LMFDB - Newform orbit 2400.2.b.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(2351,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2400.2351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.1640964851$
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} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 $ x^16 + 24x^14 + 192x^12 + 672x^10 + 1092x^8 + 880x^6 + 352x^4 + 64*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 120)
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_{5} q^{3} - \beta_{15} q^{7} + ( - \beta_{13} + \beta_{8} + 1) q^{9}+O(q^{10}) $ q - b5 * q^3 - b15 * q^7 + (-b13 + b8 + 1) * q^9	$ q - \beta_{5} q^{3} - \beta_{15} q^{7} + ( - \beta_{13} + \beta_{8} + 1) q^{9} + (\beta_{13} - \beta_{7}) q^{11} - \beta_{9} q^{13} + ( - \beta_{14} - \beta_{5} + \beta_{4}) q^{17} + (\beta_{8} - 2) q^{19} + (\beta_{11} - 2 \beta_{10}) q^{21} + \beta_{3} q^{23} + ( - \beta_{14} + \beta_{6} + \cdots + \beta_{4}) q^{27}+ \cdots + ( - \beta_{13} - 2 \beta_{8} + \cdots + 4) q^{99}+O(q^{100}) $ q - b5 * q^3 - b15 * q^7 + (-b13 + b8 + 1) * q^9 + (b13 - b7) * q^11 - b9 * q^13 + (-b14 - b5 + b4) * q^17 + (b8 - 2) * q^19 + (b11 - 2b10) q^21 + b3 * q^23 + (-b14 + b6 - b5 + b4) * q^27 + (b12 + b11 - b2) * q^29 + (b12 - b11 - 2b10) q^31 + (-2b14 - b6 - b4) q^33 + (-2b15 - b9) q^37 + (2b12 - 2b11 + b10 + b2) * q^39 + (b8 - 2b7) q^41 + (2b6 + b5 - b4) q^43 + b3 * q^47 + (-b8 - 1) * q^49 + (-b13 + b8 - b7 - 2) * q^51 + (-3b3 + 3b1) * q^53 + (-b14 + b6 + 3b5 - 2b4) * q^57 + (b13 - b7) * q^59 + (3b12 - 3b11) * q^61 + (-b15 + b9 - b3 - 3b1) q^63 + (-b5 - b4) * q^67 + (-b12 + b2) * q^69 + (2b12 + 2b11) * q^71 + (2b5 + 2b4) * q^73 + (-4b3 + 2b1) * q^77 + (-3b12 + 3b11 - 4b10) q^79 + (3b8 - 2b7 - 3) * q^81 + (4b14 - b5 + b4) q^83 + (b15 - 3b3 - 2b1) * q^87 + (2b13 - 2b7) * q^89 + (4b8 + 4) q^91 + (-2b15 - b9 + b3 - 3b1) * q^93 + (-2b6 - 2b5) * q^97 + (-b13 - 2b8 - b7 + 4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^9	$ 16 q + 16 q^{9} - 32 q^{19} - 16 q^{49} - 32 q^{51} - 48 q^{81} + 64 q^{91} + 64 q^{99}+O(q^{100}) $ 16 * q + 16 * q^9 - 32 * q^19 - 16 * q^49 - 32 * q^51 - 48 * q^81 + 64 * q^91 + 64 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 $ x^16 + 24*x^14 + 192*x^12 + 672*x^10 + 1092*x^8 + 880*x^6 + 352*x^4 + 64*x^2 + 4 :

