LMFDB - Newform orbit 4608.2.k.bl

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1153,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("4608.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.k (of order $4$, 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:	$36.7950652514$
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} + 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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{23} $
Twist minimal:	yes
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_{3} q^{5} - \beta_{14} q^{7}+O(q^{10}) $ q + b3 * q^5 - b14 * q^7	\( q + \beta_{3} q^{5} - \beta_{14} q^{7} - \beta_1 q^{11} + (\beta_{9} - \beta_{5} + 1) q^{13} - \beta_{10} q^{17} + (\beta_{14} + \beta_{11} + \beta_{7}) q^{19} - \beta_{15} q^{23} + ( - \beta_{13} - \beta_{9} + \beta_{5}) q^{25} + ( - \beta_{12} + \beta_{10} + \beta_{4}) q^{29} + (\beta_{11} + \beta_{8} + \beta_{7}) q^{31} + ( - \beta_{15} - \beta_{6} + \beta_{2}) q^{35} + ( - \beta_{13} - \beta_{5} - 1) q^{37} + ( - \beta_{12} + 2 \beta_{4} - 2 \beta_{3}) q^{41} + ( - \beta_{14} - \beta_{8} + \beta_{7}) q^{43} + (\beta_{6} - 2 \beta_{2} - 2 \beta_1) q^{47} + ( - \beta_{13} + \beta_{9} - 1) q^{49} + ( - \beta_{12} - \beta_{10} - \beta_{3}) q^{53} + ( - 2 \beta_{14} - \beta_{11} + \beta_{8}) q^{55} + (\beta_{15} - \beta_{6} - 2 \beta_1) q^{59} + ( - 3 \beta_{9} + \beta_{5} - 1) q^{61} + (\beta_{10} + 2 \beta_{4} + 2 \beta_{3}) q^{65} + (2 \beta_{14} + 2 \beta_{7}) q^{67} + ( - 2 \beta_{15} - 2 \beta_{2} + 2 \beta_1) q^{71} + (3 \beta_{13} + 3 \beta_{9} - 2 \beta_{5}) q^{73} + (\beta_{12} - \beta_{10} - 2 \beta_{4}) q^{77} + \beta_{7} q^{79} + ( - 2 \beta_{15} - 2 \beta_{6} + \beta_{2}) q^{83} + (4 \beta_{13} + 2 \beta_{5} + 2) q^{85} + (2 \beta_{12} - 2 \beta_{4} + 2 \beta_{3}) q^{89} + ( - \beta_{14} + \beta_{8} + \beta_{7}) q^{91} + (\beta_{6} - 4 \beta_{2} - 4 \beta_1) q^{95} + (4 \beta_{13} - 4 \beta_{9} - 4) q^{97}+O(q^{100}) \) q + b3 * q^5 - b14 * q^7 - b1 * q^11 + (b9 - b5 + 1) * q^13 - b10 * q^17 + (b14 + b11 + b7) * q^19 - b15 * q^23 + (-b13 - b9 + b5) * q^25 + (-b12 + b10 + b4) * q^29 + (b11 + b8 + b7) * q^31 + (-b15 - b6 + b2) * q^35 + (-b13 - b5 - 1) * q^37 + (-b12 + 2b4 - 2b3) * q^41 + (-b14 - b8 + b7) * q^43 + (b6 - 2b2 - 2b1) * q^47 + (-b13 + b9 - 1) * q^49 + (-b12 - b10 - b3) * q^53 + (-2b14 - b11 + b8) q^55 + (b15 - b6 - 2b1) q^59 + (-3b9 + b5 - 1) q^61 + (b10 + 2b4 + 2b3) * q^65 + (2b14 + 2b7) * q^67 + (-2b15 - 2b2 + 2b1) q^71 + (3b13 + 3b9 - 2b5) q^73 + (b12 - b10 - 2b4) q^77 + b7 * q^79 + (-2b15 - 2b6 + b2) * q^83 + (4b13 + 2b5 + 2) * q^85 + (2b12 - 2b4 + 2b3) q^89 + (-b14 + b8 + b7) * q^91 + (b6 - 4b2 - 4b1) * q^95 + (4b13 - 4b9 - 4) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{13} - 16 q^{37} - 16 q^{49} - 16 q^{61} + 32 q^{85} - 64 q^{97}+O(q^{100}) $ 16 * q + 16 * q^13 - 16 * q^37 - 16 * q^49 - 16 * q^61 + 32 * q^85 - 64 * q^97

