LMFDB - Newform orbit 798.2.bo.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [798,2,Mod(43,798)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(798, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 16])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("798.43"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 798 = 2 \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	798.bo (of order $9$, degree $6$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,9,0,6,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)

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

Self dual:	no
Analytic conductor:	$6.37206208130$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{9})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 18x^{10} + 153x^{8} - 773x^{6} + 2448x^{4} - 4608x^{2} + 4096 $ x^12 - 18x^10 + 153x^8 - 773x^6 + 2448x^4 - 4608*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{2} - \beta_{7} q^{3} + (\beta_{8} - \beta_1) q^{4} + ( - \beta_{11} - \beta_{9} + \beta_{2} + 1) q^{5} + \beta_{8} q^{6} + ( - \beta_{9} + 1) q^{7} + \beta_{9} q^{8} + \beta_1 q^{9} + ( - \beta_{9} - \beta_{4} + \beta_{3} + 1) q^{10}+ \cdots + (\beta_{11} + \beta_{9} - \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) $ q - b4 * q^2 - b7 * q^3 + (b8 - b1) * q^4 + (-b11 - b9 + b2 + 1) * q^5 + b8 * q^6 + (-b9 + 1) * q^7 + b9 * q^8 + b1 * q^9 + (-b9 - b4 + b3 + 1) * q^10 + (b10 - b4 + b3 - b2 + b1) * q^11 + (b9 - 1) * q^12 + (-b11 + b10 - 2b9 - b7 - b5 + b3 + b2 + b1) q^13 + (b7 - b4) * q^14 + (-b10 + b9 - b4) * q^15 - b7 * q^16 + (-b10 - b9 + 2b8 + b7 + b6 + b4 - b3 + b2 + 2) q^17 - q^18 + (b11 + b10 - b9 + 2b8 - b7 - b6 - 2b4 - b1 + 2) * q^19 + (b7 + b6 + b5 - b4 - b1) * q^20 - b4 * q^21 + (b10 - b7 + b6 + b4 - b1 - 1) * q^22 + (-b11 - b10 + b9 - 2b8 - b6 - b5 - b4 + 3b1 - 1) * q^23 + (-b7 + b4) * q^24 + (-2b11 - b9 + b6 - b5 + b4 + b2 + 2b1) * q^25 + (-b11 - b9 + b6 + b4 + b3) * q^26 - b9 * q^27 - b1 * q^28 + (-b11 + b7 + b5 - b4 - b3 - b1) * q^29 + (b8 + b5 - b4 - b1) * q^30 + (b10 - 2b9 + 2b8 - 3b7 - b6 + b4 + b2 + b1 + 2) q^31 + b8 * q^32 + (b10 - 2b9 + b8 - b7 + b6 + b5 + b4 + b3 - b1) q^33 + (-b11 - b10 + 2b9 + 2b7 - b6 - 3b4 + b2 + b1 - 2) q^34 + (-b11 - b9 - b8 + b1) * q^35 + b4 * q^36 + (-b11 - b10 + b9 - 2b8 - 3b7 - 2b6 - 2b5 - b3 + b1 - 1) * q^37 + (b11 - b10 + 3b9 + 2b8 + b7 - b5 - b4 - b3 - b2 - 2b1 - 2) q^38 + (-b11 - b10 + b8 + 2b7 + b6 + b5 - b4 + b2) q^39 + (b8 + b2 - b1 + 1) * q^40 + (-b10 + 2b9 + b7 + b6 + 2b5 - b4 - 2b3 - 3b1 - 3) * q^41 + (b8 - b1) * q^42 + (-b11 + 2b9 + b8 - 3b6 - 3b3 + b2) q^43 + (-b11 + b8 - b7 - b5 + 2b4 + b2 + b1 + 1) q^44 + (b8 - b7 - b6) * q^45 + (b10 - 3b9 - b7 + b5 + b3 - b2 - b1) q^46 + (2b9 - b6 + b4 + b3 - b2 + b1 - 2) q^47 + b1 * q^48 - b9 * q^49 + (-2b11 + b10 - 2b9 - b7 + 2b3 + b2 + b1) q^50 + (-b10 + b9 - b6 - b5 - 2b4 - b3 + b2 - 2) q^51 + (-b11 + b10 - b9 - b8 + b6 + b5 + b3 + b2 + b1 + 1) * q^52 + (3b9 + 4b8 + 3b7 + b5 - 4b4 + b2 - 4b1 - 2) q^53 + b7 * q^54 + (2b10 - 6b9 + 2b8 - 2b7 - 2b6 - b4 + 2b3 - b2 + 2b1 + 2) q^55 + q^56 + (b9 + b8 + b6 - b3 - b2 + b1 - 2) * q^57 + (b11 + b10 - b9 + b8 - b7 - b6 - b5 + b3 - b1 + 2) * q^58 + (-b10 - b9 - b8 + b7 + b6 + b5 + 3b2 + 2b1 - 1) * q^59 + (b11 + b9 + b8 - b1) * q^60 + (b11 + 2b10 - 2b8 + 2b7 + b6 + 2b5 - b4 + b2 + b1) * q^61 + (b11 - b10 + 2b9 + b8 + b7 - b5 - b4 - b2 + b1 - 3) q^62 + b8 * q^63 + (b9 - 1) * q^64 + (-b11 + 2b10 - 5b9 - 4b8 - 2b7 - 2b5 + 5b4 + b3 - b2 + 2b1) q^65 + (b9 + b7 + b6 + b4 + b2 + b1) * q^66 + (-b9 + 3b7 + 2b6 - 2b4 - b3 + 2b2 - 2b1 + 4) q^67 + (b11 - b9 - 2b7 + b5 + b4 + b3 - b2 - 3b1) * q^68 + (-b11 - 3b9 + b8 - b7 - b6 + b3 + 2) q^69 + (b10 - 2b9 + b3 + 1) q^70 + (-3b11 + 4b10 - 4b9 + 3b8 - b7 + 2b6 + 4b5 + b4 + 2b3 + 3b2 + b1 - 1) * q^71 + (-b8 + b1) * q^72 + (3b11 - 4b9 - 3b8 + b7 - b6 - b5 + b3 + 3b1 + 6) * q^73 + (b10 - 3b9 + 2b8 - b7 - b6 + b3 - 2b2 + 2) q^74 + (-b11 - b10 - b9 + b8 - b4 + b3 + 2b2) q^75 + (-b11 + 3b9 + b8 - b7 - b6 + b4 - b3 - b1 - 1) q^76 + (b11 + b10 + b8 + b7 - b2 - b1 + 1) * q^77 + (b5 - b4 + b3 + b2 - b1) * q^78 + (2b11 - b10 + b9 + b8 - b6 - b4 + b1 + 3) q^79 + (-b10 + b9 - b4) * q^80 + (b7 - b4) * q^81 + (b11 + 3b8 - b7 - 2b6 - b5 + 2b4 + b2 - b1 + 3) q^82 + (2b11 - 2b10 + 9b9 + 4b7 - b6 - b4 - 2b3 - 2b2 - 4b1 - 7) q^83 + b9 * q^84 + (-2b11 + 2b9 + 4b7 + 2b6 + 2b5 - b3 + 2b2 - 3b1) q^85 + (3b11 - 3b7 - 3b6 - 3b5 + b3 - 3b2) q^86 + (b11 + b9 - b7 - b5 + b3 - b2) * q^87 + (-b11 - b8 - b7 - b4 + b3 + 2b1 - 1) q^88 + (5b11 + b9 + 2b8 + b7 - 3b6 + 2b5 - 3b4 - 2b2 - 5b1 + 1) q^89 + (b11 + b9 - b2 - 1) * q^90 + (-b11 + b10 - b6 - b5 + b1 - 1) * q^91 + (b11 + b10 + 2b7 + b6 + b4 + 1) q^92 + (-b10 - 2b8 + b7 + b6 - b4 - b3 - b2 + 3b1 - 2) * q^93 + (b11 + b10 - 3b8 - 2b7 + b6 + b5 + 2b4 - b2 + b1 - 1) q^94 + (-b10 - 2b9 + 4b8 - 2b7 + 3b5 - 5b4 + b3 - 6b1 - 1) * q^95 - q^96 + (b10 + 5b9 - b8 - b7 - b6 - 3b5 + 7b4 - 2b3 - 4b2 - 2b1 - 1) * q^97 + b7 * q^98 + (b11 + b9 - b8 + 2b7 + b6 + b5 - b4 - 1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 9 q^{5} + 6 q^{7} + 6 q^{8} + 9 q^{10} + 3 q^{11} - 6 q^{12} - 9 q^{13} + 9 q^{15} + 15 q^{17} - 12 q^{18} + 9 q^{19} - 15 q^{22} + 3 q^{23} + 3 q^{25} + 3 q^{26} - 6 q^{27} + 3 q^{29} + 6 q^{31}+ \cdots - 12 q^{99}+O(q^{100}) $ 12 * q + 9 * q^5 + 6 * q^7 + 6 * q^8 + 9 * q^10 + 3 * q^11 - 6 * q^12 - 9 * q^13 + 9 * q^15 + 15 * q^17 - 12 * q^18 + 9 * q^19 - 15 * q^22 + 3 * q^23 + 3 * q^25 + 3 * q^26 - 6 * q^27 + 3 * q^29 + 6 * q^31 - 12 * q^33 - 6 * q^34 - 9 * q^38 + 6 * q^39 + 9 * q^40 - 27 * q^41 + 6 * q^43 + 15 * q^44 - 15 * q^46 - 6 * q^47 - 6 * q^49 - 21 * q^51 + 9 * q^52 - 9 * q^53 - 9 * q^55 + 12 * q^56 - 18 * q^57 + 12 * q^58 - 24 * q^59 - 15 * q^61 - 24 * q^62 - 6 * q^64 - 24 * q^65 + 3 * q^66 + 33 * q^67 - 6 * q^68 + 15 * q^69 - 33 * q^71 + 33 * q^73 + 12 * q^74 + 9 * q^76 + 6 * q^77 + 33 * q^79 + 9 * q^80 + 27 * q^82 - 36 * q^83 + 6 * q^84 + 15 * q^85 - 6 * q^86 + 6 * q^87 - 3 * q^88 - 6 * q^89 - 9 * q^90 - 9 * q^91 + 3 * q^92 - 21 * q^93 - 18 * q^94 - 18 * q^95 - 12 * q^96 + 21 * q^97 - 12 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 18x^{10} + 153x^{8} - 773x^{6} + 2448x^{4} - 4608x^{2} + 4096 $ x^12 - 18*x^10 + 153*x^8 - 773*x^6 + 2448*x^4 - 4608*x^2 + 4096 :

