LMFDB - Newform orbit 192.3.l.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(79,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("192.79");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	192.l (of order $4$, degree $2$, 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:	$5.23162107572$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 48)
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_{2} q^{3} - \beta_{9} q^{5} + \beta_{5} q^{7} - 3 \beta_{4} q^{9}+O(q^{10}) $ q - b2 * q^3 - b9 * q^5 + b5 * q^7 - 3b4 q^9	$ q - \beta_{2} q^{3} - \beta_{9} q^{5} + \beta_{5} q^{7} - 3 \beta_{4} q^{9} + ( - \beta_{10} + \beta_{7} - 2 \beta_{4} - 2) q^{11} + (\beta_{13} + \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{13}+ \cdots + ( - 3 \beta_{11} - 3 \beta_{9} + \cdots - 6) q^{99}+O(q^{100}) $ q - b2 * q^3 - b9 * q^5 + b5 * q^7 - 3b4 q^9 + (-b10 + b7 - 2b4 - 2) q^11 + (b13 + b8 + b7 - b6 + b5 - b1) * q^13 - b14 * q^15 + (-b9 + b7 - b6 + b5 - 2b3 - 2b2 + b1) * q^17 + (b15 + 2b12 + b11 - b9 - 2b4 + b1 + 2) * q^19 + b12 * q^21 + (-2b15 + 2b14 + 2b13 - 2b12 - 2b9 + 2b7 - 2b6 + 2b5 - 2b3 - 2b2 - 2b1 + 8) q^23 + (-b15 - b14 - b13 - b12 + b11 - b10 - 2b8 - 5b4 + 4b3 - 4b2) * q^25 + 3b3 q^27 + (-b15 + 3b14 + 2b13 - 2b10 + b8 + b7 + b6 + b5 + 2b4 - b1 + 2) * q^29 + (-2b15 - b11 + b10 - b9 + b8 - b7 - 8b4) * q^31 + (b15 - b14 - b13 + b12 + b11 + b10 + b9 - b7 + b6 + b5 + 2b3 + 2b2 + b1) * q^33 + (2b15 - b11 - b9 + 2b8 - 2b5 + 6b4 + 2b1 - 6) q^35 + (b15 - 3b14 + b12 - 2b11 + b9 + b8 + 3b6 - b5 + 6b4 + b1 - 6) * q^37 + (b11 + b10 + b9 - b7 - b6 + b5 - 2b1) q^39 + (b15 + b14 - 2b13 - 2b12 + b9 - 3b8 + b7 - 6b3 + 6b2) q^41 + (3b15 - 2b14 - 2b13 + b10 - 2b8 + b7 - 2b5 - 10b4 - b1 - 10) * q^43 - 3b7 q^45 + (-2b13 - 2b12 - 2b9 - 2b8 - 2b7 + 24b4 - 6b3 + 6b2) * q^47 + (b15 - b14 - b13 + b12 + b11 + b10 + 4b9 - 4b7 - 2b6 - 4b5 - 8b3 - 8b2 + 7) * q^49 + (b15 - 2b14 + 2b12 - b11 + 5b9 + 2b8 + 2b6 - 2b5 - 6b4 + b1 + 6) q^51 + (-b15 + 3b14 - 2b12 - 2b11 - 3b9 + 3b8 - 3b6 - 3b5 + 10b4 + 8b2 - b1 - 10) q^53 + (-b11 - b10 + 7b9 - 7b7 + 4b6 - 2b5 + 4b3 + 4b2 + 2b1 + 16) q^55 + (b15 - 3b14 - 2b11 + 2b10 + b9 - b8 + b7 + 2b3 - 2b2) q^57 + (-2b15 + 4b14 - 2b10 - 2b8 + 2b7 + 4b6 - 2b5 + 8b4 + 2b1 + 8) q^59 + (2b15 - b13 + 6b10 + b8 - b7 + b6 + b5 - 2b4 + 16b3 - b1 - 2) * q^61 - 3b8 q^63 + (-2b15 + 2b14 + 2b13 - 2b12 - b9 + b7 + 3b6 - 3b5 + 10b3 + 10b2 + b1 - 2) * q^65 + (-2b15 + 4b11 + 4b9 - 2b8 + 2b5 + 20b4 + 4b2 - 2b1 - 20) * q^67 + (-2b15 + 2b11 + 2b9 + 2b8 - 2b5 - 6b4 - 8b2 - 2b1 + 6) * q^69 + (-2b5 - 32) q^71 + (-5b15 + 7b14 + b13 + b12 + b11 - b10 - 4b9 - 4b7 + 6b4 - 8b3 + 8b2) q^73 + (-b15 - 2b13 - 6b7 + 2b6 - 12b4 + 5b3 + 3b1 - 12) * q^75 + (-b15 - b14 - 2b10 - 5b8 - b6 - 5b5 + 14b4 - 24b3 + b1 + 14) q^77 + (-2b11 + 2b10 + 6b9 + 5b8 + 6b7 + 8b4 + 12b3 - 12b2) * q^79 - 9 * q^81 + (-2b15 + 4b14 - 4b12 - 3b11 + b9 - 4b6 - 10b4 - 2b1 + 10) q^83 + (4b15 - 2b14 + 2b12 + 2b11 + 6b9 - 4b8 + 2b6 + 4b5 - 10b4 + 32b2 + 4b1 + 10) q^85 + (2b15 - 2b14 - 2b13 + 2b12 + 3b11 + 3b10 - 3b9 + 3b7 - b6 - 2b3 - 2b2) * q^87 + (4b15 - 4b14 - 2b11 + 2b10 + 8b8 + 10b4 + 12b3 - 12b2) * q^89 + (3b15 - 6b14 + 2b13 + 3b10 - 5b7 - 8b6 + 30b4 - 12b3 - 5b1 + 30) q^91 + (-b13 - 4b10 - 5b8 - 2b7 + b6 - 5b5 + 8b3 + b1) q^93 + (4b15 + 2b13 + 2b12 + 2b11 - 2b10 + 4b9 + 4b7 - 40b4 - 10b3 + 10b2) * q^95 + (2b15 - 2b14 - 2b13 + 2b12 - 4b11 - 4b10 + 2b6 - 2b5 + 12b3 + 12b2 - 2b1) q^97 + (-3b11 - 3b9 + 6b4 - 6) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} - 96 q^{35} - 96 q^{37} - 160 q^{43} + 112 q^{49} + 96 q^{51} - 160 q^{53} + 256 q^{55} + 128 q^{59} - 32 q^{61} - 32 q^{65} - 320 q^{67} + 96 q^{69} - 512 q^{71} - 192 q^{75} + 224 q^{77} - 144 q^{81} + 160 q^{83} + 160 q^{85} + 480 q^{91} - 96 q^{99}+O(q^{100}) $ 16 * q - 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 - 96 * q^35 - 96 * q^37 - 160 * q^43 + 112 * q^49 + 96 * q^51 - 160 * q^53 + 256 * q^55 + 128 * q^59 - 32 * q^61 - 32 * q^65 - 320 * q^67 + 96 * q^69 - 512 * q^71 - 192 * q^75 + 224 * q^77 - 144 * q^81 + 160 * q^83 + 160 * q^85 + 480 * q^91 - 96 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6*x^14 - 4*x^13 + 10*x^12 + 56*x^11 + 88*x^10 - 128*x^9 - 496*x^8 - 512*x^7 + 1408*x^6 + 3584*x^5 + 2560*x^4 - 4096*x^3 - 24576*x^2 + 65536 :

