LMFDB - Embedded newform 38.8.a.c.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17953}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4488 $ x^2 - x - 4488
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-66.4944$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} +61.4944 q^{3} +64.0000 q^{4} -235.483 q^{5} -491.955 q^{6} -1111.92 q^{7} -512.000 q^{8} +1594.56 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +61.4944 q^{3} +64.0000 q^{4} -235.483 q^{5} -491.955 q^{6} -1111.92 q^{7} -512.000 q^{8} +1594.56 q^{9} +1883.87 q^{10} +1428.49 q^{11} +3935.64 q^{12} -5515.63 q^{13} +8895.37 q^{14} -14480.9 q^{15} +4096.00 q^{16} -27453.3 q^{17} -12756.5 q^{18} +6859.00 q^{19} -15070.9 q^{20} -68377.0 q^{21} -11428.0 q^{22} -62205.4 q^{23} -31485.1 q^{24} -22672.7 q^{25} +44125.0 q^{26} -36431.6 q^{27} -71163.0 q^{28} +209139. q^{29} +115847. q^{30} -130002. q^{31} -32768.0 q^{32} +87844.4 q^{33} +219627. q^{34} +261839. q^{35} +102052. q^{36} -35499.1 q^{37} -54872.0 q^{38} -339180. q^{39} +120567. q^{40} +563790. q^{41} +547016. q^{42} -352181. q^{43} +91423.6 q^{44} -375492. q^{45} +497643. q^{46} -747727. q^{47} +251881. q^{48} +412827. q^{49} +181381. q^{50} -1.68823e6 q^{51} -353000. q^{52} +684889. q^{53} +291453. q^{54} -336386. q^{55} +569304. q^{56} +421790. q^{57} -1.67311e6 q^{58} +2.84345e6 q^{59} -926778. q^{60} -565615. q^{61} +1.04002e6 q^{62} -1.77303e6 q^{63} +262144. q^{64} +1.29884e6 q^{65} -702755. q^{66} +3.49970e6 q^{67} -1.75701e6 q^{68} -3.82528e6 q^{69} -2.09471e6 q^{70} +296273. q^{71} -816416. q^{72} -1.60675e6 q^{73} +283993. q^{74} -1.39424e6 q^{75} +438976. q^{76} -1.58837e6 q^{77} +2.71344e6 q^{78} -2.55303e6 q^{79} -964539. q^{80} -5.72765e6 q^{81} -4.51032e6 q^{82} +352021. q^{83} -4.37613e6 q^{84} +6.46480e6 q^{85} +2.81745e6 q^{86} +1.28609e7 q^{87} -731389. q^{88} +5.32675e6 q^{89} +3.00394e6 q^{90} +6.13295e6 q^{91} -3.98114e6 q^{92} -7.99439e6 q^{93} +5.98182e6 q^{94} -1.61518e6 q^{95} -2.01505e6 q^{96} +1.35294e7 q^{97} -3.30261e6 q^{98} +2.27782e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{2} - 11 q^{3} + 128 q^{4} - 69 q^{5} + 88 q^{6} - 348 q^{7} - 1024 q^{8} + 4663 q^{9}+O(q^{10}) $ 2 * q - 16 * q^2 - 11 * q^3 + 128 * q^4 - 69 * q^5 + 88 * q^6 - 348 * q^7 - 1024 * q^8 + 4663 * q^9	\( 2 q - 16 q^{2} - 11 q^{3} + 128 q^{4} - 69 q^{5} + 88 q^{6} - 348 q^{7} - 1024 q^{8} + 4663 q^{9} + 552 q^{10} + 2723 q^{11} - 704 q^{12} - 14113 q^{13} + 2784 q^{14} - 26550 q^{15} + 8192 q^{16} - 38560 q^{17} - 37304 q^{18} + 13718 q^{19} - 4416 q^{20} - 123757 q^{21} - 21784 q^{22} - 73897 q^{23} + 5632 q^{24} - 73081 q^{25} + 112904 q^{26} - 100331 q^{27} - 22272 q^{28} + 159813 q^{29} + 212400 q^{30} - 259468 q^{31} - 65536 q^{32} - 6000 q^{33} + 308480 q^{34} + 389019 q^{35} + 298432 q^{36} - 528168 q^{37} - 109744 q^{38} + 284081 q^{39} + 35328 q^{40} + 1005650 q^{41} + 990056 q^{42} + 286217 q^{43} + 174272 q^{44} + 135351 q^{45} + 591176 q^{46} - 1397509 q^{47} - 45056 q^{48} + 172860 q^{49} + 584648 q^{50} - 883053 q^{51} - 903232 q^{52} - 1385969 q^{53} + 802648 q^{54} - 120873 q^{55} + 178176 q^{56} - 75449 q^{57} - 1278504 q^{58} + 2700953 q^{59} - 1699200 q^{60} - 3975947 q^{61} + 2075744 q^{62} + 571019 q^{63} + 524288 q^{64} - 132480 q^{65} + 48000 q^{66} - 134557 q^{67} - 2467840 q^{68} - 2977707 q^{69} - 3112152 q^{70} + 4202740 q^{71} - 2387456 q^{72} + 900498 q^{73} + 4225344 q^{74} + 2260081 q^{75} + 877952 q^{76} - 599473 q^{77} - 2272648 q^{78} - 6893730 q^{79} - 282624 q^{80} - 7805978 q^{81} - 8045200 q^{82} - 2465330 q^{83} - 7920448 q^{84} + 4615719 q^{85} - 2289736 q^{86} + 16436697 q^{87} - 1394176 q^{88} + 17431724 q^{89} - 1082808 q^{90} - 434771 q^{91} - 4729408 q^{92} + 1391168 q^{93} + 11180072 q^{94} - 473271 q^{95} + 360448 q^{96} + 6351934 q^{97} - 1382880 q^{98} + 6249933 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 11 * q^3 + 128 * q^4 - 69 * q^5 + 88 * q^6 - 348 * q^7 - 1024 * q^8 + 4663 * q^9 + 552 * q^10 + 2723 * q^11 - 704 * q^12 - 14113 * q^13 + 2784 * q^14 - 26550 * q^15 + 8192 * q^16 - 38560 * q^17 - 37304 * q^18 + 13718 * q^19 - 4416 * q^20 - 123757 * q^21 - 21784 * q^22 - 73897 * q^23 + 5632 * q^24 - 73081 * q^25 + 112904 * q^26 - 100331 * q^27 - 22272 * q^28 + 159813 * q^29 + 212400 * q^30 - 259468 * q^31 - 65536 * q^32 - 6000 * q^33 + 308480 * q^34 + 389019 * q^35 + 298432 * q^36 - 528168 * q^37 - 109744 * q^38 + 284081 * q^39 + 35328 * q^40 + 1005650 * q^41 + 990056 * q^42 + 286217 * q^43 + 174272 * q^44 + 135351 * q^45 + 591176 * q^46 - 1397509 * q^47 - 45056 * q^48 + 172860 * q^49 + 584648 * q^50 - 883053 * q^51 - 903232 * q^52 - 1385969 * q^53 + 802648 * q^54 - 120873 * q^55 + 178176 * q^56 - 75449 * q^57 - 1278504 * q^58 + 2700953 * q^59 - 1699200 * q^60 - 3975947 * q^61 + 2075744 * q^62 + 571019 * q^63 + 524288 * q^64 - 132480 * q^65 + 48000 * q^66 - 134557 * q^67 - 2467840 * q^68 - 2977707 * q^69 - 3112152 * q^70 + 4202740 * q^71 - 2387456 * q^72 + 900498 * q^73 + 4225344 * q^74 + 2260081 * q^75 + 877952 * q^76 - 599473 * q^77 - 2272648 * q^78 - 6893730 * q^79 - 282624 * q^80 - 7805978 * q^81 - 8045200 * q^82 - 2465330 * q^83 - 7920448 * q^84 + 4615719 * q^85 - 2289736 * q^86 + 16436697 * q^87 - 1394176 * q^88 + 17431724 * q^89 - 1082808 * q^90 - 434771 * q^91 - 4729408 * q^92 + 1391168 * q^93 + 11180072 * q^94 - 473271 * q^95 + 360448 * q^96 + 6351934 * q^97 - 1382880 * q^98 + 6249933 * 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
$3$	61.4944	1.31496	0.657478	−	0.753474i	$-0.271623\pi$
$3$	61.4944	1.31496	0.657478	+	0.753474i	$0.271623\pi$
$4$	64.0000	0.500000
$5$	−235.483	−0.842490	−0.421245	−	0.906947i	$-0.638407\pi$
$5$	−235.483	−0.842490	−0.421245	+	0.906947i	$0.638407\pi$
$6$	−491.955	−0.929814
$7$	−1111.92	−1.22527	−0.612634	−	0.790367i	$-0.709890\pi$
$7$	−1111.92	−1.22527	−0.612634	+	0.790367i	$0.709890\pi$
$8$	−512.000	−0.353553
$9$	1594.56	0.729109
$10$	1883.87	0.595731
$11$	1428.49	0.323597	0.161798	−	0.986824i	$-0.448271\pi$
$11$	1428.49	0.323597	0.161798	+	0.986824i	$0.448271\pi$
$12$	3935.64	0.657478
$13$	−5515.63	−0.696295	−0.348148	−	0.937440i	$-0.613189\pi$
$13$	−5515.63	−0.696295	−0.348148	+	0.937440i	$0.613189\pi$
$14$	8895.37	0.866395
$15$	−14480.9	−1.10784
$16$	4096.00	0.250000
$17$	−27453.3	−1.35526	−0.677632	−	0.735402i	$-0.736994\pi$
$17$	−27453.3	−1.35526	−0.677632	+	0.735402i	$0.736994\pi$
$18$	−12756.5	−0.515558
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	−15070.9	−0.421245
$21$	−68377.0	−1.61117
$22$	−11428.0	−0.228817
$23$	−62205.4	−1.06606	−0.533029	−	0.846097i	$-0.678946\pi$
$23$	−62205.4	−1.06606	−0.533029	+	0.846097i	$0.678946\pi$
