LMFDB - Embedded newform 546.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,8,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("546.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	546.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:	$170.562223914$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 33506x + 97248 $ x^3 - x^2 - 33506*x + 97248
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-183.985$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -516.403 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -516.403 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -4131.22 q^{10} -1471.43 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -13942.9 q^{15} +4096.00 q^{16} +14563.7 q^{17} +5832.00 q^{18} +16884.1 q^{19} -33049.8 q^{20} -9261.00 q^{21} -11771.4 q^{22} +64859.5 q^{23} +13824.0 q^{24} +188547. q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +32385.8 q^{29} -111543. q^{30} -126478. q^{31} +32768.0 q^{32} -39728.5 q^{33} +116510. q^{34} +177126. q^{35} +46656.0 q^{36} +121016. q^{37} +135073. q^{38} +59319.0 q^{39} -264398. q^{40} -488537. q^{41} -74088.0 q^{42} -228861. q^{43} -94171.2 q^{44} -376458. q^{45} +518876. q^{46} -552791. q^{47} +110592. q^{48} +117649. q^{49} +1.50838e6 q^{50} +393220. q^{51} +140608. q^{52} -139750. q^{53} +157464. q^{54} +759849. q^{55} -175616. q^{56} +455871. q^{57} +259087. q^{58} -1.75981e6 q^{59} -892344. q^{60} -315225. q^{61} -1.01182e6 q^{62} -250047. q^{63} +262144. q^{64} -1.13454e6 q^{65} -317828. q^{66} -791524. q^{67} +932077. q^{68} +1.75121e6 q^{69} +1.41701e6 q^{70} +3.31922e6 q^{71} +373248. q^{72} -5.34583e6 q^{73} +968127. q^{74} +5.09077e6 q^{75} +1.08058e6 q^{76} +504699. q^{77} +474552. q^{78} -4.90728e6 q^{79} -2.11519e6 q^{80} +531441. q^{81} -3.90830e6 q^{82} -689871. q^{83} -592704. q^{84} -7.52074e6 q^{85} -1.83089e6 q^{86} +874418. q^{87} -753370. q^{88} -4.55146e6 q^{89} -3.01166e6 q^{90} -753571. q^{91} +4.15101e6 q^{92} -3.41490e6 q^{93} -4.42233e6 q^{94} -8.71901e6 q^{95} +884736. q^{96} +5.94617e6 q^{97} +941192. q^{98} -1.07267e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 24 q^{2} + 81 q^{3} + 192 q^{4} - 378 q^{5} + 648 q^{6} - 1029 q^{7} + 1536 q^{8} + 2187 q^{9}+O(q^{10}) $ 3 * q + 24 * q^2 + 81 * q^3 + 192 * q^4 - 378 * q^5 + 648 * q^6 - 1029 * q^7 + 1536 * q^8 + 2187 * q^9	\( 3 q + 24 q^{2} + 81 q^{3} + 192 q^{4} - 378 q^{5} + 648 q^{6} - 1029 q^{7} + 1536 q^{8} + 2187 q^{9} - 3024 q^{10} - 6069 q^{11} + 5184 q^{12} + 6591 q^{13} - 8232 q^{14} - 10206 q^{15} + 12288 q^{16} - 12231 q^{17} + 17496 q^{18} - 4836 q^{19} - 24192 q^{20} - 27783 q^{21} - 48552 q^{22} + 111606 q^{23} + 41472 q^{24} + 41919 q^{25} + 52728 q^{26} + 59049 q^{27} - 65856 q^{28} + 105414 q^{29} - 81648 q^{30} - 263061 q^{31} + 98304 q^{32} - 163863 q^{33} - 97848 q^{34} + 129654 q^{35} + 139968 q^{36} - 585969 q^{37} - 38688 q^{38} + 177957 q^{39} - 193536 q^{40} - 649608 q^{41} - 222264 q^{42} - 76182 q^{43} - 388416 q^{44} - 275562 q^{45} + 892848 q^{46} - 1269219 q^{47} + 331776 q^{48} + 352947 q^{49} + 335352 q^{50} - 330237 q^{51} + 421824 q^{52} + 326511 q^{53} + 472392 q^{54} + 475311 q^{55} - 526848 q^{56} - 130572 q^{57} + 843312 q^{58} - 3434316 q^{59} - 653184 q^{60} - 1450497 q^{61} - 2104488 q^{62} - 750141 q^{63} + 786432 q^{64} - 830466 q^{65} - 1310904 q^{66} - 1302372 q^{67} - 782784 q^{68} + 3013362 q^{69} + 1037232 q^{70} - 5186076 q^{71} + 1119744 q^{72} - 3662940 q^{73} - 4687752 q^{74} + 1131813 q^{75} - 309504 q^{76} + 2081667 q^{77} + 1423656 q^{78} - 2950251 q^{79} - 1548288 q^{80} + 1594323 q^{81} - 5196864 q^{82} - 13650225 q^{83} - 1778112 q^{84} - 9353769 q^{85} - 609456 q^{86} + 2846178 q^{87} - 3107328 q^{88} + 6379533 q^{89} - 2204496 q^{90} - 2260713 q^{91} + 7142784 q^{92} - 7102647 q^{93} - 10153752 q^{94} - 10607646 q^{95} + 2654208 q^{96} - 18032235 q^{97} + 2823576 q^{98} - 4424301 q^{99}+O(q^{100}) \) 3 * q + 24 * q^2 + 81 * q^3 + 192 * q^4 - 378 * q^5 + 648 * q^6 - 1029 * q^7 + 1536 * q^8 + 2187 * q^9 - 3024 * q^10 - 6069 * q^11 + 5184 * q^12 + 6591 * q^13 - 8232 * q^14 - 10206 * q^15 + 12288 * q^16 - 12231 * q^17 + 17496 * q^18 - 4836 * q^19 - 24192 * q^20 - 27783 * q^21 - 48552 * q^22 + 111606 * q^23 + 41472 * q^24 + 41919 * q^25 + 52728 * q^26 + 59049 * q^27 - 65856 * q^28 + 105414 * q^29 - 81648 * q^30 - 263061 * q^31 + 98304 * q^32 - 163863 * q^33 - 97848 * q^34 + 129654 * q^35 + 139968 * q^36 - 585969 * q^37 - 38688 * q^38 + 177957 * q^39 - 193536 * q^40 - 649608 * q^41 - 222264 * q^42 - 76182 * q^43 - 388416 * q^44 - 275562 * q^45 + 892848 * q^46 - 1269219 * q^47 + 331776 * q^48 + 352947 * q^49 + 335352 * q^50 - 330237 * q^51 + 421824 * q^52 + 326511 * q^53 + 472392 * q^54 + 475311 * q^55 - 526848 * q^56 - 130572 * q^57 + 843312 * q^58 - 3434316 * q^59 - 653184 * q^60 - 1450497 * q^61 - 2104488 * q^62 - 750141 * q^63 + 786432 * q^64 - 830466 * q^65 - 1310904 * q^66 - 1302372 * q^67 - 782784 * q^68 + 3013362 * q^69 + 1037232 * q^70 - 5186076 * q^71 + 1119744 * q^72 - 3662940 * q^73 - 4687752 * q^74 + 1131813 * q^75 - 309504 * q^76 + 2081667 * q^77 + 1423656 * q^78 - 2950251 * q^79 - 1548288 * q^80 + 1594323 * q^81 - 5196864 * q^82 - 13650225 * q^83 - 1778112 * q^84 - 9353769 * q^85 - 609456 * q^86 + 2846178 * q^87 - 3107328 * q^88 + 6379533 * q^89 - 2204496 * q^90 - 2260713 * q^91 + 7142784 * q^92 - 7102647 * q^93 - 10153752 * q^94 - 10607646 * q^95 + 2654208 * q^96 - 18032235 * q^97 + 2823576 * q^98 - 4424301 * 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
$5$	−516.403	−1.84754	−0.923770	−	0.382948i	$-0.874909\pi$
$5$	−516.403	−1.84754	−0.923770	+	0.382948i	$0.874909\pi$
$6$	216.000	0.408248
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−4131.22	−1.30641
$11$	−1471.43	−0.333322	−0.166661	−	0.986014i	$-0.553299\pi$
$11$	−1471.43	−0.333322	−0.166661	+	0.986014i	$0.553299\pi$
$12$	1728.00	0.288675
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	−2744.00	−0.267261
$15$	−13942.9	−1.06668
$16$	4096.00	0.250000
$17$	14563.7	0.718953	0.359477	−	0.933154i	$-0.382955\pi$
$17$	14563.7	0.718953	0.359477	+	0.933154i	$0.382955\pi$
$18$	5832.00	0.235702
$19$	16884.1	0.564730	0.282365	−	0.959307i	$-0.408881\pi$
$19$	16884.1	0.564730	0.282365	+	0.959307i	$0.408881\pi$
$20$	−33049.8	−0.923770
$21$	−9261.00	−0.218218
$22$	−11771.4	−0.235694
$23$	64859.5	1.11154	0.555771	−	0.831335i	$-0.312423\pi$
$23$	64859.5	1.11154	0.555771	+	0.831335i	$0.312423\pi$
$24$	13824.0	0.204124
$25$	188547.	2.41340
$26$	17576.0	0.196116
$27$	19683.0	0.192450
$28$	−21952.0	−0.188982
$29$	32385.8	0.246582	0.123291	−	0.992371i	$-0.460655\pi$
