LMFDB - Newform orbit 2592.2.f.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(1295,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2592.1295");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.f (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:	$20.6972242039$
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} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + \cdots + 256 $ x^16 - 3x^15 + 7x^14 - 12x^13 + 16x^12 - 12x^11 - 8x^10 + 36x^9 - 68x^8 + 72x^7 - 32x^6 - 96x^5 + 256x^4 - 384x^3 + 448x^2 - 384*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q - \beta_{2} q^{5} + \beta_{10} q^{7}+O(q^{10}) $ q - b2 * q^5 + b10 * q^7	$ q - \beta_{2} q^{5} + \beta_{10} q^{7} + (\beta_{7} + \beta_1) q^{11} + \beta_{4} q^{13} + (\beta_{7} + \beta_{5}) q^{17} - \beta_{11} q^{19} - \beta_{14} q^{23} + (\beta_{11} - 2 \beta_{3} + 2) q^{25} + (\beta_{14} + \beta_{12}) q^{29} + ( - \beta_{13} + \beta_{10} + \beta_{4}) q^{31} + (2 \beta_{7} - 2 \beta_{6} + \cdots + 3 \beta_1) q^{35}+ \cdots + (3 \beta_{11} - \beta_{8} + 2 \beta_{3}) q^{97}+O(q^{100}) $ q - b2 * q^5 + b10 * q^7 + (b7 + b1) * q^11 + b4 * q^13 + (b7 + b5) * q^17 - b11 * q^19 - b14 * q^23 + (b11 - 2b3 + 2) q^25 + (b14 + b12) * q^29 + (-b13 + b10 + b4) * q^31 + (2b7 - 2b6 + 3b5 + 3b1) * q^35 + (-b15 + b13 - b10) * q^37 + (b6 - 2b5 - 2b1) * q^41 + (2b11 - 2b8 - b3 + 2) * q^43 + (b12 + b9 + b2) * q^47 + (2b11 + b8 - 2b3 - 1) * q^49 + (b14 - b12 - b9 - b2) * q^53 + (-b15 + b13) * q^55 + (b7 - 3b6) q^59 + (-b15 + b10) * q^61 + (-b7 + 4b6 - 3b5 - b1) * q^65 + (3b11 - b8 - b3 - 1) q^67 + (-b14 - b12 + b2) * q^71 + (3b8 - 2b3 - 1) * q^73 + (-b9 + b2) * q^77 + (-b15 - b13) * q^79 + (-2b7 - 2b6 - b5 + b1) * q^83 + (-b15 - b13 + b10) * q^85 + (-4b7 + 3b6 - b1) * q^89 + (4b11 + b8 - b3 + 3) q^91 + b9 * q^95 + (3b11 - b8 + 2b3) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 4 q^{19} + 28 q^{25} + 16 q^{43} - 20 q^{49} - 32 q^{67} - 4 q^{73} + 36 q^{91} - 16 q^{97}+O(q^{100}) $ 16 * q + 4 * q^19 + 28 * q^25 + 16 * q^43 - 20 * q^49 - 32 * q^67 - 4 * q^73 + 36 * q^91 - 16 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + \cdots + 256 $ x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 16*x^12 - 12*x^11 - 8*x^10 + 36*x^9 - 68*x^8 + 72*x^7 - 32*x^6 - 96*x^5 + 256*x^4 - 384*x^3 + 448*x^2 - 384*x + 256 :

