LMFDB - Embedded newform 135.3.d.f.134.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [135,3,Mod(134,135)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("135.134"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	135.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,0,-6,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.67848356886$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			134.1
Root			$0.500000 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	135.134
Dual form			135.3.d.f.134.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -3.00000 q^{4} +(0.500000 - 4.97494i) q^{5} -9.94987i q^{7} -7.00000 q^{8} +(0.500000 - 4.97494i) q^{10} +9.94987i q^{11} -19.8997i q^{13} -9.94987i q^{14} +5.00000 q^{16} +22.0000 q^{17} -4.00000 q^{19} +(-1.50000 + 14.9248i) q^{20} +9.94987i q^{22} -20.0000 q^{23} +(-24.5000 - 4.97494i) q^{25} -19.8997i q^{26} +29.8496i q^{28} +39.7995i q^{29} +29.0000 q^{31} +33.0000 q^{32} +22.0000 q^{34} +(-49.5000 - 4.97494i) q^{35} -4.00000 q^{38} +(-3.50000 + 34.8246i) q^{40} -39.7995i q^{41} +19.8997i q^{43} -29.8496i q^{44} -20.0000 q^{46} +58.0000 q^{47} -50.0000 q^{49} +(-24.5000 - 4.97494i) q^{50} +59.6992i q^{52} +31.0000 q^{53} +(49.5000 + 4.97494i) q^{55} +69.6491i q^{56} +39.7995i q^{58} -39.7995i q^{59} +44.0000 q^{61} +29.0000 q^{62} +13.0000 q^{64} +(-99.0000 - 9.94987i) q^{65} -19.8997i q^{67} -66.0000 q^{68} +(-49.5000 - 4.97494i) q^{70} +59.6992i q^{71} -89.5489i q^{73} +12.0000 q^{76} +99.0000 q^{77} -10.0000 q^{79} +(2.50000 - 24.8747i) q^{80} -39.7995i q^{82} +19.0000 q^{83} +(11.0000 - 109.449i) q^{85} +19.8997i q^{86} -69.6491i q^{88} +59.6992i q^{89} -198.000 q^{91} +60.0000 q^{92} +58.0000 q^{94} +(-2.00000 + 19.8997i) q^{95} -129.348i q^{97} -50.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 6 q^{4} + q^{5} - 14 q^{8} + q^{10} + 10 q^{16} + 44 q^{17} - 8 q^{19} - 3 q^{20} - 40 q^{23} - 49 q^{25} + 58 q^{31} + 66 q^{32} + 44 q^{34} - 99 q^{35} - 8 q^{38} - 7 q^{40} - 40 q^{46}+ \cdots - 100 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 6 * q^4 + q^5 - 14 * q^8 + q^10 + 10 * q^16 + 44 * q^17 - 8 * q^19 - 3 * q^20 - 40 * q^23 - 49 * q^25 + 58 * q^31 + 66 * q^32 + 44 * q^34 - 99 * q^35 - 8 * q^38 - 7 * q^40 - 40 * q^46 + 116 * q^47 - 100 * q^49 - 49 * q^50 + 62 * q^53 + 99 * q^55 + 88 * q^61 + 58 * q^62 + 26 * q^64 - 198 * q^65 - 132 * q^68 - 99 * q^70 + 24 * q^76 + 198 * q^77 - 20 * q^79 + 5 * q^80 + 38 * q^83 + 22 * q^85 - 396 * q^91 + 120 * q^92 + 116 * q^94 - 4 * q^95 - 100 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000			0.500000			0.250000	−	0.968246i	$-0.419569\pi$
$2$	1.00000			0.500000			0.250000	+	0.968246i	$0.419569\pi$
$3$		0			0
$3$		0			0
$4$	−3.00000			−0.750000
$5$	0.500000	−	4.97494i	0.100000	−	0.994987i
$5$	0.500000	−	4.97494i	0.100000	−	0.994987i
$6$		0			0
$7$		−	9.94987i		−	1.42141i	−0.703490	−	0.710705i	$-0.748376\pi$
$7$		−	9.94987i		−	1.42141i	0.703490	−	0.710705i	$-0.251624\pi$
$8$	−7.00000			−0.875000
$9$		0			0
$10$	0.500000	−	4.97494i	0.0500000	−	0.497494i
$11$			9.94987i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			9.94987i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		0			0
$13$		−	19.8997i		−	1.53075i	−0.643585	−	0.765375i	$-0.722554\pi$
$13$		−	19.8997i		−	1.53075i	0.643585	−	0.765375i	$-0.277446\pi$
$14$		−	9.94987i		−	0.710705i
$15$		0			0
$16$	5.00000			0.312500
$17$	22.0000			1.29412			0.647059	−	0.762440i	$-0.275999\pi$
$17$	22.0000			1.29412			0.647059	+	0.762440i	$0.275999\pi$
$18$		0			0
$19$	−4.00000			−0.210526			−0.105263	−	0.994444i	$-0.533568\pi$
$19$	−4.00000			−0.210526			−0.105263	+	0.994444i	$0.533568\pi$
$20$	−1.50000	+	14.9248i	−0.0750000	+	0.746241i
$21$		0			0
$22$			9.94987i			0.452267i
$23$	−20.0000			−0.869565			−0.434783	−	0.900535i	$-0.643175\pi$
$23$	−20.0000			−0.869565			−0.434783	+	0.900535i	$0.643175\pi$
$24$		0			0
$25$	−24.5000	−	4.97494i	−0.980000	−	0.198997i
$26$		−	19.8997i		−	0.765375i
$27$		0			0
$28$			29.8496i			1.06606i
$29$			39.7995i			1.37240i	0.727415	+	0.686198i	$0.240722\pi$
$29$			39.7995i			1.37240i	−0.727415	+	0.686198i	$0.759278\pi$
$30$		0			0
$31$	29.0000			0.935484			0.467742	−	0.883865i	$-0.345068\pi$
$31$	29.0000			0.935484			0.467742	+	0.883865i	$0.345068\pi$
$32$	33.0000			1.03125
$33$		0			0
$34$	22.0000			0.647059
$35$	−49.5000	−	4.97494i	−1.41429	−	0.142141i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−4.00000			−0.105263
$39$		0			0
$40$	−3.50000	+	34.8246i	−0.0875000	+	0.870614i
$41$		−	39.7995i		−	0.970719i	−0.874315	−	0.485360i	$-0.838689\pi$
$41$		−	39.7995i		−	0.970719i	0.874315	−	0.485360i	$-0.161311\pi$
$42$		0			0
$43$			19.8997i			0.462785i	0.972860	+	0.231392i	$0.0743281\pi$
$43$			19.8997i			0.462785i	−0.972860	+	0.231392i	$0.925672\pi$
$44$		−	29.8496i		−	0.678401i
$45$		0			0
$46$	−20.0000			−0.434783
$47$	58.0000			1.23404			0.617021	−	0.786946i	$-0.288339\pi$
$47$	58.0000			1.23404			0.617021	+	0.786946i	$0.288339\pi$
$48$		0			0
$49$	−50.0000			−1.02041
$50$	−24.5000	−	4.97494i	−0.490000	−	0.0994987i
$51$		0			0
$52$			59.6992i			1.14806i
$53$	31.0000			0.584906			0.292453	−	0.956280i	$-0.405529\pi$
$53$	31.0000			0.584906			0.292453	+	0.956280i	$0.405529\pi$
$54$		0			0
$55$	49.5000	+	4.97494i	0.900000	+	0.0904534i
$56$			69.6491i			1.24373i
$57$		0			0
$58$			39.7995i			0.686198i
$59$		−	39.7995i		−	0.674568i	−0.941403	−	0.337284i	$-0.890492\pi$
$59$		−	39.7995i		−	0.674568i	0.941403	−	0.337284i	$-0.109508\pi$
$60$		0			0
$61$	44.0000			0.721311			0.360656	−	0.932699i	$-0.382553\pi$
$61$	44.0000			0.721311			0.360656	+	0.932699i	$0.382553\pi$
$62$	29.0000			0.467742
$63$		0			0
$64$	13.0000			0.203125
$65$	−99.0000	−	9.94987i	−1.52308	−	0.153075i
$66$		0			0
$67$		−	19.8997i		−	0.297011i	−0.988912	−	0.148506i	$-0.952554\pi$
$67$		−	19.8997i		−	0.297011i	0.988912	−	0.148506i	$-0.0474463\pi$
$68$	−66.0000			−0.970588
$69$		0			0
$70$	−49.5000	−	4.97494i	−0.707143	−	0.0710705i
$71$			59.6992i			0.840834i	0.907331	+	0.420417i	$0.138116\pi$
