LMFDB - Newform orbit 3460.1.bm.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3460,1,Mod(3,3460)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3460, base_ring=CyclotomicField(172))

chi = DirichletCharacter(H, H._module([86, 129, 27]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3460.3");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3460 = 2^{2} \cdot 5 \cdot 173 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3460.bm (of order $172$, degree $84$, 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:	$1.72676494371$
Analytic rank:	$0$
Dimension:	$84$
Coefficient field:	$\Q(\zeta_{172})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{84} - x^{82} + x^{80} - x^{78} + x^{76} - x^{74} + x^{72} - x^{70} + x^{68} - x^{66} + x^{64} + \cdots + 1 $ x^84 - x^82 + x^80 - x^78 + x^76 - x^74 + x^72 - x^70 + x^68 - x^66 + x^64 - x^62 + x^60 - x^58 + x^56 - x^54 + x^52 - x^50 + x^48 - x^46 + x^44 - x^42 + x^40 - x^38 + x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{172}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{172} - \cdots)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + \zeta_{172}^{78} q^{2} - \zeta_{172}^{70} q^{4} - \zeta_{172}^{52} q^{5} + \zeta_{172}^{62} q^{8} + \zeta_{172}^{84} q^{9} +O(q^{10}) $ q + z^78 * q^2 - z^70 * q^4 - z^52 * q^5 + z^62 * q^8 + z^84 * q^9	\( q + \zeta_{172}^{78} q^{2} - \zeta_{172}^{70} q^{4} - \zeta_{172}^{52} q^{5} + \zeta_{172}^{62} q^{8} + \zeta_{172}^{84} q^{9} + \zeta_{172}^{44} q^{10} + ( - \zeta_{172}^{65} + \zeta_{172}^{48}) q^{13} - \zeta_{172}^{54} q^{16} + (\zeta_{172}^{85} - \zeta_{172}^{37}) q^{17} - \zeta_{172}^{76} q^{18} - \zeta_{172}^{36} q^{20} - \zeta_{172}^{18} q^{25} + (\zeta_{172}^{57} - \zeta_{172}^{40}) q^{26} + (\zeta_{172}^{81} - \zeta_{172}^{23}) q^{29} + \zeta_{172}^{46} q^{32} + ( - \zeta_{172}^{77} + \zeta_{172}^{29}) q^{34} + \zeta_{172}^{68} q^{36} + (\zeta_{172}^{59} + \zeta_{172}^{58}) q^{37} + \zeta_{172}^{28} q^{40} + ( - \zeta_{172}^{69} - \zeta_{172}^{45}) q^{41} + \zeta_{172}^{50} q^{45} - \zeta_{172}^{28} q^{49} + \zeta_{172}^{10} q^{50} + ( - \zeta_{172}^{49} + \zeta_{172}^{32}) q^{52} + ( - \zeta_{172}^{75} - \zeta_{172}^{25}) q^{53} + ( - \zeta_{172}^{73} + \zeta_{172}^{15}) q^{58} + ( - \zeta_{172}^{63} - \zeta_{172}^{30}) q^{61} - \zeta_{172}^{38} q^{64} + ( - \zeta_{172}^{31} + \zeta_{172}^{14}) q^{65} + (\zeta_{172}^{69} - \zeta_{172}^{21}) q^{68} - \zeta_{172}^{60} q^{72} + (\zeta_{172}^{40} - \zeta_{172}^{13}) q^{73} + ( - \zeta_{172}^{51} - \zeta_{172}^{50}) q^{74} - \zeta_{172}^{20} q^{80} - \zeta_{172}^{82} q^{81} + (\zeta_{172}^{61} + \zeta_{172}^{37}) q^{82} + (\zeta_{172}^{51} - \zeta_{172}^{3}) q^{85} + (\zeta_{172}^{72} + \zeta_{172}^{8}) q^{89} - \zeta_{172}^{42} q^{90} + (\zeta_{172}^{79} + \zeta_{172}^{49}) q^{97} + \zeta_{172}^{20} q^{98} +O(q^{100}) \) q + z^78 * q^2 - z^70 * q^4 - z^52 * q^5 + z^62 * q^8 + z^84 * q^9 + z^44 * q^10 + (-z^65 + z^48) * q^13 - z^54 * q^16 + (z^85 - z^37) * q^17 - z^76 * q^18 - z^36 * q^20 - z^18 * q^25 + (z^57 - z^40) * q^26 + (z^81 - z^23) * q^29 + z^46 * q^32 + (-z^77 + z^29) * q^34 + z^68 * q^36 + (z^59 + z^58) * q^37 + z^28 * q^40 + (-z^69 - z^45) * q^41 + z^50 * q^45 - z^28 * q^49 + z^10 * q^50 + (-z^49 + z^32) * q^52 + (-z^75 - z^25) * q^53 + (-z^73 + z^15) * q^58 + (-z^63 - z^30) * q^61 - z^38 * q^64 + (-z^31 + z^14) * q^65 + (z^69 - z^21) * q^68 - z^60 * q^72 + (z^40 - z^13) * q^73 + (-z^51 - z^50) * q^74 - z^20 * q^80 - z^82 * q^81 + (z^61 + z^37) * q^82 + (z^51 - z^3) * q^85 + (z^72 + z^8) * q^89 - z^42 * q^90 + (z^79 + z^49) * q^97 + z^20 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{8} - 2 q^{9}+O(q^{10}) $ 84 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^8 - 2 * q^9	$ 84 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{8} - 2 q^{9} - 2 q^{10} - 2 q^{13} - 2 q^{16} + 2 q^{18} + 2 q^{20} - 2 q^{25} + 2 q^{26} + 2 q^{32} - 2 q^{36} + 2 q^{37} - 2 q^{40} + 2 q^{45} + 2 q^{49} + 2 q^{50} - 2 q^{52} - 2 q^{61} - 2 q^{64} + 2 q^{65} + 2 q^{72} - 2 q^{73} - 2 q^{74} + 2 q^{80} - 2 q^{81} - 4 q^{89} - 2 q^{90} - 2 q^{98}+O(q^{100}) $ 84 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^8 - 2 * q^9 - 2 * q^10 - 2 * q^13 - 2 * q^16 + 2 * q^18 + 2 * q^20 - 2 * q^25 + 2 * q^26 + 2 * q^32 - 2 * q^36 + 2 * q^37 - 2 * q^40 + 2 * q^45 + 2 * q^49 + 2 * q^50 - 2 * q^52 - 2 * q^61 - 2 * q^64 + 2 * q^65 + 2 * q^72 - 2 * q^73 - 2 * q^74 + 2 * q^80 - 2 * q^81 - 4 * q^89 - 2 * q^90 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3460\mathbb{Z}\right)^\times$.

