LMFDB - Embedded newform 138.8.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,8,Mod(1,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("138.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	138.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:	$43.1091335168$
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} - 2x^{3} - 8367x^{2} - 89140x + 11077220 $ x^4 - 2x^3 - 8367x^2 - 89140*x + 11077220
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-65.5856$ of defining polynomial
Character	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +340.987 q^{5} +216.000 q^{6} +1267.12 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +340.987 q^{5} +216.000 q^{6} +1267.12 q^{7} +512.000 q^{8} +729.000 q^{9} +2727.89 q^{10} +5101.25 q^{11} +1728.00 q^{12} -5814.29 q^{13} +10137.0 q^{14} +9206.64 q^{15} +4096.00 q^{16} +6404.73 q^{17} +5832.00 q^{18} -38566.6 q^{19} +21823.2 q^{20} +34212.2 q^{21} +40810.0 q^{22} -12167.0 q^{23} +13824.0 q^{24} +38147.0 q^{25} -46514.3 q^{26} +19683.0 q^{27} +81095.7 q^{28} -208061. q^{29} +73653.1 q^{30} +3653.78 q^{31} +32768.0 q^{32} +137734. q^{33} +51237.9 q^{34} +432071. q^{35} +46656.0 q^{36} +23930.9 q^{37} -308533. q^{38} -156986. q^{39} +174585. q^{40} -1151.50 q^{41} +273698. q^{42} -489302. q^{43} +326480. q^{44} +248579. q^{45} -97336.0 q^{46} +988907. q^{47} +110592. q^{48} +782050. q^{49} +305176. q^{50} +172928. q^{51} -372115. q^{52} -777027. q^{53} +157464. q^{54} +1.73946e6 q^{55} +648766. q^{56} -1.04130e6 q^{57} -1.66449e6 q^{58} +1.61187e6 q^{59} +589225. q^{60} -1.04092e6 q^{61} +29230.2 q^{62} +923731. q^{63} +262144. q^{64} -1.98260e6 q^{65} +1.10187e6 q^{66} -2.06066e6 q^{67} +409903. q^{68} -328509. q^{69} +3.45657e6 q^{70} +3.58392e6 q^{71} +373248. q^{72} -975850. q^{73} +191447. q^{74} +1.02997e6 q^{75} -2.46826e6 q^{76} +6.46390e6 q^{77} -1.25589e6 q^{78} +6.00359e6 q^{79} +1.39668e6 q^{80} +531441. q^{81} -9212.01 q^{82} -5.40252e6 q^{83} +2.18958e6 q^{84} +2.18393e6 q^{85} -3.91441e6 q^{86} -5.61764e6 q^{87} +2.61184e6 q^{88} -4.13911e6 q^{89} +1.98863e6 q^{90} -7.36740e6 q^{91} -778688. q^{92} +98652.1 q^{93} +7.91125e6 q^{94} -1.31507e7 q^{95} +884736. q^{96} +1.18844e7 q^{97} +6.25640e6 q^{98} +3.71881e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) $ 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9	\( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9} + 2160 q^{10} + 4120 q^{11} + 6912 q^{12} + 8036 q^{13} + 16176 q^{14} + 7290 q^{15} + 16384 q^{16} + 37182 q^{17} + 23328 q^{18} + 5702 q^{19} + 17280 q^{20} + 54594 q^{21} + 32960 q^{22} - 48668 q^{23} + 55296 q^{24} + 121480 q^{25} + 64288 q^{26} + 78732 q^{27} + 129408 q^{28} + 217716 q^{29} + 58320 q^{30} + 222852 q^{31} + 131072 q^{32} + 111240 q^{33} + 297456 q^{34} + 68440 q^{35} + 186624 q^{36} + 486428 q^{37} + 45616 q^{38} + 216972 q^{39} + 138240 q^{40} + 338336 q^{41} + 436752 q^{42} + 730974 q^{43} + 263680 q^{44} + 196830 q^{45} - 389344 q^{46} + 338248 q^{47} + 442368 q^{48} - 310552 q^{49} + 971840 q^{50} + 1003914 q^{51} + 514304 q^{52} - 375502 q^{53} + 629856 q^{54} + 424840 q^{55} + 1035264 q^{56} + 153954 q^{57} + 1741728 q^{58} + 71392 q^{59} + 466560 q^{60} + 2101164 q^{61} + 1782816 q^{62} + 1474038 q^{63} + 1048576 q^{64} + 1578780 q^{65} + 889920 q^{66} + 4337162 q^{67} + 2379648 q^{68} - 1314036 q^{69} + 547520 q^{70} + 2288016 q^{71} + 1492992 q^{72} - 1107328 q^{73} + 3891424 q^{74} + 3279960 q^{75} + 364928 q^{76} + 5826200 q^{77} + 1735776 q^{78} + 60610 q^{79} + 1105920 q^{80} + 2125764 q^{81} + 2706688 q^{82} + 1485464 q^{83} + 3494016 q^{84} - 8843820 q^{85} + 5847792 q^{86} + 5878332 q^{87} + 2109440 q^{88} + 1485090 q^{89} + 1574640 q^{90} - 2898412 q^{91} - 3114752 q^{92} + 6017004 q^{93} + 2705984 q^{94} + 8545200 q^{95} + 3538944 q^{96} + 1935444 q^{97} - 2484416 q^{98} + 3003480 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9 + 2160 * q^10 + 4120 * q^11 + 6912 * q^12 + 8036 * q^13 + 16176 * q^14 + 7290 * q^15 + 16384 * q^16 + 37182 * q^17 + 23328 * q^18 + 5702 * q^19 + 17280 * q^20 + 54594 * q^21 + 32960 * q^22 - 48668 * q^23 + 55296 * q^24 + 121480 * q^25 + 64288 * q^26 + 78732 * q^27 + 129408 * q^28 + 217716 * q^29 + 58320 * q^30 + 222852 * q^31 + 131072 * q^32 + 111240 * q^33 + 297456 * q^34 + 68440 * q^35 + 186624 * q^36 + 486428 * q^37 + 45616 * q^38 + 216972 * q^39 + 138240 * q^40 + 338336 * q^41 + 436752 * q^42 + 730974 * q^43 + 263680 * q^44 + 196830 * q^45 - 389344 * q^46 + 338248 * q^47 + 442368 * q^48 - 310552 * q^49 + 971840 * q^50 + 1003914 * q^51 + 514304 * q^52 - 375502 * q^53 + 629856 * q^54 + 424840 * q^55 + 1035264 * q^56 + 153954 * q^57 + 1741728 * q^58 + 71392 * q^59 + 466560 * q^60 + 2101164 * q^61 + 1782816 * q^62 + 1474038 * q^63 + 1048576 * q^64 + 1578780 * q^65 + 889920 * q^66 + 4337162 * q^67 + 2379648 * q^68 - 1314036 * q^69 + 547520 * q^70 + 2288016 * q^71 + 1492992 * q^72 - 1107328 * q^73 + 3891424 * q^74 + 3279960 * q^75 + 364928 * q^76 + 5826200 * q^77 + 1735776 * q^78 + 60610 * q^79 + 1105920 * q^80 + 2125764 * q^81 + 2706688 * q^82 + 1485464 * q^83 + 3494016 * q^84 - 8843820 * q^85 + 5847792 * q^86 + 5878332 * q^87 + 2109440 * q^88 + 1485090 * q^89 + 1574640 * q^90 - 2898412 * q^91 - 3114752 * q^92 + 6017004 * q^93 + 2705984 * q^94 + 8545200 * q^95 + 3538944 * q^96 + 1935444 * q^97 - 2484416 * q^98 + 3003480 * 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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	64.0000	0.500000
$5$	340.987	1.21995	0.609976	−	0.792420i	$-0.291179\pi$
$5$	340.987	1.21995	0.609976	+	0.792420i	$0.291179\pi$
$6$	216.000	0.408248
$7$	1267.12	1.39629	0.698143	−	0.715958i	$-0.254010\pi$
$7$	1267.12	1.39629	0.698143	+	0.715958i	$0.254010\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	2727.89	0.862636
$11$	5101.25	1.15559	0.577793	−	0.816183i	$-0.303914\pi$
$11$	5101.25	1.15559	0.577793	+	0.816183i	$0.303914\pi$
$12$	1728.00	0.288675
$13$	−5814.29	−0.733998	−0.366999	−	0.930221i	$-0.619615\pi$
$13$	−5814.29	−0.733998	−0.366999	+	0.930221i	$0.619615\pi$
$14$	10137.0	0.987324
$15$	9206.64	0.704339
$16$	4096.00	0.250000
$17$	6404.73	0.316177	0.158088	−	0.987425i	$-0.449467\pi$
$17$	6404.73	0.316177	0.158088	+	0.987425i	$0.449467\pi$
$18$	5832.00	0.235702
$19$	−38566.6	−1.28995	−0.644977	−	0.764202i	$-0.723133\pi$
$19$	−38566.6	−1.28995	−0.644977	+	0.764202i	$0.723133\pi$
$20$	21823.2	0.609976
$21$	34212.2	0.806147
$22$	40810.0	0.817122
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	13824.0	0.204124
$25$	38147.0	0.488281
$26$	−46514.3	−0.519015
$27$	19683.0	0.192450
