LMFDB - Newform orbit 405.2.f.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(242,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("405.242");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.f (of order $4$, 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:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 $ x^16 - 6x^15 + 18x^14 - 36x^13 + 34x^12 + 18x^11 - 72x^10 + 132x^9 - 93x^8 - 102x^7 + 144x^6 - 432x^5 + 502x^4 + 288x^3 + 72x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{3} q^{2} + ( - \beta_{14} + \beta_{6}) q^{4} - \beta_{4} q^{5} + \beta_{10} q^{7} + ( - \beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{8}+O(q^{10}) $ q + b3 * q^2 + (-b14 + b6) * q^4 - b4 * q^5 + b10 * q^7 + (-b11 - b8 - b4 + b2) * q^8	$ q + \beta_{3} q^{2} + ( - \beta_{14} + \beta_{6}) q^{4} - \beta_{4} q^{5} + \beta_{10} q^{7} + ( - \beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{8}+ \cdots + ( - \beta_{11} - \beta_{8} + \cdots - 4 \beta_{2}) q^{98}+O(q^{100}) $ q + b3 * q^2 + (-b14 + b6) * q^4 - b4 * q^5 + b10 * q^7 + (-b11 - b8 - b4 + b2) * q^8 + (-b13 + b10 + b9 + b6 + b1 - 1) * q^10 + (-b11 + b5 - b4 - b3 + b2) * q^11 + (b14 - b12 + b9 + b1) * q^13 + (-b8 + b7) * q^14 + (-b13 - b12 + 1) * q^16 + (-b15 - b11 + b8 + b5 - b4) * q^17 + (-b14 + b13 - b12 - b10 - b9) * q^19 + (b11 - 2b8 - b5) q^20 + (-b13 + b10 - b6 + 1) * q^22 + (-b11 + b8 - 2b7 + b5 + b4 + b2) q^23 + (b13 - b12 - b10 + b9 + 2b1 - 1) q^25 + (2b15 + 3b11 - b5 + 3b4 + b3 - b2) q^26 + (-b12 - b6 - 1) * q^28 + (2b8 + b7 - b3 - b2) q^29 + (2b10 - 2b9 - b1) * q^31 + (-b11 + b7 + b3) * q^32 + (-3b14 - b13 + b12 + 4b6) * q^34 + (-b15 + 2b11 + b8 - b3 - 2b2) * q^35 + (b14 - b13 + 3b10 - b1) q^37 + (2b8 - 2b7 - b5 + 2b4 - 2b2) * q^38 + (b14 - 2b10 + b9 - 3b6 + 2b1 - 1) q^40 + (-b15 + 2b11 - 2b5 + b4) * q^41 + (b14 + b12 + b9 + b1) * q^43 + (b15 - b8 + 2b7 - b5) q^44 + (b13 + b12 - b10 + b9 - b1 + 3) * q^46 + (-b15 - b8 + b7 - b5 + b4 - b3) * q^47 + (-b14 + 4b6) q^49 + (2b15 + 3b11 + b8 - 2b7 - b5 + 3b4 - b3 - 2b2) q^50 + (2b14 + b13 - b10 - 3b6 - 2b1 + 3) q^52 + (-2b11 + 2b8 - 4b7 + 2b4) * q^53 + (b13 + b12 - 2b10 + b6 + b1 - 1) q^55 + (b15 + b11 + b5) * q^56 + (-2b14 + 2b12 + b9 + 2b6 - 2b1 + 2) * q^58 + (-b15 + 2b8 + 2b7 + b5 - b3 - b2) * q^59 + (3b10 - 3b9 - 3b1 - 1) q^61 + (-b15 - b11 - b8 + 2b7 - b5 + b4) q^62 + (-3b14 - 2b13 + 2b12 - b10 - b9 + 3b6) * q^64 + (b15 + 3b11 + b8 - 2b7 + b5 + b4 + 3b3 + b2) q^65 + (2b14 + 3b13 - 2b10 - b6 - 2b1 + 1) * q^67 + (-2b11 - 3b8 + b7 - 3b4 - 2b2) * q^68 + (b14 + b12 + b10 - 3b6 - b1 - 2) q^70 + (-2b15 - b11 + b5 - 3b4 - 2b3 + 2b2) * q^71 + (3b14 - 2b6 + 3b1 - 2) q^73 + (-3b8 + 4b7 + b3 + b2) * q^74 + (-b10 + b9 - b1 - 2) * q^76 + (b15 - b11 + b8 + b5 - b4 - 2b3) q^77 + (2b14 + b13 - b12 - 3b10 - 3b9 - 4b6) * q^79 + (-b11 + b8 - 4b7 + 3b5 - 2b4 - b3 + 3b2) * q^80 + (b14 + b13 - 2b10 - 3b6 - b1 + 3) * q^82 + (-4b11 - b8 - 3b7 + b5 - b4 + 3b2) q^83 + (-2b14 - b13 - b9 + 2b6 + 2b1 - 4) q^85 + (b11 + b5 - b4 - b3 + b2) * q^86 + (b12 + b9 - 2b6 - 2) q^88 + (b15 - 4b8 + 5b7 - b5 + 2b3 + 2b2) * q^89 + (2b10 - 2b9 - 3b1 - 2) q^91 + (b15 - 4b11 + 2b8 + 2b7 + 2b5 - 2b4 - b3) q^92 + (b13 - b12 - 3b10 - 3b9 - 5b6) q^94 + (-2b15 + b11 - 3b8 + b7 - 2b4 + 2b3 + 4b2) q^95 + (-b14 + 4b6 + b1 - 4) q^97 + (-b11 - b8 - b4 - 4b2) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{7}+O(q^{10}) $ 16 * q + 4 * q^7	$ 16 q + 4 q^{7} - 8 q^{10} + 4 q^{13} + 16 q^{16} + 20 q^{22} - 8 q^{25} - 16 q^{28} + 8 q^{31} + 4 q^{37} - 12 q^{40} + 4 q^{43} + 32 q^{46} + 28 q^{52} - 16 q^{55} + 12 q^{58} - 16 q^{61} - 8 q^{67} - 36 q^{70} - 8 q^{73} - 48 q^{76} + 32 q^{82} - 44 q^{85} - 36 q^{88} - 40 q^{91} - 56 q^{97}+O(q^{100}) $ 16 * q + 4 * q^7 - 8 * q^10 + 4 * q^13 + 16 * q^16 + 20 * q^22 - 8 * q^25 - 16 * q^28 + 8 * q^31 + 4 * q^37 - 12 * q^40 + 4 * q^43 + 32 * q^46 + 28 * q^52 - 16 * q^55 + 12 * q^58 - 16 * q^61 - 8 * q^67 - 36 * q^70 - 8 * q^73 - 48 * q^76 + 32 * q^82 - 44 * q^85 - 36 * q^88 - 40 * q^91 - 56 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 $ x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 34*x^12 + 18*x^11 - 72*x^10 + 132*x^9 - 93*x^8 - 102*x^7 + 144*x^6 - 432*x^5 + 502*x^4 + 288*x^3 + 72*x^2 + 12*x + 1 :

