LMFDB - Newform orbit 448.2.j.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(111,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

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

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

chi := DirichletCharacter("448.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.j (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 $ x^16 - 4x^14 + 6x^12 - 12x^10 + 33x^8 - 48x^6 + 96x^4 - 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{6} q^{3} + \beta_{5} q^{5} + (\beta_{13} - 1) q^{7} + (\beta_{15} - 3 \beta_{8} + \cdots + \beta_1) q^{9}+O(q^{10}) $ q - b6 * q^3 + b5 * q^5 + (b13 - 1) * q^7 + (b15 - 3b8 - b4 + b1) q^9	$ q - \beta_{6} q^{3} + \beta_{5} q^{5} + (\beta_{13} - 1) q^{7} + (\beta_{15} - 3 \beta_{8} + \cdots + \beta_1) q^{9}+ \cdots + (4 \beta_{15} - 6 \beta_{8} - 2 \beta_{7} + \cdots - 6) q^{99}+O(q^{100}) $ q - b6 * q^3 + b5 * q^5 + (b13 - 1) * q^7 + (b15 - 3b8 - b4 + b1) q^9 + (-2b8 + 2) q^11 + (b12 + b2) * q^13 + (b10 + b8 - b4 - b1) * q^15 + (b15 + 2b13 - b5 - b2 - b1 - 2) q^17 + (b13 - b11 - b6) * q^19 + (-b15 + b14 + b10 + 2b8 + b7 - b5 + b4 + b3 + 2) q^21 + (-b15 + b14 + b7 + b1 + 1) * q^23 + (-b15 + b8 - b4 - b1) * q^25 + (-b15 - b13 + 2b12 - b11 + b1 + 2) q^27 + (-b15 + b14 - b10 - b7 + b4) * q^29 + (-b12 - b9 + b6 + b3) * q^31 + (-2b9 - 2b6) * q^33 + (-b14 - b10 - b8 + b6 - 2b5 - 1) q^35 + (-b15 - b14 - b10) * q^37 + (2b15 + b14 - 3b7 - 2b1 - 3) q^39 + (-b15 + b12 - 2b11 + b5 - b3 - b2 + b1 + 2) q^41 + (-b15 + b14 - b10 - b8 + b7 - b4 + 2b1 + 1) q^43 + (-b12 + b2) * q^45 + (b15 + b12 + 2b11 - b9 + b6 + 2b5 - b3 - 2b2 - b1 - 2) q^47 + (-b12 + 2b7 - b3 - 1) q^49 + (-b15 + b14 + b10 + 3b8 + 3b7 + 3b4 + 3) q^51 + (b15 + b14 + b10 - 2b7 - 2b4) * q^53 + (2b5 + 2b2) * q^55 + (b15 + 2b10 - 2b8 + b4 - b1) * q^57 + (b15 + b13 + b11 + b9 - b1 - 2) * q^59 + (-2b15 - 2b13 - 2b11 + 4b9 - b2 + 2b1 + 4) q^61 + (2b15 - b12 + b11 - b10 + b9 - 5b8 - b6 - b4 + b3 + 2b1 - 1) q^63 + (-b15 - 2b14 + b7 + b1 - 4) q^65 + (-b15 - b14 - b10 + 3b8 - b7 - b4 + 3) q^67 + (-2b13 + 2b11 + b3) * q^69 + (2b15 - 4b14 - 2b7 - 2b1 - 4) * q^71 + (-b12 + b3) * q^73 + (-b15 - b13 - 2b12 - b11 + b9 + 4b2 + b1 + 2) * q^75 + (2b15 + 2b13 + 2b11 - 4) q^77 + (2b10 + 6b8 + 4b4 - 2b1) * q^79 + (5b15 - 2b14 - 3b7 - 5b1 - 3) * q^81 + (-2b13 + 2b11 + b6 - 2b3) q^83 + (2b15 - 2b14 - 2b10 + 2b8 + b7 + b4 + 2) * q^85 + (-b12 + b9 + b6 - 2b5 - b3 - 2b2) * q^87 + (2b15 + 4b11 - 2b9 + 2b6 - 2b5 + 2b2 - 2b1 - 4) q^89 + (b15 - 2b14 - b13 - b11 + 2b10 - b9 + b7 - b4 - 2b2 + 2b1 + 2) * q^91 + (2b8 + b7 - b4 + 4b1 - 2) * q^93 + (-b15 - b10 - b8 - b4) * q^95 + (-b15 - 2b13 - b12 + 2b9 + 2b6 + b5 - b3 + b2 + b1 + 2) q^97 + (4b15 - 6b8 - 2b7 - 2b4 - 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{7}+O(q^{10}) $ 16 * q - 8 * q^7	$ 16 q - 8 q^{7} + 32 q^{11} + 16 q^{21} - 8 q^{35} - 16 q^{39} - 16 q^{49} + 32 q^{51} - 80 q^{65} + 48 q^{67} - 32 q^{71} - 16 q^{77} + 32 q^{81} + 64 q^{85} - 8 q^{91} - 64 q^{93} - 64 q^{99}+O(q^{100}) $ 16 * q - 8 * q^7 + 32 * q^11 + 16 * q^21 - 8 * q^35 - 16 * q^39 - 16 * q^49 + 32 * q^51 - 80 * q^65 + 48 * q^67 - 32 * q^71 - 16 * q^77 + 32 * q^81 + 64 * q^85 - 8 * q^91 - 64 * q^93 - 64 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 $ x^16 - 4*x^14 + 6*x^12 - 12*x^10 + 33*x^8 - 48*x^6 + 96*x^4 - 256*x^2 + 256 :

