LMFDB - Newform orbit 1560.2.ec.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(121,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1560.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1560.ec (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:	$12.4566627153$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} + 62x^{14} + 1487x^{12} + 17244x^{10} + 98079x^{8} + 239374x^{6} + 141601x^{4} + 11256x^{2} + 144 $ x^16 + 62x^14 + 1487x^12 + 17244x^10 + 98079x^8 + 239374x^6 + 141601x^4 + 11256*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q - \beta_{7} q^{3} + (\beta_{9} - \beta_{8}) q^{5} + ( - \beta_{13} - \beta_{11}) q^{7} + ( - \beta_{7} - 1) q^{9}+O(q^{10}) $ q - b7 * q^3 + (b9 - b8) * q^5 + (-b13 - b11) * q^7 + (-b7 - 1) * q^9	$ q - \beta_{7} q^{3} + (\beta_{9} - \beta_{8}) q^{5} + ( - \beta_{13} - \beta_{11}) q^{7} + ( - \beta_{7} - 1) q^{9} + ( - \beta_{9} + \beta_{2}) q^{11} + (\beta_{14} - \beta_{9} + \beta_{7} + \cdots + 1) q^{13}+ \cdots + (\beta_{9} - \beta_{8} - \beta_{2} - \beta_1) q^{99}+O(q^{100}) $ q - b7 * q^3 + (b9 - b8) * q^5 + (-b13 - b11) * q^7 + (-b7 - 1) * q^9 + (-b9 + b2) * q^11 + (b14 - b9 + b7 - b1 + 1) * q^13 + b9 * q^15 + (-b15 + 2b2 + b1) q^17 + (b12 - b10 - b8 + b7 - b1 - 1) * q^19 + (b12 - b11 + b10) * q^21 + (-b15 + b14 + b13 - b12 + b11 + b9 - 2b8 - 2b7 + b6 - 2b4) q^23 - q^25 - q^27 + (-b15 + b14 + b13 - b12 + b10 - b7 - 2b4 + b2 + 2b1) * q^29 + (-b15 + b9 - b8 + 2b6 + b5 - 2b4 - b3) * q^31 + (-b8 - b1) * q^33 - b15 * q^35 + (-b15 + b13 + b12 + b10 + 2b9 - b7 + b5 + b4 + b2 - 2) q^37 + (b14 - b8 - b4 - b2 - b1 + 1) * q^39 + (-b15 + 2b14 - b13 - b12 + b11 - 3b9 + b7 - b5 - b4 + 2) * q^41 + (-b13 - b12 + b11 - b10 - 2b9 + b8 - 2b7 - b6 + b3 + 2b2 + b1 - 2) q^43 + b8 * q^45 + (b15 - b13 + b11 - b10 - b9 + b8 - 2b6 + b4 + b3 - b2 - b1) q^47 + (b14 + b13 - b12 + 2b11 - b10 + b9 - 2b8 - 4b7 - b6 + b5 - b4 - b2 - 2b1) * q^49 + (b5 + b4 + b2 - b1) * q^51 + (-2b3 + 2) q^53 + (-b7 - b6) * q^55 + (-b13 + b11 - b10 + b9 - b8 + 2b7 - b2 - b1 + 1) q^57 + (b14 - b13 - b11 + 2b8 - b7 + b5 + b1 + 1) q^59 + (b15 + b13 + b12 - b11 + b10 + 2b9 - b8 - 3b7 - 3) * q^61 + (b13 + b12 + b10) * q^63 + (-b12 - b8 - b7 - b6 + b3) * q^65 + (-b15 - b13 - b12 + b11 - b9 + b7 - b6 + b5 + b4 + 2b3 + 2) q^67 + (-b14 - b12 - b10 + 2b9 - b8 - 2b7 + b6 + b5 - b3 - 2) * q^69 + (b14 + b13 + b11 - 5b8 + b7 + b5 - 1) q^71 + (-b15 + b13 + b12 - 2b11 + 2b10 + 4b9 - 4b8 - 4b7 + 2b6 + b5 - 2b4 - b3 - b2 - b1 - 2) q^73 + b7 * q^75 + (b12 - b11 - b10 - 2b9 - 2b8 - b5 - b4 - b3 + b2 - b1) * q^77 + (b13 + 2b12 - b11 - b10 + 3b9 + 3b8 + 2b3 + b2 - b1 - 4) * q^79 + b7 * q^81 + (b12 - b11 + b10 + 4b7 - 2b6 - b5 + b4 + b3 + b2 + b1 + 2) * q^83 + (b13 + b11 - b6 - b3) * q^85 + (-b14 + b13 - b11 - b7 + b5 + 2b2 + b1 - 1) q^87 + (-b15 + b14 + b13 + b12 + b10 + b9 - b7 - 2) * q^89 + (-2b14 - 4b13 - b11 - 3b10 - 2b9 + 3b8 + b7 + b4 + b3 - 2b1 + 1) * q^91 + (b15 - 3b14 + b9 + b6 + 2b5 + 2b4 - 2b3) * q^93 + (b14 - 2b9 + b8 - b7 - b6 - b5 + b3 - 1) q^95 + (-b15 + b14 - 2b12 + 2b10 - 3b8 - b7 - b5 - 2b4 - b1 + 1) * q^97 + (b9 - b8 - b2 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{3} - 6 q^{7} - 8 q^{9}+O(q^{10}) $ 16 * q + 8 * q^3 - 6 * q^7 - 8 * q^9	$ 16 q + 8 q^{3} - 6 q^{7} - 8 q^{9} + 2 q^{13} - 12 q^{19} + 12 q^{23} - 16 q^{25} - 16 q^{27} - 4 q^{29} - 18 q^{37} + 10 q^{39} - 20 q^{43} + 30 q^{49} + 24 q^{53} + 6 q^{55} + 12 q^{59} - 18 q^{61} + 6 q^{63} + 4 q^{65} + 18 q^{67} - 12 q^{69} - 24 q^{71} - 8 q^{75} + 8 q^{77} - 32 q^{79} - 8 q^{81} + 4 q^{87} - 24 q^{89} + 18 q^{91} + 12 q^{93} - 12 q^{95} - 6 q^{97}+O(q^{100}) $ 16 * q + 8 * q^3 - 6 * q^7 - 8 * q^9 + 2 * q^13 - 12 * q^19 + 12 * q^23 - 16 * q^25 - 16 * q^27 - 4 * q^29 - 18 * q^37 + 10 * q^39 - 20 * q^43 + 30 * q^49 + 24 * q^53 + 6 * q^55 + 12 * q^59 - 18 * q^61 + 6 * q^63 + 4 * q^65 + 18 * q^67 - 12 * q^69 - 24 * q^71 - 8 * q^75 + 8 * q^77 - 32 * q^79 - 8 * q^81 + 4 * q^87 - 24 * q^89 + 18 * q^91 + 12 * q^93 - 12 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 62x^{14} + 1487x^{12} + 17244x^{10} + 98079x^{8} + 239374x^{6} + 141601x^{4} + 11256x^{2} + 144 $ x^16 + 62*x^14 + 1487*x^12 + 17244*x^10 + 98079*x^8 + 239374*x^6 + 141601*x^4 + 11256*x^2 + 144 :

