LMFDB - Embedded newform 74.8.a.a.1.3

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} - 405x^{2} - 2998x - 4396 $ x^4 - 2x^3 - 405x^2 - 2998*x - 4396
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.83090$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} +20.4940 q^{3} +64.0000 q^{4} +375.302 q^{5} -163.952 q^{6} -980.897 q^{7} -512.000 q^{8} -1766.99 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +20.4940 q^{3} +64.0000 q^{4} +375.302 q^{5} -163.952 q^{6} -980.897 q^{7} -512.000 q^{8} -1766.99 q^{9} -3002.41 q^{10} -3376.51 q^{11} +1311.62 q^{12} +3436.44 q^{13} +7847.18 q^{14} +7691.45 q^{15} +4096.00 q^{16} -4744.28 q^{17} +14136.0 q^{18} -11268.9 q^{19} +24019.3 q^{20} -20102.6 q^{21} +27012.1 q^{22} -17737.1 q^{23} -10492.9 q^{24} +62726.4 q^{25} -27491.5 q^{26} -81033.3 q^{27} -62777.4 q^{28} -106898. q^{29} -61531.6 q^{30} -31372.3 q^{31} -32768.0 q^{32} -69198.3 q^{33} +37954.2 q^{34} -368132. q^{35} -113088. q^{36} +50653.0 q^{37} +90151.3 q^{38} +70426.5 q^{39} -192154. q^{40} +278656. q^{41} +160820. q^{42} -606906. q^{43} -216097. q^{44} -663156. q^{45} +141897. q^{46} -1.13131e6 q^{47} +83943.6 q^{48} +138617. q^{49} -501811. q^{50} -97229.5 q^{51} +219932. q^{52} -894201. q^{53} +648267. q^{54} -1.26721e6 q^{55} +502219. q^{56} -230946. q^{57} +855188. q^{58} -897569. q^{59} +492253. q^{60} +1.69191e6 q^{61} +250979. q^{62} +1.73324e6 q^{63} +262144. q^{64} +1.28970e6 q^{65} +553587. q^{66} +2.02846e6 q^{67} -303634. q^{68} -363504. q^{69} +2.94506e6 q^{70} -585796. q^{71} +904701. q^{72} +428683. q^{73} -405224. q^{74} +1.28552e6 q^{75} -721211. q^{76} +3.31201e6 q^{77} -563412. q^{78} -1.57559e6 q^{79} +1.53724e6 q^{80} +2.20372e6 q^{81} -2.22925e6 q^{82} -537839. q^{83} -1.28656e6 q^{84} -1.78054e6 q^{85} +4.85525e6 q^{86} -2.19078e6 q^{87} +1.72877e6 q^{88} -1.84127e6 q^{89} +5.30525e6 q^{90} -3.37079e6 q^{91} -1.13517e6 q^{92} -642946. q^{93} +9.05050e6 q^{94} -4.22924e6 q^{95} -671549. q^{96} -6.63828e6 q^{97} -1.10893e6 q^{98} +5.96627e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9}+O(q^{10}) $ 4 * q - 32 * q^2 - 53 * q^3 + 256 * q^4 + 111 * q^5 + 424 * q^6 - 1666 * q^7 - 2048 * q^8 + 4609 * q^9	\( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9} - 888 q^{10} - 4593 q^{11} - 3392 q^{12} + 7847 q^{13} + 13328 q^{14} + 18900 q^{15} + 16384 q^{16} + 23172 q^{17} - 36872 q^{18} + 23696 q^{19} + 7104 q^{20} + 69416 q^{21} + 36744 q^{22} + 24105 q^{23} + 27136 q^{24} - 138149 q^{25} - 62776 q^{26} - 433646 q^{27} - 106624 q^{28} - 140949 q^{29} - 151200 q^{30} - 664609 q^{31} - 131072 q^{32} - 240450 q^{33} - 185376 q^{34} - 248544 q^{35} + 294976 q^{36} + 202612 q^{37} - 189568 q^{38} - 2288827 q^{39} - 56832 q^{40} - 709737 q^{41} - 555328 q^{42} - 128962 q^{43} - 293952 q^{44} - 1755342 q^{45} - 192840 q^{46} - 445842 q^{47} - 217088 q^{48} - 1602774 q^{49} + 1105192 q^{50} - 2883630 q^{51} + 502208 q^{52} - 975870 q^{53} + 3469168 q^{54} - 644145 q^{55} + 852992 q^{56} + 3494630 q^{57} + 1127592 q^{58} - 1812858 q^{59} + 1209600 q^{60} - 2955031 q^{61} + 5316872 q^{62} - 3362482 q^{63} + 1048576 q^{64} + 666 q^{65} + 1923600 q^{66} + 2737235 q^{67} + 1483008 q^{68} - 1781673 q^{69} + 1988352 q^{70} + 4958184 q^{71} - 2359808 q^{72} - 931591 q^{73} - 1620896 q^{74} + 4945810 q^{75} + 1516544 q^{76} + 4352514 q^{77} + 18310616 q^{78} + 5813561 q^{79} + 454656 q^{80} + 16394896 q^{81} + 5677896 q^{82} + 2120460 q^{83} + 4442624 q^{84} - 4845402 q^{85} + 1031696 q^{86} + 7965333 q^{87} + 2351616 q^{88} + 8833716 q^{89} + 14042736 q^{90} - 18886274 q^{91} + 1542720 q^{92} + 3024182 q^{93} + 3566736 q^{94} - 3151794 q^{95} + 1736704 q^{96} - 22666876 q^{97} + 12822192 q^{98} - 17931894 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 53 * q^3 + 256 * q^4 + 111 * q^5 + 424 * q^6 - 1666 * q^7 - 2048 * q^8 + 4609 * q^9 - 888 * q^10 - 4593 * q^11 - 3392 * q^12 + 7847 * q^13 + 13328 * q^14 + 18900 * q^15 + 16384 * q^16 + 23172 * q^17 - 36872 * q^18 + 23696 * q^19 + 7104 * q^20 + 69416 * q^21 + 36744 * q^22 + 24105 * q^23 + 27136 * q^24 - 138149 * q^25 - 62776 * q^26 - 433646 * q^27 - 106624 * q^28 - 140949 * q^29 - 151200 * q^30 - 664609 * q^31 - 131072 * q^32 - 240450 * q^33 - 185376 * q^34 - 248544 * q^35 + 294976 * q^36 + 202612 * q^37 - 189568 * q^38 - 2288827 * q^39 - 56832 * q^40 - 709737 * q^41 - 555328 * q^42 - 128962 * q^43 - 293952 * q^44 - 1755342 * q^45 - 192840 * q^46 - 445842 * q^47 - 217088 * q^48 - 1602774 * q^49 + 1105192 * q^50 - 2883630 * q^51 + 502208 * q^52 - 975870 * q^53 + 3469168 * q^54 - 644145 * q^55 + 852992 * q^56 + 3494630 * q^57 + 1127592 * q^58 - 1812858 * q^59 + 1209600 * q^60 - 2955031 * q^61 + 5316872 * q^62 - 3362482 * q^63 + 1048576 * q^64 + 666 * q^65 + 1923600 * q^66 + 2737235 * q^67 + 1483008 * q^68 - 1781673 * q^69 + 1988352 * q^70 + 4958184 * q^71 - 2359808 * q^72 - 931591 * q^73 - 1620896 * q^74 + 4945810 * q^75 + 1516544 * q^76 + 4352514 * q^77 + 18310616 * q^78 + 5813561 * q^79 + 454656 * q^80 + 16394896 * q^81 + 5677896 * q^82 + 2120460 * q^83 + 4442624 * q^84 - 4845402 * q^85 + 1031696 * q^86 + 7965333 * q^87 + 2351616 * q^88 + 8833716 * q^89 + 14042736 * q^90 - 18886274 * q^91 + 1542720 * q^92 + 3024182 * q^93 + 3566736 * q^94 - 3151794 * q^95 + 1736704 * q^96 - 22666876 * q^97 + 12822192 * q^98 - 17931894 * 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
$3$	20.4940	0.438231	0.219116	−	0.975699i	$-0.429683\pi$
$3$	20.4940	0.438231	0.219116	+	0.975699i	$0.429683\pi$
$4$	64.0000	0.500000
$5$	375.302	1.34272	0.671360	−	0.741131i	$-0.265710\pi$
$5$	375.302	1.34272	0.671360	+	0.741131i	$0.265710\pi$
$6$	−163.952	−0.309876
$7$	−980.897	−1.08089	−0.540444	−	0.841380i	$-0.681744\pi$
$7$	−980.897	−1.08089	−0.540444	+	0.841380i	$0.681744\pi$
$8$	−512.000	−0.353553
$9$	−1766.99	−0.807953
$10$	−3002.41	−0.949447
$11$	−3376.51	−0.764881	−0.382440	−	0.923980i	$-0.624916\pi$
$11$	−3376.51	−0.764881	−0.382440	+	0.923980i	$0.624916\pi$
$12$	1311.62	0.219116
$13$	3436.44	0.433817	0.216909	−	0.976192i	$-0.430403\pi$
$13$	3436.44	0.433817	0.216909	+	0.976192i	$0.430403\pi$
$14$	7847.18	0.764303
$15$	7691.45	0.588422
$16$	4096.00	0.250000
$17$	−4744.28	−0.234207	−0.117103	−	0.993120i	$-0.537361\pi$
$17$	−4744.28	−0.234207	−0.117103	+	0.993120i	$0.537361\pi$
$18$	14136.0	0.571309
$19$	−11268.9	−0.376916	−0.188458	−	0.982081i	$-0.560349\pi$
$19$	−11268.9	−0.376916	−0.188458	+	0.982081i	$0.560349\pi$
$20$	24019.3	0.671360
$21$	−20102.6	−0.473678
$22$	27012.1	0.540852
$23$	−17737.1	−0.303973	−0.151986	−	0.988383i	$-0.548567\pi$
$23$	−17737.1	−0.303973	−0.151986	+	0.988383i	$0.548567\pi$
$24$	−10492.9	−0.154938