$28$	81095.7	0.698143
$29$	−208061.	−1.58415	−0.792077	−	0.610422i	$-0.791000\pi$
$29$	−208061.	−1.58415	−0.792077	+	0.610422i	$0.791000\pi$
$30$	73653.1	0.498043
$31$	3653.78	0.0220281	0.0110140	−	0.999939i	$-0.496494\pi$
$31$	3653.78	0.0220281	0.0110140	+	0.999939i	$0.496494\pi$
$32$	32768.0	0.176777
$33$	137734.	0.667178
$34$	51237.9	0.223571
$35$	432071.	1.70340
$36$	46656.0	0.166667
$37$	23930.9	0.0776699	0.0388350	−	0.999246i	$-0.487635\pi$
$37$	23930.9	0.0776699	0.0388350	+	0.999246i	$0.487635\pi$
$38$	−308533.	−0.912135
$39$	−156986.	−0.423774
$40$	174585.	0.431318
$41$	−1151.50	−0.00260928	−0.00130464	−	0.999999i	$-0.500415\pi$
$41$	−1151.50	−0.00260928	−0.00130464	+	0.999999i	$0.500415\pi$
$42$	273698.	0.570032
$43$	−489302.	−0.938506	−0.469253	−	0.883064i	$-0.655477\pi$
$43$	−489302.	−0.938506	−0.469253	+	0.883064i	$0.655477\pi$
$44$	326480.	0.577793
$45$	248579.	0.406650
$46$	−97336.0	−0.147442
$47$	988907.	1.38935	0.694677	−	0.719322i	$-0.255547\pi$
$47$	988907.	1.38935	0.694677	+	0.719322i	$0.255547\pi$
$48$	110592.	0.144338
$49$	782050.	0.949617
$50$	305176.	0.345267
$51$	172928.	0.182545
$52$	−372115.	−0.366999
$53$	−777027.	−0.716920	−0.358460	−	0.933545i	$-0.616698\pi$
$53$	−777027.	−0.716920	−0.358460	+	0.933545i	$0.616698\pi$
$54$	157464.	0.136083
$55$	1.73946e6	1.40976
$56$	648766.	0.493662
$57$	−1.04130e6	−0.744755
$58$	−1.66449e6	−1.12017
$59$	1.61187e6	1.02176	0.510879	−	0.859652i	$-0.329320\pi$
$59$	1.61187e6	1.02176	0.510879	+	0.859652i	$0.329320\pi$
$60$	589225.	0.352170
$61$	−1.04092e6	−0.587171	−0.293586	−	0.955933i	$-0.594849\pi$
$61$	−1.04092e6	−0.587171	−0.293586	+	0.955933i	$0.594849\pi$
$62$	29230.2	0.0155762
$63$	923731.	0.465429
$64$	262144.	0.125000
$65$	−1.98260e6	−0.895442
$66$	1.10187e6	0.471766
$67$	−2.06066e6	−0.837036	−0.418518	−	0.908209i	$-0.637450\pi$
$67$	−2.06066e6	−0.837036	−0.418518	+	0.908209i	$0.637450\pi$
$68$	409903.	0.158088
$69$	−328509.	−0.120386
$70$	3.45657e6	1.20449
$71$	3.58392e6	1.18838	0.594189	−	0.804325i	$-0.297473\pi$
$71$	3.58392e6	1.18838	0.594189	+	0.804325i	$0.297473\pi$
$72$	373248.	0.117851
$73$	−975850.	−0.293598	−0.146799	−	0.989166i	$-0.546897\pi$
$73$	−975850.	−0.293598	−0.146799	+	0.989166i	$0.546897\pi$
$74$	191447.	0.0549209
$75$	1.02997e6	0.281909
$76$	−2.46826e6	−0.644977
$77$	6.46390e6	1.61353
$78$	−1.25589e6	−0.299653
$79$	6.00359e6	1.36999	0.684993	−	0.728550i	$-0.259805\pi$
$79$	6.00359e6	1.36999	0.684993	+	0.728550i	$0.259805\pi$
$80$	1.39668e6	0.304988
$81$	531441.	0.111111
$82$	−9212.01	−0.00184504
$83$	−5.40252e6	−1.03711	−0.518553	−	0.855046i	$-0.673529\pi$
$83$	−5.40252e6	−1.03711	−0.518553	+	0.855046i	$0.673529\pi$
$84$	2.18958e6	0.403073
$85$	2.18393e6	0.385720
$86$	−3.91441e6	−0.663624
$87$	−5.61764e6	−0.914611
$88$	2.61184e6	0.408561
$89$	−4.13911e6	−0.622361	−0.311180	−	0.950351i	$-0.600724\pi$
$89$	−4.13911e6	−0.622361	−0.311180	+	0.950351i	$0.600724\pi$
$90$	1.98863e6	0.287545
$91$	−7.36740e6	−1.02487
$92$	−778688.	−0.104257
$93$	98652.1	0.0127179
$94$	7.91125e6	0.982422
$95$	−1.31507e7	−1.57368
$96$	884736.	0.102062
$97$	1.18844e7	1.32214	0.661070	−	0.750324i	$-0.270103\pi$
$97$	1.18844e7	1.32214	0.661070	+	0.750324i	$0.270103\pi$
$98$	6.25640e6	0.671481
$99$	3.71881e6	0.385195
$100$	2.44141e6	0.244141
$101$	9.54327e6	0.921663	0.460832	−	0.887488i	$-0.347551\pi$
$101$	9.54327e6	0.921663	0.460832	+	0.887488i	$0.347551\pi$
$102$	1.38342e6	0.129079
$103$	−3.12347e6	−0.281648	−0.140824	−	0.990035i	$-0.544975\pi$
$103$	−3.12347e6	−0.281648	−0.140824	+	0.990035i	$0.544975\pi$
$104$	−2.97692e6	−0.259508
$105$	1.16659e7	0.983460
$106$	−6.21621e6	−0.506939
$107$	1.02468e7	0.808620	0.404310	−	0.914622i	$-0.367512\pi$
$107$	1.02468e7	0.808620	0.404310	+	0.914622i	$0.367512\pi$
$108$	1.25971e6	0.0962250
$109$	−1.26733e7	−0.937337	−0.468668	−	0.883374i	$-0.655266\pi$
$109$	−1.26733e7	−0.937337	−0.468668	+	0.883374i	$0.655266\pi$
$110$	1.39157e7	0.996850
$111$	646134.	0.0448427
$112$	5.19012e6	0.349072
$113$	543321.	0.0354227	0.0177114	−	0.999843i	$-0.494362\pi$
$113$	543321.	0.0354227	0.0177114	+	0.999843i	$0.494362\pi$
$114$	−8.33039e6	−0.526621
$115$	−4.14879e6	−0.254377
$116$	−1.33159e7	−0.792077
$117$	−4.23862e6	−0.244666
$118$	1.28950e7	0.722493
$119$	8.11557e6	0.441473
$120$	4.71380e6	0.249022
$121$	6.53557e6	0.335378
$122$	−8.32740e6	−0.415193
$123$	−31090.5	−0.00150647
$124$	233842.	0.0110140
$125$	−1.36320e7	−0.624272
$126$	7.38984e6	0.329108
$127$	−6.08628e6	−0.263657	−0.131828	−	0.991273i	$-0.542085\pi$
$127$	−6.08628e6	−0.263657	−0.131828	+	0.991273i	$0.542085\pi$
$128$	2.09715e6	0.0883883
$129$	−1.32111e7	−0.541847
$130$	−1.58608e7	−0.633173
$131$	−2.60740e7	−1.01335	−0.506674	−	0.862138i	$-0.669125\pi$
$131$	−2.60740e7	−1.01335	−0.506674	+	0.862138i	$0.669125\pi$
$132$	8.81496e6	0.333589
$133$	−4.88686e7	−1.80115
$134$	−1.64853e7	−0.591874
$135$	6.71164e6	0.234780
$136$	3.27922e6	0.111785
$137$	4.55380e7	1.51304	0.756522	−	0.653968i	$-0.226897\pi$
$137$	4.55380e7	1.51304	0.756522	+	0.653968i	$0.226897\pi$
$138$	−2.62807e6	−0.0851257
$139$	−4.36079e7	−1.37725	−0.688626	−	0.725116i	$-0.741786\pi$
$139$	−4.36079e7	−1.37725	−0.688626	+	0.725116i	$0.741786\pi$
$140$	2.76526e7	0.851701
$141$	2.67005e7	0.802144
$142$	2.86714e7	0.840310
$143$	−2.96601e7	−0.848198
$144$	2.98598e6	0.0833333
$145$	−7.09460e7	−1.93259
$146$	−7.80680e6	−0.207605
$147$	2.11154e7	0.548262
$148$	1.53158e6	0.0388350
$149$	−7.44370e7	−1.84347	−0.921737	−	0.387816i	$-0.873230\pi$
$149$	−7.44370e7	−1.84347	−0.921737	+	0.387816i	$0.873230\pi$
$150$	8.23975e6	0.199340
$151$	2.77945e7	0.656962	0.328481	−	0.944511i	$-0.393463\pi$
$151$	2.77945e7	0.656962	0.328481	+	0.944511i	$0.393463\pi$
$152$	−1.97461e7	−0.456067
$153$	4.66905e6	0.105392
$154$	5.17112e7	1.14094
$155$	1.24589e6	0.0268732
$156$	−1.00471e7	−0.211887
$157$	3.33077e7	0.686904	0.343452	−	0.939170i	$-0.388404\pi$
$157$	3.33077e7	0.686904	0.343452	+	0.939170i	$0.388404\pi$
$158$	4.80287e7	0.968726
$159$	−2.09797e7	−0.413914
$160$	1.11735e7	0.215659
$161$	−1.54171e7	−0.291146
$162$	4.25153e6	0.0785674
$163$	−1.45963e7	−0.263989	−0.131994	−	0.991250i	$-0.542138\pi$
$163$	−1.45963e7	−0.263989	−0.131994	+	0.991250i	$0.542138\pi$
$164$	−73696.1	−0.00130464
$165$	4.69654e7	0.813924
$166$	−4.32201e7	−0.733344
$167$	−7.61604e7	−1.26538	−0.632691	−	0.774404i	$-0.718050\pi$
$167$	−7.61604e7	−1.26538	−0.632691	+	0.774404i	$0.718050\pi$
$168$	1.75167e7	0.285016
$169$	−2.89426e7	−0.461247
$170$	1.74714e7	0.272745
$171$	−2.81151e7	−0.429985
$172$	−3.13153e7	−0.469253
$173$	−1.17815e8	−1.72997	−0.864983	−	0.501801i	$-0.832671\pi$
