LMFDB - Newform orbit 416.2.ba.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(17,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("416.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.ba (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:	$3.32177672409$
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} + 5 x^{14} - 6 x^{13} + 6 x^{12} - 20 x^{10} + 48 x^{9} - 76 x^{8} + 96 x^{7} + \cdots + 256 $ x^16 - 3x^15 + 5x^14 - 6x^13 + 6x^12 - 20x^10 + 48x^9 - 76x^8 + 96x^7 - 80x^6 + 96x^4 - 192x^3 + 320x^2 - 384*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 104)
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_{12} + \beta_{8} - \beta_1) q^{3} + (\beta_{12} + \beta_{8} - \beta_{3} - \beta_1) q^{5} + (\beta_{11} + \beta_{2} - 1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{5}) q^{9}+O(q^{10}) $ q + (b12 + b8 - b1) * q^3 + (b12 + b8 - b3 - b1) * q^5 + (b11 + b2 - 1) * q^7 + (b10 - b6 - b5) * q^9	$ q + (\beta_{12} + \beta_{8} - \beta_1) q^{3} + (\beta_{12} + \beta_{8} - \beta_{3} - \beta_1) q^{5} + (\beta_{11} + \beta_{2} - 1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{5}) q^{9} + (\beta_{12} - \beta_{8} - 2 \beta_{4} + \cdots + \beta_1) q^{11}+ \cdots + (\beta_{14} + \beta_{13} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) $ q + (b12 + b8 - b1) * q^3 + (b12 + b8 - b3 - b1) * q^5 + (b11 + b2 - 1) * q^7 + (b10 - b6 - b5) * q^9 + (b12 - b8 - 2b4 + 2b3 + b1) * q^11 + (b14 + b12 + b8 - b7 + b3) * q^13 + (-b15 + b10 - 2b9 - b5 + 2) q^15 + (b15 - b6 - b5) * q^17 + (2b14 - b13 - b12 - 2b8) * q^19 + (-b8 - b7 + b3 + b1) * q^21 + (-2b11 - b9 - b2 - 1) q^23 + (-b15 + b10 - b9 + b6 - b5) * q^25 + (-b14 + b13 - b12 - b3) * q^27 + (b14 - 2b12 - 3b8 + 2b7 - b4 + 2b1) * q^29 + (2b15 + 2b10 + 2b9 - 2b6 - 2b5) q^31 + (2b11 - b10 + b9 - b6 + b5 + 2b2 - 2) * q^33 + (b13 - b12 - b8 + b3 + b1) * q^35 + (-b12 - 2b8 - b4 + b3 - b1) q^37 + (b15 - b10 + b9 - b5 + b2 + 1) * q^39 + (-b15 + b10 - 3b9 + 2b6 + b5 + 1) * q^41 + (2b12 + 3b8 - 3b3 - b1) q^43 + (-2b14 + b13 + 2b12 + b8 + 2b7 - 2b4 + b3) * q^45 + (-2b15 + 2b11 - b10 + b9 + b6 + 2b5 + 2) q^47 + (b15 - 4b11 + b10 - 4b9 - b6 - 2b2 - 5) q^49 + (b14 - b13 + 3b12 + 4b8 - 2b7 + 4b4 + b3) * q^51 + (b12 + b8 + b3 + b1) * q^53 + (b15 + b10 - 2b6 + b5 - 2) q^55 + (-b15 - 3b10 + 5b9 + 3b6 + b5 + 4) q^57 + (-2b14 + b13 - 3b12 - 2b3 + 2b1) * q^59 + (-3b13 + b12 + b8 - b7 - b4 - 4b1) * q^61 + (b15 - b10 + b5 - 2b2) q^63 + (2b15 - b10 + 5b9 + b5 + 2b2 + 1) q^65 + (b14 - 2b13 - 2b8 + 2b4 + b3 - 2b1) * q^67 + (-b12 + b7 + b4 - b3) * q^69 + (-3b15 - b11 - 2b10 - b9 + b6 - b5 - b2 + 1) * q^71 + (-b15 + 4b11 - 2b9 + b5 + 1) * q^73 + (-2b14 + b12 - b8 + 4b7 - 2b4 - 4b3 - b1) * q^75 + (-2b14 + 2b13 - 4b12 + 3b8 - 3b7 + 6b4 - 7b3 - 3b1) * q^77 + (b10 - b9 + b6 - 2) * q^79 + (-3b15 - 3b10 + b9 + 2b6 + b5 + 3) q^81 + (-2b12 + 2b8 - 2b7 + 4b3 - 2b1) q^83 + (-b12 + 2b8 - 2b7 + 2b4 + b1) q^85 + (-b15 + b11 - 2b10 + 3b9 + 3b6 + 3b5 - b2 + 1) * q^87 + (b9 - b6 - b5 - 4b2) q^89 + (-2b14 + b13 + b12 + 2b8 - 2b7 + 2b4 - 4b3 - 2b1) * q^91 + (2b12 + 2b8 + 4b4 + 2b1) * q^93 + (b15 - 2b11 - 4b10 + 3b9 + 3b6 + 3b5 + 2b2 - 2) * q^95 + (3b15 - 2b11 + b10 - 2b9 - 2b6 + 2b5 - 2b2 - 6) * q^97 + (b14 + b13 - 4b12 - 2b8 + 4b3 + 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 18 q^{7} + 2 q^{9}+O(q^{10}) $ 16 * q - 18 * q^7 + 2 * q^9	$ 16 q - 18 q^{7} + 2 q^{9} + 36 q^{15} + 8 q^{17} + 2 q^{23} - 12 q^{25} - 30 q^{33} + 14 q^{39} + 24 q^{41} - 14 q^{49} - 4 q^{55} + 6 q^{65} - 6 q^{71} - 32 q^{79} + 12 q^{81} - 34 q^{87} - 30 q^{89} - 28 q^{95} - 30 q^{97}+O(q^{100}) $ 16 * q - 18 * q^7 + 2 * q^9 + 36 * q^15 + 8 * q^17 + 2 * q^23 - 12 * q^25 - 30 * q^33 + 14 * q^39 + 24 * q^41 - 14 * q^49 - 4 * q^55 + 6 * q^65 - 6 * q^71 - 32 * q^79 + 12 * q^81 - 34 * q^87 - 30 * q^89 - 28 * q^95 - 30 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 3 x^{15} + 5 x^{14} - 6 x^{13} + 6 x^{12} - 20 x^{10} + 48 x^{9} - 76 x^{8} + 96 x^{7} + \cdots + 256 $ x^16 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 20*x^10 + 48*x^9 - 76*x^8 + 96*x^7 - 80*x^6 + 96*x^4 - 192*x^3 + 320*x^2 - 384*x + 256 :

