LMFDB - Embedded newform 33.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,8,Mod(1,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("33.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	33.a (trivial)

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:	yes
Analytic conductor:	$10.3087058410$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 510x^{2} - 1544x + 28880 $ x^4 - x^3 - 510x^2 - 1544x + 28880
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$23.3791$ of defining polynomial
Character	$\chi$	$=$	33.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.3791 q^{2} +27.0000 q^{3} +247.551 q^{4} +336.012 q^{5} -523.236 q^{6} -879.192 q^{7} -2316.79 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-19.3791 q^{2} +27.0000 q^{3} +247.551 q^{4} +336.012 q^{5} -523.236 q^{6} -879.192 q^{7} -2316.79 q^{8} +729.000 q^{9} -6511.62 q^{10} +1331.00 q^{11} +6683.87 q^{12} +460.766 q^{13} +17038.0 q^{14} +9072.33 q^{15} +13210.8 q^{16} +24960.3 q^{17} -14127.4 q^{18} -2470.32 q^{19} +83180.0 q^{20} -23738.2 q^{21} -25793.6 q^{22} +105303. q^{23} -62553.2 q^{24} +34779.2 q^{25} -8929.24 q^{26} +19683.0 q^{27} -217644. q^{28} -128131. q^{29} -175814. q^{30} +210904. q^{31} +40534.3 q^{32} +35937.0 q^{33} -483708. q^{34} -295419. q^{35} +180464. q^{36} +3948.26 q^{37} +47872.7 q^{38} +12440.7 q^{39} -778469. q^{40} +503852. q^{41} +460025. q^{42} +961195. q^{43} +329490. q^{44} +244953. q^{45} -2.04068e6 q^{46} +1.33928e6 q^{47} +356692. q^{48} -50564.9 q^{49} -673991. q^{50} +673927. q^{51} +114063. q^{52} -2.03087e6 q^{53} -381439. q^{54} +447232. q^{55} +2.03690e6 q^{56} -66698.8 q^{57} +2.48306e6 q^{58} -2.71800e6 q^{59} +2.24586e6 q^{60} -548346. q^{61} -4.08714e6 q^{62} -640931. q^{63} -2.47651e6 q^{64} +154823. q^{65} -696428. q^{66} -471831. q^{67} +6.17893e6 q^{68} +2.84318e6 q^{69} +5.72497e6 q^{70} +3.22993e6 q^{71} -1.68894e6 q^{72} +367407. q^{73} -76513.8 q^{74} +939038. q^{75} -611530. q^{76} -1.17020e6 q^{77} -241089. q^{78} -3.99092e6 q^{79} +4.43900e6 q^{80} +531441. q^{81} -9.76422e6 q^{82} -57990.0 q^{83} -5.87640e6 q^{84} +8.38695e6 q^{85} -1.86271e7 q^{86} -3.45953e6 q^{87} -3.08364e6 q^{88} +1.03128e6 q^{89} -4.74697e6 q^{90} -405101. q^{91} +2.60679e7 q^{92} +5.69441e6 q^{93} -2.59541e7 q^{94} -830059. q^{95} +1.09443e6 q^{96} -7.10206e6 q^{97} +979905. q^{98} +970299. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 15 q^{2} + 108 q^{3} + 565 q^{4} + 306 q^{5} + 405 q^{6} + 890 q^{7} + 2457 q^{8} + 2916 q^{9}+O(q^{10}) $ 4 * q + 15 * q^2 + 108 * q^3 + 565 * q^4 + 306 * q^5 + 405 * q^6 + 890 * q^7 + 2457 * q^8 + 2916 * q^9	\( 4 q + 15 q^{2} + 108 q^{3} + 565 q^{4} + 306 q^{5} + 405 q^{6} + 890 q^{7} + 2457 q^{8} + 2916 q^{9} - 8946 q^{10} + 5324 q^{11} + 15255 q^{12} - 1822 q^{13} + 35340 q^{14} + 8262 q^{15} + 65041 q^{16} + 32856 q^{17} + 10935 q^{18} - 12784 q^{19} - 4002 q^{20} + 24030 q^{21} + 19965 q^{22} + 114858 q^{23} + 66339 q^{24} + 72856 q^{25} - 221466 q^{26} + 78732 q^{27} + 10112 q^{28} - 104952 q^{29} - 241542 q^{30} - 24976 q^{31} + 337761 q^{32} + 143748 q^{33} - 741690 q^{34} - 722856 q^{35} + 411885 q^{36} - 498856 q^{37} - 897156 q^{38} - 49194 q^{39} - 2676930 q^{40} + 734556 q^{41} + 954180 q^{42} - 201916 q^{43} + 752015 q^{44} + 223074 q^{45} - 3068508 q^{46} + 1995894 q^{47} + 1756107 q^{48} - 771024 q^{49} + 1632129 q^{50} + 887112 q^{51} - 4412266 q^{52} + 929970 q^{53} + 295245 q^{54} + 407286 q^{55} + 7224888 q^{56} - 345168 q^{57} + 2864322 q^{58} + 1353156 q^{59} - 108054 q^{60} + 3998774 q^{61} - 3783264 q^{62} + 648810 q^{63} + 1480129 q^{64} + 6612108 q^{65} + 539055 q^{66} + 1722008 q^{67} + 1596906 q^{68} + 3101166 q^{69} - 2751024 q^{70} + 5571858 q^{71} + 1791153 q^{72} + 5600528 q^{73} - 10907838 q^{74} + 1967112 q^{75} - 19634884 q^{76} + 1184590 q^{77} - 5979582 q^{78} - 7710226 q^{79} - 24073794 q^{80} + 2125764 q^{81} - 11230842 q^{82} + 3431856 q^{83} + 273024 q^{84} + 5909484 q^{85} - 25687140 q^{86} - 2833704 q^{87} + 3270267 q^{88} + 4611528 q^{89} - 6521634 q^{90} - 9032696 q^{91} + 13608576 q^{92} - 674352 q^{93} - 3497436 q^{94} + 21828000 q^{95} + 9119547 q^{96} + 1401692 q^{97} - 7230081 q^{98} + 3881196 q^{99}+O(q^{100}) \) 4 * q + 15 * q^2 + 108 * q^3 + 565 * q^4 + 306 * q^5 + 405 * q^6 + 890 * q^7 + 2457 * q^8 + 2916 * q^9 - 8946 * q^10 + 5324 * q^11 + 15255 * q^12 - 1822 * q^13 + 35340 * q^14 + 8262 * q^15 + 65041 * q^16 + 32856 * q^17 + 10935 * q^18 - 12784 * q^19 - 4002 * q^20 + 24030 * q^21 + 19965 * q^22 + 114858 * q^23 + 66339 * q^24 + 72856 * q^25 - 221466 * q^26 + 78732 * q^27 + 10112 * q^28 - 104952 * q^29 - 241542 * q^30 - 24976 * q^31 + 337761 * q^32 + 143748 * q^33 - 741690 * q^34 - 722856 * q^35 + 411885 * q^36 - 498856 * q^37 - 897156 * q^38 - 49194 * q^39 - 2676930 * q^40 + 734556 * q^41 + 954180 * q^42 - 201916 * q^43 + 752015 * q^44 + 223074 * q^45 - 3068508 * q^46 + 1995894 * q^47 + 1756107 * q^48 - 771024 * q^49 + 1632129 * q^50 + 887112 * q^51 - 4412266 * q^52 + 929970 * q^53 + 295245 * q^54 + 407286 * q^55 + 7224888 * q^56 - 345168 * q^57 + 2864322 * q^58 + 1353156 * q^59 - 108054 * q^60 + 3998774 * q^61 - 3783264 * q^62 + 648810 * q^63 + 1480129 * q^64 + 6612108 * q^65 + 539055 * q^66 + 1722008 * q^67 + 1596906 * q^68 + 3101166 * q^69 - 2751024 * q^70 + 5571858 * q^71 + 1791153 * q^72 + 5600528 * q^73 - 10907838 * q^74 + 1967112 * q^75 - 19634884 * q^76 + 1184590 * q^77 - 5979582 * q^78 - 7710226 * q^79 - 24073794 * q^80 + 2125764 * q^81 - 11230842 * q^82 + 3431856 * q^83 + 273024 * q^84 + 5909484 * q^85 - 25687140 * q^86 - 2833704 * q^87 + 3270267 * q^88 + 4611528 * q^89 - 6521634 * q^90 - 9032696 * q^91 + 13608576 * q^92 - 674352 * q^93 - 3497436 * q^94 + 21828000 * q^95 + 9119547 * q^96 + 1401692 * q^97 - 7230081 * q^98 + 3881196 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−19.3791	−1.71289	−0.856445	−	0.516239i	$-0.827332\pi$
$2$	−19.3791	−1.71289	−0.856445	+	0.516239i	$0.827332\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	247.551	1.93399
$5$	336.012	1.20215	0.601077	−	0.799191i	$-0.294739\pi$
$5$	336.012	1.20215	0.601077	+	0.799191i	$0.294739\pi$
$6$	−523.236	−0.988937
$7$	−879.192	−0.968814	−0.484407	−	0.874843i	$-0.660965\pi$
$7$	−879.192	−0.968814	−0.484407	+	0.874843i	$0.660965\pi$
$8$	−2316.79	−1.59982
$9$	729.000	0.333333
$10$	−6511.62	−2.05916
$11$	1331.00	0.301511
$11$	1331.00	0.301511
$12$	6683.87	1.11659
$13$	460.766	0.0581672	0.0290836	−	0.999577i	$-0.490741\pi$
$13$	460.766	0.0581672	0.0290836	+	0.999577i	$0.490741\pi$
$14$	17038.0	1.65947
$15$	9072.33	0.694064
$16$	13210.8	0.806325
$17$	24960.3	1.23219	0.616095	−	0.787672i	$-0.288714\pi$
$17$	24960.3	1.23219	0.616095	+	0.787672i	$0.288714\pi$
$18$	−14127.4	−0.570963
