LMFDB - Embedded newform 38.8.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,8,Mod(1,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, 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("38.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	38.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:	$11.8706309684$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9097x^{2} - 110520x + 10368000 $ x^4 - 9097x^2 - 110520x + 10368000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-76.8211$ of defining polynomial
Character	$\chi$	$=$	38.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +79.8211 q^{3} +64.0000 q^{4} -330.525 q^{5} +638.569 q^{6} +1294.09 q^{7} +512.000 q^{8} +4184.42 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +79.8211 q^{3} +64.0000 q^{4} -330.525 q^{5} +638.569 q^{6} +1294.09 q^{7} +512.000 q^{8} +4184.42 q^{9} -2644.20 q^{10} +3402.95 q^{11} +5108.55 q^{12} -13955.5 q^{13} +10352.7 q^{14} -26382.9 q^{15} +4096.00 q^{16} +18722.0 q^{17} +33475.3 q^{18} +6859.00 q^{19} -21153.6 q^{20} +103296. q^{21} +27223.6 q^{22} -59529.0 q^{23} +40868.4 q^{24} +31121.8 q^{25} -111644. q^{26} +159436. q^{27} +82821.9 q^{28} -197670. q^{29} -211063. q^{30} -65838.5 q^{31} +32768.0 q^{32} +271628. q^{33} +149776. q^{34} -427730. q^{35} +267803. q^{36} +436467. q^{37} +54872.0 q^{38} -1.11395e6 q^{39} -169229. q^{40} +88301.2 q^{41} +826367. q^{42} -590778. q^{43} +217789. q^{44} -1.38305e6 q^{45} -476232. q^{46} -424919. q^{47} +326947. q^{48} +851131. q^{49} +248974. q^{50} +1.49441e6 q^{51} -893154. q^{52} -697171. q^{53} +1.27549e6 q^{54} -1.12476e6 q^{55} +662575. q^{56} +547493. q^{57} -1.58136e6 q^{58} -2.78553e6 q^{59} -1.68850e6 q^{60} -1.26287e6 q^{61} -526708. q^{62} +5.41502e6 q^{63} +262144. q^{64} +4.61265e6 q^{65} +2.17302e6 q^{66} +2.67972e6 q^{67} +1.19821e6 q^{68} -4.75167e6 q^{69} -3.42184e6 q^{70} +537682. q^{71} +2.14242e6 q^{72} +3.24778e6 q^{73} +3.49173e6 q^{74} +2.48418e6 q^{75} +438976. q^{76} +4.40374e6 q^{77} -8.91157e6 q^{78} +8.46098e6 q^{79} -1.35383e6 q^{80} +3.57505e6 q^{81} +706410. q^{82} -2.31950e6 q^{83} +6.61094e6 q^{84} -6.18810e6 q^{85} -4.72622e6 q^{86} -1.57783e7 q^{87} +1.74231e6 q^{88} +1.72405e6 q^{89} -1.10644e7 q^{90} -1.80597e7 q^{91} -3.80986e6 q^{92} -5.25530e6 q^{93} -3.39935e6 q^{94} -2.26707e6 q^{95} +2.61558e6 q^{96} -675649. q^{97} +6.80905e6 q^{98} +1.42394e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9}+O(q^{10}) $ 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9	\( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9} - 2232 q^{10} + 5269 q^{11} + 768 q^{12} + 5406 q^{13} + 19880 q^{14} + 26658 q^{15} + 16384 q^{16} + 22885 q^{17} + 75856 q^{18} + 27436 q^{19} - 17856 q^{20} + 2854 q^{21} + 42152 q^{22} + 3364 q^{23} + 6144 q^{24} + 112561 q^{25} + 43248 q^{26} - 220194 q^{27} + 159040 q^{28} - 122136 q^{29} + 213264 q^{30} + 225480 q^{31} + 131072 q^{32} + 176138 q^{33} + 183080 q^{34} - 785781 q^{35} + 606848 q^{36} + 154096 q^{37} + 219488 q^{38} - 1749220 q^{39} - 142848 q^{40} - 1054628 q^{41} + 22832 q^{42} - 840795 q^{43} + 337216 q^{44} - 4162563 q^{45} + 26912 q^{46} - 1021877 q^{47} + 49152 q^{48} - 621441 q^{49} + 900488 q^{50} + 724892 q^{51} + 345984 q^{52} - 326842 q^{53} - 1761552 q^{54} - 221553 q^{55} + 1272320 q^{56} + 82308 q^{57} - 977088 q^{58} + 421384 q^{59} + 1706112 q^{60} + 116825 q^{61} + 1803840 q^{62} + 10245825 q^{63} + 1048576 q^{64} + 4477428 q^{65} + 1409104 q^{66} + 5794566 q^{67} + 1464640 q^{68} - 2472196 q^{69} - 6286248 q^{70} + 10590626 q^{71} + 4854784 q^{72} + 3971389 q^{73} + 1232768 q^{74} - 3690042 q^{75} + 1755904 q^{76} + 5806573 q^{77} - 13993760 q^{78} + 5597800 q^{79} - 1142784 q^{80} + 20567744 q^{81} - 8437024 q^{82} + 4665800 q^{83} + 182656 q^{84} - 2014461 q^{85} - 6726360 q^{86} - 14449584 q^{87} + 2697728 q^{88} - 2794214 q^{89} - 33300504 q^{90} - 8827314 q^{91} + 215296 q^{92} - 43981204 q^{93} - 8175016 q^{94} - 1913661 q^{95} + 393216 q^{96} - 14445130 q^{97} - 4971528 q^{98} - 7940315 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9 - 2232 * q^10 + 5269 * q^11 + 768 * q^12 + 5406 * q^13 + 19880 * q^14 + 26658 * q^15 + 16384 * q^16 + 22885 * q^17 + 75856 * q^18 + 27436 * q^19 - 17856 * q^20 + 2854 * q^21 + 42152 * q^22 + 3364 * q^23 + 6144 * q^24 + 112561 * q^25 + 43248 * q^26 - 220194 * q^27 + 159040 * q^28 - 122136 * q^29 + 213264 * q^30 + 225480 * q^31 + 131072 * q^32 + 176138 * q^33 + 183080 * q^34 - 785781 * q^35 + 606848 * q^36 + 154096 * q^37 + 219488 * q^38 - 1749220 * q^39 - 142848 * q^40 - 1054628 * q^41 + 22832 * q^42 - 840795 * q^43 + 337216 * q^44 - 4162563 * q^45 + 26912 * q^46 - 1021877 * q^47 + 49152 * q^48 - 621441 * q^49 + 900488 * q^50 + 724892 * q^51 + 345984 * q^52 - 326842 * q^53 - 1761552 * q^54 - 221553 * q^55 + 1272320 * q^56 + 82308 * q^57 - 977088 * q^58 + 421384 * q^59 + 1706112 * q^60 + 116825 * q^61 + 1803840 * q^62 + 10245825 * q^63 + 1048576 * q^64 + 4477428 * q^65 + 1409104 * q^66 + 5794566 * q^67 + 1464640 * q^68 - 2472196 * q^69 - 6286248 * q^70 + 10590626 * q^71 + 4854784 * q^72 + 3971389 * q^73 + 1232768 * q^74 - 3690042 * q^75 + 1755904 * q^76 + 5806573 * q^77 - 13993760 * q^78 + 5597800 * q^79 - 1142784 * q^80 + 20567744 * q^81 - 8437024 * q^82 + 4665800 * q^83 + 182656 * q^84 - 2014461 * q^85 - 6726360 * q^86 - 14449584 * q^87 + 2697728 * q^88 - 2794214 * q^89 - 33300504 * q^90 - 8827314 * q^91 + 215296 * q^92 - 43981204 * q^93 - 8175016 * q^94 - 1913661 * q^95 + 393216 * q^96 - 14445130 * q^97 - 4971528 * q^98 - 7940315 * 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
$3$	79.8211	1.70684	0.853422	−	0.521221i	$-0.174523\pi$
$3$	79.8211	1.70684	0.853422	+	0.521221i	$0.174523\pi$
$4$	64.0000	0.500000
$5$	−330.525	−1.18252	−0.591261	−	0.806480i	$-0.701370\pi$
$5$	−330.525	−1.18252	−0.591261	+	0.806480i	$0.701370\pi$
$6$	638.569	1.20692
$7$	1294.09	1.42601	0.713004	−	0.701160i	$-0.247334\pi$
$7$	1294.09	1.42601	0.713004	+	0.701160i	$0.247334\pi$
$8$	512.000	0.353553
$9$	4184.42	1.91331
$10$	−2644.20	−0.836170
$11$	3402.95	0.770871	0.385436	−	0.922735i	$-0.374051\pi$
$11$	3402.95	0.770871	0.385436	+	0.922735i	$0.374051\pi$
$12$	5108.55	0.853422
$13$	−13955.5	−1.76175	−0.880876	−	0.473347i	$-0.843046\pi$
$13$	−13955.5	−1.76175	−0.880876	+	0.473347i	$0.843046\pi$
$14$	10352.7	1.00834
$15$	−26382.9	−2.01838
$16$	4096.00	0.250000
$17$	18722.0	0.924234	0.462117	−	0.886819i	$-0.347090\pi$
$17$	18722.0	0.924234	0.462117	+	0.886819i	$0.347090\pi$
$18$	33475.3	1.35292
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	−21153.6	−0.591261
$21$	103296.	2.43397