$n$	$521$	$1731$	$2077$
$\chi(n)$	$-\zeta_{172}^{27}$	$-1$	$\zeta_{172}^{43}$

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} $

3.1

−0.999333	−	0.0365220i
−0.994001	−	0.109371i
0.719903	+	0.694074i
0.551768	−	0.833998i
−0.667319	+	0.744772i
0.889342	−	0.457242i
−0.983366	−	0.181637i
0.145601	−	0.989343i
0.424457	+	0.905448i
−0.853593	−	0.520940i
−0.611174	+	0.791496i
−0.0729953	+	0.997332i
−0.920346	−	0.391105i
−0.424457	+	0.905448i
0.946440	−	0.322880i
−0.667319	−	0.744772i
−0.357231	+	0.934016i
−0.288099	+	0.957601i
0.288099	−	0.957601i
0.357231	−	0.934016i

−0.957601

0.288099i

0.833998

−

0.551768i

0.322880

−

0.946440i

−0.639673

0.768647i

−0.997332

0.0729953i

−0.0365220

0.999333i

27.1

−0.639673

0.768647i

−0.181637

−

0.983366i

−0.833998

0.551768i

0.872049

0.489418i

−0.976076

0.217430i

0.109371

−

0.994001i

107.1

−0.989343

−

0.145601i

0.957601

0.288099i

0.581859

−

0.813290i

−0.905448

−

0.424457i

−0.0365220

0.999333i

−0.694074

0.719903i

147.1

0.0365220

−

0.999333i

−0.997332

−

0.0729953i

−0.520940

0.853593i

−0.109371

0.994001i

0.391105

−

0.920346i

0.833998

0.551768i

203.1

−0.905448

−

0.424457i

0.639673

0.768647i

−0.957601

−

0.288099i

−0.252933

−

0.967484i

0.109371

−

0.994001i

0.744772

0.667319i

223.1

0.791496

0.611174i

0.252933

0.967484i

−0.905448

−

0.424457i

−0.391105

0.920346i

−0.581859

−

0.813290i

−0.457242

−

0.889342i

243.1

−0.109371

0.994001i

−0.976076

−

0.217430i

0.997332

0.0729953i

0.322880

−

0.946440i

−0.934016

0.357231i

−0.181637

0.983366i

267.1

−0.391105

0.920346i

−0.694074

−

0.719903i

−0.252933

−

0.967484i

0.934016

−

0.357231i

0.957601

−

0.288099i

0.989343

0.145601i

283.1

0.934016

0.357231i

0.744772

0.667319i

0.694074

−

0.719903i

0.457242

0.889342i

0.639673

0.768647i

0.905448

−

0.424457i

287.1

0.322880

−

0.946440i

−0.791496

−

0.611174i

0.976076

0.217430i

−0.833998

0.551768i

−0.457242

0.889342i

0.520940

−

0.853593i

407.1

−0.520940

−

0.853593i

−0.457242

0.889342i

0.934016

−

0.357231i

0.997332

−

0.0729953i

0.252933

−

0.967484i

−0.791496

−

0.611174i

547.1

−0.833998

0.551768i

0.391105

−

0.920346i

0.791496

0.611174i

0.181637

0.983366i

0.989343

−

0.145601i

−0.997332

−

0.0729953i

687.1

0.997332

−

0.0729953i

0.989343

−

0.145601i

0.457242

−

0.889342i

0.976076

−

0.217430i

−0.694074

0.719903i

0.391105

−

0.920346i

703.1

0.934016

−

0.357231i

0.744772

−

0.667319i

0.694074

0.719903i

0.457242

−

0.889342i

0.639673

−

0.768647i

0.905448

0.424457i

763.1

0.872049

−

0.489418i

0.520940

−

0.853593i

0.181637

−

0.983366i

0.0365220

−

0.999333i

−0.791496

−

0.611174i

−0.322880

−

0.946440i

767.1

−0.905448

0.424457i

0.639673

−

0.768647i

−0.957601

0.288099i

−0.252933

0.967484i

0.109371

0.994001i

0.744772

−

0.667319i

847.1

0.976076

0.217430i

0.905448

0.424457i

−0.989343

−

0.145601i

0.791496

0.611174i

0.744772

−

0.667319i

−0.934016

−

0.357231i

863.1

0.694074

0.719903i

−0.0365220

0.999333i

0.872049

−

0.489418i

−0.744772

0.667319i

0.833998

−

0.551768i

0.957601

0.288099i

867.1

0.694074

0.719903i

−0.0365220

0.999333i

0.872049

−

0.489418i

−0.744772

0.667319i

0.833998

−

0.551768i

0.957601

0.288099i

883.1

0.976076

0.217430i

0.905448

0.424457i

−0.989343

−

0.145601i

0.791496

0.611174i

0.744772

−

0.667319i

−0.934016

−

0.357231i

See all 84 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
865.o	even	172	1	inner
3460.bm	odd	172	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3460.1.bm.a	yes	84
4.b	odd	2	1	CM	3460.1.bm.a	yes	84
5.c	odd	4	1		3460.1.bd.a	✓	84
20.e	even	4	1		3460.1.bd.a	✓	84
173.f	odd	172	1		3460.1.bd.a	✓	84
692.k	even	172	1		3460.1.bd.a	✓	84
865.o	even	172	1	inner	3460.1.bm.a	yes	84
3460.bm	odd	172	1	inner	3460.1.bm.a	yes	84

