LMFDB - Embedded newform 69.8.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, 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("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.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:	$21.5545667584$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 775x^{5} - 474x^{4} + 167184x^{3} - 33920x^{2} - 9348928x + 28965760 $ x^7 - 775x^5 - 474x^4 + 167184x^3 - 33920x^2 - 9348928*x + 28965760
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-11.3612$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.3612 q^{2} -27.0000 q^{3} +1.07734 q^{4} +392.713 q^{5} +306.753 q^{6} -1170.86 q^{7} +1442.00 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-11.3612 q^{2} -27.0000 q^{3} +1.07734 q^{4} +392.713 q^{5} +306.753 q^{6} -1170.86 q^{7} +1442.00 q^{8} +729.000 q^{9} -4461.70 q^{10} +1763.47 q^{11} -29.0881 q^{12} -14839.2 q^{13} +13302.4 q^{14} -10603.3 q^{15} -16520.7 q^{16} -26146.8 q^{17} -8282.33 q^{18} +42426.4 q^{19} +423.084 q^{20} +31613.3 q^{21} -20035.2 q^{22} -12167.0 q^{23} -38933.9 q^{24} +76098.5 q^{25} +168591. q^{26} -19683.0 q^{27} -1261.41 q^{28} +166035. q^{29} +120466. q^{30} +90693.2 q^{31} +3120.22 q^{32} -47613.7 q^{33} +297060. q^{34} -459812. q^{35} +785.379 q^{36} -38871.7 q^{37} -482016. q^{38} +400657. q^{39} +566291. q^{40} +797288. q^{41} -359165. q^{42} +557159. q^{43} +1899.85 q^{44} +286288. q^{45} +138232. q^{46} -303537. q^{47} +446060. q^{48} +547373. q^{49} -864572. q^{50} +705965. q^{51} -15986.8 q^{52} +1.41335e6 q^{53} +223623. q^{54} +692538. q^{55} -1.68838e6 q^{56} -1.14551e6 q^{57} -1.88636e6 q^{58} -724711. q^{59} -11423.3 q^{60} -2.16696e6 q^{61} -1.03039e6 q^{62} -853558. q^{63} +2.07921e6 q^{64} -5.82753e6 q^{65} +540950. q^{66} +286204. q^{67} -28169.0 q^{68} +328509. q^{69} +5.22403e6 q^{70} +1.87366e6 q^{71} +1.05122e6 q^{72} -1.67247e6 q^{73} +441630. q^{74} -2.05466e6 q^{75} +45707.6 q^{76} -2.06478e6 q^{77} -4.55196e6 q^{78} +5.35515e6 q^{79} -6.48791e6 q^{80} +531441. q^{81} -9.05817e6 q^{82} +9.69011e6 q^{83} +34058.1 q^{84} -1.02682e7 q^{85} -6.33001e6 q^{86} -4.48295e6 q^{87} +2.54292e6 q^{88} -763797. q^{89} -3.25258e6 q^{90} +1.73746e7 q^{91} -13108.0 q^{92} -2.44872e6 q^{93} +3.44855e6 q^{94} +1.66614e7 q^{95} -84245.9 q^{96} +1.48153e7 q^{97} -6.21883e6 q^{98} +1.28557e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9}+O(q^{10}) $ 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9} - 15310 q^{10} + 9040 q^{11} - 17658 q^{12} + 3774 q^{13} + 4536 q^{14} + 13932 q^{15} + 52002 q^{16} - 40760 q^{17} + 81598 q^{19} - 88946 q^{20} - 27486 q^{21} + 245034 q^{22} - 85169 q^{23} - 38394 q^{24} + 321325 q^{25} + 412748 q^{26} - 137781 q^{27} + 965948 q^{28} + 154126 q^{29} + 413370 q^{30} + 243132 q^{31} + 1278286 q^{32} - 244080 q^{33} + 984836 q^{34} - 130296 q^{35} + 476766 q^{36} + 582114 q^{37} + 772558 q^{38} - 101898 q^{39} - 132618 q^{40} + 113062 q^{41} - 122472 q^{42} - 659778 q^{43} + 659390 q^{44} - 376164 q^{45} - 591032 q^{47} - 1404054 q^{48} + 3263235 q^{49} - 702684 q^{50} + 1100520 q^{51} + 1793280 q^{52} + 207128 q^{53} + 184664 q^{55} + 5390508 q^{56} - 2203146 q^{57} - 1142916 q^{58} + 447148 q^{59} + 2401542 q^{60} + 2248970 q^{61} - 5729060 q^{62} + 742122 q^{63} + 7212922 q^{64} - 827096 q^{65} - 6615918 q^{66} + 4467570 q^{67} - 5477620 q^{68} + 2299563 q^{69} - 12744284 q^{70} - 5154608 q^{71} + 1036638 q^{72} - 13239250 q^{73} - 2827426 q^{74} - 8675775 q^{75} - 527434 q^{76} - 18415912 q^{77} - 11144196 q^{78} + 9594446 q^{79} - 55932394 q^{80} + 3720087 q^{81} - 20889952 q^{82} - 573720 q^{83} - 26080596 q^{84} + 7477272 q^{85} - 28416910 q^{86} - 4161402 q^{87} + 26555702 q^{88} - 3810540 q^{89} - 11160990 q^{90} + 36092068 q^{91} - 7957218 q^{92} - 6564564 q^{93} + 33545768 q^{94} + 10497320 q^{95} - 34513722 q^{96} + 49497978 q^{97} - 1023376 q^{98} + 6590160 q^{99}+O(q^{100}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9 - 15310 * q^10 + 9040 * q^11 - 17658 * q^12 + 3774 * q^13 + 4536 * q^14 + 13932 * q^15 + 52002 * q^16 - 40760 * q^17 + 81598 * q^19 - 88946 * q^20 - 27486 * q^21 + 245034 * q^22 - 85169 * q^23 - 38394 * q^24 + 321325 * q^25 + 412748 * q^26 - 137781 * q^27 + 965948 * q^28 + 154126 * q^29 + 413370 * q^30 + 243132 * q^31 + 1278286 * q^32 - 244080 * q^33 + 984836 * q^34 - 130296 * q^35 + 476766 * q^36 + 582114 * q^37 + 772558 * q^38 - 101898 * q^39 - 132618 * q^40 + 113062 * q^41 - 122472 * q^42 - 659778 * q^43 + 659390 * q^44 - 376164 * q^45 - 591032 * q^47 - 1404054 * q^48 + 3263235 * q^49 - 702684 * q^50 + 1100520 * q^51 + 1793280 * q^52 + 207128 * q^53 + 184664 * q^55 + 5390508 * q^56 - 2203146 * q^57 - 1142916 * q^58 + 447148 * q^59 + 2401542 * q^60 + 2248970 * q^61 - 5729060 * q^62 + 742122 * q^63 + 7212922 * q^64 - 827096 * q^65 - 6615918 * q^66 + 4467570 * q^67 - 5477620 * q^68 + 2299563 * q^69 - 12744284 * q^70 - 5154608 * q^71 + 1036638 * q^72 - 13239250 * q^73 - 2827426 * q^74 - 8675775 * q^75 - 527434 * q^76 - 18415912 * q^77 - 11144196 * q^78 + 9594446 * q^79 - 55932394 * q^80 + 3720087 * q^81 - 20889952 * q^82 - 573720 * q^83 - 26080596 * q^84 + 7477272 * q^85 - 28416910 * q^86 - 4161402 * q^87 + 26555702 * q^88 - 3810540 * q^89 - 11160990 * q^90 + 36092068 * q^91 - 7957218 * q^92 - 6564564 * q^93 + 33545768 * q^94 + 10497320 * q^95 - 34513722 * q^96 + 49497978 * q^97 - 1023376 * q^98 + 6590160 * 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 $

$2$	−11.3612	−1.00420	−0.502100	−	0.864810i	$-0.667439\pi$
$2$	−11.3612	−1.00420	−0.502100	+	0.864810i	$0.667439\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	1.07734	0.00841670
$5$	392.713	1.40501	0.702506	−	0.711677i	$-0.252064\pi$
$5$	392.713	1.40501	0.702506	+	0.711677i	$0.252064\pi$
$6$	306.753	0.579775
$7$	−1170.86	−1.29022	−0.645108	−	0.764092i	$-0.723188\pi$
$7$	−1170.86	−1.29022	−0.645108	+	0.764092i	$0.723188\pi$
$8$	1442.00	0.995747
$9$	729.000	0.333333
$10$	−4461.70	−1.41091
$11$	1763.47	0.399479	0.199739	−	0.979849i	$-0.435990\pi$
$11$	1763.47	0.399479	0.199739	+	0.979849i	$0.435990\pi$
$12$	−29.0881	−0.00485938
$13$	−14839.2	−1.87330	−0.936651	−	0.350265i	$-0.886091\pi$
$13$	−14839.2	−1.87330	−0.936651	+	0.350265i	$0.886091\pi$
$14$	13302.4	1.29563
$15$	−10603.3	−0.811184
$16$	−16520.7	−1.00835
$17$	−26146.8	−1.29077	−0.645384	−	0.763859i	$-0.723303\pi$
$17$	−26146.8	−1.29077	−0.645384	+	0.763859i	$0.723303\pi$
$18$	−8282.33	−0.334733
$19$	42426.4	1.41905	0.709527	−	0.704678i	$-0.248909\pi$
$19$	42426.4	1.41905	0.709527	+	0.704678i	$0.248909\pi$
$20$	423.084	0.0118256
$21$	31613.3	0.744906
$22$	−20035.2	−0.401156
