LMFDB - Newform orbit 930.2.bg.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(121,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([0, 0, 16]))

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

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

chi := DirichletCharacter("930.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.bg (of order $15$, degree $8$, minimal)

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:	no
Analytic conductor:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{15})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 27x^{14} + 345x^{12} - 2652x^{10} + 13244x^{8} - 43398x^{6} + 89940x^{4} - 107433x^{2} + 58081 $ x^16 - 27x^14 + 345x^12 - 2652x^10 + 13244x^8 - 43398x^6 + 89940x^4 - 107433*x^2 + 58081
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 27x^{14} + 345x^{12} - 2652x^{10} + 13244x^{8} - 43398x^{6} + 89940x^{4} - 107433x^{2} + 58081 $ x^16 - 27*x^14 + 345*x^12 - 2652*x^10 + 13244*x^8 - 43398*x^6 + 89940*x^4 - 107433*x^2 + 58081 :

$\beta_{1}$	$=$	$ ( - 481 \nu^{14} + 10935 \nu^{12} - 120641 \nu^{10} + 792237 \nu^{8} - 3303859 \nu^{6} + \cdots + 8734563 ) / 323698 $ (-481v^14 + 10935v^12 - 120641v^10 + 792237v^8 - 3303859v^6 + 8551395v^4 - 12690614v^2 + 161849v + 8734563) / 323698
$\beta_{2}$	$=$	$ ( 481 \nu^{14} - 10935 \nu^{12} + 120641 \nu^{10} - 792237 \nu^{8} + 3303859 \nu^{6} - 8551395 \nu^{4} + \cdots - 8734563 ) / 323698 $ (481v^14 - 10935v^12 + 120641v^10 - 792237v^8 + 3303859v^6 - 8551395v^4 + 12690614v^2 + 161849v - 8734563) / 323698
$\beta_{3}$	$=$	$ ( - 879144720 \nu^{15} - 3144854790 \nu^{14} + 13836683352 \nu^{13} + 78024201152 \nu^{12} + \cdots + 59040737512289 ) / 11901469429298 $ (-879144720v^15 - 3144854790v^14 + 13836683352v^13 + 78024201152v^12 - 89278933653v^11 - 889423402658v^10 + 95062455881v^9 + 5888432618929v^8 + 2063158037274v^7 - 24026211151887v^6 - 13843669424702v^5 + 59633015429809v^4 + 36965828626322v^3 - 82401300908132v^2 - 36427645933254*v + 59040737512289) / 11901469429298
$\beta_{4}$	$=$	$ ( - 879144720 \nu^{15} + 3144854790 \nu^{14} + 13836683352 \nu^{13} - 78024201152 \nu^{12} + \cdots - 59040737512289 ) / 11901469429298 $ (-879144720v^15 + 3144854790v^14 + 13836683352v^13 - 78024201152v^12 - 89278933653v^11 + 889423402658v^10 + 95062455881v^9 - 5888432618929v^8 + 2063158037274v^7 + 24026211151887v^6 - 13843669424702v^5 - 59633015429809v^4 + 36965828626322v^3 + 82401300908132v^2 - 36427645933254*v - 59040737512289) / 11901469429298
$\beta_{5}$	$=$	$ ( - 879144720 \nu^{15} - 10804470307 \nu^{14} + 13836683352 \nu^{13} + 249798214715 \nu^{12} + \cdots + 44600586148174 ) / 11901469429298 $ (-879144720v^15 - 10804470307v^14 + 13836683352v^13 + 249798214715v^12 - 89278933653v^11 - 2731917586661v^10 + 95062455881v^9 + 17346636905054v^8 + 2063158037274v^7 - 67024837757108v^6 - 13843669424702v^5 + 147677522073294v^4 + 36965828626322v^3 - 151545537147092v^2 - 42378380647903*v + 44600586148174) / 11901469429298
$\beta_{6}$	$=$	$ ( 1761178984 \nu^{15} + 11368095320 \nu^{14} - 44779737057 \nu^{13} - 280055462260 \nu^{12} + \cdots - 230349536745774 ) / 11901469429298 $ (1761178984v^15 + 11368095320v^14 - 44779737057v^13 - 280055462260v^12 + 534721065978v^11 + 3239592599263v^10 - 3768012967205v^9 - 22120938532531v^8 + 16738334520669v^7 + 94843723502693v^6 - 46192300263954v^5 - 250479400389093v^4 + 73208859844642v^3 + 366581167649995v^2 - 52214246429899*v - 230349536745774) / 11901469429298
