LMFDB - Newform orbit 896.2.t.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(65,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("896.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.t (of order $6$, 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:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + \cdots + 1 $ x^16 - 4x^15 + 4x^14 - 24x^13 + 104x^12 - 196x^11 + 312x^10 - 236x^9 + 31x^8 + 236x^7 + 312x^6 + 196x^5 + 104x^4 + 24x^3 + 4x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{10} q^{3} + ( - 2 \beta_{6} + \beta_{3}) q^{5} + (\beta_{15} + \beta_{13} + \beta_{5}) q^{7} + ( - \beta_{12} - \beta_{2}) q^{9}+O(q^{10}) $ q - b10 * q^3 + (-2b6 + b3) q^5 + (b15 + b13 + b5) * q^7 + (-b12 - b2) * q^9	$ q - \beta_{10} q^{3} + ( - 2 \beta_{6} + \beta_{3}) q^{5} + (\beta_{15} + \beta_{13} + \beta_{5}) q^{7} + ( - \beta_{12} - \beta_{2}) q^{9} + ( - \beta_{14} - 2 \beta_{10}) q^{11} + ( - \beta_{11} - 2 \beta_{7} + \cdots + 2 \beta_{3}) q^{13}+ \cdots + ( - \beta_{14} - 3 \beta_{10} + \cdots + 3 \beta_{4}) q^{99}+O(q^{100}) $ q - b10 * q^3 + (-2b6 + b3) q^5 + (b15 + b13 + b5) * q^7 + (-b12 - b2) * q^9 + (-b14 - 2b10) q^11 + (-b11 - 2b7 - b6 + 2b3) * q^13 + (2b5 + 3b1) * q^15 + (-3b12 + b9 - 1) q^17 + (3b8 + 2b4) * q^19 + (2b11 - b7 + 3b6 - 2b3) q^21 + (b15 - 2b13 + b5 - 2b1) * q^23 + (4b12 - 4b9 + 4) * q^25 + (-b14 + 2b10 + b8 - 2b4) * q^27 + (b11 + b6) * q^29 - 3b13 q^31 + (-4b12 + 5b9 - 4b2) q^33 + (-b14 - 5b10 + 2b8 + 3b4) q^35 + (-b6 + b3) * q^37 + (-b15 - 3b13) q^39 + (-3b2 - 4) q^41 + (-2b14 + 2b8) * q^43 + (b11 + 4b7) q^45 + (2b15 - b13 + 2b5 - b1) * q^47 + (4b12 + 2b9 + 5b2 - 3) q^49 + (3b8 + 4b4) * q^51 + (-7b11 + 3b7) * q^53 + (3b5 + 8b1) * q^55 + (8b2 - 3) q^57 + (-2b14 + b10) q^59 + (-b6 - 7b3) q^61 + (-b15 + b13 - b5 - b1) * q^63 + (5b12 - 6b9 + 5b2) q^65 + (4b14 - 3b10) * q^67 + (-7b7 + 7b3) * q^69 - 2b1 q^71 + (4b12 - 3b9 + 3) * q^73 + (-4b8 - 8b4) * q^75 + (b11 + 3b7 + 5b6 - 8b3) q^77 + (3b15 - 4b13 + 3b5 - 4b1) * q^79 + (3b12 - 7b9 + 7) * q^81 + (-2b14 - 2b10 + 2b8 + 2b4) * q^83 + (5b11 + 13b7 + 5b6 - 13b3) * q^85 + (b15 + b13) * q^87 + 5b9 q^89 + (-b14 - 3b10 - b8 - b4) q^91 + (-3b6 + 9b3) * q^93 + (b15 + 12b13) q^95 + (-b2 - 10) * q^97 + (-b14 - 3b10 + b8 + 3b4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{17} + 32 q^{25} + 40 q^{33} - 64 q^{41} - 32 q^{49} - 48 q^{57} - 48 q^{65} + 24 q^{73} + 56 q^{81} + 40 q^{89} - 160 q^{97}+O(q^{100}) $ 16 * q - 8 * q^17 + 32 * q^25 + 40 * q^33 - 64 * q^41 - 32 * q^49 - 48 * q^57 - 48 * q^65 + 24 * q^73 + 56 * q^81 + 40 * q^89 - 160 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + \cdots + 1 $ x^16 - 4*x^15 + 4*x^14 - 24*x^13 + 104*x^12 - 196*x^11 + 312*x^10 - 236*x^9 + 31*x^8 + 236*x^7 + 312*x^6 + 196*x^5 + 104*x^4 + 24*x^3 + 4*x^2 + 4*x + 1 :

