LMFDB - Embedded newform 546.8.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("546.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$170.562223914$
Analytic rank:	$1$
Dimension:	$5$
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} - 5672x^{3} - 117684x^{2} + 1695035x + 39011360 $ x^5 - 5672x^3 - 117684x^2 + 1695035*x + 39011360
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$82.3485$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +455.671 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +455.671 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -3645.37 q^{10} -7839.76 q^{11} +1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +12303.1 q^{15} +4096.00 q^{16} -27423.7 q^{17} -5832.00 q^{18} +43542.6 q^{19} +29163.0 q^{20} -9261.00 q^{21} +62718.1 q^{22} -71311.8 q^{23} -13824.0 q^{24} +129511. q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +186524. q^{29} -98425.0 q^{30} +96228.1 q^{31} -32768.0 q^{32} -211674. q^{33} +219390. q^{34} -156295. q^{35} +46656.0 q^{36} +540255. q^{37} -348341. q^{38} -59319.0 q^{39} -233304. q^{40} +198469. q^{41} +74088.0 q^{42} -470197. q^{43} -501745. q^{44} +332184. q^{45} +570494. q^{46} -1.08615e6 q^{47} +110592. q^{48} +117649. q^{49} -1.03609e6 q^{50} -740441. q^{51} -140608. q^{52} +1.51874e6 q^{53} -157464. q^{54} -3.57236e6 q^{55} +175616. q^{56} +1.17565e6 q^{57} -1.49219e6 q^{58} -1.52953e6 q^{59} +787400. q^{60} -116460. q^{61} -769825. q^{62} -250047. q^{63} +262144. q^{64} -1.00111e6 q^{65} +1.69339e6 q^{66} -266395. q^{67} -1.75512e6 q^{68} -1.92542e6 q^{69} +1.25036e6 q^{70} -4.83614e6 q^{71} -373248. q^{72} -4.30653e6 q^{73} -4.32204e6 q^{74} +3.49681e6 q^{75} +2.78673e6 q^{76} +2.68904e6 q^{77} +474552. q^{78} -1.89924e6 q^{79} +1.86643e6 q^{80} +531441. q^{81} -1.58775e6 q^{82} +1.41691e6 q^{83} -592704. q^{84} -1.24962e7 q^{85} +3.76157e6 q^{86} +5.03615e6 q^{87} +4.01396e6 q^{88} -5.66367e6 q^{89} -2.65748e6 q^{90} +753571. q^{91} -4.56396e6 q^{92} +2.59816e6 q^{93} +8.68919e6 q^{94} +1.98411e7 q^{95} -884736. q^{96} -7.63738e6 q^{97} -941192. q^{98} -5.71519e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 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 + 56 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 448 q^{10} - 3679 q^{11} + 8640 q^{12} - 10985 q^{13} + 13720 q^{14} + 1512 q^{15} + 20480 q^{16} + 409 q^{17} - 29160 q^{18} + 33730 q^{19} + 3584 q^{20} - 46305 q^{21} + 29432 q^{22} - 142142 q^{23} - 69120 q^{24} + 153981 q^{25} + 87880 q^{26} + 98415 q^{27} - 109760 q^{28} + 88028 q^{29} - 12096 q^{30} + 244543 q^{31} - 163840 q^{32} - 99333 q^{33} - 3272 q^{34} - 19208 q^{35} + 233280 q^{36} + 730963 q^{37} - 269840 q^{38} - 296595 q^{39} - 28672 q^{40} + 479512 q^{41} + 370440 q^{42} - 406536 q^{43} - 235456 q^{44} + 40824 q^{45} + 1137136 q^{46} + 1138945 q^{47} + 552960 q^{48} + 588245 q^{49} - 1231848 q^{50} + 11043 q^{51} - 703040 q^{52} + 297595 q^{53} - 787320 q^{54} - 1834423 q^{55} + 878080 q^{56} + 910710 q^{57} - 704224 q^{58} + 941652 q^{59} + 96768 q^{60} - 2985259 q^{61} - 1956344 q^{62} - 1250235 q^{63} + 1310720 q^{64} - 123032 q^{65} + 794664 q^{66} - 2333504 q^{67} + 26176 q^{68} - 3837834 q^{69} + 153664 q^{70} - 11322272 q^{71} - 1866240 q^{72} - 6631604 q^{73} - 5847704 q^{74} + 4157487 q^{75} + 2158720 q^{76} + 1261897 q^{77} + 2372760 q^{78} - 10600265 q^{79} + 229376 q^{80} + 2657205 q^{81} - 3836096 q^{82} - 2425229 q^{83} - 2963520 q^{84} - 12267705 q^{85} + 3252288 q^{86} + 2376756 q^{87} + 1883648 q^{88} - 1581837 q^{89} - 326592 q^{90} + 3767855 q^{91} - 9097088 q^{92} + 6602661 q^{93} - 9111560 q^{94} - 11507718 q^{95} - 4423680 q^{96} + 5298407 q^{97} - 4705960 q^{98} - 2681991 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 56 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9 - 448 * q^10 - 3679 * q^11 + 8640 * q^12 - 10985 * q^13 + 13720 * q^14 + 1512 * q^15 + 20480 * q^16 + 409 * q^17 - 29160 * q^18 + 33730 * q^19 + 3584 * q^20 - 46305 * q^21 + 29432 * q^22 - 142142 * q^23 - 69120 * q^24 + 153981 * q^25 + 87880 * q^26 + 98415 * q^27 - 109760 * q^28 + 88028 * q^29 - 12096 * q^30 + 244543 * q^31 - 163840 * q^32 - 99333 * q^33 - 3272 * q^34 - 19208 * q^35 + 233280 * q^36 + 730963 * q^37 - 269840 * q^38 - 296595 * q^39 - 28672 * q^40 + 479512 * q^41 + 370440 * q^42 - 406536 * q^43 - 235456 * q^44 + 40824 * q^45 + 1137136 * q^46 + 1138945 * q^47 + 552960 * q^48 + 588245 * q^49 - 1231848 * q^50 + 11043 * q^51 - 703040 * q^52 + 297595 * q^53 - 787320 * q^54 - 1834423 * q^55 + 878080 * q^56 + 910710 * q^57 - 704224 * q^58 + 941652 * q^59 + 96768 * q^60 - 2985259 * q^61 - 1956344 * q^62 - 1250235 * q^63 + 1310720 * q^64 - 123032 * q^65 + 794664 * q^66 - 2333504 * q^67 + 26176 * q^68 - 3837834 * q^69 + 153664 * q^70 - 11322272 * q^71 - 1866240 * q^72 - 6631604 * q^73 - 5847704 * q^74 + 4157487 * q^75 + 2158720 * q^76 + 1261897 * q^77 + 2372760 * q^78 - 10600265 * q^79 + 229376 * q^80 + 2657205 * q^81 - 3836096 * q^82 - 2425229 * q^83 - 2963520 * q^84 - 12267705 * q^85 + 3252288 * q^86 + 2376756 * q^87 + 1883648 * q^88 - 1581837 * q^89 - 326592 * q^90 + 3767855 * q^91 - 9097088 * q^92 + 6602661 * q^93 - 9111560 * q^94 - 11507718 * q^95 - 4423680 * q^96 + 5298407 * q^97 - 4705960 * q^98 - 2681991 * 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
$5$	455.671	1.63026	0.815130	−	0.579278i	$-0.196666\pi$
$5$	455.671	1.63026	0.815130	+	0.579278i	$0.196666\pi$
$6$	−216.000	−0.408248
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	−3645.37	−1.15277
$11$	−7839.76	−1.77594	−0.887970	−	0.459901i	$-0.847885\pi$
$11$	−7839.76	−1.77594	−0.887970	+	0.459901i	$0.847885\pi$
$12$	1728.00	0.288675
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	2744.00	0.267261
$15$	12303.1	0.941231
$16$	4096.00	0.250000
$17$	−27423.7	−1.35380	−0.676901	−	0.736074i	$-0.736678\pi$
$17$	−27423.7	−1.35380	−0.676901	+	0.736074i	$0.736678\pi$
$18$	−5832.00	−0.235702
$19$	43542.6	1.45639	0.728194	−	0.685372i	$-0.240360\pi$
$19$	43542.6	1.45639	0.728194	+	0.685372i	$0.240360\pi$
$20$	29163.0	0.815130
$21$	−9261.00	−0.218218
$22$	62718.1	1.25578
$23$	−71311.8	−1.22212	−0.611060	−	0.791584i	$-0.709257\pi$
$23$	−71311.8	−1.22212	−0.611060	+	0.791584i	$0.709257\pi$
$24$	−13824.0	−0.204124
$25$	129511.	1.65775
$26$	17576.0	0.196116
$27$	19683.0	0.192450
$28$	−21952.0	−0.188982
$29$	186524.	1.42018	0.710088	−	0.704113i	$-0.248655\pi$
$29$	186524.	1.42018	0.710088	+	0.704113i	$0.248655\pi$
$30$	−98425.0	−0.665551
$31$	96228.1	0.580144	0.290072	−	0.957005i	$-0.406321\pi$