$25$	62726.4	0.802898
$26$	−27491.5	−0.306755
$27$	−81033.3	−0.792302
$28$	−62777.4	−0.540444
$29$	−106898.	−0.813914	−0.406957	−	0.913447i	$-0.633410\pi$
$29$	−106898.	−0.813914	−0.406957	+	0.913447i	$0.633410\pi$
$30$	−61531.6	−0.416077
$31$	−31372.3	−0.189139	−0.0945695	−	0.995518i	$-0.530147\pi$
$31$	−31372.3	−0.189139	−0.0945695	+	0.995518i	$0.530147\pi$
$32$	−32768.0	−0.176777
$33$	−69198.3	−0.335194
$34$	37954.2	0.165609
$35$	−368132.	−1.45133
$36$	−113088.	−0.403977
$37$	50653.0	0.164399
$37$	50653.0	0.164399
$38$	90151.3	0.266520
$39$	70426.5	0.190112
$40$	−192154.	−0.474723
$41$	278656.	0.631430	0.315715	−	0.948854i	$-0.397756\pi$
$41$	278656.	0.631430	0.315715	+	0.948854i	$0.397756\pi$
$42$	160820.	0.334941
$43$	−606906.	−1.16408	−0.582039	−	0.813161i	$-0.697745\pi$
$43$	−606906.	−1.16408	−0.582039	+	0.813161i	$0.697745\pi$
$44$	−216097.	−0.382440
$45$	−663156.	−1.08486
$46$	141897.	0.214941
$47$	−1.13131e6	−1.58943	−0.794713	−	0.606986i	$-0.792379\pi$
$47$	−1.13131e6	−1.58943	−0.794713	+	0.606986i	$0.792379\pi$
$48$	83943.6	0.109558
$49$	138617.	0.168317
$50$	−501811.	−0.567735
$51$	−97229.5	−0.102637
$52$	219932.	0.216909
$53$	−894201.	−0.825030	−0.412515	−	0.910951i	$-0.635350\pi$
$53$	−894201.	−0.825030	−0.412515	+	0.910951i	$0.635350\pi$
$54$	648267.	0.560242
$55$	−1.26721e6	−1.02702
$56$	502219.	0.382151
$57$	−230946.	−0.165176
$58$	855188.	0.575524
$59$	−897569.	−0.568966	−0.284483	−	0.958681i	$-0.591822\pi$
$59$	−897569.	−0.568966	−0.284483	+	0.958681i	$0.591822\pi$
$60$	492253.	0.294211
$61$	1.69191e6	0.954381	0.477191	−	0.878800i	$-0.341655\pi$
$61$	1.69191e6	0.954381	0.477191	+	0.878800i	$0.341655\pi$
$62$	250979.	0.133741
$63$	1.73324e6	0.873307
$64$	262144.	0.125000
$65$	1.28970e6	0.582495
$66$	553587.	0.237018
$67$	2.02846e6	0.823957	0.411979	−	0.911194i	$-0.364838\pi$
$67$	2.02846e6	0.823957	0.411979	+	0.911194i	$0.364838\pi$
$68$	−303634.	−0.117103
$69$	−363504.	−0.133210
$70$	2.94506e6	1.02624
$71$	−585796.	−0.194242	−0.0971208	−	0.995273i	$-0.530963\pi$
$71$	−585796.	−0.194242	−0.0971208	+	0.995273i	$0.530963\pi$
$72$	904701.	0.285655
$73$	428683.	0.128975	0.0644875	−	0.997919i	$-0.479459\pi$
$73$	428683.	0.128975	0.0644875	+	0.997919i	$0.479459\pi$
$74$	−405224.	−0.116248
$75$	1.28552e6	0.351855
$76$	−721211.	−0.188458
$77$	3.31201e6	0.826750
$78$	−563412.	−0.134430
$79$	−1.57559e6	−0.359540	−0.179770	−	0.983709i	$-0.557535\pi$
$79$	−1.57559e6	−0.359540	−0.179770	+	0.983709i	$0.557535\pi$
$80$	1.53724e6	0.335680
$81$	2.20372e6	0.460742
$82$	−2.22925e6	−0.446488
$83$	−537839.	−0.103247	−0.0516236	−	0.998667i	$-0.516440\pi$
$83$	−537839.	−0.103247	−0.0516236	+	0.998667i	$0.516440\pi$
$84$	−1.28656e6	−0.236839
$85$	−1.78054e6	−0.314474
$86$	4.85525e6	0.823127
$87$	−2.19078e6	−0.356682
$88$	1.72877e6	0.270426
$89$	−1.84127e6	−0.276856	−0.138428	−	0.990373i	$-0.544205\pi$
$89$	−1.84127e6	−0.276856	−0.138428	+	0.990373i	$0.544205\pi$
$90$	5.30525e6	0.767109
$91$	−3.37079e6	−0.468907
$92$	−1.13517e6	−0.151986
$93$	−642946.	−0.0828866
$94$	9.05050e6	1.12389
$95$	−4.22924e6	−0.506093
$96$	−671549.	−0.0774691
$97$	−6.63828e6	−0.738506	−0.369253	−	0.929329i	$-0.620387\pi$
$97$	−6.63828e6	−0.738506	−0.369253	+	0.929329i	$0.620387\pi$
$98$	−1.10893e6	−0.119018
$99$	5.96627e6	0.617988
$100$	4.01449e6	0.401449
$101$	2.01181e6	0.194295	0.0971475	−	0.995270i	$-0.469028\pi$
$101$	2.01181e6	0.194295	0.0971475	+	0.995270i	$0.469028\pi$
$102$	777836.	0.0725750
$103$	−8.44464e6	−0.761466	−0.380733	−	0.924685i	$-0.624328\pi$
$103$	−8.44464e6	−0.761466	−0.380733	+	0.924685i	$0.624328\pi$
$104$	−1.75946e6	−0.153378
$105$	−7.54452e6	−0.636018
$106$	7.15361e6	0.583384
$107$	2.02730e7	1.59984	0.799919	−	0.600108i	$-0.204876\pi$
$107$	2.02730e7	1.59984	0.799919	+	0.600108i	$0.204876\pi$
$108$	−5.18613e6	−0.396151
$109$	−2.49898e7	−1.84829	−0.924145	−	0.382042i	$-0.875221\pi$
$109$	−2.49898e7	−1.84829	−0.924145	+	0.382042i	$0.875221\pi$
$110$	1.01377e7	0.726213
$111$	1.03808e6	0.0720448
$112$	−4.01776e6	−0.270222
$113$	1.39234e7	0.907760	0.453880	−	0.891063i	$-0.350039\pi$
$113$	1.39234e7	0.907760	0.453880	+	0.891063i	$0.350039\pi$
$114$	1.84756e6	0.116797
$115$	−6.65675e6	−0.408150
$116$	−6.84150e6	−0.406957
$117$	−6.07217e6	−0.350504
$118$	7.18056e6	0.402319
$119$	4.65365e6	0.253151
$120$	−3.93802e6	−0.208039
$121$	−8.08635e6	−0.414958
$122$	−1.35352e7	−0.674849
$123$	5.71079e6	0.276712
$124$	−2.00783e6	−0.0945695
$125$	−5.77912e6	−0.264653
$126$	−1.38659e7	−0.617521
$127$	3.66831e6	0.158911	0.0794553	−	0.996838i	$-0.474682\pi$
$127$	3.66831e6	0.158911	0.0794553	+	0.996838i	$0.474682\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.24380e7	−0.510135
$130$	−1.03176e7	−0.411886
$131$	4.68710e7	1.82161	0.910803	−	0.412842i	$-0.135464\pi$
$131$	4.68710e7	1.82161	0.910803	+	0.412842i	$0.135464\pi$
$132$	−4.42869e6	−0.167597
$133$	1.10536e7	0.407404
$134$	−1.62277e7	−0.582626
$135$	−3.04119e7	−1.06384
$136$	2.42907e6	0.0828045
$137$	1.68178e7	0.558787	0.279393	−	0.960177i	$-0.409867\pi$
$137$	1.68178e7	0.558787	0.279393	+	0.960177i	$0.409867\pi$
$138$	2.90803e6	0.0941939
$139$	1.17238e7	0.370270	0.185135	−	0.982713i	$-0.440728\pi$
$139$	1.17238e7	0.370270	0.185135	+	0.982713i	$0.440728\pi$
$140$	−2.35605e7	−0.725665
$141$	−2.31852e7	−0.696536
$142$	4.68637e6	0.137350
$143$	−1.16032e7	−0.331818
$144$	−7.23761e6	−0.201988
$145$	−4.01192e7	−1.09286
$146$	−3.42946e6	−0.0911992
$147$	2.84081e6	0.0737619
$148$	3.24179e6	0.0821995
$149$	7.64948e7	1.89444	0.947219	−	0.320588i	$-0.103881\pi$
$149$	7.64948e7	1.89444	0.947219	+	0.320588i	$0.103881\pi$
$150$	−1.02841e7	−0.248799
$151$	−2.19120e7	−0.517921	−0.258960	−	0.965888i	$-0.583380\pi$
$151$	−2.19120e7	−0.517921	−0.258960	+	0.965888i	$0.583380\pi$
$152$	5.76968e6	0.133260
$153$	8.38311e6	0.189228
$154$	−2.64961e7	−0.584600
$155$	−1.17741e7	−0.253961
$156$	4.50730e6	0.0950561
$157$	1.91872e6	0.0395696	0.0197848	−	0.999804i	$-0.493702\pi$
$157$	1.91872e6	0.0395696	0.0197848	+	0.999804i	$0.493702\pi$
$158$	1.26047e7	0.254233
$159$	−1.83258e7	−0.361554
$160$	−1.22979e7	−0.237362
$161$	1.73982e7	0.328560
$162$	−1.76297e7	−0.325794
$163$	−2.21304e7	−0.400251	−0.200126	−	0.979770i	$-0.564135\pi$
$163$	−2.21304e7	−0.400251	−0.200126	+	0.979770i	$0.564135\pi$
$164$	1.78340e7	0.315715
$165$	−2.59703e7	−0.450072
$166$	4.30271e6	0.0730068
$167$	4.43183e7	0.736334	0.368167	−	0.929760i	$-0.379985\pi$
$167$	4.43183e7	0.736334	0.368167	+	0.929760i	$0.379985\pi$
$168$	1.02925e7	0.167471
$169$	−5.09394e7	−0.811803
$170$	1.42443e7	0.222367
$171$	1.99121e7	0.304531
$172$	−3.88420e7	−0.582039
$173$	7.47705e7	1.09792	0.548958	−	0.835850i	$-0.315025\pi$
