LMFDB - Newform orbit 7581.2.a.bt

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7581,2,Mod(1,7581)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7581, 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("7581.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7581 = 3 \cdot 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7581.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:	$60.5345897723$
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} - 30 x^{16} - x^{15} + 372 x^{14} + 27 x^{13} - 2464 x^{12} - 276 x^{11} + 9402 x^{10} + \cdots - 73 $ x^18 - 30x^16 - x^15 + 372x^14 + 27x^13 - 2464x^12 - 276x^11 + 9402x^10 + 1337x^9 - 20799x^8 - 3063x^7 + 25640x^6 + 2667x^5 - 16170x^4 + 4x^3 + 4113x^2 - 711*x - 73
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
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} + (\beta_{9} + 1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9}+O(q^{10}) $ q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + (b9 + 1) * q^5 + b1 * q^6 + q^7 + (b3 + b1) * q^8 + q^9	$ q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{9} + 1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9} + (\beta_{16} + \beta_{15} + \beta_{14} + \cdots - 1) q^{10}+ \cdots + (\beta_{5} + 1) q^{99}+O(q^{100}) $ q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + (b9 + 1) * q^5 + b1 * q^6 + q^7 + (b3 + b1) * q^8 + q^9 + (b16 + b15 + b14 + b12 + b11 + b9 - b7 - b5 + b4 + b3 + b1 - 1) * q^10 + (b5 + 1) * q^11 + (b2 + 1) * q^12 + (-b4 + 1) * q^13 + b1 * q^14 + (b9 + 1) * q^15 + (-b15 + b14 + b12 - b8 + b2 + 1) * q^16 + (b16 + b15 + b12 + b4 + 1) * q^17 + b1 * q^18 + (-b17 - b13 + b12 - b11 + b9 - b8 + b7 + b4 + b2 + 2) * q^20 + q^21 + (b13 - b12 + b11 - b10 - b7 - b5 - b4 + b1 - 1) * q^22 + (-b16 - b14 + b9 - b2 + b1 + 1) * q^23 + (b3 + b1) * q^24 + (-b16 - b12 + b10 + b9 + b8 - b3 + 2) * q^25 + (2b14 + b13 + b11 - b8 + b6 + b3 + b2 - 1) q^26 + q^27 + (b2 + 1) * q^28 + (-b17 + b13 + b11 + b9 - b7 - b4) * q^29 + (b16 + b15 + b14 + b12 + b11 + b9 - b7 - b5 + b4 + b3 + b1 - 1) * q^30 + (-b16 - b15 + b13 - b12 - b10 - b9 + b8 - b7 - b4) * q^31 + (b17 + b15 - b11 + b10 + 2b8 + b7 + b6 - b5) q^32 + (b5 + 1) * q^33 + (-b17 - b16 - b15 - b14 - b12 + b9 + 2b7 - b6 - b3 - b2 + 2b1) * q^34 + (b9 + 1) * q^35 + (b2 + 1) * q^36 + (-b15 + b14 - b9 - b7 + b3 + b2 + 1) * q^37 + (-b4 + 1) * q^39 + (b17 + b15 - b13 - b11 + b10 - b9 + b8 - b5 + b4 + b3 + 2b1 - 1) q^40 + (2b16 + b15 + b14 + b11 - b10 + b9 - b8 + b4 + b3 + b1 - 2) q^41 + b1 * q^42 + (b16 + b12 - b8 - b7 - b6 - b5 + b4 - b1 + 2) * q^43 + (-b14 - b12 - b11 + b10 - b9 + b8 + b5 - b4 - b3 + b2 - b1 + 3) * q^44 + (b9 + 1) * q^45 + (-b13 + 2b12 + b11 + 2b10 - b8 + 2b7 - b6 + b4 - b3 + b2 + 3) q^46 + (-b15 - b14 - b12 - b11 + b10 - b9 + 2b7 + b5 - b3 + b1 + 2) q^47 + (-b15 + b14 + b12 - b8 + b2 + 1) * q^48 + q^49 + (2b16 + 2b15 - b13 + 2b12 + b11 - 2b10 + b9 - 2b8 - 3b7 - b5 + b4 + b3 - 2b2 + 2b1) * q^50 + (b16 + b15 + b12 + b4 + 1) * q^51 + (b17 + b16 + b15 + b12 - b11 + b10 - b8 - 2b7 + 2b6 - b4 + b3 + b2) * q^52 + (-b16 - b15 - b12 - b11 - b10 - b9 + b7 - b6 + 2b5 - 1) q^53 + b1 * q^54 + (-b16 - b14 - b13 - b12 - b11 + b10 + b9 + 2b8 + 2b7 + b5 - b3 - b2 - b1 + 4) * q^55 + (b3 + b1) * q^56 + (b17 + b16 + b15 + b13 - 2b12 + b11 - 2b10 - b9 + b8 - b7 + b5 - b4 - 1) * q^58 + (-b17 + b15 - b14 - b13 + b10 + b9 - b8 + b7 + b5 + b4 + b3 - b2 + b1 - 1) * q^59 + (-b17 - b13 + b12 - b11 + b9 - b8 + b7 + b4 + b2 + 2) * q^60 + (-b16 - b15 - b14 - b13 + b12 - b10 - b9 - b8 - b6 + b5 - b4 + b1 + 4) * q^61 + (b17 - 2b15 - b14 - b12 - b11 - 3b10 - 3b9 + b8 + 2b5 - 2b4 - b3 + b2 - b1 + 2) q^62 + q^63 + (-b17 - b15 + b14 - b13 + 2b12 + b11 - 2b10 + b9 - 3b8 + 2b7 + b6 + b4 + b3 - b2 - b1 - 1) * q^64 + (b17 - b16 + b15 + b12 + b11 + b10 + b9 + b8 - b7 + b6 - b5 - b1 + 1) * q^65 + (b13 - b12 + b11 - b10 - b7 - b5 - b4 + b1 - 1) * q^66 + (b17 - 2b15 - 2b13 - b12 - 2b9 + b8 + b7 - b6 + b5 + 2) q^67 + (b17 + b15 - 2b14 - b13 - b10 - b9 + 3b8 - 3b7 - 2b6 - b3 - b1 + 3) * q^68 + (-b16 - b14 + b9 - b2 + b1 + 1) * q^69 + (b16 + b15 + b14 + b12 + b11 + b9 - b7 - b5 + b4 + b3 + b1 - 1) * q^70 + (-b17 + b16 + b13 + b12 - b10 + b9 - b8 - b7 - 1) * q^71 + (b3 + b1) * q^72 + (b17 + 2b13 + b12 + b11 - b10 - 2b7 - b5 - b4 - b3 - 2b1 + 1) q^73 + (b17 - b15 - b12 - 2b11 - 2b10 - b9 + b8 - b7 + b6 - b4 + b3 + 2b2 + 2b1 + 1) * q^74 + (-b16 - b12 + b10 + b9 + b8 - b3 + 2) * q^75 + (b5 + 1) * q^77 + (2b14 + b13 + b11 - b8 + b6 + b3 + b2 - 1) q^78 + (b17 + b16 + b15 - b13 - b12 - b11 - b9 + 2b7 + b6 + 2b2 - b1 + 1) * q^79 + (-b16 - 3b15 + 2b14 + 2b12 + b11 + b10 - 4b8 + 2b2 - b1 + 4) q^80 + q^81 + (-b17 + b16 + 2b15 + b13 - b12 + b10 + 3b9 + 2b8 - 4b7 - b5 + b2 - b1 + 2) * q^82 + (b17 + 2b15 - b14 - 2b12 + b9 + b8 - b5 - b3 - b1 + 3) * q^83 + (b2 + 1) * q^84 + (3b16 + 2b15 - b14 - b13 - b11 + b9 + b8 + b7 + b6 + b4 - b2 + 2b1) q^85 + (-b17 - b16 - 2b14 - b13 - b12 - 2b11 + 3b10 + b9 + 2b8 + 3b7 - 2b6 + b5 + 2b4 - b3 - 2b2 + 3b1 - 2) q^86 + (-b17 + b13 + b11 + b9 - b7 - b4) * q^87 + (-b15 + b14 - b12 - b10 - 2b9 - b8 + 2b5 + b4 + b3 - b2 + 2b1 - 3) q^88 + (b16 - b15 - b13 + 2b12 - b11 + 2b10 - b9 + 4b7 + b6 + b5 + b4 - 2b2 + 2) * q^89 + (b16 + b15 + b14 + b12 + b11 + b9 - b7 - b5 + b4 + b3 + b1 - 1) * q^90 + (-b4 + 1) * q^91 + (-b17 - b16 + b15 - 2b14 - b13 + b12 - 2b11 + 3b10 + 6b7 - b6 + b4 - 2b2 + 5b1 - 1) * q^92 + (-b16 - b15 + b13 - b12 - b10 - b9 + b8 - b7 - b4) * q^93 + (b17 - 2b14 + b13 - 2b12 - b10 - 3b9 + 3b8 - 2b7 - b6 - b4 - b3 + 2b1 + 2) * q^94 + (b17 + b15 - b11 + b10 + 2b8 + b7 + b6 - b5) q^96 + (-2b16 - b15 - 2b14 - b12 - b11 - 3b10 - 3b9 - b7 - b6 - 2b4 - 2b3 + b2 + 1) * q^97 + b1 * q^98 + (b5 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 24 q^{4} + 12 q^{5} + 18 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 24 * q^4 + 12 * q^5 + 18 * q^7 + 3 * q^8 + 18 * q^9	$ 18 q + 18 q^{3} + 24 q^{4} + 12 q^{5} + 18 q^{7} + 3 q^{8} + 18 q^{9} - 6 q^{10} + 18 q^{11} + 24 q^{12} + 9 q^{13} + 12 q^{15} + 24 q^{16} + 18 q^{17} + 36 q^{20} + 18 q^{21} - 9 q^{22} + 6 q^{23} + 3 q^{24} + 36 q^{25} + 27 q^{26} + 18 q^{27} + 24 q^{28} + 3 q^{29} - 6 q^{30} + 6 q^{31} - 9 q^{32} + 18 q^{33} - 15 q^{34} + 12 q^{35} + 24 q^{36} + 36 q^{37} + 9 q^{39} - 18 q^{40} - 21 q^{41} + 27 q^{43} + 42 q^{44} + 12 q^{45} + 54 q^{46} + 27 q^{47} + 24 q^{48} + 18 q^{49} - 12 q^{50} + 18 q^{51} - 3 q^{52} - 21 q^{53} + 45 q^{55} + 3 q^{56} - 9 q^{58} - 15 q^{59} + 36 q^{60} + 51 q^{61} + 15 q^{62} + 18 q^{63} + 3 q^{64} + 24 q^{65} - 9 q^{66} + 24 q^{67} + 12 q^{68} + 6 q^{69} - 6 q^{70} - 24 q^{71} + 3 q^{72} + 9 q^{73} + 15 q^{74} + 36 q^{75} + 18 q^{77} + 27 q^{78} + 30 q^{79} + 102 q^{80} + 18 q^{81} + 33 q^{82} + 48 q^{83} + 24 q^{84} - 27 q^{85} - 78 q^{86} + 3 q^{87} - 24 q^{88} + 15 q^{89} - 6 q^{90} + 9 q^{91} - 54 q^{92} + 6 q^{93} + 21 q^{94} - 9 q^{96} + 3 q^{97} + 18 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 24 * q^4 + 12 * q^5 + 18 * q^7 + 3 * q^8 + 18 * q^9 - 6 * q^10 + 18 * q^11 + 24 * q^12 + 9 * q^13 + 12 * q^15 + 24 * q^16 + 18 * q^17 + 36 * q^20 + 18 * q^21 - 9 * q^22 + 6 * q^23 + 3 * q^24 + 36 * q^25 + 27 * q^26 + 18 * q^27 + 24 * q^28 + 3 * q^29 - 6 * q^30 + 6 * q^31 - 9 * q^32 + 18 * q^33 - 15 * q^34 + 12 * q^35 + 24 * q^36 + 36 * q^37 + 9 * q^39 - 18 * q^40 - 21 * q^41 + 27 * q^43 + 42 * q^44 + 12 * q^45 + 54 * q^46 + 27 * q^47 + 24 * q^48 + 18 * q^49 - 12 * q^50 + 18 * q^51 - 3 * q^52 - 21 * q^53 + 45 * q^55 + 3 * q^56 - 9 * q^58 - 15 * q^59 + 36 * q^60 + 51 * q^61 + 15 * q^62 + 18 * q^63 + 3 * q^64 + 24 * q^65 - 9 * q^66 + 24 * q^67 + 12 * q^68 + 6 * q^69 - 6 * q^70 - 24 * q^71 + 3 * q^72 + 9 * q^73 + 15 * q^74 + 36 * q^75 + 18 * q^77 + 27 * q^78 + 30 * q^79 + 102 * q^80 + 18 * q^81 + 33 * q^82 + 48 * q^83 + 24 * q^84 - 27 * q^85 - 78 * q^86 + 3 * q^87 - 24 * q^88 + 15 * q^89 - 6 * q^90 + 9 * q^91 - 54 * q^92 + 6 * q^93 + 21 * q^94 - 9 * q^96 + 3 * q^97 + 18 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - 30 x^{16} - x^{15} + 372 x^{14} + 27 x^{13} - 2464 x^{12} - 276 x^{11} + 9402 x^{10} + \cdots - 73 $ x^18 - 30*x^16 - x^15 + 372*x^14 + 27*x^13 - 2464*x^12 - 276*x^11 + 9402*x^10 + 1337*x^9 - 20799*x^8 - 3063*x^7 + 25640*x^6 + 2667*x^5 - 16170*x^4 + 4*x^3 + 4113*x^2 - 711*x - 73 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ \nu^{3} - 5\nu $ v^3 - 5*v
