LMFDB - Newform orbit 1248.2.d.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,2,Mod(287,1248)]

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([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("1248.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.d (of order $2$, degree $1$, 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} - 4 x^{13} + 5 x^{12} - 4 x^{11} + 8 x^{10} - 16 x^{9} + 28 x^{8} - 32 x^{7} + 32 x^{6} + \cdots + 256 $ x^16 - 4x^13 + 5x^12 - 4x^11 + 8x^10 - 16x^9 + 28x^8 - 32x^7 + 32x^6 - 32x^5 + 80x^4 - 128*x^3 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
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_{5} q^{3} - \beta_{4} q^{5} - \beta_{2} q^{7} + ( - \beta_{9} - \beta_{4} - 1) q^{9}+O(q^{10}) $ q + b5 * q^3 - b4 * q^5 - b2 * q^7 + (-b9 - b4 - 1) * q^9	$ q + \beta_{5} q^{3} - \beta_{4} q^{5} - \beta_{2} q^{7} + ( - \beta_{9} - \beta_{4} - 1) q^{9} + (\beta_{15} - \beta_{13} + \cdots + \beta_{8}) q^{11}+ \cdots + ( - \beta_{15} - 2 \beta_{11} + \cdots - \beta_1) q^{99}+O(q^{100}) $ q + b5 * q^3 - b4 * q^5 - b2 * q^7 + (-b9 - b4 - 1) * q^9 + (b15 - b13 + b11 + b8) * q^11 - q^13 + (-b13 + b11 + 2b8 - b2) q^15 + (b9 - b7 + b4 - b3) * q^17 + (-b15 + b11 - b5 + b2) * q^19 + (-b12 + b9 + 2) * q^21 + (-b8 - b6 + b5 + b1) * q^23 + (b14 + b10 + 1) * q^25 + (b15 - b13 + b11 + b6) * q^27 + (b14 - b10 + b9 - b4 - b3) * q^29 + (-b6 - b2 - b1) * q^31 + (b14 + b7 + b4 - b3 - 1) * q^33 + (b8 - b5) * q^35 + (-2b12 + b9 + b4 + b3 + 2) q^37 - b5 * q^39 + (b9 - 2b7 - b3) q^41 + (-b6 + 2b2 - b1) q^43 + (b14 + b10 + b7 - b4 - b3 - 2) * q^45 + (3b13 - b8 + b5) q^47 + (2b12 - b9 - b4 - b3 + 1) q^49 + (-b11 - 2b8 - b6 + b1) q^51 + (-b7 - 2b4) q^53 + (-3b8 - b6 - 3b5 + 2b2 - b1) q^55 + (b14 + 3b12 + b10 - b9 + b7 - 2b4 - b3) * q^57 + (-b15 - b13 - b11 + 3b8 - 4b5) * q^59 + (-b14 - b10 - b9 - b4 - b3 + 2) * q^61 + (-b15 - b13 + b11 + b8 - b6 + b2) * q^63 + b4 * q^65 + (-b15 + b11 - b5 - 3b2) q^67 + (-b12 - b10 - 2b9 + 3b7 - b4 + 3b3 + 3) q^69 + (b15 - b13 + b11 + b8 - b6 + b1) * q^71 + (b14 + 2b12 + b10 - b9 - b4 - b3 - 4) q^73 + (b15 - 2b13 + b8 - b6 - b5 + 2b2 - b1) * q^75 + b7 * q^77 + (4b8 + 4b5 - 4b2) q^79 + (2b14 + b12 + b10 - b7 + b4 - 2b3 + 2) * q^81 + (b13 + b8 - b6 - b5 + b1) * q^83 + (-b14 - b10 + 2) * q^85 + (-2b15 - 3b13 - b11 + 2b8 - 2b5 - 2b2 + b1) q^87 + (b14 - b10 - 2b7 + 3b4) * q^89 + b2 * q^91 + (-2b14 - 2b12 - b10 + 2b9 + b7 - b3 + 1) q^93 + (-b15 - b11 + b8 - 2b5) q^95 + (b14 - 2b12 + b10 - b9 - b4 - b3 - 4) q^97 + (-b15 - 2b11 - 5b8 - 2b5 + 2b2 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 12 q^{9}+O(q^{10}) $ 16 * q - 12 * q^9	$ 16 q - 12 q^{9} - 16 q^{13} + 20 q^{21} + 8 q^{25} - 16 q^{33} + 8 q^{37} - 36 q^{45} + 40 q^{49} + 24 q^{57} + 48 q^{61} + 40 q^{69} - 48 q^{73} + 36 q^{81} + 40 q^{85} + 8 q^{93} - 80 q^{97}+O(q^{100}) $ 16 * q - 12 * q^9 - 16 * q^13 + 20 * q^21 + 8 * q^25 - 16 * q^33 + 8 * q^37 - 36 * q^45 + 40 * q^49 + 24 * q^57 + 48 * q^61 + 40 * q^69 - 48 * q^73 + 36 * q^81 + 40 * q^85 + 8 * q^93 - 80 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{13} + 5 x^{12} - 4 x^{11} + 8 x^{10} - 16 x^{9} + 28 x^{8} - 32 x^{7} + 32 x^{6} + \cdots + 256 $ x^16 - 4*x^13 + 5*x^12 - 4*x^11 + 8*x^10 - 16*x^9 + 28*x^8 - 32*x^7 + 32*x^6 - 32*x^5 + 80*x^4 - 128*x^3 + 256 :