$\beta_{1}$	$=$	$ ( \nu^{14} + 84\nu^{12} - 42\nu^{10} + 132\nu^{8} - 351\nu^{6} - 216\nu^{4} + 528\nu^{2} + 2048 ) / 1920 $ (v^14 + 84v^12 - 42v^10 + 132v^8 - 351v^6 - 216v^4 + 528v^2 + 2048) / 1920
$\beta_{2}$	$=$	$ ( -13\nu^{15} + 48\nu^{13} - 14\nu^{11} + 4\nu^{9} - 477\nu^{7} + 428\nu^{5} - 1904\nu^{3} + 5696\nu ) / 2560 $ (-13v^15 + 48v^13 - 14v^11 + 4v^9 - 477v^7 + 428v^5 - 1904v^3 + 5696v) / 2560
$\beta_{3}$	$=$	$ ( -13\nu^{15} + 88\nu^{13} - 14\nu^{11} + 244\nu^{9} - 637\nu^{7} + 468\nu^{5} - 1744\nu^{3} + 4416\nu ) / 1280 $ (-13v^15 + 88v^13 - 14v^11 + 244v^9 - 637v^7 + 468v^5 - 1744v^3 + 4416v) / 1280
$\beta_{4}$	$=$	$ ( -13\nu^{14} - 12\nu^{12} + 66\nu^{10} - 36\nu^{8} + 243\nu^{6} - 912\nu^{4} + 1776\nu^{2} - 1664 ) / 1920 $ (-13v^14 - 12v^12 + 66v^10 - 36v^8 + 243v^6 - 912v^4 + 1776*v^2 - 1664) / 1920
$\beta_{5}$	$=$	$ ( -47\nu^{15} + 112\nu^{13} - 26\nu^{11} + 236\nu^{9} - 543\nu^{7} - 188\nu^{5} - 2576\nu^{3} + 4544\nu ) / 2560 $ (-47v^15 + 112v^13 - 26v^11 + 236v^9 - 543v^7 - 188v^5 - 2576v^3 + 4544v) / 2560
$\beta_{6}$	$=$	$ ( -53\nu^{15} + 88\nu^{13} - 94\nu^{11} + 404\nu^{9} - 997\nu^{7} + 948\nu^{5} - 4144\nu^{3} + 3776\nu ) / 2560 $ (-53v^15 + 88v^13 - 94v^11 + 404v^9 - 997v^7 + 948v^5 - 4144v^3 + 3776v) / 2560
$\beta_{7}$	$=$	$ ( -\nu^{14} + 2\nu^{12} + 2\nu^{10} - 9\nu^{6} + 14\nu^{4} - 64\nu^{2} + 96 ) / 64 $ (-v^14 + 2v^12 + 2v^10 - 9v^6 + 14v^4 - 64*v^2 + 96) / 64
$\beta_{8}$	$=$	$ ( -43\nu^{14} + 108\nu^{12} - 114\nu^{10} + 324\nu^{8} - 747\nu^{6} + 528\nu^{4} - 3024\nu^{2} + 6016 ) / 1920 $ (-43v^14 + 108v^12 - 114v^10 + 324v^8 - 747v^6 + 528v^4 - 3024*v^2 + 6016) / 1920
$\beta_{9}$	$=$	$ ( -127\nu^{15} + 312\nu^{13} - 346\nu^{11} + 1116\nu^{9} - 2383\nu^{7} + 2252\nu^{5} - 8656\nu^{3} + 17984\nu ) / 2560 $ (-127v^15 + 312v^13 - 346v^11 + 1116v^9 - 2383v^7 + 2252v^5 - 8656v^3 + 17984v) / 2560
$\beta_{10}$	$=$	$ ( -71\nu^{14} + 216\nu^{12} - 138\nu^{10} + 828\nu^{8} - 2199\nu^{6} + 1356\nu^{4} - 5328\nu^{2} + 12992 ) / 1920 $ (-71v^14 + 216v^12 - 138v^10 + 828v^8 - 2199v^6 + 1356v^4 - 5328*v^2 + 12992) / 1920
$\beta_{11}$	$=$	$ ( - 13 \nu^{15} - 90 \nu^{14} + 48 \nu^{13} + 240 \nu^{12} - 14 \nu^{11} - 220 \nu^{10} + 4 \nu^{9} + \cdots + 14720 ) / 2560 $ (-13v^15 - 90v^14 + 48v^13 + 240v^12 - 14v^11 - 220v^10 + 4v^9 + 1000v^8 - 477v^7 - 1850v^6 + 428v^5 + 1640v^4 + 656v^3 - 6240v^2 + 3136*v + 14720) / 2560
$\beta_{12}$	$=$	$ ( 59\nu^{15} - 104\nu^{13} + 82\nu^{11} - 492\nu^{9} + 811\nu^{7} - 844\nu^{5} + 3872\nu^{3} - 6528\nu ) / 640 $ (59v^15 - 104v^13 + 82v^11 - 492v^9 + 811v^7 - 844v^5 + 3872v^3 - 6528v) / 640
$\beta_{13}$	$=$	$ ( - 15 \nu^{15} - 18 \nu^{14} + 32 \nu^{13} + 48 \nu^{12} - 26 \nu^{11} - 44 \nu^{10} + 140 \nu^{9} + \cdots + 2944 ) / 512 $ (-15v^15 - 18v^14 + 32v^13 + 48v^12 - 26v^11 - 44v^10 + 140v^9 + 200v^8 - 319v^7 - 370v^6 + 116v^5 + 328v^4 - 1040v^3 - 1248v^2 + 2240*v + 2944) / 512
$\beta_{14}$	$=$	$ ( 7\nu^{14} - 16\nu^{12} + 10\nu^{10} - 76\nu^{8} + 119\nu^{6} - 68\nu^{4} + 560\nu^{2} - 960 ) / 128 $ (7v^14 - 16v^12 + 10v^10 - 76v^8 + 119v^6 - 68v^4 + 560*v^2 - 960) / 128
$\beta_{15}$	$=$	$ ( 34\nu^{14} - 69\nu^{12} + 72\nu^{10} - 342\nu^{8} + 606\nu^{6} - 669\nu^{4} + 2472\nu^{2} - 4048 ) / 480 $ (34v^14 - 69v^12 + 72v^10 - 342v^8 + 606v^6 - 669v^4 + 2472*v^2 - 4048) / 480

