LMFDB - Newform orbit 192.4.c.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,4,Mod(191,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("192.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	192.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.3283667211$
Analytic rank:	$0$
Dimension:	$12$
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} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 $ x^12 - 16x^10 + 103x^8 - 260x^6 + 259x^4 + 356*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{44}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9}+O(q^{10}) $ q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9	\( q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9} - \beta_{10} q^{11} + ( - \beta_{2} - 6) q^{13} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_1) q^{15} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} - 1) q^{17} + ( - \beta_{6} - 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{19} + ( - \beta_{9} + 3 \beta_{8} + \beta_{5} - \beta_{2} - 12) q^{21} + (2 \beta_{11} - 2 \beta_{6} + 4 \beta_{4} - \beta_{3}) q^{23} + (\beta_{8} - \beta_{7} + 4 \beta_{5} - 2 \beta_{2} - 12) q^{25} + (2 \beta_{11} + \beta_{10} + 2 \beta_{6} - \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{27} + (9 \beta_{8} + 2 \beta_{7} + 4 \beta_{5} - 2) q^{29} + (2 \beta_{6} + 6 \beta_{4} + 3 \beta_{3} - 5 \beta_1) q^{31} + (\beta_{9} + 11 \beta_{8} + \beta_{7} + 2 \beta_{5} - 2 \beta_{2} + 6) q^{33} + ( - 4 \beta_{11} + 3 \beta_{10} - 3 \beta_{6} + 13 \beta_{4} + 2 \beta_{3}) q^{35} + (2 \beta_{8} - 2 \beta_{7} + 8 \beta_{5} - \beta_{2}) q^{37} + ( - 3 \beta_{11} - 3 \beta_{10} - 5 \beta_{6} + 4 \beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{39} + ( - 3 \beta_{9} + 16 \beta_{8} - 3 \beta_{5}) q^{41} + (7 \beta_{6} + 21 \beta_{4} + 5 \beta_{3} + 10 \beta_1) q^{43} + ( - 3 \beta_{9} - 17 \beta_{8} + 2 \beta_{7} - \beta_{5} + 44) q^{45} + (4 \beta_{11} - 2 \beta_{10} + 10 \beta_{6} - 34 \beta_{4} - 2 \beta_{3}) q^{47} + ( - \beta_{8} + \beta_{7} - 4 \beta_{5} - 2 \beta_{2} - 44) q^{49} + (6 \beta_{11} - 3 \beta_{10} - 11 \beta_{6} + 3 \beta_{4} - 12 \beta_{3} + 12 \beta_1) q^{51} + (2 \beta_{9} - 15 \beta_{8} + 2 \beta_{5}) q^{53} + ( - 14 \beta_{6} - 42 \beta_{4} + 11 \beta_{3} + 6 \beta_1) q^{55} + (3 \beta_{9} - 10 \beta_{8} - 2 \beta_{7} + 2 \beta_{5} - 73) q^{57} + ( - 8 \beta_{11} + 4 \beta_{10} + 17 \beta_{6} - 43 \beta_{4} + 4 \beta_{3}) q^{59} + ( - 2 \beta_{8} + 2 \beta_{7} - 8 \beta_{5} + 3 \beta_{2} - 36) q^{61} + ( - 2 \beta_{11} - 4 \beta_{10} + 12 \beta_{6} + 10 \beta_{4} - 12 \beta_{3} + \cdots + 11 \beta_1) q^{63}+ \cdots + ( - 14 \beta_{11} + 14 \beta_{10} - 31 \beta_{6} - 17 \beta_{4} - 9 \beta_{3} + \cdots - 22 \beta_1) q^{99}+O(q^{100}) \) q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9 - b10 * q^11 + (-b2 - 6) * q^13 + (-b11 + b10 + b6 + b1) * q^15 + (-b9 - b8 + b7 + b5 - 1) * q^17 + (-b6 - 3b4 + b3 - 2b1) * q^19 + (-b9 + 3b8 + b5 - b2 - 12) q^21 + (2b11 - 2b6 + 4b4 - b3) q^23 + (b8 - b7 + 4b5 - 2b2 - 12) * q^25 + (2b11 + b10 + 2b6 - b4 - 3b3 - 2b1) * q^27 + (9b8 + 2b7 + 4b5 - 2) q^29 + (2b6 + 6b4 + 3b3 - 5b1) * q^31 + (b9 + 11b8 + b7 + 2b5 - 2b2 + 6) q^33 + (-4b11 + 3b10 - 3b6 + 13b4 + 2b3) q^35 + (2b8 - 2b7 + 8b5 - b2) q^37 + (-3b11 - 3b10 - 5b6 + 4b4 - 3b3 - 6b1) * q^39 + (-3b9 + 16b8 - 3b5) q^41 + (7b6 + 21b4 + 5b3 + 10b1) * q^43 + (-3b9 - 17b8 + 2b7 - b5 + 44) q^45 + (4b11 - 2b10 + 10b6 - 34b4 - 2b3) q^47 + (-b8 + b7 - 4b5 - 2b2 - 44) * q^49 + (6b11 - 3b10 - 11b6 + 3b4 - 12b3 + 12b1) * q^51 + (2b9 - 15b8 + 2b5) q^53 + (-14b6 - 42b4 + 11b3 + 6b1) * q^55 + (3b9 - 10b8 - 2b7 + 2b5 - 73) * q^57 + (-8b11 + 4b10 + 17b6 - 43b4 + 4b3) q^59 + (-2b8 + 2b7 - 8b5 + 3b2 - 36) * q^61 + (-2b11 - 4b10 + 12b6 + 10b4 - 12b3 + 11b1) * q^63 + (b9 - 38b8 - 2b7 - 3b5 + 2) q^65 + (-19b6 - 57b4 + 14b3 - 8b1) * q^67 + (b9 + 22b8 - 4b7 - 5b5 + 4b2 - 106) * q^69 + (-2b11 + 6b10 - 16b6 + 50b4 + b3) * q^71 + (-2b8 + 2b7 - 8b5 + 10b2 + 204) * q^73 + (-3b10 + 23b6 - 6b4 - 15b3 - 18b1) q^75 + (6b9 + 22b8 - 8b7 - 10b5 + 8) * q^77 + (34b6 + 102b4 + 13b3 + 3b1) * q^79 + (13b8 - b7 - 6b5 + 12b2 + 246) q^81 + (12b11 - 17b10 - 34b6 + 90b4 - 6b3) q^83 + (-8b8 + 8b7 - 32b5 + 4b2 + 264) * q^85 + (b11 + 17b10 - 31b6 - 6b4 - 18b3 - b1) * q^87 + (8b9 - 21b8 - 3b7 + 2b5 + 3) * q^89 + (28b6 + 84b4 + 23b3 - 26b1) * q^91 + (8b9 - 27b8 - 6b7 - 8b5 - b2 + 204) * q^93 + (-2b11 + 8b10 + 18b6 - 52b4 + b3) * q^95 + (b8 - b7 + 4b5 + 12b2 - 247) * q^97 + (-14b11 + 14b10 - 31b6 - 17b4 - 9b3 - 22b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{9}+O(q^{10}) $ 12 * q - 20 * q^9	$ 12 q - 20 q^{9} - 72 q^{13} - 136 q^{21} - 132 q^{25} + 80 q^{33} + 24 q^{37} + 544 q^{45} - 540 q^{49} - 888 q^{57} - 456 q^{61} - 1312 q^{69} + 2424 q^{73} + 2924 q^{81} + 3072 q^{85} + 2360 q^{93} - 2952 q^{97}+O(q^{100}) $ 12 * q - 20 * q^9 - 72 * q^13 - 136 * q^21 - 132 * q^25 + 80 * q^33 + 24 * q^37 + 544 * q^45 - 540 * q^49 - 888 * q^57 - 456 * q^61 - 1312 * q^69 + 2424 * q^73 + 2924 * q^81 + 3072 * q^85 + 2360 * q^93 - 2952 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 $ x^12 - 16*x^10 + 103*x^8 - 260*x^6 + 259*x^4 + 356*x^2 + 169 :

