LMFDB - Embedded newform 546.8.a.k.1.2

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 148556x^{3} - 20997404x^{2} - 256427072x + 44264019648 $ x^5 - x^4 - 148556x^3 - 20997404x^2 - 256427072*x + 44264019648
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$443.950$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -78.1553 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -78.1553 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +625.242 q^{10} -6524.48 q^{11} +1728.00 q^{12} +2197.00 q^{13} +2744.00 q^{14} -2110.19 q^{15} +4096.00 q^{16} -25890.9 q^{17} -5832.00 q^{18} -12320.7 q^{19} -5001.94 q^{20} -9261.00 q^{21} +52195.8 q^{22} +63235.6 q^{23} -13824.0 q^{24} -72016.8 q^{25} -17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} -36586.2 q^{29} +16881.5 q^{30} -259298. q^{31} -32768.0 q^{32} -176161. q^{33} +207127. q^{34} +26807.3 q^{35} +46656.0 q^{36} +188688. q^{37} +98565.4 q^{38} +59319.0 q^{39} +40015.5 q^{40} -34629.2 q^{41} +74088.0 q^{42} -429043. q^{43} -417566. q^{44} -56975.2 q^{45} -505884. q^{46} +845048. q^{47} +110592. q^{48} +117649. q^{49} +576134. q^{50} -699054. q^{51} +140608. q^{52} +748941. q^{53} -157464. q^{54} +509922. q^{55} +175616. q^{56} -332658. q^{57} +292689. q^{58} -1.29989e6 q^{59} -135052. q^{60} -2.98992e6 q^{61} +2.07439e6 q^{62} -250047. q^{63} +262144. q^{64} -171707. q^{65} +1.40929e6 q^{66} -2.20826e6 q^{67} -1.65702e6 q^{68} +1.70736e6 q^{69} -214458. q^{70} +3.10404e6 q^{71} -373248. q^{72} +3.85157e6 q^{73} -1.50951e6 q^{74} -1.94445e6 q^{75} -788523. q^{76} +2.23790e6 q^{77} -474552. q^{78} -1.82590e6 q^{79} -320124. q^{80} +531441. q^{81} +277033. q^{82} +1.01110e7 q^{83} -592704. q^{84} +2.02351e6 q^{85} +3.43234e6 q^{86} -987827. q^{87} +3.34053e6 q^{88} +1.24996e7 q^{89} +455801. q^{90} -753571. q^{91} +4.04708e6 q^{92} -7.00106e6 q^{93} -6.76038e6 q^{94} +962925. q^{95} -884736. q^{96} +1.32410e7 q^{97} -941192. q^{98} -4.75634e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9}+O(q^{10}) $ 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 509 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 4072 q^{10} + 958 q^{11} + 8640 q^{12} + 10985 q^{13} + 13720 q^{14} + 13743 q^{15} + 20480 q^{16} + 7864 q^{17} - 29160 q^{18} + 60173 q^{19} + 32576 q^{20} - 46305 q^{21} - 7664 q^{22} + 122869 q^{23} - 69120 q^{24} + 73722 q^{25} - 87880 q^{26} + 98415 q^{27} - 109760 q^{28} - 17317 q^{29} - 109944 q^{30} - 177665 q^{31} - 163840 q^{32} + 25866 q^{33} - 62912 q^{34} - 174587 q^{35} + 233280 q^{36} - 55136 q^{37} - 481384 q^{38} + 296595 q^{39} - 260608 q^{40} - 237570 q^{41} + 370440 q^{42} - 970601 q^{43} + 61312 q^{44} + 371061 q^{45} - 982952 q^{46} - 384035 q^{47} + 552960 q^{48} + 588245 q^{49} - 589776 q^{50} + 212328 q^{51} + 703040 q^{52} - 1977 q^{53} - 787320 q^{54} + 1520014 q^{55} + 878080 q^{56} + 1624671 q^{57} + 138536 q^{58} + 2057936 q^{59} + 879552 q^{60} - 723756 q^{61} + 1421320 q^{62} - 1250235 q^{63} + 1310720 q^{64} + 1118273 q^{65} - 206928 q^{66} + 2695018 q^{67} + 503296 q^{68} + 3317463 q^{69} + 1396696 q^{70} + 7392916 q^{71} - 1866240 q^{72} + 8720441 q^{73} + 441088 q^{74} + 1990494 q^{75} + 3851072 q^{76} - 328594 q^{77} - 2372760 q^{78} + 4646419 q^{79} + 2084864 q^{80} + 2657205 q^{81} + 1900560 q^{82} + 17766733 q^{83} - 2963520 q^{84} + 16495320 q^{85} + 7764808 q^{86} - 467559 q^{87} - 490496 q^{88} + 4692321 q^{89} - 2968488 q^{90} - 3767855 q^{91} + 7863616 q^{92} - 4796955 q^{93} + 3072280 q^{94} - 2355945 q^{95} - 4423680 q^{96} + 15680305 q^{97} - 4705960 q^{98} + 698382 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 509 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9 - 4072 * q^10 + 958 * q^11 + 8640 * q^12 + 10985 * q^13 + 13720 * q^14 + 13743 * q^15 + 20480 * q^16 + 7864 * q^17 - 29160 * q^18 + 60173 * q^19 + 32576 * q^20 - 46305 * q^21 - 7664 * q^22 + 122869 * q^23 - 69120 * q^24 + 73722 * q^25 - 87880 * q^26 + 98415 * q^27 - 109760 * q^28 - 17317 * q^29 - 109944 * q^30 - 177665 * q^31 - 163840 * q^32 + 25866 * q^33 - 62912 * q^34 - 174587 * q^35 + 233280 * q^36 - 55136 * q^37 - 481384 * q^38 + 296595 * q^39 - 260608 * q^40 - 237570 * q^41 + 370440 * q^42 - 970601 * q^43 + 61312 * q^44 + 371061 * q^45 - 982952 * q^46 - 384035 * q^47 + 552960 * q^48 + 588245 * q^49 - 589776 * q^50 + 212328 * q^51 + 703040 * q^52 - 1977 * q^53 - 787320 * q^54 + 1520014 * q^55 + 878080 * q^56 + 1624671 * q^57 + 138536 * q^58 + 2057936 * q^59 + 879552 * q^60 - 723756 * q^61 + 1421320 * q^62 - 1250235 * q^63 + 1310720 * q^64 + 1118273 * q^65 - 206928 * q^66 + 2695018 * q^67 + 503296 * q^68 + 3317463 * q^69 + 1396696 * q^70 + 7392916 * q^71 - 1866240 * q^72 + 8720441 * q^73 + 441088 * q^74 + 1990494 * q^75 + 3851072 * q^76 - 328594 * q^77 - 2372760 * q^78 + 4646419 * q^79 + 2084864 * q^80 + 2657205 * q^81 + 1900560 * q^82 + 17766733 * q^83 - 2963520 * q^84 + 16495320 * q^85 + 7764808 * q^86 - 467559 * q^87 - 490496 * q^88 + 4692321 * q^89 - 2968488 * q^90 - 3767855 * q^91 + 7863616 * q^92 - 4796955 * q^93 + 3072280 * q^94 - 2355945 * q^95 - 4423680 * q^96 + 15680305 * q^97 - 4705960 * q^98 + 698382 * 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
$5$	−78.1553	−0.279617	−0.139808	−	0.990179i	$-0.544649\pi$
$5$	−78.1553	−0.279617	−0.139808	+	0.990179i	$0.544649\pi$
$6$	−216.000	−0.408248
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	625.242	0.197719
$11$	−6524.48	−1.47799	−0.738995	−	0.673711i	$-0.764699\pi$
$11$	−6524.48	−1.47799	−0.738995	+	0.673711i	$0.764699\pi$
$12$	1728.00	0.288675
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	2744.00	0.267261
$15$	−2110.19	−0.161437
$16$	4096.00	0.250000
$17$	−25890.9	−1.27813	−0.639066	−	0.769152i	$-0.720679\pi$
$17$	−25890.9	−1.27813	−0.639066	+	0.769152i	$0.720679\pi$
$18$	−5832.00	−0.235702
$19$	−12320.7	−0.412095	−0.206047	−	0.978542i	$-0.566060\pi$
$19$	−12320.7	−0.412095	−0.206047	+	0.978542i	$0.566060\pi$
$20$	−5001.94	−0.139808
$21$	−9261.00	−0.218218
$22$	52195.8	1.04510
$23$	63235.6	1.08371	0.541856	−	0.840471i	$-0.317722\pi$
$23$	63235.6	1.08371	0.541856	+	0.840471i	$0.317722\pi$
$24$	−13824.0	−0.204124
$25$	−72016.8	−0.921814
$26$	−17576.0	−0.196116
$27$	19683.0	0.192450
$28$	−21952.0	−0.188982
$29$	−36586.2	−0.278563	−0.139282	−	0.990253i	$-0.544479\pi$
$29$	−36586.2	−0.278563	−0.139282	+	0.990253i	$0.544479\pi$
$30$	16881.5	0.114153
$31$	−259298.	−1.56327	−0.781635	−	0.623736i	$-0.785614\pi$
$31$	−259298.	−1.56327	−0.781635	+	0.623736i	$0.785614\pi$
$32$	−32768.0	−0.176777
$33$	−176161.	−0.853317
