LMFDB - Newform orbit 6045.2.a.bi

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
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} - 4 x^{17} - 21 x^{16} + 97 x^{15} + 156 x^{14} - 935 x^{13} - 411 x^{12} + 4582 x^{11} + \cdots - 112 $ x^18 - 4x^17 - 21x^16 + 97x^15 + 156x^14 - 935x^13 - 411x^12 + 4582x^11 - 446x^10 - 12159x^9 + 4398x^8 + 17347x^7 - 7839x^6 - 12832x^5 + 4744x^4 + 5064x^3 - 746x^2 - 892*x - 112
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{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 + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} - \beta_{7} q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9}+O(q^{10}) $ q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + q^5 + b1 * q^6 - b7 * q^7 + (b3 + b1) * q^8 + q^9	$ q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} - \beta_{7} q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9} + \beta_1 q^{10} + \beta_{13} q^{11} + (\beta_{2} + 1) q^{12} - q^{13} + ( - \beta_{17} + \beta_{11} + \cdots + \beta_1) q^{14}+ \cdots + \beta_{13} q^{99}+O(q^{100}) $ q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + q^5 + b1 * q^6 - b7 * q^7 + (b3 + b1) * q^8 + q^9 + b1 * q^10 + b13 * q^11 + (b2 + 1) * q^12 - q^13 + (-b17 + b11 + b10 + b1) * q^14 + q^15 + (b4 + b2 + 1) * q^16 + (-b11 + 1) * q^17 + b1 * q^18 + (b14 + b13 + b12 - b9 + b6 - b5 + b3 - b2 + b1 + 1) * q^19 + (b2 + 1) * q^20 - b7 * q^21 + (b15 - b13 + b9 + b7 + b5 - b4 - b3 + b1 + 1) * q^22 + (b17 + b16 - b13 - b10 + b9 - b3 + 2) * q^23 + (b3 + b1) * q^24 + q^25 - b1 * q^26 + q^27 + (-b17 - b15 + b14 + 2b13 + b10 - b9 - b8 - b7 + b4 + b3 - b2 - 1) q^28 - b8 * q^29 + b1 * q^30 + q^31 + (b12 + b11 + b10 - b9 - b5 + b3 + b2 + 2b1 + 1) q^32 + b13 * q^33 + (b15 - b14 - b13 + b9 + b7 - b6 - b4 + b2 + 2b1 + 1) q^34 - b7 * q^35 + (b2 + 1) * q^36 - b12 * q^37 + (-b10 + b8 + b4 - b3 + b2 + 2) * q^38 - q^39 + (b3 + b1) * q^40 + (-b16 + b13 + b10 - b9 + b6) * q^41 + (-b17 + b11 + b10 + b1) * q^42 + (-b15 - b14 - b10 - b1 + 1) * q^43 + (-b16 + b14 + b13 - b12 - 2b11 - b7 + b5 - b2 - b1 - 1) q^44 + q^45 + (b17 + b16 - b15 - b10 + b8 - b5 + b1 + 1) * q^46 + (-b13 - b10 - b4 + b3 + 2) * q^47 + (b4 + b2 + 1) * q^48 + (b17 + b15 - b14 - 2b13 - b12 + b9 + b8 + 2b5 - b4 - 2b3 + b2 + 2) q^49 + b1 * q^50 + (-b11 + 1) * q^51 + (-b2 - 1) * q^52 + (b17 - b13 - b10 + b8 - b3 + b2 - b1 + 3) * q^53 + b1 * q^54 + b13 * q^55 + (-b17 - b13 + b11 - b10 + b9 - b8 - b6 + b5 + b3 - b2 + b1 - 1) * q^56 + (b14 + b13 + b12 - b9 + b6 - b5 + b3 - b2 + b1 + 1) * q^57 + (-b17 - b16 - b15 - b14 - b12 + b10 - b6 + b3 - 1) * q^58 + (b15 - b14 - b13 + b8 - b7 - b6 - b3 + b2 + 2) * q^59 + (b2 + 1) * q^60 + (-b15 - b14 - b13 - b10 + b9 - b6 + 1) * q^61 + b1 * q^62 - b7 * q^63 + (-b15 - b14 + b12 + b11 + b10 - b9 + b8 - b7 - 2b5 + b4 + 2b2 + b1 + 2) * q^64 - q^65 + (b15 - b13 + b9 + b7 + b5 - b4 - b3 + b1 + 1) * q^66 + (-b16 + b15 + b13 - b12 + 2b10 + b2 - b1 + 1) q^67 + (b17 - b16 - b14 + b13 - b11 + b10 - b3 + 3b2 - b1 + 2) q^68 + (b17 + b16 - b13 - b10 + b9 - b3 + 2) * q^69 + (-b17 + b11 + b10 + b1) * q^70 + (-b17 - b15 + b14 + b13 - b12 + b10 - b9 + b4 + b3 - b2 - 1) * q^71 + (b3 + b1) * q^72 + (2b13 + b12 - b9 - b5 + b3 - b2 + b1) q^73 + (b17 - b13 - b12 - b11 - b10 + b6 + b5 - b4 - b3 - b1 + 2) * q^74 + q^75 + (b17 + b16 - b13 - b10 + b5 - b4 + b3 - b2 + 3b1) q^76 + (-b17 + b13 + b12 + b11 + b10 - b5 + b2 + 1) * q^77 - b1 * q^78 + (b17 + b16 - b15 - 2b13 - b11 - 2b10 + b9 - b6 - b3 - b2 + b1 + 1) * q^79 + (b4 + b2 + 1) * q^80 + q^81 + (-b16 + 2b15 - b14 - 2b13 - b12 + b11 + b10 + b9 + b7 - b6 + b5 - b4 - b3 + 2b2 + 2) q^82 + (-b16 + b15 + b6 + b5 - b3 - b2 - b1 + 1) * q^83 + (-b17 - b15 + b14 + 2b13 + b10 - b9 - b8 - b7 + b4 + b3 - b2 - 1) q^84 + (-b11 + 1) * q^85 + (b16 - b15 - b13 - b10 - b8 - b6 - b5 + b3 - b2 + b1 - 1) * q^86 - b8 * q^87 + (-b17 - b16 + 2b15 - b12 + b11 + b9 - b8 + 2b7 + 2b5 - 2b4 - b2 - 1) * q^88 + (b17 - b10 - b7 - b4 + b2 - b1 + 1) * q^89 + b1 * q^90 + b7 * q^91 + (b17 + b16 - b13 + b12 - 2b10 - b9 + b8 + b7 + b6 - b5 - b3 + 2b2 + 4) * q^92 + q^93 + (-2b15 + b14 + 2b13 - b12 - 2b11 - 2b10 - b9 - b7 + b6 + 2b4 - b2 - 2) q^94 + (b14 + b13 + b12 - b9 + b6 - b5 + b3 - b2 + b1 + 1) * q^95 + (b12 + b11 + b10 - b9 - b5 + b3 + b2 + 2b1 + 1) q^96 + (-b17 - b15 + b14 + 3b13 + b12 - 2b9 + b6 - b5 + b4 + b3 - 2b2 + 2) q^97 + (b17 + b15 + b14 + b13 - b12 + b8 + b6 - b3 + b2 + 1) * q^98 + b13 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10}) $ 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9	\( 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9} + 4 q^{10} + 6 q^{11} + 22 q^{12} - 18 q^{13} + 5 q^{14} + 18 q^{15} + 30 q^{16} + 18 q^{17} + 4 q^{18} + 12 q^{19} + 22 q^{20} + 8 q^{21} + 7 q^{22} + 32 q^{23} + 9 q^{24} + 18 q^{25} - 4 q^{26} + 18 q^{27} + 10 q^{28} + 7 q^{29} + 4 q^{30} + 18 q^{31} + 22 q^{32} + 6 q^{33} + 15 q^{34} + 8 q^{35} + 22 q^{36} + 3 q^{37} + 32 q^{38} - 18 q^{39} + 9 q^{40} + 4 q^{41} + 5 q^{42} + 14 q^{43} - 5 q^{44} + 18 q^{45} + 10 q^{46} + 23 q^{47} + 30 q^{48} + 28 q^{49} + 4 q^{50} + 18 q^{51} - 22 q^{52} + 35 q^{53} + 4 q^{54} + 6 q^{55} - 7 q^{56} + 12 q^{57} - 6 q^{58} + 28 q^{59} + 22 q^{60} + 19 q^{61} + 4 q^{62} + 8 q^{63} + 43 q^{64} - 18 q^{65} + 7 q^{66} + 34 q^{67} + 55 q^{68} + 32 q^{69} + 5 q^{70} - 8 q^{71} + 9 q^{72} + 22 q^{74} + 18 q^{75} + 2 q^{76} + 21 q^{77} - 4 q^{78} + 4 q^{79} + 30 q^{80} + 18 q^{81} + 29 q^{82} + 11 q^{83} + 10 q^{84} + 18 q^{85} - 22 q^{86} + 7 q^{87} - 31 q^{88} + 17 q^{89} + 4 q^{90} - 8 q^{91} + 33 q^{92} + 18 q^{93} - 14 q^{94} + 12 q^{95} + 22 q^{96} + 32 q^{97} + 20 q^{98} + 6 q^{99}+O(q^{100}) \) 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9 + 4 * q^10 + 6 * q^11 + 22 * q^12 - 18 * q^13 + 5 * q^14 + 18 * q^15 + 30 * q^16 + 18 * q^17 + 4 * q^18 + 12 * q^19 + 22 * q^20 + 8 * q^21 + 7 * q^22 + 32 * q^23 + 9 * q^24 + 18 * q^25 - 4 * q^26 + 18 * q^27 + 10 * q^28 + 7 * q^29 + 4 * q^30 + 18 * q^31 + 22 * q^32 + 6 * q^33 + 15 * q^34 + 8 * q^35 + 22 * q^36 + 3 * q^37 + 32 * q^38 - 18 * q^39 + 9 * q^40 + 4 * q^41 + 5 * q^42 + 14 * q^43 - 5 * q^44 + 18 * q^45 + 10 * q^46 + 23 * q^47 + 30 * q^48 + 28 * q^49 + 4 * q^50 + 18 * q^51 - 22 * q^52 + 35 * q^53 + 4 * q^54 + 6 * q^55 - 7 * q^56 + 12 * q^57 - 6 * q^58 + 28 * q^59 + 22 * q^60 + 19 * q^61 + 4 * q^62 + 8 * q^63 + 43 * q^64 - 18 * q^65 + 7 * q^66 + 34 * q^67 + 55 * q^68 + 32 * q^69 + 5 * q^70 - 8 * q^71 + 9 * q^72 + 22 * q^74 + 18 * q^75 + 2 * q^76 + 21 * q^77 - 4 * q^78 + 4 * q^79 + 30 * q^80 + 18 * q^81 + 29 * q^82 + 11 * q^83 + 10 * q^84 + 18 * q^85 - 22 * q^86 + 7 * q^87 - 31 * q^88 + 17 * q^89 + 4 * q^90 - 8 * q^91 + 33 * q^92 + 18 * q^93 - 14 * q^94 + 12 * q^95 + 22 * q^96 + 32 * q^97 + 20 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - 4 x^{17} - 21 x^{16} + 97 x^{15} + 156 x^{14} - 935 x^{13} - 411 x^{12} + 4582 x^{11} + \cdots - 112 $ x^18 - 4*x^17 - 21*x^16 + 97*x^15 + 156*x^14 - 935*x^13 - 411*x^12 + 4582*x^11 - 446*x^10 - 12159*x^9 + 4398*x^8 + 17347*x^7 - 7839*x^6 - 12832*x^5 + 4744*x^4 + 5064*x^3 - 746*x^2 - 892*x - 112 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ \nu^{3} - 5\nu $ v^3 - 5*v
