LMFDB - Newform orbit 1248.2.g.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,2,Mod(625,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1248.625");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.g (of order $2$, degree $1$, 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:	$9.96533017226$
Analytic rank:	$0$
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} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 2x^14 - 4x^13 + 9x^12 - 10x^11 + 2x^10 - 8x^9 + 28x^8 - 16x^7 + 8x^6 - 80x^5 + 144x^4 - 128x^3 + 128x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{6} q^{3} + \beta_{12} q^{5} - \beta_{5} q^{7} - q^{9}+O(q^{10}) $ q + b6 * q^3 + b12 * q^5 - b5 * q^7 - q^9	$ q + \beta_{6} q^{3} + \beta_{12} q^{5} - \beta_{5} q^{7} - q^{9} + \beta_{10} q^{11} + \beta_{6} q^{13} + \beta_{3} q^{15} + ( - \beta_{2} + 1) q^{17} + (\beta_{6} - \beta_1) q^{19} - \beta_{9} q^{21} + ( - \beta_{8} - \beta_{5} - \beta_{3} + 1) q^{23} + (\beta_{11} + \beta_{4} - \beta_{3} + \cdots - 2) q^{25}+ \cdots - \beta_{10} q^{99}+O(q^{100}) $ q + b6 * q^3 + b12 * q^5 - b5 * q^7 - q^9 + b10 * q^11 + b6 * q^13 + b3 * q^15 + (-b2 + 1) * q^17 + (b6 - b1) * q^19 - b9 * q^21 + (-b8 - b5 - b3 + 1) * q^23 + (b11 + b4 - b3 + b2 - 2) * q^25 - b6 * q^27 + (-b14 - b12 + b1) * q^29 + b11 * q^31 + b4 * q^33 + (-b14 - b12 - b7 - b6) * q^35 + (-b14 + b10 - b6) * q^37 - q^39 + (b13 - b8 - 2) * q^41 + (b15 - b14 + b10 - 2b6) q^43 - b12 * q^45 + (b5 + b3 - 2) * q^47 + (b11 - b8 - b3 + b2 + 3) * q^49 + (b15 + b6) * q^51 + (-b15 + b12 - b10 + b9 - b7 + b1) * q^53 + (b11 + b8 - b5 - b4 + 2b3 - b2 - 2) q^55 + (-b13 - 1) * q^57 + (-2b12 + b10 - b9 - b7) q^59 + (-b15 - b12 + b10 - b7 + b6) * q^61 + b5 * q^63 + b3 * q^65 + (-b9 + b7 + 3b6 + b1) q^67 + (b14 + b12 - b9 + b6) * q^69 + (b11 + 2b4 - b3 + 2b2) * q^71 + (-2b13 + b8 + b4 + 2b3 - 3) * q^73 + (-b15 + b12 - b10 - b7 - 2b6) q^75 + (-b15 + b14 - b10 - b9 - b7 - 2b6 - 2b1) * q^77 + (b13 - b11 - b8 - b4 - 2b3 - b2 + 1) q^79 + q^81 + (b14 + 2b12 - 2b9 + 3b6) q^83 + (-b14 + 2b12 - b10 + b6) q^85 + (b13 - b8 - b3) * q^87 + (b13 - b11 - b5 - b4 - 2b2 - 3) q^89 - b9 * q^91 - b7 * q^93 + (-2b13 - b8 + b5 + 2b4 - b3 + 2b2 + 1) q^95 + (b8 - 2b5 - b4 + 3) q^97 - b10 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7} - 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^7 - 16 * q^9	$ 16 q - 4 q^{7} - 16 q^{9} + 4 q^{15} + 16 q^{17} + 8 q^{23} - 32 q^{25} + 4 q^{31} - 16 q^{39} - 36 q^{41} - 24 q^{47} + 48 q^{49} - 24 q^{55} - 12 q^{57} + 4 q^{63} + 4 q^{65} - 32 q^{73} + 16 q^{81} - 8 q^{87} - 60 q^{89} + 24 q^{95} + 40 q^{97}+O(q^{100}) $ 16 * q - 4 * q^7 - 16 * q^9 + 4 * q^15 + 16 * q^17 + 8 * q^23 - 32 * q^25 + 4 * q^31 - 16 * q^39 - 36 * q^41 - 24 * q^47 + 48 * q^49 - 24 * q^55 - 12 * q^57 + 4 * q^63 + 4 * q^65 - 32 * q^73 + 16 * q^81 - 8 * q^87 - 60 * q^89 + 24 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 $ x^16 - 2*x^15 + 2*x^14 - 4*x^13 + 9*x^12 - 10*x^11 + 2*x^10 - 8*x^9 + 28*x^8 - 16*x^7 + 8*x^6 - 80*x^5 + 144*x^4 - 128*x^3 + 128*x^2 - 256*x + 256 :