$\beta_{1}$	$=$	$ ( - 5 \nu^{11} - 4 \nu^{10} + 42 \nu^{9} + 72 \nu^{8} - 157 \nu^{7} - 612 \nu^{6} + 105 \nu^{5} + \cdots + 12288 ) / 2048 $ (-5v^11 - 4v^10 + 42v^9 + 72v^8 - 157v^7 - 612v^6 + 105v^5 + 3092v^4 + 1056v^3 - 8768v^2 - 3072*v + 12288) / 2048
$\beta_{2}$	$=$	$ ( 3 \nu^{11} - 20 \nu^{10} - 54 \nu^{9} + 168 \nu^{8} + 459 \nu^{7} - 628 \nu^{6} - 2319 \nu^{5} + \cdots - 13312 ) / 2048 $ (3v^11 - 20v^10 - 54v^9 + 168v^8 + 459v^7 - 628v^6 - 2319v^5 + 420v^4 + 6320v^3 + 4224v^2 - 8704*v - 13312) / 2048
$\beta_{3}$	$=$	$ ( 3 \nu^{11} + 20 \nu^{10} - 54 \nu^{9} - 168 \nu^{8} + 459 \nu^{7} + 628 \nu^{6} - 2319 \nu^{5} + \cdots + 13312 ) / 2048 $ (3v^11 + 20v^10 - 54v^9 - 168v^8 + 459v^7 + 628v^6 - 2319v^5 - 420v^4 + 6320v^3 - 4224v^2 - 8704*v + 13312) / 2048
$\beta_{4}$	$=$	$ ( 5 \nu^{11} - 4 \nu^{10} - 42 \nu^{9} + 72 \nu^{8} + 157 \nu^{7} - 612 \nu^{6} - 105 \nu^{5} + \cdots + 12288 ) / 2048 $ (5v^11 - 4v^10 - 42v^9 + 72v^8 + 157v^7 - 612v^6 - 105v^5 + 3092v^4 - 1056v^3 - 8768v^2 + 3072*v + 12288) / 2048
$\beta_{5}$	$=$	$ ( -\nu^{10} + 18\nu^{8} - 153\nu^{6} + 709\nu^{4} - 1872\nu^{2} + 64\nu + 2304 ) / 128 $ (-v^10 + 18v^8 - 153v^6 + 709v^4 - 1872v^2 + 64*v + 2304) / 128
$\beta_{6}$	$=$	$ ( \nu^{10} - 18\nu^{8} + 153\nu^{6} - 709\nu^{4} + 1872\nu^{2} + 64\nu - 2304 ) / 128 $ (v^10 - 18v^8 + 153v^6 - 709v^4 + 1872v^2 + 64*v - 2304) / 128
$\beta_{7}$	$=$	$ ( - \nu^{11} - 28 \nu^{10} + 18 \nu^{9} + 440 \nu^{8} - 153 \nu^{7} - 3132 \nu^{6} + 773 \nu^{5} + \cdots + 33792 ) / 2048 $ (-v^11 - 28v^10 + 18v^9 + 440v^8 - 153v^7 - 3132v^6 + 773v^5 + 12876v^4 - 2448v^3 - 30336v^2 + 3584v + 33792) / 2048
$\beta_{8}$	$=$	$ ( \nu^{11} - 28 \nu^{10} - 18 \nu^{9} + 440 \nu^{8} + 153 \nu^{7} - 3132 \nu^{6} - 773 \nu^{5} + \cdots + 33792 ) / 2048 $ (v^11 - 28v^10 - 18v^9 + 440v^8 + 153v^7 - 3132v^6 - 773v^5 + 12876v^4 + 2448v^3 - 30336v^2 - 3584v + 33792) / 2048
$\beta_{9}$	$=$	$ ( -9\nu^{11} + 146\nu^{9} - 1089\nu^{7} + 4509\nu^{5} - 10688\nu^{3} + 11520\nu + 1024 ) / 2048 $ (-9v^11 + 146v^9 - 1089v^7 + 4509v^5 - 10688v^3 + 11520v + 1024) / 2048
$\beta_{10}$	$=$	$ ( 15 \nu^{11} - 48 \nu^{10} - 222 \nu^{9} + 608 \nu^{8} + 1431 \nu^{7} - 3760 \nu^{6} - 5275 \nu^{5} + \cdots + 23552 ) / 2048 $ (15v^11 - 48v^10 - 222v^9 + 608v^8 + 1431v^7 - 3760v^6 - 5275v^5 + 13296v^4 + 10880v^3 - 27136v^2 - 9984*v + 23552) / 2048
$\beta_{11}$	$=$	$ ( 7\nu^{11} - 110\nu^{9} + 783\nu^{7} - 3219\nu^{5} + 7584\nu^{3} + 256\nu^{2} - 8448\nu - 1024 ) / 512 $ (7v^11 - 110v^9 + 783v^7 - 3219v^5 + 7584v^3 + 256v^2 - 8448*v - 1024) / 512

