LMFDB - Newform orbit 1386.2.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(881,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1386.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.g (of order $2$, degree $1$, 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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 $ x^16 - 4x^14 + 8x^12 + 80x^10 + 1189x^8 - 2028x^6 + 1800x^4 + 1080*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + \beta_{7} q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} - \beta_1 q^{11} + \beta_{11} q^{14} + q^{16} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{15} - \beta_{13} + \cdots - \beta_{10}) q^{98}+O(q^{100}) $ q + b1 * q^2 - q^4 - b4 * q^5 + b7 * q^7 - b1 * q^8 + b3 * q^10 - b1 * q^11 + b11 * q^14 + q^16 + (-b11 + b10 - b4 - b1) * q^17 + b14 * q^19 + b4 * q^20 + q^22 + (-b13 + b12) * q^23 + (-b9 + b8 - b7 + b5 - b2 + 1) * q^25 - b7 * q^28 + (b11 + b10 + b6 + b1) * q^29 + (-b14 + b9 + b8 + b7 + b5) * q^31 + b1 * q^32 + (b7 + b5 + b3 + 1) * q^34 + (b12 + 2b4) q^35 + (-b9 + b8 - 1) * q^37 + b15 * q^38 - b3 * q^40 + (b15 - b11 + b10 - b1) * q^41 + (-b7 + b5 + 3) * q^43 + b1 * q^44 + (b9 - b8 - 1) * q^46 + (b15 - b11 + b10 + 2b4 - b1) q^47 + (b14 - b9 - b7 + b5 + 1) * q^49 + (-b13 + b12 - b11 - b10 + b6) * q^50 + (-b13 + b12 + b11 + b10 - 5b1) q^53 - b3 * q^55 - b11 * q^56 + (-b7 + b5 + b2 - 1) * q^58 - 2b4 q^59 + (2b7 + 2b5 - 4b3 + 2) q^61 + (-b15 + b13 + b12 + b11 - b10 + b1) * q^62 - q^64 + (-b7 + b5 - b2 + 1) * q^67 + (b11 - b10 + b4 + b1) * q^68 + (-b8 - 2b3) q^70 + (-b13 + b12 + 2b6) q^71 + (b14 - b7 - b5 + 2b3 - 1) q^73 + (-b13 + b12 - 2b1) q^74 - b14 * q^76 - b11 * q^77 + (b9 - b8 - 2b7 + 2b5 + b2 - 1) * q^79 - b4 * q^80 + (-b14 + b7 + b5 + 1) * q^82 + (b13 + b12 + b11 - b10 - b4 + b1) * q^83 + (-2b9 + 2b8 - b7 + b5 - b2 + 7) * q^85 + (-b11 - b10 + 3b1) q^86 - q^88 + (2b15 - b11 + b10 - b1) q^89 + (b13 - b12) * q^92 + (-b14 + b7 + b5 - 2b3 + 1) q^94 + (-b11 - b10 - b6 - b1) * q^95 + (2b14 + 2b3) * q^97 + (b15 - b13 - b11 - b10) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} - 8 q^{7} + 16 q^{16} + 16 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{37} + 48 q^{43} - 16 q^{46} + 8 q^{49} - 16 q^{58} - 16 q^{64} + 16 q^{67} - 8 q^{70} - 16 q^{79} + 112 q^{85} - 16 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 - 8 * q^7 + 16 * q^16 + 16 * q^22 + 16 * q^25 + 8 * q^28 - 16 * q^37 + 48 * q^43 - 16 * q^46 + 8 * q^49 - 16 * q^58 - 16 * q^64 + 16 * q^67 - 8 * q^70 - 16 * q^79 + 112 * q^85 - 16 * q^88

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 $ x^16 - 4*x^14 + 8*x^12 + 80*x^10 + 1189*x^8 - 2028*x^6 + 1800*x^4 + 1080*x^2 + 324 :

