LMFDB - Embedded newform 169.6.b.b.168.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,6,Mod(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	169.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$27.1048655484$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 201x^{4} + 10512x^{2} + 65536 $ x^6 + 201x^4 + 10512x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.5
Root			$10.6486i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.6.b.b.168.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.64858i q^{2} +10.2870 q^{3} -42.7979 q^{4} -92.1784i q^{5} +88.9676i q^{6} +2.86088i q^{7} -93.3863i q^{8} -137.178 q^{9} +O(q^{10})$	\(q+8.64858i q^{2} +10.2870 q^{3} -42.7979 q^{4} -92.1784i q^{5} +88.9676i q^{6} +2.86088i q^{7} -93.3863i q^{8} -137.178 q^{9} +797.212 q^{10} +571.711i q^{11} -440.260 q^{12} -24.7426 q^{14} -948.236i q^{15} -561.873 q^{16} +855.571 q^{17} -1186.40i q^{18} +2355.90i q^{19} +3945.04i q^{20} +29.4298i q^{21} -4944.49 q^{22} -2509.66 q^{23} -960.662i q^{24} -5371.86 q^{25} -3910.88 q^{27} -122.440i q^{28} -5497.50 q^{29} +8200.89 q^{30} +144.924i q^{31} -7847.77i q^{32} +5881.17i q^{33} +7399.47i q^{34} +263.712 q^{35} +5870.95 q^{36} -515.975i q^{37} -20375.2 q^{38} -8608.21 q^{40} +13928.9i q^{41} -254.526 q^{42} -7753.58 q^{43} -24468.0i q^{44} +12644.9i q^{45} -21705.0i q^{46} +8344.77i q^{47} -5779.97 q^{48} +16798.8 q^{49} -46459.0i q^{50} +8801.22 q^{51} -5976.21 q^{53} -33823.6i q^{54} +52699.5 q^{55} +267.168 q^{56} +24235.1i q^{57} -47545.6i q^{58} +2110.84i q^{59} +40582.5i q^{60} -17394.2 q^{61} -1253.38 q^{62} -392.452i q^{63} +49892.1 q^{64} -50863.8 q^{66} +3121.05i q^{67} -36616.6 q^{68} -25816.8 q^{69} +2280.73i q^{70} -43323.3i q^{71} +12810.6i q^{72} +6808.31i q^{73} +4462.45 q^{74} -55260.1 q^{75} -100828. i q^{76} -1635.60 q^{77} -1280.80 q^{79} +51792.6i q^{80} -6896.72 q^{81} -120465. q^{82} -7148.44i q^{83} -1259.53i q^{84} -78865.2i q^{85} -67057.5i q^{86} -56552.6 q^{87} +53390.0 q^{88} +107123. i q^{89} -109360. q^{90} +107408. q^{92} +1490.82i q^{93} -72170.4 q^{94} +217163. q^{95} -80729.7i q^{96} -42456.2i q^{97} +145286. i q^{98} -78426.5i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9}+O(q^{10}) $ 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9	$ 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9} + 2582 q^{10} + 2182 q^{12} - 1586 q^{14} + 5570 q^{16} - 1816 q^{17} - 6820 q^{22} - 7248 q^{23} - 4046 q^{25} - 8552 q^{27} - 17516 q^{29} - 3434 q^{30} + 8696 q^{35} + 9356 q^{36} - 67572 q^{38} - 87018 q^{40} + 27206 q^{42} - 4064 q^{43} - 152446 q^{48} + 89142 q^{49} - 26936 q^{51} - 25140 q^{53} + 70624 q^{55} + 98574 q^{56} - 25508 q^{61} + 80680 q^{62} - 234786 q^{64} - 224708 q^{66} - 98350 q^{68} + 40224 q^{69} - 263038 q^{74} - 47096 q^{75} + 42104 q^{77} - 123744 q^{79} - 216442 q^{81} - 284520 q^{82} + 278144 q^{87} - 316020 q^{88} - 202516 q^{90} + 601728 q^{92} + 145686 q^{94} + 393120 q^{95}+O(q^{100}) $ 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9 + 2582 * q^10 + 2182 * q^12 - 1586 * q^14 + 5570 * q^16 - 1816 * q^17 - 6820 * q^22 - 7248 * q^23 - 4046 * q^25 - 8552 * q^27 - 17516 * q^29 - 3434 * q^30 + 8696 * q^35 + 9356 * q^36 - 67572 * q^38 - 87018 * q^40 + 27206 * q^42 - 4064 * q^43 - 152446 * q^48 + 89142 * q^49 - 26936 * q^51 - 25140 * q^53 + 70624 * q^55 + 98574 * q^56 - 25508 * q^61 + 80680 * q^62 - 234786 * q^64 - 224708 * q^66 - 98350 * q^68 + 40224 * q^69 - 263038 * q^74 - 47096 * q^75 + 42104 * q^77 - 123744 * q^79 - 216442 * q^81 - 284520 * q^82 + 278144 * q^87 - 316020 * q^88 - 202516 * q^90 + 601728 * q^92 + 145686 * q^94 + 393120 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	8.64858i	1.52887i	0.644703	+	0.764433i	$0.276981\pi$
$2$	8.64858i	1.52887i	−0.644703	+	0.764433i	$0.723019\pi$
$3$	10.2870	0.659909	0.329954	−	0.943997i	$-0.392967\pi$
$3$	10.2870	0.659909	0.329954	+	0.943997i	$0.392967\pi$
$4$	−42.7979	−1.33743
$5$	− 92.1784i	− 1.64894i	−0.565907	−	0.824469i	$-0.691474\pi$
$5$	− 92.1784i	− 1.64894i	0.565907	−	0.824469i	$-0.308526\pi$
$6$	88.9676i	1.00891i
$7$	2.86088i	0.0220676i	0.999939	+	0.0110338i	$0.00351224\pi$
$7$	2.86088i	0.0220676i	−0.999939	+	0.0110338i	$0.996488\pi$
$8$	− 93.3863i	− 0.515892i
$9$	−137.178	−0.564520
$10$	797.212	2.52101
$11$	571.711i	1.42461i	0.701871	+	0.712304i	$0.252348\pi$
$11$	571.711i	1.42461i	−0.701871	+	0.712304i	$0.747652\pi$
$12$	−440.260	−0.882585
$13$	0	0
$13$	0	0
$14$	−24.7426	−0.0337384
$15$	− 948.236i	− 1.08815i
$16$	−561.873	−0.548704
$17$	855.571	0.718015	0.359008	−	0.933335i	$-0.383115\pi$
$17$	855.571	0.718015	0.359008	+	0.933335i	$0.383115\pi$
$18$	− 1186.40i	− 0.863076i
$19$	2355.90i	1.49718i	0.663034	+	0.748589i	$0.269268\pi$
$19$	2355.90i	1.49718i	−0.663034	+	0.748589i	$0.730732\pi$
$20$	3945.04i	2.20535i
$21$	29.4298i	0.0145626i
$22$	−4944.49	−2.17804
$23$	−2509.66	−0.989227	−0.494613	−	0.869113i	$-0.664690\pi$
$23$	−2509.66	−0.989227	−0.494613	+	0.869113i	$0.664690\pi$
$24$	− 960.662i	− 0.340441i
$25$	−5371.86	−1.71900
$26$	0	0
$27$	−3910.88	−1.03244
$28$	− 122.440i	− 0.0295140i
$29$	−5497.50	−1.21387	−0.606933	−	0.794753i	$-0.707600\pi$
$29$	−5497.50	−1.21387	−0.606933	+	0.794753i	$0.707600\pi$
$30$	8200.89	1.66363
$31$	144.924i	0.0270854i	0.999908	+	0.0135427i	$0.00431090\pi$
$31$	144.924i	0.0270854i	−0.999908	+	0.0135427i	$0.995689\pi$
$32$	− 7847.77i	− 1.35479i
$33$	5881.17i	0.940111i
$34$	7399.47i	1.09775i
$35$	263.712	0.0363881
$36$	5870.95	0.755009
$37$	− 515.975i	− 0.0619618i	−0.999520	−	0.0309809i	$-0.990137\pi$
$37$	− 515.975i	− 0.0619618i	0.999520	−	0.0309809i	$-0.00986311\pi$
$38$	−20375.2	−2.28899
$39$	0	0
$40$	−8608.21	−0.850673
$41$	13928.9i	1.29407i	0.762462	+	0.647033i	$0.223991\pi$
$41$	13928.9i	1.29407i	−0.762462	+	0.647033i	$0.776009\pi$
$42$	−254.526	−0.0222643
$43$	−7753.58	−0.639486	−0.319743	−	0.947504i	$-0.603597\pi$
$43$	−7753.58	−0.639486	−0.319743	+	0.947504i	$0.603597\pi$
$44$	− 24468.0i	− 1.90532i
$45$	12644.9i	0.930859i
$46$	− 21705.0i	− 1.51240i
$47$	8344.77i	0.551023i	0.961298	+	0.275512i	$0.0888472\pi$
$47$	8344.77i	0.551023i	−0.961298	+	0.275512i	$0.911153\pi$
$48$	−5779.97	−0.362095
$49$	16798.8	0.999513
$50$	− 46459.0i	− 2.62812i
