LMFDB - Newform orbit 288.2.p.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,2,Mod(47,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("288.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	288.p (of order $6$, degree $2$, not 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:	$2.29969157821$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + \cdots + 256 $ x^16 - 3x^15 + 7x^14 - 12x^13 + 16x^12 - 12x^11 - 8x^10 + 36x^9 - 68x^8 + 72x^7 - 32x^6 - 96x^5 + 256x^4 - 384x^3 + 448x^2 - 384*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{8} + \beta_{3} - \beta_1) q^{3} - \beta_{9} q^{5} - \beta_{4} q^{7} + ( - \beta_{10} + \beta_{7} - \beta_{5} + \cdots - 2) q^{9}+O(q^{10}) $ q + (-b8 + b3 - b1) * q^3 - b9 * q^5 - b4 * q^7 + (-b10 + b7 - b5 - b2 - b1 - 2) * q^9	$ q + ( - \beta_{8} + \beta_{3} - \beta_1) q^{3} - \beta_{9} q^{5} - \beta_{4} q^{7} + ( - \beta_{10} + \beta_{7} - \beta_{5} + \cdots - 2) q^{9}+ \cdots + (\beta_{10} - \beta_{7} + \beta_{5} + \cdots + 5) q^{99}+O(q^{100}) $ q + (-b8 + b3 - b1) * q^3 - b9 * q^5 - b4 * q^7 + (-b10 + b7 - b5 - b2 - b1 - 2) * q^9 + (-b10 - b3 - b2 - 2) * q^11 - b13 * q^13 + (b14 - b11 - b6) * q^15 + (-b8 - 2b7 + b5 + b3 - b1) q^17 + (2b10 + b7 - b5 + b3 + b2 - b1 + 1) q^19 + (-b15 + b14 + b13 - 2b11 + b9 - b6 - 2b4) * q^21 + (-b15 - b14 + b12 - b9) * q^23 + (b10 - b8 + 2b7 + b5 + b1) q^25 + (-3b10 + b8 - 2b3 - b1) * q^27 + (-b15 + b13 + b12 + b11) * q^29 + (-b13 + b11 - b9 + 2b6 + b4) q^31 + (b8 - b5 - b3 + 2b2 - 2b1) * q^33 + (-3b7 + 2b5 + 2b3 - 2b2 + 4b1 + 3) q^35 + (2b15 - b14 - b13 + 2b11 + b9 + b6 + b4) * q^37 + (b14 + b13 - 2b12 - b11 + b9 - b6) q^39 + (-b10 - b7 - b5 + b3 + b1 - 2) * q^41 + (2b10 + b8 + 4b7 - b5 - 2b3 + 2b2) * q^43 + (2b15 - b12 - b11 + b4) q^45 + (b15 - b14 + 2b11 + b4) q^47 + (2b10 - b8 - 2b7 - b5 + b2 + 2b1 + 2) q^49 + (b10 - b7 - b5 + 2b3 - b2 - b1 - 1) q^51 + (-b14 - b12 + b11 - b9 + b6 + 2b4) q^53 + (2b15 - b14 - b13 + b11 + b9 + b6 + b4) q^55 + (b10 + b8 + 2b7 + b5 - 2b2 + b1 + 2) * q^57 + (-2b10 + b8 - 2b7 + 2b5 - 3b3 + b2 + 3b1) q^59 + (-b15 + b13 - b12 - b11 - 2b4) q^61 + (-b14 + b13 + b12 - b11 + 2b9 - b6 - b4) q^63 + (4b8 + 4b5 - 3b3 - 3b2 + 4b1 + 2) q^65 + (3b10 - b8 - 3b7 + b5 + 2b3 + b2 + 3b1 + 5) * q^67 + (-b15 - b13 + b12 + b11 + b9 - b6 + 2b4) q^69 + (b14 + b13 - b9 + b6 + b4) * q^71 + (3b8 + b5 - b3 + 2b2 + b1 + 2) * q^73 + (b10 + b8 + 5b7 - 2b5 - 3b3 + b2 - 2b1 - 7) * q^75 + (b15 - b14 - b13 + b12 + b11 + b9 + b4) * q^77 + (-b15 + b13 - b12 - b11 + 2b9 - b6 - b4) q^79 + (-3b10 - 3b8 - 3b7 + 3b3 + 3b2) q^81 + (5b10 - 2b8 - 2b5 + 4b3 + 4b2 - 5b1 - 2) * q^83 + (-b15 + b14 + b12 + b11 - b9 + 2b6 + b4) q^85 + (-b13 + 2b12 + 2b11 + b9 + 2b6 + b4) q^87 + (b8 + b7 - 4b5 - 4b3 + 3b2 + 2b1 - 1) * q^89 + (-8b10 + b8 - 4b7 + 4b5 - 4b3 - 3b2 + 4b1) * q^91 + (-b15 + 2b14 + 2b13 - 2b12 - b9 + b6 + b4) q^93 + (-b15 + b14 + b13 - b12 - b11 - b4) * q^95 + (3b10 + 5b8 + 6b7 - 5b5 - b3 + b2 - 3b1) q^97 + (b10 - b7 + b5 + 3b3 + b2 + 4b1 + 5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{3} - 6 q^{9}+O(q^{10}) $ 16 * q + 6 * q^3 - 6 * q^9	$ 16 q + 6 q^{3} - 6 q^{9} - 12 q^{11} + 4 q^{19} - 14 q^{25} + 36 q^{27} + 12 q^{33} - 36 q^{41} - 8 q^{43} + 10 q^{49} - 18 q^{51} + 18 q^{57} - 12 q^{59} - 6 q^{65} + 16 q^{67} - 4 q^{73} - 78 q^{75} - 6 q^{81} - 54 q^{83} + 36 q^{91} + 8 q^{97} + 6 q^{99}+O(q^{100}) $ 16 * q + 6 * q^3 - 6 * q^9 - 12 * q^11 + 4 * q^19 - 14 * q^25 + 36 * q^27 + 12 * q^33 - 36 * q^41 - 8 * q^43 + 10 * q^49 - 18 * q^51 + 18 * q^57 - 12 * q^59 - 6 * q^65 + 16 * q^67 - 4 * q^73 - 78 * q^75 - 6 * q^81 - 54 * q^83 + 36 * q^91 + 8 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + \cdots + 256 $ x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 16*x^12 - 12*x^11 - 8*x^10 + 36*x^9 - 68*x^8 + 72*x^7 - 32*x^6 - 96*x^5 + 256*x^4 - 384*x^3 + 448*x^2 - 384*x + 256 :