$19$	−2470.32	−0.0826260	−0.0413130	−	0.999146i	$-0.513154\pi$
$19$	−2470.32	−0.0826260	−0.0413130	+	0.999146i	$0.513154\pi$
$20$	83180.0	2.32495
$21$	−23738.2	−0.559345
$22$	−25793.6	−0.516456
$23$	105303.	1.80465	0.902327	−	0.431053i	$-0.141858\pi$
$23$	105303.	1.80465	0.902327	+	0.431053i	$0.141858\pi$
$24$	−62553.2	−0.923657
$25$	34779.2	0.445174
$26$	−8929.24	−0.0996340
$27$	19683.0	0.192450
$28$	−217644.	−1.87368
$29$	−128131.	−0.975574	−0.487787	−	0.872963i	$-0.662196\pi$
$29$	−128131.	−0.975574	−0.487787	+	0.872963i	$0.662196\pi$
$30$	−175814.	−1.18885
$31$	210904.	1.27151	0.635754	−	0.771892i	$-0.280689\pi$
$31$	210904.	1.27151	0.635754	+	0.771892i	$0.280689\pi$
$32$	40534.3	0.218675
$33$	35937.0	0.174078
$34$	−483708.	−2.11061
$35$	−295419.	−1.16466
$36$	180464.	0.644663
$37$	3948.26	0.0128144	0.00640722	−	0.999979i	$-0.497961\pi$
$37$	3948.26	0.0128144	0.00640722	+	0.999979i	$0.497961\pi$
$38$	47872.7	0.141529
$39$	12440.7	0.0335829
$40$	−778469.	−1.92323
$41$	503852.	1.14172	0.570860	−	0.821047i	$-0.306610\pi$
$41$	503852.	1.14172	0.570860	+	0.821047i	$0.306610\pi$
$42$	460025.	0.958096
$43$	961195.	1.84362	0.921811	−	0.387640i	$-0.126710\pi$
$43$	961195.	1.84362	0.921811	+	0.387640i	$0.126710\pi$
$44$	329490.	0.583120
$45$	244953.	0.400718
$46$	−2.04068e6	−3.09117
$47$	1.33928e6	1.88161	0.940804	−	0.338950i	$-0.110072\pi$
$47$	1.33928e6	1.88161	0.940804	+	0.338950i	$0.110072\pi$
$48$	356692.	0.465532
$49$	−50564.9	−0.0613993
$50$	−673991.	−0.762533
$51$	673927.	0.711406
$52$	114063.	0.112495
$53$	−2.03087e6	−1.87377	−0.936884	−	0.349641i	$-0.886304\pi$
$53$	−2.03087e6	−1.87377	−0.936884	+	0.349641i	$0.886304\pi$
$54$	−381439.	−0.329646
$55$	447232.	0.362463
$56$	2.03690e6	1.54993
$57$	−66698.8	−0.0477041
$58$	2.48306e6	1.67105
$59$	−2.71800e6	−1.72293	−0.861463	−	0.507820i	$-0.830451\pi$
$59$	−2.71800e6	−1.72293	−0.861463	+	0.507820i	$0.830451\pi$
$60$	2.24586e6	1.34231
$61$	−548346.	−0.309315	−0.154657	−	0.987968i	$-0.549427\pi$
$61$	−548346.	−0.309315	−0.154657	+	0.987968i	$0.549427\pi$
$62$	−4.08714e6	−2.17795
$63$	−640931.	−0.322938
$64$	−2.47651e6	−1.18089
$65$	154823.	0.0699260
$66$	−696428.	−0.298176
$67$	−471831.	−0.191657	−0.0958284	−	0.995398i	$-0.530550\pi$
$67$	−471831.	−0.191657	−0.0958284	+	0.995398i	$0.530550\pi$
$68$	6.17893e6	2.38304
$69$	2.84318e6	1.04192
$70$	5.72497e6	1.99494
$71$	3.22993e6	1.07100	0.535500	−	0.844535i	$-0.320123\pi$
$71$	3.22993e6	1.07100	0.535500	+	0.844535i	$0.320123\pi$
$72$	−1.68894e6	−0.533273
$73$	367407.	0.110540	0.0552698	−	0.998471i	$-0.482398\pi$
$73$	367407.	0.110540	0.0552698	+	0.998471i	$0.482398\pi$
$74$	−76513.8	−0.0219497
$75$	939038.	0.257021
$76$	−611530.	−0.159798
$77$	−1.17020e6	−0.292108
$78$	−241089.	−0.0575237
$79$	−3.99092e6	−0.910705	−0.455352	−	0.890311i	$-0.650487\pi$
$79$	−3.99092e6	−0.910705	−0.455352	+	0.890311i	$0.650487\pi$
$80$	4.43900e6	0.969327
$81$	531441.	0.111111
$82$	−9.76422e6	−1.95564
$83$	−57990.0	−0.0111322	−0.00556609	−	0.999985i	$-0.501772\pi$
$83$	−57990.0	−0.0111322	−0.00556609	+	0.999985i	$0.501772\pi$
$84$	−5.87640e6	−1.08177
$85$	8.38695e6	1.48128
$86$	−1.86271e7	−3.15792
$87$	−3.45953e6	−0.563248
$88$	−3.08364e6	−0.482364
$89$	1.03128e6	0.155064	0.0775319	−	0.996990i	$-0.475296\pi$
$89$	1.03128e6	0.155064	0.0775319	+	0.996990i	$0.475296\pi$
$90$	−4.74697e6	−0.686385
$91$	−405101.	−0.0563532
$92$	2.60679e7	3.49018
$93$	5.69441e6	0.734106
$94$	−2.59541e7	−3.22299
$95$	−830059.	−0.0993291
$96$	1.09443e6	0.126252
$97$	−7.10206e6	−0.790103	−0.395051	−	0.918659i	$-0.629273\pi$
$97$	−7.10206e6	−0.790103	−0.395051	+	0.918659i	$0.629273\pi$
$98$	979905.	0.105170
$99$	970299.	0.100504
$100$	8.60961e6	0.860961
$101$	8.49824e6	0.820738	0.410369	−	0.911920i	$-0.365400\pi$
$101$	8.49824e6	0.820738	0.410369	+	0.911920i	$0.365400\pi$
$102$	−1.30601e7	−1.21856
$103$	3.50239e6	0.315816	0.157908	−	0.987454i	$-0.449525\pi$
$103$	3.50239e6	0.315816	0.157908	+	0.987454i	$0.449525\pi$
$104$	−1.06750e6	−0.0930571
$105$	−7.97632e6	−0.672419
$106$	3.93564e7	3.20956
$107$	−1.62589e7	−1.28307	−0.641533	−	0.767095i	$-0.721701\pi$
$107$	−1.62589e7	−1.28307	−0.641533	+	0.767095i	$0.721701\pi$
$108$	4.87254e6	0.372196
$109$	−130881.	−0.00968017	−0.00484008	−	0.999988i	$-0.501541\pi$
$109$	−130881.	−0.00968017	−0.00484008	+	0.999988i	$0.501541\pi$
$110$	−8.66697e6	−0.620859
$111$	106603.	0.00739842
$112$	−1.16149e7	−0.781179
$113$	6.24988e6	0.407472	0.203736	−	0.979026i	$-0.434692\pi$
$113$	6.24988e6	0.407472	0.203736	+	0.979026i	$0.434692\pi$
$114$	1.29256e6	0.0817119
$115$	3.53831e7	2.16947
$116$	−3.17188e7	−1.88675
$117$	335898.	0.0193891
$118$	5.26724e7	2.95118
$119$	−2.19449e7	−1.19376
$120$	−2.10187e7	−1.11038
$121$	1.77156e6	0.0909091
$122$	1.06265e7	0.529822
$123$	1.36040e7	0.659173
$124$	5.22095e7	2.45908
$125$	−1.45647e7	−0.666986
$126$	1.24207e7	0.553157
$127$	2.32449e6	0.100697	0.0503484	−	0.998732i	$-0.483967\pi$
$127$	2.32449e6	0.100697	0.0503484	+	0.998732i	$0.483967\pi$
$128$	4.28041e7	1.80406
$129$	2.59523e7	1.06442
$130$	−3.00033e6	−0.119775
$131$	3.27479e6	0.127272	0.0636362	−	0.997973i	$-0.479730\pi$
$131$	3.27479e6	0.127272	0.0636362	+	0.997973i	$0.479730\pi$
$132$	8.89623e6	0.336664
$133$	2.17189e6	0.0800492
$134$	9.14366e6	0.328287
$135$	6.61373e6	0.231355
$136$	−5.78276e7	−1.97128
$137$	−3.07318e7	−1.02109	−0.510547	−	0.859850i	$-0.670557\pi$
$137$	−3.07318e7	−1.02109	−0.510547	+	0.859850i	$0.670557\pi$
$138$	−5.50984e7	−1.78469
$139$	−6.22939e7	−1.96740	−0.983702	−	0.179804i	$-0.942454\pi$
$139$	−6.22939e7	−1.96740	−0.983702	+	0.179804i	$0.942454\pi$
$140$	−7.31312e7	−2.25245
$141$	3.61606e7	1.08635
$142$	−6.25933e7	−1.83450
$143$	613279.	0.0175381
$144$	9.63070e6	0.268775
$145$	−4.30535e7	−1.17279
$146$	−7.12003e6	−0.189342
$147$	−1.36525e6	−0.0354489
$148$	977394.	0.0247830
$149$	−1.29966e7	−0.321867	−0.160934	−	0.986965i	$-0.551451\pi$
$149$	−1.29966e7	−0.321867	−0.160934	+	0.986965i	$0.551451\pi$
$150$	−1.81977e7	−0.440249
$151$	−2.13270e7	−0.504093	−0.252046	−	0.967715i	$-0.581104\pi$
$151$	−2.13270e7	−0.504093	−0.252046	+	0.967715i	$0.581104\pi$
$152$	5.72322e6	0.132187
$153$	1.81960e7	0.410730
$154$	2.26775e7	0.500349
$155$	7.08664e7	1.52855
$156$	3.07970e6	0.0649489
$157$	1.64853e7	0.339977	0.169988	−	0.985446i	$-0.445627\pi$
$157$	1.64853e7	0.339977	0.169988	+	0.985446i	$0.445627\pi$
$158$	7.73405e7	1.55994
$159$	−5.48334e7	−1.08182
$160$	1.36200e7	0.262880
$161$	−9.25816e7	−1.74837
$162$	−1.02989e7	−0.190321
$163$	2.62947e7	0.475567	0.237784	−	0.971318i	$-0.423579\pi$
$163$	2.62947e7	0.475567	0.237784	+	0.971318i	$0.423579\pi$
$164$	1.24729e8	2.20808
$165$	1.20753e7	0.209268
$166$	1.12380e6	0.0190682
$167$	1.04773e8	1.74078	0.870390	−	0.492364i	$-0.163867\pi$
