LMFDB - Newform orbit 1890.2.i.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1890,2,Mod(991,1890)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1890, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0, 4]))

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

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

chi := DirichletCharacter("1890.991");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1890.i (of order $3$, 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:	$15.0917259820$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
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} - x^{15} + 2 x^{14} - 4 x^{13} + 5 x^{12} + 2 x^{11} - 35 x^{10} + 81 x^{9} - 66 x^{8} + \cdots + 6561 $ x^16 - x^15 + 2x^14 - 4x^13 + 5x^12 + 2x^11 - 35x^10 + 81x^9 - 66x^8 + 243x^7 - 315x^6 + 54x^5 + 405x^4 - 972x^3 + 1458x^2 - 2187x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{5} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 + q^{2} + q^{4} + \beta_{4} q^{5} + (\beta_{13} - \beta_{4} + \beta_{2} - 1) q^{7} + q^{8}+O(q^{10}) $ q + q^2 + q^4 + b4 * q^5 + (b13 - b4 + b2 - 1) * q^7 + q^8	$ q + q^{2} + q^{4} + \beta_{4} q^{5} + (\beta_{13} - \beta_{4} + \beta_{2} - 1) q^{7} + q^{8} + \beta_{4} q^{10} + ( - \beta_{14} + \beta_{8} + \beta_{5} + \cdots - 1) q^{11}+ \cdots + (2 \beta_{15} - \beta_{13} + \cdots + \beta_1) q^{98}+O(q^{100}) $ q + q^2 + q^4 + b4 * q^5 + (b13 - b4 + b2 - 1) * q^7 + q^8 + b4 * q^10 + (-b14 + b8 + b5 - b4 - 1) * q^11 + (b15 + b14 - b13 - b10 - b9 + b7 - b5 + 2b4 - b3 + 2) q^13 + (b13 - b4 + b2 - 1) * q^14 + q^16 + (-b11 + b10 + b8 + b7 + b5 + b4) * q^17 + (b13 - b12 - b9 - b7 + b6 - 2b4 + b3 + b2 - 1) q^19 + b4 * q^20 + (-b14 + b8 + b5 - b4 - 1) * q^22 + (-b8 - b5 + 2b4 - b3) q^23 + (-b4 - 1) * q^25 + (b15 + b14 - b13 - b10 - b9 + b7 - b5 + 2b4 - b3 + 2) q^26 + (b13 - b4 + b2 - 1) * q^28 + (b14 + b13 + b11 - b8 - b5 + 3b4 + b2 - b1) q^29 + (b5 - b3 + 1) * q^31 + q^32 + (-b11 + b10 + b8 + b7 + b5 + b4) * q^34 + (b4 - b2 + 1) * q^35 + (b15 + b14 - b13 - b12 - 2b10 - b9 + b7 - b6 - b5 + b4 - 2b3 + b2) * q^37 + (b13 - b12 - b9 - b7 + b6 - 2b4 + b3 + b2 - 1) q^38 + b4 * q^40 + (b15 - b13 - b12 - 2b10 - 3b9 + b8 + b7 + b6 - b4 + b2) * q^41 + (b13 + b12 - b10 - b8 + 2b7 - b5 + 3b4 - b3 + 2b2 - 1) q^43 + (-b14 + b8 + b5 - b4 - 1) * q^44 + (-b8 - b5 + 2b4 - b3) q^46 + (-b15 - b14 + b13 - b11 + b9 + b8 + b6 + 3b5 - 2b4 - b3 + b1 - 4) * q^47 + (2b15 - b13 + b12 + b11 - b10 - b8 + b7 - b5 + 2b4 + b2 + b1) * q^49 + (-b4 - 1) * q^50 + (b15 + b14 - b13 - b10 - b9 + b7 - b5 + 2b4 - b3 + 2) q^52 + (2b10 + 2b8 + b5 + b4 + 2b3 - b1 + 1) q^53 + (-b5 + b3 + 1) * q^55 + (b13 - b4 + b2 - 1) * q^56 + (b14 + b13 + b11 - b8 - b5 + 3b4 + b2 - b1) q^58 + (-b15 + b13 - b11 - b9 + b8 + b7 + b5 - b2) * q^59 + (b15 - b13 + b11 + b7 + b6 + b4 - b2 + b1 + 4) * q^61 + (b5 - b3 + 1) * q^62 + q^64 + (-b15 - b14 + b13 - b11 + b9 + b5 - b4 - 2) * q^65 + (-b15 - b14 + 3b13 + 2b12 - b11 - b7 - b5 - 2b4 + 2b3 + 3b2) q^67 + (-b11 + b10 + b8 + b7 + b5 + b4) * q^68 + (b4 - b2 + 1) * q^70 + (b14 + 2b13 + 2b12 - b8 + b7 - 2b6 - 2b5 + 3b4 + b2 - 2b1 + 4) * q^71 + (b14 + b13 - 2b12 + b10 - b7 + b5 - b2 - 1) q^73 + (b15 + b14 - b13 - b12 - 2b10 - b9 + b7 - b6 - b5 + b4 - 2b3 + b2) * q^74 + (b13 - b12 - b9 - b7 + b6 - 2b4 + b3 + b2 - 1) q^76 + (2b15 + 2b14 - 2b13 - b12 + b11 - 2b9 - b8 + b7 + b6 + 4b4 - 2b3 + 3) * q^77 + (b14 - b13 - b12 - b6 - b5 + b4 - b2 - b1 + 5) * q^79 + b4 * q^80 + (b15 - b13 - b12 - 2b10 - 3b9 + b8 + b7 + b6 - b4 + b2) * q^82 + (-2b14 - b11 - 2b10 + 2b8 + b7 + b5 - b4 + b3 + b1 - 1) q^83 + (-b15 + b13 - b10 - b9 - b7 - 2b4 - 2) q^85 + (b13 + b12 - b10 - b8 + 2b7 - b5 + 3b4 - b3 + 2b2 - 1) q^86 + (-b14 + b8 + b5 - b4 - 1) * q^88 + (b15 + b14 - b13 + b12 + b8 + b7 - b5 + 3b4 - 2b3 - b2 + 3) * q^89 + (-b13 - b12 + b10 + b9 + b8 - 2b7 - b6 + 2b5 - b4 - b3 - 2b2 + b1) q^91 + (-b8 - b5 + 2b4 - b3) q^92 + (-b15 - b14 + b13 - b11 + b9 + b8 + b6 + 3b5 - 2b4 - b3 + b1 - 4) * q^94 + (b14 + b9 + b7 - b6 + 2b4 - b3 - b2 - b1 + 1) q^95 + (2b13 + b12 + b11 + b10 + 2b7 - b5 - 3b4 + b3 + 3b2 - b1 - 1) * q^97 + (2b15 - b13 + b12 + b11 - b10 - b8 + b7 - b5 + 2b4 + b2 + b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} - 8 q^{5} + 4 q^{7} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 - 8 * q^5 + 4 * q^7 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} - 8 q^{5} + 4 q^{7} + 16 q^{8} - 8 q^{10} - q^{11} + 2 q^{13} + 4 q^{14} + 16 q^{16} - 11 q^{17} - 2 q^{19} - 8 q^{20} - q^{22} - 11 q^{23} - 8 q^{25} + 2 q^{26} + 4 q^{28} - 17 q^{29} + 30 q^{31} + 16 q^{32} - 11 q^{34} + 4 q^{35} - 2 q^{37} - 2 q^{38} - 8 q^{40} - 7 q^{41} - 13 q^{43} - q^{44} - 11 q^{46} - 10 q^{47} - 14 q^{49} - 8 q^{50} + 2 q^{52} - 18 q^{53} + 2 q^{55} + 4 q^{56} - 17 q^{58} + 2 q^{59} + 54 q^{61} + 30 q^{62} + 16 q^{64} - 4 q^{65} + 20 q^{67} - 11 q^{68} + 4 q^{70} + 38 q^{71} - 8 q^{73} - 2 q^{74} - 2 q^{76} + 7 q^{77} + 50 q^{79} - 8 q^{80} - 7 q^{82} - 2 q^{83} - 11 q^{85} - 13 q^{86} - q^{88} + 6 q^{89} + 14 q^{91} - 11 q^{92} - 10 q^{94} + 4 q^{95} + 26 q^{97} - 14 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 - 8 * q^5 + 4 * q^7 + 16 * q^8 - 8 * q^10 - q^11 + 2 * q^13 + 4 * q^14 + 16 * q^16 - 11 * q^17 - 2 * q^19 - 8 * q^20 - q^22 - 11 * q^23 - 8 * q^25 + 2 * q^26 + 4 * q^28 - 17 * q^29 + 30 * q^31 + 16 * q^32 - 11 * q^34 + 4 * q^35 - 2 * q^37 - 2 * q^38 - 8 * q^40 - 7 * q^41 - 13 * q^43 - q^44 - 11 * q^46 - 10 * q^47 - 14 * q^49 - 8 * q^50 + 2 * q^52 - 18 * q^53 + 2 * q^55 + 4 * q^56 - 17 * q^58 + 2 * q^59 + 54 * q^61 + 30 * q^62 + 16 * q^64 - 4 * q^65 + 20 * q^67 - 11 * q^68 + 4 * q^70 + 38 * q^71 - 8 * q^73 - 2 * q^74 - 2 * q^76 + 7 * q^77 + 50 * q^79 - 8 * q^80 - 7 * q^82 - 2 * q^83 - 11 * q^85 - 13 * q^86 - q^88 + 6 * q^89 + 14 * q^91 - 11 * q^92 - 10 * q^94 + 4 * q^95 + 26 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - x^{15} + 2 x^{14} - 4 x^{13} + 5 x^{12} + 2 x^{11} - 35 x^{10} + 81 x^{9} - 66 x^{8} + \cdots + 6561 $ x^16 - x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 2*x^11 - 35*x^10 + 81*x^9 - 66*x^8 + 243*x^7 - 315*x^6 + 54*x^5 + 405*x^4 - 972*x^3 + 1458*x^2 - 2187*x + 6561 :