$173$	−1.17815e8	−1.72997	−0.864983	+	0.501801i	$0.832671\pi$
$174$	−4.49411e7	−0.646728
$175$	4.83368e7	0.681781
$176$	2.08947e7	0.288896
$177$	4.35205e7	0.589913
$178$	−3.31129e7	−0.440075
$179$	2.34279e7	0.305314	0.152657	−	0.988279i	$-0.451217\pi$
$179$	2.34279e7	0.305314	0.152657	+	0.988279i	$0.451217\pi$
$180$	1.59091e7	0.203325
$181$	−5.66147e7	−0.709667	−0.354833	−	0.934930i	$-0.615462\pi$
$181$	−5.66147e7	−0.709667	−0.354833	+	0.934930i	$0.615462\pi$
$182$	−5.89392e7	−0.724694
$183$	−2.81050e7	−0.339004
$184$	−6.22950e6	−0.0737210
$185$	8.16012e6	0.0947535
$186$	789217.	0.00899292
$187$	3.26721e7	0.365369
$188$	6.32900e7	0.694677
$189$	2.49407e7	0.268716
$190$	−1.05206e8	−1.11276
$191$	−1.26851e8	−1.31727	−0.658636	−	0.752461i	$-0.728866\pi$
$191$	−1.26851e8	−1.31727	−0.658636	+	0.752461i	$0.728866\pi$
$192$	7.07789e6	0.0721688
$193$	1.36553e8	1.36726	0.683630	−	0.729828i	$-0.260400\pi$
$193$	1.36553e8	1.36726	0.683630	+	0.729828i	$0.260400\pi$
$194$	9.50755e7	0.934894
$195$	−5.35301e7	−0.516984
$196$	5.00512e7	0.474808
$197$	−1.16049e8	−1.08146	−0.540729	−	0.841197i	$-0.681852\pi$
$197$	−1.16049e8	−1.08146	−0.540729	+	0.841197i	$0.681852\pi$
$198$	2.97505e7	0.272374
$199$	1.41367e7	0.127163	0.0635816	−	0.997977i	$-0.479748\pi$
$199$	1.41367e7	0.127163	0.0635816	+	0.997977i	$0.479748\pi$
$200$	1.95312e7	0.172633
$201$	−5.56377e7	−0.483263
$202$	7.63462e7	0.651714
$203$	−2.63638e8	−2.21193
$204$	1.10674e7	0.0912723
$205$	−392647.	−0.00318320
$206$	−2.49877e7	−0.199155
$207$	−8.86974e6	−0.0695048
$208$	−2.38153e7	−0.183500
$209$	−1.96738e8	−1.49065
$210$	9.33274e7	0.695411
$211$	1.88579e8	1.38199	0.690993	−	0.722861i	$-0.257173\pi$
$211$	1.88579e8	1.38199	0.690993	+	0.722861i	$0.257173\pi$
$212$	−4.97297e7	−0.358460
$213$	9.67660e7	0.686110
$214$	8.19743e7	0.571781
$215$	−1.66845e8	−1.14493
$216$	1.00777e7	0.0680414
$217$	4.62978e6	0.0307575
$218$	−1.01386e8	−0.662797
$219$	−2.63479e7	−0.169509
$220$	1.11325e8	0.704879
$221$	−3.72390e7	−0.232073
$222$	5.16907e6	0.0317086
$223$	3.17153e8	1.91515	0.957573	−	0.288192i	$-0.0930542\pi$
$223$	3.17153e8	1.91515	0.957573	+	0.288192i	$0.0930542\pi$
$224$	4.15210e7	0.246831
$225$	2.78091e7	0.162760
$226$	4.34657e6	0.0250476
$227$	9.71697e6	0.0551366	0.0275683	−	0.999620i	$-0.491224\pi$
$227$	9.71697e6	0.0551366	0.0275683	+	0.999620i	$0.491224\pi$
$228$	−6.66431e7	−0.372378
$229$	2.54644e8	1.40123	0.700614	−	0.713540i	$-0.252909\pi$
$229$	2.54644e8	1.40123	0.700614	+	0.713540i	$0.252909\pi$
$230$	−3.31903e7	−0.179872
$231$	1.74525e8	0.931571
$232$	−1.06527e8	−0.560083
$233$	2.64155e7	0.136808	0.0684042	−	0.997658i	$-0.478209\pi$
$233$	2.64155e7	0.136808	0.0684042	+	0.997658i	$0.478209\pi$
$234$	−3.39089e7	−0.173005
$235$	3.37204e8	1.69494
$236$	1.03160e8	0.510879
$237$	1.62097e8	0.790962
$238$	6.49245e7	0.312169
$239$	−1.27674e7	−0.0604936	−0.0302468	−	0.999542i	$-0.509629\pi$
$239$	−1.27674e7	−0.0604936	−0.0302468	+	0.999542i	$0.509629\pi$
$240$	3.77104e7	0.176085
$241$	1.47494e8	0.678757	0.339379	−	0.940650i	$-0.389783\pi$
$241$	1.47494e8	0.678757	0.339379	+	0.940650i	$0.389783\pi$
$242$	5.22845e7	0.237148
$243$	1.43489e7	0.0641500
$244$	−6.66192e7	−0.293586
$245$	2.66669e8	1.15849
$246$	−248724.	−0.00106524
$247$	2.24238e8	0.946823
$248$	1.87074e6	0.00778810
$249$	−1.45868e8	−0.598773
$250$	−1.09056e8	−0.441427
$251$	2.67374e8	1.06724	0.533618	−	0.845726i	$-0.320832\pi$
$251$	2.67374e8	1.06724	0.533618	+	0.845726i	$0.320832\pi$
$252$	5.91188e7	0.232714
$253$	−6.20669e7	−0.240956
$254$	−4.86902e7	−0.186433
$255$	5.89661e7	0.222696
$256$	1.67772e7	0.0625000
$257$	−1.37033e8	−0.503568	−0.251784	−	0.967784i	$-0.581017\pi$
$257$	−1.37033e8	−0.503568	−0.251784	+	0.967784i	$0.581017\pi$
$258$	−1.05689e8	−0.383143
$259$	3.03233e7	0.108449
$260$	−1.26886e8	−0.447721
$261$	−1.51676e8	−0.528051
$262$	−2.08592e8	−0.716545
$263$	1.60586e8	0.544329	0.272165	−	0.962251i	$-0.412260\pi$
$263$	1.60586e8	0.544329	0.272165	+	0.962251i	$0.412260\pi$
$264$	7.05197e7	0.235883
$265$	−2.64956e8	−0.874607
$266$	−3.90948e8	−1.27360
$267$	−1.11756e8	−0.359320
$268$	−1.31882e8	−0.418518
$269$	4.67109e8	1.46314	0.731568	−	0.681768i	$-0.238789\pi$
$269$	4.67109e8	1.46314	0.731568	+	0.681768i	$0.238789\pi$
$270$	5.36931e7	0.166014
$271$	5.51685e8	1.68383	0.841916	−	0.539609i	$-0.181428\pi$
$271$	5.51685e8	1.68383	0.841916	+	0.539609i	$0.181428\pi$
$272$	2.62338e7	0.0790442
$273$	−1.98920e8	−0.591710
$274$	3.64304e8	1.06988
$275$	1.94597e8	0.564251
$276$	−2.10246e7	−0.0601929
$277$	4.78153e8	1.35172	0.675862	−	0.737028i	$-0.263772\pi$
$277$	4.78153e8	1.35172	0.675862	+	0.737028i	$0.263772\pi$
$278$	−3.48863e8	−0.973865
$279$	2.66361e6	0.00734269
$280$	2.21220e8	0.602244
$281$	7.37447e8	1.98271	0.991354	−	0.131218i	$-0.0418887\pi$
$281$	7.37447e8	1.98271	0.991354	+	0.131218i	$0.0418887\pi$
$282$	2.13604e8	0.567201
$283$	−6.82950e8	−1.79117	−0.895584	−	0.444893i	$-0.853242\pi$
$283$	−6.82950e8	−1.79117	−0.895584	+	0.444893i	$0.853242\pi$
$284$	2.29371e8	0.594189
$285$	−3.55069e8	−0.908565
$286$	−2.37281e8	−0.599766
$287$	−1.45909e6	−0.00364331
$288$	2.38879e7	0.0589256
$289$	−3.69318e8	−0.900032
$290$	−5.67568e8	−1.36655
$291$	3.20880e8	0.763338
$292$	−6.24544e7	−0.146799
$293$	−4.73242e7	−0.109912	−0.0549562	−	0.998489i	$-0.517502\pi$
$293$	−4.73242e7	−0.109912	−0.0549562	+	0.998489i	$0.517502\pi$
$294$	1.68923e8	0.387680
$295$	5.49627e8	1.24650
$296$	1.22526e7	0.0274605
$297$	1.00408e8	0.222393
$298$	−5.95496e8	−1.30353
$299$	7.07425e7	0.153049
$300$	6.59180e7	0.140955
$301$	−6.20004e8	−1.31042
$302$	2.22356e8	0.464542
$303$	2.57668e8	0.532123
$304$	−1.57969e8	−0.322488
$305$	−3.54942e8	−0.716321
$306$	3.73524e7	0.0745236
$307$	3.47596e8	0.685630	0.342815	−	0.939403i	$-0.388620\pi$
$307$	3.47596e8	0.685630	0.342815	+	0.939403i	$0.388620\pi$
$308$	4.13689e8	0.806764
$309$	−8.43336e7	−0.162610
$310$	9.96713e6	0.0190022
$311$	−8.46459e8	−1.59568	−0.797838	−	0.602873i	$-0.794023\pi$
$311$	−8.46459e8	−1.59568	−0.797838	+	0.602873i	$0.794023\pi$
$312$	−8.03767e7	−0.149827
$313$	5.31447e8	0.979613	0.489806	−	0.871831i	$-0.337067\pi$
$313$	5.31447e8	0.979613	0.489806	+	0.871831i	$0.337067\pi$
$314$	2.66462e8	0.485715
$315$	3.14980e8	0.567801
$316$	3.84230e8	0.684993
$317$	−7.52971e8	−1.32761	−0.663806	−	0.747905i	$-0.731060\pi$
$317$	−7.52971e8	−1.32761	−0.663806	+	0.747905i	$0.731060\pi$
$318$	−1.67838e8	−0.292681
$319$	−1.06137e9	−1.83062
$320$	8.93876e7	0.152494
$321$	2.76663e8	0.466857
$322$	−1.23336e8	−0.205871
$323$	−2.47009e8	−0.407853
$324$	3.40122e7	0.0555556
$325$	−2.21798e8	−0.358398