$167$	1.04773e8	1.74078	0.870390	+	0.492364i	$0.163867\pi$
$168$	5.49963e7	0.894851
$169$	−6.25362e7	−0.996617
$170$	−1.62532e8	−2.53727
$171$	−1.80087e6	−0.0275420
$172$	2.37944e8	3.56554
$173$	−5.46896e7	−0.803051	−0.401525	−	0.915848i	$-0.631520\pi$
$173$	−5.46896e7	−0.803051	−0.401525	+	0.915848i	$0.631520\pi$
$174$	6.70427e7	0.964782
$175$	−3.05776e7	−0.431291
$176$	1.75836e7	0.243116
$177$	−7.33859e7	−0.994732
$178$	−1.99853e7	−0.265607
$179$	6.66307e7	0.868338	0.434169	−	0.900832i	$-0.357042\pi$
$179$	6.66307e7	0.868338	0.434169	+	0.900832i	$0.357042\pi$
$180$	6.06382e7	0.774984
$181$	−1.34921e8	−1.69123	−0.845616	−	0.533792i	$-0.820767\pi$
$181$	−1.34921e8	−1.69123	−0.845616	+	0.533792i	$0.820767\pi$
$182$	7.85051e6	0.0965268
$183$	−1.48054e7	−0.178583
$184$	−2.43965e8	−2.88712
$185$	1.32666e6	0.0154049
$186$	−1.10353e8	−1.25744
$187$	3.32221e7	0.371519
$188$	3.31540e8	3.63901
$189$	−1.73051e7	−0.186448
$190$	1.60858e7	0.170140
$191$	−8.71714e7	−0.905226	−0.452613	−	0.891707i	$-0.649508\pi$
$191$	−8.71714e7	−0.905226	−0.452613	+	0.891707i	$0.649508\pi$
$192$	−6.68657e7	−0.681787
$193$	−2.62717e6	−0.0263050	−0.0131525	−	0.999914i	$-0.504187\pi$
$193$	−2.62717e6	−0.0263050	−0.0131525	+	0.999914i	$0.504187\pi$
$194$	1.37632e8	1.35336
$195$	4.18022e6	0.0403718
$196$	−1.25174e7	−0.118746
$197$	2.01380e7	0.187666	0.0938328	−	0.995588i	$-0.470088\pi$
$197$	2.01380e7	0.187666	0.0938328	+	0.995588i	$0.470088\pi$
$198$	−1.88035e7	−0.172152
$199$	−5.33613e7	−0.480000	−0.240000	−	0.970773i	$-0.577147\pi$
$199$	−5.33613e7	−0.480000	−0.240000	+	0.970773i	$0.577147\pi$
$200$	−8.05760e7	−0.712198
$201$	−1.27394e7	−0.110653
$202$	−1.64689e8	−1.40583
$203$	1.12652e8	0.945150
$204$	1.66831e8	1.37585
$205$	1.69301e8	1.37252
$206$	−6.78733e7	−0.540958
$207$	7.67660e7	0.601551
$208$	6.08710e6	0.0469017
$209$	−3.28800e6	−0.0249127
$210$	1.54574e8	1.15178
$211$	−6.49227e7	−0.475782	−0.237891	−	0.971292i	$-0.576456\pi$
$211$	−6.49227e7	−0.475782	−0.237891	+	0.971292i	$0.576456\pi$
$212$	−5.02742e8	−3.62385
$213$	8.72082e7	0.618342
$214$	3.15084e8	2.19775
$215$	3.22973e8	2.21632
$216$	−4.56013e7	−0.307886
$217$	−1.85425e8	−1.23186
$218$	2.53635e6	0.0165811
$219$	9.92000e6	0.0638201
$220$	1.10713e8	0.701000
$221$	1.15008e7	0.0716731
$222$	−2.06587e6	−0.0126727
$223$	−1.06871e8	−0.645346	−0.322673	−	0.946510i	$-0.604581\pi$
$223$	−1.06871e8	−0.645346	−0.322673	+	0.946510i	$0.604581\pi$
$224$	−3.56375e7	−0.211855
$225$	2.53540e7	0.148391
$226$	−1.21117e8	−0.697954
$227$	8.38607e7	0.475848	0.237924	−	0.971284i	$-0.423533\pi$
$227$	8.38607e7	0.475848	0.237924	+	0.971284i	$0.423533\pi$
$228$	−1.65113e7	−0.0922592
$229$	−3.12025e8	−1.71698	−0.858490	−	0.512831i	$-0.828597\pi$
$229$	−3.12025e8	−1.71698	−0.858490	+	0.512831i	$0.828597\pi$
$230$	−6.85694e8	−3.71606
$231$	−3.15955e7	−0.168649
$232$	2.96852e8	1.56074
$233$	1.69763e8	0.879221	0.439610	−	0.898189i	$-0.355117\pi$
$233$	1.69763e8	0.879221	0.439610	+	0.898189i	$0.355117\pi$
$234$	−6.50941e6	−0.0332113
$235$	4.50015e8	2.26198
$236$	−6.72841e8	−3.33212
$237$	−1.07755e8	−0.525796
$238$	4.25272e8	2.04478
$239$	2.79109e8	1.32246	0.661228	−	0.750185i	$-0.270036\pi$
$239$	2.79109e8	1.32246	0.661228	+	0.750185i	$0.270036\pi$
$240$	1.19853e8	0.559641
$241$	1.34553e8	0.619202	0.309601	−	0.950867i	$-0.399805\pi$
$241$	1.34553e8	0.619202	0.309601	+	0.950867i	$0.399805\pi$
$242$	−3.43313e7	−0.155717
$243$	1.43489e7	0.0641500
$244$	−1.35744e8	−0.598211
$245$	−1.69904e7	−0.0738114
$246$	−2.63634e8	−1.12909
$247$	−1.13824e6	−0.00480612
$248$	−4.88620e8	−2.03418
$249$	−1.56573e6	−0.00642716
$250$	2.82252e8	1.14247
$251$	3.33150e8	1.32979	0.664894	−	0.746938i	$-0.268477\pi$
$251$	3.33150e8	1.32979	0.664894	+	0.746938i	$0.268477\pi$
$252$	−1.58663e8	−0.624559
$253$	1.40158e8	0.544124
$254$	−4.50467e7	−0.172482
$255$	2.26448e8	0.855219
$256$	−5.12514e8	−1.90926
$257$	1.39421e8	0.512343	0.256172	−	0.966631i	$-0.417539\pi$
$257$	1.39421e8	0.512343	0.256172	+	0.966631i	$0.417539\pi$
$258$	−5.02932e8	−1.82323
$259$	−3.47128e6	−0.0124148
$260$	3.83265e7	0.135236
$261$	−9.34073e7	−0.325191
$262$	−6.34626e7	−0.218003
$263$	−5.39031e8	−1.82713	−0.913564	−	0.406695i	$-0.866681\pi$
$263$	−5.39031e8	−1.82713	−0.913564	+	0.406695i	$0.866681\pi$
$264$	−8.32584e7	−0.278493
$265$	−6.82396e8	−2.25256
$266$	−4.20893e7	−0.137115
$267$	2.78445e7	0.0895261
$268$	−1.16802e8	−0.370662
$269$	2.21835e8	0.694860	0.347430	−	0.937706i	$-0.387054\pi$
$269$	2.21835e8	0.694860	0.347430	+	0.937706i	$0.387054\pi$
$270$	−1.28168e8	−0.396285
$271$	1.89715e8	0.579042	0.289521	−	0.957172i	$-0.406504\pi$
$271$	1.89715e8	0.579042	0.289521	+	0.957172i	$0.406504\pi$
$272$	3.29746e8	0.993546
$273$	−1.09377e7	−0.0325356
$274$	5.95555e8	1.74902
$275$	4.62911e7	0.134225
$276$	7.03832e8	2.01506
$277$	4.74541e8	1.34151	0.670756	−	0.741678i	$-0.265970\pi$
$277$	4.74541e8	1.34151	0.670756	+	0.741678i	$0.265970\pi$
$278$	1.20720e9	3.36995
$279$	1.53749e8	0.423836
$280$	6.84423e8	1.86325
$281$	−2.21947e8	−0.596730	−0.298365	−	0.954452i	$-0.596441\pi$
$281$	−2.21947e8	−0.596730	−0.298365	+	0.954452i	$0.596441\pi$
$282$	−7.00761e8	−1.86079
$283$	−9.83605e7	−0.257969	−0.128985	−	0.991647i	$-0.541172\pi$
$283$	−9.83605e7	−0.257969	−0.128985	+	0.991647i	$0.541172\pi$
$284$	7.99572e8	2.07130
$285$	−2.24116e7	−0.0573477
$286$	−1.18848e7	−0.0300408
$287$	−4.42983e8	−1.10611
$288$	2.95495e7	0.0728915
$289$	2.12676e8	0.518294
$290$	8.34339e8	2.00886
$291$	−1.91756e8	−0.456166
$292$	9.09519e7	0.213782
$293$	−1.33842e8	−0.310854	−0.155427	−	0.987847i	$-0.549675\pi$
$293$	−1.33842e8	−0.310854	−0.155427	+	0.987847i	$0.549675\pi$
$294$	2.64574e7	0.0607200
$295$	−9.13280e8	−2.07122
$296$	−9.14727e6	−0.0205008
$297$	2.61981e7	0.0580259
$298$	2.51862e8	0.551323
$299$	4.85201e7	0.104972
$300$	2.32460e8	0.497076
$301$	−8.45074e8	−1.78613
$302$	4.13298e8	0.863455
$303$	2.29453e8	0.473853
$304$	−3.26350e7	−0.0666234
$305$	−1.84251e8	−0.371844
$306$	−3.52623e8	−0.703535
$307$	5.34037e8	1.05338	0.526692	−	0.850056i	$-0.323432\pi$
$307$	5.34037e8	1.05338	0.526692	+	0.850056i	$0.323432\pi$
$308$	−2.89685e8	−0.564935
$309$	9.45646e7	0.182337
$310$	−1.37333e9	−2.61823
$311$	5.06144e8	0.954142	0.477071	−	0.878865i	$-0.341698\pi$
$311$	5.06144e8	0.954142	0.477071	+	0.878865i	$0.341698\pi$
$312$	−2.88224e7	−0.0537265
$313$	4.15318e8	0.765554	0.382777	−	0.923841i	$-0.374968\pi$
$313$	4.15318e8	0.765554	0.382777	+	0.923841i	$0.374968\pi$
$314$	−3.19472e8	−0.582342
$315$	−2.15361e8	−0.388221
$316$	−9.87954e8	−1.76129
$317$	−4.42808e8	−0.780743	−0.390372	−	0.920657i	$-0.627653\pi$
$317$	−4.42808e8	−0.780743	−0.390372	+	0.920657i	$0.627653\pi$
$318$	1.06262e9	1.85304
$319$	−1.70542e8	−0.294147
$320$	−8.32136e8	−1.41961
$321$	−4.38992e8	−0.740779
