LMFDB - Newform orbit 2156.2.c.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,2,Mod(1077,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2156.1077");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2156.c (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:	$17.2157466758$
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} + 2x^{14} - 17x^{12} - 343x^{10} + 490x^{8} - 16807x^{6} - 40817x^{4} + 235298x^{2} + 5764801 $ x^16 + 2x^14 - 17x^12 - 343x^10 + 490x^8 - 16807x^6 - 40817x^4 + 235298*x^2 + 5764801
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 308)
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_{2} q^{3} - \beta_{3} q^{5} + \beta_{5} q^{9}+O(q^{10}) $ q - b2 * q^3 - b3 * q^5 + b5 * q^9	$ q - \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_{5} q^{9} - \beta_{11} q^{11} - \beta_{14} q^{13} + q^{15} - \beta_{7} q^{17} - \beta_{7} q^{19} + (\beta_{11} - \beta_{10} - \beta_{5}) q^{23} + (\beta_{11} - \beta_{10} + \cdots + \beta_{4}) q^{25}+ \cdots + ( - \beta_{13} + \beta_{12} - \beta_{11} + \cdots - 3) q^{99}+O(q^{100}) $ q - b2 * q^3 - b3 * q^5 + b5 * q^9 - b11 * q^11 - b14 * q^13 + q^15 - b7 * q^17 - b7 * q^19 + (b11 - b10 - b5) * q^23 + (b11 - b10 - 2b5 + b4) q^25 + (b3 + b2 + b1) * q^27 + b13 * q^29 + (-b9 - b8 + b3 + b1) * q^31 + (-b9 + b7 - b2 + b1) * q^33 + (b11 - b10) * q^37 + (b13 + b6) * q^39 + b14 * q^41 + b6 * q^43 + (-3b3 - b2) q^45 + (-b9 - b8 + b3 - b2) * q^47 + (-b12 - b11 - b10) * q^51 + (2b5 + b4) q^53 + (b14 + b9 + b1) * q^55 + (-b12 - b11 - b10) * q^57 + (-b9 - b8 - 2b3 - b2 + b1) q^59 + (-b9 + b8) * q^61 + (-b13 + b12 - 2b11 - 2b10 - b6) * q^65 + (-b11 + b10 - 2b4 + 2) q^67 + (b9 + b8 - b3 - 2b2 - 3b1) * q^69 + (b11 - b10 - b5 + 2b4 - 5) q^71 + (-b15 - b14 - b9) * q^73 + (-b3 - 5b2) q^75 + (b13 - b11 - b10) * q^79 + (2b5 - b4 + 2) q^81 + (b9 - b8 - b7) * q^83 + (-b13 + b12 - b11 - b10) * q^85 + (b15 + b14 + b8) * q^87 + (-b9 - b8 - 3b3 + 2b2 - b1) * q^89 + (b11 - b10 - 3b4 - 1) q^93 + (-b13 + b12 - b11 - b10) * q^95 + (-b9 - b8 + 2b3 + b2 + 2b1) * q^97 + (-b13 + b12 - b11 + b10 - b6 + b5 - 2b4 - 3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} + 2 q^{11} + 16 q^{15} + 4 q^{23} + 12 q^{25} - 4 q^{37} - 16 q^{53} + 36 q^{67} - 76 q^{71} + 16 q^{81} - 20 q^{93} - 52 q^{99}+O(q^{100}) $ 16 * q - 8 * q^9 + 2 * q^11 + 16 * q^15 + 4 * q^23 + 12 * q^25 - 4 * q^37 - 16 * q^53 + 36 * q^67 - 76 * q^71 + 16 * q^81 - 20 * q^93 - 52 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 2x^{14} - 17x^{12} - 343x^{10} + 490x^{8} - 16807x^{6} - 40817x^{4} + 235298x^{2} + 5764801 $ x^16 + 2*x^14 - 17*x^12 - 343*x^10 + 490*x^8 - 16807*x^6 - 40817*x^4 + 235298*x^2 + 5764801 :

