LMFDB - Newform orbit 1098.2.k.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1098,2,Mod(217,1098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1098, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 8]))

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

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

chi := DirichletCharacter("1098.217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1098 = 2 \cdot 3^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1098.k (of order $5$, degree $4$, 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:	$8.76757414194$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$3$ over $\Q(\zeta_{5})$
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} - 2 x^{11} + 6 x^{10} - 2 x^{9} + 23 x^{8} - 24 x^{7} + 211 x^{6} - 43 x^{5} + 587 x^{4} + \cdots + 16 $ x^12 - 2x^11 + 6x^10 - 2x^9 + 23x^8 - 24x^7 + 211x^6 - 43x^5 + 587x^4 - 386x^3 + 432x^2 - 128*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 366)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 2 x^{11} + 6 x^{10} - 2 x^{9} + 23 x^{8} - 24 x^{7} + 211 x^{6} - 43 x^{5} + 587 x^{4} + \cdots + 16 $ x^12 - 2*x^11 + 6*x^10 - 2*x^9 + 23*x^8 - 24*x^7 + 211*x^6 - 43*x^5 + 587*x^4 - 386*x^3 + 432*x^2 - 128*x + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 9369074079 \nu^{11} - 1560695776 \nu^{10} - 12613298554 \nu^{9} + 218682686966 \nu^{8} + \cdots + 51463556834218 ) / 17095024943558 $ (9369074079v^11 - 1560695776v^10 - 12613298554v^9 + 218682686966v^8 - 15411299411v^7 + 579013376554v^6 + 929070857795v^5 + 6357177943309v^4 - 419015971645v^3 + 20551706548244v^2 - 7237359569116*v + 51463556834218) / 17095024943558
$\beta_{3}$	$=$	$ ( 22310250443 \nu^{11} - 63358649044 \nu^{10} + 136982894210 \nu^{9} - 19393903778 \nu^{8} + \cdots + 11619007081528 ) / 34190049887116 $ (22310250443v^11 - 63358649044v^10 + 136982894210v^9 - 19393903778v^8 + 75770386257v^7 - 504623411810v^6 + 3549436090365v^5 - 2817482484639v^4 + 381761123423v^3 - 7773724727708v^2 - 31465384905112*v + 11619007081528) / 34190049887116
$\beta_{4}$	$=$	$ ( 32861360229 \nu^{11} - 182341612548 \nu^{10} + 334860780442 \nu^{9} - 504636743934 \nu^{8} + \cdots - 10491976871728 ) / 34190049887116 $ (32861360229v^11 - 182341612548v^10 + 334860780442v^9 - 504636743934v^8 + 374092038847v^7 - 3290055396442v^6 + 9365503243983v^5 - 23396797775185v^4 + 3816610606721v^3 - 65356928166948v^2 + 22858719688716*v - 10491976871728) / 34190049887116
$\beta_{5}$	$=$	$ ( - 137371135969 \nu^{11} + 288800455453 \nu^{10} - 805189858548 \nu^{9} + \cdots + 5064286827228 ) / 17095024943558 $ (-137371135969v^11 + 288800455453v^10 - 805189858548v^9 + 224598509440v^8 - 2762591515335v^7 + 3374496406429v^6 - 28501043693739v^5 + 7629887587600v^4 - 71219542830116v^3 + 53656264484007v^2 - 41137506955818*v + 5064286827228) / 17095024943558
$\beta_{6}$	$=$	$ ( - 693350409171 \nu^{11} + 1457930342622 \nu^{10} - 4837394394338 \nu^{9} + \cdots + 691967230520 ) / 68380099774232 $ (-693350409171v^11 + 1457930342622v^10 - 4837394394338v^9 + 2038165464558v^8 - 17805939750045v^7 + 15389905403936v^6 - 159628140201793v^5 + 39288270670473v^4 - 509042028841697v^3 + 191826350970246v^2 - 673364918378136*v + 691967230520) / 68380099774232
$\beta_{7}$	$=$	$ ( - 1266071706807 \nu^{11} + 1982658869738 \nu^{10} - 6441228419030 \nu^{9} + \cdots - 2492849351976 ) / 68380099774232 $ (-1266071706807v^11 + 1982658869738v^10 - 6441228419030v^9 - 688616020578v^8 - 28221255218801v^7 + 19335354902028v^6 - 253643144510561v^5 - 59563091382255v^4 - 712664541545309v^3 + 203825507507038v^2 - 332317919404596*v - 2492849351976) / 68380099774232
$\beta_{8}$	$=$	$ ( - 655748554483 \nu^{11} + 1278635748737 \nu^{10} - 3752149714350 \nu^{9} + \cdots + 61077095285108 ) / 34190049887116 $ (-655748554483v^11 + 1278635748737v^10 - 3752149714350v^9 + 976636328524v^8 - 14577580009175v^7 + 15363873268745v^6 - 135072889599471v^5 + 18831684598786v^4 - 361527603706336v^3 + 249302331423717v^2 - 217926447369708*v + 61077095285108) / 34190049887116
$\beta_{9}$	$=$	$ ( - 1452375885191 \nu^{11} + 2949372271268 \nu^{10} - 8840972609234 \nu^{9} + \cdots + 122973343494224 ) / 68380099774232 $ (-1452375885191v^11 + 2949372271268v^10 - 8840972609234v^9 + 3178717558802v^8 - 33443433166949v^7 + 35008562017098v^6 - 307460558598921v^5 + 69551035243943v^4 - 858179609576395v^3 + 561380613930572v^2 - 642973831857928*v + 122973343494224) / 68380099774232
$\beta_{10}$	$=$	$ ( - 3564977336241 \nu^{11} + 5672591371078 \nu^{10} - 18445857210138 \nu^{9} + \cdots - 6426984528600 ) / 68380099774232 $ (-3564977336241v^11 + 5672591371078v^10 - 18445857210138v^9 - 1302409568894v^8 - 79661000154511v^7 + 53142552234756v^6 - 716961914961007v^5 - 147743258451361v^4 - 2032648738398819v^3 + 594151298381538v^2 - 984382312688956*v - 6426984528600) / 68380099774232
$\beta_{11}$	$=$	$ ( - 1860004990231 \nu^{11} + 3657019321059 \nu^{10} - 10742474191686 \nu^{9} + \cdots + 179718155275068 ) / 34190049887116 $ (-1860004990231v^11 + 3657019321059v^10 - 10742474191686v^9 + 2955104382020v^8 - 41417358161747v^7 + 43849358332239v^6 - 384822625279351v^5 + 65847789563304v^4 - 1034010411943690v^3 + 709744097845155v^2 - 636929901331332*v + 179718155275068) / 34190049887116

