LMFDB - Newform orbit 4320.2.d.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(3889,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("4320.3889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.d (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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3x^{18} + 8x^{16} - 24x^{14} + 56x^{12} - 92x^{10} + 224x^{8} - 384x^{6} + 512x^{4} - 768x^{2} + 1024 $ x^20 - 3x^18 + 8x^16 - 24x^14 + 56x^12 - 92x^10 + 224x^8 - 384x^6 + 512x^4 - 768*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 1080)
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 $1,\beta_1,\ldots,\beta_{19}$ 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_{6} q^{11} + \beta_{19} q^{13} + (\beta_{18} - \beta_{11} + \cdots - \beta_{2}) q^{17}+ \cdots + ( - \beta_{12} - 2 \beta_{5} + \cdots - \beta_1) q^{97}+O(q^{100}) $ q + b2 * q^5 + b10 * q^7 + b6 * q^11 + b19 * q^13 + (b18 - b11 - b6 - b2) * q^17 + b3 * q^19 + (b18 + b13 + b11 + b2) * q^23 + (-b12 + b7 + 1) * q^25 + (-b18 + b16) * q^29 + (b1 - 1) * q^31 + (b16 + 2b13 + b9 + b6) q^35 + (b17 - b12 + 2) * q^37 - b4 * q^41 + (-b19 + b17 - b12 - 2b1 + 2) q^43 + (b18 - b16 - 2b13 - b11 - 2b6 - b2) * q^47 + (-b19 - b17 - b12 - 2) * q^49 + (b18 - b15 - b13 - b9 - b6) * q^53 + (-b19 + b17 + b14 - b10 - b5 + b3 - b1 + 2) * q^55 + (2b18 - 2b16 + b13 + b11 - b6 + b2) * q^59 + (-b12 + b7 - b5 - b1) * q^61 + (-b18 - b16 + b15 + b13 + b11 + b8 + b6 + b2) * q^65 + (b19 - b17 - 2b12 - b1 + 1) q^67 + (b18 - b13 - b11 - 2b9 + b8 - b6 + b4 + b2) q^71 + (-b19 + b17 + 2b14 - b12 - b10 + b7 - b5 + b3 - b1 + 1) q^73 + (b15 - b9 + b8 - b4) * q^77 + (-b19 - 2b12 - 2b1) * q^79 + (-b15 + b11 + b9 - b8 - b2) * q^83 + (b19 - b14 + b10 - b7 - b3 + 2) * q^85 + (-b18 + b15 + b13 - b11 + b9 - b8 + b6 + b2) * q^89 + (b19 - b17 - 2b14 + b12 + 3b10 - b7 + b5 - b3 + b1 - 1) * q^91 + (-2b16 - b15 + 2b11 + b9 + b6 + b2) * q^95 + (-b12 - 2b5 - b3 - b1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 4 q^{13} + 20 q^{25} - 12 q^{31} + 32 q^{37} + 12 q^{43} - 52 q^{49} + 28 q^{55} - 36 q^{79} + 44 q^{85}+O(q^{100}) $ 20 * q + 4 * q^13 + 20 * q^25 - 12 * q^31 + 32 * q^37 + 12 * q^43 - 52 * q^49 + 28 * q^55 - 36 * q^79 + 44 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 3x^{18} + 8x^{16} - 24x^{14} + 56x^{12} - 92x^{10} + 224x^{8} - 384x^{6} + 512x^{4} - 768x^{2} + 1024 $ x^20 - 3*x^18 + 8*x^16 - 24*x^14 + 56*x^12 - 92*x^10 + 224*x^8 - 384*x^6 + 512*x^4 - 768*x^2 + 1024 :

