LMFDB - Newform orbit 8030.2.a.bj

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8030,2,Mod(1,8030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8030, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8030 = 2 \cdot 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8030.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1198728231$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 19 x^{16} + 169 x^{15} + 40 x^{14} - 1802 x^{13} + 1266 x^{12} + 9247 x^{11} + \cdots + 624 $ x^18 - 6x^17 - 19x^16 + 169x^15 + 40x^14 - 1802x^13 + 1266x^12 + 9247x^11 - 10582x^10 - 24226x^9 + 33432x^8 + 32951x^7 - 49245x^6 - 22852x^5 + 33519x^4 + 7814x^3 - 9238x^2 - 1136*x + 624
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 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 $1,\beta_1,\ldots,\beta_{17}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - q^{2} - \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + \beta_{7} q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) $ q - q^2 - b1 * q^3 + q^4 - q^5 + b1 * q^6 + b7 * q^7 - q^8 + (b2 + 1) * q^9	$ q - q^{2} - \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + \beta_{7} q^{7} - q^{8} + (\beta_{2} + 1) q^{9} + q^{10} + q^{11} - \beta_1 q^{12} + \beta_{11} q^{13} - \beta_{7} q^{14} + \beta_1 q^{15} + q^{16} + ( - \beta_{10} - 1) q^{17} + ( - \beta_{2} - 1) q^{18} + ( - \beta_{8} + 1) q^{19} - q^{20} + ( - \beta_{7} + \beta_{4} - \beta_{2} + \cdots - 1) q^{21}+ \cdots + (\beta_{2} + 1) q^{99}+O(q^{100}) $ q - q^2 - b1 * q^3 + q^4 - q^5 + b1 * q^6 + b7 * q^7 - q^8 + (b2 + 1) * q^9 + q^10 + q^11 - b1 * q^12 + b11 * q^13 - b7 * q^14 + b1 * q^15 + q^16 + (-b10 - 1) * q^17 + (-b2 - 1) * q^18 + (-b8 + 1) * q^19 - q^20 + (-b7 + b4 - b2 + b1 - 1) * q^21 - q^22 + (-b16 + b6 - 1) * q^23 + b1 * q^24 + q^25 - b11 * q^26 + (b10 - b7 + b5 - b2 - 1) * q^27 + b7 * q^28 + (-b11 - b8 - b6 + b4 - b3 - b2) * q^29 - b1 * q^30 + (b11 - b9 + b8 + b6 + b3 + b2 + 3) * q^31 - q^32 - b1 * q^33 + (b10 + 1) * q^34 - b7 * q^35 + (b2 + 1) * q^36 + (-b14 - b12 - b8 + b7 - b3) * q^37 + (b8 - 1) * q^38 + (b17 + b16 - b15 - b14 - b8 + b7 + b2 + b1) * q^39 + q^40 + (-b10 + b9 + b2 - b1 + 1) * q^41 + (b7 - b4 + b2 - b1 + 1) * q^42 + (b14 + b13 - b12 + b11 + b8 - b7 + b5 + b3 - b2 + 2b1) q^43 + q^44 + (-b2 - 1) * q^45 + (b16 - b6 + 1) * q^46 + (-b16 + b15 - b13 + 2b12 - b11 - b8 + 2b7 - b5 - b3 + b2 - 3b1 - 1) q^47 - b1 * q^48 + (-b17 - b16 + b14 + b12 + b6 - b5 - b4 + b2 - b1 + 2) * q^49 - q^50 + (b14 + b13 + b10 - b7 - b6 + b3 - 2b2 + 3b1 - 2) * q^51 + b11 * q^52 + (-b17 + b15 - b13 - b10 - b9 + 2b8 + b4 - b1 + 1) q^53 + (-b10 + b7 - b5 + b2 + 1) * q^54 - q^55 - b7 * q^56 + (b16 - 2b15 + b14 + b13 - b12 + 2b11 + b9 + b8 - 2b7 + b6 + b5 - b4 + b3 + b2) q^57 + (b11 + b8 + b6 - b4 + b3 + b2) * q^58 + (-b16 + b15 - b14 + b12 - b11 - b8 + b7 - b2 - b1) * q^59 + b1 * q^60 + (-b12 + b11 - b6 + b4 + 3) * q^61 + (-b11 + b9 - b8 - b6 - b3 - b2 - 3) * q^62 + (b17 + b16 - b15 - b14 - b13 + b11 - b10 + 3b7 - b5 - b4 + 2b2 - 1) * q^63 + q^64 - b11 * q^65 + b1 * q^66 + (b16 + b15 - b14 - b13 - b9 + b7 - b6 - b3 + b2 - b1 + 1) * q^67 + (-b10 - 1) * q^68 + (2b16 - 2b12 + b9 + b8 - 2b6 + b5 + 2b1 - 1) * q^69 + b7 * q^70 + (-2b16 + b15 - b13 + 2b12 - b11 - b8 + 2b7 + b6 - 2b5 - b3 + b2 - 4b1 + 2) q^71 + (-b2 - 1) * q^72 - q^73 + (b14 + b12 + b8 - b7 + b3) * q^74 - b1 * q^75 + (-b8 + 1) * q^76 + b7 * q^77 + (-b17 - b16 + b15 + b14 + b8 - b7 - b2 - b1) * q^78 + (b15 + b14 - b13 + b12 + b8 + b7 + b2 - 2b1 + 3) q^79 - q^80 + (b16 - b15 + b14 + b13 - b12 + b11 - b10 + b9 + b8 + b6 - 2b4 + 3b2 + b1 + 2) * q^81 + (b10 - b9 - b2 + b1 - 1) * q^82 + (b15 + b14 + 2b13 - b11 + b9 - b6 + b3 - b2 + b1) q^83 + (-b7 + b4 - b2 + b1 - 1) * q^84 + (b10 + 1) * q^85 + (-b14 - b13 + b12 - b11 - b8 + b7 - b5 - b3 + b2 - 2b1) q^86 + (-b15 + b14 + b12 + b11 - b9 + b8 + 3b7 - b5 - b4 + b3 + 2b2 - b1 + 2) * q^87 - q^88 + (-b15 + b13 + b12 + b6 - b4 + 2b2 + 2) q^89 + (b2 + 1) * q^90 + (-b16 - b14 - b13 + b12 + b10 - b9 - b8 + b7 - 3b1 + 2) q^91 + (-b16 + b6 - 1) * q^92 + (2b16 - b14 - 2b12 + b9 - b7 - 2b6 + 2b5 + b4 - b3 - 2b2 - 2b1) * q^93 + (b16 - b15 + b13 - 2b12 + b11 + b8 - 2b7 + b5 + b3 - b2 + 3b1 + 1) q^94 + (b8 - 1) * q^95 + b1 * q^96 + (b17 - b16 + b15 - b14 - b11 - 2b8 - b6 - b5 + b4 - 3b1) * q^97 + (b17 + b16 - b14 - b12 - b6 + b5 + b4 - b2 + b1 - 2) * q^98 + (b2 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 18 q^{2} - 6 q^{3} + 18 q^{4} - 18 q^{5} + 6 q^{6} - 6 q^{7} - 18 q^{8} + 20 q^{9}+O(q^{10}) $ 18 * q - 18 * q^2 - 6 * q^3 + 18 * q^4 - 18 * q^5 + 6 * q^6 - 6 * q^7 - 18 * q^8 + 20 * q^9	\( 18 q - 18 q^{2} - 6 q^{3} + 18 q^{4} - 18 q^{5} + 6 q^{6} - 6 q^{7} - 18 q^{8} + 20 q^{9} + 18 q^{10} + 18 q^{11} - 6 q^{12} - 8 q^{13} + 6 q^{14} + 6 q^{15} + 18 q^{16} - 10 q^{17} - 20 q^{18} + 24 q^{19} - 18 q^{20} - 13 q^{21} - 18 q^{22} - 14 q^{23} + 6 q^{24} + 18 q^{25} + 8 q^{26} - 15 q^{27} - 6 q^{28} + 11 q^{29} - 6 q^{30} + 32 q^{31} - 18 q^{32} - 6 q^{33} + 10 q^{34} + 6 q^{35} + 20 q^{36} - 3 q^{37} - 24 q^{38} + 7 q^{39} + 18 q^{40} + 28 q^{41} + 13 q^{42} - 6 q^{43} + 18 q^{44} - 20 q^{45} + 14 q^{46} - 16 q^{47} - 6 q^{48} + 36 q^{49} - 18 q^{50} - 29 q^{51} - 8 q^{52} - q^{53} + 15 q^{54} - 18 q^{55} + 6 q^{56} - 11 q^{57} - 11 q^{58} + 14 q^{59} + 6 q^{60} + 39 q^{61} - 32 q^{62} - 29 q^{63} + 18 q^{64} + 8 q^{65} + 6 q^{66} + 4 q^{67} - 10 q^{68} - 19 q^{69} - 6 q^{70} + 29 q^{71} - 20 q^{72} - 18 q^{73} + 3 q^{74} - 6 q^{75} + 24 q^{76} - 6 q^{77} - 7 q^{78} + 38 q^{79} - 18 q^{80} + 34 q^{81} - 28 q^{82} + 4 q^{83} - 13 q^{84} + 10 q^{85} + 6 q^{86} - 18 q^{88} + 44 q^{89} + 20 q^{90} + 27 q^{91} - 14 q^{92} - 12 q^{93} + 16 q^{94} - 24 q^{95} + 6 q^{96} + 3 q^{97} - 36 q^{98} + 20 q^{99}+O(q^{100}) \) 18 * q - 18 * q^2 - 6 * q^3 + 18 * q^4 - 18 * q^5 + 6 * q^6 - 6 * q^7 - 18 * q^8 + 20 * q^9 + 18 * q^10 + 18 * q^11 - 6 * q^12 - 8 * q^13 + 6 * q^14 + 6 * q^15 + 18 * q^16 - 10 * q^17 - 20 * q^18 + 24 * q^19 - 18 * q^20 - 13 * q^21 - 18 * q^22 - 14 * q^23 + 6 * q^24 + 18 * q^25 + 8 * q^26 - 15 * q^27 - 6 * q^28 + 11 * q^29 - 6 * q^30 + 32 * q^31 - 18 * q^32 - 6 * q^33 + 10 * q^34 + 6 * q^35 + 20 * q^36 - 3 * q^37 - 24 * q^38 + 7 * q^39 + 18 * q^40 + 28 * q^41 + 13 * q^42 - 6 * q^43 + 18 * q^44 - 20 * q^45 + 14 * q^46 - 16 * q^47 - 6 * q^48 + 36 * q^49 - 18 * q^50 - 29 * q^51 - 8 * q^52 - q^53 + 15 * q^54 - 18 * q^55 + 6 * q^56 - 11 * q^57 - 11 * q^58 + 14 * q^59 + 6 * q^60 + 39 * q^61 - 32 * q^62 - 29 * q^63 + 18 * q^64 + 8 * q^65 + 6 * q^66 + 4 * q^67 - 10 * q^68 - 19 * q^69 - 6 * q^70 + 29 * q^71 - 20 * q^72 - 18 * q^73 + 3 * q^74 - 6 * q^75 + 24 * q^76 - 6 * q^77 - 7 * q^78 + 38 * q^79 - 18 * q^80 + 34 * q^81 - 28 * q^82 + 4 * q^83 - 13 * q^84 + 10 * q^85 + 6 * q^86 - 18 * q^88 + 44 * q^89 + 20 * q^90 + 27 * q^91 - 14 * q^92 - 12 * q^93 + 16 * q^94 - 24 * q^95 + 6 * q^96 + 3 * q^97 - 36 * q^98 + 20 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - 6 x^{17} - 19 x^{16} + 169 x^{15} + 40 x^{14} - 1802 x^{13} + 1266 x^{12} + 9247 x^{11} + \cdots + 624 $ x^18 - 6*x^17 - 19*x^16 + 169*x^15 + 40*x^14 - 1802*x^13 + 1266*x^12 + 9247*x^11 - 10582*x^10 - 24226*x^9 + 33432*x^8 + 32951*x^7 - 49245*x^6 - 22852*x^5 + 33519*x^4 + 7814*x^3 - 9238*x^2 - 1136*x + 624 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 4 $ v^2 - 4
$\beta_{3}$	$=$	$ ( - 228908810602 \nu^{17} + 411817852383 \nu^{16} + 17940201788599 \nu^{15} + \cdots + 555269137026984 ) / 458553592515710 $ (-228908810602v^17 + 411817852383v^16 + 17940201788599v^15 - 54740679928215v^14 - 345091901988560v^13 + 1236480444135744v^12 + 2704468482734111v^11 - 11225278351545172v^10 - 9698874963407970v^9 + 48774348098535567v^8 + 16013633215034340v^7 - 104181503235510582v^6 - 14440198071326389v^5 + 102810448170732661v^4 + 15089472753225789v^3 - 36195299792332800v^2 - 11562778652928424*v + 555269137026984) / 458553592515710