$173$	7.47705e7	1.09792	0.548958	+	0.835850i	$0.315025\pi$
$174$	1.75262e7	0.252213
$175$	−6.15282e7	−0.867842
$176$	−1.38302e7	−0.191220
$177$	−1.83948e7	−0.249338
$178$	1.47302e7	0.195767
$179$	−1.20143e8	−1.56571	−0.782855	−	0.622204i	$-0.786237\pi$
$179$	−1.20143e8	−1.56571	−0.782855	+	0.622204i	$0.786237\pi$
$180$	−4.24420e7	−0.542428
$181$	1.49404e8	1.87278	0.936390	−	0.350962i	$-0.114146\pi$
$181$	1.49404e8	1.87278	0.936390	+	0.350962i	$0.114146\pi$
$182$	2.69663e7	0.331568
$183$	3.46740e7	0.418240
$184$	9.08138e6	0.107471
$185$	1.90102e7	0.220742
$186$	5.14357e6	0.0586096
$187$	1.60191e7	0.179140
$188$	−7.24040e7	−0.794713
$189$	7.94854e7	0.856389
$190$	3.38339e7	0.357862
$191$	−2.34648e7	−0.243669	−0.121835	−	0.992550i	$-0.538878\pi$
$191$	−2.34648e7	−0.243669	−0.121835	+	0.992550i	$0.538878\pi$
$192$	5.37239e6	0.0547789
$193$	1.01584e8	1.01713	0.508565	−	0.861024i	$-0.330176\pi$
$193$	1.01584e8	1.01713	0.508565	+	0.861024i	$0.330176\pi$
$194$	5.31062e7	0.522203
$195$	2.64312e7	0.255268
$196$	8.87146e6	0.0841587
$197$	1.85514e8	1.72880	0.864402	−	0.502801i	$-0.167697\pi$
$197$	1.85514e8	1.72880	0.864402	+	0.502801i	$0.167697\pi$
$198$	−4.77302e7	−0.436983
$199$	−2.86267e7	−0.257505	−0.128752	−	0.991677i	$-0.541097\pi$
$199$	−2.86267e7	−0.257505	−0.128752	+	0.991677i	$0.541097\pi$
$200$	−3.21159e7	−0.283867
$201$	4.15713e7	0.361084
$202$	−1.60945e7	−0.137387
$203$	1.04856e8	0.879749
$204$	−6.22269e6	−0.0513183
$205$	1.04580e8	0.847834
$206$	6.75571e7	0.538438
$207$	3.13413e7	0.245596
$208$	1.40756e7	0.108454
$209$	3.80496e7	0.288296
$210$	6.03562e7	0.449732
$211$	1.57443e7	0.115381	0.0576907	−	0.998335i	$-0.481626\pi$
$211$	1.57443e7	0.115381	0.0576907	+	0.998335i	$0.481626\pi$
$212$	−5.72289e7	−0.412515
$213$	−1.20053e7	−0.0851227
$214$	−1.62184e8	−1.13126
$215$	−2.27773e8	−1.56303
$216$	4.14891e7	0.280121
$217$	3.07730e7	0.204438
$218$	1.99918e8	1.30694
$219$	8.78544e6	0.0565209
$220$	−8.11014e7	−0.513510
$221$	−1.63034e7	−0.101603
$222$	−8.30468e6	−0.0509433
$223$	−3.23566e7	−0.195387	−0.0976937	−	0.995217i	$-0.531147\pi$
$223$	−3.23566e7	−0.195387	−0.0976937	+	0.995217i	$0.531147\pi$
$224$	3.21420e7	0.191076
$225$	−1.10837e8	−0.648704
$226$	−1.11387e8	−0.641883
$227$	2.94511e8	1.67113	0.835566	−	0.549390i	$-0.185140\pi$
$227$	2.94511e8	1.67113	0.835566	+	0.549390i	$0.185140\pi$
$228$	−1.47805e7	−0.0825882
$229$	−2.66064e8	−1.46407	−0.732034	−	0.681268i	$-0.761429\pi$
$229$	−2.66064e8	−1.46407	−0.732034	+	0.681268i	$0.761429\pi$
$230$	5.32540e7	0.288606
$231$	6.78765e7	0.362307
$232$	5.47320e7	0.287762
$233$	−2.14468e8	−1.11075	−0.555377	−	0.831599i	$-0.687426\pi$
$233$	−2.14468e8	−1.11075	−0.555377	+	0.831599i	$0.687426\pi$
$234$	4.85773e7	0.247844
$235$	−4.24584e8	−2.13415
$236$	−5.74444e7	−0.284483
$237$	−3.22901e7	−0.157562
$238$	−3.72292e7	−0.179005
$239$	−3.30136e8	−1.56423	−0.782115	−	0.623134i	$-0.785859\pi$
$239$	−3.30136e8	−1.56423	−0.782115	+	0.623134i	$0.785859\pi$
$240$	3.15042e7	0.147105
$241$	3.56063e8	1.63858	0.819290	−	0.573379i	$-0.194368\pi$
$241$	3.56063e8	1.63858	0.819290	+	0.573379i	$0.194368\pi$
$242$	6.46908e7	0.293419
$243$	2.22383e8	0.994213
$244$	1.08282e8	0.477191
$245$	5.20230e7	0.226003
$246$	−4.56863e7	−0.195665
$247$	−3.87249e7	−0.163513
$248$	1.60626e7	0.0668707
$249$	−1.10225e7	−0.0452462
$250$	4.62330e7	0.187138
$251$	−6.90749e7	−0.275716	−0.137858	−	0.990452i	$-0.544022\pi$
$251$	−6.90749e7	−0.275716	−0.137858	+	0.990452i	$0.544022\pi$
$252$	1.10927e8	0.436653
$253$	5.98894e7	0.232503
$254$	−2.93465e7	−0.112367
$255$	−3.64904e7	−0.137812
$256$	1.67772e7	0.0625000
$257$	−1.82080e8	−0.669110	−0.334555	−	0.942376i	$-0.608586\pi$
$257$	−1.82080e8	−0.669110	−0.334555	+	0.942376i	$0.608586\pi$
$258$	9.95037e7	0.360720
$259$	−4.96854e7	−0.177697
$260$	8.25409e7	0.291248
$261$	1.88889e8	0.657604
$262$	−3.74968e8	−1.28807
$263$	−3.32466e8	−1.12694	−0.563471	−	0.826136i	$-0.690534\pi$
$263$	−3.32466e8	−1.12694	−0.563471	+	0.826136i	$0.690534\pi$
$264$	3.54295e7	0.118509
$265$	−3.35595e8	−1.10778
$266$	−8.84292e7	−0.288078
$267$	−3.77352e7	−0.121327
$268$	1.29821e8	0.411979
$269$	3.56196e8	1.11572	0.557862	−	0.829934i	$-0.311622\pi$
$269$	3.56196e8	1.11572	0.557862	+	0.829934i	$0.311622\pi$
$270$	2.43296e8	0.752248
$271$	−6.50894e8	−1.98663	−0.993316	−	0.115423i	$-0.963178\pi$
$271$	−6.50894e8	−1.98663	−0.993316	+	0.115423i	$0.963178\pi$
$272$	−1.94326e7	−0.0585516
$273$	−6.90812e7	−0.205490
$274$	−1.34542e8	−0.395122
$275$	−2.11796e8	−0.614121
$276$	−2.32643e7	−0.0666051
$277$	1.11215e8	0.314402	0.157201	−	0.987567i	$-0.449753\pi$
$277$	1.11215e8	0.314402	0.157201	+	0.987567i	$0.449753\pi$
$278$	−9.37908e7	−0.261820
$279$	5.54347e7	0.152815
$280$	1.88484e8	0.513122
$281$	−5.93017e7	−0.159439	−0.0797196	−	0.996817i	$-0.525403\pi$
$281$	−5.93017e7	−0.159439	−0.0797196	+	0.996817i	$0.525403\pi$
$282$	1.85481e8	0.492525
$283$	−6.90811e8	−1.81179	−0.905893	−	0.423508i	$-0.860799\pi$
$283$	−6.90811e8	−1.81179	−0.905893	+	0.423508i	$0.860799\pi$
$284$	−3.74909e7	−0.0971208
$285$	−8.66743e7	−0.221786
$286$	9.28253e7	0.234631
$287$	−2.73333e8	−0.682504
$288$	5.79009e7	0.142827
$289$	−3.87830e8	−0.945147
$290$	3.20953e8	0.772768
$291$	−1.36045e8	−0.323637
$292$	2.74357e7	0.0644875
$293$	−6.33150e8	−1.47052	−0.735258	−	0.677787i	$-0.762939\pi$
$293$	−6.33150e8	−1.47052	−0.735258	+	0.677787i	$0.762939\pi$
$294$	−2.27265e7	−0.0521575
$295$	−3.36859e8	−0.763962
$296$	−2.59343e7	−0.0581238
$297$	2.73610e8	0.606016
$298$	−6.11959e8	−1.33957
$299$	−6.09523e7	−0.131869
$300$	8.22731e7	0.175927
$301$	5.95313e8	1.25824
$302$	1.75296e8	0.366225
$303$	4.12301e7	0.0851461
$304$	−4.61575e7	−0.0942290
$305$	6.34975e8	1.28147
$306$	−6.70649e7	−0.133804
$307$	3.19674e8	0.630554	0.315277	−	0.949000i	$-0.397903\pi$
$307$	3.19674e8	0.630554	0.315277	+	0.949000i	$0.397903\pi$
$308$	2.11969e8	0.413375
$309$	−1.73065e8	−0.333698
$310$	9.41927e7	0.179577
$311$	9.05480e8	1.70694	0.853469	−	0.521144i	$-0.174495\pi$
$311$	9.05480e8	1.70694	0.853469	+	0.521144i	$0.174495\pi$
$312$	−3.60584e7	−0.0672148
$313$	4.16880e8	0.768433	0.384216	−	0.923243i	$-0.374472\pi$
$313$	4.16880e8	0.768433	0.384216	+	0.923243i	$0.374472\pi$
$314$	−1.53497e7	−0.0279800
$315$	6.50488e8	1.17261
$316$	−1.00838e8	−0.179770
$317$	2.87485e8	0.506883	0.253442	−	0.967351i	$-0.418437\pi$
$317$	2.87485e8	0.506883	0.253442	+	0.967351i	$0.418437\pi$
$318$	1.46606e8	0.255657
$319$	3.60944e8	0.622547
$320$	9.83831e7	0.167840
$321$	4.15477e8	0.701099
$322$	−1.39186e8	−0.232327
$323$	5.34629e7	0.0882762
$324$	1.41038e8	0.230371
$325$	2.15555e8	0.348311
$326$	1.77043e8	0.283021
$327$	−5.12142e8	−0.809978
$328$	−1.42672e8	−0.223244
