LMFDB - Newform orbit 2010.2.i.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2010,2,Mod(841,2010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2010, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2010.841");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2010 = 2 \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2010.i (of order $3$, 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:	$16.0499308063$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} + 27 x^{10} - 8 x^{9} + 360 x^{8} - 263 x^{7} + 1822 x^{6} + 111 x^{5} + 4490 x^{4} + \cdots + 144 $ x^12 - 3x^11 + 27x^10 - 8x^9 + 360x^8 - 263x^7 + 1822x^6 + 111x^5 + 4490x^4 - 1935x^3 + 2893x^2 + 564*x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{4} + 1) q^{2} - q^{3} + \beta_{4} q^{4} + q^{5} + ( - \beta_{4} - 1) q^{6} + \beta_1 q^{7} - q^{8} + q^{9}+O(q^{10}) $ q + (b4 + 1) * q^2 - q^3 + b4 * q^4 + q^5 + (-b4 - 1) * q^6 + b1 * q^7 - q^8 + q^9	$ q + (\beta_{4} + 1) q^{2} - q^{3} + \beta_{4} q^{4} + q^{5} + ( - \beta_{4} - 1) q^{6} + \beta_1 q^{7} - q^{8} + q^{9} + (\beta_{4} + 1) q^{10} + (\beta_{9} + \beta_{5}) q^{11} - \beta_{4} q^{12} + (\beta_{10} - \beta_{7} + \beta_{4} + \cdots + 1) q^{13}+ \cdots + (\beta_{9} + \beta_{5}) q^{99}+O(q^{100}) $ q + (b4 + 1) * q^2 - q^3 + b4 * q^4 + q^5 + (-b4 - 1) * q^6 + b1 * q^7 - q^8 + q^9 + (b4 + 1) * q^10 + (b9 + b5) * q^11 - b4 * q^12 + (b10 - b7 + b4 + b3 + 1) * q^13 + b2 * q^14 - q^15 + (-b4 - 1) * q^16 + (b11 + b10 + b9 - b6) * q^17 + (b4 + 1) * q^18 + (-b10 + b9 + b7 - b3 - b2 + b1) * q^19 + b4 * q^20 - b1 * q^21 + b5 * q^22 + (b11 - b7 - b6 - 2b4 + b3 - b2 + b1 - 2) q^23 + q^24 + q^25 + (b10 + b8 + b4 + b3) * q^26 - q^27 + (b2 - b1) * q^28 + (-b9 - b5 - b4 + b3 - 2b1) q^29 + (-b4 - 1) * q^30 + (-2b11 - b10 - b8 - 2b4 - b1) * q^31 - b4 * q^32 + (-b9 - b5) * q^33 + (b11 + b10 + b9 + b8 + b5) * q^34 + b1 * q^35 + b4 * q^36 + (-b11 - b10 + b6 - 2b4 + b2 - b1 - 2) q^37 + (-b10 + b9 - b8 + b5 - b3 + b1) * q^38 + (-b10 + b7 - b4 - b3 - 1) * q^39 - q^40 + (-b11 + b10 + 2b9 + b8 + 2b5 + b4 - b3 + 2b1) q^41 - b2 * q^42 + (2b8 + b6 - 1) q^43 - b9 * q^44 + q^45 + (b11 - 2b4 + b3 + b1) q^46 + (b11 + b10 + b9 + b8 + b5 + 2b4 + 2b3 + b1) * q^47 + (b4 + 1) * q^48 + (-2b11 - 2b10 - b9 + 2b6 - b2 + b1) q^49 + (b4 + 1) * q^50 + (-b11 - b10 - b9 + b6) * q^51 + (b8 + b7 - 1) * q^52 + (-b6 - b5 - 3) * q^53 + (-b4 - 1) * q^54 + (b9 + b5) * q^55 - b1 * q^56 + (b10 - b9 - b7 + b3 + b2 - b1) * q^57 + (b7 - b5 - 2b2 + 1) q^58 + (2b8 - b7 + 2b6 + 3b5 + b2) q^59 - b4 * q^60 + (2b11 + b10 + 2b9 - 2b6 + 2b4 + 2) * q^61 + (-b8 - 2b6 - b2 + 2) q^62 + b1 * q^63 + q^64 + (b10 - b7 + b4 + b3 + 1) * q^65 - b5 * q^66 + (b10 + 2b9 + 2b5 + b3 + b1) * q^67 + (b8 + b6 + b5) * q^68 + (-b11 + b7 + b6 + 2b4 - b3 + b2 - b1 + 2) q^69 + b2 * q^70 + (-b10 - b8 + b4 + b3) * q^71 - q^72 + (2b10 + 3b9 + b7 - b3 - b2 + b1) * q^73 + (-b11 - b10 - b8 - 2b4 - b1) q^74 - q^75 + (-b8 - b7 + b5 + b2) * q^76 + (-b11 - b10 + b9 + b7 + b6 + 2b4 - b3 + 2) q^77 + (-b10 - b8 - b4 - b3) * q^78 + (-2b10 - 2b8 + b3 + b1) * q^79 + (-b4 - 1) * q^80 + q^81 + (b8 - b7 - b6 + 2b5 + 2b2 - 1) * q^82 + (b11 - b10 + b7 - b6 + b4 - b3 + 1) * q^83 + (-b2 + b1) * q^84 + (b11 + b10 + b9 - b6) * q^85 + (-b11 - 2b10 + b6 - b4 - 1) q^86 + (b9 + b5 + b4 - b3 + 2b1) q^87 + (-b9 - b5) * q^88 + (-b8 + b7 + b5 + b2 - 4) * q^89 + (b4 + 1) * q^90 + (3b8 + b7 + b6 + 2b5 + b2 - 3) * q^91 + (b7 + b6 + b2 + 2) * q^92 + (2b11 + b10 + b8 + 2b4 + b1) * q^93 + (b8 + 2b7 + b6 + b5 + b2 - 2) q^94 + (-b10 + b9 + b7 - b3 - b2 + b1) * q^95 + b4 * q^96 + (b11 - 3b10 - b9 - b6 + b4 - b2 + b1 + 1) q^97 + (-2b11 - 2b10 - b9 - 2b8 - b5 + b1) q^98 + (b9 + b5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} - 12 q^{3} - 6 q^{4} + 12 q^{5} - 6 q^{6} + 3 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q + 6 * q^2 - 12 * q^3 - 6 * q^4 + 12 * q^5 - 6 * q^6 + 3 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q + 6 q^{2} - 12 q^{3} - 6 q^{4} + 12 q^{5} - 6 q^{6} + 3 q^{7} - 12 q^{8} + 12 q^{9} + 6 q^{10} - 2 q^{11} + 6 q^{12} + 4 q^{13} + 6 q^{14} - 12 q^{15} - 6 q^{16} + q^{17} + 6 q^{18} + q^{19} - 6 q^{20} - 3 q^{21} - 4 q^{22} - 16 q^{23} + 12 q^{24} + 12 q^{25} - 4 q^{26} - 12 q^{27} + 3 q^{28} + 3 q^{29} - 6 q^{30} + 8 q^{31} + 6 q^{32} + 2 q^{33} - q^{34} + 3 q^{35} - 6 q^{36} - 8 q^{37} - q^{38} - 4 q^{39} - 12 q^{40} - 4 q^{41} - 6 q^{42} - 8 q^{43} - 2 q^{44} + 12 q^{45} + 16 q^{46} - 8 q^{47} + 6 q^{48} - 3 q^{49} + 6 q^{50} - q^{51} - 8 q^{52} - 32 q^{53} - 6 q^{54} - 2 q^{55} - 3 q^{56} - q^{57} + 6 q^{58} - 4 q^{59} + 6 q^{60} + 15 q^{61} + 16 q^{62} + 3 q^{63} + 12 q^{64} + 4 q^{65} + 4 q^{66} - q^{67} - 2 q^{68} + 16 q^{69} + 6 q^{70} - 6 q^{71} - 12 q^{72} + 2 q^{73} + 8 q^{74} - 12 q^{75} - 2 q^{76} + 16 q^{77} + 4 q^{78} + 2 q^{79} - 6 q^{80} + 12 q^{81} - 8 q^{82} + 8 q^{83} - 3 q^{84} + q^{85} - 4 q^{86} - 3 q^{87} + 2 q^{88} - 46 q^{89} + 6 q^{90} - 30 q^{91} + 32 q^{92} - 8 q^{93} - 16 q^{94} + q^{95} - 6 q^{96} + 4 q^{97} + 3 q^{98} - 2 q^{99}+O(q^{100}) \) 12 * q + 6 * q^2 - 12 * q^3 - 6 * q^4 + 12 * q^5 - 6 * q^6 + 3 * q^7 - 12 * q^8 + 12 * q^9 + 6 * q^10 - 2 * q^11 + 6 * q^12 + 4 * q^13 + 6 * q^14 - 12 * q^15 - 6 * q^16 + q^17 + 6 * q^18 + q^19 - 6 * q^20 - 