$\beta_{4}$	$=$	$ ( 60687629959 \nu^{17} - 337791681891 \nu^{16} - 1158984744611 \nu^{15} + \cdots - 382130524556314 ) / 91710718503142 $ (60687629959v^17 - 337791681891v^16 - 1158984744611v^15 + 7174877002527v^14 + 15414588942197v^13 - 73300933171233v^12 - 182732638334377v^11 + 617563723620284v^10 + 978638152319253v^9 - 3348562944219762v^8 - 1401466621346000v^7 + 8156666882637314v^6 - 1446859114193701v^5 - 7814541869839508v^4 + 3563007113645037v^3 + 2228412814161882v^2 - 1175933107443670*v - 382130524556314) / 91710718503142
$\beta_{5}$	$=$	$ ( 791109685779 \nu^{17} - 7210164301611 \nu^{16} - 9959130721243 \nu^{15} + \cdots - 28\!\cdots\!08 ) / 458553592515710 $ (791109685779v^17 - 7210164301611v^16 - 9959130721243v^15 + 219019548828250v^14 - 129967228763690v^13 - 2604115350616358v^12 + 3018929077464028v^11 + 15618836553807274v^10 - 20915771163704460v^9 - 50747240195472899v^8 + 67152166001800775v^7 + 89286097034850934v^6 - 104981772913763742v^5 - 78607939825078217v^4 + 73875327880245182v^3 + 27723146854673405v^2 - 15848111673606532*v - 2818723187680008) / 458553592515710
$\beta_{6}$	$=$	$ ( - 1311747399192 \nu^{17} + 7143588109603 \nu^{16} + 34650950134304 \nu^{15} + \cdots + 32\!\cdots\!04 ) / 458553592515710 $ (-1311747399192v^17 + 7143588109603v^16 + 34650950134304v^15 - 234658873965880v^14 - 261017524340570v^13 + 2891437929917789v^12 - 92512139262469v^11 - 16770069937789482v^10 + 7835223946356190v^9 + 48584918588855527v^8 - 26159365519600605v^7 - 73344057878002822v^6 + 26682919256091476v^5 + 58009621012678766v^4 - 2482979630434371v^3 - 22824828059440880v^2 - 3849606469617184*v + 3246123201026504) / 458553592515710
$\beta_{7}$	$=$	$ ( - 349917812484 \nu^{17} + 1688901432461 \nu^{16} + 8675131551548 \nu^{15} + \cdots + 111350345943128 ) / 91710718503142 $ (-349917812484v^17 + 1688901432461v^16 + 8675131551548v^15 - 49301994013637v^14 - 70473583515524v^13 + 544663725638447v^12 + 174968708204936v^11 - 2877988665500235v^10 + 207277902585169v^9 + 7705748675503300v^8 - 644140687242026v^7 - 10772815905056310v^6 - 1697780111919044v^5 + 7745400853159025v^4 + 3831047566347337v^3 - 2466217334047676v^2 - 1370378677979224*v + 111350345943128) / 91710718503142
$\beta_{8}$	$=$	$ ( 385407419981 \nu^{17} - 1200438988418 \nu^{16} - 13353344278772 \nu^{15} + \cdots - 136002453593002 ) / 91710718503142 $ (385407419981v^17 - 1200438988418v^16 - 13353344278772v^15 + 42197325915668v^14 + 177742287831319v^13 - 568288428837998v^12 - 1172046809992002v^11 + 3728917750761656v^10 + 4263394278218904v^9 - 12758117245734990v^8 - 9280687221504063v^7 + 22830823176478817v^6 + 12014809520507465v^5 - 19298811526607061v^4 - 8318028569739792v^3 + 5660831254005481v^2 + 2523717296614826*v - 136002453593002) / 91710718503142
$\beta_{9}$	$=$	$ ( - 2090924919828 \nu^{17} + 4551609805402 \nu^{16} + 81333032343416 \nu^{15} + \cdots + 25\!\cdots\!36 ) / 458553592515710 $ (-2090924919828v^17 + 4551609805402v^16 + 81333032343416v^15 - 196046498641865v^14 - 1158799035342505v^13 + 3050159888606616v^12 + 7642579061352369v^11 - 22191981165178958v^10 - 24451213054902220v^9 + 81478462441445898v^8 + 36692993083035020v^7 - 151616473032496463v^6 - 23997253964553076v^5 + 133740769526157199v^4 + 8317904722863821v^3 - 44555955289165880v^2 - 2235573586976146*v + 2580984130668436) / 458553592515710
$\beta_{10}$	$=$	$ ( - 2540698748199 \nu^{17} + 15654671463916 \nu^{16} + 53334788478983 \nu^{15} + \cdots + 19\!\cdots\!18 ) / 458553592515710 $ (-2540698748199v^17 + 15654671463916v^16 + 53334788478983v^15 - 465529518896435v^14 - 222400688813930v^13 + 5327433978808593v^12 - 2144085536439348v^11 - 30008779881308449v^10 + 21952160676630305v^9 + 89275983572989399v^8 - 70372869438010905v^7 - 143150176560132484v^6 + 96492872354168522v^5 + 117334944090873342v^4 - 55178643641024207v^3 - 39595679932396075v^2 + 11747539838804672*v + 1999814139848518) / 458553592515710
$\beta_{11}$	$=$	$ ( - 4327658562844 \nu^{17} + 23558355958501 \nu^{16} + 87916111010143 \nu^{15} + \cdots - 36\!\cdots\!32 ) / 458553592515710 $ (-4327658562844v^17 + 23558355958501v^16 + 87916111010143v^15 - 656053916407865v^14 - 312789490230190v^13 + 6751943842055928v^12 - 4090134542592553v^11 - 31621443818304624v^10 + 37684367706099700v^9 + 65847768782781669v^8 - 114313569412095830v^7 - 45988636897213304v^6 + 151142355686823027v^5 - 19252420667947313v^4 - 85638123473034087v^3 + 26564970713687000v^2 + 17232125425205292*v - 3638882800560732) / 458553592515710
$\beta_{12}$	$=$	$ ( - 5078289803572 \nu^{17} + 27812943199083 \nu^{16} + 104695665815899 \nu^{15} + \cdots + 13\!\cdots\!04 ) / 458553592515710 $ (-5078289803572v^17 + 27812943199083v^16 + 104695665815899v^15 - 777633919970010v^14 - 446588046770345v^13 + 8180806311266194v^12 - 3586706544269294v^11 - 40936414385815632v^10 + 36759361401501230v^9 + 102295002903682167v^8 - 113569326608039810v^7 - 128014863301212047v^6 + 146380390992679441v^5 + 77140017563607716v^4 - 70970266968786406v^3 - 20628970190782720v^2 + 8981997232336056*v + 1337342782210304) / 458553592515710
$\beta_{13}$	$=$	$ ( - 5110122742321 \nu^{17} + 37169485310269 \nu^{16} + 60549319874012 \nu^{15} + \cdots - 50\!\cdots\!48 ) / 458553592515710 $ (-5110122742321v^17 + 37169485310269v^16 + 60549319874012v^15 - 979978630598465v^14 + 745605964456930v^13 + 9393960495162337v^12 - 15440326834845312v^11 - 39931596578357406v^10 + 91712873092483690v^9 + 70122823695075001v^8 - 236206412296014350v^7 - 23745053234547846v^6 + 271759235006550893v^5 - 51342037173284612v^4 - 121372849181961728v^3 + 37546147441934705v^2 + 14908527004380998*v - 5079890635549848) / 458553592515710
$\beta_{14}$	$=$	$ ( 275203832328 \nu^{17} - 1715071952182 \nu^{16} - 4734164878966 \nu^{15} + \cdots + 271671408063074 ) / 24134399606090 $ (275203832328v^17 - 1715071952182v^16 - 4734164878966v^15 + 47279328763320v^14 - 2231788474665v^13 - 487930488485056v^12 + 481438775452656v^11 + 2372217266795203v^10 - 3545764718279135v^9 - 5604827488865783v^8 + 10618629866050155v^7 + 6065513585579753v^6 - 14544368120835104v^5 - 2214931163683654v^4 + 8336400966162809v^3 - 406179497837440v^2 - 1452542888047764*v + 271671408063074) / 24134399606090
$\beta_{15}$	$=$	$ ( - 8063567818571 \nu^{17} + 43989724734789 \nu^{16} + 171368371623552 \nu^{15} + \cdots + 11\!\cdots\!82 ) / 458553592515710 $ (-8063567818571v^17 + 43989724734789v^16 + 171368371623552v^15 - 1240040196415655v^14 - 859519239768965v^13 + 13198464010583012v^12 - 3999449483541752v^11 - 67077838206536431v^10 + 48948479437738570v^9 + 170178699214246201v^8 - 152742882271005790v^7 - 210493715781614816v^6 + 190414168430129598v^5 + 112399366880723048v^4 - 85593694947205243v^3 - 21163269474751630v^2 + 7883794302521828*v + 1154096244498282) / 458553592515710
$\beta_{16}$	$=$	$ ( - 9391136360704 \nu^{17} + 50244945428651 \nu^{16} + 210075419204663 \nu^{15} + \cdots + 77\!\cdots\!98 ) / 458553592515710 $ (-9391136360704v^17 + 50244945428651v^16 + 210075419204663v^15 - 1454010821724175v^14 - 1233669458612895v^13 + 15998314238608853v^12 - 2864276175886653v^11 - 85049020560744339v^10 + 50717689605234320v^9 + 232512391257686879v^8 - 166063921733273370v^7 - 334796739782735164v^6 + 209500531343457882v^5 + 246609280474745427v^4 - 91737962366393532v^3 - 81923515184631115v^2 + 5319702723550572*v + 7792387797974698) / 458553592515710
$\beta_{17}$	$=$	$ ( 12640662937253 \nu^{17} - 58978888205822 \nu^{16} - 314118940145836 \nu^{15} + \cdots - 35\!\cdots\!76 ) / 458553592515710 $ (12640662937253v^17 - 58978888205822v^16 - 314118940145836v^15 + 1709457619174555v^14 + 2619322482748470v^13 - 19023971479592081v^12 - 7119329077353169v^11 + 104188757710541728v^10 - 9864070375953565v^9 - 301513455598986783v^8 + 75280233838851000v^7 + 469539167775241043v^6 - 108718334901004314v^5 - 370935331497776184v^4 + 43409210372095999v^3 + 115966308977065780v^2 + 2532805914469796*v - 3579543371110476) / 458553592515710

