LMFDB - Newform orbit 1280.3.b.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(511,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1280.511");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.b (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:	$34.8774738381$
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} - x^{12} + 4 x^{11} - 6 x^{10} + 14 x^{9} - 15 x^{8} + 28 x^{7} + \cdots + 256 $ x^16 - 2x^15 + 2x^14 - x^12 + 4x^11 - 6x^10 + 14x^9 - 15x^8 + 28x^7 - 24x^6 + 32x^5 - 16x^4 + 128x^2 - 256x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{48} $
Twist minimal:	no (minimal twist has level 40)
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_{4} q^{3} - \beta_1 q^{5} + \beta_{12} q^{7} + (\beta_{2} - 3) q^{9}+O(q^{10}) $ q + b4 * q^3 - b1 * q^5 + b12 * q^7 + (b2 - 3) * q^9	$ q + \beta_{4} q^{3} - \beta_1 q^{5} + \beta_{12} q^{7} + (\beta_{2} - 3) q^{9} + \beta_{6} q^{11} - \beta_{3} q^{13} + \beta_{11} q^{15} + \beta_{9} q^{17} + (\beta_{14} - \beta_{6} - 2 \beta_{4}) q^{19} + (\beta_{7} - \beta_{3} + 2 \beta_1) q^{21} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_{10}) q^{23}+ \cdots + ( - 5 \beta_{14} + 4 \beta_{13} + \cdots - 4 \beta_{4}) q^{99}+O(q^{100}) $ q + b4 * q^3 - b1 * q^5 + b12 * q^7 + (b2 - 3) * q^9 + b6 * q^11 - b3 * q^13 + b11 * q^15 + b9 * q^17 + (b14 - b6 - 2b4) q^19 + (b7 - b3 + 2b1) q^21 + (-b15 - b12 - b11 - b10) * q^23 + 5 * q^25 + (2b14 - b13 - b6 - 2b4) * q^27 - b8 * q^29 + (-2b11 - b10) q^31 + (-b9 + b5 + b2 + 2) * q^33 + (b13 + b4) * q^35 + (-b8 - b7 - b3 - 2b1) q^37 + (2b15 - 2b12 + 2b11 - b10) q^39 + (-b2 - 6) * q^41 + (3b13 - 3b6 + b4) * q^43 + (b7 + 3b1) q^45 + (-b12 + 2b11 - 2b10) * q^47 + (-2b9 - 2b5 - b2 - 11) * q^49 + (-b14 - 2b13 + 4b6 - 2b4) q^51 + (-b8 + b7 + 8b1) q^53 + (b15 + b12) * q^55 + (b9 - 4b2 + 22) q^57 + (-5b14 - 2b13 + 3b6 + 6b4) * q^59 + (b8 + 3b7 - 3b3 - 6b1) q^61 + (b15 + b12 - 7b11 - 3b10) * q^63 + (b9 - b5 - b2) * q^65 + (2b14 - b13 + b6 - b4) q^67 + (b8 - 2b7 - 2b3 - 10b1) q^69 + (-2b15 - 4b10) * q^71 + (-b9 + b5 - 5b2 - 20) q^73 + 5b4 q^75 + (-b8 - 3b7 + 6b1) * q^77 + (-2b15 + 2b12 + 8b11 - 3b10) * q^79 + (2b5 + b2 - 7) q^81 + (-6b14 + b13 - 3b6 + 9b4) q^83 + (b8 + b7 - 2b3) q^85 + (-b15 - 4b12 - 3b11 - 5b10) q^87 + (-2b9 - 2b5 + 10b2 + 6) q^89 + (b14 - 2b13 - 2b6 - 14b4) q^91 + (b8 - 3b7 + 3b3 - 24b1) q^93 + (-b15 - b12 - 2b11 - b10) q^95 + (-b9 + b5 - 3b2 + 28) q^97 + (-5b14 + 4b13 - b6 - 4b4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9}+O(q^{10}) $ 16 * q - 48 * q^9	$ 16 q - 48 q^{9} + 80 q^{25} + 32 q^{33} - 96 q^{41} - 176 q^{49} + 352 q^{57} - 320 q^{73} - 112 q^{81} + 96 q^{89} + 448 q^{97}+O(q^{100}) $ 16 * q - 48 * q^9 + 80 * q^25 + 32 * q^33 - 96 * q^41 - 176 * q^49 + 352 * q^57 - 320 * q^73 - 112 * q^81 + 96 * q^89 + 448 * 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} - x^{12} + 4 x^{11} - 6 x^{10} + 14 x^{9} - 15 x^{8} + 28 x^{7} + \cdots + 256 $ x^16 - 2*x^15 + 2*x^14 - x^12 + 4*x^11 - 6*x^10 + 14*x^9 - 15*x^8 + 28*x^7 - 24*x^6 + 32*x^5 - 16*x^4 + 128*x^2 - 256*x + 256 :

