LMFDB - Embedded newform 546.8.a.q.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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 367021x^{4} - 17702143x^{3} + 34815194576x^{2} + 1422988371620x - 933871993059968 $ x^6 - x^5 - 367021x^4 - 17702143x^3 + 34815194576x^2 + 1422988371620x - 933871993059968
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-419.580$ 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)

$f(q)$	$=$	$q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -417.580 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -417.580 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -3340.64 q^{10} +5214.78 q^{11} -1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +11274.7 q^{15} +4096.00 q^{16} -9289.46 q^{17} +5832.00 q^{18} -28641.6 q^{19} -26725.1 q^{20} -9261.00 q^{21} +41718.2 q^{22} -70760.2 q^{23} -13824.0 q^{24} +96248.1 q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} +40590.0 q^{29} +90197.3 q^{30} +239911. q^{31} +32768.0 q^{32} -140799. q^{33} -74315.7 q^{34} -143230. q^{35} +46656.0 q^{36} +334019. q^{37} -229132. q^{38} +59319.0 q^{39} -213801. q^{40} -347373. q^{41} -74088.0 q^{42} -454134. q^{43} +333746. q^{44} -304416. q^{45} -566082. q^{46} -87902.0 q^{47} -110592. q^{48} +117649. q^{49} +769985. q^{50} +250815. q^{51} -140608. q^{52} -2.03091e6 q^{53} -157464. q^{54} -2.17759e6 q^{55} +175616. q^{56} +773322. q^{57} +324720. q^{58} -385876. q^{59} +721578. q^{60} -2.16065e6 q^{61} +1.91928e6 q^{62} +250047. q^{63} +262144. q^{64} +917423. q^{65} -1.12639e6 q^{66} +1.94749e6 q^{67} -594525. q^{68} +1.91053e6 q^{69} -1.14584e6 q^{70} +2.74370e6 q^{71} +373248. q^{72} -801961. q^{73} +2.67215e6 q^{74} -2.59870e6 q^{75} -1.83306e6 q^{76} +1.78867e6 q^{77} +474552. q^{78} -7.49135e6 q^{79} -1.71041e6 q^{80} +531441. q^{81} -2.77899e6 q^{82} +2.59240e6 q^{83} -592704. q^{84} +3.87909e6 q^{85} -3.63307e6 q^{86} -1.09593e6 q^{87} +2.66997e6 q^{88} +712280. q^{89} -2.43533e6 q^{90} -753571. q^{91} -4.52865e6 q^{92} -6.47759e6 q^{93} -703216. q^{94} +1.19601e7 q^{95} -884736. q^{96} +1.71130e7 q^{97} +941192. q^{98} +3.80157e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9}+O(q^{10}) $ 6 * q + 48 * q^2 - 162 * q^3 + 384 * q^4 + 13 * q^5 - 1296 * q^6 + 2058 * q^7 + 3072 * q^8 + 4374 * q^9	\( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 104 q^{10} + 10054 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} - 351 q^{15} + 24576 q^{16} + 21222 q^{17} + 34992 q^{18} + 9527 q^{19} + 832 q^{20} - 55566 q^{21} + 80432 q^{22} + 33229 q^{23} - 82944 q^{24} + 265321 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} + 174185 q^{29} - 2808 q^{30} + 119045 q^{31} + 196608 q^{32} - 271458 q^{33} + 169776 q^{34} + 4459 q^{35} + 279936 q^{36} + 56562 q^{37} + 76216 q^{38} + 355914 q^{39} + 6656 q^{40} + 101632 q^{41} - 444528 q^{42} + 441323 q^{43} + 643456 q^{44} + 9477 q^{45} + 265832 q^{46} - 892849 q^{47} - 663552 q^{48} + 705894 q^{49} + 2122568 q^{50} - 572994 q^{51} - 843648 q^{52} + 2093965 q^{53} - 944784 q^{54} - 331222 q^{55} + 1053696 q^{56} - 257229 q^{57} + 1393480 q^{58} - 136204 q^{59} - 22464 q^{60} - 3989946 q^{61} + 952360 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 28561 q^{65} - 2171664 q^{66} - 2218250 q^{67} + 1358208 q^{68} - 897183 q^{69} + 35672 q^{70} + 2045928 q^{71} + 2239488 q^{72} - 8557479 q^{73} + 452496 q^{74} - 7163667 q^{75} + 609728 q^{76} + 3448522 q^{77} + 2847312 q^{78} - 8559709 q^{79} + 53248 q^{80} + 3188646 q^{81} + 813056 q^{82} + 2496351 q^{83} - 3556224 q^{84} + 5335304 q^{85} + 3530584 q^{86} - 4702995 q^{87} + 5147648 q^{88} - 2446683 q^{89} + 75816 q^{90} - 4521426 q^{91} + 2126656 q^{92} - 3214215 q^{93} - 7142792 q^{94} + 16410211 q^{95} - 5308416 q^{96} + 5786889 q^{97} + 5647152 q^{98} + 7329366 q^{99}+O(q^{100}) \) 6 * q + 48 * q^2 - 162 * q^3 + 384 * q^4 + 13 * q^5 - 1296 * q^6 + 2058 * q^7 + 3072 * q^8 + 4374 * q^9 + 104 * q^10 + 10054 * q^11 - 10368 * q^12 - 13182 * q^13 + 16464 * q^14 - 351 * q^15 + 24576 * q^16 + 21222 * q^17 + 34992 * q^18 + 9527 * q^19 + 832 * q^20 - 55566 * q^21 + 80432 * q^22 + 33229 * q^23 - 82944 * q^24 + 265321 * q^25 - 105456 * q^26 - 118098 * q^27 + 131712 * q^28 + 174185 * q^29 - 2808 * q^30 + 119045 * q^31 + 196608 * q^32 - 271458 * q^33 + 169776 * q^34 + 4459 * q^35 + 279936 * q^36 + 56562 * q^37 + 76216 * q^38 + 355914 * q^39 + 6656 * q^40 + 101632 * q^41 - 444528 * q^42 + 441323 * q^43 + 643456 * q^44 + 9477 * q^45 + 265832 * q^46 - 892849 * q^47 - 663552 * q^48 + 705894 * q^49 + 2122568 * q^50 - 572994 * q^51 - 843648 * q^52 + 2093965 * q^53 - 944784 * q^54 - 331222 * q^55 + 1053696 * q^56 - 257229 * q^57 + 1393480 * q^58 - 136204 * q^59 - 22464 * q^60 - 3989946 * q^61 + 952360 * q^62 + 1500282 * q^63 + 1572864 * q^64 - 28561 * q^65 - 2171664 * q^66 - 2218250 * q^67 + 1358208 * q^68 - 897183 * q^69 + 35672 * q^70 + 2045928 * q^71 + 2239488 * q^72 - 8557479 * q^73 + 452496 * q^74 - 7163667 * q^75 + 609728 * q^76 + 3448522 * q^77 + 2847312 * q^78 - 8559709 * q^79 + 53248 * q^80 + 3188646 * q^81 + 813056 * q^82 + 2496351 * q^83 - 3556224 * q^84 + 5335304 * q^85 + 3530584 * q^86 - 4702995 * q^87 + 5147648 * q^88 - 2446683 * q^89 + 75816 * q^90 - 4521426 * q^91 + 2126656 * q^92 - 3214215 * q^93 - 7142792 * q^94 + 16410211 * q^95 - 5308416 * q^96 + 5786889 * q^97 + 5647152 * q^98 + 7329366 * 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
$5$	−417.580	−1.49398	−0.746990	−	0.664835i	$-0.768502\pi$
$5$	−417.580	−1.49398	−0.746990	+	0.664835i	$0.768502\pi$
$6$	−216.000	−0.408248
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−3340.64	−1.05640
$11$	5214.78	1.18130	0.590651	−	0.806927i	$-0.298871\pi$
$11$	5214.78	1.18130	0.590651	+	0.806927i	$0.298871\pi$
$12$	−1728.00	−0.288675
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	2744.00	0.267261
$15$	11274.7	0.862550
$16$	4096.00	0.250000
$17$	−9289.46	−0.458584	−0.229292	−	0.973358i	$-0.573641\pi$
$17$	−9289.46	−0.458584	−0.229292	+	0.973358i	$0.573641\pi$
$18$	5832.00	0.235702
$19$	−28641.6	−0.957986	−0.478993	−	0.877819i	$-0.658998\pi$
$19$	−28641.6	−0.957986	−0.478993	+	0.877819i	$0.658998\pi$
$20$	−26725.1	−0.746990
$21$	−9261.00	−0.218218
$22$	41718.2	0.835307
$23$	−70760.2	−1.21267	−0.606334	−	0.795210i	$-0.707360\pi$
$23$	−70760.2	−1.21267	−0.606334	+	0.795210i	$0.707360\pi$
$24$	−13824.0	−0.204124
$25$	96248.1	1.23198
$26$	−17576.0	−0.196116
$27$	−19683.0	−0.192450
$28$	21952.0	0.188982
$29$	40590.0	0.309048	0.154524	−	0.987989i	$-0.450616\pi$
$29$	40590.0	0.309048	0.154524	+	0.987989i	$0.450616\pi$