$329$	1.10970e9	1.71799
$330$	2.07762e8	0.318249
$331$	−1.43202e7	−0.0217046	−0.0108523	−	0.999941i	$-0.503454\pi$
$331$	−1.43202e7	−0.0217046	−0.0108523	+	0.999941i	$0.503454\pi$
$332$	−3.44217e7	−0.0516236
$333$	−8.95036e7	−0.132827
$334$	−3.54546e8	−0.520667
$335$	7.61285e8	1.10634
$336$	−8.23400e7	−0.118420
$337$	−7.28640e8	−1.03707	−0.518535	−	0.855056i	$-0.673522\pi$
$337$	−7.28640e8	−1.03707	−0.518535	+	0.855056i	$0.673522\pi$
$338$	4.07515e8	0.574031
$339$	2.85347e8	0.397809
$340$	−1.13954e8	−0.157237
$341$	1.05929e8	0.144669
$342$	−1.59297e8	−0.215336
$343$	6.71843e8	0.898955
$344$	3.10736e8	0.411564
$345$	−1.36424e8	−0.178864
$346$	−5.98164e8	−0.776343
$347$	−5.87298e8	−0.754581	−0.377290	−	0.926095i	$-0.623144\pi$
$347$	−5.87298e8	−0.754581	−0.377290	+	0.926095i	$0.623144\pi$
$348$	−1.40210e8	−0.178341
$349$	−1.89025e8	−0.238029	−0.119015	−	0.992892i	$-0.537974\pi$
$349$	−1.89025e8	−0.238029	−0.119015	+	0.992892i	$0.537974\pi$
$350$	4.92225e8	0.613657
$351$	−2.78466e8	−0.343714
$352$	1.10641e8	0.135213
$353$	1.09082e9	1.31990	0.659951	−	0.751309i	$-0.270577\pi$
$353$	1.09082e9	1.31990	0.659951	+	0.751309i	$0.270577\pi$
$354$	1.47159e8	0.176309
$355$	−2.19850e8	−0.260812
$356$	−1.17842e8	−0.138428
$357$	9.53721e7	0.110939
$358$	9.61140e8	1.10712
$359$	−1.29066e9	−1.47225	−0.736123	−	0.676848i	$-0.763345\pi$
$359$	−1.29066e9	−1.47225	−0.736123	+	0.676848i	$0.763345\pi$
$360$	3.39536e8	0.383554
$361$	−7.66883e8	−0.857934
$362$	−1.19523e9	−1.32425
$363$	−1.65722e8	−0.181847
$364$	−2.15731e8	−0.234454
$365$	1.60885e8	0.173177
$366$	−2.77392e8	−0.295740
$367$	−6.31310e8	−0.666671	−0.333335	−	0.942808i	$-0.608174\pi$
$367$	−6.31310e8	−0.666671	−0.333335	+	0.942808i	$0.608174\pi$
$368$	−7.26510e7	−0.0759931
$369$	−4.92384e8	−0.510166
$370$	−1.52081e8	−0.156088
$371$	8.77120e8	0.891765
$372$	−4.11485e7	−0.0414433
$373$	−1.48854e8	−0.148518	−0.0742589	−	0.997239i	$-0.523659\pi$
$373$	−1.48854e8	−0.148518	−0.0742589	+	0.997239i	$0.523659\pi$
$374$	−1.28153e8	−0.126671
$375$	−1.18438e8	−0.115979
$376$	5.79232e8	0.561947
$377$	−3.67350e8	−0.353090
$378$	−6.35883e8	−0.605558
$379$	1.20416e8	0.113618	0.0568092	−	0.998385i	$-0.481907\pi$
$379$	1.20416e8	0.113618	0.0568092	+	0.998385i	$0.481907\pi$
$380$	−2.70672e8	−0.253046
$381$	7.51785e7	0.0696396
$382$	1.87719e8	0.172300
$383$	7.76564e8	0.706287	0.353144	−	0.935569i	$-0.385113\pi$
$383$	7.76564e8	0.706287	0.353144	+	0.935569i	$0.385113\pi$
$384$	−4.29791e7	−0.0387345
$385$	1.24300e9	1.11009
$386$	−8.12675e8	−0.719219
$387$	1.07240e9	0.940521
$388$	−4.24850e8	−0.369253
$389$	−3.59623e8	−0.309759	−0.154880	−	0.987933i	$-0.549499\pi$
$389$	−3.59623e8	−0.309759	−0.154880	+	0.987933i	$0.549499\pi$
$390$	−2.11450e8	−0.180501
$391$	8.41496e7	0.0711924
$392$	−7.09717e7	−0.0595092
$393$	9.60575e8	0.798284
$394$	−1.48412e9	−1.22245
$395$	−5.91320e8	−0.482762
$396$	3.81842e8	0.308994
$397$	−6.04729e8	−0.485058	−0.242529	−	0.970144i	$-0.577977\pi$
$397$	−6.04729e8	−0.485058	−0.242529	+	0.970144i	$0.577977\pi$
$398$	2.29013e8	0.182083
$399$	2.26534e8	0.178537
$400$	2.56927e8	0.200724
$401$	−1.22031e9	−0.945068	−0.472534	−	0.881312i	$-0.656661\pi$
$401$	−1.22031e9	−0.945068	−0.472534	+	0.881312i	$0.656661\pi$
$402$	−3.32571e8	−0.255325
$403$	−1.07809e8	−0.0820517
$404$	1.28756e8	0.0971475
$405$	8.27058e8	0.618648
$406$	−8.38851e8	−0.622077
$407$	−1.71030e8	−0.125746
$408$	4.97815e7	0.0362875
$409$	1.32264e9	0.955895	0.477948	−	0.878388i	$-0.341381\pi$
$409$	1.32264e9	0.955895	0.477948	+	0.878388i	$0.341381\pi$
$410$	−8.36641e8	−0.599509
$411$	3.44664e8	0.244878
$412$	−5.40457e8	−0.380733
$413$	8.80423e8	0.614988
$414$	−2.50730e8	−0.173662
$415$	−2.01852e8	−0.138632
$416$	−1.12605e8	−0.0766888
$417$	2.40269e8	0.162264
$418$	−3.04397e8	−0.203856
$419$	−2.21575e9	−1.47154	−0.735769	−	0.677232i	$-0.763179\pi$
$419$	−2.21575e9	−1.47154	−0.735769	+	0.677232i	$0.763179\pi$
$420$	−4.82849e8	−0.318009
$421$	1.14567e9	0.748291	0.374146	−	0.927370i	$-0.377936\pi$
$421$	1.14567e9	0.748291	0.374146	+	0.927370i	$0.377936\pi$
$422$	−1.25955e8	−0.0815869
$423$	1.99902e9	1.28418
$424$	4.57831e8	0.291692
$425$	−2.97592e8	−0.188044
$426$	9.60426e7	0.0601908
$427$	−1.65959e9	−1.03158
$428$	1.29748e9	0.799919
$429$	−2.37796e8	−0.145413
$430$	1.82218e9	1.10523
$431$	−9.03397e8	−0.543511	−0.271755	−	0.962366i	$-0.587604\pi$
$431$	−9.03397e8	−0.543511	−0.271755	+	0.962366i	$0.587604\pi$
$432$	−3.31912e8	−0.198075
$433$	2.69931e8	0.159788	0.0798940	−	0.996803i	$-0.474542\pi$
$433$	2.69931e8	0.159788	0.0798940	+	0.996803i	$0.474542\pi$
$434$	−2.46184e8	−0.144559
$435$	−8.22204e8	−0.478925
$436$	−1.59935e9	−0.924145
$437$	1.99878e8	0.114572
$438$	−7.02835e7	−0.0399663
$439$	−1.79656e9	−1.01348	−0.506741	−	0.862098i	$-0.669150\pi$
$439$	−1.79656e9	−1.01348	−0.506741	+	0.862098i	$0.669150\pi$
$440$	6.48812e8	0.363107
$441$	−2.44935e8	−0.135993
$442$	1.30427e8	0.0718440
$443$	−1.78755e7	−0.00976891	−0.00488445	−	0.999988i	$-0.501555\pi$
$443$	−1.78755e7	−0.00976891	−0.00488445	+	0.999988i	$0.501555\pi$
$444$	6.64374e7	0.0360224
$445$	−6.91034e8	−0.371740
$446$	2.58853e8	0.138160
$447$	1.56769e9	0.830201
$448$	−2.57136e8	−0.135111
$449$	−2.18310e9	−1.13818	−0.569090	−	0.822275i	$-0.692704\pi$
$449$	−2.18310e9	−1.13818	−0.569090	+	0.822275i	$0.692704\pi$
$450$	8.86697e8	0.458703
$451$	−9.40885e8	−0.482968
$452$	8.91098e8	0.453880
$453$	−4.49066e8	−0.226969
$454$	−2.35609e9	−1.18167
$455$	−1.26506e9	−0.629612
$456$	1.18244e8	0.0583986
$457$	1.51733e9	0.743658	0.371829	−	0.928301i	$-0.378731\pi$
$457$	1.51733e9	0.743658	0.371829	+	0.928301i	$0.378731\pi$
$458$	2.12851e9	1.03525
$459$	3.84445e8	0.185562
$460$	−4.26032e8	−0.204075
$461$	2.03360e9	0.966748	0.483374	−	0.875414i	$-0.339411\pi$
$461$	2.03360e9	0.966748	0.483374	+	0.875414i	$0.339411\pi$
$462$	−5.43012e8	−0.256190
$463$	−2.70950e9	−1.26869	−0.634345	−	0.773050i	$-0.718730\pi$
$463$	−2.70950e9	−1.26869	−0.634345	+	0.773050i	$0.718730\pi$
$464$	−4.37856e8	−0.203478
$465$	−2.41299e8	−0.111293
$466$	1.71575e9	0.785421
$467$	−8.79465e8	−0.399585	−0.199793	−	0.979838i	$-0.564027\pi$
$467$	−8.79465e8	−0.399585	−0.199793	+	0.979838i	$0.564027\pi$
$468$	−3.88619e8	−0.175252
$469$	−1.98971e9	−0.890605
$470$	3.39667e9	1.50907
$471$	3.93223e7	0.0173406
$472$	4.59556e8	0.201160
$473$	2.04922e9	0.890380
$474$	2.58321e8	0.111413
$475$	−7.06858e8	−0.302625
$476$	2.97834e8	0.126575
$477$	1.58005e9	0.666586
$478$	2.64109e9	1.10608
$479$	−6.53811e8	−0.271818	−0.135909	−	0.990721i	$-0.543395\pi$
$479$	−6.53811e8	−0.271818	−0.135909	+	0.990721i	$0.543395\pi$
$480$	−2.52033e8	−0.104019
$481$	1.74066e8	0.0713191
$482$	−2.84851e9	−1.15865