$\beta_{1}$	$=$	$ ( -16388\nu^{10} + 259515\nu^{8} - 1685594\nu^{6} + 4399856\nu^{4} - 5980224\nu^{2} - 2762095 ) / 217953 $ (-16388v^10 + 259515v^8 - 1685594v^6 + 4399856v^4 - 5980224*v^2 - 2762095) / 217953
$\beta_{2}$	$=$	$ ( 5904\nu^{10} - 97064\nu^{8} + 571840\nu^{6} - 896416\nu^{4} - 1055120\nu^{2} + 1560520 ) / 72651 $ (5904v^10 - 97064v^8 + 571840v^6 - 896416v^4 - 1055120*v^2 + 1560520) / 72651
$\beta_{3}$	$=$	$ ( -544\nu^{10} + 8496\nu^{8} - 53344\nu^{6} + 135616\nu^{4} - 180960\nu^{2} - 83504 ) / 3573 $ (-544v^10 + 8496v^8 - 53344v^6 + 135616v^4 - 180960*v^2 - 83504) / 3573
$\beta_{4}$	$=$	$ ( 2707 \nu^{11} + 10504 \nu^{10} - 41518 \nu^{9} - 183781 \nu^{8} + 254446 \nu^{7} + 1315990 \nu^{6} - 571012 \nu^{5} - 4058808 \nu^{4} + 500393 \nu^{3} + \cdots + 2381353 ) / 629642 $ (2707v^11 + 10504v^10 - 41518v^9 - 183781v^8 + 254446v^7 + 1315990v^6 - 571012v^5 - 4058808v^4 + 500393v^3 + 5081284v^2 + 95344*v + 2381353) / 629642
$\beta_{5}$	$=$	$ ( - 2631 \nu^{11} - 7176 \nu^{10} + 57332 \nu^{9} + 97500 \nu^{8} - 464316 \nu^{7} - 531232 \nu^{6} + 1492582 \nu^{5} + 802880 \nu^{4} - 515419 \nu^{3} + \cdots - 7299877 ) / 314821 $ (-2631v^11 - 7176v^10 + 57332v^9 + 97500v^8 - 464316v^7 - 531232v^6 + 1492582v^5 + 802880v^4 - 515419v^3 + 954824v^2 - 2325140*v - 7299877) / 314821
$\beta_{6}$	$=$	$ ( - 8121 \nu^{11} + 31512 \nu^{10} + 124554 \nu^{9} - 551343 \nu^{8} - 763338 \nu^{7} + 3947970 \nu^{6} + 1713036 \nu^{5} - 12176424 \nu^{4} - 1501179 \nu^{3} + \cdots + 7144059 ) / 629642 $ (-8121v^11 + 31512v^10 + 124554v^9 - 551343v^8 - 763338v^7 + 3947970v^6 + 1713036v^5 - 12176424v^4 - 1501179v^3 + 15243852v^2 - 286032*v + 7144059) / 629642
$\beta_{7}$	$=$	$ ( 18023 \nu^{11} + 43056 \nu^{10} - 102208 \nu^{9} - 585000 \nu^{8} - 549788 \nu^{7} + 3187392 \nu^{6} + 5801042 \nu^{5} - 4817280 \nu^{4} + 2774507 \nu^{3} + \cdots + 44743725 ) / 944463 $ (18023v^11 + 43056v^10 - 102208v^9 - 585000v^8 - 549788v^7 + 3187392v^6 + 5801042v^5 - 4817280v^4 + 2774507v^3 - 5728944v^2 - 1691184*v + 44743725) / 944463
$\beta_{8}$	$=$	$ ( 3815\nu^{11} - 60784\nu^{9} + 386308\nu^{7} - 931534\nu^{5} + 689195\nu^{3} + 2016192\nu ) / 72651 $ (3815v^11 - 60784v^9 + 386308v^7 - 931534v^5 + 689195v^3 + 2016192v) / 72651
$\beta_{9}$	$=$	$ ( 168575 \nu^{11} + 21528 \nu^{10} - 2577340 \nu^{9} - 292500 \nu^{8} + 15775756 \nu^{7} + 1593696 \nu^{6} - 35075014 \nu^{5} - 2408640 \nu^{4} + \cdots + 21899631 ) / 944463 $ (168575v^11 + 21528v^10 - 2577340v^9 - 292500v^8 + 15775756v^7 + 1593696v^6 - 35075014v^5 - 2408640v^4 + 31664915v^3 - 2864472v^2 + 84598716*v + 21899631) / 944463
$\beta_{10}$	$=$	$ ( - 525547 \nu^{11} + 8893158 \nu^{9} - 62629246 \nu^{7} + 194548900 \nu^{5} - 288501633 \nu^{3} - 73378376 \nu ) / 2833389 $ (-525547v^11 + 8893158v^9 - 62629246v^7 + 194548900v^5 - 288501633v^3 - 73378376v) / 2833389
$\beta_{11}$	$=$	$ ( 1307467 \nu^{11} - 336856 \nu^{10} - 21563934 \nu^{9} + 5083299 \nu^{8} + 143863390 \nu^{7} - 30457882 \nu^{6} - 397870564 \nu^{5} + 71014216 \nu^{4} + \cdots - 44786495 ) / 5666778 $ (1307467v^11 - 336856v^10 - 21563934v^9 + 5083299v^8 + 143863390v^7 - 30457882v^6 - 397870564v^5 + 71014216v^4 + 510897681v^3 - 97769724v^2 + 122714864*v - 44786495) / 5666778