$71$			59.6992i			0.840834i	−0.907331	+	0.420417i	$0.861884\pi$
$72$		0			0
$73$		−	89.5489i		−	1.22670i	−0.789812	−	0.613348i	$-0.789822\pi$
$73$		−	89.5489i		−	1.22670i	0.789812	−	0.613348i	$-0.210178\pi$
$74$		0			0
$75$		0			0
$76$	12.0000			0.157895
$77$	99.0000			1.28571
$78$		0			0
$79$	−10.0000			−0.126582			−0.0632911	−	0.997995i	$-0.520160\pi$
$79$	−10.0000			−0.126582			−0.0632911	+	0.997995i	$0.520160\pi$
$80$	2.50000	−	24.8747i	0.0312500	−	0.310934i
$81$		0			0
$82$		−	39.7995i		−	0.485360i
$83$	19.0000			0.228916			0.114458	−	0.993428i	$-0.463487\pi$
$83$	19.0000			0.228916			0.114458	+	0.993428i	$0.463487\pi$
$84$		0			0
$85$	11.0000	−	109.449i	0.129412	−	1.28763i
$86$			19.8997i			0.231392i
$87$		0			0
$88$		−	69.6491i		−	0.791467i
$89$			59.6992i			0.670778i	0.942080	+	0.335389i	$0.108868\pi$
$89$			59.6992i			0.670778i	−0.942080	+	0.335389i	$0.891132\pi$
$90$		0			0
$91$	−198.000			−2.17582
$92$	60.0000			0.652174
$93$		0			0
$94$	58.0000			0.617021
$95$	−2.00000	+	19.8997i	−0.0210526	+	0.209471i
$96$		0			0
$97$		−	129.348i		−	1.33349i	−0.745287	−	0.666744i	$-0.767687\pi$
$97$		−	129.348i		−	1.33349i	0.745287	−	0.666744i	$-0.232313\pi$
$98$	−50.0000			−0.510204
$99$		0			0
$100$	73.5000	+	14.9248i	0.735000	+	0.149248i
$101$		−	169.148i		−	1.67473i	−0.546643	−	0.837366i	$-0.684095\pi$
$101$		−	169.148i		−	1.67473i	0.546643	−	0.837366i	$-0.315905\pi$
$102$		0			0
$103$			39.7995i			0.386403i	0.981159	+	0.193201i	$0.0618871\pi$
$103$			39.7995i			0.386403i	−0.981159	+	0.193201i	$0.938113\pi$
$104$			139.298i			1.33941i
$105$		0			0
$106$	31.0000			0.292453
$107$	−29.0000			−0.271028			−0.135514	−	0.990775i	$-0.543269\pi$
$107$	−29.0000			−0.271028			−0.135514	+	0.990775i	$0.543269\pi$
$108$		0			0
$109$	104.000			0.954128			0.477064	−	0.878868i	$-0.341701\pi$
$109$	104.000			0.954128			0.477064	+	0.878868i	$0.341701\pi$
$110$	49.5000	+	4.97494i	0.450000	+	0.0452267i
$111$		0			0
$112$		−	49.7494i		−	0.444191i
$113$	−80.0000			−0.707965			−0.353982	−	0.935252i	$-0.615173\pi$
$113$	−80.0000			−0.707965			−0.353982	+	0.935252i	$0.615173\pi$
$114$		0			0
$115$	−10.0000	+	99.4987i	−0.0869565	+	0.865206i
$116$		−	119.398i		−	1.02930i
$117$		0			0
$118$		−	39.7995i		−	0.337284i
$119$		−	218.897i		−	1.83947i
$120$		0			0
$121$	22.0000			0.181818
$122$	44.0000			0.360656
$123$		0			0
$124$	−87.0000			−0.701613
$125$	−37.0000	+	119.398i	−0.296000	+	0.955188i
$126$		0			0
$127$			149.248i			1.17518i	0.809158	+	0.587591i	$0.199924\pi$
$127$			149.248i			1.17518i	−0.809158	+	0.587591i	$0.800076\pi$
$128$	−119.000			−0.929688
$129$		0			0
$130$	−99.0000	−	9.94987i	−0.761538	−	0.0765375i
$131$			169.148i			1.29121i	0.763674	+	0.645603i	$0.223394\pi$
$131$			169.148i			1.29121i	−0.763674	+	0.645603i	$0.776606\pi$
$132$		0			0
$133$			39.7995i			0.299244i
$134$		−	19.8997i		−	0.148506i
$135$		0			0
$136$	−154.000			−1.13235
$137$	−98.0000			−0.715328			−0.357664	−	0.933850i	$-0.616427\pi$
$137$	−98.0000			−0.715328			−0.357664	+	0.933850i	$0.616427\pi$
$138$		0			0
$139$	−64.0000			−0.460432			−0.230216	−	0.973140i	$-0.573943\pi$
$139$	−64.0000			−0.460432			−0.230216	+	0.973140i	$0.573943\pi$
$140$	148.500	+	14.9248i	1.06071	+	0.106606i
$141$		0			0
$142$			59.6992i			0.420417i
$143$	198.000			1.38462
$144$		0			0
$145$	198.000	+	19.8997i	1.36552	+	0.137240i
$146$		−	89.5489i		−	0.613348i
$147$		0			0
$148$		0			0
$149$		−	9.94987i		−	0.0667777i	−0.999442	−	0.0333888i	$-0.989370\pi$
$149$		−	9.94987i		−	0.0667777i	0.999442	−	0.0333888i	$-0.0106300\pi$
$150$		0			0
$151$	257.000			1.70199			0.850993	−	0.525176i	$-0.176001\pi$
$151$	257.000			1.70199			0.850993	+	0.525176i	$0.176001\pi$
$152$	28.0000			0.184211
$153$		0			0
$154$	99.0000			0.642857
$155$	14.5000	−	144.273i	0.0935484	−	0.930795i
$156$		0			0
$157$			159.198i			1.01400i	0.861946	+	0.507000i	$0.169246\pi$
$157$			159.198i			1.01400i	−0.861946	+	0.507000i	$0.830754\pi$
$158$	−10.0000			−0.0632911
$159$		0			0
$160$	16.5000	−	164.173i	0.103125	−	1.02608i
$161$			198.997i			1.23601i
$162$		0			0
$163$			298.496i			1.83127i	0.402016	+	0.915633i	$0.368310\pi$
$163$			298.496i			1.83127i	−0.402016	+	0.915633i	$0.631690\pi$
$164$			119.398i			0.728040i
$165$		0			0
$166$	19.0000			0.114458
$167$	−254.000			−1.52096			−0.760479	−	0.649362i	$-0.775036\pi$
$167$	−254.000			−1.52096			−0.760479	+	0.649362i	$0.775036\pi$
$168$		0			0
$169$	−227.000			−1.34320
$170$	11.0000	−	109.449i	0.0647059	−	0.643815i
$171$		0			0
$172$		−	59.6992i		−	0.347089i
$173$	235.000			1.35838			0.679191	−	0.733962i	$-0.262331\pi$
$173$	235.000			1.35838			0.679191	+	0.733962i	$0.262331\pi$
$174$		0			0
$175$	−49.5000	+	243.772i	−0.282857	+	1.39298i
$176$			49.7494i			0.282667i
$177$		0			0
$178$			59.6992i			0.335389i
$179$			149.248i			0.833788i	0.908955	+	0.416894i	$0.136881\pi$
$179$			149.248i			0.833788i	−0.908955	+	0.416894i	$0.863119\pi$
$180$		0			0
$181$	−292.000			−1.61326			−0.806630	−	0.591057i	$-0.798711\pi$
$181$	−292.000			−1.61326			−0.806630	+	0.591057i	$0.798711\pi$
$182$	−198.000			−1.08791
$183$		0			0
$184$	140.000			0.760870
$185$		0			0
$186$		0			0
$187$			218.897i			1.17057i
$188$	−174.000			−0.925532
$189$		0			0
$190$	−2.00000	+	19.8997i	−0.0105263	+	0.104736i
$191$			99.4987i			0.520936i	0.965483	+	0.260468i	$0.0838768\pi$
$191$			99.4987i			0.520936i	−0.965483	+	0.260468i	$0.916123\pi$
$192$		0			0
$193$		−	169.148i		−	0.876414i	−0.898874	−	0.438207i	$-0.855614\pi$
$193$		−	169.148i		−	0.876414i	0.898874	−	0.438207i	$-0.144386\pi$
$194$		−	129.348i		−	0.666744i
$195$		0			0
$196$	150.000			0.765306
$197$	−203.000			−1.03046			−0.515228	−	0.857053i	$-0.672293\pi$
$197$	−203.000			−1.03046			−0.515228	+	0.857053i	$0.672293\pi$
$198$		0			0
$199$	155.000			0.778894			0.389447	−	0.921049i	$-0.372666\pi$
$199$	155.000			0.778894			0.389447	+	0.921049i	$0.372666\pi$
$200$	171.500	+	34.8246i	0.857500	+	0.174123i
$201$		0			0
$202$		−	169.148i		−	0.837366i
$203$	396.000			1.95074
$204$		0			0