$\beta_{1}$	$=$	$ ( 73568941 \nu^{15} + 2828429898 \nu^{14} - 10706428236 \nu^{13} + 20127265496 \nu^{12} + \cdots + 10982429182739 ) / 3707507912227 $ (73568941v^15 + 2828429898v^14 - 10706428236v^13 + 20127265496v^12 - 35741021050v^11 + 17927851065v^10 - 58772639836v^9 + 308269732922v^8 - 50382616322v^7 + 35812385940v^6 + 245517720058v^5 - 1525793905272v^4 - 747026251899v^3 - 178512937629v^2 - 28424647318*v + 10982429182739) / 3707507912227
$\beta_{2}$	$=$	$ ( - 94386116202 \nu^{15} + 619740656932 \nu^{14} - 2033008548312 \nu^{13} + \cdots + 80029143512 ) / 3707507912227 $ (-94386116202v^15 + 619740656932v^14 - 2033008548312v^13 + 4441986378774v^12 - 5386888154268v^11 + 640680728681v^10 + 7214824090311v^9 - 16443022873810v^8 + 16824695358916v^7 + 2692744559593v^6 - 17484917305182v^5 + 49690281510088v^4 - 74318822560500v^3 + 6157027329225v^2 + 993781902738*v + 80029143512) / 3707507912227
$\beta_{3}$	$=$	$ ( 196858530 \nu^{15} - 1212076232 \nu^{14} + 3734556994 \nu^{13} - 7676123862 \nu^{12} + \cdots + 785074438 ) / 1973128213 $ (196858530v^15 - 1212076232v^14 + 3734556994v^13 - 7676123862v^12 + 7906540260v^11 + 2288455604v^10 - 14534291253v^9 + 28282811704v^8 - 22763827488v^7 - 16547104520v^6 + 31379154942v^5 - 90050852224v^4 + 112985327352v^3 + 38916218628v^2 + 6077223668*v + 785074438) / 1973128213
$\beta_{4}$	$=$	$ ( - 458282837172 \nu^{15} + 3278355328030 \nu^{14} - 11567438085651 \nu^{13} + \cdots + 3479823499409 ) / 3707507912227 $ (-458282837172v^15 + 3278355328030v^14 - 11567438085651v^13 + 26927968536797v^12 - 37449835526942v^11 + 15597997864898v^10 + 36317007615948v^9 - 100050764445960v^8 + 123309140658256v^7 - 23651454139949v^6 - 102232920253345v^5 + 287003871762252v^4 - 480932440039679v^3 + 198024659235246v^2 + 31299923543934*v + 3479823499409) / 3707507912227
$\beta_{5}$	$=$	$ ( - 1263290532917 \nu^{15} + 8307388188784 \nu^{14} - 27291768145635 \nu^{13} + \cdots + 1071789225554 ) / 3707507912227 $ (-1263290532917v^15 + 8307388188784v^14 - 27291768145635v^13 + 59721708540564v^12 - 72662176309554v^11 + 9183517518119v^10 + 96760918658508v^9 - 221515642871632v^8 + 227173488204868v^7 + 35108102916679v^6 - 235176348728859v^5 + 665141988015058v^4 - 975637136257126v^3 + 82449382716699v^2 + 13308754547460*v + 1071789225554) / 3707507912227
$\beta_{6}$	$=$	$ ( 1616321538 \nu^{15} - 9938916556 \nu^{14} + 30594542475 \nu^{13} - 62871582510 \nu^{12} + \cdots + 10233974547 ) / 3810388399 $ (1616321538v^15 - 9938916556v^14 + 30594542475v^13 - 62871582510v^12 + 64714538406v^11 + 18641341380v^10 - 118271636061v^9 + 231117756606v^8 - 186162436800v^7 - 134499720676v^6 + 250255719435v^5 - 736991238906v^4 + 924410484114v^3 + 318408624800v^2 + 82003950372*v + 10233974547) / 3810388399
$\beta_{7}$	$=$	$ ( - 854458 \nu^{15} + 5149288 \nu^{14} - 15489103 \nu^{13} + 31002870 \nu^{12} - 29353426 \nu^{11} + \cdots - 4877403 ) / 1828267 $ (-854458v^15 + 5149288v^14 - 15489103v^13 + 31002870v^12 - 29353426v^11 - 15677868v^10 + 63062803v^9 - 114195864v^8 + 80446252v^7 + 89178588v^6 - 128785525v^5 + 370469382v^4 - 434157674v^3 - 248804552v^2 - 38956914*v - 4877403) / 1828267
$\beta_{8}$	$=$	$ ( - 315552650318 \nu^{15} + 1900582212134 \nu^{14} - 5713442004763 \nu^{13} + \cdots - 1796844761346 ) / 529643987461 $ (-315552650318v^15 + 1900582212134v^14 - 5713442004763v^13 + 11427384011211v^12 - 10792414939398v^11 - 5844691005419v^10 + 23294405344177v^9 - 42097274125416v^8 + 29554626469000v^7 + 32994050005382v^6 - 47477278412835v^5 + 136625993542464v^4 - 159897884475264v^3 - 91655155082751v^2 - 14350917254916*v - 1796844761346) / 529643987461
$\beta_{9}$	$=$	$ ( - 2712751885095 \nu^{15} + 16678491631658 \nu^{14} - 51337065544716 \nu^{13} + \cdots - 23594947704644 ) / 3707507912227 $ (-2712751885095v^15 + 16678491631658v^14 - 51337065544716v^13 + 105497880078934v^12 - 108587037241966v^11 - 31285400547667v^10 + 198545816545737v^9 - 388380189945330v^8 + 313419238953178v^7 + 224631606730413v^6 - 419929403639154v^5 + 1238858757336820v^4 - 1552826796501673v^3 - 534895302063361v^2 - 137764843521504*v - 23594947704644) / 3707507912227
$\beta_{10}$	$=$	$ ( - 2717282793429 \nu^{15} + 16707385417880 \nu^{14} - 51423315159482 \nu^{13} + \cdots - 10795341151078 ) / 3707507912227 $ (-2717282793429v^15 + 16707385417880v^14 - 51423315159482v^13 + 105664904992298v^12 - 108755702488842v^11 - 31280777936752v^10 + 198470922157507v^9 - 387930687122496v^8 + 312810984796866v^7 + 224984020905618v^6 - 418380709675378v^5 + 1236842558873240v^4 - 1553943585818787v^3 - 535172136374176v^2 - 137810381709166*v - 10795341151078) / 3707507912227
$\beta_{11}$	$=$	$ ( 1570148876 \nu^{15} - 9814610316 \nu^{14} + 30686832232 \nu^{13} - 63994473524 \nu^{12} + \cdots + 4714210963 ) / 1973128213 $ (1570148876v^15 - 9814610316v^14 + 30686832232v^13 - 63994473524v^12 + 68737309508v^11 + 12449599248v^10 - 117627630280v^9 + 236328234138v^8 - 202589468876v^7 - 114627530376v^6 + 259195647184v^5 - 741062624316v^4 + 968316440200v^3 + 226232221584v^2 + 35218281816*v + 4714210963) / 1973128213
$\beta_{12}$	$=$	$ ( 3757574245950 \nu^{15} - 23096381861488 \nu^{14} + 71089447066218 \nu^{13} + \cdots + 32644912196796 ) / 3707507912227 $ (3757574245950v^15 - 23096381861488v^14 + 71089447066218v^13 - 146083562570344v^12 + 150316650857126v^11 + 43353340898378v^10 - 274940334918912v^9 + 537780099499056v^8 - 433167757411736v^7 - 310933164972742v^6 + 581459639319606v^5 - 1716023307015740v^4 + 2147688630225299v^3 + 739900598877122v^2 + 190586429788452*v + 32644912196796) / 3707507912227
$\beta_{13}$	$=$	$ ( - 3758547730578 \nu^{15} + 23113179926128 \nu^{14} - 71146859123410 \nu^{13} + \cdots - 14937505605495 ) / 3707507912227 $ (-3758547730578v^15 + 23113179926128v^14 - 71146859123410v^13 + 146190483542135v^12 - 150476465892702v^11 - 43294531756298v^10 + 274731591096056v^9 - 536670080913120v^8 + 432843551243472v^7 + 311142270044362v^6 - 580257664123502v^5 + 1710584928995308v^4 - 2150385871006647v^3 - 740547803922626v^2 - 190689873599252*v - 14937505605495) / 3707507912227
$\beta_{14}$	$=$	$ ( 4659934396714 \nu^{15} - 28652614036898 \nu^{14} + 88198331151147 \nu^{13} + \cdots + 29503074524823 ) / 3707507912227 $ (4659934396714v^15 - 28652614036898v^14 + 88198331151147v^13 - 181242264245730v^12 + 186543596485370v^11 + 53736739030473v^10 - 340917929949083v^9 + 666259937224872v^8 - 536770650269540v^7 - 386908446130506v^6 + 721086474541709v^5 - 2124666379562946v^4 + 2664947390704218v^3 + 917928790168959v^2 + 236405919870430*v + 29503074524823) / 3707507912227
$\beta_{15}$	$=$	$ ( - 29169335295 \nu^{15} + 179860916164 \nu^{14} - 554518308281 \nu^{13} + 1139653845381 \nu^{12} + \cdots - 116597819081 ) / 22200646181 $ (-29169335295v^15 + 179860916164v^14 - 554518308281v^13 + 1139653845381v^12 - 1173093973890v^11 - 346731406534v^10 + 2175410366414v^9 - 4207778503040v^8 + 3380869821552v^7 + 2471943196468v^6 - 4657440308511v^5 + 13333326030116v^4 - 16790149734936v^3 - 5781324089814v^2 - 902821290514*v - 116597819081) / 22200646181