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	−38933.9	−0.574895
$25$	76098.5	0.974061
$26$	168591.	1.88117
$27$	−19683.0	−0.192450
$28$	−1261.41	−0.0108594
$29$	166035.	1.26417	0.632087	−	0.774898i	$-0.282199\pi$
$29$	166035.	1.26417	0.632087	+	0.774898i	$0.282199\pi$
$30$	120466.	0.814591
$31$	90693.2	0.546775	0.273388	−	0.961904i	$-0.411856\pi$
$31$	90693.2	0.546775	0.273388	+	0.961904i	$0.411856\pi$
$32$	3120.22	0.0168329
$33$	−47613.7	−0.230639
$34$	297060.	1.29619
$35$	−459812.	−1.81277
$36$	785.379	0.00280557
$37$	−38871.7	−0.126162	−0.0630808	−	0.998008i	$-0.520093\pi$
$37$	−38871.7	−0.126162	−0.0630808	+	0.998008i	$0.520093\pi$
$38$	−482016.	−1.42501
$39$	400657.	1.08155
$40$	566291.	1.39904
$41$	797288.	1.80664	0.903320	−	0.428967i	$-0.141122\pi$
$41$	797288.	1.80664	0.903320	+	0.428967i	$0.141122\pi$
$42$	−359165.	−0.748035
$43$	557159.	1.06866	0.534330	−	0.845276i	$-0.320564\pi$
$43$	557159.	1.06866	0.534330	+	0.845276i	$0.320564\pi$
$44$	1899.85	0.00336229
$45$	286288.	0.468338
$46$	138232.	0.209390
$47$	−303537.	−0.426451	−0.213225	−	0.977003i	$-0.568397\pi$
$47$	−303537.	−0.426451	−0.213225	+	0.977003i	$0.568397\pi$
$48$	446060.	0.582169
$49$	547373.	0.664656
$50$	−864572.	−0.978151
$51$	705965.	0.745225
$52$	−15986.8	−0.0157670
$53$	1.41335e6	1.30402	0.652009	−	0.758211i	$-0.273926\pi$
$53$	1.41335e6	1.30402	0.652009	+	0.758211i	$0.273926\pi$
$54$	223623.	0.193258
$55$	692538.	0.561273
$56$	−1.68838e6	−1.28473
$57$	−1.14551e6	−0.819291
$58$	−1.88636e6	−1.26948
$59$	−724711.	−0.459391	−0.229696	−	0.973263i	$-0.573773\pi$
$59$	−724711.	−0.459391	−0.229696	+	0.973263i	$0.573773\pi$
$60$	−11423.3	−0.00682749
$61$	−2.16696e6	−1.22235	−0.611177	−	0.791494i	$-0.709304\pi$
$61$	−2.16696e6	−1.22235	−0.611177	+	0.791494i	$0.709304\pi$
$62$	−1.03039e6	−0.549072
$63$	−853558.	−0.430072
$64$	2.07921e6	0.991442
$65$	−5.82753e6	−2.63201
$66$	540950.	0.231608
$67$	286204.	0.116256	0.0581278	−	0.998309i	$-0.481487\pi$
$67$	286204.	0.116256	0.0581278	+	0.998309i	$0.481487\pi$
$68$	−28169.0	−0.0108640
$69$	328509.	0.120386
$70$	5.22403e6	1.82038
$71$	1.87366e6	0.621279	0.310639	−	0.950528i	$-0.399457\pi$
$71$	1.87366e6	0.621279	0.310639	+	0.950528i	$0.399457\pi$
$72$	1.05122e6	0.331916
$73$	−1.67247e6	−0.503186	−0.251593	−	0.967833i	$-0.580954\pi$
$73$	−1.67247e6	−0.503186	−0.251593	+	0.967833i	$0.580954\pi$
$74$	441630.	0.126691
$75$	−2.05466e6	−0.562374
$76$	45707.6	0.0119437
$77$	−2.06478e6	−0.515414
$78$	−4.55196e6	−1.08609
$79$	5.35515e6	1.22202	0.611008	−	0.791624i	$-0.290764\pi$
$79$	5.35515e6	1.22202	0.611008	+	0.791624i	$0.290764\pi$
$80$	−6.48791e6	−1.41674
$81$	531441.	0.111111
$82$	−9.05817e6	−1.81423
$83$	9.69011e6	1.86018	0.930091	−	0.367330i	$-0.119728\pi$
$83$	9.69011e6	1.86018	0.930091	+	0.367330i	$0.119728\pi$
$84$	34058.1	0.00626965
$85$	−1.02682e7	−1.81354
$86$	−6.33001e6	−1.07315
$87$	−4.48295e6	−0.729871
$88$	2.54292e6	0.397780
$89$	−763797.	−0.114845	−0.0574226	−	0.998350i	$-0.518288\pi$
$89$	−763797.	−0.114845	−0.0574226	+	0.998350i	$0.518288\pi$
$90$	−3.25258e6	−0.470304
$91$	1.73746e7	2.41696
$92$	−13108.0	−0.00175500
$93$	−2.44872e6	−0.315681
$94$	3.44855e6	0.428242
$95$	1.66614e7	1.99379
$96$	−84245.9	−0.00971851
$97$	1.48153e7	1.64819	0.824097	−	0.566449i	$-0.191683\pi$
$97$	1.48153e7	1.64819	0.824097	+	0.566449i	$0.191683\pi$
$98$	−6.21883e6	−0.667448
$99$	1.28557e6	0.133160
$100$	81983.7	0.00819837
$101$	3.67489e6	0.354911	0.177455	−	0.984129i	$-0.443213\pi$
$101$	3.67489e6	0.354911	0.177455	+	0.984129i	$0.443213\pi$
$102$	−8.02062e6	−0.748355
$103$	−1.01045e6	−0.0911136	−0.0455568	−	0.998962i	$-0.514506\pi$
$103$	−1.01045e6	−0.0911136	−0.0455568	+	0.998962i	$0.514506\pi$
$104$	−2.13980e7	−1.86533
$105$	1.24149e7	1.04660
$106$	−1.60574e7	−1.30949
$107$	7.31751e6	0.577458	0.288729	−	0.957411i	$-0.406767\pi$
$107$	7.31751e6	0.577458	0.288729	+	0.957411i	$0.406767\pi$
$108$	−21205.2	−0.00161979
$109$	8.27606e6	0.612112	0.306056	−	0.952014i	$-0.400991\pi$
$109$	8.27606e6	0.612112	0.306056	+	0.952014i	$0.400991\pi$
$110$	−7.86807e6	−0.563630
$111$	1.04954e6	0.0728395
$112$	1.93435e7	1.30098
$113$	−7.73720e6	−0.504440	−0.252220	−	0.967670i	$-0.581161\pi$
$113$	−7.73720e6	−0.504440	−0.252220	+	0.967670i	$0.581161\pi$
$114$	1.30144e7	0.822732
$115$	−4.77814e6	−0.292965
$116$	178876.	0.0106402
$117$	−1.08177e7	−0.624434
$118$	8.23360e6	0.461320
$119$	3.06143e7	1.66537
$120$	−1.52898e7	−0.807735
$121$	−1.63773e7	−0.840417
$122$	2.46194e7	1.22749
$123$	−2.15268e7	−1.04306
$124$	97707.2	0.00460204
$125$	−795831.	−0.0364448
$126$	9.69746e6	0.431878
$127$	3.87137e7	1.67707	0.838537	−	0.544845i	$-0.183412\pi$
$127$	3.87137e7	1.67707	0.838537	+	0.544845i	$0.183412\pi$
$128$	−2.40217e7	−1.01244
$129$	−1.50433e7	−0.616991
$130$	6.62079e7	2.64307
$131$	−1.95939e7	−0.761504	−0.380752	−	0.924677i	$-0.624335\pi$
$131$	−1.95939e7	−0.761504	−0.380752	+	0.924677i	$0.624335\pi$
$132$	−51296.0	−0.00194122
$133$	−4.96755e7	−1.83089
$134$	−3.25162e6	−0.116744
$135$	−7.72977e6	−0.270395
$136$	−3.77037e7	−1.28528
$137$	3.83160e7	1.27309	0.636544	−	0.771241i	$-0.280364\pi$
$137$	3.83160e7	1.27309	0.636544	+	0.771241i	$0.280364\pi$
$138$	−3.73226e6	−0.120891
$139$	1.71827e7	0.542674	0.271337	−	0.962484i	$-0.412534\pi$
$139$	1.71827e7	0.542674	0.271337	+	0.962484i	$0.412534\pi$
$140$	−495373.	−0.0152575
$141$	8.19550e6	0.246211
$142$	−2.12871e7	−0.623888
$143$	−2.61684e7	−0.748344
$144$	−1.20436e7	−0.336115
$145$	6.52041e7	1.77618
$146$	1.90013e7	0.505299
$147$	−1.47791e7	−0.383740
$148$	−41877.9	−0.00106186
$149$	−1.49325e7	−0.369811	−0.184905	−	0.982756i	$-0.559198\pi$
$149$	−1.49325e7	−0.369811	−0.184905	+	0.982756i	$0.559198\pi$
$150$	2.33434e7	0.564736
$151$	1.92788e7	0.455680	0.227840	−	0.973699i	$-0.426834\pi$
$151$	1.92788e7	0.455680	0.227840	+	0.973699i	$0.426834\pi$
$152$	6.11788e7	1.41302
$153$	−1.90610e7	−0.430256
$154$	2.34584e7	0.517578
$155$	3.56164e7	0.768227
$156$	431643.	0.00910309
$157$	−7.87019e7	−1.62307	−0.811534	−	0.584306i	$-0.801367\pi$
$157$	−7.87019e7	−1.62307	−0.811534	+	0.584306i	$0.801367\pi$
$158$	−6.08411e7	−1.22715
$159$	−3.81604e7	−0.752876
$160$	1.22535e6	0.0236505
$161$	1.42459e7	0.269029
$162$	−6.03782e6	−0.111578
$163$	8.12119e7	1.46880	0.734401	−	0.678716i	$-0.237463\pi$
$163$	8.12119e7	1.46880	0.734401	+	0.678716i	$0.237463\pi$
$164$	858948.	0.0152059
$165$	−1.86985e7	−0.324051
$166$	−1.10091e8	−1.86799
$167$	−9.86019e6	−0.163824	−0.0819120	−	0.996640i	$-0.526103\pi$
$167$	−9.86019e6	−0.163824	−0.0819120	+	0.996640i	$0.526103\pi$
$168$	4.55862e7	0.741739
$169$	1.57452e8	2.50926
$170$	1.16659e8	1.82116
$171$	3.09289e7	0.473018