$322$	1.79415e9	2.99477
$323$	−6.16600e7	−0.101811
$324$	1.31559e8	0.214888
$325$	1.60251e7	0.0258945
$326$	−5.09569e8	−0.814594
$327$	−3.53378e6	−0.00558885
$328$	−1.16732e9	−1.82655
$329$	−1.17748e9	−1.82293
$330$	−2.34008e8	−0.358453
$331$	−3.06232e8	−0.464144	−0.232072	−	0.972699i	$-0.574551\pi$
$331$	−3.06232e8	−0.464144	−0.232072	+	0.972699i	$0.574551\pi$
$332$	−1.43555e7	−0.0215295
$333$	2.87828e6	0.00427148
$334$	−2.03042e9	−2.98176
$335$	−1.58541e8	−0.230401
$336$	−3.13601e8	−0.451014
$337$	8.11501e7	0.115501	0.0577503	−	0.998331i	$-0.481607\pi$
$337$	8.11501e7	0.115501	0.0577503	+	0.998331i	$0.481607\pi$
$338$	1.21190e9	1.70709
$339$	1.68747e8	0.235254
$340$	2.07620e9	2.86478
$341$	2.80713e8	0.383374
$342$	3.48992e7	0.0471764
$343$	7.68508e8	1.02830
$344$	−2.22688e9	−2.94946
$345$	9.55345e8	1.25254
$346$	1.05984e9	1.37554
$347$	−1.03404e9	−1.32857	−0.664284	−	0.747481i	$-0.731263\pi$
$347$	−1.03404e9	−1.32857	−0.664284	+	0.747481i	$0.731263\pi$
$348$	−8.56409e8	−1.08932
$349$	−2.16551e8	−0.272691	−0.136346	−	0.990661i	$-0.543536\pi$
$349$	−2.16551e8	−0.272691	−0.136346	+	0.990661i	$0.543536\pi$
$350$	5.92567e8	0.738753
$351$	9.06925e6	0.0111943
$352$	5.39512e7	0.0659328
$353$	4.52788e8	0.547878	0.273939	−	0.961747i	$-0.411673\pi$
$353$	4.52788e8	0.547878	0.273939	+	0.961747i	$0.411673\pi$
$354$	1.42215e9	1.70387
$355$	1.08530e9	1.28751
$356$	2.55293e8	0.299892
$357$	−5.92511e8	−0.689220
$358$	−1.29124e9	−1.48737
$359$	−1.49122e9	−1.70103	−0.850515	−	0.525950i	$-0.823710\pi$
$359$	−1.49122e9	−1.70103	−0.850515	+	0.525950i	$0.823710\pi$
$360$	−5.67504e8	−0.641077
$361$	−8.87769e8	−0.993173
$362$	2.61464e9	2.89689
$363$	4.78321e7	0.0524864
$364$	−1.00283e8	−0.108987
$365$	1.23453e8	0.132886
$366$	2.86915e8	0.305893
$367$	1.50796e9	1.59243	0.796213	−	0.605016i	$-0.206833\pi$
$367$	1.50796e9	1.59243	0.796213	+	0.605016i	$0.206833\pi$
$368$	1.39114e9	1.45514
$369$	3.67308e8	0.380573
$370$	−2.57096e7	−0.0263869
$371$	1.78552e9	1.81533
$372$	1.40966e9	1.41975
$373$	−1.41874e9	−1.41554	−0.707769	−	0.706444i	$-0.750298\pi$
$373$	−1.41874e9	−1.41554	−0.707769	+	0.706444i	$0.750298\pi$
$374$	−6.43816e8	−0.636372
$375$	−3.93247e8	−0.385085
$376$	−3.10283e9	−3.01024
$377$	−5.90383e7	−0.0567465
$378$	3.35358e8	0.319365
$379$	−5.57982e8	−0.526481	−0.263241	−	0.964730i	$-0.584791\pi$
$379$	−5.57982e8	−0.526481	−0.263241	+	0.964730i	$0.584791\pi$
$380$	−2.05482e8	−0.192101
$381$	6.27613e7	0.0581373
$382$	1.68931e9	1.55055
$383$	−8.53420e6	−0.00776188	−0.00388094	−	0.999992i	$-0.501235\pi$
$383$	−8.53420e6	−0.00776188	−0.00388094	+	0.999992i	$0.501235\pi$
$384$	1.15571e9	1.04157
$385$	−3.93203e8	−0.351159
$386$	5.09123e7	0.0450575
$387$	7.00711e8	0.614540
$388$	−1.75812e9	−1.52805
$389$	3.19743e8	0.275408	0.137704	−	0.990473i	$-0.456028\pi$
$389$	3.19743e8	0.275408	0.137704	+	0.990473i	$0.456028\pi$
$390$	−8.10090e7	−0.0691524
$391$	2.62839e9	2.22368
$392$	1.17148e8	0.0982278
$393$	8.84194e7	0.0734807
$394$	−3.90257e8	−0.321450
$395$	−1.34100e9	−1.09481
$396$	2.40198e8	0.194373
$397$	2.61242e8	0.209545	0.104772	−	0.994496i	$-0.466589\pi$
$397$	2.61242e8	0.209545	0.104772	+	0.994496i	$0.466589\pi$
$398$	1.03410e9	0.822186
$399$	5.86410e7	0.0462164
$400$	4.59462e8	0.358955
$401$	−1.35042e9	−1.04584	−0.522920	−	0.852382i	$-0.675157\pi$
$401$	−1.35042e9	−1.04584	−0.522920	+	0.852382i	$0.675157\pi$
$402$	2.46879e8	0.189536
$403$	9.71774e7	0.0739601
$404$	2.10375e9	1.58730
$405$	1.78571e8	0.133573
$406$	−2.18309e9	−1.61894
$407$	5.25513e6	0.00386370
$408$	−1.56135e9	−1.13812
$409$	1.20771e9	0.872836	0.436418	−	0.899744i	$-0.356247\pi$
$409$	1.20771e9	0.872836	0.436418	+	0.899744i	$0.356247\pi$
$410$	−3.28090e9	−2.35098
$411$	−8.29758e8	−0.589529
$412$	8.67020e8	0.610785
$413$	2.38964e9	1.66920
$414$	−1.48766e9	−1.03039
$415$	−1.94854e7	−0.0133826
$416$	1.86768e7	0.0127197
$417$	−1.68194e9	−1.13588
$418$	6.37186e7	0.0426726
$419$	−1.08041e9	−0.717532	−0.358766	−	0.933427i	$-0.616802\pi$
$419$	−1.08041e9	−0.717532	−0.358766	+	0.933427i	$0.616802\pi$
$420$	−1.97454e9	−1.30045
$421$	1.12814e9	0.736845	0.368423	−	0.929658i	$-0.379898\pi$
$421$	1.12814e9	0.736845	0.368423	+	0.929658i	$0.379898\pi$
$422$	1.25814e9	0.814962
$423$	9.76336e8	0.627203
$424$	4.70508e9	2.99769
$425$	8.68098e8	0.548539
$426$	−1.69002e9	−1.05915
$427$	4.82102e8	0.299668
$428$	−4.02491e9	−2.48144
$429$	1.65585e7	0.0101256
$430$	−6.25894e9	−3.79630
$431$	2.87215e9	1.72797	0.863985	−	0.503518i	$-0.167961\pi$
$431$	2.87215e9	1.72797	0.863985	+	0.503518i	$0.167961\pi$
$432$	2.60029e8	0.155177
$433$	−1.19052e9	−0.704740	−0.352370	−	0.935861i	$-0.614624\pi$
$433$	−1.19052e9	−0.704740	−0.352370	+	0.935861i	$0.614624\pi$
$434$	3.59338e9	2.11003
$435$	−1.16244e9	−0.677111
$436$	−3.23996e7	−0.0187213
$437$	−2.60133e8	−0.149111
$438$	−1.92241e8	−0.109317
$439$	6.09735e8	0.343966	0.171983	−	0.985100i	$-0.444983\pi$
$439$	6.09735e8	0.343966	0.171983	+	0.985100i	$0.444983\pi$
$440$	−1.03614e9	−0.579876
$441$	−3.68618e7	−0.0204664
$442$	−2.22876e8	−0.122768
$443$	−3.00323e9	−1.64125	−0.820626	−	0.571466i	$-0.806375\pi$
$443$	−3.00323e9	−1.64125	−0.820626	+	0.571466i	$0.806375\pi$
$444$	2.63896e7	0.0143085
$445$	3.46522e8	0.186411
$446$	2.07107e9	1.10541
$447$	−3.50908e8	−0.185830
$448$	2.17732e9	1.14406
$449$	−2.75859e9	−1.43822	−0.719109	−	0.694898i	$-0.755450\pi$
$449$	−2.75859e9	−1.43822	−0.719109	+	0.694898i	$0.755450\pi$
$450$	−4.91339e8	−0.254178
$451$	6.70627e8	0.344242
$452$	1.54716e9	0.788046
$453$	−5.75829e8	−0.291038
$454$	−1.62515e9	−0.815075
$455$	−1.36119e8	−0.0677453
$456$	1.54527e8	0.0763180
$457$	−1.88695e9	−0.924812	−0.462406	−	0.886668i	$-0.653014\pi$
$457$	−1.88695e9	−0.924812	−0.462406	+	0.886668i	$0.653014\pi$
$458$	6.04677e9	2.94100
$459$	4.91293e8	0.237135
$460$	8.75912e9	4.19573
$461$	2.58243e9	1.22765	0.613827	−	0.789441i	$-0.289629\pi$
$461$	2.58243e9	1.22765	0.613827	+	0.789441i	$0.289629\pi$
$462$	6.12293e8	0.288877
$463$	−1.15752e8	−0.0541995	−0.0270998	−	0.999633i	$-0.508627\pi$
$463$	−1.15752e8	−0.0541995	−0.0270998	+	0.999633i	$0.508627\pi$
$464$	−1.69271e9	−0.786630
$465$	1.91339e9	0.882508
$466$	−3.28986e9	−1.50601
$467$	−1.97950e9	−0.899388	−0.449694	−	0.893183i	$-0.648467\pi$
$467$	−1.97950e9	−0.899388	−0.449694	+	0.893183i	$0.648467\pi$
$468$	8.31518e7	0.0374983
$469$	4.14829e8	0.185680
$470$	−8.72090e9	−3.87453
$471$	4.45104e8	0.196286
$472$	6.29702e9	2.75637
$473$	1.27935e9	0.555873
$474$	2.08819e9	0.900630
$475$	−8.59159e7	−0.0367829
$476$	−5.43246e9	−2.30873
$477$	−1.48050e9	−0.624589
$478$	−5.40889e9	−2.26522
$479$	2.83797e9	1.17987	0.589934	−	0.807451i	$-0.299154\pi$
$479$	2.83797e9	1.17987	0.589934	+	0.807451i	$0.299154\pi$
$480$	3.67741e8	0.151774
$481$	1.81922e6	0.000745380 0