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

$\nu$	$=$	$ ( 3\beta_{9} + 6\beta_{8} + 32\beta_{6} + 3\beta_{5} - 96\beta_{4} ) / 384 $ (3b9 + 6b8 + 32b6 + 3b5 - 96*b4) / 384
$\nu^{2}$	$=$	$ ( 4\beta_{8} - 4\beta_{7} - 8\beta_{6} + 16\beta_{5} - 24\beta_{4} + 3\beta_{3} - 24\beta _1 + 508 ) / 192 $ (4b8 - 4b7 - 8b6 + 16b5 - 24b4 + 3b3 - 24*b1 + 508) / 192
$\nu^{3}$	$=$	$ ( 24 \beta_{11} + 12 \beta_{10} + 39 \beta_{9} - 126 \beta_{8} - 12 \beta_{7} + 124 \beta_{6} + 15 \beta_{5} - 396 \beta_{4} - 12 \beta_{3} + 12 ) / 384 $ (24b11 + 12b10 + 39b9 - 126b8 - 12b7 + 124b6 + 15b5 - 396b4 - 12*b3 + 12) / 384
$\nu^{4}$	$=$	$ ( 7\beta_{8} - 7\beta_{7} - 24\beta_{6} + 28\beta_{5} - 72\beta_{4} + 42\beta_{3} + 9\beta_{2} - 120\beta _1 + 793 ) / 96 $ (7b8 - 7b7 - 24b6 + 28b5 - 72b4 + 42b3 + 9b2 - 120b1 + 793) / 96
$\nu^{5}$	$=$	$ ( 90 \beta_{11} + 72 \beta_{10} + 372 \beta_{9} - 1216 \beta_{8} - 200 \beta_{7} + 232 \beta_{6} - 28 \beta_{5} - 786 \beta_{4} - 45 \beta_{3} + 200 ) / 384 $ (90b11 + 72b10 + 372b9 - 1216b8 - 200b7 + 232b6 - 28b5 - 786b4 - 45*b3 + 200) / 384
$\nu^{6}$	$=$	$ ( - 4 \beta_{8} + 4 \beta_{7} - 80 \beta_{6} - 16 \beta_{5} - 240 \beta_{4} + 411 \beta_{3} + 18 \beta_{2} - 912 \beta _1 - 1084 ) / 96 $ (-4b8 + 4b7 - 80b6 - 16b5 - 240b4 + 411b3 + 18b2 - 912b1 - 1084) / 96
$\nu^{7}$	$=$	$ ( - 450 \beta_{11} - 252 \beta_{10} + 2622 \beta_{9} - 8728 \beta_{8} - 1652 \beta_{7} - 2764 \beta_{6} - 682 \beta_{5} + 8742 \beta_{4} + 225 \beta_{3} + 1652 ) / 384 $ (-450b11 - 252b10 + 2622b9 - 8728b8 - 1652b7 - 2764b6 - 682b5 + 8742b4 + 225*b3 + 1652) / 384
$\nu^{8}$	$=$	$ ( - 34 \beta_{8} + 34 \beta_{7} - 20 \beta_{6} - 136 \beta_{5} - 60 \beta_{4} + 223 \beta_{3} - 33 \beta_{2} - 452 \beta _1 - 4118 ) / 8 $ (-34b8 + 34b7 - 20b6 - 136b5 - 60b4 + 223b3 - 33b2 - 452b1 - 4118) / 8
$\nu^{9}$	$=$	$ ( - 10680 \beta_{11} - 7176 \beta_{10} + 13431 \beta_{9} - 45018 \beta_{8} - 8904 \beta_{7} - 46024 \beta_{6} - 4377 \beta_{5} + 148752 \beta_{4} + 5340 \beta_{3} + 8904 ) / 384 $ (-10680b11 - 7176b10 + 13431b9 - 45018b8 - 8904b7 - 46024b6 - 4377b5 + 148752b4 + 5340*b3 + 8904) / 384
$\nu^{10}$	$=$	$ ( - 9800 \beta_{8} + 9800 \beta_{7} - 1112 \beta_{6} - 39200 \beta_{5} - 3336 \beta_{4} + 21663 \beta_{3} - 11412 \beta_{2} - 42408 \beta _1 - 1134008 ) / 192 $ (-9800b8 + 9800b7 - 1112b6 - 39200b5 - 3336b4 + 21663b3 - 11412b2 - 42408b1 - 1134008) / 192
$\nu^{11}$	$=$	$ ( - 106956 \beta_{11} - 73404 \beta_{10} + 30693 \beta_{9} - 103482 \beta_{8} - 21252 \beta_{7} - 436076 \beta_{6} - 11811 \beta_{5} + 1415184 \beta_{4} + 53478 \beta_{3} + 21252 ) / 384 $ (-106956b11 - 73404b10 + 30693b9 - 103482b8 - 21252b7 - 436076b6 - 11811b5 + 1415184b4 + 53478*b3 + 21252) / 384