$\beta_{1}$	$=$	$ ( - 44 \nu^{15} - 409 \nu^{14} + 2669 \nu^{13} - 5491 \nu^{12} + 9953 \nu^{11} - 25312 \nu^{10} + \cdots + 2823417 ) / 393660 $ (-44v^15 - 409v^14 + 2669v^13 - 5491v^12 + 9953v^11 - 25312v^10 + 43771v^9 - 51357v^8 + 25890v^7 + 59193v^6 - 206649v^5 + 624456v^4 - 1425843v^3 + 1659447v^2 - 2396223*v + 2823417) / 393660
$\beta_{2}$	$=$	$ ( 125 \nu^{15} - 518 \nu^{14} - 2660 \nu^{13} + 3988 \nu^{12} - 5975 \nu^{11} + 12361 \nu^{10} + \cdots - 684531 ) / 787320 $ (125v^15 - 518v^14 - 2660v^13 + 3988v^12 - 5975v^11 + 12361v^10 - 31180v^9 + 9957v^8 + 42609v^7 - 114894v^6 + 40275v^5 - 403893v^4 + 742770v^3 - 238626v^2 - 641520*v - 684531) / 787320
$\beta_{3}$	$=$	$ ( - 263 \nu^{15} + 1142 \nu^{14} - 5932 \nu^{13} + 12008 \nu^{12} - 20419 \nu^{11} + 46061 \nu^{10} + \cdots - 2477871 ) / 787320 $ (-263v^15 + 1142v^14 - 5932v^13 + 12008v^12 - 20419v^11 + 46061v^10 - 63728v^9 + 40641v^8 + 70125v^7 - 235674v^6 + 613287v^5 - 1265193v^4 + 1912734v^3 - 1910466v^2 + 1892484*v - 2477871) / 787320
$\beta_{4}$	$=$	$ ( 53 \nu^{15} - 173 \nu^{14} + 37 \nu^{13} - 317 \nu^{12} + 1474 \nu^{11} - 899 \nu^{10} + \cdots - 170586 ) / 131220 $ (53v^15 - 173v^14 + 37v^13 - 317v^12 + 1474v^11 - 899v^10 - 1447v^9 - 12v^8 - 717v^7 + 5841v^6 - 36702v^5 + 36207v^4 + 13041v^3 + 88209v^2 - 175689*v - 170586) / 131220
$\beta_{5}$	$=$	$ ( - 461 \nu^{15} + 1052 \nu^{14} - 6094 \nu^{13} + 12278 \nu^{12} - 16963 \nu^{11} + 46871 \nu^{10} + \cdots - 2215431 ) / 787320 $ (-461v^15 + 1052v^14 - 6094v^13 + 12278v^12 - 16963v^11 + 46871v^10 - 61046v^9 + 37005v^8 + 59811v^7 - 281304v^6 + 466839v^5 - 1201203v^4 + 2339928v^3 - 1422036v^2 + 1721898*v - 2215431) / 787320
$\beta_{6}$	$=$	$ ( - 559 \nu^{15} - 86 \nu^{14} - 6476 \nu^{13} + 2116 \nu^{12} + 5653 \nu^{11} + 32197 \nu^{10} + \cdots - 9843687 ) / 787320 $ (-559v^15 - 86v^14 - 6476v^13 + 2116v^12 + 5653v^11 + 32197v^10 - 50404v^9 - 33279v^8 - 27699v^7 + 39762v^6 - 238689v^5 - 513081v^4 + 1973322v^3 - 22842v^2 + 1930392*v - 9843687) / 787320
$\beta_{7}$	$=$	$ ( - 667 \nu^{15} + 1570 \nu^{14} - 1328 \nu^{13} + 2620 \nu^{12} - 9611 \nu^{11} + 17905 \nu^{10} + \cdots - 2810295 ) / 787320 $ (-667v^15 + 1570v^14 - 1328v^13 + 2620v^12 - 9611v^11 + 17905v^10 - 6052v^9 - 12399v^8 - 43791v^7 + 3690v^6 + 259623v^5 - 662445v^4 + 381186v^3 - 789750v^2 + 2350296*v - 2810295) / 787320
$\beta_{8}$	$=$	$ ( - 397 \nu^{15} + 697 \nu^{14} + 2137 \nu^{13} - 287 \nu^{12} - 476 \nu^{11} - 27149 \nu^{10} + \cdots + 3875364 ) / 393660 $ (-397v^15 + 697v^14 + 2137v^13 - 287v^12 - 476v^11 - 27149v^10 + 40343v^9 - 41082v^8 + 46173v^7 - 53469v^6 + 62388v^5 + 615897v^4 - 1306449v^3 + 574209v^2 - 1815939*v + 3875364) / 393660
$\beta_{9}$	$=$	$ ( - 29 \nu^{15} + 5 \nu^{14} - 7 \nu^{13} - 67 \nu^{12} + 32 \nu^{11} - 421 \nu^{10} + 643 \nu^{9} + \cdots - 26244 ) / 26244 $ (-29v^15 + 5v^14 - 7v^13 - 67v^12 + 32v^11 - 421v^10 + 643v^9 - 942v^8 - 867v^7 + 1071v^6 - 1476v^5 + 9477v^4 - 16929v^3 + 12393v^2 - 8019*v - 26244) / 26244
$\beta_{10}$	$=$	$ ( 107 \nu^{15} - 312 \nu^{14} + 578 \nu^{13} - 1078 \nu^{12} + 1781 \nu^{11} - 2041 \nu^{10} + \cdots + 64881 ) / 87480 $ (107v^15 - 312v^14 + 578v^13 - 1078v^12 + 1781v^11 - 2041v^10 + 582v^9 + 5117v^8 - 12573v^7 + 21684v^6 - 58833v^5 + 59013v^4 - 87156v^3 + 38556v^2 - 53946*v + 64881) / 87480
$\beta_{11}$	$=$	$ ( - 905 \nu^{15} + 5402 \nu^{14} - 5380 \nu^{13} + 7448 \nu^{12} - 31405 \nu^{11} + 37811 \nu^{10} + \cdots - 443961 ) / 787320 $ (-905v^15 + 5402v^14 - 5380v^13 + 7448v^12 - 31405v^11 + 37811v^10 + 5560v^9 - 54993v^8 + 55419v^7 - 175734v^6 + 782145v^5 - 1303263v^4 + 331290v^3 - 962766v^2 + 3484620*v - 443961) / 787320
$\beta_{12}$	$=$	$ ( - 937 \nu^{15} - 482 \nu^{14} + 292 \nu^{13} + 8452 \nu^{12} - 3941 \nu^{11} + 4459 \nu^{10} + \cdots - 1983609 ) / 787320 $ (-937v^15 - 482v^14 + 292v^13 + 8452v^12 - 3941v^11 + 4459v^10 - 24412v^9 + 11199v^8 - 22245v^7 - 138546v^6 + 218313v^5 + 179793v^4 + 949806v^3 - 1209654v^2 - 798984*v - 1983609) / 787320
$\beta_{13}$	$=$	$ ( - 503 \nu^{15} + 905 \nu^{14} + 1193 \nu^{13} - 55 \nu^{12} + 4346 \nu^{11} - 24115 \nu^{10} + \cdots + 3171150 ) / 393660 $ (-503v^15 + 905v^14 + 1193v^13 - 55v^12 + 4346v^11 - 24115v^10 + 41647v^9 - 55416v^8 + 63771v^7 - 44865v^6 - 19458v^5 + 556335v^4 - 1062801v^3 + 786105v^2 - 2186271*v + 3171150) / 393660
$\beta_{14}$	$=$	$ ( - 773 \nu^{15} - 427 \nu^{14} - 2227 \nu^{13} + 6677 \nu^{12} - 1504 \nu^{11} + 7619 \nu^{10} + \cdots - 2169504 ) / 393660 $ (-773v^15 - 427v^14 - 2227v^13 + 6677v^12 - 1504v^11 + 7619v^10 - 16313v^9 - 12198v^8 + 12957v^7 - 147141v^6 + 102852v^5 - 52947v^4 + 701379v^3 - 231579v^2 - 611631*v - 2169504) / 393660
$\beta_{15}$	$=$	$ ( 94 \nu^{15} - 21 \nu^{14} + 271 \nu^{13} - 899 \nu^{12} + 1057 \nu^{11} - 1718 \nu^{10} + \cdots + 376893 ) / 43740 $ (94v^15 - 21v^14 + 271v^13 - 899v^12 + 1057v^11 - 1718v^10 + 3279v^9 + 727v^8 + 3780v^7 + 17457v^6 - 32481v^5 + 43074v^4 - 123687v^3 + 141993v^2 - 75087*v + 376893) / 43740