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

$\nu$	$=$	$ ( \beta_{14} + \beta_{10} - \beta_{6} - \beta_{3} - \beta _1 + 1 ) / 2 $ (b14 + b10 - b6 - b3 - b1 + 1) / 2
$\nu^{2}$	$=$	$ ( \beta_{15} + \beta_{14} - \beta_{8} - \beta_{7} - 3\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} ) / 2 $ (b15 + b14 - b8 - b7 - 3*b6 - b5 - b3 - b2) / 2
$\nu^{3}$	$=$	$ -\beta_{11} - \beta_{8} - \beta_{4} - 3\beta_{2} $ -b11 - b8 - b4 - 3*b2
$\nu^{4}$	$=$	$ ( - 5 \beta_{15} + \beta_{13} + \beta_{12} - 9 \beta_{11} + 3 \beta_{5} - 8 \beta_{4} + 5 \beta_{3} + \cdots + 13 ) / 2 $ (-5b15 + b13 + b12 - 9b11 + 3b5 - 8b4 + 5b3 - 5b2 - 6*b1 + 13) / 2
$\nu^{5}$	$=$	$ ( - 8 \beta_{15} + 19 \beta_{14} + 8 \beta_{13} - 7 \beta_{11} + 3 \beta_{10} + 8 \beta_{8} - \beta_{7} + \cdots + 32 ) / 2 $ (-8b15 + 19b14 + 8b13 - 7b11 + 3b10 + 8b8 - b7 - 32b6 + 8b5 - 8b4 + 11b3 - 19*b1 + 32) / 2
$\nu^{6}$	$=$	$ 33\beta_{14} + 8\beta_{13} - 8\beta_{12} - \beta_{10} - \beta_{9} - 63\beta_{6} $ 33b14 + 8b13 - 8b12 - b10 - b9 - 63b6
$\nu^{7}$	$=$	$ ( 97 \beta_{14} - 51 \beta_{12} - 41 \beta_{11} - 2 \beta_{9} - 50 \beta_{8} + 9 \beta_{7} - 169 \beta_{6} + \cdots - 169 ) / 2 $ (97b14 - 51b12 - 41b11 - 2b9 - 50b8 + 9b7 - 169b6 + b5 - 50b4 - 47b2 + 97b1 - 169) / 2
$\nu^{8}$	$=$	$ ( - 148 \beta_{15} - 50 \beta_{13} - 50 \beta_{12} - 245 \beta_{11} + 10 \beta_{10} - 10 \beta_{9} + \cdots - 323 ) / 2 $ (-148b15 - 50b13 - 50b12 - 245b11 + 10b10 - 10b9 + 150b5 - 298b4 + 118b3 - 118b2 + 178*b1 - 323) / 2
$\nu^{9}$	$=$	$ -278\beta_{15} - 228\beta_{11} + 288\beta_{8} - 60\beta_{7} + 288\beta_{5} - 288\beta_{4} + 223\beta_{3} $ -278b15 - 228b11 + 288b8 - 60b7 + 288b5 - 288b4 + 223*b3
$\nu^{10}$	$=$	$ ( - 809 \beta_{15} + 957 \beta_{14} + 288 \beta_{13} - 288 \beta_{12} - 70 \beta_{10} + \cdots + 599 \beta_{2} ) / 2 $ (-809b15 + 957b14 + 288b13 - 288b12 - 70b10 - 70b9 + 1673b8 - 353b7 - 1701b6 + 809b5 + 599b3 + 599b2) / 2
$\nu^{11}$	$=$	$ ( 2728 \beta_{14} - 1673 \beta_{12} + 1245 \beta_{11} - 408 \beta_{9} + 1603 \beta_{8} - 358 \beta_{7} + \cdots - 4790 ) / 2 $ (2728b14 - 1673b12 + 1245b11 - 408b9 + 1603b8 - 358b7 - 4790b6 - 70b5 + 1603b4 + 1125b2 + 2728*b1 - 4790) / 2
$\nu^{12}$	$=$	$ -1603\beta_{13} - 1603\beta_{12} + 428\beta_{10} - 428\beta_{9} + 5148\beta _1 - 9071 $ -1603b13 - 1603b12 + 428b10 - 428b9 + 5148*b1 - 9071
$\nu^{13}$	$=$	$ ( - 8354 \beta_{15} - 14647 \beta_{14} - 9210 \beta_{13} - 6751 \beta_{11} + 2489 \beta_{10} + \cdots - 25690 ) / 2 $ (-8354b15 - 14647b14 - 9210b13 - 6751b11 + 2489b10 + 8782b8 - 2031b7 + 25690b6 + 8782b5 - 8782b4 + 5865b3 + 14647b1 - 25690) / 2
$\nu^{14}$	$=$	$ ( - 23857 \beta_{15} - 27721 \beta_{14} - 8782 \beta_{13} + 8782 \beta_{12} + 2459 \beta_{10} + \cdots + 16480 \beta_{2} ) / 2 $ (-23857b15 - 27721b14 - 8782b13 + 8782b12 + 2459b10 + 2459b9 + 50203b8 - 11699b7 + 48661b6 + 23857b5 + 16480b3 + 16480b2) / 2
$\nu^{15}$	$=$	$ 36503\beta_{11} + 47744\beta_{8} - 11241\beta_{7} - 2459\beta_{5} + 47744\beta_{4} + 31097\beta_{2} $ 36503b11 + 47744b8 - 11241b7 - 2459b5 + 47744b4 + 31097b2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-\beta_{6}$	$-1$