$\beta_{1}$	$=$	$ ( - 83 \nu^{14} - 5048 \nu^{12} - 118322 \nu^{10} - 1331740 \nu^{8} - 7227123 \nu^{6} + \cdots + 133824 ) / 2033408 $ (-83v^14 - 5048v^12 - 118322v^10 - 1331740v^8 - 7227123v^6 - 15912660v^4 - 5798128v^2 + 1016704v + 133824) / 2033408
$\beta_{2}$	$=$	$ ( 83 \nu^{14} + 5048 \nu^{12} + 118322 \nu^{10} + 1331740 \nu^{8} + 7227123 \nu^{6} + 15912660 \nu^{4} + \cdots - 133824 ) / 2033408 $ (83v^14 + 5048v^12 + 118322v^10 + 1331740v^8 + 7227123v^6 + 15912660v^4 + 5798128v^2 + 1016704v - 133824) / 2033408
$\beta_{3}$	$=$	$ ( 49 \nu^{14} + 2972 \nu^{12} + 69698 \nu^{10} + 789816 \nu^{8} + 4379181 \nu^{6} + 10258996 \nu^{4} + \cdots + 147936 ) / 584064 $ (49v^14 + 2972v^12 + 69698v^10 + 789816v^8 + 4379181v^6 + 10258996v^4 + 4842184*v^2 + 147936) / 584064
$\beta_{4}$	$=$	$ ( 16575 \nu^{15} + 162893 \nu^{14} + 1009515 \nu^{13} + 10098145 \nu^{12} + 23441574 \nu^{11} + \cdots + 878686800 ) / 219608064 $ (16575v^15 + 162893v^14 + 1009515v^13 + 10098145v^12 + 23441574v^11 + 242065234v^10 + 254802366v^9 + 2802832746v^8 + 1239614883v^7 + 15873838473v^6 + 1621246263v^5 + 38208565973v^4 - 3556902648v^3 + 20749512008v^2 - 2216479824*v + 878686800) / 219608064
$\beta_{5}$	$=$	$ ( - 16575 \nu^{15} + 162893 \nu^{14} - 1009515 \nu^{13} + 10098145 \nu^{12} - 23441574 \nu^{11} + \cdots + 878686800 ) / 219608064 $ (-16575v^15 + 162893v^14 - 1009515v^13 + 10098145v^12 - 23441574v^11 + 242065234v^10 - 254802366v^9 + 2802832746v^8 - 1239614883v^7 + 15873838473v^6 - 1621246263v^5 + 38208565973v^4 + 3556902648v^3 + 20749512008v^2 + 2216479824*v + 878686800) / 219608064
$\beta_{6}$	$=$	$ ( 27209 \nu^{15} + 4606 \nu^{14} + 1687465 \nu^{13} + 279368 \nu^{12} + 40480570 \nu^{11} + \cdots + 13905984 ) / 109804032 $ (27209v^15 + 4606v^14 + 1687465v^13 + 279368v^12 + 40480570v^11 + 6551612v^10 + 469509378v^9 + 74242704v^8 + 2670709197v^7 + 411643014v^6 + 6512359061v^5 + 964345624v^4 + 3777569120v^3 + 455165296v^2 + 153454704*v + 13905984) / 109804032
$\beta_{7}$	$=$	$ ( - 2788 \nu^{15} - 173105 \nu^{13} - 4160900 \nu^{11} - 48431238 \nu^{9} - 277439472 \nu^{7} + \cdots - 3050112 ) / 6100224 $ (-2788v^15 - 173105v^13 - 4160900v^11 - 48431238v^9 - 277439472v^7 - 689056081v^5 - 442521568v^3 - 48776112v - 3050112) / 6100224
$\beta_{8}$	$=$	$ ( - 289708 \nu^{15} - 81627 \nu^{14} - 17948078 \nu^{13} - 5062395 \nu^{12} - 429957692 \nu^{11} + \cdots - 460364112 ) / 329412096 $ (-289708v^15 - 81627v^14 - 17948078v^13 - 5062395v^12 - 429957692v^11 - 121441710v^10 - 4976069916v^9 - 1408528134v^8 - 28191542820v^7 - 8012127591v^6 - 68113633750v^5 - 19537077183v^4 - 38129905636v^3 - 11332707360v^2 - 1895457360*v - 460364112) / 329412096
$\beta_{9}$	$=$	$ ( 289708 \nu^{15} - 81627 \nu^{14} + 17948078 \nu^{13} - 5062395 \nu^{12} + 429957692 \nu^{11} + \cdots - 460364112 ) / 329412096 $ (289708v^15 - 81627v^14 + 17948078v^13 - 5062395v^12 + 429957692v^11 - 121441710v^10 + 4976069916v^9 - 1408528134v^8 + 28191542820v^7 - 8012127591v^6 + 68113633750v^5 - 19537077183v^4 + 38129905636v^3 - 11332707360v^2 + 1895457360*v - 460364112) / 329412096
$\beta_{10}$	$=$	$ ( - 1099625 \nu^{15} + 353127 \nu^{14} - 68204113 \nu^{13} + 21807951 \nu^{12} + \cdots + 1556315568 ) / 658824192 $ (-1099625v^15 + 353127v^14 - 68204113v^13 + 21807951v^12 - 1636741402v^11 + 520632870v^10 - 18997772874v^9 + 6003768582v^8 - 108233748885v^7 + 33909723243v^6 - 265211117861v^5 + 82060206699v^4 - 160250269496v^3 + 47880318072v^2 - 15779265936*v + 1556315568) / 658824192
$\beta_{11}$	$=$	$ ( - 558943 \nu^{15} + 82743 \nu^{14} - 34590617 \nu^{13} + 5123379 \nu^{12} - 827223362 \nu^{11} + \cdots + 1144674576 ) / 329412096 $ (-558943v^15 + 82743v^14 - 34590617v^13 + 5123379v^12 - 827223362v^11 + 122901726v^10 - 9545957778v^9 + 1429486830v^8 - 53783025315v^7 + 8204532939v^6 - 128219646805v^5 + 20564988063v^4 - 66784275196v^3 + 13846600176v^2 - 508718784*v + 1144674576) / 329412096
$\beta_{12}$	$=$	$ ( - 2217511 \nu^{15} - 187641 \nu^{14} - 137385347 \nu^{13} - 11561193 \nu^{12} + \cdots + 733033584 ) / 658824192 $ (-2217511v^15 - 187641v^14 - 137385347v^13 - 11561193v^12 - 3291188126v^11 - 274829418v^10 - 38089688430v^9 - 3144794922v^8 - 215799799515v^7 - 17500657365v^6 - 521650411471v^5 - 40930230573v^4 - 293818819888v^3 - 20187117720v^2 - 16796703504*v + 733033584) / 658824192
$\beta_{13}$	$=$	$ ( 2217511 \nu^{15} - 187641 \nu^{14} + 137385347 \nu^{13} - 11561193 \nu^{12} + 3291188126 \nu^{11} + \cdots + 733033584 ) / 658824192 $ (2217511v^15 - 187641v^14 + 137385347v^13 - 11561193v^12 + 3291188126v^11 - 274829418v^10 + 38089688430v^9 - 3144794922v^8 + 215799799515v^7 - 17500657365v^6 + 521650411471v^5 - 40930230573v^4 + 293818819888v^3 - 20187117720v^2 + 16796703504*v + 733033584) / 658824192
$\beta_{14}$	$=$	$ ( 796727 \nu^{15} + 53335 \nu^{14} + 49369231 \nu^{13} + 3294995 \nu^{12} + 1182941758 \nu^{11} + \cdots + 606275760 ) / 219608064 $ (796727v^15 + 53335v^14 + 49369231v^13 + 3294995v^12 + 1182941758v^11 + 78506582v^10 + 13693991334v^9 + 899779566v^8 + 77598342507v^7 + 5010266427v^6 + 187460551595v^5 + 11714718943v^4 + 104509559504v^3 + 5928512680v^2 + 4372588752*v + 606275760) / 219608064
$\beta_{15}$	$=$	$ ( 1576879 \nu^{15} - 162893 \nu^{14} + 97728947 \nu^{13} - 10098145 \nu^{12} + 2342441942 \nu^{11} + \cdots - 878686800 ) / 219608064 $ (1576879v^15 - 162893v^14 + 97728947v^13 - 10098145v^12 + 2342441942v^11 - 242065234v^10 + 27133180302v^9 - 2802832746v^8 + 153957070131v^7 - 15873838473v^6 + 373299856927v^5 - 38208565973v^4 + 212576021656v^3 - 20749512008v^2 + 10961657328*v - 878686800) / 219608064

