LMFDB - Newform orbit 3168.2.f.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3168,2,Mod(1585,3168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3168.1585"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	20.0.74831334220841134637329678336.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2x^{18} + 5x^{16} - 8x^{14} + 28x^{12} - 64x^{10} + 112x^{8} - 128x^{6} + 320x^{4} - 512x^{2} + 1024 $ x^20 - 2x^18 + 5x^16 - 8x^14 + 28x^12 - 64x^10 + 112x^8 - 128x^6 + 320x^4 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{29} $
Twist minimal:	no (minimal twist has level 792)
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_{3} + \beta_1) q^{5} + \beta_{14} q^{7} - \beta_1 q^{11} + \beta_{15} q^{13} - \beta_{19} q^{17} + ( - \beta_{5} + \beta_{4}) q^{19} - \beta_{10} q^{23} + ( - \beta_{13} - 2 \beta_{2} - 1) q^{25}+ \cdots + (\beta_{14} + \beta_{11} + 3 \beta_{2} + 1) q^{97}+O(q^{100}) $ q + (b3 + b1) * q^5 + b14 * q^7 - b1 * q^11 + b15 * q^13 - b19 * q^17 + (-b5 + b4) * q^19 - b10 * q^23 + (-b13 - 2b2 - 1) q^25 + (-b8 - 2b1) q^29 + (-b13 - 2) * q^31 + (b17 - 2b9) q^35 + (b6 - b4) * q^37 + (-b18 - b16 + b12 + b10) * q^41 + (b15 + b7 - b5 + b4) * q^43 + (-b19 + b18 + 2b16 + 2b12 - b10) * q^47 + (-b14 - 2b13 - b11 + b2 + 2) q^49 + (b17 + b9 - b8 + 2b3 + 2b1) * q^53 + (b2 + 1) * q^55 + (-b17 + b9 + b8 - b3 - b1) * q^59 + (-2b15 + b7 - 2b6 - 2b5) q^61 + (b19 + b18 - b16 - 3b12 + b10) q^65 + (-b15 - b7 - b6 - b5 - b4) * q^67 + (-b19 - b18 + b12) * q^71 + (-b14 - b13 + b11 - b2 - 3) * q^73 + b9 * q^77 + (-b13 - b11 + b2 + 1) * q^79 + (-b17 - 2b9 + 2b8) * q^83 + (-b15 - b7 - 2b5) q^85 + (2b18 - b12 - b10) q^89 + (-b15 - b7 + b6 - b5 - b4) * q^91 + (2b19 - 2b12) * q^95 + (b14 + b11 + 3b2 + 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{25} - 40 q^{31} + 36 q^{49} + 16 q^{55} - 56 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) $ 20 * q - 12 * q^25 - 40 * q^31 + 36 * q^49 + 16 * q^55 - 56 * q^73 + 16 * q^79 + 8 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 2x^{18} + 5x^{16} - 8x^{14} + 28x^{12} - 64x^{10} + 112x^{8} - 128x^{6} + 320x^{4} - 512x^{2} + 1024 $ x^20 - 2*x^18 + 5*x^16 - 8*x^14 + 28*x^12 - 64*x^10 + 112*x^8 - 128*x^6 + 320*x^4 - 512*x^2 + 1024 :

