LMFDB - Newform orbit 1152.2.r.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(193,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1152.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.19876631285$
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^{4} $
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_{2} q^{3} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{13} + \beta_{11} + \beta_{6}) q^{7} + (\beta_{3} + 2 \beta_1 - 1) q^{9}+O(q^{10}) $ q + b2 * q^3 + (b7 + b4) * q^5 + (b13 + b11 + b6) * q^7 + (b3 + 2b1 - 1) q^9	$ q + \beta_{2} q^{3} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{13} + \beta_{11} + \beta_{6}) q^{7} + (\beta_{3} + 2 \beta_1 - 1) q^{9} + (\beta_{10} + \beta_{5} - \beta_{2}) q^{11} - \beta_{7} q^{13} + (2 \beta_{12} - \beta_{11} + \beta_{6}) q^{15} + ( - \beta_{3} + 4) q^{17} + (3 \beta_{9} - \beta_{8} + \beta_{4}) q^{21} + ( - 2 \beta_{13} - \beta_{12} + \cdots + \beta_{6}) q^{23}+ \cdots + ( - 3 \beta_{14} - 3 \beta_{10} + 3 \beta_{2}) q^{99}+O(q^{100}) $ q + b2 * q^3 + (b7 + b4) * q^5 + (b13 + b11 + b6) * q^7 + (b3 + 2b1 - 1) q^9 + (b10 + b5 - b2) * q^11 - b7 * q^13 + (2b12 - b11 + b6) q^15 + (-b3 + 4) * q^17 + (3b9 - b8 + b4) q^21 + (-2b13 - b12 - b11 + b6) q^23 + (-2b15 - 2b1 + 2) * q^25 + (-b14 + 2b10 + 2b5 - b2) * q^27 + (b9 - 2b8) q^29 + (2b13 - b12 + 3b11 - b6) * q^31 + (-2b15 + b3 - 3b1 + 3) * q^33 + (-2b14 - 2b10 + 3b5 - 3b2) * q^35 + (6b9 + b8 - 6b7 + b4) * q^37 + b11 * q^39 + (-3b15 + 3b3 - b1) * q^41 + (b14 - 3b10 - 4b5 + b2) * q^43 + (2b9 - 2b8 - 7b7 - 2b4) * q^45 + (-b13 - b11 - b6) * q^47 + (-3b15 + 3b3 - 2b1) q^49 + (b14 - 2b5 + 4b2) * q^51 + (4b9 + 3b8 - 4b7 + 3b4) * q^53 + (-3b13 - 2b12 + 2b11 - 3b6) * q^55 + (b14 - b10 + b5 + 2b2) q^59 + (-3b9 - 2b8) * q^61 + (-3b13 + 3b12 - 3b11) q^63 + (b15 + b1 - 1) * q^65 + (3b14 + 7b10 - 2b5 - 4b2) * q^67 + (6b9 + 3b8 - 3b7 + 3b4) * q^69 + (2b13 - b12 - 6b11 + 2b6) q^71 + (b3 + 6) * q^73 + (-2b14 - 2b10 - 2b5 + 2b2) * q^75 + (-9b7 - 3b4) * q^77 + (7b13 - 3b12 + b11 - 2b6) q^79 + (-4b15 + 2b3 + 3) * q^81 + (b14 - b10 - 2b5 - b2) q^83 + (10b7 + 5b4) * q^85 + (-b13 + 2b12 - b11 - 2b6) * q^87 + (-b3 + 2) * q^89 + (b14 - b5 + b2) * q^91 + (-6b9 - b8 + 9b7 + b4) * q^93 + (3b15 + 9b1 - 9) * q^97 + (-3b14 - 3b10 + 3b2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 64 q^{17} + 16 q^{25} + 24 q^{33} - 8 q^{41} - 16 q^{49} - 8 q^{65} + 96 q^{73} + 48 q^{81} + 32 q^{89} - 72 q^{97}+O(q^{100}) $ 16 * q + 64 * q^17 + 16 * q^25 + 24 * q^33 - 8 * q^41 - 16 * q^49 - 8 * q^65 + 96 * q^73 + 48 * q^81 + 32 * q^89 - 72 * q^97

Display 100 coefficients 10 coefficients

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 :

