LMFDB - Newform orbit 1183.2.e.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1183,2,Mod(170,1183)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1183, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 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("1183.170"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1183 = 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1183.e (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$9.44630255912$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 11x^{14} + 85x^{12} + 334x^{10} + 952x^{8} + 1050x^{6} + 853x^{4} + 93x^{2} + 9 $ x^16 + 11x^14 + 85x^12 + 334x^10 + 952x^8 + 1050x^6 + 853x^4 + 93*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{9} q^{2} + ( - \beta_{6} - \beta_{4} - 1) q^{3} + ( - \beta_{10} - \beta_{6} - 1) q^{4} + \beta_{13} q^{5} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{6} + (\beta_{14} + \beta_{13} + \cdots + \beta_{9}) q^{7}+ \cdots + (2 \beta_{15} + \beta_{14} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) $ q + b9 * q^2 + (-b6 - b4 - 1) * q^3 + (-b10 - b6 - 1) * q^4 + b13 * q^5 + (-b13 - b12 - b9 + b8 - b1) * q^6 + (b14 + b13 - b12 + b9) * q^7 - b12 * q^8 + (-b11 + b7 + 2b6 + b5 + b4 + 1) q^9 + (b11 + b10 - 2b4) q^10 + (b15 - b14 - b8 - b3 - b1) * q^11 + (-b11 + b10 + b7 + 3b6 + b5 + b4 - b2 + 1) q^12 + (b11 + b10 - b4 - b2 - 2) * q^14 + (-b13 - 2b9 + b8 - 2b1) * q^15 + (-b10 + b5 + b4 + b2 + 1) * q^16 + (-b11 - b6 - 1) * q^17 + (b15 - b14 - 4b8 - b3 + 3b1) * q^18 + (-b12 + 2b9 - b3) q^19 + (-2b15 - b14 - 3b13 + b12 - 2b9 + 3b8 + b3 - b1) * q^20 + (-b14 - b13 - 2b12 - b9 - b3) q^21 + (-b7 - b5 - b2 - 3) * q^22 + (-b6 + b5 + b4 + 1) * q^23 + (b15 - b14 - 2b8 + 3b1) * q^24 + (-b11 - 2b10 + b4) q^25 + (-b7 - b5 + 2b2 + 2) q^27 + (b14 - 2b13 - b12 - b9 + 2b8 + b1) * q^28 + (-3b5 - 2b2 - 3) * q^29 + (-b11 + b10 + b7 + 6b6 + 2b5 + 2b4 - b2 + 2) q^30 + (b15 - b14 - b1) * q^31 + (-b8 + 2b3 - b1) q^32 + (b15 + 2b14 + b9 + b3 + b1) q^33 + (2b15 + b14 + 3b13 - 2b12 + 2b9 - 3b8 - b3 + b1) q^34 + (-b11 - 2b10 + b7 - 3b6 + b5 + b4 - 1) * q^35 + (-2b7 - 5b5 + 1) * q^36 + (-b15 - 2b14 - 3b9 - b3 - b1) * q^37 + (-b10 - 4b6 + b4 - 4) q^38 + (-b11 + b7 + b6 + b5 + b4 + 1) * q^40 + (-2b15 - b14 + b13 + 2b12 - b9 - b8 + b3) * q^41 + (-b11 + 2b10 + 6b6 + 2b5 + 4b4 - b2 + 6) * q^42 + (b7 - 2b5 - 2) q^43 + (-b15 - 2b14 + 2b13 - 2b9 - b3 - b1) q^44 + (-2b3 + 4b1) * q^45 + (-b8 - b3 - b1) * q^46 + (-b15 - 2b14 + b13 + 3b12 - b9 + 2b3 - b1) q^47 + (-b2 + 2) * q^48 + (-2b10 + b7 + 3b5 + 3b4 + 2b2) * q^49 + (2b15 + b14 + 4b13 - 3b12 + b9 - 4b8 - b3) * q^50 + (-2b10 + b6 - b5 - b4 + 2b2 - 1) * q^51 + (-2b10 - 3b6 - 3) * q^53 + (b15 + 2b14 + 4b13 + b12 + 8b9 + 2b3 + b1) * q^54 + (b7 + 3b5 + 4b2 + 4) * q^55 + (2b6 + 2b5 + 3b4 + 5) q^56 + (2b15 + b14 - 4b12 + 3b9 - b3 + 2b1) * q^57 + (3b13 + b12 - 2b9 + b3) * q^58 + (-b15 + b14 + 3b8 - b3 - 3b1) * q^59 + (b15 - b14 - 3b8 - 3b3 + 4b1) q^60 + (-2b10 + b6 + 2b2) * q^61 + (2b5 - b2 - 2) q^62 + (b15 + 3b13 + 3b9 - 3b8 + 7b1) * q^63 + (-b7 + 2b5 - 2b2 - 5) * q^64 - b10 * q^66 + (-2b8 - 3b3) * q^67 + (b11 + 3b10 - b7 - b6 - 4b5 - 4b4 - 3b2 - 4) * q^68 + (-b7 + b5 + 4) * q^69 + (b15 - b14 - 2b12 - 4b9 - 4b8 - b3 - 4b1) * q^70 + (b13 + 3b12 - 4b9 - b8 - 4b1) q^71 + (3b13 + 3b12 + 2b9 + 3b3) * q^72 + (2b15 - 2b14 - 3b8 - 2b3 + 2b1) q^73 + (3b10 + 6b6 + 6) * q^74 + (-b11 + b7 - 2b5 - 2b4 - 2) * q^75 + (b13 - 2b12 - b9 - b8 - b1) q^76 + (4b10 + b7 - 3b6 - b5 - 3b4 - 2b2 + 2) * q^77 + (b11 - b7 - 3b6 - b5 - b4 - 1) q^79 + (-b15 + b14 + 2b8 + b3 + 2b1) * q^80 + (-4b10 - 5b6 - 2b4 - 5) q^81 + (b11 - b7 - 4b6 - 4b5 - 4b4 - 4) q^82 + (2b15 + b14 + b13 - 2b12 + 7b9 - b8 - b3 + 6b1) * q^83 + (-b14 + 3b13 + b12 + 4b9 - 5b8 + 8b1) * q^84 + (-3b13 + 2b12 + 2b9 + 3b8 + 2b1) q^85 + (-b15 - 2b14 - b13 + 4b12 - 3b9 + 3b3 - b1) * q^86 + (b11 + 2b10 - 8b6 - 8) * q^87 + (2b10 - b6 - 2b4 - 1) * q^88 + (-b15 - 2b14 - 2b13 - 2b12 + 3b9 - 3b3 - b1) q^89 + (-2b5 + 6b2 + 14) * q^90 + (-b7 - b5 - b2) * q^92 + (-3b13 + 2b12 - 6b9 + 2b3) * q^93 + (b11 - b10 - 6b6 - 5b4 - 6) * q^94 + (b11 + 2b10 + b6 - 3b4 + 1) * q^95 + (-2b15 - 4b14 - 4b13 + 3b12 - 9b9 + b3 - 2b1) * q^96 + (2b15 + b14 + b13 + 3b9 - b8 - b3 + 2b1) q^97 + (-b15 - 2b14 - 3b13 + 2b12 - 6b9 - 3b8 - 3b1) * q^98 + (2b15 + b14 + 2b13 - b12 - b9 - 2b8 - b3 - 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 6 q^{4} - 12 q^{9} + 6 q^{10} - 18 q^{12} - 26 q^{14} + 2 q^{16} - 8 q^{17} - 36 q^{22} + 12 q^{23} + 32 q^{27} - 16 q^{29} - 38 q^{30} + 56 q^{36} - 34 q^{38} - 4 q^{40} + 16 q^{42} - 16 q^{43}+ \cdots + 16 q^{95}+O(q^{100}) $ 16 * q - 4 * q^3 - 6 * q^4 - 12 * q^9 + 6 * q^10 - 18 * q^12 - 26 * q^14 + 2 * q^16 - 8 * q^17 - 36 * q^22 + 12 * q^23 + 32 * q^27 - 16 * q^29 - 38 * q^30 + 56 * q^36 - 34 * q^38 - 4 * q^40 + 16 * q^42 - 16 * q^43 + 36 * q^48 - 40 * q^49 - 16 * q^51 - 20 * q^53 + 24 * q^55 + 36 * q^56 - 12 * q^61 - 44 * q^62 - 88 * q^64 + 2 * q^66 - 2 * q^68 + 56 * q^69 + 42 * q^74 - 8 * q^75 + 76 * q^77 + 20 * q^79 - 24 * q^81 + 16 * q^82 - 68 * q^87 - 4 * q^88 + 216 * q^90 + 12 * q^92 - 26 * q^94 + 16 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 11x^{14} + 85x^{12} + 334x^{10} + 952x^{8} + 1050x^{6} + 853x^{4} + 93x^{2} + 9 $ x^16 + 11*x^14 + 85*x^12 + 334*x^10 + 952*x^8 + 1050*x^6 + 853*x^4 + 93*x^2 + 9 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 24498 \nu^{14} + 246060 \nu^{12} + 1852321 \nu^{10} + 6411671 \nu^{8} + 17193085 \nu^{6} + \cdots - 40872159 ) / 14163622 $ (24498v^14 + 246060v^12 + 1852321v^10 + 6411671v^8 + 17193085v^6 + 6321845v^4 + 690027*v^2 - 40872159) / 14163622
$\beta_{3}$	$=$	$ ( - 24498 \nu^{15} - 246060 \nu^{13} - 1852321 \nu^{11} - 6411671 \nu^{9} - 17193085 \nu^{7} + \cdots + 55035781 \nu ) / 14163622 $ (-24498v^15 - 246060v^13 - 1852321v^11 - 6411671v^9 - 17193085v^7 - 6321845v^5 - 690027v^3 + 55035781v) / 14163622
$\beta_{4}$	$=$	$ ( 172099 \nu^{14} + 2170865 \nu^{12} + 17340370 \nu^{10} + 78484018 \nu^{8} + 236538400 \nu^{6} + \cdots + 23829231 ) / 42490866 $ (172099v^14 + 2170865v^12 + 17340370v^10 + 78484018v^8 + 236538400v^6 + 377649654v^4 + 218482087*v^2 + 23829231) / 42490866
$\beta_{5}$	$=$	$ ( 99072 \nu^{14} + 1000291 \nu^{12} + 7490944 \nu^{10} + 25929344 \nu^{8} + 66564370 \nu^{6} + \cdots - 20454191 ) / 14163622 $ (99072v^14 + 1000291v^12 + 7490944v^10 + 25929344v^8 + 66564370v^6 + 25566080v^4 + 2790528*v^2 - 20454191) / 14163622
$\beta_{6}$	$=$	$ ( 539569 \nu^{14} + 5861765 \nu^{12} + 45125185 \nu^{10} + 174659083 \nu^{8} + 494434675 \nu^{6} + \cdots + 5618970 ) / 42490866 $ (539569v^14 + 5861765v^12 + 45125185v^10 + 174659083v^8 + 494434675v^6 + 514968195v^4 + 441286822*v^2 + 5618970) / 42490866
$\beta_{7}$	$=$	$ ( - 102312 \nu^{14} - 1048444 \nu^{12} - 7735924 \nu^{10} - 26777324 \nu^{8} - 67022271 \nu^{6} + \cdots + 23341327 ) / 7081811 $ (-102312v^14 - 1048444v^12 - 7735924v^10 - 26777324v^8 - 67022271v^6 - 26402180v^4 - 2881788*v^2 + 23341327) / 7081811
$\beta_{8}$	$=$	$ ( 123570 \nu^{15} + 1246351 \nu^{13} + 9343265 \nu^{11} + 32341015 \nu^{9} + 83757455 \nu^{7} + \cdots - 61326350 \nu ) / 14163622 $ (123570v^15 + 1246351v^13 + 9343265v^11 + 32341015v^9 + 83757455v^7 + 31887925v^5 + 3480555v^3 - 61326350v) / 14163622
$\beta_{9}$	$=$	$ ( - 539569 \nu^{15} - 5861765 \nu^{13} - 45125185 \nu^{11} - 174659083 \nu^{9} + \cdots - 48109836 \nu ) / 42490866 $ (-539569v^15 - 5861765v^13 - 45125185v^11 - 174659083v^9 - 494434675v^7 - 514968195v^5 - 441286822v^3 - 48109836v) / 42490866
$\beta_{10}$	$=$	$ ( - 515071 \nu^{14} - 5615705 \nu^{12} - 43272864 \nu^{10} - 168247412 \nu^{8} - 477241590 \nu^{6} + \cdots - 46491129 ) / 14163622 $ (-515071v^14 - 5615705v^12 - 43272864v^10 - 168247412v^8 - 477241590v^6 - 508646350v^4 - 426433173*v^2 - 46491129) / 14163622
$\beta_{11}$	$=$	$ ( 3598 \nu^{14} + 40712 \nu^{12} + 317920 \nu^{10} + 1287475 \nu^{8} + 3722089 \nu^{6} + 4460568 \nu^{4} + \cdots + 366555 ) / 95271 $ (3598v^14 + 40712v^12 + 317920v^10 + 1287475v^8 + 3722089v^6 + 4460568v^4 + 3361729*v^2 + 366555) / 95271
$\beta_{12}$	$=$	$ ( 1079138 \nu^{15} + 11723530 \nu^{13} + 90250370 \nu^{11} + 349318166 \nu^{9} + \cdots + 11237940 \nu ) / 21245433 $ (1079138v^15 + 11723530v^13 + 90250370v^11 + 349318166v^9 + 988869350v^7 + 1029936390v^5 + 861328211v^3 + 11237940v) / 21245433
$\beta_{13}$	$=$	$ ( - 205171 \nu^{15} - 2261795 \nu^{13} - 17480377 \nu^{11} - 68898667 \nu^{9} - 196608895 \nu^{7} + \cdots - 19219314 \nu ) / 3862806 $ (-205171v^15 - 2261795v^13 - 17480377v^11 - 68898667v^9 - 196608895v^7 - 219868809v^5 - 176278948v^3 - 19219314v) / 3862806
$\beta_{14}$	$=$	$ ( 1137374 \nu^{15} + 12639109 \nu^{13} + 98087264 \nu^{11} + 390741062 \nu^{9} + \cdots + 167644965 \nu ) / 11588418 $ (1137374v^15 + 12639109v^13 + 98087264v^11 + 390741062v^9 + 1125399698v^7 + 1313214930v^5 + 1094093084v^3 + 167644965v) / 11588418
$\beta_{15}$	$=$	$ ( 17689120 \nu^{15} + 191553668 \nu^{13} + 1470474691 \nu^{11} + 5653350919 \nu^{9} + \cdots - 359333319 \nu ) / 127472598 $ (17689120v^15 + 191553668v^13 + 1470474691v^11 + 5653350919v^9 + 15847248841v^7 + 15781577445v^5 + 12180871093v^3 - 359333319v) / 127472598