$482$	−2.60751e9	−1.06062
$483$	−2.49970e9	−1.00942
$484$	4.38551e8	0.175817
$485$	−2.38638e9	−0.949825
$486$	−2.78069e8	−0.109882
$487$	−1.13328e9	−0.444618	−0.222309	−	0.974976i	$-0.571359\pi$
$487$	−1.13328e9	−0.444618	−0.222309	+	0.974976i	$0.571359\pi$
$488$	1.27040e9	0.494848
$489$	7.09958e8	0.274569
$490$	3.29260e8	0.126431
$491$	−2.65492e9	−1.01220	−0.506099	−	0.862475i	$-0.668913\pi$
$491$	−2.65492e9	−1.01220	−0.506099	+	0.862475i	$0.668913\pi$
$492$	3.36768e9	1.27483
$493$	−3.19818e9	−1.20209
$494$	2.20581e7	0.00823236
$495$	3.26032e8	0.120821
$496$	2.78622e9	1.02525
$497$	−2.83973e9	−1.03760
$498$	3.03425e7	0.0110090
$499$	2.34660e9	0.845448	0.422724	−	0.906259i	$-0.361074\pi$
$499$	2.34660e9	0.845448	0.422724	+	0.906259i	$0.361074\pi$
$500$	−3.60550e9	−1.28994
$501$	2.82888e9	1.00504
$502$	−6.45616e9	−2.27778
$503$	−8.19889e8	−0.287255	−0.143627	−	0.989632i	$-0.545877\pi$
$503$	−8.19889e8	−0.287255	−0.143627	+	0.989632i	$0.545877\pi$
$504$	1.48490e9	0.516643
$505$	2.85551e9	0.986653
$506$	−2.71615e9	−0.932023
$507$	−1.68848e9	−0.575397
$508$	5.75430e8	0.194746
$509$	5.08839e9	1.71028	0.855141	−	0.518395i	$-0.173470\pi$
$509$	5.08839e9	1.71028	0.855141	+	0.518395i	$0.173470\pi$
$510$	−4.38836e9	−1.46490
$511$	−3.23021e8	−0.107092
$512$	4.45315e9	1.46630
$513$	−4.86234e7	−0.0159014
$514$	−2.70185e9	−0.877588
$515$	1.17685e9	0.379660
$516$	6.42450e9	2.05857
$517$	1.78258e9	0.567326
$518$	6.72703e7	0.0212652
$519$	−1.47662e9	−0.463642
$520$	−3.58692e8	−0.111869
$521$	2.41340e9	0.747649	0.373825	−	0.927499i	$-0.378046\pi$
$521$	2.41340e9	0.747649	0.373825	+	0.927499i	$0.378046\pi$
$522$	1.81015e9	0.557017
$523$	3.32686e9	1.01690	0.508451	−	0.861091i	$-0.330218\pi$
$523$	3.32686e9	1.01690	0.508451	+	0.861091i	$0.330218\pi$
$524$	8.10677e8	0.246143
$525$	−8.25595e8	−0.249006
$526$	1.04460e10	3.12967
$527$	5.26422e9	1.56674
$528$	4.74758e8	0.140363
$529$	7.68392e9	2.25677
$530$	1.32242e10	3.85838
$531$	−1.98142e9	−0.574309
$532$	5.37653e8	0.154814
$533$	2.32158e8	0.0664107
$534$	−5.39602e8	−0.153348
$535$	−5.46320e9	−1.54244
$536$	1.09313e9	0.306616
$537$	1.79903e9	0.501335
$538$	−4.29897e9	−1.19022
$539$	−6.73019e7	−0.0185126
$540$	1.63723e9	0.447437
$541$	−2.52256e9	−0.684939	−0.342469	−	0.939529i	$-0.611263\pi$
$541$	−2.52256e9	−0.684939	−0.342469	+	0.939529i	$0.611263\pi$
$542$	−3.67652e9	−0.991834
$543$	−3.64286e9	−0.976433
$544$	1.01175e9	0.269449
$545$	−4.39775e7	−0.0116371
$546$	2.11964e8	0.0557298
$547$	5.62619e8	0.146980	0.0734901	−	0.997296i	$-0.476586\pi$
$547$	5.62619e8	0.146980	0.0734901	+	0.997296i	$0.476586\pi$
$548$	−7.60767e9	−1.97478
$549$	−3.99745e8	−0.103105
$550$	−8.97081e8	−0.229912
$551$	3.16525e8	0.0806078
$552$	−6.58705e9	−1.66688
$553$	3.50878e9	0.882304
$554$	−9.19620e9	−2.29786
$555$	3.58199e7	0.00889404
$556$	−1.54209e10	−3.80494
$557$	6.87397e9	1.68545	0.842723	−	0.538348i	$-0.180951\pi$
$557$	6.87397e9	1.68545	0.842723	+	0.538348i	$0.180951\pi$
$558$	−2.97952e9	−0.725984
$559$	4.42885e8	0.107238
$560$	−3.90273e9	−0.939097
$561$	8.96997e8	0.214497
$562$	4.30114e9	1.02213
$563$	−3.37523e9	−0.797120	−0.398560	−	0.917142i	$-0.630490\pi$
$563$	−3.37523e9	−0.797120	−0.398560	+	0.917142i	$0.630490\pi$
$564$	8.95158e9	2.10098
$565$	2.10004e9	0.489844
$566$	1.90614e9	0.441873
$567$	−4.67239e8	−0.107646
$568$	−7.48307e9	−1.71341
$569$	6.59246e8	0.150022	0.0750109	−	0.997183i	$-0.476101\pi$
$569$	6.59246e8	0.150022	0.0750109	+	0.997183i	$0.476101\pi$
$570$	4.34317e8	0.0982302
$571$	−7.92258e9	−1.78090	−0.890451	−	0.455079i	$-0.849611\pi$
$571$	−7.92258e9	−1.78090	−0.890451	+	0.455079i	$0.849611\pi$
$572$	1.51818e8	0.0339185
$573$	−2.35363e9	−0.522633
$574$	8.58462e9	1.89465
$575$	3.66236e9	0.803384
$576$	−1.80537e9	−0.393630
$577$	−6.35060e9	−1.37626	−0.688128	−	0.725589i	$-0.741567\pi$
$577$	−6.35060e9	−1.37626	−0.688128	+	0.725589i	$0.741567\pi$
$578$	−4.12148e9	−0.887780
$579$	−7.09336e7	−0.0151872
$580$	−1.06579e10	−2.26816
$581$	5.09843e7	0.0107850
$582$	3.71606e9	0.781362
$583$	−2.70308e9	−0.564962
$584$	−8.51204e8	−0.176843
$585$	1.12866e8	0.0233087
$586$	2.59375e9	0.532459
$587$	−5.18008e8	−0.105707	−0.0528534	−	0.998602i	$-0.516832\pi$
$587$	−5.18008e8	−0.105707	−0.0528534	+	0.998602i	$0.516832\pi$
$588$	−3.37969e8	−0.0685578
$589$	−5.21002e8	−0.105060
$590$	1.76986e10	3.54777
$591$	5.43726e8	0.108349
$592$	5.21598e7	0.0103326
$593$	−3.50248e9	−0.689739	−0.344870	−	0.938651i	$-0.612077\pi$
$593$	−3.50248e9	−0.689739	−0.344870	+	0.938651i	$0.612077\pi$
$594$	−5.07696e8	−0.0993919
$595$	−7.37374e9	−1.43509
$596$	−3.21731e9	−0.622488
$597$	−1.44076e9	−0.277128
$598$	−9.40277e8	−0.179805
$599$	−2.23154e9	−0.424239	−0.212120	−	0.977244i	$-0.568037\pi$
$599$	−2.23154e9	−0.424239	−0.212120	+	0.977244i	$0.568037\pi$
$600$	−2.17555e9	−0.411188
$601$	−3.31848e9	−0.623560	−0.311780	−	0.950154i	$-0.600925\pi$
$601$	−3.31848e9	−0.623560	−0.311780	+	0.950154i	$0.600925\pi$
$602$	1.63768e10	3.05944
$603$	−3.43964e8	−0.0638856
$604$	−5.27951e9	−0.974909
$605$	5.95266e8	0.109287
$606$	−4.44659e9	−0.811658
$607$	−5.90514e9	−1.07169	−0.535846	−	0.844316i	$-0.680007\pi$
$607$	−5.90514e9	−1.07169	−0.535846	+	0.844316i	$0.680007\pi$
$608$	−1.00133e8	−0.0180682
$609$	3.04159e9	0.545683
$610$	3.57063e9	0.636927
$611$	6.17095e8	0.109448
$612$	4.50444e9	0.794348
$613$	−4.56186e8	−0.0799891	−0.0399945	−	0.999200i	$-0.512734\pi$
$613$	−4.56186e8	−0.0799891	−0.0399945	+	0.999200i	$0.512734\pi$
$614$	−1.03492e10	−1.80433
$615$	4.57111e9	0.792427
$616$	2.71111e9	0.467321
$617$	1.36012e9	0.233120	0.116560	−	0.993184i	$-0.462813\pi$
$617$	1.36012e9	0.233120	0.116560	+	0.993184i	$0.462813\pi$
$618$	−1.83258e9	−0.312322
$619$	8.59252e9	1.45614	0.728071	−	0.685502i	$-0.240417\pi$
$619$	8.59252e9	1.45614	0.728071	+	0.685502i	$0.240417\pi$
$620$	1.75430e10	2.95620
$621$	2.07268e9	0.347306
$622$	−9.80864e9	−1.63434
$623$	−9.06690e8	−0.150228
$624$	1.64352e8	0.0270787
$625$	−7.61105e9	−1.24699
$626$	−8.04850e9	−1.31131
$627$	−8.87761e7	−0.0143833
$628$	4.08096e9	0.657511
$629$	9.85496e7	0.0157898
$630$	4.17350e9	0.664980
$631$	8.39016e9	1.32944	0.664718	−	0.747094i	$-0.268552\pi$
$631$	8.39016e9	1.32944	0.664718	+	0.747094i	$0.268552\pi$
$632$	9.24610e9	1.45696
$633$	−1.75291e9	−0.274693
$634$	8.58124e9	1.33733
$635$	7.81058e8	0.121053
$636$	−1.35740e10	−2.09223
$637$	−2.32986e7	−0.00357143
$638$	3.30496e9	0.503841
$639$	2.35462e9	0.357000
$640$	1.43827e10	2.16876
$641$	−1.17180e10	−1.75731	−0.878656	−	0.477456i	$-0.841559\pi$
$641$	−1.17180e10	−1.75731	−0.878656	+	0.477456i	$0.841559\pi$
$642$	8.50727e9	1.26887
$643$	6.76104e9	1.00294	0.501470	−	0.865175i	$-0.332793\pi$
$643$	6.76104e9	1.00294	0.501470	+	0.865175i	$0.332793\pi$
$644$	−2.29186e10	−3.38134
