LMFDB - Embedded newform 61.8.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [61,8,Mod(1,61)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(61, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("61.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 61 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	61.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:	$19.0554865545$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 1853 x^{17} + 7710 x^{16} + 1430821 x^{15} - 6206876 x^{14} - 595839157 x^{13} + \cdots + 26\!\cdots\!12 $ x^19 - 4x^18 - 1853x^17 + 7710x^16 + 1430821x^15 - 6206876x^14 - 595839157x^13 + 2700452562x^12 + 144549098447x^11 - 684286581560x^10 - 20557296029519x^9 + 100969689451234x^8 + 1631786322213439x^7 - 8141606682319020x^6 - 63693090720294879x^5 + 306768753471031726x^4 + 905054963486192972x^3 - 3710179436322944828x^2 - 581545816296452448x + 264343180497275712
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{13}\cdot 3\cdot 7 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.340895$ of defining polynomial
Character	$\chi$	$=$	61.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.659105 q^{2} +52.6301 q^{3} -127.566 q^{4} +68.7263 q^{5} +34.6888 q^{6} +248.822 q^{7} -168.445 q^{8} +582.927 q^{9} +O(q^{10})$	\(q+0.659105 q^{2} +52.6301 q^{3} -127.566 q^{4} +68.7263 q^{5} +34.6888 q^{6} +248.822 q^{7} -168.445 q^{8} +582.927 q^{9} +45.2978 q^{10} +3412.37 q^{11} -6713.79 q^{12} +11965.7 q^{13} +164.000 q^{14} +3617.07 q^{15} +16217.4 q^{16} +7630.02 q^{17} +384.210 q^{18} -18115.6 q^{19} -8767.11 q^{20} +13095.5 q^{21} +2249.11 q^{22} +96490.7 q^{23} -8865.25 q^{24} -73401.7 q^{25} +7886.64 q^{26} -84422.5 q^{27} -31741.1 q^{28} +73043.4 q^{29} +2384.03 q^{30} +257315. q^{31} +32249.9 q^{32} +179593. q^{33} +5028.98 q^{34} +17100.6 q^{35} -74361.4 q^{36} +394936. q^{37} -11940.1 q^{38} +629755. q^{39} -11576.6 q^{40} +62815.8 q^{41} +8631.33 q^{42} +627894. q^{43} -435301. q^{44} +40062.4 q^{45} +63597.5 q^{46} -1.07241e6 q^{47} +853522. q^{48} -761631. q^{49} -48379.4 q^{50} +401568. q^{51} -1.52641e6 q^{52} -97183.7 q^{53} -55643.3 q^{54} +234520. q^{55} -41912.7 q^{56} -953424. q^{57} +48143.3 q^{58} -2.32444e6 q^{59} -461414. q^{60} -226981. q^{61} +169598. q^{62} +145045. q^{63} -2.05457e6 q^{64} +822357. q^{65} +118371. q^{66} +850187. q^{67} -973328. q^{68} +5.07831e6 q^{69} +11271.1 q^{70} +108707. q^{71} -98190.8 q^{72} -487519. q^{73} +260304. q^{74} -3.86314e6 q^{75} +2.31092e6 q^{76} +849073. q^{77} +415074. q^{78} +838970. q^{79} +1.11456e6 q^{80} -5.71803e6 q^{81} +41402.2 q^{82} -9.58150e6 q^{83} -1.67054e6 q^{84} +524383. q^{85} +413848. q^{86} +3.84428e6 q^{87} -574795. q^{88} -2.25354e6 q^{89} +26405.3 q^{90} +2.97732e6 q^{91} -1.23089e7 q^{92} +1.35425e7 q^{93} -706830. q^{94} -1.24502e6 q^{95} +1.69731e6 q^{96} -1.21224e7 q^{97} -501995. q^{98} +1.98916e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 23 q^{2} + 134 q^{3} + 1317 q^{4} + 429 q^{5} + q^{6} + 1017 q^{7} + 4479 q^{8} + 18249 q^{9}+O(q^{10}) $ 19 * q + 23 * q^2 + 134 * q^3 + 1317 * q^4 + 429 * q^5 + q^6 + 1017 * q^7 + 4479 * q^8 + 18249 * q^9	\( 19 q + 23 q^{2} + 134 q^{3} + 1317 q^{4} + 429 q^{5} + q^{6} + 1017 q^{7} + 4479 q^{8} + 18249 q^{9} + 5029 q^{10} + 21895 q^{11} + 13639 q^{12} + 4293 q^{13} + 28184 q^{14} + 28006 q^{15} + 30149 q^{16} - 3512 q^{17} - 52668 q^{18} + 62964 q^{19} + 128631 q^{20} + 151802 q^{21} + 380355 q^{22} + 194883 q^{23} + 643935 q^{24} + 594054 q^{25} + 412571 q^{26} + 544388 q^{27} + 818132 q^{28} + 343584 q^{29} + 1287286 q^{30} + 376400 q^{31} + 1182835 q^{32} + 545950 q^{33} + 952570 q^{34} + 1101957 q^{35} + 1334624 q^{36} + 630086 q^{37} + 263947 q^{38} + 1043878 q^{39} + 78765 q^{40} + 1584539 q^{41} + 510094 q^{42} - 93664 q^{43} + 2646833 q^{44} + 206061 q^{45} - 559318 q^{46} + 1752508 q^{47} - 2563869 q^{48} + 2287550 q^{49} + 411602 q^{50} + 2278880 q^{51} - 3538347 q^{52} + 127548 q^{53} - 8008280 q^{54} - 2282221 q^{55} + 1035726 q^{56} - 2815944 q^{57} - 5418817 q^{58} + 5826729 q^{59} - 8433754 q^{60} - 4312639 q^{61} - 6139425 q^{62} - 10627967 q^{63} - 7157059 q^{64} - 10672623 q^{65} - 15073362 q^{66} - 10206319 q^{67} - 10901542 q^{68} - 11593480 q^{69} - 29382493 q^{70} + 4358512 q^{71} - 23635494 q^{72} - 15042353 q^{73} - 10739915 q^{74} + 9338192 q^{75} - 12059835 q^{76} + 6632395 q^{77} - 41176024 q^{78} + 3473903 q^{79} - 6140121 q^{80} + 21257391 q^{81} - 14428970 q^{82} + 19386052 q^{83} - 10704930 q^{84} - 1181128 q^{85} + 3163940 q^{86} + 23797004 q^{87} + 13465595 q^{88} + 10913876 q^{89} - 18268219 q^{90} + 17042997 q^{91} + 20668398 q^{92} + 23643110 q^{93} - 43306680 q^{94} + 27353940 q^{95} + 24794935 q^{96} + 55835432 q^{97} + 16445433 q^{98} + 26225127 q^{99}+O(q^{100}) \) 19 * q + 23 * q^2 + 134 * q^3 + 1317 * q^4 + 429 * q^5 + q^6 + 1017 * q^7 + 4479 * q^8 + 18249 * q^9 + 5029 * q^10 + 21895 * q^11 + 13639 * q^12 + 4293 * q^13 + 28184 * q^14 + 28006 * q^15 + 30149 * q^16 - 3512 * q^17 - 52668 * q^18 + 62964 * q^19 + 128631 * q^20 + 151802 * q^21 + 380355 * q^22 + 194883 * q^23 + 643935 * q^24 + 594054 * q^25 + 412571 * q^26 + 544388 * q^27 + 818132 * q^28 + 343584 * q^29 + 1287286 * q^30 + 376400 * q^31 + 1182835 * q^32 + 545950 * q^33 + 952570 * q^34 + 1101957 * q^35 + 1334624 * q^36 + 630086 * q^37 + 263947 * q^38 + 1043878 * q^39 + 78765 * q^40 + 1584539 * q^41 + 510094 * q^42 - 93664 * q^43 + 2646833 * q^44 + 206061 * q^45 - 559318 * q^46 + 1752508 * q^47 - 2563869 * q^48 + 2287550 * q^49 + 411602 * q^50 + 2278880 * q^51 - 3538347 * q^52 + 127548 * q^53 - 8008280 * q^54 - 2282221 * q^55 + 1035726 * q^56 - 2815944 * q^57 - 5418817 * q^58 + 5826729 * q^59 - 8433754 * q^60 - 4312639 * q^61 - 6139425 * q^62 - 10627967 * q^63 - 7157059 * q^64 - 10672623 * q^65 - 15073362 * q^66 - 10206319 * q^67 - 10901542 * q^68 - 11593480 * q^69 - 29382493 * q^70 + 4358512 * q^71 - 23635494 * q^72 - 15042353 * q^73 - 10739915 * q^74 + 9338192 * q^75 - 12059835 * q^76 + 6632395 * q^77 - 41176024 * q^78 + 3473903 * q^79 - 6140121 * q^80 + 21257391 * q^81 - 14428970 * q^82 + 19386052 * q^83 - 10704930 * q^84 - 1181128 * q^85 + 3163940 * q^86 + 23797004 * q^87 + 13465595 * q^88 + 10913876 * q^89 - 18268219 * q^90 + 17042997 * q^91 + 20668398 * q^92 + 23643110 * q^93 - 43306680 * q^94 + 27353940 * q^95 + 24794935 * q^96 + 55835432 * q^97 + 16445433 * q^98 + 26225127 * 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 $

$2$	0.659105	0.0582572	0.0291286	−	0.999576i	$-0.490727\pi$
$2$	0.659105	0.0582572	0.0291286	+	0.999576i	$0.490727\pi$
$3$	52.6301	1.12541	0.562704	−	0.826659i	$-0.309761\pi$
$3$	52.6301	1.12541	0.562704	+	0.826659i	$0.309761\pi$
$4$	−127.566	−0.996606
$5$	68.7263	0.245883	0.122941	−	0.992414i	$-0.460767\pi$
$5$	68.7263	0.245883	0.122941	+	0.992414i	$0.460767\pi$