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{10} + 3\beta_{6} - \beta_{2} $ b10 + 3*b6 - b2
$\nu^{3}$	$=$	$ -\beta_{12} - 4\beta_{9} - 4\beta_1 $ -b12 - 4b9 - 4b1
$\nu^{4}$	$=$	$ -5\beta_{10} - 14\beta_{6} - \beta_{4} - 14 $ -5b10 - 14b6 - b4 - 14
$\nu^{5}$	$=$	$ \beta_{13} + 6\beta_{12} + 19\beta_{9} + 6\beta_{3} $ b13 + 6b12 + 19b9 + 6*b3
$\nu^{6}$	$=$	$ -\beta_{7} - 8\beta_{5} + 24\beta_{2} + 61 $ -b7 - 8b5 + 24b2 + 61
$\nu^{7}$	$=$	$ \beta_{15} - \beta_{14} - 11\beta_{8} - 32\beta_{3} + 94\beta_1 $ b15 - b14 - 11b8 - 32b3 + 94*b1
$\nu^{8}$	$=$	$ -11\beta_{11} + 115\beta_{10} + 11\beta_{7} + 345\beta_{6} + 52\beta_{5} + 52\beta_{4} - 115\beta_{2} + 52 $ -11b11 + 115b10 + 11b7 + 345b6 + 52b5 + 52b4 - 115*b2 + 52
$\nu^{9}$	$=$	$ -22\beta_{15} - 11\beta_{14} - 85\beta_{13} - 145\beta_{12} - 493\beta_{9} + 85\beta_{8} + 11\beta_{3} - 482\beta_1 $ -22b15 - 11b14 - 85b13 - 145b12 - 493b9 + 85b8 + 11b3 - 482b1
$\nu^{10}$	$=$	$ 85\beta_{11} - 553\beta_{10} - 1736\beta_{6} - 315\beta_{4} - 1736 $ 85b11 - 553b10 - 1736b6 - 315b4 - 1736
$\nu^{11}$	$=$	$ 85\beta_{15} + 170\beta_{14} + 570\beta_{13} + 698\beta_{12} + 2544\beta_{9} + 783\beta_{3} + 85\beta_1 $ 85b15 + 170b14 + 570b13 + 698b12 + 2544b9 + 783b3 + 85*b1
$\nu^{12}$	$=$	$ -570\beta_{7} - 1838\beta_{5} + 2672\beta_{2} + 6935 $ -570b7 - 1838b5 + 2672*b2 + 6935
$\nu^{13}$	$=$	$ 570\beta_{15} - 570\beta_{14} - 3548\beta_{8} - 4510\beta_{3} + 12015\beta_1 $ 570b15 - 570b14 - 3548b8 - 4510b3 + 12015*b1
$\nu^{14}$	$=$	$ - 3548 \beta_{11} + 12977 \beta_{10} + 3548 \beta_{7} + 44495 \beta_{6} + 10466 \beta_{5} + \cdots + 10466 $ -3548b11 + 12977b10 + 3548b7 + 44495b6 + 10466b5 + 10466b4 - 12977*b2 + 10466
$\nu^{15}$	$=$	$ - 7096 \beta_{15} - 3548 \beta_{14} - 21110 \beta_{13} - 16347 \beta_{12} - 68116 \beta_{9} + \cdots - 64568 \beta_1 $ -7096b15 - 3548b14 - 21110b13 - 16347b12 - 68116b9 + 21110b8 + 3548b3 - 64568b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$.