$29$	32385.8	0.246582	0.123291	+	0.992371i	$0.460655\pi$
$30$	−111543.	−0.754255
$31$	−126478.	−0.762515	−0.381257	−	0.924469i	$-0.624509\pi$
$31$	−126478.	−0.762515	−0.381257	+	0.924469i	$0.624509\pi$
$32$	32768.0	0.176777
$33$	−39728.5	−0.192444
$34$	116510.	0.508377
$35$	177126.	0.698304
$36$	46656.0	0.166667
$37$	121016.	0.392768	0.196384	−	0.980527i	$-0.437080\pi$
$37$	121016.	0.392768	0.196384	+	0.980527i	$0.437080\pi$
$38$	135073.	0.399324
$39$	59319.0	0.160128
$40$	−264398.	−0.653204
$41$	−488537.	−1.10702	−0.553509	−	0.832844i	$-0.686711\pi$
$41$	−488537.	−1.10702	−0.553509	+	0.832844i	$0.686711\pi$
$42$	−74088.0	−0.154303
$43$	−228861.	−0.438967	−0.219483	−	0.975616i	$-0.570437\pi$
$43$	−228861.	−0.438967	−0.219483	+	0.975616i	$0.570437\pi$
$44$	−94171.2	−0.166661
$45$	−376458.	−0.615847
$46$	518876.	0.785979
$47$	−552791.	−0.776638	−0.388319	−	0.921525i	$-0.626944\pi$
$47$	−552791.	−0.776638	−0.388319	+	0.921525i	$0.626944\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	1.50838e6	1.70653
$51$	393220.	0.415088
$52$	140608.	0.138675
$53$	−139750.	−0.128940	−0.0644698	−	0.997920i	$-0.520536\pi$
$53$	−139750.	−0.128940	−0.0644698	+	0.997920i	$0.520536\pi$
$54$	157464.	0.136083
$55$	759849.	0.615825
$56$	−175616.	−0.133631
$57$	455871.	0.326047
$58$	259087.	0.174360
$59$	−1.75981e6	−1.11553	−0.557767	−	0.829998i	$-0.688342\pi$
$59$	−1.75981e6	−1.11553	−0.557767	+	0.829998i	$0.688342\pi$
$60$	−892344.	−0.533339
$61$	−315225.	−0.177814	−0.0889070	−	0.996040i	$-0.528337\pi$
$61$	−315225.	−0.177814	−0.0889070	+	0.996040i	$0.528337\pi$
$62$	−1.01182e6	−0.539179
$63$	−250047.	−0.125988
$64$	262144.	0.125000
$65$	−1.13454e6	−0.512415
$66$	−317828.	−0.136078
$67$	−791524.	−0.321516	−0.160758	−	0.986994i	$-0.551394\pi$
$67$	−791524.	−0.321516	−0.160758	+	0.986994i	$0.551394\pi$
$68$	932077.	0.359477
$69$	1.75121e6	0.641749
$70$	1.41701e6	0.493776
$71$	3.31922e6	1.10060	0.550302	−	0.834966i	$-0.314513\pi$
$71$	3.31922e6	1.10060	0.550302	+	0.834966i	$0.314513\pi$
$72$	373248.	0.117851
$73$	−5.34583e6	−1.60837	−0.804184	−	0.594381i	$-0.797397\pi$
$73$	−5.34583e6	−1.60837	−0.804184	+	0.594381i	$0.797397\pi$
$74$	968127.	0.277729
$75$	5.09077e6	1.39338
$76$	1.08058e6	0.282365
$77$	504699.	0.125984
$78$	474552.	0.113228
$79$	−4.90728e6	−1.11981	−0.559907	−	0.828556i	$-0.689163\pi$
$79$	−4.90728e6	−1.11981	−0.559907	+	0.828556i	$0.689163\pi$
$80$	−2.11519e6	−0.461885
$81$	531441.	0.111111
$82$	−3.90830e6	−0.782779
$83$	−689871.	−0.132432	−0.0662162	−	0.997805i	$-0.521093\pi$
$83$	−689871.	−0.132432	−0.0662162	+	0.997805i	$0.521093\pi$
$84$	−592704.	−0.109109
$85$	−7.52074e6	−1.32829
$86$	−1.83089e6	−0.310396
$87$	874418.	0.142364
$88$	−753370.	−0.117847
$89$	−4.55146e6	−0.684361	−0.342181	−	0.939634i	$-0.611165\pi$
$89$	−4.55146e6	−0.684361	−0.342181	+	0.939634i	$0.611165\pi$
$90$	−3.01166e6	−0.435469
$91$	−753571.	−0.104828
$92$	4.15101e6	0.555771
$93$	−3.41490e6	−0.440238
$94$	−4.42233e6	−0.549166
$95$	−8.71901e6	−1.04336
$96$	884736.	0.102062
$97$	5.94617e6	0.661510	0.330755	−	0.943717i	$-0.392697\pi$
$97$	5.94617e6	0.661510	0.330755	+	0.943717i	$0.392697\pi$
$98$	941192.	0.101015
$99$	−1.07267e6	−0.111107
$100$	1.20670e7	1.20670
$101$	−5.39373e6	−0.520912	−0.260456	−	0.965486i	$-0.583873\pi$
$101$	−5.39373e6	−0.520912	−0.260456	+	0.965486i	$0.583873\pi$
$102$	3.14576e6	0.293511
$103$	9.61893e6	0.867355	0.433677	−	0.901068i	$-0.357216\pi$
$103$	9.61893e6	0.867355	0.433677	+	0.901068i	$0.357216\pi$
$104$	1.12486e6	0.0980581
$105$	4.78241e6	0.403166
$106$	−1.11800e6	−0.0911741
$107$	−5.45241e6	−0.430274	−0.215137	−	0.976584i	$-0.569020\pi$
$107$	−5.45241e6	−0.430274	−0.215137	+	0.976584i	$0.569020\pi$
$108$	1.25971e6	0.0962250
$109$	−9.84661e6	−0.728273	−0.364136	−	0.931346i	$-0.618636\pi$
$109$	−9.84661e6	−0.728273	−0.364136	+	0.931346i	$0.618636\pi$
$110$	6.07879e6	0.435454
$111$	3.26743e6	0.226765
$112$	−1.40493e6	−0.0944911
$113$	−9.15345e6	−0.596775	−0.298387	−	0.954445i	$-0.596449\pi$
$113$	−9.15345e6	−0.596775	−0.298387	+	0.954445i	$0.596449\pi$
$114$	3.64697e6	0.230550
$115$	−3.34936e7	−2.05362
$116$	2.07269e6	0.123291
$117$	1.60161e6	0.0924500
$118$	−1.40784e7	−0.788801
$119$	−4.99535e6	−0.271739
$120$	−7.13876e6	−0.377127
$121$	−1.73221e7	−0.888896
$122$	−2.52180e6	−0.125733
$123$	−1.31905e7	−0.639137
$124$	−8.09457e6	−0.381257
$125$	−5.70223e7	−2.61132
$126$	−2.00038e6	−0.0890871
$127$	−1.08728e7	−0.471009	−0.235505	−	0.971873i	$-0.575674\pi$
$127$	−1.08728e7	−0.471009	−0.235505	+	0.971873i	$0.575674\pi$
$128$	2.09715e6	0.0883883
$129$	−6.17924e6	−0.253438
$130$	−9.07630e6	−0.362332
$131$	−2.14062e7	−0.831934	−0.415967	−	0.909380i	$-0.636557\pi$
$131$	−2.14062e7	−0.831934	−0.415967	+	0.909380i	$0.636557\pi$
$132$	−2.54262e6	−0.0962218
$133$	−5.79125e6	−0.213448
$134$	−6.33219e6	−0.227346
$135$	−1.01644e7	−0.355559
$136$	7.45662e6	0.254188
$137$	−3.60615e7	−1.19818	−0.599090	−	0.800682i	$-0.704471\pi$
$137$	−3.60615e7	−1.19818	−0.599090	+	0.800682i	$0.704471\pi$
$138$	1.40096e7	0.453785
$139$	3.97329e7	1.25487	0.627434	−	0.778669i	$-0.284105\pi$
$139$	3.97329e7	1.25487	0.627434	+	0.778669i	$0.284105\pi$
$140$	1.13361e7	0.349152
$141$	−1.49254e7	−0.448392
$142$	2.65537e7	0.778245
$143$	−3.23272e6	−0.0924469
$144$	2.98598e6	0.0833333
$145$	−1.67241e7	−0.455571
$146$	−4.27667e7	−1.13729
$147$	3.17652e6	0.0824786
$148$	7.74502e6	0.196384
$149$	−1.80502e7	−0.447022	−0.223511	−	0.974701i	$-0.571752\pi$
$149$	−1.80502e7	−0.447022	−0.223511	+	0.974701i	$0.571752\pi$
$150$	4.07262e7	0.985267
$151$	7.44196e6	0.175901	0.0879505	−	0.996125i	$-0.471968\pi$
$151$	7.44196e6	0.175901	0.0879505	+	0.996125i	$0.471968\pi$
$152$	8.64467e6	0.199662
$153$	1.06169e7	0.239651
$154$	4.03759e6	0.0890840
$155$	6.53135e7	1.40878
$156$	3.79642e6	0.0800641
$157$	3.83696e7	0.791296	0.395648	−	0.918402i	$-0.370520\pi$
$157$	3.83696e7	0.791296	0.395648	+	0.918402i	$0.370520\pi$
$158$	−3.92582e7	−0.791828
$159$	−3.77325e6	−0.0744433
$160$	−1.69215e7	−0.326602
$161$	−2.22468e7	−0.420123
$162$	4.25153e6	0.0785674
$163$	2.03325e7	0.367734	0.183867	−	0.982951i	$-0.441138\pi$
$163$	2.03325e7	0.367734	0.183867	+	0.982951i	$0.441138\pi$
$164$	−3.12664e7	−0.553509
$165$	2.05159e7	0.355547
$166$	−5.51897e6	−0.0936439
$167$	−5.12204e7	−0.851011	−0.425505	−	0.904956i	$-0.639904\pi$
$167$	−5.12204e7	−0.851011	−0.425505	+	0.904956i	$0.639904\pi$
$168$	−4.74163e6	−0.0771517
$169$	4.82681e6	0.0769231
$170$	−6.01659e7	−0.939246
$171$	1.23085e7	0.188243
$172$	−1.46471e7	−0.219483
$173$	8.22786e7	1.20816	0.604081	−	0.796923i	$-0.293540\pi$
$173$	8.22786e7	1.20816	0.604081	+	0.796923i	$0.293540\pi$