$34$	207127.	0.903776
$35$	26807.3	0.105685
$36$	46656.0	0.166667
$37$	188688.	0.612405	0.306202	−	0.951966i	$-0.400942\pi$
$37$	188688.	0.612405	0.306202	+	0.951966i	$0.400942\pi$
$38$	98565.4	0.291395
$39$	59319.0	0.160128
$40$	40015.5	0.0988595
$41$	−34629.2	−0.0784690	−0.0392345	−	0.999230i	$-0.512492\pi$
$41$	−34629.2	−0.0784690	−0.0392345	+	0.999230i	$0.512492\pi$
$42$	74088.0	0.154303
$43$	−429043.	−0.822926	−0.411463	−	0.911426i	$-0.634982\pi$
$43$	−429043.	−0.822926	−0.411463	+	0.911426i	$0.634982\pi$
$44$	−417566.	−0.738995
$45$	−56975.2	−0.0932056
$46$	−505884.	−0.766300
$47$	845048.	1.18724	0.593621	−	0.804745i	$-0.297698\pi$
$47$	845048.	1.18724	0.593621	+	0.804745i	$0.297698\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	576134.	0.651821
$51$	−699054.	−0.737930
$52$	140608.	0.138675
$53$	748941.	0.691007	0.345503	−	0.938418i	$-0.387708\pi$
$53$	748941.	0.691007	0.345503	+	0.938418i	$0.387708\pi$
$54$	−157464.	−0.136083
$55$	509922.	0.413271
$56$	175616.	0.133631
$57$	−332658.	−0.237923
$58$	292689.	0.196974
$59$	−1.29989e6	−0.823996	−0.411998	−	0.911185i	$-0.635169\pi$
$59$	−1.29989e6	−0.823996	−0.411998	+	0.911185i	$0.635169\pi$
$60$	−135052.	−0.0807184
$61$	−2.98992e6	−1.68657	−0.843287	−	0.537464i	$-0.819382\pi$
$61$	−2.98992e6	−1.68657	−0.843287	+	0.537464i	$0.819382\pi$
$62$	2.07439e6	1.10540
$63$	−250047.	−0.125988
$64$	262144.	0.125000
$65$	−171707.	−0.0775517
$66$	1.40929e6	0.603387
$67$	−2.20826e6	−0.896991	−0.448495	−	0.893785i	$-0.648040\pi$
$67$	−2.20826e6	−0.896991	−0.448495	+	0.893785i	$0.648040\pi$
$68$	−1.65702e6	−0.639066
$69$	1.70736e6	0.625681
$70$	−214458.	−0.0747307
$71$	3.10404e6	1.02926	0.514628	−	0.857414i	$-0.327930\pi$
$71$	3.10404e6	1.02926	0.514628	+	0.857414i	$0.327930\pi$
$72$	−373248.	−0.117851
$73$	3.85157e6	1.15880	0.579399	−	0.815044i	$-0.303287\pi$
$73$	3.85157e6	1.15880	0.579399	+	0.815044i	$0.303287\pi$
$74$	−1.50951e6	−0.433036
$75$	−1.94445e6	−0.532210
$76$	−788523.	−0.206047
$77$	2.23790e6	0.558627
$78$	−474552.	−0.113228
$79$	−1.82590e6	−0.416660	−0.208330	−	0.978059i	$-0.566803\pi$
$79$	−1.82590e6	−0.416660	−0.208330	+	0.978059i	$0.566803\pi$
$80$	−320124.	−0.0699042
$81$	531441.	0.111111
$82$	277033.	0.0554860
$83$	1.01110e7	1.94098	0.970491	−	0.241137i	$-0.0775203\pi$
$83$	1.01110e7	1.94098	0.970491	+	0.241137i	$0.0775203\pi$
$84$	−592704.	−0.109109
$85$	2.02351e6	0.357387
$86$	3.43234e6	0.581896
$87$	−987827.	−0.160829
$88$	3.34053e6	0.522548
$89$	1.24996e7	1.87945	0.939724	−	0.341934i	$-0.111082\pi$
$89$	1.24996e7	1.87945	0.939724	+	0.341934i	$0.111082\pi$
$90$	455801.	0.0659063
$91$	−753571.	−0.104828
$92$	4.04708e6	0.541856
$93$	−7.00106e6	−0.902554
$94$	−6.76038e6	−0.839506
$95$	962925.	0.115229
$96$	−884736.	−0.102062
$97$	1.32410e7	1.47306	0.736530	−	0.676405i	$-0.236463\pi$
$97$	1.32410e7	1.47306	0.736530	+	0.676405i	$0.236463\pi$
$98$	−941192.	−0.101015
$99$	−4.75634e6	−0.492663
$100$	−4.60907e6	−0.460907
$101$	−1.65888e7	−1.60210	−0.801050	−	0.598598i	$-0.795725\pi$
$101$	−1.65888e7	−1.60210	−0.801050	+	0.598598i	$0.795725\pi$
$102$	5.59243e6	0.521795
$103$	4.37254e6	0.394278	0.197139	−	0.980376i	$-0.436835\pi$
$103$	4.37254e6	0.394278	0.197139	+	0.980376i	$0.436835\pi$
$104$	−1.12486e6	−0.0980581
$105$	723796.	0.0610174
$106$	−5.99153e6	−0.488615
$107$	−1.57384e7	−1.24199	−0.620996	−	0.783814i	$-0.713272\pi$
$107$	−1.57384e7	−1.24199	−0.620996	+	0.783814i	$0.713272\pi$
$108$	1.25971e6	0.0962250
$109$	−9.89194e6	−0.731625	−0.365813	−	0.930688i	$-0.619209\pi$
$109$	−9.89194e6	−0.731625	−0.365813	+	0.930688i	$0.619209\pi$
$110$	−4.07938e6	−0.292226
$111$	5.09458e6	0.353572
$112$	−1.40493e6	−0.0944911
$113$	−1.66864e7	−1.08790	−0.543950	−	0.839118i	$-0.683072\pi$
$113$	−1.66864e7	−1.08790	−0.543950	+	0.839118i	$0.683072\pi$
$114$	2.66127e6	0.168237
$115$	−4.94219e6	−0.303024
$116$	−2.34152e6	−0.139282
$117$	1.60161e6	0.0924500
$118$	1.03991e7	0.582653
$119$	8.88058e6	0.483089
$120$	1.08042e6	0.0570765
$121$	2.30816e7	1.18445
$122$	2.39194e7	1.19259
$123$	−934987.	−0.0453041
$124$	−1.65951e7	−0.781635
$125$	1.17344e7	0.537372
$126$	2.00038e6	0.0890871
$127$	2.90307e7	1.25761	0.628803	−	0.777565i	$-0.283545\pi$
$127$	2.90307e7	1.25761	0.628803	+	0.777565i	$0.283545\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.15841e7	−0.475116
$130$	1.37366e6	0.0548374
$131$	−1.02337e7	−0.397727	−0.198863	−	0.980027i	$-0.563725\pi$
$131$	−1.02337e7	−0.397727	−0.198863	+	0.980027i	$0.563725\pi$
$132$	−1.12743e7	−0.426659
$133$	4.22599e6	0.155757
$134$	1.76661e7	0.634268
$135$	−1.53833e6	−0.0538123
$136$	1.32561e7	0.451888
$137$	2.94105e7	0.977194	0.488597	−	0.872510i	$-0.337509\pi$
$137$	2.94105e7	0.977194	0.488597	+	0.872510i	$0.337509\pi$
$138$	−1.36589e7	−0.442424
$139$	−2.66421e7	−0.841429	−0.420714	−	0.907193i	$-0.638221\pi$
$139$	−2.66421e7	−0.841429	−0.420714	+	0.907193i	$0.638221\pi$
$140$	1.71566e6	0.0528426
$141$	2.28163e7	0.685454
$142$	−2.48323e7	−0.727793
$143$	−1.43343e7	−0.409920
$144$	2.98598e6	0.0833333
$145$	2.85940e6	0.0778910
$146$	−3.08126e7	−0.819394
$147$	3.17652e6	0.0824786
$148$	1.20760e7	0.306202
$149$	−3.36738e7	−0.833950	−0.416975	−	0.908918i	$-0.636910\pi$
$149$	−3.36738e7	−0.833950	−0.416975	+	0.908918i	$0.636910\pi$
$150$	1.55556e7	0.376329
$151$	−4.26859e7	−1.00894	−0.504469	−	0.863430i	$-0.668312\pi$
$151$	−4.26859e7	−1.00894	−0.504469	+	0.863430i	$0.668312\pi$
$152$	6.30818e6	0.145697
$153$	−1.88745e7	−0.426044
$154$	−1.79032e7	−0.395009
$155$	2.02655e7	0.437116
$156$	3.79642e6	0.0800641
$157$	1.37486e7	0.283536	0.141768	−	0.989900i	$-0.454721\pi$
$157$	1.37486e7	0.283536	0.141768	+	0.989900i	$0.454721\pi$
$158$	1.46072e7	0.294623
$159$	2.02214e7	0.398953
$160$	2.56099e6	0.0494297
$161$	−2.16898e7	−0.409605
$162$	−4.25153e6	−0.0785674
$163$	2.39518e7	0.433193	0.216596	−	0.976261i	$-0.430504\pi$
$163$	2.39518e7	0.433193	0.216596	+	0.976261i	$0.430504\pi$
$164$	−2.21627e6	−0.0392345
$165$	1.37679e7	0.238602
$166$	−8.08881e7	−1.37248
$167$	−1.38679e7	−0.230412	−0.115206	−	0.993342i	$-0.536753\pi$
$167$	−1.38679e7	−0.230412	−0.115206	+	0.993342i	$0.536753\pi$
$168$	4.74163e6	0.0771517
$169$	4.82681e6	0.0769231
$170$	−1.61881e7	−0.252711
$171$	−8.98177e6	−0.137365
$172$	−2.74587e7	−0.411463
$173$	−7.98090e7	−1.17190	−0.585950	−	0.810348i	$-0.699278\pi$
$173$	−7.98090e7	−1.17190	−0.585950	+	0.810348i	$0.699278\pi$
$174$	7.90262e6	0.113723
$175$	2.47017e7	0.348413
$176$	−2.67243e7	−0.369497
$177$	−3.50971e7	−0.475734
$178$	−9.99966e7	−1.32897
$179$	1.23699e8	1.61206	0.806032	−	0.591872i	$-0.201611\pi$