$\beta_{1}$	$=$	$ ( -\nu^{14} - 20\nu^{12} - 100\nu^{10} + 4\nu^{8} + 918\nu^{6} + 1440\nu^{4} + 664\nu^{2} + 72 ) / 8 $ (-v^14 - 20v^12 - 100v^10 + 4v^8 + 918v^6 + 1440v^4 + 664v^2 + 72) / 8
$\beta_{2}$	$=$	$ ( -3\nu^{14} - 69\nu^{12} - 506\nu^{10} - 1488\nu^{8} - 1638\nu^{6} - 594\nu^{4} - 36\nu^{2} - 8 ) / 4 $ (-3v^14 - 69v^12 - 506v^10 - 1488v^8 - 1638v^6 - 594v^4 - 36*v^2 - 8) / 4
$\beta_{3}$	$=$	$ ( -7\nu^{14} - 164\nu^{12} - 1250\nu^{10} - 3984\nu^{8} - 5334\nu^{6} - 3024\nu^{4} - 596\nu^{2} - 16 ) / 8 $ (-7v^14 - 164v^12 - 1250v^10 - 3984v^8 - 5334v^6 - 3024v^4 - 596*v^2 - 16) / 8
$\beta_{4}$	$=$	$ ( - 15 \nu^{14} - 3 \nu^{13} - 352 \nu^{12} - 69 \nu^{11} - 2692 \nu^{10} - 506 \nu^{9} - 8638 \nu^{8} + \cdots - 76 ) / 16 $ (-15v^14 - 3v^13 - 352v^12 - 69v^11 - 2692v^10 - 506v^9 - 8638v^8 - 1488v^7 - 11726v^6 - 1638v^5 - 6808v^4 - 594v^3 - 1496v^2 - 28v - 76) / 16
$\beta_{5}$	$=$	$ ( - 15 \nu^{14} + 3 \nu^{13} - 352 \nu^{12} + 69 \nu^{11} - 2692 \nu^{10} + 506 \nu^{9} - 8638 \nu^{8} + \cdots - 76 ) / 16 $ (-15v^14 + 3v^13 - 352v^12 + 69v^11 - 2692v^10 + 506v^9 - 8638v^8 + 1488v^7 - 11726v^6 + 1638v^5 - 6808v^4 + 594v^3 - 1496v^2 + 28v - 76) / 16
$\beta_{6}$	$=$	$ ( - 27 \nu^{14} - 3 \nu^{13} - 636 \nu^{12} - 69 \nu^{11} - 4904 \nu^{10} - 506 \nu^{9} - 16030 \nu^{8} + \cdots - 428 ) / 16 $ (-27v^14 - 3v^13 - 636v^12 - 69v^11 - 4904v^10 - 506v^9 - 16030v^8 - 1488v^7 - 22886v^6 - 1638v^5 - 15248v^4 - 594v^3 - 4576v^2 - 28v - 428) / 16
$\beta_{7}$	$=$	$ ( - 5 \nu^{15} + 12 \nu^{14} - 110 \nu^{13} + 282 \nu^{12} - 728 \nu^{11} + 2164 \nu^{10} - 1625 \nu^{9} + \cdots + 136 ) / 8 $ (-5v^15 + 12v^14 - 110v^13 + 282v^12 - 728v^11 + 2164v^10 - 1625v^9 + 7004v^8 - 118v^7 + 9752v^6 + 2276v^5 + 6092v^4 + 1648v^3 + 1608v^2 + 270*v + 136) / 8
$\beta_{8}$	$=$	$ ( 6\nu^{14} + 141\nu^{12} + 1082\nu^{10} + 3502\nu^{8} + 4876\nu^{6} + 3046\nu^{4} + 804\nu^{2} + 68 ) / 2 $ (6v^14 + 141v^12 + 1082v^10 + 3502v^8 + 4876v^6 + 3046v^4 + 804*v^2 + 68) / 2
$\beta_{9}$	$=$	$ ( -11\nu^{15} - 260\nu^{13} - 2018\nu^{11} - 6672\nu^{9} - 9702\nu^{7} - 6544\nu^{5} - 2004\nu^{3} - 272\nu ) / 8 $ (-11v^15 - 260v^13 - 2018v^11 - 6672v^9 - 9702v^7 - 6544v^5 - 2004v^3 - 272v) / 8
$\beta_{10}$	$=$	$ ( -11\nu^{15} - 261\nu^{13} - 2042\nu^{11} - 6862\nu^{9} - 10334\nu^{7} - 7410\nu^{5} - 2412\nu^{3} - 252\nu ) / 8 $ (-11v^15 - 261v^13 - 2042v^11 - 6862v^9 - 10334v^7 - 7410v^5 - 2412v^3 - 252v) / 8
$\beta_{11}$	$=$	$ ( - 11 \nu^{15} + 8 \nu^{14} - 258 \nu^{13} + 192 \nu^{12} - 1971 \nu^{11} + 1534 \nu^{10} - 6311 \nu^{9} + \cdots + 164 ) / 8 $ (-11v^15 + 8v^14 - 258v^13 + 192v^12 - 1971v^11 + 1534v^10 - 6311v^9 + 5330v^8 - 8530v^7 + 8400v^6 - 4916v^5 + 6072v^4 - 1046v^3 + 1820v^2 - 38*v + 164) / 8
$\beta_{12}$	$=$	$ ( 11 \nu^{15} + 8 \nu^{14} + 258 \nu^{13} + 192 \nu^{12} + 1971 \nu^{11} + 1534 \nu^{10} + 6311 \nu^{9} + \cdots + 164 ) / 8 $ (11v^15 + 8v^14 + 258v^13 + 192v^12 + 1971v^11 + 1534v^10 + 6311v^9 + 5330v^8 + 8530v^7 + 8400v^6 + 4916v^5 + 6072v^4 + 1046v^3 + 1820v^2 + 38*v + 164) / 8
$\beta_{13}$	$=$	$ ( - 11 \nu^{15} + 12 \nu^{14} - 261 \nu^{13} + 282 \nu^{12} - 2040 \nu^{11} + 2164 \nu^{10} - 6817 \nu^{9} + \cdots + 136 ) / 8 $ (-11v^15 + 12v^14 - 261v^13 + 282v^12 - 2040v^11 + 2164v^10 - 6817v^9 + 7004v^8 - 10018v^7 + 9752v^6 - 6554v^5 + 6092v^4 - 1640v^3 + 1608v^2 - 82*v + 136) / 8
$\beta_{14}$	$=$	$ ( -19\nu^{15} - 451\nu^{13} - 3529\nu^{11} - 11832\nu^{9} - 17582\nu^{7} - 11966\nu^{5} - 3498\nu^{3} - 344\nu ) / 8 $ (-19v^15 - 451v^13 - 3529v^11 - 11832v^9 - 17582v^7 - 11966v^5 - 3498v^3 - 344v) / 8
$\beta_{15}$	$=$	$ ( 38\nu^{15} + 896\nu^{13} + 6921\nu^{11} + 22674\nu^{9} + 32332\nu^{7} + 20960\nu^{5} + 5842\nu^{3} + 540\nu ) / 8 $ (38v^15 + 896v^13 + 6921v^11 + 22674v^9 + 32332v^7 + 20960v^5 + 5842v^3 + 540v) / 8