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

$\nu$	$=$	$ \beta_{6} + \beta_{5} $ b6 + b5
$\nu^{2}$	$=$	$ \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{3} + 3 $ b11 - b10 + b9 + b7 - b3 + 3
$\nu^{3}$	$=$	$ 3\beta_{8} - 3\beta_{7} + 2\beta_{6} + 2\beta_{5} - \beta_{3} - \beta_{2} $ 3b8 - 3b7 + 2b6 + 2b5 - b3 - b2
$\nu^{4}$	$=$	$ 5\beta_{11} - 5\beta_{10} + 5\beta_{9} + 5\beta_{7} + \beta_{6} - \beta_{5} + 4\beta_{4} - 5\beta_{3} + 4\beta _1 + 3 $ 5b11 - 5b10 + 5b9 + 5b7 + b6 - b5 + 4b4 - 5b3 + 4*b1 + 3
$\nu^{5}$	$=$	$ - \beta_{11} - \beta_{10} - 7 \beta_{9} + 13 \beta_{8} - 12 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots + 3 $ -b11 - b10 - 7b9 + 13b8 - 12b7 - 2b6 - 2b5 - 8b3 - 7*b2 + 3
$\nu^{6}$	$=$	$ 11 \beta_{11} - 11 \beta_{10} + 11 \beta_{9} + 3 \beta_{8} + 14 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} + \cdots - 19 $ 11b11 - 11b10 + 11b9 + 3b8 + 14b7 + 11b6 - 11b5 + 28b4 - 10b3 - b2 + 28b1 - 19
$\nu^{7}$	$=$	$ - 14 \beta_{11} - 14 \beta_{10} - 74 \beta_{9} + 27 \beta_{8} - 13 \beta_{7} - 29 \beta_{6} - 29 \beta_{5} + \cdots + 30 $ -14b11 - 14b10 - 74b9 + 27b8 - 13b7 - 29b6 - 29b5 + 4b4 - 31b3 - 17b2 - 4*b1 + 30
$\nu^{8}$	$=$	$ - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 38 \beta_{8} + 36 \beta_{7} + 61 \beta_{6} - 61 \beta_{5} + \cdots - 114 $ -2b11 + 2b10 - 2b9 + 38b8 + 36b7 + 61b6 - 61b5 + 68b4 + 20b3 - 18b2 + 68*b1 - 114
$\nu^{9}$	$=$	$ - 99 \beta_{11} - 99 \beta_{10} - 389 \beta_{9} + 48 \beta_{8} + 51 \beta_{7} - 94 \beta_{6} + \cdots + 145 $ -99b11 - 99b10 - 389b9 + 48b8 + 51b7 - 94b6 - 94b5 + 72b4 - 43b3 + 56b2 - 72*b1 + 145
$\nu^{10}$	$=$	$ - 46 \beta_{11} + 46 \beta_{10} - 46 \beta_{9} + 225 \beta_{8} + 179 \beta_{7} + 188 \beta_{6} + \cdots - 330 $ -46b11 + 46b10 - 46b9 + 225b8 + 179b7 + 188b6 - 188b5 - 224b4 + 217b3 - 171b2 - 224*b1 - 330
$\nu^{11}$	$=$	$ - 413 \beta_{11} - 413 \beta_{10} - 1091 \beta_{9} + 462 \beta_{8} - 49 \beta_{7} - 113 \beta_{6} + \cdots + 339 $ -413b11 - 413b10 - 1091b9 + 462b8 - 49b7 - 113b6 - 113b5 + 684b4 + 233b3 + 646b2 - 684*b1 + 339

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/798\mathbb{Z}\right)^\times$.

$n$	$115$	$211$	$533$
$\chi(n)$	$1$	$-\beta_{1} + \beta_{8}$	$1$

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} $

43.1

−1.74073	−	0.984808i
1.74073	−	0.984808i
1.89389	−	0.642788i
−1.89389	−	0.642788i
1.89389	+	0.642788i
−1.89389	+	0.642788i
−1.97054	−	0.342020i
1.97054	−	0.342020i
−1.74073	+	0.984808i
1.74073	+	0.984808i
−1.97054	+	0.342020i
1.97054	+	0.342020i