$\beta_{1}$	$=$	$ ( - \nu^{15} - 3 \nu^{14} + 9 \nu^{13} - 14 \nu^{12} + 12 \nu^{11} - 10 \nu^{10} - 4 \nu^{9} + \cdots + 32 \nu ) / 320 $ (-v^15 - 3v^14 + 9v^13 - 14v^12 + 12v^11 - 10v^10 - 4v^9 + 28v^8 - 68v^7 + 80v^6 - 144v^5 + 56v^4 + 48v^3 + 128v^2 + 32v) / 320
$\beta_{2}$	$=$	$ ( \nu^{15} + \nu^{14} - 3 \nu^{13} + 2 \nu^{12} - 6 \nu^{11} + 8 \nu^{10} + 12 \nu^{9} - 32 \nu^{8} + \cdots + 512 ) / 128 $ (v^15 + v^14 - 3v^13 + 2v^12 - 6v^11 + 8v^10 + 12v^9 - 32v^8 + 52v^7 - 80v^6 + 96v^5 - 128v^3 + 128v^2 - 192v + 512) / 128
$\beta_{3}$	$=$	$ ( 9 \nu^{15} + 5 \nu^{14} - 7 \nu^{13} - 6 \nu^{12} - 26 \nu^{11} + 44 \nu^{10} - 4 \nu^{9} + \cdots + 1664 ) / 640 $ (9v^15 + 5v^14 - 7v^13 - 6v^12 - 26v^11 + 44v^10 - 4v^9 - 40v^8 - 12v^7 - 176v^6 + 416v^5 - 112v^4 - 160v^3 - 576v^2 - 512*v + 1664) / 640
$\beta_{4}$	$=$	$ ( - \nu^{15} + 10 \nu^{14} - 12 \nu^{13} + 19 \nu^{12} - 26 \nu^{11} + 14 \nu^{10} + 36 \nu^{9} + \cdots + 384 ) / 320 $ (-v^15 + 10v^14 - 12v^13 + 19v^12 - 26v^11 + 14v^10 + 36v^9 - 100v^8 + 108v^7 - 196v^6 + 216v^5 - 32v^4 - 80v^3 + 64v^2 - 192v + 384) / 320
$\beta_{5}$	$=$	$ ( 3 \nu^{15} + \nu^{14} - 3 \nu^{13} - 8 \nu^{12} + 6 \nu^{11} + 8 \nu^{10} - 4 \nu^{9} - 16 \nu^{8} + \cdots + 512 ) / 128 $ (3v^15 + v^14 - 3v^13 - 8v^12 + 6v^11 + 8v^10 - 4v^9 - 16v^8 - 20v^7 + 24v^6 + 32v^5 + 80v^4 - 128v^3 - 128v^2 + 128v + 512) / 128
$\beta_{6}$	$=$	$ ( - \nu^{15} + \nu^{14} - \nu^{13} + 4 \nu^{12} - 6 \nu^{11} - 2 \nu^{10} + 8 \nu^{9} - 12 \nu^{8} + \cdots - 64 ) / 64 $ (-v^15 + v^14 - v^13 + 4v^12 - 6v^11 - 2v^10 + 8v^9 - 12v^8 + 36v^7 - 32v^6 + 24v^5 - 24v^4 + 16v^3 + 128v^2 - 160v - 64) / 64
$\beta_{7}$	$=$	$ ( 9 \nu^{15} - 15 \nu^{14} + 13 \nu^{13} - 26 \nu^{12} + 54 \nu^{11} + 4 \nu^{10} - 124 \nu^{9} + \cdots - 1536 ) / 640 $ (9v^15 - 15v^14 + 13v^13 - 26v^12 + 54v^11 + 4v^10 - 124v^9 + 200v^8 - 412v^7 + 544v^6 - 224v^5 - 272v^4 + 320v^3 - 1216v^2 + 2688*v - 1536) / 640
$\beta_{8}$	$=$	$ ( 3 \nu^{15} - 10 \nu^{14} + 16 \nu^{13} - 27 \nu^{12} + 28 \nu^{11} - 12 \nu^{10} - 68 \nu^{9} + \cdots - 832 ) / 320 $ (3v^15 - 10v^14 + 16v^13 - 27v^12 + 28v^11 - 12v^10 - 68v^9 + 160v^8 - 244v^7 + 308v^6 - 248v^5 - 24v^4 + 320v^3 - 672v^2 + 1056*v - 832) / 320
$\beta_{9}$	$=$	$ ( - \nu^{15} + 2 \nu^{14} - 2 \nu^{13} + 3 \nu^{12} - 2 \nu^{11} - 8 \nu^{10} + 20 \nu^{9} - 32 \nu^{8} + \cdots + 64 ) / 64 $ (-v^15 + 2v^14 - 2v^13 + 3v^12 - 2v^11 - 8v^10 + 20v^9 - 32v^8 + 44v^7 - 44v^6 + 56v^4 - 80v^3 + 160v^2 - 224*v + 64) / 64
$\beta_{10}$	$=$	$ ( \nu^{15} - 2 \nu^{14} + 2 \nu^{13} - 5 \nu^{12} + 4 \nu^{11} + 2 \nu^{10} - 12 \nu^{9} + 12 \nu^{8} + \cdots + 128 \nu ) / 64 $ (v^15 - 2v^14 + 2v^13 - 5v^12 + 4v^11 + 2v^10 - 12v^9 + 12v^8 - 28v^7 + 52v^6 - 16v^5 - 64v^2 + 128v) / 64
$\beta_{11}$	$=$	$ ( - \nu^{15} + 2 \nu^{14} - 2 \nu^{13} + 3 \nu^{12} - 6 \nu^{11} + 4 \nu^{10} - 8 \nu^{8} + 20 \nu^{7} + \cdots + 64 ) / 64 $ (-v^15 + 2v^14 - 2v^13 + 3v^12 - 6v^11 + 4v^10 - 8v^8 + 20v^7 - 44v^6 + 48v^5 - 40v^4 + 64v^3 - 32v^2 - 96*v + 64) / 64
$\beta_{12}$	$=$	$ ( 7 \nu^{15} - 29 \nu^{14} + 27 \nu^{13} - 22 \nu^{12} + 46 \nu^{11} + 20 \nu^{10} - 172 \nu^{9} + \cdots - 1920 ) / 640 $ (7v^15 - 29v^14 + 27v^13 - 22v^12 + 46v^11 + 20v^10 - 172v^9 + 264v^8 - 404v^7 + 560v^6 - 272v^5 - 272v^4 + 384v^3 - 256v^2 + 2176*v - 1920) / 640
$\beta_{13}$	$=$	$ ( 5 \nu^{15} - 16 \nu^{14} + 2 \nu^{13} - \nu^{12} + 6 \nu^{11} + 42 \nu^{10} - 120 \nu^{9} + \cdots - 1088 ) / 320 $ (5v^15 - 16v^14 + 2v^13 - v^12 + 6v^11 + 42v^10 - 120v^9 + 156v^8 - 212v^7 + 252v^6 + 40v^5 - 384v^4 + 576v^3 - 672v^2 + 1728v - 1088) / 320
$\beta_{14}$	$=$	$ ( - 12 \nu^{15} + 7 \nu^{14} + 7 \nu^{13} - 5 \nu^{12} + 26 \nu^{11} - 86 \nu^{10} + 112 \nu^{9} + \cdots - 576 ) / 320 $ (-12v^15 + 7v^14 + 7v^13 - 5v^12 + 26v^11 - 86v^10 + 112v^9 - 52v^8 + 80v^7 + 44v^6 - 368v^5 + 464v^4 + 128v^3 + 512v^2 - 640*v - 576) / 320
$\beta_{15}$	$=$	$ ( - 7 \nu^{15} + 17 \nu^{14} - 19 \nu^{13} + 22 \nu^{12} - 26 \nu^{11} - 28 \nu^{10} + 132 \nu^{9} + \cdots + 1024 ) / 128 $ (-7v^15 + 17v^14 - 19v^13 + 22v^12 - 26v^11 - 28v^10 + 132v^9 - 200v^8 + 292v^7 - 352v^6 + 160v^5 + 240v^4 - 448v^3 + 768v^2 - 1408*v + 1024) / 128

