LMFDB - Newform orbit 3460.1.bd.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3460,1,Mod(7,3460)]

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, 43, 95]))

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

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

chi := DirichletCharacter("3460.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3460 = 2^{2} \cdot 5 \cdot 173 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3460.bd (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:	$\Z[a_1, a_2]$
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}^{69} q^{2} - \zeta_{172}^{52} q^{4} - \zeta_{172}^{47} q^{5} - \zeta_{172}^{35} q^{8} - \zeta_{172}^{28} q^{9} +O(q^{10}) $ q - z^69 * q^2 - z^52 * q^4 - z^47 * q^5 - z^35 * q^8 - z^28 * q^9	\( q - \zeta_{172}^{69} q^{2} - \zeta_{172}^{52} q^{4} - \zeta_{172}^{47} q^{5} - \zeta_{172}^{35} q^{8} - \zeta_{172}^{28} q^{9} - \zeta_{172}^{30} q^{10} + (\zeta_{172}^{59} - \zeta_{172}^{36}) q^{13} - \zeta_{172}^{18} q^{16} + (\zeta_{172}^{84} + \zeta_{172}^{14}) q^{17} - \zeta_{172}^{11} q^{18} - \zeta_{172}^{13} q^{20} - \zeta_{172}^{8} q^{25} + (\zeta_{172}^{42} - \zeta_{172}^{19}) q^{26} + ( - \zeta_{172}^{65} + \zeta_{172}^{27}) q^{29} - \zeta_{172} q^{32} + ( - \zeta_{172}^{83} + \zeta_{172}^{67}) q^{34} + \zeta_{172}^{80} q^{36} + ( - \zeta_{172}^{34} - \zeta_{172}^{5}) q^{37} + \zeta_{172}^{82} q^{40} + (\zeta_{172}^{23} + \zeta_{172}^{15}) q^{41} + \zeta_{172}^{75} q^{45} - \zeta_{172}^{38} q^{49} + \zeta_{172}^{77} q^{50} + (\zeta_{172}^{25} - \zeta_{172}^{2}) q^{52} + ( - \zeta_{172}^{80} + \zeta_{172}^{68}) q^{53} + ( - \zeta_{172}^{48} + \zeta_{172}^{10}) q^{58} + (\zeta_{172}^{21} - \zeta_{172}^{10}) q^{61} + \zeta_{172}^{70} q^{64} + (\zeta_{172}^{83} + \zeta_{172}^{20}) q^{65} + ( - \zeta_{172}^{66} + \zeta_{172}^{50}) q^{