$\beta_{1}$	$=$	$ ( -\nu^{18} - \nu^{16} + 4\nu^{14} - 8\nu^{12} + 40\nu^{10} - 132\nu^{8} + 144\nu^{6} - 256\nu^{4} + 512\nu^{2} - 512 ) / 256 $ (-v^18 - v^16 + 4v^14 - 8v^12 + 40v^10 - 132v^8 + 144v^6 - 256v^4 + 512*v^2 - 512) / 256
$\beta_{2}$	$=$	$ ( -\nu^{19} - 2\nu^{17} - 3\nu^{15} + 6\nu^{13} - 8\nu^{11} + 4\nu^{9} - 36\nu^{7} - 72\nu^{5} - 32\nu^{3} - 64\nu ) / 384 $ (-v^19 - 2v^17 - 3v^15 + 6v^13 - 8v^11 + 4v^9 - 36v^7 - 72v^5 - 32v^3 - 64*v) / 384
$\beta_{3}$	$=$	$ ( \nu^{18} + \nu^{16} - 4\nu^{14} + 40\nu^{12} - 8\nu^{10} + 132\nu^{8} - 272\nu^{6} + 512\nu^{4} - 384\nu^{2} + 2560 ) / 256 $ (v^18 + v^16 - 4v^14 + 40v^12 - 8v^10 + 132v^8 - 272v^6 + 512v^4 - 384*v^2 + 2560) / 256
$\beta_{4}$	$=$	$ ( \nu^{19} + 7\nu^{17} + 2\nu^{15} - 40\nu^{11} + 20\nu^{9} + 136\nu^{7} + 160\nu^{5} + 576\nu^{3} - 1024\nu ) / 256 $ (v^19 + 7v^17 + 2v^15 - 40v^11 + 20v^9 + 136v^7 + 160v^5 + 576v^3 - 1024v) / 256
$\beta_{5}$	$=$	$ ( \nu^{18} - 4 \nu^{16} + 3 \nu^{14} - 24 \nu^{12} + 32 \nu^{10} - 52 \nu^{8} + 252 \nu^{6} + \cdots - 1280 ) / 192 $ (v^18 - 4v^16 + 3v^14 - 24v^12 + 32v^10 - 52v^8 + 252v^6 - 384v^4 + 512v^2 - 1280) / 192
$\beta_{6}$	$=$	$ ( - 5 \nu^{19} + 23 \nu^{17} - 48 \nu^{15} + 168 \nu^{13} - 184 \nu^{11} + 524 \nu^{9} + \cdots + 6400 \nu ) / 1536 $ (-5v^19 + 23v^17 - 48v^15 + 168v^13 - 184v^11 + 524v^9 - 960v^7 + 1728v^5 - 1408v^3 + 6400v) / 1536
$\beta_{7}$	$=$	$ ( -5\nu^{18} + 23\nu^{16} + 120\nu^{12} - 88\nu^{10} + 140\nu^{8} - 576\nu^{6} + 1152\nu^{4} + 1280\nu^{2} + 4096 ) / 768 $ (-5v^18 + 23v^16 + 120v^12 - 88v^10 + 140v^8 - 576v^6 + 1152v^4 + 1280v^2 + 4096) / 768
$\beta_{8}$	$=$	$ ( - \nu^{19} + 3 \nu^{17} - 8 \nu^{15} + 24 \nu^{13} - 56 \nu^{11} + 92 \nu^{9} - 224 \nu^{7} + \cdots + 1280 \nu ) / 256 $ (-v^19 + 3v^17 - 8v^15 + 24v^13 - 56v^11 + 92v^9 - 224v^7 + 384v^5 - 512v^3 + 1280*v) / 256
$\beta_{9}$	$=$	$ ( - 7 \nu^{19} + 13 \nu^{17} - 72 \nu^{15} - 296 \nu^{11} + 580 \nu^{9} - 384 \nu^{7} + \cdots - 2560 \nu ) / 1536 $ (-7v^19 + 13v^17 - 72v^15 - 296v^11 + 580v^9 - 384v^7 + 480v^5 - 3968v^3 - 2560*v) / 1536
$\beta_{10}$	$=$	$ ( 7 \nu^{18} - 25 \nu^{16} + 36 \nu^{14} - 168 \nu^{12} + 296 \nu^{10} - 484 \nu^{8} + 912 \nu^{6} + \cdots - 3584 ) / 768 $ (7v^18 - 25v^16 + 36v^14 - 168v^12 + 296v^10 - 484v^8 + 912v^6 - 1920v^4 + 1280*v^2 - 3584) / 768
$\beta_{11}$	$=$	$ ( \nu^{19} - \nu^{17} + 9 \nu^{15} - 21 \nu^{13} + 32 \nu^{11} - 76 \nu^{9} + 192 \nu^{7} + \cdots - 608 \nu ) / 192 $ (v^19 - v^17 + 9v^15 - 21v^13 + 32v^11 - 76v^9 + 192v^7 - 132v^5 + 416v^3 - 608v) / 192
$\beta_{12}$	$=$	$ ( 3\nu^{18} + 3\nu^{16} + 4\nu^{14} - 24\nu^{12} + 8\nu^{10} + 12\nu^{8} + 208\nu^{6} + 64\nu^{4} - 512 ) / 256 $ (3v^18 + 3v^16 + 4v^14 - 24v^12 + 8v^10 + 12v^8 + 208v^6 + 64v^4 - 512) / 256
$\beta_{13}$	$=$	$ ( - 3 \nu^{19} + 5 \nu^{17} - 12 \nu^{15} + 40 \nu^{13} - 72 \nu^{11} + 52 \nu^{9} - 304 \nu^{7} + \cdots + 256 \nu ) / 512 $ (-3v^19 + 5v^17 - 12v^15 + 40v^13 - 72v^11 + 52v^9 - 304v^7 + 256v^5 - 512v^3 + 256v) / 512
$\beta_{14}$	$=$	$ ( \nu^{18} - 3 \nu^{16} + 8 \nu^{14} - 40 \nu^{12} + 40 \nu^{10} - 92 \nu^{8} + 288 \nu^{6} + \cdots - 1536 ) / 128 $ (v^18 - 3v^16 + 8v^14 - 40v^12 + 40v^10 - 92v^8 + 288v^6 - 256v^4 + 448v^2 - 1536) / 128
$\beta_{15}$	$=$	$ ( - 13 \nu^{19} - 17 \nu^{17} - 24 \nu^{15} - 96 \nu^{13} + 136 \nu^{11} - 404 \nu^{9} + \cdots - 7168 \nu ) / 1536 $ (-13v^19 - 17v^17 - 24v^15 - 96v^13 + 136v^11 - 404v^9 - 2016v^5 + 2176v^3 - 7168*v) / 1536
$\beta_{16}$	$=$	$ ( 11 \nu^{19} - 41 \nu^{17} + 96 \nu^{15} - 312 \nu^{13} + 520 \nu^{11} - 1076 \nu^{9} + \cdots - 7936 \nu ) / 1536 $ (11v^19 - 41v^17 + 96v^15 - 312v^13 + 520v^11 - 1076v^9 + 2304v^7 - 4032v^5 + 4480v^3 - 7936v) / 1536
$\beta_{17}$	$=$	$ ( -\nu^{18} + \nu^{16} - 6\nu^{14} + 12\nu^{12} - 16\nu^{10} + 44\nu^{8} - 104\nu^{6} + 48\nu^{4} - 288\nu^{2} + 256 ) / 64 $ (-v^18 + v^16 - 6v^14 + 12v^12 - 16v^10 + 44v^8 - 104v^6 + 48v^4 - 288*v^2 + 256) / 64
$\beta_{18}$	$=$	$ ( 7 \nu^{19} - 13 \nu^{17} + 24 \nu^{15} - 144 \nu^{13} + 200 \nu^{11} - 388 \nu^{9} + 1152 \nu^{7} + \cdots - 4352 \nu ) / 768 $ (7v^19 - 13v^17 + 24v^15 - 144v^13 + 200v^11 - 388v^9 + 1152v^7 - 1440v^5 + 1280v^3 - 4352v) / 768
$\beta_{19}$	$=$	$ ( - 3 \nu^{18} + 5 \nu^{16} - 20 \nu^{14} + 48 \nu^{12} - 88 \nu^{10} + 180 \nu^{8} - 432 \nu^{6} + \cdots + 1152 ) / 128 $ (-3v^18 + 5v^16 - 20v^14 + 48v^12 - 88v^10 + 180v^8 - 432v^6 + 480v^4 - 832*v^2 + 1152) / 128