$\beta_{1}$	$=$	$ ( - 6 \nu^{15} + 23 \nu^{14} - 18 \nu^{13} + 50 \nu^{12} - 50 \nu^{11} + 7 \nu^{10} - 26 \nu^{9} + \cdots + 384 ) / 448 $ (-6v^15 + 23v^14 - 18v^13 + 50v^12 - 50v^11 + 7v^10 - 26v^9 + 42v^8 - 112v^7 + 124v^6 - 224v^5 + 1096v^4 - 624v^3 + 400v^2 - 1856*v + 384) / 448
$\beta_{2}$	$=$	$ ( \nu^{15} - 8 \nu^{14} + 2 \nu^{13} + 25 \nu^{11} - 16 \nu^{10} - 14 \nu^{9} - 4 \nu^{8} + 68 \nu^{7} + \cdots - 448 ) / 64 $ (v^15 - 8v^14 + 2v^13 + 25v^11 - 16v^10 - 14v^9 - 4v^8 + 68v^7 - 24v^6 - 40v^5 - 192v^4 + 400v^3 + 32v^2 + 320*v - 448) / 64
$\beta_{3}$	$=$	$ ( - 3 \nu^{15} + 10 \nu^{14} - 2 \nu^{13} + 12 \nu^{12} - 51 \nu^{11} + 18 \nu^{10} + 30 \nu^{9} + \cdots + 1280 ) / 128 $ (-3v^15 + 10v^14 - 2v^13 + 12v^12 - 51v^11 + 18v^10 + 30v^9 + 56v^8 - 124v^7 - 64v^6 + 24v^5 + 528v^4 - 528v^3 - 832v + 1280) / 128
$\beta_{4}$	$=$	$ ( - 5 \nu^{15} + 22 \nu^{14} - 6 \nu^{13} + 20 \nu^{12} - 117 \nu^{11} + 46 \nu^{10} + 58 \nu^{9} + \cdots + 2816 ) / 128 $ (-5v^15 + 22v^14 - 6v^13 + 20v^12 - 117v^11 + 46v^10 + 58v^9 + 136v^8 - 260v^7 - 64v^6 + 104v^5 + 1072v^4 - 1264v^3 - 256v^2 - 1856*v + 2816) / 128
$\beta_{5}$	$=$	$ ( 3 \nu^{15} - 8 \nu^{14} + 2 \nu^{13} - 8 \nu^{12} + 43 \nu^{11} - 16 \nu^{10} - 30 \nu^{9} + \cdots - 1024 ) / 64 $ (3v^15 - 8v^14 + 2v^13 - 8v^12 + 43v^11 - 16v^10 - 30v^9 - 52v^8 + 108v^7 + 24v^6 - 56v^5 - 416v^4 + 496v^3 + 32v^2 + 512*v - 1024) / 64
$\beta_{6}$	$=$	$ ( - 25 \nu^{15} + 13 \nu^{14} + 16 \nu^{13} + 94 \nu^{12} - 101 \nu^{11} - 91 \nu^{10} - 8 \nu^{9} + \cdots + 256 ) / 448 $ (-25v^15 + 13v^14 + 16v^13 + 94v^12 - 101v^11 - 91v^10 - 8v^9 + 322v^8 - 140v^7 - 356v^6 - 616v^5 + 1720v^4 + 256v^3 + 528v^2 - 2880*v + 256) / 448
$\beta_{7}$	$=$	$ ( 39 \nu^{15} - 34 \nu^{14} - 2 \nu^{13} - 164 \nu^{12} + 199 \nu^{11} + 70 \nu^{10} + 78 \nu^{9} + \cdots - 1152 ) / 448 $ (39v^15 - 34v^14 - 2v^13 - 164v^12 + 199v^11 + 70v^10 + 78v^9 - 448v^8 + 252v^7 + 384v^6 + 840v^5 - 2672v^4 + 80v^3 - 1984v^2 + 6016*v - 1152) / 448
$\beta_{8}$	$=$	$ ( 13 \nu^{15} - 38 \nu^{14} + 6 \nu^{13} - 20 \nu^{12} + 157 \nu^{11} - 62 \nu^{10} - 122 \nu^{9} + \cdots - 3456 ) / 128 $ (13v^15 - 38v^14 + 6v^13 - 20v^12 + 157v^11 - 62v^10 - 122v^9 - 168v^8 + 452v^7 + 64v^6 - 168v^5 - 1584v^4 + 1776v^3 + 256v^2 + 2112*v - 3456) / 128
$\beta_{9}$	$=$	$ ( 23 \nu^{15} - 10 \nu^{14} - 8 \nu^{13} - 96 \nu^{12} + 103 \nu^{11} + 70 \nu^{10} + 32 \nu^{9} + \cdots - 1472 ) / 224 $ (23v^15 - 10v^14 - 8v^13 - 96v^12 + 103v^11 + 70v^10 + 32v^9 - 364v^8 + 140v^7 + 416v^6 + 616v^5 - 1728v^4 - 16v^3 - 544v^2 + 3232*v - 1472) / 224
$\beta_{10}$	$=$	$ ( 107 \nu^{15} - 38 \nu^{14} - 134 \nu^{13} - 404 \nu^{12} + 411 \nu^{11} + 434 \nu^{10} - 38 \nu^{9} + \cdots - 1024 ) / 896 $ (107v^15 - 38v^14 - 134v^13 - 404v^12 + 411v^11 + 434v^10 - 38v^9 - 1568v^8 + 588v^7 + 2096v^6 + 2184v^5 - 7216v^4 - 2032v^3 - 320v^2 + 12864*v - 1024) / 896
$\beta_{11}$	$=$	$ ( - 4 \nu^{15} + 13 \nu^{14} - 4 \nu^{13} + 10 \nu^{12} - 56 \nu^{11} + 29 \nu^{10} + 28 \nu^{9} + \cdots + 1280 ) / 32 $ (-4v^15 + 13v^14 - 4v^13 + 10v^12 - 56v^11 + 29v^10 + 28v^9 + 58v^8 - 152v^7 - 4v^6 + 32v^5 + 536v^4 - 656v^3 + 16v^2 - 736*v + 1280) / 32
$\beta_{12}$	$=$	$ ( - 117 \nu^{15} + 74 \nu^{14} + 34 \nu^{13} + 492 \nu^{12} - 485 \nu^{11} - 350 \nu^{10} - 94 \nu^{9} + \cdots + 2560 ) / 896 $ (-117v^15 + 74v^14 + 34v^13 + 492v^12 - 485v^11 - 350v^10 - 94v^9 + 1568v^8 - 532v^7 - 1712v^6 - 3192v^5 + 8016v^4 + 656v^3 + 3264v^2 - 14912*v + 2560) / 896
$\beta_{13}$	$=$	$ ( 7 \nu^{15} - 23 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} + 107 \nu^{11} - 39 \nu^{10} - 78 \nu^{9} + \cdots - 2464 ) / 32 $ (7v^15 - 23v^14 + 2v^13 - 14v^12 + 107v^11 - 39v^10 - 78v^9 - 122v^8 + 308v^7 + 60v^6 - 152v^5 - 1016v^4 + 1184v^3 + 304v^2 + 1312*v - 2464) / 32
$\beta_{14}$	$=$	$ ( 199 \nu^{15} - 64 \nu^{14} - 194 \nu^{13} - 816 \nu^{12} + 767 \nu^{11} + 952 \nu^{10} + 174 \nu^{9} + \cdots - 1536 ) / 896 $ (199v^15 - 64v^14 - 194v^13 - 816v^12 + 767v^11 + 952v^10 + 174v^9 - 3052v^8 + 588v^7 + 4264v^6 + 4872v^5 - 13792v^4 - 5456v^3 - 1824v^2 + 24896*v - 1536) / 896
$\beta_{15}$	$=$	$ ( - 104 \nu^{15} + 65 \nu^{14} + 66 \nu^{13} + 470 \nu^{12} - 428 \nu^{11} - 399 \nu^{10} - 166 \nu^{9} + \cdots + 384 ) / 448 $ (-104v^15 + 65v^14 + 66v^13 + 470v^12 - 428v^11 - 399v^10 - 166v^9 + 1526v^8 - 392v^7 - 1612v^6 - 3024v^5 + 7704v^4 + 1616v^3 + 3312v^2 - 15744*v + 384) / 448