$\beta_{1}$	$=$	$ ( - 96575 \nu^{14} + 438755 \nu^{12} - 939814 \nu^{10} - 7435564 \nu^{8} - 110537567 \nu^{6} + \cdots - 51358644 ) / 86730966 $ (-96575v^14 + 438755v^12 - 939814v^10 - 7435564v^8 - 110537567v^6 + 264348225v^4 - 229027878*v^2 - 51358644) / 86730966
$\beta_{2}$	$=$	$ ( - 3833 \nu^{14} + 11990 \nu^{12} - 16306 \nu^{10} - 332152 \nu^{8} - 4851221 \nu^{6} + \cdots - 7232976 ) / 1867860 $ (-3833v^14 + 11990v^12 - 16306v^10 - 332152v^8 - 4851221v^6 + 3917118v^4 + 2396970*v^2 - 7232976) / 1867860
$\beta_{3}$	$=$	$ ( - 3079634 \nu^{15} + 11597979 \nu^{13} - 22413962 \nu^{11} - 248772494 \nu^{9} + \cdots - 5487054570 \nu ) / 2601928980 $ (-3079634v^15 + 11597979v^13 - 22413962v^11 - 248772494v^9 - 3735647454v^7 + 5354283043v^5 - 4822618578v^3 - 5487054570v) / 2601928980
$\beta_{4}$	$=$	$ ( 12591283 \nu^{15} - 57579817 \nu^{13} + 129110888 \nu^{11} + 952830506 \nu^{9} + \cdots + 3202119162 \nu ) / 7805786940 $ (12591283v^15 - 57579817v^13 + 129110888v^11 + 952830506v^9 + 14397572539v^7 - 34397644005v^5 + 38419343964v^3 + 3202119162v) / 7805786940
$\beta_{5}$	$=$	$ ( 14545077 \nu^{15} - 3765719 \nu^{14} - 55196585 \nu^{13} + 11685650 \nu^{12} + \cdots - 9136826568 ) / 5203857960 $ (14545077v^15 - 3765719v^14 - 55196585v^13 + 11685650v^12 + 105758504v^11 - 14001178v^10 + 1189121858v^9 - 333984496v^8 + 17514628229v^7 - 4702927103v^6 - 25836351497v^5 + 3800759034v^4 + 22083943080v^3 + 2325320910v^2 + 25498762434*v - 9136826568) / 5203857960
$\beta_{6}$	$=$	$ ( - 11395 \nu^{14} + 45868 \nu^{12} - 96938 \nu^{10} - 893276 \nu^{8} - 13578607 \nu^{6} + \cdots - 6275124 ) / 2256660 $ (-11395v^14 + 45868v^12 - 96938v^10 - 893276v^8 - 13578607v^6 + 23130228v^4 - 27991926*v^2 - 6275124) / 2256660
$\beta_{7}$	$=$	$ ( 14545077 \nu^{15} + 3765719 \nu^{14} - 55196585 \nu^{13} - 11685650 \nu^{12} + \cdots + 3932968608 ) / 5203857960 $ (14545077v^15 + 3765719v^14 - 55196585v^13 - 11685650v^12 + 105758504v^11 + 14001178v^10 + 1189121858v^9 + 333984496v^8 + 17514628229v^7 + 4702927103v^6 - 25836351497v^5 - 3800759034v^4 + 22083943080v^3 - 2325320910v^2 + 25498762434*v + 3932968608) / 5203857960
$\beta_{8}$	$=$	$ ( 18506049 \nu^{15} - 1441114 \nu^{14} - 72148880 \nu^{13} + 4446220 \nu^{12} + \cdots - 13615971048 ) / 5203857960 $ (18506049v^15 - 1441114v^14 - 72148880v^13 + 4446220v^12 + 136369838v^11 - 4803548v^10 + 1512241796v^9 - 147925256v^8 + 22114467473v^7 - 1782429418v^6 - 35425496144v^5 + 1441445244v^4 + 25614327990v^3 + 881758260v^2 + 31120049628*v - 13615971048) / 5203857960
$\beta_{9}$	$=$	$ ( 18506049 \nu^{15} + 1441114 \nu^{14} - 72148880 \nu^{13} - 4446220 \nu^{12} + \cdots + 18819829008 ) / 5203857960 $ (18506049v^15 + 1441114v^14 - 72148880v^13 - 4446220v^12 + 136369838v^11 + 4803548v^10 + 1512241796v^9 + 147925256v^8 + 22114467473v^7 + 1782429418v^6 - 35425496144v^5 - 1441445244v^4 + 25614327990v^3 - 881758260v^2 + 31120049628*v + 18819829008) / 5203857960
$\beta_{10}$	$=$	$ ( 60462680 \nu^{15} - 34694355 \nu^{14} - 274567721 \nu^{13} + 143608152 \nu^{12} + \cdots - 18889526916 ) / 15611573880 $ (60462680v^15 - 34694355v^14 - 274567721v^13 + 143608152v^12 + 617221546v^11 - 309169302v^10 + 4576799302v^9 - 2703731964v^8 + 69289811864v^7 - 40803930363v^6 - 161200544481v^5 + 75040099632v^4 + 178659278442v^3 - 84253473054v^2 + 14747147118*v - 18889526916) / 15611573880
$\beta_{11}$	$=$	$ ( - 60462680 \nu^{15} - 17310855 \nu^{14} + 274567721 \nu^{13} + 64632252 \nu^{12} + \cdots - 9644970996 ) / 15611573880 $ (-60462680v^15 - 17310855v^14 + 274567721v^13 + 64632252v^12 - 617221546v^11 - 140002782v^10 - 4576799302v^9 - 1365330444v^8 - 69289811864v^7 - 20907168303v^6 + 161200544481v^5 + 27457419132v^4 - 178659278442v^3 - 43028455014v^2 - 14747147118*v - 9644970996) / 15611573880
$\beta_{12}$	$=$	$ ( - 81296765 \nu^{15} + 47163030 \nu^{14} + 357230942 \nu^{13} - 210264972 \nu^{12} + \cdots + 25206770376 ) / 15611573880 $ (-81296765v^15 + 47163030v^14 + 357230942v^13 - 210264972v^12 - 785602642v^11 + 450833772v^10 - 6207754144v^9 + 3640498104v^8 - 94072523573v^7 + 54294184518v^6 + 202259564982v^5 - 121172690412v^4 - 218620670994v^3 + 112411745244v^2 - 17473193856*v + 25206770376) / 15611573880
$\beta_{13}$	$=$	$ ( - 81296765 \nu^{15} - 47163030 \nu^{14} + 357230942 \nu^{13} + 210264972 \nu^{12} + \cdots - 25206770376 ) / 15611573880 $ (-81296765v^15 - 47163030v^14 + 357230942v^13 + 210264972v^12 - 785602642v^11 - 450833772v^10 - 6207754144v^9 - 3640498104v^8 - 94072523573v^7 - 54294184518v^6 + 202259564982v^5 + 121172690412v^4 - 218620670994v^3 - 112411745244v^2 - 17473193856*v - 25206770376) / 15611573880
$\beta_{14}$	$=$	$ ( 23298266 \nu^{15} - 93544963 \nu^{13} + 176425470 \nu^{11} + 1897812810 \nu^{9} + \cdots + 37162486386 \nu ) / 2601928980 $ (23298266v^15 - 93544963v^13 + 176425470v^11 + 1897812810v^9 + 27613656994v^7 - 48610532639v^5 + 28420002066v^3 + 37162486386v) / 2601928980
$\beta_{15}$	$=$	$ ( - 15573353 \nu^{15} + 65944679 \nu^{13} - 141046900 \nu^{11} - 1206672370 \nu^{9} + \cdots - 2860280478 \nu ) / 1115112420 $ (-15573353v^15 + 65944679v^13 - 141046900v^11 - 1206672370v^9 - 18241907237v^7 + 35778301767v^5 - 37432034028v^3 - 2860280478v) / 1115112420