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

$\nu$	$=$	$ ( \beta_{16} + \beta_{8} + \beta_{6} ) / 4 $ (b16 + b8 + b6) / 4
$\nu^{2}$	$=$	$ ( \beta_{19} - \beta_{17} + \beta_{10} + \beta_{7} + \beta_{5} + 1 ) / 4 $ (b19 - b17 + b10 + b7 + b5 + 1) / 4
$\nu^{3}$	$=$	$ ( \beta_{15} - 2\beta_{13} - \beta_{11} - \beta_{9} + \beta_{8} + \beta_{4} + \beta_{2} ) / 4 $ (b15 - 2*b13 - b11 - b9 + b8 + b4 + b2) / 4
$\nu^{4}$	$=$	$ ( \beta_{19} - \beta_{17} + 2\beta_{14} - \beta_{10} - \beta_{7} - \beta_{5} + 2\beta_{3} + 2\beta _1 - 3 ) / 4 $ (b19 - b17 + 2b14 - b10 - b7 - b5 + 2b3 + 2*b1 - 3) / 4
$\nu^{5}$	$=$	$ ( 2 \beta_{18} - 6 \beta_{16} + \beta_{15} - 2 \beta_{13} + 5 \beta_{11} - \beta_{9} + \beta_{8} + \cdots + 3 \beta_{2} ) / 4 $ (2b18 - 6b16 + b15 - 2b13 + 5b11 - b9 + b8 - 2b6 - b4 + 3b2) / 4
$\nu^{6}$	$=$	$ ( -3\beta_{19} + 3\beta_{17} + 2\beta_{14} - 7\beta_{10} - 3\beta_{7} + \beta_{5} + 2\beta_{3} + 2\beta _1 + 13 ) / 4 $ (-3b19 + 3b17 + 2b14 - 7b10 - 3b7 + b5 + 2b3 + 2*b1 + 13) / 4
$\nu^{7}$	$=$	$ ( 2\beta_{18} - 3\beta_{15} + 2\beta_{13} + 9\beta_{11} + 3\beta_{9} - \beta_{8} + 8\beta_{6} - \beta_{4} + 15\beta_{2} ) / 4 $ (2b18 - 3b15 + 2b13 + 9b11 + 3b9 - b8 + 8b6 - b4 + 15*b2) / 4
$\nu^{8}$	$=$	$ ( - 5 \beta_{19} + 9 \beta_{17} + 6 \beta_{14} - 4 \beta_{12} - \beta_{10} + 7 \beta_{7} + 3 \beta_{5} + \cdots + 3 ) / 4 $ (-5b19 + 9b17 + 6b14 - 4b12 - b10 + 7b7 + 3b5 + 2b3 - 14b1 + 3) / 4
$\nu^{9}$	$=$	$ ( - 14 \beta_{18} + 4 \beta_{16} - \beta_{15} - 18 \beta_{13} - \beta_{11} + 9 \beta_{9} + \cdots + \beta_{2} ) / 4 $ (-14b18 + 4b16 - b15 - 18b13 - b11 + 9b9 - 11b8 + 4b6 + b4 + b2) / 4
$\nu^{10}$	$=$	$ ( - 3 \beta_{19} + 15 \beta_{17} + 10 \beta_{14} + 4 \beta_{12} + 9 \beta_{10} + 9 \beta_{7} - 3 \beta_{5} + \cdots - 59 ) / 4 $ (-3b19 + 15b17 + 10b14 + 4b12 + 9b10 + 9b7 - 3b5 + 14b3 + 6*b1 - 59) / 4
$\nu^{11}$	$=$	$ ( - 2 \beta_{18} - 16 \beta_{16} + 5 \beta_{15} - 6 \beta_{13} + 13 \beta_{11} + 3 \beta_{9} + \cdots - 29 \beta_{2} ) / 4 $ (-2b18 - 16b16 + 5b15 - 6b13 + 13b11 + 3b9 - 29b8 + 16b6 - 13b4 - 29b2) / 4
$\nu^{12}$	$=$	$ ( - 21 \beta_{19} + 9 \beta_{17} - 18 \beta_{14} - 4 \beta_{12} - 33 \beta_{10} - 17 \beta_{7} + \cdots - 125 ) / 4 $ (-21b19 + 9b17 - 18b14 - 4b12 - 33b10 - 17b7 + 11b5 + 10b3 + 18*b1 - 125) / 4
$\nu^{13}$	$=$	$ ( 10 \beta_{18} - 16 \beta_{16} - 29 \beta_{15} + 38 \beta_{13} + 3 \beta_{11} - 11 \beta_{9} + \cdots + 77 \beta_{2} ) / 4 $ (10b18 - 16b16 - 29b15 + 38b13 + 3b11 - 11b9 - 51b8 + 16b6 - 3b4 + 77b2) / 4
$\nu^{14}$	$=$	$ ( - 91 \beta_{19} + 55 \beta_{17} + 18 \beta_{14} - 76 \beta_{12} - 55 \beta_{10} + 25 \beta_{7} + \cdots - 67 ) / 4 $ (-91b19 + 55b17 + 18b14 - 76b12 - 55b10 + 25b7 - 51b5 - 26b3 - 130*b1 - 67) / 4
$\nu^{15}$	$=$	$ ( - 154 \beta_{18} + 72 \beta_{16} - 51 \beta_{15} + 10 \beta_{13} + 61 \beta_{11} + 59 \beta_{9} + \cdots - 45 \beta_{2} ) / 4 $ (-154b18 + 72b16 - 51b15 + 10b13 + 61b11 + 59b9 - 69b8 - 40b6 - 13b4 - 45b2) / 4
$\nu^{16}$	$=$	$ ( - 13 \beta_{19} + 65 \beta_{17} - 34 \beta_{14} + 124 \beta_{12} + 79 \beta_{10} + 111 \beta_{7} + \cdots - 69 ) / 4 $ (-13b19 + 65b17 - 34b14 + 124b12 + 79b10 + 111b7 - 37b5 - 70b3 + 2*b1 - 69) / 4
$\nu^{17}$	$=$	$ ( - 70 \beta_{18} + 184 \beta_{16} + 11 \beta_{15} + 198 \beta_{13} - 21 \beta_{11} + \cdots - 539 \beta_{2} ) / 4 $ (-70b18 + 184b16 + 11b15 + 198b13 - 21b11 + 77b9 - 67b8 + 200b6 + 5b4 - 539b2) / 4
$\nu^{18}$	$=$	$ ( 181 \beta_{19} - 329 \beta_{17} - 366 \beta_{14} + 292 \beta_{12} + 217 \beta_{10} - 103 \beta_{7} + \cdots - 851 ) / 4 $ (181b19 - 329b17 - 366b14 + 292b12 + 217b10 - 103b7 + 141b5 - 42b3 + 174*b1 - 851) / 4
$\nu^{19}$	$=$	$ ( 406 \beta_{18} - 168 \beta_{16} - 83 \beta_{15} - 86 \beta_{13} - 883 \beta_{11} - 389 \beta_{9} + \cdots - 413 \beta_{2} ) / 4 $ (406b18 - 168b16 - 83b15 - 86b13 - 883b11 - 389b9 + 91b8 - 504b6 + 195b4 - 413b2) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Embeddings

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

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

3889.1

0.564088	+	1.29684i
0.564088	−	1.29684i
−1.37859	−	0.315404i
−1.37859	+	0.315404i
−0.847166	−	1.13239i
−0.847166	+	1.13239i
1.35662	−	0.399488i
1.35662	+	0.399488i
0.986501	+	1.01332i
0.986501	−	1.01332i
−0.986501	+	1.01332i
−0.986501	−	1.01332i
−1.35662	−	0.399488i
−1.35662	+	0.399488i
0.847166	−	1.13239i
0.847166	+	1.13239i
1.37859	−	0.315404i
1.37859	+	0.315404i
−0.564088	+	1.29684i
−0.564088	−	1.29684i

