LMFDB - Newform orbit 1280.2.q.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(449,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("1280.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.q (of order $4$, degree $2$, 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:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.9349208943630483456.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 $ x^16 - 8x^15 + 48x^14 - 196x^13 + 642x^12 - 1668x^11 + 3580x^10 - 6328x^9 + 9297x^8 - 11276x^7 + 11224x^6 - 9024x^5 + 5736x^4 - 2780x^3 + 972x^2 - 220*x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{15} + \beta_{9} - \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{7} + \beta_{3} q^{9}+O(q^{10}) $ q - b5 * q^3 + (-b15 + b9 - b4) * q^5 + (-b6 - b5 + b1) * q^7 + b3 * q^9	$ q - \beta_{5} q^{3} + ( - \beta_{15} + \beta_{9} - \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{7} + \beta_{3} q^{9} + (2 \beta_{14} + 3 \beta_{8} - 2 \beta_{2}) q^{11} - \beta_{10} q^{13} + ( - \beta_{12} - \beta_{8} + \cdots + \beta_{5}) q^{15}+ \cdots + (5 \beta_{14} - \beta_{7} + 4 \beta_{2}) q^{99}+O(q^{100}) $ q - b5 * q^3 + (-b15 + b9 - b4) * q^5 + (-b6 - b5 + b1) * q^7 + b3 * q^9 + (2b14 + 3b8 - 2b2) q^11 - b10 * q^13 + (-b12 - b8 - b7 - b6 + b5) * q^15 + (-b15 + b13 + b11 - b10 + b9) * q^17 + (b14 - b7) * q^19 + (-3b9 + 2b4 + 2b3 + 3) q^21 + 3b1 q^23 + (b11 + b10 + b9 + 2b3) q^25 + (b12 - 2b6 - b1) q^27 + (b9 + 2b4 - 2b3 + 1) * q^29 + (b8 + b7 + b2) * q^31 + (-3b15 + 3b13 - 2b11 + 2b10 + 3b9) q^33 + (2b14 - b12 - b7 + 3b5 + b2 - b1) * q^35 + (-4b15 - 2b11 + 2b9 - 2b4 + 2b3 + 2) q^37 + (-4b14 - b8 + b7 + b2) q^39 + (-5b9 - b3) q^41 + (3b12 + b6 - 3b1) * q^43 + (b13 - b10 + 2b9 + b4 - b3 + 1) q^45 + (-3b12 + b6 - b5) q^47 + (5b4 + 2) q^49 + (-3b14 - 3b2) * q^51 + (-2b15 + 3b11 + b9 - b4 + b3 + 1) * q^53 + (-7b14 - 2b8 + 2b7 + 3b6 + 3b5 + 2b2 + 2b1) q^55 + (-b15 - b13 - b11 - b10 - b4 + b3 + 1) * q^57 + (3b14 + b8 - 3b2) * q^59 + (-2b9 - 2) q^61 + (b12 + 2b6 - 2b5) * q^63 + (-b15 + b13 - b11 + b10 + b9 + 2b4) q^65 + (-3b12 - 5b5 - 3b1) q^67 + (3b4 + 3b3) * q^69 + (4b14 - 2b8 + 2b7 + 2b2) * q^71 + (-3b15 - 3b13 - 3b4 + 3b3 + 3) * q^73 + (4b14 + 2b12 + 2b8 + b6 - 4b2 - 2b1) q^75 + (4b13 + 7b10 + 2b9 + 2b4 - 2b3 - 2) q^77 + (-b8 - b7 + b2) * q^79 + (3b4 + 6) q^81 + (b12 + 3b5 + b1) q^83 + (-2b15 - b11 + 4b9 + 2b4 + 4b3 - 2) * q^85 + (-3b6 - 3b5 + 4b1) q^87 + (4b9 - 2b3) * q^89 + (-5b14 - 4b8 + 5b2) q^91 + (2b13 + b9 + b4 - b3 - 1) q^93 + (b8 + b7 - b6 + b5 - 2b2) q^95 + (-b15 + b13 + 4b11 - 4b10 + b9) * q^97 + (5b14 - b7 + 4b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{5}+O(q^{10}) $ 16 * q - 8 * q^5	$ 16 q - 8 q^{5} + 48 q^{21} + 16 q^{29} + 24 q^{45} + 32 q^{49} - 32 q^{61} + 96 q^{81} - 48 q^{85}+O(q^{100}) $ 16 * q - 8 * q^5 + 48 * q^21 + 16 * q^29 + 24 * q^45 + 32 * q^49 - 32 * q^61 + 96 * q^81 - 48 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 $ x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25 :