$174$	6.99534e6	0.100667
$175$	−6.46716e7	−0.912180
$176$	−6.02696e6	−0.0833305
$177$	−4.75147e7	−0.644053
$178$	−3.64117e7	−0.483916
$179$	2.87723e7	0.374963	0.187482	−	0.982268i	$-0.439968\pi$
$179$	2.87723e7	0.374963	0.187482	+	0.982268i	$0.439968\pi$
$180$	−2.40933e7	−0.307923
$181$	3.37373e7	0.422898	0.211449	−	0.977389i	$-0.432182\pi$
$181$	3.37373e7	0.422898	0.211449	+	0.977389i	$0.432182\pi$
$182$	−6.02857e6	−0.0741249
$183$	−8.51107e6	−0.102661
$184$	3.32080e7	0.392989
$185$	−6.24930e7	−0.725655
$186$	−2.73192e7	−0.311295
$187$	−2.14294e7	−0.239643
$188$	−3.53786e7	−0.388319
$189$	−6.75127e6	−0.0727393
$190$	−6.97520e7	−0.737767
$191$	−2.22000e7	−0.230535	−0.115267	−	0.993334i	$-0.536773\pi$
$191$	−2.22000e7	−0.230535	−0.115267	+	0.993334i	$0.536773\pi$
$192$	7.07789e6	0.0721688
$193$	1.88081e8	1.88319	0.941593	−	0.336752i	$-0.109328\pi$
$193$	1.88081e8	1.88319	0.941593	+	0.336752i	$0.109328\pi$
$194$	4.75694e7	0.467758
$195$	−3.06325e7	−0.295843
$196$	7.52954e6	0.0714286
$197$	−1.93760e8	−1.80564	−0.902822	−	0.430015i	$-0.858508\pi$
$197$	−1.93760e8	−1.80564	−0.902822	+	0.430015i	$0.858508\pi$
$198$	−8.58135e6	−0.0785647
$199$	1.40804e8	1.26657	0.633283	−	0.773920i	$-0.281707\pi$
$199$	1.40804e8	1.26657	0.633283	+	0.773920i	$0.281707\pi$
$200$	9.65361e7	0.853267
$201$	−2.13711e7	−0.185627
$202$	−4.31498e7	−0.368340
$203$	−1.11083e7	−0.0931994
$204$	2.51661e7	0.207544
$205$	2.52282e8	2.04526
$206$	7.69515e7	0.613312
$207$	4.72825e7	0.370514
$208$	8.99891e6	0.0693375
$209$	−2.48437e7	−0.188237
$210$	3.82593e7	0.285082
$211$	6.12082e7	0.448560	0.224280	−	0.974525i	$-0.427997\pi$
$211$	6.12082e7	0.448560	0.224280	+	0.974525i	$0.427997\pi$
$212$	−8.94401e6	−0.0644698
$213$	8.96188e7	0.635434
$214$	−4.36193e7	−0.304250
$215$	1.18184e8	0.811008
$216$	1.00777e7	0.0680414
$217$	4.33819e7	0.288203
$218$	−7.87729e7	−0.514967
$219$	−1.44337e8	−0.928592
$220$	4.86303e7	0.307913
$221$	3.19965e7	0.199402
$222$	2.61394e7	0.160347
$223$	−9.82487e7	−0.593280	−0.296640	−	0.954989i	$-0.595866\pi$
$223$	−9.82487e7	−0.593280	−0.296640	+	0.954989i	$0.595866\pi$
$224$	−1.12394e7	−0.0668153
$225$	1.37451e8	0.804467
$226$	−7.32276e7	−0.421983
$227$	1.36719e7	0.0775779	0.0387889	−	0.999247i	$-0.487650\pi$
$227$	1.36719e7	0.0775779	0.0387889	+	0.999247i	$0.487650\pi$
$228$	2.91757e7	0.163023
$229$	7.63779e7	0.420285	0.210142	−	0.977671i	$-0.432607\pi$
$229$	7.63779e7	0.420285	0.210142	+	0.977671i	$0.432607\pi$
$230$	−2.67949e8	−1.45213
$231$	1.36269e7	0.0727368
$232$	1.65816e7	0.0871801
$233$	−2.93560e8	−1.52038	−0.760188	−	0.649703i	$-0.774893\pi$
$233$	−2.93560e8	−1.52038	−0.760188	+	0.649703i	$0.774893\pi$
$234$	1.28129e7	0.0653720
$235$	2.85463e8	1.43487
$236$	−1.12628e8	−0.557767
$237$	−1.32496e8	−0.646525
$238$	−3.99628e7	−0.192148
$239$	1.33482e8	0.632457	0.316228	−	0.948683i	$-0.397583\pi$
$239$	1.33482e8	0.632457	0.316228	+	0.948683i	$0.397583\pi$
$240$	−5.71100e7	−0.266669
$241$	1.26943e7	0.0584183	0.0292092	−	0.999573i	$-0.490701\pi$
$241$	1.26943e7	0.0584183	0.0292092	+	0.999573i	$0.490701\pi$
$242$	−1.38577e8	−0.628545
$243$	1.43489e7	0.0641500
$244$	−2.01744e7	−0.0889070
$245$	−6.07543e7	−0.263934
$246$	−1.05524e8	−0.451938
$247$	3.70944e7	0.156628
$248$	−6.47566e7	−0.269590
$249$	−1.86265e7	−0.0764599
$250$	−4.56178e8	−1.84648
$251$	−3.77216e8	−1.50568	−0.752839	−	0.658205i	$-0.771316\pi$
$251$	−3.77216e8	−1.50568	−0.752839	+	0.658205i	$0.771316\pi$
$252$	−1.60030e7	−0.0629941
$253$	−9.54359e7	−0.370501
$254$	−8.69826e7	−0.333054
$255$	−2.03060e8	−0.766891
$256$	1.67772e7	0.0625000
$257$	−3.68438e8	−1.35394	−0.676969	−	0.736012i	$-0.736707\pi$
$257$	−3.68438e8	−1.35394	−0.676969	+	0.736012i	$0.736707\pi$
$258$	−4.94339e7	−0.179207
$259$	−4.15084e7	−0.148452
$260$	−7.26104e7	−0.256208
$261$	2.36093e7	0.0821942
$262$	−1.71249e8	−0.588266
$263$	4.67280e8	1.58392	0.791958	−	0.610576i	$-0.209062\pi$
$263$	4.67280e8	1.58392	0.791958	+	0.610576i	$0.209062\pi$
$264$	−2.03410e7	−0.0680391
$265$	7.21674e7	0.238221
$266$	−4.63300e7	−0.150930
$267$	−1.22889e8	−0.395116
$268$	−5.06575e7	−0.160758
$269$	−1.41244e8	−0.442422	−0.221211	−	0.975226i	$-0.571001\pi$
$269$	−1.41244e8	−0.442422	−0.221211	+	0.975226i	$0.571001\pi$
$270$	−8.13149e7	−0.251418
$271$	−5.22279e8	−1.59408	−0.797040	−	0.603926i	$-0.793602\pi$
$271$	−5.22279e8	−1.59408	−0.797040	+	0.603926i	$0.793602\pi$
$272$	5.96529e7	0.179738
$273$	−2.03464e7	−0.0605228
$274$	−2.88492e8	−0.847241
$275$	−2.77433e8	−0.804440
$276$	1.12077e8	0.320875
$277$	−7.81608e7	−0.220958	−0.110479	−	0.993878i	$-0.535238\pi$
$277$	−7.81608e7	−0.220958	−0.110479	+	0.993878i	$0.535238\pi$
$278$	3.17863e8	0.887326
$279$	−9.22023e7	−0.254172
$280$	9.06886e7	0.246888
$281$	5.92454e8	1.59288	0.796440	−	0.604718i	$-0.206714\pi$
$281$	5.92454e8	1.59288	0.796440	+	0.604718i	$0.206714\pi$
$282$	−1.19403e8	−0.317061
$283$	−4.74028e8	−1.24323	−0.621615	−	0.783323i	$-0.713523\pi$
$283$	−4.74028e8	−1.24323	−0.621615	+	0.783323i	$0.713523\pi$
$284$	2.12430e8	0.550302
$285$	−2.35413e8	−0.602384
$286$	−2.58618e7	−0.0653698
$287$	1.67568e8	0.418413
$288$	2.38879e7	0.0589256
$289$	−1.98237e8	−0.483106
$290$	−1.33793e8	−0.322137
$291$	1.60547e8	0.381923
$292$	−3.42133e8	−0.804184
$293$	−7.28785e8	−1.69263	−0.846316	−	0.532681i	$-0.821184\pi$
$293$	−7.28785e8	−1.69263	−0.846316	+	0.532681i	$0.821184\pi$
$294$	2.54122e7	0.0583212
$295$	9.08769e8	2.06099
$296$	6.19601e7	0.138865
$297$	−2.89621e7	−0.0641478
$298$	−1.44401e8	−0.316092
$299$	1.42496e8	0.308286
$300$	3.25809e8	0.696689
$301$	7.84992e7	0.165914
$302$	5.95357e7	0.124381
$303$	−1.45631e8	−0.300749
$304$	6.91573e7	0.141182
$305$	1.62783e8	0.328518
$306$	8.49355e7	0.169459
$307$	−4.52400e8	−0.892357	−0.446178	−	0.894944i	$-0.647215\pi$
$307$	−4.52400e8	−0.892357	−0.446178	+	0.894944i	$0.647215\pi$
$308$	3.23007e7	0.0629919
$309$	2.59711e8	0.500767
$310$	5.22508e8	0.996155
$311$	−3.11733e8	−0.587654	−0.293827	−	0.955859i	$-0.594929\pi$
$311$	−3.11733e8	−0.587654	−0.293827	+	0.955859i	$0.594929\pi$
$312$	3.03713e7	0.0566139
$313$	−2.67275e8	−0.492667	−0.246334	−	0.969185i	$-0.579226\pi$
$313$	−2.67275e8	−0.492667	−0.246334	+	0.969185i	$0.579226\pi$
$314$	3.06957e8	0.559531
$315$	1.29125e8	0.232768
$316$	−3.14066e8	−0.559907
$317$	4.74561e8	0.836728	0.418364	−	0.908279i	$-0.362604\pi$
$317$	4.74561e8	0.836728	0.418364	+	0.908279i	$0.362604\pi$
$318$	−3.01860e7	−0.0526394
$319$	−4.76534e7	−0.0821913
$320$	−1.35372e8	−0.230942
$321$	−1.47215e8	−0.248419
$322$	−1.77974e8	−0.297072
$323$	2.45895e8	0.406014
$324$	3.40122e7	0.0555556
$325$	4.14238e8	0.669357
$326$	1.62660e8	0.260028