$22$	27223.6	0.545088
$23$	−59529.0	−1.02019	−0.510095	−	0.860118i	$-0.670390\pi$
$23$	−59529.0	−1.02019	−0.510095	+	0.860118i	$0.670390\pi$
$24$	40868.4	0.603460
$25$	31121.8	0.398359
$26$	−111644.	−1.24575
$27$	159436.	1.55888
$28$	82821.9	0.713004
$29$	−197670.	−1.50504	−0.752521	−	0.658568i	$-0.771162\pi$
$29$	−197670.	−1.50504	−0.752521	+	0.658568i	$0.771162\pi$
$30$	−211063.	−1.42721
$31$	−65838.5	−0.396930	−0.198465	−	0.980108i	$-0.563596\pi$
$31$	−65838.5	−0.396930	−0.198465	+	0.980108i	$0.563596\pi$
$32$	32768.0	0.176777
$33$	271628.	1.31576
$34$	149776.	0.653532
$35$	−427730.	−1.68629
$36$	267803.	0.956657
$37$	436467.	1.41659	0.708297	−	0.705915i	$-0.249464\pi$
$37$	436467.	1.41659	0.708297	+	0.705915i	$0.249464\pi$
$38$	54872.0	0.162221
$39$	−1.11395e6	−3.00703
$40$	−169229.	−0.418085
$41$	88301.2	0.200089	0.100045	−	0.994983i	$-0.468101\pi$
$41$	88301.2	0.200089	0.100045	+	0.994983i	$0.468101\pi$
$42$	826367.	1.72108
$43$	−590778.	−1.13314	−0.566572	−	0.824013i	$-0.691731\pi$
$43$	−590778.	−1.13314	−0.566572	+	0.824013i	$0.691731\pi$
$44$	217789.	0.385436
$45$	−1.38305e6	−2.26254
$46$	−476232.	−0.721383
$47$	−424919.	−0.596986	−0.298493	−	0.954412i	$-0.596484\pi$
$47$	−424919.	−0.596986	−0.298493	+	0.954412i	$0.596484\pi$
$48$	326947.	0.426711
$49$	851131.	1.03350
$50$	248974.	0.281682
$51$	1.49441e6	1.57752
$52$	−893154.	−0.880876
$53$	−697171.	−0.643241	−0.321620	−	0.946869i	$-0.604227\pi$
$53$	−697171.	−0.643241	−0.321620	+	0.946869i	$0.604227\pi$
$54$	1.27549e6	1.10230
$55$	−1.12476e6	−0.911572
$56$	662575.	0.504170
$57$	547493.	0.391577
$58$	−1.58136e6	−1.06423
$59$	−2.78553e6	−1.76574	−0.882868	−	0.469621i	$-0.844390\pi$
$59$	−2.78553e6	−1.76574	−0.882868	+	0.469621i	$0.844390\pi$
$60$	−1.68850e6	−1.00919
$61$	−1.26287e6	−0.712369	−0.356184	−	0.934416i	$-0.615922\pi$
$61$	−1.26287e6	−0.712369	−0.356184	+	0.934416i	$0.615922\pi$
$62$	−526708.	−0.280672
$63$	5.41502e6	2.72840
$64$	262144.	0.125000
$65$	4.61265e6	2.08331
$66$	2.17302e6	0.930380
$67$	2.67972e6	1.08850	0.544249	−	0.838924i	$-0.316815\pi$
$67$	2.67972e6	1.08850	0.544249	+	0.838924i	$0.316815\pi$
$68$	1.19821e6	0.462117
$69$	−4.75167e6	−1.74130
$70$	−3.42184e6	−1.19238
$71$	537682.	0.178288	0.0891439	−	0.996019i	$-0.471587\pi$
$71$	537682.	0.178288	0.0891439	+	0.996019i	$0.471587\pi$
$72$	2.14242e6	0.676458
$73$	3.24778e6	0.977139	0.488569	−	0.872525i	$-0.337519\pi$
$73$	3.24778e6	0.977139	0.488569	+	0.872525i	$0.337519\pi$
$74$	3.49173e6	1.00168
$75$	2.48418e6	0.679936
$76$	438976.	0.114708
$77$	4.40374e6	1.09927
$78$	−8.91157e6	−2.12629
$79$	8.46098e6	1.93075	0.965375	−	0.260868i	$-0.0840087\pi$
$79$	8.46098e6	1.93075	0.965375	+	0.260868i	$0.0840087\pi$
$80$	−1.35383e6	−0.295631
$81$	3.57505e6	0.747454
$82$	706410.	0.141484
$83$	−2.31950e6	−0.445268	−0.222634	−	0.974902i	$-0.571465\pi$
$83$	−2.31950e6	−0.445268	−0.222634	+	0.974902i	$0.571465\pi$
$84$	6.61094e6	1.21699
$85$	−6.18810e6	−1.09293
$86$	−4.72622e6	−0.801253
$87$	−1.57783e7	−2.56887
$88$	1.74231e6	0.272544
$89$	1.72405e6	0.259229	0.129615	−	0.991564i	$-0.458626\pi$
$89$	1.72405e6	0.259229	0.129615	+	0.991564i	$0.458626\pi$
$90$	−1.10644e7	−1.59985
$91$	−1.80597e7	−2.51227
$92$	−3.80986e6	−0.510095
$93$	−5.25530e6	−0.677497
$94$	−3.39935e6	−0.422133
$95$	−2.26707e6	−0.271289
$96$	2.61558e6	0.301730
$97$	−675649.	−0.0751658	−0.0375829	−	0.999294i	$-0.511966\pi$
$97$	−675649.	−0.0751658	−0.0375829	+	0.999294i	$0.511966\pi$
$98$	6.80905e6	0.730795
$99$	1.42394e7	1.47492
$100$	1.99180e6	0.199180
$101$	7.66002e6	0.739784	0.369892	−	0.929075i	$-0.379395\pi$
$101$	7.66002e6	0.739784	0.369892	+	0.929075i	$0.379395\pi$
$102$	1.19553e7	1.11548
$103$	1.61486e7	1.45615	0.728073	−	0.685500i	$-0.240416\pi$
$103$	1.61486e7	1.45615	0.728073	+	0.685500i	$0.240416\pi$
$104$	−7.14523e6	−0.622873
$105$	−3.41419e7	−2.87823
$106$	−5.57737e6	−0.454840
$107$	2.96082e6	0.233652	0.116826	−	0.993152i	$-0.462728\pi$
$107$	2.96082e6	0.233652	0.116826	+	0.993152i	$0.462728\pi$
$108$	1.02039e7	0.779441
$109$	1.44071e7	1.06557	0.532786	−	0.846250i	$-0.321145\pi$
$109$	1.44071e7	1.06557	0.532786	+	0.846250i	$0.321145\pi$
$110$	−8.99809e6	−0.644579
$111$	3.48393e7	2.41790
$112$	5.30060e6	0.356502
$113$	−2.80082e6	−0.182604	−0.0913020	−	0.995823i	$-0.529103\pi$
$113$	−2.80082e6	−0.182604	−0.0913020	+	0.995823i	$0.529103\pi$
$114$	4.37995e6	0.276887
$115$	1.96758e7	1.20640
$116$	−1.26509e7	−0.752521
$117$	−5.83958e7	−3.37078
$118$	−2.22842e7	−1.24856
$119$	2.42280e7	1.31797
$120$	−1.35080e7	−0.713605
$121$	−7.90707e6	−0.405758
$122$	−1.01030e7	−0.503721
$123$	7.04831e6	0.341521
$124$	−4.21366e6	−0.198465
$125$	1.55357e7	0.711454
$126$	4.33202e7	1.92927
$127$	1.05226e7	0.455838	0.227919	−	0.973680i	$-0.426808\pi$
$127$	1.05226e7	0.455838	0.227919	+	0.973680i	$0.426808\pi$
$128$	2.09715e6	0.0883883
$129$	−4.71566e7	−1.93410
$130$	3.69012e7	1.47312
$131$	−7.33827e6	−0.285196	−0.142598	−	0.989781i	$-0.545546\pi$
$131$	−7.33827e6	−0.285196	−0.142598	+	0.989781i	$0.545546\pi$
$132$	1.73842e7	0.657878
$133$	8.87618e6	0.327149
$134$	2.14377e7	0.769684
$135$	−5.26976e7	−1.84341
$136$	9.58568e6	0.326766
$137$	2.76807e7	0.919719	0.459860	−	0.887992i	$-0.347900\pi$
$137$	2.76807e7	0.919719	0.459860	+	0.887992i	$0.347900\pi$
$138$	−3.80134e7	−1.23129
$139$	1.56327e7	0.493721	0.246860	−	0.969051i	$-0.420601\pi$
$139$	1.56327e7	0.493721	0.246860	+	0.969051i	$0.420601\pi$
$140$	−2.73747e7	−0.843143
$141$	−3.39175e7	−1.01896
$142$	4.30146e6	0.126068
$143$	−4.74900e7	−1.35808
$144$	1.71394e7	0.478328
$145$	6.53350e7	1.77975
$146$	2.59822e7	0.690941
$147$	6.79383e7	1.76402
$148$	2.79339e7	0.708297
$149$	−3.06822e7	−0.759862	−0.379931	−	0.925015i	$-0.624052\pi$
$149$	−3.06822e7	−0.759862	−0.379931	+	0.925015i	$0.624052\pi$
$150$	1.98734e7	0.480788
$151$	1.88077e7	0.444545	0.222273	−	0.974985i	$-0.428652\pi$
$151$	1.88077e7	0.444545	0.222273	+	0.974985i	$0.428652\pi$
$152$	3.51181e6	0.0811107
$153$	7.83408e7	1.76835
$154$	3.52299e7	0.777300
$155$	2.17613e7	0.469378
$156$	−7.12926e7	−1.50352
$157$	−8.56960e7	−1.76731	−0.883653	−	0.468143i	$-0.844923\pi$
$157$	−8.56960e7	−1.76731	−0.883653	+	0.468143i	$0.844923\pi$
$158$	6.76878e7	1.36525
$159$	−5.56490e7	−1.09791
$160$	−1.08306e7	−0.209042
$161$	−7.70360e7	−1.45480
$162$	2.86004e7	0.528530
$163$	1.98916e7	0.359761	0.179880	−	0.983688i	$-0.442429\pi$
$163$	1.98916e7	0.359761	0.179880	+	0.983688i	$0.442429\pi$
$164$	5.65128e6	0.100045
$165$	−8.97798e7	−1.55591
$166$	−1.85560e7	−0.314852
$167$	−9.93383e7	−1.65048	−0.825238	−	0.564785i	$-0.808959\pi$
$167$	−9.93383e7	−1.65048	−0.825238	+	0.564785i	$0.808959\pi$
$168$	5.28875e7	0.860539
$169$	1.32008e8	2.10377