$179$	1.23699e8	1.61206	0.806032	+	0.591872i	$0.201611\pi$
$180$	−3.64641e6	−0.0466028
$181$	−1.01324e8	−1.27010	−0.635052	−	0.772470i	$-0.719021\pi$
$181$	−1.01324e8	−1.27010	−0.635052	+	0.772470i	$0.719021\pi$
$182$	6.02857e6	0.0741249
$183$	−8.07278e7	−0.973743
$184$	−3.23766e7	−0.383150
$185$	−1.47470e7	−0.171239
$186$	5.60085e7	0.638202
$187$	1.68925e8	1.88907
$188$	5.40831e7	0.593621
$189$	−6.75127e6	−0.0727393
$190$	−7.70340e6	−0.0814789
$191$	6.21589e7	0.645486	0.322743	−	0.946487i	$-0.395395\pi$
$191$	6.21589e7	0.645486	0.322743	+	0.946487i	$0.395395\pi$
$192$	7.07789e6	0.0721688
$193$	−1.18401e7	−0.118551	−0.0592754	−	0.998242i	$-0.518879\pi$
$193$	−1.18401e7	−0.118551	−0.0592754	+	0.998242i	$0.518879\pi$
$194$	−1.05928e8	−1.04161
$195$	−4.63609e6	−0.0447745
$196$	7.52954e6	0.0714286
$197$	−5.96857e7	−0.556209	−0.278105	−	0.960551i	$-0.589706\pi$
$197$	−5.96857e7	−0.556209	−0.278105	+	0.960551i	$0.589706\pi$
$198$	3.80507e7	0.348365
$199$	−1.83161e8	−1.64758	−0.823792	−	0.566892i	$-0.808146\pi$
$199$	−1.83161e8	−1.64758	−0.823792	+	0.566892i	$0.808146\pi$
$200$	3.68726e7	0.325911
$201$	−5.96230e7	−0.517878
$202$	1.32710e8	1.13286
$203$	1.25491e7	0.105287
$204$	−4.47395e7	−0.368965
$205$	2.70645e6	0.0219413
$206$	−3.49803e7	−0.278797
$207$	4.60987e7	0.361237
$208$	8.99891e6	0.0693375
$209$	8.03859e7	0.609071
$210$	−5.79037e6	−0.0431458
$211$	9.45083e7	0.692598	0.346299	−	0.938124i	$-0.387438\pi$
$211$	9.45083e7	0.692598	0.346299	+	0.938124i	$0.387438\pi$
$212$	4.79322e7	0.345503
$213$	8.38091e7	0.594241
$214$	1.25908e8	0.878220
$215$	3.35319e7	0.230104
$216$	−1.00777e7	−0.0680414
$217$	8.89394e7	0.590861
$218$	7.91355e7	0.517337
$219$	1.03992e8	0.669033
$220$	3.26350e7	0.206635
$221$	−5.68823e7	−0.354490
$222$	−4.07566e7	−0.250013
$223$	1.32876e8	0.802380	0.401190	−	0.915995i	$-0.368597\pi$
$223$	1.32876e8	0.802380	0.401190	+	0.915995i	$0.368597\pi$
$224$	1.12394e7	0.0668153
$225$	−5.25002e7	−0.307271
$226$	1.33491e8	0.769261
$227$	2.88792e8	1.63868	0.819342	−	0.573305i	$-0.194339\pi$
$227$	2.88792e8	1.63868	0.819342	+	0.573305i	$0.194339\pi$
$228$	−2.12901e7	−0.118961
$229$	1.99283e8	1.09660	0.548298	−	0.836283i	$-0.315276\pi$
$229$	1.99283e8	1.09660	0.548298	+	0.836283i	$0.315276\pi$
$230$	3.95375e7	0.214270
$231$	6.04232e7	0.322524
$232$	1.87321e7	0.0984871
$233$	7.75364e7	0.401569	0.200784	−	0.979635i	$-0.435651\pi$
$233$	7.75364e7	0.401569	0.200784	+	0.979635i	$0.435651\pi$
$234$	−1.28129e7	−0.0653720
$235$	−6.60450e7	−0.331973
$236$	−8.31930e7	−0.411998
$237$	−4.92992e7	−0.240559
$238$	−7.10446e7	−0.341595
$239$	3.63349e8	1.72160	0.860799	−	0.508945i	$-0.169964\pi$
$239$	3.63349e8	1.72160	0.860799	+	0.508945i	$0.169964\pi$
$240$	−8.64335e6	−0.0403592
$241$	−4.70275e7	−0.216417	−0.108209	−	0.994128i	$-0.534511\pi$
$241$	−4.70275e7	−0.216417	−0.108209	+	0.994128i	$0.534511\pi$
$242$	−1.84653e8	−0.837534
$243$	1.43489e7	0.0641500
$244$	−1.91355e8	−0.843287
$245$	−9.19489e6	−0.0399453
$246$	7.47990e6	0.0320349
$247$	−2.70685e7	−0.114294
$248$	1.32761e8	0.552699
$249$	2.72997e8	1.12063
$250$	−9.38749e7	−0.379979
$251$	3.29585e8	1.31556	0.657778	−	0.753212i	$-0.271496\pi$
$251$	3.29585e8	1.31556	0.657778	+	0.753212i	$0.271496\pi$
$252$	−1.60030e7	−0.0629941
$253$	−4.12579e8	−1.60171
$254$	−2.32246e8	−0.889262
$255$	5.46348e7	0.206338
$256$	1.67772e7	0.0625000
$257$	3.33946e8	1.22719	0.613593	−	0.789622i	$-0.289724\pi$
$257$	3.33946e8	1.22719	0.613593	+	0.789622i	$0.289724\pi$
$258$	9.26732e7	0.335958
$259$	−6.47200e7	−0.231467
$260$	−1.09893e7	−0.0387759
$261$	−2.66713e7	−0.0928545
$262$	8.18699e7	0.281235
$263$	1.04563e8	0.354433	0.177217	−	0.984172i	$-0.443291\pi$
$263$	1.04563e8	0.354433	0.177217	+	0.984172i	$0.443291\pi$
$264$	9.01944e7	0.301693
$265$	−5.85337e7	−0.193217
$266$	−3.38079e7	−0.110137
$267$	3.37489e8	1.08510
$268$	−1.41328e8	−0.448495
$269$	−3.26993e8	−1.02425	−0.512125	−	0.858911i	$-0.671141\pi$
$269$	−3.26993e8	−1.02425	−0.512125	+	0.858911i	$0.671141\pi$
$270$	1.23066e7	0.0380510
$271$	−2.72217e8	−0.830851	−0.415425	−	0.909627i	$-0.636367\pi$
$271$	−2.72217e8	−0.830851	−0.415425	+	0.909627i	$0.636367\pi$
$272$	−1.06049e8	−0.319533
$273$	−2.03464e7	−0.0605228
$274$	−2.35284e8	−0.690980
$275$	4.69872e8	1.36243
$276$	1.09271e8	0.312841
$277$	5.51945e8	1.56033	0.780165	−	0.625574i	$-0.215135\pi$
$277$	5.51945e8	1.56033	0.780165	+	0.625574i	$0.215135\pi$
$278$	2.13137e8	0.594980
$279$	−1.89029e8	−0.521090
$280$	−1.37253e7	−0.0373654
$281$	2.31882e8	0.623440	0.311720	−	0.950174i	$-0.399095\pi$
$281$	2.31882e8	0.623440	0.311720	+	0.950174i	$0.399095\pi$
$282$	−1.82530e8	−0.484689
$283$	1.10147e8	0.288882	0.144441	−	0.989513i	$-0.453862\pi$
$283$	1.10147e8	0.288882	0.144441	+	0.989513i	$0.453862\pi$
$284$	1.98659e8	0.514628
$285$	2.59990e7	0.0665272
$286$	1.14674e8	0.289858
$287$	1.18778e7	0.0296585
$288$	−2.38879e7	−0.0589256
$289$	2.60000e8	0.633623
$290$	−2.28752e7	−0.0550773
$291$	3.57508e8	0.850472
$292$	2.46501e8	0.579399
$293$	7.01475e7	0.162920	0.0814602	−	0.996677i	$-0.474042\pi$
$293$	7.01475e7	0.162920	0.0814602	+	0.996677i	$0.474042\pi$
$294$	−2.54122e7	−0.0583212
$295$	1.01593e8	0.230403
$296$	−9.66083e7	−0.216518
$297$	−1.28421e8	−0.284439
$298$	2.69390e8	0.589691
$299$	1.38929e8	0.300568
$300$	−1.24445e8	−0.266105
$301$	1.47162e8	0.311037
$302$	3.41487e8	0.713428
$303$	−4.47897e8	−0.924973
$304$	−5.04655e7	−0.103024
$305$	2.33678e8	0.471594
$306$	1.50996e8	0.301259
$307$	−2.91604e8	−0.575187	−0.287593	−	0.957753i	$-0.592855\pi$
$307$	−2.91604e8	−0.575187	−0.287593	+	0.957753i	$0.592855\pi$
$308$	1.43225e8	0.279314
$309$	1.18058e8	0.227637
$310$	−1.62124e8	−0.309088
$311$	9.89726e8	1.86575	0.932875	−	0.360200i	$-0.117291\pi$
$311$	9.89726e8	1.86575	0.932875	+	0.360200i	$0.117291\pi$
$312$	−3.03713e7	−0.0566139
$313$	9.32843e8	1.71951	0.859753	−	0.510711i	$-0.170618\pi$
$313$	9.32843e8	1.71951	0.859753	+	0.510711i	$0.170618\pi$
$314$	−1.09988e8	−0.200490
$315$	1.95425e7	0.0352284
$316$	−1.16857e8	−0.208330
$317$	3.66693e8	0.646540	0.323270	−	0.946307i	$-0.395218\pi$
$317$	3.66693e8	0.646540	0.323270	+	0.946307i	$0.395218\pi$
$318$	−1.61771e8	−0.282102
$319$	2.38706e8	0.411714
$320$	−2.04879e7	−0.0349521
$321$	−4.24938e8	−0.717064
$322$	1.73518e8	0.289634
$323$	3.18993e8	0.526711
$324$	3.40122e7	0.0555556
$325$	−1.58221e8	−0.255665
$326$	−1.91614e8	−0.306313
$327$	−2.67082e8	−0.422404
$328$	1.77301e7	0.0277430
$329$	−2.89852e8	−0.448735
$330$	−1.10143e8	−0.168717
$331$	3.22513e8	0.488821	0.244410	−	0.969672i	$-0.421406\pi$