$\beta_{1}$	$=$	$ ( 14 \nu^{15} - 15 \nu^{14} - 80 \nu^{13} - 126 \nu^{12} + 80 \nu^{11} + 1258 \nu^{10} + 1392 \nu^{9} + \cdots - 163840 ) / 61440 $ (14v^15 - 15v^14 - 80v^13 - 126v^12 + 80v^11 + 1258v^10 + 1392v^9 - 2184v^8 - 10752v^7 - 16752v^6 + 16960v^5 + 82304v^4 + 84480v^3 - 166400v^2 - 620544*v - 163840) / 61440
$\beta_{2}$	$=$	$ ( - 81 \nu^{15} - 268 \nu^{14} - 218 \nu^{13} + 588 \nu^{12} + 2310 \nu^{11} + 1616 \nu^{10} + \cdots + 180224 ) / 245760 $ (-81v^15 - 268v^14 - 218v^13 + 588v^12 + 2310v^11 + 1616v^10 - 9208v^9 - 30752v^8 - 38416v^7 + 22336v^6 + 142976v^5 + 146432v^4 - 195072v^3 - 976896v^2 - 966656*v + 180224) / 245760
$\beta_{3}$	$=$	$ ( 131 \nu^{15} + 88 \nu^{14} - 1122 \nu^{13} - 2268 \nu^{12} - 610 \nu^{11} + 9944 \nu^{10} + \cdots - 7716864 ) / 245760 $ (131v^15 + 88v^14 - 1122v^13 - 2268v^12 - 610v^11 + 9944v^10 + 27688v^9 + 1472v^8 - 117584v^7 - 278656v^6 - 125056v^5 + 588288v^4 + 1316352v^3 + 741376v^2 - 4317184*v - 7716864) / 245760
$\beta_{4}$	$=$	$ ( - 347 \nu^{15} - 626 \nu^{14} + 1234 \nu^{13} + 4536 \nu^{12} + 5530 \nu^{11} - 11868 \nu^{10} + \cdots + 10231808 ) / 368640 $ (-347v^15 - 626v^14 + 1234v^13 + 4536v^12 + 5530v^11 - 11868v^10 - 59096v^9 - 66544v^8 + 88528v^7 + 450272v^6 + 454272v^5 - 499456v^4 - 2271744v^3 - 3177472v^2 + 3719168*v + 10231808) / 368640
$\beta_{5}$	$=$	$ ( - 47 \nu^{15} - 110 \nu^{14} + 90 \nu^{13} + 528 \nu^{12} + 610 \nu^{11} - 1684 \nu^{10} + \cdots + 1228800 ) / 40960 $ (-47v^15 - 110v^14 + 90v^13 + 528v^12 + 610v^11 - 1684v^10 - 8376v^9 - 11728v^8 + 5136v^7 + 52256v^6 + 60800v^5 - 73472v^4 - 350720v^3 - 537600v^2 + 172032*v + 1228800) / 40960
$\beta_{6}$	$=$	$ ( 91 \nu^{15} + 1260 \nu^{14} + 3590 \nu^{13} + 2316 \nu^{12} - 9170 \nu^{11} - 28288 \nu^{10} + \cdots + 14909440 ) / 122880 $ (91v^15 + 1260v^14 + 3590v^13 + 2316v^12 - 9170v^11 - 28288v^10 - 4872v^9 + 162144v^8 + 452592v^7 + 476352v^6 - 428800v^5 - 1854464v^4 - 1497600v^3 + 4352000v^2 + 14905344*v + 14909440) / 122880
$\beta_{7}$	$=$	$ ( - 751 \nu^{15} - 448 \nu^{14} + 6626 \nu^{13} + 14076 \nu^{12} + 5306 \nu^{11} - 54888 \nu^{10} + \cdots + 41107456 ) / 368640 $ (-751v^15 - 448v^14 + 6626v^13 + 14076v^12 + 5306v^11 - 54888v^10 - 156376v^9 - 19520v^8 + 641360v^7 + 1598464v^6 + 829440v^5 - 3026432v^4 - 7073280v^3 - 4517888v^2 + 22466560*v + 41107456) / 368640
$\beta_{8}$	$=$	$ ( 1063 \nu^{15} + 2242 \nu^{14} - 2666 \nu^{13} - 14184 \nu^{12} - 21122 \nu^{11} + 26652 \nu^{10} + \cdots - 26214400 ) / 368640 $ (1063v^15 + 2242v^14 - 2666v^13 - 14184v^12 - 21122v^11 + 26652v^10 + 189400v^9 + 278960v^8 - 106640v^7 - 1269472v^6 - 1748352v^5 + 775424v^4 + 6971904v^3 + 12336128v^2 - 4268032*v - 26214400) / 368640
$\beta_{9}$	$=$	$ ( - 545 \nu^{15} - 1574 \nu^{14} - 302 \nu^{13} + 5256 \nu^{12} + 12838 \nu^{11} - 1188 \nu^{10} + \cdots + 6053888 ) / 184320 $ (-545v^15 - 1574v^14 - 302v^13 + 5256v^12 + 12838v^11 - 1188v^10 - 82544v^9 - 190768v^8 - 127664v^7 + 372128v^6 + 897600v^5 + 303872v^4 - 2511360v^3 - 7066624v^2 - 3770368*v + 6053888) / 184320
$\beta_{10}$	$=$	$ ( - 134 \nu^{15} - 20 \nu^{14} + 1153 \nu^{13} + 2232 \nu^{12} + 622 \nu^{11} - 9756 \nu^{10} + \cdots + 6232064 ) / 46080 $ (-134v^15 - 20v^14 + 1153v^13 + 2232v^12 + 622v^11 - 9756v^10 - 25118v^9 + 1448v^8 + 113704v^7 + 257408v^6 + 102288v^5 - 502912v^4 - 1184256v^3 - 538624v^2 + 3969536*v + 6232064) / 46080
$\beta_{11}$	$=$	$ ( 1417 \nu^{15} + 4300 \nu^{14} + 1186 \nu^{13} - 13356 \nu^{12} - 35366 \nu^{11} + 528 \nu^{10} + \cdots - 13975552 ) / 368640 $ (1417v^15 + 4300v^14 + 1186v^13 - 13356v^12 - 35366v^11 + 528v^10 + 219304v^9 + 508256v^8 + 401488v^7 - 933184v^6 - 2421504v^5 - 922624v^4 + 6455808v^3 + 18839552v^2 + 11743232*v - 13975552) / 368640
$\beta_{12}$	$=$	$ ( 411 \nu^{15} - 178 \nu^{14} - 4586 \nu^{13} - 7776 \nu^{12} + 198 \nu^{11} + 38228 \nu^{10} + \cdots - 25722880 ) / 122880 $ (411v^15 - 178v^14 - 4586v^13 - 7776v^12 + 198v^11 + 38228v^10 + 84104v^9 - 37808v^8 - 478864v^7 - 985376v^6 - 282112v^5 + 2124032v^4 + 4176384v^3 + 1102848v^2 - 16609280*v - 25722880) / 122880
$\beta_{13}$	$=$	$ ( - 1229 \nu^{15} + 844 \nu^{14} + 13750 \nu^{13} + 23436 \nu^{12} - 2786 \nu^{11} - 118848 \nu^{10} + \cdots + 79364096 ) / 368640 $ (-1229v^15 + 844v^14 + 13750v^13 + 23436v^12 - 2786v^11 - 118848v^10 - 248264v^9 + 154016v^8 + 1503088v^7 + 2985920v^6 + 638208v^5 - 6613504v^4 - 12980736v^3 - 1914880v^2 + 52957184*v + 79364096) / 368640
$\beta_{14}$	$=$	$ ( - 151 \nu^{15} - 271 \nu^{14} + 512 \nu^{13} + 1878 \nu^{12} + 2402 \nu^{11} - 4854 \nu^{10} + \cdots + 4022272 ) / 30720 $ (-151v^15 - 271v^14 + 512v^13 + 1878v^12 + 2402v^11 - 4854v^10 - 25204v^9 - 30296v^8 + 32192v^7 + 186640v^6 + 194400v^5 - 199808v^4 - 975360v^3 - 1435136v^2 + 1352704*v + 4022272) / 30720
$\beta_{15}$	$=$	$ ( - 4331 \nu^{15} - 7634 \nu^{14} + 15322 \nu^{13} + 56088 \nu^{12} + 66634 \nu^{11} + \cdots + 128368640 ) / 368640 $ (-4331v^15 - 7634v^14 + 15322v^13 + 56088v^12 + 66634v^11 - 143484v^10 - 729800v^9 - 826480v^8 + 1085200v^7 + 5587424v^6 + 5662464v^5 - 6193408v^4 - 28205568v^3 - 39095296v^2 + 45215744*v + 128368640) / 368640