$170$	−4.95048e7	−0.772816
$171$	2.87009e7	0.438944
$172$	−3.78098e7	−0.566572
$173$	4.62203e7	0.678689	0.339345	−	0.940662i	$-0.389795\pi$
$173$	4.62203e7	0.678689	0.339345	+	0.940662i	$0.389795\pi$
$174$	−1.26226e8	−1.81647
$175$	4.02745e7	0.568063
$176$	1.39385e7	0.192718
$177$	−2.22344e8	−3.01383
$178$	1.37924e7	0.183303
$179$	9.68920e7	1.26271	0.631353	−	0.775495i	$-0.282500\pi$
$179$	9.68920e7	1.26271	0.631353	+	0.775495i	$0.282500\pi$
$180$	−8.85155e7	−1.13127
$181$	1.14940e8	1.44078	0.720389	−	0.693570i	$-0.243963\pi$
$181$	1.14940e8	1.44078	0.720389	+	0.693570i	$0.243963\pi$
$182$	−1.44478e8	−1.77644
$183$	−1.00804e8	−1.21590
$184$	−3.04788e7	−0.360692
$185$	−1.44263e8	−1.67515
$186$	−4.20424e7	−0.479063
$187$	6.37102e7	0.712465
$188$	−2.71948e7	−0.298493
$189$	2.06325e8	2.22298
$190$	−1.81366e7	−0.191830
$191$	5.03031e7	0.522369	0.261185	−	0.965289i	$-0.415887\pi$
$191$	5.03031e7	0.522369	0.261185	+	0.965289i	$0.415887\pi$
$192$	2.09246e7	0.213355
$193$	−7.04906e6	−0.0705799	−0.0352899	−	0.999377i	$-0.511235\pi$
$193$	−7.04906e6	−0.0705799	−0.0352899	+	0.999377i	$0.511235\pi$
$194$	−5.40519e6	−0.0531502
$195$	3.68187e8	3.55588
$196$	5.44724e7	0.516750
$197$	−8.45761e7	−0.788163	−0.394081	−	0.919076i	$-0.628937\pi$
$197$	−8.45761e7	−0.788163	−0.394081	+	0.919076i	$0.628937\pi$
$198$	1.13915e8	1.04292
$199$	−1.23496e8	−1.11088	−0.555439	−	0.831557i	$-0.687450\pi$
$199$	−1.23496e8	−1.11088	−0.555439	+	0.831557i	$0.687450\pi$
$200$	1.59344e7	0.140841
$201$	2.13898e8	1.85789
$202$	6.12802e7	0.523106
$203$	−2.55804e8	−2.14620
$204$	9.56425e7	0.788761
$205$	−2.91858e7	−0.236610
$206$	1.29189e8	1.02965
$207$	−2.49094e8	−1.95194
$208$	−5.71619e7	−0.440438
$209$	2.33409e7	0.176850
$210$	−2.73135e8	−2.03521
$211$	1.86177e7	0.136438	0.0682192	−	0.997670i	$-0.478268\pi$
$211$	1.86177e7	0.136438	0.0682192	+	0.997670i	$0.478268\pi$
$212$	−4.46189e7	−0.321620
$213$	4.29184e7	0.304309
$214$	2.36866e7	0.165217
$215$	1.95267e8	1.33997
$216$	8.16312e7	0.551148
$217$	−8.52010e7	−0.566025
$218$	1.15257e8	0.753474
$219$	2.59241e8	1.66782
$220$	−7.19847e7	−0.455786
$221$	−2.61276e8	−1.62827
$222$	2.78714e8	1.70972
$223$	−2.18672e7	−0.132046	−0.0660230	−	0.997818i	$-0.521031\pi$
$223$	−2.18672e7	−0.132046	−0.0660230	+	0.997818i	$0.521031\pi$
$224$	4.24048e7	0.252085
$225$	1.30227e8	0.762186
$226$	−2.24065e7	−0.129120
$227$	−2.01807e8	−1.14511	−0.572553	−	0.819868i	$-0.694047\pi$
$227$	−2.01807e8	−1.14511	−0.572553	+	0.819868i	$0.694047\pi$
$228$	3.50396e7	0.195788
$229$	−7.36441e7	−0.405242	−0.202621	−	0.979257i	$-0.564946\pi$
$229$	−7.36441e7	−0.405242	−0.202621	+	0.979257i	$0.564946\pi$
$230$	1.57407e8	0.853052
$231$	3.51511e8	1.87628
$232$	−1.01207e8	−0.532113
$233$	−5.30771e7	−0.274892	−0.137446	−	0.990509i	$-0.543889\pi$
$233$	−5.30771e7	−0.274892	−0.137446	+	0.990509i	$0.543889\pi$
$234$	−4.67166e8	−2.38350
$235$	1.40446e8	0.705949
$236$	−1.78274e8	−0.882868
$237$	6.75365e8	3.29549
$238$	1.93824e8	0.931942
$239$	−1.20670e8	−0.571749	−0.285875	−	0.958267i	$-0.592284\pi$
$239$	−1.20670e8	−0.571749	−0.285875	+	0.958267i	$0.592284\pi$
$240$	−1.08064e8	−0.504595
$241$	1.81290e8	0.834285	0.417142	−	0.908841i	$-0.363032\pi$
$241$	1.81290e8	0.834285	0.417142	+	0.908841i	$0.363032\pi$
$242$	−6.32566e7	−0.286914
$243$	−6.33221e7	−0.283096
$244$	−8.08238e7	−0.356184
$245$	−2.81320e8	−1.22214
$246$	5.63864e7	0.241491
$247$	−9.57210e7	−0.404174
$248$	−3.37093e7	−0.140336
$249$	−1.85145e8	−0.760002
$250$	1.24286e8	0.503074
$251$	−3.14723e7	−0.125623	−0.0628116	−	0.998025i	$-0.520007\pi$
$251$	−3.14723e7	−0.125623	−0.0628116	+	0.998025i	$0.520007\pi$
$252$	3.46561e8	1.36420
$253$	−2.02574e8	−0.786435
$254$	8.41808e7	0.322326
$255$	−4.93941e8	−1.86546
$256$	1.67772e7	0.0625000
$257$	4.78878e8	1.75978	0.879890	−	0.475177i	$-0.157616\pi$
$257$	4.78878e8	1.75978	0.879890	+	0.475177i	$0.157616\pi$
$258$	−3.77253e8	−1.36761
$259$	5.64828e8	2.02007
$260$	2.95210e8	1.04166
$261$	−8.27135e8	−2.87962
$262$	−5.87062e7	−0.201664
$263$	−4.72709e8	−1.60232	−0.801158	−	0.598453i	$-0.795783\pi$
$263$	−4.72709e8	−1.60232	−0.801158	+	0.598453i	$0.795783\pi$
$264$	1.39073e8	0.465190
$265$	2.30432e8	0.760647
$266$	7.10094e7	0.231329
$267$	1.37615e8	0.442463
$268$	1.71502e8	0.544249
$269$	2.00270e8	0.627311	0.313655	−	0.949537i	$-0.398446\pi$
$269$	2.00270e8	0.627311	0.313655	+	0.949537i	$0.398446\pi$
$270$	−4.21581e8	−1.30349
$271$	−2.49114e8	−0.760335	−0.380167	−	0.924918i	$-0.624134\pi$
$271$	−2.49114e8	−0.760335	−0.380167	+	0.924918i	$0.624134\pi$
$272$	7.66855e7	0.231058
$273$	−1.44155e9	−4.28805
$274$	2.21446e8	0.650340
$275$	1.05906e8	0.307083
$276$	−3.04107e8	−0.870652
$277$	5.12716e8	1.44943	0.724716	−	0.689048i	$-0.241971\pi$
$277$	5.12716e8	1.44943	0.724716	+	0.689048i	$0.241971\pi$
$278$	1.25061e8	0.349113
$279$	−2.75496e8	−0.759451
$280$	−2.18998e8	−0.596192
$281$	−2.63422e8	−0.708240	−0.354120	−	0.935200i	$-0.615220\pi$
$281$	−2.63422e8	−0.708240	−0.354120	+	0.935200i	$0.615220\pi$
$282$	−2.71340e8	−0.720514
$283$	−1.24481e8	−0.326474	−0.163237	−	0.986587i	$-0.552194\pi$
$283$	−1.24481e8	−0.326474	−0.163237	+	0.986587i	$0.552194\pi$
$284$	3.44117e7	0.0891439
$285$	−1.80960e8	−0.463048
$286$	−3.79920e8	−0.960310
$287$	1.14270e8	0.285329
$288$	1.37115e8	0.338229
$289$	−5.98239e7	−0.145792
$290$	5.22680e8	1.25847
$291$	−5.39311e7	−0.128296
$292$	2.07858e8	0.488569
$293$	−7.81779e8	−1.81571	−0.907857	−	0.419280i	$-0.862283\pi$
$293$	−7.81779e8	−1.81571	−0.907857	+	0.419280i	$0.862283\pi$
$294$	5.43506e8	1.24735
$295$	9.20687e8	2.08802
$296$	2.23471e8	0.500841
$297$	5.42554e8	1.20170
$298$	−2.45458e8	−0.537303
$299$	8.30759e8	1.79732
$300$	1.58987e8	0.339968
$301$	−7.64521e8	−1.61587
$302$	1.50462e8	0.314341
$303$	6.11432e8	1.26270
$304$	2.80945e7	0.0573539
$305$	4.17411e8	0.842392
$306$	6.26726e8	1.25041
$307$	−3.30905e8	−0.652708	−0.326354	−	0.945248i	$-0.605820\pi$
$307$	−3.30905e8	−0.652708	−0.326354	+	0.945248i	$0.605820\pi$
$308$	2.81839e8	0.549634
$309$	1.28900e9	2.48541
$310$	1.74090e8	0.331901
$311$	6.01012e8	1.13298	0.566490	−	0.824069i	$-0.308301\pi$
$311$	6.01012e8	1.13298	0.566490	+	0.824069i	$0.308301\pi$
$312$	−5.70341e8	−1.06315
$313$	−1.37125e8	−0.252761	−0.126381	−	0.991982i	$-0.540336\pi$
$313$	−1.37125e8	−0.252761	−0.126381	+	0.991982i	$0.540336\pi$
$314$	−6.85568e8	−1.24967
$315$	−1.78980e9	−3.22639
$316$	5.41503e8	0.965375
$317$	7.75395e8	1.36715	0.683574	−	0.729881i	$-0.260425\pi$
$317$	7.75395e8	1.36715	0.683574	+	0.729881i	$0.260425\pi$
$318$	−4.45192e8	−0.776340
$319$	−6.72663e8	−1.16019
$320$	−8.66452e7	−0.147815
$321$	2.36336e8	0.398807
$322$	−6.16288e8	−1.02870