$\beta_{1}$	$=$	$ ( - 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
$\beta_{2}$	$=$	$ ( - 13659 \nu^{15} + 79956 \nu^{14} - 449175 \nu^{13} + 1476715 \nu^{12} - 4234710 \nu^{11} + \cdots - 486085 ) / 17095 $ (-13659v^15 + 79956v^14 - 449175v^13 + 1476715v^12 - 4234710v^11 + 8664904v^10 - 14862759v^9 + 18235965v^8 - 16679044v^7 + 6722523v^6 + 5047356v^5 - 13391846v^4 + 12973260v^3 - 7799187v^2 + 2711261*v - 486085) / 17095
$\beta_{3}$	$=$	$ ( - 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
$\beta_{4}$	$=$	$ ( 22230 \nu^{15} - 144633 \nu^{14} + 820486 \nu^{13} - 2915351 \nu^{12} + 8665028 \nu^{11} + \cdots + 599315 ) / 17095 $ (22230v^15 - 144633v^14 + 820486v^13 - 2915351v^12 + 8665028v^11 - 19406977v^10 + 35943838v^9 - 51803400v^8 + 59433982v^7 - 50057927v^6 + 27990758v^5 - 4701824v^4 - 7461590v^3 + 7646662v^2 - 3312642*v + 599315) / 17095
$\beta_{5}$	$=$	$ ( 13659 \nu^{15} - 141235 \nu^{14} + 878128 \nu^{13} - 3969692 \nu^{12} + 13616183 \nu^{11} + \cdots - 3568060 ) / 17095 $ (13659v^15 - 141235v^14 + 878128v^13 - 3969692v^12 + 13616183v^11 - 37825818v^10 + 84893873v^9 - 158094894v^8 + 241598485v^7 - 305000028v^6 + 311903675v^5 - 255430974v^4 + 161068568v^3 - 74625539v^2 + 22634049*v - 3568060) / 17095
$\beta_{6}$	$=$	$ ( 16755 \nu^{15} - 161299 \nu^{14} + 987248 \nu^{13} - 4333114 \nu^{12} + 14582345 \nu^{11} + \cdots - 2399970 ) / 17095 $ (16755v^15 - 161299v^14 + 987248v^13 - 4333114v^12 + 14582345v^11 - 39558898v^10 + 86882269v^9 - 157807206v^8 + 234915641v^7 - 287693722v^6 + 284271347v^5 - 223504414v^4 + 134281042v^3 - 58660817v^2 + 16543083*v - 2399970) / 17095
$\beta_{7}$	$=$	$ ( - 26630 \nu^{15} + 182630 \nu^{14} - 1056740 \nu^{13} + 3923413 \nu^{12} - 12097498 \nu^{11} + \cdots - 218885 ) / 17095 $ (-26630v^15 + 182630v^14 - 1056740v^13 + 3923413v^12 - 12097498v^11 + 28666706v^10 - 56590650v^9 + 89528958v^8 - 116711850v^7 + 121073088v^6 - 99711814v^5 + 61611091v^4 - 27082796v^3 + 6949514v^2 - 510382*v - 218885) / 17095
$\beta_{8}$	$=$	$ ( 22230 \nu^{15} - 188817 \nu^{14} + 1129774 \nu^{13} - 4704014 \nu^{12} + 15376262 \nu^{11} + \cdots - 1317955 ) / 17095 $ (22230v^15 - 188817v^14 + 1129774v^13 - 4704014v^12 + 15376262v^11 - 40133218v^10 + 85426762v^9 - 149690685v^8 + 215272528v^7 - 253642808v^6 + 240583652v^5 - 180367151v^4 + 102603910v^3 - 41706077v^2 + 10736292*v - 1317955) / 17095
$\beta_{9}$	$=$	$ ( 26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} + \cdots - 2071845 ) / 17095 $ (26630v^15 - 216820v^14 + 1296070v^13 - 5311527v^12 + 17314892v^11 - 44845414v^10 + 95362110v^9 - 166733397v^8 + 240513840v^7 - 284726942v^6 + 273082466v^5 - 208344314v^4 + 122001074v^3 - 52073476v^2 + 14507768*v - 2071845) / 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
$\beta_{11}$	$=$	$ ( - 31151 \nu^{15} + 280841 \nu^{14} - 1704486 \nu^{13} + 7300282 \nu^{12} - 24305077 \nu^{11} + \cdots + 3720725 ) / 17095 $ (-31151v^15 + 280841v^14 - 1704486v^13 + 7300282v^12 - 24305077v^11 + 64922646v^10 - 141130150v^9 + 253301046v^8 - 373571776v^7 + 453043752v^6 - 444138430v^5 + 346560318v^4 - 207159008v^3 + 90144910v^2 - 25539997*v + 3720725) / 17095
$\beta_{12}$	$=$	$ ( 62302 \nu^{15} - 444121 \nu^{14} + 2586045 \nu^{13} - 9837108 \nu^{12} + 30727469 \nu^{11} + \cdots - 650790 ) / 17095 $ (62302v^15 - 444121v^14 + 2586045v^13 - 9837108v^12 + 30727469v^11 - 74534025v^10 + 150015484v^9 - 244231188v^8 + 328041743v^7 - 355265301v^6 + 308644322v^5 - 207543210v^4 + 104230233v^3 - 35964362v^2 + 7564277*v - 650790) / 17095
$\beta_{13}$	$=$	$ ( - 79057 \nu^{15} + 557291 \nu^{14} - 3236390 \nu^{13} + 12221918 \nu^{12} - 37999729 \nu^{11} + \cdots + 386570 ) / 17095 $ (-79057v^15 + 557291v^14 - 3236390v^13 + 12221918v^12 - 37999729v^11 + 91514110v^10 - 182983279v^9 + 295319043v^8 - 392920783v^7 + 420375811v^6 - 359811142v^5 + 237122470v^4 - 115715523v^3 + 38043622v^2 - 7221182*v + 386570) / 17095
$\beta_{14}$	$=$	$ ( - 99326 \nu^{15} + 728639 \nu^{14} - 4265385 \nu^{13} + 16500310 \nu^{12} - 52023445 \nu^{11} + \cdots + 1544670 ) / 17095 $ (-99326v^15 + 728639v^14 - 4265385v^13 + 16500310v^12 - 52023445v^11 + 128096911v^10 - 261028126v^9 + 432001290v^8 - 589819051v^7 + 652095757v^6 - 579766186v^5 + 401566936v^4 - 209122475v^3 + 75774262v^2 - 16856521*v + 1544670) / 17095
$\beta_{15}$	$=$	$ ( - 162552 \nu^{15} + 1183635 \nu^{14} - 6913620 \nu^{13} + 26621820 \nu^{12} - 83662434 \nu^{11} + \cdots + 2721635 ) / 17095 $ (-162552v^15 + 1183635v^14 - 6913620v^13 + 26621820v^12 - 83662434v^11 + 205121880v^10 - 416302560v^9 + 685838345v^8 - 931919096v^7 + 1025248280v^6 - 906867386v^5 + 625568415v^4 - 324976348v^3 + 118357235v^2 - 26928904*v + 2721635) / 17095