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{10} - 3\beta_{9} + 3\beta_{8} - 3\beta_{7} + \beta_{6} + \beta_{2} + \beta _1 - 3 $ b10 - 3b9 + 3b8 - 3*b7 + b6 + b2 + b1 - 3
$\nu^{3}$	$=$	$ \beta_{10} - \beta_{7} - 5\beta_{5} - 4\beta_{4} - 5\beta_{3} + \beta_{2} - 4\beta _1 - 1 $ b10 - b7 - 5b5 - 4b4 - 5b3 + b2 - 4b1 - 1
$\nu^{4}$	$=$	$ 6\beta_{11} - 2\beta_{9} - 14\beta_{8} - 2\beta_{5} - \beta_{3} - 2\beta _1 - 2 $ 6b11 - 2b9 - 14b8 - 2b5 - b3 - 2*b1 - 2
$\nu^{5}$	$=$	$ 8\beta_{11} - 8\beta_{10} - 12\beta_{8} + 12\beta_{7} + 20\beta_{4} - 8\beta_{3} - 7\beta_{2} - 8\beta _1 + 9 $ 8b11 - 8b10 - 12b8 + 12b7 + 20b4 - 8b3 - 7b2 - 8b1 + 9
$\nu^{6}$	$=$	$ -36\beta_{10} + 17\beta_{9} - 17\beta_{8} + 92\beta_{7} - \beta_{6} + 21\beta_{5} + 21\beta_{4} + 7\beta_1 $ -36b10 + 17b9 - 17b8 + 92b7 - b6 + 21b5 + 21b4 + 7*b1
$\nu^{7}$	$=$	$ - 56 \beta_{11} - 14 \beta_{10} + 40 \beta_{9} + 58 \beta_{8} + 40 \beta_{7} - 14 \beta_{6} + 165 \beta_{5} + \cdots - 58 $ -56b11 - 14b10 + 40b9 + 58b8 + 40b7 - 14b6 + 165b5 + 54b4 + 54b3 + 42b2 - 14*b1 - 58
$\nu^{8}$	$=$	$ - 221 \beta_{11} - 12 \beta_{10} + 551 \beta_{9} + 120 \beta_{7} - 221 \beta_{6} + 124 \beta_{5} + \cdots + 120 $ -221b11 - 12b10 + 551b9 + 120b7 - 221b6 + 124b5 - 44b4 + 124b3 - 12b2 - 265b1 + 120
$\nu^{9}$	$=$	$ - 136 \beta_{11} + 713 \beta_{9} - 329 \beta_{8} - 377 \beta_{6} + 353 \beta_{5} + 1005 \beta_{3} + \cdots + 713 $ -136b11 + 713b9 - 329b8 - 377b6 + 353b5 + 1005b3 - 377b2 - 24b1 + 713
$\nu^{10}$	$=$	$ - 112 \beta_{11} + 112 \beta_{10} + 818 \beta_{8} - 818 \beta_{7} + 193 \beta_{4} + 1226 \beta_{3} + \cdots + 2574 $ -112b11 + 112b10 + 818b8 - 818b7 + 193b4 + 1226b3 - 1270b2 + 1226b1 + 2574
$\nu^{11}$	$=$	$ 1145 \beta_{10} - 4948 \beta_{9} + 4948 \beta_{8} - 3211 \beta_{7} + 2496 \beta_{6} - 2312 \beta_{5} + \cdots + 6452 \beta_1 $ 1145b10 - 4948b9 + 4948b8 - 3211b7 + 2496b6 - 2312b5 - 2312b4 + 6452b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1098\mathbb{Z}\right)^\times$.

