LMFDB - Newform orbit 1152.2.p.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,2,Mod(191,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1152.191"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.p (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.9349208943630483456.9
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 $ x^16 - 8x^15 + 48x^14 - 196x^13 + 642x^12 - 1668x^11 + 3580x^10 - 6328x^9 + 9297x^8 - 11276x^7 + 11224x^6 - 9024x^5 + 5736x^4 - 2780x^3 + 972x^2 - 220*x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{11} + \beta_{8} + \beta_{4}) q^{3} + ( - \beta_{14} + \beta_{7} - \beta_1) q^{5} + (\beta_{13} + \beta_{12} + \cdots - \beta_{2}) q^{7} + (\beta_{6} + 2 \beta_{3} + 1) q^{9} - \beta_{9} q^{11}+ \cdots + (\beta_{11} - 2 \beta_{9} + 2 \beta_{8}) q^{99}+O(q^{100}) $ q + (-b11 + b8 + b4) * q^3 + (-b14 + b7 - b1) * q^5 + (b13 + b12 - b10 - b2) * q^7 + (b6 + 2b3 + 1) q^9 - b9 * q^11 + (b14 - 2b5) q^13 + (-b13 + 2b12 - 2b10 - b2) * q^15 + (2b15 + b6) q^17 + (2b9 - 2b8 - 2b4) q^19 + (b14 + 2b7 - 3b5 - b1) * q^21 + (2b13 - b12 - b2) q^23 + (-2b15 - 2b6) * q^25 + (b11 + b9 + 2b8 + 3b4) * q^27 + (3b14 + 2b7 - 3b5) q^29 + (-2b13 + 3b12 - 2b10 + 5b2) * q^31 + (b6 - b3 + 1) * q^33 + (2b11 - 4b9 - 4b8 - 5b4) * q^35 + (6b7 - 3b1) * q^37 + 3b10 q^39 + (-b15 - 2b6 - 3b3 - 6) * q^41 + 3b11 q^43 + (-b14 + 2b7 - 2b5 - 4b1) q^45 + (5b13 - 7b12 + 3b10 - 7b2) * q^47 + (b15 - 2b3) q^49 + (3b9 + 3b8) * q^51 + (-4b5 - b1) q^53 + (-3b13 + 3b12 - 4b10 + b2) q^55 + (-2b15 - 4b6 - 6) * q^57 + (-b11 - b4) * q^59 + (-3b14 - 3b5) * q^61 + (-3b13 + 2b12 + 2b10) q^63 + (-b15 + b6 - 3b3 + 3) q^65 + (-6b11 + 2b9 + b8 + 7b4) q^67 + (b14 - 2b5 + 3b1) * q^69 + (-5b13 - 5b12 + 4b2) q^71 + (b6 - 4) * q^73 + (-2b11 - 2b9 - 4b8) q^75 + (-b14 + 3b7 - 3b1) * q^77 + (5b13 - 2b12 + 2b10 + 2b2) * q^79 + (4b15 + 2b6 + 3) * q^81 + (b11 - 2b9 - 2b4) * q^83 + (2b14 - 3b7 - 4b5 - 3b1) * q^85 + (-4b13 - b12 + 4b10 + 2b2) q^87 + (-2b15 - b6 + 12b3 + 6) * q^89 + (-3b9 + 3b8 - 3b4) q^91 + (-b14 + 8b7 + 4b5 - b1) * q^93 + (8b13 - 10b12 + 12b10 + 2b2) * q^95 + (5b15 + 5b6 + b3 + 1) * q^97 + (b11 - 2b9 + 2b8) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{33} - 72 q^{41} + 16 q^{49} - 96 q^{57} + 72 q^{65} - 64 q^{73} + 48 q^{81} + 8 q^{97}+O(q^{100}) $ 16 * q + 24 * q^33 - 72 * q^41 + 16 * q^49 - 96 * q^57 + 72 * q^65 - 64 * q^73 + 48 * q^81 + 8 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 $ x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25 :