$\beta_{4}$	$=$	$ \nu^{4} - 7\nu^{2} + 6 $ v^4 - 7*v^2 + 6
$\beta_{5}$	$=$	$ ( - 228355 \nu^{17} - 336512 \nu^{16} + 8567893 \nu^{15} + 6198769 \nu^{14} - 123630228 \nu^{13} + \cdots + 317132228 ) / 17310212 $ (-228355v^17 - 336512v^16 + 8567893v^15 + 6198769v^14 - 123630228v^13 - 31566349v^12 + 891219629v^11 - 59637650v^10 - 3413604116v^9 + 1071437101v^8 + 6645499448v^7 - 3608265451v^6 - 5229581577v^5 + 4661942660v^4 - 103549980v^3 - 1874991824v^2 + 727071450*v + 317132228) / 17310212
$\beta_{6}$	$=$	$ ( 446299 \nu^{17} + 1332792 \nu^{16} - 19095959 \nu^{15} - 27974025 \nu^{14} + 299322856 \nu^{13} + \cdots + 152855836 ) / 17310212 $ (446299v^17 + 1332792v^16 - 19095959v^15 - 27974025v^14 + 299322856v^13 + 211957035v^12 - 2313633925v^11 - 639195438v^10 + 9631313854v^9 + 168689975v^8 - 21628657114v^7 + 2831804025v^6 + 24385040543v^5 - 4515463168v^4 - 11484187796v^3 + 1183377184v^2 + 1777245698*v + 152855836) / 17310212
$\beta_{7}$	$=$	$ ( - 740009 \nu^{17} + 2872262 \nu^{16} + 14791069 \nu^{15} - 67180803 \nu^{14} - 98215806 \nu^{13} + \cdots + 222862852 ) / 17310212 $ (-740009v^17 + 2872262v^16 + 14791069v^15 - 67180803v^14 - 98215806v^13 + 613552943v^12 + 151793377v^11 - 2759586728v^10 + 976237358v^9 + 6315037203v^8 - 4581147060v^7 - 6769396543v^6 + 6955418053v^5 + 2731690702v^4 - 3873662520v^3 - 728115164v^2 + 862479718*v + 222862852) / 17310212
$\beta_{8}$	$=$	$ ( - 610666 \nu^{17} + 354827 \nu^{16} + 18681397 \nu^{15} - 10014793 \nu^{14} - 234498267 \nu^{13} + \cdots - 175204944 ) / 8655106 $ (-610666v^17 + 354827v^16 + 18681397v^15 - 10014793v^14 - 234498267v^13 + 116139789v^12 + 1554418411v^11 - 711950221v^10 - 5830999753v^9 + 2463590886v^8 + 12312429494v^7 - 4696943709v^6 - 13698626968v^5 + 4235562551v^4 + 6955732814v^3 - 939434104v^2 - 1389743616*v - 175204944) / 8655106
$\beta_{9}$	$=$	$ ( 658857 \nu^{17} - 2572882 \nu^{16} - 14528758 \nu^{15} + 64014013 \nu^{14} + 119878466 \nu^{13} + \cdots - 101408846 ) / 8655106 $ (658857v^17 - 2572882v^16 - 14528758v^15 + 64014013v^14 + 119878466v^13 - 639146514v^12 - 440926905v^11 + 3285967890v^10 + 584398593v^9 - 9283925003v^8 + 383710967v^7 + 14213130072v^6 - 1242628076v^5 - 10798785650v^4 + 121132724v^3 + 3275128816v^2 + 323408598*v - 101408846) / 8655106
$\beta_{10}$	$=$	$ ( 870813 \nu^{17} - 3033149 \nu^{16} - 19402563 \nu^{15} + 73271914 \nu^{14} + 163062311 \nu^{13} + \cdots + 39621994 ) / 8655106 $ (870813v^17 - 3033149v^16 - 19402563v^15 + 73271914v^14 + 163062311v^13 - 701176099v^12 - 623550578v^11 + 3385844207v^10 + 942614544v^9 - 8697601105v^8 + 175276449v^7 + 11465269759v^6 - 1412926912v^5 - 6833192315v^4 + 246169214v^3 + 1478840262v^2 + 262624594*v + 39621994) / 8655106
$\beta_{11}$	$=$	$ ( 2172011 \nu^{17} - 4934604 \nu^{16} - 57254415 \nu^{15} + 122228351 \nu^{14} + 620355732 \nu^{13} + \cdots + 547325784 ) / 17310212 $ (2172011v^17 - 4934604v^16 - 57254415v^15 + 122228351v^14 + 620355732v^13 - 1212202237v^12 - 3589511029v^11 + 6172190758v^10 + 12072754710v^9 - 17208963633v^8 - 24057580418v^7 + 25833441061v^6 + 27680230503v^5 - 18654446728v^4 - 16961256348v^3 + 4248411632v^2 + 4385523658*v + 547325784) / 17310212
$\beta_{12}$	$=$	$ ( - 1412139 \nu^{17} + 2759313 \nu^{16} + 37784959 \nu^{15} - 67272692 \nu^{14} - 415176825 \nu^{13} + \cdots - 238817406 ) / 8655106 $ (-1412139v^17 + 2759313v^16 + 37784959v^15 - 67272692v^14 - 415176825v^13 + 652347529v^12 + 2422989002v^11 - 3215790521v^10 - 8101395364v^9 + 8553876469v^8 + 15559974451v^7 - 11972992943v^6 - 16275952098v^5 + 7692601359v^4 + 8225920740v^3 - 1274068280v^2 - 1638615510*v - 238817406) / 8655106
$\beta_{13}$	$=$	$ ( - 1412019 \nu^{17} + 5494307 \nu^{16} + 28955834 \nu^{15} - 129885278 \nu^{14} - 204157033 \nu^{13} + \cdots + 239590942 ) / 8655106 $ (-1412019v^17 + 5494307v^16 + 28955834v^15 - 129885278v^14 - 204157033v^13 + 1204272526v^12 + 436001482v^11 - 5541738709v^10 + 1252562487v^9 + 13173455103v^8 - 7494401204v^7 - 15183703738v^6 + 12043662899v^5 + 7115561525v^4 - 6641795912v^3 - 1548008000v^2 + 1216991632*v + 239590942) / 8655106
$\beta_{14}$	$=$	$ ( 1613979 \nu^{17} - 7015797 \nu^{16} - 31883760 \nu^{15} + 167766567 \nu^{14} + 205502471 \nu^{13} + \cdots - 297472862 ) / 8655106 $ (1613979v^17 - 7015797v^16 - 31883760v^15 + 167766567v^14 + 205502471v^13 - 1582940924v^12 - 250642325v^11 + 7492954365v^10 - 2503879729v^9 - 18709447578v^8 + 10879951810v^7 + 23716389037v^6 - 15972796554v^5 - 13649055788v^4 + 8029654174v^3 + 3521009816v^2 - 1072984312*v - 297472862) / 8655106
$\beta_{15}$	$=$	$ ( - 1740463 \nu^{17} + 6102749 \nu^{16} + 38885676 \nu^{15} - 147290343 \nu^{14} - 329077721 \nu^{13} + \cdots - 43868350 ) / 8655106 $ (-1740463v^17 + 6102749v^16 + 38885676v^15 - 147290343v^14 - 329077721v^13 + 1408087416v^12 + 1283554233v^11 - 6795292397v^10 - 2108436645v^9 + 17479801312v^8 + 400301490v^7 - 23233156855v^6 + 1419906454v^5 + 14283007824v^4 + 836788936v^3 - 3418543606v^2 - 973053192*v - 43868350) / 8655106
$\beta_{16}$	$=$	$ ( - 1814482 \nu^{17} + 6186612 \nu^{16} + 40308086 \nu^{15} - 148177865 \nu^{14} - 336742404 \nu^{13} + \cdots + 194458614 ) / 8655106 $ (-1814482v^17 + 6186612v^16 + 40308086v^15 - 148177865v^14 - 336742404v^13 + 1402238426v^12 + 1266225703v^11 - 6674822568v^10 - 1758453438v^9 + 16857338927v^8 - 1134599382v^7 - 21905743275v^6 + 4577755507v^5 + 13292663783v^4 - 2217653358v^3 - 3544809292v^2 + 50031272*v + 194458614) / 8655106
$\beta_{17}$	$=$	$ ( 3825863 \nu^{17} - 11750022 \nu^{16} - 91459471 \nu^{15} + 285997777 \nu^{14} + 868124882 \nu^{13} + \cdots + 543688764 ) / 17310212 $ (3825863v^17 - 11750022v^16 - 91459471v^15 + 285997777v^14 + 868124882v^13 - 2766904757v^12 - 4205477675v^11 + 13590072516v^10 + 11275251570v^9 - 35930753321v^8 - 17639487940v^7 + 49918468081v^6 + 18090271893v^5 - 32683891182v^4 - 13449627508v^3 + 7516525160v^2 + 4490857882*v + 543688764) / 17310212