$n$	$245$	$307$
$\chi(n)$	$1$	$\beta_{7}$

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} $

217.1

0.759652	−	2.33797i
0.247744	−	0.762478i
−0.507396	+	1.56160i
0.759652	+	2.33797i
0.247744	+	0.762478i
−0.507396	−	1.56160i
2.10151	−	1.52684i
0.175285	−	0.127352i
−1.77680	+	1.29092i
2.10151	+	1.52684i
0.175285	+	0.127352i
−1.77680	−	1.29092i

−0.809017

0.587785i

0.309017

−

0.951057i

−0.759652

2.33797i

2.38365

1.73182i

0.309017

0.951057i

−0.759652

−

2.33797i

217.2

−0.809017

0.587785i

0.309017

−

0.951057i

−0.247744

0.762478i

−2.51711

−

1.82879i

0.309017

0.951057i

−0.247744

−

0.762478i

217.3

−0.809017

0.587785i

0.309017

−

0.951057i

0.507396

−

1.56160i

3.36953

2.44811i

0.309017

0.951057i

0.507396

1.56160i

253.1

−0.809017

−

0.587785i

0.309017

0.951057i

−0.759652

−

2.33797i

2.38365

−

1.73182i

0.309017

−

0.951057i

−0.759652

2.33797i

253.2

−0.809017

−

0.587785i

0.309017

0.951057i

−0.247744

−

0.762478i

−2.51711

1.82879i

0.309017

−

0.951057i

−0.247744

0.762478i

253.3

−0.809017

−

0.587785i

0.309017

0.951057i

0.507396

1.56160i

3.36953

−

2.44811i

0.309017

−

0.951057i

0.507396

−

1.56160i

325.1

0.309017

0.951057i

−0.809017

0.587785i

−2.10151

1.52684i

0.593794

−

1.82751i

−0.809017

−

0.587785i

−2.10151

−

1.52684i

325.2

0.309017

0.951057i

−0.809017

0.587785i

−0.175285

0.127352i

−1.14064

3.51052i

−0.809017

−

0.587785i

−0.175285

−

0.127352i

325.3

0.309017

0.951057i

−0.809017

0.587785i

1.77680

−

1.29092i

−0.689224

2.12121i

−0.809017

−

0.587785i

1.77680

1.29092i

973.1

0.309017

−

0.951057i

−0.809017

−

0.587785i

−2.10151

−

1.52684i

0.593794

1.82751i

−0.809017

0.587785i

−2.10151

1.52684i

973.2

0.309017

−

0.951057i

−0.809017

−

0.587785i

−0.175285

−

0.127352i

−1.14064

−

3.51052i

−0.809017

0.587785i

−0.175285

0.127352i

973.3

0.309017

−

0.951057i

−0.809017

−

0.587785i

1.77680

1.29092i

−0.689224

−

2.12121i

−0.809017

0.587785i

1.77680

−

1.29092i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.e	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1098.2.k.h		12
3.b	odd	2	1		366.2.h.d	✓	12
61.e	even	5	1	inner	1098.2.k.h		12
183.n	odd	10	1		366.2.h.d	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
366.2.h.d	✓	12	3.b	odd	2	1
366.2.h.d	✓	12	183.n	odd	10	1
1098.2.k.h		12	1.a	even	1	1	trivial
1098.2.k.h		12	61.e	even	5	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1098, [\chi])$:

$ T_{5}^{12} + 2 T_{5}^{11} + 6 T_{5}^{10} + 2 T_{5}^{9} + 23 T_{5}^{8} + 24 T_{5}^{7} + 211 T_{5}^{6} + \cdots + 16 $