$326$	−1.16770e8	−0.186668
$327$	−3.42178e8	−0.541172
$328$	−589569.	−0.000922521 0
$329$	1.25306e9	1.93994
$330$	3.75723e8	0.575531
$331$	−1.86489e8	−0.282654	−0.141327	−	0.989963i	$-0.545137\pi$
$331$	−1.86489e8	−0.282654	−0.141327	+	0.989963i	$0.545137\pi$
$332$	−3.45761e8	−0.518553
$333$	1.74456e7	0.0258900
$334$	−6.09284e8	−0.894760
$335$	−7.02657e8	−1.02114
$336$	1.40133e8	0.201537
$337$	−7.20597e8	−1.02562	−0.512811	−	0.858501i	$-0.671396\pi$
$337$	−7.20597e8	−1.02562	−0.512811	+	0.858501i	$0.671396\pi$
$338$	−2.31540e8	−0.326151
$339$	1.46697e7	0.0204513
$340$	1.39771e8	0.192860
$341$	1.86388e7	0.0254553
$342$	−2.24921e8	−0.304045
$343$	−5.25761e7	−0.0703491
$344$	−2.50522e8	−0.331812
$345$	−1.12017e8	−0.146865
$346$	−9.42517e8	−1.22327
$347$	−8.14326e8	−1.04627	−0.523137	−	0.852249i	$-0.675238\pi$
$347$	−8.14326e8	−1.04627	−0.523137	+	0.852249i	$0.675238\pi$
$348$	−3.59529e8	−0.457306
$349$	−4.25035e8	−0.535224	−0.267612	−	0.963527i	$-0.586234\pi$
$349$	−4.25035e8	−0.535224	−0.267612	+	0.963527i	$0.586234\pi$
$350$	3.86694e8	0.482092
$351$	−1.14443e8	−0.141258
$352$	1.67158e8	0.204281
$353$	−2.10866e8	−0.255149	−0.127575	−	0.991829i	$-0.540719\pi$
$353$	−2.10866e8	−0.255149	−0.127575	+	0.991829i	$0.540719\pi$
$354$	3.48164e8	0.417131
$355$	1.22207e9	1.44976
$356$	−2.64903e8	−0.311180
$357$	2.19120e8	0.254885
$358$	1.87423e8	0.215890
$359$	−9.62505e8	−1.09792	−0.548962	−	0.835847i	$-0.684977\pi$
$359$	−9.62505e8	−1.09792	−0.548962	+	0.835847i	$0.684977\pi$
$360$	1.27273e8	0.143773
$361$	5.93513e8	0.663980
$362$	−4.52918e8	−0.501810
$363$	1.76460e8	0.193631
$364$	−4.71514e8	−0.512436
$365$	−3.32752e8	−0.358175
$366$	−2.24840e8	−0.239712
$367$	−5.91557e7	−0.0624691	−0.0312345	−	0.999512i	$-0.509944\pi$
$367$	−5.91557e7	−0.0624691	−0.0312345	+	0.999512i	$0.509944\pi$
$368$	−4.98360e7	−0.0521286
$369$	−839445.	−0.000869761 0
$370$	6.52809e7	0.0670009
$371$	−9.84586e8	−1.00103
$372$	6.31373e6	0.00635896
$373$	−7.57694e8	−0.755984	−0.377992	−	0.925809i	$-0.623385\pi$
$373$	−7.57694e8	−0.755984	−0.377992	+	0.925809i	$0.623385\pi$
$374$	2.61377e8	0.258355
$375$	−3.68063e8	−0.360424
$376$	5.06320e8	0.491211
$377$	1.20973e9	1.16277
$378$	1.99526e8	0.190011
$379$	−2.47156e8	−0.233203	−0.116601	−	0.993179i	$-0.537200\pi$
$379$	−2.47156e8	−0.233203	−0.116601	+	0.993179i	$0.537200\pi$
$380$	−8.41645e8	−0.786840
$381$	−1.64330e8	−0.152222
$382$	−1.01480e9	−0.931452
$383$	−1.95064e9	−1.77411	−0.887056	−	0.461662i	$-0.847253\pi$
$383$	−1.95064e9	−1.77411	−0.887056	+	0.461662i	$0.847253\pi$
$384$	5.66231e7	0.0510310
$385$	2.20410e9	1.96843
$386$	1.09243e9	0.966799
$387$	−3.56701e8	−0.312835
$388$	7.60604e8	0.661070
$389$	1.70660e9	1.46997	0.734987	−	0.678082i	$-0.237188\pi$
$389$	1.70660e9	1.46997	0.734987	+	0.678082i	$0.237188\pi$
$390$	−4.28241e8	−0.365563
$391$	−7.79264e7	−0.0659274
$392$	4.00410e8	0.335740
$393$	−7.03998e8	−0.585056
$394$	−9.28392e8	−0.764706
$395$	2.04714e9	1.67132
$396$	2.38004e8	0.192598
$397$	9.51729e8	0.763390	0.381695	−	0.924288i	$-0.375340\pi$
$397$	9.51729e8	0.763390	0.381695	+	0.924288i	$0.375340\pi$
$398$	1.13093e8	0.0899179
$399$	−1.31945e9	−1.03989
$400$	1.56250e8	0.122070
$401$	1.03883e9	0.804522	0.402261	−	0.915525i	$-0.368225\pi$
$401$	1.03883e9	0.804522	0.402261	+	0.915525i	$0.368225\pi$
$402$	−4.45102e8	−0.341718
$403$	−2.12441e7	−0.0161686
$404$	6.10769e8	0.460832
$405$	1.81214e8	0.135550
$406$	−2.10910e9	−1.56407
$407$	1.22077e8	0.0897542
$408$	8.85390e7	0.0645393
$409$	2.67559e9	1.93370	0.966849	−	0.255348i	$-0.0821899\pi$
$409$	2.67559e9	1.93370	0.966849	+	0.255348i	$0.0821899\pi$
$410$	−3.14117e6	−0.00225086
$411$	1.22953e9	0.873557
$412$	−1.99902e8	−0.140824
$413$	2.04243e9	1.42667
$414$	−7.09579e7	−0.0491473
$415$	−1.84219e9	−1.26522
$416$	−1.90523e8	−0.129754
$417$	−1.17741e9	−0.795157
$418$	−1.57390e9	−1.05405
$419$	3.41769e8	0.226978	0.113489	−	0.993539i	$-0.463797\pi$
$419$	3.41769e8	0.226978	0.113489	+	0.993539i	$0.463797\pi$
$420$	7.46619e8	0.491730
$421$	1.83184e9	1.19646	0.598232	−	0.801323i	$-0.295870\pi$
$421$	1.83184e9	1.19646	0.598232	+	0.801323i	$0.295870\pi$
$422$	1.50863e9	0.977212
$423$	7.20913e8	0.463118
$424$	−3.97838e8	−0.253469
$425$	2.44321e8	0.154383
$426$	7.74128e8	0.485153
$427$	−1.31898e9	−0.819860
$428$	6.55794e8	0.404310
$429$	−8.00824e8	−0.489707
$430$	−1.33476e9	−0.809589
$431$	−1.20261e9	−0.723529	−0.361764	−	0.932270i	$-0.617826\pi$
$431$	−1.20261e9	−0.723529	−0.361764	+	0.932270i	$0.617826\pi$
$432$	8.06216e7	0.0481125
$433$	1.83198e9	1.08446	0.542228	−	0.840231i	$-0.317581\pi$
$433$	1.83198e9	1.08446	0.542228	+	0.840231i	$0.317581\pi$
$434$	3.70382e7	0.0217488
$435$	−1.91554e9	−1.11578
$436$	−8.11089e8	−0.468668
$437$	4.69240e8	0.268974
$438$	−2.10784e8	−0.119861
$439$	−1.50903e9	−0.851277	−0.425639	−	0.904893i	$-0.639951\pi$
$439$	−1.50903e9	−0.851277	−0.425639	+	0.904893i	$0.639951\pi$
$440$	8.90603e8	0.498425
$441$	5.70115e8	0.316539
$442$	−2.97912e8	−0.164100
$443$	2.98792e9	1.63289	0.816444	−	0.577425i	$-0.195942\pi$
$443$	2.98792e9	1.63289	0.816444	+	0.577425i	$0.195942\pi$
$444$	4.13526e7	0.0224214
$445$	−1.41138e9	−0.759250
$446$	2.53722e9	1.35421
$447$	−2.00980e9	−1.06433
$448$	3.32168e8	0.174536
$449$	1.91998e9	1.00100	0.500500	−	0.865737i	$-0.333150\pi$
$449$	1.91998e9	1.00100	0.500500	+	0.865737i	$0.333150\pi$
$450$	2.22473e8	0.115089
$451$	−5.87410e6	−0.00301525
$452$	3.47725e7	0.0177114
$453$	7.50452e8	0.379297
$454$	7.77358e7	0.0389875
$455$	−2.51219e9	−1.25029
$456$	−5.33145e8	−0.263311
$457$	−1.94014e9	−0.950883	−0.475441	−	0.879747i	$-0.657712\pi$
$457$	−1.94014e9	−0.950883	−0.475441	+	0.879747i	$0.657712\pi$
$458$	2.03715e9	0.990818
$459$	1.26064e8	0.0608482
$460$	−2.65522e8	−0.127189
$461$	−8.14499e8	−0.387202	−0.193601	−	0.981080i	$-0.562017\pi$
$461$	−8.14499e8	−0.387202	−0.193601	+	0.981080i	$0.562017\pi$
$462$	1.39620e9	0.658720
$463$	−6.61468e7	−0.0309724	−0.0154862	−	0.999880i	$-0.504930\pi$
$463$	−6.61468e7	−0.0309724	−0.0154862	+	0.999880i	$0.504930\pi$
$464$	−8.52217e8	−0.396038
$465$	3.36390e7	0.0155152
$466$	2.11324e8	0.0967382
$467$	−2.88027e9	−1.30865	−0.654327	−	0.756212i	$-0.727048\pi$
$467$	−2.88027e9	−1.30865	−0.654327	+	0.756212i	$0.727048\pi$
$468$	−2.71272e8	−0.122333
$469$	−2.61110e9	−1.16874
$470$	2.69763e9	1.19851
$471$	8.99308e8	0.396584
$472$	8.25278e8	0.361246
$473$	−2.49605e9	−1.08452
$474$	1.29677e9	0.559294
$475$	−1.47120e9	−0.629860
$476$	5.19396e8	0.220737
$477$	−5.66453e8	−0.238973
$478$	−1.02139e8	−0.0427754
$479$	−3.40043e9	−1.41371	−0.706854	−	0.707360i	$-0.749886\pi$
$479$	−3.40043e9	−1.41371	−0.706854	+	0.707360i	$0.749886\pi$