$\beta_{1}$	$=$	$ ( -3\nu^{14} - 5\nu^{12} + 298\nu^{10} - 409\nu^{8} + 2205\nu^{6} + 25774\nu^{4} + 328937\nu^{2} - 5332621 ) / 316932 $ (-3v^14 - 5v^12 + 298v^10 - 409v^8 + 2205v^6 + 25774v^4 + 328937*v^2 - 5332621) / 316932
$\beta_{2}$	$=$	$ ( 445 \nu^{14} + 2311 \nu^{12} - 18786 \nu^{10} + 37583 \nu^{8} - 234367 \nu^{6} - 4048086 \nu^{4} + \cdots + 342476239 ) / 46589004 $ (445v^14 + 2311v^12 - 18786v^10 + 37583v^8 - 234367v^6 - 4048086v^4 - 61223099*v^2 + 342476239) / 46589004
$\beta_{3}$	$=$	$ ( 541 \nu^{14} + 1327 \nu^{12} - 39234 \nu^{10} + 20531 \nu^{8} - 253183 \nu^{6} - 5517498 \nu^{4} + \cdots + 696835027 ) / 46589004 $ (541v^14 + 1327v^12 - 39234v^10 + 20531v^8 - 253183v^6 - 5517498v^4 - 43964711*v^2 + 696835027) / 46589004
$\beta_{4}$	$=$	$ ( 87 \nu^{14} - 512 \nu^{12} + 579 \nu^{10} - 11319 \nu^{8} + 51548 \nu^{6} - 1663893 \nu^{4} + \cdots + 14941423 ) / 3882417 $ (87v^14 - 512v^12 + 579v^10 - 11319v^8 + 51548v^6 - 1663893v^4 + 5575122*v^2 + 14941423) / 3882417
$\beta_{5}$	$=$	$ ( - 100 \nu^{14} + 731 \nu^{12} - 554 \nu^{10} + 10241 \nu^{8} - 96677 \nu^{6} + 1716715 \nu^{4} + \cdots - 21294469 ) / 3882417 $ (-100v^14 + 731v^12 - 554v^10 + 10241v^8 - 96677v^6 + 1716715v^4 - 9716847*v^2 - 21294469) / 3882417
$\beta_{6}$	$=$	$ ( - 87 \nu^{15} - 811 \nu^{13} - 4597 \nu^{11} + 14259 \nu^{9} + 106232 \nu^{7} + \cdots + 53294997 \nu ) / 27176919 $ (-87v^15 - 811v^13 - 4597v^11 + 14259v^9 + 106232v^7 + 1200500v^5 + 5668761v^3 + 53294997v) / 27176919
$\beta_{7}$	$=$	$ ( 1349 \nu^{15} + 2159 \nu^{13} - 69630 \nu^{11} - 23765 \nu^{9} - 983675 \nu^{7} + \cdots + 1253550095 \nu ) / 326123028 $ (1349v^15 + 2159v^13 - 69630v^11 - 23765v^9 - 983675v^7 - 7793646v^5 - 144710671v^3 + 1253550095v) / 326123028
$\beta_{8}$	$=$	$ ( 3603 \nu^{15} + 1127 \nu^{14} - 879 \nu^{13} + 2597 \nu^{12} - 127842 \nu^{11} + \cdots + 4367248529 ) / 652246056 $ (3603v^15 + 1127v^14 - 879v^13 + 2597v^12 - 127842v^11 - 251370v^10 - 39543v^9 - 34643v^8 - 402633v^7 - 5371037v^6 - 65878638v^5 + 25109658v^4 - 125180937v^3 - 120119629v^2 + 2388392349*v + 4367248529) / 652246056
$\beta_{9}$	$=$	$ ( - 3603 \nu^{15} + 1127 \nu^{14} + 879 \nu^{13} + 2597 \nu^{12} + 127842 \nu^{11} + \cdots + 4367248529 ) / 652246056 $ (-3603v^15 + 1127v^14 + 879v^13 + 2597v^12 + 127842v^11 - 251370v^10 + 39543v^9 - 34643v^8 + 402633v^7 - 5371037v^6 + 65878638v^5 + 25109658v^4 + 125180937v^3 - 120119629v^2 - 2388392349*v + 4367248529) / 652246056
$\beta_{10}$	$=$	$ ( 687 \nu^{15} + 1022 \nu^{14} - 5045 \nu^{13} - 6188 \nu^{12} + 30706 \nu^{11} - 224 \nu^{10} + \cdots + 220709524 ) / 108707676 $ (687v^15 + 1022v^14 - 5045v^13 - 6188v^12 + 30706v^11 - 224v^10 - 100107v^9 - 143374v^8 + 843241v^7 + 1105832v^6 - 14439614v^5 - 16269176v^4 + 75725139v^3 + 60404358v^2 - 315887565*v + 220709524) / 108707676
$\beta_{11}$	$=$	$ ( 687 \nu^{15} - 1022 \nu^{14} - 5045 \nu^{13} + 6188 \nu^{12} + 30706 \nu^{11} + 224 \nu^{10} + \cdots - 220709524 ) / 108707676 $ (687v^15 - 1022v^14 - 5045v^13 + 6188v^12 + 30706v^11 + 224v^10 - 100107v^9 + 143374v^8 + 843241v^7 - 1105832v^6 - 14439614v^5 + 16269176v^4 + 75725139v^3 - 60404358v^2 - 315887565*v - 220709524) / 108707676
$\beta_{12}$	$=$	$ ( 277 \nu^{15} - 1798 \nu^{13} + 7394 \nu^{11} - 55027 \nu^{9} + 354221 \nu^{7} + \cdots - 6000099 \nu ) / 27176919 $ (277v^15 - 1798v^13 + 7394v^11 - 55027v^9 + 354221v^7 - 5707177v^5 + 23988391v^3 - 6000099v) / 27176919
$\beta_{13}$	$=$	$ ( 281 \nu^{15} - 3799 \nu^{13} + 10511 \nu^{11} - 75068 \nu^{9} + 317765 \nu^{7} + \cdots - 21882714 \nu ) / 27176919 $ (281v^15 - 3799v^13 + 10511v^11 - 75068v^9 + 317765v^7 - 6557131v^5 + 63355187v^3 - 21882714v) / 27176919
$\beta_{14}$	$=$	$ ( - 2795 \nu^{15} + 1417 \nu^{13} + 135960 \nu^{11} - 36799 \nu^{9} + 87857 \nu^{7} + \cdots - 2658279155 \nu ) / 163061514 $ (-2795v^15 + 1417v^13 + 135960v^11 - 36799v^9 + 87857v^7 + 31131366v^5 + 93612589v^3 - 2658279155v) / 163061514
$\beta_{15}$	$=$	$ ( - 14071 \nu^{15} - 1127 \nu^{14} + 3659 \nu^{13} - 2597 \nu^{12} + 689370 \nu^{11} + \cdots - 4367248529 ) / 652246056 $ (-14071v^15 - 1127v^14 + 3659v^13 - 2597v^12 + 689370v^11 + 251370v^10 - 874013v^9 + 34643v^8 + 5290285v^7 + 5371037v^6 + 170811942v^5 - 25109658v^4 + 758574341v^3 + 120119629v^2 - 15484608433*v - 4367248529) / 652246056

