LMFDB - Embedded newform 74.8.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,8,Mod(1,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, 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("74.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 74 = 2 \cdot 37 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	74.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:	$23.1164918858$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 2177x^{2} - 14018x + 634476 $ x^4 - 2x^3 - 2177x^2 - 14018*x + 634476
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$47.8378$ of defining polynomial
Character	$\chi$	$=$	74.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +22.3907 q^{3} +64.0000 q^{4} -504.619 q^{5} +179.126 q^{6} +892.385 q^{7} +512.000 q^{8} -1685.66 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +22.3907 q^{3} +64.0000 q^{4} -504.619 q^{5} +179.126 q^{6} +892.385 q^{7} +512.000 q^{8} -1685.66 q^{9} -4036.95 q^{10} +4703.14 q^{11} +1433.01 q^{12} -11843.3 q^{13} +7139.08 q^{14} -11298.8 q^{15} +4096.00 q^{16} -22642.5 q^{17} -13485.2 q^{18} -51600.8 q^{19} -32295.6 q^{20} +19981.1 q^{21} +37625.1 q^{22} -72804.0 q^{23} +11464.0 q^{24} +176515. q^{25} -94746.6 q^{26} -86711.5 q^{27} +57112.7 q^{28} +162871. q^{29} -90390.2 q^{30} -20354.2 q^{31} +32768.0 q^{32} +105307. q^{33} -181140. q^{34} -450314. q^{35} -107882. q^{36} -50653.0 q^{37} -412806. q^{38} -265180. q^{39} -258365. q^{40} -540030. q^{41} +159849. q^{42} +158658. q^{43} +301001. q^{44} +850613. q^{45} -582432. q^{46} +470210. q^{47} +91712.4 q^{48} -27191.7 q^{49} +1.41212e6 q^{50} -506981. q^{51} -757973. q^{52} +1.52235e6 q^{53} -693692. q^{54} -2.37329e6 q^{55} +456901. q^{56} -1.15538e6 q^{57} +1.30297e6 q^{58} +1.33575e6 q^{59} -723121. q^{60} -3.08165e6 q^{61} -162833. q^{62} -1.50425e6 q^{63} +262144. q^{64} +5.97636e6 q^{65} +842454. q^{66} +202860. q^{67} -1.44912e6 q^{68} -1.63013e6 q^{69} -3.60251e6 q^{70} -1.68686e6 q^{71} -863056. q^{72} +3.90554e6 q^{73} -405224. q^{74} +3.95230e6 q^{75} -3.30245e6 q^{76} +4.19702e6 q^{77} -2.12144e6 q^{78} +4.25515e6 q^{79} -2.06692e6 q^{80} +1.74500e6 q^{81} -4.32024e6 q^{82} +2.69308e6 q^{83} +1.27879e6 q^{84} +1.14258e7 q^{85} +1.26926e6 q^{86} +3.64679e6 q^{87} +2.40801e6 q^{88} +6.50359e6 q^{89} +6.80491e6 q^{90} -1.05688e7 q^{91} -4.65946e6 q^{92} -455744. q^{93} +3.76168e6 q^{94} +2.60387e7 q^{95} +733699. q^{96} -7.24155e6 q^{97} -217533. q^{98} -7.92788e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} - 41 q^{3} + 256 q^{4} - 363 q^{5} - 328 q^{6} - 774 q^{7} + 2048 q^{8} - 1079 q^{9}+O(q^{10}) $ 4 * q + 32 * q^2 - 41 * q^3 + 256 * q^4 - 363 * q^5 - 328 * q^6 - 774 * q^7 + 2048 * q^8 - 1079 * q^9	\( 4 q + 32 q^{2} - 41 q^{3} + 256 q^{4} - 363 q^{5} - 328 q^{6} - 774 q^{7} + 2048 q^{8} - 1079 q^{9} - 2904 q^{10} - 309 q^{11} - 2624 q^{12} - 20827 q^{13} - 6192 q^{14} - 22940 q^{15} + 16384 q^{16} - 48756 q^{17} - 8632 q^{18} - 69068 q^{19} - 23232 q^{20} - 640 q^{21} - 2472 q^{22} - 50237 q^{23} - 20992 q^{24} - 3581 q^{25} - 166616 q^{26} - 368414 q^{27} - 49536 q^{28} - 205195 q^{29} - 183520 q^{30} - 172283 q^{31} + 131072 q^{32} - 205234 q^{33} - 390048 q^{34} - 584964 q^{35} - 69056 q^{36} - 202612 q^{37} - 552544 q^{38} + 329055 q^{39} - 185856 q^{40} - 1018945 q^{41} - 5120 q^{42} + 1263046 q^{43} - 19776 q^{44} + 1279606 q^{45} - 401896 q^{46} - 420930 q^{47} - 167936 q^{48} - 482790 q^{49} - 28648 q^{50} + 728262 q^{51} - 1332928 q^{52} + 2051230 q^{53} - 2947312 q^{54} - 1442891 q^{55} - 396288 q^{56} - 926198 q^{57} - 1641560 q^{58} + 357914 q^{59} - 1468160 q^{60} - 2507513 q^{61} - 1378264 q^{62} + 2879054 q^{63} + 1048576 q^{64} + 3097954 q^{65} - 1641872 q^{66} + 586879 q^{67} - 3120384 q^{68} - 252895 q^{69} - 4679712 q^{70} - 130272 q^{71} - 552448 q^{72} + 3517417 q^{73} - 1620896 q^{74} + 8154290 q^{75} - 4420352 q^{76} + 8777590 q^{77} + 2632440 q^{78} + 3790171 q^{79} - 1486848 q^{80} + 15888376 q^{81} - 8151560 q^{82} + 12973460 q^{83} - 40960 q^{84} + 14870322 q^{85} + 10104368 q^{86} + 23063695 q^{87} - 158208 q^{88} + 18852848 q^{89} + 10236848 q^{90} - 2046622 q^{91} - 3215168 q^{92} + 4005314 q^{93} - 3367440 q^{94} + 30367150 q^{95} - 1343488 q^{96} + 14580104 q^{97} - 3862320 q^{98} + 16949258 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 - 41 * q^3 + 256 * q^4 - 363 * q^5 - 328 * q^6 - 774 * q^7 + 2048 * q^8 - 1079 * q^9 - 2904 * q^10 - 309 * q^11 - 2624 * q^12 - 20827 * q^13 - 6192 * q^14 - 22940 * q^15 + 16384 * q^16 - 48756 * q^17 - 8632 * q^18 - 69068 * q^19 - 23232 * q^20 - 640 * q^21 - 2472 * q^22 - 50237 * q^23 - 20992 * q^24 - 3581 * q^25 - 166616 * q^26 - 368414 * q^27 - 49536 * q^28 - 205195 * q^29 - 183520 * q^30 - 172283 * q^31 + 131072 * q^32 - 205234 * q^33 - 390048 * q^34 - 584964 * q^35 - 69056 * q^36 - 202612 * q^37 - 552544 * q^38 + 329055 * q^39 - 185856 * q^40 - 1018945 * q^41 - 5120 * q^42 + 1263046 * q^43 - 19776 * q^44 + 1279606 * q^45 - 401896 * q^46 - 420930 * q^47 - 167936 * q^48 - 482790 * q^49 - 28648 * q^50 + 728262 * q^51 - 1332928 * q^52 + 2051230 * q^53 - 2947312 * q^54 - 1442891 * q^55 - 396288 * q^56 - 926198 * q^57 - 1641560 * q^58 + 357914 * q^59 - 1468160 * q^60 - 2507513 * q^61 - 1378264 * q^62 + 2879054 * q^63 + 1048576 * q^64 + 3097954 * q^65 - 1641872 * q^66 + 586879 * q^67 - 3120384 * q^68 - 252895 * q^69 - 4679712 * q^70 - 130272 * q^71 - 552448 * q^72 + 3517417 * q^73 - 1620896 * q^74 + 8154290 * q^75 - 4420352 * q^76 + 8777590 * q^77 + 2632440 * q^78 + 3790171 * q^79 - 1486848 * q^80 + 15888376 * q^81 - 8151560 * q^82 + 12973460 * q^83 - 40960 * q^84 + 14870322 * q^85 + 10104368 * q^86 + 23063695 * q^87 - 158208 * q^88 + 18852848 * q^89 + 10236848 * q^90 - 2046622 * q^91 - 3215168 * q^92 + 4005314 * q^93 - 3367440 * q^94 + 30367150 * q^95 - 1343488 * q^96 + 14580104 * q^97 - 3862320 * q^98 + 16949258 * 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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	22.3907	0.478788	0.239394	−	0.970922i	$-0.423051\pi$
$3$	22.3907	0.478788	0.239394	+	0.970922i	$0.423051\pi$
$4$	64.0000	0.500000
$5$	−504.619	−1.80538	−0.902689	−	0.430293i	$-0.858410\pi$
$5$	−504.619	−1.80538	−0.902689	+	0.430293i	$0.858410\pi$
$6$	179.126	0.338554
$7$	892.385	0.983352	0.491676	−	0.870778i	$-0.336384\pi$
$7$	892.385	0.983352	0.491676	+	0.870778i	$0.336384\pi$
$8$	512.000	0.353553
$9$	−1685.66	−0.770762
$10$	−4036.95	−1.27660
$11$	4703.14	1.06540	0.532701	−	0.846303i	$-0.321177\pi$
$11$	4703.14	1.06540	0.532701	+	0.846303i	$0.321177\pi$
$12$	1433.01	0.239394
$13$	−11843.3	−1.49511	−0.747553	−	0.664202i	$-0.768771\pi$
$13$	−11843.3	−1.49511	−0.747553	+	0.664202i	$0.768771\pi$
$14$	7139.08	0.695335
