LMFDB - Newform orbit 8030.2.a.bk

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} - 20 x^{16} + 171 x^{15} + 79 x^{14} - 1897 x^{13} + 800 x^{12} + 10423 x^{11} + \cdots - 224 $ x^18 - 6x^17 - 20x^16 + 171x^15 + 79x^14 - 1897x^13 + 800x^12 + 10423x^11 - 7942x^10 - 30513x^9 + 25074x^8 + 49703x^7 - 34959x^6 - 43987x^5 + 21684x^4 + 19388x^3 - 4543x^2 - 3292*x - 224
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_{5} + 1) q^{7} + q^{8} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) $ q + q^2 + b1 * q^3 + q^4 + q^5 + b1 * q^6 + (b5 + 1) * q^7 + q^8 + (b2 + b1 + 1) * q^9	$ q + q^{2} + \beta_1 q^{3} + q^{4} + q^{5} + \beta_1 q^{6} + (\beta_{5} + 1) q^{7} + q^{8} + (\beta_{2} + \beta_1 + 1) q^{9} + q^{10} - q^{11} + \beta_1 q^{12} + ( - \beta_{4} + 1) q^{13} + (\beta_{5} + 1) q^{14} + \beta_1 q^{15} + q^{16} + ( - \beta_{7} + 1) q^{17} + (\beta_{2} + \beta_1 + 1) q^{18} + (\beta_{9} + 1) q^{19} + q^{20} + (\beta_{13} - \beta_{11} + \cdots + \beta_1) q^{21}+ \cdots + ( - \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) $ q + q^2 + b1 * q^3 + q^4 + q^5 + b1 * q^6 + (b5 + 1) * q^7 + q^8 + (b2 + b1 + 1) * q^9 + q^10 - q^11 + b1 * q^12 + (-b4 + 1) * q^13 + (b5 + 1) * q^14 + b1 * q^15 + q^16 + (-b7 + 1) * q^17 + (b2 + b1 + 1) * q^18 + (b9 + 1) * q^19 + q^20 + (b13 - b11 + b10 + b8 - b6 + b1) * q^21 - q^22 + (b16 - b11 - b7) * q^23 + b1 * q^24 + q^25 + (-b4 + 1) * q^26 + (-b16 + b12 + b10 - b5 + b1 + 1) * q^27 + (b5 + 1) * q^28 + (-b17 + b16) * q^29 + b1 * q^30 + (b17 - b16 - b14 - b13 + b11 - b10 + b7 + b6 - b2 + 1) * q^31 + q^32 - b1 * q^33 + (-b7 + 1) * q^34 + (b5 + 1) * q^35 + (b2 + b1 + 1) * q^36 + (-b14 + b7 - b3 + 1) * q^37 + (b9 + 1) * q^38 + (-b17 + b16 + b15 + b13 + b10 + b8 - b3 + b2 + b1 + 2) * q^39 + q^40 + (-b16 - b14 - b13 + b11 + b7 - b5 - b2) * q^41 + (b13 - b11 + b10 + b8 - b6 + b1) * q^42 + (-b17 + b11 - b8 + b7 - b3 + 2) * q^43 - q^44 + (b2 + b1 + 1) * q^45 + (b16 - b11 - b7) * q^46 + (b16 - b15 + b14 + b13 - b12 - b11 - b10 - b7 - b4 + b3) * q^47 + b1 * q^48 + (-b16 + b15 + b14 + b13 + b12 + b10 + b5 + b4 - b3 + 3) * q^49 + q^50 + (b17 - b13 - b12 - b10 - b9 - b8 + b3 + b1) * q^51 + (-b4 + 1) * q^52 + (b17 - b15 + b14 - b13 - 2b10 - b7 + b5 + b4 + 2b3) * q^53 + (-b16 + b12 + b10 - b5 + b1 + 1) * q^54 - q^55 + (b5 + 1) * q^56 + (b17 - b16 + b14 - b13 + b11 - b10 - b8 + b7 + b6 + b5 + b4 + b3 - b2 + 2b1 - 1) q^57 + (-b17 + b16) * q^58 + (-b13 - b10 - b8 + b3 - b1 + 1) * q^59 + b1 * q^60 + (b17 - b15 + b14 - b12 - b10 - b9 - b8 - b7 + b6 + b5 + b3 + 1) * q^61 + (b17 - b16 - b14 - b13 + b11 - b10 + b7 + b6 - b2 + 1) * q^62 + (-b16 + b15 + b14 + 2b13 + 2b10 - b9 + b8 + b7 - b6 + b5 + 2b4 - b3 + 2b2 + b1 + 4) * q^63 + q^64 + (-b4 + 1) * q^65 - b1 * q^66 + (b16 - b13 - 2b12 + b11 - b10 - b9 + b7 - b6 + b3 + b2 - b1 + 1) q^67 + (-b7 + 1) * q^68 + (2b16 + b14 - b13 - b12 + b11 - b10 - b9 - 2b8 - b7 + b5 + b2 + 1) * q^69 + (b5 + 1) * q^70 + (-b16 - b15 + b14 + b13 - b7 - b4 + b3 - b2 - b1 + 1) * q^71 + (b2 + b1 + 1) * q^72 + q^73 + (-b14 + b7 - b3 + 1) * q^74 + b1 * q^75 + (b9 + 1) * q^76 + (-b5 - 1) * q^77 + (-b17 + b16 + b15 + b13 + b10 + b8 - b3 + b2 + b1 + 2) * q^78 + (b17 - 2b15 + b14 - b13 - 2b10 - b9 + b3 + b2 + 3) * q^79 + q^80 + (-b16 - b15 + b14 + b12 + b4 + b1 + 1) * q^81 + (-b16 - b14 - b13 + b11 + b7 - b5 - b2) * q^82 + (b17 - b16 + b14 - b13 + b11 - b8 - b7 + b6 + b5 + 2b4 + b3 + b1 + 2) q^83 + (b13 - b11 + b10 + b8 - b6 + b1) * q^84 + (-b7 + 1) * q^85 + (-b17 + b11 - b8 + b7 - b3 + 2) * q^86 + (b16 - b15 - 2b13 - 2b10 - b8 - b5 - b4 - b1) * q^87 - q^88 + (b17 + b16 - b15 + b14 - 2b12 - b11 - 2b10 + 2b5 + b3 + b2 - 1) q^89 + (b2 + b1 + 1) * q^90 + (-b17 + b15 + b13 + 2b12 - b11 + 2b10 + b8 - b7 - b6 - 2b3 - b2 - b1 + 3) q^91 + (b16 - b11 - b7) * q^92 + (-b17 - b16 + b15 - 3b14 + b13 + b11 + b10 + b9 + b8 + 2b7 + b6 - 2b5 - b4 - 2b3 - b2 + b1 + 1) * q^93 + (b16 - b15 + b14 + b13 - b12 - b11 - b10 - b7 - b4 + b3) * q^94 + (b9 + 1) * q^95 + b1 * q^96 + (-b17 + 3b15 - 2b14 + b11 + b10 + 3b7 + b6 + b4 - b3 - b2 + 1) q^97 + (-b16 + b15 + b14 + b13 + b12 + b10 + b5 + b4 - b3 + 3) * q^98 + (-b2 - b1 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{2} + 6 q^{3} + 18 q^{4} + 18 q^{5} + 6 q^{6} + 12 q^{7} + 18 q^{8} + 22 q^{9}+O(q^{10}) $ 18 * q + 18 * q^2 + 6 * q^3 + 18 * q^4 + 18 * q^5 + 6 * q^6 + 12 * q^7 + 18 * q^8 + 22 * q^9	\( 18 q + 18 q^{2} + 6 q^{3} + 18 q^{4} + 18 q^{5} + 6 q^{6} + 12 q^{7} + 18 q^{8} + 22 q^{9} + 18 q^{10} - 18 q^{11} + 6 q^{12} + 12 q^{13} + 12 q^{14} + 6 q^{15} + 18 q^{16} + 17 q^{17} + 22 q^{18} + 27 q^{19} + 18 q^{20} + 5 q^{21} - 18 q^{22} + 7 q^{23} + 6 q^{24} + 18 q^{25} + 12 q^{26} + 27 q^{27} + 12 q^{28} + 2 q^{29} + 6 q^{30} + 15 q^{31} + 18 q^{32} - 6 q^{33} + 17 q^{34} + 12 q^{35} + 22 q^{36} + 12 q^{37} + 27 q^{38} + 33 q^{39} + 18 q^{40} - 7 q^{41} + 5 q^{42} + 29 q^{43} - 18 q^{44} + 22 q^{45} + 7 q^{46} + 14 q^{47} + 6 q^{48} + 50 q^{49} + 18 q^{50} + q^{51} + 12 q^{52} + 19 q^{53} + 27 q^{54} - 18 q^{55} + 12 q^{56} + 3 q^{57} + 2 q^{58} + 18 q^{59} + 6 q^{60} + 19 q^{61} + 15 q^{62} + 59 q^{63} + 18 q^{64} + 12 q^{65} - 6 q^{66} - 6 q^{67} + 17 q^{68} + 11 q^{69} + 12 q^{70} + 17 q^{71} + 22 q^{72} + 18 q^{73} + 12 q^{74} + 6 q^{75} + 27 q^{76} - 12 q^{77} + 33 q^{78} + 60 q^{79} + 18 q^{80} + 34 q^{81} - 7 q^{82} + 50 q^{83} + 5 q^{84} + 17 q^{85} + 29 q^{86} - 18 q^{88} - 17 q^{89} + 22 q^{90} + 39 q^{91} + 7 q^{92} + 14 q^{93} + 14 q^{94} + 27 q^{95} + 6 q^{96} + 5 q^{97} + 50 q^{98} - 22 q^{99}+O(q^{100}) \) 18 * q + 18 * q^2 + 6 * q^3 + 18 * q^4 + 18 * q^5 + 6 * q^6 + 12 * q^7 + 18 * q^8 + 22 * q^9 + 18 * q^10 - 18 * q^11 + 6 * q^12 + 12 * q^13 + 12 * q^14 + 6 * q^15 + 18 * q^16 + 17 * q^17 + 22 * q^18 + 27 * q^19 + 18 * q^20 + 5 * q^21 - 18 * q^22 + 7 * q^23 + 6 * q^24 + 18 * q^25 + 12 * q^26 + 27 * q^27 + 12 * q^28 + 2 * q^29 + 6 * q^30 + 15 * q^31 + 18 * q^32 - 6 * q^33 + 17 * q^34 + 12 * q^35 + 22 * q^36 + 12 * q^37 + 27 * q^38 + 33 * q^39 + 18 * q^40 - 7 * q^41 + 5 * q^42 + 29 * q^43 - 18 * q^44 + 22 * q^45 + 7 * q^46 + 14 * q^47 + 6 * q^48 + 50 * q^49 + 18 * q^50 + q^51 + 12 * q^52 + 19 * q^53 + 27 * q^54 - 18 * q^55 + 12 * q^56 + 3 * q^57 + 2 * q^58 + 18 * q^59 + 6 * q^60 + 19 * q^61 + 15 * q^62 + 59 * q^63 + 18 * q^64 + 12 * q^65 - 6 * q^66 - 6 * q^67 + 17 * q^68 + 11 * q^69 + 12 * q^70 + 17 * q^71 + 22 * q^72 + 18 * q^73 + 12 * q^74 + 6 * q^75 + 27 * q^76 - 12 * q^77 + 33 * q^78 + 60 * q^79 + 18 * q^80 + 34 * q^81 - 7 * q^82 + 50 * q^83 + 5 * q^84 + 17 * q^85 + 29 * q^86 - 18 * q^88 - 17 * q^89 + 22 * q^90 + 39 * q^91 + 7 * q^92 + 14 * q^93 + 14 * q^94 + 27 * q^95 + 6 * q^96 + 5 * q^97 + 50 * q^98 - 22 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - 6 x^{17} - 20 x^{16} + 171 x^{15} + 79 x^{14} - 1897 x^{13} + 800 x^{12} + 10423 x^{11} + \cdots - 224 $ x^18 - 6*x^17 - 20*x^16 + 171*x^15 + 79*x^14 - 1897*x^13 + 800*x^12 + 10423*x^11 - 7942*x^10 - 30513*x^9 + 25074*x^8 + 49703*x^7 - 34959*x^6 - 43987*x^5 + 21684*x^4 + 19388*x^3 - 4543*x^2 - 3292*x - 224 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - \nu - 4 $ v^2 - v - 4
$\beta_{3}$	$=$	$ ( - 1196617721 \nu^{17} + 30359377824 \nu^{16} - 104343432452 \nu^{15} - 679711043955 \nu^{14} + \cdots + 2333730940764 ) / 740401001108 $ (-1196617721v^17 + 30359377824v^16 - 104343432452v^15 - 679711043955v^14 + 3431558705101v^13 + 4593369273769v^12 - 37921329986004v^11 - 1924463096147v^10 + 194418003841664v^9 - 84799100814417v^8 - 491641033155700v^7 + 274324522004777v^6 + 629357488658751v^5 - 295873912085091v^4 - 375300356668818v^3 + 99197222277010v^2 + 78416784339933*v + 2333730940764) / 740401001108