$327$	−2.65859e8	−0.420469
$328$	−2.50131e8	−0.391390
$329$	1.89607e8	0.293542
$330$	1.64127e8	0.251410
$331$	−7.27641e8	−1.10286	−0.551428	−	0.834222i	$-0.685917\pi$
$331$	−7.27641e8	−1.10286	−0.551428	+	0.834222i	$0.685917\pi$
$332$	−4.41517e7	−0.0662162
$333$	8.82206e7	0.130923
$334$	−4.09763e8	−0.601755
$335$	4.08745e8	0.594013
$336$	−3.79331e7	−0.0545545
$337$	5.32192e8	0.757466	0.378733	−	0.925506i	$-0.376360\pi$
$337$	5.32192e8	0.757466	0.378733	+	0.925506i	$0.376360\pi$
$338$	3.86145e7	0.0543928
$339$	−2.47143e8	−0.344548
$340$	−4.81327e8	−0.664147
$341$	1.86103e8	0.254163
$342$	9.84681e7	0.133108
$343$	−4.03536e7	−0.0539949
$344$	−1.17177e8	−0.155198
$345$	−9.04328e8	−1.18566
$346$	6.58228e8	0.854300
$347$	−1.27560e9	−1.63893	−0.819466	−	0.573128i	$-0.805730\pi$
$347$	−1.27560e9	−1.63893	−0.819466	+	0.573128i	$0.805730\pi$
$348$	5.59627e7	0.0711822
$349$	−4.19065e8	−0.527706	−0.263853	−	0.964563i	$-0.584993\pi$
$349$	−4.19065e8	−0.527706	−0.263853	+	0.964563i	$0.584993\pi$
$350$	−5.17373e8	−0.645009
$351$	4.32436e7	0.0533761
$352$	−4.82157e7	−0.0589235
$353$	8.15330e8	0.986556	0.493278	−	0.869872i	$-0.335799\pi$
$353$	8.15330e8	0.986556	0.493278	+	0.869872i	$0.335799\pi$
$354$	−3.80118e8	−0.455415
$355$	−1.71405e9	−2.03341
$356$	−2.91293e8	−0.342181
$357$	−1.34874e8	−0.156888
$358$	2.30178e8	0.265139
$359$	9.32785e8	1.06402	0.532011	−	0.846737i	$-0.321436\pi$
$359$	9.32785e8	1.06402	0.532011	+	0.846737i	$0.321436\pi$
$360$	−1.92746e8	−0.217735
$361$	−6.08799e8	−0.681080
$362$	2.69898e8	0.299034
$363$	−4.67696e8	−0.513205
$364$	−4.82285e7	−0.0524142
$365$	2.76060e9	2.97152
$366$	−6.80885e7	−0.0725922
$367$	−4.86420e8	−0.513665	−0.256832	−	0.966456i	$-0.582679\pi$
$367$	−4.86420e8	−0.513665	−0.256832	+	0.966456i	$0.582679\pi$
$368$	2.65664e8	0.277886
$369$	−3.56144e8	−0.369006
$370$	−4.99944e8	−0.513115
$371$	4.79343e7	0.0487346
$372$	−2.18554e8	−0.220119
$373$	1.04915e9	1.04678	0.523392	−	0.852092i	$-0.324666\pi$
$373$	1.04915e9	1.04678	0.523392	+	0.852092i	$0.324666\pi$
$374$	−1.71435e8	−0.169453
$375$	−1.53960e9	−1.50764
$376$	−2.83029e8	−0.274583
$377$	7.11517e7	0.0683897
$378$	−5.40102e7	−0.0514344
$379$	−1.33603e8	−0.126061	−0.0630303	−	0.998012i	$-0.520076\pi$
$379$	−1.33603e8	−0.126061	−0.0630303	+	0.998012i	$0.520076\pi$
$380$	−5.58016e8	−0.521680
$381$	−2.93566e8	−0.271937
$382$	−1.77600e8	−0.163013
$383$	−1.49318e9	−1.35806	−0.679028	−	0.734112i	$-0.737598\pi$
$383$	−1.49318e9	−1.35806	−0.679028	+	0.734112i	$0.737598\pi$
$384$	5.66231e7	0.0510310
$385$	−2.60628e8	−0.232760
$386$	1.50464e9	1.33161
$387$	−1.66839e8	−0.146322
$388$	3.80555e8	0.330755
$389$	1.43828e7	0.0123886	0.00619428	−	0.999981i	$-0.498028\pi$
$389$	1.43828e7	0.0123886	0.00619428	+	0.999981i	$0.498028\pi$
$390$	−2.45060e8	−0.209193
$391$	9.44594e8	0.799147
$392$	6.02363e7	0.0505076
$393$	−5.77966e8	−0.480317
$394$	−1.55008e9	−1.27678
$395$	2.53413e9	2.06890
$396$	−6.86508e7	−0.0555537
$397$	1.71630e9	1.37666	0.688329	−	0.725399i	$-0.258345\pi$
$397$	1.71630e9	1.37666	0.688329	+	0.725399i	$0.258345\pi$
$398$	1.12643e9	0.895598
$399$	−1.56364e8	−0.123234
$400$	7.72289e8	0.603351
$401$	−2.41226e9	−1.86818	−0.934091	−	0.357034i	$-0.883788\pi$
$401$	−2.41226e9	−1.86818	−0.934091	+	0.357034i	$0.883788\pi$
$402$	−1.70969e8	−0.131258
$403$	−2.77872e8	−0.211483
$404$	−3.45199e8	−0.260456
$405$	−2.74438e8	−0.205282
$406$	−8.88667e7	−0.0659019
$407$	−1.78066e8	−0.130918
$408$	2.01329e8	0.146756
$409$	1.80504e9	1.30454	0.652268	−	0.757989i	$-0.273818\pi$
$409$	1.80504e9	1.30454	0.652268	+	0.757989i	$0.273818\pi$
$410$	2.01826e9	1.44622
$411$	−9.73661e8	−0.691770
$412$	6.15612e8	0.433677
$413$	6.03613e8	0.421632
$414$	3.78260e8	0.261993
$415$	3.56251e8	0.244674
$416$	7.19913e7	0.0490290
$417$	1.07279e9	0.724499
$418$	−1.98750e8	−0.133104
$419$	−2.25732e8	−0.149915	−0.0749574	−	0.997187i	$-0.523882\pi$
$419$	−2.25732e8	−0.149915	−0.0749574	+	0.997187i	$0.523882\pi$
$420$	3.06074e8	0.201583
$421$	−4.16774e8	−0.272216	−0.136108	−	0.990694i	$-0.543459\pi$
$421$	−4.16774e8	−0.272216	−0.136108	+	0.990694i	$0.543459\pi$
$422$	4.89665e8	0.317180
$423$	−4.02985e8	−0.258879
$424$	−7.15520e7	−0.0455871
$425$	2.74594e9	1.73512
$426$	7.16950e8	0.449320
$427$	1.08122e8	0.0672074
$428$	−3.48954e8	−0.215137
$429$	−8.72835e7	−0.0533742
$430$	9.45475e8	0.573469
$431$	−6.69778e8	−0.402958	−0.201479	−	0.979493i	$-0.564575\pi$
$431$	−6.69778e8	−0.402958	−0.201479	+	0.979493i	$0.564575\pi$
$432$	8.06216e7	0.0481125
$433$	−2.64084e9	−1.56327	−0.781635	−	0.623736i	$-0.785614\pi$
$433$	−2.64084e9	−1.56327	−0.781635	+	0.623736i	$0.785614\pi$
$434$	3.47055e8	0.203791
$435$	−4.51552e8	−0.263024
$436$	−6.30183e8	−0.364136
$437$	1.09509e9	0.627721
$438$	−1.15470e9	−0.656613
$439$	5.35810e8	0.302263	0.151132	−	0.988514i	$-0.451708\pi$
$439$	5.35810e8	0.302263	0.151132	+	0.988514i	$0.451708\pi$
$440$	3.89042e8	0.217727
$441$	8.57661e7	0.0476190
$442$	2.55972e8	0.140998
$443$	−2.84328e9	−1.55384	−0.776920	−	0.629599i	$-0.783219\pi$
$443$	−2.84328e9	−1.55384	−0.776920	+	0.629599i	$0.783219\pi$
$444$	2.09115e8	0.113382
$445$	2.35039e9	1.26438
$446$	−7.85989e8	−0.419512
$447$	−4.87354e8	−0.258088
$448$	−8.99154e7	−0.0472456
$449$	3.32803e9	1.73510	0.867551	−	0.497347i	$-0.165693\pi$
$449$	3.32803e9	1.73510	0.867551	+	0.497347i	$0.165693\pi$
$450$	1.09961e9	0.568844
$451$	7.18846e8	0.368993
$452$	−5.85821e8	−0.298387
$453$	2.00933e8	0.101556
$454$	1.09375e8	0.0548558
$455$	3.89146e8	0.193675
$456$	2.33406e8	0.115275
$457$	−1.89202e9	−0.927296	−0.463648	−	0.886020i	$-0.653460\pi$
$457$	−1.89202e9	−0.927296	−0.463648	+	0.886020i	$0.653460\pi$
$458$	6.11023e8	0.297186
$459$	2.86657e8	0.138363
$460$	−2.14359e9	−1.02681
$461$	−5.31966e8	−0.252890	−0.126445	−	0.991974i	$-0.540357\pi$
$461$	−5.31966e8	−0.252890	−0.126445	+	0.991974i	$0.540357\pi$
$462$	1.09015e8	0.0514327
$463$	−1.10994e9	−0.519716	−0.259858	−	0.965647i	$-0.583676\pi$
$463$	−1.10994e9	−0.519716	−0.259858	+	0.965647i	$0.583676\pi$
$464$	1.32652e8	0.0616456
$465$	1.76346e9	0.813357
$466$	−2.34848e9	−1.07507
$467$	−4.17172e9	−1.89542	−0.947712	−	0.319127i	$-0.896610\pi$
$467$	−4.17172e9	−1.89542	−0.947712	+	0.319127i	$0.896610\pi$
$468$	1.02503e8	0.0462250
$469$	2.71493e8	0.121522
$470$	2.28370e9	1.01461
$471$	1.03598e9	0.456855
$472$	−9.01020e8	−0.394401
$473$	3.36751e8	0.146317
$474$	−1.05997e9	−0.457162
$475$	3.18345e9	1.36292
$476$	−3.19702e8	−0.135869
$477$	−1.01878e8	−0.0429799
$478$	1.06786e9	0.447214
$479$	−2.42758e9	−1.00925	−0.504625	−	0.863339i	$-0.668369\pi$
$479$	−2.42758e9	−1.00925	−0.504625	+	0.863339i	$0.668369\pi$
$480$	−4.56880e8	−0.188564
$481$	2.65872e8	0.108934
