LMFDB - Newform orbit 1176.2.k.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(881,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1176.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.k (of order $2$, degree $1$, 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:	$9.39040727770$
Analytic rank:	$0$
Dimension:	$16$
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} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + \cdots + 6561 $ x^16 - 6x^15 + 19x^14 - 42x^13 + 65x^12 - 48x^11 - 94x^10 + 444x^9 - 962x^8 + 1332x^7 - 846x^6 - 1296x^5 + 5265x^4 - 10206x^3 + 13851x^2 - 13122*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 168)
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{3} + \beta_{8} q^{5} + \beta_{5} q^{9}+O(q^{10}) $ q - b1 * q^3 + b8 * q^5 + b5 * q^9	$ q - \beta_1 q^{3} + \beta_{8} q^{5} + \beta_{5} q^{9} - \beta_{6} q^{11} + ( - \beta_{12} - \beta_{7} + \beta_1) q^{13} + (\beta_{11} - \beta_{10} - \beta_{6} + \cdots + 1) q^{15}+ \cdots + ( - \beta_{14} + \beta_{13} + \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) $ q - b1 * q^3 + b8 * q^5 + b5 * q^9 - b6 * q^11 + (-b12 - b7 + b1) * q^13 + (b11 - b10 - b6 + b5 + 1) * q^15 + (-b15 - b12 - b9 - b2) * q^17 - b12 * q^19 + (b14 - b13 + b11) * q^23 + (b13 + b11 - b10 + b5 + b3 + 3) * q^25 + (-b15 + b8 + b7 - b4) * q^27 + (b13 - b11 - b6) * q^29 + (-b15 - b12 + 2b4 - b1) q^31 + (-b15 - b12 - b9 - b8 - b7 - b1) * q^33 + (-b13 - b11 + 2b10 - 2b5 - b3 - 1) * q^37 + (-b14 + b13 - b11 - b10 - b5 + b3 + 3) * q^39 + (-2b7 - 2b1) * q^41 + (b13 + b11 + b10 - b5 + b3 + 1) * q^43 + (-2b15 - b12 - b9 - 3b7 + 2b4 - 2b1) * q^45 + (b9 - 2b8 - b4 - b2 + b1) q^47 + (-2b14 + b13 - b10 + b6 - b5 - b3 - 1) q^51 + (-b14 + b13 - b11 - b10 - b6 - b5) * q^53 + (b15 + 3b7 + 2b4 - 2b1) q^55 + (-b14 + b13 - b11 - b10 + b3) * q^57 + (-b15 - b12 - b8 - b4 - 2b2 + b1) q^59 + (-b15 + b12 - b7 + b4) * q^61 + (b14 + 2b13 - 2b11 + 2b6) q^65 + (-b10 + b5 + b3 - 1) * q^67 + (b12 - b8 - 2b7 + b4 - b2) q^69 + (-b14 - 2b10 - 2b5) * q^71 + (-b15 - 2b12 - b4 - b1) q^73 + (b12 - b9 + 2b8 + 3b4 + b2 - 3b1) q^75 + (b13 + b11 - b10 + b5 - 3) * q^79 + (2b11 + b10 - 2b6 - 4) * q^81 + (-b15 - b12 - 2b9 - b8 + 3b7 + b4 + 2b1) q^83 + (-2b13 - 2b11 - b10 + b5 + 2b3 + 2) q^85 + (-b15 - 2b12 - 2b9 - 2b4 - b1) q^87 + (-b15 - b12 + b7 - b4 - 2b2 + 2b1) * q^89 + (-b14 - 2b13 + b10 - b6 + b3 - 2) q^93 + (b14 + b13 - b11) * q^95 + (-b15 + 2b12 - 2b7 - 2b4 + b1) q^97 + (-b14 + b13 + b10 + 2b6 - b5 + b3 + 1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^9	$ 16 q - 4 q^{9} + 8 q^{15} + 36 q^{25} + 4 q^{37} + 44 q^{39} + 20 q^{43} - 12 q^{51} - 8 q^{57} - 28 q^{67} - 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} + 20 q^{99}+O(q^{100}) $ 16 * q - 4 * q^9 + 8 * q^15 + 36 * q^25 + 4 * q^37 + 44 * q^39 + 20 * q^43 - 12 * q^51 - 8 * q^57 - 28 * q^67 - 56 * q^79 - 60 * q^81 + 16 * q^85 - 32 * q^93 + 20 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + \cdots + 6561 $ x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 65*x^12 - 48*x^11 - 94*x^10 + 444*x^9 - 962*x^8 + 1332*x^7 - 846*x^6 - 1296*x^5 + 5265*x^4 - 10206*x^3 + 13851*x^2 - 13122*x + 6561 :