$480$	3.01683e8	0.124511
$481$	−1.39141e8	−0.0570096
$482$	1.17995e9	0.479954
$483$	−4.16260e8	−0.168093
$484$	4.18276e8	0.167689
$485$	4.05243e9	1.61295
$486$	1.14791e8	0.0453609
$487$	3.01213e9	1.18174	0.590871	−	0.806766i	$-0.298784\pi$
$487$	3.01213e9	1.18174	0.590871	+	0.806766i	$0.298784\pi$
$488$	−5.32954e8	−0.207596
$489$	−3.94100e8	−0.152414
$490$	2.13335e9	0.819174
$491$	−835191.	−0.000318420 0	−0.000159210	−	1.00000i	$-0.500051\pi$
$491$	−835191.	−0.000318420 0	−0.000159210		1.00000i	$0.500051\pi$
$492$	−1.98979e6	−0.000753235 0
$493$	−1.33257e9	−0.500872
$494$	1.79390e9	0.669505
$495$	1.26807e9	0.469919
$496$	1.49659e7	0.00550702
$497$	4.54126e9	1.65932
$498$	−1.16694e9	−0.423396
$499$	−4.35806e9	−1.57015	−0.785075	−	0.619401i	$-0.787375\pi$
$499$	−4.35806e9	−1.57015	−0.785075	+	0.619401i	$0.787375\pi$
$500$	−8.72447e8	−0.312136
$501$	−2.05633e9	−0.730569
$502$	2.13899e9	0.754650
$503$	−2.40286e8	−0.0841862	−0.0420931	−	0.999114i	$-0.513403\pi$
$503$	−2.40286e8	−0.0841862	−0.0420931	+	0.999114i	$0.513403\pi$
$504$	4.72950e8	0.164554
$505$	3.25413e9	1.12438
$506$	−4.96535e8	−0.170382
$507$	−7.81449e8	−0.266301
$508$	−3.89522e8	−0.131828
$509$	−3.36572e9	−1.13127	−0.565634	−	0.824656i	$-0.691369\pi$
$509$	−3.36572e9	−1.13127	−0.565634	+	0.824656i	$0.691369\pi$
$510$	4.71729e8	0.157470
$511$	−1.23652e9	−0.409947
$512$	1.34218e8	0.0441942
$513$	−7.59107e8	−0.248252
$514$	−1.09626e9	−0.356076
$515$	−1.06506e9	−0.343597
$516$	−8.45513e8	−0.270923
$517$	5.04466e9	1.60552
$518$	2.42587e8	0.0766854
$519$	−3.18099e9	−0.998797
$520$	−1.01509e9	−0.316587
$521$	6.69390e8	0.207371	0.103685	−	0.994610i	$-0.466937\pi$
$521$	6.69390e8	0.207371	0.103685	+	0.994610i	$0.466937\pi$
$522$	−1.21341e9	−0.373388
$523$	6.04759e9	1.84853	0.924265	−	0.381752i	$-0.124679\pi$
$523$	6.04759e9	1.84853	0.924265	+	0.381752i	$0.124679\pi$
$524$	−1.66874e9	−0.506674
$525$	1.30509e9	0.393626
$526$	1.28469e9	0.384899
$527$	2.34015e7	0.00696476
$528$	5.64157e8	0.166794
$529$	1.48036e8	0.0434783
$530$	−2.11965e9	−0.618441
$531$	1.17505e9	0.340586
$532$	−3.12759e9	−0.900573
$533$	6.69516e6	0.00191521
$534$	−8.94048e8	−0.254078
$535$	3.49402e9	0.986477
$536$	−1.05506e9	−0.295937
$537$	6.32553e8	0.176273
$538$	3.73687e9	1.03459
$539$	3.98943e9	1.09736
$540$	4.29545e8	0.117390
$541$	9.56170e8	0.259624	0.129812	−	0.991539i	$-0.458563\pi$
$541$	9.56170e8	0.259624	0.129812	+	0.991539i	$0.458563\pi$
$542$	4.41348e9	1.19065
$543$	−1.52860e9	−0.409726
$544$	2.09870e8	0.0558927
$545$	−4.32141e9	−1.14351
$546$	−1.59136e9	−0.418402
$547$	−7.50590e9	−1.96086	−0.980431	−	0.196864i	$-0.936924\pi$
$547$	−7.50590e9	−1.96086	−0.980431	+	0.196864i	$0.936924\pi$
$548$	2.91443e9	0.756522
$549$	−7.58834e8	−0.195724
$550$	1.55678e9	0.398986
$551$	8.02420e9	2.04348
$552$	−1.68197e8	−0.0425628
$553$	7.60727e9	1.91289
$554$	3.82523e9	0.955813
$555$	2.20323e8	0.0547060
$556$	−2.79091e9	−0.688626
$557$	2.30925e9	0.566209	0.283105	−	0.959089i	$-0.408636\pi$
$557$	2.30925e9	0.566209	0.283105	+	0.959089i	$0.408636\pi$
$558$	2.13088e7	0.00519207
$559$	2.84494e9	0.688862
$560$	1.76976e9	0.425851
$561$	8.82148e8	0.210946
$562$	5.89957e9	1.40199
$563$	2.20676e9	0.521166	0.260583	−	0.965451i	$-0.416085\pi$
$563$	2.20676e9	0.521166	0.260583	+	0.965451i	$0.416085\pi$
$564$	1.70883e9	0.401072
$565$	1.85265e8	0.0432140
$566$	−5.46360e9	−1.26655
$567$	6.73400e8	0.155143
$568$	1.83497e9	0.420155
$569$	−2.56775e9	−0.584332	−0.292166	−	0.956368i	$-0.594376\pi$
$569$	−2.56775e9	−0.584332	−0.292166	+	0.956368i	$0.594376\pi$
$570$	−2.84055e9	−0.642452
$571$	−3.30249e9	−0.742360	−0.371180	−	0.928561i	$-0.621047\pi$
$571$	−3.30249e9	−0.742360	−0.371180	+	0.928561i	$0.621047\pi$
$572$	−1.89825e9	−0.424099
$573$	−3.42497e9	−0.760528
$574$	−1.16727e7	−0.00257621
$575$	−4.64134e8	−0.101814
$576$	1.91103e8	0.0416667
$577$	−4.83666e9	−1.04817	−0.524083	−	0.851667i	$-0.675592\pi$
$577$	−4.83666e9	−1.04817	−0.524083	+	0.851667i	$0.675592\pi$
$578$	−2.95454e9	−0.636419
$579$	3.68694e9	0.789388
$580$	−4.54054e9	−0.966295
$581$	−6.84564e9	−1.44810
$582$	2.56704e9	0.539761
$583$	−3.96381e9	−0.828462
$584$	−4.99635e8	−0.103803
$585$	−1.44531e9	−0.298481
$586$	−3.78593e8	−0.0777198
$587$	8.36248e9	1.70648	0.853241	−	0.521517i	$-0.174634\pi$
$587$	8.36248e9	1.70648	0.853241	+	0.521517i	$0.174634\pi$
$588$	1.35138e9	0.274131
$589$	−1.40914e8	−0.0284152
$590$	4.39702e9	0.881406
$591$	−3.13332e9	−0.624380
$592$	9.80209e7	0.0194175
$593$	−7.18982e9	−1.41588	−0.707941	−	0.706272i	$-0.750376\pi$
$593$	−7.18982e9	−1.41588	−0.707941	+	0.706272i	$0.750376\pi$
$594$	8.03263e8	0.157255
$595$	2.76730e9	0.538576
$596$	−4.76397e9	−0.921737
$597$	3.81690e8	0.0734177
$598$	5.65940e8	0.108222
$599$	7.64624e9	1.45363	0.726815	−	0.686833i	$-0.241000\pi$
$599$	7.64624e9	1.45363	0.726815	+	0.686833i	$0.241000\pi$
$600$	5.27344e8	0.0996700
$601$	1.72237e9	0.323643	0.161821	−	0.986820i	$-0.448263\pi$
$601$	1.72237e9	0.323643	0.161821	+	0.986820i	$0.448263\pi$
$602$	−4.96003e9	−0.926609
$603$	−1.50222e9	−0.279012
$604$	1.77885e9	0.328481
$605$	2.22854e9	0.409145
$606$	2.06135e9	0.376268
$607$	−4.96333e9	−0.900768	−0.450384	−	0.892835i	$-0.648713\pi$
$607$	−4.96333e9	−0.900768	−0.450384	+	0.892835i	$0.648713\pi$
$608$	−1.26375e9	−0.228034
$609$	−7.11823e9	−1.27706
$610$	−2.83953e9	−0.506515
$611$	−5.74979e9	−1.01978
$612$	2.98819e8	0.0526961
$613$	5.74407e9	1.00718	0.503591	−	0.863942i	$-0.332012\pi$
$613$	5.74407e9	1.00718	0.503591	+	0.863942i	$0.332012\pi$
$614$	2.78076e9	0.484814
$615$	−1.06015e7	−0.00183782
$616$	3.30951e9	0.570469
$617$	−1.05462e10	−1.80758	−0.903788	−	0.427981i	$-0.859225\pi$
$617$	−1.05462e10	−1.80758	−0.903788	+	0.427981i	$0.859225\pi$
$618$	−6.74669e8	−0.114982
$619$	3.60119e9	0.610279	0.305140	−	0.952308i	$-0.401297\pi$
$619$	3.60119e9	0.610279	0.305140	+	0.952308i	$0.401297\pi$
$620$	7.97370e7	0.0134366
$621$	−2.39483e8	−0.0401286
$622$	−6.77167e9	−1.12831
$623$	−5.24475e9	−0.868994
$624$	−6.43014e8	−0.105944
$625$	−7.62856e9	−1.24986
$626$	4.25157e9	0.692691
$627$	−5.31193e9	−0.860628
$628$	2.13169e9	0.343452
$629$	1.53271e8	0.0245574
$630$	2.51984e9	0.401496
$631$	−2.99729e9	−0.474926	−0.237463	−	0.971397i	$-0.576316\pi$
$631$	−2.99729e9	−0.474926	−0.237463	+	0.971397i	$0.576316\pi$
$632$	3.07384e9	0.484363
$633$	5.09162e9	0.797890
$634$	−6.02377e9	−0.938763
$635$	−2.07534e9	−0.321648
$636$	−1.34270e9	−0.206957
$637$	−4.54707e9	−0.697017
$638$	−8.49096e9	−1.29445
$639$	2.61268e9	0.396126
$640$	7.15101e8	0.107829
$641$	1.30075e9	0.195069	0.0975347	−	0.995232i	$-0.468904\pi$
$641$	1.30075e9	0.195069	0.0975347	+	0.995232i	$0.468904\pi$