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

$\nu$	$=$	$ ( -2\beta_{13} - \beta_{12} + \beta_{11} + \beta_{8} - 2\beta_{5} + 2\beta_{4} ) / 4 $ (-2b13 - b12 + b11 + b8 - 2b5 + 2*b4) / 4
$\nu^{2}$	$=$	$ ( \beta_{8} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 6 ) / 2 $ (b8 + b6 + 2*b5 + b4 - b3 - b2 + b1 - 6) / 2
$\nu^{3}$	$=$	$ ( - 3 \beta_{15} - 2 \beta_{14} + 8 \beta_{13} + 4 \beta_{12} - 4 \beta_{11} - 3 \beta_{10} + \cdots - 9 \beta_{4} ) / 2 $ (-3b15 - 2b14 + 8b13 + 4b12 - 4b11 - 3b10 - 3b9 - 3b8 - 2b7 + 9b5 - 9*b4) / 2
$\nu^{4}$	$=$	$ ( 4 \beta_{12} + 4 \beta_{11} - 13 \beta_{8} - 12 \beta_{6} - 20 \beta_{5} - 8 \beta_{4} + 12 \beta_{3} + \cdots + 48 ) / 2 $ (4b12 + 4b11 - 13b8 - 12b6 - 20b5 - 8b4 + 12b3 + 10b2 - 20*b1 + 48) / 2
$\nu^{5}$	$=$	$ ( 50 \beta_{15} + 31 \beta_{14} - 83 \beta_{13} - 47 \beta_{12} + 47 \beta_{11} + 55 \beta_{10} + \cdots + 95 \beta_{4} ) / 2 $ (50b15 + 31b14 - 83b13 - 47b12 + 47b11 + 55b10 + 40b9 + 27b8 + 29b7 - 95b5 + 95*b4) / 2
$\nu^{6}$	$=$	$ ( - 67 \beta_{12} - 67 \beta_{11} + 161 \beta_{8} + 144 \beta_{6} + 218 \beta_{5} + 74 \beta_{4} + \cdots - 504 ) / 2 $ (-67b12 - 67b11 + 161b8 + 144b6 + 218b5 + 74b4 - 134b3 - 114b2 + 280*b1 - 504) / 2
$\nu^{7}$	$=$	$ ( - 672 \beta_{15} - 413 \beta_{14} + 956 \beta_{13} + 571 \beta_{12} - 571 \beta_{11} + \cdots - 1091 \beta_{4} ) / 2 $ (-672b15 - 413b14 + 956b13 + 571b12 - 571b11 - 756b10 - 497b9 - 292b8 - 372b7 + 1091b5 - 1091*b4) / 2
$\nu^{8}$	$=$	$ 450 \beta_{12} + 450 \beta_{11} - 990 \beta_{8} - 876 \beta_{6} - 1272 \beta_{5} - 396 \beta_{4} + \cdots + 2910 $ 450b12 + 450b11 - 990b8 - 876b6 - 1272b5 - 396b4 + 784b3 + 680b2 - 1792*b1 + 2910
$\nu^{9}$	$=$	$ ( 8496 \beta_{15} + 5220 \beta_{14} - 11446 \beta_{13} - 6971 \beta_{12} + 6971 \beta_{11} + \cdots + 13034 \beta_{4} ) / 2 $ (8496b15 + 5220b14 - 11446b13 - 6971b12 + 6971b11 + 9624b10 + 6108b9 + 3407b8 + 4632b7 - 13034b5 + 13034*b4) / 2
$\nu^{10}$	$=$	$ - 5688 \beta_{12} - 5688 \beta_{11} + 12147 \beta_{8} + 10707 \beta_{6} + 15294 \beta_{5} + \cdots - 34858 $ -5688b12 - 5688b11 + 12147b8 + 10707b6 + 15294b5 + 4587b4 - 9423b3 - 8243b2 + 22287*b1 - 34858
$\nu^{11}$	$=$	$ - 52613 \beta_{15} - 32346 \beta_{14} + 69508 \beta_{13} + 42622 \beta_{12} - 42622 \beta_{11} + \cdots - 79079 \beta_{4} $ -52613b15 - 32346b14 + 69508b13 + 42622b12 - 42622b11 - 59741b10 - 37433b9 - 20483b8 - 28542b7 + 79079b5 - 79079*b4
$\nu^{12}$	$=$	$ 70448 \beta_{12} + 70448 \beta_{11} - 148879 \beta_{8} - 131044 \beta_{6} - 186020 \beta_{5} + \cdots + 423384 $ 70448b12 + 70448b11 - 148879b8 - 131044b6 - 186020b5 - 54976b4 + 114548b3 + 100538b2 - 274428*b1 + 423384
$\nu^{13}$	$=$	$ 646906 \beta_{15} + 397897 \beta_{14} - 848621 \beta_{13} - 521589 \beta_{12} + 521589 \beta_{11} + \cdots + 965133 \beta_{4} $ 646906b15 + 397897b14 - 848621b13 - 521589b12 + 521589b11 + 735189b10 + 458484b9 + 249125b8 + 350371b7 - 965133b5 + 965133*b4
$\nu^{14}$	$=$	$ - 866233 \beta_{12} - 866233 \beta_{11} + 1823791 \beta_{8} + 1604464 \beta_{6} + 2272238 \beta_{5} + \cdots - 5168936 $ -866233b12 - 866233b11 + 1823791b8 + 1604464b6 + 2272238b5 + 667774b4 - 1398722b3 - 1229198b2 + 3367136*b1 - 5168936
$\nu^{15}$	$=$	$ - 7933264 \beta_{15} - 4880807 \beta_{14} + 10380392 \beta_{13} + 6385193 \beta_{12} + \cdots - 11804009 \beta_{4} $ -7933264b15 - 4880807b14 + 10380392b13 + 6385193b12 - 6385193b11 - 9018824b10 - 5614479b9 - 3042952b8 - 4294488b7 + 11804009b5 - 11804009*b4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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} $