$205$	−198.000	−	19.8997i	−0.965854	−	0.0970719i
$206$			39.7995i			0.193201i
$207$		0			0
$208$		−	99.4987i		−	0.478359i
$209$		−	39.7995i		−	0.190428i
$210$		0			0
$211$	308.000			1.45972			0.729858	−	0.683599i	$-0.239586\pi$
$211$	308.000			1.45972			0.729858	+	0.683599i	$0.239586\pi$
$212$	−93.0000			−0.438679
$213$		0			0
$214$	−29.0000			−0.135514
$215$	99.0000	+	9.94987i	0.460465	+	0.0462785i
$216$		0			0
$217$		−	288.546i		−	1.32971i
$218$	104.000			0.477064
$219$		0			0
$220$	−148.500	−	14.9248i	−0.675000	−	0.0678401i
$221$		−	437.794i		−	1.98097i
$222$		0			0
$223$		−	39.7995i		−	0.178473i	−0.996010	−	0.0892365i	$-0.971557\pi$
$223$		−	39.7995i		−	0.178473i	0.996010	−	0.0892365i	$-0.0284427\pi$
$224$		−	328.346i		−	1.46583i
$225$		0			0
$226$	−80.0000			−0.353982
$227$	−242.000			−1.06608			−0.533040	−	0.846090i	$-0.678950\pi$
$227$	−242.000			−1.06608			−0.533040	+	0.846090i	$0.678950\pi$
$228$		0			0
$229$	−106.000			−0.462882			−0.231441	−	0.972849i	$-0.574344\pi$
$229$	−106.000			−0.462882			−0.231441	+	0.972849i	$0.574344\pi$
$230$	−10.0000	+	99.4987i	−0.0434783	+	0.432603i
$231$		0			0
$232$		−	278.596i		−	1.20085i
$233$	142.000			0.609442			0.304721	−	0.952442i	$-0.401437\pi$
$233$	142.000			0.609442			0.304721	+	0.952442i	$0.401437\pi$
$234$		0			0
$235$	29.0000	−	288.546i	0.123404	−	1.22786i
$236$			119.398i			0.505926i
$237$		0			0
$238$		−	218.897i		−	0.919736i
$239$			198.997i			0.832625i	0.909221	+	0.416313i	$0.136678\pi$
$239$			198.997i			0.832625i	−0.909221	+	0.416313i	$0.863322\pi$
$240$		0			0
$241$	170.000			0.705394			0.352697	−	0.935738i	$-0.385265\pi$
$241$	170.000			0.705394			0.352697	+	0.935738i	$0.385265\pi$
$242$	22.0000			0.0909091
$243$		0			0
$244$	−132.000			−0.540984
$245$	−25.0000	+	248.747i	−0.102041	+	1.01529i
$246$		0			0
$247$			79.5990i			0.322263i
$248$	−203.000			−0.818548
$249$		0			0
$250$	−37.0000	+	119.398i	−0.148000	+	0.477594i
$251$		−	358.195i		−	1.42707i	−0.700618	−	0.713537i	$-0.747092\pi$
$251$		−	358.195i		−	1.42707i	0.700618	−	0.713537i	$-0.252908\pi$
$252$		0			0
$253$		−	198.997i		−	0.786551i
$254$			149.248i			0.587591i
$255$		0			0
$256$	−171.000			−0.667969
$257$	−290.000			−1.12840			−0.564202	−	0.825637i	$-0.690816\pi$
$257$	−290.000			−1.12840			−0.564202	+	0.825637i	$0.690816\pi$
$258$		0			0
$259$		0			0
$260$	297.000	+	29.8496i	1.14231	+	0.114806i
$261$		0			0
$262$			169.148i			0.645603i
$263$	−104.000			−0.395437			−0.197719	−	0.980259i	$-0.563353\pi$
$263$	−104.000			−0.395437			−0.197719	+	0.980259i	$0.563353\pi$
$264$		0			0
$265$	15.5000	−	154.223i	0.0584906	−	0.581974i
$266$			39.7995i			0.149622i
$267$		0			0
$268$			59.6992i			0.222758i
$269$			119.398i			0.443861i	0.975063	+	0.221930i	$0.0712357\pi$
$269$			119.398i			0.443861i	−0.975063	+	0.221930i	$0.928764\pi$
$270$		0			0
$271$	−121.000			−0.446494			−0.223247	−	0.974762i	$-0.571666\pi$
$271$	−121.000			−0.446494			−0.223247	+	0.974762i	$0.571666\pi$
$272$	110.000			0.404412
$273$		0			0
$274$	−98.0000			−0.357664
$275$	49.5000	−	243.772i	0.180000	−	0.886443i
$276$		0			0
$277$		−	159.198i		−	0.574722i	−0.957822	−	0.287361i	$-0.907222\pi$
$277$		−	159.198i		−	0.574722i	0.957822	−	0.287361i	$-0.0927779\pi$
$278$	−64.0000			−0.230216
$279$		0			0
$280$	346.500	+	34.8246i	1.23750	+	0.124373i
$281$			218.897i			0.778994i	0.921028	+	0.389497i	$0.127351\pi$
$281$			218.897i			0.778994i	−0.921028	+	0.389497i	$0.872649\pi$
$282$		0			0
$283$			278.596i			0.984440i	0.870471	+	0.492220i	$0.163814\pi$
$283$			278.596i			0.984440i	−0.870471	+	0.492220i	$0.836186\pi$
$284$		−	179.098i		−	0.630626i
$285$		0			0
$286$	198.000			0.692308
$287$	−396.000			−1.37979
$288$		0			0
$289$	195.000			0.674740
$290$	198.000	+	19.8997i	0.682759	+	0.0686198i
$291$		0			0
$292$			268.647i			0.920023i
$293$	286.000			0.976109			0.488055	−	0.872813i	$-0.337707\pi$
$293$	286.000			0.976109			0.488055	+	0.872813i	$0.337707\pi$
$294$		0			0
$295$	−198.000	−	19.8997i	−0.671186	−	0.0674568i
$296$		0			0
$297$		0			0
$298$		−	9.94987i		−	0.0333888i
$299$			397.995i			1.33109i
$300$		0			0
$301$	198.000			0.657807
$302$	257.000			0.850993
$303$		0			0
$304$	−20.0000			−0.0657895
$305$	22.0000	−	218.897i	0.0721311	−	0.717696i
$306$		0			0
$307$			59.6992i			0.194460i	0.995262	+	0.0972300i	$0.0309983\pi$
$307$			59.6992i			0.194460i	−0.995262	+	0.0972300i	$0.969002\pi$
$308$	−297.000			−0.964286
$309$		0			0
$310$	14.5000	−	144.273i	0.0467742	−	0.465397i
$311$			198.997i			0.639863i	0.947441	+	0.319932i	$0.103660\pi$
$311$			198.997i			0.639863i	−0.947441	+	0.319932i	$0.896340\pi$
$312$		0			0
$313$		−	69.6491i		−	0.222521i	−0.993791	−	0.111261i	$-0.964511\pi$
$313$		−	69.6491i		−	0.222521i	0.993791	−	0.111261i	$-0.0354888\pi$
$314$			159.198i			0.507000i
$315$		0			0
$316$	30.0000			0.0949367
$317$	541.000			1.70662			0.853312	−	0.521400i	$-0.174590\pi$
$317$	541.000			1.70662			0.853312	+	0.521400i	$0.174590\pi$
$318$		0			0
$319$	−396.000			−1.24138
$320$	6.50000	−	64.6742i	0.0203125	−	0.202107i
$321$		0			0
$322$			198.997i			0.618005i
$323$	−88.0000			−0.272446
$324$		0			0
$325$	−99.0000	+	487.544i	−0.304615	+	1.50013i
$326$			298.496i			0.915633i
$327$		0			0
$328$			278.596i			0.849380i
$329$		−	577.093i		−	1.75408i
$330$		0			0
$331$	−274.000			−0.827795			−0.413897	−	0.910324i	$-0.635833\pi$
$331$	−274.000			−0.827795			−0.413897	+	0.910324i	$0.635833\pi$
$332$	−57.0000			−0.171687
$333$		0			0
$334$	−254.000			−0.760479
$335$	−99.0000	−	9.94987i	−0.295522	−	0.0297011i
$336$		0			0
$337$		−	437.794i		−	1.29909i	−0.760322	−	0.649547i	$-0.774959\pi$
$337$		−	437.794i		−	1.29909i	0.760322	−	0.649547i	$-0.225041\pi$
$338$	−227.000			−0.671598
$339$		0			0
$340$	−33.0000	+	328.346i	−0.0970588	+	0.965723i
$341$			288.546i			0.846177i
$342$		0			0
$343$			9.94987i			0.0290084i
$344$		−	139.298i		−	0.404937i
$345$		0			0
$346$	235.000			0.679191
$347$	−179.000			−0.515850			−0.257925	−	0.966165i	$-0.583039\pi$
$347$	−179.000			−0.515850			−0.257925	+	0.966165i	$0.583039\pi$
$348$		0			0