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

$\nu$	$=$	$ ( 2 \beta_{15} - 2 \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + \cdots + 3 ) / 6 $ (2b15 - 2b14 - b13 + b12 - 2b11 + 2b10 - 3b9 + 3b7 - b6 + b5 - b3 - b2 + 3) / 6
$\nu^{2}$	$=$	$ ( 2 \beta_{15} - 4 \beta_{14} - 4 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{8} + \cdots - 12 ) / 6 $ (2b15 - 4b14 - 4b13 - 2b12 - 2b11 - 2b10 - b8 + 6b7 - 4b6 + 2b5 + b4 - b3 + 4b2 - 12) / 6
$\nu^{3}$	$=$	$ ( - 10 \beta_{15} + 6 \beta_{14} + 3 \beta_{13} - 9 \beta_{12} + 9 \beta_{11} - 9 \beta_{10} + 21 \beta_{9} + \cdots - 15 ) / 6 $ (-10b15 + 6b14 + 3b13 - 9b12 + 9b11 - 9b10 + 21b9 - 6b8 - 12b7 - 3b6 - 6b5 - 3b4 + 5b3 + 15b2 - 9*b1 - 15) / 6
$\nu^{4}$	$=$	$ ( - 11 \beta_{15} + 16 \beta_{14} + 16 \beta_{13} + 2 \beta_{12} + 8 \beta_{11} + 8 \beta_{10} + \cdots + 27 ) / 3 $ (-11b15 + 16b14 + 16b13 + 2b12 + 8b11 + 8b10 + 9b9 - 27b7 + 10b6 - 8b5 - 9b4 + b3 - 4b2 - 9*b1 + 27) / 3
$\nu^{5}$	$=$	$ ( 43 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 68 \beta_{12} - 61 \beta_{11} + 61 \beta_{10} - 138 \beta_{9} + \cdots + 108 ) / 6 $ (43b15 + 2b14 + 4b13 + 68b12 - 61b11 + 61b10 - 138b9 + 55b8 + 33b7 + 40b6 + 35b5 + 5b4 - 44b3 - 110b2 + 75*b1 + 108) / 6
$\nu^{6}$	$=$	$ ( 185 \beta_{15} - 204 \beta_{14} - 246 \beta_{13} + 42 \beta_{12} - 174 \beta_{11} - 114 \beta_{10} + \cdots - 300 ) / 6 $ (185b15 - 204b14 - 246b13 + 42b12 - 174b11 - 114b10 - 258b9 + 70b8 + 438b7 - 108b6 + 132b5 + 155b4 - 25b3 - 66b2 + 270*b1 - 300) / 6
$\nu^{7}$	$=$	$ ( - 128 \beta_{15} - 229 \beta_{14} - 281 \beta_{13} - 460 \beta_{12} + 332 \beta_{11} - 542 \beta_{10} + \cdots - 921 ) / 6 $ (-128b15 - 229b14 - 281b13 - 460b12 + 332b11 - 542b10 + 801b9 - 364b8 + 207b7 - 395b6 - 193b5 + 112b4 + 379b3 + 697b2 - 378*b1 - 921) / 6
$\nu^{8}$	$=$	$ ( - 714 \beta_{15} + 596 \beta_{14} + 800 \beta_{13} - 392 \beta_{12} + 832 \beta_{11} + 280 \beta_{10} + \cdots + 801 ) / 3 $ (-714b15 + 596b14 + 800b13 - 392b12 + 832b11 + 280b10 + 1338b9 - 486b8 - 1560b7 + 236b6 - 580b5 - 564b4 + 252b3 + 592b2 - 1407*b1 + 801) / 3
$\nu^{9}$	$=$	$ ( - 379 \beta_{15} + 2835 \beta_{14} + 3789 \beta_{13} + 2682 \beta_{12} - 1071 \beta_{11} + 4779 \beta_{10} + \cdots + 8175 ) / 6 $ (-379b15 + 2835b14 + 3789b13 + 2682b12 - 1071b11 + 4779b10 - 3675b9 + 1806b8 - 4830b7 + 3483b6 + 630b5 - 1968b4 - 2740b3 - 3879b2 + 27*b1 + 8175) / 6
$\nu^{10}$	$=$	$ ( 10223 \beta_{15} - 5966 \beta_{14} - 8390 \beta_{13} + 8666 \beta_{12} - 14032 \beta_{11} - 400 \beta_{10} + \cdots - 5592 ) / 6 $ (10223b15 - 5966b14 - 8390b13 + 8666b12 - 14032b11 - 400b10 - 24006b9 + 9581b8 + 18930b7 - 434b6 + 9724b5 + 7024b4 - 6664b3 - 12610b2 + 23220*b1 - 5592) / 6
$\nu^{11}$	$=$	$ ( 12443 \beta_{15} - 27044 \beta_{14} - 37504 \beta_{13} - 11792 \beta_{12} - 4961 \beta_{11} + \cdots - 68274 ) / 6 $ (12443b15 - 27044b14 - 37504b13 - 11792b12 - 4961b11 - 38335b10 + 5778b9 - 3993b8 + 56559b7 - 27583b6 + 3724b5 + 22341b4 + 15806b3 + 16604b2 + 24783*b1 - 68274) / 6
$\nu^{12}$	$=$	$ ( - 32900 \beta_{15} + 9264 \beta_{14} + 13392 \beta_{13} - 39132 \beta_{12} + 52512 \beta_{11} + \cdots - 9414 ) / 3 $ (-32900b15 + 9264b14 + 13392b13 - 39132b12 + 52512b11 - 16332b10 + 95841b9 - 39501b8 - 44046b7 - 11526b6 - 36684b5 - 16731b4 + 34369b3 + 55908b2 - 79695*b1 - 9414) / 3
$\nu^{13}$	$=$	$ ( - 158381 \beta_{15} + 224389 \beta_{14} + 315527 \beta_{13} + 12610 \beta_{12} + 140842 \beta_{11} + \cdots + 514725 ) / 6 $ (-158381b15 + 224389b14 + 315527b13 + 12610b12 + 140842b11 + 271076b10 + 140085b9 - 50609b8 - 525717b7 + 193280b6 - 99368b5 - 208546b4 - 57875b3 - 16423b2 - 362934*b1 + 514725) / 6
$\nu^{14}$	$=$	$ ( 352742 \beta_{15} + 79100 \beta_{14} + 108212 \beta_{13} + 616330 \beta_{12} - 683318 \beta_{11} + \cdots + 639942 ) / 6 $ (352742b15 + 79100b14 + 108212b13 + 616330b12 - 683318b11 + 520114b10 - 1355046b9 + 563485b8 + 149892b7 + 369170b6 + 480440b5 + 56225b4 - 600613b3 - 873992b2 + 881244*b1 + 639942) / 6
$\nu^{15}$	$=$	$ ( 1567315 \beta_{15} - 1650306 \beta_{14} - 2331621 \beta_{13} + 516459 \beta_{12} - 1773543 \beta_{11} + \cdots - 3396279 ) / 6 $ (1567315b15 - 1650306b14 - 2331621b13 + 516459b12 - 1773543b11 - 1616775b10 - 2418363b9 + 968762b8 + 4219254b7 - 1147140b6 + 1250523b5 + 1689964b4 - 142790b3 - 737517b2 + 3753099*b1 - 3396279) / 6