$\beta_{1}$	$=$	$ ( \nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + \cdots + 384 ) / 896 $ (v^15 - 13v^14 + 11v^13 + 4v^12 - 38v^11 + 60v^10 - 104v^9 + 68v^8 + 148v^7 - 344v^6 + 440v^5 - 240v^4 - 32v^3 + 608v^2 - 1152v + 384) / 896
$\beta_{2}$	$=$	$ ( - \nu^{15} - \nu^{14} - 25 \nu^{13} + 38 \nu^{12} - 46 \nu^{11} + 24 \nu^{10} - 8 \nu^{9} + \cdots + 512 ) / 896 $ (-v^15 - v^14 - 25v^13 + 38v^12 - 46v^11 + 24v^10 - 8v^9 - 68v^8 + 244v^7 - 272v^6 + 8v^5 + 128v^4 - 416v^3 + 1184v^2 - 1088*v + 512) / 896
$\beta_{3}$	$=$	$ ( 3 \nu^{15} - 25 \nu^{14} + 33 \nu^{13} - 44 \nu^{12} + 40 \nu^{11} + 40 \nu^{10} - 144 \nu^{9} + \cdots - 640 ) / 896 $ (3v^15 - 25v^14 + 33v^13 - 44v^12 + 40v^11 + 40v^10 - 144v^9 + 316v^8 - 172v^7 - 24v^6 + 480v^5 - 720v^4 + 1024v^3 - 1088v^2 + 128*v - 640) / 896
$\beta_{4}$	$=$	$ ( - \nu^{15} - 29 \nu^{14} + 59 \nu^{13} - 102 \nu^{12} + 66 \nu^{11} + 24 \nu^{10} - 176 \nu^{9} + \cdots - 384 ) / 896 $ (-v^15 - 29v^14 + 59v^13 - 102v^12 + 66v^11 + 24v^10 - 176v^9 + 492v^8 - 540v^7 + 288v^6 + 456v^5 - 1216v^4 + 2496v^3 - 2400v^2 + 1600v - 384) / 896
$\beta_{5}$	$=$	$ ( - 3 \nu^{15} + 11 \nu^{14} - 19 \nu^{13} + 16 \nu^{12} + 2 \nu^{11} - 54 \nu^{10} + 116 \nu^{9} + \cdots - 704 ) / 448 $ (-3v^15 + 11v^14 - 19v^13 + 16v^12 + 2v^11 - 54v^10 + 116v^9 - 92v^8 + 4v^7 + 192v^6 - 424v^5 + 440v^4 - 352v^3 - 256v^2 + 544*v - 704) / 448
$\beta_{6}$	$=$	$ ( 6 \nu^{15} - 8 \nu^{14} + 3 \nu^{13} + 17 \nu^{12} - 39 \nu^{11} + 80 \nu^{10} - 92 \nu^{9} + \cdots + 64 ) / 448 $ (6v^15 - 8v^14 + 3v^13 + 17v^12 - 39v^11 + 80v^10 - 92v^9 + 16v^8 + 272v^7 - 300v^6 + 260v^5 - 152v^4 - 304v^3 + 1072v^2 - 1312*v + 64) / 448
$\beta_{7}$	$=$	$ ( - 13 \nu^{15} + 15 \nu^{14} - 31 \nu^{13} + 4 \nu^{12} + 32 \nu^{11} - 136 \nu^{10} + 176 \nu^{9} + \cdots - 1408 ) / 896 $ (-13v^15 + 15v^14 - 31v^13 + 4v^12 + 32v^11 - 136v^10 + 176v^9 - 100v^8 - 76v^7 + 552v^6 - 736v^5 + 880v^4 - 256v^3 - 512v^2 + 1536*v - 1408) / 896
$\beta_{8}$	$=$	$ ( - 3 \nu^{15} + 18 \nu^{14} - 47 \nu^{13} + 100 \nu^{12} - 117 \nu^{11} + 100 \nu^{10} + 60 \nu^{9} + \cdots + 1536 ) / 448 $ (-3v^15 + 18v^14 - 47v^13 + 100v^12 - 117v^11 + 100v^10 + 60v^9 - 316v^8 + 592v^7 - 648v^6 + 52v^5 + 888v^4 - 2144v^3 + 2768v^2 - 3040*v + 1536) / 448
$\beta_{9}$	$=$	$ ( 6 \nu^{15} - 15 \nu^{14} + 24 \nu^{13} - 32 \nu^{12} + 45 \nu^{11} + 24 \nu^{10} - 64 \nu^{9} + \cdots - 1728 ) / 448 $ (6v^15 - 15v^14 + 24v^13 - 32v^12 + 45v^11 + 24v^10 - 64v^9 + 128v^8 - 92v^7 + 176v^6 - 20v^5 - 600v^4 + 592v^3 - 944v^2 + 928*v - 1728) / 448
$\beta_{10}$	$=$	$ ( 5 \nu^{15} - 16 \nu^{14} + 48 \nu^{13} - 71 \nu^{12} + 76 \nu^{11} - 22 \nu^{10} - 100 \nu^{9} + \cdots - 768 ) / 448 $ (5v^15 - 16v^14 + 48v^13 - 71v^12 + 76v^11 - 22v^10 - 100v^9 + 284v^8 - 408v^7 + 212v^6 + 184v^5 - 920v^4 + 1520v^3 - 1888v^2 + 1184*v - 768) / 448
$\beta_{11}$	$=$	$ ( - 19 \nu^{15} + 37 \nu^{14} - 13 \nu^{13} - 48 \nu^{12} + 232 \nu^{11} - 384 \nu^{10} + 464 \nu^{9} + \cdots - 4608 ) / 896 $ (-19v^15 + 37v^14 - 13v^13 - 48v^12 + 232v^11 - 384v^10 + 464v^9 - 60v^8 - 740v^7 + 1608v^6 - 1696v^5 + 752v^4 + 1280v^3 - 4608v^2 + 5760*v - 4608) / 896
$\beta_{12}$	$=$	$ ( - 13 \nu^{15} + 29 \nu^{14} - 59 \nu^{13} + 116 \nu^{12} - 150 \nu^{11} + 88 \nu^{10} + 120 \nu^{9} + \cdots + 3072 ) / 448 $ (-13v^15 + 29v^14 - 59v^13 + 116v^12 - 150v^11 + 88v^10 + 120v^9 - 324v^8 + 652v^7 - 680v^6 + 104v^5 + 1216v^4 - 2720v^3 + 3968v^2 - 4288*v + 3072) / 448
$\beta_{13}$	$=$	$ ( - 16 \nu^{15} + 40 \nu^{14} - 99 \nu^{13} + 153 \nu^{12} - 113 \nu^{11} - 36 \nu^{10} + 376 \nu^{9} + \cdots - 320 ) / 448 $ (-16v^15 + 40v^14 - 99v^13 + 153v^12 - 113v^11 - 36v^10 + 376v^9 - 640v^8 + 768v^7 - 124v^6 - 964v^5 + 2440v^4 - 3072v^3 + 3152v^2 - 2400*v - 320) / 448
$\beta_{14}$	$=$	$ ( - 37 \nu^{15} + 103 \nu^{14} - 183 \nu^{13} + 244 \nu^{12} - 232 \nu^{11} - 64 \nu^{10} + 712 \nu^{9} + \cdots + 2816 ) / 896 $ (-37v^15 + 103v^14 - 183v^13 + 244v^12 - 232v^11 - 64v^10 + 712v^9 - 1172v^8 + 1300v^7 - 600v^6 - 1216v^5 + 4176v^4 - 6432v^3 + 6848v^2 - 3968*v + 2816) / 896
$\beta_{15}$	$=$	$ ( - 57 \nu^{15} + 167 \nu^{14} - 375 \nu^{13} + 528 \nu^{12} - 536 \nu^{11} + 24 \nu^{10} + 1112 \nu^{9} + \cdots + 5888 ) / 896 $ (-57v^15 + 167v^14 - 375v^13 + 528v^12 - 536v^11 + 24v^10 + 1112v^9 - 2308v^8 + 2932v^7 - 1448v^6 - 1952v^5 + 7856v^4 - 12512v^3 + 14400v^2 - 12288*v + 5888) / 896