$349$	620.000			1.77650			0.888252	−	0.459356i	$-0.151920\pi$
$349$	620.000			1.77650			0.888252	+	0.459356i	$0.151920\pi$
$350$	−49.5000	+	243.772i	−0.141429	+	0.696491i
$351$		0			0
$352$			328.346i			0.932801i
$353$	−32.0000			−0.0906516			−0.0453258	−	0.998972i	$-0.514433\pi$
$353$	−32.0000			−0.0906516			−0.0453258	+	0.998972i	$0.514433\pi$
$354$		0			0
$355$	297.000	+	29.8496i	0.836620	+	0.0840834i
$356$		−	179.098i		−	0.503084i
$357$		0			0
$358$			149.248i			0.416894i
$359$		−	537.293i		−	1.49664i	−0.663339	−	0.748319i	$-0.730861\pi$
$359$		−	537.293i		−	1.49664i	0.663339	−	0.748319i	$-0.269139\pi$
$360$		0			0
$361$	−345.000			−0.955679
$362$	−292.000			−0.806630
$363$		0			0
$364$	594.000			1.63187
$365$	−445.500	−	44.7744i	−1.22055	−	0.122670i
$366$		0			0
$367$			348.246i			0.948898i	0.880283	+	0.474449i	$0.157353\pi$
$367$			348.246i			0.948898i	−0.880283	+	0.474449i	$0.842647\pi$
$368$	−100.000			−0.271739
$369$		0			0
$370$		0			0
$371$		−	308.446i		−	0.831391i
$372$		0			0
$373$			338.296i			0.906959i	0.891267	+	0.453480i	$0.149817\pi$
$373$			338.296i			0.906959i	−0.891267	+	0.453480i	$0.850183\pi$
$374$			218.897i			0.585287i
$375$		0			0
$376$	−406.000			−1.07979
$377$	792.000			2.10080
$378$		0			0
$379$	248.000			0.654354			0.327177	−	0.944963i	$-0.393903\pi$
$379$	248.000			0.654354			0.327177	+	0.944963i	$0.393903\pi$
$380$	6.00000	−	59.6992i	0.0157895	−	0.157103i
$381$		0			0
$382$			99.4987i			0.260468i
$383$	394.000			1.02872			0.514360	−	0.857574i	$-0.328029\pi$
$383$	394.000			1.02872			0.514360	+	0.857574i	$0.328029\pi$
$384$		0			0
$385$	49.5000	−	492.519i	0.128571	−	1.27927i
$386$		−	169.148i		−	0.438207i
$387$		0			0
$388$			388.045i			1.00012i
$389$			9.94987i			0.0255781i	0.999918	+	0.0127890i	$0.00407099\pi$
$389$			9.94987i			0.0255781i	−0.999918	+	0.0127890i	$0.995929\pi$
$390$		0			0
$391$	−440.000			−1.12532
$392$	350.000			0.892857
$393$		0			0
$394$	−203.000			−0.515228
$395$	−5.00000	+	49.7494i	−0.0126582	+	0.125948i
$396$		0			0
$397$			59.6992i			0.150376i	0.997169	+	0.0751880i	$0.0239557\pi$
$397$			59.6992i			0.150376i	−0.997169	+	0.0751880i	$0.976044\pi$
$398$	155.000			0.389447
$399$		0			0
$400$	−122.500	−	24.8747i	−0.306250	−	0.0621867i
$401$			198.997i			0.496253i	0.968728	+	0.248127i	$0.0798149\pi$
$401$			198.997i			0.496253i	−0.968728	+	0.248127i	$0.920185\pi$
$402$		0			0
$403$		−	577.093i		−	1.43199i
$404$			507.444i			1.25605i
$405$		0			0
$406$	396.000			0.975369
$407$		0			0
$408$		0			0
$409$	−331.000			−0.809291			−0.404645	−	0.914474i	$-0.632605\pi$
$409$	−331.000			−0.809291			−0.404645	+	0.914474i	$0.632605\pi$
$410$	−198.000	−	19.8997i	−0.482927	−	0.0485360i
$411$		0			0
$412$		−	119.398i		−	0.289802i
$413$	−396.000			−0.958838
$414$		0			0
$415$	9.50000	−	94.5238i	0.0228916	−	0.227768i
$416$		−	656.692i		−	1.57859i
$417$		0			0
$418$		−	39.7995i		−	0.0952141i
$419$			676.591i			1.61478i	0.590020	+	0.807388i	$0.299120\pi$
$419$			676.591i			1.61478i	−0.590020	+	0.807388i	$0.700880\pi$
$420$		0			0
$421$	8.00000			0.0190024			0.00950119	−	0.999955i	$-0.496976\pi$
$421$	8.00000			0.0190024			0.00950119	+	0.999955i	$0.496976\pi$
$422$	308.000			0.729858
$423$		0			0
$424$	−217.000			−0.511792
$425$	−539.000	−	109.449i	−1.26824	−	0.257526i
$426$		0			0
$427$		−	437.794i		−	1.02528i
$428$	87.0000			0.203271
$429$		0			0
$430$	99.0000	+	9.94987i	0.230233	+	0.0231392i
$431$		−	298.496i		−	0.692567i	−0.938130	−	0.346283i	$-0.887444\pi$
$431$		−	298.496i		−	0.692567i	0.938130	−	0.346283i	$-0.112556\pi$
$432$		0			0
$433$		−	686.541i		−	1.58555i	−0.609517	−	0.792773i	$-0.708637\pi$
$433$		−	686.541i		−	1.58555i	0.609517	−	0.792773i	$-0.291363\pi$
$434$		−	288.546i		−	0.664853i
$435$		0			0
$436$	−312.000			−0.715596
$437$	80.0000			0.183066
$438$		0			0
$439$	215.000			0.489749			0.244875	−	0.969555i	$-0.421253\pi$
$439$	215.000			0.489749			0.244875	+	0.969555i	$0.421253\pi$
$440$	−346.500	−	34.8246i	−0.787500	−	0.0791467i
$441$		0			0
$442$		−	437.794i		−	0.990485i
$443$	190.000			0.428894			0.214447	−	0.976736i	$-0.431205\pi$
$443$	190.000			0.428894			0.214447	+	0.976736i	$0.431205\pi$
$444$		0			0
$445$	297.000	+	29.8496i	0.667416	+	0.0670778i
$446$		−	39.7995i		−	0.0892365i
$447$		0			0
$448$		−	129.348i		−	0.288724i
$449$			119.398i			0.265921i	0.991121	+	0.132960i	$0.0424483\pi$
$449$			119.398i			0.265921i	−0.991121	+	0.132960i	$0.957552\pi$
$450$		0			0
$451$	396.000			0.878049
$452$	240.000			0.530973
$453$		0			0
$454$	−242.000			−0.533040
$455$	−99.0000	+	985.038i	−0.217582	+	2.16492i
$456$		0			0
$457$		−	129.348i		−	0.283038i	−0.989936	−	0.141519i	$-0.954801\pi$
$457$		−	129.348i		−	0.283038i	0.989936	−	0.141519i	$-0.0451986\pi$
$458$	−106.000			−0.231441
$459$		0			0
$460$	30.0000	−	298.496i	0.0652174	−	0.648905i
$461$			189.048i			0.410082i	0.978753	+	0.205041i	$0.0657327\pi$
$461$			189.048i			0.410082i	−0.978753	+	0.205041i	$0.934267\pi$
$462$		0			0
$463$		−	825.840i		−	1.78367i	−0.452360	−	0.891835i	$-0.649418\pi$
$463$		−	825.840i		−	1.78367i	0.452360	−	0.891835i	$-0.350582\pi$
$464$			198.997i			0.428874i
$465$		0			0
$466$	142.000			0.304721
$467$	193.000			0.413276			0.206638	−	0.978417i	$-0.433748\pi$
$467$	193.000			0.413276			0.206638	+	0.978417i	$0.433748\pi$
$468$		0			0
$469$	−198.000			−0.422175
$470$	29.0000	−	288.546i	0.0617021	−	0.613928i
$471$		0			0
$472$			278.596i			0.590247i
$473$	−198.000			−0.418605
$474$		0			0
$475$	98.0000	+	19.8997i	0.206316	+	0.0418942i
$476$			656.692i			1.37960i
$477$		0			0
$478$			198.997i			0.416313i
$479$		−	736.291i		−	1.53714i	−0.639765	−	0.768571i	$-0.720968\pi$
$479$		−	736.291i		−	1.53714i	0.639765	−	0.768571i	$-0.279032\pi$
$480$		0			0
$481$		0			0
$482$	170.000			0.352697
$483$		0			0
$484$	−66.0000			−0.136364
$485$	−643.500	−	64.6742i	−1.32680	−	0.133349i
$486$		0			0
$487$		−	477.594i		−	0.980686i	−0.871530	−	0.490343i	$-0.836872\pi$
$487$		−	477.594i		−	0.980686i	0.871530	−	0.490343i	$-0.163128\pi$
$488$	−308.000			−0.631148
$489$		0			0
