28223592960 \) x^10 - 4x^9 - 1411x^8 - 10836x^7 + 510263x^6 + 7885642x^5 - 20087358x^4 - 1125633336x^3 - 8611887056x^2 - 26186766720*x - 28223592960
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2\cdot 3^{2} \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( ( 52\!\cdots\!84 \nu^{9} + \cdots + 54\!\cdots\!40 ) / 65\!\cdots\!20 \) (52673262242122489684v^9 - 7249567385006809086157v^8 + 51525944050085515225824v^7 + 7785326420953314203900935v^6 - 15639003380722476554915988v^5 - 2800392627370805345245497015v^4 - 11016361185581660427343939822v^3 + 305683227540181653106372162854v^2 + 2697408234425127043458342618360*v + 5470899035689027104808707923240) / 6581884742775441832319264920
\(\beta_{2}\)	\(=\)	\( ( 91\!\cdots\!43 \nu^{9} + \cdots - 11\!\cdots\!80 ) / 52\!\cdots\!60 \) (912715035205246271643v^9 - 4852791171636913706896v^8 - 1299000634192230000646513v^7 - 7718516085447619542085808v^6 + 491917060346576092665505021v^5 + 6325270955478490374551901666v^4 - 31526773634154529684649154418v^3 - 980939310186016569925580070320v^2 - 6113986151225373725187734804880*v - 11301765413797659968311564433280) / 52655077942203534658554119360
\(\beta_{3}\)	\(=\)	\( ( 75\!\cdots\!83 \nu^{9} + \cdots + 20\!\cdots\!20 ) / 52\!\cdots\!60 \) (7592351900021275439383v^9 - 162227809534674363198416v^8 - 8581295581066119753538453v^7 + 79267007089157322712058592v^6 + 3309107754122370964210266081v^5 - 2148632464691253621604381894v^4 - 425904792018002102722119079978v^3 - 1949011445991183546768236430320v^2 + 3782751806620884388017320442000*v + 20404704751862914588025412827520) / 52655077942203534658554119360
\(\beta_{4}\)	\(=\)	\( ( - 49\!\cdots\!15 \nu^{9} + \cdots + 10\!\cdots\!40 ) / 26\!\cdots\!80 \) (-4929007902923026423315v^9 + 66231119248052767491246v^8 + 6376406979413789569137313v^7 - 7568718353496918554561666v^6 - 2499856360988579427907389821v^5 - 15017936110272512637426358688v^4 + 262477513556466249576678021782v^3 + 3144087981919701916855006601164v^2 + 10313003273494343312660785054800*v + 10322240804292173276651280178240) / 26327538971101767329277059680
\(\beta_{5}\)	\(=\)	\( ( - 12\!\cdots\!95 \nu^{9} + \cdots + 41\!\cdots\!80 ) / 65\!\cdots\!20 \) (-1276233122455915919695v^9 + 15094547393005111231429v^8 + 1689202897856400784359417v^7 + 411709698531672126407801v^6 - 659677638806826374946403569v^5 - 4805625885171338337217203227v^4 + 64885042030948605632579150328v^3 + 922119557062721010235804434426v^2 + 3597241134771925785888900135240*v + 4183324647908152391647649840080) / 6581884742775441832319264920
\(\beta_{6}\)	\(=\)	\( ( - 11\!\cdots\!99 \nu^{9} + \cdots + 31\!\cdots\!60 ) / 52\!\cdots\!60 \) (-11162658891261166940699v^9 + 164813828687695098225796v^8 + 13968519938958356246147313v^7 - 29307113911554243827174364v^6 - 5368760086616069059512198221v^5 - 30255717849686768333125625382v^4 + 544285224694795290304364148170v^3 + 6675241397840186105148218752312v^2 + 25121759945873929322650478389200*v + 31640038453882658495562518856960) / 52655077942203534658554119360