$31$	96228.1	0.580144	0.290072	+	0.957005i	$0.406321\pi$
$32$	−32768.0	−0.176777
$33$	−211674.	−1.02534
$34$	219390.	0.957283
$35$	−156295.	−0.616180
$36$	46656.0	0.166667
$37$	540255.	1.75345	0.876724	−	0.480994i	$-0.159724\pi$
$37$	540255.	1.75345	0.876724	+	0.480994i	$0.159724\pi$
$38$	−348341.	−1.02982
$39$	−59319.0	−0.160128
$40$	−233304.	−0.576384
$41$	198469.	0.449727	0.224863	−	0.974390i	$-0.427806\pi$
$41$	198469.	0.449727	0.224863	+	0.974390i	$0.427806\pi$
$42$	74088.0	0.154303
$43$	−470197.	−0.901862	−0.450931	−	0.892559i	$-0.648908\pi$
$43$	−470197.	−0.901862	−0.450931	+	0.892559i	$0.648908\pi$
$44$	−501745.	−0.887970
$45$	332184.	0.543420
$46$	570494.	0.864170
$47$	−1.08615e6	−1.52597	−0.762986	−	0.646415i	$-0.776268\pi$
$47$	−1.08615e6	−1.52597	−0.762986	+	0.646415i	$0.776268\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	−1.03609e6	−1.17220
$51$	−740441.	−0.781618
$52$	−140608.	−0.138675
$53$	1.51874e6	1.40126	0.700628	−	0.713527i	$-0.252903\pi$
$53$	1.51874e6	1.40126	0.700628	+	0.713527i	$0.252903\pi$
$54$	−157464.	−0.136083
$55$	−3.57236e6	−2.89524
$56$	175616.	0.133631
$57$	1.17565e6	0.840845
$58$	−1.49219e6	−1.00422
$59$	−1.52953e6	−0.969564	−0.484782	−	0.874635i	$-0.661101\pi$
$59$	−1.52953e6	−0.969564	−0.484782	+	0.874635i	$0.661101\pi$
$60$	787400.	0.470615
$61$	−116460.	−0.0656938	−0.0328469	−	0.999460i	$-0.510457\pi$
$61$	−116460.	−0.0656938	−0.0328469	+	0.999460i	$0.510457\pi$
$62$	−769825.	−0.410224
$63$	−250047.	−0.125988
$64$	262144.	0.125000
$65$	−1.00111e6	−0.452153
$66$	1.69339e6	0.725025
$67$	−266395.	−0.108209	−0.0541047	−	0.998535i	$-0.517230\pi$
$67$	−266395.	−0.108209	−0.0541047	+	0.998535i	$0.517230\pi$
$68$	−1.75512e6	−0.676901
$69$	−1.92542e6	−0.705592
$70$	1.25036e6	0.435705
$71$	−4.83614e6	−1.60359	−0.801797	−	0.597596i	$-0.796123\pi$
$71$	−4.83614e6	−1.60359	−0.801797	+	0.597596i	$0.796123\pi$
$72$	−373248.	−0.117851
$73$	−4.30653e6	−1.29568	−0.647840	−	0.761777i	$-0.724327\pi$
$73$	−4.30653e6	−1.29568	−0.647840	+	0.761777i	$0.724327\pi$
$74$	−4.32204e6	−1.23987
$75$	3.49681e6	0.957100
$76$	2.78673e6	0.728194
$77$	2.68904e6	0.671243
$78$	474552.	0.113228
$79$	−1.89924e6	−0.433395	−0.216698	−	0.976239i	$-0.569529\pi$
$79$	−1.89924e6	−0.433395	−0.216698	+	0.976239i	$0.569529\pi$
$80$	1.86643e6	0.407565
$81$	531441.	0.111111
$82$	−1.58775e6	−0.318005
$83$	1.41691e6	0.272000	0.136000	−	0.990709i	$-0.456575\pi$
$83$	1.41691e6	0.272000	0.136000	+	0.990709i	$0.456575\pi$
$84$	−592704.	−0.109109
$85$	−1.24962e7	−2.20705
$86$	3.76157e6	0.637713
$87$	5.03615e6	0.819939
$88$	4.01396e6	0.627890
$89$	−5.66367e6	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$89$	−5.66367e6	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$90$	−2.65748e6	−0.384256
$91$	753571.	0.104828
$92$	−4.56396e6	−0.611060
$93$	2.59816e6	0.334947
$94$	8.68919e6	1.07903
$95$	1.98411e7	2.37429
$96$	−884736.	−0.102062
$97$	−7.63738e6	−0.849656	−0.424828	−	0.905274i	$-0.639665\pi$
$97$	−7.63738e6	−0.849656	−0.424828	+	0.905274i	$0.639665\pi$
$98$	−941192.	−0.101015
$99$	−5.71519e6	−0.591980
$100$	8.28873e6	0.828873
$101$	−734133.	−0.0709006	−0.0354503	−	0.999371i	$-0.511287\pi$
$101$	−734133.	−0.0709006	−0.0354503	+	0.999371i	$0.511287\pi$
$102$	5.92353e6	0.552688
$103$	−4.64919e6	−0.419225	−0.209613	−	0.977785i	$-0.567220\pi$
$103$	−4.64919e6	−0.419225	−0.209613	+	0.977785i	$0.567220\pi$
$104$	1.12486e6	0.0980581
$105$	−4.21997e6	−0.355752
$106$	−1.21499e7	−0.990837
$107$	1.38811e7	1.09542	0.547710	−	0.836668i	$-0.315500\pi$
$107$	1.38811e7	1.09542	0.547710	+	0.836668i	$0.315500\pi$
$108$	1.25971e6	0.0962250
$109$	−7.91577e6	−0.585465	−0.292732	−	0.956194i	$-0.594564\pi$
$109$	−7.91577e6	−0.585465	−0.292732	+	0.956194i	$0.594564\pi$
$110$	2.85788e7	2.04725
$111$	1.45869e7	1.01235
$112$	−1.40493e6	−0.0944911
$113$	−1.94259e7	−1.26650	−0.633252	−	0.773945i	$-0.718281\pi$
$113$	−1.94259e7	−1.26650	−0.633252	+	0.773945i	$0.718281\pi$
$114$	−9.40520e6	−0.594568
$115$	−3.24948e7	−1.99237
$116$	1.19375e7	0.710088
$117$	−1.60161e6	−0.0924500
$118$	1.22363e7	0.685585
$119$	9.40634e6	0.511689
$120$	−6.29920e6	−0.332775
$121$	4.19747e7	2.15397
$122$	931684.	0.0464525
$123$	5.35866e6	0.259650
$124$	6.15860e6	0.290072
$125$	2.34153e7	1.07230
$126$	2.00038e6	0.0890871
$127$	−5.25461e6	−0.227629	−0.113814	−	0.993502i	$-0.536307\pi$
$127$	−5.25461e6	−0.227629	−0.113814	+	0.993502i	$0.536307\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.26953e7	−0.520690
$130$	8.00888e6	0.319720
$131$	−4.99739e7	−1.94220	−0.971099	−	0.238677i	$-0.923286\pi$
$131$	−4.99739e7	−1.94220	−0.971099	+	0.238677i	$0.923286\pi$
$132$	−1.35471e7	−0.512670
$133$	−1.49351e7	−0.550463
$134$	2.13116e6	0.0765156
$135$	8.96898e6	0.313744
$136$	1.40410e7	0.478642
$137$	−8.13476e6	−0.270285	−0.135143	−	0.990826i	$-0.543149\pi$
$137$	−8.13476e6	−0.270285	−0.135143	+	0.990826i	$0.543149\pi$
$138$	1.54034e7	0.498929
$139$	−4.12095e7	−1.30150	−0.650752	−	0.759290i	$-0.725546\pi$
$139$	−4.12095e7	−1.30150	−0.650752	+	0.759290i	$0.725546\pi$
$140$	−1.00029e7	−0.308090
$141$	−2.93260e7	−0.881020
$142$	3.86891e7	1.13391
$143$	1.72240e7	0.492557
$144$	2.98598e6	0.0833333
$145$	8.49937e7	2.31526
$146$	3.44522e7	0.916184
$147$	3.17652e6	0.0824786
$148$	3.45763e7	0.876724
$149$	−2.80802e7	−0.695423	−0.347711	−	0.937602i	$-0.613041\pi$
$149$	−2.80802e7	−0.695423	−0.347711	+	0.937602i	$0.613041\pi$
$150$	−2.79745e7	−0.676772
$151$	1.13111e7	0.267353	0.133676	−	0.991025i	$-0.457322\pi$
$151$	1.13111e7	0.267353	0.133676	+	0.991025i	$0.457322\pi$
$152$	−2.22938e7	−0.514911
$153$	−1.99919e7	−0.451268
$154$	−2.15123e7	−0.474640
$155$	4.38484e7	0.945786
$156$	−3.79642e6	−0.0800641
$157$	−2.27758e7	−0.469705	−0.234853	−	0.972031i	$-0.575461\pi$
$157$	−2.27758e7	−0.469705	−0.234853	+	0.972031i	$0.575461\pi$
$158$	1.51939e7	0.306457
$159$	4.10059e7	0.809015
$160$	−1.49314e7	−0.288192
$161$	2.44600e7	0.461918
$162$	−4.25153e6	−0.0785674
$163$	−2.12998e6	−0.0385230	−0.0192615	−	0.999814i	$-0.506131\pi$
$163$	−2.12998e6	−0.0385230	−0.0192615	+	0.999814i	$0.506131\pi$
$164$	1.27020e7	0.224863
$165$	−9.64536e7	−1.67157
$166$	−1.13353e7	−0.192333
$167$	−2.07084e7	−0.344064	−0.172032	−	0.985091i	$-0.555033\pi$
$167$	−2.07084e7	−0.344064	−0.172032	+	0.985091i	$0.555033\pi$
$168$	4.74163e6	0.0771517
$169$	4.82681e6	0.0769231
$170$	9.99697e7	1.56062
$171$	3.17426e7	0.485462
$172$	−3.00926e7	−0.450931
$173$	1.10102e8	1.61671	0.808355	−	0.588695i	$-0.200358\pi$
$173$	1.10102e8	1.61671	0.808355	+	0.588695i	$0.200358\pi$
$174$	−4.02892e7	−0.579784