$172$	600248.	0.00899459
$173$	−7.27616e7	−1.06842	−0.534209	−	0.845353i	$-0.679390\pi$
$173$	−7.27616e7	−1.06842	−0.534209	+	0.845353i	$0.679390\pi$
$174$	5.09317e7	0.732936
$175$	−8.91008e7	−1.25675
$176$	−2.91338e7	−0.402813
$177$	1.95672e7	0.265230
$178$	8.67766e6	0.115327
$179$	4.97342e6	0.0648141	0.0324071	−	0.999475i	$-0.489683\pi$
$179$	4.97342e6	0.0648141	0.0324071	+	0.999475i	$0.489683\pi$
$180$	308428.	0.00394186
$181$	−6.89922e7	−0.864818	−0.432409	−	0.901678i	$-0.642336\pi$
$181$	−6.89922e7	−0.864818	−0.432409	+	0.901678i	$0.642336\pi$
$182$	−1.97397e8	−2.42711
$183$	5.85080e7	0.705727
$184$	−1.75448e7	−0.207628
$185$	−1.52654e7	−0.177259
$186$	2.78204e7	0.317007
$187$	−4.61092e7	−0.515634
$188$	−327012.	−0.00358931
$189$	2.30461e7	0.248302
$190$	−1.89294e8	−2.00216
$191$	−1.83352e8	−1.90401	−0.952004	−	0.306087i	$-0.900980\pi$
$191$	−1.83352e8	−1.90401	−0.952004	+	0.306087i	$0.900980\pi$
$192$	−5.61385e7	−0.572409
$193$	−9.29213e7	−0.930389	−0.465195	−	0.885208i	$-0.654016\pi$
$193$	−9.29213e7	−0.930389	−0.465195	+	0.885208i	$0.654016\pi$
$194$	−1.68319e8	−1.65511
$195$	1.57343e8	1.51959
$196$	589705.	0.00559421
$197$	−8.71794e7	−0.812423	−0.406212	−	0.913779i	$-0.633150\pi$
$197$	−8.71794e7	−0.812423	−0.406212	+	0.913779i	$0.633150\pi$
$198$	−1.46056e7	−0.133719
$199$	−1.27549e8	−1.14734	−0.573668	−	0.819088i	$-0.694480\pi$
$199$	−1.27549e8	−1.14734	−0.573668	+	0.819088i	$0.694480\pi$
$200$	1.09734e8	0.969919
$201$	−7.72750e6	−0.0671202
$202$	−4.17512e7	−0.356401
$203$	−1.94404e8	−1.63106
$204$	760562.	0.00627233
$205$	3.13105e8	2.53835
$206$	1.14799e7	0.0914963
$207$	−8.86974e6	−0.0695048
$208$	2.45154e8	1.88894
$209$	7.48177e7	0.566882
$210$	−1.41049e8	−1.05100
$211$	1.31901e8	0.966630	0.483315	−	0.875446i	$-0.339433\pi$
$211$	1.31901e8	0.966630	0.483315	+	0.875446i	$0.339433\pi$
$212$	1.52265e6	0.0109755
$213$	−5.05888e7	−0.358695
$214$	−8.31359e7	−0.579883
$215$	2.18804e8	1.50148
$216$	−2.83828e7	−0.191632
$217$	−1.06189e8	−0.705458
$218$	−9.40261e7	−0.614683
$219$	4.51567e7	0.290514
$220$	746096.	0.00472406
$221$	3.87997e8	2.41800
$222$	−1.19240e7	−0.0731454
$223$	−2.86007e7	−0.172707	−0.0863535	−	0.996265i	$-0.527521\pi$
$223$	−2.86007e7	−0.172707	−0.0863535	+	0.996265i	$0.527521\pi$
$224$	−3.65334e6	−0.0217181
$225$	5.54758e7	0.324687
$226$	8.79040e7	0.506558
$227$	1.68470e8	0.955942	0.477971	−	0.878376i	$-0.341372\pi$
$227$	1.68470e8	0.955942	0.477971	+	0.878376i	$0.341372\pi$
$228$	−1.23410e6	−0.00689572
$229$	−1.94092e8	−1.06803	−0.534015	−	0.845475i	$-0.679317\pi$
$229$	−1.94092e8	−1.06803	−0.534015	+	0.845475i	$0.679317\pi$
$230$	5.42855e7	0.294196
$231$	5.57490e7	0.297574
$232$	2.39422e8	1.25880
$233$	1.08444e8	0.561642	0.280821	−	0.959760i	$-0.409393\pi$
$233$	1.08444e8	0.561642	0.280821	+	0.959760i	$0.409393\pi$
$234$	1.22903e8	0.627056
$235$	−1.19203e8	−0.599169
$236$	−780758.	−0.00386656
$237$	−1.44589e8	−0.705532
$238$	−3.47816e8	−1.67236
$239$	4.09522e8	1.94037	0.970184	−	0.242369i	$-0.0779244\pi$
$239$	4.09522e8	1.94037	0.970184	+	0.242369i	$0.0779244\pi$
$240$	1.75174e8	0.817955
$241$	−1.71366e8	−0.788614	−0.394307	−	0.918979i	$-0.629015\pi$
$241$	−1.71366e8	−0.788614	−0.394307	+	0.918979i	$0.629015\pi$
$242$	1.86067e8	0.843946
$243$	−1.43489e7	−0.0641500
$244$	−2.33455e6	−0.0102882
$245$	2.14961e8	0.933851
$246$	2.44570e8	1.04744
$247$	−6.29573e8	−2.65832
$248$	1.30779e8	0.544450
$249$	−2.61633e8	−1.07398
$250$	9.04161e6	0.0365979
$251$	1.27165e8	0.507586	0.253793	−	0.967259i	$-0.418322\pi$
$251$	1.27165e8	0.507586	0.253793	+	0.967259i	$0.418322\pi$
$252$	−919570.	−0.00361978
$253$	−2.14561e7	−0.0832971
$254$	−4.39835e8	−1.68412
$255$	2.77242e8	1.04705
$256$	6.77754e6	0.0252483
$257$	−1.47963e8	−0.543735	−0.271867	−	0.962335i	$-0.587641\pi$
$257$	−1.47963e8	−0.543735	−0.271867	+	0.962335i	$0.587641\pi$
$258$	1.70910e8	0.619582
$259$	4.55134e7	0.162776
$260$	−6.27822e6	−0.0221528
$261$	1.21040e8	0.421391
$262$	2.22611e8	0.764702
$263$	3.15454e8	1.06928	0.534639	−	0.845081i	$-0.320448\pi$
$263$	3.15454e8	1.06928	0.534639	+	0.845081i	$0.320448\pi$
$264$	−6.86588e7	−0.229658
$265$	5.55040e8	1.83216
$266$	5.64374e8	1.83857
$267$	2.06225e7	0.0663059
$268$	308338.	0.000978487 0
$269$	−3.49514e8	−1.09479	−0.547396	−	0.836874i	$-0.684381\pi$
$269$	−3.49514e8	−1.09479	−0.547396	+	0.836874i	$0.684381\pi$
$270$	8.78196e7	0.271530
$271$	5.74060e8	1.75212	0.876062	−	0.482198i	$-0.160161\pi$
$271$	5.74060e8	1.75212	0.876062	+	0.482198i	$0.160161\pi$
$272$	4.31965e8	1.30154
$273$	−4.69114e8	−1.39543
$274$	−4.35316e8	−1.27843
$275$	1.34197e8	0.389117
$276$	353915.	0.00101325
$277$	−2.81324e7	−0.0795294	−0.0397647	−	0.999209i	$-0.512661\pi$
$277$	−2.81324e7	−0.0795294	−0.0397647	+	0.999209i	$0.512661\pi$
$278$	−1.95216e8	−0.544953
$279$	6.61154e7	0.182258
$280$	−6.63048e8	−1.80506
$281$	−5.97185e6	−0.0160560	−0.00802800	−	0.999968i	$-0.502555\pi$
$281$	−5.97185e6	−0.0160560	−0.00802800	+	0.999968i	$0.502555\pi$
$282$	−9.31108e7	−0.247245
$283$	−8.59689e7	−0.225470	−0.112735	−	0.993625i	$-0.535961\pi$
$283$	−8.59689e7	−0.225470	−0.112735	+	0.993625i	$0.535961\pi$
$284$	2.01856e6	0.00522911
$285$	−4.49858e8	−1.15111
$286$	2.97305e8	0.751487
$287$	−9.33514e8	−2.33096
$288$	2.27464e6	0.00561098
$289$	2.73319e8	0.666081
$290$	−7.40798e8	−1.78364
$291$	−4.00012e8	−0.951585
$292$	−1.80181e6	−0.00423516
$293$	1.62287e8	0.376918	0.188459	−	0.982081i	$-0.439651\pi$
$293$	1.62287e8	0.376918	0.188459	+	0.982081i	$0.439651\pi$
$294$	1.67908e8	0.385351
$295$	−2.84603e8	−0.645450
$296$	−5.60528e7	−0.125625
$297$	−3.47104e7	−0.0768797
$298$	1.69651e8	0.371364
$299$	1.80548e8	0.390610
$300$	−2.21356e6	−0.00473333
$301$	−6.52356e8	−1.37880
$302$	−2.19030e8	−0.457594
$303$	−9.92219e7	−0.204908
$304$	−7.00916e8	−1.43090
$305$	−8.50995e8	−1.71742
$306$	2.16557e8	0.432063
$307$	3.30003e8	0.650930	0.325465	−	0.945554i	$-0.394479\pi$
$307$	3.30003e8	0.650930	0.325465	+	0.945554i	$0.394479\pi$
$308$	−2.22446e6	−0.00433808
$309$	2.72821e7	0.0526045
$310$	−4.04646e8	−0.771453
$311$	2.46004e8	0.463746	0.231873	−	0.972746i	$-0.425515\pi$
$311$	2.46004e8	0.463746	0.231873	+	0.972746i	$0.425515\pi$
$312$	5.77746e8	1.07695
$313$	6.06687e8	1.11830	0.559152	−	0.829065i	$-0.311127\pi$
$313$	6.06687e8	1.11830	0.559152	+	0.829065i	$0.311127\pi$
$314$	8.94150e8	1.62988
$315$	−3.35203e8	−0.604256
$316$	5.76931e6	0.0102853
$317$	4.92260e8	0.867936	0.433968	−	0.900928i	$-0.357113\pi$
$317$	4.92260e8	0.867936	0.433968	+	0.900928i	$0.357113\pi$
$318$	4.33549e8	0.756037
$319$	2.92798e8	0.505010
$320$	8.16531e8	1.39299
$321$	−1.97573e8	−0.333395
$322$	−1.61850e8	−0.270158