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

$\nu$	$=$	$ \beta_{2} + \beta_1 $ b2 + b1
$\nu^{2}$	$=$	$ -\beta_{15} + 2\beta_{14} + \beta_{13} + \beta_{12} - \beta_{4} + \beta_{3} - 8 $ -b15 + 2*b14 + b13 + b12 - b4 + b3 - 8
$\nu^{3}$	$=$	$ -2\beta_{15} + 6\beta_{9} - 6\beta_{8} - 8\beta_{7} - 2\beta_{5} - 13\beta_{2} - 13\beta _1 - 4 $ -2b15 + 6b9 - 6b8 - 8b7 - 2b5 - 13b2 - 13*b1 - 4
$\nu^{4}$	$=$	$ 15 \beta_{15} - 30 \beta_{14} - 13 \beta_{13} - 15 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \cdots + 114 $ 15b15 - 30b14 - 13b13 - 15b12 + 2b11 + 2b10 + 2b9 + 2b8 + 15b4 - 21b3 + 6b2 - 6b1 + 114
$\nu^{5}$	$=$	$ 42 \beta_{15} - 2 \beta_{13} - 10 \beta_{12} + 12 \beta_{11} - 12 \beta_{10} - 128 \beta_{9} + \cdots + 110 $ 42b15 - 2b13 - 10b12 + 12b11 - 12b10 - 128b9 + 128b8 + 220b7 + 4b6 + 44b5 - 2b4 - 2b3 + 193b2 + 193b1 + 110
$\nu^{6}$	$=$	$ - 237 \beta_{15} + 474 \beta_{14} + 193 \beta_{13} + 225 \beta_{12} - 32 \beta_{11} - 32 \beta_{10} + \cdots - 1768 $ -237b15 + 474b14 + 193b13 + 225b12 - 32b11 - 32b10 - 12b9 - 12b8 + 4b5 - 233b4 + 369b3 - 168b2 + 168*b1 - 1768
$\nu^{7}$	$=$	$ - 774 \beta_{15} + 80 \beta_{13} + 268 \beta_{12} - 348 \beta_{11} + 348 \beta_{10} + 2306 \beta_{9} + \cdots - 2320 $ -774b15 + 80b13 + 268b12 - 348b11 + 348b10 + 2306b9 - 2306b8 - 4640b7 - 256b6 - 834b5 + 60b4 + 128b3 - 3033b2 - 3033b1 - 2320
$\nu^{8}$	$=$	$ 3839 \beta_{15} - 7678 \beta_{14} - 3081 \beta_{13} - 3451 \beta_{12} + 370 \beta_{11} + 370 \beta_{10} + \cdots + 28614 $ 3839b15 - 7678b14 - 3081b13 - 3451b12 + 370b11 + 370b10 - 586b9 - 586b8 - 236b5 + 3603b4 - 6241b3 + 3630b2 - 3630*b1 + 28614
$\nu^{9}$	$=$	$ 13710 \beta_{15} - 1986 \beta_{13} - 5662 \beta_{12} + 7648 \beta_{11} - 7648 \beta_{10} - 40012 \beta_{9} + \cdots + 44850 $ 13710b15 - 1986b13 - 5662b12 + 7648b11 - 7648b10 - 40012b9 + 40012b8 + 89700b7 + 8164b6 + 15156b5 - 1446b4 - 4082b3 + 49301b2 + 49301b1 + 44850
$\nu^{10}$	$=$	$ - 62901 \beta_{15} + 125802 \beta_{14} + 51397 \beta_{13} + 54173 \beta_{12} - 2776 \beta_{11} + \cdots - 474700 $ -62901b15 + 125802b14 + 51397b13 + 54173b12 - 2776b11 - 2776b10 + 25392b9 + 25392b8 + 7624b5 - 55277b4 + 105253b3 - 71736b2 + 71736*b1 - 474700
$\nu^{11}$	$=$	$ - 239890 \beta_{15} + 40888 \beta_{13} + 111312 \beta_{12} - 152200 \beta_{11} + 152200 \beta_{10} + \cdots - 834788 $ -239890b15 + 40888b13 + 111312b12 - 152200b11 + 152200b10 + 693614b9 - 693614b8 - 1669576b7 - 203952b6 - 270946b5 + 31056b4 + 101976b3 - 818949b2 - 818949b1 - 834788
$\nu^{12}$	$=$	$ 1038639 \beta_{15} - 2077278 \beta_{14} - 880037 \beta_{13} - 866775 \beta_{12} - 13262 \beta_{11} + \cdots + 7997930 $ 1038639b15 - 2077278b14 - 880037b13 - 866775b12 - 13262b11 - 13262b10 - 695542b9 - 695542b8 - 194120b5 + 844519b4 - 1791821b3 + 1359910b2 - 1359910*b1 + 7997930
$\nu^{13}$	$=$	$ 4190370 \beta_{15} - 765330 \beta_{13} - 2126354 \beta_{12} + 2891684 \beta_{11} - 2891684 \beta_{10} + \cdots + 15236950 $ 4190370b15 - 765330b13 - 2126354b12 + 2891684b11 - 2891684b10 - 12151736b9 + 12151736b8 + 30473900b7 + 4525668b6 + 4806092b5 - 615722b4 - 2262834b3 + 13807961b2 + 13807961b1 + 15236950
$\nu^{14}$	$=$	$ - 17266157 \beta_{15} + 34532314 \beta_{14} + 15305465 \beta_{13} + 14075257 \beta_{12} + \cdots - 136174416 $ -17266157b15 + 34532314b14 + 15305465b13 + 14075257b12 + 1230208b11 + 1230208b10 + 16168236b9 + 16168236b8 + 4376060b5 - 12890097b4 + 30895369b3 - 25197560b2 + 25197560*b1 - 136174416
$\nu^{15}$	$=$	$ - 73359086 \beta_{15} + 13529040 \beta_{13} + 40056980 \beta_{12} - 53586020 \beta_{11} + \cdots - 274976120 $ -73359086b15 + 13529040b13 + 40056980b12 - 53586020b11 + 53586020b10 + 215590778b9 - 215590778b8 - 549952240b7 - 93944128b6 - 84882386b5 + 11523300b4 + 46972064b3 - 235358881b2 - 235358881b1 - 274976120

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$.

$n$	$391$	$521$	$781$	$937$	$1081$
$\chi(n)$	$1$	$1$	$1$	$1$	$1 + \beta_{7}$

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} $

121.1

	−	0.126022i
	−	2.40689i
	−	2.69881i
		4.23172i
	−	0.274343i
		4.09764i
	−	3.66426i
		0.840962i
		0.274343i
	−	4.09764i
		3.66426i
	−	0.840962i
		0.126022i
		2.40689i
		2.69881i
	−	4.23172i