$\beta_{4}$	$=$	$ ( - 1597579710 \nu^{17} - 3642664769 \nu^{16} + 113447465521 \nu^{15} - 44033883253 \nu^{14} + \cdots - 926856105442 ) / 370200500554 $ (-1597579710v^17 - 3642664769v^16 + 113447465521v^15 - 44033883253v^14 - 2295415181824v^13 + 2748272401674v^12 + 20429427011097v^11 - 31958985457380v^10 - 88746872713406v^9 + 152117846663121v^8 + 191732494314002v^7 - 310635178559258v^6 - 228636323902555v^5 + 263566343532477v^4 + 148259545736427v^3 - 75053073070624v^2 - 38352701683688*v - 926856105442) / 370200500554
$\beta_{5}$	$=$	$ ( - 2112411542 \nu^{17} + 7345039349 \nu^{16} + 60606895279 \nu^{15} - 206536681061 \nu^{14} + \cdots - 1624240109598 ) / 370200500554 $ (-2112411542v^17 + 7345039349v^16 + 60606895279v^15 - 206536681061v^14 - 686047319621v^13 + 2221942318706v^12 + 3935638317620v^11 - 11391520664073v^10 - 12312923043601v^9 + 28448836764561v^8 + 20263297034798v^7 - 32686086969480v^6 - 10692605863994v^5 + 14327738705160v^4 - 4358351288236v^3 + 1687808013032v^2 + 3332787175017*v - 1624240109598) / 370200500554
$\beta_{6}$	$=$	$ ( 2525923078 \nu^{17} - 5490603097 \nu^{16} - 74065800282 \nu^{15} + 101551438228 \nu^{14} + \cdots - 3425929544914 ) / 370200500554 $ (2525923078v^17 - 5490603097v^16 - 74065800282v^15 + 101551438228v^14 + 934030477630v^13 - 265784153970v^12 - 6873126688980v^11 - 5113962234178v^10 + 31917576152587v^9 + 38217203544749v^8 - 83632298337541v^7 - 85700725504468v^6 + 86537189598044v^5 + 58673453165511v^4 - 18225435211157v^3 - 3776383270541v^2 - 8604078147039*v - 3425929544914) / 370200500554
$\beta_{7}$	$=$	$ ( - 11531076211 \nu^{17} + 86841377152 \nu^{16} + 110330240524 \nu^{15} - 2209122509287 \nu^{14} + \cdots - 2701913424648 ) / 740401001108 $ (-11531076211v^17 + 86841377152v^16 + 110330240524v^15 - 2209122509287v^14 + 2197354459301v^13 + 20368512950237v^12 - 38686056642704v^11 - 79995247403097v^10 + 214254383734802v^9 + 114520598170765v^8 - 497877476212984v^7 - 13303957170899v^6 + 511406363468605v^5 - 79124153940469v^4 - 205345303936768v^3 + 42126470941756v^2 + 18001353308241*v - 2701913424648) / 740401001108
$\beta_{8}$	$=$	$ ( 5912542223 \nu^{17} - 36172904739 \nu^{16} - 101462522256 \nu^{15} + 939150760888 \nu^{14} + \cdots + 4393984359332 ) / 370200500554 $ (5912542223v^17 - 36172904739v^16 - 101462522256v^15 + 939150760888v^14 + 161231412628v^13 - 9030598189030v^12 + 5518450873948v^11 + 39418326736791v^10 - 32174068672612v^9 - 80470903905306v^8 + 40164075432279v^7 + 90357968890278v^6 + 35802858657415v^5 - 57819893544688v^4 - 80236035832764v^3 + 10052157695564v^2 + 31280139926105*v + 4393984359332) / 370200500554
$\beta_{9}$	$=$	$ ( 12872643469 \nu^{17} - 81590528270 \nu^{16} - 176136654710 \nu^{15} + 2043980178961 \nu^{14} + \cdots - 7959434299208 ) / 740401001108 $ (12872643469v^17 - 81590528270v^16 - 176136654710v^15 + 2043980178961v^14 - 1068358614811v^13 - 18135719019665v^12 + 29979790286684v^11 + 62891399429393v^10 - 185468599242570v^9 - 35539969498285v^8 + 483246566015676v^7 - 189620931949213v^6 - 619663073380783v^5 + 302197266818659v^4 + 379887798704466v^3 - 109851993398338v^2 - 84864001775635*v - 7959434299208) / 740401001108
$\beta_{10}$	$=$	$ ( - 7368792221 \nu^{17} + 36032856239 \nu^{16} + 190778072729 \nu^{15} - 1071242030927 \nu^{14} + \cdots - 5171470402916 ) / 370200500554 $ (-7368792221v^17 + 36032856239v^16 + 190778072729v^15 - 1071242030927v^14 - 1813871052343v^13 + 12543888417156v^12 + 7691377412900v^11 - 73588600546308v^10 - 15482463378540v^9 + 229133287576338v^8 + 28278704278146v^7 - 372560927987140v^6 - 73745518814393v^5 + 278303376942237v^4 + 88730894435189v^3 - 66673825169979v^2 - 32768558308519*v - 5171470402916) / 370200500554
$\beta_{11}$	$=$	$ ( - 21055108899 \nu^{17} + 133731830584 \nu^{16} + 355648919466 \nu^{15} + \cdots + 9743683016604 ) / 740401001108 $ (-21055108899v^17 + 133731830584v^16 + 355648919466v^15 - 3605369321091v^14 - 105115230565v^13 + 36808766678391v^12 - 29395736837166v^11 - 177925358841801v^10 + 200464961966626v^9 + 431159938833053v^8 - 498087869992494v^7 - 572467851918489v^6 + 532455028203587v^5 + 411887543338771v^4 - 240027877267986v^3 - 133197435928964v^2 + 34489244582237*v + 9743683016604) / 740401001108
$\beta_{12}$	$=$	$ ( - 21893357459 \nu^{17} + 132283003946 \nu^{16} + 388113009668 \nu^{15} + \cdots + 3949574073272 ) / 740401001108 $ (-21893357459v^17 + 132283003946v^16 + 388113009668v^15 - 3540814213357v^14 - 566374260951v^13 + 35586205958015v^12 - 26608557701118v^11 - 165658011283115v^10 + 196077973138472v^9 + 363943274570195v^8 - 519444270595486v^7 - 384557389333353v^6 + 609264737893023v^5 + 190681808244883v^4 - 294033969050270v^3 - 39800976708996v^2 + 35718200056539*v + 3949574073272) / 740401001108
$\beta_{13}$	$=$	$ ( - 23749622553 \nu^{17} + 159748050268 \nu^{16} + 338257603198 \nu^{15} + \cdots + 3500435643128 ) / 740401001108 $ (-23749622553v^17 + 159748050268v^16 + 338257603198v^15 - 4176417520163v^14 + 1497620251189v^13 + 40501753016639v^12 - 48309357112436v^11 - 177992126711453v^10 + 287598825141134v^9 + 356729386658299v^8 - 657623818283334v^7 - 348544186655245v^6 + 624232912128043v^5 + 171161844051679v^4 - 208181978916494v^3 - 40034664441794v^2 + 3101327241263*v + 3500435643128) / 740401001108
$\beta_{14}$	$=$	$ ( 24869169061 \nu^{17} - 117005657678 \nu^{16} - 624027602860 \nu^{15} + 3362151234965 \nu^{14} + \cdots + 4596964704100 ) / 740401001108 $ (24869169061v^17 - 117005657678v^16 - 624027602860v^15 + 3362151234965v^14 + 5593082987259v^13 - 37586049707303v^12 - 20457399399566v^11 + 207004495417565v^10 + 22610482495938v^9 - 594608807451241v^8 + 2417802109494v^7 + 892330180929937v^6 + 37363968000639v^5 - 634692500203877v^4 - 94163873982724v^3 + 158455481262598v^2 + 44875794924525*v + 4596964704100) / 740401001108
$\beta_{15}$	$=$	$ ( 32186770999 \nu^{17} - 181666620996 \nu^{16} - 657475026718 \nu^{15} + 5003494168191 \nu^{14} + \cdots + 3174104069880 ) / 740401001108 $ (32186770999v^17 - 181666620996v^16 - 657475026718v^15 + 5003494168191v^14 + 3257900089055v^13 - 52733397100855v^12 + 12889976432068v^11 + 267480684267275v^10 - 148544182260996v^9 - 691742015748553v^8 + 369851976250802v^7 + 950809505846741v^6 - 293802853903673v^5 - 636251490614185v^4 + 16917127044388v^3 + 151736210497344v^2 + 35930676517573*v + 3174104069880) / 740401001108
$\beta_{16}$	$=$	$ ( - 32406118817 \nu^{17} + 189658637726 \nu^{16} + 648455364568 \nu^{15} + \cdots - 2404485512256 ) / 740401001108 $ (-32406118817v^17 + 189658637726v^16 + 648455364568v^15 - 5270224913089v^14 - 2822021726395v^13 + 56230098154915v^12 - 19097079510558v^11 - 290052171047585v^10 + 189738892468594v^9 + 765312176193749v^8 - 503413456108790v^7 - 1064307071368673v^6 + 483158911992225v^5 + 718633084719037v^4 - 108595878604528v^3 - 176524243075018v^2 - 31301683902777*v - 2404485512256) / 740401001108
$\beta_{17}$	$=$	$ ( - 26070569352 \nu^{17} + 150046167408 \nu^{16} + 535258484380 \nu^{15} - 4183016265326 \nu^{14} + \cdots - 907583269738 ) / 370200500554 $ (-26070569352v^17 + 150046167408v^16 + 535258484380v^15 - 4183016265326v^14 - 2684005993537v^13 + 44923323405688v^12 - 10395148955663v^11 - 234924699777551v^10 + 122902130674273v^9 + 638001915592530v^8 - 312560027015514v^7 - 934872209172284v^6 + 247466685450651v^5 + 678093622686427v^4 + 2135474045373v^3 - 183871932928318v^2 - 43387939635667*v - 907583269738) / 370200500554

