LMFDB - Newform orbit 1152.2.p.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(191,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("1152.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.p (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 16x^{14} + 178x^{12} - 1024x^{10} + 4267x^{8} - 7936x^{6} + 10594x^{4} - 2800x^{2} + 625 $ x^16 - 16x^14 + 178x^12 - 1024x^10 + 4267x^8 - 7936x^6 + 10594x^4 - 2800*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q + ( - \beta_{13} - \beta_{11} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{6} - 2 \beta_{4} + \beta_{3} - 1) q^{9}+O(q^{10}) $ q + (-b13 - b11 + b9 - b1) * q^3 + (b8 + b2) * q^5 - b5 * q^7 + (-b6 - 2b4 + b3 - 1) q^9	$ q + ( - \beta_{13} - \beta_{11} + \cdots - \beta_1) q^{3}+ \cdots + ( - 2 \beta_{13} + 2 \beta_{11} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) $ q + (-b13 - b11 + b9 - b1) * q^3 + (b8 + b2) * q^5 - b5 * q^7 + (-b6 - 2b4 + b3 - 1) q^9 + (-b13 - 2b11 + b9 - b1) q^11 + (-b8 - b7) * q^13 + (b15 - b14 - b12 + b5) * q^15 + (-b6 - 4b4 - b3 - 2) q^17 + (-2b11 + 2b1) * q^19 + (-2b10 - b8 - b7 - b2) q^21 + (-b15 + b14) * q^23 + (2b6 - 4b4 + 4b3 - 4) q^25 + (b13 + 2b11 - 2b9 - 2b1) q^27 + (b10 + b8 + 2b7 + b2) q^29 + (-b15 + b14 + 2b12) q^31 + (-b6 - 5b4 + b3 - 1) q^33 + (-4b13 - 2b11 + 7b9 - 2b1) * q^35 + (b10 + b7) * q^37 + (-b14 + b12 + 2b5) q^39 + (-b4 + 5b3 - 2) q^41 + (2b13 - 4b9 + 3b1) q^43 + (2b10 + b8 + b7 - 2b2) * q^45 + (2b14 - b5) q^47 + (-6b6 - 8b4 - 3b3) q^49 + (-2b13 + b11 - 3b9 - b1) * q^51 + (b10 - b7 - 2b2) q^53 + (2b15 - b14 - b12 + b5) q^55 + (4b6 - 4b4 + 2b3 - 2) q^57 + (2b13 - 2b11 + 3b9 - b1) q^59 + (-b10 - b8 - b2) * q^61 + (-b15 - 2b14 - b12 - b5) q^63 + (-3b6 - 5b4 + 5) * q^65 + (7b13 + 7b11 - 3b9 + 4b1) * q^67 + (b10 - b8 + 2b7 + 2b2) * q^69 + (-b12 - b5) * q^71 + (-b6 + b3 - 2) * q^73 + (2b13 + 8b11 - 2b9 - 4b1) * q^75 + (-4b10 - 3b8 - 2b7 - b2) q^77 + (-b15 - b12 - b5) * q^79 + (-6b6 - 6b3 + 3) * q^81 + 3b1 q^83 + (2b8 + b7 - b2) q^85 + (b15 + 2b14 - 2b5) * q^87 + (5b6 - 8b4 + 5b3 - 4) q^89 + (3b11 - 8b9 + 13b1) q^91 + (-b10 + b8 - 2b7 + 4b2) * q^93 + (2b15 - 2b14) * q^95 + (b6 - b4 + 2b3 - 1) q^97 + (-2b13 + 2b11 - 2b9 - 5b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 32 q^{25} + 24 q^{33} - 24 q^{41} + 64 q^{49} + 120 q^{65} - 32 q^{73} + 48 q^{81} - 8 q^{97}+O(q^{100}) $ 16 * q - 32 * q^25 + 24 * q^33 - 24 * q^41 + 64 * q^49 + 120 * q^65 - 32 * q^73 + 48 * q^81 - 8 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 16x^{14} + 178x^{12} - 1024x^{10} + 4267x^{8} - 7936x^{6} + 10594x^{4} - 2800x^{2} + 625 $ x^16 - 16*x^14 + 178*x^12 - 1024*x^10 + 4267*x^8 - 7936*x^6 + 10594*x^4 - 2800*x^2 + 625 :