$n$	$339$	$1016$
$\chi(n)$	$\beta_{6}$	$1$

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} $

170.1

−1.14241	−	1.97871i
−1.06275	−	1.84073i
−0.536527	−	0.929293i
−0.166188	−	0.287846i
0.166188	+	0.287846i
0.536527	+	0.929293i
1.06275	+	1.84073i
1.14241	+	1.97871i
−1.14241	+	1.97871i
−1.06275	+	1.84073i
−0.536527	+	0.929293i
−0.166188	+	0.287846i
0.166188	−	0.287846i
0.536527	−	0.929293i
1.06275	−	1.84073i
1.14241	−	1.97871i

−1.14241

1.97871i

−1.57521

−

2.72835i

−1.61019

−

2.78892i

−1.06250

1.84030i

7.19813

0.331665

2.62488i

2.78832

−3.46258

5.99736i

−2.42760

−

4.20473i

170.2

−1.06275

1.84073i

0.0894272

0.154892i

−1.25885

−

2.18040i

1.80301

−

3.12291i

−0.380153

2.35320

−

1.20931i

1.10038

1.48401

−

2.57037i

3.83229

6.63772i

170.3

−0.536527

0.929293i

1.21570

2.10566i

0.424277

0.734868i

0.312716

−

0.541640i

−2.60903

−1.21561

2.34996i

−3.05665

−1.45586

2.52163i

0.335561

0.581209i

170.4

−0.166188

0.287846i

−0.729919

−

1.26426i

0.944763

1.63638i

−0.722811

1.25195i

0.485214

−1.36920

−

2.26391i

−1.29278

0.434437

−

0.752468i

−0.240245

−

0.416116i

170.5

0.166188

−

0.287846i

−0.729919

−

1.26426i

0.944763

1.63638i

0.722811

−

1.25195i

−0.485214

1.36920

2.26391i

1.29278

0.434437

−

0.752468i

−0.240245

−

0.416116i

170.6

0.536527

−

0.929293i

1.21570

2.10566i

0.424277

0.734868i

−0.312716

0.541640i

2.60903

1.21561

−

2.34996i

3.05665

−1.45586

2.52163i

0.335561

0.581209i

170.7

1.06275

−

1.84073i

0.0894272

0.154892i

−1.25885

−

2.18040i

−1.80301

3.12291i

0.380153

−2.35320

1.20931i

−1.10038

1.48401

−

2.57037i

3.83229

6.63772i

170.8

1.14241

−

1.97871i

−1.57521

−

2.72835i

−1.61019

−

2.78892i

1.06250

−

1.84030i

−7.19813

−0.331665

−

2.62488i

−2.78832

−3.46258

5.99736i

−2.42760

−

4.20473i

508.1

−1.14241

−

1.97871i

−1.57521

2.72835i

−1.61019

2.78892i

−1.06250

−

1.84030i

7.19813

0.331665

−

2.62488i

2.78832

−3.46258

−

5.99736i

−2.42760

4.20473i

508.2

−1.06275

−

1.84073i

0.0894272

−

0.154892i

−1.25885

2.18040i

1.80301

3.12291i

−0.380153

2.35320

1.20931i

1.10038

1.48401

2.57037i

3.83229

−

6.63772i

508.3

−0.536527

−

0.929293i

1.21570

−

2.10566i

0.424277

−

0.734868i

0.312716

0.541640i

−2.60903

−1.21561

−

2.34996i

−3.05665

−1.45586

−

2.52163i

0.335561

−

0.581209i

508.4

−0.166188

−

0.287846i

−0.729919

1.26426i

0.944763

−

1.63638i

−0.722811

−

1.25195i

0.485214

−1.36920

2.26391i

−1.29278

0.434437

0.752468i

−0.240245

0.416116i

508.5

0.166188

0.287846i

−0.729919

1.26426i

0.944763

−

1.63638i

0.722811

1.25195i

−0.485214

1.36920

−

2.26391i

1.29278

0.434437

0.752468i

−0.240245

0.416116i

508.6

0.536527

0.929293i

1.21570

−

2.10566i

0.424277

−

0.734868i

−0.312716

−

0.541640i

2.60903

1.21561

2.34996i

3.05665

−1.45586

−

2.52163i

0.335561

−

0.581209i

508.7

1.06275

1.84073i

0.0894272

−

0.154892i

−1.25885

2.18040i

−1.80301

−

3.12291i

0.380153

−2.35320

−

1.20931i

−1.10038

1.48401

2.57037i

3.83229

−

6.63772i

508.8

1.14241

1.97871i

−1.57521

2.72835i

−1.61019

2.78892i

1.06250

1.84030i

−7.19813

−0.331665

2.62488i

−2.78832

−3.46258

−

5.99736i

−2.42760

4.20473i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
13.b	even	2	1	inner
91.r	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1183.2.e.i		16
7.c	even	3	1	inner	1183.2.e.i		16
7.c	even	3	1		8281.2.a.ck		8
7.d	odd	6	1		8281.2.a.cj		8
13.b	even	2	1	inner	1183.2.e.i		16
13.d	odd	4	2		91.2.r.a	✓	16
39.f	even	4	2		819.2.dl.e		16
91.i	even	4	2		637.2.r.f		16
91.r	even	6	1	inner	1183.2.e.i		16
91.r	even	6	1		8281.2.a.ck		8
91.s	odd	6	1		8281.2.a.cj		8
91.z	odd	12	2		91.2.r.a	✓	16
91.z	odd	12	2		637.2.c.f		8
91.bb	even	12	2		637.2.c.e		8
91.bb	even	12	2		637.2.r.f		16
273.cd	even	12	2		819.2.dl.e		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.r.a	✓	16	13.d	odd	4	2
91.2.r.a	✓	16	91.z	odd	12	2
637.2.c.e		8	91.bb	even	12	2
637.2.c.f		8	91.z	odd	12	2
637.2.r.f		16	91.i	even	4	2
637.2.r.f		16	91.bb	even	12	2
819.2.dl.e		16	39.f	even	4	2
819.2.dl.e		16	273.cd	even	12	2
1183.2.e.i		16	1.a	even	1	1	trivial
1183.2.e.i		16	7.c	even	3	1	inner
1183.2.e.i		16	13.b	even	2	1	inner
1183.2.e.i		16	91.r	even	6	1	inner
8281.2.a.cj		8	7.d	odd	6	1
8281.2.a.cj		8	91.s	odd	6	1
8281.2.a.ck		8	7.c	even	3	1
8281.2.a.ck		8	91.r	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1183, [\chi])$:

$ T_{2}^{16} + 11T_{2}^{14} + 85T_{2}^{12} + 334T_{2}^{10} + 952T_{2}^{8} + 1050T_{2}^{6} + 853T_{2}^{4} + 93T_{2}^{2} + 9 $