$483$	3.56560e8	0.143985
$484$	−5.17527e8	−0.207479
$485$	−2.49136e9	−0.991608
$486$	−1.77906e9	−0.703015
$487$	−1.56097e9	−0.612413	−0.306207	−	0.951965i	$-0.599060\pi$
$487$	−1.56097e9	−0.612413	−0.306207	+	0.951965i	$0.599060\pi$
$488$	−8.66256e8	−0.337425
$489$	−4.53542e8	−0.175403
$490$	−4.16184e8	−0.159808
$491$	−1.64069e9	−0.625520	−0.312760	−	0.949832i	$-0.601254\pi$
$491$	−1.64069e9	−0.625520	−0.312760	+	0.949832i	$0.601254\pi$
$492$	3.65491e8	0.138356
$493$	5.07156e8	0.190624
$494$	3.09799e8	0.115621
$495$	2.23915e9	0.829785
$496$	−1.28501e8	−0.0472847
$497$	5.74606e8	0.209953
$498$	8.81799e7	0.0319939
$499$	2.04350e9	0.736245	0.368123	−	0.929777i	$-0.380001\pi$
$499$	2.04350e9	0.736245	0.368123	+	0.929777i	$0.380001\pi$
$500$	−3.69864e8	−0.132326
$501$	9.08260e8	0.322685
$502$	5.52599e8	0.194961
$503$	5.03945e9	1.76561	0.882805	−	0.469739i	$-0.155652\pi$
$503$	5.03945e9	1.76561	0.882805	+	0.469739i	$0.155652\pi$
$504$	−8.87419e8	−0.308761
$505$	7.55035e8	0.260884
$506$	−4.79115e8	−0.164404
$507$	−1.04395e9	−0.355757
$508$	2.34772e8	0.0794553
$509$	1.43927e9	0.483761	0.241881	−	0.970306i	$-0.422236\pi$
$509$	1.43927e9	0.483761	0.241881	+	0.970306i	$0.422236\pi$
$510$	2.91923e8	0.0974480
$511$	−4.20494e8	−0.139408
$512$	−1.34218e8	−0.0441942
$513$	9.13158e8	0.298631
$514$	1.45664e9	0.473132
$515$	−3.16929e9	−1.02244
$516$	−7.96030e8	−0.255068
$517$	3.81989e9	1.21572
$518$	3.97483e8	0.125651
$519$	1.53235e9	0.481141
$520$	−6.60327e8	−0.205943
$521$	3.31327e8	0.102642	0.0513210	−	0.998682i	$-0.483657\pi$
$521$	3.31327e8	0.102642	0.0513210	+	0.998682i	$0.483657\pi$
$522$	−1.51111e9	−0.464997
$523$	−3.41666e9	−1.04435	−0.522175	−	0.852838i	$-0.674879\pi$
$523$	−3.41666e9	−1.04435	−0.522175	+	0.852838i	$0.674879\pi$
$524$	2.99974e9	0.910803
$525$	−1.26096e9	−0.380315
$526$	2.65972e9	0.796868
$527$	1.48839e8	0.0442976
$528$	−2.83436e8	−0.0837986
$529$	−3.09022e9	−0.907601
$530$	2.68476e9	0.783322
$531$	1.58600e9	0.459698
$532$	7.07434e8	0.203702
$533$	9.57584e8	0.273925
$534$	3.01881e8	0.0857910
$535$	7.60851e9	2.14813
$536$	−1.03857e9	−0.291313
$537$	−2.46221e9	−0.686143
$538$	−2.84957e9	−0.788935
$539$	−4.68040e8	−0.128743
$540$	−1.94636e9	−0.531920
$541$	−3.31862e9	−0.901087	−0.450544	−	0.892754i	$-0.648770\pi$
$541$	−3.31862e9	−0.901087	−0.450544	+	0.892754i	$0.648770\pi$
$542$	5.20715e9	1.40476
$543$	3.06189e9	0.820710
$544$	1.55461e8	0.0414023
$545$	−9.37872e9	−2.48174
$546$	5.52649e8	0.145303
$547$	3.77978e9	0.987440	0.493720	−	0.869621i	$-0.335637\pi$
$547$	3.77978e9	0.987440	0.493720	+	0.869621i	$0.335637\pi$
$548$	1.07634e9	0.279393
$549$	−2.98959e9	−0.771096
$550$	1.69437e9	0.434249
$551$	1.20463e9	0.306777
$552$	1.86114e8	0.0470969
$553$	1.54549e9	0.388622
$554$	−8.89722e8	−0.222316
$555$	3.89595e8	0.0967360
$556$	7.50326e8	0.185135
$557$	−2.29803e9	−0.563460	−0.281730	−	0.959494i	$-0.590908\pi$
$557$	−2.29803e9	−0.563460	−0.281730	+	0.959494i	$0.590908\pi$
$558$	−4.43478e8	−0.108057
$559$	−2.08560e9	−0.504997
$560$	−1.50787e9	−0.362832
$561$	3.28296e8	0.0785047
$562$	4.74414e8	0.112741
$563$	4.61923e9	1.09091	0.545456	−	0.838139i	$-0.316356\pi$
$563$	4.61923e9	1.09091	0.545456	+	0.838139i	$0.316356\pi$
$564$	−1.48385e9	−0.348268
$565$	5.22548e9	1.21887
$566$	5.52649e9	1.28113
$567$	−2.16162e9	−0.498010
$568$	2.99928e8	0.0686748
$569$	−1.50413e8	−0.0342289	−0.0171145	−	0.999854i	$-0.505448\pi$
$569$	−1.50413e8	−0.0342289	−0.0171145	+	0.999854i	$0.505448\pi$
$570$	6.93394e8	0.156826
$571$	−1.41430e9	−0.317918	−0.158959	−	0.987285i	$-0.550814\pi$
$571$	−1.41430e9	−0.317918	−0.158959	+	0.987285i	$0.550814\pi$
$572$	−7.42603e8	−0.165909
$573$	−4.80890e8	−0.106784
$574$	2.18666e9	0.482603
$575$	−1.11258e9	−0.244059
$576$	−4.63207e8	−0.100994
$577$	−5.23894e9	−1.13535	−0.567673	−	0.823254i	$-0.692156\pi$
$577$	−5.23894e9	−1.13535	−0.567673	+	0.823254i	$0.692156\pi$
$578$	3.10264e9	0.668320
$579$	2.08187e9	0.445738
$580$	−2.56763e9	−0.546429
$581$	5.27564e8	0.111599
$582$	1.08836e9	0.228846
$583$	3.01928e9	0.631049
$584$	−2.19485e8	−0.0455996
$585$	−2.27889e9	−0.470629
$586$	5.06520e9	1.03981
$587$	9.01754e8	0.184016	0.0920078	−	0.995758i	$-0.470672\pi$
$587$	9.01754e8	0.184016	0.0920078	+	0.995758i	$0.470672\pi$
$588$	1.81812e8	0.0368809
$589$	3.53532e8	0.0712895
$590$	2.69487e9	0.540202
$591$	3.80194e9	0.757616
$592$	2.07475e8	0.0410997
$593$	1.90592e9	0.375329	0.187665	−	0.982233i	$-0.439908\pi$
$593$	1.90592e9	0.375329	0.187665	+	0.982233i	$0.439908\pi$
$594$	−2.18888e9	−0.428518
$595$	1.74652e9	0.339911
$596$	4.89567e9	0.947219
$597$	−5.86676e8	−0.112847
$598$	4.87619e8	0.0932451
$599$	1.30391e9	0.247887	0.123944	−	0.992289i	$-0.460446\pi$
$599$	1.30391e9	0.247887	0.123944	+	0.992289i	$0.460446\pi$
$600$	−6.58185e8	−0.124399
$601$	−2.06025e9	−0.387131	−0.193566	−	0.981087i	$-0.562005\pi$
$601$	−2.06025e9	−0.387131	−0.193566	+	0.981087i	$0.562005\pi$
$602$	−4.76250e9	−0.889708
$603$	−3.58428e9	−0.665719
$604$	−1.40237e9	−0.258960
$605$	−3.03482e9	−0.557172
$606$	−3.29841e8	−0.0602074
$607$	3.47284e9	0.630267	0.315134	−	0.949047i	$-0.397951\pi$
$607$	3.47284e9	0.630267	0.315134	+	0.949047i	$0.397951\pi$
$608$	3.69260e8	0.0666300
$609$	2.14893e9	0.385533
$610$	−5.07980e9	−0.906134
$611$	−3.88769e9	−0.689520
$612$	5.36519e8	0.0946140
$613$	−6.62090e9	−1.16093	−0.580464	−	0.814286i	$-0.697129\pi$
$613$	−6.62090e9	−1.16093	−0.580464	+	0.814286i	$0.697129\pi$
$614$	−2.55739e9	−0.445869
$615$	2.14327e9	0.371547
$616$	−1.69575e9	−0.292300
$617$	−7.31262e9	−1.25336	−0.626679	−	0.779278i	$-0.715586\pi$
$617$	−7.31262e9	−1.25336	−0.626679	+	0.779278i	$0.715586\pi$
$618$	1.38452e9	0.235960
$619$	−1.16269e10	−1.97037	−0.985185	−	0.171494i	$-0.945140\pi$
$619$	−1.16269e10	−1.97037	−0.985185	+	0.171494i	$0.945140\pi$
$620$	−7.53542e8	−0.126980
$621$	1.43729e9	0.240838
$622$	−7.24384e9	−1.20699
$623$	1.80610e9	0.299250
$624$	2.88467e8	0.0475280
$625$	−7.06941e9	−1.15825
$626$	−3.33504e9	−0.543364
$627$	7.79790e8	0.126340
$628$	1.22798e8	0.0197848
$629$	−2.40312e8	−0.0385033
$630$	−5.20390e9	−0.829158
$631$	−1.21058e10	−1.91818	−0.959091	−	0.283097i	$-0.908638\pi$
$631$	−1.21058e10	−1.91818	−0.959091	+	0.283097i	$0.908638\pi$
$632$	8.06700e8	0.127117
$633$	3.22665e8	0.0505637
$634$	−2.29988e9	−0.358421
$635$	1.37672e9	0.213373
$636$	−1.17285e9	−0.180777
$637$	4.76347e8	0.0730189
$638$	−2.88755e9	−0.440207
$639$	1.03510e9	0.156938
$640$	−7.87065e8	−0.118681
$641$	−1.02354e10	−1.53498	−0.767489	−	0.641062i	$-0.778494\pi$
$641$	−1.02354e10	−1.53498	−0.767489	+	0.641062i	$0.778494\pi$
$642$	−3.32381e9	−0.495752
$643$	5.79264e9	0.859287	0.429643	−	0.902999i	$-0.358639\pi$