$\beta_{1}$	$=$	$ ( -\nu^{17} - 5\nu^{13} + 14\nu^{11} - 32\nu^{9} + 16\nu^{7} - 80\nu^{5} + 96\nu^{3} - 384\nu ) / 512 $ (-v^17 - 5v^13 + 14v^11 - 32v^9 + 16v^7 - 80v^5 + 96v^3 - 384*v) / 512
$\beta_{2}$	$=$	$ ( \nu^{18} + 2\nu^{16} - 3\nu^{14} + 12\nu^{12} - 4\nu^{10} + 48\nu^{8} - 144\nu^{6} + 64\nu^{4} + 64\nu^{2} + 256 ) / 512 $ (v^18 + 2v^16 - 3v^14 + 12v^12 - 4v^10 + 48v^8 - 144v^6 + 64v^4 + 64v^2 + 256) / 512
$\beta_{3}$	$=$	$ ( \nu^{19} - 4 \nu^{17} - 3 \nu^{15} + 14 \nu^{13} + 16 \nu^{11} + 16 \nu^{9} - 48 \nu^{7} + \cdots - 512 \nu ) / 1024 $ (v^19 - 4v^17 - 3v^15 + 14v^13 + 16v^11 + 16v^9 - 48v^7 + 224v^5 + 128v^3 - 512*v) / 1024
$\beta_{4}$	$=$	$ ( \nu^{16} - \nu^{14} + 3\nu^{12} + \nu^{10} + 12\nu^{8} + 48\nu^{4} + 48\nu^{2} + 64 ) / 64 $ (v^16 - v^14 + 3v^12 + v^10 + 12v^8 + 48v^4 + 48v^2 + 64) / 64
$\beta_{5}$	$=$	$ ( \nu^{18} + 2\nu^{16} + 5\nu^{14} - 4\nu^{12} + 4\nu^{10} + 48\nu^{8} + 48\nu^{6} + 64\nu^{4} + 192\nu^{2} + 768 ) / 256 $ (v^18 + 2v^16 + 5v^14 - 4v^12 + 4v^10 + 48v^8 + 48v^6 + 64v^4 + 192v^2 + 768) / 256
$\beta_{6}$	$=$	$ ( -\nu^{18} - 2\nu^{16} - \nu^{14} + 12\nu^{12} + 32\nu^{8} - 48\nu^{6} + 192\nu^{4} - 512\nu^{2} ) / 256 $ (-v^18 - 2v^16 - v^14 + 12v^12 + 32v^8 - 48v^6 + 192v^4 - 512v^2) / 256
$\beta_{7}$	$=$	$ ( \nu^{18} + 6\nu^{16} - 3\nu^{14} - 60\nu^{10} + 144\nu^{8} - 272\nu^{6} + 512\nu^{4} - 1088\nu^{2} + 2304 ) / 512 $ (v^18 + 6v^16 - 3v^14 - 60v^10 + 144v^8 - 272v^6 + 512v^4 - 1088*v^2 + 2304) / 512
$\beta_{8}$	$=$	$ ( -\nu^{19} + 3\nu^{15} - 2\nu^{13} + 40\nu^{11} + 16\nu^{9} + 176\nu^{7} - 32\nu^{5} + 256\nu^{3} ) / 512 $ (-v^19 + 3v^15 - 2v^13 + 40v^11 + 16v^9 + 176v^7 - 32v^5 + 256*v^3) / 512
$\beta_{9}$	$=$	$ ( -\nu^{19} - 6\nu^{17} + 3\nu^{15} - 4\nu^{11} - 16\nu^{9} + 208\nu^{7} - 256\nu^{5} - 704\nu^{3} - 256\nu ) / 1024 $ (-v^19 - 6v^17 + 3v^15 - 4v^11 - 16v^9 + 208v^7 - 256v^5 - 704v^3 - 256v) / 1024
$\beta_{10}$	$=$	$ ( - 3 \nu^{19} + 2 \nu^{17} - 23 \nu^{15} + 36 \nu^{13} - 68 \nu^{11} - 48 \nu^{9} + 48 \nu^{7} + \cdots - 2304 \nu ) / 1024 $ (-3v^19 + 2v^17 - 23v^15 + 36v^13 - 68v^11 - 48v^9 + 48v^7 + 320v^5 + 64v^3 - 2304v) / 1024
$\beta_{11}$	$=$	$ ( -\nu^{18} - 9\nu^{14} + 6\nu^{12} - 36\nu^{10} + 64\nu^{8} - 80\nu^{6} + 224\nu^{4} - 64\nu^{2} + 256 ) / 256 $ (-v^18 - 9v^14 + 6v^12 - 36v^10 + 64v^8 - 80v^6 + 224v^4 - 64*v^2 + 256) / 256
$\beta_{12}$	$=$	$ ( 3 \nu^{19} - 2 \nu^{17} - 9 \nu^{15} - 36 \nu^{13} + 36 \nu^{11} - 16 \nu^{9} + 80 \nu^{7} + \cdots + 256 \nu ) / 1024 $ (3v^19 - 2v^17 - 9v^15 - 36v^13 + 36v^11 - 16v^9 + 80v^7 - 64v^5 + 960v^3 + 256v) / 1024
$\beta_{13}$	$=$	$ ( \nu^{18} - 2 \nu^{16} + 5 \nu^{14} - 24 \nu^{12} + 60 \nu^{10} - 80 \nu^{8} + 112 \nu^{6} - 256 \nu^{4} + \cdots - 768 ) / 256 $ (v^18 - 2v^16 + 5v^14 - 24v^12 + 60v^10 - 80v^8 + 112v^6 - 256v^4 + 576v^2 - 768) / 256
$\beta_{14}$	$=$	$ ( 3\nu^{18} - 6\nu^{16} - \nu^{14} - 8\nu^{12} + 36\nu^{10} - 80\nu^{8} + 144\nu^{6} + 192\nu^{2} - 768 ) / 512 $ (3v^18 - 6v^16 - v^14 - 8v^12 + 36v^10 - 80v^8 + 144v^6 + 192*v^2 - 768) / 512
$\beta_{15}$	$=$	$ ( 3 \nu^{18} - 6 \nu^{16} + 7 \nu^{14} - 24 \nu^{12} + 44 \nu^{10} - 208 \nu^{8} + 80 \nu^{6} + \cdots - 1280 ) / 512 $ (3v^18 - 6v^16 + 7v^14 - 24v^12 + 44v^10 - 208v^8 + 80v^6 - 128v^4 + 832*v^2 - 1280) / 512
$\beta_{16}$	$=$	$ ( \nu^{19} - 2 \nu^{17} + 5 \nu^{15} - 8 \nu^{13} + 28 \nu^{11} - 64 \nu^{9} + 112 \nu^{7} + \cdots - 1024 \nu ) / 256 $ (v^19 - 2v^17 + 5v^15 - 8v^13 + 28v^11 - 64v^9 + 112v^7 - 128v^5 + 320v^3 - 1024*v) / 256
$\beta_{17}$	$=$	$ ( \nu^{19} - 2\nu^{17} + 5\nu^{15} - 8\nu^{13} + 28\nu^{11} - 64\nu^{9} + 112\nu^{7} - 128\nu^{5} + 320\nu^{3} ) / 256 $ (v^19 - 2v^17 + 5v^15 - 8v^13 + 28v^11 - 64v^9 + 112v^7 - 128v^5 + 320v^3) / 256
$\beta_{18}$	$=$	$ ( - 3 \nu^{19} + 2 \nu^{17} - 7 \nu^{15} + 4 \nu^{13} - 52 \nu^{11} + 80 \nu^{9} - 80 \nu^{7} + \cdots + 768 \nu ) / 512 $ (-3v^19 + 2v^17 - 7v^15 + 4v^13 - 52v^11 + 80v^9 - 80v^7 - 64v^5 + 64v^3 + 768v) / 512
$\beta_{19}$	$=$	$ ( \nu^{19} - 4 \nu^{17} + 5 \nu^{15} - 18 \nu^{13} + 24 \nu^{11} - 64 \nu^{9} + 112 \nu^{7} + \cdots - 512 \nu ) / 256 $ (v^19 - 4v^17 + 5v^15 - 18v^13 + 24v^11 - 64v^9 + 112v^7 - 32v^5 + 256v^3 - 512*v) / 256