$\beta_{1}$	$=$	$ ( 11 \nu^{15} + 30 \nu^{14} - 142 \nu^{13} + 363 \nu^{12} - 662 \nu^{11} + 258 \nu^{10} + 1288 \nu^{9} + \cdots - 19683 ) / 69984 $ (11v^15 + 30v^14 - 142v^13 + 363v^12 - 662v^11 + 258v^10 + 1288v^9 - 3546v^8 + 8138v^7 - 7392v^6 - 846v^5 + 28890v^4 - 41067v^3 + 38880v^2 - 34992*v - 19683) / 69984
$\beta_{2}$	$=$	$ ( \nu^{15} + 246 \nu^{14} - 962 \nu^{13} + 1965 \nu^{12} - 2374 \nu^{11} - 462 \nu^{10} + \cdots + 150903 ) / 69984 $ (v^15 + 246v^14 - 962v^13 + 1965v^12 - 2374v^11 - 462v^10 + 7988v^9 - 24594v^8 + 38170v^7 - 30276v^6 - 19998v^5 + 114858v^4 - 199989v^3 + 244944v^2 - 195372v + 150903) / 69984
$\beta_{3}$	$=$	$ ( 35 \nu^{15} - 162 \nu^{14} + 512 \nu^{13} - 963 \nu^{12} + 1042 \nu^{11} + 306 \nu^{10} + \cdots - 54675 ) / 34992 $ (35v^15 - 162v^14 + 512v^13 - 963v^12 + 1042v^11 + 306v^10 - 5162v^9 + 13134v^8 - 21322v^7 + 14538v^6 + 16074v^5 - 72198v^4 + 144099v^3 - 184680v^2 + 173502*v - 54675) / 34992
$\beta_{4}$	$=$	$ ( - 22 \nu^{15} + 75 \nu^{14} - 238 \nu^{13} + 489 \nu^{12} - 737 \nu^{11} + 429 \nu^{10} + \cdots + 113724 ) / 17496 $ (-22v^15 + 75v^14 - 238v^13 + 489v^12 - 737v^11 + 429v^10 + 1483v^9 - 5787v^8 + 11651v^7 - 14727v^6 + 5787v^5 + 23193v^4 - 70227v^3 + 126846v^2 - 147987*v + 113724) / 17496
$\beta_{5}$	$=$	$ ( 61 \nu^{15} - 234 \nu^{14} + 565 \nu^{13} - 729 \nu^{12} + 563 \nu^{11} + 1251 \nu^{10} + \cdots - 113724 ) / 34992 $ (61v^15 - 234v^14 + 565v^13 - 729v^12 + 563v^11 + 1251v^10 - 5437v^9 + 10815v^8 - 13421v^7 + 6351v^6 + 19359v^5 - 58185v^4 + 98010v^3 - 136323v^2 + 129762*v - 113724) / 34992
$\beta_{6}$	$=$	$ ( - 131 \nu^{15} + 660 \nu^{14} - 1742 \nu^{13} + 3567 \nu^{12} - 5176 \nu^{11} + 2040 \nu^{10} + \cdots + 754515 ) / 69984 $ (-131v^15 + 660v^14 - 1742v^13 + 3567v^12 - 5176v^11 + 2040v^10 + 12350v^9 - 42648v^8 + 81940v^7 - 96318v^6 + 29736v^5 + 167832v^4 - 496287v^3 + 862164v^2 - 1106622*v + 754515) / 69984
$\beta_{7}$	$=$	$ ( - 19 \nu^{15} + 82 \nu^{14} - 226 \nu^{13} + 433 \nu^{12} - 542 \nu^{11} + 74 \nu^{10} + \cdots + 75087 ) / 7776 $ (-19v^15 + 82v^14 - 226v^13 + 433v^12 - 542v^11 + 74v^10 + 1912v^9 - 5482v^8 + 9482v^7 - 10040v^6 + 42v^5 + 24354v^4 - 60237v^3 + 95580v^2 - 105948*v + 75087) / 7776
$\beta_{8}$	$=$	$ ( 61 \nu^{15} - 302 \nu^{14} + 784 \nu^{13} - 1463 \nu^{12} + 1772 \nu^{11} - 100 \nu^{10} + \cdots - 255879 ) / 23328 $ (61v^15 - 302v^14 + 784v^13 - 1463v^12 + 1772v^11 - 100v^10 - 6358v^9 + 18728v^8 - 32084v^7 + 33490v^6 + 1116v^5 - 85716v^4 + 197181v^3 - 331290v^2 + 374220*v - 255879) / 23328
$\beta_{9}$	$=$	$ ( - 37 \nu^{15} + 178 \nu^{14} - 562 \nu^{13} + 1105 \nu^{12} - 1544 \nu^{11} + 572 \nu^{10} + \cdots + 244215 ) / 11664 $ (-37v^15 + 178v^14 - 562v^13 + 1105v^12 - 1544v^11 + 572v^10 + 4048v^9 - 13408v^8 + 25460v^7 - 30032v^6 + 8184v^5 + 55692v^4 - 157923v^3 + 266814v^2 - 310554*v + 244215) / 11664
$\beta_{10}$	$=$	$ ( 2 \nu^{15} - 8 \nu^{14} + 21 \nu^{13} - 38 \nu^{12} + 41 \nu^{11} + 17 \nu^{10} - 195 \nu^{9} + \cdots - 4617 ) / 432 $ (2v^15 - 8v^14 + 21v^13 - 38v^12 + 41v^11 + 17v^10 - 195v^9 + 497v^8 - 779v^7 + 661v^6 + 349v^5 - 2499v^4 + 5247v^3 - 7803v^2 + 7452*v - 4617) / 432
$\beta_{11}$	$=$	$ ( 107 \nu^{15} - 468 \nu^{14} + 1268 \nu^{13} - 2511 \nu^{12} + 3166 \nu^{11} - 90 \nu^{10} + \cdots - 365229 ) / 23328 $ (107v^15 - 468v^14 + 1268v^13 - 2511v^12 + 3166v^11 - 90v^10 - 11048v^9 + 32070v^8 - 54982v^7 + 57756v^6 + 5526v^5 - 151794v^4 + 356805v^3 - 554526v^2 + 581742*v - 365229) / 23328
$\beta_{12}$	$=$	$ ( - 30 \nu^{15} + 137 \nu^{14} - 378 \nu^{13} + 731 \nu^{12} - 939 \nu^{11} + 175 \nu^{10} + \cdots + 125388 ) / 5832 $ (-30v^15 + 137v^14 - 378v^13 + 731v^12 - 939v^11 + 175v^10 + 2997v^9 - 9161v^8 + 16161v^7 - 17341v^6 + 681v^5 + 41427v^4 - 104409v^3 + 162810v^2 - 181521*v + 125388) / 5832
$\beta_{13}$	$=$	$ ( - 125 \nu^{15} + 512 \nu^{14} - 1364 \nu^{13} + 2537 \nu^{12} - 3082 \nu^{11} + 142 \nu^{10} + \cdots + 465831 ) / 23328 $ (-125v^15 + 512v^14 - 1364v^13 + 2537v^12 - 3082v^11 + 142v^10 + 11108v^9 - 31994v^8 + 56962v^7 - 59656v^6 - 786v^5 + 141174v^4 - 357399v^3 + 579150v^2 - 650754*v + 465831) / 23328
$\beta_{14}$	$=$	$ ( - 45 \nu^{15} + 188 \nu^{14} - 498 \nu^{13} + 917 \nu^{12} - 1020 \nu^{11} - 236 \nu^{10} + \cdots + 140697 ) / 5832 $ (-45v^15 + 188v^14 - 498v^13 + 917v^12 - 1020v^11 - 236v^10 + 4386v^9 - 11588v^8 + 19032v^7 - 18226v^6 - 5868v^5 + 55620v^4 - 124497v^3 + 190512v^2 - 202662*v + 140697) / 5832
$\beta_{15}$	$=$	$ ( 243 \nu^{15} - 1090 \nu^{14} + 3042 \nu^{13} - 5725 \nu^{12} + 7074 \nu^{11} - 326 \nu^{10} + \cdots - 993627 ) / 23328 $ (243v^15 - 1090v^14 + 3042v^13 - 5725v^12 + 7074v^11 - 326v^10 - 24984v^9 + 72382v^8 - 125070v^7 + 129248v^6 + 3594v^5 - 332046v^4 + 796365v^3 - 1239624v^2 + 1397736*v - 993627) / 23328