$175$	−4.44224e7	−0.626569
$176$	−3.21117e7	−0.443985
$177$	−4.12974e7	−0.559778
$178$	4.53093e7	0.602168
$179$	−7.05849e7	−0.919870	−0.459935	−	0.887953i	$-0.652127\pi$
$179$	−7.05849e7	−0.919870	−0.459935	+	0.887953i	$0.652127\pi$
$180$	2.12598e7	0.271710
$181$	−1.49938e8	−1.87948	−0.939740	−	0.341890i	$-0.888933\pi$
$181$	−1.49938e8	−1.87948	−0.939740	+	0.341890i	$0.888933\pi$
$182$	−6.02857e6	−0.0741249
$183$	−3.14443e6	−0.0379283
$184$	3.65116e7	0.432085
$185$	2.46179e8	2.85857
$186$	−2.07853e7	−0.236843
$187$	2.14996e8	2.40427
$188$	−6.95135e7	−0.762986
$189$	−6.75127e6	−0.0727393
$190$	−1.58729e8	−1.67888
$191$	2.59002e7	0.268959	0.134479	−	0.990916i	$-0.457064\pi$
$191$	2.59002e7	0.268959	0.134479	+	0.990916i	$0.457064\pi$
$192$	7.07789e6	0.0721688
$193$	1.53175e8	1.53369	0.766845	−	0.641833i	$-0.221826\pi$
$193$	1.53175e8	1.53369	0.766845	+	0.641833i	$0.221826\pi$
$194$	6.10990e7	0.600798
$195$	−2.70300e7	−0.261050
$196$	7.52954e6	0.0714286
$197$	−3.22995e6	−0.0300998	−0.0150499	−	0.999887i	$-0.504791\pi$
$197$	−3.22995e6	−0.0300998	−0.0150499	+	0.999887i	$0.504791\pi$
$198$	4.57215e7	0.418593
$199$	−5.47172e7	−0.492196	−0.246098	−	0.969245i	$-0.579149\pi$
$199$	−5.47172e7	−0.492196	−0.246098	+	0.969245i	$0.579149\pi$
$200$	−6.63098e7	−0.586102
$201$	−7.19268e6	−0.0624747
$202$	5.87307e6	0.0501343
$203$	−6.39778e7	−0.536776
$204$	−4.73882e7	−0.390809
$205$	9.04366e7	0.733172
$206$	3.71935e7	0.296437
$207$	−5.19863e7	−0.407374
$208$	−8.99891e6	−0.0693375
$209$	−3.41364e8	−2.58646
$210$	3.37598e7	0.251555
$211$	−1.57636e8	−1.15522	−0.577611	−	0.816312i	$-0.696015\pi$
$211$	−1.57636e8	−1.15522	−0.577611	+	0.816312i	$0.696015\pi$
$212$	9.71992e7	0.700628
$213$	−1.30576e8	−0.925836
$214$	−1.11049e8	−0.774579
$215$	−2.14255e8	−1.47027
$216$	−1.00777e7	−0.0680414
$217$	−3.30062e7	−0.219274
$218$	6.33262e7	0.413986
$219$	−1.16276e8	−0.748061
$220$	−2.28631e8	−1.44762
$221$	6.02500e7	0.375477
$222$	−1.16695e8	−0.715842
$223$	2.03339e8	1.22787	0.613935	−	0.789356i	$-0.289585\pi$
$223$	2.03339e8	1.22787	0.613935	+	0.789356i	$0.289585\pi$
$224$	1.12394e7	0.0668153
$225$	9.44138e7	0.552582
$226$	1.55407e8	0.895554
$227$	−1.01690e8	−0.577016	−0.288508	−	0.957477i	$-0.593159\pi$
$227$	−1.01690e8	−0.577016	−0.288508	+	0.957477i	$0.593159\pi$
$228$	7.52416e7	0.420423
$229$	−3.47526e8	−1.91233	−0.956165	−	0.292827i	$-0.905404\pi$
$229$	−3.47526e8	−1.91233	−0.956165	+	0.292827i	$0.905404\pi$
$230$	2.59958e8	1.40882
$231$	7.26040e7	0.387542
$232$	−9.55004e7	−0.502108
$233$	−2.23390e8	−1.15696	−0.578479	−	0.815697i	$-0.696354\pi$
$233$	−2.23390e8	−1.15696	−0.578479	+	0.815697i	$0.696354\pi$
$234$	1.28129e7	0.0653720
$235$	−4.94927e8	−2.48773
$236$	−9.78901e7	−0.484782
$237$	−5.12794e7	−0.250221
$238$	−7.52507e7	−0.361819
$239$	−1.54192e8	−0.730584	−0.365292	−	0.930893i	$-0.619031\pi$
$239$	−1.54192e8	−0.730584	−0.365292	+	0.930893i	$0.619031\pi$
$240$	5.03936e7	0.235308
$241$	−2.79403e8	−1.28579	−0.642897	−	0.765953i	$-0.722268\pi$
$241$	−2.79403e8	−1.28579	−0.642897	+	0.765953i	$0.722268\pi$
$242$	−3.35798e8	−1.52308
$243$	1.43489e7	0.0641500
$244$	−7.45347e6	−0.0328469
$245$	5.36093e7	0.232894
$246$	−4.28693e7	−0.183600
$247$	−9.56631e7	−0.403929
$248$	−4.92688e7	−0.205112
$249$	3.82566e7	0.157039
$250$	−1.87323e8	−0.758228
$251$	−1.22743e7	−0.0489935	−0.0244967	−	0.999700i	$-0.507798\pi$
$251$	−1.22743e7	−0.0489935	−0.0244967	+	0.999700i	$0.507798\pi$
$252$	−1.60030e7	−0.0629941
$253$	5.59068e8	2.17041
$254$	4.20369e7	0.160958
$255$	−3.37398e8	−1.27424
$256$	1.67772e7	0.0625000
$257$	−3.63199e8	−1.33469	−0.667343	−	0.744751i	$-0.732568\pi$
$257$	−3.63199e8	−1.33469	−0.667343	+	0.744751i	$0.732568\pi$
$258$	1.01563e8	0.368184
$259$	−1.85308e8	−0.662741
$260$	−6.40710e7	−0.226076
$261$	1.35976e8	0.473392
$262$	3.99791e8	1.37334
$263$	4.08272e8	1.38390	0.691950	−	0.721945i	$-0.256752\pi$
$263$	4.08272e8	1.38390	0.691950	+	0.721945i	$0.256752\pi$
$264$	1.08377e8	0.362512
$265$	6.92045e8	2.28441
$266$	1.19481e8	0.389236
$267$	−1.52919e8	−0.491668
$268$	−1.70493e7	−0.0541047
$269$	1.22404e8	0.383410	0.191705	−	0.981453i	$-0.438598\pi$
$269$	1.22404e8	0.383410	0.191705	+	0.981453i	$0.438598\pi$
$270$	−7.17518e7	−0.221850
$271$	3.17537e8	0.969175	0.484587	−	0.874743i	$-0.338970\pi$
$271$	3.17537e8	0.969175	0.484587	+	0.874743i	$0.338970\pi$
$272$	−1.12328e8	−0.338451
$273$	2.03464e7	0.0605228
$274$	6.50781e7	0.191121
$275$	−1.01534e9	−2.94406
$276$	−1.23227e8	−0.352796
$277$	5.97381e8	1.68878	0.844389	−	0.535731i	$-0.179964\pi$
$277$	5.97381e8	1.68878	0.844389	+	0.535731i	$0.179964\pi$
$278$	3.29676e8	0.920302
$279$	7.01503e7	0.193381
$280$	8.00232e7	0.217853
$281$	−3.00006e8	−0.806600	−0.403300	−	0.915068i	$-0.632137\pi$
$281$	−3.00006e8	−0.806600	−0.403300	+	0.915068i	$0.632137\pi$
$282$	2.34608e8	0.622976
$283$	6.73506e7	0.176640	0.0883199	−	0.996092i	$-0.471850\pi$
$283$	6.73506e7	0.176640	0.0883199	+	0.996092i	$0.471850\pi$
$284$	−3.09513e8	−0.801797
$285$	5.35710e8	1.37080
$286$	−1.37792e8	−0.348291
$287$	−6.80748e7	−0.169981
$288$	−2.38879e7	−0.0589256
$289$	3.41723e8	0.832782
$290$	−6.79950e8	−1.63713
$291$	−2.06209e8	−0.490549
$292$	−2.75618e8	−0.647840
$293$	5.90976e8	1.37257	0.686283	−	0.727334i	$-0.259241\pi$
$293$	5.90976e8	1.37257	0.686283	+	0.727334i	$0.259241\pi$
$294$	−2.54122e7	−0.0583212
$295$	−6.96964e8	−1.58064
$296$	−2.76611e8	−0.619937
$297$	−1.54310e8	−0.341780
$298$	2.24642e8	0.491738
$299$	1.56672e8	0.338955
$300$	2.23796e8	0.478550
$301$	1.61278e8	0.340872
$302$	−9.04886e7	−0.189047
$303$	−1.98216e7	−0.0409345
$304$	1.78351e8	0.364097
$305$	−5.30677e7	−0.107098
$306$	1.59935e8	0.319094
$307$	−1.08502e8	−0.214020	−0.107010	−	0.994258i	$-0.534128\pi$
$307$	−1.08502e8	−0.214020	−0.107010	+	0.994258i	$0.534128\pi$
$308$	1.72098e8	0.335621
$309$	−1.25528e8	−0.242040
$310$	−3.50787e8	−0.668772
$311$	−4.43839e8	−0.836689	−0.418345	−	0.908288i	$-0.637390\pi$
$311$	−4.43839e8	−0.836689	−0.418345	+	0.908288i	$0.637390\pi$
$312$	3.03713e7	0.0566139
$313$	3.88413e8	0.715961	0.357980	−	0.933729i	$-0.383465\pi$
$313$	3.88413e8	0.715961	0.357980	+	0.933729i	$0.383465\pi$
$314$	1.82207e8	0.332132
$315$	−1.13939e8	−0.205393
$316$	−1.21551e8	−0.216698
$317$	−8.14915e8	−1.43683	−0.718414	−	0.695615i	$-0.755132\pi$
$317$	−8.14915e8	−1.43683	−0.718414	+	0.695615i	$0.755132\pi$
$318$	−3.28047e8	−0.572060
$319$	−1.46231e9	−2.52215
$320$	1.19452e8	0.203782
$321$	3.74790e8	0.632441
$322$	−1.95680e8	−0.326625
$323$	−1.19410e9	−1.97166
$324$	3.40122e7	0.0555556
$325$	−2.84537e8	−0.459776
$326$	1.70399e7	0.0272398
