LMFDB - Newform orbit 1026.2.h.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1026,2,Mod(505,1026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1026, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 4]))

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

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

chi := DirichletCharacter("1026.505");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1026 = 2 \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1026.h (of order $3$, degree $2$, not 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:	$8.19265124738$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 4 x^{16} - 6 x^{15} + x^{14} - 21 x^{13} - 12 x^{12} + 9 x^{10} + 135 x^{9} + 27 x^{8} + \cdots + 19683 $ x^18 + 4x^16 - 6x^15 + x^14 - 21x^13 - 12x^12 + 9x^10 + 135x^9 + 27x^8 - 324x^6 - 1701x^5 + 243x^4 - 4374x^3 + 8748x^2 + 19683
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 342)
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 $1,\beta_1,\ldots,\beta_{17}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{8} q^{2} + (\beta_{8} - 1) q^{4} + \beta_{5} q^{5} + ( - \beta_{8} + \beta_{6} - \beta_{4} + 1) q^{7} + q^{8}+O(q^{10}) $ q - b8 * q^2 + (b8 - 1) * q^4 + b5 * q^5 + (-b8 + b6 - b4 + 1) * q^7 + q^8	$ q - \beta_{8} q^{2} + (\beta_{8} - 1) q^{4} + \beta_{5} q^{5} + ( - \beta_{8} + \beta_{6} - \beta_{4} + 1) q^{7} + q^{8} - \beta_{14} q^{10} + ( - \beta_{13} + \beta_{3}) q^{11} + ( - \beta_{14} - \beta_{8} + \beta_{6} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{17} - \beta_{16} + \cdots - \beta_{3}) q^{98}+O(q^{100}) $ q - b8 * q^2 + (b8 - 1) * q^4 + b5 * q^5 + (-b8 + b6 - b4 + 1) * q^7 + q^8 - b14 * q^10 + (-b13 + b3) * q^11 + (-b14 - b8 + b6 + b5 - b4 - b2 + b1 + 1) * q^13 + (b4 - 1) * q^14 - b8 * q^16 + (b13 - b6 + b4 - b3) * q^17 + (b16 - b11 - b9 + 1) * q^19 + (b14 - b5) * q^20 - b3 * q^22 + (-b16 + b13 + b10 + b9 + b6 - b4 - b3) * q^23 + (b12 + 2b9 - b4 - 2b3 - b2 + 1) * q^25 + (-b5 + b4 + b2 - 1) * q^26 + (b8 - b6) * q^28 + (-b9 - b7 - b4) * q^29 + (b17 - b15 + b14 + b13 - b12 + b11 + b10 + b8) * q^31 + (b8 - 1) * q^32 + (-b4 + b3) * q^34 + (b16 - b14 - b13 - b10 - b9 + b5 + b3) * q^35 + (-b7 + b5 - b4 + 2) * q^37 + (-b15 + b11 + b9 - b8) * q^38 + b5 * q^40 + (b9 + b5) * q^41 + (b17 + b16 + b14 - b13 + b8 + b6 - b1) * q^43 + b13 * q^44 + (-b12 - b9 + b4 + b3) * q^46 + (-b12 - b5 - b2 + 3) * q^47 + (b17 + b16 - b15 - b13 + b11 + b6) * q^49 + (-2b16 + 2b13 - b12 + b10 - b8 + b6 + b1) * q^50 + (b14 + b8 - b6 - b1) * q^52 + (2b16 - b15 - b14 + b11 + b6) q^53 + (b17 - b16 - 2b15 + 2b14 + b13 + b10 + b9 + b7 - 2b5 - b3 + b2 - b1) q^55 + (-b8 + b6 - b4 + 1) * q^56 + (-b17 + b16 + b6) * q^58 + (2b9 - 2b5 - b4 - b3 - b2) * q^59 + (-b12 - b7 + b5 + b3 - 1) * q^61 + (-b17 + b15 - b14 - b13 - b10 - b8 - b7 + b5 + b3 + 1) * q^62 + q^64 + (-b16 - b15 - b14 + 2b13 + b10 + b9 - 6b8 + 2b6 + b5 - 2b4 - 2b3 - b2 + b1 + 6) q^65 + (b16 - b15 + 2b13 + b10 - b9 - b8 - b6 + b4 - 2b3 + 2b2 - 2b1 + 1) * q^67 + (-b13 + b6) * q^68 + (b12 + b9 - b5 - b3) * q^70 + (-b17 + b13 + b10 - b7 + 2b6 - 2b4 - b3 - b2 + b1) * q^71 + (-b15 - b14 + b13 - 7b8 + 2b6 + b5 - 2b4 - b3 - 2b2 + 2b1 + 7) q^73 + (-b17 - b14 - 2b8 + b6) q^74 + (-b16 + b15 + b8 - 1) * q^76 + (b17 - 2b13 + b6) q^77 + (b17 - 2b16 + b15 + b14 + b13 - 2b12 - b11 + 2b10 + b8 + b6 + 2b1) * q^79 - b14 * q^80 + (-b16 - b14) * q^82 + (b17 + b16 - 2b15 - b14 - b10 - b9 + b7 - 2b6 + b5 + 2b4 + 2b2 - 2b1) q^83 + (-b17 + 2b15 - 2b14 - b7 + 2b5 - b2 + b1) q^85 + (-b17 - b16 - b14 + b13 + b9 - b8 - b7 - b6 + b5 + b4 - b3 - b2 + b1 + 1) * q^86 + (-b13 + b3) * q^88 + (b17 - b16 - b15 - 2b12 + b11 + 2b10 + b6 + b1) * q^89 + (b16 - 2b14 - 2b13 + 2b12 - 2b10 - 4b8 + b6 - b1) q^91 + (b16 - b13 + b12 - b10 - b6) * q^92 + (b14 + b12 - b10 - 3b8 + b1) q^94 + (-b17 + b16 + b15 + b14 + b11 - 2b10 - b9 - b7 - b4 + 2b3 - b2) * q^95 + (-2b17 - b16 + 2b15 - b14 + b13 - 2b11 - 2b8 + 2b6 + 3b1) * q^97 + (-b17 - b16 + b15 + b13 + b9 - b7 - b6 + b4 - b3) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 9 q^{2} - 9 q^{4} + 5 q^{7} + 18 q^{8}+O(q^{10}) $ 18 * q - 9 * q^2 - 9 * q^4 + 5 * q^7 + 18 * q^8	$ 18 q - 9 q^{2} - 9 q^{4} + 5 q^{7} + 18 q^{8} - q^{11} + q^{13} - 10 q^{14} - 9 q^{16} + 5 q^{17} + 9 q^{19} + 2 q^{22} + 2 q^{23} + 18 q^{25} - 2 q^{26} + 5 q^{28} - 18 q^{29} + 4 q^{31} - 9 q^{32} - 10 q^{34} - 6 q^{35} + 20 q^{37} - 3 q^{38} + 2 q^{41} + 7 q^{43} - q^{44} - 4 q^{46} + 38 q^{47} + 6 q^{49} - 9 q^{50} + q^{52} + 10 q^{53} + 6 q^{55} + 5 q^{56} + 9 q^{58} - 10 q^{59} - 36 q^{61} + 4 q^{62} + 18 q^{64} + 45 q^{65} + 22 q^{67} + 5 q^{68} + 12 q^{70} - 11 q^{71} + 44 q^{73} - 10 q^{74} - 6 q^{76} + 2 q^{77} + 2 q^{79} - q^{82} + 7 q^{83} + 7 q^{86} - q^{88} - q^{89} - 25 q^{91} + 2 q^{92} - 19 q^{94} - 21 q^{95} + 6 q^{98}+O(q^{100}) $ 18 * q - 9 * q^2 - 9 * q^4 + 5 * q^7 + 18 * q^8 - q^11 + q^13 - 10 * q^14 - 9 * q^16 + 5 * q^17 + 9 * q^19 + 2 * q^22 + 2 * q^23 + 18 * q^25 - 2 * q^26 + 5 * q^28 - 18 * q^29 + 4 * q^31 - 9 * q^32 - 10 * q^34 - 6 * q^35 + 20 * q^37 - 3 * q^38 + 2 * q^41 + 7 * q^43 - q^44 - 4 * q^46 + 38 * q^47 + 6 * q^49 - 9 * q^50 + q^52 + 10 * q^53 + 6 * q^55 + 5 * q^56 + 9 * q^58 - 10 * q^59 - 36 * q^61 + 4 * q^62 + 18 * q^64 + 45 * q^65 + 22 * q^67 + 5 * q^68 + 12 * q^70 - 11 * q^71 + 44 * q^73 - 10 * q^74 - 6 * q^76 + 2 * q^77 + 2 * q^79 - q^82 + 7 * q^83 + 7 * q^86 - q^88 - q^89 - 25 * q^91 + 2 * q^92 - 19 * q^94 - 21 * q^95 + 6 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} + 4 x^{16} - 6 x^{15} + x^{14} - 21 x^{13} - 12 x^{12} + 9 x^{10} + 135 x^{9} + 27 x^{8} + \cdots + 19683 $ x^18 + 4*x^16 - 6*x^15 + x^14 - 21*x^13 - 12*x^12 + 9*x^10 + 135*x^9 + 27*x^8 - 324*x^6 - 1701*x^5 + 243*x^4 - 4374*x^3 + 8748*x^2 + 19683 :