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

$\nu$	$=$	$ ( \beta_{17} - \beta_{16} ) / 4 $ (b17 - b16) / 4
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{14} + \beta_{11} - \beta_{7} + \beta_{5} + \beta_{2} + 1 ) / 4 $ (b15 - b14 + b11 - b7 + b5 + b2 + 1) / 4
$\nu^{3}$	$=$	$ ( 2\beta_{18} + \beta_{17} + \beta_{16} - 2\beta_{9} + 2\beta_{3} - 2\beta_1 ) / 4 $ (2b18 + b17 + b16 - 2b9 + 2b3 - 2b1) / 4
$\nu^{4}$	$=$	$ ( 3\beta_{15} - \beta_{14} + \beta_{11} + \beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} - 3\beta_{2} - 3 ) / 4 $ (3b15 - b14 + b11 + b7 + 2b6 + b5 + 2b4 - 3b2 - 3) / 4
$\nu^{5}$	$=$	$ ( 4 \beta_{19} - 2 \beta_{18} - \beta_{17} - 3 \beta_{16} - 2 \beta_{12} + 2 \beta_{10} - 4 \beta_{9} + \cdots - 4 \beta_1 ) / 4 $ (4b19 - 2b18 - b17 - 3b16 - 2b12 + 2b10 - 4b9 + 2b8 - 4b1) / 4
$\nu^{6}$	$=$	$ ( - 3 \beta_{15} + 5 \beta_{14} - 2 \beta_{13} - \beta_{11} - \beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots - 1 ) / 4 $ (-3b15 + 5b14 - 2b13 - b11 - b7 - 2b6 + b5 + 2b4 - 9b2 - 1) / 4
$\nu^{7}$	$=$	$ ( -2\beta_{18} + \beta_{17} + \beta_{16} + 4\beta_{12} + 4\beta_{10} + 6\beta_{9} + 4\beta_{8} - 6\beta_{3} - 18\beta_1 ) / 4 $ (-2b18 + b17 + b16 + 4b12 + 4b10 + 6b9 + 4b8 - 6b3 - 18*b1) / 4
$\nu^{8}$	$=$	$ ( - 9 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} + 5 \beta_{11} - 3 \beta_{7} + 2 \beta_{6} + 9 \beta_{5} + \cdots - 23 ) / 4 $ (-9b15 + 3b14 + 4b13 + 5b11 - 3b7 + 2b6 + 9b5 - 6b4 + 9*b2 - 23) / 4
$\nu^{9}$	$=$	$ ( - 4 \beta_{19} + 6 \beta_{18} - 13 \beta_{17} + 13 \beta_{16} + 10 \beta_{12} - 10 \beta_{10} + \cdots - 24 \beta_1 ) / 4 $ (-4b19 + 6b18 - 13b17 + 13b16 + 10b12 - 10b10 + 8b9 + 6b8 + 12b3 - 24b1) / 4
$\nu^{10}$	$=$	$ ( - 7 \beta_{15} + \beta_{14} + 18 \beta_{13} - 5 \beta_{11} + 3 \beta_{7} + 14 \beta_{6} - 7 \beta_{5} + \cdots + 35 ) / 4 $ (-7b15 + b14 + 18b13 - 5b11 + 3b7 + 14b6 - 7b5 + 10b4 + 11b2 + 35) / 4
$\nu^{11}$	$=$	$ ( - 8 \beta_{19} - 18 \beta_{18} - 3 \beta_{17} - 15 \beta_{16} - 8 \beta_{10} - 6 \beta_{9} + \cdots + 82 \beta_1 ) / 4 $ (-8b19 - 18b18 - 3b17 - 15b16 - 8b10 - 6b9 + 24b8 + 14b3 + 82*b1) / 4
$\nu^{12}$	$=$	$ ( 3 \beta_{15} + 7 \beta_{14} - 32 \beta_{13} - 23 \beta_{11} - 31 \beta_{7} + 10 \beta_{6} + \beta_{5} + \cdots + 53 ) / 4 $ (3b15 + 7b14 - 32b13 - 23b11 - 31b7 + 10b6 + b5 + 10b4 + 37b2 + 53) / 4
$\nu^{13}$	$=$	$ ( - 44 \beta_{19} + 6 \beta_{18} + 47 \beta_{17} + 13 \beta_{16} - 26 \beta_{12} + 10 \beta_{10} + \cdots - 36 \beta_1 ) / 4 $ (-44b19 + 6b18 + 47b17 + 13b16 - 26b12 + 10b10 + 12b9 - 6b8 + 80b3 - 36b1) / 4
$\nu^{14}$	$=$	$ ( 69 \beta_{15} - 91 \beta_{14} - 34 \beta_{13} - 33 \beta_{11} - 25 \beta_{7} + 54 \beta_{6} + 97 \beta_{5} + \cdots - 177 ) / 4 $ (69b15 - 91b14 - 34b13 - 33b11 - 25b7 + 54b6 + 97b5 - 38b4 + 7*b2 - 177) / 4
$\nu^{15}$	$=$	$ ( 48 \beta_{19} + 46 \beta_{18} - 7 \beta_{17} + 65 \beta_{16} - 148 \beta_{12} - 68 \beta_{10} + \cdots - 202 \beta_1 ) / 4 $ (48b19 + 46b18 - 7b17 + 65b16 - 148b12 - 68b10 - 82b9 - 4b8 + 2b3 - 202b1) / 4
$\nu^{16}$	$=$	$ ( - 17 \beta_{15} - 53 \beta_{14} - 4 \beta_{13} - 115 \beta_{11} + 101 \beta_{7} - 110 \beta_{6} + \cdots - 255 ) / 4 $ (-17b15 - 53b14 - 4b13 - 115b11 + 101b7 - 110b6 - 103b5 + 154b4 - 127*b2 - 255) / 4
$\nu^{17}$	$=$	$ ( - 84 \beta_{19} - 154 \beta_{18} - 53 \beta_{17} + 45 \beta_{16} + 34 \beta_{12} + 62 \beta_{10} + \cdots - 112 \beta_1 ) / 4 $ (-84b19 - 154b18 - 53b17 + 45b16 + 34b12 + 62b10 - 176b9 + 78b8 - 492b3 - 112b1) / 4
$\nu^{18}$	$=$	$ ( - 79 \beta_{15} + 457 \beta_{14} - 118 \beta_{13} - 125 \beta_{11} + 107 \beta_{7} - 194 \beta_{6} + \cdots - 453 ) / 4 $ (-79b15 + 457b14 - 118b13 - 125b11 + 107b7 - 194b6 + 41b5 - 54b4 + 323*b2 - 453) / 4
$\nu^{19}$	$=$	$ ( - 280 \beta_{19} - 274 \beta_{18} + 21 \beta_{17} + 305 \beta_{16} + 536 \beta_{12} - 64 \beta_{10} + \cdots - 1190 \beta_1 ) / 4 $ (-280b19 - 274b18 + 21b17 + 305b16 + 536b12 - 64b10 + 290b9 - 352b8 + 54b3 - 1190b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$1$	$-1$	$1$