$\beta_{1}$	$=$	$ ( 3898973 \nu^{15} - 17115576 \nu^{14} + 21290192 \nu^{13} - 97537769 \nu^{12} + 437864868 \nu^{11} + \cdots - 67550280 ) / 55881261 $ (3898973v^15 - 17115576v^14 + 21290192v^13 - 97537769v^12 + 437864868v^11 - 909810341v^10 + 1459662827v^9 - 1252213188v^8 + 221491124v^7 + 1187365943v^6 + 627279588v^5 + 295531268v^4 + 69097952v^3 + 15174693v^2 + 12576548*v - 67550280) / 55881261
$\beta_{2}$	$=$	$ ( 1882 \nu^{15} - 8264 \nu^{14} + 10280 \nu^{13} - 47068 \nu^{12} + 211436 \nu^{11} - 439319 \nu^{10} + \cdots + 12732 ) / 8037 $ (1882v^15 - 8264v^14 + 10280v^13 - 47068v^12 + 211436v^11 - 439319v^10 + 704290v^9 - 605132v^8 + 107996v^7 + 574999v^6 + 303892v^5 + 143180v^4 + 33472v^3 + 7351v^2 + 6092*v + 12732) / 8037
$\beta_{3}$	$=$	$ ( - 13834008 \nu^{15} + 64112846 \nu^{14} - 93912434 \nu^{13} + 381979504 \nu^{12} + \cdots - 46147796 ) / 55881261 $ (-13834008v^15 + 64112846v^14 - 93912434v^13 + 381979504v^12 - 1668044366v^11 + 3715509806v^10 - 6426826240v^9 + 6809102612v^8 - 3880394738v^7 - 1631311526v^6 - 2993148016v^5 - 323683055v^4 - 801585146v^3 + 66863192v^2 - 152483858*v - 46147796) / 55881261
$\beta_{4}$	$=$	$ ( - 15919536 \nu^{15} + 74254123 \nu^{14} - 109689207 \nu^{13} + 440117880 \nu^{12} + \cdots - 53568309 ) / 55881261 $ (-15919536v^15 + 74254123v^14 - 109689207v^13 + 440117880v^12 - 1929304513v^11 + 4319223459v^10 - 7457325012v^9 + 7916862349v^8 - 4492614528v^7 - 1884513747v^6 - 3470054828v^5 + 42825432v^4 - 930266175v^3 + 77785720v^2 - 176970420*v - 53568309) / 55881261
$\beta_{5}$	$=$	$ ( 18093492 \nu^{15} - 79447396 \nu^{14} + 98828257 \nu^{13} - 452520737 \nu^{12} + 2032659628 \nu^{11} + \cdots + 183016684 ) / 55881261 $ (18093492v^15 - 79447396v^14 + 98828257v^13 - 452520737v^12 + 2032659628v^11 - 4223442586v^10 + 6771223100v^9 - 5817088348v^8 + 1037320804v^7 + 5526371458v^6 + 2920629623v^5 + 1376061028v^4 + 321693715v^3 + 70649078v^2 + 58549408*v + 183016684) / 55881261
$\beta_{6}$	$=$	$ ( - 24578404 \nu^{15} + 112409521 \nu^{14} - 161766434 \nu^{13} + 676914588 \nu^{12} + \cdots - 80530744 ) / 55881261 $ (-24578404v^15 + 112409521v^14 - 161766434v^13 + 676914588v^12 - 2932745926v^11 + 6464372519v^10 - 11224587756v^9 + 11842175116v^8 - 6808155680v^7 - 2875070153v^6 - 5236873676v^5 - 1970820116v^4 - 1399477502v^3 + 116148784v^2 - 266195456*v - 80530744) / 55881261
$\beta_{7}$	$=$	$ ( 41399790 \nu^{15} - 174606650 \nu^{14} + 204752252 \nu^{13} - 1042462561 \nu^{12} + \cdots + 127576424 ) / 55881261 $ (41399790v^15 - 174606650v^14 + 204752252v^13 - 1042462561v^12 + 4535116370v^11 - 9125208020v^10 + 15012471622v^9 - 13215897836v^8 + 4393187486v^7 + 8766927572v^6 + 10778117002v^5 + 6198193409v^4 + 3399573596v^3 + 397372258v^2 + 119497190*v + 127576424) / 55881261
$\beta_{8}$	$=$	$ ( 16331260 \nu^{15} - 74991976 \nu^{14} + 108507436 \nu^{13} - 450127912 \nu^{12} + 1954868524 \nu^{11} + \cdots + 53802093 ) / 18627087 $ (16331260v^15 - 74991976v^14 + 108507436v^13 - 450127912v^12 + 1954868524v^11 - 4322761180v^10 + 7497148468v^9 - 7919869405v^8 + 4541015683v^7 + 1915043380v^6 + 3495942116v^5 + 1025333980v^4 + 934847368v^3 - 77706604v^2 + 177822799*v + 53802093) / 18627087
$\beta_{9}$	$=$	$ ( 16361188 \nu^{15} - 72139004 \nu^{14} + 94401360 \nu^{13} - 428892200 \nu^{12} + 1874038268 \nu^{11} + \cdots + 51319535 ) / 18627087 $ (16361188v^15 - 72139004v^14 + 94401360v^13 - 428892200v^12 + 1874038268v^11 - 3958996896v^10 + 6661892204v^9 - 6457440527v^8 + 2929890732v^7 + 2870483616v^6 + 3833618044v^5 + 1548767784v^4 + 916399440v^3 - 54074084v^2 + 25648164*v + 51319535) / 18627087
$\beta_{10}$	$=$	$ ( 52321296 \nu^{15} - 220783753 \nu^{14} + 259359264 \nu^{13} - 1318143672 \nu^{12} + \cdots + 164345664 ) / 55881261 $ (52321296v^15 - 220783753v^14 + 259359264v^13 - 1318143672v^12 + 5733009463v^11 - 11544111072v^10 + 18998772912v^9 - 16718120539v^8 + 5501388816v^7 + 11262556896v^6 + 13325872505v^5 + 7996180536v^4 + 4374375072v^3 + 480961151v^2 + 154864368*v + 164345664) / 55881261
$\beta_{11}$	$=$	$ ( - 57962620 \nu^{15} + 244034719 \nu^{14} - 284439440 \nu^{13} + 1456967568 \nu^{12} + \cdots - 168001120 ) / 55881261 $ (-57962620v^15 + 244034719v^14 - 284439440v^13 + 1456967568v^12 - 6343717582v^11 + 12731116736v^10 - 20919252180v^9 + 18436863304v^8 - 6312521300v^7 - 11645397440v^6 - 16068699614v^5 - 8122374800v^4 - 4493625488v^3 - 627232847v^2 - 154199804*v - 168001120) / 55881261
$\beta_{12}$	$=$	$ ( 63542546 \nu^{15} - 274353511 \nu^{14} + 341089720 \nu^{13} - 1633918160 \nu^{12} + \cdots + 146658816 ) / 55881261 $ (63542546v^15 - 274353511v^14 + 341089720v^13 - 1633918160v^12 + 7132592440v^11 - 14721390352v^10 + 24513948218v^9 - 22896231172v^8 + 9485875894v^7 + 11528263472v^6 + 16589208392v^5 + 6999046528v^4 + 4022989640v^3 - 253368535v^2 + 123390718*v + 146658816) / 55881261
$\beta_{13}$	$=$	$ ( - 63563905 \nu^{15} + 281986149 \nu^{14} - 374337712 \nu^{13} + 1675704013 \nu^{12} + \cdots - 121136064 ) / 55881261 $ (-63563905v^15 + 281986149v^14 - 374337712v^13 + 1675704013v^12 - 7323794952v^11 + 15575202244v^10 - 26285370847v^9 + 25732865256v^8 - 11945659951v^7 - 11037565252v^6 - 14385938376v^5 - 5723333389v^4 - 3421775728v^3 + 197155395v^2 - 92547601*v - 121136064) / 55881261
$\beta_{14}$	$=$	$ ( - 41682527 \nu^{15} + 175603960 \nu^{14} - 205133396 \nu^{13} + 1048417004 \nu^{12} + \cdots - 123561984 ) / 18627087 $ (-41682527v^15 + 175603960v^14 - 205133396v^13 + 1048417004v^12 - 4563459364v^11 + 9167107496v^10 - 15069743507v^9 + 13276126963v^8 - 4498087181v^7 - 8537548760v^6 - 11303294252v^5 - 5984840948v^4 - 3300359084v^3 - 432772592v^2 - 114281729*v - 123561984) / 18627087
$\beta_{15}$	$=$	$ ( 141101244 \nu^{15} - 614100430 \nu^{14} + 778827436 \nu^{13} - 3654890939 \nu^{12} + \cdots + 309173416 ) / 55881261 $ (141101244v^15 - 614100430v^14 + 778827436v^13 - 3654890939v^12 + 15960688813v^11 - 33238515580v^10 + 55574479568v^9 - 52673856436v^8 + 22650778144v^7 + 25281468604v^6 + 35413383053v^5 + 14717640379v^4 + 8544748204v^3 - 526297546v^2 + 254078860*v + 309173416) / 55881261

