LMFDB - Newform orbit 1596.2.r.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1596,2,Mod(457,1596)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1596, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

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

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

chi := DirichletCharacter("1596.457");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1596 = 2^{2} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1596.r (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.7441241626$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} + 8 x^{12} + 7 x^{11} - 57 x^{10} + 47 x^{9} + 645 x^{8} - 2490 x^{7} + 4515 x^{6} + \cdots + 823543 $ x^14 - 3x^13 + 8x^12 + 7x^11 - 57x^10 + 47x^9 + 645x^8 - 2490x^7 + 4515x^6 + 2303x^5 - 19551x^4 + 16807x^3 + 134456x^2 - 352947*x + 823543
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{8} + 1) q^{3} - \beta_{5} q^{5} - \beta_{12} q^{7} - \beta_{8} q^{9} + (\beta_{12} - \beta_{11} + \beta_{8} + \cdots - 1) q^{11} + ( - \beta_{12} - \beta_{10} + \cdots + \beta_{2}) q^{13}+ \cdots + (\beta_{9} + \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) $ q + (-b8 + 1) * q^3 - b5 * q^5 - b12 * q^7 - b8 * q^9 + (b12 - b11 + b8 - b2 - 1) * q^11 + (-b12 - b10 - b4 + b2) * q^13 + (b7 - b5) * q^15 + (-b13 - b12 - 3b8 - b6 - b3 - b1 + 3) q^17 - b8 * q^19 - b2 * q^21 + (b12 - b10 + b8 + b6 + b1) * q^23 + (b13 + 2b12 + b11 - 2b10 + 3b8 + b6 + b3 - b2 + 3b1 - 3) * q^25 - q^27 + (b9 + b4 - 1) * q^29 + (b13 - b10 - b8 - 2b7 + b6 - b3 - b2 + 1) q^31 + (b12 - b11 + b9 + b8 - b1) * q^33 + (-b13 - b12 + b11 - b10 - b9 + b8 + b7 - b6 + b5 + b2 + 2b1 + 1) q^35 + (b12 - b11 - 2b10 + b9 + b5 - b4 - b3 + b2 - b1 + 1) q^37 + (b8 + b3 - 1) * q^39 + (-b13 + b12 + 2b10 + 2b4 - 3b2 + b1 - 4) q^41 + (-b13 - 2b12 - b10 - b9 - b7 + b5 + b1 + 1) q^43 + b7 * q^45 + (-b12 - b11 - b10 + b9 - b6 - b5 - b4 - b3 + 2b2 - b1 + 1) q^47 + (b13 + 2b12 - b10 + 2b8 + b7 + b6 + b4 + 2b3 + 2b1) * q^49 + (-b12 - b10 - 3b8 - b6 - b4 - b3 + b2 - b1 + 1) q^51 + (3b8 - 2b7 + b3 - 3) * q^53 + (2b13 - b12 - 3b10 - 2b9 - 2b7 + 2b5 + b4 + 3b1 - 1) * q^55 - q^57 + (b13 + 2b12 + 2b11 - b10 - b8 + b6 + 3b3 + b2 + 4b1 + 1) * q^59 + (b12 + 2b11 - 2b10 - 2b9 + b8 - b6 + b4 + b3 + 2b1 - 1) * q^61 + (b12 - b2) * q^63 + (-b12 - b10 - b6 + 2b5 + 2b2 + b1) * q^65 + (-b12 + b8 + b7 - b3 - b1 - 1) * q^67 + (-b13 - b12 + b2 - b1 + 1) * q^69 + (b13 - b12 - 2b10 + b7 - b5 - b4 + b2 + b1 - 2) q^71 + (-2b13 - 2b12 + b11 - b8 - b7 - 2b6 + b2 - b1 + 1) q^73 + (2b12 + b11 - b10 - b9 + 3b8 + b6 + b4 + b3 - b2 + 2b1 - 1) q^75 + (-b13 - b12 - 2b11 + 2b10 - b8 + b7 - b4 - 2b3 - 4b1 + 5) * q^77 + (b12 - b11 + b9 + 3b5 + b4 + b3 + b2 + b1 - 1) q^79 + (b8 - 1) * q^81 + (b13 + b12 + b9 - b7 + b5 - b2 + b1 - 2) * q^83 + (-b13 + b10 + 3b7 - 3b5 + b4 + b2 - 2b1 + 1) q^85 + (-b12 + b11 - b10 - b3) * q^87 + (b12 + b11 - b9 + 2b8 + 2b6 + b5 + b4 + b3 - b2 + 3b1 - 1) q^89 + (b13 + b11 - 3b10 - b8 + b7 + b6 - b5 - 2b4 + 2b2 + 5) q^91 + (-b8 + b6 - 2b5 - b4 - b3 - b2 - b1 + 1) q^93 + b7 * q^95 + (-b13 + b12 + 2b10 - b9 + 2b7 - 2b5 + b4 - b2 - b1 - 7) q^97 + (b9 + b2 - b1 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 7 q^{3} + 2 q^{5} - 7 q^{9} - 4 q^{11} - 12 q^{13} + 4 q^{15} + 18 q^{17} - 7 q^{19} + 3 q^{21} + 2 q^{23} - 15 q^{25} - 14 q^{27} - 8 q^{29} + q^{31} + 4 q^{33} + 22 q^{35} - 7 q^{37} - 6 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) $ 14 * q + 7 * q^3 + 2 * q^5 - 7 * q^9 - 4 * q^11 - 12 * q^13 + 4 * q^15 + 18 * q^17 - 7 * q^19 + 3 * q^21 + 2 * q^23 - 15 * q^25 - 14 * q^27 - 8 * q^29 + q^31 + 4 * q^33 + 22 * q^35 - 7 * q^37 - 6 * q^39 - 30 * q^41 + 6 * q^43 + 2 * q^45 + 2 * q^47 + 26 * q^49 - 18 * q^51 - 24 * q^53 - 8 * q^55 - 14 * q^57 + 15 * q^59 + 5 * q^61 + 3 * q^63 - 5 * q^65 - 9 * q^67 + 4 * q^69 - 32 * q^71 + q^73 + 15 * q^75 + 47 * q^77 - 13 * q^79 - 7 * q^81 - 22 * q^83 + 22 * q^85 - 4 * q^87 + 7 * q^89 + 39 * q^91 - q^93 + 2 * q^95 - 82 * q^97 + 8 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{14} - 3 x^{13} + 8 x^{12} + 7 x^{11} - 57 x^{10} + 47 x^{9} + 645 x^{8} - 2490 x^{7} + 4515 x^{6} + \cdots + 823543 $ x^14 - 3*x^13 + 8*x^12 + 7*x^11 - 57*x^10 + 47*x^9 + 645*x^8 - 2490*x^7 + 4515*x^6 + 2303*x^5 - 19551*x^4 + 16807*x^3 + 134456*x^2 - 352947*x + 823543 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{13} - 3 \nu^{12} + 8 \nu^{11} + 7 \nu^{10} - 57 \nu^{9} + 47 \nu^{8} + 645 \nu^{7} + \cdots - 352947 ) / 117649 $ (v^13 - 3v^12 + 8v^11 + 7v^10 - 57v^9 + 47v^8 + 645v^7 - 2490v^6 + 4515v^5 + 2303v^4 - 19551v^3 + 16807v^2 + 134456v - 352947) / 117649