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

$\nu$	$=$	$ ( -\beta_{15} + \beta_{14} - 2\beta_{10} ) / 4 $ (-b15 + b14 - 2*b10) / 4
$\nu^{2}$	$=$	$ ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{8} - 2 \beta_{5} - \beta_{4} + \cdots - 4 ) / 4 $ (-b15 + b14 + b13 + b12 - b11 - 2b8 - 2b5 - b4 + 2b2 - 4b1 - 4) / 4
$\nu^{3}$	$=$	$ ( \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + \cdots - 2 ) / 4 $ (b13 - b12 + b11 - 2b9 - 2b7 + 2b6 - 2b5 + 3b4 - 2b3 + 2b2 - 2b1 - 2) / 4
$\nu^{4}$	$=$	$ ( - \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + \cdots - 6 \beta_1 ) / 4 $ (-b15 + b14 + 2b13 - 2b12 - 2b11 + 2b10 - 4b9 + 2b8 + 4b7 + 2b6 - 2b5 + 4b3 - 4b2 - 6b1) / 4
$\nu^{5}$	$=$	$ ( \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + 3 \beta_{11} - 10 \beta_{10} - 2 \beta_{9} + \cdots - 10 ) / 4 $ (b15 - b14 - b13 + b12 + 3b11 - 10b10 - 2b9 - 6b8 - 10b7 - b4 + 6b3 - 6b2 + 4b1 - 10) / 4
$\nu^{6}$	$=$	$ ( 2 \beta_{15} - 6 \beta_{14} - \beta_{13} + 5 \beta_{12} + 3 \beta_{11} - 12 \beta_{10} + 2 \beta_{9} + \cdots - 2 ) / 4 $ (2b15 - 6b14 - b13 + 5b12 + 3b11 - 12b10 + 2b9 - 8b8 - 6b7 + 2b6 + 2b5 + b4 - 2b3 - 10b2 + 2*b1 - 2) / 4
$\nu^{7}$	$=$	$ ( 3 \beta_{15} + \beta_{14} - 4 \beta_{13} - 4 \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} + \cdots + 28 ) / 4 $ (3b15 + b14 - 4b13 - 4b12 + 8b11 - 2b10 + 2b8 + 14b6 + 2b5 + 6b4 - 4b3 - 4b2 + 18b1 + 28) / 4
$\nu^{8}$	$=$	$ ( - 5 \beta_{15} + 5 \beta_{14} - \beta_{13} + 5 \beta_{12} - \beta_{11} - 2 \beta_{10} - 6 \beta_{9} + \cdots + 2 ) / 4 $ (-5b15 + 5b14 - b13 + 5b12 - b11 - 2b10 - 6b9 - 2b8 + 2b7 + 12b6 + 16b5 - b4 + 18b3 + 2b2 - 16b1 + 2) / 4
$\nu^{9}$	$=$	$ ( 12 \beta_{14} + 9 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 6 \beta_{9} - 28 \beta_{8} - 50 \beta_{7} + \cdots + 10 ) / 4 $ (12b14 + 9b13 + 3b12 - 3b11 + 6b9 - 28b8 - 50b7 - 6b6 + 18b5 + 3b4 + 18b3 + 10b2 + 2*b1 + 10) / 4
$\nu^{10}$	$=$	$ ( - \beta_{15} + 5 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} - 2 \beta_{10} + \cdots - 58 \beta_1 ) / 4 $ (-b15 + 5b14 + 6b13 - 6b12 - 6b11 - 2b10 + 12b9 + 18b8 - 4b7 - 6b6 - 18b5 + 28b4 + 8b3 - 8b2 - 58b1) / 4
$\nu^{11}$	$=$	$ ( - 3 \beta_{15} + 11 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} + 7 \beta_{11} + 26 \beta_{10} + \cdots + 34 ) / 4 $ (-3b15 + 11b14 + 7b13 - 11b12 + 7b11 + 26b10 + 2b9 + 26b8 + 26b7 - 76b5 - 21b4 + 50b3 + 26b2 - 84b1 + 34) / 4
$\nu^{12}$	$=$	$ ( - 38 \beta_{15} - 30 \beta_{14} + 19 \beta_{13} + 49 \beta_{12} + 39 \beta_{11} - 108 \beta_{10} + \cdots - 74 ) / 4 $ (-38b15 - 30b14 + 19b13 + 49b12 + 39b11 - 108b10 - 14b9 - 8b8 - 54b7 - 14b6 + 2b5 - 19b4 - 2b3 - 10b2 + 2*b1 - 74) / 4
$\nu^{13}$	$=$	$ ( - 41 \beta_{15} + 5 \beta_{14} + 36 \beta_{13} + 36 \beta_{12} + 16 \beta_{11} - 2 \beta_{10} + \cdots + 44 ) / 4 $ (-41b15 + 5b14 + 36b13 + 36b12 + 16b11 - 2b10 - 6b8 + 22b6 - 6b5 - 10b4 - 108b3 - 108b2 + 130*b1 + 44) / 4
$\nu^{14}$	$=$	$ ( 7 \beta_{15} - 7 \beta_{14} - 77 \beta_{13} - 7 \beta_{12} + 91 \beta_{11} - 34 \beta_{10} - 86 \beta_{9} + \cdots - 190 ) / 4 $ (7b15 - 7b14 - 77b13 - 7b12 + 91b11 - 34b10 - 86b9 + 102b8 + 34b7 + 172b6 + 8b5 + 91b4 - 94b3 - 102b2 - 96*b1 - 190) / 4
$\nu^{15}$	$=$	$ ( 12 \beta_{14} + 29 \beta_{13} - 17 \beta_{12} - 79 \beta_{11} - 130 \beta_{9} - 156 \beta_{8} + \cdots - 62 ) / 4 $ (12b14 + 29b13 - 17b12 - 79b11 - 130b9 - 156b8 - 218b7 + 130b6 + 202b5 - 129b4 + 202b3 - 46b2 - 326*b1 - 62) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1 + \beta_{1}$	$-1$

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} $

47.1

−0.533474	+	1.30973i
0.867527	−	1.11687i
1.12063	−	0.862658i
−0.186766	+	1.40183i
−0.409484	−	1.35363i
−1.37702	+	0.322193i
0.608741	+	1.27649i
1.40985	−	0.111062i
−0.533474	−	1.30973i
0.867527	+	1.11687i
1.12063	+	0.862658i
−0.186766	−	1.40183i
−0.409484	+	1.35363i
−1.37702	−	0.322193i
0.608741	−	1.27649i
1.40985	+	0.111062i