$327$	−2.13726e8	−0.338018
$328$	−1.01616e8	−0.159002
$329$	3.72549e8	0.576763
$330$	7.71629e8	1.18198
$331$	6.30032e8	0.954914	0.477457	−	0.878655i	$-0.341559\pi$
$331$	6.30032e8	0.954914	0.477457	+	0.878655i	$0.341559\pi$
$332$	9.06823e7	0.136000
$333$	3.93846e8	0.584483
$334$	1.65668e8	0.243290
$335$	−1.21389e8	−0.176409
$336$	−3.79331e7	−0.0545545
$337$	−1.00120e7	−0.0142500	−0.00712499	−	0.999975i	$-0.502268\pi$
$337$	−1.00120e7	−0.0142500	−0.00712499	+	0.999975i	$0.502268\pi$
$338$	−3.86145e7	−0.0543928
$339$	−5.24500e8	−0.731217
$340$	−7.99758e8	−1.10352
$341$	−7.54406e8	−1.03030
$342$	−2.53940e8	−0.343274
$343$	−4.03536e7	−0.0539949
$344$	2.40741e8	0.318856
$345$	−8.77358e8	−1.15030
$346$	−8.80813e8	−1.14319
$347$	−8.08247e8	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$347$	−8.08247e8	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$348$	3.22314e8	0.409969
$349$	−4.87238e8	−0.613553	−0.306776	−	0.951782i	$-0.599250\pi$
$349$	−4.87238e8	−0.613553	−0.306776	+	0.951782i	$0.599250\pi$
$350$	3.55379e8	0.443051
$351$	−4.32436e7	−0.0533761
$352$	2.56893e8	0.313945
$353$	−3.29070e8	−0.398177	−0.199089	−	0.979982i	$-0.563798\pi$
$353$	−3.29070e8	−0.398177	−0.199089	+	0.979982i	$0.563798\pi$
$354$	3.30379e8	0.395823
$355$	−2.20369e9	−2.61428
$356$	−3.62475e8	−0.425797
$357$	2.53971e8	0.295424
$358$	5.64680e8	0.650446
$359$	−2.85510e8	−0.325680	−0.162840	−	0.986653i	$-0.552065\pi$
$359$	−2.85510e8	−0.325680	−0.162840	+	0.986653i	$0.552065\pi$
$360$	−1.70078e8	−0.192128
$361$	1.00209e9	1.12106
$362$	1.19951e9	1.32899
$363$	1.13332e9	1.24359
$364$	4.82285e7	0.0524142
$365$	−1.96236e9	−2.11229
$366$	2.51555e7	0.0268194
$367$	4.23025e8	0.446719	0.223360	−	0.974736i	$-0.428298\pi$
$367$	4.23025e8	0.446719	0.223360	+	0.974736i	$0.428298\pi$
$368$	−2.92093e8	−0.305530
$369$	1.44684e8	0.149909
$370$	−1.96943e9	−2.02132
$371$	−5.20927e8	−0.529625
$372$	1.66282e8	0.167473
$373$	4.93805e8	0.492691	0.246345	−	0.969182i	$-0.420770\pi$
$373$	4.93805e8	0.492691	0.246345	+	0.969182i	$0.420770\pi$
$374$	−1.71996e9	−1.70008
$375$	6.32214e8	0.619091
$376$	5.56108e8	0.539513
$377$	−4.09794e8	−0.393886
$378$	5.40102e7	0.0514344
$379$	1.05951e9	0.999696	0.499848	−	0.866113i	$-0.333389\pi$
$379$	1.05951e9	0.999696	0.499848	+	0.866113i	$0.333389\pi$
$380$	1.26983e9	1.18714
$381$	−1.41875e8	−0.131422
$382$	−2.07201e8	−0.190183
$383$	−2.41116e8	−0.219296	−0.109648	−	0.993970i	$-0.534972\pi$
$383$	−2.41116e8	−0.219296	−0.109648	+	0.993970i	$0.534972\pi$
$384$	−5.66231e7	−0.0510310
$385$	1.22532e9	1.09430
$386$	−1.22540e9	−1.08448
$387$	−3.42773e8	−0.300621
$388$	−4.88792e8	−0.424828
$389$	3.51696e7	0.0302931	0.0151465	−	0.999885i	$-0.495179\pi$
$389$	3.51696e7	0.0302931	0.0151465	+	0.999885i	$0.495179\pi$
$390$	2.16240e8	0.184591
$391$	1.95564e9	1.65451
$392$	−6.02363e7	−0.0505076
$393$	−1.34929e9	−1.12133
$394$	2.58396e7	0.0212838
$395$	−8.65428e8	−0.706547
$396$	−3.65772e8	−0.295990
$397$	1.19041e9	0.954840	0.477420	−	0.878675i	$-0.341572\pi$
$397$	1.19041e9	0.954840	0.477420	+	0.878675i	$0.341572\pi$
$398$	4.37738e8	0.348035
$399$	−4.03248e8	−0.317810
$400$	5.30479e8	0.414437
$401$	−1.29570e9	−1.00346	−0.501729	−	0.865025i	$-0.667302\pi$
$401$	−1.29570e9	−1.00346	−0.501729	+	0.865025i	$0.667302\pi$
$402$	5.75414e7	0.0441763
$403$	−2.11413e8	−0.160903
$404$	−4.69845e7	−0.0354503
$405$	2.42162e8	0.181140
$406$	5.11822e8	0.379558
$407$	−4.23547e9	−3.11402
$408$	3.79106e8	0.276344
$409$	1.12797e8	0.0815202	0.0407601	−	0.999169i	$-0.487022\pi$
$409$	1.12797e8	0.0815202	0.0407601	+	0.999169i	$0.487022\pi$
$410$	−7.23493e8	−0.518431
$411$	−2.19638e8	−0.156049
$412$	−2.97548e8	−0.209613
$413$	5.24630e8	0.366461
$414$	4.15890e8	0.288057
$415$	6.45646e8	0.443431
$416$	7.19913e7	0.0490290
$417$	−1.11266e9	−0.751424
$418$	2.73091e9	1.82890
$419$	1.95798e9	1.30035	0.650174	−	0.759785i	$-0.274696\pi$
$419$	1.95798e9	1.30035	0.650174	+	0.759785i	$0.274696\pi$
$420$	−2.70078e8	−0.177876
$421$	−1.47827e9	−0.965533	−0.482766	−	0.875749i	$-0.660368\pi$
$421$	−1.47827e9	−0.965533	−0.482766	+	0.875749i	$0.660368\pi$
$422$	1.26108e9	0.816866
$423$	−7.91802e8	−0.508657
$424$	−7.77594e8	−0.495419
$425$	−3.55169e9	−2.24426
$426$	1.04461e9	0.654665
$427$	3.99459e7	0.0248299
$428$	8.88390e8	0.547710
$429$	4.65047e8	0.284378
$430$	1.71404e9	1.03964
$431$	1.50909e9	0.907912	0.453956	−	0.891024i	$-0.350012\pi$
$431$	1.50909e9	0.907912	0.453956	+	0.891024i	$0.350012\pi$
$432$	8.06216e7	0.0481125
$433$	−1.48071e9	−0.876521	−0.438260	−	0.898848i	$-0.644405\pi$
$433$	−1.48071e9	−0.876521	−0.438260	+	0.898848i	$0.644405\pi$
$434$	2.64050e8	0.155050
$435$	2.29483e9	1.33671
$436$	−5.06610e8	−0.292732
$437$	−3.10510e9	−1.77988
$438$	9.30211e8	0.528959
$439$	−7.01901e8	−0.395959	−0.197979	−	0.980206i	$-0.563438\pi$
$439$	−7.01901e8	−0.395959	−0.197979	+	0.980206i	$0.563438\pi$
$440$	1.82905e9	1.02362
$441$	8.57661e7	0.0476190
$442$	−4.82000e8	−0.265503
$443$	−6.31057e8	−0.344870	−0.172435	−	0.985021i	$-0.555163\pi$
$443$	−6.31057e8	−0.344870	−0.172435	+	0.985021i	$0.555163\pi$
$444$	9.33561e8	0.506177
$445$	−2.58077e9	−1.38832
$446$	−1.62671e9	−0.868236
$447$	−7.58167e8	−0.401503
$448$	−8.99154e7	−0.0472456
$449$	2.70227e9	1.40886	0.704428	−	0.709776i	$-0.251204\pi$
$449$	2.70227e9	1.40886	0.704428	+	0.709776i	$0.251204\pi$
$450$	−7.55311e8	−0.390734
$451$	−1.55595e9	−0.798688
$452$	−1.24326e9	−0.633252
$453$	3.05399e8	0.154356
$454$	8.13521e8	0.408012
$455$	3.43381e8	0.170898
$456$	−6.01933e8	−0.297284
$457$	2.60138e9	1.27496	0.637481	−	0.770466i	$-0.279976\pi$
$457$	2.60138e9	1.27496	0.637481	+	0.770466i	$0.279976\pi$
$458$	2.78021e9	1.35222
$459$	−5.39781e8	−0.260539
$460$	−2.07966e9	−0.996187
$461$	7.58390e8	0.360528	0.180264	−	0.983618i	$-0.442305\pi$
$461$	7.58390e8	0.360528	0.180264	+	0.983618i	$0.442305\pi$
$462$	−5.80832e8	−0.274034
$463$	−9.74202e8	−0.456158	−0.228079	−	0.973643i	$-0.573244\pi$
$463$	−9.74202e8	−0.456158	−0.228079	+	0.973643i	$0.573244\pi$
$464$	7.64003e8	0.355044
$465$	1.18391e9	0.546050
$466$	1.78712e9	0.818093
$467$	3.23896e9	1.47162	0.735812	−	0.677186i	$-0.236801\pi$
$467$	3.23896e9	1.47162	0.735812	+	0.677186i	$0.236801\pi$
$468$	−1.02503e8	−0.0462250
$469$	9.13737e7	0.0408993
$470$	3.95941e9	1.75909
$471$	−6.14948e8	−0.271185
$472$	7.83121e8	0.342793
$473$	3.68623e9	1.60165
$474$	4.10235e8	0.176933
$475$	5.63926e9	2.41432
$476$	6.02006e8	0.255845
$477$	1.10716e9	0.467085
$478$	1.23354e9	0.516601
$479$	2.74471e9	1.14110	0.570548	−	0.821264i	$-0.306731\pi$
$479$	2.74471e9	1.14110	0.570548	+	0.821264i	$0.306731\pi$