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{13} - \beta_{12} - 3\beta_{3} ) / 6 $ (b15 - b13 - b12 - 3*b3) / 6
$\nu^{2}$	$=$	$ ( \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 + 1 ) / 2 $ (b11 + b10 + b7 - b6 - b5 + b2 + b1 + 1) / 2
$\nu^{3}$	$=$	$ ( -\beta_{14} + \beta_{11} - \beta_{10} + 2\beta_{9} + 2\beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + \beta _1 - 3 ) / 2 $ (-b14 + b11 - b10 + 2b9 + 2b8 - b7 - b5 + 5*b4 + b1 - 3) / 2
$\nu^{4}$	$=$	$ ( -7\beta_{13} + 7\beta_{12} + 2\beta_{11} + 2\beta_{10} + 32\beta_1 ) / 2 $ (-7b13 + 7b12 + 2b11 + 2b10 + 32*b1) / 2
$\nu^{5}$	$=$	$ ( - 5 \beta_{14} + \beta_{13} + \beta_{12} - 7 \beta_{11} + 7 \beta_{10} + 13 \beta_{9} + 13 \beta_{8} + \cdots - 22 ) / 2 $ (-5b14 + b13 + b12 - 7b11 + 7b10 + 13b9 + 13b8 - 9b7 - 9b5 - 27b4 - 2b3 - 7b1 - 22) / 2
$\nu^{6}$	$=$	$ ( - 11 \beta_{13} + 11 \beta_{12} + 45 \beta_{11} + 45 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} + \cdots - 68 ) / 2 $ (-11b13 + 11b12 + 45b11 + 45b10 + 11b9 - 11b8 - 45b7 - 29b6 + 45b5 - 29b2 + 57*b1 - 68) / 2
$\nu^{7}$	$=$	$ ( - 31 \beta_{15} - 2 \beta_{14} + 87 \beta_{13} + 87 \beta_{12} - 67 \beta_{11} + 67 \beta_{10} + \cdots - 58 ) / 2 $ (-31b15 - 2b14 + 87b13 + 87b12 - 67b11 + 67b10 + 13b9 + 13b8 - 45b7 - 45b5 - 28b4 - 149b3 - 67*b1 - 58) / 2
$\nu^{8}$	$=$	$ ( 289\beta_{9} - 289\beta_{8} - 190\beta_{7} + 190\beta_{5} - 56\beta_{2} - 1541 ) / 2 $ (289b9 - 289b8 - 190b7 + 190b5 - 56*b2 - 1541) / 2
$\nu^{9}$	$=$	$ ( - 205 \beta_{15} + 28 \beta_{14} + 589 \beta_{13} + 589 \beta_{12} - 479 \beta_{11} + 479 \beta_{10} + \cdots + 412 ) / 2 $ (-205b15 + 28b14 + 589b13 + 589b12 - 479b11 + 479b10 - 123b9 - 123b8 + 289b7 + 289b5 - 274b4 + 835b3 - 479*b1 + 412) / 2
$\nu^{10}$	$=$	$ ( 753 \beta_{13} - 753 \beta_{12} - 1877 \beta_{11} - 1877 \beta_{10} + 753 \beta_{9} - 753 \beta_{8} + \cdots - 4274 ) / 2 $ (753b13 - 753b12 - 1877b11 - 1877b10 + 753b9 - 753b8 - 1877b7 + 1109b6 + 1877b5 - 1109b2 - 3521*b1 - 4274) / 2
$\nu^{11}$	$=$	$ ( - 274 \beta_{15} + 1383 \beta_{14} + 1027 \beta_{13} + 1027 \beta_{12} - 1877 \beta_{11} + 1877 \beta_{10} + \cdots + 7396 ) / 2 $ (-274b15 + 1383b14 + 1027b13 + 1027b12 - 1877b11 + 1877b10 - 4013b9 - 4013b8 + 3383b7 + 3383b5 - 4753b4 + 2328b3 - 1877*b1 + 7396) / 2
$\nu^{12}$	$=$	$ ( 12341\beta_{13} - 12341\beta_{12} - 11422\beta_{11} - 11422\beta_{10} + 4656\beta_{6} - 53908\beta_1 ) / 2 $ (12341b13 - 12341b12 - 11422b11 - 11422b10 + 4656b6 - 53908b1) / 2
$\nu^{13}$	$=$	$ ( 2328 \beta_{15} + 9409 \beta_{14} - 8039 \beta_{13} - 8039 \beta_{12} + 12341 \beta_{11} - 12341 \beta_{10} + \cdots + 51224 ) / 2 $ (2328b15 + 9409b14 - 8039b13 - 8039b12 + 12341b11 - 12341b10 - 27461b9 - 27461b8 + 23763b7 + 23763b5 + 27519b4 + 18406b3 + 12341*b1 + 51224) / 2
$\nu^{14}$	$=$	$ ( 42169 \beta_{13} - 42169 \beta_{12} - 82029 \beta_{11} - 82029 \beta_{10} - 42169 \beta_{9} + \cdots + 234634 ) / 2 $ (42169b13 - 42169b12 - 82029b11 - 82029b10 - 42169b9 + 42169b8 + 82029b7 + 45925b6 - 82029b5 + 45925b2 - 192465*b1 + 234634) / 2
$\nu^{15}$	$=$	$ ( 64331 \beta_{15} + 18406 \beta_{14} - 188529 \beta_{13} - 188529 \beta_{12} + 166367 \beta_{11} + \cdots + 142604 ) / 2 $ (64331b15 + 18406b14 - 188529b13 - 188529b12 + 166367b11 - 166367b10 - 60575b9 - 60575b8 + 82029b7 + 82029b5 + 139556b4 + 162277b3 + 166367*b1 + 142604) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$-1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