$323$	−1.10932e9	−1.83167
$324$	572541.	0.000935188 0
$325$	−1.12924e9	−1.82471
$326$	−9.22666e8	−1.47497
$327$	−2.23454e8	−0.353403
$328$	1.14969e9	1.79896
$329$	3.55400e8	0.550214
$330$	2.12438e8	0.325412
$331$	−6.11729e8	−0.927174	−0.463587	−	0.886051i	$-0.653438\pi$
$331$	−6.11729e8	−0.927174	−0.463587	+	0.886051i	$0.653438\pi$
$332$	1.04395e7	0.0156566
$333$	−2.83375e7	−0.0420539
$334$	1.12024e8	0.164512
$335$	1.12396e8	0.163341
$336$	−5.22274e8	−0.751123
$337$	3.71805e8	0.529189	0.264595	−	0.964360i	$-0.414762\pi$
$337$	3.71805e8	0.529189	0.264595	+	0.964360i	$0.414762\pi$
$338$	−1.78885e9	−2.51980
$339$	2.08904e8	0.291238
$340$	−1.10623e7	−0.0152641
$341$	1.59935e8	0.218425
$342$	−3.51390e8	−0.475004
$343$	3.23357e8	0.432665
$344$	8.03421e8	1.06412
$345$	1.29010e8	0.169144
$346$	8.26661e8	1.07290
$347$	−6.18438e8	−0.794590	−0.397295	−	0.917691i	$-0.630051\pi$
$347$	−6.18438e8	−0.794590	−0.397295	+	0.917691i	$0.630051\pi$
$348$	−4.82964e6	−0.00614310
$349$	4.47955e8	0.564086	0.282043	−	0.959402i	$-0.408988\pi$
$349$	4.47955e8	0.564086	0.282043	+	0.959402i	$0.408988\pi$
$350$	1.01229e9	1.26203
$351$	2.92079e8	0.360517
$352$	5.50241e6	0.00672440
$353$	9.25109e7	0.111939	0.0559695	−	0.998432i	$-0.482175\pi$
$353$	9.25109e7	0.111939	0.0559695	+	0.998432i	$0.482175\pi$
$354$	−2.22307e8	−0.266343
$355$	7.35811e8	0.872905
$356$	−822867.	−0.000966617 0
$357$	−8.26587e8	−0.961501
$358$	−5.65041e7	−0.0650863
$359$	−1.30614e8	−0.148990	−0.0744952	−	0.997221i	$-0.523735\pi$
$359$	−1.30614e8	−0.148990	−0.0744952	+	0.997221i	$0.523735\pi$
$360$	4.12826e8	0.466346
$361$	9.06130e8	1.01371
$362$	7.83835e8	0.868450
$363$	4.42188e8	0.485215
$364$	1.87183e7	0.0203428
$365$	−6.56801e8	−0.706983
$366$	−6.64723e8	−0.708691
$367$	1.16358e9	1.22875	0.614377	−	0.789012i	$-0.289407\pi$
$367$	1.16358e9	1.22875	0.614377	+	0.789012i	$0.289407\pi$
$368$	2.01008e8	0.210255
$369$	5.81223e8	0.602213
$370$	1.73434e8	0.178003
$371$	−1.65484e9	−1.68247
$372$	−2.63809e6	−0.00265699
$373$	−1.52608e9	−1.52264	−0.761320	−	0.648377i	$-0.775448\pi$
$373$	−1.52608e9	−1.52264	−0.761320	+	0.648377i	$0.775448\pi$
$374$	5.23856e8	0.517800
$375$	2.14874e7	0.0210414
$376$	−4.37699e8	−0.424637
$377$	−2.46382e9	−2.36818
$378$	−2.61831e8	−0.249345
$379$	−1.27741e8	−0.120530	−0.0602648	−	0.998182i	$-0.519195\pi$
$379$	−1.27741e8	−0.120530	−0.0602648	+	0.998182i	$0.519195\pi$
$380$	1.79500e7	0.0167811
$381$	−1.04527e9	−0.968259
$382$	2.08310e9	1.91200
$383$	−1.33134e9	−1.21086	−0.605429	−	0.795899i	$-0.706999\pi$
$383$	−1.33134e9	−1.21086	−0.605429	+	0.795899i	$0.706999\pi$
$384$	6.48586e8	0.584532
$385$	−8.10865e8	−0.724163
$386$	1.05570e9	0.934296
$387$	4.06169e8	0.356220
$388$	1.59610e7	0.0138723
$389$	2.18772e8	0.188438	0.0942188	−	0.995552i	$-0.469965\pi$
$389$	2.18772e8	0.188438	0.0942188	+	0.995552i	$0.469965\pi$
$390$	−1.78761e9	−1.52597
$391$	3.18129e8	0.269144
$392$	7.89310e8	0.661830
$393$	5.29036e8	0.439654
$394$	9.90465e8	0.815835
$395$	2.10304e9	1.71695
$396$	1.38499e6	0.00112076
$397$	1.72787e9	1.38594	0.692970	−	0.720967i	$-0.256302\pi$
$397$	1.72787e9	1.38594	0.692970	+	0.720967i	$0.256302\pi$
$398$	1.44911e9	1.15215
$399$	1.34124e9	1.05706
$400$	−1.25720e9	−0.982190
$401$	6.96045e7	0.0539054	0.0269527	−	0.999637i	$-0.491420\pi$
$401$	6.96045e7	0.0539054	0.0269527	+	0.999637i	$0.491420\pi$
$402$	8.77939e7	0.0674020
$403$	−1.34581e9	−1.02428
$404$	3.95909e6	0.00298717
$405$	2.08704e8	0.156113
$406$	2.20867e9	1.63791
$407$	−6.85490e7	−0.0503989
$408$	1.01800e9	0.742056
$409$	8.08384e8	0.584233	0.292117	−	0.956383i	$-0.405640\pi$
$409$	8.08384e8	0.584233	0.292117	+	0.956383i	$0.405640\pi$
$410$	−3.55726e9	−2.54901
$411$	−1.03453e9	−0.735017
$412$	−1.08859e6	−0.000766876 0
$413$	8.48536e8	0.592714
$414$	1.00771e8	0.0697967
$415$	3.80543e9	2.61358
$416$	−4.63014e7	−0.0315332
$417$	−4.63932e8	−0.313313
$418$	−8.50021e8	−0.569262
$419$	−8.78688e8	−0.583560	−0.291780	−	0.956485i	$-0.594248\pi$
$419$	−8.78688e8	−0.583560	−0.291780	+	0.956485i	$0.594248\pi$
$420$	1.33751e7	0.00880894
$421$	−2.74747e9	−1.79451	−0.897255	−	0.441512i	$-0.854442\pi$
$421$	−2.74747e9	−1.79451	−0.897255	+	0.441512i	$0.854442\pi$
$422$	−1.49856e9	−0.970690
$423$	−2.21278e8	−0.142150
$424$	2.03804e9	1.29847
$425$	−1.98974e9	−1.25729
$426$	5.74751e8	0.360202
$427$	2.53721e9	1.57710
$428$	7.88343e6	0.00486029
$429$	7.06547e8	0.432057
$430$	−2.48588e9	−1.50779
$431$	1.38074e9	0.830692	0.415346	−	0.909664i	$-0.363661\pi$
$431$	1.38074e9	0.830692	0.415346	+	0.909664i	$0.363661\pi$
$432$	3.25178e8	0.194056
$433$	1.29261e9	0.765173	0.382587	−	0.923920i	$-0.375033\pi$
$433$	1.29261e9	0.765173	0.382587	+	0.923920i	$0.375033\pi$
$434$	1.20644e9	0.708421
$435$	−1.76051e9	−1.02548
$436$	8.91611e6	0.00515196
$437$	−5.16202e8	−0.295893
$438$	−5.13035e8	−0.291735
$439$	2.12976e9	1.20145	0.600723	−	0.799457i	$-0.294880\pi$
$439$	2.12976e9	1.20145	0.600723	+	0.799457i	$0.294880\pi$
$440$	9.98637e8	0.558886
$441$	3.99035e8	0.221552
$442$	−4.40812e9	−2.42815
$443$	−2.58813e9	−1.41440	−0.707200	−	0.707013i	$-0.750042\pi$
$443$	−2.58813e9	−1.41440	−0.707200	+	0.707013i	$0.750042\pi$
$444$	1.13070e6	0.000613068 0
$445$	−2.99953e8	−0.161359
$446$	3.24939e8	0.173432
$447$	4.03176e8	0.213510
$448$	−2.43446e9	−1.27917
$449$	−6.85574e8	−0.357431	−0.178715	−	0.983901i	$-0.557194\pi$
$449$	−6.85574e8	−0.357431	−0.178715	+	0.983901i	$0.557194\pi$
$450$	−6.30273e8	−0.326050
$451$	1.40599e9	0.721714
$452$	−8.33557e6	−0.00424571
$453$	−5.20527e8	−0.263087
$454$	−1.91402e9	−0.959956
$455$	6.82323e9	3.39586
$456$	−1.65183e9	−0.815807
$457$	1.12411e9	0.550939	0.275470	−	0.961310i	$-0.411167\pi$
$457$	1.12411e9	0.550939	0.275470	+	0.961310i	$0.411167\pi$
$458$	2.20512e9	1.07251
$459$	5.14648e8	0.248408
$460$	−5.14767e6	−0.00246580
$461$	5.81992e8	0.276671	0.138336	−	0.990385i	$-0.455825\pi$
$461$	5.81992e8	0.276671	0.138336	+	0.990385i	$0.455825\pi$
$462$	−6.33377e8	−0.298824
$463$	−3.38927e9	−1.58698	−0.793492	−	0.608581i	$-0.791739\pi$
$463$	−3.38927e9	−1.58698	−0.793492	+	0.608581i	$0.791739\pi$
$464$	−2.74302e9	−1.27472
$465$	−9.61643e8	−0.443536
$466$	−1.23206e9	−0.564001
$467$	−2.93656e9	−1.33423	−0.667113	−	0.744957i	$-0.732470\pi$
$467$	−2.93656e9	−1.33423	−0.667113	+	0.744957i	$0.732470\pi$
$468$	−1.16544e7	−0.00525567
$469$	−3.35105e8	−0.149995
$470$	1.35429e9	0.601685
$471$	2.12495e9	0.937078
$472$	−1.04503e9	−0.457438
$473$	9.82533e8	0.426907
$474$	1.64271e9	0.708495
$475$	3.22859e9	1.38224
$476$	3.29819e7	0.0140169
$477$	1.03033e9	0.434673
$478$	−4.65267e9	−1.94852
$479$	−1.16240e9	−0.483261	−0.241631	−	0.970368i	$-0.577682\pi$
$479$	−1.16240e9	−0.483261	−0.241631	+	0.970368i	$0.577682\pi$