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

$\nu$	$=$	$ ( \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} ) / 2 $ (b8 + b4 - b3 + b2) / 2
$\nu^{2}$	$=$	$ ( \beta_{12} - \beta_{10} + \beta_{6} + \beta_{5} + \beta_1 ) / 2 $ (b12 - b10 + b6 + b5 + b1) / 2
$\nu^{3}$	$=$	$ ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} + \cdots + 1 ) / 2 $ (-b15 + b14 + b13 - b12 - b10 + b9 - b8 + b6 + b5 + b3 + b1 + 1) / 2
$\nu^{4}$	$=$	$ ( \beta_{14} + 3\beta_{12} + 2\beta_{8} - 4\beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{4} - 3\beta _1 - 1 ) / 2 $ (b14 + 3b12 + 2b8 - 4b7 - b6 + b5 + 2b4 - 3*b1 - 1) / 2
$\nu^{5}$	$=$	$ ( \beta_{15} + 2 \beta_{14} - \beta_{13} + 6 \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \cdots - 4 ) / 2 $ (b15 + 2b14 - b13 + 6b12 + 2b11 + b10 - b9 + b8 + 3b3 + 2b2 - 4b1 - 4) / 2
$\nu^{6}$	$=$	$ ( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 3 \beta_{10} - 5 \beta_{9} - 3 \beta_{8} + \cdots + 5 ) / 2 $ (b15 + b14 - b13 + b12 - 2b11 + 3b10 - 5b9 - 3b8 + 3b6 + b5 + b3 - 4b2 - 3*b1 + 5) / 2
$\nu^{7}$	$=$	$ ( - 2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 5 \beta_{12} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots - 1 ) / 2 $ (-2b15 + b14 - 2b13 - 5b12 + 2b10 - 2b9 + 2b8 + 3b6 + b5 - 8b3 + 6b2 - 7b1 - 1) / 2
$\nu^{8}$	$=$	$ ( 3 \beta_{15} - 3 \beta_{13} + 6 \beta_{12} - 2 \beta_{11} - 7 \beta_{10} - 11 \beta_{9} + 7 \beta_{8} + \cdots - 2 ) / 2 $ (3b15 - 3b13 + 6b12 - 2b11 - 7b10 - 11b9 + 7b8 - 4b7 + 4b6 + 4b5 - 4b4 - 3b3 + 2b2 - 8b1 - 2) / 2
$\nu^{9}$	$=$	$ ( 3 \beta_{15} + 5 \beta_{14} + 5 \beta_{13} - 11 \beta_{12} - 14 \beta_{11} - 3 \beta_{10} - 19 \beta_{9} + \cdots - 15 ) / 2 $ (3b15 + 5b14 + 5b13 - 11b12 - 14b11 - 3b10 - 19b9 - 9b8 + 4b7 + 3b6 - 3b5 + 7b3 + 9*b1 - 15) / 2
$\nu^{10}$	$=$	$ ( 3 \beta_{14} - 11 \beta_{12} - 12 \beta_{11} - 32 \beta_{9} - 24 \beta_{8} - 8 \beta_{7} - 3 \beta_{6} + \cdots - 35 ) / 2 $ (3b14 - 11b12 - 12b11 - 32b9 - 24b8 - 8b7 - 3b6 + 3b5 + 4b4 - 10b3 - 6b2 + 11b1 - 35) / 2
$\nu^{11}$	$=$	$ ( - 9 \beta_{15} + 14 \beta_{14} - 7 \beta_{13} + 4 \beta_{12} + 14 \beta_{11} - 15 \beta_{10} + \cdots - 24 ) / 2 $ (-9b15 + 14b14 - 7b13 + 4b12 + 14b11 - 15b10 + b9 - 37b8 + 28b7 - 6b6 + 6b5 - 28b4 + 5b3 + 14b2 - 6b1 - 24) / 2
$\nu^{12}$	$=$	$ ( - 7 \beta_{15} + 9 \beta_{14} - 9 \beta_{13} + 21 \beta_{12} - 10 \beta_{11} - 9 \beta_{10} - \beta_{9} + \cdots + 81 ) / 2 $ (-7b15 + 9b14 - 9b13 + 21b12 - 10b11 - 9b10 - b9 - 27b8 - 4b7 - 9b6 - 7b5 + 8b4 + 17b3 - 20b2 - 35b1 + 81) / 2
$\nu^{13}$	$=$	$ ( - 22 \beta_{15} + 21 \beta_{14} - 42 \beta_{13} + 15 \beta_{12} + 22 \beta_{10} - 10 \beta_{9} + \cdots + 15 ) / 2 $ (-22b15 + 21b14 - 42b13 + 15b12 + 22b10 - 10b9 + 2b8 - 5b6 - 27b5 + 60b4 - 8b3 + 42b2 - 27*b1 + 15) / 2
$\nu^{14}$	$=$	$ ( - 13 \beta_{15} - 67 \beta_{13} + 34 \beta_{12} + 14 \beta_{11} - 51 \beta_{10} - 43 \beta_{9} + \cdots + 14 ) / 2 $ (-13b15 - 67b13 + 34b12 + 14b11 - 51b10 - 43b9 + 7b8 + 28b7 + 64b6 + 64b5 + 28b4 - 35b3 - 14b2 - 12b1 + 14) / 2
$\nu^{15}$	$=$	$ ( - 73 \beta_{15} + 9 \beta_{14} + 9 \beta_{13} - 87 \beta_{12} - 62 \beta_{11} - 103 \beta_{10} + \cdots + 61 ) / 2 $ (-73b15 + 9b14 + 9b13 - 87b12 - 62b11 - 103b10 + 81b9 - 21b8 + 36b7 + 103b6 + 73b5 + 51b3 + 21*b1 + 61) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$-\beta_{9}$

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} $

17.1

−0.608487	+	1.27661i
1.21425	−	0.724984i
−1.41366	−	0.0395346i
0.867327	+	1.11702i
1.40104	+	0.192615i
−0.741068	−	1.20450i
−0.0207305	+	1.41406i
0.801337	−	1.16527i
−0.608487	−	1.27661i
1.21425	+	0.724984i
−1.41366	+	0.0395346i
0.867327	−	1.11702i
1.40104	−	0.192615i
−0.741068	+	1.20450i
−0.0207305	−	1.41406i
0.801337	+	1.16527i