$\beta_{1}$	$=$	$ ( - 110578 \nu^{15} + 1716459 \nu^{13} - 17177867 \nu^{11} + 85536336 \nu^{9} + \cdots + 3946547785 \nu ) / 2764740900 $ (-110578v^15 + 1716459v^13 - 17177867v^11 + 85536336v^9 - 257319563v^7 + 236330704v^5 - 62625425v^3 + 3946547785v) / 2764740900
$\beta_{2}$	$=$	$ ( - 105014 \nu^{14} + 1510912 \nu^{12} - 16313521 \nu^{10} + 81232368 \nu^{8} - 317125394 \nu^{6} + \cdots - 2566590900 ) / 691185225 $ (-105014v^14 + 1510912v^12 - 16313521v^10 + 81232368v^8 - 317125394v^6 + 224439152v^4 - 59474275*v^2 - 2566590900) / 691185225
$\beta_{3}$	$=$	$ ( 708368 \nu^{14} - 12977761 \nu^{12} + 149741088 \nu^{10} - 989599529 \nu^{8} + \cdots - 7531194400 ) / 4147111350 $ (708368v^14 - 12977761v^12 + 149741088v^10 - 989599529v^8 + 4435986832v^6 - 11471975481v^4 + 15554797256*v^2 - 7531194400) / 4147111350
$\beta_{4}$	$=$	$ ( - 317040 \nu^{14} + 4967626 \nu^{12} - 54922208 \nu^{10} + 308335439 \nu^{8} - 1271577312 \nu^{6} + \cdots + 137052500 ) / 691185225 $ (-317040v^14 + 4967626v^12 - 54922208v^10 + 308335439v^8 - 1271577312v^6 + 2198904046v^4 - 3134282608*v^2 + 137052500) / 691185225
$\beta_{5}$	$=$	$ ( 530634 \nu^{15} - 7760107 \nu^{13} + 82431951 \nu^{11} - 410465808 \nu^{9} + \cdots + 14614038515 \nu ) / 2764740900 $ (530634v^15 - 7760107v^13 + 82431951v^11 - 410465808v^9 + 1525821139v^7 - 1134087312v^5 + 300522525v^3 + 14614038515v) / 2764740900
$\beta_{6}$	$=$	$ ( 2720768 \nu^{14} - 42102736 \nu^{12} + 462359688 \nu^{10} - 2546268329 \nu^{8} + \cdots + 2424996575 ) / 4147111350 $ (2720768v^14 - 42102736v^12 + 462359688v^10 - 2546268329v^8 + 10220140432v^6 - 15772938681v^4 + 16694512256*v^2 + 2424996575) / 4147111350
$\beta_{7}$	$=$	$ ( 1225959 \nu^{14} - 20897028 \nu^{12} + 240426024 \nu^{10} - 1489932092 \nu^{8} + \cdots - 4194007400 ) / 1382370450 $ (1225959v^14 - 20897028v^12 + 240426024v^10 - 1489932092v^8 + 6559330186v^6 - 14385193188v^4 + 19360217233*v^2 - 4194007400) / 1382370450
$\beta_{8}$	$=$	$ ( - 1163146 \nu^{14} + 18359592 \nu^{12} - 203375311 \nu^{10} + 1152109388 \nu^{8} + \cdots + 3114800900 ) / 691185225 $ (-1163146v^14 + 18359592v^12 - 203375311v^10 + 1152109388v^8 - 4769183854v^6 + 8571177032v^4 - 11786470932*v^2 + 3114800900) / 691185225
$\beta_{9}$	$=$	$ ( 205217 \nu^{15} - 3176477 \nu^{13} + 35119216 \nu^{11} - 195787803 \nu^{9} + 813091824 \nu^{7} + \cdots - 87636250 \nu ) / 321481500 $ (205217v^15 - 3176477v^13 + 35119216v^11 - 195787803v^9 + 813091824v^7 - 1406057567v^5 + 2061211663v^3 - 87636250v) / 321481500
$\beta_{10}$	$=$	$ ( 2869705 \nu^{14} - 45231296 \nu^{12} + 495775643 \nu^{10} - 2761432844 \nu^{8} + \cdots - 2047429700 ) / 1382370450 $ (2869705v^14 - 45231296v^12 + 495775643v^10 - 2761432844v^8 + 11042480777v^6 - 17898257716v^4 + 20291146458*v^2 - 2047429700) / 1382370450
$\beta_{11}$	$=$	$ ( 11893313 \nu^{15} - 195102928 \nu^{13} + 2198135399 \nu^{11} - 13085459367 \nu^{9} + \cdots - 59224367875 \nu ) / 13823704500 $ (11893313v^15 - 195102928v^13 + 2198135399v^11 - 13085459367v^9 + 56099912911v^7 - 115740004063v^5 + 162649041682v^3 - 59224367875v) / 13823704500
$\beta_{12}$	$=$	$ ( 12073728 \nu^{15} - 219796993 \nu^{13} + 2561121169 \nu^{11} - 16879437652 \nu^{9} + \cdots - 205930758375 \nu ) / 13823704500 $ (12073728v^15 - 219796993v^13 + 2561121169v^11 - 16879437652v^9 + 76131189941v^7 - 194314591928v^5 + 272754233717v^3 - 205930758375v) / 13823704500
$\beta_{13}$	$=$	$ ( - 4975271 \nu^{15} + 79616685 \nu^{13} - 885251120 \nu^{11} + 5084189985 \nu^{9} + \cdots + 5631124020 \nu ) / 2764740900 $ (-4975271v^15 + 79616685v^13 - 885251120v^11 + 5084189985v^9 - 21086618580v^7 + 38441633365v^5 - 48456154321v^3 + 5631124020v) / 2764740900
$\beta_{14}$	$=$	$ ( 35561269 \nu^{15} - 558879129 \nu^{13} + 6178982832 \nu^{11} - 34748085931 \nu^{9} + \cdots - 15418991250 \nu ) / 13823704500 $ (35561269v^15 - 558879129v^13 + 6178982832v^11 - 34748085931v^9 + 143057875248v^7 - 247386091059v^5 + 336343759111v^3 - 15418991250v) / 13823704500
$\beta_{15}$	$=$	$ ( - 47122248 \nu^{15} + 730796303 \nu^{13} - 8005773949 \nu^{11} + 43990815892 \nu^{9} + \cdots - 116841670475 \nu ) / 13823704500 $ (-47122248v^15 + 730796303v^13 - 8005773949v^11 + 43990815892v^9 - 175978047861v^7 + 269221367288v^5 - 292603828217v^3 - 116841670475v) / 13823704500