$\beta_{3}$	$=$	$ ( 15452 \nu^{13} - 2762783 \nu^{12} + 6344404 \nu^{11} + 2075745 \nu^{10} - 62842930 \nu^{9} + \cdots - 321216005859 ) / 1109665368 $ (15452v^13 - 2762783v^12 + 6344404v^11 + 2075745v^10 - 62842930v^9 + 99663530v^8 + 393922679v^7 - 2235177148v^6 + 2601425477v^5 + 5203585086v^4 - 21360176138v^3 + 2974927837v^2 + 108960923876*v - 321216005859) / 1109665368
$\beta_{4}$	$=$	$ ( - 168 \nu^{13} + 31 \nu^{12} + 908 \nu^{11} - 3637 \nu^{10} - 2310 \nu^{9} + 35578 \nu^{8} + \cdots - 40723361 ) / 1210104 $ (-168v^13 + 31v^12 + 908v^11 - 3637v^10 - 2310v^9 + 35578v^8 - 87863v^7 - 123092v^6 + 662143v^5 - 647318v^4 - 2519874v^3 + 7163555v^2 - 6866860*v - 40723361) / 1210104
$\beta_{5}$	$=$	$ ( 26483 \nu^{13} - 151788 \nu^{12} + 197027 \nu^{11} + 613822 \nu^{10} - 2917294 \nu^{9} + \cdots - 9955189468 ) / 158523624 $ (26483v^13 - 151788v^12 + 197027v^11 + 613822v^10 - 2917294v^9 - 166379v^8 + 32570500v^7 - 95436065v^6 + 40577038v^5 + 345649598v^4 - 695627177v^3 - 848909908v^2 + 6226712583*v - 9955189468) / 158523624
$\beta_{6}$	$=$	$ ( - 194486 \nu^{13} - 3086439 \nu^{12} + 8196022 \nu^{11} - 2151079 \nu^{10} + \cdots - 417380886671 ) / 1109665368 $ (-194486v^13 - 3086439v^12 + 8196022v^11 - 2151079v^10 - 73260350v^9 + 157114436v^8 + 335020247v^7 - 2676676978v^6 + 3745308413v^5 + 4967713786v^4 - 27433618828v^3 + 12220180021v^2 + 114608042262*v - 417380886671) / 1109665368
$\beta_{7}$	$=$	$ ( 197957 \nu^{13} - 1029635 \nu^{12} + 1197361 \nu^{11} + 4891887 \nu^{10} - 20714236 \nu^{9} + \cdots - 63598578771 ) / 1109665368 $ (197957v^13 - 1029635v^12 + 1197361v^11 + 4891887v^10 - 20714236v^9 - 4104031v^8 + 232439771v^7 - 641775427v^6 + 201777947v^5 + 2551894128v^4 - 4792015613v^3 - 6257214887v^2 + 42803176829*v - 63598578771) / 1109665368
$\beta_{8}$	$=$	$ ( - 227294 \nu^{13} + 1199119 \nu^{12} - 1426086 \nu^{11} - 5576333 \nu^{10} + \cdots + 79705197467 ) / 1109665368 $ (-227294v^13 + 1199119v^12 - 1426086v^11 - 5576333v^10 + 23762806v^9 + 6493992v^8 - 276748959v^7 + 765945018v^6 - 234990273v^5 - 2995058458v^4 + 5539315544v^3 + 7945665315v^2 - 52828871662*v + 79705197467) / 1109665368
$\beta_{9}$	$=$	$ ( - 3936 \nu^{13} - 449 \nu^{12} + 23756 \nu^{11} - 84301 \nu^{10} - 73470 \nu^{9} + \cdots - 1131430433 ) / 8470728 $ (-3936v^13 - 449v^12 + 23756v^11 - 84301v^10 - 73470v^9 + 902962v^8 - 1879607v^7 - 3960380v^6 + 16355143v^5 - 13034294v^4 - 69710634v^3 + 169152851v^2 - 87194716*v - 1131430433) / 8470728
$\beta_{10}$	$=$	$ ( 73891 \nu^{13} + 56038 \nu^{12} - 569325 \nu^{11} + 1543864 \nu^{10} + 2453830 \nu^{9} + \cdots + 26740911806 ) / 158523624 $ (73891v^13 + 56038v^12 - 569325v^11 + 1543864v^10 + 2453830v^9 - 18592047v^8 + 28568994v^7 + 113034591v^6 - 353085768v^5 + 156498650v^4 + 1680827939v^3 - 3181118514v^2 - 73933993*v + 26740911806) / 158523624
$\beta_{11}$	$=$	$ ( 327108 \nu^{13} - 820345 \nu^{12} - 44858 \nu^{11} + 7382977 \nu^{10} - 13879122 \nu^{9} + \cdots - 7534124311 ) / 554832684 $ (327108v^13 - 820345v^12 - 44858v^11 + 7382977v^10 - 13879122v^9 - 41782282v^8 + 272802689v^7 - 370812334v^6 - 527452303v^5 + 2644191998v^4 - 1059427530v^3 - 12386089121v^2 + 40569913090*v - 7534124311) / 554832684
$\beta_{12}$	$=$	$ ( 677483 \nu^{13} - 441391 \nu^{12} - 2973969 \nu^{11} + 14724983 \nu^{10} + 417800 \nu^{9} + \cdots + 130686509233 ) / 1109665368 $ (677483v^13 - 441391v^12 - 2973969v^11 + 14724983v^10 + 417800v^9 - 134497941v^8 + 391518591v^7 + 250310043v^6 - 2302779381v^5 + 3205175260v^4 + 7719939073v^3 - 27388752027v^2 + 35471997043*v + 130686509233) / 1109665368
$\beta_{13}$	$=$	$ ( 875653 \nu^{13} + 396327 \nu^{12} - 6095423 \nu^{11} + 18272345 \nu^{10} + 23052160 \nu^{9} + \cdots + 285684596071 ) / 1109665368 $ (875653v^13 + 396327v^12 - 6095423v^11 + 18272345v^10 + 23052160v^9 - 212248579v^8 + 375025289v^7 + 1125978053v^6 - 3926658379v^5 + 2383000732v^4 + 17949025703v^3 - 38048793461v^2 + 8932231413*v + 285684596071) / 1109665368

