LMFDB - Newform orbit 576.2.p.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,2,Mod(95,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 5]))

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

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

chi := DirichletCharacter("576.95");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	576.p (of order $6$, 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:	$4.59938315643$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 $ x^16 + 11x^14 + 85x^12 + 332x^10 + 940x^8 + 1064x^6 + 880x^4 + 128*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{12} q^{3} + (\beta_{7} + \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{10} - \beta_{6}) q^{7} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{9}+O(q^{10}) $ q - b12 * q^3 + (b7 + b4 - b3) * q^5 + (-b10 - b6) * q^7 + (-b7 + b3 + b1) * q^9	$ q - \beta_{12} q^{3} + (\beta_{7} + \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{10} - \beta_{6}) q^{7} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{9} + (\beta_{14} - \beta_{13} + \cdots - \beta_{6}) q^{11}+ \cdots + ( - 3 \beta_{15} - 3 \beta_{13} + \cdots + 3 \beta_{6}) q^{99}+O(q^{100}) $ q - b12 * q^3 + (b7 + b4 - b3) * q^5 + (-b10 - b6) * q^7 + (-b7 + b3 + b1) * q^9 + (b14 - b13 + b11 - b10 - b9 - b6) * q^11 + (-b8 + 2b1) q^13 + (-b15 - 2b14 + b13 - b12 - b11 - b10 + b9) q^15 + (2b8 - 2b4 - b1 + 1) * q^17 + (-b15 - 2b12 - b10 + 2b9 - 2b6) q^19 + (-b8 + 2b7 - b5 - b4 - b3 + b1 - 1) q^21 + (-b15 + 2b14 - 2b12 + b9) * q^23 + (b8 + b7 + b5 + 2b4 + b2 - b1 - 2) q^25 + (2b14 - b13 - b12 + 2b11 + b10 + b9 - b6) * q^27 + (b5 - 2b4 + b2 + 2) q^29 + (3b15 - 2b14 + b9 + 2b6) q^31 + (-b8 + 2b7 - 2b5 + b2 + b1 - 2) * q^33 + (2b15 + b14 - 2b12 - b11 + b10 + b9) * q^35 + (-2b8 + 2b7 + 2b4 - b3 + b1 - 1) q^37 + (-b14 - b13 - b12 + 2b11 + b10 + b9 - b6) q^39 + (b7 + b4 + b3 - 3b2 - 2) q^41 + (b15 - 2b14 - b12 - b9) q^43 + (b8 + b5 - 3b4 - 2b2 + b1 + 4) * q^45 + (-2b15 + 2b14 + b13 + 2b12 + b11 - b10 + b6) q^47 + (-2b8 - 2b7 + 2b5 + 2b4 + 2b3 - b2 - 2) q^49 + (-3b15 + 2b14 + 2b13 - b11 - 2b10 - b9 - b6) * q^51 + (b5 + b4 + b3 - 2b2 - 5) q^53 + (6b15 - b14 - 3b12 + b11 + 2b10 - b9) q^55 + (-b8 + b7 - 2b5 - 3b4 - 2b3 + b2 + 2b1 + 4) * q^57 + (2b15 - b14 + 3b13 - b12 + 2b9) q^59 + (-3b5 - 4b4 + 3b2 + 2) q^61 + (-b15 - 3b14 + 3b12 + b11 - 2b9 + 2b6) * q^63 + (2b8 + 3b5 + 4b4 - 3b2 + 2b1 - 2) q^65 + (3b15 - 2b14 - b13 + 3b12 + 2b11 + 2b10 - 2b9 + b6) * q^67 + (b8 - 2b7 + 6b4 + b3 + b1 - 3) * q^69 + (-4b15 + 6b14 - 2b13 + b12 + b11 - b10 - 4b9 - 2b6) q^71 + (-b5 - b4 - 2b3 + 2b2 - 4b1 + 5) q^73 + (-6b15 + 2b14 - b13 + b12 - 4b11 - 2b10 + b9 - b6) * q^75 + (-3b8 - b7 + 2b5 + 10b4 + b3 - b2 - 2) q^77 + (3b15 - 3b14 + 2b13 + 3b12 - 2b11) q^79 + (-b8 - b7 - 3b5 - b3 + 3b2 - b1 + 3) * q^81 + (2b15 - 2b14 - 2b13 + 2b12 + 2b11 + b10 + b6) q^83 + (2b8 - b7 - b3 + 3b2 - 4b1) q^85 + (-3b15 + b14 - 2b13 - 2b11 - b10 - 2b9 - 2b6) q^87 + (2b8 - 2b7 - 10b4 + b3 - b1 + 5) q^89 + (2b15 - 3b14 + b12 + 2b10 - b9 + 4b6) * q^91 + (5b8 - 2b7 + 2b5 - 2b4 + b3 - 2b2 - 3b1 - 7) * q^93 + (-4b15 + 2b14 + 3b13 - 2b12 - 6b11 + 4b9) * q^95 + (-2b8 + b7 + b5 + b2 + 2b1) * q^97 + (-3b15 - 3b13 + 4b12 + 3b11 - 2b9 + 3b6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{5} + 6 q^{9}+O(q^{10}) $ 16 * q + 6 * q^5 + 6 * q^9	$ 16 q + 6 q^{5} + 6 q^{9} + 6 q^{13} - 30 q^{21} - 14 q^{25} + 18 q^{29} - 48 q^{33} + 66 q^{45} + 6 q^{49} - 48 q^{53} + 18 q^{57} - 42 q^{61} + 54 q^{65} + 6 q^{69} + 28 q^{73} + 66 q^{77} - 6 q^{81} - 36 q^{85} - 102 q^{93} + 8 q^{97}+O(q^{100}) $ 16 * q + 6 * q^5 + 6 * q^9 + 6 * q^13 - 30 * q^21 - 14 * q^25 + 18 * q^29 - 48 * q^33 + 66 * q^45 + 6 * q^49 - 48 * q^53 + 18 * q^57 - 42 * q^61 + 54 * q^65 + 6 * q^69 + 28 * q^73 + 66 * q^77 - 6 * q^81 - 36 * q^85 - 102 * q^93 + 8 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 $ x^16 + 11*x^14 + 85*x^12 + 332*x^10 + 940*x^8 + 1064*x^6 + 880*x^4 + 128*x^2 + 16 :