$30$	90197.3	0.609915
$31$	239911.	1.44638	0.723192	−	0.690647i	$-0.242674\pi$
$31$	239911.	1.44638	0.723192	+	0.690647i	$0.242674\pi$
$32$	32768.0	0.176777
$33$	−140799.	−0.682026
$34$	−74315.7	−0.324268
$35$	−143230.	−0.564671
$36$	46656.0	0.166667
$37$	334019.	1.08409	0.542045	−	0.840350i	$-0.317650\pi$
$37$	334019.	1.08409	0.542045	+	0.840350i	$0.317650\pi$
$38$	−229132.	−0.677398
$39$	59319.0	0.160128
$40$	−213801.	−0.528202
$41$	−347373.	−0.787141	−0.393571	−	0.919294i	$-0.628760\pi$
$41$	−347373.	−0.787141	−0.393571	+	0.919294i	$0.628760\pi$
$42$	−74088.0	−0.154303
$43$	−454134.	−0.871052	−0.435526	−	0.900176i	$-0.643438\pi$
$43$	−454134.	−0.871052	−0.435526	+	0.900176i	$0.643438\pi$
$44$	333746.	0.590651
$45$	−304416.	−0.497993
$46$	−566082.	−0.857485
$47$	−87902.0	−0.123497	−0.0617485	−	0.998092i	$-0.519668\pi$
$47$	−87902.0	−0.123497	−0.0617485	+	0.998092i	$0.519668\pi$
$48$	−110592.	−0.144338
$49$	117649.	0.142857
$50$	769985.	0.871138
$51$	250815.	0.264764
$52$	−140608.	−0.138675
$53$	−2.03091e6	−1.87380	−0.936902	−	0.349592i	$-0.886320\pi$
$53$	−2.03091e6	−1.87380	−0.936902	+	0.349592i	$0.886320\pi$
$54$	−157464.	−0.136083
$55$	−2.17759e6	−1.76484
$56$	175616.	0.133631
$57$	773322.	0.553093
$58$	324720.	0.218530
$59$	−385876.	−0.244605	−0.122303	−	0.992493i	$-0.539028\pi$
$59$	−385876.	−0.244605	−0.122303	+	0.992493i	$0.539028\pi$
$60$	721578.	0.431275
$61$	−2.16065e6	−1.21879	−0.609397	−	0.792865i	$-0.708589\pi$
$61$	−2.16065e6	−1.21879	−0.609397	+	0.792865i	$0.708589\pi$
$62$	1.91928e6	1.02275
$63$	250047.	0.125988
$64$	262144.	0.125000
$65$	917423.	0.414355
$66$	−1.12639e6	−0.482265
$67$	1.94749e6	0.791066	0.395533	−	0.918452i	$-0.370560\pi$
$67$	1.94749e6	0.791066	0.395533	+	0.918452i	$0.370560\pi$
$68$	−594525.	−0.229292
$69$	1.91053e6	0.700134
$70$	−1.14584e6	−0.399283
$71$	2.74370e6	0.909771	0.454885	−	0.890550i	$-0.349680\pi$
$71$	2.74370e6	0.909771	0.454885	+	0.890550i	$0.349680\pi$
$72$	373248.	0.117851
$73$	−801961.	−0.241281	−0.120641	−	0.992696i	$-0.538495\pi$
$73$	−801961.	−0.241281	−0.120641	+	0.992696i	$0.538495\pi$
$74$	2.67215e6	0.766567
$75$	−2.59870e6	−0.711281
$76$	−1.83306e6	−0.478993
$77$	1.78867e6	0.446491
$78$	474552.	0.113228
$79$	−7.49135e6	−1.70948	−0.854742	−	0.519053i	$-0.826285\pi$
$79$	−7.49135e6	−1.70948	−0.854742	+	0.519053i	$0.826285\pi$
$80$	−1.71041e6	−0.373495
$81$	531441.	0.111111
$82$	−2.77899e6	−0.556593
$83$	2.59240e6	0.497656	0.248828	−	0.968548i	$-0.419955\pi$
$83$	2.59240e6	0.497656	0.248828	+	0.968548i	$0.419955\pi$
$84$	−592704.	−0.109109
$85$	3.87909e6	0.685116
$86$	−3.63307e6	−0.615927
$87$	−1.09593e6	−0.178429
$88$	2.66997e6	0.417654
$89$	712280.	0.107099	0.0535495	−	0.998565i	$-0.482946\pi$
$89$	712280.	0.107099	0.0535495	+	0.998565i	$0.482946\pi$
$90$	−2.43533e6	−0.352134
$91$	−753571.	−0.104828
$92$	−4.52865e6	−0.606334
$93$	−6.47759e6	−0.835070
$94$	−703216.	−0.0873255
$95$	1.19601e7	1.43121
$96$	−884736.	−0.102062
$97$	1.71130e7	1.90381	0.951907	−	0.306387i	$-0.0991202\pi$
$97$	1.71130e7	1.90381	0.951907	+	0.306387i	$0.0991202\pi$
$98$	941192.	0.101015
$99$	3.80157e6	0.393768
$100$	6.15988e6	0.615988
$101$	−8.28018e6	−0.799678	−0.399839	−	0.916585i	$-0.630934\pi$
$101$	−8.28018e6	−0.799678	−0.399839	+	0.916585i	$0.630934\pi$
$102$	2.00652e6	0.187216
$103$	−1.79135e7	−1.61529	−0.807645	−	0.589669i	$-0.799258\pi$
$103$	−1.79135e7	−1.61529	−0.807645	+	0.589669i	$0.799258\pi$
$104$	−1.12486e6	−0.0980581
$105$	3.86721e6	0.326013
$106$	−1.62472e7	−1.32498
$107$	2.05021e7	1.61791	0.808957	−	0.587868i	$-0.200033\pi$
$107$	2.05021e7	1.61791	0.808957	+	0.587868i	$0.200033\pi$
$108$	−1.25971e6	−0.0962250
$109$	2.18594e7	1.61676	0.808380	−	0.588661i	$-0.200345\pi$
$109$	2.18594e7	1.61676	0.808380	+	0.588661i	$0.200345\pi$
$110$	−1.74207e7	−1.24793
$111$	−9.01851e6	−0.625899
$112$	1.40493e6	0.0944911
$113$	1.74187e7	1.13564	0.567822	−	0.823151i	$-0.307786\pi$
$113$	1.74187e7	1.13564	0.567822	+	0.823151i	$0.307786\pi$
$114$	6.18658e6	0.391096
$115$	2.95481e7	1.81170
$116$	2.59776e6	0.154524
$117$	−1.60161e6	−0.0924500
$118$	−3.08701e6	−0.172962
$119$	−3.18628e6	−0.173329
$120$	5.77263e6	0.304957
$121$	7.70672e6	0.395476
$122$	−1.72852e7	−0.861818
$123$	9.37908e6	0.454456
$124$	1.53543e7	0.723192
$125$	−7.56784e6	−0.346567
$126$	2.00038e6	0.0890871
$127$	2.64428e7	1.14550	0.572748	−	0.819731i	$-0.305877\pi$
$127$	2.64428e7	1.14550	0.572748	+	0.819731i	$0.305877\pi$
$128$	2.09715e6	0.0883883
$129$	1.22616e7	0.502902
$130$	7.33939e6	0.292994
$131$	2.78528e7	1.08248	0.541238	−	0.840869i	$-0.317956\pi$
$131$	2.78528e7	1.08248	0.541238	+	0.840869i	$0.317956\pi$
$132$	−9.01113e6	−0.341013
$133$	−9.82406e6	−0.362085
$134$	1.55799e7	0.559368
$135$	8.21923e6	0.287517
$136$	−4.75620e6	−0.162134
$137$	1.34607e7	0.447244	0.223622	−	0.974676i	$-0.428212\pi$
$137$	1.34607e7	0.447244	0.223622	+	0.974676i	$0.428212\pi$
$138$	1.52842e7	0.495069
$139$	1.64036e7	0.518068	0.259034	−	0.965868i	$-0.416596\pi$
$139$	1.64036e7	0.518068	0.259034	+	0.965868i	$0.416596\pi$
$140$	−9.16672e6	−0.282336
$141$	2.37335e6	0.0713010
$142$	2.19496e7	0.643305
$143$	−1.14569e7	−0.327634
$144$	2.98598e6	0.0833333
$145$	−1.69496e7	−0.461711
$146$	−6.41569e6	−0.170612
$147$	−3.17652e6	−0.0824786
$148$	2.13772e7	0.542045
$149$	6.94840e6	0.172081	0.0860405	−	0.996292i	$-0.472579\pi$
$149$	6.94840e6	0.172081	0.0860405	+	0.996292i	$0.472579\pi$
$150$	−2.07896e7	−0.502952
$151$	7.14181e7	1.68806	0.844032	−	0.536293i	$-0.180176\pi$
$151$	7.14181e7	1.68806	0.844032	+	0.536293i	$0.180176\pi$
$152$	−1.46645e7	−0.338699
$153$	−6.77201e6	−0.152861
$154$	1.43093e7	0.315716
$155$	−1.00182e8	−2.16087
$156$	3.79642e6	0.0800641
$157$	3.30011e7	0.680580	0.340290	−	0.940321i	$-0.389475\pi$
$157$	3.30011e7	0.680580	0.340290	+	0.940321i	$0.389475\pi$
$158$	−5.99308e7	−1.20879
$159$	5.48344e7	1.08184
$160$	−1.36833e7	−0.264101
$161$	−2.42708e7	−0.458345
$162$	4.25153e6	0.0785674
$163$	4.27115e7	0.772481	0.386241	−	0.922398i	$-0.373773\pi$
$163$	4.27115e7	0.772481	0.386241	+	0.922398i	$0.373773\pi$
$164$	−2.22319e7	−0.393571
$165$	5.87948e7	1.01893
$166$	2.07392e7	0.351896
$167$	−7.77575e6	−0.129192	−0.0645958	−	0.997912i	$-0.520576\pi$
$167$	−7.77575e6	−0.129192	−0.0645958	+	0.997912i	$0.520576\pi$
$168$	−4.74163e6	−0.0771517
$169$	4.82681e6	0.0769231
$170$	3.10327e7	0.484450
$171$	−2.08797e7	−0.319329
$172$	−2.90645e7	−0.435526
$173$	−6.03299e7	−0.885873	−0.442937	−	0.896553i	$-0.646063\pi$
$173$	−6.03299e7	−0.885873	−0.442937	+	0.896553i	$0.646063\pi$
$174$	−8.76743e6	−0.126168
$175$	3.30131e7	0.465643