$51$	8801.22	0.473825
$52$	0	0
$53$	−5976.21	−0.292238	−0.146119	−	0.989267i	$-0.546678\pi$
$53$	−5976.21	−0.292238	−0.146119	+	0.989267i	$0.546678\pi$
$54$	− 33823.6i	− 1.57846i
$55$	52699.5	2.34909
$56$	267.168	0.0113845
$57$	24235.1i	0.988001i
$58$	− 47545.6i	− 1.85584i
$59$	2110.84i	0.0789451i	0.999221	+	0.0394726i	$0.0125678\pi$
$59$	2110.84i	0.0789451i	−0.999221	+	0.0394726i	$0.987432\pi$
$60$	40582.5i	1.45533i
$61$	−17394.2	−0.598522	−0.299261	−	0.954171i	$-0.596740\pi$
$61$	−17394.2	−0.598522	−0.299261	+	0.954171i	$0.596740\pi$
$62$	−1253.38	−0.0414099
$63$	− 392.452i	− 0.0124576i
$64$	49892.1	1.52259
$65$	0	0
$66$	−50863.8	−1.43730
$67$	3121.05i	0.0849402i	0.999098	+	0.0424701i	$0.0135227\pi$
$67$	3121.05i	0.0849402i	−0.999098	+	0.0424701i	$0.986477\pi$
$68$	−36616.6	−0.960298
$69$	−25816.8	−0.652800
$70$	2280.73i	0.0556326i
$71$	− 43323.3i	− 1.01994i	−0.860191	−	0.509971i	$-0.829656\pi$
$71$	− 43323.3i	− 1.01994i	0.860191	−	0.509971i	$-0.170344\pi$
$72$	12810.6i	0.291231i
$73$	6808.31i	0.149531i	0.997201	+	0.0747657i	$0.0238209\pi$
$73$	6808.31i	0.149531i	−0.997201	+	0.0747657i	$0.976179\pi$
$74$	4462.45	0.0947314
$75$	−55260.1	−1.13438
$76$	− 100828.i	− 2.00238i
$77$	−1635.60	−0.0314377
$78$	0	0
$79$	−1280.80	−0.0230895	−0.0115447	−	0.999933i	$-0.503675\pi$
$79$	−1280.80	−0.0230895	−0.0115447	+	0.999933i	$0.503675\pi$
$80$	51792.6i	0.904779i
$81$	−6896.72	−0.116797
$82$	−120465.	−1.97846
$83$	− 7148.44i	− 0.113898i	−0.998377	−	0.0569490i	$-0.981863\pi$
$83$	− 7148.44i	− 0.113898i	0.998377	−	0.0569490i	$-0.0181372\pi$
$84$	− 1259.53i	− 0.0194765i
$85$	− 78865.2i	− 1.18396i
$86$	− 67057.5i	− 0.977690i
$87$	−56552.6	−0.801041
$88$	53390.0	0.734943
$89$	107123.i	1.43353i	0.697315	+	0.716765i	$0.254378\pi$
$89$	107123.i	1.43353i	−0.697315	+	0.716765i	$0.745622\pi$
$90$	−109360.	−1.42316
$91$	0	0
$92$	107408.	1.32303
$93$	1490.82i	0.0178739i
$94$	−72170.4	−0.842441
$95$	217163.	2.46875
$96$	− 80729.7i	− 0.894036i
$97$	− 42456.2i	− 0.458155i	−0.973408	−	0.229077i	$-0.926429\pi$
$97$	− 42456.2i	− 0.458155i	0.973408	−	0.229077i	$-0.0735709\pi$
$98$	145286.i	1.52812i
$99$	− 78426.5i	− 0.804220i
$100$	229904.	2.29904
$101$	87947.3	0.857866	0.428933	−	0.903336i	$-0.358890\pi$
$101$	87947.3	0.857866	0.428933	+	0.903336i	$0.358890\pi$
$102$	76118.1i	0.724415i
$103$	−96885.8	−0.899844	−0.449922	−	0.893068i	$-0.648548\pi$
$103$	−96885.8	−0.899844	−0.449922	+	0.893068i	$0.648548\pi$
$104$	0	0
$105$	2712.79	0.0240128
$106$	− 51685.7i	− 0.446792i
$107$	106509.	0.899346	0.449673	−	0.893193i	$-0.351540\pi$
$107$	106509.	0.899346	0.449673	+	0.893193i	$0.351540\pi$
$108$	167377.	1.38082
$109$	142520.i	1.14897i	0.818515	+	0.574485i	$0.194798\pi$
$109$	142520.i	1.14897i	−0.818515	+	0.574485i	$0.805202\pi$
$110$	455775.i	3.59144i
$111$	− 5307.81i	− 0.0408892i
$112$	− 1607.45i	− 0.0121086i
$113$	−209540.	−1.54373	−0.771864	−	0.635787i	$-0.780676\pi$
$113$	−209540.	−1.54373	−0.771864	+	0.635787i	$0.780676\pi$
$114$	−209599.	−1.51052
$115$	231337.i	1.63117i
$116$	235282.	1.62346
$117$	0	0
$118$	−18255.8	−0.120697
$119$	2447.69i	0.0158449i
$120$	−88552.3	−0.561367
$121$	−165803.	−1.02951
$122$	− 150435.i	− 0.915061i
$123$	143286.i	0.853966i
$124$	− 6202.42i	− 0.0362249i
$125$	207112.i	1.18558i
$126$	3394.15	0.0190460
$127$	54468.2	0.299663	0.149832	−	0.988712i	$-0.452127\pi$
$127$	54468.2	0.299663	0.149832	+	0.988712i	$0.452127\pi$
$128$	180367.i	0.973043i
$129$	−79760.8	−0.422003
$130$	0	0
$131$	258252.	1.31482	0.657408	−	0.753535i	$-0.271653\pi$
$131$	258252.	1.31482	0.657408	+	0.753535i	$0.271653\pi$
$132$	− 251702.i	− 1.25734i
$133$	−6739.97	−0.0330391
$134$	−26992.6	−0.129862
$135$	360499.i	1.70243i
$136$	− 79898.6i	− 0.370418i
$137$	293754.i	1.33716i	0.743641	+	0.668580i	$0.233097\pi$
$137$	293754.i	1.33716i	−0.743641	+	0.668580i	$0.766903\pi$
$138$	− 223279.i	− 0.998044i
$139$	180400.	0.791955	0.395977	−	0.918260i	$-0.370406\pi$
$139$	180400.	0.791955	0.395977	+	0.918260i	$0.370406\pi$
$140$	−11286.3	−0.0486667
$141$	85842.4i	0.363625i
$142$	374685.	1.55936
$143$	0	0
$144$	77076.9	0.309755
$145$	506751.i	2.00159i
$146$	−58882.2	−0.228614
$147$	172809.	0.659588
$148$	22082.6i	0.0828698i
$149$	− 516009.i	− 1.90411i	−0.305930	−	0.952054i	$-0.598968\pi$
$149$	− 516009.i	− 1.90411i	0.305930	−	0.952054i	$-0.401032\pi$
$150$	− 477922.i	− 1.73432i
$151$	− 333770.i	− 1.19126i	−0.803260	−	0.595628i	$-0.796903\pi$
$151$	− 333770.i	− 1.19126i	0.803260	−	0.595628i	$-0.203097\pi$
$152$	220009.	0.772381
$153$	−117366.	−0.405334
$154$	− 14145.6i	− 0.0480640i
$155$	13358.8	0.0446621
$156$	0	0
$157$	−265860.	−0.860804	−0.430402	−	0.902637i	$-0.641628\pi$
$157$	−265860.	−0.860804	−0.430402	+	0.902637i	$0.641628\pi$
$158$	− 11077.1i	− 0.0353007i
$159$	−61477.0	−0.192850
$160$	−723395.	−2.23396
$161$	− 7179.86i	− 0.0218299i
$162$	− 59646.8i	− 0.178566i
$163$	− 231417.i	− 0.682223i	−0.940023	−	0.341112i	$-0.889197\pi$
$163$	− 231417.i	− 0.682223i	0.940023	−	0.341112i	$-0.110803\pi$
$164$	− 596127.i	− 1.73073i
$165$	542117.	1.55018
$166$	61823.8	0.174135
$167$	− 656226.i	− 1.82080i	−0.413730	−	0.910400i	$-0.635774\pi$
$167$	− 656226.i	− 1.82080i	0.413730	−	0.910400i	$-0.364226\pi$
$168$	2748.34	0.00751273
$169$	0	0
$170$	682071.	1.81012
$171$	− 323179.i	− 0.845187i
$172$	331837.	0.855271
$173$	−45111.6	−0.114597	−0.0572985	−	0.998357i	$-0.518249\pi$
$173$	−45111.6	−0.114597	−0.0572985	+	0.998357i	$0.518249\pi$
$174$	− 489100.i	− 1.22468i
$175$	− 15368.3i	− 0.0379341i
$176$	− 321229.i	− 0.781688i
$177$	21714.1i	0.0520966i
$178$	−926460.	−2.19168
$179$	−521846.	−1.21733	−0.608667	−	0.793426i	$-0.708295\pi$
$179$	−521846.	−1.21733	−0.608667	+	0.793426i	$0.708295\pi$
$180$	− 541175.i	− 1.24496i
$181$	122780.	0.278569	0.139284	−	0.990252i	$-0.455520\pi$
$181$	122780.	0.278569	0.139284	+	0.990252i	$0.455520\pi$
$182$	0	0
$183$	−178934.	−0.394970
$184$	234368.i	0.510334i
$185$	−47561.7	−0.102171
$186$	−12893.5	−0.0273268
$187$	489140.i	1.02289i
$188$	− 357139.i	− 0.736957i
$189$	− 11188.6i	− 0.0227835i
$190$	1.87815e6i	3.77440i
$191$	350208.	0.694613	0.347306	−	0.937752i	$-0.387096\pi$
$191$	350208.	0.694613	0.347306	+	0.937752i	$0.387096\pi$
$192$	513238.	1.00477
