LMFDB - Newform orbit 567.2.e.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,2,Mod(163,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("567.163");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	567.e (of order $3$, 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:	$4.52751779461$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 14x^{12} - 39x^{10} + 77x^{8} - 156x^{6} + 224x^{4} - 256x^{2} + 256 $ x^16 - 4x^14 + 14x^12 - 39x^10 + 77x^8 - 156x^6 + 224x^4 - 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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_1 q^{2} + ( - \beta_{12} - \beta_{8} + \cdots - \beta_{5}) q^{4}+ \cdots + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{2}) q^{8}+O(q^{10}) $ q - b1 * q^2 + (-b12 - b8 - b7 - b5) * q^4 + (-b10 - b1) * q^5 + (b12 + b9 + b7 - b3 + 1) * q^7 + (-b15 - b14 + b10 - b6 + b4 + b2) * q^8	$ q - \beta_1 q^{2} + ( - \beta_{12} - \beta_{8} + \cdots - \beta_{5}) q^{4}+ \cdots + (\beta_{15} + 2 \beta_{14} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) $ q - b1 * q^2 + (-b12 - b8 - b7 - b5) * q^4 + (-b10 - b1) * q^5 + (b12 + b9 + b7 - b3 + 1) * q^7 + (-b15 - b14 + b10 - b6 + b4 + b2) * q^8 + (-b12 + b9 - 2b8 - 2b7 - b5) * q^10 + b2 * q^11 + (-b13 + b12 + b9 - b7 + b5 - b3 + 1) * q^13 + (b14 - b10 - b2 - 2b1) q^14 + (2b13 + b12 - b9 + b8 - b5 + b3 - 1) q^16 + (-b11 + b6 - b4 + b2) * q^17 + (b13 + b12 - 3b9 + 2b8 - b5 - 3) * q^19 + (-3b15 - b14 + b2) q^20 + (b13 - b12 - 2b9 - b7 - 2b5 + 2b3 - 1) q^22 + (-b14 - 2b10 + b1) q^23 + (-b8 - b7) * q^25 + (b14 + 3b11 + b10 - b1) q^26 + (-2b13 - b12 + 3b9 - 2b8 - b7 - b5 - b3) q^28 + (2b10 - 2b6 - b4) * q^29 + (b13 - b12 + 2b9 - b8 - b7 - 2b5 - b3 + 1) * q^31 + (2b15 - b6 - b2 + 2b1) * q^32 + (-b9 - b5 + b3 - 1) * q^34 + (-b15 - 2b11 - 3b4 - 3b1) q^35 + (2b13 + b8 + 2b3) * q^37 + (6b15 + b11 + 2b6 + b4 - 2b2 + 6b1) * q^38 + (3b13 + b12 - 5b9 - b5 + 2b3 - 5) q^40 + (b15 + 2b14 - 2b10 + 2b6 - 2b2) * q^41 + (b13 - b12 - 3b7 + 2) q^43 + (-b14 - 2b11 + b10 + 3b1) * q^44 + (b13 + 3b12 - 2b9 + 2b5 - b3 + 1) q^46 + (b14 - 3b11 + b10 - 3b1) * q^47 + (-b13 + 2b12 + 2b9 - b8 + b7 + 3b5 - 1) q^49 + (-b15 + b10 - b6 - b4) * q^50 + (-b13 - 4b12 + b9 - 3b8 - 3b7 - 3b5 + b3 - 1) * q^52 + (5b15 + 3b11 + b6 + 3b4 + 5b1) * q^53 + (2b13 - 2b12 - 2b9 + b7 - 2b5 + 2b3) q^55 + (-5b15 - 2b14 + 2b11 - b10 + 3b4 + b2 - 3b1) q^56 + (-b13 + 3b9 + 2b8 - b3 + 3) * q^58 + (-b15 + b11 + b6 + b4 - b1) * q^59 + (-3b13 - b12 - 4b9 - b8 + b5 - 2b3 - 4) q^61 + (-5b15 + b10 - b6 + 2b4) * q^62 + (-b13 + b12 + 4) * q^64 + (3b14 + 2b11 + b1) * q^65 + (b13 - 2b9 + 4b8 + 4b7 - b5 - b3 + 1) q^67 + (-3b11 + 2b10 + b1) * q^68 + (-2b13 - b12 + 6b9 - 2b8 - 3b7 - b5 - 2b3) q^70 + (b15 + 2b14 + 5b4 - 2b2) q^71 + (b13 + b12 + 5b9 + 2b8 + 2b7 - b3 + 1) q^73 + (3b15 - 3b11 + b6 - 3b4 - 2b2 + 3b1) q^74 + (-7b13 + 7b12 + 7b9 + 6b7 + 7b5 - 7b3 + 9) * q^76 + (b15 - b14 + 2b11 + b10 + 3b4 + b2 - 2b1) q^77 + (-b13 - b12 + 4b9 + b5 + 4) q^79 + (2b15 - 5b11 - 5b4 - 2b2 + 2b1) q^80 + (-5b13 - 2b12 + b9 - 5b8 + 2b5 - 3b3 + 1) q^82 + (-3b15 - 2b4) * q^83 + (3b13 - 3b12 - 2b9 - 2b5 + 2b3 + 4) q^85 + (b14 + 2b11 + 3b10 + 2b1) q^86 + (-b13 + 3b12 - 7b9 + 3b8 + 3b7 + 4b5 + b3 - 1) q^88 + (-4b14 + 5b11 - b10 + b1) * q^89 + (-b13 + 2b12 + 2b9 + 5b8 + 4b7 + 3b5 - 3b3) * q^91 + (6b15 + 2b14 - 4b10 + 4b6 - 5b4 - 2b2) * q^92 + (-b13 - 2b12 + 5b9 - 3b8 - 3b7 - b5 + b3 - 1) * q^94 + (6b15 - 3b11 + 3b6 - 3b4 + b2 + 6b1) q^95 + (3b13 - 3b12 - b9 + b7 - b5 + b3 + 3) * q^97 + (b15 + 2b14 - b11 - b10 - b6 - 5b4 - b2 - 2b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{4} + 6 q^{7}+O(q^{10}) $ 16 * q - 6 * q^4 + 6 * q^7	$ 16 q - 6 q^{4} + 6 q^{7} - 14 q^{10} + 12 q^{13} - 6 q^{16} - 24 q^{19} + 4 q^{22} - 26 q^{28} - 20 q^{31} + 4 q^{37} - 36 q^{40} + 20 q^{43} + 36 q^{46} - 14 q^{49} - 34 q^{52} + 8 q^{55} + 22 q^{58} - 36 q^{61} + 76 q^{64} + 18 q^{67} - 58 q^{70} - 32 q^{73} + 116 q^{76} + 32 q^{79} + 2 q^{82} + 60 q^{85} + 72 q^{88} - 22 q^{91} - 54 q^{94} + 28 q^{97}+O(q^{100}) $ 16 * q - 6 * q^4 + 6 * q^7 - 14 * q^10 + 12 * q^13 - 6 * q^16 - 24 * q^19 + 4 * q^22 - 26 * q^28 - 20 * q^31 + 4 * q^37 - 36 * q^40 + 20 * q^43 + 36 * q^46 - 14 * q^49 - 34 * q^52 + 8 * q^55 + 22 * q^58 - 36 * q^61 + 76 * q^64 + 18 * q^67 - 58 * q^70 - 32 * q^73 + 116 * q^76 + 32 * q^79 + 2 * q^82 + 60 * q^85 + 72 * q^88 - 22 * q^91 - 54 * q^94 + 28 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4x^{14} + 14x^{12} - 39x^{10} + 77x^{8} - 156x^{6} + 224x^{4} - 256x^{2} + 256 $ x^16 - 4*x^14 + 14*x^12 - 39*x^10 + 77*x^8 - 156*x^6 + 224*x^4 - 256*x^2 + 256 :