−2.22935

−

0.173169i

−

3.43285i

3889.2

−2.22935

0.173169i

3.43285i

3889.3

−2.03021

−

0.937151i

−

3.36920i

3889.4

−2.03021

0.937151i

3.36920i

3889.5

−1.82935

−

1.28588i

1.09701i

3889.6

−1.82935

1.28588i

−

1.09701i

3889.7

−1.49578

−

1.66212i

−

1.43534i

3889.8

−1.49578

1.66212i

1.43534i

3889.9

−0.569524

−

2.16232i

4.64762i

3889.10

−0.569524

2.16232i

−

4.64762i

3889.11

0.569524

−

2.16232i

−

4.64762i

3889.12

0.569524

2.16232i

4.64762i

3889.13

1.49578

−

1.66212i

1.43534i

3889.14

1.49578

1.66212i

−

1.43534i

3889.15

1.82935

−

1.28588i

−

1.09701i

3889.16

1.82935

1.28588i

1.09701i

3889.17

2.03021

−

0.937151i

3.36920i

3889.18

2.03021

0.937151i

−

3.36920i

3889.19

2.22935

−

0.173169i

3.43285i

3889.20

2.22935

0.173169i

−

3.43285i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
40.f	even	2	1	inner
120.i	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4320.2.d.j		20
3.b	odd	2	1	inner	4320.2.d.j		20
4.b	odd	2	1		1080.2.d.j	yes	20
5.b	even	2	1		4320.2.d.i		20
8.b	even	2	1		4320.2.d.i		20
8.d	odd	2	1		1080.2.d.i	✓	20
12.b	even	2	1		1080.2.d.j	yes	20
15.d	odd	2	1		4320.2.d.i		20
20.d	odd	2	1		1080.2.d.i	✓	20
24.f	even	2	1		1080.2.d.i	✓	20
24.h	odd	2	1		4320.2.d.i		20
40.e	odd	2	1		1080.2.d.j	yes	20
40.f	even	2	1	inner	4320.2.d.j		20
60.h	even	2	1		1080.2.d.i	✓	20
120.i	odd	2	1	inner	4320.2.d.j		20
120.m	even	2	1		1080.2.d.j	yes	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1080.2.d.i	✓	20	8.d	odd	2	1
1080.2.d.i	✓	20	20.d	odd	2	1
1080.2.d.i	✓	20	24.f	even	2	1
1080.2.d.i	✓	20	60.h	even	2	1
1080.2.d.j	yes	20	4.b	odd	2	1
1080.2.d.j	yes	20	12.b	even	2	1
1080.2.d.j	yes	20	40.e	odd	2	1
1080.2.d.j	yes	20	120.m	even	2	1
4320.2.d.i		20	5.b	even	2	1
4320.2.d.i		20	8.b	even	2	1
4320.2.d.i		20	15.d	odd	2	1
4320.2.d.i		20	24.h	odd	2	1
4320.2.d.j		20	1.a	even	1	1	trivial
4320.2.d.j		20	3.b	odd	2	1	inner
4320.2.d.j		20	40.f	even	2	1	inner
4320.2.d.j		20	120.i	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4320, [\chi])$:

$ T_{7}^{10} + 48T_{7}^{8} + 782T_{7}^{6} + 5068T_{7}^{4} + 11001T_{7}^{2} + 7164 $

$ T_{11}^{10} + 53T_{11}^{8} + 972T_{11}^{6} + 7000T_{11}^{4} + 14928T_{11}^{2} + 144 $

$ T_{13}^{5} - T_{13}^{4} - 34T_{13}^{3} - 22T_{13}^{2} + 189T_{13} + 243 $

$ T_{53}^{10} - 300T_{53}^{8} + 27832T_{53}^{6} - 866704T_{53}^{4} + 9083600T_{53}^{2} - 12239296 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ T^{20} $ T^20
$5$	$ T^{20} - 10 T^{18} + \cdots + 9765625 $ T^20 - 10T^18 + 61T^16 - 232T^14 + 426T^12 - 1084T^10 + 10650T^8 - 145000T^6 + 953125T^4 - 3906250*T^2 + 9765625
$7$	$ (T^{10} + 48 T^{8} + \cdots + 7164)^{2} $ (T^10 + 48T^8 + 782T^6 + 5068T^4 + 11001T^2 + 7164)^2
$11$	$ (T^{10} + 53 T^{8} + \cdots + 144)^{2} $ (T^10 + 53T^8 + 972T^6 + 7000T^4 + 14928T^2 + 144)^2
$13$	$ (T^{5} - T^{4} - 34 T^{3} + \cdots + 243)^{4} $ (T^5 - T^4 - 34T^3 - 22T^2 + 189*T + 243)^4
$17$	$ (T^{10} + 69 T^{8} + \cdots + 576)^{2} $ (T^10 + 69T^8 + 788T^6 + 3184T^4 + 4416T^2 + 576)^2
$19$	$ (T^{10} + 132 T^{8} + \cdots + 114624)^{2} $ (T^10 + 132T^8 + 5798T^6 + 89716T^4 + 198705T^2 + 114624)^2
$23$	$ (T^{10} + 101 T^{8} + \cdots + 41616)^{2} $ (T^10 + 101T^8 + 3516T^6 + 49672T^4 + 240144T^2 + 41616)^2
$29$	$ (T^{10} + 81 T^{8} + \cdots + 11664)^{2} $ (T^10 + 81T^8 + 1608T^6 + 11032T^4 + 21600T^2 + 11664)^2
$31$	$ (T^{5} + 3 T^{4} - 40 T^{3} + \cdots - 32)^{4} $ (T^5 + 3T^4 - 40T^3 - 168T^2 - 176T - 32)^4
$37$	$ (T^{5} - 8 T^{4} - 48 T^{3} + \cdots + 18)^{4} $ (T^5 - 8T^4 - 48T^3 + 566T^2 - 1245T + 18)^4
$41$	$ (T^{10} - 252 T^{8} + \cdots - 83560896)^{2} $ (T^10 - 252T^8 + 21704T^6 - 806608T^4 + 13484880T^2 - 83560896)^2
$43$	$ (T^{5} - 3 T^{4} + \cdots + 6192)^{4} $ (T^5 - 3T^4 - 124T^3 + 16T^2 + 3552T + 6192)^4
$47$	$ (T^{10} + 213 T^{8} + \cdots + 3779136)^{2} $ (T^10 + 213T^8 + 14900T^6 + 368272T^4 + 2270592T^2 + 3779136)^2
$53$	$ (T^{10} - 300 T^{8} + \cdots - 12239296)^{2} $ (T^10 - 300T^8 + 27832T^6 - 866704T^4 + 9083600T^2 - 12239296)^2
$59$	$ (T^{10} + 308 T^{8} + \cdots + 60466176)^{2} $ (T^10 + 308T^8 + 30288T^6 + 1101568T^4 + 14354496T^2 + 60466176)^2
$61$	$ (T^{10} + 236 T^{8} + \cdots + 114624)^{2} $ (T^10 + 236T^8 + 12406T^6 + 115468T^4 + 260049T^2 + 114624)^2
$67$	$ (T^{5} - 226 T^{3} + \cdots - 12564)^{4} $ (T^5 - 226T^3 - 4T^2 + 7449*T - 12564)^4
$71$	$ (T^{10} - 464 T^{8} + \cdots - 16505856)^{2} $ (T^10 - 464T^8 + 72312T^6 - 3889840T^4 + 16350480T^2 - 16505856)^2
$73$	$ (T^{10} + 456 T^{8} + \cdots + 549686556)^{2} $ (T^10 + 456T^8 + 68430T^6 + 3831220T^4 + 83540073T^2 + 549686556)^2
$79$	$ (T^{5} + 9 T^{4} + \cdots - 547)^{4} $ (T^5 + 9T^4 - 154T^3 - 1362T^2 - 2555T - 547)^4
$83$	$ (T^{10} - 352 T^{8} + \cdots - 50944)^{2} $ (T^10 - 352T^8 + 32808T^6 - 333168T^4 + 902928T^2 - 50944)^2
$89$	$ (T^{10} - 428 T^{8} + \cdots - 16505856)^{2} $ (T^10 - 428T^8 + 61968T^6 - 3438784T^4 + 58559040T^2 - 16505856)^2
$97$	$ (T^{10} + 708 T^{8} + \cdots + 3391036416)^{2} $ (T^10 + 708T^8 + 180822T^6 + 19682164T^4 + 789241089T^2 + 3391036416)^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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.d (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	\( q + \beta_{2} q^{5} + \beta_{10} q^{7} + \beta_{6} q^{11} + \beta_{19} q^{13} + (\beta_{18} - \beta_{11} + \cdots - \beta_{2}) q^{17}+ \cdots + ( - \beta_{12} - 2 \beta_{5} + \cdots - \beta_1) q^{97}+O(q^{100}) \) q + b2 * q^5 + b10 * q^7 + b6 * q^11 + b19 * q^13 + (b18 - b11 - b6 - b2) * q^17 + b3 * q^19 + (b18 + b13 + b11 + b2) * q^23 + (-b12 + b7 + 1) * q^25 + (-b18 + b16) * q^29 + (b1 - 1) * q^31 + (b16 + 2b13 + b9 + b6) q^35 + (b17 - b12 + 2) * q^37 - b4 * q^41 + (-b19 + b17 - b12 - 2b1 + 2) q^43 + (b18 - b16 - 2b13 - b11 - 2b6 - b2) * q^47 + (-b19 - b17 - b12 - 2) * q^49 + (b18 - b15 - b13 - b9 - b6) * q^53 + (-b19 + b17 + b14 - b10 - b5 + b3 - b1 + 2) * q^55 + (2b18 - 2b16 + b13 + b11 - b6 + b2) * q^59 + (-b12 + b7 - b5 - b1) * q^61 + (-b18 - b16 + b15 + b13 + b11 + b8 + b6 + b2) * q^65 + (b19 - b17 - 2b12 - b1 + 1) q^67 + (b18 - b13 - b11 - 2b9 + b8 - b6 + b4 + b2) q^71 + (-b19 + b17 + 2b14 - b12 - b10 + b7 - b5 + b3 - b1 + 1) q^73 + (b15 - b9 + b8 - b4) * q^77 + (-b19 - 2b12 - 2b1) * q^79 + (-b15 + b11 + b9 - b8 - b2) * q^83 + (b19 - b14 + b10 - b7 - b3 + 2) * q^85 + (-b18 + b15 + b13 - b11 + b9 - b8 + b6 + b2) * q^89 + (b19 - b17 - 2b14 + b12 + 3b10 - b7 + b5 - b3 + b1 - 1) * q^91 + (-2b16 - b15 + 2b11 + b9 + b6 + b2) * q^95 + (-b12 - 2b5 - b3 - b1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 4 q^{13} + 20 q^{25} - 12 q^{31} + 32 q^{37} + 12 q^{43} - 52 q^{49} + 28 q^{55} - 36 q^{79} + 44 q^{85}+O(q^{100}) \) 20 * q + 4 * q^13 + 20 * q^25 - 12 * q^31 + 32 * q^37 + 12 * q^43 - 52 * q^49 + 28 * q^55 - 36 * q^79 + 44 * q^85

Introduction

L-functions

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Varieties

Fields

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Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4320.2.d.j