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

$\nu$	$=$	$ ( \beta_{15} + 2\beta_{13} - \beta_{12} - 6\beta_{6} - 2\beta_{5} - \beta_{3} + 4\beta_{2} - \beta _1 - 2 ) / 12 $ (b15 + 2b13 - b12 - 6b6 - 2b5 - b3 + 4b2 - b1 - 2) / 12
$\nu^{2}$	$=$	$ ( -\beta_{15} + \beta_{14} - \beta_{8} + \beta_{4} + \beta _1 + 2 ) / 2 $ (-b15 + b14 - b8 + b4 + b1 + 2) / 2
$\nu^{3}$	$=$	$ ( 7 \beta_{15} + 2 \beta_{13} - \beta_{12} + 12 \beta_{11} - 6 \beta_{6} - 2 \beta_{5} - \beta_{3} + \cdots - 14 ) / 12 $ (7b15 + 2b13 - b12 + 12b11 - 6b6 - 2b5 - b3 - 8b2 - 7*b1 - 14) / 12
$\nu^{4}$	$=$	$ ( -3\beta_{15} + 2\beta_{14} - \beta_{10} - 4\beta_{8} + 2\beta_{7} - \beta_{4} + 2\beta _1 + 3 ) / 2 $ (-3b15 + 2b14 - b10 - 4b8 + 2b7 - b4 + 2*b1 + 3) / 2
$\nu^{5}$	$=$	$ ( - 5 \beta_{15} - 22 \beta_{13} - 7 \beta_{12} + 12 \beta_{11} + 6 \beta_{6} - 14 \beta_{5} + 5 \beta_{3} + \cdots + 10 ) / 12 $ (-5b15 - 22b13 - 7b12 + 12b11 + 6b6 - 14b5 + 5b3 + 4b2 + 5*b1 + 10) / 12
$\nu^{6}$	$=$	$ ( -3\beta_{15} - \beta_{14} - 6\beta_{10} - 3\beta_{8} + 2\beta_{7} - \beta_{4} + 5\beta _1 + 8 ) / 2 $ (-3b15 - b14 - 6b10 - 3b8 + 2b7 - b4 + 5*b1 + 8) / 2
$\nu^{7}$	$=$	$ ( - 29 \beta_{15} - 34 \beta_{13} - 13 \beta_{12} - 24 \beta_{11} + 12 \beta_{9} - 42 \beta_{6} + \cdots + 58 ) / 12 $ (-29b15 - 34b13 - 13b12 - 24b11 + 12b9 - 42b6 + 10b5 - b3 + 4b2 + 29*b1 + 58) / 12
$\nu^{8}$	$=$	$ ( -5\beta_{15} - 4\beta_{14} - 3\beta_{10} - 18\beta_{8} - 4\beta_{7} + \beta_{4} + 6\beta _1 + 5 ) / 2 $ (-5b15 - 4b14 - 3b10 - 18b8 - 4b7 + b4 + 6b1 + 5) / 2
$\nu^{9}$	$=$	$ ( 7 \beta_{15} + 38 \beta_{13} - 19 \beta_{12} - 24 \beta_{11} - 12 \beta_{9} - 54 \beta_{6} - 50 \beta_{5} + \cdots - 14 ) / 12 $ (7b15 + 38b13 - 19b12 - 24b11 - 12b9 - 54b6 - 50b5 + 29b3 - 80b2 - 7b1 - 14) / 12
$\nu^{10}$	$=$	$ ( -\beta_{15} - 7\beta_{14} + 4\beta_{10} - 41\beta_{8} + 20\beta_{7} + \beta_{4} + 5\beta _1 + 6 ) / 2 $ (-b15 - 7b14 + 4b10 - 41b8 + 20b7 + b4 + 5*b1 + 6) / 2
$\nu^{11}$	$=$	$ ( 31 \beta_{15} - 22 \beta_{13} - 73 \beta_{12} + 84 \beta_{11} - 168 \beta_{9} + 18 \beta_{6} + \cdots - 62 ) / 12 $ (31b15 - 22b13 - 73b12 + 84b11 - 168b9 + 18b6 + 22b5 + 95b3 + 16b2 - 31b1 - 62) / 12
$\nu^{12}$	$=$	$ ( -7\beta_{15} - 2\beta_{14} - 21\beta_{10} - 8\beta_{8} + 30\beta_{7} - 13\beta_{4} + 58\beta _1 - 25 ) / 2 $ (-7b15 - 2b14 - 21b10 - 8b8 + 30b7 - 13b4 + 58*b1 - 25) / 2
$\nu^{13}$	$=$	$ ( - 149 \beta_{15} - 286 \beta_{13} + 41 \beta_{12} - 12 \beta_{11} + 120 \beta_{9} - 18 \beta_{6} + \cdots + 298 ) / 12 $ (-149b15 - 286b13 + 41b12 - 12b11 + 120b9 - 18b6 + 298b5 + 173b3 - 116b2 + 149b1 + 298) / 12
$\nu^{14}$	$=$	$ ( 33\beta_{15} - 45\beta_{14} + 6\beta_{10} - 63\beta_{8} - 18\beta_{7} - 93\beta_{4} + 45\beta _1 - 4 ) / 2 $ (33b15 - 45b14 + 6b10 - 63b8 - 18b7 - 93b4 + 45*b1 - 4) / 2
$\nu^{15}$	$=$	$ ( - 269 \beta_{15} + 86 \beta_{13} + 179 \beta_{12} - 624 \beta_{11} + 180 \beta_{9} - 114 \beta_{6} + \cdots + 538 ) / 12 $ (-269b15 + 86b13 + 179b12 - 624b11 + 180b9 - 114b6 - 350b5 + 455b3 + 76b2 + 269b1 + 538) / 12

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$\beta_{8}$

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} $

111.1

−1.40927	−	0.118126i
0.517174	−	1.31626i
−0.944649	−	1.05244i
−1.36166	−	0.381939i
1.36166	+	0.381939i
0.944649	+	1.05244i
−0.517174	+	1.31626i
1.40927	+	0.118126i
−1.40927	+	0.118126i
0.517174	+	1.31626i
−0.944649	+	1.05244i
−1.36166	+	0.381939i
1.36166	−	0.381939i
0.944649	−	1.05244i
−0.517174	−	1.31626i
1.40927	−	0.118126i