$\beta_{1}$	$=$	$ ( -5\nu^{15} - 96\nu^{13} + 122\nu^{11} - 469\nu^{9} + 1099\nu^{7} - 872\nu^{5} + 2304\nu^{3} - 1984\nu ) / 2688 $ (-5v^15 - 96v^13 + 122v^11 - 469v^9 + 1099v^7 - 872v^5 + 2304v^3 - 1984v) / 2688
$\beta_{2}$	$=$	$ ( -\nu^{15} - 8\nu^{13} - 54\nu^{11} + 63\nu^{9} - 329\nu^{7} + 744\nu^{5} - 424\nu^{3} + 2112\nu ) / 448 $ (-v^15 - 8v^13 - 54v^11 + 63v^9 - 329v^7 + 744v^5 - 424v^3 + 2112*v) / 448
$\beta_{3}$	$=$	$ ( -5\nu^{14} + 72\nu^{12} - 214\nu^{10} + 539\nu^{8} - 749\nu^{6} + 1648\nu^{4} - 2064\nu^{2} + 2048 ) / 1344 $ (-5v^14 + 72v^12 - 214v^10 + 539v^8 - 749v^6 + 1648v^4 - 2064*v^2 + 2048) / 1344
$\beta_{4}$	$=$	$ ( -2\nu^{15} - 9\nu^{13} + 32\nu^{11} - 112\nu^{9} + 301\nu^{7} - 269\nu^{5} + 804\nu^{3} - 592\nu ) / 672 $ (-2v^15 - 9v^13 + 32v^11 - 112v^9 + 301v^7 - 269v^5 + 804v^3 - 592v) / 672
$\beta_{5}$	$=$	$ ( 2\nu^{14} + 9\nu^{12} - 32\nu^{10} + 112\nu^{8} - 301\nu^{6} + 269\nu^{4} - 804\nu^{2} + 592 ) / 336 $ (2v^14 + 9v^12 - 32v^10 + 112v^8 - 301v^6 + 269v^4 - 804*v^2 + 592) / 336
$\beta_{6}$	$=$	$ ( 4\nu^{15} - 45\nu^{13} + 104\nu^{11} - 322\nu^{9} + 679\nu^{7} - 1037\nu^{5} + 1752\nu^{3} - 496\nu ) / 672 $ (4v^15 - 45v^13 + 104v^11 - 322v^9 + 679v^7 - 1037v^5 + 1752v^3 - 496v) / 672
$\beta_{7}$	$=$	$ ( -\nu^{14} + 4\nu^{12} - 14\nu^{10} + 39\nu^{8} - 77\nu^{6} + 156\nu^{4} - 160\nu^{2} + 192 ) / 64 $ (-v^14 + 4v^12 - 14v^10 + 39v^8 - 77v^6 + 156v^4 - 160v^2 + 192) / 64
$\beta_{8}$	$=$	$ ( -23\nu^{14} + 12\nu^{12} - 178\nu^{10} + 161\nu^{8} - 539\nu^{6} + 1348\nu^{4} - 1296\nu^{2} + 3776 ) / 1344 $ (-23v^14 + 12v^12 - 178v^10 + 161v^8 - 539v^6 + 1348v^4 - 1296*v^2 + 3776) / 1344
$\beta_{9}$	$=$	$ ( 29\nu^{14} - 48\nu^{12} + 166\nu^{10} - 371\nu^{8} + 413\nu^{6} - 856\nu^{4} + 144\nu^{2} - 320 ) / 1344 $ (29v^14 - 48v^12 + 166v^10 - 371v^8 + 413v^6 - 856v^4 + 144*v^2 - 320) / 1344
$\beta_{10}$	$=$	$ ( -37\nu^{15} + 12\nu^{13} - 38\nu^{11} - 77\nu^{9} + 791\nu^{7} - 1564\nu^{5} + 4416\nu^{3} - 4736\nu ) / 2688 $ (-37v^15 + 12v^13 - 38v^11 - 77v^9 + 791v^7 - 1564v^5 + 4416v^3 - 4736v) / 2688
$\beta_{11}$	$=$	$ ( -19\nu^{15} + 114\nu^{13} - 410\nu^{11} + 889\nu^{9} - 2065\nu^{7} + 3146\nu^{5} - 4416\nu^{3} + 5632\nu ) / 1344 $ (-19v^15 + 114v^13 - 410v^11 + 889v^9 - 2065v^7 + 3146v^5 - 4416v^3 + 5632v) / 1344
$\beta_{12}$	$=$	$ ( -19\nu^{14} + 72\nu^{12} - 74\nu^{10} + 301\nu^{8} - 91\nu^{6} + 80\nu^{4} - 384\nu^{2} - 2432 ) / 672 $ (-19v^14 + 72v^12 - 74v^10 + 301v^8 - 91v^6 + 80v^4 - 384*v^2 - 2432) / 672
$\beta_{13}$	$=$	$ ( 23\nu^{14} - 96\nu^{12} + 346\nu^{10} - 665\nu^{8} + 1463\nu^{6} - 2608\nu^{4} + 2808\nu^{2} - 4448 ) / 672 $ (23v^14 - 96v^12 + 346v^10 - 665v^8 + 1463v^6 - 2608v^4 + 2808*v^2 - 4448) / 672
$\beta_{14}$	$=$	$ ( -19\nu^{15} + 44\nu^{13} - 186\nu^{11} + 357\nu^{9} - 1015\nu^{7} + 1732\nu^{5} - 2064\nu^{3} + 3392\nu ) / 896 $ (-19v^15 + 44v^13 - 186v^11 + 357v^9 - 1015v^7 + 1732v^5 - 2064v^3 + 3392v) / 896
$\beta_{15}$	$=$	$ ( -37\nu^{15} + 138\nu^{13} - 374\nu^{11} + 1015\nu^{9} - 1771\nu^{7} + 2930\nu^{5} - 3648\nu^{3} + 2656\nu ) / 1344 $ (-37v^15 + 138v^13 - 374v^11 + 1015v^9 - 1771v^7 + 2930v^5 - 3648v^3 + 2656v) / 1344