$\beta_{1}$	$=$	$ ( 56 \nu^{14} - 392 \nu^{13} + 2267 \nu^{12} - 8506 \nu^{11} + 26269 \nu^{10} - 62716 \nu^{9} + \cdots + 2430 ) / 65 $ (56v^14 - 392v^13 + 2267v^12 - 8506v^11 + 26269v^10 - 62716v^9 + 124065v^8 - 197514v^7 + 258029v^6 - 269446v^5 + 222643v^4 - 139500v^3 + 62551v^2 - 17806v + 2430) / 65
$\beta_{2}$	$=$	$ ( - 62 \nu^{14} + 434 \nu^{13} - 2541 \nu^{12} + 9604 \nu^{11} - 30137 \nu^{10} + 72992 \nu^{9} + \cdots - 5945 ) / 65 $ (-62v^14 + 434v^13 - 2541v^12 + 9604v^11 - 30137v^10 + 72992v^9 - 147747v^8 + 240948v^7 - 326020v^6 + 354283v^5 - 310030v^4 + 208129v^3 - 103708v^2 + 33855v - 5945) / 65
$\beta_{3}$	$=$	$ ( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} + \cdots - 142555 ) / 17095 $ (3456v^15 - 25920v^14 + 151876v^13 - 594074v^12 + 1879372v^11 - 4666596v^10 + 9554736v^9 - 15945783v^8 + 21928484v^7 - 24527176v^6 + 22138664v^5 - 15739188v^4 + 8598668v^3 - 3418428v^2 + 929924*v - 142555) / 17095
$\beta_{4}$	$=$	$ ( - 88 \nu^{14} + 616 \nu^{13} - 3568 \nu^{12} + 13400 \nu^{11} - 41486 \nu^{10} + 99278 \nu^{9} + \cdots - 5230 ) / 65 $ (-88v^14 + 616v^13 - 3568v^12 + 13400v^11 - 41486v^10 + 99278v^9 - 197277v^8 + 315672v^7 - 416019v^6 + 439199v^5 - 369648v^4 + 237379v^3 - 111209v^2 + 33751v - 5230) / 65
$\beta_{5}$	$=$	$ ( - 10 \nu^{14} + 70 \nu^{13} - 406 \nu^{12} + 1526 \nu^{11} - 4732 \nu^{10} + 11340 \nu^{9} - 22581 \nu^{8} + \cdots - 670 ) / 5 $ (-10v^14 + 70v^13 - 406v^12 + 1526v^11 - 4732v^10 + 11340v^9 - 22581v^8 + 36210v^7 - 47866v^6 + 50708v^5 - 42917v^4 + 27762v^3 - 13198v^2 + 4094v - 670) / 5
$\beta_{6}$	$=$	$ ( 54 \nu^{14} - 378 \nu^{13} + 2193 \nu^{12} - 8244 \nu^{11} + 25569 \nu^{10} - 61284 \nu^{9} + \cdots + 3308 ) / 13 $ (54v^14 - 378v^13 + 2193v^12 - 8244v^11 + 25569v^10 - 61284v^9 + 122021v^8 - 195620v^7 + 258293v^6 - 273134v^5 + 230109v^4 - 147802v^3 + 69203v^2 - 20980v + 3308) / 13
$\beta_{7}$	$=$	$ ( 22230 \nu^{15} - 159361 \nu^{14} + 923582 \nu^{13} - 3511572 \nu^{12} + 10902106 \nu^{11} + \cdots - 39775 ) / 17095 $ (22230v^15 - 159361v^14 + 923582v^13 - 3511572v^12 + 10902106v^11 - 26315724v^10 + 52438146v^9 - 84432495v^8 + 111380164v^7 - 117919554v^6 + 98855056v^5 - 63256933v^4 + 29226910v^3 - 8804251v^2 + 1370336*v - 39775) / 17095
$\beta_{8}$	$=$	$ ( 31151 \nu^{15} - 186424 \nu^{14} + 1043567 \nu^{13} - 3475210 \nu^{12} + 9946592 \nu^{11} + \cdots + 1694445 ) / 17095 $ (31151v^15 - 186424v^14 + 1043567v^13 - 3475210v^12 + 9946592v^11 - 20522197v^10 + 34995448v^9 - 42813993v^8 + 37491703v^7 - 11634546v^6 - 19984771v^5 + 41799684v^4 - 40498098v^3 + 24932581v^2 - 9099207*v + 1694445) / 17095
$\beta_{9}$	$=$	$ ( 31151 \nu^{15} - 257697 \nu^{14} + 1542478 \nu^{13} - 6361898 \nu^{12} + 20780877 \nu^{11} + \cdots - 2345235 ) / 17095 $ (31151v^15 - 257697v^14 + 1542478v^13 - 6361898v^12 + 20780877v^11 - 54011828v^10 + 115020036v^9 - 201417195v^8 + 290550040v^7 - 343630755v^6 + 328629093v^5 - 249342894v^4 + 144728331v^3 - 60896943v^2 + 16663484*v - 2345235) / 17095
$\beta_{10}$	$=$	$ ( - 36004 \nu^{15} + 278183 \nu^{14} - 1640254 \nu^{13} + 6529376 \nu^{12} - 20875054 \nu^{11} + \cdots + 1576650 ) / 17095 $ (-36004v^15 + 278183v^14 - 1640254v^13 + 6529376v^12 - 20875054v^11 + 52692141v^10 - 109431091v^9 + 186121407v^8 - 260824747v^7 + 298801493v^6 - 276219577v^5 + 202070739v^4 - 112620474v^3 + 45386761v^2 - 11822664*v + 1576650) / 17095
$\beta_{11}$	$=$	$ ( - 47906 \nu^{15} + 347723 \nu^{14} - 2030815 \nu^{13} + 7808324 \nu^{12} - 24528937 \nu^{11} + \cdots + 705525 ) / 17095 $ (-47906v^15 + 347723v^14 - 2030815v^13 + 7808324v^12 - 24528937v^11 + 60081095v^10 - 121877717v^9 + 200621199v^8 - 272407344v^7 + 299328268v^6 - 264286576v^5 + 181704730v^4 - 93782944v^3 + 33728236v^2 - 7443876*v + 705525) / 17095
$\beta_{12}$	$=$	$ ( - 49663 \nu^{15} + 358139 \nu^{14} - 2089429 \nu^{13} + 8006091 \nu^{12} - 25109764 \nu^{11} + \cdots + 1090565 ) / 17095 $ (-49663v^15 + 358139v^14 - 2089429v^13 + 8006091v^12 - 25109764v^11 + 61357045v^10 - 124293850v^9 + 204357372v^8 - 277503791v^7 + 305524016v^6 - 271172221v^5 + 188678893v^4 - 99647214v^3 + 37587574v^2 - 9111403*v + 1090565) / 17095
$\beta_{13}$	$=$	$ ( 49663 \nu^{15} - 403112 \nu^{14} + 2404240 \nu^{13} - 9830785 \nu^{12} + 31965385 \nu^{11} + \cdots - 3581175 ) / 17095 $ (49663v^15 - 403112v^14 + 2404240v^13 - 9830785v^12 + 31965385v^11 - 82591928v^10 + 175128068v^9 - 305358840v^8 + 439053908v^7 - 518058261v^6 + 494946823v^5 - 375963823v^4 + 218951115v^3 - 92737096v^2 + 25552848*v - 3581175) / 17095
$\beta_{14}$	$=$	$ ( 79890 \nu^{15} - 616270 \nu^{14} + 3648880 \nu^{13} - 14546467 \nu^{12} + 46727282 \nu^{11} + \cdots - 3924805 ) / 17095 $ (79890v^15 - 616270v^14 + 3648880v^13 - 14546467v^12 + 46727282v^11 - 118357534v^10 + 247314870v^9 - 422995752v^8 + 597739530v^7 - 690526972v^6 + 645876746v^5 - 478299719v^4 + 271084944v^3 - 111096466v^2 + 29525918*v - 3924805) / 17095
$\beta_{15}$	$=$	$ ( 162552 \nu^{15} - 1254645 \nu^{14} + 7410690 \nu^{13} - 29505615 \nu^{12} + 94503294 \nu^{11} + \cdots - 7071655 ) / 17095 $ (162552v^15 - 1254645v^14 + 7410690v^13 - 29505615v^12 + 94503294v^11 - 238745115v^10 + 496891020v^9 - 846295960v^8 + 1189159396v^7 - 1364903575v^6 + 1266038596v^5 - 928161750v^4 + 519335978v^3 - 209359180v^2 + 54517604*v - 7071655) / 17095