−2.23450

−

2.23450i

0.584038

0.584038i

1.82596

1.91465i

6.98602i

111.2

−1.50619

−

1.50619i

−2.54054

−

2.54054i

−2.59286

0.526369i

1.53721i

111.3

−1.28999

−

1.28999i

0.599312

0.599312i

0.152445

−

2.64136i

0.328129i

111.4

−1.03649

−

1.03649i

1.68683

1.68683i

−1.38554

2.25395i

−

0.851361i

111.5

1.03649

1.03649i

−1.68683

−

1.68683i

−1.38554

−

2.25395i

−

0.851361i

111.6

1.28999

1.28999i

−0.599312

−

0.599312i

0.152445

2.64136i

0.328129i

111.7

1.50619

1.50619i

2.54054

2.54054i

−2.59286

−

0.526369i

1.53721i

111.8

2.23450

2.23450i

−0.584038

−

0.584038i

1.82596

−

1.91465i

6.98602i

335.1

−2.23450

2.23450i

0.584038

−

0.584038i

1.82596

−

1.91465i

−

6.98602i

335.2

−1.50619

1.50619i

−2.54054

2.54054i

−2.59286

−

0.526369i

−

1.53721i

335.3

−1.28999

1.28999i

0.599312

−

0.599312i

0.152445

2.64136i

−

0.328129i

335.4

−1.03649

1.03649i

1.68683

−

1.68683i

−1.38554

−

2.25395i

0.851361i

335.5

1.03649

−

1.03649i

−1.68683

1.68683i

−1.38554

2.25395i

0.851361i

335.6

1.28999

−

1.28999i

−0.599312

0.599312i

0.152445

−

2.64136i

−

0.328129i

335.7

1.50619

−

1.50619i

2.54054

−

2.54054i

−2.59286

0.526369i

−

1.53721i

335.8

2.23450

−

2.23450i

−0.584038

0.584038i

1.82596

1.91465i

−

6.98602i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
16.f	odd	4	1	inner
112.j	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	448.2.j.d		16
4.b	odd	2	1		112.2.j.d	✓	16
7.b	odd	2	1	inner	448.2.j.d		16
8.b	even	2	1		896.2.j.g		16
8.d	odd	2	1		896.2.j.h		16
16.e	even	4	1		112.2.j.d	✓	16
16.e	even	4	1		896.2.j.h		16
16.f	odd	4	1	inner	448.2.j.d		16
16.f	odd	4	1		896.2.j.g		16
28.d	even	2	1		112.2.j.d	✓	16
28.f	even	6	2		784.2.w.e		32
28.g	odd	6	2		784.2.w.e		32
56.e	even	2	1		896.2.j.h		16
56.h	odd	2	1		896.2.j.g		16
112.j	even	4	1	inner	448.2.j.d		16
112.j	even	4	1		896.2.j.g		16
112.l	odd	4	1		112.2.j.d	✓	16
112.l	odd	4	1		896.2.j.h		16
112.w	even	12	2		784.2.w.e		32
112.x	odd	12	2		784.2.w.e		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.j.d	✓	16	4.b	odd	2	1
112.2.j.d	✓	16	16.e	even	4	1
112.2.j.d	✓	16	28.d	even	2	1
112.2.j.d	✓	16	112.l	odd	4	1
448.2.j.d		16	1.a	even	1	1	trivial
448.2.j.d		16	7.b	odd	2	1	inner
448.2.j.d		16	16.f	odd	4	1	inner
448.2.j.d		16	112.j	even	4	1	inner
784.2.w.e		32	28.f	even	6	2
784.2.w.e		32	28.g	odd	6	2
784.2.w.e		32	112.w	even	12	2
784.2.w.e		32	112.x	odd	12	2
896.2.j.g		16	8.b	even	2	1
896.2.j.g		16	16.f	odd	4	1
896.2.j.g		16	56.h	odd	2	1
896.2.j.g		16	112.j	even	4	1
896.2.j.h		16	8.d	odd	2	1
896.2.j.h		16	16.e	even	4	1
896.2.j.h		16	56.e	even	2	1
896.2.j.h		16	112.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(448, [\chi])$:

$ T_{3}^{16} + 136T_{3}^{12} + 3992T_{3}^{8} + 38368T_{3}^{4} + 104976 $

$ T_{5}^{16} + 200T_{5}^{12} + 5592T_{5}^{8} + 5344T_{5}^{4} + 1296 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 136 T^{12} + \cdots + 104976 $ T^16 + 136T^12 + 3992T^8 + 38368*T^4 + 104976
$5$	$ T^{16} + 200 T^{12} + \cdots + 1296 $ T^16 + 200T^12 + 5592T^8 + 5344*T^4 + 1296
$7$	$ (T^{8} + 4 T^{7} + \cdots + 2401)^{2} $ (T^8 + 4T^7 + 12T^6 + 36T^5 + 86T^4 + 252T^3 + 588T^2 + 1372*T + 2401)^2
$11$	$ (T^{2} - 4 T + 8)^{8} $ (T^2 - 4*T + 8)^8
$13$	$ T^{16} + 2440 T^{12} + \cdots + 18974736 $ T^16 + 2440T^12 + 644120T^8 + 7263712*T^4 + 18974736
$17$	$ (T^{8} + 88 T^{6} + \cdots + 14400)^{2} $ (T^8 + 88T^6 + 2256T^4 + 13952*T^2 + 14400)^2
$19$	$ T^{16} + 968 T^{12} + \cdots + 1296 $ T^16 + 968T^12 + 11992T^8 + 31456*T^4 + 1296
$23$	$ (T^{4} - 16 T^{2} + 16 T + 4)^{4} $ (T^4 - 16T^2 + 16T + 4)^4
$29$	$ (T^{8} + 128 T^{5} + \cdots + 1296)^{2} $ (T^8 + 128T^5 + 1864T^4 + 5632T^3 + 8192T^2 - 4608*T + 1296)^2
$31$	$ (T^{8} - 120 T^{6} + \cdots + 166464)^{2} $ (T^8 - 120T^6 + 4176T^4 - 48128*T^2 + 166464)^2
$37$	$ (T^{8} - 64 T^{5} + \cdots + 400)^{2} $ (T^8 - 64T^5 + 744T^4 - 1792T^3 + 2048T^2 + 1280*T + 400)^2
$41$	$ (T^{8} - 216 T^{6} + \cdots + 419904)^{2} $ (T^8 - 216T^6 + 11664T^4 - 186624*T^2 + 419904)^2
$43$	$ (T^{8} - 64 T^{5} + \cdots + 256)^{2} $ (T^8 - 64T^5 + 992T^4 - 2048T^3 + 2048T^2 + 1024*T + 256)^2
$47$	$ (T^{8} - 312 T^{6} + \cdots + 3968064)^{2} $ (T^8 - 312T^6 + 32592T^4 - 1179008*T^2 + 3968064)^2
$53$	$ (T^{8} - 128 T^{5} + \cdots + 1948816)^{2} $ (T^8 - 128T^5 + 8872T^4 - 13824T^3 + 8192T^2 + 178688*T + 1948816)^2
$59$	$ T^{16} + \cdots + 506250000 $ T^16 + 1736T^12 + 691800T^8 + 46871776*T^4 + 506250000
$61$	$ T^{16} + \cdots + 56528804622096 $ T^16 + 52040T^12 + 566435160T^8 + 841803181792*T^4 + 56528804622096
$67$	$ (T^{8} - 24 T^{7} + \cdots + 256)^{2} $ (T^8 - 24T^7 + 288T^6 - 1312T^5 + 2272T^4 + 8064T^3 + 12800T^2 + 2560*T + 256)^2
$71$	$ (T^{4} + 8 T^{3} + \cdots - 1728)^{4} $ (T^4 + 8T^3 - 144T^2 - 1152*T - 1728)^4
$73$	$ (T^{8} - 96 T^{6} + \cdots + 9216)^{2} $ (T^8 - 96T^6 + 2624T^4 - 15872*T^2 + 9216)^2
$79$	$ (T^{8} + 384 T^{6} + \cdots + 36192256)^{2} $ (T^8 + 384T^6 + 51456T^4 + 2674688*T^2 + 36192256)^2
$83$	$ T^{16} + \cdots + 57524882954256 $ T^16 + 43592T^12 + 298592536T^8 + 567338412256*T^4 + 57524882954256
$89$	$ (T^{8} - 512 T^{6} + \cdots + 5308416)^{2} $ (T^8 - 512T^6 + 86528T^4 - 4849664*T^2 + 5308416)^2
$97$	$ (T^{8} + 312 T^{6} + \cdots + 1742400)^{2} $ (T^8 + 312T^6 + 26192T^4 + 407552*T^2 + 1742400)^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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{3} + \beta_{5} q^{5} + (\beta_{13} - 1) q^{7} + (\beta_{15} - 3 \beta_{8} + \cdots + \beta_1) q^{9}+O(q^{10}) \) q - b6 * q^3 + b5 * q^5 + (b13 - 1) * q^7 + (b15 - 3b8 - b4 + b1) q^9	\( q - \beta_{6} q^{3} + \beta_{5} q^{5} + (\beta_{13} - 1) q^{7} + (\beta_{15} - 3 \beta_{8} + \cdots + \beta_1) q^{9}+ \cdots + (4 \beta_{15} - 6 \beta_{8} - 2 \beta_{7} + \cdots - 6) q^{99}+O(q^{100}) \) q - b6 * q^3 + b5 * q^5 + (b13 - 1) * q^7 + (b15 - 3b8 - b4 + b1) q^9 + (-2b8 + 2) q^11 + (b12 + b2) * q^13 + (b10 + b8 - b4 - b1) * q^15 + (b15 + 2b13 - b5 - b2 - b1 - 2) q^17 + (b13 - b11 - b6) * q^19 + (-b15 + b14 + b10 + 2b8 + b7 - b5 + b4 + b3 + 2) q^21 + (-b15 + b14 + b7 + b1 + 1) * q^23 + (-b15 + b8 - b4 - b1) * q^25 + (-b15 - b13 + 2b12 - b11 + b1 + 2) q^27 + (-b15 + b14 - b10 - b7 + b4) * q^29 + (-b12 - b9 + b6 + b3) * q^31 + (-2b9 - 2b6) * q^33 + (-b14 - b10 - b8 + b6 - 2b5 - 1) q^35 + (-b15 - b14 - b10) * q^37 + (2b15 + b14 - 3b7 - 2b1 - 3) q^39 + (-b15 + b12 - 2b11 + b5 - b3 - b2 + b1 + 2) q^41 + (-b15 + b14 - b10 - b8 + b7 - b4 + 2b1 + 1) q^43 + (-b12 + b2) * q^45 + (b15 + b12 + 2b11 - b9 + b6 + 2b5 - b3 - 2b2 - b1 - 2) q^47 + (-b12 + 2b7 - b3 - 1) q^49 + (-b15 + b14 + b10 + 3b8 + 3b7 + 3b4 + 3) q^51 + (b15 + b14 + b10 - 2b7 - 2b4) * q^53 + (2b5 + 2b2) * q^55 + (b15 + 2b10 - 2b8 + b4 - b1) * q^57 + (b15 + b13 + b11 + b9 - b1 - 2) * q^59 + (-2b15 - 2b13 - 2b11 + 4b9 - b2 + 2b1 + 4) q^61 + (2b15 - b12 + b11 - b10 + b9 - 5b8 - b6 - b4 + b3 + 2b1 - 1) q^63 + (-b15 - 2b14 + b7 + b1 - 4) q^65 + (-b15 - b14 - b10 + 3b8 - b7 - b4 + 3) q^67 + (-2b13 + 2b11 + b3) * q^69 + (2b15 - 4b14 - 2b7 - 2b1 - 4) * q^71 + (-b12 + b3) * q^73 + (-b15 - b13 - 2b12 - b11 + b9 + 4b2 + b1 + 2) * q^75 + (2b15 + 2b13 + 2b11 - 4) q^77 + (2b10 + 6b8 + 4b4 - 2b1) * q^79 + (5b15 - 2b14 - 3b7 - 5b1 - 3) * q^81 + (-2b13 + 2b11 + b6 - 2b3) q^83 + (2b15 - 2b14 - 2b10 + 2b8 + b7 + b4 + 2) * q^85 + (-b12 + b9 + b6 - 2b5 - b3 - 2b2) * q^87 + (2b15 + 4b11 - 2b9 + 2b6 - 2b5 + 2b2 - 2b1 - 4) q^89 + (b15 - 2b14 - b13 - b11 + 2b10 - b9 + b7 - b4 - 2b2 + 2b1 + 2) * q^91 + (2b8 + b7 - b4 + 4b1 - 2) * q^93 + (-b15 - b10 - b8 - b4) * q^95 + (-b15 - 2b13 - b12 + 2b9 + 2b6 + b5 - b3 + b2 + b1 + 2) q^97 + (4b15 - 6b8 - 2b7 - 2b4 - 6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{7}+O(q^{10}) \) 16 * q - 8 * q^7	\( 16 q - 8 q^{7} + 32 q^{11} + 16 q^{21} - 8 q^{35} - 16 q^{39} - 16 q^{49} + 32 q^{51} - 80 q^{65} + 48 q^{67} - 32 q^{71} - 16 q^{77} + 32 q^{81} + 64 q^{85} - 8 q^{91} - 64 q^{93} - 64 q^{99}+O(q^{100}) \) 16 * q - 8 * q^7 + 32 * q^11 + 16 * q^21 - 8 * q^35 - 16 * q^39 - 16 * q^49 + 32 * q^51 - 80 * q^65 + 48 * q^67 - 32 * q^71 - 16 * q^77 + 32 * q^81 + 64 * q^85 - 8 * q^91 - 64 * q^93 - 64 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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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	448.2.j.d

