LMFDB - Newform orbit 6069.2.a.bj

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6069,2,Mod(1,6069)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6069, 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("6069.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6069 = 3 \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6069.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.4612089867$
Analytic rank:	$1$
Dimension:	$16$
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} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 63 x^{12} - 448 x^{11} - 52 x^{10} + 1440 x^{9} - 350 x^{8} + \cdots - 2 $ x^16 - 4x^15 - 14x^14 + 68x^13 + 63x^12 - 448x^11 - 52x^10 + 1440x^9 - 350x^8 - 2336x^7 + 952x^6 + 1784x^5 - 814x^4 - 496x^3 + 212x^2 - 8*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 357)
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 $1,\beta_1,\ldots,\beta_{15}$ 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_{14} q^{5} + \beta_1 q^{6} + q^{7} + ( - \beta_{15} + \beta_{14} - \beta_{12} + \cdots - 1) q^{8}+ \cdots + q^{9}+O(q^{10}) $ q - b1 * q^2 - q^3 + (b2 + 1) * q^4 - b14 * q^5 + b1 * q^6 + q^7 + (-b15 + b14 - b12 - b8 - b2 - 1) * q^8 + q^9	$ q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{14} q^{5} + \beta_1 q^{6} + q^{7} + ( - \beta_{15} + \beta_{14} - \beta_{12} + \cdots - 1) q^{8}+ \cdots + (\beta_{9} - \beta_{5}) q^{99}+O(q^{100}) $ q - b1 * q^2 - q^3 + (b2 + 1) * q^4 - b14 * q^5 + b1 * q^6 + q^7 + (-b15 + b14 - b12 - b8 - b2 - 1) * q^8 + q^9 + (b14 - b12 - b11 + b10 - b5 + b4 - 1) * q^10 + (b9 - b5) * q^11 + (-b2 - 1) * q^12 + (b8 + b4 - b3 + b2 - 1) * q^13 - b1 * q^14 + b14 * q^15 + (b15 - b14 + b13 + b12 - b11 + b10 + b8 + b1) * q^16 - b1 * q^18 + (-b13 - b12 - b8 - b6 + b3 - b2 + b1 - 3) * q^19 + (b11 - 2b10 - b9 - b8 + 2b5 - b4 + 1) * q^20 - q^21 + (b12 - b6 - b4 - b3 + 1) * q^22 + (2b14 - b9 + b7 + b6 + b5 + b4 + b2 - 2b1) * q^23 + (b15 - b14 + b12 + b8 + b2 + 1) * q^24 + (b15 - b14 + 2b12 + b10 + b9 + b8 + b7 + b6 - b4 + b3 - b2 + b1) q^25 + (-b14 - 2b10 - b8 - b7 - 2b5 - b4 - 2b2 + 2b1 - 2) * q^26 - q^27 + (b2 + 1) * q^28 + (-2b15 + b11 - 2b10 - b7 + b6 + 2b5 - b4 - b2 + 2) q^29 + (-b14 + b12 + b11 - b10 + b5 - b4 + 1) * q^30 + (b15 + b13 + b11 + b10 - b9 + b8 - b6 + b5 + b3 + b2 + 2) * q^31 + (-b14 - b13 + 2b11 - b10 + b6 + 2b5 - 1) * q^32 + (-b9 + b5) * q^33 - b14 * q^35 + (b2 + 1) * q^36 + (-b13 + 2b12 + b9 - b4 + 2b1 + 1) * q^37 + (2b15 - b14 + b12 + b10 - b9 + 2b8 - 2b5 + b4 + b2 + 2b1 - 1) * q^38 + (-b8 - b4 + b3 - b2 + 1) * q^39 + (2b12 + b11 + 2b10 + b9 + 2b8 + b6 + b5 + b3 + b2 - b1 + 1) q^40 + (b14 - b13 - b12 - b11 + b10 - b9 - b8 - b7 - 2b6 - b5 - b4 - b2 + b1 - 3) q^41 + b1 * q^42 + (b13 + b12 + b11 - 2b10 - b9 - b7 + b6 - b5 - b4 - b3) q^43 + (-b15 + b14 - b12 + 2b11 + b10 - 2b7 - 3b5 + 2) q^44 - b14 * q^45 + (-2b14 + 2b12 + b11 - b10 - b9 - b8 + b6 + 4b5 - b4 + 2b3 + b2 - b1 + 4) * q^46 + (b15 - b14 - b13 + 2b12 + 2b11 - b9 - b8 + 2b5 - b4 + b3 + 2b2 - b1) * q^47 + (-b15 + b14 - b13 - b12 + b11 - b10 - b8 - b1) * q^48 + q^49 + (-b13 + 2b12 + b11 - b10 + b7 + 4b5 - b4 - b2 + b1 - 2) * q^50 + (b15 + b14 - b12 - b11 + 3b10 + b9 + 2b8 + b7 - 2b6 + b3 - b2 + 3b1 - 2) * q^52 + (-b15 - b14 + 2b13 - b12 - b11 - 3b10 + b9 + b8 - 2b5 - 2b2 + b1 - 1) * q^53 + b1 * q^54 + (-b15 - b14 - b12 + b11 + b10 - b9 - b8 - b7 - b4 - 2b2 + b1 - 1) q^55 + (-b15 + b14 - b12 - b8 - b2 - 1) * q^56 + (b13 + b12 + b8 + b6 - b3 + b2 - b1 + 3) * q^57 + (b15 + 2b13 - b12 - 4b11 - 2b10 + b9 + 3b8 + 2b7 + b6 + b5 - b3 + b2) q^58 + (b15 - b14 + b13 - b11 - b10 + b8 + 2b6 + b5 + b4 - b3 + b2 - b1 - 4) q^59 + (-b11 + 2b10 + b9 + b8 - 2b5 + b4 - 1) * q^60 + (b15 + b14 - b12 - b11 - b10 + 2b8 + b7 + b4 - 3b3 + 2b2 - 2) q^61 + (-b15 + b14 - b13 + b12 + b11 - b10 - b9 - 3b8 + b3 - 2b2 - 2b1 - 2) q^62 + q^63 + (b15 + 2b14 - 2b13 + b12 - 2b11 + b7 - b5 + b4 + b1 - 4) q^64 + (b15 + 2b14 + b7 - b6 + 3b5 - b2 - b1) * q^65 + (-b12 + b6 + b4 + b3 - 1) * q^66 + (2b14 - b12 - 2b11 + 2b10 + b9 - 3b8 - b7 - 2b6 - 2b5 + b4 - 2b2 + 2b1 - 6) * q^67 + (-2b14 + b9 - b7 - b6 - b5 - b4 - b2 + 2b1) * q^69 + (b14 - b12 - b11 + b10 - b5 + b4 - 1) * q^70 + (3b15 + b14 + 2b13 + b12 - 2b11 - b9 + b7 - b6 - 2b5 + 2b4 - b3 + b2 - 3b1 + 2) * q^71 + (-b15 + b14 - b12 - b8 - b2 - 1) * q^72 + (2b15 - b14 - b13 + b12 + b11 - 2b9 - b7 + b5 - b4 + 2) * q^73 + (2b11 - b10 + b9 + b8 - b5 - b4 - b3 - 2b2 - b1 - 4) * q^74 + (-b15 + b14 - 2b12 - b10 - b9 - b8 - b7 - b6 + b4 - b3 + b2 - b1) q^75 + (-b14 + 2b12 + 2b11 - 3b10 - b9 + b5 - b3 - 3b2 - 4) * q^76 + (b9 - b5) * q^77 + (b14 + 2b10 + b8 + b7 + 2b5 + b4 + 2b2 - 2b1 + 2) * q^78 + (-b15 + 2b14 + b12 + 2b11 - b9 - 3b8 - b5 + b4 - b2 + 1) q^79 + (-b14 + 2b12 - 2b11 - 2b10 + b9 - b8 + 2b7 - b4 - b3 - 2b2 - 1) q^80 + q^81 + (-b14 - 3b12 + 3b11 - b10 + b9 + b8 - b7 - 4b5 + b4 + b2 + 2b1 + 2) * q^82 + (-b14 + b12 - 3b11 + b10 + 2b9 + b8 - b6 - 3b4 + b3 - 3b2 + 4b1 - 5) q^83 + (-b2 - 1) * q^84 + (-b15 + 2b14 - b12 - b11 + 2b10 + 2b9 + b8 + b7 - 2b6 + 2b5 + b4 + 2b1 - 2) * q^86 + (2b15 - b11 + 2b10 + b7 - b6 - 2b5 + b4 + b2 - 2) q^87 + (2b15 - b14 + b13 - 3b11 - 2b10 - b8 + 2b7 - 3b6 + b4 - 3) q^88 + (b14 + b13 - 2b12 - b11 + 2b10 + b9 - b8 + 2b7 + 2b5 - b4 + b3 - b2 + b1 - 2) * q^89 + (b14 - b12 - b11 + b10 - b5 + b4 - 1) * q^90 + (b8 + b4 - b3 + b2 - 1) * q^91 + (-b15 + b14 - 3b12 - 2b11 + 4b10 + b9 + b7 + b6 + 2b5 + 2b4 + b3 - b2 - 3b1 - 2) * q^92 + (-b15 - b13 - b11 - b10 + b9 - b8 + b6 - b5 - b3 - b2 - 2) * q^93 + (4b14 - b13 + 5b10 - b8 + b7 - b5 + 3b4 + b3 - 2b1 - 2) * q^94 + (b15 + b14 - b13 + 2b12 + b11 + b10 + 2b8 + b6 - b5 + 2b2 - b1 + 3) q^95 + (b14 + b13 - 2b11 + b10 - b6 - 2b5 + 1) * q^96 + (-b15 + b14 - b13 + b12 + b11 - 4b10 - 2b7 + 2b6 + b4 - b3 + 1) q^97 - b1 * q^98 + (b9 - b5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} - 16 q^{3} + 12 q^{4} - 4 q^{5} + 4 q^{6} + 16 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^2 - 16 * q^3 + 12 * q^4 - 4 * q^5 + 4 * q^6 + 16 * q^7 - 12 * q^8 + 16 * q^9	$ 16 q - 4 q^{2} - 16 q^{3} + 12 q^{4} - 4 q^{5} + 4 q^{6} + 16 q^{7} - 12 q^{8} + 16 q^{9} + 4 q^{11} - 12 q^{12} - 12 q^{13} - 4 q^{14} + 4 q^{15} + 4 q^{16} - 4 q^{18} - 36 q^{19} - 16 q^{21} + 16 q^{22} - 4 q^{23} + 12 q^{24} + 4 q^{25} - 24 q^{26} - 16 q^{27} + 12 q^{28} + 16 q^{29} + 8 q^{31} - 28 q^{32} - 4 q^{33} - 4 q^{35} + 12 q^{36} + 32 q^{37} - 8 q^{38} + 12 q^{39} - 28 q^{41} + 4 q^{42} - 20 q^{43} + 16 q^{44} - 4 q^{45} + 