$480$	−3.30845e7	−0.0136546
$481$	5.76823e8	0.236339
$482$	1.94692e9	0.791926
$483$	−3.84638e8	−0.155324
$484$	−1.76439e7	−0.00707353
$485$	5.81814e9	2.31573
$486$	1.63021e8	0.0644194
$487$	4.38111e9	1.71883	0.859416	−	0.511278i	$-0.170828\pi$
$487$	4.38111e9	1.71883	0.859416	+	0.511278i	$0.170828\pi$
$488$	−3.12475e9	−1.21716
$489$	−2.19272e9	−0.848013
$490$	−2.44221e9	−0.937773
$491$	−3.82021e8	−0.145647	−0.0728235	−	0.997345i	$-0.523201\pi$
$491$	−3.82021e8	−0.145647	−0.0728235	+	0.997345i	$0.523201\pi$
$492$	−2.31916e7	−0.00877916
$493$	−4.34129e9	−1.63175
$494$	7.15271e9	2.66948
$495$	5.04860e8	0.187091
$496$	−1.49832e9	−0.551339
$497$	−2.19380e9	−0.801584
$498$	2.97247e9	1.07849
$499$	−1.92036e9	−0.691881	−0.345940	−	0.938257i	$-0.612440\pi$
$499$	−1.92036e9	−0.691881	−0.345940	+	0.938257i	$0.612440\pi$
$500$	−857378.	−0.000306745 0
$501$	2.66225e8	0.0945839
$502$	−1.44475e9	−0.509717
$503$	−3.49937e9	−1.22603	−0.613016	−	0.790070i	$-0.710044\pi$
$503$	−3.49937e9	−1.22603	−0.613016	+	0.790070i	$0.710044\pi$
$504$	−1.23083e9	−0.428243
$505$	1.44318e9	0.498654
$506$	2.43768e8	0.0836469
$507$	−4.25121e9	−1.44872
$508$	4.17077e7	0.0141154
$509$	5.14542e9	1.72945	0.864727	−	0.502243i	$-0.167492\pi$
$509$	5.14542e9	1.72945	0.864727	+	0.502243i	$0.167492\pi$
$510$	−3.14980e9	−1.05145
$511$	1.95823e9	0.649218
$512$	2.99778e9	0.987084
$513$	−8.35079e8	−0.273097
$514$	1.68104e9	0.546018
$515$	−3.96816e8	−0.128016
$516$	−1.62067e7	−0.00519303
$517$	−5.35278e8	−0.170358
$518$	−5.17087e8	−0.163459
$519$	1.96456e9	0.616851
$520$	−8.40328e9	−2.62082
$521$	1.79927e9	0.557396	0.278698	−	0.960379i	$-0.410097\pi$
$521$	1.79927e9	0.557396	0.278698	+	0.960379i	$0.410097\pi$
$522$	−1.37516e9	−0.423161
$523$	−4.44596e8	−0.135897	−0.0679484	−	0.997689i	$-0.521645\pi$
$523$	−4.44596e8	−0.135897	−0.0679484	+	0.997689i	$0.521645\pi$
$524$	−2.11093e7	−0.00640935
$525$	2.40572e9	0.725584
$526$	−3.58394e9	−1.07377
$527$	−2.37134e9	−0.705760
$528$	7.86613e8	0.232564
$529$	1.48036e8	0.0434783
$530$	−6.30594e9	−1.83986
$531$	−5.28314e8	−0.153130
$532$	−5.35172e7	−0.0154100
$533$	−1.18311e10	−3.38438
$534$	−2.34297e8	−0.0665844
$535$	2.87368e9	0.811335
$536$	4.12705e8	0.115761
$537$	−1.34282e8	−0.0374204
$538$	3.97091e9	1.09939
$539$	9.65276e8	0.265516
$540$	−8.32757e6	−0.00227583
$541$	4.48022e9	1.21649	0.608246	−	0.793749i	$-0.291874\pi$
$541$	4.48022e9	1.21649	0.608246	+	0.793749i	$0.291874\pi$
$542$	−6.52203e9	−1.75948
$543$	1.86279e9	0.499303
$544$	−8.15839e7	−0.0217274
$545$	3.25012e9	0.860025
$546$	5.32971e9	1.40129
$547$	−6.94497e8	−0.181432	−0.0907162	−	0.995877i	$-0.528916\pi$
$547$	−6.94497e8	−0.181432	−0.0907162	+	0.995877i	$0.528916\pi$
$548$	4.12792e7	0.0107152
$549$	−1.57972e9	−0.407452
$550$	−1.52465e9	−0.390751
$551$	7.04427e9	1.79393
$552$	4.73709e8	0.119874
$553$	−6.27014e9	−1.57667
$554$	3.19618e8	0.0798634
$555$	4.12166e8	0.102340
$556$	1.85115e7	0.00456752
$557$	6.59351e9	1.61668	0.808339	−	0.588717i	$-0.200367\pi$
$557$	6.59351e9	1.61668	0.808339	+	0.588717i	$0.200367\pi$
$558$	−7.51151e8	−0.183024
$559$	−8.26777e9	−2.00192
$560$	7.59644e9	1.82790
$561$	1.24495e9	0.297701
$562$	6.78476e7	0.0161234
$563$	−4.28123e9	−1.01109	−0.505544	−	0.862801i	$-0.668708\pi$
$563$	−4.28123e9	−1.01109	−0.505544	+	0.862801i	$0.668708\pi$
$564$	8.82931e6	0.00207229
$565$	−3.03850e9	−0.708744
$566$	9.76711e8	0.226417
$567$	−6.22244e8	−0.143357
$568$	2.70181e9	0.618637
$569$	−4.87363e9	−1.10907	−0.554536	−	0.832160i	$-0.687104\pi$
$569$	−4.87363e9	−1.10907	−0.554536	+	0.832160i	$0.687104\pi$
$570$	5.11094e9	1.15595
$571$	−9.55591e8	−0.214806	−0.107403	−	0.994216i	$-0.534253\pi$
$571$	−9.55591e8	−0.214806	−0.107403	+	0.994216i	$0.534253\pi$
$572$	−2.81922e7	−0.00629858
$573$	4.95050e9	1.09928
$574$	1.06059e10	2.34074
$575$	−9.25891e8	−0.203106
$576$	1.51574e9	0.330481
$577$	4.90266e9	1.06247	0.531235	−	0.847225i	$-0.321728\pi$
$577$	4.90266e9	1.06247	0.531235	+	0.847225i	$0.321728\pi$
$578$	−3.10523e9	−0.668878
$579$	2.50887e9	0.537160
$580$	7.02468e7	0.0149496
$581$	−1.13458e10	−2.40003
$582$	4.54462e9	0.955581
$583$	2.49240e9	0.520928
$584$	−2.41170e9	−0.501046
$585$	−4.24827e9	−0.877337
$586$	−1.84378e9	−0.378501
$587$	7.28679e9	1.48697	0.743486	−	0.668751i	$-0.233171\pi$
$587$	7.28679e9	1.48697	0.743486	+	0.668751i	$0.233171\pi$
$588$	−1.59220e7	−0.00322982
$589$	3.84779e9	0.775904
$590$	3.23344e9	0.648161
$591$	2.35384e9	0.469053
$592$	6.42189e8	0.127215
$593$	2.72582e9	0.536791	0.268395	−	0.963309i	$-0.413507\pi$
$593$	2.72582e9	0.536791	0.268395	+	0.963309i	$0.413507\pi$
$594$	3.94352e8	0.0772026
$595$	1.20226e10	2.33986
$596$	−1.60873e7	−0.00311258
$597$	3.44382e9	0.662415
$598$	−2.05125e9	−0.392251
$599$	2.88173e9	0.547847	0.273923	−	0.961751i	$-0.411679\pi$
$599$	2.88173e9	0.547847	0.273923	+	0.961751i	$0.411679\pi$
$600$	−2.96281e9	−0.559983
$601$	−9.69030e9	−1.82086	−0.910430	−	0.413664i	$-0.864249\pi$
$601$	−9.69030e9	−1.82086	−0.910430	+	0.413664i	$0.864249\pi$
$602$	7.41156e9	1.38459
$603$	2.08643e8	0.0387518
$604$	2.07697e7	0.00383532
$605$	−6.43160e9	−1.18080
$606$	1.12728e9	0.205768
$607$	3.94372e9	0.715724	0.357862	−	0.933775i	$-0.383506\pi$
$607$	3.94372e9	0.715724	0.357862	+	0.933775i	$0.383506\pi$
$608$	1.32380e8	0.0238869
$609$	5.24891e9	0.941691
$610$	9.66834e9	1.72464
$611$	4.50423e9	0.798871
$612$	−2.05352e7	−0.00362133
$613$	−4.73921e9	−0.830988	−0.415494	−	0.909596i	$-0.636391\pi$
$613$	−4.73921e9	−0.830988	−0.415494	+	0.909596i	$0.636391\pi$
$614$	−3.74924e9	−0.653663
$615$	−8.45384e9	−1.46552
$616$	−2.97740e9	−0.513222
$617$	6.42115e9	1.10056	0.550282	−	0.834979i	$-0.314520\pi$
$617$	6.42115e9	1.10056	0.550282	+	0.834979i	$0.314520\pi$
$618$	−3.09958e8	−0.0528254
$619$	1.41855e9	0.240396	0.120198	−	0.992750i	$-0.461647\pi$
$619$	1.41855e9	0.240396	0.120198	+	0.992750i	$0.461647\pi$
$620$	3.83709e7	0.00646593
$621$	2.39483e8	0.0401286
$622$	−2.79490e9	−0.465694
$623$	8.94300e8	0.148175
$624$	−6.61916e9	−1.09058
$625$	−6.25773e9	−1.02527
$626$	−6.89271e9	−1.12300
$627$	−2.02008e9	−0.327289
$628$	−8.47885e7	−0.0136609
$629$	1.01637e9	0.162845
$630$	3.80832e9	0.606794
$631$	4.52238e9	0.716579	0.358289	−	0.933611i	$-0.383360\pi$
$631$	4.52238e9	0.716579	0.358289	+	0.933611i	$0.383360\pi$
$632$	7.72211e9	1.21682
$633$	−3.56133e9	−0.558084
$634$	−5.59268e9	−0.871581
$635$	1.52034e10	2.35631
$636$	−4.11116e7	−0.00633672
$637$	−8.12256e9	−1.24510
$638$	−3.32654e9	−0.507131
$639$	1.36590e9	0.207093
$640$	−9.43363e9	−1.42249
$641$	−8.31176e9	−1.24649	−0.623247	−	0.782025i	$-0.714187\pi$
$641$	−8.31176e9	−1.24649	−0.623247	+	0.782025i	$0.714187\pi$