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + \beta _1 + 4 $ b2 + b1 + 4
$\nu^{3}$	$=$	$ -\beta_{16} + \beta_{12} + \beta_{10} - \beta_{5} + 7\beta _1 + 1 $ -b16 + b12 + b10 - b5 + 7*b1 + 1
$\nu^{4}$	$=$	$ -\beta_{16} - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{4} + 9\beta_{2} + 10\beta _1 + 28 $ -b16 - b15 + b14 + b12 + b4 + 9b2 + 10b1 + 28
$\nu^{5}$	$=$	$ - 11 \beta_{16} - 3 \beta_{15} + 2 \beta_{14} + 11 \beta_{12} - \beta_{11} + 10 \beta_{10} + \beta_{9} + \cdots + 12 $ -11b16 - 3b15 + 2b14 + 11b12 - b11 + 10b10 + b9 + b8 - b6 - 10b5 + b4 + b3 + 2b2 + 57b1 + 12
$\nu^{6}$	$=$	$ - 17 \beta_{16} - 15 \beta_{15} + 16 \beta_{14} + 16 \beta_{12} + 4 \beta_{10} + \beta_{9} + \beta_{8} + \cdots + 224 $ -17b16 - 15b15 + 16b14 + 16b12 + 4b10 + b9 + b8 + 2b7 - b6 + b5 + 15b4 + b3 + 77b2 + 96*b1 + 224
$\nu^{7}$	$=$	$ 2 \beta_{17} - 111 \beta_{16} - 49 \beta_{15} + 37 \beta_{14} + 6 \beta_{13} + 110 \beta_{12} + \cdots + 140 $ 2b17 - 111b16 - 49b15 + 37b14 + 6b13 + 110b12 - 20b11 + 94b10 + 17b9 + 20b8 - 2b7 - 17b6 - 86b5 + 18b4 + 16b3 + 39b2 + 493*b1 + 140
$\nu^{8}$	$=$	$ 6 \beta_{17} - 221 \beta_{16} - 173 \beta_{15} + 202 \beta_{14} + 13 \beta_{13} + 196 \beta_{12} + \cdots + 1892 $ 6b17 - 221b16 - 173b15 + 202b14 + 13b13 + 196b12 - 8b11 + 86b10 + 18b9 + 29b8 + 33b7 - 24b6 + 21b5 + 177b4 + 20b3 + 672b2 + 942*b1 + 1892
$\nu^{9}$	$=$	$ 53 \beta_{17} - 1116 \beta_{16} - 595 \beta_{15} + 509 \beta_{14} + 143 \beta_{13} + 1074 \beta_{12} + \cdots + 1636 $ 53b17 - 1116b16 - 595b15 + 509b14 + 143b13 + 1074b12 - 283b11 + 909b10 + 211b9 + 291b8 - 42b7 - 223b6 - 696b5 + 243b4 + 201b3 + 551b2 + 4426*b1 + 1636
$\nu^{10}$	$=$	$ 169 \beta_{17} - 2608 \beta_{16} - 1837 \beta_{15} + 2350 \beta_{14} + 344 \beta_{13} + 2178 \beta_{12} + \cdots + 16558 $ 169b17 - 2608b16 - 1837b15 + 2350b14 + 344b13 + 2178b12 - 228b11 + 1289b10 + 244b9 + 523b8 + 376b7 - 390b6 + 349b5 + 1923b4 + 309b3 + 6004b2 + 9397*b1 + 16558
$\nu^{11}$	$=$	$ 937 \beta_{17} - 11313 \beta_{16} - 6496 \beta_{15} + 6281 \beta_{14} + 2336 \beta_{13} + 10395 \beta_{12} + \cdots + 18748 $ 937b17 - 11313b16 - 6496b15 + 6281b14 + 2336b13 + 10395b12 - 3516b11 + 9060b10 + 2336b9 + 3730b8 - 602b7 - 2673b6 - 5308b5 + 2988b4 + 2351b3 + 6828b2 + 40836*b1 + 18748
$\nu^{12}$	$=$	$ 3089 \beta_{17} - 29405 \beta_{16} - 18879 \beta_{15} + 26328 \beta_{14} + 6033 \beta_{13} + \cdots + 148967 $ 3089b17 - 29405b16 - 18879b15 + 26328b14 + 6033b13 + 23117b12 - 4181b11 + 16747b10 + 3006b9 + 7739b8 + 3652b7 - 5424b6 + 5162b5 + 20186b4 + 4313b3 + 54776b2 + 94534*b1 + 148967
$\nu^{13}$	$=$	$ 13911 \beta_{17} - 115784 \beta_{16} - 67594 \beta_{15} + 73523 \beta_{14} + 32563 \beta_{13} + \cdots + 209539 $ 13911b17 - 115784b16 - 67594b15 + 73523b14 + 32563b13 + 100424b12 - 41050b11 + 92293b10 + 24556b9 + 44781b8 - 7430b7 - 30697b6 - 37056b5 + 35218b4 + 26762b3 + 78950b2 + 384714*b1 + 209539
$\nu^{14}$	$=$	$ 46651 \beta_{17} - 323437 \beta_{16} - 191457 \beta_{15} + 289208 \beta_{14} + 88567 \beta_{13} + \cdots + 1370719 $ 46651b17 - 323437b16 - 191457b15 + 289208b14 + 88567b13 + 239538b12 - 63154b11 + 202282b10 + 35396b9 + 103066b8 + 32028b7 - 69509b6 + 70656b5 + 208900b4 + 56563b3 + 508915b2 + 954643*b1 + 1370719
$\nu^{15}$	$=$	$ 187479 \beta_{17} - 1194832 \beta_{16} - 687120 \beta_{15} + 835409 \beta_{14} + 416826 \beta_{13} + \cdots + 2290611 $ 187479b17 - 1194832b16 - 687120b15 + 835409b14 + 416826b13 + 971938b12 - 463596b11 + 953813b10 + 251727b9 + 517536b8 - 85426b7 - 344307b6 - 217224b5 + 404916b4 + 300764b3 + 876154b2 + 3683505*b1 + 2290611
$\nu^{16}$	$=$	$ 633631 \beta_{17} - 3505574 \beta_{16} - 1932646 \beta_{15} + 3141707 \beta_{14} + 1179156 \beta_{13} + \cdots + 12850089 $ 633631b17 - 3505574b16 - 1932646b15 + 3141707b14 + 1179156b13 + 2449786b12 - 856675b11 + 2341972b10 + 405676b9 + 1287720b8 + 255931b7 - 847294b6 + 915716b5 + 2151523b4 + 709001b3 + 4803934b2 + 9657046*b1 + 12850089
$\nu^{17}$	$=$	$ 2377642 \beta_{17} - 12408386 \beta_{16} - 6907767 \beta_{15} + 9317906 \beta_{14} + 5066246 \beta_{13} + \cdots + 24609081 $ 2377642b17 - 12408386b16 - 6907767b15 + 9317906b14 + 5066246b13 + 9442297b12 - 5137044b11 + 9948089b10 + 2550344b9 + 5840448b8 - 948245b7 - 3807120b6 - 711853b5 + 4575057b4 + 3353767b3 + 9475359b2 + 35720852*b1 + 24609081