Level	$4320$
Weight	$2$
Character orbit	4320.d
Analytic conductor	$34.495$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(34.4953736732\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 3x^{18} + 8x^{16} - 24x^{14} + 56x^{12} - 92x^{10} + 224x^{8} - 384x^{6} + 512x^{4} - 768x^{2} + 1024 \) x^20 - 3x^18 + 8x^16 - 24x^14 + 56x^12 - 92x^10 + 224x^8 - 384x^6 + 512x^4 - 768*x^2 + 1024
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{20}\cdot 3^{4} \)
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{18} - \nu^{16} + 4\nu^{14} - 8\nu^{12} + 40\nu^{10} - 132\nu^{8} + 144\nu^{6} - 256\nu^{4} + 512\nu^{2} - 512 ) / 256 \) (-v^18 - v^16 + 4v^14 - 8v^12 + 40v^10 - 132v^8 + 144v^6 - 256v^4 + 512*v^2 - 512) / 256
\(\beta_{2}\)	\(=\)	\( ( -\nu^{19} - 2\nu^{17} - 3\nu^{15} + 6\nu^{13} - 8\nu^{11} + 4\nu^{9} - 36\nu^{7} - 72\nu^{5} - 32\nu^{3} - 64\nu ) / 384 \) (-v^19 - 2v^17 - 3v^15 + 6v^13 - 8v^11 + 4v^9 - 36v^7 - 72v^5 - 32v^3 - 64*v) / 384
\(\beta_{3}\)	\(=\)	\( ( \nu^{18} + \nu^{16} - 4\nu^{14} + 40\nu^{12} - 8\nu^{10} + 132\nu^{8} - 272\nu^{6} + 512\nu^{4} - 384\nu^{2} + 2560 ) / 256 \) (v^18 + v^16 - 4v^14 + 40v^12 - 8v^10 + 132v^8 - 272v^6 + 512v^4 - 384*v^2 + 2560) / 256
\(\beta_{4}\)	\(=\)	\( ( \nu^{19} + 7\nu^{17} + 2\nu^{15} - 40\nu^{11} + 20\nu^{9} + 136\nu^{7} + 160\nu^{5} + 576\nu^{3} - 1024\nu ) / 256 \) (v^19 + 7v^17 + 2v^15 - 40v^11 + 20v^9 + 136v^7 + 160v^5 + 576v^3 - 1024v) / 256
\(\beta_{5}\)	\(=\)	\( ( \nu^{18} - 4 \nu^{16} + 3 \nu^{14} - 24 \nu^{12} + 32 \nu^{10} - 52 \nu^{8} + 252 \nu^{6} + \cdots - 1280 ) / 192 \) (v^18 - 4v^16 + 3v^14 - 24v^12 + 32v^10 - 52v^8 + 252v^6 - 384v^4 + 512v^2 - 1280) / 192
\(\beta_{6}\)	\(=\)	\( ( - 5 \nu^{19} + 23 \nu^{17} - 48 \nu^{15} + 168 \nu^{13} - 184 \nu^{11} + 524 \nu^{9} + \cdots + 6400 \nu ) / 1536 \) (-5v^19 + 23v^17 - 48v^15 + 168v^13 - 184v^11 + 524v^9 - 960v^7 + 1728v^5 - 1408v^3 + 6400v) / 1536
\(\beta_{7}\)	\(=\)	\( ( -5\nu^{18} + 23\nu^{16} + 120\nu^{12} - 88\nu^{10} + 140\nu^{8} - 576\nu^{6} + 1152\nu^{4} + 1280\nu^{2} + 4096 ) / 768 \) (-5v^18 + 23v^16 + 120v^12 - 88v^10 + 140v^8 - 576v^6 + 1152v^4 + 1280v^2 + 4096) / 768
\(\beta_{8}\)	\(=\)	\( ( - \nu^{19} + 3 \nu^{17} - 8 \nu^{15} + 24 \nu^{13} - 56 \nu^{11} + 92 \nu^{9} - 224 \nu^{7} + \cdots + 1280 \nu ) / 256 \) (-v^19 + 3v^17 - 8v^15 + 24v^13 - 56v^11 + 92v^9 - 224v^7 + 384v^5 - 512v^3 + 1280*v) / 256
\(\beta_{9}\)	\(=\)	\( ( - 7 \nu^{19} + 13 \nu^{17} - 72 \nu^{15} - 296 \nu^{11} + 580 \nu^{9} - 384 \nu^{7} + \cdots - 2560 \nu ) / 1536 \) (-7v^19 + 13v^17 - 72v^15 - 296v^11 + 580v^9 - 384v^7 + 480v^5 - 3968v^3 - 2560*v) / 1536
\(\beta_{10}\)	\(=\)	\( ( 7 \nu^{18} - 25 \nu^{16} + 36 \nu^{14} - 168 \nu^{12} + 296 \nu^{10} - 484 \nu^{8} + 912 \nu^{6} + \cdots - 3584 ) / 768 \) (7v^18 - 25v^16 + 36v^14 - 168v^12 + 296v^10 - 484v^8 + 912v^6 - 1920v^4 + 1280*v^2 - 3584) / 768
\(\beta_{11}\)	\(=\)	\( ( \nu^{19} - \nu^{17} + 9 \nu^{15} - 21 \nu^{13} + 32 \nu^{11} - 76 \nu^{9} + 192 \nu^{7} + \cdots - 608 \nu ) / 192 \) (v^19 - v^17 + 9v^15 - 21v^13 + 32v^11 - 76v^9 + 192v^7 - 132v^5 + 416v^3 - 608v) / 192
\(\beta_{12}\)	\(=\)	\( ( 3\nu^{18} + 3\nu^{16} + 4\nu^{14} - 24\nu^{12} + 8\nu^{10} + 12\nu^{8} + 208\nu^{6} + 64\nu^{4} - 512 ) / 256 \) (3v^18 + 3v^16 + 4v^14 - 24v^12 + 8v^10 + 12v^8 + 208v^6 + 64v^4 - 512) / 256
\(\beta_{13}\)	\(=\)	\( ( - 3 \nu^{19} + 5 \nu^{17} - 12 \nu^{15} + 40 \nu^{13} - 72 \nu^{11} + 52 \nu^{9} - 304 \nu^{7} + \cdots + 256 \nu ) / 512 \) (-3v^19 + 5v^17 - 12v^15 + 40v^13 - 72v^11 + 52v^9 - 304v^7 + 256v^5 - 512v^3 + 256v) / 512
\(\beta_{14}\)	\(=\)	\( ( \nu^{18} - 3 \nu^{16} + 8 \nu^{14} - 40 \nu^{12} + 40 \nu^{10} - 92 \nu^{8} + 288 \nu^{6} + \cdots - 1536 ) / 128 \) (v^18 - 3v^16 + 8v^14 - 40v^12 + 40v^10 - 92v^8 + 288v^6 - 256v^4 + 448v^2 - 1536) / 128
\(\beta_{15}\)	\(=\)	\( ( - 13 \nu^{19} - 17 \nu^{17} - 24 \nu^{15} - 96 \nu^{13} + 136 \nu^{11} - 404 \nu^{9} + \cdots - 7168 \nu ) / 1536 \) (-13v^19 - 17v^17 - 24v^15 - 96v^13 + 136v^11 - 404v^9 - 2016v^5 + 2176v^3 - 7168*v) / 1536
\(\beta_{16}\)	\(=\)	\( ( 11 \nu^{19} - 41 \nu^{17} + 96 \nu^{15} - 312 \nu^{13} + 520 \nu^{11} - 1076 \nu^{9} + \cdots - 7936 \nu ) / 1536 \) (11v^19 - 41v^17 + 96v^15 - 312v^13 + 520v^11 - 1076v^9 + 2304v^7 - 4032v^5 + 4480v^3 - 7936v) / 1536
\(\beta_{17}\)	\(=\)	\( ( -\nu^{18} + \nu^{16} - 6\nu^{14} + 12\nu^{12} - 16\nu^{10} + 44\nu^{8} - 104\nu^{6} + 48\nu^{4} - 288\nu^{2} + 256 ) / 64 \) (-v^18 + v^16 - 6v^14 + 12v^12 - 16v^10 + 44v^8 - 104v^6 + 48v^4 - 288*v^2 + 256) / 64
\(\beta_{18}\)	\(=\)	\( ( 7 \nu^{19} - 13 \nu^{17} + 24 \nu^{15} - 144 \nu^{13} + 200 \nu^{11} - 388 \nu^{9} + 1152 \nu^{7} + \cdots - 4352 \nu ) / 768 \) (7v^19 - 13v^17 + 24v^15 - 144v^13 + 200v^11 - 388v^9 + 1152v^7 - 1440v^5 + 1280v^3 - 4352v) / 768
\(\beta_{19}\)	\(=\)	\( ( - 3 \nu^{18} + 5 \nu^{16} - 20 \nu^{14} + 48 \nu^{12} - 88 \nu^{10} + 180 \nu^{8} - 432 \nu^{6} + \cdots + 1152 ) / 128 \) (-3v^18 + 5v^16 - 20v^14 + 48v^12 - 88v^10 + 180v^8 - 432v^6 + 480v^4 - 832*v^2 + 1152) / 128