$\beta_{1}$	$=$	$ ( \nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + \cdots + 832 ) / 448 $ (v^15 - 13v^14 + 11v^13 + 4v^12 - 38v^11 + 60v^10 - 104v^9 + 68v^8 + 148v^7 - 344v^6 + 440v^5 - 240v^4 - 32v^3 + 608v^2 - 1152v + 832) / 448
$\beta_{2}$	$=$	$ ( \nu^{14} - 3 \nu^{13} + 7 \nu^{12} - 12 \nu^{11} + 8 \nu^{10} - 4 \nu^{9} - 16 \nu^{8} + 52 \nu^{7} + \cdots + 256 ) / 64 $ (v^14 - 3v^13 + 7v^12 - 12v^11 + 8v^10 - 4v^9 - 16v^8 + 52v^7 - 68v^6 + 40v^5 + 64v^4 - 128v^3 + 288v^2 - 320*v + 256) / 64
$\beta_{3}$	$=$	$ ( - \nu^{15} + 5 \nu^{14} - 5 \nu^{13} + 6 \nu^{12} - 4 \nu^{11} - 16 \nu^{10} + 40 \nu^{9} + \cdots - 128 ) / 128 $ (-v^15 + 5v^14 - 5v^13 + 6v^12 - 4v^11 - 16v^10 + 40v^9 - 52v^8 + 12v^7 + 48v^6 - 128v^5 + 176v^4 - 192v^3 + 128*v - 128) / 128
$\beta_{4}$	$=$	$ ( - 2 \nu^{15} - 2 \nu^{14} + 13 \nu^{13} - 15 \nu^{12} + 41 \nu^{11} - 36 \nu^{10} - 2 \nu^{9} + \cdots - 320 ) / 224 $ (-2v^15 - 2v^14 + 13v^13 - 15v^12 + 41v^11 - 36v^10 - 2v^9 + 116v^8 - 184v^7 + 212v^6 - 68v^5 - 248v^4 + 568v^3 - 880v^2 + 288*v - 320) / 224
$\beta_{5}$	$=$	$ ( 13 \nu^{15} - 15 \nu^{14} + 31 \nu^{13} - 4 \nu^{12} - 32 \nu^{11} + 136 \nu^{10} - 176 \nu^{9} + \cdots + 1408 ) / 896 $ (13v^15 - 15v^14 + 31v^13 - 4v^12 - 32v^11 + 136v^10 - 176v^9 + 100v^8 + 76v^7 - 552v^6 + 736v^5 - 880v^4 + 256v^3 + 512v^2 - 1536*v + 1408) / 896
$\beta_{6}$	$=$	$ ( 9 \nu^{15} - 26 \nu^{14} + 50 \nu^{13} - 83 \nu^{12} + 78 \nu^{11} - 20 \nu^{10} - 152 \nu^{9} + \cdots - 576 ) / 448 $ (9v^15 - 26v^14 + 50v^13 - 83v^12 + 78v^11 - 20v^10 - 152v^9 + 332v^8 - 432v^7 + 236v^6 + 320v^5 - 1264v^4 + 2064v^3 - 1920v^2 + 1728*v - 576) / 448
$\beta_{7}$	$=$	$ ( 15 \nu^{15} - 41 \nu^{14} + 109 \nu^{13} - 220 \nu^{12} + 284 \nu^{11} - 192 \nu^{10} - 104 \nu^{9} + \cdots - 4096 ) / 896 $ (15v^15 - 41v^14 + 109v^13 - 220v^12 + 284v^11 - 192v^10 - 104v^9 + 796v^8 - 1420v^7 + 1448v^6 - 176v^5 - 2256v^4 + 4896v^3 - 7232v^2 + 6912*v - 4096) / 896
$\beta_{8}$	$=$	$ ( - \nu^{15} + 5 \nu^{14} - 11 \nu^{13} + 20 \nu^{12} - 22 \nu^{11} + 8 \nu^{10} + 28 \nu^{9} + \cdots + 256 ) / 64 $ (-v^15 + 5v^14 - 11v^13 + 20v^12 - 22v^11 + 8v^10 + 28v^9 - 76v^8 + 108v^7 - 88v^6 - 56v^5 + 224v^4 - 432v^3 + 480v^2 - 448v + 256) / 64
$\beta_{9}$	$=$	$ ( - \nu^{15} - \nu^{14} + 9 \nu^{13} - 16 \nu^{12} + 16 \nu^{11} - 8 \nu^{10} - 16 \nu^{9} + 76 \nu^{8} + \cdots + 128 ) / 64 $ (-v^15 - v^14 + 9v^13 - 16v^12 + 16v^11 - 8v^10 - 16v^9 + 76v^8 - 108v^7 + 56v^6 + 64v^5 - 208v^4 + 320v^3 - 384v^2 + 256*v + 128) / 64
$\beta_{10}$	$=$	$ ( 19 \nu^{15} - 37 \nu^{14} + 13 \nu^{13} + 48 \nu^{12} - 232 \nu^{11} + 384 \nu^{10} - 464 \nu^{9} + \cdots + 4608 ) / 896 $ (19v^15 - 37v^14 + 13v^13 + 48v^12 - 232v^11 + 384v^10 - 464v^9 + 60v^8 + 740v^7 - 1608v^6 + 1696v^5 - 752v^4 - 1280v^3 + 4608v^2 - 5760*v + 4608) / 896
$\beta_{11}$	$=$	$ ( - \nu^{15} + 6 \nu^{14} - 14 \nu^{13} + 23 \nu^{12} - 26 \nu^{11} + 8 \nu^{10} + 36 \nu^{9} + \cdots + 192 ) / 64 $ (-v^15 + 6v^14 - 14v^13 + 23v^12 - 26v^11 + 8v^10 + 36v^9 - 100v^8 + 128v^7 - 76v^6 - 64v^5 + 288v^4 - 512v^3 + 576v^2 - 384v + 192) / 64
$\beta_{12}$	$=$	$ ( 3 \nu^{15} + \nu^{14} - 9 \nu^{13} + 22 \nu^{12} - 28 \nu^{11} + 32 \nu^{10} - 8 \nu^{9} - 100 \nu^{8} + \cdots + 256 ) / 128 $ (3v^15 + v^14 - 9v^13 + 22v^12 - 28v^11 + 32v^10 - 8v^9 - 100v^8 + 156v^7 - 176v^6 + 32v^5 + 112v^4 - 640v^3 + 768v^2 - 640v + 256) / 128
$\beta_{13}$	$=$	$ ( 12 \nu^{15} - 23 \nu^{14} + 27 \nu^{13} - 15 \nu^{12} + 6 \nu^{11} + 104 \nu^{10} - 156 \nu^{9} + \cdots - 1664 ) / 448 $ (12v^15 - 23v^14 + 27v^13 - 15v^12 + 6v^11 + 104v^10 - 156v^9 + 144v^8 + 180v^7 - 124v^6 + 240v^5 - 752v^4 + 288v^3 + 128v^2 - 384*v - 1664) / 448
$\beta_{14}$	$=$	$ ( 7 \nu^{15} - 23 \nu^{14} + 55 \nu^{13} - 86 \nu^{12} + 84 \nu^{11} - 16 \nu^{10} - 176 \nu^{9} + \cdots - 512 ) / 128 $ (7v^15 - 23v^14 + 55v^13 - 86v^12 + 84v^11 - 16v^10 - 176v^9 + 380v^8 - 484v^7 + 256v^6 + 352v^5 - 1232v^4 + 2080v^3 - 2304v^2 + 1664*v - 512) / 128
$\beta_{15}$	$=$	$ ( 65 \nu^{15} - 159 \nu^{14} + 323 \nu^{13} - 356 \nu^{12} + 316 \nu^{11} + 176 \nu^{10} - 1048 \nu^{9} + \cdots - 4608 ) / 896 $ (65v^15 - 159v^14 + 323v^13 - 356v^12 + 316v^11 + 176v^10 - 1048v^9 + 1732v^8 - 1748v^7 + 152v^6 + 2000v^5 - 6192v^4 + 8672v^3 - 8640v^2 + 8448*v - 4608) / 896