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

$\nu$	$=$	$ ( -\beta_{14} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 1 ) / 8 $ (-b14 + b11 + b10 + b9 + b8 + b7 + b6 - b5 - b4 + 1) / 8
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{12} + \beta_{10} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 ) / 4 $ (b15 - b12 + b10 - b5 - b4 + b3 - b1) / 4
$\nu^{3}$	$=$	$ ( - \beta_{14} + \beta_{11} + \beta_{10} + 3 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} + \cdots + 3 ) / 8 $ (-b14 + b11 + b10 + 3b9 - b8 - b7 + b6 + 3b5 + b4 + 4*b2 + 3) / 8
$\nu^{4}$	$=$	$ ( - \beta_{14} + \beta_{13} - 3 \beta_{12} + \beta_{11} - \beta_{8} + \beta_{7} + 3 \beta_{6} - \beta_{5} + \cdots - 3 ) / 4 $ (-b14 + b13 - 3b12 + b11 - b8 + b7 + 3b6 - b5 + b3 + b2 + 2*b1 - 3) / 4
$\nu^{5}$	$=$	$ ( - 3 \beta_{14} - 4 \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - 5 \beta_{7} + \cdots - 1 ) / 8 $ (-3b14 - 4b12 - b11 - b10 - b9 - b8 - 5b7 - 17b6 - 7b5 - 3b4 - 4b3 + 4b1 - 1) / 8
$\nu^{6}$	$=$	$ ( \beta_{15} + 2 \beta_{14} - \beta_{12} - \beta_{10} + 2 \beta_{9} + 8 \beta_{6} - \beta_{5} + 3 \beta_{4} + \cdots + 14 ) / 4 $ (b15 + 2b14 - b12 - b10 + 2b9 + 8b6 - b5 + 3b4 - 7b3 + 2b2 + b1 + 14) / 4
$\nu^{7}$	$=$	$ ( - 7 \beta_{14} + 8 \beta_{13} + 3 \beta_{11} - \beta_{10} + 9 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + \cdots + 29 ) / 8 $ (-7b14 + 8b13 + 3b11 - b10 + 9b9 - 7b8 + 5b7 - 9b6 - 15b5 + 7b4 - 8b3 - 4*b2 + 29) / 8
$\nu^{8}$	$=$	$ ( 8 \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + \cdots - 11 ) / 4 $ (8b15 - b14 - b13 - b12 + b11 + 6b10 - 2b9 - b8 + 5b7 - 21b6 - 5b5 - 2b4 - 13b3 + 3b2 - 6b1 - 11) / 4
$\nu^{9}$	$=$	$ ( 8 \beta_{15} + 15 \beta_{14} + 28 \beta_{12} - 11 \beta_{11} - 3 \beta_{10} + 21 \beta_{9} - 11 \beta_{8} + \cdots + 5 ) / 8 $ (8b15 + 15b14 + 28b12 - 11b11 - 3b10 + 21b9 - 11b8 + 9b7 + 5b6 - 5b5 + 7b4 + 12b3 + 16b2 - 12b1 + 5) / 8
$\nu^{10}$	$=$	$ ( - 5 \beta_{15} + 8 \beta_{13} + 5 \beta_{12} + 14 \beta_{11} - 21 \beta_{10} + 8 \beta_{9} - 6 \beta_{8} + \cdots - 44 ) / 4 $ (-5b15 + 8b13 + 5b12 + 14b11 - 21b10 + 8b9 - 6b8 + 2b7 - 26b6 + 7b5 + 7b4 + 3b3 + 14b2 + 3b1 - 44) / 4
$\nu^{11}$	$=$	$ ( 16 \beta_{15} + 11 \beta_{14} - 8 \beta_{12} - 7 \beta_{11} - 27 \beta_{10} - 45 \beta_{9} - 21 \beta_{8} + \cdots + 39 ) / 8 $ (16b15 + 11b14 - 8b12 - 7b11 - 27b10 - 45b9 - 21b8 + 15b7 - 139b6 - 13b5 - 35b4 - 4b2 + 32*b1 + 39) / 8
$\nu^{12}$	$=$	$ ( - 8 \beta_{15} + 11 \beta_{14} - 11 \beta_{13} + 45 \beta_{12} - 3 \beta_{11} - 28 \beta_{10} + \cdots + 49 ) / 4 $ (-8b15 + 11b14 - 11b13 + 45b12 - 3b11 - 28b10 + 20b9 + 31b8 - 7b7 - 53b6 - 9b5 - 8b4 + 9b3 + b2 + 22b1 + 49) / 4
$\nu^{13}$	$=$	$ ( 32 \beta_{15} + 21 \beta_{14} + 16 \beta_{13} - 68 \beta_{12} + 39 \beta_{11} - 89 \beta_{10} + \cdots + 295 ) / 8 $ (32b15 + 21b14 + 16b13 - 68b12 + 39b11 - 89b10 + 39b9 + 39b8 + 35b7 + 55b6 - 31b5 - 11b4 + 28b3 - 48b2 - 28*b1 + 295) / 8
$\nu^{14}$	$=$	$ ( 41 \beta_{15} - 42 \beta_{14} - 8 \beta_{13} + 23 \beta_{12} + 12 \beta_{11} + 51 \beta_{10} + \cdots + 194 ) / 4 $ (41b15 - 42b14 - 8b13 + 23b12 + 12b11 + 51b10 + 38b9 - 4b8 + 28b7 - 196b6 + 3b5 - 49b4 - 15b3 - 30b2 - 15*b1 + 194) / 4
$\nu^{15}$	$=$	$ ( 192 \beta_{15} + \beta_{14} - 72 \beta_{13} - 48 \beta_{12} + 11 \beta_{11} + 199 \beta_{10} + \cdots - 315 ) / 8 $ (192b15 + b14 - 72b13 - 48b12 + 11b11 + 199b10 + 113b9 + 161b8 + 157b7 + 47b6 + 41b5 - 49b4 + 232b3 + 76b2 - 64b1 - 315) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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} $

625.1

−0.879244	−	1.10767i
0.791485	−	1.17199i
0.802079	+	1.16476i
1.40722	+	0.140463i
−0.414573	+	1.35208i
−0.654116	−	1.25385i
1.27276	−	0.616518i
−1.32561	+	0.492712i
−1.32561	−	0.492712i
1.27276	+	0.616518i
−0.654116	+	1.25385i
−0.414573	−	1.35208i
1.40722	−	0.140463i
0.802079	−	1.16476i
0.791485	+	1.17199i
−0.879244	+	1.10767i