$643$	5.79264e9	0.859287	0.429643	+	0.902999i	$0.358639\pi$
$644$	1.11349e9	0.164280
$645$	−4.66799e9	−0.684969
$646$	−4.27703e8	−0.0624207
$647$	5.49161e9	0.797140	0.398570	−	0.917138i	$-0.369507\pi$
$647$	5.49161e9	0.797140	0.398570	+	0.917138i	$0.369507\pi$
$648$	−1.12830e9	−0.162897
$649$	3.03065e9	0.435191
$650$	−1.72444e9	−0.246293
$651$	6.30664e8	0.0895910
$652$	−1.41635e9	−0.200126
$653$	1.23545e10	1.73632	0.868159	−	0.496286i	$-0.165303\pi$
$653$	1.23545e10	1.73632	0.868159	+	0.496286i	$0.165303\pi$
$654$	4.09714e9	0.572741
$655$	1.75908e10	2.44591
$656$	1.14138e9	0.157857
$657$	−7.57480e8	−0.104206
$658$	−8.87761e9	−1.21480
$659$	−1.28561e9	−0.174989	−0.0874943	−	0.996165i	$-0.527886\pi$
$659$	−1.28561e9	−0.174989	−0.0874943	+	0.996165i	$0.527886\pi$
$660$	−1.66210e9	−0.225036
$661$	3.05232e9	0.411078	0.205539	−	0.978649i	$-0.434105\pi$
$661$	3.05232e9	0.411078	0.205539	+	0.978649i	$0.434105\pi$
$662$	1.14562e8	0.0153475
$663$	−3.34123e8	−0.0445255
$664$	2.75373e8	0.0365034
$665$	4.14845e9	0.547029
$666$	7.16028e8	0.0939227
$667$	1.89607e9	0.247407
$668$	2.83637e9	0.368167
$669$	−6.63118e8	−0.0856248
$670$	−6.09028e9	−0.782303
$671$	−5.71274e9	−0.729988
$672$	6.58720e8	0.0837353
$673$	1.07819e10	1.36346	0.681730	−	0.731604i	$-0.261228\pi$
$673$	1.07819e10	1.36346	0.681730	+	0.731604i	$0.261228\pi$
$674$	5.82912e9	0.733319
$675$	−5.08293e9	−0.636137
$676$	−3.26012e9	−0.405901
$677$	1.22831e10	1.52142	0.760709	−	0.649093i	$-0.224851\pi$
$677$	1.22831e10	1.52142	0.760709	+	0.649093i	$0.224851\pi$
$678$	−2.28278e9	−0.281293
$679$	6.51147e9	0.798242
$680$	9.11635e8	0.111183
$681$	6.03572e9	0.732342
$682$	−8.47432e8	−0.102296
$683$	7.76468e8	0.0932505	0.0466252	−	0.998912i	$-0.485153\pi$
$683$	7.76468e8	0.0932505	0.0466252	+	0.998912i	$0.485153\pi$
$684$	1.27437e9	0.152265
$685$	6.31173e9	0.750294
$686$	−5.37474e9	−0.635657
$687$	−5.45272e9	−0.641601
$688$	−2.48589e9	−0.291019
$689$	−3.07287e9	−0.357912
$690$	1.09139e9	0.126476
$691$	−1.01602e10	−1.17147	−0.585733	−	0.810504i	$-0.699193\pi$
$691$	−1.01602e10	−1.17147	−0.585733	+	0.810504i	$0.699193\pi$
$692$	4.78531e9	0.548958
$693$	−5.85230e9	−0.667975
$694$	4.69839e9	0.533569
$695$	4.39998e9	0.497169
$696$	1.12168e9	0.126106
$697$	−1.32202e9	−0.147885
$698$	1.51220e9	0.168312
$699$	−4.39532e9	−0.486767
$700$	−3.93780e9	−0.433921
$701$	−1.47467e10	−1.61689	−0.808446	−	0.588571i	$-0.799691\pi$
$701$	−1.47467e10	−1.61689	−0.808446	+	0.588571i	$0.799691\pi$
$702$	2.22773e9	0.243043
$703$	−5.70804e8	−0.0619646
$704$	−8.85132e8	−0.0956101
$705$	−8.70144e9	−0.935253
$706$	−8.72656e9	−0.933311
$707$	−1.97338e9	−0.210011
$708$	−1.17727e9	−0.124669
$709$	1.12511e10	1.18559	0.592795	−	0.805353i	$-0.298024\pi$
$709$	1.12511e10	1.18559	0.592795	+	0.805353i	$0.298024\pi$
$710$	1.75880e9	0.184422
$711$	2.78405e9	0.290492
$712$	9.42733e8	0.0978833
$713$	5.56453e8	0.0574930
$714$	−7.62977e8	−0.0784454
$715$	−4.35469e9	−0.445539
$716$	−7.68912e9	−0.782855
$717$	−6.76583e9	−0.685495
$718$	1.03253e10	1.04103
$719$	1.05695e10	1.06048	0.530242	−	0.847846i	$-0.322101\pi$
$719$	1.05695e10	1.06048	0.530242	+	0.847846i	$0.322101\pi$
$720$	−2.71629e9	−0.271214
$721$	8.28332e9	0.823059
$722$	6.13507e9	0.606651
$723$	7.29718e9	0.718077
$724$	9.56184e9	0.936390
$725$	−6.70535e9	−0.653490
$726$	1.32578e9	0.128586
$727$	1.28794e10	1.24316	0.621578	−	0.783352i	$-0.286492\pi$
$727$	1.28794e10	1.24316	0.621578	+	0.783352i	$0.286492\pi$
$728$	1.72585e9	0.165784
$729$	−2.62002e8	−0.0250471
$730$	−1.28708e9	−0.122455
$731$	2.87933e9	0.272635
$732$	2.21914e9	0.209120
$733$	4.04814e9	0.379657	0.189828	−	0.981817i	$-0.439207\pi$
$733$	4.04814e9	0.379657	0.189828	+	0.981817i	$0.439207\pi$
$734$	5.05048e9	0.471407
$735$	1.06616e9	0.0990416
$736$	5.81208e8	0.0537353
$737$	−6.84912e9	−0.630229
$738$	3.93907e9	0.360742
$739$	2.07897e10	1.89493	0.947465	−	0.319860i	$-0.103636\pi$
$739$	2.07897e10	1.89493	0.947465	+	0.319860i	$0.103636\pi$
$740$	1.21665e9	0.110371
$741$	−7.93630e8	−0.0716563
$742$	−7.01696e9	−0.630573
$743$	−1.77336e9	−0.158612	−0.0793059	−	0.996850i	$-0.525270\pi$
$743$	−1.77336e9	−0.158612	−0.0793059	+	0.996850i	$0.525270\pi$
$744$	3.29188e8	0.0293048
$745$	2.87086e10	2.54370
$746$	1.19083e9	0.105018
$747$	9.50358e8	0.0834190
$748$	1.02522e9	0.0895700
$749$	−1.98858e10	−1.72924
$750$	9.47500e8	0.0820097
$751$	1.57353e9	0.135561	0.0677805	−	0.997700i	$-0.478408\pi$
$751$	1.57353e9	0.135561	0.0677805	+	0.997700i	$0.478408\pi$
$752$	−4.63386e9	−0.397356
$753$	−1.41562e9	−0.120827
$754$	2.93880e9	0.249672
$755$	−8.22363e9	−0.695423
$756$	5.08706e9	0.428194
$757$	3.09511e9	0.259322	0.129661	−	0.991558i	$-0.458611\pi$
$757$	3.09511e9	0.259322	0.129661	+	0.991558i	$0.458611\pi$
$758$	−9.63331e8	−0.0803403
$759$	1.22738e9	0.101890
$760$	2.16537e9	0.178931
$761$	8.84425e9	0.727470	0.363735	−	0.931503i	$-0.381501\pi$
$761$	8.84425e9	0.727470	0.363735	+	0.931503i	$0.381501\pi$
$762$	−6.01428e8	−0.0492426
$763$	2.45124e10	1.99779
$764$	−1.50175e9	−0.121835
$765$	3.14620e9	0.254080
$766$	−6.21251e9	−0.499420
$767$	−3.08444e9	−0.246827
$768$	3.43833e8	0.0273894
$769$	−1.74163e10	−1.38106	−0.690532	−	0.723302i	$-0.742624\pi$
$769$	−1.74163e10	−1.38106	−0.690532	+	0.723302i	$0.742624\pi$
$770$	−9.94402e9	−0.784955
$771$	−3.73156e9	−0.293225
$772$	6.50140e9	0.508565
$773$	9.23354e9	0.719018	0.359509	−	0.933142i	$-0.382944\pi$
$773$	9.23354e9	0.719018	0.359509	+	0.933142i	$0.382944\pi$
$774$	−8.57920e9	−0.665049
$775$	−1.96787e9	−0.151859
$776$	3.39880e9	0.261101
$777$	−1.01825e9	−0.0778723
$778$	2.87699e9	0.219033
$779$	−3.14015e9	−0.237996
$780$	1.69160e9	0.127634
$781$	1.97795e9	0.148572
$782$	−6.73197e8	−0.0503406
$783$	8.66234e9	0.644865
$784$	5.67773e8	0.0420793
$785$	7.20098e8	0.0531309
$786$	−7.68460e9	−0.564472
$787$	1.14265e10	0.835609	0.417805	−	0.908537i	$-0.362800\pi$
$787$	1.14265e10	0.835609	0.417805	+	0.908537i	$0.362800\pi$
$788$	1.18729e10	0.864402
$789$	−6.81356e9	−0.493861
$790$	4.73056e9	0.341364
$791$	−1.36574e10	−0.981186
$792$	−3.05473e9	−0.218492
$793$	5.81413e9	0.414027
$794$	4.83783e9	0.342988
$795$	−6.87770e9	−0.485466
$796$	−1.83211e9	−0.128752
$797$	−2.63234e10	−1.84178	−0.920891	−	0.389820i	$-0.872537\pi$
$797$	−2.63234e10	−1.84178	−0.920891	+	0.389820i	$0.872537\pi$
$798$	−1.81227e9	−0.126245
$799$	5.36726e9	0.372254
$800$	−2.05542e9	−0.141934
$801$	3.25352e9	0.223687
$802$	9.76244e9	0.668264
$803$	−1.44745e9	−0.0986505
$804$	2.66057e9	0.180542
$805$	6.52959e9	0.441164
$806$	8.62473e8	0.0580193
$807$	7.29990e9	0.488945
$808$	−1.03005e9	−0.0686937
$809$	−7.47328e9	−0.496240	−0.248120	−	0.968729i	$-0.579813\pi$
$809$	−7.47328e9	−0.496240	−0.248120	+	0.968729i	$0.579813\pi$