$\beta_{4}$	$=$	$ ( - 21082 \nu^{17} + 895597 \nu^{16} - 198070 \nu^{15} - 26519371 \nu^{14} + 13343331 \nu^{13} + \cdots + 131029873 ) / 10631878 $ (-21082v^17 + 895597v^16 - 198070v^15 - 26519371v^14 + 13343331v^13 + 321631338v^12 - 161517411v^11 - 2054095906v^10 + 888273417v^9 + 7367916988v^8 - 2557687788v^7 - 14576988270v^6 + 4079180810v^5 + 14429497614v^4 - 3827061619v^3 - 5620961057v^2 + 1640968884*v + 131029873) / 10631878
$\beta_{5}$	$=$	$ ( - 73770 \nu^{17} - 229885 \nu^{16} + 2975775 \nu^{15} + 6062098 \nu^{14} - 45853139 \nu^{13} + \cdots + 42206054 ) / 21263756 $ (-73770v^17 - 229885v^16 + 2975775v^15 + 6062098v^14 - 45853139v^13 - 62881162v^12 + 351895690v^11 + 319261203v^10 - 1436331921v^9 - 765500678v^8 + 3020590269v^7 + 461784323v^6 - 2828887865v^5 + 1113437269v^4 + 760908392v^3 - 1230676966v^2 + 179254149*v + 42206054) / 21263756
$\beta_{6}$	$=$	$ ( 121846 \nu^{17} + 1026809 \nu^{16} - 3700143 \nu^{15} - 29317769 \nu^{14} + 47090291 \nu^{13} + \cdots + 70641825 ) / 10631878 $ (121846v^17 + 1026809v^16 - 3700143v^15 - 29317769v^14 + 47090291v^13 + 341737237v^12 - 324246257v^11 - 2088523097v^10 + 1305195806v^9 + 7136926421v^8 - 3124106164v^7 - 13411143512v^6 + 4428788937v^5 + 12602437854v^4 - 3683752024v^3 - 4598968566v^2 + 1423008712*v + 70641825) / 10631878
$\beta_{7}$	$=$	$ ( 322745 \nu^{17} - 883328 \nu^{16} - 9035895 \nu^{15} + 25269891 \nu^{14} + 102571089 \nu^{13} + \cdots - 106956766 ) / 21263756 $ (322745v^17 - 883328v^16 - 9035895v^15 + 25269891v^14 + 102571089v^13 - 294009186v^12 - 607050171v^11 + 1787311070v^10 + 2000856991v^9 - 6053117735v^8 - 3600546686v^7 + 11224753484v^6 + 3042700234v^5 - 10372655595v^4 - 336036346v^3 + 3777320133v^2 - 648832661*v - 106956766) / 21263756
$\beta_{8}$	$=$	$ ( - 465048 \nu^{17} - 206149 \nu^{16} + 13279125 \nu^{15} + 5954208 \nu^{14} - 154847379 \nu^{13} + \cdots - 41498894 ) / 21263756 $ (-465048v^17 - 206149v^16 + 13279125v^15 + 5954208v^14 - 154847379v^13 - 71868892v^12 + 946879506v^11 + 461945801v^10 - 3239383737v^9 - 1667957746v^8 + 6126330053v^7 + 3275065591v^6 - 5994963845v^5 - 3117142337v^4 + 2781799410v^3 + 1165078830v^2 - 509056065*v - 41498894) / 21263756
$\beta_{9}$	$=$	$ ( 498578 \nu^{17} - 807750 \nu^{16} - 15109441 \nu^{15} + 20826079 \nu^{14} + 189739036 \nu^{13} + \cdots + 122424649 ) / 21263756 $ (498578v^17 - 807750v^16 - 15109441v^15 + 20826079v^14 + 189739036v^13 - 210806935v^12 - 1276282036v^11 + 1053686784v^10 + 4944010631v^9 - 2647690845v^8 - 10971110616v^7 + 2863134679v^6 + 12944556289v^5 - 356658119v^4 - 6731483635v^3 - 952449274v^2 + 838575398*v + 122424649) / 21263756
$\beta_{10}$	$=$	$ ( 568478 \nu^{17} + 465048 \nu^{16} - 16848191 \nu^{15} - 13847603 \nu^{14} + 205519608 \nu^{13} + \cdots + 104868207 ) / 21263756 $ (568478v^17 + 465048v^16 - 16848191v^15 - 13847603v^14 + 205519608v^13 + 170196285v^12 - 1328860900v^11 - 1103779434v^10 + 4882884355v^9 + 3999438823v^8 - 10155816176v^7 - 7867578167v^6 + 11300710329v^5 + 7511094671v^4 - 6075146923v^3 - 2779525498v^2 + 1173071184*v + 104868207) / 21263756
$\beta_{11}$	$=$	$ ( 313682 \nu^{17} + 211500 \nu^{16} - 9415057 \nu^{15} - 5734413 \nu^{14} + 116223708 \nu^{13} + \cdots + 29692336 ) / 10631878 $ (313682v^17 + 211500v^16 - 9415057v^15 - 5734413v^14 + 116223708v^13 + 64111552v^12 - 760326239v^11 - 382170061v^10 + 2831095669v^9 + 1306229065v^8 - 5991410659v^7 - 2534842111v^6 + 6777624852v^5 + 2535342606v^4 - 3485085038v^3 - 999250720v^2 + 495637232*v + 29692336) / 10631878
$\beta_{12}$	$=$	$ ( - 883328 \nu^{17} + 646455 \nu^{16} + 25592636 \nu^{15} - 17490051 \nu^{14} - 302723301 \nu^{13} + \cdots + 23560385 ) / 21263756 $ (-883328v^17 + 646455v^16 + 25592636v^15 - 17490051v^14 - 302723301v^13 + 188193509v^12 + 1876388690v^11 - 1033591499v^10 - 6484627800v^9 + 3112226569v^8 + 12213321419v^7 - 5232481566v^6 - 11233416510v^5 + 4882750304v^4 + 3776029153v^3 - 1976282846v^2 + 122514929*v + 23560385) / 21263756
$\beta_{13}$	$=$	$ ( - 965288 \nu^{17} + 2129159 \nu^{16} + 27106627 \nu^{15} - 59986280 \nu^{14} - 307135869 \nu^{13} + \cdots + 282231500 ) / 21263756 $ (-965288v^17 + 2129159v^16 + 27106627v^15 - 59986280v^14 - 307135869v^13 + 686665428v^12 + 1788891648v^11 - 4117396731v^10 - 5589733067v^9 + 13902670864v^8 + 8580821381v^7 - 26435239309v^6 - 3805600037v^5 + 26490984157v^4 - 3788580772v^3 - 10969915692v^2 + 3106426489*v + 282231500) / 21263756
$\beta_{14}$	$=$	$ ( - 1081005 \nu^{17} - 1575433 \nu^{16} + 32741902 \nu^{15} + 46836591 \nu^{14} - 409370306 \nu^{13} + \cdots - 218132860 ) / 21263756 $ (-1081005v^17 - 1575433v^16 + 32741902v^15 + 46836591v^14 - 409370306v^13 - 571652490v^12 + 2723480963v^11 + 3673878043v^10 - 10357220432v^9 - 13232297485v^8 + 22529833595v^7 + 26181414783v^6 - 26786819001v^5 - 25773886148v^4 + 16064272004v^3 + 9927405519v^2 - 3778636524*v - 218132860) / 21263756
$\beta_{15}$	$=$	$ ( - 1499285 \nu^{17} - 722829 \nu^{16} + 45055413 \nu^{15} + 23392332 \nu^{14} - 557246228 \nu^{13} + \cdots - 280656117 ) / 21263756 $ (-1499285v^17 - 722829v^16 + 45055413v^15 + 23392332v^14 - 557246228v^13 - 311590089v^12 + 3652990147v^11 + 2178340743v^10 - 13602464495v^9 - 8452113170v^8 + 28616824961v^7 + 17673867626v^6 - 32025271666v^5 - 17795257263v^4 + 17058501747v^3 + 6934890135v^2 - 3147065530*v - 280656117) / 21263756
$\beta_{16}$	$=$	$ ( 1821065 \nu^{17} - 1019951 \nu^{16} - 53789719 \nu^{15} + 26542626 \nu^{14} + 652916130 \nu^{13} + \cdots + 24268763 ) / 21263756 $ (1821065v^17 - 1019951v^16 - 53789719v^15 + 26542626v^14 + 652916130v^13 - 270305967v^12 - 4196750633v^11 + 1363170411v^10 + 15311799363v^9 - 3548222382v^8 - 31480349789v^7 + 4585279416v^6 + 33914786288v^5 - 2815324915v^4 - 16093861251v^3 + 641427271v^2 + 1805053370*v + 24268763) / 21263756
$\beta_{17}$	$=$	$ ( 924030 \nu^{17} - 153548 \nu^{16} - 27091815 \nu^{15} + 3252887 \nu^{14} + 325631992 \nu^{13} + \cdots + 163024770 ) / 10631878 $ (924030v^17 - 153548v^16 - 27091815v^15 + 3252887v^14 + 325631992v^13 - 19495879v^12 - 2065551181v^11 - 26898353v^10 + 7406479214v^9 + 707406092v^8 - 14913570463v^7 - 2683393500v^6 + 15780918257v^5 + 3935174817v^4 - 7611392491v^3 - 2147041852v^2 + 1110307699*v + 163024770) / 10631878