−

1.00000i

−

3.31390i

−4.17825

−1.00000

625.2

−

1.00000i

−

3.05343i

0.397397

−1.00000

625.3

−

1.00000i

−

1.47174i

2.93973

−1.00000

625.4

−

1.00000i

−

0.218531i

4.47783

−1.00000

625.5

−

1.00000i

0.550135i

−4.37841

−1.00000

625.6

−

1.00000i

1.87654i

−0.584696

−1.00000

625.7

−

1.00000i

3.29521i

−2.97802

−1.00000

625.8

−

1.00000i

4.33571i

2.30442

−1.00000

625.9

1.00000i

−

4.33571i

2.30442

−1.00000

625.10

1.00000i

−

3.29521i

−2.97802

−1.00000

625.11

1.00000i

−

1.87654i

−0.584696

−1.00000

625.12

1.00000i

−

0.550135i

−4.37841

−1.00000

625.13

1.00000i

0.218531i

4.47783

−1.00000

625.14

1.00000i

1.47174i

2.93973

−1.00000

625.15

1.00000i

3.05343i

0.397397

−1.00000

625.16

1.00000i

3.31390i

−4.17825

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1248.2.g.b		16
3.b	odd	2	1		3744.2.g.e		16
4.b	odd	2	1		312.2.g.b	✓	16
8.b	even	2	1	inner	1248.2.g.b		16
8.d	odd	2	1		312.2.g.b	✓	16
12.b	even	2	1		936.2.g.e		16
16.e	even	4	1		9984.2.a.bs		8
16.e	even	4	1		9984.2.a.bv		8
16.f	odd	4	1		9984.2.a.bt		8
16.f	odd	4	1		9984.2.a.bu		8
24.f	even	2	1		936.2.g.e		16
24.h	odd	2	1		3744.2.g.e		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.g.b	✓	16	4.b	odd	2	1
312.2.g.b	✓	16	8.d	odd	2	1
936.2.g.e		16	12.b	even	2	1
936.2.g.e		16	24.f	even	2	1
1248.2.g.b		16	1.a	even	1	1	trivial
1248.2.g.b		16	8.b	even	2	1	inner
3744.2.g.e		16	3.b	odd	2	1
3744.2.g.e		16	24.h	odd	2	1
9984.2.a.bs		8	16.e	even	4	1
9984.2.a.bt		8	16.f	odd	4	1
9984.2.a.bu		8	16.f	odd	4	1
9984.2.a.bv		8	16.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{16} + 56 T_{5}^{14} + 1220 T_{5}^{12} + 13152 T_{5}^{10} + 73152 T_{5}^{8} + 197888 T_{5}^{6} + \cdots + 2304 $ T5^16 + 56*T5^14 + 1220*T5^12 + 13152*T5^10 + 73152*T5^8 + 197888*T5^6 + 221248*T5^4 + 58368*T5^2 + 2304 acting on $S_{2}^{\mathrm{new}}(1248, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$5$	$ T^{16} + 56 T^{14} + \cdots + 2304 $ T^16 + 56T^14 + 1220T^12 + 13152T^10 + 73152T^8 + 197888T^6 + 221248T^4 + 58368*T^2 + 2304
$7$	$ (T^{8} + 2 T^{7} + \cdots + 384)^{2} $ (T^8 + 2T^7 - 38T^6 - 60T^5 + 444T^4 + 416T^3 - 1696T^2 - 384*T + 384)^2
$11$	$ T^{16} + 128 T^{14} + \cdots + 1679616 $ T^16 + 128T^14 + 6560T^12 + 171776T^10 + 2420448T^8 + 17463808T^6 + 52308480T^4 + 20772864*T^2 + 1679616
$13$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$17$	$ (T^{8} - 8 T^{7} + \cdots - 64256)^{2} $ (T^8 - 8T^7 - 68T^6 + 720T^5 - 14720T^3 + 37056T^2 + 4864T - 64256)^2
$19$	$ T^{16} + 224 T^{14} + \cdots + 22429696 $ T^16 + 224T^14 + 20204T^12 + 943712T^10 + 24400016T^8 + 343948032T^6 + 2373474304T^4 + 5707161600*T^2 + 22429696
$23$	$ (T^{8} - 4 T^{7} + \cdots - 166912)^{2} $ (T^8 - 4T^7 - 120T^6 + 624T^5 + 3536T^4 - 24448T^3 + 3584T^2 + 153600*T - 166912)^2
$29$	$ T^{16} + \cdots + 4671995904 $ T^16 + 248T^14 + 23952T^12 + 1138304T^10 + 28005376T^8 + 355358720T^6 + 2247888896T^4 + 6215335936*T^2 + 4671995904
$31$	$ (T^{8} - 2 T^{7} + \cdots + 110464)^{2} $ (T^8 - 2T^7 - 110T^6 + 140T^5 + 3804T^4 - 2496T^3 - 45408T^2 + 1920*T + 110464)^2
$37$	$ T^{16} + \cdots + 157351936 $ T^16 + 256T^14 + 24512T^12 + 1103872T^10 + 24667648T^8 + 262275072T^6 + 1151582208T^4 + 1165492224*T^2 + 157351936
$41$	$ (T^{8} + 18 T^{7} + \cdots - 139792)^{2} $ (T^8 + 18T^7 + 14T^6 - 1116T^5 - 4008T^4 + 15400T^3 + 77912T^2 + 14928*T - 139792)^2
$43$	$ T^{16} + \cdots + 561468473344 $ T^16 + 328T^14 + 45200T^12 + 3392384T^10 + 149857280T^8 + 3908044800T^6 + 56422666240T^4 + 373267726336*T^2 + 561468473344
$47$	$ (T^{8} + 12 T^{7} + \cdots + 48)^{2} $ (T^8 + 12T^7 - 368T^5 - 480T^4 + 2960T^3 + 2016T^2 - 1600T + 48)^2
$53$	$ T^{16} + \cdots + 462422016 $ T^16 + 496T^14 + 95136T^12 + 8962304T^10 + 425390336T^8 + 8906956800T^6 + 45540622336T^4 + 47784656896*T^2 + 462422016