0.500000

−

0.866025i

−

1.00000i

−4.40668

2.54420i

−0.500000

−

0.866025i

121.2

0.500000

−

0.866025i

−

1.00000i

−1.67646

0.967902i

−0.500000

−

0.866025i

121.3

0.500000

−

0.866025i

−

1.00000i

1.26295

−

0.729162i

−0.500000

−

0.866025i

121.4

0.500000

−

0.866025i

−

1.00000i

3.32019

−

1.91691i

−0.500000

−

0.866025i

121.5

0.500000

−

0.866025i

1.00000i

−3.73246

2.15494i

−0.500000

−

0.866025i

121.6

0.500000

−

0.866025i

1.00000i

−2.23858

1.29244i

−0.500000

−

0.866025i

121.7

0.500000

−

0.866025i

1.00000i

1.63787

−

0.945627i

−0.500000

−

0.866025i

121.8

0.500000

−

0.866025i

1.00000i

2.83316

−

1.63573i

−0.500000

−

0.866025i

361.1

0.500000

0.866025i

−

1.00000i

−3.73246

−

2.15494i

−0.500000

0.866025i

361.2

0.500000

0.866025i

−

1.00000i

−2.23858

−

1.29244i

−0.500000

0.866025i

361.3

0.500000

0.866025i

−

1.00000i

1.63787

0.945627i

−0.500000

0.866025i

361.4

0.500000

0.866025i

−

1.00000i

2.83316

1.63573i

−0.500000

0.866025i

361.5

0.500000

0.866025i

1.00000i

−4.40668

−

2.54420i

−0.500000

0.866025i

361.6

0.500000

0.866025i

1.00000i

−1.67646

−

0.967902i

−0.500000

0.866025i

361.7

0.500000

0.866025i

1.00000i

1.26295

0.729162i

−0.500000

0.866025i

361.8

0.500000

0.866025i

1.00000i

3.32019

1.91691i

−0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1560.2.ec.e	✓	16
13.e	even	6	1	inner	1560.2.ec.e	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1560.2.ec.e	✓	16	1.a	even	1	1	trivial
1560.2.ec.e	✓	16	13.e	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{16} + 6 T_{7}^{15} - 25 T_{7}^{14} - 222 T_{7}^{13} + 560 T_{7}^{12} + 4878 T_{7}^{11} + \cdots + 14409616 $ T7^16 + 6*T7^15 - 25*T7^14 - 222*T7^13 + 560*T7^12 + 4878*T7^11 - 6143*T7^10 - 62634*T7^9 + 71232*T7^8 + 500874*T7^7 - 381709*T7^6 - 2457066*T7^5 + 2237869*T7^4 + 6827976*T7^3 - 5875380*T7^2 - 11205792*T7 + 14409616 acting on $S_{2}^{\mathrm{new}}(1560, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} - T + 1)^{8} $ (T^2 - T + 1)^8
$5$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$7$	$ T^{16} + 6 T^{15} + \cdots + 14409616 $ T^16 + 6T^15 - 25T^14 - 222T^13 + 560T^12 + 4878T^11 - 6143T^10 - 62634T^9 + 71232T^8 + 500874T^7 - 381709T^6 - 2457066T^5 + 2237869T^4 + 6827976T^3 - 5875380T^2 - 11205792*T + 14409616
$11$	$ T^{16} - 33 T^{14} + \cdots + 1024 $ T^16 - 33T^14 + 877T^12 + 684T^11 - 6388T^10 - 7248T^9 + 36024T^8 + 94176T^7 + 59296T^6 - 45504T^5 - 49984T^4 + 33792T^3 + 51968T^2 + 12288*T + 1024
$13$	$ T^{16} + \cdots + 815730721 $ T^16 - 2T^15 - 31T^14 - 26T^13 + 291T^12 + 2276T^11 + 2408T^10 - 20932T^9 - 80922T^8 - 272116T^7 + 406952T^6 + 5000372T^5 + 8311251T^4 - 9653618T^3 - 149631079T^2 - 125497034*T + 815730721
$17$	$ T^{16} + \cdots + 11246178304 $ T^16 + 82T^14 - 56T^13 + 4358T^12 - 3492T^11 + 139156T^10 - 117192T^9 + 3244156T^8 - 2313544T^7 + 49504952T^6 - 23873264T^5 + 538236816T^4 - 99758464T^3 + 3111646976T^2 + 1347233792T + 11246178304
$19$	$ T^{16} + 12 T^{15} + \cdots + 24920064 $ T^16 + 12T^15 - 8T^14 - 672T^13 - 84T^12 + 22752T^11 + 10368T^10 - 495744T^9 + 311696T^8 + 5811840T^7 - 7188928T^6 - 44828160T^5 + 136801024T^4 - 103234560T^3 - 29964288T^2 + 45047808*T + 24920064
$23$	$ T^{16} + \cdots + 36021003264 $ T^16 - 12T^15 + 203T^14 - 1100T^13 + 13197T^12 - 47088T^11 + 595892T^10 - 1066608T^9 + 15161080T^8 - 1661184T^7 + 295036128T^6 + 205726400T^5 + 3396275392T^4 + 5045722368T^3 + 27913925376T^2 + 29643992064*T + 36021003264
$29$	$ T^{16} + \cdots + 76788843664 $ T^16 + 4T^15 + 145T^14 - 4T^13 + 12525T^12 - 4220T^11 + 577688T^10 - 602700T^9 + 18272710T^8 - 11560612T^7 + 287406432T^6 - 168255800T^5 + 3221474244T^4 - 1060847096T^3 + 19018689416T^2 - 4450354480*T + 76788843664
$31$	$ T^{16} + \cdots + 335714789281 $ T^16 + 416T^14 + 71644T^12 + 6614624T^10 + 353358310T^8 + 10917933920T^6 + 180744663196T^4 + 1247553923744*T^2 + 335714789281
$37$	$ T^{16} + \cdots + 2390818816 $ T^16 + 18T^15 - 9T^14 - 2106T^13 - 671T^12 + 216204T^11 + 1118240T^10 - 3338016T^9 - 18636240T^8 + 39245568T^7 + 254817280T^6 - 60955392T^5 - 1192773376T^4 - 157065216T^3 + 4597821440T^2 + 5783027712*T + 2390818816
$41$	$ T^{16} - 118 T^{14} + \cdots + 85229824 $ T^16 - 118T^14 + 11222T^12 - 42372T^11 - 219452T^10 + 1141680T^9 + 3859212T^8 - 20229096T^7 - 27053848T^6 + 155942928T^5 + 247193968T^4 - 475555392T^3 + 46575936T^2 + 236191488*T + 85229824
$43$	$ T^{16} + \cdots + 66725605969 $ T^16 + 20T^15 + 430T^14 + 4576T^13 + 63786T^12 + 567392T^11 + 6123008T^10 + 38542968T^9 + 270376279T^8 + 1182399940T^7 + 6998115216T^6 + 25803739700T^5 + 85723182762T^4 + 151105329728T^3 + 200054347454T^2 + 137763489160*T + 66725605969
$47$	$ T^{16} + \cdots + 15913317904 $ T^16 + 270T^14 + 27681T^12 + 1415476T^10 + 39059732T^8 + 582517800T^6 + 4435974916T^4 + 14860658176*T^2 + 15913317904
$53$	$ (T^{8} - 12 T^{7} + \cdots - 8192)^{2} $ (T^8 - 12T^7 - 60T^6 + 736T^5 + 1472T^4 - 8448T^3 - 6656T^2 + 28672*T - 8192)^2
$59$	$ T^{16} + \cdots + 429484176 $ T^16 - 12T^15 - 123T^14 + 2052T^13 + 16645T^12 - 219660T^11 - 626344T^10 + 9126516T^9 + 46018006T^8 - 23641188T^7 - 297966272T^6 + 60242376T^5 + 1865125060T^4 + 2445765720T^3 + 82708872T^2 - 1138244976*T + 429484176