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{13} + 2\beta_{12} + \beta_{10} + 3\beta_{8} - \beta_{7} + \beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 3 $ -b13 + 2b12 + b10 + 3b8 - b7 + b5 + 2*b4 + b3 - b2 + b1 - 3
$\nu^{3}$	$=$	$ - 2 \beta_{13} - 3 \beta_{12} - \beta_{11} + 6 \beta_{10} + 2 \beta_{9} - 10 \beta_{8} - \beta_{7} + \cdots - 1 $ -2b13 - 3b12 - b11 + 6b10 + 2b9 - 10b8 - b7 - 3b6 + 2b4 + b3 - 6b1 - 1
$\nu^{4}$	$=$	$ 8 \beta_{13} + 9 \beta_{12} - 5 \beta_{11} - 7 \beta_{10} + 7 \beta_{9} + 16 \beta_{8} + 4 \beta_{7} + \cdots - 5 $ 8b13 + 9b12 - 5b11 - 7b10 + 7b9 + 16b8 + 4b7 - b4 + 7b3 - 2*b2 - b1 - 5
$\nu^{5}$	$=$	$ 15 \beta_{13} - 8 \beta_{12} - 14 \beta_{11} + 9 \beta_{10} + 7 \beta_{9} - 5 \beta_{8} + 14 \beta_{7} + \cdots + 54 $ 15b13 - 8b12 - 14b11 + 9b10 + 7b9 - 5b8 + 14b7 + 9b6 - 10b5 + b4 - b3 + 28b2 - 8*b1 + 54
$\nu^{6}$	$=$	$ 21 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} - 6 \beta_{10} - 38 \beta_{9} + 214 \beta_{8} + 26 \beta_{7} + \cdots - 317 $ 21b13 + 13b12 + 13b11 - 6b10 - 38b9 + 214b8 + 26b7 + 36b6 + 20b5 + 37b4 + 29b3 - 16b2 + 133*b1 - 317
$\nu^{7}$	$=$	$ - 87 \beta_{13} - 29 \beta_{12} + 85 \beta_{11} + 255 \beta_{10} - 83 \beta_{9} + 397 \beta_{8} + \cdots - 318 $ -87b13 - 29b12 + 85b11 + 255b10 - 83b9 + 397b8 - 109b7 + 51b6 + 56b5 + 160b4 + 116b3 - 28b2 - 136*b1 - 318
$\nu^{8}$	$=$	$ - 4 \beta_{13} - 498 \beta_{12} + 67 \beta_{11} + 397 \beta_{10} + 178 \beta_{9} - 227 \beta_{8} + \cdots - 207 $ -4b13 - 498b12 + 67b11 + 397b10 + 178b9 - 227b8 + 238b7 - 141b6 - 105b5 - 480b4 + 96b3 + 148b2 - 550*b1 - 207
$\nu^{9}$	$=$	$ 649 \beta_{13} + 1351 \beta_{12} - 831 \beta_{11} - 708 \beta_{10} + 489 \beta_{9} + 2040 \beta_{8} + \cdots + 2198 $ 649b13 + 1351b12 - 831b11 - 708b10 + 489b9 + 2040b8 + 22b7 + 630b6 + 380b5 - 1194b4 + 273b3 + 176b2 - 449*b1 + 2198
$\nu^{10}$	$=$	$ - 1516 \beta_{13} - 1286 \beta_{12} - 2176 \beta_{11} + 2288 \beta_{10} - 1063 \beta_{9} - 7165 \beta_{8} + \cdots + 4492 $ -1516b13 - 1286b12 - 2176b11 + 2288b10 - 1063b9 - 7165b8 + 1938b7 - 93b6 - 1660b5 - 1176b4 - 1485b3 + 1059b2 - 576*b1 + 4492
$\nu^{11}$	$=$	$ - 3684 \beta_{13} + 3857 \beta_{12} + 2579 \beta_{11} - 6453 \beta_{10} - 9808 \beta_{9} + 12032 \beta_{8} + \cdots - 27336 $ -3684b13 + 3857b12 + 2579b11 - 6453b10 - 9808b9 + 12032b8 - 2918b7 - 2580b6 + 3916b5 + 4796b4 + 4393b3 - 4041b2 + 12308*b1 - 27336
$\nu^{12}$	$=$	$ - 10059 \beta_{13} - 20743 \beta_{12} + 12419 \beta_{11} + 7941 \beta_{10} - 4567 \beta_{9} + \cdots + 23704 $ -10059b13 - 20743b12 + 12419b11 + 7941b10 - 4567b9 - 43078b8 - 6134b7 - 15741b6 - 21305b5 - 1897b4 + 4423b3 + 23830b2 - 31297*b1 + 23704
$\nu^{13}$	$=$	$ 47049 \beta_{13} - 9469 \beta_{12} + 14060 \beta_{11} - 103014 \beta_{10} + 42002 \beta_{9} + \cdots - 39327 $ 47049b13 - 9469b12 + 14060b11 - 103014b10 + 42002b9 - 31795b8 + 41323b7 - 25938b6 - 15005b5 - 89119b4 - 11903b3 + 90343b2 + 678*b1 - 39327