3 * q^21 - 4 * q^22 - 16 * q^23 + 12 * q^24 + 12 * q^25 - 4 * q^26 - 12 * q^27 + 3 * q^28 + 3 * q^29 - 6 * q^30 + 8 * q^31 + 6 * q^32 + 2 * q^33 - q^34 + 3 * q^35 - 6 * q^36 - 8 * q^37 - q^38 - 4 * q^39 - 12 * q^40 - 4 * q^41 - 6 * q^42 - 8 * q^43 - 2 * q^44 + 12 * q^45 + 16 * q^46 - 8 * q^47 + 6 * q^48 - 3 * q^49 + 6 * q^50 - q^51 - 8 * q^52 - 32 * q^53 - 6 * q^54 - 2 * q^55 - 3 * q^56 - q^57 + 6 * q^58 - 4 * q^59 + 6 * q^60 + 15 * q^61 + 16 * q^62 + 3 * q^63 + 12 * q^64 + 4 * q^65 + 4 * q^66 - q^67 - 2 * q^68 + 16 * q^69 + 6 * q^70 - 6 * q^71 - 12 * q^72 + 2 * q^73 + 8 * q^74 - 12 * q^75 - 2 * q^76 + 16 * q^77 + 4 * q^78 + 2 * q^79 - 6 * q^80 + 12 * q^81 - 8 * q^82 + 8 * q^83 - 3 * q^84 + q^85 - 4 * q^86 - 3 * q^87 + 2 * q^88 - 46 * q^89 + 6 * q^90 - 30 * q^91 + 32 * q^92 - 8 * q^93 - 16 * q^94 + q^95 - 6 * q^96 + 4 * q^97 + 3 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 3 x^{11} + 27 x^{10} - 8 x^{9} + 360 x^{8} - 263 x^{7} + 1822 x^{6} + 111 x^{5} + 4490 x^{4} + \cdots + 144 $ x^12 - 3*x^11 + 27*x^10 - 8*x^9 + 360*x^8 - 263*x^7 + 1822*x^6 + 111*x^5 + 4490*x^4 - 1935*x^3 + 2893*x^2 + 564*x + 144 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 35938258794 \nu^{11} + 62517608115 \nu^{10} - 845526637926 \nu^{9} + \cdots - 34144252570668 ) / 158642928495899 $ (-35938258794v^11 + 62517608115v^10 - 845526637926v^9 - 844903952451v^8 - 13093304027391v^7 - 5298093822722v^6 - 58557326859300v^5 - 67236121931613v^4 - 224768674096063v^3 - 68889189297216v^2 - 13897734837696*v - 34144252570668) / 158642928495899
$\beta_{3}$	$=$	$ ( 244833882592285 \nu^{11} - 162129464925423 \nu^{10} + \cdots + 16\!\cdots\!00 ) / 83\!\cdots\!32 $ (244833882592285v^11 - 162129464925423v^10 + 4071132288602235v^9 + 14419260419711020v^8 + 64981197186284064v^7 + 112319436792220405v^6 - 23321915921366270v^5 + 808398492565489419v^4 - 293206296948072250v^3 + 322903144821347205v^2 - 6127278622354023971*v + 16771413043630800) / 835730947316395932
$\beta_{4}$	$=$	$ ( 2845354380889 \nu^{11} - 8967322248195 \nu^{10} + 77574779581383 \nu^{9} + \cdots - 465708089181744 ) / 19\!\cdots\!88 $ (2845354380889v^11 - 8967322248195v^10 + 77574779581383v^9 - 32909154702224v^8 + 1014188729690628v^7 - 905447850502499v^6 + 5120658556107094v^5 - 386853586032921v^4 + 11968807707012254v^3 - 8202984816172971v^2 + 7404939952345285*v - 465708089181744) / 1903715141950788
$\beta_{5}$	$=$	$ ( 110851837905447 \nu^{11} - 232584154978189 \nu^{10} + \cdots + 28\!\cdots\!43 ) / 69\!\cdots\!61 $ (110851837905447v^11 - 232584154978189v^10 + 2675991401838361v^9 + 1820449033010994v^8 + 38871780300962625v^7 + 5767913077539826v^6 + 171497673776687153v^5 + 188954267750559052v^4 + 516538358576311471v^3 + 196715677012045163v^2 + 39721337340028716*v + 285991718869466143) / 69644245609699661
$\beta_{6}$	$=$	$ ( 136756740569394 \nu^{11} - 300855210794880 \nu^{10} + \cdots + 39\!\cdots\!82 ) / 69\!\cdots\!61 $ (136756740569394v^11 - 300855210794880v^10 + 3335889300490609v^9 + 1970752798358586v^8 + 47425330724951936v^7 + 4396021133521239v^6 + 208380316469012730v^5 + 226655122218600418v^4 + 542753586570887254v^3 + 237162622127548166v^2 + 47902042155265176*v + 397777575583131882) / 69644245609699661
$\beta_{7}$	$=$	$ ( 144125220126750 \nu^{11} - 267806694775783 \nu^{10} + \cdots + 11\!\cdots\!53 ) / 69\!\cdots\!61 $ (144125220126750v^11 - 267806694775783v^10 + 3433700508280683v^9 + 3050587162836867v^8 + 51857637283522819v^7 + 17771106757362150v^6 + 230913572015302496v^5 + 261714621421022181v^4 + 729899638135358276v^3 + 269488697487487860v^2 + 54382171361029776*v + 112987933824386653) / 69644245609699661
$\beta_{8}$	$=$	$ ( - 190128378892794 \nu^{11} + 403474909547333 \nu^{10} + \cdots - 30\!\cdots\!18 ) / 69\!\cdots\!61 $ (-190128378892794v^11 + 403474909547333v^10 - 4610914098429288v^9 - 3032294708316633v^8 - 66497607742568665v^7 - 9962327921987907v^6 - 293099775545370016v^5 - 321972991539503839v^4 - 819974166269167301v^3 - 335590241583269276v^2 - 67767684745071792*v - 305649175307138518) / 69644245609699661
$\beta_{9}$	$=$	$ ( - 48\!\cdots\!33 \nu^{11} + \cdots - 28\!\cdots\!84 ) / 83\!\cdots\!32 $ (-4867758870859433v^11 + 14362097539353135v^10 - 130620723605746383v^9 + 32734282668919180v^8 - 1750370038870981968v^7 + 1207183542615352459v^6 - 8791814020009619054v^5 - 843963964718300379v^4 - 22126927517964137674v^3 + 8652801813442203951v^2 - 14390281601159680625*v - 2807873705990386884) / 835730947316395932
$\beta_{10}$	$=$	$ ( 25\!\cdots\!79 \nu^{11} + \cdots + 15\!\cdots\!92 ) / 41\!\cdots\!66 $ (2539472529633679v^11 - 7323986887728015v^10 + 68148934038644409v^9 - 13550230531227980v^8 + 923017082582555874v^7 - 556978179050304797v^6 + 4694423921340997804v^5 + 783510273321659967v^4 + 11997307098679814432v^3 - 2947137049755184131v^2 + 7948924701118948885*v + 1553670816886823892) / 417865473658197966
$\beta_{11}$	$=$	$ ( - 880201728262911 \nu^{11} + \cdots + 15\!\cdots\!08 ) / 13\!\cdots\!22 $ (-880201728262911v^11 + 2911713621802935v^10 - 24653894388504369v^9 + 14528791086503752v^8 - 320926399565283260v^7 + 328048773636887949v^6 - 1701004817926697842v^5 + 391053897074414505v^4 - 4036023292297035218v^3 + 2796883246553380533v^2 - 3175183704225452753*v + 158394041818438608) / 139288491219399322