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

$\nu$	$=$	$ ( \beta_{15} - 3\beta_{13} + \beta_{12} - 6\beta_{11} + 3\beta_{9} + 2\beta_{5} ) / 6 $ (b15 - 3b13 + b12 - 6b11 + 3b9 + 2b5) / 6
$\nu^{2}$	$=$	$ \beta_{8} - 4\beta_{4} + \beta_{2} $ b8 - 4*b4 + b2
$\nu^{3}$	$=$	$ ( 14\beta_{15} - 6\beta_{14} - 42\beta_{13} - 7\beta_{12} - 21\beta_{11} + 15\beta_{9} + 7\beta_{5} - 21\beta_1 ) / 6 $ (14b15 - 6b14 - 42b13 - 7b12 - 21b11 + 15b9 + 7b5 - 21b1) / 6
$\nu^{4}$	$=$	$ 8\beta_{8} + 2\beta_{6} - 25\beta_{4} + 4\beta_{3} - 25 $ 8b8 + 2b6 - 25b4 + 4b3 - 25
$\nu^{5}$	$=$	$ ( 49 \beta_{15} - 60 \beta_{14} - 159 \beta_{13} - 98 \beta_{12} + 159 \beta_{11} - 36 \beta_{9} + \cdots - 123 \beta_1 ) / 6 $ (49b15 - 60b14 - 159b13 - 98b12 + 159b11 - 36b9 + 11b5 - 123b1) / 6
$\nu^{6}$	$=$	$ 2\beta_{10} - 2\beta_{7} - 24\beta_{6} + 24\beta_{3} - 61\beta_{2} - 172 $ 2b10 - 2b7 - 24b6 + 24b3 - 61*b2 - 172
$\nu^{7}$	$=$	$ ( -355\beta_{15} + 1197\beta_{13} - 355\beta_{12} + 2394\beta_{11} - 1197\beta_{9} - 188\beta_{5} + 138\beta_1 ) / 6 $ (-355b15 + 1197b13 - 355b12 + 2394b11 - 1197b9 - 188b5 + 138*b1) / 6
$\nu^{8}$	$=$	$ 64\beta_{10} - 400\beta_{8} + 32\beta_{7} - 456\beta_{6} + 1225\beta_{4} - 228\beta_{3} - 432\beta_{2} $ 64b10 - 400b8 + 32b7 - 456b6 + 1225b4 - 228b3 - 432*b2
$\nu^{9}$	$=$	$ ( - 5234 \beta_{15} + 4344 \beta_{14} + 18054 \beta_{13} + 2617 \beta_{12} + 9027 \beta_{11} + \cdots + 9027 \beta_1 ) / 6 $ (-5234b15 + 4344b14 + 18054b13 + 2617b12 + 9027b11 - 8955b9 - 2617b5 + 9027b1) / 6
$\nu^{10}$	$=$	$ 356\beta_{10} - 2825\beta_{8} + 712\beta_{7} - 2000\beta_{6} + 8884\beta_{4} - 4000\beta_{3} + 356\beta_{2} + 8884 $ 356b10 - 2825b8 + 712b7 - 2000b6 + 8884b4 - 4000b3 + 356*b2 + 8884
$\nu^{11}$	$=$	$ ( - 19495 \beta_{15} + 35358 \beta_{14} + 68349 \beta_{13} + 38990 \beta_{12} - 68349 \beta_{11} + \cdots + 74079 \beta_1 ) / 6 $ (-19495b15 + 35358b14 + 68349b13 + 38990b12 - 68349b11 - 5730b9 - 15863b5 + 74079b1) / 6
$\nu^{12}$	$=$	$ -3424\beta_{10} + 3424\beta_{7} + 16854\beta_{6} - 16854\beta_{3} + 27032\beta_{2} + 65233 $ -3424b10 + 3424b7 + 16854b6 - 16854b3 + 27032*b2 + 65233
$\nu^{13}$	$=$	$ ( 146329 \beta_{15} - 519567 \beta_{13} + 146329 \beta_{12} - 1039134 \beta_{11} + 519567 \beta_{9} + \cdots + 82740 \beta_1 ) / 6 $ (146329b15 - 519567b13 + 146329b12 - 1039134b11 + 519567b9 + 8798b5 + 82740*b1) / 6
$\nu^{14}$	$=$	$ - 61100 \beta_{10} + 145969 \beta_{8} - 30550 \beta_{7} + 277200 \beta_{6} - 483676 \beta_{4} + \cdots + 176519 \beta_{2} $ -61100b10 + 145969b8 - 30550b7 + 277200b6 - 483676b4 + 138600b3 + 176519*b2
$\nu^{15}$	$=$	$ ( 2209766 \beta_{15} - 2257314 \beta_{14} - 7925898 \beta_{13} - 1104883 \beta_{12} - 3962949 \beta_{11} + \cdots - 3962949 \beta_1 ) / 6 $ (2209766b15 - 2257314b14 - 7925898b13 - 1104883b12 - 3962949b11 + 4835235b9 + 1104883b5 - 3962949b1) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-\beta_{4}$	$-1$

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} $

191.1

1.21704	+	0.702656i
−2.12575	−	1.22730i
−0.451553	−	0.260704i
−2.40751	−	1.38998i
0.451553	+	0.260704i
2.40751	+	1.38998i
−1.21704	−	0.702656i
2.12575	+	1.22730i
1.21704	−	0.702656i
−2.12575	+	1.22730i
−0.451553	+	0.260704i
−2.40751	+	1.38998i
0.451553	−	0.260704i
2.40751	−	1.38998i
−1.21704	+	0.702656i
2.12575	−	1.22730i