$\beta_{1}$	$=$	$ ( - 678 \nu^{14} - 6789 \nu^{12} - 51076 \nu^{10} - 174698 \nu^{8} - 464941 \nu^{6} - 179896 \nu^{4} + \cdots + 1117832 ) / 392916 $ (-678v^14 - 6789v^12 - 51076v^10 - 174698v^8 - 464941v^6 - 179896v^4 - 26216*v^2 + 1117832) / 392916
$\beta_{2}$	$=$	$ ( - 2936 \nu^{14} - 36643 \nu^{12} - 297579 \nu^{10} - 1334969 \nu^{8} - 4109118 \nu^{6} + \cdots - 1460696 ) / 392916 $ (-2936v^14 - 36643v^12 - 297579v^10 - 1334969v^8 - 4109118v^6 - 6585444v^4 - 5439720*v^2 - 1460696) / 392916
$\beta_{3}$	$=$	$ ( 3399 \nu^{14} + 34180 \nu^{12} + 256058 \nu^{10} + 875809 \nu^{8} + 2248150 \nu^{6} + \cdots - 1217016 ) / 392916 $ (3399v^14 + 34180v^12 + 256058v^10 + 875809v^8 + 2248150v^6 + 901868v^4 + 131428*v^2 - 1217016) / 392916
$\beta_{4}$	$=$	$ ( - 15229 \nu^{14} - 164807 \nu^{12} - 1267309 \nu^{10} - 4851724 \nu^{8} - 13616468 \nu^{6} + \cdots - 272784 ) / 1571664 $ (-15229v^14 - 164807v^12 - 1267309v^10 - 4851724v^8 - 13616468v^6 - 14343892v^4 - 12681936*v^2 - 272784) / 1571664
$\beta_{5}$	$=$	$ ( - 41127 \nu^{14} - 462669 \nu^{12} - 3589379 \nu^{10} - 14297016 \nu^{8} - 40389328 \nu^{6} + \cdots - 1994224 ) / 1571664 $ (-41127v^14 - 462669v^12 - 3589379v^10 - 14297016v^8 - 40389328v^6 - 47060636v^4 - 32106720*v^2 - 1994224) / 1571664
$\beta_{6}$	$=$	$ ( - 28442 \nu^{15} - 276974 \nu^{13} - 2022573 \nu^{11} - 6400837 \nu^{9} - 14945299 \nu^{7} + \cdots + 16825348 \nu ) / 2357496 $ (-28442v^15 - 276974v^13 - 2022573v^11 - 6400837v^9 - 14945299v^7 + 2363604v^5 + 8242912v^3 + 16825348v) / 2357496
$\beta_{7}$	$=$	$ ( - 11161 \nu^{14} - 124073 \nu^{12} - 960853 \nu^{10} - 3803536 \nu^{8} - 10826822 \nu^{6} + \cdots - 1478952 ) / 392916 $ (-11161v^14 - 124073v^12 - 960853v^10 - 3803536v^8 - 10826822v^6 - 12871600v^4 - 10167144*v^2 - 1478952) / 392916
$\beta_{8}$	$=$	$ ( - 15229 \nu^{14} - 164807 \nu^{12} - 1267309 \nu^{10} - 4851724 \nu^{8} - 13616468 \nu^{6} + \cdots - 272784 ) / 523888 $ (-15229v^14 - 164807v^12 - 1267309v^10 - 4851724v^8 - 13616468v^6 - 14343892v^4 - 12158048*v^2 - 272784) / 523888
$\beta_{9}$	$=$	$ ( 14088 \nu^{15} + 160481 \nu^{13} + 1257754 \nu^{11} + 5136186 \nu^{9} + 15012495 \nu^{7} + \cdots + 4461244 \nu ) / 785832 $ (14088v^15 + 160481v^13 + 1257754v^11 + 5136186v^9 + 15012495v^7 + 19847572v^5 + 17571096v^3 + 4461244v) / 785832
$\beta_{10}$	$=$	$ ( 53485 \nu^{15} + 552511 \nu^{13} + 4149261 \nu^{11} + 14709020 \nu^{9} + 38382152 \nu^{7} + \cdots - 15145376 \nu ) / 2357496 $ (53485v^15 + 552511v^13 + 4149261v^11 + 14709020v^9 + 38382152v^7 + 24101568v^5 + 11410756v^3 - 15145376v) / 2357496
$\beta_{11}$	$=$	$ ( - 427 \nu^{15} - 4723 \nu^{13} - 36549 \nu^{11} - 143882 \nu^{9} - 409832 \nu^{7} + \cdots - 112384 \nu ) / 17208 $ (-427v^15 - 4723v^13 - 36549v^11 - 143882v^9 - 409832v^7 - 483588v^5 - 417712v^3 - 112384v) / 17208
$\beta_{12}$	$=$	$ ( - 26391 \nu^{15} - 285268 \nu^{13} - 2184580 \nu^{11} - 8306259 \nu^{9} - 22932244 \nu^{7} + \cdots + 1749840 \nu ) / 785832 $ (-26391v^15 - 285268v^13 - 2184580v^11 - 8306259v^9 - 22932244v^7 - 22719052v^5 - 16082232v^3 + 1749840v) / 785832
$\beta_{13}$	$=$	$ ( 82961 \nu^{15} + 892865 \nu^{13} + 6850017 \nu^{11} + 26014876 \nu^{9} + 72425530 \nu^{7} + \cdots - 5684968 \nu ) / 2357496 $ (82961v^15 + 892865v^13 + 6850017v^11 + 26014876v^9 + 72425530v^7 + 72742140v^5 + 58172408v^3 - 5684968v) / 2357496
$\beta_{14}$	$=$	$ ( - 29781 \nu^{15} - 319213 \nu^{13} - 2439960 \nu^{11} - 9179749 \nu^{9} - 25256949 \nu^{7} + \cdots + 1445260 \nu ) / 785832 $ (-29781v^15 - 319213v^13 - 2439960v^11 - 9179749v^9 - 25256949v^7 - 24011448v^5 - 18177892v^3 + 1445260v) / 785832
$\beta_{15}$	$=$	$ ( - 10800 \nu^{15} - 118285 \nu^{13} - 911829 \nu^{11} - 3535889 \nu^{9} - 9931987 \nu^{7} + \cdots - 751284 \nu ) / 196458 $ (-10800v^15 - 118285v^13 - 911829v^11 - 3535889v^9 - 9931987v^7 - 10822149v^5 - 8439635v^3 - 751284v) / 196458