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} $

242.1

2.24352	+	0.601150i
1.60599	+	0.430324i
−0.347596	−	1.29724i
−0.0499037	−	0.186243i
−0.186243	−	0.0499037i
−1.29724	−	0.347596i
0.430324	+	1.60599i
0.601150	+	2.24352i
2.24352	−	0.601150i
1.60599	−	0.430324i
−0.347596	+	1.29724i
−0.0499037	+	0.186243i
−0.186243	+	0.0499037i
−1.29724	+	0.347596i
0.430324	−	1.60599i
0.601150	−	2.24352i

−1.64237

1.64237i

−

3.39477i

0.394116

2.20106i

−0.550102

−

0.550102i

2.29074

2.29074i

−4.26225

−

2.96768i

242.2

−1.17567

1.17567i

−

0.764383i

0.981675

−

2.00906i

1.27193

1.27193i

−1.45267

−

1.45267i

1.20786

3.51610i

242.3

−0.949649

0.949649i

0.196335i

−2.15519

0.595953i

−1.44851

−

1.44851i

−2.08575

−

2.08575i

1.48073

−

2.61262i

242.4

−0.136339

0.136339i

1.96282i

1.79893

1.32810i

1.72668

1.72668i

−0.540289

−

0.540289i

−0.426337

0.0641924i

242.5

0.136339

−

0.136339i

1.96282i

−1.79893

−

1.32810i

1.72668

1.72668i

0.540289

0.540289i

−0.426337

0.0641924i

242.6

0.949649

−

0.949649i

0.196335i

2.15519

−

0.595953i

−1.44851

−

1.44851i

2.08575

2.08575i

1.48073

−

2.61262i

242.7

1.17567

−

1.17567i

−

0.764383i

−0.981675

2.00906i

1.27193

1.27193i

1.45267

1.45267i

1.20786

3.51610i

242.8

1.64237

−

1.64237i

−

3.39477i

−0.394116

−

2.20106i

−0.550102

−

0.550102i

−2.29074

−

2.29074i

−4.26225

−

2.96768i

323.1

−1.64237

−

1.64237i

3.39477i

0.394116

−

2.20106i

−0.550102

0.550102i

2.29074

−

2.29074i

−4.26225

2.96768i

323.2

−1.17567

−

1.17567i

0.764383i

0.981675

2.00906i

1.27193

−

1.27193i

−1.45267

1.45267i

1.20786

−

3.51610i

323.3

−0.949649

−

0.949649i

−

0.196335i

−2.15519

−

0.595953i

−1.44851

1.44851i

−2.08575

2.08575i

1.48073

2.61262i

323.4

−0.136339

−

0.136339i

−

1.96282i

1.79893

−

1.32810i

1.72668

−

1.72668i

−0.540289

0.540289i

−0.426337

−

0.0641924i

323.5

0.136339

0.136339i

−

1.96282i

−1.79893

1.32810i

1.72668

−

1.72668i

0.540289

−

0.540289i

−0.426337

−

0.0641924i

323.6

0.949649

0.949649i

−

0.196335i

2.15519

0.595953i

−1.44851

1.44851i

2.08575

−

2.08575i

1.48073

2.61262i

323.7

1.17567

1.17567i

0.764383i

−0.981675

−

2.00906i

1.27193

−

1.27193i

1.45267

−

1.45267i

1.20786

−

3.51610i

323.8

1.64237

1.64237i

3.39477i

−0.394116

2.20106i

−0.550102

0.550102i

−2.29074

2.29074i

−4.26225

2.96768i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	405.2.f.a		16
3.b	odd	2	1	inner	405.2.f.a		16
5.c	odd	4	1	inner	405.2.f.a		16
9.c	even	3	1		45.2.l.a	✓	16
9.c	even	3	1		135.2.m.a		16
9.d	odd	6	1		45.2.l.a	✓	16
9.d	odd	6	1		135.2.m.a		16
15.e	even	4	1	inner	405.2.f.a		16
36.f	odd	6	1		720.2.cu.c		16
36.h	even	6	1		720.2.cu.c		16
45.h	odd	6	1		225.2.p.b		16
45.h	odd	6	1		675.2.q.a		16
45.j	even	6	1		225.2.p.b		16
45.j	even	6	1		675.2.q.a		16
45.k	odd	12	1		45.2.l.a	✓	16
45.k	odd	12	1		135.2.m.a		16
45.k	odd	12	1		225.2.p.b		16
45.k	odd	12	1		675.2.q.a		16
45.l	even	12	1		45.2.l.a	✓	16
45.l	even	12	1		135.2.m.a		16
45.l	even	12	1		225.2.p.b		16
45.l	even	12	1		675.2.q.a		16
180.v	odd	12	1		720.2.cu.c		16
180.x	even	12	1		720.2.cu.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.2.l.a	✓	16	9.c	even	3	1
45.2.l.a	✓	16	9.d	odd	6	1
45.2.l.a	✓	16	45.k	odd	12	1
45.2.l.a	✓	16	45.l	even	12	1
135.2.m.a		16	9.c	even	3	1
135.2.m.a		16	9.d	odd	6	1
135.2.m.a		16	45.k	odd	12	1
135.2.m.a		16	45.l	even	12	1
225.2.p.b		16	45.h	odd	6	1
225.2.p.b		16	45.j	even	6	1
225.2.p.b		16	45.k	odd	12	1
225.2.p.b		16	45.l	even	12	1
405.2.f.a		16	1.a	even	1	1	trivial
405.2.f.a		16	3.b	odd	2	1	inner
405.2.f.a		16	5.c	odd	4	1	inner
405.2.f.a		16	15.e	even	4	1	inner
675.2.q.a		16	45.h	odd	6	1
675.2.q.a		16	45.j	even	6	1
675.2.q.a		16	45.k	odd	12	1
675.2.q.a		16	45.l	even	12	1
720.2.cu.c		16	36.f	odd	6	1
720.2.cu.c		16	36.h	even	6	1
720.2.cu.c		16	180.v	odd	12	1
720.2.cu.c		16	180.x	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{16} + 40T_{2}^{12} + 342T_{2}^{8} + 724T_{2}^{4} + 1 $ T2^16 + 40*T2^12 + 342*T2^8 + 724*T2^4 + 1 acting on $S_{2}^{\mathrm{new}}(405, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 40 T^{12} + \cdots + 1 $ T^16 + 40T^12 + 342T^8 + 724*T^4 + 1
$3$	$ T^{16} $ T^16
$5$	$ T^{16} + 4 T^{14} + \cdots + 390625 $ T^16 + 4T^14 + 4T^12 + 28T^10 + 406T^8 + 700T^6 + 2500T^4 + 62500*T^2 + 390625
$7$	$ (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 49)^{2} $ (T^8 - 2T^7 + 2T^6 + 4T^5 + 22T^4 - 34T^3 + 32T^2 + 56*T + 49)^2
$11$	$ (T^{8} + 20 T^{6} + 96 T^{4} + \cdots + 4)^{2} $ (T^8 + 20T^6 + 96T^4 + 68*T^2 + 4)^2
$13$	$ (T^{8} - 2 T^{7} + \cdots + 2116)^{2} $ (T^8 - 2T^7 + 2T^6 + 52T^5 + 484T^4 + 188T^3 + 8T^2 - 184*T + 2116)^2
$17$	$ T^{16} + 964 T^{12} + \cdots + 16 $ T^16 + 964T^12 + 6504T^8 + 832*T^4 + 16
$19$	$ (T^{8} + 60 T^{6} + \cdots + 324)^{2} $ (T^8 + 60T^6 + 864T^4 + 1836*T^2 + 324)^2
$23$	$ T^{16} + 2680 T^{12} + \cdots + 62742241 $ T^16 + 2680T^12 + 476262T^8 + 12797524*T^4 + 62742241
$29$	$ (T^{8} - 84 T^{6} + \cdots + 31329)^{2} $ (T^8 - 84T^6 + 2166T^4 - 17316*T^2 + 31329)^2
$31$	$ (T^{4} - 2 T^{3} - 42 T^{2} + \cdots - 26)^{4} $ (T^4 - 2T^3 - 42T^2 + 142*T - 26)^4