$59$	$ T^{16} + \cdots + 276642337024 $ T^16 + 544T^14 + 115808T^12 + 12457856T^10 + 716910560T^8 + 21081421824T^6 + 264285707776T^4 + 855183167488*T^2 + 276642337024
$61$	$ T^{16} + \cdots + 329127100416 $ T^16 + 520T^14 + 94736T^12 + 7458944T^10 + 278176768T^8 + 5171939328T^6 + 48761671680T^4 + 215901306880*T^2 + 329127100416
$67$	$ T^{16} + \cdots + 409763856384 $ T^16 + 528T^14 + 107500T^12 + 10552416T^10 + 509533712T^8 + 11087418624T^6 + 94434490368T^4 + 333682647040*T^2 + 409763856384
$71$	$ (T^{8} - 408 T^{6} + \cdots + 21431088)^{2} $ (T^8 - 408T^6 - 600T^5 + 49664T^4 + 112384T^3 - 1886592T^2 - 3108384T + 21431088)^2
$73$	$ (T^{8} + 16 T^{7} + \cdots + 3326208)^{2} $ (T^8 + 16T^7 - 320T^6 - 5312T^5 + 22240T^4 + 392960T^3 - 618496T^2 - 7414784*T + 3326208)^2
$79$	$ (T^{8} - 336 T^{6} + \cdots + 2166016)^{2} $ (T^8 - 336T^6 - 320T^5 + 35360T^4 + 53760T^3 - 1101568T^2 - 1229824T + 2166016)^2
$83$	$ T^{16} + \cdots + 6293836597504 $ T^16 + 560T^14 + 127424T^12 + 15118656T^10 + 996178016T^8 + 35690143488T^6 + 621437020160T^4 + 3852663542784*T^2 + 6293836597504
$89$	$ (T^{8} + 30 T^{7} + \cdots + 37581936)^{2} $ (T^8 + 30T^7 + 86T^6 - 5916T^5 - 69736T^4 - 43112T^3 + 3470840T^2 + 20801808*T + 37581936)^2
$97$	$ (T^{8} - 20 T^{7} + \cdots - 637696)^{2} $ (T^8 - 20T^7 - 128T^6 + 4272T^5 - 15584T^4 - 91328T^3 + 564736T^2 - 549632*T - 637696)^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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + \beta_{12} q^{5} - \beta_{5} q^{7} - q^{9}+O(q^{10}) \) q + b6 * q^3 + b12 * q^5 - b5 * q^7 - q^9	\( q + \beta_{6} q^{3} + \beta_{12} q^{5} - \beta_{5} q^{7} - q^{9} + \beta_{10} q^{11} + \beta_{6} q^{13} + \beta_{3} q^{15} + ( - \beta_{2} + 1) q^{17} + (\beta_{6} - \beta_1) q^{19} - \beta_{9} q^{21} + ( - \beta_{8} - \beta_{5} - \beta_{3} + 1) q^{23} + (\beta_{11} + \beta_{4} - \beta_{3} + \cdots - 2) q^{25}+ \cdots - \beta_{10} q^{99}+O(q^{100}) \) q + b6 * q^3 + b12 * q^5 - b5 * q^7 - q^9 + b10 * q^11 + b6 * q^13 + b3 * q^15 + (-b2 + 1) * q^17 + (b6 - b1) * q^19 - b9 * q^21 + (-b8 - b5 - b3 + 1) * q^23 + (b11 + b4 - b3 + b2 - 2) * q^25 - b6 * q^27 + (-b14 - b12 + b1) * q^29 + b11 * q^31 + b4 * q^33 + (-b14 - b12 - b7 - b6) * q^35 + (-b14 + b10 - b6) * q^37 - q^39 + (b13 - b8 - 2) * q^41 + (b15 - b14 + b10 - 2b6) q^43 - b12 * q^45 + (b5 + b3 - 2) * q^47 + (b11 - b8 - b3 + b2 + 3) * q^49 + (b15 + b6) * q^51 + (-b15 + b12 - b10 + b9 - b7 + b1) * q^53 + (b11 + b8 - b5 - b4 + 2b3 - b2 - 2) q^55 + (-b13 - 1) * q^57 + (-2b12 + b10 - b9 - b7) q^59 + (-b15 - b12 + b10 - b7 + b6) * q^61 + b5 * q^63 + b3 * q^65 + (-b9 + b7 + 3b6 + b1) q^67 + (b14 + b12 - b9 + b6) * q^69 + (b11 + 2b4 - b3 + 2b2) * q^71 + (-2b13 + b8 + b4 + 2b3 - 3) * q^73 + (-b15 + b12 - b10 - b7 - 2b6) q^75 + (-b15 + b14 - b10 - b9 - b7 - 2b6 - 2b1) * q^77 + (b13 - b11 - b8 - b4 - 2b3 - b2 + 1) q^79 + q^81 + (b14 + 2b12 - 2b9 + 3b6) q^83 + (-b14 + 2b12 - b10 + b6) q^85 + (b13 - b8 - b3) * q^87 + (b13 - b11 - b5 - b4 - 2b2 - 3) q^89 - b9 * q^91 - b7 * q^93 + (-2b13 - b8 + b5 + 2b4 - b3 + 2b2 + 1) q^95 + (b8 - 2b5 - b4 + 3) q^97 - b10 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} - 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^7 - 16 * q^9	\( 16 q - 4 q^{7} - 16 q^{9} + 4 q^{15} + 16 q^{17} + 8 q^{23} - 32 q^{25} + 4 q^{31} - 16 q^{39} - 36 q^{41} - 24 q^{47} + 48 q^{49} - 24 q^{55} - 12 q^{57} + 4 q^{63} + 4 q^{65} - 32 q^{73} + 16 q^{81} - 8 q^{87} - 60 q^{89} + 24 q^{95} + 40 q^{97}+O(q^{100}) \) 16 * q - 4 * q^7 - 16 * q^9 + 4 * q^15 + 16 * q^17 + 8 * q^23 - 32 * q^25 + 4 * q^31 - 16 * q^39 - 36 * q^41 - 24 * q^47 + 48 * q^49 - 24 * q^55 - 12 * q^57 + 4 * q^63 + 4 * q^65 - 32 * q^73 + 16 * q^81 - 8 * q^87 - 60 * q^89 + 24 * q^95 + 40 * q^97

Introduction

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Newspace parameters

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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	1248.2.g.b