$\beta_{1}$	$=$	$ ( - 5 \nu^{15} + 30 \nu^{14} - 10 \nu^{13} + 12 \nu^{12} - 29 \nu^{11} + 58 \nu^{10} - 82 \nu^{9} + \cdots + 512 ) / 1088 $ (-5v^15 + 30v^14 - 10v^13 + 12v^12 - 29v^11 + 58v^10 - 82v^9 + 96v^8 - 104v^7 + 512v^6 - 376v^5 + 240v^4 - 208v^3 + 1344v^2 - 448*v + 512) / 1088
$\beta_{2}$	$=$	$ ( 11 \nu^{15} - 100 \nu^{14} + 192 \nu^{13} - 244 \nu^{12} + 567 \nu^{11} - 1168 \nu^{10} + 1656 \nu^{9} + \cdots - 8960 ) / 2176 $ (11v^15 - 100v^14 + 192v^13 - 244v^12 + 567v^11 - 1168v^10 + 1656v^9 - 2360v^8 + 3044v^7 - 4608v^6 + 7056v^5 - 7872v^4 + 7312v^3 - 7744v^2 + 10560*v - 8960) / 2176
$\beta_{3}$	$=$	$ ( 5 \nu^{15} + 21 \nu^{14} - 58 \nu^{13} + 56 \nu^{12} - 73 \nu^{11} + 197 \nu^{10} - 394 \nu^{9} + \cdots + 2752 ) / 816 $ (5v^15 + 21v^14 - 58v^13 + 56v^12 - 73v^11 + 197v^10 - 394v^9 + 584v^8 - 474v^7 + 780v^6 - 1188v^5 + 2004v^4 - 1560v^3 + 288v^2 - 2272*v + 2752) / 816
$\beta_{4}$	$=$	$ ( 43 \nu^{15} - 156 \nu^{14} + 52 \nu^{13} - 212 \nu^{12} + 535 \nu^{11} - 968 \nu^{10} + 1324 \nu^{9} + \cdots - 1792 ) / 6528 $ (43v^15 - 156v^14 + 52v^13 - 212v^12 + 535v^11 - 968v^10 + 1324v^9 - 1832v^8 + 1860v^7 - 3696v^6 + 5328v^5 - 6144v^4 + 48v^3 - 6336v^2 + 10816*v - 1792) / 6528
$\beta_{5}$	$=$	$ ( - 29 \nu^{15} - 13 \nu^{14} + 44 \nu^{13} + 56 \nu^{12} - 73 \nu^{11} - 109 \nu^{10} + 320 \nu^{9} + \cdots - 9216 ) / 3264 $ (-29v^15 - 13v^14 + 44v^13 + 56v^12 - 73v^11 - 109v^10 + 320v^9 - 436v^8 + 376v^7 - 172v^6 + 2008v^5 - 3232v^4 + 2656v^3 - 1072v^2 + 5888*v - 9216) / 3264
$\beta_{6}$	$=$	$ ( 6 \nu^{15} - 19 \nu^{14} + 46 \nu^{13} - 62 \nu^{12} + 96 \nu^{11} - 175 \nu^{10} + 350 \nu^{9} + \cdots - 1920 ) / 544 $ (6v^15 - 19v^14 + 46v^13 - 62v^12 + 96v^11 - 175v^10 + 350v^9 - 462v^8 + 526v^7 - 764v^6 + 1308v^5 - 1376v^4 + 1256v^3 - 416v^2 + 864*v - 1920) / 544
$\beta_{7}$	$=$	$ ( 53 \nu^{15} - 12 \nu^{14} - 64 \nu^{13} - 100 \nu^{12} + 185 \nu^{11} - 64 \nu^{10} - 280 \nu^{9} + \cdots + 10240 ) / 3264 $ (53v^15 - 12v^14 - 64v^13 - 100v^12 + 185v^11 - 64v^10 - 280v^9 - 256v^8 + 300v^7 - 96v^6 - 720v^5 + 3168v^4 - 2256v^3 - 2496v^2 - 3520*v + 10240) / 3264
$\beta_{8}$	$=$	$ ( 56 \nu^{15} - 47 \nu^{14} + 10 \nu^{13} - 80 \nu^{12} + 352 \nu^{11} - 347 \nu^{10} + 286 \nu^{9} + \cdots + 3840 ) / 3264 $ (56v^15 - 47v^14 + 10v^13 - 80v^12 + 352v^11 - 347v^10 + 286v^9 - 368v^8 + 716v^7 - 1532v^6 + 1736v^5 + 304v^4 - 64v^3 - 4880v^2 + 3712*v + 3840) / 3264
$\beta_{9}$	$=$	$ ( 121 \nu^{15} - 12 \nu^{14} + 4 \nu^{13} - 644 \nu^{12} + 1069 \nu^{11} - 1424 \nu^{10} + 1420 \nu^{9} + \cdots + 5888 ) / 6528 $ (121v^15 - 12v^14 + 4v^13 - 644v^12 + 1069v^11 - 1424v^10 + 1420v^9 - 2432v^8 + 6012v^7 - 7440v^6 + 7440v^5 - 4992v^4 + 12432v^3 - 22080v^2 + 7360*v + 5888) / 6528
$\beta_{10}$	$=$	$ ( - 169 \nu^{15} + 708 \nu^{14} - 916 \nu^{13} + 1412 \nu^{12} - 3469 \nu^{11} + 6632 \nu^{10} + \cdots + 29056 ) / 6528 $ (-169v^15 + 708v^14 - 916v^13 + 1412v^12 - 3469v^11 + 6632v^10 - 10204v^9 + 11840v^8 - 17292v^7 + 27696v^6 - 37488v^5 + 36672v^4 - 33360v^3 + 42816v^2 - 66496*v + 29056) / 6528
$\beta_{11}$	$=$	$ ( 125 \nu^{15} - 104 \nu^{14} + 46 \nu^{13} - 572 \nu^{12} + 997 \nu^{11} - 668 \nu^{10} + 622 \nu^{9} + \cdots + 30720 ) / 3264 $ (125v^15 - 104v^14 + 46v^13 - 572v^12 + 997v^11 - 668v^10 + 622v^9 - 1040v^8 + 2192v^7 - 1784v^6 - 664v^5 + 4336v^4 - 2416v^3 - 3680v^2 - 14912*v + 30720) / 3264
$\beta_{12}$	$=$	$ ( - 3 \nu^{15} + 4 \nu^{14} - 4 \nu^{13} + 8 \nu^{12} - 15 \nu^{11} + 16 \nu^{10} - 12 \nu^{9} + \cdots - 384 ) / 64 $ (-3v^15 + 4v^14 - 4v^13 + 8v^12 - 15v^11 + 16v^10 - 12v^9 + 12v^8 - 20v^7 + 16v^6 + 16v^5 - 80v^4 + 144v^3 - 64v^2 + 192*v - 384) / 64
$\beta_{13}$	$=$	$ ( 11 \nu^{15} - 8 \nu^{14} - 8 \nu^{13} - 20 \nu^{12} + 55 \nu^{11} - 20 \nu^{10} - 32 \nu^{9} + \cdots + 2304 ) / 192 $ (11v^15 - 8v^14 - 8v^13 - 20v^12 + 55v^11 - 20v^10 - 32v^9 + 40v^8 + 20v^7 - 32v^6 - 208v^5 + 640v^4 - 496v^3 - 128v^2 - 704*v + 2304) / 192
$\beta_{14}$	$=$	$ ( 373 \nu^{15} - 708 \nu^{14} + 916 \nu^{13} - 1820 \nu^{12} + 4489 \nu^{11} - 7448 \nu^{10} + \cdots - 16000 ) / 6528 $ (373v^15 - 708v^14 + 916v^13 - 1820v^12 + 4489v^11 - 7448v^10 + 10204v^9 - 13064v^8 + 21372v^7 - 30960v^6 + 37488v^5 - 38304v^4 + 36624v^3 - 55872v^2 + 53440*v - 16000) / 6528
$\beta_{15}$	$=$	$ ( 193 \nu^{15} - 19 \nu^{14} - 328 \nu^{13} - 28 \nu^{12} + 113 \nu^{11} + 845 \nu^{10} - 2404 \nu^{9} + \cdots + 43776 ) / 3264 $ (193v^15 - 19v^14 - 328v^13 - 28v^12 + 113v^11 + 845v^10 - 2404v^9 + 3584v^8 - 3860v^7 + 5084v^6 - 11408v^5 + 20384v^4 - 17648v^3 + 5840v^2 - 25792*v + 43776) / 3264

