LMFDB - Newform orbit 847.2.l.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(118,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("847.118");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.l (of order $10$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.76332905120$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
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} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 $ x^16 + 2x^14 - 16x^12 - 72x^10 + 26x^8 + 360x^6 + 725x^4 + 1000*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-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_{15} - \beta_{10} + \beta_{8}) q^{2} + (\beta_{7} - \beta_{2}) q^{3} + ( - 2 \beta_{4} + 2) q^{4} + ( - \beta_{6} - \beta_{2}) q^{5} + (\beta_{15} + 2 \beta_{13} + \cdots - 2 \beta_1) q^{6}+ \cdots + ( - 2 \beta_{4} - 2 \beta_{3}) q^{9}+O(q^{10}) $ q + (b15 - b10 + b8) * q^2 + (b7 - b2) * q^3 + (-2b4 + 2) q^4 + (-b6 - b2) * q^5 + (b15 + 2b13 + 2b12 - b10 - 2b1) q^6 + (b15 - b14 - b13 + b12 - b11 + b10 + b8 + b1) * q^7 + (2b15 - 2b14 - 2b10 + 2b8) * q^8 + (-2b4 - 2b3) * q^9	$ q + (\beta_{15} - \beta_{10} + \beta_{8}) q^{2} + (\beta_{7} - \beta_{2}) q^{3} + ( - 2 \beta_{4} + 2) q^{4} + ( - \beta_{6} - \beta_{2}) q^{5} + (\beta_{15} + 2 \beta_{13} + \cdots - 2 \beta_1) q^{6}+ \cdots + ( - 3 \beta_{15} + 3 \beta_{14} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) $ q + (b15 - b10 + b8) * q^2 + (b7 - b2) * q^3 + (-2b4 + 2) q^4 + (-b6 - b2) * q^5 + (b15 + 2b13 + 2b12 - b10 - 2b1) q^6 + (b15 - b14 - b13 + b12 - b11 + b10 + b8 + b1) * q^7 + (2b15 - 2b14 - 2b10 + 2b8) * q^8 + (-2b4 - 2b3) * q^9 + (b15 - b14 + 2b13 - 2b11 - 2b10 + b8 - 2b1) * q^10 + (2b9 + 2b7) * q^12 + (4b15 - 4b14 - 4b13 - 4b11 - 2b10 + 2b8) * q^13 + (-b9 + b6 + 3b5 - b3 - 3) q^14 + (-3b5 - 5b4 - 3b3 + 5) q^15 + (-2b15 + b14 - 2b13 - 2b12 + 2b10 + 4b1) q^17 + (-2b14 + 2b10 + 2b8) q^18 + (b14 + 2b12 + b10 + 2b1) * q^19 + (2b7 - 2b6 - 4b2) q^20 + (-2b15 + 2b14 - b13 + 3b11 + 3b10 - 3b8 + 3b1) * q^21 + (4b5 + 4b4 - 3) * q^23 + (2b15 + 4b12) * q^24 + (-2b5 - 2b3 + 2) * q^25 + (-4b9 - 4b2) * q^26 + (2b9 - 2b7 - b6 + b2) * q^27 + (-4b15 + 4b10 + 2b1) q^28 + (-3b14 - 3b10 + 3b8) q^29 + (7b15 - 2b14 - 2b10 + 7b8) * q^30 + (b7 - b6 + b2) * q^31 - 4b8 q^32 + (-6b9 - 2b7) * q^34 + (-b15 + b14 + b13 + 3b12 + b11) q^35 + (4b5 - 4b3 - 4) * q^36 + (-3b5 - 3b4 - 3b3 + 3) q^37 + (2b6 + 2b2) * q^38 + (6b15 - 2b14 - 6b10) q^39 + (4b13 - 2b10 - 4b1) q^40 + (b14 + 2b12 + b10 + 2b1) * q^41 + (-4b7 + 4b6 + b5 + 3b4 + 3b3 + 7b2) q^42 + (b15 - b14 + 7b8) q^43 + (2b9 + 2b7) * q^45 + (-7b15 + 3b10 - 3b8) q^46 + (-2b9 + 4b7 + 2b6 - 4b2) * q^47 + (b9 - b7 - 2b6 + 3b4 + 5b3 - b2 - 5) q^49 + 2b14 q^50 + (8b14 - b10 - 8b8) * q^51 + (4b15 - 4b14 - 8b11 - 4b10 + 4b8) q^52 + 2b5 q^53 + (-3b15 + 3b14 + 2b13 + 2b11 + 2b10 - b8 + 2b1) * q^54 + (-2b9 + 2b7 + 2b5 + 2b4 - 6) * q^56 + (-2b15 + 3b10 - 3b8) q^57 - 6b3 q^58 + (b9 + 4b6 + b2) q^59 + (-14b4 - 10b3 + 10) * q^60 + (b15 - b14 - b10 - 2b1) q^61 + (2b15 - b14 - 2b13 + 2b12 - 2b11 + b8 + 2b1) q^62 + (-4b15 - 2b12 - 4b8 - 2b1) * q^63 + (8b4 + 8b3) * q^64 + (-6b15 + 6b14 - 4b8) q^65 + (-8b5 - 8b4 + 1) * q^67 + (-2b15 - 4b14 - 4b13 - 12b12 - 4b11 - 2b10 + 2b8) q^68 + (-4b9 - 3b7 + 4b6 + 3b2) * q^69 + (4b9 + 3b6 - b5 + 2b4 - b3 + 4b2 - 2) * q^70 - b4 * q^71 + (-4b15 + 4b10) * q^72 + (-2b15 + b14 + 4b13 - 2b12 + 2b11 - b10 - b8 - 4b1) q^73 + (3b15 + 3b8) * q^74 - 2b2 q^75 + (-4b15 + 4b14 + 4b11 + 4b10 - 2b8 + 4b1) * q^76 + (-8b5 - 8b4 + 20) * q^78 - 3b15 q^79 + (-5b5 - 7b4 - 5b3 + 7) q^81 + (2b6 + 2b2) * q^82 + (-b15 + 4b14 + 6b13 + 6b12 + b10 + 2b1) * q^83 + (-6b15 + 6b14 - 8b13 - 6b12 + 6b11 + 8b10 - 6b8 + 8b1) * q^84 + (-9b15 + b14 + b10 - 9b8) * q^85 + (2b5 - 14b4 - 14b3) q^86 + (3b15 - 3b14 - 6b13) q^87 + (b9 - 8b7) q^89 + (4b14 + 4b13 + 8b12 + 4b11 + 2b10 - 2b8) * q^90 + (4b9 + 2b7 - 4b6 + 8b5 + 10b3 - 2b2 - 8) * q^91 + (8b5 + 14b4 + 8b3 - 14) q^92 + (-b4 + 4b3 - 4) q^93 + (4b15 - 2b14 + 4b13 + 4b12 - 4b10 - 8b1) * q^94 + (-3b14 - b10 + 3b8) * q^95 + (4b15 - 4b14 - 8b11 - 4b10 + 4b8) q^96 + (b7 - b6 + b2) * q^97 + (-3b15 + 3b14 + 4b13 - 2b11 - 2b10 - 7b8 - 2b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{4} - 16 q^{9}+O(q^{10}) $ 16 * q + 24 * q^4 - 16 * q^9	$ 16 q + 24 q^{4} - 16 q^{9} - 40 q^{14} + 36 q^{15} - 16 q^{23} + 16 q^{25} - 64 q^{36} + 12 q^{37} + 28 q^{42} - 48 q^{49} + 8 q^{53} - 80 q^{56} - 24 q^{58} + 64 q^{60} + 64 q^{64} - 48 q^{67} - 32 q^{70} - 4 q^{71} + 256 q^{78} + 44 q^{81} - 104 q^{86} - 56 q^{91} - 104 q^{92} - 52 q^{93}+O(q^{100}) $ 16 * q + 24 * q^4 - 16 * q^9 - 40 * q^14 + 36 * q^15 - 16 * q^23 + 16 * q^25 - 64 * q^36 + 12 * q^37 + 28 * q^42 - 48 * q^49 + 8 * q^53 - 80 * q^56 - 24 * q^58 + 64 * q^60 + 64 * q^64 - 48 * q^67 - 32 * q^70 - 4 * q^71 + 256 * q^78 + 44 * q^81 - 104 * q^86 - 56 * q^91 - 104 * q^92 - 52 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 $ x^16 + 2*x^14 - 16*x^12 - 72*x^10 + 26*x^8 + 360*x^6 + 725*x^4 + 1000*x^2 + 625 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 155 \nu^{14} - 1408 \nu^{12} - 4611 \nu^{10} + 9188 \nu^{8} + 65146 \nu^{6} - 26953 \nu^{4} + \cdots + 108500 ) / 176275 $ (155v^14 - 1408v^12 - 4611v^10 + 9188v^8 + 65146v^6 - 26953v^4 + 145025*v^2 + 108500) / 176275
$\beta_{3}$	$=$	$ ( 5996 \nu^{14} + 4807 \nu^{12} - 95306 \nu^{10} - 277252 \nu^{8} + 500966 \nu^{6} + 1108500 \nu^{4} + \cdots + 2787000 ) / 881375 $ (5996v^14 + 4807v^12 - 95306v^10 - 277252v^8 + 500966v^6 + 1108500v^4 + 1748000*v^2 + 2787000) / 881375
$\beta_{4}$	$=$	$ ( 8474 \nu^{14} + 2743 \nu^{12} - 149594 \nu^{10} - 389798 \nu^{8} + 955934 \nu^{6} + 2009635 \nu^{4} + \cdots + 3175500 ) / 881375 $ (8474v^14 + 2743v^12 - 149594v^10 - 389798v^8 + 955934v^6 + 2009635v^4 + 3037000*v^2 + 3175500) / 881375
$\beta_{5}$	$=$	$ ( - 8672 \nu^{14} - 3649 \nu^{12} + 153342 \nu^{10} + 402539 \nu^{8} - 938462 \nu^{6} + \cdots - 3730750 ) / 881375 $ (-8672v^14 - 3649v^12 + 153342v^10 + 402539v^8 - 938462v^6 - 1990750v^4 - 3034300*v^2 - 3730750) / 881375
$\beta_{6}$	$=$	$ ( - 12916 \nu^{14} - 10637 \nu^{12} + 225196 \nu^{10} + 690882 \nu^{8} - 1135131 \nu^{6} + \cdots - 6749625 ) / 881375 $ (-12916v^14 - 10637v^12 + 225196v^10 + 690882v^8 - 1135131v^6 - 3696115v^4 - 6351875*v^2 - 6749625) / 881375
$\beta_{7}$	$=$	$ ( - 14403 \nu^{14} - 14916 \nu^{12} + 224078 \nu^{10} + 748351 \nu^{8} - 1023533 \nu^{6} + \cdots - 6732875 ) / 881375 $ (-14403v^14 - 14916v^12 + 224078v^10 + 748351v^8 - 1023533v^6 - 3087765v^4 - 6177175*v^2 - 6732875) / 881375
$\beta_{8}$	$=$	$ ( - 1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} + \cdots - 598500 \nu ) / 400625 $ (-1496v^15 - 2572v^13 + 24401v^11 + 91992v^9 - 100886v^7 - 478015v^5 - 412275v^3 - 598500v) / 400625
$\beta_{9}$	$=$	$ ( - 22098 \nu^{14} - 13706 \nu^{12} + 377023 \nu^{10} + 1116041 \nu^{8} - 1983428 \nu^{6} + \cdots - 11238000 ) / 881375 $ (-22098v^14 - 13706v^12 + 377023v^10 + 1116041v^8 - 1983428v^6 - 5540615v^4 - 10624800*v^2 - 11238000) / 881375
$\beta_{10}$	$=$	$ ( 22194 \nu^{15} + 11523 \nu^{13} - 339409 \nu^{11} - 1024903 \nu^{9} + 1669899 \nu^{7} + \cdots + 9814875 \nu ) / 4406875 $ (22194v^15 + 11523v^13 - 339409v^11 - 1024903v^9 + 1669899v^7 + 3769000v^5 + 11127550v^3 + 9814875v) / 4406875
$\beta_{11}$	$=$	$ ( 25903 \nu^{15} - 3944 \nu^{13} - 422373 \nu^{11} - 826866 \nu^{9} + 3248903 \nu^{7} + \cdots + 4286500 \nu ) / 4406875 $ (25903v^15 - 3944v^13 - 422373v^11 - 826866v^9 + 3248903v^7 + 2357930v^5 + 4320250v^3 + 4286500v) / 4406875
$\beta_{12}$	$=$	$ ( 8474 \nu^{15} + 2743 \nu^{13} - 149594 \nu^{11} - 389798 \nu^{9} + 955934 \nu^{7} + \cdots + 2294125 \nu ) / 881375 $ (8474v^15 + 2743v^13 - 149594v^11 - 389798v^9 + 955934v^7 + 2009635v^5 + 3037000v^3 + 2294125v) / 881375
$\beta_{13}$	$=$	$ ( - 8672 \nu^{15} - 3649 \nu^{13} + 153342 \nu^{11} + 402539 \nu^{9} - 938462 \nu^{7} + \cdots - 3730750 \nu ) / 881375 $ (-8672v^15 - 3649v^13 + 153342v^11 + 402539v^9 - 938462v^7 - 1990750v^5 - 3034300v^3 - 3730750v) / 881375
$\beta_{14}$	$=$	$ ( - 50508 \nu^{15} - 41726 \nu^{13} + 817683 \nu^{11} + 2587261 \nu^{9} - 4065438 \nu^{7} + \cdots - 22711875 \nu ) / 4406875 $ (-50508v^15 - 41726v^13 + 817683v^11 + 2587261v^9 - 4065438v^7 - 11753915v^5 - 22984550v^3 - 22711875v) / 4406875
$\beta_{15}$	$=$	$ ( - 11661 \nu^{15} - 8721 \nu^{13} + 202398 \nu^{11} + 611766 \nu^{9} - 1064278 \nu^{7} + \cdots - 5486525 \nu ) / 881375 $ (-11661v^15 - 8721v^13 + 202398v^11 + 611766v^9 - 1064278v^7 - 3191899v^5 - 5385345v^3 - 5486525v) / 881375