$323$	1.28414e8	0.212034
$324$	2.28803e8	0.373727
$325$	−4.34321e8	−0.701810
$326$	1.59133e8	0.254389
$327$	1.14999e9	1.81876
$328$	4.52102e7	0.0707421
$329$	−5.49885e8	−0.851306
$330$	−7.18238e8	−1.10020
$331$	−1.01426e9	−1.53727	−0.768636	−	0.639687i	$-0.779064\pi$
$331$	−1.01426e9	−1.53727	−0.768636	+	0.639687i	$0.779064\pi$
$332$	−1.48448e8	−0.222634
$333$	1.82636e9	2.71039
$334$	−7.94707e8	−1.16706
$335$	−8.85714e8	−1.28717
$336$	4.23100e8	0.608493
$337$	9.79983e8	1.39481	0.697403	−	0.716679i	$-0.254339\pi$
$337$	9.79983e8	1.39481	0.697403	+	0.716679i	$0.254339\pi$
$338$	1.05607e9	1.48759
$339$	−2.23564e8	−0.311676
$340$	−3.96039e8	−0.546464
$341$	−2.24045e8	−0.305982
$342$	2.29607e8	0.310380
$343$	3.57019e7	0.0477707
$344$	−3.02478e8	−0.400627
$345$	1.57055e9	2.05913
$346$	3.69762e8	0.479906
$347$	5.23903e8	0.673128	0.336564	−	0.941661i	$-0.390735\pi$
$347$	5.23903e8	0.673128	0.336564	+	0.941661i	$0.390735\pi$
$348$	−1.00981e9	−1.28443
$349$	6.94676e8	0.874769	0.437384	−	0.899275i	$-0.355905\pi$
$349$	6.94676e8	0.874769	0.437384	+	0.899275i	$0.355905\pi$
$350$	3.22196e8	0.401681
$351$	−2.22501e9	−2.74636
$352$	1.11508e8	0.136272
$353$	7.99575e7	0.0967492	0.0483746	−	0.998829i	$-0.484596\pi$
$353$	7.99575e7	0.0967492	0.0483746	+	0.998829i	$0.484596\pi$
$354$	−1.77875e9	−2.13110
$355$	−1.77717e8	−0.210829
$356$	1.10339e8	0.129615
$357$	1.93391e9	2.24956
$358$	7.75136e8	0.892868
$359$	−8.52281e7	−0.0972192	−0.0486096	−	0.998818i	$-0.515479\pi$
$359$	−8.52281e7	−0.0972192	−0.0486096	+	0.998818i	$0.515479\pi$
$360$	−7.08124e8	−0.799927
$361$	4.70459e7	0.0526316
$362$	9.19522e8	1.01878
$363$	−6.31152e8	−0.692565
$364$	−1.15582e9	−1.25614
$365$	−1.07347e9	−1.15549
$366$	−8.06431e8	−0.859772
$367$	−1.59386e9	−1.68313	−0.841567	−	0.540152i	$-0.818367\pi$
$367$	−1.59386e9	−1.68313	−0.841567	+	0.540152i	$0.818367\pi$
$368$	−2.43831e8	−0.255048
$369$	3.69489e8	0.382833
$370$	−1.15411e9	−1.18451
$371$	−9.02203e8	−0.917267
$372$	−3.36339e8	−0.338749
$373$	4.50474e8	0.449458	0.224729	−	0.974421i	$-0.427850\pi$
$373$	4.50474e8	0.449458	0.224729	+	0.974421i	$0.427850\pi$
$374$	5.09682e8	0.503789
$375$	1.24008e9	1.21434
$376$	−2.17559e8	−0.211066
$377$	2.75859e9	2.65151
$378$	1.65060e9	1.57188
$379$	8.55260e8	0.806977	0.403488	−	0.914985i	$-0.367798\pi$
$379$	8.55260e8	0.806977	0.403488	+	0.914985i	$0.367798\pi$
$380$	−1.45093e8	−0.135645
$381$	8.39926e8	0.778043
$382$	4.02425e8	0.369371
$383$	1.86910e9	1.69996	0.849978	−	0.526819i	$-0.176615\pi$
$383$	1.86910e9	1.69996	0.849978	+	0.526819i	$0.176615\pi$
$384$	1.67397e8	0.150865
$385$	−1.45555e9	−1.29991
$386$	−5.63925e7	−0.0499075
$387$	−2.47206e9	−2.16806
$388$	−4.32415e7	−0.0375829
$389$	−9.69679e8	−0.835227	−0.417613	−	0.908625i	$-0.637133\pi$
$389$	−9.69679e8	−0.835227	−0.417613	+	0.908625i	$0.637133\pi$
$390$	2.94550e9	2.51439
$391$	−1.11450e9	−0.942894
$392$	4.35779e8	0.365397
$393$	−5.85749e8	−0.486786
$394$	−6.76609e8	−0.557315
$395$	−2.79657e9	−2.28315
$396$	9.11320e8	0.737459
$397$	−7.95454e8	−0.638041	−0.319020	−	0.947748i	$-0.603354\pi$
$397$	−7.95454e8	−0.638041	−0.319020	+	0.947748i	$0.603354\pi$
$398$	−9.87966e8	−0.785509
$399$	7.08507e8	0.558392
$400$	1.27475e8	0.0995898
$401$	−1.04256e9	−0.807412	−0.403706	−	0.914889i	$-0.632278\pi$
$401$	−1.04256e9	−0.807412	−0.403706	+	0.914889i	$0.632278\pi$
$402$	1.71119e9	1.31373
$403$	9.18811e8	0.699292
$404$	4.90241e8	0.369892
$405$	−1.18164e9	−0.883881
$406$	−2.04643e9	−1.51759
$407$	1.48528e9	1.09201
$408$	7.65140e8	0.557738
$409$	−1.54551e9	−1.11697	−0.558485	−	0.829515i	$-0.688617\pi$
$409$	−1.54551e9	−1.11697	−0.558485	+	0.829515i	$0.688617\pi$
$410$	−2.33486e8	−0.167308
$411$	2.20951e9	1.56982
$412$	1.03351e9	0.728073
$413$	−3.60473e9	−2.51795
$414$	−1.99275e9	−1.38023
$415$	7.66653e8	0.526539
$416$	−4.57295e8	−0.311437
$417$	1.24782e9	0.842704
$418$	1.86727e8	0.125052
$419$	−1.72415e9	−1.14505	−0.572527	−	0.819886i	$-0.694037\pi$
$419$	−1.72415e9	−1.14505	−0.572527	+	0.819886i	$0.694037\pi$
$420$	−2.18508e9	−1.43911
$421$	−6.56525e8	−0.428809	−0.214404	−	0.976745i	$-0.568781\pi$
$421$	−6.56525e8	−0.428809	−0.214404	+	0.976745i	$0.568781\pi$
$422$	1.48941e8	0.0964765
$423$	−1.77804e9	−1.14222
$424$	−3.56951e8	−0.227420
$425$	5.82663e8	0.368177
$426$	3.43347e8	0.215179
$427$	−1.63427e9	−1.01584
$428$	1.89492e8	0.116826
$429$	−3.79071e9	−2.31804
$430$	1.56214e9	0.947500
$431$	1.29848e9	0.781204	0.390602	−	0.920560i	$-0.372267\pi$
$431$	1.29848e9	0.781204	0.390602	+	0.920560i	$0.372267\pi$
$432$	6.53050e8	0.389721
$433$	2.91145e9	1.72346	0.861731	−	0.507366i	$-0.169381\pi$
$433$	2.91145e9	1.72346	0.861731	+	0.507366i	$0.169381\pi$
$434$	−6.81608e8	−0.400240
$435$	5.21511e9	3.03775
$436$	9.22053e8	0.532786
$437$	−4.08309e8	−0.234048
$438$	2.07393e9	1.17933
$439$	2.61576e9	1.47561	0.737806	−	0.675013i	$-0.235862\pi$
$439$	2.61576e9	1.47561	0.737806	+	0.675013i	$0.235862\pi$
$440$	−5.75878e8	−0.322289
$441$	3.56149e9	1.97741
$442$	−2.09021e9	−1.15136
$443$	2.73901e8	0.149686	0.0748430	−	0.997195i	$-0.476154\pi$
$443$	2.73901e8	0.149686	0.0748430	+	0.997195i	$0.476154\pi$
$444$	2.22971e9	1.20895
$445$	−5.69840e8	−0.306544
$446$	−1.74937e8	−0.0933707
$447$	−2.44909e9	−1.29696
$448$	3.39238e8	0.178251
$449$	−1.17866e9	−0.614506	−0.307253	−	0.951628i	$-0.599410\pi$
$449$	−1.17866e9	−0.614506	−0.307253	+	0.951628i	$0.599410\pi$
$450$	1.04181e9	0.538947
$451$	3.00485e8	0.154243
$452$	−1.79252e8	−0.0913020
$453$	1.50125e9	0.758769
$454$	−1.61445e9	−0.809712
$455$	5.96920e9	2.97082
$456$	2.80317e8	0.138443
$457$	2.63721e9	1.29252	0.646262	−	0.763116i	$-0.276331\pi$
$457$	2.63721e9	1.29252	0.646262	+	0.763116i	$0.276331\pi$
$458$	−5.89153e8	−0.286549
$459$	2.98497e9	1.44077
$460$	1.25925e9	0.603199
$461$	−9.72641e8	−0.462380	−0.231190	−	0.972909i	$-0.574262\pi$
$461$	−9.72641e8	−0.462380	−0.231190	+	0.972909i	$0.574262\pi$
$462$	2.81209e9	1.32673
$463$	1.95954e8	0.0917531	0.0458766	−	0.998947i	$-0.485392\pi$
$463$	1.95954e8	0.0917531	0.0458766	+	0.998947i	$0.485392\pi$
$464$	−8.09658e8	−0.376260
$465$	1.73701e9	0.801155
$466$	−4.24617e8	−0.194378
$467$	−1.68202e9	−0.764228	−0.382114	−	0.924115i	$-0.624804\pi$
$467$	−1.68202e9	−0.764228	−0.382114	+	0.924115i	$0.624804\pi$
$468$	−3.73733e9	−1.68539
$469$	3.46780e9	1.55221
$470$	1.12357e9	0.499181
$471$	−6.84035e9	−3.01651
$472$	−1.42619e9	−0.624282
$473$	−2.01039e9	−0.873507
$474$	5.40292e9	2.33026
$475$	2.13464e8	0.0913898
$476$	1.55059e9	0.658983
$477$	−2.91725e9	−1.23072
$478$	−9.65358e8	−0.404288
$479$	1.28606e9	0.534671	0.267336	−	0.963603i	$-0.413857\pi$
$479$	1.28606e9	0.534671	0.267336	+	0.963603i	$0.413857\pi$
$480$	−8.64514e8	−0.356803