$\beta_{1}$	$=$	$ ( - 164 \nu^{17} + 831 \nu^{16} - 2204 \nu^{15} + 1587 \nu^{14} - 6086 \nu^{13} + 2337 \nu^{12} + \cdots + 7466418 ) / 2574099 $ (-164v^17 + 831v^16 - 2204v^15 + 1587v^14 - 6086v^13 + 2337v^12 - 3711v^11 - 5502v^10 + 30060v^9 - 6813v^8 - 5211v^7 - 248670v^6 - 2592v^5 + 683964v^4 - 768852v^3 + 2225394v^2 - 4400973*v + 7466418) / 2574099
$\beta_{2}$	$=$	$ ( - 118 \nu^{17} + 474 \nu^{16} - 409 \nu^{15} + 1893 \nu^{14} - 1981 \nu^{13} - 1053 \nu^{12} + \cdots + 2821230 ) / 702027 $ (-118v^17 + 474v^16 - 409v^15 + 1893v^14 - 1981v^13 - 1053v^12 - 3237v^11 - 5103v^10 + 10764v^9 + 2565v^8 + 23868v^7 - 25110v^6 - 45603v^5 + 119799v^4 - 712476v^3 + 589032v^2 - 1961739*v + 2821230) / 702027
$\beta_{3}$	$=$	$ ( 44 \nu^{17} + 351 \nu^{16} - 397 \nu^{15} + 774 \nu^{14} - 2455 \nu^{13} - 282 \nu^{12} + \cdots + 2755620 ) / 234009 $ (44v^17 + 351v^16 - 397v^15 + 774v^14 - 2455v^13 - 282v^12 - 2460v^11 - 2058v^10 + 11835v^9 + 3222v^8 + 34533v^7 - 20898v^6 - 60264v^5 - 19845v^4 - 504225v^3 + 659988v^2 - 889380*v + 2755620) / 234009
$\beta_{4}$	$=$	$ ( - 128 \nu^{17} - 1809 \nu^{16} + 1972 \nu^{15} - 4236 \nu^{14} + 11023 \nu^{13} + 762 \nu^{12} + \cdots - 12419973 ) / 702027 $ (-128v^17 - 1809v^16 + 1972v^15 - 4236v^14 + 11023v^13 + 762v^12 + 9987v^11 + 7767v^10 - 42921v^9 - 16902v^8 - 120420v^7 + 157707v^6 + 283257v^5 + 195615v^4 + 2180682v^3 - 3307473v^2 + 4253715*v - 12419973) / 702027
$\beta_{5}$	$=$	$ ( 18 \nu^{17} - 85 \nu^{16} + 144 \nu^{15} - 267 \nu^{14} + 411 \nu^{13} - 57 \nu^{12} + 15 \nu^{11} + \cdots - 475308 ) / 78003 $ (18v^17 - 85v^16 + 144v^15 - 267v^14 + 411v^13 - 57v^12 + 15v^11 + 802v^10 - 1272v^9 - 141v^8 - 5877v^7 + 1431v^6 + 3186v^5 - 23220v^4 + 115587v^3 - 197235v^2 + 309825*v - 475308) / 78003
$\beta_{6}$	$=$	$ ( 1832 \nu^{17} - 15738 \nu^{16} + 19523 \nu^{15} - 39879 \nu^{14} + 80051 \nu^{13} + \cdots - 86769225 ) / 7722297 $ (1832v^17 - 15738v^16 + 19523v^15 - 39879v^14 + 80051v^13 - 12477v^12 + 47199v^11 + 83412v^10 - 298899v^9 + 74061v^8 - 794556v^7 + 674892v^6 + 1846638v^5 - 2418336v^4 + 20080062v^3 - 29845989v^2 + 46510929*v - 86769225) / 7722297
$\beta_{7}$	$=$	$ ( 82 \nu^{17} + 445 \nu^{16} - 521 \nu^{15} + 1102 \nu^{14} - 3014 \nu^{13} + 418 \nu^{12} + \cdots + 3667599 ) / 234009 $ (82v^17 + 445v^16 - 521v^15 + 1102v^14 - 3014v^13 + 418v^12 - 4176v^11 - 1530v^10 + 9756v^9 + 2853v^8 + 35073v^7 - 35532v^6 - 44550v^5 - 122472v^4 - 593406v^3 + 620136v^2 - 998001*v + 3667599) / 234009
$\beta_{8}$	$=$	$ ( - 3248 \nu^{17} + 3348 \nu^{16} - 6368 \nu^{15} + 15600 \nu^{14} + 9073 \nu^{13} + 16881 \nu^{12} + \cdots - 5006043 ) / 7722297 $ (-3248v^17 + 3348v^16 - 6368v^15 + 15600v^14 + 9073v^13 + 16881v^12 + 24135v^11 - 71901v^10 - 21078v^9 - 196371v^8 + 201555v^7 + 406539v^6 + 112914v^5 + 4144122v^4 - 5330205v^3 + 5697135v^2 - 19081575*v - 5006043) / 7722297
$\beta_{9}$	$=$	$ ( 323 \nu^{17} - 699 \nu^{16} + 1193 \nu^{15} - 3204 \nu^{14} + 2150 \nu^{13} - 3252 \nu^{12} + \cdots - 2427570 ) / 702027 $ (323v^17 - 699v^16 + 1193v^15 - 3204v^14 + 2150v^13 - 3252v^12 - 2067v^11 + 12834v^10 + 2070v^9 + 31050v^8 - 38745v^7 - 162v^6 + 38232v^5 - 425979v^4 + 1153035v^3 - 1482786v^2 + 3483891*v - 2427570) / 702027
$\beta_{10}$	$=$	$ ( - 4568 \nu^{17} - 13647 \nu^{16} + 14389 \nu^{15} - 12879 \nu^{14} + 94180 \nu^{13} + \cdots - 112540833 ) / 7722297 $ (-4568v^17 - 13647v^16 + 14389v^15 - 12879v^14 + 94180v^13 + 32160v^12 + 103038v^11 + 24624v^10 - 433314v^9 - 359721v^8 - 728649v^7 + 2013903v^6 + 2240622v^5 + 4849794v^4 + 15310701v^3 - 23708538v^2 + 2377269*v - 112540833) / 7722297
$\beta_{11}$	$=$	$ ( - 460 \nu^{17} - 1014 \nu^{16} + 707 \nu^{15} - 738 \nu^{14} + 6902 \nu^{13} + 3561 \nu^{12} + \cdots - 8194689 ) / 702027 $ (-460v^17 - 1014v^16 + 707v^15 - 738v^14 + 6902v^13 + 3561v^12 + 8355v^11 + 738v^10 - 31194v^9 - 34560v^8 - 67662v^7 + 115263v^6 + 202986v^5 + 627426v^4 + 1182195v^3 - 1548396v^2 + 1121931*v - 8194689) / 702027
$\beta_{12}$	$=$	$ ( - 481 \nu^{17} - 2502 \nu^{16} + 3107 \nu^{15} - 5295 \nu^{14} + 16106 \nu^{13} + 1434 \nu^{12} + \cdots - 19210608 ) / 702027 $ (-481v^17 - 2502v^16 + 3107v^15 - 5295v^14 + 16106v^13 + 1434v^12 + 16095v^11 + 6579v^10 - 70263v^9 - 28350v^8 - 150525v^7 + 255636v^6 + 347571v^5 + 589032v^4 + 3154869v^3 - 5089878v^2 + 4907628*v - 19210608) / 702027
$\beta_{13}$	$=$	$ ( - 6235 \nu^{17} + 14928 \nu^{16} - 21007 \nu^{15} + 52014 \nu^{14} - 30850 \nu^{13} + \cdots + 31893021 ) / 7722297 $ (-6235v^17 + 14928v^16 - 21007v^15 + 52014v^14 - 30850v^13 + 21834v^12 + 10695v^11 - 257778v^10 + 162882v^9 - 469125v^8 + 919485v^7 + 969246v^6 - 645813v^5 + 9312975v^4 - 19877157v^3 + 24380676v^2 - 58158891*v + 31893021) / 7722297
$\beta_{14}$	$=$	$ ( - 3391 \nu^{17} + 477 \nu^{16} - 5620 \nu^{15} + 8307 \nu^{14} + 17675 \nu^{13} + 14505 \nu^{12} + \cdots - 22447368 ) / 2574099 $ (-3391v^17 + 477v^16 - 5620v^15 + 8307v^14 + 17675v^13 + 14505v^12 + 43011v^11 - 45996v^10 - 62559v^9 - 253791v^8 - 93069v^7 + 583848v^6 + 143208v^5 + 4659120v^4 - 1277937v^3 + 3176496v^2 - 10493226*v - 22447368) / 2574099
$\beta_{15}$	$=$	$ ( - 10619 \nu^{17} - 783 \nu^{16} - 12623 \nu^{15} + 29226 \nu^{14} + 49204 \nu^{13} + \cdots - 61981767 ) / 7722297 $ (-10619v^17 - 783v^16 - 12623v^15 + 29226v^14 + 49204v^13 + 49326v^12 + 106089v^11 - 128340v^10 - 268344v^9 - 887409v^8 - 29943v^7 + 1744011v^6 + 2057886v^5 + 13311054v^4 - 1701243v^3 + 10763685v^2 - 35000748*v - 61981767) / 7722297
$\beta_{16}$	$=$	$ ( - 3837 \nu^{17} + 1541 \nu^{16} - 5079 \nu^{15} + 13019 \nu^{14} + 23655 \nu^{13} + \cdots - 29043360 ) / 2574099 $ (-3837v^17 + 1541v^16 - 5079v^15 + 13019v^14 + 23655v^13 + 14948v^12 + 40989v^11 - 62055v^10 - 96102v^9 - 292212v^8 + 11556v^7 + 821637v^6 + 620298v^5 + 5560083v^4 - 2197935v^3 + 2991330v^2 - 17807283*v - 29043360) / 2574099
$\beta_{17}$	$=$	$ ( 12686 \nu^{17} - 19122 \nu^{16} + 40061 \nu^{15} - 82359 \nu^{14} + 27392 \nu^{13} + \cdots - 36131427 ) / 7722297 $ (12686v^17 - 19122v^16 + 40061v^15 - 82359v^14 + 27392v^13 - 75126v^12 - 16314v^11 + 311778v^10 - 8325v^9 + 716715v^8 - 974943v^7 - 1808082v^6 + 1132704v^5 - 16656435v^4 + 29707965v^3 - 42408117v^2 + 92094570*v - 36131427) / 7722297