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

191.1

−2.59708	−	0.707107i
−2.59708	+	0.707107i
−1.64111	+	0.707107i
−1.64111	−	0.707107i
−0.248859	−	0.707107i
−0.248859	+	0.707107i
0.248859	+	0.707107i
0.248859	−	0.707107i
1.64111	−	0.707107i
1.64111	+	0.707107i
2.59708	+	0.707107i
2.59708	−	0.707107i

−5.19416

−

0.143987i

7.96714i

−

25.6706i

26.9585

1.49578i

191.2

−5.19416

0.143987i

−

7.96714i

25.6706i

26.9585

−

1.49578i

191.3

−3.28223

−

4.02827i

−

2.00080i

14.5105i

−5.45398

26.4434i

191.4

−3.28223

4.02827i

2.00080i

−

14.5105i

−5.45398

−

26.4434i

191.5

−0.497717

−

5.17226i

−

18.4532i

−

17.1600i

−26.5046

5.14865i

191.6

−0.497717

5.17226i

18.4532i

17.1600i

−26.5046

−

5.14865i

191.7

0.497717

−

5.17226i

18.4532i

−

17.1600i

−26.5046

−

5.14865i

191.8

0.497717

5.17226i

−

18.4532i

17.1600i

−26.5046

5.14865i

191.9

3.28223

−

4.02827i

2.00080i

14.5105i

−5.45398

−

26.4434i

191.10

3.28223

4.02827i

−

2.00080i

−

14.5105i

−5.45398

26.4434i

191.11

5.19416

−

0.143987i

−

7.96714i

−

25.6706i

26.9585

−

1.49578i

191.12

5.19416

0.143987i

7.96714i

25.6706i

26.9585

1.49578i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	192.4.c.d		12
3.b	odd	2	1	inner	192.4.c.d		12
4.b	odd	2	1	inner	192.4.c.d		12
8.b	even	2	1		96.4.c.a	✓	12
8.d	odd	2	1		96.4.c.a	✓	12
12.b	even	2	1	inner	192.4.c.d		12
16.e	even	4	1		768.4.f.d		12
16.e	even	4	1		768.4.f.i		12
16.f	odd	4	1		768.4.f.d		12
16.f	odd	4	1		768.4.f.i		12
24.f	even	2	1		96.4.c.a	✓	12
24.h	odd	2	1		96.4.c.a	✓	12
48.i	odd	4	1		768.4.f.d		12
48.i	odd	4	1		768.4.f.i		12
48.k	even	4	1		768.4.f.d		12
48.k	even	4	1		768.4.f.i		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.4.c.a	✓	12	8.b	even	2	1
96.4.c.a	✓	12	8.d	odd	2	1
96.4.c.a	✓	12	24.f	even	2	1
96.4.c.a	✓	12	24.h	odd	2	1
192.4.c.d		12	1.a	even	1	1	trivial
192.4.c.d		12	3.b	odd	2	1	inner
192.4.c.d		12	4.b	odd	2	1	inner
192.4.c.d		12	12.b	even	2	1	inner
768.4.f.d		12	16.e	even	4	1
768.4.f.d		12	16.f	odd	4	1
768.4.f.d		12	48.i	odd	4	1
768.4.f.d		12	48.k	even	4	1
768.4.f.i		12	16.e	even	4	1
768.4.f.i		12	16.f	odd	4	1
768.4.f.i		12	48.i	odd	4	1
768.4.f.i		12	48.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + 408T_{5}^{4} + 23232T_{5}^{2} + 86528 $ T5^6 + 408*T5^4 + 23232*T5^2 + 86528 acting on $S_{4}^{\mathrm{new}}(192, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + 10 T^{10} + \cdots + 387420489 $ T^12 + 10T^10 - 681T^8 - 16596T^6 - 496449T^4 + 5314410*T^2 + 387420489
$5$	$ (T^{6} + 408 T^{4} + 23232 T^{2} + \cdots + 86528)^{2} $ (T^6 + 408T^4 + 23232T^2 + 86528)^2
$7$	$ (T^{6} + 1164 T^{4} + 394800 T^{2} + \cdots + 40857664)^{2} $ (T^6 + 1164T^4 + 394800T^2 + 40857664)^2
$11$	$ (T^{6} - 5400 T^{4} + \cdots - 3202560512)^{2} $ (T^6 - 5400T^4 + 8526528T^2 - 3202560512)^2
$13$	$ (T^{3} + 18 T^{2} - 4500 T - 133928)^{4} $ (T^3 + 18T^2 - 4500T - 133928)^4
$17$	$ (T^{6} + 24576 T^{4} + \cdots + 391378894848)^{2} $ (T^6 + 24576T^4 + 179306496T^2 + 391378894848)^2
$19$	$ (T^{6} + 8076 T^{4} + \cdots + 2518433856)^{2} $ (T^6 + 8076T^4 + 9033264T^2 + 2518433856)^2
$23$	$ (T^{6} - 26976 T^{4} + \cdots - 124728147968)^{2} $ (T^6 - 26976T^4 + 177859584T^2 - 124728147968)^2
$29$	$ (T^{6} + 118296 T^{4} + \cdots + 19385223967232)^{2} $ (T^6 + 118296T^4 + 2933334720T^2 + 19385223967232)^2
$31$	$ (T^{6} + 58956 T^{4} + \cdots + 708600302656)^{2} $ (T^6 + 58956T^4 + 414766128T^2 + 708600302656)^2
$37$	$ (T^{3} - 6 T^{2} - 96756 T + 3249144)^{4} $ (T^3 - 6T^2 - 96756T + 3249144)^4
$41$	$ (T^{6} + 317280 T^{4} + \cdots + 359281246896128)^{2} $ (T^6 + 317280T^4 + 25285684224T^2 + 359281246896128)^2
$43$	$ (T^{6} + 246252 T^{4} + \cdots + 24182238991936)^{2} $ (T^6 + 246252T^4 + 4898641200T^2 + 24182238991936)^2
$47$	$ (T^{6} - 273792 T^{4} + \cdots - 165630483365888)^{2} $ (T^6 - 273792T^4 + 15725936640T^2 - 165630483365888)^2
$53$	$ (T^{6} + 187224 T^{4} + \cdots + 152455972372992)^{2} $ (T^6 + 187224T^4 + 9858320064T^2 + 152455972372992)^2
$59$	$ (T^{6} - 680472 T^{4} + \cdots - 61\!\cdots\!48)^{2} $ (T^6 - 680472T^4 + 131795888832T^2 - 6138947587240448)^2
$61$	$ (T^{3} + 114 T^{2} - 104724 T - 17941928)^{4} $ (T^3 + 114T^2 - 104724T - 17941928)^4
$67$	$ (T^{6} + 1232556 T^{4} + \cdots + 53\!\cdots\!56)^{2} $ (T^6 + 1232556T^4 + 348595981104T^2 + 5330258010235456)^2
$71$	$ (T^{6} - 589152 T^{4} + \cdots - 61\!\cdots\!68)^{2} $ (T^6 - 589152T^4 + 108701346816T^2 - 6162070084026368)^2
$73$	$ (T^{3} - 606 T^{2} - 319956 T + 145079064)^{4} $ (T^3 - 606T^2 - 319956T + 145079064)^4
$79$	$ (T^{6} + 2122764 T^{4} + \cdots + 76\!\cdots\!44)^{2} $ (T^6 + 2122764T^4 + 836589949488T^2 + 76931405354772544)^2
$83$	$ (T^{6} - 3104664 T^{4} + \cdots - 22\!\cdots\!92)^{2} $ (T^6 - 3104664T^4 + 2166924624576T^2 - 227939504096072192)^2