$176$	2.13597e7	0.295326
$177$	1.04187e7	0.141223
$178$	5.69824e6	0.0757305
$179$	1.35910e8	1.77119	0.885594	−	0.464459i	$-0.153751\pi$
$179$	1.35910e8	1.77119	0.885594	+	0.464459i	$0.153751\pi$
$180$	−1.94826e7	−0.248997
$181$	−2.57944e7	−0.323334	−0.161667	−	0.986845i	$-0.551687\pi$
$181$	−2.57944e7	−0.323334	−0.161667	+	0.986845i	$0.551687\pi$
$182$	−6.02857e6	−0.0741249
$183$	5.83376e7	0.703672
$184$	−3.62292e7	−0.428743
$185$	−1.39480e8	−1.61961
$186$	−5.18207e7	−0.590484
$187$	−4.84424e7	−0.541727
$188$	−5.62573e6	−0.0617485
$189$	−6.75127e6	−0.0727393
$190$	9.56812e7	1.01202
$191$	−8.97084e7	−0.931572	−0.465786	−	0.884897i	$-0.654228\pi$
$191$	−8.97084e7	−0.931572	−0.465786	+	0.884897i	$0.654228\pi$
$192$	−7.07789e6	−0.0721688
$193$	−5.89536e7	−0.590282	−0.295141	−	0.955454i	$-0.595367\pi$
$193$	−5.89536e7	−0.590282	−0.295141	+	0.955454i	$0.595367\pi$
$194$	1.36904e8	1.34620
$195$	−2.47704e7	−0.239228
$196$	7.52954e6	0.0714286
$197$	8.91164e7	0.830474	0.415237	−	0.909713i	$-0.363699\pi$
$197$	8.91164e7	0.830474	0.415237	+	0.909713i	$0.363699\pi$
$198$	3.04126e7	0.278436
$199$	1.64764e8	1.48210	0.741048	−	0.671453i	$-0.234329\pi$
$199$	1.64764e8	1.48210	0.741048	+	0.671453i	$0.234329\pi$
$200$	4.92790e7	0.435569
$201$	−5.25821e7	−0.456722
$202$	−6.62415e7	−0.565458
$203$	1.39224e7	0.116809
$204$	1.60522e7	0.132382
$205$	1.45056e8	1.17597
$206$	−1.43308e8	−1.14218
$207$	−5.15842e7	−0.404222
$208$	−8.99891e6	−0.0693375
$209$	−1.49359e8	−1.13167
$210$	3.09377e7	0.230526
$211$	2.06727e8	1.51499	0.757494	−	0.652842i	$-0.226424\pi$
$211$	2.06727e8	1.51499	0.757494	+	0.652842i	$0.226424\pi$
$212$	−1.29978e8	−0.936902
$213$	−7.40798e7	−0.525256
$214$	1.64017e8	1.14404
$215$	1.89637e8	1.30133
$216$	−1.00777e7	−0.0680414
$217$	8.22893e7	0.546682
$218$	1.74875e8	1.14322
$219$	2.16530e7	0.139304
$220$	−1.39366e8	−0.882421
$221$	2.04089e7	0.127188
$222$	−7.21481e7	−0.442578
$223$	−8.56828e7	−0.517400	−0.258700	−	0.965958i	$-0.583294\pi$
$223$	−8.56828e7	−0.517400	−0.258700	+	0.965958i	$0.583294\pi$
$224$	1.12394e7	0.0668153
$225$	7.01649e7	0.410659
$226$	1.39350e8	0.803021
$227$	3.19553e8	1.81323	0.906613	−	0.421964i	$-0.138659\pi$
$227$	3.19553e8	1.81323	0.906613	+	0.421964i	$0.138659\pi$
$228$	4.94926e7	0.276547
$229$	8.92189e7	0.490945	0.245473	−	0.969404i	$-0.421057\pi$
$229$	8.92189e7	0.490945	0.245473	+	0.969404i	$0.421057\pi$
$230$	2.36384e8	1.28107
$231$	−4.82940e7	−0.257781
$232$	2.07821e7	0.109265
$233$	−2.98130e8	−1.54405	−0.772023	−	0.635594i	$-0.780755\pi$
$233$	−2.98130e8	−1.54405	−0.772023	+	0.635594i	$0.780755\pi$
$234$	−1.28129e7	−0.0653720
$235$	3.67061e7	0.184502
$236$	−2.46961e7	−0.122303
$237$	2.02266e8	0.986971
$238$	−2.54903e7	−0.122562
$239$	−1.47766e8	−0.700135	−0.350067	−	0.936724i	$-0.613841\pi$
$239$	−1.47766e8	−0.700135	−0.350067	+	0.936724i	$0.613841\pi$
$240$	4.61810e7	0.215637
$241$	−6.66547e7	−0.306741	−0.153370	−	0.988169i	$-0.549013\pi$
$241$	−6.66547e7	−0.306741	−0.153370	+	0.988169i	$0.549013\pi$
$242$	6.16537e7	0.279644
$243$	−1.43489e7	−0.0641500
$244$	−1.38282e8	−0.609397
$245$	−4.91279e7	−0.213426
$246$	7.50326e7	0.321349
$247$	6.29255e7	0.265697
$248$	1.22834e8	0.511374
$249$	−6.99949e7	−0.287322
$250$	−6.05427e7	−0.245060
$251$	−1.62351e8	−0.648034	−0.324017	−	0.946051i	$-0.605033\pi$
$251$	−1.62351e8	−0.648034	−0.324017	+	0.946051i	$0.605033\pi$
$252$	1.60030e7	0.0629941
$253$	−3.68999e8	−1.43253
$254$	2.11542e8	0.809988
$255$	−1.04735e8	−0.395552
$256$	1.67772e7	0.0625000
$257$	−4.07344e8	−1.49691	−0.748454	−	0.663186i	$-0.769204\pi$
$257$	−4.07344e8	−1.49691	−0.748454	+	0.663186i	$0.769204\pi$
$258$	9.80928e7	0.355605
$259$	1.14569e8	0.409747
$260$	5.87151e7	0.207178
$261$	2.95901e7	0.103016
$262$	2.22822e8	0.765427
$263$	5.04618e8	1.71048	0.855240	−	0.518232i	$-0.173410\pi$
$263$	5.04618e8	1.71048	0.855240	+	0.518232i	$0.173410\pi$
$264$	−7.20891e7	−0.241132
$265$	8.48065e8	2.79942
$266$	−7.85924e7	−0.256032
$267$	−1.92316e7	−0.0618337
$268$	1.24639e8	0.395533
$269$	1.19771e8	0.375161	0.187581	−	0.982249i	$-0.439935\pi$
$269$	1.19771e8	0.375161	0.187581	+	0.982249i	$0.439935\pi$
$270$	6.57538e7	0.203305
$271$	4.65730e8	1.42148	0.710741	−	0.703454i	$-0.248360\pi$
$271$	4.65730e8	1.42148	0.710741	+	0.703454i	$0.248360\pi$
$272$	−3.80496e7	−0.114646
$273$	2.03464e7	0.0605228
$274$	1.07685e8	0.316249
$275$	5.01912e8	1.45534
$276$	1.22274e8	0.350067
$277$	−1.74789e8	−0.494122	−0.247061	−	0.969000i	$-0.579465\pi$
$277$	−1.74789e8	−0.494122	−0.247061	+	0.969000i	$0.579465\pi$
$278$	1.31229e8	0.366330
$279$	1.74895e8	0.482128
$280$	−7.33337e7	−0.199641
$281$	4.02412e8	1.08193	0.540965	−	0.841045i	$-0.318059\pi$
$281$	4.02412e8	1.08193	0.540965	+	0.841045i	$0.318059\pi$
$282$	1.89868e7	0.0504174
$283$	1.34858e8	0.353690	0.176845	−	0.984239i	$-0.443411\pi$
$283$	1.34858e8	0.353690	0.176845	+	0.984239i	$0.443411\pi$
$284$	1.75597e8	0.454885
$285$	−3.22924e8	−0.826310
$286$	−9.16549e7	−0.231673
$287$	−1.19149e8	−0.297512
$288$	2.38879e7	0.0589256
$289$	−3.24045e8	−0.789701
$290$	−1.35596e8	−0.326479
$291$	−4.62050e8	−1.09917
$292$	−5.13255e7	−0.120641
$293$	1.37180e8	0.318605	0.159303	−	0.987230i	$-0.449075\pi$
$293$	1.37180e8	0.318605	0.159303	+	0.987230i	$0.449075\pi$
$294$	−2.54122e7	−0.0583212
$295$	1.61134e8	0.365435
$296$	1.71018e8	0.383284
$297$	−1.02642e8	−0.227342
$298$	5.55872e7	0.121680
$299$	1.55460e8	0.336333
$300$	−1.66317e8	−0.355641
$301$	−1.55768e8	−0.329227
$302$	5.71345e8	1.19364
$303$	2.23565e8	0.461694
$304$	−1.17316e8	−0.239496
$305$	9.02246e8	1.82086
$306$	−5.41761e7	−0.108089
$307$	−5.62841e8	−1.11020	−0.555101	−	0.831783i	$-0.687320\pi$
$307$	−5.62841e8	−1.11020	−0.555101	+	0.831783i	$0.687320\pi$
$308$	1.14475e8	0.223245
$309$	4.83665e8	0.932588
$310$	−8.01455e8	−1.52796
$311$	−3.69566e8	−0.696676	−0.348338	−	0.937369i	$-0.613254\pi$
$311$	−3.69566e8	−0.696676	−0.348338	+	0.937369i	$0.613254\pi$
$312$	3.03713e7	0.0566139
$313$	1.69417e7	0.0312286	0.0156143	−	0.999878i	$-0.495030\pi$
$313$	1.69417e7	0.0312286	0.0156143	+	0.999878i	$0.495030\pi$
$314$	2.64008e8	0.481243
$315$	−1.04415e8	−0.188224
$316$	−4.79446e8	−0.854742
$317$	−2.87449e8	−0.506820	−0.253410	−	0.967359i	$-0.581552\pi$
$317$	−2.87449e8	−0.506820	−0.253410	+	0.967359i	$0.581552\pi$
$318$	4.38676e8	0.764977
$319$	2.11668e8	0.365079
$320$	−1.09466e8	−0.186747
$321$	−5.53557e8	−0.934103
$322$	−1.94166e8	−0.324099
$323$	2.66065e8	0.439317
$324$	3.40122e7	0.0555556
$325$	−2.11457e8	−0.341689
$326$	3.41692e8	0.546227
$327$	−5.90204e8	−0.933437
$328$	−1.77855e8	−0.278297
$329$	−3.01504e7	−0.0466774
$330$	4.70359e8	0.720494
