LMFDB - Embedded newform 91.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,8,Mod(1,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("91.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 91 = 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	91.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:	$28.4270373191$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 764 x^{7} + 1562 x^{6} + 176422 x^{5} + 56746 x^{4} - 13204236 x^{3} + \cdots + 176334338 $ x^9 - 4x^8 - 764x^7 + 1562x^6 + 176422x^5 + 56746x^4 - 13204236x^3 - 22500802x^2 + 176026849x + 176334338
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-20.0538$ of defining polynomial
Character	$\chi$	$=$	91.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.0538 q^{2} -33.4331 q^{3} +315.264 q^{4} +271.446 q^{5} +703.895 q^{6} -343.000 q^{7} -3942.62 q^{8} -1069.23 q^{9} +O(q^{10})$	\(q-21.0538 q^{2} -33.4331 q^{3} +315.264 q^{4} +271.446 q^{5} +703.895 q^{6} -343.000 q^{7} -3942.62 q^{8} -1069.23 q^{9} -5714.98 q^{10} -610.654 q^{11} -10540.2 q^{12} +2197.00 q^{13} +7221.46 q^{14} -9075.28 q^{15} +42653.4 q^{16} +8454.94 q^{17} +22511.3 q^{18} +3641.40 q^{19} +85577.0 q^{20} +11467.6 q^{21} +12856.6 q^{22} +29005.6 q^{23} +131814. q^{24} -4442.13 q^{25} -46255.3 q^{26} +108866. q^{27} -108135. q^{28} -85317.7 q^{29} +191069. q^{30} -46589.0 q^{31} -393363. q^{32} +20416.1 q^{33} -178009. q^{34} -93105.9 q^{35} -337089. q^{36} +465957. q^{37} -76665.3 q^{38} -73452.5 q^{39} -1.07021e6 q^{40} -113709. q^{41} -241436. q^{42} +941191. q^{43} -192517. q^{44} -290237. q^{45} -610679. q^{46} -1.04289e6 q^{47} -1.42604e6 q^{48} +117649. q^{49} +93523.8 q^{50} -282675. q^{51} +692634. q^{52} -1.46860e6 q^{53} -2.29204e6 q^{54} -165760. q^{55} +1.35232e6 q^{56} -121743. q^{57} +1.79626e6 q^{58} +861156. q^{59} -2.86111e6 q^{60} -2.05917e6 q^{61} +980876. q^{62} +366745. q^{63} +2.82216e6 q^{64} +596367. q^{65} -429836. q^{66} +2.17291e6 q^{67} +2.66554e6 q^{68} -969747. q^{69} +1.96024e6 q^{70} -1.10976e6 q^{71} +4.21556e6 q^{72} -2.49953e6 q^{73} -9.81018e6 q^{74} +148514. q^{75} +1.14800e6 q^{76} +209454. q^{77} +1.54646e6 q^{78} -4.91564e6 q^{79} +1.15781e7 q^{80} -1.30132e6 q^{81} +2.39401e6 q^{82} -871288. q^{83} +3.61530e6 q^{84} +2.29506e6 q^{85} -1.98157e7 q^{86} +2.85243e6 q^{87} +2.40758e6 q^{88} -1.09846e7 q^{89} +6.11061e6 q^{90} -753571. q^{91} +9.14441e6 q^{92} +1.55761e6 q^{93} +2.19568e7 q^{94} +988442. q^{95} +1.31513e7 q^{96} +1.31931e7 q^{97} -2.47696e6 q^{98} +652928. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9}+O(q^{10}) $ 9 * q - 5 * q^2 - 26 * q^3 + 393 * q^4 - 181 * q^5 + 703 * q^6 - 3087 * q^7 + 1197 * q^8 + 3211 * q^9	\( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9} - 5124 q^{10} - 9826 q^{11} - 20919 q^{12} + 19773 q^{13} + 1715 q^{14} - 20346 q^{15} + 31113 q^{16} - 22766 q^{17} - 12978 q^{18} - 17769 q^{19} - 44204 q^{20} + 8918 q^{21} - 203553 q^{22} - 49103 q^{23} + 52737 q^{24} + 227466 q^{25} - 10985 q^{26} + 103624 q^{27} - 134799 q^{28} - 487455 q^{29} - 287992 q^{30} - 63843 q^{31} - 587099 q^{32} - 314392 q^{33} - 576240 q^{34} + 62083 q^{35} - 1514926 q^{36} - 796926 q^{37} - 766702 q^{38} - 57122 q^{39} - 2887296 q^{40} - 1567546 q^{41} - 241129 q^{42} - 277899 q^{43} - 1281195 q^{44} - 1650593 q^{45} - 1907445 q^{46} + 1077367 q^{47} - 1110835 q^{48} + 1058841 q^{49} - 267459 q^{50} - 3054368 q^{51} + 863421 q^{52} - 7322659 q^{53} - 3355387 q^{54} - 2613324 q^{55} - 410571 q^{56} - 3751946 q^{57} - 2992332 q^{58} - 169804 q^{59} - 2754416 q^{60} - 6352284 q^{61} + 6001087 q^{62} - 1101373 q^{63} + 1657017 q^{64} - 397657 q^{65} - 5962713 q^{66} + 921120 q^{67} + 5615224 q^{68} - 5202780 q^{69} + 1757532 q^{70} + 3786654 q^{71} + 2229758 q^{72} + 5792889 q^{73} - 1991961 q^{74} + 145628 q^{75} - 2806026 q^{76} + 3370318 q^{77} + 1544491 q^{78} + 3464037 q^{79} + 15422512 q^{80} - 5010363 q^{81} - 12539943 q^{82} + 6834945 q^{83} + 7175217 q^{84} + 3880662 q^{85} - 7977524 q^{86} + 3727078 q^{87} + 7013709 q^{88} - 20408371 q^{89} + 34910060 q^{90} - 6782139 q^{91} - 3544371 q^{92} + 3121742 q^{93} + 61343967 q^{94} + 3360807 q^{95} + 23547905 q^{96} + 41644125 q^{97} - 588245 q^{98} + 50754068 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 - 26 * q^3 + 393 * q^4 - 181 * q^5 + 703 * q^6 - 3087 * q^7 + 1197 * q^8 + 3211 * q^9 - 5124 * q^10 - 9826 * q^11 - 20919 * q^12 + 19773 * q^13 + 1715 * q^14 - 20346 * q^15 + 31113 * q^16 - 22766 * q^17 - 12978 * q^18 - 17769 * q^19 - 44204 * q^20 + 8918 * q^21 - 203553 * q^22 - 49103 * q^23 + 52737 * q^24 + 227466 * q^25 - 10985 * q^26 + 103624 * q^27 - 134799 * q^28 - 487455 * q^29 - 287992 * q^30 - 63843 * q^31 - 587099 * q^32 - 314392 * q^33 - 576240 * q^34 + 62083 * q^35 - 1514926 * q^36 - 796926 * q^37 - 766702 * q^38 - 57122 * q^39 - 2887296 * q^40 - 1567546 * q^41 - 241129 * q^42 - 277899 * q^43 - 1281195 * q^44 - 1650593 * q^45 - 1907445 * q^46 + 1077367 * q^47 - 1110835 * q^48 + 1058841 * q^49 - 267459 * q^50 - 3054368 * q^51 + 863421 * q^52 - 7322659 * q^53 - 3355387 * q^54 - 2613324 * q^55 - 410571 * q^56 - 3751946 * q^57 - 2992332 * q^58 - 169804 * q^59 - 2754416 * q^60 - 6352284 * q^61 + 6001087 * q^62 - 1101373 * q^63 + 1657017 * q^64 - 397657 * q^65 - 5962713 * q^66 + 921120 * q^67 + 5615224 * q^68 - 5202780 * q^69 + 1757532 * q^70 + 3786654 * q^71 + 2229758 * q^72 + 5792889 * q^73 - 1991961 * q^74 + 145628 * q^75 - 2806026 * q^76 + 3370318 * q^77 + 1544491 * q^78 + 3464037 * q^79 + 15422512 * q^80 - 5010363 * q^81 - 12539943 * q^82 + 6834945 * q^83 + 7175217 * q^84 + 3880662 * q^85 - 7977524 * q^86 + 3727078 * q^87 + 7013709 * q^88 - 20408371 * q^89 + 34910060 * q^90 - 6782139 * q^91 - 3544371 * q^92 + 3121742 * q^93 + 61343967 * q^94 + 3360807 * q^95 + 23547905 * q^96 + 41644125 * q^97 - 588245 * q^98 + 50754068 * 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 $

$2$	−21.0538	−1.86091	−0.930457	−	0.366402i	$-0.880589\pi$
$2$	−21.0538	−1.86091	−0.930457	+	0.366402i	$0.880589\pi$
$3$	−33.4331	−0.714911	−0.357456	−	0.933930i	$-0.616356\pi$
$3$	−33.4331	−0.714911	−0.357456	+	0.933930i	$0.616356\pi$
$4$	315.264	2.46300
$5$	271.446	0.971154	0.485577	−	0.874194i	$-0.338609\pi$
$5$	271.446	0.971154	0.485577	+	0.874194i	$0.338609\pi$
$6$	703.895	1.33039
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−3942.62	−2.72251
$9$	−1069.23	−0.488902
$10$	−5714.98	−1.80723
$11$	−610.654	−0.138331	−0.0691657	−	0.997605i	$-0.522034\pi$
$11$	−610.654	−0.138331	−0.0691657	+	0.997605i	$0.522034\pi$
$12$	−10540.2	−1.76083
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	7221.46	0.703359
$15$	−9075.28	−0.694289
$16$	42653.4	2.60336
$17$	8454.94	0.417387	0.208694	−	0.977981i	$-0.433079\pi$
$17$	8454.94	0.417387	0.208694	+	0.977981i	$0.433079\pi$
$18$	22511.3	0.909803
$19$	3641.40	0.121795	0.0608976	−	0.998144i	$-0.480604\pi$
$19$	3641.40	0.121795	0.0608976	+	0.998144i	$0.480604\pi$
$20$	85577.0	2.39195
$21$	11467.6	0.270211
$22$	12856.6	0.257423
$23$	29005.6	0.497089	0.248545	−	0.968620i	$-0.420048\pi$
$23$	29005.6	0.497089	0.248545	+	0.968620i	$0.420048\pi$
$24$	131814.	1.94635
$25$	−4442.13	−0.0568592
$26$	−46255.3	−0.516124
$27$	108866.	1.06443
$28$	−108135.	−0.930925
$29$	−85317.7	−0.649600	−0.324800	−	0.945783i	$-0.605297\pi$
$29$	−85317.7	−0.649600	−0.324800	+	0.945783i	$0.605297\pi$
$30$	191069.	1.29201
$31$	−46589.0	−0.280878	−0.140439	−	0.990089i	$-0.544851\pi$
$31$	−46589.0	−0.280878	−0.140439	+	0.990089i	$0.544851\pi$
$32$	−393363.	−2.12211
$33$	20416.1	0.0988948
$34$	−178009.	−0.776722
$35$	−93105.9	−0.367062
$36$	−337089.	−1.20416
$37$	465957.	1.51231	0.756153	−	0.654395i	$-0.227077\pi$
$37$	465957.	1.51231	0.756153	+	0.654395i	$0.227077\pi$
$38$	−76665.3	−0.226650
$39$	−73452.5	−0.198281
$40$	−1.07021e6	−2.64398
$41$	−113709.	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$41$	−113709.	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$42$	−241436.	−0.502839
$43$	941191.	1.80525	0.902626	−	0.430425i	$-0.141636\pi$
$43$	941191.	1.80525	0.902626	+	0.430425i	$0.141636\pi$
$44$	−192517.	−0.340710
$45$	−290237.	−0.474799
$46$	−610679.	−0.925040
$47$	−1.04289e6	−1.46520	−0.732599	−	0.680661i	$-0.761693\pi$
$47$	−1.04289e6	−1.46520	−0.732599	+	0.680661i	$0.761693\pi$
$48$	−1.42604e6	−1.86117
$49$	117649.	0.142857
$50$	93523.8	0.105810
$51$	−282675.	−0.298395
$52$	692634.	0.683113
$53$	−1.46860e6	−1.35499	−0.677497	−	0.735525i	$-0.736935\pi$
$53$	−1.46860e6	−1.35499	−0.677497	+	0.735525i	$0.736935\pi$
$54$	−2.29204e6	−1.98082
$55$	−165760.	−0.134341
$56$	1.35232e6	1.02901
$57$	−121743.	−0.0870728
$58$	1.79626e6	1.20885
$59$	861156.	0.545884	0.272942	−	0.962031i	$-0.412003\pi$
$59$	861156.	0.545884	0.272942	+	0.962031i	$0.412003\pi$
$60$	−2.86111e6	−1.71003
$61$	−2.05917e6	−1.16155	−0.580775	−	0.814064i	$-0.697250\pi$
$61$	−2.05917e6	−1.16155	−0.580775	+	0.814064i	$0.697250\pi$
$62$	980876.	0.522689
$63$	366745.	0.184787
$64$	2.82216e6	1.34571
$65$	596367.	0.269350
$66$	−429836.	−0.184035
$67$	2.17291e6	0.882632	0.441316	−	0.897352i	$-0.354512\pi$
$67$	2.17291e6	0.882632	0.441316	+	0.897352i	$0.354512\pi$
$68$	2.66554e6	1.02802
$69$	−969747.	−0.355375
$70$	1.96024e6	0.683070
$71$	−1.10976e6	−0.367981	−0.183991	−	0.982928i	$-0.558902\pi$