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{10} + 2\beta_{6} - \beta_1 ) / 3 $ (b15 - b10 + 2*b6 - b1) / 3
$\nu^{2}$	$=$	$ ( -\beta_{13} - \beta_{12} - 2\beta_{9} - 3\beta_{8} - \beta_{5} - \beta_{3} + 1 ) / 3 $ (-b13 - b12 - 2b9 - 3b8 - b5 - b3 + 1) / 3
$\nu^{3}$	$=$	$ ( -\beta_{14} - \beta_{11} + 2\beta_{10} - \beta_{6} + \beta_{4} + 2\beta_{2} - 3\beta_1 ) / 3 $ (-b14 - b11 + 2b10 - b6 + b4 + 2b2 - 3*b1) / 3
$\nu^{4}$	$=$	$ \beta_{9} + \beta_{7} - \beta_{5} - 1 $ b9 + b7 - b5 - 1
$\nu^{5}$	$=$	$ ( -\beta_{14} - \beta_{11} - \beta_{10} - 4\beta_{6} + 7\beta_{4} + 2\beta_{2} ) / 3 $ (-b14 - b11 - b10 - 4b6 + 7b4 + 2*b2) / 3
$\nu^{6}$	$=$	$ ( 4\beta_{13} + \beta_{12} + 2\beta_{9} + 6\beta_{8} - 2\beta_{5} + 10\beta_{3} + 2 ) / 3 $ (4b13 + b12 + 2b9 + 6b8 - 2b5 + 10*b3 + 2) / 3
$\nu^{7}$	$=$	$ ( 7\beta_{15} - 6\beta_{14} + 6\beta_{11} - 10\beta_{10} + 8\beta_{6} + 9\beta_{4} - 3\beta_{2} + 8\beta_1 ) / 3 $ (7b15 - 6b14 + 6b11 - 10b10 + 8b6 + 9b4 - 3b2 + 8b1) / 3
$\nu^{8}$	$=$	$ 3\beta_{13} - \beta_{12} - 6\beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{5} + 4\beta_{3} + 5 $ 3b13 - b12 - 6b9 - b8 + b7 + 2b5 + 4b3 + 5
$\nu^{9}$	$=$	$ ( 25\beta_{15} - 15\beta_{14} - 9\beta_{11} - 7\beta_{10} + 17\beta_{6} - 21\beta_{4} + 6\beta_{2} + 2\beta_1 ) / 3 $ (25b15 - 15b14 - 9b11 - 7b10 + 17b6 - 21b4 + 6b2 + 2b1) / 3
$\nu^{10}$	$=$	$ ( 5\beta_{13} + 11\beta_{12} + 19\beta_{9} - 9\beta_{8} + 33\beta_{7} - 4\beta_{5} - 31\beta_{3} + 58 ) / 3 $ (5b13 + 11b12 + 19b9 - 9b8 + 33b7 - 4b5 - 31*b3 + 58) / 3
$\nu^{11}$	$=$	$ ( 30\beta_{15} + 14\beta_{14} - 58\beta_{11} - 13\beta_{10} + 5\beta_{6} - 14\beta_{4} + 14\beta_{2} - 45\beta_1 ) / 3 $ (30b15 + 14b14 - 58b11 - 13b10 + 5b6 - 14b4 + 14b2 - 45b1) / 3
$\nu^{12}$	$=$	$ -14\beta_{13} + 11\beta_{12} + 29\beta_{9} - 12\beta_{8} + \beta_{7} - 13\beta_{5} - 14\beta_{3} + 29 $ -14b13 + 11b12 + 29b9 - 12b8 + b7 - 13b5 - 14b3 + 29
$\nu^{13}$	$=$	$ ( - 21 \beta_{15} + 17 \beta_{14} + 17 \beta_{11} - 19 \beta_{10} - 10 \beta_{6} + 142 \beta_{4} + \cdots - 108 \beta_1 ) / 3 $ (-21b15 + 17b14 + 17b11 - 19b10 - 10b6 + 142b4 - 19b2 - 108b1) / 3
$\nu^{14}$	$=$	$ ( -35\beta_{13} - 56\beta_{12} + 14\beta_{9} - 117\beta_{8} - 57\beta_{7} - 20\beta_{5} + 124\beta_{3} - 85 ) / 3 $ (-35b13 - 56b12 + 14b9 - 117b8 - 57b7 - 20b5 + 124*b3 - 85) / 3
$\nu^{15}$	$=$	$ ( -68\beta_{15} - 183\beta_{14} + 135\beta_{11} - \beta_{10} - 79\beta_{6} + 120\beta_{4} + 57\beta_{2} - 52\beta_1 ) / 3 $ (-68b15 - 183b14 + 135b11 - b10 - 79b6 + 120b4 + 57b2 - 52*b1) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$-1 - \beta_{9}$	$1$

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} $

163.1

−1.14160	+	0.834713i
−1.41264	+	0.0667052i
0.776749	+	1.18180i
1.04779	+	0.949812i
−1.04779	−	0.949812i
−0.776749	−	1.18180i
1.41264	−	0.0667052i
1.14160	−	0.834713i
−1.14160	−	0.834713i
−1.41264	−	0.0667052i
0.776749	−	1.18180i
1.04779	−	0.949812i
−1.04779	+	0.949812i
−0.776749	+	1.18180i
1.41264	+	0.0667052i
1.14160	+	0.834713i