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

$\nu$	$=$	$ ( - \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots + \beta_1 ) / 8 $ (-b15 - b14 + b13 - b12 - b11 - b10 + 2b9 + 2b8 - b6 + b5 - 2b3 - 2b2 + b1) / 8
$\nu^{2}$	$=$	$ ( -\beta_{15} + 4\beta_{13} - 3\beta_{11} - 5\beta_{8} + 2\beta_{6} - 2\beta_{5} + 2\beta_1 ) / 8 $ (-b15 + 4b13 - 3b11 - 5b8 + 2b6 - 2b5 + 2b1) / 8
$\nu^{3}$	$=$	$ ( 3 \beta_{15} + \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - \beta_{11} + \beta_{10} - 4 \beta_{8} + \cdots + 4 ) / 8 $ (3b15 + b14 + 3b13 + 3b12 - b11 + b10 - 4b8 - 2b7 + b6 + 7b5 - 6b4 - 4b3 + 2*b2 - b1 + 4) / 8
$\nu^{4}$	$=$	$ ( -3\beta_{14} - 5\beta_{10} - \beta_{9} + 7\beta_{7} - 9\beta_{4} + \beta_{3} - 12 ) / 8 $ (-3b14 - 5b10 - b9 + 7b7 - 9b4 + b3 - 12) / 8
$\nu^{5}$	$=$	$ ( - \beta_{15} - 5 \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots + 12 ) / 8 $ (-b15 - 5b14 + b13 - b12 - b11 - b10 + 2b9 + 2b8 + 7b6 + b5 + 4b4 + 10b3 + 6*b2 + b1 + 12) / 8
$\nu^{6}$	$=$	$ ( 5\beta_{15} - 4\beta_{13} + 7\beta_{11} + \beta_{8} - 2\beta_{6} + 34\beta_{5} + 6\beta_1 ) / 8 $ (5b15 - 4b13 + 7b11 + b8 - 2b6 + 34b5 + 6b1) / 8
$\nu^{7}$	$=$	$ ( - 15 \beta_{15} + 3 \beta_{14} + 5 \beta_{13} - 11 \beta_{12} - 3 \beta_{11} - 5 \beta_{10} + 4 \beta_{9} + \cdots - 4 ) / 8 $ (-15b15 + 3b14 + 5b13 - 11b12 - 3b11 - 5b10 + 4b9 - 16b8 + 14b7 - b6 + 9b5 - 6b4 + 16b3 - 18*b2 + b1 - 4) / 8
$\nu^{8}$	$=$	$ ( -5\beta_{14} - 8\beta_{12} - 3\beta_{10} + 17\beta_{9} - 39\beta_{7} + 9\beta_{4} + 15\beta_{3} + 12 ) / 8 $ (-5b14 - 8b12 - 3b10 + 17b9 - 39b7 + 9b4 + 15*b3 + 12) / 8
$\nu^{9}$	$=$	$ ( 21 \beta_{15} + 9 \beta_{14} - 33 \beta_{13} + 17 \beta_{12} + 29 \beta_{11} - 19 \beta_{10} - 22 \beta_{9} + \cdots - 12 ) / 8 $ (21b15 + 9b14 - 33b13 + 17b12 + 29b11 - 19b10 - 22b9 + 18b8 - 4b7 + b6 + 23b5 - 8b4 - 6b3 - 38b2 + 7b1 - 12) / 8
$\nu^{10}$	$=$	$ ( -45\beta_{15} + 12\beta_{13} + 17\beta_{11} + 79\beta_{8} + 26\beta_{6} + 6\beta_{5} - 80\beta_{2} - 30\beta_1 ) / 8 $ (-45b15 + 12b13 + 17b11 + 79b8 + 26b6 + 6b5 - 80b2 - 30b1) / 8
$\nu^{11}$	$=$	$ ( - 29 \beta_{15} + 9 \beta_{14} + 43 \beta_{13} + 27 \beta_{12} + 31 \beta_{11} + 25 \beta_{10} + \cdots - 172 ) / 8 $ (-29b15 + 9b14 + 43b13 + 27b12 + 31b11 + 25b10 - 48b9 + 84b8 - 58b7 - 7b6 - b5 - 6b4 - 4b3 + 18b2 + 23b1 - 172) / 8
$\nu^{12}$	$=$	$ ( 93\beta_{14} + 48\beta_{12} + 27\beta_{10} - 177\beta_{9} + 87\beta_{7} - 121\beta_{4} + 49\beta_{3} + 84 ) / 8 $ (93b14 + 48b12 + 27b10 - 177b9 + 87b7 - 121b4 + 49*b3 + 84) / 8
$\nu^{13}$	$=$	$ ( - 73 \beta_{15} - 29 \beta_{14} - 63 \beta_{13} - 113 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + \cdots + 188 ) / 8 $ (-73b15 - 29b14 - 63b13 - 113b12 + 7b11 + 7b10 + 26b9 + 250b8 - 24b7 - 41b6 - 111b5 - 148b4 + 2b3 + 6b2 - 47*b1 + 188) / 8
$\nu^{14}$	$=$	$ ( 21 \beta_{15} - 148 \beta_{13} + 23 \beta_{11} + 257 \beta_{8} - 18 \beta_{6} - 334 \beta_{5} + \cdots + 214 \beta_1 ) / 8 $ (21b15 - 148b13 + 23b11 + 257b8 - 18b6 - 334b5 + 32b2 + 214b1) / 8
$\nu^{15}$	$=$	$ ( 169 \beta_{15} + 203 \beta_{14} - 59 \beta_{13} + 21 \beta_{12} - 155 \beta_{11} + 147 \beta_{10} + \cdots + 508 ) / 8 $ (169b15 + 203b14 - 59b13 + 21b12 - 155b11 + 147b10 + 28b9 + 104b8 + 38b7 + 175b6 + 217b5 - 14b4 - 168b3 - 210b2 + 49*b1 + 508) / 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} $