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{9} - \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} $ b9 - b7 - b6 - b3 + b2
$\nu^{3}$	$=$	$ -\beta_{15} - 2\beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 3\beta_{10} + 2\beta_{8} - \beta_1 $ -b15 - 2b14 + b13 - b12 + b11 - 3b10 + 2*b8 - b1
$\nu^{4}$	$=$	$ 2\beta_{7} - 2\beta_{6} + 8\beta_{5} + 5\beta_{4} + 5\beta_{3} $ 2b7 - 2b6 + 8b5 + 5b4 + 5*b3
$\nu^{5}$	$=$	$ -13\beta_{15} + 13\beta_{14} + 12\beta_{13} + 7\beta_{11} + 7\beta_{10} - 5\beta_{8} + 7\beta_1 $ -13b15 + 13b14 + 12b13 + 7b11 + 7b10 - 5b8 + 7*b1
$\nu^{6}$	$=$	$ 11\beta_{9} - 11\beta_{7} - 6\beta_{6} + 11\beta_{4} - 15\beta_{3} + 5\beta_{2} + 15 $ 11b9 - 11b7 - 6b6 + 11b4 - 15b3 + 5b2 + 15
$\nu^{7}$	$=$	$ 20\beta_{15} - 43\beta_{14} + 9\beta_{12} - 4\beta_{11} - 43\beta_{10} + 20\beta_{8} + 9\beta_1 $ 20b15 - 43b14 + 9b12 - 4b11 - 43b10 + 20b8 + 9*b1
$\nu^{8}$	$=$	$ 32\beta_{9} - 52\beta_{6} + 53\beta_{5} + 18\beta_{4} + 53\beta_{3} + 32\beta_{2} - 18 $ 32b9 - 52b6 + 53b5 + 18b4 + 53b3 + 32b2 - 18
$\nu^{9}$	$=$	$ -209\beta_{15} + 137\beta_{14} + 137\beta_{13} - 87\beta_{12} + 137\beta_{11} + 21\beta_{10} - 21\beta_{8} $ -209b15 + 137b14 + 137b13 - 87b12 + 137b11 + 21b10 - 21*b8
$\nu^{10}$	$=$	$ 21\beta_{9} - 36\beta_{7} + 275\beta_{5} + 275\beta_{4} + 166 $ 21b9 - 36b7 + 275b5 + 275b4 + 166
$\nu^{11}$	$=$	$ 78\beta_{15} - 129\beta_{14} + 239\beta_{13} + 239\beta_{12} - 78\beta_{10} + 384\beta_1 $ 78b15 - 129b14 + 239b13 + 239b12 - 78b10 + 384b1
$\nu^{12}$	$=$	$ 752\beta_{9} - 462\beta_{7} - 752\beta_{6} - 110\beta_{5} - 177\beta_{3} + 462\beta_{2} + 110 $ 752b9 - 462b7 - 752b6 - 110b5 - 177b3 + 462b2 + 110
$\nu^{13}$	$=$	$ - 1037 \beta_{15} - 639 \beta_{14} + 642 \beta_{13} - 1037 \beta_{12} + 1037 \beta_{11} + \cdots - 642 \beta_1 $ -1037b15 - 639b14 + 642b13 - 1037b12 + 1037b11 - 1681b10 + 639b8 - 642b1
$\nu^{14}$	$=$	$ 639\beta_{7} - 639\beta_{6} + 5431\beta_{5} + 3360\beta_{4} + 3360\beta_{3} + 400\beta_{2} $ 639b7 - 639b6 + 5431b5 + 3360b4 + 3360b3 + 400b2
$\nu^{15}$	$=$	$ -6716\beta_{15} + 6716\beta_{14} + 7109\beta_{13} + 4399\beta_{11} + 4399\beta_{10} - 2960\beta_{8} + 4399\beta_1 $ -6716b15 + 6716b14 + 7109b13 + 4399b11 + 4399b10 - 2960b8 + 4399*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$.