$480$	−4.03149e8	−0.166388
$481$	−1.18694e9	−0.486319
$482$	2.23522e9	0.909193
$483$	6.60419e8	0.266689
$484$	2.68638e9	1.07698
$485$	−3.48013e9	−1.38516
$486$	−1.14791e8	−0.0453609
$487$	2.10877e9	0.827329	0.413665	−	0.910429i	$-0.364249\pi$
$487$	2.10877e9	0.827329	0.413665	+	0.910429i	$0.364249\pi$
$488$	5.96278e7	0.0232263
$489$	−5.75095e7	−0.0222412
$490$	−4.28874e8	−0.164681
$491$	−1.30974e9	−0.499344	−0.249672	−	0.968330i	$-0.580323\pi$
$491$	−1.30974e9	−0.499344	−0.249672	+	0.968330i	$0.580323\pi$
$492$	3.42954e8	0.129825
$493$	−5.11519e9	−1.92264
$494$	7.65305e8	0.285621
$495$	−2.60425e9	−0.965081
$496$	3.94150e8	0.145036
$497$	1.65880e9	0.606102
$498$	−3.06053e8	−0.111044
$499$	4.96201e8	0.178775	0.0893874	−	0.995997i	$-0.471509\pi$
$499$	4.96201e8	0.178775	0.0893874	+	0.995997i	$0.471509\pi$
$500$	1.49858e9	0.536148
$501$	−5.59128e8	−0.198646
$502$	9.81943e7	0.0346436
$503$	3.62740e9	1.27089	0.635443	−	0.772147i	$-0.280817\pi$
$503$	3.62740e9	1.27089	0.635443	+	0.772147i	$0.280817\pi$
$504$	1.28024e8	0.0445435
$505$	−3.34523e8	−0.115586
$506$	−4.47254e9	−1.53471
$507$	1.30324e8	0.0444116
$508$	−3.36295e8	−0.113814
$509$	3.07915e9	1.03495	0.517474	−	0.855699i	$-0.326872\pi$
$509$	3.07915e9	1.03495	0.517474	+	0.855699i	$0.326872\pi$
$510$	2.69918e9	0.901024
$511$	1.47714e9	0.489721
$512$	−1.34218e8	−0.0441942
$513$	8.57049e8	0.280282
$514$	2.90559e9	0.943765
$515$	−2.11850e9	−0.683446
$516$	−8.12500e8	−0.260345
$517$	8.51514e9	2.71004
$518$	1.48246e9	0.468629
$519$	2.97274e9	0.933408
$520$	5.12568e8	0.159860
$521$	−3.09970e9	−0.960259	−0.480129	−	0.877198i	$-0.659410\pi$
$521$	−3.09970e9	−0.960259	−0.480129	+	0.877198i	$0.659410\pi$
$522$	−1.08781e9	−0.334739
$523$	5.80954e8	0.177577	0.0887883	−	0.996051i	$-0.471701\pi$
$523$	5.80954e8	0.177577	0.0887883	+	0.996051i	$0.471701\pi$
$524$	−3.19833e9	−0.971099
$525$	−1.19941e9	−0.361750
$526$	−3.26618e9	−0.978565
$527$	−2.63893e9	−0.785401
$528$	−8.67015e8	−0.256335
$529$	1.68055e9	0.493579
$530$	−5.53636e9	−1.61532
$531$	−1.11503e9	−0.323188
$532$	−9.55847e8	−0.275231
$533$	−4.36036e8	−0.124732
$534$	1.22335e9	0.347662
$535$	6.32522e9	1.78582
$536$	1.36394e8	0.0382578
$537$	−1.90579e9	−0.531087
$538$	−9.79233e8	−0.271112
$539$	−9.22340e8	−0.253706
$540$	5.74015e8	0.156872
$541$	6.14647e9	1.66892	0.834459	−	0.551069i	$-0.185780\pi$
$541$	6.14647e9	1.66892	0.834459	+	0.551069i	$0.185780\pi$
$542$	−2.54030e9	−0.685310
$543$	−4.04834e9	−1.08512
$544$	8.98621e8	0.239321
$545$	−3.60699e9	−0.954459
$546$	−1.62771e8	−0.0427960
$547$	−5.68006e9	−1.48388	−0.741938	−	0.670469i	$-0.766093\pi$
$547$	−5.68006e9	−1.48388	−0.741938	+	0.670469i	$0.766093\pi$
$548$	−5.20625e8	−0.135143
$549$	−8.48997e7	−0.0218979
$550$	8.12271e9	2.08176
$551$	8.12175e9	2.06833
$552$	9.85814e8	0.249464
$553$	6.51438e8	0.163808
$554$	−4.77905e9	−1.19415
$555$	6.64683e9	1.65040
$556$	−2.63741e9	−0.650752
$557$	−5.78253e9	−1.41783	−0.708915	−	0.705294i	$-0.750815\pi$
$557$	−5.78253e9	−1.41783	−0.708915	+	0.705294i	$0.750815\pi$
$558$	−5.61202e8	−0.136741
$559$	1.03302e9	0.250132
$560$	−6.40185e8	−0.154045
$561$	5.80488e9	1.38811
$562$	2.40005e9	0.570352
$563$	4.74751e9	1.12121	0.560604	−	0.828084i	$-0.310569\pi$
$563$	4.74751e9	1.12121	0.560604	+	0.828084i	$0.310569\pi$
$564$	−1.87686e9	−0.440510
$565$	−8.85183e9	−2.06473
$566$	−5.38805e8	−0.124903
$567$	−1.82284e8	−0.0419961
$568$	2.47610e9	0.566956
$569$	2.46178e9	0.560216	0.280108	−	0.959968i	$-0.409630\pi$
$569$	2.46178e9	0.560216	0.280108	+	0.959968i	$0.409630\pi$
$570$	−4.28568e9	−0.969299
$571$	4.75691e9	1.06930	0.534649	−	0.845074i	$-0.320444\pi$
$571$	4.75691e9	1.06930	0.534649	+	0.845074i	$0.320444\pi$
$572$	1.10233e9	0.246279
$573$	6.99304e8	0.155283
$574$	5.44599e8	0.120195
$575$	−9.23569e9	−2.02597
$576$	1.91103e8	0.0416667
$577$	−1.17874e9	−0.255447	−0.127724	−	0.991810i	$-0.540767\pi$
$577$	−1.17874e9	−0.255447	−0.127724	+	0.991810i	$0.540767\pi$
$578$	−2.73378e9	−0.588866
$579$	4.13573e9	0.885476
$580$	5.43960e9	1.15763
$581$	−4.86001e8	−0.102806
$582$	1.64967e9	0.346871
$583$	−1.19065e10	−2.48855
$584$	2.20494e9	0.458092
$585$	−7.29809e8	−0.150718
$586$	−4.72781e9	−0.970551
$587$	2.48459e7	0.00507016	0.00253508	−	0.999997i	$-0.499193\pi$
$587$	2.48459e7	0.00507016	0.00253508	+	0.999997i	$0.499193\pi$
$588$	2.03297e8	0.0412393
$589$	4.19002e9	0.844915
$590$	5.57571e9	1.11768
$591$	−8.72085e7	−0.0173781
$592$	2.21288e9	0.438362
$593$	6.43685e9	1.26760	0.633800	−	0.773497i	$-0.281494\pi$
$593$	6.43685e9	1.26760	0.633800	+	0.773497i	$0.281494\pi$
$594$	1.23448e9	0.241675
$595$	4.28620e9	0.834186
$596$	−1.79714e9	−0.347711
$597$	−1.47736e9	−0.284169
$598$	−1.25338e9	−0.239678
$599$	8.32987e9	1.58360	0.791798	−	0.610783i	$-0.209145\pi$
$599$	8.32987e9	1.58360	0.791798	+	0.610783i	$0.209145\pi$
$600$	−1.79037e9	−0.338386
$601$	1.50211e9	0.282255	0.141128	−	0.989991i	$-0.454927\pi$
$601$	1.50211e9	0.282255	0.141128	+	0.989991i	$0.454927\pi$
$602$	−1.29022e9	−0.241033
$603$	−1.94202e8	−0.0360698
$604$	7.23909e8	0.133676
$605$	1.91267e10	3.51152
$606$	1.58573e8	0.0289451
$607$	−6.53302e8	−0.118564	−0.0592821	−	0.998241i	$-0.518881\pi$
$607$	−6.53302e8	−0.118564	−0.0592821	+	0.998241i	$0.518881\pi$
$608$	−1.42680e9	−0.257455
$609$	−1.72740e9	−0.309908
$610$	4.24542e8	0.0757296
$611$	2.38627e9	0.423229
$612$	−1.27948e9	−0.225634
$613$	9.20520e9	1.61407	0.807033	−	0.590506i	$-0.201072\pi$
$613$	9.20520e9	1.61407	0.807033	+	0.590506i	$0.201072\pi$
$614$	8.68016e8	0.151335
$615$	2.44179e9	0.423297
$616$	−1.37679e9	−0.237320
$617$	6.13406e9	1.05136	0.525678	−	0.850684i	$-0.323812\pi$
$617$	6.13406e9	1.05136	0.525678	+	0.850684i	$0.323812\pi$
$618$	1.00423e9	0.171148
$619$	−1.06482e10	−1.80450	−0.902252	−	0.431210i	$-0.858087\pi$
$619$	−1.06482e10	−1.80450	−0.902252	+	0.431210i	$0.858087\pi$
$620$	2.80630e9	0.472893
$621$	−1.40363e9	−0.235197
$622$	3.55071e9	0.591629
$623$	1.94264e9	0.321872
$624$	−2.42971e8	−0.0400320
$625$	5.51611e8	0.0903759
$626$	−3.10731e9	−0.506261
$627$	−9.21682e9	−1.49329
$628$	−1.45765e9	−0.234853
$629$	−1.48158e10	−2.37382
$630$	9.11514e8	0.145235
$631$	−6.66975e9	−1.05683	−0.528417	−	0.848985i	$-0.677214\pi$
$631$	−6.66975e9	−1.05683	−0.528417	+	0.848985i	$0.677214\pi$
$632$	9.72409e8	0.153228
$633$	−4.25616e9	−0.666968
$634$	6.51932e9	1.01599
$635$	−2.39438e9	−0.371094
$636$	2.62438e9	0.404508
$637$	−2.58475e8	−0.0396214
$638$	1.16984e10	1.78343
$639$	−3.52555e9	−0.534532
$640$	−9.55612e8	−0.144096
$641$	−7.73704e9	−1.16030	−0.580152	−	0.814509i	$-0.697007\pi$
$641$	−7.73704e9	−1.16030	−0.580152	+	0.814509i	$0.697007\pi$