$482$	1.01554e8	0.0413080
$483$	−6.00663e8	−0.242558
$484$	−1.10861e9	−0.444448
$485$	−3.07062e9	−1.22217
$486$	1.14791e8	0.0453609
$487$	3.33023e9	1.30654	0.653270	−	0.757125i	$-0.273397\pi$
$487$	3.33023e9	1.30654	0.653270	+	0.757125i	$0.273397\pi$
$488$	−1.61395e8	−0.0628667
$489$	5.48978e8	0.212312
$490$	−4.86034e8	−0.186630
$491$	4.05374e9	1.54550	0.772752	−	0.634708i	$-0.218880\pi$
$491$	4.05374e9	1.54550	0.772752	+	0.634708i	$0.218880\pi$
$492$	−8.44193e8	−0.319568
$493$	4.71658e8	0.177281
$494$	2.96755e8	0.110753
$495$	5.53930e8	0.205275
$496$	−5.18053e8	−0.190629
$497$	−1.13849e9	−0.415989
$498$	−1.49012e8	−0.0540653
$499$	−4.75453e9	−1.71299	−0.856496	−	0.516154i	$-0.827363\pi$
$499$	−4.75453e9	−1.71299	−0.856496	+	0.516154i	$0.827363\pi$
$500$	−3.64943e9	−1.30566
$501$	−1.38295e9	−0.491331
$502$	−3.01773e9	−1.06467
$503$	−7.64636e8	−0.267896	−0.133948	−	0.990988i	$-0.542766\pi$
$503$	−7.64636e8	−0.267896	−0.133948	+	0.990988i	$0.542766\pi$
$504$	−1.28024e8	−0.0445435
$505$	2.78534e9	0.962405
$506$	−7.63487e8	−0.261984
$507$	1.30324e8	0.0444116
$508$	−6.95861e8	−0.235505
$509$	2.58118e9	0.867573	0.433787	−	0.901016i	$-0.357177\pi$
$509$	2.58118e9	0.867573	0.433787	+	0.901016i	$0.357177\pi$
$510$	−1.62448e9	−0.542274
$511$	1.83362e9	0.607906
$512$	1.34218e8	0.0441942
$513$	3.32330e8	0.108682
$514$	−2.94750e9	−0.957378
$515$	−4.96725e9	−1.60247
$516$	−3.95471e8	−0.126719
$517$	8.13391e8	0.258870
$518$	−3.32068e8	−0.104972
$519$	2.22152e9	0.697533
$520$	−5.80883e8	−0.181166
$521$	1.42264e9	0.440721	0.220361	−	0.975418i	$-0.429277\pi$
$521$	1.42264e9	0.440721	0.220361	+	0.975418i	$0.429277\pi$
$522$	1.88874e8	0.0581200
$523$	−2.51651e9	−0.769208	−0.384604	−	0.923082i	$-0.625662\pi$
$523$	−2.51651e9	−0.769208	−0.384604	+	0.923082i	$0.625662\pi$
$524$	−1.36999e9	−0.415967
$525$	−1.74613e9	−0.526648
$526$	3.73824e9	1.12000
$527$	−1.84198e9	−0.548212
$528$	−1.62728e8	−0.0481109
$529$	8.01924e8	0.235526
$530$	5.77339e8	0.168448
$531$	−1.28290e9	−0.371844
$532$	−3.70640e8	−0.106724
$533$	−1.07332e9	−0.307031
$534$	−9.83115e8	−0.279389
$535$	2.81564e9	0.794948
$536$	−4.05260e8	−0.113673
$537$	7.76851e8	0.216485
$538$	−1.12995e9	−0.312840
$539$	−1.73112e8	−0.0476174
$540$	−6.50519e8	−0.177780
$541$	−1.69431e9	−0.460047	−0.230023	−	0.973185i	$-0.573880\pi$
$541$	−1.69431e9	−0.460047	−0.230023	+	0.973185i	$0.573880\pi$
$542$	−4.17823e9	−1.12719
$543$	9.10907e8	0.244160
$544$	4.77224e8	0.127094
$545$	5.08482e9	1.34551
$546$	−1.62771e8	−0.0427960
$547$	1.50857e9	0.394102	0.197051	−	0.980393i	$-0.436864\pi$
$547$	1.50857e9	0.394102	0.197051	+	0.980393i	$0.436864\pi$
$548$	−2.30794e9	−0.599090
$549$	−2.29799e8	−0.0592713
$550$	−2.21946e9	−0.568825
$551$	5.46806e8	0.139252
$552$	8.96617e8	0.226893
$553$	1.68320e9	0.423250
$554$	−6.25286e8	−0.156241
$555$	−1.68731e9	−0.418957
$556$	2.54290e9	0.627434
$557$	−5.30475e9	−1.30068	−0.650342	−	0.759641i	$-0.725375\pi$
$557$	−5.30475e9	−1.30068	−0.650342	+	0.759641i	$0.725375\pi$
$558$	−7.37618e8	−0.179726
$559$	−5.02807e8	−0.121747
$560$	7.25509e8	0.174576
$561$	−5.78594e8	−0.138358
$562$	4.73964e9	1.12634
$563$	−3.76471e9	−0.889103	−0.444552	−	0.895753i	$-0.646637\pi$
$563$	−3.76471e9	−0.889103	−0.444552	+	0.895753i	$0.646637\pi$
$564$	−9.55223e8	−0.224196
$565$	4.72687e9	1.10256
$566$	−3.79222e9	−0.879096
$567$	−1.82284e8	−0.0419961
$568$	1.69944e9	0.389122
$569$	1.67981e9	0.382266	0.191133	−	0.981564i	$-0.438784\pi$
$569$	1.67981e9	0.382266	0.191133	+	0.981564i	$0.438784\pi$
$570$	−1.88331e9	−0.425950
$571$	6.99441e9	1.57226	0.786131	−	0.618060i	$-0.212081\pi$
$571$	6.99441e9	1.57226	0.786131	+	0.618060i	$0.212081\pi$
$572$	−2.06894e8	−0.0462234
$573$	−5.99401e8	−0.133099
$574$	1.34055e9	0.295863
$575$	1.22291e10	2.68260
$576$	1.91103e8	0.0416667
$577$	3.49967e9	0.758423	0.379211	−	0.925310i	$-0.376195\pi$
$577$	3.49967e9	0.758423	0.379211	+	0.925310i	$0.376195\pi$
$578$	−1.58590e9	−0.341608
$579$	5.07818e9	1.08726
$580$	−1.07035e9	−0.227785
$581$	2.36626e8	0.0500548
$582$	1.28437e9	0.270060
$583$	2.05632e8	0.0429784
$584$	−2.73707e9	−0.568644
$585$	−8.27078e8	−0.170805
$586$	−5.83028e9	−1.19687
$587$	2.96654e9	0.605363	0.302682	−	0.953092i	$-0.402118\pi$
$587$	2.96654e9	0.605363	0.302682	+	0.953092i	$0.402118\pi$
$588$	2.03297e8	0.0412393
$589$	−2.13546e9	−0.430615
$590$	7.27015e9	1.45734
$591$	−5.23152e9	−1.04249
$592$	4.95681e8	0.0981921
$593$	1.03690e9	0.204194	0.102097	−	0.994774i	$-0.467445\pi$
$593$	1.03690e9	0.204194	0.102097	+	0.994774i	$0.467445\pi$
$594$	−2.31697e8	−0.0453594
$595$	2.57961e9	0.502048
$596$	−1.15521e9	−0.223511
$597$	3.80170e9	0.731253
$598$	1.13997e9	0.217991
$599$	−9.97018e7	−0.0189544	−0.00947718	−	0.999955i	$-0.503017\pi$
$599$	−9.97018e7	−0.0189544	−0.00947718	+	0.999955i	$0.503017\pi$
$600$	2.60647e9	0.492634
$601$	−3.53369e9	−0.664000	−0.332000	−	0.943279i	$-0.607723\pi$
$601$	−3.53369e9	−0.664000	−0.332000	+	0.943279i	$0.607723\pi$
$602$	6.27994e8	0.117319
$603$	−5.77021e8	−0.107172
$604$	4.76286e8	0.0879505
$605$	8.94517e9	1.64227
$606$	−1.16504e9	−0.212661
$607$	−1.38816e9	−0.251929	−0.125964	−	0.992035i	$-0.540203\pi$
$607$	−1.38816e9	−0.251929	−0.125964	+	0.992035i	$0.540203\pi$
$608$	5.53259e8	0.0998310
$609$	−2.99925e8	−0.0538087
$610$	1.30226e9	0.232298
$611$	−1.21448e9	−0.215401
$612$	6.79484e8	0.119826
$613$	7.02191e9	1.23124	0.615622	−	0.788042i	$-0.288905\pi$
$613$	7.02191e9	1.23124	0.615622	+	0.788042i	$0.288905\pi$
$614$	−3.61920e9	−0.630991
$615$	6.81162e9	1.18083
$616$	2.58406e8	0.0445420
$617$	−4.83999e9	−0.829557	−0.414779	−	0.909922i	$-0.636141\pi$
$617$	−4.83999e9	−0.829557	−0.414779	+	0.909922i	$0.636141\pi$
$618$	2.07769e9	0.354096
$619$	4.72038e9	0.799945	0.399972	−	0.916527i	$-0.369020\pi$
$619$	4.72038e9	0.799945	0.399972	+	0.916527i	$0.369020\pi$
$620$	4.18006e9	0.704388
$621$	1.27663e9	0.213916
$622$	−2.49386e9	−0.415534
$623$	1.56115e9	0.258664
$624$	2.42971e8	0.0400320
$625$	1.47162e10	2.41111
$626$	−2.13820e9	−0.348368
$627$	−6.70780e8	−0.108679
$628$	2.45566e9	0.395648
$629$	1.76244e9	0.282382
$630$	1.03300e9	0.164592
$631$	−4.52977e8	−0.0717750	−0.0358875	−	0.999356i	$-0.511426\pi$
$631$	−4.52977e8	−0.0717750	−0.0358875	+	0.999356i	$0.511426\pi$
$632$	−2.51253e9	−0.395914
$633$	1.65262e9	0.258976
$634$	3.79649e9	0.591656
$635$	5.61476e9	0.870209
$636$	−2.41488e8	−0.0372217
$637$	2.58475e8	0.0396214
$638$	−3.81227e8	−0.0581181
$639$	2.41971e9	0.366868
$640$	−1.08298e9	−0.163301
$641$	6.93374e8	0.103983	0.0519917	−	0.998648i	$-0.483443\pi$
$641$	6.93374e8	0.103983	0.0519917	+	0.998648i	$0.483443\pi$
$642$	−1.17772e9	−0.175659