$6$	34.6888	0.0655631
$7$	248.822	0.274186	0.137093	−	0.990558i	$-0.456224\pi$
$7$	248.822	0.274186	0.137093	+	0.990558i	$0.456224\pi$
$8$	−168.445	−0.116317
$9$	582.927	0.266542
$10$	45.2978	0.0143244
$11$	3412.37	0.773004	0.386502	−	0.922289i	$-0.373683\pi$
$11$	3412.37	0.773004	0.386502	+	0.922289i	$0.373683\pi$
$12$	−6713.79	−1.12159
$13$	11965.7	1.51055	0.755276	−	0.655407i	$-0.227503\pi$
$13$	11965.7	1.51055	0.755276	+	0.655407i	$0.227503\pi$
$14$	164.000	0.0159733
$15$	3617.07	0.276718
$16$	16217.4	0.989830
$17$	7630.02	0.376664	0.188332	−	0.982105i	$-0.439692\pi$
$17$	7630.02	0.376664	0.188332	+	0.982105i	$0.439692\pi$
$18$	384.210	0.0155280
$19$	−18115.6	−0.605919	−0.302959	−	0.953003i	$-0.597975\pi$
$19$	−18115.6	−0.605919	−0.302959	+	0.953003i	$0.597975\pi$
$20$	−8767.11	−0.245048
$21$	13095.5	0.308571
$22$	2249.11	0.0450330
$23$	96490.7	1.65363	0.826814	−	0.562475i	$-0.190151\pi$
$23$	96490.7	1.65363	0.826814	+	0.562475i	$0.190151\pi$
$24$	−8865.25	−0.130904
$25$	−73401.7	−0.939542
$26$	7886.64	0.0880005
$27$	−84422.5	−0.825439
$28$	−31741.1	−0.273256
$29$	73043.4	0.556145	0.278073	−	0.960560i	$-0.410304\pi$
$29$	73043.4	0.556145	0.278073	+	0.960560i	$0.410304\pi$
$30$	2384.03	0.0161208
$31$	257315.	1.55131	0.775657	−	0.631155i	$-0.217419\pi$
$31$	257315.	1.55131	0.775657	+	0.631155i	$0.217419\pi$
$32$	32249.9	0.173981
$33$	179593.	0.869944
$34$	5028.98	0.0219434
$35$	17100.6	0.0674177
$36$	−74361.4	−0.265637
$37$	394936.	1.28180	0.640901	−	0.767624i	$-0.278561\pi$
$37$	394936.	1.28180	0.640901	+	0.767624i	$0.278561\pi$
$38$	−11940.1	−0.0352991
$39$	629755.	1.69999
$40$	−11576.6	−0.0286003
$41$	62815.8	0.142339	0.0711697	−	0.997464i	$-0.477327\pi$
$41$	62815.8	0.142339	0.0711697	+	0.997464i	$0.477327\pi$
$42$	8631.33	0.0179765
$43$	627894.	1.20433	0.602167	−	0.798370i	$-0.294304\pi$
$43$	627894.	1.20433	0.602167	+	0.798370i	$0.294304\pi$
$44$	−435301.	−0.770380
$45$	40062.4	0.0655380
$46$	63597.5	0.0963358
$47$	−1.07241e6	−1.50667	−0.753334	−	0.657638i	$-0.771556\pi$
$47$	−1.07241e6	−1.50667	−0.753334	+	0.657638i	$0.771556\pi$
$48$	853522.	1.11396
$49$	−761631.	−0.924822
$50$	−48379.4	−0.0547351
$51$	401568.	0.423901
$52$	−1.52641e6	−1.50542
$53$	−97183.7	−0.0896660	−0.0448330	−	0.998994i	$-0.514276\pi$
$53$	−97183.7	−0.0896660	−0.0448330	+	0.998994i	$0.514276\pi$
$54$	−55643.3	−0.0480878
$55$	234520.	0.190068
$56$	−41912.7	−0.0318924
$57$	−953424.	−0.681906
$58$	48143.3	0.0323995
$59$	−2.32444e6	−1.47345	−0.736726	−	0.676191i	$-0.763629\pi$
$59$	−2.32444e6	−1.47345	−0.736726	+	0.676191i	$0.763629\pi$
$60$	−461414.	−0.275779
$61$	−226981.	−0.128037
$61$	−226981.	−0.128037
$62$	169598.	0.0903752
$63$	145045.	0.0730821
$64$	−2.05457e6	−0.979694
$65$	822357.	0.371418
$66$	118371.	0.0506805
$67$	850187.	0.345345	0.172672	−	0.984979i	$-0.444760\pi$
$67$	850187.	0.345345	0.172672	+	0.984979i	$0.444760\pi$
$68$	−973328.	−0.375386
$69$	5.07831e6	1.86101
$70$	11271.1	0.00392756
$71$	108707.	0.0360457	0.0180229	−	0.999838i	$-0.494263\pi$
$71$	108707.	0.0360457	0.0180229	+	0.999838i	$0.494263\pi$
$72$	−98190.8	−0.0310032
$73$	−487519.	−0.146677	−0.0733384	−	0.997307i	$-0.523365\pi$
$73$	−487519.	−0.146677	−0.0733384	+	0.997307i	$0.523365\pi$
$74$	260304.	0.0746742
$75$	−3.86314e6	−1.05737
$76$	2.31092e6	0.603862
$77$	849073.	0.211947
$78$	415074.	0.0990364
$79$	838970.	0.191448	0.0957242	−	0.995408i	$-0.469483\pi$
$79$	838970.	0.191448	0.0957242	+	0.995408i	$0.469483\pi$
$80$	1.11456e6	0.243382
$81$	−5.71803e6	−1.19550
$82$	41402.2	0.00829230
$83$	−9.58150e6	−1.83933	−0.919666	−	0.392702i	$-0.871540\pi$
$83$	−9.58150e6	−1.83933	−0.919666	+	0.392702i	$0.871540\pi$
$84$	−1.67054e6	−0.307524
$85$	524383.	0.0926152
$86$	413848.	0.0701611
$87$	3.84428e6	0.625890
$88$	−574795.	−0.0899133
$89$	−2.25354e6	−0.338844	−0.169422	−	0.985544i	$-0.554190\pi$
$89$	−2.25354e6	−0.338844	−0.169422	+	0.985544i	$0.554190\pi$
$90$	26405.3	0.00381806
$91$	2.97732e6	0.414173
$92$	−1.23089e7	−1.64802
$93$	1.35425e7	1.74586
$94$	−706830.	−0.0877743
$95$	−1.24502e6	−0.148985
$96$	1.69731e6	0.195800
$97$	−1.21224e7	−1.34862	−0.674309	−	0.738450i	$-0.735558\pi$
$97$	−1.21224e7	−1.34862	−0.674309	+	0.738450i	$0.735558\pi$
$98$	−501995.	−0.0538775
$99$	1.98916e6	0.206038
$100$	9.36353e6	0.936353
$101$	4.41488e6	0.426378	0.213189	−	0.977011i	$-0.431615\pi$
$101$	4.41488e6	0.426378	0.213189	+	0.977011i	$0.431615\pi$
$102$	264676.	0.0246953
$103$	−5.36668e6	−0.483922	−0.241961	−	0.970286i	$-0.577791\pi$
$103$	−5.36668e6	−0.483922	−0.241961	+	0.970286i	$0.577791\pi$
$104$	−2.01555e6	−0.175702
$105$	900007.	0.0758723
$106$	−64054.2	−0.00522369
$107$	1.97231e7	1.55644	0.778221	−	0.627991i	$-0.216123\pi$
$107$	1.97231e7	1.55644	0.778221	+	0.627991i	$0.216123\pi$
$108$	1.07694e7	0.822638
$109$	1.14510e7	0.846940	0.423470	−	0.905910i	$-0.360812\pi$
$109$	1.14510e7	0.846940	0.423470	+	0.905910i	$0.360812\pi$
$110$	154573.	0.0110728
$111$	2.07855e7	1.44255
$112$	4.03524e6	0.271398
$113$	1.57483e7	1.02674	0.513370	−	0.858167i	$-0.328397\pi$
$113$	1.57483e7	1.02674	0.513370	+	0.858167i	$0.328397\pi$
$114$	−628407.	−0.0397259
$115$	6.63145e6	0.406599
$116$	−9.31783e6	−0.554258
$117$	6.97511e6	0.402625
$118$	−1.53205e6	−0.0858392
$119$	1.89852e6	0.103276
$120$	−609276.	−0.0321869
$121$	−7.84290e6	−0.402465
$122$	−149604.	−0.00745907
$123$	3.30600e6	0.160190
$124$	−3.28246e7	−1.54605
$125$	−1.04139e7	−0.476900
$126$	95599.9	0.00425756
$127$	1.48908e7	0.645065	0.322533	−	0.946558i	$-0.395466\pi$
$127$	1.48908e7	0.645065	0.322533	+	0.946558i	$0.395466\pi$
$128$	−5.48216e6	−0.231056
$129$	3.30461e7	1.35537
$130$	542019.	0.0216378
$131$	−1.73504e7	−0.674309	−0.337154	−	0.941449i	$-0.609464\pi$
$131$	−1.73504e7	−0.674309	−0.337154	+	0.941449i	$0.609464\pi$
$132$	−2.29099e7	−0.866992
$133$	−4.50755e6	−0.166135
$134$	560363.	0.0201188
$135$	−5.80205e6	−0.202961
$136$	−1.28523e6	−0.0438123
$137$	5.87697e7	1.95268	0.976341	−	0.216234i	$-0.0693776\pi$
$137$	5.87697e7	1.95268	0.976341	+	0.216234i	$0.0693776\pi$
$138$	3.34714e6	0.108417
$139$	−4.02545e7	−1.27134	−0.635671	−	0.771960i	$-0.719277\pi$
$139$	−4.02545e7	−1.27134	−0.635671	+	0.771960i	$0.719277\pi$
$140$	−2.18145e6	−0.0671889
$141$	−5.64410e7	−1.69562
$142$	71649.4	0.00209992
$143$	4.08313e7	1.16766
$144$	9.45354e6	0.263831
$145$	5.02001e6	0.136746
$146$	−321326.	−0.00854499
$147$	−4.00847e7	−1.04080
$148$	−5.03803e7	−1.27745
$149$	5.97128e6	0.147882	0.0739410	−	0.997263i	$-0.476442\pi$
$149$	5.97128e6	0.147882	0.0739410	+	0.997263i	$0.476442\pi$
$150$	−2.54621e6	−0.0615993
$151$	−7.01490e7	−1.65807	−0.829034	−	0.559198i	$-0.811109\pi$
$151$	−7.01490e7	−1.65807	−0.829034	+	0.559198i	$0.811109\pi$
$152$	3.05147e6	0.0704785
$153$	4.44774e6	0.100397
$154$	559628.	0.0123474
$155$	1.76843e7	0.381441
$156$	−8.03350e7	−1.69422
$157$	2.90140e7	0.598355	0.299178	−	0.954197i	$-0.403288\pi$