$\beta_{7}$	$=$	$ ( 11502471 \nu^{15} - 100754402 \nu^{14} - 302430222 \nu^{13} + 2300277459 \nu^{12} + \cdots + 1379612638329 ) / 49383690578 $ (11502471v^15 - 100754402v^14 - 302430222v^13 + 2300277459v^12 + 3745368043v^11 - 25050172064v^10 - 27330788147v^9 + 160572232380v^8 + 125474461758v^7 - 646303508562v^6 - 353492024798v^5 + 1593931115935v^4 + 543750044164v^3 - 2207888439990v^2 - 325676753988*v + 1379612638329) / 49383690578
$\beta_{8}$	$=$	$ ( 13049190 \nu^{15} - 41079768 \nu^{14} - 323751872 \nu^{13} + 888074667 \nu^{12} + \cdots + 211873877520 ) / 49383690578 $ (13049190v^15 - 41079768v^14 - 323751872v^13 + 888074667v^12 + 3690553538v^11 - 9279789799v^10 - 24433330369v^9 + 56873654394v^8 + 99693822207v^7 - 215754323582v^6 - 247439898049v^5 + 481477613042v^4 + 341914111652v^3 - 543057264054v^2 - 244982313329*v + 211873877520) / 49383690578
$\beta_{9}$	$=$	$ ( 158509246 \nu^{15} + 204032769 \nu^{14} - 3519708183 \nu^{13} - 5203789126 \nu^{12} + \cdots - 4537175879612 ) / 410395497562 $ (158509246v^15 + 204032769v^14 - 3519708183v^13 - 5203789126v^12 + 37026555855v^11 + 61795072449v^10 - 225774923500v^9 - 430177674564v^8 + 842415020205v^7 + 1871226083985v^6 - 1818260803239v^5 - 4993951496763v^4 + 1975431901078v^3 + 7360174536264v^2 - 809518665534*v - 4537175879612) / 410395497562
$\beta_{10}$	$=$	$ ( 167082381 \nu^{15} + 204032769 \nu^{14} - 4015307501 \nu^{13} - 5203789126 \nu^{12} + \cdots - 4742373628393 ) / 410395497562 $ (167082381v^15 + 204032769v^14 - 4015307501v^13 - 5203789126v^12 + 45907529277v^11 + 61795072449v^10 - 312045159727v^9 - 430177674564v^8 + 1351068112770v^7 + 1871226083985v^6 - 3673028978218v^5 - 4993951496763v^4 + 5782517133443v^3 + 7360174536264v^2 - 4114840286229*v - 4742373628393) / 410395497562
$\beta_{11}$	$=$	$ ( - 167082381 \nu^{15} + 204032769 \nu^{14} + 4015307501 \nu^{13} - 5203789126 \nu^{12} + \cdots - 4742373628393 ) / 410395497562 $ (-167082381v^15 + 204032769v^14 + 4015307501v^13 - 5203789126v^12 - 45907529277v^11 + 61795072449v^10 + 312045159727v^9 - 430177674564v^8 - 1351068112770v^7 + 1871226083985v^6 + 3673028978218v^5 - 4993951496763v^4 - 5782517133443v^3 + 7360174536264v^2 + 4114840286229*v - 4742373628393) / 410395497562
$\beta_{12}$	$=$	$ ( 36243 \nu^{15} - 862640 \nu^{13} + 9868500 \nu^{11} - 67041955 \nu^{9} + 289073175 \nu^{7} + \cdots + 39005609 ) / 78011218 $ (36243v^15 - 862640v^13 + 9868500v^11 - 67041955v^9 + 289073175v^7 - 776643695v^5 + 1198809225v^3 - 835256245v + 39005609) / 78011218
$\beta_{13}$	$=$	$ ( 846609 \nu^{15} + 2057746 \nu^{14} - 21592486 \nu^{13} - 48696648 \nu^{12} + 256411089 \nu^{11} + \cdots - 40266853821 ) / 1702885882 $ (846609v^15 + 2057746v^14 - 21592486v^13 - 48696648v^12 + 256411089v^11 + 543806285v^10 - 1784969604v^9 - 3575813034v^8 + 7764423585v^7 + 14846523620v^6 - 20721790443v^5 - 38360465617v^4 + 30540143304v^3 + 57407971584v^2 - 19677898873*v - 40266853821) / 1702885882
$\beta_{14}$	$=$	$ ( 6135024293 \nu^{15} + 13352156499 \nu^{14} - 144306881117 \nu^{13} - 288319318279 \nu^{12} + \cdots - 33115752912402 ) / 11901469429298 $ (6135024293v^15 + 13352156499v^14 - 144306881117v^13 - 288319318279v^12 + 1621765377709v^11 + 2948465510484v^10 - 10824464034847v^9 - 17387689074968v^8 + 45498484447973v^7 + 62256078466158v^6 - 117622505367539v^5 - 126129629458253v^4 + 168334643403717v^3 + 119928992729707v^2 - 106868595929830*v - 33115752912402) / 11901469429298
$\beta_{15}$	$=$	$ ( - 6229064855 \nu^{15} - 4449079069 \nu^{14} + 149688381624 \nu^{13} + \cdots - 41759097040057 ) / 11901469429298 $ (-6229064855v^15 - 4449079069v^14 + 149688381624v^13 + 84880417141v^12 - 1686020685713v^11 - 788893843317v^10 + 11169729283807v^9 + 4060764738779v^8 - 46468851901548v^7 - 11418624916134v^6 + 119958555534177v^5 + 10381297760156v^4 - 176885437933337v^3 + 22259305461325v^2 + 126156859383031*v - 41759097040057) / 11901469429298