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

$\nu$	$=$	$ ( \beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} + 2\beta_{4} - 2\beta_{3} - \beta_1 ) / 8 $ (b11 - b9 - b8 - b6 + 2b4 - 2b3 - b1) / 8
$\nu^{2}$	$=$	$ ( 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} - 3 \beta_{7} + \cdots + 6 ) / 8 $ (2b15 - 2b14 - 2b13 - b11 + b10 - b9 - 3b7 + 2b6 - 2b4 - 4*b2 + 6) / 8
$\nu^{3}$	$=$	$ ( - \beta_{15} - 3 \beta_{14} - 4 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} + \cdots + 6 ) / 8 $ (-b15 - 3b14 - 4b12 - b11 - b10 + b9 + b8 + 3b7 - 3b6 - b5 + 10b4 + 6b3 - 6b2 - 3b1 + 6) / 8
$\nu^{4}$	$=$	$ ( 2 \beta_{15} - 3 \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + \cdots + 8 ) / 4 $ (2b15 - 3b14 - b13 + b12 + b11 + 2b10 - 4b9 - b8 - 5b7 + b5 + 22b4 + 4b3 - 12b2 + 3*b1 + 8) / 4
$\nu^{5}$	$=$	$ ( - 4 \beta_{14} - 4 \beta_{13} - 8 \beta_{12} - \beta_{11} - 7 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} + \cdots - 40 ) / 4 $ (-4b14 - 4b13 - 8b12 - b11 - 7b9 - 7b8 - 4b7 - b6 + 6b5 - 14b4 - 6b3 + 40b2 - b1 - 40) / 4
$\nu^{6}$	$=$	$ ( 8 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} - 13 \beta_{10} - 7 \beta_{9} + \cdots - 102 ) / 4 $ (8b15 + 2b14 + 4b13 + 2b12 - 9b11 - 13b10 - 7b9 + 8b8 - 7b7 + 8b6 - 4b5 - 2b4 + 36b3 - 48b2 + 8*b1 - 102) / 4
$\nu^{7}$	$=$	$ ( - 11 \beta_{15} - 13 \beta_{14} - 4 \beta_{13} - 8 \beta_{12} + 3 \beta_{11} + 5 \beta_{10} + \cdots - 182 ) / 4 $ (-11b15 - 13b14 - 4b13 - 8b12 + 3b11 + 5b10 + 57b9 + 13b8 - 19b7 + 17b6 - 11b5 + 90b4 - 42b3 + 22b2 + 21*b1 - 182) / 4
$\nu^{8}$	$=$	$ ( - 8 \beta_{15} + 3 \beta_{14} + 21 \beta_{13} + 9 \beta_{12} - 26 \beta_{11} + 3 \beta_{10} + \cdots - 146 ) / 2 $ (-8b15 + 3b14 + 21b13 + 9b12 - 26b11 + 3b10 - 35b9 + 9b8 + 24b7 - 12b6 + 9b5 - 16b4 + 24b3 + 72b2 + 17*b1 - 146) / 2
$\nu^{9}$	$=$	$ ( - 19 \beta_{15} + 71 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} + 42 \beta_{11} - 45 \beta_{10} + \cdots - 138 ) / 2 $ (-19b15 + 71b14 + 12b13 + 16b12 + 42b11 - 45b10 + 38b9 + 12b8 + 23b7 + 38b6 + 35b5 + 20b4 + 108b3 + 270b2 + 30*b1 - 138) / 2
$\nu^{10}$	$=$	$ ( 50 \beta_{15} + 42 \beta_{14} + 38 \beta_{13} + 80 \beta_{12} - 5 \beta_{11} + 17 \beta_{10} + \cdots - 786 ) / 2 $ (50b15 + 42b14 + 38b13 + 80b12 - 5b11 + 17b10 + 107b9 + 56b8 + 53b7 + 6b6 - 128b5 - 1018b4 - 320b3 - 116b2 + 32*b1 - 786) / 2
$\nu^{11}$	$=$	$ ( - 215 \beta_{15} + 219 \beta_{14} + 48 \beta_{13} + 36 \beta_{12} - 143 \beta_{11} - 75 \beta_{10} + \cdots + 1210 ) / 2 $ (-215b15 + 219b14 + 48b13 + 36b12 - 143b11 - 75b10 + 175b9 + 59b8 - 47b7 + 183b6 - 199b5 + 1006b4 - 198b3 + 446b2 + 39*b1 + 1210) / 2
$\nu^{12}$	$=$	$ - 38 \beta_{15} - 13 \beta_{14} + 165 \beta_{13} + 227 \beta_{12} + 175 \beta_{11} + 42 \beta_{10} + \cdots + 8 $ -38b15 - 13b14 + 165b13 + 227b12 + 175b11 + 42b10 + 152b9 + 17b8 + 541b7 - 132b6 + 71b5 - 238b4 + 516b3 + 180b2 - 123*b1 + 8
$\nu^{13}$	$=$	$ 192 \beta_{15} + 300 \beta_{14} - 452 \beta_{13} + 16 \beta_{12} + 161 \beta_{11} + 304 \beta_{10} + \cdots + 3224 $ 192b15 + 300b14 - 452b13 + 16b12 + 161b11 + 304b10 - 353b9 - 9b8 - 52b7 + 33b6 + 10b5 - 1394b4 - 250b3 - 312b2 + 73*b1 + 3224
$\nu^{14}$	$=$	$ - 280 \beta_{15} + 390 \beta_{14} + 484 \beta_{13} - 426 \beta_{12} + 25 \beta_{11} - 371 \beta_{10} + \cdots + 7174 $ -280b15 + 390b14 + 484b13 - 426b12 + 25b11 - 371b10 + 495b9 - 888b8 + 63b7 - 416b6 - 692b5 - 4926b4 - 500b3 - 112b2 - 1096*b1 + 7174
$\nu^{15}$	$=$	$ 211 \beta_{15} - 779 \beta_{14} - 1244 \beta_{13} + 920 \beta_{12} + 565 \beta_{11} + 11 \beta_{10} + \cdots + 4502 $ 211b15 - 779b14 - 1244b13 + 920b12 + 565b11 + 11b10 - 977b9 - 1245b8 - 781b7 + 591b6 - 605b5 + 14390b4 - 2838b3 - 10182b2 - 149*b1 + 4502

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$-1$	$-\beta_{4}$

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} $

79.1

−1.87459	+	0.697079i
−1.25564	−	1.55672i
1.78012	−	0.911682i
0.125358	+	1.99607i
1.84258	−	0.777752i
−0.455024	−	1.94755i
−1.96679	−	0.362960i
1.80398	+	0.863518i
−1.87459	−	0.697079i
−1.25564	+	1.55672i
1.78012	+	0.911682i
0.125358	−	1.99607i
1.84258	+	0.777752i
−0.455024	+	1.94755i
−1.96679	+	0.362960i
1.80398	−	0.863518i

−1.22474

−

1.22474i

−5.24354

−

5.24354i

5.32796

3.00000i

79.2

−1.22474

−

1.22474i

0.909023

0.909023i

0.654713

3.00000i

79.3

−1.22474

−

1.22474i

1.00772

1.00772i

−10.0236

3.00000i

79.4

−1.22474

−

1.22474i

3.32679

3.32679i

4.04088

3.00000i

79.5

1.22474

1.22474i

−4.78830

−

4.78830i

10.3302

3.00000i

79.6

1.22474

1.22474i

−3.40572

−

3.40572i

−12.1303

3.00000i

79.7

1.22474

1.22474i

1.69930

1.69930i

5.74280

3.00000i

79.8

1.22474

1.22474i

6.49473

6.49473i

−3.94273

3.00000i

175.1

−1.22474

1.22474i

−5.24354

5.24354i

5.32796

−

3.00000i

175.2

−1.22474

1.22474i

0.909023

−

0.909023i

0.654713

−

3.00000i

175.3

−1.22474

1.22474i

1.00772

−

1.00772i

−10.0236

−

3.00000i

175.4

−1.22474

1.22474i

3.32679

−

3.32679i

4.04088

−

3.00000i

175.5

1.22474

−

1.22474i

−4.78830

4.78830i

10.3302

−

3.00000i

175.6

1.22474

−

1.22474i

−3.40572

3.40572i

−12.1303

−

3.00000i

175.7

1.22474

−

1.22474i

1.69930

−

1.69930i

5.74280

−

3.00000i

175.8

1.22474

−

1.22474i

6.49473

−

6.49473i

−3.94273

−

3.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.f	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	192.3.l.a		16
3.b	odd	2	1		576.3.m.c		16
4.b	odd	2	1		48.3.l.a	✓	16
8.b	even	2	1		384.3.l.b		16
8.d	odd	2	1		384.3.l.a		16
12.b	even	2	1		144.3.m.c		16
16.e	even	4	1		48.3.l.a	✓	16
16.e	even	4	1		384.3.l.a		16
16.f	odd	4	1	inner	192.3.l.a		16
16.f	odd	4	1		384.3.l.b		16
24.f	even	2	1		1152.3.m.f		16
24.h	odd	2	1		1152.3.m.c		16
48.i	odd	4	1		144.3.m.c		16
48.i	odd	4	1		1152.3.m.f		16
48.k	even	4	1		576.3.m.c		16
48.k	even	4	1		1152.3.m.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
48.3.l.a	✓	16	4.b	odd	2	1
48.3.l.a	✓	16	16.e	even	4	1
144.3.m.c		16	12.b	even	2	1
144.3.m.c		16	48.i	odd	4	1
192.3.l.a		16	1.a	even	1	1	trivial
192.3.l.a		16	16.f	odd	4	1	inner
384.3.l.a		16	8.d	odd	2	1
384.3.l.a		16	16.e	even	4	1
384.3.l.b		16	8.b	even	2	1
384.3.l.b		16	16.f	odd	4	1
576.3.m.c		16	3.b	odd	2	1
576.3.m.c		16	48.k	even	4	1
1152.3.m.c		16	24.h	odd	2	1
1152.3.m.c		16	48.k	even	4	1
1152.3.m.f		16	24.f	even	2	1
1152.3.m.f		16	48.i	odd	4	1