68} + \zeta_{172}^{63} q^{72} + ( - \zeta_{172}^{85} - \zeta_{172}^{76}) q^{73} + (\zeta_{172}^{74} - \zeta_{172}^{17}) q^{74} + \zeta_{172}^{65} q^{80} + \zeta_{172}^{56} q^{81} + ( - \zeta_{172}^{84} + \zeta_{172}^{6}) q^{82} + ( - \zeta_{172}^{61} + \zeta_{172}^{45}) q^{85} + ( - \zeta_{172}^{60} - \zeta_{172}^{24}) q^{89} + \zeta_{172}^{58} q^{90} + (\zeta_{172}^{12} + \zeta_{172}^{2}) q^{97} - \zeta_{172}^{21} q^{98} +O(q^{100}) \) q - z^69 * q^2 - z^52 * q^4 - z^47 * q^5 - z^35 * q^8 - z^28 * q^9 - z^30 * q^10 + (z^59 - z^36) * q^13 - z^18 * q^16 + (z^84 + z^14) * q^17 - z^11 * q^18 - z^13 * q^20 - z^8 * q^25 + (z^42 - z^19) * q^26 + (-z^65 + z^27) * q^29 - z * q^32 + (-z^83 + z^67) * q^34 + z^80 * q^36 + (-z^34 - z^5) * q^37 + z^82 * q^40 + (z^23 + z^15) * q^41 + z^75 * q^45 - z^38 * q^49 + z^77 * q^50 + (z^25 - z^2) * q^52 + (-z^80 + z^68) * q^53 + (-z^48 + z^10) * q^58 + (z^21 - z^10) * q^61 + z^70 * q^64 + (z^83 + z^20) * q^65 + (-z^66 + z^50) * q^68 + z^63 * q^72 + (-z^85 - z^76) * q^73 + (z^74 - z^17) * q^74 + z^65 * q^80 + z^56 * q^81 + (-z^84 + z^6) * q^82 + (-z^61 + z^45) * q^85 + (-z^60 - z^24) * q^89 + z^58 * q^90 + (z^12 + z^2) * q^97 - z^21 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q + 2 q^{4} + 2 q^{9}+O(q^{10}) $ 84 * q + 2 * q^4 + 2 * q^9	$ 84 q + 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{13} - 2 q^{16} + 2 q^{25} + 2 q^{26} - 2 q^{36} - 2 q^{37} + 2 q^{40} - 2 q^{49} - 2 q^{52} + 4 q^{58} - 2 q^{61} + 2 q^{64} - 2 q^{65} + 2 q^{73} + 2 q^{74} - 2 q^{81} + 4 q^{82} + 4 q^{89} + 2 q^{90}+O(q^{100}) $ 84 * q + 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^13 - 2 * q^16 + 2 * q^25 + 2 * q^26 - 2 * q^36 - 2 * q^37 + 2 * q^40 - 2 * q^49 - 2 * q^52 + 4 * q^58 - 2 * q^61 + 2 * q^64 - 2 * q^65 + 2 * q^73 + 2 * q^74 - 2 * q^81 + 4 * q^82 + 4 * q^89 + 2 * q^90