$157$	2.90140e7	0.598355	0.299178	+	0.954197i	$0.403288\pi$
$158$	552969.	0.0111532
$159$	−5.11479e6	−0.100911
$160$	2.21641e6	0.0427790
$161$	2.40090e7	0.453402
$162$	−3.76878e6	−0.0696463
$163$	7.29723e7	1.31978	0.659890	−	0.751362i	$-0.270603\pi$
$163$	7.29723e7	1.31978	0.659890	+	0.751362i	$0.270603\pi$
$164$	−8.01313e6	−0.141856
$165$	1.23428e7	0.213904
$166$	−6.31521e6	−0.107154
$167$	3.07624e7	0.511109	0.255554	−	0.966795i	$-0.417742\pi$
$167$	3.07624e7	0.511109	0.255554	+	0.966795i	$0.417742\pi$
$168$	−2.20587e6	−0.0358920
$169$	8.04289e7	1.28177
$170$	345623.	0.00539550
$171$	−1.05601e7	−0.161503
$172$	−8.00977e7	−1.20025
$173$	−1.91780e7	−0.281605	−0.140803	−	0.990038i	$-0.544968\pi$
$173$	−1.91780e7	−0.281605	−0.140803	+	0.990038i	$0.544968\pi$
$174$	2.53379e6	0.0364626
$175$	−1.82640e7	−0.257609
$176$	5.53397e7	0.765142
$177$	−1.22335e8	−1.65823
$178$	−1.48532e6	−0.0197401
$179$	−1.16925e8	−1.52378	−0.761888	−	0.647708i	$-0.775728\pi$
$179$	−1.16925e8	−1.52378	−0.761888	+	0.647708i	$0.775728\pi$
$180$	−5.11058e6	−0.0653156
$181$	−1.03177e7	−0.129332	−0.0646662	−	0.997907i	$-0.520598\pi$
$181$	−1.03177e7	−0.129332	−0.0646662	+	0.997907i	$0.520598\pi$
$182$	1.96237e6	0.0241285
$183$	−1.19460e7	−0.144094
$184$	−1.62533e7	−0.192345
$185$	2.71425e7	0.315173
$186$	8.92594e6	0.101709
$187$	2.60364e7	0.291163
$188$	1.36802e8	1.50155
$189$	−2.10062e7	−0.226324
$190$	−820597.	−0.00867945
$191$	5.68340e7	0.590189	0.295095	−	0.955468i	$-0.404649\pi$
$191$	5.68340e7	0.590189	0.295095	+	0.955468i	$0.404649\pi$
$192$	−1.08132e8	−1.10255
$193$	1.22301e8	1.22456	0.612281	−	0.790640i	$-0.290252\pi$
$193$	1.22301e8	1.22456	0.612281	+	0.790640i	$0.290252\pi$
$194$	−7.98996e6	−0.0785667
$195$	4.32807e7	0.417997
$196$	9.71578e7	0.921683
$197$	−7.57684e6	−0.0706084	−0.0353042	−	0.999377i	$-0.511240\pi$
$197$	−7.57684e6	−0.0706084	−0.0353042	+	0.999377i	$0.511240\pi$
$198$	1.31107e6	0.0120032
$199$	−1.28993e8	−1.16033	−0.580164	−	0.814499i	$-0.697012\pi$
$199$	−1.28993e8	−1.16033	−0.580164	+	0.814499i	$0.697012\pi$
$200$	1.23641e7	0.109284
$201$	4.47454e7	0.388653
$202$	2.90987e6	0.0248396
$203$	1.81748e7	0.152487
$204$	−5.12263e7	−0.422462
$205$	4.31710e6	0.0349988
$206$	−3.53720e6	−0.0281919
$207$	5.62470e7	0.440761
$208$	1.94052e8	1.49519
$209$	−6.18170e7	−0.468378
$210$	593199.	0.00442011
$211$	−1.64906e8	−1.20850	−0.604250	−	0.796795i	$-0.706527\pi$
$211$	−1.64906e8	−1.20850	−0.604250	+	0.796795i	$0.706527\pi$
$212$	1.23973e7	0.0893617
$213$	5.72127e6	0.0405661
$214$	1.29996e7	0.0906739
$215$	4.31528e7	0.296125
$216$	1.42205e7	0.0960124
$217$	6.40257e7	0.425349
$218$	7.54744e6	0.0493403
$219$	−2.56582e7	−0.165071
$220$	−2.99166e7	−0.189423
$221$	9.12983e7	0.568971
$222$	1.36998e7	0.0840389
$223$	1.39539e7	0.0842612	0.0421306	−	0.999112i	$-0.486585\pi$
$223$	1.39539e7	0.0842612	0.0421306	+	0.999112i	$0.486585\pi$
$224$	8.02448e6	0.0477033
$225$	−4.27878e7	−0.250427
$226$	1.03798e7	0.0598150
$227$	1.41762e8	0.804396	0.402198	−	0.915553i	$-0.368246\pi$
$227$	1.41762e8	0.804396	0.402198	+	0.915553i	$0.368246\pi$
$228$	1.21624e8	0.679591
$229$	−1.31816e8	−0.725343	−0.362672	−	0.931917i	$-0.618135\pi$
$229$	−1.31816e8	−0.725343	−0.362672	+	0.931917i	$0.618135\pi$
$230$	4.37082e6	0.0236873
$231$	4.46868e7	0.238527
$232$	−1.23038e7	−0.0646890
$233$	2.54812e8	1.31970	0.659848	−	0.751399i	$-0.270621\pi$
$233$	2.54812e8	1.31970	0.659848	+	0.751399i	$0.270621\pi$
$234$	4.59733e6	0.0234558
$235$	−7.37027e7	−0.370464
$236$	2.96518e8	1.46845
$237$	4.41551e7	0.215457
$238$	1.25132e6	0.00601658
$239$	−1.03289e8	−0.489398	−0.244699	−	0.969599i	$-0.578689\pi$
$239$	−1.03289e8	−0.489398	−0.244699	+	0.969599i	$0.578689\pi$
$240$	5.86594e7	0.273904
$241$	−9.69982e7	−0.446379	−0.223190	−	0.974775i	$-0.571647\pi$
$241$	−9.69982e7	−0.446379	−0.223190	+	0.974775i	$0.571647\pi$
$242$	−5.16930e6	−0.0234465
$243$	−1.16308e8	−0.519982
$244$	2.89550e7	0.127602
$245$	−5.23441e7	−0.227398
$246$	2.17900e6	0.00933222
$247$	−2.16765e8	−0.915272
$248$	−4.33433e7	−0.180444
$249$	−5.04275e8	−2.07000
$250$	−6.86383e6	−0.0277828
$251$	1.48707e8	0.593573	0.296787	−	0.954944i	$-0.404085\pi$
$251$	1.48707e8	0.593573	0.296787	+	0.954944i	$0.404085\pi$
$252$	−1.85028e7	−0.0728341
$253$	3.29262e8	1.27826
$254$	9.81457e6	0.0375797
$255$	2.75983e7	0.104230
$256$	2.59371e8	0.966233
$257$	−1.45823e8	−0.535870	−0.267935	−	0.963437i	$-0.586341\pi$
$257$	−1.45823e8	−0.535870	−0.267935	+	0.963437i	$0.586341\pi$
$258$	2.17809e7	0.0789598
$259$	9.82689e7	0.351453
$260$	−1.04904e8	−0.370158
$261$	4.25790e7	0.148236
$262$	−1.14357e7	−0.0392833
$263$	3.45638e8	1.17159	0.585796	−	0.810459i	$-0.300782\pi$
$263$	3.45638e8	1.17159	0.585796	+	0.810459i	$0.300782\pi$
$264$	−3.02515e7	−0.101189
$265$	−6.67907e6	−0.0220473
$266$	−2.97095e6	−0.00967854
$267$	−1.18604e8	−0.381338
$268$	−1.08455e8	−0.344173
$269$	−2.85963e8	−0.895729	−0.447864	−	0.894102i	$-0.647815\pi$
$269$	−2.85963e8	−0.895729	−0.447864	+	0.894102i	$0.647815\pi$
$270$	−3.82416e6	−0.0118240
$271$	−3.71652e8	−1.13434	−0.567171	−	0.823600i	$-0.691962\pi$
$271$	−3.71652e8	−1.13434	−0.567171	+	0.823600i	$0.691962\pi$
$272$	1.23739e8	0.372833
$273$	1.56697e8	0.466113
$274$	3.87354e7	0.113758
$275$	−2.50474e8	−0.726269
$276$	−6.47818e8	−1.85469
$277$	−5.30764e8	−1.50045	−0.750226	−	0.661181i	$-0.770056\pi$
$277$	−5.30764e8	−1.50045	−0.750226	+	0.661181i	$0.770056\pi$
$278$	−2.65319e7	−0.0740648
$279$	1.49996e8	0.413490
$280$	−2.88051e6	−0.00784180
$281$	1.06042e8	0.285105	0.142553	−	0.989787i	$-0.454469\pi$
$281$	1.06042e8	0.285105	0.142553	+	0.989787i	$0.454469\pi$
$282$	−3.72005e7	−0.0987818
$283$	−2.47113e7	−0.0648102	−0.0324051	−	0.999475i	$-0.510317\pi$
$283$	−2.47113e7	−0.0648102	−0.0324051	+	0.999475i	$0.510317\pi$
$284$	−1.38673e7	−0.0359234
$285$	−6.55253e7	−0.167669
$286$	2.69121e7	0.0680247
$287$	1.56300e7	0.0390275
$288$	1.87993e7	0.0463733
$289$	−3.52122e8	−0.858124
$290$	3.30871e6	0.00796647
$291$	−6.38005e8	−1.51774
$292$	6.21907e7	0.146179
$293$	−2.32653e8	−0.540346	−0.270173	−	0.962812i	$-0.587081\pi$
$293$	−2.32653e8	−0.540346	−0.270173	+	0.962812i	$0.587081\pi$
$294$	−2.64200e7	−0.0606342
$295$	−1.59750e8	−0.362296
$296$	−6.65249e7	−0.149095
$297$	−2.88081e8	−0.638068
$298$	3.93570e6	0.00861519
$299$	1.15458e9	2.49789
$300$	4.92803e8	1.05378
$301$	1.56234e8	0.330212
$302$	−4.62356e7	−0.0965944
$303$	2.32356e8	0.479849
$304$	−2.93787e8	−0.599757
$305$	−1.55996e7	−0.0314821
$306$	2.93153e6	0.00584883
$307$	3.89005e8	0.767309	0.383655	−	0.923477i	$-0.374665\pi$
$307$	3.89005e8	0.767309	0.383655	+	0.923477i	$0.374665\pi$
$308$	−1.08312e8	−0.211228
$309$	−2.82449e8	−0.544609
$310$	1.16558e7	0.0222217
$311$	3.59123e8	0.676989	0.338495	−	0.940968i	$-0.390082\pi$
$311$	3.59123e8	0.676989	0.338495	+	0.940968i	$0.390082\pi$
$312$	−1.06079e8	−0.197737