$24$	−31485.1	−0.464907
$25$	−22672.7	−0.290210
$26$	44125.0	0.492355
$27$	−36431.6	−0.356210
$28$	−71163.0	−0.612634
$29$	209139.	1.59236	0.796180	−	0.605059i	$-0.206851\pi$
$29$	209139.	1.59236	0.796180	+	0.605059i	$0.206851\pi$
$30$	115847.	0.783360
$31$	−130002.	−0.783762	−0.391881	−	0.920016i	$-0.628175\pi$
$31$	−130002.	−0.783762	−0.391881	+	0.920016i	$0.628175\pi$
$32$	−32768.0	−0.176777
$33$	87844.4	0.425515
$34$	219627.	0.958316
$35$	261839.	1.03228
$36$	102052.	0.364555
$37$	−35499.1	−0.115216	−0.0576078	−	0.998339i	$-0.518347\pi$
$37$	−35499.1	−0.115216	−0.0576078	+	0.998339i	$0.518347\pi$
$38$	−54872.0	−0.162221
$39$	−339180.	−0.915597
$40$	120567.	0.297865
$41$	563790.	1.27754	0.638769	−	0.769399i	$-0.279444\pi$
$41$	563790.	1.27754	0.638769	+	0.769399i	$0.279444\pi$
$42$	547016.	1.13927
$43$	−352181.	−0.675502	−0.337751	−	0.941236i	$-0.609666\pi$
$43$	−352181.	−0.675502	−0.337751	+	0.941236i	$0.609666\pi$
$44$	91423.6	0.161798
$45$	−375492.	−0.614267
$46$	497643.	0.753816
$47$	−747727.	−1.05051	−0.525256	−	0.850944i	$-0.676030\pi$
$47$	−747727.	−1.05051	−0.525256	+	0.850944i	$0.676030\pi$
$48$	251881.	0.328739
$49$	412827.	0.501281
$50$	181381.	0.205209
$51$	−1.68823e6	−1.78211
$52$	−353000.	−0.348148
$53$	684889.	0.631910	0.315955	−	0.948774i	$-0.397675\pi$
$53$	684889.	0.631910	0.315955	+	0.948774i	$0.397675\pi$
$54$	291453.	0.251878
$55$	−336386.	−0.272627
$56$	569304.	0.433198
$57$	421790.	0.301672
$58$	−1.67311e6	−1.12597
$59$	2.84345e6	1.80245	0.901225	−	0.433352i	$-0.142669\pi$
$59$	2.84345e6	1.80245	0.901225	+	0.433352i	$0.142669\pi$
$60$	−926778.	−0.553919
$61$	−565615.	−0.319056	−0.159528	−	0.987193i	$-0.550997\pi$
$61$	−565615.	−0.319056	−0.159528	+	0.987193i	$0.550997\pi$
$62$	1.04002e6	0.554203
$63$	−1.77303e6	−0.893354
$64$	262144.	0.125000
$65$	1.29884e6	0.586622
$66$	−702755.	−0.300885
$67$	3.49970e6	1.42157	0.710787	−	0.703407i	$-0.248339\pi$
$67$	3.49970e6	1.42157	0.710787	+	0.703407i	$0.248339\pi$
$68$	−1.75701e6	−0.677632
$69$	−3.82528e6	−1.40182
$70$	−2.09471e6	−0.729930
$71$	296273.	0.0982398	0.0491199	−	0.998793i	$-0.484358\pi$
$71$	296273.	0.0982398	0.0491199	+	0.998793i	$0.484358\pi$
$72$	−816416.	−0.257779
$73$	−1.60675e6	−0.483412	−0.241706	−	0.970350i	$-0.577707\pi$
$73$	−1.60675e6	−0.483412	−0.241706	+	0.970350i	$0.577707\pi$
$74$	283993.	0.0814697
$75$	−1.39424e6	−0.381613
$76$	438976.	0.114708
$77$	−1.58837e6	−0.396493
$78$	2.71344e6	0.647425
$79$	−2.55303e6	−0.582586	−0.291293	−	0.956634i	$-0.594086\pi$
$79$	−2.55303e6	−0.582586	−0.291293	+	0.956634i	$0.594086\pi$
$80$	−964539.	−0.210623
$81$	−5.72765e6	−1.19751
$82$	−4.51032e6	−0.903356
$83$	352021.	0.0675764	0.0337882	−	0.999429i	$-0.489243\pi$
$83$	352021.	0.0675764	0.0337882	+	0.999429i	$0.489243\pi$
$84$	−4.37613e6	−0.805587
$85$	6.46480e6	1.14180
$86$	2.81745e6	0.477652
$87$	1.28609e7	2.09388
$88$	−731389.	−0.114409
$89$	5.32675e6	0.800935	0.400467	−	0.916311i	$-0.368848\pi$
$89$	5.32675e6	0.800935	0.400467	+	0.916311i	$0.368848\pi$
$90$	3.00394e6	0.434353
$91$	6.13295e6	0.853148
$92$	−3.98114e6	−0.533029
$93$	−7.99439e6	−1.03061
$94$	5.98182e6	0.742824
$95$	−1.61518e6	−0.193281
$96$	−2.01505e6	−0.232454
$97$	1.35294e7	1.50514	0.752572	−	0.658510i	$-0.228813\pi$
$97$	1.35294e7	1.50514	0.752572	+	0.658510i	$0.228813\pi$
$98$	−3.30261e6	−0.354459
$99$	2.27782e6	0.235937
$100$	−1.45105e6	−0.145105
$101$	−3.78746e6	−0.365783	−0.182892	−	0.983133i	$-0.558546\pi$
$101$	−3.78746e6	−0.365783	−0.182892	+	0.983133i	$0.558546\pi$
$102$	1.35058e7	1.26014
$103$	−1.37733e7	−1.24196	−0.620980	−	0.783826i	$-0.713265\pi$
$103$	−1.37733e7	−1.24196	−0.620980	+	0.783826i	$0.713265\pi$
$104$	2.82400e6	0.246177
$105$	1.61016e7	1.35740
$106$	−5.47912e6	−0.446827
$107$	1.75990e7	1.38882	0.694408	−	0.719581i	$-0.255666\pi$
$107$	1.75990e7	1.38882	0.694408	+	0.719581i	$0.255666\pi$
$108$	−2.33163e6	−0.178105
$109$	−2.28268e7	−1.68831	−0.844155	−	0.536099i	$-0.819897\pi$
$109$	−2.28268e7	−1.68831	−0.844155	+	0.536099i	$0.819897\pi$
$110$	2.69109e6	0.192776
$111$	−2.18300e6	−0.151503
$112$	−4.55443e6	−0.306317
$113$	−1.61795e7	−1.05485	−0.527425	−	0.849601i	$-0.676843\pi$
$113$	−1.61795e7	−1.05485	−0.527425	+	0.849601i	$0.676843\pi$
$114$	−3.37432e6	−0.213314
$115$	1.46483e7	0.898143
$116$	1.33849e7	0.796180
$117$	−8.79501e6	−0.507675
$118$	−2.27476e7	−1.27452
$119$	3.05259e7	1.66056
$120$	7.41422e6	0.391680
$121$	−1.74466e7	−0.895285
$122$	4.52492e6	0.225607
$123$	3.46699e7	1.67991
$124$	−8.32013e6	−0.391881
$125$	2.37362e7	1.08699
$126$	1.41842e7	0.631697
$127$	−4.46993e7	−1.93637	−0.968183	−	0.250245i	$-0.919489\pi$
$127$	−4.46993e7	−1.93637	−0.968183	+	0.250245i	$0.919489\pi$
$128$	−2.09715e6	−0.0883883
$129$	−2.16572e7	−0.888255
$130$	−1.03907e7	−0.414804
$131$	−1.51806e7	−0.589984	−0.294992	−	0.955500i	$-0.595317\pi$
$131$	−1.51806e7	−0.589984	−0.294992	+	0.955500i	$0.595317\pi$
$132$	5.62204e6	0.212758
$133$	−7.62667e6	−0.281096
$134$	−2.79976e7	−1.00520
$135$	8.57904e6	0.300103
$136$	1.40561e7	0.479158
$137$	−2.46831e7	−0.820120	−0.410060	−	0.912059i	$-0.634492\pi$
$137$	−2.46831e7	−0.820120	−0.410060	+	0.912059i	$0.634492\pi$
$138$	3.06023e7	0.991235
$139$	−3.71264e7	−1.17255	−0.586275	−	0.810112i	$-0.699406\pi$
$139$	−3.71264e7	−1.17255	−0.586275	+	0.810112i	$0.699406\pi$
$140$	1.67577e7	0.516138
$141$	−4.59811e7	−1.38138
$142$	−2.37018e6	−0.0694660
$143$	−7.87904e6	−0.225319
$144$	6.53132e6	0.182277
$145$	−4.92487e7	−1.34155
$146$	1.28540e7	0.341824
$147$	2.53865e7	0.659163
$148$	−2.27194e6	−0.0576078
$149$	−5.61635e7	−1.39092	−0.695460	−	0.718565i	$-0.744799\pi$
$149$	−5.61635e7	−1.39092	−0.695460	+	0.718565i	$0.744799\pi$
$150$	1.11539e7	0.269841
$151$	−2.90803e7	−0.687352	−0.343676	−	0.939088i	$-0.611672\pi$
$151$	−2.90803e7	−0.687352	−0.343676	+	0.939088i	$0.611672\pi$
$152$	−3.51181e6	−0.0811107
$153$	−4.37760e7	−0.988135
$154$	1.27070e7	0.280363
$155$	3.06133e7	0.660312
$156$	−2.17075e7	−0.457799
$157$	7.42283e7	1.53081	0.765404	−	0.643550i	$-0.222539\pi$
$157$	7.42283e7	1.53081	0.765404	+	0.643550i	$0.222539\pi$
$158$	2.04242e7	0.411951
$159$	4.21169e7	0.830933
$160$	7.71631e6	0.148933
$161$	6.91675e7	1.30621
$162$	4.58212e7	0.846767
$163$	6.25286e7	1.13089	0.565447	−	0.824785i	$-0.308704\pi$
$163$	6.25286e7	1.13089	0.565447	+	0.824785i	$0.308704\pi$
$164$	3.60826e7	0.638769
$165$	−2.06859e7	−0.358493
$166$	−2.81616e6	−0.0477837
$167$	4.65406e7	0.773258	0.386629	−	0.922235i	$-0.373639\pi$
$167$	4.65406e7	0.773258	0.386629	+	0.922235i	$0.373639\pi$
$168$	3.50090e7	0.569636
$169$	−3.23264e7	−0.515173
$170$	−5.17184e7	−0.807372
$171$	1.09371e7	0.167269
$172$	−2.25396e7	−0.337751