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

$\nu$	$=$	$ ( -4\beta_{14} - \beta_{13} - \beta_{11} - 2\beta_{10} - 4\beta_{9} + 2\beta_{6} ) / 6 $ (-4b14 - b13 - b11 - 2b10 - 4b9 + 2b6) / 6
$\nu^{2}$	$=$	$ \beta_{8} - 3\beta_{4} $ b8 - 3*b4
$\nu^{3}$	$=$	$ ( -3\beta_{15} + 8\beta_{14} + 2\beta_{13} - \beta_{11} - 5\beta_{10} + 2\beta_{9} - 10\beta_{6} ) / 3 $ (-3b15 + 8b14 + 2b13 - b11 - 5b10 + 2b9 - 10b6) / 3
$\nu^{4}$	$=$	$ -6\beta_{8} + \beta_{7} + 14\beta_{4} + 6\beta _1 - 14 $ -6b8 + b7 + 14b4 + 6*b1 - 14
$\nu^{5}$	$=$	$ ( 4\beta_{14} - 5\beta_{13} + 21\beta_{12} + 10\beta_{11} + 50\beta_{10} + 25\beta_{9} + 25\beta_{6} ) / 3 $ (4b14 - 5b13 + 21b12 + 10b11 + 50b10 + 25b9 + 25*b6) / 3
$\nu^{6}$	$=$	$ -\beta_{5} - \beta_{4} - 9\beta_{3} + 2\beta_{2} - 33\beta _1 + 72 $ -b5 - b4 - 9b3 + 2b2 - 33*b1 + 72
$\nu^{7}$	$=$	$ ( 129 \beta_{15} - 256 \beta_{14} - 31 \beta_{13} - 129 \beta_{12} - 31 \beta_{11} - 125 \beta_{10} + \cdots + 125 \beta_{6} ) / 3 $ (129b15 - 256b14 - 31b13 - 129b12 - 31b11 - 125b10 - 127b9 + 125b6) / 3
$\nu^{8}$	$=$	$ 179\beta_{8} - 63\beta_{7} + 22\beta_{5} - 347\beta_{4} + 63\beta_{3} - 11\beta_{2} - 22 $ 179b8 - 63b7 + 22b5 - 347b4 + 63b3 - 11b2 - 22
$\nu^{9}$	$=$	$ ( - 693 \beta_{15} + 1390 \beta_{14} + 382 \beta_{13} + 66 \beta_{12} - 191 \beta_{11} + \cdots - 1262 \beta_{6} ) / 3 $ (-693b15 + 1390b14 + 382b13 + 66b12 - 191b11 - 631b10 - 128b9 - 1262b6) / 3
$\nu^{10}$	$=$	$ -969\beta_{8} + 401\beta_{7} - 85\beta_{5} + 1854\beta_{4} - 85\beta_{2} + 969\beta _1 - 1854 $ -969b8 + 401b7 - 85b5 + 1854b4 - 85b2 + 969b1 - 1854
$\nu^{11}$	$=$	$ ( - 510 \beta_{15} - 634 \beta_{14} - 1141 \beta_{13} + 3855 \beta_{12} + 2282 \beta_{11} + \cdots + 3221 \beta_{6} ) / 3 $ (-510b15 - 634b14 - 1141b13 + 3855b12 + 2282b11 + 6442b10 + 3731b9 + 3221b6) / 3
$\nu^{12}$	$=$	$ -571\beta_{5} - 571\beta_{4} - 2427\beta_{3} + 1142\beta_{2} - 5247\beta _1 + 10832 $ -571b5 - 571b4 - 2427b3 + 1142b2 - 5247*b1 + 10832
$\nu^{13}$	$=$	$ ( 24735 \beta_{15} - 36656 \beta_{14} - 6647 \beta_{13} - 24735 \beta_{12} - 6647 \beta_{11} + \cdots + 16615 \beta_{6} ) / 3 $ (24735b15 - 36656b14 - 6647b13 - 24735b12 - 6647b11 - 16615b10 - 11921b9 + 16615b6) / 3
$\nu^{14}$	$=$	$ 28429\beta_{8} - 14241\beta_{7} + 7138\beta_{5} - 47453\beta_{4} + 14241\beta_{3} - 3569\beta_{2} - 7138 $ 28429b8 - 14241b7 + 7138b5 - 47453b4 + 14241b3 - 3569b2 - 7138
$\nu^{15}$	$=$	$ ( - 117303 \beta_{15} + 225218 \beta_{14} + 76058 \beta_{13} + 21414 \beta_{12} - 38029 \beta_{11} + \cdots - 173002 \beta_{6} ) / 3 $ (-117303b15 + 225218b14 + 76058b13 + 21414b12 - 38029b11 - 86501b10 - 52216b9 - 173002b6) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$\beta_{4}$	$-1$	$-1$

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} $

95.1

−0.539169	−	0.933868i
0.192865	+	0.334053i
−1.03144	−	1.78651i
−1.16543	−	2.01859i
1.16543	+	2.01859i
1.03144	+	1.78651i
−0.192865	−	0.334053i
0.539169	+	0.933868i
−0.539169	+	0.933868i
0.192865	−	0.334053i
−1.03144	+	1.78651i
−1.16543	+	2.01859i
1.16543	−	2.01859i
1.03144	−	1.78651i
−0.192865	+	0.334053i
0.539169	−	0.933868i

