LMFDB - Newform orbit 810.3.d.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(161,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 $ x^16 - 12x^14 + 138x^12 - 1040x^10 + 5541x^8 - 26220x^6 + 99328x^4 - 202728*x^2 + 181476
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{14} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{7} q^{2} - 2 q^{4} - \beta_{9} q^{5} + ( - \beta_{5} - \beta_1 + 1) q^{7} + 2 \beta_{7} q^{8} + \beta_1 q^{10} + ( - \beta_{12} - \beta_{11} - \beta_{7}) q^{11} + (\beta_{5} - \beta_{2} - 3) q^{13}+ \cdots + (2 \beta_{15} - 5 \beta_{12} + \cdots - 6 \beta_{6}) q^{98}+O(q^{100}) $ q - b7 * q^2 - 2 * q^4 - b9 * q^5 + (-b5 - b1 + 1) * q^7 + 2b7 q^8 + b1 * q^10 + (-b12 - b11 - b7) * q^11 + (b5 - b2 - 3) * q^13 + (-b10 - 2b9) q^14 + 4 * q^16 + (b11 + b9 + 3b7 + b6) q^17 + (-b13 + b5 - 2b3 + b1 + 4) q^19 + 2b9 q^20 + (b14 + b13 - 2) * q^22 + (-b11 + b10 + b9 + 2b8) q^23 - 5 * q^25 + (-b15 - b12 + b10 + 2b7 - b6) q^26 + (2b5 + 2b1 - 2) * q^28 + (-b12 - b10 + 2b9 - 3b6) * q^29 + (-2b14 + b13 - b5 - 2b3 - 2b2 - b1 + 2) q^31 - 4b7 q^32 + (-b14 + b13 + b4 + 2b3 - b2 - 3b1 + 6) * q^34 + (b11 - b9 - b8 - 4b7 - b6) q^35 + (-b5 - 4b3 + b2 - 2b1 - 5) * q^37 + (-3b12 + b10 - b9 + b8 - 5b7) * q^38 - 2b1 q^40 + (-2b12 + 3b11 - 4b9 + 2b8 - 7b7 - b6) q^41 + (-2b13 - 3b5 + b4 - 2b3 - 2b2 + 4b1 - 11) q^43 + (2b12 + 2b11 + 2b7) q^44 + (b14 - 2b13 - 2b5 + b3 + 2) * q^46 + (-2b15 - 2b12 + b10 - 8b9 + 2b8 + 4b7 - 3b6) * q^47 + (b14 - b13 + b5 + 4b4 + b3 - 2b2 - b1 + 10) * q^49 + 5b7 q^50 + (-2b5 + 2b2 + 6) * q^52 + (2b15 - 6b12 - 2b10 - 4b9 + 2b7 - 2b6) * q^53 + (b14 + 2b13 - b5 + b1 + 2) q^55 + (2b10 + 4b9) * q^56 + (-2b13 + 2b5 - 3b4 - 2b3 + 3b2 - 2) q^58 + (-8b12 + 2b10 + 8b7 - 2b6) * q^59 + (4b14 + 5b5 - b4 - 3b3 - b2 + 9) q^61 + (-2b15 - 5b12 - 4b11 - b10 - b9 + 5b8 - b7 - 2b6) q^62 - 8 * q^64 + (-b15 - 3b12 - b11 - b10 + b9 + 2b8 + b7) * q^65 + (-b14 + b13 - 2b5 + b4 + 2b2 + 9b1 - 6) q^67 + (-2b11 - 2b9 - 6b7 - 2b6) * q^68 + (-b14 - b4 - b3 + b2 + b1 - 8) * q^70 + (-2b15 - 5b12 - 5b11 - 2b10 + 2b9 + 8b8 + b7 - 2b6) q^71 + (8b13 + 4b4 + 6b3 - 6b2 - 6b1 + 30) q^73 + (b15 - 3b12 - b10 - 8b9 + 4b8 + 6b7 + b6) * q^74 + (2b13 - 2b5 + 4b3 - 2b1 - 8) * q^76 + (-4b15 - 10b12 - b11 - 2b10 + 5b9 + 2b8 + 5b7 - 5b6) q^77 + (-2b14 + 6b5 - 2b3 + 2b2 + 2b1 - 14) q^79 - 4b9 q^80 + (-3b14 - b13 - b4 + 6b3 + b2 - 14) * q^82 + (-2b15 - 5b12 + 5b11 - 2b10 + 5b8 + b7 + 2b6) * q^83 + (b14 - 3b13 - b5 - b4 - 2b3 + 2b2 - b1 + 7) q^85 + (-b15 - 7b12 - 3b10 + 4b9 + 14b7 - 3b6) q^86 + (-2b14 - 2b13 + 4) * q^88 + (b15 - 8b12 - 2b11 + 4b10 - 10b9 + 4b8 - 34b7 - 2b6) q^89 + (-4b14 - 5b13 + b5 - 4b4 - 2b2 - 11b1 - 46) q^91 + (2b11 - 2b10 - 2b9 - 4b8) * q^92 + (-3b13 - 2b5 - b4 + b3 + 5b2 + 9b1 + 10) * q^94 + (-2b15 - 3b11 + b10 - 3b9 + b8) q^95 + (2b14 + 6b13 + 8b5 - 2b3 - 2b2 + 4b1 + 2) * q^97 + (2b15 - 5b12 + 2b11 + b10 - 3b9 - 3b8 - 11b7 - 6b6) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 8 q^{7} - 40 q^{13} + 64 q^{16} + 80 q^{19} - 48 q^{22} - 80 q^{25} - 16 q^{28} + 32 q^{31} + 96 q^{34} - 88 q^{37} - 184 q^{43} + 24 q^{46} + 168 q^{49} + 80 q^{52} + 152 q^{61} - 128 q^{64}+ \cdots + 32 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 8 * q^7 - 40 * q^13 + 64 * q^16 + 80 * q^19 - 48 * q^22 - 80 * q^25 - 16 * q^28 + 32 * q^31 + 96 * q^34 - 88 * q^37 - 184 * q^43 + 24 * q^46 + 168 * q^49 + 80 * q^52 + 152 * q^61 - 128 * q^64 - 112 * q^67 - 120 * q^70 + 416 * q^73 - 160 * q^76 - 160 * q^79 - 192 * q^82 + 120 * q^85 + 96 * q^88 - 656 * q^91 + 168 * q^94 + 32 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 $ x^16 - 12*x^14 + 138*x^12 - 1040*x^10 + 5541*x^8 - 26220*x^6 + 99328*x^4 - 202728*x^2 + 181476 :