24 q^{46} - 32 q^{47} - 4 q^{48} + 16 q^{49} - 28 q^{50} - 16 q^{53} + 4 q^{54} - 28 q^{55} - 12 q^{56} + 36 q^{57} + 24 q^{58} - 64 q^{59} - 8 q^{61} - 40 q^{62} + 16 q^{63} - 12 q^{64} + 12 q^{65} - 16 q^{66} - 48 q^{67} + 4 q^{69} + 40 q^{71} - 12 q^{72} + 24 q^{73} - 72 q^{74} - 4 q^{75} - 72 q^{76} + 4 q^{77} + 24 q^{78} + 8 q^{79} + 8 q^{80} + 16 q^{81} + 16 q^{82} - 40 q^{83} - 12 q^{84} - 16 q^{87} - 24 q^{88} - 24 q^{89} - 12 q^{91} - 16 q^{92} - 8 q^{93} - 8 q^{94} + 44 q^{95} + 28 q^{96} + 24 q^{97} - 4 q^{98} + 4 q^{99}+O(q^{100}) $ 16 * q - 4 * q^2 - 16 * q^3 + 12 * q^4 - 4 * q^5 + 4 * q^6 + 16 * q^7 - 12 * q^8 + 16 * q^9 + 4 * q^11 - 12 * q^12 - 12 * q^13 - 4 * q^14 + 4 * q^15 + 4 * q^16 - 4 * q^18 - 36 * q^19 - 16 * q^21 + 16 * q^22 - 4 * q^23 + 12 * q^24 + 4 * q^25 - 24 * q^26 - 16 * q^27 + 12 * q^28 + 16 * q^29 + 8 * q^31 - 28 * q^32 - 4 * q^33 - 4 * q^35 + 12 * q^36 + 32 * q^37 - 8 * q^38 + 12 * q^39 - 28 * q^41 + 4 * q^42 - 20 * q^43 + 16 * q^44 - 4 * q^45 + 24 * q^46 - 32 * q^47 - 4 * q^48 + 16 * q^49 - 28 * q^50 - 16 * q^53 + 4 * q^54 - 28 * q^55 - 12 * q^56 + 36 * q^57 + 24 * q^58 - 64 * q^59 - 8 * q^61 - 40 * q^62 + 16 * q^63 - 12 * q^64 + 12 * q^65 - 16 * q^66 - 48 * q^67 + 4 * q^69 + 40 * q^71 - 12 * q^72 + 24 * q^73 - 72 * q^74 - 4 * q^75 - 72 * q^76 + 4 * q^77 + 24 * q^78 + 8 * q^79 + 8 * q^80 + 16 * q^81 + 16 * q^82 - 40 * q^83 - 12 * q^84 - 16 * q^87 - 24 * q^88 - 24 * q^89 - 12 * q^91 - 16 * q^92 - 8 * q^93 - 8 * q^94 + 44 * q^95 + 28 * q^96 + 24 * q^97 - 4 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 63 x^{12} - 448 x^{11} - 52 x^{10} + 1440 x^{9} - 350 x^{8} + \cdots - 2 $ x^16 - 4*x^15 - 14*x^14 + 68*x^13 + 63*x^12 - 448*x^11 - 52*x^10 + 1440*x^9 - 350*x^8 - 2336*x^7 + 952*x^6 + 1784*x^5 - 814*x^4 - 496*x^3 + 212*x^2 - 8*x - 2 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ ( - 433 \nu^{15} + 2618 \nu^{14} + 7133 \nu^{13} - 46166 \nu^{12} - 52432 \nu^{11} + 308616 \nu^{10} + \cdots - 78986 ) / 41854 $ (-433v^15 + 2618v^14 + 7133v^13 - 46166v^12 - 52432v^11 + 308616v^10 + 247248v^9 - 944718v^8 - 825246v^7 + 1197510v^6 + 1737622v^5 - 200282v^4 - 1828998v^3 - 349368v^2 + 598332*v - 78986) / 41854
$\beta_{4}$	$=$	$ ( 4209 \nu^{15} - 21437 \nu^{14} - 43915 \nu^{13} + 361523 \nu^{12} + 29942 \nu^{11} - 2391634 \nu^{10} + \cdots + 459634 ) / 83708 $ (4209v^15 - 21437v^14 - 43915v^13 + 361523v^12 + 29942v^11 - 2391634v^10 + 1139510v^9 + 7936842v^8 - 4913164v^7 - 14135172v^6 + 7212712v^5 + 13447036v^4 - 2829146v^3 - 5938358v^2 - 751978*v + 459634) / 83708
$\beta_{5}$	$=$	$ ( - 219 \nu^{15} + 1290 \nu^{14} + 2027 \nu^{13} - 22400 \nu^{12} + 3338 \nu^{11} + 153082 \nu^{10} + \cdots - 2712 ) / 2462 $ (-219v^15 + 1290v^14 + 2027v^13 - 22400v^12 + 3338v^11 + 153082v^10 - 95306v^9 - 522772v^8 + 385628v^7 + 933894v^6 - 619764v^5 - 826200v^4 + 388182v^3 + 289248v^2 - 62614*v - 2712) / 2462
$\beta_{6}$	$=$	$ ( - 414 \nu^{15} + 1039 \nu^{14} + 7508 \nu^{13} - 17135 \nu^{12} - 54970 \nu^{11} + 106694 \nu^{10} + \cdots + 438 ) / 2462 $ (-414v^15 + 1039v^14 + 7508v^13 - 17135v^12 - 54970v^11 + 106694v^10 + 207412v^9 - 308978v^8 - 422310v^7 + 411276v^6 + 435504v^5 - 209916v^4 - 180624v^3 + 16186v^2 + 4464*v + 438) / 2462
$\beta_{7}$	$=$	$ ( 15491 \nu^{15} - 107049 \nu^{14} - 115129 \nu^{13} + 1881251 \nu^{12} - 704546 \nu^{11} + \cdots + 330414 ) / 83708 $ (15491v^15 - 107049v^14 - 115129v^13 + 1881251v^12 - 704546v^11 - 13054962v^10 + 9718594v^9 + 45450058v^8 - 36359720v^7 - 83110380v^6 + 57413292v^5 + 75457384v^4 - 36281654v^3 - 27066966v^2 + 5988930*v + 330414) / 83708
$\beta_{8}$	$=$	$ ( 17989 \nu^{15} - 49657 \nu^{14} - 313063 \nu^{13} + 841315 \nu^{12} + 2156154 \nu^{11} + \cdots + 49922 ) / 83708 $ (17989v^15 - 49657v^14 - 313063v^13 + 841315v^12 + 2156154v^11 - 5472074v^10 - 7489554v^9 + 17066302v^8 + 13791976v^7 - 26005264v^6 - 12717932v^5 + 17496608v^4 + 4424918v^3 - 3665886v^2 + 301750*v + 49922) / 83708
$\beta_{9}$	$=$	$ ( - 11295 \nu^{15} + 32334 \nu^{14} + 194187 \nu^{13} - 538270 \nu^{12} - 1331078 \nu^{11} + \cdots + 21102 ) / 41854 $ (-11295v^15 + 32334v^14 + 194187v^13 - 538270v^12 - 1331078v^11 + 3415226v^10 + 4682718v^9 - 10253158v^8 - 9047462v^7 + 14605504v^6 + 9328056v^5 - 8449814v^4 - 4095952v^3 + 968946v^2 - 71638*v + 21102) / 41854
$\beta_{10}$	$=$	$ ( 24961 \nu^{15} - 81855 \nu^{14} - 399111 \nu^{13} + 1384285 \nu^{12} + 2413858 \nu^{11} + \cdots + 102062 ) / 83708 $ (24961v^15 - 81855v^14 - 399111v^13 + 1384285v^12 + 2413858v^11 - 9026374v^10 - 6770046v^9 + 28454286v^8 + 8329952v^7 - 44516920v^6 - 2242392v^5 + 31812492v^4 - 2821646v^3 - 7955738v^2 + 1625846*v + 102062) / 83708
$\beta_{11}$	$=$	$ ( - 27825 \nu^{15} + 82739 \nu^{14} + 466203 \nu^{13} - 1385741 \nu^{12} - 3042522 \nu^{11} + \cdots - 58070 ) / 83708 $ (-27825v^15 + 82739v^14 + 466203v^13 - 1385741v^12 - 3042522v^11 + 8879458v^10 + 9761570v^9 - 27157826v^8 - 15923816v^7 + 40354296v^6 + 12171804v^5 - 26414096v^4 - 3118462v^3 + 5572218v^2 - 495414*v - 58070) / 83708
$\beta_{12}$	$=$	$ ( 29035 \nu^{15} - 88315 \nu^{14} - 489229 \nu^{13} + 1508177 \nu^{12} + 3214946 \nu^{11} + \cdots + 263134 ) / 83708 $ (29035v^15 - 88315v^14 - 489229v^13 + 1508177v^12 + 3214946v^11 - 9965158v^10 - 10389278v^9 + 32048830v^8 + 16995576v^7 - 51901944v^6 - 12712976v^5 + 39626636v^4 + 2779606v^3 - 11282898v^2 + 583202*v + 263134) / 83708
$\beta_{13}$	$=$	$ ( - 26393 \nu^{15} + 82297 \nu^{14} + 432657 \nu^{13} - 1385013 \nu^{12} - 2728190 \nu^{11} + \cdots + 3642 ) / 41854 $ (-26393v^15 + 82297v^14 + 432657v^13 - 1385013v^12 - 2728190v^11 + 8952916v^10 + 8265808v^9 - 27806056v^8 - 12126884v^7 + 42435608v^6 + 7207098v^5 - 29071440v^4 - 190262v^3 + 6554708v^2 - 935068*v + 3642) / 41854
$\beta_{14}$	$=$	$ ( - 56793 \nu^{15} + 205785 \nu^{14} + 859311 \nu^{13} - 3491531 \nu^{12} - 4684518 \nu^{11} + \cdots - 478718 ) / 83708 $ (-56793v^15 + 205785v^14 + 859311v^13 - 3491531v^12 - 4684518v^11 + 22916630v^10 + 10332770v^9 - 73211574v^8 - 4059652v^7 + 117842472v^6 - 15081304v^5 - 89816548v^4 + 16550742v^3 + 26209946v^2 - 3848018*v - 478718) / 83708
$\beta_{15}$	$=$	$ ( - 103817 \nu^{15} + 343757 \nu^{14} + 1661603 \nu^{13} - 5841023 \nu^{12} - 10055618 \nu^{11} + \cdots - 624358 ) / 83708 $ (-103817v^15 + 343757v^14 + 1661603v^13 - 5841023v^12 - 10055618v^11 + 38353862v^10 + 28211602v^9 - 122326706v^8 - 34847204v^7 + 195749680v^6 + 10349604v^5 - 146939792v^4 + 9429926v^3 + 41075022v^2 - 5067802*v - 624358) / 83708