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 3 $ b2 + 3
$\nu^{3}$	$=$	$ \beta_{3} + 5\beta_1 $ b3 + 5*b1
$\nu^{4}$	$=$	$ \beta_{4} + 7\beta_{2} + 15 $ b4 + 7*b2 + 15
$\nu^{5}$	$=$	$ \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} + 9\beta_{3} + \beta_{2} + 30\beta _1 + 1 $ b12 + b11 + b10 - b9 - b5 + 9b3 + b2 + 30b1 + 1
$\nu^{6}$	$=$	$ - \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 88 $ -b15 - b14 + b12 + b11 + b10 - b9 + b8 - b7 - 2b5 + 11b4 + 48*b2 + b1 + 88
$\nu^{7}$	$=$	$ \beta_{17} + 2 \beta_{16} - 2 \beta_{14} - 3 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} + 12 \beta_{10} + \cdots + 16 $ b17 + 2b16 - 2b14 - 3b13 + 13b12 + 13b11 + 12b10 - 11b9 - b7 - b6 - 14b5 + b4 + 67b3 + 14b2 + 193*b1 + 16
$\nu^{8}$	$=$	$ \beta_{17} + 2 \beta_{16} - 16 \beta_{15} - 16 \beta_{14} - \beta_{13} + 15 \beta_{12} + 14 \beta_{11} + \cdots + 555 $ b17 + 2b16 - 16b15 - 16b14 - b13 + 15b12 + 14b11 + 16b10 - 15b9 + 14b8 - 15b7 - 2b6 - 32b5 + 95b4 + 3b3 + 334b2 + 15*b1 + 555
$\nu^{9}$	$=$	$ 17 \beta_{17} + 32 \beta_{16} - \beta_{15} - 37 \beta_{14} - 52 \beta_{13} + 126 \beta_{12} + 126 \beta_{11} + \cdots + 174 $ 17b17 + 32b16 - b15 - 37b14 - 52b13 + 126b12 + 126b11 + 110b10 - 92b9 + 2b8 - 13b7 - 18b6 - 144b5 + 18b4 + 474b3 + 152b2 + 1285b1 + 174
$\nu^{10}$	$=$	$ 24 \beta_{17} + 35 \beta_{16} - 173 \beta_{15} - 180 \beta_{14} - 22 \beta_{13} + 164 \beta_{12} + \cdots + 3646 $ 24b17 + 35b16 - 173b15 - 180b14 - 22b13 + 164b12 + 141b11 + 177b10 - 160b9 + 140b8 - 158b7 - 36b6 - 357b5 + 758b4 + 58b3 + 2360b2 + 168*b1 + 3646
$\nu^{11}$	$=$	$ 199 \beta_{17} + 348 \beta_{16} - 17 \beta_{15} - 457 \beta_{14} - 608 \beta_{13} + 1098 \beta_{12} + \cdots + 1658 $ 199b17 + 348b16 - 17b15 - 457b14 - 608b13 + 1098b12 + 1097b11 + 929b10 - 707b9 + 46b8 - 114b7 - 217b6 - 1319b5 + 223b4 + 3304b3 + 1491b2 + 8735*b1 + 1658
$\nu^{12}$	$=$	$ 352 \beta_{17} + 411 \beta_{16} - 1586 \beta_{15} - 1761 \beta_{14} - 317 \beta_{13} + 1584 \beta_{12} + \cdots + 24615 $ 352b17 + 411b16 - 1586b15 - 1761b14 - 317b13 + 1584b12 + 1261b11 + 1703b10 - 1490b9 + 1237b8 - 1438b7 - 434b6 - 3441b5 + 5843b4 + 734b3 + 16914b2 + 1705*b1 + 24615
$\nu^{13}$	$=$	$ 2008 \beta_{17} + 3234 \beta_{16} - 187 \beta_{15} - 4759 \beta_{14} - 6033 \beta_{13} + 9098 \beta_{12} + \cdots + 14882 $ 2008b17 + 3234b16 - 187b15 - 4759b14 - 6033b13 + 9098b12 + 9065b11 + 7605b10 - 5270b9 + 667b8 - 851b7 - 2212b6 - 11406b5 + 2356b4 + 22982b3 + 13815b2 + 60272*b1 + 14882
$\nu^{14}$	$=$	$ 4132 \beta_{17} + 4088 \beta_{16} - 13322 \beta_{15} - 16046 \beta_{14} - 3777 \beta_{13} + \cdots + 169687 $ 4132b17 + 4088b16 - 13322b15 - 16046b14 - 3777b13 + 14333b12 + 10699b11 + 15311b10 - 12924b9 + 10325b8 - 12104b7 - 4435b6 - 30750b5 + 44276b4 + 7752b3 + 122797b2 + 16463*b1 + 169687
$\nu^{15}$	$=$	$ 18770 \beta_{17} + 27735 \beta_{16} - 1720 \beta_{15} - 45215 \beta_{14} - 54884 \beta_{13} + \cdots + 129274 $ 18770b17 + 27735b16 - 1720b15 - 45215b14 - 54884b13 + 73369b12 + 72728b11 + 61411b10 - 38911b9 + 7822b8 - 5865b7 - 20629b6 - 95444b5 + 22823b4 + 160374b3 + 123295b2 + 420988*b1 + 129274
$\nu^{16}$	$=$	$ 42802 \beta_{17} + 37277 \beta_{16} - 106255 \beta_{15} - 140290 \beta_{14} - 40349 \beta_{13} + \cdots + 1190213 $ 42802b17 + 37277b16 - 106255b15 - 140290b14 - 40349b13 + 124643b12 + 88514b11 + 132495b10 - 107445b9 + 83696b8 - 97178b7 - 41625b6 - 262937b5 + 332557b4 + 74348b3 + 901853b2 + 153689*b1 + 1190213
$\nu^{17}$	$=$	$ 167614 \beta_{17} + 227228 \beta_{16} - 14592 \beta_{15} - 406314 \beta_{14} - 474075 \beta_{13} + \cdots + 1099782 $ 167614b17 + 227228b16 - 14592b15 - 406314b14 - 474075b13 + 582521b12 + 572993b11 + 492490b10 - 287313b9 + 81269b8 - 38911b7 - 182403b6 - 782561b5 + 209639b4 + 1125602b3 + 1071334b2 + 2972215*b1 + 1099782