$331$	3.22513e8	0.488821	0.244410	+	0.969672i	$0.421406\pi$
$332$	6.47105e8	0.970491
$333$	1.37554e8	0.204135
$334$	1.10944e8	0.162926
$335$	1.72587e8	0.250814
$336$	−3.79331e7	−0.0545545
$337$	−1.25722e9	−1.78939	−0.894695	−	0.446677i	$-0.852607\pi$
$337$	−1.25722e9	−1.78939	−0.894695	+	0.446677i	$0.852607\pi$
$338$	−3.86145e7	−0.0543928
$339$	−4.50534e8	−0.628099
$340$	1.29505e8	0.178694
$341$	1.69179e9	2.31050
$342$	7.18542e7	0.0971316
$343$	−4.03536e7	−0.0539949
$344$	2.19670e8	0.290948
$345$	−1.33439e8	−0.174951
$346$	6.38472e8	0.828658
$347$	−7.02728e8	−0.902888	−0.451444	−	0.892299i	$-0.649091\pi$
$347$	−7.02728e8	−0.902888	−0.451444	+	0.892299i	$0.649091\pi$
$348$	−6.32209e7	−0.0804143
$349$	1.08255e9	1.36320	0.681599	−	0.731726i	$-0.261285\pi$
$349$	1.08255e9	1.36320	0.681599	+	0.731726i	$0.261285\pi$
$350$	−1.97614e8	−0.246365
$351$	4.32436e7	0.0533761
$352$	2.13794e8	0.261274
$353$	7.24807e8	0.877023	0.438511	−	0.898726i	$-0.355506\pi$
$353$	7.24807e8	0.877023	0.438511	+	0.898726i	$0.355506\pi$
$354$	2.80776e8	0.336395
$355$	−2.42597e8	−0.287797
$356$	7.99973e8	0.939724
$357$	2.39776e8	0.278911
$358$	−9.89595e8	−1.13990
$359$	−1.16985e8	−0.133444	−0.0667220	−	0.997772i	$-0.521254\pi$
$359$	−1.16985e8	−0.133444	−0.0667220	+	0.997772i	$0.521254\pi$
$360$	2.91713e7	0.0329532
$361$	−7.42073e8	−0.830178
$362$	8.10595e8	0.898099
$363$	6.23204e8	0.683844
$364$	−4.82285e7	−0.0524142
$365$	−3.01021e8	−0.324020
$366$	6.45823e8	0.688541
$367$	1.00546e9	1.06178	0.530888	−	0.847442i	$-0.321858\pi$
$367$	1.00546e9	1.06178	0.530888	+	0.847442i	$0.321858\pi$
$368$	2.59013e8	0.270928
$369$	−2.52447e7	−0.0261563
$370$	1.17976e8	0.121084
$371$	−2.56887e8	−0.261176
$372$	−4.48068e8	−0.451277
$373$	1.48418e9	1.48083	0.740413	−	0.672152i	$-0.234630\pi$
$373$	1.48418e9	1.48083	0.740413	+	0.672152i	$0.234630\pi$
$374$	−1.35140e9	−1.33577
$375$	3.16828e8	0.310252
$376$	−4.32665e8	−0.419753
$377$	−8.03798e7	−0.0772596
$378$	5.40102e7	0.0514344
$379$	5.79727e8	0.546999	0.273500	−	0.961872i	$-0.411819\pi$
$379$	5.79727e8	0.546999	0.273500	+	0.961872i	$0.411819\pi$
$380$	6.16272e7	0.0576143
$381$	7.83829e8	0.726079
$382$	−4.97271e8	−0.456427
$383$	2.43624e7	0.0221577	0.0110789	−	0.999939i	$-0.496473\pi$
$383$	2.43624e7	0.0221577	0.0110789	+	0.999939i	$0.496473\pi$
$384$	−5.66231e7	−0.0510310
$385$	−1.74903e8	−0.156202
$386$	9.47208e7	0.0838281
$387$	−3.12772e8	−0.274309
$388$	8.47426e8	0.736530
$389$	−5.36368e8	−0.461997	−0.230998	−	0.972954i	$-0.574199\pi$
$389$	−5.36368e8	−0.461997	−0.230998	+	0.972954i	$0.574199\pi$
$390$	3.70887e7	0.0316604
$391$	−1.63723e9	−1.38513
$392$	−6.02363e7	−0.0505076
$393$	−2.76311e8	−0.229628
$394$	4.77485e8	0.393299
$395$	1.42703e8	0.116505
$396$	−3.04406e8	−0.246332
$397$	−1.00310e7	−0.00804598	−0.00402299	−	0.999992i	$-0.501281\pi$
$397$	−1.00310e7	−0.00804598	−0.00402299	+	0.999992i	$0.501281\pi$
$398$	1.46529e9	1.16502
$399$	1.14102e8	0.0899264
$400$	−2.94981e8	−0.230454
$401$	2.43982e8	0.188953	0.0944764	−	0.995527i	$-0.469882\pi$
$401$	2.43982e8	0.188953	0.0944764	+	0.995527i	$0.469882\pi$
$402$	4.76984e8	0.366195
$403$	−5.69679e8	−0.433573
$404$	−1.06168e9	−0.801050
$405$	−4.15349e7	−0.0310685
$406$	−1.00392e8	−0.0744492
$407$	−1.23109e9	−0.905128
$408$	3.57916e8	0.260898
$409$	1.81215e9	1.30967	0.654836	−	0.755771i	$-0.272738\pi$
$409$	1.81215e9	1.30967	0.654836	+	0.755771i	$0.272738\pi$
$410$	−2.16516e7	−0.0155148
$411$	7.94084e8	0.564183
$412$	2.79842e8	0.197139
$413$	4.45863e8	0.311441
$414$	−3.68790e8	−0.255433
$415$	−7.90229e8	−0.542731
$416$	−7.19913e7	−0.0490290
$417$	−7.19338e8	−0.485799
$418$	−6.43087e8	−0.430678
$419$	−2.22170e9	−1.47549	−0.737746	−	0.675078i	$-0.764110\pi$
$419$	−2.22170e9	−1.47549	−0.737746	+	0.675078i	$0.764110\pi$
$420$	4.63229e7	0.0305087
$421$	−8.70212e8	−0.568379	−0.284189	−	0.958768i	$-0.591724\pi$
$421$	−8.70212e8	−0.568379	−0.284189	+	0.958768i	$0.591724\pi$
$422$	−7.56066e8	−0.489741
$423$	6.16040e8	0.395747
$424$	−3.83458e8	−0.244308
$425$	1.86458e9	1.17820
$426$	−6.70473e8	−0.420192
$427$	1.02554e9	0.637465
$428$	−1.00726e9	−0.620996
$429$	−3.87025e8	−0.236668
$430$	−2.68255e8	−0.162708
$431$	−2.81414e9	−1.69307	−0.846535	−	0.532333i	$-0.821316\pi$
$431$	−2.81414e9	−1.69307	−0.846535	+	0.532333i	$0.821316\pi$
$432$	8.06216e7	0.0481125
$433$	−2.63219e9	−1.55815	−0.779076	−	0.626930i	$-0.784311\pi$
$433$	−2.63219e9	−1.55815	−0.779076	+	0.626930i	$0.784311\pi$
$434$	−7.11515e8	−0.417801
$435$	7.72039e7	0.0449704
$436$	−6.33084e8	−0.365813
$437$	−7.79105e8	−0.446592
$438$	−8.31940e8	−0.473078
$439$	7.29249e8	0.411387	0.205693	−	0.978616i	$-0.434055\pi$
$439$	7.29249e8	0.411387	0.205693	+	0.978616i	$0.434055\pi$
$440$	−2.61080e8	−0.146113
$441$	8.57661e7	0.0476190
$442$	4.55058e8	0.250662
$443$	2.22205e9	1.21434	0.607170	−	0.794572i	$-0.292305\pi$
$443$	2.22205e9	1.21434	0.607170	+	0.794572i	$0.292305\pi$
$444$	3.26053e8	0.176786
$445$	−9.76908e8	−0.525525
$446$	−1.06301e9	−0.567368
$447$	−9.09192e8	−0.481481
$448$	−8.99154e7	−0.0472456
$449$	1.32233e9	0.689408	0.344704	−	0.938711i	$-0.387979\pi$
$449$	1.32233e9	0.689408	0.344704	+	0.938711i	$0.387979\pi$
$450$	4.20002e8	0.217274
$451$	2.25937e8	0.115976
$452$	−1.06793e9	−0.543950
$453$	−1.15252e9	−0.582511
$454$	−2.31034e9	−1.15872
$455$	5.88955e7	0.0293118
$456$	1.70321e8	0.0841184
$457$	2.96532e9	1.45333	0.726667	−	0.686990i	$-0.241068\pi$
$457$	2.96532e9	1.45333	0.726667	+	0.686990i	$0.241068\pi$
$458$	−1.59427e9	−0.775410
$459$	−5.09611e8	−0.245977
$460$	−3.16300e8	−0.151512
$461$	3.38065e9	1.60711	0.803557	−	0.595228i	$-0.202938\pi$
$461$	3.38065e9	1.60711	0.803557	+	0.595228i	$0.202938\pi$
$462$	−4.83385e8	−0.228059
$463$	−1.73405e9	−0.811949	−0.405975	−	0.913884i	$-0.633068\pi$
$463$	−1.73405e9	−0.811949	−0.405975	+	0.913884i	$0.633068\pi$
$464$	−1.49857e8	−0.0696409
$465$	5.47170e8	0.252369
$466$	−6.20291e8	−0.283952
$467$	1.81693e9	0.825522	0.412761	−	0.910839i	$-0.364565\pi$
$467$	1.81693e9	0.825522	0.412761	+	0.910839i	$0.364565\pi$
$468$	1.02503e8	0.0462250
$469$	7.57432e8	0.339031
$470$	5.28360e8	0.234740
$471$	3.71211e8	0.163700
$472$	6.65544e8	0.291326
$473$	2.79928e9	1.21628
$474$	3.94394e8	0.170101
$475$	8.87295e8	0.379875
$476$	5.68357e8	0.241544
$477$	5.45978e8	0.230336
$478$	−2.90679e9	−1.21735
$479$	−1.40689e9	−0.584905	−0.292452	−	0.956280i	$-0.594471\pi$
$479$	−1.40689e9	−0.584905	−0.292452	+	0.956280i	$0.594471\pi$
$480$	6.91468e7	0.0285383
$481$	4.14548e8	0.169851
$482$	3.76220e8	0.153030
$483$	−5.85624e8	−0.236485
$484$	1.47722e9	0.592226
$485$	−1.03486e9	−0.411892