$\beta_{1}$	$=$	$ ( 5 \nu^{15} + 16 \nu^{14} + 16 \nu^{13} + 32 \nu^{12} + 55 \nu^{11} + 74 \nu^{10} + 96 \nu^{9} + \cdots + 1728 ) / 192 $ (5v^15 + 16v^14 + 16v^13 + 32v^12 + 55v^11 + 74v^10 + 96v^9 + 114v^8 + 189v^7 + 242v^6 + 458v^5 + 536v^4 + 776v^3 + 928v^2 + 1120*v + 1728) / 192
$\beta_{2}$	$=$	$ ( - 3 \nu^{15} - 3 \nu^{14} - 6 \nu^{13} - 6 \nu^{12} - 9 \nu^{11} - 19 \nu^{10} - 16 \nu^{9} + \cdots - 192 ) / 64 $ (-3v^15 - 3v^14 - 6v^13 - 6v^12 - 9v^11 - 19v^10 - 16v^9 - 68v^8 - 45v^7 - 73v^6 - 102v^5 - 136v^4 - 152v^3 - 160v^2 - 288*v - 192) / 64
$\beta_{3}$	$=$	$ ( - 5 \nu^{15} - 17 \nu^{14} - 14 \nu^{13} - 42 \nu^{12} - 55 \nu^{11} - 73 \nu^{10} - 100 \nu^{9} + \cdots - 1856 ) / 64 $ (-5v^15 - 17v^14 - 14v^13 - 42v^12 - 55v^11 - 73v^10 - 100v^9 - 132v^8 - 219v^7 - 163v^6 - 566v^5 - 520v^4 - 1016v^3 - 672v^2 - 1440*v - 1856) / 64
$\beta_{4}$	$=$	$ ( 9 \nu^{15} - 21 \nu^{14} + 42 \nu^{13} - 50 \nu^{12} + 67 \nu^{11} - 73 \nu^{10} + 60 \nu^{9} + \cdots + 320 ) / 128 $ (9v^15 - 21v^14 + 42v^13 - 50v^12 + 67v^11 - 73v^10 + 60v^9 - 85v^7 + 293v^6 - 442v^5 + 840v^4 - 1224v^3 + 1568v^2 - 1376v + 320) / 128
$\beta_{5}$	$=$	$ ( - 11 \nu^{15} - 31 \nu^{14} - 38 \nu^{13} - 78 \nu^{12} - 113 \nu^{11} - 135 \nu^{10} - 232 \nu^{9} + \cdots - 3520 ) / 64 $ (-11v^15 - 31v^14 - 38v^13 - 78v^12 - 113v^11 - 135v^10 - 232v^9 - 268v^8 - 397v^7 - 517v^6 - 798v^5 - 1480v^4 - 1656v^3 - 1952v^2 - 2208*v - 3520) / 64
$\beta_{6}$	$=$	$ ( \nu^{15} - 13 \nu^{14} + 28 \nu^{13} - 46 \nu^{12} + 55 \nu^{11} - 65 \nu^{10} + 74 \nu^{9} + \cdots + 1280 ) / 32 $ (v^15 - 13v^14 + 28v^13 - 46v^12 + 55v^11 - 65v^10 + 74v^9 - 48v^8 - 17v^7 + 105v^6 - 328v^5 + 584v^4 - 992v^3 + 1376v^2 - 1600v + 1280) / 32
$\beta_{7}$	$=$	$ ( - 55 \nu^{15} - 161 \nu^{14} - 122 \nu^{13} - 322 \nu^{12} - 485 \nu^{11} - 637 \nu^{10} + \cdots - 16704 ) / 192 $ (-55v^15 - 161v^14 - 122v^13 - 322v^12 - 485v^11 - 637v^10 - 984v^9 - 1080v^8 - 1989v^7 - 2191v^6 - 4294v^5 - 4984v^4 - 7096v^3 - 7136v^2 - 12320*v - 16704) / 192
$\beta_{8}$	$=$	$ ( - 3 \nu^{15} + 22 \nu^{14} + 6 \nu^{13} + 16 \nu^{12} + 43 \nu^{11} + 20 \nu^{10} + 102 \nu^{9} + \cdots + 2432 ) / 32 $ (-3v^15 + 22v^14 + 6v^13 + 16v^12 + 43v^11 + 20v^10 + 102v^9 + 30v^8 + 197v^7 + 20v^6 + 356v^5 + 312v^4 + 528v^3 + 928v^2 + 256*v + 2432) / 32
$\beta_{9}$	$=$	$ ( - 15 \nu^{15} - 61 \nu^{14} - 54 \nu^{13} - 114 \nu^{12} - 165 \nu^{11} - 241 \nu^{10} - 372 \nu^{9} + \cdots - 6464 ) / 64 $ (-15v^15 - 61v^14 - 54v^13 - 114v^12 - 165v^11 - 241v^10 - 372v^9 - 368v^8 - 733v^7 - 755v^6 - 1562v^5 - 1672v^4 - 3048v^3 - 2848v^2 - 3680*v - 6464) / 64
$\beta_{10}$	$=$	$ ( 8 \nu^{15} - 25 \nu^{14} + 60 \nu^{13} - 86 \nu^{12} + 116 \nu^{11} - 119 \nu^{10} + 130 \nu^{9} + \cdots + 2240 ) / 48 $ (8v^15 - 25v^14 + 60v^13 - 86v^12 + 116v^11 - 119v^10 + 130v^9 - 66v^8 + 6v^7 + 323v^6 - 594v^5 + 1392v^4 - 1800v^3 + 2816v^2 - 2976*v + 2240) / 48
$\beta_{11}$	$=$	$ ( - 161 \nu^{15} + 229 \nu^{14} - 330 \nu^{13} + 290 \nu^{12} - 395 \nu^{11} + 233 \nu^{10} + \cdots + 15040 ) / 384 $ (-161v^15 + 229v^14 - 330v^13 + 290v^12 - 395v^11 + 233v^10 + 68v^9 - 672v^8 + 1341v^7 - 3173v^6 + 4458v^5 - 6648v^4 + 9000v^3 - 7904v^2 + 864*v + 15040) / 384
$\beta_{12}$	$=$	$ ( 71 \nu^{15} - 67 \nu^{14} + 70 \nu^{13} + 18 \nu^{12} - 19 \nu^{11} + 97 \nu^{10} - 284 \nu^{9} + \cdots - 11840 ) / 128 $ (71v^15 - 67v^14 + 70v^13 + 18v^12 - 19v^11 + 97v^10 - 284v^9 + 416v^8 - 587v^7 + 1027v^6 - 998v^5 + 1096v^4 - 1560v^3 - 992v^2 + 4192*v - 11840) / 128
$\beta_{13}$	$=$	$ ( 53 \nu^{15} - 23 \nu^{14} - 2 \nu^{13} + 90 \nu^{12} - 121 \nu^{11} + 209 \nu^{10} - 348 \nu^{9} + \cdots - 11456 ) / 64 $ (53v^15 - 23v^14 - 2v^13 + 90v^12 - 121v^11 + 209v^10 - 348v^9 + 404v^8 - 485v^7 + 539v^6 - 170v^5 - 248v^4 + 632v^3 - 3936v^2 + 6048*v - 11456) / 64
$\beta_{14}$	$=$	$ ( 12 \nu^{15} - 13 \nu^{14} + 16 \nu^{13} - 6 \nu^{12} + 8 \nu^{11} + 5 \nu^{10} - 26 \nu^{9} + \cdots - 1600 ) / 16 $ (12v^15 - 13v^14 + 16v^13 - 6v^12 + 8v^11 + 5v^10 - 26v^9 + 62v^8 - 102v^7 + 191v^6 - 206v^5 + 304v^4 - 408v^3 + 128v^2 + 416*v - 1600) / 16
$\beta_{15}$	$=$	$ ( 427 \nu^{15} - 35 \nu^{14} - 450 \nu^{13} + 1538 \nu^{12} - 2039 \nu^{11} + 3017 \nu^{10} + \cdots - 121664 ) / 384 $ (427v^15 - 35v^14 - 450v^13 + 1538v^12 - 2039v^11 + 3017v^10 - 4180v^9 + 4488v^8 - 4023v^7 + 2971v^6 + 3930v^5 - 11352v^4 + 21480v^3 - 54368v^2 + 86880*v - 121664) / 384