\(\nu\)	\(=\)	\( ( \beta_{16} + \beta_{8} + \beta_{6} ) / 4 \) (b16 + b8 + b6) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{19} - \beta_{17} + \beta_{10} + \beta_{7} + \beta_{5} + 1 ) / 4 \) (b19 - b17 + b10 + b7 + b5 + 1) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} - 2\beta_{13} - \beta_{11} - \beta_{9} + \beta_{8} + \beta_{4} + \beta_{2} ) / 4 \) (b15 - 2*b13 - b11 - b9 + b8 + b4 + b2) / 4
\(\nu^{4}\)	\(=\)	\( ( \beta_{19} - \beta_{17} + 2\beta_{14} - \beta_{10} - \beta_{7} - \beta_{5} + 2\beta_{3} + 2\beta _1 - 3 ) / 4 \) (b19 - b17 + 2b14 - b10 - b7 - b5 + 2b3 + 2*b1 - 3) / 4
\(\nu^{5}\)	\(=\)	\( ( 2 \beta_{18} - 6 \beta_{16} + \beta_{15} - 2 \beta_{13} + 5 \beta_{11} - \beta_{9} + \beta_{8} + \cdots + 3 \beta_{2} ) / 4 \) (2b18 - 6b16 + b15 - 2b13 + 5b11 - b9 + b8 - 2b6 - b4 + 3b2) / 4
\(\nu^{6}\)	\(=\)	\( ( -3\beta_{19} + 3\beta_{17} + 2\beta_{14} - 7\beta_{10} - 3\beta_{7} + \beta_{5} + 2\beta_{3} + 2\beta _1 + 13 ) / 4 \) (-3b19 + 3b17 + 2b14 - 7b10 - 3b7 + b5 + 2b3 + 2*b1 + 13) / 4
\(\nu^{7}\)	\(=\)	\( ( 2\beta_{18} - 3\beta_{15} + 2\beta_{13} + 9\beta_{11} + 3\beta_{9} - \beta_{8} + 8\beta_{6} - \beta_{4} + 15\beta_{2} ) / 4 \) (2b18 - 3b15 + 2b13 + 9b11 + 3b9 - b8 + 8b6 - b4 + 15*b2) / 4
\(\nu^{8}\)	\(=\)	\( ( - 5 \beta_{19} + 9 \beta_{17} + 6 \beta_{14} - 4 \beta_{12} - \beta_{10} + 7 \beta_{7} + 3 \beta_{5} + \cdots + 3 ) / 4 \) (-5b19 + 9b17 + 6b14 - 4b12 - b10 + 7b7 + 3b5 + 2b3 - 14b1 + 3) / 4
\(\nu^{9}\)	\(=\)	\( ( - 14 \beta_{18} + 4 \beta_{16} - \beta_{15} - 18 \beta_{13} - \beta_{11} + 9 \beta_{9} + \cdots + \beta_{2} ) / 4 \) (-14b18 + 4b16 - b15 - 18b13 - b11 + 9b9 - 11b8 + 4b6 + b4 + b2) / 4
\(\nu^{10}\)	\(=\)	\( ( - 3 \beta_{19} + 15 \beta_{17} + 10 \beta_{14} + 4 \beta_{12} + 9 \beta_{10} + 9 \beta_{7} - 3 \beta_{5} + \cdots - 59 ) / 4 \) (-3b19 + 15b17 + 10b14 + 4b12 + 9b10 + 9b7 - 3b5 + 14b3 + 6*b1 - 59) / 4
\(\nu^{11}\)	\(=\)	\( ( - 2 \beta_{18} - 16 \beta_{16} + 5 \beta_{15} - 6 \beta_{13} + 13 \beta_{11} + 3 \beta_{9} + \cdots - 29 \beta_{2} ) / 4 \) (-2b18 - 16b16 + 5b15 - 6b13 + 13b11 + 3b9 - 29b8 + 16b6 - 13b4 - 29b2) / 4
\(\nu^{12}\)	\(=\)	\( ( - 21 \beta_{19} + 9 \beta_{17} - 18 \beta_{14} - 4 \beta_{12} - 33 \beta_{10} - 17 \beta_{7} + \cdots - 125 ) / 4 \) (-21b19 + 9b17 - 18b14 - 4b12 - 33b10 - 17b7 + 11b5 + 10b3 + 18*b1 - 125) / 4
\(\nu^{13}\)	\(=\)	\( ( 10 \beta_{18} - 16 \beta_{16} - 29 \beta_{15} + 38 \beta_{13} + 3 \beta_{11} - 11 \beta_{9} + \cdots + 77 \beta_{2} ) / 4 \) (10b18 - 16b16 - 29b15 + 38b13 + 3b11 - 11b9 - 51b8 + 16b6 - 3b4 + 77b2) / 4
\(\nu^{14}\)	\(=\)	\( ( - 91 \beta_{19} + 55 \beta_{17} + 18 \beta_{14} - 76 \beta_{12} - 55 \beta_{10} + 25 \beta_{7} + \cdots - 67 ) / 4 \) (-91b19 + 55b17 + 18b14 - 76b12 - 55b10 + 25b7 - 51b5 - 26b3 - 130*b1 - 67) / 4
\(\nu^{15}\)	\(=\)	\( ( - 154 \beta_{18} + 72 \beta_{16} - 51 \beta_{15} + 10 \beta_{13} + 61 \beta_{11} + 59 \beta_{9} + \cdots - 45 \beta_{2} ) / 4 \) (-154b18 + 72b16 - 51b15 + 10b13 + 61b11 + 59b9 - 69b8 - 40b6 - 13b4 - 45b2) / 4
\(\nu^{16}\)	\(=\)	\( ( - 13 \beta_{19} + 65 \beta_{17} - 34 \beta_{14} + 124 \beta_{12} + 79 \beta_{10} + 111 \beta_{7} + \cdots - 69 ) / 4 \) (-13b19 + 65b17 - 34b14 + 124b12 + 79b10 + 111b7 - 37b5 - 70b3 + 2*b1 - 69) / 4
\(\nu^{17}\)	\(=\)	\( ( - 70 \beta_{18} + 184 \beta_{16} + 11 \beta_{15} + 198 \beta_{13} - 21 \beta_{11} + \cdots - 539 \beta_{2} ) / 4 \) (-70b18 + 184b16 + 11b15 + 198b13 - 21b11 + 77b9 - 67b8 + 200b6 + 5b4 - 539b2) / 4
\(\nu^{18}\)	\(=\)	\( ( 181 \beta_{19} - 329 \beta_{17} - 366 \beta_{14} + 292 \beta_{12} + 217 \beta_{10} - 103 \beta_{7} + \cdots - 851 ) / 4 \) (181b19 - 329b17 - 366b14 + 292b12 + 217b10 - 103b7 + 141b5 - 42b3 + 174*b1 - 851) / 4
\(\nu^{19}\)	\(=\)	\( ( 406 \beta_{18} - 168 \beta_{16} - 83 \beta_{15} - 86 \beta_{13} - 883 \beta_{11} - 389 \beta_{9} + \cdots - 413 \beta_{2} ) / 4 \) (406b18 - 168b16 - 83b15 - 86b13 - 883b11 - 389b9 + 91b8 - 504b6 + 195b4 - 413b2) / 4