$193$	876316.i	1.69343i	0.532046	+	0.846716i	$0.321423\pi$
$193$	876316.i	1.69343i	−0.532046	+	0.846716i	$0.678577\pi$
$194$	367186.	0.700457
$195$	0	0
$196$	−718954.	−1.33678
$197$	− 306582.i	− 0.562835i	−0.959585	−	0.281418i	$-0.909195\pi$
$197$	− 306582.i	− 0.562835i	0.959585	−	0.281418i	$-0.0908046\pi$
$198$	678277.	1.22954
$199$	327849.	0.586869	0.293434	−	0.955979i	$-0.405202\pi$
$199$	327849.	0.586869	0.293434	+	0.955979i	$0.405202\pi$
$200$	501659.i	0.886815i
$201$	32106.1i	0.0560528i
$202$	760619.i	1.31156i
$203$	− 15727.7i	− 0.0267871i
$204$	−376674.	−0.633709
$205$	1.28394e6	2.13384
$206$	− 837924.i	− 1.37574i
$207$	344272.	0.558439
$208$	0	0
$209$	−1.34690e6	−2.13289
$210$	23461.8i	0.0367124i
$211$	−509483.	−0.787814	−0.393907	−	0.919150i	$-0.628877\pi$
$211$	−509483.	−0.787814	−0.393907	+	0.919150i	$0.628877\pi$
$212$	255769.	0.390849
$213$	− 445665.i	− 0.673069i
$214$	921151.i	1.37498i
$215$	714713.i	1.05447i
$216$	365223.i	0.532627i
$217$	−414.610	−0.000597709 0
$218$	−1.23259e6	−1.75662
$219$	70036.8i	0.0986771i
$220$	−2.25543e6	−3.14175
$221$	0	0
$222$	45905.0	0.0625141
$223$	15061.4i	0.0202816i	0.999949	+	0.0101408i	$0.00322798\pi$
$223$	15061.4i	0.0202816i	−0.999949	+	0.0101408i	$0.996772\pi$
$224$	22451.6	0.0298969
$225$	736904.	0.970408
$226$	− 1.81222e6i	− 2.36016i
$227$	− 273713.i	− 0.352558i	−0.984340	−	0.176279i	$-0.943594\pi$
$227$	− 273713.i	− 0.352558i	0.984340	−	0.176279i	$-0.0564061\pi$
$228$	− 1.03721e6i	− 1.32139i
$229$	743056.i	0.936339i	0.883639	+	0.468169i	$0.155086\pi$
$229$	743056.i	0.936339i	−0.883639	+	0.468169i	$0.844914\pi$
$230$	−2.00073e6	−2.49385
$231$	−16825.4	−0.0207460
$232$	513392.i	0.626223i
$233$	−1.12122e6	−1.35301	−0.676506	−	0.736437i	$-0.736507\pi$
$233$	−1.12122e6	−1.35301	−0.676506	+	0.736437i	$0.736507\pi$
$234$	0	0
$235$	769208.	0.908603
$236$	− 90339.5i	− 0.105584i
$237$	−13175.6	−0.0152369
$238$	−21169.0	−0.0242247
$239$	− 279838.i	− 0.316892i	−0.987368	−	0.158446i	$-0.949352\pi$
$239$	− 279838.i	− 0.316892i	0.987368	−	0.158446i	$-0.0506484\pi$
$240$	532788.i	0.597072i
$241$	816889.i	0.905983i	0.891515	+	0.452992i	$0.149643\pi$
$241$	816889.i	0.905983i	−0.891515	+	0.452992i	$0.850357\pi$
$242$	− 1.43396e6i	− 1.57398i
$243$	879398.	0.955366
$244$	744436.	0.800484
$245$	− 1.54849e6i	− 1.64813i
$246$	−1.23922e6	−1.30560
$247$	0	0
$248$	13533.9	0.0139731
$249$	− 73535.7i	− 0.0751623i
$250$	−1.79123e6	−1.81259
$251$	766697.	0.768139	0.384069	−	0.923304i	$-0.374522\pi$
$251$	766697.	0.768139	0.384069	+	0.923304i	$0.374522\pi$
$252$	16796.1i	0.0166612i
$253$	− 1.43480e6i	− 1.40926i
$254$	471072.i	0.458145i
$255$	− 811283.i	− 0.781307i
$256$	36629.4	0.0349325
$257$	1.82413e6	1.72276	0.861379	−	0.507963i	$-0.169601\pi$
$257$	1.82413e6	1.72276	0.861379	+	0.507963i	$0.169601\pi$
$258$	− 689817.i	− 0.645186i
$259$	1476.14	0.00136735
$260$	0	0
$261$	754139.	0.685252
$262$	2.23351e6i	2.01018i
$263$	−294257.	−0.262323	−0.131162	−	0.991361i	$-0.541871\pi$
$263$	−294257.	−0.262323	−0.131162	+	0.991361i	$0.541871\pi$
$264$	549221.	0.484995
$265$	550878.i	0.481882i
$266$	− 58291.1i	− 0.0505124i
$267$	1.10197e6i	0.945999i
$268$	− 133574.i	− 0.113602i
$269$	289738.	0.244132	0.122066	−	0.992522i	$-0.461048\pi$
$269$	289738.	0.244132	0.122066	+	0.992522i	$0.461048\pi$
$270$	−3.11780e6	−2.60279
$271$	1.63179e6i	1.34971i	0.737950	+	0.674855i	$0.235794\pi$
$271$	1.63179e6i	1.34971i	−0.737950	+	0.674855i	$0.764206\pi$
$272$	−480722.	−0.393978
$273$	0	0
$274$	−2.54056e6	−2.04434
$275$	− 3.07116e6i	− 2.44889i
$276$	1.10490e6	0.873076
$277$	−398868.	−0.312341	−0.156171	−	0.987730i	$-0.549915\pi$
$277$	−398868.	−0.312341	−0.156171	+	0.987730i	$0.549915\pi$
$278$	1.56021e6i	1.21079i
$279$	− 19880.4i	− 0.0152902i
$280$	− 24627.1i	− 0.0187723i
$281$	274611.i	0.207468i	0.994605	+	0.103734i	$0.0330791\pi$
$281$	274611.i	0.207468i	−0.994605	+	0.103734i	$0.966921\pi$
$282$	−742414.	−0.555934
$283$	794683.	0.589831	0.294916	−	0.955523i	$-0.404708\pi$
$283$	794683.	0.589831	0.294916	+	0.955523i	$0.404708\pi$
$284$	1.85415e6i	1.36411i
$285$	2.23395e6	1.62915
$286$	0	0
$287$	−39848.9	−0.0285570
$288$	1.07654e6i	0.764805i
$289$	−687856.	−0.484454
$290$	−4.38268e6	−3.06016
$291$	− 436746.i	− 0.302340i
$292$	− 291381.i	− 0.199988i
$293$	− 1.55323e6i	− 1.05698i	−0.848940	−	0.528490i	$-0.822759\pi$
$293$	− 1.55323e6i	− 1.05698i	0.848940	−	0.528490i	$-0.177241\pi$
$294$	1.49455e6i	1.00842i
$295$	194574.	0.130176
$296$	−48185.0	−0.0319656
$297$	− 2.23590e6i	− 1.47082i
$298$	4.46274e6	2.91113
$299$	0	0
$300$	2.36502e6	1.51716
$301$	− 22182.1i	− 0.0141119i
$302$	2.88664e6	1.82127
$303$	904711.	0.566113
$304$	− 1.32372e6i	− 0.821508i
$305$	1.60337e6i	0.986926i
$306$	− 1.01505e6i	− 0.619702i
$307$	2.07136e6i	1.25433i	0.778888	+	0.627163i	$0.215784\pi$
$307$	2.07136e6i	1.25433i	−0.778888	+	0.627163i	$0.784216\pi$
$308$	70000.2	0.0420458
$309$	−996660.	−0.593815
$310$	115535.i	0.0682824i
$311$	2.50827e6	1.47053	0.735263	−	0.677782i	$-0.237059\pi$
$311$	2.50827e6	1.47053	0.735263	+	0.677782i	$0.237059\pi$
$312$	0	0
$313$	832384.	0.480245	0.240123	−	0.970743i	$-0.422812\pi$
$313$	832384.	0.480245	0.240123	+	0.970743i	$0.422812\pi$
$314$	− 2.29931e6i	− 1.31606i
$315$	−36175.6	−0.0205418
$316$	54815.6	0.0308806
$317$	1.76537e6i	0.986708i	0.869829	+	0.493354i	$0.164229\pi$
$317$	1.76537e6i	0.986708i	−0.869829	+	0.493354i	$0.835771\pi$
$318$	− 531689.i	− 0.294842i
$319$	− 3.14299e6i	− 1.72928i
$320$	− 4.59897e6i	− 2.51065i
$321$	1.09565e6	0.593487
$322$	62095.5	0.0333750
$323$	2.01564e6i	1.07500i
$324$	295165.	0.156208
$325$	0	0
$326$	2.00143e6	1.04303
$327$	1.46609e6i	0.758215i
$328$	1.30077e6	0.667598
$329$	−23873.4	−0.0121598
$330$	4.68854e6i	2.37003i
$331$	285068.i	0.143014i	0.997440	+	0.0715070i	$0.0227808\pi$
$331$	285068.i	0.143014i	−0.997440	+	0.0715070i	$0.977219\pi$
$332$	305938.i	0.152331i
$333$	70780.6i	0.0349787i
$334$	5.67542e6	2.78376
$335$	287693.	0.140061
$336$	− 16535.8i	− 0.00799057i
$337$	−1.96932e6	−0.944584	−0.472292	−	0.881442i	$-0.656573\pi$
$337$	−1.96932e6	−0.944584	−0.472292	+	0.881442i	$0.656573\pi$
$338$	0	0
$339$	−2.15553e6	−1.01872
$340$	3.37526e6i	1.58347i
$341$	−82854.5	−0.0385860
$342$	2.79504e6	1.29218
$343$	96142.4i	0.0441245i
$344$	724079.i	0.329906i