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

$\nu$	$=$	$ ( \beta_{11} - \beta_{10} - \beta_1 ) / 3 $ (b11 - b10 - b1) / 3
$\nu^{2}$	$=$	$ ( -\beta_{17} + \beta_{15} - 2\beta_{14} - \beta_{12} - 2\beta_{7} + \beta_{5} - \beta_{2} ) / 3 $ (-b17 + b15 - 2b14 - b12 - 2b7 + b5 - b2) / 3
$\nu^{3}$	$=$	$ ( 3 \beta_{13} + \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - 3 \beta_{8} + 3 \beta_{6} - 6 \beta_{4} + \cdots + 6 ) / 3 $ (3b13 + b11 + 2b10 + 3b9 - 3b8 + 3b6 - 6b4 - 3b3 - 3b2 + 2*b1 + 6) / 3
$\nu^{4}$	$=$	$ ( - \beta_{17} + 6 \beta_{16} - 3 \beta_{15} - 2 \beta_{14} - 3 \beta_{13} + \beta_{12} + 5 \beta_{11} + \cdots + 9 ) / 3 $ (-b17 + 6b16 - 3b15 - 2b14 - 3b13 + b12 + 5b11 - 4b10 - 3b9 - 9b8 - 2b7 + b5 + 6b3 + b1 + 9) / 3
$\nu^{5}$	$=$	$ ( 3 \beta_{17} + 4 \beta_{15} - 6 \beta_{14} + 3 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 3 \beta_{10} + \cdots - 12 ) / 3 $ (3b17 + 4b15 - 6b14 + 3b13 - 2b12 + 2b11 - 3b10 + 3b9 + 6b8 + 3b7 - 6b6 - 9b5 + 12b4 - 3b3 + 2*b2 + b1 - 12) / 3
$\nu^{6}$	$=$	$ ( - 4 \beta_{17} - 4 \beta_{15} + \beta_{14} + 9 \beta_{13} - 3 \beta_{12} + \beta_{11} + 7 \beta_{10} + \cdots + 27 ) / 3 $ (-4b17 - 4b15 + b14 + 9b13 - 3b12 + b11 + 7b10 + 9b9 - 27b8 + b7 - 9b6 - 5b5 + 9b4 - 5b2 - 10b1 + 27) / 3
$\nu^{7}$	$=$	$ ( 6 \beta_{17} + 9 \beta_{16} + 8 \beta_{15} - 3 \beta_{14} - 15 \beta_{13} + 14 \beta_{12} - 5 \beta_{10} + \cdots - 3 ) / 3 $ (6b17 + 9b16 + 8b15 - 3b14 - 15b13 + 14b12 - 5b10 + 3b9 + 42b8 + 6b7 + 30b6 - 9b5 - 33b4 + 24b3 - 20b2 - 6b1 - 3) / 3
$\nu^{8}$	$=$	$ ( - 22 \beta_{17} - 6 \beta_{16} - 35 \beta_{15} - 17 \beta_{14} - 6 \beta_{13} + 5 \beta_{12} + \cdots - 63 ) / 3 $ (-22b17 - 6b16 - 35b15 - 17b14 - 6b13 + 5b12 + 33b11 + 21b9 + 36b8 - 17b7 + 9b6 - 59b5 - 9b4 - 6b3 + 2b2 + 6b1 - 63) / 3
$\nu^{9}$	$=$	$ ( 36 \beta_{17} - 72 \beta_{16} - 15 \beta_{15} + 36 \beta_{14} + 84 \beta_{13} - 78 \beta_{12} + \cdots - 120 ) / 3 $ (36b17 - 72b16 - 15b15 + 36b14 + 84b13 - 78b12 + 58b11 + 68b10 + 93b9 + 168b8 - 9b7 - 24b6 + 9b5 + 48b4 - 3b3 + 12b2 + 17*b1 - 120) / 3
$\nu^{10}$	$=$	$ ( 53 \beta_{17} + 150 \beta_{16} - 39 \beta_{15} + 52 \beta_{14} - 210 \beta_{13} - 8 \beta_{12} + \cdots + 144 ) / 3 $ (53b17 + 150b16 - 39b15 + 52b14 - 210b13 - 8b12 + 77b11 - 13b10 + 33b9 - 9b8 + 52b7 - 135b6 - 26b5 + 96b3 + 36b2 - 125b1 + 144) / 3
$\nu^{11}$	$=$	$ ( 210 \beta_{17} - 140 \beta_{15} + 219 \beta_{14} + 3 \beta_{13} - 2 \beta_{12} + 83 \beta_{11} + \cdots + 312 ) / 3 $ (210b17 - 140b15 + 219b14 + 3b13 - 2b12 + 83b11 + 42b10 - 132b9 + 87b8 + 12b7 - 87b6 - 324b5 + 12b4 + 24b3 - 43b2 - 161b1 + 312) / 3
$\nu^{12}$	$=$	$ ( - 103 \beta_{17} - 243 \beta_{16} - 184 \beta_{15} - 197 \beta_{14} + 198 \beta_{13} - 165 \beta_{12} + \cdots - 405 ) / 3 $ (-103b17 - 243b16 - 184b15 - 197b14 + 198b13 - 165b12 + 406b11 + 205b10 + 117b9 + 837b8 - 35b7 - 225b6 + 175b5 + 36b4 - 108b3 - 527b2 + 152*b1 - 405) / 3
$\nu^{13}$	$=$	$ ( - 111 \beta_{17} + 144 \beta_{16} - 91 \beta_{15} - 183 \beta_{14} - 150 \beta_{13} - 454 \beta_{12} + \cdots + 1725 ) / 3 $ (-111b17 + 144b16 - 91b15 - 183b14 - 150b13 - 454b12 - 36b11 + 139b10 + 192b9 + 1446b8 - 255b7 + 570b6 + 459b5 - 141b4 - 57b3 - 83b2 + 57*b1 + 1725) / 3
$\nu^{14}$	$=$	$ ( 230 \beta_{17} + 534 \beta_{16} - 656 \beta_{15} + 379 \beta_{14} - 6 \beta_{13} - 733 \beta_{12} + \cdots + 477 ) / 3 $ (230b17 + 534b16 - 656b15 + 379b14 - 6b13 - 733b12 + 1698b11 + 108b10 - 222b9 - 1368b8 - 26b7 + 1008b6 - 365b5 - 873b4 - 141b3 + 812b2 - 1092*b1 + 477) / 3
$\nu^{15}$	$=$	$ ( - 135 \beta_{17} + 171 \beta_{16} + 462 \beta_{15} - 2493 \beta_{14} + 381 \beta_{13} - 186 \beta_{12} + \cdots + 1527 ) / 3 $ (-135b17 + 171b16 + 462b15 - 2493b14 + 381b13 - 186b12 + 1057b11 - 211b10 - 663b9 + 924b8 - 1890b7 - 2400b6 + 1800b5 + 345b4 + 429b3 - 114b2 - 766*b1 + 1527) / 3
$\nu^{16}$	$=$	$ ( 791 \beta_{17} + 1554 \beta_{16} - 831 \beta_{15} - 1199 \beta_{14} + 2436 \beta_{13} + 1135 \beta_{12} + \cdots + 4248 ) / 3 $ (791b17 + 1554b16 - 831b15 - 1199b14 + 2436b13 + 1135b12 + 437b11 - 859b10 + 2841b9 + 1746b8 + 826b7 - 2619b6 + 1072b5 - 3051b4 - 4359b3 - 3546b2 + 55*b1 + 4248) / 3
$\nu^{17}$	$=$	$ ( - 231 \beta_{17} + 8694 \beta_{16} - 6026 \beta_{15} - 1122 \beta_{14} - 1617 \beta_{13} - 3800 \beta_{12} + \cdots + 9897 ) / 3 $ (-231b17 + 8694b16 - 6026b15 - 1122b14 - 1617b13 - 3800b12 + 4142b11 - 390b10 - 6612b9 - 14871b8 - 1896b7 + 4827b6 + 1224b5 + 1443b4 + 5667b3 - 2446b2 + 640*b1 + 9897) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1026\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$-\beta_{8}$	$-\beta_{8}$

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} $

505.1

−1.24302	−	1.20619i
1.73011	+	0.0819856i
−0.238928	+	1.71549i
0.837220	+	1.51627i
−1.68875	+	0.384872i
0.339544	−	1.69844i
−0.614525	−	1.61937i
1.55117	−	0.770640i
−0.672818	+	1.59603i
−1.24302	+	1.20619i
1.73011	−	0.0819856i
−0.238928	−	1.71549i
0.837220	−	1.51627i
−1.68875	−	0.384872i
0.339544	+	1.69844i
−0.614525	+	1.61937i
1.55117	+	0.770640i
−0.672818	−	1.59603i