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

−2.86101
−2.74468
−2.64792
−1.32681
−1.30717
−0.963783
−0.806903
−0.355341
−0.0793867
0.794907
0.863008
1.06366
2.27290
2.33066
2.43998
3.03856
3.04979
3.23953

1.00000

−2.86101

1.00000

−2.86101

4.73806

1.00000

5.18537

1.00000

1.2

1.00000

−2.74468

1.00000

−2.74468

1.95232

1.00000

4.53326

1.00000

1.3

1.00000

−2.64792

1.00000

−2.64792

−1.05998

1.00000

4.01147

1.00000

1.4

1.00000

−1.32681

1.00000

−1.32681

−2.12605

1.00000

−1.23957

1.00000

1.5

1.00000

−1.30717

1.00000

−1.30717

−1.36400

1.00000

−1.29131

1.00000

1.6

1.00000

−0.963783

1.00000

−0.963783

4.37868

1.00000

−2.07112

1.00000

1.7

1.00000

−0.806903

1.00000

−0.806903

2.79489

1.00000

−2.34891

1.00000

1.8

1.00000

−0.355341

1.00000

−0.355341

−4.89733

1.00000

−2.87373

1.00000

1.9

1.00000

−0.0793867

1.00000

−0.0793867

−4.06592

1.00000

−2.99370

1.00000

1.10

1.00000

0.794907

1.00000

0.794907

1.78277

1.00000

−2.36812

1.00000

1.11

1.00000

0.863008

1.00000

0.863008

1.45490

1.00000

−2.25522

1.00000

1.12

1.00000

1.06366

1.00000

1.06366

3.29008

1.00000

−1.86864

1.00000

1.13

1.00000

2.27290

1.00000

2.27290

3.58623

1.00000

2.16610

1.00000

1.14

1.00000

2.33066

1.00000

2.33066

−3.11416

1.00000

2.43196

1.00000

1.15

1.00000

2.43998

1.00000

2.43998

2.17573

1.00000

2.95349

1.00000

1.16

1.00000

3.03856

1.00000

3.03856

−2.20328

1.00000

6.23286

1.00000

1.17

1.00000

3.04979

1.00000

3.04979

−0.448123

1.00000

6.30123

1.00000

1.18

1.00000

3.23953

1.00000

3.23953

5.12519

1.00000

7.49458

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.bk	✓	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8030.2.a.bk	✓	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} - 20 T_{3}^{16} + 171 T_{3}^{15} + 79 T_{3}^{14} - 1897 T_{3}^{13} + \cdots - 224 $