287.1

1.24942	−	0.662534i
1.24942	+	0.662534i
0.254324	−	1.39116i
0.254324	+	1.39116i
−1.12277	−	0.859879i
−1.12277	+	0.859879i
−0.810702	−	1.15878i
−0.810702	+	1.15878i
−1.15878	−	0.810702i
−1.15878	+	0.810702i
0.859879	+	1.12277i
0.859879	−	1.12277i
1.39116	−	0.254324i
1.39116	+	0.254324i
−0.662534	+	1.24942i
−0.662534	−	1.24942i

−1.71165

−

0.265070i

−

2.00908i

1.70391i

2.85948

0.907412i

287.2

−1.71165

0.265070i

2.00908i

−

1.70391i

2.85948

−

0.907412i

287.3

−1.09547

−

1.34162i

3.60516i

−

0.607725i

−0.599886

2.93941i

287.4

−1.09547

1.34162i

−

3.60516i

0.607725i

−0.599886

−

2.93941i

287.5

−0.559912

−

1.63905i

−

0.294389i

3.80389i

−2.37300

1.83545i

287.6

−0.559912

1.63905i

0.294389i

−

3.80389i

−2.37300

−

1.83545i

287.7

−0.238125

−

1.71560i

0.937964i

−

0.507749i

−2.88659

0.817056i

287.8

−0.238125

1.71560i

−

0.937964i

0.507749i

−2.88659

−

0.817056i

287.9

0.238125

−

1.71560i

−

0.937964i

−

0.507749i

−2.88659

−

0.817056i

287.10

0.238125

1.71560i

0.937964i

0.507749i

−2.88659

0.817056i

287.11

0.559912

−

1.63905i

0.294389i

3.80389i

−2.37300

−

1.83545i

287.12

0.559912

1.63905i

−

0.294389i

−

3.80389i

−2.37300

1.83545i

287.13

1.09547

−

1.34162i

−

3.60516i

−

0.607725i

−0.599886

−

2.93941i

287.14

1.09547

1.34162i

3.60516i

0.607725i

−0.599886

2.93941i

287.15

1.71165

−

0.265070i

2.00908i

1.70391i

2.85948

−

0.907412i

287.16

1.71165

0.265070i

−

2.00908i

−

1.70391i

2.85948

0.907412i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.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.d.c	✓	16
3.b	odd	2	1	inner	1248.2.d.c	✓	16
4.b	odd	2	1	inner	1248.2.d.c	✓	16
8.b	even	2	1		2496.2.d.p		16
8.d	odd	2	1		2496.2.d.p		16
12.b	even	2	1	inner	1248.2.d.c	✓	16
24.f	even	2	1		2496.2.d.p		16
24.h	odd	2	1		2496.2.d.p		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1248.2.d.c	✓	16	1.a	even	1	1	trivial
1248.2.d.c	✓	16	3.b	odd	2	1	inner
1248.2.d.c	✓	16	4.b	odd	2	1	inner
1248.2.d.c	✓	16	12.b	even	2	1	inner
2496.2.d.p		16	8.b	even	2	1
2496.2.d.p		16	8.d	odd	2	1
2496.2.d.p		16	24.f	even	2	1
2496.2.d.p		16	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} + 18T_{5}^{6} + 69T_{5}^{4} + 52T_{5}^{2} + 4 $ T5^8 + 18*T5^6 + 69*T5^4 + 52*T5^2 + 4 acting on $S_{2}^{\mathrm{new}}(1248, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 6 T^{14} + \cdots + 6561 $ T^16 + 6T^14 + 9T^12 - 34T^10 - 188T^8 - 306T^6 + 729T^4 + 4374*T^2 + 6561
$5$	$ (T^{8} + 18 T^{6} + 69 T^{4} + \cdots + 4)^{2} $ (T^8 + 18T^6 + 69T^4 + 52*T^2 + 4)^2
$7$	$ (T^{8} + 18 T^{6} + 53 T^{4} + \cdots + 4)^{2} $ (T^8 + 18T^6 + 53T^4 + 28*T^2 + 4)^2
$11$	$ (T^{8} - 56 T^{6} + \cdots + 1024)^{2} $ (T^8 - 56T^6 + 848T^4 - 2304*T^2 + 1024)^2
$13$	$ (T + 1)^{16} $ (T + 1)^16
$17$	$ (T^{8} + 54 T^{6} + \cdots + 576)^{2} $ (T^8 + 54T^6 + 521T^4 + 1040*T^2 + 576)^2
$19$	$ (T^{8} + 92 T^{6} + \cdots + 5184)^{2} $ (T^8 + 92T^6 + 2004T^4 + 11296*T^2 + 5184)^2
$23$	$ (T^{8} - 144 T^{6} + \cdots + 43264)^{2} $ (T^8 - 144T^6 + 5600T^4 - 62720*T^2 + 43264)^2
$29$	$ (T^{8} + 240 T^{6} + \cdots + 1048576)^{2} $ (T^8 + 240T^6 + 19008T^4 + 507904*T^2 + 1048576)^2
$31$	$ (T^{8} + 140 T^{6} + \cdots + 179776)^{2} $ (T^8 + 140T^6 + 6612T^4 + 109152*T^2 + 179776)^2
$37$	$ (T^{4} - 2 T^{3} + \cdots - 156)^{4} $ (T^4 - 2T^3 - 67T^2 - 212*T - 156)^4
$41$	$ (T^{8} + 132 T^{6} + \cdots + 87616)^{2} $ (T^8 + 132T^6 + 4052T^4 + 38240*T^2 + 87616)^2
$43$	$ (T^{8} + 206 T^{6} + \cdots + 602176)^{2} $ (T^8 + 206T^6 + 12873T^4 + 235152*T^2 + 602176)^2
$47$	$ (T^{8} - 258 T^{6} + \cdots + 10163344)^{2} $ (T^8 - 258T^6 + 22929T^4 - 831592*T^2 + 10163344)^2