$345$	2.37975e6i	1.07643i
$346$	− 390151.i	− 0.175204i
$347$	1.89753e6	0.845988	0.422994	−	0.906133i	$-0.360979\pi$
$347$	1.89753e6	0.845988	0.422994	+	0.906133i	$0.360979\pi$
$348$	2.42033e6	1.07134
$349$	− 365808.i	− 0.160764i	−0.996764	−	0.0803822i	$-0.974386\pi$
$349$	− 365808.i	− 0.160764i	0.996764	−	0.0803822i	$-0.0256141\pi$
$350$	132914.	0.0579962
$351$	0	0
$352$	4.48666e6	1.93004
$353$	− 1.56672e6i	− 0.669196i	−0.942361	−	0.334598i	$-0.891399\pi$
$353$	− 1.56672e6i	− 0.669196i	0.942361	−	0.334598i	$-0.108601\pi$
$354$	−187796.	−0.0796487
$355$	−3.99348e6	−1.68182
$356$	− 4.58463e6i	− 1.91725i
$357$	25179.3i	0.0104562i
$358$	− 4.51322e6i	− 1.86114i
$359$	− 4.03307e6i	− 1.65158i	−0.563979	−	0.825789i	$-0.690730\pi$
$359$	− 4.03307e6i	− 1.65158i	0.563979	−	0.825789i	$-0.309270\pi$
$360$	1.18086e6	0.480222
$361$	−3.07418e6	−1.24154
$362$	1.06187e6i	0.425894i
$363$	−1.70561e6	−0.679380
$364$	0	0
$365$	627579.	0.246568
$366$	− 1.54752e6i	− 0.603857i
$367$	683844.	0.265028	0.132514	−	0.991181i	$-0.457695\pi$
$367$	683844.	0.265028	0.132514	+	0.991181i	$0.457695\pi$
$368$	1.41011e6	0.542793
$369$	− 1.91074e6i	− 0.730527i
$370$	− 411341.i	− 0.156206i
$371$	− 17097.2i	− 0.00644899i
$372$	− 63804.1i	− 0.0239051i
$373$	−3.09158e6	−1.15056	−0.575279	−	0.817957i	$-0.695107\pi$
$373$	−3.09158e6	−1.15056	−0.575279	+	0.817957i	$0.695107\pi$
$374$	−4.23036e6	−1.56386
$375$	2.13056e6i	0.782374i
$376$	779288.	0.284268
$377$	0	0
$378$	96765.3	0.0348329
$379$	5.24600e6i	1.87599i	0.346649	+	0.937995i	$0.387319\pi$
$379$	5.24600e6i	1.87599i	−0.346649	+	0.937995i	$0.612681\pi$
$380$	−9.29414e6	−3.30179
$381$	560312.	0.197751
$382$	3.02880e6i	1.06197i
$383$	− 1.57009e6i	− 0.546926i	−0.961883	−	0.273463i	$-0.911831\pi$
$383$	− 1.57009e6i	− 0.546926i	0.961883	−	0.273463i	$-0.0881691\pi$
$384$	1.85543e6i	0.642120i
$385$	150767.i	0.0518388i
$386$	−7.57889e6	−2.58903
$387$	1.06362e6	0.361003
$388$	1.81704e6i	0.612751i
$389$	−1.54482e6	−0.517611	−0.258805	−	0.965929i	$-0.583329\pi$
$389$	−1.54482e6	−0.517611	−0.258805	+	0.965929i	$0.583329\pi$
$390$	0	0
$391$	−2.14719e6	−0.710280
$392$	− 1.56878e6i	− 0.515640i
$393$	2.65663e6	0.867659
$394$	2.65150e6	0.860500
$395$	118062.i	0.0380731i
$396$	3.35649e6i	1.07559i
$397$	5.06711e6i	1.61356i	0.590855	+	0.806778i	$0.298790\pi$
$397$	5.06711e6i	1.61356i	−0.590855	+	0.806778i	$0.701210\pi$
$398$	2.83543e6i	0.897244i
$399$	−69333.8	−0.0218028
$400$	3.01831e6	0.943221
$401$	− 138120.i	− 0.0428939i	−0.999770	−	0.0214469i	$-0.993173\pi$
$401$	− 138120.i	− 0.0428939i	0.999770	−	0.0214469i	$-0.00682729\pi$
$402$	−277672.	−0.0856973
$403$	0	0
$404$	−3.76396e6	−1.14734
$405$	635729.i	0.192590i
$406$	136022.	0.0409539
$407$	294989.	0.0882713
$408$	− 821914.i	− 0.244442i
$409$	4.13317e6i	1.22173i	0.791735	+	0.610865i	$0.209178\pi$
$409$	4.13317e6i	1.22173i	−0.791735	+	0.610865i	$0.790822\pi$
$410$	1.11043e7i	3.26235i
$411$	3.02184e6i	0.882403i
$412$	4.14651e6	1.20348
$413$	−6038.87	−0.00174213
$414$	2.97746e6i	0.853778i
$415$	−658932.	−0.187811
$416$	0	0
$417$	1.85577e6	0.522618
$418$	− 1.16487e7i	− 3.26091i
$419$	−2.96703e6	−0.825633	−0.412817	−	0.910814i	$-0.635455\pi$
$419$	−2.96703e6	−0.825633	−0.412817	+	0.910814i	$0.635455\pi$
$420$	−116102.	−0.0321156
$421$	2.98125e6i	0.819773i	0.912137	+	0.409887i	$0.134432\pi$
$421$	2.98125e6i	0.819773i	−0.912137	+	0.409887i	$0.865568\pi$
$422$	− 4.40630e6i	− 1.20446i
$423$	− 1.14472e6i	− 0.311064i
$424$	558096.i	0.150763i
$425$	−4.59601e6	−1.23426
$426$	3.85437e6	1.02903
$427$	− 49762.8i	− 0.0132080i
$428$	−4.55836e6	−1.20282
$429$	0	0
$430$	−6.18125e6	−1.61215
$431$	3.34790e6i	0.868119i	0.900884	+	0.434059i	$0.142919\pi$
$431$	3.34790e6i	0.868119i	−0.900884	+	0.434059i	$0.857081\pi$
$432$	2.19742e6	0.566505
$433$	3.76782e6	0.965764	0.482882	−	0.875686i	$-0.339590\pi$
$433$	3.76782e6	0.965764	0.482882	+	0.875686i	$0.339590\pi$
$434$	− 3585.78i	0 0.000913818i
$435$	5.21293e6i	1.32087i
$436$	− 6.09954e6i	− 1.53667i
$437$	− 5.91252e6i	− 1.48105i
$438$	−605719.	−0.150864
$439$	−5.63075e6	−1.39446	−0.697228	−	0.716849i	$-0.745584\pi$
$439$	−5.63075e6	−1.39446	−0.697228	+	0.716849i	$0.745584\pi$
$440$	− 4.92141e6i	− 1.21188i
$441$	−2.30444e6	−0.564245
$442$	0	0
$443$	3.69803e6	0.895284	0.447642	−	0.894213i	$-0.352264\pi$
$443$	3.69803e6	0.895284	0.447642	+	0.894213i	$0.352264\pi$
$444$	227163.i	0.0546865i
$445$	9.87441e6	2.36380
$446$	−130260.	−0.0310079
$447$	− 5.30816e6i	− 1.25654i
$448$	142735.i	0.0335998i
$449$	− 505514.i	− 0.118336i	−0.998248	−	0.0591681i	$-0.981155\pi$
$449$	− 505514.i	− 0.118336i	0.998248	−	0.0591681i	$-0.0188448\pi$
$450$	6.37317e6i	1.48362i
$451$	−7.96330e6	−1.84354
$452$	8.96787e6	2.06464
$453$	− 3.43348e6i	− 0.786121i
$454$	2.36723e6	0.539015
$455$	0	0
$456$	2.26323e6	0.509701
$457$	− 783627.i	− 0.175517i	−0.996142	−	0.0877584i	$-0.972030\pi$
$457$	− 783627.i	− 0.175517i	0.996142	−	0.0877584i	$-0.0279704\pi$
$458$	−6.42638e6	−1.43154
$459$	−3.34603e6	−0.741308
$460$	− 9.90073e6i	− 2.18159i
$461$	− 4.64856e6i	− 1.01875i	−0.860546	−	0.509373i	$-0.829877\pi$
$461$	− 4.64856e6i	− 1.01875i	0.860546	−	0.509373i	$-0.170123\pi$
$462$	− 145515.i	− 0.0317179i
$463$	2.64872e6i	0.574226i	0.957897	+	0.287113i	$0.0926956\pi$
$463$	2.64872e6i	0.574226i	−0.957897	+	0.287113i	$0.907304\pi$
$464$	3.08890e6	0.666053
$465$	137422.	0.0294729
$466$	− 9.69698e6i	− 2.06858i
$467$	2.42172e6	0.513845	0.256923	−	0.966432i	$-0.417291\pi$
$467$	2.42172e6	0.513845	0.256923	+	0.966432i	$0.417291\pi$
$468$	0	0
$469$	−8928.96	−0.00187443
$470$	6.65255e6i	1.38913i
$471$	−2.73490e6	−0.568052
$472$	197124.	0.0407271
$473$	− 4.43281e6i	− 0.911017i
$474$	− 113950.i	− 0.0232953i
$475$	− 1.26556e7i	− 2.57364i
$476$	− 104756.i	− 0.0211915i
$477$	819807.	0.164974
$478$	2.42020e6	0.484486
$479$	5.58315e6i	1.11184i	0.831237	+	0.555918i	$0.187633\pi$
$479$	5.58315e6i	1.11184i	−0.831237	+	0.555918i	$0.812367\pi$
$480$	−7.44154e6	−1.47421
$481$	0	0
$482$	−7.06492e6	−1.38513
$483$	− 73858.9i	− 0.0144057i
$484$	7.09602e6	1.37690
$485$	−3.91355e6	−0.755468
$486$	7.60554e6i	1.46063i
$487$	− 7.25183e6i	− 1.38556i	−0.721149	−	0.692780i	$-0.756386\pi$
$487$	− 7.25183e6i	− 1.38556i	0.721149	−	0.692780i	$-0.243614\pi$