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

$\nu$	$=$	$ \beta_{2} + \beta_1 $ b2 + b1
$\nu^{2}$	$=$	$ \beta_{15} - \beta_{11} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta _1 + 4 $ b15 - b11 + b8 - b6 - b5 + b4 - b1 + 4
$\nu^{3}$	$=$	$ \beta_{15} - \beta_{13} + 4 \beta_{12} - 5 \beta_{10} + 6 \beta_{9} - \beta_{7} - \beta_{6} + 5 \beta_{4} + \cdots - 5 $ b15 - b13 + 4b12 - 5b10 + 6b9 - b7 - b6 + 5b4 + 5b3 + 2b2 + 2*b1 - 5
$\nu^{4}$	$=$	$ 8 \beta_{15} + \beta_{13} - \beta_{12} - 7 \beta_{11} + 2 \beta_{10} - \beta_{9} + 7 \beta_{8} + \cdots + 9 $ 8b15 + b13 - b12 - 7b11 + 2b10 - b9 + 7b8 - b7 - 6b6 - 9b5 + b4 + 6b3 - 2b2 - 7*b1 + 9
$\nu^{5}$	$=$	$ 10 \beta_{15} - 4 \beta_{13} + 21 \beta_{12} - 5 \beta_{11} - 36 \beta_{10} + 51 \beta_{9} - 6 \beta_{8} + \cdots - 36 $ 10b15 - 4b13 + 21b12 - 5b11 - 36b10 + 51b9 - 6b8 - 4b7 - 10b6 + 38b4 + 38b3 - 5b2 - 5*b1 - 36
$\nu^{6}$	$=$	$ 42 \beta_{15} + 15 \beta_{13} - 4 \beta_{12} - 21 \beta_{11} + 25 \beta_{10} - 4 \beta_{9} + 27 \beta_{8} + \cdots - 28 $ 42b15 + 15b13 - 4b12 - 21b11 + 25b10 - 4b9 + 27b8 - 15b7 - 34b6 - 44b5 - 33b4 + 69b3 - 19b2 - 25b1 - 28
$\nu^{7}$	$=$	$ 56 \beta_{15} - 26 \beta_{14} + 15 \beta_{13} + 39 \beta_{12} - 77 \beta_{11} - 138 \beta_{10} + \cdots - 155 $ 56b15 - 26b14 + 15b13 + 39b12 - 77b11 - 138b10 + 245b9 - 71b8 - 11b7 - 56b6 - 13b5 + 156b4 + 169b3 - 93b2 - 80*b1 - 155
$\nu^{8}$	$=$	$ 138 \beta_{15} + 120 \beta_{13} + 67 \beta_{12} + 110 \beta_{11} + 181 \beta_{10} + 67 \beta_{9} + \cdots - 438 $ 138b15 + 120b13 + 67b12 + 110b11 + 181b10 + 67b9 + 18b8 - 120b7 - 272b6 - 105b5 - 245b4 + 484b3 - 101b2 - 4b1 - 438
$\nu^{9}$	$=$	$ 153 \beta_{15} - 402 \beta_{14} + 345 \beta_{13} - 172 \beta_{12} - 614 \beta_{11} - 211 \beta_{10} + \cdots - 403 $ 153b15 - 402b14 + 345b13 - 172b12 - 614b11 - 211b10 + 576b9 - 498b8 - 57b7 - 153b6 - 201b5 + 158b4 + 359b3 - 661b2 - 460*b1 - 403
$\nu^{10}$	$=$	$ - 33 \beta_{15} + 566 \beta_{13} + 1149 \beta_{12} + 2089 \beta_{11} + 907 \beta_{10} + 1149 \beta_{9} + \cdots - 2578 $ -33b15 + 566b13 + 1149b12 + 2089b11 + 907b10 + 1149b9 - 599b8 - 566b7 - 2265b6 + 325b5 - 499b4 + 2472b3 - 230b2 + 555b1 - 2578
$\nu^{11}$	$=$	$ - 413 \beta_{15} - 3342 \beta_{14} + 2841 \beta_{13} - 948 \beta_{12} - 2786 \beta_{11} + 963 \beta_{10} + \cdots - 231 $ -413b15 - 3342b14 + 2841b13 - 948b12 - 2786b11 + 963b10 - 1932b9 - 2428b8 - 501b7 + 413b6 - 1671b5 - 3057b4 - 1386b3 - 3095b2 - 1424*b1 - 231
$\nu^{12}$	$=$	$ - 4536 \beta_{15} + 702 \beta_{13} + 9525 \beta_{12} + 16543 \beta_{11} + 2482 \beta_{10} + 9525 \beta_{9} + \cdots - 8380 $ -4536b15 + 702b13 + 9525b12 + 16543b11 + 2482b10 + 9525b9 - 5238b8 - 702b7 - 14514b6 + 5283b5 + 5073b4 + 8694b3 + 1662b2 + 3621b1 - 8380
$\nu^{13}$	$=$	$ - 6889 \beta_{15} - 17220 \beta_{14} + 13966 \beta_{13} + 7188 \beta_{12} - 1893 \beta_{11} + \cdots + 3249 $ -6889b15 - 17220b14 + 13966b13 + 7188b12 - 1893b11 + 8690b10 - 30906b9 - 7077b8 - 3254b7 + 6889b6 - 8610b5 - 28599b4 - 19989b3 - 8131b2 + 479*b1 + 3249
$\nu^{14}$	$=$	$ - 40193 \beta_{15} - 13606 \beta_{13} + 48406 \beta_{12} + 79494 \beta_{11} - 9105 \beta_{10} + \cdots - 374 $ -40193b15 - 13606b13 + 48406b12 + 79494b11 - 9105b10 + 48406b9 - 26587b8 + 13606b7 - 56619b6 + 34250b5 + 55017b4 + 7545b3 + 24403b2 + 9847b1 - 374
$\nu^{15}$	$=$	$ - 33940 \beta_{15} - 36120 \beta_{14} + 24745 \beta_{13} + 119836 \beta_{12} + 84770 \beta_{11} + \cdots + 23912 $ -33940b15 - 36120b14 + 24745b13 + 119836b12 + 84770b11 + 48950b10 - 203780b9 + 9195b8 - 11375b7 + 33940b6 - 18060b5 - 151395b4 - 133335b3 + 18000b2 + 36060*b1 + 23912