$643$	−1.12837e10	−1.67384	−0.836918	−	0.547329i	$-0.815645\pi$
$643$	−1.12837e10	−1.67384	−0.836918	+	0.547329i	$0.815645\pi$
$644$	−1.42379e9	−0.210062
$645$	3.19098e9	0.468236
$646$	1.96716e9	0.287095
$647$	9.48336e9	1.37657	0.688283	−	0.725442i	$-0.258365\pi$
$647$	9.48336e9	1.37657	0.688283	+	0.725442i	$0.258365\pi$
$648$	2.72098e8	0.0392837
$649$	2.58942e9	0.371832
$650$	3.31390e9	0.473307
$651$	1.17131e9	0.166394
$652$	1.30128e9	0.183867
$653$	1.15311e10	1.62060	0.810300	−	0.586016i	$-0.199304\pi$
$653$	1.15311e10	1.62060	0.810300	+	0.586016i	$0.199304\pi$
$654$	−2.12687e9	−0.297316
$655$	1.10542e10	1.53703
$656$	−2.00105e9	−0.276754
$657$	−3.89711e9	−0.536123
$658$	1.51686e9	0.207565
$659$	−6.26341e9	−0.852535	−0.426267	−	0.904597i	$-0.640172\pi$
$659$	−6.26341e9	−0.852535	−0.426267	+	0.904597i	$0.640172\pi$
$660$	1.31302e9	0.177773
$661$	2.97321e9	0.400424	0.200212	−	0.979753i	$-0.435837\pi$
$661$	2.97321e9	0.400424	0.200212	+	0.979753i	$0.435837\pi$
$662$	−5.82113e9	−0.779837
$663$	8.63905e8	0.115125
$664$	−3.53214e8	−0.0468220
$665$	2.99062e9	0.394353
$666$	7.05765e8	0.0925764
$667$	2.10053e9	0.274087
$668$	−3.27810e9	−0.425505
$669$	−2.65271e9	−0.342530
$670$	3.26996e9	0.420031
$671$	4.63830e8	0.0592693
$672$	−3.03464e8	−0.0385758
$673$	8.10354e9	1.02476	0.512380	−	0.858759i	$-0.328764\pi$
$673$	8.10354e9	1.02476	0.512380	+	0.858759i	$0.328764\pi$
$674$	4.25753e9	0.535610
$675$	3.71117e9	0.464459
$676$	3.08916e8	0.0384615
$677$	−1.96307e9	−0.243150	−0.121575	−	0.992582i	$-0.538795\pi$
$677$	−1.96307e9	−0.243150	−0.121575	+	0.992582i	$0.538795\pi$
$678$	−1.97715e9	−0.243632
$679$	−2.03954e9	−0.250027
$680$	−3.85062e9	−0.469623
$681$	3.69141e8	0.0447896
$682$	1.48882e9	0.179720
$683$	7.66062e9	0.920008	0.460004	−	0.887917i	$-0.347848\pi$
$683$	7.66062e9	0.920008	0.460004	+	0.887917i	$0.347848\pi$
$684$	7.87745e8	0.0941216
$685$	1.86223e10	2.21369
$686$	−3.22829e8	−0.0381802
$687$	2.06220e9	0.242651
$688$	−9.37413e8	−0.109742
$689$	−3.07031e8	−0.0357614
$690$	−7.23462e9	−0.838386
$691$	5.05861e9	0.583255	0.291627	−	0.956532i	$-0.405803\pi$
$691$	5.05861e9	0.583255	0.291627	+	0.956532i	$0.405803\pi$
$692$	5.26583e9	0.604081
$693$	3.67926e8	0.0419946
$694$	−1.02048e10	−1.15890
$695$	−2.05182e10	−2.31842
$696$	4.47702e8	0.0503334
$697$	−7.11492e9	−0.795894
$698$	−3.35252e9	−0.373145
$699$	−7.92612e9	−0.877790
$700$	−4.13898e9	−0.456090
$701$	1.07307e9	0.117657	0.0588284	−	0.998268i	$-0.481264\pi$
$701$	1.07307e9	0.117657	0.0588284	+	0.998268i	$0.481264\pi$
$702$	3.45948e8	0.0377426
$703$	2.04325e9	0.221808
$704$	−3.85725e8	−0.0416652
$705$	7.70750e9	0.828422
$706$	6.52264e9	0.697601
$707$	1.85005e9	0.196886
$708$	−3.04094e9	−0.322027
$709$	1.67237e10	1.76226	0.881129	−	0.472875i	$-0.156784\pi$
$709$	1.67237e10	1.76226	0.881129	+	0.472875i	$0.156784\pi$
$710$	−1.37124e10	−1.43784
$711$	−3.57740e9	−0.373271
$712$	−2.33035e9	−0.241958
$713$	−8.20328e9	−0.847567
$714$	−1.07900e9	−0.110937
$715$	1.66939e9	0.170799
$716$	1.84143e9	0.187482
$717$	3.60402e9	0.365149
$718$	7.46228e9	0.752378
$719$	−1.12950e10	−1.13327	−0.566637	−	0.823968i	$-0.691756\pi$
$719$	−1.12950e10	−1.13327	−0.566637	+	0.823968i	$0.691756\pi$
$720$	−1.54197e9	−0.153962
$721$	−3.29929e9	−0.327829
$722$	−4.87039e9	−0.481597
$723$	3.42746e8	0.0337278
$724$	2.15919e9	0.211449
$725$	6.10625e9	0.595103
$726$	−3.74157e9	−0.362890
$727$	1.21172e10	1.16958	0.584792	−	0.811183i	$-0.301176\pi$
$727$	1.21172e10	1.16958	0.584792	+	0.811183i	$0.301176\pi$
$728$	−3.85828e8	−0.0370625
$729$	3.87420e8	0.0370370
$730$	2.20848e10	2.10118
$731$	−3.33306e9	−0.315597
$732$	−5.44708e8	−0.0513305
$733$	1.15331e10	1.08164	0.540820	−	0.841138i	$-0.318114\pi$
$733$	1.15331e10	1.08164	0.540820	+	0.841138i	$0.318114\pi$
$734$	−3.89136e9	−0.363216
$735$	−1.64037e9	−0.152382
$736$	2.12531e9	0.196495
$737$	1.16467e9	0.107168
$738$	−2.84915e9	−0.260926
$739$	−5.39886e9	−0.492092	−0.246046	−	0.969258i	$-0.579131\pi$
$739$	−5.39886e9	−0.492092	−0.246046	+	0.969258i	$0.579131\pi$
$740$	−3.99955e9	−0.362827
$741$	1.00155e9	0.0904291
$742$	3.83474e8	0.0344606
$743$	6.90501e9	0.617594	0.308797	−	0.951128i	$-0.400074\pi$
$743$	6.90501e9	0.617594	0.308797	+	0.951128i	$0.400074\pi$
$744$	−1.74843e9	−0.155648
$745$	9.32116e9	0.825891
$746$	8.39321e9	0.740189
$747$	−5.02916e8	−0.0441442
$748$	−1.37148e9	−0.119821
$749$	1.87018e9	0.162628
$750$	−1.23168e10	−1.06607
$751$	8.82226e9	0.760046	0.380023	−	0.924977i	$-0.375916\pi$
$751$	8.82226e9	0.760046	0.380023	+	0.924977i	$0.375916\pi$
$752$	−2.26423e9	−0.194159
$753$	−1.01848e10	−0.869303
$754$	5.69214e8	0.0483588
$755$	−3.84305e9	−0.324984
$756$	−4.32081e8	−0.0363696
$757$	−9.08213e9	−0.760943	−0.380472	−	0.924793i	$-0.624238\pi$
$757$	−9.08213e9	−0.760943	−0.380472	+	0.924793i	$0.624238\pi$
$758$	−1.06882e9	−0.0891383
$759$	−2.57677e9	−0.213909
$760$	−4.46413e9	−0.368884
$761$	−2.05660e9	−0.169163	−0.0845813	−	0.996417i	$-0.526955\pi$
$761$	−2.05660e9	−0.169163	−0.0845813	+	0.996417i	$0.526955\pi$
$762$	−2.34853e9	−0.192289
$763$	3.37739e9	0.275261
$764$	−1.42080e9	−0.115267
$765$	−5.48262e9	−0.442765
$766$	−1.19455e10	−0.960290
$767$	−3.86629e9	−0.309393
$768$	4.52985e8	0.0360844
$769$	3.50458e9	0.277903	0.138952	−	0.990299i	$-0.455627\pi$
$769$	3.50458e9	0.277903	0.138952	+	0.990299i	$0.455627\pi$
$770$	−2.08502e9	−0.164586
$771$	−9.94783e9	−0.781696
$772$	1.20372e10	0.941593
$773$	−2.08982e10	−1.62735	−0.813673	−	0.581323i	$-0.802535\pi$
$773$	−2.08982e10	−1.62735	−0.813673	+	0.581323i	$0.802535\pi$
$774$	−1.33472e9	−0.103465
$775$	−2.38470e10	−1.84025
$776$	3.04444e9	0.233879
$777$	−1.12073e9	−0.0857090
$778$	1.15063e8	0.00876003
$779$	−8.24852e9	−0.625165
$780$	−1.96048e9	−0.147922
$781$	−4.88398e9	−0.366856
$782$	7.55675e9	0.565082
$783$	6.37450e8	0.0474548
$784$	4.81890e8	0.0357143
$785$	−1.98142e10	−1.46195
$786$	−4.62373e9	−0.339636
$787$	−1.52566e10	−1.11570	−0.557850	−	0.829942i	$-0.688374\pi$
$787$	−1.52566e10	−1.11570	−0.557850	+	0.829942i	$0.688374\pi$
$788$	−1.24006e10	−0.902822
$789$	1.26166e10	0.914474
$790$	2.02731e10	1.46293
$791$	3.13963e9	0.225560
$792$	−5.49207e8	−0.0392824
$793$	−6.92549e8	−0.0493167
$794$	1.37304e10	0.973444
$795$	1.94852e9	0.137537
$796$	9.01143e9	0.633283
$797$	−2.46279e10	−1.72315	−0.861576	−	0.507629i	$-0.830522\pi$
$797$	−2.46279e10	−1.72315	−0.861576	+	0.507629i	$0.830522\pi$
$798$	−1.25091e9	−0.0871397
$799$	−8.05069e9	−0.558366
$800$	6.17831e9	0.426633
$801$	−3.31801e9	−0.228120
$802$	−1.92981e10	−1.32100
$803$	7.86600e9	0.536104
$804$	−1.36775e9	−0.0928136
$805$	1.14883e10	0.776195
$806$	−2.22297e9	−0.149541
$807$	−3.81359e9	−0.255433
$808$	−2.76159e9	−0.184170