T2^16 + 11*T2^14 + 85*T2^12 + 334*T2^10 + 952*T2^8 + 1050*T2^6 + 853*T2^4 + 93*T2^2 + 9

$ T_{3}^{8} + 2T_{3}^{7} + 11T_{3}^{6} + 6T_{3}^{5} + 67T_{3}^{4} + 62T_{3}^{3} + 114T_{3}^{2} - 20T_{3} + 4 $

T3^8 + 2*T3^7 + 11*T3^6 + 6*T3^5 + 67*T3^4 + 62*T3^3 + 114*T3^2 - 20*T3 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 11 T^{14} + \cdots + 9 $ T^16 + 11T^14 + 85T^12 + 334T^10 + 952T^8 + 1050T^6 + 853T^4 + 93*T^2 + 9
$3$	$ (T^{8} + 2 T^{7} + 11 T^{6} + \cdots + 4)^{2} $ (T^8 + 2T^7 + 11T^6 + 6T^5 + 67T^4 + 62T^3 + 114T^2 - 20*T + 4)^2
$5$	$ T^{16} + 20 T^{14} + \cdots + 2304 $ T^16 + 20T^14 + 297T^12 + 1740T^10 + 7361T^8 + 14560T^6 + 20656T^4 + 7680*T^2 + 2304
$7$	$ T^{16} + 20 T^{14} + \cdots + 5764801 $ T^16 + 20T^14 + 217T^12 + 1612T^10 + 10652T^8 + 78988T^6 + 521017T^4 + 2352980*T^2 + 5764801
$11$	$ T^{16} + 52 T^{14} + \cdots + 729 $ T^16 + 52T^14 + 2108T^12 + 30312T^10 + 337509T^8 + 199832T^6 + 99508T^4 + 9180*T^2 + 729
$13$	$ T^{16} $ T^16
$17$	$ (T^{8} + 4 T^{7} + \cdots + 15129)^{2} $ (T^8 + 4T^7 + 36T^6 + 24T^5 + 485T^4 + 56T^3 + 5164T^2 - 6396*T + 15129)^2
$19$	$ T^{16} + 44 T^{14} + \cdots + 10673289 $ T^16 + 44T^14 + 1396T^12 + 19096T^10 + 185725T^8 + 971784T^6 + 3674044T^4 + 7618644*T^2 + 10673289
$23$	$ (T^{8} - 6 T^{7} + 31 T^{6} + \cdots + 36)^{2} $ (T^8 - 6T^7 + 31T^6 - 50T^5 + 91T^4 - 22T^3 + 130T^2 - 60*T + 36)^2
$29$	$ (T^{4} + 4 T^{3} + \cdots + 624)^{4} $ (T^4 + 4T^3 - 63T^2 - 208*T + 624)^4
$31$	$ T^{16} + \cdots + 1136229264 $ T^16 + 80T^14 + 4309T^12 + 128296T^10 + 2779213T^8 + 35364492T^6 + 309454636T^4 + 657036336*T^2 + 1136229264
$37$	$ T^{16} + 120 T^{14} + \cdots + 76527504 $ T^16 + 120T^14 + 10053T^12 + 422496T^10 + 12939021T^8 + 213389964T^6 + 2419355628T^4 + 433655856*T^2 + 76527504
$41$	$ (T^{8} - 132 T^{6} + \cdots + 292032)^{2} $ (T^8 - 132T^6 + 5732T^4 - 88192*T^2 + 292032)^2
$43$	$ (T^{4} + 4 T^{3} + \cdots - 104)^{4} $ (T^4 + 4T^3 - 66T^2 + 156*T - 104)^4
$47$	$ T^{16} + \cdots + 57728231289 $ T^16 + 196T^14 + 26204T^12 + 1905768T^10 + 101089845T^8 + 2884224440T^6 + 56549167060T^4 + 58599199164*T^2 + 57728231289
$53$	$ (T^{8} + 10 T^{7} + \cdots + 7569)^{2} $ (T^8 + 10T^7 + 100T^6 + 260T^5 + 1213T^4 - 1740T^3 + 16900T^2 - 11310*T + 7569)^2
$59$	$ T^{16} + \cdots + 12487392009 $ T^16 + 188T^14 + 24868T^12 + 1646008T^10 + 79227709T^8 + 1652371368T^6 + 24989166028T^4 + 18073959780*T^2 + 12487392009
$61$	$ (T^{8} + 6 T^{7} + \cdots + 49729)^{2} $ (T^8 + 6T^7 + 60T^6 + 44T^5 + 917T^4 - 420T^3 + 14188T^2 - 20962*T + 49729)^2
$67$	$ T^{16} + \cdots + 66330457209 $ T^16 + 284T^14 + 59004T^12 + 5578872T^10 + 387569525T^8 + 6027737800T^6 + 75732974260T^4 + 73439011956*T^2 + 66330457209
$71$	$ (T^{8} - 292 T^{6} + \cdots + 397488)^{2} $ (T^8 - 292T^6 + 19829T^4 - 253084*T^2 + 397488)^2
$73$	$ T^{16} + \cdots + 8437677133824 $ T^16 + 260T^14 + 47625T^12 + 4249020T^10 + 273313457T^8 + 7922514640T^6 + 164987876800T^4 + 1371747640320*T^2 + 8437677133824
$79$	$ (T^{8} - 10 T^{7} + \cdots + 64)^{2} $ (T^8 - 10T^7 + 91T^6 - 210T^5 + 689T^4 + 380T^3 + 3672T^2 - 480*T + 64)^2
$83$	$ (T^{8} - 296 T^{6} + \cdots + 5483712)^{2} $ (T^8 - 296T^6 + 28692T^4 - 959920*T^2 + 5483712)^2
$89$	$ T^{16} + \cdots + 58102628210064 $ T^16 + 440T^14 + 130701T^12 + 21797952T^10 + 2655587933T^8 + 178140025756T^6 + 8157120819724T^4 + 22401057000432*T^2 + 58102628210064
$97$	$ (T^{8} - 104 T^{6} + \cdots + 192)^{2} $ (T^8 - 104T^6 + 2740T^4 - 4816*T^2 + 192)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1183 = 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1183.