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

$\nu$	$=$	$ ( -2\beta_{15} + \beta_{14} + 3\beta_{12} - 2\beta_{11} - 2\beta_{10} + \beta_{9} - 3\beta_{8} - 3\beta_{7} - \beta_{6} ) / 14 $ (-2b15 + b14 + 3b12 - 2b11 - 2b10 + b9 - 3b8 - 3b7 - b6) / 14
$\nu^{2}$	$=$	$ ( 2\beta_{11} - 2\beta_{10} - 2\beta_{5} - \beta_{4} + 4\beta_{3} - \beta_{2} + 3\beta _1 - 1 ) / 2 $ (2b11 - 2b10 - 2b5 - b4 + 4b3 - b2 + 3*b1 - 1) / 2
$\nu^{3}$	$=$	$ ( 5 \beta_{15} - 20 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + \cdots - 15 \beta_{6} ) / 14 $ (5b15 - 20b14 + 7b13 - 11b12 - 2b11 - 2b10 + 8b9 - 3b8 - 38b7 - 15b6) / 14
$\nu^{4}$	$=$	$ ( - 8 \beta_{11} + 8 \beta_{10} + 11 \beta_{9} + 11 \beta_{8} - 7 \beta_{5} - 14 \beta_{4} - 13 \beta_{3} + \cdots + 4 ) / 2 $ (-8b11 + 8b10 + 11b9 + 11b8 - 7b5 - 14b4 - 13b3 + 15b2 + 5*b1 + 4) / 2
$\nu^{5}$	$=$	$ ( 19 \beta_{15} - 90 \beta_{14} - 39 \beta_{12} + 40 \beta_{11} + 40 \beta_{10} + 169 \beta_{9} + \cdots - 22 \beta_{6} ) / 14 $ (19b15 - 90b14 - 39b12 + 40b11 + 40b10 + 169b9 - 150b8 + 109b7 - 22*b6) / 14
$\nu^{6}$	$=$	$ ( - 111 \beta_{11} + 111 \beta_{10} - 71 \beta_{9} - 71 \beta_{8} - 6 \beta_{5} - 108 \beta_{4} + \cdots + 199 ) / 2 $ (-111b11 + 111b10 - 71b9 - 71b8 - 6b5 - 108b4 + 80b3 - 40b2 + 13*b1 + 199) / 2
$\nu^{7}$	$=$	$ ( 243 \beta_{15} - 797 \beta_{14} - 714 \beta_{13} + 1116 \beta_{12} - 16 \beta_{11} + \cdots + 475 \beta_{6} ) / 14 $ (243b15 - 797b14 - 714b13 + 1116b12 - 16b11 - 16b10 + 582b9 - 339b8 - 1354b7 + 475b6) / 14
$\nu^{8}$	$=$	$ ( 772 \beta_{11} - 772 \beta_{10} - 259 \beta_{9} - 259 \beta_{8} - 793 \beta_{5} - 365 \beta_{4} + \cdots - 1163 ) / 2 $ (772b11 - 772b10 - 259b9 - 259b8 - 793b5 - 365b4 - 675b3 + 482b2 - 676*b1 - 1163) / 2
$\nu^{9}$	$=$	$ ( - 1821 \beta_{15} - 213 \beta_{14} - 1001 \beta_{13} - 3383 \beta_{12} + 3247 \beta_{11} + \cdots + 612 \beta_{6} ) / 7 $ (-1821b15 - 213b14 - 1001b13 - 3383b12 + 3247b11 + 3247b10 + 438b9 - 2259b8 - 5388b7 + 612b6) / 7
$\nu^{10}$	$=$	$ ( 734 \beta_{11} - 734 \beta_{10} + 683 \beta_{9} + 683 \beta_{8} - 623 \beta_{5} + 413 \beta_{4} + \cdots + 30209 ) / 2 $ (734b11 - 734b10 + 683b9 + 683b8 - 623b5 + 413b4 - 4395b3 + 4334b2 + 446*b1 + 30209) / 2
$\nu^{11}$	$=$	$ ( - 39741 \beta_{15} + 33461 \beta_{14} - 6762 \beta_{13} + 36522 \beta_{12} - 8164 \beta_{11} + \cdots - 23381 \beta_{6} ) / 14 $ (-39741b15 + 33461b14 - 6762b13 + 36522b12 - 8164b11 - 8164b10 + 21372b9 - 61113b8 - 24692b7 - 23381b6) / 14
$\nu^{12}$	$=$	$ ( 20721 \beta_{11} - 20721 \beta_{10} - 9101 \beta_{9} - 9101 \beta_{8} - 18066 \beta_{5} - 15606 \beta_{4} + \cdots + 46133 ) / 2 $ (20721b11 - 20721b10 - 9101b9 - 9101b8 - 18066b5 - 15606b4 + 86348b3 - 7138b2 + 66457*b1 + 46133) / 2
$\nu^{13}$	$=$	$ ( 77201 \beta_{15} - 296232 \beta_{14} - 30114 \beta_{13} + 67371 \beta_{12} - 114452 \beta_{11} + \cdots - 337366 \beta_{6} ) / 14 $ (77201b15 - 296232b14 - 30114b13 + 67371b12 - 114452b11 - 114452b10 + 96167b9 - 18966b8 - 673249b7 - 337366b6) / 14
$\nu^{14}$	$=$	$ ( 2102 \beta_{11} - 2102 \beta_{10} + 160643 \beta_{9} + 160643 \beta_{8} - 207503 \beta_{5} + \cdots - 401668 ) / 2 $ (2102b11 - 2102b10 + 160643b9 + 160643b8 - 207503b5 - 192508b4 - 102763b3 + 366519b2 + 195863*b1 - 401668) / 2
$\nu^{15}$	$=$	$ ( 442825 \beta_{15} - 1982098 \beta_{14} - 105833 \beta_{13} - 1721171 \beta_{12} + \cdots - 1105917 \beta_{6} ) / 14 $ (442825b15 - 1982098b14 - 105833b13 - 1721171b12 + 1739554b11 + 1739554b10 + 2294356b9 - 1851531b8 + 1351466b7 - 1105917b6) / 14

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times$.