$\beta_{1}$	$=$	$ ( 50 \nu^{14} - 350 \nu^{13} + 2019 \nu^{12} - 7564 \nu^{11} + 23298 \nu^{10} - 55495 \nu^{9} + \cdots + 2945 ) / 65 $ (50v^14 - 350v^13 + 2019v^12 - 7564v^11 + 23298v^10 - 55495v^9 + 109509v^8 - 173970v^7 + 227114v^6 - 237337v^5 + 197448v^4 - 125328v^3 + 58327v^2 - 17721v + 2945) / 65
$\beta_{2}$	$=$	$ ( 56 \nu^{14} - 392 \nu^{13} + 2267 \nu^{12} - 8506 \nu^{11} + 26269 \nu^{10} - 62716 \nu^{9} + \cdots + 2430 ) / 65 $ (56v^14 - 392v^13 + 2267v^12 - 8506v^11 + 26269v^10 - 62716v^9 + 124065v^8 - 197514v^7 + 258029v^6 - 269446v^5 + 222643v^4 - 139500v^3 + 62551v^2 - 17806v + 2430) / 65
$\beta_{3}$	$=$	$ ( 6912 \nu^{15} - 51840 \nu^{14} + 303752 \nu^{13} - 1188148 \nu^{12} + 3758744 \nu^{11} + \cdots - 268015 ) / 17095 $ (6912v^15 - 51840v^14 + 303752v^13 - 1188148v^12 + 3758744v^11 - 9333192v^10 + 19109472v^9 - 31891566v^8 + 43856968v^7 - 49054352v^6 + 44277328v^5 - 31478376v^4 + 17197336v^3 - 6836856v^2 + 1859848*v - 268015) / 17095
$\beta_{4}$	$=$	$ ( - 10 \nu^{14} + 70 \nu^{13} - 406 \nu^{12} + 1526 \nu^{11} - 4732 \nu^{10} + 11340 \nu^{9} - 22581 \nu^{8} + \cdots - 670 ) / 5 $ (-10v^14 + 70v^13 - 406v^12 + 1526v^11 - 4732v^10 + 11340v^9 - 22581v^8 + 36210v^7 - 47866v^6 + 50708v^5 - 42917v^4 + 27762v^3 - 13198v^2 + 4094v - 670) / 5
$\beta_{5}$	$=$	$ ( 18512 \nu^{15} - 119115 \nu^{14} + 677662 \nu^{13} - 2393217 \nu^{12} + 7123788 \nu^{11} + \cdots + 928550 ) / 17095 $ (18512v^15 - 119115v^14 + 677662v^13 - 2393217v^12 + 7123788v^11 - 15870625v^10 + 29396207v^9 - 42001200v^8 + 47866918v^7 - 39002231v^6 + 20051595v^5 + 205561v^4 - 9984556v^3 + 9343017v^2 - 4230391*v + 928550) / 17095
$\beta_{6}$	$=$	$ ( - 18512 \nu^{15} + 158565 \nu^{14} - 953812 \nu^{13} + 3999884 \nu^{12} - 13173840 \nu^{11} + \cdots + 2010475 ) / 17095 $ (-18512v^15 + 158565v^14 - 953812v^13 + 3999884v^12 - 13173840v^11 + 34707474v^10 - 74703217v^9 + 132742512v^8 - 194257978v^7 + 234158488v^6 - 228737361v^5 + 178549753v^4 - 107184048v^3 + 47180154v^2 - 13549987*v + 2010475) / 17095
$\beta_{7}$	$=$	$ ( - 31954 \nu^{15} + 196786 \nu^{14} - 1112172 \nu^{13} + 3804882 \nu^{12} - 11134674 \nu^{11} + \cdots - 1221660 ) / 17095 $ (-31954v^15 + 196786v^14 - 1112172v^13 + 3804882v^12 - 11134674v^11 + 23941744v^10 - 42972246v^9 + 58065320v^8 - 61566586v^7 + 42749814v^6 - 12846038v^5 - 14153682v^4 + 22376504v^3 - 16136734v^2 + 6273246*v - 1221660) / 17095
$\beta_{8}$	$=$	$ ( 31954 \nu^{15} - 267796 \nu^{14} + 1609242 \nu^{13} - 6688677 \nu^{12} + 21975534 \nu^{11} + \cdots - 3128360 ) / 17095 $ (31954v^15 - 267796v^14 + 1609242v^13 - 6688677v^12 + 21975534v^11 - 57564979v^10 + 123560706v^9 - 218522935v^8 + 318806886v^7 - 382405109v^6 + 372017248v^5 - 288439653v^4 + 171983126v^3 - 74865211v^2 + 21315454*v - 3128360) / 17095
$\beta_{9}$	$=$	$ ( 10652 \nu^{15} - 79890 \nu^{14} + 470562 \nu^{13} - 1846988 \nu^{12} + 5882478 \nu^{11} + \cdots - 370592 ) / 3419 $ (10652v^15 - 79890v^14 + 470562v^13 - 1846988v^12 + 5882478v^11 - 14702424v^10 + 30390552v^9 - 51252471v^8 + 71445138v^7 - 81160006v^6 + 74558856v^5 - 53991081v^4 + 29816774v^3 - 11804598v^2 + 3003630*v - 370592) / 3419
$\beta_{10}$	$=$	$ ( - 65398 \nu^{15} + 432362 \nu^{14} - 2472404 \nu^{13} + 8920509 \nu^{12} - 26909398 \nu^{11} + \cdots - 1617955 ) / 17095 $ (-65398v^15 + 432362v^14 - 2472404v^13 + 8920509v^12 - 26909398v^11 + 61614323v^10 - 117286302v^9 + 176081610v^8 - 214691622v^7 + 201119043v^6 - 141083896v^5 + 63229386v^4 - 9384882v^3 - 9306713v^2 + 6509002*v - 1617955) / 17095
$\beta_{11}$	$=$	$ ( 65398 \nu^{15} - 548608 \nu^{14} + 3286126 \nu^{13} - 13631891 \nu^{12} + 44599304 \nu^{11} + \cdots - 4912335 ) / 17095 $ (65398v^15 - 548608v^14 + 3286126v^13 - 13631891v^12 + 44599304v^11 - 116338837v^10 + 248145108v^9 - 435686280v^8 + 629300076v^7 - 745649497v^6 + 713472362v^5 - 541656894v^4 + 313915212v^3 - 131672333v^2 + 35695134*v - 4912335) / 17095
$\beta_{12}$	$=$	$ ( - 7132 \nu^{15} + 53490 \nu^{14} - 314802 \nu^{13} + 1234948 \nu^{12} - 3929184 \nu^{11} + \cdots + 243510 ) / 1315 $ (-7132v^15 + 53490v^14 - 314802v^13 + 1234948v^12 - 3929184v^11 + 9810867v^10 - 20251957v^9 + 34104411v^8 - 47454158v^7 + 53797197v^6 - 49304463v^5 + 35611106v^4 - 19615956v^3 + 7748766v^2 - 1970153*v + 243510) / 1315
$\beta_{13}$	$=$	$ ( - 103988 \nu^{15} + 793586 \nu^{14} - 4684051 \nu^{13} + 18554445 \nu^{12} - 59336829 \nu^{11} + \cdots + 4607650 ) / 17095 $ (-103988v^15 + 793586v^14 - 4684051v^13 + 18554445v^12 - 59336829v^11 + 149420317v^10 - 310526075v^9 + 527761314v^8 - 740885077v^7 + 849623555v^6 - 788325485v^5 + 578698089v^4 - 324816687v^3 + 131799564v^2 - 34692108*v + 4607650) / 17095
$\beta_{14}$	$=$	$ ( - 8336 \nu^{15} + 62520 \nu^{14} - 367290 \nu^{13} + 1439165 \nu^{12} - 4568352 \nu^{11} + \cdots + 251110 ) / 1315 $ (-8336v^15 + 62520v^14 - 367290v^13 + 1439165v^12 - 4568352v^11 + 11381205v^10 - 23415220v^9 + 39285495v^8 - 54386628v^7 + 61285945v^6 - 55715538v^5 + 39839235v^4 - 21649034v^3 + 8402985v^2 - 2088372*v + 251110) / 1315
$\beta_{15}$	$=$	$ ( 110900 \nu^{15} - 818074 \nu^{14} + 4796339 \nu^{13} - 18624580 \nu^{12} + 58876527 \nu^{11} + \cdots - 2379795 ) / 17095 $ (110900v^15 - 818074v^14 + 4796339v^13 - 18624580v^12 + 58876527v^11 - 145559588v^10 + 297777305v^9 - 495474831v^8 + 680592467v^7 - 758693790v^6 + 681565069v^5 - 479122776v^4 + 254788495v^3 - 95485221v^2 + 22527218*v - 2379795) / 17095