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$.

$n$	$187$	$311$	$871$
$\chi(n)$	$1$	$1$	$-\beta_{4}$

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} $

121.1

−2.08547	−	0.994522i
2.08547	−	0.994522i
2.29670	−	0.743145i
−2.29670	−	0.743145i
1.37050	−	0.406737i
−1.37050	−	0.406737i
1.37050	+	0.406737i
−1.37050	+	0.406737i
2.29670	+	0.743145i
−2.29670	+	0.743145i
−2.08547	+	0.994522i
2.08547	+	0.994522i
−1.93590	+	0.207912i
1.93590	+	0.207912i
−1.93590	−	0.207912i
1.93590	−	0.207912i

0.809017

−

0.587785i

0.104528

−

0.994522i

0.309017

−

0.951057i

0.500000

−

0.866025i

−0.500000

−

0.866025i

−4.44421

0.944645i

−0.309017

−

0.951057i

−0.978148

−

0.207912i

−0.104528

−

0.994522i

121.2

0.809017

−

0.587785i

0.104528

−

0.994522i

0.309017

−

0.951057i

0.500000

−

0.866025i

−0.500000

−

0.866025i

2.15704

−

0.458493i

−0.309017

−

0.951057i

−0.978148

−

0.207912i

−0.104528

−

0.994522i

361.1

−0.309017

−

0.951057i

−0.669131

−

0.743145i

−0.809017

0.587785i

0.500000

0.866025i

−0.500000

0.866025i

−0.296127

−

2.81746i

0.809017

0.587785i

−0.104528

0.994522i

0.669131

−

0.743145i

361.2

−0.309017

−

0.951057i

−0.669131

−

0.743145i

−0.809017

0.587785i

0.500000

0.866025i

−0.500000

0.866025i

0.000615948

0.00586036i

0.809017

0.587785i

−0.104528

0.994522i

0.669131

−

0.743145i

391.1

0.809017

0.587785i

−0.913545

−

0.406737i

0.309017

0.951057i

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.80375

2.00327i

−0.309017

0.951057i

0.669131

0.743145i

0.913545

−

0.406737i

391.2

0.809017

0.587785i

−0.913545

−

0.406737i

0.309017

0.951057i

0.500000

−

0.866025i

−0.500000

−

0.866025i

1.16386

−

1.29260i

−0.309017

0.951057i

0.669131

0.743145i

0.913545

−

0.406737i

421.1

0.809017

−

0.587785i

−0.913545

0.406737i

0.309017

−

0.951057i

0.500000

0.866025i

−0.500000

0.866025i

−1.80375

−

2.00327i

−0.309017

−

0.951057i

0.669131

−

0.743145i

0.913545

0.406737i

421.2

0.809017

−

0.587785i

−0.913545

0.406737i

0.309017

−

0.951057i

0.500000

0.866025i

−0.500000

0.866025i

1.16386

1.29260i

−0.309017

−

0.951057i

0.669131

−

0.743145i

0.913545

0.406737i

541.1

−0.309017

0.951057i

−0.669131

0.743145i

−0.809017

−

0.587785i

0.500000

−

0.866025i

−0.500000

−

0.866025i

−0.296127

2.81746i

0.809017

−

0.587785i

−0.104528

−

0.994522i

0.669131

0.743145i

541.2

−0.309017

0.951057i

−0.669131

0.743145i

−0.809017

−

0.587785i

0.500000

−

0.866025i

−0.500000

−

0.866025i

0.000615948

−

0.00586036i

0.809017

−

0.587785i

−0.104528

−

0.994522i

0.669131

0.743145i

661.1

0.809017

0.587785i

0.104528

0.994522i

0.309017

0.951057i

0.500000

0.866025i

−0.500000

0.866025i

−4.44421

−

0.944645i

−0.309017

0.951057i

−0.978148

0.207912i

−0.104528

0.994522i

661.2

0.809017

0.587785i

0.104528

0.994522i

0.309017

0.951057i

0.500000

0.866025i

−0.500000

0.866025i

2.15704

0.458493i

−0.309017

0.951057i

−0.978148

0.207912i

−0.104528

0.994522i

691.1

−0.309017

0.951057i

0.978148

0.207912i

−0.809017

−

0.587785i

0.500000

0.866025i

−0.500000

0.866025i

−0.731733

0.325789i

0.809017

−

0.587785i

0.913545

0.406737i

−0.978148

0.207912i

691.2

−0.309017

0.951057i

0.978148

0.207912i

−0.809017

−

0.587785i

0.500000

0.866025i

−0.500000

0.866025i

1.45430

−

0.647494i

0.809017

−

0.587785i

0.913545

0.406737i

−0.978148

0.207912i

751.1

−0.309017

−

0.951057i

0.978148

−

0.207912i

−0.809017

0.587785i

0.500000

−

0.866025i

−0.500000

−

0.866025i

−0.731733

−

0.325789i

0.809017

0.587785i

0.913545

−

0.406737i

−0.978148

−

0.207912i

751.2

−0.309017

−

0.951057i

0.978148

−

0.207912i

−0.809017

0.587785i

0.500000

−

0.866025i

−0.500000

−

0.866025i

1.45430

0.647494i

0.809017

0.587785i

0.913545

−

0.406737i

−0.978148

−

0.207912i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.g	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.bg.e	✓	16
31.g	even	15	1	inner	930.2.bg.e	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.bg.e	✓	16	1.a	even	1	1	trivial
930.2.bg.e	✓	16	31.g	even	15	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{16} + 5 T_{7}^{15} - 4 T_{7}^{14} - 30 T_{7}^{13} - 35 T_{7}^{12} - 120 T_{7}^{11} + 824 T_{7}^{10} + \cdots + 1 $ T7^16 + 5*T7^15 - 4*T7^14 - 30*T7^13 - 35*T7^12 - 120*T7^11 + 824*T7^10 - 205*T7^9 + 1339*T7^8 - 15485*T7^7 + 32714*T7^6 - 7650*T7^5 - 36110*T7^4 + 13800*T7^3 + 28781*T7^2 - 35*T7 + 1 acting on $S_{2}^{\mathrm{new}}(930, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} $ (T^4 - T^3 + T^2 - T + 1)^4
$3$	$ (T^{8} + T^{7} - T^{5} + \cdots + 1)^{2} $ (T^8 + T^7 - T^5 - T^4 - T^3 + T + 1)^2
$5$	$ (T^{2} - T + 1)^{8} $ (T^2 - T + 1)^8
$7$	$ T^{16} + 5 T^{15} + \cdots + 1 $ T^16 + 5T^15 - 4T^14 - 30T^13 - 35T^12 - 120T^11 + 824T^10 - 205T^9 + 1339T^8 - 15485T^7 + 32714T^6 - 7650T^5 - 36110T^4 + 13800T^3 + 28781T^2 - 35*T + 1
$11$	$ T^{16} + 4 T^{15} + \cdots + 12952801 $ T^16 + 4T^15 + 8T^14 - 13T^13 - 232T^12 - 373T^11 + 5901T^10 + 52261T^9 + 224508T^8 + 595013T^7 + 796569T^6 - 1002986T^5 - 5690047T^4 - 7737749T^3 + 4906397T^2 + 19297838*T + 12952801