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

$\nu$	$=$	$ ( 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots + 8 ) / 64 $ (2b15 - 2b14 - 2b13 - 2b11 + b8 - b7 + 2b6 + b5 + 4b4 - b3 + 3b2 - 8b1 + 8) / 64
$\nu^{2}$	$=$	$ ( 3\beta_{14} - 2\beta_{13} + \beta_{8} + \beta_{7} - 2\beta_{6} - 4\beta_{4} + \beta_{3} + 8\beta_1 ) / 32 $ (3b14 - 2b13 + b8 + b7 - 2b6 - 4b4 + b3 + 8*b1) / 32
$\nu^{3}$	$=$	$ ( \beta_{14} - 4 \beta_{12} - 4 \beta_{11} + \beta_{10} - 2 \beta_{9} + 2 \beta_{7} - 4 \beta_{6} + \cdots - 8 ) / 32 $ (b14 - 4b12 - 4b11 + b10 - 2b9 + 2b7 - 4b6 - 2b3 + 2b2 - 8b1 - 8) / 32
$\nu^{4}$	$=$	$ ( 2\beta_{15} - 4\beta_{12} + 2\beta_{11} + 5\beta_{10} + 2\beta_{9} - 3\beta_{5} + \beta_{2} - 8 ) / 32 $ (2b15 - 4b12 + 2b11 + 5b10 + 2b9 - 3b5 + b2 - 8) / 32
$\nu^{5}$	$=$	$ ( 2 \beta_{15} + 12 \beta_{14} - 2 \beta_{13} - 8 \beta_{12} + 22 \beta_{11} + 6 \beta_{10} + 4 \beta_{9} + \cdots - 72 ) / 64 $ (2b15 + 12b14 - 2b13 - 8b12 + 22b11 + 6b10 + 4b9 - 3b8 - b7 - 6b6 + 5b5 + 36b4 - 9b3 - 5b2 + 40b1 - 72) / 64
$\nu^{6}$	$=$	$ ( -\beta_{14} - \beta_{8} - \beta_{7} - 2\beta_{6} + 8\beta_{4} + 3\beta_{3} + 4\beta_1 ) / 8 $ (-b14 - b8 - b7 - 2b6 + 8b4 + 3b3 + 4b1) / 8
$\nu^{7}$	$=$	$ ( 2 \beta_{15} - 28 \beta_{14} - 10 \beta_{13} + 40 \beta_{12} - 10 \beta_{11} + 14 \beta_{10} + \cdots - 216 ) / 64 $ (2b15 - 28b14 - 10b13 + 40b12 - 10b11 + 14b10 - 4b9 - 7b8 - 13b7 + 2b6 + 5b5 - 60b4 + 3b3 - 21b2 - 104*b1 - 216) / 64
$\nu^{8}$	$=$	$ ( -2\beta_{15} - 4\beta_{12} - 10\beta_{11} - 9\beta_{10} + 6\beta_{9} - 5\beta_{5} - 49\beta_{2} - 8 ) / 32 $ (-2b15 - 4b12 - 10b11 - 9b10 + 6b9 - 5b5 - 49*b2 - 8) / 32
$\nu^{9}$	$=$	$ ( - 4 \beta_{15} + 29 \beta_{14} + 4 \beta_{13} - 28 \beta_{12} + 8 \beta_{11} - 19 \beta_{10} + \cdots - 296 ) / 32 $ (-4b15 + 29b14 + 4b13 - 28b12 + 8b11 - 19b10 - 10b9 - 6b8 - 8b7 + 16b6 - 2b5 - 8b4 + 4b3 + 4b2 - 136*b1 - 296) / 32
$\nu^{10}$	$=$	$ ( -3\beta_{14} + 10\beta_{13} - 9\beta_{8} + 7\beta_{7} + 50\beta_{6} - 140\beta_{4} + 23\beta_{3} + 200\beta_1 ) / 32 $ (-3b14 + 10b13 - 9b8 + 7b7 + 50b6 - 140b4 + 23b3 + 200b1) / 32
$\nu^{11}$	$=$	$ ( - 42 \beta_{15} - 94 \beta_{14} + 66 \beta_{13} + 32 \beta_{12} - 134 \beta_{11} - 64 \beta_{10} + \cdots - 696 ) / 64 $ (-42b15 - 94b14 + 66b13 + 32b12 - 134b11 - 64b10 + 40b9 - 5b8 - 11b7 + 14b6 + 23b5 - 52b4 - 83b3 + 69b2 + 344*b1 - 696) / 64
$\nu^{12}$	$=$	$ ( -4\beta_{15} + 15\beta_{12} + 7\beta_{11} - 6\beta_{10} + 2\beta_{9} - 5\beta_{5} + 7\beta_{2} + 74 ) / 4 $ (-4b15 + 15b12 + 7b11 - 6b10 + 2b9 - 5b5 + 7*b2 + 74) / 4
$\nu^{13}$	$=$	$ ( 54 \beta_{15} - 254 \beta_{14} + 10 \beta_{13} + 288 \beta_{12} + 138 \beta_{11} - 24 \beta_{10} + \cdots + 1048 ) / 64 $ (54b15 - 254b14 + 10b13 + 288b12 + 138b11 - 24b10 - 128b9 + 39b8 + 105b7 + 246b6 + 55b5 + 460b4 - 135b3 + 53b2 - 408*b1 + 1048) / 64
$\nu^{14}$	$=$	$ ( - 69 \beta_{14} + 70 \beta_{13} + 101 \beta_{8} + 13 \beta_{7} + 22 \beta_{6} + 172 \beta_{4} + \cdots + 104 \beta_1 ) / 32 $ (-69b14 + 70b13 + 101b8 + 13b7 + 22b6 + 172b4 + 109b3 + 104b1) / 32
$\nu^{15}$	$=$	$ ( - 96 \beta_{15} + 317 \beta_{14} + 48 \beta_{13} - 116 \beta_{12} + 44 \beta_{11} + 37 \beta_{10} + \cdots + 1752 ) / 32 $ (-96b15 + 317b14 + 48b13 - 116b12 + 44b11 + 37b10 - 74b9 + 56b8 + 2b7 - 196b6 + 80b5 - 544b4 - 82b3 + 90b2 - 680*b1 + 1752) / 32