$645$	8.72027e9	1.27959
$646$	1.19492e9	0.174391
$647$	−3.99383e9	−0.579728	−0.289864	−	0.957068i	$-0.593610\pi$
$647$	−3.99383e9	−0.579728	−0.289864	+	0.957068i	$0.593610\pi$
$648$	−1.23124e9	−0.177758
$649$	−3.61765e9	−0.519482
$650$	−3.10552e8	−0.0443545
$651$	−5.00648e9	−0.711212
$652$	6.50928e9	0.919742
$653$	8.99014e9	1.26349	0.631743	−	0.775178i	$-0.282340\pi$
$653$	8.99014e9	1.26349	0.631743	+	0.775178i	$0.282340\pi$
$654$	6.84816e7	0.00957308
$655$	1.10037e9	0.153001
$656$	6.65631e9	0.920598
$657$	2.67840e8	0.0368465
$658$	2.28186e10	3.12248
$659$	−1.01242e10	−1.37804	−0.689022	−	0.724740i	$-0.741960\pi$
$659$	−1.01242e10	−1.37804	−0.689022	+	0.724740i	$0.741960\pi$
$660$	2.98924e9	0.404722
$661$	7.76430e9	1.04568	0.522838	−	0.852432i	$-0.324873\pi$
$661$	7.76430e9	1.04568	0.522838	+	0.852432i	$0.324873\pi$
$662$	5.93451e9	0.795028
$663$	3.10522e8	0.0413805
$664$	1.34350e8	0.0178095
$665$	7.29781e8	0.0962314
$666$	−5.57786e7	−0.00731657
$667$	−1.34926e10	−1.76057
$668$	2.59367e10	3.36665
$669$	−2.88552e9	−0.372591
$670$	3.07238e9	0.394651
$671$	−7.29849e8	−0.0932619
$672$	−9.62211e8	−0.122315
$673$	4.03659e9	0.510460	0.255230	−	0.966880i	$-0.417849\pi$
$673$	4.03659e9	0.510460	0.255230	+	0.966880i	$0.417849\pi$
$674$	−1.57262e9	−0.197840
$675$	6.84559e8	0.0856737
$676$	−1.54809e10	−1.92745
$677$	−8.86216e9	−1.09769	−0.548844	−	0.835925i	$-0.684932\pi$
$677$	−8.86216e9	−1.09769	−0.548844	+	0.835925i	$0.684932\pi$
$678$	−3.27017e9	−0.402964
$679$	6.24408e9	0.765463
$680$	−1.94308e10	−2.36979
$681$	2.26424e9	0.274731
$682$	−5.43998e9	−0.656677
$683$	−6.04044e9	−0.725432	−0.362716	−	0.931900i	$-0.618150\pi$
$683$	−6.04044e9	−0.725432	−0.362716	+	0.931900i	$0.618150\pi$
$684$	−4.45806e8	−0.0532659
$685$	−1.03262e10	−1.22751
$686$	−1.48930e10	−1.76136
$687$	−8.42467e9	−0.991299
$688$	1.26982e10	1.48656
$689$	−9.35753e8	−0.108992
$690$	−1.85137e10	−2.14547
$691$	2.98957e8	0.0344696	0.0172348	−	0.999851i	$-0.494514\pi$
$691$	2.98957e8	0.0344696	0.0172348	+	0.999851i	$0.494514\pi$
$692$	−1.35384e10	−1.55309
$693$	−8.53079e8	−0.0973695
$694$	2.00388e10	2.27569
$695$	−2.09315e10	−2.36512
$696$	8.01499e9	0.901096
$697$	1.25763e10	1.40682
$698$	4.19657e9	0.467090
$699$	4.58361e9	0.507618
$700$	−7.56950e9	−0.834111
$701$	−6.71671e9	−0.736450	−0.368225	−	0.929737i	$-0.620034\pi$
$701$	−6.71671e9	−0.736450	−0.368225	+	0.929737i	$0.620034\pi$
$702$	−1.75754e8	−0.0191746
$703$	−9.75348e6	−0.00105881
$704$	−3.29623e9	−0.356052
$705$	1.21504e10	1.30596
$706$	−8.77464e9	−0.938454
$707$	−7.47159e9	−0.795142
$708$	−1.81667e10	−1.92380
$709$	−7.82020e8	−0.0824055	−0.0412028	−	0.999151i	$-0.513119\pi$
$709$	−7.82020e8	−0.0824055	−0.0412028	+	0.999151i	$0.513119\pi$
$710$	−2.10321e10	−2.20536
$711$	−2.90938e9	−0.303568
$712$	−2.38925e9	−0.248074
$713$	2.22089e10	2.29463
$714$	1.14823e10	1.18056
$715$	2.06069e8	0.0210835
$716$	1.64945e10	1.67936
$717$	7.53595e9	0.763520
$718$	2.88986e10	2.91368
$719$	−4.32158e9	−0.433602	−0.216801	−	0.976216i	$-0.569562\pi$
$719$	−4.32158e9	−0.433602	−0.216801	+	0.976216i	$0.569562\pi$
$720$	3.23603e9	0.323109
$721$	−3.07928e9	−0.305967
$722$	1.72042e10	1.70120
$723$	3.63292e9	0.357496
$724$	−3.33997e10	−3.27082
$725$	−4.45629e9	−0.434300
$726$	−9.26945e8	−0.0899034
$727$	1.09540e10	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$727$	1.09540e10	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$728$	9.38534e8	0.0901550
$729$	3.87420e8	0.0370370
$730$	−2.39242e9	−0.227618
$731$	2.39917e10	2.27169
$732$	−3.66507e9	−0.345378
$733$	−7.81174e9	−0.732629	−0.366314	−	0.930491i	$-0.619381\pi$
$733$	−7.81174e9	−0.732629	−0.366314	+	0.930491i	$0.619381\pi$
$734$	−2.92230e10	−2.72765
$735$	−4.58742e8	−0.0426150
$736$	4.26839e9	0.394632
$737$	−6.28006e8	−0.0577867
$738$	−7.11812e9	−0.651880
$739$	2.45930e9	0.224159	0.112079	−	0.993699i	$-0.464249\pi$
$739$	2.45930e9	0.224159	0.112079	+	0.993699i	$0.464249\pi$
$740$	3.28416e8	0.0297930
$741$	−3.07325e7	−0.00277482
$742$	−3.46018e10	−3.10946
$743$	1.96680e10	1.75914	0.879568	−	0.475773i	$-0.157832\pi$
$743$	1.96680e10	1.75914	0.879568	+	0.475773i	$0.157832\pi$
$744$	−1.31927e10	−1.17444
$745$	−4.36701e9	−0.386934
$746$	2.74939e10	2.42466
$747$	−4.22747e7	−0.00371072
$748$	8.22415e9	0.718515
$749$	1.42947e10	1.24305
$750$	7.62079e9	0.659608
$751$	−6.82664e9	−0.588122	−0.294061	−	0.955787i	$-0.595007\pi$
$751$	−6.82664e9	−0.588122	−0.294061	+	0.955787i	$0.595007\pi$
$752$	1.76930e10	1.51719
$753$	8.99506e9	0.767753
$754$	1.14411e9	0.0972004
$755$	−7.16613e9	−0.605997
$756$	−4.28390e9	−0.360589
$757$	−1.16891e10	−0.979370	−0.489685	−	0.871899i	$-0.662888\pi$
$757$	−1.16891e10	−0.979370	−0.489685	+	0.871899i	$0.662888\pi$
$758$	1.08132e10	0.901804
$759$	3.78428e9	0.314150
$760$	1.92307e9	0.158909
$761$	1.15956e9	0.0953776	0.0476888	−	0.998862i	$-0.484814\pi$
$761$	1.15956e9	0.0953776	0.0476888	+	0.998862i	$0.484814\pi$
$762$	−1.21626e9	−0.0995827
$763$	1.15069e8	0.00937828
$764$	−2.15793e10	−1.75070
$765$	6.11409e9	0.493761
$766$	1.65385e8	0.0132952
$767$	−1.25236e9	−0.100218
$768$	−1.38379e10	−1.10231
$769$	3.98333e9	0.315867	0.157934	−	0.987450i	$-0.449517\pi$
$769$	3.98333e9	0.315867	0.157934	+	0.987450i	$0.449517\pi$
$770$	7.61993e9	0.601497
$771$	3.76436e9	0.295802
$772$	−6.50358e8	−0.0508735
$773$	1.53208e10	1.19304	0.596519	−	0.802599i	$-0.296550\pi$
$773$	1.53208e10	1.19304	0.596519	+	0.802599i	$0.296550\pi$
$774$	−1.35792e10	−1.05264
$775$	7.33508e9	0.566042
$776$	1.64540e10	1.26402
$777$	−9.37245e7	−0.00716769
$778$	−6.19633e9	−0.471744
$779$	−1.24468e9	−0.0943357
$780$	1.03482e9	0.0780786
$781$	4.29904e9	0.322919
$782$	−5.09360e10	−3.80891
$783$	−2.52200e9	−0.187749
$784$	−6.68005e8	−0.0495078
$785$	5.53928e9	0.408704
$786$	−1.71349e9	−0.125864
$787$	−1.83797e10	−1.34408	−0.672042	−	0.740513i	$-0.734583\pi$
$787$	−1.83797e10	−1.34408	−0.672042	+	0.740513i	$0.734583\pi$
$788$	4.98518e9	0.362943
$789$	−1.45538e10	−1.05489
$790$	2.59873e10	1.87528
$791$	−5.49485e9	−0.394764
$792$	−2.24798e9	−0.160788
$793$	−2.52659e8	−0.0179920
$794$	−5.06265e9	−0.358927
$795$	−1.84247e10	−1.30051
$796$	−1.32096e10	−0.928314
$797$	5.52162e9	0.386334	0.193167	−	0.981166i	$-0.438124\pi$
$797$	5.52162e9	0.386334	0.193167	+	0.981166i	$0.438124\pi$
$798$	−1.13641e9	−0.0791636
$799$	3.34288e10	2.31850
$800$	1.40975e9	0.0973482
$801$	7.51801e8	0.0516879
$802$	2.61701e10	1.79141
$803$	4.89019e8	0.0333289
$804$	−3.15365e9	−0.214002
$805$	−3.11086e10	−2.10181
$806$	−1.88321e9	−0.126685
$807$	5.98955e9	0.401178
$808$	−1.96886e10	−1.31303
$809$	2.79525e10	1.85610	0.928049	−	0.372458i	$-0.121485\pi$
$809$	2.79525e10	1.85610	0.928049	+	0.372458i	$0.121485\pi$
$810$	−3.46054e9	−0.228795
$811$	9.41377e9	0.619713	0.309857	−	0.950783i	$-0.399719\pi$