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} - 3 \nu^{13} + 20 \nu^{12} - 69 \nu^{11} + 100 \nu^{10} - 506 \nu^{9} - 4 \nu^{8} - 1488 \nu^{7} + \cdots - 72 ) / 8 $ (v^14 - 3v^13 + 20v^12 - 69v^11 + 100v^10 - 506v^9 - 4v^8 - 1488v^7 - 918v^6 - 1638v^5 - 1440v^4 - 594v^3 - 664v^2 - 28*v - 72) / 8
$\beta_{2}$	$=$	$ ( \nu^{14} + 3 \nu^{13} + 20 \nu^{12} + 69 \nu^{11} + 100 \nu^{10} + 506 \nu^{9} - 4 \nu^{8} + 1488 \nu^{7} + \cdots - 72 ) / 8 $ (v^14 + 3v^13 + 20v^12 + 69v^11 + 100v^10 + 506v^9 - 4v^8 + 1488v^7 - 918v^6 + 1638v^5 - 1440v^4 + 594v^3 - 664v^2 + 28*v - 72) / 8
$\beta_{3}$	$=$	$ ( 6 \nu^{15} - 6 \nu^{14} + 151 \nu^{13} - 138 \nu^{12} + 1312 \nu^{11} - 1012 \nu^{10} + 5192 \nu^{9} + \cdots - 16 ) / 16 $ (6v^15 - 6v^14 + 151v^13 - 138v^12 + 1312v^11 - 1012v^10 + 5192v^9 - 2976v^8 + 9900v^7 - 3276v^6 + 8830v^5 - 1188v^4 + 3288v^3 - 72v^2 + 352*v - 16) / 16
$\beta_{4}$	$=$	$ ( - 6 \nu^{15} - 6 \nu^{14} - 151 \nu^{13} - 138 \nu^{12} - 1312 \nu^{11} - 1012 \nu^{10} - 5192 \nu^{9} + \cdots - 16 ) / 16 $ (-6v^15 - 6v^14 - 151v^13 - 138v^12 - 1312v^11 - 1012v^10 - 5192v^9 - 2976v^8 - 9900v^7 - 3276v^6 - 8830v^5 - 1188v^4 - 3288v^3 - 72v^2 - 352*v - 16) / 16
$\beta_{5}$	$=$	$ ( -11\nu^{15} - 261\nu^{13} - 2042\nu^{11} - 6862\nu^{9} - 10334\nu^{7} - 7410\nu^{5} - 2412\nu^{3} - 252\nu ) / 16 $ (-11v^15 - 261v^13 - 2042v^11 - 6862v^9 - 10334v^7 - 7410v^5 - 2412v^3 - 252v) / 16
$\beta_{6}$	$=$	$ ( 7\nu^{14} + 164\nu^{12} + 1250\nu^{10} + 3984\nu^{8} + 5334\nu^{6} + 3024\nu^{4} + 596\nu^{2} + 16 ) / 4 $ (7v^14 + 164v^12 + 1250v^10 + 3984v^8 + 5334v^6 + 3024v^4 + 596*v^2 + 16) / 4
$\beta_{7}$	$=$	$ ( -15\nu^{14} - 352\nu^{12} - 2692\nu^{10} - 8638\nu^{8} - 11726\nu^{6} - 6808\nu^{4} - 1496\nu^{2} - 76 ) / 8 $ (-15v^14 - 352v^12 - 2692v^10 - 8638v^8 - 11726v^6 - 6808v^4 - 1496*v^2 - 76) / 8
$\beta_{8}$	$=$	$ ( 11 \nu^{15} - 6 \nu^{14} + 260 \nu^{13} - 142 \nu^{12} + 2018 \nu^{11} - 1106 \nu^{10} + 6672 \nu^{9} + \cdots - 176 ) / 8 $ (11v^15 - 6v^14 + 260v^13 - 142v^12 + 2018v^11 - 1106v^10 + 6672v^9 - 3696v^8 + 9702v^7 - 5580v^6 + 6544v^5 - 4220v^4 + 2004v^3 - 1540v^2 + 272*v - 176) / 8
$\beta_{9}$	$=$	$ ( 11 \nu^{15} - 12 \nu^{14} + 258 \nu^{13} - 282 \nu^{12} + 1971 \nu^{11} - 2164 \nu^{10} + 6311 \nu^{9} + \cdots - 136 ) / 8 $ (11v^15 - 12v^14 + 258v^13 - 282v^12 + 1971v^11 - 2164v^10 + 6311v^9 - 7004v^8 + 8530v^7 - 9752v^6 + 4916v^5 - 6092v^4 + 1046v^3 - 1608v^2 + 38*v - 136) / 8
$\beta_{10}$	$=$	$ ( 11\nu^{14} + 261\nu^{12} + 2040\nu^{10} + 6818\nu^{8} + 10038\nu^{6} + 6666\nu^{4} + 1856\nu^{2} + 172 ) / 4 $ (11v^14 + 261v^12 + 2040v^10 + 6818v^8 + 10038v^6 + 6666v^4 + 1856*v^2 + 172) / 4
$\beta_{11}$	$=$	$ ( - 11 \nu^{15} - 6 \nu^{14} - 260 \nu^{13} - 142 \nu^{12} - 2018 \nu^{11} - 1106 \nu^{10} - 6672 \nu^{9} + \cdots - 176 ) / 8 $ (-11v^15 - 6v^14 - 260v^13 - 142v^12 - 2018v^11 - 1106v^10 - 6672v^9 - 3696v^8 - 9702v^7 - 5580v^6 - 6544v^5 - 4220v^4 - 2004v^3 - 1540v^2 - 272*v - 176) / 8
$\beta_{12}$	$=$	$ ( 16\nu^{15} + 371\nu^{13} + 2768\nu^{11} + 8442\nu^{9} + 10136\nu^{7} + 4278\nu^{5} - 8\nu^{3} - 188\nu ) / 8 $ (16v^15 + 371v^13 + 2768v^11 + 8442v^9 + 10136v^7 + 4278v^5 - 8v^3 - 188v) / 8
$\beta_{13}$	$=$	$ ( 11 \nu^{15} + 12 \nu^{14} + 258 \nu^{13} + 282 \nu^{12} + 1971 \nu^{11} + 2164 \nu^{10} + 6311 \nu^{9} + \cdots + 136 ) / 8 $ (11v^15 + 12v^14 + 258v^13 + 282v^12 + 1971v^11 + 2164v^10 + 6311v^9 + 7004v^8 + 8530v^7 + 9752v^6 + 4916v^5 + 6092v^4 + 1046v^3 + 1608v^2 + 38*v + 136) / 8
$\beta_{14}$	$=$	$ ( 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
$\beta_{15}$	$=$	$ ( -19\nu^{15} - 451\nu^{13} - 3529\nu^{11} - 11832\nu^{9} - 17582\nu^{7} - 11966\nu^{5} - 3498\nu^{3} - 344\nu ) / 4 $ (-19v^15 - 451v^13 - 3529v^11 - 11832v^9 - 17582v^7 - 11966v^5 - 3498v^3 - 344v) / 4