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 4 $ b2 + 4
$\nu^{3}$	$=$	$ -\beta_{10} + \beta_{7} - \beta_{5} + \beta_{2} + 6\beta _1 + 1 $ -b10 + b7 - b5 + b2 + 6*b1 + 1
$\nu^{4}$	$=$	$ \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 29 $ b16 - b15 + b14 + b13 - b12 + b11 - b10 + b9 + b8 + b6 - 2b4 + 12b2 + b1 + 29
$\nu^{5}$	$=$	$ \beta_{17} - 2 \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{11} - 13 \beta_{10} + \cdots + 13 $ b17 - 2b15 + b14 + b13 + 2b12 + b11 - 13b10 + 2b9 + 18b7 - 14b5 - 3b4 - b3 + 18b2 + 44*b1 + 13
$\nu^{6}$	$=$	$ - \beta_{17} + 17 \beta_{16} - 19 \beta_{15} + 15 \beta_{14} + 17 \beta_{13} - 21 \beta_{12} + \cdots + 253 $ -b17 + 17b16 - 19b15 + 15b14 + 17b13 - 21b12 + 19b11 - 17b10 + 17b9 + 20b8 + 6b7 + 17b6 + b5 - 35b4 + 2b3 + 133b2 + 19*b1 + 253
$\nu^{7}$	$=$	$ 19 \beta_{17} + 7 \beta_{16} - 41 \beta_{15} + 20 \beta_{14} + 19 \beta_{13} + 31 \beta_{12} + \cdots + 160 $ 19b17 + 7b16 - 41b15 + 20b14 + 19b13 + 31b12 + 25b11 - 150b10 + 37b9 + 7b8 + 242b7 - 2b6 - 161b5 - 61b4 - 11b3 + 247b2 + 364*b1 + 160
$\nu^{8}$	$=$	$ - 13 \beta_{17} + 236 \beta_{16} - 265 \beta_{15} + 180 \beta_{14} + 215 \beta_{13} - 306 \beta_{12} + \cdots + 2414 $ -13b17 + 236b16 - 265b15 + 180b14 + 215b13 - 306b12 + 267b11 - 228b10 + 222b9 + 291b8 + 149b7 + 209b6 + 18b5 - 470b4 + 43b3 + 1469b2 + 270*b1 + 2414
$\nu^{9}$	$=$	$ 274 \beta_{17} + 193 \beta_{16} - 609 \beta_{15} + 274 \beta_{14} + 259 \beta_{13} + 329 \beta_{12} + \cdots + 1974 $ 274b17 + 193b16 - 609b15 + 274b14 + 259b13 + 329b12 + 409b11 - 1685b10 + 514b9 + 173b8 + 2979b7 - 64b6 - 1756b5 - 902b4 - 92b3 + 3101b2 + 3288*b1 + 1974
$\nu^{10}$	$=$	$ - 76 \beta_{17} + 3048 \beta_{16} - 3334 \beta_{15} + 2006 \beta_{14} + 2480 \beta_{13} - 3912 \beta_{12} + \cdots + 24303 $ -76b17 + 3048b16 - 3334b15 + 2006b14 + 2480b13 - 3912b12 + 3355b11 - 2848b10 + 2679b9 + 3709b8 + 2590b7 + 2266b6 + 199b5 - 5776b4 + 632b3 + 16332b2 + 3466*b1 + 24303
$\nu^{11}$	$=$	$ 3578 \beta_{17} + 3558 \beta_{16} - 8058 \beta_{15} + 3277 \beta_{14} + 3180 \beta_{13} + 2825 \beta_{12} + \cdots + 24322 $ 3578b17 + 3558b16 - 8058b15 + 3277b14 + 3180b13 + 2825b12 + 5701b11 - 18823b10 + 6502b9 + 2948b8 + 35336b7 - 1223b6 - 18821b5 - 11914b4 - 686b3 + 37529b2 + 31762*b1 + 24322
$\nu^{12}$	$=$	$ 464 \beta_{17} + 37845 \beta_{16} - 40134 \beta_{15} + 21658 \beta_{14} + 27593 \beta_{13} + \cdots + 253444 $ 464b17 + 37845b16 - 40134b15 + 21658b14 + 27593b13 - 47035b12 + 40019b11 - 34639b10 + 31338b9 + 44357b8 + 38946b7 + 23158b6 + 1428b5 - 68343b4 + 7983b3 + 182929b2 + 42487*b1 + 253444
$\nu^{13}$	$=$	$ 44485 \beta_{17} + 55141 \beta_{16} - 101140 \beta_{15} + 37005 \beta_{14} + 37584 \beta_{13} + \cdots + 298541 $ 44485b17 + 55141b16 - 101140b15 + 37005b14 + 37584b13 + 18965b12 + 73757b11 - 210549b10 + 79051b9 + 43284b8 + 411631b7 - 18604b6 - 200858b5 - 149602b4 - 4596b3 + 446755b2 + 322355*b1 + 298541
$\nu^{14}$	$=$	$ 22314 \beta_{17} + 458201 \beta_{16} - 472938 \beta_{15} + 230756 \beta_{14} + 302624 \beta_{13} + \cdots + 2707214 $ 22314b17 + 458201b16 - 472938b15 + 230756b14 + 302624b13 - 547414b12 + 465094b11 - 416557b10 + 361581b9 + 513597b8 + 542090b7 + 229803b6 + 565b5 - 794497b4 + 93600b3 + 2062252b2 + 509183*b1 + 2707214
$\nu^{15}$	$=$	$ 537550 \beta_{17} + 777568 \beta_{16} - 1235148 \beta_{15} + 408398 \beta_{14} + 438556 \beta_{13} + \cdots + 3648872 $ 537550b17 + 777568b16 - 1235148b15 + 408398b14 + 438556b13 + 61245b12 + 918056b11 - 2363460b10 + 942877b9 + 589024b8 + 4750220b7 - 249924b6 - 2146764b5 - 1830684b4 - 24991b3 + 5276966b2 + 3387715*b1 + 3648872
$\nu^{16}$	$=$	$ 446056 \beta_{17} + 5454344 \beta_{16} - 5512652 \beta_{15} + 2450347 \beta_{14} + 3304989 \beta_{13} + \cdots + 29405517 $ 446056b17 + 5454344b16 - 5512652b15 + 2450347b14 + 3304989b13 - 6256413b12 + 5335356b11 - 4981103b10 + 4146857b9 + 5849445b8 + 7195932b7 + 2247640b6 - 216425b5 - 9159712b4 + 1054841b3 + 23371435b2 + 6035579*b1 + 29405517
$\nu^{17}$	$=$	$ 6379968 \beta_{17} + 10353960 \beta_{16} - 14851037 \beta_{15} + 4482626 \beta_{14} + 5105978 \beta_{13} + \cdots + 44409364 $ 6379968b17 + 10353960b16 - 14851037b15 + 4482626b14 + 5105978b13 - 980262b12 + 11183711b11 - 26635216b10 + 11134974b9 + 7671146b8 + 54544947b7 - 3114422b6 - 23039936b5 - 22079443b4 - 49724b3 + 62086348b2 + 36471671*b1 + 44409364

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

3.40653
2.94825
2.70913
2.43342
2.35083
1.76835
1.41957
0.966609
0.740413
0.244353
−0.382125
−0.665502
−0.897994
−1.32012
−1.50485
−2.50583
−2.50785
−3.20319