2351.1

		1.05636i
	−	0.357857i
		0.357857i
	−	1.05636i
	−	0.528036i
		0.886177i
	−	0.886177i
		0.528036i
	−	2.08509i
	−	3.49930i
		3.49930i
		2.08509i
	−	2.13875i
	−	0.724535i
		0.724535i
		2.13875i

−1.64533

−

0.541196i

−

3.29066i

2.41421

1.78089i

2351.2

−1.64533

−

0.541196i

3.29066i

2.41421

1.78089i

2351.3

−1.64533

0.541196i

−

3.29066i

2.41421

−

1.78089i

2351.4

−1.64533

0.541196i

3.29066i

2.41421

−

1.78089i

2351.5

−1.13705

−

1.30656i

−

2.27411i

−0.414214

2.97127i

2351.6

−1.13705

−

1.30656i

2.27411i

−0.414214

2.97127i

2351.7

−1.13705

1.30656i

−

2.27411i

−0.414214

−

2.97127i

2351.8

−1.13705

1.30656i

2.27411i

−0.414214

−

2.97127i

2351.9

1.13705

−

1.30656i

−

2.27411i

−0.414214

−

2.97127i

2351.10

1.13705

−

1.30656i

2.27411i

−0.414214

−

2.97127i

2351.11

1.13705

1.30656i

−

2.27411i

−0.414214

2.97127i

2351.12

1.13705

1.30656i

2.27411i

−0.414214

2.97127i

2351.13

1.64533

−

0.541196i

−

3.29066i

2.41421

−

1.78089i

2351.14

1.64533

−

0.541196i

3.29066i

2.41421

−

1.78089i

2351.15

1.64533

0.541196i

−

3.29066i

2.41421

1.78089i

2351.16

1.64533

0.541196i

3.29066i

2.41421

1.78089i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
8.d	odd	2	1	inner
15.d	odd	2	1	inner
24.f	even	2	1	inner
40.e	odd	2	1	inner
120.m	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2400.2.b.i		16
3.b	odd	2	1	inner	2400.2.b.i		16
4.b	odd	2	1		600.2.b.i		16
5.b	even	2	1	inner	2400.2.b.i		16
5.c	odd	4	2		480.2.m.b		16
8.b	even	2	1		600.2.b.i		16
8.d	odd	2	1	inner	2400.2.b.i		16
12.b	even	2	1		600.2.b.i		16
15.d	odd	2	1	inner	2400.2.b.i		16
15.e	even	4	2		480.2.m.b		16
20.d	odd	2	1		600.2.b.i		16
20.e	even	4	2		120.2.m.b	✓	16
24.f	even	2	1	inner	2400.2.b.i		16
24.h	odd	2	1		600.2.b.i		16
40.e	odd	2	1	inner	2400.2.b.i		16
40.f	even	2	1		600.2.b.i		16
40.i	odd	4	2		120.2.m.b	✓	16
40.k	even	4	2		480.2.m.b		16
60.h	even	2	1		600.2.b.i		16
60.l	odd	4	2		120.2.m.b	✓	16
120.i	odd	2	1		600.2.b.i		16
120.m	even	2	1	inner	2400.2.b.i		16
120.q	odd	4	2		480.2.m.b		16
120.w	even	4	2		120.2.m.b	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.2.m.b	✓	16	20.e	even	4	2
120.2.m.b	✓	16	40.i	odd	4	2
120.2.m.b	✓	16	60.l	odd	4	2
120.2.m.b	✓	16	120.w	even	4	2
480.2.m.b		16	5.c	odd	4	2
480.2.m.b		16	15.e	even	4	2
480.2.m.b		16	40.k	even	4	2
480.2.m.b		16	120.q	odd	4	2
600.2.b.i		16	4.b	odd	2	1
600.2.b.i		16	8.b	even	2	1
600.2.b.i		16	12.b	even	2	1
600.2.b.i		16	20.d	odd	2	1
600.2.b.i		16	24.h	odd	2	1
600.2.b.i		16	40.f	even	2	1
600.2.b.i		16	60.h	even	2	1
600.2.b.i		16	120.i	odd	2	1
2400.2.b.i		16	1.a	even	1	1	trivial
2400.2.b.i		16	3.b	odd	2	1	inner
2400.2.b.i		16	5.b	even	2	1	inner
2400.2.b.i		16	8.d	odd	2	1	inner
2400.2.b.i		16	15.d	odd	2	1	inner
2400.2.b.i		16	24.f	even	2	1	inner
2400.2.b.i		16	40.e	odd	2	1	inner
2400.2.b.i		16	120.m	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2400, [\chi])$:

$ T_{7}^{4} + 16T_{7}^{2} + 56 $

$ T_{11}^{4} + 24T_{11}^{2} + 112 $

$ T_{23}^{4} - 8T_{23}^{2} + 8 $

$ T_{43}^{4} - 112T_{43}^{2} + 2744 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 81)^{2} $ (T^8 - 4T^6 + 14T^4 - 36*T^2 + 81)^2
$5$	$ T^{16} $ T^16
$7$	$ (T^{4} + 16 T^{2} + 56)^{4} $ (T^4 + 16*T^2 + 56)^4
$11$	$ (T^{4} + 24 T^{2} + 112)^{4} $ (T^4 + 24*T^2 + 112)^4
$13$	$ (T^{4} + 32 T^{2} + 224)^{4} $ (T^4 + 32*T^2 + 224)^4
$17$	$ (T^{4} + 16 T^{2} + 32)^{4} $ (T^4 + 16*T^2 + 32)^4
$19$	$ (T^{2} + 4 T - 4)^{8} $ (T^2 + 4*T - 4)^8
$23$	$ (T^{4} - 8 T^{2} + 8)^{4} $ (T^4 - 8*T^2 + 8)^4
$29$	$ (T^{4} - 40 T^{2} + 112)^{4} $ (T^4 - 40*T^2 + 112)^4
$31$	$ (T^{4} + 48 T^{2} + 64)^{4} $ (T^4 + 48*T^2 + 64)^4
$37$	$ (T^{4} + 64 T^{2} + 224)^{4} $ (T^4 + 64*T^2 + 224)^4
$41$	$ (T^{4} + 80 T^{2} + 448)^{4} $ (T^4 + 80*T^2 + 448)^4
$43$	$ (T^{4} - 112 T^{2} + 2744)^{4} $ (T^4 - 112*T^2 + 2744)^4
$47$	$ (T^{4} - 8 T^{2} + 8)^{4} $ (T^4 - 8*T^2 + 8)^4
$53$	$ (T^{4} - 144 T^{2} + 2592)^{4} $ (T^4 - 144*T^2 + 2592)^4
$59$	$ (T^{4} + 24 T^{2} + 112)^{4} $ (T^4 + 24*T^2 + 112)^4
$61$	$ (T^{2} + 72)^{8} $ (T^2 + 72)^8
$67$	$ (T^{4} - 16 T^{2} + 56)^{4} $ (T^4 - 16*T^2 + 56)^4
$71$	$ (T^{4} - 192 T^{2} + 7168)^{4} $ (T^4 - 192*T^2 + 7168)^4
$73$	$ (T^{4} - 64 T^{2} + 896)^{4} $ (T^4 - 64*T^2 + 896)^4
$79$	$ (T^{4} + 272 T^{2} + 64)^{4} $ (T^4 + 272*T^2 + 64)^4
$83$	$ (T^{4} + 136 T^{2} + 4232)^{4} $ (T^4 + 136*T^2 + 4232)^4
$89$	$ (T^{4} + 96 T^{2} + 1792)^{4} $ (T^4 + 96*T^2 + 1792)^4
$97$	$ (T^{4} - 128 T^{2} + 896)^{4} $ (T^4 - 128*T^2 + 896)^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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{3} - \beta_{15} q^{7} + ( - \beta_{13} + \beta_{8} + 1) q^{9}+O(q^{10}) \) q - b5 * q^3 - b15 * q^7 + (-b13 + b8 + 1) * q^9	\( q - \beta_{5} q^{3} - \beta_{15} q^{7} + ( - \beta_{13} + \beta_{8} + 1) q^{9} + (\beta_{13} - \beta_{7}) q^{11} - \beta_{9} q^{13} + ( - \beta_{14} - \beta_{5} + \beta_{4}) q^{17} + (\beta_{8} - 2) q^{19} + (\beta_{11} - 2 \beta_{10}) q^{21} + \beta_{3} q^{23} + ( - \beta_{14} + \beta_{6} + \cdots + \beta_{4}) q^{27}+ \cdots + ( - \beta_{13} - 2 \beta_{8} + \cdots + 4) q^{99}+O(q^{100}) \) q - b5 * q^3 - b15 * q^7 + (-b13 + b8 + 1) * q^9 + (b13 - b7) * q^11 - b9 * q^13 + (-b14 - b5 + b4) * q^17 + (b8 - 2) * q^19 + (b11 - 2b10) q^21 + b3 * q^23 + (-b14 + b6 - b5 + b4) * q^27 + (b12 + b11 - b2) * q^29 + (b12 - b11 - 2b10) q^31 + (-2b14 - b6 - b4) q^33 + (-2b15 - b9) q^37 + (2b12 - 2b11 + b10 + b2) * q^39 + (b8 - 2b7) q^41 + (2b6 + b5 - b4) q^43 + b3 * q^47 + (-b8 - 1) * q^49 + (-b13 + b8 - b7 - 2) * q^51 + (-3b3 + 3b1) * q^53 + (-b14 + b6 + 3b5 - 2b4) * q^57 + (b13 - b7) * q^59 + (3b12 - 3b11) * q^61 + (-b15 + b9 - b3 - 3b1) q^63 + (-b5 - b4) * q^67 + (-b12 + b2) * q^69 + (2b12 + 2b11) * q^71 + (2b5 + 2b4) * q^73 + (-4b3 + 2b1) * q^77 + (-3b12 + 3b11 - 4b10) q^79 + (3b8 - 2b7 - 3) * q^81 + (4b14 - b5 + b4) q^83 + (b15 - 3b3 - 2b1) * q^87 + (2b13 - 2b7) * q^89 + (4b8 + 4) q^91 + (-2b15 - b9 + b3 - 3b1) * q^93 + (-2b6 - 2b5) * q^97 + (-b13 - 2b8 - b7 + 4) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^9	\( 16 q + 16 q^{9} - 32 q^{19} - 16 q^{49} - 32 q^{51} - 48 q^{81} + 64 q^{91} + 64 q^{99}+O(q^{100}) \) 16 * q + 16 * q^9 - 32 * q^19 - 16 * q^49 - 32 * q^51 - 48 * q^81 + 64 * q^91 + 64 * q^99

Introduction

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Modular forms

Varieties

Fields

Representations

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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	2400.2.b.i