$642$	2.21331e9	0.330118
$643$	−9.31677e9	−1.38206	−0.691030	−	0.722826i	$-0.742843\pi$
$643$	−9.31677e9	−1.38206	−0.691030	+	0.722826i	$0.742843\pi$
$644$	−9.86691e8	−0.145573
$645$	−4.50482e9	−0.661027
$646$	−1.97607e9	−0.288396
$647$	−9.62984e9	−1.39783	−0.698914	−	0.715205i	$-0.746333\pi$
$647$	−9.62984e9	−1.39783	−0.698914	+	0.715205i	$0.746333\pi$
$648$	2.72098e8	0.0392837
$649$	8.22256e9	1.18073
$650$	−1.77438e9	−0.253425
$651$	1.25004e8	0.0177579
$652$	−9.34162e8	−0.131994
$653$	4.51671e9	0.634784	0.317392	−	0.948294i	$-0.397193\pi$
$653$	4.51671e9	0.634784	0.317392	+	0.948294i	$0.397193\pi$
$654$	−2.73742e9	−0.382666
$655$	−8.89089e9	−1.23623
$656$	−4.71655e6	−0.000652321 0
$657$	−7.11395e8	−0.0978660
$658$	1.00245e10	1.37174
$659$	6.60804e9	0.899443	0.449722	−	0.893169i	$-0.351523\pi$
$659$	6.60804e9	0.899443	0.449722	+	0.893169i	$0.351523\pi$
$660$	3.00578e9	0.406962
$661$	1.31054e10	1.76500	0.882502	−	0.470308i	$-0.155857\pi$
$661$	1.31054e10	1.76500	0.882502	+	0.470308i	$0.155857\pi$
$662$	−1.49191e9	−0.199867
$663$	−1.00545e9	−0.133987
$664$	−2.76609e9	−0.366672
$665$	−1.66635e10	−2.19731
$666$	1.39565e8	0.0183070
$667$	2.53148e9	0.330319
$668$	−4.87427e9	−0.632691
$669$	8.56313e9	1.10571
$670$	−5.62125e9	−0.722057
$671$	−5.31002e9	−0.678527
$672$	1.12107e9	0.142508
$673$	−1.48862e9	−0.188249	−0.0941243	−	0.995560i	$-0.530005\pi$
$673$	−1.48862e9	−0.188249	−0.0941243	+	0.995560i	$0.530005\pi$
$674$	−5.76477e9	−0.725225
$675$	7.50847e8	0.0939698
$676$	−1.85232e9	−0.230623
$677$	1.38673e10	1.71763	0.858817	−	0.512282i	$-0.171200\pi$
$677$	1.38673e10	1.71763	0.858817	+	0.512282i	$0.171200\pi$
$678$	1.17357e8	0.0144613
$679$	1.50590e10	1.84609
$680$	1.11817e9	0.136373
$681$	2.62358e8	0.0318332
$682$	1.49111e8	0.0179996
$683$	−6.45650e9	−0.775398	−0.387699	−	0.921786i	$-0.626730\pi$
$683$	−6.45650e9	−0.775398	−0.387699	+	0.921786i	$0.626730\pi$
$684$	−1.79936e9	−0.214992
$685$	1.55279e10	1.84584
$686$	−4.20609e8	−0.0497443
$687$	6.87538e9	0.809000
$688$	−2.00418e9	−0.234626
$689$	4.51786e9	0.526218
$690$	−8.96138e8	−0.103849
$691$	−8.22490e8	−0.0948325	−0.0474163	−	0.998875i	$-0.515099\pi$
$691$	−8.22490e8	−0.0948325	−0.0474163	+	0.998875i	$0.515099\pi$
$692$	−7.54013e9	−0.864983
$693$	4.71218e9	0.537843
$694$	−6.51461e9	−0.739827
$695$	−1.48697e10	−1.68018
$696$	−2.87623e9	−0.323364
$697$	−7.37506e6	−0.000824994 0
$698$	−3.40028e9	−0.378460
$699$	7.13218e8	0.0789864
$700$	3.09355e9	0.340890
$701$	1.78860e10	1.96111	0.980553	−	0.196252i	$-0.0628772\pi$
$701$	1.78860e10	1.96111	0.980553	+	0.196252i	$0.0628772\pi$
$702$	−9.15541e8	−0.0998845
$703$	−9.22934e8	−0.100191
$704$	1.33726e9	0.144448
$705$	9.10451e9	0.978576
$706$	−1.68693e9	−0.180418
$707$	1.20925e10	1.28691
$708$	2.78531e9	0.294956
$709$	1.14149e10	1.20285	0.601424	−	0.798930i	$-0.294600\pi$
$709$	1.14149e10	1.20285	0.601424	+	0.798930i	$0.294600\pi$
$710$	9.77657e9	1.02514
$711$	4.37661e9	0.456662
$712$	−2.11923e9	−0.220038
$713$	−4.44555e7	−0.00459317
$714$	1.75296e9	0.180231
$715$	−1.01137e10	−1.03476
$716$	1.49938e9	0.152657
$717$	−3.44720e8	−0.0349260
$718$	−7.70004e9	−0.776349
$719$	−4.68698e9	−0.470264	−0.235132	−	0.971963i	$-0.575552\pi$
$719$	−4.68698e9	−0.470264	−0.235132	+	0.971963i	$0.575552\pi$
$720$	1.01818e9	0.101663
$721$	−3.95781e9	−0.393262
$722$	4.74811e9	0.469505
$723$	3.98234e9	0.391881
$724$	−3.62334e9	−0.354833
$725$	−7.93689e9	−0.773512
$726$	1.41168e9	0.136917
$727$	−1.85231e10	−1.78790	−0.893951	−	0.448164i	$-0.852078\pi$
$727$	−1.85231e10	−1.78790	−0.893951	+	0.448164i	$0.852078\pi$
$728$	−3.77211e9	−0.362347
$729$	3.87420e8	0.0370370
$730$	−2.66201e9	−0.253268
$731$	−3.13385e9	−0.296734
$732$	−1.79872e9	−0.169502
$733$	1.69649e9	0.159107	0.0795534	−	0.996831i	$-0.474651\pi$
$733$	1.69649e9	0.159107	0.0795534	+	0.996831i	$0.474651\pi$
$734$	−4.73245e8	−0.0441723
$735$	7.20006e9	0.668853
$736$	−3.98688e8	−0.0368605
$737$	−1.05119e10	−0.967266
$738$	−6.71556e6	−0.000615014 0
$739$	−8.17586e9	−0.745209	−0.372604	−	0.927990i	$-0.621535\pi$
$739$	−8.17586e9	−0.745209	−0.372604	+	0.927990i	$0.621535\pi$
$740$	5.22247e8	0.0473768
$741$	6.05441e9	0.546649
$742$	−7.87669e9	−0.707832
$743$	1.95551e9	0.174904	0.0874519	−	0.996169i	$-0.472128\pi$
$743$	1.95551e9	0.174904	0.0874519	+	0.996169i	$0.472128\pi$
$744$	5.05099e7	0.00449646
$745$	−2.53820e10	−2.24895
$746$	−6.06155e9	−0.534562
$747$	−3.93843e9	−0.345702
$748$	2.09102e9	0.182685
$749$	1.29839e10	1.12907
$750$	−2.94451e9	−0.254858
$751$	1.41485e10	1.21891	0.609453	−	0.792823i	$-0.291389\pi$
$751$	1.41485e10	1.21891	0.609453	+	0.792823i	$0.291389\pi$
$752$	4.05056e9	0.347338
$753$	7.21909e9	0.616169
$754$	9.67780e9	0.822199
$755$	9.47757e9	0.801461
$756$	1.59621e9	0.134358
$757$	−3.50371e9	−0.293557	−0.146779	−	0.989169i	$-0.546890\pi$
$757$	−3.50371e9	−0.293557	−0.146779	+	0.989169i	$0.546890\pi$
$758$	−1.97725e9	−0.164899
$759$	−1.67581e9	−0.139116
$760$	−6.73316e9	−0.556380
$761$	1.18352e9	0.0973485	0.0486743	−	0.998815i	$-0.484500\pi$
$761$	1.18352e9	0.0973485	0.0486743	+	0.998815i	$0.484500\pi$
$762$	−1.31464e9	−0.107637
$763$	−1.60585e10	−1.30879
$764$	−8.11844e9	−0.658636
$765$	1.59208e9	0.128573
$766$	−1.56051e10	−1.25449
$767$	−9.37189e9	−0.749969
$768$	4.52985e8	0.0360844
$769$	1.26642e10	1.00424	0.502118	−	0.864799i	$-0.332554\pi$
$769$	1.26642e10	1.00424	0.502118	+	0.864799i	$0.332554\pi$
$770$	1.76328e10	1.39189
$771$	−3.69988e9	−0.290735
$772$	8.73941e9	0.683630
$773$	−6.91080e9	−0.538146	−0.269073	−	0.963120i	$-0.586717\pi$
$773$	−6.91080e9	−0.538146	−0.269073	+	0.963120i	$0.586717\pi$
$774$	−2.85361e9	−0.221208
$775$	1.39381e8	0.0107559
$776$	6.08483e9	0.467447
$777$	8.18729e8	0.0626133
$778$	1.36528e10	1.03943
$779$	4.44095e7	0.00336585
$780$	−3.42593e9	−0.258492
$781$	1.82825e10	1.37327
$782$	−6.23411e8	−0.0466177
$783$	−4.09526e9	−0.304870
$784$	3.20328e9	0.237404
$785$	1.13575e10	0.837990
$786$	−5.63199e9	−0.413697
$787$	1.12082e10	0.819644	0.409822	−	0.912166i	$-0.365591\pi$
$787$	1.12082e10	0.819644	0.409822	+	0.912166i	$0.365591\pi$
$788$	−7.42714e9	−0.540729
$789$	4.33581e9	0.314269
$790$	1.63771e10	1.18180
$791$	6.88453e8	0.0494603
$792$	1.90403e9	0.136187
$793$	6.05224e9	0.430983
$794$	7.61383e9	0.539798
$795$	−7.15381e9	−0.504955
$796$	9.04747e8	0.0635816
$797$	2.00641e9	0.140383	0.0701917	−	0.997534i	$-0.477639\pi$
$797$	2.00641e9	0.140383	0.0701917	+	0.997534i	$0.477639\pi$
$798$	−1.05556e10	−0.735314
$799$	6.33368e9	0.439281
$800$	1.25000e9	0.0863167
$801$	−3.01741e9	−0.207454
$802$	8.31061e9	0.568883
$803$	−4.97805e9	−0.339278