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} $

1585.1

1.13849	−	0.838954i
−1.13849	−	0.838954i
−1.25295	+	0.655834i
1.25295	+	0.655834i
0.484785	+	1.32853i
−0.484785	+	1.32853i
−1.38682	−	0.277013i
1.38682	−	0.277013i
−0.689696	+	1.23463i
0.689696	+	1.23463i
0.689696	−	1.23463i
−0.689696	−	1.23463i
1.38682	+	0.277013i
−1.38682	+	0.277013i
−0.484785	−	1.32853i
0.484785	−	1.32853i
1.25295	−	0.655834i
−1.25295	−	0.655834i
−1.13849	+	0.838954i
1.13849	+	0.838954i

−

3.89064i

1.77496

1585.2

−

3.89064i

1.77496

1585.3

−

2.54164i

−4.84329

1585.4

−

2.54164i

−4.84329

1585.5

−

2.21156i

−1.26179

1585.6

−

2.21156i

−1.26179

1585.7

−

0.972802i

0.372730

1585.8

−

0.972802i

0.372730

1585.9

−

0.752075i

3.95740

1585.10

−

0.752075i

3.95740

1585.11

0.752075i

3.95740

1585.12

0.752075i

3.95740

1585.13

0.972802i

0.372730

1585.14

0.972802i

0.372730

1585.15

2.21156i

−1.26179

1585.16

2.21156i

−1.26179

1585.17

2.54164i

−4.84329

1585.18

2.54164i

−4.84329

1585.19

3.89064i

1.77496

1585.20

3.89064i

1.77496

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3168.2.f.h		20
3.b	odd	2	1	inner	3168.2.f.h		20
4.b	odd	2	1		792.2.f.h	✓	20
8.b	even	2	1	inner	3168.2.f.h		20
8.d	odd	2	1		792.2.f.h	✓	20
12.b	even	2	1		792.2.f.h	✓	20
24.f	even	2	1		792.2.f.h	✓	20
24.h	odd	2	1	inner	3168.2.f.h		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
792.2.f.h	✓	20	4.b	odd	2	1
792.2.f.h	✓	20	8.d	odd	2	1
792.2.f.h	✓	20	12.b	even	2	1
792.2.f.h	✓	20	24.f	even	2	1
3168.2.f.h		20	1.a	even	1	1	trivial
3168.2.f.h		20	3.b	odd	2	1	inner
3168.2.f.h		20	8.b	even	2	1	inner
3168.2.f.h		20	24.h	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 $S_{2}^{\mathrm{new}}(3168, [\chi])$:

$ T_{5}^{10} + 28T_{5}^{8} + 244T_{5}^{6} + 800T_{5}^{4} + 832T_{5}^{2} + 256 $

T5^10 + 28*T5^8 + 244*T5^6 + 800*T5^4 + 832*T5^2 + 256

$ T_{7}^{5} - 22T_{7}^{3} + 16T_{7}^{2} + 40T_{7} - 16 $

T7^5 - 22*T7^3 + 16*T7^2 + 40*T7 - 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ T^{20} $ T^20
$5$	$ (T^{10} + 28 T^{8} + \cdots + 256)^{2} $ (T^10 + 28T^8 + 244T^6 + 800T^4 + 832T^2 + 256)^2
$7$	$ (T^{5} - 22 T^{3} + \cdots - 16)^{4} $ (T^5 - 22T^3 + 16T^2 + 40*T - 16)^4
$11$	$ (T^{2} + 1)^{10} $ (T^2 + 1)^10
$13$	$ (T^{10} + 76 T^{8} + \cdots + 1792)^{2} $ (T^10 + 76T^8 + 1940T^6 + 17152T^4 + 13120T^2 + 1792)^2
$17$	$ (T^{10} - 116 T^{8} + \cdots - 1835008)^{2} $ (T^10 - 116T^8 + 4704T^6 - 82880T^4 + 647168T^2 - 1835008)^2
$19$	$ (T^{10} + 108 T^{8} + \cdots + 351232)^{2} $ (T^10 + 108T^8 + 3856T^6 + 52224T^4 + 240640T^2 + 351232)^2
$23$	$ (T^{10} - 140 T^{8} + \cdots - 2453248)^{2} $ (T^10 - 140T^8 + 6548T^6 - 120704T^4 + 929856T^2 - 2453248)^2
$29$	$ (T^{10} + 132 T^{8} + \cdots + 1478656)^{2} $ (T^10 + 132T^8 + 6048T^6 + 112448T^4 + 775168T^2 + 1478656)^2
$31$	$ (T^{5} + 10 T^{4} + \cdots + 64)^{4} $ (T^5 + 10T^4 + 4T^3 - 120T^2 - 32T + 64)^4
$37$	$ (T^{10} + 224 T^{8} + \cdots + 1835008)^{2} $ (T^10 + 224T^8 + 14912T^6 + 271360T^4 + 1622016T^2 + 1835008)^2
$41$	$ (T^{10} - 188 T^{8} + \cdots - 6028288)^{2} $ (T^10 - 188T^8 + 10896T^6 - 248576T^4 + 2137088T^2 - 6028288)^2
$43$	$ (T^{10} + 188 T^{8} + \cdots + 6028288)^{2} $ (T^10 + 188T^8 + 10896T^6 + 248576T^4 + 2137088T^2 + 6028288)^2
$47$	$ (T^{10} - 348 T^{8} + \cdots - 1507072)^{2} $ (T^10 - 348T^8 + 41748T^6 - 1992896T^4 + 32381248T^2 - 1507072)^2
$53$	$ (T^{10} + 220 T^{8} + \cdots + 33362176)^{2} $ (T^10 + 220T^8 + 16820T^6 + 540448T^4 + 7349824T^2 + 33362176)^2
$59$	$ (T^{10} + 208 T^{8} + \cdots + 4194304)^{2} $ (T^10 + 208T^8 + 14848T^6 + 413696T^4 + 3407872T^2 + 4194304)^2
$61$	$ (T^{10} + 428 T^{8} + \cdots + 143519488)^{2} $ (T^10 + 428T^8 + 60500T^6 + 3149312T^4 + 44513088T^2 + 143519488)^2
$67$	$ (T^{10} + 408 T^{8} + \cdots + 1101463552)^{2} $ (T^10 + 408T^8 + 61456T^6 + 4200960T^4 + 124383232T^2 + 1101463552)^2
$71$	$ (T^{10} - 156 T^{8} + \cdots - 646912)^{2} $ (T^10 - 156T^8 + 8404T^6 - 173376T^4 + 916288T^2 - 646912)^2
$73$	$ (T^{5} + 14 T^{4} + \cdots + 224)^{4} $ (T^5 + 14T^4 - 40T^3 - 1008T^2 - 2160T + 224)^4
$79$	$ (T^{5} - 4 T^{4} + \cdots - 304)^{4} $ (T^5 - 4T^4 - 86T^3 + 312T^2 + 1000T - 304)^4
$83$	$ (T^{10} + 472 T^{8} + \cdots + 65536)^{2} $ (T^10 + 472T^8 + 57488T^6 + 453888T^4 + 712704T^2 + 65536)^2
$89$	$ (T^{10} - 416 T^{8} + \cdots - 7340032)^{2} $ (T^10 - 416T^8 + 52736T^6 - 2068480T^4 + 11993088T^2 - 7340032)^2
$97$	$ (T^{5} - 2 T^{4} + \cdots - 4864)^{4} $ (T^5 - 2T^4 - 196T^3 + 968T^2 + 384T - 4864)^4
show more
show less