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$1$	$-1$

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} $

511.1

−1.06626	−	0.929025i
0.929025	+	1.06626i
−1.41421	−	0.00487727i
−0.00487727	−	1.41421i
1.28251	−	0.595955i
−0.595955	+	1.28251i
0.579920	−	1.28984i
1.28984	−	0.579920i
0.579920	+	1.28984i
1.28984	+	0.579920i
1.28251	+	0.595955i
−0.595955	−	1.28251i
−1.41421	+	0.00487727i
−0.00487727	+	1.41421i
−1.06626	+	0.929025i
0.929025	−	1.06626i

−

5.01222i

−2.23607

8.92613i

−16.1223

511.2

−

5.01222i

2.23607

−

8.92613i

−16.1223

511.3

−

4.13348i

−2.23607

−

2.28473i

−8.08569

511.4

−

4.13348i

2.23607

2.28473i

−8.08569

511.5

−

2.08343i

−2.23607

−

1.18656i

4.65931

511.6

−

2.08343i

2.23607

1.18656i

4.65931

511.7

−

1.20470i

−2.23607

−

12.3974i

7.54870

511.8

−

1.20470i

2.23607

12.3974i

7.54870

511.9

1.20470i

−2.23607

12.3974i

7.54870

511.10

1.20470i

2.23607

−

12.3974i

7.54870

511.11

2.08343i

−2.23607

1.18656i

4.65931

511.12

2.08343i

2.23607

−

1.18656i

4.65931

511.13

4.13348i

−2.23607

2.28473i

−8.08569

511.14

4.13348i

2.23607

−

2.28473i

−8.08569

511.15

5.01222i

−2.23607

−

8.92613i

−16.1223

511.16

5.01222i

2.23607

8.92613i

−16.1223

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1280.3.b.i		16
4.b	odd	2	1	inner	1280.3.b.i		16
8.b	even	2	1	inner	1280.3.b.i		16
8.d	odd	2	1	inner	1280.3.b.i		16
16.e	even	4	1		40.3.g.a	✓	8
16.e	even	4	1		160.3.g.a		8
16.f	odd	4	1		40.3.g.a	✓	8
16.f	odd	4	1		160.3.g.a		8
48.i	odd	4	1		360.3.g.a		8
48.i	odd	4	1		1440.3.g.a		8
48.k	even	4	1		360.3.g.a		8
48.k	even	4	1		1440.3.g.a		8
80.i	odd	4	1		200.3.e.d		16
80.i	odd	4	1		800.3.e.d		16
80.j	even	4	1		200.3.e.d		16
80.j	even	4	1		800.3.e.d		16
80.k	odd	4	1		200.3.g.g		8
80.k	odd	4	1		800.3.g.g		8
80.q	even	4	1		200.3.g.g		8
80.q	even	4	1		800.3.g.g		8
80.s	even	4	1		200.3.e.d		16
80.s	even	4	1		800.3.e.d		16
80.t	odd	4	1		200.3.e.d		16
80.t	odd	4	1		800.3.e.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.3.g.a	✓	8	16.e	even	4	1
40.3.g.a	✓	8	16.f	odd	4	1
160.3.g.a		8	16.e	even	4	1
160.3.g.a		8	16.f	odd	4	1
200.3.e.d		16	80.i	odd	4	1
200.3.e.d		16	80.j	even	4	1
200.3.e.d		16	80.s	even	4	1
200.3.e.d		16	80.t	odd	4	1
200.3.g.g		8	80.k	odd	4	1
200.3.g.g		8	80.q	even	4	1
360.3.g.a		8	48.i	odd	4	1
360.3.g.a		8	48.k	even	4	1
800.3.e.d		16	80.i	odd	4	1
800.3.e.d		16	80.j	even	4	1
800.3.e.d		16	80.s	even	4	1
800.3.e.d		16	80.t	odd	4	1
800.3.g.g		8	80.k	odd	4	1
800.3.g.g		8	80.q	even	4	1
1280.3.b.i		16	1.a	even	1	1	trivial
1280.3.b.i		16	4.b	odd	2	1	inner
1280.3.b.i		16	8.b	even	2	1	inner
1280.3.b.i		16	8.d	odd	2	1	inner
1440.3.g.a		8	48.i	odd	4	1
1440.3.g.a		8	48.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(1280, [\chi])$:

$ T_{3}^{8} + 48T_{3}^{6} + 680T_{3}^{4} + 2752T_{3}^{2} + 2704 $

$ T_{13}^{8} - 880T_{13}^{6} + 170080T_{13}^{4} - 10622720T_{13}^{2} + 138297600 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 48 T^{6} + \cdots + 2704)^{2} $ (T^8 + 48T^6 + 680T^4 + 2752*T^2 + 2704)^2
$5$	$ (T^{2} - 5)^{8} $ (T^2 - 5)^8
$7$	$ (T^{8} + 240 T^{6} + \cdots + 90000)^{2} $ (T^8 + 240T^6 + 13800T^4 + 82880*T^2 + 90000)^2
$11$	$ (T^{8} + 432 T^{6} + \cdots + 60715264)^{2} $ (T^8 + 432T^6 + 63840T^4 + 3646208*T^2 + 60715264)^2
$13$	$ (T^{8} - 880 T^{6} + \cdots + 138297600)^{2} $ (T^8 - 880T^6 + 170080T^4 - 10622720*T^2 + 138297600)^2
$17$	$ (T^{4} - 664 T^{2} + \cdots + 72592)^{4} $ (T^4 - 664T^2 + 1024T + 72592)^4
$19$	$ (T^{8} + 592 T^{6} + \cdots + 7139584)^{2} $ (T^8 + 592T^6 + 115040T^4 + 7396608*T^2 + 7139584)^2
$23$	$ (T^{8} + 2320 T^{6} + \cdots + 47610000)^{2} $ (T^8 + 2320T^6 + 1295400T^4 + 206888000*T^2 + 47610000)^2
$29$	$ (T^{8} - 4480 T^{6} + \cdots + 184968806400)^{2} $ (T^8 - 4480T^6 + 5877760T^4 - 2561146880*T^2 + 184968806400)^2
$31$	$ (T^{8} + 2240 T^{6} + \cdots + 23040000)^{2} $ (T^8 + 2240T^6 + 1040000T^4 + 65024000*T^2 + 23040000)^2
$37$	$ (T^{8} - 6480 T^{6} + \cdots + 134307590400)^{2} $ (T^8 - 6480T^6 + 12541280T^4 - 6469479680*T^2 + 134307590400)^2
$41$	$ (T^{4} + 24 T^{3} + \cdots - 1472)^{4} $ (T^4 + 24T^3 + 32T^2 - 1600*T - 1472)^4
$43$	$ (T^{8} + \cdots + 31337648784016)^{2} $ (T^8 + 12144T^6 + 50920616T^4 + 81214435264*T^2 + 31337648784016)^2
$47$	$ (T^{8} + \cdots + 1101366291600)^{2} $ (T^8 + 6160T^6 + 11390760T^4 + 6478038080*T^2 + 1101366291600)^2
$53$	$ (T^{8} + \cdots + 3957553209600)^{2} $ (T^8 - 9200T^6 + 23243360T^4 - 17849511680*T^2 + 3957553209600)^2
$59$	$ (T^{8} + \cdots + 18557347614976)^{2} $ (T^8 + 11216T^6 + 39463776T^4 + 50950169856*T^2 + 18557347614976)^2
$61$	$ (T^{8} + \cdots + 39126526214400)^{2} $ (T^8 - 19600T^6 + 110516320T^4 - 173208561920*T^2 + 39126526214400)^2
$67$	$ (T^{8} + 2928 T^{6} + \cdots + 485232784)^{2} $ (T^8 + 2928T^6 + 2147880T^4 + 102859712*T^2 + 485232784)^2
$71$	$ (T^{8} + \cdots + 2536502169600)^{2} $ (T^8 + 18880T^6 + 61456000T^4 + 43757957120*T^2 + 2536502169600)^2
$73$	$ (T^{4} + 80 T^{3} + \cdots - 12225008)^{4} $ (T^4 + 80T^3 - 4984T^2 - 566976*T - 12225008)^4
$79$	$ (T^{8} + \cdots + 976317515366400)^{2} $ (T^8 + 33600T^6 + 327226880T^4 + 1057224212480*T^2 + 976317515366400)^2
$83$	$ (T^{8} + 22288 T^{6} + \cdots + 520862437264)^{2} $ (T^8 + 22288T^6 + 118590440T^4 + 184986142272*T^2 + 520862437264)^2
$89$	$ (T^{4} - 24 T^{3} + \cdots - 3224432)^{4} $ (T^4 - 24T^3 - 23688T^2 + 781600*T - 3224432)^4
$97$	$ (T^{4} - 112 T^{3} + \cdots + 83984)^{4} $ (T^4 - 112T^3 + 520T^2 + 22592*T + 83984)^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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{3} - \beta_1 q^{5} + \beta_{12} q^{7} + (\beta_{2} - 3) q^{9}+O(q^{10}) \) q + b4 * q^3 - b1 * q^5 + b12 * q^7 + (b2 - 3) * q^9	\( q + \beta_{4} q^{3} - \beta_1 q^{5} + \beta_{12} q^{7} + (\beta_{2} - 3) q^{9} + \beta_{6} q^{11} - \beta_{3} q^{13} + \beta_{11} q^{15} + \beta_{9} q^{17} + (\beta_{14} - \beta_{6} - 2 \beta_{4}) q^{19} + (\beta_{7} - \beta_{3} + 2 \beta_1) q^{21} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_{10}) q^{23}+ \cdots + ( - 5 \beta_{14} + 4 \beta_{13} + \cdots - 4 \beta_{4}) q^{99}+O(q^{100}) \) q + b4 * q^3 - b1 * q^5 + b12 * q^7 + (b2 - 3) * q^9 + b6 * q^11 - b3 * q^13 + b11 * q^15 + b9 * q^17 + (b14 - b6 - 2b4) q^19 + (b7 - b3 + 2b1) q^21 + (-b15 - b12 - b11 - b10) * q^23 + 5 * q^25 + (2b14 - b13 - b6 - 2b4) * q^27 - b8 * q^29 + (-2b11 - b10) q^31 + (-b9 + b5 + b2 + 2) * q^33 + (b13 + b4) * q^35 + (-b8 - b7 - b3 - 2b1) q^37 + (2b15 - 2b12 + 2b11 - b10) q^39 + (-b2 - 6) * q^41 + (3b13 - 3b6 + b4) * q^43 + (b7 + 3b1) q^45 + (-b12 + 2b11 - 2b10) * q^47 + (-2b9 - 2b5 - b2 - 11) * q^49 + (-b14 - 2b13 + 4b6 - 2b4) q^51 + (-b8 + b7 + 8b1) q^53 + (b15 + b12) * q^55 + (b9 - 4b2 + 22) q^57 + (-5b14 - 2b13 + 3b6 + 6b4) * q^59 + (b8 + 3b7 - 3b3 - 6b1) q^61 + (b15 + b12 - 7b11 - 3b10) * q^63 + (b9 - b5 - b2) * q^65 + (2b14 - b13 + b6 - b4) q^67 + (b8 - 2b7 - 2b3 - 10b1) q^69 + (-2b15 - 4b10) * q^71 + (-b9 + b5 - 5b2 - 20) q^73 + 5b4 q^75 + (-b8 - 3b7 + 6b1) * q^77 + (-2b15 + 2b12 + 8b11 - 3b10) * q^79 + (2b5 + b2 - 7) q^81 + (-6b14 + b13 - 3b6 + 9b4) q^83 + (b8 + b7 - 2b3) q^85 + (-b15 - 4b12 - 3b11 - 5b10) q^87 + (-2b9 - 2b5 + 10b2 + 6) q^89 + (b14 - 2b13 - 2b6 - 14b4) q^91 + (b8 - 3b7 + 3b3 - 24b1) q^93 + (-b15 - b12 - 2b11 - b10) q^95 + (-b9 + b5 - 3b2 + 28) q^97 + (-5b14 + 4b13 - b6 - 4b4) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9}+O(q^{10}) \) 16 * q - 48 * q^9	\( 16 q - 48 q^{9} + 80 q^{25} + 32 q^{33} - 96 q^{41} - 176 q^{49} + 352 q^{57} - 320 q^{73} - 112 q^{81} + 96 q^{89} + 448 q^{97}+O(q^{100}) \) 16 * q - 48 * q^9 + 80 * q^25 + 32 * q^33 - 96 * q^41 - 176 * q^49 + 352 * q^57 - 320 * q^73 - 112 * q^81 + 96 * q^89 + 448 * 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	1280.3.b.i