Level	$2400$
Weight	$2$
Character orbit	2400.b
Analytic conductor	$19.164$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(19.1640964851\)
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:	\( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) x^16 + 24x^14 + 192x^12 + 672x^10 + 1092x^8 + 880x^6 + 352x^4 + 64*x^2 + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index:	\( 2^{19} \)
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{14} - 20\nu^{12} - 100\nu^{10} + 4\nu^{8} + 918\nu^{6} + 1440\nu^{4} + 664\nu^{2} + 72 ) / 8 \) (-v^14 - 20v^12 - 100v^10 + 4v^8 + 918v^6 + 1440v^4 + 664v^2 + 72) / 8
\(\beta_{2}\)	\(=\)	\( ( -3\nu^{14} - 69\nu^{12} - 506\nu^{10} - 1488\nu^{8} - 1638\nu^{6} - 594\nu^{4} - 36\nu^{2} - 8 ) / 4 \) (-3v^14 - 69v^12 - 506v^10 - 1488v^8 - 1638v^6 - 594v^4 - 36*v^2 - 8) / 4
\(\beta_{3}\)	\(=\)	\( ( -7\nu^{14} - 164\nu^{12} - 1250\nu^{10} - 3984\nu^{8} - 5334\nu^{6} - 3024\nu^{4} - 596\nu^{2} - 16 ) / 8 \) (-7v^14 - 164v^12 - 1250v^10 - 3984v^8 - 5334v^6 - 3024v^4 - 596*v^2 - 16) / 8
\(\beta_{4}\)	\(=\)	\( ( - 15 \nu^{14} - 3 \nu^{13} - 352 \nu^{12} - 69 \nu^{11} - 2692 \nu^{10} - 506 \nu^{9} - 8638 \nu^{8} + \cdots - 76 ) / 16 \) (-15v^14 - 3v^13 - 352v^12 - 69v^11 - 2692v^10 - 506v^9 - 8638v^8 - 1488v^7 - 11726v^6 - 1638v^5 - 6808v^4 - 594v^3 - 1496v^2 - 28v - 76) / 16
\(\beta_{5}\)	\(=\)	\( ( - 15 \nu^{14} + 3 \nu^{13} - 352 \nu^{12} + 69 \nu^{11} - 2692 \nu^{10} + 506 \nu^{9} - 8638 \nu^{8} + \cdots - 76 ) / 16 \) (-15v^14 + 3v^13 - 352v^12 + 69v^11 - 2692v^10 + 506v^9 - 8638v^8 + 1488v^7 - 11726v^6 + 1638v^5 - 6808v^4 + 594v^3 - 1496v^2 + 28v - 76) / 16
\(\beta_{6}\)	\(=\)	\( ( - 27 \nu^{14} - 3 \nu^{13} - 636 \nu^{12} - 69 \nu^{11} - 4904 \nu^{10} - 506 \nu^{9} - 16030 \nu^{8} + \cdots - 428 ) / 16 \) (-27v^14 - 3v^13 - 636v^12 - 69v^11 - 4904v^10 - 506v^9 - 16030v^8 - 1488v^7 - 22886v^6 - 1638v^5 - 15248v^4 - 594v^3 - 4576v^2 - 28v - 428) / 16
\(\beta_{7}\)	\(=\)	\( ( - 5 \nu^{15} + 12 \nu^{14} - 110 \nu^{13} + 282 \nu^{12} - 728 \nu^{11} + 2164 \nu^{10} - 1625 \nu^{9} + \cdots + 136 ) / 8 \) (-5v^15 + 12v^14 - 110v^13 + 282v^12 - 728v^11 + 2164v^10 - 1625v^9 + 7004v^8 - 118v^7 + 9752v^6 + 2276v^5 + 6092v^4 + 1648v^3 + 1608v^2 + 270*v + 136) / 8
\(\beta_{8}\)	\(=\)	\( ( 6\nu^{14} + 141\nu^{12} + 1082\nu^{10} + 3502\nu^{8} + 4876\nu^{6} + 3046\nu^{4} + 804\nu^{2} + 68 ) / 2 \) (6v^14 + 141v^12 + 1082v^10 + 3502v^8 + 4876v^6 + 3046v^4 + 804*v^2 + 68) / 2
\(\beta_{9}\)	\(=\)	\( ( -11\nu^{15} - 260\nu^{13} - 2018\nu^{11} - 6672\nu^{9} - 9702\nu^{7} - 6544\nu^{5} - 2004\nu^{3} - 272\nu ) / 8 \) (-11v^15 - 260v^13 - 2018v^11 - 6672v^9 - 9702v^7 - 6544v^5 - 2004v^3 - 272v) / 8
\(\beta_{10}\)	\(=\)	\( ( -11\nu^{15} - 261\nu^{13} - 2042\nu^{11} - 6862\nu^{9} - 10334\nu^{7} - 7410\nu^{5} - 2412\nu^{3} - 252\nu ) / 8 \) (-11v^15 - 261v^13 - 2042v^11 - 6862v^9 - 10334v^7 - 7410v^5 - 2412v^3 - 252v) / 8
\(\beta_{11}\)	\(=\)	\( ( - 11 \nu^{15} + 8 \nu^{14} - 258 \nu^{13} + 192 \nu^{12} - 1971 \nu^{11} + 1534 \nu^{10} - 6311 \nu^{9} + \cdots + 164 ) / 8 \) (-11v^15 + 8v^14 - 258v^13 + 192v^12 - 1971v^11 + 1534v^10 - 6311v^9 + 5330v^8 - 8530v^7 + 8400v^6 - 4916v^5 + 6072v^4 - 1046v^3 + 1820v^2 - 38*v + 164) / 8
\(\beta_{12}\)	\(=\)	\( ( 11 \nu^{15} + 8 \nu^{14} + 258 \nu^{13} + 192 \nu^{12} + 1971 \nu^{11} + 1534 \nu^{10} + 6311 \nu^{9} + \cdots + 164 ) / 8 \) (11v^15 + 8v^14 + 258v^13 + 192v^12 + 1971v^11 + 1534v^10 + 6311v^9 + 5330v^8 + 8530v^7 + 8400v^6 + 4916v^5 + 6072v^4 + 1046v^3 + 1820v^2 + 38*v + 164) / 8
\(\beta_{13}\)	\(=\)	\( ( - 11 \nu^{15} + 12 \nu^{14} - 261 \nu^{13} + 282 \nu^{12} - 2040 \nu^{11} + 2164 \nu^{10} - 6817 \nu^{9} + \cdots + 136 ) / 8 \) (-11v^15 + 12v^14 - 261v^13 + 282v^12 - 2040v^11 + 2164v^10 - 6817v^9 + 7004v^8 - 10018v^7 + 9752v^6 - 6554v^5 + 6092v^4 - 1640v^3 + 1608v^2 - 82*v + 136) / 8
\(\beta_{14}\)	\(=\)	\( ( -19\nu^{15} - 451\nu^{13} - 3529\nu^{11} - 11832\nu^{9} - 17582\nu^{7} - 11966\nu^{5} - 3498\nu^{3} - 344\nu ) / 8 \) (-19v^15 - 451v^13 - 3529v^11 - 11832v^9 - 17582v^7 - 11966v^5 - 3498v^3 - 344v) / 8
\(\beta_{15}\)	\(=\)	\( ( 38\nu^{15} + 896\nu^{13} + 6921\nu^{11} + 22674\nu^{9} + 32332\nu^{7} + 20960\nu^{5} + 5842\nu^{3} + 540\nu ) / 8 \) (38v^15 + 896v^13 + 6921v^11 + 22674v^9 + 32332v^7 + 20960v^5 + 5842v^3 + 540v) / 8