$15$	−11298.8	−0.864394
$16$	4096.00	0.250000
$17$	−22642.5	−1.11777	−0.558885	−	0.829245i	$-0.688771\pi$
$17$	−22642.5	−1.11777	−0.558885	+	0.829245i	$0.688771\pi$
$18$	−13485.2	−0.545011
$19$	−51600.8	−1.72591	−0.862956	−	0.505279i	$-0.831390\pi$
$19$	−51600.8	−1.72591	−0.862956	+	0.505279i	$0.831390\pi$
$20$	−32295.6	−0.902689
$21$	19981.1	0.470818
$22$	37625.1	0.753353
$23$	−72804.0	−1.24769	−0.623847	−	0.781547i	$-0.714431\pi$
$23$	−72804.0	−1.24769	−0.623847	+	0.781547i	$0.714431\pi$
$24$	11464.0	0.169277
$25$	176515.	2.25939
$26$	−94746.6	−1.05720
$27$	−86711.5	−0.847820
$28$	57112.7	0.491676
$29$	162871.	1.24008	0.620041	−	0.784570i	$-0.287116\pi$
$29$	162871.	1.24008	0.620041	+	0.784570i	$0.287116\pi$
$30$	−90390.2	−0.611219
$31$	−20354.2	−0.122712	−0.0613560	−	0.998116i	$-0.519543\pi$
$31$	−20354.2	−0.122712	−0.0613560	+	0.998116i	$0.519543\pi$
$32$	32768.0	0.176777
$33$	105307.	0.510102
$34$	−181140.	−0.790383
$35$	−450314.	−1.77532
$36$	−107882.	−0.385381
$37$	−50653.0	−0.164399
$37$	−50653.0	−0.164399
$38$	−412806.	−1.22040
$39$	−265180.	−0.715839
$40$	−258365.	−0.638298
$41$	−540030.	−1.22370	−0.611849	−	0.790975i	$-0.709574\pi$
$41$	−540030.	−1.22370	−0.611849	+	0.790975i	$0.709574\pi$
$42$	159849.	0.332918
$43$	158658.	0.304314	0.152157	−	0.988356i	$-0.451378\pi$
$43$	158658.	0.304314	0.152157	+	0.988356i	$0.451378\pi$
$44$	301001.	0.532701
$45$	850613.	1.39152
$46$	−582432.	−0.882252
$47$	470210.	0.660617	0.330308	−	0.943873i	$-0.392847\pi$
$47$	470210.	0.660617	0.330308	+	0.943873i	$0.392847\pi$
$48$	91712.4	0.119697
$49$	−27191.7	−0.0330179
$50$	1.41212e6	1.59763
$51$	−506981.	−0.535175
$52$	−757973.	−0.747553
$53$	1.52235e6	1.40458	0.702292	−	0.711889i	$-0.252160\pi$
$53$	1.52235e6	1.40458	0.702292	+	0.711889i	$0.252160\pi$
$54$	−693692.	−0.599499
$55$	−2.37329e6	−1.92346
$56$	456901.	0.347668
$57$	−1.15538e6	−0.826346
$58$	1.30297e6	0.876870
$59$	1.33575e6	0.846725	0.423362	−	0.905960i	$-0.360850\pi$
$59$	1.33575e6	0.846725	0.423362	+	0.905960i	$0.360850\pi$
$60$	−723121.	−0.432197
$61$	−3.08165e6	−1.73832	−0.869159	−	0.494532i	$-0.835339\pi$
$61$	−3.08165e6	−1.73832	−0.869159	+	0.494532i	$0.835339\pi$
$62$	−162833.	−0.0867705
$63$	−1.50425e6	−0.757930
$64$	262144.	0.125000
$65$	5.97636e6	2.69923
$66$	842454.	0.360697
$67$	202860.	0.0824015	0.0412008	−	0.999151i	$-0.486882\pi$
$67$	202860.	0.0824015	0.0412008	+	0.999151i	$0.486882\pi$
$68$	−1.44912e6	−0.558885
$69$	−1.63013e6	−0.597381
$70$	−3.60251e6	−1.25534
$71$	−1.68686e6	−0.559338	−0.279669	−	0.960096i	$-0.590225\pi$
$71$	−1.68686e6	−0.559338	−0.279669	+	0.960096i	$0.590225\pi$
$72$	−863056.	−0.272505
$73$	3.90554e6	1.17504	0.587518	−	0.809211i	$-0.300105\pi$
$73$	3.90554e6	1.17504	0.587518	+	0.809211i	$0.300105\pi$
$74$	−405224.	−0.116248
$75$	3.95230e6	1.08177
$76$	−3.30245e6	−0.862956
$77$	4.19702e6	1.04767
$78$	−2.12144e6	−0.506175
$79$	4.25515e6	0.971001	0.485501	−	0.874236i	$-0.338637\pi$
$79$	4.25515e6	0.971001	0.485501	+	0.874236i	$0.338637\pi$
$80$	−2.06692e6	−0.451345
$81$	1.74500e6	0.364835
$82$	−4.32024e6	−0.865285
$83$	2.69308e6	0.516982	0.258491	−	0.966014i	$-0.416775\pi$
$83$	2.69308e6	0.516982	0.258491	+	0.966014i	$0.416775\pi$
$84$	1.27879e6	0.235409
$85$	1.14258e7	2.01800
$86$	1.26926e6	0.215182
$87$	3.64679e6	0.593736
$88$	2.40801e6	0.376677
$89$	6.50359e6	0.977885	0.488943	−	0.872316i	$-0.337383\pi$
$89$	6.50359e6	0.977885	0.488943	+	0.872316i	$0.337383\pi$
$90$	6.80491e6	0.983951
$91$	−1.05688e7	−1.47022
$92$	−4.65946e6	−0.623847
$93$	−455744.	−0.0587531
$94$	3.76168e6	0.467127
$95$	2.60387e7	3.11592
$96$	733699.	0.0846386
$97$	−7.24155e6	−0.805620	−0.402810	−	0.915284i	$-0.631966\pi$
$97$	−7.24155e6	−0.805620	−0.402810	+	0.915284i	$0.631966\pi$
$98$	−217533.	−0.0233472
$99$	−7.92788e6	−0.821172
$100$	1.12970e7	1.12970
$101$	−1.45545e7	−1.40563	−0.702815	−	0.711372i	$-0.748074\pi$
$101$	−1.45545e7	−1.40563	−0.702815	+	0.711372i	$0.748074\pi$
$102$	−4.05585e6	−0.378426
$103$	−1.89626e7	−1.70989	−0.854946	−	0.518717i	$-0.826410\pi$
$103$	−1.89626e7	−1.70989	−0.854946	+	0.518717i	$0.826410\pi$
$104$	−6.06378e6	−0.528600
$105$	−1.00829e7	−0.850004
$106$	1.21788e7	0.993191
$107$	1.11941e6	0.0883374	0.0441687	−	0.999024i	$-0.485936\pi$
$107$	1.11941e6	0.0883374	0.0441687	+	0.999024i	$0.485936\pi$
$108$	−5.54954e6	−0.423910
$109$	4.69508e6	0.347256	0.173628	−	0.984811i	$-0.444451\pi$
$109$	4.69508e6	0.347256	0.173628	+	0.984811i	$0.444451\pi$
$110$	−1.89863e7	−1.36009
$111$	−1.13416e6	−0.0787123
$112$	3.65521e6	0.245838
$113$	1.57905e7	1.02949	0.514745	−	0.857343i	$-0.327887\pi$
$113$	1.57905e7	1.02949	0.514745	+	0.857343i	$0.327887\pi$
$114$	−9.24303e6	−0.584315
$115$	3.67383e7	2.25256
$116$	1.04237e7	0.620041
$117$	1.99638e7	1.15237
$118$	1.06860e7	0.598725
$119$	−2.02058e7	−1.09916
$120$	−5.78497e6	−0.305609
$121$	2.63238e6	0.135083
$122$	−2.46532e7	−1.22918
$123$	−1.20916e7	−0.585892
$124$	−1.30267e6	−0.0613560
$125$	−4.96494e7	−2.27368
$126$	−1.20340e7	−0.535938
$127$	−2.06929e7	−0.896415	−0.448208	−	0.893930i	$-0.647937\pi$
$127$	−2.06929e7	−0.896415	−0.448208	+	0.893930i	$0.647937\pi$
$128$	2.09715e6	0.0883883
$129$	3.55246e6	0.145702
$130$	4.78109e7	1.90864
$131$	1.60306e7	0.623017	0.311508	−	0.950243i	$-0.399166\pi$
$131$	1.60306e7	0.623017	0.311508	+	0.950243i	$0.399166\pi$
$132$	6.73963e6	0.255051
$133$	−4.60478e7	−1.69718
$134$	1.62288e6	0.0582667
$135$	4.37563e7	1.53064
$136$	−1.15929e7	−0.395192
$137$	−3.82499e7	−1.27089	−0.635446	−	0.772145i	$-0.719184\pi$
$137$	−3.82499e7	−1.27089	−0.635446	+	0.772145i	$0.719184\pi$
$138$	−1.30411e7	−0.422412
$139$	2.59639e7	0.820009	0.410004	−	0.912084i	$-0.365527\pi$
$139$	2.59639e7	0.820009	0.410004	+	0.912084i	$0.365527\pi$
$140$	−2.88201e7	−0.887662
$141$	1.05283e7	0.316296
$142$	−1.34949e7	−0.395512
$143$	−5.57008e7	−1.59289
$144$	−6.90445e6	−0.192690
$145$	−8.21876e7	−2.23882
$146$	3.12443e7	0.830876
$147$	−608840.	−0.0158086
$148$	−3.24179e6	−0.0821995
$149$	2.14478e7	0.531168	0.265584	−	0.964088i	$-0.414435\pi$
$149$	2.14478e7	0.531168	0.265584	+	0.964088i	$0.414435\pi$
$150$	3.16184e7	0.764927
$151$	−7.61861e7	−1.80076	−0.900382	−	0.435101i	$-0.856713\pi$
$151$	−7.61861e7	−1.80076	−0.900382	+	0.435101i	$0.856713\pi$
$152$	−2.64196e7	−0.610202
$153$	3.81674e7	0.861535
$154$	3.35761e7	0.740812
$155$	1.02711e7	0.221542
$156$	−1.69715e7	−0.357920
$157$	4.99325e6	0.102976	0.0514878	−	0.998674i	$-0.483604\pi$
$157$	4.99325e6	0.102976	0.0514878	+	0.998674i	$0.483604\pi$
$158$	3.40412e7	0.686602
$159$	3.40864e7	0.672499
$160$	−1.65353e7	−0.319149
$161$	−6.49692e7	−1.22692
$162$	1.39600e7	0.257978
$163$	−7.53673e7	−1.36310	−0.681548	−	0.731774i	$-0.738693\pi$
$163$	−7.53673e7	−1.36310	−0.681548	+	0.731774i	$0.738693\pi$