Level	$1280$
Weight	$3$
Character orbit	1280.b
Analytic conductor	$34.877$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(34.8774738381\)
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:	\( x^{16} - 2 x^{15} + 2 x^{14} - x^{12} + 4 x^{11} - 6 x^{10} + 14 x^{9} - 15 x^{8} + 28 x^{7} + \cdots + 256 \) x^16 - 2x^15 + 2x^14 - x^12 + 4x^11 - 6x^10 + 14x^9 - 15x^8 + 28x^7 - 24x^6 + 32x^5 - 16x^4 + 128x^2 - 256x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{48} \)
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 5 \nu^{15} + 16 \nu^{14} + 16 \nu^{13} + 32 \nu^{12} + 55 \nu^{11} + 74 \nu^{10} + 96 \nu^{9} + \cdots + 1728 ) / 192 \) (5v^15 + 16v^14 + 16v^13 + 32v^12 + 55v^11 + 74v^10 + 96v^9 + 114v^8 + 189v^7 + 242v^6 + 458v^5 + 536v^4 + 776v^3 + 928v^2 + 1120*v + 1728) / 192
\(\beta_{2}\)	\(=\)	\( ( - 3 \nu^{15} - 3 \nu^{14} - 6 \nu^{13} - 6 \nu^{12} - 9 \nu^{11} - 19 \nu^{10} - 16 \nu^{9} + \cdots - 192 ) / 64 \) (-3v^15 - 3v^14 - 6v^13 - 6v^12 - 9v^11 - 19v^10 - 16v^9 - 68v^8 - 45v^7 - 73v^6 - 102v^5 - 136v^4 - 152v^3 - 160v^2 - 288*v - 192) / 64
\(\beta_{3}\)	\(=\)	\( ( - 5 \nu^{15} - 17 \nu^{14} - 14 \nu^{13} - 42 \nu^{12} - 55 \nu^{11} - 73 \nu^{10} - 100 \nu^{9} + \cdots - 1856 ) / 64 \) (-5v^15 - 17v^14 - 14v^13 - 42v^12 - 55v^11 - 73v^10 - 100v^9 - 132v^8 - 219v^7 - 163v^6 - 566v^5 - 520v^4 - 1016v^3 - 672v^2 - 1440*v - 1856) / 64
\(\beta_{4}\)	\(=\)	\( ( 9 \nu^{15} - 21 \nu^{14} + 42 \nu^{13} - 50 \nu^{12} + 67 \nu^{11} - 73 \nu^{10} + 60 \nu^{9} + \cdots + 320 ) / 128 \) (9v^15 - 21v^14 + 42v^13 - 50v^12 + 67v^11 - 73v^10 + 60v^9 - 85v^7 + 293v^6 - 442v^5 + 840v^4 - 1224v^3 + 1568v^2 - 1376v + 320) / 128
\(\beta_{5}\)	\(=\)	\( ( - 11 \nu^{15} - 31 \nu^{14} - 38 \nu^{13} - 78 \nu^{12} - 113 \nu^{11} - 135 \nu^{10} - 232 \nu^{9} + \cdots - 3520 ) / 64 \) (-11v^15 - 31v^14 - 38v^13 - 78v^12 - 113v^11 - 135v^10 - 232v^9 - 268v^8 - 397v^7 - 517v^6 - 798v^5 - 1480v^4 - 1656v^3 - 1952v^2 - 2208*v - 3520) / 64
\(\beta_{6}\)	\(=\)	\( ( \nu^{15} - 13 \nu^{14} + 28 \nu^{13} - 46 \nu^{12} + 55 \nu^{11} - 65 \nu^{10} + 74 \nu^{9} + \cdots + 1280 ) / 32 \) (v^15 - 13v^14 + 28v^13 - 46v^12 + 55v^11 - 65v^10 + 74v^9 - 48v^8 - 17v^7 + 105v^6 - 328v^5 + 584v^4 - 992v^3 + 1376v^2 - 1600v + 1280) / 32
\(\beta_{7}\)	\(=\)	\( ( - 55 \nu^{15} - 161 \nu^{14} - 122 \nu^{13} - 322 \nu^{12} - 485 \nu^{11} - 637 \nu^{10} + \cdots - 16704 ) / 192 \) (-55v^15 - 161v^14 - 122v^13 - 322v^12 - 485v^11 - 637v^10 - 984v^9 - 1080v^8 - 1989v^7 - 2191v^6 - 4294v^5 - 4984v^4 - 7096v^3 - 7136v^2 - 12320*v - 16704) / 192
\(\beta_{8}\)	\(=\)	\( ( - 3 \nu^{15} + 22 \nu^{14} + 6 \nu^{13} + 16 \nu^{12} + 43 \nu^{11} + 20 \nu^{10} + 102 \nu^{9} + \cdots + 2432 ) / 32 \) (-3v^15 + 22v^14 + 6v^13 + 16v^12 + 43v^11 + 20v^10 + 102v^9 + 30v^8 + 197v^7 + 20v^6 + 356v^5 + 312v^4 + 528v^3 + 928v^2 + 256*v + 2432) / 32
\(\beta_{9}\)	\(=\)	\( ( - 15 \nu^{15} - 61 \nu^{14} - 54 \nu^{13} - 114 \nu^{12} - 165 \nu^{11} - 241 \nu^{10} - 372 \nu^{9} + \cdots - 6464 ) / 64 \) (-15v^15 - 61v^14 - 54v^13 - 114v^12 - 165v^11 - 241v^10 - 372v^9 - 368v^8 - 733v^7 - 755v^6 - 1562v^5 - 1672v^4 - 3048v^3 - 2848v^2 - 3680*v - 6464) / 64
\(\beta_{10}\)	\(=\)	\( ( 8 \nu^{15} - 25 \nu^{14} + 60 \nu^{13} - 86 \nu^{12} + 116 \nu^{11} - 119 \nu^{10} + 130 \nu^{9} + \cdots + 2240 ) / 48 \) (8v^15 - 25v^14 + 60v^13 - 86v^12 + 116v^11 - 119v^10 + 130v^9 - 66v^8 + 6v^7 + 323v^6 - 594v^5 + 1392v^4 - 1800v^3 + 2816v^2 - 2976*v + 2240) / 48
\(\beta_{11}\)	\(=\)	\( ( - 161 \nu^{15} + 229 \nu^{14} - 330 \nu^{13} + 290 \nu^{12} - 395 \nu^{11} + 233 \nu^{10} + \cdots + 15040 ) / 384 \) (-161v^15 + 229v^14 - 330v^13 + 290v^12 - 395v^11 + 233v^10 + 68v^9 - 672v^8 + 1341v^7 - 3173v^6 + 4458v^5 - 6648v^4 + 9000v^3 - 7904v^2 + 864*v + 15040) / 384
\(\beta_{12}\)	\(=\)	\( ( 71 \nu^{15} - 67 \nu^{14} + 70 \nu^{13} + 18 \nu^{12} - 19 \nu^{11} + 97 \nu^{10} - 284 \nu^{9} + \cdots - 11840 ) / 128 \) (71v^15 - 67v^14 + 70v^13 + 18v^12 - 19v^11 + 97v^10 - 284v^9 + 416v^8 - 587v^7 + 1027v^6 - 998v^5 + 1096v^4 - 1560v^3 - 992v^2 + 4192*v - 11840) / 128
\(\beta_{13}\)	\(=\)	\( ( 53 \nu^{15} - 23 \nu^{14} - 2 \nu^{13} + 90 \nu^{12} - 121 \nu^{11} + 209 \nu^{10} - 348 \nu^{9} + \cdots - 11456 ) / 64 \) (53v^15 - 23v^14 - 2v^13 + 90v^12 - 121v^11 + 209v^10 - 348v^9 + 404v^8 - 485v^7 + 539v^6 - 170v^5 - 248v^4 + 632v^3 - 3936v^2 + 6048*v - 11456) / 64
\(\beta_{14}\)	\(=\)	\( ( 12 \nu^{15} - 13 \nu^{14} + 16 \nu^{13} - 6 \nu^{12} + 8 \nu^{11} + 5 \nu^{10} - 26 \nu^{9} + \cdots - 1600 ) / 16 \) (12v^15 - 13v^14 + 16v^13 - 6v^12 + 8v^11 + 5v^10 - 26v^9 + 62v^8 - 102v^7 + 191v^6 - 206v^5 + 304v^4 - 408v^3 + 128v^2 + 416*v - 1600) / 16
\(\beta_{15}\)	\(=\)	\( ( 427 \nu^{15} - 35 \nu^{14} - 450 \nu^{13} + 1538 \nu^{12} - 2039 \nu^{11} + 3017 \nu^{10} + \cdots - 121664 ) / 384 \) (427v^15 - 35v^14 - 450v^13 + 1538v^12 - 2039v^11 + 3017v^10 - 4180v^9 + 4488v^8 - 4023v^7 + 2971v^6 + 3930v^5 - 11352v^4 + 21480v^3 - 54368v^2 + 86880*v - 121664) / 384