$173$	−3.78118e7	−0.555221	−0.277611	−	0.960694i	$-0.589542\pi$
$173$	−3.78118e7	−0.555221	−0.277611	+	0.960694i	$0.589542\pi$
$174$	−1.02887e8	−1.48060
$175$	2.52102e7	0.355585
$176$	5.85111e6	0.0808992
$177$	1.74856e8	2.37014
$178$	−4.26140e7	−0.566347
$179$	1.36978e8	1.78511	0.892554	−	0.450941i	$-0.148912\pi$
$179$	1.36978e8	1.78511	0.892554	+	0.450941i	$0.148912\pi$
$180$	−2.40315e7	−0.307134
$181$	3.29158e7	0.412600	0.206300	−	0.978489i	$-0.433858\pi$
$181$	3.29158e7	0.412600	0.206300	+	0.978489i	$0.433858\pi$
$182$	−4.90636e7	−0.603267
$183$	−3.47822e7	−0.419544
$184$	3.18492e7	0.376908
$185$	8.35944e6	0.0970680
$186$	6.39552e7	0.728753
$187$	−3.92169e7	−0.438559
$188$	−4.78546e7	−0.525256
$189$	4.05091e7	0.436452
$190$	1.29214e7	0.136670
$191$	−2.19935e7	−0.228390	−0.114195	−	0.993458i	$-0.536429\pi$
$191$	−2.19935e7	−0.228390	−0.114195	+	0.993458i	$0.536429\pi$
$192$	1.61204e7	0.164369
$193$	−1.22938e8	−1.23094	−0.615468	−	0.788162i	$-0.711033\pi$
$193$	−1.22938e8	−1.23094	−0.615468	+	0.788162i	$0.711033\pi$
$194$	−1.08235e8	−1.06430
$195$	7.98713e7	0.771382
$196$	2.64209e7	0.250641
$197$	−1.51613e8	−1.41288	−0.706439	−	0.707774i	$-0.749699\pi$
$197$	−1.51613e8	−1.41288	−0.706439	+	0.707774i	$0.749699\pi$
$198$	−1.82226e7	−0.166833
$199$	−6.05312e7	−0.544495	−0.272247	−	0.962227i	$-0.587767\pi$
$199$	−6.05312e7	−0.544495	−0.272247	+	0.962227i	$0.587767\pi$
$200$	1.16084e7	0.102605
$201$	2.15212e8	1.86931
$202$	3.02997e7	0.258648
$203$	−2.32546e8	−1.95107
$204$	−1.08046e8	−0.891056
$205$	−1.32763e8	−1.07631
$206$	1.10186e8	0.878198
$207$	−9.91903e7	−0.777272
$208$	−2.25920e7	−0.174074
$209$	9.79804e6	0.0742382
$210$	−1.28813e8	−0.959825
$211$	1.35344e7	0.0991862	0.0495931	−	0.998770i	$-0.484208\pi$
$211$	1.35344e7	0.0991862	0.0495931	+	0.998770i	$0.484208\pi$
$212$	4.38329e7	0.315955
$213$	1.82191e7	0.129181
$214$	−1.40792e8	−0.982042
$215$	8.29327e7	0.569104
$216$	1.86530e7	0.125939
$217$	1.44552e8	0.960318
$218$	1.82614e8	1.19382
$219$	−9.88060e7	−0.635666
$220$	−2.15287e7	−0.136314
$221$	1.51422e8	0.943663
$222$	1.74640e7	0.107129
$223$	−4.84849e6	−0.0292778	−0.0146389	−	0.999893i	$-0.504660\pi$
$223$	−4.84849e6	−0.0292778	−0.0146389	+	0.999893i	$0.504660\pi$
$224$	3.64354e7	0.216599
$225$	−3.61530e7	−0.211595
$226$	1.29436e8	0.745892
$227$	3.45214e8	1.95883	0.979417	−	0.201848i	$-0.0646948\pi$
$227$	3.45214e8	1.95883	0.979417	+	0.201848i	$0.0646948\pi$
$228$	2.69946e7	0.150836
$229$	1.36402e8	0.750581	0.375290	−	0.926907i	$-0.377543\pi$
$229$	1.36402e8	0.750581	0.375290	+	0.926907i	$0.377543\pi$
$230$	−1.17187e8	−0.635083
$231$	−9.76761e7	−0.521370
$232$	−1.07079e8	−0.562984
$233$	−3.27111e8	−1.69414	−0.847069	−	0.531482i	$-0.821635\pi$
$233$	−3.27111e8	−1.69414	−0.847069	+	0.531482i	$0.821635\pi$
$234$	7.03601e7	0.358980
$235$	1.76077e8	0.885046
$236$	1.81981e8	0.901225
$237$	−1.56997e8	−0.766075
$238$	−2.44207e8	−1.17419
$239$	−1.78511e8	−0.845807	−0.422903	−	0.906175i	$-0.638989\pi$
$239$	−1.78511e8	−0.845807	−0.422903	+	0.906175i	$0.638989\pi$
$240$	−5.93138e7	−0.276959
$241$	−2.24108e8	−1.03133	−0.515665	−	0.856791i	$-0.672455\pi$
$241$	−2.24108e8	−1.03133	−0.515665	+	0.856791i	$0.672455\pi$
$242$	1.39573e8	0.633062
$243$	−2.72542e8	−1.21846
$244$	−3.61994e7	−0.159528
$245$	−9.72138e7	−0.422325
$246$	−2.77359e8	−1.18787
$247$	−3.78317e7	−0.159741
$248$	6.65610e7	0.277102
$249$	2.16473e7	0.0888599
$250$	−1.89889e8	−0.768618
$251$	2.31812e8	0.925290	0.462645	−	0.886544i	$-0.346901\pi$
$251$	2.31812e8	0.925290	0.462645	+	0.886544i	$0.346901\pi$
$252$	−1.13474e8	−0.446677
$253$	−8.88601e7	−0.344973
$254$	3.57594e8	1.36922
$255$	3.97549e8	1.50141
$256$	1.67772e7	0.0625000
$257$	−3.98240e8	−1.46345	−0.731727	−	0.681598i	$-0.761285\pi$
$257$	−3.98240e8	−1.46345	−0.731727	+	0.681598i	$0.761285\pi$
$258$	1.73257e8	0.628091
$259$	3.94722e7	0.141170
$260$	8.31256e7	0.293311
$261$	3.33485e8	1.16100
$262$	1.21445e8	0.417182
$263$	3.10207e8	1.05149	0.525746	−	0.850642i	$-0.323786\pi$
$263$	3.10207e8	1.05149	0.525746	+	0.850642i	$0.323786\pi$
$264$	−4.49763e7	−0.150442
$265$	−1.61280e8	−0.532378
$266$	6.10134e7	0.198765
$267$	3.27565e8	1.05319
$268$	2.23981e8	0.710787
$269$	−1.86836e8	−0.585230	−0.292615	−	0.956230i	$-0.594525\pi$
$269$	−1.86836e8	−0.585230	−0.292615	+	0.956230i	$0.594525\pi$
$270$	−6.86323e7	−0.212205
$271$	−1.10277e8	−0.336584	−0.168292	−	0.985737i	$-0.553825\pi$
$271$	−1.10277e8	−0.336584	−0.168292	+	0.985737i	$0.553825\pi$
$272$	−1.12449e8	−0.338816
$273$	3.77142e8	1.12185
$274$	1.97465e8	0.579912
$275$	−3.23878e7	−0.0939110
$276$	−2.44818e8	−0.700909
$277$	6.68855e8	1.89083	0.945415	−	0.325868i	$-0.105656\pi$
$277$	6.68855e8	1.89083	0.945415	+	0.325868i	$0.105656\pi$
$278$	2.97011e8	0.829118
$279$	−2.07296e8	−0.571448
$280$	−1.34062e8	−0.364965
$281$	−2.93967e8	−0.790364	−0.395182	−	0.918603i	$-0.629318\pi$
$281$	−2.93967e8	−0.790364	−0.395182	+	0.918603i	$0.629318\pi$
$282$	3.67848e8	0.976781
$283$	−3.03430e8	−0.795804	−0.397902	−	0.917428i	$-0.630262\pi$
$283$	−3.03430e8	−0.795804	−0.397902	+	0.917428i	$0.630262\pi$
$284$	1.89615e7	0.0491199
$285$	−9.93245e7	−0.254155
$286$	6.30324e7	0.159324
$287$	−6.26890e8	−1.56533
$288$	−5.22506e7	−0.128889
$289$	3.43346e8	0.836738
$290$	3.93989e8	0.948618
$291$	8.31983e8	1.97920
$292$	−1.02832e8	−0.241706
$293$	−4.38613e8	−1.01870	−0.509348	−	0.860561i	$-0.670113\pi$
$293$	−4.38613e8	−1.01870	−0.509348	+	0.860561i	$0.670113\pi$
$294$	−2.03092e8	−0.466099
$295$	−6.69584e8	−1.51855
$296$	1.81755e7	0.0407349
$297$	−5.20424e7	−0.115268
$298$	4.49308e8	0.983529
$299$	3.43102e8	0.742290
$300$	−8.92315e7	−0.190807
$301$	3.91598e8	0.827671
$302$	2.32642e8	0.486031
$303$	−2.32908e8	−0.480989
$304$	2.80945e7	0.0573539
$305$	1.33193e8	0.268801
$306$	3.50208e8	0.698717
$307$	−7.09332e8	−1.39915	−0.699577	−	0.714557i	$-0.746628\pi$
$307$	−7.09332e8	−1.39915	−0.699577	+	0.714557i	$0.746628\pi$
$308$	−1.01656e8	−0.198246
$309$	−8.46980e8	−1.63312
$310$	−2.44906e8	−0.466911
$311$	8.87968e7	0.167393	0.0836963	−	0.996491i	$-0.473327\pi$
$311$	8.87968e7	0.167393	0.0836963	+	0.996491i	$0.473327\pi$
$312$	1.73660e8	0.323713
$313$	−3.18179e8	−0.586498	−0.293249	−	0.956036i	$-0.594737\pi$
$313$	−3.18179e8	−0.586498	−0.293249	+	0.956036i	$0.594737\pi$
$314$	−5.93826e8	−1.08244
$315$	4.17518e8	0.752642
$316$	−1.63394e8	−0.291293
$317$	−3.59359e8	−0.633609	−0.316804	−	0.948491i	$-0.602610\pi$
$317$	−3.59359e8	−0.633609	−0.316804	+	0.948491i	$0.602610\pi$
$318$	−3.36935e8	−0.587558
$319$	2.98753e8	0.515283
$320$	−6.17305e7	−0.105311
$321$	1.08224e9	1.82623
$322$	−5.53340e8	−0.923627
$323$	−1.88302e8	−0.310919
$324$	−3.66570e8	−0.598755
$325$	1.25054e8	0.202072
$326$	−5.00228e8	−0.799663