$164$	−3.45619e7	−0.611849
$165$	−5.31397e7	−0.920928
$166$	2.15446e7	0.365562
$167$	1.76591e7	0.293401	0.146701	−	0.989181i	$-0.453135\pi$
$167$	1.76591e7	0.293401	0.146701	+	0.989181i	$0.453135\pi$
$168$	1.02303e7	0.166459
$169$	7.75158e7	1.23534
$170$	9.14065e7	1.42694
$171$	8.69812e7	1.33027
$172$	1.01541e7	0.152157
$173$	5.17704e7	0.760187	0.380093	−	0.924948i	$-0.375892\pi$
$173$	5.17704e7	0.760187	0.380093	+	0.924948i	$0.375892\pi$
$174$	2.91743e7	0.419835
$175$	1.57519e8	2.22178
$176$	1.92641e7	0.266351
$177$	2.99083e7	0.405402
$178$	5.20287e7	0.691469
$179$	3.40585e7	0.443853	0.221927	−	0.975063i	$-0.428765\pi$
$179$	3.40585e7	0.443853	0.221927	+	0.975063i	$0.428765\pi$
$180$	5.44393e7	0.695758
$181$	9.39883e7	1.17814	0.589072	−	0.808080i	$-0.299493\pi$
$181$	9.39883e7	1.17814	0.589072	+	0.808080i	$0.299493\pi$
$182$	−8.45505e7	−1.03960
$183$	−6.90004e7	−0.832287
$184$	−3.72756e7	−0.441126
$185$	2.55604e7	0.296802
$186$	−3.64595e6	−0.0415447
$187$	−1.06491e8	−1.19088
$188$	3.00935e7	0.330308
$189$	−7.73801e7	−0.833706
$190$	2.08310e8	2.20329
$191$	−7.50093e7	−0.778930	−0.389465	−	0.921041i	$-0.627340\pi$
$191$	−7.50093e7	−0.778930	−0.389465	+	0.921041i	$0.627340\pi$
$192$	5.86959e6	0.0598485
$193$	−4.91473e7	−0.492095	−0.246047	−	0.969258i	$-0.579132\pi$
$193$	−4.91473e7	−0.492095	−0.246047	+	0.969258i	$0.579132\pi$
$194$	−5.79324e7	−0.569659
$195$	1.33815e8	1.29236
$196$	−1.74027e6	−0.0165089
$197$	−1.33349e8	−1.24268	−0.621339	−	0.783542i	$-0.713411\pi$
$197$	−1.33349e8	−1.24268	−0.621339	+	0.783542i	$0.713411\pi$
$198$	−6.34231e7	−0.580656
$199$	−8.34957e7	−0.751066	−0.375533	−	0.926809i	$-0.622540\pi$
$199$	−8.34957e7	−0.751066	−0.375533	+	0.926809i	$0.622540\pi$
$200$	9.03756e7	0.798815
$201$	4.54219e6	0.0394529
$202$	−1.16436e8	−0.993931
$203$	1.45343e8	1.21944
$204$	−3.24468e7	−0.267588
$205$	2.72509e8	2.20924
$206$	−1.51701e8	−1.20908
$207$	1.22722e8	0.961674
$208$	−4.85103e7	−0.373776
$209$	−2.42686e8	−1.83879
$210$	−8.06628e7	−0.601044
$211$	−5.17239e7	−0.379056	−0.189528	−	0.981875i	$-0.560696\pi$
$211$	−5.17239e7	−0.379056	−0.189528	+	0.981875i	$0.560696\pi$
$212$	9.74302e7	0.702292
$213$	−3.77700e7	−0.267805
$214$	8.95525e6	0.0624640
$215$	−8.00617e7	−0.549402
$216$	−4.43963e7	−0.299750
$217$	−1.81637e7	−0.120669
$218$	3.75606e7	0.245547
$219$	8.74478e7	0.562593
$220$	−1.51891e8	−0.961728
$221$	2.68162e8	1.67119
$222$	−9.07325e6	−0.0556580
$223$	−1.88593e8	−1.13883	−0.569414	−	0.822051i	$-0.692830\pi$
$223$	−1.88593e8	−1.13883	−0.569414	+	0.822051i	$0.692830\pi$
$224$	2.92417e7	0.173834
$225$	−2.97543e8	−1.74145
$226$	1.26324e8	0.727959
$227$	3.12875e8	1.77533	0.887667	−	0.460486i	$-0.152325\pi$
$227$	3.12875e8	1.77533	0.887667	+	0.460486i	$0.152325\pi$
$228$	−7.39442e7	−0.413173
$229$	−2.14457e8	−1.18009	−0.590046	−	0.807370i	$-0.700890\pi$
$229$	−2.14457e8	−1.18009	−0.590046	+	0.807370i	$0.700890\pi$
$230$	2.93906e8	1.59280
$231$	9.39742e7	0.501610
$232$	8.33898e7	0.438435
$233$	7.68437e7	0.397981	0.198991	−	0.980001i	$-0.436234\pi$
$233$	7.68437e7	0.397981	0.198991	+	0.980001i	$0.436234\pi$
$234$	1.59710e8	0.814849
$235$	−2.37277e8	−1.19266
$236$	8.54878e7	0.423362
$237$	9.52758e7	0.464904
$238$	−1.61646e8	−0.777225
$239$	−1.96745e8	−0.932206	−0.466103	−	0.884730i	$-0.654342\pi$
$239$	−1.96745e8	−0.932206	−0.466103	+	0.884730i	$0.654342\pi$
$240$	−4.62798e7	−0.216099
$241$	1.30039e8	0.598430	0.299215	−	0.954186i	$-0.403275\pi$
$241$	1.30039e8	0.598430	0.299215	+	0.954186i	$0.403275\pi$
$242$	2.10591e7	0.0955181
$243$	2.28710e8	1.02250
$244$	−1.97226e8	−0.869159
$245$	1.37214e7	0.0596098
$246$	−9.67332e7	−0.414288
$247$	6.11125e8	2.58042
$248$	−1.04213e7	−0.0433853
$249$	6.03000e7	0.247525
$250$	−3.97195e8	−1.60773
$251$	−4.56690e8	−1.82290	−0.911452	−	0.411406i	$-0.865038\pi$
$251$	−4.56690e8	−1.82290	−0.911452	+	0.411406i	$0.865038\pi$
$252$	−9.62723e7	−0.378965
$253$	−3.42408e8	−1.32930
$254$	−1.65544e8	−0.633861
$255$	2.55832e8	0.966194
$256$	1.67772e7	0.0625000
$257$	9.55782e7	0.351231	0.175616	−	0.984459i	$-0.443808\pi$
$257$	9.55782e7	0.351231	0.175616	+	0.984459i	$0.443808\pi$
$258$	2.84197e7	0.103027
$259$	−4.52020e7	−0.161662
$260$	3.82487e8	1.34962
$261$	−2.74544e8	−0.955807
$262$	1.28245e8	0.440539
$263$	−3.48809e8	−1.18234	−0.591170	−	0.806547i	$-0.701334\pi$
$263$	−3.48809e8	−1.18234	−0.591170	+	0.806547i	$0.701334\pi$
$264$	5.39170e7	0.180348
$265$	−7.68204e8	−2.53581
$266$	−3.68382e8	−1.20009
$267$	1.45620e8	0.468200
$268$	1.29831e7	0.0412008
$269$	5.90362e7	0.184921	0.0924604	−	0.995716i	$-0.470527\pi$
$269$	5.90362e7	0.184921	0.0924604	+	0.995716i	$0.470527\pi$
$270$	3.50050e8	1.08232
$271$	2.36805e8	0.722767	0.361384	−	0.932417i	$-0.382304\pi$
$271$	2.36805e8	0.722767	0.361384	+	0.932417i	$0.382304\pi$
$272$	−9.27436e7	−0.279443
$273$	−2.36643e8	−0.703922
$274$	−3.05999e8	−0.898656
$275$	8.30175e8	2.40716
$276$	−1.04329e8	−0.298690
$277$	8.74196e7	0.247132	0.123566	−	0.992336i	$-0.460567\pi$
$277$	8.74196e7	0.247132	0.123566	+	0.992336i	$0.460567\pi$
$278$	2.07711e8	0.579834
$279$	3.43101e7	0.0945818
$280$	−2.30561e8	−0.627672
$281$	6.81033e6	0.0183103	0.00915516	−	0.999958i	$-0.497086\pi$
$281$	6.81033e6	0.0183103	0.00915516	+	0.999958i	$0.497086\pi$
$282$	8.42268e7	0.223655
$283$	3.29675e8	0.864636	0.432318	−	0.901721i	$-0.357696\pi$
$283$	3.29675e8	0.864636	0.432318	+	0.901721i	$0.357696\pi$
$284$	−1.07959e8	−0.279669
$285$	5.83025e8	1.49187
$286$	−4.45607e8	−1.12634
$287$	−4.81914e8	−1.20333
$288$	−5.52356e7	−0.136253
$289$	1.02343e8	0.249411
$290$	−6.57501e8	−1.58308
$291$	−1.62143e8	−0.385721
$292$	2.49955e8	0.587518
$293$	−5.11505e6	−0.0118799	−0.00593995	−	0.999982i	$-0.501891\pi$
$293$	−5.11505e6	−0.0118799	−0.00593995	+	0.999982i	$0.501891\pi$
$294$	−4.87072e6	−0.0111784
$295$	−6.74043e8	−1.52866
$296$	−2.59343e7	−0.0581238
$297$	−4.07817e8	−0.903270
$298$	1.71583e8	0.375592
$299$	8.62241e8	1.86543
$300$	2.52947e8	0.540885
$301$	1.41584e8	0.299248
$302$	−6.09489e8	−1.27333
$303$	−3.25885e8	−0.672999
$304$	−2.11357e8	−0.431478
$305$	1.55506e9	3.13832
$306$	3.05339e8	0.609197
$307$	4.03860e7	0.0796611	0.0398305	−	0.999206i	$-0.487318\pi$
$307$	4.03860e7	0.0796611	0.0398305	+	0.999206i	$0.487318\pi$
$308$	2.68609e8	0.523833
$309$	−4.24587e8	−0.818676
$310$	8.21687e7	0.156654
$311$	−2.00657e8	−0.378262	−0.189131	−	0.981952i	$-0.560567\pi$
$311$	−2.00657e8	−0.378262	−0.189131	+	0.981952i	$0.560567\pi$
$312$	−1.35772e8	−0.253087
$313$	−5.11656e8	−0.943133	−0.471566	−	0.881831i	$-0.656311\pi$
$313$	−5.11656e8	−0.943133	−0.471566	+	0.881831i	$0.656311\pi$
$314$	3.99460e7	0.0728148
$315$	7.59075e8	1.36835
$316$	2.72329e8	0.485501
$317$	−9.61366e8	−1.69504	−0.847522	−	0.530760i	$-0.821907\pi$