$61$	$ T^{16} + \cdots + 1590733456 $ T^16 + 18T^15 + 269T^14 + 2966T^13 + 31378T^12 + 281178T^11 + 2174165T^10 + 13554114T^9 + 69002482T^8 + 278657886T^7 + 900275897T^6 + 2249615458T^5 + 4345339805T^4 + 6080931272T^3 + 6230063996T^2 + 4002758240*T + 1590733456
$67$	$ T^{16} + \cdots + 682084381456 $ T^16 - 18T^15 - 137T^14 + 4410T^13 + 19960T^12 - 957654T^11 + 4075857T^10 + 56331198T^9 - 413031832T^8 - 3030807474T^7 + 50851457539T^6 - 272970117978T^5 + 696906295797T^4 - 662407319340T^3 - 382610249172T^2 + 714663853488*T + 682084381456
$71$	$ T^{16} + \cdots + 21041243136 $ T^16 + 24T^15 + 66T^14 - 3024T^13 - 10562T^12 + 613452T^11 + 9402212T^10 + 60502008T^9 + 169268092T^8 - 60878616T^7 - 1337935208T^6 + 411455568T^5 + 18886878736T^4 + 44295380160T^3 + 12314732928T^2 - 45501166080*T + 21041243136
$73$	$ T^{16} + \cdots + 216645564304 $ T^16 + 630T^14 + 154027T^12 + 18336452T^10 + 1083314979T^8 + 27954565702T^6 + 199077842441T^4 + 398915511656*T^2 + 216645564304
$79$	$ (T^{8} + 16 T^{7} + \cdots + 28089769)^{2} $ (T^8 + 16T^7 - 272T^6 - 4296T^5 + 25622T^4 + 356800T^3 - 1125648T^2 - 8773800*T + 28089769)^2
$83$	$ T^{16} + \cdots + 637544759296 $ T^16 + 504T^14 + 102016T^12 + 10652480T^10 + 613209408T^8 + 19208834560T^6 + 299500516352T^4 + 1827336814592*T^2 + 637544759296
$89$	$ T^{16} + \cdots + 1448868096 $ T^16 + 24T^15 + 162T^14 - 720T^13 - 10010T^12 + 39036T^11 + 894212T^10 + 3924984T^9 + 1358524T^8 - 30550488T^7 - 28359320T^6 + 199760880T^5 + 319570480T^4 - 508321920T^3 - 688006848T^2 + 858723840*T + 1448868096
$97$	$ T^{16} + \cdots + 5766702748816 $ T^16 + 6T^15 - 433T^14 - 2670T^13 + 146272T^12 + 526986T^11 - 18785311T^10 - 76618338T^9 + 1693762504T^8 + 11502410118T^7 - 22858254109T^6 - 295365828834T^5 + 425277346645T^4 + 5640136911780T^3 + 7443755905148T^2 - 13104100987728*T + 5766702748816
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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1560.ec (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{3} + (\beta_{9} - \beta_{8}) q^{5} + ( - \beta_{13} - \beta_{11}) q^{7} + ( - \beta_{7} - 1) q^{9}+O(q^{10}) \) q - b7 * q^3 + (b9 - b8) * q^5 + (-b13 - b11) * q^7 + (-b7 - 1) * q^9	\( q - \beta_{7} q^{3} + (\beta_{9} - \beta_{8}) q^{5} + ( - \beta_{13} - \beta_{11}) q^{7} + ( - \beta_{7} - 1) q^{9} + ( - \beta_{9} + \beta_{2}) q^{11} + (\beta_{14} - \beta_{9} + \beta_{7} + \cdots + 1) q^{13}+ \cdots + (\beta_{9} - \beta_{8} - \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) q - b7 * q^3 + (b9 - b8) * q^5 + (-b13 - b11) * q^7 + (-b7 - 1) * q^9 + (-b9 + b2) * q^11 + (b14 - b9 + b7 - b1 + 1) * q^13 + b9 * q^15 + (-b15 + 2b2 + b1) q^17 + (b12 - b10 - b8 + b7 - b1 - 1) * q^19 + (b12 - b11 + b10) * q^21 + (-b15 + b14 + b13 - b12 + b11 + b9 - 2b8 - 2b7 + b6 - 2b4) q^23 - q^25 - q^27 + (-b15 + b14 + b13 - b12 + b10 - b7 - 2b4 + b2 + 2b1) * q^29 + (-b15 + b9 - b8 + 2b6 + b5 - 2b4 - b3) * q^31 + (-b8 - b1) * q^33 - b15 * q^35 + (-b15 + b13 + b12 + b10 + 2b9 - b7 + b5 + b4 + b2 - 2) q^37 + (b14 - b8 - b4 - b2 - b1 + 1) * q^39 + (-b15 + 2b14 - b13 - b12 + b11 - 3b9 + b7 - b5 - b4 + 2) * q^41 + (-b13 - b12 + b11 - b10 - 2b9 + b8 - 2b7 - b6 + b3 + 2b2 + b1 - 2) q^43 + b8 * q^45 + (b15 - b13 + b11 - b10 - b9 + b8 - 2b6 + b4 + b3 - b2 - b1) q^47 + (b14 + b13 - b12 + 2b11 - b10 + b9 - 2b8 - 4b7 - b6 + b5 - b4 - b2 - 2b1) * q^49 + (b5 + b4 + b2 - b1) * q^51 + (-2b3 + 2) q^53 + (-b7 - b6) * q^55 + (-b13 + b11 - b10 + b9 - b8 + 2b7 - b2 - b1 + 1) q^57 + (b14 - b13 - b11 + 2b8 - b7 + b5 + b1 + 1) q^59 + (b15 + b13 + b12 - b11 + b10 + 2b9 - b8 - 3b7 - 3) * q^61 + (b13 + b12 + b10) * q^63 + (-b12 - b8 - b7 - b6 + b3) * q^65 + (-b15 - b13 - b12 + b11 - b9 + b7 - b6 + b5 + b4 + 2b3 + 2) q^67 + (-b14 - b12 - b10 + 2b9 - b8 - 2b7 + b6 + b5 - b3 - 2) * q^69 + (b14 + b13 + b11 - 5b8 + b7 + b5 - 1) q^71 + (-b15 + b13 + b12 - 2b11 + 2b10 + 4b9 - 4b8 - 4b7 + 2b6 + b5 - 2b4 - b3 - b2 - b1 - 2) q^73 + b7 * q^75 + (b12 - b11 - b10 - 2b9 - 2b8 - b5 - b4 - b3 + b2 - b1) * q^77 + (b13 + 2b12 - b11 - b10 + 3b9 + 3b8 + 2b3 + b2 - b1 - 4) * q^79 + b7 * q^81 + (b12 - b11 + b10 + 4b7 - 2b6 - b5 + b4 + b3 + b2 + b1 + 2) * q^83 + (b13 + b11 - b6 - b3) * q^85 + (-b14 + b13 - b11 - b7 + b5 + 2b2 + b1 - 1) q^87 + (-b15 + b14 + b13 + b12 + b10 + b9 - b7 - 2) * q^89 + (-2b14 - 4b13 - b11 - 3b10 - 2b9 + 3b8 + b7 + b4 + b3 - 2b1 + 1) * q^91 + (b15 - 3b14 + b9 + b6 + 2b5 + 2b4 - 2b3) * q^93 + (b14 - 2b9 + b8 - b7 - b6 - b5 + b3 - 1) q^95 + (-b15 + b14 - 2b12 + 2b10 - 3b8 - b7 - b5 - 2b4 - b1 + 1) * q^97 + (b9 - b8 - b2 - b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} - 6 q^{7} - 8 q^{9}+O(q^{10}) \) 16 * q + 8 * q^3 - 6 * q^7 - 8 * q^9	\( 16 q + 8 q^{3} - 6 q^{7} - 8 q^{9} + 2 q^{13} - 12 q^{19} + 12 q^{23} - 16 q^{25} - 16 q^{27} - 4 q^{29} - 18 q^{37} + 10 q^{39} - 20 q^{43} + 30 q^{49} + 24 q^{53} + 6 q^{55} + 12 q^{59} - 18 q^{61} + 6 q^{63} + 4 q^{65} + 18 q^{67} - 12 q^{69} - 24 q^{71} - 8 q^{75} + 8 q^{77} - 32 q^{79} - 8 q^{81} + 4 q^{87} - 24 q^{89} + 18 q^{91} + 12 q^{93} - 12 q^{95} - 6 q^{97}+O(q^{100}) \) 16 * q + 8 * q^3 - 6 * q^7 - 8 * q^9 + 2 * q^13 - 12 * q^19 + 12 * q^23 - 16 * q^25 - 16 * q^27 - 4 * q^29 - 18 * q^37 + 10 * q^39 - 20 * q^43 + 30 * q^49 + 24 * q^53 + 6 * q^55 + 12 * q^59 - 18 * q^61 + 6 * q^63 + 4 * q^65 + 18 * q^67 - 12 * q^69 - 24 * q^71 - 8 * q^75 + 8 * q^77 - 32 * q^79 - 8 * q^81 + 4 * q^87 - 24 * q^89 + 18 * q^91 + 12 * q^93 - 12 * q^95 - 6 * 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	1560.2.ec.e