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 3 $ b2 + 3
$\nu^{3}$	$=$	$ \beta_{15} - \beta_{14} + \beta_{12} + \beta_{8} + \beta_{2} + 4\beta _1 + 1 $ b15 - b14 + b12 + b8 + b2 + 4*b1 + 1
$\nu^{4}$	$=$	$ \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 6\beta_{2} + \beta _1 + 14 $ b15 - b14 + b13 + b12 - b11 + b10 + b8 + 6*b2 + b1 + 14
$\nu^{5}$	$=$	$ 8 \beta_{15} - 7 \beta_{14} + \beta_{13} + 8 \beta_{12} - 2 \beta_{11} + \beta_{10} + 8 \beta_{8} + \cdots + 9 $ 8b15 - 7b14 + b13 + 8b12 - 2b11 + b10 + 8b8 - b6 - 2b5 + 8b2 + 20b1 + 9
$\nu^{6}$	$=$	$ 11 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} + 11 \beta_{12} - 12 \beta_{11} + 10 \beta_{10} + 10 \beta_{8} + \cdots + 72 $ 11b15 - 8b14 + 8b13 + 11b12 - 12b11 + 10b10 + 10b8 + b7 - b5 + b4 + 36b2 + 11*b1 + 72
$\nu^{7}$	$=$	$ 55 \beta_{15} - 41 \beta_{14} + 11 \beta_{13} + 55 \beta_{12} - 26 \beta_{11} + 14 \beta_{10} + \cdots + 66 $ 55b15 - 41b14 + 11b13 + 55b12 - 26b11 + 14b10 + b9 + 54b8 + b7 - 11b6 - 22b5 + 2b4 - b3 + 57b2 + 110b1 + 66
$\nu^{8}$	$=$	$ 92 \beta_{15} - 53 \beta_{14} + 55 \beta_{13} + 91 \beta_{12} - 106 \beta_{11} + 79 \beta_{10} + \cdots + 392 $ 92b15 - 53b14 + 55b13 + 91b12 - 106b11 + 79b10 + b9 + 82b8 + 13b7 - 4b6 - 15b5 + 14b4 + 221b2 + 94*b1 + 392
$\nu^{9}$	$=$	$ 368 \beta_{15} - 233 \beta_{14} + 92 \beta_{13} + 366 \beta_{12} - 240 \beta_{11} + 139 \beta_{10} + \cdots + 459 $ 368b15 - 233b14 + 92b13 + 366b12 - 240b11 + 139b10 + 14b9 + 355b8 + 17b7 - 91b6 - 175b5 + 30b4 - 12b3 + 397b2 + 637*b1 + 459
$\nu^{10}$	$=$	$ 699 \beta_{15} - 337 \beta_{14} + 368 \beta_{13} + 681 \beta_{12} - 832 \beta_{11} + 583 \beta_{10} + \cdots + 2226 $ 699b15 - 337b14 + 368b13 + 681b12 - 832b11 + 583b10 + 18b9 + 628b8 + 120b7 - 65b6 - 153b5 + 136b4 - 3b3 + 1389b2 + 729*b1 + 2226
$\nu^{11}$	$=$	$ 2453 \beta_{15} - 1330 \beta_{14} + 699 \beta_{13} + 2417 \beta_{12} - 1940 \beta_{11} + 1189 \beta_{10} + \cdots + 3134 $ 2453b15 - 1330b14 + 699b13 + 2417b12 - 1940b11 + 1189b10 + 133b9 + 2340b8 + 188b7 - 679b6 - 1241b5 + 309b4 - 102b3 + 2746b2 + 3816*b1 + 3134
$\nu^{12}$	$=$	$ 5076 \beta_{15} - 2126 \beta_{14} + 2453 \beta_{13} + 4873 \beta_{12} - 6159 \beta_{11} + 4188 \beta_{10} + \cdots + 13081 $ 5076b15 - 2126b14 + 2453b13 + 4873b12 - 6159b11 + 4188b10 + 207b9 + 4634b8 + 969b7 - 699b6 - 1333b5 + 1143b4 - 55b3 + 8902b2 + 5378*b1 + 13081
$\nu^{13}$	$=$	$ 16371 \beta_{15} - 7704 \beta_{14} + 5076 \beta_{13} + 15951 \beta_{12} - 14681 \beta_{11} + \cdots + 21240 $ 16371b15 - 7704b14 + 5076b13 + 15951b12 - 14681b11 + 9382b10 + 1088b9 + 15543b8 + 1723b7 - 4826b6 - 8386b5 + 2715b4 - 762b3 + 18914b2 + 23453*b1 + 21240
$\nu^{14}$	$=$	$ 35956 \beta_{15} - 13436 \beta_{14} + 16371 \beta_{13} + 34083 \beta_{12} - 44125 \beta_{11} + \cdots + 79097 $ 35956b15 - 13436b14 + 16371b13 + 34083b12 - 44125b11 + 29674b10 + 1953b9 + 33372b8 + 7311b7 - 6295b6 - 10699b5 + 8927b4 - 635b3 + 57910b2 + 38530*b1 + 79097
$\nu^{15}$	$=$	$ 109479 \beta_{15} - 45418 \beta_{14} + 35956 \beta_{13} + 105449 \beta_{12} - 107023 \beta_{11} + \cdots + 143490 $ 109479b15 - 45418b14 + 35956b13 + 105449b12 - 107023b11 + 70532b10 + 8292b9 + 103955b8 + 14241b7 - 33426b6 - 55550b5 + 21889b4 - 5358b3 + 129812b2 + 147125*b1 + 143490