−0.925606

−

1.46399i

−0.895377

−

1.55084i

−2.08793

−

1.20546i

−1.28651

2.71015i

47.2

−0.925606

−

1.46399i

0.895377

1.55084i

2.08793

1.20546i

−1.28651

2.71015i

47.3

−0.418594

1.68071i

−1.60936

−

2.78750i

−1.82223

−

1.05206i

−2.64956

−

1.40707i

47.4

−0.418594

1.68071i

1.60936

2.78750i

1.82223

1.05206i

−2.64956

−

1.40707i

47.5

1.12774

−

1.31461i

−0.565188

−

0.978934i

−3.71499

−

2.14485i

−0.456412

−

2.96508i

47.6

1.12774

−

1.31461i

0.565188

0.978934i

3.71499

2.14485i

−0.456412

−

2.96508i

47.7

1.71646

0.231865i

−1.74322

−

3.01934i

1.80802

1.04386i

2.89248

0.795973i

47.8

1.71646

0.231865i

1.74322

3.01934i

−1.80802

−

1.04386i

2.89248

0.795973i

239.1

−0.925606

1.46399i

−0.895377

1.55084i

−2.08793

1.20546i

−1.28651

−

2.71015i

239.2

−0.925606

1.46399i

0.895377

−

1.55084i

2.08793

−

1.20546i

−1.28651

−

2.71015i

239.3

−0.418594

−

1.68071i

−1.60936

2.78750i

−1.82223

1.05206i

−2.64956

1.40707i

239.4

−0.418594

−

1.68071i

1.60936

−

2.78750i

1.82223

−

1.05206i

−2.64956

1.40707i

239.5

1.12774

1.31461i

−0.565188

0.978934i

−3.71499

2.14485i

−0.456412

2.96508i

239.6

1.12774

1.31461i

0.565188

−

0.978934i

3.71499

−

2.14485i

−0.456412

2.96508i

239.7

1.71646

−

0.231865i

−1.74322

3.01934i

1.80802

−

1.04386i

2.89248

−

0.795973i

239.8

1.71646

−

0.231865i

1.74322

−

3.01934i

−1.80802

1.04386i

2.89248

−

0.795973i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
9.d	odd	6	1	inner
72.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.2.p.b		16
3.b	odd	2	1		864.2.p.b		16
4.b	odd	2	1		72.2.l.b	✓	16
8.b	even	2	1		72.2.l.b	✓	16
8.d	odd	2	1	inner	288.2.p.b		16
9.c	even	3	1		864.2.p.b		16
9.c	even	3	1		2592.2.f.b		16
9.d	odd	6	1	inner	288.2.p.b		16
9.d	odd	6	1		2592.2.f.b		16
12.b	even	2	1		216.2.l.b		16
24.f	even	2	1		864.2.p.b		16
24.h	odd	2	1		216.2.l.b		16
36.f	odd	6	1		216.2.l.b		16
36.f	odd	6	1		648.2.f.b		16
36.h	even	6	1		72.2.l.b	✓	16
36.h	even	6	1		648.2.f.b		16
72.j	odd	6	1		72.2.l.b	✓	16
72.j	odd	6	1		648.2.f.b		16
72.l	even	6	1	inner	288.2.p.b		16
72.l	even	6	1		2592.2.f.b		16
72.n	even	6	1		216.2.l.b		16
72.n	even	6	1		648.2.f.b		16
72.p	odd	6	1		864.2.p.b		16
72.p	odd	6	1		2592.2.f.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.2.l.b	✓	16	4.b	odd	2	1
72.2.l.b	✓	16	8.b	even	2	1
72.2.l.b	✓	16	36.h	even	6	1
72.2.l.b	✓	16	72.j	odd	6	1
216.2.l.b		16	12.b	even	2	1
216.2.l.b		16	24.h	odd	2	1
216.2.l.b		16	36.f	odd	6	1
216.2.l.b		16	72.n	even	6	1
288.2.p.b		16	1.a	even	1	1	trivial
288.2.p.b		16	8.d	odd	2	1	inner
288.2.p.b		16	9.d	odd	6	1	inner
288.2.p.b		16	72.l	even	6	1	inner
648.2.f.b		16	36.f	odd	6	1
648.2.f.b		16	36.h	even	6	1
648.2.f.b		16	72.j	odd	6	1
648.2.f.b		16	72.n	even	6	1
864.2.p.b		16	3.b	odd	2	1
864.2.p.b		16	9.c	even	3	1
864.2.p.b		16	24.f	even	2	1
864.2.p.b		16	72.p	odd	6	1
2592.2.f.b		16	9.c	even	3	1
2592.2.f.b		16	9.d	odd	6	1
2592.2.f.b		16	72.l	even	6	1
2592.2.f.b		16	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{16} + 27 T_{5}^{14} + 498 T_{5}^{12} + 4923 T_{5}^{10} + 35106 T_{5}^{8} + 123903 T_{5}^{6} + \cdots + 266256 $ T5^16 + 27*T5^14 + 498*T5^12 + 4923*T5^10 + 35106*T5^8 + 123903*T5^6 + 312453*T5^4 + 339012*T5^2 + 266256 acting on $S_{2}^{\mathrm{new}}(288, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 3 T^{7} + 6 T^{6} + \cdots + 81)^{2} $ (T^8 - 3T^7 + 6T^6 - 15T^5 + 30T^4 - 45T^3 + 54T^2 - 81*T + 81)^2
$5$	$ T^{16} + 27 T^{14} + \cdots + 266256 $ T^16 + 27T^14 + 498T^12 + 4923T^10 + 35106T^8 + 123903T^6 + 312453T^4 + 339012*T^2 + 266256
$7$	$ T^{16} - 33 T^{14} + \cdots + 4260096 $ T^16 - 33T^14 + 750T^12 - 8373T^10 + 66426T^8 - 340749T^6 + 1279953T^4 - 2904048*T^2 + 4260096
$11$	$ (T^{8} + 6 T^{7} + 8 T^{6} + \cdots + 1)^{2} $ (T^8 + 6T^7 + 8T^6 - 24T^5 - 9T^4 + 48T^3 + 44T^2 - 12*T + 1)^2
$13$	$ T^{16} - 57 T^{14} + \cdots + 266256 $ T^16 - 57T^14 + 2442T^12 - 39129T^10 + 454938T^8 - 2713221T^6 + 11382813T^4 - 1772460*T^2 + 266256
$17$	$ (T^{8} + 35 T^{6} + \cdots + 784)^{2} $ (T^8 + 35T^6 + 360T^4 + 992*T^2 + 784)^2
$19$	$ (T^{4} - T^{3} - 12 T^{2} + \cdots + 16)^{4} $ (T^4 - T^3 - 12T^2 + 8T + 16)^4
$23$	$ T^{16} + \cdots + 639280656 $ T^16 + 99T^14 + 7158T^12 + 225735T^10 + 5182026T^8 + 42464691T^6 + 255771909T^4 + 454125924*T^2 + 639280656
$29$	$ T^{16} + \cdots + 17449353216 $ T^16 + 135T^14 + 12654T^12 + 599547T^10 + 20607630T^8 + 389228679T^6 + 5081053545T^4 + 10074829824*T^2 + 17449353216
$31$	$ T^{16} + \cdots + 279189651456 $ T^16 - 117T^14 + 8826T^12 - 398493T^10 + 13147422T^8 - 290875401T^6 + 4696155729T^4 - 45038923776*T^2 + 279189651456
$37$	$ (T^{8} + 156 T^{6} + \cdots + 74304)^{2} $ (T^8 + 156T^6 + 4896T^4 + 40176*T^2 + 74304)^2
$41$	$ (T^{8} + 18 T^{7} + \cdots + 7921)^{2} $ (T^8 + 18T^7 + 128T^6 + 360T^5 - 51T^4 - 1800T^3 + 920T^2 + 8010*T + 7921)^2