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

$\nu$	$=$	$ ( -\beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} + 2\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 1 ) / 3 $ (-b13 - b12 - b11 - b9 + 2*b8 + b7 + b6 + b5 - b4 - b2 + 1) / 3
$\nu^{2}$	$=$	$ ( \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{10} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1 ) / 3 $ (b15 + b14 - 2b13 - 2b10 + b8 + 2b7 - b6 + b5 + 4b4 - 2*b3 + 1) / 3
$\nu^{3}$	$=$	$ ( - 2 \beta_{15} - 3 \beta_{14} - 2 \beta_{11} - 3 \beta_{10} - \beta_{8} - \beta_{7} + 3 \beta_{5} + \cdots - 4 ) / 3 $ (-2b15 - 3b14 - 2b11 - 3b10 - b8 - b7 + 3b5 - 3b4 - 3*b3 + b1 - 4) / 3
$\nu^{4}$	$=$	$ ( \beta_{15} + \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + \beta_{10} + \beta_{8} - 7 \beta_{7} + \cdots + 4 ) / 3 $ (b15 + b14 - 2b13 + 3b12 + 3b11 + b10 + b8 - 7b7 + 2b6 - 2b5 - 2b4 + b3 - 3b2 + 4) / 3
$\nu^{5}$	$=$	$ ( - 5 \beta_{15} - 3 \beta_{14} - 3 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 6 \beta_{10} + 3 \beta_{9} + \cdots - 1 ) / 3 $ (-5b15 - 3b14 - 3b13 - 3b12 - 2b11 - 6b10 + 3b9 - b8 - 4b7 - 3b6 + 9b4 + 6b3 - 12b2 + 4*b1 - 1) / 3
$\nu^{6}$	$=$	$ ( 7 \beta_{15} + 4 \beta_{14} + \beta_{13} + 9 \beta_{12} - 3 \beta_{11} - 20 \beta_{10} - 14 \beta_{8} + \cdots + 40 ) / 3 $ (7b15 + 4b14 + b13 + 9b12 - 3b11 - 20b10 - 14b8 + 11b7 - 7b6 - 23b5 - 2b4 - 2b3 + 15b2 - 3*b1 + 40) / 3
$\nu^{7}$	$=$	$ ( 7 \beta_{15} - 18 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} - 26 \beta_{11} + 3 \beta_{10} - 12 \beta_{9} + \cdots - 61 ) / 3 $ (7b15 - 18b14 - 3b13 + 3b12 - 26b11 + 3b10 - 12b9 + 26b8 + 14b7 + 27b6 + 18b5 - 57b4 + 18b3 + 3b2 + 10*b1 - 61) / 3
$\nu^{8}$	$=$	$ ( 31 \beta_{15} + 28 \beta_{14} + 10 \beta_{13} + 9 \beta_{12} + 18 \beta_{11} - 2 \beta_{10} + 48 \beta_{9} + \cdots - 92 ) / 3 $ (31b15 + 28b14 + 10b13 + 9b12 + 18b11 - 2b10 + 48b9 - 29b8 - b7 - 40b6 + b5 + 58b4 - 11b3 - 27b1 - 92) / 3
$\nu^{9}$	$=$	$ ( - 41 \beta_{15} + 15 \beta_{14} + 60 \beta_{13} - 39 \beta_{12} - 8 \beta_{11} - 39 \beta_{10} + \cdots - 295 ) / 3 $ (-41b15 + 15b14 + 60b13 - 39b12 - 8b11 - 39b10 - 3b9 - 127b8 - 94b7 - 42b6 - 21b5 - 213b4 - 57b3 + 6b2 + 7*b1 - 295) / 3
$\nu^{10}$	$=$	$ ( 64 \beta_{15} + 73 \beta_{14} + 25 \beta_{13} + 129 \beta_{12} + 138 \beta_{11} + 145 \beta_{10} + \cdots + 112 ) / 3 $ (64b15 + 73b14 + 25b13 + 129b12 + 138b11 + 145b10 - 27b9 - 89b8 - 178b7 + 47b6 - 92b5 - 191b4 + 91b3 - 30b2 + 54*b1 + 112) / 3
$\nu^{11}$	$=$	$ ( - 32 \beta_{15} + 42 \beta_{14} + 105 \beta_{13} - 129 \beta_{12} - 20 \beta_{11} + 102 \beta_{10} + \cdots + 251 ) / 3 $ (-32b15 + 42b14 + 105b13 - 129b12 - 20b11 + 102b10 - 6b9 - 10b8 + 50b7 - 21b6 + 45b5 + 342b4 + 195b3 - 336b2 - 14*b1 + 251) / 3
$\nu^{12}$	$=$	$ ( 295 \beta_{15} + 616 \beta_{14} - 17 \beta_{13} - 27 \beta_{12} + 24 \beta_{11} - 389 \beta_{10} + \cdots + 949 ) / 3 $ (295b15 + 616b14 - 17b13 - 27b12 + 24b11 - 389b10 - 450b9 - 140b8 + 587b7 - 295b6 - 572b5 + 322b4 - 344b3 + 366b2 - 327*b1 + 949) / 3
$\nu^{13}$	$=$	$ ( 151 \beta_{15} - 486 \beta_{14} - 318 \beta_{13} + 507 \beta_{12} - 404 \beta_{11} + 192 \beta_{10} + \cdots - 1204 ) / 3 $ (151b15 - 486b14 - 318b13 + 507b12 - 404b11 + 192b10 - 687b9 + 773b8 + 941b7 + 459b6 + 837b5 - 471b4 - 198b3 + 120b2 + 379*b1 - 1204) / 3
$\nu^{14}$	$=$	$ ( 679 \beta_{15} + 100 \beta_{14} + 487 \beta_{13} + 846 \beta_{12} + 1170 \beta_{11} + 385 \beta_{10} + \cdots + 205 ) / 3 $ (679b15 + 100b14 + 487b13 + 846b12 + 1170b11 + 385b10 + 552b9 - 182b8 + 314b7 - 499b6 + 226b5 + 3631b4 - 488b3 - 72b2 - 828*b1 + 205) / 3
$\nu^{15}$	$=$	$ ( - 1256 \beta_{15} + 465 \beta_{14} + 6 \beta_{13} - 1434 \beta_{12} + 649 \beta_{11} - 1173 \beta_{10} + \cdots - 2338 ) / 3 $ (-1256b15 + 465b14 + 6b13 - 1434b12 + 649b11 - 1173b10 - 2172b9 - 946b8 - 1876b7 - 816b6 - 1596b5 + 795b4 - 948b3 - 318b2 - 443*b1 - 2338) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times$.