$13$	$ T^{16} - 4 T^{15} + \cdots + 625 $ T^16 - 4T^15 + 10T^14 - 36T^13 - 81T^12 + 1244T^11 + 170T^10 - 8569T^9 + 16666T^8 + 50480T^7 - 51740T^6 - 59925T^5 + 89825T^4 - 43000T^3 + 13375T^2 - 2500*T + 625
$17$	$ T^{16} + \cdots + 417344041 $ T^16 + 22T^15 + 202T^14 + 1124T^13 + 5608T^12 + 27436T^11 + 112484T^10 + 462212T^9 + 1786973T^8 + 4843854T^7 + 8903786T^6 - 2631222T^5 - 19837017T^4 - 151858132T^3 + 375591843T^2 + 885270286*T + 417344041
$19$	$ T^{16} + 3 T^{15} + \cdots + 1113025 $ T^16 + 3T^15 - 50T^14 - 117T^13 + 1164T^12 + 1407T^11 - 15355T^10 + 552T^9 + 115346T^8 - 79290T^7 - 312415T^6 - 393915T^5 + 2833905T^4 - 1811250T^3 + 5015500T^2 - 822900*T + 1113025
$23$	$ T^{16} - 9 T^{15} + \cdots + 841 $ T^16 - 9T^15 + 78T^14 - 528T^13 + 3266T^12 - 14889T^11 + 67300T^10 - 242094T^9 + 801909T^8 - 2032758T^7 + 5104870T^6 - 8031207T^5 + 15942666T^4 - 5947956T^3 + 967492T^2 - 44457*T + 841
$29$	$ T^{16} + 3 T^{15} + \cdots + 72361 $ T^16 + 3T^15 + 42T^14 + 24T^13 + 836T^12 - 1077T^11 + 19600T^10 - 52383T^9 + 418614T^8 - 1043499T^7 + 5686000T^6 - 13247964T^5 + 37619541T^4 - 63497388T^3 + 170027863T^2 + 5683701*T + 72361
$31$	$ T^{16} + \cdots + 852891037441 $ T^16 + 15T^15 + 117T^14 + 960T^13 + 7473T^12 + 47730T^11 + 313284T^10 + 1963845T^9 + 11016725T^8 + 60879195T^7 + 301065924T^6 + 1421924430T^5 + 6901472433T^4 + 27483984960T^3 + 103837930677T^2 + 412689211665*T + 852891037441
$37$	$ T^{16} + \cdots + 1579983001 $ T^16 + 17T^15 + 256T^14 + 1733T^13 + 12836T^12 + 47816T^11 + 328159T^10 + 899584T^9 + 5620207T^8 + 7952964T^7 + 57208369T^6 + 54418476T^5 + 401320516T^4 + 91303733T^3 + 1269425326T^2 + 939785607*T + 1579983001
$41$	$ T^{16} - 28 T^{15} + \cdots + 20475625 $ T^16 - 28T^15 + 325T^14 - 2423T^13 + 19579T^12 - 180302T^11 + 1457860T^10 - 9674607T^9 + 45608156T^8 - 117143970T^7 + 59865690T^6 + 529228475T^5 - 333119175T^4 - 450039250T^3 + 446154750T^2 - 42648125*T + 20475625
$43$	$ T^{16} + 28 T^{15} + \cdots + 32041 $ T^16 + 28T^15 + 247T^14 + 266T^13 - 6197T^12 - 19826T^11 + 84464T^10 + 1045238T^9 + 6075308T^8 + 17952246T^7 + 32318561T^6 + 29383257T^5 + 16335408T^4 + 8076047T^3 + 3061263T^2 - 406151*T + 32041
$47$	$ T^{16} + \cdots + 367450561 $ T^16 - 9T^15 + 125T^14 - 705T^13 + 6710T^12 - 28887T^11 + 286443T^10 - 970550T^9 + 12538415T^8 - 53224430T^7 + 373432473T^6 - 993503737T^5 + 1456941950T^4 - 1296660380T^3 + 1195473360T^2 - 874700639*T + 367450561
$53$	$ T^{16} + \cdots + 787221435025 $ T^16 - 2T^15 + 190T^14 - 397T^13 - 5516T^12 - 34298T^11 - 997210T^10 + 3526337T^9 + 108488516T^8 + 64526340T^7 - 3981085045T^6 - 13177946355T^5 + 38833564155T^4 + 353802165575T^3 + 989688694875T^2 + 1259245531300*T + 787221435025
$59$	$ T^{16} + \cdots + 1596721681 $ T^16 - 24T^15 + 348T^14 - 3462T^13 + 22988T^12 - 85182T^11 + 12361T^10 + 1592589T^9 - 1172367T^8 - 110255193T^7 + 1144584754T^6 - 6101155929T^5 + 19678570203T^4 - 38255103501T^3 + 39297649492T^2 - 13901416428*T + 1596721681
$61$	$ (T^{8} + 20 T^{7} + \cdots - 8339)^{2} $ (T^8 + 20T^7 + 9T^6 - 1630T^5 - 8439T^4 - 8815T^3 + 14969T^2 + 16485*T - 8339)^2
$67$	$ T^{16} + \cdots + 479998166761 $ T^16 + T^15 + 234T^14 - 859T^13 + 39236T^12 - 149157T^11 + 3224971T^10 - 16536032T^9 + 192956202T^8 - 768100148T^7 + 4864276891T^6 - 15876306783T^5 + 79529870201T^4 - 208567024141T^3 + 536655909024T^2 - 562568335181T + 479998166761
$71$	$ T^{16} + \cdots + 14311278150625 $ T^16 - 39T^15 + 750T^14 - 10321T^13 + 103209T^12 - 694576T^11 + 3251455T^10 - 6706914T^9 - 12793574T^8 - 43791945T^7 + 1597349965T^6 - 80764675T^5 - 87255820425T^4 + 377356986875T^3 + 673624346625T^2 - 7615513051875*T + 14311278150625
$73$	$ T^{16} + \cdots + 708376405801 $ T^16 - 81T^15 + 3023T^14 - 67548T^13 + 990983T^12 - 9993903T^11 + 72606531T^10 - 406496264T^9 + 1811937233T^8 - 6227272472T^7 + 20525523579T^6 - 68819108791T^5 + 190762605443T^4 - 471947964974T^3 + 1215812729577T^2 - 1566848642687*T + 708376405801
$79$	$ T^{16} + \cdots + 49171383597361 $ T^16 - 64T^15 + 1628T^14 - 15227T^13 - 166897T^12 + 6221998T^11 - 64086024T^10 + 763859T^9 + 7537641843T^8 - 101606472263T^7 + 782089982844T^6 - 4030651837414T^5 + 12650926852793T^4 - 11274043893541T^3 + 43561638775652T^2 + 94913295379552*T + 49171383597361
$83$	$ T^{16} + \cdots + 19217999641 $ T^16 + 40T^15 + 679T^14 + 5760T^13 + 9900T^12 - 451040T^11 - 6311194T^10 - 32501240T^9 + 86892289T^8 + 2348990170T^7 + 15168971926T^6 + 60027140160T^5 + 140860084880T^4 - 444697653845T^3 + 375553834009T^2 - 119737333025*T + 19217999641
$89$	$ T^{16} + \cdots + 12\!\cdots\!25 $ T^16 + 7T^15 + 78T^14 - 101T^13 + 39965T^12 - 350521T^11 + 6514473T^10 - 38623178T^9 + 557084366T^8 - 2755186700T^7 + 35298347760T^6 - 222110283315T^5 + 2478297239990T^4 - 16669084599150T^3 + 109758607329050T^2 - 402544001313225*T + 1215209490621025
$97$	$ T^{16} + \cdots + 663205420200625 $ T^16 + 17T^15 + 573T^14 + 7879T^13 + 147975T^12 + 1411944T^11 + 16692383T^10 + 58701852T^9 + 954430946T^8 + 77066200T^7 + 57197276705T^6 + 2968115900T^5 + 1636051676650T^4 + 6508971100000T^3 + 64335812472625T^2 - 343914146098750*T + 663205420200625
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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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.bg (of order \(15\), degree \(8\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	930.2.bg.e