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

$\nu$	$=$	$ ( -\beta_{13} + \beta_{12} - \beta_{9} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 ) / 4 $ (-b13 + b12 - b9 - b4 + b3 - b2 + b1) / 4
$\nu^{2}$	$=$	$ ( 2 \beta_{13} + \beta_{11} - 2 \beta_{9} + \beta_{8} + 4 \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + \cdots - 12 ) / 4 $ (2b13 + b11 - 2b9 + b8 + 4b7 + b6 - 2b4 - 2*b3 - b2 - b1 - 12) / 4
$\nu^{3}$	$=$	$ ( - 2 \beta_{15} - 6 \beta_{14} + 8 \beta_{13} - 6 \beta_{12} - 3 \beta_{11} + 8 \beta_{9} + \cdots - 9 \beta_1 ) / 4 $ (-2b15 - 6b14 + 8b13 - 6b12 - 3b11 + 8b9 + 3b8 - 12b5 + 10b4 - 10b3 + 9b2 - 9b1) / 4
$\nu^{4}$	$=$	$ ( - 13 \beta_{13} - 6 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} - 6 \beta_{8} - 20 \beta_{7} - 6 \beta_{6} + \cdots + 48 ) / 2 $ (-13b13 - 6b11 + 4b10 + 13b9 - 6b8 - 20b7 - 6b6 + 14b4 + 14b3 + 10b2 + 10*b1 + 48) / 2
$\nu^{5}$	$=$	$ ( 31 \beta_{15} + 100 \beta_{14} - 94 \beta_{13} + 54 \beta_{12} + 40 \beta_{11} - 94 \beta_{9} + \cdots + 95 \beta_1 ) / 4 $ (31b15 + 100b14 - 94b13 + 54b12 + 40b11 - 94b9 - 40b8 + 220b5 - 112b4 + 112b3 - 95b2 + 95b1) / 4
$\nu^{6}$	$=$	$ ( 161 \beta_{13} + 72 \beta_{11} - 67 \beta_{10} - 161 \beta_{9} + 72 \beta_{8} + 218 \beta_{7} + \cdots - 504 ) / 2 $ (161b13 + 72b11 - 67b10 - 161b9 + 72b8 + 218b7 + 67b6 - 181b4 - 181b3 - 140b2 - 140*b1 - 504) / 2
$\nu^{7}$	$=$	$ ( - 413 \beta_{15} - 1344 \beta_{14} + 1142 \beta_{13} - 584 \beta_{12} - 497 \beta_{11} + \cdots - 1091 \beta_1 ) / 4 $ (-413b15 - 1344b14 + 1142b13 - 584b12 - 497b11 + 1142b9 + 497b8 - 3024b5 + 1328b4 - 1328b3 + 1091b2 - 1091b1) / 4
$\nu^{8}$	$=$	$ - 990 \beta_{13} - 438 \beta_{11} + 450 \beta_{10} + 990 \beta_{9} - 438 \beta_{8} - 1272 \beta_{7} + \cdots + 2910 $ -990b13 - 438b11 + 450b10 + 990b9 - 438b8 - 1272b7 - 392b6 + 1130b4 + 1130b3 + 896b2 + 896*b1 + 2910
$\nu^{9}$	$=$	$ ( 2610 \beta_{15} + 8496 \beta_{14} - 6971 \beta_{13} + 3407 \beta_{12} + 3054 \beta_{11} + \cdots + 6517 \beta_1 ) / 2 $ (2610b15 + 8496b14 - 6971b13 + 3407b12 + 3054b11 - 6971b9 - 3054b8 + 19248b5 - 8039b4 + 8039b3 - 6517b2 + 6517b1) / 2
$\nu^{10}$	$=$	$ ( 24294 \beta_{13} + 10707 \beta_{11} - 11376 \beta_{10} - 24294 \beta_{9} + 10707 \beta_{8} + \cdots - 69716 ) / 2 $ (24294b13 + 10707b11 - 11376b10 - 24294b9 + 10707b8 + 30588b7 + 9423b6 - 27862b4 - 27862b3 - 22287b2 - 22287*b1 - 69716) / 2
$\nu^{11}$	$=$	$ ( - 32346 \beta_{15} - 105226 \beta_{14} + 85244 \beta_{13} - 40966 \beta_{12} - 37433 \beta_{11} + \cdots - 79079 \beta_1 ) / 2 $ (-32346b15 - 105226b14 + 85244b13 - 40966b12 - 37433b11 + 85244b9 + 37433b8 - 238964b5 + 98050b4 - 98050b3 + 79079b2 - 79079b1) / 2
$\nu^{12}$	$=$	$ - 148879 \beta_{13} - 65522 \beta_{11} + 70448 \beta_{10} + 148879 \beta_{9} - 65522 \beta_{8} + \cdots + 423384 $ -148879b13 - 65522b11 + 70448b10 + 148879b9 - 65522b8 - 186020b7 - 57274b6 + 170986b4 + 170986b3 + 137214b2 + 137214*b1 + 423384
$\nu^{13}$	$=$	$ ( 397897 \beta_{15} + 1293812 \beta_{14} - 1043178 \beta_{13} + 498250 \beta_{12} + 458484 \beta_{11} + \cdots + 965133 \beta_1 ) / 2 $ (397897b15 + 1293812b14 - 1043178b13 + 498250b12 + 458484b11 - 1043178b9 - 458484b8 + 2940756b5 - 1198992b4 + 1198992b3 - 965133b2 + 965133b1) / 2
$\nu^{14}$	$=$	$ 1823791 \beta_{13} + 802232 \beta_{11} - 866233 \beta_{10} - 1823791 \beta_{9} + 802232 \beta_{8} + \cdots - 5168936 $ 1823791b13 + 802232b11 - 866233b10 - 1823791b9 + 802232b8 + 2272238b7 + 699361b6 - 2095431b4 - 2095431b3 - 1683568b2 - 1683568*b1 - 5168936
$\nu^{15}$	$=$	$ ( - 4880807 \beta_{15} - 15866528 \beta_{14} + 12770386 \beta_{13} - 6085904 \beta_{12} + \cdots - 11804009 \beta_1 ) / 2 $ (-4880807b15 - 15866528b14 + 12770386b13 - 6085904b12 - 5614479b11 + 12770386b9 + 5614479b8 - 36075296b5 + 14674880b4 - 14674880b3 + 11804009b2 - 11804009b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$\beta_{5}$	$1$	$1$