Level	$448$
Weight	$2$
Character orbit	448.j
Analytic conductor	$3.577$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.57729801055\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) x^16 - 4x^14 + 6x^12 - 12x^10 + 33x^8 - 48x^6 + 96x^4 - 256*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{12}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{14} + 84\nu^{12} - 42\nu^{10} + 132\nu^{8} - 351\nu^{6} - 216\nu^{4} + 528\nu^{2} + 2048 ) / 1920 \) (v^14 + 84v^12 - 42v^10 + 132v^8 - 351v^6 - 216v^4 + 528v^2 + 2048) / 1920
\(\beta_{2}\)	\(=\)	\( ( -13\nu^{15} + 48\nu^{13} - 14\nu^{11} + 4\nu^{9} - 477\nu^{7} + 428\nu^{5} - 1904\nu^{3} + 5696\nu ) / 2560 \) (-13v^15 + 48v^13 - 14v^11 + 4v^9 - 477v^7 + 428v^5 - 1904v^3 + 5696v) / 2560
\(\beta_{3}\)	\(=\)	\( ( -13\nu^{15} + 88\nu^{13} - 14\nu^{11} + 244\nu^{9} - 637\nu^{7} + 468\nu^{5} - 1744\nu^{3} + 4416\nu ) / 1280 \) (-13v^15 + 88v^13 - 14v^11 + 244v^9 - 637v^7 + 468v^5 - 1744v^3 + 4416v) / 1280
\(\beta_{4}\)	\(=\)	\( ( -13\nu^{14} - 12\nu^{12} + 66\nu^{10} - 36\nu^{8} + 243\nu^{6} - 912\nu^{4} + 1776\nu^{2} - 1664 ) / 1920 \) (-13v^14 - 12v^12 + 66v^10 - 36v^8 + 243v^6 - 912v^4 + 1776*v^2 - 1664) / 1920
\(\beta_{5}\)	\(=\)	\( ( -47\nu^{15} + 112\nu^{13} - 26\nu^{11} + 236\nu^{9} - 543\nu^{7} - 188\nu^{5} - 2576\nu^{3} + 4544\nu ) / 2560 \) (-47v^15 + 112v^13 - 26v^11 + 236v^9 - 543v^7 - 188v^5 - 2576v^3 + 4544v) / 2560
\(\beta_{6}\)	\(=\)	\( ( -53\nu^{15} + 88\nu^{13} - 94\nu^{11} + 404\nu^{9} - 997\nu^{7} + 948\nu^{5} - 4144\nu^{3} + 3776\nu ) / 2560 \) (-53v^15 + 88v^13 - 94v^11 + 404v^9 - 997v^7 + 948v^5 - 4144v^3 + 3776v) / 2560
\(\beta_{7}\)	\(=\)	\( ( -\nu^{14} + 2\nu^{12} + 2\nu^{10} - 9\nu^{6} + 14\nu^{4} - 64\nu^{2} + 96 ) / 64 \) (-v^14 + 2v^12 + 2v^10 - 9v^6 + 14v^4 - 64*v^2 + 96) / 64
\(\beta_{8}\)	\(=\)	\( ( -43\nu^{14} + 108\nu^{12} - 114\nu^{10} + 324\nu^{8} - 747\nu^{6} + 528\nu^{4} - 3024\nu^{2} + 6016 ) / 1920 \) (-43v^14 + 108v^12 - 114v^10 + 324v^8 - 747v^6 + 528v^4 - 3024*v^2 + 6016) / 1920
\(\beta_{9}\)	\(=\)	\( ( -127\nu^{15} + 312\nu^{13} - 346\nu^{11} + 1116\nu^{9} - 2383\nu^{7} + 2252\nu^{5} - 8656\nu^{3} + 17984\nu ) / 2560 \) (-127v^15 + 312v^13 - 346v^11 + 1116v^9 - 2383v^7 + 2252v^5 - 8656v^3 + 17984v) / 2560
\(\beta_{10}\)	\(=\)	\( ( -71\nu^{14} + 216\nu^{12} - 138\nu^{10} + 828\nu^{8} - 2199\nu^{6} + 1356\nu^{4} - 5328\nu^{2} + 12992 ) / 1920 \) (-71v^14 + 216v^12 - 138v^10 + 828v^8 - 2199v^6 + 1356v^4 - 5328*v^2 + 12992) / 1920
\(\beta_{11}\)	\(=\)	\( ( - 13 \nu^{15} - 90 \nu^{14} + 48 \nu^{13} + 240 \nu^{12} - 14 \nu^{11} - 220 \nu^{10} + 4 \nu^{9} + \cdots + 14720 ) / 2560 \) (-13v^15 - 90v^14 + 48v^13 + 240v^12 - 14v^11 - 220v^10 + 4v^9 + 1000v^8 - 477v^7 - 1850v^6 + 428v^5 + 1640v^4 + 656v^3 - 6240v^2 + 3136*v + 14720) / 2560
\(\beta_{12}\)	\(=\)	\( ( 59\nu^{15} - 104\nu^{13} + 82\nu^{11} - 492\nu^{9} + 811\nu^{7} - 844\nu^{5} + 3872\nu^{3} - 6528\nu ) / 640 \) (59v^15 - 104v^13 + 82v^11 - 492v^9 + 811v^7 - 844v^5 + 3872v^3 - 6528v) / 640
\(\beta_{13}\)	\(=\)	\( ( - 15 \nu^{15} - 18 \nu^{14} + 32 \nu^{13} + 48 \nu^{12} - 26 \nu^{11} - 44 \nu^{10} + 140 \nu^{9} + \cdots + 2944 ) / 512 \) (-15v^15 - 18v^14 + 32v^13 + 48v^12 - 26v^11 - 44v^10 + 140v^9 + 200v^8 - 319v^7 - 370v^6 + 116v^5 + 328v^4 - 1040v^3 - 1248v^2 + 2240*v + 2944) / 512
\(\beta_{14}\)	\(=\)	\( ( 7\nu^{14} - 16\nu^{12} + 10\nu^{10} - 76\nu^{8} + 119\nu^{6} - 68\nu^{4} + 560\nu^{2} - 960 ) / 128 \) (7v^14 - 16v^12 + 10v^10 - 76v^8 + 119v^6 - 68v^4 + 560*v^2 - 960) / 128
\(\beta_{15}\)	\(=\)	\( ( 34\nu^{14} - 69\nu^{12} + 72\nu^{10} - 342\nu^{8} + 606\nu^{6} - 669\nu^{4} + 2472\nu^{2} - 4048 ) / 480 \) (34v^14 - 69v^12 + 72v^10 - 342v^8 + 606v^6 - 669v^4 + 2472*v^2 - 4048) / 480