$490$	−25.0000	+	248.747i	−0.0510204	+	0.507647i
$491$			288.546i			0.587671i	0.955856	+	0.293835i	$0.0949318\pi$
$491$			288.546i			0.587671i	−0.955856	+	0.293835i	$0.905068\pi$
$492$		0			0
$493$			875.589i			1.77604i
$494$			79.5990i			0.161132i
$495$		0			0
$496$	145.000			0.292339
$497$	594.000			1.19517
$498$		0			0
$499$	290.000			0.581162			0.290581	−	0.956850i	$-0.406151\pi$
$499$	290.000			0.581162			0.290581	+	0.956850i	$0.406151\pi$
$500$	111.000	−	358.195i	0.222000	−	0.716391i
$501$		0			0
$502$		−	358.195i		−	0.713537i
$503$	220.000			0.437376			0.218688	−	0.975795i	$-0.429822\pi$
$503$	220.000			0.437376			0.218688	+	0.975795i	$0.429822\pi$
$504$		0			0
$505$	−841.500	−	84.5739i	−1.66634	−	0.167473i
$506$		−	198.997i		−	0.393276i
$507$		0			0
$508$		−	447.744i		−	0.881387i
$509$		−	9.94987i		−	0.0195479i	−0.999952	−	0.00977394i	$-0.996889\pi$
$509$		−	9.94987i		−	0.0195479i	0.999952	−	0.00977394i	$-0.00311119\pi$
$510$		0			0
$511$	−891.000			−1.74364
$512$	305.000			0.595703
$513$		0			0
$514$	−290.000			−0.564202
$515$	198.000	+	19.8997i	0.384466	+	0.0386403i
$516$		0			0
$517$			577.093i			1.11623i
$518$		0			0
$519$		0			0
$520$	693.000	+	69.6491i	1.33269	+	0.133941i
$521$			298.496i			0.572929i	0.958091	+	0.286465i	$0.0924801\pi$
$521$			298.496i			0.572929i	−0.958091	+	0.286465i	$0.907520\pi$
$522$		0			0
$523$			537.293i			1.02733i	0.857991	+	0.513665i	$0.171712\pi$
$523$			537.293i			1.02733i	−0.857991	+	0.513665i	$0.828288\pi$
$524$		−	507.444i		−	0.968404i
$525$		0			0
$526$	−104.000			−0.197719
$527$	638.000			1.21063
$528$		0			0
$529$	−129.000			−0.243856
$530$	15.5000	−	154.223i	0.0292453	−	0.290987i
$531$		0			0
$532$		−	119.398i		−	0.224433i
$533$	−792.000			−1.48593
$534$		0			0
$535$	−14.5000	+	144.273i	−0.0271028	+	0.269669i
$536$			139.298i			0.259885i
$537$		0			0
$538$			119.398i			0.221930i
$539$		−	497.494i		−	0.922994i
$540$		0			0
$541$	704.000			1.30129			0.650647	−	0.759380i	$-0.274498\pi$
$541$	704.000			1.30129			0.650647	+	0.759380i	$0.274498\pi$
$542$	−121.000			−0.223247
$543$		0			0
$544$	726.000			1.33456
$545$	52.0000	−	517.393i	0.0954128	−	0.949346i
$546$		0			0
$547$		−	99.4987i		−	0.181899i	−0.995856	−	0.0909495i	$-0.971010\pi$
$547$		−	99.4987i		−	0.181899i	0.995856	−	0.0909495i	$-0.0289902\pi$
$548$	294.000			0.536496
$549$		0			0
$550$	49.5000	−	243.772i	0.0900000	−	0.443222i
$551$		−	159.198i		−	0.288926i
$552$		0			0
$553$			99.4987i			0.179925i
$554$		−	159.198i		−	0.287361i
$555$		0			0
$556$	192.000			0.345324
$557$	−101.000			−0.181329			−0.0906643	−	0.995882i	$-0.528899\pi$
$557$	−101.000			−0.181329			−0.0906643	+	0.995882i	$0.528899\pi$
$558$		0			0
$559$	396.000			0.708408
$560$	−247.500	−	24.8747i	−0.441964	−	0.0444191i
$561$		0			0
$562$			218.897i			0.389497i
$563$	−935.000			−1.66075			−0.830373	−	0.557208i	$-0.811873\pi$
$563$	−935.000			−1.66075			−0.830373	+	0.557208i	$0.811873\pi$
$564$		0			0
$565$	−40.0000	+	397.995i	−0.0707965	+	0.704416i
$566$			278.596i			0.492220i
$567$		0			0
$568$		−	417.895i		−	0.735730i
$569$			756.190i			1.32898i	0.747297	+	0.664491i	$0.231351\pi$
$569$			756.190i			1.32898i	−0.747297	+	0.664491i	$0.768649\pi$
$570$		0			0
$571$	506.000			0.886165			0.443082	−	0.896481i	$-0.353885\pi$
$571$	506.000			0.886165			0.443082	+	0.896481i	$0.353885\pi$
$572$	−594.000			−1.03846
$573$		0			0
$574$	−396.000			−0.689895
$575$	490.000	+	99.4987i	0.852174	+	0.173041i
$576$		0			0
$577$			1074.59i			1.86237i	0.364549	+	0.931184i	$0.381223\pi$
$577$			1074.59i			1.86237i	−0.364549	+	0.931184i	$0.618777\pi$
$578$	195.000			0.337370
$579$		0			0
$580$	−594.000	−	59.6992i	−1.02414	−	0.102930i
$581$		−	189.048i		−	0.325383i
$582$		0			0
$583$			308.446i			0.529067i
$584$			626.842i			1.07336i
$585$		0			0
$586$	286.000			0.488055
$587$	−125.000			−0.212947			−0.106474	−	0.994316i	$-0.533956\pi$
$587$	−125.000			−0.212947			−0.106474	+	0.994316i	$0.533956\pi$
$588$		0			0
$589$	−116.000			−0.196944
$590$	−198.000	−	19.8997i	−0.335593	−	0.0337284i
$591$		0			0
$592$		0			0
$593$	−50.0000			−0.0843170			−0.0421585	−	0.999111i	$-0.513423\pi$
$593$	−50.0000			−0.0843170			−0.0421585	+	0.999111i	$0.513423\pi$
$594$		0			0
$595$	−1089.00	−	109.449i	−1.83025	−	0.183947i
$596$			29.8496i			0.0500833i
$597$		0			0
$598$			397.995i			0.665543i
$599$			139.298i			0.232551i	0.993217	+	0.116276i	$0.0370956\pi$
$599$			139.298i			0.232551i	−0.993217	+	0.116276i	$0.962904\pi$
$600$		0			0
$601$	53.0000			0.0881864			0.0440932	−	0.999027i	$-0.485960\pi$
$601$	53.0000			0.0881864			0.0440932	+	0.999027i	$0.485960\pi$
$602$	198.000			0.328904
$603$		0			0
$604$	−771.000			−1.27649
$605$	11.0000	−	109.449i	0.0181818	−	0.180907i
$606$		0			0
$607$			397.995i			0.655675i	0.944734	+	0.327838i	$0.106320\pi$
$607$			397.995i			0.655675i	−0.944734	+	0.327838i	$0.893680\pi$
$608$	−132.000			−0.217105
$609$		0			0
$610$	22.0000	−	218.897i	0.0360656	−	0.358848i
$611$		−	1154.19i		−	1.88901i
$612$		0			0
$613$			895.489i			1.46083i	0.683004	+	0.730415i	$0.260673\pi$
$613$			895.489i			1.46083i	−0.683004	+	0.730415i	$0.739327\pi$
$614$			59.6992i			0.0972300i
$615$		0			0
$616$	−693.000			−1.12500
$617$	1132.00			1.83468			0.917342	−	0.398100i	$-0.130330\pi$
$617$	1132.00			1.83468			0.917342	+	0.398100i	$0.130330\pi$
$618$		0			0
$619$	302.000			0.487884			0.243942	−	0.969790i	$-0.421559\pi$
$619$	302.000			0.487884			0.243942	+	0.969790i	$0.421559\pi$
$620$	−43.5000	+	432.820i	−0.0701613	+	0.698096i
$621$		0			0
$622$			198.997i			0.319932i
$623$	594.000			0.953451
$624$		0			0
$625$	575.500	+	243.772i	0.920800	+	0.390035i
$626$		−	69.6491i		−	0.111261i
$627$		0			0
$628$		−	477.594i		−	0.760500i
$629$		0			0
$630$		0			0
$631$	1103.00			1.74802			0.874010	−	0.485909i	$-0.161511\pi$
$631$	1103.00			1.74802			0.874010	+	0.485909i	$0.161511\pi$
$632$	70.0000			0.110759
$633$		0			0
$634$	541.000			0.853312
$635$	742.500	+	74.6241i	1.16929	+	0.117518i
$636$		0			0
$637$			994.987i			1.56199i
$638$	−396.000			−0.620690