$488$	1.62438e6i	0.308773i
$489$	− 2.38058e6i	− 0.450205i
$490$	1.33922e7	2.51978
$491$	−4.17410e6	−0.781373	−0.390687	−	0.920524i	$-0.627762\pi$
$491$	−4.17410e6	−0.781373	−0.390687	+	0.920524i	$0.627762\pi$
$492$	− 6.13233e6i	− 1.14212i
$493$	−4.70350e6	−0.871574
$494$	0	0
$495$	−7.22923e6	−1.32611
$496$	− 81428.7i	− 0.0148619i
$497$	123943.	0.0225077
$498$	635979.	0.114913
$499$	7.83551e6i	1.40869i	0.709857	+	0.704345i	$0.248759\pi$
$499$	7.83551e6i	1.40869i	−0.709857	+	0.704345i	$0.751241\pi$
$500$	− 8.86396e6i	− 1.58563i
$501$	− 6.75057e6i	− 1.20156i
$502$	6.63084e6i	1.17438i
$503$	−3.72420e6	−0.656315	−0.328158	−	0.944623i	$-0.606428\pi$
$503$	−3.72420e6	−0.656315	−0.328158	+	0.944623i	$0.606428\pi$
$504$	−36649.6	−0.00642678
$505$	− 8.10685e6i	− 1.41457i
$506$	1.24090e7	2.15457
$507$	0	0
$508$	−2.33112e6	−0.400780
$509$	6.76675e6i	1.15767i	0.815444	+	0.578836i	$0.196493\pi$
$509$	6.76675e6i	1.15767i	−0.815444	+	0.578836i	$0.803507\pi$
$510$	7.01644e6	1.19451
$511$	−19477.8	−0.00329980
$512$	6.08853e6i	1.02645i
$513$	− 9.21366e6i	− 1.54575i
$514$	1.57762e7i	2.63387i
$515$	8.93078e6i	1.48379i
$516$	3.41359e6	0.564401
$517$	−4.77080e6	−0.784992
$518$	12766.5i	0.00209049i
$519$	−464062.	−0.0756236
$520$	0	0
$521$	9.08380e6	1.46613	0.733066	−	0.680157i	$-0.238088\pi$
$521$	9.08380e6	1.46613	0.733066	+	0.680157i	$0.238088\pi$
$522$	6.52223e6i	1.04766i
$523$	−1.09284e7	−1.74703	−0.873515	−	0.486796i	$-0.838165\pi$
$523$	−1.09284e7	−1.74703	−0.873515	+	0.486796i	$0.838165\pi$
$524$	−1.10526e7	−1.75848
$525$	− 158093.i	− 0.0250331i
$526$	− 2.54490e6i	− 0.401057i
$527$	123992.i	0.0194477i
$528$	− 3.30447e6i	− 0.515843i
$529$	−137934.	−0.0214305
$530$	−4.76431e6	−0.736733
$531$	− 289562.i	− 0.0445661i
$532$	288456.	0.0441877
$533$	0	0
$534$	−9.53045e6	−1.44631
$535$	− 9.81783e6i	− 1.48297i
$536$	291463.	0.0438200
$537$	−5.36821e6	−0.803329
$538$	2.50582e6i	0.373245i
$539$	9.60408e6i	1.42391i
$540$	− 1.54286e7i	− 2.27689i
$541$	1.17853e6i	0.173120i	0.996247	+	0.0865598i	$0.0275874\pi$
$541$	1.17853e6i	0.173120i	−0.996247	+	0.0865598i	$0.972413\pi$
$542$	−1.41126e7	−2.06353
$543$	1.26304e6	0.183830
$544$	− 6.71432e6i	− 0.972758i
$545$	1.31372e7	1.89458
$546$	0	0
$547$	9.57702e6	1.36855	0.684277	−	0.729222i	$-0.260118\pi$
$547$	9.57702e6	1.36855	0.684277	+	0.729222i	$0.260118\pi$
$548$	− 1.25721e7i	− 1.78836i
$549$	2.38611e6	0.337878
$550$	2.65611e7	3.74403
$551$	− 1.29516e7i	− 1.81737i
$552$	2.41094e6i	0.336774i
$553$	− 3664.22i	0 0.000509529i
$554$	− 3.44964e6i	− 0.477528i
$555$	−489266.	−0.0674237
$556$	−7.72075e6	−1.05919
$557$	2.46873e6i	0.337159i	0.985688	+	0.168580i	$0.0539181\pi$
$557$	2.46873e6i	0.337159i	−0.985688	+	0.168580i	$0.946082\pi$
$558$	171937.	0.0233767
$559$	0	0
$560$	−148173.	−0.0199663
$561$	5.03176e6i	0.675014i
$562$	−2.37499e6	−0.317191
$563$	−7.81681e6	−1.03934	−0.519671	−	0.854367i	$-0.673945\pi$
$563$	−7.81681e6	−1.03934	−0.519671	+	0.854367i	$0.673945\pi$
$564$	− 3.67387e6i	− 0.486325i
$565$	1.93151e7i	2.54551i
$566$	6.87288e6i	0.901774i
$567$	− 19730.7i	− 0.00257742i
$568$	−4.04581e6	−0.526180
$569$	9.64303e6	1.24863	0.624314	−	0.781174i	$-0.285379\pi$
$569$	9.64303e6	1.24863	0.624314	+	0.781174i	$0.285379\pi$
$570$	1.93205e7i	2.49076i
$571$	−1.12378e7	−1.44242	−0.721211	−	0.692715i	$-0.756414\pi$
$571$	−1.12378e7	−1.44242	−0.721211	+	0.692715i	$0.756414\pi$
$572$	0	0
$573$	3.60258e6	0.458381
$574$	− 344636.i	− 0.0436598i
$575$	1.34816e7	1.70048
$576$	−6.84412e6	−0.859530
$577$	5.44674e6i	0.681078i	0.940230	+	0.340539i	$0.110610\pi$
$577$	5.44674e6i	0.681078i	−0.940230	+	0.340539i	$0.889390\pi$
$578$	− 5.94897e6i	− 0.740666i
$579$	9.01463e6i	1.11751i
$580$	− 2.16879e7i	− 2.67699i
$581$	20450.9	0.00251346
$582$	3.77723e6	0.462238
$583$	− 3.41667e6i	− 0.416324i
$584$	635803.	0.0771420
$585$	0	0
$586$	1.34332e7	1.61598
$587$	− 9.96240e6i	− 1.19335i	−0.802482	−	0.596676i	$-0.796488\pi$
$587$	− 9.96240e6i	− 1.19335i	0.802482	−	0.596676i	$-0.203512\pi$
$588$	−7.39585e6	−0.882155
$589$	−341426.	−0.0405516
$590$	1.68279e6i	0.199021i
$591$	− 3.15380e6i	− 0.371420i
$592$	289912.i	0.0339987i
$593$	− 3.17929e6i	− 0.371273i	−0.982619	−	0.185636i	$-0.940565\pi$
$593$	− 3.17929e6i	− 0.371273i	0.982619	−	0.185636i	$-0.0594346\pi$
$594$	1.93373e7	2.24869
$595$	225624.	0.0261272
$596$	2.20841e7i	2.54662i
$597$	3.37257e6	0.387280
$598$	0	0
$599$	−1.66017e7	−1.89054	−0.945271	−	0.326285i	$-0.894203\pi$
$599$	−1.66017e7	−1.89054	−0.945271	+	0.326285i	$0.894203\pi$
$600$	5.16054e6i	0.585217i
$601$	1.05771e6	0.119449	0.0597244	−	0.998215i	$-0.480978\pi$
$601$	1.05771e6	0.119449	0.0597244	+	0.998215i	$0.480978\pi$
$602$	191844.	0.0215753
$603$	− 428141.i	− 0.0479505i
$604$	1.42847e7i	1.59323i
$605$	1.52835e7i	1.69759i
$606$	7.82446e6i	0.865512i
$607$	2.42026e6	0.266619	0.133309	−	0.991074i	$-0.457440\pi$
$607$	2.42026e6	0.266619	0.133309	+	0.991074i	$0.457440\pi$
$608$	1.84886e7	2.02836
$609$	− 161790.i	− 0.0176770i
$610$	−1.38669e7	−1.50888
$611$	0	0
$612$	5.02301e6	0.542108
$613$	− 110410.i	− 0.0118675i	−0.999982	−	0.00593373i	$-0.998111\pi$
$613$	− 110410.i	− 0.0118675i	0.999982	−	0.00593373i	$-0.00188877\pi$
$614$	−1.79144e7	−1.91770
$615$	1.32079e7	1.40814
$616$	152743.i	0.0162184i
$617$	1.06544e7i	1.12672i	0.826211	+	0.563361i	$0.190492\pi$
$617$	1.06544e7i	1.12672i	−0.826211	+	0.563361i	$0.809508\pi$
$618$	− 8.61969e6i	− 0.907864i
$619$	− 1.30456e7i	− 1.36848i	−0.729256	−	0.684240i	$-0.760134\pi$
$619$	− 1.30456e7i	− 1.36848i	0.729256	−	0.684240i	$-0.239866\pi$
$620$	−571730.	−0.0597326
$621$	9.81499e6	1.02132
$622$	2.16929e7i	2.24824i
$623$	−306466.	−0.0316346
$624$	0	0
$625$	2.30421e6	0.235951
$626$	7.19894e6i	0.734231i
$627$	−1.38555e7	−1.40751
$628$	1.13783e7	1.15127
$629$	− 441453.i	− 0.0444895i
$630$	− 312867.i	− 0.0314057i
$631$	− 1.53156e6i	− 0.153130i	−0.997065	−	0.0765651i	$-0.975605\pi$
$631$	− 1.53156e6i	− 0.153130i	0.997065	−	0.0765651i	$-0.0243953\pi$
$632$	119609.i	0.0119117i
$633$	−5.24103e6	−0.519885
$634$	−1.52680e7	−1.50855
$635$	− 5.02079e6i	− 0.494126i
$636$	2.63109e6	0.257924
$637$	0	0
$638$	2.71824e7	2.64384
$639$	5.94302e6i	0.575778i
$640$	1.66259e7	1.60449
$641$	−8.09995e6	−0.778641	−0.389321	−	0.921102i	$-0.627290\pi$