$n$	$757$	$1081$	$1541$
$\chi(n)$	$1$	$-1 - \beta_{4}$	$-1 - \beta_{4}$

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} $

991.1

0.702194	+	1.58333i
−1.03843	−	1.38624i
1.67659	−	0.434811i
1.52536	+	0.820531i
−0.803168	−	1.53458i
0.748691	−	1.56188i
−0.579249	+	1.63232i
−1.73198	+	0.0153002i
0.702194	−	1.58333i
−1.03843	+	1.38624i
1.67659	+	0.434811i
1.52536	−	0.820531i
−0.803168	+	1.53458i
0.748691	+	1.56188i
−0.579249	−	1.63232i
−1.73198	−	0.0153002i

1.00000

−0.500000

0.866025i

−2.52336

−

0.795395i

1.00000

−0.500000

0.866025i

991.2

1.00000

−0.500000

0.866025i

−1.52280

−

2.16358i

1.00000

−0.500000

0.866025i

991.3

1.00000

−0.500000

0.866025i

−1.09594

2.40809i

1.00000

−0.500000

0.866025i

991.4

1.00000

−0.500000

0.866025i

0.226513

−

2.63604i

1.00000

−0.500000

0.866025i

991.5

1.00000

−0.500000

0.866025i

0.832221

2.51146i

1.00000

−0.500000

0.866025i

991.6

1.00000

−0.500000

0.866025i

1.01860

−

2.44181i

1.00000

−0.500000

0.866025i

991.7

1.00000

−0.500000

0.866025i

2.48140

−

0.917950i

1.00000

−0.500000

0.866025i

991.8

1.00000

−0.500000

0.866025i

2.58337

0.571125i

1.00000

−0.500000

0.866025i

1171.1

1.00000

−0.500000

−

0.866025i

−2.52336

0.795395i

1.00000

−0.500000

−

0.866025i

1171.2

1.00000

−0.500000

−

0.866025i

−1.52280

2.16358i

1.00000

−0.500000

−

0.866025i

1171.3

1.00000

−0.500000

−

0.866025i

−1.09594

−

2.40809i

1.00000

−0.500000

−

0.866025i

1171.4

1.00000

−0.500000

−

0.866025i

0.226513

2.63604i

1.00000

−0.500000

−

0.866025i

1171.5

1.00000

−0.500000

−

0.866025i

0.832221

−

2.51146i

1.00000

−0.500000

−

0.866025i

1171.6

1.00000

−0.500000

−

0.866025i

1.01860

2.44181i

1.00000

−0.500000

−

0.866025i

1171.7

1.00000

−0.500000

−

0.866025i

2.48140

0.917950i

1.00000

−0.500000

−

0.866025i

1171.8

1.00000

−0.500000

−

0.866025i

2.58337

−

0.571125i

1.00000

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1890.2.i.i		16
3.b	odd	2	1		630.2.i.i	✓	16
7.c	even	3	1		1890.2.l.i		16
9.c	even	3	1		1890.2.l.i		16
9.d	odd	6	1		630.2.l.i	yes	16
21.h	odd	6	1		630.2.l.i	yes	16
63.h	even	3	1	inner	1890.2.i.i		16
63.j	odd	6	1		630.2.i.i	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
630.2.i.i	✓	16	3.b	odd	2	1
630.2.i.i	✓	16	63.j	odd	6	1
630.2.l.i	yes	16	9.d	odd	6	1
630.2.l.i	yes	16	21.h	odd	6	1
1890.2.i.i		16	1.a	even	1	1	trivial
1890.2.i.i		16	63.h	even	3	1	inner
1890.2.l.i		16	7.c	even	3	1
1890.2.l.i		16	9.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1890, [\chi])$:

$ T_{11}^{16} + T_{11}^{15} + 53 T_{11}^{14} - 116 T_{11}^{13} + 2018 T_{11}^{12} - 4688 T_{11}^{11} + \cdots + 2916 $