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

$\nu$	$=$	$ ( \beta_{13} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} + \beta_1 ) / 2 $ (b13 + b9 + b8 + b6 + b3 + b1) / 2
$\nu^{2}$	$=$	$ ( 2\beta_{15} + 2\beta_{14} - 3\beta_{12} - 3\beta_{11} - 2\beta_{9} + 2\beta_{7} + 2 ) / 2 $ (2b15 + 2b14 - 3b12 - 3b11 - 2b9 + 2b7 + 2) / 2
$\nu^{3}$	$=$	$ ( - 2 \beta_{14} + 2 \beta_{11} + 3 \beta_{10} + 2 \beta_{8} - 7 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} + \cdots + 11 ) / 2 $ (-2b14 + 2b11 + 3b10 + 2b8 - 7b7 + 2b6 - 7b5 - 3b4 + 7b3 + 9b2 + 2*b1 + 11) / 2
$\nu^{4}$	$=$	$ 10\beta_{8} + 12\beta_{6} - 6\beta_{4} + 21\beta_{3} $ 10b8 + 12b6 - 6b4 + 21b3
$\nu^{5}$	$=$	$ ( 59 \beta_{15} + 22 \beta_{14} - 22 \beta_{13} - 75 \beta_{12} - 26 \beta_{11} - 15 \beta_{10} + \cdots + 101 ) / 2 $ (59b15 + 22b14 - 22b13 - 75b12 - 26b11 - 15b10 - 101b9 + 49b7 + 101) / 2
$\nu^{6}$	$=$	$ ( - 134 \beta_{14} + 171 \beta_{11} + 48 \beta_{10} + 134 \beta_{8} - 226 \beta_{7} + 171 \beta_{6} + \cdots + 226 ) / 2 $ (-134b14 + 171b11 + 48b10 + 134b8 - 226b7 + 171b6 - 134b5 - 48b4 + 226b3 + 171b2 + 48*b1 + 226) / 2
$\nu^{7}$	$=$	$ ( 204 \beta_{15} - 101 \beta_{13} - 252 \beta_{12} - 391 \beta_{9} + 509 \beta_{8} + 643 \beta_{6} + \cdots - 101 \beta_1 ) / 2 $ (204b15 - 101b13 - 252b12 - 391b9 + 509b8 + 643b6 + 204b5 - 204b4 + 895b3 - 252b2 - 101*b1) / 2
$\nu^{8}$	$=$	$ 804\beta_{15} - 348\beta_{13} - 1008\beta_{12} - 1457\beta_{9} + 1457 $ 804b15 - 348b13 - 1008b12 - 1457b9 + 1457
$\nu^{9}$	$=$	$ ( - 4421 \beta_{14} + 5573 \beta_{11} + 1812 \beta_{10} + 4421 \beta_{8} - 7841 \beta_{7} + 5573 \beta_{6} + \cdots + 3305 ) / 2 $ (-4421b14 + 5573b11 + 1812b10 + 4421b8 - 7841b7 + 5573b6 - 1812b5 - 1812b4 + 7841b3 + 2268b2 + 797*b1 + 3305) / 2
$\nu^{10}$	$=$	$ ( 9994 \beta_{15} - 4080 \beta_{13} - 12603 \beta_{12} - 17698 \beta_{9} + 9994 \beta_{8} + \cdots - 4080 \beta_1 ) / 2 $ (9994b15 - 4080b13 - 12603b12 - 17698b9 + 9994b8 + 12603b6 + 9994b5 - 4080b4 + 17698b3 - 12603b2 - 4080*b1) / 2
$\nu^{11}$	$=$	$ ( 38483 \beta_{15} - 15886 \beta_{14} - 15886 \beta_{13} - 48477 \beta_{12} + 19966 \beta_{11} + \cdots + 68443 ) / 2 $ (38483b15 - 15886b14 - 15886b13 - 48477b12 + 19966b11 + 6711b10 - 68443b9 - 28511b7 + 68443) / 2
$\nu^{12}$	$=$	$ - 61406 \beta_{14} + 77292 \beta_{11} + 25554 \beta_{10} + 61406 \beta_{8} - 109557 \beta_{7} + \cdots + 109557 \beta_{3} $ -61406b14 + 77292b11 + 25554b10 + 61406b8 - 109557b7 + 77292b6 - 25554b4 + 109557b3
$\nu^{13}$	$=$	$ ( 335215 \beta_{15} - 138698 \beta_{13} - 422175 \beta_{12} - 596725 \beta_{9} + 138698 \beta_{8} + \cdots - 138698 \beta_1 ) / 2 $ (335215b15 - 138698b13 - 422175b12 - 596725b9 + 138698b8 + 174550b6 + 335215b5 - 57819b4 + 247625b3 - 422175b2 - 138698*b1) / 2
$\nu^{14}$	$=$	$ ( 757390 \beta_{15} - 757390 \beta_{14} - 313248 \beta_{13} - 953907 \beta_{12} + 953907 \beta_{11} + \cdots + 1348034 ) / 2 $ (757390b15 - 757390b14 - 313248b13 - 953907b12 + 953907b11 + 313248b10 - 1348034b9 - 1348034b7 + 1348034) / 2
$\nu^{15}$	$=$	$ ( - 2920633 \beta_{14} + 3678023 \beta_{11} + 1209336 \beta_{10} + 2920633 \beta_{8} - 5200607 \beta_{7} + \cdots - 2155439 ) / 2 $ (-2920633b14 + 3678023b11 + 1209336b10 + 2920633b8 - 5200607b7 + 3678023b6 + 1209336b5 - 1209336b4 + 5200607b3 - 1522584b2 - 501961*b1 - 2155439) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$1$	$-\beta_{9}$	$-1$