−2.28316

−

1.31818i

−3.40672

−1.30715

0.754684i

1.97521

3.42116i

17.2

−1.95589

−

1.12924i

−0.642566

−0.306483

0.176948i

1.05034

1.81925i

17.3

−0.779193

−

0.449867i

0.893415

−3.65473

2.11006i

−1.09524

−

1.89701i

17.4

−0.323315

−

0.186666i

−2.04528

0.768362

−

0.443614i

−1.43031

−

2.47737i

17.5

0.323315

0.186666i

2.04528

0.768362

−

0.443614i

−1.43031

−

2.47737i

17.6

0.779193

0.449867i

−0.893415

−3.65473

2.11006i

−1.09524

−

1.89701i

17.7

1.95589

1.12924i

0.642566

−0.306483

0.176948i

1.05034

1.81925i

17.8

2.28316

1.31818i

3.40672

−1.30715

0.754684i

1.97521

3.42116i

49.1

−2.28316

1.31818i

−3.40672

−1.30715

−

0.754684i

1.97521

−

3.42116i

49.2

−1.95589

1.12924i

−0.642566

−0.306483

−

0.176948i

1.05034

−

1.81925i

49.3

−0.779193

0.449867i

0.893415

−3.65473

−

2.11006i

−1.09524

1.89701i

49.4

−0.323315

0.186666i

−2.04528

0.768362

0.443614i

−1.43031

2.47737i

49.5

0.323315

−

0.186666i

2.04528

0.768362

0.443614i

−1.43031

2.47737i

49.6

0.779193

−

0.449867i

−0.893415

−3.65473

−

2.11006i

−1.09524

1.89701i

49.7

1.95589

−

1.12924i

0.642566

−0.306483

−

0.176948i

1.05034

−

1.81925i

49.8

2.28316

−

1.31818i

3.40672

−1.30715

−

0.754684i

1.97521

−

3.42116i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
13.e	even	6	1	inner
104.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.2.ba.c		16
4.b	odd	2	1		104.2.s.c	✓	16
8.b	even	2	1	inner	416.2.ba.c		16
8.d	odd	2	1		104.2.s.c	✓	16
12.b	even	2	1		936.2.dg.d		16
13.e	even	6	1	inner	416.2.ba.c		16
24.f	even	2	1		936.2.dg.d		16
52.i	odd	6	1		104.2.s.c	✓	16
104.p	odd	6	1		104.2.s.c	✓	16
104.s	even	6	1	inner	416.2.ba.c		16
156.r	even	6	1		936.2.dg.d		16
312.ba	even	6	1		936.2.dg.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.s.c	✓	16	4.b	odd	2	1
104.2.s.c	✓	16	8.d	odd	2	1
104.2.s.c	✓	16	52.i	odd	6	1
104.2.s.c	✓	16	104.p	odd	6	1
416.2.ba.c		16	1.a	even	1	1	trivial
416.2.ba.c		16	8.b	even	2	1	inner
416.2.ba.c		16	13.e	even	6	1	inner
416.2.ba.c		16	104.s	even	6	1	inner
936.2.dg.d		16	12.b	even	2	1
936.2.dg.d		16	24.f	even	2	1
936.2.dg.d		16	156.r	even	6	1
936.2.dg.d		16	312.ba	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{16} - 13T_{3}^{14} + 122T_{3}^{12} - 541T_{3}^{10} + 1750T_{3}^{8} - 1541T_{3}^{6} + 1037T_{3}^{4} - 140T_{3}^{2} + 16 $ T3^16 - 13*T3^14 + 122*T3^12 - 541*T3^10 + 1750*T3^8 - 1541*T3^6 + 1037*T3^4 - 140*T3^2 + 16 acting on $S_{2}^{\mathrm{new}}(416, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} - 13 T^{14} + \cdots + 16 $ T^16 - 13T^14 + 122T^12 - 541T^10 + 1750T^8 - 1541T^6 + 1037T^4 - 140*T^2 + 16
$5$	$ (T^{8} - 17 T^{6} + \cdots + 16)^{2} $ (T^8 - 17T^6 + 68T^4 - 64*T^2 + 16)^2
$7$	$ (T^{8} + 9 T^{7} + 30 T^{6} + \cdots + 4)^{2} $ (T^8 + 9T^7 + 30T^6 + 27T^5 - 16T^4 - 27T^3 + 21T^2 + 18*T + 4)^2
$11$	$ T^{16} + 55 T^{14} + \cdots + 14776336 $ T^16 + 55T^14 + 2258T^12 + 35383T^10 + 397390T^8 + 2185727T^6 + 8618453T^4 + 13073444*T^2 + 14776336
$13$	$ T^{16} + \cdots + 815730721 $ T^16 - 9T^14 - 54T^12 + 801T^10 - 3766T^8 + 135369T^6 - 1542294T^4 - 43441281*T^2 + 815730721
$17$	$ (T^{8} - 4 T^{7} + \cdots + 5329)^{2} $ (T^8 - 4T^7 + 38T^6 + 56T^5 + 475T^4 + 232T^3 + 1862T^2 + 1168*T + 5329)^2
$19$	$ T^{16} + \cdots + 31713911056 $ T^16 + 107T^14 + 7742T^12 + 306263T^10 + 8728114T^8 + 129420475T^6 + 1382249861T^4 + 8048150212*T^2 + 31713911056
$23$	$ (T^{8} - T^{7} + 22 T^{6} + \cdots + 256)^{2} $ (T^8 - T^7 + 22T^6 + 47T^5 + 412T^4 + 305T^3 + 505T^2 - 208T + 256)^2
$29$	$ T^{16} + \cdots + 12594450625 $ T^16 - 92T^14 + 5738T^12 - 190064T^10 + 4525363T^8 - 62122864T^6 + 616047146T^4 - 3407599900*T^2 + 12594450625
$31$	$ (T^{8} + 132 T^{6} + \cdots + 16384)^{2} $ (T^8 + 132T^6 + 4784T^4 + 27840*T^2 + 16384)^2
$37$	$ T^{16} + \cdots + 2947295521 $ T^16 + 76T^14 + 3770T^12 + 111568T^10 + 2416003T^8 + 32758736T^6 + 309053402T^4 + 1109884316*T^2 + 2947295521
$41$	$ (T^{8} - 12 T^{7} + \cdots + 81)^{2} $ (T^8 - 12T^7 - 18T^6 + 792T^5 + 1179T^4 - 52272T^3 + 208494T^2 + 7128*T + 81)^2
$43$	$ T^{16} - 145 T^{14} + \cdots + 4096 $ T^16 - 145T^14 + 15182T^12 - 726469T^10 + 25385050T^8 - 352799309T^6 + 3645732737T^4 - 3864512*T^2 + 4096
$47$	$ (T^{8} + 204 T^{6} + \cdots + 64)^{2} $ (T^8 + 204T^6 + 7520T^4 + 57264*T^2 + 64)^2
$53$	$ (T^{8} + 31 T^{6} + \cdots + 400)^{2} $ (T^8 + 31T^6 + 320T^4 + 1136*T^2 + 400)^2