\(n\)	\(2081\)	\(2431\)	\(3457\)	\(3781\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3889.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( T^{20} \) T^20
$5$	\( T^{20} - 10 T^{18} + \cdots + 9765625 \) T^20 - 10T^18 + 61T^16 - 232T^14 + 426T^12 - 1084T^10 + 10650T^8 - 145000T^6 + 953125T^4 - 3906250*T^2 + 9765625
$7$	\( (T^{10} + 48 T^{8} + \cdots + 7164)^{2} \) (T^10 + 48T^8 + 782T^6 + 5068T^4 + 11001T^2 + 7164)^2
$11$	\( (T^{10} + 53 T^{8} + \cdots + 144)^{2} \) (T^10 + 53T^8 + 972T^6 + 7000T^4 + 14928T^2 + 144)^2
$13$	\( (T^{5} - T^{4} - 34 T^{3} + \cdots + 243)^{4} \) (T^5 - T^4 - 34T^3 - 22T^2 + 189*T + 243)^4
$17$	\( (T^{10} + 69 T^{8} + \cdots + 576)^{2} \) (T^10 + 69T^8 + 788T^6 + 3184T^4 + 4416T^2 + 576)^2
$19$	\( (T^{10} + 132 T^{8} + \cdots + 114624)^{2} \) (T^10 + 132T^8 + 5798T^6 + 89716T^4 + 198705T^2 + 114624)^2
$23$	\( (T^{10} + 101 T^{8} + \cdots + 41616)^{2} \) (T^10 + 101T^8 + 3516T^6 + 49672T^4 + 240144T^2 + 41616)^2
$29$	\( (T^{10} + 81 T^{8} + \cdots + 11664)^{2} \) (T^10 + 81T^8 + 1608T^6 + 11032T^4 + 21600T^2 + 11664)^2
$31$	\( (T^{5} + 3 T^{4} - 40 T^{3} + \cdots - 32)^{4} \) (T^5 + 3T^4 - 40T^3 - 168T^2 - 176T - 32)^4
$37$	\( (T^{5} - 8 T^{4} - 48 T^{3} + \cdots + 18)^{4} \) (T^5 - 8T^4 - 48T^3 + 566T^2 - 1245T + 18)^4
$41$	\( (T^{10} - 252 T^{8} + \cdots - 83560896)^{2} \) (T^10 - 252T^8 + 21704T^6 - 806608T^4 + 13484880T^2 - 83560896)^2
$43$	\( (T^{5} - 3 T^{4} + \cdots + 6192)^{4} \) (T^5 - 3T^4 - 124T^3 + 16T^2 + 3552T + 6192)^4
$47$	\( (T^{10} + 213 T^{8} + \cdots + 3779136)^{2} \) (T^10 + 213T^8 + 14900T^6 + 368272T^4 + 2270592T^2 + 3779136)^2
$53$	\( (T^{10} - 300 T^{8} + \cdots - 12239296)^{2} \) (T^10 - 300T^8 + 27832T^6 - 866704T^4 + 9083600T^2 - 12239296)^2
$59$	\( (T^{10} + 308 T^{8} + \cdots + 60466176)^{2} \) (T^10 + 308T^8 + 30288T^6 + 1101568T^4 + 14354496T^2 + 60466176)^2
$61$	\( (T^{10} + 236 T^{8} + \cdots + 114624)^{2} \) (T^10 + 236T^8 + 12406T^6 + 115468T^4 + 260049T^2 + 114624)^2
$67$	\( (T^{5} - 226 T^{3} + \cdots - 12564)^{4} \) (T^5 - 226T^3 - 4T^2 + 7449*T - 12564)^4
$71$	\( (T^{10} - 464 T^{8} + \cdots - 16505856)^{2} \) (T^10 - 464T^8 + 72312T^6 - 3889840T^4 + 16350480T^2 - 16505856)^2
$73$	\( (T^{10} + 456 T^{8} + \cdots + 549686556)^{2} \) (T^10 + 456T^8 + 68430T^6 + 3831220T^4 + 83540073T^2 + 549686556)^2
$79$	\( (T^{5} + 9 T^{4} + \cdots - 547)^{4} \) (T^5 + 9T^4 - 154T^3 - 1362T^2 - 2555T - 547)^4
$83$	\( (T^{10} - 352 T^{8} + \cdots - 50944)^{2} \) (T^10 - 352T^8 + 32808T^6 - 333168T^4 + 902928T^2 - 50944)^2
$89$	\( (T^{10} - 428 T^{8} + \cdots - 16505856)^{2} \) (T^10 - 428T^8 + 61968T^6 - 3438784T^4 + 58559040T^2 - 16505856)^2
$97$	\( (T^{10} + 708 T^{8} + \cdots + 3391036416)^{2} \) (T^10 + 708T^8 + 180822T^6 + 19682164T^4 + 789241089T^2 + 3391036416)^2
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