Level	$930$
Weight	$2$
Character orbit	930.bg
Analytic conductor	$7.426$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(7.42608738798\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{15})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 27x^{14} + 345x^{12} - 2652x^{10} + 13244x^{8} - 43398x^{6} + 89940x^{4} - 107433x^{2} + 58081 \) x^16 - 27x^14 + 345x^12 - 2652x^10 + 13244x^8 - 43398x^6 + 89940x^4 - 107433*x^2 + 58081
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

\(\beta_{1}\)	\(=\)	\( ( - 481 \nu^{14} + 10935 \nu^{12} - 120641 \nu^{10} + 792237 \nu^{8} - 3303859 \nu^{6} + \cdots + 8734563 ) / 323698 \) (-481v^14 + 10935v^12 - 120641v^10 + 792237v^8 - 3303859v^6 + 8551395v^4 - 12690614v^2 + 161849v + 8734563) / 323698
\(\beta_{2}\)	\(=\)	\( ( 481 \nu^{14} - 10935 \nu^{12} + 120641 \nu^{10} - 792237 \nu^{8} + 3303859 \nu^{6} - 8551395 \nu^{4} + \cdots - 8734563 ) / 323698 \) (481v^14 - 10935v^12 + 120641v^10 - 792237v^8 + 3303859v^6 - 8551395v^4 + 12690614v^2 + 161849v - 8734563) / 323698
\(\beta_{3}\)	\(=\)	\( ( - 879144720 \nu^{15} - 3144854790 \nu^{14} + 13836683352 \nu^{13} + 78024201152 \nu^{12} + \cdots + 59040737512289 ) / 11901469429298 \) (-879144720v^15 - 3144854790v^14 + 13836683352v^13 + 78024201152v^12 - 89278933653v^11 - 889423402658v^10 + 95062455881v^9 + 5888432618929v^8 + 2063158037274v^7 - 24026211151887v^6 - 13843669424702v^5 + 59633015429809v^4 + 36965828626322v^3 - 82401300908132v^2 - 36427645933254*v + 59040737512289) / 11901469429298
\(\beta_{4}\)	\(=\)	\( ( - 879144720 \nu^{15} + 3144854790 \nu^{14} + 13836683352 \nu^{13} - 78024201152 \nu^{12} + \cdots - 59040737512289 ) / 11901469429298 \) (-879144720v^15 + 3144854790v^14 + 13836683352v^13 - 78024201152v^12 - 89278933653v^11 + 889423402658v^10 + 95062455881v^9 - 5888432618929v^8 + 2063158037274v^7 + 24026211151887v^6 - 13843669424702v^5 - 59633015429809v^4 + 36965828626322v^3 + 82401300908132v^2 - 36427645933254*v - 59040737512289) / 11901469429298
\(\beta_{5}\)	\(=\)	\( ( - 879144720 \nu^{15} - 10804470307 \nu^{14} + 13836683352 \nu^{13} + 249798214715 \nu^{12} + \cdots + 44600586148174 ) / 11901469429298 \) (-879144720v^15 - 10804470307v^14 + 13836683352v^13 + 249798214715v^12 - 89278933653v^11 - 2731917586661v^10 + 95062455881v^9 + 17346636905054v^8 + 2063158037274v^7 - 67024837757108v^6 - 13843669424702v^5 + 147677522073294v^4 + 36965828626322v^3 - 151545537147092v^2 - 42378380647903*v + 44600586148174) / 11901469429298
\(\beta_{6}\)	\(=\)	\( ( 1761178984 \nu^{15} + 11368095320 \nu^{14} - 44779737057 \nu^{13} - 280055462260 \nu^{12} + \cdots - 230349536745774 ) / 11901469429298 \) (1761178984v^15 + 11368095320v^14 - 44779737057v^13 - 280055462260v^12 + 534721065978v^11 + 3239592599263v^10 - 3768012967205v^9 - 22120938532531v^8 + 16738334520669v^7 + 94843723502693v^6 - 46192300263954v^5 - 250479400389093v^4 + 73208859844642v^3 + 366581167649995v^2 - 52214246429899*v - 230349536745774) / 11901469429298
\(\beta_{7}\)	\(=\)	\( ( 11502471 \nu^{15} - 100754402 \nu^{14} - 302430222 \nu^{13} + 2300277459 \nu^{12} + \cdots + 1379612638329 ) / 49383690578 \) (11502471v^15 - 100754402v^14 - 302430222v^13 + 2300277459v^12 + 3745368043v^11 - 25050172064v^10 - 27330788147v^9 + 160572232380v^8 + 125474461758v^7 - 646303508562v^6 - 353492024798v^5 + 1593931115935v^4 + 543750044164v^3 - 2207888439990v^2 - 325676753988*v + 1379612638329) / 49383690578
\(\beta_{8}\)	\(=\)	\( ( 13049190 \nu^{15} - 41079768 \nu^{14} - 323751872 \nu^{13} + 888074667 \nu^{12} + \cdots + 211873877520 ) / 49383690578 \) (13049190v^15 - 41079768v^14 - 323751872v^13 + 888074667v^12 + 3690553538v^11 - 9279789799v^10 - 24433330369v^9 + 56873654394v^8 + 99693822207v^7 - 215754323582v^6 - 247439898049v^5 + 481477613042v^4 + 341914111652v^3 - 543057264054v^2 - 244982313329*v + 211873877520) / 49383690578
\(\beta_{9}\)	\(=\)	\( ( 158509246 \nu^{15} + 204032769 \nu^{14} - 3519708183 \nu^{13} - 5203789126 \nu^{12} + \cdots - 4537175879612 ) / 410395497562 \) (158509246v^15 + 204032769v^14 - 3519708183v^13 - 5203789126v^12 + 37026555855v^11 + 61795072449v^10 - 225774923500v^9 - 430177674564v^8 + 842415020205v^7 + 1871226083985v^6 - 1818260803239v^5 - 4993951496763v^4 + 1975431901078v^3 + 7360174536264v^2 - 809518665534*v - 4537175879612) / 410395497562
\(\beta_{10}\)	\(=\)	\( ( 167082381 \nu^{15} + 204032769 \nu^{14} - 4015307501 \nu^{13} - 5203789126 \nu^{12} + \cdots - 4742373628393 ) / 410395497562 \) (167082381v^15 + 204032769v^14 - 4015307501v^13 - 5203789126v^12 + 45907529277v^11 + 61795072449v^10 - 312045159727v^9 - 430177674564v^8 + 1351068112770v^7 + 1871226083985v^6 - 3673028978218v^5 - 4993951496763v^4 + 5782517133443v^3 + 7360174536264v^2 - 4114840286229*v - 4742373628393) / 410395497562
\(\beta_{11}\)	\(=\)	\( ( - 167082381 \nu^{15} + 204032769 \nu^{14} + 4015307501 \nu^{13} - 5203789126 \nu^{12} + \cdots - 4742373628393 ) / 410395497562 \) (-167082381v^15 + 204032769v^14 + 4015307501v^13 - 5203789126v^12 - 45907529277v^11 + 61795072449v^10 + 312045159727v^9 - 430177674564v^8 - 1351068112770v^7 + 1871226083985v^6 + 3673028978218v^5 - 4993951496763v^4 - 5782517133443v^3 + 7360174536264v^2 + 4114840286229*v - 4742373628393) / 410395497562
\(\beta_{12}\)	\(=\)	\( ( 36243 \nu^{15} - 862640 \nu^{13} + 9868500 \nu^{11} - 67041955 \nu^{9} + 289073175 \nu^{7} + \cdots + 39005609 ) / 78011218 \) (36243v^15 - 862640v^13 + 9868500v^11 - 67041955v^9 + 289073175v^7 - 776643695v^5 + 1198809225v^3 - 835256245v + 39005609) / 78011218
\(\beta_{13}\)	\(=\)	\( ( 846609 \nu^{15} + 2057746 \nu^{14} - 21592486 \nu^{13} - 48696648 \nu^{12} + 256411089 \nu^{11} + \cdots - 40266853821 ) / 1702885882 \) (846609v^15 + 2057746v^14 - 21592486v^13 - 48696648v^12 + 256411089v^11 + 543806285v^10 - 1784969604v^9 - 3575813034v^8 + 7764423585v^7 + 14846523620v^6 - 20721790443v^5 - 38360465617v^4 + 30540143304v^3 + 57407971584v^2 - 19677898873*v - 40266853821) / 1702885882
\(\beta_{14}\)	\(=\)	\( ( 6135024293 \nu^{15} + 13352156499 \nu^{14} - 144306881117 \nu^{13} - 288319318279 \nu^{12} + \cdots - 33115752912402 ) / 11901469429298 \) (6135024293v^15 + 13352156499v^14 - 144306881117v^13 - 288319318279v^12 + 1621765377709v^11 + 2948465510484v^10 - 10824464034847v^9 - 17387689074968v^8 + 45498484447973v^7 + 62256078466158v^6 - 117622505367539v^5 - 126129629458253v^4 + 168334643403717v^3 + 119928992729707v^2 - 106868595929830*v - 33115752912402) / 11901469429298
\(\beta_{15}\)	\(=\)	\( ( - 6229064855 \nu^{15} - 4449079069 \nu^{14} + 149688381624 \nu^{13} + \cdots - 41759097040057 ) / 11901469429298 \) (-6229064855v^15 - 4449079069v^14 + 149688381624v^13 + 84880417141v^12 - 1686020685713v^11 - 788893843317v^10 + 11169729283807v^9 + 4060764738779v^8 - 46468851901548v^7 - 11418624916134v^6 + 119958555534177v^5 + 10381297760156v^4 - 176885437933337v^3 + 22259305461325v^2 + 126156859383031*v - 41759097040057) / 11901469429298