$642$	2.24467e9	0.334795
$643$	6.83021e9	1.01320	0.506601	−	0.862181i	$-0.330902\pi$
$643$	6.83021e9	1.01320	0.506601	+	0.862181i	$0.330902\pi$
$644$	1.53476e7	0.00226433
$645$	−5.90770e9	−0.866881
$646$	1.26032e10	1.83936
$647$	−4.33983e9	−0.629952	−0.314976	−	0.949100i	$-0.601996\pi$
$647$	−4.33983e9	−0.629952	−0.314976	+	0.949100i	$0.601996\pi$
$648$	7.66336e8	0.110639
$649$	−1.27801e9	−0.183517
$650$	1.28295e10	1.83237
$651$	2.86711e9	0.407297
$652$	8.74926e7	0.0123625
$653$	5.82673e9	0.818896	0.409448	−	0.912333i	$-0.365721\pi$
$653$	5.82673e9	0.818896	0.409448	+	0.912333i	$0.365721\pi$
$654$	2.53871e9	0.354887
$655$	−7.69479e9	−1.06992
$656$	−1.31718e10	−1.82172
$657$	−1.21923e9	−0.167729
$658$	−4.03777e9	−0.552524
$659$	1.11625e9	0.151937	0.0759684	−	0.997110i	$-0.475795\pi$
$659$	1.11625e9	0.151937	0.0759684	+	0.997110i	$0.475795\pi$
$660$	−2.01446e7	−0.00272744
$661$	1.16388e10	1.56748	0.783739	−	0.621090i	$-0.213310\pi$
$661$	1.16388e10	1.56748	0.783739	+	0.621090i	$0.213310\pi$
$662$	6.94999e9	0.931068
$663$	−1.04759e10	−1.39603
$664$	1.39731e10	1.85227
$665$	−1.95082e10	−2.57242
$666$	3.21948e8	0.0422305
$667$	−2.02015e9	−0.263598
$668$	−1.06228e7	−0.00137886
$669$	7.72219e8	0.0997124
$670$	−1.27696e9	−0.164026
$671$	−3.82138e9	−0.488305
$672$	9.86403e7	0.0125390
$673$	1.09171e10	1.38056	0.690281	−	0.723542i	$-0.257487\pi$
$673$	1.09171e10	1.38056	0.690281	+	0.723542i	$0.257487\pi$
$674$	−4.22416e9	−0.531412
$675$	−1.49785e9	−0.187458
$676$	1.69629e8	0.0211197
$677$	4.80668e8	0.0595367	0.0297683	−	0.999557i	$-0.490523\pi$
$677$	4.80668e8	0.0595367	0.0297683	+	0.999557i	$0.490523\pi$
$678$	−2.37341e9	−0.292461
$679$	−1.73466e10	−2.12652
$680$	−1.48067e10	−1.80583
$681$	−4.54868e9	−0.551913
$682$	−1.81705e9	−0.219342
$683$	−2.68059e9	−0.321927	−0.160963	−	0.986960i	$-0.551460\pi$
$683$	−2.68059e9	−0.321927	−0.160963	+	0.986960i	$0.551460\pi$
$684$	3.33208e7	0.00398125
$685$	1.50472e10	1.78870
$686$	−3.67372e9	−0.434482
$687$	5.24048e9	0.616627
$688$	−9.20468e9	−1.07758
$689$	−2.09729e10	−2.44282
$690$	−1.46571e9	−0.169854
$691$	1.44729e9	0.166871	0.0834356	−	0.996513i	$-0.473411\pi$
$691$	1.44729e9	0.166871	0.0834356	+	0.996513i	$0.473411\pi$
$692$	−7.83888e7	−0.00899255
$693$	−1.50522e9	−0.171805
$694$	7.02621e9	0.797927
$695$	6.74786e9	0.762464
$696$	−6.46439e9	−0.726767
$697$	−2.08466e10	−2.33195
$698$	−5.08932e9	−0.566455
$699$	−2.92799e9	−0.324264
$700$	−9.59916e7	−0.0105777
$701$	1.36631e10	1.49809	0.749043	−	0.662522i	$-0.230514\pi$
$701$	1.36631e10	1.49809	0.749043	+	0.662522i	$0.230514\pi$
$702$	−3.31838e9	−0.362031
$703$	−1.64919e9	−0.179030
$704$	3.66662e9	0.396060
$705$	3.21848e9	0.345930
$706$	−1.05104e9	−0.112409
$707$	−4.30278e9	−0.457911
$708$	2.10805e7	0.00223236
$709$	3.34871e8	0.0352870	0.0176435	−	0.999844i	$-0.494384\pi$
$709$	3.34871e8	0.0352870	0.0176435	+	0.999844i	$0.494384\pi$
$710$	−8.35971e9	−0.876570
$711$	3.90391e9	0.407339
$712$	−1.10139e9	−0.114357
$713$	−1.10346e9	−0.114011
$714$	9.39103e9	0.965539
$715$	−1.02767e10	−1.05143
$716$	5.35805e6	0.000545521 0
$717$	−1.10571e10	−1.12027
$718$	1.48393e9	0.149616
$719$	6.19348e9	0.621418	0.310709	−	0.950505i	$-0.399434\pi$
$719$	6.19348e9	0.621418	0.310709	+	0.950505i	$0.399434\pi$
$720$	−4.72969e9	−0.472246
$721$	1.18309e9	0.117556
$722$	−1.02947e10	−1.01797
$723$	4.62688e9	0.455306
$724$	−7.43278e7	−0.00727891
$725$	1.26350e10	1.23138
$726$	−5.02380e9	−0.487253
$727$	−3.64052e9	−0.351393	−0.175697	−	0.984444i	$-0.556218\pi$
$727$	−3.64052e9	−0.351393	−0.175697	+	0.984444i	$0.556218\pi$
$728$	2.50541e10	2.40668
$729$	3.87420e8	0.0370370
$730$	7.46206e9	0.709952
$731$	−1.45680e10	−1.37939
$732$	6.30329e7	0.00593989
$733$	2.07157e9	0.194283	0.0971415	−	0.995271i	$-0.469030\pi$
$733$	2.07157e9	0.194283	0.0971415	+	0.995271i	$0.469030\pi$
$734$	−1.32197e10	−1.23391
$735$	−5.80394e9	−0.539159
$736$	−3.79637e7	−0.00350991
$737$	5.04712e8	0.0464416
$738$	−6.60340e9	−0.604742
$739$	−5.37794e9	−0.490185	−0.245093	−	0.969500i	$-0.578818\pi$
$739$	−5.37794e9	−0.490185	−0.245093	+	0.969500i	$0.578818\pi$
$740$	−1.64460e7	−0.00149193
$741$	1.69985e10	1.53478
$742$	1.88009e10	1.68953
$743$	6.20890e9	0.555333	0.277667	−	0.960678i	$-0.410439\pi$
$743$	6.20890e9	0.555333	0.277667	+	0.960678i	$0.410439\pi$
$744$	−3.53104e9	−0.314339
$745$	−5.86417e9	−0.519589
$746$	1.73382e10	1.52903
$747$	7.06409e9	0.620060
$748$	−4.96751e7	−0.00433994
$749$	−8.56779e9	−0.745045
$750$	−2.44124e8	−0.0211298
$751$	−7.13921e9	−0.615050	−0.307525	−	0.951540i	$-0.599501\pi$
$751$	−7.13921e9	−0.615050	−0.307525	+	0.951540i	$0.599501\pi$
$752$	5.01465e9	0.430010
$753$	−3.43345e9	−0.293055
$754$	2.79920e10	2.37812
$755$	7.57103e9	0.640236
$756$	2.48284e7	0.00208988
$757$	−9.00503e9	−0.754483	−0.377242	−	0.926115i	$-0.623127\pi$
$757$	−9.00503e9	−0.754483	−0.377242	+	0.926115i	$0.623127\pi$
$758$	1.45130e9	0.121036
$759$	5.79316e8	0.0480916
$760$	2.40257e10	1.98531
$761$	−4.51473e9	−0.371352	−0.185676	−	0.982611i	$-0.559447\pi$
$761$	−4.51473e9	−0.371352	−0.185676	+	0.982611i	$0.559447\pi$
$762$	1.18756e10	0.972325
$763$	−9.69012e9	−0.789756
$764$	−1.97532e8	−0.0160254
$765$	−7.48552e9	−0.604515
$766$	1.51257e10	1.21594
$767$	1.07541e10	0.860578
$768$	−1.82994e8	−0.0145771
$769$	1.15299e10	0.914285	0.457142	−	0.889394i	$-0.348873\pi$
$769$	1.15299e10	0.914285	0.457142	+	0.889394i	$0.348873\pi$
$770$	9.21242e9	0.727204
$771$	3.99500e9	0.313925
$772$	−1.00108e8	−0.00783080
$773$	−1.73559e10	−1.35151	−0.675754	−	0.737127i	$-0.736182\pi$
$773$	−1.73559e10	−1.35151	−0.675754	+	0.737127i	$0.736182\pi$
$774$	−4.61458e9	−0.357716
$775$	6.90162e9	0.532593
$776$	2.13635e10	1.64118
$777$	−1.22886e9	−0.0939786
$778$	−2.48551e9	−0.189229
$779$	3.38261e10	2.56372
$780$	1.69512e8	0.0127900
$781$	3.30414e9	0.248188
$782$	−3.61433e9	−0.270274
$783$	−3.26807e9	−0.243290
$784$	−9.04301e9	−0.670204
$785$	−3.09073e10	−2.28043
$786$	−6.01050e9	−0.441501
$787$	−1.78942e9	−0.130858	−0.0654292	−	0.997857i	$-0.520842\pi$
$787$	−1.78942e9	−0.130858	−0.0654292	+	0.997857i	$0.520842\pi$
$788$	−9.39216e7	−0.00683792
$789$	−8.51725e9	−0.617348
$790$	−2.38931e10	−1.72416
$791$	9.05919e9	0.650836
$792$	1.85379e9	0.132593
$793$	3.21559e10	2.28984
$794$	−1.96307e10	−1.39176
$795$	−1.49861e10	−1.05780
$796$	−1.37413e8	−0.00965678
$797$	−1.34591e9	−0.0941700	−0.0470850	−	0.998891i	$-0.514993\pi$
$797$	−1.34591e9	−0.0941700	−0.0470850	+	0.998891i	$0.514993\pi$
$798$	−1.52381e10	−1.06150
$799$	7.93653e9	0.550449
$800$	2.37444e8	0.0163963
$801$	−5.56808e8	−0.0382817
$802$	−7.90792e8	−0.0541317
$803$	−2.94935e9	−0.201012
$804$	−8.32513e6	−0.000564930 0
$805$	5.59454e9	0.377989