$331$	−2.57572e8	−0.390391	−0.195196	−	0.980764i	$-0.562534\pi$
$331$	−2.57572e8	−0.390391	−0.195196	+	0.980764i	$0.562534\pi$
$332$	1.65914e8	0.248828
$333$	2.43500e8	0.361363
$334$	−6.22060e7	−0.0913523
$335$	−8.13232e8	−1.18184
$336$	−3.79331e7	−0.0545545
$337$	−7.89868e8	−1.12422	−0.562108	−	0.827064i	$-0.690010\pi$
$337$	−7.89868e8	−1.12422	−0.562108	+	0.827064i	$0.690010\pi$
$338$	3.86145e7	0.0543928
$339$	−4.70306e8	−0.655664
$340$	2.48262e8	0.342558
$341$	1.25108e9	1.70862
$342$	−1.67038e8	−0.225799
$343$	4.03536e7	0.0539949
$344$	−2.32516e8	−0.307963
$345$	−7.97797e8	−1.04599
$346$	−4.82640e8	−0.626407
$347$	−5.09917e8	−0.655158	−0.327579	−	0.944824i	$-0.606233\pi$
$347$	−5.09917e8	−0.655158	−0.327579	+	0.944824i	$0.606233\pi$
$348$	−7.01395e7	−0.0892144
$349$	4.13805e8	0.521082	0.260541	−	0.965463i	$-0.416099\pi$
$349$	4.13805e8	0.521082	0.260541	+	0.965463i	$0.416099\pi$
$350$	2.64105e8	0.329259
$351$	4.32436e7	0.0533761
$352$	1.70878e8	0.208827
$353$	−3.76671e7	−0.0455776	−0.0227888	−	0.999740i	$-0.507255\pi$
$353$	−3.76671e7	−0.0455776	−0.0227888	+	0.999740i	$0.507255\pi$
$354$	8.33492e7	0.0998597
$355$	−1.14571e9	−1.35918
$356$	4.55859e7	0.0535495
$357$	8.60297e7	0.100071
$358$	1.08728e9	1.25242
$359$	−1.17417e8	−0.133937	−0.0669683	−	0.997755i	$-0.521333\pi$
$359$	−1.17417e8	−0.133937	−0.0669683	+	0.997755i	$0.521333\pi$
$360$	−1.55861e8	−0.176067
$361$	−7.35326e7	−0.0822631
$362$	−2.06356e8	−0.228632
$363$	−2.08081e8	−0.228328
$364$	−4.82285e7	−0.0524142
$365$	3.34883e8	0.360469
$366$	4.66701e8	0.497571
$367$	7.55687e8	0.798015	0.399007	−	0.916948i	$-0.369355\pi$
$367$	7.55687e8	0.798015	0.399007	+	0.916948i	$0.369355\pi$
$368$	−2.89834e8	−0.303167
$369$	−2.53235e8	−0.262380
$370$	−1.11584e9	−1.14524
$371$	−6.96600e8	−0.708231
$372$	−4.14565e8	−0.417535
$373$	4.77037e8	0.475961	0.237980	−	0.971270i	$-0.423515\pi$
$373$	4.77037e8	0.475961	0.237980	+	0.971270i	$0.423515\pi$
$374$	−3.87540e8	−0.383059
$375$	2.04332e8	0.200090
$376$	−4.50058e7	−0.0436627
$377$	−8.91761e7	−0.0857144
$378$	−5.40102e7	−0.0514344
$379$	1.36069e9	1.28387	0.641935	−	0.766759i	$-0.278132\pi$
$379$	1.36069e9	1.28387	0.641935	+	0.766759i	$0.278132\pi$
$380$	7.65449e8	0.715606
$381$	−7.13954e8	−0.661353
$382$	−7.17667e8	−0.658721
$383$	−1.97066e9	−1.79233	−0.896163	−	0.443726i	$-0.853656\pi$
$383$	−1.97066e9	−1.79233	−0.896163	+	0.443726i	$0.853656\pi$
$384$	−5.66231e7	−0.0510310
$385$	−7.46912e8	−0.667048
$386$	−4.71629e8	−0.417392
$387$	−3.31063e8	−0.290351
$388$	1.09523e9	0.951907
$389$	−1.56134e9	−1.34485	−0.672425	−	0.740165i	$-0.734747\pi$
$389$	−1.56134e9	−1.34485	−0.672425	+	0.740165i	$0.734747\pi$
$390$	−1.98163e8	−0.169160
$391$	6.57324e8	0.556110
$392$	6.02363e7	0.0505076
$393$	−7.52024e8	−0.624968
$394$	7.12931e8	0.587234
$395$	3.12824e9	2.55394
$396$	2.43301e8	0.196884
$397$	8.83178e8	0.708405	0.354203	−	0.935169i	$-0.384752\pi$
$397$	8.83178e8	0.708405	0.354203	+	0.935169i	$0.384752\pi$
$398$	1.31811e9	1.04800
$399$	2.65250e8	0.209050
$400$	3.94232e8	0.307994
$401$	−5.92174e7	−0.0458611	−0.0229305	−	0.999737i	$-0.507300\pi$
$401$	−5.92174e7	−0.0458611	−0.0229305	+	0.999737i	$0.507300\pi$
$402$	−4.20657e8	−0.322951
$403$	−5.27084e8	−0.401155
$404$	−5.29932e8	−0.399839
$405$	−2.21919e8	−0.165998
$406$	1.11379e8	0.0825965
$407$	1.74183e9	1.28064
$408$	1.28417e8	0.0936081
$409$	4.68060e8	0.338275	0.169137	−	0.985592i	$-0.445902\pi$
$409$	4.68060e8	0.338275	0.169137	+	0.985592i	$0.445902\pi$
$410$	1.16045e9	0.831539
$411$	−3.63438e8	−0.258216
$412$	−1.14646e9	−0.807645
$413$	−1.32355e8	−0.0924521
$414$	−4.12674e8	−0.285828
$415$	−1.08254e9	−0.743488
$416$	−7.19913e7	−0.0490290
$417$	−4.42897e8	−0.299107
$418$	−1.19487e9	−0.800213
$419$	−1.92159e9	−1.27618	−0.638090	−	0.769962i	$-0.720275\pi$
$419$	−1.92159e9	−1.27618	−0.638090	+	0.769962i	$0.720275\pi$
$420$	2.47501e8	0.163007
$421$	2.94243e9	1.92185	0.960924	−	0.276813i	$-0.0892784\pi$
$421$	2.94243e9	1.92185	0.960924	+	0.276813i	$0.0892784\pi$
$422$	1.65382e9	1.07126
$423$	−6.40805e7	−0.0411656
$424$	−1.03982e9	−0.662490
$425$	−8.94093e8	−0.564965
$426$	−5.92639e8	−0.371412
$427$	−7.41104e8	−0.460661
$428$	1.31213e9	0.808957
$429$	3.09335e8	0.189160
$430$	1.51710e9	0.920182
$431$	2.27957e9	1.37146	0.685728	−	0.727858i	$-0.259484\pi$
$431$	2.27957e9	1.37146	0.685728	+	0.727858i	$0.259484\pi$
$432$	−8.06216e7	−0.0481125
$433$	5.87675e8	0.347880	0.173940	−	0.984756i	$-0.444350\pi$
$433$	5.87675e8	0.347880	0.173940	+	0.984756i	$0.444350\pi$
$434$	6.58315e8	0.386562
$435$	4.57638e8	0.266569
$436$	1.39900e9	0.808380
$437$	2.02668e9	1.16172
$438$	1.73224e8	0.0985026
$439$	1.73709e9	0.979932	0.489966	−	0.871741i	$-0.337009\pi$
$439$	1.73709e9	0.979932	0.489966	+	0.871741i	$0.337009\pi$
$440$	−1.11492e9	−0.623966
$441$	8.57661e7	0.0476190
$442$	1.63272e8	0.0899358
$443$	1.11318e9	0.608350	0.304175	−	0.952616i	$-0.401619\pi$
$443$	1.11318e9	0.608350	0.304175	+	0.952616i	$0.401619\pi$
$444$	−5.77185e8	−0.312950
$445$	−2.97434e8	−0.160004
$446$	−6.85462e8	−0.365857
$447$	−1.87607e8	−0.0993511
$448$	8.99154e7	0.0472456
$449$	−3.34446e8	−0.174367	−0.0871833	−	0.996192i	$-0.527787\pi$
$449$	−3.34446e8	−0.174367	−0.0871833	+	0.996192i	$0.527787\pi$
$450$	5.61319e8	0.290379
$451$	−1.81147e9	−0.929852
$452$	1.11480e9	0.567822
$453$	−1.92829e9	−0.974604
$454$	2.55642e9	1.28214
$455$	3.14676e8	0.156612
$456$	3.95941e8	0.195548
$457$	2.60831e8	0.127836	0.0639179	−	0.997955i	$-0.479640\pi$
$457$	2.60831e8	0.127836	0.0639179	+	0.997955i	$0.479640\pi$
$458$	7.13752e8	0.347151
$459$	1.82844e8	0.0882546
$460$	1.89108e9	0.905850
$461$	4.44614e8	0.211363	0.105682	−	0.994400i	$-0.466298\pi$
$461$	4.44614e8	0.211363	0.105682	+	0.994400i	$0.466298\pi$
$462$	−3.86352e8	−0.182279
$463$	−1.06605e9	−0.499167	−0.249583	−	0.968353i	$-0.580294\pi$
$463$	−1.06605e9	−0.499167	−0.249583	+	0.968353i	$0.580294\pi$
$464$	1.66256e8	0.0772620
$465$	2.70491e9	1.24758
$466$	−2.38504e9	−1.09181
$467$	3.94192e9	1.79101	0.895507	−	0.445047i	$-0.146813\pi$
$467$	3.94192e9	1.79101	0.895507	+	0.445047i	$0.146813\pi$
$468$	−1.02503e8	−0.0462250
$469$	6.67988e8	0.298995
$470$	2.93649e8	0.130463
$471$	−8.91029e8	−0.392933
$472$	−1.97569e8	−0.0864810
$473$	−2.36820e9	−1.02898
$474$	1.61813e9	0.697894
$475$	−2.75670e9	−1.18022
$476$	−2.03922e8	−0.0866643
$477$	−1.48053e9	−0.624601
$478$	−1.18213e9	−0.495070
$479$	−1.27553e9	−0.530296	−0.265148	−	0.964208i	$-0.585421\pi$
$479$	−1.27553e9	−0.530296	−0.265148	+	0.964208i	$0.585421\pi$
$480$	3.69448e8	0.152479
$481$	−7.33840e8	−0.300672
$482$	−5.33238e8	−0.216898