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3460.1.bd.a	✓	84	5.c	odd	4	1
3460.1.bd.a	✓	84	20.e	even	4	1
3460.1.bd.a	✓	84	173.f	odd	172	1
3460.1.bd.a	✓	84	692.k	even	172	1
3460.1.bm.a	yes	84	1.a	even	1	1	trivial
3460.1.bm.a	yes	84	4.b	odd	2	1	CM
3460.1.bm.a	yes	84	865.o	even	172	1	inner
3460.1.bm.a	yes	84	3460.bm	odd	172	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3460, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{42} - T^{41} + T^{40} + \cdots + 1)^{2} $ (T^42 - T^41 + T^40 - T^39 + T^38 - T^37 + T^36 - T^35 + T^34 - T^33 + T^32 - T^31 + T^30 - T^29 + T^28 - T^27 + T^26 - T^25 + T^24 - T^23 + T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	$ T^{84} $ T^84
$5$	$ (T^{42} - T^{41} + T^{40} + \cdots + 1)^{2} $ (T^42 - T^41 + T^40 - T^39 + T^38 - T^37 + T^36 - T^35 + T^34 - T^33 + T^32 - T^31 + T^30 - T^29 + T^28 - T^27 + T^26 - T^25 + T^24 - T^23 + T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$7$	$ T^{84} $ T^84
$11$	$ T^{84} $ T^84
$13$	$ T^{84} + 2 T^{83} + \cdots + 1 $ T^84 + 2T^83 + 2T^82 - 4T^80 - 8T^79 - 8T^78 + 16T^76 + 32T^75 + 32T^74 - 64T^72 - 128T^71 + 18319T^70 + 36894T^69 + 37150T^68 + 512T^67 - 73276T^66 - 146802T^65 - 147052T^64 - 500T^63 + 293104T^62 - 329466T^61 - 1245140T^60 - 1831348T^59 - 1172416T^58 + 1317864T^57 + 81688217T^56 + 160740706T^55 + 158104978T^54 - 5271456T^53 - 324317477T^52 - 659997016T^51 - 671359078T^50 - 22724124T^49 + 1297269908T^48 + 7495540066T^47 + 12396672799T^46 + 9802265466T^45 - 5188814666T^44 - 29983732172T^43 - 3490563400T^42 + 52986337542T^41 + 112953801884T^40 + 119934928684T^39 + 30566809810T^38 - 161696009064T^37 - 384525637748T^36 - 445659257368T^35 - 122266843124T^34 - 7358784736T^33 + 230418312039T^32 + 475554193550T^31 + 490271763022T^30 + 37101037474T^29 + 90714936024T^28 + 107230491136T^27 + 33031110224T^26 - 148398809210T^25 + 794758168T^24 + 121065065262T^23 + 240540615263T^22 + 238951100002T^21 - 2543530469T^20 - 2703930042T^19 + 6767325283T^18 + 18942510650T^17 + 24350373615T^16 - 125571954T^15 + 676211852T^14 + 1509022652T^13 + 1665621600T^12 + 321613598T^11 + 2465659T^10 + 152470T^9 - 4110636T^8 - 8526298T^7 + 527454T^6 + 5794T^5 + 10266T^4 + 7740T^3 + 1097T^2 + 44T + 1
$17$	$ T^{84} + 86 T^{76} + \cdots + 1849 $ T^84 + 86T^76 + 3053T^68 - 40936T^66 + 57792T^60 + 28424204T^58 + 10062T^56 + 629090T^52 - 2072239488T^50 + 231519224T^48 - 86T^46 + 3961160T^44 + 32382350474T^42 + 53688410842T^40 + 75925487T^38 + 13797238T^36 - 127854870182T^34 + 844480468294T^32 - 30657788260T^30 + 28210193T^28 + 120259930725T^26 + 1229317538452T^24 + 229648503238T^22 + 1284783197T^20 - 20463230612T^18 + 135506350210T^16 - 42804686518T^14 + 1380232275T^12 + 314679461T^10 + 478515653T^8 + 84467867T^6 + 3884749T^4 + 20339T^2 + 1849
$19$	$ T^{84} $ T^84
$23$	$ T^{84} $ T^84
$29$	$ T^{84} - 4 T^{82} + \cdots + 1 $ T^84 - 4T^82 + 16T^80 - 64T^78 + 256T^76 - 1024T^74 + 4096T^72 - 16384T^70 + 4046T^68 - 16184T^66 + 64736T^64 + 407728T^62 - 1630912T^60 + 7285178T^58 - 29442572T^56 + 117770288T^54 + 152445723T^52 - 609767756T^50 + 2438170432T^48 - 413769298T^46 + 1655077106T^44 - 1194147884T^42 + 20877387696T^40 - 83509245699T^38 + 19690131010T^36 - 72125446708T^34 + 292523371077T^32 - 23744303440T^30 + 95950678892T^28 - 133561340932T^26 - 30168616632T^24 + 121172100927T^22 + 14722607845T^20 - 5953997701T^18 + 15465172794T^16 - 364008804T^14 + 424259504T^12 + 121616970T^10 + 2631277T^8 - 992653T^6 + 30995T^4 + 118*T^2 + 1
$31$	$ T^{84} $ T^84
$37$	$ T^{84} - 2 T^{83} + \cdots + 1 $ T^84 - 2T^83 + 2T^82 - 4T^80 + 8T^79 - 8T^78 + 16T^76 - 32T^75 + 32T^74 - 64T^72 + 128T^71 - 128T^70 + 256T^68 - 512T^67 + 512T^66 + 4386T^65 - 9796T^64 + 10820T^63 + 1247962T^62 - 2517564T^61 + 2539204T^60 + 17845752T^59 - 40769912T^58 + 45848320T^57 + 30149879T^56 - 151996398T^55 + 243693038T^54 - 161508430T^53 - 164369216T^52 + 651755292T^51 - 971370336T^50 + 639230002T^49 + 664280668T^48 - 2606867830T^47 + 3888544965T^46 - 2563354270T^45 - 2650379713T^44 + 9282847698T^43 - 13264935970T^42 + 7964176546T^41 + 55837283509T^40 - 127602920110T^39 + 143531273202T^38 - 460652109430T^37 + 634241672456T^36 - 347179126052T^35 + 850671362318T^34 - 1006984472532T^33 + 312626220428T^32 - 705779582036T^31 + 786306849679T^30 - 161054535286T^29 + 338115095679T^28 - 353995868322T^27 + 31761545286T^26 - 46192010834T^25 + 36370296726T^24 + 19643428216T^23 + 73069193803T^22 - 114112272352T^21 + 82086157098T^20 + 31921284866T^19 - 61723521315T^18 + 59604472898T^17 + 7966934413T^16 - 15447625816T^15 + 14961383537T^14 + 678419942T^13 - 1353651049T^12 + 1350128878T^11 + 22821945T^10 - 43048238T^9 + 42980384T^8 - 430430T^7 - 373267T^6 + 367188T^5 + 25144T^4 - 774T^3 + 710T^2 - 44*T + 1