$n$	$122$	$365$
$\chi(n)$	$-1$	$1 - \beta_{3} - \beta_{4} - \beta_{5}$

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} $

118.1

−1.90184	+	0.0324487i
0.556839	+	1.81878i
1.90184	−	0.0324487i
−0.556839	−	1.81878i
−0.752864	−	0.902863i
−0.0783900	+	1.17295i
0.752864	+	0.902863i
0.0783900	−	1.17295i
−1.90184	−	0.0324487i
0.556839	−	1.81878i
1.90184	+	0.0324487i
−0.556839	+	1.81878i
−0.752864	+	0.902863i
−0.0783900	−	1.17295i
0.752864	−	0.902863i
0.0783900	+	1.17295i

−0.513743

−

0.707107i

−1.26313

0.410415i

0.381966

−

1.17557i

0.780656

−

1.07448i

0.939130

0.682318i

1.68914

−

2.03637i

−2.68999

0.874032i

−1.00000

0.726543i

−1.16083

118.2

−0.513743

−

0.707107i

1.26313

−

0.410415i

0.381966

−

1.17557i

−0.780656

1.07448i

−0.939130

−

0.682318i

0.169598

2.64031i

−2.68999

0.874032i

−1.00000

0.726543i

1.16083

118.3

0.513743

0.707107i

−1.26313

0.410415i

0.381966

−

1.17557i

0.780656

−

1.07448i

−0.939130

−

0.682318i

−1.68914

2.03637i

2.68999

−

0.874032i

−1.00000

0.726543i

1.16083

118.4

0.513743

0.707107i

1.26313

−

0.410415i

0.381966

−

1.17557i

−0.780656

1.07448i

0.939130

0.682318i

−0.169598

−

2.64031i

2.68999

−

0.874032i

−1.00000

0.726543i

−1.16083

475.1

−2.17625

−

0.707107i

−1.46782

−

2.02029i

2.61803

1.90211i

−2.37499

0.771681i

1.76580

5.43456i

0.958092

−

2.46618i

−1.66251

−

2.28825i

−1.00000

3.07768i

5.71423

475.2

−2.17625

−

0.707107i

1.46782

2.02029i

2.61803

1.90211i

2.37499

−

0.771681i

−1.76580

−

5.43456i

2.04941

−

1.67329i

−1.66251

−

2.28825i

−1.00000

3.07768i

−5.71423

475.3

2.17625

0.707107i

−1.46782

−

2.02029i

2.61803

1.90211i

−2.37499

0.771681i

−1.76580

−

5.43456i

−0.958092

2.46618i

1.66251

2.28825i

−1.00000

3.07768i

−5.71423

475.4

2.17625

0.707107i

1.46782

2.02029i

2.61803

1.90211i

2.37499

−

0.771681i

1.76580

5.43456i

−2.04941

1.67329i

1.66251

2.28825i

−1.00000

3.07768i

5.71423

524.1

−0.513743

0.707107i

−1.26313

−

0.410415i

0.381966

1.17557i

0.780656

1.07448i

0.939130

−

0.682318i

1.68914

2.03637i

−2.68999

−

0.874032i

−1.00000

−

0.726543i

−1.16083

524.2

−0.513743

0.707107i

1.26313

0.410415i

0.381966

1.17557i

−0.780656

−

1.07448i

−0.939130

0.682318i

0.169598

−

2.64031i

−2.68999

−

0.874032i

−1.00000

−

0.726543i

1.16083

524.3

0.513743

−

0.707107i

−1.26313

−

0.410415i

0.381966

1.17557i

0.780656

1.07448i

−0.939130

0.682318i

−1.68914

−

2.03637i

2.68999

0.874032i

−1.00000

−

0.726543i

1.16083

524.4

0.513743

−

0.707107i

1.26313

0.410415i

0.381966

1.17557i

−0.780656

−

1.07448i

0.939130

−

0.682318i

−0.169598

2.64031i

2.68999

0.874032i

−1.00000

−

0.726543i

−1.16083

699.1

−2.17625

0.707107i

−1.46782

2.02029i

2.61803

−

1.90211i

−2.37499

−

0.771681i

1.76580

−

5.43456i

0.958092

2.46618i

−1.66251

2.28825i

−1.00000

−

3.07768i

5.71423

699.2

−2.17625

0.707107i

1.46782

−

2.02029i

2.61803

−

1.90211i

2.37499

0.771681i

−1.76580

5.43456i

2.04941

1.67329i

−1.66251

2.28825i

−1.00000

−

3.07768i

−5.71423

699.3

2.17625

−

0.707107i

−1.46782

2.02029i

2.61803

−

1.90211i

−2.37499

−

0.771681i

−1.76580

5.43456i

−0.958092

−

2.46618i

1.66251

−

2.28825i

−1.00000

−

3.07768i

−5.71423

699.4

2.17625

−

0.707107i

1.46782

−

2.02029i

2.61803

−

1.90211i

2.37499

0.771681i

1.76580

−

5.43456i

−2.04941

−

1.67329i

1.66251

−

2.28825i

−1.00000

−

3.07768i

5.71423

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
11.b	odd	2	1	inner
11.c	even	5	1	inner
11.d	odd	10	1	inner
77.b	even	2	1	inner
77.j	odd	10	1	inner
77.l	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	847.2.l.h		16
7.b	odd	2	1	inner	847.2.l.h		16
11.b	odd	2	1	inner	847.2.l.h		16
11.c	even	5	1		847.2.b.c	✓	8