$59$	$ T^{16} + \cdots + 121173610000 $ T^16 + 131T^14 + 11582T^12 + 558143T^10 + 19464898T^8 + 390561187T^6 + 5514790709T^4 + 30059479300*T^2 + 121173610000
$61$	$ T^{16} + \cdots + 18\!\cdots\!81 $ T^16 - 372T^14 + 91602T^12 - 12589776T^10 + 1250030475T^8 - 80381145744T^6 + 3766673705634T^4 - 104163889593924*T^2 + 1873438951464081
$67$	$ T^{16} + \cdots + 2897022976 $ T^16 + 191T^14 + 31226T^12 + 925343T^10 + 20077630T^8 + 185335387T^6 + 1252305641T^4 + 2108878144*T^2 + 2897022976
$71$	$ (T^{8} + 3 T^{7} + \cdots + 1295044)^{2} $ (T^8 + 3T^7 - 104T^6 - 321T^5 + 10614T^4 - 32421T^3 - 91163T^2 + 344814*T + 1295044)^2
$73$	$ (T^{8} + 307 T^{6} + \cdots + 9610000)^{2} $ (T^8 + 307T^6 + 29904T^4 + 1005232*T^2 + 9610000)^2
$79$	$ (T^{4} + 8 T^{3} - 4 T^{2} + \cdots + 40)^{4} $ (T^4 + 8T^3 - 4T^2 - 44*T + 40)^4
$83$	$ (T^{8} - 560 T^{6} + \cdots + 189337600)^{2} $ (T^8 - 560T^6 + 102848T^4 - 7520512*T^2 + 189337600)^2
$89$	$ (T^{8} + 15 T^{7} + \cdots + 67600)^{2} $ (T^8 + 15T^7 - 32T^6 - 1605T^5 + 8664T^4 + 65163T^3 + 151447T^2 + 158340*T + 67600)^2
$97$	$ (T^{8} + 15 T^{7} + \cdots + 473344)^{2} $ (T^8 + 15T^7 - 76T^6 - 2265T^5 + 14268T^4 + 236919T^3 + 716699T^2 - 1079472*T + 473344)^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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.ba (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{12} + \beta_{8} - \beta_1) q^{3} + (\beta_{12} + \beta_{8} - \beta_{3} - \beta_1) q^{5} + (\beta_{11} + \beta_{2} - 1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{5}) q^{9}+O(q^{10}) \) q + (b12 + b8 - b1) * q^3 + (b12 + b8 - b3 - b1) * q^5 + (b11 + b2 - 1) * q^7 + (b10 - b6 - b5) * q^9	\( q + (\beta_{12} + \beta_{8} - \beta_1) q^{3} + (\beta_{12} + \beta_{8} - \beta_{3} - \beta_1) q^{5} + (\beta_{11} + \beta_{2} - 1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{5}) q^{9} + (\beta_{12} - \beta_{8} - 2 \beta_{4} + \cdots + \beta_1) q^{11}+ \cdots + (\beta_{14} + \beta_{13} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) q + (b12 + b8 - b1) * q^3 + (b12 + b8 - b3 - b1) * q^5 + (b11 + b2 - 1) * q^7 + (b10 - b6 - b5) * q^9 + (b12 - b8 - 2b4 + 2b3 + b1) * q^11 + (b14 + b12 + b8 - b7 + b3) * q^13 + (-b15 + b10 - 2b9 - b5 + 2) q^15 + (b15 - b6 - b5) * q^17 + (2b14 - b13 - b12 - 2b8) * q^19 + (-b8 - b7 + b3 + b1) * q^21 + (-2b11 - b9 - b2 - 1) q^23 + (-b15 + b10 - b9 + b6 - b5) * q^25 + (-b14 + b13 - b12 - b3) * q^27 + (b14 - 2b12 - 3b8 + 2b7 - b4 + 2b1) * q^29 + (2b15 + 2b10 + 2b9 - 2b6 - 2b5) q^31 + (2b11 - b10 + b9 - b6 + b5 + 2b2 - 2) * q^33 + (b13 - b12 - b8 + b3 + b1) * q^35 + (-b12 - 2b8 - b4 + b3 - b1) q^37 + (b15 - b10 + b9 - b5 + b2 + 1) * q^39 + (-b15 + b10 - 3b9 + 2b6 + b5 + 1) * q^41 + (2b12 + 3b8 - 3b3 - b1) q^43 + (-2b14 + b13 + 2b12 + b8 + 2b7 - 2b4 + b3) * q^45 + (-2b15 + 2b11 - b10 + b9 + b6 + 2b5 + 2) q^47 + (b15 - 4b11 + b10 - 4b9 - b6 - 2b2 - 5) q^49 + (b14 - b13 + 3b12 + 4b8 - 2b7 + 4b4 + b3) * q^51 + (b12 + b8 + b3 + b1) * q^53 + (b15 + b10 - 2b6 + b5 - 2) q^55 + (-b15 - 3b10 + 5b9 + 3b6 + b5 + 4) q^57 + (-2b14 + b13 - 3b12 - 2b3 + 2b1) * q^59 + (-3b13 + b12 + b8 - b7 - b4 - 4b1) * q^61 + (b15 - b10 + b5 - 2b2) q^63 + (2b15 - b10 + 5b9 + b5 + 2b2 + 1) q^65 + (b14 - 2b13 - 2b8 + 2b4 + b3 - 2b1) * q^67 + (-b12 + b7 + b4 - b3) * q^69 + (-3b15 - b11 - 2b10 - b9 + b6 - b5 - b2 + 1) * q^71 + (-b15 + 4b11 - 2b9 + b5 + 1) * q^73 + (-2b14 + b12 - b8 + 4b7 - 2b4 - 4b3 - b1) * q^75 + (-2b14 + 2b13 - 4b12 + 3b8 - 3b7 + 6b4 - 7b3 - 3b1) * q^77 + (b10 - b9 + b6 - 2) * q^79 + (-3b15 - 3b10 + b9 + 2b6 + b5 + 3) q^81 + (-2b12 + 2b8 - 2b7 + 4b3 - 2b1) q^83 + (-b12 + 2b8 - 2b7 + 2b4 + b1) q^85 + (-b15 + b11 - 2b10 + 3b9 + 3b6 + 3b5 - b2 + 1) * q^87 + (b9 - b6 - b5 - 4b2) q^89 + (-2b14 + b13 + b12 + 2b8 - 2b7 + 2b4 - 4b3 - 2b1) * q^91 + (2b12 + 2b8 + 4b4 + 2b1) * q^93 + (b15 - 2b11 - 4b10 + 3b9 + 3b6 + 3b5 + 2b2 - 2) * q^95 + (3b15 - 2b11 + b10 - 2b9 - 2b6 + 2b5 - 2b2 - 6) * q^97 + (b14 + b13 - 4b12 - 2b8 + 4b3 + 2b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 18 q^{7} + 2 q^{9}+O(q^{10}) \) 16 * q - 18 * q^7 + 2 * q^9	\( 16 q - 18 q^{7} + 2 q^{9} + 36 q^{15} + 8 q^{17} + 2 q^{23} - 12 q^{25} - 30 q^{33} + 14 q^{39} + 24 q^{41} - 14 q^{49} - 4 q^{55} + 6 q^{65} - 6 q^{71} - 32 q^{79} + 12 q^{81} - 34 q^{87} - 30 q^{89} - 28 q^{95} - 30 q^{97}+O(q^{100}) \) 16 * q - 18 * q^7 + 2 * q^9 + 36 * q^15 + 8 * q^17 + 2 * q^23 - 12 * q^25 - 30 * q^33 + 14 * q^39 + 24 * q^41 - 14 * q^49 - 4 * q^55 + 6 * q^65 - 6 * q^71 - 32 * q^79 + 12 * q^81 - 34 * q^87 - 30 * q^89 - 28 * q^95 - 30 * q^97