$486$	−1.14791e8	−0.0453609
$487$	2.56706e9	1.00713	0.503565	−	0.863957i	$-0.332021\pi$
$487$	2.56706e9	1.00713	0.503565	+	0.863957i	$0.332021\pi$
$488$	1.53084e9	0.596294
$489$	6.46698e8	0.250104
$490$	7.35591e7	0.0282456
$491$	4.12564e9	1.57292	0.786459	−	0.617642i	$-0.211912\pi$
$491$	4.12564e9	1.57292	0.786459	+	0.617642i	$0.211912\pi$
$492$	−5.98392e7	−0.0226521
$493$	9.47249e8	0.356041
$494$	2.16548e8	0.0808184
$495$	3.71733e8	0.137757
$496$	−1.06209e9	−0.390817
$497$	−1.06469e9	−0.389022
$498$	−2.18398e9	−0.792403
$499$	−3.03885e9	−1.09486	−0.547429	−	0.836852i	$-0.684393\pi$
$499$	−3.03885e9	−1.09486	−0.547429	+	0.836852i	$0.684393\pi$
$500$	7.51000e8	0.268686
$501$	−3.74434e8	−0.133028
$502$	−2.63668e9	−0.930239
$503$	−4.52601e9	−1.58572	−0.792862	−	0.609401i	$-0.791410\pi$
$503$	−4.52601e9	−1.58572	−0.792862	+	0.609401i	$0.791410\pi$
$504$	1.28024e8	0.0445435
$505$	1.29650e9	0.447974
$506$	3.30063e9	1.13258
$507$	1.30324e8	0.0444116
$508$	1.85796e9	0.628803
$509$	−4.99671e9	−1.67947	−0.839734	−	0.542998i	$-0.817289\pi$
$509$	−4.99671e9	−1.67947	−0.839734	+	0.542998i	$0.817289\pi$
$510$	−4.37078e8	−0.145903
$511$	−1.32109e9	−0.437985
$512$	−1.34218e8	−0.0441942
$513$	−2.42508e8	−0.0793076
$514$	−2.67157e9	−0.867752
$515$	−3.41737e8	−0.110247
$516$	−7.41385e8	−0.237558
$517$	−5.51350e9	−1.75473
$518$	5.17760e8	0.163672
$519$	−2.15484e9	−0.676596
$520$	8.79140e7	0.0274187
$521$	−3.97826e9	−1.23243	−0.616213	−	0.787579i	$-0.711334\pi$
$521$	−3.97826e9	−1.23243	−0.616213	+	0.787579i	$0.711334\pi$
$522$	2.13371e8	0.0656580
$523$	−2.36286e9	−0.722240	−0.361120	−	0.932519i	$-0.617605\pi$
$523$	−2.36286e9	−0.722240	−0.361120	+	0.932519i	$0.617605\pi$
$524$	−6.54959e8	−0.198863
$525$	6.66947e8	0.201156
$526$	−8.36507e8	−0.250622
$527$	6.71347e9	1.99807
$528$	−7.21555e8	−0.213329
$529$	5.93910e8	0.174432
$530$	4.68270e8	0.136625
$531$	−9.47620e8	−0.274665
$532$	2.70463e8	0.0778785
$533$	−7.60802e7	−0.0217634
$534$	−2.69991e9	−0.767281
$535$	1.23004e9	0.347282
$536$	1.13063e9	0.317134
$537$	3.33988e9	0.930725
$538$	2.61594e9	0.724253
$539$	−7.67598e8	−0.211141
$540$	−9.84531e7	−0.0269061
$541$	−4.59709e9	−1.24822	−0.624112	−	0.781335i	$-0.714539\pi$
$541$	−4.59709e9	−1.24822	−0.624112	+	0.781335i	$0.714539\pi$
$542$	2.17774e9	0.587500
$543$	−2.73576e9	−0.733295
$544$	8.48393e8	0.225944
$545$	7.73107e8	0.204575
$546$	1.62771e8	0.0427960
$547$	−3.13179e9	−0.818157	−0.409078	−	0.912499i	$-0.634150\pi$
$547$	−3.13179e9	−0.818157	−0.409078	+	0.912499i	$0.634150\pi$
$548$	1.88227e9	0.488597
$549$	−2.17965e9	−0.562191
$550$	−3.75897e9	−0.963385
$551$	4.50766e8	0.114794
$552$	−8.74168e8	−0.221212
$553$	6.26283e8	0.157483
$554$	−4.41556e9	−1.10332
$555$	−3.98168e8	−0.0988647
$556$	−1.70510e9	−0.420714
$557$	4.04796e9	0.992529	0.496265	−	0.868171i	$-0.334705\pi$
$557$	4.04796e9	0.992529	0.496265	+	0.868171i	$0.334705\pi$
$558$	1.51223e9	0.368466
$559$	−9.42606e8	−0.228239
$560$	1.09803e8	0.0264213
$561$	4.56096e9	1.09065
$562$	−1.85505e9	−0.440838
$563$	1.58389e9	0.374064	0.187032	−	0.982354i	$-0.440113\pi$
$563$	1.58389e9	0.374064	0.187032	+	0.982354i	$0.440113\pi$
$564$	1.46024e9	0.342727
$565$	1.30413e9	0.304195
$566$	−8.81176e8	−0.204270
$567$	−1.82284e8	−0.0419961
$568$	−1.58927e9	−0.363897
$569$	6.15000e9	1.39953	0.699765	−	0.714373i	$-0.253288\pi$
$569$	6.15000e9	1.39953	0.699765	+	0.714373i	$0.253288\pi$
$570$	−2.07992e8	−0.0470419
$571$	−1.74647e9	−0.392586	−0.196293	−	0.980545i	$-0.562890\pi$
$571$	−1.74647e9	−0.392586	−0.196293	+	0.980545i	$0.562890\pi$
$572$	−9.17393e8	−0.204960
$573$	1.67829e9	0.372671
$574$	−9.50224e7	−0.0209717
$575$	−4.55402e9	−0.998981
$576$	1.91103e8	0.0416667
$577$	−7.54881e9	−1.63593	−0.817963	−	0.575271i	$-0.804896\pi$
$577$	−7.54881e9	−1.63593	−0.817963	+	0.575271i	$0.804896\pi$
$578$	−2.08000e9	−0.448039
$579$	−3.19683e8	−0.0684454
$580$	1.83002e8	0.0389455
$581$	−3.46808e9	−0.733622
$582$	−2.86006e9	−0.601374
$583$	−4.88645e9	−1.02130
$584$	−1.97201e9	−0.409697
$585$	−1.25174e8	−0.0258506
$586$	−5.61180e8	−0.115202
$587$	−2.22579e9	−0.454204	−0.227102	−	0.973871i	$-0.572925\pi$
$587$	−2.22579e9	−0.454204	−0.227102	+	0.973871i	$0.572925\pi$
$588$	2.03297e8	0.0412393
$589$	3.19473e9	0.644215
$590$	−8.12746e8	−0.162919
$591$	−1.61151e9	−0.321128
$592$	7.72867e8	0.153101
$593$	4.48628e9	0.883477	0.441739	−	0.897144i	$-0.354362\pi$
$593$	4.48628e9	0.883477	0.441739	+	0.897144i	$0.354362\pi$
$594$	1.02737e9	0.201129
$595$	−6.94064e8	−0.135080
$596$	−2.15512e9	−0.416975
$597$	−4.94535e9	−0.951233
$598$	−1.11143e9	−0.212533
$599$	6.27638e9	1.19321	0.596603	−	0.802537i	$-0.296517\pi$
$599$	6.27638e9	1.19321	0.596603	+	0.802537i	$0.296517\pi$
$600$	9.95560e8	0.188165
$601$	−2.51940e9	−0.473409	−0.236705	−	0.971582i	$-0.576067\pi$
$601$	−2.51940e9	−0.473409	−0.236705	+	0.971582i	$0.576067\pi$
$602$	−1.17729e9	−0.219936
$603$	−1.60982e9	−0.298997
$604$	−2.73190e9	−0.504469
$605$	−1.80395e9	−0.331193
$606$	3.58318e9	0.654054
$607$	6.87475e8	0.124766	0.0623830	−	0.998052i	$-0.480130\pi$
$607$	6.87475e8	0.124766	0.0623830	+	0.998052i	$0.480130\pi$
$608$	4.03724e8	0.0728487
$609$	3.38825e8	0.0607875
$610$	−1.86942e9	−0.333467
$611$	1.85657e9	0.329281
$612$	−1.20797e9	−0.213022
$613$	−3.94694e9	−0.692067	−0.346034	−	0.938222i	$-0.612472\pi$
$613$	−3.94694e9	−0.692067	−0.346034	+	0.938222i	$0.612472\pi$
$614$	2.33283e9	0.406719
$615$	7.30742e7	0.0126678
$616$	−1.14580e9	−0.197505
$617$	1.00788e10	1.72747	0.863736	−	0.503944i	$-0.168118\pi$
$617$	1.00788e10	1.72747	0.863736	+	0.503944i	$0.168118\pi$
$618$	−9.44468e8	−0.160963
$619$	−2.41567e9	−0.409374	−0.204687	−	0.978827i	$-0.565618\pi$
$619$	−2.41567e9	−0.409374	−0.204687	+	0.978827i	$0.565618\pi$
$620$	1.29699e9	0.218558
$621$	1.24467e9	0.208560
$622$	−7.91781e9	−1.31928
$623$	−4.28735e9	−0.710364
$624$	2.42971e8	0.0400320
$625$	4.70921e9	0.771556
$626$	−7.46275e9	−1.21587
$627$	2.17042e9	0.351647
$628$	8.79908e8	0.141768
$629$	−4.88531e9	−0.782735
$630$	−1.56340e8	−0.0249102
$631$	4.08225e9	0.646841	0.323420	−	0.946255i	$-0.395167\pi$
$631$	4.08225e9	0.646841	0.323420	+	0.946255i	$0.395167\pi$
$632$	9.34859e8	0.147311
$633$	2.55172e9	0.399872
$634$	−2.93355e9	−0.457173
$635$	−2.26890e9	−0.351648
$636$	1.29417e9	0.199476
$637$	2.58475e8	0.0396214
$638$	−1.90965e9	−0.291126
$639$	2.26284e9	0.343085
$640$	1.63903e8	0.0247149
$641$	−3.51474e9	−0.527097	−0.263549	−	0.964646i	$-0.584893\pi$
$641$	−3.51474e9	−0.527097	−0.263549	+	0.964646i	$0.584893\pi$
$642$	3.39950e9	0.507041
$643$	−6.85151e9	−1.01636	−0.508180	−	0.861251i	$-0.669682\pi$