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

$\nu$	$=$	$ ( \beta_{15} + 2\beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{5} - \beta_{4} - 2\beta_{3} + 2 ) / 8 $ (b15 + 2b11 - b10 + b9 - 2b5 - b4 - 2*b3 + 2) / 8
$\nu^{2}$	$=$	$ ( \beta_{15} - 2 \beta_{14} + \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + \cdots - 2 ) / 8 $ (b15 - 2b14 + b10 + b9 - 2b8 - 2b7 + 4b6 - 2b5 + b4 + 2b2 - 4*b1 - 2) / 8
$\nu^{3}$	$=$	$ ( -2\beta_{12} - \beta_{9} - 2\beta_{7} + 2\beta_{6} + 2\beta_{5} - 2\beta_1 ) / 4 $ (-2b12 - b9 - 2b7 + 2b6 + 2b5 - 2*b1) / 4
$\nu^{4}$	$=$	$ ( \beta_{15} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + \beta_{10} - 3 \beta_{9} - 4 \beta_{8} + \cdots + 4 ) / 8 $ (b15 - 2b13 - 2b12 - 2b11 + b10 - 3b9 - 4b8 + 2b7 - 10b6 + 2b5 - b4 + 4b2 - 2b1 + 4) / 8
$\nu^{5}$	$=$	$ ( - \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + 10 \beta_{11} - 5 \beta_{10} + \beta_{9} - 12 \beta_{8} + \cdots - 8 ) / 8 $ (-b15 - 2b13 + 2b12 + 10b11 - 5b10 + b9 - 12b8 + 6b7 - 6b6 + 10b5 + b4 - 4b3 + 4b2 + 10*b1 - 8) / 8
$\nu^{6}$	$=$	$ ( -\beta_{15} + 2\beta_{13} + 6\beta_{11} + 3\beta_{10} - 8\beta_{8} + 5\beta_{4} + 4\beta_{3} + 12 ) / 4 $ (-b15 + 2b13 + 6b11 + 3b10 - 8b8 + 5b4 + 4b3 + 12) / 4
$\nu^{7}$	$=$	$ ( - 3 \beta_{15} + 4 \beta_{14} + 14 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 15 \beta_{10} + \beta_{9} + \cdots + 44 ) / 8 $ (-3b15 + 4b14 + 14b13 - 2b12 + 2b11 - 15b10 + b9 - 2b7 - 2b6 + 2b5 - 9b4 + 4b3 + 8b2 + 14*b1 + 44) / 8
$\nu^{8}$	$=$	$ ( 5 \beta_{15} - 4 \beta_{14} + 6 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + \beta_{10} + 11 \beta_{9} + \cdots - 12 ) / 8 $ (5b15 - 4b14 + 6b13 + 2b12 + 2b11 + b10 + 11b9 + 16b8 + 34b7 - 14b6 - 26b5 - b4 - 20b3 + 24b2 + 2*b1 - 12) / 8
$\nu^{9}$	$=$	$ ( -12\beta_{14} - 6\beta_{12} + 3\beta_{9} + 18\beta_{7} + 10\beta_{6} + 22\beta_{5} + 10\beta_1 ) / 4 $ (-12b14 - 6b12 + 3b9 + 18b7 + 10b6 + 22b5 + 10*b1) / 4
$\nu^{10}$	$=$	$ ( \beta_{15} - 4 \beta_{14} + 6 \beta_{13} - 34 \beta_{12} + 2 \beta_{11} + 29 \beta_{10} - 7 \beta_{9} + \cdots + 60 ) / 8 $ (b15 - 4b14 + 6b13 - 34b12 + 2b11 + 29b10 - 7b9 - 8b8 - 10b7 - 34b6 + 50b5 - 5b4 - 20b3 - 48b2 - 50b1 + 60) / 8
$\nu^{11}$	$=$	$ ( 3 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} + 14 \beta_{12} - 26 \beta_{11} - 57 \beta_{10} - 7 \beta_{9} + \cdots + 100 ) / 8 $ (3b15 - 4b14 + 2b13 + 14b12 - 26b11 - 57b10 - 7b9 - 24b8 - 26b7 - 50b6 + 50b5 - 15b4 - 76b3 + 16b2 - 34*b1 + 100) / 8
$\nu^{12}$	$=$	$ ( 19\beta_{15} - 14\beta_{13} + 54\beta_{11} - 9\beta_{10} - 8\beta_{8} + 49\beta_{4} - 44\beta_{3} - 12 ) / 4 $ (19b15 - 14b13 + 54b11 - 9b10 - 8b8 + 49b4 - 44*b3 - 12) / 4
$\nu^{13}$	$=$	$ ( 41 \beta_{15} - 36 \beta_{14} + 22 \beta_{13} - 26 \beta_{12} + 2 \beta_{11} - 11 \beta_{10} + \cdots + 188 ) / 8 $ (41b15 - 36b14 + 22b13 - 26b12 + 2b11 - 11b10 + 5b9 - 120b8 - 114b7 + 6b6 + 106b5 + 67b4 + 220b3 + 32b2 + 22*b1 + 188) / 8
$\nu^{14}$	$=$	$ ( - 7 \beta_{15} + 84 \beta_{14} + 86 \beta_{13} - 14 \beta_{12} + 34 \beta_{11} - 91 \beta_{10} + \cdots - 164 ) / 8 $ (-7b15 + 84b14 + 86b13 - 14b12 + 34b11 - 91b10 + 63b9 + 8b8 - 86b7 - 110b6 + 94b5 - 77b4 + 60b3 + 160b2 - 190*b1 - 164) / 8
$\nu^{15}$	$=$	$ ( -12\beta_{14} + 50\beta_{12} - 17\beta_{9} + 202\beta_{7} - 46\beta_{6} + 62\beta_{5} + 192\beta_{2} - 62\beta_1 ) / 4 $ (-12b14 + 50b12 - 17b9 + 202b7 - 46b6 + 62b5 + 192b2 - 62b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$-1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1295.1