$804$	−3.56082e9	−0.241631
$805$	−5.25701e9	−0.355184
$806$	−1.69953e8	−0.0114329
$807$	1.26119e10	0.844742
$808$	4.88615e9	0.325857
$809$	−2.85624e9	−0.189660	−0.0948300	−	0.995493i	$-0.530231\pi$
$809$	−2.85624e9	−0.189660	−0.0948300	+	0.995493i	$0.530231\pi$
$810$	1.44971e9	0.0958484
$811$	1.97522e10	1.30030	0.650148	−	0.759808i	$-0.274707\pi$
$811$	1.97522e10	1.30030	0.650148	+	0.759808i	$0.274707\pi$
$812$	−1.68728e10	−1.10597
$813$	1.48955e10	0.972161
$814$	9.76619e8	0.0634658
$815$	−4.97714e9	−0.322054
$816$	7.08312e8	0.0456362
$817$	1.88707e10	1.21063
$818$	2.14048e10	1.36733
$819$	−5.37084e9	−0.341624
$820$	−2.51294e7	−0.00159160
$821$	1.99137e10	1.25589	0.627943	−	0.778260i	$-0.283897\pi$
$821$	1.99137e10	1.25589	0.627943	+	0.778260i	$0.283897\pi$
$822$	9.83621e9	0.617698
$823$	2.60455e10	1.62867	0.814336	−	0.580393i	$-0.197101\pi$
$823$	2.60455e10	1.62867	0.814336	+	0.580393i	$0.197101\pi$
$824$	−1.59922e9	−0.0995776
$825$	5.25412e9	0.325770
$826$	1.63395e10	1.00881
$827$	9.95427e9	0.611984	0.305992	−	0.952034i	$-0.401012\pi$
$827$	9.95427e9	0.611984	0.305992	+	0.952034i	$0.401012\pi$
$828$	−5.67664e8	−0.0347524
$829$	−3.29779e9	−0.201040	−0.100520	−	0.994935i	$-0.532051\pi$
$829$	−3.29779e9	−0.201040	−0.100520	+	0.994935i	$0.532051\pi$
$830$	−1.47375e10	−0.894644
$831$	1.29101e10	0.780418
$832$	−1.52418e9	−0.0917498
$833$	5.00882e9	0.300247
$834$	−9.41931e9	−0.562261
$835$	−2.59697e10	−1.54370
$836$	−1.25912e10	−0.745326
$837$	7.19174e7	0.00423930
$838$	2.73415e9	0.160498
$839$	1.21672e9	0.0711252	0.0355626	−	0.999367i	$-0.488678\pi$
$839$	1.21672e9	0.0711252	0.0355626	+	0.999367i	$0.488678\pi$
$840$	5.97295e9	0.347705
$841$	2.60394e10	1.50954
$842$	1.46547e10	0.846028
$843$	1.99111e10	1.14472
$844$	1.20690e10	0.690993
$845$	−9.86903e9	−0.562699
$846$	5.76730e9	0.327474
$847$	8.28135e9	0.468284
$848$	−3.18270e9	−0.179230
$849$	−1.84396e10	−1.03413
$850$	1.95457e9	0.109165
$851$	−2.91167e8	−0.0161953
$852$	6.19302e9	0.343055
$853$	−1.25896e10	−0.694527	−0.347264	−	0.937768i	$-0.612889\pi$
$853$	−1.25896e10	−0.694527	−0.347264	+	0.937768i	$0.612889\pi$
$854$	−1.05518e10	−0.579728
$855$	−9.58687e9	−0.524560
$856$	5.24636e9	0.285890
$857$	−3.98461e9	−0.216248	−0.108124	−	0.994137i	$-0.534484\pi$
$857$	−3.98461e9	−0.216248	−0.108124	+	0.994137i	$0.534484\pi$
$858$	−6.40659e9	−0.346275
$859$	−1.29735e10	−0.698361	−0.349181	−	0.937055i	$-0.613540\pi$
$859$	−1.29735e10	−0.698361	−0.349181	+	0.937055i	$0.613540\pi$
$860$	−1.06781e10	−0.572466
$861$	−3.93955e7	−0.00210346
$862$	−9.62091e9	−0.511612
$863$	−2.90007e10	−1.53593	−0.767963	−	0.640495i	$-0.778729\pi$
$863$	−2.90007e10	−1.53593	−0.767963	+	0.640495i	$0.778729\pi$
$864$	6.44973e8	0.0340207
$865$	−4.01732e10	−2.11048
$866$	1.46558e10	0.766826
$867$	−9.97159e9	−0.519634
$868$	2.96306e8	0.0153788
$869$	3.06258e10	1.58314
$870$	−1.53243e10	−0.788976
$871$	1.19813e10	0.614383
$872$	−6.48871e9	−0.331399
$873$	8.66375e9	0.440713
$874$	3.75392e9	0.190193
$875$	−1.72734e10	−0.871663
$876$	−1.68627e9	−0.0847544
$877$	−2.05862e10	−1.03057	−0.515286	−	0.857018i	$-0.672314\pi$
$877$	−2.05862e10	−1.03057	−0.515286	+	0.857018i	$0.672314\pi$
$878$	−1.20722e10	−0.601944
$879$	−1.27775e9	−0.0634579
$880$	7.12482e9	0.352440
$881$	4.47154e9	0.220314	0.110157	−	0.993914i	$-0.464865\pi$
$881$	4.47154e9	0.220314	0.110157	+	0.993914i	$0.464865\pi$
$882$	4.56092e9	0.223827
$883$	−6.15100e9	−0.300665	−0.150333	−	0.988635i	$-0.548034\pi$
$883$	−6.15100e9	−0.300665	−0.150333	+	0.988635i	$0.548034\pi$
$884$	−2.38329e9	−0.116037
$885$	1.48399e10	0.719665
$886$	2.39034e10	1.15463
$887$	2.96262e10	1.42542	0.712712	−	0.701457i	$-0.247467\pi$
$887$	2.96262e10	1.42542	0.712712	+	0.701457i	$0.247467\pi$
$888$	3.30821e8	0.0158543
$889$	−7.71205e9	−0.368140
$890$	−1.12911e10	−0.536871
$891$	2.71101e9	0.128398
$892$	2.02978e10	0.957573
$893$	−3.81388e10	−1.79220
$894$	−1.60784e10	−0.752595
$895$	7.98860e9	0.372469
$896$	2.65734e9	0.123415
$897$	1.91005e9	0.0883630
$898$	1.53598e10	0.707813
$899$	−7.60208e8	−0.0348958
$900$	1.77979e9	0.0813802
$901$	−4.97665e9	−0.226673
$902$	−4.69928e7	−0.00213210
$903$	−1.67401e10	−0.756573
$904$	2.78180e8	0.0125238
$905$	−1.93049e10	−0.865759
$906$	6.00362e9	0.268204
$907$	3.02880e10	1.34786	0.673930	−	0.738795i	$-0.264605\pi$
$907$	3.02880e10	1.34786	0.673930	+	0.738795i	$0.264605\pi$
$908$	6.21886e8	0.0275683
$909$	6.95704e9	0.307221
$910$	−2.00975e10	−0.884091
$911$	9.96481e9	0.436671	0.218336	−	0.975874i	$-0.429937\pi$
$911$	9.96481e9	0.436671	0.218336	+	0.975874i	$0.429937\pi$
$912$	−4.26516e9	−0.186189
$913$	−2.75596e10	−1.19846
$914$	−1.55211e10	−0.672376
$915$	−9.58342e9	−0.413568
$916$	1.62972e10	0.700614
$917$	−3.30389e10	−1.41492
$918$	1.00851e9	0.0430262
$919$	−3.72161e10	−1.58171	−0.790855	−	0.612003i	$-0.790364\pi$
$919$	−3.72161e10	−1.58171	−0.790855	+	0.612003i	$0.790364\pi$
$920$	−2.12418e9	−0.0899360
$921$	9.38508e9	0.395849
$922$	−6.51599e9	−0.273793
$923$	−2.08380e10	−0.872267
$924$	1.11696e10	0.465786
$925$	9.12891e8	0.0379248
$926$	−5.29174e8	−0.0219008
$927$	−2.27701e9	−0.0938827
$928$	−6.81773e9	−0.280041
$929$	2.36879e9	0.0969328	0.0484664	−	0.998825i	$-0.484567\pi$
$929$	2.36879e9	0.0969328	0.0484664	+	0.998825i	$0.484567\pi$
$930$	2.69112e8	0.0109709
$931$	−3.01610e10	−1.22496
$932$	1.69059e9	0.0684042
$933$	−2.28544e10	−0.921263
$934$	−2.30422e10	−0.925357
$935$	1.11408e10	0.445733
$936$	−2.17017e9	−0.0865025
$937$	2.37255e10	0.942166	0.471083	−	0.882089i	$-0.343863\pi$
$937$	2.37255e10	0.942166	0.471083	+	0.882089i	$0.343863\pi$
$938$	−2.08888e10	−0.826425
$939$	1.43491e10	0.565580
$940$	2.15811e10	0.847472
$941$	−2.53391e10	−0.991351	−0.495676	−	0.868508i	$-0.665079\pi$
$941$	−2.53391e10	−0.991351	−0.495676	+	0.868508i	$0.665079\pi$
$942$	7.19447e9	0.280427
$943$	1.40103e7	0.000544073 0
$944$	6.60223e9	0.255440
$945$	8.50446e9	0.327820
$946$	−1.99684e10	−0.766874
$947$	3.19577e10	1.22278	0.611392	−	0.791328i	$-0.290610\pi$
$947$	3.19577e10	1.22278	0.611392	+	0.791328i	$0.290610\pi$
$948$	1.03742e10	0.395481
$949$	5.67387e9	0.215500
$950$	−1.17696e10	−0.445378
$951$	−2.03302e10	−0.766497
$952$	4.15517e9	0.156084
$953$	1.06573e10	0.398861	0.199431	−	0.979912i	$-0.436091\pi$
$953$	1.06573e10	0.398861	0.199431	+	0.979912i	$0.436091\pi$
$954$	−4.53162e9	−0.168980
$955$	−4.32544e10	−1.60701
$956$	−8.17113e8	−0.0302468
$957$	−2.86570e10	−1.05691
$958$	−2.72034e10	−0.999642
$959$	5.77021e10	2.11264
$960$	2.41347e9	0.0880424
$961$	−2.74993e10	−0.999515
$962$	−1.11313e9	−0.0403119
$963$	7.46991e9	0.269540
$964$	9.43961e9	0.339379
$965$	4.65628e10	1.66799