Introduction

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Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	416.2.ba.c

Level	$416$
Weight	$2$
Character orbit	416.ba
Analytic conductor	$3.322$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.32177672409\)
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:	\( x^{16} - 3 x^{15} + 5 x^{14} - 6 x^{13} + 6 x^{12} - 20 x^{10} + 48 x^{9} - 76 x^{8} + 96 x^{7} + \cdots + 256 \) x^16 - 3x^15 + 5x^14 - 6x^13 + 6x^12 - 20x^10 + 48x^9 - 76x^8 + 96x^7 - 80x^6 + 96x^4 - 192x^3 + 320x^2 - 384*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - \nu^{15} - 3 \nu^{14} + 9 \nu^{13} - 14 \nu^{12} + 12 \nu^{11} - 10 \nu^{10} - 4 \nu^{9} + \cdots + 32 \nu ) / 320 \) (-v^15 - 3v^14 + 9v^13 - 14v^12 + 12v^11 - 10v^10 - 4v^9 + 28v^8 - 68v^7 + 80v^6 - 144v^5 + 56v^4 + 48v^3 + 128v^2 + 32v) / 320
\(\beta_{2}\)	\(=\)	\( ( \nu^{15} + \nu^{14} - 3 \nu^{13} + 2 \nu^{12} - 6 \nu^{11} + 8 \nu^{10} + 12 \nu^{9} - 32 \nu^{8} + \cdots + 512 ) / 128 \) (v^15 + v^14 - 3v^13 + 2v^12 - 6v^11 + 8v^10 + 12v^9 - 32v^8 + 52v^7 - 80v^6 + 96v^5 - 128v^3 + 128v^2 - 192v + 512) / 128
\(\beta_{3}\)	\(=\)	\( ( 9 \nu^{15} + 5 \nu^{14} - 7 \nu^{13} - 6 \nu^{12} - 26 \nu^{11} + 44 \nu^{10} - 4 \nu^{9} + \cdots + 1664 ) / 640 \) (9v^15 + 5v^14 - 7v^13 - 6v^12 - 26v^11 + 44v^10 - 4v^9 - 40v^8 - 12v^7 - 176v^6 + 416v^5 - 112v^4 - 160v^3 - 576v^2 - 512*v + 1664) / 640
\(\beta_{4}\)	\(=\)	\( ( - \nu^{15} + 10 \nu^{14} - 12 \nu^{13} + 19 \nu^{12} - 26 \nu^{11} + 14 \nu^{10} + 36 \nu^{9} + \cdots + 384 ) / 320 \) (-v^15 + 10v^14 - 12v^13 + 19v^12 - 26v^11 + 14v^10 + 36v^9 - 100v^8 + 108v^7 - 196v^6 + 216v^5 - 32v^4 - 80v^3 + 64v^2 - 192v + 384) / 320
\(\beta_{5}\)	\(=\)	\( ( 3 \nu^{15} + \nu^{14} - 3 \nu^{13} - 8 \nu^{12} + 6 \nu^{11} + 8 \nu^{10} - 4 \nu^{9} - 16 \nu^{8} + \cdots + 512 ) / 128 \) (3v^15 + v^14 - 3v^13 - 8v^12 + 6v^11 + 8v^10 - 4v^9 - 16v^8 - 20v^7 + 24v^6 + 32v^5 + 80v^4 - 128v^3 - 128v^2 + 128v + 512) / 128
\(\beta_{6}\)	\(=\)	\( ( - \nu^{15} + \nu^{14} - \nu^{13} + 4 \nu^{12} - 6 \nu^{11} - 2 \nu^{10} + 8 \nu^{9} - 12 \nu^{8} + \cdots - 64 ) / 64 \) (-v^15 + v^14 - v^13 + 4v^12 - 6v^11 - 2v^10 + 8v^9 - 12v^8 + 36v^7 - 32v^6 + 24v^5 - 24v^4 + 16v^3 + 128v^2 - 160v - 64) / 64
\(\beta_{7}\)	\(=\)	\( ( 9 \nu^{15} - 15 \nu^{14} + 13 \nu^{13} - 26 \nu^{12} + 54 \nu^{11} + 4 \nu^{10} - 124 \nu^{9} + \cdots - 1536 ) / 640 \) (9v^15 - 15v^14 + 13v^13 - 26v^12 + 54v^11 + 4v^10 - 124v^9 + 200v^8 - 412v^7 + 544v^6 - 224v^5 - 272v^4 + 320v^3 - 1216v^2 + 2688*v - 1536) / 640
\(\beta_{8}\)	\(=\)	\( ( 3 \nu^{15} - 10 \nu^{14} + 16 \nu^{13} - 27 \nu^{12} + 28 \nu^{11} - 12 \nu^{10} - 68 \nu^{9} + \cdots - 832 ) / 320 \) (3v^15 - 10v^14 + 16v^13 - 27v^12 + 28v^11 - 12v^10 - 68v^9 + 160v^8 - 244v^7 + 308v^6 - 248v^5 - 24v^4 + 320v^3 - 672v^2 + 1056*v - 832) / 320
\(\beta_{9}\)	\(=\)	\( ( - \nu^{15} + 2 \nu^{14} - 2 \nu^{13} + 3 \nu^{12} - 2 \nu^{11} - 8 \nu^{10} + 20 \nu^{9} - 32 \nu^{8} + \cdots + 64 ) / 64 \) (-v^15 + 2v^14 - 2v^13 + 3v^12 - 2v^11 - 8v^10 + 20v^9 - 32v^8 + 44v^7 - 44v^6 + 56v^4 - 80v^3 + 160v^2 - 224*v + 64) / 64
\(\beta_{10}\)	\(=\)	\( ( \nu^{15} - 2 \nu^{14} + 2 \nu^{13} - 5 \nu^{12} + 4 \nu^{11} + 2 \nu^{10} - 12 \nu^{9} + 12 \nu^{8} + \cdots + 128 \nu ) / 64 \) (v^15 - 2v^14 + 2v^13 - 5v^12 + 4v^11 + 2v^10 - 12v^9 + 12v^8 - 28v^7 + 52v^6 - 16v^5 - 64v^2 + 128v) / 64
\(\beta_{11}\)	\(=\)	\( ( - \nu^{15} + 2 \nu^{14} - 2 \nu^{13} + 3 \nu^{12} - 6 \nu^{11} + 4 \nu^{10} - 8 \nu^{8} + 20 \nu^{7} + \cdots + 64 ) / 64 \) (-v^15 + 2v^14 - 2v^13 + 3v^12 - 6v^11 + 4v^10 - 8v^8 + 20v^7 - 44v^6 + 48v^5 - 40v^4 + 64v^3 - 32v^2 - 96*v + 64) / 64
\(\beta_{12}\)	\(=\)	\( ( 7 \nu^{15} - 29 \nu^{14} + 27 \nu^{13} - 22 \nu^{12} + 46 \nu^{11} + 20 \nu^{10} - 172 \nu^{9} + \cdots - 1920 ) / 640 \) (7v^15 - 29v^14 + 27v^13 - 22v^12 + 46v^11 + 20v^10 - 172v^9 + 264v^8 - 404v^7 + 560v^6 - 272v^5 - 272v^4 + 384v^3 - 256v^2 + 2176*v - 1920) / 640
\(\beta_{13}\)	\(=\)	\( ( 5 \nu^{15} - 16 \nu^{14} + 2 \nu^{13} - \nu^{12} + 6 \nu^{11} + 42 \nu^{10} - 120 \nu^{9} + \cdots - 1088 ) / 320 \) (5v^15 - 16v^14 + 2v^13 - v^12 + 6v^11 + 42v^10 - 120v^9 + 156v^8 - 212v^7 + 252v^6 + 40v^5 - 384v^4 + 576v^3 - 672v^2 + 1728v - 1088) / 320
\(\beta_{14}\)	\(=\)	\( ( - 12 \nu^{15} + 7 \nu^{14} + 7 \nu^{13} - 5 \nu^{12} + 26 \nu^{11} - 86 \nu^{10} + 112 \nu^{9} + \cdots - 576 ) / 320 \) (-12v^15 + 7v^14 + 7v^13 - 5v^12 + 26v^11 - 86v^10 + 112v^9 - 52v^8 + 80v^7 + 44v^6 - 368v^5 + 464v^4 + 128v^3 + 512v^2 - 640*v - 576) / 320
\(\beta_{15}\)	\(=\)	\( ( - 7 \nu^{15} + 17 \nu^{14} - 19 \nu^{13} + 22 \nu^{12} - 26 \nu^{11} - 28 \nu^{10} + 132 \nu^{9} + \cdots + 1024 ) / 128 \) (-7v^15 + 17v^14 - 19v^13 + 22v^12 - 26v^11 - 28v^10 + 132v^9 - 200v^8 + 292v^7 - 352v^6 + 160v^5 + 240v^4 - 448v^3 + 768v^2 - 1408*v + 1024) / 128