$641$	−8.09995e6	−0.778641	−0.389321	+	0.921102i	$0.627290\pi$
$642$	9.47585e6i	0.907362i
$643$	5.29175e6i	0.504745i	0.967630	+	0.252373i	$0.0812108\pi$
$643$	5.29175e6i	0.504745i	−0.967630	+	0.252373i	$0.918789\pi$
$644$	307283.i	0.0291960i
$645$	7.35223e6i	0.695856i
$646$	−1.74324e7	−1.64353
$647$	−1.80626e7	−1.69637	−0.848185	−	0.529700i	$-0.822305\pi$
$647$	−1.80626e7	−1.69637	−0.848185	+	0.529700i	$0.822305\pi$
$648$	644060.i	0.0602544i
$649$	−1.20679e6	−0.112466
$650$	0	0
$651$	−4265.07	−0.000394434 0
$652$	9.90416e6i	0.912428i
$653$	−7.10212e6	−0.651786	−0.325893	−	0.945407i	$-0.605665\pi$
$653$	−7.10212e6	−0.651786	−0.325893	+	0.945407i	$0.605665\pi$
$654$	−1.26796e7	−1.15921
$655$	− 2.38052e7i	− 2.16805i
$656$	− 7.82627e6i	− 0.710060i
$657$	− 933954.i	− 0.0844135i
$658$	− 206471.i	− 0.0185907i
$659$	1.41430e7	1.26861	0.634303	−	0.773084i	$-0.281287\pi$
$659$	1.41430e7	1.26861	0.634303	+	0.773084i	$0.281287\pi$
$660$	−2.32015e7	−2.07327
$661$	− 675807.i	− 0.0601615i	−0.999547	−	0.0300808i	$-0.990424\pi$
$661$	− 675807.i	− 0.0601615i	0.999547	−	0.0300808i	$-0.00957645\pi$
$662$	−2.46543e6	−0.218649
$663$	0	0
$664$	−667566.	−0.0587590
$665$	621280.i	0.0544795i
$666$	−612152.	−0.0534778
$667$	1.37969e7	1.20079
$668$	2.80851e7i	2.43520i
$669$	154936.i	0.0133840i
$670$	2.48814e6i	0.214135i
$671$	− 9.94447e6i	− 0.852659i
$672$	230958.	0.0197292
$673$	−2.66714e6	−0.226991	−0.113496	−	0.993539i	$-0.536205\pi$
$673$	−2.66714e6	−0.226991	−0.113496	+	0.993539i	$0.536205\pi$
$674$	− 1.70318e7i	− 1.44414i
$675$	2.10087e7	1.77476
$676$	0	0
$677$	−1.20319e7	−1.00893	−0.504465	−	0.863432i	$-0.668310\pi$
$677$	−1.20319e7	−1.00893	−0.504465	+	0.863432i	$0.668310\pi$
$678$	− 1.86423e7i	− 1.55749i
$679$	121462.	0.0101104
$680$	−7.36493e6	−0.610796
$681$	− 2.81568e6i	− 0.232656i
$682$	− 716573.i	− 0.0589929i
$683$	− 1.83758e7i	− 1.50728i	−0.657287	−	0.753640i	$-0.728296\pi$
$683$	− 1.83758e7i	− 1.50728i	0.657287	−	0.753640i	$-0.271704\pi$
$684$	1.38314e7i	1.13038i
$685$	2.70778e7	2.20489
$686$	−831495.	−0.0674604
$687$	7.64379e6i	0.617898i
$688$	4.35653e6	0.350889
$689$	0	0
$690$	−2.05815e7	−1.64571
$691$	− 1.33412e7i	− 1.06292i	−0.847084	−	0.531459i	$-0.821644\pi$
$691$	− 1.33412e7i	− 1.06292i	0.847084	−	0.531459i	$-0.178356\pi$
$692$	1.93068e6	0.153266
$693$	224369.	0.0177472
$694$	1.64109e7i	1.29340i
$695$	− 1.66290e7i	− 1.30588i
$696$	5.28124e6i	0.413250i
$697$	1.19171e7i	0.929159i
$698$	3.16372e6	0.245787
$699$	−1.15340e7	−0.892865
$700$	657730.i	0.0507344i
$701$	−2.51952e6	−0.193652	−0.0968262	−	0.995301i	$-0.530869\pi$
$701$	−2.51952e6	−0.193652	−0.0968262	+	0.995301i	$0.530869\pi$
$702$	0	0
$703$	1.21559e6	0.0927679
$704$	2.85239e7i	2.16909i
$705$	7.91281e6	0.599595
$706$	1.35499e7	1.02311
$707$	251607.i	0.0189310i
$708$	− 929319.i	− 0.0696757i
$709$	2.51441e7i	1.87854i	0.343174	+	0.939272i	$0.388498\pi$
$709$	2.51441e7i	1.87854i	−0.343174	+	0.939272i	$0.611502\pi$
$710$	− 3.45379e7i	− 2.57128i
$711$	175698.	0.0130345
$712$	1.00038e7	0.739546
$713$	− 363709.i	− 0.0267936i
$714$	−217765.	−0.0159861
$715$	0	0
$716$	2.23339e7	1.62810
$717$	− 2.87868e6i	− 0.209120i
$718$	3.48803e7	2.52504
$719$	−415865.	−0.0300006	−0.0150003	−	0.999887i	$-0.504775\pi$
$719$	−415865.	−0.0300006	−0.0150003	+	0.999887i	$0.504775\pi$
$720$	− 7.10483e6i	− 0.510766i
$721$	− 277179.i	− 0.0198574i
$722$	− 2.65873e7i	− 1.89815i
$723$	8.40330e6i	0.597866i
$724$	−5.25474e6	−0.372567
$725$	2.95318e7	2.08663
$726$	− 1.47511e7i	− 1.03868i
$727$	1.53987e7	1.08056	0.540278	−	0.841486i	$-0.318319\pi$
$727$	1.53987e7	1.08056	0.540278	+	0.841486i	$0.318319\pi$
$728$	0	0
$729$	1.07222e7	0.747251
$730$	5.42767e6i	0.376970i
$731$	−6.63374e6	−0.459161
$732$	7.65798e6	0.528246
$733$	1.82125e7i	1.25202i	0.779817	+	0.626008i	$0.215312\pi$
$733$	1.82125e7i	1.25202i	−0.779817	+	0.626008i	$0.784688\pi$
$734$	5.91428e6i	0.405193i
$735$	− 1.59292e7i	− 1.08762i
$736$	1.96953e7i	1.34019i
$737$	−1.78434e6	−0.121007
$738$	1.65252e7	1.11688
$739$	− 1.88107e7i	− 1.26705i	−0.773723	−	0.633524i	$-0.781608\pi$
$739$	− 1.88107e7i	− 1.26705i	0.773723	−	0.633524i	$-0.218392\pi$
$740$	2.03554e6	0.136647
$741$	0	0
$742$	147867.	0.00985964
$743$	− 2.61292e7i	− 1.73642i	−0.496199	−	0.868209i	$-0.665271\pi$
$743$	− 2.61292e7i	− 1.73642i	0.496199	−	0.868209i	$-0.334729\pi$
$744$	139223.	0.00922098
$745$	−4.75649e7	−3.13976
$746$	− 2.67378e7i	− 1.75905i
$747$	980611.i	0.0642977i
$748$	− 2.09341e7i	− 1.36805i
$749$	304710.i	0.0198464i
$750$	−1.84263e7	−1.19615
$751$	1.90282e7	1.23111	0.615557	−	0.788092i	$-0.288931\pi$
$751$	1.90282e7	1.23111	0.615557	+	0.788092i	$0.288931\pi$
$752$	− 4.68871e6i	− 0.302349i
$753$	7.88699e6	0.506902
$754$	0	0
$755$	−3.07664e7	−1.96431
$756$	478847.i	0.0304714i
$757$	−1.91639e7	−1.21547	−0.607734	−	0.794140i	$-0.707921\pi$
$757$	−1.91639e7	−1.21547	−0.607734	+	0.794140i	$0.707921\pi$
$758$	−4.53705e7	−2.86814
$759$	− 1.47598e7i	− 0.929983i
$760$	− 2.02801e7i	− 1.27361i
$761$	− 468091.i	− 0.0293001i	−0.999893	−	0.0146500i	$-0.995337\pi$
$761$	− 468091.i	− 0.0293001i	0.999893	−	0.0146500i	$-0.00466342\pi$
$762$	4.84590e6i	0.302334i
$763$	−407732.	−0.0253550
$764$	−1.49882e7	−0.928999
$765$	1.08186e7i	0.668371i
$766$	1.35791e7	0.836177
$767$	0	0
$768$	376805.	0.0230523
$769$	1.92729e7i	1.17525i	0.809132	+	0.587627i	$0.199938\pi$
$769$	1.92729e7i	1.17525i	−0.809132	+	0.587627i	$0.800062\pi$
$770$	−1.30392e6	−0.0792546
$771$	1.87648e7	1.13686
$772$	− 3.75045e7i	− 2.26485i
$773$	− 2.42083e7i	− 1.45719i	−0.684947	−	0.728593i	$-0.740175\pi$
$773$	− 2.42083e7i	− 1.45719i	0.684947	−	0.728593i	$-0.259825\pi$
$774$	9.19884e6i	0.551926i
$775$	− 778509.i	− 0.0465596i
$776$	−3.96483e6	−0.236358
$777$	15185.0	0.000902326 0
$778$	− 1.33605e7i	− 0.791358i
$779$	−3.28151e7	−1.93745
$780$	0	0
$781$	2.47684e7	1.45302
$782$	− 1.85702e7i	− 1.08592i
$783$	2.15001e7	1.25324
$784$	−9.43881e6	−0.548437
$785$	2.45066e7i	1.41941i
$786$	2.29760e7i	1.32653i
$787$	− 1.48087e7i	− 0.852276i	−0.904658	−	0.426138i	$-0.859874\pi$
$787$	− 1.48087e7i	− 0.852276i	0.904658	−	0.426138i	$-0.140126\pi$
$788$	1.31211e7i	0.752755i
$789$	−3.02701e6	−0.173110
$790$	−1.02107e6	−0.0582087
$791$	− 599470.i	− 0.0340664i
$792$	−7.32396e6	−0.414890
$793$	0	0