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

$\nu$	$=$	$ ( \beta_{15} + \beta_{14} - \beta_{13} - 2\beta_{9} + 1 ) / 2 $ (b15 + b14 - b13 - 2*b9 + 1) / 2
$\nu^{2}$	$=$	$ ( 2\beta_{15} + \beta_{14} - 2\beta_{9} + 2\beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 - 5 ) / 2 $ (2b15 + b14 - 2b9 + 2b4 - b3 + b2 + 2b1 - 5) / 2
$\nu^{3}$	$=$	$ ( - 4 \beta_{15} - 7 \beta_{14} + 10 \beta_{13} + 6 \beta_{12} + \beta_{11} - \beta_{10} + 18 \beta_{9} + \cdots - 16 ) / 4 $ (-4b15 - 7b14 + 10b13 + 6b12 + b11 - b10 + 18b9 + 3b8 - 3b7 + 6b4 + 6*b1 - 16) / 4
$\nu^{4}$	$=$	$ ( - 14 \beta_{15} - 8 \beta_{14} + 2 \beta_{13} + 6 \beta_{12} + 2 \beta_{11} + 20 \beta_{9} - 6 \beta_{7} + \cdots + 21 ) / 2 $ (-14b15 - 8b14 + 2b13 + 6b12 + 2b11 + 20b9 - 6b7 - 2b6 - 2b5 - 12b4 + 9b3 - 12b2 - 10*b1 + 21) / 2
$\nu^{5}$	$=$	$ ( - 6 \beta_{15} + 23 \beta_{14} - 64 \beta_{13} - 40 \beta_{12} - 3 \beta_{11} + 13 \beta_{10} + \cdots + 132 ) / 4 $ (-6b15 + 23b14 - 64b13 - 40b12 - 3b11 + 13b10 - 86b9 - 35b8 + 5b7 - 20b5 - 70b4 + 10b3 - 30b2 - 60b1 + 132) / 4
$\nu^{6}$	$=$	$ ( 93 \beta_{15} + 55 \beta_{14} - 35 \beta_{13} - 75 \beta_{12} - 24 \beta_{11} + 5 \beta_{10} - 180 \beta_{9} + \cdots - 84 ) / 2 $ (93b15 + 55b14 - 35b13 - 75b12 - 24b11 + 5b10 - 180b9 - 15b8 + 60b7 + 29b6 - b5 + 59b4 - 74b3 + 80b2 + 41b1 - 84) / 2
$\nu^{7}$	$=$	$ ( 238 \beta_{15} - 44 \beta_{14} + 420 \beta_{13} + 154 \beta_{12} - 45 \beta_{11} - 123 \beta_{10} + \cdots - 1069 ) / 4 $ (238b15 - 44b14 + 420b13 + 154b12 - 45b11 - 123b10 + 289b9 + 294b8 + 126b7 + 70b6 + 196b5 + 665b4 - 203b3 + 392b2 + 504*b1 - 1069) / 4
$\nu^{8}$	$=$	$ ( - 560 \beta_{15} - 364 \beta_{14} + 440 \beta_{13} + 672 \beta_{12} + 184 \beta_{11} - 112 \beta_{10} + \cdots + 125 ) / 2 $ (-560b15 - 364b14 + 440b13 + 672b12 + 184b11 - 112b10 + 1466b9 + 280b8 - 420b7 - 240b6 + 152b5 - 134b4 + 529b3 - 428b2 - 64*b1 + 125) / 2
$\nu^{9}$	$=$	$ ( - 2904 \beta_{15} - 325 \beta_{14} - 2430 \beta_{13} + 192 \beta_{12} + 843 \beta_{11} + 855 \beta_{10} + \cdots + 8126 ) / 4 $ (-2904b15 - 325b14 - 2430b13 + 192b12 + 843b11 + 855b10 + 622b9 - 1899b8 - 2007b7 - 1176b6 - 1296b5 - 5502b4 + 2658b3 - 3828b2 - 3864*b1 + 8126) / 4
$\nu^{10}$	$=$	$ ( 2779 \beta_{15} + 2344 \beta_{14} - 4515 \beta_{13} - 5115 \beta_{12} - 1038 \beta_{11} + 1425 \beta_{10} + \cdots + 2874 ) / 2 $ (2779b15 + 2344b14 - 4515b13 - 5115b12 - 1038b11 + 1425b10 - 10803b9 - 3285b8 + 2250b7 + 1407b6 - 2043b5 - 1714b4 - 3002b3 + 1489b2 - 1349*b1 + 2874) / 2
$\nu^{11}$	$=$	$ ( 27636 \beta_{15} + 6327 \beta_{14} + 10204 \beta_{13} - 11792 \beta_{12} - 9222 \beta_{11} + \cdots - 56175 ) / 4 $ (27636b15 + 6327b14 + 10204b13 - 11792b12 - 9222b11 - 4044b10 - 25643b9 + 8569b8 + 20801b7 + 12870b6 + 5940b5 + 39699b4 - 27181b3 + 32450b2 + 26818*b1 - 56175) / 4
$\nu^{12}$	$=$	$ ( - 7610 \beta_{15} - 14156 \beta_{14} + 39918 \beta_{13} + 33726 \beta_{12} + 3320 \beta_{11} + \cdots - 48995 ) / 2 $ (-7610b15 - 14156b14 + 39918b13 + 33726b12 + 3320b11 - 13794b10 + 70393b9 + 30690b8 - 6798b7 - 4574b6 + 19714b5 + 33419b4 + 10411b3 + 4026b2 + 23290*b1 - 48995) / 2
$\nu^{13}$	$=$	$ ( - 113922 \beta_{15} - 35998 \beta_{14} - 608 \beta_{13} + 79378 \beta_{12} + 40329 \beta_{11} + \cdots + 168351 ) / 2 $ (-113922b15 - 35998b14 - 608b13 + 79378b12 + 40329b11 + 1700b10 + 167624b9 - 2496b8 - 88881b7 - 56628b6 - 2548b5 - 121537b4 + 117754b3 - 122161b2 - 80626*b1 + 168351) / 2
$\nu^{14}$	$=$	$ ( - 54082 \beta_{15} + 73086 \beta_{14} - 310674 \beta_{13} - 182546 \beta_{12} + 16149 \beta_{11} + \cdots + 542566 ) / 2 $ (-54082b15 + 73086b14 - 310674b13 - 182546b12 + 16149b11 + 111709b10 - 379119b9 - 244699b8 - 38129b7 - 22844b6 - 158798b5 - 383237b4 + 35504b3 - 150345b2 - 259042*b1 + 542566) / 2
$\nu^{15}$	$=$	$ ( 1656924 \beta_{15} + 681073 \beta_{14} - 604602 \beta_{13} - 1596432 \beta_{12} - 602091 \beta_{11} + \cdots - 1539314 ) / 4 $ (1656924b15 + 681073b14 - 604602b13 - 1596432b12 - 602091b11 + 204687b10 - 3343302b9 - 458997b8 + 1313487b7 + 848770b6 - 290198b5 + 1126458b4 - 1777194b3 + 1607444b2 + 739408*b1 - 1539314) / 4