$639$		0			0
$640$	−59.5000	+	592.018i	−0.0929688	+	0.925027i
$641$			397.995i			0.620897i	0.950590	+	0.310448i	$0.100479\pi$
$641$			397.995i			0.620897i	−0.950590	+	0.310448i	$0.899521\pi$
$642$		0			0
$643$			278.596i			0.433276i	0.976252	+	0.216638i	$0.0695091\pi$
$643$			278.596i			0.433276i	−0.976252	+	0.216638i	$0.930491\pi$
$644$		−	596.992i		−	0.927007i
$645$		0			0
$646$	−88.0000			−0.136223
$647$	−1256.00			−1.94127			−0.970634	−	0.240562i	$-0.922668\pi$
$647$	−1256.00			−1.94127			−0.970634	+	0.240562i	$0.922668\pi$
$648$		0			0
$649$	396.000			0.610169
$650$	−99.0000	+	487.544i	−0.152308	+	0.750067i
$651$		0			0
$652$		−	895.489i		−	1.37345i
$653$	7.00000			0.0107198			0.00535988	−	0.999986i	$-0.498294\pi$
$653$	7.00000			0.0107198			0.00535988	+	0.999986i	$0.498294\pi$
$654$		0			0
$655$	841.500	+	84.5739i	1.28473	+	0.129121i
$656$		−	198.997i		−	0.303350i
$657$		0			0
$658$		−	577.093i		−	0.877041i
$659$			965.138i			1.46455i	0.681010	+	0.732275i	$0.261541\pi$
$659$			965.138i			1.46455i	−0.681010	+	0.732275i	$0.738459\pi$
$660$		0			0
$661$	−640.000			−0.968230			−0.484115	−	0.875004i	$-0.660858\pi$
$661$	−640.000			−0.968230			−0.484115	+	0.875004i	$0.660858\pi$
$662$	−274.000			−0.413897
$663$		0			0
$664$	−133.000			−0.200301
$665$	198.000	+	19.8997i	0.297744	+	0.0299244i
$666$		0			0
$667$		−	795.990i		−	1.19339i
$668$	762.000			1.14072
$669$		0			0
$670$	−99.0000	−	9.94987i	−0.147761	−	0.0148506i
$671$			437.794i			0.652451i
$672$		0			0
$673$		−	308.446i		−	0.458315i	−0.973389	−	0.229158i	$-0.926403\pi$
$673$		−	308.446i		−	0.458315i	0.973389	−	0.229158i	$-0.0735971\pi$
$674$		−	437.794i		−	0.649547i
$675$		0			0
$676$	681.000			1.00740
$677$	22.0000			0.0324963			0.0162482	−	0.999868i	$-0.494828\pi$
$677$	22.0000			0.0324963			0.0162482	+	0.999868i	$0.494828\pi$
$678$		0			0
$679$	−1287.00			−1.89543
$680$	−77.0000	+	766.140i	−0.113235	+	1.12668i
$681$		0			0
$682$			288.546i			0.423088i
$683$	166.000			0.243045			0.121523	−	0.992589i	$-0.461222\pi$
$683$	166.000			0.243045			0.121523	+	0.992589i	$0.461222\pi$
$684$		0			0
$685$	−49.0000	+	487.544i	−0.0715328	+	0.711743i
$686$			9.94987i			0.0145042i
$687$		0			0
$688$			99.4987i			0.144620i
$689$		−	616.892i		−	0.895344i
$690$		0			0
$691$	−460.000			−0.665702			−0.332851	−	0.942979i	$-0.608011\pi$
$691$	−460.000			−0.665702			−0.332851	+	0.942979i	$0.608011\pi$
$692$	−705.000			−1.01879
$693$		0			0
$694$	−179.000			−0.257925
$695$	−32.0000	+	318.396i	−0.0460432	+	0.458124i
$696$		0			0
$697$		−	875.589i		−	1.25623i
$698$	620.000			0.888252
$699$		0			0
$700$	148.500	−	731.316i	0.212143	−	1.04474i
$701$		−	268.647i		−	0.383233i	−0.981470	−	0.191617i	$-0.938627\pi$
$701$		−	268.647i		−	0.383233i	0.981470	−	0.191617i	$-0.0613730\pi$
$702$		0			0
$703$		0			0
$704$			129.348i			0.183733i
$705$		0			0
$706$	−32.0000			−0.0453258
$707$	−1683.00			−2.38048
$708$		0			0
$709$	−544.000			−0.767278			−0.383639	−	0.923483i	$-0.625329\pi$
$709$	−544.000			−0.767278			−0.383639	+	0.923483i	$0.625329\pi$
$710$	297.000	+	29.8496i	0.418310	+	0.0420417i
$711$		0			0
$712$		−	417.895i		−	0.586931i
$713$	−580.000			−0.813464
$714$		0			0
$715$	99.0000	−	985.038i	0.138462	−	1.37767i
$716$		−	447.744i		−	0.625341i
$717$		0			0
$718$		−	537.293i		−	0.748319i
$719$		−	358.195i		−	0.498186i	−0.968480	−	0.249093i	$-0.919868\pi$
$719$		−	358.195i		−	0.498186i	0.968480	−	0.249093i	$-0.0801324\pi$
$720$		0			0
$721$	396.000			0.549237
$722$	−345.000			−0.477839
$723$		0			0
$724$	876.000			1.20994
$725$	198.000	−	975.088i	0.273103	−	1.34495i
$726$		0			0
$727$		−	1383.03i		−	1.90238i	−0.308601	−	0.951192i	$-0.599861\pi$
$727$		−	1383.03i		−	1.90238i	0.308601	−	0.951192i	$-0.400139\pi$
$728$	1386.00			1.90385
$729$		0			0
$730$	−445.500	−	44.7744i	−0.610274	−	0.0613348i
$731$			437.794i			0.598898i
$732$		0			0
$733$			636.792i			0.868748i	0.900733	+	0.434374i	$0.143030\pi$
$733$			636.792i			0.868748i	−0.900733	+	0.434374i	$0.856970\pi$
$734$			348.246i			0.474449i
$735$		0			0
$736$	−660.000			−0.896739
$737$	198.000			0.268657
$738$		0			0
$739$	−958.000			−1.29635			−0.648173	−	0.761493i	$-0.724467\pi$
$739$	−958.000			−1.29635			−0.648173	+	0.761493i	$0.724467\pi$
$740$		0			0
$741$		0			0
$742$		−	308.446i		−	0.415696i
$743$	−602.000			−0.810229			−0.405114	−	0.914266i	$-0.632768\pi$
$743$	−602.000			−0.810229			−0.405114	+	0.914266i	$0.632768\pi$
$744$		0			0
$745$	−49.5000	−	4.97494i	−0.0664430	−	0.00667777i
$746$			338.296i			0.453480i
$747$		0			0
$748$		−	656.692i		−	0.877930i
$749$			288.546i			0.385242i
$750$		0			0
$751$	533.000			0.709720			0.354860	−	0.934919i	$-0.384528\pi$
$751$	533.000			0.709720			0.354860	+	0.934919i	$0.384528\pi$
$752$	290.000			0.385638
$753$		0			0
$754$	792.000			1.05040
$755$	128.500	−	1278.56i	0.170199	−	1.69346i
$756$		0			0
$757$			895.489i			1.18294i	0.806325	+	0.591472i	$0.201453\pi$
$757$			895.489i			1.18294i	−0.806325	+	0.591472i	$0.798547\pi$
$758$	248.000			0.327177
$759$		0			0
$760$	14.0000	−	139.298i	0.0184211	−	0.183287i
$761$		−	994.987i		−	1.30747i	−0.756722	−	0.653737i	$-0.773200\pi$
$761$		−	994.987i		−	1.30747i	0.756722	−	0.653737i	$-0.226800\pi$
$762$		0			0
$763$		−	1034.79i		−	1.35621i
$764$		−	298.496i		−	0.390702i
$765$		0			0
$766$	394.000			0.514360
$767$	−792.000			−1.03259
$768$		0			0
$769$	605.000			0.786736			0.393368	−	0.919381i	$-0.371310\pi$
$769$	605.000			0.786736			0.393368	+	0.919381i	$0.371310\pi$
$770$	49.5000	−	492.519i	0.0642857	−	0.639635i
$771$		0			0
$772$			507.444i			0.657310i
$773$	886.000			1.14618			0.573092	−	0.819491i	$-0.305744\pi$
$773$	886.000			1.14618			0.573092	+	0.819491i	$0.305744\pi$
$774$		0			0
$775$	−710.500	−	144.273i	−0.916774	−	0.186159i
$776$			905.439i			1.16680i
$777$		0			0
$778$			9.94987i			0.0127890i
$779$			159.198i			0.204362i
$780$		0			0
$781$	−594.000			−0.760563
$782$	−440.000			−0.562660
$783$		0			0
$784$	−250.000			−0.318878
$785$	792.000	+	79.5990i	1.00892	+	0.101400i
$786$		0			0