Level	$1248$
Weight	$2$
Character orbit	1248.g
Analytic conductor	$9.965$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(9.96533017226\)
Analytic rank:	\(0\)
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:	\( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 \) x^16 - 2x^15 + 2x^14 - 4x^13 + 9x^12 - 10x^11 + 2x^10 - 8x^9 + 28x^8 - 16x^7 + 8x^6 - 80x^5 + 144x^4 - 128x^3 + 128x^2 - 256*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{20} \)
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 6 \nu^{15} + 23 \nu^{14} - 18 \nu^{13} + 50 \nu^{12} - 50 \nu^{11} + 7 \nu^{10} - 26 \nu^{9} + \cdots + 384 ) / 448 \) (-6v^15 + 23v^14 - 18v^13 + 50v^12 - 50v^11 + 7v^10 - 26v^9 + 42v^8 - 112v^7 + 124v^6 - 224v^5 + 1096v^4 - 624v^3 + 400v^2 - 1856*v + 384) / 448
\(\beta_{2}\)	\(=\)	\( ( \nu^{15} - 8 \nu^{14} + 2 \nu^{13} + 25 \nu^{11} - 16 \nu^{10} - 14 \nu^{9} - 4 \nu^{8} + 68 \nu^{7} + \cdots - 448 ) / 64 \) (v^15 - 8v^14 + 2v^13 + 25v^11 - 16v^10 - 14v^9 - 4v^8 + 68v^7 - 24v^6 - 40v^5 - 192v^4 + 400v^3 + 32v^2 + 320*v - 448) / 64
\(\beta_{3}\)	\(=\)	\( ( - 3 \nu^{15} + 10 \nu^{14} - 2 \nu^{13} + 12 \nu^{12} - 51 \nu^{11} + 18 \nu^{10} + 30 \nu^{9} + \cdots + 1280 ) / 128 \) (-3v^15 + 10v^14 - 2v^13 + 12v^12 - 51v^11 + 18v^10 + 30v^9 + 56v^8 - 124v^7 - 64v^6 + 24v^5 + 528v^4 - 528v^3 - 832v + 1280) / 128
\(\beta_{4}\)	\(=\)	\( ( - 5 \nu^{15} + 22 \nu^{14} - 6 \nu^{13} + 20 \nu^{12} - 117 \nu^{11} + 46 \nu^{10} + 58 \nu^{9} + \cdots + 2816 ) / 128 \) (-5v^15 + 22v^14 - 6v^13 + 20v^12 - 117v^11 + 46v^10 + 58v^9 + 136v^8 - 260v^7 - 64v^6 + 104v^5 + 1072v^4 - 1264v^3 - 256v^2 - 1856*v + 2816) / 128
\(\beta_{5}\)	\(=\)	\( ( 3 \nu^{15} - 8 \nu^{14} + 2 \nu^{13} - 8 \nu^{12} + 43 \nu^{11} - 16 \nu^{10} - 30 \nu^{9} + \cdots - 1024 ) / 64 \) (3v^15 - 8v^14 + 2v^13 - 8v^12 + 43v^11 - 16v^10 - 30v^9 - 52v^8 + 108v^7 + 24v^6 - 56v^5 - 416v^4 + 496v^3 + 32v^2 + 512*v - 1024) / 64
\(\beta_{6}\)	\(=\)	\( ( - 25 \nu^{15} + 13 \nu^{14} + 16 \nu^{13} + 94 \nu^{12} - 101 \nu^{11} - 91 \nu^{10} - 8 \nu^{9} + \cdots + 256 ) / 448 \) (-25v^15 + 13v^14 + 16v^13 + 94v^12 - 101v^11 - 91v^10 - 8v^9 + 322v^8 - 140v^7 - 356v^6 - 616v^5 + 1720v^4 + 256v^3 + 528v^2 - 2880*v + 256) / 448
\(\beta_{7}\)	\(=\)	\( ( 39 \nu^{15} - 34 \nu^{14} - 2 \nu^{13} - 164 \nu^{12} + 199 \nu^{11} + 70 \nu^{10} + 78 \nu^{9} + \cdots - 1152 ) / 448 \) (39v^15 - 34v^14 - 2v^13 - 164v^12 + 199v^11 + 70v^10 + 78v^9 - 448v^8 + 252v^7 + 384v^6 + 840v^5 - 2672v^4 + 80v^3 - 1984v^2 + 6016*v - 1152) / 448
\(\beta_{8}\)	\(=\)	\( ( 13 \nu^{15} - 38 \nu^{14} + 6 \nu^{13} - 20 \nu^{12} + 157 \nu^{11} - 62 \nu^{10} - 122 \nu^{9} + \cdots - 3456 ) / 128 \) (13v^15 - 38v^14 + 6v^13 - 20v^12 + 157v^11 - 62v^10 - 122v^9 - 168v^8 + 452v^7 + 64v^6 - 168v^5 - 1584v^4 + 1776v^3 + 256v^2 + 2112*v - 3456) / 128
\(\beta_{9}\)	\(=\)	\( ( 23 \nu^{15} - 10 \nu^{14} - 8 \nu^{13} - 96 \nu^{12} + 103 \nu^{11} + 70 \nu^{10} + 32 \nu^{9} + \cdots - 1472 ) / 224 \) (23v^15 - 10v^14 - 8v^13 - 96v^12 + 103v^11 + 70v^10 + 32v^9 - 364v^8 + 140v^7 + 416v^6 + 616v^5 - 1728v^4 - 16v^3 - 544v^2 + 3232*v - 1472) / 224
\(\beta_{10}\)	\(=\)	\( ( 107 \nu^{15} - 38 \nu^{14} - 134 \nu^{13} - 404 \nu^{12} + 411 \nu^{11} + 434 \nu^{10} - 38 \nu^{9} + \cdots - 1024 ) / 896 \) (107v^15 - 38v^14 - 134v^13 - 404v^12 + 411v^11 + 434v^10 - 38v^9 - 1568v^8 + 588v^7 + 2096v^6 + 2184v^5 - 7216v^4 - 2032v^3 - 320v^2 + 12864*v - 1024) / 896
\(\beta_{11}\)	\(=\)	\( ( - 4 \nu^{15} + 13 \nu^{14} - 4 \nu^{13} + 10 \nu^{12} - 56 \nu^{11} + 29 \nu^{10} + 28 \nu^{9} + \cdots + 1280 ) / 32 \) (-4v^15 + 13v^14 - 4v^13 + 10v^12 - 56v^11 + 29v^10 + 28v^9 + 58v^8 - 152v^7 - 4v^6 + 32v^5 + 536v^4 - 656v^3 + 16v^2 - 736*v + 1280) / 32
\(\beta_{12}\)	\(=\)	\( ( - 117 \nu^{15} + 74 \nu^{14} + 34 \nu^{13} + 492 \nu^{12} - 485 \nu^{11} - 350 \nu^{10} - 94 \nu^{9} + \cdots + 2560 ) / 896 \) (-117v^15 + 74v^14 + 34v^13 + 492v^12 - 485v^11 - 350v^10 - 94v^9 + 1568v^8 - 532v^7 - 1712v^6 - 3192v^5 + 8016v^4 + 656v^3 + 3264v^2 - 14912*v + 2560) / 896
\(\beta_{13}\)	\(=\)	\( ( 7 \nu^{15} - 23 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} + 107 \nu^{11} - 39 \nu^{10} - 78 \nu^{9} + \cdots - 2464 ) / 32 \) (7v^15 - 23v^14 + 2v^13 - 14v^12 + 107v^11 - 39v^10 - 78v^9 - 122v^8 + 308v^7 + 60v^6 - 152v^5 - 1016v^4 + 1184v^3 + 304v^2 + 1312*v - 2464) / 32
\(\beta_{14}\)	\(=\)	\( ( 199 \nu^{15} - 64 \nu^{14} - 194 \nu^{13} - 816 \nu^{12} + 767 \nu^{11} + 952 \nu^{10} + 174 \nu^{9} + \cdots - 1536 ) / 896 \) (199v^15 - 64v^14 - 194v^13 - 816v^12 + 767v^11 + 952v^10 + 174v^9 - 3052v^8 + 588v^7 + 4264v^6 + 4872v^5 - 13792v^4 - 5456v^3 - 1824v^2 + 24896*v - 1536) / 896
\(\beta_{15}\)	\(=\)	\( ( - 104 \nu^{15} + 65 \nu^{14} + 66 \nu^{13} + 470 \nu^{12} - 428 \nu^{11} - 399 \nu^{10} - 166 \nu^{9} + \cdots + 384 ) / 448 \) (-104v^15 + 65v^14 + 66v^13 + 470v^12 - 428v^11 - 399v^10 - 166v^9 + 1526v^8 - 392v^7 - 1612v^6 - 3024v^5 + 7704v^4 + 1616v^3 + 3312v^2 - 15744*v + 384) / 448