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{9} q^{2} + ( - \beta_{6} - \beta_{4} - 1) q^{3} + ( - \beta_{10} - \beta_{6} - 1) q^{4} + \beta_{13} q^{5} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{6} + (\beta_{14} + \beta_{13} + \cdots + \beta_{9}) q^{7}+ \cdots + (2 \beta_{15} + \beta_{14} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) q + b9 * q^2 + (-b6 - b4 - 1) * q^3 + (-b10 - b6 - 1) * q^4 + b13 * q^5 + (-b13 - b12 - b9 + b8 - b1) * q^6 + (b14 + b13 - b12 + b9) * q^7 - b12 * q^8 + (-b11 + b7 + 2b6 + b5 + b4 + 1) q^9 + (b11 + b10 - 2b4) q^10 + (b15 - b14 - b8 - b3 - b1) * q^11 + (-b11 + b10 + b7 + 3b6 + b5 + b4 - b2 + 1) q^12 + (b11 + b10 - b4 - b2 - 2) * q^14 + (-b13 - 2b9 + b8 - 2b1) * q^15 + (-b10 + b5 + b4 + b2 + 1) * q^16 + (-b11 - b6 - 1) * q^17 + (b15 - b14 - 4b8 - b3 + 3b1) * q^18 + (-b12 + 2b9 - b3) q^19 + (-2b15 - b14 - 3b13 + b12 - 2b9 + 3b8 + b3 - b1) * q^20 + (-b14 - b13 - 2b12 - b9 - b3) q^21 + (-b7 - b5 - b2 - 3) * q^22 + (-b6 + b5 + b4 + 1) * q^23 + (b15 - b14 - 2b8 + 3b1) * q^24 + (-b11 - 2b10 + b4) q^25 + (-b7 - b5 + 2b2 + 2) q^27 + (b14 - 2b13 - b12 - b9 + 2b8 + b1) * q^28 + (-3b5 - 2b2 - 3) * q^29 + (-b11 + b10 + b7 + 6b6 + 2b5 + 2b4 - b2 + 2) q^30 + (b15 - b14 - b1) * q^31 + (-b8 + 2b3 - b1) q^32 + (b15 + 2b14 + b9 + b3 + b1) q^33 + (2b15 + b14 + 3b13 - 2b12 + 2b9 - 3b8 - b3 + b1) q^34 + (-b11 - 2b10 + b7 - 3b6 + b5 + b4 - 1) * q^35 + (-2b7 - 5b5 + 1) * q^36 + (-b15 - 2b14 - 3b9 - b3 - b1) * q^37 + (-b10 - 4b6 + b4 - 4) q^38 + (-b11 + b7 + b6 + b5 + b4 + 1) * q^40 + (-2b15 - b14 + b13 + 2b12 - b9 - b8 + b3) * q^41 + (-b11 + 2b10 + 6b6 + 2b5 + 4b4 - b2 + 6) * q^42 + (b7 - 2b5 - 2) q^43 + (-b15 - 2b14 + 2b13 - 2b9 - b3 - b1) q^44 + (-2b3 + 4b1) * q^45 + (-b8 - b3 - b1) * q^46 + (-b15 - 2b14 + b13 + 3b12 - b9 + 2b3 - b1) q^47 + (-b2 + 2) * q^48 + (-2b10 + b7 + 3b5 + 3b4 + 2b2) * q^49 + (2b15 + b14 + 4b13 - 3b12 + b9 - 4b8 - b3) * q^50 + (-2b10 + b6 - b5 - b4 + 2b2 - 1) * q^51 + (-2b10 - 3b6 - 3) * q^53 + (b15 + 2b14 + 4b13 + b12 + 8b9 + 2b3 + b1) * q^54 + (b7 + 3b5 + 4b2 + 4) * q^55 + (2b6 + 2b5 + 3b4 + 5) q^56 + (2b15 + b14 - 4b12 + 3b9 - b3 + 2b1) * q^57 + (3b13 + b12 - 2b9 + b3) * q^58 + (-b15 + b14 + 3b8 - b3 - 3b1) * q^59 + (b15 - b14 - 3b8 - 3b3 + 4b1) q^60 + (-2b10 + b6 + 2b2) * q^61 + (2b5 - b2 - 2) q^62 + (b15 + 3b13 + 3b9 - 3b8 + 7b1) * q^63 + (-b7 + 2b5 - 2b2 - 5) * q^64 - b10 * q^66 + (-2b8 - 3b3) * q^67 + (b11 + 3b10 - b7 - b6 - 4b5 - 4b4 - 3b2 - 4) * q^68 + (-b7 + b5 + 4) * q^69 + (b15 - b14 - 2b12 - 4b9 - 4b8 - b3 - 4b1) * q^70 + (b13 + 3b12 - 4b9 - b8 - 4b1) q^71 + (3b13 + 3b12 + 2b9 + 3b3) * q^72 + (2b15 - 2b14 - 3b8 - 2b3 + 2b1) q^73 + (3b10 + 6b6 + 6) * q^74 + (-b11 + b7 - 2b5 - 2b4 - 2) * q^75 + (b13 - 2b12 - b9 - b8 - b1) q^76 + (4b10 + b7 - 3b6 - b5 - 3b4 - 2b2 + 2) * q^77 + (b11 - b7 - 3b6 - b5 - b4 - 1) q^79 + (-b15 + b14 + 2b8 + b3 + 2b1) * q^80 + (-4b10 - 5b6 - 2b4 - 5) q^81 + (b11 - b7 - 4b6 - 4b5 - 4b4 - 4) q^82 + (2b15 + b14 + b13 - 2b12 + 7b9 - b8 - b3 + 6b1) * q^83 + (-b14 + 3b13 + b12 + 4b9 - 5b8 + 8b1) * q^84 + (-3b13 + 2b12 + 2b9 + 3b8 + 2b1) q^85 + (-b15 - 2b14 - b13 + 4b12 - 3b9 + 3b3 - b1) * q^86 + (b11 + 2b10 - 8b6 - 8) * q^87 + (2b10 - b6 - 2b4 - 1) * q^88 + (-b15 - 2b14 - 2b13 - 2b12 + 3b9 - 3b3 - b1) q^89 + (-2b5 + 6b2 + 14) * q^90 + (-b7 - b5 - b2) * q^92 + (-3b13 + 2b12 - 6b9 + 2b3) * q^93 + (b11 - b10 - 6b6 - 5b4 - 6) * q^94 + (b11 + 2b10 + b6 - 3b4 + 1) * q^95 + (-2b15 - 4b14 - 4b13 + 3b12 - 9b9 + b3 - 2b1) * q^96 + (2b15 + b14 + b13 + 3b9 - b8 - b3 + 2b1) q^97 + (-b15 - 2b14 - 3b13 + 2b12 - 6b9 - 3b8 - 3b1) * q^98 + (2b15 + b14 + 2b13 - b12 - b9 - 2b8 - b3 - 2b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 6 q^{4} - 12 q^{9} + 6 q^{10} - 18 q^{12} - 26 q^{14} + 2 q^{16} - 8 q^{17} - 36 q^{22} + 12 q^{23} + 32 q^{27} - 16 q^{29} - 38 q^{30} + 56 q^{36} - 34 q^{38} - 4 q^{40} + 16 q^{42} - 16 q^{43}+ \cdots + 16 q^{95}+O(q^{100}) \) 16 * q - 4 * q^3 - 6 * q^4 - 12 * q^9 + 6 * q^10 - 18 * q^12 - 26 * q^14 + 2 * q^16 - 8 * q^17 - 36 * q^22 + 12 * q^23 + 32 * q^27 - 16 * q^29 - 38 * q^30 + 56 * q^36 - 34 * q^38 - 4 * q^40 + 16 * q^42 - 16 * q^43 + 36 * q^48 - 40 * q^49 - 16 * q^51 - 20 * q^53 + 24 * q^55 + 36 * q^56 - 12 * q^61 - 44 * q^62 - 88 * q^64 + 2 * q^66 - 2 * q^68 + 56 * q^69 + 42 * q^74 - 8 * q^75 + 76 * q^77 + 20 * q^79 - 24 * q^81 + 16 * q^82 - 68 * q^87 - 4 * q^88 + 216 * q^90 + 12 * q^92 - 26 * q^94 + 16 * q^95