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

$\nu$	$=$	$ ( - 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} - 3 \beta_{12} + 3 \beta_{9} + 3 \beta_{8} - \beta_{6} + \cdots + 3 ) / 6 $ (-2b15 - 2b14 + 3b13 - 3b12 + 3b9 + 3b8 - b6 + b5 + 3b4 - 3b2 + 3) / 6
$\nu^{2}$	$=$	$ ( - 2 \beta_{15} - 2 \beta_{14} + 6 \beta_{13} + 6 \beta_{8} - \beta_{6} + 4 \beta_{5} + 6 \beta_{4} + \cdots - 12 ) / 6 $ (-2b15 - 2b14 + 6b13 + 6b8 - b6 + 4b5 + 6b4 - 6b2 + 3b1 - 12) / 6
$\nu^{3}$	$=$	$ ( 10 \beta_{15} + 11 \beta_{14} - 12 \beta_{13} + 21 \beta_{12} + 6 \beta_{11} - 12 \beta_{10} - 18 \beta_{9} + \cdots - 15 ) / 6 $ (10b15 + 11b14 - 12b13 + 21b12 + 6b11 - 12b10 - 18b9 - 9b8 - 9b7 + 5b6 - b5 - 12b4 + 9b3 + 6b2 + 9b1 - 15) / 6
$\nu^{4}$	$=$	$ ( 11 \beta_{15} + 12 \beta_{14} - 24 \beta_{13} + 12 \beta_{12} + 6 \beta_{11} - 12 \beta_{10} - 6 \beta_{9} + \cdots + 27 ) / 3 $ (11b15 + 12b14 - 24b13 + 12b12 + 6b11 - 12b10 - 6b9 - 24b8 - 9b7 + 10b6 - 15b5 - 24b4 + 9b3 + 24b2 - 9*b1 + 27) / 3
$\nu^{5}$	$=$	$ ( - 43 \beta_{15} - 57 \beta_{14} + 33 \beta_{13} - 108 \beta_{12} - 36 \beta_{11} + 84 \beta_{10} + \cdots + 108 ) / 6 $ (-43b15 - 57b14 + 33b13 - 108b12 - 36b11 + 84b10 + 129b9 + 24b8 + 60b7 + b6 - 24b5 + 57b4 - 75b3 + 39b2 - 105b1 + 108) / 6
$\nu^{6}$	$=$	$ ( - 185 \beta_{15} - 232 \beta_{14} + 336 \beta_{13} - 270 \beta_{12} - 138 \beta_{11} + 312 \beta_{10} + \cdots - 300 ) / 6 $ (-185b15 - 232b14 + 336b13 - 270b12 - 138b11 + 312b10 + 216b9 + 396b8 + 225b7 - 160b6 + 206b5 + 420b4 - 270b3 - 276b2 + 15*b1 - 300) / 6
$\nu^{7}$	$=$	$ ( 128 \beta_{15} + 198 \beta_{14} + 36 \beta_{13} + 468 \beta_{12} + 114 \beta_{11} - 348 \beta_{10} + \cdots - 921 ) / 6 $ (128b15 + 198b14 + 36b13 + 468b12 + 114b11 - 348b10 - 792b9 + 216b8 - 252b7 - 251b6 + 405b5 - 51b4 + 378b3 - 651b2 + 840*b1 - 921) / 6
$\nu^{8}$	$=$	$ 238 \beta_{15} + 322 \beta_{14} - 392 \beta_{13} + 396 \beta_{12} + 188 \beta_{11} - 484 \beta_{10} + \cdots + 267 $ 238b15 + 322b14 - 392b13 + 396b12 + 188b11 - 484b10 - 408b9 - 476b8 - 350b7 + 154b6 - 198b5 - 544b4 + 469b3 + 204b2 + 136*b1 + 267
$\nu^{9}$	$=$	$ ( 379 \beta_{15} + 385 \beta_{14} - 2205 \beta_{13} - 1044 \beta_{12} + 306 \beta_{11} - 270 \beta_{10} + \cdots + 8175 ) / 6 $ (379b15 + 385b14 - 2205b13 - 1044b12 + 306b11 - 270b10 + 3753b9 - 4590b8 - 162b7 + 3119b6 - 4445b5 - 2718b4 - 27b3 + 6300b2 - 5580*b1 + 8175) / 6
$\nu^{10}$	$=$	$ ( - 10223 \beta_{15} - 14312 \beta_{14} + 15690 \beta_{13} - 18576 \beta_{12} - 7956 \beta_{11} + \cdots - 5592 ) / 6 $ (-10223b15 - 14312b14 + 15690b13 - 18576b12 - 7956b11 + 22734b10 + 22698b9 + 17490b8 + 16605b7 - 3559b6 + 4618b5 + 22422b4 - 23220b3 - 2208b2 - 12138*b1 - 5592) / 6
$\nu^{11}$	$=$	$ ( - 12443 \beta_{15} - 16927 \beta_{14} + 30768 \beta_{13} - 10440 \beta_{12} - 9921 \beta_{11} + \cdots - 68274 ) / 6 $ (-12443b15 - 16927b14 + 30768b13 - 10440b12 - 9921b11 + 25455b10 - 6831b9 + 53295b8 + 18348b7 - 28249b6 + 39632b5 + 42465b4 - 24783b3 - 50364b2 + 30327*b1 - 68274) / 6
$\nu^{12}$	$=$	$ ( 32900 \beta_{15} + 46629 \beta_{14} - 45948 \beta_{13} + 65172 \beta_{12} + 24918 \beta_{11} + \cdots - 9414 ) / 3 $ (32900b15 + 46629b14 - 45948b13 + 65172b12 + 24918b11 - 76260b10 - 91644b9 - 40998b8 - 56232b7 - 1469b6 + 2583b5 - 65904b4 + 79695b3 - 18156b2 + 62271*b1 - 9414) / 3
$\nu^{13}$	$=$	$ ( 158381 \beta_{15} + 221934 \beta_{14} - 323757 \beta_{13} + 207966 \beta_{12} + 122247 \beta_{11} + \cdots + 514725 ) / 6 $ (158381b15 + 221934b14 - 323757b13 + 207966b12 + 122247b11 - 354021b10 - 128232b9 - 496197b8 - 259155b7 + 216256b6 - 303783b5 - 456369b4 + 362934b3 + 349410b2 - 107328*b1 + 514725) / 6
$\nu^{14}$	$=$	$ ( - 352742 \beta_{15} - 501646 \beta_{14} + 401340 \beta_{13} - 794892 \beta_{12} - 260958 \beta_{11} + \cdots + 639942 ) / 6 $ (-352742b15 - 501646b14 + 401340b13 - 794892b12 - 260958b11 + 834318b10 + 1299648b9 + 138948b8 + 619710b7 + 247871b6 - 351754b5 + 575106b4 - 881244b3 + 638640b2 - 1070745*b1 + 639942) / 6
$\nu^{15}$	$=$	$ ( - 1567315 \beta_{15} - 2212539 \beta_{14} + 2900829 \beta_{13} - 2387823 \beta_{12} - 1184481 \beta_{11} + \cdots - 3396279 ) / 6 $ (-1567315b15 - 2212539b14 + 2900829b13 - 2387823b12 - 1184481b11 + 3604815b10 + 2290002b9 + 3995220b8 + 2658726b7 - 1424525b6 + 2006715b5 + 4105164b4 - 3753099b3 - 2050089b2 - 247560*b1 - 3396279) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-\beta_{3}$	$-1$