$ T_{7}^{12} - 4 T_{7}^{11} + 15 T_{7}^{10} - 54 T_{7}^{9} + 308 T_{7}^{8} - 274 T_{7}^{7} + 1890 T_{7}^{6} + \cdots + 364816 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} $ (T^4 + T^3 + T^2 + T + 1)^3
$3$	$ T^{12} $ T^12
$5$	$ T^{12} + 2 T^{11} + \cdots + 16 $ T^12 + 2T^11 + 6T^10 + 2T^9 + 23T^8 + 24T^7 + 211T^6 + 43T^5 + 587T^4 + 386T^3 + 432T^2 + 128*T + 16
$7$	$ T^{12} - 4 T^{11} + \cdots + 364816 $ T^12 - 4T^11 + 15T^10 - 54T^9 + 308T^8 - 274T^7 + 1890T^6 - 5990T^5 + 23917T^4 - 32238T^3 + 145636T^2 - 107512*T + 364816
$11$	$ (T^{6} - 2 T^{5} + \cdots + 704)^{2} $ (T^6 - 2T^5 - 47T^4 - 16T^3 + 620T^2 + 1376*T + 704)^2
$13$	$ (T^{6} - 52 T^{4} + \cdots - 89)^{2} $ (T^6 - 52T^4 + 12T^3 + 496T^2 - 612T - 89)^2
$17$	$ T^{12} + 2 T^{11} + \cdots + 4096 $ T^12 + 2T^11 + 74T^10 + 255T^9 + 2419T^8 + 5554T^7 + 19108T^6 + 1864T^5 - 17744T^4 - 18752T^3 + 479744T^2 - 71680*T + 4096
$19$	$ T^{12} + 7 T^{11} + \cdots + 1936 $ T^12 + 7T^11 + 56T^10 + 225T^9 + 1405T^8 + 1667T^7 + 16654T^6 - 61033T^5 + 166375T^4 + 314630T^3 + 205316T^2 + 14872*T + 1936
$23$	$ T^{12} - 3 T^{11} + \cdots + 160000 $ T^12 - 3T^11 + 79T^10 - 307T^9 + 2871T^8 - 4830T^7 + 15720T^6 + 151400T^5 + 592400T^4 + 1084000T^3 + 1408000T^2 + 720000*T + 160000
$29$	$ (T^{6} - 9 T^{5} - 18 T^{4} + \cdots - 20)^{2} $ (T^6 - 9T^5 - 18T^4 + 129T^3 + 151T^2 + 10*T - 20)^2
$31$	$ T^{12} - 3 T^{11} + \cdots + 41783296 $ T^12 - 3T^11 - 19T^10 + 10T^9 + 7595T^8 - 29348T^7 + 353724T^6 - 1473248T^5 + 7665785T^4 - 22683680T^3 + 45346176T^2 - 50626048*T + 41783296
$37$	$ T^{12} + \cdots + 450967696 $ T^12 - 4T^11 + 38T^10 - 122T^9 + 4452T^8 - 41444T^7 + 460539T^6 - 2536448T^5 + 18502873T^4 - 81246986T^3 + 235446724T^2 - 375325064*T + 450967696
$41$	$ T^{12} + \cdots + 661724176 $ T^12 - 7T^11 + 97T^10 - 164T^9 + 3017T^8 - 31102T^7 + 432266T^6 - 2749906T^5 + 16019393T^4 - 51792788T^3 + 210127696T^2 - 516074888*T + 661724176
$43$	$ T^{12} + T^{11} + \cdots + 4096 $ T^12 + T^11 + 12T^10 + 360T^9 + 6329T^8 - 32798T^7 + 464016T^6 + 2794016T^5 + 8462736T^4 + 7150784T^3 + 88187136T^2 + 371712T + 4096
$47$	$ (T^{6} + T^{5} - 113 T^{4} + \cdots + 3856)^{2} $ (T^6 + T^5 - 113T^4 - 338T^3 + 2520T^2 + 9992T + 3856)^2