$41$	$ T^{84} - 4 T^{82} + \cdots + 1 $ T^84 - 4T^82 + 16T^80 - 64T^78 + 256T^76 - 1024T^74 + 4096T^72 + 2063T^70 - 11262T^68 + 45048T^66 - 180192T^64 + 720768T^62 - 2883072T^60 + 11749868T^58 + 29708185T^56 - 17759950T^54 + 73134459T^52 - 292537750T^50 + 1170151000T^48 - 4680436945T^46 + 17765072944T^44 - 24906430288T^42 + 3878610238T^40 + 231683726T^38 - 1144587782T^36 + 4578522096T^34 - 17725782809T^32 + 168069693564T^30 + 350849688828T^28 + 114602206452T^26 + 9173860300T^24 - 1064261115T^22 + 4898513687T^20 - 15037167373T^18 + 19595328857T^16 - 9535571770T^14 + 2001721986T^12 - 208336435T^10 + 12145930T^8 - 93566T^6 + 33360T^4 + 333*T^2 + 1
$43$	$ T^{84} $ T^84
$47$	$ T^{84} $ T^84
$53$	$ T^{84} + 1247 T^{54} + \cdots + 1849 $ T^84 + 1247T^54 + 541198T^52 + 45924516T^50 + 1435257096T^48 + 21883727700T^46 + 190774614420T^44 + 1052049481860T^42 + 3933750236520T^40 + 10484957361186T^38 + 20679144147860T^36 + 31052069680093T^34 + 36294626898810T^32 + 33595619161195T^30 + 24960347444340T^28 + 15039532815214T^26 + 7405837658121T^24 + 2996334133923T^22 + 999380208471T^20 + 273420342488T^18 + 67899514210T^16 + 1622319996T^14 + 10828479902T^12 - 4090479834T^10 + 1090776872T^8 - 134095027T^6 + 8340839T^4 - 208937*T^2 + 1849
$59$	$ T^{84} $ T^84
$61$	$ T^{84} + 2 T^{83} + \cdots + 1 $ T^84 + 2T^83 + 2T^82 - 4T^80 - 8T^79 - 8T^78 + 16T^76 + 32T^75 + 32T^74 - 64T^72 - 128T^71 - 128T^70 + 602T^69 + 1460T^68 - 121264T^67 - 245448T^66 - 195736T^65 + 99424T^64 + 590234T^63 + 981620T^62 + 782772T^61 - 397696T^60 - 2360936T^59 - 3926480T^58 - 3131088T^57 + 1590784T^56 + 9443744T^55 + 15808561T^54 + 12729634T^53 + 93184831T^52 + 160910394T^51 + 2154449820T^50 + 3987078852T^49 + 6261455132T^48 + 4548752560T^47 - 3099636284T^46 - 15296777688T^45 - 24390556643T^44 - 18187557910T^43 + 12405999401T^42 + 61187114620T^41 + 97562230438T^40 + 72754865230T^39 - 49614730416T^38 - 248598187420T^37 - 397966914008T^36 - 139929239480T^35 + 516075349056T^34 + 431675696466T^33 - 168799305180T^32 - 159094398562T^31 + 19409813236T^30 + 22736625776T^29 + 6653625080T^28 + 226825602T^27 - 12853597709T^26 - 27107580288T^25 - 28481408874T^24 - 2740460434T^23 + 55974374340T^22 + 117429660948T^21 + 198702780684T^20 + 162546239472T^19 + 205519909658T^18 + 85947340372T^17 + 97150916713T^16 + 22407152682T^15 + 24030595581T^14 + 3246885798T^13 + 3386034142T^12 + 278284648T^11 + 283862389T^10 + 12599938T^9 + 17438126T^8 + 2648542T^7 + 186808T^6 - 294948T^5 + 18866T^4 + 9116T^3 + 65T^2 - 42T + 1
$67$	$ T^{84} $ T^84
$71$	$ T^{84} $ T^84
$73$	$ T^{84} + 2 T^{83} + \cdots + 1 $ T^84 + 2T^83 + 2T^82 - 4T^80 - 8T^79 - 8T^78 - 430T^77 - 844T^76 - 828T^75 + 32T^74 + 1720T^73 + 3376T^72 + 3312T^71 + 73875T^70 + 141126T^69 + 134502T^68 - 13248T^67 - 295500T^66 - 564504T^65 - 538008T^64 - 6469678T^63 - 11863340T^62 - 10787324T^61 + 2152032T^60 + 25879744T^59 + 47455424T^58 + 43151360T^57 + 304771529T^56 + 523240338T^55 + 436937618T^54 - 172605354T^53 - 1217094313T^52 - 2088977918T^51 - 1743767210T^50 - 7415693570T^49 - 11343852720T^48 - 7856318300T^47 + 6975198915T^46 + 30094087500T^45 + 46237777170T^44 + 32287379340T^43 + 77470929832T^42 + 90367100982T^41 + 25792342300T^40 - 129128261260T^39 - 290696694402T^38 - 323136866284T^37 - 64880343764T^36 - 86637108124T^35 - 43513391464T^34 + 86247433320T^33 + 260195194043T^32 + 541322821002T^31 + 562255253918T^30 + 41864865832T^29 + 183032244904T^28 + 282296171320T^27 + 198527852832T^26 - 163189283656T^25 - 327093135296T^24 - 327807703280T^23 - 1429134893T^22 + 26362228338T^21 + 56160617867T^20 + 59596779058T^19 + 11499102387T^18 + 22950235006T^17 + 22902266055T^16 - 96040844T^15 + 927674304T^14 + 1598650218T^13 + 1341951828T^12 - 47301806T^11 - 92263513T^10 - 89860806T^9 + 5014480T^8 + 856646T^7 + 1043110T^6 + 372928T^5 + 2354T^4 - 344T^3 + 753T^2 - 42T + 1
$79$	$ T^{84} $ T^84
$83$	$ T^{84} $ T^84
$89$	$ (T^{42} + 2 T^{41} + \cdots + 1)^{2} $ (T^42 + 2T^41 + 4T^40 + 8T^39 + 16T^38 + 32T^37 + 64T^36 + 128T^35 + 256T^34 + 512T^33 + 1024T^32 + 758T^31 + 1516T^30 + 3032T^29 - 85T^28 - 170T^27 - 340T^26 - 2400T^25 - 4800T^24 - 11793T^23 - 23629T^22 - 47258T^21 + 112529T^20 + 225058T^19 + 450116T^18 - 112074T^17 - 224148T^16 - 448855T^15 + 72843T^14 + 145686T^13 + 202362T^12 + 94049T^11 + 188098T^10 - 1817T^9 + 39065T^8 + 78087T^7 + 2019T^6 + 1243T^5 + 4292T^4 + 27T^3 + 140T^2 + 22*T + 1)^2