$787$			517.393i			0.657425i	0.944430	+	0.328712i	$0.106615\pi$
$787$			517.393i			0.657425i	−0.944430	+	0.328712i	$0.893385\pi$
$788$	609.000			0.772843
$789$		0			0
$790$	−5.00000	+	49.7494i	−0.00632911	+	0.0629739i
$791$			795.990i			1.00631i
$792$		0			0
$793$		−	875.589i		−	1.10415i
$794$			59.6992i			0.0751880i
$795$		0			0
$796$	−465.000			−0.584171
$797$	−335.000			−0.420326			−0.210163	−	0.977666i	$-0.567400\pi$
$797$	−335.000			−0.420326			−0.210163	+	0.977666i	$0.567400\pi$
$798$		0			0
$799$	1276.00			1.59700
$800$	−808.500	−	164.173i	−1.01063	−	0.205216i
$801$		0			0
$802$			198.997i			0.248127i
$803$	891.000			1.10959
$804$		0			0
$805$	990.000	+	99.4987i	1.22981	+	0.123601i
$806$		−	577.093i		−	0.715996i
$807$		0			0
$808$			1184.04i			1.46539i
$809$			417.895i			0.516557i	0.966070	+	0.258279i	$0.0831552\pi$
$809$			417.895i			0.516557i	−0.966070	+	0.258279i	$0.916845\pi$
$810$		0			0
$811$	−700.000			−0.863132			−0.431566	−	0.902081i	$-0.642039\pi$
$811$	−700.000			−0.863132			−0.431566	+	0.902081i	$0.642039\pi$
$812$	−1188.00			−1.46305
$813$		0			0
$814$		0			0
$815$	1485.00	+	149.248i	1.82209	+	0.183127i
$816$		0			0
$817$		−	79.5990i		−	0.0974284i
$818$	−331.000			−0.404645
$819$		0			0
$820$	594.000	+	59.6992i	0.724390	+	0.0728040i
$821$			1114.39i			1.35735i	0.734438	+	0.678676i	$0.237446\pi$
$821$			1114.39i			1.35735i	−0.734438	+	0.678676i	$0.762554\pi$
$822$		0			0
$823$		−	945.238i		−	1.14853i	−0.818670	−	0.574264i	$-0.805288\pi$
$823$		−	945.238i		−	1.14853i	0.818670	−	0.574264i	$-0.194712\pi$
$824$		−	278.596i		−	0.338103i
$825$		0			0
$826$	−396.000			−0.479419
$827$	−230.000			−0.278114			−0.139057	−	0.990284i	$-0.544407\pi$
$827$	−230.000			−0.278114			−0.139057	+	0.990284i	$0.544407\pi$
$828$		0			0
$829$	686.000			0.827503			0.413752	−	0.910390i	$-0.364218\pi$
$829$	686.000			0.827503			0.413752	+	0.910390i	$0.364218\pi$
$830$	9.50000	−	94.5238i	0.0114458	−	0.113884i
$831$		0			0
$832$		−	258.697i		−	0.310934i
$833$	−1100.00			−1.32053
$834$		0			0
$835$	−127.000	+	1263.63i	−0.152096	+	1.51333i
$836$			119.398i			0.142821i
$837$		0			0
$838$			676.591i			0.807388i
$839$		−	1213.88i		−	1.44682i	−0.690417	−	0.723412i	$-0.742573\pi$
$839$		−	1213.88i		−	1.44682i	0.690417	−	0.723412i	$-0.257427\pi$
$840$		0			0
$841$	−743.000			−0.883472
$842$	8.00000			0.00950119
$843$		0			0
$844$	−924.000			−1.09479
$845$	−113.500	+	1129.31i	−0.134320	+	1.33646i
$846$		0			0
$847$		−	218.897i		−	0.258438i
$848$	155.000			0.182783
$849$		0			0
$850$	−539.000	−	109.449i	−0.634118	−	0.128763i
$851$		0			0
$852$		0			0
$853$			736.291i			0.863178i	0.902070	+	0.431589i	$0.142047\pi$
$853$			736.291i			0.863178i	−0.902070	+	0.431589i	$0.857953\pi$
$854$		−	437.794i		−	0.512640i
$855$		0			0
$856$	203.000			0.237150
$857$	154.000			0.179697			0.0898483	−	0.995955i	$-0.471362\pi$
$857$	154.000			0.179697			0.0898483	+	0.995955i	$0.471362\pi$
$858$		0			0
$859$	−1132.00			−1.31781			−0.658906	−	0.752226i	$-0.728980\pi$
$859$	−1132.00			−1.31781			−0.658906	+	0.752226i	$0.728980\pi$
$860$	−297.000	−	29.8496i	−0.345349	−	0.0347089i
$861$		0			0
$862$		−	298.496i		−	0.346283i
$863$	−1040.00			−1.20510			−0.602549	−	0.798082i	$-0.705848\pi$
$863$	−1040.00			−1.20510			−0.602549	+	0.798082i	$0.705848\pi$
$864$		0			0
$865$	117.500	−	1169.11i	0.135838	−	1.35157i
$866$		−	686.541i		−	0.792773i
$867$		0			0
$868$			865.639i			0.997280i
$869$		−	99.4987i		−	0.114498i
$870$		0			0
$871$	−396.000			−0.454650
$872$	−728.000			−0.834862
$873$		0			0
$874$	80.0000			0.0915332
$875$	1188.00	+	368.145i	1.35771	+	0.420738i
$876$		0			0
$877$			1651.68i			1.88333i	0.336554	+	0.941664i	$0.390739\pi$
$877$			1651.68i			1.88333i	−0.336554	+	0.941664i	$0.609261\pi$
$878$	215.000			0.244875
$879$		0			0
$880$	247.500	+	24.8747i	0.281250	+	0.0282667i
$881$		−	417.895i		−	0.474341i	−0.971468	−	0.237171i	$-0.923780\pi$
$881$		−	417.895i		−	0.474341i	0.971468	−	0.237171i	$-0.0762201\pi$
$882$		0			0
$883$		−	238.797i		−	0.270438i	−0.990816	−	0.135219i	$-0.956826\pi$
$883$		−	238.797i		−	0.270438i	0.990816	−	0.135219i	$-0.0431738\pi$
$884$			1313.38i			1.48573i
$885$		0			0
$886$	190.000			0.214447
$887$	−614.000			−0.692221			−0.346110	−	0.938194i	$-0.612498\pi$
$887$	−614.000			−0.692221			−0.346110	+	0.938194i	$0.612498\pi$
$888$		0			0
$889$	1485.00			1.67042
$890$	297.000	+	29.8496i	0.333708	+	0.0335389i
$891$		0			0
$892$			119.398i			0.133855i
$893$	−232.000			−0.259798
$894$		0			0
$895$	742.500	+	74.6241i	0.829609	+	0.0833788i
$896$			1184.04i			1.32147i
$897$		0			0
$898$			119.398i			0.132960i
$899$			1154.19i			1.28385i
$900$		0			0
$901$	682.000			0.756937
$902$	396.000			0.439024
$903$		0			0
$904$	560.000			0.619469
$905$	−146.000	+	1452.68i	−0.161326	+	1.60517i
$906$		0			0
$907$		−	397.995i		−	0.438804i	−0.975635	−	0.219402i	$-0.929589\pi$
$907$		−	397.995i		−	0.438804i	0.975635	−	0.219402i	$-0.0704106\pi$
$908$	726.000			0.799559
$909$		0			0
$910$	−99.0000	+	985.038i	−0.108791	+	1.08246i
$911$		−	1452.68i		−	1.59460i	−0.603582	−	0.797301i	$-0.706261\pi$
$911$		−	1452.68i		−	1.59460i	0.603582	−	0.797301i	$-0.293739\pi$
$912$		0			0
$913$			189.048i			0.207062i
$914$		−	129.348i		−	0.141519i
$915$		0			0
$916$	318.000			0.347162
$917$	1683.00			1.83533
$918$		0			0
$919$	−187.000			−0.203482			−0.101741	−	0.994811i	$-0.532441\pi$
$919$	−187.000			−0.203482			−0.101741	+	0.994811i	$0.532441\pi$
$920$	70.0000	−	696.491i	0.0760870	−	0.757056i
$921$		0			0
$922$			189.048i			0.205041i
$923$	1188.00			1.28711
$924$		0			0
$925$		0			0
$926$		−	825.840i		−	0.891835i
$927$		0			0
$928$			1313.38i			1.41528i
$929$		−	676.591i		−	0.728301i	−0.931340	−	0.364150i	$-0.881359\pi$
$929$		−	676.591i		−	0.728301i	0.931340	−	0.364150i	$-0.118641\pi$
$930$		0			0
$931$	200.000			0.214823
$932$	−426.000			−0.457082
$933$		0			0
$934$	193.000			0.206638
$935$	1089.00	+	109.449i	1.16471	+	0.117057i
$936$		0			0
$937$		−	985.038i		−	1.05127i	−0.850711	−	0.525634i	$-0.823828\pi$