$n$	$981$	$1079$	$1277$
$\chi(n)$	$-1$	$1$	$-1$

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} $

1077.1

2.64407	+	0.0942693i
−2.64407	−	0.0942693i
−0.938723	+	2.47362i
0.938723	−	2.47362i
2.16092	−	1.52658i
−2.16092	+	1.52658i
0.978837	−	2.45802i
−0.978837	+	2.45802i
−0.978837	−	2.45802i
0.978837	+	2.45802i
−2.16092	−	1.52658i
2.16092	+	1.52658i
0.938723	+	2.47362i
−0.938723	−	2.47362i
−2.64407	+	0.0942693i
2.64407	−	0.0942693i

−

2.73846i

0.365169i

−4.49914

1077.2

−

2.73846i

0.365169i

−4.49914

1077.3

−

2.47749i

0.403634i

−3.13796

1077.4

−

2.47749i

0.403634i

−3.13796

1077.5

−

0.535970i

1.86578i

2.71274

1077.6

−

0.535970i

1.86578i

2.71274

1077.7

−

0.275005i

3.63629i

2.92437

1077.8

−

0.275005i

3.63629i

2.92437

1077.9

0.275005i

−

3.63629i

2.92437

1077.10

0.275005i

−

3.63629i

2.92437

1077.11

0.535970i

−

1.86578i

2.71274

1077.12

0.535970i

−

1.86578i

2.71274

1077.13

2.47749i

−

0.403634i

−3.13796

1077.14

2.47749i

−

0.403634i

−3.13796

1077.15

2.73846i

−

0.365169i

−4.49914

1077.16

2.73846i

−

0.365169i

−4.49914

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
11.b	odd	2	1	inner
77.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2156.2.c.c		16
7.b	odd	2	1	inner	2156.2.c.c		16
7.c	even	3	1		308.2.q.a	✓	16
7.c	even	3	1		2156.2.q.c		16
7.d	odd	6	1		308.2.q.a	✓	16
7.d	odd	6	1		2156.2.q.c		16
11.b	odd	2	1	inner	2156.2.c.c		16
21.g	even	6	1		2772.2.cs.a		16
21.h	odd	6	1		2772.2.cs.a		16
28.f	even	6	1		1232.2.bn.c		16
28.g	odd	6	1		1232.2.bn.c		16
77.b	even	2	1	inner	2156.2.c.c		16
77.h	odd	6	1		308.2.q.a	✓	16
77.h	odd	6	1		2156.2.q.c		16
77.i	even	6	1		308.2.q.a	✓	16
77.i	even	6	1		2156.2.q.c		16
231.k	odd	6	1		2772.2.cs.a		16
231.l	even	6	1		2772.2.cs.a		16
308.m	odd	6	1		1232.2.bn.c		16
308.n	even	6	1		1232.2.bn.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
308.2.q.a	✓	16	7.c	even	3	1
308.2.q.a	✓	16	7.d	odd	6	1
308.2.q.a	✓	16	77.h	odd	6	1
308.2.q.a	✓	16	77.i	even	6	1
1232.2.bn.c		16	28.f	even	6	1
1232.2.bn.c		16	28.g	odd	6	1
1232.2.bn.c		16	308.m	odd	6	1
1232.2.bn.c		16	308.n	even	6	1
2156.2.c.c		16	1.a	even	1	1	trivial
2156.2.c.c		16	7.b	odd	2	1	inner
2156.2.c.c		16	11.b	odd	2	1	inner
2156.2.c.c		16	77.b	even	2	1	inner
2156.2.q.c		16	7.c	even	3	1
2156.2.q.c		16	7.d	odd	6	1
2156.2.q.c		16	77.h	odd	6	1
2156.2.q.c		16	77.i	even	6	1
2772.2.cs.a		16	21.g	even	6	1
2772.2.cs.a		16	21.h	odd	6	1
2772.2.cs.a		16	231.k	odd	6	1
2772.2.cs.a		16	231.l	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 14T_{3}^{6} + 51T_{3}^{4} + 17T_{3}^{2} + 1 $ T3^8 + 14*T3^6 + 51*T3^4 + 17*T3^2 + 1 acting on $S_{2}^{\mathrm{new}}(2156, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 14 T^{6} + 51 T^{4} + \cdots + 1)^{2} $ (T^8 + 14T^6 + 51T^4 + 17*T^2 + 1)^2
$5$	$ (T^{8} + 17 T^{6} + 51 T^{4} + \cdots + 1)^{2} $ (T^8 + 17T^6 + 51T^4 + 14*T^2 + 1)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} - T^{7} + \cdots + 14641)^{2} $ (T^8 - T^7 + 8T^6 + 11T^5 - 26T^4 + 121T^3 + 968T^2 - 1331T + 14641)^2
$13$	$ (T^{8} - 95 T^{6} + \cdots + 240100)^{2} $ (T^8 - 95T^6 + 3234T^4 - 46697*T^2 + 240100)^2
$17$	$ (T^{8} - 92 T^{6} + \cdots + 117649)^{2} $ (T^8 - 92T^6 + 2667T^4 - 30233*T^2 + 117649)^2
$19$	$ (T^{8} - 92 T^{6} + \cdots + 117649)^{2} $ (T^8 - 92T^6 + 2667T^4 - 30233*T^2 + 117649)^2
$23$	$ (T^{4} - T^{3} - 33 T^{2} + \cdots + 211)^{4} $ (T^4 - T^3 - 33T^2 + 20T + 211)^4