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	1183.2.e.i

Level	$1183$
Weight	$2$
Character orbit	1183.e
Analytic conductor	$9.446$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(9.44630255912\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 11x^{14} + 85x^{12} + 334x^{10} + 952x^{8} + 1050x^{6} + 853x^{4} + 93x^{2} + 9 \) x^16 + 11x^14 + 85x^12 + 334x^10 + 952x^8 + 1050x^6 + 853x^4 + 93*x^2 + 9
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 24498 \nu^{14} + 246060 \nu^{12} + 1852321 \nu^{10} + 6411671 \nu^{8} + 17193085 \nu^{6} + \cdots - 40872159 ) / 14163622 \) (24498v^14 + 246060v^12 + 1852321v^10 + 6411671v^8 + 17193085v^6 + 6321845v^4 + 690027*v^2 - 40872159) / 14163622
\(\beta_{3}\)	\(=\)	\( ( - 24498 \nu^{15} - 246060 \nu^{13} - 1852321 \nu^{11} - 6411671 \nu^{9} - 17193085 \nu^{7} + \cdots + 55035781 \nu ) / 14163622 \) (-24498v^15 - 246060v^13 - 1852321v^11 - 6411671v^9 - 17193085v^7 - 6321845v^5 - 690027v^3 + 55035781v) / 14163622
\(\beta_{4}\)	\(=\)	\( ( 172099 \nu^{14} + 2170865 \nu^{12} + 17340370 \nu^{10} + 78484018 \nu^{8} + 236538400 \nu^{6} + \cdots + 23829231 ) / 42490866 \) (172099v^14 + 2170865v^12 + 17340370v^10 + 78484018v^8 + 236538400v^6 + 377649654v^4 + 218482087*v^2 + 23829231) / 42490866
\(\beta_{5}\)	\(=\)	\( ( 99072 \nu^{14} + 1000291 \nu^{12} + 7490944 \nu^{10} + 25929344 \nu^{8} + 66564370 \nu^{6} + \cdots - 20454191 ) / 14163622 \) (99072v^14 + 1000291v^12 + 7490944v^10 + 25929344v^8 + 66564370v^6 + 25566080v^4 + 2790528*v^2 - 20454191) / 14163622
\(\beta_{6}\)	\(=\)	\( ( 539569 \nu^{14} + 5861765 \nu^{12} + 45125185 \nu^{10} + 174659083 \nu^{8} + 494434675 \nu^{6} + \cdots + 5618970 ) / 42490866 \) (539569v^14 + 5861765v^12 + 45125185v^10 + 174659083v^8 + 494434675v^6 + 514968195v^4 + 441286822*v^2 + 5618970) / 42490866
\(\beta_{7}\)	\(=\)	\( ( - 102312 \nu^{14} - 1048444 \nu^{12} - 7735924 \nu^{10} - 26777324 \nu^{8} - 67022271 \nu^{6} + \cdots + 23341327 ) / 7081811 \) (-102312v^14 - 1048444v^12 - 7735924v^10 - 26777324v^8 - 67022271v^6 - 26402180v^4 - 2881788*v^2 + 23341327) / 7081811
\(\beta_{8}\)	\(=\)	\( ( 123570 \nu^{15} + 1246351 \nu^{13} + 9343265 \nu^{11} + 32341015 \nu^{9} + 83757455 \nu^{7} + \cdots - 61326350 \nu ) / 14163622 \) (123570v^15 + 1246351v^13 + 9343265v^11 + 32341015v^9 + 83757455v^7 + 31887925v^5 + 3480555v^3 - 61326350v) / 14163622
\(\beta_{9}\)	\(=\)	\( ( - 539569 \nu^{15} - 5861765 \nu^{13} - 45125185 \nu^{11} - 174659083 \nu^{9} + \cdots - 48109836 \nu ) / 42490866 \) (-539569v^15 - 5861765v^13 - 45125185v^11 - 174659083v^9 - 494434675v^7 - 514968195v^5 - 441286822v^3 - 48109836v) / 42490866
\(\beta_{10}\)	\(=\)	\( ( - 515071 \nu^{14} - 5615705 \nu^{12} - 43272864 \nu^{10} - 168247412 \nu^{8} - 477241590 \nu^{6} + \cdots - 46491129 ) / 14163622 \) (-515071v^14 - 5615705v^12 - 43272864v^10 - 168247412v^8 - 477241590v^6 - 508646350v^4 - 426433173*v^2 - 46491129) / 14163622
\(\beta_{11}\)	\(=\)	\( ( 3598 \nu^{14} + 40712 \nu^{12} + 317920 \nu^{10} + 1287475 \nu^{8} + 3722089 \nu^{6} + 4460568 \nu^{4} + \cdots + 366555 ) / 95271 \) (3598v^14 + 40712v^12 + 317920v^10 + 1287475v^8 + 3722089v^6 + 4460568v^4 + 3361729*v^2 + 366555) / 95271
\(\beta_{12}\)	\(=\)	\( ( 1079138 \nu^{15} + 11723530 \nu^{13} + 90250370 \nu^{11} + 349318166 \nu^{9} + \cdots + 11237940 \nu ) / 21245433 \) (1079138v^15 + 11723530v^13 + 90250370v^11 + 349318166v^9 + 988869350v^7 + 1029936390v^5 + 861328211v^3 + 11237940v) / 21245433
\(\beta_{13}\)	\(=\)	\( ( - 205171 \nu^{15} - 2261795 \nu^{13} - 17480377 \nu^{11} - 68898667 \nu^{9} - 196608895 \nu^{7} + \cdots - 19219314 \nu ) / 3862806 \) (-205171v^15 - 2261795v^13 - 17480377v^11 - 68898667v^9 - 196608895v^7 - 219868809v^5 - 176278948v^3 - 19219314v) / 3862806
\(\beta_{14}\)	\(=\)	\( ( 1137374 \nu^{15} + 12639109 \nu^{13} + 98087264 \nu^{11} + 390741062 \nu^{9} + \cdots + 167644965 \nu ) / 11588418 \) (1137374v^15 + 12639109v^13 + 98087264v^11 + 390741062v^9 + 1125399698v^7 + 1313214930v^5 + 1094093084v^3 + 167644965v) / 11588418
\(\beta_{15}\)	\(=\)	\( ( 17689120 \nu^{15} + 191553668 \nu^{13} + 1470474691 \nu^{11} + 5653350919 \nu^{9} + \cdots - 359333319 \nu ) / 127472598 \) (17689120v^15 + 191553668v^13 + 1470474691v^11 + 5653350919v^9 + 15847248841v^7 + 15781577445v^5 + 12180871093v^3 - 359333319v) / 127472598