$643$	−6.85151e9	−1.01636	−0.508180	+	0.861251i	$0.669682\pi$
$644$	−1.38815e9	−0.204802
$645$	9.05362e8	0.132851
$646$	−2.55195e9	−0.372441
$647$	−4.62798e9	−0.671780	−0.335890	−	0.941901i	$-0.609037\pi$
$647$	−4.62798e9	−0.671780	−0.335890	+	0.941901i	$0.609037\pi$
$648$	−2.72098e8	−0.0392837
$649$	8.48111e9	1.21786
$650$	1.26577e9	0.180783
$651$	2.40136e9	0.341133
$652$	1.53291e9	0.216596
$653$	−1.24384e10	−1.74810	−0.874052	−	0.485833i	$-0.838516\pi$
$653$	−1.24384e10	−1.74810	−0.874052	+	0.485833i	$0.838516\pi$
$654$	2.13666e9	0.298685
$655$	7.99820e8	0.111211
$656$	−1.41841e8	−0.0196173
$657$	2.80780e9	0.386266
$658$	2.31881e9	0.317304
$659$	−1.80418e9	−0.245573	−0.122787	−	0.992433i	$-0.539183\pi$
$659$	−1.80418e9	−0.245573	−0.122787	+	0.992433i	$0.539183\pi$
$660$	8.81145e8	0.119301
$661$	7.78788e8	0.104885	0.0524426	−	0.998624i	$-0.483299\pi$
$661$	7.78788e8	0.104885	0.0524426	+	0.998624i	$0.483299\pi$
$662$	−2.58011e9	−0.345648
$663$	−1.53582e9	−0.204665
$664$	−5.17684e9	−0.686241
$665$	−3.30283e8	−0.0435523
$666$	−1.10043e9	−0.144345
$667$	−2.31355e9	−0.301883
$668$	−8.87548e8	−0.115206
$669$	3.58766e9	0.463254
$670$	−1.38070e9	−0.177352
$671$	1.95077e10	2.49274
$672$	3.03464e8	0.0385758
$673$	−1.36063e9	−0.172063	−0.0860316	−	0.996292i	$-0.527419\pi$
$673$	−1.36063e9	−0.172063	−0.0860316	+	0.996292i	$0.527419\pi$
$674$	1.00577e10	1.26529
$675$	−1.41751e9	−0.177403
$676$	3.08916e8	0.0384615
$677$	4.75381e9	0.588818	0.294409	−	0.955679i	$-0.404877\pi$
$677$	4.75381e9	0.588818	0.294409	+	0.955679i	$0.404877\pi$
$678$	3.60427e9	0.444133
$679$	−4.54167e9	−0.556765
$680$	−1.03604e9	−0.126355
$681$	7.79740e9	0.946095
$682$	−1.35343e10	−1.63377
$683$	2.21855e9	0.266438	0.133219	−	0.991087i	$-0.457469\pi$
$683$	2.21855e9	0.266438	0.133219	+	0.991087i	$0.457469\pi$
$684$	−5.74833e8	−0.0686824
$685$	−2.29859e9	−0.273240
$686$	3.22829e8	0.0381802
$687$	5.38065e9	0.633120
$688$	−1.75736e9	−0.205731
$689$	1.64542e9	0.191651
$690$	1.06751e9	0.123709
$691$	1.42944e10	1.64813	0.824066	−	0.566494i	$-0.191701\pi$
$691$	1.42944e10	1.64813	0.824066	+	0.566494i	$0.191701\pi$
$692$	−5.10777e9	−0.585950
$693$	1.63143e9	0.186209
$694$	5.62182e9	0.638438
$695$	2.08222e9	0.235278
$696$	5.05767e8	0.0568615
$697$	8.96580e8	0.100294
$698$	−8.66040e9	−0.963927
$699$	2.09348e9	0.231846
$700$	1.58091e9	0.174207
$701$	1.73952e10	1.90729	0.953645	−	0.300934i	$-0.0972985\pi$
$701$	1.73952e10	1.90729	0.953645	+	0.300934i	$0.0972985\pi$
$702$	−3.45948e8	−0.0377426
$703$	−2.32477e9	−0.252369
$704$	−1.71035e9	−0.184749
$705$	−1.78321e9	−0.191664
$706$	−5.79846e9	−0.620149
$707$	5.68995e9	0.605537
$708$	−2.24621e9	−0.237867
$709$	−1.49260e9	−0.157283	−0.0786417	−	0.996903i	$-0.525058\pi$
$709$	−1.49260e9	−0.157283	−0.0786417	+	0.996903i	$0.525058\pi$
$710$	1.94078e9	0.203503
$711$	−1.33108e9	−0.138887
$712$	−6.39978e9	−0.664485
$713$	−1.63969e10	−1.69413
$714$	−1.91821e9	−0.197220
$715$	1.12030e9	0.114621
$716$	7.91676e9	0.806032
$717$	9.81043e9	0.993965
$718$	9.35878e8	0.0943591
$719$	−4.16119e8	−0.0417509	−0.0208755	−	0.999782i	$-0.506645\pi$
$719$	−4.16119e8	−0.0417509	−0.0208755	+	0.999782i	$0.506645\pi$
$720$	−2.33370e8	−0.0233014
$721$	−1.49978e9	−0.149023
$722$	5.93658e9	0.587025
$723$	−1.26974e9	−0.124949
$724$	−6.48476e9	−0.635052
$725$	2.63482e9	0.256784
$726$	−4.98563e9	−0.483550
$727$	1.97494e10	1.90626	0.953132	−	0.302555i	$-0.0978396\pi$
$727$	1.97494e10	1.90626	0.953132	+	0.302555i	$0.0978396\pi$
$728$	3.85828e8	0.0370625
$729$	3.87420e8	0.0370370
$730$	2.40817e9	0.229116
$731$	1.11083e10	1.05181
$732$	−5.16658e9	−0.486872
$733$	−6.45203e9	−0.605108	−0.302554	−	0.953132i	$-0.597839\pi$
$733$	−6.45203e9	−0.605108	−0.302554	+	0.953132i	$0.597839\pi$
$734$	−8.04367e9	−0.750789
$735$	−2.48262e8	−0.0230624
$736$	−2.07210e9	−0.191575
$737$	1.44077e10	1.32574
$738$	2.01957e8	0.0184953
$739$	−3.00766e9	−0.274141	−0.137070	−	0.990561i	$-0.543769\pi$
$739$	−3.00766e9	−0.274141	−0.137070	+	0.990561i	$0.543769\pi$
$740$	−9.43806e8	−0.0856193
$741$	−7.30850e8	−0.0659879
$742$	2.05509e9	0.184679
$743$	−1.48884e10	−1.33164	−0.665819	−	0.746113i	$-0.731918\pi$
$743$	−1.48884e10	−1.33164	−0.665819	+	0.746113i	$0.731918\pi$
$744$	3.58454e9	0.319101
$745$	2.63178e9	0.233186
$746$	−1.18734e10	−1.04710
$747$	7.37093e9	0.646994
$748$	1.08112e10	0.944533
$749$	5.39829e9	0.469429
$750$	−2.53462e9	−0.219381
$751$	−2.73695e9	−0.235791	−0.117895	−	0.993026i	$-0.537615\pi$
$751$	−2.73695e9	−0.235791	−0.117895	+	0.993026i	$0.537615\pi$
$752$	3.46132e9	0.296810
$753$	8.89880e9	0.759537
$754$	6.43039e8	0.0546308
$755$	3.33613e9	0.282116
$756$	−4.32081e8	−0.0363696
$757$	−1.39749e10	−1.17088	−0.585442	−	0.810714i	$-0.699079\pi$
$757$	−1.39749e10	−1.17088	−0.585442	+	0.810714i	$0.699079\pi$
$758$	−4.63782e9	−0.386787
$759$	−1.11396e10	−0.924750
$760$	−4.93018e8	−0.0407394
$761$	−2.16850e10	−1.78366	−0.891831	−	0.452369i	$-0.850579\pi$
$761$	−2.16850e10	−1.78366	−0.891831	+	0.452369i	$0.850579\pi$
$762$	−6.27063e9	−0.513415
$763$	3.39294e9	0.276528
$764$	3.97817e9	0.322743
$765$	1.47514e9	0.119129
$766$	−1.94900e8	−0.0156679
$767$	−2.85586e9	−0.228535
$768$	4.52985e8	0.0360844
$769$	−2.42150e10	−1.92018	−0.960091	−	0.279689i	$-0.909769\pi$
$769$	−2.42150e10	−1.92018	−0.960091	+	0.279689i	$0.909769\pi$
$770$	1.39923e9	0.110451
$771$	9.01655e9	0.708516
$772$	−7.57766e8	−0.0592754
$773$	−1.74396e9	−0.135803	−0.0679013	−	0.997692i	$-0.521630\pi$
$773$	−1.74396e9	−0.135803	−0.0679013	+	0.997692i	$0.521630\pi$
$774$	2.50218e9	0.193965
$775$	1.86738e10	1.44104
$776$	−6.77941e9	−0.520806
$777$	−1.74744e9	−0.133638
$778$	4.29094e9	0.326681
$779$	4.26654e8	0.0323367
$780$	−2.96710e8	−0.0223873
$781$	−2.02522e10	−1.52123
$782$	1.30978e10	0.979433
$783$	−7.20126e8	−0.0536096
$784$	4.81890e8	0.0357143
$785$	−1.07452e9	−0.0792815
$786$	2.21049e9	0.162371
$787$	−4.62315e9	−0.338086	−0.169043	−	0.985609i	$-0.554068\pi$
$787$	−4.62315e9	−0.338086	−0.169043	+	0.985609i	$0.554068\pi$
$788$	−3.81988e9	−0.278105
$789$	2.82321e9	0.204632
$790$	−1.14163e9	−0.0823815
$791$	5.72345e9	0.411188
$792$	2.43525e9	0.174183
$793$	−6.56885e9	−0.467771
$794$	8.02483e7	0.00568937
$795$	−1.58041e9	−0.111554
$796$	−1.17223e10	−0.823792
$797$	1.68675e10	1.18018	0.590088	−	0.807339i	$-0.299093\pi$
$797$	1.68675e10	1.18018	0.590088	+	0.807339i	$0.299093\pi$
$798$	−9.12814e8	−0.0635876
$799$	−2.18791e10	−1.51745
$800$	2.35985e9	0.162955
$801$	9.11219e9	0.626483
$802$	−1.95186e9	−0.133610
$803$	−2.51295e10	−1.71269
$804$	−3.81587e9	−0.258939