$37$	$ (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 4)^{2} $ (T^8 - 2T^7 + 2T^6 + 124T^5 + 2308T^4 + 1340T^3 + 392T^2 + 56*T + 4)^2
$41$	$ (T^{8} + 92 T^{6} + \cdots + 32761)^{2} $ (T^8 + 92T^6 + 2334T^4 + 19496*T^2 + 32761)^2
$43$	$ (T^{8} - 2 T^{7} + \cdots + 3364)^{2} $ (T^8 - 2T^7 + 2T^6 + 28T^5 + 460T^4 - 364T^3 + 200T^2 + 1160*T + 3364)^2
$47$	$ T^{16} + \cdots + 33243864241 $ T^16 + 5380T^12 + 3314742T^8 + 639487144*T^4 + 33243864241
$53$	$ T^{16} + \cdots + 409600000000 $ T^16 + 20032T^12 + 104540160T^8 + 132589551616*T^4 + 409600000000
$59$	$ (T^{8} - 144 T^{6} + \cdots + 24336)^{2} $ (T^8 - 144T^6 + 6360T^4 - 79920*T^2 + 24336)^2
$61$	$ (T^{4} + 4 T^{3} + \cdots - 107)^{4} $ (T^4 + 4T^3 - 102T^2 + 274*T - 107)^4
$67$	$ (T^{8} + 4 T^{7} + \cdots + 734449)^{2} $ (T^8 + 4T^7 + 8T^6 + 466T^5 + 26050T^4 + 173468T^3 + 594050T^2 + 934130*T + 734449)^2
$71$	$ (T^{8} + 116 T^{6} + \cdots + 128164)^{2} $ (T^8 + 116T^6 + 3972T^4 + 43580*T^2 + 128164)^2
$73$	$ (T^{8} + 4 T^{7} + \cdots + 270400)^{2} $ (T^8 + 4T^7 + 8T^6 - 584T^5 + 16384T^4 - 9472T^3 + 1568T^2 + 29120*T + 270400)^2
$79$	$ (T^{8} + 372 T^{6} + \cdots + 41679936)^{2} $ (T^8 + 372T^6 + 45456T^4 + 2298816*T^2 + 41679936)^2
$83$	$ T^{16} + \cdots + 13841287201 $ T^16 + 25456T^12 + 165625782T^8 + 121914011548*T^4 + 13841287201
$89$	$ (T^{8} - 300 T^{6} + \cdots + 3969)^{2} $ (T^8 - 300T^6 + 8262T^4 - 49356*T^2 + 3969)^2
$97$	$ (T^{8} + 28 T^{7} + \cdots + 78400)^{2} $ (T^8 + 28T^7 + 392T^6 + 3208T^5 + 16864T^4 + 56576T^3 + 119072T^2 + 136640*T + 78400)^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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{2} + ( - \beta_{14} + \beta_{6}) q^{4} - \beta_{4} q^{5} + \beta_{10} q^{7} + ( - \beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{8}+O(q^{10}) \) q + b3 * q^2 + (-b14 + b6) * q^4 - b4 * q^5 + b10 * q^7 + (-b11 - b8 - b4 + b2) * q^8	\( q + \beta_{3} q^{2} + ( - \beta_{14} + \beta_{6}) q^{4} - \beta_{4} q^{5} + \beta_{10} q^{7} + ( - \beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{8}+ \cdots + ( - \beta_{11} - \beta_{8} + \cdots - 4 \beta_{2}) q^{98}+O(q^{100}) \) q + b3 * q^2 + (-b14 + b6) * q^4 - b4 * q^5 + b10 * q^7 + (-b11 - b8 - b4 + b2) * q^8 + (-b13 + b10 + b9 + b6 + b1 - 1) * q^10 + (-b11 + b5 - b4 - b3 + b2) * q^11 + (b14 - b12 + b9 + b1) * q^13 + (-b8 + b7) * q^14 + (-b13 - b12 + 1) * q^16 + (-b15 - b11 + b8 + b5 - b4) * q^17 + (-b14 + b13 - b12 - b10 - b9) * q^19 + (b11 - 2b8 - b5) q^20 + (-b13 + b10 - b6 + 1) * q^22 + (-b11 + b8 - 2b7 + b5 + b4 + b2) q^23 + (b13 - b12 - b10 + b9 + 2b1 - 1) q^25 + (2b15 + 3b11 - b5 + 3b4 + b3 - b2) q^26 + (-b12 - b6 - 1) * q^28 + (2b8 + b7 - b3 - b2) q^29 + (2b10 - 2b9 - b1) * q^31 + (-b11 + b7 + b3) * q^32 + (-3b14 - b13 + b12 + 4b6) * q^34 + (-b15 + 2b11 + b8 - b3 - 2b2) * q^35 + (b14 - b13 + 3b10 - b1) q^37 + (2b8 - 2b7 - b5 + 2b4 - 2b2) * q^38 + (b14 - 2b10 + b9 - 3b6 + 2b1 - 1) q^40 + (-b15 + 2b11 - 2b5 + b4) * q^41 + (b14 + b12 + b9 + b1) * q^43 + (b15 - b8 + 2b7 - b5) q^44 + (b13 + b12 - b10 + b9 - b1 + 3) * q^46 + (-b15 - b8 + b7 - b5 + b4 - b3) * q^47 + (-b14 + 4b6) q^49 + (2b15 + 3b11 + b8 - 2b7 - b5 + 3b4 - b3 - 2b2) q^50 + (2b14 + b13 - b10 - 3b6 - 2b1 + 3) q^52 + (-2b11 + 2b8 - 4b7 + 2b4) * q^53 + (b13 + b12 - 2b10 + b6 + b1 - 1) q^55 + (b15 + b11 + b5) * q^56 + (-2b14 + 2b12 + b9 + 2b6 - 2b1 + 2) * q^58 + (-b15 + 2b8 + 2b7 + b5 - b3 - b2) * q^59 + (3b10 - 3b9 - 3b1 - 1) q^61 + (-b15 - b11 - b8 + 2b7 - b5 + b4) q^62 + (-3b14 - 2b13 + 2b12 - b10 - b9 + 3b6) * q^64 + (b15 + 3b11 + b8 - 2b7 + b5 + b4 + 3b3 + b2) q^65 + (2b14 + 3b13 - 2b10 - b6 - 2b1 + 1) * q^67 + (-2b11 - 3b8 + b7 - 3b4 - 2b2) * q^68 + (b14 + b12 + b10 - 3b6 - b1 - 2) q^70 + (-2b15 - b11 + b5 - 3b4 - 2b3 + 2b2) * q^71 + (3b14 - 2b6 + 3b1 - 2) q^73 + (-3b8 + 4b7 + b3 + b2) * q^74 + (-b10 + b9 - b1 - 2) * q^76 + (b15 - b11 + b8 + b5 - b4 - 2b3) q^77 + (2b14 + b13 - b12 - 3b10 - 3b9 - 4b6) * q^79 + (-b11 + b8 - 4b7 + 3b5 - 2b4 - b3 + 3b2) * q^80 + (b14 + b13 - 2b10 - 3b6 - b1 + 3) * q^82 + (-4b11 - b8 - 3b7 + b5 - b4 + 3b2) q^83 + (-2b14 - b13 - b9 + 2b6 + 2b1 - 4) q^85 + (b11 + b5 - b4 - b3 + b2) * q^86 + (b12 + b9 - 2b6 - 2) q^88 + (b15 - 4b8 + 5b7 - b5 + 2b3 + 2b2) * q^89 + (2b10 - 2b9 - 3b1 - 2) q^91 + (b15 - 4b11 + 2b8 + 2b7 + 2b5 - 2b4 - b3) q^92 + (b13 - b12 - 3b10 - 3b9 - 5b6) q^94 + (-2b15 + b11 - 3b8 + b7 - 2b4 + 2b3 + 4b2) q^95 + (-b14 + 4b6 + b1 - 4) q^97 + (-b11 - b8 - b4 - 4b2) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{7}+O(q^{10}) \) 16 * q + 4 * q^7	\( 16 q + 4 q^{7} - 8 q^{10} + 4 q^{13} + 16 q^{16} + 20 q^{22} - 8 q^{25} - 16 q^{28} + 8 q^{31} + 4 q^{37} - 12 q^{40} + 4 q^{43} + 32 q^{46} + 28 q^{52} - 16 q^{55} + 12 q^{58} - 16 q^{61} - 8 q^{67} - 36 q^{70} - 8 q^{73} - 48 q^{76} + 32 q^{82} - 44 q^{85} - 36 q^{88} - 40 q^{91} - 56 q^{97}+O(q^{100}) \) 16 * q + 4 * q^7 - 8 * q^10 + 4 * q^13 + 16 * q^16 + 20 * q^22 - 8 * q^25 - 16 * q^28 + 8 * q^31 + 4 * q^37 - 12 * q^40 + 4 * q^43 + 32 * q^46 + 28 * q^52 - 16 * q^55 + 12 * q^58 - 16 * q^61 - 8 * q^67 - 36 * q^70 - 8 * q^73 - 48 * q^76 + 32 * q^82 - 44 * q^85 - 36 * q^88 - 40 * q^91 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	405.2.f.a