$966$	−3.33008e9	−0.118860
$967$	5.33089e10	1.89586	0.947932	−	0.318472i	$-0.103170\pi$
$967$	5.33089e10	1.89586	0.947932	+	0.318472i	$0.103170\pi$
$968$	3.34621e9	0.118574
$969$	−6.66924e9	−0.235474
$970$	3.24195e10	1.14053
$971$	−5.29648e10	−1.85661	−0.928304	−	0.371823i	$-0.878733\pi$
$971$	−5.29648e10	−1.85661	−0.928304	+	0.371823i	$0.878733\pi$
$972$	9.18330e8	0.0320750
$973$	−5.52565e10	−1.92304
$974$	2.40970e10	0.835617
$975$	−5.98853e9	−0.206921
$976$	−4.26363e9	−0.146793
$977$	−1.79920e10	−0.617233	−0.308616	−	0.951187i	$-0.599866\pi$
$977$	−1.79920e10	−0.617233	−0.308616	+	0.951187i	$0.599866\pi$
$978$	−3.15280e9	−0.107773
$979$	−2.11146e10	−0.719191
$980$	1.70668e10	0.579243
$981$	−9.23881e9	−0.312446
$982$	−6.68153e6	−0.000225157 0
$983$	1.56089e10	0.524124	0.262062	−	0.965051i	$-0.415597\pi$
$983$	1.56089e10	0.524124	0.262062	+	0.965051i	$0.415597\pi$
$984$	−1.59184e7	−0.000532618 0
$985$	−3.95712e10	−1.31933
$986$	−1.06606e10	−0.354170
$987$	3.38327e10	1.12002
$988$	1.43512e10	0.473412
$989$	5.95333e9	0.195692
$990$	1.01445e10	0.332283
$991$	4.73053e10	1.54402	0.772009	−	0.635612i	$-0.219252\pi$
$991$	4.73053e10	1.54402	0.772009	+	0.635612i	$0.219252\pi$
$992$	1.19727e8	0.00389405
$993$	−5.03521e9	−0.163191
$994$	3.63301e10	1.17331
$995$	4.82042e9	0.155133
$996$	−9.33555e9	−0.299386
$997$	3.63716e10	1.16233	0.581166	−	0.813785i	$-0.302597\pi$
$997$	3.63716e10	1.16233	0.581166	+	0.813785i	$0.302597\pi$
$998$	−3.48645e10	−1.11026
$999$	4.71032e8	0.0149476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.8.a.h.1.3	✓	4
3.2	odd	2		414.8.a.i.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.h.1.3	✓	4	1.1	even	1	trivial
414.8.a.i.1.2		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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +340.987 q^{5} +216.000 q^{6} +1267.12 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +340.987 q^{5} +216.000 q^{6} +1267.12 q^{7} +512.000 q^{8} +729.000 q^{9} +2727.89 q^{10} +5101.25 q^{11} +1728.00 q^{12} -5814.29 q^{13} +10137.0 q^{14} +9206.64 q^{15} +4096.00 q^{16} +6404.73 q^{17} +5832.00 q^{18} -38566.6 q^{19} +21823.2 q^{20} +34212.2 q^{21} +40810.0 q^{22} -12167.0 q^{23} +13824.0 q^{24} +38147.0 q^{25} -46514.3 q^{26} +19683.0 q^{27} +81095.7 q^{28} -208061. q^{29} +73653.1 q^{30} +3653.78 q^{31} +32768.0 q^{32} +137734. q^{33} +51237.9 q^{34} +432071. q^{35} +46656.0 q^{36} +23930.9 q^{37} -308533. q^{38} -156986. q^{39} +174585. q^{40} -1151.50 q^{41} +273698. q^{42} -489302. q^{43} +326480. q^{44} +248579. q^{45} -97336.0 q^{46} +988907. q^{47} +110592. q^{48} +782050. q^{49} +305176. q^{50} +172928. q^{51} -372115. q^{52} -777027. q^{53} +157464. q^{54} +1.73946e6 q^{55} +648766. q^{56} -1.04130e6 q^{57} -1.66449e6 q^{58} +1.61187e6 q^{59} +589225. q^{60} -1.04092e6 q^{61} +29230.2 q^{62} +923731. q^{63} +262144. q^{64} -1.98260e6 q^{65} +1.10187e6 q^{66} -2.06066e6 q^{67} +409903. q^{68} -328509. q^{69} +3.45657e6 q^{70} +3.58392e6 q^{71} +373248. q^{72} -975850. q^{73} +191447. q^{74} +1.02997e6 q^{75} -2.46826e6 q^{76} +6.46390e6 q^{77} -1.25589e6 q^{78} +6.00359e6 q^{79} +1.39668e6 q^{80} +531441. q^{81} -9212.01 q^{82} -5.40252e6 q^{83} +2.18958e6 q^{84} +2.18393e6 q^{85} -3.91441e6 q^{86} -5.61764e6 q^{87} +2.61184e6 q^{88} -4.13911e6 q^{89} +1.98863e6 q^{90} -7.36740e6 q^{91} -778688. q^{92} +98652.1 q^{93} +7.91125e6 q^{94} -1.31507e7 q^{95} +884736. q^{96} +1.18844e7 q^{97} +6.25640e6 q^{98} +3.71881e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9	\( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9} + 2160 q^{10} + 4120 q^{11} + 6912 q^{12} + 8036 q^{13} + 16176 q^{14} + 7290 q^{15} + 16384 q^{16} + 37182 q^{17} + 23328 q^{18} + 5702 q^{19} + 17280 q^{20} + 54594 q^{21} + 32960 q^{22} - 48668 q^{23} + 55296 q^{24} + 121480 q^{25} + 64288 q^{26} + 78732 q^{27} + 129408 q^{28} + 217716 q^{29} + 58320 q^{30} + 222852 q^{31} + 131072 q^{32} + 111240 q^{33} + 297456 q^{34} + 68440 q^{35} + 186624 q^{36} + 486428 q^{37} + 45616 q^{38} + 216972 q^{39} + 138240 q^{40} + 338336 q^{41} + 436752 q^{42} + 730974 q^{43} + 263680 q^{44} + 196830 q^{45} - 389344 q^{46} + 338248 q^{47} + 442368 q^{48} - 310552 q^{49} + 971840 q^{50} + 1003914 q^{51} + 514304 q^{52} - 375502 q^{53} + 629856 q^{54} + 424840 q^{55} + 1035264 q^{56} + 153954 q^{57} + 1741728 q^{58} + 71392 q^{59} + 466560 q^{60} + 2101164 q^{61} + 1782816 q^{62} + 1474038 q^{63} + 1048576 q^{64} + 1578780 q^{65} + 889920 q^{66} + 4337162 q^{67} + 2379648 q^{68} - 1314036 q^{69} + 547520 q^{70} + 2288016 q^{71} + 1492992 q^{72} - 1107328 q^{73} + 3891424 q^{74} + 3279960 q^{75} + 364928 q^{76} + 5826200 q^{77} + 1735776 q^{78} + 60610 q^{79} + 1105920 q^{80} + 2125764 q^{81} + 2706688 q^{82} + 1485464 q^{83} + 3494016 q^{84} - 8843820 q^{85} + 5847792 q^{86} + 5878332 q^{87} + 2109440 q^{88} + 1485090 q^{89} + 1574640 q^{90} - 2898412 q^{91} - 3114752 q^{92} + 6017004 q^{93} + 2705984 q^{94} + 8545200 q^{95} + 3538944 q^{96} + 1935444 q^{97} - 2484416 q^{98} + 3003480 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9 + 2160 * q^10 + 4120 * q^11 + 6912 * q^12 + 8036 * q^13 + 16176 * q^14 + 7290 * q^15 + 16384 * q^16 + 37182 * q^17 + 23328 * q^18 + 5702 * q^19 + 17280 * q^20 + 54594 * q^21 + 32960 * q^22 - 48668 * q^23 + 55296 * q^24 + 121480 * q^25 + 64288 * q^26 + 78732 * q^27 + 129408 * q^28 + 217716 * q^29 + 58320 * q^30 + 222852 * q^31 + 131072 * q^32 + 111240 * q^33 + 297456 * q^34 + 68440 * q^35 + 186624 * q^36 + 486428 * q^37 + 45616 * q^38 + 216972 * q^39 + 138240 * q^40 + 338336 * q^41 + 436752 * q^42 + 730974 * q^43 + 263680 * q^44 + 196830 * q^45 - 389344 * q^46 + 338248 * q^47 + 442368 * q^48 - 310552 * q^49 + 971840 * q^50 + 1003914 * q^51 + 514304 * q^52 - 375502 * q^53 + 629856 * q^54 + 424840 * q^55 + 1035264 * q^56 + 153954 * q^57 + 1741728 * q^58 + 71392 * q^59 + 466560 * q^60 + 2101164 * q^61 + 1782816 * q^62 + 1474038 * q^63 + 1048576 * q^64 + 1578780 * q^65 + 889920 * q^66 + 4337162 * q^67 + 2379648 * q^68 - 1314036 * q^69 + 547520 * q^70 + 2288016 * q^71 + 1492992 * q^72 - 1107328 * q^73 + 3891424 * q^74 + 3279960 * q^75 + 364928 * q^76 + 5826200 * q^77 + 1735776 * q^78 + 60610 * q^79 + 1105920 * q^80 + 2125764 * q^81 + 2706688 * q^82 + 1485464 * q^83 + 3494016 * q^84 - 8843820 * q^85 + 5847792 * q^86 + 5878332 * q^87 + 2109440 * q^88 + 1485090 * q^89 + 1574640 * q^90 - 2898412 * q^91 - 3114752 * q^92 + 6017004 * q^93 + 2705984 * q^94 + 8545200 * q^95 + 3538944 * q^96 + 1935444 * q^97 - 2484416 * q^98 + 3003480 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more