$89$	$ (T^{6} + 1258080 T^{4} + \cdots + 761571160915968)^{2} $ (T^6 + 1258080T^4 + 130935794688T^2 + 761571160915968)^2
$97$	$ (T^{3} + 738 T^{2} - 581844 T - 38714344)^{4} $ (T^3 + 738T^2 - 581844T - 38714344)^4
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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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9}+O(q^{10}) \) q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9	\( q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9} - \beta_{10} q^{11} + ( - \beta_{2} - 6) q^{13} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_1) q^{15} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} - 1) q^{17} + ( - \beta_{6} - 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{19} + ( - \beta_{9} + 3 \beta_{8} + \beta_{5} - \beta_{2} - 12) q^{21} + (2 \beta_{11} - 2 \beta_{6} + 4 \beta_{4} - \beta_{3}) q^{23} + (\beta_{8} - \beta_{7} + 4 \beta_{5} - 2 \beta_{2} - 12) q^{25} + (2 \beta_{11} + \beta_{10} + 2 \beta_{6} - \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{27} + (9 \beta_{8} + 2 \beta_{7} + 4 \beta_{5} - 2) q^{29} + (2 \beta_{6} + 6 \beta_{4} + 3 \beta_{3} - 5 \beta_1) q^{31} + (\beta_{9} + 11 \beta_{8} + \beta_{7} + 2 \beta_{5} - 2 \beta_{2} + 6) q^{33} + ( - 4 \beta_{11} + 3 \beta_{10} - 3 \beta_{6} + 13 \beta_{4} + 2 \beta_{3}) q^{35} + (2 \beta_{8} - 2 \beta_{7} + 8 \beta_{5} - \beta_{2}) q^{37} + ( - 3 \beta_{11} - 3 \beta_{10} - 5 \beta_{6} + 4 \beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{39} + ( - 3 \beta_{9} + 16 \beta_{8} - 3 \beta_{5}) q^{41} + (7 \beta_{6} + 21 \beta_{4} + 5 \beta_{3} + 10 \beta_1) q^{43} + ( - 3 \beta_{9} - 17 \beta_{8} + 2 \beta_{7} - \beta_{5} + 44) q^{45} + (4 \beta_{11} - 2 \beta_{10} + 10 \beta_{6} - 34 \beta_{4} - 2 \beta_{3}) q^{47} + ( - \beta_{8} + \beta_{7} - 4 \beta_{5} - 2 \beta_{2} - 44) q^{49} + (6 \beta_{11} - 3 \beta_{10} - 11 \beta_{6} + 3 \beta_{4} - 12 \beta_{3} + 12 \beta_1) q^{51} + (2 \beta_{9} - 15 \beta_{8} + 2 \beta_{5}) q^{53} + ( - 14 \beta_{6} - 42 \beta_{4} + 11 \beta_{3} + 6 \beta_1) q^{55} + (3 \beta_{9} - 10 \beta_{8} - 2 \beta_{7} + 2 \beta_{5} - 73) q^{57} + ( - 8 \beta_{11} + 4 \beta_{10} + 17 \beta_{6} - 43 \beta_{4} + 4 \beta_{3}) q^{59} + ( - 2 \beta_{8} + 2 \beta_{7} - 8 \beta_{5} + 3 \beta_{2} - 36) q^{61} + ( - 2 \beta_{11} - 4 \beta_{10} + 12 \beta_{6} + 10 \beta_{4} - 12 \beta_{3} + \cdots + 11 \beta_1) q^{63}+ \cdots + ( - 14 \beta_{11} + 14 \beta_{10} - 31 \beta_{6} - 17 \beta_{4} - 9 \beta_{3} + \cdots - 22 \beta_1) q^{99}+O(q^{100}) \) q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9 - b10 * q^11 + (-b2 - 6) * q^13 + (-b11 + b10 + b6 + b1) * q^15 + (-b9 - b8 + b7 + b5 - 1) * q^17 + (-b6 - 3b4 + b3 - 2b1) * q^19 + (-b9 + 3b8 + b5 - b2 - 12) q^21 + (2b11 - 2b6 + 4b4 - b3) q^23 + (b8 - b7 + 4b5 - 2b2 - 12) * q^25 + (2b11 + b10 + 2b6 - b4 - 3b3 - 2b1) * q^27 + (9b8 + 2b7 + 4b5 - 2) q^29 + (2b6 + 6b4 + 3b3 - 5b1) * q^31 + (b9 + 11b8 + b7 + 2b5 - 2b2 + 6) q^33 + (-4b11 + 3b10 - 3b6 + 13b4 + 2b3) q^35 + (2b8 - 2b7 + 8b5 - b2) q^37 + (-3b11 - 3b10 - 5b6 + 4b4 - 3b3 - 6b1) * q^39 + (-3b9 + 16b8 - 3b5) q^41 + (7b6 + 21b4 + 5b3 + 10b1) * q^43 + (-3b9 - 17b8 + 2b7 - b5 + 44) q^45 + (4b11 - 2b10 + 10b6 - 34b4 - 2b3) q^47 + (-b8 + b7 - 4b5 - 2b2 - 44) * q^49 + (6b11 - 3b10 - 11b6 + 3b4 - 12b3 + 12b1) * q^51 + (2b9 - 15b8 + 2b5) q^53 + (-14b6 - 42b4 + 11b3 + 6b1) * q^55 + (3b9 - 10b8 - 2b7 + 2b5 - 73) * q^57 + (-8b11 + 4b10 + 17b6 - 43b4 + 4b3) q^59 + (-2b8 + 2b7 - 8b5 + 3b2 - 36) * q^61 + (-2b11 - 4b10 + 12b6 + 10b4 - 12b3 + 11b1) * q^63 + (b9 - 38b8 - 2b7 - 3b5 + 2) q^65 + (-19b6 - 57b4 + 14b3 - 8b1) * q^67 + (b9 + 22b8 - 4b7 - 5b5 + 4b2 - 106) * q^69 + (-2b11 + 6b10 - 16b6 + 50b4 + b3) * q^71 + (-2b8 + 2b7 - 8b5 + 10b2 + 204) * q^73 + (-3b10 + 23b6 - 6b4 - 15b3 - 18b1) q^75 + (6b9 + 22b8 - 8b7 - 10b5 + 8) * q^77 + (34b6 + 102b4 + 13b3 + 3b1) * q^79 + (13b8 - b7 - 6b5 + 12b2 + 246) q^81 + (12b11 - 17b10 - 34b6 + 90b4 - 6b3) q^83 + (-8b8 + 8b7 - 32b5 + 4b2 + 264) * q^85 + (b11 + 17b10 - 31b6 - 6b4 - 18b3 - b1) * q^87 + (8b9 - 21b8 - 3b7 + 2b5 + 3) * q^89 + (28b6 + 84b4 + 23b3 - 26b1) * q^91 + (8b9 - 27b8 - 6b7 - 8b5 - b2 + 204) * q^93 + (-2b11 + 8b10 + 18b6 - 52b4 + b3) * q^95 + (b8 - b7 + 4b5 + 12b2 - 247) * q^97 + (-14b11 + 14b10 - 31b6 - 17b4 - 9b3 - 22b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{9}+O(q^{10}) \) 12 * q - 20 * q^9	\( 12 q - 20 q^{9} - 72 q^{13} - 136 q^{21} - 132 q^{25} + 80 q^{33} + 24 q^{37} + 544 q^{45} - 540 q^{49} - 888 q^{57} - 456 q^{61} - 1312 q^{69} + 2424 q^{73} + 2924 q^{81} + 3072 q^{85} + 2360 q^{93} - 2952 q^{97}+O(q^{100}) \) 12 * q - 20 * q^9 - 72 * q^13 - 136 * q^21 - 132 * q^25 + 80 * q^33 + 24 * q^37 + 544 * q^45 - 540 * q^49 - 888 * q^57 - 456 * q^61 - 1312 * q^69 + 2424 * q^73 + 2924 * q^81 + 3072 * q^85 + 2360 * q^93 - 2952 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	192.4.c.d