$483$	6.55310e8	0.264626
$484$	4.93230e8	0.197738
$485$	−7.14604e9	−2.84426
$486$	−1.14791e8	−0.0453609
$487$	8.31710e7	0.0326302	0.0163151	−	0.999867i	$-0.494807\pi$
$487$	8.31710e7	0.0326302	0.0163151	+	0.999867i	$0.494807\pi$
$488$	−1.10625e9	−0.430909
$489$	−1.15321e9	−0.445992
$490$	−3.93023e8	−0.150915
$491$	3.42177e8	0.130456	0.0652282	−	0.997870i	$-0.479222\pi$
$491$	3.42177e8	0.130456	0.0652282	+	0.997870i	$0.479222\pi$
$492$	6.00261e8	0.227228
$493$	−3.77059e8	−0.141724
$494$	5.03404e8	0.187876
$495$	−1.58746e9	−0.588281
$496$	9.82674e8	0.361596
$497$	9.41088e8	0.343861
$498$	−5.59959e8	−0.203167
$499$	4.03473e9	1.45366	0.726830	−	0.686818i	$-0.240993\pi$
$499$	4.03473e9	1.45366	0.726830	+	0.686818i	$0.240993\pi$
$500$	−4.84342e8	−0.173283
$501$	2.09945e8	0.0745888
$502$	−1.29881e9	−0.458229
$503$	4.57881e9	1.60422	0.802111	−	0.597174i	$-0.203710\pi$
$503$	4.57881e9	1.60422	0.802111	+	0.597174i	$0.203710\pi$
$504$	1.28024e8	0.0445435
$505$	3.45764e9	1.19470
$506$	−2.95199e9	−1.01295
$507$	−1.30324e8	−0.0444116
$508$	1.69234e9	0.572748
$509$	−9.10655e8	−0.306085	−0.153042	−	0.988220i	$-0.548907\pi$
$509$	−9.10655e8	−0.306085	−0.153042	+	0.988220i	$0.548907\pi$
$510$	−8.37884e8	−0.279697
$511$	−2.75073e8	−0.0911957
$512$	1.34218e8	0.0441942
$513$	5.63752e8	0.184364
$514$	−3.25875e9	−1.05847
$515$	7.48032e9	2.41321
$516$	7.84743e8	0.251451
$517$	−4.58389e8	−0.145887
$518$	9.16548e8	0.289735
$519$	1.62891e9	0.511459
$520$	4.69721e8	0.146497
$521$	3.78933e9	1.17390	0.586949	−	0.809624i	$-0.300329\pi$
$521$	3.78933e9	1.17390	0.586949	+	0.809624i	$0.300329\pi$
$522$	2.36721e8	0.0728433
$523$	7.58145e8	0.231738	0.115869	−	0.993265i	$-0.463035\pi$
$523$	7.58145e8	0.231738	0.115869	+	0.993265i	$0.463035\pi$
$524$	1.78258e9	0.541238
$525$	−8.91354e8	−0.268839
$526$	4.03695e9	1.20949
$527$	−2.22864e9	−0.663289
$528$	−5.76713e8	−0.170506
$529$	1.60218e9	0.470562
$530$	6.78452e9	1.97949
$531$	−2.81304e8	−0.0815351
$532$	−6.28740e8	−0.181042
$533$	7.63179e8	0.218314
$534$	−1.53853e8	−0.0437230
$535$	−8.56127e9	−2.41713
$536$	9.97113e8	0.279684
$537$	−3.66956e9	−1.02260
$538$	9.58167e8	0.265279
$539$	6.13513e8	0.168758
$540$	5.26031e8	0.143758
$541$	−2.69793e9	−0.732554	−0.366277	−	0.930506i	$-0.619368\pi$
$541$	−2.69793e9	−0.732554	−0.366277	+	0.930506i	$0.619368\pi$
$542$	3.72584e9	1.00514
$543$	6.96450e8	0.186677
$544$	−3.04397e8	−0.0810670
$545$	−9.12805e9	−2.41541
$546$	1.62771e8	0.0427960
$547$	4.73299e9	1.23646	0.618230	−	0.785997i	$-0.287850\pi$
$547$	4.73299e9	1.23646	0.618230	+	0.785997i	$0.287850\pi$
$548$	8.61483e8	0.223622
$549$	−1.57512e9	−0.406265
$550$	4.01530e9	1.02908
$551$	−1.16256e9	−0.296063
$552$	9.78189e8	0.247535
$553$	−2.56953e9	−0.646124
$554$	−1.39831e9	−0.349397
$555$	3.76595e9	0.935081
$556$	1.04983e9	0.259034
$557$	−7.53882e9	−1.84846	−0.924231	−	0.381834i	$-0.875293\pi$
$557$	−7.53882e9	−1.84846	−0.924231	+	0.381834i	$0.875293\pi$
$558$	1.39916e9	0.340916
$559$	9.97731e8	0.241586
$560$	−5.86670e8	−0.141168
$561$	1.30795e9	0.312766
$562$	3.21930e9	0.765040
$563$	−5.85241e9	−1.38215	−0.691076	−	0.722782i	$-0.742863\pi$
$563$	−5.85241e9	−1.38215	−0.691076	+	0.722782i	$0.742863\pi$
$564$	1.51895e8	0.0356505
$565$	−7.27372e9	−1.69663
$566$	1.07886e9	0.250097
$567$	1.82284e8	0.0419961
$568$	1.40477e9	0.321653
$569$	1.36162e8	0.0309857	0.0154929	−	0.999880i	$-0.495068\pi$
$569$	1.36162e8	0.0309857	0.0154929	+	0.999880i	$0.495068\pi$
$570$	−2.58339e9	−0.584290
$571$	2.69163e9	0.605046	0.302523	−	0.953142i	$-0.402171\pi$
$571$	2.69163e9	0.605046	0.302523	+	0.953142i	$0.402171\pi$
$572$	−7.33239e8	−0.163817
$573$	2.42213e9	0.537843
$574$	−9.53192e8	−0.210372
$575$	−6.81054e9	−1.49398
$576$	1.91103e8	0.0416667
$577$	−1.66238e9	−0.360259	−0.180130	−	0.983643i	$-0.557652\pi$
$577$	−1.66238e9	−0.360259	−0.180130	+	0.983643i	$0.557652\pi$
$578$	−2.59236e9	−0.558403
$579$	1.59175e9	0.340800
$580$	−1.08477e9	−0.230856
$581$	8.89195e8	0.188096
$582$	−3.69640e9	−0.777229
$583$	−1.05907e10	−2.21353
$584$	−4.10604e8	−0.0853058
$585$	6.68802e8	0.138118
$586$	1.09744e9	0.225288
$587$	−5.21278e9	−1.06374	−0.531871	−	0.846825i	$-0.678511\pi$
$587$	−5.21278e9	−1.06374	−0.531871	+	0.846825i	$0.678511\pi$
$588$	−2.03297e8	−0.0412393
$589$	−6.87141e9	−1.38561
$590$	1.28907e9	0.258402
$591$	−2.40614e9	−0.479474
$592$	1.36814e9	0.271022
$593$	2.40644e9	0.473897	0.236948	−	0.971522i	$-0.423853\pi$
$593$	2.40644e9	0.473897	0.236948	+	0.971522i	$0.423853\pi$
$594$	−8.21139e8	−0.160755
$595$	1.33053e9	0.258949
$596$	4.44698e8	0.0860405
$597$	−4.44862e9	−0.855688
$598$	1.24368e9	0.237824
$599$	3.57730e9	0.680082	0.340041	−	0.940411i	$-0.389559\pi$
$599$	3.57730e9	0.680082	0.340041	+	0.940411i	$0.389559\pi$
$600$	−1.33053e9	−0.251476
$601$	7.94216e9	1.49238	0.746188	−	0.665736i	$-0.231882\pi$
$601$	7.94216e9	1.49238	0.746188	+	0.665736i	$0.231882\pi$
$602$	−1.24614e9	−0.232798
$603$	1.41972e9	0.263689
$604$	4.57076e9	0.844032
$605$	−3.21817e9	−0.590834
$606$	1.78852e9	0.326467
$607$	1.49061e9	0.270523	0.135261	−	0.990810i	$-0.456813\pi$
$607$	1.49061e9	0.270523	0.135261	+	0.990810i	$0.456813\pi$
$608$	−9.38527e8	−0.169350
$609$	−3.75904e8	−0.0674398
$610$	7.21797e9	1.28754
$611$	1.93121e8	0.0342519
$612$	−4.33409e8	−0.0764307
$613$	1.08594e10	1.90411	0.952056	−	0.305923i	$-0.0989650\pi$
$613$	1.08594e10	1.90411	0.952056	+	0.305923i	$0.0989650\pi$
$614$	−4.50273e9	−0.785031
$615$	−3.91651e9	−0.678949
$616$	9.15798e8	0.157858
$617$	−5.50961e9	−0.944328	−0.472164	−	0.881511i	$-0.656527\pi$
$617$	−5.50961e9	−0.944328	−0.472164	+	0.881511i	$0.656527\pi$
$618$	3.86932e9	0.659439
$619$	−3.31953e9	−0.562548	−0.281274	−	0.959627i	$-0.590757\pi$
$619$	−3.31953e9	−0.562548	−0.281274	+	0.959627i	$0.590757\pi$
$620$	−6.41164e9	−1.08043
$621$	1.39277e9	0.233378
$622$	−2.95653e9	−0.492625
$623$	2.44312e8	0.0404797
$624$	2.42971e8	0.0400320
$625$	−4.35920e9	−0.714212
$626$	1.35534e8	0.0220819
$627$	4.03270e9	0.653371
$628$	2.11207e9	0.340290
$629$	−3.10286e9	−0.497146
$630$	−8.35317e8	−0.133094
$631$	−6.04983e9	−0.958607	−0.479303	−	0.877649i	$-0.659111\pi$
$631$	−6.04983e9	−0.958607	−0.479303	+	0.877649i	$0.659111\pi$
$632$	−3.83557e9	−0.604394
$633$	−5.58163e9	−0.874678
$634$	−2.29959e9	−0.358376
$635$	−1.10420e10	−1.71135
$636$	3.50940e9	0.540921
$637$	−2.58475e8	−0.0396214
$638$	1.69334e9	0.258150
$639$	2.00016e9	0.303257
$640$	−8.75729e8	−0.132050
$641$	5.52551e8	0.0828647	0.0414323	−	0.999141i	$-0.486808\pi$
$641$	5.52551e8	0.0828647	0.0414323	+	0.999141i	$0.486808\pi$
$642$	−4.42846e9	−0.660510
$643$	−1.22576e10	−1.81831	−0.909156	−	0.416456i	$-0.863272\pi$