Level	$1560$
Weight	$2$
Character orbit	1560.ec
Analytic conductor	$12.457$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(12.4566627153\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 62x^{14} + 1487x^{12} + 17244x^{10} + 98079x^{8} + 239374x^{6} + 141601x^{4} + 11256x^{2} + 144 \) x^16 + 62x^14 + 1487x^12 + 17244x^10 + 98079x^8 + 239374x^6 + 141601x^4 + 11256*x^2 + 144
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 83 \nu^{14} - 5048 \nu^{12} - 118322 \nu^{10} - 1331740 \nu^{8} - 7227123 \nu^{6} + \cdots + 133824 ) / 2033408 \) (-83v^14 - 5048v^12 - 118322v^10 - 1331740v^8 - 7227123v^6 - 15912660v^4 - 5798128v^2 + 1016704v + 133824) / 2033408
\(\beta_{2}\)	\(=\)	\( ( 83 \nu^{14} + 5048 \nu^{12} + 118322 \nu^{10} + 1331740 \nu^{8} + 7227123 \nu^{6} + 15912660 \nu^{4} + \cdots - 133824 ) / 2033408 \) (83v^14 + 5048v^12 + 118322v^10 + 1331740v^8 + 7227123v^6 + 15912660v^4 + 5798128v^2 + 1016704v - 133824) / 2033408
\(\beta_{3}\)	\(=\)	\( ( 49 \nu^{14} + 2972 \nu^{12} + 69698 \nu^{10} + 789816 \nu^{8} + 4379181 \nu^{6} + 10258996 \nu^{4} + \cdots + 147936 ) / 584064 \) (49v^14 + 2972v^12 + 69698v^10 + 789816v^8 + 4379181v^6 + 10258996v^4 + 4842184*v^2 + 147936) / 584064
\(\beta_{4}\)	\(=\)	\( ( 16575 \nu^{15} + 162893 \nu^{14} + 1009515 \nu^{13} + 10098145 \nu^{12} + 23441574 \nu^{11} + \cdots + 878686800 ) / 219608064 \) (16575v^15 + 162893v^14 + 1009515v^13 + 10098145v^12 + 23441574v^11 + 242065234v^10 + 254802366v^9 + 2802832746v^8 + 1239614883v^7 + 15873838473v^6 + 1621246263v^5 + 38208565973v^4 - 3556902648v^3 + 20749512008v^2 - 2216479824*v + 878686800) / 219608064
\(\beta_{5}\)	\(=\)	\( ( - 16575 \nu^{15} + 162893 \nu^{14} - 1009515 \nu^{13} + 10098145 \nu^{12} - 23441574 \nu^{11} + \cdots + 878686800 ) / 219608064 \) (-16575v^15 + 162893v^14 - 1009515v^13 + 10098145v^12 - 23441574v^11 + 242065234v^10 - 254802366v^9 + 2802832746v^8 - 1239614883v^7 + 15873838473v^6 - 1621246263v^5 + 38208565973v^4 + 3556902648v^3 + 20749512008v^2 + 2216479824*v + 878686800) / 219608064
\(\beta_{6}\)	\(=\)	\( ( 27209 \nu^{15} + 4606 \nu^{14} + 1687465 \nu^{13} + 279368 \nu^{12} + 40480570 \nu^{11} + \cdots + 13905984 ) / 109804032 \) (27209v^15 + 4606v^14 + 1687465v^13 + 279368v^12 + 40480570v^11 + 6551612v^10 + 469509378v^9 + 74242704v^8 + 2670709197v^7 + 411643014v^6 + 6512359061v^5 + 964345624v^4 + 3777569120v^3 + 455165296v^2 + 153454704*v + 13905984) / 109804032
\(\beta_{7}\)	\(=\)	\( ( - 2788 \nu^{15} - 173105 \nu^{13} - 4160900 \nu^{11} - 48431238 \nu^{9} - 277439472 \nu^{7} + \cdots - 3050112 ) / 6100224 \) (-2788v^15 - 173105v^13 - 4160900v^11 - 48431238v^9 - 277439472v^7 - 689056081v^5 - 442521568v^3 - 48776112v - 3050112) / 6100224
\(\beta_{8}\)	\(=\)	\( ( - 289708 \nu^{15} - 81627 \nu^{14} - 17948078 \nu^{13} - 5062395 \nu^{12} - 429957692 \nu^{11} + \cdots - 460364112 ) / 329412096 \) (-289708v^15 - 81627v^14 - 17948078v^13 - 5062395v^12 - 429957692v^11 - 121441710v^10 - 4976069916v^9 - 1408528134v^8 - 28191542820v^7 - 8012127591v^6 - 68113633750v^5 - 19537077183v^4 - 38129905636v^3 - 11332707360v^2 - 1895457360*v - 460364112) / 329412096
\(\beta_{9}\)	\(=\)	\( ( 289708 \nu^{15} - 81627 \nu^{14} + 17948078 \nu^{13} - 5062395 \nu^{12} + 429957692 \nu^{11} + \cdots - 460364112 ) / 329412096 \) (289708v^15 - 81627v^14 + 17948078v^13 - 5062395v^12 + 429957692v^11 - 121441710v^10 + 4976069916v^9 - 1408528134v^8 + 28191542820v^7 - 8012127591v^6 + 68113633750v^5 - 19537077183v^4 + 38129905636v^3 - 11332707360v^2 + 1895457360*v - 460364112) / 329412096
\(\beta_{10}\)	\(=\)	\( ( - 1099625 \nu^{15} + 353127 \nu^{14} - 68204113 \nu^{13} + 21807951 \nu^{12} + \cdots + 1556315568 ) / 658824192 \) (-1099625v^15 + 353127v^14 - 68204113v^13 + 21807951v^12 - 1636741402v^11 + 520632870v^10 - 18997772874v^9 + 6003768582v^8 - 108233748885v^7 + 33909723243v^6 - 265211117861v^5 + 82060206699v^4 - 160250269496v^3 + 47880318072v^2 - 15779265936*v + 1556315568) / 658824192
\(\beta_{11}\)	\(=\)	\( ( - 558943 \nu^{15} + 82743 \nu^{14} - 34590617 \nu^{13} + 5123379 \nu^{12} - 827223362 \nu^{11} + \cdots + 1144674576 ) / 329412096 \) (-558943v^15 + 82743v^14 - 34590617v^13 + 5123379v^12 - 827223362v^11 + 122901726v^10 - 9545957778v^9 + 1429486830v^8 - 53783025315v^7 + 8204532939v^6 - 128219646805v^5 + 20564988063v^4 - 66784275196v^3 + 13846600176v^2 - 508718784*v + 1144674576) / 329412096
\(\beta_{12}\)	\(=\)	\( ( - 2217511 \nu^{15} - 187641 \nu^{14} - 137385347 \nu^{13} - 11561193 \nu^{12} + \cdots + 733033584 ) / 658824192 \) (-2217511v^15 - 187641v^14 - 137385347v^13 - 11561193v^12 - 3291188126v^11 - 274829418v^10 - 38089688430v^9 - 3144794922v^8 - 215799799515v^7 - 17500657365v^6 - 521650411471v^5 - 40930230573v^4 - 293818819888v^3 - 20187117720v^2 - 16796703504*v + 733033584) / 658824192
\(\beta_{13}\)	\(=\)	\( ( 2217511 \nu^{15} - 187641 \nu^{14} + 137385347 \nu^{13} - 11561193 \nu^{12} + 3291188126 \nu^{11} + \cdots + 733033584 ) / 658824192 \) (2217511v^15 - 187641v^14 + 137385347v^13 - 11561193v^12 + 3291188126v^11 - 274829418v^10 + 38089688430v^9 - 3144794922v^8 + 215799799515v^7 - 17500657365v^6 + 521650411471v^5 - 40930230573v^4 + 293818819888v^3 - 20187117720v^2 + 16796703504*v + 733033584) / 658824192
\(\beta_{14}\)	\(=\)	\( ( 796727 \nu^{15} + 53335 \nu^{14} + 49369231 \nu^{13} + 3294995 \nu^{12} + 1182941758 \nu^{11} + \cdots + 606275760 ) / 219608064 \) (796727v^15 + 53335v^14 + 49369231v^13 + 3294995v^12 + 1182941758v^11 + 78506582v^10 + 13693991334v^9 + 899779566v^8 + 77598342507v^7 + 5010266427v^6 + 187460551595v^5 + 11714718943v^4 + 104509559504v^3 + 5928512680v^2 + 4372588752*v + 606275760) / 219608064
\(\beta_{15}\)	\(=\)	\( ( 1576879 \nu^{15} - 162893 \nu^{14} + 97728947 \nu^{13} - 10098145 \nu^{12} + 2342441942 \nu^{11} + \cdots - 878686800 ) / 219608064 \) (1576879v^15 - 162893v^14 + 97728947v^13 - 10098145v^12 + 2342441942v^11 - 242065234v^10 + 27133180302v^9 - 2802832746v^8 + 153957070131v^7 - 15873838473v^6 + 373299856927v^5 - 38208565973v^4 + 212576021656v^3 - 20749512008v^2 + 10961657328*v - 878686800) / 219608064