\(\nu\)	\(=\)	\( ( 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots + 8 ) / 64 \) (2b15 - 2b14 - 2b13 - 2b11 + b8 - b7 + 2b6 + b5 + 4b4 - b3 + 3b2 - 8b1 + 8) / 64
\(\nu^{2}\)	\(=\)	\( ( 3\beta_{14} - 2\beta_{13} + \beta_{8} + \beta_{7} - 2\beta_{6} - 4\beta_{4} + \beta_{3} + 8\beta_1 ) / 32 \) (3b14 - 2b13 + b8 + b7 - 2b6 - 4b4 + b3 + 8*b1) / 32
\(\nu^{3}\)	\(=\)	\( ( \beta_{14} - 4 \beta_{12} - 4 \beta_{11} + \beta_{10} - 2 \beta_{9} + 2 \beta_{7} - 4 \beta_{6} + \cdots - 8 ) / 32 \) (b14 - 4b12 - 4b11 + b10 - 2b9 + 2b7 - 4b6 - 2b3 + 2b2 - 8b1 - 8) / 32
\(\nu^{4}\)	\(=\)	\( ( 2\beta_{15} - 4\beta_{12} + 2\beta_{11} + 5\beta_{10} + 2\beta_{9} - 3\beta_{5} + \beta_{2} - 8 ) / 32 \) (2b15 - 4b12 + 2b11 + 5b10 + 2b9 - 3b5 + b2 - 8) / 32
\(\nu^{5}\)	\(=\)	\( ( 2 \beta_{15} + 12 \beta_{14} - 2 \beta_{13} - 8 \beta_{12} + 22 \beta_{11} + 6 \beta_{10} + 4 \beta_{9} + \cdots - 72 ) / 64 \) (2b15 + 12b14 - 2b13 - 8b12 + 22b11 + 6b10 + 4b9 - 3b8 - b7 - 6b6 + 5b5 + 36b4 - 9b3 - 5b2 + 40b1 - 72) / 64
\(\nu^{6}\)	\(=\)	\( ( -\beta_{14} - \beta_{8} - \beta_{7} - 2\beta_{6} + 8\beta_{4} + 3\beta_{3} + 4\beta_1 ) / 8 \) (-b14 - b8 - b7 - 2b6 + 8b4 + 3b3 + 4b1) / 8
\(\nu^{7}\)	\(=\)	\( ( 2 \beta_{15} - 28 \beta_{14} - 10 \beta_{13} + 40 \beta_{12} - 10 \beta_{11} + 14 \beta_{10} + \cdots - 216 ) / 64 \) (2b15 - 28b14 - 10b13 + 40b12 - 10b11 + 14b10 - 4b9 - 7b8 - 13b7 + 2b6 + 5b5 - 60b4 + 3b3 - 21b2 - 104*b1 - 216) / 64
\(\nu^{8}\)	\(=\)	\( ( -2\beta_{15} - 4\beta_{12} - 10\beta_{11} - 9\beta_{10} + 6\beta_{9} - 5\beta_{5} - 49\beta_{2} - 8 ) / 32 \) (-2b15 - 4b12 - 10b11 - 9b10 + 6b9 - 5b5 - 49*b2 - 8) / 32
\(\nu^{9}\)	\(=\)	\( ( - 4 \beta_{15} + 29 \beta_{14} + 4 \beta_{13} - 28 \beta_{12} + 8 \beta_{11} - 19 \beta_{10} + \cdots - 296 ) / 32 \) (-4b15 + 29b14 + 4b13 - 28b12 + 8b11 - 19b10 - 10b9 - 6b8 - 8b7 + 16b6 - 2b5 - 8b4 + 4b3 + 4b2 - 136*b1 - 296) / 32
\(\nu^{10}\)	\(=\)	\( ( -3\beta_{14} + 10\beta_{13} - 9\beta_{8} + 7\beta_{7} + 50\beta_{6} - 140\beta_{4} + 23\beta_{3} + 200\beta_1 ) / 32 \) (-3b14 + 10b13 - 9b8 + 7b7 + 50b6 - 140b4 + 23b3 + 200b1) / 32
\(\nu^{11}\)	\(=\)	\( ( - 42 \beta_{15} - 94 \beta_{14} + 66 \beta_{13} + 32 \beta_{12} - 134 \beta_{11} - 64 \beta_{10} + \cdots - 696 ) / 64 \) (-42b15 - 94b14 + 66b13 + 32b12 - 134b11 - 64b10 + 40b9 - 5b8 - 11b7 + 14b6 + 23b5 - 52b4 - 83b3 + 69b2 + 344*b1 - 696) / 64
\(\nu^{12}\)	\(=\)	\( ( -4\beta_{15} + 15\beta_{12} + 7\beta_{11} - 6\beta_{10} + 2\beta_{9} - 5\beta_{5} + 7\beta_{2} + 74 ) / 4 \) (-4b15 + 15b12 + 7b11 - 6b10 + 2b9 - 5b5 + 7*b2 + 74) / 4
\(\nu^{13}\)	\(=\)	\( ( 54 \beta_{15} - 254 \beta_{14} + 10 \beta_{13} + 288 \beta_{12} + 138 \beta_{11} - 24 \beta_{10} + \cdots + 1048 ) / 64 \) (54b15 - 254b14 + 10b13 + 288b12 + 138b11 - 24b10 - 128b9 + 39b8 + 105b7 + 246b6 + 55b5 + 460b4 - 135b3 + 53b2 - 408*b1 + 1048) / 64
\(\nu^{14}\)	\(=\)	\( ( - 69 \beta_{14} + 70 \beta_{13} + 101 \beta_{8} + 13 \beta_{7} + 22 \beta_{6} + 172 \beta_{4} + \cdots + 104 \beta_1 ) / 32 \) (-69b14 + 70b13 + 101b8 + 13b7 + 22b6 + 172b4 + 109b3 + 104b1) / 32
\(\nu^{15}\)	\(=\)	\( ( - 96 \beta_{15} + 317 \beta_{14} + 48 \beta_{13} - 116 \beta_{12} + 44 \beta_{11} + 37 \beta_{10} + \cdots + 1752 ) / 32 \) (-96b15 + 317b14 + 48b13 - 116b12 + 44b11 + 37b10 - 74b9 + 56b8 + 2b7 - 196b6 + 80b5 - 544b4 - 82b3 + 90b2 - 680*b1 + 1752) / 32