$643$	−1.22576e10	−1.81831	−0.909156	+	0.416456i	$0.863272\pi$
$644$	−1.55333e9	−0.229173
$645$	−5.12020e9	−0.751325
$646$	2.12852e9	0.310644
$647$	−9.44182e9	−1.37054	−0.685269	−	0.728290i	$-0.740315\pi$
$647$	−9.44182e9	−1.37054	−0.685269	+	0.728290i	$0.740315\pi$
$648$	2.72098e8	0.0392837
$649$	−2.01226e9	−0.288953
$650$	−1.69166e9	−0.241610
$651$	−2.22181e9	−0.315627
$652$	2.73353e9	0.386241
$653$	−1.47622e9	−0.207470	−0.103735	−	0.994605i	$-0.533079\pi$
$653$	−1.47622e9	−0.207470	−0.103735	+	0.994605i	$0.533079\pi$
$654$	−4.72163e9	−0.660040
$655$	−1.16308e10	−1.61720
$656$	−1.42284e9	−0.196785
$657$	−5.84630e8	−0.0804271
$658$	−2.41203e8	−0.0330059
$659$	−1.04164e10	−1.41781	−0.708906	−	0.705303i	$-0.750811\pi$
$659$	−1.04164e10	−1.41781	−0.708906	+	0.705303i	$0.750811\pi$
$660$	3.76287e9	0.509466
$661$	−1.98552e9	−0.267405	−0.133702	−	0.991022i	$-0.542687\pi$
$661$	−1.98552e9	−0.267405	−0.133702	+	0.991022i	$0.542687\pi$
$662$	−2.06057e9	−0.276048
$663$	−5.51041e8	−0.0734322
$664$	1.32731e9	0.175948
$665$	4.10233e9	0.540947
$666$	1.94800e9	0.255522
$667$	−2.87215e9	−0.374772
$668$	−4.97648e8	−0.0645958
$669$	2.31343e9	0.298721
$670$	−6.50585e9	−0.835685
$671$	−1.12673e10	−1.43977
$672$	−3.03464e8	−0.0385758
$673$	4.00058e9	0.505907	0.252953	−	0.967478i	$-0.418598\pi$
$673$	4.00058e9	0.505907	0.252953	+	0.967478i	$0.418598\pi$
$674$	−6.31895e9	−0.794941
$675$	−1.89445e9	−0.237094
$676$	3.08916e8	0.0384615
$677$	4.72498e9	0.585247	0.292624	−	0.956228i	$-0.405472\pi$
$677$	4.72498e9	0.585247	0.292624	+	0.956228i	$0.405472\pi$
$678$	−3.76245e9	−0.463625
$679$	5.86975e9	0.719574
$680$	1.98610e9	0.242225
$681$	−8.62792e9	−1.04687
$682$	1.00086e10	1.20817
$683$	−7.14183e9	−0.857704	−0.428852	−	0.903375i	$-0.641082\pi$
$683$	−7.14183e9	−0.857704	−0.428852	+	0.903375i	$0.641082\pi$
$684$	−1.33630e9	−0.159664
$685$	−5.62091e9	−0.668174
$686$	3.22829e8	0.0381802
$687$	−2.40891e9	−0.283447
$688$	−1.86013e9	−0.217763
$689$	4.46190e9	0.519700
$690$	−6.38238e9	−0.739624
$691$	1.21613e10	1.40219	0.701096	−	0.713067i	$-0.252694\pi$
$691$	1.21613e10	1.40219	0.701096	+	0.713067i	$0.252694\pi$
$692$	−3.86112e9	−0.442937
$693$	1.30394e9	0.148830
$694$	−4.07933e9	−0.463267
$695$	−6.84981e9	−0.773983
$696$	−5.61116e8	−0.0630841
$697$	3.22691e9	0.360971
$698$	3.31044e9	0.368461
$699$	8.04952e9	0.891456
$700$	2.11284e9	0.232821
$701$	−5.36586e9	−0.588338	−0.294169	−	0.955753i	$-0.595043\pi$
$701$	−5.36586e9	−0.588338	−0.294169	+	0.955753i	$0.595043\pi$
$702$	3.45948e8	0.0377426
$703$	−9.56683e9	−1.03854
$704$	1.36702e9	0.147663
$705$	−9.91065e8	−0.106522
$706$	−3.01337e8	−0.0322282
$707$	−2.84010e9	−0.302250
$708$	6.66794e8	0.0706115
$709$	−1.47913e8	−0.0155864	−0.00779319	−	0.999970i	$-0.502481\pi$
$709$	−1.47913e8	−0.0155864	−0.00779319	+	0.999970i	$0.502481\pi$
$710$	−9.16571e9	−0.961085
$711$	−5.46119e9	−0.569828
$712$	3.64687e8	0.0378652
$713$	−1.69761e10	−1.75398
$714$	6.88237e8	0.0707611
$715$	4.78416e9	0.489479
$716$	8.69822e9	0.885594
$717$	3.98968e9	0.404223
$718$	−9.39333e8	−0.0947075
$719$	1.37582e10	1.38042	0.690209	−	0.723610i	$-0.257518\pi$
$719$	1.37582e10	1.38042	0.690209	+	0.723610i	$0.257518\pi$
$720$	−1.24689e9	−0.124498
$721$	−6.14433e9	−0.610522
$722$	−5.88261e8	−0.0581688
$723$	1.79968e9	0.177097
$724$	−1.65084e9	−0.161667
$725$	3.90671e9	0.380739
$726$	−1.66465e9	−0.161453
$727$	1.08983e10	1.05193	0.525966	−	0.850506i	$-0.323704\pi$
$727$	1.08983e10	1.05193	0.525966	+	0.850506i	$0.323704\pi$
$728$	−3.85828e8	−0.0370625
$729$	3.87420e8	0.0370370
$730$	2.67906e9	0.254890
$731$	4.21865e9	0.399451
$732$	3.73361e9	0.351836
$733$	1.46717e10	1.37599	0.687996	−	0.725714i	$-0.258490\pi$
$733$	1.46717e10	1.37599	0.687996	+	0.725714i	$0.258490\pi$
$734$	6.04550e9	0.564282
$735$	1.32645e9	0.123221
$736$	−2.31867e9	−0.214371
$737$	1.01557e10	0.934488
$738$	−2.02588e9	−0.185531
$739$	1.07386e10	0.978797	0.489398	−	0.872060i	$-0.337216\pi$
$739$	1.07386e10	0.978797	0.489398	+	0.872060i	$0.337216\pi$
$740$	−8.92670e9	−0.809804
$741$	−1.69899e9	−0.153401
$742$	−5.57280e9	−0.500795
$743$	1.26609e10	1.13241	0.566205	−	0.824264i	$-0.308411\pi$
$743$	1.26609e10	1.13241	0.566205	+	0.824264i	$0.308411\pi$
$744$	−3.31652e9	−0.295242
$745$	−2.90151e9	−0.257086
$746$	3.81630e9	0.336555
$747$	1.88986e9	0.165885
$748$	−3.10032e9	−0.270863
$749$	7.03222e9	0.611514
$750$	1.63465e9	0.141485
$751$	−6.84407e7	−0.00589623	−0.00294812	−	0.999996i	$-0.500938\pi$
$751$	−6.84407e7	−0.00589623	−0.00294812	+	0.999996i	$0.500938\pi$
$752$	−3.60046e8	−0.0308742
$753$	4.38348e9	0.374142
$754$	−7.13409e8	−0.0606093
$755$	−2.98228e10	−2.52193
$756$	−4.32081e8	−0.0363696
$757$	1.57268e10	1.31766	0.658830	−	0.752292i	$-0.271051\pi$
$757$	1.57268e10	1.31766	0.658830	+	0.752292i	$0.271051\pi$
$758$	1.08855e10	0.907833
$759$	9.96296e9	0.827070
$760$	6.12359e9	0.506010
$761$	1.32261e10	1.08789	0.543945	−	0.839121i	$-0.316930\pi$
$761$	1.32261e10	1.08789	0.543945	+	0.839121i	$0.316930\pi$
$762$	−5.71164e9	−0.467647
$763$	7.49778e9	0.611078
$764$	−5.74134e9	−0.465786
$765$	2.82786e9	0.228372
$766$	−1.57653e10	−1.26737
$767$	8.47770e8	0.0678413
$768$	−4.52985e8	−0.0360844
$769$	1.28832e10	1.02160	0.510801	−	0.859699i	$-0.329349\pi$
$769$	1.28832e10	1.02160	0.510801	+	0.859699i	$0.329349\pi$
$770$	−5.97530e9	−0.471674
$771$	1.09983e10	0.864241
$772$	−3.77303e9	−0.295141
$773$	7.09192e9	0.552249	0.276125	−	0.961122i	$-0.410950\pi$
$773$	7.09192e9	0.552249	0.276125	+	0.961122i	$0.410950\pi$
$774$	−2.64851e9	−0.205309
$775$	2.30909e10	1.78191
$776$	8.76184e9	0.673100
$777$	−3.09335e9	−0.236568
$778$	−1.24907e10	−0.950952
$779$	9.94931e9	0.754070
$780$	−1.58531e9	−0.119614
$781$	1.43078e10	1.07471
$782$	5.25859e9	0.393229
$783$	−7.98932e8	−0.0594763
$784$	4.81890e8	0.0357143
$785$	−1.37806e10	−1.01677
$786$	−6.01620e9	−0.441919
$787$	−3.46049e9	−0.253061	−0.126531	−	0.991963i	$-0.540384\pi$
$787$	−3.46049e9	−0.253061	−0.126531	+	0.991963i	$0.540384\pi$
$788$	5.70345e9	0.415237
$789$	−1.36247e10	−0.987546
$790$	2.50259e10	1.80591
$791$	5.97463e9	0.429233
$792$	1.94640e9	0.139218
$793$	4.74696e9	0.338033
$794$	7.06543e9	0.500918
$795$	−2.28978e10	−1.61625
$796$	1.05449e10	0.741048
$797$	−1.22093e10	−0.854253	−0.427126	−	0.904192i	$-0.640474\pi$
$797$	−1.22093e10	−0.854253	−0.427126	+	0.904192i	$0.640474\pi$
$798$	2.12200e9	0.147820
$799$	8.16562e8	0.0566337
$800$	3.15386e9	0.217785
$801$	5.19252e8	0.0356997
$802$	−4.73739e8	−0.0324287
$803$	−4.18205e9	−0.285026
$804$	−3.36526e9	−0.228361
$805$	1.01350e10	0.684759
$806$	−4.21667e9	−0.283659
$807$	−3.23381e9	−0.216599