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

−2.62138
−2.48986
−2.10676
−1.73135
−1.55076
−0.698229
−0.603996
−0.396252
−0.173863
0.844677
0.925440
1.33141
1.46257
1.69075
2.34662
2.43391
2.53544
2.80164

−2.62138

1.00000

4.87165

1.00000

−2.62138

2.00076

−7.52769

1.00000

−2.62138

1.2

−2.48986

1.00000

4.19938

1.00000

−2.48986

−2.76131

−5.47615

1.00000

−2.48986

1.3

−2.10676

1.00000

2.43846

1.00000

−2.10676

3.31927

−0.923726

1.00000

−2.10676

1.4

−1.73135

1.00000

0.997580

1.00000

−1.73135

0.816002

1.73554

1.00000

−1.73135

1.5

−1.55076

1.00000

0.404854

1.00000

−1.55076

−3.31793

2.47369

1.00000

−1.55076

1.6

−0.698229

1.00000

−1.51248

1.00000

−0.698229

4.99716

2.45251

1.00000

−0.698229

1.7

−0.603996

1.00000

−1.63519

1.00000

−0.603996

0.896809

2.19564

1.00000

−0.603996

1.8

−0.396252

1.00000

−1.84298

1.00000

−0.396252

0.499526

1.52279

1.00000

−0.396252

1.9

−0.173863

1.00000

−1.96977

1.00000

−0.173863

−4.18769

0.690195

1.00000

−0.173863

1.10

0.844677

1.00000

−1.28652

1.00000

0.844677

−2.49791

−2.77605

1.00000

0.844677

1.11

0.925440

1.00000

−1.14356

1.00000

0.925440

−0.122708

−2.90918

1.00000

0.925440

1.12

1.33141

1.00000

−0.227354

1.00000

1.33141

4.86785

−2.96552

1.00000

1.33141

1.13

1.46257

1.00000

0.139121

1.00000

1.46257

2.97636

−2.72167

1.00000

1.46257

1.14

1.69075

1.00000

0.858621

1.00000

1.69075

0.189464

−1.92978

1.00000

1.69075

1.15

2.34662

1.00000

3.50662

1.00000

2.34662

−4.52573

3.53546

1.00000

2.34662

1.16

2.43391

1.00000

3.92391

1.00000

2.43391

2.58128

4.68261

1.00000

2.43391

1.17

2.53544

1.00000

4.42846

1.00000

2.53544

3.07784

6.15722

1.00000

2.53544

1.18

2.80164

1.00000

5.84921

1.00000

2.80164

−0.809034

10.7841

1.00000

2.80164

Atkin-Lehner signs

$ p $	Sign
$3$	$-1$
$5$	$-1$
$13$	$1$
$31$	$-1$

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	6045.2.a.bi	✓	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6045.2.a.bi	✓	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 $S_{2}^{\mathrm{new}}(\Gamma_0(6045))$:

$ T_{2}^{18} - 4 T_{2}^{17} - 21 T_{2}^{16} + 97 T_{2}^{15} + 156 T_{2}^{14} - 935 T_{2}^{13} - 411 T_{2}^{12} + \cdots - 112 $

$ T_{7}^{18} - 8 T_{7}^{17} - 45 T_{7}^{16} + 491 T_{7}^{15} + 368 T_{7}^{14} - 11282 T_{7}^{13} + \cdots - 11392 $