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{3} + \beta_1) q^{5} + \beta_{14} q^{7} - \beta_1 q^{11} + \beta_{15} q^{13} - \beta_{19} q^{17} + ( - \beta_{5} + \beta_{4}) q^{19} - \beta_{10} q^{23} + ( - \beta_{13} - 2 \beta_{2} - 1) q^{25}+ \cdots + (\beta_{14} + \beta_{11} + 3 \beta_{2} + 1) q^{97}+O(q^{100}) \) q + (b3 + b1) * q^5 + b14 * q^7 - b1 * q^11 + b15 * q^13 - b19 * q^17 + (-b5 + b4) * q^19 - b10 * q^23 + (-b13 - 2b2 - 1) q^25 + (-b8 - 2b1) q^29 + (-b13 - 2) * q^31 + (b17 - 2b9) q^35 + (b6 - b4) * q^37 + (-b18 - b16 + b12 + b10) * q^41 + (b15 + b7 - b5 + b4) * q^43 + (-b19 + b18 + 2b16 + 2b12 - b10) * q^47 + (-b14 - 2b13 - b11 + b2 + 2) q^49 + (b17 + b9 - b8 + 2b3 + 2b1) * q^53 + (b2 + 1) * q^55 + (-b17 + b9 + b8 - b3 - b1) * q^59 + (-2b15 + b7 - 2b6 - 2b5) q^61 + (b19 + b18 - b16 - 3b12 + b10) q^65 + (-b15 - b7 - b6 - b5 - b4) * q^67 + (-b19 - b18 + b12) * q^71 + (-b14 - b13 + b11 - b2 - 3) * q^73 + b9 * q^77 + (-b13 - b11 + b2 + 1) * q^79 + (-b17 - 2b9 + 2b8) * q^83 + (-b15 - b7 - 2b5) q^85 + (2b18 - b12 - b10) q^89 + (-b15 - b7 + b6 - b5 - b4) * q^91 + (2b19 - 2b12) * q^95 + (b14 + b11 + 3b2 + 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{25} - 40 q^{31} + 36 q^{49} + 16 q^{55} - 56 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) \) 20 * q - 12 * q^25 - 40 * q^31 + 36 * q^49 + 16 * q^55 - 56 * q^73 + 16 * q^79 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	3168.2.f.h