$808$	−4.23945e9	−0.282729
$809$	−1.33753e10	−0.888142	−0.444071	−	0.895992i	$-0.646466\pi$
$809$	−1.33753e10	−0.888142	−0.444071	+	0.895992i	$0.646466\pi$
$810$	−1.77535e9	−0.117378
$811$	−2.49686e10	−1.64369	−0.821847	−	0.569709i	$-0.807056\pi$
$811$	−2.49686e10	−1.64369	−0.821847	+	0.569709i	$0.807056\pi$
$812$	8.91031e8	0.0584045
$813$	−1.25747e10	−0.820693
$814$	1.39347e10	0.905548
$815$	−1.78355e10	−1.15407
$816$	1.02734e9	0.0661909
$817$	1.30071e10	0.834455
$818$	3.74448e9	0.239196
$819$	−5.49353e8	−0.0349428
$820$	9.28359e9	0.587987
$821$	2.22509e10	1.40329	0.701643	−	0.712529i	$-0.252450\pi$
$821$	2.22509e10	1.40329	0.701643	+	0.712529i	$0.252450\pi$
$822$	−2.90750e9	−0.182587
$823$	5.96037e9	0.372712	0.186356	−	0.982482i	$-0.440332\pi$
$823$	5.96037e9	0.372712	0.186356	+	0.982482i	$0.440332\pi$
$824$	−9.17172e9	−0.571091
$825$	−1.35516e10	−0.840239
$826$	−1.05884e9	−0.0653735
$827$	−2.18600e10	−1.34395	−0.671973	−	0.740576i	$-0.734553\pi$
$827$	−2.18600e10	−1.34395	−0.671973	+	0.740576i	$0.734553\pi$
$828$	−3.30139e9	−0.202111
$829$	1.31510e10	0.801714	0.400857	−	0.916141i	$-0.368712\pi$
$829$	1.31510e10	0.801714	0.400857	+	0.916141i	$0.368712\pi$
$830$	−8.66029e9	−0.525726
$831$	4.71929e9	0.285281
$832$	−5.75930e8	−0.0346688
$833$	−1.09290e9	−0.0655120
$834$	−3.54317e9	−0.211500
$835$	3.24700e9	0.193010
$836$	−9.55900e9	−0.565836
$837$	−4.72216e9	−0.278357
$838$	−1.53727e10	−0.902395
$839$	−1.98047e9	−0.115772	−0.0578858	−	0.998323i	$-0.518436\pi$
$839$	−1.98047e9	−0.115772	−0.0578858	+	0.998323i	$0.518436\pi$
$840$	1.98001e9	0.115263
$841$	−1.56023e10	−0.904489
$842$	2.35394e10	1.35895
$843$	−1.08651e10	−0.624653
$844$	1.32305e10	0.757494
$845$	−2.01558e9	−0.114922
$846$	−5.12644e8	−0.0291085
$847$	2.64340e9	0.149476
$848$	−8.31859e9	−0.468451
$849$	−3.64116e9	−0.204203
$850$	−7.15274e9	−0.399490
$851$	−2.36353e10	−1.31464
$852$	−4.74111e9	−0.262628
$853$	−1.15271e10	−0.635916	−0.317958	−	0.948105i	$-0.602997\pi$
$853$	−1.15271e10	−0.635916	−0.317958	+	0.948105i	$0.602997\pi$
$854$	−5.92883e9	−0.325737
$855$	8.71895e9	0.477071
$856$	1.04971e10	0.572019
$857$	1.48622e10	0.806586	0.403293	−	0.915071i	$-0.367866\pi$
$857$	1.48622e10	0.806586	0.403293	+	0.915071i	$0.367866\pi$
$858$	2.47468e9	0.133756
$859$	3.35830e10	1.80777	0.903885	−	0.427776i	$-0.140703\pi$
$859$	3.35830e10	1.80777	0.903885	+	0.427776i	$0.140703\pi$
$860$	1.21368e10	0.650667
$861$	3.21702e9	0.171768
$862$	1.82365e10	0.969765
$863$	2.97813e10	1.57727	0.788635	−	0.614862i	$-0.210788\pi$
$863$	2.97813e10	1.57727	0.788635	+	0.614862i	$0.210788\pi$
$864$	−6.44973e8	−0.0340207
$865$	2.51926e10	1.32348
$866$	4.70140e9	0.245988
$867$	8.74921e9	0.455934
$868$	5.26652e9	0.273341
$869$	−3.90657e10	−2.01942
$870$	3.66110e9	0.188493
$871$	−4.27863e9	−0.219402
$872$	1.11920e10	0.571611
$873$	1.24754e10	0.634605
$874$	1.62135e10	0.821459
$875$	−2.59577e9	−0.130990
$876$	1.38579e9	0.0696519
$877$	1.84914e9	0.0925704	0.0462852	−	0.998928i	$-0.485262\pi$
$877$	1.84914e9	0.0925704	0.0462852	+	0.998928i	$0.485262\pi$
$878$	1.38967e10	0.692917
$879$	−3.70385e9	−0.183947
$880$	−8.91939e9	−0.441211
$881$	1.54420e10	0.760831	0.380415	−	0.924816i	$-0.375781\pi$
$881$	1.54420e10	0.760831	0.380415	+	0.924816i	$0.375781\pi$
$882$	6.86129e8	0.0336718
$883$	−2.81407e10	−1.37554	−0.687768	−	0.725931i	$-0.741409\pi$
$883$	−2.81407e10	−1.37554	−0.687768	+	0.725931i	$0.741409\pi$
$884$	1.30617e9	0.0635942
$885$	−4.35062e9	−0.210984
$886$	8.90546e9	0.430168
$887$	7.47660e9	0.359726	0.179863	−	0.983692i	$-0.442435\pi$
$887$	7.47660e9	0.359726	0.179863	+	0.983692i	$0.442435\pi$
$888$	−4.61748e9	−0.221289
$889$	9.06986e9	0.432957
$890$	−2.37947e9	−0.113140
$891$	2.77135e9	0.131256
$892$	−5.48370e9	−0.258700
$893$	2.51765e9	0.118308
$894$	−1.50086e9	−0.0702518
$895$	−5.67532e10	−2.64612
$896$	7.19323e8	0.0334077
$897$	−4.19743e9	−0.194182
$898$	−2.67557e9	−0.123296
$899$	9.73796e9	0.447002
$900$	4.49055e9	0.205329
$901$	1.88660e10	0.859297
$902$	−1.44918e10	−0.657505
$903$	4.20573e9	0.190079
$904$	8.91839e9	0.401511
$905$	1.07712e10	0.483054
$906$	−1.54263e10	−0.689149
$907$	1.22547e10	0.545350	0.272675	−	0.962106i	$-0.412092\pi$
$907$	1.22547e10	0.545350	0.272675	+	0.962106i	$0.412092\pi$
$908$	2.04514e10	0.906613
$909$	−6.03625e9	−0.266559
$910$	2.51741e9	0.110741
$911$	−1.25825e10	−0.551381	−0.275691	−	0.961246i	$-0.588907\pi$
$911$	−1.25825e10	−0.551381	−0.275691	+	0.961246i	$0.588907\pi$
$912$	3.16753e9	0.138273
$913$	1.35188e10	0.587883
$914$	2.08665e9	0.0903936
$915$	−2.43606e10	−1.05127
$916$	5.71001e9	0.245473
$917$	9.55350e9	0.409138
$918$	1.46276e9	0.0624054
$919$	−1.59682e8	−0.00678658	−0.00339329	−	0.999994i	$-0.501080\pi$
$919$	−1.59682e8	−0.00678658	−0.00339329	+	0.999994i	$0.501080\pi$
$920$	1.51286e10	0.640533
$921$	1.51967e10	0.640975
$922$	3.55691e9	0.149456
$923$	−6.02790e9	−0.252325
$924$	−3.09082e9	−0.128891
$925$	3.21487e10	1.33557
$926$	−8.52843e9	−0.352964
$927$	−1.30589e10	−0.538430
$928$	1.33005e9	0.0546324
$929$	−5.05924e9	−0.207028	−0.103514	−	0.994628i	$-0.533009\pi$
$929$	−5.05924e9	−0.207028	−0.103514	+	0.994628i	$0.533009\pi$
$930$	2.16393e10	0.882171
$931$	−3.36965e9	−0.136855
$932$	−1.90803e10	−0.772023
$933$	9.97829e9	0.402226
$934$	3.15354e10	1.26644
$935$	2.02286e10	0.809329
$936$	−8.20026e8	−0.0326860
$937$	2.59671e9	0.103118	0.0515591	−	0.998670i	$-0.483581\pi$
$937$	2.59671e9	0.103118	0.0515591	+	0.998670i	$0.483581\pi$
$938$	5.34390e9	0.211421
$939$	−4.57426e8	−0.0180298
$940$	2.34919e9	0.0922509
$941$	4.20695e10	1.64590	0.822950	−	0.568114i	$-0.192327\pi$
$941$	4.20695e10	1.64590	0.822950	+	0.568114i	$0.192327\pi$
$942$	−7.12823e9	−0.277846
$943$	2.45802e10	0.954541
$944$	−1.58055e9	−0.0611513
$945$	2.81920e9	0.108671
$946$	−1.89456e10	−0.727596
$947$	−3.01808e10	−1.15480	−0.577398	−	0.816463i	$-0.695932\pi$
$947$	−3.01808e10	−1.15480	−0.577398	+	0.816463i	$0.695932\pi$
$948$	1.29451e10	0.493486
$949$	1.76191e9	0.0669194
$950$	−2.20536e10	−0.834538
$951$	7.76113e9	0.292613
$952$	−1.63138e9	−0.0612809
$953$	−4.11878e9	−0.154150	−0.0770751	−	0.997025i	$-0.524558\pi$
$953$	−4.11878e9	−0.154150	−0.0770751	+	0.997025i	$0.524558\pi$
$954$	−1.18442e10	−0.441660
$955$	3.74604e10	1.39175
$956$	−9.45702e9	−0.350067
$957$	−5.71502e9	−0.210778
$958$	−1.02043e10	−0.374976
$959$	4.61701e9	0.169042
$960$	2.95558e9	0.107819
$961$	3.00445e10	1.09203
$962$	−5.87072e9	−0.212607
$963$	1.49460e10	0.539304
$964$	−4.26590e9	−0.153370
$965$	2.46178e10	0.881870
$966$	5.24248e9	0.187119
$967$	−5.03708e10	−1.79138	−0.895688	−	0.444684i	$-0.853316\pi$