$ T_{13}^{16} - 2 T_{13}^{15} + 54 T_{13}^{14} - 8 T_{13}^{13} + 1834 T_{13}^{12} + 477 T_{13}^{11} + \cdots + 2298256 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T - 1)^{16} $ (T - 1)^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{2} + T + 1)^{8} $ (T^2 + T + 1)^8
$7$	$ T^{16} - 4 T^{15} + \cdots + 5764801 $ T^16 - 4T^15 + 15T^14 - 46T^13 + 85T^12 - 234T^11 + 295T^10 + 350T^9 - 39T^8 + 2450T^7 + 14455T^6 - 80262T^5 + 204085T^4 - 773122T^3 + 1764735T^2 - 3294172*T + 5764801
$11$	$ T^{16} + T^{15} + \cdots + 2916 $ T^16 + T^15 + 53T^14 - 116T^13 + 2018T^12 - 4688T^11 + 39862T^10 - 156501T^9 + 562494T^8 - 1194714T^7 + 1937466T^6 - 1987308T^5 + 1493181T^4 - 627426T^3 + 188406T^2 - 28188T + 2916
$13$	$ T^{16} - 2 T^{15} + \cdots + 2298256 $ T^16 - 2T^15 + 54T^14 - 8T^13 + 1834T^12 + 477T^11 + 34540T^10 + 52582T^9 + 433476T^8 + 684178T^7 + 3205075T^6 + 6671493T^5 + 14639257T^4 + 16719280T^3 + 15234300T^2 + 6900832*T + 2298256
$17$	$ T^{16} + \cdots + 8707129344 $ T^16 + 11T^15 + 172T^14 + 999T^13 + 10767T^12 + 48267T^11 + 453195T^10 + 1438209T^9 + 11605383T^8 + 26237682T^7 + 215458947T^6 + 330564807T^5 + 2370323817T^4 + 2178566928T^3 + 17265612672T^2 + 11972302848*T + 8707129344
$19$	$ T^{16} + \cdots + 1174158756 $ T^16 + 2T^15 + 98T^14 + 280T^13 + 6918T^12 + 19245T^11 + 235788T^10 + 637458T^9 + 5715978T^8 + 14051506T^7 + 75895571T^6 + 153664667T^5 + 652539913T^4 + 1231454664T^3 + 2158008918T^2 + 1769359176*T + 1174158756
$23$	$ T^{16} + \cdots + 927567936 $ T^16 + 11T^15 + 176T^14 + 1007T^13 + 11912T^12 + 56534T^11 + 516169T^10 + 1244817T^9 + 8326524T^8 + 101754T^7 + 97089705T^6 - 44990154T^5 + 547995969T^4 - 642142404T^3 + 2201299416T^2 - 1413280224*T + 927567936
$29$	$ T^{16} + \cdots + 17639292969 $ T^16 + 17T^15 + 278T^14 + 2735T^13 + 28451T^12 + 225260T^11 + 1821232T^10 + 11447856T^9 + 70777128T^8 + 351438732T^7 + 1743315804T^6 + 6789615372T^5 + 24575037471T^4 + 58028658471T^3 + 110947789962T^2 + 48483252837*T + 17639292969
$31$	$ (T^{8} - 15 T^{7} + \cdots + 27294)^{2} $ (T^8 - 15T^7 + 46T^6 + 292T^5 - 1946T^4 + 1420T^3 + 13629T^2 - 36302*T + 27294)^2
$37$	$ T^{16} + \cdots + 147622500 $ T^16 + 2T^15 + 187T^14 - 50T^13 + 23858T^12 - 11501T^11 + 1541301T^10 - 1916286T^9 + 71688606T^8 - 89262833T^7 + 1552868072T^6 - 3683050217T^5 + 23529518011T^4 - 29017050460T^3 + 34261002250T^2 - 2308986000*T + 147622500
$41$	$ T^{16} + \cdots + 86812992387801 $ T^16 + 7T^15 + 270T^14 + 901T^13 + 39076T^12 + 84663T^11 + 3702942T^10 + 2777112T^9 + 245168532T^8 + 9845874T^7 + 11620565451T^6 - 9476498448T^5 + 381671616141T^4 - 402751302138T^3 + 8136771273864T^2 - 12647350897902*T + 86812992387801
$43$	$ T^{16} + \cdots + 285914281 $ T^16 + 13T^15 + 234T^14 + 1787T^13 + 23078T^12 + 158697T^11 + 1454068T^10 + 7267054T^9 + 48039750T^8 + 205695460T^7 + 1045232041T^6 + 2987533494T^5 + 7532421827T^4 + 6805283222T^3 + 4526436618T^2 + 1330433938*T + 285914281
$47$	$ (T^{8} + 5 T^{7} + \cdots - 423063)^{2} $ (T^8 + 5T^7 - 209T^6 - 966T^5 + 9951T^4 + 50121T^3 - 37818T^2 - 406404*T - 423063)^2
$53$	$ T^{16} + \cdots + 1238515248996 $ T^16 + 18T^15 + 399T^14 + 2630T^13 + 38448T^12 + 106899T^11 + 2838763T^10 + 1726038T^9 + 108882912T^8 - 66347235T^7 + 3097006380T^6 - 3808368603T^5 + 45774547983T^4 - 92330519760T^3 + 506947936338T^2 - 716382524376*T + 1238515248996
$59$	$ (T^{8} - T^{7} - 155 T^{6} + \cdots + 1458)^{2} $ (T^8 - T^7 - 155T^6 - 246T^5 + 5748T^4 + 22311T^3 + 25713T^2 + 11124T + 1458)^2
$61$	$ (T^{8} - 27 T^{7} + \cdots - 502368)^{2} $ (T^8 - 27T^7 + 156T^6 + 2006T^5 - 33888T^4 + 203910T^3 - 611453T^2 + 898440*T - 502368)^2
$67$	$ (T^{8} - 10 T^{7} + \cdots + 40406508)^{2} $ (T^8 - 10T^7 - 353T^6 + 2779T^5 + 44516T^4 - 227405T^3 - 2320437T^2 + 5028960*T + 40406508)^2
$71$	$ (T^{8} - 19 T^{7} + \cdots - 7422678)^{2} $ (T^8 - 19T^7 - 168T^6 + 5670T^5 - 28107T^4 - 131949T^3 + 1116099T^2 - 14580*T - 7422678)^2
$73$	$ T^{16} + 8 T^{15} + \cdots + 2143296 $ T^16 + 8T^15 + 194T^14 + 2596T^13 + 38910T^12 + 345057T^11 + 2413146T^10 + 10754460T^9 + 35783472T^8 + 66554746T^7 + 92478587T^6 + 76189157T^5 + 52769077T^4 + 21750732T^3 + 12384024T^2 + 3882528*T + 2143296
$79$	$ (T^{8} - 25 T^{7} + \cdots - 406534)^{2} $ (T^8 - 25T^7 + 175T^6 + 726T^5 - 18231T^4 + 117846T^3 - 377959T^2 + 615718*T - 406534)^2
$83$	$ T^{16} + \cdots + 2857962683601 $ T^16 + 2T^15 + 285T^14 + 950T^13 + 59479T^12 + 181731T^11 + 5425806T^10 + 12293469T^9 + 344131362T^8 + 646842591T^7 + 11108005821T^6 + 8778869019T^5 + 229179273264T^4 + 211647489975T^3 + 1490295654990T^2 - 1411385241717*T + 2857962683601
$89$	$ T^{16} + \cdots + 99588211776 $ T^16 - 6T^15 + 351T^14 - 110T^13 + 80562T^12 - 66627T^11 + 7690357T^10 - 2106420T^9 + 520688268T^8 - 643546485T^7 + 11015185194T^6 - 21704323815T^5 + 188866782837T^4 - 298622599884T^3 + 812199516120T^2 + 261580946400*T + 99588211776
$97$	$ T^{16} + \cdots + 1545178274704 $ T^16 - 26T^15 + 720T^14 - 12640T^13 + 253103T^12 - 3815676T^11 + 50715262T^10 - 482307254T^9 + 3675013929T^8 - 19206109952T^7 + 76868712919T^6 - 161539486848T^5 + 267155797613T^4 + 299843403512T^3 + 2147276158380T^2 + 1714109041504*T + 1545178274704
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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:	\( N \)	\(=\)	\( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1890.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + q^{2} + q^{4} + \beta_{4} q^{5} + (\beta_{13} - \beta_{4} + \beta_{2} - 1) q^{7} + q^{8}+O(q^{10}) \) q + q^2 + q^4 + b4 * q^5 + (b13 - b4 + b2 - 1) * q^7 + q^8	\( q + q^{2} + q^{4} + \beta_{4} q^{5} + (\beta_{13} - \beta_{4} + \beta_{2} - 1) q^{7} + q^{8} + \beta_{4} q^{10} + ( - \beta_{14} + \beta_{8} + \beta_{5} + \cdots - 1) q^{11}+ \cdots + (2 \beta_{15} - \beta_{13} + \cdots + \beta_1) q^{98}+O(q^{100}) \) q + q^2 + q^4 + b4 * q^5 + (b13 - b4 + b2 - 1) * q^7 + q^8 + b4 * q^10 + (-b14 + b8 + b5 - b4 - 1) * q^11 + (b15 + b14 - b13 - b10 - b9 + b7 - b5 + 2b4 - b3 + 2) q^13 + (b13 - b4 + b2 - 1) * q^14 + q^16 + (-b11 + b10 + b8 + b7 + b5 + b4) * q^17 + (b13 - b12 - b9 - b7 + b6 - 2b4 + b3 + b2 - 1) q^19 + b4 * q^20 + (-b14 + b8 + b5 - b4 - 1) * q^22 + (-b8 - b5 + 2b4 - b3) q^23 + (-b4 - 1) * q^25 + (b15 + b14 - b13 - b10 - b9 + b7 - b5 + 2b4 - b3 + 2) q^26 + (b13 - b4 + b2 - 1) * q^28 + (b14 + b13 + b11 - b8 - b5 + 3b4 + b2 - b1) q^29 + (b5 - b3 + 1) * q^31 + q^32 + (-b11 + b10 + b8 + b7 + b5 + b4) * q^34 + (b4 - b2 + 1) * q^35 + (b15 + b14 - b13 - b12 - 2b10 - b9 + b7 - b6 - b5 + b4 - 2b3 + b2) * q^37 + (b13 - b12 - b9 - b7 + b6 - 2b4 + b3 + b2 - 1) q^38 + b4 * q^40 + (b15 - b13 - b12 - 2b10 - 3b9 + b8 + b7 + b6 - b4 + b2) * q^41 + (b13 + b12 - b10 - b8 + 2b7 - b5 + 3b4 - b3 + 2b2 - 1) q^43 + (-b14 + b8 + b5 - b4 - 1) * q^44 + (-b8 - b5 + 2b4 - b3) q^46 + (-b15 - b14 + b13 - b11 + b9 + b8 + b6 + 3b5 - 2b4 - b3 + b1 - 4) * q^47 + (2b15 - b13 + b12 + b11 - b10 - b8 + b7 - b5 + 2b4 + b2 + b1) * q^49 + (-b4 - 1) * q^50 + (b15 + b14 - b13 - b10 - b9 + b7 - b5 + 2b4 - b3 + 2) q^52 + (2b10 + 2b8 + b5 + b4 + 2b3 - b1 + 1) q^53 + (-b5 + b3 + 1) * q^55 + (b13 - b4 + b2 - 1) * q^56 + (b14 + b13 + b11 - b8 - b5 + 3b4 + b2 - b1) q^58 + (-b15 + b13 - b11 - b9 + b8 + b7 + b5 - b2) * q^59 + (b15 - b13 + b11 + b7 + b6 + b4 - b2 + b1 + 4) * q^61 + (b5 - b3 + 1) * q^62 + q^64 + (-b15 - b14 + b13 - b11 + b9 + b5 - b4 - 2) * q^65 + (-b15 - b14 + 3b13 + 2b12 - b11 - b7 - b5 - 2b4 + 2b3 + 3b2) q^67 + (-b11 + b10 + b8 + b7 + b5 + b4) * q^68 + (b4 - b2 + 1) * q^70 + (b14 + 2b13 + 2b12 - b8 + b7 - 2b6 - 2b5 + 3b4 + b2 - 2b1 + 4) * q^71 + (b14 + b13 - 2b12 + b10 - b7 + b5 - b2 - 1) q^73 + (b15 + b14 - b13 - b12 - 2b10 - b9 + b7 - b6 - b5 + b4 - 2b3 + b2) * q^74 + (b13 - b12 - b9 - b7 + b6 - 2b4 + b3 + b2 - 1) q^76 + (2b15 + 2b14 - 2b13 - b12 + b11 - 2b9 - b8 + b7 + b6 + 4b4 - 2b3 + 3) * q^77 + (b14 - b13 - b12 - b6 - b5 + b4 - b2 - b1 + 5) * q^79 + b4 * q^80 + (b15 - b13 - b12 - 2b10 - 3b9 + b8 + b7 + b6 - b4 + b2) * q^82 + (-2b14 - b11 - 2b10 + 2b8 + b7 + b5 - b4 + b3 + b1 - 1) q^83 + (-b15 + b13 - b10 - b9 - b7 - 2b4 - 2) q^85 + (b13 + b12 - b10 - b8 + 2b7 - b5 + 3b4 - b3 + 2b2 - 1) q^86 + (-b14 + b8 + b5 - b4 - 1) * q^88 + (b15 + b14 - b13 + b12 + b8 + b7 - b5 + 3b4 - 2b3 - b2 + 3) * q^89 + (-b13 - b12 + b10 + b9 + b8 - 2b7 - b6 + 2b5 - b4 - b3 - 2b2 + b1) q^91 + (-b8 - b5 + 2b4 - b3) q^92 + (-b15 - b14 + b13 - b11 + b9 + b8 + b6 + 3b5 - 2b4 - b3 + b1 - 4) * q^94 + (b14 + b9 + b7 - b6 + 2b4 - b3 - b2 - b1 + 1) q^95 + (2b13 + b12 + b11 + b10 + 2b7 - b5 - 3b4 + b3 + 3b2 - b1 - 1) * q^97 + (2b15 - b13 + b12 + b11 - b10 - b8 + b7 - b5 + 2b4 + b2 + b1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} - 8 q^{5} + 4 q^{7} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 - 8 * q^5 + 4 * q^7 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} - 8 q^{5} + 4 q^{7} + 16 q^{8} - 8 q^{10} - q^{11} + 2 q^{13} + 4 q^{14} + 16 q^{16} - 11 q^{17} - 2 q^{19} - 8 q^{20} - q^{22} - 11 q^{23} - 8 q^{25} + 2 q^{26} + 4 q^{28} - 17 q^{29} + 30 q^{31} + 16 q^{32} - 11 q^{34} + 4 q^{35} - 2 q^{37} - 2 q^{38} - 8 q^{40} - 7 q^{41} - 13 q^{43} - q^{44} - 11 q^{46} - 10 q^{47} - 14 q^{49} - 8 q^{50} + 2 q^{52} - 18 q^{53} + 2 q^{55} + 4 q^{56} - 17 q^{58} + 2 q^{59} + 54 q^{61} + 30 q^{62} + 16 q^{64} - 4 q^{65} + 20 q^{67} - 11 q^{68} + 4 q^{70} + 38 q^{71} - 8 q^{73} - 2 q^{74} - 2 q^{76} + 7 q^{77} + 50 q^{79} - 8 q^{80} - 7 q^{82} - 2 q^{83} - 11 q^{85} - 13 q^{86} - q^{88} + 6 q^{89} + 14 q^{91} - 11 q^{92} - 10 q^{94} + 4 q^{95} + 26 q^{97} - 14 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 - 8 * q^5 + 4 * q^7 + 16 * q^8 - 8 * q^10 - q^11 + 2 * q^13 + 4 * q^14 + 16 * q^16 - 11 * q^17 - 2 * q^19 - 8 * q^20 - q^22 - 11 * q^23 - 8 * q^25 + 2 * q^26 + 4 * q^28 - 17 * q^29 + 30 * q^31 + 16 * q^32 - 11 * q^34 + 4 * q^35 - 2 * q^37 - 2 * q^38 - 8 * q^40 - 7 * q^41 - 13 * q^43 - q^44 - 11 * q^46 - 10 * q^47 - 14 * q^49 - 8 * q^50 + 2 * q^52 - 18 * q^53 + 2 * q^55 + 4 * q^56 - 17 * q^58 + 2 * q^59 + 54 * q^61 + 30 * q^62 + 16 * q^64 - 4 * q^65 + 20 * q^67 - 11 * q^68 + 4 * q^70 + 38 * q^71 - 8 * q^73 - 2 * q^74 - 2 * q^76 + 7 * q^77 + 50 * q^79 - 8 * q^80 - 7 * q^82 - 2 * q^83 - 11 * q^85 - 13 * q^86 - q^88 + 6 * q^89 + 14 * q^91 - 11 * q^92 - 10 * q^94 + 4 * q^95 + 26 * q^97 - 14 * q^98