$805$	1.69517e9	0.114532
$806$	4.55743e9	0.306582
$807$	−8.82881e9	−0.591350
$808$	8.49345e9	0.566428
$809$	1.76998e10	1.17530	0.587649	−	0.809116i	$-0.300054\pi$
$809$	1.76998e10	1.17530	0.587649	+	0.809116i	$0.300054\pi$
$810$	3.32279e8	0.0219688
$811$	−7.30027e9	−0.480580	−0.240290	−	0.970701i	$-0.577243\pi$
$811$	−7.30027e9	−0.480580	−0.240290	+	0.970701i	$0.577243\pi$
$812$	8.03140e8	0.0526435
$813$	−7.34986e9	−0.479692
$814$	9.84873e9	0.640022
$815$	−1.87196e9	−0.121128
$816$	−2.86333e9	−0.184483
$817$	5.28609e9	0.339123
$818$	−1.44972e10	−0.926078
$819$	−5.49353e8	−0.0349428
$820$	1.73213e8	0.0109706
$821$	5.46698e9	0.344783	0.172392	−	0.985028i	$-0.444851\pi$
$821$	5.46698e9	0.344783	0.172392	+	0.985028i	$0.444851\pi$
$822$	−6.35267e9	−0.398938
$823$	2.01402e10	1.25940	0.629701	−	0.776837i	$-0.283177\pi$
$823$	2.01402e10	1.25940	0.629701	+	0.776837i	$0.283177\pi$
$824$	−2.23874e9	−0.139398
$825$	1.26865e10	0.786600
$826$	−3.56690e9	−0.220222
$827$	1.18009e10	0.725515	0.362757	−	0.931884i	$-0.381835\pi$
$827$	1.18009e10	0.725515	0.362757	+	0.931884i	$0.381835\pi$
$828$	2.95032e9	0.180619
$829$	1.63838e10	0.998791	0.499396	−	0.866374i	$-0.333555\pi$
$829$	1.63838e10	0.998791	0.499396	+	0.866374i	$0.333555\pi$
$830$	6.32183e9	0.383769
$831$	1.49025e10	0.900857
$832$	5.75930e8	0.0346688
$833$	−3.04604e9	−0.182590
$834$	5.75470e9	0.343512
$835$	1.08385e9	0.0644269
$836$	5.14470e9	0.304536
$837$	−5.10377e9	−0.300851
$838$	1.77736e10	1.04333
$839$	−1.80326e10	−1.05413	−0.527063	−	0.849826i	$-0.676707\pi$
$839$	−1.80326e10	−1.05413	−0.527063	+	0.849826i	$0.676707\pi$
$840$	−3.70583e8	−0.0215729
$841$	−1.59113e10	−0.922402
$842$	6.96170e9	0.401904
$843$	6.26081e9	0.359943
$844$	6.04853e9	0.346299
$845$	−3.77241e8	−0.0215090
$846$	−4.92832e9	−0.279835
$847$	−7.91699e9	−0.447681
$848$	3.06766e9	0.172752
$849$	2.97397e9	0.166786
$850$	−1.49166e10	−0.833114
$851$	1.19318e10	0.663671
$852$	5.36378e9	0.297120
$853$	−1.79290e10	−0.989089	−0.494545	−	0.869152i	$-0.664665\pi$
$853$	−1.79290e10	−0.989089	−0.494545	+	0.869152i	$0.664665\pi$
$854$	−8.20434e9	−0.450756
$855$	7.01973e8	0.0384095
$856$	8.05808e9	0.439110
$857$	1.93790e10	1.05172	0.525858	−	0.850572i	$-0.323744\pi$
$857$	1.93790e10	1.05172	0.525858	+	0.850572i	$0.323744\pi$
$858$	3.09620e9	0.167349
$859$	−2.16954e10	−1.16786	−0.583930	−	0.811804i	$-0.698486\pi$
$859$	−2.16954e10	−1.16786	−0.583930	+	0.811804i	$0.698486\pi$
$860$	2.14604e9	0.115052
$861$	3.20701e8	0.0171233
$862$	2.25131e10	1.19718
$863$	−9.19005e9	−0.486721	−0.243361	−	0.969936i	$-0.578250\pi$
$863$	−9.19005e9	−0.486721	−0.243361	+	0.969936i	$0.578250\pi$
$864$	−6.44973e8	−0.0340207
$865$	6.23749e9	0.327683
$866$	2.10575e10	1.10178
$867$	7.02000e9	0.365822
$868$	5.69212e9	0.295430
$869$	1.19130e10	0.615819
$870$	−6.17631e8	−0.0317989
$871$	−4.85154e9	−0.248780
$872$	5.06467e9	0.258669
$873$	9.65271e9	0.491020
$874$	6.23284e9	0.315788
$875$	−4.02489e9	−0.203107
$876$	6.65552e9	0.334516
$877$	−2.31492e9	−0.115888	−0.0579440	−	0.998320i	$-0.518454\pi$
$877$	−2.31492e9	−0.115888	−0.0579440	+	0.998320i	$0.518454\pi$
$878$	−5.83400e9	−0.290894
$879$	1.89398e9	0.0940621
$880$	2.08864e9	0.103318
$881$	−8.52180e8	−0.0419871	−0.0209935	−	0.999780i	$-0.506683\pi$
$881$	−8.52180e8	−0.0419871	−0.0209935	+	0.999780i	$0.506683\pi$
$882$	−6.86129e8	−0.0336718
$883$	4.34591e9	0.212431	0.106216	−	0.994343i	$-0.466127\pi$
$883$	4.34591e9	0.212431	0.106216	+	0.994343i	$0.466127\pi$
$884$	−3.64047e9	−0.177245
$885$	2.74302e9	0.133023
$886$	−1.77764e10	−0.858669
$887$	2.45947e10	1.18334	0.591668	−	0.806181i	$-0.298469\pi$
$887$	2.45947e10	1.18334	0.591668	+	0.806181i	$0.298469\pi$
$888$	−2.60843e9	−0.125007
$889$	−9.95753e9	−0.475330
$890$	7.81526e9	0.371602
$891$	−3.46737e9	−0.164221
$892$	8.50407e9	0.401190
$893$	−1.04116e10	−0.489256
$894$	7.27353e9	0.340459
$895$	−9.66776e9	−0.450760
$896$	7.19323e8	0.0334077
$897$	3.75107e9	0.173533
$898$	−1.05786e10	−0.487485
$899$	9.48674e9	0.435470
$900$	−3.36001e9	−0.153636
$901$	−1.93908e10	−0.883198
$902$	−1.80750e9	−0.0820077
$903$	3.97336e9	0.179577
$904$	8.54345e9	0.384631
$905$	7.91904e9	0.355142
$906$	9.22015e9	0.411898
$907$	1.11880e10	0.497882	0.248941	−	0.968519i	$-0.419917\pi$
$907$	1.11880e10	0.497882	0.248941	+	0.968519i	$0.419917\pi$
$908$	1.84827e10	0.819342
$909$	−1.20932e10	−0.534033
$910$	−4.71164e8	−0.0207266
$911$	−2.12068e10	−0.929311	−0.464655	−	0.885492i	$-0.653822\pi$
$911$	−2.12068e10	−0.929311	−0.464655	+	0.885492i	$0.653822\pi$
$912$	−1.36257e9	−0.0594807
$913$	−6.59691e10	−2.86875
$914$	−2.37226e10	−1.02766
$915$	6.30931e9	0.272275
$916$	1.27541e10	0.548298
$917$	3.51017e9	0.150327
$918$	4.07688e9	0.173932
$919$	8.50251e9	0.361362	0.180681	−	0.983542i	$-0.442170\pi$
$919$	8.50251e9	0.361362	0.180681	+	0.983542i	$0.442170\pi$
$920$	2.53040e9	0.107135
$921$	−7.87331e9	−0.332084
$922$	−2.70452e10	−1.13640
$923$	6.81957e9	0.285464
$924$	3.86708e9	0.161262
$925$	−1.35887e10	−0.564524
$926$	1.38724e10	0.574135
$927$	3.18758e9	0.131426
$928$	1.19886e9	0.0492435
$929$	−4.36689e9	−0.178697	−0.0893486	−	0.996000i	$-0.528479\pi$
$929$	−4.36689e9	−0.178697	−0.0893486	+	0.996000i	$0.528479\pi$
$930$	−4.37736e9	−0.178452
$931$	−1.44951e9	−0.0588706
$932$	4.96233e9	0.200784
$933$	2.67226e10	1.07719
$934$	−1.45354e10	−0.583732
$935$	−1.32023e10	−0.528215
$936$	−8.20026e8	−0.0326860
$937$	4.33762e10	1.72252	0.861258	−	0.508168i	$-0.169677\pi$
$937$	4.33762e10	1.72252	0.861258	+	0.508168i	$0.169677\pi$
$938$	−6.05946e9	−0.239731
$939$	2.51868e10	0.992757
$940$	−4.22688e9	−0.165986
$941$	−5.65734e9	−0.221334	−0.110667	−	0.993858i	$-0.535299\pi$
$941$	−5.65734e9	−0.221334	−0.110667	+	0.993858i	$0.535299\pi$
$942$	−2.96969e9	−0.115753
$943$	−2.18979e9	−0.0850378
$944$	−5.32435e9	−0.205999
$945$	5.27647e8	0.0203391
$946$	−2.23942e10	−0.860037
$947$	−6.50917e9	−0.249058	−0.124529	−	0.992216i	$-0.539742\pi$
$947$	−6.50917e9	−0.249058	−0.124529	+	0.992216i	$0.539742\pi$
$948$	−3.15515e9	−0.120279
$949$	8.46190e9	0.321393
$950$	−7.09836e9	−0.268612
$951$	9.90072e9	0.373280
$952$	−4.54686e9	−0.170798
$953$	−2.12326e10	−0.794653	−0.397326	−	0.917677i	$-0.630062\pi$
$953$	−2.12326e10	−0.794653	−0.397326	+	0.917677i	$0.630062\pi$
$954$	−4.36783e9	−0.162872
$955$	−4.85805e9	−0.180489
$956$	2.32544e10	0.860799
$957$	6.44505e9	0.237703
$958$	1.12551e10	0.413590
$959$	−1.00878e10	−0.369345
$960$	−5.53174e8	−0.0201796
$961$	3.97231e10	1.44381
$962$	−3.31638e9	−0.120102
$963$	−1.14733e10	−0.413997
$964$	−3.00976e9	−0.108209
$965$	9.25366e8	0.0331488