Level	$3168$
Weight	$2$
Character orbit	3168.f
Analytic conductor	$25.297$
Analytic rank	$0$
Dimension	$20$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(25.2966073603\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	20.0.74831334220841134637329678336.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 2x^{18} + 5x^{16} - 8x^{14} + 28x^{12} - 64x^{10} + 112x^{8} - 128x^{6} + 320x^{4} - 512x^{2} + 1024 \) x^20 - 2x^18 + 5x^16 - 8x^14 + 28x^12 - 64x^10 + 112x^8 - 128x^6 + 320x^4 - 512*x^2 + 1024
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{29} \)
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{17} - 5\nu^{13} + 14\nu^{11} - 32\nu^{9} + 16\nu^{7} - 80\nu^{5} + 96\nu^{3} - 384\nu ) / 512 \) (-v^17 - 5v^13 + 14v^11 - 32v^9 + 16v^7 - 80v^5 + 96v^3 - 384*v) / 512
\(\beta_{2}\)	\(=\)	\( ( \nu^{18} + 2\nu^{16} - 3\nu^{14} + 12\nu^{12} - 4\nu^{10} + 48\nu^{8} - 144\nu^{6} + 64\nu^{4} + 64\nu^{2} + 256 ) / 512 \) (v^18 + 2v^16 - 3v^14 + 12v^12 - 4v^10 + 48v^8 - 144v^6 + 64v^4 + 64v^2 + 256) / 512
\(\beta_{3}\)	\(=\)	\( ( \nu^{19} - 4 \nu^{17} - 3 \nu^{15} + 14 \nu^{13} + 16 \nu^{11} + 16 \nu^{9} - 48 \nu^{7} + \cdots - 512 \nu ) / 1024 \) (v^19 - 4v^17 - 3v^15 + 14v^13 + 16v^11 + 16v^9 - 48v^7 + 224v^5 + 128v^3 - 512*v) / 1024
\(\beta_{4}\)	\(=\)	\( ( \nu^{16} - \nu^{14} + 3\nu^{12} + \nu^{10} + 12\nu^{8} + 48\nu^{4} + 48\nu^{2} + 64 ) / 64 \) (v^16 - v^14 + 3v^12 + v^10 + 12v^8 + 48v^4 + 48v^2 + 64) / 64
\(\beta_{5}\)	\(=\)	\( ( \nu^{18} + 2\nu^{16} + 5\nu^{14} - 4\nu^{12} + 4\nu^{10} + 48\nu^{8} + 48\nu^{6} + 64\nu^{4} + 192\nu^{2} + 768 ) / 256 \) (v^18 + 2v^16 + 5v^14 - 4v^12 + 4v^10 + 48v^8 + 48v^6 + 64v^4 + 192v^2 + 768) / 256
\(\beta_{6}\)	\(=\)	\( ( -\nu^{18} - 2\nu^{16} - \nu^{14} + 12\nu^{12} + 32\nu^{8} - 48\nu^{6} + 192\nu^{4} - 512\nu^{2} ) / 256 \) (-v^18 - 2v^16 - v^14 + 12v^12 + 32v^8 - 48v^6 + 192v^4 - 512v^2) / 256
\(\beta_{7}\)	\(=\)	\( ( \nu^{18} + 6\nu^{16} - 3\nu^{14} - 60\nu^{10} + 144\nu^{8} - 272\nu^{6} + 512\nu^{4} - 1088\nu^{2} + 2304 ) / 512 \) (v^18 + 6v^16 - 3v^14 - 60v^10 + 144v^8 - 272v^6 + 512v^4 - 1088*v^2 + 2304) / 512
\(\beta_{8}\)	\(=\)	\( ( -\nu^{19} + 3\nu^{15} - 2\nu^{13} + 40\nu^{11} + 16\nu^{9} + 176\nu^{7} - 32\nu^{5} + 256\nu^{3} ) / 512 \) (-v^19 + 3v^15 - 2v^13 + 40v^11 + 16v^9 + 176v^7 - 32v^5 + 256*v^3) / 512
\(\beta_{9}\)	\(=\)	\( ( -\nu^{19} - 6\nu^{17} + 3\nu^{15} - 4\nu^{11} - 16\nu^{9} + 208\nu^{7} - 256\nu^{5} - 704\nu^{3} - 256\nu ) / 1024 \) (-v^19 - 6v^17 + 3v^15 - 4v^11 - 16v^9 + 208v^7 - 256v^5 - 704v^3 - 256v) / 1024
\(\beta_{10}\)	\(=\)	\( ( - 3 \nu^{19} + 2 \nu^{17} - 23 \nu^{15} + 36 \nu^{13} - 68 \nu^{11} - 48 \nu^{9} + 48 \nu^{7} + \cdots - 2304 \nu ) / 1024 \) (-3v^19 + 2v^17 - 23v^15 + 36v^13 - 68v^11 - 48v^9 + 48v^7 + 320v^5 + 64v^3 - 2304v) / 1024
\(\beta_{11}\)	\(=\)	\( ( -\nu^{18} - 9\nu^{14} + 6\nu^{12} - 36\nu^{10} + 64\nu^{8} - 80\nu^{6} + 224\nu^{4} - 64\nu^{2} + 256 ) / 256 \) (-v^18 - 9v^14 + 6v^12 - 36v^10 + 64v^8 - 80v^6 + 224v^4 - 64*v^2 + 256) / 256
\(\beta_{12}\)	\(=\)	\( ( 3 \nu^{19} - 2 \nu^{17} - 9 \nu^{15} - 36 \nu^{13} + 36 \nu^{11} - 16 \nu^{9} + 80 \nu^{7} + \cdots + 256 \nu ) / 1024 \) (3v^19 - 2v^17 - 9v^15 - 36v^13 + 36v^11 - 16v^9 + 80v^7 - 64v^5 + 960v^3 + 256v) / 1024
\(\beta_{13}\)	\(=\)	\( ( \nu^{18} - 2 \nu^{16} + 5 \nu^{14} - 24 \nu^{12} + 60 \nu^{10} - 80 \nu^{8} + 112 \nu^{6} - 256 \nu^{4} + \cdots - 768 ) / 256 \) (v^18 - 2v^16 + 5v^14 - 24v^12 + 60v^10 - 80v^8 + 112v^6 - 256v^4 + 576v^2 - 768) / 256
\(\beta_{14}\)	\(=\)	\( ( 3\nu^{18} - 6\nu^{16} - \nu^{14} - 8\nu^{12} + 36\nu^{10} - 80\nu^{8} + 144\nu^{6} + 192\nu^{2} - 768 ) / 512 \) (3v^18 - 6v^16 - v^14 - 8v^12 + 36v^10 - 80v^8 + 144v^6 + 192*v^2 - 768) / 512
\(\beta_{15}\)	\(=\)	\( ( 3 \nu^{18} - 6 \nu^{16} + 7 \nu^{14} - 24 \nu^{12} + 44 \nu^{10} - 208 \nu^{8} + 80 \nu^{6} + \cdots - 1280 ) / 512 \) (3v^18 - 6v^16 + 7v^14 - 24v^12 + 44v^10 - 208v^8 + 80v^6 - 128v^4 + 832*v^2 - 1280) / 512
\(\beta_{16}\)	\(=\)	\( ( \nu^{19} - 2 \nu^{17} + 5 \nu^{15} - 8 \nu^{13} + 28 \nu^{11} - 64 \nu^{9} + 112 \nu^{7} + \cdots - 1024 \nu ) / 256 \) (v^19 - 2v^17 + 5v^15 - 8v^13 + 28v^11 - 64v^9 + 112v^7 - 128v^5 + 320v^3 - 1024*v) / 256
\(\beta_{17}\)	\(=\)	\( ( \nu^{19} - 2\nu^{17} + 5\nu^{15} - 8\nu^{13} + 28\nu^{11} - 64\nu^{9} + 112\nu^{7} - 128\nu^{5} + 320\nu^{3} ) / 256 \) (v^19 - 2v^17 + 5v^15 - 8v^13 + 28v^11 - 64v^9 + 112v^7 - 128v^5 + 320v^3) / 256
\(\beta_{18}\)	\(=\)	\( ( - 3 \nu^{19} + 2 \nu^{17} - 7 \nu^{15} + 4 \nu^{13} - 52 \nu^{11} + 80 \nu^{9} - 80 \nu^{7} + \cdots + 768 \nu ) / 512 \) (-3v^19 + 2v^17 - 7v^15 + 4v^13 - 52v^11 + 80v^9 - 80v^7 - 64v^5 + 64v^3 + 768v) / 512
\(\beta_{19}\)	\(=\)	\( ( \nu^{19} - 4 \nu^{17} + 5 \nu^{15} - 18 \nu^{13} + 24 \nu^{11} - 64 \nu^{9} + 112 \nu^{7} + \cdots - 512 \nu ) / 256 \) (v^19 - 4v^17 + 5v^15 - 18v^13 + 24v^11 - 64v^9 + 112v^7 - 32v^5 + 256v^3 - 512*v) / 256