$794$	−4.38233e7	−2.46691
$795$	5.66686e6i	0.317998i
$796$	−1.40312e7	−0.784898
$797$	2.90994e7	1.62270	0.811351	−	0.584560i	$-0.198733\pi$
$797$	2.90994e7	1.62270	0.811351	+	0.584560i	$0.198733\pi$
$798$	− 599639.i	− 0.0333336i
$799$	7.13954e6i	0.395643i
$800$	4.21571e7i	2.32887i
$801$	− 1.46949e7i	− 0.809257i
$802$	1.19454e6	0.0655790
$803$	−3.89239e6	−0.213023
$804$	− 1.37407e6i	− 0.0749669i
$805$	−661828.	−0.0359961
$806$	0	0
$807$	2.98052e6	0.161105
$808$	− 8.21308e6i	− 0.442566i
$809$	3.37792e7	1.81459	0.907295	−	0.420496i	$-0.138144\pi$
$809$	3.37792e7	1.81459	0.907295	+	0.420496i	$0.138144\pi$
$810$	−5.49815e6	−0.294445
$811$	2.42986e7i	1.29727i	0.761101	+	0.648633i	$0.224659\pi$
$811$	2.42986e7i	1.29727i	−0.761101	+	0.648633i	$0.775341\pi$
$812$	673113.i	0.0358260i
$813$	1.67861e7i	0.890686i
$814$	2.55123e6i	0.134955i
$815$	−2.13317e7	−1.12494
$816$	−4.94517e6	−0.259990
$817$	− 1.82667e7i	− 0.957425i
$818$	−3.57460e7	−1.86786
$819$	0	0
$820$	−5.49500e7	−2.85386
$821$	7.42442e6i	0.384419i	0.981354	+	0.192209i	$0.0615653\pi$
$821$	7.42442e6i	0.384419i	−0.981354	+	0.192209i	$0.938435\pi$
$822$	−2.61346e7	−1.34908
$823$	2.98540e7	1.53640	0.768198	−	0.640212i	$-0.221153\pi$
$823$	2.98540e7	1.53640	0.768198	+	0.640212i	$0.221153\pi$
$824$	9.04781e6i	0.464222i
$825$	− 3.15929e7i	− 1.61605i
$826$	− 52227.6i	− 0.00266348i
$827$	2.47154e7i	1.25662i	0.777962	+	0.628311i	$0.216253\pi$
$827$	2.47154e7i	1.25662i	−0.777962	+	0.628311i	$0.783747\pi$
$828$	−1.47341e7	−0.746875
$829$	1.14800e7	0.580169	0.290084	−	0.957001i	$-0.406317\pi$
$829$	1.14800e7	0.580169	0.290084	+	0.957001i	$0.406317\pi$
$830$	− 5.69882e6i	− 0.287137i
$831$	−4.10314e6	−0.206117
$832$	0	0
$833$	1.43726e7	0.717665
$834$	1.60498e7i	0.799013i
$835$	−6.04899e7	−3.00239
$836$	5.76443e7	2.85260
$837$	− 566779.i	− 0.0279640i
$838$	− 2.56606e7i	− 1.26228i
$839$	2.41188e7i	1.18291i	0.806339	+	0.591454i	$0.201446\pi$
$839$	2.41188e7i	1.18291i	−0.806339	+	0.591454i	$0.798554\pi$
$840$	− 253338.i	− 0.0123880i
$841$	9.71140e6	0.473469
$842$	−2.57836e7	−1.25332
$843$	2.82491e6i	0.136910i
$844$	2.18048e7	1.05365
$845$	0	0
$846$	9.90022e6	0.475575
$847$	− 474343.i	− 0.0227187i
$848$	3.35787e6	0.160352
$849$	8.17487e6	0.389235
$850$	− 3.97489e7i	− 1.88703i
$851$	1.29492e6i	0.0612943i
$852$	1.90735e7i	0.900186i
$853$	− 2.02681e6i	− 0.0953764i	−0.998862	−	0.0476882i	$-0.984815\pi$
$853$	− 2.02681e6i	− 0.0953764i	0.998862	−	0.0476882i	$-0.0151854\pi$
$854$	430378.	0.0201932
$855$	−2.97901e7	−1.39366
$856$	− 9.94649e6i	− 0.463965i
$857$	−1.52859e7	−0.710950	−0.355475	−	0.934686i	$-0.615681\pi$
$857$	−1.52859e7	−0.710950	−0.355475	+	0.934686i	$0.615681\pi$
$858$	0	0
$859$	1.78567e7	0.825693	0.412846	−	0.910801i	$-0.364535\pi$
$859$	1.78567e7	0.825693	0.412846	+	0.910801i	$0.364535\pi$
$860$	− 3.05882e7i	− 1.41029i
$861$	−409924.	−0.0188450
$862$	−2.89546e7	−1.32724
$863$	1.12315e7i	0.513347i	0.966498	+	0.256673i	$0.0826264\pi$
$863$	1.12315e7i	0.513347i	−0.966498	+	0.256673i	$0.917374\pi$
$864$	3.06917e7i	1.39874i
$865$	4.15832e6i	0.188963i
$866$	3.25863e7i	1.47652i
$867$	−7.07595e6	−0.319696
$868$	17744.4	0.000799397 0
$869$	− 732249.i	− 0.0328934i
$870$	−4.50844e7	−2.01943
$871$	0	0
$872$	1.33094e7	0.592744
$873$	5.82408e6i	0.258638i
$874$	5.11349e7	2.26433
$875$	−592524.	−0.0261629
$876$	− 2.99743e6i	− 0.131974i
$877$	− 1.75233e6i	− 0.0769336i	−0.999260	−	0.0384668i	$-0.987753\pi$
$877$	− 1.75233e6i	− 0.0769336i	0.999260	−	0.0384668i	$-0.0122474\pi$
$878$	− 4.86980e7i	− 2.13194i
$879$	− 1.59780e7i	− 0.697510i
$880$	−2.96104e7	−1.28896
$881$	1.04009e7	0.451474	0.225737	−	0.974188i	$-0.427521\pi$
$881$	1.04009e7	0.451474	0.225737	+	0.974188i	$0.427521\pi$
$882$	− 1.99301e7i	− 0.862656i
$883$	−4.25645e7	−1.83715	−0.918577	−	0.395241i	$-0.870661\pi$
$883$	−4.25645e7	−1.83715	−0.918577	+	0.395241i	$0.870661\pi$
$884$	0	0
$885$	2.00157e6	0.0859040
$886$	3.19827e7i	1.36877i
$887$	201500.	0.00859935	0.00429968	−	0.999991i	$-0.498631\pi$
$887$	201500.	0.00859935	0.00429968	+	0.999991i	$0.498631\pi$
$888$	−495677.	−0.0210944
$889$	155827.i	0.00661285i
$890$	8.53996e7i	3.61394i
$891$	− 3.94294e6i	− 0.166389i
$892$	− 644596.i	− 0.0271253i
$893$	−1.96595e7	−0.824980
$894$	4.59081e7	1.92108
$895$	4.81029e7i	2.00731i
$896$	−516009.	−0.0214727
$897$	0	0
$898$	4.37198e6	0.180920
$899$	− 796718.i	− 0.0328780i
$900$	−3.15379e7	−1.29786
$901$	−5.11307e6	−0.209831
$902$	− 6.88712e7i	− 2.81852i
$903$	− 228186.i	− 0.00931259i
$904$	1.95682e7i	0.796397i
$905$	− 1.13177e7i	− 0.459342i
$906$	2.96947e7	1.20187
$907$	−1.47930e7	−0.597088	−0.298544	−	0.954396i	$-0.596501\pi$
$907$	−1.47930e7	−0.597088	−0.298544	+	0.954396i	$0.596501\pi$
$908$	1.17143e7i	0.471523i
$909$	−1.20645e7	−0.484283
$910$	0	0
$911$	−3.50070e7	−1.39752	−0.698762	−	0.715354i	$-0.746265\pi$
$911$	−3.50070e7	−1.39752	−0.698762	+	0.715354i	$0.746265\pi$
$912$	− 1.36170e7i	− 0.542121i
$913$	4.08684e6	0.162260
$914$	6.77726e6	0.268342
$915$	1.64938e7i	0.651281i
$916$	− 3.18012e7i	− 1.25229i
$917$	738828.i	0.0290148i
$918$	− 2.89384e7i	− 1.13336i
$919$	4.24526e7	1.65812	0.829060	−	0.559160i	$-0.188876\pi$
$919$	4.24526e7	1.65812	0.829060	+	0.559160i	$0.188876\pi$
$920$	2.16037e7	0.841509
$921$	2.13080e7i	0.827741i
$922$	4.02034e7	1.55753
$923$	0	0
$924$	720090.	0.0277464
$925$	2.77175e6i	0.106512i
$926$	−2.29076e7	−0.877916
$927$	1.32906e7	0.507980
$928$	4.31431e7i	1.64453i
$929$	3.95487e7i	1.50347i	0.659468	+	0.751733i	$0.270782\pi$
$929$	3.95487e7i	1.50347i	−0.659468	+	0.751733i	$0.729218\pi$
$930$	1.18850e6i	0.0450602i
$931$	3.95764e7i	1.49645i
$932$	4.79859e7	1.80957
$933$	2.58024e7	0.970413
$934$	2.09445e7i	0.785601i
$935$	4.50881e7	1.68668
$936$	0	0
$937$	−4.62850e7	−1.72223	−0.861116	−	0.508409i	$-0.830234\pi$
$937$	−4.62850e7	−1.72223	−0.861116	+	0.508409i	$0.830234\pi$
$938$	− 77222.8i	− 0.00286575i
$939$	8.56270e6	0.316918
$940$	−3.29205e7	−1.21520
$941$	− 5.74465e6i	− 0.211490i	−0.994393	−	0.105745i	$-0.966277\pi$
$941$	− 5.74465e6i	− 0.211490i	0.994393	−	0.105745i	$-0.0337227\pi$
$942$	− 2.36530e7i	− 0.868477i
$943$	− 3.49568e7i	− 1.28013i
$944$	− 1.18602e6i	− 0.0433175i
$945$	−1.03135e6	−0.0375686
$946$	3.83375e7	1.39282
$947$	5.15966e7i	1.86959i	0.355187	+	0.934795i	$0.384417\pi$