\(\nu\)	\(=\)	\( ( -\beta_{14} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 1 ) / 8 \) (-b14 + b11 + b10 + b9 + b8 + b7 + b6 - b5 - b4 + 1) / 8
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{12} + \beta_{10} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 ) / 4 \) (b15 - b12 + b10 - b5 - b4 + b3 - b1) / 4
\(\nu^{3}\)	\(=\)	\( ( - \beta_{14} + \beta_{11} + \beta_{10} + 3 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} + \cdots + 3 ) / 8 \) (-b14 + b11 + b10 + 3b9 - b8 - b7 + b6 + 3b5 + b4 + 4*b2 + 3) / 8
\(\nu^{4}\)	\(=\)	\( ( - \beta_{14} + \beta_{13} - 3 \beta_{12} + \beta_{11} - \beta_{8} + \beta_{7} + 3 \beta_{6} - \beta_{5} + \cdots - 3 ) / 4 \) (-b14 + b13 - 3b12 + b11 - b8 + b7 + 3b6 - b5 + b3 + b2 + 2*b1 - 3) / 4
\(\nu^{5}\)	\(=\)	\( ( - 3 \beta_{14} - 4 \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - 5 \beta_{7} + \cdots - 1 ) / 8 \) (-3b14 - 4b12 - b11 - b10 - b9 - b8 - 5b7 - 17b6 - 7b5 - 3b4 - 4b3 + 4b1 - 1) / 8
\(\nu^{6}\)	\(=\)	\( ( \beta_{15} + 2 \beta_{14} - \beta_{12} - \beta_{10} + 2 \beta_{9} + 8 \beta_{6} - \beta_{5} + 3 \beta_{4} + \cdots + 14 ) / 4 \) (b15 + 2b14 - b12 - b10 + 2b9 + 8b6 - b5 + 3b4 - 7b3 + 2b2 + b1 + 14) / 4
\(\nu^{7}\)	\(=\)	\( ( - 7 \beta_{14} + 8 \beta_{13} + 3 \beta_{11} - \beta_{10} + 9 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + \cdots + 29 ) / 8 \) (-7b14 + 8b13 + 3b11 - b10 + 9b9 - 7b8 + 5b7 - 9b6 - 15b5 + 7b4 - 8b3 - 4*b2 + 29) / 8
\(\nu^{8}\)	\(=\)	\( ( 8 \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + \cdots - 11 ) / 4 \) (8b15 - b14 - b13 - b12 + b11 + 6b10 - 2b9 - b8 + 5b7 - 21b6 - 5b5 - 2b4 - 13b3 + 3b2 - 6b1 - 11) / 4
\(\nu^{9}\)	\(=\)	\( ( 8 \beta_{15} + 15 \beta_{14} + 28 \beta_{12} - 11 \beta_{11} - 3 \beta_{10} + 21 \beta_{9} - 11 \beta_{8} + \cdots + 5 ) / 8 \) (8b15 + 15b14 + 28b12 - 11b11 - 3b10 + 21b9 - 11b8 + 9b7 + 5b6 - 5b5 + 7b4 + 12b3 + 16b2 - 12b1 + 5) / 8
\(\nu^{10}\)	\(=\)	\( ( - 5 \beta_{15} + 8 \beta_{13} + 5 \beta_{12} + 14 \beta_{11} - 21 \beta_{10} + 8 \beta_{9} - 6 \beta_{8} + \cdots - 44 ) / 4 \) (-5b15 + 8b13 + 5b12 + 14b11 - 21b10 + 8b9 - 6b8 + 2b7 - 26b6 + 7b5 + 7b4 + 3b3 + 14b2 + 3b1 - 44) / 4
\(\nu^{11}\)	\(=\)	\( ( 16 \beta_{15} + 11 \beta_{14} - 8 \beta_{12} - 7 \beta_{11} - 27 \beta_{10} - 45 \beta_{9} - 21 \beta_{8} + \cdots + 39 ) / 8 \) (16b15 + 11b14 - 8b12 - 7b11 - 27b10 - 45b9 - 21b8 + 15b7 - 139b6 - 13b5 - 35b4 - 4b2 + 32*b1 + 39) / 8
\(\nu^{12}\)	\(=\)	\( ( - 8 \beta_{15} + 11 \beta_{14} - 11 \beta_{13} + 45 \beta_{12} - 3 \beta_{11} - 28 \beta_{10} + \cdots + 49 ) / 4 \) (-8b15 + 11b14 - 11b13 + 45b12 - 3b11 - 28b10 + 20b9 + 31b8 - 7b7 - 53b6 - 9b5 - 8b4 + 9b3 + b2 + 22b1 + 49) / 4
\(\nu^{13}\)	\(=\)	\( ( 32 \beta_{15} + 21 \beta_{14} + 16 \beta_{13} - 68 \beta_{12} + 39 \beta_{11} - 89 \beta_{10} + \cdots + 295 ) / 8 \) (32b15 + 21b14 + 16b13 - 68b12 + 39b11 - 89b10 + 39b9 + 39b8 + 35b7 + 55b6 - 31b5 - 11b4 + 28b3 - 48b2 - 28*b1 + 295) / 8
\(\nu^{14}\)	\(=\)	\( ( 41 \beta_{15} - 42 \beta_{14} - 8 \beta_{13} + 23 \beta_{12} + 12 \beta_{11} + 51 \beta_{10} + \cdots + 194 ) / 4 \) (41b15 - 42b14 - 8b13 + 23b12 + 12b11 + 51b10 + 38b9 - 4b8 + 28b7 - 196b6 + 3b5 - 49b4 - 15b3 - 30b2 - 15*b1 + 194) / 4
\(\nu^{15}\)	\(=\)	\( ( 192 \beta_{15} + \beta_{14} - 72 \beta_{13} - 48 \beta_{12} + 11 \beta_{11} + 199 \beta_{10} + \cdots - 315 ) / 8 \) (192b15 + b14 - 72b13 - 48b12 + 11b11 + 199b10 + 113b9 + 161b8 + 157b7 + 47b6 + 41b5 - 49b4 + 232b3 + 76b2 - 64b1 - 315) / 8