$53$	$ T^{12} - 4 T^{11} + \cdots + 256 $ T^12 - 4T^11 + 3T^10 + 68T^9 + 1862T^8 + 11416T^7 + 35444T^6 + 55432T^5 + 53233T^4 + 44364T^3 + 55984T^2 + 2496*T + 256
$59$	$ T^{12} - 15 T^{11} + \cdots + 65536 $ T^12 - 15T^11 + 156T^10 - 1156T^9 + 7641T^8 - 37642T^7 + 167172T^6 - 499688T^5 + 1252304T^4 - 1672320T^3 + 1222656T^2 - 425984*T + 65536
$61$	$ T^{12} + \cdots + 51520374361 $ T^12 - 32T^11 + 521T^10 - 4830T^9 + 19930T^8 + 86538T^7 - 1648136T^6 + 5278818T^5 + 74159530T^4 - 1096318230T^3 + 7213683161T^2 - 27027081632*T + 51520374361
$67$	$ T^{12} + \cdots + 64882278400 $ T^12 + 26T^11 + 394T^10 + 2577T^9 + 14279T^8 + 71994T^7 + 5093196T^6 + 51556648T^5 + 353961936T^4 + 1496385920T^3 + 7183984640T^2 + 15486976000*T + 64882278400
$71$	$ T^{12} + \cdots + 5126560000 $ T^12 + 16T^11 + 304T^10 + 4317T^9 + 50959T^8 + 490154T^7 + 4032156T^6 + 27962848T^5 + 162802736T^4 + 743691840T^3 + 2451897600T^2 + 5057824000*T + 5126560000
$73$	$ T^{12} - 17 T^{11} + \cdots + 85008400 $ T^12 - 17T^11 + 231T^10 - 1436T^9 + 5879T^8 - 2838T^7 + 238354T^6 - 50344T^5 + 1830521T^4 - 11338490T^3 + 42833340T^2 - 62419400*T + 85008400
$79$	$ T^{12} + \cdots + 315986176 $ T^12 - 29T^11 + 648T^10 - 11348T^9 + 164225T^8 - 1906306T^7 + 17043928T^6 - 109806264T^5 + 477740240T^4 - 1249789472T^3 + 1494484608T^2 + 216582784*T + 315986176
$83$	$ T^{12} + \cdots + 2170742542336 $ T^12 - 22T^11 + 348T^10 - 4456T^9 + 75488T^8 - 515744T^7 + 5024512T^6 - 36214656T^5 + 526412288T^4 - 43999744T^3 + 45642482688T^2 - 29042556928*T + 2170742542336
$89$	$ T^{12} + \cdots + 34352398336 $ T^12 + 454T^10 + 2798T^9 + 86971T^8 + 532514T^7 + 8935063T^6 + 25260709T^5 + 446469009T^4 - 3656319520T^3 + 14315090944T^2 + 38308380672T + 34352398336
$97$	$ T^{12} - 9 T^{11} + \cdots + 16 $ T^12 - 9T^11 + 112T^10 - 1295T^9 + 10509T^8 - 19993T^7 + 30416T^6 - 25419T^5 + 14011T^4 - 5246T^3 + 1336T^2 - 208*T + 16
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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:	\( N \)	\(=\)	\( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1098.k (of order \(5\), degree \(4\), minimal)