$327$	−1.40372e9	−2.22005
$328$	−2.88660e8	−0.451678
$329$	8.31414e8	1.28716
$330$	1.65487e8	0.253493
$331$	3.76118e8	0.570068	0.285034	−	0.958517i	$-0.407995\pi$
$331$	3.76118e8	0.570068	0.285034	+	0.958517i	$0.407995\pi$
$332$	2.25293e7	0.0337882
$333$	−5.66055e7	−0.0840047
$334$	−3.72325e8	−0.546776
$335$	−8.24122e8	−1.19766
$336$	−2.80072e8	−0.402793
$337$	−1.77732e8	−0.252965	−0.126483	−	0.991969i	$-0.540369\pi$
$337$	−1.77732e8	−0.252965	−0.126483	+	0.991969i	$0.540369\pi$
$338$	2.58611e8	0.364282
$339$	−9.94950e8	−1.38708
$340$	4.13747e8	0.570898
$341$	−1.85707e8	−0.253623
$342$	−8.74968e7	−0.118277
$343$	4.56684e8	0.611064
$344$	1.80317e8	0.238826
$345$	9.00790e8	1.18102
$346$	3.02494e8	0.392601
$347$	−1.02645e9	−1.31881	−0.659406	−	0.751787i	$-0.729192\pi$
$347$	−1.02645e9	−1.31881	−0.659406	+	0.751787i	$0.729192\pi$
$348$	8.23095e8	1.04694
$349$	9.58510e8	1.20700	0.603501	−	0.797362i	$-0.293772\pi$
$349$	9.58510e8	1.20700	0.603501	+	0.797362i	$0.293772\pi$
$350$	−2.01682e8	−0.251437
$351$	2.00943e8	0.248027
$352$	−4.68089e7	−0.0572044
$353$	1.57619e9	1.90721	0.953603	−	0.301065i	$-0.0973423\pi$
$353$	1.57619e9	1.90721	0.953603	+	0.301065i	$0.0973423\pi$
$354$	−1.39885e9	−1.67594
$355$	−6.97673e7	−0.0827661
$356$	3.40912e8	0.400467
$357$	1.87717e9	2.18356
$358$	−1.09582e9	−1.26226
$359$	6.66897e8	0.760726	0.380363	−	0.924837i	$-0.375799\pi$
$359$	6.66897e8	0.760726	0.380363	+	0.924837i	$0.375799\pi$
$360$	1.92252e8	0.217176
$361$	4.70459e7	0.0526316
$362$	−2.63326e8	−0.291752
$363$	−1.07287e9	−1.17726
$364$	3.92509e8	0.426574
$365$	3.78362e8	0.407270
$366$	2.78257e8	0.296663
$367$	−3.96513e8	−0.418723	−0.209361	−	0.977838i	$-0.567138\pi$
$367$	−3.96513e8	−0.418723	−0.209361	+	0.977838i	$0.567138\pi$
$368$	−2.54793e8	−0.266514
$369$	8.98998e8	0.931465
$370$	−6.68755e7	−0.0686375
$371$	−7.61543e8	−0.774258
$372$	−5.11641e8	−0.515306
$373$	7.83723e8	0.781955	0.390977	−	0.920400i	$-0.372137\pi$
$373$	7.83723e8	0.781955	0.390977	+	0.920400i	$0.372137\pi$
$374$	3.13735e8	0.310108
$375$	1.45964e9	1.42934
$376$	3.82836e8	0.371412
$377$	−1.15353e9	−1.10875
$378$	−3.24073e8	−0.308618
$379$	1.45545e9	1.37328	0.686641	−	0.726997i	$-0.259085\pi$
$379$	1.45545e9	1.37328	0.686641	+	0.726997i	$0.259085\pi$
$380$	−1.03371e8	−0.0966403
$381$	−2.74875e9	−2.54623
$382$	1.75948e8	0.161496
$383$	−2.76212e8	−0.251215	−0.125608	−	0.992080i	$-0.540088\pi$
$383$	−2.76212e8	−0.251215	−0.125608	+	0.992080i	$0.540088\pi$
$384$	−1.28963e8	−0.116227
$385$	3.74035e8	0.334041
$386$	9.83504e8	0.870403
$387$	−5.61574e8	−0.492514
$388$	8.65883e8	0.752572
$389$	1.23552e9	1.06421	0.532104	−	0.846679i	$-0.321401\pi$
$389$	1.23552e9	1.06421	0.532104	+	0.846679i	$0.321401\pi$
$390$	−6.38970e8	−0.545449
$391$	1.70774e9	1.44479
$392$	−2.11367e8	−0.177230
$393$	−9.33523e8	−0.775803
$394$	1.21290e9	0.999055
$395$	6.01195e8	0.490823
$396$	1.45781e8	0.117969
$397$	−9.34170e8	−0.749306	−0.374653	−	0.927165i	$-0.622238\pi$
$397$	−9.34170e8	−0.749306	−0.374653	+	0.927165i	$0.622238\pi$
$398$	4.84250e8	0.385016
$399$	−4.68998e8	−0.369628
$400$	−9.28672e7	−0.0725525
$401$	3.41301e8	0.264321	0.132161	−	0.991228i	$-0.457809\pi$
$401$	3.41301e8	0.264321	0.132161	+	0.991228i	$0.457809\pi$
$402$	−1.72170e9	−1.32180
$403$	7.17043e8	0.545729
$404$	−2.42398e8	−0.182892
$405$	1.34877e9	1.00889
$406$	1.86037e9	1.37961
$407$	−5.07103e7	−0.0372834
$408$	8.64371e8	0.630071
$409$	5.77116e8	0.417091	0.208546	−	0.978013i	$-0.433127\pi$
$409$	5.77116e8	0.417091	0.208546	+	0.978013i	$0.433127\pi$
$410$	1.06210e9	0.761068
$411$	−1.51787e9	−1.07842
$412$	−8.81491e8	−0.620980
$413$	−3.16169e9	−2.20848
$414$	7.93523e8	0.549614
$415$	−8.28949e7	−0.0569324
$416$	1.80736e8	0.123089
$417$	−2.28307e9	−1.54185
$418$	−7.83843e7	−0.0524943
$419$	−2.35846e9	−1.56631	−0.783157	−	0.621824i	$-0.786392\pi$
$419$	−2.35846e9	−1.56631	−0.783157	+	0.621824i	$0.786392\pi$
$420$	1.03050e9	0.678699
$421$	1.38293e9	0.903258	0.451629	−	0.892206i	$-0.350843\pi$
$421$	1.38293e9	0.903258	0.451629	+	0.892206i	$0.350843\pi$
$422$	−1.08275e8	−0.0701352
$423$	−1.19230e9	−0.765937
$424$	−3.50663e8	−0.223414
$425$	6.22440e8	0.393311
$426$	−1.45753e8	−0.0913448
$427$	6.28920e8	0.390929
$428$	1.12634e9	0.694408
$429$	−4.84517e8	−0.296284
$430$	−6.63462e8	−0.402417
$431$	2.15391e9	1.29586	0.647929	−	0.761701i	$-0.275636\pi$
$431$	2.15391e9	1.29586	0.647929	+	0.761701i	$0.275636\pi$
$432$	−1.49224e8	−0.0890524
$433$	2.51495e9	1.48875	0.744375	−	0.667762i	$-0.232748\pi$
$433$	2.51495e9	1.48875	0.744375	+	0.667762i	$0.232748\pi$
$434$	−1.15642e9	−0.679047
$435$	−3.02852e9	−1.76408
$436$	−1.46091e9	−0.844155
$437$	−4.26667e8	−0.244570
$438$	7.90448e8	0.449483
$439$	2.29307e9	1.29357	0.646787	−	0.762671i	$-0.276112\pi$
$439$	2.29307e9	1.29357	0.646787	+	0.762671i	$0.276112\pi$
$440$	1.72230e8	0.0963882
$441$	6.58278e8	0.365489
$442$	−1.21138e9	−0.667270
$443$	−1.06452e9	−0.581757	−0.290879	−	0.956760i	$-0.593948\pi$
$443$	−1.06452e9	−0.581757	−0.290879	+	0.956760i	$0.593948\pi$
$444$	−1.39712e8	−0.0757517
$445$	−1.25436e9	−0.674780
$446$	3.87879e7	0.0207026
$447$	−3.45374e9	−1.82900
$448$	−2.91484e8	−0.153158
$449$	−1.22692e9	−0.639665	−0.319832	−	0.947474i	$-0.603627\pi$
$449$	−1.22692e9	−0.639665	−0.319832	+	0.947474i	$0.603627\pi$
$450$	2.89224e8	0.149620
$451$	8.05371e8	0.413407
$452$	−1.03549e9	−0.527425
$453$	−1.78827e9	−0.903837
$454$	−2.76171e9	−1.38510
$455$	−1.44421e9	−0.718769
$456$	−2.15957e8	−0.106657
$457$	−5.19768e7	−0.0254743	−0.0127372	−	0.999919i	$-0.504054\pi$
$457$	−5.19768e7	−0.0254743	−0.0127372	+	0.999919i	$0.504054\pi$
$458$	−1.09122e9	−0.530741
$459$	1.00017e9	0.482758
$460$	9.37493e8	0.449072
$461$	−1.80708e9	−0.859060	−0.429530	−	0.903053i	$-0.641321\pi$
$461$	−1.80708e9	−0.859060	−0.429530	+	0.903053i	$0.641321\pi$
$462$	7.81409e8	0.368665
$463$	2.65408e9	1.24274	0.621372	−	0.783516i	$-0.286576\pi$
$463$	2.65408e9	1.24274	0.621372	+	0.783516i	$0.286576\pi$
$464$	8.56632e8	0.398090
$465$	1.88255e9	0.868281
$466$	2.61689e9	1.19794
$467$	−1.15242e9	−0.523602	−0.261801	−	0.965122i	$-0.584316\pi$
$467$	−1.15242e9	−0.523602	−0.261801	+	0.965122i	$0.584316\pi$
$468$	−5.62881e8	−0.253838
$469$	−3.89140e9	−1.74181
$470$	−1.40862e9	−0.625822
$471$	4.56462e9	2.01294
$472$	−1.45584e9	−0.637262
$473$	−5.03089e8	−0.218590
$474$	1.25597e9	0.541697
$475$	−1.55512e8	−0.0665787
$476$	1.95366e9	0.830280
$477$	1.09210e9	0.460731
$478$	1.42808e9	0.598076
$479$	−1.63103e9	−0.678089	−0.339044	−	0.940770i	$-0.610104\pi$
$479$	−1.63103e9	−0.678089	−0.339044	+	0.940770i	$0.610104\pi$
$480$	4.74510e8	0.195840
$481$	1.95800e8	0.0802240
$482$	1.79286e9	0.729260
$483$	4.25342e9	1.71760