$71$	−1.10976e6	−0.367981	−0.183991	+	0.982928i	$0.558902\pi$
$72$	4.21556e6	1.33104
$73$	−2.49953e6	−0.752019	−0.376009	−	0.926616i	$-0.622704\pi$
$73$	−2.49953e6	−0.752019	−0.376009	+	0.926616i	$0.622704\pi$
$74$	−9.81018e6	−2.81427
$75$	148514.	0.0406493
$76$	1.14800e6	0.299981
$77$	209454.	0.0522844
$78$	1.54646e6	0.368983
$79$	−4.91564e6	−1.12172	−0.560861	−	0.827910i	$-0.689530\pi$
$79$	−4.91564e6	−1.12172	−0.560861	+	0.827910i	$0.689530\pi$
$80$	1.15781e7	2.52826
$81$	−1.30132e6	−0.272074
$82$	2.39401e6	0.479488
$83$	−871288.	−0.167259	−0.0836293	−	0.996497i	$-0.526651\pi$
$83$	−871288.	−0.167259	−0.0836293	+	0.996497i	$0.526651\pi$
$84$	3.61530e6	0.665529
$85$	2.29506e6	0.405348
$86$	−1.98157e7	−3.35942
$87$	2.85243e6	0.464407
$88$	2.40758e6	0.376609
$89$	−1.09846e7	−1.65165	−0.825827	−	0.563923i	$-0.809291\pi$
$89$	−1.09846e7	−1.65165	−0.825827	+	0.563923i	$0.809291\pi$
$90$	6.11061e6	0.883559
$91$	−753571.	−0.104828
$92$	9.14441e6	1.22433
$93$	1.55761e6	0.200803
$94$	2.19568e7	2.72661
$95$	988442.	0.118282
$96$	1.31513e7	1.51712
$97$	1.31931e7	1.46773	0.733866	−	0.679294i	$-0.237714\pi$
$97$	1.31931e7	1.46773	0.733866	+	0.679294i	$0.237714\pi$
$98$	−2.47696e6	−0.265845
$99$	652928.	0.0676305
$100$	−1.40044e6	−0.140044
$101$	5.27510e6	0.509455	0.254728	−	0.967013i	$-0.418014\pi$
$101$	5.27510e6	0.509455	0.254728	+	0.967013i	$0.418014\pi$
$102$	5.95139e6	0.555287
$103$	−6.10984e6	−0.550934	−0.275467	−	0.961311i	$-0.588832\pi$
$103$	−6.10984e6	−0.550934	−0.275467	+	0.961311i	$0.588832\pi$
$104$	−8.66193e6	−0.755089
$105$	3.11282e6	0.262417
$106$	3.09196e7	2.52153
$107$	−8.24940e6	−0.650997	−0.325498	−	0.945543i	$-0.605532\pi$
$107$	−8.24940e6	−0.650997	−0.325498	+	0.945543i	$0.605532\pi$
$108$	3.43214e7	2.62170
$109$	−1.33508e7	−0.987445	−0.493723	−	0.869619i	$-0.664364\pi$
$109$	−1.33508e7	−0.987445	−0.493723	+	0.869619i	$0.664364\pi$
$110$	3.48987e6	0.249997
$111$	−1.55784e7	−1.08117
$112$	−1.46301e7	−0.983977
$113$	−1.58765e7	−1.03509	−0.517547	−	0.855655i	$-0.673155\pi$
$113$	−1.58765e7	−1.03509	−0.517547	+	0.855655i	$0.673155\pi$
$114$	2.56316e6	0.162035
$115$	7.87345e6	0.482750
$116$	−2.68976e7	−1.59996
$117$	−2.34909e6	−0.135597
$118$	−1.81306e7	−1.01584
$119$	−2.90004e6	−0.157758
$120$	3.57803e7	1.89021
$121$	−1.91143e7	−0.980864
$122$	4.33534e7	2.16154
$123$	3.80165e6	0.184206
$124$	−1.46878e7	−0.691801
$125$	−2.24125e7	−1.02637
$126$	−7.72139e6	−0.343873
$127$	−2.10470e7	−0.911754	−0.455877	−	0.890043i	$-0.650674\pi$
$127$	−2.10470e7	−0.911754	−0.455877	+	0.890043i	$0.650674\pi$
$128$	−9.06674e6	−0.382135
$129$	−3.14669e7	−1.29060
$130$	−1.25558e7	−0.501236
$131$	−3.92524e7	−1.52552	−0.762759	−	0.646683i	$-0.776156\pi$
$131$	−3.92524e7	−1.52552	−0.762759	+	0.646683i	$0.776156\pi$
$132$	6.43644e6	0.243578
$133$	−1.24900e6	−0.0460343
$134$	−4.57480e7	−1.64250
$135$	2.95512e7	1.03373
$136$	−3.33346e7	−1.13634
$137$	−1.25748e7	−0.417812	−0.208906	−	0.977936i	$-0.566990\pi$
$137$	−1.25748e7	−0.417812	−0.208906	+	0.977936i	$0.566990\pi$
$138$	2.04169e7	0.661322
$139$	6.10572e6	0.192835	0.0964174	−	0.995341i	$-0.469262\pi$
$139$	6.10572e6	0.192835	0.0964174	+	0.995341i	$0.469262\pi$
$140$	−2.93529e7	−0.904072
$141$	3.48671e7	1.04749
$142$	2.33647e7	0.684781
$143$	−1.34161e6	−0.0383662
$144$	−4.56062e7	−1.27279
$145$	−2.31591e7	−0.630862
$146$	5.26247e7	1.39944
$147$	−3.93337e6	−0.102130
$148$	1.46899e8	3.72481
$149$	−3.23100e7	−0.800176	−0.400088	−	0.916477i	$-0.631020\pi$
$149$	−3.23100e7	−0.800176	−0.400088	+	0.916477i	$0.631020\pi$
$150$	−3.12679e6	−0.0756448
$151$	6.12681e7	1.44816	0.724078	−	0.689718i	$-0.242266\pi$
$151$	6.12681e7	1.44816	0.724078	+	0.689718i	$0.242266\pi$
$152$	−1.43566e7	−0.331589
$153$	−9.04026e6	−0.204061
$154$	−4.40982e6	−0.0972967
$155$	−1.26464e7	−0.272776
$156$	−2.31569e7	−0.488365
$157$	−8.20826e7	−1.69279	−0.846394	−	0.532558i	$-0.821231\pi$
$157$	−8.20826e7	−1.69279	−0.846394	+	0.532558i	$0.821231\pi$
$158$	1.03493e8	2.08743
$159$	4.90998e7	0.968701
$160$	−1.06777e8	−2.06090
$161$	−9.94892e6	−0.187882
$162$	2.73978e7	0.506306
$163$	8.24993e7	1.49208	0.746042	−	0.665899i	$-0.231952\pi$
$163$	8.24993e7	1.49208	0.746042	+	0.665899i	$0.231952\pi$
$164$	−3.58484e7	−0.634623
$165$	5.54186e6	0.0960421
$166$	1.83439e7	0.311254
$167$	−4.27606e7	−0.710455	−0.355227	−	0.934780i	$-0.615597\pi$
$167$	−4.27606e7	−0.710455	−0.355227	+	0.934780i	$0.615597\pi$
$168$	−4.52122e7	−0.735653
$169$	4.82681e6	0.0769231
$170$	−4.83198e7	−0.754317
$171$	−3.89348e6	−0.0595459
$172$	2.96723e8	4.44633
$173$	−6.52843e7	−0.958623	−0.479311	−	0.877645i	$-0.659113\pi$