$97$	$ T^{84} + 86 T^{76} + \cdots + 1849 $ T^84 + 86T^76 + 3053T^68 - 40936T^66 + 57792T^60 + 28424204T^58 + 10062T^56 + 629090T^52 - 2072239488T^50 + 231519224T^48 - 86T^46 + 3961160T^44 + 32382350474T^42 + 53688410842T^40 + 75925487T^38 + 13797238T^36 - 127854870182T^34 + 844480468294T^32 - 30657788260T^30 + 28210193T^28 + 120259930725T^26 + 1229317538452T^24 + 229648503238T^22 + 1284783197T^20 - 20463230612T^18 + 135506350210T^16 - 42804686518T^14 + 1380232275T^12 + 314679461T^10 + 478515653T^8 + 84467867T^6 + 3884749T^4 + 20339T^2 + 1849
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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:	\( N \)	\(=\)	\( 3460 = 2^{2} \cdot 5 \cdot 173 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3460.bm (of order \(172\), degree \(84\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{172}^{78} q^{2} - \zeta_{172}^{70} q^{4} - \zeta_{172}^{52} q^{5} + \zeta_{172}^{62} q^{8} + \zeta_{172}^{84} q^{9} +O(q^{10}) \) q + z^78 * q^2 - z^70 * q^4 - z^52 * q^5 + z^62 * q^8 + z^84 * q^9	\( q + \zeta_{172}^{78} q^{2} - \zeta_{172}^{70} q^{4} - \zeta_{172}^{52} q^{5} + \zeta_{172}^{62} q^{8} + \zeta_{172}^{84} q^{9} + \zeta_{172}^{44} q^{10} + ( - \zeta_{172}^{65} + \zeta_{172}^{48}) q^{13} - \zeta_{172}^{54} q^{16} + (\zeta_{172}^{85} - \zeta_{172}^{37}) q^{17} - \zeta_{172}^{76} q^{18} - \zeta_{172}^{36} q^{20} - \zeta_{172}^{18} q^{25} + (\zeta_{172}^{57} - \zeta_{172}^{40}) q^{26} + (\zeta_{172}^{81} - \zeta_{172}^{23}) q^{29} + \zeta_{172}^{46} q^{32} + ( - \zeta_{172}^{77} + \zeta_{172}^{29}) q^{34} + \zeta_{172}^{68} q^{36} + (\zeta_{172}^{59} + \zeta_{172}^{58}) q^{37} + \zeta_{172}^{28} q^{40} + ( - \zeta_{172}^{69} - \zeta_{172}^{45}) q^{41} + \zeta_{172}^{50} q^{45} - \zeta_{172}^{28} q^{49} + \zeta_{172}^{10} q^{50} + ( - \zeta_{172}^{49} + \zeta_{172}^{32}) q^{52} + ( - \zeta_{172}^{75} - \zeta_{172}^{25}) q^{53} + ( - \zeta_{172}^{73} + \zeta_{172}^{15}) q^{58} + ( - \zeta_{172}^{63} - \zeta_{172}^{30}) q^{61} - \zeta_{172}^{38} q^{64} + ( - \zeta_{172}^{31} + \zeta_{172}^{14}) q^{65} + (\zeta_{172}^{69} - \zeta_{172}^{21}) q^{68} - \zeta_{172}^{60} q^{72} + (\zeta_{172}^{40} - \zeta_{172}^{13}) q^{73} + ( - \zeta_{172}^{51} - \zeta_{172}^{50}) q^{74} - \zeta_{172}^{20} q^{80} - \zeta_{172}^{82} q^{81} + (\zeta_{172}^{61} + \zeta_{172}^{37}) q^{82} + (\zeta_{172}^{51} - \zeta_{172}^{3}) q^{85} + (\zeta_{172}^{72} + \zeta_{172}^{8}) q^{89} - \zeta_{172}^{42} q^{90} + (\zeta_{172}^{79} + \zeta_{172}^{49}) q^{97} + \zeta_{172}^{20} q^{98} +O(q^{100}) \) q + z^78 * q^2 - z^70 * q^4 - z^52 * q^5 + z^62 * q^8 + z^84 * q^9 + z^44 * q^10 + (-z^65 + z^48) * q^13 - z^54 * q^16 + (z^85 - z^37) * q^17 - z^76 * q^18 - z^36 * q^20 - z^18 * q^25 + (z^57 - z^40) * q^26 + (z^81 - z^23) * q^29 + z^46 * q^32 + (-z^77 + z^29) * q^34 + z^68 * q^36 + (z^59 + z^58) * q^37 + z^28 * q^40 + (-z^69 - z^45) * q^41 + z^50 * q^45 - z^28 * q^49 + z^10 * q^50 + (-z^49 + z^32) * q^52 + (-z^75 - z^25) * q^53 + (-z^73 + z^15) * q^58 + (-z^63 - z^30) * q^61 - z^38 * q^64 + (-z^31 + z^14) * q^65 + (z^69 - z^21) * q^68 - z^60 * q^72 + (z^40 - z^13) * q^73 + (-z^51 - z^50) * q^74 - z^20 * q^80 - z^82 * q^81 + (z^61 + z^37) * q^82 + (z^51 - z^3) * q^85 + (z^72 + z^8) * q^89 - z^42 * q^90 + (z^79 + z^49) * q^97 + z^20 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 84 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^8 - 2 * q^9	\( 84 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{8} - 2 q^{9} - 2 q^{10} - 2 q^{13} - 2 q^{16} + 2 q^{18} + 2 q^{20} - 2 q^{25} + 2 q^{26} + 2 q^{32} - 2 q^{36} + 2 q^{37} - 2 q^{40} + 2 q^{45} + 2 q^{49} + 2 q^{50} - 2 q^{52} - 2 q^{61} - 2 q^{64} + 2 q^{65} + 2 q^{72} - 2 q^{73} - 2 q^{74} + 2 q^{80} - 2 q^{81} - 4 q^{89} - 2 q^{90} - 2 q^{98}+O(q^{100}) \) 84 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^8 - 2 * q^9 - 2 * q^10 - 2 * q^13 - 2 * q^16 + 2 * q^18 + 2 * q^20 - 2 * q^25 + 2 * q^26 + 2 * q^32 - 2 * q^36 + 2 * q^37 - 2 * q^40 + 2 * q^45 + 2 * q^49 + 2 * q^50 - 2 * q^52 - 2 * q^61 - 2 * q^64 + 2 * q^65 + 2 * q^72 - 2 * q^73 - 2 * q^74 + 2 * q^80 - 2 * q^81 - 4 * q^89 - 2 * q^90 - 2 * 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	3460.1.bm.a