$481$	−6.09113e9	−2.49569
$482$	1.45032e9	0.589929
$483$	−6.14910e9	−2.48311
$484$	−5.06053e8	−0.202879
$485$	2.23319e8	0.0888852
$486$	−5.06576e8	−0.200179
$487$	−3.31949e8	−0.130233	−0.0651163	−	0.997878i	$-0.520742\pi$
$487$	−3.31949e8	−0.130233	−0.0651163	+	0.997878i	$0.520742\pi$
$488$	−6.46590e8	−0.251860
$489$	1.58777e9	0.614055
$490$	−2.25056e9	−0.864181
$491$	2.94088e9	1.12122	0.560611	−	0.828079i	$-0.310566\pi$
$491$	2.94088e9	1.12122	0.560611	+	0.828079i	$0.310566\pi$
$492$	4.51092e8	0.170760
$493$	−3.70079e9	−1.39101
$494$	−7.65768e8	−0.285794
$495$	−4.70647e9	−1.74412
$496$	−2.69674e8	−0.0992325
$497$	6.95810e8	0.254240
$498$	−1.48116e9	−0.537403
$499$	2.16559e9	0.780232	0.390116	−	0.920766i	$-0.372435\pi$
$499$	2.16559e9	0.780232	0.390116	+	0.920766i	$0.372435\pi$
$500$	9.94287e8	0.355727
$501$	−7.92930e9	−2.81710
$502$	−2.51778e8	−0.0888291
$503$	3.55297e9	1.24481	0.622406	−	0.782695i	$-0.286155\pi$
$503$	3.55297e9	1.24481	0.622406	+	0.782695i	$0.286155\pi$
$504$	2.77249e9	0.964635
$505$	−2.53183e9	−0.874811
$506$	−1.62060e9	−0.556094
$507$	1.05371e10	3.59080
$508$	6.73447e8	0.227919
$509$	−3.01169e9	−1.01228	−0.506138	−	0.862453i	$-0.668927\pi$
$509$	−3.01169e9	−1.01228	−0.506138	+	0.862453i	$0.668927\pi$
$510$	−3.95153e9	−1.31908
$511$	4.20292e9	1.39341
$512$	1.34218e8	0.0441942
$513$	1.09357e9	0.357632
$514$	3.83102e9	1.24435
$515$	−5.33752e9	−1.72192
$516$	−3.01802e9	−0.967049
$517$	−1.44598e9	−0.460199
$518$	4.51863e9	1.42841
$519$	3.68935e9	1.15842
$520$	2.36168e9	0.736562
$521$	2.71528e9	0.841167	0.420583	−	0.907254i	$-0.361825\pi$
$521$	2.71528e9	0.841167	0.420583	+	0.907254i	$0.361825\pi$
$522$	−6.61708e9	−2.03620
$523$	−2.42694e9	−0.741828	−0.370914	−	0.928667i	$-0.620956\pi$
$523$	−2.42694e9	−0.741828	−0.370914	+	0.928667i	$0.620956\pi$
$524$	−4.69649e8	−0.142598
$525$	3.21475e9	0.969595
$526$	−3.78167e9	−1.13301
$527$	−1.23263e9	−0.366856
$528$	1.11259e9	0.328939
$529$	1.38876e8	0.0407879
$530$	1.84346e9	0.537858
$531$	−1.16558e10	−3.37841
$532$	5.68075e8	0.163574
$533$	−1.23229e9	−0.352507
$534$	1.10092e9	0.312869
$535$	−9.78625e8	−0.276298
$536$	1.37202e9	0.384842
$537$	7.73403e9	2.15524
$538$	1.60216e9	0.443576
$539$	2.89636e9	0.796695
$540$	−3.37265e9	−0.921706
$541$	4.62943e9	1.25700	0.628502	−	0.777808i	$-0.283668\pi$
$541$	4.62943e9	1.25700	0.628502	+	0.777808i	$0.283668\pi$
$542$	−1.99291e9	−0.537638
$543$	9.17466e9	2.45918
$544$	6.13484e8	0.163383
$545$	−4.76190e9	−1.26006
$546$	−1.15324e10	−3.03211
$547$	1.73198e9	0.452466	0.226233	−	0.974073i	$-0.427359\pi$
$547$	1.73198e9	0.452466	0.226233	+	0.974073i	$0.427359\pi$
$548$	1.77157e9	0.459860
$549$	−5.28438e9	−1.36298
$550$	8.47248e8	0.217141
$551$	−1.35582e9	−0.345280
$552$	−2.43286e9	−0.615644
$553$	1.09493e10	2.75326
$554$	4.10173e9	1.02490
$555$	−1.15153e10	−2.85922
$556$	1.00049e9	0.246860
$557$	−2.73961e8	−0.0671731	−0.0335865	−	0.999436i	$-0.510693\pi$
$557$	−2.73961e8	−0.0671731	−0.0335865	+	0.999436i	$0.510693\pi$
$558$	−2.20396e9	−0.537013
$559$	8.24462e9	1.99632
$560$	−1.75198e9	−0.421572
$561$	5.08543e9	1.21607
$562$	−2.10738e9	−0.500802
$563$	4.29589e7	0.0101455	0.00507275	−	0.999987i	$-0.498385\pi$
$563$	4.29589e7	0.0101455	0.00507275	+	0.999987i	$0.498385\pi$
$564$	−2.17072e9	−0.509480
$565$	9.25740e8	0.215933
$566$	−9.95845e8	−0.230852
$567$	4.62644e9	1.06588
$568$	2.75293e8	0.0630342
$569$	−5.00452e9	−1.13886	−0.569429	−	0.822040i	$-0.692836\pi$
$569$	−5.00452e9	−1.13886	−0.569429	+	0.822040i	$0.692836\pi$
$570$	−1.44768e9	−0.327424
$571$	4.83434e9	1.08670	0.543352	−	0.839505i	$-0.317155\pi$
$571$	4.83434e9	1.08670	0.543352	+	0.839505i	$0.317155\pi$
$572$	−3.03936e9	−0.679042
$573$	4.01525e9	0.891603
$574$	9.14159e8	0.201758
$575$	−1.85265e9	−0.406402
$576$	1.09692e9	0.239164
$577$	−2.99513e9	−0.649082	−0.324541	−	0.945872i	$-0.605210\pi$
$577$	−2.99513e9	−0.649082	−0.324541	+	0.945872i	$0.605210\pi$
$578$	−4.78591e8	−0.103090
$579$	−5.62664e8	−0.120469
$580$	4.18144e9	0.889873
$581$	−3.00165e9	−0.634956
$582$	−4.31449e8	−0.0907191
$583$	−2.37244e9	−0.495856
$584$	1.66286e9	0.345471
$585$	1.93013e10	3.98603
$586$	−6.25423e9	−1.28390
$587$	1.88112e9	0.383868	0.191934	−	0.981408i	$-0.438524\pi$
$587$	1.88112e9	0.383868	0.191934	+	0.981408i	$0.438524\pi$
$588$	4.34805e9	0.882011
$589$	−4.51586e8	−0.0910620
$590$	7.36550e9	1.47645
$591$	−6.75096e9	−1.34527
$592$	1.78777e9	0.354148
$593$	−3.10239e9	−0.610949	−0.305474	−	0.952200i	$-0.598815\pi$
$593$	−3.10239e9	−0.610949	−0.305474	+	0.952200i	$0.598815\pi$
$594$	4.34043e9	0.849728
$595$	−8.00797e9	−1.55852
$596$	−1.96366e9	−0.379931
$597$	−9.85757e9	−1.89609
$598$	6.64607e9	1.27090
$599$	4.60481e9	0.875423	0.437712	−	0.899115i	$-0.355789\pi$
$599$	4.60481e9	0.875423	0.437712	+	0.899115i	$0.355789\pi$
$600$	1.27190e9	0.240394
$601$	9.59555e8	0.180306	0.0901528	−	0.995928i	$-0.471264\pi$
$601$	9.59555e8	0.180306	0.0901528	+	0.995928i	$0.471264\pi$
$602$	−6.11617e9	−1.14259
$603$	1.12131e10	2.08264
$604$	1.20369e9	0.222273
$605$	2.61349e9	0.479818
$606$	4.89145e9	0.892861
$607$	−1.69208e9	−0.307086	−0.153543	−	0.988142i	$-0.549068\pi$
$607$	−1.69208e9	−0.307086	−0.153543	+	0.988142i	$0.549068\pi$
$608$	2.24756e8	0.0405554
$609$	−2.04185e10	−3.66323
$610$	3.33929e9	0.595661
$611$	5.92997e9	1.05174
$612$	5.01381e9	0.884174
$613$	3.11132e9	0.545548	0.272774	−	0.962078i	$-0.412059\pi$
$613$	3.11132e9	0.545548	0.272774	+	0.962078i	$0.412059\pi$
$614$	−2.64724e9	−0.461534
$615$	−2.32964e9	−0.403856
$616$	2.25471e9	0.388650
$617$	−2.66014e9	−0.455938	−0.227969	−	0.973668i	$-0.573209\pi$
$617$	−2.66014e9	−0.455938	−0.227969	+	0.973668i	$0.573209\pi$
$618$	1.03120e10	1.75745
$619$	5.21764e9	0.884212	0.442106	−	0.896963i	$-0.354231\pi$
$619$	5.21764e9	0.884212	0.442106	+	0.896963i	$0.354231\pi$
$620$	1.39272e9	0.234689
$621$	−9.49107e9	−1.59036
$622$	4.80810e9	0.801137
$623$	2.23107e9	0.369663
$624$	−4.56273e9	−0.751758
$625$	−7.56634e9	−1.23967
$626$	−1.09700e9	−0.178729
$627$	1.86309e9	0.301855
$628$	−5.48454e9	−0.883653
$629$	8.17155e9	1.30926
$630$	−1.43184e10	−2.28141
$631$	1.03639e9	0.164218	0.0821092	−	0.996623i	$-0.473834\pi$
$631$	1.03639e9	0.164218	0.0821092	+	0.996623i	$0.473834\pi$
$632$	4.33202e9	0.682623
$633$	1.48608e9	0.232879
$634$	6.20316e9	0.966719
$635$	−3.47798e9	−0.539038
$636$	−3.56153e9	−0.548956
$637$	−1.18780e10	−1.82077
$638$	−5.38131e9	−0.820380
$639$	2.24989e9	0.341120
$640$	−6.93161e8	−0.104521
$641$	3.09588e8	0.0464281	0.0232140	−	0.999731i	$-0.492610\pi$
$641$	3.09588e8	0.0464281	0.0232140	+	0.999731i	$0.492610\pi$
$642$	1.89069e9	0.281999
$643$	−3.23958e9	−0.480562	−0.240281	−	0.970703i	$-0.577240\pi$