−0.500000

−

0.866025i

−0.500000

0.866025i

−3.94363

1.02646

−

1.77787i

1.00000

1.97181

3.41528i

505.2

−0.500000

−

0.866025i

−0.500000

0.866025i

−2.93688

0.360554

−

0.624499i

1.00000

1.46844

2.54341i

505.3

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.41232

1.53389

−

2.65678i

1.00000

0.706161

1.22311i

505.4

−0.500000

−

0.866025i

−0.500000

0.866025i

−0.730937

−1.79034

3.10095i

1.00000

0.365468

0.633010i

505.5

−0.500000

−

0.866025i

−0.500000

0.866025i

−0.149615

−0.733568

1.27058i

1.00000

0.0748074

0.129570i

505.6

−0.500000

−

0.866025i

−0.500000

0.866025i

0.935186

−0.568176

0.984110i

1.00000

−0.467593

−

0.809895i

505.7

−0.500000

−

0.866025i

−0.500000

0.866025i

1.57955

2.31561

−

4.01075i

1.00000

−0.789777

−

1.36793i

505.8

−0.500000

−

0.866025i

−0.500000

0.866025i

2.39847

0.959469

−

1.66185i

1.00000

−1.19924

−

2.07714i

505.9

−0.500000

−

0.866025i

−0.500000

0.866025i

4.26017

−0.603898

1.04598i

1.00000

−2.13008

−

3.68941i

577.1

−0.500000

0.866025i

−0.500000

−

0.866025i

−3.94363

1.02646

1.77787i

1.00000

1.97181

−

3.41528i

577.2

−0.500000

0.866025i

−0.500000

−

0.866025i

−2.93688

0.360554

0.624499i

1.00000

1.46844

−

2.54341i

577.3

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.41232

1.53389

2.65678i

1.00000

0.706161

−

1.22311i

577.4

−0.500000

0.866025i

−0.500000

−

0.866025i

−0.730937

−1.79034

−

3.10095i

1.00000

0.365468

−

0.633010i

577.5

−0.500000

0.866025i

−0.500000

−

0.866025i

−0.149615

−0.733568

−

1.27058i

1.00000

0.0748074

−

0.129570i

577.6

−0.500000

0.866025i

−0.500000

−

0.866025i

0.935186

−0.568176

−

0.984110i

1.00000

−0.467593

0.809895i

577.7

−0.500000

0.866025i

−0.500000

−

0.866025i

1.57955

2.31561

4.01075i

1.00000

−0.789777

1.36793i

577.8

−0.500000

0.866025i

−0.500000

−

0.866025i

2.39847

0.959469

1.66185i

1.00000

−1.19924

2.07714i

577.9

−0.500000

0.866025i

−0.500000

−

0.866025i

4.26017

−0.603898

−

1.04598i

1.00000

−2.13008

3.68941i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1026.2.h.g		18
3.b	odd	2	1		342.2.h.g	yes	18
9.c	even	3	1		1026.2.f.g		18
9.d	odd	6	1		342.2.f.g	✓	18
19.c	even	3	1		1026.2.f.g		18
57.h	odd	6	1		342.2.f.g	✓	18
171.g	even	3	1	inner	1026.2.h.g		18
171.n	odd	6	1		342.2.h.g	yes	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
342.2.f.g	✓	18	9.d	odd	6	1
342.2.f.g	✓	18	57.h	odd	6	1
342.2.h.g	yes	18	3.b	odd	2	1
342.2.h.g	yes	18	171.n	odd	6	1
1026.2.f.g		18	9.c	even	3	1
1026.2.f.g		18	19.c	even	3	1
1026.2.h.g		18	1.a	even	1	1	trivial
1026.2.h.g		18	171.g	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{9} - 27T_{5}^{7} - 2T_{5}^{6} + 192T_{5}^{5} - 9T_{5}^{4} - 387T_{5}^{3} + 189T_{5} + 27 $ T5^9 - 27*T5^7 - 2*T5^6 + 192*T5^5 - 9*T5^4 - 387*T5^3 + 189*T5 + 27 acting on $S_{2}^{\mathrm{new}}(1026, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + T + 1)^{9} $ (T^2 + T + 1)^9
$3$	$ T^{18} $ T^18
$5$	$ (T^{9} - 27 T^{7} + \cdots + 27)^{2} $ (T^9 - 27T^7 - 2T^6 + 192T^5 - 9T^4 - 387T^3 + 189T + 27)^2
$7$	$ T^{18} - 5 T^{17} + \cdots + 84681 $ T^18 - 5T^17 + 41T^16 - 102T^15 + 670T^14 - 1450T^13 + 6829T^12 - 8320T^11 + 31093T^10 - 18876T^9 + 101282T^8 - 24344T^7 + 211204T^6 + 22185T^5 + 317517T^4 + 25638T^3 + 197271T^2 - 22698*T + 84681
$11$	$ T^{18} + T^{17} + \cdots + 81 $ T^18 + T^17 + 45T^16 - 114T^15 + 1380T^14 - 3822T^13 + 26565T^12 - 89952T^11 + 341493T^10 - 702452T^9 + 1227106T^8 - 1091364T^7 + 819804T^6 - 25029T^5 + 160785T^4 - 20466T^3 + 11205T^2 + 810T + 81
$13$	$ T^{18} - T^{17} + \cdots + 3154176 $ T^18 - T^17 + 67T^16 - 98T^15 + 3221T^14 - 4112T^13 + 71352T^12 - 40302T^11 + 1080141T^10 - 344030T^9 + 8788532T^8 + 3469552T^7 + 44580400T^6 + 1038224T^5 + 67628320T^4 - 5438976T^3 + 82110336T^2 - 15685632T + 3154176
$17$	$ T^{18} + \cdots + 130439241 $ T^18 - 5T^17 + 84T^16 - 455T^15 + 4990T^14 - 23505T^13 + 144597T^12 - 462564T^11 + 2119446T^10 - 5428512T^9 + 19996578T^8 - 32480028T^7 + 79424145T^6 - 57024567T^5 + 173879622T^4 - 58180761T^3 + 267373872T^2 + 119338029*T + 130439241
$19$	$ T^{18} + \cdots + 322687697779 $ T^18 - 9T^17 + 21T^16 - 56T^15 + 681T^14 - 1680T^13 - 2773T^12 - 20712T^11 + 147222T^10 - 217310T^9 + 2797218T^8 - 7477032T^7 - 19020007T^6 - 218939280T^5 + 1686223419T^4 - 2634569336T^3 + 18771306519T^2 - 152852067369*T + 322687697779
$23$	$ T^{18} - 2 T^{17} + \cdots + 2985984 $ T^18 - 2T^17 + 93T^16 - 114T^15 + 6204T^14 - 4926T^13 + 168165T^12 + 302466T^11 + 2737065T^10 + 7316176T^9 + 39198184T^8 + 112040928T^7 + 289066560T^6 + 454749120T^5 + 557551872T^4 + 405292032T^3 + 210760704T^2 + 27869184*T + 2985984
$29$	$ (T^{9} + 9 T^{8} + \cdots + 800577)^{2} $ (T^9 + 9T^8 - 117T^7 - 1069T^6 + 4476T^5 + 41796T^4 - 55737T^3 - 558225T^2 + 6993T + 800577)^2
$31$	$ T^{18} + \cdots + 2690808129 $ T^18 - 4T^17 + 133T^16 - 672T^15 + 12579T^14 - 60273T^13 + 640000T^12 - 2813896T^11 + 22609930T^10 - 82415622T^9 + 380223249T^8 - 719077392T^7 + 2195978641T^6 - 2143522498T^5 + 9132292384T^4 - 4884501024T^3 + 7641570399T^2 + 2546082459*T + 2690808129
$37$	$ (T^{9} - 10 T^{8} + \cdots - 2416)^{2} $ (T^9 - 10T^8 - 61T^7 + 1060T^6 - 4173T^5 + 4530T^4 + 6428T^3 - 16880T^2 + 11440T - 2416)^2
$41$	$ (T^{9} - T^{8} - 86 T^{7} + \cdots - 1296)^{2} $ (T^9 - T^8 - 86T^7 - 82T^6 + 1741T^5 + 2675T^4 - 9024T^3 - 11712T^2 + 11448*T - 1296)^2
$43$	$ T^{18} + \cdots + 89913620736 $ T^18 - 7T^17 + 244T^16 - 1485T^15 + 39237T^14 - 230253T^13 + 3260470T^12 - 16313413T^11 + 180426277T^10 - 841210860T^9 + 4932342312T^8 - 14419743312T^7 + 58194561568T^6 - 138624967312T^5 + 450383424064T^4 - 674297670912T^3 + 889610072064T^2 - 311639141376*T + 89913620736
$47$	$ (T^{9} - 19 T^{8} + \cdots + 1256751)^{2} $ (T^9 - 19T^8 - 26T^7 + 2146T^6 - 5740T^5 - 74732T^4 + 292482T^3 + 775374T^2 - 3660381T + 1256751)^2
$53$	$ T^{18} + \cdots + 20372138361 $ T^18 - 10T^17 + 261T^16 - 1166T^15 + 32819T^14 - 121461T^13 + 2383418T^12 - 883334T^11 + 75446700T^10 + 123579344T^9 + 2197599211T^8 + 4724897376T^7 + 21739573425T^6 + 11592184698T^5 + 51886176804T^4 - 11334420504T^3 + 127770590727T^2 - 48417067089*T + 20372138361
$59$	$ (T^{9} + 5 T^{8} + \cdots - 3722193)^{2} $ (T^9 + 5T^8 - 287T^7 - 2547T^6 + 15624T^5 + 260472T^4 + 925515T^3 + 89703T^2 - 4132809T - 3722193)^2
$61$	$ (T^{9} + 18 T^{8} + \cdots - 225347)^{2} $ (T^9 + 18T^8 + 6T^7 - 1434T^6 - 6741T^5 + 10191T^4 + 92520T^3 + 47916T^2 - 262218T - 225347)^2
$67$	$ T^{18} + \cdots + 344768247300096 $ T^18 - 22T^17 + 712T^16 - 11434T^15 + 258781T^14 - 3681724T^13 + 58734925T^12 - 659534263T^11 + 8085305077T^10 - 75918711588T^9 + 708408945696T^8 - 4616971819008T^7 + 25677858097152T^6 - 80532701077248T^5 + 199019284273152T^4 - 184802862624768T^3 + 249721518219264T^2 - 45882112573440*T + 344768247300096