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	1890.2.i.i

Level	$1890$
Weight	$2$
Character orbit	1890.i
Analytic conductor	$15.092$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(15.0917259820\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{15} + 2 x^{14} - 4 x^{13} + 5 x^{12} + 2 x^{11} - 35 x^{10} + 81 x^{9} - 66 x^{8} + \cdots + 6561 \) x^16 - x^15 + 2x^14 - 4x^13 + 5x^12 + 2x^11 - 35x^10 + 81x^9 - 66x^8 + 243x^7 - 315x^6 + 54x^5 + 405x^4 - 972x^3 + 1458x^2 - 2187x + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 3^{5} \)
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - 44 \nu^{15} - 409 \nu^{14} + 2669 \nu^{13} - 5491 \nu^{12} + 9953 \nu^{11} - 25312 \nu^{10} + \cdots + 2823417 ) / 393660 \) (-44v^15 - 409v^14 + 2669v^13 - 5491v^12 + 9953v^11 - 25312v^10 + 43771v^9 - 51357v^8 + 25890v^7 + 59193v^6 - 206649v^5 + 624456v^4 - 1425843v^3 + 1659447v^2 - 2396223*v + 2823417) / 393660
\(\beta_{2}\)	\(=\)	\( ( 125 \nu^{15} - 518 \nu^{14} - 2660 \nu^{13} + 3988 \nu^{12} - 5975 \nu^{11} + 12361 \nu^{10} + \cdots - 684531 ) / 787320 \) (125v^15 - 518v^14 - 2660v^13 + 3988v^12 - 5975v^11 + 12361v^10 - 31180v^9 + 9957v^8 + 42609v^7 - 114894v^6 + 40275v^5 - 403893v^4 + 742770v^3 - 238626v^2 - 641520*v - 684531) / 787320
\(\beta_{3}\)	\(=\)	\( ( - 263 \nu^{15} + 1142 \nu^{14} - 5932 \nu^{13} + 12008 \nu^{12} - 20419 \nu^{11} + 46061 \nu^{10} + \cdots - 2477871 ) / 787320 \) (-263v^15 + 1142v^14 - 5932v^13 + 12008v^12 - 20419v^11 + 46061v^10 - 63728v^9 + 40641v^8 + 70125v^7 - 235674v^6 + 613287v^5 - 1265193v^4 + 1912734v^3 - 1910466v^2 + 1892484*v - 2477871) / 787320
\(\beta_{4}\)	\(=\)	\( ( 53 \nu^{15} - 173 \nu^{14} + 37 \nu^{13} - 317 \nu^{12} + 1474 \nu^{11} - 899 \nu^{10} + \cdots - 170586 ) / 131220 \) (53v^15 - 173v^14 + 37v^13 - 317v^12 + 1474v^11 - 899v^10 - 1447v^9 - 12v^8 - 717v^7 + 5841v^6 - 36702v^5 + 36207v^4 + 13041v^3 + 88209v^2 - 175689*v - 170586) / 131220
\(\beta_{5}\)	\(=\)	\( ( - 461 \nu^{15} + 1052 \nu^{14} - 6094 \nu^{13} + 12278 \nu^{12} - 16963 \nu^{11} + 46871 \nu^{10} + \cdots - 2215431 ) / 787320 \) (-461v^15 + 1052v^14 - 6094v^13 + 12278v^12 - 16963v^11 + 46871v^10 - 61046v^9 + 37005v^8 + 59811v^7 - 281304v^6 + 466839v^5 - 1201203v^4 + 2339928v^3 - 1422036v^2 + 1721898*v - 2215431) / 787320
\(\beta_{6}\)	\(=\)	\( ( - 559 \nu^{15} - 86 \nu^{14} - 6476 \nu^{13} + 2116 \nu^{12} + 5653 \nu^{11} + 32197 \nu^{10} + \cdots - 9843687 ) / 787320 \) (-559v^15 - 86v^14 - 6476v^13 + 2116v^12 + 5653v^11 + 32197v^10 - 50404v^9 - 33279v^8 - 27699v^7 + 39762v^6 - 238689v^5 - 513081v^4 + 1973322v^3 - 22842v^2 + 1930392*v - 9843687) / 787320
\(\beta_{7}\)	\(=\)	\( ( - 667 \nu^{15} + 1570 \nu^{14} - 1328 \nu^{13} + 2620 \nu^{12} - 9611 \nu^{11} + 17905 \nu^{10} + \cdots - 2810295 ) / 787320 \) (-667v^15 + 1570v^14 - 1328v^13 + 2620v^12 - 9611v^11 + 17905v^10 - 6052v^9 - 12399v^8 - 43791v^7 + 3690v^6 + 259623v^5 - 662445v^4 + 381186v^3 - 789750v^2 + 2350296*v - 2810295) / 787320
\(\beta_{8}\)	\(=\)	\( ( - 397 \nu^{15} + 697 \nu^{14} + 2137 \nu^{13} - 287 \nu^{12} - 476 \nu^{11} - 27149 \nu^{10} + \cdots + 3875364 ) / 393660 \) (-397v^15 + 697v^14 + 2137v^13 - 287v^12 - 476v^11 - 27149v^10 + 40343v^9 - 41082v^8 + 46173v^7 - 53469v^6 + 62388v^5 + 615897v^4 - 1306449v^3 + 574209v^2 - 1815939*v + 3875364) / 393660
\(\beta_{9}\)	\(=\)	\( ( - 29 \nu^{15} + 5 \nu^{14} - 7 \nu^{13} - 67 \nu^{12} + 32 \nu^{11} - 421 \nu^{10} + 643 \nu^{9} + \cdots - 26244 ) / 26244 \) (-29v^15 + 5v^14 - 7v^13 - 67v^12 + 32v^11 - 421v^10 + 643v^9 - 942v^8 - 867v^7 + 1071v^6 - 1476v^5 + 9477v^4 - 16929v^3 + 12393v^2 - 8019*v - 26244) / 26244
\(\beta_{10}\)	\(=\)	\( ( 107 \nu^{15} - 312 \nu^{14} + 578 \nu^{13} - 1078 \nu^{12} + 1781 \nu^{11} - 2041 \nu^{10} + \cdots + 64881 ) / 87480 \) (107v^15 - 312v^14 + 578v^13 - 1078v^12 + 1781v^11 - 2041v^10 + 582v^9 + 5117v^8 - 12573v^7 + 21684v^6 - 58833v^5 + 59013v^4 - 87156v^3 + 38556v^2 - 53946*v + 64881) / 87480
\(\beta_{11}\)	\(=\)	\( ( - 905 \nu^{15} + 5402 \nu^{14} - 5380 \nu^{13} + 7448 \nu^{12} - 31405 \nu^{11} + 37811 \nu^{10} + \cdots - 443961 ) / 787320 \) (-905v^15 + 5402v^14 - 5380v^13 + 7448v^12 - 31405v^11 + 37811v^10 + 5560v^9 - 54993v^8 + 55419v^7 - 175734v^6 + 782145v^5 - 1303263v^4 + 331290v^3 - 962766v^2 + 3484620*v - 443961) / 787320
\(\beta_{12}\)	\(=\)	\( ( - 937 \nu^{15} - 482 \nu^{14} + 292 \nu^{13} + 8452 \nu^{12} - 3941 \nu^{11} + 4459 \nu^{10} + \cdots - 1983609 ) / 787320 \) (-937v^15 - 482v^14 + 292v^13 + 8452v^12 - 3941v^11 + 4459v^10 - 24412v^9 + 11199v^8 - 22245v^7 - 138546v^6 + 218313v^5 + 179793v^4 + 949806v^3 - 1209654v^2 - 798984*v - 1983609) / 787320
\(\beta_{13}\)	\(=\)	\( ( - 503 \nu^{15} + 905 \nu^{14} + 1193 \nu^{13} - 55 \nu^{12} + 4346 \nu^{11} - 24115 \nu^{10} + \cdots + 3171150 ) / 393660 \) (-503v^15 + 905v^14 + 1193v^13 - 55v^12 + 4346v^11 - 24115v^10 + 41647v^9 - 55416v^8 + 63771v^7 - 44865v^6 - 19458v^5 + 556335v^4 - 1062801v^3 + 786105v^2 - 2186271*v + 3171150) / 393660
\(\beta_{14}\)	\(=\)	\( ( - 773 \nu^{15} - 427 \nu^{14} - 2227 \nu^{13} + 6677 \nu^{12} - 1504 \nu^{11} + 7619 \nu^{10} + \cdots - 2169504 ) / 393660 \) (-773v^15 - 427v^14 - 2227v^13 + 6677v^12 - 1504v^11 + 7619v^10 - 16313v^9 - 12198v^8 + 12957v^7 - 147141v^6 + 102852v^5 - 52947v^4 + 701379v^3 - 231579v^2 - 611631*v - 2169504) / 393660
\(\beta_{15}\)	\(=\)	\( ( 94 \nu^{15} - 21 \nu^{14} + 271 \nu^{13} - 899 \nu^{12} + 1057 \nu^{11} - 1718 \nu^{10} + \cdots + 376893 ) / 43740 \) (94v^15 - 21v^14 + 271v^13 - 899v^12 + 1057v^11 - 1718v^10 + 3279v^9 + 727v^8 + 3780v^7 + 17457v^6 - 32481v^5 + 43074v^4 - 123687v^3 + 141993v^2 - 75087*v + 376893) / 43740