Display 100 coefficients 10 coefficients

Character values

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

$n$	$521$	$1731$	$2077$
$\chi(n)$	$\zeta_{172}^{9}$	$-1$	$-\zeta_{172}^{43}$

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} $

7.1

−0.999333	−	0.0365220i
−0.967484	−	0.252933i
0.768647	+	0.639673i
−0.853593	−	0.520940i
0.145601	−	0.989343i
0.424457	+	0.905448i
0.813290	+	0.581859i
0.999333	−	0.0365220i
0.551768	+	0.833998i
0.611174	−	0.791496i
−0.994001	−	0.109371i
0.920346	−	0.391105i
−0.667319	−	0.744772i
0.288099	+	0.957601i
0.946440	−	0.322880i
−0.288099	+	0.957601i
−0.489418	−	0.872049i
0.0729953	+	0.997332i
−0.889342	−	0.457242i
−0.217430	−	0.976076i

−0.813290

0.581859i

0.322880

−

0.946440i

−0.145601

0.989343i

0.288099

0.957601i

−0.520940

−

0.853593i

−0.457242

−

0.889342i

63.1

0.357231

−

0.934016i

−0.744772

−

0.667319i

0.853593

−

0.520940i

−0.889342

0.457242i

−0.639673

−

0.768647i

−0.181637

−

0.983366i

87.1

0.719903

0.694074i

0.0365220

0.999333i

−0.357231

−

0.934016i

−0.667319

0.744772i

−0.833998

−

0.551768i

0.391105

−

0.920346i

103.1

0.994001

0.109371i

0.976076

0.217430i

0.813290

0.581859i

0.946440

0.322880i

0.934016

−

0.357231i

0.744772

0.667319i

123.1

0.611174

−

0.791496i

−0.252933

−

0.967484i

0.551768

−

0.833998i

−0.920346

−

0.391105i

0.581859

0.813290i

−0.322880

−

0.946440i

127.1

0.920346

−

0.391105i

0.694074

−

0.719903i

0.983366

−

0.181637i

0.357231

−

0.934016i

−0.957601

−

0.288099i

0.833998

−

0.551768i

143.1

−0.424457

0.905448i

−0.639673

−

0.768647i

0.611174

0.791496i

0.967484

−

0.252933i

−0.109371

0.994001i

−0.976076

0.217430i

247.1

0.813290

0.581859i

0.322880

0.946440i

0.145601

0.989343i

−0.288099

0.957601i

−0.520940

0.853593i

−0.457242

0.889342i

307.1

−0.489418

0.872049i

−0.520940

−

0.853593i

0.719903

−

0.694074i

0.999333

−

0.0365220i

0.791496

−

0.611174i

0.252933

0.967484i

327.1

−0.983366

0.181637i

0.934016

−

0.357231i

−0.489418

−

0.872049i

−0.853593

0.520940i

−0.905448

0.424457i

0.639673

0.768647i

343.1

0.288099

0.957601i

−0.833998

0.551768i

0.424457

−

0.905448i

−0.768647

−

0.639673i

0.997332

−

0.0729953i

0.989343

0.145601i

363.1

0.853593

0.520940i

0.457242

0.889342i

−0.999333

0.0365220i

−0.0729953

0.997332i

−0.252933

−

0.967484i

−0.872049

−

0.489418i

443.1

0.145601

0.989343i

−0.957601

0.288099i

−0.217430

0.976076i

−0.424457

−

0.905448i

0.0365220

0.999333i

−0.997332

−

0.0729953i

447.1

−0.967484

−

0.252933i

0.872049

0.489418i

0.920346

0.391105i

−0.719903

−

0.694074i

0.322880

0.946440i

−0.791496

−

0.611174i

487.1

0.768647

−

0.639673i

0.181637

−

0.983366i

0.967484

0.252933i

−0.489418

−

0.872049i

0.976076

0.217430i

0.905448

−

0.424457i

507.1

0.967484

−

0.252933i

0.872049

−

0.489418i

−0.920346

0.391105i

0.719903

−

0.694074i

0.322880

−

0.946440i

−0.791496

0.611174i

527.1

−0.667319

−

0.744772i

−0.109371

0.994001i

0.889342

−

0.457242i

0.813290

−

0.581859i

0.181637

0.983366i

−0.934016

−

0.357231i

563.1

0.946440

−

0.322880i

0.791496

−

0.611174i

−0.288099

−

0.957601i

0.551768

−

0.833998i

0.457242

0.889342i

−0.581859

−

0.813290i

567.1

0.217430

0.976076i

−0.905448

0.424457i

−0.946440

−

0.322880i

−0.611174

−

0.791496i

−0.744772

−

0.667319i

0.109371

−

0.994001i

587.1

0.551768

−

0.833998i

−0.391105

−

0.920346i

0.768647

0.639673i

−0.983366

−

0.181637i

−0.989343

−

0.145601i

0.957601

−

0.288099i

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.t	even	172	1	inner
3460.bd	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.bd.a	✓	84
4.b	odd	2	1	CM	3460.1.bd.a	✓	84
5.c	odd	4	1		3460.1.bm.a	yes	84
20.e	even	4	1		3460.1.bm.a	yes	84
173.f	odd	172	1		3460.1.bm.a	yes	84
692.k	even	172	1		3460.1.bm.a	yes	84
865.t	even	172	1	inner	3460.1.bd.a	✓	84
3460.bd	odd	172	1	inner	3460.1.bd.a	✓	84