$71$	$ T^{18} + \cdots + 120758121 $ T^18 + 11T^17 + 291T^16 + 1006T^15 + 33818T^14 + 45420T^13 + 2891432T^12 - 4788308T^11 + 132752949T^10 - 427129117T^9 + 5083570705T^8 - 16988165916T^7 + 73173306732T^6 - 80451237168T^5 + 105368305446T^4 + 49394551734T^3 + 34651559187T^2 + 2116778103*T + 120758121
$73$	$ T^{18} + \cdots + 490005727861329 $ T^18 - 44T^17 + 1409T^16 - 29568T^15 + 540085T^14 - 8090347T^13 + 112628434T^12 - 1318092754T^11 + 13470713512T^10 - 108710987430T^9 + 720799832855T^8 - 3551777870840T^7 + 14296218390253T^6 - 41171015991294T^5 + 127051061574072T^4 - 288100799541078T^3 + 751261583672037T^2 - 662101299235449*T + 490005727861329
$79$	$ T^{18} + \cdots + 28\!\cdots\!84 $ T^18 - 2T^17 + 374T^16 - 598T^15 + 96069T^14 - 151854T^13 + 12207275T^12 - 18719611T^11 + 1102487977T^10 - 1808396632T^9 + 60850235128T^8 - 113414724352T^7 + 2450760870656T^6 - 4554089038128T^5 + 57711517345152T^4 - 104361417799936T^3 + 914124712666112T^2 - 1438413349872128*T + 2818816529430784
$83$	$ T^{18} + \cdots + 25101031868649 $ T^18 - 7T^17 + 435T^16 - 5644T^15 + 150580T^14 - 1769706T^13 + 28959861T^12 - 324883044T^11 + 3985324065T^10 - 33745825422T^9 + 245625382314T^8 - 1164734144604T^7 + 4336480494144T^6 - 9259157312991T^5 + 16576329818907T^4 - 15350705121888T^3 + 23582756116197T^2 - 16925837666364*T + 25101031868649
$89$	$ T^{18} + \cdots + 28991407990689 $ T^18 + T^17 + 285T^16 - 20T^15 + 59926T^14 + 1482T^13 + 4909866T^12 + 229512T^11 + 286600059T^10 - 72026613T^9 + 8456006835T^8 - 12326847192T^7 + 178146913842T^6 - 286830152190T^5 + 2192868734310T^4 - 4509941331564T^3 + 18078647419701T^2 - 22029259928679T + 28991407990689
$97$	$ T^{18} + \cdots + 469410009520384 $ T^18 + 504T^16 - 1570T^15 + 178725T^14 - 586566T^13 + 32643257T^12 - 129619533T^11 + 4299730839T^10 - 11651571850T^9 + 229021292628T^8 + 13748902128T^7 + 7463856715328T^6 + 950805829488T^5 + 98362892789280T^4 + 229473838173824T^3 + 655402935502464T^2 + 587870677921536T + 469410009520384
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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:	\( N \)	\(=\)	\( 1026 = 2 \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1026.h (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{8} q^{2} + (\beta_{8} - 1) q^{4} + \beta_{5} q^{5} + ( - \beta_{8} + \beta_{6} - \beta_{4} + 1) q^{7} + q^{8}+O(q^{10}) \) q - b8 * q^2 + (b8 - 1) * q^4 + b5 * q^5 + (-b8 + b6 - b4 + 1) * q^7 + q^8	\( q - \beta_{8} q^{2} + (\beta_{8} - 1) q^{4} + \beta_{5} q^{5} + ( - \beta_{8} + \beta_{6} - \beta_{4} + 1) q^{7} + q^{8} - \beta_{14} q^{10} + ( - \beta_{13} + \beta_{3}) q^{11} + ( - \beta_{14} - \beta_{8} + \beta_{6} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{17} - \beta_{16} + \cdots - \beta_{3}) q^{98}+O(q^{100}) \) q - b8 * q^2 + (b8 - 1) * q^4 + b5 * q^5 + (-b8 + b6 - b4 + 1) * q^7 + q^8 - b14 * q^10 + (-b13 + b3) * q^11 + (-b14 - b8 + b6 + b5 - b4 - b2 + b1 + 1) * q^13 + (b4 - 1) * q^14 - b8 * q^16 + (b13 - b6 + b4 - b3) * q^17 + (b16 - b11 - b9 + 1) * q^19 + (b14 - b5) * q^20 - b3 * q^22 + (-b16 + b13 + b10 + b9 + b6 - b4 - b3) * q^23 + (b12 + 2b9 - b4 - 2b3 - b2 + 1) * q^25 + (-b5 + b4 + b2 - 1) * q^26 + (b8 - b6) * q^28 + (-b9 - b7 - b4) * q^29 + (b17 - b15 + b14 + b13 - b12 + b11 + b10 + b8) * q^31 + (b8 - 1) * q^32 + (-b4 + b3) * q^34 + (b16 - b14 - b13 - b10 - b9 + b5 + b3) * q^35 + (-b7 + b5 - b4 + 2) * q^37 + (-b15 + b11 + b9 - b8) * q^38 + b5 * q^40 + (b9 + b5) * q^41 + (b17 + b16 + b14 - b13 + b8 + b6 - b1) * q^43 + b13 * q^44 + (-b12 - b9 + b4 + b3) * q^46 + (-b12 - b5 - b2 + 3) * q^47 + (b17 + b16 - b15 - b13 + b11 + b6) * q^49 + (-2b16 + 2b13 - b12 + b10 - b8 + b6 + b1) * q^50 + (b14 + b8 - b6 - b1) * q^52 + (2b16 - b15 - b14 + b11 + b6) q^53 + (b17 - b16 - 2b15 + 2b14 + b13 + b10 + b9 + b7 - 2b5 - b3 + b2 - b1) q^55 + (-b8 + b6 - b4 + 1) * q^56 + (-b17 + b16 + b6) * q^58 + (2b9 - 2b5 - b4 - b3 - b2) * q^59 + (-b12 - b7 + b5 + b3 - 1) * q^61 + (-b17 + b15 - b14 - b13 - b10 - b8 - b7 + b5 + b3 + 1) * q^62 + q^64 + (-b16 - b15 - b14 + 2b13 + b10 + b9 - 6b8 + 2b6 + b5 - 2b4 - 2b3 - b2 + b1 + 6) q^65 + (b16 - b15 + 2b13 + b10 - b9 - b8 - b6 + b4 - 2b3 + 2b2 - 2b1 + 1) * q^67 + (-b13 + b6) * q^68 + (b12 + b9 - b5 - b3) * q^70 + (-b17 + b13 + b10 - b7 + 2b6 - 2b4 - b3 - b2 + b1) * q^71 + (-b15 - b14 + b13 - 7b8 + 2b6 + b5 - 2b4 - b3 - 2b2 + 2b1 + 7) q^73 + (-b17 - b14 - 2b8 + b6) q^74 + (-b16 + b15 + b8 - 1) * q^76 + (b17 - 2b13 + b6) q^77 + (b17 - 2b16 + b15 + b14 + b13 - 2b12 - b11 + 2b10 + b8 + b6 + 2b1) * q^79 - b14 * q^80 + (-b16 - b14) * q^82 + (b17 + b16 - 2b15 - b14 - b10 - b9 + b7 - 2b6 + b5 + 2b4 + 2b2 - 2b1) q^83 + (-b17 + 2b15 - 2b14 - b7 + 2b5 - b2 + b1) q^85 + (-b17 - b16 - b14 + b13 + b9 - b8 - b7 - b6 + b5 + b4 - b3 - b2 + b1 + 1) * q^86 + (-b13 + b3) * q^88 + (b17 - b16 - b15 - 2b12 + b11 + 2b10 + b6 + b1) * q^89 + (b16 - 2b14 - 2b13 + 2b12 - 2b10 - 4b8 + b6 - b1) q^91 + (b16 - b13 + b12 - b10 - b6) * q^92 + (b14 + b12 - b10 - 3b8 + b1) q^94 + (-b17 + b16 + b15 + b14 + b11 - 2b10 - b9 - b7 - b4 + 2b3 - b2) * q^95 + (-2b17 - b16 + 2b15 - b14 + b13 - 2b11 - 2b8 + 2b6 + 3b1) * q^97 + (-b17 - b16 + b15 + b13 + b9 - b7 - b6 + b4 - b3) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 9 q^{2} - 9 q^{4} + 5 q^{7} + 18 q^{8}+O(q^{10}) \) 18 * q - 9 * q^2 - 9 * q^4 + 5 * q^7 + 18 * q^8	\( 18 q - 9 q^{2} - 9 q^{4} + 5 q^{7} + 18 q^{8} - q^{11} + q^{13} - 10 q^{14} - 9 q^{16} + 5 q^{17} + 9 q^{19} + 2 q^{22} + 2 q^{23} + 18 q^{25} - 2 q^{26} + 5 q^{28} - 18 q^{29} + 4 q^{31} - 9 q^{32} - 10 q^{34} - 6 q^{35} + 20 q^{37} - 3 q^{38} + 2 q^{41} + 7 q^{43} - q^{44} - 4 q^{46} + 38 q^{47} + 6 q^{49} - 9 q^{50} + q^{52} + 10 q^{53} + 6 q^{55} + 5 q^{56} + 9 q^{58} - 10 q^{59} - 36 q^{61} + 4 q^{62} + 18 q^{64} + 45 q^{65} + 22 q^{67} + 5 q^{68} + 12 q^{70} - 11 q^{71} + 44 q^{73} - 10 q^{74} - 6 q^{76} + 2 q^{77} + 2 q^{79} - q^{82} + 7 q^{83} + 7 q^{86} - q^{88} - q^{89} - 25 q^{91} + 2 q^{92} - 19 q^{94} - 21 q^{95} + 6 q^{98}+O(q^{100}) \) 18 * q - 9 * q^2 - 9 * q^4 + 5 * q^7 + 18 * q^8 - q^11 + q^13 - 10 * q^14 - 9 * q^16 + 5 * q^17 + 9 * q^19 + 2 * q^22 + 2 * q^23 + 18 * q^25 - 2 * q^26 + 5 * q^28 - 18 * q^29 + 4 * q^31 - 9 * q^32 - 10 * q^34 - 6 * q^35 + 20 * q^37 - 3 * q^38 + 2 * q^41 + 7 * q^43 - q^44 - 4 * q^46 + 38 * q^47 + 6 * q^49 - 9 * q^50 + q^52 + 10 * q^53 + 6 * q^55 + 5 * q^56 + 9 * q^58 - 10 * q^59 - 36 * q^61 + 4 * q^62 + 18 * q^64 + 45 * q^65 + 22 * q^67 + 5 * q^68 + 12 * q^70 - 11 * q^71 + 44 * q^73 - 10 * q^74 - 6 * q^76 + 2 * q^77 + 2 * q^79 - q^82 + 7 * q^83 + 7 * q^86 - q^88 - q^89 - 25 * q^91 + 2 * q^92 - 19 * q^94 - 21 * q^95 + 6 * 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	1026.2.h.g