$43$	$ (T^{8} + 4 T^{7} + \cdots + 6889)^{2} $ (T^8 + 4T^7 + 76T^6 - 524T^5 + 3115T^4 - 7856T^3 + 15184T^2 - 11786*T + 6889)^2
$47$	$ T^{16} + 111 T^{14} + \cdots + 266256 $ T^16 + 111T^14 + 10818T^12 + 160239T^10 + 1892526T^8 + 4840839T^6 + 10094661T^4 + 1701252*T^2 + 266256
$53$	$ (T^{8} - 228 T^{6} + \cdots + 297216)^{2} $ (T^8 - 228T^6 + 15408T^4 - 331344*T^2 + 297216)^2
$59$	$ (T^{8} + 6 T^{7} + \cdots + 528529)^{2} $ (T^8 + 6T^7 - 58T^6 - 420T^5 + 3717T^4 + 15960T^3 - 33562T^2 - 165756*T + 528529)^2
$61$	$ T^{16} + \cdots + 1192149524736 $ T^16 - 189T^14 + 24750T^12 - 1679649T^10 + 82050270T^8 - 1747852317T^6 + 26804642049T^4 - 215024661360*T^2 + 1192149524736
$67$	$ (T^{8} - 8 T^{7} + \cdots + 582169)^{2} $ (T^8 - 8T^7 + 130T^6 - 20T^5 + 5785T^4 - 5876T^3 + 125434T^2 + 209062*T + 582169)^2
$71$	$ (T^{8} - 168 T^{6} + \cdots + 74304)^{2} $ (T^8 - 168T^6 + 3744T^4 - 28944*T^2 + 74304)^2
$73$	$ (T^{4} + T^{3} - 78 T^{2} + \cdots + 172)^{4} $ (T^4 + T^3 - 78T^2 - 224T + 172)^4
$79$	$ T^{16} + \cdots + 74509345296 $ T^16 - 249T^14 + 46758T^12 - 3530925T^10 + 199135626T^8 - 1880575641T^6 + 13340118429T^4 - 36110680524*T^2 + 74509345296
$83$	$ (T^{8} + 27 T^{7} + \cdots + 432964)^{2} $ (T^8 + 27T^7 + 212T^6 - 837T^5 - 6366T^4 + 22971T^3 + 162629T^2 - 487578*T + 432964)^2
$89$	$ (T^{8} + 272 T^{6} + \cdots + 891136)^{2} $ (T^8 + 272T^6 + 23808T^4 + 689456*T^2 + 891136)^2
$97$	$ (T^{8} - 4 T^{7} + \cdots + 1018081)^{2} $ (T^8 - 4T^7 + 190T^6 - 1096T^5 + 32851T^4 - 147832T^3 + 978382T^2 + 904064*T + 1018081)^2
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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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{8} + \beta_{3} - \beta_1) q^{3} - \beta_{9} q^{5} - \beta_{4} q^{7} + ( - \beta_{10} + \beta_{7} - \beta_{5} + \cdots - 2) q^{9}+O(q^{10}) \) q + (-b8 + b3 - b1) * q^3 - b9 * q^5 - b4 * q^7 + (-b10 + b7 - b5 - b2 - b1 - 2) * q^9	\( q + ( - \beta_{8} + \beta_{3} - \beta_1) q^{3} - \beta_{9} q^{5} - \beta_{4} q^{7} + ( - \beta_{10} + \beta_{7} - \beta_{5} + \cdots - 2) q^{9}+ \cdots + (\beta_{10} - \beta_{7} + \beta_{5} + \cdots + 5) q^{99}+O(q^{100}) \) q + (-b8 + b3 - b1) * q^3 - b9 * q^5 - b4 * q^7 + (-b10 + b7 - b5 - b2 - b1 - 2) * q^9 + (-b10 - b3 - b2 - 2) * q^11 - b13 * q^13 + (b14 - b11 - b6) * q^15 + (-b8 - 2b7 + b5 + b3 - b1) q^17 + (2b10 + b7 - b5 + b3 + b2 - b1 + 1) q^19 + (-b15 + b14 + b13 - 2b11 + b9 - b6 - 2b4) * q^21 + (-b15 - b14 + b12 - b9) * q^23 + (b10 - b8 + 2b7 + b5 + b1) q^25 + (-3b10 + b8 - 2b3 - b1) * q^27 + (-b15 + b13 + b12 + b11) * q^29 + (-b13 + b11 - b9 + 2b6 + b4) q^31 + (b8 - b5 - b3 + 2b2 - 2b1) * q^33 + (-3b7 + 2b5 + 2b3 - 2b2 + 4b1 + 3) q^35 + (2b15 - b14 - b13 + 2b11 + b9 + b6 + b4) * q^37 + (b14 + b13 - 2b12 - b11 + b9 - b6) q^39 + (-b10 - b7 - b5 + b3 + b1 - 2) * q^41 + (2b10 + b8 + 4b7 - b5 - 2b3 + 2b2) * q^43 + (2b15 - b12 - b11 + b4) q^45 + (b15 - b14 + 2b11 + b4) q^47 + (2b10 - b8 - 2b7 - b5 + b2 + 2b1 + 2) q^49 + (b10 - b7 - b5 + 2b3 - b2 - b1 - 1) q^51 + (-b14 - b12 + b11 - b9 + b6 + 2b4) q^53 + (2b15 - b14 - b13 + b11 + b9 + b6 + b4) q^55 + (b10 + b8 + 2b7 + b5 - 2b2 + b1 + 2) * q^57 + (-2b10 + b8 - 2b7 + 2b5 - 3b3 + b2 + 3b1) q^59 + (-b15 + b13 - b12 - b11 - 2b4) q^61 + (-b14 + b13 + b12 - b11 + 2b9 - b6 - b4) q^63 + (4b8 + 4b5 - 3b3 - 3b2 + 4b1 + 2) q^65 + (3b10 - b8 - 3b7 + b5 + 2b3 + b2 + 3b1 + 5) * q^67 + (-b15 - b13 + b12 + b11 + b9 - b6 + 2b4) q^69 + (b14 + b13 - b9 + b6 + b4) * q^71 + (3b8 + b5 - b3 + 2b2 + b1 + 2) * q^73 + (b10 + b8 + 5b7 - 2b5 - 3b3 + b2 - 2b1 - 7) * q^75 + (b15 - b14 - b13 + b12 + b11 + b9 + b4) * q^77 + (-b15 + b13 - b12 - b11 + 2b9 - b6 - b4) q^79 + (-3b10 - 3b8 - 3b7 + 3b3 + 3b2) q^81 + (5b10 - 2b8 - 2b5 + 4b3 + 4b2 - 5b1 - 2) * q^83 + (-b15 + b14 + b12 + b11 - b9 + 2b6 + b4) q^85 + (-b13 + 2b12 + 2b11 + b9 + 2b6 + b4) q^87 + (b8 + b7 - 4b5 - 4b3 + 3b2 + 2b1 - 1) * q^89 + (-8b10 + b8 - 4b7 + 4b5 - 4b3 - 3b2 + 4b1) * q^91 + (-b15 + 2b14 + 2b13 - 2b12 - b9 + b6 + b4) q^93 + (-b15 + b14 + b13 - b12 - b11 - b4) * q^95 + (3b10 + 5b8 + 6b7 - 5b5 - b3 + b2 - 3b1) q^97 + (b10 - b7 + b5 + 3b3 + b2 + 4b1 + 5) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{3} - 6 q^{9}+O(q^{10}) \) 16 * q + 6 * q^3 - 6 * q^9	\( 16 q + 6 q^{3} - 6 q^{9} - 12 q^{11} + 4 q^{19} - 14 q^{25} + 36 q^{27} + 12 q^{33} - 36 q^{41} - 8 q^{43} + 10 q^{49} - 18 q^{51} + 18 q^{57} - 12 q^{59} - 6 q^{65} + 16 q^{67} - 4 q^{73} - 78 q^{75} - 6 q^{81} - 54 q^{83} + 36 q^{91} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) 16 * q + 6 * q^3 - 6 * q^9 - 12 * q^11 + 4 * q^19 - 14 * q^25 + 36 * q^27 + 12 * q^33 - 36 * q^41 - 8 * q^43 + 10 * q^49 - 18 * q^51 + 18 * q^57 - 12 * q^59 - 6 * q^65 + 16 * q^67 - 4 * q^73 - 78 * q^75 - 6 * q^81 - 54 * q^83 + 36 * q^91 + 8 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	288.2.p.b