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -2\beta_{11} - 2\beta_{10} - \beta_{9} + 2\beta_{6} - 7\beta_{4} - \beta_{2} + \beta _1 - 7 $ -2b11 - 2b10 - b9 + 2b6 - 7b4 - b2 + b1 - 7
$\nu^{3}$	$=$	$ 3\beta_{8} - \beta_{7} + 5\beta_{6} - 2\beta_{5} - 16\beta_{2} - 9 $ 3b8 - b7 + 5b6 - 2b5 - 16b2 - 9
$\nu^{4}$	$=$	$ 45\beta_{11} + 43\beta_{10} + 7\beta_{9} + 43\beta_{8} + 7\beta_{5} + 113\beta_{4} + 4\beta_{3} - 44\beta_1 $ 45b11 + 43b10 + 7b9 + 43b8 + 7b5 + 113b4 + 4b3 - 44b1
$\nu^{5}$	$=$	$ 167 \beta_{11} + 120 \beta_{10} - 41 \beta_{9} + 5 \beta_{7} - 167 \beta_{6} + 322 \beta_{4} + \cdots + 322 $ 167b11 + 120b10 - 41b9 + 5b7 - 167b6 + 322b4 - 5b3 + 331b2 - 331*b1 + 322
$\nu^{6}$	$=$	$ -927\beta_{8} - 88\beta_{7} - 1042\beta_{6} - 8\beta_{5} + 1279\beta_{2} + 2334 $ -927b8 - 88b7 - 1042b6 - 8b5 + 1279*b2 + 2334
$\nu^{7}$	$=$	$ - 4546 \beta_{11} - 3531 \beta_{10} + 770 \beta_{9} - 3531 \beta_{8} + 770 \beta_{5} + \cdots + 7578 \beta_1 $ -4546b11 - 3531b10 + 770b9 - 3531b8 + 770b5 - 9111b4 - 107b3 + 7578b1
$\nu^{8}$	$=$	$ - 24911 \beta_{11} - 21273 \beta_{10} + 1376 \beta_{9} + 1785 \beta_{7} + 24911 \beta_{6} + \cdots - 53429 $ -24911b11 - 21273b10 + 1376b9 + 1785b7 + 24911b6 - 53429b4 - 1785b3 - 33751b2 + 33751*b1 - 53429
$\nu^{9}$	$=$	$ 93857\beta_{8} + 5014\beta_{7} + 116915\beta_{6} - 15594\beta_{5} - 181401\beta_{2} - 238996 $ 93857b8 + 5014b7 + 116915b6 - 15594b5 - 181401*b2 - 238996
$\nu^{10}$	$=$	$ 607212 \beta_{11} + 508341 \beta_{10} - 49979 \beta_{9} + 508341 \beta_{8} - 49979 \beta_{5} + \cdots - 859985 \beta_1 $ 607212b11 + 508341b10 - 49979b9 + 508341b8 - 49979b5 + 1279721b4 + 38652b3 - 859985b1
$\nu^{11}$	$=$	$ 2945721 \beta_{11} + 2398728 \beta_{10} - 344199 \beta_{9} - 148850 \beta_{7} - 2945721 \beta_{6} + \cdots + 6079027 $ 2945721b11 + 2398728b10 - 344199b9 - 148850b7 - 2945721b6 + 6079027b4 + 148850b3 + 4431031b2 - 4431031*b1 + 6079027