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_{15} + \beta_{14} + \beta_{12} - \beta_{8} + 7\beta_{2} + 15 $ -b15 + b14 + b12 - b8 + 7*b2 + 15
$\nu^{5}$	$=$	$ \beta_{17} + \beta_{15} - \beta_{11} + \beta_{10} + 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 8\beta_{3} + 28\beta_1 $ b17 + b15 - b11 + b10 + 2b8 + b7 + b6 - b5 + 8b3 + 28*b1
$\nu^{6}$	$=$	$ - \beta_{17} - 11 \beta_{15} + 11 \beta_{14} - \beta_{13} + 12 \beta_{12} + \beta_{11} - 2 \beta_{10} + \cdots + 85 $ -b17 - 11b15 + 11b14 - b13 + 12b12 + b11 - 2b10 + b9 - 13b8 + 2b7 + b6 + b4 + b3 + 45*b2 - b1 + 85
$\nu^{7}$	$=$	$ 13 \beta_{17} + 14 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - 12 \beta_{11} + 15 \beta_{10} + \cdots - 4 $ 13b17 + 14b15 - b14 - b13 + b12 - 12b11 + 15b10 + 27b8 + 11b7 + 12b6 - 12b5 + 56b3 - b2 + 166b1 - 4
$\nu^{8}$	$=$	$ - 15 \beta_{17} - 3 \beta_{16} - 97 \beta_{15} + 94 \beta_{14} - 12 \beta_{13} + 105 \beta_{12} + \cdots + 510 $ -15b17 - 3b16 - 97b15 + 94b14 - 12b13 + 105b12 + 14b11 - 28b10 + 15b9 - 125b8 + 30b7 + 12b6 + 13b4 + 11b3 + 290b2 - 17b1 + 510
$\nu^{9}$	$=$	$ 124 \beta_{17} + 5 \beta_{16} + 145 \beta_{15} - 19 \beta_{14} - 16 \beta_{13} + 18 \beta_{12} + \cdots - 63 $ 124b17 + 5b16 + 145b15 - 19b14 - 16b13 + 18b12 - 109b11 + 157b10 - 3b9 + 262b8 + 87b7 + 105b6 - 107b5 + b4 + 384b3 - 22b2 + 1023b1 - 63
$\nu^{10}$	$=$	$ - 164 \beta_{17} - 55 \beta_{16} - 794 \beta_{15} + 739 \beta_{14} - 103 \beta_{13} + 819 \beta_{12} + \cdots + 3174 $ -164b17 - 55b16 - 794b15 + 739b14 - 103b13 + 819b12 + 143b11 - 272b10 + 158b9 - 1074b8 + 320b7 + 101b6 + 4b5 + 124b4 + 80b3 + 1899b2 - 196*b1 + 3174
$\nu^{11}$	$=$	$ 1059 \beta_{17} + 99 \beta_{16} + 1339 \beta_{15} - 248 \beta_{14} - 175 \beta_{13} + 201 \beta_{12} + \cdots - 695 $ 1059b17 + 99b16 + 1339b15 - 248b14 - 175b13 + 201b12 - 899b11 + 1412b10 - 59b9 + 2257b8 + 584b7 + 819b6 - 861b5 + 13b4 + 2632b3 - 303b2 + 6500*b1 - 695
$\nu^{12}$	$=$	$ - 1579 \beta_{17} - 668 \beta_{16} - 6281 \beta_{15} + 5611 \beta_{14} - 775 \beta_{13} + 6050 \beta_{12} + \cdots + 20319 $ -1579b17 - 668b16 - 6281b15 + 5611b14 - 775b13 + 6050b12 + 1305b11 - 2304b10 + 1447b9 - 8741b8 + 2964b7 + 733b6 + 94b5 + 1057b4 + 451b3 + 12650b2 - 1939*b1 + 20319
$\nu^{13}$	$=$	$ 8593 \beta_{17} + 1282 \beta_{16} + 11652 \beta_{15} - 2721 \beta_{14} - 1631 \beta_{13} + 1817 \beta_{12} + \cdots - 6682 $ 8593b17 + 1282b16 + 11652b15 - 2721b14 - 1631b13 + 1817b12 - 7102b11 + 11747b10 - 760b9 + 18423b8 + 3427b7 + 6062b6 - 6622b5 + 90b4 + 18121b3 - 3399b2 + 42370*b1 - 6682
$\nu^{14}$	$=$	$ - 14183 \beta_{17} - 6789 \beta_{16} - 48804 \beta_{15} + 41947 \beta_{14} - 5430 \beta_{13} + \cdots + 133160 $ -14183b17 - 6789b16 - 48804b15 + 41947b14 - 5430b13 + 43410b12 + 11282b11 - 18370b10 + 12333b9 - 69082b8 + 25446b7 + 4882b6 + 1370b5 + 8533b4 + 1763b3 + 85624b2 - 17771*b1 + 133160
$\nu^{15}$	$=$	$ 67869 \beta_{17} + 13771 \beta_{16} + 97802 \beta_{15} - 26961 \beta_{14} - 13952 \beta_{13} + \cdots - 60025 $ 67869b17 + 13771b16 + 97802b15 - 26961b14 - 13952b13 + 14546b12 - 54916b11 + 93548b10 - 8159b9 + 146374b8 + 16908b7 + 43726b6 - 49798b5 + 229b4 + 125519b3 - 34060b2 + 282276*b1 - 60025
$\nu^{16}$	$=$	$ - 121945 \beta_{17} - 62619 \beta_{16} - 375476 \beta_{15} + 311527 \beta_{14} - 36256 \beta_{13} + \cdots + 890244 $ -121945b17 - 62619b16 - 375476b15 + 311527b14 - 36256b13 + 306424b12 + 94532b11 - 142594b10 + 100777b9 - 536800b8 + 208616b7 + 30488b6 + 16050b5 + 66781b4 - 541b3 + 588001b2 - 155871*b1 + 890244
$\nu^{17}$	$=$	$ 527909 \beta_{17} + 133293 \beta_{16} + 801422 \beta_{15} - 250083 \beta_{14} - 113350 \beta_{13} + \cdots - 518709 $ 527909b17 + 133293b16 + 801422b15 - 250083b14 - 113350b13 + 107126b12 - 419962b11 + 726076b10 - 79349b9 + 1146544b8 + 56396b7 + 311490b6 - 370078b5 - 4037b4 + 875248b3 - 318266b2 + 1916032*b1 - 518709

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

−2.76457
−2.45063
−2.22586
−2.15017
−1.57330
−1.28240
−1.13000
−1.03221
−0.0727208
0.377366
0.433645
1.04079
1.11167
2.09089
2.18259
2.30937
2.54287
2.59268

−2.76457

1.00000

5.64283

3.51734

−2.76457

1.00000

−10.0708

1.00000

−9.72391

1.2

−2.45063

1.00000

4.00561

1.33799

−2.45063

1.00000

−4.91500

1.00000

−3.27892

1.3

−2.22586

1.00000

2.95447

−3.44173

−2.22586

1.00000

−2.12453

1.00000

7.66082

1.4

−2.15017

1.00000

2.62321

4.25014

−2.15017

1.00000

−1.34001

1.00000

−9.13851

1.5

−1.57330

1.00000

0.475274

0.685193

−1.57330

1.00000

2.39885

1.00000

−1.07801

1.6

−1.28240

1.00000

−0.355439

0.804877

−1.28240

1.00000

3.02063

1.00000

−1.03218

1.7

−1.13000

1.00000

−0.723093

−1.31627

−1.13000

1.00000

3.07710

1.00000

1.48740

1.8

−1.03221

1.00000

−0.934545

−0.923018

−1.03221

1.00000

3.02906

1.00000

0.952747

1.9

−0.0727208

1.00000

−1.99471

3.77274

−0.0727208

1.00000

0.290499

1.00000

−0.274357

1.10

0.377366

1.00000

−1.85760

2.38028

0.377366

1.00000

−1.45572

1.00000

0.898235

1.11

0.433645

1.00000

−1.81195

−2.25399

0.433645

1.00000

−1.65303

1.00000

−0.977433

1.12

1.04079

1.00000

−0.916763

−3.39936

1.04079

1.00000

−3.03573

1.00000

−3.53801

1.13

1.11167

1.00000

−0.764196

3.59094

1.11167

1.00000

−3.07287

1.00000

3.99193

1.14

2.09089

1.00000

2.37184

−2.28283

2.09089

1.00000

0.777479

1.00000

−4.77315

1.15

2.18259

1.00000

2.76369

3.28475

2.18259

1.00000

1.66682

1.00000

7.16926

1.16

2.30937

1.00000

3.33319

−1.90652

2.30937

1.00000

3.07882

1.00000

−4.40287

1.17

2.54287

1.00000

4.46619

1.06712

2.54287

1.00000

6.27120

1.00000

2.71354

1.18

2.59268

1.00000

4.72200

2.83236

2.59268

1.00000

7.05728

1.00000

7.34341

Atkin-Lehner signs

$ p $	Sign
$3$	$-1$
$7$	$-1$
$19$	$-1$

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	7581.2.a.bt		18
19.b	odd	2	1		7581.2.a.br		18
19.f	odd	18	2		399.2.bo.d	✓	36