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

0.500000	+	1.00333i
0.500000	+	2.74530i
0.500000	−	0.410882i
0.500000	−	1.74530i
0.500000	−	2.00333i
0.500000	+	1.33108i
0.500000	−	0.589118i
0.500000	−	0.331082i
0.500000	−	1.00333i
0.500000	−	2.74530i
0.500000	+	0.410882i
0.500000	+	1.74530i
0.500000	+	2.00333i
0.500000	−	1.33108i
0.500000	+	0.589118i
0.500000	+	0.331082i

−1.65068

0.524648i

−1.57313

−

2.72474i

2.21650

1.27970i

2.44949

−

1.73205i

191.2

−1.65068

0.524648i

1.57313

2.72474i

−2.21650

−

1.27970i

2.44949

−

1.73205i

191.3

−0.524648

1.65068i

−0.158919

−

0.275255i

2.93038

1.69185i

−2.44949

−

1.73205i

191.4

−0.524648

1.65068i

0.158919

0.275255i

−2.93038

−

1.69185i

−2.44949

−

1.73205i

191.5

0.524648

−

1.65068i

−0.158919

−

0.275255i

−2.93038

−

1.69185i

−2.44949

−

1.73205i

191.6

0.524648

−

1.65068i

0.158919

0.275255i

2.93038

1.69185i

−2.44949

−

1.73205i

191.7

1.65068

−

0.524648i

−1.57313

−

2.72474i

−2.21650

−

1.27970i

2.44949

−

1.73205i

191.8

1.65068

−

0.524648i

1.57313

2.72474i

2.21650

1.27970i

2.44949

−

1.73205i

959.1

−1.65068

−

0.524648i

−1.57313

2.72474i

2.21650

−

1.27970i

2.44949

1.73205i

959.2

−1.65068

−

0.524648i

1.57313

−

2.72474i

−2.21650

1.27970i

2.44949

1.73205i

959.3

−0.524648

−

1.65068i

−0.158919

0.275255i

2.93038

−

1.69185i

−2.44949

1.73205i

959.4

−0.524648

−

1.65068i

0.158919

−

0.275255i

−2.93038

1.69185i

−2.44949

1.73205i

959.5

0.524648

1.65068i

−0.158919

0.275255i

−2.93038

1.69185i

−2.44949

1.73205i

959.6

0.524648

1.65068i

0.158919

−

0.275255i

2.93038

−

1.69185i

−2.44949

1.73205i

959.7

1.65068

0.524648i

−1.57313

2.72474i

−2.21650

1.27970i

2.44949

1.73205i

959.8

1.65068

0.524648i

1.57313

−

2.72474i

2.21650

−

1.27970i

2.44949

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
9.d	odd	6	1	inner
36.h	even	6	1	inner
72.j	odd	6	1	inner
72.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1152.2.p.e	✓	16
3.b	odd	2	1		3456.2.p.e		16
4.b	odd	2	1	inner	1152.2.p.e	✓	16
8.b	even	2	1	inner	1152.2.p.e	✓	16
8.d	odd	2	1	inner	1152.2.p.e	✓	16
9.c	even	3	1		3456.2.p.e		16
9.d	odd	6	1	inner	1152.2.p.e	✓	16
12.b	even	2	1		3456.2.p.e		16
24.f	even	2	1		3456.2.p.e		16
24.h	odd	2	1		3456.2.p.e		16
36.f	odd	6	1		3456.2.p.e		16
36.h	even	6	1	inner	1152.2.p.e	✓	16
72.j	odd	6	1	inner	1152.2.p.e	✓	16
72.l	even	6	1	inner	1152.2.p.e	✓	16
72.n	even	6	1		3456.2.p.e		16
72.p	odd	6	1		3456.2.p.e		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1152.2.p.e	✓	16	1.a	even	1	1	trivial
1152.2.p.e	✓	16	4.b	odd	2	1	inner
1152.2.p.e	✓	16	8.b	even	2	1	inner
1152.2.p.e	✓	16	8.d	odd	2	1	inner
1152.2.p.e	✓	16	9.d	odd	6	1	inner
1152.2.p.e	✓	16	36.h	even	6	1	inner
1152.2.p.e	✓	16	72.j	odd	6	1	inner
1152.2.p.e	✓	16	72.l	even	6	1	inner
3456.2.p.e		16	3.b	odd	2	1
3456.2.p.e		16	9.c	even	3	1
3456.2.p.e		16	12.b	even	2	1
3456.2.p.e		16	24.f	even	2	1
3456.2.p.e		16	24.h	odd	2	1
3456.2.p.e		16	36.f	odd	6	1
3456.2.p.e		16	72.n	even	6	1
3456.2.p.e		16	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1152, [\chi])$:

$ T_{5}^{8} + 10T_{5}^{6} + 99T_{5}^{4} + 10T_{5}^{2} + 1 $

T5^8 + 10*T5^6 + 99*T5^4 + 10*T5^2 + 1

$ T_{7}^{8} - 18T_{7}^{6} + 249T_{7}^{4} - 1350T_{7}^{2} + 5625 $

T7^8 - 18*T7^6 + 249*T7^4 - 1350*T7^2 + 5625

$ T_{11}^{8} - 6T_{11}^{6} + 33T_{11}^{4} - 18T_{11}^{2} + 9 $

T11^8 - 6*T11^6 + 33*T11^4 - 18*T11^2 + 9

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 6 T^{4} + 81)^{2} $ (T^8 - 6*T^4 + 81)^2
$5$	$ (T^{8} + 10 T^{6} + 99 T^{4} + \cdots + 1)^{2} $ (T^8 + 10T^6 + 99T^4 + 10*T^2 + 1)^2
$7$	$ (T^{8} - 18 T^{6} + \cdots + 5625)^{2} $ (T^8 - 18T^6 + 249T^4 - 1350*T^2 + 5625)^2
$11$	$ (T^{8} - 6 T^{6} + 33 T^{4} + \cdots + 9)^{2} $ (T^8 - 6T^6 + 33T^4 - 18*T^2 + 9)^2
$13$	$ (T^{4} - 9 T^{2} + 81)^{4} $ (T^4 - 9*T^2 + 81)^4
$17$	$ (T^{2} + 18)^{8} $ (T^2 + 18)^8
$19$	$ (T^{4} - 72 T^{2} + 432)^{4} $ (T^4 - 72*T^2 + 432)^4
$23$	$ (T^{8} + 18 T^{6} + \cdots + 729)^{2} $ (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$29$	$ (T^{8} + 70 T^{6} + \cdots + 130321)^{2} $ (T^8 + 70T^6 + 4539T^4 + 25270*T^2 + 130321)^2
$31$	$ (T^{8} - 102 T^{6} + \cdots + 3515625)^{2} $ (T^8 - 102T^6 + 8529T^4 - 191250*T^2 + 3515625)^2
$37$	$ (T^{2} + 54)^{8} $ (T^2 + 54)^8
$41$	$ (T^{4} + 18 T^{3} + \cdots + 81)^{4} $ (T^4 + 18T^3 + 117T^2 + 162*T + 81)^4
$43$	$ (T^{8} + 54 T^{6} + \cdots + 59049)^{2} $ (T^8 + 54T^6 + 2673T^4 + 13122*T^2 + 59049)^2
$47$	$ (T^{8} + 198 T^{6} + \cdots + 95004009)^{2} $ (T^8 + 198T^6 + 29457T^4 + 1929906*T^2 + 95004009)^2
$53$	$ (T^{4} - 100 T^{2} + 2116)^{4} $ (T^4 - 100*T^2 + 2116)^4
$59$	$ (T^{8} - 18 T^{6} + \cdots + 729)^{2} $ (T^8 - 18T^6 + 297T^4 - 486*T^2 + 729)^2
$61$	$ (T^{4} - 81 T^{2} + 6561)^{4} $ (T^4 - 81*T^2 + 6561)^4
$67$	$ (T^{8} + 162 T^{6} + \cdots + 729)^{2} $ (T^8 + 162T^6 + 26217T^4 + 4374*T^2 + 729)^2
$71$	$ (T^{4} - 396 T^{2} + 38988)^{4} $ (T^4 - 396*T^2 + 38988)^4
$73$	$ (T^{2} + 8 T + 10)^{8} $ (T^2 + 8*T + 10)^8
$79$	$ (T^{8} - 198 T^{6} + \cdots + 95004009)^{2} $ (T^8 - 198T^6 + 29457T^4 - 1929906*T^2 + 95004009)^2
$83$	$ (T^{8} - 18 T^{6} + \cdots + 5625)^{2} $ (T^8 - 18T^6 + 249T^4 - 1350*T^2 + 5625)^2
$89$	$ (T^{4} + 252 T^{2} + 8100)^{4} $ (T^4 + 252*T^2 + 8100)^4
$97$	$ (T^{4} - 2 T^{3} + \cdots + 22201)^{4} $ (T^4 - 2T^3 + 153T^2 + 298*T + 22201)^4
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{11} + \beta_{8} + \beta_{4}) q^{3} + ( - \beta_{14} + \beta_{7} - \beta_1) q^{5} + (\beta_{13} + \beta_{12} + \cdots - \beta_{2}) q^{7} + (\beta_{6} + 2 \beta_{3} + 1) q^{9} - \beta_{9} q^{11}+ \cdots + (\beta_{11} - 2 \beta_{9} + 2 \beta_{8}) q^{99}+O(q^{100}) \) q + (-b11 + b8 + b4) * q^3 + (-b14 + b7 - b1) * q^5 + (b13 + b12 - b10 - b2) * q^7 + (b6 + 2b3 + 1) q^9 - b9 * q^11 + (b14 - 2b5) q^13 + (-b13 + 2b12 - 2b10 - b2) * q^15 + (2b15 + b6) q^17 + (2b9 - 2b8 - 2b4) q^19 + (b14 + 2b7 - 3b5 - b1) * q^21 + (2b13 - b12 - b2) q^23 + (-2b15 - 2b6) * q^25 + (b11 + b9 + 2b8 + 3b4) * q^27 + (3b14 + 2b7 - 3b5) q^29 + (-2b13 + 3b12 - 2b10 + 5b2) * q^31 + (b6 - b3 + 1) * q^33 + (2b11 - 4b9 - 4b8 - 5b4) * q^35 + (6b7 - 3b1) * q^37 + 3b10 q^39 + (-b15 - 2b6 - 3b3 - 6) * q^41 + 3b11 q^43 + (-b14 + 2b7 - 2b5 - 4b1) q^45 + (5b13 - 7b12 + 3b10 - 7b2) * q^47 + (b15 - 2b3) q^49 + (3b9 + 3b8) * q^51 + (-4b5 - b1) q^53 + (-3b13 + 3b12 - 4b10 + b2) q^55 + (-2b15 - 4b6 - 6) * q^57 + (-b11 - b4) * q^59 + (-3b14 - 3b5) * q^61 + (-3b13 + 2b12 + 2b10) q^63 + (-b15 + b6 - 3b3 + 3) q^65 + (-6b11 + 2b9 + b8 + 7b4) q^67 + (b14 - 2b5 + 3b1) * q^69 + (-5b13 - 5b12 + 4b2) q^71 + (b6 - 4) * q^73 + (-2b11 - 2b9 - 4b8) q^75 + (-b14 + 3b7 - 3b1) * q^77 + (5b13 - 2b12 + 2b10 + 2b2) * q^79 + (4b15 + 2b6 + 3) * q^81 + (b11 - 2b9 - 2b4) * q^83 + (2b14 - 3b7 - 4b5 - 3b1) * q^85 + (-4b13 - b12 + 4b10 + 2b2) q^87 + (-2b15 - b6 + 12b3 + 6) * q^89 + (-3b9 + 3b8 - 3b4) q^91 + (-b14 + 8b7 + 4b5 - b1) * q^93 + (8b13 - 10b12 + 12b10 + 2b2) * q^95 + (5b15 + 5b6 + b3 + 1) * q^97 + (b11 - 2b9 + 2b8) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{33} - 72 q^{41} + 16 q^{49} - 96 q^{57} + 72 q^{65} - 64 q^{73} + 48 q^{81} + 8 q^{97}+O(q^{100}) \) 16 * q + 24 * q^33 - 72 * q^41 + 16 * q^49 - 96 * q^57 + 72 * q^65 - 64 * q^73 + 48 * q^81 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1152.2.p.e