\(\nu\)	\(=\)	\( ( \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} ) / 2 \) (b8 + b4 - b3 + b2) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{12} - \beta_{10} + \beta_{6} + \beta_{5} + \beta_1 ) / 2 \) (b12 - b10 + b6 + b5 + b1) / 2
\(\nu^{3}\)	\(=\)	\( ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} + \cdots + 1 ) / 2 \) (-b15 + b14 + b13 - b12 - b10 + b9 - b8 + b6 + b5 + b3 + b1 + 1) / 2
\(\nu^{4}\)	\(=\)	\( ( \beta_{14} + 3\beta_{12} + 2\beta_{8} - 4\beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{4} - 3\beta _1 - 1 ) / 2 \) (b14 + 3b12 + 2b8 - 4b7 - b6 + b5 + 2b4 - 3*b1 - 1) / 2
\(\nu^{5}\)	\(=\)	\( ( \beta_{15} + 2 \beta_{14} - \beta_{13} + 6 \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \cdots - 4 ) / 2 \) (b15 + 2b14 - b13 + 6b12 + 2b11 + b10 - b9 + b8 + 3b3 + 2b2 - 4b1 - 4) / 2
\(\nu^{6}\)	\(=\)	\( ( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 3 \beta_{10} - 5 \beta_{9} - 3 \beta_{8} + \cdots + 5 ) / 2 \) (b15 + b14 - b13 + b12 - 2b11 + 3b10 - 5b9 - 3b8 + 3b6 + b5 + b3 - 4b2 - 3*b1 + 5) / 2
\(\nu^{7}\)	\(=\)	\( ( - 2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 5 \beta_{12} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots - 1 ) / 2 \) (-2b15 + b14 - 2b13 - 5b12 + 2b10 - 2b9 + 2b8 + 3b6 + b5 - 8b3 + 6b2 - 7b1 - 1) / 2
\(\nu^{8}\)	\(=\)	\( ( 3 \beta_{15} - 3 \beta_{13} + 6 \beta_{12} - 2 \beta_{11} - 7 \beta_{10} - 11 \beta_{9} + 7 \beta_{8} + \cdots - 2 ) / 2 \) (3b15 - 3b13 + 6b12 - 2b11 - 7b10 - 11b9 + 7b8 - 4b7 + 4b6 + 4b5 - 4b4 - 3b3 + 2b2 - 8b1 - 2) / 2
\(\nu^{9}\)	\(=\)	\( ( 3 \beta_{15} + 5 \beta_{14} + 5 \beta_{13} - 11 \beta_{12} - 14 \beta_{11} - 3 \beta_{10} - 19 \beta_{9} + \cdots - 15 ) / 2 \) (3b15 + 5b14 + 5b13 - 11b12 - 14b11 - 3b10 - 19b9 - 9b8 + 4b7 + 3b6 - 3b5 + 7b3 + 9*b1 - 15) / 2
\(\nu^{10}\)	\(=\)	\( ( 3 \beta_{14} - 11 \beta_{12} - 12 \beta_{11} - 32 \beta_{9} - 24 \beta_{8} - 8 \beta_{7} - 3 \beta_{6} + \cdots - 35 ) / 2 \) (3b14 - 11b12 - 12b11 - 32b9 - 24b8 - 8b7 - 3b6 + 3b5 + 4b4 - 10b3 - 6b2 + 11b1 - 35) / 2
\(\nu^{11}\)	\(=\)	\( ( - 9 \beta_{15} + 14 \beta_{14} - 7 \beta_{13} + 4 \beta_{12} + 14 \beta_{11} - 15 \beta_{10} + \cdots - 24 ) / 2 \) (-9b15 + 14b14 - 7b13 + 4b12 + 14b11 - 15b10 + b9 - 37b8 + 28b7 - 6b6 + 6b5 - 28b4 + 5b3 + 14b2 - 6b1 - 24) / 2
\(\nu^{12}\)	\(=\)	\( ( - 7 \beta_{15} + 9 \beta_{14} - 9 \beta_{13} + 21 \beta_{12} - 10 \beta_{11} - 9 \beta_{10} - \beta_{9} + \cdots + 81 ) / 2 \) (-7b15 + 9b14 - 9b13 + 21b12 - 10b11 - 9b10 - b9 - 27b8 - 4b7 - 9b6 - 7b5 + 8b4 + 17b3 - 20b2 - 35b1 + 81) / 2
\(\nu^{13}\)	\(=\)	\( ( - 22 \beta_{15} + 21 \beta_{14} - 42 \beta_{13} + 15 \beta_{12} + 22 \beta_{10} - 10 \beta_{9} + \cdots + 15 ) / 2 \) (-22b15 + 21b14 - 42b13 + 15b12 + 22b10 - 10b9 + 2b8 - 5b6 - 27b5 + 60b4 - 8b3 + 42b2 - 27*b1 + 15) / 2
\(\nu^{14}\)	\(=\)	\( ( - 13 \beta_{15} - 67 \beta_{13} + 34 \beta_{12} + 14 \beta_{11} - 51 \beta_{10} - 43 \beta_{9} + \cdots + 14 ) / 2 \) (-13b15 - 67b13 + 34b12 + 14b11 - 51b10 - 43b9 + 7b8 + 28b7 + 64b6 + 64b5 + 28b4 - 35b3 - 14b2 - 12b1 + 14) / 2
\(\nu^{15}\)	\(=\)	\( ( - 73 \beta_{15} + 9 \beta_{14} + 9 \beta_{13} - 87 \beta_{12} - 62 \beta_{11} - 103 \beta_{10} + \cdots + 61 ) / 2 \) (-73b15 + 9b14 + 9b13 - 87b12 - 62b11 - 103b10 + 81b9 - 21b8 + 36b7 + 103b6 + 73b5 + 51b3 + 21*b1 + 61) / 2