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2010\mathbb{Z}\right)^\times$.

$n$	$671$	$1141$	$1207$
$\chi(n)$	$1$	$\beta_{4}$	$1$

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} $

841.1

−1.65988	+	2.87500i
−0.840274	+	1.45540i
−0.104525	+	0.181042i
0.440049	−	0.762187i
1.17300	−	2.03170i
2.49163	−	4.31563i
−1.65988	−	2.87500i
−0.840274	−	1.45540i
−0.104525	−	0.181042i
0.440049	+	0.762187i
1.17300	+	2.03170i
2.49163	+	4.31563i

0.500000

0.866025i

−1.00000

−0.500000

0.866025i

1.00000

−0.500000

−

0.866025i

−1.65988

2.87500i

−1.00000

1.00000

0.500000

0.866025i

841.2

0.500000

0.866025i

−1.00000

−0.500000

0.866025i

1.00000

−0.500000

−

0.866025i

−0.840274

1.45540i

−1.00000

1.00000

0.500000

0.866025i

841.3

0.500000

0.866025i

−1.00000

−0.500000

0.866025i

1.00000

−0.500000

−

0.866025i

−0.104525

0.181042i

−1.00000

1.00000

0.500000

0.866025i

841.4

0.500000

0.866025i

−1.00000

−0.500000

0.866025i

1.00000

−0.500000

−

0.866025i

0.440049

−

0.762187i

−1.00000

1.00000

0.500000

0.866025i

841.5

0.500000

0.866025i

−1.00000

−0.500000

0.866025i

1.00000

−0.500000

−

0.866025i

1.17300

−

2.03170i

−1.00000

1.00000

0.500000

0.866025i

841.6

0.500000

0.866025i

−1.00000

−0.500000

0.866025i

1.00000

−0.500000

−

0.866025i

2.49163

−

4.31563i

−1.00000

1.00000

0.500000

0.866025i

1771.1

0.500000

−

0.866025i

−1.00000

−0.500000

−

0.866025i

1.00000

−0.500000

0.866025i

−1.65988

−

2.87500i

−1.00000

1.00000

0.500000

−

0.866025i

1771.2

0.500000

−

0.866025i

−1.00000

−0.500000

−

0.866025i

1.00000

−0.500000

0.866025i

−0.840274

−

1.45540i

−1.00000

1.00000

0.500000

−

0.866025i

1771.3

0.500000

−

0.866025i

−1.00000

−0.500000

−

0.866025i

1.00000

−0.500000

0.866025i

−0.104525

−

0.181042i

−1.00000

1.00000

0.500000

−

0.866025i

1771.4

0.500000

−

0.866025i

−1.00000

−0.500000

−

0.866025i

1.00000

−0.500000

0.866025i

0.440049

0.762187i

−1.00000

1.00000

0.500000

−

0.866025i

1771.5

0.500000

−

0.866025i

−1.00000

−0.500000

−

0.866025i

1.00000

−0.500000

0.866025i

1.17300

2.03170i

−1.00000

1.00000

0.500000

−

0.866025i

1771.6

0.500000

−

0.866025i

−1.00000

−0.500000

−

0.866025i

1.00000

−0.500000

0.866025i

2.49163

4.31563i

−1.00000

1.00000

0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
67.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2010.2.i.h	✓	12
67.c	even	3	1	inner	2010.2.i.h	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2010.2.i.h	✓	12	1.a	even	1	1	trivial
2010.2.i.h	✓	12	67.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{12} - 3 T_{7}^{11} + 27 T_{7}^{10} - 8 T_{7}^{9} + 360 T_{7}^{8} - 263 T_{7}^{7} + 1822 T_{7}^{6} + \cdots + 144 $ T7^12 - 3*T7^11 + 27*T7^10 - 8*T7^9 + 360*T7^8 - 263*T7^7 + 1822*T7^6 + 111*T7^5 + 4490*T7^4 - 1935*T7^3 + 2893*T7^2 + 564*T7 + 144 acting on $S_{2}^{\mathrm{new}}(2010, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$3$	$ (T + 1)^{12} $ (T + 1)^12
$5$	$ (T - 1)^{12} $ (T - 1)^12
$7$	$ T^{12} - 3 T^{11} + \cdots + 144 $ T^12 - 3T^11 + 27T^10 - 8T^9 + 360T^8 - 263T^7 + 1822T^6 + 111T^5 + 4490T^4 - 1935T^3 + 2893T^2 + 564*T + 144
$11$	$ T^{12} + 2 T^{11} + \cdots + 1089 $ T^12 + 2T^11 + 34T^10 - 50T^9 + 801T^8 - 272T^7 + 3201T^6 + 229T^5 + 10961T^4 + 1196T^3 + 3793T^2 - 462*T + 1089
$13$	$ T^{12} - 4 T^{11} + \cdots + 106276 $ T^12 - 4T^11 + 52T^10 - 142T^9 + 1728T^8 - 4531T^7 + 22825T^6 - 14892T^5 + 79793T^4 + 22816T^3 + 298649T^2 + 163978*T + 106276
$17$	$ T^{12} - T^{11} + \cdots + 962361 $ T^12 - T^11 + 49T^10 + 26T^9 + 1784T^8 + 781T^7 + 23894T^6 + 30534T^5 + 232156T^4 + 152739T^3 + 581920T^2 - 242307T + 962361
$19$	$ T^{12} - T^{11} + \cdots + 211600 $ T^12 - T^11 + 71T^10 + 50T^9 + 3765T^8 + 3021T^7 + 82601T^6 + 211330T^5 + 1328915T^4 + 1629295T^3 + 1521061T^2 + 658260T + 211600
$23$	$ T^{12} + 16 T^{11} + \cdots + 20820969 $ T^12 + 16T^11 + 235T^10 + 1394T^9 + 9885T^8 + 24177T^7 + 246731T^6 + 426536T^5 + 2941431T^4 + 980174T^3 + 19885216T^2 + 17914338*T + 20820969