−1.00000

−3.40653

1.00000

−1.00000

3.40653

3.89438

−1.00000

8.60447

1.00000

1.2

−1.00000

−2.94825

1.00000

−1.00000

2.94825

−4.64558

−1.00000

5.69219

1.00000

1.3

−1.00000

−2.70913

1.00000

−1.00000

2.70913

−0.626205

−1.00000

4.33941

1.00000

1.4

−1.00000

−2.43342

1.00000

−1.00000

2.43342

−1.94600

−1.00000

2.92154

1.00000

1.5

−1.00000

−2.35083

1.00000

−1.00000

2.35083

−0.774003

−1.00000

2.52639

1.00000

1.6

−1.00000

−1.76835

1.00000

−1.00000

1.76835

4.06465

−1.00000

0.127049

1.00000

1.7

−1.00000

−1.41957

1.00000

−1.00000

1.41957

−1.86698

−1.00000

−0.984814

1.00000

1.8

−1.00000

−0.966609

1.00000

−1.00000

0.966609

3.98367

−1.00000

−2.06567

1.00000

1.9

−1.00000

−0.740413

1.00000

−1.00000

0.740413

−0.188188

−1.00000

−2.45179

1.00000

1.10

−1.00000

−0.244353

1.00000

−1.00000

0.244353

−3.17260

−1.00000

−2.94029

1.00000

1.11

−1.00000

0.382125

1.00000

−1.00000

−0.382125

2.29644

−1.00000

−2.85398

1.00000

1.12

−1.00000

0.665502

1.00000

−1.00000

−0.665502

−1.21475

−1.00000

−2.55711

1.00000

1.13

−1.00000

0.897994

1.00000

−1.00000

−0.897994

−4.78725

−1.00000

−2.19361

1.00000

1.14

−1.00000

1.32012

1.00000

−1.00000

−1.32012

3.05333

−1.00000

−1.25729

1.00000

1.15

−1.00000

1.50485

1.00000

−1.00000

−1.50485

1.32574

−1.00000

−0.735415

1.00000

1.16

−1.00000

2.50583

1.00000

−1.00000

−2.50583

−4.02834

−1.00000

3.27917

1.00000

1.17

−1.00000

2.50785

1.00000

−1.00000

−2.50785

2.19558

−1.00000

3.28930

1.00000

1.18

−1.00000

3.20319

1.00000

−1.00000

−3.20319

−3.56388

−1.00000

7.26044

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$1$
$5$	$1$
$11$	$-1$
$73$	$1$

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	8030.2.a.bj	✓	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8030.2.a.bj	✓	18	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8030))$:

$ T_{3}^{18} + 6 T_{3}^{17} - 19 T_{3}^{16} - 169 T_{3}^{15} + 40 T_{3}^{14} + 1802 T_{3}^{13} + \cdots + 624 $

$ T_{7}^{18} + 6 T_{7}^{17} - 63 T_{7}^{16} - 408 T_{7}^{15} + 1518 T_{7}^{14} + 11182 T_{7}^{13} + \cdots - 524800 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T + 1)^{18} $ (T + 1)^18
$3$	$ T^{18} + 6 T^{17} + \cdots + 624 $ T^18 + 6T^17 - 19T^16 - 169T^15 + 40T^14 + 1802T^13 + 1266T^12 - 9247T^11 - 10582T^10 + 24226T^9 + 33432T^8 - 32951T^7 - 49245T^6 + 22852T^5 + 33519T^4 - 7814T^3 - 9238T^2 + 1136*T + 624
$5$	$ (T + 1)^{18} $ (T + 1)^18
$7$	$ T^{18} + 6 T^{17} + \cdots - 524800 $ T^18 + 6T^17 - 63T^16 - 408T^15 + 1518T^14 + 11182T^13 - 16535T^12 - 157854T^11 + 57777T^10 + 1215862T^9 + 371934T^8 - 4958716T^7 - 3893832T^6 + 9223080T^5 + 11496656T^4 - 3580256T^3 - 10227488T^2 - 4522240*T - 524800
$11$	$ (T - 1)^{18} $ (T - 1)^18
$13$	$ T^{18} + \cdots - 172583880 $ T^18 + 8T^17 - 124T^16 - 1087T^15 + 5674T^14 + 58821T^13 - 108257T^12 - 1618889T^11 + 316006T^10 + 23864561T^9 + 17498921T^8 - 180613099T^7 - 224943459T^6 + 617326835T^5 + 830690075T^4 - 928892928T^3 - 736656822T^2 + 836431960*T - 172583880
$17$	$ T^{18} + 10 T^{17} + \cdots + 60418752 $ T^18 + 10T^17 - 128T^16 - 1373T^15 + 6109T^14 + 69694T^13 - 154169T^12 - 1649994T^11 + 2805582T^10 + 18798951T^9 - 38520105T^8 - 86552002T^7 + 266961088T^6 - 27478596T^5 - 462073776T^4 + 361040640T^3 + 127855152T^2 - 219946112*T + 60418752
$19$	$ T^{18} + \cdots - 13941911200 $ T^18 - 24T^17 + 37T^16 + 3517T^15 - 28021T^14 - 114736T^13 + 2190198T^12 - 4634490T^11 - 54481379T^10 + 317070151T^9 - 25926057T^8 - 4419542948T^7 + 12956702511T^6 - 4358611480T^5 - 35091956772T^4 + 40369680472T^3 + 23151536428T^2 - 30581238240*T - 13941911200
$23$	$ T^{18} + \cdots + 39058865408 $ T^18 + 14T^17 - 148T^16 - 2868T^15 + 3567T^14 + 218924T^13 + 428425T^12 - 7686252T^11 - 29758916T^10 + 119178266T^9 + 722250255T^8 - 483516896T^7 - 7585105760T^6 - 4182927860T^5 + 35901141536T^4 + 37489017664T^3 - 69395991504T^2 - 73038890368*T + 39058865408
$29$	$ T^{18} + \cdots + 180241745088 $ T^18 - 11T^17 - 352T^16 + 4000T^15 + 50323T^14 - 602156T^13 - 3716379T^12 + 48503659T^11 + 146694156T^10 - 2252437510T^9 - 2685243269T^8 + 60333906396T^7 + 1974625376T^6 - 873314513492T^5 + 619544313856T^4 + 5652926621232T^3 - 6079106357072T^2 - 7484975811776*T + 180241745088
$31$	$ T^{18} + \cdots - 14163675856896 $ T^18 - 32T^17 + 123T^16 + 6532T^15 - 77402T^14 - 252668T^13 + 8558724T^12 - 26090912T^11 - 339578080T^10 + 2378421488T^9 + 2613391648T^8 - 64136941056T^7 + 114777628224T^6 + 576472539520T^5 - 1917906651648T^4 - 1499494087680T^3 + 9441723218944T^2 - 432622100480*T - 14163675856896
$37$	$ T^{18} + \cdots + 410404544 $ T^18 + 3T^17 - 383T^16 - 1125T^15 + 58411T^14 + 177118T^13 - 4499803T^12 - 14689935T^11 + 182210563T^10 + 649804813T^9 - 3573293851T^8 - 13665046874T^7 + 25548196124T^6 + 93485958516T^5 - 53173561696T^4 - 59593934992T^3 + 13485870096T^2 + 5889480896*T + 410404544
$41$	$ T^{18} + \cdots - 261750610176 $ T^18 - 28T^17 + 63T^16 + 4670T^15 - 36659T^14 - 231988T^13 + 3216029T^12 + 458842T^11 - 112295854T^10 + 246361056T^9 + 1640176644T^8 - 5911516216T^7 - 9778498400T^6 + 55112499120T^5 + 10211303520T^4 - 227508687040T^3 + 102355288384T^2 + 344491283200*T - 261750610176
$43$	$ T^{18} + \cdots + 7913538708224 $ T^18 + 6T^17 - 418T^16 - 3065T^15 + 64817T^14 + 583121T^13 - 4442578T^12 - 53000350T^11 + 104839108T^10 + 2421438025T^9 + 2118235469T^8 - 52700934997T^7 - 135018298369T^6 + 457291974140T^5 + 1889101474060T^4 - 689890188480T^3 - 8445986314000T^2 - 3956622566016*T + 7913538708224
$47$	$ T^{18} + \cdots + 12420879360 $ T^18 + 16T^17 - 277T^16 - 5307T^15 + 21666T^14 + 646006T^13 + 205040T^12 - 35119120T^11 - 93224546T^10 + 809583197T^9 + 3849827987T^8 - 3914689376T^7 - 51780883144T^6 - 82899392288T^5 + 85960773104T^4 + 382383485888T^3 + 391400997376T^2 + 138822003200*T + 12420879360
$53$	$ T^{18} + \cdots + 2341113982464 $ T^18 + T^17 - 650T^16 - 172T^15 + 171393T^14 - 81840T^13 - 23261858T^12 + 27929600T^11 + 1689168372T^10 - 3133398328T^9 - 60765010632T^8 + 140814467936T^7 + 839747570656T^6 - 1834589820960T^5 - 3305153463552T^4 + 7257822028928T^3 + 2173499981184T^2 - 7005658890752T + 2341113982464
$59$	$ T^{18} + \cdots - 233947460856064 $ T^18 - 14T^17 - 602T^16 + 7869T^15 + 154263T^14 - 1794959T^13 - 21940417T^12 + 212152097T^11 + 1885661044T^10 - 13735173542T^9 - 99215593097T^8 + 472136455678T^7 + 3050891910818T^6 - 7591326254940T^5 - 47798157503772T^4 + 39093469765504T^3 + 267698309621040T^2 - 43254688668288*T - 233947460856064
$61$	$ T^{18} + \cdots + 2241677481600 $ T^18 - 39T^17 + 240T^16 + 7886T^15 - 96962T^14 - 537466T^13 + 10923004T^12 + 8836203T^11 - 587097557T^10 + 584603220T^9 + 16311354967T^8 - 30635304732T^7 - 221921158016T^6 + 526613163184T^5 + 1236857771292T^4 - 3111145310656T^3 - 2721285964176T^2 + 5608991967040*T + 2241677481600
$67$	$ T^{18} + \cdots + 11080034081904 $ T^18 - 4T^17 - 712T^16 + 2845T^15 + 197622T^14 - 781795T^13 - 27238093T^12 + 105670365T^11 + 1962652948T^10 - 7365479053T^9 - 71353433899T^8 + 251433826687T^7 + 1165016652101T^6 - 3540542188837T^5 - 7875305113987T^4 + 18132570325856T^3 + 14158448565834T^2 - 32118640667072*T + 11080034081904
$71$	$ T^{18} + \cdots + 300602579161088 $ T^18 - 29T^17 - 163T^16 + 11379T^15 - 35053T^14 - 1723093T^13 + 11040218T^12 + 127270595T^11 - 1130351789T^10 - 4690308505T^9 + 57548230442T^8 + 71708006975T^7 - 1575495659830T^6 + 215188480348T^5 + 22694570281853T^4 - 17593154698688T^3 - 151219217550720T^2 + 131981029818368*T + 300602579161088
$73$	$ (T + 1)^{18} $ (T + 1)^18
$79$	$ T^{18} + \cdots + 5397118976 $ T^18 - 38T^17 + 49T^16 + 13710T^15 - 124005T^14 - 1569116T^13 + 22606814T^12 + 49334736T^11 - 1583814880T^10 + 1968256856T^9 + 45506018112T^8 - 125322374816T^7 - 428948141856T^6 + 1551345363968T^5 - 172671465728T^4 - 659148831744T^3 + 49259881984T^2 + 72647122944*T + 5397118976
$83$	$ T^{18} + \cdots + 38\!\cdots\!36 $ T^18 - 4T^17 - 1053T^16 + 1207T^15 + 474606T^14 + 671168T^13 - 116559436T^12 - 423627855T^11 + 16524507250T^10 + 92635654050T^9 - 1321938308222T^8 - 10076413086231T^7 + 53073912032891T^6 + 568541994783132T^5 - 637812434702837T^4 - 15703287969927472T^3 - 16656470061068608T^2 + 165567768356209664*T + 381454654376084736
$89$	$ T^{18} + \cdots - 127389733163796 $ T^18 - 44T^17 + 219T^16 + 14511T^15 - 166444T^14 - 1844264T^13 + 29733865T^12 + 113306318T^11 - 2561356589T^10 - 3350335726T^9 + 120085833735T^8 + 29500951142T^7 - 3048822113036T^6 + 973291212556T^5 + 38003222923852T^4 - 32730725824619T^3 - 170613906968185T^2 + 299707334175496*T - 127389733163796
$97$	$ T^{18} + \cdots + 74\!\cdots\!16 $ T^18 - 3T^17 - 1234T^16 + 2608T^15 + 628305T^14 - 726110T^13 - 169993108T^12 + 25229983T^11 + 26153967155T^10 + 23117020134T^9 - 2261267223468T^8 - 4330188180676T^7 + 100495910816344T^6 + 294360495272928T^5 - 1767434837471888T^4 - 7147575221188608T^3 - 439870974394048T^2 + 14078604869679616*T + 7458432499823616
show more
show less