$53$	$ (T^{8} + 136 T^{6} + \cdots + 36864)^{2} $ (T^8 + 136T^6 + 3856T^4 + 31232*T^2 + 36864)^2
$59$	$ (T^{8} - 280 T^{6} + \cdots + 2876416)^{2} $ (T^8 - 280T^6 + 26448T^4 - 873216*T^2 + 2876416)^2
$61$	$ (T^{4} - 12 T^{3} + \cdots + 2624)^{4} $ (T^4 - 12T^3 - 68T^2 + 608*T + 2624)^4
$67$	$ (T^{8} + 252 T^{6} + \cdots + 3655744)^{2} $ (T^8 + 252T^6 + 20244T^4 + 546848*T^2 + 3655744)^2
$71$	$ (T^{8} - 210 T^{6} + \cdots + 44944)^{2} $ (T^8 - 210T^6 + 9825T^4 - 62824*T^2 + 44944)^2
$73$	$ (T^{4} + 12 T^{3} + \cdots - 288)^{4} $ (T^4 + 12T^3 - 4T^2 - 320*T - 288)^4
$79$	$ (T^{8} + 448 T^{6} + \cdots + 4194304)^{2} $ (T^8 + 448T^6 + 54272T^4 + 1179648*T^2 + 4194304)^2
$83$	$ (T^{8} - 200 T^{6} + \cdots + 1048576)^{2} $ (T^8 - 200T^6 + 12176T^4 - 221184*T^2 + 1048576)^2
$89$	$ (T^{8} + 404 T^{6} + \cdots + 1498176)^{2} $ (T^8 + 404T^6 + 42836T^4 + 1291040*T^2 + 1498176)^2
$97$	$ (T^{4} + 20 T^{3} + \cdots + 5792)^{4} $ (T^4 + 20T^3 - 84T^2 - 1952*T + 5792)^4
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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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{3} - \beta_{4} q^{5} - \beta_{2} q^{7} + ( - \beta_{9} - \beta_{4} - 1) q^{9}+O(q^{10}) \) q + b5 * q^3 - b4 * q^5 - b2 * q^7 + (-b9 - b4 - 1) * q^9	\( q + \beta_{5} q^{3} - \beta_{4} q^{5} - \beta_{2} q^{7} + ( - \beta_{9} - \beta_{4} - 1) q^{9} + (\beta_{15} - \beta_{13} + \cdots + \beta_{8}) q^{11}+ \cdots + ( - \beta_{15} - 2 \beta_{11} + \cdots - \beta_1) q^{99}+O(q^{100}) \) q + b5 * q^3 - b4 * q^5 - b2 * q^7 + (-b9 - b4 - 1) * q^9 + (b15 - b13 + b11 + b8) * q^11 - q^13 + (-b13 + b11 + 2b8 - b2) q^15 + (b9 - b7 + b4 - b3) * q^17 + (-b15 + b11 - b5 + b2) * q^19 + (-b12 + b9 + 2) * q^21 + (-b8 - b6 + b5 + b1) * q^23 + (b14 + b10 + 1) * q^25 + (b15 - b13 + b11 + b6) * q^27 + (b14 - b10 + b9 - b4 - b3) * q^29 + (-b6 - b2 - b1) * q^31 + (b14 + b7 + b4 - b3 - 1) * q^33 + (b8 - b5) * q^35 + (-2b12 + b9 + b4 + b3 + 2) q^37 - b5 * q^39 + (b9 - 2b7 - b3) q^41 + (-b6 + 2b2 - b1) q^43 + (b14 + b10 + b7 - b4 - b3 - 2) * q^45 + (3b13 - b8 + b5) q^47 + (2b12 - b9 - b4 - b3 + 1) q^49 + (-b11 - 2b8 - b6 + b1) q^51 + (-b7 - 2b4) q^53 + (-3b8 - b6 - 3b5 + 2b2 - b1) q^55 + (b14 + 3b12 + b10 - b9 + b7 - 2b4 - b3) * q^57 + (-b15 - b13 - b11 + 3b8 - 4b5) * q^59 + (-b14 - b10 - b9 - b4 - b3 + 2) * q^61 + (-b15 - b13 + b11 + b8 - b6 + b2) * q^63 + b4 * q^65 + (-b15 + b11 - b5 - 3b2) q^67 + (-b12 - b10 - 2b9 + 3b7 - b4 + 3b3 + 3) q^69 + (b15 - b13 + b11 + b8 - b6 + b1) * q^71 + (b14 + 2b12 + b10 - b9 - b4 - b3 - 4) q^73 + (b15 - 2b13 + b8 - b6 - b5 + 2b2 - b1) * q^75 + b7 * q^77 + (4b8 + 4b5 - 4b2) q^79 + (2b14 + b12 + b10 - b7 + b4 - 2b3 + 2) * q^81 + (b13 + b8 - b6 - b5 + b1) * q^83 + (-b14 - b10 + 2) * q^85 + (-2b15 - 3b13 - b11 + 2b8 - 2b5 - 2b2 + b1) q^87 + (b14 - b10 - 2b7 + 3b4) * q^89 + b2 * q^91 + (-2b14 - 2b12 - b10 + 2b9 + b7 - b3 + 1) q^93 + (-b15 - b11 + b8 - 2b5) q^95 + (b14 - 2b12 + b10 - b9 - b4 - b3 - 4) q^97 + (-b15 - 2b11 - 5b8 - 2b5 + 2b2 - b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 12 q^{9}+O(q^{10}) \) 16 * q - 12 * q^9	\( 16 q - 12 q^{9} - 16 q^{13} + 20 q^{21} + 8 q^{25} - 16 q^{33} + 8 q^{37} - 36 q^{45} + 40 q^{49} + 24 q^{57} + 48 q^{61} + 40 q^{69} - 48 q^{73} + 36 q^{81} + 40 q^{85} + 8 q^{93} - 80 q^{97}+O(q^{100}) \) 16 * q - 12 * q^9 - 16 * q^13 + 20 * q^21 + 8 * q^25 - 16 * q^33 + 8 * q^37 - 36 * q^45 + 40 * q^49 + 24 * q^57 + 48 * q^61 + 40 * q^69 - 48 * q^73 + 36 * q^81 + 40 * q^85 + 8 * q^93 - 80 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1248.2.d.c

Level	$1248$
Weight	$2$
Character orbit	1248.d
Analytic conductor	$9.965$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