−1.68071

0.418594i

−0.312371

0.541042i

0.751481

−

0.433868i

2.64956

−

1.40707i

95.2

−1.46399

−

0.925606i

2.06470

−

3.57617i

0.287429

−

0.165947i

1.28651

2.71015i

95.3

−1.31461

1.12774i

−1.21189

2.09905i

3.96035

−

2.28651i

0.456412

−

2.96508i

95.4

−0.231865

−

1.71646i

0.959555

−

1.66200i

2.63027

−

1.51859i

−2.89248

0.795973i

95.5

0.231865

1.71646i

0.959555

−

1.66200i

−2.63027

1.51859i

−2.89248

0.795973i

95.6

1.31461

−

1.12774i

−1.21189

2.09905i

−3.96035

2.28651i

0.456412

−

2.96508i

95.7

1.46399

0.925606i

2.06470

−

3.57617i

−0.287429

0.165947i

1.28651

2.71015i

95.8

1.68071

−

0.418594i

−0.312371

0.541042i

−0.751481

0.433868i

2.64956

−

1.40707i

479.1

−1.68071

−

0.418594i

−0.312371

−

0.541042i

0.751481

0.433868i

2.64956

1.40707i

479.2

−1.46399

0.925606i

2.06470

3.57617i

0.287429

0.165947i

1.28651

−

2.71015i

479.3

−1.31461

−

1.12774i

−1.21189

−

2.09905i

3.96035

2.28651i

0.456412

2.96508i

479.4

−0.231865

1.71646i

0.959555

1.66200i

2.63027

1.51859i

−2.89248

−

0.795973i

479.5

0.231865

−

1.71646i

0.959555

1.66200i

−2.63027

−

1.51859i

−2.89248

−

0.795973i

479.6

1.31461

1.12774i

−1.21189

−

2.09905i

−3.96035

−

2.28651i

0.456412

2.96508i

479.7

1.46399

−

0.925606i

2.06470

3.57617i

−0.287429

−

0.165947i

1.28651

−

2.71015i

479.8

1.68071

0.418594i

−0.312371

−

0.541042i

−0.751481

−

0.433868i

2.64956

1.40707i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
72.j	odd	6	1	inner
72.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	576.2.p.c	yes	16
3.b	odd	2	1		1728.2.p.a		16
4.b	odd	2	1	inner	576.2.p.c	yes	16
8.b	even	2	1		576.2.p.a	✓	16
8.d	odd	2	1		576.2.p.a	✓	16
9.c	even	3	1		1728.2.p.c		16
9.c	even	3	1		5184.2.f.a		16
9.d	odd	6	1		576.2.p.a	✓	16
9.d	odd	6	1		5184.2.f.f		16
12.b	even	2	1		1728.2.p.a		16
24.f	even	2	1		1728.2.p.c		16
24.h	odd	2	1		1728.2.p.c		16
36.f	odd	6	1		1728.2.p.c		16
36.f	odd	6	1		5184.2.f.a		16
36.h	even	6	1		576.2.p.a	✓	16
36.h	even	6	1		5184.2.f.f		16
72.j	odd	6	1	inner	576.2.p.c	yes	16
72.j	odd	6	1		5184.2.f.a		16
72.l	even	6	1	inner	576.2.p.c	yes	16
72.l	even	6	1		5184.2.f.a		16
72.n	even	6	1		1728.2.p.a		16
72.n	even	6	1		5184.2.f.f		16
72.p	odd	6	1		1728.2.p.a		16
72.p	odd	6	1		5184.2.f.f		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
576.2.p.a	✓	16	8.b	even	2	1
576.2.p.a	✓	16	8.d	odd	2	1
576.2.p.a	✓	16	9.d	odd	6	1
576.2.p.a	✓	16	36.h	even	6	1
576.2.p.c	yes	16	1.a	even	1	1	trivial
576.2.p.c	yes	16	4.b	odd	2	1	inner
576.2.p.c	yes	16	72.j	odd	6	1	inner
576.2.p.c	yes	16	72.l	even	6	1	inner
1728.2.p.a		16	3.b	odd	2	1
1728.2.p.a		16	12.b	even	2	1
1728.2.p.a		16	72.n	even	6	1
1728.2.p.a		16	72.p	odd	6	1
1728.2.p.c		16	9.c	even	3	1
1728.2.p.c		16	24.f	even	2	1
1728.2.p.c		16	24.h	odd	2	1
1728.2.p.c		16	36.f	odd	6	1
5184.2.f.a		16	9.c	even	3	1
5184.2.f.a		16	36.f	odd	6	1
5184.2.f.a		16	72.j	odd	6	1
5184.2.f.a		16	72.l	even	6	1
5184.2.f.f		16	9.d	odd	6	1
5184.2.f.f		16	36.h	even	6	1
5184.2.f.f		16	72.n	even	6	1
5184.2.f.f		16	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 3T_{5}^{7} + 18T_{5}^{6} - 3T_{5}^{5} + 114T_{5}^{4} - 63T_{5}^{3} + 333T_{5}^{2} + 180T_{5} + 144 $ T5^8 - 3*T5^7 + 18*T5^6 - 3*T5^5 + 114*T5^4 - 63*T5^3 + 333*T5^2 + 180*T5 + 144 acting on $S_{2}^{\mathrm{new}}(576, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} - 3 T^{14} + \cdots + 6561 $ T^16 - 3T^14 + 6T^12 + 27T^10 - 126T^8 + 243T^6 + 486T^4 - 2187*T^2 + 6561
$5$	$ (T^{8} - 3 T^{7} + \cdots + 144)^{2} $ (T^8 - 3T^7 + 18T^6 - 3T^5 + 114T^4 - 63T^3 + 333T^2 + 180*T + 144)^2
$7$	$ T^{16} - 31 T^{14} + \cdots + 256 $ T^16 - 31T^14 + 742T^12 - 6451T^10 + 42706T^8 - 36019T^6 + 25057T^4 - 2704*T^2 + 256
$11$	$ T^{16} + \cdots + 639128961 $ T^16 - 60T^14 + 2334T^12 - 54792T^10 + 942435T^8 - 10365624T^6 + 80015310T^4 - 267574104*T^2 + 639128961
$13$	$ (T^{8} - 3 T^{7} + \cdots + 20736)^{2} $ (T^8 - 3T^7 - 24T^6 + 81T^5 + 540T^4 - 1215T^3 - 3213T^2 + 6480*T + 20736)^2
$17$	$ (T^{8} + 63 T^{6} + \cdots + 11664)^{2} $ (T^8 + 63T^6 + 1188T^4 + 7776*T^2 + 11664)^2
$19$	$ (T^{8} - 75 T^{6} + \cdots + 20736)^{2} $ (T^8 - 75T^6 + 1728T^4 - 12096*T^2 + 20736)^2
$23$	$ T^{16} + \cdots + 2176782336 $ T^16 + 81T^14 + 4482T^12 + 132921T^10 + 2838726T^8 + 29321109T^6 + 217674297T^4 + 827630784*T^2 + 2176782336
$29$	$ (T^{8} - 9 T^{7} + \cdots + 11664)^{2} $ (T^8 - 9T^7 + 90T^6 - 81T^5 + 702T^4 + 1215T^3 + 7533T^2 + 8748*T + 11664)^2
$31$	$ T^{16} + \cdots + 18339659776 $ T^16 - 187T^14 + 24994T^12 - 1605643T^10 + 75084934T^8 - 1244515399T^6 + 15507830881T^4 - 17583587584*T^2 + 18339659776
$37$	$ (T^{8} + 84 T^{6} + \cdots + 20736)^{2} $ (T^8 + 84T^6 + 1728T^4 + 10800*T^2 + 20736)^2
$41$	$ (T^{8} - 126 T^{6} + \cdots + 700569)^{2} $ (T^8 - 126T^6 + 15039T^4 + 68040T^3 - 8262T^2 - 451980*T + 700569)^2