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

$\nu$	$=$	$ ( 2 \beta_{15} - \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{10} + \beta_{9} + \beta_{7} + \cdots + 4 ) / 8 $ (2b15 - b14 + 2b13 + 2b12 - 2b10 + b9 + b7 - 2b6 - b4 + 2b3 + b2 + 2*b1 + 4) / 8
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{10} + \beta_{9} - \beta_{8} ) / 2 $ (b15 - b10 + b9 - b8) / 2
$\nu^{3}$	$=$	$ ( -4\beta_{12} - \beta_{9} - 4\beta_{8} + 3\beta_{7} + 5\beta_{4} + \beta_{2} + 2\beta_1 ) / 4 $ (-4b12 - b9 - 4b8 + 3b7 + 5b4 + b2 + 2*b1) / 4
$\nu^{4}$	$=$	$ ( - 2 \beta_{15} - 2 \beta_{13} - \beta_{12} - \beta_{9} - 3 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} + \cdots + 4 ) / 2 $ (-2b15 - 2b13 - b12 - b9 - 3b8 - 3b7 - 2b6 + b5 + 3b4 - 2b2 + 4b1 + 4) / 2
$\nu^{5}$	$=$	$ ( - 16 \beta_{15} - 15 \beta_{14} - 8 \beta_{12} + 10 \beta_{11} - 24 \beta_{10} + \beta_{9} + \beta_{7} + \cdots - 12 ) / 8 $ (-16b15 - 15b14 - 8b12 + 10b11 - 24b10 + b9 + b7 - 6b6 + 18b5 - 15b4 + 18b3 - 7b2 + 8*b1 - 12) / 8
$\nu^{6}$	$=$	$ -2\beta_{14} + 4\beta_{11} - 6\beta_{10} + 4\beta_{6} + 2\beta_{5} - 2\beta_{3} + 7 $ -2b14 + 4b11 - 6b10 + 4b6 + 2b5 - 2b3 + 7
$\nu^{7}$	$=$	$ ( - 2 \beta_{15} - 35 \beta_{14} + 94 \beta_{13} - 26 \beta_{12} + 16 \beta_{11} - 22 \beta_{10} + \cdots + 92 ) / 8 $ (-2b15 - 35b14 + 94b13 - 26b12 + 16b11 - 22b10 - 13b9 - 29b7 + 2b6 + 24b5 - 11b4 - 26b3 + 11b2 + 70b1 + 92) / 8
$\nu^{8}$	$=$	$ ( 11 \beta_{15} + 16 \beta_{14} + 12 \beta_{13} + 2 \beta_{12} + 14 \beta_{11} - 5 \beta_{10} + 15 \beta_{9} + \cdots + 4 ) / 2 $ (11b15 + 16b14 + 12b13 + 2b12 + 14b11 - 5b10 + 15b9 + 5b8 - 12b7 - 2b6 + 8b5 - 12b4 + 4b3 - 14b2 + 70*b1 + 4) / 2
$\nu^{9}$	$=$	$ ( 16\beta_{15} - 68\beta_{12} + 37\beta_{9} + 20\beta_{8} + 241\beta_{7} - 97\beta_{4} - 37\beta_{2} + 102\beta_1 ) / 4 $ (16b15 - 68b12 + 37b9 + 20b8 + 241b7 - 97b4 - 37b2 + 102b1) / 4
$\nu^{10}$	$=$	$ ( 40 \beta_{15} + 16 \beta_{14} - 32 \beta_{13} + 57 \beta_{12} + 60 \beta_{11} + 24 \beta_{10} + \cdots + 32 ) / 2 $ (40b15 + 16b14 - 32b13 + 57b12 + 60b11 + 24b10 - 23b9 - 49b8 + 183b7 - 28b6 - 3b5 + 51b4 - 44b3 - 12b2 + 36*b1 + 32) / 2
$\nu^{11}$	$=$	$ ( - 176 \beta_{15} + 163 \beta_{14} + 112 \beta_{13} + 80 \beta_{12} + 422 \beta_{11} + 80 \beta_{10} + \cdots - 436 ) / 8 $ (-176b15 + 163b14 + 112b13 + 80b12 + 422b11 + 80b10 + 3b9 - 512b8 - 93b7 - 682b6 + 774b5 - 349b4 - 74b3 - 173b2 - 448*b1 - 436) / 8
$\nu^{12}$	$=$	$ 148\beta_{14} - 76\beta_{13} + 140\beta_{11} - 76\beta_{10} + 24\beta_{6} + 244\beta_{5} + 64\beta_{3} - 39 $ 148b14 - 76b13 + 140b11 - 76b10 + 24b6 + 244b5 + 64*b3 - 39
$\nu^{13}$	$=$	$ ( - 174 \beta_{15} - 201 \beta_{14} + 114 \beta_{13} - 942 \beta_{12} - 848 \beta_{11} - 594 \beta_{10} + \cdots + 5604 ) / 8 $ (-174b15 - 201b14 + 114b13 - 942b12 - 848b11 - 594b10 - 855b9 + 1001b7 + 1182b6 + 1776b5 - 1209b4 + 66b3 - 327b2 - 414b1 + 5604) / 8
$\nu^{14}$	$=$	$ ( 405 \beta_{15} - 176 \beta_{14} + 420 \beta_{13} + 764 \beta_{12} + 32 \beta_{11} - 141 \beta_{10} + \cdots + 1120 ) / 2 $ (405b15 - 176b14 + 420b13 + 764b12 + 32b11 - 141b10 - 191b9 - 377b8 - 48b7 + 196b6 - 40b5 - 984b4 + 460b3 - 20b2 + 712*b1 + 1120) / 2
$\nu^{15}$	$=$	$ ( 1456 \beta_{15} + 1548 \beta_{12} + 2919 \beta_{9} - 2276 \beta_{8} - 1301 \beta_{7} - 6899 \beta_{4} + \cdots + 962 \beta_1 ) / 4 $ (1456b15 + 1548b12 + 2919b9 - 2276b8 - 1301b7 - 6899b4 + 153b2 + 962b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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} $