\(\nu\)	\(=\)	\( ( -2\beta_{13} - \beta_{12} + \beta_{11} + \beta_{8} - 2\beta_{5} + 2\beta_{4} ) / 4 \) (-2b13 - b12 + b11 + b8 - 2b5 + 2*b4) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{8} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 6 ) / 2 \) (b8 + b6 + 2*b5 + b4 - b3 - b2 + b1 - 6) / 2
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{15} - 2 \beta_{14} + 8 \beta_{13} + 4 \beta_{12} - 4 \beta_{11} - 3 \beta_{10} + \cdots - 9 \beta_{4} ) / 2 \) (-3b15 - 2b14 + 8b13 + 4b12 - 4b11 - 3b10 - 3b9 - 3b8 - 2b7 + 9b5 - 9*b4) / 2
\(\nu^{4}\)	\(=\)	\( ( 4 \beta_{12} + 4 \beta_{11} - 13 \beta_{8} - 12 \beta_{6} - 20 \beta_{5} - 8 \beta_{4} + 12 \beta_{3} + \cdots + 48 ) / 2 \) (4b12 + 4b11 - 13b8 - 12b6 - 20b5 - 8b4 + 12b3 + 10b2 - 20*b1 + 48) / 2
\(\nu^{5}\)	\(=\)	\( ( 50 \beta_{15} + 31 \beta_{14} - 83 \beta_{13} - 47 \beta_{12} + 47 \beta_{11} + 55 \beta_{10} + \cdots + 95 \beta_{4} ) / 2 \) (50b15 + 31b14 - 83b13 - 47b12 + 47b11 + 55b10 + 40b9 + 27b8 + 29b7 - 95b5 + 95*b4) / 2
\(\nu^{6}\)	\(=\)	\( ( - 67 \beta_{12} - 67 \beta_{11} + 161 \beta_{8} + 144 \beta_{6} + 218 \beta_{5} + 74 \beta_{4} + \cdots - 504 ) / 2 \) (-67b12 - 67b11 + 161b8 + 144b6 + 218b5 + 74b4 - 134b3 - 114b2 + 280*b1 - 504) / 2
\(\nu^{7}\)	\(=\)	\( ( - 672 \beta_{15} - 413 \beta_{14} + 956 \beta_{13} + 571 \beta_{12} - 571 \beta_{11} + \cdots - 1091 \beta_{4} ) / 2 \) (-672b15 - 413b14 + 956b13 + 571b12 - 571b11 - 756b10 - 497b9 - 292b8 - 372b7 + 1091b5 - 1091*b4) / 2
\(\nu^{8}\)	\(=\)	\( 450 \beta_{12} + 450 \beta_{11} - 990 \beta_{8} - 876 \beta_{6} - 1272 \beta_{5} - 396 \beta_{4} + \cdots + 2910 \) 450b12 + 450b11 - 990b8 - 876b6 - 1272b5 - 396b4 + 784b3 + 680b2 - 1792*b1 + 2910
\(\nu^{9}\)	\(=\)	\( ( 8496 \beta_{15} + 5220 \beta_{14} - 11446 \beta_{13} - 6971 \beta_{12} + 6971 \beta_{11} + \cdots + 13034 \beta_{4} ) / 2 \) (8496b15 + 5220b14 - 11446b13 - 6971b12 + 6971b11 + 9624b10 + 6108b9 + 3407b8 + 4632b7 - 13034b5 + 13034*b4) / 2
\(\nu^{10}\)	\(=\)	\( - 5688 \beta_{12} - 5688 \beta_{11} + 12147 \beta_{8} + 10707 \beta_{6} + 15294 \beta_{5} + \cdots - 34858 \) -5688b12 - 5688b11 + 12147b8 + 10707b6 + 15294b5 + 4587b4 - 9423b3 - 8243b2 + 22287*b1 - 34858
\(\nu^{11}\)	\(=\)	\( - 52613 \beta_{15} - 32346 \beta_{14} + 69508 \beta_{13} + 42622 \beta_{12} - 42622 \beta_{11} + \cdots - 79079 \beta_{4} \) -52613b15 - 32346b14 + 69508b13 + 42622b12 - 42622b11 - 59741b10 - 37433b9 - 20483b8 - 28542b7 + 79079b5 - 79079*b4
\(\nu^{12}\)	\(=\)	\( 70448 \beta_{12} + 70448 \beta_{11} - 148879 \beta_{8} - 131044 \beta_{6} - 186020 \beta_{5} + \cdots + 423384 \) 70448b12 + 70448b11 - 148879b8 - 131044b6 - 186020b5 - 54976b4 + 114548b3 + 100538b2 - 274428*b1 + 423384
\(\nu^{13}\)	\(=\)	\( 646906 \beta_{15} + 397897 \beta_{14} - 848621 \beta_{13} - 521589 \beta_{12} + 521589 \beta_{11} + \cdots + 965133 \beta_{4} \) 646906b15 + 397897b14 - 848621b13 - 521589b12 + 521589b11 + 735189b10 + 458484b9 + 249125b8 + 350371b7 - 965133b5 + 965133*b4
\(\nu^{14}\)	\(=\)	\( - 866233 \beta_{12} - 866233 \beta_{11} + 1823791 \beta_{8} + 1604464 \beta_{6} + 2272238 \beta_{5} + \cdots - 5168936 \) -866233b12 - 866233b11 + 1823791b8 + 1604464b6 + 2272238b5 + 667774b4 - 1398722b3 - 1229198b2 + 3367136*b1 - 5168936
\(\nu^{15}\)	\(=\)	\( - 7933264 \beta_{15} - 4880807 \beta_{14} + 10380392 \beta_{13} + 6385193 \beta_{12} + \cdots - 11804009 \beta_{4} \) -7933264b15 - 4880807b14 + 10380392b13 + 6385193b12 - 6385193b11 - 9018824b10 - 5614479b9 - 3042952b8 - 4294488b7 + 11804009b5 - 11804009*b4