$317$	−9.61366e8	−1.69504	−0.847522	+	0.530760i	$0.821907\pi$
$318$	2.72691e8	0.475528
$319$	7.66004e8	1.32119
$320$	−1.32283e8	−0.225672
$321$	2.50643e7	0.0422949
$322$	−5.19754e8	−0.867565
$323$	1.16837e9	1.92917
$324$	1.11680e8	0.182418
$325$	−2.09052e9	−3.37803
$326$	−6.02938e8	−0.963854
$327$	1.05126e8	0.166262
$328$	−2.76495e8	−0.432642
$329$	4.19609e8	0.649619
$330$	−4.25118e8	−0.651194
$331$	−4.03400e8	−0.611417	−0.305709	−	0.952125i	$-0.598893\pi$
$331$	−4.03400e8	−0.611417	−0.305709	+	0.952125i	$0.598893\pi$
$332$	1.72357e8	0.258491
$333$	8.53835e7	0.126712
$334$	1.41273e8	0.207466
$335$	−1.02367e8	−0.148766
$336$	8.18428e7	0.117704
$337$	−3.99976e8	−0.569285	−0.284643	−	0.958634i	$-0.591875\pi$
$337$	−3.99976e8	−0.569285	−0.284643	+	0.958634i	$0.591875\pi$
$338$	6.20126e8	0.873518
$339$	3.53561e8	0.492908
$340$	7.31252e8	1.00900
$341$	−9.57285e7	−0.130738
$342$	6.95849e8	0.940641
$343$	−7.59183e8	−1.01582
$344$	8.12328e7	0.107591
$345$	8.22596e8	1.07850
$346$	4.14163e8	0.537533
$347$	7.68302e8	0.987140	0.493570	−	0.869706i	$-0.335692\pi$
$347$	7.68302e8	0.987140	0.493570	+	0.869706i	$0.335692\pi$
$348$	2.33395e8	0.296868
$349$	1.39255e8	0.175357	0.0876783	−	0.996149i	$-0.472055\pi$
$349$	1.39255e8	0.175357	0.0876783	+	0.996149i	$0.472055\pi$
$350$	1.26015e9	1.57103
$351$	1.02695e9	1.26758
$352$	1.54113e8	0.188338
$353$	3.62036e8	0.438067	0.219034	−	0.975717i	$-0.429710\pi$
$353$	3.62036e8	0.438067	0.219034	+	0.975717i	$0.429710\pi$
$354$	2.39267e8	0.286662
$355$	8.51220e8	1.00982
$356$	4.16230e8	0.488943
$357$	−4.52422e8	−0.526266
$358$	2.72468e8	0.313852
$359$	−6.39521e8	−0.729498	−0.364749	−	0.931106i	$-0.618845\pi$
$359$	−6.39521e8	−0.729498	−0.364749	+	0.931106i	$0.618845\pi$
$360$	4.35514e8	0.491975
$361$	1.76877e9	1.97877
$362$	7.51906e8	0.833074
$363$	5.89410e7	0.0646761
$364$	−6.76404e8	−0.735108
$365$	−1.97081e9	−2.12138
$366$	−5.52003e8	−0.588516
$367$	−1.56742e9	−1.65521	−0.827605	−	0.561311i	$-0.810297\pi$
$367$	−1.56742e9	−1.65521	−0.827605	+	0.561311i	$0.810297\pi$
$368$	−2.98205e8	−0.311923
$369$	9.10304e8	0.943179
$370$	2.04484e8	0.209871
$371$	1.35852e9	1.38120
$372$	−2.91676e7	−0.0293765
$373$	−6.28913e8	−0.627494	−0.313747	−	0.949507i	$-0.601584\pi$
$373$	−6.28913e8	−0.627494	−0.313747	+	0.949507i	$0.601584\pi$
$374$	−8.51926e8	−0.842076
$375$	−1.11169e9	−1.08861
$376$	2.40748e8	0.233563
$377$	−1.92893e9	−1.85405
$378$	−6.19041e8	−0.589519
$379$	−4.65760e8	−0.439466	−0.219733	−	0.975560i	$-0.570519\pi$
$379$	−4.65760e8	−0.439466	−0.219733	+	0.975560i	$0.570519\pi$
$380$	1.66648e9	1.55796
$381$	−4.63330e8	−0.429193
$382$	−6.00074e8	−0.550787
$383$	1.35682e9	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$383$	1.35682e9	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$384$	4.69567e7	0.0423193
$385$	−2.11789e9	−1.89143
$386$	−3.93178e8	−0.347963
$387$	−2.67442e8	−0.234553
$388$	−4.63459e8	−0.402810
$389$	9.47602e8	0.816211	0.408105	−	0.912935i	$-0.366190\pi$
$389$	9.47602e8	0.816211	0.408105	+	0.912935i	$0.366190\pi$
$390$	1.07052e9	0.913837
$391$	1.64846e9	1.39463
$392$	−1.39221e7	−0.0116736
$393$	3.58936e8	0.298293
$394$	−1.06679e9	−0.878706
$395$	−2.14723e9	−1.75302
$396$	−5.07384e8	−0.410586
$397$	1.61811e9	1.29790	0.648951	−	0.760830i	$-0.275208\pi$
$397$	1.61811e9	1.29790	0.648951	+	0.760830i	$0.275208\pi$
$398$	−6.67965e8	−0.531084
$399$	−1.03104e9	−0.812590
$400$	7.23005e8	0.564848
$401$	1.47928e9	1.14563	0.572816	−	0.819684i	$-0.305851\pi$
$401$	1.47928e9	1.14563	0.572816	+	0.819684i	$0.305851\pi$
$402$	3.63375e7	0.0278974
$403$	2.41061e8	0.183467
$404$	−9.31485e8	−0.702815
$405$	−8.80558e8	−0.658666
$406$	1.16275e9	0.862272
$407$	−2.38228e8	−0.175151
$408$	−2.59574e8	−0.189213
$409$	1.53241e9	1.10750	0.553750	−	0.832683i	$-0.313196\pi$
$409$	1.53241e9	1.10750	0.553750	+	0.832683i	$0.313196\pi$
$410$	2.18007e9	1.56217
$411$	−8.56443e8	−0.608488
$412$	−1.21361e9	−0.854946
$413$	1.19200e9	0.832629
$414$	9.81780e8	0.680006
$415$	−1.35898e9	−0.933349
$416$	−3.88082e8	−0.264300
$417$	5.81351e8	0.392611
$418$	−1.94149e9	−1.30022
$419$	5.61884e8	0.373162	0.186581	−	0.982440i	$-0.440259\pi$
$419$	5.61884e8	0.373162	0.186581	+	0.982440i	$0.440259\pi$
$420$	−6.45303e8	−0.425002
$421$	−8.61751e6	−0.00562853	−0.00281426	−	0.999996i	$-0.500896\pi$
$421$	−8.61751e6	−0.00562853	−0.00281426	+	0.999996i	$0.500896\pi$
$422$	−4.13791e8	−0.268033
$423$	−7.92613e8	−0.509178
$424$	7.79441e8	0.496596
$425$	−3.99674e9	−2.52548
$426$	−3.02160e8	−0.189366
$427$	−2.75002e9	−1.70938
$428$	7.16420e7	0.0441687
$429$	−1.24718e9	−0.762657
$430$	−6.40493e8	−0.388486
$431$	−4.27833e8	−0.257397	−0.128699	−	0.991684i	$-0.541080\pi$
$431$	−4.27833e8	−0.257397	−0.128699	+	0.991684i	$0.541080\pi$
$432$	−3.55170e8	−0.211955
$433$	1.56113e9	0.924127	0.462064	−	0.886847i	$-0.347109\pi$
$433$	1.56113e9	0.924127	0.462064	+	0.886847i	$0.347109\pi$
$434$	−1.45310e8	−0.0853260
$435$	−1.84024e9	−1.07192
$436$	3.00485e8	0.173628
$437$	3.75674e9	2.15341
$438$	6.99582e8	0.397814
$439$	1.63407e9	0.921817	0.460908	−	0.887448i	$-0.347524\pi$
$439$	1.63407e9	0.921817	0.460908	+	0.887448i	$0.347524\pi$
$440$	−1.21513e9	−0.680044
$441$	4.58358e7	0.0254489
$442$	2.14530e9	1.18171
$443$	−6.21106e8	−0.339432	−0.169716	−	0.985493i	$-0.554285\pi$
$443$	−6.21106e8	−0.339432	−0.169716	+	0.985493i	$0.554285\pi$
$444$	−7.25860e7	−0.0393562
$445$	−3.28183e9	−1.76545
$446$	−1.50874e9	−0.805273
$447$	4.80233e8	0.254317
$448$	2.33933e8	0.122919
$449$	2.16788e8	0.113024	0.0565122	−	0.998402i	$-0.482002\pi$
$449$	2.16788e8	0.113024	0.0565122	+	0.998402i	$0.482002\pi$
$450$	−2.38035e9	−1.23139
$451$	−2.53984e9	−1.30373
$452$	1.01059e9	0.514745
$453$	−1.70586e9	−0.862184
$454$	2.50300e9	1.25535
$455$	5.33322e9	2.65430
$456$	−5.91554e8	−0.292158
$457$	−3.59868e9	−1.76375	−0.881874	−	0.471485i	$-0.843718\pi$
$457$	−3.59868e9	−1.76375	−0.881874	+	0.471485i	$0.843718\pi$
$458$	−1.71565e9	−0.834451
$459$	1.96336e9	0.947668
$460$	2.35125e9	1.12628
$461$	−4.44676e8	−0.211393	−0.105696	−	0.994398i	$-0.533707\pi$
$461$	−4.44676e8	−0.211393	−0.105696	+	0.994398i	$0.533707\pi$
$462$	7.51793e8	0.354692
$463$	−1.08602e9	−0.508517	−0.254258	−	0.967136i	$-0.581831\pi$
$463$	−1.08602e9	−0.508517	−0.254258	+	0.967136i	$0.581831\pi$
$464$	6.67119e8	0.310020
$465$	2.29977e8	0.106072
$466$	6.14750e8	0.281415
$467$	3.61227e9	1.64124	0.820619	−	0.571476i	$-0.193629\pi$
$467$	3.61227e9	1.64124	0.820619	+	0.571476i	$0.193629\pi$
$468$	1.27768e9	0.576185
$469$	1.81030e8	0.0810298
$470$	−1.89822e9	−0.843340
$471$	1.11802e8	0.0493036
$472$	6.83903e8	0.299362
$473$	7.46190e8	0.324217
$474$	7.62206e8	0.328737
$475$	−9.10831e9	−3.89951
$476$	−1.29317e9	−0.549581
$477$	−2.56615e9	−1.08260
$478$	−1.57396e9	−0.659169