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

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	1098.2.k.h

Level	$1098$
Weight	$2$
Character orbit	1098.k
Analytic conductor	$8.768$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(8.76757414194\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 2 x^{11} + 6 x^{10} - 2 x^{9} + 23 x^{8} - 24 x^{7} + 211 x^{6} - 43 x^{5} + 587 x^{4} + \cdots + 16 \) x^12 - 2x^11 + 6x^10 - 2x^9 + 23x^8 - 24x^7 + 211x^6 - 43x^5 + 587x^4 - 386x^3 + 432x^2 - 128*x + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 366)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 9369074079 \nu^{11} - 1560695776 \nu^{10} - 12613298554 \nu^{9} + 218682686966 \nu^{8} + \cdots + 51463556834218 ) / 17095024943558 \) (9369074079v^11 - 1560695776v^10 - 12613298554v^9 + 218682686966v^8 - 15411299411v^7 + 579013376554v^6 + 929070857795v^5 + 6357177943309v^4 - 419015971645v^3 + 20551706548244v^2 - 7237359569116*v + 51463556834218) / 17095024943558
\(\beta_{3}\)	\(=\)	\( ( 22310250443 \nu^{11} - 63358649044 \nu^{10} + 136982894210 \nu^{9} - 19393903778 \nu^{8} + \cdots + 11619007081528 ) / 34190049887116 \) (22310250443v^11 - 63358649044v^10 + 136982894210v^9 - 19393903778v^8 + 75770386257v^7 - 504623411810v^6 + 3549436090365v^5 - 2817482484639v^4 + 381761123423v^3 - 7773724727708v^2 - 31465384905112*v + 11619007081528) / 34190049887116
\(\beta_{4}\)	\(=\)	\( ( 32861360229 \nu^{11} - 182341612548 \nu^{10} + 334860780442 \nu^{9} - 504636743934 \nu^{8} + \cdots - 10491976871728 ) / 34190049887116 \) (32861360229v^11 - 182341612548v^10 + 334860780442v^9 - 504636743934v^8 + 374092038847v^7 - 3290055396442v^6 + 9365503243983v^5 - 23396797775185v^4 + 3816610606721v^3 - 65356928166948v^2 + 22858719688716*v - 10491976871728) / 34190049887116
\(\beta_{5}\)	\(=\)	\( ( - 137371135969 \nu^{11} + 288800455453 \nu^{10} - 805189858548 \nu^{9} + \cdots + 5064286827228 ) / 17095024943558 \) (-137371135969v^11 + 288800455453v^10 - 805189858548v^9 + 224598509440v^8 - 2762591515335v^7 + 3374496406429v^6 - 28501043693739v^5 + 7629887587600v^4 - 71219542830116v^3 + 53656264484007v^2 - 41137506955818*v + 5064286827228) / 17095024943558
\(\beta_{6}\)	\(=\)	\( ( - 693350409171 \nu^{11} + 1457930342622 \nu^{10} - 4837394394338 \nu^{9} + \cdots + 691967230520 ) / 68380099774232 \) (-693350409171v^11 + 1457930342622v^10 - 4837394394338v^9 + 2038165464558v^8 - 17805939750045v^7 + 15389905403936v^6 - 159628140201793v^5 + 39288270670473v^4 - 509042028841697v^3 + 191826350970246v^2 - 673364918378136*v + 691967230520) / 68380099774232
\(\beta_{7}\)	\(=\)	\( ( - 1266071706807 \nu^{11} + 1982658869738 \nu^{10} - 6441228419030 \nu^{9} + \cdots - 2492849351976 ) / 68380099774232 \) (-1266071706807v^11 + 1982658869738v^10 - 6441228419030v^9 - 688616020578v^8 - 28221255218801v^7 + 19335354902028v^6 - 253643144510561v^5 - 59563091382255v^4 - 712664541545309v^3 + 203825507507038v^2 - 332317919404596*v - 2492849351976) / 68380099774232
\(\beta_{8}\)	\(=\)	\( ( - 655748554483 \nu^{11} + 1278635748737 \nu^{10} - 3752149714350 \nu^{9} + \cdots + 61077095285108 ) / 34190049887116 \) (-655748554483v^11 + 1278635748737v^10 - 3752149714350v^9 + 976636328524v^8 - 14577580009175v^7 + 15363873268745v^6 - 135072889599471v^5 + 18831684598786v^4 - 361527603706336v^3 + 249302331423717v^2 - 217926447369708*v + 61077095285108) / 34190049887116
\(\beta_{9}\)	\(=\)	\( ( - 1452375885191 \nu^{11} + 2949372271268 \nu^{10} - 8840972609234 \nu^{9} + \cdots + 122973343494224 ) / 68380099774232 \) (-1452375885191v^11 + 2949372271268v^10 - 8840972609234v^9 + 3178717558802v^8 - 33443433166949v^7 + 35008562017098v^6 - 307460558598921v^5 + 69551035243943v^4 - 858179609576395v^3 + 561380613930572v^2 - 642973831857928*v + 122973343494224) / 68380099774232
\(\beta_{10}\)	\(=\)	\( ( - 3564977336241 \nu^{11} + 5672591371078 \nu^{10} - 18445857210138 \nu^{9} + \cdots - 6426984528600 ) / 68380099774232 \) (-3564977336241v^11 + 5672591371078v^10 - 18445857210138v^9 - 1302409568894v^8 - 79661000154511v^7 + 53142552234756v^6 - 716961914961007v^5 - 147743258451361v^4 - 2032648738398819v^3 + 594151298381538v^2 - 984382312688956*v - 6426984528600) / 68380099774232
\(\beta_{11}\)	\(=\)	\( ( - 1860004990231 \nu^{11} + 3657019321059 \nu^{10} - 10742474191686 \nu^{9} + \cdots + 179718155275068 ) / 34190049887116 \) (-1860004990231v^11 + 3657019321059v^10 - 10742474191686v^9 + 2955104382020v^8 - 41417358161747v^7 + 43849358332239v^6 - 384822625279351v^5 + 65847789563304v^4 - 1034010411943690v^3 + 709744097845155v^2 - 636929901331332*v + 179718155275068) / 34190049887116