881.1

1.73018	+	0.0805675i
1.73018	−	0.0805675i
−0.601642	+	1.62420i
−0.601642	−	1.62420i
0.247636	−	1.71426i
0.247636	+	1.71426i
1.22961	−	1.21986i
1.22961	+	1.21986i
−0.441628	+	1.67480i
−0.441628	−	1.67480i
1.60841	+	0.642670i
1.60841	−	0.642670i
−1.70742	−	0.291063i
−1.70742	+	0.291063i
0.934861	+	1.45809i
0.934861	−	1.45809i

−1.53866

−

0.795315i

−3.80034

1.73495

2.44744i

881.2

−1.53866

0.795315i

−3.80034

1.73495

−

2.44744i

881.3

−1.33314

−

1.10578i

−0.145339

0.554510

2.94831i

881.4

−1.33314

1.10578i

−0.145339

0.554510

−

2.94831i

881.5

−1.07159

−

1.36077i

2.57910

−0.703402

2.91637i

881.6

−1.07159

1.36077i

2.57910

−0.703402

−

2.91637i

881.7

−0.454941

−

1.67124i

2.80795

−2.58606

1.52063i

881.8

−0.454941

1.67124i

2.80795

−2.58606

−

1.52063i

881.9

0.454941

−

1.67124i

−2.80795

−2.58606

−

1.52063i

881.10

0.454941

1.67124i

−2.80795

−2.58606

1.52063i

881.11

1.07159

−

1.36077i

−2.57910

−0.703402

−

2.91637i

881.12

1.07159

1.36077i

−2.57910

−0.703402

2.91637i

881.13

1.33314

−

1.10578i

0.145339

0.554510

−

2.94831i

881.14

1.33314

1.10578i

0.145339

0.554510

2.94831i

881.15

1.53866

−

0.795315i

3.80034

1.73495

−

2.44744i

881.16

1.53866

0.795315i

3.80034

1.73495

2.44744i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1176.2.k.a		16
3.b	odd	2	1	inner	1176.2.k.a		16
4.b	odd	2	1		2352.2.k.i		16
7.b	odd	2	1	inner	1176.2.k.a		16
7.c	even	3	1		168.2.u.a	✓	16
7.c	even	3	1		1176.2.u.b		16
7.d	odd	6	1		168.2.u.a	✓	16
7.d	odd	6	1		1176.2.u.b		16
12.b	even	2	1		2352.2.k.i		16
21.c	even	2	1	inner	1176.2.k.a		16
21.g	even	6	1		168.2.u.a	✓	16
21.g	even	6	1		1176.2.u.b		16
21.h	odd	6	1		168.2.u.a	✓	16
21.h	odd	6	1		1176.2.u.b		16
28.d	even	2	1		2352.2.k.i		16
28.f	even	6	1		336.2.bc.f		16
28.g	odd	6	1		336.2.bc.f		16
84.h	odd	2	1		2352.2.k.i		16
84.j	odd	6	1		336.2.bc.f		16
84.n	even	6	1		336.2.bc.f		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.2.u.a	✓	16	7.c	even	3	1
168.2.u.a	✓	16	7.d	odd	6	1
168.2.u.a	✓	16	21.g	even	6	1
168.2.u.a	✓	16	21.h	odd	6	1
336.2.bc.f		16	28.f	even	6	1
336.2.bc.f		16	28.g	odd	6	1
336.2.bc.f		16	84.j	odd	6	1
336.2.bc.f		16	84.n	even	6	1
1176.2.k.a		16	1.a	even	1	1	trivial
1176.2.k.a		16	3.b	odd	2	1	inner
1176.2.k.a		16	7.b	odd	2	1	inner
1176.2.k.a		16	21.c	even	2	1	inner
1176.2.u.b		16	7.c	even	3	1
1176.2.u.b		16	7.d	odd	6	1
1176.2.u.b		16	21.g	even	6	1
1176.2.u.b		16	21.h	odd	6	1
2352.2.k.i		16	4.b	odd	2	1
2352.2.k.i		16	12.b	even	2	1
2352.2.k.i		16	28.d	even	2	1
2352.2.k.i		16	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 29T_{5}^{6} + 263T_{5}^{4} - 763T_{5}^{2} + 16 $ T5^8 - 29*T5^6 + 263*T5^4 - 763*T5^2 + 16 acting on $S_{2}^{\mathrm{new}}(1176, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 2 T^{14} + \cdots + 6561 $ T^16 + 2T^14 + 17T^12 + 46T^10 + 172T^8 + 414T^6 + 1377T^4 + 1458*T^2 + 6561
$5$	$ (T^{8} - 29 T^{6} + \cdots + 16)^{2} $ (T^8 - 29T^6 + 263T^4 - 763*T^2 + 16)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} + 39 T^{6} + \cdots + 64)^{2} $ (T^8 + 39T^6 + 287T^4 + 537*T^2 + 64)^2
$13$	$ (T^{8} + 55 T^{6} + \cdots + 4096)^{2} $ (T^8 + 55T^6 + 836T^4 + 3584*T^2 + 4096)^2
$17$	$ (T^{8} - 94 T^{6} + \cdots + 64)^{2} $ (T^8 - 94T^6 + 2201T^4 - 5948*T^2 + 64)^2
$19$	$ (T^{8} + 37 T^{6} + \cdots + 3844)^{2} $ (T^8 + 37T^6 + 455T^4 + 2255*T^2 + 3844)^2
$23$	$ (T^{8} + 82 T^{6} + \cdots + 64)^{2} $ (T^8 + 82T^6 + 585T^4 + 820*T^2 + 64)^2
$29$	$ (T^{8} + 129 T^{6} + \cdots + 262144)^{2} $ (T^8 + 129T^6 + 5504T^4 + 82944*T^2 + 262144)^2
$31$	$ (T^{8} + 124 T^{6} + \cdots + 368449)^{2} $ (T^8 + 124T^6 + 4934T^4 + 73724*T^2 + 368449)^2