$29$	$ (T^{8} + 195 T^{6} + \cdots + 1382976)^{2} $ (T^8 + 195T^6 + 12012T^4 + 255339*T^2 + 1382976)^2
$31$	$ (T^{8} + 134 T^{6} + \cdots + 790321)^{2} $ (T^8 + 134T^6 + 6135T^4 + 116885*T^2 + 790321)^2
$37$	$ (T^{4} + T^{3} - 36 T^{2} + \cdots + 40)^{4} $ (T^4 + T^3 - 36T^2 - 44T + 40)^4
$41$	$ (T^{8} - 95 T^{6} + \cdots + 240100)^{2} $ (T^8 - 95T^6 + 3234T^4 - 46697*T^2 + 240100)^2
$43$	$ (T^{8} + 208 T^{6} + \cdots + 9604)^{2} $ (T^8 + 208T^6 + 8211T^4 + 17983*T^2 + 9604)^2
$47$	$ (T^{8} + 158 T^{6} + \cdots + 39601)^{2} $ (T^8 + 158T^6 + 7803T^4 + 118841*T^2 + 39601)^2
$53$	$ (T^{4} + 4 T^{3} + \cdots + 931)^{4} $ (T^4 + 4T^3 - 123T^2 + 217*T + 931)^4
$59$	$ (T^{8} + 209 T^{6} + \cdots + 2337841)^{2} $ (T^8 + 209T^6 + 14211T^4 + 339542*T^2 + 2337841)^2
$61$	$ (T^{8} - 309 T^{6} + \cdots + 12446784)^{2} $ (T^8 - 309T^6 + 30996T^4 - 1100736*T^2 + 12446784)^2
$67$	$ (T^{4} - 9 T^{3} + \cdots + 264)^{4} $ (T^4 - 9T^3 - 48T^2 + 276*T + 264)^4
$71$	$ (T^{4} + 19 T^{3} + \cdots + 16)^{4} $ (T^4 + 19T^3 + 78T^2 - 113*T + 16)^4
$73$	$ (T^{8} - 441 T^{6} + \cdots + 26471025)^{2} $ (T^8 - 441T^6 + 64239T^4 - 3284568*T^2 + 26471025)^2
$79$	$ (T^{8} + 334 T^{6} + \cdots + 5764801)^{2} $ (T^8 + 334T^6 + 32781T^4 + 823543*T^2 + 5764801)^2
$83$	$ (T^{8} - 263 T^{6} + \cdots + 38416)^{2} $ (T^8 - 263T^6 + 9996T^4 - 39641*T^2 + 38416)^2
$89$	$ (T^{8} + 378 T^{6} + \cdots + 3381921)^{2} $ (T^8 + 378T^6 + 33891T^4 + 896661*T^2 + 3381921)^2
$97$	$ (T^{8} + 191 T^{6} + \cdots + 283024)^{2} $ (T^8 + 191T^6 + 11898T^4 + 247715*T^2 + 283024)^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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_{5} q^{9}+O(q^{10}) \) q - b2 * q^3 - b3 * q^5 + b5 * q^9	\( q - \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_{5} q^{9} - \beta_{11} q^{11} - \beta_{14} q^{13} + q^{15} - \beta_{7} q^{17} - \beta_{7} q^{19} + (\beta_{11} - \beta_{10} - \beta_{5}) q^{23} + (\beta_{11} - \beta_{10} + \cdots + \beta_{4}) q^{25}+ \cdots + ( - \beta_{13} + \beta_{12} - \beta_{11} + \cdots - 3) q^{99}+O(q^{100}) \) q - b2 * q^3 - b3 * q^5 + b5 * q^9 - b11 * q^11 - b14 * q^13 + q^15 - b7 * q^17 - b7 * q^19 + (b11 - b10 - b5) * q^23 + (b11 - b10 - 2b5 + b4) q^25 + (b3 + b2 + b1) * q^27 + b13 * q^29 + (-b9 - b8 + b3 + b1) * q^31 + (-b9 + b7 - b2 + b1) * q^33 + (b11 - b10) * q^37 + (b13 + b6) * q^39 + b14 * q^41 + b6 * q^43 + (-3b3 - b2) q^45 + (-b9 - b8 + b3 - b2) * q^47 + (-b12 - b11 - b10) * q^51 + (2b5 + b4) q^53 + (b14 + b9 + b1) * q^55 + (-b12 - b11 - b10) * q^57 + (-b9 - b8 - 2b3 - b2 + b1) q^59 + (-b9 + b8) * q^61 + (-b13 + b12 - 2b11 - 2b10 - b6) * q^65 + (-b11 + b10 - 2b4 + 2) q^67 + (b9 + b8 - b3 - 2b2 - 3b1) * q^69 + (b11 - b10 - b5 + 2b4 - 5) q^71 + (-b15 - b14 - b9) * q^73 + (-b3 - 5b2) q^75 + (b13 - b11 - b10) * q^79 + (2b5 - b4 + 2) q^81 + (b9 - b8 - b7) * q^83 + (-b13 + b12 - b11 - b10) * q^85 + (b15 + b14 + b8) * q^87 + (-b9 - b8 - 3b3 + 2b2 - b1) * q^89 + (b11 - b10 - 3b4 - 1) q^93 + (-b13 + b12 - b11 - b10) * q^95 + (-b9 - b8 + 2b3 + b2 + 2b1) * q^97 + (-b13 + b12 - b11 + b10 - b6 + b5 - 2b4 - 3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} + 2 q^{11} + 16 q^{15} + 4 q^{23} + 12 q^{25} - 4 q^{37} - 16 q^{53} + 36 q^{67} - 76 q^{71} + 16 q^{81} - 20 q^{93} - 52 q^{99}+O(q^{100}) \) 16 * q - 8 * q^9 + 2 * q^11 + 16 * q^15 + 4 * q^23 + 12 * q^25 - 4 * q^37 - 16 * q^53 + 36 * q^67 - 76 * q^71 + 16 * q^81 - 20 * q^93 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	2156.2.c.c