$642$	−2.99832e9	−0.447203
$643$	3.00464e9	0.445712	0.222856	−	0.974851i	$-0.428462\pi$
$643$	3.00464e9	0.445712	0.222856	+	0.974851i	$0.428462\pi$
$644$	1.56544e9	0.230959
$645$	−5.78489e9	−0.848860
$646$	9.55281e9	1.39417
$647$	8.20416e9	1.19088	0.595442	−	0.803398i	$-0.296977\pi$
$647$	8.20416e9	1.19088	0.595442	+	0.803398i	$0.296977\pi$
$648$	−2.72098e8	−0.0392837
$649$	1.19912e10	1.72189
$650$	2.27629e9	0.325111
$651$	−8.91169e8	−0.126598
$652$	−1.36319e8	−0.0192615
$653$	−1.71794e9	−0.241441	−0.120721	−	0.992687i	$-0.538521\pi$
$653$	−1.71794e9	−0.241441	−0.120721	+	0.992687i	$0.538521\pi$
$654$	1.70981e9	0.239015
$655$	−2.27717e10	−3.16629
$656$	8.12928e8	0.112432
$657$	−3.13946e9	−0.431893
$658$	−2.98039e9	−0.407833
$659$	−1.31549e10	−1.79056	−0.895279	−	0.445505i	$-0.853024\pi$
$659$	−1.31549e10	−1.79056	−0.895279	+	0.445505i	$0.853024\pi$
$660$	−6.17303e9	−0.835785
$661$	1.31159e10	1.76642	0.883210	−	0.468977i	$-0.155377\pi$
$661$	1.31159e10	1.76642	0.883210	+	0.468977i	$0.155377\pi$
$662$	−5.04025e9	−0.675226
$663$	1.62675e9	0.216782
$664$	−7.25459e8	−0.0961666
$665$	−6.80550e9	−0.897397
$666$	−3.15077e9	−0.413292
$667$	−1.33014e10	−1.73563
$668$	−1.32534e9	−0.172032
$669$	5.49014e9	0.708912
$670$	9.71110e8	0.124740
$671$	9.13022e8	0.116668
$672$	3.03464e8	0.0385758
$673$	2.51835e9	0.318466	0.159233	−	0.987241i	$-0.449098\pi$
$673$	2.51835e9	0.318466	0.159233	+	0.987241i	$0.449098\pi$
$674$	8.00957e7	0.0100763
$675$	2.54917e9	0.319033
$676$	3.08916e8	0.0384615
$677$	1.31567e10	1.62962	0.814808	−	0.579731i	$-0.196842\pi$
$677$	1.31567e10	1.62962	0.814808	+	0.579731i	$0.196842\pi$
$678$	4.19600e9	0.517048
$679$	2.61962e9	0.321140
$680$	6.39806e9	0.780310
$681$	−2.74563e9	−0.333141
$682$	6.03525e9	0.728534
$683$	−7.42900e9	−0.892192	−0.446096	−	0.894985i	$-0.647186\pi$
$683$	−7.42900e9	−0.892192	−0.446096	+	0.894985i	$0.647186\pi$
$684$	2.03152e9	0.242731
$685$	−3.70678e9	−0.440635
$686$	3.22829e8	0.0381802
$687$	−9.38320e9	−1.10408
$688$	−1.92593e9	−0.225466
$689$	−3.33667e9	−0.388638
$690$	7.01887e9	0.813383
$691$	−1.08648e10	−1.25270	−0.626352	−	0.779541i	$-0.715453\pi$
$691$	−1.08648e10	−1.25270	−0.626352	+	0.779541i	$0.715453\pi$
$692$	7.04650e9	0.808355
$693$	1.96031e9	0.223748
$694$	6.46597e9	0.734304
$695$	−1.87780e10	−2.12179
$696$	−2.57851e9	−0.289892
$697$	−5.44276e9	−0.608842
$698$	3.89790e9	0.433847
$699$	−6.03152e9	−0.667970
$700$	−2.84303e9	−0.313285
$701$	4.25335e9	0.466357	0.233178	−	0.972434i	$-0.425087\pi$
$701$	4.25335e9	0.466357	0.233178	+	0.972434i	$0.425087\pi$
$702$	3.45948e8	0.0377426
$703$	2.35241e10	2.55370
$704$	−2.05515e9	−0.221993
$705$	−1.33630e10	−1.43629
$706$	2.63256e9	0.281554
$707$	2.51808e8	0.0267979
$708$	−2.64303e9	−0.279889
$709$	−1.49676e10	−1.57722	−0.788609	−	0.614895i	$-0.789198\pi$
$709$	−1.49676e10	−1.57722	−0.788609	+	0.614895i	$0.789198\pi$
$710$	1.76295e10	1.84857
$711$	−1.38454e9	−0.144465
$712$	2.89980e9	0.301084
$713$	−6.86220e9	−0.709006
$714$	−2.03177e9	−0.208896
$715$	7.84846e9	0.802996
$716$	−4.51744e9	−0.459935
$717$	−4.16319e9	−0.421803
$718$	2.28408e9	0.230290
$719$	−1.59995e10	−1.60530	−0.802650	−	0.596451i	$-0.796577\pi$
$719$	−1.59995e10	−1.60530	−0.802650	+	0.596451i	$0.796577\pi$
$720$	1.36063e9	0.135855
$721$	1.59467e9	0.158452
$722$	−8.01669e9	−0.792711
$723$	−7.54388e9	−0.742353
$724$	−9.59606e9	−0.939740
$725$	2.41570e10	2.35429
$726$	−9.06654e9	−0.879353
$727$	−1.63389e10	−1.57708	−0.788540	−	0.614984i	$-0.789162\pi$
$727$	−1.63389e10	−1.57708	−0.788540	+	0.614984i	$0.789162\pi$
$728$	−3.85828e8	−0.0370625
$729$	3.87420e8	0.0370370
$730$	1.56989e10	1.49362
$731$	1.28946e10	1.22094
$732$	−2.01244e8	−0.0189642
$733$	−3.70430e9	−0.347410	−0.173705	−	0.984798i	$-0.555574\pi$
$733$	−3.70430e9	−0.347410	−0.173705	+	0.984798i	$0.555574\pi$
$734$	−3.38420e9	−0.315878
$735$	1.44745e9	0.134462
$736$	2.33675e9	0.216042
$737$	2.08848e9	0.192174
$738$	−1.15747e9	−0.106002
$739$	−2.95037e9	−0.268918	−0.134459	−	0.990919i	$-0.542930\pi$
$739$	−2.95037e9	−0.268918	−0.134459	+	0.990919i	$0.542930\pi$
$740$	1.57554e10	1.42929
$741$	−2.58290e9	−0.233209
$742$	4.16742e9	0.374501
$743$	−7.39550e9	−0.661464	−0.330732	−	0.943725i	$-0.607296\pi$
$743$	−7.39550e9	−0.661464	−0.330732	+	0.943725i	$0.607296\pi$
$744$	−1.33026e9	−0.118421
$745$	−1.27954e10	−1.13372
$746$	−3.95044e9	−0.348385
$747$	1.03293e9	0.0906667
$748$	1.37597e10	1.20214
$749$	−4.76122e9	−0.414030
$750$	−5.05771e9	−0.437763
$751$	1.89326e10	1.63106	0.815531	−	0.578714i	$-0.196445\pi$
$751$	1.89326e10	1.63106	0.815531	+	0.578714i	$0.196445\pi$
$752$	−4.44886e9	−0.381493
$753$	−3.31406e8	−0.0282864
$754$	3.27835e9	0.278519
$755$	5.15413e9	0.435854
$756$	−4.32081e8	−0.0363696
$757$	−3.38282e9	−0.283428	−0.141714	−	0.989908i	$-0.545261\pi$
$757$	−3.38282e9	−0.283428	−0.141714	+	0.989908i	$0.545261\pi$
$758$	−8.47608e9	−0.706892
$759$	1.50948e10	1.25309
$760$	−1.01587e10	−0.839438
$761$	−1.26419e10	−1.03984	−0.519920	−	0.854215i	$-0.674038\pi$
$761$	−1.26419e10	−1.03984	−0.519920	+	0.854215i	$0.674038\pi$
$762$	1.13500e9	0.0929291
$763$	2.71511e9	0.221285
$764$	1.65761e9	0.134479
$765$	−9.10974e9	−0.735683
$766$	1.92893e9	0.155066
$767$	3.36038e9	0.268909
$768$	4.52985e8	0.0360844
$769$	1.11312e10	0.882673	0.441336	−	0.897342i	$-0.354505\pi$
$769$	1.11312e10	0.882673	0.441336	+	0.897342i	$0.354505\pi$
$770$	−9.80254e9	−0.773787
$771$	−9.80638e9	−0.770581
$772$	9.80320e9	0.766845
$773$	8.15580e9	0.635094	0.317547	−	0.948242i	$-0.397141\pi$
$773$	8.15580e9	0.635094	0.317547	+	0.948242i	$0.397141\pi$
$774$	2.74219e9	0.212571
$775$	1.24626e10	0.961732
$776$	3.91034e9	0.300399
$777$	−5.00330e9	−0.382634
$778$	−2.81357e8	−0.0214205
$779$	8.64185e9	0.654976
$780$	−1.72992e9	−0.130525
$781$	3.79142e10	2.84789
$782$	−1.56451e10	−1.16992
$783$	3.67136e9	0.273313
$784$	4.81890e8	0.0357143
$785$	−1.03783e10	−0.765742
$786$	1.07944e10	0.792899
$787$	2.78710e9	0.203817	0.101909	−	0.994794i	$-0.467505\pi$
$787$	2.78710e9	0.203817	0.101909	+	0.994794i	$0.467505\pi$
$788$	−2.06716e8	−0.0150499
$789$	1.10234e10	0.798995
$790$	6.92342e9	0.499604
$791$	6.66309e9	0.478694
$792$	2.92618e9	0.209297
$793$	2.55864e8	0.0182202
$794$	−9.52329e9	−0.675174
$795$	1.86852e10	1.31890
$796$	−3.50190e9	−0.246098
$797$	−1.81706e10	−1.27135	−0.635675	−	0.771957i	$-0.719278\pi$
$797$	−1.81706e10	−1.27135	−0.635675	+	0.771957i	$0.719278\pi$
$798$	3.22598e9	0.224725
$799$	2.97862e10	2.06587
$800$	−4.24383e9	−0.293051
$801$	−4.12881e9	−0.283865
$802$	1.03656e10	0.709552
$803$	3.37622e10	2.30105