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8030.a (trivial)

\(f(q)\)	\(=\)	\( q - q^{2} - \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + \beta_{7} q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) q - q^2 - b1 * q^3 + q^4 - q^5 + b1 * q^6 + b7 * q^7 - q^8 + (b2 + 1) * q^9	\( q - q^{2} - \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + \beta_{7} q^{7} - q^{8} + (\beta_{2} + 1) q^{9} + q^{10} + q^{11} - \beta_1 q^{12} + \beta_{11} q^{13} - \beta_{7} q^{14} + \beta_1 q^{15} + q^{16} + ( - \beta_{10} - 1) q^{17} + ( - \beta_{2} - 1) q^{18} + ( - \beta_{8} + 1) q^{19} - q^{20} + ( - \beta_{7} + \beta_{4} - \beta_{2} + \cdots - 1) q^{21}+ \cdots + (\beta_{2} + 1) q^{99}+O(q^{100}) \) q - q^2 - b1 * q^3 + q^4 - q^5 + b1 * q^6 + b7 * q^7 - q^8 + (b2 + 1) * q^9 + q^10 + q^11 - b1 * q^12 + b11 * q^13 - b7 * q^14 + b1 * q^15 + q^16 + (-b10 - 1) * q^17 + (-b2 - 1) * q^18 + (-b8 + 1) * q^19 - q^20 + (-b7 + b4 - b2 + b1 - 1) * q^21 - q^22 + (-b16 + b6 - 1) * q^23 + b1 * q^24 + q^25 - b11 * q^26 + (b10 - b7 + b5 - b2 - 1) * q^27 + b7 * q^28 + (-b11 - b8 - b6 + b4 - b3 - b2) * q^29 - b1 * q^30 + (b11 - b9 + b8 + b6 + b3 + b2 + 3) * q^31 - q^32 - b1 * q^33 + (b10 + 1) * q^34 - b7 * q^35 + (b2 + 1) * q^36 + (-b14 - b12 - b8 + b7 - b3) * q^37 + (b8 - 1) * q^38 + (b17 + b16 - b15 - b14 - b8 + b7 + b2 + b1) * q^39 + q^40 + (-b10 + b9 + b2 - b1 + 1) * q^41 + (b7 - b4 + b2 - b1 + 1) * q^42 + (b14 + b13 - b12 + b11 + b8 - b7 + b5 + b3 - b2 + 2b1) q^43 + q^44 + (-b2 - 1) * q^45 + (b16 - b6 + 1) * q^46 + (-b16 + b15 - b13 + 2b12 - b11 - b8 + 2b7 - b5 - b3 + b2 - 3b1 - 1) q^47 - b1 * q^48 + (-b17 - b16 + b14 + b12 + b6 - b5 - b4 + b2 - b1 + 2) * q^49 - q^50 + (b14 + b13 + b10 - b7 - b6 + b3 - 2b2 + 3b1 - 2) * q^51 + b11 * q^52 + (-b17 + b15 - b13 - b10 - b9 + 2b8 + b4 - b1 + 1) q^53 + (-b10 + b7 - b5 + b2 + 1) * q^54 - q^55 - b7 * q^56 + (b16 - 2b15 + b14 + b13 - b12 + 2b11 + b9 + b8 - 2b7 + b6 + b5 - b4 + b3 + b2) q^57 + (b11 + b8 + b6 - b4 + b3 + b2) * q^58 + (-b16 + b15 - b14 + b12 - b11 - b8 + b7 - b2 - b1) * q^59 + b1 * q^60 + (-b12 + b11 - b6 + b4 + 3) * q^61 + (-b11 + b9 - b8 - b6 - b3 - b2 - 3) * q^62 + (b17 + b16 - b15 - b14 - b13 + b11 - b10 + 3b7 - b5 - b4 + 2b2 - 1) * q^63 + q^64 - b11 * q^65 + b1 * q^66 + (b16 + b15 - b14 - b13 - b9 + b7 - b6 - b3 + b2 - b1 + 1) * q^67 + (-b10 - 1) * q^68 + (2b16 - 2b12 + b9 + b8 - 2b6 + b5 + 2b1 - 1) * q^69 + b7 * q^70 + (-2b16 + b15 - b13 + 2b12 - b11 - b8 + 2b7 + b6 - 2b5 - b3 + b2 - 4b1 + 2) q^71 + (-b2 - 1) * q^72 - q^73 + (b14 + b12 + b8 - b7 + b3) * q^74 - b1 * q^75 + (-b8 + 1) * q^76 + b7 * q^77 + (-b17 - b16 + b15 + b14 + b8 - b7 - b2 - b1) * q^78 + (b15 + b14 - b13 + b12 + b8 + b7 + b2 - 2b1 + 3) q^79 - q^80 + (b16 - b15 + b14 + b13 - b12 + b11 - b10 + b9 + b8 + b6 - 2b4 + 3b2 + b1 + 2) * q^81 + (b10 - b9 - b2 + b1 - 1) * q^82 + (b15 + b14 + 2b13 - b11 + b9 - b6 + b3 - b2 + b1) q^83 + (-b7 + b4 - b2 + b1 - 1) * q^84 + (b10 + 1) * q^85 + (-b14 - b13 + b12 - b11 - b8 + b7 - b5 - b3 + b2 - 2b1) q^86 + (-b15 + b14 + b12 + b11 - b9 + b8 + 3b7 - b5 - b4 + b3 + 2b2 - b1 + 2) * q^87 - q^88 + (-b15 + b13 + b12 + b6 - b4 + 2b2 + 2) q^89 + (b2 + 1) * q^90 + (-b16 - b14 - b13 + b12 + b10 - b9 - b8 + b7 - 3b1 + 2) q^91 + (-b16 + b6 - 1) * q^92 + (2b16 - b14 - 2b12 + b9 - b7 - 2b6 + 2b5 + b4 - b3 - 2b2 - 2b1) * q^93 + (b16 - b15 + b13 - 2b12 + b11 + b8 - 2b7 + b5 + b3 - b2 + 3b1 + 1) q^94 + (b8 - 1) * q^95 + b1 * q^96 + (b17 - b16 + b15 - b14 - b11 - 2b8 - b6 - b5 + b4 - 3b1) * q^97 + (b17 + b16 - b14 - b12 - b6 + b5 + b4 - b2 + b1 - 2) * q^98 + (b2 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 18 q^{2} - 6 q^{3} + 18 q^{4} - 18 q^{5} + 6 q^{6} - 6 q^{7} - 18 q^{8} + 20 q^{9}+O(q^{10}) \) 18 * q - 18 * q^2 - 6 * q^3 + 18 * q^4 - 18 * q^5 + 6 * q^6 - 6 * q^7 - 18 * q^8 + 20 * q^9	\( 18 q - 18 q^{2} - 6 q^{3} + 18 q^{4} - 18 q^{5} + 6 q^{6} - 6 q^{7} - 18 q^{8} + 20 q^{9} + 18 q^{10} + 18 q^{11} - 6 q^{12} - 8 q^{13} + 6 q^{14} + 6 q^{15} + 18 q^{16} - 10 q^{17} - 20 q^{18} + 24 q^{19} - 18 q^{20} - 13 q^{21} - 18 q^{22} - 14 q^{23} + 6 q^{24} + 18 q^{25} + 8 q^{26} - 15 q^{27} - 6 q^{28} + 11 q^{29} - 6 q^{30} + 32 q^{31} - 18 q^{32} - 6 q^{33} + 10 q^{34} + 6 q^{35} + 20 q^{36} - 3 q^{37} - 24 q^{38} + 7 q^{39} + 18 q^{40} + 28 q^{41} + 13 q^{42} - 6 q^{43} + 18 q^{44} - 20 q^{45} + 14 q^{46} - 16 q^{47} - 6 q^{48} + 36 q^{49} - 18 q^{50} - 29 q^{51} - 8 q^{52} - q^{53} + 15 q^{54} - 18 q^{55} + 6 q^{56} - 11 q^{57} - 11 q^{58} + 14 q^{59} + 6 q^{60} + 39 q^{61} - 32 q^{62} - 29 q^{63} + 18 q^{64} + 8 q^{65} + 6 q^{66} + 4 q^{67} - 10 q^{68} - 19 q^{69} - 6 q^{70} + 29 q^{71} - 20 q^{72} - 18 q^{73} + 3 q^{74} - 6 q^{75} + 24 q^{76} - 6 q^{77} - 7 q^{78} + 38 q^{79} - 18 q^{80} + 34 q^{81} - 28 q^{82} + 4 q^{83} - 13 q^{84} + 10 q^{85} + 6 q^{86} - 18 q^{88} + 44 q^{89} + 20 q^{90} + 27 q^{91} - 14 q^{92} - 12 q^{93} + 16 q^{94} - 24 q^{95} + 6 q^{96} + 3 q^{97} - 36 q^{98} + 20 q^{99}+O(q^{100}) \) 18 * q - 18 * q^2 - 6 * q^3 + 18 * q^4 - 18 * q^5 + 6 * q^6 - 6 * q^7 - 18 * q^8 + 20 * q^9 + 18 * q^10 + 18 * q^11 - 6 * q^12 - 8 * q^13 + 6 * q^14 + 6 * q^15 + 18 * q^16 - 10 * q^17 - 20 * q^18 + 24 * q^19 - 18 * q^20 - 13 * q^21 - 18 * q^22 - 14 * q^23 + 6 * q^24 + 18 * q^25 + 8 * q^26 - 15 * q^27 - 6 * q^28 + 11 * q^29 - 6 * q^30 + 32 * q^31 - 18 * q^32 - 6 * q^33 + 10 * q^34 + 6 * q^35 + 20 * q^36 - 3 * q^37 - 24 * q^38 + 7 * q^39 + 18 * q^40 + 28 * q^41 + 13 * q^42 - 6 * q^43 + 18 * q^44 - 20 * q^45 + 14 * q^46 - 16 * q^47 - 6 * q^48 + 36 * q^49 - 18 * q^50 - 29 * q^51 - 8 * q^52 - q^53 + 15 * q^54 - 18 * q^55 + 6 * q^56 - 11 * q^57 - 11 * q^58 + 14 * q^59 + 6 * q^60 + 39 * q^61 - 32 * q^62 - 29 * q^63 + 18 * q^64 + 8 * q^65 + 6 * q^66 + 4 * q^67 - 10 * q^68 - 19 * q^69 - 6 * q^70 + 29 * q^71 - 20 * q^72 - 18 * q^73 + 3 * q^74 - 6 * q^75 + 24 * q^76 - 6 * q^77 - 7 * q^78 + 38 * q^79 - 18 * q^80 + 34 * q^81 - 28 * q^82 + 4 * q^83 - 13 * q^84 + 10 * q^85 + 6 * q^86 - 18 * q^88 + 44 * q^89 + 20 * q^90 + 27 * q^91 - 14 * q^92 - 12 * q^93 + 16 * q^94 - 24 * q^95 + 6 * q^96 + 3 * q^97 - 36 * q^98 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8030.2.a.bj