Level	$192$
Weight	$4$
Character orbit	192.c
Analytic conductor	$11.328$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(11.3283667211\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 \) x^12 - 16x^10 + 103x^8 - 260x^6 + 259x^4 + 356*x^2 + 169
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{44}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -16388\nu^{10} + 259515\nu^{8} - 1685594\nu^{6} + 4399856\nu^{4} - 5980224\nu^{2} - 2762095 ) / 217953 \) (-16388v^10 + 259515v^8 - 1685594v^6 + 4399856v^4 - 5980224*v^2 - 2762095) / 217953
\(\beta_{2}\)	\(=\)	\( ( 5904\nu^{10} - 97064\nu^{8} + 571840\nu^{6} - 896416\nu^{4} - 1055120\nu^{2} + 1560520 ) / 72651 \) (5904v^10 - 97064v^8 + 571840v^6 - 896416v^4 - 1055120*v^2 + 1560520) / 72651
\(\beta_{3}\)	\(=\)	\( ( -544\nu^{10} + 8496\nu^{8} - 53344\nu^{6} + 135616\nu^{4} - 180960\nu^{2} - 83504 ) / 3573 \) (-544v^10 + 8496v^8 - 53344v^6 + 135616v^4 - 180960*v^2 - 83504) / 3573
\(\beta_{4}\)	\(=\)	\( ( 2707 \nu^{11} + 10504 \nu^{10} - 41518 \nu^{9} - 183781 \nu^{8} + 254446 \nu^{7} + 1315990 \nu^{6} - 571012 \nu^{5} - 4058808 \nu^{4} + 500393 \nu^{3} + \cdots + 2381353 ) / 629642 \) (2707v^11 + 10504v^10 - 41518v^9 - 183781v^8 + 254446v^7 + 1315990v^6 - 571012v^5 - 4058808v^4 + 500393v^3 + 5081284v^2 + 95344*v + 2381353) / 629642
\(\beta_{5}\)	\(=\)	\( ( - 2631 \nu^{11} - 7176 \nu^{10} + 57332 \nu^{9} + 97500 \nu^{8} - 464316 \nu^{7} - 531232 \nu^{6} + 1492582 \nu^{5} + 802880 \nu^{4} - 515419 \nu^{3} + \cdots - 7299877 ) / 314821 \) (-2631v^11 - 7176v^10 + 57332v^9 + 97500v^8 - 464316v^7 - 531232v^6 + 1492582v^5 + 802880v^4 - 515419v^3 + 954824v^2 - 2325140*v - 7299877) / 314821
\(\beta_{6}\)	\(=\)	\( ( - 8121 \nu^{11} + 31512 \nu^{10} + 124554 \nu^{9} - 551343 \nu^{8} - 763338 \nu^{7} + 3947970 \nu^{6} + 1713036 \nu^{5} - 12176424 \nu^{4} - 1501179 \nu^{3} + \cdots + 7144059 ) / 629642 \) (-8121v^11 + 31512v^10 + 124554v^9 - 551343v^8 - 763338v^7 + 3947970v^6 + 1713036v^5 - 12176424v^4 - 1501179v^3 + 15243852v^2 - 286032*v + 7144059) / 629642
\(\beta_{7}\)	\(=\)	\( ( 18023 \nu^{11} + 43056 \nu^{10} - 102208 \nu^{9} - 585000 \nu^{8} - 549788 \nu^{7} + 3187392 \nu^{6} + 5801042 \nu^{5} - 4817280 \nu^{4} + 2774507 \nu^{3} + \cdots + 44743725 ) / 944463 \) (18023v^11 + 43056v^10 - 102208v^9 - 585000v^8 - 549788v^7 + 3187392v^6 + 5801042v^5 - 4817280v^4 + 2774507v^3 - 5728944v^2 - 1691184*v + 44743725) / 944463
\(\beta_{8}\)	\(=\)	\( ( 3815\nu^{11} - 60784\nu^{9} + 386308\nu^{7} - 931534\nu^{5} + 689195\nu^{3} + 2016192\nu ) / 72651 \) (3815v^11 - 60784v^9 + 386308v^7 - 931534v^5 + 689195v^3 + 2016192v) / 72651
\(\beta_{9}\)	\(=\)	\( ( 168575 \nu^{11} + 21528 \nu^{10} - 2577340 \nu^{9} - 292500 \nu^{8} + 15775756 \nu^{7} + 1593696 \nu^{6} - 35075014 \nu^{5} - 2408640 \nu^{4} + \cdots + 21899631 ) / 944463 \) (168575v^11 + 21528v^10 - 2577340v^9 - 292500v^8 + 15775756v^7 + 1593696v^6 - 35075014v^5 - 2408640v^4 + 31664915v^3 - 2864472v^2 + 84598716*v + 21899631) / 944463
\(\beta_{10}\)	\(=\)	\( ( - 525547 \nu^{11} + 8893158 \nu^{9} - 62629246 \nu^{7} + 194548900 \nu^{5} - 288501633 \nu^{3} - 73378376 \nu ) / 2833389 \) (-525547v^11 + 8893158v^9 - 62629246v^7 + 194548900v^5 - 288501633v^3 - 73378376v) / 2833389
\(\beta_{11}\)	\(=\)	\( ( 1307467 \nu^{11} - 336856 \nu^{10} - 21563934 \nu^{9} + 5083299 \nu^{8} + 143863390 \nu^{7} - 30457882 \nu^{6} - 397870564 \nu^{5} + 71014216 \nu^{4} + \cdots - 44786495 ) / 5666778 \) (1307467v^11 - 336856v^10 - 21563934v^9 + 5083299v^8 + 143863390v^7 - 30457882v^6 - 397870564v^5 + 71014216v^4 + 510897681v^3 - 97769724v^2 + 122714864*v - 44786495) / 5666778