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} $

65.1

1.33145	−	1.02165i
−0.0777751	+	0.590761i
−0.335868	−	0.0442178i
−1.79700	−	2.34189i
2.92664	+	0.385299i
0.206228	+	0.268761i
−0.472726	+	0.362736i
0.219056	−	1.66389i
1.33145	+	1.02165i
−0.0777751	−	0.590761i
−0.335868	+	0.0442178i
−1.79700	+	2.34189i
2.92664	−	0.385299i
0.206228	−	0.268761i
−0.472726	−	0.362736i
0.219056	+	1.66389i

−1.81952

1.05050i

−3.31552

−

1.91421i

−0.615370

−

2.57319i

0.707107

−

1.22474i

65.2

−1.81952

1.05050i

3.31552

1.91421i

0.615370

2.57319i

0.707107

−

1.22474i

65.3

−1.09057

0.629640i

−1.58346

−

0.914214i

2.14973

−

1.54230i

−0.707107

1.22474i

65.4

−1.09057

0.629640i

1.58346

0.914214i

−2.14973

1.54230i

−0.707107

1.22474i

65.5

1.09057

−

0.629640i

−1.58346

−

0.914214i

−2.14973

1.54230i

−0.707107

1.22474i

65.6

1.09057

−

0.629640i

1.58346

0.914214i

2.14973

−

1.54230i

−0.707107

1.22474i

65.7

1.81952

−

1.05050i

−3.31552

−

1.91421i

0.615370

2.57319i

0.707107

−

1.22474i

65.8

1.81952

−

1.05050i

3.31552

1.91421i

−0.615370

−

2.57319i

0.707107

−

1.22474i

193.1

−1.81952

−

1.05050i

−3.31552

1.91421i

−0.615370

2.57319i

0.707107

1.22474i

193.2

−1.81952

−

1.05050i

3.31552

−

1.91421i

0.615370

−

2.57319i

0.707107

1.22474i

193.3

−1.09057

−

0.629640i

−1.58346

0.914214i

2.14973

1.54230i

−0.707107

−

1.22474i

193.4

−1.09057

−

0.629640i

1.58346

−

0.914214i

−2.14973

−

1.54230i

−0.707107

−

1.22474i

193.5

1.09057

0.629640i

−1.58346

0.914214i

−2.14973

−

1.54230i

−0.707107

−

1.22474i

193.6

1.09057

0.629640i

1.58346

−

0.914214i

2.14973

1.54230i

−0.707107

−

1.22474i

193.7

1.81952

1.05050i

−3.31552

1.91421i

0.615370

−

2.57319i

0.707107

1.22474i

193.8

1.81952

1.05050i

3.31552

−

1.91421i

−0.615370

2.57319i

0.707107

1.22474i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.c	even	3	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
28.g	odd	6	1	inner
56.k	odd	6	1	inner
56.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	896.2.t.a	✓	16
4.b	odd	2	1	inner	896.2.t.a	✓	16
7.c	even	3	1	inner	896.2.t.a	✓	16
8.b	even	2	1	inner	896.2.t.a	✓	16
8.d	odd	2	1	inner	896.2.t.a	✓	16
28.g	odd	6	1	inner	896.2.t.a	✓	16
56.k	odd	6	1	inner	896.2.t.a	✓	16
56.p	even	6	1	inner	896.2.t.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
896.2.t.a	✓	16	1.a	even	1	1	trivial
896.2.t.a	✓	16	4.b	odd	2	1	inner
896.2.t.a	✓	16	7.c	even	3	1	inner
896.2.t.a	✓	16	8.b	even	2	1	inner
896.2.t.a	✓	16	8.d	odd	2	1	inner
896.2.t.a	✓	16	28.g	odd	6	1	inner
896.2.t.a	✓	16	56.k	odd	6	1	inner
896.2.t.a	✓	16	56.p	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 6T_{3}^{6} + 29T_{3}^{4} - 42T_{3}^{2} + 49 $ T3^8 - 6*T3^6 + 29*T3^4 - 42*T3^2 + 49 acting on $S_{2}^{\mathrm{new}}(896, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 6 T^{6} + 29 T^{4} + \cdots + 49)^{2} $ (T^8 - 6T^6 + 29T^4 - 42*T^2 + 49)^2
$5$	$ (T^{8} - 18 T^{6} + \cdots + 2401)^{2} $ (T^8 - 18T^6 + 275T^4 - 882*T^2 + 2401)^2
$7$	$ (T^{8} + 8 T^{6} + \cdots + 2401)^{2} $ (T^8 + 8T^6 + 42T^4 + 392*T^2 + 2401)^2
$11$	$ (T^{8} - 26 T^{6} + \cdots + 49)^{2} $ (T^8 - 26T^6 + 669T^4 - 182*T^2 + 49)^2
$13$	$ (T^{4} + 12 T^{2} + 4)^{4} $ (T^4 + 12*T^2 + 4)^4
$17$	$ (T^{4} + 2 T^{3} + \cdots + 289)^{4} $ (T^4 + 2T^3 + 21T^2 - 34*T + 289)^4
$19$	$ (T^{8} - 90 T^{6} + \cdots + 4092529)^{2} $ (T^8 - 90T^6 + 6077T^4 - 182070*T^2 + 4092529)^2
$23$	$ (T^{8} + 42 T^{6} + \cdots + 117649)^{2} $ (T^8 + 42T^6 + 1421T^4 + 14406*T^2 + 117649)^2
$29$	$ (T^{2} + 2)^{8} $ (T^2 + 2)^8