$\beta_{1}$	$=$	$ ( - 41 \nu^{14} + 225 \nu^{12} - 1333 \nu^{10} - 31 \nu^{8} + 114072 \nu^{6} - 798716 \nu^{4} + \cdots - 15616332 ) / 4341600 $ (-41v^14 + 225v^12 - 1333v^10 - 31v^8 + 114072v^6 - 798716v^4 + 1960260*v^2 - 15616332) / 4341600
$\beta_{2}$	$=$	$ ( - 1060102 \nu^{15} - 43681543 \nu^{14} - 9413665 \nu^{13} + 327317100 \nu^{12} + \cdots + 117564866064 ) / 93092587200 $ (-1060102v^15 - 43681543v^14 - 9413665v^13 + 327317100v^12 + 52993694v^11 - 4312009559v^10 - 1212202337v^9 + 24621647812v^8 + 8924351884v^7 - 110059483044v^6 - 38018411372v^5 + 547821046832v^4 + 181482674640v^3 - 1365110469420v^2 - 679629522564*v + 117564866064) / 93092587200
$\beta_{3}$	$=$	$ ( 1060102 \nu^{15} - 13081679 \nu^{14} + 9413665 \nu^{13} + 103890750 \nu^{12} + \cdots - 103455424008 ) / 93092587200 $ (1060102v^15 - 13081679v^14 + 9413665v^13 + 103890750v^12 - 52993694v^11 - 1276944727v^10 + 1212202337v^9 + 7013398886v^8 - 8924351884v^7 - 27954197532v^6 + 38018411372v^5 + 129439132696v^4 - 181482674640v^3 - 319703589660v^2 + 679629522564*v - 103455424008) / 93092587200
$\beta_{4}$	$=$	$ ( 1060102 \nu^{15} - 18397307 \nu^{14} + 9413665 \nu^{13} + 124360050 \nu^{12} + \cdots - 10369676664 ) / 93092587200 $ (1060102v^15 - 18397307v^14 + 9413665v^13 + 124360050v^12 - 52993694v^11 - 1786702291v^10 + 1212202337v^9 + 10594185338v^8 - 8924351884v^7 - 51705156156v^6 + 38018411372v^5 + 271816712968v^4 - 181482674640v^3 - 683671395180v^2 + 679629522564*v - 10369676664) / 93092587200
$\beta_{5}$	$=$	$ ( 2689399 \nu^{15} - 4953457 \nu^{14} - 30849380 \nu^{13} + 36433650 \nu^{12} + \cdots + 139190573736 ) / 93092587200 $ (2689399v^15 - 4953457v^14 - 30849380v^13 + 36433650v^12 + 387255127v^11 - 501080441v^10 - 2894713276v^9 + 2934818938v^8 + 16169623892v^7 - 14091869556v^6 - 76843428016v^5 + 72584663768v^4 + 269835314700v^3 - 181239795780v^2 - 715006917072*v + 139190573736) / 93092587200
$\beta_{6}$	$=$	$ ( - 2838939 \nu^{15} - 7781884 \nu^{14} + 28237600 \nu^{13} + 46768055 \nu^{12} + \cdots + 671904709452 ) / 93092587200 $ (-2838939v^15 - 7781884v^14 + 28237600v^13 + 46768055v^12 - 322227307v^11 - 762866032v^10 + 2381066976v^9 + 3735375391v^8 - 11733928612v^7 - 22414325972v^6 + 61473817336v^5 + 110203703356v^4 - 214675371660v^3 - 488552349000v^2 + 257974806672*v + 671904709452) / 93092587200
$\beta_{7}$	$=$	$ ( - 9042 \nu^{15} + 90896 \nu^{13} - 1066817 \nu^{11} + 7184007 \nu^{9} - 35214371 \nu^{7} + \cdots + 642245346 \nu ) / 232731468 $ (-9042v^15 + 90896v^13 - 1066817v^11 + 7184007v^9 - 35214371v^7 + 157110461v^5 - 545114616v^3 + 642245346v) / 232731468
$\beta_{8}$	$=$	$ ( 2838939 \nu^{15} - 59262990 \nu^{14} - 28237600 \nu^{13} + 645699915 \nu^{12} + \cdots + 5382715429980 ) / 93092587200 $ (2838939v^15 - 59262990v^14 - 28237600v^13 + 645699915v^12 + 322227307v^11 - 7345162290v^10 - 2381066976v^9 + 52674059715v^8 + 11733928612v^7 - 255415827420v^6 - 61473817336v^5 + 1188238063980v^4 + 214675371660v^3 - 4143288831120v^2 - 257974806672*v + 5382715429980) / 93092587200
$\beta_{9}$	$=$	$ ( - 13164 \nu^{15} + 130775 \nu^{13} - 1568132 \nu^{11} + 10603551 \nu^{9} + \cdots + 1021382172 \nu ) / 231573600 $ (-13164v^15 + 130775v^13 - 1568132v^11 + 10603551v^9 - 53262512v^7 + 246273836v^5 - 864329760v^3 + 1021382172v) / 231573600
$\beta_{10}$	$=$	$ ( - 6467061 \nu^{15} - 31030621 \nu^{14} + 69900025 \nu^{13} + 303684395 \nu^{12} + \cdots + 2493018301788 ) / 93092587200 $ (-6467061v^15 - 31030621v^14 + 69900025v^13 + 303684395v^12 - 778359193v^11 - 3632588833v^10 + 5532153849v^9 + 23934863179v^8 - 25549357888v^7 - 122484386768v^6 + 118632917764v^5 + 545037498364v^4 - 398000748540v^3 - 1925446791900v^2 + 472440570228*v + 2493018301788) / 93092587200
$\beta_{11}$	$=$	$ ( 105600 \nu^{15} - 288126 \nu^{14} - 1126175 \nu^{13} + 3290145 \nu^{12} + 12496100 \nu^{11} + \cdots + 29621850228 ) / 1311163200 $ (105600v^15 - 288126v^14 - 1126175v^13 + 3290145v^12 + 12496100v^11 - 34322298v^10 - 91217175v^9 + 264050649v^8 + 425921300v^7 - 1191287508v^6 - 2094149900v^5 + 6131359284v^4 + 7093707000v^3 - 22259844000v^2 - 8478398700*v + 29621850228) / 1311163200
$\beta_{12}$	$=$	$ ( - 9306000 \nu^{15} - 25740553 \nu^{14} + 98137625 \nu^{13} + 299465930 \nu^{12} + \cdots + 2355405360264 ) / 93092587200 $ (-9306000v^15 - 25740553v^14 + 98137625v^13 + 299465930v^12 - 1100586500v^11 - 3291148129v^10 + 7913220825v^9 + 24469342162v^8 - 37283286500v^7 - 116500750724v^6 + 180106735100v^5 + 539017180312v^4 - 612676120200v^3 - 1827368241060v^2 + 730415376900*v + 2355405360264) / 93092587200
$\beta_{13}$	$=$	$ ( - 10517273 \nu^{15} - 5393018 \nu^{14} + 45055270 \nu^{13} + 38845875 \nu^{12} + \cdots - 121324708836 ) / 93092587200 $ (-10517273v^15 - 5393018v^14 + 45055270v^13 + 38845875v^12 - 787830809v^11 - 515371534v^10 + 2644962422v^9 + 2934486587v^8 - 7241191084v^7 - 12868903644v^6 + 41999431712v^5 + 64021629532v^4 + 125281693020v^3 - 160223848320v^2 - 1323673869816*v - 121324708836) / 93092587200
$\beta_{14}$	$=$	$ ( - 12715625 \nu^{15} + 5393018 \nu^{14} + 86403430 \nu^{13} - 38845875 \nu^{12} + \cdots + 28232121636 ) / 93092587200 $ (-12715625v^15 + 5393018v^14 + 86403430v^13 - 38845875v^12 - 1177758065v^11 + 515371534v^10 + 5991736310v^9 - 2934486587v^8 - 26343432100v^7 + 12868903644v^6 + 136083621440v^5 - 64021629532v^4 - 323074851540v^3 + 160223848320v^2 + 234501529320*v + 28232121636) / 93092587200
$\beta_{15}$	$=$	$ ( - 215 \nu^{15} + 2296 \nu^{13} - 25907 \nu^{11} + 183272 \nu^{9} - 859816 \nu^{7} + \cdots + 16167456 \nu ) / 456672 $ (-215v^15 + 2296v^13 - 25907v^11 + 183272v^9 - 859816v^7 + 4023896v^5 - 13619076v^3 + 16167456v) / 456672