881.1

−0.876932	−	2.11710i
2.43348	−	1.00798i
−0.221383	−	0.534465i
1.12256	−	0.464978i
−1.12256	+	0.464978i
0.221383	+	0.534465i
−2.43348	+	1.00798i
0.876932	+	2.11710i
−0.876932	+	2.11710i
2.43348	+	1.00798i
−0.221383	+	0.534465i
1.12256	+	0.464978i
−1.12256	−	0.464978i
0.221383	−	0.534465i
−2.43348	−	1.00798i
0.876932	−	2.11710i

−

1.00000i

−1.00000

−4.23420

−2.59178

0.531652i

1.00000i

4.23420i

881.2

−

1.00000i

−1.00000

−2.01596

1.78450

1.95335i

1.00000i

2.01596i

881.3

−

1.00000i

−1.00000

−1.06893

0.884677

−

2.49346i

1.00000i

1.06893i

881.4

−

1.00000i

−1.00000

−0.929956

−2.07739

−

1.63843i

1.00000i

0.929956i

881.5

−

1.00000i

−1.00000

0.929956

−2.07739

1.63843i

1.00000i

−

0.929956i

881.6

−

1.00000i

−1.00000

1.06893

0.884677

2.49346i

1.00000i

−

1.06893i

881.7

−

1.00000i

−1.00000

2.01596

1.78450

−

1.95335i

1.00000i

−

2.01596i

881.8

−

1.00000i

−1.00000

4.23420

−2.59178

−

0.531652i

1.00000i

−

4.23420i

881.9

1.00000i

−1.00000

−4.23420

−2.59178

−

0.531652i

−

1.00000i

−

4.23420i

881.10

1.00000i

−1.00000

−2.01596

1.78450

−

1.95335i

−

1.00000i

−

2.01596i

881.11

1.00000i

−1.00000

−1.06893

0.884677

2.49346i

−

1.00000i

−

1.06893i

881.12

1.00000i

−1.00000

−0.929956

−2.07739

1.63843i

−

1.00000i

−

0.929956i

881.13

1.00000i

−1.00000

0.929956

−2.07739

−

1.63843i

−

1.00000i

0.929956i

881.14

1.00000i

−1.00000

1.06893

0.884677

−

2.49346i

−

1.00000i

1.06893i

881.15

1.00000i

−1.00000

2.01596

1.78450

1.95335i

−

1.00000i

2.01596i

881.16

1.00000i

−1.00000

4.23420

−2.59178

0.531652i

−

1.00000i

4.23420i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1386.2.g.a	✓	16
3.b	odd	2	1	inner	1386.2.g.a	✓	16
7.b	odd	2	1	inner	1386.2.g.a	✓	16
21.c	even	2	1	inner	1386.2.g.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1386.2.g.a	✓	16	1.a	even	1	1	trivial
1386.2.g.a	✓	16	3.b	odd	2	1	inner
1386.2.g.a	✓	16	7.b	odd	2	1	inner
1386.2.g.a	✓	16	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 24T_{5}^{6} + 118T_{5}^{4} - 168T_{5}^{2} + 72 $ T5^8 - 24*T5^6 + 118*T5^4 - 168*T5^2 + 72 acting on $S_{2}^{\mathrm{new}}(1386, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 24 T^{6} + \cdots + 72)^{2} $ (T^8 - 24T^6 + 118T^4 - 168*T^2 + 72)^2
$7$	$ (T^{8} + 4 T^{7} + \cdots + 2401)^{2} $ (T^8 + 4T^7 + 6T^6 + 28T^5 + 122T^4 + 196T^3 + 294T^2 + 1372*T + 2401)^2
$11$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$13$	$ T^{16} $ T^16
$17$	$ (T^{8} - 84 T^{6} + \cdots + 83232)^{2} $ (T^8 - 84T^6 + 2386T^4 - 25872*T^2 + 83232)^2
$19$	$ (T^{8} + 104 T^{6} + \cdots + 72)^{2} $ (T^8 + 104T^6 + 2710T^4 + 1320*T^2 + 72)^2
$23$	$ (T^{8} + 120 T^{6} + \cdots + 331776)^{2} $ (T^8 + 120T^6 + 4624T^4 + 69120*T^2 + 331776)^2
$29$	$ (T^{8} + 108 T^{6} + \cdots + 5184)^{2} $ (T^8 + 108T^6 + 2932T^4 + 7776*T^2 + 5184)^2
$31$	$ (T^{8} + 268 T^{6} + \cdots + 6139008)^{2} $ (T^8 + 268T^6 + 23458T^4 + 727008*T^2 + 6139008)^2
$37$	$ (T^{4} + 4 T^{3} + \cdots + 544)^{4} $ (T^4 + 4T^3 - 52T^2 - 128*T + 544)^4
$41$	$ (T^{8} - 132 T^{6} + \cdots + 23328)^{2} $ (T^8 - 132T^6 + 3154T^4 - 18576*T^2 + 23328)^2
$43$	$ (T^{4} - 12 T^{3} + \cdots - 16)^{4} $ (T^4 - 12T^3 + 26T^2 + 64*T - 16)^4