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

2.60823
2.59452
2.27516
1.93879
1.34647
1.27865
1.08084
0.262952
0.176763
−0.0756003
−0.780448
−1.01763
−1.44229
−1.87689
−2.10046
−2.26906

−2.60823

−1.00000

4.80284

−3.57522

2.60823

1.00000

−7.31043

1.00000

9.32497

1.2

−2.59452

−1.00000

4.73152

1.62893

2.59452

1.00000

−7.08698

1.00000

−4.22628

1.3

−2.27516

−1.00000

3.17634

2.78313

2.27516

1.00000

−2.67637

1.00000

−6.33206

1.4

−1.93879

−1.00000

1.75892

0.837008

1.93879

1.00000

0.467397

1.00000

−1.62279

1.5

−1.34647

−1.00000

−0.187018

−0.670349

1.34647

1.00000

2.94475

1.00000

0.902605

1.6

−1.27865

−1.00000

−0.365044

−2.75223

1.27865

1.00000

3.02407

1.00000

3.51915

1.7

−1.08084

−1.00000

−0.831793

−1.71327

1.08084

1.00000

3.06070

1.00000

1.85176

1.8

−0.262952

−1.00000

−1.93086

−2.52392

0.262952

1.00000

1.03363

1.00000

0.663669

1.9

−0.176763

−1.00000

−1.96875

4.00611

0.176763

1.00000

0.701529

1.00000

−0.708131

1.10

0.0756003

−1.00000

−1.99428

0.573803

−0.0756003

1.00000

−0.301969

1.00000

0.0433797

1.11

0.780448

−1.00000

−1.39090

1.82143

−0.780448

1.00000

−2.64642

1.00000

1.42153

1.12

1.01763

−1.00000

−0.964432

−4.38014

−1.01763

1.00000

−3.01669

1.00000

−4.45736

1.13

1.44229

−1.00000

0.0802040

1.11820

−1.44229

1.00000

−2.76890

1.00000

1.61277

1.14

1.87689

−1.00000

1.52271

−1.38373

−1.87689

1.00000

−0.895823

1.00000

−2.59711

1.15

2.10046

−1.00000

2.41192

−0.482991

−2.10046

1.00000

0.865226

1.00000

−1.01450

1.16

2.26906

−1.00000

3.14862

0.713243

−2.26906

1.00000

2.60628

1.00000

1.61839

Atkin-Lehner signs

$ p $	Sign
$3$	$1$
$7$	$-1$
$17$	$-1$

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	6069.2.a.bj		16
17.b	even	2	1		6069.2.a.bk		16
17.e	odd	16	2		357.2.u.a	✓	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
357.2.u.a	✓	32	17.e	odd	16	2
6069.2.a.bj		16	1.a	even	1	1	trivial
6069.2.a.bk		16	17.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6069))$:

$ T_{2}^{16} + 4 T_{2}^{15} - 14 T_{2}^{14} - 68 T_{2}^{13} + 63 T_{2}^{12} + 448 T_{2}^{11} - 52 T_{2}^{10} + \cdots - 2 $

$ T_{5}^{16} + 4 T_{5}^{15} - 34 T_{5}^{14} - 132 T_{5}^{13} + 413 T_{5}^{12} + 1496 T_{5}^{11} + \cdots + 1058 $

$ T_{11}^{16} - 4 T_{11}^{15} - 70 T_{11}^{14} + 308 T_{11}^{13} + 1581 T_{11}^{12} - 7696 T_{11}^{11} + \cdots + 134912 $