\(\beta_{7}\)	\(=\)	\( ( 20\!\cdots\!62 \nu^{9} + \cdots - 12\!\cdots\!00 ) / 65\!\cdots\!20 \) (2002701688005076523262v^9 - 20961523599004946249047v^8 - 2656090721404321450295166v^7 - 5197089260656381321980259v^6 + 1016728834681071147127627552v^5 + 9480644506951203442312219713v^4 - 86980121647592672283526265228v^3 - 1664249477684670898380701955102v^2 - 8137380338581405985910337131060*v - 12346329722758702442887704194200) / 6581884742775441832319264920
\(\beta_{8}\)	\(=\)	\( ( - 21\!\cdots\!55 \nu^{9} + \cdots + 52\!\cdots\!00 ) / 52\!\cdots\!60 \) (-21412602747317087609755v^9 + 324774866203753282744696v^8 + 26638039243724482491648113v^7 - 66332743908556107196434536v^6 - 10245603112885562026358891421v^5 - 54186164743895046291699349098v^4 + 1057043675755436050924079141922v^3 + 12369543485494355640370857434304v^2 + 43971568506695874316137559615440*v + 52030966571909860557472027400000) / 52655077942203534658554119360
\(\beta_{9}\)	\(=\)	\( ( 28\!\cdots\!67 \nu^{9} + \cdots - 69\!\cdots\!80 ) / 52\!\cdots\!60 \) (28033772602746220517067v^9 - 427541615297420793787864v^8 - 34779891529195066980739777v^7 + 88334611735822626970298648v^6 + 13348564889220241611470199709v^5 + 70474408369895069857505063834v^4 - 1371150548099608850632379859522v^3 - 16117647622990303799294161970400v^2 - 58069778320552098946604021782800*v - 69696602340670227052149527142080) / 52655077942203534658554119360

\(\nu\)	\(=\)	\( ( -\beta_{8} + 2\beta_{6} + \beta_{2} + 1 ) / 3 \) (-b8 + 2*b6 + b2 + 1) / 3
\(\nu^{2}\)	\(=\)	\( ( - 5 \beta_{9} - 14 \beta_{8} + \beta_{7} + 27 \beta_{6} + 18 \beta_{5} - 19 \beta_{4} + 15 \beta_{3} + \cdots + 843 ) / 3 \) (-5b9 - 14b8 + b7 + 27b6 + 18b5 - 19b4 + 15b3 + 17b2 - 17b1 + 843) / 3
\(\nu^{3}\)	\(=\)	\( ( - 233 \beta_{9} - 515 \beta_{8} + 199 \beta_{7} + 1359 \beta_{6} + 549 \beta_{5} - 619 \beta_{4} + \cdots + 14514 ) / 3 \) (-233b9 - 515b8 + 199b7 + 1359b6 + 549b5 - 619b4 + 789b3 + 1310b2 - 467*b1 + 14514) / 3
\(\nu^{4}\)	\(=\)	\( ( - 9617 \beta_{9} - 12797 \beta_{8} + 7771 \beta_{7} + 39411 \beta_{6} + 25824 \beta_{5} - 28963 \beta_{4} + \cdots + 648825 ) / 3 \) (-9617b9 - 12797b8 + 7771b7 + 39411b6 + 25824b5 - 28963b4 + 34665b3 + 39476b2 - 23888*b1 + 648825) / 3
\(\nu^{5}\)	\(=\)	\( ( - 350234 \beta_{9} - 428738 \beta_{8} + 332974 \beta_{7} + 1522134 \beta_{6} + 926541 \beta_{5} + \cdots + 19921317 ) / 3 \) (-350234b9 - 428738b8 + 332974b7 + 1522134b6 + 926541b5 - 1057006b4 + 1335294b3 + 1687940b2 - 817511*b1 + 19921317) / 3
\(\nu^{6}\)	\(=\)	\( ( - 12981281 \beta_{9} - 13665023 \beta_{8} + 12404227 \beta_{7} + 53203143 \beta_{6} + 35669361 \beta_{5} + \cdots + 750888216 ) / 3 \) (-12981281b9 - 13665023b8 + 12404227b7 + 53203143b6 + 35669361b5 - 40721287b4 + 51209421b3 + 59253026b2 - 32439197*b1 + 750888216) / 3
\(\nu^{7}\)	\(=\)	\( ( - 471509249 \beta_{9} - 478066619 \beta_{8} + 471747415 \beta_{7} + 1969732359 \beta_{6} + \cdots + 26473222281 ) / 3 \) (-471509249b9 - 478066619b8 + 471747415b7 + 1969732359b6 + 1307443344b5 - 1498770943b4 + 1899713229b3 + 2251321934b2 - 1170817490*b1 + 26473222281) / 3