$313$	−2.24233e8	−0.413328	−0.206664	−	0.978412i	$-0.566261\pi$
$313$	−2.24233e8	−0.413328	−0.206664	+	0.978412i	$0.566261\pi$
$314$	1.91233e7	0.0348585
$315$	9.96841e6	0.0179696
$316$	−1.07024e8	−0.190799
$317$	−6.20262e8	−1.09362	−0.546811	−	0.837256i	$-0.684158\pi$
$317$	−6.20262e8	−1.09362	−0.546811	+	0.837256i	$0.684158\pi$
$318$	−3.37118e6	−0.00587878
$319$	2.49251e8	0.429902
$320$	−1.41203e8	−0.240890
$321$	1.03803e9	1.75163
$322$	1.58245e7	0.0264139
$323$	−1.38222e8	−0.228228
$324$	7.29423e8	1.19144
$325$	−8.78301e8	−1.41923
$326$	4.80964e7	0.0768867
$327$	6.02670e8	0.953152
$328$	−1.05810e7	−0.0165565
$329$	−2.66839e8	−0.413108
$330$	8.13519e6	0.0124615
$331$	−5.44995e8	−0.826028	−0.413014	−	0.910725i	$-0.635524\pi$
$331$	−5.44995e8	−0.826028	−0.413014	+	0.910725i	$0.635524\pi$
$332$	1.22227e9	1.83309
$333$	2.30219e8	0.341654
$334$	2.02757e7	0.0297758
$335$	5.84302e7	0.0849143
$336$	2.12375e8	0.305433
$337$	−3.37275e8	−0.480043	−0.240021	−	0.970768i	$-0.577154\pi$
$337$	−3.37275e8	−0.480043	−0.240021	+	0.970768i	$0.577154\pi$
$338$	5.30111e7	0.0746721
$339$	8.28837e8	1.15550
$340$	−6.68932e7	−0.0923009
$341$	8.78055e8	1.19917
$342$	−6.96018e6	−0.00940869
$343$	−3.94426e8	−0.527760
$344$	−1.05765e8	−0.140084
$345$	3.49014e8	0.457589
$346$	−1.26403e7	−0.0164055
$347$	1.39097e9	1.78716	0.893581	−	0.448903i	$-0.148185\pi$
$347$	1.39097e9	1.78716	0.893581	+	0.448903i	$0.148185\pi$
$348$	−4.90398e8	−0.623766
$349$	5.78730e8	0.728764	0.364382	−	0.931250i	$-0.381280\pi$
$349$	5.78730e8	0.728764	0.364382	+	0.931250i	$0.381280\pi$
$350$	−1.20379e7	−0.0150076
$351$	−1.01017e9	−1.24687
$352$	1.10048e8	0.134488
$353$	−1.01229e9	−1.22488	−0.612441	−	0.790516i	$-0.709812\pi$
$353$	−1.01229e9	−1.22488	−0.612441	+	0.790516i	$0.709812\pi$
$354$	−8.06319e7	−0.0966041
$355$	7.47104e6	0.00886302
$356$	2.87474e8	0.337694
$357$	9.99191e7	0.116228
$358$	−7.70658e7	−0.0887710
$359$	−3.79580e8	−0.432985	−0.216493	−	0.976284i	$-0.569462\pi$
$359$	−3.79580e8	−0.432985	−0.216493	+	0.976284i	$0.569462\pi$
$360$	−6.74829e6	−0.00762316
$361$	−5.65698e8	−0.632862
$362$	−6.80043e6	−0.00753454
$363$	−4.12773e8	−0.452937
$364$	−3.79804e8	−0.412767
$365$	−3.35054e7	−0.0360653
$366$	−7.87369e6	−0.00839449
$367$	5.14162e8	0.542961	0.271481	−	0.962444i	$-0.412487\pi$
$367$	5.14162e8	0.542961	0.271481	+	0.962444i	$0.412487\pi$
$368$	1.56483e9	1.63681
$369$	3.66170e7	0.0379394
$370$	1.78898e7	0.0183611
$371$	−2.41814e7	−0.0245852
$372$	−1.72756e9	−1.73993
$373$	3.13176e8	0.312470	0.156235	−	0.987720i	$-0.450064\pi$
$373$	3.13176e8	0.312470	0.156235	+	0.987720i	$0.450064\pi$
$374$	1.71607e7	0.0169623
$375$	−5.48083e8	−0.536706
$376$	1.80641e8	0.175251
$377$	8.74014e8	0.840086
$378$	−1.38453e7	−0.0131850
$379$	−1.19538e9	−1.12790	−0.563948	−	0.825810i	$-0.690718\pi$
$379$	−1.19538e9	−1.12790	−0.563948	+	0.825810i	$0.690718\pi$
$380$	1.58821e8	0.148479
$381$	7.83702e8	0.725961
$382$	3.74596e7	0.0343828
$383$	1.85413e9	1.68634	0.843170	−	0.537647i	$-0.180687\pi$
$383$	1.85413e9	1.68634	0.843170	+	0.537647i	$0.180687\pi$
$384$	−2.88526e8	−0.260032
$385$	5.83536e7	0.0521141
$386$	8.06094e7	0.0713395
$387$	3.66016e8	0.321005
$388$	1.54641e9	1.34404
$389$	−1.19523e9	−1.02950	−0.514750	−	0.857340i	$-0.672115\pi$
$389$	−1.19523e9	−1.02950	−0.514750	+	0.857340i	$0.672115\pi$
$390$	2.85265e7	0.0243513
$391$	7.36225e8	0.622862
$392$	1.28293e8	0.107572
$393$	−9.13151e8	−0.758872
$394$	−4.99393e6	−0.00411345
$395$	5.76593e7	0.0470738
$396$	−2.53749e8	−0.205339
$397$	−1.83415e9	−1.47118	−0.735592	−	0.677424i	$-0.763096\pi$
$397$	−1.83415e9	−1.47118	−0.735592	+	0.677424i	$0.763096\pi$
$398$	−8.50201e7	−0.0675975
$399$	−2.37233e8	−0.186969
$400$	−1.19038e9	−0.929986
$401$	−1.16686e9	−0.903676	−0.451838	−	0.892100i	$-0.649232\pi$
$401$	−1.16686e9	−0.903676	−0.451838	+	0.892100i	$0.649232\pi$
$402$	2.94919e7	0.0226419
$403$	3.07895e9	2.34334
$404$	−5.63187e8	−0.424931
$405$	−3.92979e8	−0.293952
$406$	1.19791e7	0.00888349
$407$	1.34767e9	0.990838
$408$	−6.76420e7	−0.0493067
$409$	−1.34584e9	−0.972665	−0.486332	−	0.873774i	$-0.661666\pi$
$409$	−1.34584e9	−0.972665	−0.486332	+	0.873774i	$0.661666\pi$
$410$	2.84542e6	0.00203893
$411$	3.09306e9	2.19756
$412$	6.84603e8	0.482279
$413$	−5.78372e8	−0.404001
$414$	3.70727e7	0.0256775
$415$	−6.58501e8	−0.452260
$416$	3.85891e8	0.262808
$417$	−2.11860e9	−1.43078
$418$	−4.07439e7	−0.0272864
$419$	1.75544e9	1.16583	0.582917	−	0.812532i	$-0.301911\pi$
$419$	1.75544e9	1.16583	0.582917	+	0.812532i	$0.301911\pi$
$420$	−1.14810e8	−0.0756148
$421$	−4.90093e7	−0.0320104	−0.0160052	−	0.999872i	$-0.505095\pi$
$421$	−4.90093e7	−0.0320104	−0.0160052	+	0.999872i	$0.505095\pi$
$422$	−1.08690e8	−0.0704039
$423$	−6.25135e8	−0.401590
$424$	1.63701e7	0.0104296
$425$	−5.60056e8	−0.353892
$426$	3.77092e6	0.00236327
$427$	−5.64779e7	−0.0351060
$428$	−2.51599e9	−1.55116
$429$	2.14896e9	1.31410
$430$	2.84423e7	0.0172514
$431$	8.94126e8	0.537933	0.268966	−	0.963150i	$-0.413318\pi$
$431$	8.94126e8	0.537933	0.268966	+	0.963150i	$0.413318\pi$
$432$	−1.36911e9	−0.817044
$433$	−1.92286e9	−1.13826	−0.569129	−	0.822248i	$-0.692720\pi$
$433$	−1.92286e9	−1.13826	−0.569129	+	0.822248i	$0.692720\pi$
$434$	4.21997e7	0.0247796
$435$	2.64203e8	0.153896
$436$	−1.46076e9	−0.844065
$437$	−1.74798e9	−1.00196
$438$	−1.69114e7	−0.00961659
$439$	1.36084e9	0.767681	0.383841	−	0.923399i	$-0.374601\pi$
$439$	1.36084e9	0.767681	0.383841	+	0.923399i	$0.374601\pi$
$440$	−3.95035e7	−0.0221081
$441$	−4.43975e8	−0.246504
$442$	6.01752e7	0.0331466
$443$	−1.47833e9	−0.807901	−0.403951	−	0.914781i	$-0.632363\pi$
$443$	−1.47833e9	−0.807901	−0.403951	+	0.914781i	$0.632363\pi$
$444$	−2.65152e9	−1.43765
$445$	−1.54877e8	−0.0833159
$446$	9.19706e6	0.00490882
$447$	3.14269e8	0.166428
$448$	−5.11222e8	−0.268619
$449$	7.90470e8	0.412120	0.206060	−	0.978539i	$-0.433936\pi$
$449$	7.90470e8	0.412120	0.206060	+	0.978539i	$0.433936\pi$
$450$	−2.82017e7	−0.0145892
$451$	2.14351e8	0.110029
$452$	−2.00895e9	−1.02326
$453$	−3.69195e9	−1.86600
$454$	9.34361e7	0.0468619
$455$	2.04620e8	0.101838
$456$	1.60599e8	0.0793170
$457$	3.55222e9	1.74098	0.870488	−	0.492189i	$-0.163803\pi$
$457$	3.55222e9	1.74098	0.870488	+	0.492189i	$0.163803\pi$
$458$	−8.68805e7	−0.0422565
$459$	−6.44145e8	−0.310913
$460$	−8.45944e8	−0.405219
$461$	−2.84020e9	−1.35019	−0.675095	−	0.737730i	$-0.735898\pi$
$461$	−2.84020e9	−1.35019	−0.675095	+	0.737730i	$0.735898\pi$
$462$	2.94533e7	0.0138959
$463$	−9.23496e8	−0.432416	−0.216208	−	0.976347i	$-0.569369\pi$
$463$	−9.23496e8	−0.432416	−0.216208	+	0.976347i	$0.569369\pi$
$464$	1.18457e9	0.550489
$465$	9.30727e8	0.429277
$466$	1.67948e8	0.0768818
$467$	−3.35482e9	−1.52426	−0.762132	−	0.647422i	$-0.775847\pi$
$467$	−3.35482e9	−1.52426	−0.762132	+	0.647422i	$0.775847\pi$
$468$	−8.89784e8	−0.401259
$469$	2.11545e8	0.0946888
$470$	−4.85778e7	−0.0215822
$471$	1.52701e9	0.673393