Level	$1152$
Weight	$2$
Character orbit	1152.p
Analytic conductor	$9.199$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(9.19876631285\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	16.0.9349208943630483456.9
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) x^16 - 8x^15 + 48x^14 - 196x^13 + 642x^12 - 1668x^11 + 3580x^10 - 6328x^9 + 9297x^8 - 11276x^7 + 11224x^6 - 9024x^5 + 5736x^4 - 2780x^3 + 972x^2 - 220*x + 25
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 56 \nu^{14} - 392 \nu^{13} + 2267 \nu^{12} - 8506 \nu^{11} + 26269 \nu^{10} - 62716 \nu^{9} + \cdots + 2430 ) / 65 \) (56v^14 - 392v^13 + 2267v^12 - 8506v^11 + 26269v^10 - 62716v^9 + 124065v^8 - 197514v^7 + 258029v^6 - 269446v^5 + 222643v^4 - 139500v^3 + 62551v^2 - 17806v + 2430) / 65
\(\beta_{2}\)	\(=\)	\( ( - 62 \nu^{14} + 434 \nu^{13} - 2541 \nu^{12} + 9604 \nu^{11} - 30137 \nu^{10} + 72992 \nu^{9} + \cdots - 5945 ) / 65 \) (-62v^14 + 434v^13 - 2541v^12 + 9604v^11 - 30137v^10 + 72992v^9 - 147747v^8 + 240948v^7 - 326020v^6 + 354283v^5 - 310030v^4 + 208129v^3 - 103708v^2 + 33855v - 5945) / 65
\(\beta_{3}\)	\(=\)	\( ( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} + \cdots - 142555 ) / 17095 \) (3456v^15 - 25920v^14 + 151876v^13 - 594074v^12 + 1879372v^11 - 4666596v^10 + 9554736v^9 - 15945783v^8 + 21928484v^7 - 24527176v^6 + 22138664v^5 - 15739188v^4 + 8598668v^3 - 3418428v^2 + 929924*v - 142555) / 17095
\(\beta_{4}\)	\(=\)	\( ( - 88 \nu^{14} + 616 \nu^{13} - 3568 \nu^{12} + 13400 \nu^{11} - 41486 \nu^{10} + 99278 \nu^{9} + \cdots - 5230 ) / 65 \) (-88v^14 + 616v^13 - 3568v^12 + 13400v^11 - 41486v^10 + 99278v^9 - 197277v^8 + 315672v^7 - 416019v^6 + 439199v^5 - 369648v^4 + 237379v^3 - 111209v^2 + 33751v - 5230) / 65
\(\beta_{5}\)	\(=\)	\( ( - 10 \nu^{14} + 70 \nu^{13} - 406 \nu^{12} + 1526 \nu^{11} - 4732 \nu^{10} + 11340 \nu^{9} - 22581 \nu^{8} + \cdots - 670 ) / 5 \) (-10v^14 + 70v^13 - 406v^12 + 1526v^11 - 4732v^10 + 11340v^9 - 22581v^8 + 36210v^7 - 47866v^6 + 50708v^5 - 42917v^4 + 27762v^3 - 13198v^2 + 4094v - 670) / 5
\(\beta_{6}\)	\(=\)	\( ( 54 \nu^{14} - 378 \nu^{13} + 2193 \nu^{12} - 8244 \nu^{11} + 25569 \nu^{10} - 61284 \nu^{9} + \cdots + 3308 ) / 13 \) (54v^14 - 378v^13 + 2193v^12 - 8244v^11 + 25569v^10 - 61284v^9 + 122021v^8 - 195620v^7 + 258293v^6 - 273134v^5 + 230109v^4 - 147802v^3 + 69203v^2 - 20980v + 3308) / 13
\(\beta_{7}\)	\(=\)	\( ( 22230 \nu^{15} - 159361 \nu^{14} + 923582 \nu^{13} - 3511572 \nu^{12} + 10902106 \nu^{11} + \cdots - 39775 ) / 17095 \) (22230v^15 - 159361v^14 + 923582v^13 - 3511572v^12 + 10902106v^11 - 26315724v^10 + 52438146v^9 - 84432495v^8 + 111380164v^7 - 117919554v^6 + 98855056v^5 - 63256933v^4 + 29226910v^3 - 8804251v^2 + 1370336*v - 39775) / 17095
\(\beta_{8}\)	\(=\)	\( ( 31151 \nu^{15} - 186424 \nu^{14} + 1043567 \nu^{13} - 3475210 \nu^{12} + 9946592 \nu^{11} + \cdots + 1694445 ) / 17095 \) (31151v^15 - 186424v^14 + 1043567v^13 - 3475210v^12 + 9946592v^11 - 20522197v^10 + 34995448v^9 - 42813993v^8 + 37491703v^7 - 11634546v^6 - 19984771v^5 + 41799684v^4 - 40498098v^3 + 24932581v^2 - 9099207*v + 1694445) / 17095
\(\beta_{9}\)	\(=\)	\( ( 31151 \nu^{15} - 257697 \nu^{14} + 1542478 \nu^{13} - 6361898 \nu^{12} + 20780877 \nu^{11} + \cdots - 2345235 ) / 17095 \) (31151v^15 - 257697v^14 + 1542478v^13 - 6361898v^12 + 20780877v^11 - 54011828v^10 + 115020036v^9 - 201417195v^8 + 290550040v^7 - 343630755v^6 + 328629093v^5 - 249342894v^4 + 144728331v^3 - 60896943v^2 + 16663484*v - 2345235) / 17095
\(\beta_{10}\)	\(=\)	\( ( - 36004 \nu^{15} + 278183 \nu^{14} - 1640254 \nu^{13} + 6529376 \nu^{12} - 20875054 \nu^{11} + \cdots + 1576650 ) / 17095 \) (-36004v^15 + 278183v^14 - 1640254v^13 + 6529376v^12 - 20875054v^11 + 52692141v^10 - 109431091v^9 + 186121407v^8 - 260824747v^7 + 298801493v^6 - 276219577v^5 + 202070739v^4 - 112620474v^3 + 45386761v^2 - 11822664*v + 1576650) / 17095
\(\beta_{11}\)	\(=\)	\( ( - 47906 \nu^{15} + 347723 \nu^{14} - 2030815 \nu^{13} + 7808324 \nu^{12} - 24528937 \nu^{11} + \cdots + 705525 ) / 17095 \) (-47906v^15 + 347723v^14 - 2030815v^13 + 7808324v^12 - 24528937v^11 + 60081095v^10 - 121877717v^9 + 200621199v^8 - 272407344v^7 + 299328268v^6 - 264286576v^5 + 181704730v^4 - 93782944v^3 + 33728236v^2 - 7443876*v + 705525) / 17095
\(\beta_{12}\)	\(=\)	\( ( - 49663 \nu^{15} + 358139 \nu^{14} - 2089429 \nu^{13} + 8006091 \nu^{12} - 25109764 \nu^{11} + \cdots + 1090565 ) / 17095 \) (-49663v^15 + 358139v^14 - 2089429v^13 + 8006091v^12 - 25109764v^11 + 61357045v^10 - 124293850v^9 + 204357372v^8 - 277503791v^7 + 305524016v^6 - 271172221v^5 + 188678893v^4 - 99647214v^3 + 37587574v^2 - 9111403*v + 1090565) / 17095
\(\beta_{13}\)	\(=\)	\( ( 49663 \nu^{15} - 403112 \nu^{14} + 2404240 \nu^{13} - 9830785 \nu^{12} + 31965385 \nu^{11} + \cdots - 3581175 ) / 17095 \) (49663v^15 - 403112v^14 + 2404240v^13 - 9830785v^12 + 31965385v^11 - 82591928v^10 + 175128068v^9 - 305358840v^8 + 439053908v^7 - 518058261v^6 + 494946823v^5 - 375963823v^4 + 218951115v^3 - 92737096v^2 + 25552848*v - 3581175) / 17095
\(\beta_{14}\)	\(=\)	\( ( 79890 \nu^{15} - 616270 \nu^{14} + 3648880 \nu^{13} - 14546467 \nu^{12} + 46727282 \nu^{11} + \cdots - 3924805 ) / 17095 \) (79890v^15 - 616270v^14 + 3648880v^13 - 14546467v^12 + 46727282v^11 - 118357534v^10 + 247314870v^9 - 422995752v^8 + 597739530v^7 - 690526972v^6 + 645876746v^5 - 478299719v^4 + 271084944v^3 - 111096466v^2 + 29525918*v - 3924805) / 17095
\(\beta_{15}\)	\(=\)	\( ( 162552 \nu^{15} - 1254645 \nu^{14} + 7410690 \nu^{13} - 29505615 \nu^{12} + 94503294 \nu^{11} + \cdots - 7071655 ) / 17095 \) (162552v^15 - 1254645v^14 + 7410690v^13 - 29505615v^12 + 94503294v^11 - 238745115v^10 + 496891020v^9 - 846295960v^8 + 1189159396v^7 - 1364903575v^6 + 1266038596v^5 - 928161750v^4 + 519335978v^3 - 209359180v^2 + 54517604*v - 7071655) / 17095