\(\nu\)	\(=\)	\( \beta_{2} + \beta_1 \) b2 + b1
\(\nu^{2}\)	\(=\)	\( -\beta_{15} + 2\beta_{14} + \beta_{13} + \beta_{12} - \beta_{4} + \beta_{3} - 8 \) -b15 + 2*b14 + b13 + b12 - b4 + b3 - 8
\(\nu^{3}\)	\(=\)	\( -2\beta_{15} + 6\beta_{9} - 6\beta_{8} - 8\beta_{7} - 2\beta_{5} - 13\beta_{2} - 13\beta _1 - 4 \) -2b15 + 6b9 - 6b8 - 8b7 - 2b5 - 13b2 - 13*b1 - 4
\(\nu^{4}\)	\(=\)	\( 15 \beta_{15} - 30 \beta_{14} - 13 \beta_{13} - 15 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \cdots + 114 \) 15b15 - 30b14 - 13b13 - 15b12 + 2b11 + 2b10 + 2b9 + 2b8 + 15b4 - 21b3 + 6b2 - 6b1 + 114
\(\nu^{5}\)	\(=\)	\( 42 \beta_{15} - 2 \beta_{13} - 10 \beta_{12} + 12 \beta_{11} - 12 \beta_{10} - 128 \beta_{9} + \cdots + 110 \) 42b15 - 2b13 - 10b12 + 12b11 - 12b10 - 128b9 + 128b8 + 220b7 + 4b6 + 44b5 - 2b4 - 2b3 + 193b2 + 193b1 + 110
\(\nu^{6}\)	\(=\)	\( - 237 \beta_{15} + 474 \beta_{14} + 193 \beta_{13} + 225 \beta_{12} - 32 \beta_{11} - 32 \beta_{10} + \cdots - 1768 \) -237b15 + 474b14 + 193b13 + 225b12 - 32b11 - 32b10 - 12b9 - 12b8 + 4b5 - 233b4 + 369b3 - 168b2 + 168*b1 - 1768
\(\nu^{7}\)	\(=\)	\( - 774 \beta_{15} + 80 \beta_{13} + 268 \beta_{12} - 348 \beta_{11} + 348 \beta_{10} + 2306 \beta_{9} + \cdots - 2320 \) -774b15 + 80b13 + 268b12 - 348b11 + 348b10 + 2306b9 - 2306b8 - 4640b7 - 256b6 - 834b5 + 60b4 + 128b3 - 3033b2 - 3033b1 - 2320
\(\nu^{8}\)	\(=\)	\( 3839 \beta_{15} - 7678 \beta_{14} - 3081 \beta_{13} - 3451 \beta_{12} + 370 \beta_{11} + 370 \beta_{10} + \cdots + 28614 \) 3839b15 - 7678b14 - 3081b13 - 3451b12 + 370b11 + 370b10 - 586b9 - 586b8 - 236b5 + 3603b4 - 6241b3 + 3630b2 - 3630*b1 + 28614
\(\nu^{9}\)	\(=\)	\( 13710 \beta_{15} - 1986 \beta_{13} - 5662 \beta_{12} + 7648 \beta_{11} - 7648 \beta_{10} - 40012 \beta_{9} + \cdots + 44850 \) 13710b15 - 1986b13 - 5662b12 + 7648b11 - 7648b10 - 40012b9 + 40012b8 + 89700b7 + 8164b6 + 15156b5 - 1446b4 - 4082b3 + 49301b2 + 49301b1 + 44850
\(\nu^{10}\)	\(=\)	\( - 62901 \beta_{15} + 125802 \beta_{14} + 51397 \beta_{13} + 54173 \beta_{12} - 2776 \beta_{11} + \cdots - 474700 \) -62901b15 + 125802b14 + 51397b13 + 54173b12 - 2776b11 - 2776b10 + 25392b9 + 25392b8 + 7624b5 - 55277b4 + 105253b3 - 71736b2 + 71736*b1 - 474700
\(\nu^{11}\)	\(=\)	\( - 239890 \beta_{15} + 40888 \beta_{13} + 111312 \beta_{12} - 152200 \beta_{11} + 152200 \beta_{10} + \cdots - 834788 \) -239890b15 + 40888b13 + 111312b12 - 152200b11 + 152200b10 + 693614b9 - 693614b8 - 1669576b7 - 203952b6 - 270946b5 + 31056b4 + 101976b3 - 818949b2 - 818949b1 - 834788
\(\nu^{12}\)	\(=\)	\( 1038639 \beta_{15} - 2077278 \beta_{14} - 880037 \beta_{13} - 866775 \beta_{12} - 13262 \beta_{11} + \cdots + 7997930 \) 1038639b15 - 2077278b14 - 880037b13 - 866775b12 - 13262b11 - 13262b10 - 695542b9 - 695542b8 - 194120b5 + 844519b4 - 1791821b3 + 1359910b2 - 1359910*b1 + 7997930
\(\nu^{13}\)	\(=\)	\( 4190370 \beta_{15} - 765330 \beta_{13} - 2126354 \beta_{12} + 2891684 \beta_{11} - 2891684 \beta_{10} + \cdots + 15236950 \) 4190370b15 - 765330b13 - 2126354b12 + 2891684b11 - 2891684b10 - 12151736b9 + 12151736b8 + 30473900b7 + 4525668b6 + 4806092b5 - 615722b4 - 2262834b3 + 13807961b2 + 13807961b1 + 15236950
\(\nu^{14}\)	\(=\)	\( - 17266157 \beta_{15} + 34532314 \beta_{14} + 15305465 \beta_{13} + 14075257 \beta_{12} + \cdots - 136174416 \) -17266157b15 + 34532314b14 + 15305465b13 + 14075257b12 + 1230208b11 + 1230208b10 + 16168236b9 + 16168236b8 + 4376060b5 - 12890097b4 + 30895369b3 - 25197560b2 + 25197560*b1 - 136174416
\(\nu^{15}\)	\(=\)	\( - 73359086 \beta_{15} + 13529040 \beta_{13} + 40056980 \beta_{12} - 53586020 \beta_{11} + \cdots - 274976120 \) -73359086b15 + 13529040b13 + 40056980b12 - 53586020b11 + 53586020b10 + 215590778b9 - 215590778b8 - 549952240b7 - 93944128b6 - 84882386b5 + 11523300b4 + 46972064b3 - 235358881b2 - 235358881b1 - 274976120