$472$	3.91539e8	0.171387
$473$	2.14261e9	0.930955
$474$	2.91028e7	0.0125519
$475$	1.32971e9	0.569286
$476$	−2.42185e8	−0.102926
$477$	−5.66510e7	−0.0238997
$478$	−6.80784e7	−0.0285109
$479$	3.22711e9	1.34165	0.670826	−	0.741615i	$-0.265940\pi$
$479$	3.22711e9	1.34165	0.670826	+	0.741615i	$0.265940\pi$
$480$	1.16650e8	0.0481438
$481$	4.72568e9	1.93623
$482$	−6.39320e7	−0.0260048
$483$	1.26360e9	0.510262
$484$	1.00048e9	0.401099
$485$	−8.33130e8	−0.331602
$486$	−7.66593e7	−0.0302927
$487$	−2.44833e9	−0.960547	−0.480274	−	0.877119i	$-0.659463\pi$
$487$	−2.44833e9	−0.960547	−0.480274	+	0.877119i	$0.659463\pi$
$488$	3.82337e7	0.0148928
$489$	3.84054e9	1.48529
$490$	−3.45002e7	−0.0132476
$491$	−2.69890e9	−1.02897	−0.514484	−	0.857500i	$-0.672017\pi$
$491$	−2.69890e9	−1.02897	−0.514484	+	0.857500i	$0.672017\pi$
$492$	−4.21732e8	−0.159646
$493$	5.57323e8	0.209480
$494$	−1.42871e8	−0.0533212
$495$	1.36708e8	0.0506611
$496$	4.17298e9	1.53554
$497$	2.70487e7	0.00988325
$498$	−3.32370e8	−0.120592
$499$	−3.42909e8	−0.123545	−0.0617727	−	0.998090i	$-0.519675\pi$
$499$	−3.42909e8	−0.123545	−0.0617727	+	0.998090i	$0.519675\pi$
$500$	1.32845e9	0.475281
$501$	1.61903e9	0.575205
$502$	9.80137e7	0.0345799
$503$	3.97521e9	1.39275	0.696373	−	0.717680i	$-0.254796\pi$
$503$	3.97521e9	1.39275	0.696373	+	0.717680i	$0.254796\pi$
$504$	−2.44320e7	−0.00850067
$505$	3.03419e8	0.104839
$506$	2.17018e8	0.0744679
$507$	4.23298e9	1.44251
$508$	−1.89955e9	−0.642876
$509$	4.69451e9	1.57789	0.788947	−	0.614462i	$-0.210627\pi$
$509$	4.69451e9	1.57789	0.788947	+	0.614462i	$0.210627\pi$
$510$	1.81902e7	0.00607214
$511$	−1.21306e8	−0.0402168
$512$	8.72669e8	0.287346
$513$	1.52936e9	0.500149
$514$	−9.61125e7	−0.0312183
$515$	−3.68832e8	−0.118988
$516$	−4.21555e9	−1.35077
$517$	−3.65945e9	−1.16466
$518$	6.47695e7	0.0204746
$519$	−1.00934e9	−0.316921
$520$	−1.38522e8	−0.0432022
$521$	−3.24734e9	−1.00599	−0.502997	−	0.864288i	$-0.667769\pi$
$521$	−3.24734e9	−1.00599	−0.502997	+	0.864288i	$0.667769\pi$
$522$	2.80640e7	0.00863581
$523$	−1.02001e9	−0.311780	−0.155890	−	0.987774i	$-0.549825\pi$
$523$	−1.02001e9	−0.311780	−0.155890	+	0.987774i	$0.549825\pi$
$524$	2.21331e9	0.672020
$525$	−9.61234e8	−0.289916
$526$	2.27812e8	0.0682537
$527$	1.96332e9	0.584324
$528$	2.91253e9	0.861097
$529$	5.90562e9	1.73449
$530$	−4.40221e6	−0.00128441
$531$	−1.35498e9	−0.392737
$532$	5.75009e8	0.165571
$533$	7.51634e8	0.215011
$534$	−7.81724e7	−0.0222157
$535$	1.35550e9	0.382702
$536$	−1.43209e8	−0.0401694
$537$	−6.15377e9	−1.71487
$538$	−1.88479e8	−0.0521826
$539$	−2.59897e9	−0.714891
$540$	7.40142e8	0.202272
$541$	−4.49607e9	−1.22079	−0.610397	−	0.792096i	$-0.708990\pi$
$541$	−4.49607e9	−1.22079	−0.610397	+	0.792096i	$0.708990\pi$
$542$	−2.44958e8	−0.0660836
$543$	−5.43020e8	−0.145552
$544$	2.46067e8	0.0655326
$545$	7.86988e8	0.208248
$546$	1.03280e8	0.0271544
$547$	−4.16557e9	−1.08822	−0.544112	−	0.839013i	$-0.683133\pi$
$547$	−4.16557e9	−1.08822	−0.544112	+	0.839013i	$0.683133\pi$
$548$	−7.49700e9	−1.94606
$549$	−1.32313e8	−0.0341272
$550$	−1.65088e8	−0.0423104
$551$	−1.32322e9	−0.336979
$552$	−8.55414e8	−0.216466
$553$	2.08754e8	0.0524925
$554$	−3.49829e8	−0.0874122
$555$	1.42851e9	0.354698
$556$	5.13509e9	1.26703
$557$	4.09073e9	1.00301	0.501507	−	0.865153i	$-0.332779\pi$
$557$	4.09073e9	1.00301	0.501507	+	0.865153i	$0.332779\pi$
$558$	9.88630e7	0.0240888
$559$	7.51318e9	1.81921
$560$	2.77327e8	0.0667320
$561$	1.37030e9	0.327677
$562$	6.98927e7	0.0166094
$563$	−7.16735e9	−1.69270	−0.846349	−	0.532629i	$-0.821204\pi$
$563$	−7.16735e9	−1.69270	−0.846349	+	0.532629i	$0.821204\pi$
$564$	7.19992e9	1.68986
$565$	1.08233e9	0.252458
$566$	−1.62874e7	−0.00377566
$567$	−1.42277e9	−0.327789
$568$	−1.83111e7	−0.00419272
$569$	−7.81496e9	−1.77842	−0.889209	−	0.457501i	$-0.848744\pi$
$569$	−7.81496e9	−1.77842	−0.889209	+	0.457501i	$0.848744\pi$
$570$	−4.31881e7	−0.00976791
$571$	5.78188e9	1.29970	0.649849	−	0.760063i	$-0.274832\pi$
$571$	5.78188e9	1.29970	0.649849	+	0.760063i	$0.274832\pi$
$572$	−5.20867e9	−1.16370
$573$	2.99118e9	0.664203
$574$	1.03018e7	0.00227364
$575$	−7.08258e9	−1.55365
$576$	−1.19766e9	−0.261129
$577$	−6.15005e9	−1.33279	−0.666397	−	0.745597i	$-0.732165\pi$
$577$	−6.15005e9	−1.33279	−0.666397	+	0.745597i	$0.732165\pi$
$578$	−2.32085e8	−0.0499919
$579$	6.43673e9	1.37813
$580$	−6.40380e8	−0.136282
$581$	−2.38409e9	−0.504319
$582$	−4.20512e8	−0.0884195
$583$	−3.31627e8	−0.0693122
$584$	8.21200e7	0.0170610
$585$	4.79374e8	0.0989985
$586$	−1.53343e8	−0.0314791
$587$	2.21938e9	0.452896	0.226448	−	0.974023i	$-0.427289\pi$
$587$	2.21938e9	0.452896	0.226448	+	0.974023i	$0.427289\pi$
$588$	5.11343e9	1.03727
$589$	−4.66141e9	−0.939970
$590$	−1.05292e8	−0.0211064
$591$	−3.98770e8	−0.0794632
$592$	6.40483e9	1.26877
$593$	8.17752e9	1.61039	0.805193	−	0.593013i	$-0.202062\pi$
$593$	8.17752e9	1.61039	0.805193	+	0.593013i	$0.202062\pi$
$594$	−1.89876e8	−0.0371720
$595$	1.30478e8	0.0253938
$596$	−7.61730e8	−0.147380
$597$	−6.78892e9	−1.30584
$598$	7.60987e8	0.145520
$599$	3.39958e9	0.646296	0.323148	−	0.946348i	$-0.395259\pi$
$599$	3.39958e9	0.646296	0.323148	+	0.946348i	$0.395259\pi$
$600$	6.50725e8	0.122989
$601$	−3.03424e8	−0.0570150	−0.0285075	−	0.999594i	$-0.509075\pi$
$601$	−3.03424e8	−0.0570150	−0.0285075	+	0.999594i	$0.509075\pi$
$602$	1.02975e8	0.0192372
$603$	4.95597e8	0.0920488
$604$	8.94860e9	1.65244
$605$	−5.39014e8	−0.0989591
$606$	1.53147e8	0.0279546
$607$	−1.29895e9	−0.235740	−0.117870	−	0.993029i	$-0.537607\pi$
$607$	−1.29895e9	−0.235740	−0.117870	+	0.993029i	$0.537607\pi$
$608$	−5.84225e8	−0.105419
$609$	9.56543e8	0.171610
$610$	−1.02818e7	−0.00183406
$611$	−1.28321e10	−2.27590
$612$	−5.67379e8	−0.100056
$613$	−6.30888e9	−1.10622	−0.553108	−	0.833109i	$-0.686558\pi$
$613$	−6.30888e9	−1.10622	−0.553108	+	0.833109i	$0.686558\pi$
$614$	2.56395e8	0.0447013
$615$	2.27209e8	0.0393879
$616$	−1.43022e8	−0.0246530
$617$	1.64974e9	0.282759	0.141380	−	0.989955i	$-0.454846\pi$
$617$	1.64974e9	0.282759	0.141380	+	0.989955i	$0.454846\pi$
$618$	−1.86163e8	−0.0317274
$619$	−7.46962e9	−1.26585	−0.632924	−	0.774214i	$-0.718145\pi$
$619$	−7.46962e9	−1.26585	−0.632924	+	0.774214i	$0.718145\pi$
$620$	−2.25591e9	−0.380146
$621$	−8.14599e9	−1.36497
$622$	2.36700e8	0.0394395
$623$	−5.60730e8	−0.0929064
$624$	1.02130e10	1.68270
$625$	5.01880e9	0.822280
$626$	−1.47793e8	−0.0240794
$627$	−3.25344e9	−0.527116
$628$	−3.70119e9	−0.596325
$629$	3.01337e9	0.482809
$630$	6.57023e6	0.00104686
$631$	5.07756e9	0.804549	0.402275	−	0.915519i	$-0.368220\pi$
$631$	5.07756e9	0.804549	0.402275	+	0.915519i	$0.368220\pi$
$632$	−1.41320e8	−0.0222686
$633$	−8.67900e9	−1.36006
$634$	−4.08818e8	−0.0637114
$635$	1.02339e9	0.158610
$636$	6.52471e8	0.100568
$637$	−9.11343e9	−1.39699
$638$	1.64283e8	0.0250449
$639$	6.33683e7	0.00960769
$640$	−3.76768e8	−0.0568126