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

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{84} - T^{82} + \cdots + 1 $ T^84 - T^82 + T^80 - T^78 + T^76 - T^74 + T^72 - T^70 + T^68 - T^66 + T^64 - T^62 + T^60 - T^58 + T^56 - T^54 + T^52 - T^50 + T^48 - T^46 + T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	$ T^{84} $ T^84
$5$	$ T^{84} - T^{82} + \cdots + 1 $ T^84 - T^82 + T^80 - T^78 + T^76 - T^74 + T^72 - T^70 + T^68 - T^66 + T^64 - T^62 + T^60 - T^58 + T^56 - T^54 + T^52 - T^50 + T^48 - T^46 + T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$7$	$ 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 + 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
$17$	$ (T^{42} + 43 T^{34} + \cdots + 43)^{2} $ (T^42 + 43T^34 + 5676T^29 + 602T^26 + 20468T^24 - 172602T^21 + 16555T^19 + 3010T^18 + 1016262T^16 + 5031T^14 + 349977T^13 - 560462T^11 + 3913T^10 + 688T^9 + 409790T^8 + 39947T^6 - 23779T^5 + 43T^4 + 8815T^3 + 301T^2 - 215T + 43)^2
$19$	$ 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 - 2580T^73 - 5224T^72 - 5288T^71 - 128T^70 + 10320T^69 + 20896T^68 + 24592T^67 + 7392T^66 - 34400T^65 - 83584T^64 - 98368T^63 + 2048622T^62 + 4293980T^61 + 4490716T^60 + 393472T^59 - 8194488T^58 - 17174802T^57 + 13470867T^56 + 61291338T^55 + 95640942T^54 + 68699208T^53 - 53883468T^52 - 807066992T^51 - 1503852064T^50 - 1393570144T^49 + 220563840T^48 + 3228267968T^47 + 6018771673T^46 + 21246813850T^45 + 30456086031T^44 + 18418544362T^43 - 24075083338T^42 - 84987255402T^41 - 83514134699T^40 - 4410363800T^39 + 158207541798T^38 + 325235811196T^37 + 334056538796T^36 + 18268517664T^35 - 6765264578T^34 - 49859367084T^33 - 86188205012T^32 - 72657675856T^31 + 27061184775T^30 - 139074987334T^29 - 1610931533T^28 + 274927563094T^27 + 553076989254T^26 + 556298852320T^25 + 13284227622T^24 + 18735795740T^23 + 30161993651T^22 + 22852395908T^21 - 14619195486T^20 - 74953270846T^19 - 2250859435T^18 + 144294992T^17 + 5041750709T^16 + 9794911434T^15 + 9506322181T^14 + 849332046T^13 - 5555985T^12 - 25384620T^11 - 38568123T^10 - 26367006T^9 + 28397708T^8 - 1011532T^7 + 14765T^6 + 32110T^5 + 36496T^4 + 12384T^3 + 1226T^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^{42} + 43 T^{27} + \cdots + 43)^{2} $ (T^42 + 43T^27 - 6020T^25 + 114036T^23 - 722228T^21 + 2138906T^19 - 3513917T^17 + 3535675T^15 - 2314260T^13 + 301T^12 + 1021293T^11 + 11739T^10 - 309256T^9 + 61490T^8 + 64328T^7 + 76024T^6 - 9030T^5 + 26316T^4 + 817T^3 + 2451T^2 - 43T + 43)^2
$59$	$ 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 + 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
$79$	$ 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^{42} + 43 T^{34} + \cdots + 43)^{2} $ (T^42 + 43T^34 + 5676T^29 + 602T^26 + 20468T^24 - 172602T^21 + 16555T^19 + 3010T^18 + 1016262T^16 + 5031T^14 + 349977T^13 - 560462T^11 + 3913T^10 + 688T^9 + 409790T^8 + 39947T^6 - 23779T^5 + 43T^4 + 8815T^3 + 301T^2 - 215T + 43)^2