$37$	$ (T^{4} - T^{3} - 87 T^{2} + \cdots - 482)^{4} $ (T^4 - T^3 - 87T^2 + 425T - 482)^4
$41$	$ (T^{8} - 88 T^{6} + \cdots + 65536)^{2} $ (T^8 - 88T^6 + 2576T^4 - 27392*T^2 + 65536)^2
$43$	$ (T^{4} - 5 T^{3} + \cdots - 128)^{4} $ (T^4 - 5T^3 - 96T^2 + 304*T - 128)^4
$47$	$ (T^{8} - 218 T^{6} + \cdots + 256)^{2} $ (T^8 - 218T^6 + 11417T^4 - 3424*T^2 + 256)^2
$53$	$ (T^{8} + 159 T^{6} + \cdots + 2085136)^{2} $ (T^8 + 159T^6 + 9167T^4 + 228513*T^2 + 2085136)^2
$59$	$ (T^{8} - 285 T^{6} + \cdots + 4511376)^{2} $ (T^8 - 285T^6 + 26487T^4 - 848043*T^2 + 4511376)^2
$61$	$ (T^{8} + 150 T^{6} + \cdots + 82944)^{2} $ (T^8 + 150T^6 + 6345T^4 + 85104*T^2 + 82944)^2
$67$	$ (T^{4} + 7 T^{3} + \cdots + 226)^{4} $ (T^4 + 7T^3 - 19T^2 - 103*T + 226)^4
$71$	$ (T^{8} + 344 T^{6} + \cdots + 2166784)^{2} $ (T^8 + 344T^6 + 26640T^4 + 608768*T^2 + 2166784)^2
$73$	$ (T^{8} + 185 T^{6} + \cdots + 16)^{2} $ (T^8 + 185T^6 + 2507T^4 + 427*T^2 + 16)^2
$79$	$ (T^{4} + 14 T^{3} + \cdots - 401)^{4} $ (T^4 + 14T^3 + 32T^2 - 158*T - 401)^4
$83$	$ (T^{8} - 523 T^{6} + \cdots + 131239936)^{2} $ (T^8 - 523T^6 + 89840T^4 - 6003968*T^2 + 131239936)^2
$89$	$ (T^{8} - 334 T^{6} + \cdots + 984064)^{2} $ (T^8 - 334T^6 + 29657T^4 - 778892*T^2 + 984064)^2
$97$	$ (T^{8} + 347 T^{6} + \cdots + 5914624)^{2} $ (T^8 + 347T^6 + 25760T^4 + 680704*T^2 + 5914624)^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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} + \beta_{8} q^{5} + \beta_{5} q^{9}+O(q^{10}) \) q - b1 * q^3 + b8 * q^5 + b5 * q^9	\( q - \beta_1 q^{3} + \beta_{8} q^{5} + \beta_{5} q^{9} - \beta_{6} q^{11} + ( - \beta_{12} - \beta_{7} + \beta_1) q^{13} + (\beta_{11} - \beta_{10} - \beta_{6} + \cdots + 1) q^{15}+ \cdots + ( - \beta_{14} + \beta_{13} + \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) \) q - b1 * q^3 + b8 * q^5 + b5 * q^9 - b6 * q^11 + (-b12 - b7 + b1) * q^13 + (b11 - b10 - b6 + b5 + 1) * q^15 + (-b15 - b12 - b9 - b2) * q^17 - b12 * q^19 + (b14 - b13 + b11) * q^23 + (b13 + b11 - b10 + b5 + b3 + 3) * q^25 + (-b15 + b8 + b7 - b4) * q^27 + (b13 - b11 - b6) * q^29 + (-b15 - b12 + 2b4 - b1) q^31 + (-b15 - b12 - b9 - b8 - b7 - b1) * q^33 + (-b13 - b11 + 2b10 - 2b5 - b3 - 1) * q^37 + (-b14 + b13 - b11 - b10 - b5 + b3 + 3) * q^39 + (-2b7 - 2b1) * q^41 + (b13 + b11 + b10 - b5 + b3 + 1) * q^43 + (-2b15 - b12 - b9 - 3b7 + 2b4 - 2b1) * q^45 + (b9 - 2b8 - b4 - b2 + b1) q^47 + (-2b14 + b13 - b10 + b6 - b5 - b3 - 1) q^51 + (-b14 + b13 - b11 - b10 - b6 - b5) * q^53 + (b15 + 3b7 + 2b4 - 2b1) q^55 + (-b14 + b13 - b11 - b10 + b3) * q^57 + (-b15 - b12 - b8 - b4 - 2b2 + b1) q^59 + (-b15 + b12 - b7 + b4) * q^61 + (b14 + 2b13 - 2b11 + 2b6) q^65 + (-b10 + b5 + b3 - 1) * q^67 + (b12 - b8 - 2b7 + b4 - b2) q^69 + (-b14 - 2b10 - 2b5) * q^71 + (-b15 - 2b12 - b4 - b1) q^73 + (b12 - b9 + 2b8 + 3b4 + b2 - 3b1) q^75 + (b13 + b11 - b10 + b5 - 3) * q^79 + (2b11 + b10 - 2b6 - 4) * q^81 + (-b15 - b12 - 2b9 - b8 + 3b7 + b4 + 2b1) q^83 + (-2b13 - 2b11 - b10 + b5 + 2b3 + 2) q^85 + (-b15 - 2b12 - 2b9 - 2b4 - b1) q^87 + (-b15 - b12 + b7 - b4 - 2b2 + 2b1) * q^89 + (-b14 - 2b13 + b10 - b6 + b3 - 2) q^93 + (b14 + b13 - b11) * q^95 + (-b15 + 2b12 - 2b7 - 2b4 + b1) q^97 + (-b14 + b13 + b10 + 2b6 - b5 + b3 + 1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^9	\( 16 q - 4 q^{9} + 8 q^{15} + 36 q^{25} + 4 q^{37} + 44 q^{39} + 20 q^{43} - 12 q^{51} - 8 q^{57} - 28 q^{67} - 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} + 20 q^{99}+O(q^{100}) \) 16 * q - 4 * q^9 + 8 * q^15 + 36 * q^25 + 4 * q^37 + 44 * q^39 + 20 * q^43 - 12 * q^51 - 8 * q^57 - 28 * q^67 - 56 * q^79 - 60 * q^81 + 16 * q^85 - 32 * q^93 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	1176.2.k.a