\(n\)	\(391\)	\(521\)	\(781\)	\(937\)	\(1081\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1 + \beta_{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 121.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} - T + 1)^{8} \) (T^2 - T + 1)^8
$5$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$7$	\( T^{16} + 6 T^{15} + \cdots + 14409616 \) T^16 + 6T^15 - 25T^14 - 222T^13 + 560T^12 + 4878T^11 - 6143T^10 - 62634T^9 + 71232T^8 + 500874T^7 - 381709T^6 - 2457066T^5 + 2237869T^4 + 6827976T^3 - 5875380T^2 - 11205792*T + 14409616
$11$	\( T^{16} - 33 T^{14} + \cdots + 1024 \) T^16 - 33T^14 + 877T^12 + 684T^11 - 6388T^10 - 7248T^9 + 36024T^8 + 94176T^7 + 59296T^6 - 45504T^5 - 49984T^4 + 33792T^3 + 51968T^2 + 12288*T + 1024
$13$	\( T^{16} + \cdots + 815730721 \) T^16 - 2T^15 - 31T^14 - 26T^13 + 291T^12 + 2276T^11 + 2408T^10 - 20932T^9 - 80922T^8 - 272116T^7 + 406952T^6 + 5000372T^5 + 8311251T^4 - 9653618T^3 - 149631079T^2 - 125497034*T + 815730721
$17$	\( T^{16} + \cdots + 11246178304 \) T^16 + 82T^14 - 56T^13 + 4358T^12 - 3492T^11 + 139156T^10 - 117192T^9 + 3244156T^8 - 2313544T^7 + 49504952T^6 - 23873264T^5 + 538236816T^4 - 99758464T^3 + 3111646976T^2 + 1347233792T + 11246178304
$19$	\( T^{16} + 12 T^{15} + \cdots + 24920064 \) T^16 + 12T^15 - 8T^14 - 672T^13 - 84T^12 + 22752T^11 + 10368T^10 - 495744T^9 + 311696T^8 + 5811840T^7 - 7188928T^6 - 44828160T^5 + 136801024T^4 - 103234560T^3 - 29964288T^2 + 45047808*T + 24920064
$23$	\( T^{16} + \cdots + 36021003264 \) T^16 - 12T^15 + 203T^14 - 1100T^13 + 13197T^12 - 47088T^11 + 595892T^10 - 1066608T^9 + 15161080T^8 - 1661184T^7 + 295036128T^6 + 205726400T^5 + 3396275392T^4 + 5045722368T^3 + 27913925376T^2 + 29643992064*T + 36021003264
$29$	\( T^{16} + \cdots + 76788843664 \) T^16 + 4T^15 + 145T^14 - 4T^13 + 12525T^12 - 4220T^11 + 577688T^10 - 602700T^9 + 18272710T^8 - 11560612T^7 + 287406432T^6 - 168255800T^5 + 3221474244T^4 - 1060847096T^3 + 19018689416T^2 - 4450354480*T + 76788843664
$31$	\( T^{16} + \cdots + 335714789281 \) T^16 + 416T^14 + 71644T^12 + 6614624T^10 + 353358310T^8 + 10917933920T^6 + 180744663196T^4 + 1247553923744*T^2 + 335714789281
$37$	\( T^{16} + \cdots + 2390818816 \) T^16 + 18T^15 - 9T^14 - 2106T^13 - 671T^12 + 216204T^11 + 1118240T^10 - 3338016T^9 - 18636240T^8 + 39245568T^7 + 254817280T^6 - 60955392T^5 - 1192773376T^4 - 157065216T^3 + 4597821440T^2 + 5783027712*T + 2390818816
$41$	\( T^{16} - 118 T^{14} + \cdots + 85229824 \) T^16 - 118T^14 + 11222T^12 - 42372T^11 - 219452T^10 + 1141680T^9 + 3859212T^8 - 20229096T^7 - 27053848T^6 + 155942928T^5 + 247193968T^4 - 475555392T^3 + 46575936T^2 + 236191488*T + 85229824
$43$	\( T^{16} + \cdots + 66725605969 \) T^16 + 20T^15 + 430T^14 + 4576T^13 + 63786T^12 + 567392T^11 + 6123008T^10 + 38542968T^9 + 270376279T^8 + 1182399940T^7 + 6998115216T^6 + 25803739700T^5 + 85723182762T^4 + 151105329728T^3 + 200054347454T^2 + 137763489160*T + 66725605969
$47$	\( T^{16} + \cdots + 15913317904 \) T^16 + 270T^14 + 27681T^12 + 1415476T^10 + 39059732T^8 + 582517800T^6 + 4435974916T^4 + 14860658176*T^2 + 15913317904
$53$	\( (T^{8} - 12 T^{7} + \cdots - 8192)^{2} \) (T^8 - 12T^7 - 60T^6 + 736T^5 + 1472T^4 - 8448T^3 - 6656T^2 + 28672*T - 8192)^2
$59$	\( T^{16} + \cdots + 429484176 \) T^16 - 12T^15 - 123T^14 + 2052T^13 + 16645T^12 - 219660T^11 - 626344T^10 + 9126516T^9 + 46018006T^8 - 23641188T^7 - 297966272T^6 + 60242376T^5 + 1865125060T^4 + 2445765720T^3 + 82708872T^2 - 1138244976*T + 429484176
$61$	\( T^{16} + \cdots + 1590733456 \) T^16 + 18T^15 + 269T^14 + 2966T^13 + 31378T^12 + 281178T^11 + 2174165T^10 + 13554114T^9 + 69002482T^8 + 278657886T^7 + 900275897T^6 + 2249615458T^5 + 4345339805T^4 + 6080931272T^3 + 6230063996T^2 + 4002758240*T + 1590733456
$67$	\( T^{16} + \cdots + 682084381456 \) T^16 - 18T^15 - 137T^14 + 4410T^13 + 19960T^12 - 957654T^11 + 4075857T^10 + 56331198T^9 - 413031832T^8 - 3030807474T^7 + 50851457539T^6 - 272970117978T^5 + 696906295797T^4 - 662407319340T^3 - 382610249172T^2 + 714663853488*T + 682084381456
$71$	\( T^{16} + \cdots + 21041243136 \) T^16 + 24T^15 + 66T^14 - 3024T^13 - 10562T^12 + 613452T^11 + 9402212T^10 + 60502008T^9 + 169268092T^8 - 60878616T^7 - 1337935208T^6 + 411455568T^5 + 18886878736T^4 + 44295380160T^3 + 12314732928T^2 - 45501166080*T + 21041243136
$73$	\( T^{16} + \cdots + 216645564304 \) T^16 + 630T^14 + 154027T^12 + 18336452T^10 + 1083314979T^8 + 27954565702T^6 + 199077842441T^4 + 398915511656*T^2 + 216645564304
$79$	\( (T^{8} + 16 T^{7} + \cdots + 28089769)^{2} \) (T^8 + 16T^7 - 272T^6 - 4296T^5 + 25622T^4 + 356800T^3 - 1125648T^2 - 8773800*T + 28089769)^2
$83$	\( T^{16} + \cdots + 637544759296 \) T^16 + 504T^14 + 102016T^12 + 10652480T^10 + 613209408T^8 + 19208834560T^6 + 299500516352T^4 + 1827336814592*T^2 + 637544759296
$89$	\( T^{16} + \cdots + 1448868096 \) T^16 + 24T^15 + 162T^14 - 720T^13 - 10010T^12 + 39036T^11 + 894212T^10 + 3924984T^9 + 1358524T^8 - 30550488T^7 - 28359320T^6 + 199760880T^5 + 319570480T^4 - 508321920T^3 - 688006848T^2 + 858723840*T + 1448868096
$97$	\( T^{16} + \cdots + 5766702748816 \) T^16 + 6T^15 - 433T^14 - 2670T^13 + 146272T^12 + 526986T^11 - 18785311T^10 - 76618338T^9 + 1693762504T^8 + 11502410118T^7 - 22858254109T^6 - 295365828834T^5 + 425277346645T^4 + 5640136911780T^3 + 7443755905148T^2 - 13104100987728*T + 5766702748816
show more
show less