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

$\nu$	$=$	$ ( \beta_{14} - \beta_{13} - 9\beta_{9} + 9\beta_{7} - 4\beta_{5} - \beta_{3} + \beta_{2} + 2 ) / 18 $ (b14 - b13 - 9b9 + 9b7 - 4*b5 - b3 + b2 + 2) / 18
$\nu^{2}$	$=$	$ ( - \beta_{15} + 3 \beta_{12} + 3 \beta_{10} + 3 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} - 6 \beta_{6} + \cdots + 27 ) / 18 $ (-b15 + 3b12 + 3b10 + 3b9 - 3b8 - 3b7 - 6b6 + b4 + b2 - 9*b1 + 27) / 18
$\nu^{3}$	$=$	$ ( - 3 \beta_{15} + 2 \beta_{14} + 22 \beta_{13} + 18 \beta_{12} - 117 \beta_{9} - 9 \beta_{8} + \cdots + 10 ) / 36 $ (-3b15 + 2b14 + 22b13 + 18b12 - 117b9 - 9b8 + 99b7 + 9b6 + 4b5 + 15b4 + 34b3 - 19b2 - 21*b1 + 10) / 36
$\nu^{4}$	$=$	$ ( 2 \beta_{15} + 9 \beta_{12} + 24 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} - 18 \beta_{8} - 6 \beta_{7} + \cdots - 297 ) / 18 $ (2b15 + 9b12 + 24b11 + 3b10 - 6b9 - 18b8 - 6b7 - 3b6 - 10b4 + 18b3 + 8b2 - 144b1 - 297) / 18
$\nu^{5}$	$=$	$ ( 25 \beta_{15} + 86 \beta_{14} + 58 \beta_{13} + 30 \beta_{12} + 375 \beta_{9} - 15 \beta_{8} + \cdots - 44 ) / 36 $ (25b15 + 86b14 + 58b13 + 30b12 + 375b9 - 15b8 - 1143b7 + 15b6 + 232b5 + 75b4 + 184b3 - 109b2 - 27*b1 - 44) / 36
$\nu^{6}$	$=$	$ ( 59 \beta_{15} - 57 \beta_{12} - 6 \beta_{11} - 303 \beta_{10} - 177 \beta_{9} + 138 \beta_{8} + \cdots - 270 ) / 18 $ (59b15 - 57b12 - 6b11 - 303b10 - 177b9 + 138b8 + 180b7 + 189b6 - 99b4 + 135b3 + 36b2 - 198b1 - 270) / 18
$\nu^{7}$	$=$	$ ( 665 \beta_{15} + 218 \beta_{14} - 494 \beta_{13} - 1218 \beta_{12} + 3531 \beta_{9} + 609 \beta_{8} + \cdots + 160 ) / 36 $ (665b15 + 218b14 - 494b13 - 1218b12 + 3531b9 + 609b8 - 9405b7 - 609b6 - 596b5 - 315b4 - 848b3 + 533b2 + 453*b1 + 160) / 36
$\nu^{8}$	$=$	$ ( 128 \beta_{15} - 51 \beta_{12} - 828 \beta_{11} - 1545 \beta_{10} - 384 \beta_{9} + 1050 \beta_{8} + \cdots + 15957 ) / 18 $ (128b15 - 51b12 - 828b11 - 1545b10 - 384b9 + 1050b8 + 798b7 + 273b6 + 880b4 - 1008b3 - 128b2 + 5040b1 + 15957) / 18
$\nu^{9}$	$=$	$ ( 945 \beta_{15} - 5470 \beta_{14} - 698 \beta_{13} + 486 \beta_{12} - 16245 \beta_{9} - 243 \beta_{8} + \cdots + 5824 ) / 36 $ (945b15 - 5470b14 - 698b13 + 486b12 - 16245b9 - 243b8 + 13221b7 + 243b6 - 17816b5 - 3585b4 - 9344b3 + 5759b2 - 975*b1 + 5824) / 36
$\nu^{10}$	$=$	$ ( - 1807 \beta_{15} + 6057 \beta_{12} + 7386 \beta_{11} + 12171 \beta_{10} + 5421 \beta_{9} + \cdots + 65232 ) / 18 $ (-1807b15 + 6057b12 + 7386b11 + 12171b10 + 5421b9 - 9936b8 - 9114b7 - 8607b6 + 7811b4 - 9405b3 - 1594b2 + 22860b1 + 65232) / 18
$\nu^{11}$	$=$	$ ( - 31097 \beta_{15} - 11218 \beta_{14} + 9742 \beta_{13} + 99066 \beta_{12} - 270219 \beta_{9} + \cdots + 244 ) / 36 $ (-31097b15 - 11218b14 + 9742b13 + 99066b12 - 270219b9 - 49533b8 + 454725b7 + 49533b6 - 1964b5 + 3135b4 + 9436b3 - 6301b2 - 10449*b1 + 244) / 36
$\nu^{12}$	$=$	$ ( - 11056 \beta_{15} + 38175 \beta_{12} + 77772 \beta_{11} + 105933 \beta_{10} + 33168 \beta_{9} + \cdots - 687393 ) / 18 $ (-11056b15 + 38175b12 + 77772b11 + 105933b10 + 33168b9 - 89538b8 - 72054b7 - 49941b6 - 47664b4 + 52272b3 + 4608b2 - 198000b1 - 687393) / 18
$\nu^{13}$	$=$	$ ( - 47125 \beta_{15} + 309566 \beta_{14} + 17722 \beta_{13} + 172770 \beta_{12} - 246111 \beta_{9} + \cdots - 394400 ) / 36 $ (-47125b15 + 309566b14 + 17722b13 + 172770b12 - 246111b9 - 86385b8 + 360927b7 + 86385b6 + 1116088b5 + 204165b4 + 545704b3 - 341539b2 + 79131*b1 - 394400) / 36
$\nu^{14}$	$=$	$ ( 75359 \beta_{15} - 177873 \beta_{12} - 296946 \beta_{11} - 571227 \beta_{10} - 226077 \beta_{9} + \cdots - 6435612 ) / 18 $ (75359b15 - 177873b12 - 296946b11 - 571227b10 - 226077b9 + 422052b8 + 374550b7 + 302979b6 - 549011b4 + 618345b3 + 69334b2 - 1915632b1 - 6435612) / 18
$\nu^{15}$	$=$	$ ( 1971441 \beta_{15} + 756290 \beta_{14} - 62510 \beta_{13} - 5385690 \beta_{12} + 15040035 \beta_{9} + \cdots - 1072700 ) / 36 $ (1971441b15 + 756290b14 - 62510b13 - 5385690b12 + 15040035b9 + 2692845b8 - 28366749b7 - 2692845b6 + 2839180b5 + 459825b4 + 1256380b3 - 796555b2 + 286305*b1 - 1072700) / 36

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$-1$

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} $

161.1

−2.11536	−	0.410927i
−1.42311	−	0.410927i
2.11536	−	0.410927i
1.42311	−	0.410927i
−2.11536	+	1.82514i
1.42311	+	1.82514i
−1.42311	+	1.82514i
2.11536	+	1.82514i
−2.11536	−	1.82514i
1.42311	−	1.82514i
−1.42311	−	1.82514i
2.11536	−	1.82514i
−2.11536	+	0.410927i
−1.42311	+	0.410927i
2.11536	+	0.410927i
1.42311	+	0.410927i