Level	$405$
Weight	$2$
Character orbit	405.f
Analytic conductor	$3.234$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.23394128186\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \) x^16 - 6x^15 + 18x^14 - 36x^13 + 34x^12 + 18x^11 - 72x^10 + 132x^9 - 93x^8 - 102x^7 + 144x^6 - 432x^5 + 502x^4 + 288x^3 + 72x^2 + 12*x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 73568941 \nu^{15} + 2828429898 \nu^{14} - 10706428236 \nu^{13} + 20127265496 \nu^{12} + \cdots + 10982429182739 ) / 3707507912227 \) (73568941v^15 + 2828429898v^14 - 10706428236v^13 + 20127265496v^12 - 35741021050v^11 + 17927851065v^10 - 58772639836v^9 + 308269732922v^8 - 50382616322v^7 + 35812385940v^6 + 245517720058v^5 - 1525793905272v^4 - 747026251899v^3 - 178512937629v^2 - 28424647318*v + 10982429182739) / 3707507912227
\(\beta_{2}\)	\(=\)	\( ( - 94386116202 \nu^{15} + 619740656932 \nu^{14} - 2033008548312 \nu^{13} + \cdots + 80029143512 ) / 3707507912227 \) (-94386116202v^15 + 619740656932v^14 - 2033008548312v^13 + 4441986378774v^12 - 5386888154268v^11 + 640680728681v^10 + 7214824090311v^9 - 16443022873810v^8 + 16824695358916v^7 + 2692744559593v^6 - 17484917305182v^5 + 49690281510088v^4 - 74318822560500v^3 + 6157027329225v^2 + 993781902738*v + 80029143512) / 3707507912227
\(\beta_{3}\)	\(=\)	\( ( 196858530 \nu^{15} - 1212076232 \nu^{14} + 3734556994 \nu^{13} - 7676123862 \nu^{12} + \cdots + 785074438 ) / 1973128213 \) (196858530v^15 - 1212076232v^14 + 3734556994v^13 - 7676123862v^12 + 7906540260v^11 + 2288455604v^10 - 14534291253v^9 + 28282811704v^8 - 22763827488v^7 - 16547104520v^6 + 31379154942v^5 - 90050852224v^4 + 112985327352v^3 + 38916218628v^2 + 6077223668*v + 785074438) / 1973128213
\(\beta_{4}\)	\(=\)	\( ( - 458282837172 \nu^{15} + 3278355328030 \nu^{14} - 11567438085651 \nu^{13} + \cdots + 3479823499409 ) / 3707507912227 \) (-458282837172v^15 + 3278355328030v^14 - 11567438085651v^13 + 26927968536797v^12 - 37449835526942v^11 + 15597997864898v^10 + 36317007615948v^9 - 100050764445960v^8 + 123309140658256v^7 - 23651454139949v^6 - 102232920253345v^5 + 287003871762252v^4 - 480932440039679v^3 + 198024659235246v^2 + 31299923543934*v + 3479823499409) / 3707507912227
\(\beta_{5}\)	\(=\)	\( ( - 1263290532917 \nu^{15} + 8307388188784 \nu^{14} - 27291768145635 \nu^{13} + \cdots + 1071789225554 ) / 3707507912227 \) (-1263290532917v^15 + 8307388188784v^14 - 27291768145635v^13 + 59721708540564v^12 - 72662176309554v^11 + 9183517518119v^10 + 96760918658508v^9 - 221515642871632v^8 + 227173488204868v^7 + 35108102916679v^6 - 235176348728859v^5 + 665141988015058v^4 - 975637136257126v^3 + 82449382716699v^2 + 13308754547460*v + 1071789225554) / 3707507912227
\(\beta_{6}\)	\(=\)	\( ( 1616321538 \nu^{15} - 9938916556 \nu^{14} + 30594542475 \nu^{13} - 62871582510 \nu^{12} + \cdots + 10233974547 ) / 3810388399 \) (1616321538v^15 - 9938916556v^14 + 30594542475v^13 - 62871582510v^12 + 64714538406v^11 + 18641341380v^10 - 118271636061v^9 + 231117756606v^8 - 186162436800v^7 - 134499720676v^6 + 250255719435v^5 - 736991238906v^4 + 924410484114v^3 + 318408624800v^2 + 82003950372*v + 10233974547) / 3810388399
\(\beta_{7}\)	\(=\)	\( ( - 854458 \nu^{15} + 5149288 \nu^{14} - 15489103 \nu^{13} + 31002870 \nu^{12} - 29353426 \nu^{11} + \cdots - 4877403 ) / 1828267 \) (-854458v^15 + 5149288v^14 - 15489103v^13 + 31002870v^12 - 29353426v^11 - 15677868v^10 + 63062803v^9 - 114195864v^8 + 80446252v^7 + 89178588v^6 - 128785525v^5 + 370469382v^4 - 434157674v^3 - 248804552v^2 - 38956914*v - 4877403) / 1828267
\(\beta_{8}\)	\(=\)	\( ( - 315552650318 \nu^{15} + 1900582212134 \nu^{14} - 5713442004763 \nu^{13} + \cdots - 1796844761346 ) / 529643987461 \) (-315552650318v^15 + 1900582212134v^14 - 5713442004763v^13 + 11427384011211v^12 - 10792414939398v^11 - 5844691005419v^10 + 23294405344177v^9 - 42097274125416v^8 + 29554626469000v^7 + 32994050005382v^6 - 47477278412835v^5 + 136625993542464v^4 - 159897884475264v^3 - 91655155082751v^2 - 14350917254916*v - 1796844761346) / 529643987461
\(\beta_{9}\)	\(=\)	\( ( - 2712751885095 \nu^{15} + 16678491631658 \nu^{14} - 51337065544716 \nu^{13} + \cdots - 23594947704644 ) / 3707507912227 \) (-2712751885095v^15 + 16678491631658v^14 - 51337065544716v^13 + 105497880078934v^12 - 108587037241966v^11 - 31285400547667v^10 + 198545816545737v^9 - 388380189945330v^8 + 313419238953178v^7 + 224631606730413v^6 - 419929403639154v^5 + 1238858757336820v^4 - 1552826796501673v^3 - 534895302063361v^2 - 137764843521504*v - 23594947704644) / 3707507912227
\(\beta_{10}\)	\(=\)	\( ( - 2717282793429 \nu^{15} + 16707385417880 \nu^{14} - 51423315159482 \nu^{13} + \cdots - 10795341151078 ) / 3707507912227 \) (-2717282793429v^15 + 16707385417880v^14 - 51423315159482v^13 + 105664904992298v^12 - 108755702488842v^11 - 31280777936752v^10 + 198470922157507v^9 - 387930687122496v^8 + 312810984796866v^7 + 224984020905618v^6 - 418380709675378v^5 + 1236842558873240v^4 - 1553943585818787v^3 - 535172136374176v^2 - 137810381709166*v - 10795341151078) / 3707507912227
\(\beta_{11}\)	\(=\)	\( ( 1570148876 \nu^{15} - 9814610316 \nu^{14} + 30686832232 \nu^{13} - 63994473524 \nu^{12} + \cdots + 4714210963 ) / 1973128213 \) (1570148876v^15 - 9814610316v^14 + 30686832232v^13 - 63994473524v^12 + 68737309508v^11 + 12449599248v^10 - 117627630280v^9 + 236328234138v^8 - 202589468876v^7 - 114627530376v^6 + 259195647184v^5 - 741062624316v^4 + 968316440200v^3 + 226232221584v^2 + 35218281816*v + 4714210963) / 1973128213
\(\beta_{12}\)	\(=\)	\( ( 3757574245950 \nu^{15} - 23096381861488 \nu^{14} + 71089447066218 \nu^{13} + \cdots + 32644912196796 ) / 3707507912227 \) (3757574245950v^15 - 23096381861488v^14 + 71089447066218v^13 - 146083562570344v^12 + 150316650857126v^11 + 43353340898378v^10 - 274940334918912v^9 + 537780099499056v^8 - 433167757411736v^7 - 310933164972742v^6 + 581459639319606v^5 - 1716023307015740v^4 + 2147688630225299v^3 + 739900598877122v^2 + 190586429788452*v + 32644912196796) / 3707507912227
\(\beta_{13}\)	\(=\)	\( ( - 3758547730578 \nu^{15} + 23113179926128 \nu^{14} - 71146859123410 \nu^{13} + \cdots - 14937505605495 ) / 3707507912227 \) (-3758547730578v^15 + 23113179926128v^14 - 71146859123410v^13 + 146190483542135v^12 - 150476465892702v^11 - 43294531756298v^10 + 274731591096056v^9 - 536670080913120v^8 + 432843551243472v^7 + 311142270044362v^6 - 580257664123502v^5 + 1710584928995308v^4 - 2150385871006647v^3 - 740547803922626v^2 - 190689873599252*v - 14937505605495) / 3707507912227
\(\beta_{14}\)	\(=\)	\( ( 4659934396714 \nu^{15} - 28652614036898 \nu^{14} + 88198331151147 \nu^{13} + \cdots + 29503074524823 ) / 3707507912227 \) (4659934396714v^15 - 28652614036898v^14 + 88198331151147v^13 - 181242264245730v^12 + 186543596485370v^11 + 53736739030473v^10 - 340917929949083v^9 + 666259937224872v^8 - 536770650269540v^7 - 386908446130506v^6 + 721086474541709v^5 - 2124666379562946v^4 + 2664947390704218v^3 + 917928790168959v^2 + 236405919870430*v + 29503074524823) / 3707507912227
\(\beta_{15}\)	\(=\)	\( ( - 29169335295 \nu^{15} + 179860916164 \nu^{14} - 554518308281 \nu^{13} + 1139653845381 \nu^{12} + \cdots - 116597819081 ) / 22200646181 \) (-29169335295v^15 + 179860916164v^14 - 554518308281v^13 + 1139653845381v^12 - 1173093973890v^11 - 346731406534v^10 + 2175410366414v^9 - 4207778503040v^8 + 3380869821552v^7 + 2471943196468v^6 - 4657440308511v^5 + 13333326030116v^4 - 16790149734936v^3 - 5781324089814v^2 - 902821290514*v - 116597819081) / 22200646181