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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.bd (of order \(172\), degree \(84\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{172}^{69} q^{2} - \zeta_{172}^{52} q^{4} - \zeta_{172}^{47} q^{5} - \zeta_{172}^{35} q^{8} - \zeta_{172}^{28} q^{9} +O(q^{10}) \) q - z^69 * q^2 - z^52 * q^4 - z^47 * q^5 - z^35 * q^8 - z^28 * q^9	\( q - \zeta_{172}^{69} q^{2} - \zeta_{172}^{52} q^{4} - \zeta_{172}^{47} q^{5} - \zeta_{172}^{35} q^{8} - \zeta_{172}^{28} q^{9} - \zeta_{172}^{30} q^{10} + (\zeta_{172}^{59} - \zeta_{172}^{36}) q^{13} - \zeta_{172}^{18} q^{16} + (\zeta_{172}^{84} + \zeta_{172}^{14}) q^{17} - \zeta_{172}^{11} q^{18} - \zeta_{172}^{13} q^{20} - \zeta_{172}^{8} q^{25} + (\zeta_{172}^{42} - \zeta_{172}^{19}) q^{26} + ( - \zeta_{172}^{65} + \zeta_{172}^{27}) q^{29} - \zeta_{172} q^{32} + ( - \zeta_{172}^{83} + \zeta_{172}^{67}) q^{34} + \zeta_{172}^{80} q^{36} + ( - \zeta_{172}^{34} - \zeta_{172}^{5}) q^{37} + \zeta_{172}^{82} q^{40} + (\zeta_{172}^{23} + \zeta_{172}^{15}) q^{41} + \zeta_{172}^{75} q^{45} - \zeta_{172}^{38} q^{49} + \zeta_{172}^{77} q^{50} + (\zeta_{172}^{25} - \zeta_{172}^{2}) q^{52} + ( - \zeta_{172}^{80} + \zeta_{172}^{68}) q^{53} + ( - \zeta_{172}^{48} + \zeta_{172}^{10}) q^{58} + (\zeta_{172}^{21} - \zeta_{172}^{10}) q^{61} + \zeta_{172}^{70} q^{64} + (\zeta_{172}^{83} + \zeta_{172}^{20}) q^{65} + ( - \zeta_{172}^{66} + \zeta_{172}^{50}) q^{68} + \zeta_{172}^{63} q^{72} + ( - \zeta_{172}^{85} - \zeta_{172}^{76}) q^{73} + (\zeta_{172}^{74} - \zeta_{172}^{17}) q^{74} + \zeta_{172}^{65} q^{80} + \zeta_{172}^{56} q^{81} + ( - \zeta_{172}^{84} + \zeta_{172}^{6}) q^{82} + ( - \zeta_{172}^{61} + \zeta_{172}^{45}) q^{85} + ( - \zeta_{172}^{60} - \zeta_{172}^{24}) q^{89} + \zeta_{172}^{58} q^{90} + (\zeta_{172}^{12} + \zeta_{172}^{2}) q^{97} - \zeta_{172}^{21} q^{98} +O(q^{100}) \) q - z^69 * q^2 - z^52 * q^4 - z^47 * q^5 - z^35 * q^8 - z^28 * q^9 - z^30 * q^10 + (z^59 - z^36) * q^13 - z^18 * q^16 + (z^84 + z^14) * q^17 - z^11 * q^18 - z^13 * q^20 - z^8 * q^25 + (z^42 - z^19) * q^26 + (-z^65 + z^27) * q^29 - z * q^32 + (-z^83 + z^67) * q^34 + z^80 * q^36 + (-z^34 - z^5) * q^37 + z^82 * q^40 + (z^23 + z^15) * q^41 + z^75 * q^45 - z^38 * q^49 + z^77 * q^50 + (z^25 - z^2) * q^52 + (-z^80 + z^68) * q^53 + (-z^48 + z^10) * q^58 + (z^21 - z^10) * q^61 + z^70 * q^64 + (z^83 + z^20) * q^65 + (-z^66 + z^50) * q^68 + z^63 * q^72 + (-z^85 - z^76) * q^73 + (z^74 - z^17) * q^74 + z^65 * q^80 + z^56 * q^81 + (-z^84 + z^6) * q^82 + (-z^61 + z^45) * q^85 + (-z^60 - z^24) * q^89 + z^58 * q^90 + (z^12 + z^2) * q^97 - z^21 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) 84 * q + 2 * q^4 + 2 * q^9	\( 84 q + 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{13} - 2 q^{16} + 2 q^{25} + 2 q^{26} - 2 q^{36} - 2 q^{37} + 2 q^{40} - 2 q^{49} - 2 q^{52} + 4 q^{58} - 2 q^{61} + 2 q^{64} - 2 q^{65} + 2 q^{73} + 2 q^{74} - 2 q^{81} + 4 q^{82} + 4 q^{89} + 2 q^{90}+O(q^{100}) \) 84 * q + 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^13 - 2 * q^16 + 2 * q^25 + 2 * q^26 - 2 * q^36 - 2 * q^37 + 2 * q^40 - 2 * q^49 - 2 * q^52 + 4 * q^58 - 2 * q^61 + 2 * q^64 - 2 * q^65 + 2 * q^73 + 2 * q^74 - 2 * q^81 + 4 * q^82 + 4 * q^89 + 2 * q^90