\(\nu\)	\(=\)	\( ( \beta_{15} + 2\beta_{13} - \beta_{12} - 6\beta_{6} - 2\beta_{5} - \beta_{3} + 4\beta_{2} - \beta _1 - 2 ) / 12 \) (b15 + 2b13 - b12 - 6b6 - 2b5 - b3 + 4b2 - b1 - 2) / 12
\(\nu^{2}\)	\(=\)	\( ( -\beta_{15} + \beta_{14} - \beta_{8} + \beta_{4} + \beta _1 + 2 ) / 2 \) (-b15 + b14 - b8 + b4 + b1 + 2) / 2
\(\nu^{3}\)	\(=\)	\( ( 7 \beta_{15} + 2 \beta_{13} - \beta_{12} + 12 \beta_{11} - 6 \beta_{6} - 2 \beta_{5} - \beta_{3} + \cdots - 14 ) / 12 \) (7b15 + 2b13 - b12 + 12b11 - 6b6 - 2b5 - b3 - 8b2 - 7*b1 - 14) / 12
\(\nu^{4}\)	\(=\)	\( ( -3\beta_{15} + 2\beta_{14} - \beta_{10} - 4\beta_{8} + 2\beta_{7} - \beta_{4} + 2\beta _1 + 3 ) / 2 \) (-3b15 + 2b14 - b10 - 4b8 + 2b7 - b4 + 2*b1 + 3) / 2
\(\nu^{5}\)	\(=\)	\( ( - 5 \beta_{15} - 22 \beta_{13} - 7 \beta_{12} + 12 \beta_{11} + 6 \beta_{6} - 14 \beta_{5} + 5 \beta_{3} + \cdots + 10 ) / 12 \) (-5b15 - 22b13 - 7b12 + 12b11 + 6b6 - 14b5 + 5b3 + 4b2 + 5*b1 + 10) / 12
\(\nu^{6}\)	\(=\)	\( ( -3\beta_{15} - \beta_{14} - 6\beta_{10} - 3\beta_{8} + 2\beta_{7} - \beta_{4} + 5\beta _1 + 8 ) / 2 \) (-3b15 - b14 - 6b10 - 3b8 + 2b7 - b4 + 5*b1 + 8) / 2
\(\nu^{7}\)	\(=\)	\( ( - 29 \beta_{15} - 34 \beta_{13} - 13 \beta_{12} - 24 \beta_{11} + 12 \beta_{9} - 42 \beta_{6} + \cdots + 58 ) / 12 \) (-29b15 - 34b13 - 13b12 - 24b11 + 12b9 - 42b6 + 10b5 - b3 + 4b2 + 29*b1 + 58) / 12
\(\nu^{8}\)	\(=\)	\( ( -5\beta_{15} - 4\beta_{14} - 3\beta_{10} - 18\beta_{8} - 4\beta_{7} + \beta_{4} + 6\beta _1 + 5 ) / 2 \) (-5b15 - 4b14 - 3b10 - 18b8 - 4b7 + b4 + 6b1 + 5) / 2
\(\nu^{9}\)	\(=\)	\( ( 7 \beta_{15} + 38 \beta_{13} - 19 \beta_{12} - 24 \beta_{11} - 12 \beta_{9} - 54 \beta_{6} - 50 \beta_{5} + \cdots - 14 ) / 12 \) (7b15 + 38b13 - 19b12 - 24b11 - 12b9 - 54b6 - 50b5 + 29b3 - 80b2 - 7b1 - 14) / 12
\(\nu^{10}\)	\(=\)	\( ( -\beta_{15} - 7\beta_{14} + 4\beta_{10} - 41\beta_{8} + 20\beta_{7} + \beta_{4} + 5\beta _1 + 6 ) / 2 \) (-b15 - 7b14 + 4b10 - 41b8 + 20b7 + b4 + 5*b1 + 6) / 2
\(\nu^{11}\)	\(=\)	\( ( 31 \beta_{15} - 22 \beta_{13} - 73 \beta_{12} + 84 \beta_{11} - 168 \beta_{9} + 18 \beta_{6} + \cdots - 62 ) / 12 \) (31b15 - 22b13 - 73b12 + 84b11 - 168b9 + 18b6 + 22b5 + 95b3 + 16b2 - 31b1 - 62) / 12
\(\nu^{12}\)	\(=\)	\( ( -7\beta_{15} - 2\beta_{14} - 21\beta_{10} - 8\beta_{8} + 30\beta_{7} - 13\beta_{4} + 58\beta _1 - 25 ) / 2 \) (-7b15 - 2b14 - 21b10 - 8b8 + 30b7 - 13b4 + 58*b1 - 25) / 2
\(\nu^{13}\)	\(=\)	\( ( - 149 \beta_{15} - 286 \beta_{13} + 41 \beta_{12} - 12 \beta_{11} + 120 \beta_{9} - 18 \beta_{6} + \cdots + 298 ) / 12 \) (-149b15 - 286b13 + 41b12 - 12b11 + 120b9 - 18b6 + 298b5 + 173b3 - 116b2 + 149b1 + 298) / 12
\(\nu^{14}\)	\(=\)	\( ( 33\beta_{15} - 45\beta_{14} + 6\beta_{10} - 63\beta_{8} - 18\beta_{7} - 93\beta_{4} + 45\beta _1 - 4 ) / 2 \) (33b15 - 45b14 + 6b10 - 63b8 - 18b7 - 93b4 + 45*b1 - 4) / 2
\(\nu^{15}\)	\(=\)	\( ( - 269 \beta_{15} + 86 \beta_{13} + 179 \beta_{12} - 624 \beta_{11} + 180 \beta_{9} - 114 \beta_{6} + \cdots + 538 ) / 12 \) (-269b15 + 86b13 + 179b12 - 624b11 + 180b9 - 114b6 - 350b5 + 455b3 + 76b2 + 269b1 + 538) / 12