−1.65068

−

0.524648i

−1.01255

−

1.75379i

−4.09406

−

2.36371i

2.44949

1.73205i

191.2

−1.65068

−

0.524648i

1.01255

1.75379i

4.09406

2.36371i

2.44949

1.73205i

191.3

−0.524648

−

1.65068i

−1.86407

−

3.22866i

2.39555

1.38307i

−2.44949

1.73205i

191.4

−0.524648

−

1.65068i

1.86407

3.22866i

−2.39555

−

1.38307i

−2.44949

1.73205i

191.5

0.524648

1.65068i

−1.86407

−

3.22866i

−2.39555

−

1.38307i

−2.44949

1.73205i

191.6

0.524648

1.65068i

1.86407

3.22866i

2.39555

1.38307i

−2.44949

1.73205i

191.7

1.65068

0.524648i

−1.01255

−

1.75379i

4.09406

2.36371i

2.44949

1.73205i

191.8

1.65068

0.524648i

1.01255

1.75379i

−4.09406

−

2.36371i

2.44949

1.73205i

959.1

−1.65068

0.524648i

−1.01255

1.75379i

−4.09406

2.36371i

2.44949

−

1.73205i

959.2

−1.65068

0.524648i

1.01255

−

1.75379i

4.09406

−

2.36371i

2.44949

−

1.73205i

959.3

−0.524648

1.65068i

−1.86407

3.22866i

2.39555

−

1.38307i

−2.44949

−

1.73205i

959.4

−0.524648

1.65068i

1.86407

−

3.22866i

−2.39555

1.38307i

−2.44949

−

1.73205i

959.5

0.524648

−

1.65068i

−1.86407

3.22866i

−2.39555

1.38307i

−2.44949

−

1.73205i

959.6

0.524648

−

1.65068i

1.86407

−

3.22866i

2.39555

−

1.38307i

−2.44949

−

1.73205i

959.7

1.65068

−

0.524648i

−1.01255

1.75379i

4.09406

−

2.36371i

2.44949

−

1.73205i

959.8

1.65068

−

0.524648i

1.01255

−

1.75379i

−4.09406

2.36371i

2.44949

−

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
9.d	odd	6	1	inner
36.h	even	6	1	inner
72.j	odd	6	1	inner
72.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1152.2.p.d	✓	16
3.b	odd	2	1		3456.2.p.d		16
4.b	odd	2	1	inner	1152.2.p.d	✓	16
8.b	even	2	1	inner	1152.2.p.d	✓	16
8.d	odd	2	1	inner	1152.2.p.d	✓	16
9.c	even	3	1		3456.2.p.d		16
9.d	odd	6	1	inner	1152.2.p.d	✓	16
12.b	even	2	1		3456.2.p.d		16
24.f	even	2	1		3456.2.p.d		16
24.h	odd	2	1		3456.2.p.d		16
36.f	odd	6	1		3456.2.p.d		16
36.h	even	6	1	inner	1152.2.p.d	✓	16
72.j	odd	6	1	inner	1152.2.p.d	✓	16
72.l	even	6	1	inner	1152.2.p.d	✓	16
72.n	even	6	1		3456.2.p.d		16
72.p	odd	6	1		3456.2.p.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1152.2.p.d	✓	16	1.a	even	1	1	trivial
1152.2.p.d	✓	16	4.b	odd	2	1	inner
1152.2.p.d	✓	16	8.b	even	2	1	inner
1152.2.p.d	✓	16	8.d	odd	2	1	inner
1152.2.p.d	✓	16	9.d	odd	6	1	inner
1152.2.p.d	✓	16	36.h	even	6	1	inner
1152.2.p.d	✓	16	72.j	odd	6	1	inner
1152.2.p.d	✓	16	72.l	even	6	1	inner
3456.2.p.d		16	3.b	odd	2	1
3456.2.p.d		16	9.c	even	3	1
3456.2.p.d		16	12.b	even	2	1
3456.2.p.d		16	24.f	even	2	1
3456.2.p.d		16	24.h	odd	2	1
3456.2.p.d		16	36.f	odd	6	1
3456.2.p.d		16	72.n	even	6	1
3456.2.p.d		16	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1152, [\chi])$:

$ T_{5}^{8} + 18T_{5}^{6} + 267T_{5}^{4} + 1026T_{5}^{2} + 3249 $

$ T_{7}^{8} - 30T_{7}^{6} + 729T_{7}^{4} - 5130T_{7}^{2} + 29241 $

$ T_{11}^{8} - 18T_{11}^{6} + 249T_{11}^{4} - 1350T_{11}^{2} + 5625 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 6 T^{4} + 81)^{2} $ (T^8 - 6*T^4 + 81)^2
$5$	$ (T^{8} + 18 T^{6} + \cdots + 3249)^{2} $ (T^8 + 18T^6 + 267T^4 + 1026*T^2 + 3249)^2
$7$	$ (T^{8} - 30 T^{6} + \cdots + 29241)^{2} $ (T^8 - 30T^6 + 729T^4 - 5130*T^2 + 29241)^2
$11$	$ (T^{8} - 18 T^{6} + \cdots + 5625)^{2} $ (T^8 - 18T^6 + 249T^4 - 1350*T^2 + 5625)^2
$13$	$ (T^{8} - 42 T^{6} + \cdots + 3249)^{2} $ (T^8 - 42T^6 + 1707T^4 - 2394*T^2 + 3249)^2
$17$	$ (T^{4} + 28 T^{2} + 100)^{4} $ (T^4 + 28*T^2 + 100)^4
$19$	$ (T^{4} - 24 T^{2} + 48)^{4} $ (T^4 - 24*T^2 + 48)^4
$23$	$ (T^{8} + 78 T^{6} + \cdots + 29241)^{2} $ (T^8 + 78T^6 + 5913T^4 + 13338*T^2 + 29241)^2
$29$	$ (T^{8} + 78 T^{6} + \cdots + 2030625)^{2} $ (T^8 + 78T^6 + 4659T^4 + 111150*T^2 + 2030625)^2
$31$	$ (T^{8} - 138 T^{6} + \cdots + 18275625)^{2} $ (T^8 - 138T^6 + 14769T^4 - 589950*T^2 + 18275625)^2
$37$	$ (T^{4} + 36 T^{2} + 228)^{4} $ (T^4 + 36*T^2 + 228)^4
$41$	$ (T^{4} + 6 T^{3} + \cdots + 2209)^{4} $ (T^4 + 6T^3 - 35T^2 - 282*T + 2209)^4
$43$	$ (T^{8} + 66 T^{6} + \cdots + 1172889)^{2} $ (T^8 + 66T^6 + 3273T^4 + 71478*T^2 + 1172889)^2
$47$	$ (T^{8} + 90 T^{6} + \cdots + 2368521)^{2} $ (T^8 + 90T^6 + 6561T^4 + 138510*T^2 + 2368521)^2
$53$	$ (T^{4} - 108 T^{2} + 2052)^{4} $ (T^4 - 108*T^2 + 2052)^4
$59$	$ (T^{8} - 102 T^{6} + \cdots + 2518569)^{2} $ (T^8 - 102T^6 + 8817T^4 - 161874*T^2 + 2518569)^2
$61$	$ (T^{8} - 42 T^{6} + \cdots + 3249)^{2} $ (T^8 - 42T^6 + 1707T^4 - 2394*T^2 + 3249)^2
$67$	$ (T^{8} + 246 T^{6} + \cdots + 228705129)^{2} $ (T^8 + 246T^6 + 45393T^4 + 3720258*T^2 + 228705129)^2
$71$	$ (T^{4} - 60 T^{2} + 684)^{4} $ (T^4 - 60*T^2 + 684)^4
$73$	$ (T^{2} + 4 T - 2)^{8} $ (T^2 + 4*T - 2)^8
$79$	$ (T^{8} - 138 T^{6} + \cdots + 18275625)^{2} $ (T^8 - 138T^6 + 14769T^4 - 589950*T^2 + 18275625)^2
$83$	$ (T^{8} - 54 T^{6} + \cdots + 59049)^{2} $ (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$89$	$ (T^{4} + 196 T^{2} + 4)^{4} $ (T^4 + 196*T^2 + 4)^4
$97$	$ (T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 25)^{4} $ (T^4 + 2T^3 + 9T^2 - 10*T + 25)^4
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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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{13} - \beta_{11} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{6} - 2 \beta_{4} + \beta_{3} - 1) q^{9}+O(q^{10}) \) q + (-b13 - b11 + b9 - b1) * q^3 + (b8 + b2) * q^5 - b5 * q^7 + (-b6 - 2b4 + b3 - 1) q^9	\( q + ( - \beta_{13} - \beta_{11} + \cdots - \beta_1) q^{3}+ \cdots + ( - 2 \beta_{13} + 2 \beta_{11} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) q + (-b13 - b11 + b9 - b1) * q^3 + (b8 + b2) * q^5 - b5 * q^7 + (-b6 - 2b4 + b3 - 1) q^9 + (-b13 - 2b11 + b9 - b1) q^11 + (-b8 - b7) * q^13 + (b15 - b14 - b12 + b5) * q^15 + (-b6 - 4b4 - b3 - 2) q^17 + (-2b11 + 2b1) * q^19 + (-2b10 - b8 - b7 - b2) q^21 + (-b15 + b14) * q^23 + (2b6 - 4b4 + 4b3 - 4) q^25 + (b13 + 2b11 - 2b9 - 2b1) q^27 + (b10 + b8 + 2b7 + b2) q^29 + (-b15 + b14 + 2b12) q^31 + (-b6 - 5b4 + b3 - 1) q^33 + (-4b13 - 2b11 + 7b9 - 2b1) * q^35 + (b10 + b7) * q^37 + (-b14 + b12 + 2b5) q^39 + (-b4 + 5b3 - 2) q^41 + (2b13 - 4b9 + 3b1) q^43 + (2b10 + b8 + b7 - 2b2) * q^45 + (2b14 - b5) q^47 + (-6b6 - 8b4 - 3b3) q^49 + (-2b13 + b11 - 3b9 - b1) * q^51 + (b10 - b7 - 2b2) q^53 + (2b15 - b14 - b12 + b5) q^55 + (4b6 - 4b4 + 2b3 - 2) q^57 + (2b13 - 2b11 + 3b9 - b1) q^59 + (-b10 - b8 - b2) * q^61 + (-b15 - 2b14 - b12 - b5) q^63 + (-3b6 - 5b4 + 5) * q^65 + (7b13 + 7b11 - 3b9 + 4b1) * q^67 + (b10 - b8 + 2b7 + 2b2) * q^69 + (-b12 - b5) * q^71 + (-b6 + b3 - 2) * q^73 + (2b13 + 8b11 - 2b9 - 4b1) * q^75 + (-4b10 - 3b8 - 2b7 - b2) q^77 + (-b15 - b12 - b5) * q^79 + (-6b6 - 6b3 + 3) * q^81 + 3b1 q^83 + (2b8 + b7 - b2) q^85 + (b15 + 2b14 - 2b5) * q^87 + (5b6 - 8b4 + 5b3 - 4) q^89 + (3b11 - 8b9 + 13b1) q^91 + (-b10 + b8 - 2b7 + 4b2) * q^93 + (2b15 - 2b14) * q^95 + (b6 - b4 + 2b3 - 1) q^97 + (-2b13 + 2b11 - 2b9 - 5b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 32 q^{25} + 24 q^{33} - 24 q^{41} + 64 q^{49} + 120 q^{65} - 32 q^{73} + 48 q^{81} - 8 q^{97}+O(q^{100}) \) 16 * q - 32 * q^25 + 24 * q^33 - 24 * q^41 + 64 * q^49 + 120 * q^65 - 32 * q^73 + 48 * q^81 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	1152.2.p.d