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
399.2.bo.d	✓	36	19.f	odd	18	2
7581.2.a.br		18	19.b	odd	2	1
7581.2.a.bt		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 $S_{2}^{\mathrm{new}}(\Gamma_0(7581))$:

$ T_{2}^{18} - 30 T_{2}^{16} - T_{2}^{15} + 372 T_{2}^{14} + 27 T_{2}^{13} - 2464 T_{2}^{12} - 276 T_{2}^{11} + \cdots - 73 $

$ T_{5}^{18} - 12 T_{5}^{17} + 9 T_{5}^{16} + 407 T_{5}^{15} - 1308 T_{5}^{14} - 4632 T_{5}^{13} + \cdots - 492456 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} - 30 T^{16} + \cdots - 73 $ T^18 - 30T^16 - T^15 + 372T^14 + 27T^13 - 2464T^12 - 276T^11 + 9402T^10 + 1337T^9 - 20799T^8 - 3063T^7 + 25640T^6 + 2667T^5 - 16170T^4 + 4T^3 + 4113T^2 - 711*T - 73
$3$	$ (T - 1)^{18} $ (T - 1)^18
$5$	$ T^{18} - 12 T^{17} + \cdots - 492456 $ T^18 - 12T^17 + 9T^16 + 407T^15 - 1308T^14 - 4632T^13 + 24846T^12 + 14346T^11 - 206889T^10 + 99012T^9 + 868599T^8 - 902865T^7 - 1781105T^6 + 2648016T^5 + 1355631T^4 - 3225215T^3 + 276132T^2 + 1326204*T - 492456
$7$	$ (T - 1)^{18} $ (T - 1)^18
$11$	$ T^{18} - 18 T^{17} + \cdots + 107 $ T^18 - 18T^17 + 66T^16 + 624T^15 - 5529T^14 + 7914T^13 + 52947T^12 - 180270T^11 - 68817T^10 + 892962T^9 - 525942T^8 - 1563984T^7 + 1278274T^6 + 1012482T^5 - 656277T^4 - 36250T^3 + 33279T^2 - 3480*T + 107
$13$	$ T^{18} + \cdots - 245258688 $ T^18 - 9T^17 - 90T^16 + 900T^15 + 3393T^14 - 37233T^13 - 73879T^12 + 825525T^11 + 1105905T^10 - 10537699T^9 - 12221022T^8 + 76584312T^7 + 91865256T^6 - 289354224T^5 - 389766816T^4 + 443930768T^3 + 715311840T^2 - 59752512*T - 245258688
$17$	$ T^{18} + \cdots - 121528873 $ T^18 - 18T^17 + 3T^16 + 1666T^15 - 7116T^14 - 51936T^13 + 368951T^12 + 513846T^11 - 7830285T^10 + 3930238T^9 + 80436789T^8 - 113997096T^7 - 388564585T^6 + 773422482T^5 + 709305006T^4 - 1829538436T^3 - 267414066T^2 + 1305886242*T - 121528873
$19$	$ T^{18} $ T^18
$23$	$ T^{18} + \cdots + 7380080901 $ T^18 - 6T^17 - 216T^16 + 1524T^15 + 16962T^14 - 151626T^13 - 504979T^12 + 7317336T^11 - 2877456T^10 - 165997540T^9 + 472780695T^8 + 1112435160T^7 - 7776802366T^6 + 11054302242T^5 + 13616310903T^4 - 62658189404T^3 + 79633226115T^2 - 43002810774*T + 7380080901
$29$	$ T^{18} - 3 T^{17} + \cdots + 10233792 $ T^18 - 3T^17 - 255T^16 + 784T^15 + 24150T^14 - 70167T^13 - 1074085T^12 + 2556591T^11 + 24358554T^10 - 36047247T^9 - 289131198T^8 + 111214500T^7 + 1636938896T^6 + 903628944T^5 - 3221277456T^4 - 4332300112T^3 - 1316520864T^2 + 189138816*T + 10233792
$31$	$ T^{18} + \cdots - 334890393 $ T^18 - 6T^17 - 207T^16 + 1369T^15 + 15000T^14 - 113628T^13 - 435304T^12 + 4216101T^11 + 3488511T^10 - 71867797T^9 + 42525552T^8 + 535596135T^7 - 729017418T^6 - 1399716330T^5 + 2991307896T^4 - 367539245T^3 - 2348997798T^2 + 1675535913*T - 334890393
$37$	$ T^{18} + \cdots - 430180957 $ T^18 - 36T^17 + 387T^16 + 996T^15 - 51462T^14 + 350742T^13 + 53421T^12 - 10992234T^11 + 44894091T^10 + 2579008T^9 - 399157131T^8 + 624956280T^7 + 965900143T^6 - 2294235948T^5 - 1371578430T^4 + 2968212426T^3 + 1809364482T^2 - 660502320*T - 430180957
$41$	$ T^{18} + \cdots + 89666023861 $ T^18 + 21T^17 - 123T^16 - 5346T^15 - 17061T^14 + 401706T^13 + 2941190T^12 - 6577674T^11 - 116439258T^10 - 160841706T^9 + 1728810840T^8 + 5885617446T^7 - 6508067834T^6 - 56905090650T^5 - 57877206975T^4 + 137744794537T^3 + 369722359827T^2 + 309976214190*T + 89666023861
$43$	$ T^{18} + \cdots - 2086279018048 $ T^18 - 27T^17 - 72T^16 + 8399T^15 - 58305T^14 - 709299T^13 + 10435927T^12 - 9419580T^11 - 530498856T^10 + 3107749911T^9 + 1713440736T^8 - 78218602248T^7 + 282871453480T^6 - 93830046864T^5 - 2059856576208T^4 + 6603854127888T^3 - 9605959442880T^2 + 7022229160896*T - 2086279018048
$47$	$ T^{18} + \cdots - 660718651904 $ T^18 - 27T^17 - 30T^16 + 6223T^15 - 30525T^14 - 501309T^13 + 3688775T^12 + 18586368T^11 - 175758840T^10 - 352531537T^9 + 4154106570T^8 + 4233955044T^7 - 51518755664T^6 - 46827308736T^5 + 319852909152T^4 + 401747370688T^3 - 712591436928T^2 - 1462609089024*T - 660718651904
$53$	$ T^{18} + \cdots + 1341501888 $ T^18 + 21T^17 - 216T^16 - 7647T^15 - 14043T^14 + 895269T^13 + 6047221T^12 - 31420074T^11 - 437447940T^10 - 665747875T^9 + 8608971762T^8 + 43227694128T^7 + 57484017584T^6 - 46416797808T^5 - 127527551040T^4 - 8013788624T^3 + 54442157568T^2 + 18176476608*T + 1341501888
$59$	$ T^{18} + \cdots + 4022070144704 $ T^18 + 15T^17 - 375T^16 - 5990T^15 + 57378T^14 + 986967T^13 - 4612657T^12 - 86675199T^11 + 206313198T^10 + 4354765035T^9 - 4771025640T^8 - 123953329452T^7 + 31341300012T^6 + 1822022432256T^5 + 797408294064T^4 - 10381198192224T^3 - 12766710021696T^2 + 906531655680*T + 4022070144704
$61$	$ T^{18} + \cdots + 2363332672 $ T^18 - 51T^17 + 789T^16 + 1462T^15 - 157938T^14 + 1235415T^13 + 4505185T^12 - 100904829T^11 + 337999176T^10 + 1237452593T^9 - 10631580588T^8 + 22424244312T^7 - 610593556T^6 - 54959673744T^5 + 53782960656T^4 + 11654715232T^3 - 27432195648T^2 + 1321210368*T + 2363332672
$67$	$ T^{18} + \cdots + 76974743780864 $ T^18 - 24T^17 - 369T^16 + 12234T^15 + 26583T^14 - 2324736T^13 + 3579969T^12 + 223489170T^11 - 665856216T^10 - 12010782196T^9 + 41102164512T^8 + 367682561232T^7 - 1180477813040T^6 - 6119052424032T^5 + 15066165296256T^4 + 47546290329024T^3 - 63684628449024T^2 - 117977652153600*T + 76974743780864
$71$	$ T^{18} + 24 T^{17} + \cdots + 4316509 $ T^18 + 24T^17 - 129T^16 - 5507T^15 + 3696T^14 + 469980T^13 - 146906T^12 - 17101881T^11 + 16157211T^10 + 221036559T^9 - 323460816T^8 - 840790827T^7 + 1227818390T^6 + 1247898600T^5 - 1464145098T^4 - 469115081T^3 + 523464942T^2 - 100690491*T + 4316509
$73$	$ T^{18} + \cdots + 293125719226688 $ T^18 - 9T^17 - 720T^16 + 6391T^15 + 213789T^14 - 1839723T^13 - 34076177T^12 + 276287316T^11 + 3188189610T^10 - 23126613087T^9 - 180724941792T^8 + 1058705683080T^7 + 6216122641312T^6 - 23523376914360T^5 - 123973192500096T^4 + 158223306253104T^3 + 1113447144986304T^2 + 1099343282413440*T + 293125719226688
$79$	$ T^{18} + \cdots + 3004224270336 $ T^18 - 30T^17 - 189T^16 + 13995T^15 - 80730T^14 - 1782276T^13 + 21168588T^12 + 24999216T^11 - 1206450288T^10 + 2484428736T^9 + 28300586208T^8 - 94741821312T^7 - 313055558464T^6 + 1215997202688T^5 + 1744416615936T^4 - 5682023770112T^3 - 5599498761216T^2 + 4962039091200*T + 3004224270336
$83$	$ T^{18} + \cdots + 59763388484928 $ T^18 - 48T^17 + 540T^16 + 8955T^15 - 245703T^14 + 975987T^13 + 21838549T^12 - 245766516T^11 + 77717439T^10 + 11797410137T^9 - 59050982190T^8 - 74064621564T^7 + 1413138320172T^6 - 3251473709928T^5 - 4892476526352T^4 + 27574938217792T^3 - 15453712382304T^2 - 55123905529152*T + 59763388484928
$89$	$ T^{18} + \cdots + 800346452619097 $ T^18 - 15T^17 - 702T^16 + 10488T^15 + 189618T^14 - 2954634T^13 - 24711103T^12 + 433562430T^11 + 1481072172T^10 - 35392285947T^9 - 13930613667T^8 + 1566005366007T^7 - 2601407472800T^6 - 32588650482087T^5 + 105988445475423T^4 + 186618537788243T^3 - 1068503895621111T^2 + 740423213204019*T + 800346452619097
$97$	$ T^{18} + \cdots + 12\!\cdots\!52 $ T^18 - 3T^17 - 825T^16 + 1330T^15 + 270864T^14 - 57165T^13 - 46678411T^12 - 50365101T^11 + 4585851846T^10 + 10022025277T^9 - 258317182488T^8 - 797981317644T^7 + 7906095549272T^6 + 30350875276176T^5 - 111980673567120T^4 - 514525621469200T^3 + 401955800741568T^2 + 2705170550833152*T + 1255509442089152