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1596\mathbb{Z}\right)^\times$.

$n$	$533$	$799$	$913$	$1009$
$\chi(n)$	$1$	$1$	$-\beta_{8}$	$1$

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} $

457.1

−2.40447	+	1.10388i
2.03854	+	1.68652i
2.53222	−	0.766728i
0.339065	−	2.62394i
0.345480	+	2.62310i
0.945388	−	2.47108i
−2.29623	+	1.31428i
−2.40447	−	1.10388i
2.03854	−	1.68652i
2.53222	+	0.766728i
0.339065	+	2.62394i
0.345480	−	2.62310i
0.945388	+	2.47108i
−2.29623	−	1.31428i

0.500000

0.866025i

−1.84846

3.20163i

−2.15822

1.53039i

−0.500000

0.866025i

457.2

0.500000

0.866025i

−1.32289

2.29132i

−0.441293

−

2.60869i

−0.500000

0.866025i

457.3

0.500000

0.866025i

−0.353880

0.612939i

1.93011

−

1.80960i

−0.500000

0.866025i

457.4

0.500000

0.866025i

0.392110

−

0.679154i

2.44193

1.01833i

−0.500000

0.866025i

457.5

0.500000

0.866025i

0.454780

−

0.787703i

−2.09893

−

1.61074i

−0.500000

0.866025i

457.6

0.500000

0.866025i

1.63640

−

2.83433i

2.61271

0.416811i

−0.500000

0.866025i

457.7

0.500000

0.866025i

2.04195

−

3.53675i

−2.28631

1.33145i

−0.500000

0.866025i

1369.1

0.500000

−

0.866025i

−1.84846

−

3.20163i

−2.15822

−

1.53039i

−0.500000

−

0.866025i

1369.2

0.500000

−

0.866025i

−1.32289

−

2.29132i

−0.441293

2.60869i

−0.500000

−

0.866025i

1369.3

0.500000

−

0.866025i

−0.353880

−

0.612939i

1.93011

1.80960i

−0.500000

−

0.866025i

1369.4

0.500000

−

0.866025i

0.392110

0.679154i

2.44193

−

1.01833i

−0.500000

−

0.866025i

1369.5

0.500000

−

0.866025i

0.454780

0.787703i

−2.09893

1.61074i

−0.500000

−

0.866025i

1369.6

0.500000

−

0.866025i

1.63640

2.83433i

2.61271

−

0.416811i

−0.500000

−

0.866025i

1369.7

0.500000

−

0.866025i

2.04195

3.53675i

−2.28631

−

1.33145i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1596.2.r.h	✓	14
7.c	even	3	1	inner	1596.2.r.h	✓	14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1596.2.r.h	✓	14	1.a	even	1	1	trivial
1596.2.r.h	✓	14	7.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1596, [\chi])$:

$ T_{5}^{14} - 2 T_{5}^{13} + 27 T_{5}^{12} - 28 T_{5}^{11} + 474 T_{5}^{10} - 482 T_{5}^{9} + 3928 T_{5}^{8} + \cdots + 4356 $

$ T_{11}^{14} + 4 T_{11}^{13} + 67 T_{11}^{12} + 306 T_{11}^{11} + 3222 T_{11}^{10} + 13326 T_{11}^{9} + \cdots + 38416 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{14} $ T^14
$3$	$ (T^{2} - T + 1)^{7} $ (T^2 - T + 1)^7
$5$	$ T^{14} - 2 T^{13} + \cdots + 4356 $ T^14 - 2T^13 + 27T^12 - 28T^11 + 474T^10 - 482T^9 + 3928T^8 - 1809T^7 + 20505T^6 - 16263T^5 + 26520T^4 - 8649T^3 + 12951T^2 - 3762*T + 4356
$7$	$ T^{14} - 13 T^{12} + \cdots + 823543 $ T^14 - 13T^12 - 22T^11 + 123T^10 + 202T^9 - 234T^8 - 2220T^7 - 1638T^6 + 9898T^5 + 42189T^4 - 52822T^3 - 218491*T^2 + 823543
$11$	$ T^{14} + 4 T^{13} + \cdots + 38416 $ T^14 + 4T^13 + 67T^12 + 306T^11 + 3222T^10 + 13326T^9 + 80044T^8 + 273829T^7 + 1275859T^6 + 3635793T^5 + 10551150T^4 + 15476601T^3 + 19706869T^2 - 854756*T + 38416
$13$	$ (T^{7} + 6 T^{6} + \cdots - 432)^{2} $ (T^7 + 6T^6 - 26T^5 - 195T^4 - 105T^3 + 780T^2 + 732T - 432)^2
$17$	$ T^{14} - 18 T^{13} + \cdots + 324 $ T^14 - 18T^13 + 235T^12 - 1742T^11 + 10642T^10 - 43556T^9 + 183259T^8 - 582182T^7 + 2037298T^6 - 3946308T^5 + 6251215T^4 - 3108654T^3 + 1215045T^2 - 20250*T + 324
$19$	$ (T^{2} + T + 1)^{7} $ (T^2 + T + 1)^7
$23$	$ T^{14} - 2 T^{13} + \cdots + 677329 $ T^14 - 2T^13 + 78T^12 - 308T^11 + 5132T^10 - 17025T^9 + 109168T^8 - 317711T^7 + 1574722T^6 - 3812622T^5 + 9178965T^4 - 9179585T^3 + 8258544T^2 + 1929935*T + 677329
$29$	$ (T^{7} + 4 T^{6} + \cdots - 7776)^{2} $ (T^7 + 4T^6 - 111T^5 - 584T^4 + 333T^3 + 4212T^2 + 324T - 7776)^2
$31$	$ T^{14} - T^{13} + \cdots + 2985984 $ T^14 - T^13 + 170T^12 - 101T^11 + 21967T^10 - 9855T^9 + 1140000T^8 + 749283T^7 + 44081799T^6 - 1627794T^5 + 50239980T^4 - 19538352T^3 + 56571840T^2 - 12897792T + 2985984
$37$	$ T^{14} + \cdots + 43945575424 $ T^14 + 7T^13 + 202T^12 + 927T^11 + 24735T^10 + 112845T^9 + 1574620T^8 + 6113071T^7 + 67323085T^6 + 239925888T^5 + 1380627708T^4 + 2079375648T^3 + 8302935952T^2 + 1580625280*T + 43945575424
$41$	$ (T^{7} + 15 T^{6} + \cdots + 7464)^{2} $ (T^7 + 15T^6 - 83T^5 - 2343T^4 - 11571T^3 - 12910T^2 + 9500T + 7464)^2
$43$	$ (T^{7} - 3 T^{6} + \cdots + 16572)^{2} $ (T^7 - 3T^6 - 180T^5 + 730T^4 + 6456T^3 - 21357T^2 - 37983T + 16572)^2
$47$	$ T^{14} - 2 T^{13} + \cdots + 576 $ T^14 - 2T^13 + 141T^12 - 54T^11 + 17506T^10 - 19538T^9 + 240781T^8 - 682802T^7 + 3285394T^6 - 5917110T^5 + 9580029T^4 - 4707194T^3 + 1858033T^2 - 33432*T + 576
$53$	$ T^{14} + \cdots + 21941904384 $ T^14 + 24T^13 + 476T^12 + 4566T^11 + 42313T^10 + 211682T^9 + 1565485T^8 + 5281979T^7 + 42880939T^6 + 51831522T^5 + 614493508T^4 - 561034112T^3 + 8717579968T^2 - 11722257408*T + 21941904384
$59$	$ T^{14} + \cdots + 24943780096 $ T^14 - 15T^13 + 460T^12 - 3873T^11 + 105563T^10 - 813595T^9 + 12905118T^8 - 23732523T^7 + 337022409T^6 + 1400327852T^5 + 11635798572T^4 + 26603458544T^3 + 48396184656T^2 + 39952122304*T + 24943780096
$61$	$ T^{14} + \cdots + 13107860948484 $ T^14 - 5T^13 + 361T^12 - 984T^11 + 83764T^10 - 202627T^9 + 10914571T^8 - 18835243T^7 + 1005479104T^6 - 2169505368T^5 + 49478251169T^4 - 128049464127T^3 + 1755612467635T^2 - 4201184568810*T + 13107860948484
$67$	$ T^{14} + \cdots + 166719744 $ T^14 + 9T^13 + 147T^12 + 154T^11 + 6789T^10 + 12558T^9 + 155338T^8 + 235854T^7 + 2248473T^6 + 3672540T^5 + 18875652T^4 + 14515056T^3 + 64483920T^2 + 26495424*T + 166719744
$71$	$ (T^{7} + 16 T^{6} + \cdots - 44584)^{2} $ (T^7 + 16T^6 - 14T^5 - 1187T^4 - 2793T^3 + 23134T^2 + 70068T - 44584)^2
$73$	$ T^{14} + \cdots + 35570714404 $ T^14 - T^13 + 266T^12 - 1265T^11 + 54631T^10 - 263467T^9 + 4983066T^8 - 36908646T^7 + 359428572T^6 - 1698358660T^5 + 7313131273T^4 - 13468317416T^3 + 23723697809T^2 + 14726609966T + 35570714404
$79$	$ T^{14} + \cdots + 7929675777024 $ T^14 + 13T^13 + 390T^12 + 3537T^11 + 79663T^10 + 642061T^9 + 10056310T^8 + 59006461T^7 + 756778363T^6 + 3575421630T^5 + 36450556740T^4 + 81744179776T^3 + 616522522816T^2 - 269972484096*T + 7929675777024
$83$	$ (T^{7} + 11 T^{6} + \cdots + 53437)^{2} $ (T^7 + 11T^6 - 145T^5 - 1632T^4 + 2022T^3 + 38672T^2 + 84722T + 53437)^2
$89$	$ T^{14} - 7 T^{13} + \cdots + 14930496 $ T^14 - 7T^13 + 218T^12 - 1851T^11 + 41911T^10 - 300317T^9 + 1833244T^8 - 6193895T^7 + 19002235T^6 - 34517862T^5 + 84160880T^4 - 116644888T^3 + 257896384T^2 - 64652448*T + 14930496
$97$	$ (T^{7} + 41 T^{6} + \cdots - 72128)^{2} $ (T^7 + 41T^6 + 478T^5 - 1162T^4 - 61521T^3 - 426998T^2 - 959336T - 72128)^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:	\( N \)	\(=\)	\( 1596 = 2^{2} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1596.r (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{8} + 1) q^{3} - \beta_{5} q^{5} - \beta_{12} q^{7} - \beta_{8} q^{9} + (\beta_{12} - \beta_{11} + \beta_{8} + \cdots - 1) q^{11} + ( - \beta_{12} - \beta_{10} + \cdots + \beta_{2}) q^{13}+ \cdots + (\beta_{9} + \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) q + (-b8 + 1) * q^3 - b5 * q^5 - b12 * q^7 - b8 * q^9 + (b12 - b11 + b8 - b2 - 1) * q^11 + (-b12 - b10 - b4 + b2) * q^13 + (b7 - b5) * q^15 + (-b13 - b12 - 3b8 - b6 - b3 - b1 + 3) q^17 - b8 * q^19 - b2 * q^21 + (b12 - b10 + b8 + b6 + b1) * q^23 + (b13 + 2b12 + b11 - 2b10 + 3b8 + b6 + b3 - b2 + 3b1 - 3) * q^25 - q^27 + (b9 + b4 - 1) * q^29 + (b13 - b10 - b8 - 2b7 + b6 - b3 - b2 + 1) q^31 + (b12 - b11 + b9 + b8 - b1) * q^33 + (-b13 - b12 + b11 - b10 - b9 + b8 + b7 - b6 + b5 + b2 + 2b1 + 1) q^35 + (b12 - b11 - 2b10 + b9 + b5 - b4 - b3 + b2 - b1 + 1) q^37 + (b8 + b3 - 1) * q^39 + (-b13 + b12 + 2b10 + 2b4 - 3b2 + b1 - 4) q^41 + (-b13 - 2b12 - b10 - b9 - b7 + b5 + b1 + 1) q^43 + b7 * q^45 + (-b12 - b11 - b10 + b9 - b6 - b5 - b4 - b3 + 2b2 - b1 + 1) q^47 + (b13 + 2b12 - b10 + 2b8 + b7 + b6 + b4 + 2b3 + 2b1) * q^49 + (-b12 - b10 - 3b8 - b6 - b4 - b3 + b2 - b1 + 1) q^51 + (3b8 - 2b7 + b3 - 3) * q^53 + (2b13 - b12 - 3b10 - 2b9 - 2b7 + 2b5 + b4 + 3b1 - 1) * q^55 - q^57 + (b13 + 2b12 + 2b11 - b10 - b8 + b6 + 3b3 + b2 + 4b1 + 1) * q^59 + (b12 + 2b11 - 2b10 - 2b9 + b8 - b6 + b4 + b3 + 2b1 - 1) * q^61 + (b12 - b2) * q^63 + (-b12 - b10 - b6 + 2b5 + 2b2 + b1) * q^65 + (-b12 + b8 + b7 - b3 - b1 - 1) * q^67 + (-b13 - b12 + b2 - b1 + 1) * q^69 + (b13 - b12 - 2b10 + b7 - b5 - b4 + b2 + b1 - 2) q^71 + (-2b13 - 2b12 + b11 - b8 - b7 - 2b6 + b2 - b1 + 1) q^73 + (2b12 + b11 - b10 - b9 + 3b8 + b6 + b4 + b3 - b2 + 2b1 - 1) q^75 + (-b13 - b12 - 2b11 + 2b10 - b8 + b7 - b4 - 2b3 - 4b1 + 5) * q^77 + (b12 - b11 + b9 + 3b5 + b4 + b3 + b2 + b1 - 1) q^79 + (b8 - 1) * q^81 + (b13 + b12 + b9 - b7 + b5 - b2 + b1 - 2) * q^83 + (-b13 + b10 + 3b7 - 3b5 + b4 + b2 - 2b1 + 1) q^85 + (-b12 + b11 - b10 - b3) * q^87 + (b12 + b11 - b9 + 2b8 + 2b6 + b5 + b4 + b3 - b2 + 3b1 - 1) q^89 + (b13 + b11 - 3b10 - b8 + b7 + b6 - b5 - 2b4 + 2b2 + 5) q^91 + (-b8 + b6 - 2b5 - b4 - b3 - b2 - b1 + 1) q^93 + b7 * q^95 + (-b13 + b12 + 2b10 - b9 + 2b7 - 2b5 + b4 - b2 - b1 - 7) q^97 + (b9 + b2 - b1 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 7 q^{3} + 2 q^{5} - 7 q^{9} - 4 q^{11} - 12 q^{13} + 4 q^{15} + 18 q^{17} - 7 q^{19} + 3 q^{21} + 2 q^{23} - 15 q^{25} - 14 q^{27} - 8 q^{29} + q^{31} + 4 q^{33} + 22 q^{35} - 7 q^{37} - 6 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) 14 * q + 7 * q^3 + 2 * q^5 - 7 * q^9 - 4 * q^11 - 12 * q^13 + 4 * q^15 + 18 * q^17 - 7 * q^19 + 3 * q^21 + 2 * q^23 - 15 * q^25 - 14 * q^27 - 8 * q^29 + q^31 + 4 * q^33 + 22 * q^35 - 7 * q^37 - 6 * q^39 - 30 * q^41 + 6 * q^43 + 2 * q^45 + 2 * q^47 + 26 * q^49 - 18 * q^51 - 24 * q^53 - 8 * q^55 - 14 * q^57 + 15 * q^59 + 5 * q^61 + 3 * q^63 - 5 * q^65 - 9 * q^67 + 4 * q^69 - 32 * q^71 + q^73 + 15 * q^75 + 47 * q^77 - 13 * q^79 - 7 * q^81 - 22 * q^83 + 22 * q^85 - 4 * q^87 + 7 * q^89 + 39 * q^91 - q^93 + 2 * q^95 - 82 * q^97 + 8 * q^99