$643$	−3.23958e9	−0.480562	−0.240281	+	0.970703i	$0.577240\pi$
$644$	−4.93030e9	−0.727400
$645$	1.55864e10	2.28711
$646$	1.02732e9	0.149931
$647$	−8.64171e9	−1.25440	−0.627198	−	0.778860i	$-0.715798\pi$
$647$	−8.64171e9	−1.25440	−0.627198	+	0.778860i	$0.715798\pi$
$648$	1.83043e9	0.264265
$649$	−9.47903e9	−1.36115
$650$	−3.47457e9	−0.496254
$651$	−6.80085e9	−0.966116
$652$	1.27306e9	0.179880
$653$	9.57361e9	1.34549	0.672743	−	0.739876i	$-0.265116\pi$
$653$	9.57361e9	1.34549	0.672743	+	0.739876i	$0.265116\pi$
$654$	9.19991e9	1.28606
$655$	2.42548e9	0.337251
$656$	3.61682e8	0.0500223
$657$	1.35901e10	1.86957
$658$	−4.39908e9	−0.601965
$659$	−3.61692e9	−0.492312	−0.246156	−	0.969230i	$-0.579168\pi$
$659$	−3.61692e9	−0.492312	−0.246156	+	0.969230i	$0.579168\pi$
$660$	−5.74591e9	−0.777955
$661$	−1.11304e10	−1.49901	−0.749507	−	0.661996i	$-0.769709\pi$
$661$	−1.11304e10	−1.49901	−0.749507	+	0.661996i	$0.769709\pi$
$662$	−8.11407e9	−1.08702
$663$	−2.08554e10	−2.77920
$664$	−1.18758e9	−0.157426
$665$	−2.93380e9	−0.386861
$666$	1.46109e10	1.91653
$667$	1.17671e10	1.53543
$668$	−6.35765e9	−0.825238
$669$	−1.74546e9	−0.225382
$670$	−7.08571e9	−0.910168
$671$	−4.29749e9	−0.549144
$672$	3.38480e9	0.430270
$673$	−1.32797e10	−1.67933	−0.839667	−	0.543102i	$-0.817250\pi$
$673$	−1.32797e10	−1.67933	−0.839667	+	0.543102i	$0.817250\pi$
$674$	7.83986e9	0.986277
$675$	4.96194e9	0.620995
$676$	8.44854e9	1.05188
$677$	1.47364e10	1.82529	0.912645	−	0.408754i	$-0.134036\pi$
$677$	1.47364e10	1.82529	0.912645	+	0.408754i	$0.134036\pi$
$678$	−1.78851e9	−0.220388
$679$	−8.74352e8	−0.107187
$680$	−3.16831e9	−0.386408
$681$	−1.61085e10	−1.95451
$682$	−1.79236e9	−0.216362
$683$	−5.73114e9	−0.688285	−0.344143	−	0.938917i	$-0.611830\pi$
$683$	−5.73114e9	−0.688285	−0.344143	+	0.938917i	$0.611830\pi$
$684$	1.83686e9	0.219472
$685$	−9.14917e9	−1.08759
$686$	2.85615e8	0.0337790
$687$	−5.87836e9	−0.691684
$688$	−2.41983e9	−0.283286
$689$	9.72939e9	1.13323
$690$	1.25644e10	1.45603
$691$	−2.18526e9	−0.251959	−0.125980	−	0.992033i	$-0.540207\pi$
$691$	−2.18526e9	−0.251959	−0.125980	+	0.992033i	$0.540207\pi$
$692$	2.95810e9	0.339345
$693$	1.84271e10	2.10324
$694$	4.19123e9	0.475974
$695$	−5.16699e9	−0.583836
$696$	−8.07848e9	−0.908233
$697$	1.65318e9	0.184929
$698$	5.55741e9	0.618555
$699$	−4.23668e9	−0.469197
$700$	2.57757e9	0.284032
$701$	−1.33627e10	−1.46514	−0.732572	−	0.680690i	$-0.761680\pi$
$701$	−1.33627e10	−1.46514	−0.732572	+	0.680690i	$0.761680\pi$
$702$	−1.78001e10	−1.94197
$703$	2.99373e9	0.324989
$704$	8.92064e8	0.0963589
$705$	1.12106e10	1.20494
$706$	6.39660e8	0.0684120
$707$	9.91277e9	1.05494
$708$	−1.42300e10	−1.50692
$709$	1.23774e10	1.30427	0.652137	−	0.758101i	$-0.273873\pi$
$709$	1.23774e10	1.30427	0.652137	+	0.758101i	$0.273873\pi$
$710$	−1.42174e9	−0.149079
$711$	3.54043e10	3.69413
$712$	8.82711e8	0.0916513
$713$	3.91930e9	0.404944
$714$	1.54713e10	1.59068
$715$	1.56966e10	1.60596
$716$	6.20109e9	0.631353
$717$	−9.63200e9	−0.975887
$718$	−6.81825e8	−0.0687444
$719$	1.67853e9	0.168414	0.0842071	−	0.996448i	$-0.473164\pi$
$719$	1.67853e9	0.168414	0.0842071	+	0.996448i	$0.473164\pi$
$720$	−5.66499e9	−0.565634
$721$	2.08978e10	2.07648
$722$	3.76367e8	0.0372161
$723$	1.44708e10	1.42399
$724$	7.35618e9	0.720389
$725$	−6.15186e9	−0.599547
$726$	−5.04921e9	−0.489717
$727$	−1.41549e10	−1.36627	−0.683136	−	0.730291i	$-0.739385\pi$
$727$	−1.41549e10	−1.36627	−0.683136	+	0.730291i	$0.739385\pi$
$728$	−9.24659e9	−0.888222
$729$	−1.28731e10	−1.23065
$730$	−8.58777e9	−0.817054
$731$	−1.10606e10	−1.04729
$732$	−6.45145e9	−0.607951
$733$	−1.14667e10	−1.07542	−0.537708	−	0.843131i	$-0.680710\pi$
$733$	−1.14667e10	−1.07542	−0.537708	+	0.843131i	$0.680710\pi$
$734$	−1.27509e10	−1.19016
$735$	−2.24553e10	−2.08599
$736$	−1.95065e9	−0.180346
$737$	9.11896e9	0.839091
$738$	2.95591e9	0.270704
$739$	1.16971e10	1.06616	0.533079	−	0.846065i	$-0.321035\pi$
$739$	1.16971e10	1.06616	0.533079	+	0.846065i	$0.321035\pi$
$740$	−9.23284e9	−0.837577
$741$	−7.64056e9	−0.689861
$742$	−7.21762e9	−0.648606
$743$	−1.06678e10	−0.954141	−0.477070	−	0.878865i	$-0.658301\pi$
$743$	−1.06678e10	−0.954141	−0.477070	+	0.878865i	$0.658301\pi$
$744$	−2.69071e9	−0.239531
$745$	1.01412e10	0.898553
$746$	3.60379e9	0.317815
$747$	−9.70576e9	−0.851937
$748$	4.07746e9	0.356233
$749$	3.83157e9	0.333189
$750$	9.92064e9	0.858668
$751$	−1.68517e10	−1.45179	−0.725894	−	0.687806i	$-0.758574\pi$
$751$	−1.68517e10	−1.45179	−0.725894	+	0.687806i	$0.758574\pi$
$752$	−1.74047e9	−0.149246
$753$	−2.51215e9	−0.214419
$754$	2.20688e10	1.87490
$755$	−6.21641e9	−0.525685
$756$	1.32048e10	1.11149
$757$	1.07110e10	0.897415	0.448708	−	0.893679i	$-0.351884\pi$
$757$	1.07110e10	0.897415	0.448708	+	0.893679i	$0.351884\pi$
$758$	6.84208e9	0.570619
$759$	−1.61697e10	−1.34232
$760$	−1.16074e9	−0.0959152
$761$	−7.23743e9	−0.595303	−0.297652	−	0.954675i	$-0.596203\pi$
$761$	−7.23743e9	−0.595303	−0.297652	+	0.954675i	$0.596203\pi$
$762$	6.71941e9	0.550160
$763$	1.86441e10	1.51952
$764$	3.21940e9	0.261185
$765$	−2.58936e10	−2.09111
$766$	1.49528e10	1.20205
$767$	3.88735e10	3.11079
$768$	1.33918e9	0.106678
$769$	3.53668e9	0.280449	0.140224	−	0.990120i	$-0.455218\pi$
$769$	3.53668e9	0.280449	0.140224	+	0.990120i	$0.455218\pi$
$770$	−1.16444e10	−0.919175
$771$	3.82246e10	3.00367
$772$	−4.51140e8	−0.0352899
$773$	−9.65082e9	−0.751512	−0.375756	−	0.926719i	$-0.622617\pi$
$773$	−9.65082e9	−0.751512	−0.375756	+	0.926719i	$0.622617\pi$
$774$	−1.97765e10	−1.53305
$775$	−2.04901e9	−0.158121
$776$	−3.45932e8	−0.0265751
$777$	4.50852e10	3.44795
$778$	−7.75743e9	−0.590594
$779$	6.05658e8	0.0459036
$780$	2.35640e10	1.77794
$781$	1.82971e9	0.137437
$782$	−8.91603e9	−0.666727
$783$	−3.15158e10	−2.34618
$784$	3.48623e9	0.258375
$785$	2.83247e10	2.08988
$786$	−4.68599e9	−0.344209
$787$	−4.02187e9	−0.294114	−0.147057	−	0.989128i	$-0.546980\pi$
$787$	−4.02187e9	−0.294114	−0.147057	+	0.989128i	$0.546980\pi$
$788$	−5.41287e9	−0.394081
$789$	−3.77321e10	−2.73490
$790$	−2.23725e10	−1.61443
$791$	−3.62451e9	−0.260395
$792$	7.29056e9	0.521462
$793$	1.76240e10	1.25502
$794$	−6.36363e9	−0.451163
$795$	1.83934e10	1.29830
$796$	−7.90373e9	−0.555439
$797$	−9.48149e9	−0.663396	−0.331698	−	0.943386i	$-0.607621\pi$
$797$	−9.48149e9	−0.663396	−0.331698	+	0.943386i	$0.607621\pi$
$798$	5.66805e9	0.394842
$799$	−7.95535e9	−0.551754
$800$	1.01980e9	0.0704206
$801$	7.21412e9	0.495986
$802$	−8.34047e9	−0.570926
$803$	1.10520e10	0.753248
$804$	1.36895e10	0.928947
$805$	2.54623e10	1.72033
$806$	7.35049e9	0.494474
$807$	1.59858e10	1.07072
$808$	3.92193e9	0.261553
$809$	1.11567e10	0.740826	0.370413	−	0.928867i	$-0.379216\pi$