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{10} - 3\beta_{9} + 3\beta_{8} - 3\beta_{7} + \beta_{6} + \beta_{2} + \beta _1 - 3 \) b10 - 3b9 + 3b8 - 3*b7 + b6 + b2 + b1 - 3
\(\nu^{3}\)	\(=\)	\( \beta_{10} - \beta_{7} - 5\beta_{5} - 4\beta_{4} - 5\beta_{3} + \beta_{2} - 4\beta _1 - 1 \) b10 - b7 - 5b5 - 4b4 - 5b3 + b2 - 4b1 - 1
\(\nu^{4}\)	\(=\)	\( 6\beta_{11} - 2\beta_{9} - 14\beta_{8} - 2\beta_{5} - \beta_{3} - 2\beta _1 - 2 \) 6b11 - 2b9 - 14b8 - 2b5 - b3 - 2*b1 - 2
\(\nu^{5}\)	\(=\)	\( 8\beta_{11} - 8\beta_{10} - 12\beta_{8} + 12\beta_{7} + 20\beta_{4} - 8\beta_{3} - 7\beta_{2} - 8\beta _1 + 9 \) 8b11 - 8b10 - 12b8 + 12b7 + 20b4 - 8b3 - 7b2 - 8b1 + 9
\(\nu^{6}\)	\(=\)	\( -36\beta_{10} + 17\beta_{9} - 17\beta_{8} + 92\beta_{7} - \beta_{6} + 21\beta_{5} + 21\beta_{4} + 7\beta_1 \) -36b10 + 17b9 - 17b8 + 92b7 - b6 + 21b5 + 21b4 + 7*b1
\(\nu^{7}\)	\(=\)	\( - 56 \beta_{11} - 14 \beta_{10} + 40 \beta_{9} + 58 \beta_{8} + 40 \beta_{7} - 14 \beta_{6} + 165 \beta_{5} + \cdots - 58 \) -56b11 - 14b10 + 40b9 + 58b8 + 40b7 - 14b6 + 165b5 + 54b4 + 54b3 + 42b2 - 14*b1 - 58
\(\nu^{8}\)	\(=\)	\( - 221 \beta_{11} - 12 \beta_{10} + 551 \beta_{9} + 120 \beta_{7} - 221 \beta_{6} + 124 \beta_{5} + \cdots + 120 \) -221b11 - 12b10 + 551b9 + 120b7 - 221b6 + 124b5 - 44b4 + 124b3 - 12b2 - 265b1 + 120
\(\nu^{9}\)	\(=\)	\( - 136 \beta_{11} + 713 \beta_{9} - 329 \beta_{8} - 377 \beta_{6} + 353 \beta_{5} + 1005 \beta_{3} + \cdots + 713 \) -136b11 + 713b9 - 329b8 - 377b6 + 353b5 + 1005b3 - 377b2 - 24b1 + 713
\(\nu^{10}\)	\(=\)	\( - 112 \beta_{11} + 112 \beta_{10} + 818 \beta_{8} - 818 \beta_{7} + 193 \beta_{4} + 1226 \beta_{3} + \cdots + 2574 \) -112b11 + 112b10 + 818b8 - 818b7 + 193b4 + 1226b3 - 1270b2 + 1226b1 + 2574
\(\nu^{11}\)	\(=\)	\( 1145 \beta_{10} - 4948 \beta_{9} + 4948 \beta_{8} - 3211 \beta_{7} + 2496 \beta_{6} - 2312 \beta_{5} + \cdots + 6452 \beta_1 \) 1145b10 - 4948b9 + 4948b8 - 3211b7 + 2496b6 - 2312b5 - 2312b4 + 6452b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 217.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) (T^4 + T^3 + T^2 + T + 1)^3
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + 2 T^{11} + \cdots + 16 \) T^12 + 2T^11 + 6T^10 + 2T^9 + 23T^8 + 24T^7 + 211T^6 + 43T^5 + 587T^4 + 386T^3 + 432T^2 + 128*T + 16
$7$	\( T^{12} - 4 T^{11} + \cdots + 364816 \) T^12 - 4T^11 + 15T^10 - 54T^9 + 308T^8 - 274T^7 + 1890T^6 - 5990T^5 + 23917T^4 - 32238T^3 + 145636T^2 - 107512*T + 364816
$11$	\( (T^{6} - 2 T^{5} + \cdots + 704)^{2} \) (T^6 - 2T^5 - 47T^4 - 16T^3 + 620T^2 + 1376*T + 704)^2
$13$	\( (T^{6} - 52 T^{4} + \cdots - 89)^{2} \) (T^6 - 52T^4 + 12T^3 + 496T^2 - 612T - 89)^2
$17$	\( T^{12} + 2 T^{11} + \cdots + 4096 \) T^12 + 2T^11 + 74T^10 + 255T^9 + 2419T^8 + 5554T^7 + 19108T^6 + 1864T^5 - 17744T^4 - 18752T^3 + 479744T^2 - 71680*T + 4096
$19$	\( T^{12} + 7 T^{11} + \cdots + 1936 \) T^12 + 7T^11 + 56T^10 + 225T^9 + 1405T^8 + 1667T^7 + 16654T^6 - 61033T^5 + 166375T^4 + 314630T^3 + 205316T^2 + 14872*T + 1936