$806$	1.52901e10	1.02858
$807$	9.43688e9	0.632079
$808$	5.29917e9	0.353401
$809$	8.51154e9	0.565182	0.282591	−	0.959240i	$-0.408806\pi$
$809$	8.51154e9	0.565182	0.282591	+	0.959240i	$0.408806\pi$
$810$	−2.37113e9	−0.156768
$811$	9.09661e9	0.598834	0.299417	−	0.954122i	$-0.403208\pi$
$811$	9.09661e9	0.598834	0.299417	+	0.954122i	$0.403208\pi$
$812$	−2.09439e8	−0.0137281
$813$	−1.54996e10	−1.01159
$814$	7.78801e8	0.0506105
$815$	3.18930e10	2.06368
$816$	−1.16631e10	−0.751444
$817$	2.36383e10	1.51649
$818$	−9.18423e9	−0.586687
$819$	1.26661e10	0.805654
$820$	3.37320e8	0.0213645
$821$	−2.41553e10	−1.52339	−0.761695	−	0.647935i	$-0.775633\pi$
$821$	−2.41553e10	−1.52339	−0.761695	+	0.647935i	$0.775633\pi$
$822$	1.17535e10	0.738104
$823$	−2.48269e10	−1.55247	−0.776233	−	0.630446i	$-0.782872\pi$
$823$	−2.48269e10	−1.55247	−0.776233	+	0.630446i	$0.782872\pi$
$824$	−1.45706e9	−0.0907262
$825$	−3.62333e9	−0.224657
$826$	−9.64040e9	−0.595203
$827$	−1.17861e9	−0.0724602	−0.0362301	−	0.999343i	$-0.511535\pi$
$827$	−1.17861e9	−0.0724602	−0.0362301	+	0.999343i	$0.511535\pi$
$828$	−9.55570e6	−0.000585001 0
$829$	−2.31094e10	−1.40879	−0.704396	−	0.709807i	$-0.748782\pi$
$829$	−2.31094e10	−1.40879	−0.704396	+	0.709807i	$0.748782\pi$
$830$	−4.32343e10	−2.62455
$831$	7.59575e8	0.0459163
$832$	−3.08537e10	−1.85727
$833$	−1.43121e10	−0.857917
$834$	5.27084e9	0.314629
$835$	−3.87223e9	−0.230175
$836$	8.06039e7	0.00477127
$837$	−1.78512e9	−0.105227
$838$	9.98297e9	0.586011
$839$	−6.66669e9	−0.389712	−0.194856	−	0.980832i	$-0.562424\pi$
$839$	−6.66669e9	−0.389712	−0.194856	+	0.980832i	$0.562424\pi$
$840$	1.79023e10	1.04215
$841$	1.03178e10	0.598135
$842$	3.12146e10	1.80205
$843$	1.61240e8	0.00926993
$844$	1.42102e8	0.00813583
$845$	6.18335e10	3.52554
$846$	2.51399e9	0.142747
$847$	1.91756e10	1.08432
$848$	−2.33496e10	−1.31490
$849$	2.32116e9	0.130175
$850$	2.26058e10	1.26257
$851$	4.72952e8	0.0263065
$852$	−5.45012e7	−0.00301903
$853$	−2.12290e10	−1.17114	−0.585570	−	0.810622i	$-0.699129\pi$
$853$	−2.12290e10	−1.17114	−0.585570	+	0.810622i	$0.699129\pi$
$854$	−2.88258e10	−1.58372
$855$	1.21462e10	0.664596
$856$	1.05518e10	0.575002
$857$	−3.89583e9	−0.211430	−0.105715	−	0.994396i	$-0.533713\pi$
$857$	−3.89583e9	−0.211430	−0.105715	+	0.994396i	$0.533713\pi$
$858$	−8.02724e9	−0.433871
$859$	3.53189e9	0.190122	0.0950608	−	0.995471i	$-0.469695\pi$
$859$	3.53189e9	0.190122	0.0950608	+	0.995471i	$0.469695\pi$
$860$	2.35725e8	0.0126375
$861$	2.52049e10	1.34578
$862$	−1.56868e10	−0.834181
$863$	1.47909e10	0.783349	0.391675	−	0.920104i	$-0.371896\pi$
$863$	1.47909e10	0.783349	0.391675	+	0.920104i	$0.371896\pi$
$864$	−6.14153e7	−0.00323950
$865$	−2.85744e10	−1.50114
$866$	−1.46856e10	−0.768387
$867$	−7.37960e9	−0.384562
$868$	−1.14402e8	−0.00593763
$869$	9.44365e9	0.488170
$870$	2.00016e10	1.02978
$871$	−4.24702e9	−0.217782
$872$	1.19340e10	0.609509
$873$	1.08003e10	0.549398
$874$	5.86469e9	0.297136
$875$	9.31808e8	0.0470217
$876$	4.86490e7	0.00244517
$877$	2.99364e10	1.49865	0.749326	−	0.662201i	$-0.230378\pi$
$877$	2.99364e10	1.49865	0.749326	+	0.662201i	$0.230378\pi$
$878$	−2.41966e10	−1.20649
$879$	−4.38175e9	−0.217614
$880$	−1.14412e10	−0.565957
$881$	−8.65058e9	−0.426216	−0.213108	−	0.977029i	$-0.568359\pi$
$881$	−8.65058e9	−0.426216	−0.213108	+	0.977029i	$0.568359\pi$
$882$	−4.53353e9	−0.222483
$883$	6.35296e9	0.310537	0.155269	−	0.987872i	$-0.450376\pi$
$883$	6.35296e9	0.310537	0.155269	+	0.987872i	$0.450376\pi$
$884$	4.18004e8	0.0203515
$885$	7.68429e9	0.372651
$886$	2.94043e10	1.42034
$887$	−2.63325e10	−1.26695	−0.633475	−	0.773763i	$-0.718372\pi$
$887$	−2.63325e10	−1.26695	−0.633475	+	0.773763i	$0.718372\pi$
$888$	1.51343e9	0.0725297
$889$	−4.53284e10	−2.16379
$890$	3.40783e9	0.162037
$891$	9.37180e8	0.0443865
$892$	−3.08126e7	−0.00145362
$893$	−1.28780e10	−0.605157
$894$	−4.58058e9	−0.214407
$895$	1.95313e9	0.0910647
$896$	2.81261e10	1.30626
$897$	−4.87480e9	−0.225519
$898$	7.78895e9	0.358932
$899$	1.50583e10	0.691219
$900$	5.97661e7	0.00273279
$901$	−3.69546e10	−1.68318
$902$	−1.59738e10	−0.724745
$903$	1.76136e10	0.796052
$904$	−1.11570e10	−0.502294
$905$	−2.70941e10	−1.21508
$906$	5.91382e9	0.264192
$907$	1.26762e9	0.0564110	0.0282055	−	0.999602i	$-0.491021\pi$
$907$	1.26762e9	0.0564110	0.0282055	+	0.999602i	$0.491021\pi$
$908$	1.81499e8	0.00804587
$909$	2.67899e9	0.118304
$910$	−7.75202e10	−3.41012
$911$	−2.38159e10	−1.04364	−0.521821	−	0.853055i	$-0.674747\pi$
$911$	−2.38159e10	−1.04364	−0.521821	+	0.853055i	$0.674747\pi$
$912$	1.89247e10	0.826129
$913$	1.70882e10	0.743103
$914$	−1.27713e10	−0.553253
$915$	2.29769e10	0.991555
$916$	−2.09102e8	−0.00898928
$917$	2.29418e10	0.982504
$918$	−5.84703e9	−0.249452
$919$	1.46627e10	0.623176	0.311588	−	0.950217i	$-0.399139\pi$
$919$	1.46627e10	0.623176	0.311588	+	0.950217i	$0.399139\pi$
$920$	−6.89006e9	−0.291720
$921$	−8.91009e9	−0.375814
$922$	−6.61214e9	−0.277833
$923$	−2.78035e10	−1.16384
$924$	6.00605e7	0.00250459
$925$	−2.95808e9	−0.122889
$926$	3.85062e10	1.59365
$927$	−7.36616e8	−0.0303712
$928$	5.18066e8	0.0212798
$929$	3.68901e10	1.50958	0.754788	−	0.655968i	$-0.227740\pi$
$929$	3.68901e10	1.50958	0.754788	+	0.655968i	$0.227740\pi$
$930$	1.09254e10	0.445398
$931$	2.32231e10	0.943183
$932$	1.16831e8	0.00472717
$933$	−6.64210e9	−0.267744
$934$	3.33629e10	1.33983
$935$	−1.81077e10	−0.724473
$936$	−1.55992e10	−0.621778
$937$	2.46605e10	0.979294	0.489647	−	0.871921i	$-0.337126\pi$
$937$	2.46605e10	0.979294	0.489647	+	0.871921i	$0.337126\pi$
$938$	3.80720e9	0.150625
$939$	−1.63806e10	−0.645653
$940$	−1.28422e8	−0.00504302
$941$	2.85433e10	1.11671	0.558355	−	0.829602i	$-0.311433\pi$
$941$	2.85433e10	1.11671	0.558355	+	0.829602i	$0.311433\pi$
$942$	−2.41420e10	−0.941014
$943$	−9.70060e9	−0.376711
$944$	1.19728e10	0.463225
$945$	9.05049e9	0.348868
$946$	−1.11628e10	−0.428700
$947$	4.45286e10	1.70378	0.851891	−	0.523719i	$-0.175456\pi$
$947$	4.45286e10	1.70378	0.851891	+	0.523719i	$0.175456\pi$
$948$	−1.55771e8	−0.00593825
$949$	2.48181e10	0.942619
$950$	−3.66807e10	−1.38805
$951$	−1.32910e10	−0.501103
$952$	4.41457e10	1.65829
$953$	1.29354e10	0.484122	0.242061	−	0.970261i	$-0.422177\pi$
$953$	1.29354e10	0.484122	0.242061	+	0.970261i	$0.422177\pi$
$954$	−1.17058e10	−0.436498
$955$	−7.20047e10	−2.67515
$956$	4.41193e8	0.0163315
$957$	−7.90554e9	−0.291568
$958$	1.32063e10	0.485291
$959$	−4.48627e10	−1.64256
$960$	−2.20463e10	−0.804243
$961$	−1.92873e10	−0.701037
$962$	−6.55342e9	−0.237331
$963$	5.33447e9	0.192486
$964$	−1.84619e8	−0.00663752
$965$	−3.64914e10	−1.30721
$966$	4.36996e9	0.155976
$967$	−1.49195e10	−0.530595	−0.265297	−	0.964167i	$-0.585470\pi$