$ T_{7}^{18} - 12 T_{7}^{17} - 16 T_{7}^{16} + 696 T_{7}^{15} - 1473 T_{7}^{14} - 14090 T_{7}^{13} + \cdots + 7270400 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T - 1)^{18} $ (T - 1)^18
$3$	$ T^{18} - 6 T^{17} + \cdots - 224 $ T^18 - 6T^17 - 20T^16 + 171T^15 + 79T^14 - 1897T^13 + 800T^12 + 10423T^11 - 7942T^10 - 30513T^9 + 25074T^8 + 49703T^7 - 34959T^6 - 43987T^5 + 21684T^4 + 19388T^3 - 4543T^2 - 3292*T - 224
$5$	$ (T - 1)^{18} $ (T - 1)^18
$7$	$ T^{18} - 12 T^{17} + \cdots + 7270400 $ T^18 - 12T^17 - 16T^16 + 696T^15 - 1473T^14 - 14090T^13 + 53117T^12 + 114078T^11 - 697932T^10 - 191382T^9 + 4426359T^8 - 2173750T^7 - 14437312T^6 + 12047436T^5 + 24018872T^4 - 21785600T^3 - 19860144T^2 + 13147360*T + 7270400
$11$	$ (T + 1)^{18} $ (T + 1)^18
$13$	$ T^{18} - 12 T^{17} + \cdots - 1241728 $ T^18 - 12T^17 - 56T^16 + 1047T^15 + 178T^14 - 33311T^13 + 37267T^12 + 484425T^11 - 781644T^10 - 3382359T^9 + 6633033T^8 + 10787151T^7 - 26679981T^6 - 10769485T^5 + 48329859T^4 - 10305992T^3 - 31034708T^2 + 17947200*T - 1241728
$17$	$ T^{18} - 17 T^{17} + \cdots - 2025312 $ T^18 - 17T^17 - 6T^16 + 1411T^15 - 4416T^14 - 38201T^13 + 171301T^12 + 382921T^11 - 2126174T^10 - 1724261T^9 + 11177958T^8 + 5149247T^7 - 28768756T^6 - 12598940T^5 + 35578052T^4 + 19577016T^3 - 15524256T^2 - 12190224*T - 2025312
$19$	$ T^{18} + \cdots - 307765264 $ T^18 - 27T^17 + 169T^16 + 1562T^15 - 22638T^14 + 38697T^13 + 579082T^12 - 2302616T^11 - 5713095T^10 + 34809672T^9 + 32486956T^8 - 256261759T^7 - 194746849T^6 + 972934375T^5 + 1106999146T^4 - 1279122488T^3 - 2762060624T^2 - 1601400540*T - 307765264
$23$	$ T^{18} - 7 T^{17} + \cdots + 36550656 $ T^18 - 7T^17 - 189T^16 + 1405T^15 + 12562T^14 - 103358T^13 - 350065T^12 + 3547089T^11 + 3354371T^10 - 60268581T^9 + 18972948T^8 + 477217196T^7 - 471680428T^6 - 1348333696T^5 + 1703742256T^4 + 320838384T^3 - 617531904T^2 - 45239040*T + 36550656
$29$	$ T^{18} + \cdots - 430122720 $ T^18 - 2T^17 - 191T^16 + 186T^15 + 14425T^14 - 465T^13 - 549123T^12 - 425380T^11 + 11050339T^10 + 15561358T^9 - 113399181T^8 - 210641057T^7 + 546917646T^6 + 1203047152T^5 - 1021319972T^4 - 2586968376T^3 + 607047792T^2 + 1489595568*T - 430122720
$31$	$ T^{18} + \cdots + 1193156608 $ T^18 - 15T^17 - 207T^16 + 3717T^15 + 13982T^14 - 348974T^13 - 283884T^12 + 15663364T^11 - 3543200T^10 - 353945888T^9 + 160225840T^8 + 3881584416T^7 - 1577330432T^6 - 17442195136T^5 + 6395207552T^4 + 16894742528T^3 - 7170315264T^2 - 2715993088*T + 1193156608
$37$	$ T^{18} + \cdots - 4092275872 $ T^18 - 12T^17 - 146T^16 + 1908T^15 + 8098T^14 - 116915T^13 - 223755T^12 + 3618624T^11 + 3338942T^10 - 61820728T^9 - 27404732T^8 + 591543167T^7 + 134542560T^6 - 3045848208T^5 - 604882508T^4 + 7606332184T^3 + 2548289104T^2 - 7083310448*T - 4092275872
$41$	$ T^{18} + \cdots + 3278041529856 $ T^18 + 7T^17 - 318T^16 - 2352T^15 + 38645T^14 + 304443T^13 - 2332464T^12 - 20329334T^11 + 72677304T^10 + 768571556T^9 - 976163632T^8 - 16654842688T^7 - 2756673936T^6 + 195986962304T^5 + 218048516032T^4 - 1058811871296T^3 - 1873902903168T^2 + 1428667848960*T + 3278041529856
$43$	$ T^{18} + \cdots + 141637684480 $ T^18 - 29T^17 + 9T^16 + 6552T^15 - 36967T^14 - 583958T^13 + 4632682T^12 + 26966794T^11 - 254692006T^10 - 734062539T^9 + 7311433618T^8 + 13190432479T^7 - 109208563716T^6 - 160165027968T^5 + 720902425104T^4 + 978772400640T^3 - 959395110400T^2 - 190614732864*T + 141637684480
$47$	$ T^{18} + \cdots + 128855212032 $ T^18 - 14T^17 - 371T^16 + 5351T^15 + 51008T^14 - 784520T^13 - 2999598T^12 + 55012216T^11 + 50538966T^10 - 1872219815T^9 + 1391952487T^8 + 27875523272T^7 - 41359708544T^6 - 167670529520T^5 + 270117200144T^4 + 394802549760T^3 - 457015290624T^2 - 363630741504*T + 128855212032
$53$	$ T^{18} + \cdots - 19970297401344 $ T^18 - 19T^17 - 337T^16 + 7817T^15 + 36179T^14 - 1264966T^13 - 537421T^12 + 103479846T^11 - 164602532T^10 - 4587590656T^9 + 13006898932T^8 + 108335239680T^7 - 423622148224T^6 - 1150774195280T^5 + 6501526065536T^4 + 730930198080T^3 - 37578660262848T^2 + 53715426804864*T - 19970297401344
$59$	$ T^{18} + \cdots - 364754978304 $ T^18 - 18T^17 - 170T^16 + 4579T^15 + 1917T^14 - 434473T^13 + 1206337T^12 + 18590077T^11 - 89557410T^10 - 329309380T^9 + 2515276307T^8 + 791521920T^7 - 30912617614T^6 + 39486920468T^5 + 138755546468T^4 - 353949948024T^3 + 33746975520T^2 + 515742211968*T - 364754978304
$61$	$ T^{18} + \cdots + 15405370675232 $ T^18 - 19T^17 - 347T^16 + 8147T^15 + 33964T^14 - 1308430T^13 + 95716T^12 + 103710615T^11 - 203025765T^10 - 4378350675T^9 + 12569503712T^8 + 98483600496T^7 - 312579040227T^6 - 1142061237944T^5 + 3312322271732T^4 + 6264785384900T^3 - 12763269158820T^2 - 9520881937928*T + 15405370675232
$67$	$ T^{18} + \cdots + 272942978988544 $ T^18 + 6T^17 - 732T^16 - 4051T^15 + 215506T^14 + 1095005T^13 - 32660093T^12 - 153183007T^11 + 2708269882T^10 + 11998851055T^9 - 121216922587T^8 - 531435593085T^7 + 2735251558959T^6 + 12594589801957T^5 - 25195453065695T^4 - 136983219261732T^3 + 24649497710064T^2 + 461869224361152*T + 272942978988544
$71$	$ T^{18} + \cdots - 98\!\cdots\!24 $ T^18 - 17T^17 - 574T^16 + 10174T^15 + 138460T^14 - 2547804T^13 - 18575823T^12 + 348109766T^11 + 1559009749T^10 - 28346330132T^9 - 87783494067T^8 + 1406223938126T^7 + 3437507667240T^6 - 41383887751567T^5 - 91309026000133T^4 + 657024321397692T^3 + 1447896249669609T^2 - 4262105773892892*T - 9837315231078624
$73$	$ (T - 1)^{18} $ (T - 1)^18
$79$	$ T^{18} + \cdots - 11\!\cdots\!80 $ T^18 - 60T^17 + 849T^16 + 16112T^15 - 501035T^14 + 447028T^13 + 93908280T^12 - 588865676T^11 - 8125984608T^10 + 80837426752T^9 + 323592191664T^8 - 4907237861536T^7 - 3880666084032T^6 + 141853355848320T^5 - 59254607843072T^4 - 1711475709268480T^3 + 826538487573504T^2 + 6270815031064576*T - 1195949870612480
$83$	$ T^{18} + \cdots - 28\!\cdots\!40 $ T^18 - 50T^17 + 275T^16 + 25083T^15 - 422970T^14 - 3073454T^13 + 115293032T^12 - 278075399T^11 - 12723853402T^10 + 88966771040T^9 + 555907611126T^8 - 6982522324251T^7 - 1687434649041T^6 + 221947337370080T^5 - 451534076911873T^4 - 2680619825671188T^3 + 8798371270334268T^2 + 5637353333513472*T - 28262990326587840
$89$	$ T^{18} + \cdots - 613008689833050 $ T^18 + 17T^17 - 663T^16 - 12230T^15 + 157393T^14 + 3397158T^13 - 15138279T^12 - 463637901T^11 + 247296679T^10 + 33057643409T^9 + 54658237923T^8 - 1197820015991T^7 - 3761261665076T^6 + 18526880779720T^5 + 88682163940340T^4 - 30530091205239T^3 - 667933441577424T^2 - 1151089507267395*T - 613008689833050
$97$	$ T^{18} + \cdots + 15\!\cdots\!56 $ T^18 - 5T^17 - 1150T^16 + 4686T^15 + 536497T^14 - 1661404T^13 - 130780866T^12 + 281540251T^11 + 17869085435T^10 - 24205378252T^9 - 1369126488348T^8 + 1112367777700T^7 + 56360407184704T^6 - 31144783062720T^5 - 1112077230484016T^4 + 671704839496736T^3 + 7579449007097344T^2 - 9397091622854656*T + 1582105883181056
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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 $\F_{q}$