$23$	\( T^{12} - 3 T^{11} + \cdots + 160000 \) T^12 - 3T^11 + 79T^10 - 307T^9 + 2871T^8 - 4830T^7 + 15720T^6 + 151400T^5 + 592400T^4 + 1084000T^3 + 1408000T^2 + 720000*T + 160000
$29$	\( (T^{6} - 9 T^{5} - 18 T^{4} + \cdots - 20)^{2} \) (T^6 - 9T^5 - 18T^4 + 129T^3 + 151T^2 + 10*T - 20)^2
$31$	\( T^{12} - 3 T^{11} + \cdots + 41783296 \) T^12 - 3T^11 - 19T^10 + 10T^9 + 7595T^8 - 29348T^7 + 353724T^6 - 1473248T^5 + 7665785T^4 - 22683680T^3 + 45346176T^2 - 50626048*T + 41783296
$37$	\( T^{12} + \cdots + 450967696 \) T^12 - 4T^11 + 38T^10 - 122T^9 + 4452T^8 - 41444T^7 + 460539T^6 - 2536448T^5 + 18502873T^4 - 81246986T^3 + 235446724T^2 - 375325064*T + 450967696
$41$	\( T^{12} + \cdots + 661724176 \) T^12 - 7T^11 + 97T^10 - 164T^9 + 3017T^8 - 31102T^7 + 432266T^6 - 2749906T^5 + 16019393T^4 - 51792788T^3 + 210127696T^2 - 516074888*T + 661724176
$43$	\( T^{12} + T^{11} + \cdots + 4096 \) T^12 + T^11 + 12T^10 + 360T^9 + 6329T^8 - 32798T^7 + 464016T^6 + 2794016T^5 + 8462736T^4 + 7150784T^3 + 88187136T^2 + 371712T + 4096
$47$	\( (T^{6} + T^{5} - 113 T^{4} + \cdots + 3856)^{2} \) (T^6 + T^5 - 113T^4 - 338T^3 + 2520T^2 + 9992T + 3856)^2
$53$	\( T^{12} - 4 T^{11} + \cdots + 256 \) T^12 - 4T^11 + 3T^10 + 68T^9 + 1862T^8 + 11416T^7 + 35444T^6 + 55432T^5 + 53233T^4 + 44364T^3 + 55984T^2 + 2496*T + 256
$59$	\( T^{12} - 15 T^{11} + \cdots + 65536 \) T^12 - 15T^11 + 156T^10 - 1156T^9 + 7641T^8 - 37642T^7 + 167172T^6 - 499688T^5 + 1252304T^4 - 1672320T^3 + 1222656T^2 - 425984*T + 65536
$61$	\( T^{12} + \cdots + 51520374361 \) T^12 - 32T^11 + 521T^10 - 4830T^9 + 19930T^8 + 86538T^7 - 1648136T^6 + 5278818T^5 + 74159530T^4 - 1096318230T^3 + 7213683161T^2 - 27027081632*T + 51520374361
$67$	\( T^{12} + \cdots + 64882278400 \) T^12 + 26T^11 + 394T^10 + 2577T^9 + 14279T^8 + 71994T^7 + 5093196T^6 + 51556648T^5 + 353961936T^4 + 1496385920T^3 + 7183984640T^2 + 15486976000*T + 64882278400
$71$	\( T^{12} + \cdots + 5126560000 \) T^12 + 16T^11 + 304T^10 + 4317T^9 + 50959T^8 + 490154T^7 + 4032156T^6 + 27962848T^5 + 162802736T^4 + 743691840T^3 + 2451897600T^2 + 5057824000*T + 5126560000
$73$	\( T^{12} - 17 T^{11} + \cdots + 85008400 \) T^12 - 17T^11 + 231T^10 - 1436T^9 + 5879T^8 - 2838T^7 + 238354T^6 - 50344T^5 + 1830521T^4 - 11338490T^3 + 42833340T^2 - 62419400*T + 85008400
$79$	\( T^{12} + \cdots + 315986176 \) T^12 - 29T^11 + 648T^10 - 11348T^9 + 164225T^8 - 1906306T^7 + 17043928T^6 - 109806264T^5 + 477740240T^4 - 1249789472T^3 + 1494484608T^2 + 216582784*T + 315986176
$83$	\( T^{12} + \cdots + 2170742542336 \) T^12 - 22T^11 + 348T^10 - 4456T^9 + 75488T^8 - 515744T^7 + 5024512T^6 - 36214656T^5 + 526412288T^4 - 43999744T^3 + 45642482688T^2 - 29042556928*T + 2170742542336
$89$	\( T^{12} + \cdots + 34352398336 \) T^12 + 454T^10 + 2798T^9 + 86971T^8 + 532514T^7 + 8935063T^6 + 25260709T^5 + 446469009T^4 - 3656319520T^3 + 14315090944T^2 + 38308380672T + 34352398336
$97$	\( T^{12} - 9 T^{11} + \cdots + 16 \) T^12 - 9T^11 + 112T^10 - 1295T^9 + 10509T^8 - 19993T^7 + 30416T^6 - 25419T^5 + 14011T^4 - 5246T^3 + 1336T^2 - 208*T + 16
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\(n\)	\(245\)	\(307\)
\(\chi(n)\)	\(1\)	\(\beta_{7}\)