\(\nu\)	\(=\)	\( ( \beta_{14} + \beta_{10} - \beta_{6} - \beta_{3} - \beta _1 + 1 ) / 2 \) (b14 + b10 - b6 - b3 - b1 + 1) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} + \beta_{14} - \beta_{8} - \beta_{7} - 3\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} ) / 2 \) (b15 + b14 - b8 - b7 - 3*b6 - b5 - b3 - b2) / 2
\(\nu^{3}\)	\(=\)	\( -\beta_{11} - \beta_{8} - \beta_{4} - 3\beta_{2} \) -b11 - b8 - b4 - 3*b2
\(\nu^{4}\)	\(=\)	\( ( - 5 \beta_{15} + \beta_{13} + \beta_{12} - 9 \beta_{11} + 3 \beta_{5} - 8 \beta_{4} + 5 \beta_{3} + \cdots + 13 ) / 2 \) (-5b15 + b13 + b12 - 9b11 + 3b5 - 8b4 + 5b3 - 5b2 - 6*b1 + 13) / 2
\(\nu^{5}\)	\(=\)	\( ( - 8 \beta_{15} + 19 \beta_{14} + 8 \beta_{13} - 7 \beta_{11} + 3 \beta_{10} + 8 \beta_{8} - \beta_{7} + \cdots + 32 ) / 2 \) (-8b15 + 19b14 + 8b13 - 7b11 + 3b10 + 8b8 - b7 - 32b6 + 8b5 - 8b4 + 11b3 - 19*b1 + 32) / 2
\(\nu^{6}\)	\(=\)	\( 33\beta_{14} + 8\beta_{13} - 8\beta_{12} - \beta_{10} - \beta_{9} - 63\beta_{6} \) 33b14 + 8b13 - 8b12 - b10 - b9 - 63b6
\(\nu^{7}\)	\(=\)	\( ( 97 \beta_{14} - 51 \beta_{12} - 41 \beta_{11} - 2 \beta_{9} - 50 \beta_{8} + 9 \beta_{7} - 169 \beta_{6} + \cdots - 169 ) / 2 \) (97b14 - 51b12 - 41b11 - 2b9 - 50b8 + 9b7 - 169b6 + b5 - 50b4 - 47b2 + 97b1 - 169) / 2
\(\nu^{8}\)	\(=\)	\( ( - 148 \beta_{15} - 50 \beta_{13} - 50 \beta_{12} - 245 \beta_{11} + 10 \beta_{10} - 10 \beta_{9} + \cdots - 323 ) / 2 \) (-148b15 - 50b13 - 50b12 - 245b11 + 10b10 - 10b9 + 150b5 - 298b4 + 118b3 - 118b2 + 178*b1 - 323) / 2
\(\nu^{9}\)	\(=\)	\( -278\beta_{15} - 228\beta_{11} + 288\beta_{8} - 60\beta_{7} + 288\beta_{5} - 288\beta_{4} + 223\beta_{3} \) -278b15 - 228b11 + 288b8 - 60b7 + 288b5 - 288b4 + 223*b3
\(\nu^{10}\)	\(=\)	\( ( - 809 \beta_{15} + 957 \beta_{14} + 288 \beta_{13} - 288 \beta_{12} - 70 \beta_{10} + \cdots + 599 \beta_{2} ) / 2 \) (-809b15 + 957b14 + 288b13 - 288b12 - 70b10 - 70b9 + 1673b8 - 353b7 - 1701b6 + 809b5 + 599b3 + 599b2) / 2
\(\nu^{11}\)	\(=\)	\( ( 2728 \beta_{14} - 1673 \beta_{12} + 1245 \beta_{11} - 408 \beta_{9} + 1603 \beta_{8} - 358 \beta_{7} + \cdots - 4790 ) / 2 \) (2728b14 - 1673b12 + 1245b11 - 408b9 + 1603b8 - 358b7 - 4790b6 - 70b5 + 1603b4 + 1125b2 + 2728*b1 - 4790) / 2
\(\nu^{12}\)	\(=\)	\( -1603\beta_{13} - 1603\beta_{12} + 428\beta_{10} - 428\beta_{9} + 5148\beta _1 - 9071 \) -1603b13 - 1603b12 + 428b10 - 428b9 + 5148*b1 - 9071
\(\nu^{13}\)	\(=\)	\( ( - 8354 \beta_{15} - 14647 \beta_{14} - 9210 \beta_{13} - 6751 \beta_{11} + 2489 \beta_{10} + \cdots - 25690 ) / 2 \) (-8354b15 - 14647b14 - 9210b13 - 6751b11 + 2489b10 + 8782b8 - 2031b7 + 25690b6 + 8782b5 - 8782b4 + 5865b3 + 14647b1 - 25690) / 2
\(\nu^{14}\)	\(=\)	\( ( - 23857 \beta_{15} - 27721 \beta_{14} - 8782 \beta_{13} + 8782 \beta_{12} + 2459 \beta_{10} + \cdots + 16480 \beta_{2} ) / 2 \) (-23857b15 - 27721b14 - 8782b13 + 8782b12 + 2459b10 + 2459b9 + 50203b8 - 11699b7 + 48661b6 + 23857b5 + 16480b3 + 16480b2) / 2