\(\nu\)	\(=\)	\( ( -\beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} + 2\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 1 ) / 3 \) (-b13 - b12 - b11 - b9 + 2*b8 + b7 + b6 + b5 - b4 - b2 + 1) / 3
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{10} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1 ) / 3 \) (b15 + b14 - 2b13 - 2b10 + b8 + 2b7 - b6 + b5 + 4b4 - 2*b3 + 1) / 3
\(\nu^{3}\)	\(=\)	\( ( - 2 \beta_{15} - 3 \beta_{14} - 2 \beta_{11} - 3 \beta_{10} - \beta_{8} - \beta_{7} + 3 \beta_{5} + \cdots - 4 ) / 3 \) (-2b15 - 3b14 - 2b11 - 3b10 - b8 - b7 + 3b5 - 3b4 - 3*b3 + b1 - 4) / 3
\(\nu^{4}\)	\(=\)	\( ( \beta_{15} + \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + \beta_{10} + \beta_{8} - 7 \beta_{7} + \cdots + 4 ) / 3 \) (b15 + b14 - 2b13 + 3b12 + 3b11 + b10 + b8 - 7b7 + 2b6 - 2b5 - 2b4 + b3 - 3b2 + 4) / 3
\(\nu^{5}\)	\(=\)	\( ( - 5 \beta_{15} - 3 \beta_{14} - 3 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 6 \beta_{10} + 3 \beta_{9} + \cdots - 1 ) / 3 \) (-5b15 - 3b14 - 3b13 - 3b12 - 2b11 - 6b10 + 3b9 - b8 - 4b7 - 3b6 + 9b4 + 6b3 - 12b2 + 4*b1 - 1) / 3
\(\nu^{6}\)	\(=\)	\( ( 7 \beta_{15} + 4 \beta_{14} + \beta_{13} + 9 \beta_{12} - 3 \beta_{11} - 20 \beta_{10} - 14 \beta_{8} + \cdots + 40 ) / 3 \) (7b15 + 4b14 + b13 + 9b12 - 3b11 - 20b10 - 14b8 + 11b7 - 7b6 - 23b5 - 2b4 - 2b3 + 15b2 - 3*b1 + 40) / 3
\(\nu^{7}\)	\(=\)	\( ( 7 \beta_{15} - 18 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} - 26 \beta_{11} + 3 \beta_{10} - 12 \beta_{9} + \cdots - 61 ) / 3 \) (7b15 - 18b14 - 3b13 + 3b12 - 26b11 + 3b10 - 12b9 + 26b8 + 14b7 + 27b6 + 18b5 - 57b4 + 18b3 + 3b2 + 10*b1 - 61) / 3
\(\nu^{8}\)	\(=\)	\( ( 31 \beta_{15} + 28 \beta_{14} + 10 \beta_{13} + 9 \beta_{12} + 18 \beta_{11} - 2 \beta_{10} + 48 \beta_{9} + \cdots - 92 ) / 3 \) (31b15 + 28b14 + 10b13 + 9b12 + 18b11 - 2b10 + 48b9 - 29b8 - b7 - 40b6 + b5 + 58b4 - 11b3 - 27b1 - 92) / 3
\(\nu^{9}\)	\(=\)	\( ( - 41 \beta_{15} + 15 \beta_{14} + 60 \beta_{13} - 39 \beta_{12} - 8 \beta_{11} - 39 \beta_{10} + \cdots - 295 ) / 3 \) (-41b15 + 15b14 + 60b13 - 39b12 - 8b11 - 39b10 - 3b9 - 127b8 - 94b7 - 42b6 - 21b5 - 213b4 - 57b3 + 6b2 + 7*b1 - 295) / 3
\(\nu^{10}\)	\(=\)	\( ( 64 \beta_{15} + 73 \beta_{14} + 25 \beta_{13} + 129 \beta_{12} + 138 \beta_{11} + 145 \beta_{10} + \cdots + 112 ) / 3 \) (64b15 + 73b14 + 25b13 + 129b12 + 138b11 + 145b10 - 27b9 - 89b8 - 178b7 + 47b6 - 92b5 - 191b4 + 91b3 - 30b2 + 54*b1 + 112) / 3
\(\nu^{11}\)	\(=\)	\( ( - 32 \beta_{15} + 42 \beta_{14} + 105 \beta_{13} - 129 \beta_{12} - 20 \beta_{11} + 102 \beta_{10} + \cdots + 251 ) / 3 \) (-32b15 + 42b14 + 105b13 - 129b12 - 20b11 + 102b10 - 6b9 - 10b8 + 50b7 - 21b6 + 45b5 + 342b4 + 195b3 - 336b2 - 14*b1 + 251) / 3
\(\nu^{12}\)	\(=\)	\( ( 295 \beta_{15} + 616 \beta_{14} - 17 \beta_{13} - 27 \beta_{12} + 24 \beta_{11} - 389 \beta_{10} + \cdots + 949 ) / 3 \) (295b15 + 616b14 - 17b13 - 27b12 + 24b11 - 389b10 - 450b9 - 140b8 + 587b7 - 295b6 - 572b5 + 322b4 - 344b3 + 366b2 - 327*b1 + 949) / 3
\(\nu^{13}\)	\(=\)	\( ( 151 \beta_{15} - 486 \beta_{14} - 318 \beta_{13} + 507 \beta_{12} - 404 \beta_{11} + 192 \beta_{10} + \cdots - 1204 ) / 3 \) (151b15 - 486b14 - 318b13 + 507b12 - 404b11 + 192b10 - 687b9 + 773b8 + 941b7 + 459b6 + 837b5 - 471b4 - 198b3 + 120b2 + 379*b1 - 1204) / 3
\(\nu^{14}\)	\(=\)	\( ( 679 \beta_{15} + 100 \beta_{14} + 487 \beta_{13} + 846 \beta_{12} + 1170 \beta_{11} + 385 \beta_{10} + \cdots + 205 ) / 3 \) (679b15 + 100b14 + 487b13 + 846b12 + 1170b11 + 385b10 + 552b9 - 182b8 + 314b7 - 499b6 + 226b5 + 3631b4 - 488b3 - 72b2 - 828*b1 + 205) / 3
\(\nu^{15}\)	\(=\)	\( ( - 1256 \beta_{15} + 465 \beta_{14} + 6 \beta_{13} - 1434 \beta_{12} + 649 \beta_{11} - 1173 \beta_{10} + \cdots - 2338 ) / 3 \) (-1256b15 + 465b14 + 6b13 - 1434b12 + 649b11 - 1173b10 - 2172b9 - 946b8 - 1876b7 - 816b6 - 1596b5 + 795b4 - 948b3 - 318b2 - 443*b1 - 2338) / 3