$47$	$ (T^{8} - 196 T^{6} + \cdots + 288)^{2} $ (T^8 - 196T^6 + 7474T^4 - 3120*T^2 + 288)^2
$53$	$ (T^{8} + 260 T^{6} + \cdots + 8714304)^{2} $ (T^8 + 260T^6 + 22516T^4 + 776736*T^2 + 8714304)^2
$59$	$ (T^{8} - 96 T^{6} + \cdots + 18432)^{2} $ (T^8 - 96T^6 + 1888T^4 - 10752*T^2 + 18432)^2
$61$	$ (T^{8} + 528 T^{6} + \cdots + 294912)^{2} $ (T^8 + 528T^6 + 89152T^4 + 4706304*T^2 + 294912)^2
$67$	$ (T^{4} - 4 T^{3} - 70 T^{2} + \cdots + 16)^{4} $ (T^4 - 4T^3 - 70T^2 + 176*T + 16)^4
$71$	$ (T^{8} + 312 T^{6} + \cdots + 82944)^{2} $ (T^8 + 312T^6 + 25552T^4 + 232704*T^2 + 82944)^2
$73$	$ (T^{8} + 196 T^{6} + \cdots + 288)^{2} $ (T^8 + 196T^6 + 7474T^4 + 3120*T^2 + 288)^2
$79$	$ (T^{4} + 4 T^{3} + \cdots - 292)^{4} $ (T^4 + 4T^3 - 176T^2 - 840*T - 292)^4
$83$	$ (T^{8} - 348 T^{6} + \cdots + 1492992)^{2} $ (T^8 - 348T^6 + 33282T^4 - 549504*T^2 + 1492992)^2
$89$	$ (T^{8} - 420 T^{6} + \cdots + 1936512)^{2} $ (T^8 - 420T^6 + 40708T^4 - 889152*T^2 + 1936512)^2
$97$	$ (T^{8} + 480 T^{6} + \cdots + 18874368)^{2} $ (T^8 + 480T^6 + 69760T^4 + 3047424*T^2 + 18874368)^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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + \beta_{7} q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} - \beta_1 q^{11} + \beta_{11} q^{14} + q^{16} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{15} - \beta_{13} + \cdots - \beta_{10}) q^{98}+O(q^{100}) \) q + b1 * q^2 - q^4 - b4 * q^5 + b7 * q^7 - b1 * q^8 + b3 * q^10 - b1 * q^11 + b11 * q^14 + q^16 + (-b11 + b10 - b4 - b1) * q^17 + b14 * q^19 + b4 * q^20 + q^22 + (-b13 + b12) * q^23 + (-b9 + b8 - b7 + b5 - b2 + 1) * q^25 - b7 * q^28 + (b11 + b10 + b6 + b1) * q^29 + (-b14 + b9 + b8 + b7 + b5) * q^31 + b1 * q^32 + (b7 + b5 + b3 + 1) * q^34 + (b12 + 2b4) q^35 + (-b9 + b8 - 1) * q^37 + b15 * q^38 - b3 * q^40 + (b15 - b11 + b10 - b1) * q^41 + (-b7 + b5 + 3) * q^43 + b1 * q^44 + (b9 - b8 - 1) * q^46 + (b15 - b11 + b10 + 2b4 - b1) q^47 + (b14 - b9 - b7 + b5 + 1) * q^49 + (-b13 + b12 - b11 - b10 + b6) * q^50 + (-b13 + b12 + b11 + b10 - 5b1) q^53 - b3 * q^55 - b11 * q^56 + (-b7 + b5 + b2 - 1) * q^58 - 2b4 q^59 + (2b7 + 2b5 - 4b3 + 2) q^61 + (-b15 + b13 + b12 + b11 - b10 + b1) * q^62 - q^64 + (-b7 + b5 - b2 + 1) * q^67 + (b11 - b10 + b4 + b1) * q^68 + (-b8 - 2b3) q^70 + (-b13 + b12 + 2b6) q^71 + (b14 - b7 - b5 + 2b3 - 1) q^73 + (-b13 + b12 - 2b1) q^74 - b14 * q^76 - b11 * q^77 + (b9 - b8 - 2b7 + 2b5 + b2 - 1) * q^79 - b4 * q^80 + (-b14 + b7 + b5 + 1) * q^82 + (b13 + b12 + b11 - b10 - b4 + b1) * q^83 + (-2b9 + 2b8 - b7 + b5 - b2 + 7) * q^85 + (-b11 - b10 + 3b1) q^86 - q^88 + (2b15 - b11 + b10 - b1) q^89 + (b13 - b12) * q^92 + (-b14 + b7 + b5 - 2b3 + 1) q^94 + (-b11 - b10 - b6 - b1) * q^95 + (2b14 + 2b3) * q^97 + (b15 - b13 - b11 - b10) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} - 8 q^{7} + 16 q^{16} + 16 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{37} + 48 q^{43} - 16 q^{46} + 8 q^{49} - 16 q^{58} - 16 q^{64} + 16 q^{67} - 8 q^{70} - 16 q^{79} + 112 q^{85} - 16 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 - 8 * q^7 + 16 * q^16 + 16 * q^22 + 16 * q^25 + 8 * q^28 - 16 * q^37 + 48 * q^43 - 16 * q^46 + 8 * q^49 - 16 * q^58 - 16 * q^64 + 16 * q^67 - 8 * q^70 - 16 * q^79 + 112 * q^85 - 16 * q^88