$811$	9.41377e9	0.619713	0.309857	+	0.950783i	$0.399719\pi$
$812$	2.78869e10	1.82791
$813$	5.12231e9	0.334310
$814$	−1.01840e8	−0.00661809
$815$	8.83535e9	0.571705
$816$	8.90314e9	0.573624
$817$	−2.37446e9	−0.152331
$818$	−2.34045e10	−1.49507
$819$	−2.95319e8	−0.0187844
$820$	4.19105e10	2.65445
$821$	1.04271e10	0.657600	0.328800	−	0.944400i	$-0.393356\pi$
$821$	1.04271e10	0.657600	0.328800	+	0.944400i	$0.393356\pi$
$822$	1.60800e10	1.00980
$823$	7.88032e9	0.492770	0.246385	−	0.969172i	$-0.420757\pi$
$823$	7.88032e9	0.492770	0.246385	+	0.969172i	$0.420757\pi$
$824$	−8.11430e9	−0.505249
$825$	1.24986e9	0.0774948
$826$	−4.63091e10	−2.85915
$827$	−1.66909e10	−1.02615	−0.513075	−	0.858344i	$-0.671494\pi$
$827$	−1.66909e10	−1.02615	−0.513075	+	0.858344i	$0.671494\pi$
$828$	1.90035e10	1.16339
$829$	−1.34679e10	−0.821029	−0.410515	−	0.911854i	$-0.634651\pi$
$829$	−1.34679e10	−0.821029	−0.410515	+	0.911854i	$0.634651\pi$
$830$	3.77609e8	0.0229229
$831$	1.28126e10	0.774523
$832$	−1.14109e9	−0.0686891
$833$	−1.26211e9	−0.0756556
$834$	3.25944e10	1.94564
$835$	3.52052e10	2.09268
$836$	−8.13947e8	−0.0481808
$837$	4.15123e9	0.244702
$838$	2.09375e10	1.22905
$839$	−2.24933e10	−1.31488	−0.657440	−	0.753507i	$-0.728361\pi$
$839$	−2.24933e10	−1.31488	−0.657440	+	0.753507i	$0.728361\pi$
$840$	1.84794e10	1.07575
$841$	−8.32384e8	−0.0482545
$842$	−2.18624e10	−1.26213
$843$	−5.99258e9	−0.344522
$844$	−1.60716e10	−0.920157
$845$	−2.10129e10	−1.19809
$846$	−1.89205e10	−1.07433
$847$	−1.55754e9	−0.0880740
$848$	−2.68294e10	−1.51087
$849$	−2.65573e9	−0.148939
$850$	−1.68230e10	−0.939586
$851$	4.15764e8	0.0231256
$852$	2.15884e10	1.19587
$853$	−2.72012e10	−1.50060	−0.750302	−	0.661095i	$-0.770092\pi$
$853$	−2.72012e10	−1.50060	−0.750302	+	0.661095i	$0.770092\pi$
$854$	−9.34271e9	−0.513299
$855$	−6.05113e8	−0.0331097
$856$	3.76685e10	2.05268
$857$	−9.82425e9	−0.533171	−0.266586	−	0.963811i	$-0.585895\pi$
$857$	−9.82425e9	−0.533171	−0.266586	+	0.963811i	$0.585895\pi$
$858$	−3.20890e8	−0.0173441
$859$	3.07089e9	0.165306	0.0826530	−	0.996578i	$-0.473661\pi$
$859$	3.07089e9	0.165306	0.0826530	+	0.996578i	$0.473661\pi$
$860$	7.99522e10	4.28633
$861$	−1.19605e10	−0.638616
$862$	−5.56597e10	−2.95982
$863$	9.85691e9	0.522039	0.261019	−	0.965334i	$-0.415941\pi$
$863$	9.85691e9	0.522039	0.261019	+	0.965334i	$0.415941\pi$
$864$	7.97837e8	0.0420839
$865$	−1.83764e10	−0.965391
$866$	2.30712e10	1.20714
$867$	5.74225e9	0.299237
$868$	−4.59021e10	−2.38239
$869$	−5.31191e9	−0.274588
$870$	2.25272e10	1.15982
$871$	−2.17403e8	−0.0111481
$872$	3.03223e8	0.0154865
$873$	−5.17740e9	−0.263368
$874$	5.04115e9	0.255411
$875$	1.28052e10	0.646186
$876$	2.45570e9	0.123427
$877$	3.56921e9	0.178679	0.0893395	−	0.996001i	$-0.471524\pi$
$877$	3.56921e9	0.178679	0.0893395	+	0.996001i	$0.471524\pi$
$878$	−1.18161e10	−0.589176
$879$	−3.61374e9	−0.179472
$880$	5.90831e9	0.292263
$881$	2.59339e10	1.27777	0.638883	−	0.769304i	$-0.279397\pi$
$881$	2.59339e10	1.27777	0.638883	+	0.769304i	$0.279397\pi$
$882$	7.14350e8	0.0350567
$883$	−2.42366e9	−0.118470	−0.0592352	−	0.998244i	$-0.518866\pi$
$883$	−2.42366e9	−0.118470	−0.0592352	+	0.998244i	$0.518866\pi$
$884$	2.84704e9	0.138615
$885$	−2.46585e10	−1.19582
$886$	5.81999e10	2.81128
$887$	4.66569e9	0.224483	0.112241	−	0.993681i	$-0.464197\pi$
$887$	4.66569e9	0.224483	0.112241	+	0.993681i	$0.464197\pi$
$888$	−2.46976e8	−0.0118361
$889$	−2.04368e9	−0.0975564
$890$	−6.71529e9	−0.319301
$891$	7.07348e8	0.0335013
$892$	−2.64560e10	−1.24809
$893$	−3.30846e9	−0.155470
$894$	6.80028e9	0.318307
$895$	2.23887e10	1.04388
$896$	−3.76330e10	−1.74780
$897$	1.31004e9	0.0606054
$898$	5.34590e10	2.46351
$899$	−2.70233e10	−1.24045
$900$	6.27641e9	0.286987
$901$	−5.06909e10	−2.30884
$902$	−1.29962e10	−0.589648
$903$	−2.28170e10	−1.03122
$904$	−1.44796e10	−0.651881
$905$	−4.53350e10	−2.03312
$906$	1.11591e10	0.498516
$907$	3.02139e10	1.34456	0.672281	−	0.740296i	$-0.265315\pi$
$907$	3.02139e10	1.34456	0.672281	+	0.740296i	$0.265315\pi$
$908$	2.07598e10	0.920285
$909$	6.19522e9	0.273579
$910$	2.63787e9	0.116040
$911$	2.02757e10	0.888507	0.444254	−	0.895901i	$-0.353469\pi$
$911$	2.02757e10	0.888507	0.444254	+	0.895901i	$0.353469\pi$
$912$	−8.81146e8	−0.0384650
$913$	−7.71847e7	−0.00335648
$914$	3.65674e10	1.58410
$915$	−4.97478e9	−0.214684
$916$	−7.72420e10	−3.32062
$917$	−2.87917e9	−0.123303
$918$	−9.52083e9	−0.406186
$919$	−2.30171e10	−0.978244	−0.489122	−	0.872215i	$-0.662683\pi$
$919$	−2.30171e10	−0.978244	−0.489122	+	0.872215i	$0.662683\pi$
$920$	−8.19752e10	−3.47076
$921$	1.44190e10	0.608172
$922$	−5.00453e10	−2.10284
$923$	1.48824e9	0.0622971
$924$	−7.82149e9	−0.326165
$925$	1.37317e8	0.00570465
$926$	2.24317e9	0.0928378
$927$	2.55324e9	0.105272
$928$	−5.19370e9	−0.213333
$929$	−3.32058e10	−1.35881	−0.679406	−	0.733762i	$-0.737763\pi$
$929$	−3.32058e10	−1.35881	−0.679406	+	0.733762i	$0.737763\pi$
$930$	−3.70799e10	−1.51164
$931$	1.24912e8	0.00507317
$932$	4.20250e10	1.70040
$933$	1.36659e10	0.550874
$934$	3.83610e10	1.54055
$935$	1.11630e10	0.446624
$936$	−7.78205e8	−0.0310190
$937$	−4.08091e10	−1.62057	−0.810286	−	0.586034i	$-0.800688\pi$
$937$	−4.08091e10	−1.62057	−0.810286	+	0.586034i	$0.800688\pi$
$938$	−8.03903e9	−0.318049
$939$	1.12136e10	0.441993
$940$	1.11401e11	4.37465
$941$	4.32999e9	0.169404	0.0847020	−	0.996406i	$-0.473006\pi$
$941$	4.32999e9	0.169404	0.0847020	+	0.996406i	$0.473006\pi$
$942$	−8.62573e9	−0.336216
$943$	5.30572e10	2.06041
$944$	−3.59070e10	−1.38924
$945$	−5.81473e9	−0.224140
$946$	−2.47927e10	−0.952148
$947$	−5.46744e8	−0.0209199	−0.0104599	−	0.999945i	$-0.503330\pi$
$947$	−5.46744e8	−0.0209199	−0.0104599	+	0.999945i	$0.503330\pi$
$948$	−2.66747e10	−1.01688
$949$	1.69289e8	0.00642978
$950$	1.66498e9	0.0630050
$951$	−1.19558e10	−0.450762
$952$	5.08416e10	1.90981
$953$	−2.09719e10	−0.784896	−0.392448	−	0.919774i	$-0.628372\pi$
$953$	−2.09719e10	−0.784896	−0.392448	+	0.919774i	$0.628372\pi$
$954$	2.86908e10	1.06985
$955$	−2.92906e10	−1.08822
$956$	6.90936e10	2.55762
$957$	−4.60464e9	−0.169826
$958$	−5.49974e10	−2.02098
$959$	2.70191e10	0.989250
$960$	−2.24677e10	−0.819613
$961$	1.69679e10	0.616733
$962$	−3.52549e7	−0.00127675
$963$	−1.18528e10	−0.427689
$964$	3.33086e10	1.19753
$965$	−8.82762e8	−0.0316226
$966$	4.84421e10	1.72903
$967$	4.39210e9	0.156200	0.0780998	−	0.996946i	$-0.475115\pi$
$967$	4.39210e9	0.156200	0.0780998	+	0.996946i	$0.475115\pi$
$968$	−4.10433e9	−0.145438
$969$	−1.66482e9	−0.0587806
$970$	4.62460e10	1.62694
$971$	5.21763e8	0.0182897	0.00914484	−	0.999958i	$-0.497089\pi$
$971$	5.21763e8	0.0182897	0.00914484	+	0.999958i	$0.497089\pi$
$972$	3.55208e9	0.124065
$973$	5.47683e10	1.90605
$974$	2.19620e10	0.761581
$975$	4.32677e8	0.0149502
$976$	−7.24411e9	−0.249408