$810$	−6.61647e9	−0.437450
$811$	−1.32084e10	−0.869513	−0.434756	−	0.900548i	$-0.643166\pi$
$811$	−1.32084e10	−0.869513	−0.434756	+	0.900548i	$0.643166\pi$
$812$	6.71081e9	0.439875
$813$	−1.33394e10	−0.870604
$814$	1.36824e9	0.0889156
$815$	−8.30558e9	−0.537426
$816$	−3.98252e8	−0.0256591
$817$	6.83917e9	0.438759
$818$	−1.05811e10	−0.675920
$819$	5.95617e9	0.378855
$820$	6.69313e9	0.423917
$821$	−2.22151e10	−1.40103	−0.700516	−	0.713637i	$-0.747047\pi$
$821$	−2.22151e10	−1.40103	−0.700516	+	0.713637i	$0.747047\pi$
$822$	−2.75731e9	−0.173155
$823$	−5.39596e9	−0.337419	−0.168710	−	0.985666i	$-0.553960\pi$
$823$	−5.39596e9	−0.337419	−0.168710	+	0.985666i	$0.553960\pi$
$824$	4.32365e9	0.269219
$825$	−4.34056e9	−0.269127
$826$	−7.04339e9	−0.434862
$827$	−1.31682e10	−0.809575	−0.404787	−	0.914411i	$-0.632654\pi$
$827$	−1.31682e10	−0.809575	−0.404787	+	0.914411i	$0.632654\pi$
$828$	2.00584e9	0.122798
$829$	−1.33819e10	−0.815790	−0.407895	−	0.913029i	$-0.633737\pi$
$829$	−1.33819e10	−0.815790	−0.407895	+	0.913029i	$0.633737\pi$
$830$	1.61481e9	0.0980278
$831$	2.27925e9	0.137781
$832$	9.00842e8	0.0542271
$833$	−6.57636e8	−0.0394210
$834$	−1.92215e9	−0.114738
$835$	1.66327e10	0.988691
$836$	2.43517e9	0.144148
$837$	2.54220e9	0.149855
$838$	1.77260e10	1.04053
$839$	−2.87088e9	−0.167822	−0.0839108	−	0.996473i	$-0.526741\pi$
$839$	−2.87088e9	−0.167822	−0.0839108	+	0.996473i	$0.526741\pi$
$840$	3.86280e9	0.224866
$841$	−5.82260e9	−0.337544
$842$	−9.16533e9	−0.529122
$843$	−1.21533e9	−0.0698713
$844$	1.00764e9	0.0576907
$845$	−1.91177e10	−1.09002
$846$	−1.59922e10	−0.908054
$847$	7.93188e9	0.448523
$848$	−3.66265e9	−0.206258
$849$	−1.41575e10	−0.793981
$850$	2.38073e9	0.132967
$851$	−8.98436e8	−0.0499728
$852$	−7.68341e8	−0.0425613
$853$	2.37600e9	0.131077	0.0655383	−	0.997850i	$-0.479124\pi$
$853$	2.37600e9	0.131077	0.0655383	+	0.997850i	$0.479124\pi$
$854$	1.32767e10	0.729436
$855$	7.47305e9	0.408899
$856$	−1.03798e10	−0.565628
$857$	−3.42331e10	−1.85786	−0.928931	−	0.370253i	$-0.879271\pi$
$857$	−3.42331e10	−1.85786	−0.928931	+	0.370253i	$0.879271\pi$
$858$	1.90237e9	0.102823
$859$	−3.38131e9	−0.182016	−0.0910078	−	0.995850i	$-0.529009\pi$
$859$	−3.38131e9	−0.182016	−0.0910078	+	0.995850i	$0.529009\pi$
$860$	−1.45775e10	−0.781515
$861$	−5.60170e9	−0.299095
$862$	7.22717e9	0.384320
$863$	−1.21792e10	−0.645030	−0.322515	−	0.946564i	$-0.604528\pi$
$863$	−1.21792e10	−0.645030	−0.322515	+	0.946564i	$0.604528\pi$
$864$	2.65530e9	0.140060
$865$	2.80615e10	1.47419
$866$	−2.15944e9	−0.112987
$867$	−7.94821e9	−0.414193
$868$	1.96947e9	0.102219
$869$	5.31998e9	0.275005
$870$	6.57763e9	0.338651
$871$	6.97068e9	0.357447
$872$	1.27948e10	0.653469
$873$	1.17298e10	0.596679
$874$	−1.59902e9	−0.0810147
$875$	5.66872e9	0.286060
$876$	5.62268e8	0.0282605
$877$	1.71298e10	0.857539	0.428770	−	0.903414i	$-0.358947\pi$
$877$	1.71298e10	0.857539	0.428770	+	0.903414i	$0.358947\pi$
$878$	1.43725e10	0.716641
$879$	−1.29758e10	−0.644426
$880$	−5.19049e9	−0.256755
$881$	−1.99520e10	−0.983039	−0.491520	−	0.870867i	$-0.663558\pi$
$881$	−1.99520e10	−0.983039	−0.491520	+	0.870867i	$0.663558\pi$
$882$	1.95948e9	0.0961613
$883$	−3.12819e10	−1.52908	−0.764542	−	0.644574i	$-0.777035\pi$
$883$	−3.12819e10	−1.52908	−0.764542	+	0.644574i	$0.777035\pi$
$884$	−1.04342e9	−0.0508014
$885$	−6.90361e9	−0.334792
$886$	1.43004e8	0.00690766
$887$	1.92928e10	0.928244	0.464122	−	0.885771i	$-0.346370\pi$
$887$	1.92928e10	0.928244	0.464122	+	0.885771i	$0.346370\pi$
$888$	−5.31499e8	−0.0254717
$889$	−3.59824e9	−0.171765
$890$	5.52827e9	0.262860
$891$	−7.44087e9	−0.352413
$892$	−2.07082e9	−0.0976937
$893$	1.27487e10	0.599080
$894$	−1.25415e10	−0.587041
$895$	−4.50897e10	−2.10231
$896$	2.05709e9	0.0955378
$897$	−1.24916e9	−0.0577889
$898$	1.74648e10	0.804815
$899$	3.35365e9	0.153943
$900$	−7.09358e9	−0.324352
$901$	4.24234e9	0.193227
$902$	7.52708e9	0.341510
$903$	1.22004e10	0.551399
$904$	−7.12878e9	−0.320942
$905$	5.60715e10	2.51462
$906$	3.59253e9	0.160491
$907$	−2.33637e10	−1.03972	−0.519860	−	0.854251i	$-0.674016\pi$
$907$	−2.33637e10	−1.03972	−0.519860	+	0.854251i	$0.674016\pi$
$908$	1.88487e10	0.835566
$909$	−3.55485e9	−0.156981
$910$	1.01205e10	0.445203
$911$	−1.56760e10	−0.686943	−0.343471	−	0.939163i	$-0.611603\pi$
$911$	−1.56760e10	−0.686943	−0.343471	+	0.939163i	$0.611603\pi$
$912$	−9.45953e8	−0.0412941
$913$	1.81602e9	0.0789718
$914$	−1.21386e10	−0.525846
$915$	1.30132e10	0.561579
$916$	−1.70281e10	−0.732034
$917$	−4.59756e10	−1.96895
$918$	−3.07556e9	−0.131212
$919$	−2.15624e10	−0.916418	−0.458209	−	0.888845i	$-0.651509\pi$
$919$	−2.15624e10	−0.916418	−0.458209	+	0.888845i	$0.651509\pi$
$920$	3.40826e9	0.144303
$921$	6.55141e9	0.276329
$922$	−1.62688e10	−0.683594
$923$	−2.01305e9	−0.0842653
$924$	4.34409e9	0.181154
$925$	3.17728e9	0.131996
$926$	2.16760e10	0.897099
$927$	1.49216e10	0.615229
$928$	3.50285e9	0.143881
$929$	8.87728e9	0.363266	0.181633	−	0.983366i	$-0.441862\pi$
$929$	8.87728e9	0.363266	0.181633	+	0.983366i	$0.441862\pi$
$930$	1.93039e9	0.0786964
$931$	−1.56206e9	−0.0634415
$932$	−1.37260e10	−0.555377
$933$	1.85570e10	0.748033
$934$	7.03572e9	0.282550
$935$	6.01200e9	0.240535
$936$	3.10895e9	0.123922
$937$	−5.24977e9	−0.208474	−0.104237	−	0.994552i	$-0.533240\pi$
$937$	−5.24977e9	−0.208474	−0.104237	+	0.994552i	$0.533240\pi$
$938$	1.59177e10	0.629753
$939$	8.54356e9	0.336751
$940$	−2.71734e10	−1.06708
$941$	1.88471e10	0.737363	0.368682	−	0.929556i	$-0.379809\pi$
$941$	1.88471e10	0.737363	0.368682	+	0.929556i	$0.379809\pi$
$942$	−3.14578e8	−0.0122617
$943$	−4.94254e9	−0.191937
$944$	−3.67644e9	−0.142241
$945$	2.98310e10	1.14989
$946$	−1.63938e10	−0.629594
$947$	3.54978e9	0.135824	0.0679119	−	0.997691i	$-0.478366\pi$
$947$	3.54978e9	0.135824	0.0679119	+	0.997691i	$0.478366\pi$
$948$	−2.06657e9	−0.0787808
$949$	1.47314e9	0.0559516
$950$	5.65487e9	0.213988
$951$	5.89173e9	0.222132
$952$	−2.38267e9	−0.0895023
$953$	−5.00673e9	−0.187383	−0.0936913	−	0.995601i	$-0.529867\pi$
$953$	−5.00673e9	−0.187383	−0.0936913	+	0.995601i	$0.529867\pi$
$954$	−1.26404e10	−0.471347
$955$	−8.80640e9	−0.327180
$956$	−2.11287e10	−0.782115
$957$	7.39719e9	0.272819
$958$	5.23049e9	0.192204
$959$	−1.64965e10	−0.603985
$960$	2.01627e9	0.0735527
$961$	−2.65284e10	−0.964226
$962$	−1.39253e9	−0.0504302
$963$	−3.58224e10	−1.29259
$964$	2.27881e10	0.819290
$965$	3.81248e10	1.36572
$966$	−2.85248e9	−0.101813
$967$	1.13815e10	0.404770	0.202385	−	0.979306i	$-0.435131\pi$
$967$	1.13815e10	0.404770	0.202385	+	0.979306i	$0.435131\pi$
$968$	4.14021e9	0.146710
$969$	1.09567e9	0.0386854
$970$	1.99309e10	0.701173
$971$	2.87865e10	1.00907	0.504536	−	0.863391i	$-0.331664\pi$