$479$	3.95907e9	1.64596	0.822978	−	0.568073i	$-0.192311\pi$
$479$	3.95907e9	1.64596	0.822978	+	0.568073i	$0.192311\pi$
$480$	−3.70238e8	−0.152805
$481$	5.99900e8	0.245794
$482$	1.04031e9	0.423154
$483$	−1.45471e9	−0.587436
$484$	1.68473e8	0.0675415
$485$	3.65422e9	1.45445
$486$	1.82968e9	0.723016
$487$	4.30023e9	1.68710	0.843549	−	0.537052i	$-0.180462\pi$
$487$	4.30023e9	1.68710	0.843549	+	0.537052i	$0.180462\pi$
$488$	−1.57781e9	−0.614588
$489$	−1.68753e9	−0.652634
$490$	1.09771e8	0.0421505
$491$	−2.54704e9	−0.971072	−0.485536	−	0.874217i	$-0.661375\pi$
$491$	−2.54704e9	−0.971072	−0.485536	+	0.874217i	$0.661375\pi$
$492$	−7.73865e8	−0.292946
$493$	−3.68780e9	−1.38613
$494$	4.88900e9	1.82463
$495$	4.00056e9	1.48253
$496$	−8.33706e7	−0.0306780
$497$	−1.50533e9	−0.550027
$498$	4.82400e8	0.175027
$499$	−3.13518e9	−1.12956	−0.564782	−	0.825240i	$-0.691040\pi$
$499$	−3.13518e9	−1.12956	−0.564782	+	0.825240i	$0.691040\pi$
$500$	−3.17756e9	−1.13684
$501$	3.95401e8	0.140477
$502$	−3.65352e9	−1.28899
$503$	8.10436e7	0.0283943	0.0141971	−	0.999899i	$-0.495481\pi$
$503$	8.10436e7	0.0283943	0.0141971	+	0.999899i	$0.495481\pi$
$504$	−7.70178e8	−0.267969
$505$	7.34445e9	2.53769
$506$	−2.73926e9	−0.939954
$507$	1.73563e9	0.591467
$508$	−1.32435e9	−0.448208
$509$	1.83771e9	0.617683	0.308841	−	0.951114i	$-0.400059\pi$
$509$	1.83771e9	0.617683	0.308841	+	0.951114i	$0.400059\pi$
$510$	2.04666e9	0.683202
$511$	3.48525e9	1.15547
$512$	1.34218e8	0.0441942
$513$	4.47438e9	1.46326
$514$	7.64626e8	0.248358
$515$	9.56891e9	3.08700
$516$	2.27357e8	0.0728510
$517$	2.21147e9	0.703823
$518$	−3.61616e8	−0.114312
$519$	1.15918e9	0.363968
$520$	3.05990e9	0.954322
$521$	−3.26147e9	−1.01037	−0.505186	−	0.863011i	$-0.668576\pi$
$521$	−3.26147e9	−1.01037	−0.505186	+	0.863011i	$0.668576\pi$
$522$	−2.19635e9	−0.675858
$523$	1.51622e9	0.463454	0.231727	−	0.972781i	$-0.425562\pi$
$523$	1.51622e9	0.463454	0.231727	+	0.972781i	$0.425562\pi$
$524$	1.02596e9	0.311508
$525$	3.52697e9	1.06376
$526$	−2.79047e9	−0.836041
$527$	4.60868e8	0.137164
$528$	4.31336e8	0.127526
$529$	1.89560e9	0.556738
$530$	−6.14563e9	−1.79309
$531$	−2.25161e9	−0.652623
$532$	−2.94706e9	−0.848590
$533$	6.39575e9	1.82956
$534$	1.16496e9	0.331067
$535$	−5.64873e8	−0.159482
$536$	1.03864e8	0.0291333
$537$	7.62593e8	0.212512
$538$	4.72290e8	0.130759
$539$	−1.27886e8	−0.0351774
$540$	2.80040e9	0.765318
$541$	−8.73876e8	−0.237279	−0.118640	−	0.992937i	$-0.537853\pi$
$541$	−8.73876e8	−0.237279	−0.118640	+	0.992937i	$0.537853\pi$
$542$	1.89444e9	0.511074
$543$	2.10446e9	0.564082
$544$	−7.41949e8	−0.197596
$545$	−2.36922e9	−0.626929
$546$	−1.89314e9	−0.497748
$547$	−1.08535e9	−0.283541	−0.141770	−	0.989900i	$-0.545279\pi$
$547$	−1.08535e9	−0.283541	−0.141770	+	0.989900i	$0.545279\pi$
$548$	−2.44800e9	−0.635446
$549$	5.19461e9	1.33983
$550$	6.64140e9	1.70212
$551$	−8.40426e9	−2.14027
$552$	−8.34628e8	−0.211206
$553$	3.79723e9	0.954837
$554$	6.99356e8	0.174749
$555$	5.72317e8	0.142106
$556$	1.66169e9	0.410004
$557$	−6.40646e9	−1.57081	−0.785407	−	0.618980i	$-0.787546\pi$
$557$	−6.40646e9	−1.57081	−0.785407	+	0.618980i	$0.787546\pi$
$558$	2.74481e8	0.0668794
$559$	−1.87904e9	−0.454981
$560$	−1.84449e9	−0.443831
$561$	−2.38440e9	−0.570177
$562$	5.44826e7	0.0129473
$563$	−7.36885e8	−0.174028	−0.0870142	−	0.996207i	$-0.527733\pi$
$563$	−7.36885e8	−0.174028	−0.0870142	+	0.996207i	$0.527733\pi$
$564$	6.73814e8	0.158148
$565$	−7.96820e9	−1.85862
$566$	2.63740e9	0.611390
$567$	1.55721e9	0.358762
$568$	−8.63672e8	−0.197756
$569$	2.26466e9	0.515360	0.257680	−	0.966230i	$-0.417042\pi$
$569$	2.26466e9	0.515360	0.257680	+	0.966230i	$0.417042\pi$
$570$	4.66420e9	1.05491
$571$	1.77146e9	0.398202	0.199101	−	0.979979i	$-0.436198\pi$
$571$	1.77146e9	0.398202	0.199101	+	0.979979i	$0.436198\pi$
$572$	−3.56485e9	−0.796445
$573$	−1.67951e9	−0.372942
$574$	−3.85532e9	−0.850880
$575$	−1.28510e10	−2.81903
$576$	−4.41885e8	−0.0963452
$577$	4.19705e9	0.909554	0.454777	−	0.890605i	$-0.349719\pi$
$577$	4.19705e9	0.909554	0.454777	+	0.890605i	$0.349719\pi$
$578$	8.18744e8	0.176360
$579$	−1.10044e9	−0.235609
$580$	−5.26001e9	−1.11941
$581$	2.40326e9	0.508376
$582$	−1.29715e9	−0.272746
$583$	7.15981e9	1.49645
$584$	1.99964e9	0.415438
$585$	−1.00741e10	−2.08046
$586$	−4.09204e7	−0.00840036
$587$	6.47404e9	1.32112	0.660560	−	0.750773i	$-0.270319\pi$
$587$	6.47404e9	1.32112	0.660560	+	0.750773i	$0.270319\pi$
$588$	−3.89658e7	−0.00790429
$589$	1.05029e9	0.211790
$590$	−5.39234e9	−1.08092
$591$	−2.98578e9	−0.594979
$592$	−2.07475e8	−0.0410997
$593$	−9.00131e9	−1.77261	−0.886307	−	0.463098i	$-0.846738\pi$
$593$	−9.00131e9	−1.77261	−0.886307	+	0.463098i	$0.846738\pi$
$594$	−3.26253e9	−0.638708
$595$	1.01962e10	1.98440
$596$	1.37266e9	0.265584
$597$	−1.86953e9	−0.359602
$598$	6.89793e9	1.31906
$599$	2.07040e9	0.393604	0.196802	−	0.980443i	$-0.436944\pi$
$599$	2.07040e9	0.393604	0.196802	+	0.980443i	$0.436944\pi$
$600$	2.02358e9	0.382463
$601$	−4.27425e9	−0.803155	−0.401577	−	0.915825i	$-0.631538\pi$
$601$	−4.27425e9	−0.803155	−0.401577	+	0.915825i	$0.631538\pi$
$602$	1.13267e9	0.211600
$603$	−3.41953e8	−0.0635120
$604$	−4.87591e9	−0.900382
$605$	−1.32835e9	−0.243876
$606$	−2.60708e9	−0.475882
$607$	−9.36384e9	−1.69939	−0.849696	−	0.527272i	$-0.823215\pi$
$607$	−9.36384e9	−1.69939	−0.849696	+	0.527272i	$0.823215\pi$
$608$	−1.69085e9	−0.305101
$609$	3.25434e9	0.583852
$610$	1.24405e10	2.21913
$611$	−5.56885e9	−0.987692
$612$	2.44272e9	0.430767
$613$	−6.35668e8	−0.111460	−0.0557299	−	0.998446i	$-0.517749\pi$
$613$	−6.35668e8	−0.111460	−0.0557299	+	0.998446i	$0.517749\pi$
$614$	3.23088e8	0.0563289
$615$	6.10167e9	1.05776
$616$	2.14887e9	0.370406
$617$	−3.37951e9	−0.579237	−0.289618	−	0.957142i	$-0.593528\pi$
$617$	−3.37951e9	−0.579237	−0.289618	+	0.957142i	$0.593528\pi$
$618$	−3.39670e9	−0.578892
$619$	−7.37657e9	−1.25008	−0.625039	−	0.780593i	$-0.714917\pi$
$619$	−7.37657e9	−1.25008	−0.625039	+	0.780593i	$0.714917\pi$
$620$	6.57350e8	0.110771
$621$	6.31295e9	1.05782
$622$	−1.60526e9	−0.267472
$623$	5.80370e9	0.961606
$624$	−1.08618e9	−0.178960
$625$	1.12638e10	1.84546
$626$	−4.09325e9	−0.666896
$627$	−5.43391e9	−0.880392
$628$	3.19568e8	0.0514878
$629$	1.14691e9	0.183760
$630$	6.07260e9	0.967571
$631$	−8.76066e9	−1.38814	−0.694071	−	0.719906i	$-0.744185\pi$
$631$	−8.76066e9	−1.38814	−0.694071	+	0.719906i	$0.744185\pi$
$632$	2.17864e9	0.343301
$633$	−1.15814e9	−0.181487
$634$	−7.69092e9	−1.19858
$635$	1.04420e10	1.61837
$636$	2.18153e9	0.336249
$637$	3.22040e8	0.0493652
$638$	6.12804e9	0.934219
$639$	2.84346e9	0.431117
$640$	−1.05826e9	−0.159574
$641$	−5.91406e9	−0.886916	−0.443458	−	0.896295i	$-0.646248\pi$
$641$	−5.91406e9	−0.886916	−0.443458	+	0.896295i	$0.646248\pi$
$642$	2.00514e8	0.0299070