$809$	2.89844e10	1.92461	0.962307	−	0.271964i	$-0.0876733\pi$
$809$	2.89844e10	1.92461	0.962307	+	0.271964i	$0.0876733\pi$
$810$	−2.19550e9	−0.145156
$811$	−9.03805e9	−0.594979	−0.297489	−	0.954725i	$-0.596149\pi$
$811$	−9.03805e9	−0.594979	−0.297489	+	0.954725i	$0.596149\pi$
$812$	−7.10934e8	−0.0465997
$813$	−1.41015e10	−0.920343
$814$	−1.42453e9	−0.0925732
$815$	−1.04998e10	−0.679404
$816$	1.61063e9	0.103772
$817$	−3.86411e9	−0.247897
$818$	1.44403e10	0.922446
$819$	−5.49353e8	−0.0349428
$820$	1.61461e10	1.02263
$821$	8.81176e9	0.555727	0.277864	−	0.960621i	$-0.410374\pi$
$821$	8.81176e9	0.555727	0.277864	+	0.960621i	$0.410374\pi$
$822$	−7.78929e9	−0.489155
$823$	−1.47730e10	−0.923781	−0.461891	−	0.886937i	$-0.652829\pi$
$823$	−1.47730e10	−0.923781	−0.461891	+	0.886937i	$0.652829\pi$
$824$	4.92489e9	0.306656
$825$	−7.49069e9	−0.464444
$826$	4.82891e9	0.298139
$827$	2.13992e10	1.31561	0.657806	−	0.753187i	$-0.271485\pi$
$827$	2.13992e10	1.31561	0.657806	+	0.753187i	$0.271485\pi$
$828$	3.02608e9	0.185257
$829$	4.84592e9	0.295417	0.147708	−	0.989031i	$-0.452810\pi$
$829$	4.84592e9	0.295417	0.147708	+	0.989031i	$0.452810\pi$
$830$	2.85001e9	0.173011
$831$	−2.11034e9	−0.127570
$832$	5.75930e8	0.0346688
$833$	1.71341e9	0.102708
$834$	8.58230e9	0.512298
$835$	2.64503e10	1.57228
$836$	−1.59000e9	−0.0941184
$837$	−2.48946e9	−0.146746
$838$	−1.80586e9	−0.106006
$839$	7.67077e9	0.448407	0.224203	−	0.974542i	$-0.428022\pi$
$839$	7.67077e9	0.448407	0.224203	+	0.974542i	$0.428022\pi$
$840$	2.44859e9	0.142541
$841$	−1.62010e10	−0.939197
$842$	−3.33419e9	−0.192485
$843$	1.59963e10	0.919649
$844$	3.91732e9	0.224280
$845$	−2.49258e9	−0.142118
$846$	−3.22388e9	−0.183055
$847$	5.94147e9	0.335971
$848$	−5.72416e8	−0.0322349
$849$	−1.27988e10	−0.717779
$850$	2.19676e10	1.22692
$851$	7.84902e9	0.436578
$852$	5.73560e9	0.317717
$853$	−1.07715e10	−0.594229	−0.297114	−	0.954842i	$-0.596024\pi$
$853$	−1.07715e10	−0.594229	−0.297114	+	0.954842i	$0.596024\pi$
$854$	8.64977e8	0.0475228
$855$	−6.35616e9	−0.347787
$856$	−2.79163e9	−0.152125
$857$	−1.05105e10	−0.570415	−0.285207	−	0.958466i	$-0.592062\pi$
$857$	−1.05105e10	−0.570415	−0.285207	+	0.958466i	$0.592062\pi$
$858$	−6.98268e8	−0.0377413
$859$	1.30325e9	0.0701537	0.0350768	−	0.999385i	$-0.488832\pi$
$859$	1.30325e9	0.0701537	0.0350768	+	0.999385i	$0.488832\pi$
$860$	7.56380e9	0.405504
$861$	4.52434e9	0.241571
$862$	−5.35822e9	−0.284935
$863$	−1.42548e10	−0.754961	−0.377480	−	0.926018i	$-0.623209\pi$
$863$	−1.42548e10	−0.754961	−0.377480	+	0.926018i	$0.623209\pi$
$864$	6.44973e8	0.0340207
$865$	−4.24889e10	−2.23213
$866$	−2.11267e10	−1.10540
$867$	−5.35240e9	−0.278921
$868$	2.77644e9	0.144102
$869$	7.22069e9	0.373258
$870$	−3.61242e9	−0.185986
$871$	−1.73898e9	−0.0891724
$872$	−5.04147e9	−0.257483
$873$	4.33476e9	0.220503
$874$	8.76075e9	0.443866
$875$	1.95586e10	0.986985
$876$	−9.23760e9	−0.464296
$877$	1.16163e10	0.581529	0.290764	−	0.956795i	$-0.406090\pi$
$877$	1.16163e10	0.581529	0.290764	+	0.956795i	$0.406090\pi$
$878$	4.28648e9	0.213732
$879$	−1.96772e10	−0.977241
$880$	3.11234e9	0.153956
$881$	−3.03148e9	−0.149362	−0.0746808	−	0.997207i	$-0.523794\pi$
$881$	−3.03148e9	−0.149362	−0.0746808	+	0.997207i	$0.523794\pi$
$882$	6.86129e8	0.0336718
$883$	1.95378e10	0.955021	0.477511	−	0.878626i	$-0.341539\pi$
$883$	1.95378e10	0.955021	0.477511	+	0.878626i	$0.341539\pi$
$884$	2.04777e9	0.0997009
$885$	2.45368e10	1.18991
$886$	−2.27462e10	−1.09873
$887$	5.10399e9	0.245571	0.122785	−	0.992433i	$-0.460817\pi$
$887$	5.10399e9	0.245571	0.122785	+	0.992433i	$0.460817\pi$
$888$	1.67292e9	0.0801735
$889$	3.72938e9	0.178025
$890$	1.88031e10	0.894055
$891$	−7.81976e8	−0.0370358
$892$	−6.28792e9	−0.296640
$893$	−9.33339e9	−0.438591
$894$	−3.89883e9	−0.182496
$895$	−1.48581e10	−0.692759
$896$	−7.19323e8	−0.0334077
$897$	3.84740e9	0.177989
$898$	2.66243e10	1.22690
$899$	−4.09609e9	−0.188023
$900$	8.79685e9	0.402234
$901$	−2.03528e9	−0.0927016
$902$	5.75077e9	0.260917
$903$	2.11948e9	0.0957904
$904$	−4.68657e9	−0.210992
$905$	−1.74220e10	−0.781320
$906$	1.60746e9	0.0718113
$907$	1.63787e10	0.728876	0.364438	−	0.931228i	$-0.381261\pi$
$907$	1.63787e10	0.728876	0.364438	+	0.931228i	$0.381261\pi$
$908$	8.75001e8	0.0387889
$909$	−3.93203e9	−0.173637
$910$	3.11317e9	0.136949
$911$	2.59951e10	1.13914	0.569569	−	0.821943i	$-0.307110\pi$
$911$	2.59951e10	1.13914	0.569569	+	0.821943i	$0.307110\pi$
$912$	1.86725e9	0.0815117
$913$	1.01509e9	0.0441427
$914$	−1.51361e10	−0.655697
$915$	4.39514e9	0.189670
$916$	4.88818e9	0.210142
$917$	7.34231e9	0.314442
$918$	2.29326e9	0.0978372
$919$	2.90735e8	0.0123564	0.00617822	−	0.999981i	$-0.498033\pi$
$919$	2.90735e8	0.0123564	0.00617822	+	0.999981i	$0.498033\pi$
$920$	−1.71487e10	−0.726064
$921$	−1.22148e10	−0.515202
$922$	−4.25573e9	−0.178820
$923$	7.29232e9	0.305253
$924$	8.72120e8	0.0363684
$925$	2.28172e10	0.947908
$926$	−8.87953e9	−0.367495
$927$	7.01220e9	0.289118
$928$	1.06122e9	0.0435900
$929$	−6.45086e9	−0.263975	−0.131987	−	0.991251i	$-0.542136\pi$
$929$	−6.45086e9	−0.263975	−0.131987	+	0.991251i	$0.542136\pi$
$930$	1.41077e10	0.575130
$931$	1.98640e9	0.0806757
$932$	−1.87878e10	−0.760188
$933$	−8.41679e9	−0.339282
$934$	−3.33738e10	−1.34027
$935$	1.10662e10	0.442750
$936$	8.20026e8	0.0326860
$937$	3.07845e10	1.22249	0.611243	−	0.791443i	$-0.290670\pi$
$937$	3.07845e10	1.22249	0.611243	+	0.791443i	$0.290670\pi$
$938$	2.17194e9	0.0859287
$939$	−7.21643e9	−0.284441
$940$	1.82696e10	0.717435
$941$	−3.70287e10	−1.44869	−0.724343	−	0.689439i	$-0.757857\pi$
$941$	−3.70287e10	−1.44869	−0.724343	+	0.689439i	$0.757857\pi$
$942$	8.28784e9	0.323045
$943$	−3.16863e10	−1.23050
$944$	−7.20816e9	−0.278883
$945$	3.48638e9	0.134389
$946$	2.69401e9	0.103462
$947$	−1.60901e9	−0.0615648	−0.0307824	−	0.999526i	$-0.509800\pi$
$947$	−1.60901e9	−0.0615648	−0.0307824	+	0.999526i	$0.509800\pi$
$948$	−8.47977e9	−0.323262
$949$	−1.17448e10	−0.446081
$950$	2.54676e10	0.963730
$951$	1.28131e10	0.483085
$952$	−2.55762e9	−0.0960742
$953$	−6.33867e9	−0.237232	−0.118616	−	0.992940i	$-0.537846\pi$
$953$	−6.33867e9	−0.237232	−0.118616	+	0.992940i	$0.537846\pi$
$954$	−8.15022e8	−0.0303914
$955$	1.14642e10	0.425922
$956$	8.54286e9	0.316228
$957$	−1.28664e9	−0.0474532
$958$	−1.94206e10	−0.713648
$959$	1.23691e10	0.452870
$960$	−3.65504e9	−0.133335
$961$	−1.15160e10	−0.418572
$962$	2.12698e9	0.0770282
$963$	−3.97480e9	−0.143425
$964$	8.12435e8	0.0292092
$965$	−9.71254e10	−3.47926
$966$	−4.80531e9	−0.171515
$967$	−2.49228e10	−0.886349	−0.443175	−	0.896435i	$-0.646148\pi$
$967$	−2.49228e10	−0.886349	−0.443175	+	0.896435i	$0.646148\pi$