$43$	$ T^{16} + \cdots + 1275989841 $ T^16 + 72T^14 + 3510T^12 + 91368T^10 + 1716795T^8 + 19263096T^6 + 152779446T^4 + 520812180*T^2 + 1275989841
$47$	$ T^{16} + \cdots + 142657607172096 $ T^16 + 261T^14 + 44334T^12 + 4403889T^10 + 318387834T^8 + 15227300241T^6 + 529960897449T^4 + 10776523751424*T^2 + 142657607172096
$53$	$ (T^{4} + 12 T^{3} + \cdots - 384)^{4} $ (T^4 + 12T^3 + 12T^2 - 204*T - 384)^4
$59$	$ T^{16} - 144 T^{14} + \cdots + 531441 $ T^16 - 144T^14 + 16470T^12 - 600696T^10 + 17218251T^8 - 28815912T^6 + 43184502T^4 - 4960116*T^2 + 531441
$61$	$ (T^{8} + 21 T^{7} + \cdots + 82944)^{2} $ (T^8 + 21T^7 + 84T^6 - 1323T^5 - 6264T^4 + 94689T^3 + 771147T^2 + 432864*T + 82944)^2
$67$	$ T^{16} + \cdots + 5103121662081 $ T^16 + 228T^14 + 37134T^12 + 2712312T^10 + 141485859T^8 + 3970540296T^6 + 79850237886T^4 + 760707726696*T^2 + 5103121662081
$71$	$ (T^{8} - 432 T^{6} + \cdots + 9144576)^{2} $ (T^8 - 432T^6 + 52704T^4 - 1963440*T^2 + 9144576)^2
$73$	$ (T^{4} - 7 T^{3} + \cdots - 188)^{4} $ (T^4 - 7T^3 - 132T^2 - 340*T - 188)^4
$79$	$ T^{16} - 235 T^{14} + \cdots + 256 $ T^16 - 235T^14 + 47506T^12 - 1803499T^10 + 58353190T^8 - 40386007T^6 + 27260785T^4 - 83728*T^2 + 256
$83$	$ T^{16} + \cdots + 306402103296 $ T^16 - 183T^14 + 22998T^12 - 1511307T^10 + 72125586T^8 - 1940433867T^6 + 35920312353T^4 - 113072459328*T^2 + 306402103296
$89$	$ (T^{8} + 324 T^{6} + \cdots + 2985984)^{2} $ (T^8 + 324T^6 + 26784T^4 + 594864*T^2 + 2985984)^2
$97$	$ (T^{8} - 4 T^{7} + \cdots + 36481)^{2} $ (T^8 - 4T^7 + 94T^6 + 776T^5 + 5347T^4 + 16568T^3 + 38926T^2 + 44312*T + 36481)^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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{12} q^{3} + (\beta_{7} + \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{10} - \beta_{6}) q^{7} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) q - b12 * q^3 + (b7 + b4 - b3) * q^5 + (-b10 - b6) * q^7 + (-b7 + b3 + b1) * q^9	\( q - \beta_{12} q^{3} + (\beta_{7} + \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{10} - \beta_{6}) q^{7} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{9} + (\beta_{14} - \beta_{13} + \cdots - \beta_{6}) q^{11}+ \cdots + ( - 3 \beta_{15} - 3 \beta_{13} + \cdots + 3 \beta_{6}) q^{99}+O(q^{100}) \) q - b12 * q^3 + (b7 + b4 - b3) * q^5 + (-b10 - b6) * q^7 + (-b7 + b3 + b1) * q^9 + (b14 - b13 + b11 - b10 - b9 - b6) * q^11 + (-b8 + 2b1) q^13 + (-b15 - 2b14 + b13 - b12 - b11 - b10 + b9) q^15 + (2b8 - 2b4 - b1 + 1) * q^17 + (-b15 - 2b12 - b10 + 2b9 - 2b6) q^19 + (-b8 + 2b7 - b5 - b4 - b3 + b1 - 1) q^21 + (-b15 + 2b14 - 2b12 + b9) * q^23 + (b8 + b7 + b5 + 2b4 + b2 - b1 - 2) q^25 + (2b14 - b13 - b12 + 2b11 + b10 + b9 - b6) * q^27 + (b5 - 2b4 + b2 + 2) q^29 + (3b15 - 2b14 + b9 + 2b6) q^31 + (-b8 + 2b7 - 2b5 + b2 + b1 - 2) * q^33 + (2b15 + b14 - 2b12 - b11 + b10 + b9) * q^35 + (-2b8 + 2b7 + 2b4 - b3 + b1 - 1) q^37 + (-b14 - b13 - b12 + 2b11 + b10 + b9 - b6) q^39 + (b7 + b4 + b3 - 3b2 - 2) q^41 + (b15 - 2b14 - b12 - b9) q^43 + (b8 + b5 - 3b4 - 2b2 + b1 + 4) * q^45 + (-2b15 + 2b14 + b13 + 2b12 + b11 - b10 + b6) q^47 + (-2b8 - 2b7 + 2b5 + 2b4 + 2b3 - b2 - 2) q^49 + (-3b15 + 2b14 + 2b13 - b11 - 2b10 - b9 - b6) * q^51 + (b5 + b4 + b3 - 2b2 - 5) q^53 + (6b15 - b14 - 3b12 + b11 + 2b10 - b9) q^55 + (-b8 + b7 - 2b5 - 3b4 - 2b3 + b2 + 2b1 + 4) * q^57 + (2b15 - b14 + 3b13 - b12 + 2b9) q^59 + (-3b5 - 4b4 + 3b2 + 2) q^61 + (-b15 - 3b14 + 3b12 + b11 - 2b9 + 2b6) * q^63 + (2b8 + 3b5 + 4b4 - 3b2 + 2b1 - 2) q^65 + (3b15 - 2b14 - b13 + 3b12 + 2b11 + 2b10 - 2b9 + b6) * q^67 + (b8 - 2b7 + 6b4 + b3 + b1 - 3) * q^69 + (-4b15 + 6b14 - 2b13 + b12 + b11 - b10 - 4b9 - 2b6) q^71 + (-b5 - b4 - 2b3 + 2b2 - 4b1 + 5) q^73 + (-6b15 + 2b14 - b13 + b12 - 4b11 - 2b10 + b9 - b6) * q^75 + (-3b8 - b7 + 2b5 + 10b4 + b3 - b2 - 2) q^77 + (3b15 - 3b14 + 2b13 + 3b12 - 2b11) q^79 + (-b8 - b7 - 3b5 - b3 + 3b2 - b1 + 3) * q^81 + (2b15 - 2b14 - 2b13 + 2b12 + 2b11 + b10 + b6) q^83 + (2b8 - b7 - b3 + 3b2 - 4b1) q^85 + (-3b15 + b14 - 2b13 - 2b11 - b10 - 2b9 - 2b6) q^87 + (2b8 - 2b7 - 10b4 + b3 - b1 + 5) q^89 + (2b15 - 3b14 + b12 + 2b10 - b9 + 4b6) * q^91 + (5b8 - 2b7 + 2b5 - 2b4 + b3 - 2b2 - 3b1 - 7) * q^93 + (-4b15 + 2b14 + 3b13 - 2b12 - 6b11 + 4b9) * q^95 + (-2b8 + b7 + b5 + b2 + 2b1) * q^97 + (-3b15 - 3b13 + 4b12 + 3b11 - 2b9 + 3b6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{5} + 6 q^{9}+O(q^{10}) \) 16 * q + 6 * q^5 + 6 * q^9	\( 16 q + 6 q^{5} + 6 q^{9} + 6 q^{13} - 30 q^{21} - 14 q^{25} + 18 q^{29} - 48 q^{33} + 66 q^{45} + 6 q^{49} - 48 q^{53} + 18 q^{57} - 42 q^{61} + 54 q^{65} + 6 q^{69} + 28 q^{73} + 66 q^{77} - 6 q^{81} - 36 q^{85} - 102 q^{93} + 8 q^{97}+O(q^{100}) \) 16 * q + 6 * q^5 + 6 * q^9 + 6 * q^13 - 30 * q^21 - 14 * q^25 + 18 * q^29 - 48 * q^33 + 66 * q^45 + 6 * q^49 - 48 * q^53 + 18 * q^57 - 42 * q^61 + 54 * q^65 + 6 * q^69 + 28 * q^73 + 66 * q^77 - 6 * q^81 - 36 * q^85 - 102 * q^93 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	576.2.p.c