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

Character values

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

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$-\beta_{9}$	$1$

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} $

449.1

0.500000	−	1.00333i
0.500000	+	2.00333i
0.500000	−	0.331082i
0.500000	+	1.33108i
0.500000	−	1.74530i
0.500000	+	2.74530i
0.500000	+	0.410882i
0.500000	+	0.589118i
0.500000	+	1.00333i
0.500000	−	2.00333i
0.500000	+	0.331082i
0.500000	−	1.33108i
0.500000	+	1.74530i
0.500000	−	2.74530i
0.500000	−	0.410882i
0.500000	−	0.589118i

−1.53819

−

1.53819i

−2.16225

−

0.569800i

−4.20241

1.73205i

449.2

−1.53819

−

1.53819i

−0.569800

−

2.16225i

−4.20241

1.73205i

449.3

−0.796225

−

0.796225i

−1.17216

1.90421i

0.582877

−

1.73205i

449.4

−0.796225

−

0.796225i

1.90421

−

1.17216i

0.582877

−

1.73205i

449.5

0.796225

0.796225i

−1.17216

1.90421i

−0.582877

−

1.73205i

449.6

0.796225

0.796225i

1.90421

−

1.17216i

−0.582877

−

1.73205i

449.7

1.53819

1.53819i

−2.16225

−

0.569800i

4.20241

1.73205i

449.8

1.53819

1.53819i

−0.569800

−

2.16225i

4.20241

1.73205i

1089.1

−1.53819

1.53819i

−2.16225

0.569800i

−4.20241

−

1.73205i

1089.2

−1.53819

1.53819i

−0.569800

2.16225i

−4.20241

−

1.73205i

1089.3

−0.796225

0.796225i

−1.17216

−

1.90421i

0.582877

1.73205i

1089.4

−0.796225

0.796225i

1.90421

1.17216i

0.582877

1.73205i

1089.5

0.796225

−

0.796225i

−1.17216

−

1.90421i

−0.582877

1.73205i

1089.6

0.796225

−

0.796225i

1.90421

1.17216i

−0.582877

1.73205i

1089.7

1.53819

−

1.53819i

−2.16225

0.569800i

4.20241

−

1.73205i

1089.8

1.53819

−

1.53819i

−0.569800

2.16225i

4.20241

−

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
16.e	even	4	1	inner
16.f	odd	4	1	inner
20.d	odd	2	1	inner
80.k	odd	4	1	inner
80.q	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1280.2.q.a	✓	16
4.b	odd	2	1	inner	1280.2.q.a	✓	16
5.b	even	2	1	inner	1280.2.q.a	✓	16
8.b	even	2	1		1280.2.q.b	yes	16
8.d	odd	2	1		1280.2.q.b	yes	16
16.e	even	4	1	inner	1280.2.q.a	✓	16
16.e	even	4	1		1280.2.q.b	yes	16
16.f	odd	4	1	inner	1280.2.q.a	✓	16
16.f	odd	4	1		1280.2.q.b	yes	16
20.d	odd	2	1	inner	1280.2.q.a	✓	16
40.e	odd	2	1		1280.2.q.b	yes	16
40.f	even	2	1		1280.2.q.b	yes	16
80.k	odd	4	1	inner	1280.2.q.a	✓	16
80.k	odd	4	1		1280.2.q.b	yes	16
80.q	even	4	1	inner	1280.2.q.a	✓	16
80.q	even	4	1		1280.2.q.b	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1280.2.q.a	✓	16	1.a	even	1	1	trivial
1280.2.q.a	✓	16	4.b	odd	2	1	inner
1280.2.q.a	✓	16	5.b	even	2	1	inner
1280.2.q.a	✓	16	16.e	even	4	1	inner
1280.2.q.a	✓	16	16.f	odd	4	1	inner
1280.2.q.a	✓	16	20.d	odd	2	1	inner
1280.2.q.a	✓	16	80.k	odd	4	1	inner
1280.2.q.a	✓	16	80.q	even	4	1	inner
1280.2.q.b	yes	16	8.b	even	2	1
1280.2.q.b	yes	16	8.d	odd	2	1
1280.2.q.b	yes	16	16.e	even	4	1
1280.2.q.b	yes	16	16.f	odd	4	1
1280.2.q.b	yes	16	40.e	odd	2	1
1280.2.q.b	yes	16	40.f	even	2	1
1280.2.q.b	yes	16	80.k	odd	4	1
1280.2.q.b	yes	16	80.q	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1280, [\chi])$:

$ T_{3}^{8} + 24T_{3}^{4} + 36 $

$ T_{29}^{4} - 4T_{29}^{3} + 8T_{29}^{2} + 88T_{29} + 484 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 24 T^{4} + 36)^{2} $ (T^8 + 24*T^4 + 36)^2
$5$	$ (T^{8} + 4 T^{7} + \cdots + 625)^{2} $ (T^8 + 4T^7 + 8T^6 + 4T^5 - 14T^4 + 20T^3 + 200T^2 + 500*T + 625)^2
$7$	$ (T^{4} - 18 T^{2} + 6)^{4} $ (T^4 - 18*T^2 + 6)^4
$11$	$ (T^{8} + 1784 T^{4} + 456976)^{2} $ (T^8 + 1784*T^4 + 456976)^2
$13$	$ (T^{8} + 96 T^{4} + 576)^{2} $ (T^8 + 96*T^4 + 576)^2
$17$	$ (T^{4} + 36 T^{2} + 216)^{4} $ (T^4 + 36*T^2 + 216)^4
$19$	$ (T^{8} + 56 T^{4} + 16)^{2} $ (T^8 + 56*T^4 + 16)^2
$23$	$ (T^{4} - 54 T^{2} + 486)^{4} $ (T^4 - 54*T^2 + 486)^4
$29$	$ (T^{4} - 4 T^{3} + \cdots + 484)^{4} $ (T^4 - 4T^3 + 8T^2 + 88*T + 484)^4
$31$	$ (T^{4} - 16 T^{2} + 16)^{4} $ (T^4 - 16*T^2 + 16)^4
$37$	$ (T^{8} + 13824 T^{4} + 11943936)^{2} $ (T^8 + 13824*T^4 + 11943936)^2
$41$	$ (T^{4} + 56 T^{2} + 484)^{4} $ (T^4 + 56*T^2 + 484)^4
$43$	$ (T^{8} + 6648 T^{4} + 10074276)^{2} $ (T^8 + 6648*T^4 + 10074276)^2
$47$	$ (T^{4} + 66 T^{2} + 6)^{4} $ (T^4 + 66*T^2 + 6)^4
$53$	$ (T^{8} + 9312 T^{4} + 16451136)^{2} $ (T^8 + 9312*T^4 + 16451136)^2
$59$	$ (T^{8} + 2168 T^{4} + 234256)^{2} $ (T^8 + 2168*T^4 + 234256)^2
$61$	$ (T^{2} + 4 T + 8)^{8} $ (T^2 + 4*T + 8)^8
$67$	$ (T^{8} + 50136 T^{4} + 67469796)^{2} $ (T^8 + 50136*T^4 + 67469796)^2
$71$	$ (T^{4} + 192 T^{2} + 2304)^{4} $ (T^4 + 192*T^2 + 2304)^4
$73$	$ (T^{4} - 108 T^{2} + 1944)^{4} $ (T^4 - 108*T^2 + 1944)^4
$79$	$ (T^{4} - 16 T^{2} + 16)^{4} $ (T^4 - 16*T^2 + 16)^4
$83$	$ (T^{8} + 2328 T^{4} + 1028196)^{2} $ (T^8 + 2328*T^4 + 1028196)^2
$89$	$ (T^{4} + 56 T^{2} + 16)^{4} $ (T^4 + 56*T^2 + 16)^4
$97$	$ (T^{4} + 396 T^{2} + 36504)^{4} $ (T^4 + 396*T^2 + 36504)^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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.q (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{3} + ( - \beta_{15} + \beta_{9} - \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{7} + \beta_{3} q^{9}+O(q^{10}) \) q - b5 * q^3 + (-b15 + b9 - b4) * q^5 + (-b6 - b5 + b1) * q^7 + b3 * q^9	\( q - \beta_{5} q^{3} + ( - \beta_{15} + \beta_{9} - \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{7} + \beta_{3} q^{9} + (2 \beta_{14} + 3 \beta_{8} - 2 \beta_{2}) q^{11} - \beta_{10} q^{13} + ( - \beta_{12} - \beta_{8} + \cdots + \beta_{5}) q^{15}+ \cdots + (5 \beta_{14} - \beta_{7} + 4 \beta_{2}) q^{99}+O(q^{100}) \) q - b5 * q^3 + (-b15 + b9 - b4) * q^5 + (-b6 - b5 + b1) * q^7 + b3 * q^9 + (2b14 + 3b8 - 2b2) q^11 - b10 * q^13 + (-b12 - b8 - b7 - b6 + b5) * q^15 + (-b15 + b13 + b11 - b10 + b9) * q^17 + (b14 - b7) * q^19 + (-3b9 + 2b4 + 2b3 + 3) q^21 + 3b1 q^23 + (b11 + b10 + b9 + 2b3) q^25 + (b12 - 2b6 - b1) q^27 + (b9 + 2b4 - 2b3 + 1) * q^29 + (b8 + b7 + b2) * q^31 + (-3b15 + 3b13 - 2b11 + 2b10 + 3b9) q^33 + (2b14 - b12 - b7 + 3b5 + b2 - b1) * q^35 + (-4b15 - 2b11 + 2b9 - 2b4 + 2b3 + 2) q^37 + (-4b14 - b8 + b7 + b2) q^39 + (-5b9 - b3) q^41 + (3b12 + b6 - 3b1) * q^43 + (b13 - b10 + 2b9 + b4 - b3 + 1) q^45 + (-3b12 + b6 - b5) q^47 + (5b4 + 2) q^49 + (-3b14 - 3b2) * q^51 + (-2b15 + 3b11 + b9 - b4 + b3 + 1) * q^53 + (-7b14 - 2b8 + 2b7 + 3b6 + 3b5 + 2b2 + 2b1) q^55 + (-b15 - b13 - b11 - b10 - b4 + b3 + 1) * q^57 + (3b14 + b8 - 3b2) * q^59 + (-2b9 - 2) q^61 + (b12 + 2b6 - 2b5) * q^63 + (-b15 + b13 - b11 + b10 + b9 + 2b4) q^65 + (-3b12 - 5b5 - 3b1) q^67 + (3b4 + 3b3) * q^69 + (4b14 - 2b8 + 2b7 + 2b2) * q^71 + (-3b15 - 3b13 - 3b4 + 3b3 + 3) * q^73 + (4b14 + 2b12 + 2b8 + b6 - 4b2 - 2b1) q^75 + (4b13 + 7b10 + 2b9 + 2b4 - 2b3 - 2) q^77 + (-b8 - b7 + b2) * q^79 + (3b4 + 6) q^81 + (b12 + 3b5 + b1) q^83 + (-2b15 - b11 + 4b9 + 2b4 + 4b3 - 2) * q^85 + (-3b6 - 3b5 + 4b1) q^87 + (4b9 - 2b3) * q^89 + (-5b14 - 4b8 + 5b2) q^91 + (2b13 + b9 + b4 - b3 - 1) q^93 + (b8 + b7 - b6 + b5 - 2b2) q^95 + (-b15 + b13 + 4b11 - 4b10 + b9) * q^97 + (5b14 - b7 + 4b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{5}+O(q^{10}) \) 16 * q - 8 * q^5	\( 16 q - 8 q^{5} + 48 q^{21} + 16 q^{29} + 24 q^{45} + 32 q^{49} - 32 q^{61} + 96 q^{81} - 48 q^{85}+O(q^{100}) \) 16 * q - 8 * q^5 + 48 * q^21 + 16 * q^29 + 24 * q^45 + 32 * q^49 - 32 * q^61 + 96 * q^81 - 48 * q^85

Introduction

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Modular forms

Varieties

Fields

Representations

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Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1280.2.q.a