11.c	even	5	2		847.2.l.f		16
11.c	even	5	1	inner	847.2.l.h		16
11.d	odd	10	1		847.2.b.c	✓	8
11.d	odd	10	2		847.2.l.f		16
11.d	odd	10	1	inner	847.2.l.h		16
77.b	even	2	1	inner	847.2.l.h		16
77.j	odd	10	1		847.2.b.c	✓	8
77.j	odd	10	2		847.2.l.f		16
77.j	odd	10	1	inner	847.2.l.h		16
77.l	even	10	1		847.2.b.c	✓	8
77.l	even	10	2		847.2.l.f		16
77.l	even	10	1	inner	847.2.l.h		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
847.2.b.c	✓	8	11.c	even	5	1
847.2.b.c	✓	8	11.d	odd	10	1
847.2.b.c	✓	8	77.j	odd	10	1
847.2.b.c	✓	8	77.l	even	10	1
847.2.l.f		16	11.c	even	5	2
847.2.l.f		16	11.d	odd	10	2
847.2.l.f		16	77.j	odd	10	2
847.2.l.f		16	77.l	even	10	2
847.2.l.h		16	1.a	even	1	1	trivial
847.2.l.h		16	7.b	odd	2	1	inner
847.2.l.h		16	11.b	odd	2	1	inner
847.2.l.h		16	11.c	even	5	1	inner
847.2.l.h		16	11.d	odd	10	1	inner
847.2.l.h		16	77.b	even	2	1	inner
847.2.l.h		16	77.j	odd	10	1	inner
847.2.l.h		16	77.l	even	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - 8T_{2}^{6} + 24T_{2}^{4} + 8T_{2}^{2} + 16 $ T2^8 - 8*T2^6 + 24*T2^4 + 8*T2^2 + 16 acting on $S_{2}^{\mathrm{new}}(847, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - 8 T^{6} + 24 T^{4} + \cdots + 16)^{2} $ (T^8 - 8T^6 + 24T^4 + 8*T^2 + 16)^2
$3$	$ (T^{8} + T^{6} + 31 T^{4} + \cdots + 121)^{2} $ (T^8 + T^6 + 31T^4 - 99T^2 + 121)^2
$5$	$ (T^{8} - 9 T^{6} + \cdots + 121)^{2} $ (T^8 - 9T^6 + 31T^4 + 11*T^2 + 121)^2
$7$	$ T^{16} + 24 T^{14} + \cdots + 5764801 $ T^16 + 24T^14 + 327T^12 + 3322T^10 + 26205T^8 + 162778T^6 + 785127T^4 + 2823576*T^2 + 5764801
$11$	$ T^{16} $ T^16
$13$	$ (T^{8} + 8 T^{6} + \cdots + 495616)^{2} $ (T^8 + 8T^6 + 1024T^4 + 33792*T^2 + 495616)^2
$17$	$ (T^{8} + 90 T^{6} + \cdots + 1210000)^{2} $ (T^8 + 90T^6 + 3100T^4 - 11000*T^2 + 1210000)^2
$19$	$ (T^{8} + 12 T^{6} + \cdots + 1936)^{2} $ (T^8 + 12T^6 + 64T^4 + 88*T^2 + 1936)^2
$23$	$ (T^{2} + 2 T - 19)^{8} $ (T^2 + 2*T - 19)^8
$29$	$ (T^{8} - 72 T^{6} + \cdots + 104976)^{2} $ (T^8 - 72T^6 + 1944T^4 + 5832*T^2 + 104976)^2
$31$	$ (T^{8} + 19 T^{6} + \cdots + 121)^{2} $ (T^8 + 19T^6 + 991T^4 - 561*T^2 + 121)^2
$37$	$ (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{4} $ (T^4 - 3T^3 + 9T^2 - 27*T + 81)^4
$41$	$ (T^{8} + 12 T^{6} + \cdots + 1936)^{2} $ (T^8 + 12T^6 + 64T^4 + 88*T^2 + 1936)^2
$43$	$ (T^{4} + 174 T^{2} + 6724)^{4} $ (T^4 + 174*T^2 + 6724)^4
$47$	$ (T^{8} - 120 T^{6} + \cdots + 19360000)^{2} $ (T^8 - 120T^6 + 6400T^4 - 88000*T^2 + 19360000)^2
$53$	$ (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{4} $ (T^4 - 2T^3 + 4T^2 - 8*T + 16)^4
$59$	$ (T^{8} - 39 T^{6} + \cdots + 15768841)^{2} $ (T^8 - 39T^6 + 4471T^4 - 353419*T^2 + 15768841)^2
$61$	$ (T^{8} + 2 T^{6} + \cdots + 1936)^{2} $ (T^8 + 2T^6 + 64T^4 + 528*T^2 + 1936)^2
$67$	$ (T^{2} + 6 T - 71)^{8} $ (T^2 + 6*T - 71)^8
$71$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} $ (T^4 + T^3 + T^2 + T + 1)^4
$73$	$ (T^{8} + 60 T^{6} + \cdots + 1210000)^{2} $ (T^8 + 60T^6 + 1600T^4 + 11000*T^2 + 1210000)^2
$79$	$ (T^{8} - 18 T^{6} + \cdots + 104976)^{2} $ (T^8 - 18T^6 + 324T^4 - 5832*T^2 + 104976)^2
$83$	$ (T^{8} + 68 T^{6} + \cdots + 28344976)^{2} $ (T^8 + 68T^6 + 5344T^4 + 415272*T^2 + 28344976)^2
$89$	$ (T^{4} + 440 T^{2} + 33275)^{4} $ (T^4 + 440*T^2 + 33275)^4
$97$	$ (T^{8} + 19 T^{6} + \cdots + 121)^{2} $ (T^8 + 19T^6 + 991T^4 - 561*T^2 + 121)^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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.l (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{15} - \beta_{10} + \beta_{8}) q^{2} + (\beta_{7} - \beta_{2}) q^{3} + ( - 2 \beta_{4} + 2) q^{4} + ( - \beta_{6} - \beta_{2}) q^{5} + (\beta_{15} + 2 \beta_{13} + \cdots - 2 \beta_1) q^{6}+ \cdots + ( - 2 \beta_{4} - 2 \beta_{3}) q^{9}+O(q^{10}) \) q + (b15 - b10 + b8) * q^2 + (b7 - b2) * q^3 + (-2b4 + 2) q^4 + (-b6 - b2) * q^5 + (b15 + 2b13 + 2b12 - b10 - 2b1) q^6 + (b15 - b14 - b13 + b12 - b11 + b10 + b8 + b1) * q^7 + (2b15 - 2b14 - 2b10 + 2b8) * q^8 + (-2b4 - 2b3) * q^9	\( q + (\beta_{15} - \beta_{10} + \beta_{8}) q^{2} + (\beta_{7} - \beta_{2}) q^{3} + ( - 2 \beta_{4} + 2) q^{4} + ( - \beta_{6} - \beta_{2}) q^{5} + (\beta_{15} + 2 \beta_{13} + \cdots - 2 \beta_1) q^{6}+ \cdots + ( - 3 \beta_{15} + 3 \beta_{14} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) q + (b15 - b10 + b8) * q^2 + (b7 - b2) * q^3 + (-2b4 + 2) q^4 + (-b6 - b2) * q^5 + (b15 + 2b13 + 2b12 - b10 - 2b1) q^6 + (b15 - b14 - b13 + b12 - b11 + b10 + b8 + b1) * q^7 + (2b15 - 2b14 - 2b10 + 2b8) * q^8 + (-2b4 - 2b3) * q^9 + (b15 - b14 + 2b13 - 2b11 - 2b10 + b8 - 2b1) * q^10 + (2b9 + 2b7) * q^12 + (4b15 - 4b14 - 4b13 - 4b11 - 2b10 + 2b8) * q^13 + (-b9 + b6 + 3b5 - b3 - 3) q^14 + (-3b5 - 5b4 - 3b3 + 5) q^15 + (-2b15 + b14 - 2b13 - 2b12 + 2b10 + 4b1) q^17 + (-2b14 + 2b10 + 2b8) q^18 + (b14 + 2b12 + b10 + 2b1) * q^19 + (2b7 - 2b6 - 4b2) q^20 + (-2b15 + 2b14 - b13 + 3b11 + 3b10 - 3b8 + 3b1) * q^21 + (4b5 + 4b4 - 3) * q^23 + (2b15 + 4b12) * q^24 + (-2b5 - 2b3 + 2) * q^25 + (-4b9 - 4b2) * q^26 + (2b9 - 2b7 - b6 + b2) * q^27 + (-4b15 + 4b10 + 2b1) q^28 + (-3b14 - 3b10 + 3b8) q^29 + (7b15 - 2b14 - 2b10 + 7b8) * q^30 + (b7 - b6 + b2) * q^31 - 4b8 q^32 + (-6b9 - 2b7) * q^34 + (-b15 + b14 + b13 + 3b12 + b11) q^35 + (4b5 - 4b3 - 4) * q^36 + (-3b5 - 3b4 - 3b3 + 3) q^37 + (2b6 + 2b2) * q^38 + (6b15 - 2b14 - 6b10) q^39 + (4b13 - 2b10 - 4b1) q^40 + (b14 + 2b12 + b10 + 2b1) * q^41 + (-4b7 + 4b6 + b5 + 3b4 + 3b3 + 7b2) q^42 + (b15 - b14 + 7b8) q^43 + (2b9 + 2b7) * q^45 + (-7b15 + 3b10 - 3b8) q^46 + (-2b9 + 4b7 + 2b6 - 4b2) * q^47 + (b9 - b7 - 2b6 + 3b4 + 5b3 - b2 - 5) q^49 + 2b14 q^50 + (8b14 - b10 - 8b8) * q^51 + (4b15 - 4b14 - 8b11 - 4b10 + 4b8) q^52 + 2b5 q^53 + (-3b15 + 3b14 + 2b13 + 2b11 + 2b10 - b8 + 2b1) * q^54 + (-2b9 + 2b7 + 2b5 + 2b4 - 6) * q^56 + (-2b15 + 3b10 - 3b8) q^57 - 6b3 q^58 + (b9 + 4b6 + b2) q^59 + (-14b4 - 10b3 + 10) * q^60 + (b15 - b14 - b10 - 2b1) q^61 + (2b15 - b14 - 2b13 + 2b12 - 2b11 + b8 + 2b1) q^62 + (-4b15 - 2b12 - 4b8 - 2b1) * q^63 + (8b4 + 8b3) * q^64 + (-6b15 + 6b14 - 4b8) q^65 + (-8b5 - 8b4 + 1) * q^67 + (-2b15 - 4b14 - 4b13 - 12b12 - 4b11 - 2b10 + 2b8) q^68 + (-4b9 - 3b7 + 4b6 + 3b2) * q^69 + (4b9 + 3b6 - b5 + 2b4 - b3 + 4b2 - 2) * q^70 - b4 * q^71 + (-4b15 + 4b10) * q^72 + (-2b15 + b14 + 4b13 - 2b12 + 2b11 - b10 - b8 - 4b1) q^73 + (3b15 + 3b8) * q^74 - 2b2 q^75 + (-4b15 + 4b14 + 4b11 + 4b10 - 2b8 + 4b1) * q^76 + (-8b5 - 8b4 + 20) * q^78 - 3b15 q^79 + (-5b5 - 7b4 - 5b3 + 7) q^81 + (2b6 + 2b2) * q^82 + (-b15 + 4b14 + 6b13 + 6b12 + b10 + 2b1) * q^83 + (-6b15 + 6b14 - 8b13 - 6b12 + 6b11 + 8b10 - 6b8 + 8b1) * q^84 + (-9b15 + b14 + b10 - 9b8) * q^85 + (2b5 - 14b4 - 14b3) q^86 + (3b15 - 3b14 - 6b13) q^87 + (b9 - 8b7) q^89 + (4b14 + 4b13 + 8b12 + 4b11 + 2b10 - 2b8) * q^90 + (4b9 + 2b7 - 4b6 + 8b5 + 10b3 - 2b2 - 8) * q^91 + (8b5 + 14b4 + 8b3 - 14) q^92 + (-b4 + 4b3 - 4) q^93 + (4b15 - 2b14 + 4b13 + 4b12 - 4b10 - 8b1) * q^94 + (-3b14 - b10 + 3b8) * q^95 + (4b15 - 4b14 - 8b11 - 4b10 + 4b8) q^96 + (b7 - b6 + b2) * q^97 + (-3b15 + 3b14 + 4b13 - 2b11 - 2b10 - 7b8 - 2b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{4} - 16 q^{9}+O(q^{10}) \) 16 * q + 24 * q^4 - 16 * q^9	\( 16 q + 24 q^{4} - 16 q^{9} - 40 q^{14} + 36 q^{15} - 16 q^{23} + 16 q^{25} - 64 q^{36} + 12 q^{37} + 28 q^{42} - 48 q^{49} + 8 q^{53} - 80 q^{56} - 24 q^{58} + 64 q^{60} + 64 q^{64} - 48 q^{67} - 32 q^{70} - 4 q^{71} + 256 q^{78} + 44 q^{81} - 104 q^{86} - 56 q^{91} - 104 q^{92} - 52 q^{93}+O(q^{100}) \) 16 * q + 24 * q^4 - 16 * q^9 - 40 * q^14 + 36 * q^15 - 16 * q^23 + 16 * q^25 - 64 * q^36 + 12 * q^37 + 28 * q^42 - 48 * q^49 + 8 * q^53 - 80 * q^56 - 24 * q^58 + 64 * q^60 + 64 * q^64 - 48 * q^67 - 32 * q^70 - 4 * q^71 + 256 * q^78 + 44 * q^81 - 104 * q^86 - 56 * q^91 - 104 * q^92 - 52 * q^93