Level	$576$
Weight	$2$
Character orbit	576.p
Analytic conductor	$4.599$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 \) x^16 + 11x^14 + 85x^12 + 332x^10 + 940x^8 + 1064x^6 + 880x^4 + 128*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 678 \nu^{14} - 6789 \nu^{12} - 51076 \nu^{10} - 174698 \nu^{8} - 464941 \nu^{6} - 179896 \nu^{4} + \cdots + 1117832 ) / 392916 \) (-678v^14 - 6789v^12 - 51076v^10 - 174698v^8 - 464941v^6 - 179896v^4 - 26216*v^2 + 1117832) / 392916
\(\beta_{2}\)	\(=\)	\( ( - 2936 \nu^{14} - 36643 \nu^{12} - 297579 \nu^{10} - 1334969 \nu^{8} - 4109118 \nu^{6} + \cdots - 1460696 ) / 392916 \) (-2936v^14 - 36643v^12 - 297579v^10 - 1334969v^8 - 4109118v^6 - 6585444v^4 - 5439720*v^2 - 1460696) / 392916
\(\beta_{3}\)	\(=\)	\( ( 3399 \nu^{14} + 34180 \nu^{12} + 256058 \nu^{10} + 875809 \nu^{8} + 2248150 \nu^{6} + \cdots - 1217016 ) / 392916 \) (3399v^14 + 34180v^12 + 256058v^10 + 875809v^8 + 2248150v^6 + 901868v^4 + 131428*v^2 - 1217016) / 392916
\(\beta_{4}\)	\(=\)	\( ( - 15229 \nu^{14} - 164807 \nu^{12} - 1267309 \nu^{10} - 4851724 \nu^{8} - 13616468 \nu^{6} + \cdots - 272784 ) / 1571664 \) (-15229v^14 - 164807v^12 - 1267309v^10 - 4851724v^8 - 13616468v^6 - 14343892v^4 - 12681936*v^2 - 272784) / 1571664
\(\beta_{5}\)	\(=\)	\( ( - 41127 \nu^{14} - 462669 \nu^{12} - 3589379 \nu^{10} - 14297016 \nu^{8} - 40389328 \nu^{6} + \cdots - 1994224 ) / 1571664 \) (-41127v^14 - 462669v^12 - 3589379v^10 - 14297016v^8 - 40389328v^6 - 47060636v^4 - 32106720*v^2 - 1994224) / 1571664
\(\beta_{6}\)	\(=\)	\( ( - 28442 \nu^{15} - 276974 \nu^{13} - 2022573 \nu^{11} - 6400837 \nu^{9} - 14945299 \nu^{7} + \cdots + 16825348 \nu ) / 2357496 \) (-28442v^15 - 276974v^13 - 2022573v^11 - 6400837v^9 - 14945299v^7 + 2363604v^5 + 8242912v^3 + 16825348v) / 2357496
\(\beta_{7}\)	\(=\)	\( ( - 11161 \nu^{14} - 124073 \nu^{12} - 960853 \nu^{10} - 3803536 \nu^{8} - 10826822 \nu^{6} + \cdots - 1478952 ) / 392916 \) (-11161v^14 - 124073v^12 - 960853v^10 - 3803536v^8 - 10826822v^6 - 12871600v^4 - 10167144*v^2 - 1478952) / 392916
\(\beta_{8}\)	\(=\)	\( ( - 15229 \nu^{14} - 164807 \nu^{12} - 1267309 \nu^{10} - 4851724 \nu^{8} - 13616468 \nu^{6} + \cdots - 272784 ) / 523888 \) (-15229v^14 - 164807v^12 - 1267309v^10 - 4851724v^8 - 13616468v^6 - 14343892v^4 - 12158048*v^2 - 272784) / 523888
\(\beta_{9}\)	\(=\)	\( ( 14088 \nu^{15} + 160481 \nu^{13} + 1257754 \nu^{11} + 5136186 \nu^{9} + 15012495 \nu^{7} + \cdots + 4461244 \nu ) / 785832 \) (14088v^15 + 160481v^13 + 1257754v^11 + 5136186v^9 + 15012495v^7 + 19847572v^5 + 17571096v^3 + 4461244v) / 785832
\(\beta_{10}\)	\(=\)	\( ( 53485 \nu^{15} + 552511 \nu^{13} + 4149261 \nu^{11} + 14709020 \nu^{9} + 38382152 \nu^{7} + \cdots - 15145376 \nu ) / 2357496 \) (53485v^15 + 552511v^13 + 4149261v^11 + 14709020v^9 + 38382152v^7 + 24101568v^5 + 11410756v^3 - 15145376v) / 2357496
\(\beta_{11}\)	\(=\)	\( ( - 427 \nu^{15} - 4723 \nu^{13} - 36549 \nu^{11} - 143882 \nu^{9} - 409832 \nu^{7} + \cdots - 112384 \nu ) / 17208 \) (-427v^15 - 4723v^13 - 36549v^11 - 143882v^9 - 409832v^7 - 483588v^5 - 417712v^3 - 112384v) / 17208
\(\beta_{12}\)	\(=\)	\( ( - 26391 \nu^{15} - 285268 \nu^{13} - 2184580 \nu^{11} - 8306259 \nu^{9} - 22932244 \nu^{7} + \cdots + 1749840 \nu ) / 785832 \) (-26391v^15 - 285268v^13 - 2184580v^11 - 8306259v^9 - 22932244v^7 - 22719052v^5 - 16082232v^3 + 1749840v) / 785832
\(\beta_{13}\)	\(=\)	\( ( 82961 \nu^{15} + 892865 \nu^{13} + 6850017 \nu^{11} + 26014876 \nu^{9} + 72425530 \nu^{7} + \cdots - 5684968 \nu ) / 2357496 \) (82961v^15 + 892865v^13 + 6850017v^11 + 26014876v^9 + 72425530v^7 + 72742140v^5 + 58172408v^3 - 5684968v) / 2357496
\(\beta_{14}\)	\(=\)	\( ( - 29781 \nu^{15} - 319213 \nu^{13} - 2439960 \nu^{11} - 9179749 \nu^{9} - 25256949 \nu^{7} + \cdots + 1445260 \nu ) / 785832 \) (-29781v^15 - 319213v^13 - 2439960v^11 - 9179749v^9 - 25256949v^7 - 24011448v^5 - 18177892v^3 + 1445260v) / 785832
\(\beta_{15}\)	\(=\)	\( ( - 10800 \nu^{15} - 118285 \nu^{13} - 911829 \nu^{11} - 3535889 \nu^{9} - 9931987 \nu^{7} + \cdots - 751284 \nu ) / 196458 \) (-10800v^15 - 118285v^13 - 911829v^11 - 3535889v^9 - 9931987v^7 - 10822149v^5 - 8439635v^3 - 751284v) / 196458