Level	$1026$
Weight	$2$
Character orbit	1026.h
Analytic conductor	$8.193$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(8.19265124738\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Relative dimension:	\(9\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{18} + 4 x^{16} - 6 x^{15} + x^{14} - 21 x^{13} - 12 x^{12} + 9 x^{10} + 135 x^{9} + 27 x^{8} + \cdots + 19683 \) x^18 + 4x^16 - 6x^15 + x^14 - 21x^13 - 12x^12 + 9x^10 + 135x^9 + 27x^8 - 324x^6 - 1701x^5 + 243x^4 - 4374x^3 + 8748x^2 + 19683
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{6} \)
Twist minimal:	no (minimal twist has level 342)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - 164 \nu^{17} + 831 \nu^{16} - 2204 \nu^{15} + 1587 \nu^{14} - 6086 \nu^{13} + 2337 \nu^{12} + \cdots + 7466418 ) / 2574099 \) (-164v^17 + 831v^16 - 2204v^15 + 1587v^14 - 6086v^13 + 2337v^12 - 3711v^11 - 5502v^10 + 30060v^9 - 6813v^8 - 5211v^7 - 248670v^6 - 2592v^5 + 683964v^4 - 768852v^3 + 2225394v^2 - 4400973*v + 7466418) / 2574099
\(\beta_{2}\)	\(=\)	\( ( - 118 \nu^{17} + 474 \nu^{16} - 409 \nu^{15} + 1893 \nu^{14} - 1981 \nu^{13} - 1053 \nu^{12} + \cdots + 2821230 ) / 702027 \) (-118v^17 + 474v^16 - 409v^15 + 1893v^14 - 1981v^13 - 1053v^12 - 3237v^11 - 5103v^10 + 10764v^9 + 2565v^8 + 23868v^7 - 25110v^6 - 45603v^5 + 119799v^4 - 712476v^3 + 589032v^2 - 1961739*v + 2821230) / 702027
\(\beta_{3}\)	\(=\)	\( ( 44 \nu^{17} + 351 \nu^{16} - 397 \nu^{15} + 774 \nu^{14} - 2455 \nu^{13} - 282 \nu^{12} + \cdots + 2755620 ) / 234009 \) (44v^17 + 351v^16 - 397v^15 + 774v^14 - 2455v^13 - 282v^12 - 2460v^11 - 2058v^10 + 11835v^9 + 3222v^8 + 34533v^7 - 20898v^6 - 60264v^5 - 19845v^4 - 504225v^3 + 659988v^2 - 889380*v + 2755620) / 234009
\(\beta_{4}\)	\(=\)	\( ( - 128 \nu^{17} - 1809 \nu^{16} + 1972 \nu^{15} - 4236 \nu^{14} + 11023 \nu^{13} + 762 \nu^{12} + \cdots - 12419973 ) / 702027 \) (-128v^17 - 1809v^16 + 1972v^15 - 4236v^14 + 11023v^13 + 762v^12 + 9987v^11 + 7767v^10 - 42921v^9 - 16902v^8 - 120420v^7 + 157707v^6 + 283257v^5 + 195615v^4 + 2180682v^3 - 3307473v^2 + 4253715*v - 12419973) / 702027
\(\beta_{5}\)	\(=\)	\( ( 18 \nu^{17} - 85 \nu^{16} + 144 \nu^{15} - 267 \nu^{14} + 411 \nu^{13} - 57 \nu^{12} + 15 \nu^{11} + \cdots - 475308 ) / 78003 \) (18v^17 - 85v^16 + 144v^15 - 267v^14 + 411v^13 - 57v^12 + 15v^11 + 802v^10 - 1272v^9 - 141v^8 - 5877v^7 + 1431v^6 + 3186v^5 - 23220v^4 + 115587v^3 - 197235v^2 + 309825*v - 475308) / 78003
\(\beta_{6}\)	\(=\)	\( ( 1832 \nu^{17} - 15738 \nu^{16} + 19523 \nu^{15} - 39879 \nu^{14} + 80051 \nu^{13} + \cdots - 86769225 ) / 7722297 \) (1832v^17 - 15738v^16 + 19523v^15 - 39879v^14 + 80051v^13 - 12477v^12 + 47199v^11 + 83412v^10 - 298899v^9 + 74061v^8 - 794556v^7 + 674892v^6 + 1846638v^5 - 2418336v^4 + 20080062v^3 - 29845989v^2 + 46510929*v - 86769225) / 7722297
\(\beta_{7}\)	\(=\)	\( ( 82 \nu^{17} + 445 \nu^{16} - 521 \nu^{15} + 1102 \nu^{14} - 3014 \nu^{13} + 418 \nu^{12} + \cdots + 3667599 ) / 234009 \) (82v^17 + 445v^16 - 521v^15 + 1102v^14 - 3014v^13 + 418v^12 - 4176v^11 - 1530v^10 + 9756v^9 + 2853v^8 + 35073v^7 - 35532v^6 - 44550v^5 - 122472v^4 - 593406v^3 + 620136v^2 - 998001*v + 3667599) / 234009
\(\beta_{8}\)	\(=\)	\( ( - 3248 \nu^{17} + 3348 \nu^{16} - 6368 \nu^{15} + 15600 \nu^{14} + 9073 \nu^{13} + 16881 \nu^{12} + \cdots - 5006043 ) / 7722297 \) (-3248v^17 + 3348v^16 - 6368v^15 + 15600v^14 + 9073v^13 + 16881v^12 + 24135v^11 - 71901v^10 - 21078v^9 - 196371v^8 + 201555v^7 + 406539v^6 + 112914v^5 + 4144122v^4 - 5330205v^3 + 5697135v^2 - 19081575*v - 5006043) / 7722297
\(\beta_{9}\)	\(=\)	\( ( 323 \nu^{17} - 699 \nu^{16} + 1193 \nu^{15} - 3204 \nu^{14} + 2150 \nu^{13} - 3252 \nu^{12} + \cdots - 2427570 ) / 702027 \) (323v^17 - 699v^16 + 1193v^15 - 3204v^14 + 2150v^13 - 3252v^12 - 2067v^11 + 12834v^10 + 2070v^9 + 31050v^8 - 38745v^7 - 162v^6 + 38232v^5 - 425979v^4 + 1153035v^3 - 1482786v^2 + 3483891*v - 2427570) / 702027
\(\beta_{10}\)	\(=\)	\( ( - 4568 \nu^{17} - 13647 \nu^{16} + 14389 \nu^{15} - 12879 \nu^{14} + 94180 \nu^{13} + \cdots - 112540833 ) / 7722297 \) (-4568v^17 - 13647v^16 + 14389v^15 - 12879v^14 + 94180v^13 + 32160v^12 + 103038v^11 + 24624v^10 - 433314v^9 - 359721v^8 - 728649v^7 + 2013903v^6 + 2240622v^5 + 4849794v^4 + 15310701v^3 - 23708538v^2 + 2377269*v - 112540833) / 7722297
\(\beta_{11}\)	\(=\)	\( ( - 460 \nu^{17} - 1014 \nu^{16} + 707 \nu^{15} - 738 \nu^{14} + 6902 \nu^{13} + 3561 \nu^{12} + \cdots - 8194689 ) / 702027 \) (-460v^17 - 1014v^16 + 707v^15 - 738v^14 + 6902v^13 + 3561v^12 + 8355v^11 + 738v^10 - 31194v^9 - 34560v^8 - 67662v^7 + 115263v^6 + 202986v^5 + 627426v^4 + 1182195v^3 - 1548396v^2 + 1121931*v - 8194689) / 702027
\(\beta_{12}\)	\(=\)	\( ( - 481 \nu^{17} - 2502 \nu^{16} + 3107 \nu^{15} - 5295 \nu^{14} + 16106 \nu^{13} + 1434 \nu^{12} + \cdots - 19210608 ) / 702027 \) (-481v^17 - 2502v^16 + 3107v^15 - 5295v^14 + 16106v^13 + 1434v^12 + 16095v^11 + 6579v^10 - 70263v^9 - 28350v^8 - 150525v^7 + 255636v^6 + 347571v^5 + 589032v^4 + 3154869v^3 - 5089878v^2 + 4907628*v - 19210608) / 702027
\(\beta_{13}\)	\(=\)	\( ( - 6235 \nu^{17} + 14928 \nu^{16} - 21007 \nu^{15} + 52014 \nu^{14} - 30850 \nu^{13} + \cdots + 31893021 ) / 7722297 \) (-6235v^17 + 14928v^16 - 21007v^15 + 52014v^14 - 30850v^13 + 21834v^12 + 10695v^11 - 257778v^10 + 162882v^9 - 469125v^8 + 919485v^7 + 969246v^6 - 645813v^5 + 9312975v^4 - 19877157v^3 + 24380676v^2 - 58158891*v + 31893021) / 7722297
\(\beta_{14}\)	\(=\)	\( ( - 3391 \nu^{17} + 477 \nu^{16} - 5620 \nu^{15} + 8307 \nu^{14} + 17675 \nu^{13} + 14505 \nu^{12} + \cdots - 22447368 ) / 2574099 \) (-3391v^17 + 477v^16 - 5620v^15 + 8307v^14 + 17675v^13 + 14505v^12 + 43011v^11 - 45996v^10 - 62559v^9 - 253791v^8 - 93069v^7 + 583848v^6 + 143208v^5 + 4659120v^4 - 1277937v^3 + 3176496v^2 - 10493226*v - 22447368) / 2574099
\(\beta_{15}\)	\(=\)	\( ( - 10619 \nu^{17} - 783 \nu^{16} - 12623 \nu^{15} + 29226 \nu^{14} + 49204 \nu^{13} + \cdots - 61981767 ) / 7722297 \) (-10619v^17 - 783v^16 - 12623v^15 + 29226v^14 + 49204v^13 + 49326v^12 + 106089v^11 - 128340v^10 - 268344v^9 - 887409v^8 - 29943v^7 + 1744011v^6 + 2057886v^5 + 13311054v^4 - 1701243v^3 + 10763685v^2 - 35000748*v - 61981767) / 7722297
\(\beta_{16}\)	\(=\)	\( ( - 3837 \nu^{17} + 1541 \nu^{16} - 5079 \nu^{15} + 13019 \nu^{14} + 23655 \nu^{13} + \cdots - 29043360 ) / 2574099 \) (-3837v^17 + 1541v^16 - 5079v^15 + 13019v^14 + 23655v^13 + 14948v^12 + 40989v^11 - 62055v^10 - 96102v^9 - 292212v^8 + 11556v^7 + 821637v^6 + 620298v^5 + 5560083v^4 - 2197935v^3 + 2991330v^2 - 17807283*v - 29043360) / 2574099
\(\beta_{17}\)	\(=\)	\( ( 12686 \nu^{17} - 19122 \nu^{16} + 40061 \nu^{15} - 82359 \nu^{14} + 27392 \nu^{13} + \cdots - 36131427 ) / 7722297 \) (12686v^17 - 19122v^16 + 40061v^15 - 82359v^14 + 27392v^13 - 75126v^12 - 16314v^11 + 311778v^10 - 8325v^9 + 716715v^8 - 974943v^7 - 1808082v^6 + 1132704v^5 - 16656435v^4 + 29707965v^3 - 42408117v^2 + 92094570*v - 36131427) / 7722297