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:	\( x^{16} - 4 x^{13} + 5 x^{12} - 4 x^{11} + 8 x^{10} - 16 x^{9} + 28 x^{8} - 32 x^{7} + 32 x^{6} + \cdots + 256 \) x^16 - 4x^13 + 5x^12 - 4x^11 + 8x^10 - 16x^9 + 28x^8 - 32x^7 + 32x^6 - 32x^5 + 80x^4 - 128*x^3 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{18} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 5 \nu^{15} + 30 \nu^{14} - 10 \nu^{13} + 12 \nu^{12} - 29 \nu^{11} + 58 \nu^{10} - 82 \nu^{9} + \cdots + 512 ) / 1088 \) (-5v^15 + 30v^14 - 10v^13 + 12v^12 - 29v^11 + 58v^10 - 82v^9 + 96v^8 - 104v^7 + 512v^6 - 376v^5 + 240v^4 - 208v^3 + 1344v^2 - 448*v + 512) / 1088
\(\beta_{2}\)	\(=\)	\( ( 11 \nu^{15} - 100 \nu^{14} + 192 \nu^{13} - 244 \nu^{12} + 567 \nu^{11} - 1168 \nu^{10} + 1656 \nu^{9} + \cdots - 8960 ) / 2176 \) (11v^15 - 100v^14 + 192v^13 - 244v^12 + 567v^11 - 1168v^10 + 1656v^9 - 2360v^8 + 3044v^7 - 4608v^6 + 7056v^5 - 7872v^4 + 7312v^3 - 7744v^2 + 10560*v - 8960) / 2176
\(\beta_{3}\)	\(=\)	\( ( 5 \nu^{15} + 21 \nu^{14} - 58 \nu^{13} + 56 \nu^{12} - 73 \nu^{11} + 197 \nu^{10} - 394 \nu^{9} + \cdots + 2752 ) / 816 \) (5v^15 + 21v^14 - 58v^13 + 56v^12 - 73v^11 + 197v^10 - 394v^9 + 584v^8 - 474v^7 + 780v^6 - 1188v^5 + 2004v^4 - 1560v^3 + 288v^2 - 2272*v + 2752) / 816
\(\beta_{4}\)	\(=\)	\( ( 43 \nu^{15} - 156 \nu^{14} + 52 \nu^{13} - 212 \nu^{12} + 535 \nu^{11} - 968 \nu^{10} + 1324 \nu^{9} + \cdots - 1792 ) / 6528 \) (43v^15 - 156v^14 + 52v^13 - 212v^12 + 535v^11 - 968v^10 + 1324v^9 - 1832v^8 + 1860v^7 - 3696v^6 + 5328v^5 - 6144v^4 + 48v^3 - 6336v^2 + 10816*v - 1792) / 6528
\(\beta_{5}\)	\(=\)	\( ( - 29 \nu^{15} - 13 \nu^{14} + 44 \nu^{13} + 56 \nu^{12} - 73 \nu^{11} - 109 \nu^{10} + 320 \nu^{9} + \cdots - 9216 ) / 3264 \) (-29v^15 - 13v^14 + 44v^13 + 56v^12 - 73v^11 - 109v^10 + 320v^9 - 436v^8 + 376v^7 - 172v^6 + 2008v^5 - 3232v^4 + 2656v^3 - 1072v^2 + 5888*v - 9216) / 3264
\(\beta_{6}\)	\(=\)	\( ( 6 \nu^{15} - 19 \nu^{14} + 46 \nu^{13} - 62 \nu^{12} + 96 \nu^{11} - 175 \nu^{10} + 350 \nu^{9} + \cdots - 1920 ) / 544 \) (6v^15 - 19v^14 + 46v^13 - 62v^12 + 96v^11 - 175v^10 + 350v^9 - 462v^8 + 526v^7 - 764v^6 + 1308v^5 - 1376v^4 + 1256v^3 - 416v^2 + 864*v - 1920) / 544
\(\beta_{7}\)	\(=\)	\( ( 53 \nu^{15} - 12 \nu^{14} - 64 \nu^{13} - 100 \nu^{12} + 185 \nu^{11} - 64 \nu^{10} - 280 \nu^{9} + \cdots + 10240 ) / 3264 \) (53v^15 - 12v^14 - 64v^13 - 100v^12 + 185v^11 - 64v^10 - 280v^9 - 256v^8 + 300v^7 - 96v^6 - 720v^5 + 3168v^4 - 2256v^3 - 2496v^2 - 3520*v + 10240) / 3264
\(\beta_{8}\)	\(=\)	\( ( 56 \nu^{15} - 47 \nu^{14} + 10 \nu^{13} - 80 \nu^{12} + 352 \nu^{11} - 347 \nu^{10} + 286 \nu^{9} + \cdots + 3840 ) / 3264 \) (56v^15 - 47v^14 + 10v^13 - 80v^12 + 352v^11 - 347v^10 + 286v^9 - 368v^8 + 716v^7 - 1532v^6 + 1736v^5 + 304v^4 - 64v^3 - 4880v^2 + 3712*v + 3840) / 3264
\(\beta_{9}\)	\(=\)	\( ( 121 \nu^{15} - 12 \nu^{14} + 4 \nu^{13} - 644 \nu^{12} + 1069 \nu^{11} - 1424 \nu^{10} + 1420 \nu^{9} + \cdots + 5888 ) / 6528 \) (121v^15 - 12v^14 + 4v^13 - 644v^12 + 1069v^11 - 1424v^10 + 1420v^9 - 2432v^8 + 6012v^7 - 7440v^6 + 7440v^5 - 4992v^4 + 12432v^3 - 22080v^2 + 7360*v + 5888) / 6528
\(\beta_{10}\)	\(=\)	\( ( - 169 \nu^{15} + 708 \nu^{14} - 916 \nu^{13} + 1412 \nu^{12} - 3469 \nu^{11} + 6632 \nu^{10} + \cdots + 29056 ) / 6528 \) (-169v^15 + 708v^14 - 916v^13 + 1412v^12 - 3469v^11 + 6632v^10 - 10204v^9 + 11840v^8 - 17292v^7 + 27696v^6 - 37488v^5 + 36672v^4 - 33360v^3 + 42816v^2 - 66496*v + 29056) / 6528
\(\beta_{11}\)	\(=\)	\( ( 125 \nu^{15} - 104 \nu^{14} + 46 \nu^{13} - 572 \nu^{12} + 997 \nu^{11} - 668 \nu^{10} + 622 \nu^{9} + \cdots + 30720 ) / 3264 \) (125v^15 - 104v^14 + 46v^13 - 572v^12 + 997v^11 - 668v^10 + 622v^9 - 1040v^8 + 2192v^7 - 1784v^6 - 664v^5 + 4336v^4 - 2416v^3 - 3680v^2 - 14912*v + 30720) / 3264
\(\beta_{12}\)	\(=\)	\( ( - 3 \nu^{15} + 4 \nu^{14} - 4 \nu^{13} + 8 \nu^{12} - 15 \nu^{11} + 16 \nu^{10} - 12 \nu^{9} + \cdots - 384 ) / 64 \) (-3v^15 + 4v^14 - 4v^13 + 8v^12 - 15v^11 + 16v^10 - 12v^9 + 12v^8 - 20v^7 + 16v^6 + 16v^5 - 80v^4 + 144v^3 - 64v^2 + 192*v - 384) / 64
\(\beta_{13}\)	\(=\)	\( ( 11 \nu^{15} - 8 \nu^{14} - 8 \nu^{13} - 20 \nu^{12} + 55 \nu^{11} - 20 \nu^{10} - 32 \nu^{9} + \cdots + 2304 ) / 192 \) (11v^15 - 8v^14 - 8v^13 - 20v^12 + 55v^11 - 20v^10 - 32v^9 + 40v^8 + 20v^7 - 32v^6 - 208v^5 + 640v^4 - 496v^3 - 128v^2 - 704*v + 2304) / 192
\(\beta_{14}\)	\(=\)	\( ( 373 \nu^{15} - 708 \nu^{14} + 916 \nu^{13} - 1820 \nu^{12} + 4489 \nu^{11} - 7448 \nu^{10} + \cdots - 16000 ) / 6528 \) (373v^15 - 708v^14 + 916v^13 - 1820v^12 + 4489v^11 - 7448v^10 + 10204v^9 - 13064v^8 + 21372v^7 - 30960v^6 + 37488v^5 - 38304v^4 + 36624v^3 - 55872v^2 + 53440*v - 16000) / 6528
\(\beta_{15}\)	\(=\)	\( ( 193 \nu^{15} - 19 \nu^{14} - 328 \nu^{13} - 28 \nu^{12} + 113 \nu^{11} + 845 \nu^{10} - 2404 \nu^{9} + \cdots + 43776 ) / 3264 \) (193v^15 - 19v^14 - 328v^13 - 28v^12 + 113v^11 + 845v^10 - 2404v^9 + 3584v^8 - 3860v^7 + 5084v^6 - 11408v^5 + 20384v^4 - 17648v^3 + 5840v^2 - 25792*v + 43776) / 3264