$ T_{23}^{16} + 4 T_{23}^{15} - 218 T_{23}^{14} - 900 T_{23}^{13} + 18553 T_{23}^{12} + 78072 T_{23}^{11} + \cdots - 573924896 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 4 T^{15} + \cdots - 2 $ T^16 + 4T^15 - 14T^14 - 68T^13 + 63T^12 + 448T^11 - 52T^10 - 1440T^9 - 350T^8 + 2336T^7 + 952T^6 - 1784T^5 - 814T^4 + 496T^3 + 212T^2 + 8*T - 2
$3$	$ (T + 1)^{16} $ (T + 1)^16
$5$	$ T^{16} + 4 T^{15} + \cdots + 1058 $ T^16 + 4T^15 - 34T^14 - 132T^13 + 413T^12 + 1496T^11 - 2496T^10 - 7568T^9 + 8787T^8 + 18236T^7 - 18062T^6 - 19828T^5 + 19123T^4 + 7896T^3 - 8008T^2 - 920*T + 1058
$7$	$ (T - 1)^{16} $ (T - 1)^16
$11$	$ T^{16} - 4 T^{15} + \cdots + 134912 $ T^16 - 4T^15 - 70T^14 + 308T^13 + 1581T^12 - 7696T^11 - 14640T^10 + 82616T^9 + 59583T^8 - 421340T^7 - 108570T^6 + 1030068T^5 + 118527T^4 - 1116912T^3 - 187104T^2 + 420096*T + 134912
$13$	$ T^{16} + 12 T^{15} + \cdots - 1018436 $ T^16 + 12T^15 - 38T^14 - 972T^13 - 1799T^12 + 22872T^11 + 90596T^10 - 136344T^9 - 1116971T^8 - 695316T^7 + 4181966T^6 + 5567988T^5 - 4846357T^4 - 8706288T^3 + 1898908T^2 + 3301264*T - 1018436
$17$	$ T^{16} $ T^16
$19$	$ T^{16} + 36 T^{15} + \cdots - 4624 $ T^16 + 36T^15 + 470T^14 + 1892T^13 - 14581T^12 - 187760T^11 - 610764T^10 + 1314224T^9 + 15773915T^8 + 47917948T^7 + 60928406T^6 + 12849172T^5 - 35405281T^4 - 17496488T^3 + 5542488T^2 + 1564000*T - 4624
$23$	$ T^{16} + \cdots - 573924896 $ T^16 + 4T^15 - 218T^14 - 900T^13 + 18553T^12 + 78072T^11 - 775732T^10 - 3272568T^9 + 16408151T^8 + 67586388T^7 - 160982266T^6 - 612451412T^5 + 564091663T^4 + 1717738176T^3 - 201914992T^2 - 1517272448*T - 573924896
$29$	$ T^{16} - 16 T^{15} + \cdots - 58262536 $ T^16 - 16T^15 - 100T^14 + 2472T^13 + 1058T^12 - 142640T^11 + 172028T^10 + 3786528T^9 - 6345664T^8 - 46521632T^7 + 72457944T^6 + 253049504T^5 - 271763748T^4 - 499034784T^3 + 292289008T^2 + 113287808*T - 58262536
$31$	$ T^{16} - 8 T^{15} + \cdots - 13666304 $ T^16 - 8T^15 - 212T^14 + 1352T^13 + 17860T^12 - 81696T^11 - 745904T^10 + 2257824T^9 + 16450766T^8 - 28757472T^7 - 189014740T^6 + 138552792T^5 + 1016579262T^4 - 13323840T^3 - 1981697664T^2 - 1090103296*T - 13666304
$37$	$ T^{16} - 32 T^{15} + \cdots + 1932932 $ T^16 - 32T^15 + 252T^14 + 1944T^13 - 35798T^12 + 78416T^11 + 1059508T^10 - 5402096T^9 - 5214936T^8 + 75620768T^7 - 93359336T^6 - 244056992T^5 + 699278066T^4 - 630035808T^3 + 236604056T^2 - 37270528*T + 1932932
$41$	$ T^{16} + \cdots + 3023243940386 $ T^16 + 28T^15 - 66T^14 - 8028T^13 - 35173T^12 + 892424T^11 + 6251412T^10 - 50187344T^9 - 434613003T^8 + 1615437948T^7 + 15153788942T^6 - 34926130244T^5 - 265165449201T^4 + 583871800248T^3 + 1813532351696T^2 - 5433826069912*T + 3023243940386
$43$	$ T^{16} + 20 T^{15} + \cdots + 3063872 $ T^16 + 20T^15 - 78T^14 - 3772T^13 - 12049T^12 + 197672T^11 + 1147052T^10 - 2976728T^9 - 27343313T^8 + 4590532T^7 + 229625170T^6 + 85443476T^5 - 676860543T^4 - 226903040T^3 + 416417136T^2 + 154694400*T + 3063872
$47$	$ T^{16} + \cdots - 375931093504 $ T^16 + 32T^15 + 92T^14 - 7264T^13 - 78852T^12 + 321008T^11 + 8868460T^10 + 26483608T^9 - 294422098T^8 - 2180338680T^7 - 1129877852T^6 + 34360493048T^5 + 122471645410T^4 + 81920148256T^3 - 326023329216T^2 - 674378050048*T - 375931093504
$53$	$ T^{16} + \cdots - 7158842248 $ T^16 + 16T^15 - 300T^14 - 6072T^13 + 22120T^12 + 842408T^11 + 1228896T^10 - 50453664T^9 - 218310692T^8 + 1085551448T^7 + 8321866400T^6 + 3279295616T^5 - 84962017714T^4 - 212466046400T^3 - 115292999064T^2 + 69974636928*T - 7158842248
$59$	$ T^{16} + \cdots - 37801420672 $ T^16 + 64T^15 + 1584T^14 + 16480T^13 - 8492T^12 - 1836392T^11 - 14631988T^10 + 6970720T^9 + 675465914T^8 + 3050890600T^7 - 3236311620T^6 - 63591307144T^5 - 169395015006T^4 - 37314166368T^3 + 441485099808T^2 + 457974126848*T - 37801420672
$61$	$ T^{16} + \cdots - 2864566057988 $ T^16 + 8T^15 - 520T^14 - 4104T^13 + 105152T^12 + 828064T^11 - 10433508T^10 - 84560392T^9 + 516403930T^8 + 4611115576T^7 - 10740691004T^6 - 126646959352T^5 + 13481060842T^4 + 1416259635904T^3 + 1251868825184T^2 - 4099822619856*T - 2864566057988
$67$	$ T^{16} + \cdots + 9763690651904 $ T^16 + 48T^15 + 516T^14 - 9640T^13 - 232896T^12 - 386408T^11 + 25139172T^10 + 163454688T^9 - 953754372T^8 - 10871948096T^7 + 5866971032T^6 + 286424891440T^5 + 341907151876T^4 - 2886983700288T^3 - 5147109881024T^2 + 5147029705216*T + 9763690651904
$71$	$ T^{16} + \cdots - 14932552212608 $ T^16 - 40T^15 + 84T^14 + 14520T^13 - 134692T^12 - 1750752T^11 + 23247372T^10 + 87450368T^9 - 1618578040T^8 - 2200460320T^7 + 55330403392T^6 + 41506576928T^5 - 935849135876T^4 - 675760872320T^3 + 6667193084224T^2 + 3834445637120*T - 14932552212608
$73$	$ T^{16} + \cdots - 7577792581888 $ T^16 - 24T^15 - 268T^14 + 8960T^13 + 27874T^12 - 1417088T^11 - 1962944T^10 + 122727840T^9 + 175307488T^8 - 6109809088T^7 - 13852481408T^6 + 159486980608T^5 + 580060274240T^4 - 1317994829312T^3 - 9199078763776T^2 - 14758812153856*T - 7577792581888
$79$	$ T^{16} + \cdots + 642231364672 $ T^16 - 8T^15 - 648T^14 + 5576T^13 + 154144T^12 - 1451856T^11 - 16452412T^10 + 179281296T^9 + 710146848T^8 - 10721385248T^7 - 71445648T^6 + 264899496336T^5 - 625378821374T^4 - 1066925165152T^3 + 5020268418336T^2 - 4757589110272*T + 642231364672
$83$	$ T^{16} + \cdots - 70755062775808 $ T^16 + 40T^15 + 16T^14 - 17360T^13 - 143942T^12 + 2730400T^11 + 33266728T^10 - 190562592T^9 - 3136499620T^8 + 5954030848T^7 + 143366411568T^6 - 71780854080T^5 - 3146450909048T^4 + 188985741824T^3 + 29135283001344T^2 + 565316943872*T - 70755062775808
$89$	$ T^{16} + \cdots + 294804791296 $ T^16 + 24T^15 - 284T^14 - 9920T^13 + 11636T^12 + 1539488T^11 + 3024224T^10 - 117274112T^9 - 372114112T^8 + 4751224832T^7 + 16358979584T^6 - 103002073088T^5 - 304740779008T^4 + 1113446760448T^3 + 1881486737408T^2 - 4561256808448*T + 294804791296
$97$	$ T^{16} + \cdots + 80975422001648 $ T^16 - 24T^15 - 504T^14 + 15048T^13 + 68440T^12 - 3591184T^11 + 3720424T^10 + 405694192T^9 - 1630916952T^8 - 21574492544T^7 + 137030237824T^6 + 423556338976T^5 - 4491790657808T^4 + 2315765798400T^3 + 48634446641120T^2 - 121104412160064*T + 80975422001648
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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:	\( N \)	\(=\)	\( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6069.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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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	6069.2.a.bj