Level	$8030$
Weight	$2$
Character orbit	8030.a
Self dual	yes
Analytic conductor	$64.120$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(64.1198728231\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{18} - 6 x^{17} - 19 x^{16} + 169 x^{15} + 40 x^{14} - 1802 x^{13} + 1266 x^{12} + 9247 x^{11} + \cdots + 624 \) x^18 - 6x^17 - 19x^16 + 169x^15 + 40x^14 - 1802x^13 + 1266x^12 + 9247x^11 - 10582x^10 - 24226x^9 + 33432x^8 + 32951x^7 - 49245x^6 - 22852x^5 + 33519x^4 + 7814x^3 - 9238x^2 - 1136*x + 624
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 4 \) v^2 - 4
\(\beta_{3}\)	\(=\)	\( ( - 228908810602 \nu^{17} + 411817852383 \nu^{16} + 17940201788599 \nu^{15} + \cdots + 555269137026984 ) / 458553592515710 \) (-228908810602v^17 + 411817852383v^16 + 17940201788599v^15 - 54740679928215v^14 - 345091901988560v^13 + 1236480444135744v^12 + 2704468482734111v^11 - 11225278351545172v^10 - 9698874963407970v^9 + 48774348098535567v^8 + 16013633215034340v^7 - 104181503235510582v^6 - 14440198071326389v^5 + 102810448170732661v^4 + 15089472753225789v^3 - 36195299792332800v^2 - 11562778652928424*v + 555269137026984) / 458553592515710
\(\beta_{4}\)	\(=\)	\( ( 60687629959 \nu^{17} - 337791681891 \nu^{16} - 1158984744611 \nu^{15} + \cdots - 382130524556314 ) / 91710718503142 \) (60687629959v^17 - 337791681891v^16 - 1158984744611v^15 + 7174877002527v^14 + 15414588942197v^13 - 73300933171233v^12 - 182732638334377v^11 + 617563723620284v^10 + 978638152319253v^9 - 3348562944219762v^8 - 1401466621346000v^7 + 8156666882637314v^6 - 1446859114193701v^5 - 7814541869839508v^4 + 3563007113645037v^3 + 2228412814161882v^2 - 1175933107443670*v - 382130524556314) / 91710718503142
\(\beta_{5}\)	\(=\)	\( ( 791109685779 \nu^{17} - 7210164301611 \nu^{16} - 9959130721243 \nu^{15} + \cdots - 28\!\cdots\!08 ) / 458553592515710 \) (791109685779v^17 - 7210164301611v^16 - 9959130721243v^15 + 219019548828250v^14 - 129967228763690v^13 - 2604115350616358v^12 + 3018929077464028v^11 + 15618836553807274v^10 - 20915771163704460v^9 - 50747240195472899v^8 + 67152166001800775v^7 + 89286097034850934v^6 - 104981772913763742v^5 - 78607939825078217v^4 + 73875327880245182v^3 + 27723146854673405v^2 - 15848111673606532*v - 2818723187680008) / 458553592515710
\(\beta_{6}\)	\(=\)	\( ( - 1311747399192 \nu^{17} + 7143588109603 \nu^{16} + 34650950134304 \nu^{15} + \cdots + 32\!\cdots\!04 ) / 458553592515710 \) (-1311747399192v^17 + 7143588109603v^16 + 34650950134304v^15 - 234658873965880v^14 - 261017524340570v^13 + 2891437929917789v^12 - 92512139262469v^11 - 16770069937789482v^10 + 7835223946356190v^9 + 48584918588855527v^8 - 26159365519600605v^7 - 73344057878002822v^6 + 26682919256091476v^5 + 58009621012678766v^4 - 2482979630434371v^3 - 22824828059440880v^2 - 3849606469617184*v + 3246123201026504) / 458553592515710
\(\beta_{7}\)	\(=\)	\( ( - 349917812484 \nu^{17} + 1688901432461 \nu^{16} + 8675131551548 \nu^{15} + \cdots + 111350345943128 ) / 91710718503142 \) (-349917812484v^17 + 1688901432461v^16 + 8675131551548v^15 - 49301994013637v^14 - 70473583515524v^13 + 544663725638447v^12 + 174968708204936v^11 - 2877988665500235v^10 + 207277902585169v^9 + 7705748675503300v^8 - 644140687242026v^7 - 10772815905056310v^6 - 1697780111919044v^5 + 7745400853159025v^4 + 3831047566347337v^3 - 2466217334047676v^2 - 1370378677979224*v + 111350345943128) / 91710718503142
\(\beta_{8}\)	\(=\)	\( ( 385407419981 \nu^{17} - 1200438988418 \nu^{16} - 13353344278772 \nu^{15} + \cdots - 136002453593002 ) / 91710718503142 \) (385407419981v^17 - 1200438988418v^16 - 13353344278772v^15 + 42197325915668v^14 + 177742287831319v^13 - 568288428837998v^12 - 1172046809992002v^11 + 3728917750761656v^10 + 4263394278218904v^9 - 12758117245734990v^8 - 9280687221504063v^7 + 22830823176478817v^6 + 12014809520507465v^5 - 19298811526607061v^4 - 8318028569739792v^3 + 5660831254005481v^2 + 2523717296614826*v - 136002453593002) / 91710718503142
\(\beta_{9}\)	\(=\)	\( ( - 2090924919828 \nu^{17} + 4551609805402 \nu^{16} + 81333032343416 \nu^{15} + \cdots + 25\!\cdots\!36 ) / 458553592515710 \) (-2090924919828v^17 + 4551609805402v^16 + 81333032343416v^15 - 196046498641865v^14 - 1158799035342505v^13 + 3050159888606616v^12 + 7642579061352369v^11 - 22191981165178958v^10 - 24451213054902220v^9 + 81478462441445898v^8 + 36692993083035020v^7 - 151616473032496463v^6 - 23997253964553076v^5 + 133740769526157199v^4 + 8317904722863821v^3 - 44555955289165880v^2 - 2235573586976146*v + 2580984130668436) / 458553592515710
\(\beta_{10}\)	\(=\)	\( ( - 2540698748199 \nu^{17} + 15654671463916 \nu^{16} + 53334788478983 \nu^{15} + \cdots + 19\!\cdots\!18 ) / 458553592515710 \) (-2540698748199v^17 + 15654671463916v^16 + 53334788478983v^15 - 465529518896435v^14 - 222400688813930v^13 + 5327433978808593v^12 - 2144085536439348v^11 - 30008779881308449v^10 + 21952160676630305v^9 + 89275983572989399v^8 - 70372869438010905v^7 - 143150176560132484v^6 + 96492872354168522v^5 + 117334944090873342v^4 - 55178643641024207v^3 - 39595679932396075v^2 + 11747539838804672*v + 1999814139848518) / 458553592515710
\(\beta_{11}\)	\(=\)	\( ( - 4327658562844 \nu^{17} + 23558355958501 \nu^{16} + 87916111010143 \nu^{15} + \cdots - 36\!\cdots\!32 ) / 458553592515710 \) (-4327658562844v^17 + 23558355958501v^16 + 87916111010143v^15 - 656053916407865v^14 - 312789490230190v^13 + 6751943842055928v^12 - 4090134542592553v^11 - 31621443818304624v^10 + 37684367706099700v^9 + 65847768782781669v^8 - 114313569412095830v^7 - 45988636897213304v^6 + 151142355686823027v^5 - 19252420667947313v^4 - 85638123473034087v^3 + 26564970713687000v^2 + 17232125425205292*v - 3638882800560732) / 458553592515710
\(\beta_{12}\)	\(=\)	\( ( - 5078289803572 \nu^{17} + 27812943199083 \nu^{16} + 104695665815899 \nu^{15} + \cdots + 13\!\cdots\!04 ) / 458553592515710 \) (-5078289803572v^17 + 27812943199083v^16 + 104695665815899v^15 - 777633919970010v^14 - 446588046770345v^13 + 8180806311266194v^12 - 3586706544269294v^11 - 40936414385815632v^10 + 36759361401501230v^9 + 102295002903682167v^8 - 113569326608039810v^7 - 128014863301212047v^6 + 146380390992679441v^5 + 77140017563607716v^4 - 70970266968786406v^3 - 20628970190782720v^2 + 8981997232336056*v + 1337342782210304) / 458553592515710
\(\beta_{13}\)	\(=\)	\( ( - 5110122742321 \nu^{17} + 37169485310269 \nu^{16} + 60549319874012 \nu^{15} + \cdots - 50\!\cdots\!48 ) / 458553592515710 \) (-5110122742321v^17 + 37169485310269v^16 + 60549319874012v^15 - 979978630598465v^14 + 745605964456930v^13 + 9393960495162337v^12 - 15440326834845312v^11 - 39931596578357406v^10 + 91712873092483690v^9 + 70122823695075001v^8 - 236206412296014350v^7 - 23745053234547846v^6 + 271759235006550893v^5 - 51342037173284612v^4 - 121372849181961728v^3 + 37546147441934705v^2 + 14908527004380998*v - 5079890635549848) / 458553592515710
\(\beta_{14}\)	\(=\)	\( ( 275203832328 \nu^{17} - 1715071952182 \nu^{16} - 4734164878966 \nu^{15} + \cdots + 271671408063074 ) / 24134399606090 \) (275203832328v^17 - 1715071952182v^16 - 4734164878966v^15 + 47279328763320v^14 - 2231788474665v^13 - 487930488485056v^12 + 481438775452656v^11 + 2372217266795203v^10 - 3545764718279135v^9 - 5604827488865783v^8 + 10618629866050155v^7 + 6065513585579753v^6 - 14544368120835104v^5 - 2214931163683654v^4 + 8336400966162809v^3 - 406179497837440v^2 - 1452542888047764*v + 271671408063074) / 24134399606090
\(\beta_{15}\)	\(=\)	\( ( - 8063567818571 \nu^{17} + 43989724734789 \nu^{16} + 171368371623552 \nu^{15} + \cdots + 11\!\cdots\!82 ) / 458553592515710 \) (-8063567818571v^17 + 43989724734789v^16 + 171368371623552v^15 - 1240040196415655v^14 - 859519239768965v^13 + 13198464010583012v^12 - 3999449483541752v^11 - 67077838206536431v^10 + 48948479437738570v^9 + 170178699214246201v^8 - 152742882271005790v^7 - 210493715781614816v^6 + 190414168430129598v^5 + 112399366880723048v^4 - 85593694947205243v^3 - 21163269474751630v^2 + 7883794302521828*v + 1154096244498282) / 458553592515710
\(\beta_{16}\)	\(=\)	\( ( - 9391136360704 \nu^{17} + 50244945428651 \nu^{16} + 210075419204663 \nu^{15} + \cdots + 77\!\cdots\!98 ) / 458553592515710 \) (-9391136360704v^17 + 50244945428651v^16 + 210075419204663v^15 - 1454010821724175v^14 - 1233669458612895v^13 + 15998314238608853v^12 - 2864276175886653v^11 - 85049020560744339v^10 + 50717689605234320v^9 + 232512391257686879v^8 - 166063921733273370v^7 - 334796739782735164v^6 + 209500531343457882v^5 + 246609280474745427v^4 - 91737962366393532v^3 - 81923515184631115v^2 + 5319702723550572*v + 7792387797974698) / 458553592515710
\(\beta_{17}\)	\(=\)	\( ( 12640662937253 \nu^{17} - 58978888205822 \nu^{16} - 314118940145836 \nu^{15} + \cdots - 35\!\cdots\!76 ) / 458553592515710 \) (12640662937253v^17 - 58978888205822v^16 - 314118940145836v^15 + 1709457619174555v^14 + 2619322482748470v^13 - 19023971479592081v^12 - 7119329077353169v^11 + 104188757710541728v^10 - 9864070375953565v^9 - 301513455598986783v^8 + 75280233838851000v^7 + 469539167775241043v^6 - 108718334901004314v^5 - 370935331497776184v^4 + 43409210372095999v^3 + 115966308977065780v^2 + 2532805914469796*v - 3579543371110476) / 458553592515710