\(\nu\)	\(=\)	\( ( - \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots + \beta_1 ) / 8 \) (-b15 - b14 + b13 - b12 - b11 - b10 + 2b9 + 2b8 - b6 + b5 - 2b3 - 2b2 + b1) / 8
\(\nu^{2}\)	\(=\)	\( ( -\beta_{15} + 4\beta_{13} - 3\beta_{11} - 5\beta_{8} + 2\beta_{6} - 2\beta_{5} + 2\beta_1 ) / 8 \) (-b15 + 4b13 - 3b11 - 5b8 + 2b6 - 2b5 + 2b1) / 8
\(\nu^{3}\)	\(=\)	\( ( 3 \beta_{15} + \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - \beta_{11} + \beta_{10} - 4 \beta_{8} + \cdots + 4 ) / 8 \) (3b15 + b14 + 3b13 + 3b12 - b11 + b10 - 4b8 - 2b7 + b6 + 7b5 - 6b4 - 4b3 + 2*b2 - b1 + 4) / 8
\(\nu^{4}\)	\(=\)	\( ( -3\beta_{14} - 5\beta_{10} - \beta_{9} + 7\beta_{7} - 9\beta_{4} + \beta_{3} - 12 ) / 8 \) (-3b14 - 5b10 - b9 + 7b7 - 9b4 + b3 - 12) / 8
\(\nu^{5}\)	\(=\)	\( ( - \beta_{15} - 5 \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots + 12 ) / 8 \) (-b15 - 5b14 + b13 - b12 - b11 - b10 + 2b9 + 2b8 + 7b6 + b5 + 4b4 + 10b3 + 6*b2 + b1 + 12) / 8
\(\nu^{6}\)	\(=\)	\( ( 5\beta_{15} - 4\beta_{13} + 7\beta_{11} + \beta_{8} - 2\beta_{6} + 34\beta_{5} + 6\beta_1 ) / 8 \) (5b15 - 4b13 + 7b11 + b8 - 2b6 + 34b5 + 6b1) / 8
\(\nu^{7}\)	\(=\)	\( ( - 15 \beta_{15} + 3 \beta_{14} + 5 \beta_{13} - 11 \beta_{12} - 3 \beta_{11} - 5 \beta_{10} + 4 \beta_{9} + \cdots - 4 ) / 8 \) (-15b15 + 3b14 + 5b13 - 11b12 - 3b11 - 5b10 + 4b9 - 16b8 + 14b7 - b6 + 9b5 - 6b4 + 16b3 - 18*b2 + b1 - 4) / 8
\(\nu^{8}\)	\(=\)	\( ( -5\beta_{14} - 8\beta_{12} - 3\beta_{10} + 17\beta_{9} - 39\beta_{7} + 9\beta_{4} + 15\beta_{3} + 12 ) / 8 \) (-5b14 - 8b12 - 3b10 + 17b9 - 39b7 + 9b4 + 15*b3 + 12) / 8
\(\nu^{9}\)	\(=\)	\( ( 21 \beta_{15} + 9 \beta_{14} - 33 \beta_{13} + 17 \beta_{12} + 29 \beta_{11} - 19 \beta_{10} - 22 \beta_{9} + \cdots - 12 ) / 8 \) (21b15 + 9b14 - 33b13 + 17b12 + 29b11 - 19b10 - 22b9 + 18b8 - 4b7 + b6 + 23b5 - 8b4 - 6b3 - 38b2 + 7b1 - 12) / 8
\(\nu^{10}\)	\(=\)	\( ( -45\beta_{15} + 12\beta_{13} + 17\beta_{11} + 79\beta_{8} + 26\beta_{6} + 6\beta_{5} - 80\beta_{2} - 30\beta_1 ) / 8 \) (-45b15 + 12b13 + 17b11 + 79b8 + 26b6 + 6b5 - 80b2 - 30b1) / 8
\(\nu^{11}\)	\(=\)	\( ( - 29 \beta_{15} + 9 \beta_{14} + 43 \beta_{13} + 27 \beta_{12} + 31 \beta_{11} + 25 \beta_{10} + \cdots - 172 ) / 8 \) (-29b15 + 9b14 + 43b13 + 27b12 + 31b11 + 25b10 - 48b9 + 84b8 - 58b7 - 7b6 - b5 - 6b4 - 4b3 + 18b2 + 23b1 - 172) / 8
\(\nu^{12}\)	\(=\)	\( ( 93\beta_{14} + 48\beta_{12} + 27\beta_{10} - 177\beta_{9} + 87\beta_{7} - 121\beta_{4} + 49\beta_{3} + 84 ) / 8 \) (93b14 + 48b12 + 27b10 - 177b9 + 87b7 - 121b4 + 49*b3 + 84) / 8
\(\nu^{13}\)	\(=\)	\( ( - 73 \beta_{15} - 29 \beta_{14} - 63 \beta_{13} - 113 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + \cdots + 188 ) / 8 \) (-73b15 - 29b14 - 63b13 - 113b12 + 7b11 + 7b10 + 26b9 + 250b8 - 24b7 - 41b6 - 111b5 - 148b4 + 2b3 + 6b2 - 47*b1 + 188) / 8