$947$	5.15966e7i	1.86959i	−0.355187	+	0.934795i	$0.615583\pi$
$948$	563886.	0.0203784
$949$	0	0
$950$	1.09453e8	3.93476
$951$	1.81603e7i	0.651137i
$952$	228581.	0.00817424
$953$	2.18333e7	0.778729	0.389365	−	0.921084i	$-0.372695\pi$
$953$	2.18333e7	0.778729	0.389365	+	0.921084i	$0.372695\pi$
$954$	7.09017e6i	0.252223i
$955$	− 3.22816e7i	− 1.14537i
$956$	1.19765e7i	0.423822i
$957$	− 3.23318e7i	− 1.14117i
$958$	−4.82863e7	−1.69985
$959$	−840398.	−0.0295079
$960$	− 4.73095e7i	− 1.65680i
$961$	2.86081e7	0.999266
$962$	0	0
$963$	−1.46107e7	−0.507699
$964$	− 3.49611e7i	− 1.21169i
$965$	8.07774e7	2.79236
$966$	638774.	0.0220244
$967$	7.30509e6i	0.251223i	0.992080	+	0.125611i	$0.0400893\pi$
$967$	7.30509e6i	0.251223i	−0.992080	+	0.125611i	$0.959911\pi$
$968$	1.54837e7i	0.531114i
$969$	2.07348e7i	0.709400i
$970$	− 3.38466e7i	− 1.15501i
$971$	−1.62933e6	−0.0554576	−0.0277288	−	0.999615i	$-0.508827\pi$
$971$	−1.62933e6	−0.0554576	−0.0277288	+	0.999615i	$0.508827\pi$
$972$	−3.76364e7	−1.27774
$973$	516105.i	0.0174765i
$974$	6.27180e7	2.11834
$975$	0	0
$976$	9.77335e6	0.328412
$977$	2.55515e7i	0.856407i	0.903682	+	0.428204i	$0.140853\pi$
$977$	2.55515e7i	0.856407i	−0.903682	+	0.428204i	$0.859147\pi$
$978$	2.05886e7	0.688304
$979$	−6.12433e7	−2.04222
$980$	6.62720e7i	2.20427i
$981$	− 1.95506e7i	− 0.648617i
$982$	− 3.61000e7i	− 1.19462i
$983$	− 8.48095e6i	− 0.279937i	−0.990156	−	0.139969i	$-0.955300\pi$
$983$	− 8.48095e6i	− 0.279937i	0.990156	−	0.139969i	$-0.0447002\pi$
$984$	1.33809e7	0.440554
$985$	−2.82603e7	−0.928080
$986$	− 4.06786e7i	− 1.33252i
$987$	−245585.	−0.00802434
$988$	0	0
$989$	1.94589e7	0.632597
$990$	− 6.25225e7i	− 2.02744i
$991$	−2.74483e7	−0.887833	−0.443916	−	0.896068i	$-0.646411\pi$
$991$	−2.74483e7	−0.887833	−0.443916	+	0.896068i	$0.646411\pi$
$992$	1.13733e6	0.0366949
$993$	2.93248e6i	0.0943762i
$994$	1.07193e6i	0.0344113i
$995$	− 3.02206e7i	− 0.967710i
$996$	3.14717e6i	0.100525i
$997$	−2.26765e7	−0.722501	−0.361251	−	0.932469i	$-0.617650\pi$
$997$	−2.26765e7	−0.722501	−0.361251	+	0.932469i	$0.617650\pi$
$998$	−6.77660e7	−2.15370
$999$	2.01792e6i	0.0639719i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.6.b.b.168.5		6
13.5	odd	4		169.6.a.b.1.3		3
13.8	odd	4		13.6.a.b.1.1	✓	3
13.12	even	2	inner	169.6.b.b.168.2		6
39.8	even	4		117.6.a.d.1.3		3
52.47	even	4		208.6.a.j.1.2		3
65.8	even	4		325.6.b.c.274.5		6
65.34	odd	4		325.6.a.c.1.3		3
65.47	even	4		325.6.b.c.274.2		6
91.34	even	4		637.6.a.b.1.1		3
104.21	odd	4		832.6.a.s.1.2		3
104.99	even	4		832.6.a.t.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.b.1.1	✓	3	13.8	odd	4
117.6.a.d.1.3		3	39.8	even	4
169.6.a.b.1.3		3	13.5	odd	4
169.6.b.b.168.2		6	13.12	even	2	inner
169.6.b.b.168.5		6	1.1	even	1	trivial
208.6.a.j.1.2		3	52.47	even	4
325.6.a.c.1.3		3	65.34	odd	4
325.6.b.c.274.2		6	65.47	even	4
325.6.b.c.274.5		6	65.8	even	4
637.6.a.b.1.1		3	91.34	even	4
832.6.a.s.1.2		3	104.21	odd	4
832.6.a.t.1.2		3	104.99	even	4

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.64858i q^{2} +10.2870 q^{3} -42.7979 q^{4} -92.1784i q^{5} +88.9676i q^{6} +2.86088i q^{7} -93.3863i q^{8} -137.178 q^{9} +O(q^{10})\)	\(q+8.64858i q^{2} +10.2870 q^{3} -42.7979 q^{4} -92.1784i q^{5} +88.9676i q^{6} +2.86088i q^{7} -93.3863i q^{8} -137.178 q^{9} +797.212 q^{10} +571.711i q^{11} -440.260 q^{12} -24.7426 q^{14} -948.236i q^{15} -561.873 q^{16} +855.571 q^{17} -1186.40i q^{18} +2355.90i q^{19} +3945.04i q^{20} +29.4298i q^{21} -4944.49 q^{22} -2509.66 q^{23} -960.662i q^{24} -5371.86 q^{25} -3910.88 q^{27} -122.440i q^{28} -5497.50 q^{29} +8200.89 q^{30} +144.924i q^{31} -7847.77i q^{32} +5881.17i q^{33} +7399.47i q^{34} +263.712 q^{35} +5870.95 q^{36} -515.975i q^{37} -20375.2 q^{38} -8608.21 q^{40} +13928.9i q^{41} -254.526 q^{42} -7753.58 q^{43} -24468.0i q^{44} +12644.9i q^{45} -21705.0i q^{46} +8344.77i q^{47} -5779.97 q^{48} +16798.8 q^{49} -46459.0i q^{50} +8801.22 q^{51} -5976.21 q^{53} -33823.6i q^{54} +52699.5 q^{55} +267.168 q^{56} +24235.1i q^{57} -47545.6i q^{58} +2110.84i q^{59} +40582.5i q^{60} -17394.2 q^{61} -1253.38 q^{62} -392.452i q^{63} +49892.1 q^{64} -50863.8 q^{66} +3121.05i q^{67} -36616.6 q^{68} -25816.8 q^{69} +2280.73i q^{70} -43323.3i q^{71} +12810.6i q^{72} +6808.31i q^{73} +4462.45 q^{74} -55260.1 q^{75} -100828. i q^{76} -1635.60 q^{77} -1280.80 q^{79} +51792.6i q^{80} -6896.72 q^{81} -120465. q^{82} -7148.44i q^{83} -1259.53i q^{84} -78865.2i q^{85} -67057.5i q^{86} -56552.6 q^{87} +53390.0 q^{88} +107123. i q^{89} -109360. q^{90} +107408. q^{92} +1490.82i q^{93} -72170.4 q^{94} +217163. q^{95} -80729.7i q^{96} -42456.2i q^{97} +145286. i q^{98} -78426.5i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9}+O(q^{10}) \) 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9	\( 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9} + 2582 q^{10} + 2182 q^{12} - 1586 q^{14} + 5570 q^{16} - 1816 q^{17} - 6820 q^{22} - 7248 q^{23} - 4046 q^{25} - 8552 q^{27} - 17516 q^{29} - 3434 q^{30} + 8696 q^{35} + 9356 q^{36} - 67572 q^{38} - 87018 q^{40} + 27206 q^{42} - 4064 q^{43} - 152446 q^{48} + 89142 q^{49} - 26936 q^{51} - 25140 q^{53} + 70624 q^{55} + 98574 q^{56} - 25508 q^{61} + 80680 q^{62} - 234786 q^{64} - 224708 q^{66} - 98350 q^{68} + 40224 q^{69} - 263038 q^{74} - 47096 q^{75} + 42104 q^{77} - 123744 q^{79} - 216442 q^{81} - 284520 q^{82} + 278144 q^{87} - 316020 q^{88} - 202516 q^{90} + 601728 q^{92} + 145686 q^{94} + 393120 q^{95}+O(q^{100}) \) 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9 + 2582 * q^10 + 2182 * q^12 - 1586 * q^14 + 5570 * q^16 - 1816 * q^17 - 6820 * q^22 - 7248 * q^23 - 4046 * q^25 - 8552 * q^27 - 17516 * q^29 - 3434 * q^30 + 8696 * q^35 + 9356 * q^36 - 67572 * q^38 - 87018 * q^40 + 27206 * q^42 - 4064 * q^43 - 152446 * q^48 + 89142 * q^49 - 26936 * q^51 - 25140 * q^53 + 70624 * q^55 + 98574 * q^56 - 25508 * q^61 + 80680 * q^62 - 234786 * q^64 - 224708 * q^66 - 98350 * q^68 + 40224 * q^69 - 263038 * q^74 - 47096 * q^75 + 42104 * q^77 - 123744 * q^79 - 216442 * q^81 - 284520 * q^82 + 278144 * q^87 - 316020 * q^88 - 202516 * q^90 + 601728 * q^92 + 145686 * q^94 + 393120 * q^95

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