$ T_{11}^{18} - 6 T_{11}^{17} - 93 T_{11}^{16} + 580 T_{11}^{15} + 3069 T_{11}^{14} - 20204 T_{11}^{13} + \cdots + 693248 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} - 4 T^{17} + \cdots - 112 $ T^18 - 4T^17 - 21T^16 + 97T^15 + 156T^14 - 935T^13 - 411T^12 + 4582T^11 - 446T^10 - 12159T^9 + 4398T^8 + 17347T^7 - 7839T^6 - 12832T^5 + 4744T^4 + 5064T^3 - 746T^2 - 892*T - 112
$3$	$ (T - 1)^{18} $ (T - 1)^18
$5$	$ (T - 1)^{18} $ (T - 1)^18
$7$	$ T^{18} - 8 T^{17} + \cdots - 11392 $ T^18 - 8T^17 - 45T^16 + 491T^15 + 368T^14 - 11282T^13 + 10537T^12 + 118380T^11 - 228161T^10 - 523667T^9 + 1602728T^8 + 289187T^7 - 4070968T^6 + 3080409T^5 + 1170986T^4 - 2030612T^3 + 586104T^2 + 11008*T - 11392
$11$	$ T^{18} - 6 T^{17} + \cdots + 693248 $ T^18 - 6T^17 - 93T^16 + 580T^15 + 3069T^14 - 20204T^13 - 45733T^12 + 318802T^11 + 364513T^10 - 2376168T^9 - 2364908T^8 + 8596104T^7 + 11089859T^6 - 10818984T^5 - 23369928T^4 - 8323264T^3 + 4871408T^2 + 3864832*T + 693248
$13$	$ (T + 1)^{18} $ (T + 1)^18
$17$	$ T^{18} + \cdots + 1181187584 $ T^18 - 18T^17 + 10T^16 + 1704T^15 - 9736T^14 - 36802T^13 + 461525T^12 - 437366T^11 - 8053515T^10 + 25104739T^9 + 46770822T^8 - 320884308T^7 + 181259656T^6 + 1515270424T^5 - 2794091104T^4 - 1074470016T^3 + 6908382208T^2 - 5848755072*T + 1181187584
$19$	$ T^{18} + \cdots + 613057408 $ T^18 - 12T^17 - 117T^16 + 1862T^15 + 3095T^14 - 110370T^13 + 120249T^12 + 3087960T^11 - 7977043T^10 - 40176888T^9 + 153291107T^8 + 188693824T^7 - 1169661808T^6 + 145349410T^5 + 2899479866T^4 - 874575816T^3 - 2553710696T^2 + 607271104*T + 613057408
$23$	$ T^{18} + \cdots + 42281900032 $ T^18 - 32T^17 + 273T^16 + 2041T^15 - 50955T^14 + 249676T^13 + 1311362T^12 - 19575697T^11 + 63066147T^10 + 216966281T^9 - 2370036122T^8 + 6700424791T^7 + 3193601154T^6 - 76965670080T^5 + 247992869872T^4 - 417360403152T^3 + 401033972320T^2 - 205032063488*T + 42281900032
$29$	$ T^{18} - 7 T^{17} + \cdots - 30800128 $ T^18 - 7T^17 - 202T^16 + 1308T^15 + 14591T^14 - 92054T^13 - 449307T^12 + 3077789T^11 + 5416149T^10 - 50161228T^9 - 5830670T^8 + 385663307T^7 - 327432059T^6 - 1132118811T^5 + 1772958865T^4 + 61522480T^3 - 1046369084T^2 + 418159616*T - 30800128
$31$	$ (T - 1)^{18} $ (T - 1)^18
$37$	$ T^{18} - 3 T^{17} + \cdots + 13057664 $ T^18 - 3T^17 - 274T^16 + 364T^15 + 28487T^14 - 1731T^13 - 1413571T^12 - 1193539T^11 + 34651292T^10 + 48748643T^9 - 401307851T^8 - 643902274T^7 + 2174740464T^6 + 3404991232T^5 - 4950340712T^4 - 6736769888T^3 + 2941792192T^2 + 3633390336*T + 13057664
$41$	$ T^{18} + \cdots + 34149338888 $ T^18 - 4T^17 - 288T^16 + 1498T^15 + 29944T^14 - 194609T^13 - 1331034T^12 + 11198630T^11 + 21621984T^10 - 304505105T^9 + 67926677T^8 + 3920657657T^7 - 4905783713T^6 - 22427935600T^5 + 37831953179T^4 + 53028510961T^3 - 86376979222T^2 - 48846836628*T + 34149338888
$43$	$ T^{18} + \cdots - 118767872 $ T^18 - 14T^17 - 225T^16 + 3414T^15 + 15629T^14 - 265711T^13 - 579638T^12 + 9441744T^11 + 13928859T^10 - 167634315T^9 - 202486825T^8 + 1456896510T^7 + 1382431989T^6 - 5683472088T^5 - 3047162928T^4 + 8134890848T^3 + 1177844336T^2 - 1972652928*T - 118767872
$47$	$ T^{18} + \cdots + 1087757697024 $ T^18 - 23T^17 - 105T^16 + 6170T^15 - 23246T^14 - 554948T^13 + 4194218T^12 + 18381125T^11 - 240344907T^10 - 53422444T^9 + 6202035073T^8 - 8236886712T^7 - 77164848980T^6 + 143051455638T^5 + 463265124094T^4 - 713103880568T^3 - 1378072416768T^2 + 657265577472*T + 1087757697024
$53$	$ T^{18} + \cdots - 9816052352 $ T^18 - 35T^17 + 201T^16 + 6476T^15 - 97537T^14 + 34340T^13 + 7805711T^12 - 48312411T^11 - 82581193T^10 + 1853024747T^9 - 5605074684T^8 - 8423834046T^7 + 83870505792T^6 - 173453281130T^5 + 88800355712T^4 + 132518858000T^3 - 185433517088T^2 + 75326317664*T - 9816052352
$59$	$ T^{18} + \cdots - 970411965526528 $ T^18 - 28T^17 - 281T^16 + 13417T^15 - 4613T^14 - 2499138T^13 + 8344453T^12 + 240367898T^11 - 1006874043T^10 - 13785287978T^9 + 53195913382T^8 + 504705140565T^7 - 1360604099339T^6 - 11664270665218T^5 + 13271000034140T^4 + 150483440919208T^3 + 46930278129632T^2 - 756753829993088*T - 970411965526528
$61$	$ T^{18} + \cdots - 375396992 $ T^18 - 19T^17 - 280T^16 + 6500T^15 + 20070T^14 - 762812T^13 + 279660T^12 + 36315752T^11 - 65139787T^10 - 630420435T^9 + 1790293216T^8 + 1839300756T^7 - 9520320180T^6 + 4383170058T^5 + 11881339668T^4 - 12611210952T^3 + 943477968T^2 + 2329624832*T - 375396992
$67$	$ T^{18} + \cdots - 174194192384 $ T^18 - 34T^17 - 77T^16 + 13771T^15 - 94397T^14 - 1644796T^13 + 17633020T^12 + 78867465T^11 - 1223185531T^10 - 1506474652T^9 + 39937178581T^8 + 9834960354T^7 - 614986261615T^6 - 57644280668T^5 + 3514341429588T^4 - 447143445472T^3 - 4033801515712T^2 + 1769684958208*T - 174194192384
$71$	$ T^{18} + \cdots - 10523050246144 $ T^18 + 8T^17 - 610T^16 - 5332T^15 + 140196T^14 + 1316309T^13 - 15318407T^12 - 153952565T^11 + 838415717T^10 + 9259287675T^9 - 20468227092T^8 - 282045797172T^7 + 51688711312T^6 + 3780172401536T^5 + 5319090217472T^4 - 12793396511744T^3 - 39158076600320T^2 - 35046480543744*T - 10523050246144
$73$	$ T^{18} + \cdots - 9066168448 $ T^18 - 523T^16 + 60T^15 + 105330T^14 + 13319T^13 - 10475093T^12 - 6681537T^11 + 550020604T^10 + 675914679T^9 - 14922000910T^8 - 26098292950T^7 + 191764726930T^6 + 433983581758T^5 - 834426290932T^4 - 2635745585144T^3 - 1294730448752T^2 + 308826680064T - 9066168448
$79$	$ T^{18} + \cdots + 582240185344 $ T^18 - 4T^17 - 599T^16 + 1981T^15 + 131322T^14 - 267960T^13 - 13421620T^12 + 3044031T^11 + 675376477T^10 + 1152177409T^9 - 14323322296T^8 - 49440670358T^7 + 54226519764T^6 + 398575506370T^5 + 237088612880T^4 - 802159265560T^3 - 816250424320T^2 + 472820597376*T + 582240185344
$83$	$ T^{18} + \cdots - 10567766010368 $ T^18 - 11T^17 - 585T^16 + 8375T^15 + 97665T^14 - 2022245T^13 - 1634548T^12 + 187041722T^11 - 676659403T^10 - 6396126962T^9 + 46198846023T^8 + 18342352995T^7 - 896644027485T^6 + 1931284638222T^5 + 3941646386684T^4 - 18392642352616T^3 + 14335155966432T^2 + 12090693294208*T - 10567766010368
$89$	$ T^{18} + \cdots - 199799780992 $ T^18 - 17T^17 - 389T^16 + 6948T^15 + 47190T^14 - 974691T^13 - 1622588T^12 + 58390947T^11 - 35565993T^10 - 1509677173T^9 + 2042467275T^8 + 18590409498T^7 - 25167688256T^6 - 118410335104T^5 + 111986220296T^4 + 381673118560T^3 - 116578967872T^2 - 496194984192*T - 199799780992
$97$	$ T^{18} + \cdots + 997352445824 $ T^18 - 32T^17 - 58T^16 + 10389T^15 - 44218T^14 - 1401509T^13 + 8627497T^12 + 102880803T^11 - 692471071T^10 - 4482618426T^9 + 28711007253T^8 + 115961119544T^7 - 620648894512T^6 - 1596191202271T^5 + 6376053967210T^4 + 8018181669532T^3 - 25909790826056T^2 + 743445516416*T + 997352445824
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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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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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	6045.2.a.bi