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} $

1153.1

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

−2.10100

2.10100i

−

2.27411i

1153.2

−2.10100

2.10100i

2.27411i

1153.3

−1.25928

1.25928i

−

3.29066i

1153.4

−1.25928

1.25928i

3.29066i

1153.5

1.25928

−

1.25928i

−

3.29066i

1153.6

1.25928

−

1.25928i

3.29066i

1153.7

2.10100

−

2.10100i

−

2.27411i

1153.8

2.10100

−

2.10100i

2.27411i

3457.1

−2.10100

−

2.10100i

−

2.27411i

3457.2

−2.10100

−

2.10100i

2.27411i

3457.3

−1.25928

−

1.25928i

−

3.29066i

3457.4

−1.25928

−

1.25928i

3.29066i

3457.5

1.25928

1.25928i

−

3.29066i

3457.6

1.25928

1.25928i

3.29066i

3457.7

2.10100

2.10100i

−

2.27411i

3457.8

2.10100

2.10100i

2.27411i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
16.e	even	4	1	inner
16.f	odd	4	1	inner
48.i	odd	4	1	inner
48.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4608.2.k.bl	yes	16
3.b	odd	2	1	inner	4608.2.k.bl	yes	16
4.b	odd	2	1	inner	4608.2.k.bl	yes	16
8.b	even	2	1		4608.2.k.bk	✓	16
8.d	odd	2	1		4608.2.k.bk	✓	16
12.b	even	2	1	inner	4608.2.k.bl	yes	16
16.e	even	4	1		4608.2.k.bk	✓	16
16.e	even	4	1	inner	4608.2.k.bl	yes	16
16.f	odd	4	1		4608.2.k.bk	✓	16
16.f	odd	4	1	inner	4608.2.k.bl	yes	16
24.f	even	2	1		4608.2.k.bk	✓	16
24.h	odd	2	1		4608.2.k.bk	✓	16
32.g	even	8	1		9216.2.a.br		8
32.g	even	8	1		9216.2.a.bs		8
32.h	odd	8	1		9216.2.a.br		8
32.h	odd	8	1		9216.2.a.bs		8
48.i	odd	4	1		4608.2.k.bk	✓	16
48.i	odd	4	1	inner	4608.2.k.bl	yes	16
48.k	even	4	1		4608.2.k.bk	✓	16
48.k	even	4	1	inner	4608.2.k.bl	yes	16
96.o	even	8	1		9216.2.a.br		8
96.o	even	8	1		9216.2.a.bs		8
96.p	odd	8	1		9216.2.a.br		8
96.p	odd	8	1		9216.2.a.bs		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4608.2.k.bk	✓	16	8.b	even	2	1
4608.2.k.bk	✓	16	8.d	odd	2	1
4608.2.k.bk	✓	16	16.e	even	4	1
4608.2.k.bk	✓	16	16.f	odd	4	1
4608.2.k.bk	✓	16	24.f	even	2	1
4608.2.k.bk	✓	16	24.h	odd	2	1
4608.2.k.bk	✓	16	48.i	odd	4	1
4608.2.k.bk	✓	16	48.k	even	4	1
4608.2.k.bl	yes	16	1.a	even	1	1	trivial
4608.2.k.bl	yes	16	3.b	odd	2	1	inner
4608.2.k.bl	yes	16	4.b	odd	2	1	inner
4608.2.k.bl	yes	16	12.b	even	2	1	inner
4608.2.k.bl	yes	16	16.e	even	4	1	inner
4608.2.k.bl	yes	16	16.f	odd	4	1	inner
4608.2.k.bl	yes	16	48.i	odd	4	1	inner
4608.2.k.bl	yes	16	48.k	even	4	1	inner
9216.2.a.br		8	32.g	even	8	1
9216.2.a.br		8	32.h	odd	8	1
9216.2.a.br		8	96.o	even	8	1
9216.2.a.br		8	96.p	odd	8	1
9216.2.a.bs		8	32.g	even	8	1
9216.2.a.bs		8	32.h	odd	8	1
9216.2.a.bs		8	96.o	even	8	1
9216.2.a.bs		8	96.p	odd	8	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4608, [\chi])$:

$ T_{5}^{8} + 88T_{5}^{4} + 784 $

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

$ T_{11}^{8} + 192T_{11}^{4} + 1024 $

$ T_{13}^{4} - 4T_{13}^{3} + 8T_{13}^{2} + 8T_{13} + 4 $

$ T_{19}^{8} + 3648T_{19}^{4} + 50176 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} + 88 T^{4} + 784)^{2} $ (T^8 + 88*T^4 + 784)^2
$7$	$ (T^{4} + 16 T^{2} + 56)^{4} $ (T^4 + 16*T^2 + 56)^4
$11$	$ (T^{8} + 192 T^{4} + 1024)^{2} $ (T^8 + 192*T^4 + 1024)^2
$13$	$ (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 4)^{4} $ (T^4 - 4T^3 + 8T^2 + 8*T + 4)^4
$17$	$ (T^{4} - 40 T^{2} + 112)^{4} $ (T^4 - 40*T^2 + 112)^4
$19$	$ (T^{8} + 3648 T^{4} + 50176)^{2} $ (T^8 + 3648*T^4 + 50176)^2
$23$	$ (T^{4} + 32 T^{2} + 128)^{4} $ (T^4 + 32*T^2 + 128)^4
$29$	$ (T^{8} + 3032 T^{4} + 1882384)^{2} $ (T^8 + 3032*T^4 + 1882384)^2
$31$	$ (T^{4} - 112 T^{2} + 2744)^{4} $ (T^4 - 112*T^2 + 2744)^4
$37$	$ (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 4)^{4} $ (T^4 + 4T^3 + 8T^2 - 8*T + 4)^4
$41$	$ (T^{4} + 104 T^{2} + 112)^{4} $ (T^4 + 104*T^2 + 112)^4
$43$	$ (T^{8} + 15936 T^{4} + 50176)^{2} $ (T^8 + 15936*T^4 + 50176)^2
$47$	$ (T^{4} - 160 T^{2} + 128)^{4} $ (T^4 - 160*T^2 + 128)^4
$53$	$ (T^{8} + 3032 T^{4} + 1882384)^{2} $ (T^8 + 3032*T^4 + 1882384)^2
$59$	$ (T^{8} + 12288 T^{4} + 4194304)^{2} $ (T^8 + 12288*T^4 + 4194304)^2
$61$	$ (T^{4} + 4 T^{3} + \cdots + 1156)^{4} $ (T^4 + 4T^3 + 8T^2 - 136*T + 1156)^4
$67$	$ (T^{8} + 9216 T^{4} + 12845056)^{2} $ (T^8 + 9216*T^4 + 12845056)^2
$71$	$ (T^{4} + 256 T^{2} + 8192)^{4} $ (T^4 + 256*T^2 + 8192)^4
$73$	$ (T^{4} + 152 T^{2} + 4624)^{4} $ (T^4 + 152*T^2 + 4624)^4
$79$	$ (T^{4} - 16 T^{2} + 56)^{4} $ (T^4 - 16*T^2 + 56)^4
$83$	$ (T^{8} + 40128 T^{4} + 286557184)^{2} $ (T^8 + 40128*T^4 + 286557184)^2
$89$	$ (T^{4} + 192 T^{2} + 7168)^{4} $ (T^4 + 192*T^2 + 7168)^4
$97$	$ (T^{2} + 8 T - 112)^{8} $ (T^2 + 8*T - 112)^8
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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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{5} - \beta_{14} q^{7}+O(q^{10}) \) q + b3 * q^5 - b14 * q^7	\( q + \beta_{3} q^{5} - \beta_{14} q^{7} - \beta_1 q^{11} + (\beta_{9} - \beta_{5} + 1) q^{13} - \beta_{10} q^{17} + (\beta_{14} + \beta_{11} + \beta_{7}) q^{19} - \beta_{15} q^{23} + ( - \beta_{13} - \beta_{9} + \beta_{5}) q^{25} + ( - \beta_{12} + \beta_{10} + \beta_{4}) q^{29} + (\beta_{11} + \beta_{8} + \beta_{7}) q^{31} + ( - \beta_{15} - \beta_{6} + \beta_{2}) q^{35} + ( - \beta_{13} - \beta_{5} - 1) q^{37} + ( - \beta_{12} + 2 \beta_{4} - 2 \beta_{3}) q^{41} + ( - \beta_{14} - \beta_{8} + \beta_{7}) q^{43} + (\beta_{6} - 2 \beta_{2} - 2 \beta_1) q^{47} + ( - \beta_{13} + \beta_{9} - 1) q^{49} + ( - \beta_{12} - \beta_{10} - \beta_{3}) q^{53} + ( - 2 \beta_{14} - \beta_{11} + \beta_{8}) q^{55} + (\beta_{15} - \beta_{6} - 2 \beta_1) q^{59} + ( - 3 \beta_{9} + \beta_{5} - 1) q^{61} + (\beta_{10} + 2 \beta_{4} + 2 \beta_{3}) q^{65} + (2 \beta_{14} + 2 \beta_{7}) q^{67} + ( - 2 \beta_{15} - 2 \beta_{2} + 2 \beta_1) q^{71} + (3 \beta_{13} + 3 \beta_{9} - 2 \beta_{5}) q^{73} + (\beta_{12} - \beta_{10} - 2 \beta_{4}) q^{77} + \beta_{7} q^{79} + ( - 2 \beta_{15} - 2 \beta_{6} + \beta_{2}) q^{83} + (4 \beta_{13} + 2 \beta_{5} + 2) q^{85} + (2 \beta_{12} - 2 \beta_{4} + 2 \beta_{3}) q^{89} + ( - \beta_{14} + \beta_{8} + \beta_{7}) q^{91} + (\beta_{6} - 4 \beta_{2} - 4 \beta_1) q^{95} + (4 \beta_{13} - 4 \beta_{9} - 4) q^{97}+O(q^{100}) \) q + b3 * q^5 - b14 * q^7 - b1 * q^11 + (b9 - b5 + 1) * q^13 - b10 * q^17 + (b14 + b11 + b7) * q^19 - b15 * q^23 + (-b13 - b9 + b5) * q^25 + (-b12 + b10 + b4) * q^29 + (b11 + b8 + b7) * q^31 + (-b15 - b6 + b2) * q^35 + (-b13 - b5 - 1) * q^37 + (-b12 + 2b4 - 2b3) * q^41 + (-b14 - b8 + b7) * q^43 + (b6 - 2b2 - 2b1) * q^47 + (-b13 + b9 - 1) * q^49 + (-b12 - b10 - b3) * q^53 + (-2b14 - b11 + b8) q^55 + (b15 - b6 - 2b1) q^59 + (-3b9 + b5 - 1) q^61 + (b10 + 2b4 + 2b3) * q^65 + (2b14 + 2b7) * q^67 + (-2b15 - 2b2 + 2b1) q^71 + (3b13 + 3b9 - 2b5) q^73 + (b12 - b10 - 2b4) q^77 + b7 * q^79 + (-2b15 - 2b6 + b2) * q^83 + (4b13 + 2b5 + 2) * q^85 + (2b12 - 2b4 + 2b3) q^89 + (-b14 + b8 + b7) * q^91 + (b6 - 4b2 - 4b1) * q^95 + (4b13 - 4b9 - 4) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{13} - 16 q^{37} - 16 q^{49} - 16 q^{61} + 32 q^{85} - 64 q^{97}+O(q^{100}) \) 16 * q + 16 * q^13 - 16 * q^37 - 16 * q^49 - 16 * q^61 + 32 * q^85 - 64 * q^97