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	1596.2.r.h

Level	$1596$
Weight	$2$
Character orbit	1596.r
Analytic conductor	$12.744$
Analytic rank	$0$
Dimension	$14$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(12.7441241626\)
Analytic rank:	\(0\)
Dimension:	\(14\)
Relative dimension:	\(7\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{14} - 3 x^{13} + 8 x^{12} + 7 x^{11} - 57 x^{10} + 47 x^{9} + 645 x^{8} - 2490 x^{7} + 4515 x^{6} + \cdots + 823543 \) x^14 - 3x^13 + 8x^12 + 7x^11 - 57x^10 + 47x^9 + 645x^8 - 2490x^7 + 4515x^6 + 2303x^5 - 19551x^4 + 16807x^3 + 134456x^2 - 352947*x + 823543
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{13} - 3 \nu^{12} + 8 \nu^{11} + 7 \nu^{10} - 57 \nu^{9} + 47 \nu^{8} + 645 \nu^{7} + \cdots - 352947 ) / 117649 \) (v^13 - 3v^12 + 8v^11 + 7v^10 - 57v^9 + 47v^8 + 645v^7 - 2490v^6 + 4515v^5 + 2303v^4 - 19551v^3 + 16807v^2 + 134456v - 352947) / 117649
\(\beta_{3}\)	\(=\)	\( ( 15452 \nu^{13} - 2762783 \nu^{12} + 6344404 \nu^{11} + 2075745 \nu^{10} - 62842930 \nu^{9} + \cdots - 321216005859 ) / 1109665368 \) (15452v^13 - 2762783v^12 + 6344404v^11 + 2075745v^10 - 62842930v^9 + 99663530v^8 + 393922679v^7 - 2235177148v^6 + 2601425477v^5 + 5203585086v^4 - 21360176138v^3 + 2974927837v^2 + 108960923876*v - 321216005859) / 1109665368
\(\beta_{4}\)	\(=\)	\( ( - 168 \nu^{13} + 31 \nu^{12} + 908 \nu^{11} - 3637 \nu^{10} - 2310 \nu^{9} + 35578 \nu^{8} + \cdots - 40723361 ) / 1210104 \) (-168v^13 + 31v^12 + 908v^11 - 3637v^10 - 2310v^9 + 35578v^8 - 87863v^7 - 123092v^6 + 662143v^5 - 647318v^4 - 2519874v^3 + 7163555v^2 - 6866860*v - 40723361) / 1210104
\(\beta_{5}\)	\(=\)	\( ( 26483 \nu^{13} - 151788 \nu^{12} + 197027 \nu^{11} + 613822 \nu^{10} - 2917294 \nu^{9} + \cdots - 9955189468 ) / 158523624 \) (26483v^13 - 151788v^12 + 197027v^11 + 613822v^10 - 2917294v^9 - 166379v^8 + 32570500v^7 - 95436065v^6 + 40577038v^5 + 345649598v^4 - 695627177v^3 - 848909908v^2 + 6226712583*v - 9955189468) / 158523624
\(\beta_{6}\)	\(=\)	\( ( - 194486 \nu^{13} - 3086439 \nu^{12} + 8196022 \nu^{11} - 2151079 \nu^{10} + \cdots - 417380886671 ) / 1109665368 \) (-194486v^13 - 3086439v^12 + 8196022v^11 - 2151079v^10 - 73260350v^9 + 157114436v^8 + 335020247v^7 - 2676676978v^6 + 3745308413v^5 + 4967713786v^4 - 27433618828v^3 + 12220180021v^2 + 114608042262*v - 417380886671) / 1109665368
\(\beta_{7}\)	\(=\)	\( ( 197957 \nu^{13} - 1029635 \nu^{12} + 1197361 \nu^{11} + 4891887 \nu^{10} - 20714236 \nu^{9} + \cdots - 63598578771 ) / 1109665368 \) (197957v^13 - 1029635v^12 + 1197361v^11 + 4891887v^10 - 20714236v^9 - 4104031v^8 + 232439771v^7 - 641775427v^6 + 201777947v^5 + 2551894128v^4 - 4792015613v^3 - 6257214887v^2 + 42803176829*v - 63598578771) / 1109665368
\(\beta_{8}\)	\(=\)	\( ( - 227294 \nu^{13} + 1199119 \nu^{12} - 1426086 \nu^{11} - 5576333 \nu^{10} + \cdots + 79705197467 ) / 1109665368 \) (-227294v^13 + 1199119v^12 - 1426086v^11 - 5576333v^10 + 23762806v^9 + 6493992v^8 - 276748959v^7 + 765945018v^6 - 234990273v^5 - 2995058458v^4 + 5539315544v^3 + 7945665315v^2 - 52828871662*v + 79705197467) / 1109665368
\(\beta_{9}\)	\(=\)	\( ( - 3936 \nu^{13} - 449 \nu^{12} + 23756 \nu^{11} - 84301 \nu^{10} - 73470 \nu^{9} + \cdots - 1131430433 ) / 8470728 \) (-3936v^13 - 449v^12 + 23756v^11 - 84301v^10 - 73470v^9 + 902962v^8 - 1879607v^7 - 3960380v^6 + 16355143v^5 - 13034294v^4 - 69710634v^3 + 169152851v^2 - 87194716*v - 1131430433) / 8470728
\(\beta_{10}\)	\(=\)	\( ( 73891 \nu^{13} + 56038 \nu^{12} - 569325 \nu^{11} + 1543864 \nu^{10} + 2453830 \nu^{9} + \cdots + 26740911806 ) / 158523624 \) (73891v^13 + 56038v^12 - 569325v^11 + 1543864v^10 + 2453830v^9 - 18592047v^8 + 28568994v^7 + 113034591v^6 - 353085768v^5 + 156498650v^4 + 1680827939v^3 - 3181118514v^2 - 73933993*v + 26740911806) / 158523624
\(\beta_{11}\)	\(=\)	\( ( 327108 \nu^{13} - 820345 \nu^{12} - 44858 \nu^{11} + 7382977 \nu^{10} - 13879122 \nu^{9} + \cdots - 7534124311 ) / 554832684 \) (327108v^13 - 820345v^12 - 44858v^11 + 7382977v^10 - 13879122v^9 - 41782282v^8 + 272802689v^7 - 370812334v^6 - 527452303v^5 + 2644191998v^4 - 1059427530v^3 - 12386089121v^2 + 40569913090*v - 7534124311) / 554832684
\(\beta_{12}\)	\(=\)	\( ( 677483 \nu^{13} - 441391 \nu^{12} - 2973969 \nu^{11} + 14724983 \nu^{10} + 417800 \nu^{9} + \cdots + 130686509233 ) / 1109665368 \) (677483v^13 - 441391v^12 - 2973969v^11 + 14724983v^10 + 417800v^9 - 134497941v^8 + 391518591v^7 + 250310043v^6 - 2302779381v^5 + 3205175260v^4 + 7719939073v^3 - 27388752027v^2 + 35471997043*v + 130686509233) / 1109665368
\(\beta_{13}\)	\(=\)	\( ( 875653 \nu^{13} + 396327 \nu^{12} - 6095423 \nu^{11} + 18272345 \nu^{10} + 23052160 \nu^{9} + \cdots + 285684596071 ) / 1109665368 \) (875653v^13 + 396327v^12 - 6095423v^11 + 18272345v^10 + 23052160v^9 - 212248579v^8 + 375025289v^7 + 1125978053v^6 - 3926658379v^5 + 2383000732v^4 + 17949025703v^3 - 38048793461v^2 + 8932231413*v + 285684596071) / 1109665368