−1.29368

2.24073i

−2.34723

−

4.06553i

−1.14160

1.97731i

2.45374

0.989520i

6.97158

−2.95374

−

5.11603i

163.2

−0.764088

1.32344i

−0.167661

−

0.290398i

−1.41264

2.44676i

1.65876

−

2.06119i

−2.54392

−2.15876

−

3.73909i

163.3

−0.635098

1.10002i

0.193301

0.334806i

0.776749

−

1.34537i

−1.48662

−

2.18860i

−3.03145

0.986623

1.70888i

163.4

−0.298668

0.517308i

0.821595

1.42304i

1.04779

−

1.81482i

−1.12588

2.39424i

−2.17621

0.625881

1.08406i

163.5

0.298668

−

0.517308i

0.821595

1.42304i

−1.04779

1.81482i

−1.12588

2.39424i

2.17621

0.625881

1.08406i

163.6

0.635098

−

1.10002i

0.193301

0.334806i

−0.776749

1.34537i

−1.48662

−

2.18860i

3.03145

0.986623

1.70888i

163.7

0.764088

−

1.32344i

−0.167661

−

0.290398i

1.41264

−

2.44676i

1.65876

−

2.06119i

2.54392

−2.15876

−

3.73909i

163.8

1.29368

−

2.24073i

−2.34723

−

4.06553i

1.14160

−

1.97731i

2.45374

0.989520i

−6.97158

−2.95374

−

5.11603i

487.1

−1.29368

−

2.24073i

−2.34723

4.06553i

−1.14160

−

1.97731i

2.45374

−

0.989520i

6.97158

−2.95374

5.11603i

487.2

−0.764088

−

1.32344i

−0.167661

0.290398i

−1.41264

−

2.44676i

1.65876

2.06119i

−2.54392

−2.15876

3.73909i

487.3

−0.635098

−

1.10002i

0.193301

−

0.334806i

0.776749

1.34537i

−1.48662

2.18860i

−3.03145

0.986623

−

1.70888i

487.4

−0.298668

−

0.517308i

0.821595

−

1.42304i

1.04779

1.81482i

−1.12588

−

2.39424i

−2.17621

0.625881

−

1.08406i

487.5

0.298668

0.517308i

0.821595

−

1.42304i

−1.04779

−

1.81482i

−1.12588

−

2.39424i

2.17621

0.625881

−

1.08406i

487.6

0.635098

1.10002i

0.193301

−

0.334806i

−0.776749

−

1.34537i

−1.48662

2.18860i

3.03145

0.986623

−

1.70888i

487.7

0.764088

1.32344i

−0.167661

0.290398i

1.41264

2.44676i

1.65876

2.06119i

2.54392

−2.15876

3.73909i

487.8

1.29368

2.24073i

−2.34723

4.06553i

1.14160

1.97731i

2.45374

−

0.989520i

−6.97158

−2.95374

5.11603i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
21.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	567.2.e.g	✓	16
3.b	odd	2	1	inner	567.2.e.g	✓	16
7.c	even	3	1	inner	567.2.e.g	✓	16
7.c	even	3	1		3969.2.a.bg		8
7.d	odd	6	1		3969.2.a.bf		8
9.c	even	3	1		567.2.g.l		16
9.c	even	3	1		567.2.h.l		16
9.d	odd	6	1		567.2.g.l		16
9.d	odd	6	1		567.2.h.l		16
21.g	even	6	1		3969.2.a.bf		8
21.h	odd	6	1	inner	567.2.e.g	✓	16
21.h	odd	6	1		3969.2.a.bg		8
63.g	even	3	1		567.2.h.l		16
63.h	even	3	1		567.2.g.l		16
63.j	odd	6	1		567.2.g.l		16
63.n	odd	6	1		567.2.h.l		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
567.2.e.g	✓	16	1.a	even	1	1	trivial
567.2.e.g	✓	16	3.b	odd	2	1	inner
567.2.e.g	✓	16	7.c	even	3	1	inner
567.2.e.g	✓	16	21.h	odd	6	1	inner
567.2.g.l		16	9.c	even	3	1
567.2.g.l		16	9.d	odd	6	1
567.2.g.l		16	63.h	even	3	1
567.2.g.l		16	63.j	odd	6	1
567.2.h.l		16	9.c	even	3	1
567.2.h.l		16	9.d	odd	6	1
567.2.h.l		16	63.g	even	3	1
567.2.h.l		16	63.n	odd	6	1
3969.2.a.bf		8	7.d	odd	6	1
3969.2.a.bf		8	21.g	even	6	1
3969.2.a.bg		8	7.c	even	3	1
3969.2.a.bg		8	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{16} + 11T_{2}^{14} + 87T_{2}^{12} + 302T_{2}^{10} + 751T_{2}^{8} + 1026T_{2}^{6} + 990T_{2}^{4} + 324T_{2}^{2} + 81 $ T2^16 + 11*T2^14 + 87*T2^12 + 302*T2^10 + 751*T2^8 + 1026*T2^6 + 990*T2^4 + 324*T2^2 + 81 acting on $S_{2}^{\mathrm{new}}(567, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 11 T^{14} + \cdots + 81 $ T^16 + 11T^14 + 87T^12 + 302T^10 + 751T^8 + 1026T^6 + 990T^4 + 324*T^2 + 81