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	8030.2.a.bk

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:	\( x^{18} - 6 x^{17} - 20 x^{16} + 171 x^{15} + 79 x^{14} - 1897 x^{13} + 800 x^{12} + 10423 x^{11} + \cdots - 224 \) x^18 - 6x^17 - 20x^16 + 171x^15 + 79x^14 - 1897x^13 + 800x^12 + 10423x^11 - 7942x^10 - 30513x^9 + 25074x^8 + 49703x^7 - 34959x^6 - 43987x^5 + 21684x^4 + 19388x^3 - 4543x^2 - 3292*x - 224
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
\(\beta_{2}\)	\(=\)	\( \nu^{2} - \nu - 4 \) v^2 - v - 4
\(\beta_{3}\)	\(=\)	\( ( - 1196617721 \nu^{17} + 30359377824 \nu^{16} - 104343432452 \nu^{15} - 679711043955 \nu^{14} + \cdots + 2333730940764 ) / 740401001108 \) (-1196617721v^17 + 30359377824v^16 - 104343432452v^15 - 679711043955v^14 + 3431558705101v^13 + 4593369273769v^12 - 37921329986004v^11 - 1924463096147v^10 + 194418003841664v^9 - 84799100814417v^8 - 491641033155700v^7 + 274324522004777v^6 + 629357488658751v^5 - 295873912085091v^4 - 375300356668818v^3 + 99197222277010v^2 + 78416784339933*v + 2333730940764) / 740401001108
\(\beta_{4}\)	\(=\)	\( ( - 1597579710 \nu^{17} - 3642664769 \nu^{16} + 113447465521 \nu^{15} - 44033883253 \nu^{14} + \cdots - 926856105442 ) / 370200500554 \) (-1597579710v^17 - 3642664769v^16 + 113447465521v^15 - 44033883253v^14 - 2295415181824v^13 + 2748272401674v^12 + 20429427011097v^11 - 31958985457380v^10 - 88746872713406v^9 + 152117846663121v^8 + 191732494314002v^7 - 310635178559258v^6 - 228636323902555v^5 + 263566343532477v^4 + 148259545736427v^3 - 75053073070624v^2 - 38352701683688*v - 926856105442) / 370200500554
\(\beta_{5}\)	\(=\)	\( ( - 2112411542 \nu^{17} + 7345039349 \nu^{16} + 60606895279 \nu^{15} - 206536681061 \nu^{14} + \cdots - 1624240109598 ) / 370200500554 \) (-2112411542v^17 + 7345039349v^16 + 60606895279v^15 - 206536681061v^14 - 686047319621v^13 + 2221942318706v^12 + 3935638317620v^11 - 11391520664073v^10 - 12312923043601v^9 + 28448836764561v^8 + 20263297034798v^7 - 32686086969480v^6 - 10692605863994v^5 + 14327738705160v^4 - 4358351288236v^3 + 1687808013032v^2 + 3332787175017*v - 1624240109598) / 370200500554
\(\beta_{6}\)	\(=\)	\( ( 2525923078 \nu^{17} - 5490603097 \nu^{16} - 74065800282 \nu^{15} + 101551438228 \nu^{14} + \cdots - 3425929544914 ) / 370200500554 \) (2525923078v^17 - 5490603097v^16 - 74065800282v^15 + 101551438228v^14 + 934030477630v^13 - 265784153970v^12 - 6873126688980v^11 - 5113962234178v^10 + 31917576152587v^9 + 38217203544749v^8 - 83632298337541v^7 - 85700725504468v^6 + 86537189598044v^5 + 58673453165511v^4 - 18225435211157v^3 - 3776383270541v^2 - 8604078147039*v - 3425929544914) / 370200500554
\(\beta_{7}\)	\(=\)	\( ( - 11531076211 \nu^{17} + 86841377152 \nu^{16} + 110330240524 \nu^{15} - 2209122509287 \nu^{14} + \cdots - 2701913424648 ) / 740401001108 \) (-11531076211v^17 + 86841377152v^16 + 110330240524v^15 - 2209122509287v^14 + 2197354459301v^13 + 20368512950237v^12 - 38686056642704v^11 - 79995247403097v^10 + 214254383734802v^9 + 114520598170765v^8 - 497877476212984v^7 - 13303957170899v^6 + 511406363468605v^5 - 79124153940469v^4 - 205345303936768v^3 + 42126470941756v^2 + 18001353308241*v - 2701913424648) / 740401001108
\(\beta_{8}\)	\(=\)	\( ( 5912542223 \nu^{17} - 36172904739 \nu^{16} - 101462522256 \nu^{15} + 939150760888 \nu^{14} + \cdots + 4393984359332 ) / 370200500554 \) (5912542223v^17 - 36172904739v^16 - 101462522256v^15 + 939150760888v^14 + 161231412628v^13 - 9030598189030v^12 + 5518450873948v^11 + 39418326736791v^10 - 32174068672612v^9 - 80470903905306v^8 + 40164075432279v^7 + 90357968890278v^6 + 35802858657415v^5 - 57819893544688v^4 - 80236035832764v^3 + 10052157695564v^2 + 31280139926105*v + 4393984359332) / 370200500554
\(\beta_{9}\)	\(=\)	\( ( 12872643469 \nu^{17} - 81590528270 \nu^{16} - 176136654710 \nu^{15} + 2043980178961 \nu^{14} + \cdots - 7959434299208 ) / 740401001108 \) (12872643469v^17 - 81590528270v^16 - 176136654710v^15 + 2043980178961v^14 - 1068358614811v^13 - 18135719019665v^12 + 29979790286684v^11 + 62891399429393v^10 - 185468599242570v^9 - 35539969498285v^8 + 483246566015676v^7 - 189620931949213v^6 - 619663073380783v^5 + 302197266818659v^4 + 379887798704466v^3 - 109851993398338v^2 - 84864001775635*v - 7959434299208) / 740401001108
\(\beta_{10}\)	\(=\)	\( ( - 7368792221 \nu^{17} + 36032856239 \nu^{16} + 190778072729 \nu^{15} - 1071242030927 \nu^{14} + \cdots - 5171470402916 ) / 370200500554 \) (-7368792221v^17 + 36032856239v^16 + 190778072729v^15 - 1071242030927v^14 - 1813871052343v^13 + 12543888417156v^12 + 7691377412900v^11 - 73588600546308v^10 - 15482463378540v^9 + 229133287576338v^8 + 28278704278146v^7 - 372560927987140v^6 - 73745518814393v^5 + 278303376942237v^4 + 88730894435189v^3 - 66673825169979v^2 - 32768558308519*v - 5171470402916) / 370200500554
\(\beta_{11}\)	\(=\)	\( ( - 21055108899 \nu^{17} + 133731830584 \nu^{16} + 355648919466 \nu^{15} + \cdots + 9743683016604 ) / 740401001108 \) (-21055108899v^17 + 133731830584v^16 + 355648919466v^15 - 3605369321091v^14 - 105115230565v^13 + 36808766678391v^12 - 29395736837166v^11 - 177925358841801v^10 + 200464961966626v^9 + 431159938833053v^8 - 498087869992494v^7 - 572467851918489v^6 + 532455028203587v^5 + 411887543338771v^4 - 240027877267986v^3 - 133197435928964v^2 + 34489244582237*v + 9743683016604) / 740401001108
\(\beta_{12}\)	\(=\)	\( ( - 21893357459 \nu^{17} + 132283003946 \nu^{16} + 388113009668 \nu^{15} + \cdots + 3949574073272 ) / 740401001108 \) (-21893357459v^17 + 132283003946v^16 + 388113009668v^15 - 3540814213357v^14 - 566374260951v^13 + 35586205958015v^12 - 26608557701118v^11 - 165658011283115v^10 + 196077973138472v^9 + 363943274570195v^8 - 519444270595486v^7 - 384557389333353v^6 + 609264737893023v^5 + 190681808244883v^4 - 294033969050270v^3 - 39800976708996v^2 + 35718200056539*v + 3949574073272) / 740401001108
\(\beta_{13}\)	\(=\)	\( ( - 23749622553 \nu^{17} + 159748050268 \nu^{16} + 338257603198 \nu^{15} + \cdots + 3500435643128 ) / 740401001108 \) (-23749622553v^17 + 159748050268v^16 + 338257603198v^15 - 4176417520163v^14 + 1497620251189v^13 + 40501753016639v^12 - 48309357112436v^11 - 177992126711453v^10 + 287598825141134v^9 + 356729386658299v^8 - 657623818283334v^7 - 348544186655245v^6 + 624232912128043v^5 + 171161844051679v^4 - 208181978916494v^3 - 40034664441794v^2 + 3101327241263*v + 3500435643128) / 740401001108
\(\beta_{14}\)	\(=\)	\( ( 24869169061 \nu^{17} - 117005657678 \nu^{16} - 624027602860 \nu^{15} + 3362151234965 \nu^{14} + \cdots + 4596964704100 ) / 740401001108 \) (24869169061v^17 - 117005657678v^16 - 624027602860v^15 + 3362151234965v^14 + 5593082987259v^13 - 37586049707303v^12 - 20457399399566v^11 + 207004495417565v^10 + 22610482495938v^9 - 594608807451241v^8 + 2417802109494v^7 + 892330180929937v^6 + 37363968000639v^5 - 634692500203877v^4 - 94163873982724v^3 + 158455481262598v^2 + 44875794924525*v + 4596964704100) / 740401001108
\(\beta_{15}\)	\(=\)	\( ( 32186770999 \nu^{17} - 181666620996 \nu^{16} - 657475026718 \nu^{15} + 5003494168191 \nu^{14} + \cdots + 3174104069880 ) / 740401001108 \) (32186770999v^17 - 181666620996v^16 - 657475026718v^15 + 5003494168191v^14 + 3257900089055v^13 - 52733397100855v^12 + 12889976432068v^11 + 267480684267275v^10 - 148544182260996v^9 - 691742015748553v^8 + 369851976250802v^7 + 950809505846741v^6 - 293802853903673v^5 - 636251490614185v^4 + 16917127044388v^3 + 151736210497344v^2 + 35930676517573*v + 3174104069880) / 740401001108
\(\beta_{16}\)	\(=\)	\( ( - 32406118817 \nu^{17} + 189658637726 \nu^{16} + 648455364568 \nu^{15} + \cdots - 2404485512256 ) / 740401001108 \) (-32406118817v^17 + 189658637726v^16 + 648455364568v^15 - 5270224913089v^14 - 2822021726395v^13 + 56230098154915v^12 - 19097079510558v^11 - 290052171047585v^10 + 189738892468594v^9 + 765312176193749v^8 - 503413456108790v^7 - 1064307071368673v^6 + 483158911992225v^5 + 718633084719037v^4 - 108595878604528v^3 - 176524243075018v^2 - 31301683902777*v - 2404485512256) / 740401001108
\(\beta_{17}\)	\(=\)	\( ( - 26070569352 \nu^{17} + 150046167408 \nu^{16} + 535258484380 \nu^{15} - 4183016265326 \nu^{14} + \cdots - 907583269738 ) / 370200500554 \) (-26070569352v^17 + 150046167408v^16 + 535258484380v^15 - 4183016265326v^14 - 2684005993537v^13 + 44923323405688v^12 - 10395148955663v^11 - 234924699777551v^10 + 122902130674273v^9 + 638001915592530v^8 - 312560027015514v^7 - 934872209172284v^6 + 247466685450651v^5 + 678093622686427v^4 + 2135474045373v^3 - 183871932928318v^2 - 43387939635667*v - 907583269738) / 370200500554