0.939693

−

0.342020i

0.173648

−

0.984808i

0.766044

−

0.642788i

−0.0268090

−

0.0224954i

−0.173648

−

0.984808i

0.500000

0.866025i

0.500000

−

0.866025i

−0.939693

−

0.342020i

−0.0328860

−

0.0119695i

43.2

0.939693

−

0.342020i

0.173648

−

0.984808i

0.766044

−

0.642788i

2.64015

2.21535i

−0.173648

−

0.984808i

0.500000

0.866025i

0.500000

−

0.866025i

−0.939693

−

0.342020i

3.23862

1.17876i

85.1

−0.173648

−

0.984808i

0.766044

−

0.642788i

−0.939693

0.342020i

−0.733478

−

0.266964i

−0.766044

−

0.642788i

0.500000

−

0.866025i

0.500000

0.866025i

0.173648

−

0.984808i

−0.135541

0.768692i

85.2

−0.173648

−

0.984808i

0.766044

−

0.642788i

−0.939693

0.342020i

2.82587

1.02853i

−0.766044

−

0.642788i

0.500000

−

0.866025i

0.500000

0.866025i

0.173648

−

0.984808i

0.522200

−

2.96155i

169.1

−0.173648

0.984808i

0.766044

0.642788i

−0.939693

−

0.342020i

−0.733478

0.266964i

−0.766044

0.642788i

0.500000

0.866025i

0.500000

−

0.866025i

0.173648

0.984808i

−0.135541

−

0.768692i

169.2

−0.173648

0.984808i

0.766044

0.642788i

−0.939693

−

0.342020i

2.82587

−

1.02853i

−0.766044

0.642788i

0.500000

0.866025i

0.500000

−

0.866025i

0.173648

0.984808i

0.522200

2.96155i

253.1

−0.766044

0.642788i

−0.939693

−

0.342020i

0.173648

−

0.984808i

−0.445049

−

2.52400i

0.939693

−

0.342020i

0.500000

−

0.866025i

0.500000

0.866025i

0.766044

0.642788i

1.96332

1.64742i

253.2

−0.766044

0.642788i

−0.939693

−

0.342020i

0.173648

−

0.984808i

0.239312

1.35721i

0.939693

−

0.342020i

0.500000

−

0.866025i

0.500000

0.866025i

0.766044

0.642788i

−1.05572

−

0.885853i

631.1

0.939693

0.342020i

0.173648

0.984808i

0.766044

0.642788i

−0.0268090

0.0224954i

−0.173648

0.984808i

0.500000

−

0.866025i

0.500000

0.866025i

−0.939693

0.342020i

−0.0328860

0.0119695i

631.2

0.939693

0.342020i

0.173648

0.984808i

0.766044

0.642788i

2.64015

−

2.21535i

−0.173648

0.984808i

0.500000

−

0.866025i

0.500000

0.866025i

−0.939693

0.342020i

3.23862

−

1.17876i

757.1

−0.766044

−

0.642788i

−0.939693

0.342020i

0.173648

0.984808i

−0.445049

2.52400i

0.939693

0.342020i

0.500000

0.866025i

0.500000

−

0.866025i

0.766044

−

0.642788i

1.96332

−

1.64742i

757.2

−0.766044

−

0.642788i

−0.939693

0.342020i

0.173648

0.984808i

0.239312

−

1.35721i

0.939693

0.342020i

0.500000

0.866025i

0.500000

−

0.866025i

0.766044

−

0.642788i

−1.05572

0.885853i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	798.2.bo.d	✓	12
19.e	even	9	1	inner	798.2.bo.d	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
798.2.bo.d	✓	12	1.a	even	1	1	trivial
798.2.bo.d	✓	12	19.e	even	9	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} - 9 T_{5}^{11} + 39 T_{5}^{10} - 108 T_{5}^{9} + 255 T_{5}^{8} - 459 T_{5}^{7} + 343 T_{5}^{6} + \cdots + 1 $ T5^12 - 9*T5^11 + 39*T5^10 - 108*T5^9 + 255*T5^8 - 459*T5^7 + 343*T5^6 + 207*T5^5 + 33*T5^4 + 999*T5^3 + 870*T5^2 + 45*T5 + 1 acting on $S_{2}^{\mathrm{new}}(798, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{6} - T^{3} + 1)^{2} $ (T^6 - T^3 + 1)^2
$3$	$ (T^{6} + T^{3} + 1)^{2} $ (T^6 + T^3 + 1)^2
$5$	$ T^{12} - 9 T^{11} + \cdots + 1 $ T^12 - 9T^11 + 39T^10 - 108T^9 + 255T^8 - 459T^7 + 343T^6 + 207T^5 + 33T^4 + 999T^3 + 870T^2 + 45*T + 1
$7$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$11$	$ T^{12} - 3 T^{11} + \cdots + 110889 $ T^12 - 3T^11 + 36T^10 - 155T^9 + 1119T^8 - 3735T^7 + 12619T^6 - 23229T^5 + 49581T^4 - 66600T^3 + 122877T^2 - 110889*T + 110889
$13$	$ T^{12} + 9 T^{11} + \cdots + 11449 $ T^12 + 9T^11 + 63T^10 + 385T^9 + 1593T^8 + 4212T^7 + 7473T^6 + 27T^5 - 20817T^4 - 28316T^3 + 80361T^2 - 47187*T + 11449
$17$	$ T^{12} - 15 T^{11} + \cdots + 4289041 $ T^12 - 15T^11 + 141T^10 - 956T^9 + 5445T^8 - 21267T^7 + 55317T^6 - 68490T^5 - 118323T^4 + 661939T^3 - 527415T^2 - 2093781*T + 4289041
$19$	$ T^{12} - 9 T^{11} + \cdots + 47045881 $ T^12 - 9T^11 + 54T^10 - 288T^9 + 1530T^8 - 8667T^7 + 40789T^6 - 164673T^5 + 552330T^4 - 1975392T^3 + 7037334T^2 - 22284891*T + 47045881