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

Character values

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

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

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} $

193.1

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

−1.65068

0.524648i

−1.25529

−

0.724745i

0.642559

1.11295i

2.44949

−

1.73205i

193.2

−1.65068

0.524648i

1.25529

0.724745i

−0.642559

−

1.11295i

2.44949

−

1.73205i

193.3

−0.524648

1.65068i

−2.98735

−

1.72474i

2.02166

3.50162i

−2.44949

−

1.73205i

193.4

−0.524648

1.65068i

2.98735

1.72474i

−2.02166

−

3.50162i

−2.44949

−

1.73205i

193.5

0.524648

−

1.65068i

−2.98735

−

1.72474i

−2.02166

−

3.50162i

−2.44949

−

1.73205i

193.6

0.524648

−

1.65068i

2.98735

1.72474i

2.02166

3.50162i

−2.44949

−

1.73205i

193.7

1.65068

−

0.524648i

−1.25529

−

0.724745i

−0.642559

−

1.11295i

2.44949

−

1.73205i

193.8

1.65068

−

0.524648i

1.25529

0.724745i

0.642559

1.11295i

2.44949

−

1.73205i

961.1

−1.65068

−

0.524648i

−1.25529

0.724745i

0.642559

−

1.11295i

2.44949

1.73205i

961.2

−1.65068

−

0.524648i

1.25529

−

0.724745i

−0.642559

1.11295i

2.44949

1.73205i

961.3

−0.524648

−

1.65068i

−2.98735

1.72474i

2.02166

−

3.50162i

−2.44949

1.73205i

961.4

−0.524648

−

1.65068i

2.98735

−

1.72474i

−2.02166

3.50162i

−2.44949

1.73205i

961.5

0.524648

1.65068i

−2.98735

1.72474i

−2.02166

3.50162i

−2.44949

1.73205i

961.6

0.524648

1.65068i

2.98735

−

1.72474i

2.02166

−

3.50162i

−2.44949

1.73205i

961.7

1.65068

0.524648i

−1.25529

0.724745i

−0.642559

1.11295i

2.44949

1.73205i

961.8

1.65068

0.524648i

1.25529

−

0.724745i

0.642559

−

1.11295i

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.c	even	3	1	inner
36.f	odd	6	1	inner
72.n	even	6	1	inner
72.p	odd	6	1	inner

Twists

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

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

Hecke kernels

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

$ T_{5}^{8} - 14T_{5}^{6} + 171T_{5}^{4} - 350T_{5}^{2} + 625 $

$ T_{7}^{8} + 18T_{7}^{6} + 297T_{7}^{4} + 486T_{7}^{2} + 729 $

$ T_{11}^{8} - 18T_{11}^{6} + 297T_{11}^{4} - 486T_{11}^{2} + 729 $

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} - 14 T^{6} + \cdots + 625)^{2} $ (T^8 - 14T^6 + 171T^4 - 350*T^2 + 625)^2
$7$	$ (T^{8} + 18 T^{6} + \cdots + 729)^{2} $ (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$11$	$ (T^{8} - 18 T^{6} + \cdots + 729)^{2} $ (T^8 - 18T^6 + 297T^4 - 486*T^2 + 729)^2
$13$	$ (T^{4} - T^{2} + 1)^{4} $ (T^4 - T^2 + 1)^4
$17$	$ (T^{2} - 8 T + 10)^{8} $ (T^2 - 8*T + 10)^8
$19$	$ T^{16} $ T^16
$23$	$ (T^{8} + 54 T^{6} + \cdots + 59049)^{2} $ (T^8 + 54T^6 + 2673T^4 + 13122*T^2 + 59049)^2
$29$	$ (T^{8} - 50 T^{6} + \cdots + 279841)^{2} $ (T^8 - 50T^6 + 1971T^4 - 26450*T^2 + 279841)^2
$31$	$ (T^{8} + 54 T^{6} + \cdots + 455625)^{2} $ (T^8 + 54T^6 + 2241T^4 + 36450*T^2 + 455625)^2
$37$	$ (T^{4} + 84 T^{2} + 900)^{4} $ (T^4 + 84*T^2 + 900)^4
$41$	$ (T^{4} + 2 T^{3} + \cdots + 2809)^{4} $ (T^4 + 2T^3 + 57T^2 - 106*T + 2809)^4
$43$	$ (T^{8} - 198 T^{6} + \cdots + 95004009)^{2} $ (T^8 - 198T^6 + 29457T^4 - 1929906*T^2 + 95004009)^2
$47$	$ (T^{8} + 18 T^{6} + \cdots + 729)^{2} $ (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$53$	$ (T^{4} + 140 T^{2} + 1444)^{4} $ (T^4 + 140*T^2 + 1444)^4
$59$	$ (T^{8} - 54 T^{6} + \cdots + 59049)^{2} $ (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$61$	$ (T^{8} - 66 T^{6} + \cdots + 50625)^{2} $ (T^8 - 66T^6 + 4131T^4 - 14850*T^2 + 50625)^2
$67$	$ (T^{8} - 306 T^{6} + \cdots + 204004089)^{2} $ (T^8 - 306T^6 + 79353T^4 - 4370598*T^2 + 204004089)^2
$71$	$ (T^{4} - 396 T^{2} + 38988)^{4} $ (T^4 - 396*T^2 + 38988)^4
$73$	$ (T^{2} - 12 T + 30)^{8} $ (T^2 - 12*T + 30)^8
$79$	$ (T^{8} + 342 T^{6} + \cdots + 515607849)^{2} $ (T^8 + 342T^6 + 94257T^4 + 7765794*T^2 + 515607849)^2
$83$	$ (T^{8} - 54 T^{6} + \cdots + 59049)^{2} $ (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$89$	$ (T^{2} - 4 T - 2)^{8} $ (T^2 - 4*T - 2)^8
$97$	$ (T^{4} + 18 T^{3} + \cdots + 729)^{4} $ (T^4 + 18T^3 + 297T^2 + 486*T + 729)^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 $\F_{q}$