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{13} + 2\beta_{12} + \beta_{10} + 3\beta_{8} - \beta_{7} + \beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 3 \) -b13 + 2b12 + b10 + 3b8 - b7 + b5 + 2*b4 + b3 - b2 + b1 - 3
\(\nu^{3}\)	\(=\)	\( - 2 \beta_{13} - 3 \beta_{12} - \beta_{11} + 6 \beta_{10} + 2 \beta_{9} - 10 \beta_{8} - \beta_{7} + \cdots - 1 \) -2b13 - 3b12 - b11 + 6b10 + 2b9 - 10b8 - b7 - 3b6 + 2b4 + b3 - 6b1 - 1
\(\nu^{4}\)	\(=\)	\( 8 \beta_{13} + 9 \beta_{12} - 5 \beta_{11} - 7 \beta_{10} + 7 \beta_{9} + 16 \beta_{8} + 4 \beta_{7} + \cdots - 5 \) 8b13 + 9b12 - 5b11 - 7b10 + 7b9 + 16b8 + 4b7 - b4 + 7b3 - 2*b2 - b1 - 5
\(\nu^{5}\)	\(=\)	\( 15 \beta_{13} - 8 \beta_{12} - 14 \beta_{11} + 9 \beta_{10} + 7 \beta_{9} - 5 \beta_{8} + 14 \beta_{7} + \cdots + 54 \) 15b13 - 8b12 - 14b11 + 9b10 + 7b9 - 5b8 + 14b7 + 9b6 - 10b5 + b4 - b3 + 28b2 - 8*b1 + 54
\(\nu^{6}\)	\(=\)	\( 21 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} - 6 \beta_{10} - 38 \beta_{9} + 214 \beta_{8} + 26 \beta_{7} + \cdots - 317 \) 21b13 + 13b12 + 13b11 - 6b10 - 38b9 + 214b8 + 26b7 + 36b6 + 20b5 + 37b4 + 29b3 - 16b2 + 133*b1 - 317
\(\nu^{7}\)	\(=\)	\( - 87 \beta_{13} - 29 \beta_{12} + 85 \beta_{11} + 255 \beta_{10} - 83 \beta_{9} + 397 \beta_{8} + \cdots - 318 \) -87b13 - 29b12 + 85b11 + 255b10 - 83b9 + 397b8 - 109b7 + 51b6 + 56b5 + 160b4 + 116b3 - 28b2 - 136*b1 - 318
\(\nu^{8}\)	\(=\)	\( - 4 \beta_{13} - 498 \beta_{12} + 67 \beta_{11} + 397 \beta_{10} + 178 \beta_{9} - 227 \beta_{8} + \cdots - 207 \) -4b13 - 498b12 + 67b11 + 397b10 + 178b9 - 227b8 + 238b7 - 141b6 - 105b5 - 480b4 + 96b3 + 148b2 - 550*b1 - 207
\(\nu^{9}\)	\(=\)	\( 649 \beta_{13} + 1351 \beta_{12} - 831 \beta_{11} - 708 \beta_{10} + 489 \beta_{9} + 2040 \beta_{8} + \cdots + 2198 \) 649b13 + 1351b12 - 831b11 - 708b10 + 489b9 + 2040b8 + 22b7 + 630b6 + 380b5 - 1194b4 + 273b3 + 176b2 - 449*b1 + 2198
\(\nu^{10}\)	\(=\)	\( - 1516 \beta_{13} - 1286 \beta_{12} - 2176 \beta_{11} + 2288 \beta_{10} - 1063 \beta_{9} - 7165 \beta_{8} + \cdots + 4492 \) -1516b13 - 1286b12 - 2176b11 + 2288b10 - 1063b9 - 7165b8 + 1938b7 - 93b6 - 1660b5 - 1176b4 - 1485b3 + 1059b2 - 576*b1 + 4492
\(\nu^{11}\)	\(=\)	\( - 3684 \beta_{13} + 3857 \beta_{12} + 2579 \beta_{11} - 6453 \beta_{10} - 9808 \beta_{9} + 12032 \beta_{8} + \cdots - 27336 \) -3684b13 + 3857b12 + 2579b11 - 6453b10 - 9808b9 + 12032b8 - 2918b7 - 2580b6 + 3916b5 + 4796b4 + 4393b3 - 4041b2 + 12308*b1 - 27336
\(\nu^{12}\)	\(=\)	\( - 10059 \beta_{13} - 20743 \beta_{12} + 12419 \beta_{11} + 7941 \beta_{10} - 4567 \beta_{9} + \cdots + 23704 \) -10059b13 - 20743b12 + 12419b11 + 7941b10 - 4567b9 - 43078b8 - 6134b7 - 15741b6 - 21305b5 - 1897b4 + 4423b3 + 23830b2 - 31297*b1 + 23704
\(\nu^{13}\)	\(=\)	\( 47049 \beta_{13} - 9469 \beta_{12} + 14060 \beta_{11} - 103014 \beta_{10} + 42002 \beta_{9} + \cdots - 39327 \) 47049b13 - 9469b12 + 14060b11 - 103014b10 + 42002b9 - 31795b8 + 41323b7 - 25938b6 - 15005b5 - 89119b4 - 11903b3 + 90343b2 + 678*b1 - 39327