$967$	−5.03708e10	−1.79138	−0.895688	+	0.444684i	$0.853316\pi$
$968$	3.94584e9	0.139822
$969$	−7.18374e9	−0.253640
$970$	−5.71683e10	−2.01120
$971$	2.36143e10	0.827765	0.413882	−	0.910330i	$-0.364172\pi$
$971$	2.36143e10	0.827765	0.413882	+	0.910330i	$0.364172\pi$
$972$	−9.18330e8	−0.0320750
$973$	5.62643e9	0.195811
$974$	6.65368e8	0.0230731
$975$	5.70934e9	0.197274
$976$	−8.85004e9	−0.304699
$977$	−3.85449e9	−0.132232	−0.0661159	−	0.997812i	$-0.521061\pi$
$977$	−3.85449e9	−0.132232	−0.0661159	+	0.997812i	$0.521061\pi$
$978$	−9.22568e9	−0.315364
$979$	3.71438e9	0.126516
$980$	−3.14418e9	−0.106713
$981$	1.59355e10	0.538920
$982$	2.73741e9	0.0922466
$983$	3.36188e10	1.12887	0.564436	−	0.825477i	$-0.309094\pi$
$983$	3.36188e10	1.12887	0.564436	+	0.825477i	$0.309094\pi$
$984$	4.80209e9	0.160675
$985$	−3.72132e10	−1.24071
$986$	−3.01647e9	−0.100214
$987$	8.14060e8	0.0269492
$988$	4.02723e9	0.132849
$989$	3.21346e10	1.05630
$990$	−1.26997e10	−0.415977
$991$	−2.11989e10	−0.691918	−0.345959	−	0.938250i	$-0.612446\pi$
$991$	−2.11989e10	−0.691918	−0.345959	+	0.938250i	$0.612446\pi$
$992$	7.86139e9	0.255687
$993$	6.95443e9	0.225392
$994$	7.52871e9	0.243146
$995$	−6.88021e10	−2.21422
$996$	−4.47968e9	−0.143661
$997$	−2.65730e9	−0.0849196	−0.0424598	−	0.999098i	$-0.513519\pi$
$997$	−2.65730e9	−0.0849196	−0.0424598	+	0.999098i	$0.513519\pi$
$998$	3.22779e10	1.02789
$999$	−6.57450e9	−0.208633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.q.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.q.1.1	✓	6	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)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -417.580 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -417.580 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -3340.64 q^{10} +5214.78 q^{11} -1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +11274.7 q^{15} +4096.00 q^{16} -9289.46 q^{17} +5832.00 q^{18} -28641.6 q^{19} -26725.1 q^{20} -9261.00 q^{21} +41718.2 q^{22} -70760.2 q^{23} -13824.0 q^{24} +96248.1 q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} +40590.0 q^{29} +90197.3 q^{30} +239911. q^{31} +32768.0 q^{32} -140799. q^{33} -74315.7 q^{34} -143230. q^{35} +46656.0 q^{36} +334019. q^{37} -229132. q^{38} +59319.0 q^{39} -213801. q^{40} -347373. q^{41} -74088.0 q^{42} -454134. q^{43} +333746. q^{44} -304416. q^{45} -566082. q^{46} -87902.0 q^{47} -110592. q^{48} +117649. q^{49} +769985. q^{50} +250815. q^{51} -140608. q^{52} -2.03091e6 q^{53} -157464. q^{54} -2.17759e6 q^{55} +175616. q^{56} +773322. q^{57} +324720. q^{58} -385876. q^{59} +721578. q^{60} -2.16065e6 q^{61} +1.91928e6 q^{62} +250047. q^{63} +262144. q^{64} +917423. q^{65} -1.12639e6 q^{66} +1.94749e6 q^{67} -594525. q^{68} +1.91053e6 q^{69} -1.14584e6 q^{70} +2.74370e6 q^{71} +373248. q^{72} -801961. q^{73} +2.67215e6 q^{74} -2.59870e6 q^{75} -1.83306e6 q^{76} +1.78867e6 q^{77} +474552. q^{78} -7.49135e6 q^{79} -1.71041e6 q^{80} +531441. q^{81} -2.77899e6 q^{82} +2.59240e6 q^{83} -592704. q^{84} +3.87909e6 q^{85} -3.63307e6 q^{86} -1.09593e6 q^{87} +2.66997e6 q^{88} +712280. q^{89} -2.43533e6 q^{90} -753571. q^{91} -4.52865e6 q^{92} -6.47759e6 q^{93} -703216. q^{94} +1.19601e7 q^{95} -884736. q^{96} +1.71130e7 q^{97} +941192. q^{98} +3.80157e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) 6 * q + 48 * q^2 - 162 * q^3 + 384 * q^4 + 13 * q^5 - 1296 * q^6 + 2058 * q^7 + 3072 * q^8 + 4374 * q^9	\( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 104 q^{10} + 10054 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} - 351 q^{15} + 24576 q^{16} + 21222 q^{17} + 34992 q^{18} + 9527 q^{19} + 832 q^{20} - 55566 q^{21} + 80432 q^{22} + 33229 q^{23} - 82944 q^{24} + 265321 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} + 174185 q^{29} - 2808 q^{30} + 119045 q^{31} + 196608 q^{32} - 271458 q^{33} + 169776 q^{34} + 4459 q^{35} + 279936 q^{36} + 56562 q^{37} + 76216 q^{38} + 355914 q^{39} + 6656 q^{40} + 101632 q^{41} - 444528 q^{42} + 441323 q^{43} + 643456 q^{44} + 9477 q^{45} + 265832 q^{46} - 892849 q^{47} - 663552 q^{48} + 705894 q^{49} + 2122568 q^{50} - 572994 q^{51} - 843648 q^{52} + 2093965 q^{53} - 944784 q^{54} - 331222 q^{55} + 1053696 q^{56} - 257229 q^{57} + 1393480 q^{58} - 136204 q^{59} - 22464 q^{60} - 3989946 q^{61} + 952360 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 28561 q^{65} - 2171664 q^{66} - 2218250 q^{67} + 1358208 q^{68} - 897183 q^{69} + 35672 q^{70} + 2045928 q^{71} + 2239488 q^{72} - 8557479 q^{73} + 452496 q^{74} - 7163667 q^{75} + 609728 q^{76} + 3448522 q^{77} + 2847312 q^{78} - 8559709 q^{79} + 53248 q^{80} + 3188646 q^{81} + 813056 q^{82} + 2496351 q^{83} - 3556224 q^{84} + 5335304 q^{85} + 3530584 q^{86} - 4702995 q^{87} + 5147648 q^{88} - 2446683 q^{89} + 75816 q^{90} - 4521426 q^{91} + 2126656 q^{92} - 3214215 q^{93} - 7142792 q^{94} + 16410211 q^{95} - 5308416 q^{96} + 5786889 q^{97} + 5647152 q^{98} + 7329366 q^{99}+O(q^{100}) \) 6 * q + 48 * q^2 - 162 * q^3 + 384 * q^4 + 13 * q^5 - 1296 * q^6 + 2058 * q^7 + 3072 * q^8 + 4374 * q^9 + 104 * q^10 + 10054 * q^11 - 10368 * q^12 - 13182 * q^13 + 16464 * q^14 - 351 * q^15 + 24576 * q^16 + 21222 * q^17 + 34992 * q^18 + 9527 * q^19 + 832 * q^20 - 55566 * q^21 + 80432 * q^22 + 33229 * q^23 - 82944 * q^24 + 265321 * q^25 - 105456 * q^26 - 118098 * q^27 + 131712 * q^28 + 174185 * q^29 - 2808 * q^30 + 119045 * q^31 + 196608 * q^32 - 271458 * q^33 + 169776 * q^34 + 4459 * q^35 + 279936 * q^36 + 56562 * q^37 + 76216 * q^38 + 355914 * q^39 + 6656 * q^40 + 101632 * q^41 - 444528 * q^42 + 441323 * q^43 + 643456 * q^44 + 9477 * q^45 + 265832 * q^46 - 892849 * q^47 - 663552 * q^48 + 705894 * q^49 + 2122568 * q^50 - 572994 * q^51 - 843648 * q^52 + 2093965 * q^53 - 944784 * q^54 - 331222 * q^55 + 1053696 * q^56 - 257229 * q^57 + 1393480 * q^58 - 136204 * q^59 - 22464 * q^60 - 3989946 * q^61 + 952360 * q^62 + 1500282 * q^63 + 1572864 * q^64 - 28561 * q^65 - 2171664 * q^66 - 2218250 * q^67 + 1358208 * q^68 - 897183 * q^69 + 35672 * q^70 + 2045928 * q^71 + 2239488 * q^72 - 8557479 * q^73 + 452496 * q^74 - 7163667 * q^75 + 609728 * q^76 + 3448522 * q^77 + 2847312 * q^78 - 8559709 * q^79 + 53248 * q^80 + 3188646 * q^81 + 813056 * q^82 + 2496351 * q^83 - 3556224 * q^84 + 5335304 * q^85 + 3530584 * q^86 - 4702995 * q^87 + 5147648 * q^88 - 2446683 * q^89 + 75816 * q^90 - 4521426 * q^91 + 2126656 * q^92 - 3214215 * q^93 - 7142792 * q^94 + 16410211 * q^95 - 5308416 * q^96 + 5786889 * q^97 + 5647152 * q^98 + 7329366 * q^99

Introduction

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