$937$		−	985.038i		−	1.05127i	0.850711	−	0.525634i	$-0.176172\pi$
$938$	−198.000			−0.211087
$939$		0			0
$940$	−87.0000	+	865.639i	−0.0925532	+	0.920893i
$941$		−	308.446i		−	0.327785i	−0.986478	−	0.163893i	$-0.947595\pi$
$941$		−	308.446i		−	0.327785i	0.986478	−	0.163893i	$-0.0524051\pi$
$942$		0			0
$943$			795.990i			0.844104i
$944$		−	198.997i		−	0.210802i
$945$		0			0
$946$	−198.000			−0.209302
$947$	−1385.00			−1.46251			−0.731257	−	0.682102i	$-0.761066\pi$
$947$	−1385.00			−1.46251			−0.731257	+	0.682102i	$0.761066\pi$
$948$		0			0
$949$	−1782.00			−1.87777
$950$	98.0000	+	19.8997i	0.103158	+	0.0209471i
$951$		0			0
$952$			1532.28i			1.60954i
$953$	−1496.00			−1.56978			−0.784890	−	0.619635i	$-0.787280\pi$
$953$	−1496.00			−1.56978			−0.784890	+	0.619635i	$0.787280\pi$
$954$		0			0
$955$	495.000	+	49.7494i	0.518325	+	0.0520936i
$956$		−	596.992i		−	0.624469i
$957$		0			0
$958$		−	736.291i		−	0.768571i
$959$			975.088i			1.01678i
$960$		0			0
$961$	−120.000			−0.124870
$962$		0			0
$963$		0			0
$964$	−510.000			−0.529046
$965$	−841.500	−	84.5739i	−0.872021	−	0.0876414i
$966$		0			0
$967$			69.6491i			0.0720260i	0.999351	+	0.0360130i	$0.0114658\pi$
$967$			69.6491i			0.0720260i	−0.999351	+	0.0360130i	$0.988534\pi$
$968$	−154.000			−0.159091
$969$		0			0
$970$	−643.500	−	64.6742i	−0.663402	−	0.0666744i
$971$			746.241i			0.768528i	0.923223	+	0.384264i	$0.125545\pi$
$971$			746.241i			0.768528i	−0.923223	+	0.384264i	$0.874455\pi$
$972$		0			0
$973$			636.792i			0.654462i
$974$		−	477.594i		−	0.490343i
$975$		0			0
$976$	220.000			0.225410
$977$	1054.00			1.07881			0.539406	−	0.842046i	$-0.318649\pi$
$977$	1054.00			1.07881			0.539406	+	0.842046i	$0.318649\pi$
$978$		0			0
$979$	−594.000			−0.606742
$980$	75.0000	−	746.241i	0.0765306	−	0.761470i
$981$		0			0
$982$			288.546i			0.293835i
$983$	−1394.00			−1.41811			−0.709054	−	0.705154i	$-0.750878\pi$
$983$	−1394.00			−1.41811			−0.709054	+	0.705154i	$0.750878\pi$
$984$		0			0
$985$	−101.500	+	1009.91i	−0.103046	+	1.02529i
$986$			875.589i			0.888021i
$987$		0			0
$988$		−	238.797i		−	0.241697i
$989$		−	397.995i		−	0.402422i
$990$		0			0
$991$	−265.000			−0.267407			−0.133703	−	0.991021i	$-0.542687\pi$
$991$	−265.000			−0.267407			−0.133703	+	0.991021i	$0.542687\pi$
$992$	957.000			0.964718
$993$		0			0
$994$	594.000			0.597586
$995$	77.5000	−	771.115i	0.0778894	−	0.774990i
$996$		0			0
$997$		−	457.694i		−	0.459071i	−0.973300	−	0.229536i	$-0.926279\pi$
$997$		−	457.694i		−	0.459071i	0.973300	−	0.229536i	$-0.0737208\pi$
$998$	290.000			0.290581
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.3.d.f.134.1	yes	2
3.2	odd	2		135.3.d.a.134.2	yes	2
4.3	odd	2		2160.3.c.d.1889.1		2
5.2	odd	4		675.3.c.q.26.4		4
5.3	odd	4		675.3.c.q.26.1		4
5.4	even	2		135.3.d.a.134.1	✓	2
9.2	odd	6		405.3.h.h.134.1		4
9.4	even	3		405.3.h.c.269.1		4
9.5	odd	6		405.3.h.h.269.2		4
9.7	even	3		405.3.h.c.134.2		4
12.11	even	2		2160.3.c.c.1889.2		2
15.2	even	4		675.3.c.q.26.2		4
15.8	even	4		675.3.c.q.26.3		4
15.14	odd	2	inner	135.3.d.f.134.2	yes	2
20.19	odd	2		2160.3.c.c.1889.1		2
45.4	even	6		405.3.h.h.269.1		4
45.14	odd	6		405.3.h.c.269.2		4
45.29	odd	6		405.3.h.c.134.1		4
45.34	even	6		405.3.h.h.134.2		4
60.59	even	2		2160.3.c.d.1889.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.3.d.a.134.1	✓	2	5.4	even	2
135.3.d.a.134.2	yes	2	3.2	odd	2
135.3.d.f.134.1	yes	2	1.1	even	1	trivial
135.3.d.f.134.2	yes	2	15.14	odd	2	inner
405.3.h.c.134.1		4	45.29	odd	6
405.3.h.c.134.2		4	9.7	even	3
405.3.h.c.269.1		4	9.4	even	3
405.3.h.c.269.2		4	45.14	odd	6
405.3.h.h.134.1		4	9.2	odd	6
405.3.h.h.134.2		4	45.34	even	6
405.3.h.h.269.1		4	45.4	even	6
405.3.h.h.269.2		4	9.5	odd	6
675.3.c.q.26.1		4	5.3	odd	4
675.3.c.q.26.2		4	15.2	even	4
675.3.c.q.26.3		4	15.8	even	4
675.3.c.q.26.4		4	5.2	odd	4
2160.3.c.c.1889.1		2	20.19	odd	2
2160.3.c.c.1889.2		2	12.11	even	2
2160.3.c.d.1889.1		2	4.3	odd	2
2160.3.c.d.1889.2		2	60.59	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	135.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.00000 q^{4} +(0.500000 - 4.97494i) q^{5} -9.94987i q^{7} -7.00000 q^{8} +(0.500000 - 4.97494i) q^{10} +9.94987i q^{11} -19.8997i q^{13} -9.94987i q^{14} +5.00000 q^{16} +22.0000 q^{17} -4.00000 q^{19} +(-1.50000 + 14.9248i) q^{20} +9.94987i q^{22} -20.0000 q^{23} +(-24.5000 - 4.97494i) q^{25} -19.8997i q^{26} +29.8496i q^{28} +39.7995i q^{29} +29.0000 q^{31} +33.0000 q^{32} +22.0000 q^{34} +(-49.5000 - 4.97494i) q^{35} -4.00000 q^{38} +(-3.50000 + 34.8246i) q^{40} -39.7995i q^{41} +19.8997i q^{43} -29.8496i q^{44} -20.0000 q^{46} +58.0000 q^{47} -50.0000 q^{49} +(-24.5000 - 4.97494i) q^{50} +59.6992i q^{52} +31.0000 q^{53} +(49.5000 + 4.97494i) q^{55} +69.6491i q^{56} +39.7995i q^{58} -39.7995i q^{59} +44.0000 q^{61} +29.0000 q^{62} +13.0000 q^{64} +(-99.0000 - 9.94987i) q^{65} -19.8997i q^{67} -66.0000 q^{68} +(-49.5000 - 4.97494i) q^{70} +59.6992i q^{71} -89.5489i q^{73} +12.0000 q^{76} +99.0000 q^{77} -10.0000 q^{79} +(2.50000 - 24.8747i) q^{80} -39.7995i q^{82} +19.0000 q^{83} +(11.0000 - 109.449i) q^{85} +19.8997i q^{86} -69.6491i q^{88} +59.6992i q^{89} -198.000 q^{91} +60.0000 q^{92} +58.0000 q^{94} +(-2.00000 + 19.8997i) q^{95} -129.348i q^{97} -50.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 6 q^{4} + q^{5} - 14 q^{8} + q^{10} + 10 q^{16} + 44 q^{17} - 8 q^{19} - 3 q^{20} - 40 q^{23} - 49 q^{25} + 58 q^{31} + 66 q^{32} + 44 q^{34} - 99 q^{35} - 8 q^{38} - 7 q^{40} - 40 q^{46}+ \cdots - 100 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 6 * q^4 + q^5 - 14 * q^8 + q^10 + 10 * q^16 + 44 * q^17 - 8 * q^19 - 3 * q^20 - 40 * q^23 - 49 * q^25 + 58 * q^31 + 66 * q^32 + 44 * q^34 - 99 * q^35 - 8 * q^38 - 7 * q^40 - 40 * q^46 + 116 * q^47 - 100 * q^49 - 49 * q^50 + 62 * q^53 + 99 * q^55 + 88 * q^61 + 58 * q^62 + 26 * q^64 - 198 * q^65 - 132 * q^68 - 99 * q^70 + 24 * q^76 + 198 * q^77 - 20 * q^79 + 5 * q^80 + 38 * q^83 + 22 * q^85 - 396 * q^91 + 120 * q^92 + 116 * q^94 - 4 * q^95 - 100 * q^98

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