$804$	−4.60331e8	−0.0312374
$805$	1.11457e10	0.753046
$806$	1.69131e9	0.113776
$807$	3.30491e9	0.221362
$808$	3.75876e8	0.0250672
$809$	−8.32097e9	−0.552528	−0.276264	−	0.961082i	$-0.589096\pi$
$809$	−8.32097e9	−0.552528	−0.276264	+	0.961082i	$0.589096\pi$
$810$	−1.93730e9	−0.128085
$811$	1.13006e10	0.743927	0.371963	−	0.928247i	$-0.378685\pi$
$811$	1.13006e10	0.743927	0.371963	+	0.928247i	$0.378685\pi$
$812$	−4.09458e9	−0.268388
$813$	8.57350e9	0.559553
$814$	3.38838e10	2.20194
$815$	−9.70572e8	−0.0628024
$816$	−3.03285e9	−0.195405
$817$	−2.04736e10	−1.31346
$818$	−9.02375e8	−0.0576435
$819$	5.49353e8	0.0349428
$820$	5.78794e9	0.366586
$821$	−2.69264e10	−1.69815	−0.849077	−	0.528269i	$-0.822841\pi$
$821$	−2.69264e10	−1.69815	−0.849077	+	0.528269i	$0.822841\pi$
$822$	1.75711e9	0.110344
$823$	1.51781e10	0.949112	0.474556	−	0.880225i	$-0.342609\pi$
$823$	1.51781e10	0.949112	0.474556	+	0.880225i	$0.342609\pi$
$824$	2.38039e9	0.148218
$825$	−2.74141e10	−1.69975
$826$	−4.19704e9	−0.259127
$827$	−6.95191e9	−0.427401	−0.213700	−	0.976899i	$-0.568552\pi$
$827$	−6.95191e9	−0.427401	−0.213700	+	0.976899i	$0.568552\pi$
$828$	−3.32712e9	−0.203687
$829$	−2.84068e10	−1.73173	−0.865867	−	0.500273i	$-0.833233\pi$
$829$	−2.84068e10	−1.73173	−0.865867	+	0.500273i	$0.833233\pi$
$830$	−5.16517e9	−0.313553
$831$	1.61293e10	0.975016
$832$	−5.75930e8	−0.0346688
$833$	−3.22638e9	−0.193400
$834$	8.90125e9	0.531337
$835$	−9.43624e9	−0.560914
$836$	−2.18473e10	−1.29323
$837$	1.89406e9	0.111649
$838$	−1.56639e10	−0.919485
$839$	−6.22084e9	−0.363649	−0.181825	−	0.983331i	$-0.558200\pi$
$839$	−6.22084e9	−0.363649	−0.181825	+	0.983331i	$0.558200\pi$
$840$	2.16063e9	0.125777
$841$	1.75414e10	1.01690
$842$	1.18262e10	0.682735
$843$	−8.10017e9	−0.465691
$844$	−1.00887e10	−0.577611
$845$	2.19944e9	0.125405
$846$	6.33442e9	0.359675
$847$	−1.43973e10	−0.814123
$848$	6.22075e9	0.350314
$849$	1.81847e9	0.101983
$850$	2.84135e10	1.58693
$851$	−3.85266e10	−2.14292
$852$	−8.35685e9	−0.462918
$853$	1.31427e10	0.725040	0.362520	−	0.931976i	$-0.381916\pi$
$853$	1.31427e10	0.725040	0.362520	+	0.931976i	$0.381916\pi$
$854$	−3.19568e8	−0.0175574
$855$	1.44642e10	0.791430
$856$	−7.10712e9	−0.387289
$857$	−2.82678e10	−1.53412	−0.767060	−	0.641576i	$-0.778281\pi$
$857$	−2.82678e10	−1.53412	−0.767060	+	0.641576i	$0.778281\pi$
$858$	−3.72037e9	−0.201086
$859$	−3.36693e10	−1.81242	−0.906208	−	0.422833i	$-0.861036\pi$
$859$	−3.36693e10	−1.81242	−0.906208	+	0.422833i	$0.861036\pi$
$860$	−1.37123e10	−0.735135
$861$	−1.83802e9	−0.0981385
$862$	−1.20727e10	−0.641991
$863$	−1.78511e9	−0.0945425	−0.0472713	−	0.998882i	$-0.515053\pi$
$863$	−1.78511e9	−0.0945425	−0.0472713	+	0.998882i	$0.515053\pi$
$864$	−6.44973e8	−0.0340207
$865$	5.01701e10	2.63566
$866$	1.18457e10	0.619794
$867$	9.22651e9	0.480807
$868$	−2.11240e9	−0.109637
$869$	1.48896e10	0.769685
$870$	−1.83586e10	−0.945199
$871$	5.85271e8	0.0300119
$872$	4.05288e9	0.206993
$873$	−5.56765e9	−0.283219
$874$	2.48408e10	1.25857
$875$	−8.03145e9	−0.405290
$876$	−7.44169e9	−0.374030
$877$	8.63204e9	0.432130	0.216065	−	0.976379i	$-0.430678\pi$
$877$	8.63204e9	0.432130	0.216065	+	0.976379i	$0.430678\pi$
$878$	5.61521e9	0.279985
$879$	1.59564e10	0.792452
$880$	−1.46324e10	−0.723811
$881$	−2.03612e10	−1.00320	−0.501600	−	0.865100i	$-0.667255\pi$
$881$	−2.03612e10	−1.00320	−0.501600	+	0.865100i	$0.667255\pi$
$882$	−6.86129e8	−0.0336718
$883$	1.02230e10	0.499709	0.249855	−	0.968283i	$-0.419617\pi$
$883$	1.02230e10	0.499709	0.249855	+	0.968283i	$0.419617\pi$
$884$	3.85600e9	0.187739
$885$	−1.88180e10	−0.912584
$886$	5.04845e9	0.243860
$887$	−2.84825e10	−1.37039	−0.685196	−	0.728359i	$-0.740283\pi$
$887$	−2.84825e10	−1.37039	−0.685196	+	0.728359i	$0.740283\pi$
$888$	−7.46849e9	−0.357921
$889$	1.80233e9	0.0860357
$890$	2.06462e10	0.981690
$891$	−4.16637e9	−0.197327
$892$	1.30137e10	0.613935
$893$	−4.72937e10	−2.22241
$894$	6.06533e9	0.283905
$895$	−3.21635e10	−1.49963
$896$	7.19323e8	0.0334077
$897$	4.23015e9	0.195696
$898$	−2.16181e10	−0.996211
$899$	1.79489e10	0.823907
$900$	6.04248e9	0.276291
$901$	−4.16495e10	−1.89702
$902$	1.24476e10	0.564758
$903$	4.35449e9	0.196802
$904$	9.94607e9	0.447777
$905$	−6.83226e10	−3.06404
$906$	−2.44319e9	−0.109146
$907$	−4.08623e10	−1.81843	−0.909217	−	0.416321i	$-0.863319\pi$
$907$	−4.08623e10	−1.81843	−0.909217	+	0.416321i	$0.863319\pi$
$908$	−6.50816e9	−0.288508
$909$	−5.35183e8	−0.0236335
$910$	−2.74705e9	−0.120843
$911$	−1.03566e10	−0.453838	−0.226919	−	0.973914i	$-0.572865\pi$
$911$	−1.03566e10	−0.453838	−0.226919	+	0.973914i	$0.572865\pi$
$912$	4.81546e9	0.210211
$913$	−1.11082e10	−0.483056
$914$	−2.08110e10	−0.901534
$915$	−1.43283e9	−0.0618330
$916$	−2.22417e10	−0.956165
$917$	1.71410e10	0.734082
$918$	4.31825e9	0.184229
$919$	7.36752e9	0.313124	0.156562	−	0.987668i	$-0.449959\pi$
$919$	7.36752e9	0.313124	0.156562	+	0.987668i	$0.449959\pi$
$920$	1.66373e10	0.704410
$921$	−2.92955e9	−0.123564
$922$	−6.06712e9	−0.254932
$923$	1.06250e10	0.444757
$924$	4.64666e9	0.193771
$925$	6.99692e10	2.90677
$926$	7.79361e9	0.322553
$927$	−3.38926e9	−0.139742
$928$	−6.11202e9	−0.251054
$929$	3.27765e10	1.34124	0.670621	−	0.741800i	$-0.266028\pi$
$929$	3.27765e10	1.34124	0.670621	+	0.741800i	$0.266028\pi$
$930$	−9.47125e9	−0.386115
$931$	5.12274e9	0.208055
$932$	−1.42969e10	−0.578479
$933$	−1.19837e10	−0.483063
$934$	−2.59117e10	−1.04060
$935$	9.79673e10	3.91959
$936$	8.20026e8	0.0326860
$937$	−1.42519e10	−0.565956	−0.282978	−	0.959126i	$-0.591322\pi$
$937$	−1.42519e10	−0.565956	−0.282978	+	0.959126i	$0.591322\pi$
$938$	−7.30989e8	−0.0289202
$939$	1.04872e10	0.413360
$940$	−3.16753e10	−1.24387
$941$	4.16605e10	1.62990	0.814950	−	0.579531i	$-0.196764\pi$
$941$	4.16605e10	1.62990	0.814950	+	0.579531i	$0.196764\pi$
$942$	4.91958e9	0.191756
$943$	−1.41532e10	−0.549620
$944$	−6.26496e9	−0.242391
$945$	−3.07636e9	−0.118584
$946$	−2.94899e10	−1.13254
$947$	−1.67814e10	−0.642100	−0.321050	−	0.947062i	$-0.604036\pi$
$947$	−1.67814e10	−0.642100	−0.321050	+	0.947062i	$0.604036\pi$
$948$	−3.28188e9	−0.125110
$949$	9.46145e9	0.359357
$950$	−4.51141e10	−1.70718
$951$	−2.20027e10	−0.829554
$952$	−4.81605e9	−0.180910
$953$	7.21833e9	0.270154	0.135077	−	0.990835i	$-0.456872\pi$
$953$	7.21833e9	0.270154	0.135077	+	0.990835i	$0.456872\pi$
$954$	−8.85728e9	−0.330279
$955$	1.18020e10	0.438473
$956$	−9.86830e9	−0.365292
$957$	−3.94822e10	−1.45616
$958$	−2.19577e10	−0.806876
$959$	2.79022e9	0.102158
$960$	3.22519e9	0.117654
$961$	−1.82528e10	−0.663432
$962$	9.49552e9	0.343879
$963$	1.01193e10	0.365140