Level	$1176$
Weight	$2$
Character orbit	1176.k
Analytic conductor	$9.390$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(9.39040727770\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + \cdots + 6561 \) x^16 - 6x^15 + 19x^14 - 42x^13 + 65x^12 - 48x^11 - 94x^10 + 444x^9 - 962x^8 + 1332x^7 - 846x^6 - 1296x^5 + 5265x^4 - 10206x^3 + 13851x^2 - 13122*x + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 11 \nu^{15} + 30 \nu^{14} - 142 \nu^{13} + 363 \nu^{12} - 662 \nu^{11} + 258 \nu^{10} + 1288 \nu^{9} + \cdots - 19683 ) / 69984 \) (11v^15 + 30v^14 - 142v^13 + 363v^12 - 662v^11 + 258v^10 + 1288v^9 - 3546v^8 + 8138v^7 - 7392v^6 - 846v^5 + 28890v^4 - 41067v^3 + 38880v^2 - 34992*v - 19683) / 69984
\(\beta_{2}\)	\(=\)	\( ( \nu^{15} + 246 \nu^{14} - 962 \nu^{13} + 1965 \nu^{12} - 2374 \nu^{11} - 462 \nu^{10} + \cdots + 150903 ) / 69984 \) (v^15 + 246v^14 - 962v^13 + 1965v^12 - 2374v^11 - 462v^10 + 7988v^9 - 24594v^8 + 38170v^7 - 30276v^6 - 19998v^5 + 114858v^4 - 199989v^3 + 244944v^2 - 195372v + 150903) / 69984
\(\beta_{3}\)	\(=\)	\( ( 35 \nu^{15} - 162 \nu^{14} + 512 \nu^{13} - 963 \nu^{12} + 1042 \nu^{11} + 306 \nu^{10} + \cdots - 54675 ) / 34992 \) (35v^15 - 162v^14 + 512v^13 - 963v^12 + 1042v^11 + 306v^10 - 5162v^9 + 13134v^8 - 21322v^7 + 14538v^6 + 16074v^5 - 72198v^4 + 144099v^3 - 184680v^2 + 173502*v - 54675) / 34992
\(\beta_{4}\)	\(=\)	\( ( - 22 \nu^{15} + 75 \nu^{14} - 238 \nu^{13} + 489 \nu^{12} - 737 \nu^{11} + 429 \nu^{10} + \cdots + 113724 ) / 17496 \) (-22v^15 + 75v^14 - 238v^13 + 489v^12 - 737v^11 + 429v^10 + 1483v^9 - 5787v^8 + 11651v^7 - 14727v^6 + 5787v^5 + 23193v^4 - 70227v^3 + 126846v^2 - 147987*v + 113724) / 17496
\(\beta_{5}\)	\(=\)	\( ( 61 \nu^{15} - 234 \nu^{14} + 565 \nu^{13} - 729 \nu^{12} + 563 \nu^{11} + 1251 \nu^{10} + \cdots - 113724 ) / 34992 \) (61v^15 - 234v^14 + 565v^13 - 729v^12 + 563v^11 + 1251v^10 - 5437v^9 + 10815v^8 - 13421v^7 + 6351v^6 + 19359v^5 - 58185v^4 + 98010v^3 - 136323v^2 + 129762*v - 113724) / 34992
\(\beta_{6}\)	\(=\)	\( ( - 131 \nu^{15} + 660 \nu^{14} - 1742 \nu^{13} + 3567 \nu^{12} - 5176 \nu^{11} + 2040 \nu^{10} + \cdots + 754515 ) / 69984 \) (-131v^15 + 660v^14 - 1742v^13 + 3567v^12 - 5176v^11 + 2040v^10 + 12350v^9 - 42648v^8 + 81940v^7 - 96318v^6 + 29736v^5 + 167832v^4 - 496287v^3 + 862164v^2 - 1106622*v + 754515) / 69984
\(\beta_{7}\)	\(=\)	\( ( - 19 \nu^{15} + 82 \nu^{14} - 226 \nu^{13} + 433 \nu^{12} - 542 \nu^{11} + 74 \nu^{10} + \cdots + 75087 ) / 7776 \) (-19v^15 + 82v^14 - 226v^13 + 433v^12 - 542v^11 + 74v^10 + 1912v^9 - 5482v^8 + 9482v^7 - 10040v^6 + 42v^5 + 24354v^4 - 60237v^3 + 95580v^2 - 105948*v + 75087) / 7776
\(\beta_{8}\)	\(=\)	\( ( 61 \nu^{15} - 302 \nu^{14} + 784 \nu^{13} - 1463 \nu^{12} + 1772 \nu^{11} - 100 \nu^{10} + \cdots - 255879 ) / 23328 \) (61v^15 - 302v^14 + 784v^13 - 1463v^12 + 1772v^11 - 100v^10 - 6358v^9 + 18728v^8 - 32084v^7 + 33490v^6 + 1116v^5 - 85716v^4 + 197181v^3 - 331290v^2 + 374220*v - 255879) / 23328
\(\beta_{9}\)	\(=\)	\( ( - 37 \nu^{15} + 178 \nu^{14} - 562 \nu^{13} + 1105 \nu^{12} - 1544 \nu^{11} + 572 \nu^{10} + \cdots + 244215 ) / 11664 \) (-37v^15 + 178v^14 - 562v^13 + 1105v^12 - 1544v^11 + 572v^10 + 4048v^9 - 13408v^8 + 25460v^7 - 30032v^6 + 8184v^5 + 55692v^4 - 157923v^3 + 266814v^2 - 310554*v + 244215) / 11664
\(\beta_{10}\)	\(=\)	\( ( 2 \nu^{15} - 8 \nu^{14} + 21 \nu^{13} - 38 \nu^{12} + 41 \nu^{11} + 17 \nu^{10} - 195 \nu^{9} + \cdots - 4617 ) / 432 \) (2v^15 - 8v^14 + 21v^13 - 38v^12 + 41v^11 + 17v^10 - 195v^9 + 497v^8 - 779v^7 + 661v^6 + 349v^5 - 2499v^4 + 5247v^3 - 7803v^2 + 7452*v - 4617) / 432
\(\beta_{11}\)	\(=\)	\( ( 107 \nu^{15} - 468 \nu^{14} + 1268 \nu^{13} - 2511 \nu^{12} + 3166 \nu^{11} - 90 \nu^{10} + \cdots - 365229 ) / 23328 \) (107v^15 - 468v^14 + 1268v^13 - 2511v^12 + 3166v^11 - 90v^10 - 11048v^9 + 32070v^8 - 54982v^7 + 57756v^6 + 5526v^5 - 151794v^4 + 356805v^3 - 554526v^2 + 581742*v - 365229) / 23328
\(\beta_{12}\)	\(=\)	\( ( - 30 \nu^{15} + 137 \nu^{14} - 378 \nu^{13} + 731 \nu^{12} - 939 \nu^{11} + 175 \nu^{10} + \cdots + 125388 ) / 5832 \) (-30v^15 + 137v^14 - 378v^13 + 731v^12 - 939v^11 + 175v^10 + 2997v^9 - 9161v^8 + 16161v^7 - 17341v^6 + 681v^5 + 41427v^4 - 104409v^3 + 162810v^2 - 181521*v + 125388) / 5832
\(\beta_{13}\)	\(=\)	\( ( - 125 \nu^{15} + 512 \nu^{14} - 1364 \nu^{13} + 2537 \nu^{12} - 3082 \nu^{11} + 142 \nu^{10} + \cdots + 465831 ) / 23328 \) (-125v^15 + 512v^14 - 1364v^13 + 2537v^12 - 3082v^11 + 142v^10 + 11108v^9 - 31994v^8 + 56962v^7 - 59656v^6 - 786v^5 + 141174v^4 - 357399v^3 + 579150v^2 - 650754*v + 465831) / 23328
\(\beta_{14}\)	\(=\)	\( ( - 45 \nu^{15} + 188 \nu^{14} - 498 \nu^{13} + 917 \nu^{12} - 1020 \nu^{11} - 236 \nu^{10} + \cdots + 140697 ) / 5832 \) (-45v^15 + 188v^14 - 498v^13 + 917v^12 - 1020v^11 - 236v^10 + 4386v^9 - 11588v^8 + 19032v^7 - 18226v^6 - 5868v^5 + 55620v^4 - 124497v^3 + 190512v^2 - 202662*v + 140697) / 5832
\(\beta_{15}\)	\(=\)	\( ( 243 \nu^{15} - 1090 \nu^{14} + 3042 \nu^{13} - 5725 \nu^{12} + 7074 \nu^{11} - 326 \nu^{10} + \cdots - 993627 ) / 23328 \) (243v^15 - 1090v^14 + 3042v^13 - 5725v^12 + 7074v^11 - 326v^10 - 24984v^9 + 72382v^8 - 125070v^7 + 129248v^6 + 3594v^5 - 332046v^4 + 796365v^3 - 1239624v^2 + 1397736*v - 993627) / 23328