Level	$2156$
Weight	$2$
Character orbit	2156.c
Analytic conductor	$17.216$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(17.2157466758\)
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:	\( x^{16} + 2x^{14} - 17x^{12} - 343x^{10} + 490x^{8} - 16807x^{6} - 40817x^{4} + 235298x^{2} + 5764801 \) x^16 + 2x^14 - 17x^12 - 343x^10 + 490x^8 - 16807x^6 - 40817x^4 + 235298*x^2 + 5764801
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{8}\cdot 7^{2} \)
Twist minimal:	no (minimal twist has level 308)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -3\nu^{14} - 5\nu^{12} + 298\nu^{10} - 409\nu^{8} + 2205\nu^{6} + 25774\nu^{4} + 328937\nu^{2} - 5332621 ) / 316932 \) (-3v^14 - 5v^12 + 298v^10 - 409v^8 + 2205v^6 + 25774v^4 + 328937*v^2 - 5332621) / 316932
\(\beta_{2}\)	\(=\)	\( ( 445 \nu^{14} + 2311 \nu^{12} - 18786 \nu^{10} + 37583 \nu^{8} - 234367 \nu^{6} - 4048086 \nu^{4} + \cdots + 342476239 ) / 46589004 \) (445v^14 + 2311v^12 - 18786v^10 + 37583v^8 - 234367v^6 - 4048086v^4 - 61223099*v^2 + 342476239) / 46589004
\(\beta_{3}\)	\(=\)	\( ( 541 \nu^{14} + 1327 \nu^{12} - 39234 \nu^{10} + 20531 \nu^{8} - 253183 \nu^{6} - 5517498 \nu^{4} + \cdots + 696835027 ) / 46589004 \) (541v^14 + 1327v^12 - 39234v^10 + 20531v^8 - 253183v^6 - 5517498v^4 - 43964711*v^2 + 696835027) / 46589004
\(\beta_{4}\)	\(=\)	\( ( 87 \nu^{14} - 512 \nu^{12} + 579 \nu^{10} - 11319 \nu^{8} + 51548 \nu^{6} - 1663893 \nu^{4} + \cdots + 14941423 ) / 3882417 \) (87v^14 - 512v^12 + 579v^10 - 11319v^8 + 51548v^6 - 1663893v^4 + 5575122*v^2 + 14941423) / 3882417
\(\beta_{5}\)	\(=\)	\( ( - 100 \nu^{14} + 731 \nu^{12} - 554 \nu^{10} + 10241 \nu^{8} - 96677 \nu^{6} + 1716715 \nu^{4} + \cdots - 21294469 ) / 3882417 \) (-100v^14 + 731v^12 - 554v^10 + 10241v^8 - 96677v^6 + 1716715v^4 - 9716847*v^2 - 21294469) / 3882417
\(\beta_{6}\)	\(=\)	\( ( - 87 \nu^{15} - 811 \nu^{13} - 4597 \nu^{11} + 14259 \nu^{9} + 106232 \nu^{7} + \cdots + 53294997 \nu ) / 27176919 \) (-87v^15 - 811v^13 - 4597v^11 + 14259v^9 + 106232v^7 + 1200500v^5 + 5668761v^3 + 53294997v) / 27176919
\(\beta_{7}\)	\(=\)	\( ( 1349 \nu^{15} + 2159 \nu^{13} - 69630 \nu^{11} - 23765 \nu^{9} - 983675 \nu^{7} + \cdots + 1253550095 \nu ) / 326123028 \) (1349v^15 + 2159v^13 - 69630v^11 - 23765v^9 - 983675v^7 - 7793646v^5 - 144710671v^3 + 1253550095v) / 326123028
\(\beta_{8}\)	\(=\)	\( ( 3603 \nu^{15} + 1127 \nu^{14} - 879 \nu^{13} + 2597 \nu^{12} - 127842 \nu^{11} + \cdots + 4367248529 ) / 652246056 \) (3603v^15 + 1127v^14 - 879v^13 + 2597v^12 - 127842v^11 - 251370v^10 - 39543v^9 - 34643v^8 - 402633v^7 - 5371037v^6 - 65878638v^5 + 25109658v^4 - 125180937v^3 - 120119629v^2 + 2388392349*v + 4367248529) / 652246056
\(\beta_{9}\)	\(=\)	\( ( - 3603 \nu^{15} + 1127 \nu^{14} + 879 \nu^{13} + 2597 \nu^{12} + 127842 \nu^{11} + \cdots + 4367248529 ) / 652246056 \) (-3603v^15 + 1127v^14 + 879v^13 + 2597v^12 + 127842v^11 - 251370v^10 + 39543v^9 - 34643v^8 + 402633v^7 - 5371037v^6 + 65878638v^5 + 25109658v^4 + 125180937v^3 - 120119629v^2 - 2388392349*v + 4367248529) / 652246056
\(\beta_{10}\)	\(=\)	\( ( 687 \nu^{15} + 1022 \nu^{14} - 5045 \nu^{13} - 6188 \nu^{12} + 30706 \nu^{11} - 224 \nu^{10} + \cdots + 220709524 ) / 108707676 \) (687v^15 + 1022v^14 - 5045v^13 - 6188v^12 + 30706v^11 - 224v^10 - 100107v^9 - 143374v^8 + 843241v^7 + 1105832v^6 - 14439614v^5 - 16269176v^4 + 75725139v^3 + 60404358v^2 - 315887565*v + 220709524) / 108707676
\(\beta_{11}\)	\(=\)	\( ( 687 \nu^{15} - 1022 \nu^{14} - 5045 \nu^{13} + 6188 \nu^{12} + 30706 \nu^{11} + 224 \nu^{10} + \cdots - 220709524 ) / 108707676 \) (687v^15 - 1022v^14 - 5045v^13 + 6188v^12 + 30706v^11 + 224v^10 - 100107v^9 + 143374v^8 + 843241v^7 - 1105832v^6 - 14439614v^5 + 16269176v^4 + 75725139v^3 - 60404358v^2 - 315887565*v - 220709524) / 108707676
\(\beta_{12}\)	\(=\)	\( ( 277 \nu^{15} - 1798 \nu^{13} + 7394 \nu^{11} - 55027 \nu^{9} + 354221 \nu^{7} + \cdots - 6000099 \nu ) / 27176919 \) (277v^15 - 1798v^13 + 7394v^11 - 55027v^9 + 354221v^7 - 5707177v^5 + 23988391v^3 - 6000099v) / 27176919
\(\beta_{13}\)	\(=\)	\( ( 281 \nu^{15} - 3799 \nu^{13} + 10511 \nu^{11} - 75068 \nu^{9} + 317765 \nu^{7} + \cdots - 21882714 \nu ) / 27176919 \) (281v^15 - 3799v^13 + 10511v^11 - 75068v^9 + 317765v^7 - 6557131v^5 + 63355187v^3 - 21882714v) / 27176919
\(\beta_{14}\)	\(=\)	\( ( - 2795 \nu^{15} + 1417 \nu^{13} + 135960 \nu^{11} - 36799 \nu^{9} + 87857 \nu^{7} + \cdots - 2658279155 \nu ) / 163061514 \) (-2795v^15 + 1417v^13 + 135960v^11 - 36799v^9 + 87857v^7 + 31131366v^5 + 93612589v^3 - 2658279155v) / 163061514
\(\beta_{15}\)	\(=\)	\( ( - 14071 \nu^{15} - 1127 \nu^{14} + 3659 \nu^{13} - 2597 \nu^{12} + 689370 \nu^{11} + \cdots - 4367248529 ) / 652246056 \) (-14071v^15 - 1127v^14 + 3659v^13 - 2597v^12 + 689370v^11 + 251370v^10 - 874013v^9 + 34643v^8 + 5290285v^7 + 5371037v^6 + 170811942v^5 - 25109658v^4 + 758574341v^3 + 120119629v^2 - 15484608433*v - 4367248529) / 652246056