Level	$1280$
Weight	$2$
Character orbit	1280.q
Analytic conductor	$10.221$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(10.2208514587\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	16.0.9349208943630483456.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) x^16 - 8x^15 + 48x^14 - 196x^13 + 642x^12 - 1668x^11 + 3580x^10 - 6328x^9 + 9297x^8 - 11276x^7 + 11224x^6 - 9024x^5 + 5736x^4 - 2780x^3 + 972x^2 - 220*x + 25
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 50 \nu^{14} - 350 \nu^{13} + 2019 \nu^{12} - 7564 \nu^{11} + 23298 \nu^{10} - 55495 \nu^{9} + \cdots + 2945 ) / 65 \) (50v^14 - 350v^13 + 2019v^12 - 7564v^11 + 23298v^10 - 55495v^9 + 109509v^8 - 173970v^7 + 227114v^6 - 237337v^5 + 197448v^4 - 125328v^3 + 58327v^2 - 17721v + 2945) / 65
\(\beta_{2}\)	\(=\)	\( ( 56 \nu^{14} - 392 \nu^{13} + 2267 \nu^{12} - 8506 \nu^{11} + 26269 \nu^{10} - 62716 \nu^{9} + \cdots + 2430 ) / 65 \) (56v^14 - 392v^13 + 2267v^12 - 8506v^11 + 26269v^10 - 62716v^9 + 124065v^8 - 197514v^7 + 258029v^6 - 269446v^5 + 222643v^4 - 139500v^3 + 62551v^2 - 17806v + 2430) / 65
\(\beta_{3}\)	\(=\)	\( ( 6912 \nu^{15} - 51840 \nu^{14} + 303752 \nu^{13} - 1188148 \nu^{12} + 3758744 \nu^{11} + \cdots - 268015 ) / 17095 \) (6912v^15 - 51840v^14 + 303752v^13 - 1188148v^12 + 3758744v^11 - 9333192v^10 + 19109472v^9 - 31891566v^8 + 43856968v^7 - 49054352v^6 + 44277328v^5 - 31478376v^4 + 17197336v^3 - 6836856v^2 + 1859848*v - 268015) / 17095
\(\beta_{4}\)	\(=\)	\( ( - 10 \nu^{14} + 70 \nu^{13} - 406 \nu^{12} + 1526 \nu^{11} - 4732 \nu^{10} + 11340 \nu^{9} - 22581 \nu^{8} + \cdots - 670 ) / 5 \) (-10v^14 + 70v^13 - 406v^12 + 1526v^11 - 4732v^10 + 11340v^9 - 22581v^8 + 36210v^7 - 47866v^6 + 50708v^5 - 42917v^4 + 27762v^3 - 13198v^2 + 4094v - 670) / 5
\(\beta_{5}\)	\(=\)	\( ( 18512 \nu^{15} - 119115 \nu^{14} + 677662 \nu^{13} - 2393217 \nu^{12} + 7123788 \nu^{11} + \cdots + 928550 ) / 17095 \) (18512v^15 - 119115v^14 + 677662v^13 - 2393217v^12 + 7123788v^11 - 15870625v^10 + 29396207v^9 - 42001200v^8 + 47866918v^7 - 39002231v^6 + 20051595v^5 + 205561v^4 - 9984556v^3 + 9343017v^2 - 4230391*v + 928550) / 17095
\(\beta_{6}\)	\(=\)	\( ( - 18512 \nu^{15} + 158565 \nu^{14} - 953812 \nu^{13} + 3999884 \nu^{12} - 13173840 \nu^{11} + \cdots + 2010475 ) / 17095 \) (-18512v^15 + 158565v^14 - 953812v^13 + 3999884v^12 - 13173840v^11 + 34707474v^10 - 74703217v^9 + 132742512v^8 - 194257978v^7 + 234158488v^6 - 228737361v^5 + 178549753v^4 - 107184048v^3 + 47180154v^2 - 13549987*v + 2010475) / 17095
\(\beta_{7}\)	\(=\)	\( ( - 31954 \nu^{15} + 196786 \nu^{14} - 1112172 \nu^{13} + 3804882 \nu^{12} - 11134674 \nu^{11} + \cdots - 1221660 ) / 17095 \) (-31954v^15 + 196786v^14 - 1112172v^13 + 3804882v^12 - 11134674v^11 + 23941744v^10 - 42972246v^9 + 58065320v^8 - 61566586v^7 + 42749814v^6 - 12846038v^5 - 14153682v^4 + 22376504v^3 - 16136734v^2 + 6273246*v - 1221660) / 17095
\(\beta_{8}\)	\(=\)	\( ( 31954 \nu^{15} - 267796 \nu^{14} + 1609242 \nu^{13} - 6688677 \nu^{12} + 21975534 \nu^{11} + \cdots - 3128360 ) / 17095 \) (31954v^15 - 267796v^14 + 1609242v^13 - 6688677v^12 + 21975534v^11 - 57564979v^10 + 123560706v^9 - 218522935v^8 + 318806886v^7 - 382405109v^6 + 372017248v^5 - 288439653v^4 + 171983126v^3 - 74865211v^2 + 21315454*v - 3128360) / 17095
\(\beta_{9}\)	\(=\)	\( ( 10652 \nu^{15} - 79890 \nu^{14} + 470562 \nu^{13} - 1846988 \nu^{12} + 5882478 \nu^{11} + \cdots - 370592 ) / 3419 \) (10652v^15 - 79890v^14 + 470562v^13 - 1846988v^12 + 5882478v^11 - 14702424v^10 + 30390552v^9 - 51252471v^8 + 71445138v^7 - 81160006v^6 + 74558856v^5 - 53991081v^4 + 29816774v^3 - 11804598v^2 + 3003630*v - 370592) / 3419
\(\beta_{10}\)	\(=\)	\( ( - 65398 \nu^{15} + 432362 \nu^{14} - 2472404 \nu^{13} + 8920509 \nu^{12} - 26909398 \nu^{11} + \cdots - 1617955 ) / 17095 \) (-65398v^15 + 432362v^14 - 2472404v^13 + 8920509v^12 - 26909398v^11 + 61614323v^10 - 117286302v^9 + 176081610v^8 - 214691622v^7 + 201119043v^6 - 141083896v^5 + 63229386v^4 - 9384882v^3 - 9306713v^2 + 6509002*v - 1617955) / 17095
\(\beta_{11}\)	\(=\)	\( ( 65398 \nu^{15} - 548608 \nu^{14} + 3286126 \nu^{13} - 13631891 \nu^{12} + 44599304 \nu^{11} + \cdots - 4912335 ) / 17095 \) (65398v^15 - 548608v^14 + 3286126v^13 - 13631891v^12 + 44599304v^11 - 116338837v^10 + 248145108v^9 - 435686280v^8 + 629300076v^7 - 745649497v^6 + 713472362v^5 - 541656894v^4 + 313915212v^3 - 131672333v^2 + 35695134*v - 4912335) / 17095
\(\beta_{12}\)	\(=\)	\( ( - 7132 \nu^{15} + 53490 \nu^{14} - 314802 \nu^{13} + 1234948 \nu^{12} - 3929184 \nu^{11} + \cdots + 243510 ) / 1315 \) (-7132v^15 + 53490v^14 - 314802v^13 + 1234948v^12 - 3929184v^11 + 9810867v^10 - 20251957v^9 + 34104411v^8 - 47454158v^7 + 53797197v^6 - 49304463v^5 + 35611106v^4 - 19615956v^3 + 7748766v^2 - 1970153*v + 243510) / 1315
\(\beta_{13}\)	\(=\)	\( ( - 103988 \nu^{15} + 793586 \nu^{14} - 4684051 \nu^{13} + 18554445 \nu^{12} - 59336829 \nu^{11} + \cdots + 4607650 ) / 17095 \) (-103988v^15 + 793586v^14 - 4684051v^13 + 18554445v^12 - 59336829v^11 + 149420317v^10 - 310526075v^9 + 527761314v^8 - 740885077v^7 + 849623555v^6 - 788325485v^5 + 578698089v^4 - 324816687v^3 + 131799564v^2 - 34692108*v + 4607650) / 17095
\(\beta_{14}\)	\(=\)	\( ( - 8336 \nu^{15} + 62520 \nu^{14} - 367290 \nu^{13} + 1439165 \nu^{12} - 4568352 \nu^{11} + \cdots + 251110 ) / 1315 \) (-8336v^15 + 62520v^14 - 367290v^13 + 1439165v^12 - 4568352v^11 + 11381205v^10 - 23415220v^9 + 39285495v^8 - 54386628v^7 + 61285945v^6 - 55715538v^5 + 39839235v^4 - 21649034v^3 + 8402985v^2 - 2088372*v + 251110) / 1315
\(\beta_{15}\)	\(=\)	\( ( 110900 \nu^{15} - 818074 \nu^{14} + 4796339 \nu^{13} - 18624580 \nu^{12} + 58876527 \nu^{11} + \cdots - 2379795 ) / 17095 \) (110900v^15 - 818074v^14 + 4796339v^13 - 18624580v^12 + 58876527v^11 - 145559588v^10 + 297777305v^9 - 495474831v^8 + 680592467v^7 - 758693790v^6 + 681565069v^5 - 479122776v^4 + 254788495v^3 - 95485221v^2 + 22527218*v - 2379795) / 17095