Hecke kernels

This newform subspace is the entire newspace $S_{3}^{\mathrm{new}}(192, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{4} + 9)^{4} $ (T^4 + 9)^4
$5$	$ T^{16} + \cdots + 2117472256 $ T^16 + 32T^13 + 6656T^12 + 8064T^11 + 512T^10 - 518400T^9 + 6221952T^8 - 4026368T^7 + 12517376T^6 - 283883520T^5 + 1663893504T^4 - 4258177024T^3 + 6315081728T^2 - 5171462144*T + 2117472256
$7$	$ (T^{8} - 224 T^{6} + \cdots - 400880)^{2} $ (T^8 - 224T^6 + 448T^5 + 13704T^4 - 53248T^3 - 136576T^2 + 720640T - 400880)^2
$11$	$ T^{16} + \cdots + 25620118503424 $ T^16 + 32T^15 + 512T^14 + 4608T^13 + 151552T^12 + 4242432T^11 + 68780032T^10 + 584835072T^9 + 3292530688T^8 + 21404090368T^7 + 269438418944T^6 + 2206640635904T^5 + 10536909537280T^4 + 22391525736448T^3 + 16962470019072T^2 - 29481536323584*T + 25620118503424
$13$	$ T^{16} + \cdots + 27\!\cdots\!00 $ T^16 + 3200T^13 + 271248T^12 + 1717760T^11 + 5120000T^10 + 8847872T^9 + 5634806368T^8 + 37570686976T^7 + 114873139200T^6 - 1347500722176T^5 + 18426489989376T^4 + 75155778920448T^3 + 255245502513152T^2 - 3777806164172800*T + 27957043852960000
$17$	$ (T^{8} - 1344 T^{6} + \cdots + 816881920)^{2} $ (T^8 - 1344T^6 - 2944T^5 + 508448T^4 + 1044480T^3 - 53986304T^2 + 9390080T + 816881920)^2
$19$	$ T^{16} + \cdots + 10\!\cdots\!96 $ T^16 - 32T^15 + 512T^14 + 14208T^13 + 559552T^12 - 15014400T^11 + 294903808T^10 + 9271351296T^9 + 149480318464T^8 - 1219751600128T^7 + 41927575470080T^6 + 1508224010715136T^5 + 23939487244337152T^4 + 89104253114777600T^3 + 70335181828915200T^2 - 1221045122401566720*T + 10598900522979229696
$23$	$ (T^{8} - 64 T^{7} + \cdots - 35037900800)^{2} $ (T^8 - 64T^7 - 736T^6 + 109056T^5 - 1089152T^4 - 42348544T^3 + 777799680T^2 - 425492480*T - 35037900800)^2
$29$	$ T^{16} + \cdots + 56\!\cdots\!00 $ T^16 - 32T^15 + 512T^14 + 45280T^13 + 3957248T^12 - 64121984T^11 + 1050931712T^10 + 92505709824T^9 + 4665816875136T^8 - 22033099003904T^7 + 401836469485568T^6 + 34131461984319488T^5 + 1336868519252525056T^4 + 1809064899469205504T^3 + 943555165480583168T^2 - 10320873959271219200*T + 56446323002698240000
$31$	$ T^{16} + \cdots + 41\!\cdots\!00 $ T^16 + 8064T^14 + 26696080T^12 + 46823575040T^10 + 46734989650528T^8 + 26353243415873536T^6 + 7691970571770562816T^4 + 885437391795507240960*T^2 + 4109607350375443718400
$37$	$ T^{16} + \cdots + 38\!\cdots\!04 $ T^16 + 96T^15 + 4608T^14 + 14528T^13 + 1620496T^12 + 224961664T^11 + 14234605568T^10 - 28675020544T^9 - 968313092000T^8 + 6367638655488T^7 + 6893443090702336T^6 - 97908324877949952T^5 + 703119961976168704T^4 + 2884347249537718272T^3 + 70869372871016480768T^2 - 734284911784017907712*T + 3804001007988103938304
$41$	$ T^{16} + \cdots + 94\!\cdots\!00 $ T^16 + 13056T^14 + 70874688T^12 + 207527260160T^10 + 356212041131520T^8 + 365640212557922304T^6 + 219522060142929657856T^4 + 70805170442242133852160*T^2 + 9447035993707891197542400
$43$	$ T^{16} + \cdots + 92\!\cdots\!00 $ T^16 + 160T^15 + 12800T^14 + 682624T^13 + 46122944T^12 + 4149187072T^11 + 306484011008T^10 + 15508493912064T^9 + 537795837003264T^8 + 12945845326176256T^7 + 245950489383796736T^6 + 5239225685693136896T^5 + 144279758486420045824T^4 + 3207498038265755598848T^3 + 44479868925767390855168T^2 + 286341547624904646983680*T + 921669553849870911078400
$47$	$ T^{16} + \cdots + 11\!\cdots\!00 $ T^16 + 11200T^14 + 41678592T^12 + 69921902592T^10 + 57150256209920T^8 + 22735051785764864T^6 + 4369756732867477504T^4 + 373417878874107150336*T^2 + 11024466169662170726400
$53$	$ T^{16} + \cdots + 96\!\cdots\!00 $ T^16 + 160T^15 + 12800T^14 + 153504T^13 + 16960000T^12 + 3156153984T^11 + 299678376448T^10 + 1029094838016T^9 + 100041121243264T^8 + 18190042770372608T^7 + 1685959386162102272T^6 + 5591746172921030656T^5 + 91449263140998676480T^4 + 19278049905064818974720T^3 + 2138354891263017100541952T^2 + 2030668155695661908213760*T + 964202241499962888294400
$59$	$ T^{16} + \cdots + 23\!\cdots\!00 $ T^16 - 128T^15 + 8192T^14 - 675840T^13 + 158699520T^12 - 20765343744T^11 + 1586277384192T^10 - 72789256044544T^9 + 2068957458595840T^8 - 32123905497890816T^7 + 215142153171501056T^6 - 714303909442617344T^5 + 97922953659373584384T^4 - 1624651112584394571776T^3 + 6981616736878284767232T^2 + 181321183266412167168000*T + 2354567196983689216000000
$61$	$ T^{16} + \cdots + 12\!\cdots\!00 $ T^16 + 32T^15 + 512T^14 - 157120T^13 + 115399952T^12 + 2795215232T^11 + 42705459200T^10 - 12153910400256T^9 + 3370685859892320T^8 + 49175921370646016T^7 + 735866880468475904T^6 - 203632445835649061888T^5 + 35750546801949141823744T^4 + 224140820215758491060224T^3 + 3313319495492789381660672T^2 - 903620687869229632700968960*T + 123219379938459167726987526400
$67$	$ T^{16} + \cdots + 21\!\cdots\!96 $ T^16 + 320T^15 + 51200T^14 + 4611072T^13 + 255286272T^12 + 9110568960T^11 + 475717435392T^10 + 36863866568704T^9 + 2061397582544896T^8 + 50262831949414400T^7 + 579595725034225664T^6 - 621625262253015040T^5 + 133010965887499370496T^4 + 2866476071676048048128T^3 + 30474643215075243982848T^2 - 115385479389312769327104*T + 218440766638990972420096