$31$	$ (T^{8} + 54 T^{6} + \cdots + 321489)^{2} $ (T^8 + 54T^6 + 2349T^4 + 30618*T^2 + 321489)^2
$37$	$ (T^{8} - 6 T^{6} + 35 T^{4} + \cdots + 1)^{2} $ (T^8 - 6T^6 + 35T^4 - 6*T^2 + 1)^2
$41$	$ (T^{2} + 8 T - 2)^{8} $ (T^2 + 8*T - 2)^8
$43$	$ (T^{4} + 40 T^{2} + 112)^{4} $ (T^4 + 40*T^2 + 112)^4
$47$	$ (T^{8} + 54 T^{6} + \cdots + 49)^{2} $ (T^8 + 54T^6 + 2909T^4 + 378*T^2 + 49)^2
$53$	$ (T^{8} - 214 T^{6} + \cdots + 62742241)^{2} $ (T^8 - 214T^6 + 37875T^4 - 1695094*T^2 + 62742241)^2
$59$	$ (T^{8} - 54 T^{6} + \cdots + 49)^{2} $ (T^8 - 54T^6 + 2909T^4 - 378*T^2 + 49)^2
$61$	$ (T^{8} - 102 T^{6} + \cdots + 4879681)^{2} $ (T^8 - 102T^6 + 8195T^4 - 225318*T^2 + 4879681)^2
$67$	$ (T^{8} - 262 T^{6} + \cdots + 4092529)^{2} $ (T^8 - 262T^6 + 66621T^4 - 530026*T^2 + 4092529)^2
$71$	$ (T^{4} - 24 T^{2} + 112)^{4} $ (T^4 - 24*T^2 + 112)^4
$73$	$ (T^{4} - 6 T^{3} + \cdots + 529)^{4} $ (T^4 - 6T^3 + 59T^2 + 138*T + 529)^4
$79$	$ (T^{8} + 234 T^{6} + \cdots + 45252529)^{2} $ (T^8 + 234T^6 + 48029T^4 + 1574118*T^2 + 45252529)^2
$83$	$ (T^{4} + 48 T^{2} + 448)^{4} $ (T^4 + 48*T^2 + 448)^4
$89$	$ (T^{2} - 5 T + 25)^{8} $ (T^2 - 5*T + 25)^8
$97$	$ (T^{2} + 20 T + 98)^{8} $ (T^2 + 20*T + 98)^8
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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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{10} q^{3} + ( - 2 \beta_{6} + \beta_{3}) q^{5} + (\beta_{15} + \beta_{13} + \beta_{5}) q^{7} + ( - \beta_{12} - \beta_{2}) q^{9}+O(q^{10}) \) q - b10 * q^3 + (-2b6 + b3) q^5 + (b15 + b13 + b5) * q^7 + (-b12 - b2) * q^9	\( q - \beta_{10} q^{3} + ( - 2 \beta_{6} + \beta_{3}) q^{5} + (\beta_{15} + \beta_{13} + \beta_{5}) q^{7} + ( - \beta_{12} - \beta_{2}) q^{9} + ( - \beta_{14} - 2 \beta_{10}) q^{11} + ( - \beta_{11} - 2 \beta_{7} + \cdots + 2 \beta_{3}) q^{13}+ \cdots + ( - \beta_{14} - 3 \beta_{10} + \cdots + 3 \beta_{4}) q^{99}+O(q^{100}) \) q - b10 * q^3 + (-2b6 + b3) q^5 + (b15 + b13 + b5) * q^7 + (-b12 - b2) * q^9 + (-b14 - 2b10) q^11 + (-b11 - 2b7 - b6 + 2b3) * q^13 + (2b5 + 3b1) * q^15 + (-3b12 + b9 - 1) q^17 + (3b8 + 2b4) * q^19 + (2b11 - b7 + 3b6 - 2b3) q^21 + (b15 - 2b13 + b5 - 2b1) * q^23 + (4b12 - 4b9 + 4) * q^25 + (-b14 + 2b10 + b8 - 2b4) * q^27 + (b11 + b6) * q^29 - 3b13 q^31 + (-4b12 + 5b9 - 4b2) q^33 + (-b14 - 5b10 + 2b8 + 3b4) q^35 + (-b6 + b3) * q^37 + (-b15 - 3b13) q^39 + (-3b2 - 4) q^41 + (-2b14 + 2b8) * q^43 + (b11 + 4b7) q^45 + (2b15 - b13 + 2b5 - b1) * q^47 + (4b12 + 2b9 + 5b2 - 3) q^49 + (3b8 + 4b4) * q^51 + (-7b11 + 3b7) * q^53 + (3b5 + 8b1) * q^55 + (8b2 - 3) q^57 + (-2b14 + b10) q^59 + (-b6 - 7b3) q^61 + (-b15 + b13 - b5 - b1) * q^63 + (5b12 - 6b9 + 5b2) q^65 + (4b14 - 3b10) * q^67 + (-7b7 + 7b3) * q^69 - 2b1 q^71 + (4b12 - 3b9 + 3) * q^73 + (-4b8 - 8b4) * q^75 + (b11 + 3b7 + 5b6 - 8b3) q^77 + (3b15 - 4b13 + 3b5 - 4b1) * q^79 + (3b12 - 7b9 + 7) * q^81 + (-2b14 - 2b10 + 2b8 + 2b4) * q^83 + (5b11 + 13b7 + 5b6 - 13b3) * q^85 + (b15 + b13) * q^87 + 5b9 q^89 + (-b14 - 3b10 - b8 - b4) q^91 + (-3b6 + 9b3) * q^93 + (b15 + 12b13) q^95 + (-b2 - 10) * q^97 + (-b14 - 3b10 + b8 + 3b4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{17} + 32 q^{25} + 40 q^{33} - 64 q^{41} - 32 q^{49} - 48 q^{57} - 48 q^{65} + 24 q^{73} + 56 q^{81} + 40 q^{89} - 160 q^{97}+O(q^{100}) \) 16 * q - 8 * q^17 + 32 * q^25 + 40 * q^33 - 64 * q^41 - 32 * q^49 - 48 * q^57 - 48 * q^65 + 24 * q^73 + 56 * q^81 + 40 * q^89 - 160 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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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	896.2.t.a