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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:	\( N \)	\(=\)	\( 7581 = 3 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7581.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{9} + 1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + (b9 + 1) * q^5 + b1 * q^6 + q^7 + (b3 + b1) * q^8 + q^9	\( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{9} + 1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9} + (\beta_{16} + \beta_{15} + \beta_{14} + \cdots - 1) q^{10}+ \cdots + (\beta_{5} + 1) q^{99}+O(q^{100}) \) q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + (b9 + 1) * q^5 + b1 * q^6 + q^7 + (b3 + b1) * q^8 + q^9 + (b16 + b15 + b14 + b12 + b11 + b9 - b7 - b5 + b4 + b3 + b1 - 1) * q^10 + (b5 + 1) * q^11 + (b2 + 1) * q^12 + (-b4 + 1) * q^13 + b1 * q^14 + (b9 + 1) * q^15 + (-b15 + b14 + b12 - b8 + b2 + 1) * q^16 + (b16 + b15 + b12 + b4 + 1) * q^17 + b1 * q^18 + (-b17 - b13 + b12 - b11 + b9 - b8 + b7 + b4 + b2 + 2) * q^20 + q^21 + (b13 - b12 + b11 - b10 - b7 - b5 - b4 + b1 - 1) * q^22 + (-b16 - b14 + b9 - b2 + b1 + 1) * q^23 + (b3 + b1) * q^24 + (-b16 - b12 + b10 + b9 + b8 - b3 + 2) * q^25 + (2b14 + b13 + b11 - b8 + b6 + b3 + b2 - 1) q^26 + q^27 + (b2 + 1) * q^28 + (-b17 + b13 + b11 + b9 - b7 - b4) * q^29 + (b16 + b15 + b14 + b12 + b11 + b9 - b7 - b5 + b4 + b3 + b1 - 1) * q^30 + (-b16 - b15 + b13 - b12 - b10 - b9 + b8 - b7 - b4) * q^31 + (b17 + b15 - b11 + b10 + 2b8 + b7 + b6 - b5) q^32 + (b5 + 1) * q^33 + (-b17 - b16 - b15 - b14 - b12 + b9 + 2b7 - b6 - b3 - b2 + 2b1) * q^34 + (b9 + 1) * q^35 + (b2 + 1) * q^36 + (-b15 + b14 - b9 - b7 + b3 + b2 + 1) * q^37 + (-b4 + 1) * q^39 + (b17 + b15 - b13 - b11 + b10 - b9 + b8 - b5 + b4 + b3 + 2b1 - 1) q^40 + (2b16 + b15 + b14 + b11 - b10 + b9 - b8 + b4 + b3 + b1 - 2) q^41 + b1 * q^42 + (b16 + b12 - b8 - b7 - b6 - b5 + b4 - b1 + 2) * q^43 + (-b14 - b12 - b11 + b10 - b9 + b8 + b5 - b4 - b3 + b2 - b1 + 3) * q^44 + (b9 + 1) * q^45 + (-b13 + 2b12 + b11 + 2b10 - b8 + 2b7 - b6 + b4 - b3 + b2 + 3) q^46 + (-b15 - b14 - b12 - b11 + b10 - b9 + 2b7 + b5 - b3 + b1 + 2) q^47 + (-b15 + b14 + b12 - b8 + b2 + 1) * q^48 + q^49 + (2b16 + 2b15 - b13 + 2b12 + b11 - 2b10 + b9 - 2b8 - 3b7 - b5 + b4 + b3 - 2b2 + 2b1) * q^50 + (b16 + b15 + b12 + b4 + 1) * q^51 + (b17 + b16 + b15 + b12 - b11 + b10 - b8 - 2b7 + 2b6 - b4 + b3 + b2) * q^52 + (-b16 - b15 - b12 - b11 - b10 - b9 + b7 - b6 + 2b5 - 1) q^53 + b1 * q^54 + (-b16 - b14 - b13 - b12 - b11 + b10 + b9 + 2b8 + 2b7 + b5 - b3 - b2 - b1 + 4) * q^55 + (b3 + b1) * q^56 + (b17 + b16 + b15 + b13 - 2b12 + b11 - 2b10 - b9 + b8 - b7 + b5 - b4 - 1) * q^58 + (-b17 + b15 - b14 - b13 + b10 + b9 - b8 + b7 + b5 + b4 + b3 - b2 + b1 - 1) * q^59 + (-b17 - b13 + b12 - b11 + b9 - b8 + b7 + b4 + b2 + 2) * q^60 + (-b16 - b15 - b14 - b13 + b12 - b10 - b9 - b8 - b6 + b5 - b4 + b1 + 4) * q^61 + (b17 - 2b15 - b14 - b12 - b11 - 3b10 - 3b9 + b8 + 2b5 - 2b4 - b3 + b2 - b1 + 2) q^62 + q^63 + (-b17 - b15 + b14 - b13 + 2b12 + b11 - 2b10 + b9 - 3b8 + 2b7 + b6 + b4 + b3 - b2 - b1 - 1) * q^64 + (b17 - b16 + b15 + b12 + b11 + b10 + b9 + b8 - b7 + b6 - b5 - b1 + 1) * q^65 + (b13 - b12 + b11 - b10 - b7 - b5 - b4 + b1 - 1) * q^66 + (b17 - 2b15 - 2b13 - b12 - 2b9 + b8 + b7 - b6 + b5 + 2) q^67 + (b17 + b15 - 2b14 - b13 - b10 - b9 + 3b8 - 3b7 - 2b6 - b3 - b1 + 3) * q^68 + (-b16 - b14 + b9 - b2 + b1 + 1) * q^69 + (b16 + b15 + b14 + b12 + b11 + b9 - b7 - b5 + b4 + b3 + b1 - 1) * q^70 + (-b17 + b16 + b13 + b12 - b10 + b9 - b8 - b7 - 1) * q^71 + (b3 + b1) * q^72 + (b17 + 2b13 + b12 + b11 - b10 - 2b7 - b5 - b4 - b3 - 2b1 + 1) q^73 + (b17 - b15 - b12 - 2b11 - 2b10 - b9 + b8 - b7 + b6 - b4 + b3 + 2b2 + 2b1 + 1) * q^74 + (-b16 - b12 + b10 + b9 + b8 - b3 + 2) * q^75 + (b5 + 1) * q^77 + (2b14 + b13 + b11 - b8 + b6 + b3 + b2 - 1) q^78 + (b17 + b16 + b15 - b13 - b12 - b11 - b9 + 2b7 + b6 + 2b2 - b1 + 1) * q^79 + (-b16 - 3b15 + 2b14 + 2b12 + b11 + b10 - 4b8 + 2b2 - b1 + 4) q^80 + q^81 + (-b17 + b16 + 2b15 + b13 - b12 + b10 + 3b9 + 2b8 - 4b7 - b5 + b2 - b1 + 2) * q^82 + (b17 + 2b15 - b14 - 2b12 + b9 + b8 - b5 - b3 - b1 + 3) * q^83 + (b2 + 1) * q^84 + (3b16 + 2b15 - b14 - b13 - b11 + b9 + b8 + b7 + b6 + b4 - b2 + 2b1) q^85 + (-b17 - b16 - 2b14 - b13 - b12 - 2b11 + 3b10 + b9 + 2b8 + 3b7 - 2b6 + b5 + 2b4 - b3 - 2b2 + 3b1 - 2) q^86 + (-b17 + b13 + b11 + b9 - b7 - b4) * q^87 + (-b15 + b14 - b12 - b10 - 2b9 - b8 + 2b5 + b4 + b3 - b2 + 2b1 - 3) q^88 + (b16 - b15 - b13 + 2b12 - b11 + 2b10 - b9 + 4b7 + b6 + b5 + b4 - 2b2 + 2) * q^89 + (b16 + b15 + b14 + b12 + b11 + b9 - b7 - b5 + b4 + b3 + b1 - 1) * q^90 + (-b4 + 1) * q^91 + (-b17 - b16 + b15 - 2b14 - b13 + b12 - 2b11 + 3b10 + 6b7 - b6 + b4 - 2b2 + 5b1 - 1) * q^92 + (-b16 - b15 + b13 - b12 - b10 - b9 + b8 - b7 - b4) * q^93 + (b17 - 2b14 + b13 - 2b12 - b10 - 3b9 + 3b8 - 2b7 - b6 - b4 - b3 + 2b1 + 2) * q^94 + (b17 + b15 - b11 + b10 + 2b8 + b7 + b6 - b5) q^96 + (-2b16 - b15 - 2b14 - b12 - b11 - 3b10 - 3b9 - b7 - b6 - 2b4 - 2b3 + b2 + 1) * q^97 + b1 * q^98 + (b5 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 24 q^{4} + 12 q^{5} + 18 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 24 * q^4 + 12 * q^5 + 18 * q^7 + 3 * q^8 + 18 * q^9	\( 18 q + 18 q^{3} + 24 q^{4} + 12 q^{5} + 18 q^{7} + 3 q^{8} + 18 q^{9} - 6 q^{10} + 18 q^{11} + 24 q^{12} + 9 q^{13} + 12 q^{15} + 24 q^{16} + 18 q^{17} + 36 q^{20} + 18 q^{21} - 9 q^{22} + 6 q^{23} + 3 q^{24} + 36 q^{25} + 27 q^{26} + 18 q^{27} + 24 q^{28} + 3 q^{29} - 6 q^{30} + 6 q^{31} - 9 q^{32} + 18 q^{33} - 15 q^{34} + 12 q^{35} + 24 q^{36} + 36 q^{37} + 9 q^{39} - 18 q^{40} - 21 q^{41} + 27 q^{43} + 42 q^{44} + 12 q^{45} + 54 q^{46} + 27 q^{47} + 24 q^{48} + 18 q^{49} - 12 q^{50} + 18 q^{51} - 3 q^{52} - 21 q^{53} + 45 q^{55} + 3 q^{56} - 9 q^{58} - 15 q^{59} + 36 q^{60} + 51 q^{61} + 15 q^{62} + 18 q^{63} + 3 q^{64} + 24 q^{65} - 9 q^{66} + 24 q^{67} + 12 q^{68} + 6 q^{69} - 6 q^{70} - 24 q^{71} + 3 q^{72} + 9 q^{73} + 15 q^{74} + 36 q^{75} + 18 q^{77} + 27 q^{78} + 30 q^{79} + 102 q^{80} + 18 q^{81} + 33 q^{82} + 48 q^{83} + 24 q^{84} - 27 q^{85} - 78 q^{86} + 3 q^{87} - 24 q^{88} + 15 q^{89} - 6 q^{90} + 9 q^{91} - 54 q^{92} + 6 q^{93} + 21 q^{94} - 9 q^{96} + 3 q^{97} + 18 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 24 * q^4 + 12 * q^5 + 18 * q^7 + 3 * q^8 + 18 * q^9 - 6 * q^10 + 18 * q^11 + 24 * q^12 + 9 * q^13 + 12 * q^15 + 24 * q^16 + 18 * q^17 + 36 * q^20 + 18 * q^21 - 9 * q^22 + 6 * q^23 + 3 * q^24 + 36 * q^25 + 27 * q^26 + 18 * q^27 + 24 * q^28 + 3 * q^29 - 6 * q^30 + 6 * q^31 - 9 * q^32 + 18 * q^33 - 15 * q^34 + 12 * q^35 + 24 * q^36 + 36 * q^37 + 9 * q^39 - 18 * q^40 - 21 * q^41 + 27 * q^43 + 42 * q^44 + 12 * q^45 + 54 * q^46 + 27 * q^47 + 24 * q^48 + 18 * q^49 - 12 * q^50 + 18 * q^51 - 3 * q^52 - 21 * q^53 + 45 * q^55 + 3 * q^56 - 9 * q^58 - 15 * q^59 + 36 * q^60 + 51 * q^61 + 15 * q^62 + 18 * q^63 + 3 * q^64 + 24 * q^65 - 9 * q^66 + 24 * q^67 + 12 * q^68 + 6 * q^69 - 6 * q^70 - 24 * q^71 + 3 * q^72 + 9 * q^73 + 15 * q^74 + 36 * q^75 + 18 * q^77 + 27 * q^78 + 30 * q^79 + 102 * q^80 + 18 * q^81 + 33 * q^82 + 48 * q^83 + 24 * q^84 - 27 * q^85 - 78 * q^86 + 3 * q^87 - 24 * q^88 + 15 * q^89 - 6 * q^90 + 9 * q^91 - 54 * q^92 + 6 * q^93 + 21 * q^94 - 9 * q^96 + 3 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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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	7581.2.a.bt