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

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{13} + \beta_{11} + \beta_{6}) q^{7} + (\beta_{3} + 2 \beta_1 - 1) q^{9}+O(q^{10}) \) q + b2 * q^3 + (b7 + b4) * q^5 + (b13 + b11 + b6) * q^7 + (b3 + 2b1 - 1) q^9	\( q + \beta_{2} q^{3} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{13} + \beta_{11} + \beta_{6}) q^{7} + (\beta_{3} + 2 \beta_1 - 1) q^{9} + (\beta_{10} + \beta_{5} - \beta_{2}) q^{11} - \beta_{7} q^{13} + (2 \beta_{12} - \beta_{11} + \beta_{6}) q^{15} + ( - \beta_{3} + 4) q^{17} + (3 \beta_{9} - \beta_{8} + \beta_{4}) q^{21} + ( - 2 \beta_{13} - \beta_{12} + \cdots + \beta_{6}) q^{23}+ \cdots + ( - 3 \beta_{14} - 3 \beta_{10} + 3 \beta_{2}) q^{99}+O(q^{100}) \) q + b2 * q^3 + (b7 + b4) * q^5 + (b13 + b11 + b6) * q^7 + (b3 + 2b1 - 1) q^9 + (b10 + b5 - b2) * q^11 - b7 * q^13 + (2b12 - b11 + b6) q^15 + (-b3 + 4) * q^17 + (3b9 - b8 + b4) q^21 + (-2b13 - b12 - b11 + b6) q^23 + (-2b15 - 2b1 + 2) * q^25 + (-b14 + 2b10 + 2b5 - b2) * q^27 + (b9 - 2b8) q^29 + (2b13 - b12 + 3b11 - b6) * q^31 + (-2b15 + b3 - 3b1 + 3) * q^33 + (-2b14 - 2b10 + 3b5 - 3b2) * q^35 + (6b9 + b8 - 6b7 + b4) * q^37 + b11 * q^39 + (-3b15 + 3b3 - b1) * q^41 + (b14 - 3b10 - 4b5 + b2) * q^43 + (2b9 - 2b8 - 7b7 - 2b4) * q^45 + (-b13 - b11 - b6) * q^47 + (-3b15 + 3b3 - 2b1) q^49 + (b14 - 2b5 + 4b2) * q^51 + (4b9 + 3b8 - 4b7 + 3b4) * q^53 + (-3b13 - 2b12 + 2b11 - 3b6) * q^55 + (b14 - b10 + b5 + 2b2) q^59 + (-3b9 - 2b8) * q^61 + (-3b13 + 3b12 - 3b11) q^63 + (b15 + b1 - 1) * q^65 + (3b14 + 7b10 - 2b5 - 4b2) * q^67 + (6b9 + 3b8 - 3b7 + 3b4) * q^69 + (2b13 - b12 - 6b11 + 2b6) q^71 + (b3 + 6) * q^73 + (-2b14 - 2b10 - 2b5 + 2b2) * q^75 + (-9b7 - 3b4) * q^77 + (7b13 - 3b12 + b11 - 2b6) q^79 + (-4b15 + 2b3 + 3) * q^81 + (b14 - b10 - 2b5 - b2) q^83 + (10b7 + 5b4) * q^85 + (-b13 + 2b12 - b11 - 2b6) * q^87 + (-b3 + 2) * q^89 + (b14 - b5 + b2) * q^91 + (-6b9 - b8 + 9b7 + b4) * q^93 + (3b15 + 9b1 - 9) * q^97 + (-3b14 - 3b10 + 3b2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 64 q^{17} + 16 q^{25} + 24 q^{33} - 8 q^{41} - 16 q^{49} - 8 q^{65} + 96 q^{73} + 48 q^{81} + 32 q^{89} - 72 q^{97}+O(q^{100}) \) 16 * q + 64 * q^17 + 16 * q^25 + 24 * q^33 - 8 * q^41 - 16 * q^49 - 8 * q^65 + 96 * q^73 + 48 * q^81 + 32 * q^89 - 72 * 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.r.f