$484$	−1.11658e9	−0.447643
$485$	−3.18595e9	−1.26807
$486$	2.18034e9	0.861583
$487$	4.91566e9	1.92855	0.964274	−	0.264906i	$-0.0853410\pi$
$487$	4.91566e9	1.92855	0.964274	+	0.264906i	$0.0853410\pi$
$488$	2.89595e8	0.112803
$489$	3.84516e9	1.48708
$490$	7.77710e8	0.298629
$491$	4.50903e8	0.171909	0.0859544	−	0.996299i	$-0.472606\pi$
$491$	4.50903e8	0.171909	0.0859544	+	0.996299i	$0.472606\pi$
$492$	2.21888e9	0.839953
$493$	−5.74155e9	−2.15807
$494$	3.02654e8	0.112954
$495$	−5.36389e8	−0.198775
$496$	−5.32488e8	−0.195940
$497$	−3.29432e8	−0.120370
$498$	−1.73178e8	−0.0628335
$499$	−4.04813e9	−1.45849	−0.729243	−	0.684254i	$-0.760128\pi$
$499$	−4.04813e9	−1.45849	−0.729243	+	0.684254i	$0.760128\pi$
$500$	1.51911e9	0.543495
$501$	2.86199e9	1.01680
$502$	−1.85449e9	−0.654278
$503$	3.61727e9	1.26734	0.633671	−	0.773603i	$-0.281547\pi$
$503$	3.61727e9	1.26734	0.633671	+	0.773603i	$0.281547\pi$
$504$	9.07790e8	0.315848
$505$	8.91884e8	0.308169
$506$	7.10880e8	0.243933
$507$	−1.98789e9	−0.677430
$508$	−2.86075e9	−0.968183
$509$	−2.33518e9	−0.784889	−0.392444	−	0.919776i	$-0.628370\pi$
$509$	−2.33518e9	−0.784889	−0.392444	+	0.919776i	$0.628370\pi$
$510$	−3.18039e9	−1.06166
$511$	1.78658e9	0.592309
$512$	−1.34218e8	−0.0441942
$513$	−2.49885e8	−0.0817201
$514$	3.18592e9	1.03482
$515$	3.24338e9	1.04634
$516$	−1.38606e9	−0.444128
$517$	−1.06812e9	−0.339942
$518$	−3.15778e8	−0.0998222
$519$	−2.32521e9	−0.730091
$520$	−6.65005e8	−0.207402
$521$	−2.25540e9	−0.698701	−0.349350	−	0.936992i	$-0.613598\pi$
$521$	−2.25540e9	−0.698701	−0.349350	+	0.936992i	$0.613598\pi$
$522$	−2.66788e9	−0.820954
$523$	2.97677e8	0.0909891	0.0454945	−	0.998965i	$-0.485514\pi$
$523$	2.97677e8	0.0909891	0.0454945	+	0.998965i	$0.485514\pi$
$524$	−9.71560e8	−0.294992
$525$	1.55029e9	0.467579
$526$	−2.48165e9	−0.743517
$527$	3.56899e9	1.06220
$528$	3.59811e8	0.106379
$529$	4.64685e8	0.136478
$530$	1.29024e9	0.376448
$531$	4.53405e9	1.31418
$532$	−4.88107e8	−0.140548
$533$	−3.10966e9	−0.889543
$534$	−2.62052e9	−0.744721
$535$	−4.14427e9	−1.17006
$536$	−1.79185e9	−0.502602
$537$	8.42336e9	2.34734
$538$	1.49468e9	0.413820
$539$	5.89721e8	0.162213
$540$	5.49059e8	0.150052
$541$	−3.10860e9	−0.844061	−0.422031	−	0.906582i	$-0.638683\pi$
$541$	−3.10860e9	−0.844061	−0.422031	+	0.906582i	$0.638683\pi$
$542$	8.82218e8	0.238001
$543$	2.02414e9	0.542551
$544$	8.99590e8	0.239579
$545$	5.37533e9	1.42238
$546$	−3.01714e9	−0.793269
$547$	−4.78922e9	−1.25115	−0.625575	−	0.780164i	$-0.715136\pi$
$547$	−4.78922e9	−1.25115	−0.625575	+	0.780164i	$0.715136\pi$
$548$	−1.57972e9	−0.410060
$549$	−9.01908e8	−0.232627
$550$	2.59102e8	0.0664051
$551$	1.43448e9	0.365313
$552$	1.95855e9	0.495618
$553$	2.83876e9	0.713824
$554$	−5.35084e9	−1.33702
$555$	5.14059e8	0.127640
$556$	−2.37609e9	−0.586275
$557$	3.74564e9	0.918403	0.459201	−	0.888332i	$-0.348136\pi$
$557$	3.74564e9	0.918403	0.459201	+	0.888332i	$0.348136\pi$
$558$	1.65837e9	0.404075
$559$	1.94250e9	0.470349
$560$	1.07249e9	0.258069
$561$	−2.41162e9	−0.576685
$562$	2.35174e9	0.558872
$563$	−2.35071e8	−0.0555162	−0.0277581	−	0.999615i	$-0.508837\pi$
$563$	−2.35071e8	−0.0555162	−0.0277581	+	0.999615i	$0.508837\pi$
$564$	−2.94279e9	−0.690688
$565$	3.81000e9	0.888701
$566$	2.42744e9	0.562719
$567$	6.36870e9	1.46727
$568$	−1.51692e8	−0.0347330
$569$	5.35979e8	0.121970	0.0609852	−	0.998139i	$-0.480576\pi$
$569$	5.35979e8	0.121970	0.0609852	+	0.998139i	$0.480576\pi$
$570$	7.94596e8	0.179715
$571$	1.00410e9	0.225709	0.112854	−	0.993612i	$-0.464001\pi$
$571$	1.00410e9	0.225709	0.112854	+	0.993612i	$0.464001\pi$
$572$	−5.04259e8	−0.112659
$573$	−1.35248e9	−0.300323
$574$	5.01512e9	1.10685
$575$	1.41036e9	0.309381
$576$	4.18005e8	0.0911386
$577$	−8.44095e9	−1.82926	−0.914631	−	0.404290i	$-0.867519\pi$
$577$	−8.44095e9	−1.82926	−0.914631	+	0.404290i	$0.867519\pi$
$578$	−2.74677e9	−0.591663
$579$	−7.56000e9	−1.61863
$580$	−3.15191e9	−0.670774
$581$	−3.91419e8	−0.0827991
$582$	−6.65587e9	−1.39950
$583$	9.78361e8	0.204484
$584$	8.22655e8	0.170912
$585$	2.07108e9	0.427711
$586$	3.50890e9	0.720327
$587$	3.06553e9	0.625565	0.312783	−	0.949825i	$-0.398739\pi$
$587$	3.06553e9	0.625565	0.312783	+	0.949825i	$0.398739\pi$
$588$	1.62474e9	0.329581
$589$	−8.91684e8	−0.179807
$590$	5.35667e9	1.07377
$591$	−9.32335e9	−1.85787
$592$	−1.45404e8	−0.0288039
$593$	1.66016e9	0.326933	0.163467	−	0.986549i	$-0.447732\pi$
$593$	1.66016e9	0.326933	0.163467	+	0.986549i	$0.447732\pi$
$594$	4.16339e8	0.0815070
$595$	−7.18835e9	−1.39901
$596$	−3.59446e9	−0.695460
$597$	−3.72233e9	−0.715986
$598$	−2.74481e9	−0.524879
$599$	−7.40940e9	−1.40861	−0.704303	−	0.709900i	$-0.748740\pi$
$599$	−7.40940e9	−1.40861	−0.704303	+	0.709900i	$0.748740\pi$
$600$	7.13852e8	0.134921
$601$	1.74087e9	0.327118	0.163559	−	0.986534i	$-0.447703\pi$
$601$	1.74087e9	0.327118	0.163559	+	0.986534i	$0.447703\pi$
$602$	−3.13278e9	−0.585251
$603$	5.58049e9	1.03648
$604$	−1.86114e9	−0.343676
$605$	4.10838e9	0.754269
$606$	1.86326e9	0.340110
$607$	−4.36213e8	−0.0791658	−0.0395829	−	0.999216i	$-0.512603\pi$
$607$	−4.36213e8	−0.0791658	−0.0395829	+	0.999216i	$0.512603\pi$
$608$	−2.24756e8	−0.0405554
$609$	−1.43003e10	−2.56557
$610$	−1.06554e9	−0.190071
$611$	4.12419e9	0.731466
$612$	−2.80166e9	−0.494067
$613$	−6.37962e9	−1.11862	−0.559310	−	0.828958i	$-0.688934\pi$
$613$	−6.37962e9	−1.11862	−0.559310	+	0.828958i	$0.688934\pi$
$614$	5.67466e9	0.989351
$615$	−8.16418e9	−1.41530
$616$	8.13247e8	0.140181
$617$	2.32381e9	0.398293	0.199147	−	0.979970i	$-0.436183\pi$
$617$	2.32381e9	0.398293	0.199147	+	0.979970i	$0.436183\pi$
$618$	6.77584e9	1.15479
$619$	9.45273e8	0.160192	0.0800959	−	0.996787i	$-0.474477\pi$
$619$	9.45273e8	0.160192	0.0800959	+	0.996787i	$0.474477\pi$
$620$	1.95925e9	0.330156
$621$	2.26624e9	0.379740
$622$	−7.10375e8	−0.118364
$623$	−5.92293e9	−0.981360
$624$	−1.38928e9	−0.228899
$625$	−3.81816e9	−0.625568
$626$	2.54544e9	0.414717
$627$	6.02525e8	0.0976199
$628$	4.75061e9	0.765404
$629$	9.74568e8	0.156147
$630$	−3.34015e9	−0.532198
$631$	−1.09828e10	−1.74025	−0.870124	−	0.492833i	$-0.835961\pi$
$631$	−1.09828e10	−1.74025	−0.870124	+	0.492833i	$0.835961\pi$
$632$	1.30715e9	0.205975
$633$	8.32291e8	0.130425
$634$	2.87487e9	0.448029
$635$	1.05259e10	1.63137
$636$	2.69548e9	0.415467
$637$	−2.27700e9	−0.349040
$638$	−2.39003e9	−0.364360
$639$	4.72425e8	0.0716275
$640$	4.93844e8	0.0744663
$641$	−1.79159e9	−0.268680	−0.134340	−	0.990935i	$-0.542891\pi$
$641$	−1.79159e9	−0.268680	−0.134340	+	0.990935i	$0.542891\pi$
$642$	−8.65792e9	−1.29134
$643$	1.09669e10	1.62684	0.813421	−	0.581675i	$-0.197602\pi$
$643$	1.09669e10	1.62684	0.813421	+	0.581675i	$0.197602\pi$
$644$	4.42672e9	0.653103