$173$	−6.52843e7	−0.958623	−0.479311	+	0.877645i	$0.659113\pi$
$174$	−6.00547e7	−0.864220
$175$	1.52365e6	0.0214908
$176$	−2.60465e7	−0.360126
$177$	−2.87911e7	−0.390258
$178$	2.31268e8	3.07359
$179$	1.19475e8	1.55701	0.778503	−	0.627641i	$-0.215979\pi$
$179$	1.19475e8	1.55701	0.778503	+	0.627641i	$0.215979\pi$
$180$	−9.15013e7	−1.16943
$181$	7.39477e7	0.926936	0.463468	−	0.886114i	$-0.346605\pi$
$181$	7.39477e7	0.926936	0.463468	+	0.886114i	$0.346605\pi$
$182$	1.58656e7	0.195077
$183$	6.88444e7	0.830405
$184$	−1.14358e8	−1.35333
$185$	1.26482e8	1.46868
$186$	−3.27937e7	−0.373676
$187$	−5.16304e6	−0.0577378
$188$	−3.28786e8	−3.60878
$189$	−3.73410e7	−0.402318
$190$	−2.08105e7	−0.220112
$191$	−1.64334e8	−1.70652	−0.853261	−	0.521484i	$-0.825379\pi$
$191$	−1.64334e8	−1.70652	−0.853261	+	0.521484i	$0.825379\pi$
$192$	−9.43534e7	−0.962063
$193$	9.97565e7	0.998827	0.499414	−	0.866364i	$-0.333549\pi$
$193$	9.97565e7	0.998827	0.499414	+	0.866364i	$0.333549\pi$
$194$	−2.77766e8	−2.73132
$195$	−1.99384e7	−0.192561
$196$	3.70905e7	0.351857
$197$	−9.32526e7	−0.869019	−0.434509	−	0.900667i	$-0.643078\pi$
$197$	−9.32526e7	−0.869019	−0.434509	+	0.900667i	$0.643078\pi$
$198$	−1.37466e7	−0.125854
$199$	−6.86917e7	−0.617901	−0.308950	−	0.951078i	$-0.599978\pi$
$199$	−6.86917e7	−0.617901	−0.308950	+	0.951078i	$0.599978\pi$
$200$	1.75136e7	0.154800
$201$	−7.26471e7	−0.631004
$202$	−1.11061e8	−0.948051
$203$	2.92640e7	0.245526
$204$	−8.91171e7	−0.734946
$205$	−3.08659e7	−0.250230
$206$	1.28635e8	1.02524
$207$	−3.10136e7	−0.243028
$208$	9.37096e7	0.722042
$209$	−2.22363e6	−0.0168481
$210$	−6.55368e7	−0.488335
$211$	−4.51572e7	−0.330931	−0.165466	−	0.986216i	$-0.552913\pi$
$211$	−4.51572e7	−0.330931	−0.165466	+	0.986216i	$0.552913\pi$
$212$	−4.62996e8	−3.33735
$213$	3.71028e7	0.263074
$214$	1.73681e8	1.21145
$215$	2.55482e8	1.75318
$216$	−4.29216e8	−2.89793
$217$	1.59800e7	0.106162
$218$	2.81084e8	1.83755
$219$	8.35671e7	0.537627
$220$	−5.22580e7	−0.330882
$221$	1.85755e7	0.115762
$222$	3.27985e8	2.01195
$223$	2.32510e8	1.40403	0.702013	−	0.712165i	$-0.252285\pi$
$223$	2.32510e8	1.40403	0.702013	+	0.712165i	$0.252285\pi$
$224$	1.34923e8	0.802083
$225$	4.74965e6	0.0277986
$226$	3.34261e8	1.92622
$227$	2.30488e8	1.30785	0.653926	−	0.756559i	$-0.273121\pi$
$227$	2.30488e8	1.30785	0.653926	+	0.756559i	$0.273121\pi$
$228$	−3.83812e7	−0.214460
$229$	6.58374e7	0.362284	0.181142	−	0.983457i	$-0.442021\pi$
$229$	6.58374e7	0.362284	0.181142	+	0.983457i	$0.442021\pi$
$230$	−1.65766e8	−0.898357
$231$	−7.00271e6	−0.0373787
$232$	3.36375e8	1.76854
$233$	−2.02247e8	−1.04746	−0.523728	−	0.851886i	$-0.675459\pi$
$233$	−2.02247e8	−1.04746	−0.523728	+	0.851886i	$0.675459\pi$
$234$	4.94574e7	0.252334
$235$	−2.83088e8	−1.42293
$236$	2.71491e8	1.34451
$237$	1.64345e8	0.801932
$238$	6.10570e7	0.293573
$239$	−373512.	−0.00176975	−0.000884875	−	1.00000i	$-0.500282\pi$
$239$	−373512.	−0.00176975	−0.000884875		1.00000i	$0.500282\pi$
$240$	−3.87092e8	−1.80748
$241$	−2.81836e8	−1.29699	−0.648496	−	0.761218i	$-0.724602\pi$
$241$	−2.81836e8	−1.29699	−0.648496	+	0.761218i	$0.724602\pi$
$242$	4.02429e8	1.82530
$243$	−1.94582e8	−0.869924
$244$	−6.49181e8	−2.86089
$245$	3.19353e7	0.138736
$246$	−8.00393e7	−0.342792
$247$	8.00015e6	0.0337799
$248$	1.83682e8	0.764692
$249$	2.91299e7	0.119575
$250$	4.71869e8	1.90999
$251$	4.78411e8	1.90960	0.954801	−	0.297245i	$-0.0960678\pi$
$251$	4.78411e8	1.90960	0.954801	+	0.297245i	$0.0960678\pi$
$252$	1.15621e8	0.455131
$253$	−1.77124e7	−0.0687631
$254$	4.43120e8	1.69669
$255$	−7.67309e7	−0.289788
$256$	−1.70346e8	−0.634590
$257$	−1.54346e8	−0.567192	−0.283596	−	0.958944i	$-0.591527\pi$
$257$	−1.54346e8	−0.567192	−0.283596	+	0.958944i	$0.591527\pi$
$258$	6.62499e8	2.40169
$259$	−1.59823e8	−0.571598
$260$	1.88013e8	0.663408
$261$	9.12240e7	0.317590
$262$	8.26414e8	2.83886
$263$	−4.14413e8	−1.40471	−0.702357	−	0.711825i	$-0.747869\pi$
$263$	−4.14413e8	−1.40471	−0.702357	+	0.711825i	$0.747869\pi$
$264$	−8.04927e7	−0.269242
$265$	−3.98645e8	−1.31591
$266$	2.62962e7	0.0856658
$267$	3.67249e8	1.18079
$268$	6.85039e8	2.17392
$269$	−1.80741e8	−0.566140	−0.283070	−	0.959099i	$-0.591353\pi$
$269$	−1.80741e8	−0.566140	−0.283070	+	0.959099i	$0.591353\pi$
$270$	−6.22165e8	−1.92368
$271$	−4.69927e8	−1.43429	−0.717147	−	0.696922i	$-0.754552\pi$
$271$	−4.69927e8	−1.43429	−0.717147	+	0.696922i	$0.754552\pi$
$272$	3.60632e8	1.08661
$273$	2.51942e7	0.0749431
$274$	2.64749e8	0.777511
$275$	2.71260e6	0.00786542
$276$	−3.05726e8	−0.875287
$277$	3.85982e8	1.09116	0.545579	−	0.838059i	$-0.316310\pi$
$277$	3.85982e8	1.09116	0.545579	+	0.838059i	$0.316310\pi$