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 4 \) b2 + 4
\(\nu^{3}\)	\(=\)	\( -\beta_{10} + \beta_{7} - \beta_{5} + \beta_{2} + 6\beta _1 + 1 \) -b10 + b7 - b5 + b2 + 6*b1 + 1
\(\nu^{4}\)	\(=\)	\( \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 29 \) b16 - b15 + b14 + b13 - b12 + b11 - b10 + b9 + b8 + b6 - 2b4 + 12b2 + b1 + 29
\(\nu^{5}\)	\(=\)	\( \beta_{17} - 2 \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{11} - 13 \beta_{10} + \cdots + 13 \) b17 - 2b15 + b14 + b13 + 2b12 + b11 - 13b10 + 2b9 + 18b7 - 14b5 - 3b4 - b3 + 18b2 + 44*b1 + 13
\(\nu^{6}\)	\(=\)	\( - \beta_{17} + 17 \beta_{16} - 19 \beta_{15} + 15 \beta_{14} + 17 \beta_{13} - 21 \beta_{12} + \cdots + 253 \) -b17 + 17b16 - 19b15 + 15b14 + 17b13 - 21b12 + 19b11 - 17b10 + 17b9 + 20b8 + 6b7 + 17b6 + b5 - 35b4 + 2b3 + 133b2 + 19*b1 + 253
\(\nu^{7}\)	\(=\)	\( 19 \beta_{17} + 7 \beta_{16} - 41 \beta_{15} + 20 \beta_{14} + 19 \beta_{13} + 31 \beta_{12} + \cdots + 160 \) 19b17 + 7b16 - 41b15 + 20b14 + 19b13 + 31b12 + 25b11 - 150b10 + 37b9 + 7b8 + 242b7 - 2b6 - 161b5 - 61b4 - 11b3 + 247b2 + 364*b1 + 160
\(\nu^{8}\)	\(=\)	\( - 13 \beta_{17} + 236 \beta_{16} - 265 \beta_{15} + 180 \beta_{14} + 215 \beta_{13} - 306 \beta_{12} + \cdots + 2414 \) -13b17 + 236b16 - 265b15 + 180b14 + 215b13 - 306b12 + 267b11 - 228b10 + 222b9 + 291b8 + 149b7 + 209b6 + 18b5 - 470b4 + 43b3 + 1469b2 + 270*b1 + 2414
\(\nu^{9}\)	\(=\)	\( 274 \beta_{17} + 193 \beta_{16} - 609 \beta_{15} + 274 \beta_{14} + 259 \beta_{13} + 329 \beta_{12} + \cdots + 1974 \) 274b17 + 193b16 - 609b15 + 274b14 + 259b13 + 329b12 + 409b11 - 1685b10 + 514b9 + 173b8 + 2979b7 - 64b6 - 1756b5 - 902b4 - 92b3 + 3101b2 + 3288*b1 + 1974
\(\nu^{10}\)	\(=\)	\( - 76 \beta_{17} + 3048 \beta_{16} - 3334 \beta_{15} + 2006 \beta_{14} + 2480 \beta_{13} - 3912 \beta_{12} + \cdots + 24303 \) -76b17 + 3048b16 - 3334b15 + 2006b14 + 2480b13 - 3912b12 + 3355b11 - 2848b10 + 2679b9 + 3709b8 + 2590b7 + 2266b6 + 199b5 - 5776b4 + 632b3 + 16332b2 + 3466*b1 + 24303
\(\nu^{11}\)	\(=\)	\( 3578 \beta_{17} + 3558 \beta_{16} - 8058 \beta_{15} + 3277 \beta_{14} + 3180 \beta_{13} + 2825 \beta_{12} + \cdots + 24322 \) 3578b17 + 3558b16 - 8058b15 + 3277b14 + 3180b13 + 2825b12 + 5701b11 - 18823b10 + 6502b9 + 2948b8 + 35336b7 - 1223b6 - 18821b5 - 11914b4 - 686b3 + 37529b2 + 31762*b1 + 24322
\(\nu^{12}\)	\(=\)	\( 464 \beta_{17} + 37845 \beta_{16} - 40134 \beta_{15} + 21658 \beta_{14} + 27593 \beta_{13} + \cdots + 253444 \) 464b17 + 37845b16 - 40134b15 + 21658b14 + 27593b13 - 47035b12 + 40019b11 - 34639b10 + 31338b9 + 44357b8 + 38946b7 + 23158b6 + 1428b5 - 68343b4 + 7983b3 + 182929b2 + 42487*b1 + 253444
\(\nu^{13}\)	\(=\)	\( 44485 \beta_{17} + 55141 \beta_{16} - 101140 \beta_{15} + 37005 \beta_{14} + 37584 \beta_{13} + \cdots + 298541 \) 44485b17 + 55141b16 - 101140b15 + 37005b14 + 37584b13 + 18965b12 + 73757b11 - 210549b10 + 79051b9 + 43284b8 + 411631b7 - 18604b6 - 200858b5 - 149602b4 - 4596b3 + 446755b2 + 322355*b1 + 298541
\(\nu^{14}\)	\(=\)	\( 22314 \beta_{17} + 458201 \beta_{16} - 472938 \beta_{15} + 230756 \beta_{14} + 302624 \beta_{13} + \cdots + 2707214 \) 22314b17 + 458201b16 - 472938b15 + 230756b14 + 302624b13 - 547414b12 + 465094b11 - 416557b10 + 361581b9 + 513597b8 + 542090b7 + 229803b6 + 565b5 - 794497b4 + 93600b3 + 2062252b2 + 509183*b1 + 2707214
\(\nu^{15}\)	\(=\)	\( 537550 \beta_{17} + 777568 \beta_{16} - 1235148 \beta_{15} + 408398 \beta_{14} + 438556 \beta_{13} + \cdots + 3648872 \) 537550b17 + 777568b16 - 1235148b15 + 408398b14 + 438556b13 + 61245b12 + 918056b11 - 2363460b10 + 942877b9 + 589024b8 + 4750220b7 - 249924b6 - 2146764b5 - 1830684b4 - 24991b3 + 5276966b2 + 3387715*b1 + 3648872
\(\nu^{16}\)	\(=\)	\( 446056 \beta_{17} + 5454344 \beta_{16} - 5512652 \beta_{15} + 2450347 \beta_{14} + 3304989 \beta_{13} + \cdots + 29405517 \) 446056b17 + 5454344b16 - 5512652b15 + 2450347b14 + 3304989b13 - 6256413b12 + 5335356b11 - 4981103b10 + 4146857b9 + 5849445b8 + 7195932b7 + 2247640b6 - 216425b5 - 9159712b4 + 1054841b3 + 23371435b2 + 6035579*b1 + 29405517
\(\nu^{17}\)	\(=\)	\( 6379968 \beta_{17} + 10353960 \beta_{16} - 14851037 \beta_{15} + 4482626 \beta_{14} + 5105978 \beta_{13} + \cdots + 44409364 \) 6379968b17 + 10353960b16 - 14851037b15 + 4482626b14 + 5105978b13 - 980262b12 + 11183711b11 - 26635216b10 + 11134974b9 + 7671146b8 + 54544947b7 - 3114422b6 - 23039936b5 - 22079443b4 - 49724b3 + 62086348b2 + 36471671*b1 + 44409364