\(\nu^{15}\)	\(=\)	\( 36503\beta_{11} + 47744\beta_{8} - 11241\beta_{7} - 2459\beta_{5} + 47744\beta_{4} + 31097\beta_{2} \) 36503b11 + 47744b8 - 11241b7 - 2459b5 + 47744b4 + 31097b2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 242.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 40 T^{12} + \cdots + 1 \) T^16 + 40T^12 + 342T^8 + 724*T^4 + 1
$3$	\( T^{16} \) T^16
$5$	\( T^{16} + 4 T^{14} + \cdots + 390625 \) T^16 + 4T^14 + 4T^12 + 28T^10 + 406T^8 + 700T^6 + 2500T^4 + 62500*T^2 + 390625
$7$	\( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 49)^{2} \) (T^8 - 2T^7 + 2T^6 + 4T^5 + 22T^4 - 34T^3 + 32T^2 + 56*T + 49)^2
$11$	\( (T^{8} + 20 T^{6} + 96 T^{4} + \cdots + 4)^{2} \) (T^8 + 20T^6 + 96T^4 + 68*T^2 + 4)^2
$13$	\( (T^{8} - 2 T^{7} + \cdots + 2116)^{2} \) (T^8 - 2T^7 + 2T^6 + 52T^5 + 484T^4 + 188T^3 + 8T^2 - 184*T + 2116)^2
$17$	\( T^{16} + 964 T^{12} + \cdots + 16 \) T^16 + 964T^12 + 6504T^8 + 832*T^4 + 16
$19$	\( (T^{8} + 60 T^{6} + \cdots + 324)^{2} \) (T^8 + 60T^6 + 864T^4 + 1836*T^2 + 324)^2
$23$	\( T^{16} + 2680 T^{12} + \cdots + 62742241 \) T^16 + 2680T^12 + 476262T^8 + 12797524*T^4 + 62742241
$29$	\( (T^{8} - 84 T^{6} + \cdots + 31329)^{2} \) (T^8 - 84T^6 + 2166T^4 - 17316*T^2 + 31329)^2
$31$	\( (T^{4} - 2 T^{3} - 42 T^{2} + \cdots - 26)^{4} \) (T^4 - 2T^3 - 42T^2 + 142*T - 26)^4
$37$	\( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 4)^{2} \) (T^8 - 2T^7 + 2T^6 + 124T^5 + 2308T^4 + 1340T^3 + 392T^2 + 56*T + 4)^2
$41$	\( (T^{8} + 92 T^{6} + \cdots + 32761)^{2} \) (T^8 + 92T^6 + 2334T^4 + 19496*T^2 + 32761)^2
$43$	\( (T^{8} - 2 T^{7} + \cdots + 3364)^{2} \) (T^8 - 2T^7 + 2T^6 + 28T^5 + 460T^4 - 364T^3 + 200T^2 + 1160*T + 3364)^2
$47$	\( T^{16} + \cdots + 33243864241 \) T^16 + 5380T^12 + 3314742T^8 + 639487144*T^4 + 33243864241
$53$	\( T^{16} + \cdots + 409600000000 \) T^16 + 20032T^12 + 104540160T^8 + 132589551616*T^4 + 409600000000
$59$	\( (T^{8} - 144 T^{6} + \cdots + 24336)^{2} \) (T^8 - 144T^6 + 6360T^4 - 79920*T^2 + 24336)^2
$61$	\( (T^{4} + 4 T^{3} + \cdots - 107)^{4} \) (T^4 + 4T^3 - 102T^2 + 274*T - 107)^4
$67$	\( (T^{8} + 4 T^{7} + \cdots + 734449)^{2} \) (T^8 + 4T^7 + 8T^6 + 466T^5 + 26050T^4 + 173468T^3 + 594050T^2 + 934130*T + 734449)^2
$71$	\( (T^{8} + 116 T^{6} + \cdots + 128164)^{2} \) (T^8 + 116T^6 + 3972T^4 + 43580*T^2 + 128164)^2
$73$	\( (T^{8} + 4 T^{7} + \cdots + 270400)^{2} \) (T^8 + 4T^7 + 8T^6 - 584T^5 + 16384T^4 - 9472T^3 + 1568T^2 + 29120*T + 270400)^2
$79$	\( (T^{8} + 372 T^{6} + \cdots + 41679936)^{2} \) (T^8 + 372T^6 + 45456T^4 + 2298816*T^2 + 41679936)^2
$83$	\( T^{16} + \cdots + 13841287201 \) T^16 + 25456T^12 + 165625782T^8 + 121914011548*T^4 + 13841287201
$89$	\( (T^{8} - 300 T^{6} + \cdots + 3969)^{2} \) (T^8 - 300T^6 + 8262T^4 - 49356*T^2 + 3969)^2
$97$	\( (T^{8} + 28 T^{7} + \cdots + 78400)^{2} \) (T^8 + 28T^7 + 392T^6 + 3208T^5 + 16864T^4 + 56576T^3 + 119072T^2 + 136640*T + 78400)^2
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\(n\)	\(82\)	\(326\)
\(\chi(n)\)	\(-\beta_{6}\)	\(-1\)