Level	$896$
Weight	$2$
Character orbit	896.t
Analytic conductor	$7.155$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(7.15459602111\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + \cdots + 1 \) x^16 - 4x^15 + 4x^14 - 24x^13 + 104x^12 - 196x^11 + 312x^10 - 236x^9 + 31x^8 + 236x^7 + 312x^6 + 196x^5 + 104x^4 + 24x^3 + 4x^2 + 4*x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 3898973 \nu^{15} - 17115576 \nu^{14} + 21290192 \nu^{13} - 97537769 \nu^{12} + 437864868 \nu^{11} + \cdots - 67550280 ) / 55881261 \) (3898973v^15 - 17115576v^14 + 21290192v^13 - 97537769v^12 + 437864868v^11 - 909810341v^10 + 1459662827v^9 - 1252213188v^8 + 221491124v^7 + 1187365943v^6 + 627279588v^5 + 295531268v^4 + 69097952v^3 + 15174693v^2 + 12576548*v - 67550280) / 55881261
\(\beta_{2}\)	\(=\)	\( ( 1882 \nu^{15} - 8264 \nu^{14} + 10280 \nu^{13} - 47068 \nu^{12} + 211436 \nu^{11} - 439319 \nu^{10} + \cdots + 12732 ) / 8037 \) (1882v^15 - 8264v^14 + 10280v^13 - 47068v^12 + 211436v^11 - 439319v^10 + 704290v^9 - 605132v^8 + 107996v^7 + 574999v^6 + 303892v^5 + 143180v^4 + 33472v^3 + 7351v^2 + 6092*v + 12732) / 8037
\(\beta_{3}\)	\(=\)	\( ( - 13834008 \nu^{15} + 64112846 \nu^{14} - 93912434 \nu^{13} + 381979504 \nu^{12} + \cdots - 46147796 ) / 55881261 \) (-13834008v^15 + 64112846v^14 - 93912434v^13 + 381979504v^12 - 1668044366v^11 + 3715509806v^10 - 6426826240v^9 + 6809102612v^8 - 3880394738v^7 - 1631311526v^6 - 2993148016v^5 - 323683055v^4 - 801585146v^3 + 66863192v^2 - 152483858*v - 46147796) / 55881261
\(\beta_{4}\)	\(=\)	\( ( - 15919536 \nu^{15} + 74254123 \nu^{14} - 109689207 \nu^{13} + 440117880 \nu^{12} + \cdots - 53568309 ) / 55881261 \) (-15919536v^15 + 74254123v^14 - 109689207v^13 + 440117880v^12 - 1929304513v^11 + 4319223459v^10 - 7457325012v^9 + 7916862349v^8 - 4492614528v^7 - 1884513747v^6 - 3470054828v^5 + 42825432v^4 - 930266175v^3 + 77785720v^2 - 176970420*v - 53568309) / 55881261
\(\beta_{5}\)	\(=\)	\( ( 18093492 \nu^{15} - 79447396 \nu^{14} + 98828257 \nu^{13} - 452520737 \nu^{12} + 2032659628 \nu^{11} + \cdots + 183016684 ) / 55881261 \) (18093492v^15 - 79447396v^14 + 98828257v^13 - 452520737v^12 + 2032659628v^11 - 4223442586v^10 + 6771223100v^9 - 5817088348v^8 + 1037320804v^7 + 5526371458v^6 + 2920629623v^5 + 1376061028v^4 + 321693715v^3 + 70649078v^2 + 58549408*v + 183016684) / 55881261
\(\beta_{6}\)	\(=\)	\( ( - 24578404 \nu^{15} + 112409521 \nu^{14} - 161766434 \nu^{13} + 676914588 \nu^{12} + \cdots - 80530744 ) / 55881261 \) (-24578404v^15 + 112409521v^14 - 161766434v^13 + 676914588v^12 - 2932745926v^11 + 6464372519v^10 - 11224587756v^9 + 11842175116v^8 - 6808155680v^7 - 2875070153v^6 - 5236873676v^5 - 1970820116v^4 - 1399477502v^3 + 116148784v^2 - 266195456*v - 80530744) / 55881261
\(\beta_{7}\)	\(=\)	\( ( 41399790 \nu^{15} - 174606650 \nu^{14} + 204752252 \nu^{13} - 1042462561 \nu^{12} + \cdots + 127576424 ) / 55881261 \) (41399790v^15 - 174606650v^14 + 204752252v^13 - 1042462561v^12 + 4535116370v^11 - 9125208020v^10 + 15012471622v^9 - 13215897836v^8 + 4393187486v^7 + 8766927572v^6 + 10778117002v^5 + 6198193409v^4 + 3399573596v^3 + 397372258v^2 + 119497190*v + 127576424) / 55881261
\(\beta_{8}\)	\(=\)	\( ( 16331260 \nu^{15} - 74991976 \nu^{14} + 108507436 \nu^{13} - 450127912 \nu^{12} + 1954868524 \nu^{11} + \cdots + 53802093 ) / 18627087 \) (16331260v^15 - 74991976v^14 + 108507436v^13 - 450127912v^12 + 1954868524v^11 - 4322761180v^10 + 7497148468v^9 - 7919869405v^8 + 4541015683v^7 + 1915043380v^6 + 3495942116v^5 + 1025333980v^4 + 934847368v^3 - 77706604v^2 + 177822799*v + 53802093) / 18627087
\(\beta_{9}\)	\(=\)	\( ( 16361188 \nu^{15} - 72139004 \nu^{14} + 94401360 \nu^{13} - 428892200 \nu^{12} + 1874038268 \nu^{11} + \cdots + 51319535 ) / 18627087 \) (16361188v^15 - 72139004v^14 + 94401360v^13 - 428892200v^12 + 1874038268v^11 - 3958996896v^10 + 6661892204v^9 - 6457440527v^8 + 2929890732v^7 + 2870483616v^6 + 3833618044v^5 + 1548767784v^4 + 916399440v^3 - 54074084v^2 + 25648164*v + 51319535) / 18627087
\(\beta_{10}\)	\(=\)	\( ( 52321296 \nu^{15} - 220783753 \nu^{14} + 259359264 \nu^{13} - 1318143672 \nu^{12} + \cdots + 164345664 ) / 55881261 \) (52321296v^15 - 220783753v^14 + 259359264v^13 - 1318143672v^12 + 5733009463v^11 - 11544111072v^10 + 18998772912v^9 - 16718120539v^8 + 5501388816v^7 + 11262556896v^6 + 13325872505v^5 + 7996180536v^4 + 4374375072v^3 + 480961151v^2 + 154864368*v + 164345664) / 55881261
\(\beta_{11}\)	\(=\)	\( ( - 57962620 \nu^{15} + 244034719 \nu^{14} - 284439440 \nu^{13} + 1456967568 \nu^{12} + \cdots - 168001120 ) / 55881261 \) (-57962620v^15 + 244034719v^14 - 284439440v^13 + 1456967568v^12 - 6343717582v^11 + 12731116736v^10 - 20919252180v^9 + 18436863304v^8 - 6312521300v^7 - 11645397440v^6 - 16068699614v^5 - 8122374800v^4 - 4493625488v^3 - 627232847v^2 - 154199804*v - 168001120) / 55881261
\(\beta_{12}\)	\(=\)	\( ( 63542546 \nu^{15} - 274353511 \nu^{14} + 341089720 \nu^{13} - 1633918160 \nu^{12} + \cdots + 146658816 ) / 55881261 \) (63542546v^15 - 274353511v^14 + 341089720v^13 - 1633918160v^12 + 7132592440v^11 - 14721390352v^10 + 24513948218v^9 - 22896231172v^8 + 9485875894v^7 + 11528263472v^6 + 16589208392v^5 + 6999046528v^4 + 4022989640v^3 - 253368535v^2 + 123390718*v + 146658816) / 55881261
\(\beta_{13}\)	\(=\)	\( ( - 63563905 \nu^{15} + 281986149 \nu^{14} - 374337712 \nu^{13} + 1675704013 \nu^{12} + \cdots - 121136064 ) / 55881261 \) (-63563905v^15 + 281986149v^14 - 374337712v^13 + 1675704013v^12 - 7323794952v^11 + 15575202244v^10 - 26285370847v^9 + 25732865256v^8 - 11945659951v^7 - 11037565252v^6 - 14385938376v^5 - 5723333389v^4 - 3421775728v^3 + 197155395v^2 - 92547601*v - 121136064) / 55881261
\(\beta_{14}\)	\(=\)	\( ( - 41682527 \nu^{15} + 175603960 \nu^{14} - 205133396 \nu^{13} + 1048417004 \nu^{12} + \cdots - 123561984 ) / 18627087 \) (-41682527v^15 + 175603960v^14 - 205133396v^13 + 1048417004v^12 - 4563459364v^11 + 9167107496v^10 - 15069743507v^9 + 13276126963v^8 - 4498087181v^7 - 8537548760v^6 - 11303294252v^5 - 5984840948v^4 - 3300359084v^3 - 432772592v^2 - 114281729*v - 123561984) / 18627087
\(\beta_{15}\)	\(=\)	\( ( 141101244 \nu^{15} - 614100430 \nu^{14} + 778827436 \nu^{13} - 3654890939 \nu^{12} + \cdots + 309173416 ) / 55881261 \) (141101244v^15 - 614100430v^14 + 778827436v^13 - 3654890939v^12 + 15960688813v^11 - 33238515580v^10 + 55574479568v^9 - 52673856436v^8 + 22650778144v^7 + 25281468604v^6 + 35413383053v^5 + 14717640379v^4 + 8544748204v^3 - 526297546v^2 + 254078860*v + 309173416) / 55881261