$71$	$ (T^{8} + 256 T^{7} + \cdots + 290924400640)^{2} $ (T^8 + 256T^7 + 27776T^6 + 1659392T^5 + 59283584T^4 + 1284915200T^3 + 16299499520T^2 + 109021003776*T + 290924400640)^2
$73$	$ T^{16} + \cdots + 98\!\cdots\!16 $ T^16 + 42496T^14 + 709382400T^12 + 5899646435328T^10 + 25520342188187648T^8 + 53883999480140791808T^6 + 45327992512474743046144T^4 + 13067729363675367868465152*T^2 + 985907622738458033799561216
$79$	$ T^{16} + \cdots + 18\!\cdots\!00 $ T^16 + 36928T^14 + 458363920T^12 + 2270484756224T^10 + 4201068991559776T^8 + 1719443636116761600T^6 + 114950930654421319936T^4 + 2587752408175977123840*T^2 + 18873453677143355654400
$83$	$ T^{16} + \cdots + 14\!\cdots\!76 $ T^16 - 160T^15 + 12800T^14 + 206336T^13 + 3373056T^12 - 3104705536T^11 + 474865041408T^10 + 17298459828224T^9 + 105517061441536T^8 - 5903701162754048T^7 + 6381120662228434944T^6 + 396017560781535576064T^5 + 12138187060968167571456T^4 + 137663961287507678593024T^3 + 635020045117962089136128T^2 - 4236012087763315913129984*T + 14128529127252719096037376
$89$	$ T^{16} + \cdots + 52\!\cdots\!00 $ T^16 + 45728T^14 + 743027648T^12 + 5304960677376T^10 + 18282328190707200T^8 + 30661655762820161536T^6 + 23073336363015526137856T^4 + 6855948794487416265768960*T^2 + 520469506007627359651430400
$97$	$ (T^{8} + \cdots + 409778579046400)^{2} $ (T^8 - 37056T^6 + 116224T^5 + 383621120T^4 - 1982349312T^3 - 1019025981440T^2 + 2337541980160T + 409778579046400)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{3} - \beta_{9} q^{5} + \beta_{5} q^{7} - 3 \beta_{4} q^{9}+O(q^{10}) \) q - b2 * q^3 - b9 * q^5 + b5 * q^7 - 3b4 q^9	\( q - \beta_{2} q^{3} - \beta_{9} q^{5} + \beta_{5} q^{7} - 3 \beta_{4} q^{9} + ( - \beta_{10} + \beta_{7} - 2 \beta_{4} - 2) q^{11} + (\beta_{13} + \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{13}+ \cdots + ( - 3 \beta_{11} - 3 \beta_{9} + \cdots - 6) q^{99}+O(q^{100}) \) q - b2 * q^3 - b9 * q^5 + b5 * q^7 - 3b4 q^9 + (-b10 + b7 - 2b4 - 2) q^11 + (b13 + b8 + b7 - b6 + b5 - b1) * q^13 - b14 * q^15 + (-b9 + b7 - b6 + b5 - 2b3 - 2b2 + b1) * q^17 + (b15 + 2b12 + b11 - b9 - 2b4 + b1 + 2) * q^19 + b12 * q^21 + (-2b15 + 2b14 + 2b13 - 2b12 - 2b9 + 2b7 - 2b6 + 2b5 - 2b3 - 2b2 - 2b1 + 8) q^23 + (-b15 - b14 - b13 - b12 + b11 - b10 - 2b8 - 5b4 + 4b3 - 4b2) * q^25 + 3b3 q^27 + (-b15 + 3b14 + 2b13 - 2b10 + b8 + b7 + b6 + b5 + 2b4 - b1 + 2) * q^29 + (-2b15 - b11 + b10 - b9 + b8 - b7 - 8b4) * q^31 + (b15 - b14 - b13 + b12 + b11 + b10 + b9 - b7 + b6 + b5 + 2b3 + 2b2 + b1) * q^33 + (2b15 - b11 - b9 + 2b8 - 2b5 + 6b4 + 2b1 - 6) q^35 + (b15 - 3b14 + b12 - 2b11 + b9 + b8 + 3b6 - b5 + 6b4 + b1 - 6) * q^37 + (b11 + b10 + b9 - b7 - b6 + b5 - 2b1) q^39 + (b15 + b14 - 2b13 - 2b12 + b9 - 3b8 + b7 - 6b3 + 6b2) q^41 + (3b15 - 2b14 - 2b13 + b10 - 2b8 + b7 - 2b5 - 10b4 - b1 - 10) * q^43 - 3b7 q^45 + (-2b13 - 2b12 - 2b9 - 2b8 - 2b7 + 24b4 - 6b3 + 6b2) * q^47 + (b15 - b14 - b13 + b12 + b11 + b10 + 4b9 - 4b7 - 2b6 - 4b5 - 8b3 - 8b2 + 7) * q^49 + (b15 - 2b14 + 2b12 - b11 + 5b9 + 2b8 + 2b6 - 2b5 - 6b4 + b1 + 6) q^51 + (-b15 + 3b14 - 2b12 - 2b11 - 3b9 + 3b8 - 3b6 - 3b5 + 10b4 + 8b2 - b1 - 10) q^53 + (-b11 - b10 + 7b9 - 7b7 + 4b6 - 2b5 + 4b3 + 4b2 + 2b1 + 16) q^55 + (b15 - 3b14 - 2b11 + 2b10 + b9 - b8 + b7 + 2b3 - 2b2) q^57 + (-2b15 + 4b14 - 2b10 - 2b8 + 2b7 + 4b6 - 2b5 + 8b4 + 2b1 + 8) q^59 + (2b15 - b13 + 6b10 + b8 - b7 + b6 + b5 - 2b4 + 16b3 - b1 - 2) * q^61 - 3b8 q^63 + (-2b15 + 2b14 + 2b13 - 2b12 - b9 + b7 + 3b6 - 3b5 + 10b3 + 10b2 + b1 - 2) * q^65 + (-2b15 + 4b11 + 4b9 - 2b8 + 2b5 + 20b4 + 4b2 - 2b1 - 20) * q^67 + (-2b15 + 2b11 + 2b9 + 2b8 - 2b5 - 6b4 - 8b2 - 2b1 + 6) * q^69 + (-2b5 - 32) q^71 + (-5b15 + 7b14 + b13 + b12 + b11 - b10 - 4b9 - 4b7 + 6b4 - 8b3 + 8b2) q^73 + (-b15 - 2b13 - 6b7 + 2b6 - 12b4 + 5b3 + 3b1 - 12) * q^75 + (-b15 - b14 - 2b10 - 5b8 - b6 - 5b5 + 14b4 - 24b3 + b1 + 14) q^77 + (-2b11 + 2b10 + 6b9 + 5b8 + 6b7 + 8b4 + 12b3 - 12b2) * q^79 - 9 * q^81 + (-2b15 + 4b14 - 4b12 - 3b11 + b9 - 4b6 - 10b4 - 2b1 + 10) q^83 + (4b15 - 2b14 + 2b12 + 2b11 + 6b9 - 4b8 + 2b6 + 4b5 - 10b4 + 32b2 + 4b1 + 10) q^85 + (2b15 - 2b14 - 2b13 + 2b12 + 3b11 + 3b10 - 3b9 + 3b7 - b6 - 2b3 - 2b2) * q^87 + (4b15 - 4b14 - 2b11 + 2b10 + 8b8 + 10b4 + 12b3 - 12b2) * q^89 + (3b15 - 6b14 + 2b13 + 3b10 - 5b7 - 8b6 + 30b4 - 12b3 - 5b1 + 30) q^91 + (-b13 - 4b10 - 5b8 - 2b7 + b6 - 5b5 + 8b3 + b1) q^93 + (4b15 + 2b13 + 2b12 + 2b11 - 2b10 + 4b9 + 4b7 - 40b4 - 10b3 + 10b2) * q^95 + (2b15 - 2b14 - 2b13 + 2b12 - 4b11 - 4b10 + 2b6 - 2b5 + 12b3 + 12b2 - 2b1) q^97 + (-3b11 - 3b9 + 6b4 - 6) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} - 96 q^{35} - 96 q^{37} - 160 q^{43} + 112 q^{49} + 96 q^{51} - 160 q^{53} + 256 q^{55} + 128 q^{59} - 32 q^{61} - 32 q^{65} - 320 q^{67} + 96 q^{69} - 512 q^{71} - 192 q^{75} + 224 q^{77} - 144 q^{81} + 160 q^{83} + 160 q^{85} + 480 q^{91} - 96 q^{99}+O(q^{100}) \) 16 * q - 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 - 96 * q^35 - 96 * q^37 - 160 * q^43 + 112 * q^49 + 96 * q^51 - 160 * q^53 + 256 * q^55 + 128 * q^59 - 32 * q^61 - 32 * q^65 - 320 * q^67 + 96 * q^69 - 512 * q^71 - 192 * q^75 + 224 * q^77 - 144 * q^81 + 160 * q^83 + 160 * q^85 + 480 * q^91 - 96 * q^99