$23$	$ T^{12} - 3 T^{11} + \cdots + 103041 $ T^12 - 3T^11 - 30T^10 - 112T^9 + 540T^8 + 9330T^7 + 53734T^6 + 179391T^5 + 404793T^4 + 640509T^3 + 686970T^2 + 384237*T + 103041
$29$	$ T^{12} - 3 T^{11} + \cdots + 3249 $ T^12 - 3T^11 - 12T^10 - 40T^9 + 1539T^8 - 7923T^7 + 18490T^6 - 26511T^5 + 34956T^4 - 31440T^3 + 22797T^2 - 12312*T + 3249
$31$	$ T^{12} - 6 T^{11} + \cdots + 16410601 $ T^12 - 6T^11 + 84T^10 - 108T^9 + 2688T^8 - 1293T^7 + 61072T^6 + 45546T^5 + 736494T^4 + 448848T^3 + 5321973T^2 + 5821287*T + 16410601
$37$	$ (T^{6} - 108 T^{4} + \cdots + 8691)^{2} $ (T^6 - 108T^4 - 425T^3 + 1092T^2 + 7452T + 8691)^2
$41$	$ T^{12} + 27 T^{11} + \cdots + 59049 $ T^12 + 27T^11 + 279T^10 + 1584T^9 + 9153T^8 + 33615T^7 + 78327T^6 - 29160T^5 + 98415T^4 - 737019T^3 + 570807T^2 + 295245*T + 59049
$43$	$ T^{12} + \cdots + 1868573529 $ T^12 - 6T^11 + 186T^10 - 1306T^9 + 12414T^8 - 150186T^7 + 1269847T^6 - 12335166T^5 + 135293490T^4 - 733298454T^3 + 1267561872T^2 + 3529398096*T + 1868573529
$47$	$ T^{12} + 6 T^{11} + \cdots + 4915089 $ T^12 + 6T^11 + 57T^10 + 205T^9 - 45T^8 + 3003T^7 + 23119T^6 + 22143T^5 + 181341T^4 + 626622T^3 - 439767T^2 - 1097415*T + 4915089
$53$	$ T^{12} + 9 T^{11} + \cdots + 5157441 $ T^12 + 9T^11 - 21T^10 + 34T^9 + 6909T^8 + 56505T^7 + 299041T^6 + 910767T^5 - 1478655T^4 - 4257489T^3 + 62300673T^2 - 6581358*T + 5157441
$59$	$ T^{12} + 24 T^{11} + \cdots + 5948721 $ T^12 + 24T^11 + 192T^10 + 602T^9 + 2832T^8 - 11022T^7 - 28421T^6 - 321696T^5 + 1366614T^4 - 1798758T^3 + 55872180T^2 - 27877770*T + 5948721
$61$	$ T^{12} + 15 T^{11} + \cdots + 811801 $ T^12 + 15T^11 + 129T^10 + 459T^9 - 7926T^8 - 77961T^7 + 201142T^6 + 2832816T^5 + 8917404T^4 - 2858202T^3 + 18453723T^2 - 5860104*T + 811801
$67$	$ T^{12} - 33 T^{11} + \cdots + 104329 $ T^12 - 33T^11 + 570T^10 - 7648T^9 + 83115T^8 - 665955T^7 + 4217799T^6 - 19195425T^5 + 51797151T^4 - 65840398T^3 + 40111458T^2 - 3103707*T + 104329
$71$	$ T^{12} + \cdots + 159823248841 $ T^12 + 33T^11 + 795T^10 + 13894T^9 + 186579T^8 + 1744767T^7 + 12094113T^6 + 23806782T^5 - 453159225T^4 - 1015688891T^3 + 37051974411T^2 - 138243977979*T + 159823248841
$73$	$ T^{12} + \cdots + 3013021881 $ T^12 - 33T^11 + 480T^10 - 5174T^9 + 51567T^8 - 236199T^7 - 516419T^6 + 10609785T^5 + 5626503T^4 - 93830778T^3 + 379494072T^2 - 511309665*T + 3013021881
$79$	$ T^{12} - 33 T^{11} + \cdots + 3308761 $ T^12 - 33T^11 + 549T^10 - 6201T^9 + 51462T^8 - 312207T^7 + 1446958T^6 - 5295576T^5 + 15671664T^4 - 34096068T^3 + 46487817T^2 - 21271386*T + 3308761
$83$	$ T^{12} + \cdots + 268009641 $ T^12 + 36T^11 + 873T^10 + 13014T^9 + 149967T^8 + 1194102T^7 + 7895781T^6 + 36625608T^5 + 189987606T^4 + 670468104T^3 + 3213500931T^2 + 953430669*T + 268009641
$89$	$ T^{12} + \cdots + 1067682024369 $ T^12 + 6T^11 + 333T^10 + 706T^9 + 12930T^8 - 95148T^7 - 671072T^6 - 31785591T^5 + 349913196T^4 + 2699251977T^3 - 22612802283T^2 - 62855881497*T + 1067682024369
$97$	$ T^{12} + \cdots + 63635612121 $ T^12 - 21T^11 + 27T^10 + 4143T^9 - 60552T^8 + 114669T^7 + 12824280T^6 - 242377920T^5 + 2597377266T^4 - 15668329608T^3 + 68954753331T^2 + 103527914400*T + 63635612121
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:	\( N \)	\(=\)	\( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	798.bo (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{2} - \beta_{7} q^{3} + (\beta_{8} - \beta_1) q^{4} + ( - \beta_{11} - \beta_{9} + \beta_{2} + 1) q^{5} + \beta_{8} q^{6} + ( - \beta_{9} + 1) q^{7} + \beta_{9} q^{8} + \beta_1 q^{9} + ( - \beta_{9} - \beta_{4} + \beta_{3} + 1) q^{10}+ \cdots + (\beta_{11} + \beta_{9} - \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) q - b4 * q^2 - b7 * q^3 + (b8 - b1) * q^4 + (-b11 - b9 + b2 + 1) * q^5 + b8 * q^6 + (-b9 + 1) * q^7 + b9 * q^8 + b1 * q^9 + (-b9 - b4 + b3 + 1) * q^10 + (b10 - b4 + b3 - b2 + b1) * q^11 + (b9 - 1) * q^12 + (-b11 + b10 - 2b9 - b7 - b5 + b3 + b2 + b1) q^13 + (b7 - b4) * q^14 + (-b10 + b9 - b4) * q^15 - b7 * q^16 + (-b10 - b9 + 2b8 + b7 + b6 + b4 - b3 + b2 + 2) q^17 - q^18 + (b11 + b10 - b9 + 2b8 - b7 - b6 - 2b4 - b1 + 2) * q^19 + (b7 + b6 + b5 - b4 - b1) * q^20 - b4 * q^21 + (b10 - b7 + b6 + b4 - b1 - 1) * q^22 + (-b11 - b10 + b9 - 2b8 - b6 - b5 - b4 + 3b1 - 1) * q^23 + (-b7 + b4) * q^24 + (-2b11 - b9 + b6 - b5 + b4 + b2 + 2b1) * q^25 + (-b11 - b9 + b6 + b4 + b3) * q^26 - b9 * q^27 - b1 * q^28 + (-b11 + b7 + b5 - b4 - b3 - b1) * q^29 + (b8 + b5 - b4 - b1) * q^30 + (b10 - 2b9 + 2b8 - 3b7 - b6 + b4 + b2 + b1 + 2) q^31 + b8 * q^32 + (b10 - 2b9 + b8 - b7 + b6 + b5 + b4 + b3 - b1) q^33 + (-b11 - b10 + 2b9 + 2b7 - b6 - 3b4 + b2 + b1 - 2) q^34 + (-b11 - b9 - b8 + b1) * q^35 + b4 * q^36 + (-b11 - b10 + b9 - 2b8 - 3b7 - 2b6 - 2b5 - b3 + b1 - 1) * q^37 + (b11 - b10 + 3b9 + 2b8 + b7 - b5 - b4 - b3 - b2 - 2b1 - 2) q^38 + (-b11 - b10 + b8 + 2b7 + b6 + b5 - b4 + b2) q^39 + (b8 + b2 - b1 + 1) * q^40 + (-b10 + 2b9 + b7 + b6 + 2b5 - b4 - 2b3 - 3b1 - 3) * q^41 + (b8 - b1) * q^42 + (-b11 + 2b9 + b8 - 3b6 - 3b3 + b2) q^43 + (-b11 + b8 - b7 - b5 + 2b4 + b2 + b1 + 1) q^44 + (b8 - b7 - b6) * q^45 + (b10 - 3b9 - b7 + b5 + b3 - b2 - b1) q^46 + (2b9 - b6 + b4 + b3 - b2 + b1 - 2) q^47 + b1 * q^48 - b9 * q^49 + (-2b11 + b10 - 2b9 - b7 + 2b3 + b2 + b1) q^50 + (-b10 + b9 - b6 - b5 - 2b4 - b3 + b2 - 2) q^51 + (-b11 + b10 - b9 - b8 + b6 + b5 + b3 + b2 + b1 + 1) * q^52 + (3b9 + 4b8 + 3b7 + b5 - 4b4 + b2 - 4b1 - 2) q^53 + b7 * q^54 + (2b10 - 6b9 + 2b8 - 2b7 - 2b6 - b4 + 2b3 - b2 + 2b1 + 2) q^55 + q^56 + (b9 + b8 + b6 - b3 - b2 + b1 - 2) * q^57 + (b11 + b10 - b9 + b8 - b7 - b6 - b5 + b3 - b1 + 2) * q^58 + (-b10 - b9 - b8 + b7 + b6 + b5 + 3b2 + 2b1 - 1) * q^59 + (b11 + b9 + b8 - b1) * q^60 + (b11 + 2b10 - 2b8 + 2b7 + b6 + 2b5 - b4 + b2 + b1) * q^61 + (b11 - b10 + 2b9 + b8 + b7 - b5 - b4 - b2 + b1 - 3) q^62 + b8 * q^63 + (b9 - 1) * q^64 + (-b11 + 2b10 - 5b9 - 4b8 - 2b7 - 2b5 + 5b4 + b3 - b2 + 2b1) q^65 + (b9 + b7 + b6 + b4 + b2 + b1) * q^66 + (-b9 + 3b7 + 2b6 - 2b4 - b3 + 2b2 - 2b1 + 4) q^67 + (b11 - b9 - 2b7 + b5 + b4 + b3 - b2 - 3b1) * q^68 + (-b11 - 3b9 + b8 - b7 - b6 + b3 + 2) q^69 + (b10 - 2b9 + b3 + 1) q^70 + (-3b11 + 4b10 - 4b9 + 3b8 - b7 + 2b6 + 4b5 + b4 + 2b3 + 3b2 + b1 - 1) * q^71 + (-b8 + b1) * q^72 + (3b11 - 4b9 - 3b8 + b7 - b6 - b5 + b3 + 3b1 + 6) * q^73 + (b10 - 3b9 + 2b8 - b7 - b6 + b3 - 2b2 + 2) q^74 + (-b11 - b10 - b9 + b8 - b4 + b3 + 2b2) q^75 + (-b11 + 3b9 + b8 - b7 - b6 + b4 - b3 - b1 - 1) q^76 + (b11 + b10 + b8 + b7 - b2 - b1 + 1) * q^77 + (b5 - b4 + b3 + b2 - b1) * q^78 + (2b11 - b10 + b9 + b8 - b6 - b4 + b1 + 3) q^79 + (-b10 + b9 - b4) * q^80 + (b7 - b4) * q^81 + (b11 + 3b8 - b7 - 2b6 - b5 + 2b4 + b2 - b1 + 3) q^82 + (2b11 - 2b10 + 9b9 + 4b7 - b6 - b4 - 2b3 - 2b2 - 4b1 - 7) q^83 + b9 * q^84 + (-2b11 + 2b9 + 4b7 + 2b6 + 2b5 - b3 + 2b2 - 3b1) q^85 + (3b11 - 3b7 - 3b6 - 3b5 + b3 - 3b2) q^86 + (b11 + b9 - b7 - b5 + b3 - b2) * q^87 + (-b11 - b8 - b7 - b4 + b3 + 2b1 - 1) q^88 + (5b11 + b9 + 2b8 + b7 - 3b6 + 2b5 - 3b4 - 2b2 - 5b1 + 1) q^89 + (b11 + b9 - b2 - 1) * q^90 + (-b11 + b10 - b6 - b5 + b1 - 1) * q^91 + (b11 + b10 + 2b7 + b6 + b4 + 1) q^92 + (-b10 - 2b8 + b7 + b6 - b4 - b3 - b2 + 3b1 - 2) * q^93 + (b11 + b10 - 3b8 - 2b7 + b6 + b5 + 2b4 - b2 + b1 - 1) q^94 + (-b10 - 2b9 + 4b8 - 2b7 + 3b5 - 5b4 + b3 - 6b1 - 1) * q^95 - q^96 + (b10 + 5b9 - b8 - b7 - b6 - 3b5 + 7b4 - 2b3 - 4b2 - 2b1 - 1) * q^97 + b7 * q^98 + (b11 + b9 - b8 + 2b7 + b6 + b5 - b4 - 1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 9 q^{5} + 6 q^{7} + 6 q^{8} + 9 q^{10} + 3 q^{11} - 6 q^{12} - 9 q^{13} + 9 q^{15} + 15 q^{17} - 12 q^{18} + 9 q^{19} - 15 q^{22} + 3 q^{23} + 3 q^{25} + 3 q^{26} - 6 q^{27} + 3 q^{29} + 6 q^{31}+ \cdots - 12 q^{99}+O(q^{100}) \) 12 * q + 9 * q^5 + 6 * q^7 + 6 * q^8 + 9 * q^10 + 3 * q^11 - 6 * q^12 - 9 * q^13 + 9 * q^15 + 15 * q^17 - 12 * q^18 + 9 * q^19 - 15 * q^22 + 3 * q^23 + 3 * q^25 + 3 * q^26 - 6 * q^27 + 3 * q^29 + 6 * q^31 - 12 * q^33 - 6 * q^34 - 9 * q^38 + 6 * q^39 + 9 * q^40 - 27 * q^41 + 6 * q^43 + 15 * q^44 - 15 * q^46 - 6 * q^47 - 6 * q^49 - 21 * q^51 + 9 * q^52 - 9 * q^53 - 9 * q^55 + 12 * q^56 - 18 * q^57 + 12 * q^58 - 24 * q^59 - 15 * q^61 - 24 * q^62 - 6 * q^64 - 24 * q^65 + 3 * q^66 + 33 * q^67 - 6 * q^68 + 15 * q^69 - 33 * q^71 + 33 * q^73 + 12 * q^74 + 9 * q^76 + 6 * q^77 + 33 * q^79 + 9 * q^80 + 27 * q^82 - 36 * q^83 + 6 * q^84 + 15 * q^85 - 6 * q^86 + 6 * q^87 - 3 * q^88 - 6 * q^89 - 9 * q^90 - 9 * q^91 + 3 * q^92 - 21 * q^93 - 18 * q^94 - 18 * q^95 - 12 * q^96 + 21 * q^97 - 12 * q^99