Level	$1152$
Weight	$2$
Character orbit	1152.p
Analytic conductor	$9.199$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(9.19876631285\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 16x^{14} + 178x^{12} - 1024x^{10} + 4267x^{8} - 7936x^{6} + 10594x^{4} - 2800x^{2} + 625 \) x^16 - 16x^14 + 178x^12 - 1024x^10 + 4267x^8 - 7936x^6 + 10594x^4 - 2800*x^2 + 625
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 110578 \nu^{15} + 1716459 \nu^{13} - 17177867 \nu^{11} + 85536336 \nu^{9} + \cdots + 3946547785 \nu ) / 2764740900 \) (-110578v^15 + 1716459v^13 - 17177867v^11 + 85536336v^9 - 257319563v^7 + 236330704v^5 - 62625425v^3 + 3946547785v) / 2764740900
\(\beta_{2}\)	\(=\)	\( ( - 105014 \nu^{14} + 1510912 \nu^{12} - 16313521 \nu^{10} + 81232368 \nu^{8} - 317125394 \nu^{6} + \cdots - 2566590900 ) / 691185225 \) (-105014v^14 + 1510912v^12 - 16313521v^10 + 81232368v^8 - 317125394v^6 + 224439152v^4 - 59474275*v^2 - 2566590900) / 691185225
\(\beta_{3}\)	\(=\)	\( ( 708368 \nu^{14} - 12977761 \nu^{12} + 149741088 \nu^{10} - 989599529 \nu^{8} + \cdots - 7531194400 ) / 4147111350 \) (708368v^14 - 12977761v^12 + 149741088v^10 - 989599529v^8 + 4435986832v^6 - 11471975481v^4 + 15554797256*v^2 - 7531194400) / 4147111350
\(\beta_{4}\)	\(=\)	\( ( - 317040 \nu^{14} + 4967626 \nu^{12} - 54922208 \nu^{10} + 308335439 \nu^{8} - 1271577312 \nu^{6} + \cdots + 137052500 ) / 691185225 \) (-317040v^14 + 4967626v^12 - 54922208v^10 + 308335439v^8 - 1271577312v^6 + 2198904046v^4 - 3134282608*v^2 + 137052500) / 691185225
\(\beta_{5}\)	\(=\)	\( ( 530634 \nu^{15} - 7760107 \nu^{13} + 82431951 \nu^{11} - 410465808 \nu^{9} + \cdots + 14614038515 \nu ) / 2764740900 \) (530634v^15 - 7760107v^13 + 82431951v^11 - 410465808v^9 + 1525821139v^7 - 1134087312v^5 + 300522525v^3 + 14614038515v) / 2764740900
\(\beta_{6}\)	\(=\)	\( ( 2720768 \nu^{14} - 42102736 \nu^{12} + 462359688 \nu^{10} - 2546268329 \nu^{8} + \cdots + 2424996575 ) / 4147111350 \) (2720768v^14 - 42102736v^12 + 462359688v^10 - 2546268329v^8 + 10220140432v^6 - 15772938681v^4 + 16694512256*v^2 + 2424996575) / 4147111350
\(\beta_{7}\)	\(=\)	\( ( 1225959 \nu^{14} - 20897028 \nu^{12} + 240426024 \nu^{10} - 1489932092 \nu^{8} + \cdots - 4194007400 ) / 1382370450 \) (1225959v^14 - 20897028v^12 + 240426024v^10 - 1489932092v^8 + 6559330186v^6 - 14385193188v^4 + 19360217233*v^2 - 4194007400) / 1382370450
\(\beta_{8}\)	\(=\)	\( ( - 1163146 \nu^{14} + 18359592 \nu^{12} - 203375311 \nu^{10} + 1152109388 \nu^{8} + \cdots + 3114800900 ) / 691185225 \) (-1163146v^14 + 18359592v^12 - 203375311v^10 + 1152109388v^8 - 4769183854v^6 + 8571177032v^4 - 11786470932*v^2 + 3114800900) / 691185225
\(\beta_{9}\)	\(=\)	\( ( 205217 \nu^{15} - 3176477 \nu^{13} + 35119216 \nu^{11} - 195787803 \nu^{9} + 813091824 \nu^{7} + \cdots - 87636250 \nu ) / 321481500 \) (205217v^15 - 3176477v^13 + 35119216v^11 - 195787803v^9 + 813091824v^7 - 1406057567v^5 + 2061211663v^3 - 87636250v) / 321481500
\(\beta_{10}\)	\(=\)	\( ( 2869705 \nu^{14} - 45231296 \nu^{12} + 495775643 \nu^{10} - 2761432844 \nu^{8} + \cdots - 2047429700 ) / 1382370450 \) (2869705v^14 - 45231296v^12 + 495775643v^10 - 2761432844v^8 + 11042480777v^6 - 17898257716v^4 + 20291146458*v^2 - 2047429700) / 1382370450
\(\beta_{11}\)	\(=\)	\( ( 11893313 \nu^{15} - 195102928 \nu^{13} + 2198135399 \nu^{11} - 13085459367 \nu^{9} + \cdots - 59224367875 \nu ) / 13823704500 \) (11893313v^15 - 195102928v^13 + 2198135399v^11 - 13085459367v^9 + 56099912911v^7 - 115740004063v^5 + 162649041682v^3 - 59224367875v) / 13823704500
\(\beta_{12}\)	\(=\)	\( ( 12073728 \nu^{15} - 219796993 \nu^{13} + 2561121169 \nu^{11} - 16879437652 \nu^{9} + \cdots - 205930758375 \nu ) / 13823704500 \) (12073728v^15 - 219796993v^13 + 2561121169v^11 - 16879437652v^9 + 76131189941v^7 - 194314591928v^5 + 272754233717v^3 - 205930758375v) / 13823704500
\(\beta_{13}\)	\(=\)	\( ( - 4975271 \nu^{15} + 79616685 \nu^{13} - 885251120 \nu^{11} + 5084189985 \nu^{9} + \cdots + 5631124020 \nu ) / 2764740900 \) (-4975271v^15 + 79616685v^13 - 885251120v^11 + 5084189985v^9 - 21086618580v^7 + 38441633365v^5 - 48456154321v^3 + 5631124020v) / 2764740900
\(\beta_{14}\)	\(=\)	\( ( 35561269 \nu^{15} - 558879129 \nu^{13} + 6178982832 \nu^{11} - 34748085931 \nu^{9} + \cdots - 15418991250 \nu ) / 13823704500 \) (35561269v^15 - 558879129v^13 + 6178982832v^11 - 34748085931v^9 + 143057875248v^7 - 247386091059v^5 + 336343759111v^3 - 15418991250v) / 13823704500
\(\beta_{15}\)	\(=\)	\( ( - 47122248 \nu^{15} + 730796303 \nu^{13} - 8005773949 \nu^{11} + 43990815892 \nu^{9} + \cdots - 116841670475 \nu ) / 13823704500 \) (-47122248v^15 + 730796303v^13 - 8005773949v^11 + 43990815892v^9 - 175978047861v^7 + 269221367288v^5 - 292603828217v^3 - 116841670475v) / 13823704500