\(\nu\)	\(=\)	\( ( 3\beta_{9} + 6\beta_{8} + 32\beta_{6} + 3\beta_{5} - 96\beta_{4} ) / 384 \) (3b9 + 6b8 + 32b6 + 3b5 - 96*b4) / 384
\(\nu^{2}\)	\(=\)	\( ( 4\beta_{8} - 4\beta_{7} - 8\beta_{6} + 16\beta_{5} - 24\beta_{4} + 3\beta_{3} - 24\beta _1 + 508 ) / 192 \) (4b8 - 4b7 - 8b6 + 16b5 - 24b4 + 3b3 - 24*b1 + 508) / 192
\(\nu^{3}\)	\(=\)	\( ( 24 \beta_{11} + 12 \beta_{10} + 39 \beta_{9} - 126 \beta_{8} - 12 \beta_{7} + 124 \beta_{6} + 15 \beta_{5} - 396 \beta_{4} - 12 \beta_{3} + 12 ) / 384 \) (24b11 + 12b10 + 39b9 - 126b8 - 12b7 + 124b6 + 15b5 - 396b4 - 12*b3 + 12) / 384
\(\nu^{4}\)	\(=\)	\( ( 7\beta_{8} - 7\beta_{7} - 24\beta_{6} + 28\beta_{5} - 72\beta_{4} + 42\beta_{3} + 9\beta_{2} - 120\beta _1 + 793 ) / 96 \) (7b8 - 7b7 - 24b6 + 28b5 - 72b4 + 42b3 + 9b2 - 120b1 + 793) / 96
\(\nu^{5}\)	\(=\)	\( ( 90 \beta_{11} + 72 \beta_{10} + 372 \beta_{9} - 1216 \beta_{8} - 200 \beta_{7} + 232 \beta_{6} - 28 \beta_{5} - 786 \beta_{4} - 45 \beta_{3} + 200 ) / 384 \) (90b11 + 72b10 + 372b9 - 1216b8 - 200b7 + 232b6 - 28b5 - 786b4 - 45*b3 + 200) / 384
\(\nu^{6}\)	\(=\)	\( ( - 4 \beta_{8} + 4 \beta_{7} - 80 \beta_{6} - 16 \beta_{5} - 240 \beta_{4} + 411 \beta_{3} + 18 \beta_{2} - 912 \beta _1 - 1084 ) / 96 \) (-4b8 + 4b7 - 80b6 - 16b5 - 240b4 + 411b3 + 18b2 - 912b1 - 1084) / 96
\(\nu^{7}\)	\(=\)	\( ( - 450 \beta_{11} - 252 \beta_{10} + 2622 \beta_{9} - 8728 \beta_{8} - 1652 \beta_{7} - 2764 \beta_{6} - 682 \beta_{5} + 8742 \beta_{4} + 225 \beta_{3} + 1652 ) / 384 \) (-450b11 - 252b10 + 2622b9 - 8728b8 - 1652b7 - 2764b6 - 682b5 + 8742b4 + 225*b3 + 1652) / 384
\(\nu^{8}\)	\(=\)	\( ( - 34 \beta_{8} + 34 \beta_{7} - 20 \beta_{6} - 136 \beta_{5} - 60 \beta_{4} + 223 \beta_{3} - 33 \beta_{2} - 452 \beta _1 - 4118 ) / 8 \) (-34b8 + 34b7 - 20b6 - 136b5 - 60b4 + 223b3 - 33b2 - 452b1 - 4118) / 8
\(\nu^{9}\)	\(=\)	\( ( - 10680 \beta_{11} - 7176 \beta_{10} + 13431 \beta_{9} - 45018 \beta_{8} - 8904 \beta_{7} - 46024 \beta_{6} - 4377 \beta_{5} + 148752 \beta_{4} + 5340 \beta_{3} + 8904 ) / 384 \) (-10680b11 - 7176b10 + 13431b9 - 45018b8 - 8904b7 - 46024b6 - 4377b5 + 148752b4 + 5340*b3 + 8904) / 384
\(\nu^{10}\)	\(=\)	\( ( - 9800 \beta_{8} + 9800 \beta_{7} - 1112 \beta_{6} - 39200 \beta_{5} - 3336 \beta_{4} + 21663 \beta_{3} - 11412 \beta_{2} - 42408 \beta _1 - 1134008 ) / 192 \) (-9800b8 + 9800b7 - 1112b6 - 39200b5 - 3336b4 + 21663b3 - 11412b2 - 42408b1 - 1134008) / 192
\(\nu^{11}\)	\(=\)	\( ( - 106956 \beta_{11} - 73404 \beta_{10} + 30693 \beta_{9} - 103482 \beta_{8} - 21252 \beta_{7} - 436076 \beta_{6} - 11811 \beta_{5} + 1415184 \beta_{4} + 53478 \beta_{3} + 21252 ) / 384 \) (-106956b11 - 73404b10 + 30693b9 - 103482b8 - 21252b7 - 436076b6 - 11811b5 + 1415184b4 + 53478*b3 + 21252) / 384