−

1.41421i

−2.00000

−

2.23607i

−6.63197

2.82843i

−3.16228

161.2

−

1.41421i

−2.00000

−

2.23607i

−1.37233

2.82843i

−3.16228

161.3

−

1.41421i

−2.00000

−

2.23607i

8.70115

2.82843i

−3.16228

161.4

−

1.41421i

−2.00000

−

2.23607i

10.7900

2.82843i

−3.16228

161.5

−

1.41421i

−2.00000

2.23607i

−13.9160

2.82843i

3.16228

161.6

−

1.41421i

−2.00000

2.23607i

−1.06296

2.82843i

3.16228

161.7

−

1.41421i

−2.00000

2.23607i

0.993782

2.82843i

3.16228

161.8

−

1.41421i

−2.00000

2.23607i

6.49832

2.82843i

3.16228

161.9

1.41421i

−2.00000

−

2.23607i

−13.9160

−

2.82843i

3.16228

161.10

1.41421i

−2.00000

−

2.23607i

−1.06296

−

2.82843i

3.16228

161.11

1.41421i

−2.00000

−

2.23607i

0.993782

−

2.82843i

3.16228

161.12

1.41421i

−2.00000

−

2.23607i

6.49832

−

2.82843i

3.16228

161.13

1.41421i

−2.00000

2.23607i

−6.63197

−

2.82843i

−3.16228

161.14

1.41421i

−2.00000

2.23607i

−1.37233

−

2.82843i

−3.16228

161.15

1.41421i

−2.00000

2.23607i

8.70115

−

2.82843i

−3.16228

161.16

1.41421i

−2.00000

2.23607i

10.7900

−

2.82843i

−3.16228

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	810.3.d.c		16
3.b	odd	2	1	inner	810.3.d.c		16
9.c	even	3	1		90.3.h.a	✓	16
9.c	even	3	1		270.3.h.a		16
9.d	odd	6	1		90.3.h.a	✓	16
9.d	odd	6	1		270.3.h.a		16
36.f	odd	6	1		720.3.bs.d		16
36.f	odd	6	1		2160.3.bs.d		16
36.h	even	6	1		720.3.bs.d		16
36.h	even	6	1		2160.3.bs.d		16
45.h	odd	6	1		450.3.i.g		16
45.h	odd	6	1		1350.3.i.g		16
45.j	even	6	1		450.3.i.g		16
45.j	even	6	1		1350.3.i.g		16
45.k	odd	12	2		450.3.k.c		32
45.k	odd	12	2		1350.3.k.b		32
45.l	even	12	2		450.3.k.c		32
45.l	even	12	2		1350.3.k.b		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
90.3.h.a	✓	16	9.c	even	3	1
90.3.h.a	✓	16	9.d	odd	6	1
270.3.h.a		16	9.c	even	3	1
270.3.h.a		16	9.d	odd	6	1
450.3.i.g		16	45.h	odd	6	1
450.3.i.g		16	45.j	even	6	1
450.3.k.c		32	45.k	odd	12	2
450.3.k.c		32	45.l	even	12	2
720.3.bs.d		16	36.f	odd	6	1
720.3.bs.d		16	36.h	even	6	1
810.3.d.c		16	1.a	even	1	1	trivial
810.3.d.c		16	3.b	odd	2	1	inner
1350.3.i.g		16	45.h	odd	6	1
1350.3.i.g		16	45.j	even	6	1
1350.3.k.b		32	45.k	odd	12	2
1350.3.k.b		32	45.l	even	12	2
2160.3.bs.d		16	36.f	odd	6	1
2160.3.bs.d		16	36.h	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} - 4T_{7}^{7} - 230T_{7}^{6} + 1208T_{7}^{5} + 10234T_{7}^{4} - 46180T_{7}^{3} - 90890T_{7}^{2} + 42800T_{7} + 81625 $ T7^8 - 4*T7^7 - 230*T7^6 + 1208*T7^5 + 10234*T7^4 - 46180*T7^3 - 90890*T7^2 + 42800*T7 + 81625 acting on $S_{3}^{\mathrm{new}}(810, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 2)^{8} $ (T^2 + 2)^8
$3$	$ T^{16} $ T^16
$5$	$ (T^{2} + 5)^{8} $ (T^2 + 5)^8
$7$	$ (T^{8} - 4 T^{7} + \cdots + 81625)^{2} $ (T^8 - 4T^7 - 230T^6 + 1208T^5 + 10234T^4 - 46180T^3 - 90890T^2 + 42800*T + 81625)^2
$11$	$ T^{16} + \cdots + 7901855928576 $ T^16 + 1128T^14 + 466776T^12 + 88531488T^10 + 7829647920T^8 + 300959286912T^6 + 4493793292416T^4 + 16685157537792*T^2 + 7901855928576
$13$	$ (T^{8} + 20 T^{7} + \cdots + 31486096)^{2} $ (T^8 + 20T^7 - 512T^6 - 7648T^5 + 102028T^4 + 704288T^3 - 6284096T^2 - 14923168*T + 31486096)^2
$17$	$ T^{16} + \cdots + 29\!\cdots\!00 $ T^16 + 1920T^14 + 1342872T^12 + 435421440T^10 + 72254709936T^8 + 6208640824320T^6 + 260042075222400T^4 + 4710647674368000*T^2 + 29699559027360000
$19$	$ (T^{8} - 40 T^{7} + \cdots + 4403341840)^{2} $ (T^8 - 40T^7 - 572T^6 + 45848T^5 - 444284T^4 - 9334288T^3 + 227539936T^2 - 1708733920*T + 4403341840)^2
$23$	$ T^{16} + \cdots + 15\!\cdots\!61 $ T^16 + 4404T^14 + 7259976T^12 + 5662915524T^10 + 2194475892894T^8 + 424070980245036T^6 + 38049266187389448T^4 + 1334563936728408540*T^2 + 15100141333025180961
$29$	$ T^{16} + \cdots + 35\!\cdots\!61 $ T^16 + 8904T^14 + 30908844T^12 + 53065231704T^10 + 47931920288406T^8 + 23237396222537016T^6 + 5969353855924658124T^4 + 748539640577179659816*T^2 + 35243070823066089883761
$31$	$ (T^{8} + \cdots + 1127554186384)^{2} $ (T^8 - 16T^7 - 5948T^6 + 100568T^5 + 10923556T^4 - 180362320T^3 - 6405172640T^2 + 75357852896*T + 1127554186384)^2
$37$	$ (T^{8} + 44 T^{7} + \cdots - 221671664)^{2} $ (T^8 + 44T^7 - 4688T^6 - 224800T^5 + 2573836T^4 + 216728480T^3 + 2039658112T^2 - 6537683296*T - 221671664)^2
$41$	$ T^{16} + \cdots + 23\!\cdots\!81 $ T^16 + 17424T^14 + 119958732T^12 + 415063358352T^10 + 755727785562870T^8 + 678435741282226608T^6 + 231101976010465802412T^4 + 4932517638393886927536*T^2 + 23195278290872877710481
$43$	$ (T^{8} + 92 T^{7} + \cdots - 287683097840)^{2} $ (T^8 + 92T^7 - 4760T^6 - 601864T^5 - 4606844T^4 + 638148944T^3 + 9946784656T^2 - 74572385920*T - 287683097840)^2
$47$	$ T^{16} + \cdots + 64\!\cdots\!41 $ T^16 + 15708T^14 + 94018536T^12 + 273610509228T^10 + 416405681930430T^8 + 329852265044756292T^6 + 125811895005831114216T^4 + 19298360260096824414132*T^2 + 648728698903637138483841
$53$	$ T^{16} + \cdots + 19\!\cdots\!76 $ T^16 + 26112T^14 + 271537920T^12 + 1435412054016T^10 + 4054612691017728T^8 + 5778222348087853056T^6 + 3238577172737178992640T^4 + 20539665809215598886912*T^2 + 19788806337548230066176