$3$	$ T^{16} $ T^16
$5$	$ T^{16} + 20 T^{14} + \cdots + 194481 $ T^16 + 20T^14 + 258T^12 + 1994T^10 + 11263T^8 + 42426T^6 + 116307T^4 + 186543*T^2 + 194481
$7$	$ (T^{8} - 3 T^{7} + \cdots + 2401)^{2} $ (T^8 - 3T^7 + 8T^6 - 33T^5 + 123T^4 - 231T^3 + 392T^2 - 1029*T + 2401)^2
$11$	$ T^{16} + 35 T^{14} + \cdots + 10556001 $ T^16 + 35T^14 + 795T^12 + 10802T^10 + 107311T^8 + 685890T^6 + 3114306T^4 + 6900876*T^2 + 10556001
$13$	$ (T^{4} - 3 T^{3} - 32 T^{2} + \cdots + 49)^{4} $ (T^4 - 3T^3 - 32T^2 + 102*T + 49)^4
$17$	$ T^{16} + 54 T^{14} + \cdots + 15752961 $ T^16 + 54T^14 + 2205T^12 + 32238T^10 + 335340T^8 + 1759806T^6 + 6652125T^4 + 12216582*T^2 + 15752961
$19$	$ (T^{8} + 12 T^{7} + \cdots + 625681)^{2} $ (T^8 + 12T^7 + 134T^6 + 690T^5 + 4311T^4 + 16134T^3 + 89135T^2 + 225435*T + 625681)^2
$23$	$ T^{16} + \cdots + 449920319121 $ T^16 + 132T^14 + 11466T^12 + 569754T^10 + 20524671T^8 + 468474354T^6 + 7743545163T^4 + 72677625111*T^2 + 449920319121
$29$	$ (T^{8} - 92 T^{6} + \cdots + 74529)^{2} $ (T^8 - 92T^6 + 2470T^4 - 24540*T^2 + 74529)^2
$31$	$ (T^{8} + 10 T^{7} + \cdots + 423801)^{2} $ (T^8 + 10T^7 + 105T^6 + 514T^5 + 3496T^4 + 14430T^3 + 76269T^2 + 183582*T + 423801)^2
$37$	$ (T^{8} - 2 T^{7} + \cdots + 729)^{2} $ (T^8 - 2T^7 + 48T^6 - 218T^5 + 2269T^4 - 6840T^3 + 22221T^2 - 4131*T + 729)^2
$41$	$ (T^{8} - 191 T^{6} + \cdots + 159201)^{2} $ (T^8 - 191T^6 + 9454T^4 - 92076*T^2 + 159201)^2
$43$	$ (T^{4} - 5 T^{3} + \cdots - 317)^{4} $ (T^4 - 5T^3 - 78T^2 + 448*T - 317)^4
$47$	$ T^{16} + \cdots + 103355177121 $ T^16 + 189T^14 + 26964T^12 + 1398951T^10 + 52160031T^8 + 999907335T^6 + 13584340548T^4 + 41170202829*T^2 + 103355177121
$53$	$ T^{16} + \cdots + 860013262161 $ T^16 + 288T^14 + 63414T^12 + 4854330T^10 + 269568891T^8 + 6987912606T^6 + 130232857455T^4 + 357180807195*T^2 + 860013262161
$59$	$ T^{16} + 60 T^{14} + \cdots + 15752961 $ T^16 + 60T^14 + 2610T^12 + 52002T^10 + 754191T^8 + 3185730T^6 + 9753291T^4 + 14681331*T^2 + 15752961
$61$	$ (T^{8} + 18 T^{7} + \cdots + 10686361)^{2} $ (T^8 + 18T^7 + 311T^6 + 2058T^5 + 19854T^4 + 105828T^3 + 874241T^2 + 2981328*T + 10686361)^2
$67$	$ (T^{8} - 9 T^{7} + \cdots + 7284601)^{2} $ (T^8 - 9T^7 + 215T^6 - 1434T^5 + 32535T^4 - 225462T^3 + 1380734T^2 - 3562680*T + 7284601)^2
$71$	$ (T^{8} - 357 T^{6} + \cdots + 6561)^{2} $ (T^8 - 357T^6 + 33057T^4 - 256365*T^2 + 6561)^2
$73$	$ (T^{8} + 16 T^{7} + \cdots + 441)^{2} $ (T^8 + 16T^7 + 204T^6 + 766T^5 + 2197T^4 + 2388T^3 + 2181T^2 - 693*T + 441)^2
$79$	$ (T^{8} - 16 T^{7} + \cdots + 5041)^{2} $ (T^8 - 16T^7 + 190T^6 - 1042T^5 + 4315T^4 - 2734T^3 + 4735T^2 + 497*T + 5041)^2
$83$	$ (T^{8} - 111 T^{6} + \cdots + 3969)^{2} $ (T^8 - 111T^6 + 3510T^4 - 33480*T^2 + 3969)^2
$89$	$ T^{16} + \cdots + 15\!\cdots\!21 $ T^16 + 716T^14 + 362694T^12 + 91521266T^10 + 16691276647T^8 + 1013174074758T^6 + 44450599964751T^4 + 970735476412107*T^2 + 15000980267723121