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 191.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + 10 T^{10} + \cdots + 387420489 \) T^12 + 10T^10 - 681T^8 - 16596T^6 - 496449T^4 + 5314410*T^2 + 387420489
$5$	\( (T^{6} + 408 T^{4} + 23232 T^{2} + \cdots + 86528)^{2} \) (T^6 + 408T^4 + 23232T^2 + 86528)^2
$7$	\( (T^{6} + 1164 T^{4} + 394800 T^{2} + \cdots + 40857664)^{2} \) (T^6 + 1164T^4 + 394800T^2 + 40857664)^2
$11$	\( (T^{6} - 5400 T^{4} + \cdots - 3202560512)^{2} \) (T^6 - 5400T^4 + 8526528T^2 - 3202560512)^2
$13$	\( (T^{3} + 18 T^{2} - 4500 T - 133928)^{4} \) (T^3 + 18T^2 - 4500T - 133928)^4
$17$	\( (T^{6} + 24576 T^{4} + \cdots + 391378894848)^{2} \) (T^6 + 24576T^4 + 179306496T^2 + 391378894848)^2
$19$	\( (T^{6} + 8076 T^{4} + \cdots + 2518433856)^{2} \) (T^6 + 8076T^4 + 9033264T^2 + 2518433856)^2
$23$	\( (T^{6} - 26976 T^{4} + \cdots - 124728147968)^{2} \) (T^6 - 26976T^4 + 177859584T^2 - 124728147968)^2
$29$	\( (T^{6} + 118296 T^{4} + \cdots + 19385223967232)^{2} \) (T^6 + 118296T^4 + 2933334720T^2 + 19385223967232)^2
$31$	\( (T^{6} + 58956 T^{4} + \cdots + 708600302656)^{2} \) (T^6 + 58956T^4 + 414766128T^2 + 708600302656)^2
$37$	\( (T^{3} - 6 T^{2} - 96756 T + 3249144)^{4} \) (T^3 - 6T^2 - 96756T + 3249144)^4
$41$	\( (T^{6} + 317280 T^{4} + \cdots + 359281246896128)^{2} \) (T^6 + 317280T^4 + 25285684224T^2 + 359281246896128)^2
$43$	\( (T^{6} + 246252 T^{4} + \cdots + 24182238991936)^{2} \) (T^6 + 246252T^4 + 4898641200T^2 + 24182238991936)^2
$47$	\( (T^{6} - 273792 T^{4} + \cdots - 165630483365888)^{2} \) (T^6 - 273792T^4 + 15725936640T^2 - 165630483365888)^2
$53$	\( (T^{6} + 187224 T^{4} + \cdots + 152455972372992)^{2} \) (T^6 + 187224T^4 + 9858320064T^2 + 152455972372992)^2
$59$	\( (T^{6} - 680472 T^{4} + \cdots - 61\!\cdots\!48)^{2} \) (T^6 - 680472T^4 + 131795888832T^2 - 6138947587240448)^2
$61$	\( (T^{3} + 114 T^{2} - 104724 T - 17941928)^{4} \) (T^3 + 114T^2 - 104724T - 17941928)^4
$67$	\( (T^{6} + 1232556 T^{4} + \cdots + 53\!\cdots\!56)^{2} \) (T^6 + 1232556T^4 + 348595981104T^2 + 5330258010235456)^2
$71$	\( (T^{6} - 589152 T^{4} + \cdots - 61\!\cdots\!68)^{2} \) (T^6 - 589152T^4 + 108701346816T^2 - 6162070084026368)^2
$73$	\( (T^{3} - 606 T^{2} - 319956 T + 145079064)^{4} \) (T^3 - 606T^2 - 319956T + 145079064)^4
$79$	\( (T^{6} + 2122764 T^{4} + \cdots + 76\!\cdots\!44)^{2} \) (T^6 + 2122764T^4 + 836589949488T^2 + 76931405354772544)^2
$83$	\( (T^{6} - 3104664 T^{4} + \cdots - 22\!\cdots\!92)^{2} \) (T^6 - 3104664T^4 + 2166924624576T^2 - 227939504096072192)^2
$89$	\( (T^{6} + 1258080 T^{4} + \cdots + 761571160915968)^{2} \) (T^6 + 1258080T^4 + 130935794688T^2 + 761571160915968)^2
$97$	\( (T^{3} + 738 T^{2} - 581844 T - 38714344)^{4} \) (T^3 + 738T^2 - 581844T - 38714344)^4
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\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)