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	798.2.bo.d

Level	$798$
Weight	$2$
Character orbit	798.bo
Analytic conductor	$6.372$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(6.37206208130\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 18x^{10} + 153x^{8} - 773x^{6} + 2448x^{4} - 4608x^{2} + 4096 \) x^12 - 18x^10 + 153x^8 - 773x^6 + 2448x^4 - 4608*x^2 + 4096
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

\(\beta_{1}\)	\(=\)	\( ( - 5 \nu^{11} - 4 \nu^{10} + 42 \nu^{9} + 72 \nu^{8} - 157 \nu^{7} - 612 \nu^{6} + 105 \nu^{5} + \cdots + 12288 ) / 2048 \) (-5v^11 - 4v^10 + 42v^9 + 72v^8 - 157v^7 - 612v^6 + 105v^5 + 3092v^4 + 1056v^3 - 8768v^2 - 3072*v + 12288) / 2048
\(\beta_{2}\)	\(=\)	\( ( 3 \nu^{11} - 20 \nu^{10} - 54 \nu^{9} + 168 \nu^{8} + 459 \nu^{7} - 628 \nu^{6} - 2319 \nu^{5} + \cdots - 13312 ) / 2048 \) (3v^11 - 20v^10 - 54v^9 + 168v^8 + 459v^7 - 628v^6 - 2319v^5 + 420v^4 + 6320v^3 + 4224v^2 - 8704*v - 13312) / 2048
\(\beta_{3}\)	\(=\)	\( ( 3 \nu^{11} + 20 \nu^{10} - 54 \nu^{9} - 168 \nu^{8} + 459 \nu^{7} + 628 \nu^{6} - 2319 \nu^{5} + \cdots + 13312 ) / 2048 \) (3v^11 + 20v^10 - 54v^9 - 168v^8 + 459v^7 + 628v^6 - 2319v^5 - 420v^4 + 6320v^3 - 4224v^2 - 8704*v + 13312) / 2048
\(\beta_{4}\)	\(=\)	\( ( 5 \nu^{11} - 4 \nu^{10} - 42 \nu^{9} + 72 \nu^{8} + 157 \nu^{7} - 612 \nu^{6} - 105 \nu^{5} + \cdots + 12288 ) / 2048 \) (5v^11 - 4v^10 - 42v^9 + 72v^8 + 157v^7 - 612v^6 - 105v^5 + 3092v^4 - 1056v^3 - 8768v^2 + 3072*v + 12288) / 2048
\(\beta_{5}\)	\(=\)	\( ( -\nu^{10} + 18\nu^{8} - 153\nu^{6} + 709\nu^{4} - 1872\nu^{2} + 64\nu + 2304 ) / 128 \) (-v^10 + 18v^8 - 153v^6 + 709v^4 - 1872v^2 + 64*v + 2304) / 128
\(\beta_{6}\)	\(=\)	\( ( \nu^{10} - 18\nu^{8} + 153\nu^{6} - 709\nu^{4} + 1872\nu^{2} + 64\nu - 2304 ) / 128 \) (v^10 - 18v^8 + 153v^6 - 709v^4 + 1872v^2 + 64*v - 2304) / 128
\(\beta_{7}\)	\(=\)	\( ( - \nu^{11} - 28 \nu^{10} + 18 \nu^{9} + 440 \nu^{8} - 153 \nu^{7} - 3132 \nu^{6} + 773 \nu^{5} + \cdots + 33792 ) / 2048 \) (-v^11 - 28v^10 + 18v^9 + 440v^8 - 153v^7 - 3132v^6 + 773v^5 + 12876v^4 - 2448v^3 - 30336v^2 + 3584v + 33792) / 2048
\(\beta_{8}\)	\(=\)	\( ( \nu^{11} - 28 \nu^{10} - 18 \nu^{9} + 440 \nu^{8} + 153 \nu^{7} - 3132 \nu^{6} - 773 \nu^{5} + \cdots + 33792 ) / 2048 \) (v^11 - 28v^10 - 18v^9 + 440v^8 + 153v^7 - 3132v^6 - 773v^5 + 12876v^4 + 2448v^3 - 30336v^2 - 3584v + 33792) / 2048
\(\beta_{9}\)	\(=\)	\( ( -9\nu^{11} + 146\nu^{9} - 1089\nu^{7} + 4509\nu^{5} - 10688\nu^{3} + 11520\nu + 1024 ) / 2048 \) (-9v^11 + 146v^9 - 1089v^7 + 4509v^5 - 10688v^3 + 11520v + 1024) / 2048
\(\beta_{10}\)	\(=\)	\( ( 15 \nu^{11} - 48 \nu^{10} - 222 \nu^{9} + 608 \nu^{8} + 1431 \nu^{7} - 3760 \nu^{6} - 5275 \nu^{5} + \cdots + 23552 ) / 2048 \) (15v^11 - 48v^10 - 222v^9 + 608v^8 + 1431v^7 - 3760v^6 - 5275v^5 + 13296v^4 + 10880v^3 - 27136v^2 - 9984*v + 23552) / 2048
\(\beta_{11}\)	\(=\)	\( ( 7\nu^{11} - 110\nu^{9} + 783\nu^{7} - 3219\nu^{5} + 7584\nu^{3} + 256\nu^{2} - 8448\nu - 1024 ) / 512 \) (7v^11 - 110v^9 + 783v^7 - 3219v^5 + 7584v^3 + 256v^2 - 8448*v - 1024) / 512

\(\nu\)	\(=\)	\( \beta_{6} + \beta_{5} \) b6 + b5
\(\nu^{2}\)	\(=\)	\( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{3} + 3 \) b11 - b10 + b9 + b7 - b3 + 3
\(\nu^{3}\)	\(=\)	\( 3\beta_{8} - 3\beta_{7} + 2\beta_{6} + 2\beta_{5} - \beta_{3} - \beta_{2} \) 3b8 - 3b7 + 2b6 + 2b5 - b3 - b2
\(\nu^{4}\)	\(=\)	\( 5\beta_{11} - 5\beta_{10} + 5\beta_{9} + 5\beta_{7} + \beta_{6} - \beta_{5} + 4\beta_{4} - 5\beta_{3} + 4\beta _1 + 3 \) 5b11 - 5b10 + 5b9 + 5b7 + b6 - b5 + 4b4 - 5b3 + 4*b1 + 3
\(\nu^{5}\)	\(=\)	\( - \beta_{11} - \beta_{10} - 7 \beta_{9} + 13 \beta_{8} - 12 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots + 3 \) -b11 - b10 - 7b9 + 13b8 - 12b7 - 2b6 - 2b5 - 8b3 - 7*b2 + 3
\(\nu^{6}\)	\(=\)	\( 11 \beta_{11} - 11 \beta_{10} + 11 \beta_{9} + 3 \beta_{8} + 14 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} + \cdots - 19 \) 11b11 - 11b10 + 11b9 + 3b8 + 14b7 + 11b6 - 11b5 + 28b4 - 10b3 - b2 + 28b1 - 19
\(\nu^{7}\)	\(=\)	\( - 14 \beta_{11} - 14 \beta_{10} - 74 \beta_{9} + 27 \beta_{8} - 13 \beta_{7} - 29 \beta_{6} - 29 \beta_{5} + \cdots + 30 \) -14b11 - 14b10 - 74b9 + 27b8 - 13b7 - 29b6 - 29b5 + 4b4 - 31b3 - 17b2 - 4*b1 + 30
\(\nu^{8}\)	\(=\)	\( - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 38 \beta_{8} + 36 \beta_{7} + 61 \beta_{6} - 61 \beta_{5} + \cdots - 114 \) -2b11 + 2b10 - 2b9 + 38b8 + 36b7 + 61b6 - 61b5 + 68b4 + 20b3 - 18b2 + 68*b1 - 114
\(\nu^{9}\)	\(=\)	\( - 99 \beta_{11} - 99 \beta_{10} - 389 \beta_{9} + 48 \beta_{8} + 51 \beta_{7} - 94 \beta_{6} + \cdots + 145 \) -99b11 - 99b10 - 389b9 + 48b8 + 51b7 - 94b6 - 94b5 + 72b4 - 43b3 + 56b2 - 72*b1 + 145
\(\nu^{10}\)	\(=\)	\( - 46 \beta_{11} + 46 \beta_{10} - 46 \beta_{9} + 225 \beta_{8} + 179 \beta_{7} + 188 \beta_{6} + \cdots - 330 \) -46b11 + 46b10 - 46b9 + 225b8 + 179b7 + 188b6 - 188b5 - 224b4 + 217b3 - 171b2 - 224*b1 - 330
\(\nu^{11}\)	\(=\)	\( - 413 \beta_{11} - 413 \beta_{10} - 1091 \beta_{9} + 462 \beta_{8} - 49 \beta_{7} - 113 \beta_{6} + \cdots + 339 \) -413b11 - 413b10 - 1091b9 + 462b8 - 49b7 - 113b6 - 113b5 + 684b4 + 233b3 + 646b2 - 684*b1 + 339