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	192.3.l.a

Level	$192$
Weight	$3$
Character orbit	192.l
Analytic conductor	$5.232$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(5.23162107572\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 \) x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{24} \)
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 14 \nu^{15} - 15 \nu^{14} - 80 \nu^{13} - 126 \nu^{12} + 80 \nu^{11} + 1258 \nu^{10} + 1392 \nu^{9} + \cdots - 163840 ) / 61440 \) (14v^15 - 15v^14 - 80v^13 - 126v^12 + 80v^11 + 1258v^10 + 1392v^9 - 2184v^8 - 10752v^7 - 16752v^6 + 16960v^5 + 82304v^4 + 84480v^3 - 166400v^2 - 620544*v - 163840) / 61440
\(\beta_{2}\)	\(=\)	\( ( - 81 \nu^{15} - 268 \nu^{14} - 218 \nu^{13} + 588 \nu^{12} + 2310 \nu^{11} + 1616 \nu^{10} + \cdots + 180224 ) / 245760 \) (-81v^15 - 268v^14 - 218v^13 + 588v^12 + 2310v^11 + 1616v^10 - 9208v^9 - 30752v^8 - 38416v^7 + 22336v^6 + 142976v^5 + 146432v^4 - 195072v^3 - 976896v^2 - 966656*v + 180224) / 245760
\(\beta_{3}\)	\(=\)	\( ( 131 \nu^{15} + 88 \nu^{14} - 1122 \nu^{13} - 2268 \nu^{12} - 610 \nu^{11} + 9944 \nu^{10} + \cdots - 7716864 ) / 245760 \) (131v^15 + 88v^14 - 1122v^13 - 2268v^12 - 610v^11 + 9944v^10 + 27688v^9 + 1472v^8 - 117584v^7 - 278656v^6 - 125056v^5 + 588288v^4 + 1316352v^3 + 741376v^2 - 4317184*v - 7716864) / 245760
\(\beta_{4}\)	\(=\)	\( ( - 347 \nu^{15} - 626 \nu^{14} + 1234 \nu^{13} + 4536 \nu^{12} + 5530 \nu^{11} - 11868 \nu^{10} + \cdots + 10231808 ) / 368640 \) (-347v^15 - 626v^14 + 1234v^13 + 4536v^12 + 5530v^11 - 11868v^10 - 59096v^9 - 66544v^8 + 88528v^7 + 450272v^6 + 454272v^5 - 499456v^4 - 2271744v^3 - 3177472v^2 + 3719168*v + 10231808) / 368640
\(\beta_{5}\)	\(=\)	\( ( - 47 \nu^{15} - 110 \nu^{14} + 90 \nu^{13} + 528 \nu^{12} + 610 \nu^{11} - 1684 \nu^{10} + \cdots + 1228800 ) / 40960 \) (-47v^15 - 110v^14 + 90v^13 + 528v^12 + 610v^11 - 1684v^10 - 8376v^9 - 11728v^8 + 5136v^7 + 52256v^6 + 60800v^5 - 73472v^4 - 350720v^3 - 537600v^2 + 172032*v + 1228800) / 40960
\(\beta_{6}\)	\(=\)	\( ( 91 \nu^{15} + 1260 \nu^{14} + 3590 \nu^{13} + 2316 \nu^{12} - 9170 \nu^{11} - 28288 \nu^{10} + \cdots + 14909440 ) / 122880 \) (91v^15 + 1260v^14 + 3590v^13 + 2316v^12 - 9170v^11 - 28288v^10 - 4872v^9 + 162144v^8 + 452592v^7 + 476352v^6 - 428800v^5 - 1854464v^4 - 1497600v^3 + 4352000v^2 + 14905344*v + 14909440) / 122880
\(\beta_{7}\)	\(=\)	\( ( - 751 \nu^{15} - 448 \nu^{14} + 6626 \nu^{13} + 14076 \nu^{12} + 5306 \nu^{11} - 54888 \nu^{10} + \cdots + 41107456 ) / 368640 \) (-751v^15 - 448v^14 + 6626v^13 + 14076v^12 + 5306v^11 - 54888v^10 - 156376v^9 - 19520v^8 + 641360v^7 + 1598464v^6 + 829440v^5 - 3026432v^4 - 7073280v^3 - 4517888v^2 + 22466560*v + 41107456) / 368640
\(\beta_{8}\)	\(=\)	\( ( 1063 \nu^{15} + 2242 \nu^{14} - 2666 \nu^{13} - 14184 \nu^{12} - 21122 \nu^{11} + 26652 \nu^{10} + \cdots - 26214400 ) / 368640 \) (1063v^15 + 2242v^14 - 2666v^13 - 14184v^12 - 21122v^11 + 26652v^10 + 189400v^9 + 278960v^8 - 106640v^7 - 1269472v^6 - 1748352v^5 + 775424v^4 + 6971904v^3 + 12336128v^2 - 4268032*v - 26214400) / 368640
\(\beta_{9}\)	\(=\)	\( ( - 545 \nu^{15} - 1574 \nu^{14} - 302 \nu^{13} + 5256 \nu^{12} + 12838 \nu^{11} - 1188 \nu^{10} + \cdots + 6053888 ) / 184320 \) (-545v^15 - 1574v^14 - 302v^13 + 5256v^12 + 12838v^11 - 1188v^10 - 82544v^9 - 190768v^8 - 127664v^7 + 372128v^6 + 897600v^5 + 303872v^4 - 2511360v^3 - 7066624v^2 - 3770368*v + 6053888) / 184320
\(\beta_{10}\)	\(=\)	\( ( - 134 \nu^{15} - 20 \nu^{14} + 1153 \nu^{13} + 2232 \nu^{12} + 622 \nu^{11} - 9756 \nu^{10} + \cdots + 6232064 ) / 46080 \) (-134v^15 - 20v^14 + 1153v^13 + 2232v^12 + 622v^11 - 9756v^10 - 25118v^9 + 1448v^8 + 113704v^7 + 257408v^6 + 102288v^5 - 502912v^4 - 1184256v^3 - 538624v^2 + 3969536*v + 6232064) / 46080
\(\beta_{11}\)	\(=\)	\( ( 1417 \nu^{15} + 4300 \nu^{14} + 1186 \nu^{13} - 13356 \nu^{12} - 35366 \nu^{11} + 528 \nu^{10} + \cdots - 13975552 ) / 368640 \) (1417v^15 + 4300v^14 + 1186v^13 - 13356v^12 - 35366v^11 + 528v^10 + 219304v^9 + 508256v^8 + 401488v^7 - 933184v^6 - 2421504v^5 - 922624v^4 + 6455808v^3 + 18839552v^2 + 11743232*v - 13975552) / 368640
\(\beta_{12}\)	\(=\)	\( ( 411 \nu^{15} - 178 \nu^{14} - 4586 \nu^{13} - 7776 \nu^{12} + 198 \nu^{11} + 38228 \nu^{10} + \cdots - 25722880 ) / 122880 \) (411v^15 - 178v^14 - 4586v^13 - 7776v^12 + 198v^11 + 38228v^10 + 84104v^9 - 37808v^8 - 478864v^7 - 985376v^6 - 282112v^5 + 2124032v^4 + 4176384v^3 + 1102848v^2 - 16609280*v - 25722880) / 122880
\(\beta_{13}\)	\(=\)	\( ( - 1229 \nu^{15} + 844 \nu^{14} + 13750 \nu^{13} + 23436 \nu^{12} - 2786 \nu^{11} - 118848 \nu^{10} + \cdots + 79364096 ) / 368640 \) (-1229v^15 + 844v^14 + 13750v^13 + 23436v^12 - 2786v^11 - 118848v^10 - 248264v^9 + 154016v^8 + 1503088v^7 + 2985920v^6 + 638208v^5 - 6613504v^4 - 12980736v^3 - 1914880v^2 + 52957184*v + 79364096) / 368640
\(\beta_{14}\)	\(=\)	\( ( - 151 \nu^{15} - 271 \nu^{14} + 512 \nu^{13} + 1878 \nu^{12} + 2402 \nu^{11} - 4854 \nu^{10} + \cdots + 4022272 ) / 30720 \) (-151v^15 - 271v^14 + 512v^13 + 1878v^12 + 2402v^11 - 4854v^10 - 25204v^9 - 30296v^8 + 32192v^7 + 186640v^6 + 194400v^5 - 199808v^4 - 975360v^3 - 1435136v^2 + 1352704*v + 4022272) / 30720
\(\beta_{15}\)	\(=\)	\( ( - 4331 \nu^{15} - 7634 \nu^{14} + 15322 \nu^{13} + 56088 \nu^{12} + 66634 \nu^{11} + \cdots + 128368640 ) / 368640 \) (-4331v^15 - 7634v^14 + 15322v^13 + 56088v^12 + 66634v^11 - 143484v^10 - 729800v^9 - 826480v^8 + 1085200v^7 + 5587424v^6 + 5662464v^5 - 6193408v^4 - 28205568v^3 - 39095296v^2 + 45215744*v + 128368640) / 368640