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	847.2.l.h

Level	$847$
Weight	$2$
Character orbit	847.l
Analytic conductor	$6.763$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) x^16 + 2x^14 - 16x^12 - 72x^10 + 26x^8 + 360x^6 + 725x^4 + 1000*x^2 + 625
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 155 \nu^{14} - 1408 \nu^{12} - 4611 \nu^{10} + 9188 \nu^{8} + 65146 \nu^{6} - 26953 \nu^{4} + \cdots + 108500 ) / 176275 \) (155v^14 - 1408v^12 - 4611v^10 + 9188v^8 + 65146v^6 - 26953v^4 + 145025*v^2 + 108500) / 176275
\(\beta_{3}\)	\(=\)	\( ( 5996 \nu^{14} + 4807 \nu^{12} - 95306 \nu^{10} - 277252 \nu^{8} + 500966 \nu^{6} + 1108500 \nu^{4} + \cdots + 2787000 ) / 881375 \) (5996v^14 + 4807v^12 - 95306v^10 - 277252v^8 + 500966v^6 + 1108500v^4 + 1748000*v^2 + 2787000) / 881375
\(\beta_{4}\)	\(=\)	\( ( 8474 \nu^{14} + 2743 \nu^{12} - 149594 \nu^{10} - 389798 \nu^{8} + 955934 \nu^{6} + 2009635 \nu^{4} + \cdots + 3175500 ) / 881375 \) (8474v^14 + 2743v^12 - 149594v^10 - 389798v^8 + 955934v^6 + 2009635v^4 + 3037000*v^2 + 3175500) / 881375
\(\beta_{5}\)	\(=\)	\( ( - 8672 \nu^{14} - 3649 \nu^{12} + 153342 \nu^{10} + 402539 \nu^{8} - 938462 \nu^{6} + \cdots - 3730750 ) / 881375 \) (-8672v^14 - 3649v^12 + 153342v^10 + 402539v^8 - 938462v^6 - 1990750v^4 - 3034300*v^2 - 3730750) / 881375
\(\beta_{6}\)	\(=\)	\( ( - 12916 \nu^{14} - 10637 \nu^{12} + 225196 \nu^{10} + 690882 \nu^{8} - 1135131 \nu^{6} + \cdots - 6749625 ) / 881375 \) (-12916v^14 - 10637v^12 + 225196v^10 + 690882v^8 - 1135131v^6 - 3696115v^4 - 6351875*v^2 - 6749625) / 881375
\(\beta_{7}\)	\(=\)	\( ( - 14403 \nu^{14} - 14916 \nu^{12} + 224078 \nu^{10} + 748351 \nu^{8} - 1023533 \nu^{6} + \cdots - 6732875 ) / 881375 \) (-14403v^14 - 14916v^12 + 224078v^10 + 748351v^8 - 1023533v^6 - 3087765v^4 - 6177175*v^2 - 6732875) / 881375
\(\beta_{8}\)	\(=\)	\( ( - 1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} + \cdots - 598500 \nu ) / 400625 \) (-1496v^15 - 2572v^13 + 24401v^11 + 91992v^9 - 100886v^7 - 478015v^5 - 412275v^3 - 598500v) / 400625
\(\beta_{9}\)	\(=\)	\( ( - 22098 \nu^{14} - 13706 \nu^{12} + 377023 \nu^{10} + 1116041 \nu^{8} - 1983428 \nu^{6} + \cdots - 11238000 ) / 881375 \) (-22098v^14 - 13706v^12 + 377023v^10 + 1116041v^8 - 1983428v^6 - 5540615v^4 - 10624800*v^2 - 11238000) / 881375
\(\beta_{10}\)	\(=\)	\( ( 22194 \nu^{15} + 11523 \nu^{13} - 339409 \nu^{11} - 1024903 \nu^{9} + 1669899 \nu^{7} + \cdots + 9814875 \nu ) / 4406875 \) (22194v^15 + 11523v^13 - 339409v^11 - 1024903v^9 + 1669899v^7 + 3769000v^5 + 11127550v^3 + 9814875v) / 4406875
\(\beta_{11}\)	\(=\)	\( ( 25903 \nu^{15} - 3944 \nu^{13} - 422373 \nu^{11} - 826866 \nu^{9} + 3248903 \nu^{7} + \cdots + 4286500 \nu ) / 4406875 \) (25903v^15 - 3944v^13 - 422373v^11 - 826866v^9 + 3248903v^7 + 2357930v^5 + 4320250v^3 + 4286500v) / 4406875
\(\beta_{12}\)	\(=\)	\( ( 8474 \nu^{15} + 2743 \nu^{13} - 149594 \nu^{11} - 389798 \nu^{9} + 955934 \nu^{7} + \cdots + 2294125 \nu ) / 881375 \) (8474v^15 + 2743v^13 - 149594v^11 - 389798v^9 + 955934v^7 + 2009635v^5 + 3037000v^3 + 2294125v) / 881375
\(\beta_{13}\)	\(=\)	\( ( - 8672 \nu^{15} - 3649 \nu^{13} + 153342 \nu^{11} + 402539 \nu^{9} - 938462 \nu^{7} + \cdots - 3730750 \nu ) / 881375 \) (-8672v^15 - 3649v^13 + 153342v^11 + 402539v^9 - 938462v^7 - 1990750v^5 - 3034300v^3 - 3730750v) / 881375
\(\beta_{14}\)	\(=\)	\( ( - 50508 \nu^{15} - 41726 \nu^{13} + 817683 \nu^{11} + 2587261 \nu^{9} - 4065438 \nu^{7} + \cdots - 22711875 \nu ) / 4406875 \) (-50508v^15 - 41726v^13 + 817683v^11 + 2587261v^9 - 4065438v^7 - 11753915v^5 - 22984550v^3 - 22711875v) / 4406875
\(\beta_{15}\)	\(=\)	\( ( - 11661 \nu^{15} - 8721 \nu^{13} + 202398 \nu^{11} + 611766 \nu^{9} - 1064278 \nu^{7} + \cdots - 5486525 \nu ) / 881375 \) (-11661v^15 - 8721v^13 + 202398v^11 + 611766v^9 - 1064278v^7 - 3191899v^5 - 5385345v^3 - 5486525v) / 881375