\(n\)	\(115\)	\(211\)	\(533\)
\(\chi(n)\)	\(1\)	\(-\beta_{1} + \beta_{8}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 43.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{6} - T^{3} + 1)^{2} \) (T^6 - T^3 + 1)^2
$3$	\( (T^{6} + T^{3} + 1)^{2} \) (T^6 + T^3 + 1)^2
$5$	\( T^{12} - 9 T^{11} + \cdots + 1 \) T^12 - 9T^11 + 39T^10 - 108T^9 + 255T^8 - 459T^7 + 343T^6 + 207T^5 + 33T^4 + 999T^3 + 870T^2 + 45*T + 1
$7$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$11$	\( T^{12} - 3 T^{11} + \cdots + 110889 \) T^12 - 3T^11 + 36T^10 - 155T^9 + 1119T^8 - 3735T^7 + 12619T^6 - 23229T^5 + 49581T^4 - 66600T^3 + 122877T^2 - 110889*T + 110889
$13$	\( T^{12} + 9 T^{11} + \cdots + 11449 \) T^12 + 9T^11 + 63T^10 + 385T^9 + 1593T^8 + 4212T^7 + 7473T^6 + 27T^5 - 20817T^4 - 28316T^3 + 80361T^2 - 47187*T + 11449
$17$	\( T^{12} - 15 T^{11} + \cdots + 4289041 \) T^12 - 15T^11 + 141T^10 - 956T^9 + 5445T^8 - 21267T^7 + 55317T^6 - 68490T^5 - 118323T^4 + 661939T^3 - 527415T^2 - 2093781*T + 4289041
$19$	\( T^{12} - 9 T^{11} + \cdots + 47045881 \) T^12 - 9T^11 + 54T^10 - 288T^9 + 1530T^8 - 8667T^7 + 40789T^6 - 164673T^5 + 552330T^4 - 1975392T^3 + 7037334T^2 - 22284891*T + 47045881
$23$	\( T^{12} - 3 T^{11} + \cdots + 103041 \) T^12 - 3T^11 - 30T^10 - 112T^9 + 540T^8 + 9330T^7 + 53734T^6 + 179391T^5 + 404793T^4 + 640509T^3 + 686970T^2 + 384237*T + 103041
$29$	\( T^{12} - 3 T^{11} + \cdots + 3249 \) T^12 - 3T^11 - 12T^10 - 40T^9 + 1539T^8 - 7923T^7 + 18490T^6 - 26511T^5 + 34956T^4 - 31440T^3 + 22797T^2 - 12312*T + 3249
$31$	\( T^{12} - 6 T^{11} + \cdots + 16410601 \) T^12 - 6T^11 + 84T^10 - 108T^9 + 2688T^8 - 1293T^7 + 61072T^6 + 45546T^5 + 736494T^4 + 448848T^3 + 5321973T^2 + 5821287*T + 16410601
$37$	\( (T^{6} - 108 T^{4} + \cdots + 8691)^{2} \) (T^6 - 108T^4 - 425T^3 + 1092T^2 + 7452T + 8691)^2
$41$	\( T^{12} + 27 T^{11} + \cdots + 59049 \) T^12 + 27T^11 + 279T^10 + 1584T^9 + 9153T^8 + 33615T^7 + 78327T^6 - 29160T^5 + 98415T^4 - 737019T^3 + 570807T^2 + 295245*T + 59049
$43$	\( T^{12} + \cdots + 1868573529 \) T^12 - 6T^11 + 186T^10 - 1306T^9 + 12414T^8 - 150186T^7 + 1269847T^6 - 12335166T^5 + 135293490T^4 - 733298454T^3 + 1267561872T^2 + 3529398096*T + 1868573529
$47$	\( T^{12} + 6 T^{11} + \cdots + 4915089 \) T^12 + 6T^11 + 57T^10 + 205T^9 - 45T^8 + 3003T^7 + 23119T^6 + 22143T^5 + 181341T^4 + 626622T^3 - 439767T^2 - 1097415*T + 4915089
$53$	\( T^{12} + 9 T^{11} + \cdots + 5157441 \) T^12 + 9T^11 - 21T^10 + 34T^9 + 6909T^8 + 56505T^7 + 299041T^6 + 910767T^5 - 1478655T^4 - 4257489T^3 + 62300673T^2 - 6581358*T + 5157441
$59$	\( T^{12} + 24 T^{11} + \cdots + 5948721 \) T^12 + 24T^11 + 192T^10 + 602T^9 + 2832T^8 - 11022T^7 - 28421T^6 - 321696T^5 + 1366614T^4 - 1798758T^3 + 55872180T^2 - 27877770*T + 5948721
$61$	\( T^{12} + 15 T^{11} + \cdots + 811801 \) T^12 + 15T^11 + 129T^10 + 459T^9 - 7926T^8 - 77961T^7 + 201142T^6 + 2832816T^5 + 8917404T^4 - 2858202T^3 + 18453723T^2 - 5860104*T + 811801
$67$	\( T^{12} - 33 T^{11} + \cdots + 104329 \) T^12 - 33T^11 + 570T^10 - 7648T^9 + 83115T^8 - 665955T^7 + 4217799T^6 - 19195425T^5 + 51797151T^4 - 65840398T^3 + 40111458T^2 - 3103707*T + 104329
$71$	\( T^{12} + \cdots + 159823248841 \) T^12 + 33T^11 + 795T^10 + 13894T^9 + 186579T^8 + 1744767T^7 + 12094113T^6 + 23806782T^5 - 453159225T^4 - 1015688891T^3 + 37051974411T^2 - 138243977979*T + 159823248841
$73$	\( T^{12} + \cdots + 3013021881 \) T^12 - 33T^11 + 480T^10 - 5174T^9 + 51567T^8 - 236199T^7 - 516419T^6 + 10609785T^5 + 5626503T^4 - 93830778T^3 + 379494072T^2 - 511309665*T + 3013021881
$79$	\( T^{12} - 33 T^{11} + \cdots + 3308761 \) T^12 - 33T^11 + 549T^10 - 6201T^9 + 51462T^8 - 312207T^7 + 1446958T^6 - 5295576T^5 + 15671664T^4 - 34096068T^3 + 46487817T^2 - 21271386*T + 3308761
$83$	\( T^{12} + \cdots + 268009641 \) T^12 + 36T^11 + 873T^10 + 13014T^9 + 149967T^8 + 1194102T^7 + 7895781T^6 + 36625608T^5 + 189987606T^4 + 670468104T^3 + 3213500931T^2 + 953430669*T + 268009641
$89$	\( T^{12} + \cdots + 1067682024369 \) T^12 + 6T^11 + 333T^10 + 706T^9 + 12930T^8 - 95148T^7 - 671072T^6 - 31785591T^5 + 349913196T^4 + 2699251977T^3 - 22612802283T^2 - 62855881497*T + 1067682024369
$97$	\( T^{12} + \cdots + 63635612121 \) T^12 - 21T^11 + 27T^10 + 4143T^9 - 60552T^8 + 114669T^7 + 12824280T^6 - 242377920T^5 + 2597377266T^4 - 15668329608T^3 + 68954753331T^2 + 103527914400*T + 63635612121
show more
show less