\(\nu\)	\(=\)	\( ( -4\beta_{14} - \beta_{13} - \beta_{11} - 2\beta_{10} - 4\beta_{9} + 2\beta_{6} ) / 6 \) (-4b14 - b13 - b11 - 2b10 - 4b9 + 2b6) / 6
\(\nu^{2}\)	\(=\)	\( \beta_{8} - 3\beta_{4} \) b8 - 3*b4
\(\nu^{3}\)	\(=\)	\( ( -3\beta_{15} + 8\beta_{14} + 2\beta_{13} - \beta_{11} - 5\beta_{10} + 2\beta_{9} - 10\beta_{6} ) / 3 \) (-3b15 + 8b14 + 2b13 - b11 - 5b10 + 2b9 - 10b6) / 3
\(\nu^{4}\)	\(=\)	\( -6\beta_{8} + \beta_{7} + 14\beta_{4} + 6\beta _1 - 14 \) -6b8 + b7 + 14b4 + 6*b1 - 14
\(\nu^{5}\)	\(=\)	\( ( 4\beta_{14} - 5\beta_{13} + 21\beta_{12} + 10\beta_{11} + 50\beta_{10} + 25\beta_{9} + 25\beta_{6} ) / 3 \) (4b14 - 5b13 + 21b12 + 10b11 + 50b10 + 25b9 + 25*b6) / 3
\(\nu^{6}\)	\(=\)	\( -\beta_{5} - \beta_{4} - 9\beta_{3} + 2\beta_{2} - 33\beta _1 + 72 \) -b5 - b4 - 9b3 + 2b2 - 33*b1 + 72
\(\nu^{7}\)	\(=\)	\( ( 129 \beta_{15} - 256 \beta_{14} - 31 \beta_{13} - 129 \beta_{12} - 31 \beta_{11} - 125 \beta_{10} + \cdots + 125 \beta_{6} ) / 3 \) (129b15 - 256b14 - 31b13 - 129b12 - 31b11 - 125b10 - 127b9 + 125b6) / 3
\(\nu^{8}\)	\(=\)	\( 179\beta_{8} - 63\beta_{7} + 22\beta_{5} - 347\beta_{4} + 63\beta_{3} - 11\beta_{2} - 22 \) 179b8 - 63b7 + 22b5 - 347b4 + 63b3 - 11b2 - 22
\(\nu^{9}\)	\(=\)	\( ( - 693 \beta_{15} + 1390 \beta_{14} + 382 \beta_{13} + 66 \beta_{12} - 191 \beta_{11} + \cdots - 1262 \beta_{6} ) / 3 \) (-693b15 + 1390b14 + 382b13 + 66b12 - 191b11 - 631b10 - 128b9 - 1262b6) / 3
\(\nu^{10}\)	\(=\)	\( -969\beta_{8} + 401\beta_{7} - 85\beta_{5} + 1854\beta_{4} - 85\beta_{2} + 969\beta _1 - 1854 \) -969b8 + 401b7 - 85b5 + 1854b4 - 85b2 + 969b1 - 1854
\(\nu^{11}\)	\(=\)	\( ( - 510 \beta_{15} - 634 \beta_{14} - 1141 \beta_{13} + 3855 \beta_{12} + 2282 \beta_{11} + \cdots + 3221 \beta_{6} ) / 3 \) (-510b15 - 634b14 - 1141b13 + 3855b12 + 2282b11 + 6442b10 + 3731b9 + 3221b6) / 3
\(\nu^{12}\)	\(=\)	\( -571\beta_{5} - 571\beta_{4} - 2427\beta_{3} + 1142\beta_{2} - 5247\beta _1 + 10832 \) -571b5 - 571b4 - 2427b3 + 1142b2 - 5247*b1 + 10832
\(\nu^{13}\)	\(=\)	\( ( 24735 \beta_{15} - 36656 \beta_{14} - 6647 \beta_{13} - 24735 \beta_{12} - 6647 \beta_{11} + \cdots + 16615 \beta_{6} ) / 3 \) (24735b15 - 36656b14 - 6647b13 - 24735b12 - 6647b11 - 16615b10 - 11921b9 + 16615b6) / 3
\(\nu^{14}\)	\(=\)	\( 28429\beta_{8} - 14241\beta_{7} + 7138\beta_{5} - 47453\beta_{4} + 14241\beta_{3} - 3569\beta_{2} - 7138 \) 28429b8 - 14241b7 + 7138b5 - 47453b4 + 14241b3 - 3569b2 - 7138
\(\nu^{15}\)	\(=\)	\( ( - 117303 \beta_{15} + 225218 \beta_{14} + 76058 \beta_{13} + 21414 \beta_{12} - 38029 \beta_{11} + \cdots - 173002 \beta_{6} ) / 3 \) (-117303b15 + 225218b14 + 76058b13 + 21414b12 - 38029b11 - 86501b10 - 52216b9 - 173002b6) / 3