\(\nu\)	\(=\)	\( ( \beta_{11} - \beta_{10} - \beta_1 ) / 3 \) (b11 - b10 - b1) / 3
\(\nu^{2}\)	\(=\)	\( ( -\beta_{17} + \beta_{15} - 2\beta_{14} - \beta_{12} - 2\beta_{7} + \beta_{5} - \beta_{2} ) / 3 \) (-b17 + b15 - 2b14 - b12 - 2b7 + b5 - b2) / 3
\(\nu^{3}\)	\(=\)	\( ( 3 \beta_{13} + \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - 3 \beta_{8} + 3 \beta_{6} - 6 \beta_{4} + \cdots + 6 ) / 3 \) (3b13 + b11 + 2b10 + 3b9 - 3b8 + 3b6 - 6b4 - 3b3 - 3b2 + 2*b1 + 6) / 3
\(\nu^{4}\)	\(=\)	\( ( - \beta_{17} + 6 \beta_{16} - 3 \beta_{15} - 2 \beta_{14} - 3 \beta_{13} + \beta_{12} + 5 \beta_{11} + \cdots + 9 ) / 3 \) (-b17 + 6b16 - 3b15 - 2b14 - 3b13 + b12 + 5b11 - 4b10 - 3b9 - 9b8 - 2b7 + b5 + 6b3 + b1 + 9) / 3
\(\nu^{5}\)	\(=\)	\( ( 3 \beta_{17} + 4 \beta_{15} - 6 \beta_{14} + 3 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 3 \beta_{10} + \cdots - 12 ) / 3 \) (3b17 + 4b15 - 6b14 + 3b13 - 2b12 + 2b11 - 3b10 + 3b9 + 6b8 + 3b7 - 6b6 - 9b5 + 12b4 - 3b3 + 2*b2 + b1 - 12) / 3
\(\nu^{6}\)	\(=\)	\( ( - 4 \beta_{17} - 4 \beta_{15} + \beta_{14} + 9 \beta_{13} - 3 \beta_{12} + \beta_{11} + 7 \beta_{10} + \cdots + 27 ) / 3 \) (-4b17 - 4b15 + b14 + 9b13 - 3b12 + b11 + 7b10 + 9b9 - 27b8 + b7 - 9b6 - 5b5 + 9b4 - 5b2 - 10b1 + 27) / 3
\(\nu^{7}\)	\(=\)	\( ( 6 \beta_{17} + 9 \beta_{16} + 8 \beta_{15} - 3 \beta_{14} - 15 \beta_{13} + 14 \beta_{12} - 5 \beta_{10} + \cdots - 3 ) / 3 \) (6b17 + 9b16 + 8b15 - 3b14 - 15b13 + 14b12 - 5b10 + 3b9 + 42b8 + 6b7 + 30b6 - 9b5 - 33b4 + 24b3 - 20b2 - 6b1 - 3) / 3
\(\nu^{8}\)	\(=\)	\( ( - 22 \beta_{17} - 6 \beta_{16} - 35 \beta_{15} - 17 \beta_{14} - 6 \beta_{13} + 5 \beta_{12} + \cdots - 63 ) / 3 \) (-22b17 - 6b16 - 35b15 - 17b14 - 6b13 + 5b12 + 33b11 + 21b9 + 36b8 - 17b7 + 9b6 - 59b5 - 9b4 - 6b3 + 2b2 + 6b1 - 63) / 3
\(\nu^{9}\)	\(=\)	\( ( 36 \beta_{17} - 72 \beta_{16} - 15 \beta_{15} + 36 \beta_{14} + 84 \beta_{13} - 78 \beta_{12} + \cdots - 120 ) / 3 \) (36b17 - 72b16 - 15b15 + 36b14 + 84b13 - 78b12 + 58b11 + 68b10 + 93b9 + 168b8 - 9b7 - 24b6 + 9b5 + 48b4 - 3b3 + 12b2 + 17*b1 - 120) / 3
\(\nu^{10}\)	\(=\)	\( ( 53 \beta_{17} + 150 \beta_{16} - 39 \beta_{15} + 52 \beta_{14} - 210 \beta_{13} - 8 \beta_{12} + \cdots + 144 ) / 3 \) (53b17 + 150b16 - 39b15 + 52b14 - 210b13 - 8b12 + 77b11 - 13b10 + 33b9 - 9b8 + 52b7 - 135b6 - 26b5 + 96b3 + 36b2 - 125b1 + 144) / 3
\(\nu^{11}\)	\(=\)	\( ( 210 \beta_{17} - 140 \beta_{15} + 219 \beta_{14} + 3 \beta_{13} - 2 \beta_{12} + 83 \beta_{11} + \cdots + 312 ) / 3 \) (210b17 - 140b15 + 219b14 + 3b13 - 2b12 + 83b11 + 42b10 - 132b9 + 87b8 + 12b7 - 87b6 - 324b5 + 12b4 + 24b3 - 43b2 - 161b1 + 312) / 3
\(\nu^{12}\)	\(=\)	\( ( - 103 \beta_{17} - 243 \beta_{16} - 184 \beta_{15} - 197 \beta_{14} + 198 \beta_{13} - 165 \beta_{12} + \cdots - 405 ) / 3 \) (-103b17 - 243b16 - 184b15 - 197b14 + 198b13 - 165b12 + 406b11 + 205b10 + 117b9 + 837b8 - 35b7 - 225b6 + 175b5 + 36b4 - 108b3 - 527b2 + 152*b1 - 405) / 3
\(\nu^{13}\)	\(=\)	\( ( - 111 \beta_{17} + 144 \beta_{16} - 91 \beta_{15} - 183 \beta_{14} - 150 \beta_{13} - 454 \beta_{12} + \cdots + 1725 ) / 3 \) (-111b17 + 144b16 - 91b15 - 183b14 - 150b13 - 454b12 - 36b11 + 139b10 + 192b9 + 1446b8 - 255b7 + 570b6 + 459b5 - 141b4 - 57b3 - 83b2 + 57*b1 + 1725) / 3
\(\nu^{14}\)	\(=\)	\( ( 230 \beta_{17} + 534 \beta_{16} - 656 \beta_{15} + 379 \beta_{14} - 6 \beta_{13} - 733 \beta_{12} + \cdots + 477 ) / 3 \) (230b17 + 534b16 - 656b15 + 379b14 - 6b13 - 733b12 + 1698b11 + 108b10 - 222b9 - 1368b8 - 26b7 + 1008b6 - 365b5 - 873b4 - 141b3 + 812b2 - 1092*b1 + 477) / 3
\(\nu^{15}\)	\(=\)	\( ( - 135 \beta_{17} + 171 \beta_{16} + 462 \beta_{15} - 2493 \beta_{14} + 381 \beta_{13} - 186 \beta_{12} + \cdots + 1527 ) / 3 \) (-135b17 + 171b16 + 462b15 - 2493b14 + 381b13 - 186b12 + 1057b11 - 211b10 - 663b9 + 924b8 - 1890b7 - 2400b6 + 1800b5 + 345b4 + 429b3 - 114b2 - 766*b1 + 1527) / 3
\(\nu^{16}\)	\(=\)	\( ( 791 \beta_{17} + 1554 \beta_{16} - 831 \beta_{15} - 1199 \beta_{14} + 2436 \beta_{13} + 1135 \beta_{12} + \cdots + 4248 ) / 3 \) (791b17 + 1554b16 - 831b15 - 1199b14 + 2436b13 + 1135b12 + 437b11 - 859b10 + 2841b9 + 1746b8 + 826b7 - 2619b6 + 1072b5 - 3051b4 - 4359b3 - 3546b2 + 55*b1 + 4248) / 3
\(\nu^{17}\)	\(=\)	\( ( - 231 \beta_{17} + 8694 \beta_{16} - 6026 \beta_{15} - 1122 \beta_{14} - 1617 \beta_{13} - 3800 \beta_{12} + \cdots + 9897 ) / 3 \) (-231b17 + 8694b16 - 6026b15 - 1122b14 - 1617b13 - 3800b12 + 4142b11 - 390b10 - 6612b9 - 14871b8 - 1896b7 + 4827b6 + 1224b5 + 1443b4 + 5667b3 - 2446b2 + 640*b1 + 9897) / 3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 505.9
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{9} \) (T^2 + T + 1)^9
$3$	\( T^{18} \) T^18
$5$	\( (T^{9} - 27 T^{7} + \cdots + 27)^{2} \) (T^9 - 27T^7 - 2T^6 + 192T^5 - 9T^4 - 387T^3 + 189T + 27)^2
$7$	\( T^{18} - 5 T^{17} + \cdots + 84681 \) T^18 - 5T^17 + 41T^16 - 102T^15 + 670T^14 - 1450T^13 + 6829T^12 - 8320T^11 + 31093T^10 - 18876T^9 + 101282T^8 - 24344T^7 + 211204T^6 + 22185T^5 + 317517T^4 + 25638T^3 + 197271T^2 - 22698*T + 84681
$11$	\( T^{18} + T^{17} + \cdots + 81 \) T^18 + T^17 + 45T^16 - 114T^15 + 1380T^14 - 3822T^13 + 26565T^12 - 89952T^11 + 341493T^10 - 702452T^9 + 1227106T^8 - 1091364T^7 + 819804T^6 - 25029T^5 + 160785T^4 - 20466T^3 + 11205T^2 + 810T + 81