$809$	1.11567e10	0.740826	0.370413	+	0.928867i	$0.379216\pi$
$810$	−9.45315e9	−0.624998
$811$	−1.83861e10	−1.21037	−0.605184	−	0.796086i	$-0.706900\pi$
$811$	−1.83861e10	−1.21037	−0.605184	+	0.796086i	$0.706900\pi$
$812$	−1.63714e10	−1.07310
$813$	−1.98845e10	−1.29777
$814$	1.18822e10	0.772168
$815$	−6.57468e9	−0.425425
$816$	6.12112e9	0.394381
$817$	−4.05215e9	−0.259961
$818$	−1.23641e10	−0.789817
$819$	−7.55695e10	−4.80676
$820$	−1.86789e9	−0.118305
$821$	2.33129e10	1.47026	0.735132	−	0.677924i	$-0.237120\pi$
$821$	2.33129e10	1.47026	0.735132	+	0.677924i	$0.237120\pi$
$822$	1.76760e10	1.11003
$823$	7.68337e7	0.00480455	0.00240227	−	0.999997i	$-0.499235\pi$
$823$	7.68337e7	0.00480455	0.00240227	+	0.999997i	$0.499235\pi$
$824$	8.26809e9	0.514825
$825$	8.45354e9	0.524143
$826$	−2.88379e10	−1.78046
$827$	−1.09883e9	−0.0675553	−0.0337777	−	0.999429i	$-0.510754\pi$
$827$	−1.09883e9	−0.0675553	−0.0337777	+	0.999429i	$0.510754\pi$
$828$	−1.59420e10	−0.975972
$829$	−1.11180e10	−0.677774	−0.338887	−	0.940827i	$-0.610051\pi$
$829$	−1.11180e10	−0.677774	−0.338887	+	0.940827i	$0.610051\pi$
$830$	6.13323e9	0.372319
$831$	4.09256e10	2.47395
$832$	−3.65836e9	−0.220219
$833$	1.59349e10	0.955195
$834$	9.98254e9	0.595882
$835$	3.28338e10	1.95172
$836$	1.49382e9	0.0884250
$837$	−1.04970e10	−0.618767
$838$	−1.37932e10	−0.809675
$839$	−3.07835e9	−0.179950	−0.0899749	−	0.995944i	$-0.528679\pi$
$839$	−3.07835e9	−0.179950	−0.0899749	+	0.995944i	$0.528679\pi$
$840$	−1.74806e10	−1.01761
$841$	2.18237e10	1.26515
$842$	−5.25220e9	−0.303214
$843$	−2.10267e10	−1.20885
$844$	1.19153e9	0.0682192
$845$	−4.36321e10	−2.48775
$846$	−1.42243e10	−0.807672
$847$	−1.02325e10	−0.578614
$848$	−2.85561e9	−0.160810
$849$	−9.93618e9	−0.557241
$850$	4.66131e9	0.260340
$851$	−2.59824e10	−1.44519
$852$	2.74678e9	0.152155
$853$	−1.64730e10	−0.908766	−0.454383	−	0.890806i	$-0.650140\pi$
$853$	−1.64730e10	−0.908766	−0.454383	+	0.890806i	$0.650140\pi$
$854$	−1.30742e10	−0.718310
$855$	−9.48637e9	−0.519061
$856$	1.51594e9	0.0826083
$857$	−9.92590e9	−0.538687	−0.269344	−	0.963044i	$-0.586807\pi$
$857$	−9.92590e9	−0.538687	−0.269344	+	0.963044i	$0.586807\pi$
$858$	−3.03257e10	−1.63910
$859$	−3.43497e10	−1.84904	−0.924522	−	0.381128i	$-0.875536\pi$
$859$	−3.43497e10	−1.84904	−0.924522	+	0.381128i	$0.875536\pi$
$860$	1.24971e10	0.669984
$861$	9.12116e9	0.487011
$862$	1.03878e10	0.552394
$863$	1.60357e10	0.849280	0.424640	−	0.905362i	$-0.360401\pi$
$863$	1.60357e10	0.849280	0.424640	+	0.905362i	$0.360401\pi$
$864$	5.22440e9	0.275574
$865$	−1.52770e10	−0.802565
$866$	2.32916e10	1.21867
$867$	−4.77521e9	−0.248843
$868$	−5.45287e9	−0.283013
$869$	2.87923e10	1.48836
$870$	4.17209e10	2.14801
$871$	−3.73969e10	−1.91766
$872$	7.37642e9	0.376737
$873$	−2.82720e9	−0.143816
$874$	−3.26647e9	−0.165497
$875$	2.01047e10	1.01454
$876$	1.65914e10	0.833911
$877$	−2.38061e10	−1.19176	−0.595882	−	0.803072i	$-0.703197\pi$
$877$	−2.38061e10	−1.19176	−0.595882	+	0.803072i	$0.703197\pi$
$878$	2.09261e10	1.04341
$879$	−6.24025e10	−3.09914
$880$	−4.60702e9	−0.227893
$881$	−1.87040e10	−0.921549	−0.460774	−	0.887517i	$-0.652428\pi$
$881$	−1.87040e10	−0.921549	−0.460774	+	0.887517i	$0.652428\pi$
$882$	2.84919e10	1.39824
$883$	−1.80424e9	−0.0881927	−0.0440964	−	0.999027i	$-0.514041\pi$
$883$	−1.80424e9	−0.0881927	−0.0440964	+	0.999027i	$0.514041\pi$
$884$	−1.67217e10	−0.814135
$885$	7.34903e10	3.56393
$886$	2.19121e9	0.105844
$887$	3.15658e10	1.51874	0.759371	−	0.650658i	$-0.225507\pi$
$887$	3.15658e10	1.51874	0.759371	+	0.650658i	$0.225507\pi$
$888$	1.78377e10	0.854858
$889$	1.36172e10	0.650028
$890$	−4.55872e9	−0.216759
$891$	1.21657e10	0.576191
$892$	−1.39950e9	−0.0660230
$893$	−2.91452e9	−0.136958
$894$	−1.95927e10	−0.917092
$895$	−3.20252e10	−1.49318
$896$	2.71391e9	0.126043
$897$	6.63121e10	3.06775
$898$	−9.42927e9	−0.434521
$899$	1.30143e10	0.597396
$900$	8.33450e9	0.381093
$901$	−1.30525e10	−0.594505
$902$	2.40388e9	0.109066
$903$	−6.10250e10	−2.75804
$904$	−1.43402e9	−0.0645602
$905$	−3.79906e10	−1.70375
$906$	1.20100e10	0.536531
$907$	1.40348e10	0.624569	0.312285	−	0.949989i	$-0.398906\pi$
$907$	1.40348e10	0.624569	0.312285	+	0.949989i	$0.398906\pi$
$908$	−1.29156e10	−0.572553
$909$	3.20527e10	1.41544
$910$	4.77536e10	2.10069
$911$	4.28205e10	1.87645	0.938226	−	0.346023i	$-0.112468\pi$
$911$	4.28205e10	1.87645	0.938226	+	0.346023i	$0.112468\pi$
$912$	2.24253e9	0.0978942
$913$	−7.89316e9	−0.343244
$914$	2.10977e10	0.913952
$915$	3.33182e10	1.43783
$916$	−4.71322e9	−0.202621
$917$	−9.49640e9	−0.406693
$918$	2.38797e10	1.01878
$919$	−4.21743e10	−1.79244	−0.896218	−	0.443614i	$-0.853696\pi$
$919$	−4.21743e10	−1.79244	−0.896218	+	0.443614i	$0.853696\pi$
$920$	1.00740e10	0.426526
$921$	−2.64132e10	−1.11407
$922$	−7.78113e9	−0.326952
$923$	−7.50364e9	−0.314099
$924$	2.24967e10	0.938139
$925$	1.35836e10	0.564313
$926$	1.56763e9	0.0648793
$927$	6.75725e10	2.78606
$928$	−6.47726e9	−0.266056
$929$	−3.44921e9	−0.141145	−0.0705724	−	0.997507i	$-0.522483\pi$
$929$	−3.44921e9	−0.141145	−0.0705724	+	0.997507i	$0.522483\pi$
$930$	1.38961e10	0.566502
$931$	5.83791e9	0.237101
$932$	−3.39694e9	−0.137446
$933$	4.79735e10	1.93382
$934$	−1.34562e10	−0.540391
$935$	−2.10578e10	−0.842506
$936$	−2.98986e10	−1.19175
$937$	4.27045e10	1.69584	0.847921	−	0.530122i	$-0.177854\pi$
$937$	4.27045e10	1.69584	0.847921	+	0.530122i	$0.177854\pi$
$938$	2.77424e10	1.09758
$939$	−1.09455e10	−0.431424
$940$	8.98857e9	0.352974
$941$	4.15693e10	1.62633	0.813165	−	0.582034i	$-0.197743\pi$
$941$	4.15693e10	1.62633	0.813165	+	0.582034i	$0.197743\pi$
$942$	−5.47228e10	−2.13300
$943$	−5.25648e9	−0.204129
$944$	−1.14095e10	−0.441434
$945$	−6.81955e10	−2.62872
$946$	−1.60831e10	−0.617663
$947$	−2.62829e10	−1.00565	−0.502827	−	0.864387i	$-0.667707\pi$
$947$	−2.62829e10	−1.00565	−0.502827	+	0.864387i	$0.667707\pi$
$948$	4.32234e10	1.64774
$949$	−4.53245e10	−1.72148
$950$	1.70772e9	0.0646224
$951$	6.18929e10	2.33351
$952$	1.24048e10	0.465971
$953$	2.43913e10	0.912873	0.456437	−	0.889756i	$-0.349125\pi$
$953$	2.43913e10	0.912873	0.456437	+	0.889756i	$0.349125\pi$
$954$	−2.33380e10	−0.870251
$955$	−1.66264e10	−0.617714
$956$	−7.72286e9	−0.285875
$957$	−5.36927e10	−1.98027
$958$	1.02885e10	0.378070
$959$	3.58214e10	1.31153
$960$	−6.91612e9	−0.252297
$961$	−2.31779e10	−0.842447
$962$	−4.87290e10	−1.76472
$963$	1.23893e10	0.447049
$964$	1.16026e10	0.417142
$965$	2.32989e9	0.0834623
$966$	−4.91928e10	−1.75583
$967$	−3.97629e9	−0.141412	−0.0707058	−	0.997497i	$-0.522525\pi$
$967$	−3.97629e9	−0.141412	−0.0707058	+	0.997497i	$0.522525\pi$
$968$	−4.04842e9	−0.143457
$969$	1.02502e10	0.361908