\(\nu^{8}\)	\(=\)	\( ( - 17260129388 \beta_{9} - 16779767924 \beta_{8} + 17367360592 \beta_{7} + 71550629472 \beta_{6} + \cdots + 974456368095 ) / 3 \) (-17260129388b9 - 16779767924b8 + 17367360592b7 + 71550629472b6 + 48482867337b5 - 55631255944b4 + 70514227260b3 + 81837541604b2 - 43815598715*b1 + 974456368095) / 3
\(\nu^{9}\)	\(=\)	\( ( - 630625162637 \beta_{9} - 604949656055 \beta_{8} + 643276448635 \beta_{7} + 2629585743315 \beta_{6} + \cdots + 35426401897182 ) / 3 \) (-630625162637b9 - 604949656055b8 + 643276448635b7 + 2629585743315b6 + 1780184856963b5 - 2044640356219b4 + 2595168759501b3 + 3027428672906b2 - 1601876063351*b1 + 35426401897182) / 3

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 1.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T + 2)^{10} \) (T + 2)^10
$3$	\( (T + 3)^{10} \) (T + 3)^10
$5$	\( (T + 5)^{10} \) (T + 5)^10
$7$	\( T^{10} + \cdots - 198998150736 \) T^10 - 21T^9 - 1523T^8 + 34639T^7 + 567433T^6 - 12355582T^5 - 103299467T^4 + 1454871430T^3 + 10066211710T^2 - 28939171448*T - 198998150736
$11$	\( T^{10} + \cdots - 92230570090800 \) T^10 + 74T^9 - 4994T^8 - 344142T^7 + 11827352T^6 + 528395576T^5 - 16712708279T^4 - 229669157730T^3 + 10257635048970T^2 - 70606883485800*T - 92230570090800
$13$	\( T^{10} + \cdots - 10\!\cdots\!00 \) T^10 - 76T^9 - 10793T^8 + 1021997T^7 + 25073181T^6 - 4389103359T^5 + 62851735381T^4 + 5370308101640T^3 - 213688406618020T^2 + 2695784355870000*T - 10934225416248800
$17$	\( T^{10} + \cdots - 13\!\cdots\!00 \) T^10 + 16T^9 - 23072T^8 - 293698T^7 + 176226532T^6 + 1418191556T^5 - 545636176317T^4 - 2850432073230T^3 + 608016696278310T^2 + 3998506690994400*T - 131319323659296000
$19$	\( T^{10} + \cdots + 71\!\cdots\!04 \) T^10 - 94T^9 - 21710T^8 + 1596227T^7 + 103696613T^6 - 5253436828T^5 - 96339817734T^4 + 5990388156192T^3 - 43717894462880T^2 - 671458093186816*T + 7161821928059904
$23$	\( T^{10} + \cdots - 37\!\cdots\!40 \) T^10 + 222T^9 - 23081T^8 - 6014869T^7 + 191756583T^6 + 58760750337T^5 - 820497108157T^4 - 247214105186160T^3 + 2352695008232004T^2 + 377232837308070480*T - 3749225301479233440
$29$	\( T^{10} + \cdots - 49\!\cdots\!00 \) T^10 + 439T^9 - 42996T^8 - 41227950T^7 - 3531714644T^6 + 693643540499T^5 + 126458918185089T^4 + 6556213014066420T^3 + 77420214963145080T^2 - 2450821942192819500*T - 49003591330540417500
$31$	\( T^{10} + \cdots + 18\!\cdots\!00 \) T^10 + 116T^9 - 191850T^8 - 28222298T^7 + 11127705651T^6 + 2020610060270T^5 - 163557902819970T^4 - 44852072273218744T^3 - 1850353450979095336T^2 + 4254357160378222560*T + 185508705261936750400
$37$	\( T^{10} + \cdots + 24\!\cdots\!00 \) T^10 - 512T^9 - 73165T^8 + 61146854T^7 - 874646029T^6 - 1783523042064T^5 + 26214121829756T^4 + 13643263389087960T^3 - 15083989028865500T^2 - 24236712958461530000*T + 24101474474851240400
$41$	\( T^{10} + \cdots + 34\!\cdots\!60 \) T^10 + 647T^9 - 172158T^8 - 161049694T^7 - 136926613T^6 + 11160795856018T^5 + 633418845378530T^4 - 236617103062699560T^3 - 15946366664553828024T^2 + 227992596709009318560*T + 3427893826402732061760