$966$	4.68500e9	0.167220
$967$	2.82542e10	1.00483	0.502413	−	0.864628i	$-0.332446\pi$
$967$	2.82542e10	1.00483	0.502413	+	0.864628i	$0.332446\pi$
$968$	−1.18178e10	−0.418767
$969$	8.61282e9	0.304097
$970$	8.27885e9	0.291252
$971$	2.89562e9	0.101502	0.0507510	−	0.998711i	$-0.483839\pi$
$971$	2.89562e9	0.101502	0.0507510	+	0.998711i	$0.483839\pi$
$972$	9.18330e8	0.0320750
$973$	9.13825e9	0.318030
$974$	−2.05365e10	−0.712148
$975$	−4.27196e9	−0.147608
$976$	−1.22467e10	−0.421643
$977$	−5.36216e10	−1.83954	−0.919768	−	0.392462i	$-0.871624\pi$
$977$	−5.36216e10	−1.83954	−0.919768	+	0.392462i	$0.871624\pi$
$978$	−5.17358e9	−0.176850
$979$	−8.15532e10	−2.77780
$980$	−5.88473e8	−0.0199726
$981$	−7.21122e9	−0.243875
$982$	−3.30051e10	−1.11222
$983$	−3.68484e9	−0.123732	−0.0618659	−	0.998084i	$-0.519705\pi$
$983$	−3.68484e9	−0.123732	−0.0618659	+	0.998084i	$0.519705\pi$
$984$	4.78713e8	0.0160174
$985$	4.66475e9	0.155525
$986$	−7.57799e9	−0.251759
$987$	−7.82599e9	−0.259077
$988$	−1.73239e9	−0.0571472
$989$	−2.71307e10	−0.891815
$990$	−2.97387e9	−0.0974088
$991$	1.08275e10	0.353402	0.176701	−	0.984265i	$-0.443457\pi$
$991$	1.08275e10	0.353402	0.176701	+	0.984265i	$0.443457\pi$
$992$	8.49669e9	0.276350
$993$	8.70786e9	0.282221
$994$	8.51748e9	0.275080
$995$	1.43150e10	0.460692
$996$	1.74718e10	0.560313
$997$	3.84252e10	1.22796	0.613978	−	0.789323i	$-0.289568\pi$
$997$	3.84252e10	1.22796	0.613978	+	0.789323i	$0.289568\pi$
$998$	2.43108e10	0.774182
$999$	3.71395e9	0.117857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.k.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.k.1.2	✓	5	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)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -78.1553 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -78.1553 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +625.242 q^{10} -6524.48 q^{11} +1728.00 q^{12} +2197.00 q^{13} +2744.00 q^{14} -2110.19 q^{15} +4096.00 q^{16} -25890.9 q^{17} -5832.00 q^{18} -12320.7 q^{19} -5001.94 q^{20} -9261.00 q^{21} +52195.8 q^{22} +63235.6 q^{23} -13824.0 q^{24} -72016.8 q^{25} -17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} -36586.2 q^{29} +16881.5 q^{30} -259298. q^{31} -32768.0 q^{32} -176161. q^{33} +207127. q^{34} +26807.3 q^{35} +46656.0 q^{36} +188688. q^{37} +98565.4 q^{38} +59319.0 q^{39} +40015.5 q^{40} -34629.2 q^{41} +74088.0 q^{42} -429043. q^{43} -417566. q^{44} -56975.2 q^{45} -505884. q^{46} +845048. q^{47} +110592. q^{48} +117649. q^{49} +576134. q^{50} -699054. q^{51} +140608. q^{52} +748941. q^{53} -157464. q^{54} +509922. q^{55} +175616. q^{56} -332658. q^{57} +292689. q^{58} -1.29989e6 q^{59} -135052. q^{60} -2.98992e6 q^{61} +2.07439e6 q^{62} -250047. q^{63} +262144. q^{64} -171707. q^{65} +1.40929e6 q^{66} -2.20826e6 q^{67} -1.65702e6 q^{68} +1.70736e6 q^{69} -214458. q^{70} +3.10404e6 q^{71} -373248. q^{72} +3.85157e6 q^{73} -1.50951e6 q^{74} -1.94445e6 q^{75} -788523. q^{76} +2.23790e6 q^{77} -474552. q^{78} -1.82590e6 q^{79} -320124. q^{80} +531441. q^{81} +277033. q^{82} +1.01110e7 q^{83} -592704. q^{84} +2.02351e6 q^{85} +3.43234e6 q^{86} -987827. q^{87} +3.34053e6 q^{88} +1.24996e7 q^{89} +455801. q^{90} -753571. q^{91} +4.04708e6 q^{92} -7.00106e6 q^{93} -6.76038e6 q^{94} +962925. q^{95} -884736. q^{96} +1.32410e7 q^{97} -941192. q^{98} -4.75634e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 509 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 4072 q^{10} + 958 q^{11} + 8640 q^{12} + 10985 q^{13} + 13720 q^{14} + 13743 q^{15} + 20480 q^{16} + 7864 q^{17} - 29160 q^{18} + 60173 q^{19} + 32576 q^{20} - 46305 q^{21} - 7664 q^{22} + 122869 q^{23} - 69120 q^{24} + 73722 q^{25} - 87880 q^{26} + 98415 q^{27} - 109760 q^{28} - 17317 q^{29} - 109944 q^{30} - 177665 q^{31} - 163840 q^{32} + 25866 q^{33} - 62912 q^{34} - 174587 q^{35} + 233280 q^{36} - 55136 q^{37} - 481384 q^{38} + 296595 q^{39} - 260608 q^{40} - 237570 q^{41} + 370440 q^{42} - 970601 q^{43} + 61312 q^{44} + 371061 q^{45} - 982952 q^{46} - 384035 q^{47} + 552960 q^{48} + 588245 q^{49} - 589776 q^{50} + 212328 q^{51} + 703040 q^{52} - 1977 q^{53} - 787320 q^{54} + 1520014 q^{55} + 878080 q^{56} + 1624671 q^{57} + 138536 q^{58} + 2057936 q^{59} + 879552 q^{60} - 723756 q^{61} + 1421320 q^{62} - 1250235 q^{63} + 1310720 q^{64} + 1118273 q^{65} - 206928 q^{66} + 2695018 q^{67} + 503296 q^{68} + 3317463 q^{69} + 1396696 q^{70} + 7392916 q^{71} - 1866240 q^{72} + 8720441 q^{73} + 441088 q^{74} + 1990494 q^{75} + 3851072 q^{76} - 328594 q^{77} - 2372760 q^{78} + 4646419 q^{79} + 2084864 q^{80} + 2657205 q^{81} + 1900560 q^{82} + 17766733 q^{83} - 2963520 q^{84} + 16495320 q^{85} + 7764808 q^{86} - 467559 q^{87} - 490496 q^{88} + 4692321 q^{89} - 2968488 q^{90} - 3767855 q^{91} + 7863616 q^{92} - 4796955 q^{93} + 3072280 q^{94} - 2355945 q^{95} - 4423680 q^{96} + 15680305 q^{97} - 4705960 q^{98} + 698382 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 509 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9 - 4072 * q^10 + 958 * q^11 + 8640 * q^12 + 10985 * q^13 + 13720 * q^14 + 13743 * q^15 + 20480 * q^16 + 7864 * q^17 - 29160 * q^18 + 60173 * q^19 + 32576 * q^20 - 46305 * q^21 - 7664 * q^22 + 122869 * q^23 - 69120 * q^24 + 73722 * q^25 - 87880 * q^26 + 98415 * q^27 - 109760 * q^28 - 17317 * q^29 - 109944 * q^30 - 177665 * q^31 - 163840 * q^32 + 25866 * q^33 - 62912 * q^34 - 174587 * q^35 + 233280 * q^36 - 55136 * q^37 - 481384 * q^38 + 296595 * q^39 - 260608 * q^40 - 237570 * q^41 + 370440 * q^42 - 970601 * q^43 + 61312 * q^44 + 371061 * q^45 - 982952 * q^46 - 384035 * q^47 + 552960 * q^48 + 588245 * q^49 - 589776 * q^50 + 212328 * q^51 + 703040 * q^52 - 1977 * q^53 - 787320 * q^54 + 1520014 * q^55 + 878080 * q^56 + 1624671 * q^57 + 138536 * q^58 + 2057936 * q^59 + 879552 * q^60 - 723756 * q^61 + 1421320 * q^62 - 1250235 * q^63 + 1310720 * q^64 + 1118273 * q^65 - 206928 * q^66 + 2695018 * q^67 + 503296 * q^68 + 3317463 * q^69 + 1396696 * q^70 + 7392916 * q^71 - 1866240 * q^72 + 8720441 * q^73 + 441088 * q^74 + 1990494 * q^75 + 3851072 * q^76 - 328594 * q^77 - 2372760 * q^78 + 4646419 * q^79 + 2084864 * q^80 + 2657205 * q^81 + 1900560 * q^82 + 17766733 * q^83 - 2963520 * q^84 + 16495320 * q^85 + 7764808 * q^86 - 467559 * q^87 - 490496 * q^88 + 4692321 * q^89 - 2968488 * q^90 - 3767855 * q^91 + 7863616 * q^92 - 4796955 * q^93 + 3072280 * q^94 - 2355945 * q^95 - 4423680 * q^96 + 15680305 * q^97 - 4705960 * q^98 + 698382 * q^99

Introduction

L-functions

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