$59$	$ T^{16} + \cdots + 51\!\cdots\!00 $ T^16 + 34032T^14 + 423080064T^12 + 2303140891392T^10 + 5118306441592320T^8 + 3593304713206665216T^6 + 552499284082577670144T^4 + 29750860085625086607360*T^2 + 511073466923509103001600
$61$	$ (T^{8} + \cdots + 14350669588681)^{2} $ (T^8 - 76T^7 - 14672T^6 + 864020T^5 + 71896726T^4 - 2783804980T^3 - 119298641408T^2 + 1886100339308*T + 14350669588681)^2
$67$	$ (T^{8} + 56 T^{7} + \cdots + 7216851865)^{2} $ (T^8 + 56T^7 - 8978T^6 - 536668T^5 + 4437850T^4 + 237221792T^3 - 901577294T^2 - 3272469100*T + 7216851865)^2
$71$	$ T^{16} + \cdots + 19\!\cdots\!76 $ T^16 + 51336T^14 + 1029907224T^12 + 10492222826016T^10 + 58523180310590256T^8 + 178022213932214611584T^6 + 271889888918378268968064T^4 + 158754216027259568631250944*T^2 + 196306793302399195530629376
$73$	$ (T^{8} + \cdots - 638114737319936)^{2} $ (T^8 - 208T^7 - 13760T^6 + 5048192T^5 - 126003584T^4 - 27519861760T^3 + 1440880758784T^2 + 4630144311296*T - 638114737319936)^2
$79$	$ (T^{8} + \cdots + 15082254438400)^{2} $ (T^8 + 80T^7 - 14768T^6 - 1335040T^5 + 17745856T^4 + 2746081280T^3 - 8701199360T^2 - 1478650470400*T + 15082254438400)^2
$83$	$ T^{16} + \cdots + 11\!\cdots\!21 $ T^16 + 82884T^14 + 2771373096T^12 + 47675149398324T^10 + 444255526324941534T^8 + 2152785072825910495836T^6 + 4736367860723654413428648T^4 + 3746289379848649615408807980*T^2 + 1139986890170411000063988321
$89$	$ T^{16} + \cdots + 95\!\cdots\!25 $ T^16 + 72360T^14 + 2053953612T^12 + 29442669662520T^10 + 230941220295716406T^8 + 1003326989666216304600T^6 + 2319286306124995642147500T^4 + 2546707126910554657774125000*T^2 + 957861272638838140288344140625
$97$	$ (T^{8} - 16 T^{7} + \cdots + 102974242816)^{2} $ (T^8 - 16T^7 - 31616T^6 - 308608T^5 + 204514816T^4 + 10092929024T^3 + 122269229056T^2 - 379981938688*T + 102974242816)^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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{2} - 2 q^{4} - \beta_{9} q^{5} + ( - \beta_{5} - \beta_1 + 1) q^{7} + 2 \beta_{7} q^{8} + \beta_1 q^{10} + ( - \beta_{12} - \beta_{11} - \beta_{7}) q^{11} + (\beta_{5} - \beta_{2} - 3) q^{13}+ \cdots + (2 \beta_{15} - 5 \beta_{12} + \cdots - 6 \beta_{6}) q^{98}+O(q^{100}) \) q - b7 * q^2 - 2 * q^4 - b9 * q^5 + (-b5 - b1 + 1) * q^7 + 2b7 q^8 + b1 * q^10 + (-b12 - b11 - b7) * q^11 + (b5 - b2 - 3) * q^13 + (-b10 - 2b9) q^14 + 4 * q^16 + (b11 + b9 + 3b7 + b6) q^17 + (-b13 + b5 - 2b3 + b1 + 4) q^19 + 2b9 q^20 + (b14 + b13 - 2) * q^22 + (-b11 + b10 + b9 + 2b8) q^23 - 5 * q^25 + (-b15 - b12 + b10 + 2b7 - b6) q^26 + (2b5 + 2b1 - 2) * q^28 + (-b12 - b10 + 2b9 - 3b6) * q^29 + (-2b14 + b13 - b5 - 2b3 - 2b2 - b1 + 2) q^31 - 4b7 q^32 + (-b14 + b13 + b4 + 2b3 - b2 - 3b1 + 6) * q^34 + (b11 - b9 - b8 - 4b7 - b6) q^35 + (-b5 - 4b3 + b2 - 2b1 - 5) * q^37 + (-3b12 + b10 - b9 + b8 - 5b7) * q^38 - 2b1 q^40 + (-2b12 + 3b11 - 4b9 + 2b8 - 7b7 - b6) q^41 + (-2b13 - 3b5 + b4 - 2b3 - 2b2 + 4b1 - 11) q^43 + (2b12 + 2b11 + 2b7) q^44 + (b14 - 2b13 - 2b5 + b3 + 2) * q^46 + (-2b15 - 2b12 + b10 - 8b9 + 2b8 + 4b7 - 3b6) * q^47 + (b14 - b13 + b5 + 4b4 + b3 - 2b2 - b1 + 10) * q^49 + 5b7 q^50 + (-2b5 + 2b2 + 6) * q^52 + (2b15 - 6b12 - 2b10 - 4b9 + 2b7 - 2b6) * q^53 + (b14 + 2b13 - b5 + b1 + 2) q^55 + (2b10 + 4b9) * q^56 + (-2b13 + 2b5 - 3b4 - 2b3 + 3b2 - 2) q^58 + (-8b12 + 2b10 + 8b7 - 2b6) * q^59 + (4b14 + 5b5 - b4 - 3b3 - b2 + 9) q^61 + (-2b15 - 5b12 - 4b11 - b10 - b9 + 5b8 - b7 - 2b6) q^62 - 8 * q^64 + (-b15 - 3b12 - b11 - b10 + b9 + 2b8 + b7) * q^65 + (-b14 + b13 - 2b5 + b4 + 2b2 + 9b1 - 6) q^67 + (-2b11 - 2b9 - 6b7 - 2b6) * q^68 + (-b14 - b4 - b3 + b2 + b1 - 8) * q^70 + (-2b15 - 5b12 - 5b11 - 2b10 + 2b9 + 8b8 + b7 - 2b6) q^71 + (8b13 + 4b4 + 6b3 - 6b2 - 6b1 + 30) q^73 + (b15 - 3b12 - b10 - 8b9 + 4b8 + 6b7 + b6) * q^74 + (2b13 - 2b5 + 4b3 - 2b1 - 8) * q^76 + (-4b15 - 10b12 - b11 - 2b10 + 5b9 + 2b8 + 5b7 - 5b6) q^77 + (-2b14 + 6b5 - 2b3 + 2b2 + 2b1 - 14) q^79 - 4b9 q^80 + (-3b14 - b13 - b4 + 6b3 + b2 - 14) * q^82 + (-2b15 - 5b12 + 5b11 - 2b10 + 5b8 + b7 + 2b6) * q^83 + (b14 - 3b13 - b5 - b4 - 2b3 + 2b2 - b1 + 7) q^85 + (-b15 - 7b12 - 3b10 + 4b9 + 14b7 - 3b6) q^86 + (-2b14 - 2b13 + 4) * q^88 + (b15 - 8b12 - 2b11 + 4b10 - 10b9 + 4b8 - 34b7 - 2b6) q^89 + (-4b14 - 5b13 + b5 - 4b4 - 2b2 - 11b1 - 46) q^91 + (2b11 - 2b10 - 2b9 - 4b8) * q^92 + (-3b13 - 2b5 - b4 + b3 + 5b2 + 9b1 + 10) * q^94 + (-2b15 - 3b11 + b10 - 3b9 + b8) q^95 + (2b14 + 6b13 + 8b5 - 2b3 - 2b2 + 4b1 + 2) * q^97 + (2b15 - 5b12 + 2b11 + b10 - 3b9 - 3b8 - 11b7 - 6b6) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 8 q^{7} - 40 q^{13} + 64 q^{16} + 80 q^{19} - 48 q^{22} - 80 q^{25} - 16 q^{28} + 32 q^{31} + 96 q^{34} - 88 q^{37} - 184 q^{43} + 24 q^{46} + 168 q^{49} + 80 q^{52} + 152 q^{61} - 128 q^{64}+ \cdots + 32 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 8 * q^7 - 40 * q^13 + 64 * q^16 + 80 * q^19 - 48 * q^22 - 80 * q^25 - 16 * q^28 + 32 * q^31 + 96 * q^34 - 88 * q^37 - 184 * q^43 + 24 * q^46 + 168 * q^49 + 80 * q^52 + 152 * q^61 - 128 * q^64 - 112 * q^67 - 120 * q^70 + 416 * q^73 - 160 * q^76 - 160 * q^79 - 192 * q^82 + 120 * q^85 + 96 * q^88 - 656 * q^91 + 168 * q^94 + 32 * 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	810.3.d.c