\(n\)	\(533\)	\(799\)	\(913\)	\(1009\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{8}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 457.7
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{14} \) T^14
$3$	\( (T^{2} - T + 1)^{7} \) (T^2 - T + 1)^7
$5$	\( T^{14} - 2 T^{13} + \cdots + 4356 \) T^14 - 2T^13 + 27T^12 - 28T^11 + 474T^10 - 482T^9 + 3928T^8 - 1809T^7 + 20505T^6 - 16263T^5 + 26520T^4 - 8649T^3 + 12951T^2 - 3762*T + 4356
$7$	\( T^{14} - 13 T^{12} + \cdots + 823543 \) T^14 - 13T^12 - 22T^11 + 123T^10 + 202T^9 - 234T^8 - 2220T^7 - 1638T^6 + 9898T^5 + 42189T^4 - 52822T^3 - 218491*T^2 + 823543
$11$	\( T^{14} + 4 T^{13} + \cdots + 38416 \) T^14 + 4T^13 + 67T^12 + 306T^11 + 3222T^10 + 13326T^9 + 80044T^8 + 273829T^7 + 1275859T^6 + 3635793T^5 + 10551150T^4 + 15476601T^3 + 19706869T^2 - 854756*T + 38416
$13$	\( (T^{7} + 6 T^{6} + \cdots - 432)^{2} \) (T^7 + 6T^6 - 26T^5 - 195T^4 - 105T^3 + 780T^2 + 732T - 432)^2
$17$	\( T^{14} - 18 T^{13} + \cdots + 324 \) T^14 - 18T^13 + 235T^12 - 1742T^11 + 10642T^10 - 43556T^9 + 183259T^8 - 582182T^7 + 2037298T^6 - 3946308T^5 + 6251215T^4 - 3108654T^3 + 1215045T^2 - 20250*T + 324
$19$	\( (T^{2} + T + 1)^{7} \) (T^2 + T + 1)^7
$23$	\( T^{14} - 2 T^{13} + \cdots + 677329 \) T^14 - 2T^13 + 78T^12 - 308T^11 + 5132T^10 - 17025T^9 + 109168T^8 - 317711T^7 + 1574722T^6 - 3812622T^5 + 9178965T^4 - 9179585T^3 + 8258544T^2 + 1929935*T + 677329
$29$	\( (T^{7} + 4 T^{6} + \cdots - 7776)^{2} \) (T^7 + 4T^6 - 111T^5 - 584T^4 + 333T^3 + 4212T^2 + 324T - 7776)^2
$31$	\( T^{14} - T^{13} + \cdots + 2985984 \) T^14 - T^13 + 170T^12 - 101T^11 + 21967T^10 - 9855T^9 + 1140000T^8 + 749283T^7 + 44081799T^6 - 1627794T^5 + 50239980T^4 - 19538352T^3 + 56571840T^2 - 12897792T + 2985984
$37$	\( T^{14} + \cdots + 43945575424 \) T^14 + 7T^13 + 202T^12 + 927T^11 + 24735T^10 + 112845T^9 + 1574620T^8 + 6113071T^7 + 67323085T^6 + 239925888T^5 + 1380627708T^4 + 2079375648T^3 + 8302935952T^2 + 1580625280*T + 43945575424
$41$	\( (T^{7} + 15 T^{6} + \cdots + 7464)^{2} \) (T^7 + 15T^6 - 83T^5 - 2343T^4 - 11571T^3 - 12910T^2 + 9500T + 7464)^2
$43$	\( (T^{7} - 3 T^{6} + \cdots + 16572)^{2} \) (T^7 - 3T^6 - 180T^5 + 730T^4 + 6456T^3 - 21357T^2 - 37983T + 16572)^2
$47$	\( T^{14} - 2 T^{13} + \cdots + 576 \) T^14 - 2T^13 + 141T^12 - 54T^11 + 17506T^10 - 19538T^9 + 240781T^8 - 682802T^7 + 3285394T^6 - 5917110T^5 + 9580029T^4 - 4707194T^3 + 1858033T^2 - 33432*T + 576
$53$	\( T^{14} + \cdots + 21941904384 \) T^14 + 24T^13 + 476T^12 + 4566T^11 + 42313T^10 + 211682T^9 + 1565485T^8 + 5281979T^7 + 42880939T^6 + 51831522T^5 + 614493508T^4 - 561034112T^3 + 8717579968T^2 - 11722257408*T + 21941904384
$59$	\( T^{14} + \cdots + 24943780096 \) T^14 - 15T^13 + 460T^12 - 3873T^11 + 105563T^10 - 813595T^9 + 12905118T^8 - 23732523T^7 + 337022409T^6 + 1400327852T^5 + 11635798572T^4 + 26603458544T^3 + 48396184656T^2 + 39952122304*T + 24943780096
$61$	\( T^{14} + \cdots + 13107860948484 \) T^14 - 5T^13 + 361T^12 - 984T^11 + 83764T^10 - 202627T^9 + 10914571T^8 - 18835243T^7 + 1005479104T^6 - 2169505368T^5 + 49478251169T^4 - 128049464127T^3 + 1755612467635T^2 - 4201184568810*T + 13107860948484
$67$	\( T^{14} + \cdots + 166719744 \) T^14 + 9T^13 + 147T^12 + 154T^11 + 6789T^10 + 12558T^9 + 155338T^8 + 235854T^7 + 2248473T^6 + 3672540T^5 + 18875652T^4 + 14515056T^3 + 64483920T^2 + 26495424*T + 166719744
$71$	\( (T^{7} + 16 T^{6} + \cdots - 44584)^{2} \) (T^7 + 16T^6 - 14T^5 - 1187T^4 - 2793T^3 + 23134T^2 + 70068T - 44584)^2
$73$	\( T^{14} + \cdots + 35570714404 \) T^14 - T^13 + 266T^12 - 1265T^11 + 54631T^10 - 263467T^9 + 4983066T^8 - 36908646T^7 + 359428572T^6 - 1698358660T^5 + 7313131273T^4 - 13468317416T^3 + 23723697809T^2 + 14726609966T + 35570714404
$79$	\( T^{14} + \cdots + 7929675777024 \) T^14 + 13T^13 + 390T^12 + 3537T^11 + 79663T^10 + 642061T^9 + 10056310T^8 + 59006461T^7 + 756778363T^6 + 3575421630T^5 + 36450556740T^4 + 81744179776T^3 + 616522522816T^2 - 269972484096*T + 7929675777024
$83$	\( (T^{7} + 11 T^{6} + \cdots + 53437)^{2} \) (T^7 + 11T^6 - 145T^5 - 1632T^4 + 2022T^3 + 38672T^2 + 84722T + 53437)^2
$89$	\( T^{14} - 7 T^{13} + \cdots + 14930496 \) T^14 - 7T^13 + 218T^12 - 1851T^11 + 41911T^10 - 300317T^9 + 1833244T^8 - 6193895T^7 + 19002235T^6 - 34517862T^5 + 84160880T^4 - 116644888T^3 + 257896384T^2 - 64652448*T + 14930496
$97$	\( (T^{7} + 41 T^{6} + \cdots - 72128)^{2} \) (T^7 + 41T^6 + 478T^5 - 1162T^4 - 61521T^3 - 426998T^2 - 959336T - 72128)^2
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