$643$	1.21681e9	0.180504	0.0902518	−	0.995919i	$-0.471233\pi$
$643$	1.21681e9	0.180504	0.0902518	+	0.995919i	$0.471233\pi$
$644$	−4.15803e9	−0.613461
$645$	−1.79264e9	−0.263047
$646$	9.34695e9	1.36413
$647$	−4.35488e9	−0.632137	−0.316068	−	0.948736i	$-0.602363\pi$
$647$	−4.35488e9	−0.632137	−0.316068	+	0.948736i	$0.602363\pi$
$648$	8.93438e8	0.128989
$649$	6.28221e9	0.902103
$650$	−1.67242e10	−2.38863
$651$	−4.06699e8	−0.0577750
$652$	−4.82351e9	−0.681548
$653$	−2.37363e9	−0.333594	−0.166797	−	0.985991i	$-0.553342\pi$
$653$	−2.37363e9	−0.333594	−0.166797	+	0.985991i	$0.553342\pi$
$654$	8.41009e8	0.117565
$655$	−8.08933e9	−1.12478
$656$	−2.21196e9	−0.305924
$657$	−6.58340e9	−0.905672
$658$	3.35687e9	0.459350
$659$	−3.03953e9	−0.413721	−0.206861	−	0.978370i	$-0.566325\pi$
$659$	−3.03953e9	−0.413721	−0.206861	+	0.978370i	$0.566325\pi$
$660$	−3.40094e9	−0.460464
$661$	−1.60589e9	−0.216277	−0.108138	−	0.994136i	$-0.534489\pi$
$661$	−1.60589e9	−0.216277	−0.108138	+	0.994136i	$0.534489\pi$
$662$	−3.22720e9	−0.432337
$663$	6.00434e9	0.800144
$664$	1.37886e9	0.182781
$665$	2.32366e10	3.06405
$666$	6.83068e8	0.0895992
$667$	−1.18576e10	−1.54724
$668$	1.13019e9	0.146701
$669$	−4.22273e9	−0.545257
$670$	−8.18937e8	−0.105193
$671$	−1.44935e10	−1.85201
$672$	6.54742e8	0.0832296
$673$	1.06339e10	1.34474	0.672371	−	0.740214i	$-0.265276\pi$
$673$	1.06339e10	1.34474	0.672371	+	0.740214i	$0.265276\pi$
$674$	−3.19981e9	−0.402545
$675$	−1.53059e10	−1.91556
$676$	4.96101e9	0.617670
$677$	1.12699e10	1.39591	0.697957	−	0.716139i	$-0.254093\pi$
$677$	1.12699e10	1.39591	0.697957	+	0.716139i	$0.254093\pi$
$678$	2.82849e9	0.348538
$679$	−6.46225e9	−0.792208
$680$	5.85002e9	0.713470
$681$	7.00549e9	0.850009
$682$	−7.65828e8	−0.0924455
$683$	−7.09950e9	−0.852619	−0.426310	−	0.904577i	$-0.640187\pi$
$683$	−7.09950e9	−0.852619	−0.426310	+	0.904577i	$0.640187\pi$
$684$	5.56679e9	0.665133
$685$	1.93016e10	2.29444
$686$	−6.07346e9	−0.718294
$687$	−4.80184e9	−0.565014
$688$	6.49862e8	0.0760785
$689$	−1.80296e10	−2.10000
$690$	6.58076e9	0.762614
$691$	−8.79291e9	−1.01382	−0.506909	−	0.862000i	$-0.669212\pi$
$691$	−8.79291e9	−1.01382	−0.506909	+	0.862000i	$0.669212\pi$
$692$	3.31331e9	0.380093
$693$	−7.07472e9	−0.807501
$694$	6.14641e9	0.698013
$695$	−1.31019e10	−1.48043
$696$	1.86716e9	0.209917
$697$	1.22276e10	1.36781
$698$	1.11404e9	0.123996
$699$	1.72059e9	0.190549
$700$	1.00812e10	1.11089
$701$	−2.64063e9	−0.289530	−0.144765	−	0.989466i	$-0.546243\pi$
$701$	−2.64063e9	−0.289530	−0.144765	+	0.989466i	$0.546243\pi$
$702$	8.21562e9	0.896315
$703$	2.61373e9	0.283738
$704$	1.23290e9	0.133175
$705$	−5.31280e9	−0.571033
$706$	2.89629e9	0.309760
$707$	−1.29882e10	−1.38223
$708$	1.91413e9	0.202701
$709$	9.19450e9	0.968872	0.484436	−	0.874827i	$-0.339025\pi$
$709$	9.19450e9	0.968872	0.484436	+	0.874827i	$0.339025\pi$
$710$	6.80976e9	0.714049
$711$	−7.17272e9	−0.748411
$712$	3.32984e9	0.345735
$713$	1.48186e9	0.153107
$714$	−3.61938e9	−0.372126
$715$	2.81077e10	2.87577
$716$	2.17974e9	0.221927
$717$	−4.40527e9	−0.446329
$718$	−5.11617e9	−0.515833
$719$	2.41418e9	0.242225	0.121112	−	0.992639i	$-0.461354\pi$
$719$	2.41418e9	0.242225	0.121112	+	0.992639i	$0.461354\pi$
$720$	3.48411e9	0.347879
$721$	−1.69220e10	−1.68143
$722$	1.41501e10	1.39920
$723$	2.91166e9	0.286521
$724$	6.01525e9	0.589072
$725$	2.87491e10	2.80183
$726$	4.71528e8	0.0457329
$727$	7.98519e9	0.770753	0.385376	−	0.922760i	$-0.374072\pi$
$727$	7.98519e9	0.770753	0.385376	+	0.922760i	$0.374072\pi$
$728$	−5.41123e9	−0.519800
$729$	1.30467e9	0.124725
$730$	−1.57665e10	−1.50004
$731$	−3.59240e9	−0.340153
$732$	−4.41603e9	−0.416143
$733$	6.97965e9	0.654591	0.327295	−	0.944922i	$-0.393863\pi$
$733$	6.97965e9	0.654591	0.327295	+	0.944922i	$0.393863\pi$
$734$	−1.25393e10	−1.17041
$735$	3.07232e8	0.0285405
$736$	−2.38564e9	−0.220563
$737$	9.54081e8	0.0877908
$738$	7.28243e9	0.666928
$739$	2.04231e9	0.186151	0.0930757	−	0.995659i	$-0.470330\pi$
$739$	2.04231e9	0.186151	0.0930757	+	0.995659i	$0.470330\pi$
$740$	1.63587e9	0.148401
$741$	1.36835e10	1.23548
$742$	1.08682e10	0.976657
$743$	−1.55097e10	−1.38721	−0.693604	−	0.720357i	$-0.743978\pi$
$743$	−1.55097e10	−1.38721	−0.693604	+	0.720357i	$0.743978\pi$
$744$	−2.33341e8	−0.0207724
$745$	−1.08230e10	−0.958959
$746$	−5.03131e9	−0.443706
$747$	−4.53960e9	−0.398470
$748$	−6.81541e9	−0.595438
$749$	9.98942e8	0.0868668
$750$	−8.89348e9	−0.769764
$751$	−7.67830e9	−0.661493	−0.330746	−	0.943720i	$-0.607300\pi$
$751$	−7.67830e9	−0.661493	−0.330746	+	0.943720i	$0.607300\pi$
$752$	1.92598e9	0.165154
$753$	−1.02256e10	−0.872785
$754$	−1.54314e10	−1.31101
$755$	3.84449e10	3.25106
$756$	−4.95233e9	−0.416853
$757$	1.30345e9	0.109209	0.0546047	−	0.998508i	$-0.482610\pi$
$757$	1.30345e9	0.109209	0.0546047	+	0.998508i	$0.482610\pi$
$758$	−3.72608e9	−0.310749
$759$	−7.66675e9	−0.636451
$760$	1.33318e10	1.10165
$761$	−1.93614e9	−0.159254	−0.0796269	−	0.996825i	$-0.525373\pi$
$761$	−1.93614e9	−0.159254	−0.0796269	+	0.996825i	$0.525373\pi$
$762$	−3.70664e9	−0.303485
$763$	4.18982e9	0.341475
$764$	−4.80060e9	−0.389465
$765$	−1.92600e10	−1.55540
$766$	1.08545e10	0.872590
$767$	−1.58197e10	−1.26594
$768$	3.75654e8	0.0299243
$769$	1.86998e9	0.148284	0.0741422	−	0.997248i	$-0.476378\pi$
$769$	1.86998e9	0.148284	0.0741422	+	0.997248i	$0.476378\pi$
$770$	−1.69431e10	−1.33745
$771$	2.14006e9	0.168165
$772$	−3.14542e9	−0.246047
$773$	1.89091e10	1.47246	0.736228	−	0.676734i	$-0.236605\pi$
$773$	1.89091e10	1.47246	0.736228	+	0.676734i	$0.236605\pi$
$774$	−2.13954e9	−0.165854
$775$	−3.59281e9	−0.277255
$776$	−3.70767e9	−0.284830
$777$	−1.01210e9	−0.0774019
$778$	7.58082e9	0.577148
$779$	2.78659e10	2.11199
$780$	8.56416e9	0.646180
$781$	−7.93354e9	−0.595921
$782$	1.31877e10	0.986156
$783$	−1.41228e10	−1.05137
$784$	−1.11377e8	−0.00825447
$785$	−2.51969e9	−0.185910
$786$	2.87149e9	0.210925
$787$	−6.44240e9	−0.471125	−0.235563	−	0.971859i	$-0.575693\pi$
$787$	−6.44240e9	−0.471125	−0.235563	+	0.971859i	$0.575693\pi$
$788$	−8.53434e9	−0.621339
$789$	−7.81008e9	−0.566091
$790$	−1.71778e10	−1.23958
$791$	1.40912e10	1.01235
$792$	−4.05908e9	−0.290328
$793$	3.64970e10	2.59897
$794$	1.29449e10	0.917755
$795$	−1.72006e10	−1.21411
$796$	−5.34372e9	−0.375533
$797$	−6.22130e9	−0.435288	−0.217644	−	0.976028i	$-0.569837\pi$
$797$	−6.22130e9	−0.435288	−0.217644	+	0.976028i	$0.569837\pi$
$798$	−8.24834e9	−0.574588
$799$	−1.06467e10	−0.738418
$800$	5.78404e9	0.399408
$801$	−1.09628e10	−0.753717
$802$	1.18342e10	0.810084
$803$	1.83683e10	1.25189
$804$	2.90700e8	0.0197264
$805$	3.27847e10	2.21506
$806$	1.92849e9	0.129731
$807$	1.32186e9	0.0885379
$808$	−7.45188e9	−0.496965
$809$	−5.70480e6	−0.000378809 0	−0.000189404	−	1.00000i	$-0.500060\pi$