Level	$288$
Weight	$2$
Character orbit	288.p
Analytic conductor	$2.300$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(2.29969157821\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + \cdots + 256 \) x^16 - 3x^15 + 7x^14 - 12x^13 + 16x^12 - 12x^11 - 8x^10 + 36x^9 - 68x^8 + 72x^7 - 32x^6 - 96x^5 + 256x^4 - 384x^3 + 448x^2 - 384*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{10} \)
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + \cdots + 384 ) / 896 \) (v^15 - 13v^14 + 11v^13 + 4v^12 - 38v^11 + 60v^10 - 104v^9 + 68v^8 + 148v^7 - 344v^6 + 440v^5 - 240v^4 - 32v^3 + 608v^2 - 1152v + 384) / 896
\(\beta_{2}\)	\(=\)	\( ( - \nu^{15} - \nu^{14} - 25 \nu^{13} + 38 \nu^{12} - 46 \nu^{11} + 24 \nu^{10} - 8 \nu^{9} + \cdots + 512 ) / 896 \) (-v^15 - v^14 - 25v^13 + 38v^12 - 46v^11 + 24v^10 - 8v^9 - 68v^8 + 244v^7 - 272v^6 + 8v^5 + 128v^4 - 416v^3 + 1184v^2 - 1088*v + 512) / 896
\(\beta_{3}\)	\(=\)	\( ( 3 \nu^{15} - 25 \nu^{14} + 33 \nu^{13} - 44 \nu^{12} + 40 \nu^{11} + 40 \nu^{10} - 144 \nu^{9} + \cdots - 640 ) / 896 \) (3v^15 - 25v^14 + 33v^13 - 44v^12 + 40v^11 + 40v^10 - 144v^9 + 316v^8 - 172v^7 - 24v^6 + 480v^5 - 720v^4 + 1024v^3 - 1088v^2 + 128*v - 640) / 896
\(\beta_{4}\)	\(=\)	\( ( - \nu^{15} - 29 \nu^{14} + 59 \nu^{13} - 102 \nu^{12} + 66 \nu^{11} + 24 \nu^{10} - 176 \nu^{9} + \cdots - 384 ) / 896 \) (-v^15 - 29v^14 + 59v^13 - 102v^12 + 66v^11 + 24v^10 - 176v^9 + 492v^8 - 540v^7 + 288v^6 + 456v^5 - 1216v^4 + 2496v^3 - 2400v^2 + 1600v - 384) / 896
\(\beta_{5}\)	\(=\)	\( ( - 3 \nu^{15} + 11 \nu^{14} - 19 \nu^{13} + 16 \nu^{12} + 2 \nu^{11} - 54 \nu^{10} + 116 \nu^{9} + \cdots - 704 ) / 448 \) (-3v^15 + 11v^14 - 19v^13 + 16v^12 + 2v^11 - 54v^10 + 116v^9 - 92v^8 + 4v^7 + 192v^6 - 424v^5 + 440v^4 - 352v^3 - 256v^2 + 544*v - 704) / 448
\(\beta_{6}\)	\(=\)	\( ( 6 \nu^{15} - 8 \nu^{14} + 3 \nu^{13} + 17 \nu^{12} - 39 \nu^{11} + 80 \nu^{10} - 92 \nu^{9} + \cdots + 64 ) / 448 \) (6v^15 - 8v^14 + 3v^13 + 17v^12 - 39v^11 + 80v^10 - 92v^9 + 16v^8 + 272v^7 - 300v^6 + 260v^5 - 152v^4 - 304v^3 + 1072v^2 - 1312*v + 64) / 448
\(\beta_{7}\)	\(=\)	\( ( - 13 \nu^{15} + 15 \nu^{14} - 31 \nu^{13} + 4 \nu^{12} + 32 \nu^{11} - 136 \nu^{10} + 176 \nu^{9} + \cdots - 1408 ) / 896 \) (-13v^15 + 15v^14 - 31v^13 + 4v^12 + 32v^11 - 136v^10 + 176v^9 - 100v^8 - 76v^7 + 552v^6 - 736v^5 + 880v^4 - 256v^3 - 512v^2 + 1536*v - 1408) / 896
\(\beta_{8}\)	\(=\)	\( ( - 3 \nu^{15} + 18 \nu^{14} - 47 \nu^{13} + 100 \nu^{12} - 117 \nu^{11} + 100 \nu^{10} + 60 \nu^{9} + \cdots + 1536 ) / 448 \) (-3v^15 + 18v^14 - 47v^13 + 100v^12 - 117v^11 + 100v^10 + 60v^9 - 316v^8 + 592v^7 - 648v^6 + 52v^5 + 888v^4 - 2144v^3 + 2768v^2 - 3040*v + 1536) / 448
\(\beta_{9}\)	\(=\)	\( ( 6 \nu^{15} - 15 \nu^{14} + 24 \nu^{13} - 32 \nu^{12} + 45 \nu^{11} + 24 \nu^{10} - 64 \nu^{9} + \cdots - 1728 ) / 448 \) (6v^15 - 15v^14 + 24v^13 - 32v^12 + 45v^11 + 24v^10 - 64v^9 + 128v^8 - 92v^7 + 176v^6 - 20v^5 - 600v^4 + 592v^3 - 944v^2 + 928*v - 1728) / 448
\(\beta_{10}\)	\(=\)	\( ( 5 \nu^{15} - 16 \nu^{14} + 48 \nu^{13} - 71 \nu^{12} + 76 \nu^{11} - 22 \nu^{10} - 100 \nu^{9} + \cdots - 768 ) / 448 \) (5v^15 - 16v^14 + 48v^13 - 71v^12 + 76v^11 - 22v^10 - 100v^9 + 284v^8 - 408v^7 + 212v^6 + 184v^5 - 920v^4 + 1520v^3 - 1888v^2 + 1184*v - 768) / 448
\(\beta_{11}\)	\(=\)	\( ( - 19 \nu^{15} + 37 \nu^{14} - 13 \nu^{13} - 48 \nu^{12} + 232 \nu^{11} - 384 \nu^{10} + 464 \nu^{9} + \cdots - 4608 ) / 896 \) (-19v^15 + 37v^14 - 13v^13 - 48v^12 + 232v^11 - 384v^10 + 464v^9 - 60v^8 - 740v^7 + 1608v^6 - 1696v^5 + 752v^4 + 1280v^3 - 4608v^2 + 5760*v - 4608) / 896
\(\beta_{12}\)	\(=\)	\( ( - 13 \nu^{15} + 29 \nu^{14} - 59 \nu^{13} + 116 \nu^{12} - 150 \nu^{11} + 88 \nu^{10} + 120 \nu^{9} + \cdots + 3072 ) / 448 \) (-13v^15 + 29v^14 - 59v^13 + 116v^12 - 150v^11 + 88v^10 + 120v^9 - 324v^8 + 652v^7 - 680v^6 + 104v^5 + 1216v^4 - 2720v^3 + 3968v^2 - 4288*v + 3072) / 448
\(\beta_{13}\)	\(=\)	\( ( - 16 \nu^{15} + 40 \nu^{14} - 99 \nu^{13} + 153 \nu^{12} - 113 \nu^{11} - 36 \nu^{10} + 376 \nu^{9} + \cdots - 320 ) / 448 \) (-16v^15 + 40v^14 - 99v^13 + 153v^12 - 113v^11 - 36v^10 + 376v^9 - 640v^8 + 768v^7 - 124v^6 - 964v^5 + 2440v^4 - 3072v^3 + 3152v^2 - 2400*v - 320) / 448
\(\beta_{14}\)	\(=\)	\( ( - 37 \nu^{15} + 103 \nu^{14} - 183 \nu^{13} + 244 \nu^{12} - 232 \nu^{11} - 64 \nu^{10} + 712 \nu^{9} + \cdots + 2816 ) / 896 \) (-37v^15 + 103v^14 - 183v^13 + 244v^12 - 232v^11 - 64v^10 + 712v^9 - 1172v^8 + 1300v^7 - 600v^6 - 1216v^5 + 4176v^4 - 6432v^3 + 6848v^2 - 3968*v + 2816) / 896
\(\beta_{15}\)	\(=\)	\( ( - 57 \nu^{15} + 167 \nu^{14} - 375 \nu^{13} + 528 \nu^{12} - 536 \nu^{11} + 24 \nu^{10} + 1112 \nu^{9} + \cdots + 5888 ) / 896 \) (-57v^15 + 167v^14 - 375v^13 + 528v^12 - 536v^11 + 24v^10 + 1112v^9 - 2308v^8 + 2932v^7 - 1448v^6 - 1952v^5 + 7856v^4 - 12512v^3 + 14400v^2 - 12288*v + 5888) / 896