\(\nu\)	\(=\)	\( ( \beta_{13} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} + \beta_1 ) / 2 \) (b13 + b9 + b8 + b6 + b3 + b1) / 2
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{15} + 2\beta_{14} - 3\beta_{12} - 3\beta_{11} - 2\beta_{9} + 2\beta_{7} + 2 ) / 2 \) (2b15 + 2b14 - 3b12 - 3b11 - 2b9 + 2b7 + 2) / 2
\(\nu^{3}\)	\(=\)	\( ( - 2 \beta_{14} + 2 \beta_{11} + 3 \beta_{10} + 2 \beta_{8} - 7 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} + \cdots + 11 ) / 2 \) (-2b14 + 2b11 + 3b10 + 2b8 - 7b7 + 2b6 - 7b5 - 3b4 + 7b3 + 9b2 + 2*b1 + 11) / 2
\(\nu^{4}\)	\(=\)	\( 10\beta_{8} + 12\beta_{6} - 6\beta_{4} + 21\beta_{3} \) 10b8 + 12b6 - 6b4 + 21b3
\(\nu^{5}\)	\(=\)	\( ( 59 \beta_{15} + 22 \beta_{14} - 22 \beta_{13} - 75 \beta_{12} - 26 \beta_{11} - 15 \beta_{10} + \cdots + 101 ) / 2 \) (59b15 + 22b14 - 22b13 - 75b12 - 26b11 - 15b10 - 101b9 + 49b7 + 101) / 2
\(\nu^{6}\)	\(=\)	\( ( - 134 \beta_{14} + 171 \beta_{11} + 48 \beta_{10} + 134 \beta_{8} - 226 \beta_{7} + 171 \beta_{6} + \cdots + 226 ) / 2 \) (-134b14 + 171b11 + 48b10 + 134b8 - 226b7 + 171b6 - 134b5 - 48b4 + 226b3 + 171b2 + 48*b1 + 226) / 2
\(\nu^{7}\)	\(=\)	\( ( 204 \beta_{15} - 101 \beta_{13} - 252 \beta_{12} - 391 \beta_{9} + 509 \beta_{8} + 643 \beta_{6} + \cdots - 101 \beta_1 ) / 2 \) (204b15 - 101b13 - 252b12 - 391b9 + 509b8 + 643b6 + 204b5 - 204b4 + 895b3 - 252b2 - 101*b1) / 2
\(\nu^{8}\)	\(=\)	\( 804\beta_{15} - 348\beta_{13} - 1008\beta_{12} - 1457\beta_{9} + 1457 \) 804b15 - 348b13 - 1008b12 - 1457b9 + 1457
\(\nu^{9}\)	\(=\)	\( ( - 4421 \beta_{14} + 5573 \beta_{11} + 1812 \beta_{10} + 4421 \beta_{8} - 7841 \beta_{7} + 5573 \beta_{6} + \cdots + 3305 ) / 2 \) (-4421b14 + 5573b11 + 1812b10 + 4421b8 - 7841b7 + 5573b6 - 1812b5 - 1812b4 + 7841b3 + 2268b2 + 797*b1 + 3305) / 2
\(\nu^{10}\)	\(=\)	\( ( 9994 \beta_{15} - 4080 \beta_{13} - 12603 \beta_{12} - 17698 \beta_{9} + 9994 \beta_{8} + \cdots - 4080 \beta_1 ) / 2 \) (9994b15 - 4080b13 - 12603b12 - 17698b9 + 9994b8 + 12603b6 + 9994b5 - 4080b4 + 17698b3 - 12603b2 - 4080*b1) / 2
\(\nu^{11}\)	\(=\)	\( ( 38483 \beta_{15} - 15886 \beta_{14} - 15886 \beta_{13} - 48477 \beta_{12} + 19966 \beta_{11} + \cdots + 68443 ) / 2 \) (38483b15 - 15886b14 - 15886b13 - 48477b12 + 19966b11 + 6711b10 - 68443b9 - 28511b7 + 68443) / 2
\(\nu^{12}\)	\(=\)	\( - 61406 \beta_{14} + 77292 \beta_{11} + 25554 \beta_{10} + 61406 \beta_{8} - 109557 \beta_{7} + \cdots + 109557 \beta_{3} \) -61406b14 + 77292b11 + 25554b10 + 61406b8 - 109557b7 + 77292b6 - 25554b4 + 109557b3
\(\nu^{13}\)	\(=\)	\( ( 335215 \beta_{15} - 138698 \beta_{13} - 422175 \beta_{12} - 596725 \beta_{9} + 138698 \beta_{8} + \cdots - 138698 \beta_1 ) / 2 \) (335215b15 - 138698b13 - 422175b12 - 596725b9 + 138698b8 + 174550b6 + 335215b5 - 57819b4 + 247625b3 - 422175b2 - 138698*b1) / 2
\(\nu^{14}\)	\(=\)	\( ( 757390 \beta_{15} - 757390 \beta_{14} - 313248 \beta_{13} - 953907 \beta_{12} + 953907 \beta_{11} + \cdots + 1348034 ) / 2 \) (757390b15 - 757390b14 - 313248b13 - 953907b12 + 953907b11 + 313248b10 - 1348034b9 - 1348034b7 + 1348034) / 2
\(\nu^{15}\)	\(=\)	\( ( - 2920633 \beta_{14} + 3678023 \beta_{11} + 1209336 \beta_{10} + 2920633 \beta_{8} - 5200607 \beta_{7} + \cdots - 2155439 ) / 2 \) (-2920633b14 + 3678023b11 + 1209336b10 + 2920633b8 - 5200607b7 + 3678023b6 + 1209336b5 - 1209336b4 + 5200607b3 - 1522584b2 - 501961*b1 - 2155439) / 2