Level	$6069$
Weight	$2$
Character orbit	6069.a
Self dual	yes
Analytic conductor	$48.461$
Analytic rank	$1$
Dimension	$16$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(48.4612089867\)
Analytic rank:	\(1\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 63 x^{12} - 448 x^{11} - 52 x^{10} + 1440 x^{9} - 350 x^{8} + \cdots - 2 \) x^16 - 4x^15 - 14x^14 + 68x^13 + 63x^12 - 448x^11 - 52x^10 + 1440x^9 - 350x^8 - 2336x^7 + 952x^6 + 1784x^5 - 814x^4 - 496x^3 + 212x^2 - 8*x - 2
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 357)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \) v^2 - 3
\(\beta_{3}\)	\(=\)	\( ( - 433 \nu^{15} + 2618 \nu^{14} + 7133 \nu^{13} - 46166 \nu^{12} - 52432 \nu^{11} + 308616 \nu^{10} + \cdots - 78986 ) / 41854 \) (-433v^15 + 2618v^14 + 7133v^13 - 46166v^12 - 52432v^11 + 308616v^10 + 247248v^9 - 944718v^8 - 825246v^7 + 1197510v^6 + 1737622v^5 - 200282v^4 - 1828998v^3 - 349368v^2 + 598332*v - 78986) / 41854
\(\beta_{4}\)	\(=\)	\( ( 4209 \nu^{15} - 21437 \nu^{14} - 43915 \nu^{13} + 361523 \nu^{12} + 29942 \nu^{11} - 2391634 \nu^{10} + \cdots + 459634 ) / 83708 \) (4209v^15 - 21437v^14 - 43915v^13 + 361523v^12 + 29942v^11 - 2391634v^10 + 1139510v^9 + 7936842v^8 - 4913164v^7 - 14135172v^6 + 7212712v^5 + 13447036v^4 - 2829146v^3 - 5938358v^2 - 751978*v + 459634) / 83708
\(\beta_{5}\)	\(=\)	\( ( - 219 \nu^{15} + 1290 \nu^{14} + 2027 \nu^{13} - 22400 \nu^{12} + 3338 \nu^{11} + 153082 \nu^{10} + \cdots - 2712 ) / 2462 \) (-219v^15 + 1290v^14 + 2027v^13 - 22400v^12 + 3338v^11 + 153082v^10 - 95306v^9 - 522772v^8 + 385628v^7 + 933894v^6 - 619764v^5 - 826200v^4 + 388182v^3 + 289248v^2 - 62614*v - 2712) / 2462
\(\beta_{6}\)	\(=\)	\( ( - 414 \nu^{15} + 1039 \nu^{14} + 7508 \nu^{13} - 17135 \nu^{12} - 54970 \nu^{11} + 106694 \nu^{10} + \cdots + 438 ) / 2462 \) (-414v^15 + 1039v^14 + 7508v^13 - 17135v^12 - 54970v^11 + 106694v^10 + 207412v^9 - 308978v^8 - 422310v^7 + 411276v^6 + 435504v^5 - 209916v^4 - 180624v^3 + 16186v^2 + 4464*v + 438) / 2462
\(\beta_{7}\)	\(=\)	\( ( 15491 \nu^{15} - 107049 \nu^{14} - 115129 \nu^{13} + 1881251 \nu^{12} - 704546 \nu^{11} + \cdots + 330414 ) / 83708 \) (15491v^15 - 107049v^14 - 115129v^13 + 1881251v^12 - 704546v^11 - 13054962v^10 + 9718594v^9 + 45450058v^8 - 36359720v^7 - 83110380v^6 + 57413292v^5 + 75457384v^4 - 36281654v^3 - 27066966v^2 + 5988930*v + 330414) / 83708
\(\beta_{8}\)	\(=\)	\( ( 17989 \nu^{15} - 49657 \nu^{14} - 313063 \nu^{13} + 841315 \nu^{12} + 2156154 \nu^{11} + \cdots + 49922 ) / 83708 \) (17989v^15 - 49657v^14 - 313063v^13 + 841315v^12 + 2156154v^11 - 5472074v^10 - 7489554v^9 + 17066302v^8 + 13791976v^7 - 26005264v^6 - 12717932v^5 + 17496608v^4 + 4424918v^3 - 3665886v^2 + 301750*v + 49922) / 83708
\(\beta_{9}\)	\(=\)	\( ( - 11295 \nu^{15} + 32334 \nu^{14} + 194187 \nu^{13} - 538270 \nu^{12} - 1331078 \nu^{11} + \cdots + 21102 ) / 41854 \) (-11295v^15 + 32334v^14 + 194187v^13 - 538270v^12 - 1331078v^11 + 3415226v^10 + 4682718v^9 - 10253158v^8 - 9047462v^7 + 14605504v^6 + 9328056v^5 - 8449814v^4 - 4095952v^3 + 968946v^2 - 71638*v + 21102) / 41854
\(\beta_{10}\)	\(=\)	\( ( 24961 \nu^{15} - 81855 \nu^{14} - 399111 \nu^{13} + 1384285 \nu^{12} + 2413858 \nu^{11} + \cdots + 102062 ) / 83708 \) (24961v^15 - 81855v^14 - 399111v^13 + 1384285v^12 + 2413858v^11 - 9026374v^10 - 6770046v^9 + 28454286v^8 + 8329952v^7 - 44516920v^6 - 2242392v^5 + 31812492v^4 - 2821646v^3 - 7955738v^2 + 1625846*v + 102062) / 83708
\(\beta_{11}\)	\(=\)	\( ( - 27825 \nu^{15} + 82739 \nu^{14} + 466203 \nu^{13} - 1385741 \nu^{12} - 3042522 \nu^{11} + \cdots - 58070 ) / 83708 \) (-27825v^15 + 82739v^14 + 466203v^13 - 1385741v^12 - 3042522v^11 + 8879458v^10 + 9761570v^9 - 27157826v^8 - 15923816v^7 + 40354296v^6 + 12171804v^5 - 26414096v^4 - 3118462v^3 + 5572218v^2 - 495414*v - 58070) / 83708
\(\beta_{12}\)	\(=\)	\( ( 29035 \nu^{15} - 88315 \nu^{14} - 489229 \nu^{13} + 1508177 \nu^{12} + 3214946 \nu^{11} + \cdots + 263134 ) / 83708 \) (29035v^15 - 88315v^14 - 489229v^13 + 1508177v^12 + 3214946v^11 - 9965158v^10 - 10389278v^9 + 32048830v^8 + 16995576v^7 - 51901944v^6 - 12712976v^5 + 39626636v^4 + 2779606v^3 - 11282898v^2 + 583202*v + 263134) / 83708
\(\beta_{13}\)	\(=\)	\( ( - 26393 \nu^{15} + 82297 \nu^{14} + 432657 \nu^{13} - 1385013 \nu^{12} - 2728190 \nu^{11} + \cdots + 3642 ) / 41854 \) (-26393v^15 + 82297v^14 + 432657v^13 - 1385013v^12 - 2728190v^11 + 8952916v^10 + 8265808v^9 - 27806056v^8 - 12126884v^7 + 42435608v^6 + 7207098v^5 - 29071440v^4 - 190262v^3 + 6554708v^2 - 935068*v + 3642) / 41854
\(\beta_{14}\)	\(=\)	\( ( - 56793 \nu^{15} + 205785 \nu^{14} + 859311 \nu^{13} - 3491531 \nu^{12} - 4684518 \nu^{11} + \cdots - 478718 ) / 83708 \) (-56793v^15 + 205785v^14 + 859311v^13 - 3491531v^12 - 4684518v^11 + 22916630v^10 + 10332770v^9 - 73211574v^8 - 4059652v^7 + 117842472v^6 - 15081304v^5 - 89816548v^4 + 16550742v^3 + 26209946v^2 - 3848018*v - 478718) / 83708
\(\beta_{15}\)	\(=\)	\( ( - 103817 \nu^{15} + 343757 \nu^{14} + 1661603 \nu^{13} - 5841023 \nu^{12} - 10055618 \nu^{11} + \cdots - 624358 ) / 83708 \) (-103817v^15 + 343757v^14 + 1661603v^13 - 5841023v^12 - 10055618v^11 + 38353862v^10 + 28211602v^9 - 122326706v^8 - 34847204v^7 + 195749680v^6 + 10349604v^5 - 146939792v^4 + 9429926v^3 + 41075022v^2 - 5067802*v - 624358) / 83708