\(\nu\)	\(=\)	\( ( -\beta_{15} + \beta_{14} - 2\beta_{10} ) / 4 \) (-b15 + b14 - 2*b10) / 4
\(\nu^{2}\)	\(=\)	\( ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{8} - 2 \beta_{5} - \beta_{4} + \cdots - 4 ) / 4 \) (-b15 + b14 + b13 + b12 - b11 - 2b8 - 2b5 - b4 + 2b2 - 4b1 - 4) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + \cdots - 2 ) / 4 \) (b13 - b12 + b11 - 2b9 - 2b7 + 2b6 - 2b5 + 3b4 - 2b3 + 2b2 - 2b1 - 2) / 4
\(\nu^{4}\)	\(=\)	\( ( - \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + \cdots - 6 \beta_1 ) / 4 \) (-b15 + b14 + 2b13 - 2b12 - 2b11 + 2b10 - 4b9 + 2b8 + 4b7 + 2b6 - 2b5 + 4b3 - 4b2 - 6b1) / 4
\(\nu^{5}\)	\(=\)	\( ( \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + 3 \beta_{11} - 10 \beta_{10} - 2 \beta_{9} + \cdots - 10 ) / 4 \) (b15 - b14 - b13 + b12 + 3b11 - 10b10 - 2b9 - 6b8 - 10b7 - b4 + 6b3 - 6b2 + 4b1 - 10) / 4
\(\nu^{6}\)	\(=\)	\( ( 2 \beta_{15} - 6 \beta_{14} - \beta_{13} + 5 \beta_{12} + 3 \beta_{11} - 12 \beta_{10} + 2 \beta_{9} + \cdots - 2 ) / 4 \) (2b15 - 6b14 - b13 + 5b12 + 3b11 - 12b10 + 2b9 - 8b8 - 6b7 + 2b6 + 2b5 + b4 - 2b3 - 10b2 + 2*b1 - 2) / 4
\(\nu^{7}\)	\(=\)	\( ( 3 \beta_{15} + \beta_{14} - 4 \beta_{13} - 4 \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} + \cdots + 28 ) / 4 \) (3b15 + b14 - 4b13 - 4b12 + 8b11 - 2b10 + 2b8 + 14b6 + 2b5 + 6b4 - 4b3 - 4b2 + 18b1 + 28) / 4
\(\nu^{8}\)	\(=\)	\( ( - 5 \beta_{15} + 5 \beta_{14} - \beta_{13} + 5 \beta_{12} - \beta_{11} - 2 \beta_{10} - 6 \beta_{9} + \cdots + 2 ) / 4 \) (-5b15 + 5b14 - b13 + 5b12 - b11 - 2b10 - 6b9 - 2b8 + 2b7 + 12b6 + 16b5 - b4 + 18b3 + 2b2 - 16b1 + 2) / 4
\(\nu^{9}\)	\(=\)	\( ( 12 \beta_{14} + 9 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 6 \beta_{9} - 28 \beta_{8} - 50 \beta_{7} + \cdots + 10 ) / 4 \) (12b14 + 9b13 + 3b12 - 3b11 + 6b9 - 28b8 - 50b7 - 6b6 + 18b5 + 3b4 + 18b3 + 10b2 + 2*b1 + 10) / 4
\(\nu^{10}\)	\(=\)	\( ( - \beta_{15} + 5 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} - 2 \beta_{10} + \cdots - 58 \beta_1 ) / 4 \) (-b15 + 5b14 + 6b13 - 6b12 - 6b11 - 2b10 + 12b9 + 18b8 - 4b7 - 6b6 - 18b5 + 28b4 + 8b3 - 8b2 - 58b1) / 4
\(\nu^{11}\)	\(=\)	\( ( - 3 \beta_{15} + 11 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} + 7 \beta_{11} + 26 \beta_{10} + \cdots + 34 ) / 4 \) (-3b15 + 11b14 + 7b13 - 11b12 + 7b11 + 26b10 + 2b9 + 26b8 + 26b7 - 76b5 - 21b4 + 50b3 + 26b2 - 84b1 + 34) / 4
\(\nu^{12}\)	\(=\)	\( ( - 38 \beta_{15} - 30 \beta_{14} + 19 \beta_{13} + 49 \beta_{12} + 39 \beta_{11} - 108 \beta_{10} + \cdots - 74 ) / 4 \) (-38b15 - 30b14 + 19b13 + 49b12 + 39b11 - 108b10 - 14b9 - 8b8 - 54b7 - 14b6 + 2b5 - 19b4 - 2b3 - 10b2 + 2*b1 - 74) / 4
\(\nu^{13}\)	\(=\)	\( ( - 41 \beta_{15} + 5 \beta_{14} + 36 \beta_{13} + 36 \beta_{12} + 16 \beta_{11} - 2 \beta_{10} + \cdots + 44 ) / 4 \) (-41b15 + 5b14 + 36b13 + 36b12 + 16b11 - 2b10 - 6b8 + 22b6 - 6b5 - 10b4 - 108b3 - 108b2 + 130*b1 + 44) / 4
\(\nu^{14}\)	\(=\)	\( ( 7 \beta_{15} - 7 \beta_{14} - 77 \beta_{13} - 7 \beta_{12} + 91 \beta_{11} - 34 \beta_{10} - 86 \beta_{9} + \cdots - 190 ) / 4 \) (7b15 - 7b14 - 77b13 - 7b12 + 91b11 - 34b10 - 86b9 + 102b8 + 34b7 + 172b6 + 8b5 + 91b4 - 94b3 - 102b2 - 96*b1 - 190) / 4
\(\nu^{15}\)	\(=\)	\( ( 12 \beta_{14} + 29 \beta_{13} - 17 \beta_{12} - 79 \beta_{11} - 130 \beta_{9} - 156 \beta_{8} + \cdots - 62 ) / 4 \) (12b14 + 29b13 - 17b12 - 79b11 - 130b9 - 156b8 - 218b7 + 130b6 + 202b5 - 129b4 + 202b3 - 46b2 - 326*b1 - 62) / 4