\(\nu\)	\(=\)	\( ( - 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} - 3 \beta_{12} + 3 \beta_{9} + 3 \beta_{8} - \beta_{6} + \cdots + 3 ) / 6 \) (-2b15 - 2b14 + 3b13 - 3b12 + 3b9 + 3b8 - b6 + b5 + 3b4 - 3b2 + 3) / 6
\(\nu^{2}\)	\(=\)	\( ( - 2 \beta_{15} - 2 \beta_{14} + 6 \beta_{13} + 6 \beta_{8} - \beta_{6} + 4 \beta_{5} + 6 \beta_{4} + \cdots - 12 ) / 6 \) (-2b15 - 2b14 + 6b13 + 6b8 - b6 + 4b5 + 6b4 - 6b2 + 3b1 - 12) / 6
\(\nu^{3}\)	\(=\)	\( ( 10 \beta_{15} + 11 \beta_{14} - 12 \beta_{13} + 21 \beta_{12} + 6 \beta_{11} - 12 \beta_{10} - 18 \beta_{9} + \cdots - 15 ) / 6 \) (10b15 + 11b14 - 12b13 + 21b12 + 6b11 - 12b10 - 18b9 - 9b8 - 9b7 + 5b6 - b5 - 12b4 + 9b3 + 6b2 + 9b1 - 15) / 6
\(\nu^{4}\)	\(=\)	\( ( 11 \beta_{15} + 12 \beta_{14} - 24 \beta_{13} + 12 \beta_{12} + 6 \beta_{11} - 12 \beta_{10} - 6 \beta_{9} + \cdots + 27 ) / 3 \) (11b15 + 12b14 - 24b13 + 12b12 + 6b11 - 12b10 - 6b9 - 24b8 - 9b7 + 10b6 - 15b5 - 24b4 + 9b3 + 24b2 - 9*b1 + 27) / 3
\(\nu^{5}\)	\(=\)	\( ( - 43 \beta_{15} - 57 \beta_{14} + 33 \beta_{13} - 108 \beta_{12} - 36 \beta_{11} + 84 \beta_{10} + \cdots + 108 ) / 6 \) (-43b15 - 57b14 + 33b13 - 108b12 - 36b11 + 84b10 + 129b9 + 24b8 + 60b7 + b6 - 24b5 + 57b4 - 75b3 + 39b2 - 105b1 + 108) / 6
\(\nu^{6}\)	\(=\)	\( ( - 185 \beta_{15} - 232 \beta_{14} + 336 \beta_{13} - 270 \beta_{12} - 138 \beta_{11} + 312 \beta_{10} + \cdots - 300 ) / 6 \) (-185b15 - 232b14 + 336b13 - 270b12 - 138b11 + 312b10 + 216b9 + 396b8 + 225b7 - 160b6 + 206b5 + 420b4 - 270b3 - 276b2 + 15*b1 - 300) / 6
\(\nu^{7}\)	\(=\)	\( ( 128 \beta_{15} + 198 \beta_{14} + 36 \beta_{13} + 468 \beta_{12} + 114 \beta_{11} - 348 \beta_{10} + \cdots - 921 ) / 6 \) (128b15 + 198b14 + 36b13 + 468b12 + 114b11 - 348b10 - 792b9 + 216b8 - 252b7 - 251b6 + 405b5 - 51b4 + 378b3 - 651b2 + 840*b1 - 921) / 6
\(\nu^{8}\)	\(=\)	\( 238 \beta_{15} + 322 \beta_{14} - 392 \beta_{13} + 396 \beta_{12} + 188 \beta_{11} - 484 \beta_{10} + \cdots + 267 \) 238b15 + 322b14 - 392b13 + 396b12 + 188b11 - 484b10 - 408b9 - 476b8 - 350b7 + 154b6 - 198b5 - 544b4 + 469b3 + 204b2 + 136*b1 + 267
\(\nu^{9}\)	\(=\)	\( ( 379 \beta_{15} + 385 \beta_{14} - 2205 \beta_{13} - 1044 \beta_{12} + 306 \beta_{11} - 270 \beta_{10} + \cdots + 8175 ) / 6 \) (379b15 + 385b14 - 2205b13 - 1044b12 + 306b11 - 270b10 + 3753b9 - 4590b8 - 162b7 + 3119b6 - 4445b5 - 2718b4 - 27b3 + 6300b2 - 5580*b1 + 8175) / 6
\(\nu^{10}\)	\(=\)	\( ( - 10223 \beta_{15} - 14312 \beta_{14} + 15690 \beta_{13} - 18576 \beta_{12} - 7956 \beta_{11} + \cdots - 5592 ) / 6 \) (-10223b15 - 14312b14 + 15690b13 - 18576b12 - 7956b11 + 22734b10 + 22698b9 + 17490b8 + 16605b7 - 3559b6 + 4618b5 + 22422b4 - 23220b3 - 2208b2 - 12138*b1 - 5592) / 6
\(\nu^{11}\)	\(=\)	\( ( - 12443 \beta_{15} - 16927 \beta_{14} + 30768 \beta_{13} - 10440 \beta_{12} - 9921 \beta_{11} + \cdots - 68274 ) / 6 \) (-12443b15 - 16927b14 + 30768b13 - 10440b12 - 9921b11 + 25455b10 - 6831b9 + 53295b8 + 18348b7 - 28249b6 + 39632b5 + 42465b4 - 24783b3 - 50364b2 + 30327*b1 - 68274) / 6
\(\nu^{12}\)	\(=\)	\( ( 32900 \beta_{15} + 46629 \beta_{14} - 45948 \beta_{13} + 65172 \beta_{12} + 24918 \beta_{11} + \cdots - 9414 ) / 3 \) (32900b15 + 46629b14 - 45948b13 + 65172b12 + 24918b11 - 76260b10 - 91644b9 - 40998b8 - 56232b7 - 1469b6 + 2583b5 - 65904b4 + 79695b3 - 18156b2 + 62271*b1 - 9414) / 3
\(\nu^{13}\)	\(=\)	\( ( 158381 \beta_{15} + 221934 \beta_{14} - 323757 \beta_{13} + 207966 \beta_{12} + 122247 \beta_{11} + \cdots + 514725 ) / 6 \) (158381b15 + 221934b14 - 323757b13 + 207966b12 + 122247b11 - 354021b10 - 128232b9 - 496197b8 - 259155b7 + 216256b6 - 303783b5 - 456369b4 + 362934b3 + 349410b2 - 107328*b1 + 514725) / 6
\(\nu^{14}\)	\(=\)	\( ( - 352742 \beta_{15} - 501646 \beta_{14} + 401340 \beta_{13} - 794892 \beta_{12} - 260958 \beta_{11} + \cdots + 639942 ) / 6 \) (-352742b15 - 501646b14 + 401340b13 - 794892b12 - 260958b11 + 834318b10 + 1299648b9 + 138948b8 + 619710b7 + 247871b6 - 351754b5 + 575106b4 - 881244b3 + 638640b2 - 1070745*b1 + 639942) / 6
\(\nu^{15}\)	\(=\)	\( ( - 1567315 \beta_{15} - 2212539 \beta_{14} + 2900829 \beta_{13} - 2387823 \beta_{12} - 1184481 \beta_{11} + \cdots - 3396279 ) / 6 \) (-1567315b15 - 2212539b14 + 2900829b13 - 2387823b12 - 1184481b11 + 3604815b10 + 2290002b9 + 3995220b8 + 2658726b7 - 1424525b6 + 2006715b5 + 4105164b4 - 3753099b3 - 2050089b2 - 247560*b1 - 3396279) / 6