1.40985	+	0.111062i
1.40985	−	0.111062i
−0.186766	+	1.40183i
−0.186766	−	1.40183i
0.867527	−	1.11687i
0.867527	+	1.11687i
−1.37702	+	0.322193i
−1.37702	−	0.322193i
−0.409484	+	1.35363i
−0.409484	−	1.35363i
−0.533474	−	1.30973i
−0.533474	+	1.30973i
1.12063	+	0.862658i
1.12063	−	0.862658i
0.608741	+	1.27649i
0.608741	−	1.27649i

−3.48644

−

2.08772i

1295.2

−3.48644

2.08772i

1295.3

−3.21873

−

2.10412i

1295.4

−3.21873

2.10412i

1295.5

−1.79075

−

2.41093i

1295.6

−1.79075

2.41093i

1295.7

−1.13038

−

4.28970i

1295.8

−1.13038

4.28970i

1295.9

1.13038

−

4.28970i

1295.10

1.13038

4.28970i

1295.11

1.79075

−

2.41093i

1295.12

1.79075

2.41093i

1295.13

3.21873

−

2.10412i

1295.14

3.21873

2.10412i

1295.15

3.48644

−

2.08772i

1295.16

3.48644

2.08772i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
24.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2592.2.f.b		16
3.b	odd	2	1	inner	2592.2.f.b		16
4.b	odd	2	1		648.2.f.b		16
8.b	even	2	1		648.2.f.b		16
8.d	odd	2	1	inner	2592.2.f.b		16
9.c	even	3	1		288.2.p.b		16
9.c	even	3	1		864.2.p.b		16
9.d	odd	6	1		288.2.p.b		16
9.d	odd	6	1		864.2.p.b		16
12.b	even	2	1		648.2.f.b		16
24.f	even	2	1	inner	2592.2.f.b		16
24.h	odd	2	1		648.2.f.b		16
36.f	odd	6	1		72.2.l.b	✓	16
36.f	odd	6	1		216.2.l.b		16
36.h	even	6	1		72.2.l.b	✓	16
36.h	even	6	1		216.2.l.b		16
72.j	odd	6	1		72.2.l.b	✓	16
72.j	odd	6	1		216.2.l.b		16
72.l	even	6	1		288.2.p.b		16
72.l	even	6	1		864.2.p.b		16
72.n	even	6	1		72.2.l.b	✓	16
72.n	even	6	1		216.2.l.b		16
72.p	odd	6	1		288.2.p.b		16
72.p	odd	6	1		864.2.p.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.2.l.b	✓	16	36.f	odd	6	1
72.2.l.b	✓	16	36.h	even	6	1
72.2.l.b	✓	16	72.j	odd	6	1
72.2.l.b	✓	16	72.n	even	6	1
216.2.l.b		16	36.f	odd	6	1
216.2.l.b		16	36.h	even	6	1
216.2.l.b		16	72.j	odd	6	1
216.2.l.b		16	72.n	even	6	1
288.2.p.b		16	9.c	even	3	1
288.2.p.b		16	9.d	odd	6	1
288.2.p.b		16	72.l	even	6	1
288.2.p.b		16	72.p	odd	6	1
648.2.f.b		16	4.b	odd	2	1
648.2.f.b		16	8.b	even	2	1
648.2.f.b		16	12.b	even	2	1
648.2.f.b		16	24.h	odd	2	1
864.2.p.b		16	9.c	even	3	1
864.2.p.b		16	9.d	odd	6	1
864.2.p.b		16	72.l	even	6	1
864.2.p.b		16	72.p	odd	6	1
2592.2.f.b		16	1.a	even	1	1	trivial
2592.2.f.b		16	3.b	odd	2	1	inner
2592.2.f.b		16	8.d	odd	2	1	inner
2592.2.f.b		16	24.f	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 27T_{5}^{6} + 231T_{5}^{4} - 657T_{5}^{2} + 516 $ T5^8 - 27*T5^6 + 231*T5^4 - 657*T5^2 + 516 acting on $S_{2}^{\mathrm{new}}(2592, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 27 T^{6} + \cdots + 516)^{2} $ (T^8 - 27T^6 + 231T^4 - 657*T^2 + 516)^2
$7$	$ (T^{8} + 33 T^{6} + \cdots + 2064)^{2} $ (T^8 + 33T^6 + 339T^4 + 1407*T^2 + 2064)^2
$11$	$ (T^{8} + 20 T^{6} + 66 T^{4} + \cdots + 1)^{2} $ (T^8 + 20T^6 + 66T^4 + 56*T^2 + 1)^2
$13$	$ (T^{8} + 57 T^{6} + \cdots + 516)^{2} $ (T^8 + 57T^6 + 807T^4 + 3435*T^2 + 516)^2
$17$	$ (T^{8} + 35 T^{6} + \cdots + 784)^{2} $ (T^8 + 35T^6 + 360T^4 + 992*T^2 + 784)^2
$19$	$ (T^{4} - T^{3} - 12 T^{2} + \cdots + 16)^{4} $ (T^4 - T^3 - 12T^2 + 8T + 16)^4
$23$	$ (T^{8} - 99 T^{6} + \cdots + 25284)^{2} $ (T^8 - 99T^6 + 2643T^4 - 17961*T^2 + 25284)^2