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(-1\)	\(1 + \beta_{1}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 47.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 3 T^{7} + 6 T^{6} + \cdots + 81)^{2} \) (T^8 - 3T^7 + 6T^6 - 15T^5 + 30T^4 - 45T^3 + 54T^2 - 81*T + 81)^2
$5$	\( T^{16} + 27 T^{14} + \cdots + 266256 \) T^16 + 27T^14 + 498T^12 + 4923T^10 + 35106T^8 + 123903T^6 + 312453T^4 + 339012*T^2 + 266256
$7$	\( T^{16} - 33 T^{14} + \cdots + 4260096 \) T^16 - 33T^14 + 750T^12 - 8373T^10 + 66426T^8 - 340749T^6 + 1279953T^4 - 2904048*T^2 + 4260096
$11$	\( (T^{8} + 6 T^{7} + 8 T^{6} + \cdots + 1)^{2} \) (T^8 + 6T^7 + 8T^6 - 24T^5 - 9T^4 + 48T^3 + 44T^2 - 12*T + 1)^2
$13$	\( T^{16} - 57 T^{14} + \cdots + 266256 \) T^16 - 57T^14 + 2442T^12 - 39129T^10 + 454938T^8 - 2713221T^6 + 11382813T^4 - 1772460*T^2 + 266256
$17$	\( (T^{8} + 35 T^{6} + \cdots + 784)^{2} \) (T^8 + 35T^6 + 360T^4 + 992*T^2 + 784)^2
$19$	\( (T^{4} - T^{3} - 12 T^{2} + \cdots + 16)^{4} \) (T^4 - T^3 - 12T^2 + 8T + 16)^4
$23$	\( T^{16} + \cdots + 639280656 \) T^16 + 99T^14 + 7158T^12 + 225735T^10 + 5182026T^8 + 42464691T^6 + 255771909T^4 + 454125924*T^2 + 639280656
$29$	\( T^{16} + \cdots + 17449353216 \) T^16 + 135T^14 + 12654T^12 + 599547T^10 + 20607630T^8 + 389228679T^6 + 5081053545T^4 + 10074829824*T^2 + 17449353216
$31$	\( T^{16} + \cdots + 279189651456 \) T^16 - 117T^14 + 8826T^12 - 398493T^10 + 13147422T^8 - 290875401T^6 + 4696155729T^4 - 45038923776*T^2 + 279189651456
$37$	\( (T^{8} + 156 T^{6} + \cdots + 74304)^{2} \) (T^8 + 156T^6 + 4896T^4 + 40176*T^2 + 74304)^2
$41$	\( (T^{8} + 18 T^{7} + \cdots + 7921)^{2} \) (T^8 + 18T^7 + 128T^6 + 360T^5 - 51T^4 - 1800T^3 + 920T^2 + 8010*T + 7921)^2
$43$	\( (T^{8} + 4 T^{7} + \cdots + 6889)^{2} \) (T^8 + 4T^7 + 76T^6 - 524T^5 + 3115T^4 - 7856T^3 + 15184T^2 - 11786*T + 6889)^2
$47$	\( T^{16} + 111 T^{14} + \cdots + 266256 \) T^16 + 111T^14 + 10818T^12 + 160239T^10 + 1892526T^8 + 4840839T^6 + 10094661T^4 + 1701252*T^2 + 266256
$53$	\( (T^{8} - 228 T^{6} + \cdots + 297216)^{2} \) (T^8 - 228T^6 + 15408T^4 - 331344*T^2 + 297216)^2
$59$	\( (T^{8} + 6 T^{7} + \cdots + 528529)^{2} \) (T^8 + 6T^7 - 58T^6 - 420T^5 + 3717T^4 + 15960T^3 - 33562T^2 - 165756*T + 528529)^2
$61$	\( T^{16} + \cdots + 1192149524736 \) T^16 - 189T^14 + 24750T^12 - 1679649T^10 + 82050270T^8 - 1747852317T^6 + 26804642049T^4 - 215024661360*T^2 + 1192149524736
$67$	\( (T^{8} - 8 T^{7} + \cdots + 582169)^{2} \) (T^8 - 8T^7 + 130T^6 - 20T^5 + 5785T^4 - 5876T^3 + 125434T^2 + 209062*T + 582169)^2
$71$	\( (T^{8} - 168 T^{6} + \cdots + 74304)^{2} \) (T^8 - 168T^6 + 3744T^4 - 28944*T^2 + 74304)^2
$73$	\( (T^{4} + T^{3} - 78 T^{2} + \cdots + 172)^{4} \) (T^4 + T^3 - 78T^2 - 224T + 172)^4
$79$	\( T^{16} + \cdots + 74509345296 \) T^16 - 249T^14 + 46758T^12 - 3530925T^10 + 199135626T^8 - 1880575641T^6 + 13340118429T^4 - 36110680524*T^2 + 74509345296
$83$	\( (T^{8} + 27 T^{7} + \cdots + 432964)^{2} \) (T^8 + 27T^7 + 212T^6 - 837T^5 - 6366T^4 + 22971T^3 + 162629T^2 - 487578*T + 432964)^2
$89$	\( (T^{8} + 272 T^{6} + \cdots + 891136)^{2} \) (T^8 + 272T^6 + 23808T^4 + 689456*T^2 + 891136)^2
$97$	\( (T^{8} - 4 T^{7} + \cdots + 1018081)^{2} \) (T^8 - 4T^7 + 190T^6 - 1096T^5 + 32851T^4 - 147832T^3 + 978382T^2 + 904064*T + 1018081)^2
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