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 3 \) b2 + 3
\(\nu^{3}\)	\(=\)	\( \beta_{15} - \beta_{14} + \beta_{12} + \beta_{8} + \beta_{2} + 4\beta _1 + 1 \) b15 - b14 + b12 + b8 + b2 + 4*b1 + 1
\(\nu^{4}\)	\(=\)	\( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 6\beta_{2} + \beta _1 + 14 \) b15 - b14 + b13 + b12 - b11 + b10 + b8 + 6*b2 + b1 + 14
\(\nu^{5}\)	\(=\)	\( 8 \beta_{15} - 7 \beta_{14} + \beta_{13} + 8 \beta_{12} - 2 \beta_{11} + \beta_{10} + 8 \beta_{8} + \cdots + 9 \) 8b15 - 7b14 + b13 + 8b12 - 2b11 + b10 + 8b8 - b6 - 2b5 + 8b2 + 20b1 + 9
\(\nu^{6}\)	\(=\)	\( 11 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} + 11 \beta_{12} - 12 \beta_{11} + 10 \beta_{10} + 10 \beta_{8} + \cdots + 72 \) 11b15 - 8b14 + 8b13 + 11b12 - 12b11 + 10b10 + 10b8 + b7 - b5 + b4 + 36b2 + 11*b1 + 72
\(\nu^{7}\)	\(=\)	\( 55 \beta_{15} - 41 \beta_{14} + 11 \beta_{13} + 55 \beta_{12} - 26 \beta_{11} + 14 \beta_{10} + \cdots + 66 \) 55b15 - 41b14 + 11b13 + 55b12 - 26b11 + 14b10 + b9 + 54b8 + b7 - 11b6 - 22b5 + 2b4 - b3 + 57b2 + 110b1 + 66
\(\nu^{8}\)	\(=\)	\( 92 \beta_{15} - 53 \beta_{14} + 55 \beta_{13} + 91 \beta_{12} - 106 \beta_{11} + 79 \beta_{10} + \cdots + 392 \) 92b15 - 53b14 + 55b13 + 91b12 - 106b11 + 79b10 + b9 + 82b8 + 13b7 - 4b6 - 15b5 + 14b4 + 221b2 + 94*b1 + 392
\(\nu^{9}\)	\(=\)	\( 368 \beta_{15} - 233 \beta_{14} + 92 \beta_{13} + 366 \beta_{12} - 240 \beta_{11} + 139 \beta_{10} + \cdots + 459 \) 368b15 - 233b14 + 92b13 + 366b12 - 240b11 + 139b10 + 14b9 + 355b8 + 17b7 - 91b6 - 175b5 + 30b4 - 12b3 + 397b2 + 637*b1 + 459
\(\nu^{10}\)	\(=\)	\( 699 \beta_{15} - 337 \beta_{14} + 368 \beta_{13} + 681 \beta_{12} - 832 \beta_{11} + 583 \beta_{10} + \cdots + 2226 \) 699b15 - 337b14 + 368b13 + 681b12 - 832b11 + 583b10 + 18b9 + 628b8 + 120b7 - 65b6 - 153b5 + 136b4 - 3b3 + 1389b2 + 729*b1 + 2226
\(\nu^{11}\)	\(=\)	\( 2453 \beta_{15} - 1330 \beta_{14} + 699 \beta_{13} + 2417 \beta_{12} - 1940 \beta_{11} + 1189 \beta_{10} + \cdots + 3134 \) 2453b15 - 1330b14 + 699b13 + 2417b12 - 1940b11 + 1189b10 + 133b9 + 2340b8 + 188b7 - 679b6 - 1241b5 + 309b4 - 102b3 + 2746b2 + 3816*b1 + 3134
\(\nu^{12}\)	\(=\)	\( 5076 \beta_{15} - 2126 \beta_{14} + 2453 \beta_{13} + 4873 \beta_{12} - 6159 \beta_{11} + 4188 \beta_{10} + \cdots + 13081 \) 5076b15 - 2126b14 + 2453b13 + 4873b12 - 6159b11 + 4188b10 + 207b9 + 4634b8 + 969b7 - 699b6 - 1333b5 + 1143b4 - 55b3 + 8902b2 + 5378*b1 + 13081
\(\nu^{13}\)	\(=\)	\( 16371 \beta_{15} - 7704 \beta_{14} + 5076 \beta_{13} + 15951 \beta_{12} - 14681 \beta_{11} + \cdots + 21240 \) 16371b15 - 7704b14 + 5076b13 + 15951b12 - 14681b11 + 9382b10 + 1088b9 + 15543b8 + 1723b7 - 4826b6 - 8386b5 + 2715b4 - 762b3 + 18914b2 + 23453*b1 + 21240
\(\nu^{14}\)	\(=\)	\( 35956 \beta_{15} - 13436 \beta_{14} + 16371 \beta_{13} + 34083 \beta_{12} - 44125 \beta_{11} + \cdots + 79097 \) 35956b15 - 13436b14 + 16371b13 + 34083b12 - 44125b11 + 29674b10 + 1953b9 + 33372b8 + 7311b7 - 6295b6 - 10699b5 + 8927b4 - 635b3 + 57910b2 + 38530*b1 + 79097
\(\nu^{15}\)	\(=\)	\( 109479 \beta_{15} - 45418 \beta_{14} + 35956 \beta_{13} + 105449 \beta_{12} - 107023 \beta_{11} + \cdots + 143490 \) 109479b15 - 45418b14 + 35956b13 + 105449b12 - 107023b11 + 70532b10 + 8292b9 + 103955b8 + 14241b7 - 33426b6 - 55550b5 + 21889b4 - 5358b3 + 129812b2 + 147125*b1 + 143490