Level	$6045$
Weight	$2$
Character orbit	6045.a
Self dual	yes
Analytic conductor	$48.270$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(48.2695680219\)
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:	\( x^{18} - 4 x^{17} - 21 x^{16} + 97 x^{15} + 156 x^{14} - 935 x^{13} - 411 x^{12} + 4582 x^{11} + \cdots - 112 \) x^18 - 4x^17 - 21x^16 + 97x^15 + 156x^14 - 935x^13 - 411x^12 + 4582x^11 - 446x^10 - 12159x^9 + 4398x^8 + 17347x^7 - 7839x^6 - 12832x^5 + 4744x^4 + 5064x^3 - 746x^2 - 892*x - 112
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \) v^2 - 3
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 5\nu \) v^3 - 5*v
\(\beta_{4}\)	\(=\)	\( \nu^{4} - 7\nu^{2} + 6 \) v^4 - 7*v^2 + 6
\(\beta_{5}\)	\(=\)	\( ( - 228355 \nu^{17} - 336512 \nu^{16} + 8567893 \nu^{15} + 6198769 \nu^{14} - 123630228 \nu^{13} + \cdots + 317132228 ) / 17310212 \) (-228355v^17 - 336512v^16 + 8567893v^15 + 6198769v^14 - 123630228v^13 - 31566349v^12 + 891219629v^11 - 59637650v^10 - 3413604116v^9 + 1071437101v^8 + 6645499448v^7 - 3608265451v^6 - 5229581577v^5 + 4661942660v^4 - 103549980v^3 - 1874991824v^2 + 727071450*v + 317132228) / 17310212
\(\beta_{6}\)	\(=\)	\( ( 446299 \nu^{17} + 1332792 \nu^{16} - 19095959 \nu^{15} - 27974025 \nu^{14} + 299322856 \nu^{13} + \cdots + 152855836 ) / 17310212 \) (446299v^17 + 1332792v^16 - 19095959v^15 - 27974025v^14 + 299322856v^13 + 211957035v^12 - 2313633925v^11 - 639195438v^10 + 9631313854v^9 + 168689975v^8 - 21628657114v^7 + 2831804025v^6 + 24385040543v^5 - 4515463168v^4 - 11484187796v^3 + 1183377184v^2 + 1777245698*v + 152855836) / 17310212
\(\beta_{7}\)	\(=\)	\( ( - 740009 \nu^{17} + 2872262 \nu^{16} + 14791069 \nu^{15} - 67180803 \nu^{14} - 98215806 \nu^{13} + \cdots + 222862852 ) / 17310212 \) (-740009v^17 + 2872262v^16 + 14791069v^15 - 67180803v^14 - 98215806v^13 + 613552943v^12 + 151793377v^11 - 2759586728v^10 + 976237358v^9 + 6315037203v^8 - 4581147060v^7 - 6769396543v^6 + 6955418053v^5 + 2731690702v^4 - 3873662520v^3 - 728115164v^2 + 862479718*v + 222862852) / 17310212
\(\beta_{8}\)	\(=\)	\( ( - 610666 \nu^{17} + 354827 \nu^{16} + 18681397 \nu^{15} - 10014793 \nu^{14} - 234498267 \nu^{13} + \cdots - 175204944 ) / 8655106 \) (-610666v^17 + 354827v^16 + 18681397v^15 - 10014793v^14 - 234498267v^13 + 116139789v^12 + 1554418411v^11 - 711950221v^10 - 5830999753v^9 + 2463590886v^8 + 12312429494v^7 - 4696943709v^6 - 13698626968v^5 + 4235562551v^4 + 6955732814v^3 - 939434104v^2 - 1389743616*v - 175204944) / 8655106
\(\beta_{9}\)	\(=\)	\( ( 658857 \nu^{17} - 2572882 \nu^{16} - 14528758 \nu^{15} + 64014013 \nu^{14} + 119878466 \nu^{13} + \cdots - 101408846 ) / 8655106 \) (658857v^17 - 2572882v^16 - 14528758v^15 + 64014013v^14 + 119878466v^13 - 639146514v^12 - 440926905v^11 + 3285967890v^10 + 584398593v^9 - 9283925003v^8 + 383710967v^7 + 14213130072v^6 - 1242628076v^5 - 10798785650v^4 + 121132724v^3 + 3275128816v^2 + 323408598*v - 101408846) / 8655106
\(\beta_{10}\)	\(=\)	\( ( 870813 \nu^{17} - 3033149 \nu^{16} - 19402563 \nu^{15} + 73271914 \nu^{14} + 163062311 \nu^{13} + \cdots + 39621994 ) / 8655106 \) (870813v^17 - 3033149v^16 - 19402563v^15 + 73271914v^14 + 163062311v^13 - 701176099v^12 - 623550578v^11 + 3385844207v^10 + 942614544v^9 - 8697601105v^8 + 175276449v^7 + 11465269759v^6 - 1412926912v^5 - 6833192315v^4 + 246169214v^3 + 1478840262v^2 + 262624594*v + 39621994) / 8655106
\(\beta_{11}\)	\(=\)	\( ( 2172011 \nu^{17} - 4934604 \nu^{16} - 57254415 \nu^{15} + 122228351 \nu^{14} + 620355732 \nu^{13} + \cdots + 547325784 ) / 17310212 \) (2172011v^17 - 4934604v^16 - 57254415v^15 + 122228351v^14 + 620355732v^13 - 1212202237v^12 - 3589511029v^11 + 6172190758v^10 + 12072754710v^9 - 17208963633v^8 - 24057580418v^7 + 25833441061v^6 + 27680230503v^5 - 18654446728v^4 - 16961256348v^3 + 4248411632v^2 + 4385523658*v + 547325784) / 17310212
\(\beta_{12}\)	\(=\)	\( ( - 1412139 \nu^{17} + 2759313 \nu^{16} + 37784959 \nu^{15} - 67272692 \nu^{14} - 415176825 \nu^{13} + \cdots - 238817406 ) / 8655106 \) (-1412139v^17 + 2759313v^16 + 37784959v^15 - 67272692v^14 - 415176825v^13 + 652347529v^12 + 2422989002v^11 - 3215790521v^10 - 8101395364v^9 + 8553876469v^8 + 15559974451v^7 - 11972992943v^6 - 16275952098v^5 + 7692601359v^4 + 8225920740v^3 - 1274068280v^2 - 1638615510*v - 238817406) / 8655106
\(\beta_{13}\)	\(=\)	\( ( - 1412019 \nu^{17} + 5494307 \nu^{16} + 28955834 \nu^{15} - 129885278 \nu^{14} - 204157033 \nu^{13} + \cdots + 239590942 ) / 8655106 \) (-1412019v^17 + 5494307v^16 + 28955834v^15 - 129885278v^14 - 204157033v^13 + 1204272526v^12 + 436001482v^11 - 5541738709v^10 + 1252562487v^9 + 13173455103v^8 - 7494401204v^7 - 15183703738v^6 + 12043662899v^5 + 7115561525v^4 - 6641795912v^3 - 1548008000v^2 + 1216991632*v + 239590942) / 8655106
\(\beta_{14}\)	\(=\)	\( ( 1613979 \nu^{17} - 7015797 \nu^{16} - 31883760 \nu^{15} + 167766567 \nu^{14} + 205502471 \nu^{13} + \cdots - 297472862 ) / 8655106 \) (1613979v^17 - 7015797v^16 - 31883760v^15 + 167766567v^14 + 205502471v^13 - 1582940924v^12 - 250642325v^11 + 7492954365v^10 - 2503879729v^9 - 18709447578v^8 + 10879951810v^7 + 23716389037v^6 - 15972796554v^5 - 13649055788v^4 + 8029654174v^3 + 3521009816v^2 - 1072984312*v - 297472862) / 8655106
\(\beta_{15}\)	\(=\)	\( ( - 1740463 \nu^{17} + 6102749 \nu^{16} + 38885676 \nu^{15} - 147290343 \nu^{14} - 329077721 \nu^{13} + \cdots - 43868350 ) / 8655106 \) (-1740463v^17 + 6102749v^16 + 38885676v^15 - 147290343v^14 - 329077721v^13 + 1408087416v^12 + 1283554233v^11 - 6795292397v^10 - 2108436645v^9 + 17479801312v^8 + 400301490v^7 - 23233156855v^6 + 1419906454v^5 + 14283007824v^4 + 836788936v^3 - 3418543606v^2 - 973053192*v - 43868350) / 8655106
\(\beta_{16}\)	\(=\)	\( ( - 1814482 \nu^{17} + 6186612 \nu^{16} + 40308086 \nu^{15} - 148177865 \nu^{14} - 336742404 \nu^{13} + \cdots + 194458614 ) / 8655106 \) (-1814482v^17 + 6186612v^16 + 40308086v^15 - 148177865v^14 - 336742404v^13 + 1402238426v^12 + 1266225703v^11 - 6674822568v^10 - 1758453438v^9 + 16857338927v^8 - 1134599382v^7 - 21905743275v^6 + 4577755507v^5 + 13292663783v^4 - 2217653358v^3 - 3544809292v^2 + 50031272*v + 194458614) / 8655106
\(\beta_{17}\)	\(=\)	\( ( 3825863 \nu^{17} - 11750022 \nu^{16} - 91459471 \nu^{15} + 285997777 \nu^{14} + 868124882 \nu^{13} + \cdots + 543688764 ) / 17310212 \) (3825863v^17 - 11750022v^16 - 91459471v^15 + 285997777v^14 + 868124882v^13 - 2766904757v^12 - 4205477675v^11 + 13590072516v^10 + 11275251570v^9 - 35930753321v^8 - 17639487940v^7 + 49918468081v^6 + 18090271893v^5 - 32683891182v^4 - 13449627508v^3 + 7516525160v^2 + 4490857882*v + 543688764) / 17310212