$641$	2.23958e8	0.0335864	0.0167932	−	0.999859i	$-0.494654\pi$
$641$	2.23958e8	0.0335864	0.0167932	+	0.999859i	$0.494654\pi$
$642$	6.84171e8	0.102045
$643$	−8.87712e9	−1.31684	−0.658421	−	0.752650i	$-0.728775\pi$
$643$	−8.87712e9	−1.31684	−0.658421	+	0.752650i	$0.728775\pi$
$644$	−3.06272e9	−0.451863
$645$	2.27114e9	0.333261
$646$	−9.11029e7	−0.0132959
$647$	7.31927e9	1.06244	0.531218	−	0.847235i	$-0.321735\pi$
$647$	7.31927e9	1.06244	0.531218	+	0.847235i	$0.321735\pi$
$648$	9.63170e8	0.139056
$649$	−7.93185e9	−1.13898
$650$	−5.78893e8	−0.0826801
$651$	3.36968e9	0.478691
$652$	−9.30875e9	−1.31530
$653$	−4.55826e9	−0.640623	−0.320312	−	0.947312i	$-0.603788\pi$
$653$	−4.55826e9	−0.640623	−0.320312	+	0.947312i	$0.603788\pi$
$654$	3.97223e8	0.0555280
$655$	−1.19243e9	−0.165801
$656$	1.01871e9	0.140892
$657$	−2.84188e8	−0.0390955
$658$	−1.75875e8	−0.0240665
$659$	−1.98150e9	−0.269708	−0.134854	−	0.990865i	$-0.543057\pi$
$659$	−1.98150e9	−0.269708	−0.134854	+	0.990865i	$0.543057\pi$
$660$	−1.57451e9	−0.213178
$661$	−2.02175e9	−0.272284	−0.136142	−	0.990689i	$-0.543470\pi$
$661$	−2.02175e9	−0.272284	−0.136142	+	0.990689i	$0.543470\pi$
$662$	−3.59209e8	−0.0481221
$663$	4.80504e9	0.640324
$664$	1.61395e9	0.213945
$665$	−3.09788e8	−0.0408496
$666$	1.51738e8	0.0199038
$667$	7.04801e9	0.919658
$668$	−3.92423e9	−0.509374
$669$	7.34393e8	0.0948281
$670$	3.85117e7	0.00494687
$671$	−7.74543e8	−0.0989730
$672$	4.22329e8	0.0536857
$673$	1.52706e10	1.93109	0.965546	−	0.260234i	$-0.0837995\pi$
$673$	1.52706e10	1.93109	0.965546	+	0.260234i	$0.0837995\pi$
$674$	−2.22300e8	−0.0279659
$675$	6.19676e9	0.775535
$676$	−1.02600e10	−1.27742
$677$	−5.29825e9	−0.656254	−0.328127	−	0.944634i	$-0.606417\pi$
$677$	−5.29825e9	−0.656254	−0.328127	+	0.944634i	$0.606417\pi$
$678$	5.46291e8	0.0673162
$679$	−3.01633e9	−0.369772
$680$	−8.83294e7	−0.0107727
$681$	7.46096e9	0.905273
$682$	5.78730e8	0.0698604
$683$	1.35069e10	1.62212	0.811059	−	0.584965i	$-0.198892\pi$
$683$	1.35069e10	1.62212	0.811059	+	0.584965i	$0.198892\pi$
$684$	1.34710e9	0.160955
$685$	4.03903e9	0.480131
$686$	−2.59968e8	−0.0307458
$687$	−6.93748e9	−0.816307
$688$	1.01828e10	1.19209
$689$	−1.16287e9	−0.135445
$690$	2.30037e8	0.0266579
$691$	1.33717e10	1.54174	0.770872	−	0.636990i	$-0.219821\pi$
$691$	1.33717e10	1.54174	0.770872	+	0.636990i	$0.219821\pi$
$692$	2.44645e9	0.280650
$693$	4.94947e8	0.0564927
$694$	9.16793e8	0.104115
$695$	−2.76654e9	−0.312601
$696$	−6.47549e8	−0.0728015
$697$	4.79286e8	0.0536142
$698$	3.81444e8	0.0424558
$699$	1.34108e10	1.48520
$700$	2.32985e9	0.256735
$701$	−1.78885e10	−1.96138	−0.980689	−	0.195574i	$-0.937343\pi$
$701$	−1.78885e10	−1.96138	−0.980689	+	0.195574i	$0.937343\pi$
$702$	−6.65810e8	−0.0726391
$703$	−7.15450e9	−0.776668
$704$	−7.01094e9	−0.757307
$705$	−3.87898e9	−0.416923
$706$	−6.67206e8	−0.0713582
$707$	1.09852e9	0.116907
$708$	1.56058e10	1.65261
$709$	1.23116e9	0.129734	0.0648670	−	0.997894i	$-0.479338\pi$
$709$	1.23116e9	0.129734	0.0648670	+	0.997894i	$0.479338\pi$
$710$	4.92420e6	0.000516335 0
$711$	4.89058e8	0.0510290
$712$	3.79596e8	0.0394132
$713$	2.48285e10	2.56530
$714$	6.58572e7	0.00677110
$715$	2.80619e9	0.287108
$716$	1.49156e10	1.51861
$717$	−5.43612e9	−0.550772
$718$	−2.50183e8	−0.0252245
$719$	−5.12447e8	−0.0514159	−0.0257080	−	0.999669i	$-0.508184\pi$
$719$	−5.12447e8	−0.0514159	−0.0257080	+	0.999669i	$0.508184\pi$
$720$	6.49707e8	0.0648715
$721$	−1.33535e9	−0.132685
$722$	−3.72854e8	−0.0368688
$723$	−5.10502e9	−0.502358
$724$	1.31618e9	0.128893
$725$	−5.36151e9	−0.522522
$726$	−2.72061e8	−0.0263868
$727$	−9.48545e9	−0.915561	−0.457781	−	0.889065i	$-0.651356\pi$
$727$	−9.48545e9	−0.915561	−0.457781	+	0.889065i	$0.651356\pi$
$728$	−5.01514e8	−0.0481752
$729$	6.38401e9	0.610306
$730$	−2.20836e7	−0.00210106
$731$	4.79084e9	0.453629
$732$	1.52390e9	0.143605
$733$	1.12767e10	1.05759	0.528797	−	0.848748i	$-0.322643\pi$
$733$	1.12767e10	1.05759	0.528797	+	0.848748i	$0.322643\pi$
$734$	3.38887e8	0.0316314
$735$	−2.75487e9	−0.255915
$736$	3.11181e9	0.287701
$737$	2.90115e9	0.266953
$738$	2.41345e7	0.00221024
$739$	9.98637e9	0.910232	0.455116	−	0.890432i	$-0.349598\pi$
$739$	9.98637e9	0.910232	0.455116	+	0.890432i	$0.349598\pi$
$740$	−3.46245e9	−0.314103
$741$	−1.14084e10	−1.03005
$742$	−1.59381e7	−0.00143226
$743$	−7.36552e9	−0.658783	−0.329391	−	0.944193i	$-0.606844\pi$
$743$	−7.36552e9	−0.658783	−0.329391	+	0.944193i	$0.606844\pi$
$744$	−2.28116e9	−0.203073
$745$	4.10384e8	0.0363616
$746$	2.06416e8	0.0182036
$747$	−5.58531e9	−0.490259
$748$	−3.32135e9	−0.290175
$749$	4.90755e9	0.426755
$750$	−3.61244e8	−0.0312670
$751$	2.09910e10	1.80840	0.904199	−	0.427112i	$-0.140469\pi$
$751$	2.09910e10	1.80840	0.904199	+	0.427112i	$0.140469\pi$
$752$	−1.73916e10	−1.49135
$753$	7.82648e9	0.668012
$754$	5.76067e8	0.0489411
$755$	−4.82108e9	−0.407690
$756$	2.67967e9	0.225556
$757$	−2.41033e8	−0.0201949	−0.0100974	−	0.999949i	$-0.503214\pi$
$757$	−2.41033e8	−0.0201949	−0.0100974	+	0.999949i	$0.503214\pi$
$758$	−7.87882e8	−0.0657081
$759$	1.73291e10	1.43856
$760$	2.09716e8	0.0173294
$761$	−1.31561e10	−1.08213	−0.541066	−	0.840980i	$-0.681979\pi$
$761$	−1.31561e10	−1.08213	−0.541066	+	0.840980i	$0.681979\pi$
$762$	5.16542e8	0.0422925
$763$	2.84927e9	0.232219
$764$	−7.25006e9	−0.588186
$765$	3.05677e8	0.0246858
$766$	1.22207e9	0.0982415
$767$	−2.78135e10	−2.22573
$768$	1.36507e10	1.08741
$769$	9.25270e8	0.0733713	0.0366857	−	0.999327i	$-0.488320\pi$
$769$	9.25270e8	0.0733713	0.0366857	+	0.999327i	$0.488320\pi$
$770$	3.84612e7	0.00303602
$771$	−7.67467e9	−0.603072
$772$	−1.56014e10	−1.22041
$773$	2.31916e9	0.180594	0.0902970	−	0.995915i	$-0.471218\pi$
$773$	2.31916e9	0.180594	0.0902970	+	0.995915i	$0.471218\pi$
$774$	2.41243e8	0.0187009
$775$	−1.88874e10	−1.45752
$776$	2.04196e9	0.156867
$777$	5.17190e9	0.395527
$778$	−7.87779e8	−0.0599758
$779$	−1.13794e9	−0.0862462
$780$	−5.52113e9	−0.416578
$781$	3.70949e8	0.0278635
$782$	4.85250e8	0.0362862
$783$	−6.16651e9	−0.459064
$784$	−1.23516e10	−0.915416
$785$	1.99403e9	0.147125
$786$	−6.01862e8	−0.0442098
$787$	2.62357e10	1.91859	0.959294	−	0.282409i	$-0.0911335\pi$
$787$	2.62357e10	1.91859	0.959294	+	0.282409i	$0.0911335\pi$
$788$	9.66544e8	0.0703687
$789$	1.81910e10	1.31852
$790$	3.80035e7	0.00274239
$791$	3.91854e9	0.281518
$792$	−3.35063e8	−0.0239656
$793$	−2.71598e9	−0.193406
$794$	−1.20890e9	−0.0857071
$795$	−3.51520e8	−0.0248122
$796$	1.64551e10	1.15639
$797$	−1.08177e9	−0.0756884	−0.0378442	−	0.999284i	$-0.512049\pi$
$797$	−1.08177e9	−0.0756884	−0.0378442	+	0.999284i	$0.512049\pi$
$798$	−1.56361e8	−0.0108923
$799$	−8.18249e9	−0.567508
$800$	−2.36719e9	−0.163463
$801$	−1.31365e9	−0.0903161
$802$	−7.69082e8	−0.0526457
$803$	−1.66360e9	−0.113382
$804$	−5.70798e9	−0.387334
$805$	1.65005e9	0.111484
$806$	2.02935e9	0.136516
$807$	−1.50502e10	−1.00806
$808$	−7.43663e8	−0.0495948