$645$	5.09990e9	0.748346
$646$	1.50642e9	0.219853
$647$	6.07249e9	0.881458	0.440729	−	0.897640i	$-0.354720\pi$
$647$	6.07249e9	0.881458	0.440729	+	0.897640i	$0.354720\pi$
$648$	2.93256e9	0.423383
$649$	4.06185e9	0.583267
$650$	−1.00043e9	−0.142886
$651$	8.88914e9	1.26278
$652$	4.00183e9	0.565447
$653$	−7.24093e9	−1.01765	−0.508825	−	0.860870i	$-0.669920\pi$
$653$	−7.24093e9	−1.01765	−0.508825	+	0.860870i	$0.669920\pi$
$654$	1.12298e10	1.56981
$655$	3.57478e9	0.497056
$656$	2.30928e9	0.319384
$657$	−2.56206e9	−0.352460
$658$	−6.65131e9	−0.910158
$659$	6.64606e9	0.904618	0.452309	−	0.891861i	$-0.350600\pi$
$659$	6.64606e9	0.904618	0.452309	+	0.891861i	$0.350600\pi$
$660$	−1.32390e9	−0.179246
$661$	−4.77702e9	−0.643357	−0.321679	−	0.946849i	$-0.604247\pi$
$661$	−4.77702e9	−0.643357	−0.321679	+	0.946849i	$0.604247\pi$
$662$	−3.00895e9	−0.403099
$663$	9.31162e9	1.24088
$664$	−1.80235e8	−0.0238918
$665$	1.79595e9	0.236820
$666$	4.52844e8	0.0594003
$667$	−1.30096e10	−1.69755
$668$	2.97860e9	0.386629
$669$	−2.98155e8	−0.0384991
$670$	6.59297e9	0.846875
$671$	−8.07978e8	−0.103245
$672$	2.24058e9	0.284818
$673$	−1.43274e9	−0.181182	−0.0905909	−	0.995888i	$-0.528876\pi$
$673$	−1.43274e9	−0.181182	−0.0905909	+	0.995888i	$0.528876\pi$
$674$	1.42186e9	0.178873
$675$	8.26002e8	0.103376
$676$	−2.06889e9	−0.257587
$677$	1.04906e10	1.29939	0.649697	−	0.760193i	$-0.274896\pi$
$677$	1.04906e10	1.29939	0.649697	+	0.760193i	$0.274896\pi$
$678$	7.95960e9	0.980815
$679$	−1.50437e10	−1.84420
$680$	−3.30998e9	−0.403686
$681$	2.12287e10	2.57578
$682$	1.48566e9	0.179338
$683$	1.93687e9	0.232610	0.116305	−	0.993214i	$-0.462895\pi$
$683$	1.93687e9	0.232610	0.116305	+	0.993214i	$0.462895\pi$
$684$	6.99974e8	0.0836345
$685$	5.81245e9	0.690943
$686$	−3.65347e9	−0.432087
$687$	8.38797e9	0.986980
$688$	−1.44253e9	−0.168875
$689$	−3.77760e9	−0.439995
$690$	−7.20632e9	−0.835106
$691$	1.36523e10	1.57410	0.787049	−	0.616890i	$-0.211608\pi$
$691$	1.36523e10	1.57410	0.787049	+	0.616890i	$0.211608\pi$
$692$	−2.41996e9	−0.277611
$693$	−2.53276e9	−0.289086
$694$	8.21157e9	0.932542
$695$	8.74265e9	0.987862
$696$	−6.58476e9	−0.740300
$697$	−1.54779e10	−1.73140
$698$	−7.66808e9	−0.853479
$699$	−2.01155e10	−2.22772
$700$	1.61345e9	0.177793
$701$	−1.23717e10	−1.35649	−0.678247	−	0.734834i	$-0.737260\pi$
$701$	−1.23717e10	−1.35649	−0.678247	+	0.734834i	$0.737260\pi$
$702$	−1.60755e9	−0.175382
$703$	−2.43488e8	−0.0264323
$704$	3.74471e8	0.0404496
$705$	1.08278e10	1.16380
$706$	−1.26095e10	−1.34860
$707$	4.21136e9	0.448182
$708$	1.11908e10	1.18507
$709$	−6.65641e9	−0.701421	−0.350710	−	0.936484i	$-0.614060\pi$
$709$	−6.65641e9	−0.701421	−0.350710	+	0.936484i	$0.614060\pi$
$710$	5.58138e8	0.0585245
$711$	−4.07096e9	−0.424769
$712$	−2.72730e9	−0.283173
$713$	8.08682e9	0.835535
$714$	−1.50174e10	−1.54401
$715$	1.85538e9	0.189829
$716$	8.76657e9	0.892554
$717$	−1.09774e10	−1.11220
$718$	−5.33517e9	−0.537914
$719$	−5.16131e9	−0.517855	−0.258928	−	0.965897i	$-0.583369\pi$
$719$	−5.16131e9	−0.517855	−0.258928	+	0.965897i	$0.583369\pi$
$720$	−1.53802e9	−0.153567
$721$	1.53148e10	1.52173
$722$	−3.76367e8	−0.0372161
$723$	−1.37814e10	−1.35615
$724$	2.10661e9	0.206300
$725$	−4.74173e9	−0.462119
$726$	8.58293e9	0.832449
$727$	−2.85753e9	−0.275817	−0.137908	−	0.990445i	$-0.544038\pi$
$727$	−2.85753e9	−0.275817	−0.137908	+	0.990445i	$0.544038\pi$
$728$	−3.14007e9	−0.301633
$729$	−4.23346e9	−0.404715
$730$	−3.02690e9	−0.287983
$731$	9.66854e9	0.915483
$732$	−2.22606e9	−0.209772
$733$	−1.01025e10	−0.947472	−0.473736	−	0.880667i	$-0.657095\pi$
$733$	−1.01025e10	−0.947472	−0.473736	+	0.880667i	$0.657095\pi$
$734$	3.17210e9	0.296082
$735$	−5.97810e9	−0.555338
$736$	2.03835e9	0.188454
$737$	4.99931e9	0.460017
$738$	−7.19198e9	−0.658645
$739$	−2.37969e9	−0.216903	−0.108451	−	0.994102i	$-0.534589\pi$
$739$	−2.37969e9	−0.216903	−0.108451	+	0.994102i	$0.534589\pi$
$740$	5.35004e8	0.0485340
$741$	−2.32644e9	−0.210052
$742$	6.09235e9	0.547483
$743$	−1.95551e10	−1.74904	−0.874520	−	0.484989i	$-0.838823\pi$
$743$	−1.95551e10	−1.74904	−0.874520	+	0.484989i	$0.838823\pi$
$744$	4.09313e9	0.364376
$745$	1.32256e10	1.17184
$746$	−6.26978e9	−0.552925
$747$	5.61319e8	0.0492705
$748$	−2.50988e9	−0.219279
$749$	−1.95687e10	−1.70167
$750$	−1.16771e10	−1.01070
$751$	7.74452e9	0.667198	0.333599	−	0.942715i	$-0.391737\pi$
$751$	7.74452e9	0.667198	0.333599	+	0.942715i	$0.391737\pi$
$752$	−3.06269e9	−0.262628
$753$	1.42551e10	1.21671
$754$	9.22825e9	0.784007
$755$	6.84791e9	0.579087
$756$	2.59258e9	0.218226
$757$	1.16038e9	0.0972219	0.0486110	−	0.998818i	$-0.484521\pi$
$757$	1.16038e9	0.0972219	0.0486110	+	0.998818i	$0.484521\pi$
$758$	−1.16436e10	−0.971057
$759$	−5.46440e9	−0.453624
$760$	8.26972e8	0.0683350
$761$	−1.86077e9	−0.153054	−0.0765272	−	0.997067i	$-0.524383\pi$
$761$	−1.86077e9	−0.153054	−0.0765272	+	0.997067i	$0.524383\pi$
$762$	2.19900e10	1.80046
$763$	2.53816e10	2.06863
$764$	−1.40758e9	−0.114195
$765$	1.03085e10	0.832494
$766$	2.20969e9	0.177636
$767$	−1.56834e10	−1.25504
$768$	1.03170e9	0.0821847
$769$	9.85947e9	0.781828	0.390914	−	0.920427i	$-0.372159\pi$
$769$	9.85947e9	0.781828	0.390914	+	0.920427i	$0.372159\pi$
$770$	−2.99228e9	−0.236203
$771$	−2.44895e10	−1.92438
$772$	−7.86803e9	−0.615468
$773$	1.36871e10	1.06582	0.532910	−	0.846172i	$-0.321098\pi$
$773$	1.36871e10	1.06582	0.532910	+	0.846172i	$0.321098\pi$
$774$	4.49260e9	0.348260
$775$	2.94749e9	0.227456
$776$	−6.92706e9	−0.532149
$777$	2.42732e9	0.185632
$778$	−9.88417e9	−0.752509
$779$	3.86703e9	0.293087
$780$	5.11176e9	0.385691
$781$	4.23224e8	0.0317901
$782$	−1.36620e10	−1.02162
$783$	−7.61927e9	−0.567214
$784$	1.69094e9	0.125320
$785$	−1.74795e10	−1.28969
$786$	7.46819e9	0.548575
$787$	−1.99739e10	−1.46067	−0.730335	−	0.683090i	$-0.760636\pi$
$787$	−1.99739e10	−1.46067	−0.730335	+	0.683090i	$0.760636\pi$
$788$	−9.70323e9	−0.706439
$789$	1.90760e10	1.38267
$790$	−4.80956e9	−0.347065
$791$	1.79904e10	1.29247
$792$	−1.16625e9	−0.0834164
$793$	3.11972e9	0.222157
$794$	7.47336e9	0.529839
$795$	−9.91781e9	−0.700053
$796$	−3.87400e9	−0.272247
$797$	8.76673e9	0.613386	0.306693	−	0.951809i	$-0.400778\pi$
$797$	8.76673e9	0.613386	0.306693	+	0.951809i	$0.400778\pi$
$798$	3.75198e9	0.261367
$799$	2.05276e10	1.42372
$800$	7.42938e8	0.0513024
$801$	8.49383e9	0.583969
$802$	−2.73041e9	−0.186903
$803$	−2.29523e9	−0.156431
$804$	1.37736e10	0.934654
$805$	−1.62878e10	−1.10047
$806$	−5.73634e9	−0.385889
$807$	−1.14893e10	−0.769552
$808$	1.93918e9	0.129324
$809$	2.42316e10	1.60902	0.804510	−	0.593938i	$-0.202428\pi$
$809$	2.42316e10	1.60902	0.804510	+	0.593938i	$0.202428\pi$
$810$	−1.07901e10	−0.713393