\(\nu\)	\(=\)	\( \beta_{2} + \beta_1 \) b2 + b1
\(\nu^{2}\)	\(=\)	\( \beta_{15} - \beta_{11} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta _1 + 4 \) b15 - b11 + b8 - b6 - b5 + b4 - b1 + 4
\(\nu^{3}\)	\(=\)	\( \beta_{15} - \beta_{13} + 4 \beta_{12} - 5 \beta_{10} + 6 \beta_{9} - \beta_{7} - \beta_{6} + 5 \beta_{4} + \cdots - 5 \) b15 - b13 + 4b12 - 5b10 + 6b9 - b7 - b6 + 5b4 + 5b3 + 2b2 + 2*b1 - 5
\(\nu^{4}\)	\(=\)	\( 8 \beta_{15} + \beta_{13} - \beta_{12} - 7 \beta_{11} + 2 \beta_{10} - \beta_{9} + 7 \beta_{8} + \cdots + 9 \) 8b15 + b13 - b12 - 7b11 + 2b10 - b9 + 7b8 - b7 - 6b6 - 9b5 + b4 + 6b3 - 2b2 - 7*b1 + 9
\(\nu^{5}\)	\(=\)	\( 10 \beta_{15} - 4 \beta_{13} + 21 \beta_{12} - 5 \beta_{11} - 36 \beta_{10} + 51 \beta_{9} - 6 \beta_{8} + \cdots - 36 \) 10b15 - 4b13 + 21b12 - 5b11 - 36b10 + 51b9 - 6b8 - 4b7 - 10b6 + 38b4 + 38b3 - 5b2 - 5*b1 - 36
\(\nu^{6}\)	\(=\)	\( 42 \beta_{15} + 15 \beta_{13} - 4 \beta_{12} - 21 \beta_{11} + 25 \beta_{10} - 4 \beta_{9} + 27 \beta_{8} + \cdots - 28 \) 42b15 + 15b13 - 4b12 - 21b11 + 25b10 - 4b9 + 27b8 - 15b7 - 34b6 - 44b5 - 33b4 + 69b3 - 19b2 - 25b1 - 28
\(\nu^{7}\)	\(=\)	\( 56 \beta_{15} - 26 \beta_{14} + 15 \beta_{13} + 39 \beta_{12} - 77 \beta_{11} - 138 \beta_{10} + \cdots - 155 \) 56b15 - 26b14 + 15b13 + 39b12 - 77b11 - 138b10 + 245b9 - 71b8 - 11b7 - 56b6 - 13b5 + 156b4 + 169b3 - 93b2 - 80*b1 - 155
\(\nu^{8}\)	\(=\)	\( 138 \beta_{15} + 120 \beta_{13} + 67 \beta_{12} + 110 \beta_{11} + 181 \beta_{10} + 67 \beta_{9} + \cdots - 438 \) 138b15 + 120b13 + 67b12 + 110b11 + 181b10 + 67b9 + 18b8 - 120b7 - 272b6 - 105b5 - 245b4 + 484b3 - 101b2 - 4b1 - 438
\(\nu^{9}\)	\(=\)	\( 153 \beta_{15} - 402 \beta_{14} + 345 \beta_{13} - 172 \beta_{12} - 614 \beta_{11} - 211 \beta_{10} + \cdots - 403 \) 153b15 - 402b14 + 345b13 - 172b12 - 614b11 - 211b10 + 576b9 - 498b8 - 57b7 - 153b6 - 201b5 + 158b4 + 359b3 - 661b2 - 460*b1 - 403
\(\nu^{10}\)	\(=\)	\( - 33 \beta_{15} + 566 \beta_{13} + 1149 \beta_{12} + 2089 \beta_{11} + 907 \beta_{10} + 1149 \beta_{9} + \cdots - 2578 \) -33b15 + 566b13 + 1149b12 + 2089b11 + 907b10 + 1149b9 - 599b8 - 566b7 - 2265b6 + 325b5 - 499b4 + 2472b3 - 230b2 + 555b1 - 2578
\(\nu^{11}\)	\(=\)	\( - 413 \beta_{15} - 3342 \beta_{14} + 2841 \beta_{13} - 948 \beta_{12} - 2786 \beta_{11} + 963 \beta_{10} + \cdots - 231 \) -413b15 - 3342b14 + 2841b13 - 948b12 - 2786b11 + 963b10 - 1932b9 - 2428b8 - 501b7 + 413b6 - 1671b5 - 3057b4 - 1386b3 - 3095b2 - 1424*b1 - 231
\(\nu^{12}\)	\(=\)	\( - 4536 \beta_{15} + 702 \beta_{13} + 9525 \beta_{12} + 16543 \beta_{11} + 2482 \beta_{10} + 9525 \beta_{9} + \cdots - 8380 \) -4536b15 + 702b13 + 9525b12 + 16543b11 + 2482b10 + 9525b9 - 5238b8 - 702b7 - 14514b6 + 5283b5 + 5073b4 + 8694b3 + 1662b2 + 3621b1 - 8380
\(\nu^{13}\)	\(=\)	\( - 6889 \beta_{15} - 17220 \beta_{14} + 13966 \beta_{13} + 7188 \beta_{12} - 1893 \beta_{11} + \cdots + 3249 \) -6889b15 - 17220b14 + 13966b13 + 7188b12 - 1893b11 + 8690b10 - 30906b9 - 7077b8 - 3254b7 + 6889b6 - 8610b5 - 28599b4 - 19989b3 - 8131b2 + 479*b1 + 3249
\(\nu^{14}\)	\(=\)	\( - 40193 \beta_{15} - 13606 \beta_{13} + 48406 \beta_{12} + 79494 \beta_{11} - 9105 \beta_{10} + \cdots - 374 \) -40193b15 - 13606b13 + 48406b12 + 79494b11 - 9105b10 + 48406b9 - 26587b8 + 13606b7 - 56619b6 + 34250b5 + 55017b4 + 7545b3 + 24403b2 + 9847b1 - 374
\(\nu^{15}\)	\(=\)	\( - 33940 \beta_{15} - 36120 \beta_{14} + 24745 \beta_{13} + 119836 \beta_{12} + 84770 \beta_{11} + \cdots + 23912 \) -33940b15 - 36120b14 + 24745b13 + 119836b12 + 84770b11 + 48950b10 - 203780b9 + 9195b8 - 11375b7 + 33940b6 - 18060b5 - 151395b4 - 133335b3 + 18000b2 + 36060*b1 + 23912