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 4 T^{15} + \cdots - 2 \) T^16 + 4T^15 - 14T^14 - 68T^13 + 63T^12 + 448T^11 - 52T^10 - 1440T^9 - 350T^8 + 2336T^7 + 952T^6 - 1784T^5 - 814T^4 + 496T^3 + 212T^2 + 8*T - 2
$3$	\( (T + 1)^{16} \) (T + 1)^16
$5$	\( T^{16} + 4 T^{15} + \cdots + 1058 \) T^16 + 4T^15 - 34T^14 - 132T^13 + 413T^12 + 1496T^11 - 2496T^10 - 7568T^9 + 8787T^8 + 18236T^7 - 18062T^6 - 19828T^5 + 19123T^4 + 7896T^3 - 8008T^2 - 920*T + 1058
$7$	\( (T - 1)^{16} \) (T - 1)^16
$11$	\( T^{16} - 4 T^{15} + \cdots + 134912 \) T^16 - 4T^15 - 70T^14 + 308T^13 + 1581T^12 - 7696T^11 - 14640T^10 + 82616T^9 + 59583T^8 - 421340T^7 - 108570T^6 + 1030068T^5 + 118527T^4 - 1116912T^3 - 187104T^2 + 420096*T + 134912
$13$	\( T^{16} + 12 T^{15} + \cdots - 1018436 \) T^16 + 12T^15 - 38T^14 - 972T^13 - 1799T^12 + 22872T^11 + 90596T^10 - 136344T^9 - 1116971T^8 - 695316T^7 + 4181966T^6 + 5567988T^5 - 4846357T^4 - 8706288T^3 + 1898908T^2 + 3301264*T - 1018436
$17$	\( T^{16} \) T^16
$19$	\( T^{16} + 36 T^{15} + \cdots - 4624 \) T^16 + 36T^15 + 470T^14 + 1892T^13 - 14581T^12 - 187760T^11 - 610764T^10 + 1314224T^9 + 15773915T^8 + 47917948T^7 + 60928406T^6 + 12849172T^5 - 35405281T^4 - 17496488T^3 + 5542488T^2 + 1564000*T - 4624
$23$	\( T^{16} + \cdots - 573924896 \) T^16 + 4T^15 - 218T^14 - 900T^13 + 18553T^12 + 78072T^11 - 775732T^10 - 3272568T^9 + 16408151T^8 + 67586388T^7 - 160982266T^6 - 612451412T^5 + 564091663T^4 + 1717738176T^3 - 201914992T^2 - 1517272448*T - 573924896
$29$	\( T^{16} - 16 T^{15} + \cdots - 58262536 \) T^16 - 16T^15 - 100T^14 + 2472T^13 + 1058T^12 - 142640T^11 + 172028T^10 + 3786528T^9 - 6345664T^8 - 46521632T^7 + 72457944T^6 + 253049504T^5 - 271763748T^4 - 499034784T^3 + 292289008T^2 + 113287808*T - 58262536
$31$	\( T^{16} - 8 T^{15} + \cdots - 13666304 \) T^16 - 8T^15 - 212T^14 + 1352T^13 + 17860T^12 - 81696T^11 - 745904T^10 + 2257824T^9 + 16450766T^8 - 28757472T^7 - 189014740T^6 + 138552792T^5 + 1016579262T^4 - 13323840T^3 - 1981697664T^2 - 1090103296*T - 13666304
$37$	\( T^{16} - 32 T^{15} + \cdots + 1932932 \) T^16 - 32T^15 + 252T^14 + 1944T^13 - 35798T^12 + 78416T^11 + 1059508T^10 - 5402096T^9 - 5214936T^8 + 75620768T^7 - 93359336T^6 - 244056992T^5 + 699278066T^4 - 630035808T^3 + 236604056T^2 - 37270528*T + 1932932
$41$	\( T^{16} + \cdots + 3023243940386 \) T^16 + 28T^15 - 66T^14 - 8028T^13 - 35173T^12 + 892424T^11 + 6251412T^10 - 50187344T^9 - 434613003T^8 + 1615437948T^7 + 15153788942T^6 - 34926130244T^5 - 265165449201T^4 + 583871800248T^3 + 1813532351696T^2 - 5433826069912*T + 3023243940386
$43$	\( T^{16} + 20 T^{15} + \cdots + 3063872 \) T^16 + 20T^15 - 78T^14 - 3772T^13 - 12049T^12 + 197672T^11 + 1147052T^10 - 2976728T^9 - 27343313T^8 + 4590532T^7 + 229625170T^6 + 85443476T^5 - 676860543T^4 - 226903040T^3 + 416417136T^2 + 154694400*T + 3063872
$47$	\( T^{16} + \cdots - 375931093504 \) T^16 + 32T^15 + 92T^14 - 7264T^13 - 78852T^12 + 321008T^11 + 8868460T^10 + 26483608T^9 - 294422098T^8 - 2180338680T^7 - 1129877852T^6 + 34360493048T^5 + 122471645410T^4 + 81920148256T^3 - 326023329216T^2 - 674378050048*T - 375931093504
$53$	\( T^{16} + \cdots - 7158842248 \) T^16 + 16T^15 - 300T^14 - 6072T^13 + 22120T^12 + 842408T^11 + 1228896T^10 - 50453664T^9 - 218310692T^8 + 1085551448T^7 + 8321866400T^6 + 3279295616T^5 - 84962017714T^4 - 212466046400T^3 - 115292999064T^2 + 69974636928*T - 7158842248
$59$	\( T^{16} + \cdots - 37801420672 \) T^16 + 64T^15 + 1584T^14 + 16480T^13 - 8492T^12 - 1836392T^11 - 14631988T^10 + 6970720T^9 + 675465914T^8 + 3050890600T^7 - 3236311620T^6 - 63591307144T^5 - 169395015006T^4 - 37314166368T^3 + 441485099808T^2 + 457974126848*T - 37801420672
$61$	\( T^{16} + \cdots - 2864566057988 \) T^16 + 8T^15 - 520T^14 - 4104T^13 + 105152T^12 + 828064T^11 - 10433508T^10 - 84560392T^9 + 516403930T^8 + 4611115576T^7 - 10740691004T^6 - 126646959352T^5 + 13481060842T^4 + 1416259635904T^3 + 1251868825184T^2 - 4099822619856*T - 2864566057988
$67$	\( T^{16} + \cdots + 9763690651904 \) T^16 + 48T^15 + 516T^14 - 9640T^13 - 232896T^12 - 386408T^11 + 25139172T^10 + 163454688T^9 - 953754372T^8 - 10871948096T^7 + 5866971032T^6 + 286424891440T^5 + 341907151876T^4 - 2886983700288T^3 - 5147109881024T^2 + 5147029705216*T + 9763690651904
$71$	\( T^{16} + \cdots - 14932552212608 \) T^16 - 40T^15 + 84T^14 + 14520T^13 - 134692T^12 - 1750752T^11 + 23247372T^10 + 87450368T^9 - 1618578040T^8 - 2200460320T^7 + 55330403392T^6 + 41506576928T^5 - 935849135876T^4 - 675760872320T^3 + 6667193084224T^2 + 3834445637120*T - 14932552212608
$73$	\( T^{16} + \cdots - 7577792581888 \) T^16 - 24T^15 - 268T^14 + 8960T^13 + 27874T^12 - 1417088T^11 - 1962944T^10 + 122727840T^9 + 175307488T^8 - 6109809088T^7 - 13852481408T^6 + 159486980608T^5 + 580060274240T^4 - 1317994829312T^3 - 9199078763776T^2 - 14758812153856*T - 7577792581888
$79$	\( T^{16} + \cdots + 642231364672 \) T^16 - 8T^15 - 648T^14 + 5576T^13 + 154144T^12 - 1451856T^11 - 16452412T^10 + 179281296T^9 + 710146848T^8 - 10721385248T^7 - 71445648T^6 + 264899496336T^5 - 625378821374T^4 - 1066925165152T^3 + 5020268418336T^2 - 4757589110272*T + 642231364672
$83$	\( T^{16} + \cdots - 70755062775808 \) T^16 + 40T^15 + 16T^14 - 17360T^13 - 143942T^12 + 2730400T^11 + 33266728T^10 - 190562592T^9 - 3136499620T^8 + 5954030848T^7 + 143366411568T^6 - 71780854080T^5 - 3146450909048T^4 + 188985741824T^3 + 29135283001344T^2 + 565316943872*T - 70755062775808
$89$	\( T^{16} + \cdots + 294804791296 \) T^16 + 24T^15 - 284T^14 - 9920T^13 + 11636T^12 + 1539488T^11 + 3024224T^10 - 117274112T^9 - 372114112T^8 + 4751224832T^7 + 16358979584T^6 - 103002073088T^5 - 304740779008T^4 + 1113446760448T^3 + 1881486737408T^2 - 4561256808448*T + 294804791296
$97$	\( T^{16} + \cdots + 80975422001648 \) T^16 - 24T^15 - 504T^14 + 15048T^13 + 68440T^12 - 3591184T^11 + 3720424T^10 + 405694192T^9 - 1630916952T^8 - 21574492544T^7 + 137030237824T^6 + 423556338976T^5 - 4491790657808T^4 + 2315765798400T^3 + 48634446641120T^2 - 121104412160064*T + 80975422001648
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\( p \)	Sign
\(3\)	\(1\)
\(7\)	\(-1\)
\(17\)	\(-1\)