$811$	−4.37791e8	−0.0288200	−0.0144100	−	0.999896i	$-0.504587\pi$
$811$	−4.37791e8	−0.0288200	−0.0144100	+	0.999896i	$0.504587\pi$
$812$	−1.48829e10	−0.975534
$813$	−6.78144e9	−0.442593
$814$	4.05682e8	0.0263633
$815$	−1.47244e10	−0.952767
$816$	−6.91497e9	−0.445528
$817$	−2.41561e9	−0.154971
$818$	−4.61693e9	−0.294928
$819$	9.77936e9	0.622038
$820$	−8.49684e9	−0.538157
$821$	1.06296e10	0.670371	0.335185	−	0.942152i	$-0.391201\pi$
$821$	1.06296e10	0.670371	0.335185	+	0.942152i	$0.391201\pi$
$822$	1.21430e10	0.762559
$823$	−2.11804e10	−1.32445	−0.662225	−	0.749305i	$-0.730388\pi$
$823$	−2.11804e10	−1.32445	−0.662225	+	0.749305i	$0.730388\pi$
$824$	7.05193e9	0.439099
$825$	−1.99167e9	−0.123489
$826$	2.52935e10	1.56163
$827$	2.42805e10	1.49276	0.746378	−	0.665522i	$-0.231791\pi$
$827$	2.42805e10	1.49276	0.746378	+	0.665522i	$0.231791\pi$
$828$	−6.34818e9	−0.388636
$829$	−4.44427e8	−0.0270931	−0.0135466	−	0.999908i	$-0.504312\pi$
$829$	−4.44427e8	−0.0270931	−0.0135466	+	0.999908i	$0.504312\pi$
$830$	6.63160e8	0.0402573
$831$	4.11308e10	2.48636
$832$	−1.44589e9	−0.0870369
$833$	−1.13335e10	−0.679368
$834$	1.82645e10	1.09025
$835$	−1.09595e10	−0.651462
$836$	6.27075e8	0.0371191
$837$	4.73619e9	0.279183
$838$	1.88677e10	1.10755
$839$	2.28177e10	1.33384	0.666922	−	0.745127i	$-0.267611\pi$
$839$	2.28177e10	1.33384	0.666922	+	0.745127i	$0.267611\pi$
$840$	−8.24403e9	−0.479913
$841$	2.64891e10	1.53561
$842$	−1.10634e10	−0.638700
$843$	−1.80774e10	−1.03929
$844$	8.66203e8	0.0495931
$845$	7.61231e9	0.434028
$846$	9.53838e9	0.541600
$847$	1.93992e10	1.09696
$848$	2.80531e9	0.157977
$849$	−1.86593e10	−1.04645
$850$	−4.97952e9	−0.278113
$851$	2.20824e9	0.122826
$852$	1.16602e9	0.0645905
$853$	8.62323e9	0.475717	0.237858	−	0.971300i	$-0.423555\pi$
$853$	8.62323e9	0.475717	0.237858	+	0.971300i	$0.423555\pi$
$854$	−5.03136e9	−0.276428
$855$	−2.57550e9	−0.140923
$856$	−9.01069e9	−0.491021
$857$	1.18096e10	0.640916	0.320458	−	0.947263i	$-0.396163\pi$
$857$	1.18096e10	0.640916	0.320458	+	0.947263i	$0.396163\pi$
$858$	3.87614e9	0.209505
$859$	−4.29431e9	−0.231162	−0.115581	−	0.993298i	$-0.536873\pi$
$859$	−4.29431e9	−0.231162	−0.115581	+	0.993298i	$0.536873\pi$
$860$	5.30770e9	0.284552
$861$	−3.85502e10	−2.05833
$862$	−1.72313e10	−0.916309
$863$	−1.16286e10	−0.615869	−0.307934	−	0.951408i	$-0.599638\pi$
$863$	−1.16286e10	−0.615869	−0.307934	+	0.951408i	$0.599638\pi$
$864$	1.19379e9	0.0629696
$865$	8.90405e9	0.467769
$866$	−2.01196e10	−1.05270
$867$	2.11139e10	1.10027
$868$	9.25133e9	0.480159
$869$	−3.64698e9	−0.188523
$870$	2.42281e10	1.24739
$871$	−1.93031e10	−0.989835
$872$	1.16873e10	0.596908
$873$	2.15735e10	1.09741
$874$	3.41333e9	0.172937
$875$	−2.63927e10	−1.33185
$876$	−6.32358e9	−0.317833
$877$	2.30911e10	1.15597	0.577986	−	0.816047i	$-0.303839\pi$
$877$	2.30911e10	1.15597	0.577986	+	0.816047i	$0.303839\pi$
$878$	−1.83446e10	−0.914695
$879$	−2.69722e10	−1.33954
$880$	−1.37784e9	−0.0681568
$881$	−2.75942e10	−1.35957	−0.679785	−	0.733411i	$-0.737927\pi$
$881$	−2.75942e10	−1.35957	−0.679785	+	0.733411i	$0.737927\pi$
$882$	−5.26622e9	−0.258440
$883$	−2.32713e10	−1.13752	−0.568758	−	0.822505i	$-0.692576\pi$
$883$	−2.32713e10	−1.13752	−0.568758	+	0.822505i	$0.692576\pi$
$884$	9.69103e9	0.471831
$885$	−4.11757e10	−1.99682
$886$	8.51618e9	0.411364
$887$	−8.19649e9	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$887$	−8.19649e9	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$888$	1.11769e9	0.0535646
$889$	4.97021e10	2.37257
$890$	1.00349e10	0.477141
$891$	−8.18191e9	−0.387510
$892$	−3.10303e8	−0.0146389
$893$	−5.12866e9	−0.241004
$894$	2.76299e10	1.29330
$895$	−3.22560e10	−1.50394
$896$	2.33187e9	0.108299
$897$	2.10988e10	0.976079
$898$	9.81532e9	0.452311
$899$	−2.71884e10	−1.24803
$900$	−2.31379e9	−0.105797
$901$	−1.88025e10	−0.856404
$902$	−6.44297e9	−0.292323
$903$	2.40811e10	1.08835
$904$	8.28391e9	0.372946
$905$	−7.75112e9	−0.347612
$906$	1.43062e10	0.639110
$907$	2.92890e9	0.130341	0.0651703	−	0.997874i	$-0.479241\pi$
$907$	2.92890e9	0.130341	0.0651703	+	0.997874i	$0.479241\pi$
$908$	2.20937e10	0.979417
$909$	−6.03935e9	−0.266696
$910$	1.15536e10	0.508246
$911$	−9.42256e9	−0.412909	−0.206455	−	0.978456i	$-0.566193\pi$
$911$	−9.42256e9	−0.412909	−0.206455	+	0.978456i	$0.566193\pi$
$912$	1.72765e9	0.0754179
$913$	5.02859e8	0.0218675
$914$	4.15814e8	0.0180131
$915$	8.19062e9	0.353462
$916$	8.72974e9	0.375290
$917$	1.68797e10	0.722888
$918$	−8.00136e9	−0.341361
$919$	8.47572e9	0.360224	0.180112	−	0.983646i	$-0.442354\pi$
$919$	8.47572e9	0.360224	0.180112	+	0.983646i	$0.442354\pi$
$920$	−7.49994e9	−0.317542
$921$	−4.36200e10	−1.83983
$922$	1.44566e10	0.607447
$923$	−1.63413e9	−0.0684039
$924$	−6.25127e9	−0.260685
$925$	8.04859e8	0.0334367
$926$	−2.12327e10	−0.878752
$927$	−2.19624e10	−0.905524
$928$	−6.85306e9	−0.281492
$929$	−8.77309e9	−0.359002	−0.179501	−	0.983758i	$-0.557448\pi$
$929$	−8.77309e9	−0.359002	−0.179501	+	0.983758i	$0.557448\pi$
$930$	−1.50604e10	−0.613967
$931$	2.83158e9	0.115002
$932$	−2.09351e10	−0.847069
$933$	5.46051e9	0.220114
$934$	9.21935e9	0.370242
$935$	9.23492e9	0.369481
$936$	4.50304e9	0.179490
$937$	4.37252e10	1.73638	0.868188	−	0.496236i	$-0.165285\pi$
$937$	4.37252e10	1.73638	0.868188	+	0.496236i	$0.165285\pi$
$938$	3.11312e10	1.23165
$939$	−1.95663e10	−0.771220
$940$	1.12689e10	0.442523
$941$	−3.76684e10	−1.47372	−0.736858	−	0.676047i	$-0.763692\pi$
$941$	−3.76684e10	−1.47372	−0.736858	+	0.676047i	$0.763692\pi$
$942$	−3.65170e10	−1.42337
$943$	−3.50708e10	−1.36193
$944$	1.16468e10	0.450612
$945$	−9.53922e9	−0.367707
$946$	4.02471e9	0.154567
$947$	1.23213e10	0.471445	0.235722	−	0.971820i	$-0.424254\pi$
$947$	1.23213e10	0.471445	0.235722	+	0.971820i	$0.424254\pi$
$948$	−1.00478e10	−0.383038
$949$	8.86222e9	0.336597
$950$	1.24409e9	0.0470783
$951$	−2.20986e10	−0.833167
$952$	−1.56293e10	−0.587097
$953$	−2.03050e10	−0.759938	−0.379969	−	0.924999i	$-0.624065\pi$
$953$	−2.03050e10	−0.759938	−0.379969	+	0.924999i	$0.624065\pi$
$954$	−8.73679e9	−0.325786
$955$	5.17910e9	0.192416
$956$	−1.14247e10	−0.422903
$957$	1.83717e10	0.677574
$958$	1.30482e10	0.479481
$959$	2.74456e10	1.00487
$960$	−3.79608e9	−0.138480
$961$	−1.06121e10	−0.385718
$962$	−1.56640e9	−0.0567270
$963$	2.80627e10	1.01260
$964$	−1.43429e10	−0.515665
$965$	2.89498e10	1.03705
$966$	−3.40273e10	−1.21453
$967$	−1.00646e10	−0.357933	−0.178967	−	0.983855i	$-0.557275\pi$
$967$	−1.00646e10	−0.357933	−0.178967	+	0.983855i	$0.557275\pi$
$968$	8.93265e9	0.316531
$969$	−1.15795e10	−0.408844
$970$	2.54876e10	0.896660
$971$	2.39617e10	0.839942	0.419971	−	0.907537i	$-0.362040\pi$
$971$	2.39617e10	0.839942	0.419971	+	0.907537i	$0.362040\pi$
$972$	−1.74427e10	−0.609231