Level	$7581$
Weight	$2$
Character orbit	7581.a
Self dual	yes
Analytic conductor	$60.535$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(60.5345897723\)
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:	\( x^{18} - 30 x^{16} - x^{15} + 372 x^{14} + 27 x^{13} - 2464 x^{12} - 276 x^{11} + 9402 x^{10} + \cdots - 73 \) x^18 - 30x^16 - x^15 + 372x^14 + 27x^13 - 2464x^12 - 276x^11 + 9402x^10 + 1337x^9 - 20799x^8 - 3063x^7 + 25640x^6 + 2667x^5 - 16170x^4 + 4x^3 + 4113x^2 - 711*x - 73
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 399)
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
\(\beta_{4}\)	\(=\)	\( ( - 21082 \nu^{17} + 895597 \nu^{16} - 198070 \nu^{15} - 26519371 \nu^{14} + 13343331 \nu^{13} + \cdots + 131029873 ) / 10631878 \) (-21082v^17 + 895597v^16 - 198070v^15 - 26519371v^14 + 13343331v^13 + 321631338v^12 - 161517411v^11 - 2054095906v^10 + 888273417v^9 + 7367916988v^8 - 2557687788v^7 - 14576988270v^6 + 4079180810v^5 + 14429497614v^4 - 3827061619v^3 - 5620961057v^2 + 1640968884*v + 131029873) / 10631878
\(\beta_{5}\)	\(=\)	\( ( - 73770 \nu^{17} - 229885 \nu^{16} + 2975775 \nu^{15} + 6062098 \nu^{14} - 45853139 \nu^{13} + \cdots + 42206054 ) / 21263756 \) (-73770v^17 - 229885v^16 + 2975775v^15 + 6062098v^14 - 45853139v^13 - 62881162v^12 + 351895690v^11 + 319261203v^10 - 1436331921v^9 - 765500678v^8 + 3020590269v^7 + 461784323v^6 - 2828887865v^5 + 1113437269v^4 + 760908392v^3 - 1230676966v^2 + 179254149*v + 42206054) / 21263756
\(\beta_{6}\)	\(=\)	\( ( 121846 \nu^{17} + 1026809 \nu^{16} - 3700143 \nu^{15} - 29317769 \nu^{14} + 47090291 \nu^{13} + \cdots + 70641825 ) / 10631878 \) (121846v^17 + 1026809v^16 - 3700143v^15 - 29317769v^14 + 47090291v^13 + 341737237v^12 - 324246257v^11 - 2088523097v^10 + 1305195806v^9 + 7136926421v^8 - 3124106164v^7 - 13411143512v^6 + 4428788937v^5 + 12602437854v^4 - 3683752024v^3 - 4598968566v^2 + 1423008712*v + 70641825) / 10631878
\(\beta_{7}\)	\(=\)	\( ( 322745 \nu^{17} - 883328 \nu^{16} - 9035895 \nu^{15} + 25269891 \nu^{14} + 102571089 \nu^{13} + \cdots - 106956766 ) / 21263756 \) (322745v^17 - 883328v^16 - 9035895v^15 + 25269891v^14 + 102571089v^13 - 294009186v^12 - 607050171v^11 + 1787311070v^10 + 2000856991v^9 - 6053117735v^8 - 3600546686v^7 + 11224753484v^6 + 3042700234v^5 - 10372655595v^4 - 336036346v^3 + 3777320133v^2 - 648832661*v - 106956766) / 21263756
\(\beta_{8}\)	\(=\)	\( ( - 465048 \nu^{17} - 206149 \nu^{16} + 13279125 \nu^{15} + 5954208 \nu^{14} - 154847379 \nu^{13} + \cdots - 41498894 ) / 21263756 \) (-465048v^17 - 206149v^16 + 13279125v^15 + 5954208v^14 - 154847379v^13 - 71868892v^12 + 946879506v^11 + 461945801v^10 - 3239383737v^9 - 1667957746v^8 + 6126330053v^7 + 3275065591v^6 - 5994963845v^5 - 3117142337v^4 + 2781799410v^3 + 1165078830v^2 - 509056065*v - 41498894) / 21263756
\(\beta_{9}\)	\(=\)	\( ( 498578 \nu^{17} - 807750 \nu^{16} - 15109441 \nu^{15} + 20826079 \nu^{14} + 189739036 \nu^{13} + \cdots + 122424649 ) / 21263756 \) (498578v^17 - 807750v^16 - 15109441v^15 + 20826079v^14 + 189739036v^13 - 210806935v^12 - 1276282036v^11 + 1053686784v^10 + 4944010631v^9 - 2647690845v^8 - 10971110616v^7 + 2863134679v^6 + 12944556289v^5 - 356658119v^4 - 6731483635v^3 - 952449274v^2 + 838575398*v + 122424649) / 21263756
\(\beta_{10}\)	\(=\)	\( ( 568478 \nu^{17} + 465048 \nu^{16} - 16848191 \nu^{15} - 13847603 \nu^{14} + 205519608 \nu^{13} + \cdots + 104868207 ) / 21263756 \) (568478v^17 + 465048v^16 - 16848191v^15 - 13847603v^14 + 205519608v^13 + 170196285v^12 - 1328860900v^11 - 1103779434v^10 + 4882884355v^9 + 3999438823v^8 - 10155816176v^7 - 7867578167v^6 + 11300710329v^5 + 7511094671v^4 - 6075146923v^3 - 2779525498v^2 + 1173071184*v + 104868207) / 21263756
\(\beta_{11}\)	\(=\)	\( ( 313682 \nu^{17} + 211500 \nu^{16} - 9415057 \nu^{15} - 5734413 \nu^{14} + 116223708 \nu^{13} + \cdots + 29692336 ) / 10631878 \) (313682v^17 + 211500v^16 - 9415057v^15 - 5734413v^14 + 116223708v^13 + 64111552v^12 - 760326239v^11 - 382170061v^10 + 2831095669v^9 + 1306229065v^8 - 5991410659v^7 - 2534842111v^6 + 6777624852v^5 + 2535342606v^4 - 3485085038v^3 - 999250720v^2 + 495637232*v + 29692336) / 10631878
\(\beta_{12}\)	\(=\)	\( ( - 883328 \nu^{17} + 646455 \nu^{16} + 25592636 \nu^{15} - 17490051 \nu^{14} - 302723301 \nu^{13} + \cdots + 23560385 ) / 21263756 \) (-883328v^17 + 646455v^16 + 25592636v^15 - 17490051v^14 - 302723301v^13 + 188193509v^12 + 1876388690v^11 - 1033591499v^10 - 6484627800v^9 + 3112226569v^8 + 12213321419v^7 - 5232481566v^6 - 11233416510v^5 + 4882750304v^4 + 3776029153v^3 - 1976282846v^2 + 122514929*v + 23560385) / 21263756
\(\beta_{13}\)	\(=\)	\( ( - 965288 \nu^{17} + 2129159 \nu^{16} + 27106627 \nu^{15} - 59986280 \nu^{14} - 307135869 \nu^{13} + \cdots + 282231500 ) / 21263756 \) (-965288v^17 + 2129159v^16 + 27106627v^15 - 59986280v^14 - 307135869v^13 + 686665428v^12 + 1788891648v^11 - 4117396731v^10 - 5589733067v^9 + 13902670864v^8 + 8580821381v^7 - 26435239309v^6 - 3805600037v^5 + 26490984157v^4 - 3788580772v^3 - 10969915692v^2 + 3106426489*v + 282231500) / 21263756
\(\beta_{14}\)	\(=\)	\( ( - 1081005 \nu^{17} - 1575433 \nu^{16} + 32741902 \nu^{15} + 46836591 \nu^{14} - 409370306 \nu^{13} + \cdots - 218132860 ) / 21263756 \) (-1081005v^17 - 1575433v^16 + 32741902v^15 + 46836591v^14 - 409370306v^13 - 571652490v^12 + 2723480963v^11 + 3673878043v^10 - 10357220432v^9 - 13232297485v^8 + 22529833595v^7 + 26181414783v^6 - 26786819001v^5 - 25773886148v^4 + 16064272004v^3 + 9927405519v^2 - 3778636524*v - 218132860) / 21263756
\(\beta_{15}\)	\(=\)	\( ( - 1499285 \nu^{17} - 722829 \nu^{16} + 45055413 \nu^{15} + 23392332 \nu^{14} - 557246228 \nu^{13} + \cdots - 280656117 ) / 21263756 \) (-1499285v^17 - 722829v^16 + 45055413v^15 + 23392332v^14 - 557246228v^13 - 311590089v^12 + 3652990147v^11 + 2178340743v^10 - 13602464495v^9 - 8452113170v^8 + 28616824961v^7 + 17673867626v^6 - 32025271666v^5 - 17795257263v^4 + 17058501747v^3 + 6934890135v^2 - 3147065530*v - 280656117) / 21263756
\(\beta_{16}\)	\(=\)	\( ( 1821065 \nu^{17} - 1019951 \nu^{16} - 53789719 \nu^{15} + 26542626 \nu^{14} + 652916130 \nu^{13} + \cdots + 24268763 ) / 21263756 \) (1821065v^17 - 1019951v^16 - 53789719v^15 + 26542626v^14 + 652916130v^13 - 270305967v^12 - 4196750633v^11 + 1363170411v^10 + 15311799363v^9 - 3548222382v^8 - 31480349789v^7 + 4585279416v^6 + 33914786288v^5 - 2815324915v^4 - 16093861251v^3 + 641427271v^2 + 1805053370*v + 24268763) / 21263756
\(\beta_{17}\)	\(=\)	\( ( 924030 \nu^{17} - 153548 \nu^{16} - 27091815 \nu^{15} + 3252887 \nu^{14} + 325631992 \nu^{13} + \cdots + 163024770 ) / 10631878 \) (924030v^17 - 153548v^16 - 27091815v^15 + 3252887v^14 + 325631992v^13 - 19495879v^12 - 2065551181v^11 - 26898353v^10 + 7406479214v^9 + 707406092v^8 - 14913570463v^7 - 2683393500v^6 + 15780918257v^5 + 3935174817v^4 - 7611392491v^3 - 2147041852v^2 + 1110307699*v + 163024770) / 10631878