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{10} + 3\beta_{6} - \beta_{2} \) b10 + 3*b6 - b2
\(\nu^{3}\)	\(=\)	\( -\beta_{12} - 4\beta_{9} - 4\beta_1 \) -b12 - 4b9 - 4b1
\(\nu^{4}\)	\(=\)	\( -5\beta_{10} - 14\beta_{6} - \beta_{4} - 14 \) -5b10 - 14b6 - b4 - 14
\(\nu^{5}\)	\(=\)	\( \beta_{13} + 6\beta_{12} + 19\beta_{9} + 6\beta_{3} \) b13 + 6b12 + 19b9 + 6*b3
\(\nu^{6}\)	\(=\)	\( -\beta_{7} - 8\beta_{5} + 24\beta_{2} + 61 \) -b7 - 8b5 + 24b2 + 61
\(\nu^{7}\)	\(=\)	\( \beta_{15} - \beta_{14} - 11\beta_{8} - 32\beta_{3} + 94\beta_1 \) b15 - b14 - 11b8 - 32b3 + 94*b1
\(\nu^{8}\)	\(=\)	\( -11\beta_{11} + 115\beta_{10} + 11\beta_{7} + 345\beta_{6} + 52\beta_{5} + 52\beta_{4} - 115\beta_{2} + 52 \) -11b11 + 115b10 + 11b7 + 345b6 + 52b5 + 52b4 - 115*b2 + 52
\(\nu^{9}\)	\(=\)	\( -22\beta_{15} - 11\beta_{14} - 85\beta_{13} - 145\beta_{12} - 493\beta_{9} + 85\beta_{8} + 11\beta_{3} - 482\beta_1 \) -22b15 - 11b14 - 85b13 - 145b12 - 493b9 + 85b8 + 11b3 - 482b1
\(\nu^{10}\)	\(=\)	\( 85\beta_{11} - 553\beta_{10} - 1736\beta_{6} - 315\beta_{4} - 1736 \) 85b11 - 553b10 - 1736b6 - 315b4 - 1736
\(\nu^{11}\)	\(=\)	\( 85\beta_{15} + 170\beta_{14} + 570\beta_{13} + 698\beta_{12} + 2544\beta_{9} + 783\beta_{3} + 85\beta_1 \) 85b15 + 170b14 + 570b13 + 698b12 + 2544b9 + 783b3 + 85*b1
\(\nu^{12}\)	\(=\)	\( -570\beta_{7} - 1838\beta_{5} + 2672\beta_{2} + 6935 \) -570b7 - 1838b5 + 2672*b2 + 6935
\(\nu^{13}\)	\(=\)	\( 570\beta_{15} - 570\beta_{14} - 3548\beta_{8} - 4510\beta_{3} + 12015\beta_1 \) 570b15 - 570b14 - 3548b8 - 4510b3 + 12015*b1
\(\nu^{14}\)	\(=\)	\( - 3548 \beta_{11} + 12977 \beta_{10} + 3548 \beta_{7} + 44495 \beta_{6} + 10466 \beta_{5} + \cdots + 10466 \) -3548b11 + 12977b10 + 3548b7 + 44495b6 + 10466b5 + 10466b4 - 12977*b2 + 10466
\(\nu^{15}\)	\(=\)	\( - 7096 \beta_{15} - 3548 \beta_{14} - 21110 \beta_{13} - 16347 \beta_{12} - 68116 \beta_{9} + \cdots - 64568 \beta_1 \) -7096b15 - 3548b14 - 21110b13 - 16347b12 - 68116b9 + 21110b8 + 3548b3 - 64568b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 170.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 11 T^{14} + \cdots + 9 \) T^16 + 11T^14 + 85T^12 + 334T^10 + 952T^8 + 1050T^6 + 853T^4 + 93*T^2 + 9
$3$	\( (T^{8} + 2 T^{7} + 11 T^{6} + \cdots + 4)^{2} \) (T^8 + 2T^7 + 11T^6 + 6T^5 + 67T^4 + 62T^3 + 114T^2 - 20*T + 4)^2
$5$	\( T^{16} + 20 T^{14} + \cdots + 2304 \) T^16 + 20T^14 + 297T^12 + 1740T^10 + 7361T^8 + 14560T^6 + 20656T^4 + 7680*T^2 + 2304
$7$	\( T^{16} + 20 T^{14} + \cdots + 5764801 \) T^16 + 20T^14 + 217T^12 + 1612T^10 + 10652T^8 + 78988T^6 + 521017T^4 + 2352980*T^2 + 5764801
$11$	\( T^{16} + 52 T^{14} + \cdots + 729 \) T^16 + 52T^14 + 2108T^12 + 30312T^10 + 337509T^8 + 199832T^6 + 99508T^4 + 9180*T^2 + 729
$13$	\( T^{16} \) T^16
$17$	\( (T^{8} + 4 T^{7} + \cdots + 15129)^{2} \) (T^8 + 4T^7 + 36T^6 + 24T^5 + 485T^4 + 56T^3 + 5164T^2 - 6396*T + 15129)^2
$19$	\( T^{16} + 44 T^{14} + \cdots + 10673289 \) T^16 + 44T^14 + 1396T^12 + 19096T^10 + 185725T^8 + 971784T^6 + 3674044T^4 + 7618644*T^2 + 10673289
$23$	\( (T^{8} - 6 T^{7} + 31 T^{6} + \cdots + 36)^{2} \) (T^8 - 6T^7 + 31T^6 - 50T^5 + 91T^4 - 22T^3 + 130T^2 - 60*T + 36)^2
$29$	\( (T^{4} + 4 T^{3} + \cdots + 624)^{4} \) (T^4 + 4T^3 - 63T^2 - 208*T + 624)^4
$31$	\( T^{16} + \cdots + 1136229264 \) T^16 + 80T^14 + 4309T^12 + 128296T^10 + 2779213T^8 + 35364492T^6 + 309454636T^4 + 657036336*T^2 + 1136229264
$37$	\( T^{16} + 120 T^{14} + \cdots + 76527504 \) T^16 + 120T^14 + 10053T^12 + 422496T^10 + 12939021T^8 + 213389964T^6 + 2419355628T^4 + 433655856*T^2 + 76527504
$41$	\( (T^{8} - 132 T^{6} + \cdots + 292032)^{2} \) (T^8 - 132T^6 + 5732T^4 - 88192*T^2 + 292032)^2
$43$	\( (T^{4} + 4 T^{3} + \cdots - 104)^{4} \) (T^4 + 4T^3 - 66T^2 + 156*T - 104)^4
$47$	\( T^{16} + \cdots + 57728231289 \) T^16 + 196T^14 + 26204T^12 + 1905768T^10 + 101089845T^8 + 2884224440T^6 + 56549167060T^4 + 58599199164*T^2 + 57728231289
$53$	\( (T^{8} + 10 T^{7} + \cdots + 7569)^{2} \) (T^8 + 10T^7 + 100T^6 + 260T^5 + 1213T^4 - 1740T^3 + 16900T^2 - 11310*T + 7569)^2
$59$	\( T^{16} + \cdots + 12487392009 \) T^16 + 188T^14 + 24868T^12 + 1646008T^10 + 79227709T^8 + 1652371368T^6 + 24989166028T^4 + 18073959780*T^2 + 12487392009
$61$	\( (T^{8} + 6 T^{7} + \cdots + 49729)^{2} \) (T^8 + 6T^7 + 60T^6 + 44T^5 + 917T^4 - 420T^3 + 14188T^2 - 20962*T + 49729)^2
$67$	\( T^{16} + \cdots + 66330457209 \) T^16 + 284T^14 + 59004T^12 + 5578872T^10 + 387569525T^8 + 6027737800T^6 + 75732974260T^4 + 73439011956*T^2 + 66330457209
$71$	\( (T^{8} - 292 T^{6} + \cdots + 397488)^{2} \) (T^8 - 292T^6 + 19829T^4 - 253084*T^2 + 397488)^2
$73$	\( T^{16} + \cdots + 8437677133824 \) T^16 + 260T^14 + 47625T^12 + 4249020T^10 + 273313457T^8 + 7922514640T^6 + 164987876800T^4 + 1371747640320*T^2 + 8437677133824
$79$	\( (T^{8} - 10 T^{7} + \cdots + 64)^{2} \) (T^8 - 10T^7 + 91T^6 - 210T^5 + 689T^4 + 380T^3 + 3672T^2 - 480*T + 64)^2
$83$	\( (T^{8} - 296 T^{6} + \cdots + 5483712)^{2} \) (T^8 - 296T^6 + 28692T^4 - 959920*T^2 + 5483712)^2
$89$	\( T^{16} + \cdots + 58102628210064 \) T^16 + 440T^14 + 130701T^12 + 21797952T^10 + 2655587933T^8 + 178140025756T^6 + 8157120819724T^4 + 22401057000432*T^2 + 58102628210064
$97$	\( (T^{8} - 104 T^{6} + \cdots + 192)^{2} \) (T^8 - 104T^6 + 2740T^4 - 4816*T^2 + 192)^2
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\(n\)	\(339\)	\(1016\)
\(\chi(n)\)	\(\beta_{6}\)	\(1\)