$973$	4.12817e10	1.43669
$974$	−3.93253e10	−1.36369
$975$	7.69012e9	0.265716
$976$	−2.31676e9	−0.0797640
$977$	−2.23861e10	−0.767977	−0.383988	−	0.923338i	$-0.625450\pi$
$977$	−2.23861e10	−0.767977	−0.383988	+	0.923338i	$0.625450\pi$
$978$	−3.07612e10	−1.05152
$979$	7.60923e9	0.259180
$980$	−6.22168e9	−0.211162
$981$	−3.63987e10	−1.23096
$982$	−3.60723e9	−0.121558
$983$	3.27462e10	1.09957	0.549785	−	0.835306i	$-0.314709\pi$
$983$	3.27462e10	1.09957	0.549785	+	0.835306i	$0.314709\pi$
$984$	−1.77510e10	−0.593936
$985$	3.57023e10	1.19034
$986$	4.59324e10	1.52598
$987$	5.11273e10	1.69256
$988$	−2.42123e9	−0.0798705
$989$	2.19076e10	0.720124
$990$	4.29111e9	0.140555
$991$	5.31502e10	1.73479	0.867396	−	0.497619i	$-0.165792\pi$
$991$	5.31502e10	1.73479	0.867396	+	0.497619i	$0.165792\pi$
$992$	4.25990e9	0.138551
$993$	2.31292e10	0.749614
$994$	2.63546e9	0.0851145
$995$	1.42541e10	0.458731
$996$	1.38543e9	0.0444300
$997$	3.24019e10	1.03547	0.517735	−	0.855541i	$-0.326775\pi$
$997$	3.24019e10	1.03547	0.517735	+	0.855541i	$0.326775\pi$
$998$	3.23850e10	1.03131
$999$	1.29329e9	0.0410409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.a.c.1.2	✓	2
3.2	odd	2		342.8.a.i.1.2		2
4.3	odd	2		304.8.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.c.1.2	✓	2	1.1	even	1	trivial
304.8.a.b.1.1		2	4.3	odd	2
342.8.a.i.1.2		2	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)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +61.4944 q^{3} +64.0000 q^{4} -235.483 q^{5} -491.955 q^{6} -1111.92 q^{7} -512.000 q^{8} +1594.56 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +61.4944 q^{3} +64.0000 q^{4} -235.483 q^{5} -491.955 q^{6} -1111.92 q^{7} -512.000 q^{8} +1594.56 q^{9} +1883.87 q^{10} +1428.49 q^{11} +3935.64 q^{12} -5515.63 q^{13} +8895.37 q^{14} -14480.9 q^{15} +4096.00 q^{16} -27453.3 q^{17} -12756.5 q^{18} +6859.00 q^{19} -15070.9 q^{20} -68377.0 q^{21} -11428.0 q^{22} -62205.4 q^{23} -31485.1 q^{24} -22672.7 q^{25} +44125.0 q^{26} -36431.6 q^{27} -71163.0 q^{28} +209139. q^{29} +115847. q^{30} -130002. q^{31} -32768.0 q^{32} +87844.4 q^{33} +219627. q^{34} +261839. q^{35} +102052. q^{36} -35499.1 q^{37} -54872.0 q^{38} -339180. q^{39} +120567. q^{40} +563790. q^{41} +547016. q^{42} -352181. q^{43} +91423.6 q^{44} -375492. q^{45} +497643. q^{46} -747727. q^{47} +251881. q^{48} +412827. q^{49} +181381. q^{50} -1.68823e6 q^{51} -353000. q^{52} +684889. q^{53} +291453. q^{54} -336386. q^{55} +569304. q^{56} +421790. q^{57} -1.67311e6 q^{58} +2.84345e6 q^{59} -926778. q^{60} -565615. q^{61} +1.04002e6 q^{62} -1.77303e6 q^{63} +262144. q^{64} +1.29884e6 q^{65} -702755. q^{66} +3.49970e6 q^{67} -1.75701e6 q^{68} -3.82528e6 q^{69} -2.09471e6 q^{70} +296273. q^{71} -816416. q^{72} -1.60675e6 q^{73} +283993. q^{74} -1.39424e6 q^{75} +438976. q^{76} -1.58837e6 q^{77} +2.71344e6 q^{78} -2.55303e6 q^{79} -964539. q^{80} -5.72765e6 q^{81} -4.51032e6 q^{82} +352021. q^{83} -4.37613e6 q^{84} +6.46480e6 q^{85} +2.81745e6 q^{86} +1.28609e7 q^{87} -731389. q^{88} +5.32675e6 q^{89} +3.00394e6 q^{90} +6.13295e6 q^{91} -3.98114e6 q^{92} -7.99439e6 q^{93} +5.98182e6 q^{94} -1.61518e6 q^{95} -2.01505e6 q^{96} +1.35294e7 q^{97} -3.30261e6 q^{98} +2.27782e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} - 11 q^{3} + 128 q^{4} - 69 q^{5} + 88 q^{6} - 348 q^{7} - 1024 q^{8} + 4663 q^{9}+O(q^{10}) \) 2 * q - 16 * q^2 - 11 * q^3 + 128 * q^4 - 69 * q^5 + 88 * q^6 - 348 * q^7 - 1024 * q^8 + 4663 * q^9	\( 2 q - 16 q^{2} - 11 q^{3} + 128 q^{4} - 69 q^{5} + 88 q^{6} - 348 q^{7} - 1024 q^{8} + 4663 q^{9} + 552 q^{10} + 2723 q^{11} - 704 q^{12} - 14113 q^{13} + 2784 q^{14} - 26550 q^{15} + 8192 q^{16} - 38560 q^{17} - 37304 q^{18} + 13718 q^{19} - 4416 q^{20} - 123757 q^{21} - 21784 q^{22} - 73897 q^{23} + 5632 q^{24} - 73081 q^{25} + 112904 q^{26} - 100331 q^{27} - 22272 q^{28} + 159813 q^{29} + 212400 q^{30} - 259468 q^{31} - 65536 q^{32} - 6000 q^{33} + 308480 q^{34} + 389019 q^{35} + 298432 q^{36} - 528168 q^{37} - 109744 q^{38} + 284081 q^{39} + 35328 q^{40} + 1005650 q^{41} + 990056 q^{42} + 286217 q^{43} + 174272 q^{44} + 135351 q^{45} + 591176 q^{46} - 1397509 q^{47} - 45056 q^{48} + 172860 q^{49} + 584648 q^{50} - 883053 q^{51} - 903232 q^{52} - 1385969 q^{53} + 802648 q^{54} - 120873 q^{55} + 178176 q^{56} - 75449 q^{57} - 1278504 q^{58} + 2700953 q^{59} - 1699200 q^{60} - 3975947 q^{61} + 2075744 q^{62} + 571019 q^{63} + 524288 q^{64} - 132480 q^{65} + 48000 q^{66} - 134557 q^{67} - 2467840 q^{68} - 2977707 q^{69} - 3112152 q^{70} + 4202740 q^{71} - 2387456 q^{72} + 900498 q^{73} + 4225344 q^{74} + 2260081 q^{75} + 877952 q^{76} - 599473 q^{77} - 2272648 q^{78} - 6893730 q^{79} - 282624 q^{80} - 7805978 q^{81} - 8045200 q^{82} - 2465330 q^{83} - 7920448 q^{84} + 4615719 q^{85} - 2289736 q^{86} + 16436697 q^{87} - 1394176 q^{88} + 17431724 q^{89} - 1082808 q^{90} - 434771 q^{91} - 4729408 q^{92} + 1391168 q^{93} + 11180072 q^{94} - 473271 q^{95} + 360448 q^{96} + 6351934 q^{97} - 1382880 q^{98} + 6249933 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 11 * q^3 + 128 * q^4 - 69 * q^5 + 88 * q^6 - 348 * q^7 - 1024 * q^8 + 4663 * q^9 + 552 * q^10 + 2723 * q^11 - 704 * q^12 - 14113 * q^13 + 2784 * q^14 - 26550 * q^15 + 8192 * q^16 - 38560 * q^17 - 37304 * q^18 + 13718 * q^19 - 4416 * q^20 - 123757 * q^21 - 21784 * q^22 - 73897 * q^23 + 5632 * q^24 - 73081 * q^25 + 112904 * q^26 - 100331 * q^27 - 22272 * q^28 + 159813 * q^29 + 212400 * q^30 - 259468 * q^31 - 65536 * q^32 - 6000 * q^33 + 308480 * q^34 + 389019 * q^35 + 298432 * q^36 - 528168 * q^37 - 109744 * q^38 + 284081 * q^39 + 35328 * q^40 + 1005650 * q^41 + 990056 * q^42 + 286217 * q^43 + 174272 * q^44 + 135351 * q^45 + 591176 * q^46 - 1397509 * q^47 - 45056 * q^48 + 172860 * q^49 + 584648 * q^50 - 883053 * q^51 - 903232 * q^52 - 1385969 * q^53 + 802648 * q^54 - 120873 * q^55 + 178176 * q^56 - 75449 * q^57 - 1278504 * q^58 + 2700953 * q^59 - 1699200 * q^60 - 3975947 * q^61 + 2075744 * q^62 + 571019 * q^63 + 524288 * q^64 - 132480 * q^65 + 48000 * q^66 - 134557 * q^67 - 2467840 * q^68 - 2977707 * q^69 - 3112152 * q^70 + 4202740 * q^71 - 2387456 * q^72 + 900498 * q^73 + 4225344 * q^74 + 2260081 * q^75 + 877952 * q^76 - 599473 * q^77 - 2272648 * q^78 - 6893730 * q^79 - 282624 * q^80 - 7805978 * q^81 - 8045200 * q^82 - 2465330 * q^83 - 7920448 * q^84 + 4615719 * q^85 - 2289736 * q^86 + 16436697 * q^87 - 1394176 * q^88 + 17431724 * q^89 - 1082808 * q^90 - 434771 * q^91 - 4729408 * q^92 + 1391168 * q^93 + 11180072 * q^94 - 473271 * q^95 + 360448 * q^96 + 6351934 * q^97 - 1382880 * q^98 + 6249933 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more