\(\nu\)	\(=\)	\( ( \beta_{15} - 3\beta_{13} + \beta_{12} - 6\beta_{11} + 3\beta_{9} + 2\beta_{5} ) / 6 \) (b15 - 3b13 + b12 - 6b11 + 3b9 + 2b5) / 6
\(\nu^{2}\)	\(=\)	\( \beta_{8} - 4\beta_{4} + \beta_{2} \) b8 - 4*b4 + b2
\(\nu^{3}\)	\(=\)	\( ( 14\beta_{15} - 6\beta_{14} - 42\beta_{13} - 7\beta_{12} - 21\beta_{11} + 15\beta_{9} + 7\beta_{5} - 21\beta_1 ) / 6 \) (14b15 - 6b14 - 42b13 - 7b12 - 21b11 + 15b9 + 7b5 - 21b1) / 6
\(\nu^{4}\)	\(=\)	\( 8\beta_{8} + 2\beta_{6} - 25\beta_{4} + 4\beta_{3} - 25 \) 8b8 + 2b6 - 25b4 + 4b3 - 25
\(\nu^{5}\)	\(=\)	\( ( 49 \beta_{15} - 60 \beta_{14} - 159 \beta_{13} - 98 \beta_{12} + 159 \beta_{11} - 36 \beta_{9} + \cdots - 123 \beta_1 ) / 6 \) (49b15 - 60b14 - 159b13 - 98b12 + 159b11 - 36b9 + 11b5 - 123b1) / 6
\(\nu^{6}\)	\(=\)	\( 2\beta_{10} - 2\beta_{7} - 24\beta_{6} + 24\beta_{3} - 61\beta_{2} - 172 \) 2b10 - 2b7 - 24b6 + 24b3 - 61*b2 - 172
\(\nu^{7}\)	\(=\)	\( ( -355\beta_{15} + 1197\beta_{13} - 355\beta_{12} + 2394\beta_{11} - 1197\beta_{9} - 188\beta_{5} + 138\beta_1 ) / 6 \) (-355b15 + 1197b13 - 355b12 + 2394b11 - 1197b9 - 188b5 + 138*b1) / 6
\(\nu^{8}\)	\(=\)	\( 64\beta_{10} - 400\beta_{8} + 32\beta_{7} - 456\beta_{6} + 1225\beta_{4} - 228\beta_{3} - 432\beta_{2} \) 64b10 - 400b8 + 32b7 - 456b6 + 1225b4 - 228b3 - 432*b2
\(\nu^{9}\)	\(=\)	\( ( - 5234 \beta_{15} + 4344 \beta_{14} + 18054 \beta_{13} + 2617 \beta_{12} + 9027 \beta_{11} + \cdots + 9027 \beta_1 ) / 6 \) (-5234b15 + 4344b14 + 18054b13 + 2617b12 + 9027b11 - 8955b9 - 2617b5 + 9027b1) / 6
\(\nu^{10}\)	\(=\)	\( 356\beta_{10} - 2825\beta_{8} + 712\beta_{7} - 2000\beta_{6} + 8884\beta_{4} - 4000\beta_{3} + 356\beta_{2} + 8884 \) 356b10 - 2825b8 + 712b7 - 2000b6 + 8884b4 - 4000b3 + 356*b2 + 8884
\(\nu^{11}\)	\(=\)	\( ( - 19495 \beta_{15} + 35358 \beta_{14} + 68349 \beta_{13} + 38990 \beta_{12} - 68349 \beta_{11} + \cdots + 74079 \beta_1 ) / 6 \) (-19495b15 + 35358b14 + 68349b13 + 38990b12 - 68349b11 - 5730b9 - 15863b5 + 74079b1) / 6
\(\nu^{12}\)	\(=\)	\( -3424\beta_{10} + 3424\beta_{7} + 16854\beta_{6} - 16854\beta_{3} + 27032\beta_{2} + 65233 \) -3424b10 + 3424b7 + 16854b6 - 16854b3 + 27032*b2 + 65233
\(\nu^{13}\)	\(=\)	\( ( 146329 \beta_{15} - 519567 \beta_{13} + 146329 \beta_{12} - 1039134 \beta_{11} + 519567 \beta_{9} + \cdots + 82740 \beta_1 ) / 6 \) (146329b15 - 519567b13 + 146329b12 - 1039134b11 + 519567b9 + 8798b5 + 82740*b1) / 6
\(\nu^{14}\)	\(=\)	\( - 61100 \beta_{10} + 145969 \beta_{8} - 30550 \beta_{7} + 277200 \beta_{6} - 483676 \beta_{4} + \cdots + 176519 \beta_{2} \) -61100b10 + 145969b8 - 30550b7 + 277200b6 - 483676b4 + 138600b3 + 176519*b2
\(\nu^{15}\)	\(=\)	\( ( 2209766 \beta_{15} - 2257314 \beta_{14} - 7925898 \beta_{13} - 1104883 \beta_{12} - 3962949 \beta_{11} + \cdots - 3962949 \beta_1 ) / 6 \) (2209766b15 - 2257314b14 - 7925898b13 - 1104883b12 - 3962949b11 + 4835235b9 + 1104883b5 - 3962949b1) / 6