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 121.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) (T^4 - T^3 + T^2 - T + 1)^4
$3$	\( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{2} \) (T^8 + T^7 - T^5 - T^4 - T^3 + T + 1)^2
$5$	\( (T^{2} - T + 1)^{8} \) (T^2 - T + 1)^8
$7$	\( T^{16} + 5 T^{15} + \cdots + 1 \) T^16 + 5T^15 - 4T^14 - 30T^13 - 35T^12 - 120T^11 + 824T^10 - 205T^9 + 1339T^8 - 15485T^7 + 32714T^6 - 7650T^5 - 36110T^4 + 13800T^3 + 28781T^2 - 35*T + 1
$11$	\( T^{16} + 4 T^{15} + \cdots + 12952801 \) T^16 + 4T^15 + 8T^14 - 13T^13 - 232T^12 - 373T^11 + 5901T^10 + 52261T^9 + 224508T^8 + 595013T^7 + 796569T^6 - 1002986T^5 - 5690047T^4 - 7737749T^3 + 4906397T^2 + 19297838*T + 12952801
$13$	\( T^{16} - 4 T^{15} + \cdots + 625 \) T^16 - 4T^15 + 10T^14 - 36T^13 - 81T^12 + 1244T^11 + 170T^10 - 8569T^9 + 16666T^8 + 50480T^7 - 51740T^6 - 59925T^5 + 89825T^4 - 43000T^3 + 13375T^2 - 2500*T + 625
$17$	\( T^{16} + \cdots + 417344041 \) T^16 + 22T^15 + 202T^14 + 1124T^13 + 5608T^12 + 27436T^11 + 112484T^10 + 462212T^9 + 1786973T^8 + 4843854T^7 + 8903786T^6 - 2631222T^5 - 19837017T^4 - 151858132T^3 + 375591843T^2 + 885270286*T + 417344041
$19$	\( T^{16} + 3 T^{15} + \cdots + 1113025 \) T^16 + 3T^15 - 50T^14 - 117T^13 + 1164T^12 + 1407T^11 - 15355T^10 + 552T^9 + 115346T^8 - 79290T^7 - 312415T^6 - 393915T^5 + 2833905T^4 - 1811250T^3 + 5015500T^2 - 822900*T + 1113025
$23$	\( T^{16} - 9 T^{15} + \cdots + 841 \) T^16 - 9T^15 + 78T^14 - 528T^13 + 3266T^12 - 14889T^11 + 67300T^10 - 242094T^9 + 801909T^8 - 2032758T^7 + 5104870T^6 - 8031207T^5 + 15942666T^4 - 5947956T^3 + 967492T^2 - 44457*T + 841
$29$	\( T^{16} + 3 T^{15} + \cdots + 72361 \) T^16 + 3T^15 + 42T^14 + 24T^13 + 836T^12 - 1077T^11 + 19600T^10 - 52383T^9 + 418614T^8 - 1043499T^7 + 5686000T^6 - 13247964T^5 + 37619541T^4 - 63497388T^3 + 170027863T^2 + 5683701*T + 72361
$31$	\( T^{16} + \cdots + 852891037441 \) T^16 + 15T^15 + 117T^14 + 960T^13 + 7473T^12 + 47730T^11 + 313284T^10 + 1963845T^9 + 11016725T^8 + 60879195T^7 + 301065924T^6 + 1421924430T^5 + 6901472433T^4 + 27483984960T^3 + 103837930677T^2 + 412689211665*T + 852891037441
$37$	\( T^{16} + \cdots + 1579983001 \) T^16 + 17T^15 + 256T^14 + 1733T^13 + 12836T^12 + 47816T^11 + 328159T^10 + 899584T^9 + 5620207T^8 + 7952964T^7 + 57208369T^6 + 54418476T^5 + 401320516T^4 + 91303733T^3 + 1269425326T^2 + 939785607*T + 1579983001
$41$	\( T^{16} - 28 T^{15} + \cdots + 20475625 \) T^16 - 28T^15 + 325T^14 - 2423T^13 + 19579T^12 - 180302T^11 + 1457860T^10 - 9674607T^9 + 45608156T^8 - 117143970T^7 + 59865690T^6 + 529228475T^5 - 333119175T^4 - 450039250T^3 + 446154750T^2 - 42648125*T + 20475625
$43$	\( T^{16} + 28 T^{15} + \cdots + 32041 \) T^16 + 28T^15 + 247T^14 + 266T^13 - 6197T^12 - 19826T^11 + 84464T^10 + 1045238T^9 + 6075308T^8 + 17952246T^7 + 32318561T^6 + 29383257T^5 + 16335408T^4 + 8076047T^3 + 3061263T^2 - 406151*T + 32041
$47$	\( T^{16} + \cdots + 367450561 \) T^16 - 9T^15 + 125T^14 - 705T^13 + 6710T^12 - 28887T^11 + 286443T^10 - 970550T^9 + 12538415T^8 - 53224430T^7 + 373432473T^6 - 993503737T^5 + 1456941950T^4 - 1296660380T^3 + 1195473360T^2 - 874700639*T + 367450561
$53$	\( T^{16} + \cdots + 787221435025 \) T^16 - 2T^15 + 190T^14 - 397T^13 - 5516T^12 - 34298T^11 - 997210T^10 + 3526337T^9 + 108488516T^8 + 64526340T^7 - 3981085045T^6 - 13177946355T^5 + 38833564155T^4 + 353802165575T^3 + 989688694875T^2 + 1259245531300*T + 787221435025
$59$	\( T^{16} + \cdots + 1596721681 \) T^16 - 24T^15 + 348T^14 - 3462T^13 + 22988T^12 - 85182T^11 + 12361T^10 + 1592589T^9 - 1172367T^8 - 110255193T^7 + 1144584754T^6 - 6101155929T^5 + 19678570203T^4 - 38255103501T^3 + 39297649492T^2 - 13901416428*T + 1596721681
$61$	\( (T^{8} + 20 T^{7} + \cdots - 8339)^{2} \) (T^8 + 20T^7 + 9T^6 - 1630T^5 - 8439T^4 - 8815T^3 + 14969T^2 + 16485*T - 8339)^2
$67$	\( T^{16} + \cdots + 479998166761 \) T^16 + T^15 + 234T^14 - 859T^13 + 39236T^12 - 149157T^11 + 3224971T^10 - 16536032T^9 + 192956202T^8 - 768100148T^7 + 4864276891T^6 - 15876306783T^5 + 79529870201T^4 - 208567024141T^3 + 536655909024T^2 - 562568335181T + 479998166761
$71$	\( T^{16} + \cdots + 14311278150625 \) T^16 - 39T^15 + 750T^14 - 10321T^13 + 103209T^12 - 694576T^11 + 3251455T^10 - 6706914T^9 - 12793574T^8 - 43791945T^7 + 1597349965T^6 - 80764675T^5 - 87255820425T^4 + 377356986875T^3 + 673624346625T^2 - 7615513051875*T + 14311278150625
$73$	\( T^{16} + \cdots + 708376405801 \) T^16 - 81T^15 + 3023T^14 - 67548T^13 + 990983T^12 - 9993903T^11 + 72606531T^10 - 406496264T^9 + 1811937233T^8 - 6227272472T^7 + 20525523579T^6 - 68819108791T^5 + 190762605443T^4 - 471947964974T^3 + 1215812729577T^2 - 1566848642687*T + 708376405801
$79$	\( T^{16} + \cdots + 49171383597361 \) T^16 - 64T^15 + 1628T^14 - 15227T^13 - 166897T^12 + 6221998T^11 - 64086024T^10 + 763859T^9 + 7537641843T^8 - 101606472263T^7 + 782089982844T^6 - 4030651837414T^5 + 12650926852793T^4 - 11274043893541T^3 + 43561638775652T^2 + 94913295379552*T + 49171383597361
$83$	\( T^{16} + \cdots + 19217999641 \) T^16 + 40T^15 + 679T^14 + 5760T^13 + 9900T^12 - 451040T^11 - 6311194T^10 - 32501240T^9 + 86892289T^8 + 2348990170T^7 + 15168971926T^6 + 60027140160T^5 + 140860084880T^4 - 444697653845T^3 + 375553834009T^2 - 119737333025*T + 19217999641
$89$	\( T^{16} + \cdots + 12\!\cdots\!25 \) T^16 + 7T^15 + 78T^14 - 101T^13 + 39965T^12 - 350521T^11 + 6514473T^10 - 38623178T^9 + 557084366T^8 - 2755186700T^7 + 35298347760T^6 - 222110283315T^5 + 2478297239990T^4 - 16669084599150T^3 + 109758607329050T^2 - 402544001313225*T + 1215209490621025
$97$	\( T^{16} + \cdots + 663205420200625 \) T^16 + 17T^15 + 573T^14 + 7879T^13 + 147975T^12 + 1411944T^11 + 16692383T^10 + 58701852T^9 + 954430946T^8 + 77066200T^7 + 57197276705T^6 + 2968115900T^5 + 1636051676650T^4 + 6508971100000T^3 + 64335812472625T^2 - 343914146098750*T + 663205420200625
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