Level	$3460$
Weight	$1$
Character orbit	3460.bm
Analytic conductor	$1.727$
Analytic rank	$0$
Dimension	$84$
Projective image	$D_{172}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.72676494371\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Coefficient field:	\(\Q(\zeta_{172})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{84} - x^{82} + x^{80} - x^{78} + x^{76} - x^{74} + x^{72} - x^{70} + x^{68} - x^{66} + x^{64} + \cdots + 1 \) x^84 - x^82 + x^80 - x^78 + x^76 - x^74 + x^72 - x^70 + x^68 - x^66 + x^64 - x^62 + x^60 - x^58 + x^56 - x^54 + x^52 - x^50 + x^48 - x^46 + x^44 - x^42 + x^40 - x^38 + x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{172}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{172} - \cdots)\)

\(n\)	\(521\)	\(1731\)	\(2077\)
\(\chi(n)\)	\(-\zeta_{172}^{27}\)	\(-1\)	\(\zeta_{172}^{43}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
865.o	even	172	1	inner
3460.bm	odd	172	1	inner

$p$	$F_p(T)$
$2$	\( (T^{42} - T^{41} + T^{40} + \cdots + 1)^{2} \) (T^42 - T^41 + T^40 - T^39 + T^38 - T^37 + T^36 - T^35 + T^34 - T^33 + T^32 - T^31 + T^30 - T^29 + T^28 - T^27 + T^26 - T^25 + T^24 - T^23 + T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	\( T^{84} \) T^84
$5$	\( (T^{42} - T^{41} + T^{40} + \cdots + 1)^{2} \) (T^42 - T^41 + T^40 - T^39 + T^38 - T^37 + T^36 - T^35 + T^34 - T^33 + T^32 - T^31 + T^30 - T^29 + T^28 - T^27 + T^26 - T^25 + T^24 - T^23 + T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$7$	\( T^{84} \) T^84
$11$	\( T^{84} \) T^84
$13$	\( T^{84} + 2 T^{83} + \cdots + 1 \) T^84 + 2T^83 + 2T^82 - 4T^80 - 8T^79 - 8T^78 + 16T^76 + 32T^75 + 32T^74 - 64T^72 - 128T^71 + 18319T^70 + 36894T^69 + 37150T^68 + 512T^67 - 73276T^66 - 146802T^65 - 147052T^64 - 500T^63 + 293104T^62 - 329466T^61 - 1245140T^60 - 1831348T^59 - 1172416T^58 + 1317864T^57 + 81688217T^56 + 160740706T^55 + 158104978T^54 - 5271456T^53 - 324317477T^52 - 659997016T^51 - 671359078T^50 - 22724124T^49 + 1297269908T^48 + 7495540066T^47 + 12396672799T^46 + 9802265466T^45 - 5188814666T^44 - 29983732172T^43 - 3490563400T^42 + 52986337542T^41 + 112953801884T^40 + 119934928684T^39 + 30566809810T^38 - 161696009064T^37 - 384525637748T^36 - 445659257368T^35 - 122266843124T^34 - 7358784736T^33 + 230418312039T^32 + 475554193550T^31 + 490271763022T^30 + 37101037474T^29 + 90714936024T^28 + 107230491136T^27 + 33031110224T^26 - 148398809210T^25 + 794758168T^24 + 121065065262T^23 + 240540615263T^22 + 238951100002T^21 - 2543530469T^20 - 2703930042T^19 + 6767325283T^18 + 18942510650T^17 + 24350373615T^16 - 125571954T^15 + 676211852T^14 + 1509022652T^13 + 1665621600T^12 + 321613598T^11 + 2465659T^10 + 152470T^9 - 4110636T^8 - 8526298T^7 + 527454T^6 + 5794T^5 + 10266T^4 + 7740T^3 + 1097T^2 + 44T + 1
$17$	\( T^{84} + 86 T^{76} + \cdots + 1849 \) T^84 + 86T^76 + 3053T^68 - 40936T^66 + 57792T^60 + 28424204T^58 + 10062T^56 + 629090T^52 - 2072239488T^50 + 231519224T^48 - 86T^46 + 3961160T^44 + 32382350474T^42 + 53688410842T^40 + 75925487T^38 + 13797238T^36 - 127854870182T^34 + 844480468294T^32 - 30657788260T^30 + 28210193T^28 + 120259930725T^26 + 1229317538452T^24 + 229648503238T^22 + 1284783197T^20 - 20463230612T^18 + 135506350210T^16 - 42804686518T^14 + 1380232275T^12 + 314679461T^10 + 478515653T^8 + 84467867T^6 + 3884749T^4 + 20339T^2 + 1849
$19$	\( T^{84} \) T^84
$23$	\( T^{84} \) T^84