$29$	$ T^{12} - 3 T^{11} + \cdots + 41589601 $ T^12 - 3T^11 + 92T^10 - 217T^9 + 5754T^8 - 12670T^7 + 180608T^6 - 272941T^5 + 3838344T^4 - 4945156T^3 + 30619691T^2 + 27956415*T + 41589601
$31$	$ T^{12} + \cdots + 3741524224 $ T^12 - 8T^11 + 191T^10 - 768T^9 + 18849T^8 - 61865T^7 + 1080264T^6 - 457782T^5 + 28892124T^4 + 24173120T^3 + 640180057T^2 + 1176688816*T + 3741524224
$37$	$ T^{12} + 8 T^{11} + \cdots + 3034564 $ T^12 + 8T^11 + 103T^10 + 286T^9 + 3760T^8 + 11562T^7 + 80060T^6 + 151411T^5 + 793698T^4 + 1401113T^3 + 5251275T^2 + 4091958*T + 3034564
$41$	$ T^{12} + \cdots + 26713941136 $ T^12 + 4T^11 + 273T^10 + 656T^9 + 52119T^8 + 118125T^7 + 4667202T^6 + 5400418T^5 + 290992526T^4 + 418669574T^3 + 4436467537T^2 - 6556556060*T + 26713941136
$43$	$ (T^{6} + 4 T^{5} + \cdots - 26569)^{2} $ (T^6 + 4T^5 - 120T^4 - 384T^3 + 3765T^2 + 6683*T - 26569)^2
$47$	$ T^{12} + \cdots + 437520889 $ T^12 + 8T^11 + 218T^10 - 452T^9 + 18465T^8 - 98467T^7 + 1679796T^6 - 10671382T^5 + 63704369T^4 - 189536001T^3 + 429621074T^2 - 514369947*T + 437520889
$53$	$ (T^{6} + 16 T^{5} + \cdots - 32)^{2} $ (T^6 + 16T^5 + 40T^4 - 41T^3 - 198T^2 - 149*T - 32)^2
$59$	$ (T^{6} + 2 T^{5} + \cdots - 167292)^{2} $ (T^6 + 2T^5 - 275T^4 - 929T^3 + 20105T^2 + 82551*T - 167292)^2
$61$	$ T^{12} - 15 T^{11} + \cdots + 24502500 $ T^12 - 15T^11 + 328T^10 - 3375T^9 + 59252T^8 - 588931T^7 + 5083256T^6 - 25365296T^5 + 96210259T^4 - 172212077T^3 + 222066271T^2 - 82858050*T + 24502500
$67$	$ T^{12} + \cdots + 90458382169 $ T^12 + T^11 + 95T^10 + 237T^9 + 7429T^8 + 21032T^7 + 564295T^6 + 1409144T^5 + 33348781T^4 + 71280831T^3 + 1914356495T^2 + 1350125107T + 90458382169
$71$	$ T^{12} + 6 T^{11} + \cdots + 2916 $ T^12 + 6T^11 + 116T^10 - 686T^9 + 5616T^8 - 10307T^7 + 24231T^6 + 4774T^5 + 30961T^4 - 1326T^3 + 14589T^2 + 4050*T + 2916
$73$	$ T^{12} + \cdots + 2092513536 $ T^12 - 2T^11 + 273T^10 + 916T^9 + 56320T^8 + 166180T^7 + 4445930T^6 + 25293811T^5 + 267592548T^4 + 842527713T^3 + 2059431697T^2 + 2410114128*T + 2092513536
$79$	$ T^{12} + \cdots + 107557641 $ T^12 - 2T^11 + 237T^10 - 2012T^9 + 52373T^8 - 286905T^7 + 2548077T^6 - 1936580T^5 + 41292445T^4 - 95098174T^3 + 203880262T^2 - 163799574*T + 107557641
$83$	$ T^{12} - 8 T^{11} + \cdots + 36 $ T^12 - 8T^11 + 155T^10 - 212T^9 + 9823T^8 - 7553T^7 + 420582T^6 + 993090T^5 + 5047440T^4 - 606718T^3 + 60133T^2 - 1626*T + 36
$89$	$ (T^{6} + 23 T^{5} + \cdots + 71734)^{2} $ (T^6 + 23T^5 + 40T^4 - 1426T^3 - 4623T^2 + 22475*T + 71734)^2
$97$	$ T^{12} + \cdots + 14356832400 $ T^12 - 4T^11 + 383T^10 + 1704T^9 + 99679T^8 + 484131T^7 + 13587094T^6 + 116203650T^5 + 1256260862T^4 + 5710503818T^3 + 22928816281T^2 + 19712906220*T + 14356832400
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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:	\( N \)	\(=\)	\( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2010.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{4} + 1) q^{2} - q^{3} + \beta_{4} q^{4} + q^{5} + ( - \beta_{4} - 1) q^{6} + \beta_1 q^{7} - q^{8} + q^{9}+O(q^{10}) \) q + (b4 + 1) * q^2 - q^3 + b4 * q^4 + q^5 + (-b4 - 1) * q^6 + b1 * q^7 - q^8 + q^9	\( q + (\beta_{4} + 1) q^{2} - q^{3} + \beta_{4} q^{4} + q^{5} + ( - \beta_{4} - 1) q^{6} + \beta_1 q^{7} - q^{8} + q^{9} + (\beta_{4} + 1) q^{10} + (\beta_{9} + \beta_{5}) q^{11} - \beta_{4} q^{12} + (\beta_{10} - \beta_{7} + \beta_{4} + \cdots + 1) q^{13}+ \cdots + (\beta_{9} + \beta_{5}) q^{99}+O(q^{100}) \) q + (b4 + 1) * q^2 - q^3 + b4 * q^4 + q^5 + (-b4 - 1) * q^6 + b1 * q^7 - q^8 + q^9 + (b4 + 1) * q^10 + (b9 + b5) * q^11 - b4 * q^12 + (b10 - b7 + b4 + b3 + 1) * q^13 + b2 * q^14 - q^15 + (-b4 - 1) * q^16 + (b11 + b10 + b9 - b6) * q^17 + (b4 + 1) * q^18 + (-b10 + b9 + b7 - b3 - b2 + b1) * q^19 + b4 * q^20 - b1 * q^21 + b5 * q^22 + (b11 - b7 - b6 - 2b4 + b3 - b2 + b1 - 2) q^23 + q^24 + q^25 + (b10 + b8 + b4 + b3) * q^26 - q^27 + (b2 - b1) * q^28 + (-b9 - b5 - b4 + b3 - 2b1) q^29 + (-b4 - 1) * q^30 + (-2b11 - b10 - b8 - 2b4 - b1) * q^31 - b4 * q^32 + (-b9 - b5) * q^33 + (b11 + b10 + b9 + b8 + b5) * q^34 + b1 * q^35 + b4 * q^36 + (-b11 - b10 + b6 - 2b4 + b2 - b1 - 2) q^37 + (-b10 + b9 - b8 + b5 - b3 + b1) * q^38 + (-b10 + b7 - b4 - b3 - 1) * q^39 - q^40 + (-b11 + b10 + 2b9 + b8 + 2b5 + b4 - b3 + 2b1) q^41 - b2 * q^42 + (2b8 + b6 - 1) q^43 - b9 * q^44 + q^45 + (b11 - 2b4 + b3 + b1) q^46 + (b11 + b10 + b9 + b8 + b5 + 2b4 + 2b3 + b1) * q^47 + (b4 + 1) * q^48 + (-2b11 - 2b10 - b9 + 2b6 - b2 + b1) q^49 + (b4 + 1) * q^50 + (-b11 - b10 - b9 + b6) * q^51 + (b8 + b7 - 1) * q^52 + (-b6 - b5 - 3) * q^53 + (-b4 - 1) * q^54 + (b9 + b5) * q^55 - b1 * q^56 + (b10 - b9 - b7 + b3 + b2 - b1) * q^57 + (b7 - b5 - 2b2 + 1) q^58 + (2b8 - b7 + 2b6 + 3b5 + b2) q^59 - b4 * q^60 + (2b11 + b10 + 2b9 - 2b6 + 2b4 + 2) * q^61 + (-b8 - 2b6 - b2 + 2) q^62 + b1 * q^63 + q^64 + (b10 - b7 + b4 + b3 + 1) * q^65 - b5 * q^66 + (b10 + 2b9 + 2b5 + b3 + b1) * q^67 + (b8 + b6 + b5) * q^68 + (-b11 + b7 + b6 + 2b4 - b3 + b2 - b1 + 2) q^69 + b2 * q^70 + (-b10 - b8 + b4 + b3) * q^71 - q^72 + (2b10 + 3b9 + b7 - b3 - b2 + b1) * q^73 + (-b11 - b10 - b8 - 2b4 - b1) q^74 - q^75 + (-b8 - b7 + b5 + b2) * q^76 + (-b11 - b10 + b9 + b7 + b6 + 2b4 - b3 + 2) q^77 + (-b10 - b8 - b4 - b3) * q^78 + (-2b10 - 2b8 + b3 + b1) * q^79 + (-b4 - 1) * q^80 + q^81 + (b8 - b7 - b6 + 2b5 + 2b2 - 1) * q^82 + (b11 - b10 + b7 - b6 + b4 - b3 + 1) * q^83 + (-b2 + b1) * q^84 + (b11 + b10 + b9 - b6) * q^85 + (-b11 - 2b10 + b6 - b4 - 1) q^86 + (b9 + b5 + b4 - b3 + 2b1) q^87 + (-b9 - b5) * q^88 + (-b8 + b7 + b5 + b2 - 4) * q^89 + (b4 + 1) * q^90 + (3b8 + b7 + b6 + 2b5 + b2 - 3) * q^91 + (b7 + b6 + b2 + 2) * q^92 + (2b11 + b10 + b8 + 2b4 + b1) * q^93 + (b8 + 2b7 + b6 + b5 + b2 - 2) q^94 + (-b10 + b9 + b7 - b3 - b2 + b1) * q^95 + b4 * q^96 + (b11 - 3b10 - b9 - b6 + b4 - b2 + b1 + 1) q^97 + (-2b11 - 2b10 - b9 - 2b8 - b5 + b1) q^98 + (b9 + b5) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} - 12 q^{3} - 6 q^{4} + 12 q^{5} - 6 q^{6} + 3 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q + 6 * q^2 - 12 * q^3 - 6 * q^4 + 12 * q^5 - 6 * q^6 + 3 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q + 6 q^{2} - 12 q^{3} - 6 q^{4} + 12 q^{5} - 6 q^{6} + 3 q^{7} - 12 q^{8} + 12 q^{9} + 6 q^{10} - 2 q^{11} + 6 q^{12} + 4 q^{13} + 6 q^{14} - 12 q^{15} - 6 q^{16} + q^{17} + 6 q^{18} + q^{19} - 6 q^{20} - 3 q^{21} - 4 q^{22} - 16 q^{23} + 12 q^{24} + 12 q^{25} - 4 q^{26} - 12 q^{27} + 3 q^{28} + 3 q^{29} - 6 q^{30} + 8 q^{31} + 6 q^{32} + 2 q^{33} - q^{34} + 3 q^{35} - 6 q^{36} - 8 q^{37} - q^{38} - 4 q^{39} - 12 q^{40} - 4 q^{41} - 6 q^{42} - 8 q^{43} - 2 q^{44} + 12 q^{45} + 16 q^{46} - 8 q^{47} + 6 q^{48} - 3 q^{49} + 6 q^{50} - q^{51} - 8 q^{52} - 32 q^{53} - 6 q^{54} - 2 q^{55} - 3 q^{56} - q^{57} + 6 q^{58} - 4 q^{59} + 6 q^{60} + 15 q^{61} + 16 q^{62} + 3 q^{63} + 12 q^{64} + 4 q^{65} + 4 q^{66} - q^{67} - 2 q^{68} + 16 q^{69} + 6 q^{70} - 6 q^{71} - 12 q^{72} + 2 q^{73} + 8 q^{74} - 12 q^{75} - 2 q^{76} + 16 q^{77} + 4 q^{78} + 2 q^{79} - 6 q^{80} + 12 q^{81} - 8 q^{82} + 8 q^{83} - 3 q^{84} + q^{85} - 4 q^{86} - 3 q^{87} + 2 q^{88} - 46 q^{89} + 6 q^{90} - 30 q^{91} + 32 q^{92} - 8 q^{93} - 16 q^{94} + q^{95} - 6 q^{96} + 4 q^{97} + 3 q^{98} - 2 q^{99}+O(q^{100}) \) 12 * q + 6 * q^2 - 12 * q^3 - 6 * q^4 + 12 * q^5 - 6 * q^6 + 3 * q^7 - 12 * q^8 + 12 * q^9 + 6 * q^10 - 2 * q^11 + 6 * q^12 + 4 * q^13 + 6 * q^14 - 12 * q^15 - 6 * q^16 + q^17 + 6 * q^18 + q^19 - 6 * q^20 - 3 * q^21 - 4 * q^22 - 16 * q^23 + 12 * q^24 + 12 * q^25 - 4 * q^26 - 12 * q^27 + 3 * q^28 + 3 * q^29 - 6 * q^30 + 8 * q^31 + 6 * q^32 + 2 * q^33 - q^34 + 3 * q^35 - 6 * q^36 - 8 * q^37 - q^38 - 4 * q^39 - 12 * q^40 - 4 * q^41 - 6 * q^42 - 8 * q^43 - 2 * q^44 + 12 * q^45 + 16 * q^46 - 8 * q^47 + 6 * q^48 - 3 * q^49 + 6 * q^50 - q^51 - 8 * q^52 - 32 * q^53 - 6 * q^54 - 2 * q^55 - 3 * q^56 - q^57 + 6 * q^58 - 4 * q^59 + 6 * q^60 + 15 * q^61 + 16 * q^62 + 3 * q^63 + 12 * q^64 + 4 * q^65 + 4 * q^66 - q^67 - 2 * q^68 + 16 * q^69 + 6 * q^70 - 6 * q^71 - 12 * q^72 + 2 * q^73 + 8 * q^74 - 12 * q^75 - 2 * q^76 + 16 * q^77 + 4 * q^78 + 2 * q^79 - 6 * q^80 + 12 * q^81 - 8 * q^82 + 8 * q^83 - 3 * q^84 + q^85 - 4 * q^86 - 3 * q^87 + 2 * q^88 - 46 * q^89 + 6 * q^90 - 30 * q^91 + 32 * q^92 - 8 * q^93 - 16 * q^94 + q^95 - 6 * q^96 + 4 * q^97 + 3 * q^98 - 2 * q^99