$967$	−1.49195e10	−0.530595	−0.265297	+	0.964167i	$0.585470\pi$
$968$	−2.36161e10	−0.836843
$969$	2.99516e10	1.05751
$970$	−6.61012e10	−2.32546
$971$	1.64648e10	0.577150	0.288575	−	0.957457i	$-0.406819\pi$
$971$	1.64648e10	0.577150	0.288575	+	0.957457i	$0.406819\pi$
$972$	−1.54586e7	−0.000539931 0
$973$	−2.01185e10	−0.700167
$974$	−4.97748e10	−1.72605
$975$	3.04894e10	1.05350
$976$	3.57998e10	1.23256
$977$	1.10912e10	0.380494	0.190247	−	0.981736i	$-0.439071\pi$
$977$	1.10912e10	0.380494	0.190247	+	0.981736i	$0.439071\pi$
$978$	2.49120e10	0.851574
$979$	−1.34693e9	−0.0458782
$980$	2.31585e8	0.00785994
$981$	6.03325e9	0.204037
$982$	4.34022e9	0.146259
$983$	−1.69926e10	−0.570588	−0.285294	−	0.958440i	$-0.592091\pi$
$983$	−1.69926e10	−0.570588	−0.285294	+	0.958440i	$0.592091\pi$
$984$	−3.10415e10	−1.03863
$985$	−3.42365e10	−1.14146
$986$	4.93224e10	1.63861
$987$	−9.59579e9	−0.317666
$988$	−6.78262e8	−0.0223742
$989$	−6.77895e9	−0.222831
$990$	−5.73582e9	−0.187877
$991$	−2.43624e10	−0.795176	−0.397588	−	0.917564i	$-0.630153\pi$
$991$	−2.43624e10	−0.795176	−0.397588	+	0.917564i	$0.630153\pi$
$992$	2.82983e8	0.00920384
$993$	1.65167e10	0.535304
$994$	2.49242e10	0.804950
$995$	−5.00901e10	−1.61202
$996$	−2.81867e8	−0.00903933
$997$	2.37159e10	0.757890	0.378945	−	0.925419i	$-0.376287\pi$
$997$	2.37159e10	0.757890	0.378945	+	0.925419i	$0.376287\pi$
$998$	2.18177e10	0.694786
$999$	7.65111e8	0.0242798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.c.1.3	✓	7
3.2	odd	2		207.8.a.d.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.c.1.3	✓	7	1.1	even	1	trivial
207.8.a.d.1.5		7	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q-11.3612 q^{2} -27.0000 q^{3} +1.07734 q^{4} +392.713 q^{5} +306.753 q^{6} -1170.86 q^{7} +1442.00 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-11.3612 q^{2} -27.0000 q^{3} +1.07734 q^{4} +392.713 q^{5} +306.753 q^{6} -1170.86 q^{7} +1442.00 q^{8} +729.000 q^{9} -4461.70 q^{10} +1763.47 q^{11} -29.0881 q^{12} -14839.2 q^{13} +13302.4 q^{14} -10603.3 q^{15} -16520.7 q^{16} -26146.8 q^{17} -8282.33 q^{18} +42426.4 q^{19} +423.084 q^{20} +31613.3 q^{21} -20035.2 q^{22} -12167.0 q^{23} -38933.9 q^{24} +76098.5 q^{25} +168591. q^{26} -19683.0 q^{27} -1261.41 q^{28} +166035. q^{29} +120466. q^{30} +90693.2 q^{31} +3120.22 q^{32} -47613.7 q^{33} +297060. q^{34} -459812. q^{35} +785.379 q^{36} -38871.7 q^{37} -482016. q^{38} +400657. q^{39} +566291. q^{40} +797288. q^{41} -359165. q^{42} +557159. q^{43} +1899.85 q^{44} +286288. q^{45} +138232. q^{46} -303537. q^{47} +446060. q^{48} +547373. q^{49} -864572. q^{50} +705965. q^{51} -15986.8 q^{52} +1.41335e6 q^{53} +223623. q^{54} +692538. q^{55} -1.68838e6 q^{56} -1.14551e6 q^{57} -1.88636e6 q^{58} -724711. q^{59} -11423.3 q^{60} -2.16696e6 q^{61} -1.03039e6 q^{62} -853558. q^{63} +2.07921e6 q^{64} -5.82753e6 q^{65} +540950. q^{66} +286204. q^{67} -28169.0 q^{68} +328509. q^{69} +5.22403e6 q^{70} +1.87366e6 q^{71} +1.05122e6 q^{72} -1.67247e6 q^{73} +441630. q^{74} -2.05466e6 q^{75} +45707.6 q^{76} -2.06478e6 q^{77} -4.55196e6 q^{78} +5.35515e6 q^{79} -6.48791e6 q^{80} +531441. q^{81} -9.05817e6 q^{82} +9.69011e6 q^{83} +34058.1 q^{84} -1.02682e7 q^{85} -6.33001e6 q^{86} -4.48295e6 q^{87} +2.54292e6 q^{88} -763797. q^{89} -3.25258e6 q^{90} +1.73746e7 q^{91} -13108.0 q^{92} -2.44872e6 q^{93} +3.44855e6 q^{94} +1.66614e7 q^{95} -84245.9 q^{96} +1.48153e7 q^{97} -6.21883e6 q^{98} +1.28557e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9}+O(q^{10}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9} - 15310 q^{10} + 9040 q^{11} - 17658 q^{12} + 3774 q^{13} + 4536 q^{14} + 13932 q^{15} + 52002 q^{16} - 40760 q^{17} + 81598 q^{19} - 88946 q^{20} - 27486 q^{21} + 245034 q^{22} - 85169 q^{23} - 38394 q^{24} + 321325 q^{25} + 412748 q^{26} - 137781 q^{27} + 965948 q^{28} + 154126 q^{29} + 413370 q^{30} + 243132 q^{31} + 1278286 q^{32} - 244080 q^{33} + 984836 q^{34} - 130296 q^{35} + 476766 q^{36} + 582114 q^{37} + 772558 q^{38} - 101898 q^{39} - 132618 q^{40} + 113062 q^{41} - 122472 q^{42} - 659778 q^{43} + 659390 q^{44} - 376164 q^{45} - 591032 q^{47} - 1404054 q^{48} + 3263235 q^{49} - 702684 q^{50} + 1100520 q^{51} + 1793280 q^{52} + 207128 q^{53} + 184664 q^{55} + 5390508 q^{56} - 2203146 q^{57} - 1142916 q^{58} + 447148 q^{59} + 2401542 q^{60} + 2248970 q^{61} - 5729060 q^{62} + 742122 q^{63} + 7212922 q^{64} - 827096 q^{65} - 6615918 q^{66} + 4467570 q^{67} - 5477620 q^{68} + 2299563 q^{69} - 12744284 q^{70} - 5154608 q^{71} + 1036638 q^{72} - 13239250 q^{73} - 2827426 q^{74} - 8675775 q^{75} - 527434 q^{76} - 18415912 q^{77} - 11144196 q^{78} + 9594446 q^{79} - 55932394 q^{80} + 3720087 q^{81} - 20889952 q^{82} - 573720 q^{83} - 26080596 q^{84} + 7477272 q^{85} - 28416910 q^{86} - 4161402 q^{87} + 26555702 q^{88} - 3810540 q^{89} - 11160990 q^{90} + 36092068 q^{91} - 7957218 q^{92} - 6564564 q^{93} + 33545768 q^{94} + 10497320 q^{95} - 34513722 q^{96} + 49497978 q^{97} - 1023376 q^{98} + 6590160 q^{99}+O(q^{100}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9 - 15310 * q^10 + 9040 * q^11 - 17658 * q^12 + 3774 * q^13 + 4536 * q^14 + 13932 * q^15 + 52002 * q^16 - 40760 * q^17 + 81598 * q^19 - 88946 * q^20 - 27486 * q^21 + 245034 * q^22 - 85169 * q^23 - 38394 * q^24 + 321325 * q^25 + 412748 * q^26 - 137781 * q^27 + 965948 * q^28 + 154126 * q^29 + 413370 * q^30 + 243132 * q^31 + 1278286 * q^32 - 244080 * q^33 + 984836 * q^34 - 130296 * q^35 + 476766 * q^36 + 582114 * q^37 + 772558 * q^38 - 101898 * q^39 - 132618 * q^40 + 113062 * q^41 - 122472 * q^42 - 659778 * q^43 + 659390 * q^44 - 376164 * q^45 - 591032 * q^47 - 1404054 * q^48 + 3263235 * q^49 - 702684 * q^50 + 1100520 * q^51 + 1793280 * q^52 + 207128 * q^53 + 184664 * q^55 + 5390508 * q^56 - 2203146 * q^57 - 1142916 * q^58 + 447148 * q^59 + 2401542 * q^60 + 2248970 * q^61 - 5729060 * q^62 + 742122 * q^63 + 7212922 * q^64 - 827096 * q^65 - 6615918 * q^66 + 4467570 * q^67 - 5477620 * q^68 + 2299563 * q^69 - 12744284 * q^70 - 5154608 * q^71 + 1036638 * q^72 - 13239250 * q^73 - 2827426 * q^74 - 8675775 * q^75 - 527434 * q^76 - 18415912 * q^77 - 11144196 * q^78 + 9594446 * q^79 - 55932394 * q^80 + 3720087 * q^81 - 20889952 * q^82 - 573720 * q^83 - 26080596 * q^84 + 7477272 * q^85 - 28416910 * q^86 - 4161402 * q^87 + 26555702 * q^88 - 3810540 * q^89 - 11160990 * q^90 + 36092068 * q^91 - 7957218 * q^92 - 6564564 * q^93 + 33545768 * q^94 + 10497320 * q^95 - 34513722 * q^96 + 49497978 * q^97 - 1023376 * q^98 + 6590160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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