\(\nu^{14}\)	\(=\)	\( ( 21 \beta_{15} - 148 \beta_{13} + 23 \beta_{11} + 257 \beta_{8} - 18 \beta_{6} - 334 \beta_{5} + \cdots + 214 \beta_1 ) / 8 \) (21b15 - 148b13 + 23b11 + 257b8 - 18b6 - 334b5 + 32b2 + 214b1) / 8
\(\nu^{15}\)	\(=\)	\( ( 169 \beta_{15} + 203 \beta_{14} - 59 \beta_{13} + 21 \beta_{12} - 155 \beta_{11} + 147 \beta_{10} + \cdots + 508 ) / 8 \) (169b15 + 203b14 - 59b13 + 21b12 - 155b11 + 147b10 + 28b9 + 104b8 + 38b7 + 175b6 + 217b5 - 14b4 - 168b3 - 210b2 + 49*b1 + 508) / 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 287.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 6 T^{14} + \cdots + 6561 \) T^16 + 6T^14 + 9T^12 - 34T^10 - 188T^8 - 306T^6 + 729T^4 + 4374*T^2 + 6561
$5$	\( (T^{8} + 18 T^{6} + 69 T^{4} + \cdots + 4)^{2} \) (T^8 + 18T^6 + 69T^4 + 52*T^2 + 4)^2
$7$	\( (T^{8} + 18 T^{6} + 53 T^{4} + \cdots + 4)^{2} \) (T^8 + 18T^6 + 53T^4 + 28*T^2 + 4)^2
$11$	\( (T^{8} - 56 T^{6} + \cdots + 1024)^{2} \) (T^8 - 56T^6 + 848T^4 - 2304*T^2 + 1024)^2
$13$	\( (T + 1)^{16} \) (T + 1)^16
$17$	\( (T^{8} + 54 T^{6} + \cdots + 576)^{2} \) (T^8 + 54T^6 + 521T^4 + 1040*T^2 + 576)^2
$19$	\( (T^{8} + 92 T^{6} + \cdots + 5184)^{2} \) (T^8 + 92T^6 + 2004T^4 + 11296*T^2 + 5184)^2
$23$	\( (T^{8} - 144 T^{6} + \cdots + 43264)^{2} \) (T^8 - 144T^6 + 5600T^4 - 62720*T^2 + 43264)^2
$29$	\( (T^{8} + 240 T^{6} + \cdots + 1048576)^{2} \) (T^8 + 240T^6 + 19008T^4 + 507904*T^2 + 1048576)^2
$31$	\( (T^{8} + 140 T^{6} + \cdots + 179776)^{2} \) (T^8 + 140T^6 + 6612T^4 + 109152*T^2 + 179776)^2
$37$	\( (T^{4} - 2 T^{3} + \cdots - 156)^{4} \) (T^4 - 2T^3 - 67T^2 - 212*T - 156)^4
$41$	\( (T^{8} + 132 T^{6} + \cdots + 87616)^{2} \) (T^8 + 132T^6 + 4052T^4 + 38240*T^2 + 87616)^2
$43$	\( (T^{8} + 206 T^{6} + \cdots + 602176)^{2} \) (T^8 + 206T^6 + 12873T^4 + 235152*T^2 + 602176)^2
$47$	\( (T^{8} - 258 T^{6} + \cdots + 10163344)^{2} \) (T^8 - 258T^6 + 22929T^4 - 831592*T^2 + 10163344)^2
$53$	\( (T^{8} + 136 T^{6} + \cdots + 36864)^{2} \) (T^8 + 136T^6 + 3856T^4 + 31232*T^2 + 36864)^2
$59$	\( (T^{8} - 280 T^{6} + \cdots + 2876416)^{2} \) (T^8 - 280T^6 + 26448T^4 - 873216*T^2 + 2876416)^2
$61$	\( (T^{4} - 12 T^{3} + \cdots + 2624)^{4} \) (T^4 - 12T^3 - 68T^2 + 608*T + 2624)^4
$67$	\( (T^{8} + 252 T^{6} + \cdots + 3655744)^{2} \) (T^8 + 252T^6 + 20244T^4 + 546848*T^2 + 3655744)^2
$71$	\( (T^{8} - 210 T^{6} + \cdots + 44944)^{2} \) (T^8 - 210T^6 + 9825T^4 - 62824*T^2 + 44944)^2
$73$	\( (T^{4} + 12 T^{3} + \cdots - 288)^{4} \) (T^4 + 12T^3 - 4T^2 - 320*T - 288)^4
$79$	\( (T^{8} + 448 T^{6} + \cdots + 4194304)^{2} \) (T^8 + 448T^6 + 54272T^4 + 1179648*T^2 + 4194304)^2
$83$	\( (T^{8} - 200 T^{6} + \cdots + 1048576)^{2} \) (T^8 - 200T^6 + 12176T^4 - 221184*T^2 + 1048576)^2
$89$	\( (T^{8} + 404 T^{6} + \cdots + 1498176)^{2} \) (T^8 + 404T^6 + 42836T^4 + 1291040*T^2 + 1498176)^2
$97$	\( (T^{4} + 20 T^{3} + \cdots + 5792)^{4} \) (T^4 + 20T^3 - 84T^2 - 1952*T + 5792)^4
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