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	2010.2.i.h

Level	$2010$
Weight	$2$
Character orbit	2010.i
Analytic conductor	$16.050$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(16.0499308063\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 3 x^{11} + 27 x^{10} - 8 x^{9} + 360 x^{8} - 263 x^{7} + 1822 x^{6} + 111 x^{5} + 4490 x^{4} + \cdots + 144 \) x^12 - 3x^11 + 27x^10 - 8x^9 + 360x^8 - 263x^7 + 1822x^6 + 111x^5 + 4490x^4 - 1935x^3 + 2893x^2 + 564*x + 144
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( - 35938258794 \nu^{11} + 62517608115 \nu^{10} - 845526637926 \nu^{9} + \cdots - 34144252570668 ) / 158642928495899 \) (-35938258794v^11 + 62517608115v^10 - 845526637926v^9 - 844903952451v^8 - 13093304027391v^7 - 5298093822722v^6 - 58557326859300v^5 - 67236121931613v^4 - 224768674096063v^3 - 68889189297216v^2 - 13897734837696*v - 34144252570668) / 158642928495899
\(\beta_{3}\)	\(=\)	\( ( 244833882592285 \nu^{11} - 162129464925423 \nu^{10} + \cdots + 16\!\cdots\!00 ) / 83\!\cdots\!32 \) (244833882592285v^11 - 162129464925423v^10 + 4071132288602235v^9 + 14419260419711020v^8 + 64981197186284064v^7 + 112319436792220405v^6 - 23321915921366270v^5 + 808398492565489419v^4 - 293206296948072250v^3 + 322903144821347205v^2 - 6127278622354023971*v + 16771413043630800) / 835730947316395932
\(\beta_{4}\)	\(=\)	\( ( 2845354380889 \nu^{11} - 8967322248195 \nu^{10} + 77574779581383 \nu^{9} + \cdots - 465708089181744 ) / 19\!\cdots\!88 \) (2845354380889v^11 - 8967322248195v^10 + 77574779581383v^9 - 32909154702224v^8 + 1014188729690628v^7 - 905447850502499v^6 + 5120658556107094v^5 - 386853586032921v^4 + 11968807707012254v^3 - 8202984816172971v^2 + 7404939952345285*v - 465708089181744) / 1903715141950788
\(\beta_{5}\)	\(=\)	\( ( 110851837905447 \nu^{11} - 232584154978189 \nu^{10} + \cdots + 28\!\cdots\!43 ) / 69\!\cdots\!61 \) (110851837905447v^11 - 232584154978189v^10 + 2675991401838361v^9 + 1820449033010994v^8 + 38871780300962625v^7 + 5767913077539826v^6 + 171497673776687153v^5 + 188954267750559052v^4 + 516538358576311471v^3 + 196715677012045163v^2 + 39721337340028716*v + 285991718869466143) / 69644245609699661
\(\beta_{6}\)	\(=\)	\( ( 136756740569394 \nu^{11} - 300855210794880 \nu^{10} + \cdots + 39\!\cdots\!82 ) / 69\!\cdots\!61 \) (136756740569394v^11 - 300855210794880v^10 + 3335889300490609v^9 + 1970752798358586v^8 + 47425330724951936v^7 + 4396021133521239v^6 + 208380316469012730v^5 + 226655122218600418v^4 + 542753586570887254v^3 + 237162622127548166v^2 + 47902042155265176*v + 397777575583131882) / 69644245609699661
\(\beta_{7}\)	\(=\)	\( ( 144125220126750 \nu^{11} - 267806694775783 \nu^{10} + \cdots + 11\!\cdots\!53 ) / 69\!\cdots\!61 \) (144125220126750v^11 - 267806694775783v^10 + 3433700508280683v^9 + 3050587162836867v^8 + 51857637283522819v^7 + 17771106757362150v^6 + 230913572015302496v^5 + 261714621421022181v^4 + 729899638135358276v^3 + 269488697487487860v^2 + 54382171361029776*v + 112987933824386653) / 69644245609699661
\(\beta_{8}\)	\(=\)	\( ( - 190128378892794 \nu^{11} + 403474909547333 \nu^{10} + \cdots - 30\!\cdots\!18 ) / 69\!\cdots\!61 \) (-190128378892794v^11 + 403474909547333v^10 - 4610914098429288v^9 - 3032294708316633v^8 - 66497607742568665v^7 - 9962327921987907v^6 - 293099775545370016v^5 - 321972991539503839v^4 - 819974166269167301v^3 - 335590241583269276v^2 - 67767684745071792*v - 305649175307138518) / 69644245609699661
\(\beta_{9}\)	\(=\)	\( ( - 48\!\cdots\!33 \nu^{11} + \cdots - 28\!\cdots\!84 ) / 83\!\cdots\!32 \) (-4867758870859433v^11 + 14362097539353135v^10 - 130620723605746383v^9 + 32734282668919180v^8 - 1750370038870981968v^7 + 1207183542615352459v^6 - 8791814020009619054v^5 - 843963964718300379v^4 - 22126927517964137674v^3 + 8652801813442203951v^2 - 14390281601159680625*v - 2807873705990386884) / 835730947316395932
\(\beta_{10}\)	\(=\)	\( ( 25\!\cdots\!79 \nu^{11} + \cdots + 15\!\cdots\!92 ) / 41\!\cdots\!66 \) (2539472529633679v^11 - 7323986887728015v^10 + 68148934038644409v^9 - 13550230531227980v^8 + 923017082582555874v^7 - 556978179050304797v^6 + 4694423921340997804v^5 + 783510273321659967v^4 + 11997307098679814432v^3 - 2947137049755184131v^2 + 7948924701118948885*v + 1553670816886823892) / 417865473658197966
\(\beta_{11}\)	\(=\)	\( ( - 880201728262911 \nu^{11} + \cdots + 15\!\cdots\!08 ) / 13\!\cdots\!22 \) (-880201728262911v^11 + 2911713621802935v^10 - 24653894388504369v^9 + 14528791086503752v^8 - 320926399565283260v^7 + 328048773636887949v^6 - 1701004817926697842v^5 + 391053897074414505v^4 - 4036023292297035218v^3 + 2796883246553380533v^2 - 3175183704225452753*v + 158394041818438608) / 139288491219399322

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -2\beta_{11} - 2\beta_{10} - \beta_{9} + 2\beta_{6} - 7\beta_{4} - \beta_{2} + \beta _1 - 7 \) -2b11 - 2b10 - b9 + 2b6 - 7b4 - b2 + b1 - 7
\(\nu^{3}\)	\(=\)	\( 3\beta_{8} - \beta_{7} + 5\beta_{6} - 2\beta_{5} - 16\beta_{2} - 9 \) 3b8 - b7 + 5b6 - 2b5 - 16b2 - 9
\(\nu^{4}\)	\(=\)	\( 45\beta_{11} + 43\beta_{10} + 7\beta_{9} + 43\beta_{8} + 7\beta_{5} + 113\beta_{4} + 4\beta_{3} - 44\beta_1 \) 45b11 + 43b10 + 7b9 + 43b8 + 7b5 + 113b4 + 4b3 - 44b1
\(\nu^{5}\)	\(=\)	\( 167 \beta_{11} + 120 \beta_{10} - 41 \beta_{9} + 5 \beta_{7} - 167 \beta_{6} + 322 \beta_{4} + \cdots + 322 \) 167b11 + 120b10 - 41b9 + 5b7 - 167b6 + 322b4 - 5b3 + 331b2 - 331*b1 + 322
\(\nu^{6}\)	\(=\)	\( -927\beta_{8} - 88\beta_{7} - 1042\beta_{6} - 8\beta_{5} + 1279\beta_{2} + 2334 \) -927b8 - 88b7 - 1042b6 - 8b5 + 1279*b2 + 2334
\(\nu^{7}\)	\(=\)	\( - 4546 \beta_{11} - 3531 \beta_{10} + 770 \beta_{9} - 3531 \beta_{8} + 770 \beta_{5} + \cdots + 7578 \beta_1 \) -4546b11 - 3531b10 + 770b9 - 3531b8 + 770b5 - 9111b4 - 107b3 + 7578b1
\(\nu^{8}\)	\(=\)	\( - 24911 \beta_{11} - 21273 \beta_{10} + 1376 \beta_{9} + 1785 \beta_{7} + 24911 \beta_{6} + \cdots - 53429 \) -24911b11 - 21273b10 + 1376b9 + 1785b7 + 24911b6 - 53429b4 - 1785b3 - 33751b2 + 33751*b1 - 53429
\(\nu^{9}\)	\(=\)	\( 93857\beta_{8} + 5014\beta_{7} + 116915\beta_{6} - 15594\beta_{5} - 181401\beta_{2} - 238996 \) 93857b8 + 5014b7 + 116915b6 - 15594b5 - 181401*b2 - 238996
\(\nu^{10}\)	\(=\)	\( 607212 \beta_{11} + 508341 \beta_{10} - 49979 \beta_{9} + 508341 \beta_{8} - 49979 \beta_{5} + \cdots - 859985 \beta_1 \) 607212b11 + 508341b10 - 49979b9 + 508341b8 - 49979b5 + 1279721b4 + 38652b3 - 859985b1
\(\nu^{11}\)	\(=\)	\( 2945721 \beta_{11} + 2398728 \beta_{10} - 344199 \beta_{9} - 148850 \beta_{7} - 2945721 \beta_{6} + \cdots + 6079027 \) 2945721b11 + 2398728b10 - 344199b9 - 148850b7 - 2945721b6 + 6079027b4 + 148850b3 + 4431031b2 - 4431031*b1 + 6079027