$43$	\( T^{10} + \cdots + 45\!\cdots\!04 \) T^10 - 547T^9 - 197526T^8 + 152995534T^7 - 3360274422T^6 - 9920967297387T^5 + 963618829933885T^4 + 190237073635495412T^3 - 22434154655147273064T^2 - 164294777960489936384*T + 45305654782775730469504
$47$	\( T^{10} + \cdots + 35\!\cdots\!72 \) T^10 - 81T^9 - 431584T^8 - 31270180T^7 + 52834156587T^6 + 10218195992000T^5 - 609723713571192T^4 - 239551333910406480T^3 - 13779982952783865036T^2 - 109533202637352247296*T + 3528075950150166361872
$53$	\( T^{10} + \cdots - 23\!\cdots\!84 \) T^10 - 33T^9 - 937880T^8 - 30713770T^7 + 279885366569T^6 + 17557709110084T^5 - 26461695879731772T^4 - 195317573198858304T^3 + 720370004702113125504T^2 - 18694608502043040491520*T - 2338752072776418852240384
$59$	\( T^{10} + \cdots + 13\!\cdots\!84 \) T^10 + 1261T^9 - 402135T^8 - 1075868673T^7 - 208914255439T^6 + 248359026077392T^5 + 97364048061838910T^4 - 11978278696576502208T^3 - 9091368052174939278648T^2 - 468727854449755359613824*T + 133391815742881160537261184
$61$	\( T^{10} + \cdots + 25\!\cdots\!64 \) T^10 + 130T^9 - 643900T^8 - 107488105T^7 + 123798265887T^6 + 24130503458308T^5 - 7132215381148938T^4 - 1591114914549783424T^3 - 9323373792238726072T^2 + 7654758858589062437888*T + 2527149549210891099264
$67$	\( T^{10} + \cdots - 14\!\cdots\!52 \) T^10 - 2189T^9 + 1412528T^8 + 73852360T^7 - 387081152431T^6 + 99988797348422T^5 + 13149342704476514T^4 - 5419577952532486464T^3 - 169006961709060068256T^2 + 53940369313861833218720*T - 1458455940149489296157152
$71$	\( T^{10} + \cdots + 11\!\cdots\!48 \) T^10 + 1248T^9 - 1749004T^8 - 2539300105T^7 + 906810179181T^6 + 1847989267388360T^5 - 34519338056618814T^4 - 556351995448400659992T^3 - 76677754002708405254256T^2 + 55446364437785343489259008*T + 11568245604456327412564021248
$73$	\( T^{10} + \cdots + 57\!\cdots\!40 \) T^10 - 1540T^9 - 1721418T^8 + 3300409440T^7 + 630749775514T^6 - 2243179580167760T^5 + 116245186147282057T^4 + 533538344888753041820T^3 - 37347648963682279468424T^2 - 39830829389220383244821200*T + 578606709815229877271004240
$79$	\( (T - 79)^{10} \) (T - 79)^10
$83$	\( T^{10} + \cdots - 16\!\cdots\!12 \) T^10 - 1407T^9 - 3558034T^8 + 5257644750T^7 + 3523484449086T^6 - 5870899729532363T^5 - 823976447371224671T^4 + 1904347391092869793524T^3 + 133974547851473105386440T^2 - 142625983796640885878789184*T - 16924908540211400129285897712
$89$	\( T^{10} + \cdots - 10\!\cdots\!32 \) T^10 + 280T^9 - 2837204T^8 - 1614113931T^7 + 1165700915297T^6 + 1076347664527060T^5 + 216230582921468002T^4 - 11365059086173748352T^3 - 7024705003906735967424T^2 - 561453775597785149851392*T - 10561655842218896896782432
$97$	\( T^{10} + \cdots + 42\!\cdots\!00 \) T^10 - 3721T^9 + 2837173T^8 + 4742479301T^7 - 9313800447663T^6 + 6043206316958646T^5 - 1688770584821227727T^4 + 108719493968961300344T^3 + 41449112404205429949416T^2 - 8179886140456683611277120*T + 422507295947051597317915600
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\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( +1 \)
\(5\)	\( +1 \)
\(79\)	\( -1 \)