$977$	−1.26384e10	−0.433571	−0.216786	−	0.976219i	$-0.569557\pi$
$977$	−1.26384e10	−0.433571	−0.216786	+	0.976219i	$0.569557\pi$
$978$	−1.37584e10	−0.470306
$979$	1.37263e9	0.0467535
$980$	−4.20599e9	−0.142750
$981$	−9.54120e7	−0.00322672
$982$	5.14499e10	1.73378
$983$	1.04670e10	0.351469	0.175734	−	0.984438i	$-0.443770\pi$
$983$	1.04670e10	0.351469	0.175734	+	0.984438i	$0.443770\pi$
$984$	−3.15176e10	−1.05456
$985$	6.76661e9	0.225603
$986$	6.19779e10	2.05905
$987$	−3.17921e10	−1.05247
$988$	−2.81772e8	−0.00929499
$989$	1.01217e11	3.32710
$990$	−6.31822e9	−0.206953
$991$	1.70223e9	0.0555597	0.0277799	−	0.999614i	$-0.491156\pi$
$991$	1.70223e9	0.0555597	0.0277799	+	0.999614i	$0.491156\pi$
$992$	8.54886e9	0.278046
$993$	−8.26827e9	−0.267974
$994$	5.50315e10	1.77729
$995$	−1.79301e10	−0.577033
$996$	−3.87598e8	−0.0124301
$997$	−3.16766e10	−1.01229	−0.506145	−	0.862448i	$-0.668930\pi$
$997$	−3.16766e10	−1.01229	−0.506145	+	0.862448i	$0.668930\pi$
$998$	−4.54750e10	−1.44816
$999$	7.77136e7	0.00246614

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.a.e.1.1	✓	4
3.2	odd	2		99.8.a.f.1.4		4
4.3	odd	2		528.8.a.r.1.3		4
11.10	odd	2		363.8.a.f.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.a.e.1.1	✓	4	1.1	even	1	trivial
99.8.a.f.1.4		4	3.2	odd	2
363.8.a.f.1.4		4	11.10	odd	2
528.8.a.r.1.3		4	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	33.a (trivial)

\(f(q)\)	\(=\)	\(q-19.3791 q^{2} +27.0000 q^{3} +247.551 q^{4} +336.012 q^{5} -523.236 q^{6} -879.192 q^{7} -2316.79 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-19.3791 q^{2} +27.0000 q^{3} +247.551 q^{4} +336.012 q^{5} -523.236 q^{6} -879.192 q^{7} -2316.79 q^{8} +729.000 q^{9} -6511.62 q^{10} +1331.00 q^{11} +6683.87 q^{12} +460.766 q^{13} +17038.0 q^{14} +9072.33 q^{15} +13210.8 q^{16} +24960.3 q^{17} -14127.4 q^{18} -2470.32 q^{19} +83180.0 q^{20} -23738.2 q^{21} -25793.6 q^{22} +105303. q^{23} -62553.2 q^{24} +34779.2 q^{25} -8929.24 q^{26} +19683.0 q^{27} -217644. q^{28} -128131. q^{29} -175814. q^{30} +210904. q^{31} +40534.3 q^{32} +35937.0 q^{33} -483708. q^{34} -295419. q^{35} +180464. q^{36} +3948.26 q^{37} +47872.7 q^{38} +12440.7 q^{39} -778469. q^{40} +503852. q^{41} +460025. q^{42} +961195. q^{43} +329490. q^{44} +244953. q^{45} -2.04068e6 q^{46} +1.33928e6 q^{47} +356692. q^{48} -50564.9 q^{49} -673991. q^{50} +673927. q^{51} +114063. q^{52} -2.03087e6 q^{53} -381439. q^{54} +447232. q^{55} +2.03690e6 q^{56} -66698.8 q^{57} +2.48306e6 q^{58} -2.71800e6 q^{59} +2.24586e6 q^{60} -548346. q^{61} -4.08714e6 q^{62} -640931. q^{63} -2.47651e6 q^{64} +154823. q^{65} -696428. q^{66} -471831. q^{67} +6.17893e6 q^{68} +2.84318e6 q^{69} +5.72497e6 q^{70} +3.22993e6 q^{71} -1.68894e6 q^{72} +367407. q^{73} -76513.8 q^{74} +939038. q^{75} -611530. q^{76} -1.17020e6 q^{77} -241089. q^{78} -3.99092e6 q^{79} +4.43900e6 q^{80} +531441. q^{81} -9.76422e6 q^{82} -57990.0 q^{83} -5.87640e6 q^{84} +8.38695e6 q^{85} -1.86271e7 q^{86} -3.45953e6 q^{87} -3.08364e6 q^{88} +1.03128e6 q^{89} -4.74697e6 q^{90} -405101. q^{91} +2.60679e7 q^{92} +5.69441e6 q^{93} -2.59541e7 q^{94} -830059. q^{95} +1.09443e6 q^{96} -7.10206e6 q^{97} +979905. q^{98} +970299. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 15 q^{2} + 108 q^{3} + 565 q^{4} + 306 q^{5} + 405 q^{6} + 890 q^{7} + 2457 q^{8} + 2916 q^{9}+O(q^{10}) \) 4 * q + 15 * q^2 + 108 * q^3 + 565 * q^4 + 306 * q^5 + 405 * q^6 + 890 * q^7 + 2457 * q^8 + 2916 * q^9	\( 4 q + 15 q^{2} + 108 q^{3} + 565 q^{4} + 306 q^{5} + 405 q^{6} + 890 q^{7} + 2457 q^{8} + 2916 q^{9} - 8946 q^{10} + 5324 q^{11} + 15255 q^{12} - 1822 q^{13} + 35340 q^{14} + 8262 q^{15} + 65041 q^{16} + 32856 q^{17} + 10935 q^{18} - 12784 q^{19} - 4002 q^{20} + 24030 q^{21} + 19965 q^{22} + 114858 q^{23} + 66339 q^{24} + 72856 q^{25} - 221466 q^{26} + 78732 q^{27} + 10112 q^{28} - 104952 q^{29} - 241542 q^{30} - 24976 q^{31} + 337761 q^{32} + 143748 q^{33} - 741690 q^{34} - 722856 q^{35} + 411885 q^{36} - 498856 q^{37} - 897156 q^{38} - 49194 q^{39} - 2676930 q^{40} + 734556 q^{41} + 954180 q^{42} - 201916 q^{43} + 752015 q^{44} + 223074 q^{45} - 3068508 q^{46} + 1995894 q^{47} + 1756107 q^{48} - 771024 q^{49} + 1632129 q^{50} + 887112 q^{51} - 4412266 q^{52} + 929970 q^{53} + 295245 q^{54} + 407286 q^{55} + 7224888 q^{56} - 345168 q^{57} + 2864322 q^{58} + 1353156 q^{59} - 108054 q^{60} + 3998774 q^{61} - 3783264 q^{62} + 648810 q^{63} + 1480129 q^{64} + 6612108 q^{65} + 539055 q^{66} + 1722008 q^{67} + 1596906 q^{68} + 3101166 q^{69} - 2751024 q^{70} + 5571858 q^{71} + 1791153 q^{72} + 5600528 q^{73} - 10907838 q^{74} + 1967112 q^{75} - 19634884 q^{76} + 1184590 q^{77} - 5979582 q^{78} - 7710226 q^{79} - 24073794 q^{80} + 2125764 q^{81} - 11230842 q^{82} + 3431856 q^{83} + 273024 q^{84} + 5909484 q^{85} - 25687140 q^{86} - 2833704 q^{87} + 3270267 q^{88} + 4611528 q^{89} - 6521634 q^{90} - 9032696 q^{91} + 13608576 q^{92} - 674352 q^{93} - 3497436 q^{94} + 21828000 q^{95} + 9119547 q^{96} + 1401692 q^{97} - 7230081 q^{98} + 3881196 q^{99}+O(q^{100}) \) 4 * q + 15 * q^2 + 108 * q^3 + 565 * q^4 + 306 * q^5 + 405 * q^6 + 890 * q^7 + 2457 * q^8 + 2916 * q^9 - 8946 * q^10 + 5324 * q^11 + 15255 * q^12 - 1822 * q^13 + 35340 * q^14 + 8262 * q^15 + 65041 * q^16 + 32856 * q^17 + 10935 * q^18 - 12784 * q^19 - 4002 * q^20 + 24030 * q^21 + 19965 * q^22 + 114858 * q^23 + 66339 * q^24 + 72856 * q^25 - 221466 * q^26 + 78732 * q^27 + 10112 * q^28 - 104952 * q^29 - 241542 * q^30 - 24976 * q^31 + 337761 * q^32 + 143748 * q^33 - 741690 * q^34 - 722856 * q^35 + 411885 * q^36 - 498856 * q^37 - 897156 * q^38 - 49194 * q^39 - 2676930 * q^40 + 734556 * q^41 + 954180 * q^42 - 201916 * q^43 + 752015 * q^44 + 223074 * q^45 - 3068508 * q^46 + 1995894 * q^47 + 1756107 * q^48 - 771024 * q^49 + 1632129 * q^50 + 887112 * q^51 - 4412266 * q^52 + 929970 * q^53 + 295245 * q^54 + 407286 * q^55 + 7224888 * q^56 - 345168 * q^57 + 2864322 * q^58 + 1353156 * q^59 - 108054 * q^60 + 3998774 * q^61 - 3783264 * q^62 + 648810 * q^63 + 1480129 * q^64 + 6612108 * q^65 + 539055 * q^66 + 1722008 * q^67 + 1596906 * q^68 + 3101166 * q^69 - 2751024 * q^70 + 5571858 * q^71 + 1791153 * q^72 + 5600528 * q^73 - 10907838 * q^74 + 1967112 * q^75 - 19634884 * q^76 + 1184590 * q^77 - 5979582 * q^78 - 7710226 * q^79 - 24073794 * q^80 + 2125764 * q^81 - 11230842 * q^82 + 3431856 * q^83 + 273024 * q^84 + 5909484 * q^85 - 25687140 * q^86 - 2833704 * q^87 + 3270267 * q^88 + 4611528 * q^89 - 6521634 * q^90 - 9032696 * q^91 + 13608576 * q^92 - 674352 * q^93 - 3497436 * q^94 + 21828000 * q^95 + 9119547 * q^96 + 1401692 * q^97 - 7230081 * q^98 + 3881196 * q^99

Introduction

L-functions

Modular forms

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