$809$	1.55905e10	1.03524	0.517619	−	0.855611i	$-0.326819\pi$
$809$	1.55905e10	1.03524	0.517619	+	0.855611i	$0.326819\pi$
$810$	−2.59014e8	−0.0171248
$811$	−2.54597e10	−1.67602	−0.838011	−	0.545653i	$-0.816282\pi$
$811$	−2.54597e10	−1.67602	−0.838011	+	0.545653i	$0.816282\pi$
$812$	−2.31848e9	−0.151970
$813$	−1.95601e10	−1.27660
$814$	8.88255e8	0.0577235
$815$	5.01512e9	0.324511
$816$	6.51239e9	0.419589
$817$	−1.13747e10	−0.729729
$818$	−8.87053e8	−0.0566647
$819$	1.73556e9	0.110394
$820$	−5.50713e8	−0.0348800
$821$	−7.47632e9	−0.471506	−0.235753	−	0.971813i	$-0.575756\pi$
$821$	−7.47632e9	−0.471506	−0.235753	+	0.971813i	$0.575756\pi$
$822$	2.03865e9	0.128024
$823$	1.60133e10	1.00134	0.500668	−	0.865639i	$-0.333088\pi$
$823$	1.60133e10	1.00134	0.500668	+	0.865639i	$0.333088\pi$
$824$	9.03987e8	0.0562882
$825$	−1.31825e10	−0.817349
$826$	−3.81208e8	−0.0235359
$827$	4.42892e9	0.272288	0.136144	−	0.990689i	$-0.456529\pi$
$827$	4.42892e9	0.272288	0.136144	+	0.990689i	$0.456529\pi$
$828$	−7.17518e9	−0.439265
$829$	8.07994e9	0.492569	0.246285	−	0.969198i	$-0.420790\pi$
$829$	8.07994e9	0.492569	0.246285	+	0.969198i	$0.420790\pi$
$830$	−4.34021e8	−0.0263474
$831$	−2.79342e10	−1.68862
$832$	−2.45843e10	−1.47988
$833$	−5.81125e9	−0.348347
$834$	−1.39638e9	−0.0833531
$835$	2.11419e9	0.125673
$836$	7.88573e9	0.466788
$837$	−2.17232e10	−1.28051
$838$	1.15702e9	0.0679182
$839$	8.90572e9	0.520597	0.260299	−	0.965528i	$-0.416179\pi$
$839$	8.90572e9	0.520597	0.260299	+	0.965528i	$0.416179\pi$
$840$	−1.51601e8	−0.00882522
$841$	−1.19145e10	−0.690702
$842$	−3.23023e7	−0.00186484
$843$	5.58099e9	0.320859
$844$	2.10363e10	1.20440
$845$	5.52758e9	0.315164
$846$	−4.12030e8	−0.0233955
$847$	−1.95149e9	−0.110350
$848$	−1.57606e9	−0.0887541
$849$	−1.30056e9	−0.0729379
$850$	−3.69136e8	−0.0206167
$851$	3.81077e10	2.11962
$852$	−7.29837e8	−0.0404285
$853$	−7.54486e7	−0.00416227	−0.00208113	−	0.999998i	$-0.500662\pi$
$853$	−7.54486e7	−0.00416227	−0.00208113	+	0.999998i	$0.500662\pi$
$854$	−3.72249e7	−0.00204518
$855$	−7.25753e8	−0.0397107
$856$	−3.32225e9	−0.181040
$857$	5.81681e9	0.315683	0.157842	−	0.987464i	$-0.449546\pi$
$857$	5.81681e9	0.315683	0.157842	+	0.987464i	$0.449546\pi$
$858$	1.41639e9	0.0765555
$859$	−4.75325e8	−0.0255867	−0.0127934	−	0.999918i	$-0.504072\pi$
$859$	−4.75325e8	−0.0255867	−0.0127934	+	0.999918i	$0.504072\pi$
$860$	−5.50482e9	−0.295120
$861$	8.22606e8	0.0439219
$862$	5.89323e8	0.0313385
$863$	2.63855e10	1.39742	0.698711	−	0.715404i	$-0.253757\pi$
$863$	2.63855e10	1.39742	0.698711	+	0.715404i	$0.253757\pi$
$864$	−2.72261e9	−0.143611
$865$	−1.31803e9	−0.0692419
$866$	−1.26737e9	−0.0663117
$867$	−1.85322e10	−0.965739
$868$	−8.16747e9	−0.423905
$869$	2.86288e9	0.147990
$870$	1.74138e8	0.00896552
$871$	1.01731e10	0.521661
$872$	−1.92887e9	−0.0985132
$873$	−7.06649e9	−0.359463
$874$	−1.15210e9	−0.0583717
$875$	−2.59120e9	−0.130759
$876$	3.27310e9	0.164511
$877$	−1.15679e10	−0.579102	−0.289551	−	0.957163i	$-0.593506\pi$
$877$	−1.15679e10	−0.579102	−0.289551	+	0.957163i	$0.593506\pi$
$878$	8.96936e8	0.0447229
$879$	−1.22446e10	−0.608110
$880$	3.80329e9	0.188135
$881$	−2.41012e10	−1.18747	−0.593735	−	0.804660i	$-0.702347\pi$
$881$	−2.41012e10	−1.18747	−0.593735	+	0.804660i	$0.702347\pi$
$882$	−2.92626e8	−0.0143606
$883$	2.84593e10	1.39111	0.695554	−	0.718474i	$-0.255159\pi$
$883$	2.84593e10	1.39111	0.695554	+	0.718474i	$0.255159\pi$
$884$	−1.16465e10	−0.567040
$885$	−8.40766e9	−0.407731
$886$	−9.74374e8	−0.0470661
$887$	4.19509e9	0.201841	0.100920	−	0.994895i	$-0.467821\pi$
$887$	4.19509e9	0.201841	0.100920	+	0.994895i	$0.467821\pi$
$888$	−3.50121e9	−0.167793
$889$	3.70515e9	0.176868
$890$	−1.02080e8	−0.00485375
$891$	−1.95120e10	−0.924124
$892$	−1.78003e9	−0.0839752
$893$	1.94273e10	0.912919
$894$	2.07136e8	0.00969560
$895$	−8.03581e9	−0.374670
$896$	−1.36408e9	−0.0633523
$897$	6.07655e10	2.81114
$898$	5.21003e8	0.0240089
$899$	1.87952e10	0.862755
$900$	5.45825e9	0.249577
$901$	−7.41513e8	−0.0337740
$902$	1.41280e8	0.00640998
$903$	8.22261e9	0.371623
$904$	−2.65272e9	−0.119427
$905$	−7.09096e8	−0.0318006
$906$	−2.43338e9	−0.108708
$907$	6.89168e8	0.0306690	0.0153345	−	0.999882i	$-0.495119\pi$
$907$	6.89168e8	0.0306690	0.0153345	+	0.999882i	$0.495119\pi$
$908$	−1.80840e10	−0.801666
$909$	2.57355e9	0.113647
$910$	1.34866e8	0.00593279
$911$	4.48379e10	1.96486	0.982428	−	0.186643i	$-0.0597609\pi$
$911$	4.48379e10	1.96486	0.982428	+	0.186643i	$0.0597609\pi$
$912$	−1.54620e10	−0.674970
$913$	−3.26956e10	−1.42181
$914$	2.34128e9	0.101424
$915$	−8.21007e8	−0.0354301
$916$	1.68152e10	0.722882
$917$	−4.31715e9	−0.184886
$918$	−4.24559e8	−0.0181129
$919$	1.19536e10	0.508037	0.254019	−	0.967199i	$-0.418248\pi$
$919$	1.19536e10	0.508037	0.254019	+	0.967199i	$0.418248\pi$
$920$	−1.11703e9	−0.0472942
$921$	2.04733e10	0.863535
$922$	−1.87199e9	−0.0786583
$923$	1.30075e9	0.0544489
$924$	−5.70050e9	−0.237717
$925$	−2.89890e10	−1.20431
$926$	−6.08681e8	−0.0251913
$927$	−3.12838e9	−0.128985
$928$	2.35564e9	0.0967589
$929$	1.32992e9	0.0544215	0.0272107	−	0.999630i	$-0.491337\pi$
$929$	1.32992e9	0.0544215	0.0272107	+	0.999630i	$0.491337\pi$
$930$	6.13447e8	0.0250085
$931$	1.37974e10	0.560367
$932$	−3.25052e10	−1.31522
$933$	1.89007e10	0.761889
$934$	−2.21118e9	−0.0887993
$935$	1.78939e9	0.0715919
$936$	−1.17492e9	−0.0468320
$937$	−1.18640e10	−0.471134	−0.235567	−	0.971858i	$-0.575695\pi$
$937$	−1.18640e10	−0.471134	−0.235567	+	0.971858i	$0.575695\pi$
$938$	1.39431e8	0.00551630
$939$	−1.18014e10	−0.465163
$940$	9.40192e9	0.369206
$941$	3.94074e10	1.54175	0.770875	−	0.636987i	$-0.219819\pi$
$941$	3.94074e10	1.54175	0.770875	+	0.636987i	$0.219819\pi$
$942$	1.00646e9	0.0392300
$943$	6.06114e9	0.235377
$944$	−3.76963e10	−1.45847
$945$	−1.44368e9	−0.0556492
$946$	1.41220e9	0.0542348
$947$	2.53897e10	0.971478	0.485739	−	0.874104i	$-0.338551\pi$
$947$	2.53897e10	0.971478	0.485739	+	0.874104i	$0.338551\pi$
$948$	−5.63267e9	−0.214726
$949$	−5.83350e9	−0.221563
$950$	8.76421e8	0.0331650
$951$	−3.26444e10	−1.23077
$952$	−3.19795e8	−0.0120127
$953$	3.62313e10	1.35600	0.677999	−	0.735063i	$-0.262847\pi$
$953$	3.62313e10	1.35600	0.677999	+	0.735063i	$0.262847\pi$
$954$	−3.73389e7	−0.00139233
$955$	3.90599e9	0.145117
$956$	1.31761e10	0.487737
$957$	1.31181e10	0.483815
$958$	2.12701e9	0.0781609
$959$	1.46232e10	0.535399
$960$	−7.43152e9	−0.271099
$961$	3.86985e10	1.40657
$962$	3.11472e9	0.112799
$963$	1.14971e10	0.414856
$964$	1.23736e10	0.444864
$965$	8.40532e9	0.301099
$966$	8.32843e8	0.0297264
$967$	5.68929e9	0.202332	0.101166	−	0.994870i	$-0.467743\pi$
$967$	5.68929e9	0.202332	0.101166	+	0.994870i	$0.467743\pi$
$968$	1.32109e9	0.0468134
$969$	−7.27464e9	−0.256849
$970$	−5.49120e8	−0.0193182
$971$	−2.72083e10	−0.953749	−0.476875	−	0.878971i	$-0.658230\pi$
$971$	−2.72083e10	−0.953749	−0.476875	+	0.878971i	$0.658230\pi$
$972$	1.48369e10	0.518217