\(\nu\)	\(=\)	\( ( 2 \beta_{15} - \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{10} + \beta_{9} + \beta_{7} + \cdots + 4 ) / 8 \) (2b15 - b14 + 2b13 + 2b12 - 2b10 + b9 + b7 - 2b6 - b4 + 2b3 + b2 + 2*b1 + 4) / 8
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{10} + \beta_{9} - \beta_{8} ) / 2 \) (b15 - b10 + b9 - b8) / 2
\(\nu^{3}\)	\(=\)	\( ( -4\beta_{12} - \beta_{9} - 4\beta_{8} + 3\beta_{7} + 5\beta_{4} + \beta_{2} + 2\beta_1 ) / 4 \) (-4b12 - b9 - 4b8 + 3b7 + 5b4 + b2 + 2*b1) / 4
\(\nu^{4}\)	\(=\)	\( ( - 2 \beta_{15} - 2 \beta_{13} - \beta_{12} - \beta_{9} - 3 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} + \cdots + 4 ) / 2 \) (-2b15 - 2b13 - b12 - b9 - 3b8 - 3b7 - 2b6 + b5 + 3b4 - 2b2 + 4b1 + 4) / 2
\(\nu^{5}\)	\(=\)	\( ( - 16 \beta_{15} - 15 \beta_{14} - 8 \beta_{12} + 10 \beta_{11} - 24 \beta_{10} + \beta_{9} + \beta_{7} + \cdots - 12 ) / 8 \) (-16b15 - 15b14 - 8b12 + 10b11 - 24b10 + b9 + b7 - 6b6 + 18b5 - 15b4 + 18b3 - 7b2 + 8*b1 - 12) / 8
\(\nu^{6}\)	\(=\)	\( -2\beta_{14} + 4\beta_{11} - 6\beta_{10} + 4\beta_{6} + 2\beta_{5} - 2\beta_{3} + 7 \) -2b14 + 4b11 - 6b10 + 4b6 + 2b5 - 2b3 + 7
\(\nu^{7}\)	\(=\)	\( ( - 2 \beta_{15} - 35 \beta_{14} + 94 \beta_{13} - 26 \beta_{12} + 16 \beta_{11} - 22 \beta_{10} + \cdots + 92 ) / 8 \) (-2b15 - 35b14 + 94b13 - 26b12 + 16b11 - 22b10 - 13b9 - 29b7 + 2b6 + 24b5 - 11b4 - 26b3 + 11b2 + 70b1 + 92) / 8
\(\nu^{8}\)	\(=\)	\( ( 11 \beta_{15} + 16 \beta_{14} + 12 \beta_{13} + 2 \beta_{12} + 14 \beta_{11} - 5 \beta_{10} + 15 \beta_{9} + \cdots + 4 ) / 2 \) (11b15 + 16b14 + 12b13 + 2b12 + 14b11 - 5b10 + 15b9 + 5b8 - 12b7 - 2b6 + 8b5 - 12b4 + 4b3 - 14b2 + 70*b1 + 4) / 2
\(\nu^{9}\)	\(=\)	\( ( 16\beta_{15} - 68\beta_{12} + 37\beta_{9} + 20\beta_{8} + 241\beta_{7} - 97\beta_{4} - 37\beta_{2} + 102\beta_1 ) / 4 \) (16b15 - 68b12 + 37b9 + 20b8 + 241b7 - 97b4 - 37b2 + 102b1) / 4
\(\nu^{10}\)	\(=\)	\( ( 40 \beta_{15} + 16 \beta_{14} - 32 \beta_{13} + 57 \beta_{12} + 60 \beta_{11} + 24 \beta_{10} + \cdots + 32 ) / 2 \) (40b15 + 16b14 - 32b13 + 57b12 + 60b11 + 24b10 - 23b9 - 49b8 + 183b7 - 28b6 - 3b5 + 51b4 - 44b3 - 12b2 + 36*b1 + 32) / 2
\(\nu^{11}\)	\(=\)	\( ( - 176 \beta_{15} + 163 \beta_{14} + 112 \beta_{13} + 80 \beta_{12} + 422 \beta_{11} + 80 \beta_{10} + \cdots - 436 ) / 8 \) (-176b15 + 163b14 + 112b13 + 80b12 + 422b11 + 80b10 + 3b9 - 512b8 - 93b7 - 682b6 + 774b5 - 349b4 - 74b3 - 173b2 - 448*b1 - 436) / 8
\(\nu^{12}\)	\(=\)	\( 148\beta_{14} - 76\beta_{13} + 140\beta_{11} - 76\beta_{10} + 24\beta_{6} + 244\beta_{5} + 64\beta_{3} - 39 \) 148b14 - 76b13 + 140b11 - 76b10 + 24b6 + 244b5 + 64*b3 - 39
\(\nu^{13}\)	\(=\)	\( ( - 174 \beta_{15} - 201 \beta_{14} + 114 \beta_{13} - 942 \beta_{12} - 848 \beta_{11} - 594 \beta_{10} + \cdots + 5604 ) / 8 \) (-174b15 - 201b14 + 114b13 - 942b12 - 848b11 - 594b10 - 855b9 + 1001b7 + 1182b6 + 1776b5 - 1209b4 + 66b3 - 327b2 - 414b1 + 5604) / 8
\(\nu^{14}\)	\(=\)	\( ( 405 \beta_{15} - 176 \beta_{14} + 420 \beta_{13} + 764 \beta_{12} + 32 \beta_{11} - 141 \beta_{10} + \cdots + 1120 ) / 2 \) (405b15 - 176b14 + 420b13 + 764b12 + 32b11 - 141b10 - 191b9 - 377b8 - 48b7 + 196b6 - 40b5 - 984b4 + 460b3 - 20b2 + 712*b1 + 1120) / 2
\(\nu^{15}\)	\(=\)	\( ( 1456 \beta_{15} + 1548 \beta_{12} + 2919 \beta_{9} - 2276 \beta_{8} - 1301 \beta_{7} - 6899 \beta_{4} + \cdots + 962 \beta_1 ) / 4 \) (1456b15 + 1548b12 + 2919b9 - 2276b8 - 1301b7 - 6899b4 + 153b2 + 962b1) / 4