\(n\)	\(261\)	\(287\)	\(353\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-\beta_{9}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 17.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} - 13 T^{14} + \cdots + 16 \) T^16 - 13T^14 + 122T^12 - 541T^10 + 1750T^8 - 1541T^6 + 1037T^4 - 140*T^2 + 16
$5$	\( (T^{8} - 17 T^{6} + \cdots + 16)^{2} \) (T^8 - 17T^6 + 68T^4 - 64*T^2 + 16)^2
$7$	\( (T^{8} + 9 T^{7} + 30 T^{6} + \cdots + 4)^{2} \) (T^8 + 9T^7 + 30T^6 + 27T^5 - 16T^4 - 27T^3 + 21T^2 + 18*T + 4)^2
$11$	\( T^{16} + 55 T^{14} + \cdots + 14776336 \) T^16 + 55T^14 + 2258T^12 + 35383T^10 + 397390T^8 + 2185727T^6 + 8618453T^4 + 13073444*T^2 + 14776336
$13$	\( T^{16} + \cdots + 815730721 \) T^16 - 9T^14 - 54T^12 + 801T^10 - 3766T^8 + 135369T^6 - 1542294T^4 - 43441281*T^2 + 815730721
$17$	\( (T^{8} - 4 T^{7} + \cdots + 5329)^{2} \) (T^8 - 4T^7 + 38T^6 + 56T^5 + 475T^4 + 232T^3 + 1862T^2 + 1168*T + 5329)^2
$19$	\( T^{16} + \cdots + 31713911056 \) T^16 + 107T^14 + 7742T^12 + 306263T^10 + 8728114T^8 + 129420475T^6 + 1382249861T^4 + 8048150212*T^2 + 31713911056
$23$	\( (T^{8} - T^{7} + 22 T^{6} + \cdots + 256)^{2} \) (T^8 - T^7 + 22T^6 + 47T^5 + 412T^4 + 305T^3 + 505T^2 - 208T + 256)^2
$29$	\( T^{16} + \cdots + 12594450625 \) T^16 - 92T^14 + 5738T^12 - 190064T^10 + 4525363T^8 - 62122864T^6 + 616047146T^4 - 3407599900*T^2 + 12594450625
$31$	\( (T^{8} + 132 T^{6} + \cdots + 16384)^{2} \) (T^8 + 132T^6 + 4784T^4 + 27840*T^2 + 16384)^2
$37$	\( T^{16} + \cdots + 2947295521 \) T^16 + 76T^14 + 3770T^12 + 111568T^10 + 2416003T^8 + 32758736T^6 + 309053402T^4 + 1109884316*T^2 + 2947295521
$41$	\( (T^{8} - 12 T^{7} + \cdots + 81)^{2} \) (T^8 - 12T^7 - 18T^6 + 792T^5 + 1179T^4 - 52272T^3 + 208494T^2 + 7128*T + 81)^2
$43$	\( T^{16} - 145 T^{14} + \cdots + 4096 \) T^16 - 145T^14 + 15182T^12 - 726469T^10 + 25385050T^8 - 352799309T^6 + 3645732737T^4 - 3864512*T^2 + 4096
$47$	\( (T^{8} + 204 T^{6} + \cdots + 64)^{2} \) (T^8 + 204T^6 + 7520T^4 + 57264*T^2 + 64)^2
$53$	\( (T^{8} + 31 T^{6} + \cdots + 400)^{2} \) (T^8 + 31T^6 + 320T^4 + 1136*T^2 + 400)^2
$59$	\( T^{16} + \cdots + 121173610000 \) T^16 + 131T^14 + 11582T^12 + 558143T^10 + 19464898T^8 + 390561187T^6 + 5514790709T^4 + 30059479300*T^2 + 121173610000
$61$	\( T^{16} + \cdots + 18\!\cdots\!81 \) T^16 - 372T^14 + 91602T^12 - 12589776T^10 + 1250030475T^8 - 80381145744T^6 + 3766673705634T^4 - 104163889593924*T^2 + 1873438951464081
$67$	\( T^{16} + \cdots + 2897022976 \) T^16 + 191T^14 + 31226T^12 + 925343T^10 + 20077630T^8 + 185335387T^6 + 1252305641T^4 + 2108878144*T^2 + 2897022976
$71$	\( (T^{8} + 3 T^{7} + \cdots + 1295044)^{2} \) (T^8 + 3T^7 - 104T^6 - 321T^5 + 10614T^4 - 32421T^3 - 91163T^2 + 344814*T + 1295044)^2
$73$	\( (T^{8} + 307 T^{6} + \cdots + 9610000)^{2} \) (T^8 + 307T^6 + 29904T^4 + 1005232*T^2 + 9610000)^2
$79$	\( (T^{4} + 8 T^{3} - 4 T^{2} + \cdots + 40)^{4} \) (T^4 + 8T^3 - 4T^2 - 44*T + 40)^4
$83$	\( (T^{8} - 560 T^{6} + \cdots + 189337600)^{2} \) (T^8 - 560T^6 + 102848T^4 - 7520512*T^2 + 189337600)^2
$89$	\( (T^{8} + 15 T^{7} + \cdots + 67600)^{2} \) (T^8 + 15T^7 - 32T^6 - 1605T^5 + 8664T^4 + 65163T^3 + 151447T^2 + 158340*T + 67600)^2
$97$	\( (T^{8} + 15 T^{7} + \cdots + 473344)^{2} \) (T^8 + 15T^7 - 76T^6 - 2265T^5 + 14268T^4 + 236919T^3 + 716699T^2 - 1079472*T + 473344)^2
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