Level	$810$
Weight	$3$
Character orbit	810.d
Analytic conductor	$22.071$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(22.0709014132\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 \) x^16 - 12x^14 + 138x^12 - 1040x^10 + 5541x^8 - 26220x^6 + 99328x^4 - 202728*x^2 + 181476
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{12}\cdot 3^{14} \)
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 41 \nu^{14} + 225 \nu^{12} - 1333 \nu^{10} - 31 \nu^{8} + 114072 \nu^{6} - 798716 \nu^{4} + \cdots - 15616332 ) / 4341600 \) (-41v^14 + 225v^12 - 1333v^10 - 31v^8 + 114072v^6 - 798716v^4 + 1960260*v^2 - 15616332) / 4341600
\(\beta_{2}\)	\(=\)	\( ( - 1060102 \nu^{15} - 43681543 \nu^{14} - 9413665 \nu^{13} + 327317100 \nu^{12} + \cdots + 117564866064 ) / 93092587200 \) (-1060102v^15 - 43681543v^14 - 9413665v^13 + 327317100v^12 + 52993694v^11 - 4312009559v^10 - 1212202337v^9 + 24621647812v^8 + 8924351884v^7 - 110059483044v^6 - 38018411372v^5 + 547821046832v^4 + 181482674640v^3 - 1365110469420v^2 - 679629522564*v + 117564866064) / 93092587200
\(\beta_{3}\)	\(=\)	\( ( 1060102 \nu^{15} - 13081679 \nu^{14} + 9413665 \nu^{13} + 103890750 \nu^{12} + \cdots - 103455424008 ) / 93092587200 \) (1060102v^15 - 13081679v^14 + 9413665v^13 + 103890750v^12 - 52993694v^11 - 1276944727v^10 + 1212202337v^9 + 7013398886v^8 - 8924351884v^7 - 27954197532v^6 + 38018411372v^5 + 129439132696v^4 - 181482674640v^3 - 319703589660v^2 + 679629522564*v - 103455424008) / 93092587200
\(\beta_{4}\)	\(=\)	\( ( 1060102 \nu^{15} - 18397307 \nu^{14} + 9413665 \nu^{13} + 124360050 \nu^{12} + \cdots - 10369676664 ) / 93092587200 \) (1060102v^15 - 18397307v^14 + 9413665v^13 + 124360050v^12 - 52993694v^11 - 1786702291v^10 + 1212202337v^9 + 10594185338v^8 - 8924351884v^7 - 51705156156v^6 + 38018411372v^5 + 271816712968v^4 - 181482674640v^3 - 683671395180v^2 + 679629522564*v - 10369676664) / 93092587200
\(\beta_{5}\)	\(=\)	\( ( 2689399 \nu^{15} - 4953457 \nu^{14} - 30849380 \nu^{13} + 36433650 \nu^{12} + \cdots + 139190573736 ) / 93092587200 \) (2689399v^15 - 4953457v^14 - 30849380v^13 + 36433650v^12 + 387255127v^11 - 501080441v^10 - 2894713276v^9 + 2934818938v^8 + 16169623892v^7 - 14091869556v^6 - 76843428016v^5 + 72584663768v^4 + 269835314700v^3 - 181239795780v^2 - 715006917072*v + 139190573736) / 93092587200
\(\beta_{6}\)	\(=\)	\( ( - 2838939 \nu^{15} - 7781884 \nu^{14} + 28237600 \nu^{13} + 46768055 \nu^{12} + \cdots + 671904709452 ) / 93092587200 \) (-2838939v^15 - 7781884v^14 + 28237600v^13 + 46768055v^12 - 322227307v^11 - 762866032v^10 + 2381066976v^9 + 3735375391v^8 - 11733928612v^7 - 22414325972v^6 + 61473817336v^5 + 110203703356v^4 - 214675371660v^3 - 488552349000v^2 + 257974806672*v + 671904709452) / 93092587200
\(\beta_{7}\)	\(=\)	\( ( - 9042 \nu^{15} + 90896 \nu^{13} - 1066817 \nu^{11} + 7184007 \nu^{9} - 35214371 \nu^{7} + \cdots + 642245346 \nu ) / 232731468 \) (-9042v^15 + 90896v^13 - 1066817v^11 + 7184007v^9 - 35214371v^7 + 157110461v^5 - 545114616v^3 + 642245346v) / 232731468
\(\beta_{8}\)	\(=\)	\( ( 2838939 \nu^{15} - 59262990 \nu^{14} - 28237600 \nu^{13} + 645699915 \nu^{12} + \cdots + 5382715429980 ) / 93092587200 \) (2838939v^15 - 59262990v^14 - 28237600v^13 + 645699915v^12 + 322227307v^11 - 7345162290v^10 - 2381066976v^9 + 52674059715v^8 + 11733928612v^7 - 255415827420v^6 - 61473817336v^5 + 1188238063980v^4 + 214675371660v^3 - 4143288831120v^2 - 257974806672*v + 5382715429980) / 93092587200
\(\beta_{9}\)	\(=\)	\( ( - 13164 \nu^{15} + 130775 \nu^{13} - 1568132 \nu^{11} + 10603551 \nu^{9} + \cdots + 1021382172 \nu ) / 231573600 \) (-13164v^15 + 130775v^13 - 1568132v^11 + 10603551v^9 - 53262512v^7 + 246273836v^5 - 864329760v^3 + 1021382172v) / 231573600
\(\beta_{10}\)	\(=\)	\( ( - 6467061 \nu^{15} - 31030621 \nu^{14} + 69900025 \nu^{13} + 303684395 \nu^{12} + \cdots + 2493018301788 ) / 93092587200 \) (-6467061v^15 - 31030621v^14 + 69900025v^13 + 303684395v^12 - 778359193v^11 - 3632588833v^10 + 5532153849v^9 + 23934863179v^8 - 25549357888v^7 - 122484386768v^6 + 118632917764v^5 + 545037498364v^4 - 398000748540v^3 - 1925446791900v^2 + 472440570228*v + 2493018301788) / 93092587200
\(\beta_{11}\)	\(=\)	\( ( 105600 \nu^{15} - 288126 \nu^{14} - 1126175 \nu^{13} + 3290145 \nu^{12} + 12496100 \nu^{11} + \cdots + 29621850228 ) / 1311163200 \) (105600v^15 - 288126v^14 - 1126175v^13 + 3290145v^12 + 12496100v^11 - 34322298v^10 - 91217175v^9 + 264050649v^8 + 425921300v^7 - 1191287508v^6 - 2094149900v^5 + 6131359284v^4 + 7093707000v^3 - 22259844000v^2 - 8478398700*v + 29621850228) / 1311163200
\(\beta_{12}\)	\(=\)	\( ( - 9306000 \nu^{15} - 25740553 \nu^{14} + 98137625 \nu^{13} + 299465930 \nu^{12} + \cdots + 2355405360264 ) / 93092587200 \) (-9306000v^15 - 25740553v^14 + 98137625v^13 + 299465930v^12 - 1100586500v^11 - 3291148129v^10 + 7913220825v^9 + 24469342162v^8 - 37283286500v^7 - 116500750724v^6 + 180106735100v^5 + 539017180312v^4 - 612676120200v^3 - 1827368241060v^2 + 730415376900*v + 2355405360264) / 93092587200
\(\beta_{13}\)	\(=\)	\( ( - 10517273 \nu^{15} - 5393018 \nu^{14} + 45055270 \nu^{13} + 38845875 \nu^{12} + \cdots - 121324708836 ) / 93092587200 \) (-10517273v^15 - 5393018v^14 + 45055270v^13 + 38845875v^12 - 787830809v^11 - 515371534v^10 + 2644962422v^9 + 2934486587v^8 - 7241191084v^7 - 12868903644v^6 + 41999431712v^5 + 64021629532v^4 + 125281693020v^3 - 160223848320v^2 - 1323673869816*v - 121324708836) / 93092587200
\(\beta_{14}\)	\(=\)	\( ( - 12715625 \nu^{15} + 5393018 \nu^{14} + 86403430 \nu^{13} - 38845875 \nu^{12} + \cdots + 28232121636 ) / 93092587200 \) (-12715625v^15 + 5393018v^14 + 86403430v^13 - 38845875v^12 - 1177758065v^11 + 515371534v^10 + 5991736310v^9 - 2934486587v^8 - 26343432100v^7 + 12868903644v^6 + 136083621440v^5 - 64021629532v^4 - 323074851540v^3 + 160223848320v^2 + 234501529320*v + 28232121636) / 93092587200
\(\beta_{15}\)	\(=\)	\( ( - 215 \nu^{15} + 2296 \nu^{13} - 25907 \nu^{11} + 183272 \nu^{9} - 859816 \nu^{7} + \cdots + 16167456 \nu ) / 456672 \) (-215v^15 + 2296v^13 - 25907v^11 + 183272v^9 - 859816v^7 + 4023896v^5 - 13619076v^3 + 16167456v) / 456672