\(n\)	\(671\)	\(1141\)	\(1207\)
\(\chi(n)\)	\(1\)	\(\beta_{4}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 841.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$3$	\( (T + 1)^{12} \) (T + 1)^12
$5$	\( (T - 1)^{12} \) (T - 1)^12
$7$	\( T^{12} - 3 T^{11} + \cdots + 144 \) T^12 - 3T^11 + 27T^10 - 8T^9 + 360T^8 - 263T^7 + 1822T^6 + 111T^5 + 4490T^4 - 1935T^3 + 2893T^2 + 564*T + 144
$11$	\( T^{12} + 2 T^{11} + \cdots + 1089 \) T^12 + 2T^11 + 34T^10 - 50T^9 + 801T^8 - 272T^7 + 3201T^6 + 229T^5 + 10961T^4 + 1196T^3 + 3793T^2 - 462*T + 1089
$13$	\( T^{12} - 4 T^{11} + \cdots + 106276 \) T^12 - 4T^11 + 52T^10 - 142T^9 + 1728T^8 - 4531T^7 + 22825T^6 - 14892T^5 + 79793T^4 + 22816T^3 + 298649T^2 + 163978*T + 106276
$17$	\( T^{12} - T^{11} + \cdots + 962361 \) T^12 - T^11 + 49T^10 + 26T^9 + 1784T^8 + 781T^7 + 23894T^6 + 30534T^5 + 232156T^4 + 152739T^3 + 581920T^2 - 242307T + 962361
$19$	\( T^{12} - T^{11} + \cdots + 211600 \) T^12 - T^11 + 71T^10 + 50T^9 + 3765T^8 + 3021T^7 + 82601T^6 + 211330T^5 + 1328915T^4 + 1629295T^3 + 1521061T^2 + 658260T + 211600
$23$	\( T^{12} + 16 T^{11} + \cdots + 20820969 \) T^12 + 16T^11 + 235T^10 + 1394T^9 + 9885T^8 + 24177T^7 + 246731T^6 + 426536T^5 + 2941431T^4 + 980174T^3 + 19885216T^2 + 17914338*T + 20820969
$29$	\( T^{12} - 3 T^{11} + \cdots + 41589601 \) T^12 - 3T^11 + 92T^10 - 217T^9 + 5754T^8 - 12670T^7 + 180608T^6 - 272941T^5 + 3838344T^4 - 4945156T^3 + 30619691T^2 + 27956415*T + 41589601
$31$	\( T^{12} + \cdots + 3741524224 \) T^12 - 8T^11 + 191T^10 - 768T^9 + 18849T^8 - 61865T^7 + 1080264T^6 - 457782T^5 + 28892124T^4 + 24173120T^3 + 640180057T^2 + 1176688816*T + 3741524224
$37$	\( T^{12} + 8 T^{11} + \cdots + 3034564 \) T^12 + 8T^11 + 103T^10 + 286T^9 + 3760T^8 + 11562T^7 + 80060T^6 + 151411T^5 + 793698T^4 + 1401113T^3 + 5251275T^2 + 4091958*T + 3034564
$41$	\( T^{12} + \cdots + 26713941136 \) T^12 + 4T^11 + 273T^10 + 656T^9 + 52119T^8 + 118125T^7 + 4667202T^6 + 5400418T^5 + 290992526T^4 + 418669574T^3 + 4436467537T^2 - 6556556060*T + 26713941136
$43$	\( (T^{6} + 4 T^{5} + \cdots - 26569)^{2} \) (T^6 + 4T^5 - 120T^4 - 384T^3 + 3765T^2 + 6683*T - 26569)^2
$47$	\( T^{12} + \cdots + 437520889 \) T^12 + 8T^11 + 218T^10 - 452T^9 + 18465T^8 - 98467T^7 + 1679796T^6 - 10671382T^5 + 63704369T^4 - 189536001T^3 + 429621074T^2 - 514369947*T + 437520889
$53$	\( (T^{6} + 16 T^{5} + \cdots - 32)^{2} \) (T^6 + 16T^5 + 40T^4 - 41T^3 - 198T^2 - 149*T - 32)^2
$59$	\( (T^{6} + 2 T^{5} + \cdots - 167292)^{2} \) (T^6 + 2T^5 - 275T^4 - 929T^3 + 20105T^2 + 82551*T - 167292)^2
$61$	\( T^{12} - 15 T^{11} + \cdots + 24502500 \) T^12 - 15T^11 + 328T^10 - 3375T^9 + 59252T^8 - 588931T^7 + 5083256T^6 - 25365296T^5 + 96210259T^4 - 172212077T^3 + 222066271T^2 - 82858050*T + 24502500
$67$	\( T^{12} + \cdots + 90458382169 \) T^12 + T^11 + 95T^10 + 237T^9 + 7429T^8 + 21032T^7 + 564295T^6 + 1409144T^5 + 33348781T^4 + 71280831T^3 + 1914356495T^2 + 1350125107T + 90458382169
$71$	\( T^{12} + 6 T^{11} + \cdots + 2916 \) T^12 + 6T^11 + 116T^10 - 686T^9 + 5616T^8 - 10307T^7 + 24231T^6 + 4774T^5 + 30961T^4 - 1326T^3 + 14589T^2 + 4050*T + 2916
$73$	\( T^{12} + \cdots + 2092513536 \) T^12 - 2T^11 + 273T^10 + 916T^9 + 56320T^8 + 166180T^7 + 4445930T^6 + 25293811T^5 + 267592548T^4 + 842527713T^3 + 2059431697T^2 + 2410114128*T + 2092513536
$79$	\( T^{12} + \cdots + 107557641 \) T^12 - 2T^11 + 237T^10 - 2012T^9 + 52373T^8 - 286905T^7 + 2548077T^6 - 1936580T^5 + 41292445T^4 - 95098174T^3 + 203880262T^2 - 163799574*T + 107557641
$83$	\( T^{12} - 8 T^{11} + \cdots + 36 \) T^12 - 8T^11 + 155T^10 - 212T^9 + 9823T^8 - 7553T^7 + 420582T^6 + 993090T^5 + 5047440T^4 - 606718T^3 + 60133T^2 - 1626*T + 36
$89$	\( (T^{6} + 23 T^{5} + \cdots + 71734)^{2} \) (T^6 + 23T^5 + 40T^4 - 1426T^3 - 4623T^2 + 22475*T + 71734)^2
$97$	\( T^{12} + \cdots + 14356832400 \) T^12 - 4T^11 + 383T^10 + 1704T^9 + 99679T^8 + 484131T^7 + 13587094T^6 + 116203650T^5 + 1256260862T^4 + 5710503818T^3 + 22928816281T^2 + 19712906220*T + 14356832400
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