$29$	\( T^{84} - 4 T^{82} + \cdots + 1 \) T^84 - 4T^82 + 16T^80 - 64T^78 + 256T^76 - 1024T^74 + 4096T^72 - 16384T^70 + 4046T^68 - 16184T^66 + 64736T^64 + 407728T^62 - 1630912T^60 + 7285178T^58 - 29442572T^56 + 117770288T^54 + 152445723T^52 - 609767756T^50 + 2438170432T^48 - 413769298T^46 + 1655077106T^44 - 1194147884T^42 + 20877387696T^40 - 83509245699T^38 + 19690131010T^36 - 72125446708T^34 + 292523371077T^32 - 23744303440T^30 + 95950678892T^28 - 133561340932T^26 - 30168616632T^24 + 121172100927T^22 + 14722607845T^20 - 5953997701T^18 + 15465172794T^16 - 364008804T^14 + 424259504T^12 + 121616970T^10 + 2631277T^8 - 992653T^6 + 30995T^4 + 118*T^2 + 1
$31$	\( T^{84} \) T^84
$37$	\( T^{84} - 2 T^{83} + \cdots + 1 \) T^84 - 2T^83 + 2T^82 - 4T^80 + 8T^79 - 8T^78 + 16T^76 - 32T^75 + 32T^74 - 64T^72 + 128T^71 - 128T^70 + 256T^68 - 512T^67 + 512T^66 + 4386T^65 - 9796T^64 + 10820T^63 + 1247962T^62 - 2517564T^61 + 2539204T^60 + 17845752T^59 - 40769912T^58 + 45848320T^57 + 30149879T^56 - 151996398T^55 + 243693038T^54 - 161508430T^53 - 164369216T^52 + 651755292T^51 - 971370336T^50 + 639230002T^49 + 664280668T^48 - 2606867830T^47 + 3888544965T^46 - 2563354270T^45 - 2650379713T^44 + 9282847698T^43 - 13264935970T^42 + 7964176546T^41 + 55837283509T^40 - 127602920110T^39 + 143531273202T^38 - 460652109430T^37 + 634241672456T^36 - 347179126052T^35 + 850671362318T^34 - 1006984472532T^33 + 312626220428T^32 - 705779582036T^31 + 786306849679T^30 - 161054535286T^29 + 338115095679T^28 - 353995868322T^27 + 31761545286T^26 - 46192010834T^25 + 36370296726T^24 + 19643428216T^23 + 73069193803T^22 - 114112272352T^21 + 82086157098T^20 + 31921284866T^19 - 61723521315T^18 + 59604472898T^17 + 7966934413T^16 - 15447625816T^15 + 14961383537T^14 + 678419942T^13 - 1353651049T^12 + 1350128878T^11 + 22821945T^10 - 43048238T^9 + 42980384T^8 - 430430T^7 - 373267T^6 + 367188T^5 + 25144T^4 - 774T^3 + 710T^2 - 44*T + 1
$41$	\( T^{84} - 4 T^{82} + \cdots + 1 \) T^84 - 4T^82 + 16T^80 - 64T^78 + 256T^76 - 1024T^74 + 4096T^72 + 2063T^70 - 11262T^68 + 45048T^66 - 180192T^64 + 720768T^62 - 2883072T^60 + 11749868T^58 + 29708185T^56 - 17759950T^54 + 73134459T^52 - 292537750T^50 + 1170151000T^48 - 4680436945T^46 + 17765072944T^44 - 24906430288T^42 + 3878610238T^40 + 231683726T^38 - 1144587782T^36 + 4578522096T^34 - 17725782809T^32 + 168069693564T^30 + 350849688828T^28 + 114602206452T^26 + 9173860300T^24 - 1064261115T^22 + 4898513687T^20 - 15037167373T^18 + 19595328857T^16 - 9535571770T^14 + 2001721986T^12 - 208336435T^10 + 12145930T^8 - 93566T^6 + 33360T^4 + 333*T^2 + 1
$43$	\( T^{84} \) T^84
$47$	\( T^{84} \) T^84
$53$	\( T^{84} + 1247 T^{54} + \cdots + 1849 \) T^84 + 1247T^54 + 541198T^52 + 45924516T^50 + 1435257096T^48 + 21883727700T^46 + 190774614420T^44 + 1052049481860T^42 + 3933750236520T^40 + 10484957361186T^38 + 20679144147860T^36 + 31052069680093T^34 + 36294626898810T^32 + 33595619161195T^30 + 24960347444340T^28 + 15039532815214T^26 + 7405837658121T^24 + 2996334133923T^22 + 999380208471T^20 + 273420342488T^18 + 67899514210T^16 + 1622319996T^14 + 10828479902T^12 - 4090479834T^10 + 1090776872T^8 - 134095027T^6 + 8340839T^4 - 208937*T^2 + 1849
$59$	\( T^{84} \) T^84