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 3 \) b2 + 3
\(\nu^{3}\)	\(=\)	\( \beta_{3} + 5\beta_1 \) b3 + 5*b1
\(\nu^{4}\)	\(=\)	\( -\beta_{15} + \beta_{14} + \beta_{12} - \beta_{8} + 7\beta_{2} + 15 \) -b15 + b14 + b12 - b8 + 7*b2 + 15
\(\nu^{5}\)	\(=\)	\( \beta_{17} + \beta_{15} - \beta_{11} + \beta_{10} + 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 8\beta_{3} + 28\beta_1 \) b17 + b15 - b11 + b10 + 2b8 + b7 + b6 - b5 + 8b3 + 28*b1
\(\nu^{6}\)	\(=\)	\( - \beta_{17} - 11 \beta_{15} + 11 \beta_{14} - \beta_{13} + 12 \beta_{12} + \beta_{11} - 2 \beta_{10} + \cdots + 85 \) -b17 - 11b15 + 11b14 - b13 + 12b12 + b11 - 2b10 + b9 - 13b8 + 2b7 + b6 + b4 + b3 + 45*b2 - b1 + 85
\(\nu^{7}\)	\(=\)	\( 13 \beta_{17} + 14 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - 12 \beta_{11} + 15 \beta_{10} + \cdots - 4 \) 13b17 + 14b15 - b14 - b13 + b12 - 12b11 + 15b10 + 27b8 + 11b7 + 12b6 - 12b5 + 56b3 - b2 + 166b1 - 4
\(\nu^{8}\)	\(=\)	\( - 15 \beta_{17} - 3 \beta_{16} - 97 \beta_{15} + 94 \beta_{14} - 12 \beta_{13} + 105 \beta_{12} + \cdots + 510 \) -15b17 - 3b16 - 97b15 + 94b14 - 12b13 + 105b12 + 14b11 - 28b10 + 15b9 - 125b8 + 30b7 + 12b6 + 13b4 + 11b3 + 290b2 - 17b1 + 510
\(\nu^{9}\)	\(=\)	\( 124 \beta_{17} + 5 \beta_{16} + 145 \beta_{15} - 19 \beta_{14} - 16 \beta_{13} + 18 \beta_{12} + \cdots - 63 \) 124b17 + 5b16 + 145b15 - 19b14 - 16b13 + 18b12 - 109b11 + 157b10 - 3b9 + 262b8 + 87b7 + 105b6 - 107b5 + b4 + 384b3 - 22b2 + 1023b1 - 63
\(\nu^{10}\)	\(=\)	\( - 164 \beta_{17} - 55 \beta_{16} - 794 \beta_{15} + 739 \beta_{14} - 103 \beta_{13} + 819 \beta_{12} + \cdots + 3174 \) -164b17 - 55b16 - 794b15 + 739b14 - 103b13 + 819b12 + 143b11 - 272b10 + 158b9 - 1074b8 + 320b7 + 101b6 + 4b5 + 124b4 + 80b3 + 1899b2 - 196*b1 + 3174
\(\nu^{11}\)	\(=\)	\( 1059 \beta_{17} + 99 \beta_{16} + 1339 \beta_{15} - 248 \beta_{14} - 175 \beta_{13} + 201 \beta_{12} + \cdots - 695 \) 1059b17 + 99b16 + 1339b15 - 248b14 - 175b13 + 201b12 - 899b11 + 1412b10 - 59b9 + 2257b8 + 584b7 + 819b6 - 861b5 + 13b4 + 2632b3 - 303b2 + 6500*b1 - 695
\(\nu^{12}\)	\(=\)	\( - 1579 \beta_{17} - 668 \beta_{16} - 6281 \beta_{15} + 5611 \beta_{14} - 775 \beta_{13} + 6050 \beta_{12} + \cdots + 20319 \) -1579b17 - 668b16 - 6281b15 + 5611b14 - 775b13 + 6050b12 + 1305b11 - 2304b10 + 1447b9 - 8741b8 + 2964b7 + 733b6 + 94b5 + 1057b4 + 451b3 + 12650b2 - 1939*b1 + 20319
\(\nu^{13}\)	\(=\)	\( 8593 \beta_{17} + 1282 \beta_{16} + 11652 \beta_{15} - 2721 \beta_{14} - 1631 \beta_{13} + 1817 \beta_{12} + \cdots - 6682 \) 8593b17 + 1282b16 + 11652b15 - 2721b14 - 1631b13 + 1817b12 - 7102b11 + 11747b10 - 760b9 + 18423b8 + 3427b7 + 6062b6 - 6622b5 + 90b4 + 18121b3 - 3399b2 + 42370*b1 - 6682
\(\nu^{14}\)	\(=\)	\( - 14183 \beta_{17} - 6789 \beta_{16} - 48804 \beta_{15} + 41947 \beta_{14} - 5430 \beta_{13} + \cdots + 133160 \) -14183b17 - 6789b16 - 48804b15 + 41947b14 - 5430b13 + 43410b12 + 11282b11 - 18370b10 + 12333b9 - 69082b8 + 25446b7 + 4882b6 + 1370b5 + 8533b4 + 1763b3 + 85624b2 - 17771*b1 + 133160
\(\nu^{15}\)	\(=\)	\( 67869 \beta_{17} + 13771 \beta_{16} + 97802 \beta_{15} - 26961 \beta_{14} - 13952 \beta_{13} + \cdots - 60025 \) 67869b17 + 13771b16 + 97802b15 - 26961b14 - 13952b13 + 14546b12 - 54916b11 + 93548b10 - 8159b9 + 146374b8 + 16908b7 + 43726b6 - 49798b5 + 229b4 + 125519b3 - 34060b2 + 282276*b1 - 60025
\(\nu^{16}\)	\(=\)	\( - 121945 \beta_{17} - 62619 \beta_{16} - 375476 \beta_{15} + 311527 \beta_{14} - 36256 \beta_{13} + \cdots + 890244 \) -121945b17 - 62619b16 - 375476b15 + 311527b14 - 36256b13 + 306424b12 + 94532b11 - 142594b10 + 100777b9 - 536800b8 + 208616b7 + 30488b6 + 16050b5 + 66781b4 - 541b3 + 588001b2 - 155871*b1 + 890244
\(\nu^{17}\)	\(=\)	\( 527909 \beta_{17} + 133293 \beta_{16} + 801422 \beta_{15} - 250083 \beta_{14} - 113350 \beta_{13} + \cdots - 518709 \) 527909b17 + 133293b16 + 801422b15 - 250083b14 - 113350b13 + 107126b12 - 419962b11 + 726076b10 - 79349b9 + 1146544b8 + 56396b7 + 311490b6 - 370078b5 - 4037b4 + 875248b3 - 318266b2 + 1916032*b1 - 518709