$29$	$ (T^{8} - 135 T^{6} + \cdots + 132096)^{2} $ (T^8 - 135T^6 + 5571T^4 - 76269*T^2 + 132096)^2
$31$	$ (T^{8} + 117 T^{6} + \cdots + 528384)^{2} $ (T^8 + 117T^6 + 4863T^4 + 85239*T^2 + 528384)^2
$37$	$ (T^{8} + 156 T^{6} + \cdots + 74304)^{2} $ (T^8 + 156T^6 + 4896T^4 + 40176*T^2 + 74304)^2
$41$	$ (T^{8} + 68 T^{6} + \cdots + 7921)^{2} $ (T^8 + 68T^6 + 1302T^4 + 6260*T^2 + 7921)^2
$43$	$ (T^{4} - 4 T^{3} - 60 T^{2} + \cdots - 83)^{4} $ (T^4 - 4T^3 - 60T^2 - 142*T - 83)^4
$47$	$ (T^{8} - 111 T^{6} + \cdots + 516)^{2} $ (T^8 - 111T^6 + 1503T^4 - 3297*T^2 + 516)^2
$53$	$ (T^{8} - 228 T^{6} + \cdots + 297216)^{2} $ (T^8 - 228T^6 + 15408T^4 - 331344*T^2 + 297216)^2
$59$	$ (T^{8} + 152 T^{6} + \cdots + 528529)^{2} $ (T^8 + 152T^6 + 7266T^4 + 119108*T^2 + 528529)^2
$61$	$ (T^{8} + 189 T^{6} + \cdots + 1091856)^{2} $ (T^8 + 189T^6 + 10971T^4 + 196935*T^2 + 1091856)^2
$67$	$ (T^{4} + 8 T^{3} + \cdots + 763)^{4} $ (T^4 + 8T^3 - 66T^2 - 274*T + 763)^4
$71$	$ (T^{8} - 168 T^{6} + \cdots + 74304)^{2} $ (T^8 - 168T^6 + 3744T^4 - 28944*T^2 + 74304)^2
$73$	$ (T^{4} + T^{3} - 78 T^{2} + \cdots + 172)^{4} $ (T^4 + T^3 - 78T^2 - 224T + 172)^4
$79$	$ (T^{8} + 249 T^{6} + \cdots + 272964)^{2} $ (T^8 + 249T^6 + 15243T^4 + 132291*T^2 + 272964)^2
$83$	$ (T^{8} + 305 T^{6} + \cdots + 432964)^{2} $ (T^8 + 305T^6 + 15615T^4 + 223823*T^2 + 432964)^2
$89$	$ (T^{8} + 272 T^{6} + \cdots + 891136)^{2} $ (T^8 + 272T^6 + 23808T^4 + 689456*T^2 + 891136)^2
$97$	$ (T^{4} + 4 T^{3} + \cdots + 1009)^{4} $ (T^4 + 4T^3 - 174T^2 - 896*T + 1009)^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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{5} + \beta_{10} q^{7}+O(q^{10}) \) q - b2 * q^5 + b10 * q^7	\( q - \beta_{2} q^{5} + \beta_{10} q^{7} + (\beta_{7} + \beta_1) q^{11} + \beta_{4} q^{13} + (\beta_{7} + \beta_{5}) q^{17} - \beta_{11} q^{19} - \beta_{14} q^{23} + (\beta_{11} - 2 \beta_{3} + 2) q^{25} + (\beta_{14} + \beta_{12}) q^{29} + ( - \beta_{13} + \beta_{10} + \beta_{4}) q^{31} + (2 \beta_{7} - 2 \beta_{6} + \cdots + 3 \beta_1) q^{35}+ \cdots + (3 \beta_{11} - \beta_{8} + 2 \beta_{3}) q^{97}+O(q^{100}) \) q - b2 * q^5 + b10 * q^7 + (b7 + b1) * q^11 + b4 * q^13 + (b7 + b5) * q^17 - b11 * q^19 - b14 * q^23 + (b11 - 2b3 + 2) q^25 + (b14 + b12) * q^29 + (-b13 + b10 + b4) * q^31 + (2b7 - 2b6 + 3b5 + 3b1) * q^35 + (-b15 + b13 - b10) * q^37 + (b6 - 2b5 - 2b1) * q^41 + (2b11 - 2b8 - b3 + 2) * q^43 + (b12 + b9 + b2) * q^47 + (2b11 + b8 - 2b3 - 1) * q^49 + (b14 - b12 - b9 - b2) * q^53 + (-b15 + b13) * q^55 + (b7 - 3b6) q^59 + (-b15 + b10) * q^61 + (-b7 + 4b6 - 3b5 - b1) * q^65 + (3b11 - b8 - b3 - 1) q^67 + (-b14 - b12 + b2) * q^71 + (3b8 - 2b3 - 1) * q^73 + (-b9 + b2) * q^77 + (-b15 - b13) * q^79 + (-2b7 - 2b6 - b5 + b1) * q^83 + (-b15 - b13 + b10) * q^85 + (-4b7 + 3b6 - b1) * q^89 + (4b11 + b8 - b3 + 3) q^91 + b9 * q^95 + (3b11 - b8 + 2b3) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 4 q^{19} + 28 q^{25} + 16 q^{43} - 20 q^{49} - 32 q^{67} - 4 q^{73} + 36 q^{91} - 16 q^{97}+O(q^{100}) \) 16 * q + 4 * q^19 + 28 * q^25 + 16 * q^43 - 20 * q^49 - 32 * q^67 - 4 * q^73 + 36 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