\(n\)	\(295\)	\(589\)	\(785\)	\(1081\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 881.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 2 T^{14} + \cdots + 6561 \) T^16 + 2T^14 + 17T^12 + 46T^10 + 172T^8 + 414T^6 + 1377T^4 + 1458*T^2 + 6561
$5$	\( (T^{8} - 29 T^{6} + \cdots + 16)^{2} \) (T^8 - 29T^6 + 263T^4 - 763*T^2 + 16)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} + 39 T^{6} + \cdots + 64)^{2} \) (T^8 + 39T^6 + 287T^4 + 537*T^2 + 64)^2
$13$	\( (T^{8} + 55 T^{6} + \cdots + 4096)^{2} \) (T^8 + 55T^6 + 836T^4 + 3584*T^2 + 4096)^2
$17$	\( (T^{8} - 94 T^{6} + \cdots + 64)^{2} \) (T^8 - 94T^6 + 2201T^4 - 5948*T^2 + 64)^2
$19$	\( (T^{8} + 37 T^{6} + \cdots + 3844)^{2} \) (T^8 + 37T^6 + 455T^4 + 2255*T^2 + 3844)^2
$23$	\( (T^{8} + 82 T^{6} + \cdots + 64)^{2} \) (T^8 + 82T^6 + 585T^4 + 820*T^2 + 64)^2
$29$	\( (T^{8} + 129 T^{6} + \cdots + 262144)^{2} \) (T^8 + 129T^6 + 5504T^4 + 82944*T^2 + 262144)^2
$31$	\( (T^{8} + 124 T^{6} + \cdots + 368449)^{2} \) (T^8 + 124T^6 + 4934T^4 + 73724*T^2 + 368449)^2
$37$	\( (T^{4} - T^{3} - 87 T^{2} + \cdots - 482)^{4} \) (T^4 - T^3 - 87T^2 + 425T - 482)^4
$41$	\( (T^{8} - 88 T^{6} + \cdots + 65536)^{2} \) (T^8 - 88T^6 + 2576T^4 - 27392*T^2 + 65536)^2
$43$	\( (T^{4} - 5 T^{3} + \cdots - 128)^{4} \) (T^4 - 5T^3 - 96T^2 + 304*T - 128)^4
$47$	\( (T^{8} - 218 T^{6} + \cdots + 256)^{2} \) (T^8 - 218T^6 + 11417T^4 - 3424*T^2 + 256)^2
$53$	\( (T^{8} + 159 T^{6} + \cdots + 2085136)^{2} \) (T^8 + 159T^6 + 9167T^4 + 228513*T^2 + 2085136)^2
$59$	\( (T^{8} - 285 T^{6} + \cdots + 4511376)^{2} \) (T^8 - 285T^6 + 26487T^4 - 848043*T^2 + 4511376)^2
$61$	\( (T^{8} + 150 T^{6} + \cdots + 82944)^{2} \) (T^8 + 150T^6 + 6345T^4 + 85104*T^2 + 82944)^2
$67$	\( (T^{4} + 7 T^{3} + \cdots + 226)^{4} \) (T^4 + 7T^3 - 19T^2 - 103*T + 226)^4
$71$	\( (T^{8} + 344 T^{6} + \cdots + 2166784)^{2} \) (T^8 + 344T^6 + 26640T^4 + 608768*T^2 + 2166784)^2
$73$	\( (T^{8} + 185 T^{6} + \cdots + 16)^{2} \) (T^8 + 185T^6 + 2507T^4 + 427*T^2 + 16)^2
$79$	\( (T^{4} + 14 T^{3} + \cdots - 401)^{4} \) (T^4 + 14T^3 + 32T^2 - 158*T - 401)^4
$83$	\( (T^{8} - 523 T^{6} + \cdots + 131239936)^{2} \) (T^8 - 523T^6 + 89840T^4 - 6003968*T^2 + 131239936)^2
$89$	\( (T^{8} - 334 T^{6} + \cdots + 984064)^{2} \) (T^8 - 334T^6 + 29657T^4 - 778892*T^2 + 984064)^2
$97$	\( (T^{8} + 347 T^{6} + \cdots + 5914624)^{2} \) (T^8 + 347T^6 + 25760T^4 + 680704*T^2 + 5914624)^2
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