$970$	1.78655e9	0.0628513
$971$	9.16933e9	0.321418	0.160709	−	0.987002i	$-0.448622\pi$
$971$	9.16933e9	0.321418	0.160709	+	0.987002i	$0.448622\pi$
$972$	−4.05261e9	−0.141548
$973$	2.02301e10	0.704050
$974$	−2.65559e9	−0.0920884
$975$	−3.46680e10	−1.19788
$976$	−5.17272e9	−0.178092
$977$	−3.66289e10	−1.25659	−0.628294	−	0.777976i	$-0.716246\pi$
$977$	−3.66289e10	−1.25659	−0.628294	+	0.777976i	$0.716246\pi$
$978$	1.27022e10	0.434203
$979$	5.86685e9	0.199832
$980$	−1.80045e10	−0.611068
$981$	6.02852e10	2.03877
$982$	2.35270e10	0.792824
$983$	5.79013e10	1.94425	0.972123	−	0.234470i	$-0.0753356\pi$
$983$	5.79013e10	1.94425	0.972123	+	0.234470i	$0.0753356\pi$
$984$	3.60873e9	0.120746
$985$	2.79545e10	0.932020
$986$	−2.96063e10	−0.983593
$987$	−4.38924e10	−1.45305
$988$	−6.12614e9	−0.202087
$989$	3.51684e10	1.15602
$990$	−3.76518e10	−1.23328
$991$	−9.51697e9	−0.310628	−0.155314	−	0.987865i	$-0.549639\pi$
$991$	−9.51697e9	−0.310628	−0.155314	+	0.987865i	$0.549639\pi$
$992$	−2.15740e9	−0.0701680
$993$	−8.09593e10	−2.62388
$994$	5.56648e9	0.179775
$995$	4.08184e10	1.31364
$996$	−1.18493e10	−0.380001
$997$	3.54725e9	0.113360	0.0566799	−	0.998392i	$-0.481949\pi$
$997$	3.54725e9	0.113360	0.0566799	+	0.998392i	$0.481949\pi$
$998$	1.73247e10	0.551707
$999$	6.95885e10	2.20830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.a.e.1.4	✓	4
3.2	odd	2		342.8.a.o.1.3		4
4.3	odd	2		304.8.a.e.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.e.1.4	✓	4	1.1	even	1	trivial
304.8.a.e.1.1		4	4.3	odd	2
342.8.a.o.1.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +79.8211 q^{3} +64.0000 q^{4} -330.525 q^{5} +638.569 q^{6} +1294.09 q^{7} +512.000 q^{8} +4184.42 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +79.8211 q^{3} +64.0000 q^{4} -330.525 q^{5} +638.569 q^{6} +1294.09 q^{7} +512.000 q^{8} +4184.42 q^{9} -2644.20 q^{10} +3402.95 q^{11} +5108.55 q^{12} -13955.5 q^{13} +10352.7 q^{14} -26382.9 q^{15} +4096.00 q^{16} +18722.0 q^{17} +33475.3 q^{18} +6859.00 q^{19} -21153.6 q^{20} +103296. q^{21} +27223.6 q^{22} -59529.0 q^{23} +40868.4 q^{24} +31121.8 q^{25} -111644. q^{26} +159436. q^{27} +82821.9 q^{28} -197670. q^{29} -211063. q^{30} -65838.5 q^{31} +32768.0 q^{32} +271628. q^{33} +149776. q^{34} -427730. q^{35} +267803. q^{36} +436467. q^{37} +54872.0 q^{38} -1.11395e6 q^{39} -169229. q^{40} +88301.2 q^{41} +826367. q^{42} -590778. q^{43} +217789. q^{44} -1.38305e6 q^{45} -476232. q^{46} -424919. q^{47} +326947. q^{48} +851131. q^{49} +248974. q^{50} +1.49441e6 q^{51} -893154. q^{52} -697171. q^{53} +1.27549e6 q^{54} -1.12476e6 q^{55} +662575. q^{56} +547493. q^{57} -1.58136e6 q^{58} -2.78553e6 q^{59} -1.68850e6 q^{60} -1.26287e6 q^{61} -526708. q^{62} +5.41502e6 q^{63} +262144. q^{64} +4.61265e6 q^{65} +2.17302e6 q^{66} +2.67972e6 q^{67} +1.19821e6 q^{68} -4.75167e6 q^{69} -3.42184e6 q^{70} +537682. q^{71} +2.14242e6 q^{72} +3.24778e6 q^{73} +3.49173e6 q^{74} +2.48418e6 q^{75} +438976. q^{76} +4.40374e6 q^{77} -8.91157e6 q^{78} +8.46098e6 q^{79} -1.35383e6 q^{80} +3.57505e6 q^{81} +706410. q^{82} -2.31950e6 q^{83} +6.61094e6 q^{84} -6.18810e6 q^{85} -4.72622e6 q^{86} -1.57783e7 q^{87} +1.74231e6 q^{88} +1.72405e6 q^{89} -1.10644e7 q^{90} -1.80597e7 q^{91} -3.80986e6 q^{92} -5.25530e6 q^{93} -3.39935e6 q^{94} -2.26707e6 q^{95} +2.61558e6 q^{96} -675649. q^{97} +6.80905e6 q^{98} +1.42394e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9}+O(q^{10}) \) 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9	\( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9} - 2232 q^{10} + 5269 q^{11} + 768 q^{12} + 5406 q^{13} + 19880 q^{14} + 26658 q^{15} + 16384 q^{16} + 22885 q^{17} + 75856 q^{18} + 27436 q^{19} - 17856 q^{20} + 2854 q^{21} + 42152 q^{22} + 3364 q^{23} + 6144 q^{24} + 112561 q^{25} + 43248 q^{26} - 220194 q^{27} + 159040 q^{28} - 122136 q^{29} + 213264 q^{30} + 225480 q^{31} + 131072 q^{32} + 176138 q^{33} + 183080 q^{34} - 785781 q^{35} + 606848 q^{36} + 154096 q^{37} + 219488 q^{38} - 1749220 q^{39} - 142848 q^{40} - 1054628 q^{41} + 22832 q^{42} - 840795 q^{43} + 337216 q^{44} - 4162563 q^{45} + 26912 q^{46} - 1021877 q^{47} + 49152 q^{48} - 621441 q^{49} + 900488 q^{50} + 724892 q^{51} + 345984 q^{52} - 326842 q^{53} - 1761552 q^{54} - 221553 q^{55} + 1272320 q^{56} + 82308 q^{57} - 977088 q^{58} + 421384 q^{59} + 1706112 q^{60} + 116825 q^{61} + 1803840 q^{62} + 10245825 q^{63} + 1048576 q^{64} + 4477428 q^{65} + 1409104 q^{66} + 5794566 q^{67} + 1464640 q^{68} - 2472196 q^{69} - 6286248 q^{70} + 10590626 q^{71} + 4854784 q^{72} + 3971389 q^{73} + 1232768 q^{74} - 3690042 q^{75} + 1755904 q^{76} + 5806573 q^{77} - 13993760 q^{78} + 5597800 q^{79} - 1142784 q^{80} + 20567744 q^{81} - 8437024 q^{82} + 4665800 q^{83} + 182656 q^{84} - 2014461 q^{85} - 6726360 q^{86} - 14449584 q^{87} + 2697728 q^{88} - 2794214 q^{89} - 33300504 q^{90} - 8827314 q^{91} + 215296 q^{92} - 43981204 q^{93} - 8175016 q^{94} - 1913661 q^{95} + 393216 q^{96} - 14445130 q^{97} - 4971528 q^{98} - 7940315 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9 - 2232 * q^10 + 5269 * q^11 + 768 * q^12 + 5406 * q^13 + 19880 * q^14 + 26658 * q^15 + 16384 * q^16 + 22885 * q^17 + 75856 * q^18 + 27436 * q^19 - 17856 * q^20 + 2854 * q^21 + 42152 * q^22 + 3364 * q^23 + 6144 * q^24 + 112561 * q^25 + 43248 * q^26 - 220194 * q^27 + 159040 * q^28 - 122136 * q^29 + 213264 * q^30 + 225480 * q^31 + 131072 * q^32 + 176138 * q^33 + 183080 * q^34 - 785781 * q^35 + 606848 * q^36 + 154096 * q^37 + 219488 * q^38 - 1749220 * q^39 - 142848 * q^40 - 1054628 * q^41 + 22832 * q^42 - 840795 * q^43 + 337216 * q^44 - 4162563 * q^45 + 26912 * q^46 - 1021877 * q^47 + 49152 * q^48 - 621441 * q^49 + 900488 * q^50 + 724892 * q^51 + 345984 * q^52 - 326842 * q^53 - 1761552 * q^54 - 221553 * q^55 + 1272320 * q^56 + 82308 * q^57 - 977088 * q^58 + 421384 * q^59 + 1706112 * q^60 + 116825 * q^61 + 1803840 * q^62 + 10245825 * q^63 + 1048576 * q^64 + 4477428 * q^65 + 1409104 * q^66 + 5794566 * q^67 + 1464640 * q^68 - 2472196 * q^69 - 6286248 * q^70 + 10590626 * q^71 + 4854784 * q^72 + 3971389 * q^73 + 1232768 * q^74 - 3690042 * q^75 + 1755904 * q^76 + 5806573 * q^77 - 13993760 * q^78 + 5597800 * q^79 - 1142784 * q^80 + 20567744 * q^81 - 8437024 * q^82 + 4665800 * q^83 + 182656 * q^84 - 2014461 * q^85 - 6726360 * q^86 - 14449584 * q^87 + 2697728 * q^88 - 2794214 * q^89 - 33300504 * q^90 - 8827314 * q^91 + 215296 * q^92 - 43981204 * q^93 - 8175016 * q^94 - 1913661 * q^95 + 393216 * q^96 - 14445130 * q^97 - 4971528 * q^98 - 7940315 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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