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 3 \) b2 + 3
\(\nu^{3}\)	\(=\)	\( \beta_{3} + 5\beta_1 \) b3 + 5*b1
\(\nu^{4}\)	\(=\)	\( \beta_{4} + 7\beta_{2} + 15 \) b4 + 7*b2 + 15
\(\nu^{5}\)	\(=\)	\( \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} + 9\beta_{3} + \beta_{2} + 30\beta _1 + 1 \) b12 + b11 + b10 - b9 - b5 + 9b3 + b2 + 30b1 + 1
\(\nu^{6}\)	\(=\)	\( - \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 88 \) -b15 - b14 + b12 + b11 + b10 - b9 + b8 - b7 - 2b5 + 11b4 + 48*b2 + b1 + 88
\(\nu^{7}\)	\(=\)	\( \beta_{17} + 2 \beta_{16} - 2 \beta_{14} - 3 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} + 12 \beta_{10} + \cdots + 16 \) b17 + 2b16 - 2b14 - 3b13 + 13b12 + 13b11 + 12b10 - 11b9 - b7 - b6 - 14b5 + b4 + 67b3 + 14b2 + 193*b1 + 16
\(\nu^{8}\)	\(=\)	\( \beta_{17} + 2 \beta_{16} - 16 \beta_{15} - 16 \beta_{14} - \beta_{13} + 15 \beta_{12} + 14 \beta_{11} + \cdots + 555 \) b17 + 2b16 - 16b15 - 16b14 - b13 + 15b12 + 14b11 + 16b10 - 15b9 + 14b8 - 15b7 - 2b6 - 32b5 + 95b4 + 3b3 + 334b2 + 15*b1 + 555
\(\nu^{9}\)	\(=\)	\( 17 \beta_{17} + 32 \beta_{16} - \beta_{15} - 37 \beta_{14} - 52 \beta_{13} + 126 \beta_{12} + 126 \beta_{11} + \cdots + 174 \) 17b17 + 32b16 - b15 - 37b14 - 52b13 + 126b12 + 126b11 + 110b10 - 92b9 + 2b8 - 13b7 - 18b6 - 144b5 + 18b4 + 474b3 + 152b2 + 1285b1 + 174
\(\nu^{10}\)	\(=\)	\( 24 \beta_{17} + 35 \beta_{16} - 173 \beta_{15} - 180 \beta_{14} - 22 \beta_{13} + 164 \beta_{12} + \cdots + 3646 \) 24b17 + 35b16 - 173b15 - 180b14 - 22b13 + 164b12 + 141b11 + 177b10 - 160b9 + 140b8 - 158b7 - 36b6 - 357b5 + 758b4 + 58b3 + 2360b2 + 168*b1 + 3646
\(\nu^{11}\)	\(=\)	\( 199 \beta_{17} + 348 \beta_{16} - 17 \beta_{15} - 457 \beta_{14} - 608 \beta_{13} + 1098 \beta_{12} + \cdots + 1658 \) 199b17 + 348b16 - 17b15 - 457b14 - 608b13 + 1098b12 + 1097b11 + 929b10 - 707b9 + 46b8 - 114b7 - 217b6 - 1319b5 + 223b4 + 3304b3 + 1491b2 + 8735*b1 + 1658
\(\nu^{12}\)	\(=\)	\( 352 \beta_{17} + 411 \beta_{16} - 1586 \beta_{15} - 1761 \beta_{14} - 317 \beta_{13} + 1584 \beta_{12} + \cdots + 24615 \) 352b17 + 411b16 - 1586b15 - 1761b14 - 317b13 + 1584b12 + 1261b11 + 1703b10 - 1490b9 + 1237b8 - 1438b7 - 434b6 - 3441b5 + 5843b4 + 734b3 + 16914b2 + 1705*b1 + 24615
\(\nu^{13}\)	\(=\)	\( 2008 \beta_{17} + 3234 \beta_{16} - 187 \beta_{15} - 4759 \beta_{14} - 6033 \beta_{13} + 9098 \beta_{12} + \cdots + 14882 \) 2008b17 + 3234b16 - 187b15 - 4759b14 - 6033b13 + 9098b12 + 9065b11 + 7605b10 - 5270b9 + 667b8 - 851b7 - 2212b6 - 11406b5 + 2356b4 + 22982b3 + 13815b2 + 60272*b1 + 14882
\(\nu^{14}\)	\(=\)	\( 4132 \beta_{17} + 4088 \beta_{16} - 13322 \beta_{15} - 16046 \beta_{14} - 3777 \beta_{13} + \cdots + 169687 \) 4132b17 + 4088b16 - 13322b15 - 16046b14 - 3777b13 + 14333b12 + 10699b11 + 15311b10 - 12924b9 + 10325b8 - 12104b7 - 4435b6 - 30750b5 + 44276b4 + 7752b3 + 122797b2 + 16463*b1 + 169687
\(\nu^{15}\)	\(=\)	\( 18770 \beta_{17} + 27735 \beta_{16} - 1720 \beta_{15} - 45215 \beta_{14} - 54884 \beta_{13} + \cdots + 129274 \) 18770b17 + 27735b16 - 1720b15 - 45215b14 - 54884b13 + 73369b12 + 72728b11 + 61411b10 - 38911b9 + 7822b8 - 5865b7 - 20629b6 - 95444b5 + 22823b4 + 160374b3 + 123295b2 + 420988*b1 + 129274
\(\nu^{16}\)	\(=\)	\( 42802 \beta_{17} + 37277 \beta_{16} - 106255 \beta_{15} - 140290 \beta_{14} - 40349 \beta_{13} + \cdots + 1190213 \) 42802b17 + 37277b16 - 106255b15 - 140290b14 - 40349b13 + 124643b12 + 88514b11 + 132495b10 - 107445b9 + 83696b8 - 97178b7 - 41625b6 - 262937b5 + 332557b4 + 74348b3 + 901853b2 + 153689*b1 + 1190213
\(\nu^{17}\)	\(=\)	\( 167614 \beta_{17} + 227228 \beta_{16} - 14592 \beta_{15} - 406314 \beta_{14} - 474075 \beta_{13} + \cdots + 1099782 \) 167614b17 + 227228b16 - 14592b15 - 406314b14 - 474075b13 + 582521b12 + 572993b11 + 492490b10 - 287313b9 + 81269b8 - 38911b7 - 182403b6 - 782561b5 + 209639b4 + 1125602b3 + 1071334b2 + 2972215*b1 + 1099782