\(n\)	\(127\)	\(641\)	\(901\)
\(\chi(n)\)	\(-1\)	\(-\beta_{4}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 191.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 6 T^{4} + 81)^{2} \) (T^8 - 6*T^4 + 81)^2
$5$	\( (T^{8} + 18 T^{6} + \cdots + 3249)^{2} \) (T^8 + 18T^6 + 267T^4 + 1026*T^2 + 3249)^2
$7$	\( (T^{8} - 30 T^{6} + \cdots + 29241)^{2} \) (T^8 - 30T^6 + 729T^4 - 5130*T^2 + 29241)^2
$11$	\( (T^{8} - 18 T^{6} + \cdots + 5625)^{2} \) (T^8 - 18T^6 + 249T^4 - 1350*T^2 + 5625)^2
$13$	\( (T^{8} - 42 T^{6} + \cdots + 3249)^{2} \) (T^8 - 42T^6 + 1707T^4 - 2394*T^2 + 3249)^2
$17$	\( (T^{4} + 28 T^{2} + 100)^{4} \) (T^4 + 28*T^2 + 100)^4
$19$	\( (T^{4} - 24 T^{2} + 48)^{4} \) (T^4 - 24*T^2 + 48)^4
$23$	\( (T^{8} + 78 T^{6} + \cdots + 29241)^{2} \) (T^8 + 78T^6 + 5913T^4 + 13338*T^2 + 29241)^2
$29$	\( (T^{8} + 78 T^{6} + \cdots + 2030625)^{2} \) (T^8 + 78T^6 + 4659T^4 + 111150*T^2 + 2030625)^2
$31$	\( (T^{8} - 138 T^{6} + \cdots + 18275625)^{2} \) (T^8 - 138T^6 + 14769T^4 - 589950*T^2 + 18275625)^2
$37$	\( (T^{4} + 36 T^{2} + 228)^{4} \) (T^4 + 36*T^2 + 228)^4
$41$	\( (T^{4} + 6 T^{3} + \cdots + 2209)^{4} \) (T^4 + 6T^3 - 35T^2 - 282*T + 2209)^4
$43$	\( (T^{8} + 66 T^{6} + \cdots + 1172889)^{2} \) (T^8 + 66T^6 + 3273T^4 + 71478*T^2 + 1172889)^2
$47$	\( (T^{8} + 90 T^{6} + \cdots + 2368521)^{2} \) (T^8 + 90T^6 + 6561T^4 + 138510*T^2 + 2368521)^2
$53$	\( (T^{4} - 108 T^{2} + 2052)^{4} \) (T^4 - 108*T^2 + 2052)^4
$59$	\( (T^{8} - 102 T^{6} + \cdots + 2518569)^{2} \) (T^8 - 102T^6 + 8817T^4 - 161874*T^2 + 2518569)^2
$61$	\( (T^{8} - 42 T^{6} + \cdots + 3249)^{2} \) (T^8 - 42T^6 + 1707T^4 - 2394*T^2 + 3249)^2
$67$	\( (T^{8} + 246 T^{6} + \cdots + 228705129)^{2} \) (T^8 + 246T^6 + 45393T^4 + 3720258*T^2 + 228705129)^2
$71$	\( (T^{4} - 60 T^{2} + 684)^{4} \) (T^4 - 60*T^2 + 684)^4
$73$	\( (T^{2} + 4 T - 2)^{8} \) (T^2 + 4*T - 2)^8
$79$	\( (T^{8} - 138 T^{6} + \cdots + 18275625)^{2} \) (T^8 - 138T^6 + 14769T^4 - 589950*T^2 + 18275625)^2
$83$	\( (T^{8} - 54 T^{6} + \cdots + 59049)^{2} \) (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$89$	\( (T^{4} + 196 T^{2} + 4)^{4} \) (T^4 + 196*T^2 + 4)^4
$97$	\( (T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 25)^{4} \) (T^4 + 2T^3 + 9T^2 - 10*T + 25)^4
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