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{9} - \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} \) b9 - b7 - b6 - b3 + b2
\(\nu^{3}\)	\(=\)	\( -\beta_{15} - 2\beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 3\beta_{10} + 2\beta_{8} - \beta_1 \) -b15 - 2b14 + b13 - b12 + b11 - 3b10 + 2*b8 - b1
\(\nu^{4}\)	\(=\)	\( 2\beta_{7} - 2\beta_{6} + 8\beta_{5} + 5\beta_{4} + 5\beta_{3} \) 2b7 - 2b6 + 8b5 + 5b4 + 5*b3
\(\nu^{5}\)	\(=\)	\( -13\beta_{15} + 13\beta_{14} + 12\beta_{13} + 7\beta_{11} + 7\beta_{10} - 5\beta_{8} + 7\beta_1 \) -13b15 + 13b14 + 12b13 + 7b11 + 7b10 - 5b8 + 7*b1
\(\nu^{6}\)	\(=\)	\( 11\beta_{9} - 11\beta_{7} - 6\beta_{6} + 11\beta_{4} - 15\beta_{3} + 5\beta_{2} + 15 \) 11b9 - 11b7 - 6b6 + 11b4 - 15b3 + 5b2 + 15
\(\nu^{7}\)	\(=\)	\( 20\beta_{15} - 43\beta_{14} + 9\beta_{12} - 4\beta_{11} - 43\beta_{10} + 20\beta_{8} + 9\beta_1 \) 20b15 - 43b14 + 9b12 - 4b11 - 43b10 + 20b8 + 9*b1
\(\nu^{8}\)	\(=\)	\( 32\beta_{9} - 52\beta_{6} + 53\beta_{5} + 18\beta_{4} + 53\beta_{3} + 32\beta_{2} - 18 \) 32b9 - 52b6 + 53b5 + 18b4 + 53b3 + 32b2 - 18
\(\nu^{9}\)	\(=\)	\( -209\beta_{15} + 137\beta_{14} + 137\beta_{13} - 87\beta_{12} + 137\beta_{11} + 21\beta_{10} - 21\beta_{8} \) -209b15 + 137b14 + 137b13 - 87b12 + 137b11 + 21b10 - 21*b8
\(\nu^{10}\)	\(=\)	\( 21\beta_{9} - 36\beta_{7} + 275\beta_{5} + 275\beta_{4} + 166 \) 21b9 - 36b7 + 275b5 + 275b4 + 166
\(\nu^{11}\)	\(=\)	\( 78\beta_{15} - 129\beta_{14} + 239\beta_{13} + 239\beta_{12} - 78\beta_{10} + 384\beta_1 \) 78b15 - 129b14 + 239b13 + 239b12 - 78b10 + 384b1
\(\nu^{12}\)	\(=\)	\( 752\beta_{9} - 462\beta_{7} - 752\beta_{6} - 110\beta_{5} - 177\beta_{3} + 462\beta_{2} + 110 \) 752b9 - 462b7 - 752b6 - 110b5 - 177b3 + 462b2 + 110
\(\nu^{13}\)	\(=\)	\( - 1037 \beta_{15} - 639 \beta_{14} + 642 \beta_{13} - 1037 \beta_{12} + 1037 \beta_{11} + \cdots - 642 \beta_1 \) -1037b15 - 639b14 + 642b13 - 1037b12 + 1037b11 - 1681b10 + 639b8 - 642b1
\(\nu^{14}\)	\(=\)	\( 639\beta_{7} - 639\beta_{6} + 5431\beta_{5} + 3360\beta_{4} + 3360\beta_{3} + 400\beta_{2} \) 639b7 - 639b6 + 5431b5 + 3360b4 + 3360b3 + 400b2
\(\nu^{15}\)	\(=\)	\( -6716\beta_{15} + 6716\beta_{14} + 7109\beta_{13} + 4399\beta_{11} + 4399\beta_{10} - 2960\beta_{8} + 4399\beta_1 \) -6716b15 + 6716b14 + 7109b13 + 4399b11 + 4399b10 - 2960b8 + 4399*b1

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(-1\)	\(1 - \beta_{3} - \beta_{4} - \beta_{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 118.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - 8 T^{6} + 24 T^{4} + \cdots + 16)^{2} \) (T^8 - 8T^6 + 24T^4 + 8*T^2 + 16)^2
$3$	\( (T^{8} + T^{6} + 31 T^{4} + \cdots + 121)^{2} \) (T^8 + T^6 + 31T^4 - 99T^2 + 121)^2
$5$	\( (T^{8} - 9 T^{6} + \cdots + 121)^{2} \) (T^8 - 9T^6 + 31T^4 + 11*T^2 + 121)^2
$7$	\( T^{16} + 24 T^{14} + \cdots + 5764801 \) T^16 + 24T^14 + 327T^12 + 3322T^10 + 26205T^8 + 162778T^6 + 785127T^4 + 2823576*T^2 + 5764801
$11$	\( T^{16} \) T^16
$13$	\( (T^{8} + 8 T^{6} + \cdots + 495616)^{2} \) (T^8 + 8T^6 + 1024T^4 + 33792*T^2 + 495616)^2
$17$	\( (T^{8} + 90 T^{6} + \cdots + 1210000)^{2} \) (T^8 + 90T^6 + 3100T^4 - 11000*T^2 + 1210000)^2
$19$	\( (T^{8} + 12 T^{6} + \cdots + 1936)^{2} \) (T^8 + 12T^6 + 64T^4 + 88*T^2 + 1936)^2
$23$	\( (T^{2} + 2 T - 19)^{8} \) (T^2 + 2*T - 19)^8
$29$	\( (T^{8} - 72 T^{6} + \cdots + 104976)^{2} \) (T^8 - 72T^6 + 1944T^4 + 5832*T^2 + 104976)^2
$31$	\( (T^{8} + 19 T^{6} + \cdots + 121)^{2} \) (T^8 + 19T^6 + 991T^4 - 561*T^2 + 121)^2
$37$	\( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{4} \) (T^4 - 3T^3 + 9T^2 - 27*T + 81)^4
$41$	\( (T^{8} + 12 T^{6} + \cdots + 1936)^{2} \) (T^8 + 12T^6 + 64T^4 + 88*T^2 + 1936)^2
$43$	\( (T^{4} + 174 T^{2} + 6724)^{4} \) (T^4 + 174*T^2 + 6724)^4
$47$	\( (T^{8} - 120 T^{6} + \cdots + 19360000)^{2} \) (T^8 - 120T^6 + 6400T^4 - 88000*T^2 + 19360000)^2
$53$	\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{4} \) (T^4 - 2T^3 + 4T^2 - 8*T + 16)^4
$59$	\( (T^{8} - 39 T^{6} + \cdots + 15768841)^{2} \) (T^8 - 39T^6 + 4471T^4 - 353419*T^2 + 15768841)^2
$61$	\( (T^{8} + 2 T^{6} + \cdots + 1936)^{2} \) (T^8 + 2T^6 + 64T^4 + 528*T^2 + 1936)^2
$67$	\( (T^{2} + 6 T - 71)^{8} \) (T^2 + 6*T - 71)^8
$71$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) (T^4 + T^3 + T^2 + T + 1)^4
$73$	\( (T^{8} + 60 T^{6} + \cdots + 1210000)^{2} \) (T^8 + 60T^6 + 1600T^4 + 11000*T^2 + 1210000)^2
$79$	\( (T^{8} - 18 T^{6} + \cdots + 104976)^{2} \) (T^8 - 18T^6 + 324T^4 - 5832*T^2 + 104976)^2
$83$	\( (T^{8} + 68 T^{6} + \cdots + 28344976)^{2} \) (T^8 + 68T^6 + 5344T^4 + 415272*T^2 + 28344976)^2
$89$	\( (T^{4} + 440 T^{2} + 33275)^{4} \) (T^4 + 440*T^2 + 33275)^4
$97$	\( (T^{8} + 19 T^{6} + \cdots + 121)^{2} \) (T^8 + 19T^6 + 991T^4 - 561*T^2 + 121)^2
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