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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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	4608.2.k.bl

Level	$4608$
Weight	$2$
Character orbit	4608.k
Analytic conductor	$36.795$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(36.7950652514\)
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:	\( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) x^16 + 24x^14 + 192x^12 + 672x^10 + 1092x^8 + 880x^6 + 352x^4 + 64*x^2 + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 2^{23} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{14} - 3 \nu^{13} + 20 \nu^{12} - 69 \nu^{11} + 100 \nu^{10} - 506 \nu^{9} - 4 \nu^{8} - 1488 \nu^{7} + \cdots - 72 ) / 8 \) (v^14 - 3v^13 + 20v^12 - 69v^11 + 100v^10 - 506v^9 - 4v^8 - 1488v^7 - 918v^6 - 1638v^5 - 1440v^4 - 594v^3 - 664v^2 - 28*v - 72) / 8
\(\beta_{2}\)	\(=\)	\( ( \nu^{14} + 3 \nu^{13} + 20 \nu^{12} + 69 \nu^{11} + 100 \nu^{10} + 506 \nu^{9} - 4 \nu^{8} + 1488 \nu^{7} + \cdots - 72 ) / 8 \) (v^14 + 3v^13 + 20v^12 + 69v^11 + 100v^10 + 506v^9 - 4v^8 + 1488v^7 - 918v^6 + 1638v^5 - 1440v^4 + 594v^3 - 664v^2 + 28*v - 72) / 8
\(\beta_{3}\)	\(=\)	\( ( 6 \nu^{15} - 6 \nu^{14} + 151 \nu^{13} - 138 \nu^{12} + 1312 \nu^{11} - 1012 \nu^{10} + 5192 \nu^{9} + \cdots - 16 ) / 16 \) (6v^15 - 6v^14 + 151v^13 - 138v^12 + 1312v^11 - 1012v^10 + 5192v^9 - 2976v^8 + 9900v^7 - 3276v^6 + 8830v^5 - 1188v^4 + 3288v^3 - 72v^2 + 352*v - 16) / 16
\(\beta_{4}\)	\(=\)	\( ( - 6 \nu^{15} - 6 \nu^{14} - 151 \nu^{13} - 138 \nu^{12} - 1312 \nu^{11} - 1012 \nu^{10} - 5192 \nu^{9} + \cdots - 16 ) / 16 \) (-6v^15 - 6v^14 - 151v^13 - 138v^12 - 1312v^11 - 1012v^10 - 5192v^9 - 2976v^8 - 9900v^7 - 3276v^6 - 8830v^5 - 1188v^4 - 3288v^3 - 72v^2 - 352*v - 16) / 16
\(\beta_{5}\)	\(=\)	\( ( -11\nu^{15} - 261\nu^{13} - 2042\nu^{11} - 6862\nu^{9} - 10334\nu^{7} - 7410\nu^{5} - 2412\nu^{3} - 252\nu ) / 16 \) (-11v^15 - 261v^13 - 2042v^11 - 6862v^9 - 10334v^7 - 7410v^5 - 2412v^3 - 252v) / 16
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{14} + 164\nu^{12} + 1250\nu^{10} + 3984\nu^{8} + 5334\nu^{6} + 3024\nu^{4} + 596\nu^{2} + 16 ) / 4 \) (7v^14 + 164v^12 + 1250v^10 + 3984v^8 + 5334v^6 + 3024v^4 + 596*v^2 + 16) / 4
\(\beta_{7}\)	\(=\)	\( ( -15\nu^{14} - 352\nu^{12} - 2692\nu^{10} - 8638\nu^{8} - 11726\nu^{6} - 6808\nu^{4} - 1496\nu^{2} - 76 ) / 8 \) (-15v^14 - 352v^12 - 2692v^10 - 8638v^8 - 11726v^6 - 6808v^4 - 1496*v^2 - 76) / 8
\(\beta_{8}\)	\(=\)	\( ( 11 \nu^{15} - 6 \nu^{14} + 260 \nu^{13} - 142 \nu^{12} + 2018 \nu^{11} - 1106 \nu^{10} + 6672 \nu^{9} + \cdots - 176 ) / 8 \) (11v^15 - 6v^14 + 260v^13 - 142v^12 + 2018v^11 - 1106v^10 + 6672v^9 - 3696v^8 + 9702v^7 - 5580v^6 + 6544v^5 - 4220v^4 + 2004v^3 - 1540v^2 + 272*v - 176) / 8
\(\beta_{9}\)	\(=\)	\( ( 11 \nu^{15} - 12 \nu^{14} + 258 \nu^{13} - 282 \nu^{12} + 1971 \nu^{11} - 2164 \nu^{10} + 6311 \nu^{9} + \cdots - 136 ) / 8 \) (11v^15 - 12v^14 + 258v^13 - 282v^12 + 1971v^11 - 2164v^10 + 6311v^9 - 7004v^8 + 8530v^7 - 9752v^6 + 4916v^5 - 6092v^4 + 1046v^3 - 1608v^2 + 38*v - 136) / 8
\(\beta_{10}\)	\(=\)	\( ( 11\nu^{14} + 261\nu^{12} + 2040\nu^{10} + 6818\nu^{8} + 10038\nu^{6} + 6666\nu^{4} + 1856\nu^{2} + 172 ) / 4 \) (11v^14 + 261v^12 + 2040v^10 + 6818v^8 + 10038v^6 + 6666v^4 + 1856*v^2 + 172) / 4
\(\beta_{11}\)	\(=\)	\( ( - 11 \nu^{15} - 6 \nu^{14} - 260 \nu^{13} - 142 \nu^{12} - 2018 \nu^{11} - 1106 \nu^{10} - 6672 \nu^{9} + \cdots - 176 ) / 8 \) (-11v^15 - 6v^14 - 260v^13 - 142v^12 - 2018v^11 - 1106v^10 - 6672v^9 - 3696v^8 - 9702v^7 - 5580v^6 - 6544v^5 - 4220v^4 - 2004v^3 - 1540v^2 - 272*v - 176) / 8
\(\beta_{12}\)	\(=\)	\( ( 16\nu^{15} + 371\nu^{13} + 2768\nu^{11} + 8442\nu^{9} + 10136\nu^{7} + 4278\nu^{5} - 8\nu^{3} - 188\nu ) / 8 \) (16v^15 + 371v^13 + 2768v^11 + 8442v^9 + 10136v^7 + 4278v^5 - 8v^3 - 188v) / 8
\(\beta_{13}\)	\(=\)	\( ( 11 \nu^{15} + 12 \nu^{14} + 258 \nu^{13} + 282 \nu^{12} + 1971 \nu^{11} + 2164 \nu^{10} + 6311 \nu^{9} + \cdots + 136 ) / 8 \) (11v^15 + 12v^14 + 258v^13 + 282v^12 + 1971v^11 + 2164v^10 + 6311v^9 + 7004v^8 + 8530v^7 + 9752v^6 + 4916v^5 + 6092v^4 + 1046v^3 + 1608v^2 + 38*v + 136) / 8
\(\beta_{14}\)	\(=\)	\( ( 38\nu^{15} + 896\nu^{13} + 6921\nu^{11} + 22674\nu^{9} + 32332\nu^{7} + 20960\nu^{5} + 5842\nu^{3} + 540\nu ) / 8 \) (38v^15 + 896v^13 + 6921v^11 + 22674v^9 + 32332v^7 + 20960v^5 + 5842v^3 + 540v) / 8
\(\beta_{15}\)	\(=\)	\( ( -19\nu^{15} - 451\nu^{13} - 3529\nu^{11} - 11832\nu^{9} - 17582\nu^{7} - 11966\nu^{5} - 3498\nu^{3} - 344\nu ) / 4 \) (-19v^15 - 451v^13 - 3529v^11 - 11832v^9 - 17582v^7 - 11966v^5 - 3498v^3 - 344v) / 4