$97$	$ (T^{4} - 7 T^{3} + \cdots + 987)^{4} $ (T^4 - 7T^3 - 140T^2 + 576*T + 987)^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:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + ( - \beta_{12} - \beta_{8} + \cdots - \beta_{5}) q^{4}+ \cdots + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{2}) q^{8}+O(q^{10}) \) q - b1 * q^2 + (-b12 - b8 - b7 - b5) * q^4 + (-b10 - b1) * q^5 + (b12 + b9 + b7 - b3 + 1) * q^7 + (-b15 - b14 + b10 - b6 + b4 + b2) * q^8	\( q - \beta_1 q^{2} + ( - \beta_{12} - \beta_{8} + \cdots - \beta_{5}) q^{4}+ \cdots + (\beta_{15} + 2 \beta_{14} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) q - b1 * q^2 + (-b12 - b8 - b7 - b5) * q^4 + (-b10 - b1) * q^5 + (b12 + b9 + b7 - b3 + 1) * q^7 + (-b15 - b14 + b10 - b6 + b4 + b2) * q^8 + (-b12 + b9 - 2b8 - 2b7 - b5) * q^10 + b2 * q^11 + (-b13 + b12 + b9 - b7 + b5 - b3 + 1) * q^13 + (b14 - b10 - b2 - 2b1) q^14 + (2b13 + b12 - b9 + b8 - b5 + b3 - 1) q^16 + (-b11 + b6 - b4 + b2) * q^17 + (b13 + b12 - 3b9 + 2b8 - b5 - 3) * q^19 + (-3b15 - b14 + b2) q^20 + (b13 - b12 - 2b9 - b7 - 2b5 + 2b3 - 1) q^22 + (-b14 - 2b10 + b1) q^23 + (-b8 - b7) * q^25 + (b14 + 3b11 + b10 - b1) q^26 + (-2b13 - b12 + 3b9 - 2b8 - b7 - b5 - b3) q^28 + (2b10 - 2b6 - b4) * q^29 + (b13 - b12 + 2b9 - b8 - b7 - 2b5 - b3 + 1) * q^31 + (2b15 - b6 - b2 + 2b1) * q^32 + (-b9 - b5 + b3 - 1) * q^34 + (-b15 - 2b11 - 3b4 - 3b1) q^35 + (2b13 + b8 + 2b3) * q^37 + (6b15 + b11 + 2b6 + b4 - 2b2 + 6b1) * q^38 + (3b13 + b12 - 5b9 - b5 + 2b3 - 5) q^40 + (b15 + 2b14 - 2b10 + 2b6 - 2b2) * q^41 + (b13 - b12 - 3b7 + 2) q^43 + (-b14 - 2b11 + b10 + 3b1) * q^44 + (b13 + 3b12 - 2b9 + 2b5 - b3 + 1) q^46 + (b14 - 3b11 + b10 - 3b1) * q^47 + (-b13 + 2b12 + 2b9 - b8 + b7 + 3b5 - 1) q^49 + (-b15 + b10 - b6 - b4) * q^50 + (-b13 - 4b12 + b9 - 3b8 - 3b7 - 3b5 + b3 - 1) * q^52 + (5b15 + 3b11 + b6 + 3b4 + 5b1) * q^53 + (2b13 - 2b12 - 2b9 + b7 - 2b5 + 2b3) q^55 + (-5b15 - 2b14 + 2b11 - b10 + 3b4 + b2 - 3b1) q^56 + (-b13 + 3b9 + 2b8 - b3 + 3) * q^58 + (-b15 + b11 + b6 + b4 - b1) * q^59 + (-3b13 - b12 - 4b9 - b8 + b5 - 2b3 - 4) q^61 + (-5b15 + b10 - b6 + 2b4) * q^62 + (-b13 + b12 + 4) * q^64 + (3b14 + 2b11 + b1) * q^65 + (b13 - 2b9 + 4b8 + 4b7 - b5 - b3 + 1) q^67 + (-3b11 + 2b10 + b1) * q^68 + (-2b13 - b12 + 6b9 - 2b8 - 3b7 - b5 - 2b3) q^70 + (b15 + 2b14 + 5b4 - 2b2) q^71 + (b13 + b12 + 5b9 + 2b8 + 2b7 - b3 + 1) q^73 + (3b15 - 3b11 + b6 - 3b4 - 2b2 + 3b1) q^74 + (-7b13 + 7b12 + 7b9 + 6b7 + 7b5 - 7b3 + 9) * q^76 + (b15 - b14 + 2b11 + b10 + 3b4 + b2 - 2b1) q^77 + (-b13 - b12 + 4b9 + b5 + 4) q^79 + (2b15 - 5b11 - 5b4 - 2b2 + 2b1) q^80 + (-5b13 - 2b12 + b9 - 5b8 + 2b5 - 3b3 + 1) q^82 + (-3b15 - 2b4) * q^83 + (3b13 - 3b12 - 2b9 - 2b5 + 2b3 + 4) q^85 + (b14 + 2b11 + 3b10 + 2b1) q^86 + (-b13 + 3b12 - 7b9 + 3b8 + 3b7 + 4b5 + b3 - 1) q^88 + (-4b14 + 5b11 - b10 + b1) * q^89 + (-b13 + 2b12 + 2b9 + 5b8 + 4b7 + 3b5 - 3b3) * q^91 + (6b15 + 2b14 - 4b10 + 4b6 - 5b4 - 2b2) * q^92 + (-b13 - 2b12 + 5b9 - 3b8 - 3b7 - b5 + b3 - 1) * q^94 + (6b15 - 3b11 + 3b6 - 3b4 + b2 + 6b1) q^95 + (3b13 - 3b12 - b9 + b7 - b5 + b3 + 3) * q^97 + (b15 + 2b14 - b11 - b10 - b6 - 5b4 - b2 - 2b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{4} + 6 q^{7}+O(q^{10}) \) 16 * q - 6 * q^4 + 6 * q^7	\( 16 q - 6 q^{4} + 6 q^{7} - 14 q^{10} + 12 q^{13} - 6 q^{16} - 24 q^{19} + 4 q^{22} - 26 q^{28} - 20 q^{31} + 4 q^{37} - 36 q^{40} + 20 q^{43} + 36 q^{46} - 14 q^{49} - 34 q^{52} + 8 q^{55} + 22 q^{58} - 36 q^{61} + 76 q^{64} + 18 q^{67} - 58 q^{70} - 32 q^{73} + 116 q^{76} + 32 q^{79} + 2 q^{82} + 60 q^{85} + 72 q^{88} - 22 q^{91} - 54 q^{94} + 28 q^{97}+O(q^{100}) \) 16 * q - 6 * q^4 + 6 * q^7 - 14 * q^10 + 12 * q^13 - 6 * q^16 - 24 * q^19 + 4 * q^22 - 26 * q^28 - 20 * q^31 + 4 * q^37 - 36 * q^40 + 20 * q^43 + 36 * q^46 - 14 * q^49 - 34 * q^52 + 8 * q^55 + 22 * q^58 - 36 * q^61 + 76 * q^64 + 18 * q^67 - 58 * q^70 - 32 * q^73 + 116 * q^76 + 32 * q^79 + 2 * q^82 + 60 * q^85 + 72 * q^88 - 22 * q^91 - 54 * q^94 + 28 * q^97