Level	$1152$
Weight	$2$
Character orbit	1152.r
Analytic conductor	$9.199$
Analytic rank	$0$
Dimension	$16$
CM	no
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:	\( 2^{6}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 3456 \nu^{15} + 25920 \nu^{14} - 151876 \nu^{13} + 594074 \nu^{12} - 1879372 \nu^{11} + \cdots + 142555 ) / 17095 \) (-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
\(\beta_{2}\)	\(=\)	\( ( - 13659 \nu^{15} + 79956 \nu^{14} - 449175 \nu^{13} + 1476715 \nu^{12} - 4234710 \nu^{11} + \cdots - 486085 ) / 17095 \) (-13659v^15 + 79956v^14 - 449175v^13 + 1476715v^12 - 4234710v^11 + 8664904v^10 - 14862759v^9 + 18235965v^8 - 16679044v^7 + 6722523v^6 + 5047356v^5 - 13391846v^4 + 12973260v^3 - 7799187v^2 + 2711261*v - 486085) / 17095
\(\beta_{3}\)	\(=\)	\( ( - 54 \nu^{14} + 378 \nu^{13} - 2193 \nu^{12} + 8244 \nu^{11} - 25569 \nu^{10} + 61284 \nu^{9} + \cdots - 3308 ) / 13 \) (-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
\(\beta_{4}\)	\(=\)	\( ( 22230 \nu^{15} - 144633 \nu^{14} + 820486 \nu^{13} - 2915351 \nu^{12} + 8665028 \nu^{11} + \cdots + 599315 ) / 17095 \) (22230v^15 - 144633v^14 + 820486v^13 - 2915351v^12 + 8665028v^11 - 19406977v^10 + 35943838v^9 - 51803400v^8 + 59433982v^7 - 50057927v^6 + 27990758v^5 - 4701824v^4 - 7461590v^3 + 7646662v^2 - 3312642*v + 599315) / 17095
\(\beta_{5}\)	\(=\)	\( ( 13659 \nu^{15} - 141235 \nu^{14} + 878128 \nu^{13} - 3969692 \nu^{12} + 13616183 \nu^{11} + \cdots - 3568060 ) / 17095 \) (13659v^15 - 141235v^14 + 878128v^13 - 3969692v^12 + 13616183v^11 - 37825818v^10 + 84893873v^9 - 158094894v^8 + 241598485v^7 - 305000028v^6 + 311903675v^5 - 255430974v^4 + 161068568v^3 - 74625539v^2 + 22634049*v - 3568060) / 17095
\(\beta_{6}\)	\(=\)	\( ( 16755 \nu^{15} - 161299 \nu^{14} + 987248 \nu^{13} - 4333114 \nu^{12} + 14582345 \nu^{11} + \cdots - 2399970 ) / 17095 \) (16755v^15 - 161299v^14 + 987248v^13 - 4333114v^12 + 14582345v^11 - 39558898v^10 + 86882269v^9 - 157807206v^8 + 234915641v^7 - 287693722v^6 + 284271347v^5 - 223504414v^4 + 134281042v^3 - 58660817v^2 + 16543083*v - 2399970) / 17095
\(\beta_{7}\)	\(=\)	\( ( - 26630 \nu^{15} + 182630 \nu^{14} - 1056740 \nu^{13} + 3923413 \nu^{12} - 12097498 \nu^{11} + \cdots - 218885 ) / 17095 \) (-26630v^15 + 182630v^14 - 1056740v^13 + 3923413v^12 - 12097498v^11 + 28666706v^10 - 56590650v^9 + 89528958v^8 - 116711850v^7 + 121073088v^6 - 99711814v^5 + 61611091v^4 - 27082796v^3 + 6949514v^2 - 510382*v - 218885) / 17095
\(\beta_{8}\)	\(=\)	\( ( 22230 \nu^{15} - 188817 \nu^{14} + 1129774 \nu^{13} - 4704014 \nu^{12} + 15376262 \nu^{11} + \cdots - 1317955 ) / 17095 \) (22230v^15 - 188817v^14 + 1129774v^13 - 4704014v^12 + 15376262v^11 - 40133218v^10 + 85426762v^9 - 149690685v^8 + 215272528v^7 - 253642808v^6 + 240583652v^5 - 180367151v^4 + 102603910v^3 - 41706077v^2 + 10736292*v - 1317955) / 17095
\(\beta_{9}\)	\(=\)	\( ( 26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} + \cdots - 2071845 ) / 17095 \) (26630v^15 - 216820v^14 + 1296070v^13 - 5311527v^12 + 17314892v^11 - 44845414v^10 + 95362110v^9 - 166733397v^8 + 240513840v^7 - 284726942v^6 + 273082466v^5 - 208344314v^4 + 122001074v^3 - 52073476v^2 + 14507768*v - 2071845) / 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
\(\beta_{11}\)	\(=\)	\( ( - 31151 \nu^{15} + 280841 \nu^{14} - 1704486 \nu^{13} + 7300282 \nu^{12} - 24305077 \nu^{11} + \cdots + 3720725 ) / 17095 \) (-31151v^15 + 280841v^14 - 1704486v^13 + 7300282v^12 - 24305077v^11 + 64922646v^10 - 141130150v^9 + 253301046v^8 - 373571776v^7 + 453043752v^6 - 444138430v^5 + 346560318v^4 - 207159008v^3 + 90144910v^2 - 25539997*v + 3720725) / 17095
\(\beta_{12}\)	\(=\)	\( ( 62302 \nu^{15} - 444121 \nu^{14} + 2586045 \nu^{13} - 9837108 \nu^{12} + 30727469 \nu^{11} + \cdots - 650790 ) / 17095 \) (62302v^15 - 444121v^14 + 2586045v^13 - 9837108v^12 + 30727469v^11 - 74534025v^10 + 150015484v^9 - 244231188v^8 + 328041743v^7 - 355265301v^6 + 308644322v^5 - 207543210v^4 + 104230233v^3 - 35964362v^2 + 7564277*v - 650790) / 17095
\(\beta_{13}\)	\(=\)	\( ( - 79057 \nu^{15} + 557291 \nu^{14} - 3236390 \nu^{13} + 12221918 \nu^{12} - 37999729 \nu^{11} + \cdots + 386570 ) / 17095 \) (-79057v^15 + 557291v^14 - 3236390v^13 + 12221918v^12 - 37999729v^11 + 91514110v^10 - 182983279v^9 + 295319043v^8 - 392920783v^7 + 420375811v^6 - 359811142v^5 + 237122470v^4 - 115715523v^3 + 38043622v^2 - 7221182*v + 386570) / 17095
\(\beta_{14}\)	\(=\)	\( ( - 99326 \nu^{15} + 728639 \nu^{14} - 4265385 \nu^{13} + 16500310 \nu^{12} - 52023445 \nu^{11} + \cdots + 1544670 ) / 17095 \) (-99326v^15 + 728639v^14 - 4265385v^13 + 16500310v^12 - 52023445v^11 + 128096911v^10 - 261028126v^9 + 432001290v^8 - 589819051v^7 + 652095757v^6 - 579766186v^5 + 401566936v^4 - 209122475v^3 + 75774262v^2 - 16856521*v + 1544670) / 17095
\(\beta_{15}\)	\(=\)	\( ( - 162552 \nu^{15} + 1183635 \nu^{14} - 6913620 \nu^{13} + 26621820 \nu^{12} - 83662434 \nu^{11} + \cdots + 2721635 ) / 17095 \) (-162552v^15 + 1183635v^14 - 6913620v^13 + 26621820v^12 - 83662434v^11 + 205121880v^10 - 416302560v^9 + 685838345v^8 - 931919096v^7 + 1025248280v^6 - 906867386v^5 + 625568415v^4 - 324976348v^3 + 118357235v^2 - 26928904*v + 2721635) / 17095