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(1\)	\(-\beta_{9}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 65.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 6 T^{6} + 29 T^{4} + \cdots + 49)^{2} \) (T^8 - 6T^6 + 29T^4 - 42*T^2 + 49)^2
$5$	\( (T^{8} - 18 T^{6} + \cdots + 2401)^{2} \) (T^8 - 18T^6 + 275T^4 - 882*T^2 + 2401)^2
$7$	\( (T^{8} + 8 T^{6} + \cdots + 2401)^{2} \) (T^8 + 8T^6 + 42T^4 + 392*T^2 + 2401)^2
$11$	\( (T^{8} - 26 T^{6} + \cdots + 49)^{2} \) (T^8 - 26T^6 + 669T^4 - 182*T^2 + 49)^2
$13$	\( (T^{4} + 12 T^{2} + 4)^{4} \) (T^4 + 12*T^2 + 4)^4
$17$	\( (T^{4} + 2 T^{3} + \cdots + 289)^{4} \) (T^4 + 2T^3 + 21T^2 - 34*T + 289)^4
$19$	\( (T^{8} - 90 T^{6} + \cdots + 4092529)^{2} \) (T^8 - 90T^6 + 6077T^4 - 182070*T^2 + 4092529)^2
$23$	\( (T^{8} + 42 T^{6} + \cdots + 117649)^{2} \) (T^8 + 42T^6 + 1421T^4 + 14406*T^2 + 117649)^2
$29$	\( (T^{2} + 2)^{8} \) (T^2 + 2)^8
$31$	\( (T^{8} + 54 T^{6} + \cdots + 321489)^{2} \) (T^8 + 54T^6 + 2349T^4 + 30618*T^2 + 321489)^2
$37$	\( (T^{8} - 6 T^{6} + 35 T^{4} + \cdots + 1)^{2} \) (T^8 - 6T^6 + 35T^4 - 6*T^2 + 1)^2
$41$	\( (T^{2} + 8 T - 2)^{8} \) (T^2 + 8*T - 2)^8
$43$	\( (T^{4} + 40 T^{2} + 112)^{4} \) (T^4 + 40*T^2 + 112)^4
$47$	\( (T^{8} + 54 T^{6} + \cdots + 49)^{2} \) (T^8 + 54T^6 + 2909T^4 + 378*T^2 + 49)^2
$53$	\( (T^{8} - 214 T^{6} + \cdots + 62742241)^{2} \) (T^8 - 214T^6 + 37875T^4 - 1695094*T^2 + 62742241)^2
$59$	\( (T^{8} - 54 T^{6} + \cdots + 49)^{2} \) (T^8 - 54T^6 + 2909T^4 - 378*T^2 + 49)^2
$61$	\( (T^{8} - 102 T^{6} + \cdots + 4879681)^{2} \) (T^8 - 102T^6 + 8195T^4 - 225318*T^2 + 4879681)^2
$67$	\( (T^{8} - 262 T^{6} + \cdots + 4092529)^{2} \) (T^8 - 262T^6 + 66621T^4 - 530026*T^2 + 4092529)^2
$71$	\( (T^{4} - 24 T^{2} + 112)^{4} \) (T^4 - 24*T^2 + 112)^4
$73$	\( (T^{4} - 6 T^{3} + \cdots + 529)^{4} \) (T^4 - 6T^3 + 59T^2 + 138*T + 529)^4
$79$	\( (T^{8} + 234 T^{6} + \cdots + 45252529)^{2} \) (T^8 + 234T^6 + 48029T^4 + 1574118*T^2 + 45252529)^2
$83$	\( (T^{4} + 48 T^{2} + 448)^{4} \) (T^4 + 48*T^2 + 448)^4
$89$	\( (T^{2} - 5 T + 25)^{8} \) (T^2 - 5*T + 25)^8
$97$	\( (T^{2} + 20 T + 98)^{8} \) (T^2 + 20*T + 98)^8
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