\(\nu\)	\(=\)	\( ( \beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} + 2\beta_{4} - 2\beta_{3} - \beta_1 ) / 8 \) (b11 - b9 - b8 - b6 + 2b4 - 2b3 - b1) / 8
\(\nu^{2}\)	\(=\)	\( ( 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} - 3 \beta_{7} + \cdots + 6 ) / 8 \) (2b15 - 2b14 - 2b13 - b11 + b10 - b9 - 3b7 + 2b6 - 2b4 - 4*b2 + 6) / 8
\(\nu^{3}\)	\(=\)	\( ( - \beta_{15} - 3 \beta_{14} - 4 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} + \cdots + 6 ) / 8 \) (-b15 - 3b14 - 4b12 - b11 - b10 + b9 + b8 + 3b7 - 3b6 - b5 + 10b4 + 6b3 - 6b2 - 3b1 + 6) / 8
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{15} - 3 \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + \cdots + 8 ) / 4 \) (2b15 - 3b14 - b13 + b12 + b11 + 2b10 - 4b9 - b8 - 5b7 + b5 + 22b4 + 4b3 - 12b2 + 3*b1 + 8) / 4
\(\nu^{5}\)	\(=\)	\( ( - 4 \beta_{14} - 4 \beta_{13} - 8 \beta_{12} - \beta_{11} - 7 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} + \cdots - 40 ) / 4 \) (-4b14 - 4b13 - 8b12 - b11 - 7b9 - 7b8 - 4b7 - b6 + 6b5 - 14b4 - 6b3 + 40b2 - b1 - 40) / 4
\(\nu^{6}\)	\(=\)	\( ( 8 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} - 13 \beta_{10} - 7 \beta_{9} + \cdots - 102 ) / 4 \) (8b15 + 2b14 + 4b13 + 2b12 - 9b11 - 13b10 - 7b9 + 8b8 - 7b7 + 8b6 - 4b5 - 2b4 + 36b3 - 48b2 + 8*b1 - 102) / 4
\(\nu^{7}\)	\(=\)	\( ( - 11 \beta_{15} - 13 \beta_{14} - 4 \beta_{13} - 8 \beta_{12} + 3 \beta_{11} + 5 \beta_{10} + \cdots - 182 ) / 4 \) (-11b15 - 13b14 - 4b13 - 8b12 + 3b11 + 5b10 + 57b9 + 13b8 - 19b7 + 17b6 - 11b5 + 90b4 - 42b3 + 22b2 + 21*b1 - 182) / 4
\(\nu^{8}\)	\(=\)	\( ( - 8 \beta_{15} + 3 \beta_{14} + 21 \beta_{13} + 9 \beta_{12} - 26 \beta_{11} + 3 \beta_{10} + \cdots - 146 ) / 2 \) (-8b15 + 3b14 + 21b13 + 9b12 - 26b11 + 3b10 - 35b9 + 9b8 + 24b7 - 12b6 + 9b5 - 16b4 + 24b3 + 72b2 + 17*b1 - 146) / 2
\(\nu^{9}\)	\(=\)	\( ( - 19 \beta_{15} + 71 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} + 42 \beta_{11} - 45 \beta_{10} + \cdots - 138 ) / 2 \) (-19b15 + 71b14 + 12b13 + 16b12 + 42b11 - 45b10 + 38b9 + 12b8 + 23b7 + 38b6 + 35b5 + 20b4 + 108b3 + 270b2 + 30*b1 - 138) / 2
\(\nu^{10}\)	\(=\)	\( ( 50 \beta_{15} + 42 \beta_{14} + 38 \beta_{13} + 80 \beta_{12} - 5 \beta_{11} + 17 \beta_{10} + \cdots - 786 ) / 2 \) (50b15 + 42b14 + 38b13 + 80b12 - 5b11 + 17b10 + 107b9 + 56b8 + 53b7 + 6b6 - 128b5 - 1018b4 - 320b3 - 116b2 + 32*b1 - 786) / 2
\(\nu^{11}\)	\(=\)	\( ( - 215 \beta_{15} + 219 \beta_{14} + 48 \beta_{13} + 36 \beta_{12} - 143 \beta_{11} - 75 \beta_{10} + \cdots + 1210 ) / 2 \) (-215b15 + 219b14 + 48b13 + 36b12 - 143b11 - 75b10 + 175b9 + 59b8 - 47b7 + 183b6 - 199b5 + 1006b4 - 198b3 + 446b2 + 39*b1 + 1210) / 2
\(\nu^{12}\)	\(=\)	\( - 38 \beta_{15} - 13 \beta_{14} + 165 \beta_{13} + 227 \beta_{12} + 175 \beta_{11} + 42 \beta_{10} + \cdots + 8 \) -38b15 - 13b14 + 165b13 + 227b12 + 175b11 + 42b10 + 152b9 + 17b8 + 541b7 - 132b6 + 71b5 - 238b4 + 516b3 + 180b2 - 123*b1 + 8
\(\nu^{13}\)	\(=\)	\( 192 \beta_{15} + 300 \beta_{14} - 452 \beta_{13} + 16 \beta_{12} + 161 \beta_{11} + 304 \beta_{10} + \cdots + 3224 \) 192b15 + 300b14 - 452b13 + 16b12 + 161b11 + 304b10 - 353b9 - 9b8 - 52b7 + 33b6 + 10b5 - 1394b4 - 250b3 - 312b2 + 73*b1 + 3224
\(\nu^{14}\)	\(=\)	\( - 280 \beta_{15} + 390 \beta_{14} + 484 \beta_{13} - 426 \beta_{12} + 25 \beta_{11} - 371 \beta_{10} + \cdots + 7174 \) -280b15 + 390b14 + 484b13 - 426b12 + 25b11 - 371b10 + 495b9 - 888b8 + 63b7 - 416b6 - 692b5 - 4926b4 - 500b3 - 112b2 - 1096*b1 + 7174
\(\nu^{15}\)	\(=\)	\( 211 \beta_{15} - 779 \beta_{14} - 1244 \beta_{13} + 920 \beta_{12} + 565 \beta_{11} + 11 \beta_{10} + \cdots + 4502 \) 211b15 - 779b14 - 1244b13 + 920b12 + 565b11 + 11b10 - 977b9 - 1245b8 - 781b7 + 591b6 - 605b5 + 14390b4 - 2838b3 - 10182b2 - 149*b1 + 4502