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.18
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T + 1)^{18} \) (T + 1)^18
$3$	\( T^{18} + 6 T^{17} + \cdots + 624 \) T^18 + 6T^17 - 19T^16 - 169T^15 + 40T^14 + 1802T^13 + 1266T^12 - 9247T^11 - 10582T^10 + 24226T^9 + 33432T^8 - 32951T^7 - 49245T^6 + 22852T^5 + 33519T^4 - 7814T^3 - 9238T^2 + 1136*T + 624
$5$	\( (T + 1)^{18} \) (T + 1)^18
$7$	\( T^{18} + 6 T^{17} + \cdots - 524800 \) T^18 + 6T^17 - 63T^16 - 408T^15 + 1518T^14 + 11182T^13 - 16535T^12 - 157854T^11 + 57777T^10 + 1215862T^9 + 371934T^8 - 4958716T^7 - 3893832T^6 + 9223080T^5 + 11496656T^4 - 3580256T^3 - 10227488T^2 - 4522240*T - 524800
$11$	\( (T - 1)^{18} \) (T - 1)^18
$13$	\( T^{18} + \cdots - 172583880 \) T^18 + 8T^17 - 124T^16 - 1087T^15 + 5674T^14 + 58821T^13 - 108257T^12 - 1618889T^11 + 316006T^10 + 23864561T^9 + 17498921T^8 - 180613099T^7 - 224943459T^6 + 617326835T^5 + 830690075T^4 - 928892928T^3 - 736656822T^2 + 836431960*T - 172583880
$17$	\( T^{18} + 10 T^{17} + \cdots + 60418752 \) T^18 + 10T^17 - 128T^16 - 1373T^15 + 6109T^14 + 69694T^13 - 154169T^12 - 1649994T^11 + 2805582T^10 + 18798951T^9 - 38520105T^8 - 86552002T^7 + 266961088T^6 - 27478596T^5 - 462073776T^4 + 361040640T^3 + 127855152T^2 - 219946112*T + 60418752
$19$	\( T^{18} + \cdots - 13941911200 \) T^18 - 24T^17 + 37T^16 + 3517T^15 - 28021T^14 - 114736T^13 + 2190198T^12 - 4634490T^11 - 54481379T^10 + 317070151T^9 - 25926057T^8 - 4419542948T^7 + 12956702511T^6 - 4358611480T^5 - 35091956772T^4 + 40369680472T^3 + 23151536428T^2 - 30581238240*T - 13941911200
$23$	\( T^{18} + \cdots + 39058865408 \) T^18 + 14T^17 - 148T^16 - 2868T^15 + 3567T^14 + 218924T^13 + 428425T^12 - 7686252T^11 - 29758916T^10 + 119178266T^9 + 722250255T^8 - 483516896T^7 - 7585105760T^6 - 4182927860T^5 + 35901141536T^4 + 37489017664T^3 - 69395991504T^2 - 73038890368*T + 39058865408
$29$	\( T^{18} + \cdots + 180241745088 \) T^18 - 11T^17 - 352T^16 + 4000T^15 + 50323T^14 - 602156T^13 - 3716379T^12 + 48503659T^11 + 146694156T^10 - 2252437510T^9 - 2685243269T^8 + 60333906396T^7 + 1974625376T^6 - 873314513492T^5 + 619544313856T^4 + 5652926621232T^3 - 6079106357072T^2 - 7484975811776*T + 180241745088
$31$	\( T^{18} + \cdots - 14163675856896 \) T^18 - 32T^17 + 123T^16 + 6532T^15 - 77402T^14 - 252668T^13 + 8558724T^12 - 26090912T^11 - 339578080T^10 + 2378421488T^9 + 2613391648T^8 - 64136941056T^7 + 114777628224T^6 + 576472539520T^5 - 1917906651648T^4 - 1499494087680T^3 + 9441723218944T^2 - 432622100480*T - 14163675856896
$37$	\( T^{18} + \cdots + 410404544 \) T^18 + 3T^17 - 383T^16 - 1125T^15 + 58411T^14 + 177118T^13 - 4499803T^12 - 14689935T^11 + 182210563T^10 + 649804813T^9 - 3573293851T^8 - 13665046874T^7 + 25548196124T^6 + 93485958516T^5 - 53173561696T^4 - 59593934992T^3 + 13485870096T^2 + 5889480896*T + 410404544
$41$	\( T^{18} + \cdots - 261750610176 \) T^18 - 28T^17 + 63T^16 + 4670T^15 - 36659T^14 - 231988T^13 + 3216029T^12 + 458842T^11 - 112295854T^10 + 246361056T^9 + 1640176644T^8 - 5911516216T^7 - 9778498400T^6 + 55112499120T^5 + 10211303520T^4 - 227508687040T^3 + 102355288384T^2 + 344491283200*T - 261750610176
$43$	\( T^{18} + \cdots + 7913538708224 \) T^18 + 6T^17 - 418T^16 - 3065T^15 + 64817T^14 + 583121T^13 - 4442578T^12 - 53000350T^11 + 104839108T^10 + 2421438025T^9 + 2118235469T^8 - 52700934997T^7 - 135018298369T^6 + 457291974140T^5 + 1889101474060T^4 - 689890188480T^3 - 8445986314000T^2 - 3956622566016*T + 7913538708224
$47$	\( T^{18} + \cdots + 12420879360 \) T^18 + 16T^17 - 277T^16 - 5307T^15 + 21666T^14 + 646006T^13 + 205040T^12 - 35119120T^11 - 93224546T^10 + 809583197T^9 + 3849827987T^8 - 3914689376T^7 - 51780883144T^6 - 82899392288T^5 + 85960773104T^4 + 382383485888T^3 + 391400997376T^2 + 138822003200*T + 12420879360
$53$	\( T^{18} + \cdots + 2341113982464 \) T^18 + T^17 - 650T^16 - 172T^15 + 171393T^14 - 81840T^13 - 23261858T^12 + 27929600T^11 + 1689168372T^10 - 3133398328T^9 - 60765010632T^8 + 140814467936T^7 + 839747570656T^6 - 1834589820960T^5 - 3305153463552T^4 + 7257822028928T^3 + 2173499981184T^2 - 7005658890752T + 2341113982464
$59$	\( T^{18} + \cdots - 233947460856064 \) T^18 - 14T^17 - 602T^16 + 7869T^15 + 154263T^14 - 1794959T^13 - 21940417T^12 + 212152097T^11 + 1885661044T^10 - 13735173542T^9 - 99215593097T^8 + 472136455678T^7 + 3050891910818T^6 - 7591326254940T^5 - 47798157503772T^4 + 39093469765504T^3 + 267698309621040T^2 - 43254688668288*T - 233947460856064
$61$	\( T^{18} + \cdots + 2241677481600 \) T^18 - 39T^17 + 240T^16 + 7886T^15 - 96962T^14 - 537466T^13 + 10923004T^12 + 8836203T^11 - 587097557T^10 + 584603220T^9 + 16311354967T^8 - 30635304732T^7 - 221921158016T^6 + 526613163184T^5 + 1236857771292T^4 - 3111145310656T^3 - 2721285964176T^2 + 5608991967040*T + 2241677481600
$67$	\( T^{18} + \cdots + 11080034081904 \) T^18 - 4T^17 - 712T^16 + 2845T^15 + 197622T^14 - 781795T^13 - 27238093T^12 + 105670365T^11 + 1962652948T^10 - 7365479053T^9 - 71353433899T^8 + 251433826687T^7 + 1165016652101T^6 - 3540542188837T^5 - 7875305113987T^4 + 18132570325856T^3 + 14158448565834T^2 - 32118640667072*T + 11080034081904
$71$	\( T^{18} + \cdots + 300602579161088 \) T^18 - 29T^17 - 163T^16 + 11379T^15 - 35053T^14 - 1723093T^13 + 11040218T^12 + 127270595T^11 - 1130351789T^10 - 4690308505T^9 + 57548230442T^8 + 71708006975T^7 - 1575495659830T^6 + 215188480348T^5 + 22694570281853T^4 - 17593154698688T^3 - 151219217550720T^2 + 131981029818368*T + 300602579161088
$73$	\( (T + 1)^{18} \) (T + 1)^18
$79$	\( T^{18} + \cdots + 5397118976 \) T^18 - 38T^17 + 49T^16 + 13710T^15 - 124005T^14 - 1569116T^13 + 22606814T^12 + 49334736T^11 - 1583814880T^10 + 1968256856T^9 + 45506018112T^8 - 125322374816T^7 - 428948141856T^6 + 1551345363968T^5 - 172671465728T^4 - 659148831744T^3 + 49259881984T^2 + 72647122944*T + 5397118976
$83$	\( T^{18} + \cdots + 38\!\cdots\!36 \) T^18 - 4T^17 - 1053T^16 + 1207T^15 + 474606T^14 + 671168T^13 - 116559436T^12 - 423627855T^11 + 16524507250T^10 + 92635654050T^9 - 1321938308222T^8 - 10076413086231T^7 + 53073912032891T^6 + 568541994783132T^5 - 637812434702837T^4 - 15703287969927472T^3 - 16656470061068608T^2 + 165567768356209664*T + 381454654376084736
$89$	\( T^{18} + \cdots - 127389733163796 \) T^18 - 44T^17 + 219T^16 + 14511T^15 - 166444T^14 - 1844264T^13 + 29733865T^12 + 113306318T^11 - 2561356589T^10 - 3350335726T^9 + 120085833735T^8 + 29500951142T^7 - 3048822113036T^6 + 973291212556T^5 + 38003222923852T^4 - 32730725824619T^3 - 170613906968185T^2 + 299707334175496*T - 127389733163796
$97$	\( T^{18} + \cdots + 74\!\cdots\!16 \) T^18 - 3T^17 - 1234T^16 + 2608T^15 + 628305T^14 - 726110T^13 - 169993108T^12 + 25229983T^11 + 26153967155T^10 + 23117020134T^9 - 2261267223468T^8 - 4330188180676T^7 + 100495910816344T^6 + 294360495272928T^5 - 1767434837471888T^4 - 7147575221188608T^3 - 439870974394048T^2 + 14078604869679616*T + 7458432499823616
show more
show less

\( p \)	Sign
\(2\)	\(1\)
\(5\)	\(1\)
\(11\)	\(-1\)
\(73\)	\(1\)