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	2592.2.f.b

Level	$2592$
Weight	$2$
Character orbit	2592.f
Analytic conductor	$20.697$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(20.6972242039\)
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:	\( x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + \cdots + 256 \) x^16 - 3x^15 + 7x^14 - 12x^13 + 16x^12 - 12x^11 - 8x^10 + 36x^9 - 68x^8 + 72x^7 - 32x^6 - 96x^5 + 256x^4 - 384x^3 + 448x^2 - 384*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{18} \)
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + \cdots + 832 ) / 448 \) (v^15 - 13v^14 + 11v^13 + 4v^12 - 38v^11 + 60v^10 - 104v^9 + 68v^8 + 148v^7 - 344v^6 + 440v^5 - 240v^4 - 32v^3 + 608v^2 - 1152v + 832) / 448
\(\beta_{2}\)	\(=\)	\( ( \nu^{14} - 3 \nu^{13} + 7 \nu^{12} - 12 \nu^{11} + 8 \nu^{10} - 4 \nu^{9} - 16 \nu^{8} + 52 \nu^{7} + \cdots + 256 ) / 64 \) (v^14 - 3v^13 + 7v^12 - 12v^11 + 8v^10 - 4v^9 - 16v^8 + 52v^7 - 68v^6 + 40v^5 + 64v^4 - 128v^3 + 288v^2 - 320*v + 256) / 64
\(\beta_{3}\)	\(=\)	\( ( - \nu^{15} + 5 \nu^{14} - 5 \nu^{13} + 6 \nu^{12} - 4 \nu^{11} - 16 \nu^{10} + 40 \nu^{9} + \cdots - 128 ) / 128 \) (-v^15 + 5v^14 - 5v^13 + 6v^12 - 4v^11 - 16v^10 + 40v^9 - 52v^8 + 12v^7 + 48v^6 - 128v^5 + 176v^4 - 192v^3 + 128*v - 128) / 128
\(\beta_{4}\)	\(=\)	\( ( - 2 \nu^{15} - 2 \nu^{14} + 13 \nu^{13} - 15 \nu^{12} + 41 \nu^{11} - 36 \nu^{10} - 2 \nu^{9} + \cdots - 320 ) / 224 \) (-2v^15 - 2v^14 + 13v^13 - 15v^12 + 41v^11 - 36v^10 - 2v^9 + 116v^8 - 184v^7 + 212v^6 - 68v^5 - 248v^4 + 568v^3 - 880v^2 + 288*v - 320) / 224
\(\beta_{5}\)	\(=\)	\( ( 13 \nu^{15} - 15 \nu^{14} + 31 \nu^{13} - 4 \nu^{12} - 32 \nu^{11} + 136 \nu^{10} - 176 \nu^{9} + \cdots + 1408 ) / 896 \) (13v^15 - 15v^14 + 31v^13 - 4v^12 - 32v^11 + 136v^10 - 176v^9 + 100v^8 + 76v^7 - 552v^6 + 736v^5 - 880v^4 + 256v^3 + 512v^2 - 1536*v + 1408) / 896
\(\beta_{6}\)	\(=\)	\( ( 9 \nu^{15} - 26 \nu^{14} + 50 \nu^{13} - 83 \nu^{12} + 78 \nu^{11} - 20 \nu^{10} - 152 \nu^{9} + \cdots - 576 ) / 448 \) (9v^15 - 26v^14 + 50v^13 - 83v^12 + 78v^11 - 20v^10 - 152v^9 + 332v^8 - 432v^7 + 236v^6 + 320v^5 - 1264v^4 + 2064v^3 - 1920v^2 + 1728*v - 576) / 448
\(\beta_{7}\)	\(=\)	\( ( 15 \nu^{15} - 41 \nu^{14} + 109 \nu^{13} - 220 \nu^{12} + 284 \nu^{11} - 192 \nu^{10} - 104 \nu^{9} + \cdots - 4096 ) / 896 \) (15v^15 - 41v^14 + 109v^13 - 220v^12 + 284v^11 - 192v^10 - 104v^9 + 796v^8 - 1420v^7 + 1448v^6 - 176v^5 - 2256v^4 + 4896v^3 - 7232v^2 + 6912*v - 4096) / 896
\(\beta_{8}\)	\(=\)	\( ( - \nu^{15} + 5 \nu^{14} - 11 \nu^{13} + 20 \nu^{12} - 22 \nu^{11} + 8 \nu^{10} + 28 \nu^{9} + \cdots + 256 ) / 64 \) (-v^15 + 5v^14 - 11v^13 + 20v^12 - 22v^11 + 8v^10 + 28v^9 - 76v^8 + 108v^7 - 88v^6 - 56v^5 + 224v^4 - 432v^3 + 480v^2 - 448v + 256) / 64
\(\beta_{9}\)	\(=\)	\( ( - \nu^{15} - \nu^{14} + 9 \nu^{13} - 16 \nu^{12} + 16 \nu^{11} - 8 \nu^{10} - 16 \nu^{9} + 76 \nu^{8} + \cdots + 128 ) / 64 \) (-v^15 - v^14 + 9v^13 - 16v^12 + 16v^11 - 8v^10 - 16v^9 + 76v^8 - 108v^7 + 56v^6 + 64v^5 - 208v^4 + 320v^3 - 384v^2 + 256*v + 128) / 64
\(\beta_{10}\)	\(=\)	\( ( 19 \nu^{15} - 37 \nu^{14} + 13 \nu^{13} + 48 \nu^{12} - 232 \nu^{11} + 384 \nu^{10} - 464 \nu^{9} + \cdots + 4608 ) / 896 \) (19v^15 - 37v^14 + 13v^13 + 48v^12 - 232v^11 + 384v^10 - 464v^9 + 60v^8 + 740v^7 - 1608v^6 + 1696v^5 - 752v^4 - 1280v^3 + 4608v^2 - 5760*v + 4608) / 896
\(\beta_{11}\)	\(=\)	\( ( - \nu^{15} + 6 \nu^{14} - 14 \nu^{13} + 23 \nu^{12} - 26 \nu^{11} + 8 \nu^{10} + 36 \nu^{9} + \cdots + 192 ) / 64 \) (-v^15 + 6v^14 - 14v^13 + 23v^12 - 26v^11 + 8v^10 + 36v^9 - 100v^8 + 128v^7 - 76v^6 - 64v^5 + 288v^4 - 512v^3 + 576v^2 - 384v + 192) / 64
\(\beta_{12}\)	\(=\)	\( ( 3 \nu^{15} + \nu^{14} - 9 \nu^{13} + 22 \nu^{12} - 28 \nu^{11} + 32 \nu^{10} - 8 \nu^{9} - 100 \nu^{8} + \cdots + 256 ) / 128 \) (3v^15 + v^14 - 9v^13 + 22v^12 - 28v^11 + 32v^10 - 8v^9 - 100v^8 + 156v^7 - 176v^6 + 32v^5 + 112v^4 - 640v^3 + 768v^2 - 640v + 256) / 128
\(\beta_{13}\)	\(=\)	\( ( 12 \nu^{15} - 23 \nu^{14} + 27 \nu^{13} - 15 \nu^{12} + 6 \nu^{11} + 104 \nu^{10} - 156 \nu^{9} + \cdots - 1664 ) / 448 \) (12v^15 - 23v^14 + 27v^13 - 15v^12 + 6v^11 + 104v^10 - 156v^9 + 144v^8 + 180v^7 - 124v^6 + 240v^5 - 752v^4 + 288v^3 + 128v^2 - 384*v - 1664) / 448
\(\beta_{14}\)	\(=\)	\( ( 7 \nu^{15} - 23 \nu^{14} + 55 \nu^{13} - 86 \nu^{12} + 84 \nu^{11} - 16 \nu^{10} - 176 \nu^{9} + \cdots - 512 ) / 128 \) (7v^15 - 23v^14 + 55v^13 - 86v^12 + 84v^11 - 16v^10 - 176v^9 + 380v^8 - 484v^7 + 256v^6 + 352v^5 - 1232v^4 + 2080v^3 - 2304v^2 + 1664*v - 512) / 128
\(\beta_{15}\)	\(=\)	\( ( 65 \nu^{15} - 159 \nu^{14} + 323 \nu^{13} - 356 \nu^{12} + 316 \nu^{11} + 176 \nu^{10} - 1048 \nu^{9} + \cdots - 4608 ) / 896 \) (65v^15 - 159v^14 + 323v^13 - 356v^12 + 316v^11 + 176v^10 - 1048v^9 + 1732v^8 - 1748v^7 + 152v^6 + 2000v^5 - 6192v^4 + 8672v^3 - 8640v^2 + 8448*v - 4608) / 896