$13$	\( T^{18} - T^{17} + \cdots + 3154176 \) T^18 - T^17 + 67T^16 - 98T^15 + 3221T^14 - 4112T^13 + 71352T^12 - 40302T^11 + 1080141T^10 - 344030T^9 + 8788532T^8 + 3469552T^7 + 44580400T^6 + 1038224T^5 + 67628320T^4 - 5438976T^3 + 82110336T^2 - 15685632T + 3154176
$17$	\( T^{18} + \cdots + 130439241 \) T^18 - 5T^17 + 84T^16 - 455T^15 + 4990T^14 - 23505T^13 + 144597T^12 - 462564T^11 + 2119446T^10 - 5428512T^9 + 19996578T^8 - 32480028T^7 + 79424145T^6 - 57024567T^5 + 173879622T^4 - 58180761T^3 + 267373872T^2 + 119338029*T + 130439241
$19$	\( T^{18} + \cdots + 322687697779 \) T^18 - 9T^17 + 21T^16 - 56T^15 + 681T^14 - 1680T^13 - 2773T^12 - 20712T^11 + 147222T^10 - 217310T^9 + 2797218T^8 - 7477032T^7 - 19020007T^6 - 218939280T^5 + 1686223419T^4 - 2634569336T^3 + 18771306519T^2 - 152852067369*T + 322687697779
$23$	\( T^{18} - 2 T^{17} + \cdots + 2985984 \) T^18 - 2T^17 + 93T^16 - 114T^15 + 6204T^14 - 4926T^13 + 168165T^12 + 302466T^11 + 2737065T^10 + 7316176T^9 + 39198184T^8 + 112040928T^7 + 289066560T^6 + 454749120T^5 + 557551872T^4 + 405292032T^3 + 210760704T^2 + 27869184*T + 2985984
$29$	\( (T^{9} + 9 T^{8} + \cdots + 800577)^{2} \) (T^9 + 9T^8 - 117T^7 - 1069T^6 + 4476T^5 + 41796T^4 - 55737T^3 - 558225T^2 + 6993T + 800577)^2
$31$	\( T^{18} + \cdots + 2690808129 \) T^18 - 4T^17 + 133T^16 - 672T^15 + 12579T^14 - 60273T^13 + 640000T^12 - 2813896T^11 + 22609930T^10 - 82415622T^9 + 380223249T^8 - 719077392T^7 + 2195978641T^6 - 2143522498T^5 + 9132292384T^4 - 4884501024T^3 + 7641570399T^2 + 2546082459*T + 2690808129
$37$	\( (T^{9} - 10 T^{8} + \cdots - 2416)^{2} \) (T^9 - 10T^8 - 61T^7 + 1060T^6 - 4173T^5 + 4530T^4 + 6428T^3 - 16880T^2 + 11440T - 2416)^2
$41$	\( (T^{9} - T^{8} - 86 T^{7} + \cdots - 1296)^{2} \) (T^9 - T^8 - 86T^7 - 82T^6 + 1741T^5 + 2675T^4 - 9024T^3 - 11712T^2 + 11448*T - 1296)^2
$43$	\( T^{18} + \cdots + 89913620736 \) T^18 - 7T^17 + 244T^16 - 1485T^15 + 39237T^14 - 230253T^13 + 3260470T^12 - 16313413T^11 + 180426277T^10 - 841210860T^9 + 4932342312T^8 - 14419743312T^7 + 58194561568T^6 - 138624967312T^5 + 450383424064T^4 - 674297670912T^3 + 889610072064T^2 - 311639141376*T + 89913620736
$47$	\( (T^{9} - 19 T^{8} + \cdots + 1256751)^{2} \) (T^9 - 19T^8 - 26T^7 + 2146T^6 - 5740T^5 - 74732T^4 + 292482T^3 + 775374T^2 - 3660381T + 1256751)^2
$53$	\( T^{18} + \cdots + 20372138361 \) T^18 - 10T^17 + 261T^16 - 1166T^15 + 32819T^14 - 121461T^13 + 2383418T^12 - 883334T^11 + 75446700T^10 + 123579344T^9 + 2197599211T^8 + 4724897376T^7 + 21739573425T^6 + 11592184698T^5 + 51886176804T^4 - 11334420504T^3 + 127770590727T^2 - 48417067089*T + 20372138361
$59$	\( (T^{9} + 5 T^{8} + \cdots - 3722193)^{2} \) (T^9 + 5T^8 - 287T^7 - 2547T^6 + 15624T^5 + 260472T^4 + 925515T^3 + 89703T^2 - 4132809T - 3722193)^2
$61$	\( (T^{9} + 18 T^{8} + \cdots - 225347)^{2} \) (T^9 + 18T^8 + 6T^7 - 1434T^6 - 6741T^5 + 10191T^4 + 92520T^3 + 47916T^2 - 262218T - 225347)^2
$67$	\( T^{18} + \cdots + 344768247300096 \) T^18 - 22T^17 + 712T^16 - 11434T^15 + 258781T^14 - 3681724T^13 + 58734925T^12 - 659534263T^11 + 8085305077T^10 - 75918711588T^9 + 708408945696T^8 - 4616971819008T^7 + 25677858097152T^6 - 80532701077248T^5 + 199019284273152T^4 - 184802862624768T^3 + 249721518219264T^2 - 45882112573440*T + 344768247300096
$71$	\( T^{18} + \cdots + 120758121 \) T^18 + 11T^17 + 291T^16 + 1006T^15 + 33818T^14 + 45420T^13 + 2891432T^12 - 4788308T^11 + 132752949T^10 - 427129117T^9 + 5083570705T^8 - 16988165916T^7 + 73173306732T^6 - 80451237168T^5 + 105368305446T^4 + 49394551734T^3 + 34651559187T^2 + 2116778103*T + 120758121
$73$	\( T^{18} + \cdots + 490005727861329 \) T^18 - 44T^17 + 1409T^16 - 29568T^15 + 540085T^14 - 8090347T^13 + 112628434T^12 - 1318092754T^11 + 13470713512T^10 - 108710987430T^9 + 720799832855T^8 - 3551777870840T^7 + 14296218390253T^6 - 41171015991294T^5 + 127051061574072T^4 - 288100799541078T^3 + 751261583672037T^2 - 662101299235449*T + 490005727861329
$79$	\( T^{18} + \cdots + 28\!\cdots\!84 \) T^18 - 2T^17 + 374T^16 - 598T^15 + 96069T^14 - 151854T^13 + 12207275T^12 - 18719611T^11 + 1102487977T^10 - 1808396632T^9 + 60850235128T^8 - 113414724352T^7 + 2450760870656T^6 - 4554089038128T^5 + 57711517345152T^4 - 104361417799936T^3 + 914124712666112T^2 - 1438413349872128*T + 2818816529430784
$83$	\( T^{18} + \cdots + 25101031868649 \) T^18 - 7T^17 + 435T^16 - 5644T^15 + 150580T^14 - 1769706T^13 + 28959861T^12 - 324883044T^11 + 3985324065T^10 - 33745825422T^9 + 245625382314T^8 - 1164734144604T^7 + 4336480494144T^6 - 9259157312991T^5 + 16576329818907T^4 - 15350705121888T^3 + 23582756116197T^2 - 16925837666364*T + 25101031868649
$89$	\( T^{18} + \cdots + 28991407990689 \) T^18 + T^17 + 285T^16 - 20T^15 + 59926T^14 + 1482T^13 + 4909866T^12 + 229512T^11 + 286600059T^10 - 72026613T^9 + 8456006835T^8 - 12326847192T^7 + 178146913842T^6 - 286830152190T^5 + 2192868734310T^4 - 4509941331564T^3 + 18078647419701T^2 - 22029259928679T + 28991407990689
$97$	\( T^{18} + \cdots + 469410009520384 \) T^18 + 504T^16 - 1570T^15 + 178725T^14 - 586566T^13 + 32643257T^12 - 129619533T^11 + 4299730839T^10 - 11651571850T^9 + 229021292628T^8 + 13748902128T^7 + 7463856715328T^6 + 950805829488T^5 + 98362892789280T^4 + 229473838173824T^3 + 655402935502464T^2 + 587870677921536T + 469410009520384
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\(n\)	\(191\)	\(325\)
\(\chi(n)\)	\(-\beta_{8}\)	\(-\beta_{8}\)