Introduction

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Newspace parameters

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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	567.2.e.g

Level	$567$
Weight	$2$
Character orbit	567.e
Analytic conductor	$4.528$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.52751779461\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4x^{14} + 14x^{12} - 39x^{10} + 77x^{8} - 156x^{6} + 224x^{4} - 256x^{2} + 256 \) x^16 - 4x^14 + 14x^12 - 39x^10 + 77x^8 - 156x^6 + 224x^4 - 256*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( -5\nu^{15} - 96\nu^{13} + 122\nu^{11} - 469\nu^{9} + 1099\nu^{7} - 872\nu^{5} + 2304\nu^{3} - 1984\nu ) / 2688 \) (-5v^15 - 96v^13 + 122v^11 - 469v^9 + 1099v^7 - 872v^5 + 2304v^3 - 1984v) / 2688
\(\beta_{2}\)	\(=\)	\( ( -\nu^{15} - 8\nu^{13} - 54\nu^{11} + 63\nu^{9} - 329\nu^{7} + 744\nu^{5} - 424\nu^{3} + 2112\nu ) / 448 \) (-v^15 - 8v^13 - 54v^11 + 63v^9 - 329v^7 + 744v^5 - 424v^3 + 2112*v) / 448
\(\beta_{3}\)	\(=\)	\( ( -5\nu^{14} + 72\nu^{12} - 214\nu^{10} + 539\nu^{8} - 749\nu^{6} + 1648\nu^{4} - 2064\nu^{2} + 2048 ) / 1344 \) (-5v^14 + 72v^12 - 214v^10 + 539v^8 - 749v^6 + 1648v^4 - 2064*v^2 + 2048) / 1344
\(\beta_{4}\)	\(=\)	\( ( -2\nu^{15} - 9\nu^{13} + 32\nu^{11} - 112\nu^{9} + 301\nu^{7} - 269\nu^{5} + 804\nu^{3} - 592\nu ) / 672 \) (-2v^15 - 9v^13 + 32v^11 - 112v^9 + 301v^7 - 269v^5 + 804v^3 - 592v) / 672
\(\beta_{5}\)	\(=\)	\( ( 2\nu^{14} + 9\nu^{12} - 32\nu^{10} + 112\nu^{8} - 301\nu^{6} + 269\nu^{4} - 804\nu^{2} + 592 ) / 336 \) (2v^14 + 9v^12 - 32v^10 + 112v^8 - 301v^6 + 269v^4 - 804*v^2 + 592) / 336
\(\beta_{6}\)	\(=\)	\( ( 4\nu^{15} - 45\nu^{13} + 104\nu^{11} - 322\nu^{9} + 679\nu^{7} - 1037\nu^{5} + 1752\nu^{3} - 496\nu ) / 672 \) (4v^15 - 45v^13 + 104v^11 - 322v^9 + 679v^7 - 1037v^5 + 1752v^3 - 496v) / 672
\(\beta_{7}\)	\(=\)	\( ( -\nu^{14} + 4\nu^{12} - 14\nu^{10} + 39\nu^{8} - 77\nu^{6} + 156\nu^{4} - 160\nu^{2} + 192 ) / 64 \) (-v^14 + 4v^12 - 14v^10 + 39v^8 - 77v^6 + 156v^4 - 160v^2 + 192) / 64
\(\beta_{8}\)	\(=\)	\( ( -23\nu^{14} + 12\nu^{12} - 178\nu^{10} + 161\nu^{8} - 539\nu^{6} + 1348\nu^{4} - 1296\nu^{2} + 3776 ) / 1344 \) (-23v^14 + 12v^12 - 178v^10 + 161v^8 - 539v^6 + 1348v^4 - 1296*v^2 + 3776) / 1344
\(\beta_{9}\)	\(=\)	\( ( 29\nu^{14} - 48\nu^{12} + 166\nu^{10} - 371\nu^{8} + 413\nu^{6} - 856\nu^{4} + 144\nu^{2} - 320 ) / 1344 \) (29v^14 - 48v^12 + 166v^10 - 371v^8 + 413v^6 - 856v^4 + 144*v^2 - 320) / 1344
\(\beta_{10}\)	\(=\)	\( ( -37\nu^{15} + 12\nu^{13} - 38\nu^{11} - 77\nu^{9} + 791\nu^{7} - 1564\nu^{5} + 4416\nu^{3} - 4736\nu ) / 2688 \) (-37v^15 + 12v^13 - 38v^11 - 77v^9 + 791v^7 - 1564v^5 + 4416v^3 - 4736v) / 2688
\(\beta_{11}\)	\(=\)	\( ( -19\nu^{15} + 114\nu^{13} - 410\nu^{11} + 889\nu^{9} - 2065\nu^{7} + 3146\nu^{5} - 4416\nu^{3} + 5632\nu ) / 1344 \) (-19v^15 + 114v^13 - 410v^11 + 889v^9 - 2065v^7 + 3146v^5 - 4416v^3 + 5632v) / 1344
\(\beta_{12}\)	\(=\)	\( ( -19\nu^{14} + 72\nu^{12} - 74\nu^{10} + 301\nu^{8} - 91\nu^{6} + 80\nu^{4} - 384\nu^{2} - 2432 ) / 672 \) (-19v^14 + 72v^12 - 74v^10 + 301v^8 - 91v^6 + 80v^4 - 384*v^2 - 2432) / 672
\(\beta_{13}\)	\(=\)	\( ( 23\nu^{14} - 96\nu^{12} + 346\nu^{10} - 665\nu^{8} + 1463\nu^{6} - 2608\nu^{4} + 2808\nu^{2} - 4448 ) / 672 \) (23v^14 - 96v^12 + 346v^10 - 665v^8 + 1463v^6 - 2608v^4 + 2808*v^2 - 4448) / 672
\(\beta_{14}\)	\(=\)	\( ( -19\nu^{15} + 44\nu^{13} - 186\nu^{11} + 357\nu^{9} - 1015\nu^{7} + 1732\nu^{5} - 2064\nu^{3} + 3392\nu ) / 896 \) (-19v^15 + 44v^13 - 186v^11 + 357v^9 - 1015v^7 + 1732v^5 - 2064v^3 + 3392v) / 896
\(\beta_{15}\)	\(=\)	\( ( -37\nu^{15} + 138\nu^{13} - 374\nu^{11} + 1015\nu^{9} - 1771\nu^{7} + 2930\nu^{5} - 3648\nu^{3} + 2656\nu ) / 1344 \) (-37v^15 + 138v^13 - 374v^11 + 1015v^9 - 1771v^7 + 2930v^5 - 3648v^3 + 2656v) / 1344