$973$	−1.00162e10	−0.348585
$974$	−1.61371e9	−0.0559588
$975$	−4.62251e10	−1.59721
$976$	−3.68104e9	−0.126735
$977$	2.38663e10	0.818755	0.409378	−	0.912365i	$-0.365746\pi$
$977$	2.38663e10	0.818755	0.409378	+	0.912365i	$0.365746\pi$
$978$	2.53132e9	0.0865288
$979$	−7.68990e9	−0.261928
$980$	6.67730e9	0.226626
$981$	6.67512e9	0.225745
$982$	−1.77886e9	−0.0599448
$983$	1.50265e10	0.504569	0.252284	−	0.967653i	$-0.418818\pi$
$983$	1.50265e10	0.504569	0.252284	+	0.967653i	$0.418818\pi$
$984$	−5.56878e8	−0.0186328
$985$	−5.20728e8	−0.0173614
$986$	3.67334e8	0.0122037
$987$	−1.40438e10	−0.464915
$988$	2.76518e10	0.912165
$989$	6.05859e10	1.99152
$990$	9.01047e7	0.00295138
$991$	−3.29678e10	−1.07605	−0.538025	−	0.842929i	$-0.680829\pi$
$991$	−3.29678e10	−1.07605	−0.538025	+	0.842929i	$0.680829\pi$
$992$	8.29838e9	0.269900
$993$	−2.86831e10	−0.929618
$994$	1.78280e7	0.000575770 0
$995$	−8.86523e9	−0.285305
$996$	6.43281e10	2.06297
$997$	8.15441e9	0.260591	0.130296	−	0.991475i	$-0.458407\pi$
$997$	8.15441e9	0.260591	0.130296	+	0.991475i	$0.458407\pi$
$998$	−2.26013e8	−0.00719741
$999$	−3.33415e10	−1.05805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.8.a.b.1.9	✓	19
3.2	odd	2		549.8.a.e.1.11		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.8.a.b.1.9	✓	19	1.1	even	1	trivial
549.8.a.e.1.11		19	3.2	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:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	61.a (trivial)

\(f(q)\)	\(=\)	\(q+0.659105 q^{2} +52.6301 q^{3} -127.566 q^{4} +68.7263 q^{5} +34.6888 q^{6} +248.822 q^{7} -168.445 q^{8} +582.927 q^{9} +O(q^{10})\)	\(q+0.659105 q^{2} +52.6301 q^{3} -127.566 q^{4} +68.7263 q^{5} +34.6888 q^{6} +248.822 q^{7} -168.445 q^{8} +582.927 q^{9} +45.2978 q^{10} +3412.37 q^{11} -6713.79 q^{12} +11965.7 q^{13} +164.000 q^{14} +3617.07 q^{15} +16217.4 q^{16} +7630.02 q^{17} +384.210 q^{18} -18115.6 q^{19} -8767.11 q^{20} +13095.5 q^{21} +2249.11 q^{22} +96490.7 q^{23} -8865.25 q^{24} -73401.7 q^{25} +7886.64 q^{26} -84422.5 q^{27} -31741.1 q^{28} +73043.4 q^{29} +2384.03 q^{30} +257315. q^{31} +32249.9 q^{32} +179593. q^{33} +5028.98 q^{34} +17100.6 q^{35} -74361.4 q^{36} +394936. q^{37} -11940.1 q^{38} +629755. q^{39} -11576.6 q^{40} +62815.8 q^{41} +8631.33 q^{42} +627894. q^{43} -435301. q^{44} +40062.4 q^{45} +63597.5 q^{46} -1.07241e6 q^{47} +853522. q^{48} -761631. q^{49} -48379.4 q^{50} +401568. q^{51} -1.52641e6 q^{52} -97183.7 q^{53} -55643.3 q^{54} +234520. q^{55} -41912.7 q^{56} -953424. q^{57} +48143.3 q^{58} -2.32444e6 q^{59} -461414. q^{60} -226981. q^{61} +169598. q^{62} +145045. q^{63} -2.05457e6 q^{64} +822357. q^{65} +118371. q^{66} +850187. q^{67} -973328. q^{68} +5.07831e6 q^{69} +11271.1 q^{70} +108707. q^{71} -98190.8 q^{72} -487519. q^{73} +260304. q^{74} -3.86314e6 q^{75} +2.31092e6 q^{76} +849073. q^{77} +415074. q^{78} +838970. q^{79} +1.11456e6 q^{80} -5.71803e6 q^{81} +41402.2 q^{82} -9.58150e6 q^{83} -1.67054e6 q^{84} +524383. q^{85} +413848. q^{86} +3.84428e6 q^{87} -574795. q^{88} -2.25354e6 q^{89} +26405.3 q^{90} +2.97732e6 q^{91} -1.23089e7 q^{92} +1.35425e7 q^{93} -706830. q^{94} -1.24502e6 q^{95} +1.69731e6 q^{96} -1.21224e7 q^{97} -501995. q^{98} +1.98916e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 23 q^{2} + 134 q^{3} + 1317 q^{4} + 429 q^{5} + q^{6} + 1017 q^{7} + 4479 q^{8} + 18249 q^{9}+O(q^{10}) \) 19 * q + 23 * q^2 + 134 * q^3 + 1317 * q^4 + 429 * q^5 + q^6 + 1017 * q^7 + 4479 * q^8 + 18249 * q^9	\( 19 q + 23 q^{2} + 134 q^{3} + 1317 q^{4} + 429 q^{5} + q^{6} + 1017 q^{7} + 4479 q^{8} + 18249 q^{9} + 5029 q^{10} + 21895 q^{11} + 13639 q^{12} + 4293 q^{13} + 28184 q^{14} + 28006 q^{15} + 30149 q^{16} - 3512 q^{17} - 52668 q^{18} + 62964 q^{19} + 128631 q^{20} + 151802 q^{21} + 380355 q^{22} + 194883 q^{23} + 643935 q^{24} + 594054 q^{25} + 412571 q^{26} + 544388 q^{27} + 818132 q^{28} + 343584 q^{29} + 1287286 q^{30} + 376400 q^{31} + 1182835 q^{32} + 545950 q^{33} + 952570 q^{34} + 1101957 q^{35} + 1334624 q^{36} + 630086 q^{37} + 263947 q^{38} + 1043878 q^{39} + 78765 q^{40} + 1584539 q^{41} + 510094 q^{42} - 93664 q^{43} + 2646833 q^{44} + 206061 q^{45} - 559318 q^{46} + 1752508 q^{47} - 2563869 q^{48} + 2287550 q^{49} + 411602 q^{50} + 2278880 q^{51} - 3538347 q^{52} + 127548 q^{53} - 8008280 q^{54} - 2282221 q^{55} + 1035726 q^{56} - 2815944 q^{57} - 5418817 q^{58} + 5826729 q^{59} - 8433754 q^{60} - 4312639 q^{61} - 6139425 q^{62} - 10627967 q^{63} - 7157059 q^{64} - 10672623 q^{65} - 15073362 q^{66} - 10206319 q^{67} - 10901542 q^{68} - 11593480 q^{69} - 29382493 q^{70} + 4358512 q^{71} - 23635494 q^{72} - 15042353 q^{73} - 10739915 q^{74} + 9338192 q^{75} - 12059835 q^{76} + 6632395 q^{77} - 41176024 q^{78} + 3473903 q^{79} - 6140121 q^{80} + 21257391 q^{81} - 14428970 q^{82} + 19386052 q^{83} - 10704930 q^{84} - 1181128 q^{85} + 3163940 q^{86} + 23797004 q^{87} + 13465595 q^{88} + 10913876 q^{89} - 18268219 q^{90} + 17042997 q^{91} + 20668398 q^{92} + 23643110 q^{93} - 43306680 q^{94} + 27353940 q^{95} + 24794935 q^{96} + 55835432 q^{97} + 16445433 q^{98} + 26225127 q^{99}+O(q^{100}) \) 19 * q + 23 * q^2 + 134 * q^3 + 1317 * q^4 + 429 * q^5 + q^6 + 1017 * q^7 + 4479 * q^8 + 18249 * q^9 + 5029 * q^10 + 21895 * q^11 + 13639 * q^12 + 4293 * q^13 + 28184 * q^14 + 28006 * q^15 + 30149 * q^16 - 3512 * q^17 - 52668 * q^18 + 62964 * q^19 + 128631 * q^20 + 151802 * q^21 + 380355 * q^22 + 194883 * q^23 + 643935 * q^24 + 594054 * q^25 + 412571 * q^26 + 544388 * q^27 + 818132 * q^28 + 343584 * q^29 + 1287286 * q^30 + 376400 * q^31 + 1182835 * q^32 + 545950 * q^33 + 952570 * q^34 + 1101957 * q^35 + 1334624 * q^36 + 630086 * q^37 + 263947 * q^38 + 1043878 * q^39 + 78765 * q^40 + 1584539 * q^41 + 510094 * q^42 - 93664 * q^43 + 2646833 * q^44 + 206061 * q^45 - 559318 * q^46 + 1752508 * q^47 - 2563869 * q^48 + 2287550 * q^49 + 411602 * q^50 + 2278880 * q^51 - 3538347 * q^52 + 127548 * q^53 - 8008280 * q^54 - 2282221 * q^55 + 1035726 * q^56 - 2815944 * q^57 - 5418817 * q^58 + 5826729 * q^59 - 8433754 * q^60 - 4312639 * q^61 - 6139425 * q^62 - 10627967 * q^63 - 7157059 * q^64 - 10672623 * q^65 - 15073362 * q^66 - 10206319 * q^67 - 10901542 * q^68 - 11593480 * q^69 - 29382493 * q^70 + 4358512 * q^71 - 23635494 * q^72 - 15042353 * q^73 - 10739915 * q^74 + 9338192 * q^75 - 12059835 * q^76 + 6632395 * q^77 - 41176024 * q^78 + 3473903 * q^79 - 6140121 * q^80 + 21257391 * q^81 - 14428970 * q^82 + 19386052 * q^83 - 10704930 * q^84 - 1181128 * q^85 + 3163940 * q^86 + 23797004 * q^87 + 13465595 * q^88 + 10913876 * q^89 - 18268219 * q^90 + 17042997 * q^91 + 20668398 * q^92 + 23643110 * q^93 - 43306680 * q^94 + 27353940 * q^95 + 24794935 * q^96 + 55835432 * q^97 + 16445433 * q^98 + 26225127 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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