\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-\beta_{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 79.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{4} + 9)^{4} \) (T^4 + 9)^4
$5$	\( T^{16} + \cdots + 2117472256 \) T^16 + 32T^13 + 6656T^12 + 8064T^11 + 512T^10 - 518400T^9 + 6221952T^8 - 4026368T^7 + 12517376T^6 - 283883520T^5 + 1663893504T^4 - 4258177024T^3 + 6315081728T^2 - 5171462144*T + 2117472256
$7$	\( (T^{8} - 224 T^{6} + \cdots - 400880)^{2} \) (T^8 - 224T^6 + 448T^5 + 13704T^4 - 53248T^3 - 136576T^2 + 720640T - 400880)^2
$11$	\( T^{16} + \cdots + 25620118503424 \) T^16 + 32T^15 + 512T^14 + 4608T^13 + 151552T^12 + 4242432T^11 + 68780032T^10 + 584835072T^9 + 3292530688T^8 + 21404090368T^7 + 269438418944T^6 + 2206640635904T^5 + 10536909537280T^4 + 22391525736448T^3 + 16962470019072T^2 - 29481536323584*T + 25620118503424
$13$	\( T^{16} + \cdots + 27\!\cdots\!00 \) T^16 + 3200T^13 + 271248T^12 + 1717760T^11 + 5120000T^10 + 8847872T^9 + 5634806368T^8 + 37570686976T^7 + 114873139200T^6 - 1347500722176T^5 + 18426489989376T^4 + 75155778920448T^3 + 255245502513152T^2 - 3777806164172800*T + 27957043852960000
$17$	\( (T^{8} - 1344 T^{6} + \cdots + 816881920)^{2} \) (T^8 - 1344T^6 - 2944T^5 + 508448T^4 + 1044480T^3 - 53986304T^2 + 9390080T + 816881920)^2
$19$	\( T^{16} + \cdots + 10\!\cdots\!96 \) T^16 - 32T^15 + 512T^14 + 14208T^13 + 559552T^12 - 15014400T^11 + 294903808T^10 + 9271351296T^9 + 149480318464T^8 - 1219751600128T^7 + 41927575470080T^6 + 1508224010715136T^5 + 23939487244337152T^4 + 89104253114777600T^3 + 70335181828915200T^2 - 1221045122401566720*T + 10598900522979229696
$23$	\( (T^{8} - 64 T^{7} + \cdots - 35037900800)^{2} \) (T^8 - 64T^7 - 736T^6 + 109056T^5 - 1089152T^4 - 42348544T^3 + 777799680T^2 - 425492480*T - 35037900800)^2
$29$	\( T^{16} + \cdots + 56\!\cdots\!00 \) T^16 - 32T^15 + 512T^14 + 45280T^13 + 3957248T^12 - 64121984T^11 + 1050931712T^10 + 92505709824T^9 + 4665816875136T^8 - 22033099003904T^7 + 401836469485568T^6 + 34131461984319488T^5 + 1336868519252525056T^4 + 1809064899469205504T^3 + 943555165480583168T^2 - 10320873959271219200*T + 56446323002698240000
$31$	\( T^{16} + \cdots + 41\!\cdots\!00 \) T^16 + 8064T^14 + 26696080T^12 + 46823575040T^10 + 46734989650528T^8 + 26353243415873536T^6 + 7691970571770562816T^4 + 885437391795507240960*T^2 + 4109607350375443718400
$37$	\( T^{16} + \cdots + 38\!\cdots\!04 \) T^16 + 96T^15 + 4608T^14 + 14528T^13 + 1620496T^12 + 224961664T^11 + 14234605568T^10 - 28675020544T^9 - 968313092000T^8 + 6367638655488T^7 + 6893443090702336T^6 - 97908324877949952T^5 + 703119961976168704T^4 + 2884347249537718272T^3 + 70869372871016480768T^2 - 734284911784017907712*T + 3804001007988103938304
$41$	\( T^{16} + \cdots + 94\!\cdots\!00 \) T^16 + 13056T^14 + 70874688T^12 + 207527260160T^10 + 356212041131520T^8 + 365640212557922304T^6 + 219522060142929657856T^4 + 70805170442242133852160*T^2 + 9447035993707891197542400
$43$	\( T^{16} + \cdots + 92\!\cdots\!00 \) T^16 + 160T^15 + 12800T^14 + 682624T^13 + 46122944T^12 + 4149187072T^11 + 306484011008T^10 + 15508493912064T^9 + 537795837003264T^8 + 12945845326176256T^7 + 245950489383796736T^6 + 5239225685693136896T^5 + 144279758486420045824T^4 + 3207498038265755598848T^3 + 44479868925767390855168T^2 + 286341547624904646983680*T + 921669553849870911078400
$47$	\( T^{16} + \cdots + 11\!\cdots\!00 \) T^16 + 11200T^14 + 41678592T^12 + 69921902592T^10 + 57150256209920T^8 + 22735051785764864T^6 + 4369756732867477504T^4 + 373417878874107150336*T^2 + 11024466169662170726400
$53$	\( T^{16} + \cdots + 96\!\cdots\!00 \) T^16 + 160T^15 + 12800T^14 + 153504T^13 + 16960000T^12 + 3156153984T^11 + 299678376448T^10 + 1029094838016T^9 + 100041121243264T^8 + 18190042770372608T^7 + 1685959386162102272T^6 + 5591746172921030656T^5 + 91449263140998676480T^4 + 19278049905064818974720T^3 + 2138354891263017100541952T^2 + 2030668155695661908213760*T + 964202241499962888294400
$59$	\( T^{16} + \cdots + 23\!\cdots\!00 \) T^16 - 128T^15 + 8192T^14 - 675840T^13 + 158699520T^12 - 20765343744T^11 + 1586277384192T^10 - 72789256044544T^9 + 2068957458595840T^8 - 32123905497890816T^7 + 215142153171501056T^6 - 714303909442617344T^5 + 97922953659373584384T^4 - 1624651112584394571776T^3 + 6981616736878284767232T^2 + 181321183266412167168000*T + 2354567196983689216000000
$61$	\( T^{16} + \cdots + 12\!\cdots\!00 \) T^16 + 32T^15 + 512T^14 - 157120T^13 + 115399952T^12 + 2795215232T^11 + 42705459200T^10 - 12153910400256T^9 + 3370685859892320T^8 + 49175921370646016T^7 + 735866880468475904T^6 - 203632445835649061888T^5 + 35750546801949141823744T^4 + 224140820215758491060224T^3 + 3313319495492789381660672T^2 - 903620687869229632700968960*T + 123219379938459167726987526400
$67$	\( T^{16} + \cdots + 21\!\cdots\!96 \) T^16 + 320T^15 + 51200T^14 + 4611072T^13 + 255286272T^12 + 9110568960T^11 + 475717435392T^10 + 36863866568704T^9 + 2061397582544896T^8 + 50262831949414400T^7 + 579595725034225664T^6 - 621625262253015040T^5 + 133010965887499370496T^4 + 2866476071676048048128T^3 + 30474643215075243982848T^2 - 115385479389312769327104*T + 218440766638990972420096
$71$	\( (T^{8} + 256 T^{7} + \cdots + 290924400640)^{2} \) (T^8 + 256T^7 + 27776T^6 + 1659392T^5 + 59283584T^4 + 1284915200T^3 + 16299499520T^2 + 109021003776*T + 290924400640)^2
$73$	\( T^{16} + \cdots + 98\!\cdots\!16 \) T^16 + 42496T^14 + 709382400T^12 + 5899646435328T^10 + 25520342188187648T^8 + 53883999480140791808T^6 + 45327992512474743046144T^4 + 13067729363675367868465152*T^2 + 985907622738458033799561216
$79$	\( T^{16} + \cdots + 18\!\cdots\!00 \) T^16 + 36928T^14 + 458363920T^12 + 2270484756224T^10 + 4201068991559776T^8 + 1719443636116761600T^6 + 114950930654421319936T^4 + 2587752408175977123840*T^2 + 18873453677143355654400
$83$	\( T^{16} + \cdots + 14\!\cdots\!76 \) T^16 - 160T^15 + 12800T^14 + 206336T^13 + 3373056T^12 - 3104705536T^11 + 474865041408T^10 + 17298459828224T^9 + 105517061441536T^8 - 5903701162754048T^7 + 6381120662228434944T^6 + 396017560781535576064T^5 + 12138187060968167571456T^4 + 137663961287507678593024T^3 + 635020045117962089136128T^2 - 4236012087763315913129984*T + 14128529127252719096037376
$89$	\( T^{16} + \cdots + 52\!\cdots\!00 \) T^16 + 45728T^14 + 743027648T^12 + 5304960677376T^10 + 18282328190707200T^8 + 30661655762820161536T^6 + 23073336363015526137856T^4 + 6855948794487416265768960*T^2 + 520469506007627359651430400
$97$	\( (T^{8} + \cdots + 409778579046400)^{2} \) (T^8 - 37056T^6 + 116224T^5 + 383621120T^4 - 1982349312T^3 - 1019025981440T^2 + 2337541980160T + 409778579046400)^2
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