\(\nu\)	\(=\)	\( ( \beta_{17} - \beta_{16} ) / 4 \) (b17 - b16) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{14} + \beta_{11} - \beta_{7} + \beta_{5} + \beta_{2} + 1 ) / 4 \) (b15 - b14 + b11 - b7 + b5 + b2 + 1) / 4
\(\nu^{3}\)	\(=\)	\( ( 2\beta_{18} + \beta_{17} + \beta_{16} - 2\beta_{9} + 2\beta_{3} - 2\beta_1 ) / 4 \) (2b18 + b17 + b16 - 2b9 + 2b3 - 2b1) / 4
\(\nu^{4}\)	\(=\)	\( ( 3\beta_{15} - \beta_{14} + \beta_{11} + \beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} - 3\beta_{2} - 3 ) / 4 \) (3b15 - b14 + b11 + b7 + 2b6 + b5 + 2b4 - 3b2 - 3) / 4
\(\nu^{5}\)	\(=\)	\( ( 4 \beta_{19} - 2 \beta_{18} - \beta_{17} - 3 \beta_{16} - 2 \beta_{12} + 2 \beta_{10} - 4 \beta_{9} + \cdots - 4 \beta_1 ) / 4 \) (4b19 - 2b18 - b17 - 3b16 - 2b12 + 2b10 - 4b9 + 2b8 - 4b1) / 4
\(\nu^{6}\)	\(=\)	\( ( - 3 \beta_{15} + 5 \beta_{14} - 2 \beta_{13} - \beta_{11} - \beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots - 1 ) / 4 \) (-3b15 + 5b14 - 2b13 - b11 - b7 - 2b6 + b5 + 2b4 - 9b2 - 1) / 4
\(\nu^{7}\)	\(=\)	\( ( -2\beta_{18} + \beta_{17} + \beta_{16} + 4\beta_{12} + 4\beta_{10} + 6\beta_{9} + 4\beta_{8} - 6\beta_{3} - 18\beta_1 ) / 4 \) (-2b18 + b17 + b16 + 4b12 + 4b10 + 6b9 + 4b8 - 6b3 - 18*b1) / 4
\(\nu^{8}\)	\(=\)	\( ( - 9 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} + 5 \beta_{11} - 3 \beta_{7} + 2 \beta_{6} + 9 \beta_{5} + \cdots - 23 ) / 4 \) (-9b15 + 3b14 + 4b13 + 5b11 - 3b7 + 2b6 + 9b5 - 6b4 + 9*b2 - 23) / 4
\(\nu^{9}\)	\(=\)	\( ( - 4 \beta_{19} + 6 \beta_{18} - 13 \beta_{17} + 13 \beta_{16} + 10 \beta_{12} - 10 \beta_{10} + \cdots - 24 \beta_1 ) / 4 \) (-4b19 + 6b18 - 13b17 + 13b16 + 10b12 - 10b10 + 8b9 + 6b8 + 12b3 - 24b1) / 4
\(\nu^{10}\)	\(=\)	\( ( - 7 \beta_{15} + \beta_{14} + 18 \beta_{13} - 5 \beta_{11} + 3 \beta_{7} + 14 \beta_{6} - 7 \beta_{5} + \cdots + 35 ) / 4 \) (-7b15 + b14 + 18b13 - 5b11 + 3b7 + 14b6 - 7b5 + 10b4 + 11b2 + 35) / 4
\(\nu^{11}\)	\(=\)	\( ( - 8 \beta_{19} - 18 \beta_{18} - 3 \beta_{17} - 15 \beta_{16} - 8 \beta_{10} - 6 \beta_{9} + \cdots + 82 \beta_1 ) / 4 \) (-8b19 - 18b18 - 3b17 - 15b16 - 8b10 - 6b9 + 24b8 + 14b3 + 82*b1) / 4
\(\nu^{12}\)	\(=\)	\( ( 3 \beta_{15} + 7 \beta_{14} - 32 \beta_{13} - 23 \beta_{11} - 31 \beta_{7} + 10 \beta_{6} + \beta_{5} + \cdots + 53 ) / 4 \) (3b15 + 7b14 - 32b13 - 23b11 - 31b7 + 10b6 + b5 + 10b4 + 37b2 + 53) / 4
\(\nu^{13}\)	\(=\)	\( ( - 44 \beta_{19} + 6 \beta_{18} + 47 \beta_{17} + 13 \beta_{16} - 26 \beta_{12} + 10 \beta_{10} + \cdots - 36 \beta_1 ) / 4 \) (-44b19 + 6b18 + 47b17 + 13b16 - 26b12 + 10b10 + 12b9 - 6b8 + 80b3 - 36b1) / 4
\(\nu^{14}\)	\(=\)	\( ( 69 \beta_{15} - 91 \beta_{14} - 34 \beta_{13} - 33 \beta_{11} - 25 \beta_{7} + 54 \beta_{6} + 97 \beta_{5} + \cdots - 177 ) / 4 \) (69b15 - 91b14 - 34b13 - 33b11 - 25b7 + 54b6 + 97b5 - 38b4 + 7*b2 - 177) / 4
\(\nu^{15}\)	\(=\)	\( ( 48 \beta_{19} + 46 \beta_{18} - 7 \beta_{17} + 65 \beta_{16} - 148 \beta_{12} - 68 \beta_{10} + \cdots - 202 \beta_1 ) / 4 \) (48b19 + 46b18 - 7b17 + 65b16 - 148b12 - 68b10 - 82b9 - 4b8 + 2b3 - 202b1) / 4
\(\nu^{16}\)	\(=\)	\( ( - 17 \beta_{15} - 53 \beta_{14} - 4 \beta_{13} - 115 \beta_{11} + 101 \beta_{7} - 110 \beta_{6} + \cdots - 255 ) / 4 \) (-17b15 - 53b14 - 4b13 - 115b11 + 101b7 - 110b6 - 103b5 + 154b4 - 127*b2 - 255) / 4
\(\nu^{17}\)	\(=\)	\( ( - 84 \beta_{19} - 154 \beta_{18} - 53 \beta_{17} + 45 \beta_{16} + 34 \beta_{12} + 62 \beta_{10} + \cdots - 112 \beta_1 ) / 4 \) (-84b19 - 154b18 - 53b17 + 45b16 + 34b12 + 62b10 - 176b9 + 78b8 - 492b3 - 112b1) / 4
\(\nu^{18}\)	\(=\)	\( ( - 79 \beta_{15} + 457 \beta_{14} - 118 \beta_{13} - 125 \beta_{11} + 107 \beta_{7} - 194 \beta_{6} + \cdots - 453 ) / 4 \) (-79b15 + 457b14 - 118b13 - 125b11 + 107b7 - 194b6 + 41b5 - 54b4 + 323*b2 - 453) / 4
\(\nu^{19}\)	\(=\)	\( ( - 280 \beta_{19} - 274 \beta_{18} + 21 \beta_{17} + 305 \beta_{16} + 536 \beta_{12} - 64 \beta_{10} + \cdots - 1190 \beta_1 ) / 4 \) (-280b19 - 274b18 + 21b17 + 305b16 + 536b12 - 64b10 + 290b9 - 352b8 + 54b3 - 1190b1) / 4