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	1386.2.g.a

Level	$1386$
Weight	$2$
Character orbit	1386.g
Analytic conductor	$11.067$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(11.0672657201\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 \) x^16 - 4x^14 + 8x^12 + 80x^10 + 1189x^8 - 2028x^6 + 1800x^4 + 1080*x^2 + 324
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{9}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 96575 \nu^{14} + 438755 \nu^{12} - 939814 \nu^{10} - 7435564 \nu^{8} - 110537567 \nu^{6} + \cdots - 51358644 ) / 86730966 \) (-96575v^14 + 438755v^12 - 939814v^10 - 7435564v^8 - 110537567v^6 + 264348225v^4 - 229027878*v^2 - 51358644) / 86730966
\(\beta_{2}\)	\(=\)	\( ( - 3833 \nu^{14} + 11990 \nu^{12} - 16306 \nu^{10} - 332152 \nu^{8} - 4851221 \nu^{6} + \cdots - 7232976 ) / 1867860 \) (-3833v^14 + 11990v^12 - 16306v^10 - 332152v^8 - 4851221v^6 + 3917118v^4 + 2396970*v^2 - 7232976) / 1867860
\(\beta_{3}\)	\(=\)	\( ( - 3079634 \nu^{15} + 11597979 \nu^{13} - 22413962 \nu^{11} - 248772494 \nu^{9} + \cdots - 5487054570 \nu ) / 2601928980 \) (-3079634v^15 + 11597979v^13 - 22413962v^11 - 248772494v^9 - 3735647454v^7 + 5354283043v^5 - 4822618578v^3 - 5487054570v) / 2601928980
\(\beta_{4}\)	\(=\)	\( ( 12591283 \nu^{15} - 57579817 \nu^{13} + 129110888 \nu^{11} + 952830506 \nu^{9} + \cdots + 3202119162 \nu ) / 7805786940 \) (12591283v^15 - 57579817v^13 + 129110888v^11 + 952830506v^9 + 14397572539v^7 - 34397644005v^5 + 38419343964v^3 + 3202119162v) / 7805786940
\(\beta_{5}\)	\(=\)	\( ( 14545077 \nu^{15} - 3765719 \nu^{14} - 55196585 \nu^{13} + 11685650 \nu^{12} + \cdots - 9136826568 ) / 5203857960 \) (14545077v^15 - 3765719v^14 - 55196585v^13 + 11685650v^12 + 105758504v^11 - 14001178v^10 + 1189121858v^9 - 333984496v^8 + 17514628229v^7 - 4702927103v^6 - 25836351497v^5 + 3800759034v^4 + 22083943080v^3 + 2325320910v^2 + 25498762434*v - 9136826568) / 5203857960
\(\beta_{6}\)	\(=\)	\( ( - 11395 \nu^{14} + 45868 \nu^{12} - 96938 \nu^{10} - 893276 \nu^{8} - 13578607 \nu^{6} + \cdots - 6275124 ) / 2256660 \) (-11395v^14 + 45868v^12 - 96938v^10 - 893276v^8 - 13578607v^6 + 23130228v^4 - 27991926*v^2 - 6275124) / 2256660
\(\beta_{7}\)	\(=\)	\( ( 14545077 \nu^{15} + 3765719 \nu^{14} - 55196585 \nu^{13} - 11685650 \nu^{12} + \cdots + 3932968608 ) / 5203857960 \) (14545077v^15 + 3765719v^14 - 55196585v^13 - 11685650v^12 + 105758504v^11 + 14001178v^10 + 1189121858v^9 + 333984496v^8 + 17514628229v^7 + 4702927103v^6 - 25836351497v^5 - 3800759034v^4 + 22083943080v^3 - 2325320910v^2 + 25498762434*v + 3932968608) / 5203857960
\(\beta_{8}\)	\(=\)	\( ( 18506049 \nu^{15} - 1441114 \nu^{14} - 72148880 \nu^{13} + 4446220 \nu^{12} + \cdots - 13615971048 ) / 5203857960 \) (18506049v^15 - 1441114v^14 - 72148880v^13 + 4446220v^12 + 136369838v^11 - 4803548v^10 + 1512241796v^9 - 147925256v^8 + 22114467473v^7 - 1782429418v^6 - 35425496144v^5 + 1441445244v^4 + 25614327990v^3 + 881758260v^2 + 31120049628*v - 13615971048) / 5203857960
\(\beta_{9}\)	\(=\)	\( ( 18506049 \nu^{15} + 1441114 \nu^{14} - 72148880 \nu^{13} - 4446220 \nu^{12} + \cdots + 18819829008 ) / 5203857960 \) (18506049v^15 + 1441114v^14 - 72148880v^13 - 4446220v^12 + 136369838v^11 + 4803548v^10 + 1512241796v^9 + 147925256v^8 + 22114467473v^7 + 1782429418v^6 - 35425496144v^5 - 1441445244v^4 + 25614327990v^3 - 881758260v^2 + 31120049628*v + 18819829008) / 5203857960
\(\beta_{10}\)	\(=\)	\( ( 60462680 \nu^{15} - 34694355 \nu^{14} - 274567721 \nu^{13} + 143608152 \nu^{12} + \cdots - 18889526916 ) / 15611573880 \) (60462680v^15 - 34694355v^14 - 274567721v^13 + 143608152v^12 + 617221546v^11 - 309169302v^10 + 4576799302v^9 - 2703731964v^8 + 69289811864v^7 - 40803930363v^6 - 161200544481v^5 + 75040099632v^4 + 178659278442v^3 - 84253473054v^2 + 14747147118*v - 18889526916) / 15611573880
\(\beta_{11}\)	\(=\)	\( ( - 60462680 \nu^{15} - 17310855 \nu^{14} + 274567721 \nu^{13} + 64632252 \nu^{12} + \cdots - 9644970996 ) / 15611573880 \) (-60462680v^15 - 17310855v^14 + 274567721v^13 + 64632252v^12 - 617221546v^11 - 140002782v^10 - 4576799302v^9 - 1365330444v^8 - 69289811864v^7 - 20907168303v^6 + 161200544481v^5 + 27457419132v^4 - 178659278442v^3 - 43028455014v^2 - 14747147118*v - 9644970996) / 15611573880
\(\beta_{12}\)	\(=\)	\( ( - 81296765 \nu^{15} + 47163030 \nu^{14} + 357230942 \nu^{13} - 210264972 \nu^{12} + \cdots + 25206770376 ) / 15611573880 \) (-81296765v^15 + 47163030v^14 + 357230942v^13 - 210264972v^12 - 785602642v^11 + 450833772v^10 - 6207754144v^9 + 3640498104v^8 - 94072523573v^7 + 54294184518v^6 + 202259564982v^5 - 121172690412v^4 - 218620670994v^3 + 112411745244v^2 - 17473193856*v + 25206770376) / 15611573880
\(\beta_{13}\)	\(=\)	\( ( - 81296765 \nu^{15} - 47163030 \nu^{14} + 357230942 \nu^{13} + 210264972 \nu^{12} + \cdots - 25206770376 ) / 15611573880 \) (-81296765v^15 - 47163030v^14 + 357230942v^13 + 210264972v^12 - 785602642v^11 - 450833772v^10 - 6207754144v^9 - 3640498104v^8 - 94072523573v^7 - 54294184518v^6 + 202259564982v^5 + 121172690412v^4 - 218620670994v^3 - 112411745244v^2 - 17473193856*v - 25206770376) / 15611573880
\(\beta_{14}\)	\(=\)	\( ( 23298266 \nu^{15} - 93544963 \nu^{13} + 176425470 \nu^{11} + 1897812810 \nu^{9} + \cdots + 37162486386 \nu ) / 2601928980 \) (23298266v^15 - 93544963v^13 + 176425470v^11 + 1897812810v^9 + 27613656994v^7 - 48610532639v^5 + 28420002066v^3 + 37162486386v) / 2601928980
\(\beta_{15}\)	\(=\)	\( ( - 15573353 \nu^{15} + 65944679 \nu^{13} - 141046900 \nu^{11} - 1206672370 \nu^{9} + \cdots - 2860280478 \nu ) / 1115112420 \) (-15573353v^15 + 65944679v^13 - 141046900v^11 - 1206672370v^9 - 18241907237v^7 + 35778301767v^5 - 37432034028v^3 - 2860280478v) / 1115112420