\(n\)	\(703\)	\(769\)	\(833\)	\(1093\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 625.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$5$	\( T^{16} + 56 T^{14} + \cdots + 2304 \) T^16 + 56T^14 + 1220T^12 + 13152T^10 + 73152T^8 + 197888T^6 + 221248T^4 + 58368*T^2 + 2304
$7$	\( (T^{8} + 2 T^{7} + \cdots + 384)^{2} \) (T^8 + 2T^7 - 38T^6 - 60T^5 + 444T^4 + 416T^3 - 1696T^2 - 384*T + 384)^2
$11$	\( T^{16} + 128 T^{14} + \cdots + 1679616 \) T^16 + 128T^14 + 6560T^12 + 171776T^10 + 2420448T^8 + 17463808T^6 + 52308480T^4 + 20772864*T^2 + 1679616
$13$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$17$	\( (T^{8} - 8 T^{7} + \cdots - 64256)^{2} \) (T^8 - 8T^7 - 68T^6 + 720T^5 - 14720T^3 + 37056T^2 + 4864T - 64256)^2
$19$	\( T^{16} + 224 T^{14} + \cdots + 22429696 \) T^16 + 224T^14 + 20204T^12 + 943712T^10 + 24400016T^8 + 343948032T^6 + 2373474304T^4 + 5707161600*T^2 + 22429696
$23$	\( (T^{8} - 4 T^{7} + \cdots - 166912)^{2} \) (T^8 - 4T^7 - 120T^6 + 624T^5 + 3536T^4 - 24448T^3 + 3584T^2 + 153600*T - 166912)^2
$29$	\( T^{16} + \cdots + 4671995904 \) T^16 + 248T^14 + 23952T^12 + 1138304T^10 + 28005376T^8 + 355358720T^6 + 2247888896T^4 + 6215335936*T^2 + 4671995904
$31$	\( (T^{8} - 2 T^{7} + \cdots + 110464)^{2} \) (T^8 - 2T^7 - 110T^6 + 140T^5 + 3804T^4 - 2496T^3 - 45408T^2 + 1920*T + 110464)^2
$37$	\( T^{16} + \cdots + 157351936 \) T^16 + 256T^14 + 24512T^12 + 1103872T^10 + 24667648T^8 + 262275072T^6 + 1151582208T^4 + 1165492224*T^2 + 157351936
$41$	\( (T^{8} + 18 T^{7} + \cdots - 139792)^{2} \) (T^8 + 18T^7 + 14T^6 - 1116T^5 - 4008T^4 + 15400T^3 + 77912T^2 + 14928*T - 139792)^2
$43$	\( T^{16} + \cdots + 561468473344 \) T^16 + 328T^14 + 45200T^12 + 3392384T^10 + 149857280T^8 + 3908044800T^6 + 56422666240T^4 + 373267726336*T^2 + 561468473344
$47$	\( (T^{8} + 12 T^{7} + \cdots + 48)^{2} \) (T^8 + 12T^7 - 368T^5 - 480T^4 + 2960T^3 + 2016T^2 - 1600T + 48)^2
$53$	\( T^{16} + \cdots + 462422016 \) T^16 + 496T^14 + 95136T^12 + 8962304T^10 + 425390336T^8 + 8906956800T^6 + 45540622336T^4 + 47784656896*T^2 + 462422016
$59$	\( T^{16} + \cdots + 276642337024 \) T^16 + 544T^14 + 115808T^12 + 12457856T^10 + 716910560T^8 + 21081421824T^6 + 264285707776T^4 + 855183167488*T^2 + 276642337024
$61$	\( T^{16} + \cdots + 329127100416 \) T^16 + 520T^14 + 94736T^12 + 7458944T^10 + 278176768T^8 + 5171939328T^6 + 48761671680T^4 + 215901306880*T^2 + 329127100416
$67$	\( T^{16} + \cdots + 409763856384 \) T^16 + 528T^14 + 107500T^12 + 10552416T^10 + 509533712T^8 + 11087418624T^6 + 94434490368T^4 + 333682647040*T^2 + 409763856384
$71$	\( (T^{8} - 408 T^{6} + \cdots + 21431088)^{2} \) (T^8 - 408T^6 - 600T^5 + 49664T^4 + 112384T^3 - 1886592T^2 - 3108384T + 21431088)^2
$73$	\( (T^{8} + 16 T^{7} + \cdots + 3326208)^{2} \) (T^8 + 16T^7 - 320T^6 - 5312T^5 + 22240T^4 + 392960T^3 - 618496T^2 - 7414784*T + 3326208)^2
$79$	\( (T^{8} - 336 T^{6} + \cdots + 2166016)^{2} \) (T^8 - 336T^6 - 320T^5 + 35360T^4 + 53760T^3 - 1101568T^2 - 1229824T + 2166016)^2
$83$	\( T^{16} + \cdots + 6293836597504 \) T^16 + 560T^14 + 127424T^12 + 15118656T^10 + 996178016T^8 + 35690143488T^6 + 621437020160T^4 + 3852663542784*T^2 + 6293836597504
$89$	\( (T^{8} + 30 T^{7} + \cdots + 37581936)^{2} \) (T^8 + 30T^7 + 86T^6 - 5916T^5 - 69736T^4 - 43112T^3 + 3470840T^2 + 20801808*T + 37581936)^2
$97$	\( (T^{8} - 20 T^{7} + \cdots - 637696)^{2} \) (T^8 - 20T^7 - 128T^6 + 4272T^5 - 15584T^4 - 91328T^3 + 564736T^2 - 549632*T - 637696)^2
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