$278$	−1.28549e8	−0.358849
$279$	4.98142e7	0.137322
$280$	3.67081e8	0.999330
$281$	1.62777e8	0.437644	0.218822	−	0.975765i	$-0.429779\pi$
$281$	1.62777e8	0.437644	0.218822	+	0.975765i	$0.429779\pi$
$282$	−7.34085e8	−1.94928
$283$	−1.53896e8	−0.403622	−0.201811	−	0.979424i	$-0.564683\pi$
$283$	−1.53896e8	−0.403622	−0.201811	+	0.979424i	$0.564683\pi$
$284$	−3.49868e8	−0.906337
$285$	−3.30467e7	−0.0845611
$286$	2.82460e7	0.0713963
$287$	3.90022e7	0.0973874
$288$	4.20595e8	1.03750
$289$	−3.38853e8	−0.825788
$290$	4.87588e8	1.17398
$291$	−4.41088e8	−1.04930
$292$	−7.88011e8	−1.85222
$293$	6.10561e8	1.41805	0.709027	−	0.705181i	$-0.249134\pi$
$293$	6.10561e8	1.41805	0.709027	+	0.705181i	$0.249134\pi$
$294$	8.28125e7	0.190055
$295$	2.33757e8	0.530137
$296$	−1.83709e9	−4.11727
$297$	−6.64794e7	−0.147245
$298$	6.80250e8	1.48906
$299$	6.37253e7	0.137868
$300$	4.68211e7	0.100119
$301$	−3.22828e8	−0.682321
$302$	−1.28993e9	−2.69489
$303$	−1.76363e8	−0.364215
$304$	1.55318e8	0.317077
$305$	−5.58953e8	−1.12804
$306$	1.90332e8	0.379740
$307$	−3.47148e8	−0.684747	−0.342374	−	0.939564i	$-0.611231\pi$
$307$	−3.47148e8	−0.684747	−0.342374	+	0.939564i	$0.611231\pi$
$308$	6.60334e7	0.128776
$309$	2.04271e8	0.393869
$310$	2.66255e8	0.507612
$311$	−7.23288e8	−1.36348	−0.681742	−	0.731593i	$-0.738777\pi$
$311$	−7.23288e8	−1.36348	−0.681742	+	0.731593i	$0.738777\pi$
$312$	2.89595e8	0.539822
$313$	7.00601e8	1.29141	0.645707	−	0.763585i	$-0.276563\pi$
$313$	7.00601e8	1.29141	0.645707	+	0.763585i	$0.276563\pi$
$314$	1.72815e9	3.15013
$315$	9.95515e7	0.179457
$316$	−1.54972e9	−2.76280
$317$	−4.09691e8	−0.722352	−0.361176	−	0.932498i	$-0.617625\pi$
$317$	−4.09691e8	−0.722352	−0.361176	+	0.932498i	$0.617625\pi$
$318$	−1.03374e9	−1.80267
$319$	5.20996e7	0.0898601
$320$	7.66063e8	1.30689
$321$	2.75803e8	0.465405
$322$	2.09463e8	0.349632
$323$	3.07878e7	0.0508358
$324$	−4.10259e8	−0.670117
$325$	−9.75935e6	−0.0157699
$326$	−1.73693e9	−2.77664
$327$	4.46357e8	0.705936
$328$	4.48312e8	0.701490
$329$	3.57711e8	0.553793
$330$	−1.16677e8	−0.178726
$331$	−3.92059e8	−0.594228	−0.297114	−	0.954842i	$-0.596024\pi$
$331$	−3.92059e8	−0.594228	−0.297114	+	0.954842i	$0.596024\pi$
$332$	−2.74685e8	−0.411957
$333$	−4.98214e8	−0.739369
$334$	9.00275e8	1.32209
$335$	5.89827e8	0.857172
$336$	4.89130e8	0.703456
$337$	−1.05640e9	−1.50357	−0.751783	−	0.659411i	$-0.770806\pi$
$337$	−1.05640e9	−1.50357	−0.751783	+	0.659411i	$0.770806\pi$
$338$	−1.01623e8	−0.143147
$339$	5.30800e8	0.740001
$340$	7.23549e8	0.998370
$341$	2.84498e7	0.0388542
$342$	8.19727e7	0.110810
$343$	−4.03536e7	−0.0539949
$344$	−3.71075e9	−4.91482
$345$	−2.63234e8	−0.345124
$346$	1.37449e9	1.78391
$347$	−4.53674e8	−0.582895	−0.291448	−	0.956587i	$-0.594137\pi$
$347$	−4.53674e8	−0.582895	−0.291448	+	0.956587i	$0.594137\pi$
$348$	8.99269e8	1.14383
$349$	−5.37691e8	−0.677086	−0.338543	−	0.940951i	$-0.609934\pi$
$349$	−5.37691e8	−0.677086	−0.338543	+	0.940951i	$0.609934\pi$
$350$	−3.20787e7	−0.0399925
$351$	2.39178e8	0.295221
$352$	2.40209e8	0.293555
$353$	5.59100e8	0.676516	0.338258	−	0.941053i	$-0.390162\pi$
$353$	5.59100e8	0.676516	0.338258	+	0.941053i	$0.390162\pi$
$354$	6.06163e8	0.726237
$355$	−3.01240e8	−0.357367
$356$	−3.46304e9	−4.06802
$357$	9.69575e7	0.112783
$358$	−2.51540e9	−2.89745
$359$	1.32361e9	1.50983	0.754915	−	0.655822i	$-0.227678\pi$
$359$	1.32361e9	1.50983	0.754915	+	0.655822i	$0.227678\pi$
$360$	1.14430e9	1.29265
$361$	−8.80612e8	−0.985166
$362$	−1.55688e9	−1.72495
$363$	6.39049e8	0.701231
$364$	−2.37574e8	−0.258192
$365$	−6.78488e8	−0.730326
$366$	−1.44944e9	−1.54531
$367$	−1.54360e8	−0.163006	−0.0815028	−	0.996673i	$-0.525972\pi$
$367$	−1.54360e8	−0.163006	−0.0815028	+	0.996673i	$0.525972\pi$
$368$	1.23719e9	1.29410
$369$	1.21581e8	0.125972
$370$	−2.66293e9	−2.73309
$371$	5.03729e8	0.512140
$372$	4.91059e8	0.494576
$373$	5.28771e8	0.527578	0.263789	−	0.964580i	$-0.415028\pi$
$373$	5.28771e8	0.527578	0.263789	+	0.964580i	$0.415028\pi$
$374$	1.08702e8	0.107445
$375$	7.49320e8	0.733766
$376$	4.11172e9	3.98902
$377$	−1.87443e8	−0.180167
$378$	7.86170e8	0.748678
$379$	1.38402e9	1.30589	0.652943	−	0.757407i	$-0.273534\pi$
$379$	1.38402e9	1.30589	0.652943	+	0.757407i	$0.273534\pi$
$380$	3.11620e8	0.291328
$381$	7.03667e8	0.651823
$382$	3.45987e9	3.17569
$383$	−4.54924e8	−0.413755	−0.206878	−	0.978367i	$-0.566330\pi$
$383$	−4.54924e8	−0.413755	−0.206878	+	0.978367i	$0.566330\pi$
$384$	3.03129e8	0.273192
$385$	5.68555e7	0.0507762
$386$	−2.10026e9	−1.85873
$387$	−1.00635e9	−0.882591
$388$	4.15932e9	3.61502
$389$	1.24169e9	1.06952	0.534761	−	0.845003i	$-0.320402\pi$
$389$	1.24169e9	1.06952	0.534761	+	0.845003i	$0.320402\pi$