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{13} - \beta_{12} - 3\beta_{3} ) / 6 \) (b15 - b13 - b12 - 3*b3) / 6
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 + 1 ) / 2 \) (b11 + b10 + b7 - b6 - b5 + b2 + b1 + 1) / 2
\(\nu^{3}\)	\(=\)	\( ( -\beta_{14} + \beta_{11} - \beta_{10} + 2\beta_{9} + 2\beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + \beta _1 - 3 ) / 2 \) (-b14 + b11 - b10 + 2b9 + 2b8 - b7 - b5 + 5*b4 + b1 - 3) / 2
\(\nu^{4}\)	\(=\)	\( ( -7\beta_{13} + 7\beta_{12} + 2\beta_{11} + 2\beta_{10} + 32\beta_1 ) / 2 \) (-7b13 + 7b12 + 2b11 + 2b10 + 32*b1) / 2
\(\nu^{5}\)	\(=\)	\( ( - 5 \beta_{14} + \beta_{13} + \beta_{12} - 7 \beta_{11} + 7 \beta_{10} + 13 \beta_{9} + 13 \beta_{8} + \cdots - 22 ) / 2 \) (-5b14 + b13 + b12 - 7b11 + 7b10 + 13b9 + 13b8 - 9b7 - 9b5 - 27b4 - 2b3 - 7b1 - 22) / 2
\(\nu^{6}\)	\(=\)	\( ( - 11 \beta_{13} + 11 \beta_{12} + 45 \beta_{11} + 45 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} + \cdots - 68 ) / 2 \) (-11b13 + 11b12 + 45b11 + 45b10 + 11b9 - 11b8 - 45b7 - 29b6 + 45b5 - 29b2 + 57*b1 - 68) / 2
\(\nu^{7}\)	\(=\)	\( ( - 31 \beta_{15} - 2 \beta_{14} + 87 \beta_{13} + 87 \beta_{12} - 67 \beta_{11} + 67 \beta_{10} + \cdots - 58 ) / 2 \) (-31b15 - 2b14 + 87b13 + 87b12 - 67b11 + 67b10 + 13b9 + 13b8 - 45b7 - 45b5 - 28b4 - 149b3 - 67*b1 - 58) / 2
\(\nu^{8}\)	\(=\)	\( ( 289\beta_{9} - 289\beta_{8} - 190\beta_{7} + 190\beta_{5} - 56\beta_{2} - 1541 ) / 2 \) (289b9 - 289b8 - 190b7 + 190b5 - 56*b2 - 1541) / 2
\(\nu^{9}\)	\(=\)	\( ( - 205 \beta_{15} + 28 \beta_{14} + 589 \beta_{13} + 589 \beta_{12} - 479 \beta_{11} + 479 \beta_{10} + \cdots + 412 ) / 2 \) (-205b15 + 28b14 + 589b13 + 589b12 - 479b11 + 479b10 - 123b9 - 123b8 + 289b7 + 289b5 - 274b4 + 835b3 - 479*b1 + 412) / 2
\(\nu^{10}\)	\(=\)	\( ( 753 \beta_{13} - 753 \beta_{12} - 1877 \beta_{11} - 1877 \beta_{10} + 753 \beta_{9} - 753 \beta_{8} + \cdots - 4274 ) / 2 \) (753b13 - 753b12 - 1877b11 - 1877b10 + 753b9 - 753b8 - 1877b7 + 1109b6 + 1877b5 - 1109b2 - 3521*b1 - 4274) / 2
\(\nu^{11}\)	\(=\)	\( ( - 274 \beta_{15} + 1383 \beta_{14} + 1027 \beta_{13} + 1027 \beta_{12} - 1877 \beta_{11} + 1877 \beta_{10} + \cdots + 7396 ) / 2 \) (-274b15 + 1383b14 + 1027b13 + 1027b12 - 1877b11 + 1877b10 - 4013b9 - 4013b8 + 3383b7 + 3383b5 - 4753b4 + 2328b3 - 1877*b1 + 7396) / 2
\(\nu^{12}\)	\(=\)	\( ( 12341\beta_{13} - 12341\beta_{12} - 11422\beta_{11} - 11422\beta_{10} + 4656\beta_{6} - 53908\beta_1 ) / 2 \) (12341b13 - 12341b12 - 11422b11 - 11422b10 + 4656b6 - 53908b1) / 2
\(\nu^{13}\)	\(=\)	\( ( 2328 \beta_{15} + 9409 \beta_{14} - 8039 \beta_{13} - 8039 \beta_{12} + 12341 \beta_{11} - 12341 \beta_{10} + \cdots + 51224 ) / 2 \) (2328b15 + 9409b14 - 8039b13 - 8039b12 + 12341b11 - 12341b10 - 27461b9 - 27461b8 + 23763b7 + 23763b5 + 27519b4 + 18406b3 + 12341*b1 + 51224) / 2