\(\nu\)	\(=\)	\( ( \beta_{14} - \beta_{13} - 9\beta_{9} + 9\beta_{7} - 4\beta_{5} - \beta_{3} + \beta_{2} + 2 ) / 18 \) (b14 - b13 - 9b9 + 9b7 - 4*b5 - b3 + b2 + 2) / 18
\(\nu^{2}\)	\(=\)	\( ( - \beta_{15} + 3 \beta_{12} + 3 \beta_{10} + 3 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} - 6 \beta_{6} + \cdots + 27 ) / 18 \) (-b15 + 3b12 + 3b10 + 3b9 - 3b8 - 3b7 - 6b6 + b4 + b2 - 9*b1 + 27) / 18
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{15} + 2 \beta_{14} + 22 \beta_{13} + 18 \beta_{12} - 117 \beta_{9} - 9 \beta_{8} + \cdots + 10 ) / 36 \) (-3b15 + 2b14 + 22b13 + 18b12 - 117b9 - 9b8 + 99b7 + 9b6 + 4b5 + 15b4 + 34b3 - 19b2 - 21*b1 + 10) / 36
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{15} + 9 \beta_{12} + 24 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} - 18 \beta_{8} - 6 \beta_{7} + \cdots - 297 ) / 18 \) (2b15 + 9b12 + 24b11 + 3b10 - 6b9 - 18b8 - 6b7 - 3b6 - 10b4 + 18b3 + 8b2 - 144b1 - 297) / 18
\(\nu^{5}\)	\(=\)	\( ( 25 \beta_{15} + 86 \beta_{14} + 58 \beta_{13} + 30 \beta_{12} + 375 \beta_{9} - 15 \beta_{8} + \cdots - 44 ) / 36 \) (25b15 + 86b14 + 58b13 + 30b12 + 375b9 - 15b8 - 1143b7 + 15b6 + 232b5 + 75b4 + 184b3 - 109b2 - 27*b1 - 44) / 36
\(\nu^{6}\)	\(=\)	\( ( 59 \beta_{15} - 57 \beta_{12} - 6 \beta_{11} - 303 \beta_{10} - 177 \beta_{9} + 138 \beta_{8} + \cdots - 270 ) / 18 \) (59b15 - 57b12 - 6b11 - 303b10 - 177b9 + 138b8 + 180b7 + 189b6 - 99b4 + 135b3 + 36b2 - 198b1 - 270) / 18
\(\nu^{7}\)	\(=\)	\( ( 665 \beta_{15} + 218 \beta_{14} - 494 \beta_{13} - 1218 \beta_{12} + 3531 \beta_{9} + 609 \beta_{8} + \cdots + 160 ) / 36 \) (665b15 + 218b14 - 494b13 - 1218b12 + 3531b9 + 609b8 - 9405b7 - 609b6 - 596b5 - 315b4 - 848b3 + 533b2 + 453*b1 + 160) / 36
\(\nu^{8}\)	\(=\)	\( ( 128 \beta_{15} - 51 \beta_{12} - 828 \beta_{11} - 1545 \beta_{10} - 384 \beta_{9} + 1050 \beta_{8} + \cdots + 15957 ) / 18 \) (128b15 - 51b12 - 828b11 - 1545b10 - 384b9 + 1050b8 + 798b7 + 273b6 + 880b4 - 1008b3 - 128b2 + 5040b1 + 15957) / 18
\(\nu^{9}\)	\(=\)	\( ( 945 \beta_{15} - 5470 \beta_{14} - 698 \beta_{13} + 486 \beta_{12} - 16245 \beta_{9} - 243 \beta_{8} + \cdots + 5824 ) / 36 \) (945b15 - 5470b14 - 698b13 + 486b12 - 16245b9 - 243b8 + 13221b7 + 243b6 - 17816b5 - 3585b4 - 9344b3 + 5759b2 - 975*b1 + 5824) / 36
\(\nu^{10}\)	\(=\)	\( ( - 1807 \beta_{15} + 6057 \beta_{12} + 7386 \beta_{11} + 12171 \beta_{10} + 5421 \beta_{9} + \cdots + 65232 ) / 18 \) (-1807b15 + 6057b12 + 7386b11 + 12171b10 + 5421b9 - 9936b8 - 9114b7 - 8607b6 + 7811b4 - 9405b3 - 1594b2 + 22860b1 + 65232) / 18
\(\nu^{11}\)	\(=\)	\( ( - 31097 \beta_{15} - 11218 \beta_{14} + 9742 \beta_{13} + 99066 \beta_{12} - 270219 \beta_{9} + \cdots + 244 ) / 36 \) (-31097b15 - 11218b14 + 9742b13 + 99066b12 - 270219b9 - 49533b8 + 454725b7 + 49533b6 - 1964b5 + 3135b4 + 9436b3 - 6301b2 - 10449*b1 + 244) / 36
\(\nu^{12}\)	\(=\)	\( ( - 11056 \beta_{15} + 38175 \beta_{12} + 77772 \beta_{11} + 105933 \beta_{10} + 33168 \beta_{9} + \cdots - 687393 ) / 18 \) (-11056b15 + 38175b12 + 77772b11 + 105933b10 + 33168b9 - 89538b8 - 72054b7 - 49941b6 - 47664b4 + 52272b3 + 4608b2 - 198000b1 - 687393) / 18
\(\nu^{13}\)	\(=\)	\( ( - 47125 \beta_{15} + 309566 \beta_{14} + 17722 \beta_{13} + 172770 \beta_{12} - 246111 \beta_{9} + \cdots - 394400 ) / 36 \) (-47125b15 + 309566b14 + 17722b13 + 172770b12 - 246111b9 - 86385b8 + 360927b7 + 86385b6 + 1116088b5 + 204165b4 + 545704b3 - 341539b2 + 79131*b1 - 394400) / 36
\(\nu^{14}\)	\(=\)	\( ( 75359 \beta_{15} - 177873 \beta_{12} - 296946 \beta_{11} - 571227 \beta_{10} - 226077 \beta_{9} + \cdots - 6435612 ) / 18 \) (75359b15 - 177873b12 - 296946b11 - 571227b10 - 226077b9 + 422052b8 + 374550b7 + 302979b6 - 549011b4 + 618345b3 + 69334b2 - 1915632b1 - 6435612) / 18
\(\nu^{15}\)	\(=\)	\( ( 1971441 \beta_{15} + 756290 \beta_{14} - 62510 \beta_{13} - 5385690 \beta_{12} + 15040035 \beta_{9} + \cdots - 1072700 ) / 36 \) (1971441b15 + 756290b14 - 62510b13 - 5385690b12 + 15040035b9 + 2692845b8 - 28366749b7 - 2692845b6 + 2839180b5 + 459825b4 + 1256380b3 - 796555b2 + 286305*b1 - 1072700) / 36