$390$	4.19779e8	0.358340
$391$	2.45241e8	0.207479
$392$	−4.63845e8	−0.388930
$393$	1.31233e9	1.09061
$394$	1.96332e9	1.61717
$395$	−1.33433e9	−1.08936
$396$	2.05845e8	0.166574
$397$	9.67184e8	0.775787	0.387893	−	0.921704i	$-0.373203\pi$
$397$	9.67184e8	0.775787	0.387893	+	0.921704i	$0.373203\pi$
$398$	1.44622e9	1.14986
$399$	4.17579e7	0.0329104
$400$	−1.89472e8	−0.148025
$401$	−8.43457e7	−0.0653217	−0.0326609	−	0.999466i	$-0.510398\pi$
$401$	−8.43457e7	−0.0653217	−0.0326609	+	0.999466i	$0.510398\pi$
$402$	1.52950e9	1.17424
$403$	−1.02356e8	−0.0779014
$404$	1.66305e9	1.25479
$405$	−3.53238e8	−0.264226
$406$	−6.16118e8	−0.456902
$407$	−2.84539e8	−0.209200
$408$	1.11448e9	0.812384
$409$	−1.31567e9	−0.950855	−0.475427	−	0.879755i	$-0.657707\pi$
$409$	−1.31567e9	−0.950855	−0.475427	+	0.879755i	$0.657707\pi$
$410$	6.49845e8	0.465657
$411$	4.20416e8	0.298698
$412$	−1.92621e9	−1.35695
$413$	−2.95377e8	−0.206325
$414$	6.52955e8	0.452253
$415$	−2.36508e8	−0.162434
$416$	−8.64218e8	−0.588568
$417$	−2.04133e8	−0.137860
$418$	4.68160e7	0.0313529
$419$	1.78529e8	0.118566	0.0592830	−	0.998241i	$-0.481119\pi$
$419$	1.78529e8	0.118566	0.0592830	+	0.998241i	$0.481119\pi$
$420$	9.81359e8	0.646332
$421$	7.19842e8	0.470165	0.235082	−	0.971975i	$-0.424464\pi$
$421$	7.19842e8	0.470165	0.235082	+	0.971975i	$0.424464\pi$
$422$	9.50731e8	0.615835
$423$	1.11509e9	0.716338
$424$	5.79012e9	3.68899
$425$	−3.75579e7	−0.0237323
$426$	−7.81156e8	−0.489558
$427$	7.06295e8	0.439024
$428$	−2.60073e9	−1.60340
$429$	4.48541e7	0.0274285
$430$	−5.37888e9	−3.26251
$431$	−1.80243e9	−1.08440	−0.542198	−	0.840251i	$-0.682408\pi$
$431$	−1.80243e9	−1.08440	−0.542198	+	0.840251i	$0.682408\pi$
$432$	4.64350e9	2.77110
$433$	1.84926e8	0.109469	0.0547344	−	0.998501i	$-0.482569\pi$
$433$	1.84926e8	0.109469	0.0547344	+	0.998501i	$0.482569\pi$
$434$	−3.36440e8	−0.197558
$435$	7.74282e8	0.451010
$436$	−4.20901e9	−2.43207
$437$	1.05621e8	0.0605431
$438$	−1.75941e9	−1.00048
$439$	−4.18703e8	−0.236200	−0.118100	−	0.993002i	$-0.537680\pi$
$439$	−4.18703e8	−0.236200	−0.118100	+	0.993002i	$0.537680\pi$
$440$	6.53527e8	0.365745
$441$	−1.25794e8	−0.0698431
$442$	−3.91085e8	−0.215424
$443$	2.21179e9	1.20874	0.604368	−	0.796705i	$-0.293425\pi$
$443$	2.21179e9	1.20874	0.604368	+	0.796705i	$0.293425\pi$
$444$	−4.91130e9	−2.66291
$445$	−2.98172e9	−1.60401
$446$	−4.89523e9	−2.61277
$447$	1.08022e9	0.572055
$448$	−9.68000e8	−0.508630
$449$	−7.13696e8	−0.372092	−0.186046	−	0.982541i	$-0.559567\pi$
$449$	−7.13696e8	−0.372092	−0.186046	+	0.982541i	$0.559567\pi$
$450$	−9.99982e7	−0.0517307
$451$	6.94370e7	0.0356429
$452$	−5.00528e9	−2.54943
$453$	−2.04838e9	−1.03530
$454$	−4.85266e9	−2.43380
$455$	−2.04554e8	−0.101805
$456$	4.79987e8	0.237057
$457$	−2.12992e7	−0.0104389	−0.00521947	−	0.999986i	$-0.501661\pi$
$457$	−2.12992e7	−0.0104389	−0.00521947	+	0.999986i	$0.501661\pi$
$458$	−1.38613e9	−0.674178
$459$	9.20454e8	0.444281
$460$	2.48221e9	1.18901
$461$	3.32956e9	1.58283	0.791414	−	0.611280i	$-0.209345\pi$
$461$	3.32956e9	1.58283	0.791414	+	0.611280i	$0.209345\pi$
$462$	1.47434e8	0.0695585
$463$	−4.15533e9	−1.94568	−0.972842	−	0.231472i	$-0.925646\pi$
$463$	−4.15533e9	−1.94568	−0.972842	+	0.231472i	$0.925646\pi$
$464$	−3.63909e9	−1.69114
$465$	4.22808e8	0.195010
$466$	4.25807e9	1.94922
$467$	1.73447e9	0.788056	0.394028	−	0.919098i	$-0.371081\pi$
$467$	1.73447e9	0.788056	0.394028	+	0.919098i	$0.371081\pi$
$468$	−7.40584e8	−0.333975
$469$	−7.45308e8	−0.333604
$470$	5.96009e9	2.64796
$471$	2.74428e9	1.21019
$472$	−3.39521e9	−1.48617
$473$	−5.74742e8	−0.249723
$474$	−3.46009e9	−1.49233
$475$	−1.61755e7	−0.00692518
$476$	−9.14279e8	−0.388557
$477$	1.57027e9	0.662459
$478$	7.86385e6	0.00329335
$479$	4.08127e9	1.69676	0.848381	−	0.529387i	$-0.177578\pi$
$479$	4.08127e9	1.69676	0.848381	+	0.529387i	$0.177578\pi$
$480$	3.56988e9	1.47336
$481$	1.02371e9	0.419438
$482$	5.93373e9	2.41359
$483$	3.32623e8	0.134319
$484$	−6.02604e9	−2.41587
$485$	3.58122e9	1.42540
$486$	4.09670e9	1.61885
$487$	3.34070e9	1.31065	0.655324	−	0.755348i	$-0.272532\pi$
$487$	3.34070e9	1.31065	0.655324	+	0.755348i	$0.272532\pi$
$488$	8.11852e9	3.16233
$489$	−2.75821e9	−1.06671
$490$	−6.72361e8	−0.258176
$491$	2.07997e9	0.792999	0.396500	−	0.918035i	$-0.370225\pi$
$491$	2.07997e9	0.792999	0.396500	+	0.918035i	$0.370225\pi$
$492$	1.19852e9	0.453699
$493$	−7.21356e8	−0.271135
$494$	−1.68434e8	−0.0628615
$495$	1.77235e8	0.0656796
$496$	−1.98718e9	−0.731225
$497$	3.80649e8	0.139084
$498$	−6.13295e8	−0.222519
$499$	2.61527e8	0.0942247	0.0471123	−	0.998890i	$-0.484998\pi$
$499$	2.61527e8	0.0942247	0.0471123	+	0.998890i	$0.484998\pi$