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{10} + 2\beta_{6} - \beta_1 ) / 3 \) (b15 - b10 + 2*b6 - b1) / 3
\(\nu^{2}\)	\(=\)	\( ( -\beta_{13} - \beta_{12} - 2\beta_{9} - 3\beta_{8} - \beta_{5} - \beta_{3} + 1 ) / 3 \) (-b13 - b12 - 2b9 - 3b8 - b5 - b3 + 1) / 3
\(\nu^{3}\)	\(=\)	\( ( -\beta_{14} - \beta_{11} + 2\beta_{10} - \beta_{6} + \beta_{4} + 2\beta_{2} - 3\beta_1 ) / 3 \) (-b14 - b11 + 2b10 - b6 + b4 + 2b2 - 3*b1) / 3
\(\nu^{4}\)	\(=\)	\( \beta_{9} + \beta_{7} - \beta_{5} - 1 \) b9 + b7 - b5 - 1
\(\nu^{5}\)	\(=\)	\( ( -\beta_{14} - \beta_{11} - \beta_{10} - 4\beta_{6} + 7\beta_{4} + 2\beta_{2} ) / 3 \) (-b14 - b11 - b10 - 4b6 + 7b4 + 2*b2) / 3
\(\nu^{6}\)	\(=\)	\( ( 4\beta_{13} + \beta_{12} + 2\beta_{9} + 6\beta_{8} - 2\beta_{5} + 10\beta_{3} + 2 ) / 3 \) (4b13 + b12 + 2b9 + 6b8 - 2b5 + 10*b3 + 2) / 3
\(\nu^{7}\)	\(=\)	\( ( 7\beta_{15} - 6\beta_{14} + 6\beta_{11} - 10\beta_{10} + 8\beta_{6} + 9\beta_{4} - 3\beta_{2} + 8\beta_1 ) / 3 \) (7b15 - 6b14 + 6b11 - 10b10 + 8b6 + 9b4 - 3b2 + 8b1) / 3
\(\nu^{8}\)	\(=\)	\( 3\beta_{13} - \beta_{12} - 6\beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{5} + 4\beta_{3} + 5 \) 3b13 - b12 - 6b9 - b8 + b7 + 2b5 + 4b3 + 5
\(\nu^{9}\)	\(=\)	\( ( 25\beta_{15} - 15\beta_{14} - 9\beta_{11} - 7\beta_{10} + 17\beta_{6} - 21\beta_{4} + 6\beta_{2} + 2\beta_1 ) / 3 \) (25b15 - 15b14 - 9b11 - 7b10 + 17b6 - 21b4 + 6b2 + 2b1) / 3
\(\nu^{10}\)	\(=\)	\( ( 5\beta_{13} + 11\beta_{12} + 19\beta_{9} - 9\beta_{8} + 33\beta_{7} - 4\beta_{5} - 31\beta_{3} + 58 ) / 3 \) (5b13 + 11b12 + 19b9 - 9b8 + 33b7 - 4b5 - 31*b3 + 58) / 3
\(\nu^{11}\)	\(=\)	\( ( 30\beta_{15} + 14\beta_{14} - 58\beta_{11} - 13\beta_{10} + 5\beta_{6} - 14\beta_{4} + 14\beta_{2} - 45\beta_1 ) / 3 \) (30b15 + 14b14 - 58b11 - 13b10 + 5b6 - 14b4 + 14b2 - 45b1) / 3
\(\nu^{12}\)	\(=\)	\( -14\beta_{13} + 11\beta_{12} + 29\beta_{9} - 12\beta_{8} + \beta_{7} - 13\beta_{5} - 14\beta_{3} + 29 \) -14b13 + 11b12 + 29b9 - 12b8 + b7 - 13b5 - 14b3 + 29
\(\nu^{13}\)	\(=\)	\( ( - 21 \beta_{15} + 17 \beta_{14} + 17 \beta_{11} - 19 \beta_{10} - 10 \beta_{6} + 142 \beta_{4} + \cdots - 108 \beta_1 ) / 3 \) (-21b15 + 17b14 + 17b11 - 19b10 - 10b6 + 142b4 - 19b2 - 108b1) / 3
\(\nu^{14}\)	\(=\)	\( ( -35\beta_{13} - 56\beta_{12} + 14\beta_{9} - 117\beta_{8} - 57\beta_{7} - 20\beta_{5} + 124\beta_{3} - 85 ) / 3 \) (-35b13 - 56b12 + 14b9 - 117b8 - 57b7 - 20b5 + 124*b3 - 85) / 3
\(\nu^{15}\)	\(=\)	\( ( -68\beta_{15} - 183\beta_{14} + 135\beta_{11} - \beta_{10} - 79\beta_{6} + 120\beta_{4} + 57\beta_{2} - 52\beta_1 ) / 3 \) (-68b15 - 183b14 + 135b11 - b10 - 79b6 + 120b4 + 57b2 - 52*b1) / 3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 163.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 11 T^{14} + \cdots + 81 \) T^16 + 11T^14 + 87T^12 + 302T^10 + 751T^8 + 1026T^6 + 990T^4 + 324*T^2 + 81
$3$	\( T^{16} \) T^16
$5$	\( T^{16} + 20 T^{14} + \cdots + 194481 \) T^16 + 20T^14 + 258T^12 + 1994T^10 + 11263T^8 + 42426T^6 + 116307T^4 + 186543*T^2 + 194481
$7$	\( (T^{8} - 3 T^{7} + \cdots + 2401)^{2} \) (T^8 - 3T^7 + 8T^6 - 33T^5 + 123T^4 - 231T^3 + 392T^2 - 1029*T + 2401)^2
$11$	\( T^{16} + 35 T^{14} + \cdots + 10556001 \) T^16 + 35T^14 + 795T^12 + 10802T^10 + 107311T^8 + 685890T^6 + 3114306T^4 + 6900876*T^2 + 10556001
$13$	\( (T^{4} - 3 T^{3} - 32 T^{2} + \cdots + 49)^{4} \) (T^4 - 3T^3 - 32T^2 + 102*T + 49)^4
$17$	\( T^{16} + 54 T^{14} + \cdots + 15752961 \) T^16 + 54T^14 + 2205T^12 + 32238T^10 + 335340T^8 + 1759806T^6 + 6652125T^4 + 12216582*T^2 + 15752961
$19$	\( (T^{8} + 12 T^{7} + \cdots + 625681)^{2} \) (T^8 + 12T^7 + 134T^6 + 690T^5 + 4311T^4 + 16134T^3 + 89135T^2 + 225435*T + 625681)^2
$23$	\( T^{16} + \cdots + 449920319121 \) T^16 + 132T^14 + 11466T^12 + 569754T^10 + 20524671T^8 + 468474354T^6 + 7743545163T^4 + 72677625111*T^2 + 449920319121
$29$	\( (T^{8} - 92 T^{6} + \cdots + 74529)^{2} \) (T^8 - 92T^6 + 2470T^4 - 24540*T^2 + 74529)^2
$31$	\( (T^{8} + 10 T^{7} + \cdots + 423801)^{2} \) (T^8 + 10T^7 + 105T^6 + 514T^5 + 3496T^4 + 14430T^3 + 76269T^2 + 183582*T + 423801)^2
$37$	\( (T^{8} - 2 T^{7} + \cdots + 729)^{2} \) (T^8 - 2T^7 + 48T^6 - 218T^5 + 2269T^4 - 6840T^3 + 22221T^2 - 4131*T + 729)^2
$41$	\( (T^{8} - 191 T^{6} + \cdots + 159201)^{2} \) (T^8 - 191T^6 + 9454T^4 - 92076*T^2 + 159201)^2
$43$	\( (T^{4} - 5 T^{3} + \cdots - 317)^{4} \) (T^4 - 5T^3 - 78T^2 + 448*T - 317)^4
$47$	\( T^{16} + \cdots + 103355177121 \) T^16 + 189T^14 + 26964T^12 + 1398951T^10 + 52160031T^8 + 999907335T^6 + 13584340548T^4 + 41170202829*T^2 + 103355177121
$53$	\( T^{16} + \cdots + 860013262161 \) T^16 + 288T^14 + 63414T^12 + 4854330T^10 + 269568891T^8 + 6987912606T^6 + 130232857455T^4 + 357180807195*T^2 + 860013262161
$59$	\( T^{16} + 60 T^{14} + \cdots + 15752961 \) T^16 + 60T^14 + 2610T^12 + 52002T^10 + 754191T^8 + 3185730T^6 + 9753291T^4 + 14681331*T^2 + 15752961
$61$	\( (T^{8} + 18 T^{7} + \cdots + 10686361)^{2} \) (T^8 + 18T^7 + 311T^6 + 2058T^5 + 19854T^4 + 105828T^3 + 874241T^2 + 2981328*T + 10686361)^2
$67$	\( (T^{8} - 9 T^{7} + \cdots + 7284601)^{2} \) (T^8 - 9T^7 + 215T^6 - 1434T^5 + 32535T^4 - 225462T^3 + 1380734T^2 - 3562680*T + 7284601)^2
$71$	\( (T^{8} - 357 T^{6} + \cdots + 6561)^{2} \) (T^8 - 357T^6 + 33057T^4 - 256365*T^2 + 6561)^2
$73$	\( (T^{8} + 16 T^{7} + \cdots + 441)^{2} \) (T^8 + 16T^7 + 204T^6 + 766T^5 + 2197T^4 + 2388T^3 + 2181T^2 - 693*T + 441)^2
$79$	\( (T^{8} - 16 T^{7} + \cdots + 5041)^{2} \) (T^8 - 16T^7 + 190T^6 - 1042T^5 + 4315T^4 - 2734T^3 + 4735T^2 + 497*T + 5041)^2
$83$	\( (T^{8} - 111 T^{6} + \cdots + 3969)^{2} \) (T^8 - 111T^6 + 3510T^4 - 33480*T^2 + 3969)^2
$89$	\( T^{16} + \cdots + 15\!\cdots\!21 \) T^16 + 716T^14 + 362694T^12 + 91521266T^10 + 16691276647T^8 + 1013174074758T^6 + 44450599964751T^4 + 970735476412107*T^2 + 15000980267723121
$97$	\( (T^{4} - 7 T^{3} + \cdots + 987)^{4} \) (T^4 - 7T^3 - 140T^2 + 576*T + 987)^4
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\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(-1 - \beta_{9}\)	\(1\)