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 511.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 48 T^{6} + \cdots + 2704)^{2} \) (T^8 + 48T^6 + 680T^4 + 2752*T^2 + 2704)^2
$5$	\( (T^{2} - 5)^{8} \) (T^2 - 5)^8
$7$	\( (T^{8} + 240 T^{6} + \cdots + 90000)^{2} \) (T^8 + 240T^6 + 13800T^4 + 82880*T^2 + 90000)^2
$11$	\( (T^{8} + 432 T^{6} + \cdots + 60715264)^{2} \) (T^8 + 432T^6 + 63840T^4 + 3646208*T^2 + 60715264)^2
$13$	\( (T^{8} - 880 T^{6} + \cdots + 138297600)^{2} \) (T^8 - 880T^6 + 170080T^4 - 10622720*T^2 + 138297600)^2
$17$	\( (T^{4} - 664 T^{2} + \cdots + 72592)^{4} \) (T^4 - 664T^2 + 1024T + 72592)^4
$19$	\( (T^{8} + 592 T^{6} + \cdots + 7139584)^{2} \) (T^8 + 592T^6 + 115040T^4 + 7396608*T^2 + 7139584)^2
$23$	\( (T^{8} + 2320 T^{6} + \cdots + 47610000)^{2} \) (T^8 + 2320T^6 + 1295400T^4 + 206888000*T^2 + 47610000)^2
$29$	\( (T^{8} - 4480 T^{6} + \cdots + 184968806400)^{2} \) (T^8 - 4480T^6 + 5877760T^4 - 2561146880*T^2 + 184968806400)^2
$31$	\( (T^{8} + 2240 T^{6} + \cdots + 23040000)^{2} \) (T^8 + 2240T^6 + 1040000T^4 + 65024000*T^2 + 23040000)^2
$37$	\( (T^{8} - 6480 T^{6} + \cdots + 134307590400)^{2} \) (T^8 - 6480T^6 + 12541280T^4 - 6469479680*T^2 + 134307590400)^2
$41$	\( (T^{4} + 24 T^{3} + \cdots - 1472)^{4} \) (T^4 + 24T^3 + 32T^2 - 1600*T - 1472)^4
$43$	\( (T^{8} + \cdots + 31337648784016)^{2} \) (T^8 + 12144T^6 + 50920616T^4 + 81214435264*T^2 + 31337648784016)^2
$47$	\( (T^{8} + \cdots + 1101366291600)^{2} \) (T^8 + 6160T^6 + 11390760T^4 + 6478038080*T^2 + 1101366291600)^2
$53$	\( (T^{8} + \cdots + 3957553209600)^{2} \) (T^8 - 9200T^6 + 23243360T^4 - 17849511680*T^2 + 3957553209600)^2
$59$	\( (T^{8} + \cdots + 18557347614976)^{2} \) (T^8 + 11216T^6 + 39463776T^4 + 50950169856*T^2 + 18557347614976)^2
$61$	\( (T^{8} + \cdots + 39126526214400)^{2} \) (T^8 - 19600T^6 + 110516320T^4 - 173208561920*T^2 + 39126526214400)^2
$67$	\( (T^{8} + 2928 T^{6} + \cdots + 485232784)^{2} \) (T^8 + 2928T^6 + 2147880T^4 + 102859712*T^2 + 485232784)^2
$71$	\( (T^{8} + \cdots + 2536502169600)^{2} \) (T^8 + 18880T^6 + 61456000T^4 + 43757957120*T^2 + 2536502169600)^2
$73$	\( (T^{4} + 80 T^{3} + \cdots - 12225008)^{4} \) (T^4 + 80T^3 - 4984T^2 - 566976*T - 12225008)^4
$79$	\( (T^{8} + \cdots + 976317515366400)^{2} \) (T^8 + 33600T^6 + 327226880T^4 + 1057224212480*T^2 + 976317515366400)^2
$83$	\( (T^{8} + 22288 T^{6} + \cdots + 520862437264)^{2} \) (T^8 + 22288T^6 + 118590440T^4 + 184986142272*T^2 + 520862437264)^2
$89$	\( (T^{4} - 24 T^{3} + \cdots - 3224432)^{4} \) (T^4 - 24T^3 - 23688T^2 + 781600*T - 3224432)^4
$97$	\( (T^{4} - 112 T^{3} + \cdots + 83984)^{4} \) (T^4 - 112T^3 + 520T^2 + 22592*T + 83984)^4
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\(n\)	\(257\)	\(261\)	\(511\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)