$971$	2.87865e10	1.00907	0.504536	+	0.863391i	$0.331664\pi$
$972$	1.42325e10	0.497107
$973$	−1.14999e10	−0.400220
$974$	1.24878e10	0.433042
$975$	4.41760e9	0.152641
$976$	6.93005e9	0.238595
$977$	−7.39126e9	−0.253564	−0.126782	−	0.991931i	$-0.540465\pi$
$977$	−7.39126e9	−0.253564	−0.126782	+	0.991931i	$0.540465\pi$
$978$	3.62833e9	0.124028
$979$	6.21708e9	0.211762
$980$	3.32947e9	0.113002
$981$	4.41568e10	1.49333
$982$	1.31255e10	0.442310
$983$	2.07272e10	0.695991	0.347995	−	0.937496i	$-0.386862\pi$
$983$	2.07272e10	0.695991	0.347995	+	0.937496i	$0.386862\pi$
$984$	−2.92392e9	−0.0978325
$985$	6.96239e10	2.32130
$986$	−4.05725e9	−0.134791
$987$	2.27423e10	0.752877
$988$	−2.47840e9	−0.0817563
$989$	1.07647e10	0.353848
$990$	−1.79132e10	−0.586747
$991$	−3.71476e10	−1.21248	−0.606238	−	0.795283i	$-0.707322\pi$
$991$	−3.71476e10	−1.21248	−0.606238	+	0.795283i	$0.707322\pi$
$992$	1.02801e9	0.0334354
$993$	−2.93479e8	−0.00951162
$994$	−4.59685e9	−0.148459
$995$	−1.07436e10	−0.345757
$996$	−7.05439e8	−0.0226231
$997$	−3.68049e10	−1.17618	−0.588088	−	0.808797i	$-0.700119\pi$
$997$	−3.68049e10	−1.17618	−0.588088	+	0.808797i	$0.700119\pi$
$998$	−1.63480e10	−0.520604
$999$	−4.10458e9	−0.130254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.8.a.a.1.3	✓	4
4.3	odd	2		592.8.a.b.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.8.a.a.1.3	✓	4	1.1	even	1	trivial
592.8.a.b.1.2		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)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +20.4940 q^{3} +64.0000 q^{4} +375.302 q^{5} -163.952 q^{6} -980.897 q^{7} -512.000 q^{8} -1766.99 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +20.4940 q^{3} +64.0000 q^{4} +375.302 q^{5} -163.952 q^{6} -980.897 q^{7} -512.000 q^{8} -1766.99 q^{9} -3002.41 q^{10} -3376.51 q^{11} +1311.62 q^{12} +3436.44 q^{13} +7847.18 q^{14} +7691.45 q^{15} +4096.00 q^{16} -4744.28 q^{17} +14136.0 q^{18} -11268.9 q^{19} +24019.3 q^{20} -20102.6 q^{21} +27012.1 q^{22} -17737.1 q^{23} -10492.9 q^{24} +62726.4 q^{25} -27491.5 q^{26} -81033.3 q^{27} -62777.4 q^{28} -106898. q^{29} -61531.6 q^{30} -31372.3 q^{31} -32768.0 q^{32} -69198.3 q^{33} +37954.2 q^{34} -368132. q^{35} -113088. q^{36} +50653.0 q^{37} +90151.3 q^{38} +70426.5 q^{39} -192154. q^{40} +278656. q^{41} +160820. q^{42} -606906. q^{43} -216097. q^{44} -663156. q^{45} +141897. q^{46} -1.13131e6 q^{47} +83943.6 q^{48} +138617. q^{49} -501811. q^{50} -97229.5 q^{51} +219932. q^{52} -894201. q^{53} +648267. q^{54} -1.26721e6 q^{55} +502219. q^{56} -230946. q^{57} +855188. q^{58} -897569. q^{59} +492253. q^{60} +1.69191e6 q^{61} +250979. q^{62} +1.73324e6 q^{63} +262144. q^{64} +1.28970e6 q^{65} +553587. q^{66} +2.02846e6 q^{67} -303634. q^{68} -363504. q^{69} +2.94506e6 q^{70} -585796. q^{71} +904701. q^{72} +428683. q^{73} -405224. q^{74} +1.28552e6 q^{75} -721211. q^{76} +3.31201e6 q^{77} -563412. q^{78} -1.57559e6 q^{79} +1.53724e6 q^{80} +2.20372e6 q^{81} -2.22925e6 q^{82} -537839. q^{83} -1.28656e6 q^{84} -1.78054e6 q^{85} +4.85525e6 q^{86} -2.19078e6 q^{87} +1.72877e6 q^{88} -1.84127e6 q^{89} +5.30525e6 q^{90} -3.37079e6 q^{91} -1.13517e6 q^{92} -642946. q^{93} +9.05050e6 q^{94} -4.22924e6 q^{95} -671549. q^{96} -6.63828e6 q^{97} -1.10893e6 q^{98} +5.96627e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9}+O(q^{10}) \) 4 * q - 32 * q^2 - 53 * q^3 + 256 * q^4 + 111 * q^5 + 424 * q^6 - 1666 * q^7 - 2048 * q^8 + 4609 * q^9	\( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9} - 888 q^{10} - 4593 q^{11} - 3392 q^{12} + 7847 q^{13} + 13328 q^{14} + 18900 q^{15} + 16384 q^{16} + 23172 q^{17} - 36872 q^{18} + 23696 q^{19} + 7104 q^{20} + 69416 q^{21} + 36744 q^{22} + 24105 q^{23} + 27136 q^{24} - 138149 q^{25} - 62776 q^{26} - 433646 q^{27} - 106624 q^{28} - 140949 q^{29} - 151200 q^{30} - 664609 q^{31} - 131072 q^{32} - 240450 q^{33} - 185376 q^{34} - 248544 q^{35} + 294976 q^{36} + 202612 q^{37} - 189568 q^{38} - 2288827 q^{39} - 56832 q^{40} - 709737 q^{41} - 555328 q^{42} - 128962 q^{43} - 293952 q^{44} - 1755342 q^{45} - 192840 q^{46} - 445842 q^{47} - 217088 q^{48} - 1602774 q^{49} + 1105192 q^{50} - 2883630 q^{51} + 502208 q^{52} - 975870 q^{53} + 3469168 q^{54} - 644145 q^{55} + 852992 q^{56} + 3494630 q^{57} + 1127592 q^{58} - 1812858 q^{59} + 1209600 q^{60} - 2955031 q^{61} + 5316872 q^{62} - 3362482 q^{63} + 1048576 q^{64} + 666 q^{65} + 1923600 q^{66} + 2737235 q^{67} + 1483008 q^{68} - 1781673 q^{69} + 1988352 q^{70} + 4958184 q^{71} - 2359808 q^{72} - 931591 q^{73} - 1620896 q^{74} + 4945810 q^{75} + 1516544 q^{76} + 4352514 q^{77} + 18310616 q^{78} + 5813561 q^{79} + 454656 q^{80} + 16394896 q^{81} + 5677896 q^{82} + 2120460 q^{83} + 4442624 q^{84} - 4845402 q^{85} + 1031696 q^{86} + 7965333 q^{87} + 2351616 q^{88} + 8833716 q^{89} + 14042736 q^{90} - 18886274 q^{91} + 1542720 q^{92} + 3024182 q^{93} + 3566736 q^{94} - 3151794 q^{95} + 1736704 q^{96} - 22666876 q^{97} + 12822192 q^{98} - 17931894 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 53 * q^3 + 256 * q^4 + 111 * q^5 + 424 * q^6 - 1666 * q^7 - 2048 * q^8 + 4609 * q^9 - 888 * q^10 - 4593 * q^11 - 3392 * q^12 + 7847 * q^13 + 13328 * q^14 + 18900 * q^15 + 16384 * q^16 + 23172 * q^17 - 36872 * q^18 + 23696 * q^19 + 7104 * q^20 + 69416 * q^21 + 36744 * q^22 + 24105 * q^23 + 27136 * q^24 - 138149 * q^25 - 62776 * q^26 - 433646 * q^27 - 106624 * q^28 - 140949 * q^29 - 151200 * q^30 - 664609 * q^31 - 131072 * q^32 - 240450 * q^33 - 185376 * q^34 - 248544 * q^35 + 294976 * q^36 + 202612 * q^37 - 189568 * q^38 - 2288827 * q^39 - 56832 * q^40 - 709737 * q^41 - 555328 * q^42 - 128962 * q^43 - 293952 * q^44 - 1755342 * q^45 - 192840 * q^46 - 445842 * q^47 - 217088 * q^48 - 1602774 * q^49 + 1105192 * q^50 - 2883630 * q^51 + 502208 * q^52 - 975870 * q^53 + 3469168 * q^54 - 644145 * q^55 + 852992 * q^56 + 3494630 * q^57 + 1127592 * q^58 - 1812858 * q^59 + 1209600 * q^60 - 2955031 * q^61 + 5316872 * q^62 - 3362482 * q^63 + 1048576 * q^64 + 666 * q^65 + 1923600 * q^66 + 2737235 * q^67 + 1483008 * q^68 - 1781673 * q^69 + 1988352 * q^70 + 4958184 * q^71 - 2359808 * q^72 - 931591 * q^73 - 1620896 * q^74 + 4945810 * q^75 + 1516544 * q^76 + 4352514 * q^77 + 18310616 * q^78 + 5813561 * q^79 + 454656 * q^80 + 16394896 * q^81 + 5677896 * q^82 + 2120460 * q^83 + 4442624 * q^84 - 4845402 * q^85 + 1031696 * q^86 + 7965333 * q^87 + 2351616 * q^88 + 8833716 * q^89 + 14042736 * q^90 - 18886274 * q^91 + 1542720 * q^92 + 3024182 * q^93 + 3566736 * q^94 - 3151794 * q^95 + 1736704 * q^96 - 22666876 * q^97 + 12822192 * q^98 - 17931894 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more