\(\nu\)	\(=\)	\( ( -\beta_{13} + \beta_{12} - \beta_{9} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) (-b13 + b12 - b9 - b4 + b3 - b2 + b1) / 4
\(\nu^{2}\)	\(=\)	\( ( 2 \beta_{13} + \beta_{11} - 2 \beta_{9} + \beta_{8} + 4 \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + \cdots - 12 ) / 4 \) (2b13 + b11 - 2b9 + b8 + 4b7 + b6 - 2b4 - 2*b3 - b2 - b1 - 12) / 4
\(\nu^{3}\)	\(=\)	\( ( - 2 \beta_{15} - 6 \beta_{14} + 8 \beta_{13} - 6 \beta_{12} - 3 \beta_{11} + 8 \beta_{9} + \cdots - 9 \beta_1 ) / 4 \) (-2b15 - 6b14 + 8b13 - 6b12 - 3b11 + 8b9 + 3b8 - 12b5 + 10b4 - 10b3 + 9b2 - 9b1) / 4
\(\nu^{4}\)	\(=\)	\( ( - 13 \beta_{13} - 6 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} - 6 \beta_{8} - 20 \beta_{7} - 6 \beta_{6} + \cdots + 48 ) / 2 \) (-13b13 - 6b11 + 4b10 + 13b9 - 6b8 - 20b7 - 6b6 + 14b4 + 14b3 + 10b2 + 10*b1 + 48) / 2
\(\nu^{5}\)	\(=\)	\( ( 31 \beta_{15} + 100 \beta_{14} - 94 \beta_{13} + 54 \beta_{12} + 40 \beta_{11} - 94 \beta_{9} + \cdots + 95 \beta_1 ) / 4 \) (31b15 + 100b14 - 94b13 + 54b12 + 40b11 - 94b9 - 40b8 + 220b5 - 112b4 + 112b3 - 95b2 + 95b1) / 4
\(\nu^{6}\)	\(=\)	\( ( 161 \beta_{13} + 72 \beta_{11} - 67 \beta_{10} - 161 \beta_{9} + 72 \beta_{8} + 218 \beta_{7} + \cdots - 504 ) / 2 \) (161b13 + 72b11 - 67b10 - 161b9 + 72b8 + 218b7 + 67b6 - 181b4 - 181b3 - 140b2 - 140*b1 - 504) / 2
\(\nu^{7}\)	\(=\)	\( ( - 413 \beta_{15} - 1344 \beta_{14} + 1142 \beta_{13} - 584 \beta_{12} - 497 \beta_{11} + \cdots - 1091 \beta_1 ) / 4 \) (-413b15 - 1344b14 + 1142b13 - 584b12 - 497b11 + 1142b9 + 497b8 - 3024b5 + 1328b4 - 1328b3 + 1091b2 - 1091b1) / 4
\(\nu^{8}\)	\(=\)	\( - 990 \beta_{13} - 438 \beta_{11} + 450 \beta_{10} + 990 \beta_{9} - 438 \beta_{8} - 1272 \beta_{7} + \cdots + 2910 \) -990b13 - 438b11 + 450b10 + 990b9 - 438b8 - 1272b7 - 392b6 + 1130b4 + 1130b3 + 896b2 + 896*b1 + 2910
\(\nu^{9}\)	\(=\)	\( ( 2610 \beta_{15} + 8496 \beta_{14} - 6971 \beta_{13} + 3407 \beta_{12} + 3054 \beta_{11} + \cdots + 6517 \beta_1 ) / 2 \) (2610b15 + 8496b14 - 6971b13 + 3407b12 + 3054b11 - 6971b9 - 3054b8 + 19248b5 - 8039b4 + 8039b3 - 6517b2 + 6517b1) / 2
\(\nu^{10}\)	\(=\)	\( ( 24294 \beta_{13} + 10707 \beta_{11} - 11376 \beta_{10} - 24294 \beta_{9} + 10707 \beta_{8} + \cdots - 69716 ) / 2 \) (24294b13 + 10707b11 - 11376b10 - 24294b9 + 10707b8 + 30588b7 + 9423b6 - 27862b4 - 27862b3 - 22287b2 - 22287*b1 - 69716) / 2
\(\nu^{11}\)	\(=\)	\( ( - 32346 \beta_{15} - 105226 \beta_{14} + 85244 \beta_{13} - 40966 \beta_{12} - 37433 \beta_{11} + \cdots - 79079 \beta_1 ) / 2 \) (-32346b15 - 105226b14 + 85244b13 - 40966b12 - 37433b11 + 85244b9 + 37433b8 - 238964b5 + 98050b4 - 98050b3 + 79079b2 - 79079b1) / 2
\(\nu^{12}\)	\(=\)	\( - 148879 \beta_{13} - 65522 \beta_{11} + 70448 \beta_{10} + 148879 \beta_{9} - 65522 \beta_{8} + \cdots + 423384 \) -148879b13 - 65522b11 + 70448b10 + 148879b9 - 65522b8 - 186020b7 - 57274b6 + 170986b4 + 170986b3 + 137214b2 + 137214*b1 + 423384
\(\nu^{13}\)	\(=\)	\( ( 397897 \beta_{15} + 1293812 \beta_{14} - 1043178 \beta_{13} + 498250 \beta_{12} + 458484 \beta_{11} + \cdots + 965133 \beta_1 ) / 2 \) (397897b15 + 1293812b14 - 1043178b13 + 498250b12 + 458484b11 - 1043178b9 - 458484b8 + 2940756b5 - 1198992b4 + 1198992b3 - 965133b2 + 965133b1) / 2
\(\nu^{14}\)	\(=\)	\( 1823791 \beta_{13} + 802232 \beta_{11} - 866233 \beta_{10} - 1823791 \beta_{9} + 802232 \beta_{8} + \cdots - 5168936 \) 1823791b13 + 802232b11 - 866233b10 - 1823791b9 + 802232b8 + 2272238b7 + 699361b6 - 2095431b4 - 2095431b3 - 1683568b2 - 1683568*b1 - 5168936
\(\nu^{15}\)	\(=\)	\( ( - 4880807 \beta_{15} - 15866528 \beta_{14} + 12770386 \beta_{13} - 6085904 \beta_{12} + \cdots - 11804009 \beta_1 ) / 2 \) (-4880807b15 - 15866528b14 + 12770386b13 - 6085904b12 - 5614479b11 + 12770386b9 + 5614479b8 - 36075296b5 + 14674880b4 - 14674880b3 + 11804009b2 - 11804009b1) / 2