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.bd.a

Level	$3460$
Weight	$1$
Character orbit	3460.bd
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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{172}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{172} - \cdots)\)

\(n\)	\(521\)	\(1731\)	\(2077\)
\(\chi(n)\)	\(\zeta_{172}^{9}\)	\(-1\)	\(-\zeta_{172}^{43}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.1
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.t	even	172	1	inner
3460.bd	odd	172	1	inner

$p$	$F_p(T)$
$2$	\( T^{84} - T^{82} + \cdots + 1 \) T^84 - T^82 + T^80 - T^78 + T^76 - T^74 + T^72 - T^70 + T^68 - T^66 + T^64 - T^62 + T^60 - T^58 + T^56 - T^54 + T^52 - T^50 + T^48 - T^46 + T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	\( T^{84} \) T^84
$5$	\( T^{84} - T^{82} + \cdots + 1 \) T^84 - T^82 + T^80 - T^78 + T^76 - T^74 + T^72 - T^70 + T^68 - T^66 + T^64 - T^62 + T^60 - T^58 + T^56 - T^54 + T^52 - T^50 + T^48 - T^46 + T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$7$	\( T^{84} \) 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 + 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
$17$	\( (T^{42} + 43 T^{34} + \cdots + 43)^{2} \) (T^42 + 43T^34 + 5676T^29 + 602T^26 + 20468T^24 - 172602T^21 + 16555T^19 + 3010T^18 + 1016262T^16 + 5031T^14 + 349977T^13 - 560462T^11 + 3913T^10 + 688T^9 + 409790T^8 + 39947T^6 - 23779T^5 + 43T^4 + 8815T^3 + 301T^2 - 215T + 43)^2
$19$	\( T^{84} \) 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 - 2580T^73 - 5224T^72 - 5288T^71 - 128T^70 + 10320T^69 + 20896T^68 + 24592T^67 + 7392T^66 - 34400T^65 - 83584T^64 - 98368T^63 + 2048622T^62 + 4293980T^61 + 4490716T^60 + 393472T^59 - 8194488T^58 - 17174802T^57 + 13470867T^56 + 61291338T^55 + 95640942T^54 + 68699208T^53 - 53883468T^52 - 807066992T^51 - 1503852064T^50 - 1393570144T^49 + 220563840T^48 + 3228267968T^47 + 6018771673T^46 + 21246813850T^45 + 30456086031T^44 + 18418544362T^43 - 24075083338T^42 - 84987255402T^41 - 83514134699T^40 - 4410363800T^39 + 158207541798T^38 + 325235811196T^37 + 334056538796T^36 + 18268517664T^35 - 6765264578T^34 - 49859367084T^33 - 86188205012T^32 - 72657675856T^31 + 27061184775T^30 - 139074987334T^29 - 1610931533T^28 + 274927563094T^27 + 553076989254T^26 + 556298852320T^25 + 13284227622T^24 + 18735795740T^23 + 30161993651T^22 + 22852395908T^21 - 14619195486T^20 - 74953270846T^19 - 2250859435T^18 + 144294992T^17 + 5041750709T^16 + 9794911434T^15 + 9506322181T^14 + 849332046T^13 - 5555985T^12 - 25384620T^11 - 38568123T^10 - 26367006T^9 + 28397708T^8 - 1011532T^7 + 14765T^6 + 32110T^5 + 36496T^4 + 12384T^3 + 1226T^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
$53$	\( (T^{42} + 43 T^{27} + \cdots + 43)^{2} \) (T^42 + 43T^27 - 6020T^25 + 114036T^23 - 722228T^21 + 2138906T^19 - 3513917T^17 + 3535675T^15 - 2314260T^13 + 301T^12 + 1021293T^11 + 11739T^10 - 309256T^9 + 61490T^8 + 64328T^7 + 76024T^6 - 9030T^5 + 26316T^4 + 817T^3 + 2451T^2 - 43T + 43)^2
$59$	\( T^{84} \) 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 + 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
$79$	\( T^{84} \) 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
$97$	\( (T^{42} + 43 T^{34} + \cdots + 43)^{2} \) (T^42 + 43T^34 + 5676T^29 + 602T^26 + 20468T^24 - 172602T^21 + 16555T^19 + 3010T^18 + 1016262T^16 + 5031T^14 + 349977T^13 - 560462T^11 + 3913T^10 + 688T^9 + 409790T^8 + 39947T^6 - 23779T^5 + 43T^4 + 8815T^3 + 301T^2 - 215T + 43)^2
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