\(\nu\)	\(=\)	\( ( \beta_{15} + 2\beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{5} - \beta_{4} - 2\beta_{3} + 2 ) / 8 \) (b15 + 2b11 - b10 + b9 - 2b5 - b4 - 2*b3 + 2) / 8
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - 2 \beta_{14} + \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + \cdots - 2 ) / 8 \) (b15 - 2b14 + b10 + b9 - 2b8 - 2b7 + 4b6 - 2b5 + b4 + 2b2 - 4*b1 - 2) / 8
\(\nu^{3}\)	\(=\)	\( ( -2\beta_{12} - \beta_{9} - 2\beta_{7} + 2\beta_{6} + 2\beta_{5} - 2\beta_1 ) / 4 \) (-2b12 - b9 - 2b7 + 2b6 + 2b5 - 2*b1) / 4
\(\nu^{4}\)	\(=\)	\( ( \beta_{15} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + \beta_{10} - 3 \beta_{9} - 4 \beta_{8} + \cdots + 4 ) / 8 \) (b15 - 2b13 - 2b12 - 2b11 + b10 - 3b9 - 4b8 + 2b7 - 10b6 + 2b5 - b4 + 4b2 - 2b1 + 4) / 8
\(\nu^{5}\)	\(=\)	\( ( - \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + 10 \beta_{11} - 5 \beta_{10} + \beta_{9} - 12 \beta_{8} + \cdots - 8 ) / 8 \) (-b15 - 2b13 + 2b12 + 10b11 - 5b10 + b9 - 12b8 + 6b7 - 6b6 + 10b5 + b4 - 4b3 + 4b2 + 10*b1 - 8) / 8
\(\nu^{6}\)	\(=\)	\( ( -\beta_{15} + 2\beta_{13} + 6\beta_{11} + 3\beta_{10} - 8\beta_{8} + 5\beta_{4} + 4\beta_{3} + 12 ) / 4 \) (-b15 + 2b13 + 6b11 + 3b10 - 8b8 + 5b4 + 4b3 + 12) / 4
\(\nu^{7}\)	\(=\)	\( ( - 3 \beta_{15} + 4 \beta_{14} + 14 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 15 \beta_{10} + \beta_{9} + \cdots + 44 ) / 8 \) (-3b15 + 4b14 + 14b13 - 2b12 + 2b11 - 15b10 + b9 - 2b7 - 2b6 + 2b5 - 9b4 + 4b3 + 8b2 + 14*b1 + 44) / 8
\(\nu^{8}\)	\(=\)	\( ( 5 \beta_{15} - 4 \beta_{14} + 6 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + \beta_{10} + 11 \beta_{9} + \cdots - 12 ) / 8 \) (5b15 - 4b14 + 6b13 + 2b12 + 2b11 + b10 + 11b9 + 16b8 + 34b7 - 14b6 - 26b5 - b4 - 20b3 + 24b2 + 2*b1 - 12) / 8
\(\nu^{9}\)	\(=\)	\( ( -12\beta_{14} - 6\beta_{12} + 3\beta_{9} + 18\beta_{7} + 10\beta_{6} + 22\beta_{5} + 10\beta_1 ) / 4 \) (-12b14 - 6b12 + 3b9 + 18b7 + 10b6 + 22b5 + 10*b1) / 4
\(\nu^{10}\)	\(=\)	\( ( \beta_{15} - 4 \beta_{14} + 6 \beta_{13} - 34 \beta_{12} + 2 \beta_{11} + 29 \beta_{10} - 7 \beta_{9} + \cdots + 60 ) / 8 \) (b15 - 4b14 + 6b13 - 34b12 + 2b11 + 29b10 - 7b9 - 8b8 - 10b7 - 34b6 + 50b5 - 5b4 - 20b3 - 48b2 - 50b1 + 60) / 8
\(\nu^{11}\)	\(=\)	\( ( 3 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} + 14 \beta_{12} - 26 \beta_{11} - 57 \beta_{10} - 7 \beta_{9} + \cdots + 100 ) / 8 \) (3b15 - 4b14 + 2b13 + 14b12 - 26b11 - 57b10 - 7b9 - 24b8 - 26b7 - 50b6 + 50b5 - 15b4 - 76b3 + 16b2 - 34*b1 + 100) / 8
\(\nu^{12}\)	\(=\)	\( ( 19\beta_{15} - 14\beta_{13} + 54\beta_{11} - 9\beta_{10} - 8\beta_{8} + 49\beta_{4} - 44\beta_{3} - 12 ) / 4 \) (19b15 - 14b13 + 54b11 - 9b10 - 8b8 + 49b4 - 44*b3 - 12) / 4
\(\nu^{13}\)	\(=\)	\( ( 41 \beta_{15} - 36 \beta_{14} + 22 \beta_{13} - 26 \beta_{12} + 2 \beta_{11} - 11 \beta_{10} + \cdots + 188 ) / 8 \) (41b15 - 36b14 + 22b13 - 26b12 + 2b11 - 11b10 + 5b9 - 120b8 - 114b7 + 6b6 + 106b5 + 67b4 + 220b3 + 32b2 + 22*b1 + 188) / 8
\(\nu^{14}\)	\(=\)	\( ( - 7 \beta_{15} + 84 \beta_{14} + 86 \beta_{13} - 14 \beta_{12} + 34 \beta_{11} - 91 \beta_{10} + \cdots - 164 ) / 8 \) (-7b15 + 84b14 + 86b13 - 14b12 + 34b11 - 91b10 + 63b9 + 8b8 - 86b7 - 110b6 + 94b5 - 77b4 + 60b3 + 160b2 - 190*b1 - 164) / 8
\(\nu^{15}\)	\(=\)	\( ( -12\beta_{14} + 50\beta_{12} - 17\beta_{9} + 202\beta_{7} - 46\beta_{6} + 62\beta_{5} + 192\beta_{2} - 62\beta_1 ) / 4 \) (-12b14 + 50b12 - 17b9 + 202b7 - 46b6 + 62b5 + 192b2 - 62b1) / 4