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 161.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + 2)^{8} \) (T^2 + 2)^8
$3$	\( T^{16} \) T^16
$5$	\( (T^{2} + 5)^{8} \) (T^2 + 5)^8
$7$	\( (T^{8} - 4 T^{7} + \cdots + 81625)^{2} \) (T^8 - 4T^7 - 230T^6 + 1208T^5 + 10234T^4 - 46180T^3 - 90890T^2 + 42800*T + 81625)^2
$11$	\( T^{16} + \cdots + 7901855928576 \) T^16 + 1128T^14 + 466776T^12 + 88531488T^10 + 7829647920T^8 + 300959286912T^6 + 4493793292416T^4 + 16685157537792*T^2 + 7901855928576
$13$	\( (T^{8} + 20 T^{7} + \cdots + 31486096)^{2} \) (T^8 + 20T^7 - 512T^6 - 7648T^5 + 102028T^4 + 704288T^3 - 6284096T^2 - 14923168*T + 31486096)^2
$17$	\( T^{16} + \cdots + 29\!\cdots\!00 \) T^16 + 1920T^14 + 1342872T^12 + 435421440T^10 + 72254709936T^8 + 6208640824320T^6 + 260042075222400T^4 + 4710647674368000*T^2 + 29699559027360000
$19$	\( (T^{8} - 40 T^{7} + \cdots + 4403341840)^{2} \) (T^8 - 40T^7 - 572T^6 + 45848T^5 - 444284T^4 - 9334288T^3 + 227539936T^2 - 1708733920*T + 4403341840)^2
$23$	\( T^{16} + \cdots + 15\!\cdots\!61 \) T^16 + 4404T^14 + 7259976T^12 + 5662915524T^10 + 2194475892894T^8 + 424070980245036T^6 + 38049266187389448T^4 + 1334563936728408540*T^2 + 15100141333025180961
$29$	\( T^{16} + \cdots + 35\!\cdots\!61 \) T^16 + 8904T^14 + 30908844T^12 + 53065231704T^10 + 47931920288406T^8 + 23237396222537016T^6 + 5969353855924658124T^4 + 748539640577179659816*T^2 + 35243070823066089883761
$31$	\( (T^{8} + \cdots + 1127554186384)^{2} \) (T^8 - 16T^7 - 5948T^6 + 100568T^5 + 10923556T^4 - 180362320T^3 - 6405172640T^2 + 75357852896*T + 1127554186384)^2
$37$	\( (T^{8} + 44 T^{7} + \cdots - 221671664)^{2} \) (T^8 + 44T^7 - 4688T^6 - 224800T^5 + 2573836T^4 + 216728480T^3 + 2039658112T^2 - 6537683296*T - 221671664)^2
$41$	\( T^{16} + \cdots + 23\!\cdots\!81 \) T^16 + 17424T^14 + 119958732T^12 + 415063358352T^10 + 755727785562870T^8 + 678435741282226608T^6 + 231101976010465802412T^4 + 4932517638393886927536*T^2 + 23195278290872877710481
$43$	\( (T^{8} + 92 T^{7} + \cdots - 287683097840)^{2} \) (T^8 + 92T^7 - 4760T^6 - 601864T^5 - 4606844T^4 + 638148944T^3 + 9946784656T^2 - 74572385920*T - 287683097840)^2
$47$	\( T^{16} + \cdots + 64\!\cdots\!41 \) T^16 + 15708T^14 + 94018536T^12 + 273610509228T^10 + 416405681930430T^8 + 329852265044756292T^6 + 125811895005831114216T^4 + 19298360260096824414132*T^2 + 648728698903637138483841
$53$	\( T^{16} + \cdots + 19\!\cdots\!76 \) T^16 + 26112T^14 + 271537920T^12 + 1435412054016T^10 + 4054612691017728T^8 + 5778222348087853056T^6 + 3238577172737178992640T^4 + 20539665809215598886912*T^2 + 19788806337548230066176
$59$	\( T^{16} + \cdots + 51\!\cdots\!00 \) T^16 + 34032T^14 + 423080064T^12 + 2303140891392T^10 + 5118306441592320T^8 + 3593304713206665216T^6 + 552499284082577670144T^4 + 29750860085625086607360*T^2 + 511073466923509103001600
$61$	\( (T^{8} + \cdots + 14350669588681)^{2} \) (T^8 - 76T^7 - 14672T^6 + 864020T^5 + 71896726T^4 - 2783804980T^3 - 119298641408T^2 + 1886100339308*T + 14350669588681)^2
$67$	\( (T^{8} + 56 T^{7} + \cdots + 7216851865)^{2} \) (T^8 + 56T^7 - 8978T^6 - 536668T^5 + 4437850T^4 + 237221792T^3 - 901577294T^2 - 3272469100*T + 7216851865)^2
$71$	\( T^{16} + \cdots + 19\!\cdots\!76 \) T^16 + 51336T^14 + 1029907224T^12 + 10492222826016T^10 + 58523180310590256T^8 + 178022213932214611584T^6 + 271889888918378268968064T^4 + 158754216027259568631250944*T^2 + 196306793302399195530629376
$73$	\( (T^{8} + \cdots - 638114737319936)^{2} \) (T^8 - 208T^7 - 13760T^6 + 5048192T^5 - 126003584T^4 - 27519861760T^3 + 1440880758784T^2 + 4630144311296*T - 638114737319936)^2
$79$	\( (T^{8} + \cdots + 15082254438400)^{2} \) (T^8 + 80T^7 - 14768T^6 - 1335040T^5 + 17745856T^4 + 2746081280T^3 - 8701199360T^2 - 1478650470400*T + 15082254438400)^2
$83$	\( T^{16} + \cdots + 11\!\cdots\!21 \) T^16 + 82884T^14 + 2771373096T^12 + 47675149398324T^10 + 444255526324941534T^8 + 2152785072825910495836T^6 + 4736367860723654413428648T^4 + 3746289379848649615408807980*T^2 + 1139986890170411000063988321
$89$	\( T^{16} + \cdots + 95\!\cdots\!25 \) T^16 + 72360T^14 + 2053953612T^12 + 29442669662520T^10 + 230941220295716406T^8 + 1003326989666216304600T^6 + 2319286306124995642147500T^4 + 2546707126910554657774125000*T^2 + 957861272638838140288344140625
$97$	\( (T^{8} - 16 T^{7} + \cdots + 102974242816)^{2} \) (T^8 - 16T^7 - 31616T^6 - 308608T^5 + 204514816T^4 + 10092929024T^3 + 122269229056T^2 - 379981938688*T + 102974242816)^2
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\(n\)	\(487\)	\(731\)
\(\chi(n)\)	\(1\)	\(-1\)