\(n\)	\(127\)	\(641\)	\(901\)
\(\chi(n)\)	\(-1\)	\(-\beta_{3}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 191.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 6 T^{4} + 81)^{2} \) (T^8 - 6*T^4 + 81)^2
$5$	\( (T^{8} + 10 T^{6} + 99 T^{4} + \cdots + 1)^{2} \) (T^8 + 10T^6 + 99T^4 + 10*T^2 + 1)^2
$7$	\( (T^{8} - 18 T^{6} + \cdots + 5625)^{2} \) (T^8 - 18T^6 + 249T^4 - 1350*T^2 + 5625)^2
$11$	\( (T^{8} - 6 T^{6} + 33 T^{4} + \cdots + 9)^{2} \) (T^8 - 6T^6 + 33T^4 - 18*T^2 + 9)^2
$13$	\( (T^{4} - 9 T^{2} + 81)^{4} \) (T^4 - 9*T^2 + 81)^4
$17$	\( (T^{2} + 18)^{8} \) (T^2 + 18)^8
$19$	\( (T^{4} - 72 T^{2} + 432)^{4} \) (T^4 - 72*T^2 + 432)^4
$23$	\( (T^{8} + 18 T^{6} + \cdots + 729)^{2} \) (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$29$	\( (T^{8} + 70 T^{6} + \cdots + 130321)^{2} \) (T^8 + 70T^6 + 4539T^4 + 25270*T^2 + 130321)^2
$31$	\( (T^{8} - 102 T^{6} + \cdots + 3515625)^{2} \) (T^8 - 102T^6 + 8529T^4 - 191250*T^2 + 3515625)^2
$37$	\( (T^{2} + 54)^{8} \) (T^2 + 54)^8
$41$	\( (T^{4} + 18 T^{3} + \cdots + 81)^{4} \) (T^4 + 18T^3 + 117T^2 + 162*T + 81)^4
$43$	\( (T^{8} + 54 T^{6} + \cdots + 59049)^{2} \) (T^8 + 54T^6 + 2673T^4 + 13122*T^2 + 59049)^2
$47$	\( (T^{8} + 198 T^{6} + \cdots + 95004009)^{2} \) (T^8 + 198T^6 + 29457T^4 + 1929906*T^2 + 95004009)^2
$53$	\( (T^{4} - 100 T^{2} + 2116)^{4} \) (T^4 - 100*T^2 + 2116)^4
$59$	\( (T^{8} - 18 T^{6} + \cdots + 729)^{2} \) (T^8 - 18T^6 + 297T^4 - 486*T^2 + 729)^2
$61$	\( (T^{4} - 81 T^{2} + 6561)^{4} \) (T^4 - 81*T^2 + 6561)^4
$67$	\( (T^{8} + 162 T^{6} + \cdots + 729)^{2} \) (T^8 + 162T^6 + 26217T^4 + 4374*T^2 + 729)^2
$71$	\( (T^{4} - 396 T^{2} + 38988)^{4} \) (T^4 - 396*T^2 + 38988)^4
$73$	\( (T^{2} + 8 T + 10)^{8} \) (T^2 + 8*T + 10)^8
$79$	\( (T^{8} - 198 T^{6} + \cdots + 95004009)^{2} \) (T^8 - 198T^6 + 29457T^4 - 1929906*T^2 + 95004009)^2
$83$	\( (T^{8} - 18 T^{6} + \cdots + 5625)^{2} \) (T^8 - 18T^6 + 249T^4 - 1350*T^2 + 5625)^2
$89$	\( (T^{4} + 252 T^{2} + 8100)^{4} \) (T^4 + 252*T^2 + 8100)^4
$97$	\( (T^{4} - 2 T^{3} + \cdots + 22201)^{4} \) (T^4 - 2T^3 + 153T^2 + 298*T + 22201)^4
show more
show less