\(\nu\)	\(=\)	\( ( 2 \beta_{15} - 2 \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + \cdots + 3 ) / 6 \) (2b15 - 2b14 - b13 + b12 - 2b11 + 2b10 - 3b9 + 3b7 - b6 + b5 - b3 - b2 + 3) / 6
\(\nu^{2}\)	\(=\)	\( ( 2 \beta_{15} - 4 \beta_{14} - 4 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{8} + \cdots - 12 ) / 6 \) (2b15 - 4b14 - 4b13 - 2b12 - 2b11 - 2b10 - b8 + 6b7 - 4b6 + 2b5 + b4 - b3 + 4b2 - 12) / 6
\(\nu^{3}\)	\(=\)	\( ( - 10 \beta_{15} + 6 \beta_{14} + 3 \beta_{13} - 9 \beta_{12} + 9 \beta_{11} - 9 \beta_{10} + 21 \beta_{9} + \cdots - 15 ) / 6 \) (-10b15 + 6b14 + 3b13 - 9b12 + 9b11 - 9b10 + 21b9 - 6b8 - 12b7 - 3b6 - 6b5 - 3b4 + 5b3 + 15b2 - 9*b1 - 15) / 6
\(\nu^{4}\)	\(=\)	\( ( - 11 \beta_{15} + 16 \beta_{14} + 16 \beta_{13} + 2 \beta_{12} + 8 \beta_{11} + 8 \beta_{10} + \cdots + 27 ) / 3 \) (-11b15 + 16b14 + 16b13 + 2b12 + 8b11 + 8b10 + 9b9 - 27b7 + 10b6 - 8b5 - 9b4 + b3 - 4b2 - 9*b1 + 27) / 3
\(\nu^{5}\)	\(=\)	\( ( 43 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 68 \beta_{12} - 61 \beta_{11} + 61 \beta_{10} - 138 \beta_{9} + \cdots + 108 ) / 6 \) (43b15 + 2b14 + 4b13 + 68b12 - 61b11 + 61b10 - 138b9 + 55b8 + 33b7 + 40b6 + 35b5 + 5b4 - 44b3 - 110b2 + 75*b1 + 108) / 6
\(\nu^{6}\)	\(=\)	\( ( 185 \beta_{15} - 204 \beta_{14} - 246 \beta_{13} + 42 \beta_{12} - 174 \beta_{11} - 114 \beta_{10} + \cdots - 300 ) / 6 \) (185b15 - 204b14 - 246b13 + 42b12 - 174b11 - 114b10 - 258b9 + 70b8 + 438b7 - 108b6 + 132b5 + 155b4 - 25b3 - 66b2 + 270*b1 - 300) / 6
\(\nu^{7}\)	\(=\)	\( ( - 128 \beta_{15} - 229 \beta_{14} - 281 \beta_{13} - 460 \beta_{12} + 332 \beta_{11} - 542 \beta_{10} + \cdots - 921 ) / 6 \) (-128b15 - 229b14 - 281b13 - 460b12 + 332b11 - 542b10 + 801b9 - 364b8 + 207b7 - 395b6 - 193b5 + 112b4 + 379b3 + 697b2 - 378*b1 - 921) / 6
\(\nu^{8}\)	\(=\)	\( ( - 714 \beta_{15} + 596 \beta_{14} + 800 \beta_{13} - 392 \beta_{12} + 832 \beta_{11} + 280 \beta_{10} + \cdots + 801 ) / 3 \) (-714b15 + 596b14 + 800b13 - 392b12 + 832b11 + 280b10 + 1338b9 - 486b8 - 1560b7 + 236b6 - 580b5 - 564b4 + 252b3 + 592b2 - 1407*b1 + 801) / 3
\(\nu^{9}\)	\(=\)	\( ( - 379 \beta_{15} + 2835 \beta_{14} + 3789 \beta_{13} + 2682 \beta_{12} - 1071 \beta_{11} + 4779 \beta_{10} + \cdots + 8175 ) / 6 \) (-379b15 + 2835b14 + 3789b13 + 2682b12 - 1071b11 + 4779b10 - 3675b9 + 1806b8 - 4830b7 + 3483b6 + 630b5 - 1968b4 - 2740b3 - 3879b2 + 27*b1 + 8175) / 6
\(\nu^{10}\)	\(=\)	\( ( 10223 \beta_{15} - 5966 \beta_{14} - 8390 \beta_{13} + 8666 \beta_{12} - 14032 \beta_{11} - 400 \beta_{10} + \cdots - 5592 ) / 6 \) (10223b15 - 5966b14 - 8390b13 + 8666b12 - 14032b11 - 400b10 - 24006b9 + 9581b8 + 18930b7 - 434b6 + 9724b5 + 7024b4 - 6664b3 - 12610b2 + 23220*b1 - 5592) / 6
\(\nu^{11}\)	\(=\)	\( ( 12443 \beta_{15} - 27044 \beta_{14} - 37504 \beta_{13} - 11792 \beta_{12} - 4961 \beta_{11} + \cdots - 68274 ) / 6 \) (12443b15 - 27044b14 - 37504b13 - 11792b12 - 4961b11 - 38335b10 + 5778b9 - 3993b8 + 56559b7 - 27583b6 + 3724b5 + 22341b4 + 15806b3 + 16604b2 + 24783*b1 - 68274) / 6
\(\nu^{12}\)	\(=\)	\( ( - 32900 \beta_{15} + 9264 \beta_{14} + 13392 \beta_{13} - 39132 \beta_{12} + 52512 \beta_{11} + \cdots - 9414 ) / 3 \) (-32900b15 + 9264b14 + 13392b13 - 39132b12 + 52512b11 - 16332b10 + 95841b9 - 39501b8 - 44046b7 - 11526b6 - 36684b5 - 16731b4 + 34369b3 + 55908b2 - 79695*b1 - 9414) / 3
\(\nu^{13}\)	\(=\)	\( ( - 158381 \beta_{15} + 224389 \beta_{14} + 315527 \beta_{13} + 12610 \beta_{12} + 140842 \beta_{11} + \cdots + 514725 ) / 6 \) (-158381b15 + 224389b14 + 315527b13 + 12610b12 + 140842b11 + 271076b10 + 140085b9 - 50609b8 - 525717b7 + 193280b6 - 99368b5 - 208546b4 - 57875b3 - 16423b2 - 362934*b1 + 514725) / 6
\(\nu^{14}\)	\(=\)	\( ( 352742 \beta_{15} + 79100 \beta_{14} + 108212 \beta_{13} + 616330 \beta_{12} - 683318 \beta_{11} + \cdots + 639942 ) / 6 \) (352742b15 + 79100b14 + 108212b13 + 616330b12 - 683318b11 + 520114b10 - 1355046b9 + 563485b8 + 149892b7 + 369170b6 + 480440b5 + 56225b4 - 600613b3 - 873992b2 + 881244*b1 + 639942) / 6
\(\nu^{15}\)	\(=\)	\( ( 1567315 \beta_{15} - 1650306 \beta_{14} - 2331621 \beta_{13} + 516459 \beta_{12} - 1773543 \beta_{11} + \cdots - 3396279 ) / 6 \) (1567315b15 - 1650306b14 - 2331621b13 + 516459b12 - 1773543b11 - 1616775b10 - 2418363b9 + 968762b8 + 4219254b7 - 1147140b6 + 1250523b5 + 1689964b4 - 142790b3 - 737517b2 + 3753099*b1 - 3396279) / 6