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\beta_{8}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 111.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 136 T^{12} + \cdots + 104976 \) T^16 + 136T^12 + 3992T^8 + 38368*T^4 + 104976
$5$	\( T^{16} + 200 T^{12} + \cdots + 1296 \) T^16 + 200T^12 + 5592T^8 + 5344*T^4 + 1296
$7$	\( (T^{8} + 4 T^{7} + \cdots + 2401)^{2} \) (T^8 + 4T^7 + 12T^6 + 36T^5 + 86T^4 + 252T^3 + 588T^2 + 1372*T + 2401)^2
$11$	\( (T^{2} - 4 T + 8)^{8} \) (T^2 - 4*T + 8)^8
$13$	\( T^{16} + 2440 T^{12} + \cdots + 18974736 \) T^16 + 2440T^12 + 644120T^8 + 7263712*T^4 + 18974736
$17$	\( (T^{8} + 88 T^{6} + \cdots + 14400)^{2} \) (T^8 + 88T^6 + 2256T^4 + 13952*T^2 + 14400)^2
$19$	\( T^{16} + 968 T^{12} + \cdots + 1296 \) T^16 + 968T^12 + 11992T^8 + 31456*T^4 + 1296
$23$	\( (T^{4} - 16 T^{2} + 16 T + 4)^{4} \) (T^4 - 16T^2 + 16T + 4)^4
$29$	\( (T^{8} + 128 T^{5} + \cdots + 1296)^{2} \) (T^8 + 128T^5 + 1864T^4 + 5632T^3 + 8192T^2 - 4608*T + 1296)^2
$31$	\( (T^{8} - 120 T^{6} + \cdots + 166464)^{2} \) (T^8 - 120T^6 + 4176T^4 - 48128*T^2 + 166464)^2
$37$	\( (T^{8} - 64 T^{5} + \cdots + 400)^{2} \) (T^8 - 64T^5 + 744T^4 - 1792T^3 + 2048T^2 + 1280*T + 400)^2
$41$	\( (T^{8} - 216 T^{6} + \cdots + 419904)^{2} \) (T^8 - 216T^6 + 11664T^4 - 186624*T^2 + 419904)^2
$43$	\( (T^{8} - 64 T^{5} + \cdots + 256)^{2} \) (T^8 - 64T^5 + 992T^4 - 2048T^3 + 2048T^2 + 1024*T + 256)^2
$47$	\( (T^{8} - 312 T^{6} + \cdots + 3968064)^{2} \) (T^8 - 312T^6 + 32592T^4 - 1179008*T^2 + 3968064)^2
$53$	\( (T^{8} - 128 T^{5} + \cdots + 1948816)^{2} \) (T^8 - 128T^5 + 8872T^4 - 13824T^3 + 8192T^2 + 178688*T + 1948816)^2
$59$	\( T^{16} + \cdots + 506250000 \) T^16 + 1736T^12 + 691800T^8 + 46871776*T^4 + 506250000
$61$	\( T^{16} + \cdots + 56528804622096 \) T^16 + 52040T^12 + 566435160T^8 + 841803181792*T^4 + 56528804622096
$67$	\( (T^{8} - 24 T^{7} + \cdots + 256)^{2} \) (T^8 - 24T^7 + 288T^6 - 1312T^5 + 2272T^4 + 8064T^3 + 12800T^2 + 2560*T + 256)^2
$71$	\( (T^{4} + 8 T^{3} + \cdots - 1728)^{4} \) (T^4 + 8T^3 - 144T^2 - 1152*T - 1728)^4
$73$	\( (T^{8} - 96 T^{6} + \cdots + 9216)^{2} \) (T^8 - 96T^6 + 2624T^4 - 15872*T^2 + 9216)^2
$79$	\( (T^{8} + 384 T^{6} + \cdots + 36192256)^{2} \) (T^8 + 384T^6 + 51456T^4 + 2674688*T^2 + 36192256)^2
$83$	\( T^{16} + \cdots + 57524882954256 \) T^16 + 43592T^12 + 298592536T^8 + 567338412256*T^4 + 57524882954256
$89$	\( (T^{8} - 512 T^{6} + \cdots + 5308416)^{2} \) (T^8 - 512T^6 + 86528T^4 - 4849664*T^2 + 5308416)^2
$97$	\( (T^{8} + 312 T^{6} + \cdots + 1742400)^{2} \) (T^8 + 312T^6 + 26192T^4 + 407552*T^2 + 1742400)^2
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