$809$	−5.70480e6	−0.000378809 0	−0.000189404		1.00000i	$0.500060\pi$
$810$	−7.04446e9	−0.465747
$811$	1.35122e10	0.889514	0.444757	−	0.895651i	$-0.353290\pi$
$811$	1.35122e10	0.889514	0.444757	+	0.895651i	$0.353290\pi$
$812$	9.30198e9	0.609718
$813$	5.30224e9	0.346053
$814$	−1.90583e9	−0.123851
$815$	3.80317e10	2.46090
$816$	−2.07659e9	−0.133794
$817$	−8.18686e9	−0.525219
$818$	1.22593e10	0.783120
$819$	1.78154e10	1.13319
$820$	1.74406e10	1.10462
$821$	−2.04061e10	−1.28694	−0.643472	−	0.765470i	$-0.722507\pi$
$821$	−2.04061e10	−1.28694	−0.643472	+	0.765470i	$0.722507\pi$
$822$	−6.85155e9	−0.430266
$823$	1.31164e10	0.820191	0.410095	−	0.912043i	$-0.365495\pi$
$823$	1.31164e10	0.820191	0.410095	+	0.912043i	$0.365495\pi$
$824$	−9.70888e9	−0.604538
$825$	1.85882e10	1.15252
$826$	9.53601e9	0.588758
$827$	1.00525e10	0.618021	0.309010	−	0.951059i	$-0.400002\pi$
$827$	1.00525e10	0.618021	0.309010	+	0.951059i	$0.400002\pi$
$828$	7.85424e9	0.480837
$829$	−2.39621e10	−1.46078	−0.730389	−	0.683031i	$-0.760661\pi$
$829$	−2.39621e10	−1.46078	−0.730389	+	0.683031i	$0.760661\pi$
$830$	−1.08718e10	−0.659977
$831$	1.95739e9	0.118324
$832$	−3.10466e9	−0.186888
$833$	6.15686e8	0.0369064
$834$	4.65081e9	0.277618
$835$	−8.91113e9	−0.529700
$836$	−1.55319e10	−0.919396
$837$	1.76494e9	0.104038
$838$	4.49507e9	0.263865
$839$	2.73604e9	0.159939	0.0799697	−	0.996797i	$-0.474518\pi$
$839$	2.73604e9	0.159939	0.0799697	+	0.996797i	$0.474518\pi$
$840$	−5.16242e9	−0.300522
$841$	9.27700e9	0.537801
$842$	−6.89401e7	−0.00397997
$843$	1.52488e8	0.00876676
$844$	−3.31033e9	−0.189528
$845$	−3.91159e10	−2.23026
$846$	−6.34090e9	−0.360043
$847$	2.34910e9	0.132834
$848$	6.23553e9	0.351146
$849$	7.38166e9	0.413978
$850$	−3.19739e10	−1.78578
$851$	3.68774e9	0.205119
$852$	−2.41728e9	−0.133902
$853$	−1.75500e10	−0.968181	−0.484090	−	0.875018i	$-0.660849\pi$
$853$	−1.75500e10	−0.968181	−0.484090	+	0.875018i	$0.660849\pi$
$854$	−2.20002e10	−1.20871
$855$	−4.38923e10	−2.40164
$856$	5.73136e8	0.0312320
$857$	1.35123e9	0.0733327	0.0366663	−	0.999328i	$-0.488326\pi$
$857$	1.35123e9	0.0733327	0.0366663	+	0.999328i	$0.488326\pi$
$858$	−9.97745e9	−0.539280
$859$	1.07334e10	0.577777	0.288888	−	0.957363i	$-0.406714\pi$
$859$	1.07334e10	0.577777	0.288888	+	0.957363i	$0.406714\pi$
$860$	−5.12395e9	−0.274701
$861$	−1.07904e10	−0.576138
$862$	−3.42267e9	−0.182007
$863$	−1.85843e10	−0.984257	−0.492129	−	0.870523i	$-0.663781\pi$
$863$	−1.85843e10	−0.984257	−0.492129	+	0.870523i	$0.663781\pi$
$864$	−2.84136e9	−0.149875
$865$	−2.61243e10	−1.37242
$866$	1.24891e10	0.653457
$867$	2.29153e9	0.119415
$868$	−1.16248e9	−0.0603346
$869$	2.00126e10	1.03451
$870$	−1.47219e10	−0.757961
$871$	−2.40254e9	−0.123199
$872$	2.40388e9	0.122774
$873$	1.22068e10	0.620941
$874$	3.00539e10	1.52269
$875$	−4.43064e10	−2.23583
$876$	5.59666e9	0.281297
$877$	−1.63802e10	−0.820012	−0.410006	−	0.912083i	$-0.634473\pi$
$877$	−1.63802e10	−0.820012	−0.410006	+	0.912083i	$0.634473\pi$
$878$	1.30726e10	0.651823
$879$	−1.14530e8	−0.00568796
$880$	−9.72101e9	−0.480864
$881$	2.64570e10	1.30354	0.651771	−	0.758416i	$-0.274026\pi$
$881$	2.64570e10	1.30354	0.651771	+	0.758416i	$0.274026\pi$
$882$	3.66686e8	0.0179951
$883$	1.20676e10	0.589874	0.294937	−	0.955517i	$-0.404701\pi$
$883$	1.20676e10	0.589874	0.294937	+	0.955517i	$0.404701\pi$
$884$	1.71624e10	0.835593
$885$	−1.50923e10	−0.731904
$886$	−4.96885e9	−0.240015
$887$	5.29007e9	0.254524	0.127262	−	0.991869i	$-0.459381\pi$
$887$	5.29007e9	0.254524	0.127262	+	0.991869i	$0.459381\pi$
$888$	−5.80688e8	−0.0278290
$889$	−1.84661e10	−0.881492
$890$	−2.62546e10	−1.24836
$891$	8.20697e9	0.388697
$892$	−1.20699e10	−0.569414
$893$	−2.42632e10	−1.14017
$894$	3.84186e9	0.179829
$895$	−1.71865e10	−0.801323
$896$	1.87147e9	0.0869169
$897$	1.93062e10	0.893147
$898$	1.73430e9	0.0799204
$899$	−3.31510e9	−0.152173
$900$	−1.90428e10	−0.870726
$901$	−3.44697e10	−1.57000
$902$	−2.03187e10	−0.921877
$903$	3.17016e9	0.143276
$904$	8.08475e9	0.363980
$905$	−4.74282e10	−2.12700
$906$	−1.36469e10	−0.609656
$907$	4.35182e9	0.193663	0.0968313	−	0.995301i	$-0.469129\pi$
$907$	4.35182e9	0.193663	0.0968313	+	0.995301i	$0.469129\pi$
$908$	2.00240e10	0.887667
$909$	2.45338e10	1.08341
$910$	4.26657e10	1.87687
$911$	−8.14092e9	−0.356746	−0.178373	−	0.983963i	$-0.557083\pi$
$911$	−8.14092e9	−0.356746	−0.178373	+	0.983963i	$0.557083\pi$
$912$	−4.73243e9	−0.206587
$913$	1.26659e10	0.550794
$914$	−2.87894e10	−1.24716
$915$	3.48189e10	1.50259
$916$	−1.37252e10	−0.590046
$917$	1.43055e10	0.612645
$918$	1.57069e10	0.670103
$919$	−4.36837e10	−1.85659	−0.928293	−	0.371850i	$-0.878723\pi$
$919$	−4.36837e10	−1.85659	−0.928293	+	0.371850i	$0.878723\pi$
$920$	1.88100e10	0.796400
$921$	9.04271e8	0.0381408
$922$	−3.55741e9	−0.149477
$923$	1.99780e10	0.836270
$924$	6.01435e9	0.250805
$925$	−8.94101e9	−0.371442
$926$	−8.68818e9	−0.359576
$927$	3.19645e10	1.31792
$928$	5.33695e9	0.219217
$929$	−7.35357e9	−0.300915	−0.150457	−	0.988617i	$-0.548075\pi$
$929$	−7.35357e9	−0.300915	−0.150457	+	0.988617i	$0.548075\pi$
$930$	1.83982e9	0.0750039
$931$	1.40311e9	0.0569860
$932$	4.91800e9	0.198991
$933$	−4.49285e9	−0.181107
$934$	2.88982e10	1.16053
$935$	5.37372e10	2.14998
$936$	1.02214e10	0.407424
$937$	−1.44717e10	−0.574688	−0.287344	−	0.957827i	$-0.592772\pi$
$937$	−1.44717e10	−0.574688	−0.287344	+	0.957827i	$0.592772\pi$
$938$	1.44824e9	0.0572967
$939$	−1.14563e10	−0.451561
$940$	−1.51857e10	−0.596332
$941$	1.33141e9	0.0520894	0.0260447	−	0.999661i	$-0.491709\pi$
$941$	1.33141e9	0.0520894	0.0260447	+	0.999661i	$0.491709\pi$
$942$	8.94420e8	0.0348629
$943$	3.93163e10	1.52680
$944$	5.47122e9	0.211681
$945$	3.90474e10	1.50515
$946$	5.96952e9	0.229256
$947$	1.16271e10	0.444882	0.222441	−	0.974946i	$-0.428598\pi$
$947$	1.16271e10	0.444882	0.222441	+	0.974946i	$0.428598\pi$
$948$	6.09765e9	0.232452
$949$	−4.62546e10	−1.75680
$950$	−7.28665e10	−2.75737
$951$	−2.15257e10	−0.811568
$952$	−1.03454e10	−0.388613
$953$	9.52377e7	0.00356438	0.00178219	−	0.999998i	$-0.499433\pi$
$953$	9.52377e7	0.00356438	0.00178219	+	0.999998i	$0.499433\pi$
$954$	−2.05292e10	−0.765514
$955$	3.78511e10	1.40626
$956$	−1.25917e10	−0.466103
$957$	1.71514e10	0.632568
$958$	3.16725e10	1.16387
$959$	−3.41337e10	−1.24973
$960$	−2.96190e9	−0.108049
$961$	−2.70983e10	−0.984942
$962$	4.79920e9	0.173802
$963$	−1.88693e9	−0.0680871
$964$	8.32249e9	0.299215
$965$	2.48006e10	0.888417
$966$	−1.16377e10	−0.415380
$967$	4.10715e10	1.46066	0.730328	−	0.683096i	$-0.239367\pi$
$967$	4.10715e10	1.46066	0.730328	+	0.683096i	$0.239367\pi$
$968$	1.34778e9	0.0477590
$969$	2.61606e10	0.923666
$970$	2.92338e10	1.02845
$971$	1.43880e10	0.504350	0.252175	−	0.967682i	$-0.418854\pi$
$971$	1.43880e10	0.504350	0.252175	+	0.967682i	$0.418854\pi$