\(\nu\)	\(=\)	\( ( \beta_{15} + \beta_{14} - \beta_{13} - 2\beta_{9} + 1 ) / 2 \) (b15 + b14 - b13 - 2*b9 + 1) / 2
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{15} + \beta_{14} - 2\beta_{9} + 2\beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 - 5 ) / 2 \) (2b15 + b14 - 2b9 + 2b4 - b3 + b2 + 2b1 - 5) / 2
\(\nu^{3}\)	\(=\)	\( ( - 4 \beta_{15} - 7 \beta_{14} + 10 \beta_{13} + 6 \beta_{12} + \beta_{11} - \beta_{10} + 18 \beta_{9} + \cdots - 16 ) / 4 \) (-4b15 - 7b14 + 10b13 + 6b12 + b11 - b10 + 18b9 + 3b8 - 3b7 + 6b4 + 6*b1 - 16) / 4
\(\nu^{4}\)	\(=\)	\( ( - 14 \beta_{15} - 8 \beta_{14} + 2 \beta_{13} + 6 \beta_{12} + 2 \beta_{11} + 20 \beta_{9} - 6 \beta_{7} + \cdots + 21 ) / 2 \) (-14b15 - 8b14 + 2b13 + 6b12 + 2b11 + 20b9 - 6b7 - 2b6 - 2b5 - 12b4 + 9b3 - 12b2 - 10*b1 + 21) / 2
\(\nu^{5}\)	\(=\)	\( ( - 6 \beta_{15} + 23 \beta_{14} - 64 \beta_{13} - 40 \beta_{12} - 3 \beta_{11} + 13 \beta_{10} + \cdots + 132 ) / 4 \) (-6b15 + 23b14 - 64b13 - 40b12 - 3b11 + 13b10 - 86b9 - 35b8 + 5b7 - 20b5 - 70b4 + 10b3 - 30b2 - 60b1 + 132) / 4
\(\nu^{6}\)	\(=\)	\( ( 93 \beta_{15} + 55 \beta_{14} - 35 \beta_{13} - 75 \beta_{12} - 24 \beta_{11} + 5 \beta_{10} - 180 \beta_{9} + \cdots - 84 ) / 2 \) (93b15 + 55b14 - 35b13 - 75b12 - 24b11 + 5b10 - 180b9 - 15b8 + 60b7 + 29b6 - b5 + 59b4 - 74b3 + 80b2 + 41b1 - 84) / 2
\(\nu^{7}\)	\(=\)	\( ( 238 \beta_{15} - 44 \beta_{14} + 420 \beta_{13} + 154 \beta_{12} - 45 \beta_{11} - 123 \beta_{10} + \cdots - 1069 ) / 4 \) (238b15 - 44b14 + 420b13 + 154b12 - 45b11 - 123b10 + 289b9 + 294b8 + 126b7 + 70b6 + 196b5 + 665b4 - 203b3 + 392b2 + 504*b1 - 1069) / 4
\(\nu^{8}\)	\(=\)	\( ( - 560 \beta_{15} - 364 \beta_{14} + 440 \beta_{13} + 672 \beta_{12} + 184 \beta_{11} - 112 \beta_{10} + \cdots + 125 ) / 2 \) (-560b15 - 364b14 + 440b13 + 672b12 + 184b11 - 112b10 + 1466b9 + 280b8 - 420b7 - 240b6 + 152b5 - 134b4 + 529b3 - 428b2 - 64*b1 + 125) / 2
\(\nu^{9}\)	\(=\)	\( ( - 2904 \beta_{15} - 325 \beta_{14} - 2430 \beta_{13} + 192 \beta_{12} + 843 \beta_{11} + 855 \beta_{10} + \cdots + 8126 ) / 4 \) (-2904b15 - 325b14 - 2430b13 + 192b12 + 843b11 + 855b10 + 622b9 - 1899b8 - 2007b7 - 1176b6 - 1296b5 - 5502b4 + 2658b3 - 3828b2 - 3864*b1 + 8126) / 4
\(\nu^{10}\)	\(=\)	\( ( 2779 \beta_{15} + 2344 \beta_{14} - 4515 \beta_{13} - 5115 \beta_{12} - 1038 \beta_{11} + 1425 \beta_{10} + \cdots + 2874 ) / 2 \) (2779b15 + 2344b14 - 4515b13 - 5115b12 - 1038b11 + 1425b10 - 10803b9 - 3285b8 + 2250b7 + 1407b6 - 2043b5 - 1714b4 - 3002b3 + 1489b2 - 1349*b1 + 2874) / 2
\(\nu^{11}\)	\(=\)	\( ( 27636 \beta_{15} + 6327 \beta_{14} + 10204 \beta_{13} - 11792 \beta_{12} - 9222 \beta_{11} + \cdots - 56175 ) / 4 \) (27636b15 + 6327b14 + 10204b13 - 11792b12 - 9222b11 - 4044b10 - 25643b9 + 8569b8 + 20801b7 + 12870b6 + 5940b5 + 39699b4 - 27181b3 + 32450b2 + 26818*b1 - 56175) / 4
\(\nu^{12}\)	\(=\)	\( ( - 7610 \beta_{15} - 14156 \beta_{14} + 39918 \beta_{13} + 33726 \beta_{12} + 3320 \beta_{11} + \cdots - 48995 ) / 2 \) (-7610b15 - 14156b14 + 39918b13 + 33726b12 + 3320b11 - 13794b10 + 70393b9 + 30690b8 - 6798b7 - 4574b6 + 19714b5 + 33419b4 + 10411b3 + 4026b2 + 23290*b1 - 48995) / 2
\(\nu^{13}\)	\(=\)	\( ( - 113922 \beta_{15} - 35998 \beta_{14} - 608 \beta_{13} + 79378 \beta_{12} + 40329 \beta_{11} + \cdots + 168351 ) / 2 \) (-113922b15 - 35998b14 - 608b13 + 79378b12 + 40329b11 + 1700b10 + 167624b9 - 2496b8 - 88881b7 - 56628b6 - 2548b5 - 121537b4 + 117754b3 - 122161b2 - 80626*b1 + 168351) / 2
\(\nu^{14}\)	\(=\)	\( ( - 54082 \beta_{15} + 73086 \beta_{14} - 310674 \beta_{13} - 182546 \beta_{12} + 16149 \beta_{11} + \cdots + 542566 ) / 2 \) (-54082b15 + 73086b14 - 310674b13 - 182546b12 + 16149b11 + 111709b10 - 379119b9 - 244699b8 - 38129b7 - 22844b6 - 158798b5 - 383237b4 + 35504b3 - 150345b2 - 259042*b1 + 542566) / 2
\(\nu^{15}\)	\(=\)	\( ( 1656924 \beta_{15} + 681073 \beta_{14} - 604602 \beta_{13} - 1596432 \beta_{12} - 602091 \beta_{11} + \cdots - 1539314 ) / 4 \) (1656924b15 + 681073b14 - 604602b13 - 1596432b12 - 602091b11 + 204687b10 - 3343302b9 - 458997b8 + 1313487b7 + 848770b6 - 290198b5 + 1126458b4 - 1777194b3 + 1607444b2 + 739408*b1 - 1539314) / 4