\(\nu\)	\(=\)	\( ( -2\beta_{15} + \beta_{14} + 3\beta_{12} - 2\beta_{11} - 2\beta_{10} + \beta_{9} - 3\beta_{8} - 3\beta_{7} - \beta_{6} ) / 14 \) (-2b15 + b14 + 3b12 - 2b11 - 2b10 + b9 - 3b8 - 3b7 - b6) / 14
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{11} - 2\beta_{10} - 2\beta_{5} - \beta_{4} + 4\beta_{3} - \beta_{2} + 3\beta _1 - 1 ) / 2 \) (2b11 - 2b10 - 2b5 - b4 + 4b3 - b2 + 3*b1 - 1) / 2
\(\nu^{3}\)	\(=\)	\( ( 5 \beta_{15} - 20 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + \cdots - 15 \beta_{6} ) / 14 \) (5b15 - 20b14 + 7b13 - 11b12 - 2b11 - 2b10 + 8b9 - 3b8 - 38b7 - 15b6) / 14
\(\nu^{4}\)	\(=\)	\( ( - 8 \beta_{11} + 8 \beta_{10} + 11 \beta_{9} + 11 \beta_{8} - 7 \beta_{5} - 14 \beta_{4} - 13 \beta_{3} + \cdots + 4 ) / 2 \) (-8b11 + 8b10 + 11b9 + 11b8 - 7b5 - 14b4 - 13b3 + 15b2 + 5*b1 + 4) / 2
\(\nu^{5}\)	\(=\)	\( ( 19 \beta_{15} - 90 \beta_{14} - 39 \beta_{12} + 40 \beta_{11} + 40 \beta_{10} + 169 \beta_{9} + \cdots - 22 \beta_{6} ) / 14 \) (19b15 - 90b14 - 39b12 + 40b11 + 40b10 + 169b9 - 150b8 + 109b7 - 22*b6) / 14
\(\nu^{6}\)	\(=\)	\( ( - 111 \beta_{11} + 111 \beta_{10} - 71 \beta_{9} - 71 \beta_{8} - 6 \beta_{5} - 108 \beta_{4} + \cdots + 199 ) / 2 \) (-111b11 + 111b10 - 71b9 - 71b8 - 6b5 - 108b4 + 80b3 - 40b2 + 13*b1 + 199) / 2
\(\nu^{7}\)	\(=\)	\( ( 243 \beta_{15} - 797 \beta_{14} - 714 \beta_{13} + 1116 \beta_{12} - 16 \beta_{11} + \cdots + 475 \beta_{6} ) / 14 \) (243b15 - 797b14 - 714b13 + 1116b12 - 16b11 - 16b10 + 582b9 - 339b8 - 1354b7 + 475b6) / 14
\(\nu^{8}\)	\(=\)	\( ( 772 \beta_{11} - 772 \beta_{10} - 259 \beta_{9} - 259 \beta_{8} - 793 \beta_{5} - 365 \beta_{4} + \cdots - 1163 ) / 2 \) (772b11 - 772b10 - 259b9 - 259b8 - 793b5 - 365b4 - 675b3 + 482b2 - 676*b1 - 1163) / 2
\(\nu^{9}\)	\(=\)	\( ( - 1821 \beta_{15} - 213 \beta_{14} - 1001 \beta_{13} - 3383 \beta_{12} + 3247 \beta_{11} + \cdots + 612 \beta_{6} ) / 7 \) (-1821b15 - 213b14 - 1001b13 - 3383b12 + 3247b11 + 3247b10 + 438b9 - 2259b8 - 5388b7 + 612b6) / 7
\(\nu^{10}\)	\(=\)	\( ( 734 \beta_{11} - 734 \beta_{10} + 683 \beta_{9} + 683 \beta_{8} - 623 \beta_{5} + 413 \beta_{4} + \cdots + 30209 ) / 2 \) (734b11 - 734b10 + 683b9 + 683b8 - 623b5 + 413b4 - 4395b3 + 4334b2 + 446*b1 + 30209) / 2
\(\nu^{11}\)	\(=\)	\( ( - 39741 \beta_{15} + 33461 \beta_{14} - 6762 \beta_{13} + 36522 \beta_{12} - 8164 \beta_{11} + \cdots - 23381 \beta_{6} ) / 14 \) (-39741b15 + 33461b14 - 6762b13 + 36522b12 - 8164b11 - 8164b10 + 21372b9 - 61113b8 - 24692b7 - 23381b6) / 14
\(\nu^{12}\)	\(=\)	\( ( 20721 \beta_{11} - 20721 \beta_{10} - 9101 \beta_{9} - 9101 \beta_{8} - 18066 \beta_{5} - 15606 \beta_{4} + \cdots + 46133 ) / 2 \) (20721b11 - 20721b10 - 9101b9 - 9101b8 - 18066b5 - 15606b4 + 86348b3 - 7138b2 + 66457*b1 + 46133) / 2
\(\nu^{13}\)	\(=\)	\( ( 77201 \beta_{15} - 296232 \beta_{14} - 30114 \beta_{13} + 67371 \beta_{12} - 114452 \beta_{11} + \cdots - 337366 \beta_{6} ) / 14 \) (77201b15 - 296232b14 - 30114b13 + 67371b12 - 114452b11 - 114452b10 + 96167b9 - 18966b8 - 673249b7 - 337366b6) / 14
\(\nu^{14}\)	\(=\)	\( ( 2102 \beta_{11} - 2102 \beta_{10} + 160643 \beta_{9} + 160643 \beta_{8} - 207503 \beta_{5} + \cdots - 401668 ) / 2 \) (2102b11 - 2102b10 + 160643b9 + 160643b8 - 207503b5 - 192508b4 - 102763b3 + 366519b2 + 195863*b1 - 401668) / 2
\(\nu^{15}\)	\(=\)	\( ( 442825 \beta_{15} - 1982098 \beta_{14} - 105833 \beta_{13} - 1721171 \beta_{12} + \cdots - 1105917 \beta_{6} ) / 14 \) (442825b15 - 1982098b14 - 105833b13 - 1721171b12 + 1739554b11 + 1739554b10 + 2294356b9 - 1851531b8 + 1351466b7 - 1105917b6) / 14