$500$	−7.06585e9	−2.52796
$501$	1.42962e9	0.507912
$502$	−1.00724e10	−3.55360
$503$	1.39098e9	0.487342	0.243671	−	0.969858i	$-0.421648\pi$
$503$	1.39098e9	0.487342	0.243671	+	0.969858i	$0.421648\pi$
$504$	−1.44594e9	−0.503086
$505$	1.43190e9	0.494759
$506$	3.72914e8	0.127962
$507$	−1.61375e8	−0.0549932
$508$	−6.63536e9	−2.24565
$509$	4.90885e9	1.64994	0.824969	−	0.565178i	$-0.191192\pi$
$509$	4.90885e9	1.64994	0.824969	+	0.565178i	$0.191192\pi$
$510$	1.61548e9	0.539270
$511$	8.57339e8	0.284236
$512$	4.74699e9	1.56305
$513$	3.96423e8	0.129643
$514$	3.24958e9	1.05550
$515$	−1.65849e9	−0.535042
$516$	−9.92038e9	−3.17873
$517$	6.36846e8	0.202683
$518$	3.36489e9	1.06369
$519$	2.18266e9	0.685330
$520$	−2.35125e9	−0.733308
$521$	2.66534e9	0.825697	0.412848	−	0.910800i	$-0.364534\pi$
$521$	2.66534e9	0.825697	0.412848	+	0.910800i	$0.364534\pi$
$522$	−1.92062e9	−0.591008
$523$	2.37968e8	0.0727381	0.0363690	−	0.999338i	$-0.488421\pi$
$523$	2.37968e8	0.0727381	0.0363690	+	0.999338i	$0.488421\pi$
$524$	−1.23749e10	−3.75735
$525$	−5.09403e7	−0.0153640
$526$	8.72498e9	2.61405
$527$	−3.93907e8	−0.117235
$528$	8.70815e8	0.257459
$529$	−2.56350e9	−0.752902
$530$	8.39300e9	2.44879
$531$	−9.20772e8	−0.266883
$532$	−3.93764e8	−0.113382
$533$	−2.49819e8	−0.0714628
$534$	−7.73200e9	−2.19734
$535$	−2.23926e9	−0.632218
$536$	−8.56695e9	−2.40298
$537$	−3.99441e9	−1.11312
$538$	3.80529e9	1.05354
$539$	−7.18429e7	−0.0197616
$540$	9.31641e9	2.54607
$541$	−1.44765e9	−0.393072	−0.196536	−	0.980497i	$-0.562969\pi$
$541$	−1.44765e9	−0.393072	−0.196536	+	0.980497i	$0.562969\pi$
$542$	9.89377e9	2.66910
$543$	−2.47230e9	−0.662677
$544$	−3.32586e9	−0.885743
$545$	−3.62401e9	−0.958962
$546$	−5.30435e8	−0.139463
$547$	4.49404e9	1.17403	0.587017	−	0.809574i	$-0.300302\pi$
$547$	4.49404e9	1.17403	0.587017	+	0.809574i	$0.300302\pi$
$548$	−3.96439e9	−1.02907
$549$	2.20172e9	0.567883
$550$	−5.71107e7	−0.0146369
$551$	−3.10675e8	−0.0791182
$552$	3.82334e9	0.967512
$553$	1.68606e9	0.423971
$554$	−8.12640e9	−2.03055
$555$	−4.22869e9	−1.04998
$556$	1.92491e9	0.474951
$557$	−6.14557e9	−1.50685	−0.753424	−	0.657535i	$-0.771599\pi$
$557$	−6.14557e9	−1.50685	−0.753424	+	0.657535i	$0.771599\pi$
$558$	−1.04878e9	−0.255543
$559$	2.06780e9	0.500687
$560$	−3.97129e9	−0.955594
$561$	1.72617e8	0.0412774
$562$	−3.42708e9	−0.814418
$563$	3.02755e9	0.715010	0.357505	−	0.933911i	$-0.383627\pi$
$563$	3.02755e9	0.715010	0.357505	+	0.933911i	$0.383627\pi$
$564$	1.09923e10	2.57996
$565$	−4.30961e9	−1.00524
$566$	3.24010e9	0.751106
$567$	4.46353e8	0.102834
$568$	4.37537e9	1.00183
$569$	−7.74092e9	−1.76157	−0.880784	−	0.473518i	$-0.842984\pi$
$569$	−7.74092e9	−1.76157	−0.880784	+	0.473518i	$0.842984\pi$
$570$	6.95759e8	0.157361
$571$	−7.90431e9	−1.77680	−0.888398	−	0.459075i	$-0.848181\pi$
$571$	−7.90431e9	−1.77680	−0.888398	+	0.459075i	$0.848181\pi$
$572$	−4.22960e8	−0.0944960
$573$	5.49421e9	1.22001
$574$	−8.21146e8	−0.181230
$575$	−1.28847e8	−0.0282641
$576$	−3.01753e9	−0.657919
$577$	3.88975e9	0.842959	0.421479	−	0.906838i	$-0.361511\pi$
$577$	3.88975e9	0.842959	0.421479	+	0.906838i	$0.361511\pi$
$578$	7.13415e9	1.53672
$579$	−3.33517e9	−0.714073
$580$	−7.30123e9	−1.55381
$581$	2.98852e8	0.0632178
$582$	9.28658e9	1.95265
$583$	8.96806e8	0.187438
$584$	9.85469e9	2.04738
$585$	−6.37652e8	−0.131686
$586$	−1.28547e10	−2.63887
$587$	−1.51223e9	−0.308592	−0.154296	−	0.988025i	$-0.549311\pi$
$587$	−1.51223e9	−0.308592	−0.154296	+	0.988025i	$0.549311\pi$
$588$	−1.24005e9	−0.251546
$589$	−1.69649e8	−0.0342096
$590$	−4.92149e9	−0.986539
$591$	3.11772e9	0.621271
$592$	1.98747e10	3.93707
$593$	−5.77543e9	−1.13735	−0.568673	−	0.822564i	$-0.692543\pi$
$593$	−5.77543e9	−1.13735	−0.568673	+	0.822564i	$0.692543\pi$
$594$	1.39965e9	0.274009
$595$	−7.87205e8	−0.153207
$596$	−1.01862e10	−1.97083
$597$	2.29658e9	0.441744
$598$	−1.34166e9	−0.256560
$599$	−4.70735e9	−0.894917	−0.447459	−	0.894305i	$-0.647671\pi$
$599$	−4.70735e9	−0.894917	−0.447459	+	0.894305i	$0.647671\pi$
$600$	−5.85534e8	−0.110668
$601$	−5.08280e9	−0.955085	−0.477542	−	0.878609i	$-0.658472\pi$
$601$	−5.08280e9	−0.955085	−0.477542	+	0.878609i	$0.658472\pi$
$602$	6.79677e9	1.26974
$603$	−2.32333e9	−0.431520
$604$	1.93156e10	3.56680
$605$	−5.18849e9	−0.952571
$606$	3.71311e9	0.677773
$607$	4.57064e9	0.829501	0.414751	−	0.909935i	$-0.363869\pi$
$607$	4.57064e9	0.829501	0.414751	+	0.909935i	$0.363869\pi$
$608$	−1.43239e9	−0.258463
$609$	−9.78385e8	−0.175529
$610$	1.17681e10	2.09919
$611$	−2.29123e9	−0.406373
$612$	−2.85006e9	−0.502602
$613$	−1.10371e10	−1.93528	−0.967639	−	0.252340i	$-0.918800\pi$