\(n\)	\(577\)	\(901\)	\(1601\)	\(1951\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2351.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 81)^{2} \) (T^8 - 4T^6 + 14T^4 - 36*T^2 + 81)^2
$5$	\( T^{16} \) T^16
$7$	\( (T^{4} + 16 T^{2} + 56)^{4} \) (T^4 + 16*T^2 + 56)^4
$11$	\( (T^{4} + 24 T^{2} + 112)^{4} \) (T^4 + 24*T^2 + 112)^4
$13$	\( (T^{4} + 32 T^{2} + 224)^{4} \) (T^4 + 32*T^2 + 224)^4
$17$	\( (T^{4} + 16 T^{2} + 32)^{4} \) (T^4 + 16*T^2 + 32)^4
$19$	\( (T^{2} + 4 T - 4)^{8} \) (T^2 + 4*T - 4)^8
$23$	\( (T^{4} - 8 T^{2} + 8)^{4} \) (T^4 - 8*T^2 + 8)^4
$29$	\( (T^{4} - 40 T^{2} + 112)^{4} \) (T^4 - 40*T^2 + 112)^4
$31$	\( (T^{4} + 48 T^{2} + 64)^{4} \) (T^4 + 48*T^2 + 64)^4
$37$	\( (T^{4} + 64 T^{2} + 224)^{4} \) (T^4 + 64*T^2 + 224)^4
$41$	\( (T^{4} + 80 T^{2} + 448)^{4} \) (T^4 + 80*T^2 + 448)^4
$43$	\( (T^{4} - 112 T^{2} + 2744)^{4} \) (T^4 - 112*T^2 + 2744)^4
$47$	\( (T^{4} - 8 T^{2} + 8)^{4} \) (T^4 - 8*T^2 + 8)^4
$53$	\( (T^{4} - 144 T^{2} + 2592)^{4} \) (T^4 - 144*T^2 + 2592)^4
$59$	\( (T^{4} + 24 T^{2} + 112)^{4} \) (T^4 + 24*T^2 + 112)^4
$61$	\( (T^{2} + 72)^{8} \) (T^2 + 72)^8
$67$	\( (T^{4} - 16 T^{2} + 56)^{4} \) (T^4 - 16*T^2 + 56)^4
$71$	\( (T^{4} - 192 T^{2} + 7168)^{4} \) (T^4 - 192*T^2 + 7168)^4
$73$	\( (T^{4} - 64 T^{2} + 896)^{4} \) (T^4 - 64*T^2 + 896)^4
$79$	\( (T^{4} + 272 T^{2} + 64)^{4} \) (T^4 + 272*T^2 + 64)^4
$83$	\( (T^{4} + 136 T^{2} + 4232)^{4} \) (T^4 + 136*T^2 + 4232)^4
$89$	\( (T^{4} + 96 T^{2} + 1792)^{4} \) (T^4 + 96*T^2 + 1792)^4
$97$	\( (T^{4} - 128 T^{2} + 896)^{4} \) (T^4 - 128*T^2 + 896)^4
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