\(n\)	\(2053\)	\(3583\)	\(4097\)
\(\chi(n)\)	\(\beta_{5}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1153.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} + 88 T^{4} + 784)^{2} \) (T^8 + 88*T^4 + 784)^2
$7$	\( (T^{4} + 16 T^{2} + 56)^{4} \) (T^4 + 16*T^2 + 56)^4
$11$	\( (T^{8} + 192 T^{4} + 1024)^{2} \) (T^8 + 192*T^4 + 1024)^2
$13$	\( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 4)^{4} \) (T^4 - 4T^3 + 8T^2 + 8*T + 4)^4
$17$	\( (T^{4} - 40 T^{2} + 112)^{4} \) (T^4 - 40*T^2 + 112)^4
$19$	\( (T^{8} + 3648 T^{4} + 50176)^{2} \) (T^8 + 3648*T^4 + 50176)^2
$23$	\( (T^{4} + 32 T^{2} + 128)^{4} \) (T^4 + 32*T^2 + 128)^4
$29$	\( (T^{8} + 3032 T^{4} + 1882384)^{2} \) (T^8 + 3032*T^4 + 1882384)^2
$31$	\( (T^{4} - 112 T^{2} + 2744)^{4} \) (T^4 - 112*T^2 + 2744)^4
$37$	\( (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 4)^{4} \) (T^4 + 4T^3 + 8T^2 - 8*T + 4)^4
$41$	\( (T^{4} + 104 T^{2} + 112)^{4} \) (T^4 + 104*T^2 + 112)^4
$43$	\( (T^{8} + 15936 T^{4} + 50176)^{2} \) (T^8 + 15936*T^4 + 50176)^2
$47$	\( (T^{4} - 160 T^{2} + 128)^{4} \) (T^4 - 160*T^2 + 128)^4
$53$	\( (T^{8} + 3032 T^{4} + 1882384)^{2} \) (T^8 + 3032*T^4 + 1882384)^2
$59$	\( (T^{8} + 12288 T^{4} + 4194304)^{2} \) (T^8 + 12288*T^4 + 4194304)^2
$61$	\( (T^{4} + 4 T^{3} + \cdots + 1156)^{4} \) (T^4 + 4T^3 + 8T^2 - 136*T + 1156)^4
$67$	\( (T^{8} + 9216 T^{4} + 12845056)^{2} \) (T^8 + 9216*T^4 + 12845056)^2
$71$	\( (T^{4} + 256 T^{2} + 8192)^{4} \) (T^4 + 256*T^2 + 8192)^4
$73$	\( (T^{4} + 152 T^{2} + 4624)^{4} \) (T^4 + 152*T^2 + 4624)^4
$79$	\( (T^{4} - 16 T^{2} + 56)^{4} \) (T^4 - 16*T^2 + 56)^4
$83$	\( (T^{8} + 40128 T^{4} + 286557184)^{2} \) (T^8 + 40128*T^4 + 286557184)^2
$89$	\( (T^{4} + 192 T^{2} + 7168)^{4} \) (T^4 + 192*T^2 + 7168)^4
$97$	\( (T^{2} + 8 T - 112)^{8} \) (T^2 + 8*T - 112)^8
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