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1077.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 14 T^{6} + 51 T^{4} + \cdots + 1)^{2} \) (T^8 + 14T^6 + 51T^4 + 17*T^2 + 1)^2
$5$	\( (T^{8} + 17 T^{6} + 51 T^{4} + \cdots + 1)^{2} \) (T^8 + 17T^6 + 51T^4 + 14*T^2 + 1)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} - T^{7} + \cdots + 14641)^{2} \) (T^8 - T^7 + 8T^6 + 11T^5 - 26T^4 + 121T^3 + 968T^2 - 1331T + 14641)^2
$13$	\( (T^{8} - 95 T^{6} + \cdots + 240100)^{2} \) (T^8 - 95T^6 + 3234T^4 - 46697*T^2 + 240100)^2
$17$	\( (T^{8} - 92 T^{6} + \cdots + 117649)^{2} \) (T^8 - 92T^6 + 2667T^4 - 30233*T^2 + 117649)^2
$19$	\( (T^{8} - 92 T^{6} + \cdots + 117649)^{2} \) (T^8 - 92T^6 + 2667T^4 - 30233*T^2 + 117649)^2
$23$	\( (T^{4} - T^{3} - 33 T^{2} + \cdots + 211)^{4} \) (T^4 - T^3 - 33T^2 + 20T + 211)^4
$29$	\( (T^{8} + 195 T^{6} + \cdots + 1382976)^{2} \) (T^8 + 195T^6 + 12012T^4 + 255339*T^2 + 1382976)^2
$31$	\( (T^{8} + 134 T^{6} + \cdots + 790321)^{2} \) (T^8 + 134T^6 + 6135T^4 + 116885*T^2 + 790321)^2
$37$	\( (T^{4} + T^{3} - 36 T^{2} + \cdots + 40)^{4} \) (T^4 + T^3 - 36T^2 - 44T + 40)^4
$41$	\( (T^{8} - 95 T^{6} + \cdots + 240100)^{2} \) (T^8 - 95T^6 + 3234T^4 - 46697*T^2 + 240100)^2
$43$	\( (T^{8} + 208 T^{6} + \cdots + 9604)^{2} \) (T^8 + 208T^6 + 8211T^4 + 17983*T^2 + 9604)^2
$47$	\( (T^{8} + 158 T^{6} + \cdots + 39601)^{2} \) (T^8 + 158T^6 + 7803T^4 + 118841*T^2 + 39601)^2
$53$	\( (T^{4} + 4 T^{3} + \cdots + 931)^{4} \) (T^4 + 4T^3 - 123T^2 + 217*T + 931)^4
$59$	\( (T^{8} + 209 T^{6} + \cdots + 2337841)^{2} \) (T^8 + 209T^6 + 14211T^4 + 339542*T^2 + 2337841)^2
$61$	\( (T^{8} - 309 T^{6} + \cdots + 12446784)^{2} \) (T^8 - 309T^6 + 30996T^4 - 1100736*T^2 + 12446784)^2
$67$	\( (T^{4} - 9 T^{3} + \cdots + 264)^{4} \) (T^4 - 9T^3 - 48T^2 + 276*T + 264)^4
$71$	\( (T^{4} + 19 T^{3} + \cdots + 16)^{4} \) (T^4 + 19T^3 + 78T^2 - 113*T + 16)^4
$73$	\( (T^{8} - 441 T^{6} + \cdots + 26471025)^{2} \) (T^8 - 441T^6 + 64239T^4 - 3284568*T^2 + 26471025)^2
$79$	\( (T^{8} + 334 T^{6} + \cdots + 5764801)^{2} \) (T^8 + 334T^6 + 32781T^4 + 823543*T^2 + 5764801)^2
$83$	\( (T^{8} - 263 T^{6} + \cdots + 38416)^{2} \) (T^8 - 263T^6 + 9996T^4 - 39641*T^2 + 38416)^2
$89$	\( (T^{8} + 378 T^{6} + \cdots + 3381921)^{2} \) (T^8 + 378T^6 + 33891T^4 + 896661*T^2 + 3381921)^2
$97$	\( (T^{8} + 191 T^{6} + \cdots + 283024)^{2} \) (T^8 + 191T^6 + 11898T^4 + 247715*T^2 + 283024)^2
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\(n\)	\(981\)	\(1079\)	\(1277\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)