\(n\)	\(353\)	\(991\)	\(1189\)	\(1729\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1585.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( T^{20} \) T^20
$5$	\( (T^{10} + 28 T^{8} + \cdots + 256)^{2} \) (T^10 + 28T^8 + 244T^6 + 800T^4 + 832T^2 + 256)^2
$7$	\( (T^{5} - 22 T^{3} + \cdots - 16)^{4} \) (T^5 - 22T^3 + 16T^2 + 40*T - 16)^4
$11$	\( (T^{2} + 1)^{10} \) (T^2 + 1)^10
$13$	\( (T^{10} + 76 T^{8} + \cdots + 1792)^{2} \) (T^10 + 76T^8 + 1940T^6 + 17152T^4 + 13120T^2 + 1792)^2
$17$	\( (T^{10} - 116 T^{8} + \cdots - 1835008)^{2} \) (T^10 - 116T^8 + 4704T^6 - 82880T^4 + 647168T^2 - 1835008)^2
$19$	\( (T^{10} + 108 T^{8} + \cdots + 351232)^{2} \) (T^10 + 108T^8 + 3856T^6 + 52224T^4 + 240640T^2 + 351232)^2
$23$	\( (T^{10} - 140 T^{8} + \cdots - 2453248)^{2} \) (T^10 - 140T^8 + 6548T^6 - 120704T^4 + 929856T^2 - 2453248)^2
$29$	\( (T^{10} + 132 T^{8} + \cdots + 1478656)^{2} \) (T^10 + 132T^8 + 6048T^6 + 112448T^4 + 775168T^2 + 1478656)^2
$31$	\( (T^{5} + 10 T^{4} + \cdots + 64)^{4} \) (T^5 + 10T^4 + 4T^3 - 120T^2 - 32T + 64)^4
$37$	\( (T^{10} + 224 T^{8} + \cdots + 1835008)^{2} \) (T^10 + 224T^8 + 14912T^6 + 271360T^4 + 1622016T^2 + 1835008)^2
$41$	\( (T^{10} - 188 T^{8} + \cdots - 6028288)^{2} \) (T^10 - 188T^8 + 10896T^6 - 248576T^4 + 2137088T^2 - 6028288)^2
$43$	\( (T^{10} + 188 T^{8} + \cdots + 6028288)^{2} \) (T^10 + 188T^8 + 10896T^6 + 248576T^4 + 2137088T^2 + 6028288)^2
$47$	\( (T^{10} - 348 T^{8} + \cdots - 1507072)^{2} \) (T^10 - 348T^8 + 41748T^6 - 1992896T^4 + 32381248T^2 - 1507072)^2
$53$	\( (T^{10} + 220 T^{8} + \cdots + 33362176)^{2} \) (T^10 + 220T^8 + 16820T^6 + 540448T^4 + 7349824T^2 + 33362176)^2
$59$	\( (T^{10} + 208 T^{8} + \cdots + 4194304)^{2} \) (T^10 + 208T^8 + 14848T^6 + 413696T^4 + 3407872T^2 + 4194304)^2
$61$	\( (T^{10} + 428 T^{8} + \cdots + 143519488)^{2} \) (T^10 + 428T^8 + 60500T^6 + 3149312T^4 + 44513088T^2 + 143519488)^2
$67$	\( (T^{10} + 408 T^{8} + \cdots + 1101463552)^{2} \) (T^10 + 408T^8 + 61456T^6 + 4200960T^4 + 124383232T^2 + 1101463552)^2
$71$	\( (T^{10} - 156 T^{8} + \cdots - 646912)^{2} \) (T^10 - 156T^8 + 8404T^6 - 173376T^4 + 916288T^2 - 646912)^2
$73$	\( (T^{5} + 14 T^{4} + \cdots + 224)^{4} \) (T^5 + 14T^4 - 40T^3 - 1008T^2 - 2160T + 224)^4
$79$	\( (T^{5} - 4 T^{4} + \cdots - 304)^{4} \) (T^5 - 4T^4 - 86T^3 + 312T^2 + 1000T - 304)^4
$83$	\( (T^{10} + 472 T^{8} + \cdots + 65536)^{2} \) (T^10 + 472T^8 + 57488T^6 + 453888T^4 + 712704T^2 + 65536)^2
$89$	\( (T^{10} - 416 T^{8} + \cdots - 7340032)^{2} \) (T^10 - 416T^8 + 52736T^6 - 2068480T^4 + 11993088T^2 - 7340032)^2
$97$	\( (T^{5} - 2 T^{4} + \cdots - 4864)^{4} \) (T^5 - 2T^4 - 196T^3 + 968T^2 + 384T - 4864)^4
show more
show less