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + \beta _1 + 4 \) b2 + b1 + 4
\(\nu^{3}\)	\(=\)	\( -\beta_{16} + \beta_{12} + \beta_{10} - \beta_{5} + 7\beta _1 + 1 \) -b16 + b12 + b10 - b5 + 7*b1 + 1
\(\nu^{4}\)	\(=\)	\( -\beta_{16} - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{4} + 9\beta_{2} + 10\beta _1 + 28 \) -b16 - b15 + b14 + b12 + b4 + 9b2 + 10b1 + 28
\(\nu^{5}\)	\(=\)	\( - 11 \beta_{16} - 3 \beta_{15} + 2 \beta_{14} + 11 \beta_{12} - \beta_{11} + 10 \beta_{10} + \beta_{9} + \cdots + 12 \) -11b16 - 3b15 + 2b14 + 11b12 - b11 + 10b10 + b9 + b8 - b6 - 10b5 + b4 + b3 + 2b2 + 57b1 + 12
\(\nu^{6}\)	\(=\)	\( - 17 \beta_{16} - 15 \beta_{15} + 16 \beta_{14} + 16 \beta_{12} + 4 \beta_{10} + \beta_{9} + \beta_{8} + \cdots + 224 \) -17b16 - 15b15 + 16b14 + 16b12 + 4b10 + b9 + b8 + 2b7 - b6 + b5 + 15b4 + b3 + 77b2 + 96*b1 + 224
\(\nu^{7}\)	\(=\)	\( 2 \beta_{17} - 111 \beta_{16} - 49 \beta_{15} + 37 \beta_{14} + 6 \beta_{13} + 110 \beta_{12} + \cdots + 140 \) 2b17 - 111b16 - 49b15 + 37b14 + 6b13 + 110b12 - 20b11 + 94b10 + 17b9 + 20b8 - 2b7 - 17b6 - 86b5 + 18b4 + 16b3 + 39b2 + 493*b1 + 140
\(\nu^{8}\)	\(=\)	\( 6 \beta_{17} - 221 \beta_{16} - 173 \beta_{15} + 202 \beta_{14} + 13 \beta_{13} + 196 \beta_{12} + \cdots + 1892 \) 6b17 - 221b16 - 173b15 + 202b14 + 13b13 + 196b12 - 8b11 + 86b10 + 18b9 + 29b8 + 33b7 - 24b6 + 21b5 + 177b4 + 20b3 + 672b2 + 942*b1 + 1892
\(\nu^{9}\)	\(=\)	\( 53 \beta_{17} - 1116 \beta_{16} - 595 \beta_{15} + 509 \beta_{14} + 143 \beta_{13} + 1074 \beta_{12} + \cdots + 1636 \) 53b17 - 1116b16 - 595b15 + 509b14 + 143b13 + 1074b12 - 283b11 + 909b10 + 211b9 + 291b8 - 42b7 - 223b6 - 696b5 + 243b4 + 201b3 + 551b2 + 4426*b1 + 1636
\(\nu^{10}\)	\(=\)	\( 169 \beta_{17} - 2608 \beta_{16} - 1837 \beta_{15} + 2350 \beta_{14} + 344 \beta_{13} + 2178 \beta_{12} + \cdots + 16558 \) 169b17 - 2608b16 - 1837b15 + 2350b14 + 344b13 + 2178b12 - 228b11 + 1289b10 + 244b9 + 523b8 + 376b7 - 390b6 + 349b5 + 1923b4 + 309b3 + 6004b2 + 9397*b1 + 16558
\(\nu^{11}\)	\(=\)	\( 937 \beta_{17} - 11313 \beta_{16} - 6496 \beta_{15} + 6281 \beta_{14} + 2336 \beta_{13} + 10395 \beta_{12} + \cdots + 18748 \) 937b17 - 11313b16 - 6496b15 + 6281b14 + 2336b13 + 10395b12 - 3516b11 + 9060b10 + 2336b9 + 3730b8 - 602b7 - 2673b6 - 5308b5 + 2988b4 + 2351b3 + 6828b2 + 40836*b1 + 18748
\(\nu^{12}\)	\(=\)	\( 3089 \beta_{17} - 29405 \beta_{16} - 18879 \beta_{15} + 26328 \beta_{14} + 6033 \beta_{13} + \cdots + 148967 \) 3089b17 - 29405b16 - 18879b15 + 26328b14 + 6033b13 + 23117b12 - 4181b11 + 16747b10 + 3006b9 + 7739b8 + 3652b7 - 5424b6 + 5162b5 + 20186b4 + 4313b3 + 54776b2 + 94534*b1 + 148967
\(\nu^{13}\)	\(=\)	\( 13911 \beta_{17} - 115784 \beta_{16} - 67594 \beta_{15} + 73523 \beta_{14} + 32563 \beta_{13} + \cdots + 209539 \) 13911b17 - 115784b16 - 67594b15 + 73523b14 + 32563b13 + 100424b12 - 41050b11 + 92293b10 + 24556b9 + 44781b8 - 7430b7 - 30697b6 - 37056b5 + 35218b4 + 26762b3 + 78950b2 + 384714*b1 + 209539
\(\nu^{14}\)	\(=\)	\( 46651 \beta_{17} - 323437 \beta_{16} - 191457 \beta_{15} + 289208 \beta_{14} + 88567 \beta_{13} + \cdots + 1370719 \) 46651b17 - 323437b16 - 191457b15 + 289208b14 + 88567b13 + 239538b12 - 63154b11 + 202282b10 + 35396b9 + 103066b8 + 32028b7 - 69509b6 + 70656b5 + 208900b4 + 56563b3 + 508915b2 + 954643*b1 + 1370719
\(\nu^{15}\)	\(=\)	\( 187479 \beta_{17} - 1194832 \beta_{16} - 687120 \beta_{15} + 835409 \beta_{14} + 416826 \beta_{13} + \cdots + 2290611 \) 187479b17 - 1194832b16 - 687120b15 + 835409b14 + 416826b13 + 971938b12 - 463596b11 + 953813b10 + 251727b9 + 517536b8 - 85426b7 - 344307b6 - 217224b5 + 404916b4 + 300764b3 + 876154b2 + 3683505*b1 + 2290611
\(\nu^{16}\)	\(=\)	\( 633631 \beta_{17} - 3505574 \beta_{16} - 1932646 \beta_{15} + 3141707 \beta_{14} + 1179156 \beta_{13} + \cdots + 12850089 \) 633631b17 - 3505574b16 - 1932646b15 + 3141707b14 + 1179156b13 + 2449786b12 - 856675b11 + 2341972b10 + 405676b9 + 1287720b8 + 255931b7 - 847294b6 + 915716b5 + 2151523b4 + 709001b3 + 4803934b2 + 9657046*b1 + 12850089
\(\nu^{17}\)	\(=\)	\( 2377642 \beta_{17} - 12408386 \beta_{16} - 6907767 \beta_{15} + 9317906 \beta_{14} + 5066246 \beta_{13} + \cdots + 24609081 \) 2377642b17 - 12408386b16 - 6907767b15 + 9317906b14 + 5066246b13 + 9442297b12 - 5137044b11 + 9948089b10 + 2550344b9 + 5840448b8 - 948245b7 - 3807120b6 - 711853b5 + 4575057b4 + 3353767b3 + 9475359b2 + 35720852*b1 + 24609081