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(\beta_{4}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 95.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} - 3 T^{14} + \cdots + 6561 \) T^16 - 3T^14 + 6T^12 + 27T^10 - 126T^8 + 243T^6 + 486T^4 - 2187*T^2 + 6561
$5$	\( (T^{8} - 3 T^{7} + \cdots + 144)^{2} \) (T^8 - 3T^7 + 18T^6 - 3T^5 + 114T^4 - 63T^3 + 333T^2 + 180*T + 144)^2
$7$	\( T^{16} - 31 T^{14} + \cdots + 256 \) T^16 - 31T^14 + 742T^12 - 6451T^10 + 42706T^8 - 36019T^6 + 25057T^4 - 2704*T^2 + 256
$11$	\( T^{16} + \cdots + 639128961 \) T^16 - 60T^14 + 2334T^12 - 54792T^10 + 942435T^8 - 10365624T^6 + 80015310T^4 - 267574104*T^2 + 639128961
$13$	\( (T^{8} - 3 T^{7} + \cdots + 20736)^{2} \) (T^8 - 3T^7 - 24T^6 + 81T^5 + 540T^4 - 1215T^3 - 3213T^2 + 6480*T + 20736)^2
$17$	\( (T^{8} + 63 T^{6} + \cdots + 11664)^{2} \) (T^8 + 63T^6 + 1188T^4 + 7776*T^2 + 11664)^2
$19$	\( (T^{8} - 75 T^{6} + \cdots + 20736)^{2} \) (T^8 - 75T^6 + 1728T^4 - 12096*T^2 + 20736)^2
$23$	\( T^{16} + \cdots + 2176782336 \) T^16 + 81T^14 + 4482T^12 + 132921T^10 + 2838726T^8 + 29321109T^6 + 217674297T^4 + 827630784*T^2 + 2176782336
$29$	\( (T^{8} - 9 T^{7} + \cdots + 11664)^{2} \) (T^8 - 9T^7 + 90T^6 - 81T^5 + 702T^4 + 1215T^3 + 7533T^2 + 8748*T + 11664)^2
$31$	\( T^{16} + \cdots + 18339659776 \) T^16 - 187T^14 + 24994T^12 - 1605643T^10 + 75084934T^8 - 1244515399T^6 + 15507830881T^4 - 17583587584*T^2 + 18339659776
$37$	\( (T^{8} + 84 T^{6} + \cdots + 20736)^{2} \) (T^8 + 84T^6 + 1728T^4 + 10800*T^2 + 20736)^2
$41$	\( (T^{8} - 126 T^{6} + \cdots + 700569)^{2} \) (T^8 - 126T^6 + 15039T^4 + 68040T^3 - 8262T^2 - 451980*T + 700569)^2
$43$	\( T^{16} + \cdots + 1275989841 \) T^16 + 72T^14 + 3510T^12 + 91368T^10 + 1716795T^8 + 19263096T^6 + 152779446T^4 + 520812180*T^2 + 1275989841
$47$	\( T^{16} + \cdots + 142657607172096 \) T^16 + 261T^14 + 44334T^12 + 4403889T^10 + 318387834T^8 + 15227300241T^6 + 529960897449T^4 + 10776523751424*T^2 + 142657607172096
$53$	\( (T^{4} + 12 T^{3} + \cdots - 384)^{4} \) (T^4 + 12T^3 + 12T^2 - 204*T - 384)^4
$59$	\( T^{16} - 144 T^{14} + \cdots + 531441 \) T^16 - 144T^14 + 16470T^12 - 600696T^10 + 17218251T^8 - 28815912T^6 + 43184502T^4 - 4960116*T^2 + 531441
$61$	\( (T^{8} + 21 T^{7} + \cdots + 82944)^{2} \) (T^8 + 21T^7 + 84T^6 - 1323T^5 - 6264T^4 + 94689T^3 + 771147T^2 + 432864*T + 82944)^2
$67$	\( T^{16} + \cdots + 5103121662081 \) T^16 + 228T^14 + 37134T^12 + 2712312T^10 + 141485859T^8 + 3970540296T^6 + 79850237886T^4 + 760707726696*T^2 + 5103121662081
$71$	\( (T^{8} - 432 T^{6} + \cdots + 9144576)^{2} \) (T^8 - 432T^6 + 52704T^4 - 1963440*T^2 + 9144576)^2
$73$	\( (T^{4} - 7 T^{3} + \cdots - 188)^{4} \) (T^4 - 7T^3 - 132T^2 - 340*T - 188)^4
$79$	\( T^{16} - 235 T^{14} + \cdots + 256 \) T^16 - 235T^14 + 47506T^12 - 1803499T^10 + 58353190T^8 - 40386007T^6 + 27260785T^4 - 83728*T^2 + 256
$83$	\( T^{16} + \cdots + 306402103296 \) T^16 - 183T^14 + 22998T^12 - 1511307T^10 + 72125586T^8 - 1940433867T^6 + 35920312353T^4 - 113072459328*T^2 + 306402103296
$89$	\( (T^{8} + 324 T^{6} + \cdots + 2985984)^{2} \) (T^8 + 324T^6 + 26784T^4 + 594864*T^2 + 2985984)^2
$97$	\( (T^{8} - 4 T^{7} + \cdots + 36481)^{2} \) (T^8 - 4T^7 + 94T^6 + 776T^5 + 5347T^4 + 16568T^3 + 38926T^2 + 44312*T + 36481)^2
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