\(n\)	\(257\)	\(261\)	\(511\)
\(\chi(n)\)	\(-1\)	\(-\beta_{9}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 449.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 24 T^{4} + 36)^{2} \) (T^8 + 24*T^4 + 36)^2
$5$	\( (T^{8} + 4 T^{7} + \cdots + 625)^{2} \) (T^8 + 4T^7 + 8T^6 + 4T^5 - 14T^4 + 20T^3 + 200T^2 + 500*T + 625)^2
$7$	\( (T^{4} - 18 T^{2} + 6)^{4} \) (T^4 - 18*T^2 + 6)^4
$11$	\( (T^{8} + 1784 T^{4} + 456976)^{2} \) (T^8 + 1784*T^4 + 456976)^2
$13$	\( (T^{8} + 96 T^{4} + 576)^{2} \) (T^8 + 96*T^4 + 576)^2
$17$	\( (T^{4} + 36 T^{2} + 216)^{4} \) (T^4 + 36*T^2 + 216)^4
$19$	\( (T^{8} + 56 T^{4} + 16)^{2} \) (T^8 + 56*T^4 + 16)^2
$23$	\( (T^{4} - 54 T^{2} + 486)^{4} \) (T^4 - 54*T^2 + 486)^4
$29$	\( (T^{4} - 4 T^{3} + \cdots + 484)^{4} \) (T^4 - 4T^3 + 8T^2 + 88*T + 484)^4
$31$	\( (T^{4} - 16 T^{2} + 16)^{4} \) (T^4 - 16*T^2 + 16)^4
$37$	\( (T^{8} + 13824 T^{4} + 11943936)^{2} \) (T^8 + 13824*T^4 + 11943936)^2
$41$	\( (T^{4} + 56 T^{2} + 484)^{4} \) (T^4 + 56*T^2 + 484)^4
$43$	\( (T^{8} + 6648 T^{4} + 10074276)^{2} \) (T^8 + 6648*T^4 + 10074276)^2
$47$	\( (T^{4} + 66 T^{2} + 6)^{4} \) (T^4 + 66*T^2 + 6)^4
$53$	\( (T^{8} + 9312 T^{4} + 16451136)^{2} \) (T^8 + 9312*T^4 + 16451136)^2
$59$	\( (T^{8} + 2168 T^{4} + 234256)^{2} \) (T^8 + 2168*T^4 + 234256)^2
$61$	\( (T^{2} + 4 T + 8)^{8} \) (T^2 + 4*T + 8)^8
$67$	\( (T^{8} + 50136 T^{4} + 67469796)^{2} \) (T^8 + 50136*T^4 + 67469796)^2
$71$	\( (T^{4} + 192 T^{2} + 2304)^{4} \) (T^4 + 192*T^2 + 2304)^4
$73$	\( (T^{4} - 108 T^{2} + 1944)^{4} \) (T^4 - 108*T^2 + 1944)^4
$79$	\( (T^{4} - 16 T^{2} + 16)^{4} \) (T^4 - 16*T^2 + 16)^4
$83$	\( (T^{8} + 2328 T^{4} + 1028196)^{2} \) (T^8 + 2328*T^4 + 1028196)^2
$89$	\( (T^{4} + 56 T^{2} + 16)^{4} \) (T^4 + 56*T^2 + 16)^4
$97$	\( (T^{4} + 396 T^{2} + 36504)^{4} \) (T^4 + 396*T^2 + 36504)^4
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