\(n\)	\(757\)	\(1081\)	\(1541\)
\(\chi(n)\)	\(1\)	\(-1 - \beta_{4}\)	\(-1 - \beta_{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 991.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T - 1)^{16} \) (T - 1)^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{2} + T + 1)^{8} \) (T^2 + T + 1)^8
$7$	\( T^{16} - 4 T^{15} + \cdots + 5764801 \) T^16 - 4T^15 + 15T^14 - 46T^13 + 85T^12 - 234T^11 + 295T^10 + 350T^9 - 39T^8 + 2450T^7 + 14455T^6 - 80262T^5 + 204085T^4 - 773122T^3 + 1764735T^2 - 3294172*T + 5764801
$11$	\( T^{16} + T^{15} + \cdots + 2916 \) T^16 + T^15 + 53T^14 - 116T^13 + 2018T^12 - 4688T^11 + 39862T^10 - 156501T^9 + 562494T^8 - 1194714T^7 + 1937466T^6 - 1987308T^5 + 1493181T^4 - 627426T^3 + 188406T^2 - 28188T + 2916
$13$	\( T^{16} - 2 T^{15} + \cdots + 2298256 \) T^16 - 2T^15 + 54T^14 - 8T^13 + 1834T^12 + 477T^11 + 34540T^10 + 52582T^9 + 433476T^8 + 684178T^7 + 3205075T^6 + 6671493T^5 + 14639257T^4 + 16719280T^3 + 15234300T^2 + 6900832*T + 2298256
$17$	\( T^{16} + \cdots + 8707129344 \) T^16 + 11T^15 + 172T^14 + 999T^13 + 10767T^12 + 48267T^11 + 453195T^10 + 1438209T^9 + 11605383T^8 + 26237682T^7 + 215458947T^6 + 330564807T^5 + 2370323817T^4 + 2178566928T^3 + 17265612672T^2 + 11972302848*T + 8707129344
$19$	\( T^{16} + \cdots + 1174158756 \) T^16 + 2T^15 + 98T^14 + 280T^13 + 6918T^12 + 19245T^11 + 235788T^10 + 637458T^9 + 5715978T^8 + 14051506T^7 + 75895571T^6 + 153664667T^5 + 652539913T^4 + 1231454664T^3 + 2158008918T^2 + 1769359176*T + 1174158756
$23$	\( T^{16} + \cdots + 927567936 \) T^16 + 11T^15 + 176T^14 + 1007T^13 + 11912T^12 + 56534T^11 + 516169T^10 + 1244817T^9 + 8326524T^8 + 101754T^7 + 97089705T^6 - 44990154T^5 + 547995969T^4 - 642142404T^3 + 2201299416T^2 - 1413280224*T + 927567936
$29$	\( T^{16} + \cdots + 17639292969 \) T^16 + 17T^15 + 278T^14 + 2735T^13 + 28451T^12 + 225260T^11 + 1821232T^10 + 11447856T^9 + 70777128T^8 + 351438732T^7 + 1743315804T^6 + 6789615372T^5 + 24575037471T^4 + 58028658471T^3 + 110947789962T^2 + 48483252837*T + 17639292969
$31$	\( (T^{8} - 15 T^{7} + \cdots + 27294)^{2} \) (T^8 - 15T^7 + 46T^6 + 292T^5 - 1946T^4 + 1420T^3 + 13629T^2 - 36302*T + 27294)^2
$37$	\( T^{16} + \cdots + 147622500 \) T^16 + 2T^15 + 187T^14 - 50T^13 + 23858T^12 - 11501T^11 + 1541301T^10 - 1916286T^9 + 71688606T^8 - 89262833T^7 + 1552868072T^6 - 3683050217T^5 + 23529518011T^4 - 29017050460T^3 + 34261002250T^2 - 2308986000*T + 147622500
$41$	\( T^{16} + \cdots + 86812992387801 \) T^16 + 7T^15 + 270T^14 + 901T^13 + 39076T^12 + 84663T^11 + 3702942T^10 + 2777112T^9 + 245168532T^8 + 9845874T^7 + 11620565451T^6 - 9476498448T^5 + 381671616141T^4 - 402751302138T^3 + 8136771273864T^2 - 12647350897902*T + 86812992387801
$43$	\( T^{16} + \cdots + 285914281 \) T^16 + 13T^15 + 234T^14 + 1787T^13 + 23078T^12 + 158697T^11 + 1454068T^10 + 7267054T^9 + 48039750T^8 + 205695460T^7 + 1045232041T^6 + 2987533494T^5 + 7532421827T^4 + 6805283222T^3 + 4526436618T^2 + 1330433938*T + 285914281
$47$	\( (T^{8} + 5 T^{7} + \cdots - 423063)^{2} \) (T^8 + 5T^7 - 209T^6 - 966T^5 + 9951T^4 + 50121T^3 - 37818T^2 - 406404*T - 423063)^2
$53$	\( T^{16} + \cdots + 1238515248996 \) T^16 + 18T^15 + 399T^14 + 2630T^13 + 38448T^12 + 106899T^11 + 2838763T^10 + 1726038T^9 + 108882912T^8 - 66347235T^7 + 3097006380T^6 - 3808368603T^5 + 45774547983T^4 - 92330519760T^3 + 506947936338T^2 - 716382524376*T + 1238515248996
$59$	\( (T^{8} - T^{7} - 155 T^{6} + \cdots + 1458)^{2} \) (T^8 - T^7 - 155T^6 - 246T^5 + 5748T^4 + 22311T^3 + 25713T^2 + 11124T + 1458)^2
$61$	\( (T^{8} - 27 T^{7} + \cdots - 502368)^{2} \) (T^8 - 27T^7 + 156T^6 + 2006T^5 - 33888T^4 + 203910T^3 - 611453T^2 + 898440*T - 502368)^2
$67$	\( (T^{8} - 10 T^{7} + \cdots + 40406508)^{2} \) (T^8 - 10T^7 - 353T^6 + 2779T^5 + 44516T^4 - 227405T^3 - 2320437T^2 + 5028960*T + 40406508)^2
$71$	\( (T^{8} - 19 T^{7} + \cdots - 7422678)^{2} \) (T^8 - 19T^7 - 168T^6 + 5670T^5 - 28107T^4 - 131949T^3 + 1116099T^2 - 14580*T - 7422678)^2
$73$	\( T^{16} + 8 T^{15} + \cdots + 2143296 \) T^16 + 8T^15 + 194T^14 + 2596T^13 + 38910T^12 + 345057T^11 + 2413146T^10 + 10754460T^9 + 35783472T^8 + 66554746T^7 + 92478587T^6 + 76189157T^5 + 52769077T^4 + 21750732T^3 + 12384024T^2 + 3882528*T + 2143296
$79$	\( (T^{8} - 25 T^{7} + \cdots - 406534)^{2} \) (T^8 - 25T^7 + 175T^6 + 726T^5 - 18231T^4 + 117846T^3 - 377959T^2 + 615718*T - 406534)^2
$83$	\( T^{16} + \cdots + 2857962683601 \) T^16 + 2T^15 + 285T^14 + 950T^13 + 59479T^12 + 181731T^11 + 5425806T^10 + 12293469T^9 + 344131362T^8 + 646842591T^7 + 11108005821T^6 + 8778869019T^5 + 229179273264T^4 + 211647489975T^3 + 1490295654990T^2 - 1411385241717*T + 2857962683601
$89$	\( T^{16} + \cdots + 99588211776 \) T^16 - 6T^15 + 351T^14 - 110T^13 + 80562T^12 - 66627T^11 + 7690357T^10 - 2106420T^9 + 520688268T^8 - 643546485T^7 + 11015185194T^6 - 21704323815T^5 + 188866782837T^4 - 298622599884T^3 + 812199516120T^2 + 261580946400*T + 99588211776
$97$	\( T^{16} + \cdots + 1545178274704 \) T^16 - 26T^15 + 720T^14 - 12640T^13 + 253103T^12 - 3815676T^11 + 50715262T^10 - 482307254T^9 + 3675013929T^8 - 19206109952T^7 + 76868712919T^6 - 161539486848T^5 + 267155797613T^4 + 299843403512T^3 + 2147276158380T^2 + 1714109041504*T + 1545178274704
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