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

$p$	$F_p(T)$
$2$	\( T^{18} - 4 T^{17} + \cdots - 112 \) T^18 - 4T^17 - 21T^16 + 97T^15 + 156T^14 - 935T^13 - 411T^12 + 4582T^11 - 446T^10 - 12159T^9 + 4398T^8 + 17347T^7 - 7839T^6 - 12832T^5 + 4744T^4 + 5064T^3 - 746T^2 - 892*T - 112
$3$	\( (T - 1)^{18} \) (T - 1)^18
$5$	\( (T - 1)^{18} \) (T - 1)^18
$7$	\( T^{18} - 8 T^{17} + \cdots - 11392 \) T^18 - 8T^17 - 45T^16 + 491T^15 + 368T^14 - 11282T^13 + 10537T^12 + 118380T^11 - 228161T^10 - 523667T^9 + 1602728T^8 + 289187T^7 - 4070968T^6 + 3080409T^5 + 1170986T^4 - 2030612T^3 + 586104T^2 + 11008*T - 11392
$11$	\( T^{18} - 6 T^{17} + \cdots + 693248 \) T^18 - 6T^17 - 93T^16 + 580T^15 + 3069T^14 - 20204T^13 - 45733T^12 + 318802T^11 + 364513T^10 - 2376168T^9 - 2364908T^8 + 8596104T^7 + 11089859T^6 - 10818984T^5 - 23369928T^4 - 8323264T^3 + 4871408T^2 + 3864832*T + 693248
$13$	\( (T + 1)^{18} \) (T + 1)^18
$17$	\( T^{18} + \cdots + 1181187584 \) T^18 - 18T^17 + 10T^16 + 1704T^15 - 9736T^14 - 36802T^13 + 461525T^12 - 437366T^11 - 8053515T^10 + 25104739T^9 + 46770822T^8 - 320884308T^7 + 181259656T^6 + 1515270424T^5 - 2794091104T^4 - 1074470016T^3 + 6908382208T^2 - 5848755072*T + 1181187584
$19$	\( T^{18} + \cdots + 613057408 \) T^18 - 12T^17 - 117T^16 + 1862T^15 + 3095T^14 - 110370T^13 + 120249T^12 + 3087960T^11 - 7977043T^10 - 40176888T^9 + 153291107T^8 + 188693824T^7 - 1169661808T^6 + 145349410T^5 + 2899479866T^4 - 874575816T^3 - 2553710696T^2 + 607271104*T + 613057408
$23$	\( T^{18} + \cdots + 42281900032 \) T^18 - 32T^17 + 273T^16 + 2041T^15 - 50955T^14 + 249676T^13 + 1311362T^12 - 19575697T^11 + 63066147T^10 + 216966281T^9 - 2370036122T^8 + 6700424791T^7 + 3193601154T^6 - 76965670080T^5 + 247992869872T^4 - 417360403152T^3 + 401033972320T^2 - 205032063488*T + 42281900032
$29$	\( T^{18} - 7 T^{17} + \cdots - 30800128 \) T^18 - 7T^17 - 202T^16 + 1308T^15 + 14591T^14 - 92054T^13 - 449307T^12 + 3077789T^11 + 5416149T^10 - 50161228T^9 - 5830670T^8 + 385663307T^7 - 327432059T^6 - 1132118811T^5 + 1772958865T^4 + 61522480T^3 - 1046369084T^2 + 418159616*T - 30800128
$31$	\( (T - 1)^{18} \) (T - 1)^18
$37$	\( T^{18} - 3 T^{17} + \cdots + 13057664 \) T^18 - 3T^17 - 274T^16 + 364T^15 + 28487T^14 - 1731T^13 - 1413571T^12 - 1193539T^11 + 34651292T^10 + 48748643T^9 - 401307851T^8 - 643902274T^7 + 2174740464T^6 + 3404991232T^5 - 4950340712T^4 - 6736769888T^3 + 2941792192T^2 + 3633390336*T + 13057664
$41$	\( T^{18} + \cdots + 34149338888 \) T^18 - 4T^17 - 288T^16 + 1498T^15 + 29944T^14 - 194609T^13 - 1331034T^12 + 11198630T^11 + 21621984T^10 - 304505105T^9 + 67926677T^8 + 3920657657T^7 - 4905783713T^6 - 22427935600T^5 + 37831953179T^4 + 53028510961T^3 - 86376979222T^2 - 48846836628*T + 34149338888
$43$	\( T^{18} + \cdots - 118767872 \) T^18 - 14T^17 - 225T^16 + 3414T^15 + 15629T^14 - 265711T^13 - 579638T^12 + 9441744T^11 + 13928859T^10 - 167634315T^9 - 202486825T^8 + 1456896510T^7 + 1382431989T^6 - 5683472088T^5 - 3047162928T^4 + 8134890848T^3 + 1177844336T^2 - 1972652928*T - 118767872
$47$	\( T^{18} + \cdots + 1087757697024 \) T^18 - 23T^17 - 105T^16 + 6170T^15 - 23246T^14 - 554948T^13 + 4194218T^12 + 18381125T^11 - 240344907T^10 - 53422444T^9 + 6202035073T^8 - 8236886712T^7 - 77164848980T^6 + 143051455638T^5 + 463265124094T^4 - 713103880568T^3 - 1378072416768T^2 + 657265577472*T + 1087757697024
$53$	\( T^{18} + \cdots - 9816052352 \) T^18 - 35T^17 + 201T^16 + 6476T^15 - 97537T^14 + 34340T^13 + 7805711T^12 - 48312411T^11 - 82581193T^10 + 1853024747T^9 - 5605074684T^8 - 8423834046T^7 + 83870505792T^6 - 173453281130T^5 + 88800355712T^4 + 132518858000T^3 - 185433517088T^2 + 75326317664*T - 9816052352
$59$	\( T^{18} + \cdots - 970411965526528 \) T^18 - 28T^17 - 281T^16 + 13417T^15 - 4613T^14 - 2499138T^13 + 8344453T^12 + 240367898T^11 - 1006874043T^10 - 13785287978T^9 + 53195913382T^8 + 504705140565T^7 - 1360604099339T^6 - 11664270665218T^5 + 13271000034140T^4 + 150483440919208T^3 + 46930278129632T^2 - 756753829993088*T - 970411965526528
$61$	\( T^{18} + \cdots - 375396992 \) T^18 - 19T^17 - 280T^16 + 6500T^15 + 20070T^14 - 762812T^13 + 279660T^12 + 36315752T^11 - 65139787T^10 - 630420435T^9 + 1790293216T^8 + 1839300756T^7 - 9520320180T^6 + 4383170058T^5 + 11881339668T^4 - 12611210952T^3 + 943477968T^2 + 2329624832*T - 375396992
$67$	\( T^{18} + \cdots - 174194192384 \) T^18 - 34T^17 - 77T^16 + 13771T^15 - 94397T^14 - 1644796T^13 + 17633020T^12 + 78867465T^11 - 1223185531T^10 - 1506474652T^9 + 39937178581T^8 + 9834960354T^7 - 614986261615T^6 - 57644280668T^5 + 3514341429588T^4 - 447143445472T^3 - 4033801515712T^2 + 1769684958208*T - 174194192384
$71$	\( T^{18} + \cdots - 10523050246144 \) T^18 + 8T^17 - 610T^16 - 5332T^15 + 140196T^14 + 1316309T^13 - 15318407T^12 - 153952565T^11 + 838415717T^10 + 9259287675T^9 - 20468227092T^8 - 282045797172T^7 + 51688711312T^6 + 3780172401536T^5 + 5319090217472T^4 - 12793396511744T^3 - 39158076600320T^2 - 35046480543744*T - 10523050246144
$73$	\( T^{18} + \cdots - 9066168448 \) T^18 - 523T^16 + 60T^15 + 105330T^14 + 13319T^13 - 10475093T^12 - 6681537T^11 + 550020604T^10 + 675914679T^9 - 14922000910T^8 - 26098292950T^7 + 191764726930T^6 + 433983581758T^5 - 834426290932T^4 - 2635745585144T^3 - 1294730448752T^2 + 308826680064T - 9066168448
$79$	\( T^{18} + \cdots + 582240185344 \) T^18 - 4T^17 - 599T^16 + 1981T^15 + 131322T^14 - 267960T^13 - 13421620T^12 + 3044031T^11 + 675376477T^10 + 1152177409T^9 - 14323322296T^8 - 49440670358T^7 + 54226519764T^6 + 398575506370T^5 + 237088612880T^4 - 802159265560T^3 - 816250424320T^2 + 472820597376*T + 582240185344
$83$	\( T^{18} + \cdots - 10567766010368 \) T^18 - 11T^17 - 585T^16 + 8375T^15 + 97665T^14 - 2022245T^13 - 1634548T^12 + 187041722T^11 - 676659403T^10 - 6396126962T^9 + 46198846023T^8 + 18342352995T^7 - 896644027485T^6 + 1931284638222T^5 + 3941646386684T^4 - 18392642352616T^3 + 14335155966432T^2 + 12090693294208*T - 10567766010368
$89$	\( T^{18} + \cdots - 199799780992 \) T^18 - 17T^17 - 389T^16 + 6948T^15 + 47190T^14 - 974691T^13 - 1622588T^12 + 58390947T^11 - 35565993T^10 - 1509677173T^9 + 2042467275T^8 + 18590409498T^7 - 25167688256T^6 - 118410335104T^5 + 111986220296T^4 + 381673118560T^3 - 116578967872T^2 - 496194984192*T - 199799780992
$97$	\( T^{18} + \cdots + 997352445824 \) T^18 - 32T^17 - 58T^16 + 10389T^15 - 44218T^14 - 1401509T^13 + 8627497T^12 + 102880803T^11 - 692471071T^10 - 4482618426T^9 + 28711007253T^8 + 115961119544T^7 - 620648894512T^6 - 1596191202271T^5 + 6376053967210T^4 + 8018181669532T^3 - 25909790826056T^2 + 743445516416*T + 997352445824
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\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(-1\)
\(13\)	\(1\)
\(31\)	\(-1\)