$964$	−1.78818e10	−0.642897
$965$	6.97975e10	2.50031
$966$	−5.28335e9	−0.188577
$967$	2.75587e10	0.980090	0.490045	−	0.871697i	$-0.336980\pi$
$967$	2.75587e10	0.980090	0.490045	+	0.871697i	$0.336980\pi$
$968$	−2.14910e10	−0.761542
$969$	−3.22407e10	−1.13834
$970$	2.78411e10	0.979456
$971$	−1.83427e10	−0.642976	−0.321488	−	0.946914i	$-0.604183\pi$
$971$	−1.83427e10	−0.642976	−0.321488	+	0.946914i	$0.604183\pi$
$972$	9.18330e8	0.0320750
$973$	1.41349e10	0.491922
$974$	−1.68702e10	−0.585010
$975$	−7.68249e9	−0.265452
$976$	−4.77022e8	−0.0164234
$977$	−3.63191e10	−1.24596	−0.622980	−	0.782238i	$-0.714078\pi$
$977$	−3.63191e10	−1.24596	−0.622980	+	0.782238i	$0.714078\pi$
$978$	4.60076e8	0.0157269
$979$	4.44018e10	1.51238
$980$	3.43099e9	0.116447
$981$	−5.77060e9	−0.195155
$982$	1.04779e10	0.353090
$983$	4.48078e10	1.50458	0.752292	−	0.658830i	$-0.228948\pi$
$983$	4.48078e10	1.50458	0.752292	+	0.658830i	$0.228948\pi$
$984$	−2.74363e9	−0.0918001
$985$	−1.47179e9	−0.0490705
$986$	4.09215e10	1.35951
$987$	1.00588e10	0.332994
$988$	−6.12244e9	−0.201965
$989$	3.35306e10	1.10218
$990$	2.08340e10	0.682416
$991$	2.84415e10	0.928313	0.464156	−	0.885753i	$-0.346358\pi$
$991$	2.84415e10	0.928313	0.464156	+	0.885753i	$0.346358\pi$
$992$	−3.15320e9	−0.102556
$993$	1.70109e10	0.551320
$994$	−1.32704e10	−0.428579
$995$	−2.49331e10	−0.802407
$996$	2.44842e9	0.0785197
$997$	4.93045e10	1.57563	0.787813	−	0.615914i	$-0.211213\pi$
$997$	4.93045e10	1.57563	0.787813	+	0.615914i	$0.211213\pi$
$998$	−3.96961e9	−0.126413
$999$	1.06338e10	0.337451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.j.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.j.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +455.671 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +455.671 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -3645.37 q^{10} -7839.76 q^{11} +1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +12303.1 q^{15} +4096.00 q^{16} -27423.7 q^{17} -5832.00 q^{18} +43542.6 q^{19} +29163.0 q^{20} -9261.00 q^{21} +62718.1 q^{22} -71311.8 q^{23} -13824.0 q^{24} +129511. q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +186524. q^{29} -98425.0 q^{30} +96228.1 q^{31} -32768.0 q^{32} -211674. q^{33} +219390. q^{34} -156295. q^{35} +46656.0 q^{36} +540255. q^{37} -348341. q^{38} -59319.0 q^{39} -233304. q^{40} +198469. q^{41} +74088.0 q^{42} -470197. q^{43} -501745. q^{44} +332184. q^{45} +570494. q^{46} -1.08615e6 q^{47} +110592. q^{48} +117649. q^{49} -1.03609e6 q^{50} -740441. q^{51} -140608. q^{52} +1.51874e6 q^{53} -157464. q^{54} -3.57236e6 q^{55} +175616. q^{56} +1.17565e6 q^{57} -1.49219e6 q^{58} -1.52953e6 q^{59} +787400. q^{60} -116460. q^{61} -769825. q^{62} -250047. q^{63} +262144. q^{64} -1.00111e6 q^{65} +1.69339e6 q^{66} -266395. q^{67} -1.75512e6 q^{68} -1.92542e6 q^{69} +1.25036e6 q^{70} -4.83614e6 q^{71} -373248. q^{72} -4.30653e6 q^{73} -4.32204e6 q^{74} +3.49681e6 q^{75} +2.78673e6 q^{76} +2.68904e6 q^{77} +474552. q^{78} -1.89924e6 q^{79} +1.86643e6 q^{80} +531441. q^{81} -1.58775e6 q^{82} +1.41691e6 q^{83} -592704. q^{84} -1.24962e7 q^{85} +3.76157e6 q^{86} +5.03615e6 q^{87} +4.01396e6 q^{88} -5.66367e6 q^{89} -2.65748e6 q^{90} +753571. q^{91} -4.56396e6 q^{92} +2.59816e6 q^{93} +8.68919e6 q^{94} +1.98411e7 q^{95} -884736. q^{96} -7.63738e6 q^{97} -941192. q^{98} -5.71519e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 56 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 448 q^{10} - 3679 q^{11} + 8640 q^{12} - 10985 q^{13} + 13720 q^{14} + 1512 q^{15} + 20480 q^{16} + 409 q^{17} - 29160 q^{18} + 33730 q^{19} + 3584 q^{20} - 46305 q^{21} + 29432 q^{22} - 142142 q^{23} - 69120 q^{24} + 153981 q^{25} + 87880 q^{26} + 98415 q^{27} - 109760 q^{28} + 88028 q^{29} - 12096 q^{30} + 244543 q^{31} - 163840 q^{32} - 99333 q^{33} - 3272 q^{34} - 19208 q^{35} + 233280 q^{36} + 730963 q^{37} - 269840 q^{38} - 296595 q^{39} - 28672 q^{40} + 479512 q^{41} + 370440 q^{42} - 406536 q^{43} - 235456 q^{44} + 40824 q^{45} + 1137136 q^{46} + 1138945 q^{47} + 552960 q^{48} + 588245 q^{49} - 1231848 q^{50} + 11043 q^{51} - 703040 q^{52} + 297595 q^{53} - 787320 q^{54} - 1834423 q^{55} + 878080 q^{56} + 910710 q^{57} - 704224 q^{58} + 941652 q^{59} + 96768 q^{60} - 2985259 q^{61} - 1956344 q^{62} - 1250235 q^{63} + 1310720 q^{64} - 123032 q^{65} + 794664 q^{66} - 2333504 q^{67} + 26176 q^{68} - 3837834 q^{69} + 153664 q^{70} - 11322272 q^{71} - 1866240 q^{72} - 6631604 q^{73} - 5847704 q^{74} + 4157487 q^{75} + 2158720 q^{76} + 1261897 q^{77} + 2372760 q^{78} - 10600265 q^{79} + 229376 q^{80} + 2657205 q^{81} - 3836096 q^{82} - 2425229 q^{83} - 2963520 q^{84} - 12267705 q^{85} + 3252288 q^{86} + 2376756 q^{87} + 1883648 q^{88} - 1581837 q^{89} - 326592 q^{90} + 3767855 q^{91} - 9097088 q^{92} + 6602661 q^{93} - 9111560 q^{94} - 11507718 q^{95} - 4423680 q^{96} + 5298407 q^{97} - 4705960 q^{98} - 2681991 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 56 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9 - 448 * q^10 - 3679 * q^11 + 8640 * q^12 - 10985 * q^13 + 13720 * q^14 + 1512 * q^15 + 20480 * q^16 + 409 * q^17 - 29160 * q^18 + 33730 * q^19 + 3584 * q^20 - 46305 * q^21 + 29432 * q^22 - 142142 * q^23 - 69120 * q^24 + 153981 * q^25 + 87880 * q^26 + 98415 * q^27 - 109760 * q^28 + 88028 * q^29 - 12096 * q^30 + 244543 * q^31 - 163840 * q^32 - 99333 * q^33 - 3272 * q^34 - 19208 * q^35 + 233280 * q^36 + 730963 * q^37 - 269840 * q^38 - 296595 * q^39 - 28672 * q^40 + 479512 * q^41 + 370440 * q^42 - 406536 * q^43 - 235456 * q^44 + 40824 * q^45 + 1137136 * q^46 + 1138945 * q^47 + 552960 * q^48 + 588245 * q^49 - 1231848 * q^50 + 11043 * q^51 - 703040 * q^52 + 297595 * q^53 - 787320 * q^54 - 1834423 * q^55 + 878080 * q^56 + 910710 * q^57 - 704224 * q^58 + 941652 * q^59 + 96768 * q^60 - 2985259 * q^61 - 1956344 * q^62 - 1250235 * q^63 + 1310720 * q^64 - 123032 * q^65 + 794664 * q^66 - 2333504 * q^67 + 26176 * q^68 - 3837834 * q^69 + 153664 * q^70 - 11322272 * q^71 - 1866240 * q^72 - 6631604 * q^73 - 5847704 * q^74 + 4157487 * q^75 + 2158720 * q^76 + 1261897 * q^77 + 2372760 * q^78 - 10600265 * q^79 + 229376 * q^80 + 2657205 * q^81 - 3836096 * q^82 - 2425229 * q^83 - 2963520 * q^84 - 12267705 * q^85 + 3252288 * q^86 + 2376756 * q^87 + 1883648 * q^88 - 1581837 * q^89 - 326592 * q^90 + 3767855 * q^91 - 9097088 * q^92 + 6602661 * q^93 - 9111560 * q^94 - 11507718 * q^95 - 4423680 * q^96 + 5298407 * q^97 - 4705960 * q^98 - 2681991 * q^99

Introduction

L-functions

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