$968$	−8.86890e9	−0.314272
$969$	6.63917e9	0.234412
$970$	−2.45650e10	−0.864202
$971$	1.74444e10	0.611490	0.305745	−	0.952113i	$-0.401095\pi$
$971$	1.74444e10	0.611490	0.305745	+	0.952113i	$0.401095\pi$
$972$	9.18330e8	0.0320750
$973$	−1.36284e10	−0.474296
$974$	2.66418e10	0.923863
$975$	1.11844e10	0.386454
$976$	−1.29116e9	−0.0444535
$977$	−2.31493e9	−0.0794159	−0.0397080	−	0.999211i	$-0.512643\pi$
$977$	−2.31493e9	−0.0794159	−0.0397080	+	0.999211i	$0.512643\pi$
$978$	4.39182e9	0.150127
$979$	6.69713e9	0.228113
$980$	−3.88827e9	−0.131967
$981$	−7.17818e9	−0.242758
$982$	3.24299e10	1.09284
$983$	2.29448e10	0.770453	0.385227	−	0.922822i	$-0.374123\pi$
$983$	2.29448e10	0.770453	0.385227	+	0.922822i	$0.374123\pi$
$984$	−6.75354e9	−0.225969
$985$	1.00058e11	3.33600
$986$	3.77326e9	0.125357
$987$	5.11940e9	0.169476
$988$	2.37404e9	0.0783139
$989$	−1.48438e10	−0.487930
$990$	4.43144e9	0.145151
$991$	3.43651e10	1.12166	0.560828	−	0.827932i	$-0.310483\pi$
$991$	3.43651e10	1.12166	0.560828	+	0.827932i	$0.310483\pi$
$992$	−4.14442e9	−0.134795
$993$	−1.96463e10	−0.636735
$994$	−9.10793e9	−0.294149
$995$	−7.27114e10	−2.34003
$996$	−1.19210e9	−0.0382300
$997$	3.79087e10	1.21145	0.605725	−	0.795674i	$-0.292883\pi$
$997$	3.79087e10	1.21145	0.605725	+	0.795674i	$0.292883\pi$
$998$	−3.80362e10	−1.21127
$999$	2.38196e9	0.0755883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.e.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.e.1.1	✓	3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -516.403 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -516.403 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -4131.22 q^{10} -1471.43 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -13942.9 q^{15} +4096.00 q^{16} +14563.7 q^{17} +5832.00 q^{18} +16884.1 q^{19} -33049.8 q^{20} -9261.00 q^{21} -11771.4 q^{22} +64859.5 q^{23} +13824.0 q^{24} +188547. q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +32385.8 q^{29} -111543. q^{30} -126478. q^{31} +32768.0 q^{32} -39728.5 q^{33} +116510. q^{34} +177126. q^{35} +46656.0 q^{36} +121016. q^{37} +135073. q^{38} +59319.0 q^{39} -264398. q^{40} -488537. q^{41} -74088.0 q^{42} -228861. q^{43} -94171.2 q^{44} -376458. q^{45} +518876. q^{46} -552791. q^{47} +110592. q^{48} +117649. q^{49} +1.50838e6 q^{50} +393220. q^{51} +140608. q^{52} -139750. q^{53} +157464. q^{54} +759849. q^{55} -175616. q^{56} +455871. q^{57} +259087. q^{58} -1.75981e6 q^{59} -892344. q^{60} -315225. q^{61} -1.01182e6 q^{62} -250047. q^{63} +262144. q^{64} -1.13454e6 q^{65} -317828. q^{66} -791524. q^{67} +932077. q^{68} +1.75121e6 q^{69} +1.41701e6 q^{70} +3.31922e6 q^{71} +373248. q^{72} -5.34583e6 q^{73} +968127. q^{74} +5.09077e6 q^{75} +1.08058e6 q^{76} +504699. q^{77} +474552. q^{78} -4.90728e6 q^{79} -2.11519e6 q^{80} +531441. q^{81} -3.90830e6 q^{82} -689871. q^{83} -592704. q^{84} -7.52074e6 q^{85} -1.83089e6 q^{86} +874418. q^{87} -753370. q^{88} -4.55146e6 q^{89} -3.01166e6 q^{90} -753571. q^{91} +4.15101e6 q^{92} -3.41490e6 q^{93} -4.42233e6 q^{94} -8.71901e6 q^{95} +884736. q^{96} +5.94617e6 q^{97} +941192. q^{98} -1.07267e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 24 q^{2} + 81 q^{3} + 192 q^{4} - 378 q^{5} + 648 q^{6} - 1029 q^{7} + 1536 q^{8} + 2187 q^{9}+O(q^{10}) \) 3 * q + 24 * q^2 + 81 * q^3 + 192 * q^4 - 378 * q^5 + 648 * q^6 - 1029 * q^7 + 1536 * q^8 + 2187 * q^9	\( 3 q + 24 q^{2} + 81 q^{3} + 192 q^{4} - 378 q^{5} + 648 q^{6} - 1029 q^{7} + 1536 q^{8} + 2187 q^{9} - 3024 q^{10} - 6069 q^{11} + 5184 q^{12} + 6591 q^{13} - 8232 q^{14} - 10206 q^{15} + 12288 q^{16} - 12231 q^{17} + 17496 q^{18} - 4836 q^{19} - 24192 q^{20} - 27783 q^{21} - 48552 q^{22} + 111606 q^{23} + 41472 q^{24} + 41919 q^{25} + 52728 q^{26} + 59049 q^{27} - 65856 q^{28} + 105414 q^{29} - 81648 q^{30} - 263061 q^{31} + 98304 q^{32} - 163863 q^{33} - 97848 q^{34} + 129654 q^{35} + 139968 q^{36} - 585969 q^{37} - 38688 q^{38} + 177957 q^{39} - 193536 q^{40} - 649608 q^{41} - 222264 q^{42} - 76182 q^{43} - 388416 q^{44} - 275562 q^{45} + 892848 q^{46} - 1269219 q^{47} + 331776 q^{48} + 352947 q^{49} + 335352 q^{50} - 330237 q^{51} + 421824 q^{52} + 326511 q^{53} + 472392 q^{54} + 475311 q^{55} - 526848 q^{56} - 130572 q^{57} + 843312 q^{58} - 3434316 q^{59} - 653184 q^{60} - 1450497 q^{61} - 2104488 q^{62} - 750141 q^{63} + 786432 q^{64} - 830466 q^{65} - 1310904 q^{66} - 1302372 q^{67} - 782784 q^{68} + 3013362 q^{69} + 1037232 q^{70} - 5186076 q^{71} + 1119744 q^{72} - 3662940 q^{73} - 4687752 q^{74} + 1131813 q^{75} - 309504 q^{76} + 2081667 q^{77} + 1423656 q^{78} - 2950251 q^{79} - 1548288 q^{80} + 1594323 q^{81} - 5196864 q^{82} - 13650225 q^{83} - 1778112 q^{84} - 9353769 q^{85} - 609456 q^{86} + 2846178 q^{87} - 3107328 q^{88} + 6379533 q^{89} - 2204496 q^{90} - 2260713 q^{91} + 7142784 q^{92} - 7102647 q^{93} - 10153752 q^{94} - 10607646 q^{95} + 2654208 q^{96} - 18032235 q^{97} + 2823576 q^{98} - 4424301 q^{99}+O(q^{100}) \) 3 * q + 24 * q^2 + 81 * q^3 + 192 * q^4 - 378 * q^5 + 648 * q^6 - 1029 * q^7 + 1536 * q^8 + 2187 * q^9 - 3024 * q^10 - 6069 * q^11 + 5184 * q^12 + 6591 * q^13 - 8232 * q^14 - 10206 * q^15 + 12288 * q^16 - 12231 * q^17 + 17496 * q^18 - 4836 * q^19 - 24192 * q^20 - 27783 * q^21 - 48552 * q^22 + 111606 * q^23 + 41472 * q^24 + 41919 * q^25 + 52728 * q^26 + 59049 * q^27 - 65856 * q^28 + 105414 * q^29 - 81648 * q^30 - 263061 * q^31 + 98304 * q^32 - 163863 * q^33 - 97848 * q^34 + 129654 * q^35 + 139968 * q^36 - 585969 * q^37 - 38688 * q^38 + 177957 * q^39 - 193536 * q^40 - 649608 * q^41 - 222264 * q^42 - 76182 * q^43 - 388416 * q^44 - 275562 * q^45 + 892848 * q^46 - 1269219 * q^47 + 331776 * q^48 + 352947 * q^49 + 335352 * q^50 - 330237 * q^51 + 421824 * q^52 + 326511 * q^53 + 472392 * q^54 + 475311 * q^55 - 526848 * q^56 - 130572 * q^57 + 843312 * q^58 - 3434316 * q^59 - 653184 * q^60 - 1450497 * q^61 - 2104488 * q^62 - 750141 * q^63 + 786432 * q^64 - 830466 * q^65 - 1310904 * q^66 - 1302372 * q^67 - 782784 * q^68 + 3013362 * q^69 + 1037232 * q^70 - 5186076 * q^71 + 1119744 * q^72 - 3662940 * q^73 - 4687752 * q^74 + 1131813 * q^75 - 309504 * q^76 + 2081667 * q^77 + 1423656 * q^78 - 2950251 * q^79 - 1548288 * q^80 + 1594323 * q^81 - 5196864 * q^82 - 13650225 * q^83 - 1778112 * q^84 - 9353769 * q^85 - 609456 * q^86 + 2846178 * q^87 - 3107328 * q^88 + 6379533 * q^89 - 2204496 * q^90 - 2260713 * q^91 + 7142784 * q^92 - 7102647 * q^93 - 10153752 * q^94 - 10607646 * q^95 + 2654208 * q^96 - 18032235 * q^97 + 2823576 * q^98 - 4424301 * q^99

Introduction

L-functions

Modular forms

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