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 193.8
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
$5$	\( (T^{8} - 14 T^{6} + \cdots + 625)^{2} \) (T^8 - 14T^6 + 171T^4 - 350*T^2 + 625)^2
$7$	\( (T^{8} + 18 T^{6} + \cdots + 729)^{2} \) (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$11$	\( (T^{8} - 18 T^{6} + \cdots + 729)^{2} \) (T^8 - 18T^6 + 297T^4 - 486*T^2 + 729)^2
$13$	\( (T^{4} - T^{2} + 1)^{4} \) (T^4 - T^2 + 1)^4
$17$	\( (T^{2} - 8 T + 10)^{8} \) (T^2 - 8*T + 10)^8
$19$	\( T^{16} \) T^16
$23$	\( (T^{8} + 54 T^{6} + \cdots + 59049)^{2} \) (T^8 + 54T^6 + 2673T^4 + 13122*T^2 + 59049)^2
$29$	\( (T^{8} - 50 T^{6} + \cdots + 279841)^{2} \) (T^8 - 50T^6 + 1971T^4 - 26450*T^2 + 279841)^2
$31$	\( (T^{8} + 54 T^{6} + \cdots + 455625)^{2} \) (T^8 + 54T^6 + 2241T^4 + 36450*T^2 + 455625)^2
$37$	\( (T^{4} + 84 T^{2} + 900)^{4} \) (T^4 + 84*T^2 + 900)^4
$41$	\( (T^{4} + 2 T^{3} + \cdots + 2809)^{4} \) (T^4 + 2T^3 + 57T^2 - 106*T + 2809)^4
$43$	\( (T^{8} - 198 T^{6} + \cdots + 95004009)^{2} \) (T^8 - 198T^6 + 29457T^4 - 1929906*T^2 + 95004009)^2
$47$	\( (T^{8} + 18 T^{6} + \cdots + 729)^{2} \) (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$53$	\( (T^{4} + 140 T^{2} + 1444)^{4} \) (T^4 + 140*T^2 + 1444)^4
$59$	\( (T^{8} - 54 T^{6} + \cdots + 59049)^{2} \) (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$61$	\( (T^{8} - 66 T^{6} + \cdots + 50625)^{2} \) (T^8 - 66T^6 + 4131T^4 - 14850*T^2 + 50625)^2
$67$	\( (T^{8} - 306 T^{6} + \cdots + 204004089)^{2} \) (T^8 - 306T^6 + 79353T^4 - 4370598*T^2 + 204004089)^2
$71$	\( (T^{4} - 396 T^{2} + 38988)^{4} \) (T^4 - 396*T^2 + 38988)^4
$73$	\( (T^{2} - 12 T + 30)^{8} \) (T^2 - 12*T + 30)^8
$79$	\( (T^{8} + 342 T^{6} + \cdots + 515607849)^{2} \) (T^8 + 342T^6 + 94257T^4 + 7765794*T^2 + 515607849)^2
$83$	\( (T^{8} - 54 T^{6} + \cdots + 59049)^{2} \) (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$89$	\( (T^{2} - 4 T - 2)^{8} \) (T^2 - 4*T - 2)^8
$97$	\( (T^{4} + 18 T^{3} + \cdots + 729)^{4} \) (T^4 + 18T^3 + 297T^2 + 486*T + 729)^4
show more
show less