\(\nu^{14}\)	\(=\)	\( ( 42169 \beta_{13} - 42169 \beta_{12} - 82029 \beta_{11} - 82029 \beta_{10} - 42169 \beta_{9} + \cdots + 234634 ) / 2 \) (42169b13 - 42169b12 - 82029b11 - 82029b10 - 42169b9 + 42169b8 + 82029b7 + 45925b6 - 82029b5 + 45925b2 - 192465*b1 + 234634) / 2
\(\nu^{15}\)	\(=\)	\( ( 64331 \beta_{15} + 18406 \beta_{14} - 188529 \beta_{13} - 188529 \beta_{12} + 166367 \beta_{11} + \cdots + 142604 ) / 2 \) (64331b15 + 18406b14 - 188529b13 - 188529b12 + 166367b11 - 166367b10 - 60575b9 - 60575b8 + 82029b7 + 82029b5 + 139556b4 + 162277b3 + 166367*b1 + 142604) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 881.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 24 T^{6} + \cdots + 72)^{2} \) (T^8 - 24T^6 + 118T^4 - 168*T^2 + 72)^2
$7$	\( (T^{8} + 4 T^{7} + \cdots + 2401)^{2} \) (T^8 + 4T^7 + 6T^6 + 28T^5 + 122T^4 + 196T^3 + 294T^2 + 1372*T + 2401)^2
$11$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$13$	\( T^{16} \) T^16
$17$	\( (T^{8} - 84 T^{6} + \cdots + 83232)^{2} \) (T^8 - 84T^6 + 2386T^4 - 25872*T^2 + 83232)^2
$19$	\( (T^{8} + 104 T^{6} + \cdots + 72)^{2} \) (T^8 + 104T^6 + 2710T^4 + 1320*T^2 + 72)^2
$23$	\( (T^{8} + 120 T^{6} + \cdots + 331776)^{2} \) (T^8 + 120T^6 + 4624T^4 + 69120*T^2 + 331776)^2
$29$	\( (T^{8} + 108 T^{6} + \cdots + 5184)^{2} \) (T^8 + 108T^6 + 2932T^4 + 7776*T^2 + 5184)^2
$31$	\( (T^{8} + 268 T^{6} + \cdots + 6139008)^{2} \) (T^8 + 268T^6 + 23458T^4 + 727008*T^2 + 6139008)^2
$37$	\( (T^{4} + 4 T^{3} + \cdots + 544)^{4} \) (T^4 + 4T^3 - 52T^2 - 128*T + 544)^4
$41$	\( (T^{8} - 132 T^{6} + \cdots + 23328)^{2} \) (T^8 - 132T^6 + 3154T^4 - 18576*T^2 + 23328)^2
$43$	\( (T^{4} - 12 T^{3} + \cdots - 16)^{4} \) (T^4 - 12T^3 + 26T^2 + 64*T - 16)^4
$47$	\( (T^{8} - 196 T^{6} + \cdots + 288)^{2} \) (T^8 - 196T^6 + 7474T^4 - 3120*T^2 + 288)^2
$53$	\( (T^{8} + 260 T^{6} + \cdots + 8714304)^{2} \) (T^8 + 260T^6 + 22516T^4 + 776736*T^2 + 8714304)^2
$59$	\( (T^{8} - 96 T^{6} + \cdots + 18432)^{2} \) (T^8 - 96T^6 + 1888T^4 - 10752*T^2 + 18432)^2
$61$	\( (T^{8} + 528 T^{6} + \cdots + 294912)^{2} \) (T^8 + 528T^6 + 89152T^4 + 4706304*T^2 + 294912)^2
$67$	\( (T^{4} - 4 T^{3} - 70 T^{2} + \cdots + 16)^{4} \) (T^4 - 4T^3 - 70T^2 + 176*T + 16)^4
$71$	\( (T^{8} + 312 T^{6} + \cdots + 82944)^{2} \) (T^8 + 312T^6 + 25552T^4 + 232704*T^2 + 82944)^2
$73$	\( (T^{8} + 196 T^{6} + \cdots + 288)^{2} \) (T^8 + 196T^6 + 7474T^4 + 3120*T^2 + 288)^2
$79$	\( (T^{4} + 4 T^{3} + \cdots - 292)^{4} \) (T^4 + 4T^3 - 176T^2 - 840*T - 292)^4
$83$	\( (T^{8} - 348 T^{6} + \cdots + 1492992)^{2} \) (T^8 - 348T^6 + 33282T^4 - 549504*T^2 + 1492992)^2
$89$	\( (T^{8} - 420 T^{6} + \cdots + 1936512)^{2} \) (T^8 - 420T^6 + 40708T^4 - 889152*T^2 + 1936512)^2
$97$	\( (T^{8} + 480 T^{6} + \cdots + 18874368)^{2} \) (T^8 + 480T^6 + 69760T^4 + 3047424*T^2 + 18874368)^2
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\(n\)	\(155\)	\(199\)	\(1135\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)