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1295.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 27 T^{6} + \cdots + 516)^{2} \) (T^8 - 27T^6 + 231T^4 - 657*T^2 + 516)^2
$7$	\( (T^{8} + 33 T^{6} + \cdots + 2064)^{2} \) (T^8 + 33T^6 + 339T^4 + 1407*T^2 + 2064)^2
$11$	\( (T^{8} + 20 T^{6} + 66 T^{4} + \cdots + 1)^{2} \) (T^8 + 20T^6 + 66T^4 + 56*T^2 + 1)^2
$13$	\( (T^{8} + 57 T^{6} + \cdots + 516)^{2} \) (T^8 + 57T^6 + 807T^4 + 3435*T^2 + 516)^2
$17$	\( (T^{8} + 35 T^{6} + \cdots + 784)^{2} \) (T^8 + 35T^6 + 360T^4 + 992*T^2 + 784)^2
$19$	\( (T^{4} - T^{3} - 12 T^{2} + \cdots + 16)^{4} \) (T^4 - T^3 - 12T^2 + 8T + 16)^4
$23$	\( (T^{8} - 99 T^{6} + \cdots + 25284)^{2} \) (T^8 - 99T^6 + 2643T^4 - 17961*T^2 + 25284)^2
$29$	\( (T^{8} - 135 T^{6} + \cdots + 132096)^{2} \) (T^8 - 135T^6 + 5571T^4 - 76269*T^2 + 132096)^2
$31$	\( (T^{8} + 117 T^{6} + \cdots + 528384)^{2} \) (T^8 + 117T^6 + 4863T^4 + 85239*T^2 + 528384)^2
$37$	\( (T^{8} + 156 T^{6} + \cdots + 74304)^{2} \) (T^8 + 156T^6 + 4896T^4 + 40176*T^2 + 74304)^2
$41$	\( (T^{8} + 68 T^{6} + \cdots + 7921)^{2} \) (T^8 + 68T^6 + 1302T^4 + 6260*T^2 + 7921)^2
$43$	\( (T^{4} - 4 T^{3} - 60 T^{2} + \cdots - 83)^{4} \) (T^4 - 4T^3 - 60T^2 - 142*T - 83)^4
$47$	\( (T^{8} - 111 T^{6} + \cdots + 516)^{2} \) (T^8 - 111T^6 + 1503T^4 - 3297*T^2 + 516)^2
$53$	\( (T^{8} - 228 T^{6} + \cdots + 297216)^{2} \) (T^8 - 228T^6 + 15408T^4 - 331344*T^2 + 297216)^2
$59$	\( (T^{8} + 152 T^{6} + \cdots + 528529)^{2} \) (T^8 + 152T^6 + 7266T^4 + 119108*T^2 + 528529)^2
$61$	\( (T^{8} + 189 T^{6} + \cdots + 1091856)^{2} \) (T^8 + 189T^6 + 10971T^4 + 196935*T^2 + 1091856)^2
$67$	\( (T^{4} + 8 T^{3} + \cdots + 763)^{4} \) (T^4 + 8T^3 - 66T^2 - 274*T + 763)^4
$71$	\( (T^{8} - 168 T^{6} + \cdots + 74304)^{2} \) (T^8 - 168T^6 + 3744T^4 - 28944*T^2 + 74304)^2
$73$	\( (T^{4} + T^{3} - 78 T^{2} + \cdots + 172)^{4} \) (T^4 + T^3 - 78T^2 - 224T + 172)^4
$79$	\( (T^{8} + 249 T^{6} + \cdots + 272964)^{2} \) (T^8 + 249T^6 + 15243T^4 + 132291*T^2 + 272964)^2
$83$	\( (T^{8} + 305 T^{6} + \cdots + 432964)^{2} \) (T^8 + 305T^6 + 15615T^4 + 223823*T^2 + 432964)^2
$89$	\( (T^{8} + 272 T^{6} + \cdots + 891136)^{2} \) (T^8 + 272T^6 + 23808T^4 + 689456*T^2 + 891136)^2
$97$	\( (T^{4} + 4 T^{3} + \cdots + 1009)^{4} \) (T^4 + 4T^3 - 174T^2 - 896*T + 1009)^4
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\(n\)	\(325\)	\(1217\)	\(2431\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)