$61$	\( T^{84} + 2 T^{83} + \cdots + 1 \) T^84 + 2T^83 + 2T^82 - 4T^80 - 8T^79 - 8T^78 + 16T^76 + 32T^75 + 32T^74 - 64T^72 - 128T^71 - 128T^70 + 602T^69 + 1460T^68 - 121264T^67 - 245448T^66 - 195736T^65 + 99424T^64 + 590234T^63 + 981620T^62 + 782772T^61 - 397696T^60 - 2360936T^59 - 3926480T^58 - 3131088T^57 + 1590784T^56 + 9443744T^55 + 15808561T^54 + 12729634T^53 + 93184831T^52 + 160910394T^51 + 2154449820T^50 + 3987078852T^49 + 6261455132T^48 + 4548752560T^47 - 3099636284T^46 - 15296777688T^45 - 24390556643T^44 - 18187557910T^43 + 12405999401T^42 + 61187114620T^41 + 97562230438T^40 + 72754865230T^39 - 49614730416T^38 - 248598187420T^37 - 397966914008T^36 - 139929239480T^35 + 516075349056T^34 + 431675696466T^33 - 168799305180T^32 - 159094398562T^31 + 19409813236T^30 + 22736625776T^29 + 6653625080T^28 + 226825602T^27 - 12853597709T^26 - 27107580288T^25 - 28481408874T^24 - 2740460434T^23 + 55974374340T^22 + 117429660948T^21 + 198702780684T^20 + 162546239472T^19 + 205519909658T^18 + 85947340372T^17 + 97150916713T^16 + 22407152682T^15 + 24030595581T^14 + 3246885798T^13 + 3386034142T^12 + 278284648T^11 + 283862389T^10 + 12599938T^9 + 17438126T^8 + 2648542T^7 + 186808T^6 - 294948T^5 + 18866T^4 + 9116T^3 + 65T^2 - 42T + 1
$67$	\( T^{84} \) T^84
$71$	\( T^{84} \) T^84
$73$	\( T^{84} + 2 T^{83} + \cdots + 1 \) T^84 + 2T^83 + 2T^82 - 4T^80 - 8T^79 - 8T^78 - 430T^77 - 844T^76 - 828T^75 + 32T^74 + 1720T^73 + 3376T^72 + 3312T^71 + 73875T^70 + 141126T^69 + 134502T^68 - 13248T^67 - 295500T^66 - 564504T^65 - 538008T^64 - 6469678T^63 - 11863340T^62 - 10787324T^61 + 2152032T^60 + 25879744T^59 + 47455424T^58 + 43151360T^57 + 304771529T^56 + 523240338T^55 + 436937618T^54 - 172605354T^53 - 1217094313T^52 - 2088977918T^51 - 1743767210T^50 - 7415693570T^49 - 11343852720T^48 - 7856318300T^47 + 6975198915T^46 + 30094087500T^45 + 46237777170T^44 + 32287379340T^43 + 77470929832T^42 + 90367100982T^41 + 25792342300T^40 - 129128261260T^39 - 290696694402T^38 - 323136866284T^37 - 64880343764T^36 - 86637108124T^35 - 43513391464T^34 + 86247433320T^33 + 260195194043T^32 + 541322821002T^31 + 562255253918T^30 + 41864865832T^29 + 183032244904T^28 + 282296171320T^27 + 198527852832T^26 - 163189283656T^25 - 327093135296T^24 - 327807703280T^23 - 1429134893T^22 + 26362228338T^21 + 56160617867T^20 + 59596779058T^19 + 11499102387T^18 + 22950235006T^17 + 22902266055T^16 - 96040844T^15 + 927674304T^14 + 1598650218T^13 + 1341951828T^12 - 47301806T^11 - 92263513T^10 - 89860806T^9 + 5014480T^8 + 856646T^7 + 1043110T^6 + 372928T^5 + 2354T^4 - 344T^3 + 753T^2 - 42T + 1
$79$	\( T^{84} \) T^84
$83$	\( T^{84} \) T^84
$89$	\( (T^{42} + 2 T^{41} + \cdots + 1)^{2} \) (T^42 + 2T^41 + 4T^40 + 8T^39 + 16T^38 + 32T^37 + 64T^36 + 128T^35 + 256T^34 + 512T^33 + 1024T^32 + 758T^31 + 1516T^30 + 3032T^29 - 85T^28 - 170T^27 - 340T^26 - 2400T^25 - 4800T^24 - 11793T^23 - 23629T^22 - 47258T^21 + 112529T^20 + 225058T^19 + 450116T^18 - 112074T^17 - 224148T^16 - 448855T^15 + 72843T^14 + 145686T^13 + 202362T^12 + 94049T^11 + 188098T^10 - 1817T^9 + 39065T^8 + 78087T^7 + 2019T^6 + 1243T^5 + 4292T^4 + 27T^3 + 140T^2 + 22*T + 1)^2
$97$	\( T^{84} + 86 T^{76} + \cdots + 1849 \) T^84 + 86T^76 + 3053T^68 - 40936T^66 + 57792T^60 + 28424204T^58 + 10062T^56 + 629090T^52 - 2072239488T^50 + 231519224T^48 - 86T^46 + 3961160T^44 + 32382350474T^42 + 53688410842T^40 + 75925487T^38 + 13797238T^36 - 127854870182T^34 + 844480468294T^32 - 30657788260T^30 + 28210193T^28 + 120259930725T^26 + 1229317538452T^24 + 229648503238T^22 + 1284783197T^20 - 20463230612T^18 + 135506350210T^16 - 42804686518T^14 + 1380232275T^12 + 314679461T^10 + 478515653T^8 + 84467867T^6 + 3884749T^4 + 20339T^2 + 1849
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