\(n\):		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
$3$	\( T^{18} - 6 T^{17} + \cdots - 224 \) T^18 - 6T^17 - 20T^16 + 171T^15 + 79T^14 - 1897T^13 + 800T^12 + 10423T^11 - 7942T^10 - 30513T^9 + 25074T^8 + 49703T^7 - 34959T^6 - 43987T^5 + 21684T^4 + 19388T^3 - 4543T^2 - 3292*T - 224
$5$	\( (T - 1)^{18} \) (T - 1)^18
$7$	\( T^{18} - 12 T^{17} + \cdots + 7270400 \) T^18 - 12T^17 - 16T^16 + 696T^15 - 1473T^14 - 14090T^13 + 53117T^12 + 114078T^11 - 697932T^10 - 191382T^9 + 4426359T^8 - 2173750T^7 - 14437312T^6 + 12047436T^5 + 24018872T^4 - 21785600T^3 - 19860144T^2 + 13147360*T + 7270400
$11$	\( (T + 1)^{18} \) (T + 1)^18
$13$	\( T^{18} - 12 T^{17} + \cdots - 1241728 \) T^18 - 12T^17 - 56T^16 + 1047T^15 + 178T^14 - 33311T^13 + 37267T^12 + 484425T^11 - 781644T^10 - 3382359T^9 + 6633033T^8 + 10787151T^7 - 26679981T^6 - 10769485T^5 + 48329859T^4 - 10305992T^3 - 31034708T^2 + 17947200*T - 1241728
$17$	\( T^{18} - 17 T^{17} + \cdots - 2025312 \) T^18 - 17T^17 - 6T^16 + 1411T^15 - 4416T^14 - 38201T^13 + 171301T^12 + 382921T^11 - 2126174T^10 - 1724261T^9 + 11177958T^8 + 5149247T^7 - 28768756T^6 - 12598940T^5 + 35578052T^4 + 19577016T^3 - 15524256T^2 - 12190224*T - 2025312
$19$	\( T^{18} + \cdots - 307765264 \) T^18 - 27T^17 + 169T^16 + 1562T^15 - 22638T^14 + 38697T^13 + 579082T^12 - 2302616T^11 - 5713095T^10 + 34809672T^9 + 32486956T^8 - 256261759T^7 - 194746849T^6 + 972934375T^5 + 1106999146T^4 - 1279122488T^3 - 2762060624T^2 - 1601400540*T - 307765264
$23$	\( T^{18} - 7 T^{17} + \cdots + 36550656 \) T^18 - 7T^17 - 189T^16 + 1405T^15 + 12562T^14 - 103358T^13 - 350065T^12 + 3547089T^11 + 3354371T^10 - 60268581T^9 + 18972948T^8 + 477217196T^7 - 471680428T^6 - 1348333696T^5 + 1703742256T^4 + 320838384T^3 - 617531904T^2 - 45239040*T + 36550656
$29$	\( T^{18} + \cdots - 430122720 \) T^18 - 2T^17 - 191T^16 + 186T^15 + 14425T^14 - 465T^13 - 549123T^12 - 425380T^11 + 11050339T^10 + 15561358T^9 - 113399181T^8 - 210641057T^7 + 546917646T^6 + 1203047152T^5 - 1021319972T^4 - 2586968376T^3 + 607047792T^2 + 1489595568*T - 430122720
$31$	\( T^{18} + \cdots + 1193156608 \) T^18 - 15T^17 - 207T^16 + 3717T^15 + 13982T^14 - 348974T^13 - 283884T^12 + 15663364T^11 - 3543200T^10 - 353945888T^9 + 160225840T^8 + 3881584416T^7 - 1577330432T^6 - 17442195136T^5 + 6395207552T^4 + 16894742528T^3 - 7170315264T^2 - 2715993088*T + 1193156608
$37$	\( T^{18} + \cdots - 4092275872 \) T^18 - 12T^17 - 146T^16 + 1908T^15 + 8098T^14 - 116915T^13 - 223755T^12 + 3618624T^11 + 3338942T^10 - 61820728T^9 - 27404732T^8 + 591543167T^7 + 134542560T^6 - 3045848208T^5 - 604882508T^4 + 7606332184T^3 + 2548289104T^2 - 7083310448*T - 4092275872
$41$	\( T^{18} + \cdots + 3278041529856 \) T^18 + 7T^17 - 318T^16 - 2352T^15 + 38645T^14 + 304443T^13 - 2332464T^12 - 20329334T^11 + 72677304T^10 + 768571556T^9 - 976163632T^8 - 16654842688T^7 - 2756673936T^6 + 195986962304T^5 + 218048516032T^4 - 1058811871296T^3 - 1873902903168T^2 + 1428667848960*T + 3278041529856
$43$	\( T^{18} + \cdots + 141637684480 \) T^18 - 29T^17 + 9T^16 + 6552T^15 - 36967T^14 - 583958T^13 + 4632682T^12 + 26966794T^11 - 254692006T^10 - 734062539T^9 + 7311433618T^8 + 13190432479T^7 - 109208563716T^6 - 160165027968T^5 + 720902425104T^4 + 978772400640T^3 - 959395110400T^2 - 190614732864*T + 141637684480
$47$	\( T^{18} + \cdots + 128855212032 \) T^18 - 14T^17 - 371T^16 + 5351T^15 + 51008T^14 - 784520T^13 - 2999598T^12 + 55012216T^11 + 50538966T^10 - 1872219815T^9 + 1391952487T^8 + 27875523272T^7 - 41359708544T^6 - 167670529520T^5 + 270117200144T^4 + 394802549760T^3 - 457015290624T^2 - 363630741504*T + 128855212032
$53$	\( T^{18} + \cdots - 19970297401344 \) T^18 - 19T^17 - 337T^16 + 7817T^15 + 36179T^14 - 1264966T^13 - 537421T^12 + 103479846T^11 - 164602532T^10 - 4587590656T^9 + 13006898932T^8 + 108335239680T^7 - 423622148224T^6 - 1150774195280T^5 + 6501526065536T^4 + 730930198080T^3 - 37578660262848T^2 + 53715426804864*T - 19970297401344
$59$	\( T^{18} + \cdots - 364754978304 \) T^18 - 18T^17 - 170T^16 + 4579T^15 + 1917T^14 - 434473T^13 + 1206337T^12 + 18590077T^11 - 89557410T^10 - 329309380T^9 + 2515276307T^8 + 791521920T^7 - 30912617614T^6 + 39486920468T^5 + 138755546468T^4 - 353949948024T^3 + 33746975520T^2 + 515742211968*T - 364754978304
$61$	\( T^{18} + \cdots + 15405370675232 \) T^18 - 19T^17 - 347T^16 + 8147T^15 + 33964T^14 - 1308430T^13 + 95716T^12 + 103710615T^11 - 203025765T^10 - 4378350675T^9 + 12569503712T^8 + 98483600496T^7 - 312579040227T^6 - 1142061237944T^5 + 3312322271732T^4 + 6264785384900T^3 - 12763269158820T^2 - 9520881937928*T + 15405370675232
$67$	\( T^{18} + \cdots + 272942978988544 \) T^18 + 6T^17 - 732T^16 - 4051T^15 + 215506T^14 + 1095005T^13 - 32660093T^12 - 153183007T^11 + 2708269882T^10 + 11998851055T^9 - 121216922587T^8 - 531435593085T^7 + 2735251558959T^6 + 12594589801957T^5 - 25195453065695T^4 - 136983219261732T^3 + 24649497710064T^2 + 461869224361152*T + 272942978988544
$71$	\( T^{18} + \cdots - 98\!\cdots\!24 \) T^18 - 17T^17 - 574T^16 + 10174T^15 + 138460T^14 - 2547804T^13 - 18575823T^12 + 348109766T^11 + 1559009749T^10 - 28346330132T^9 - 87783494067T^8 + 1406223938126T^7 + 3437507667240T^6 - 41383887751567T^5 - 91309026000133T^4 + 657024321397692T^3 + 1447896249669609T^2 - 4262105773892892*T - 9837315231078624
$73$	\( (T - 1)^{18} \) (T - 1)^18
$79$	\( T^{18} + \cdots - 11\!\cdots\!80 \) T^18 - 60T^17 + 849T^16 + 16112T^15 - 501035T^14 + 447028T^13 + 93908280T^12 - 588865676T^11 - 8125984608T^10 + 80837426752T^9 + 323592191664T^8 - 4907237861536T^7 - 3880666084032T^6 + 141853355848320T^5 - 59254607843072T^4 - 1711475709268480T^3 + 826538487573504T^2 + 6270815031064576*T - 1195949870612480
$83$	\( T^{18} + \cdots - 28\!\cdots\!40 \) T^18 - 50T^17 + 275T^16 + 25083T^15 - 422970T^14 - 3073454T^13 + 115293032T^12 - 278075399T^11 - 12723853402T^10 + 88966771040T^9 + 555907611126T^8 - 6982522324251T^7 - 1687434649041T^6 + 221947337370080T^5 - 451534076911873T^4 - 2680619825671188T^3 + 8798371270334268T^2 + 5637353333513472*T - 28262990326587840
$89$	\( T^{18} + \cdots - 613008689833050 \) T^18 + 17T^17 - 663T^16 - 12230T^15 + 157393T^14 + 3397158T^13 - 15138279T^12 - 463637901T^11 + 247296679T^10 + 33057643409T^9 + 54658237923T^8 - 1197820015991T^7 - 3761261665076T^6 + 18526880779720T^5 + 88682163940340T^4 - 30530091205239T^3 - 667933441577424T^2 - 1151089507267395*T - 613008689833050
$97$	\( T^{18} + \cdots + 15\!\cdots\!56 \) T^18 - 5T^17 - 1150T^16 + 4686T^15 + 536497T^14 - 1661404T^13 - 130780866T^12 + 281540251T^11 + 17869085435T^10 - 24205378252T^9 - 1369126488348T^8 + 1112367777700T^7 + 56360407184704T^6 - 31144783062720T^5 - 1112077230484016T^4 + 671704839496736T^3 + 7579449007097344T^2 - 9397091622854656*T + 1582105883181056
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\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(-1\)
\(11\)	\(1\)
\(73\)	\(-1\)