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

$p$	$F_p(T)$
$2$	\( T^{18} - 30 T^{16} + \cdots - 73 \) T^18 - 30T^16 - T^15 + 372T^14 + 27T^13 - 2464T^12 - 276T^11 + 9402T^10 + 1337T^9 - 20799T^8 - 3063T^7 + 25640T^6 + 2667T^5 - 16170T^4 + 4T^3 + 4113T^2 - 711*T - 73
$3$	\( (T - 1)^{18} \) (T - 1)^18
$5$	\( T^{18} - 12 T^{17} + \cdots - 492456 \) T^18 - 12T^17 + 9T^16 + 407T^15 - 1308T^14 - 4632T^13 + 24846T^12 + 14346T^11 - 206889T^10 + 99012T^9 + 868599T^8 - 902865T^7 - 1781105T^6 + 2648016T^5 + 1355631T^4 - 3225215T^3 + 276132T^2 + 1326204*T - 492456
$7$	\( (T - 1)^{18} \) (T - 1)^18
$11$	\( T^{18} - 18 T^{17} + \cdots + 107 \) T^18 - 18T^17 + 66T^16 + 624T^15 - 5529T^14 + 7914T^13 + 52947T^12 - 180270T^11 - 68817T^10 + 892962T^9 - 525942T^8 - 1563984T^7 + 1278274T^6 + 1012482T^5 - 656277T^4 - 36250T^3 + 33279T^2 - 3480*T + 107
$13$	\( T^{18} + \cdots - 245258688 \) T^18 - 9T^17 - 90T^16 + 900T^15 + 3393T^14 - 37233T^13 - 73879T^12 + 825525T^11 + 1105905T^10 - 10537699T^9 - 12221022T^8 + 76584312T^7 + 91865256T^6 - 289354224T^5 - 389766816T^4 + 443930768T^3 + 715311840T^2 - 59752512*T - 245258688
$17$	\( T^{18} + \cdots - 121528873 \) T^18 - 18T^17 + 3T^16 + 1666T^15 - 7116T^14 - 51936T^13 + 368951T^12 + 513846T^11 - 7830285T^10 + 3930238T^9 + 80436789T^8 - 113997096T^7 - 388564585T^6 + 773422482T^5 + 709305006T^4 - 1829538436T^3 - 267414066T^2 + 1305886242*T - 121528873
$19$	\( T^{18} \) T^18
$23$	\( T^{18} + \cdots + 7380080901 \) T^18 - 6T^17 - 216T^16 + 1524T^15 + 16962T^14 - 151626T^13 - 504979T^12 + 7317336T^11 - 2877456T^10 - 165997540T^9 + 472780695T^8 + 1112435160T^7 - 7776802366T^6 + 11054302242T^5 + 13616310903T^4 - 62658189404T^3 + 79633226115T^2 - 43002810774*T + 7380080901
$29$	\( T^{18} - 3 T^{17} + \cdots + 10233792 \) T^18 - 3T^17 - 255T^16 + 784T^15 + 24150T^14 - 70167T^13 - 1074085T^12 + 2556591T^11 + 24358554T^10 - 36047247T^9 - 289131198T^8 + 111214500T^7 + 1636938896T^6 + 903628944T^5 - 3221277456T^4 - 4332300112T^3 - 1316520864T^2 + 189138816*T + 10233792
$31$	\( T^{18} + \cdots - 334890393 \) T^18 - 6T^17 - 207T^16 + 1369T^15 + 15000T^14 - 113628T^13 - 435304T^12 + 4216101T^11 + 3488511T^10 - 71867797T^9 + 42525552T^8 + 535596135T^7 - 729017418T^6 - 1399716330T^5 + 2991307896T^4 - 367539245T^3 - 2348997798T^2 + 1675535913*T - 334890393
$37$	\( T^{18} + \cdots - 430180957 \) T^18 - 36T^17 + 387T^16 + 996T^15 - 51462T^14 + 350742T^13 + 53421T^12 - 10992234T^11 + 44894091T^10 + 2579008T^9 - 399157131T^8 + 624956280T^7 + 965900143T^6 - 2294235948T^5 - 1371578430T^4 + 2968212426T^3 + 1809364482T^2 - 660502320*T - 430180957
$41$	\( T^{18} + \cdots + 89666023861 \) T^18 + 21T^17 - 123T^16 - 5346T^15 - 17061T^14 + 401706T^13 + 2941190T^12 - 6577674T^11 - 116439258T^10 - 160841706T^9 + 1728810840T^8 + 5885617446T^7 - 6508067834T^6 - 56905090650T^5 - 57877206975T^4 + 137744794537T^3 + 369722359827T^2 + 309976214190*T + 89666023861
$43$	\( T^{18} + \cdots - 2086279018048 \) T^18 - 27T^17 - 72T^16 + 8399T^15 - 58305T^14 - 709299T^13 + 10435927T^12 - 9419580T^11 - 530498856T^10 + 3107749911T^9 + 1713440736T^8 - 78218602248T^7 + 282871453480T^6 - 93830046864T^5 - 2059856576208T^4 + 6603854127888T^3 - 9605959442880T^2 + 7022229160896*T - 2086279018048
$47$	\( T^{18} + \cdots - 660718651904 \) T^18 - 27T^17 - 30T^16 + 6223T^15 - 30525T^14 - 501309T^13 + 3688775T^12 + 18586368T^11 - 175758840T^10 - 352531537T^9 + 4154106570T^8 + 4233955044T^7 - 51518755664T^6 - 46827308736T^5 + 319852909152T^4 + 401747370688T^3 - 712591436928T^2 - 1462609089024*T - 660718651904
$53$	\( T^{18} + \cdots + 1341501888 \) T^18 + 21T^17 - 216T^16 - 7647T^15 - 14043T^14 + 895269T^13 + 6047221T^12 - 31420074T^11 - 437447940T^10 - 665747875T^9 + 8608971762T^8 + 43227694128T^7 + 57484017584T^6 - 46416797808T^5 - 127527551040T^4 - 8013788624T^3 + 54442157568T^2 + 18176476608*T + 1341501888
$59$	\( T^{18} + \cdots + 4022070144704 \) T^18 + 15T^17 - 375T^16 - 5990T^15 + 57378T^14 + 986967T^13 - 4612657T^12 - 86675199T^11 + 206313198T^10 + 4354765035T^9 - 4771025640T^8 - 123953329452T^7 + 31341300012T^6 + 1822022432256T^5 + 797408294064T^4 - 10381198192224T^3 - 12766710021696T^2 + 906531655680*T + 4022070144704
$61$	\( T^{18} + \cdots + 2363332672 \) T^18 - 51T^17 + 789T^16 + 1462T^15 - 157938T^14 + 1235415T^13 + 4505185T^12 - 100904829T^11 + 337999176T^10 + 1237452593T^9 - 10631580588T^8 + 22424244312T^7 - 610593556T^6 - 54959673744T^5 + 53782960656T^4 + 11654715232T^3 - 27432195648T^2 + 1321210368*T + 2363332672
$67$	\( T^{18} + \cdots + 76974743780864 \) T^18 - 24T^17 - 369T^16 + 12234T^15 + 26583T^14 - 2324736T^13 + 3579969T^12 + 223489170T^11 - 665856216T^10 - 12010782196T^9 + 41102164512T^8 + 367682561232T^7 - 1180477813040T^6 - 6119052424032T^5 + 15066165296256T^4 + 47546290329024T^3 - 63684628449024T^2 - 117977652153600*T + 76974743780864
$71$	\( T^{18} + 24 T^{17} + \cdots + 4316509 \) T^18 + 24T^17 - 129T^16 - 5507T^15 + 3696T^14 + 469980T^13 - 146906T^12 - 17101881T^11 + 16157211T^10 + 221036559T^9 - 323460816T^8 - 840790827T^7 + 1227818390T^6 + 1247898600T^5 - 1464145098T^4 - 469115081T^3 + 523464942T^2 - 100690491*T + 4316509
$73$	\( T^{18} + \cdots + 293125719226688 \) T^18 - 9T^17 - 720T^16 + 6391T^15 + 213789T^14 - 1839723T^13 - 34076177T^12 + 276287316T^11 + 3188189610T^10 - 23126613087T^9 - 180724941792T^8 + 1058705683080T^7 + 6216122641312T^6 - 23523376914360T^5 - 123973192500096T^4 + 158223306253104T^3 + 1113447144986304T^2 + 1099343282413440*T + 293125719226688
$79$	\( T^{18} + \cdots + 3004224270336 \) T^18 - 30T^17 - 189T^16 + 13995T^15 - 80730T^14 - 1782276T^13 + 21168588T^12 + 24999216T^11 - 1206450288T^10 + 2484428736T^9 + 28300586208T^8 - 94741821312T^7 - 313055558464T^6 + 1215997202688T^5 + 1744416615936T^4 - 5682023770112T^3 - 5599498761216T^2 + 4962039091200*T + 3004224270336
$83$	\( T^{18} + \cdots + 59763388484928 \) T^18 - 48T^17 + 540T^16 + 8955T^15 - 245703T^14 + 975987T^13 + 21838549T^12 - 245766516T^11 + 77717439T^10 + 11797410137T^9 - 59050982190T^8 - 74064621564T^7 + 1413138320172T^6 - 3251473709928T^5 - 4892476526352T^4 + 27574938217792T^3 - 15453712382304T^2 - 55123905529152*T + 59763388484928
$89$	\( T^{18} + \cdots + 800346452619097 \) T^18 - 15T^17 - 702T^16 + 10488T^15 + 189618T^14 - 2954634T^13 - 24711103T^12 + 433562430T^11 + 1481072172T^10 - 35392285947T^9 - 13930613667T^8 + 1566005366007T^7 - 2601407472800T^6 - 32588650482087T^5 + 105988445475423T^4 + 186618537788243T^3 - 1068503895621111T^2 + 740423213204019*T + 800346452619097
$97$	\( T^{18} + \cdots + 12\!\cdots\!52 \) T^18 - 3T^17 - 825T^16 + 1330T^15 + 270864T^14 - 57165T^13 - 46678411T^12 - 50365101T^11 + 4585851846T^10 + 10022025277T^9 - 258317182488T^8 - 797981317644T^7 + 7906095549272T^6 + 30350875276176T^5 - 111980673567120T^4 - 514525621469200T^3 + 401955800741568T^2 + 2705170550833152*T + 1255509442089152
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