$972$	1.46374e10	0.511249
$973$	2.31698e10	0.806358
$974$	3.44019e10	1.19296
$975$	−4.68083e10	−1.61736
$976$	−1.26225e10	−0.434580
$977$	6.76985e9	0.232246	0.116123	−	0.993235i	$-0.462953\pi$
$977$	6.76985e9	0.232246	0.116123	+	0.993235i	$0.462953\pi$
$978$	−1.35002e10	−0.461482
$979$	3.05873e10	1.04184
$980$	8.78170e8	0.0298049
$981$	−7.91428e9	−0.267652
$982$	−2.03764e10	−0.686651
$983$	4.81556e10	1.61700	0.808498	−	0.588498i	$-0.200281\pi$
$983$	4.81556e10	1.61700	0.808498	+	0.588498i	$0.200281\pi$
$984$	−6.19092e9	−0.207144
$985$	6.72905e10	2.24350
$986$	−2.95024e10	−0.980139
$987$	9.39534e9	0.311030
$988$	3.91120e10	1.29021
$989$	−1.15509e10	−0.379690
$990$	3.20045e10	1.04830
$991$	1.81773e10	0.593296	0.296648	−	0.954987i	$-0.404131\pi$
$991$	1.81773e10	0.593296	0.296648	+	0.954987i	$0.404131\pi$
$992$	−6.66965e8	−0.0216926
$993$	−9.03241e9	−0.292740
$994$	−1.20426e10	−0.388928
$995$	4.21335e10	1.35596
$996$	3.85920e9	0.123763
$997$	3.32870e10	1.06376	0.531878	−	0.846821i	$-0.321487\pi$
$997$	3.32870e10	1.06376	0.531878	+	0.846821i	$0.321487\pi$
$998$	−2.50815e10	−0.798723
$999$	4.39220e9	0.139381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.8.a.b.1.4	✓	4
4.3	odd	2		592.8.a.a.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.8.a.b.1.4	✓	4	1.1	even	1	trivial
592.8.a.a.1.1		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	74.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +22.3907 q^{3} +64.0000 q^{4} -504.619 q^{5} +179.126 q^{6} +892.385 q^{7} +512.000 q^{8} -1685.66 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +22.3907 q^{3} +64.0000 q^{4} -504.619 q^{5} +179.126 q^{6} +892.385 q^{7} +512.000 q^{8} -1685.66 q^{9} -4036.95 q^{10} +4703.14 q^{11} +1433.01 q^{12} -11843.3 q^{13} +7139.08 q^{14} -11298.8 q^{15} +4096.00 q^{16} -22642.5 q^{17} -13485.2 q^{18} -51600.8 q^{19} -32295.6 q^{20} +19981.1 q^{21} +37625.1 q^{22} -72804.0 q^{23} +11464.0 q^{24} +176515. q^{25} -94746.6 q^{26} -86711.5 q^{27} +57112.7 q^{28} +162871. q^{29} -90390.2 q^{30} -20354.2 q^{31} +32768.0 q^{32} +105307. q^{33} -181140. q^{34} -450314. q^{35} -107882. q^{36} -50653.0 q^{37} -412806. q^{38} -265180. q^{39} -258365. q^{40} -540030. q^{41} +159849. q^{42} +158658. q^{43} +301001. q^{44} +850613. q^{45} -582432. q^{46} +470210. q^{47} +91712.4 q^{48} -27191.7 q^{49} +1.41212e6 q^{50} -506981. q^{51} -757973. q^{52} +1.52235e6 q^{53} -693692. q^{54} -2.37329e6 q^{55} +456901. q^{56} -1.15538e6 q^{57} +1.30297e6 q^{58} +1.33575e6 q^{59} -723121. q^{60} -3.08165e6 q^{61} -162833. q^{62} -1.50425e6 q^{63} +262144. q^{64} +5.97636e6 q^{65} +842454. q^{66} +202860. q^{67} -1.44912e6 q^{68} -1.63013e6 q^{69} -3.60251e6 q^{70} -1.68686e6 q^{71} -863056. q^{72} +3.90554e6 q^{73} -405224. q^{74} +3.95230e6 q^{75} -3.30245e6 q^{76} +4.19702e6 q^{77} -2.12144e6 q^{78} +4.25515e6 q^{79} -2.06692e6 q^{80} +1.74500e6 q^{81} -4.32024e6 q^{82} +2.69308e6 q^{83} +1.27879e6 q^{84} +1.14258e7 q^{85} +1.26926e6 q^{86} +3.64679e6 q^{87} +2.40801e6 q^{88} +6.50359e6 q^{89} +6.80491e6 q^{90} -1.05688e7 q^{91} -4.65946e6 q^{92} -455744. q^{93} +3.76168e6 q^{94} +2.60387e7 q^{95} +733699. q^{96} -7.24155e6 q^{97} -217533. q^{98} -7.92788e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} - 41 q^{3} + 256 q^{4} - 363 q^{5} - 328 q^{6} - 774 q^{7} + 2048 q^{8} - 1079 q^{9}+O(q^{10}) \) 4 * q + 32 * q^2 - 41 * q^3 + 256 * q^4 - 363 * q^5 - 328 * q^6 - 774 * q^7 + 2048 * q^8 - 1079 * q^9	\( 4 q + 32 q^{2} - 41 q^{3} + 256 q^{4} - 363 q^{5} - 328 q^{6} - 774 q^{7} + 2048 q^{8} - 1079 q^{9} - 2904 q^{10} - 309 q^{11} - 2624 q^{12} - 20827 q^{13} - 6192 q^{14} - 22940 q^{15} + 16384 q^{16} - 48756 q^{17} - 8632 q^{18} - 69068 q^{19} - 23232 q^{20} - 640 q^{21} - 2472 q^{22} - 50237 q^{23} - 20992 q^{24} - 3581 q^{25} - 166616 q^{26} - 368414 q^{27} - 49536 q^{28} - 205195 q^{29} - 183520 q^{30} - 172283 q^{31} + 131072 q^{32} - 205234 q^{33} - 390048 q^{34} - 584964 q^{35} - 69056 q^{36} - 202612 q^{37} - 552544 q^{38} + 329055 q^{39} - 185856 q^{40} - 1018945 q^{41} - 5120 q^{42} + 1263046 q^{43} - 19776 q^{44} + 1279606 q^{45} - 401896 q^{46} - 420930 q^{47} - 167936 q^{48} - 482790 q^{49} - 28648 q^{50} + 728262 q^{51} - 1332928 q^{52} + 2051230 q^{53} - 2947312 q^{54} - 1442891 q^{55} - 396288 q^{56} - 926198 q^{57} - 1641560 q^{58} + 357914 q^{59} - 1468160 q^{60} - 2507513 q^{61} - 1378264 q^{62} + 2879054 q^{63} + 1048576 q^{64} + 3097954 q^{65} - 1641872 q^{66} + 586879 q^{67} - 3120384 q^{68} - 252895 q^{69} - 4679712 q^{70} - 130272 q^{71} - 552448 q^{72} + 3517417 q^{73} - 1620896 q^{74} + 8154290 q^{75} - 4420352 q^{76} + 8777590 q^{77} + 2632440 q^{78} + 3790171 q^{79} - 1486848 q^{80} + 15888376 q^{81} - 8151560 q^{82} + 12973460 q^{83} - 40960 q^{84} + 14870322 q^{85} + 10104368 q^{86} + 23063695 q^{87} - 158208 q^{88} + 18852848 q^{89} + 10236848 q^{90} - 2046622 q^{91} - 3215168 q^{92} + 4005314 q^{93} - 3367440 q^{94} + 30367150 q^{95} - 1343488 q^{96} + 14580104 q^{97} - 3862320 q^{98} + 16949258 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 - 41 * q^3 + 256 * q^4 - 363 * q^5 - 328 * q^6 - 774 * q^7 + 2048 * q^8 - 1079 * q^9 - 2904 * q^10 - 309 * q^11 - 2624 * q^12 - 20827 * q^13 - 6192 * q^14 - 22940 * q^15 + 16384 * q^16 - 48756 * q^17 - 8632 * q^18 - 69068 * q^19 - 23232 * q^20 - 640 * q^21 - 2472 * q^22 - 50237 * q^23 - 20992 * q^24 - 3581 * q^25 - 166616 * q^26 - 368414 * q^27 - 49536 * q^28 - 205195 * q^29 - 183520 * q^30 - 172283 * q^31 + 131072 * q^32 - 205234 * q^33 - 390048 * q^34 - 584964 * q^35 - 69056 * q^36 - 202612 * q^37 - 552544 * q^38 + 329055 * q^39 - 185856 * q^40 - 1018945 * q^41 - 5120 * q^42 + 1263046 * q^43 - 19776 * q^44 + 1279606 * q^45 - 401896 * q^46 - 420930 * q^47 - 167936 * q^48 - 482790 * q^49 - 28648 * q^50 + 728262 * q^51 - 1332928 * q^52 + 2051230 * q^53 - 2947312 * q^54 - 1442891 * q^55 - 396288 * q^56 - 926198 * q^57 - 1641560 * q^58 + 357914 * q^59 - 1468160 * q^60 - 2507513 * q^61 - 1378264 * q^62 + 2879054 * q^63 + 1048576 * q^64 + 3097954 * q^65 - 1641872 * q^66 + 586879 * q^67 - 3120384 * q^68 - 252895 * q^69 - 4679712 * q^70 - 130272 * q^71 - 552448 * q^72 + 3517417 * q^73 - 1620896 * q^74 + 8154290 * q^75 - 4420352 * q^76 + 8777590 * q^77 + 2632440 * q^78 + 3790171 * q^79 - 1486848 * q^80 + 15888376 * q^81 - 8151560 * q^82 + 12973460 * q^83 - 40960 * q^84 + 14870322 * q^85 + 10104368 * q^86 + 23063695 * q^87 - 158208 * q^88 + 18852848 * q^89 + 10236848 * q^90 - 2046622 * q^91 - 3215168 * q^92 + 4005314 * q^93 - 3367440 * q^94 + 30367150 * q^95 - 1343488 * q^96 + 14580104 * q^97 - 3862320 * q^98 + 16949258 * q^99

Introduction

L-functions

Modular forms

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