$613$	−1.10371e10	−1.93528	−0.967639	+	0.252340i	$0.918800\pi$
$614$	7.30879e9	1.27426
$615$	1.03194e9	0.178893
$616$	−8.25798e8	−0.142345
$617$	−1.83201e9	−0.314000	−0.157000	−	0.987599i	$-0.550182\pi$
$617$	−1.83201e9	−0.314000	−0.157000	+	0.987599i	$0.550182\pi$
$618$	−4.30068e9	−0.732956
$619$	8.05408e9	1.36489	0.682447	−	0.730935i	$-0.260916\pi$
$619$	8.05408e9	1.36489	0.682447	+	0.730935i	$0.260916\pi$
$620$	−3.98695e9	−0.671845
$621$	3.15772e9	0.529118
$622$	1.52280e10	2.53732
$623$	3.76772e9	0.624267
$624$	−3.13300e9	−0.516196
$625$	−5.73674e9	−0.939908
$626$	−1.47503e10	−2.40321
$627$	7.43430e7	0.0120449
$628$	−2.58777e10	−4.16933
$629$	3.93964e9	0.631217
$630$	−2.09594e9	−0.333954
$631$	6.29926e8	0.0998130	0.0499065	−	0.998754i	$-0.484108\pi$
$631$	6.29926e8	0.0998130	0.0499065	+	0.998754i	$0.484108\pi$
$632$	1.93805e10	3.05390
$633$	1.50974e9	0.236587
$634$	8.62556e9	1.34423
$635$	−5.71313e9	−0.885454
$636$	1.54794e10	2.38591
$637$	2.58475e8	0.0396214
$638$	−1.09690e9	−0.167222
$639$	1.18659e9	0.179907
$640$	−2.46113e9	−0.371112
$641$	3.52082e9	0.528008	0.264004	−	0.964522i	$-0.414957\pi$
$641$	3.52082e9	0.528008	0.264004	+	0.964522i	$0.414957\pi$
$642$	−5.80671e9	−0.866078
$643$	−1.34924e9	−0.200148	−0.100074	−	0.994980i	$-0.531908\pi$
$643$	−1.34924e9	−0.200148	−0.100074	+	0.994980i	$0.531908\pi$
$644$	−3.13653e9	−0.462753
$645$	−8.54157e9	−1.25337
$646$	−6.48201e8	−0.0946010
$647$	−8.17274e9	−1.18632	−0.593161	−	0.805084i	$-0.702120\pi$
$647$	−8.17274e9	−1.18632	−0.593161	+	0.805084i	$0.702120\pi$
$648$	5.13061e9	0.740724
$649$	−5.25869e8	−0.0755129
$650$	2.05472e8	0.0293464
$651$	−5.34261e8	−0.0758963
$652$	2.60090e10	3.67500
$653$	−5.45809e9	−0.767087	−0.383544	−	0.923523i	$-0.625296\pi$
$653$	−5.45809e9	−0.767087	−0.383544	+	0.923523i	$0.625296\pi$
$654$	−9.39752e9	−1.31369
$655$	−1.06549e10	−1.48151
$656$	−4.85008e9	−0.670789
$657$	2.67257e9	0.367663
$658$	−7.53120e9	−1.03056
$659$	1.78517e9	0.242986	0.121493	−	0.992592i	$-0.461232\pi$
$659$	1.78517e9	0.242986	0.121493	+	0.992592i	$0.461232\pi$
$660$	1.74715e9	0.236551
$661$	3.31766e9	0.446814	0.223407	−	0.974725i	$-0.428282\pi$
$661$	3.31766e9	0.446814	0.223407	+	0.974725i	$0.428282\pi$
$662$	8.25434e9	1.10581
$663$	−6.21037e8	−0.0827599
$664$	3.43515e9	0.455363
$665$	−3.39036e8	−0.0447064
$666$	1.04893e10	1.37590
$667$	−2.47469e9	−0.322909
$668$	−1.34809e10	−1.74985
$669$	−7.77354e9	−1.00375
$670$	−1.24181e10	−1.59512
$671$	1.25744e9	0.160679
$672$	−4.51091e9	−0.573419
$673$	2.52303e9	0.319058	0.159529	−	0.987193i	$-0.449002\pi$
$673$	2.52303e9	0.319058	0.159529	+	0.987193i	$0.449002\pi$
$674$	2.22412e10	2.79800
$675$	−4.83596e8	−0.0605228
$676$	1.52172e9	0.189461
$677$	9.97526e9	1.23556	0.617780	−	0.786351i	$-0.288032\pi$
$677$	9.97526e9	1.23556	0.617780	+	0.786351i	$0.288032\pi$
$678$	−1.11754e10	−1.37708
$679$	−4.52525e9	−0.554751
$680$	−9.04854e9	−1.10356
$681$	−7.70594e9	−0.934998
$682$	−5.98976e8	−0.0723043
$683$	7.38641e8	0.0887076	0.0443538	−	0.999016i	$-0.485877\pi$
$683$	7.38641e8	0.0887076	0.0443538	+	0.999016i	$0.485877\pi$
$684$	−1.22747e9	−0.146661
$685$	−3.41339e9	−0.405760
$686$	8.49598e8	0.100480
$687$	−2.20115e9	−0.259001
$688$	4.01450e10	4.69972
$689$	−3.22651e9	−0.375808
$690$	5.54208e9	0.642245
$691$	−4.88499e9	−0.563236	−0.281618	−	0.959527i	$-0.590871\pi$
$691$	−4.88499e9	−0.563236	−0.281618	+	0.959527i	$0.590871\pi$
$692$	−2.05818e10	−2.36108
$693$	−2.23954e8	−0.0255619
$694$	9.55157e9	1.08472
$695$	1.65737e9	0.187272
$696$	−1.12461e10	−1.26435
$697$	−9.61404e8	−0.107545
$698$	1.13205e10	1.26000
$699$	6.76173e9	0.748838
$700$	4.80351e8	0.0529317
$701$	1.72434e9	0.189064	0.0945321	−	0.995522i	$-0.469864\pi$
$701$	1.72434e9	0.189064	0.0945321	+	0.995522i	$0.469864\pi$
$702$	−5.03562e9	−0.549380
$703$	1.69673e9	0.184192
$704$	−1.72336e9	−0.186154
$705$	9.46452e9	1.01727
$706$	−1.17712e10	−1.25894
$707$	−1.80936e9	−0.192556
$708$	−9.07680e9	−0.961205
$709$	−1.14207e10	−1.20346	−0.601728	−	0.798701i	$-0.705521\pi$
$709$	−1.14207e10	−1.20346	−0.601728	+	0.798701i	$0.705521\pi$
$710$	6.34227e9	0.665028
$711$	5.25594e9	0.548412
$712$	4.33081e10	4.49665
$713$	−1.35134e9	−0.139621
$714$	−2.04133e9	−0.209879
$715$	−3.64174e8	−0.0372595
$716$	3.76660e10	3.83490
$717$	1.24877e7	0.00126521
$718$	−2.78670e10	−2.80966
$719$	2.12933e9	0.213644	0.106822	−	0.994278i	$-0.465932\pi$
$719$	2.12933e9	0.213644	0.106822	+	0.994278i	$0.465932\pi$
$720$	−1.23796e10	−1.23607
$721$	2.09567e9	0.208233
$722$	1.85403e10	1.83331
$723$	9.42266e9	0.927235
$724$	2.33130e10	2.28304
$725$	3.78992e8	0.0369358
$726$	−1.34544e10	−1.30493
$727$	1.14030e10	1.10065	0.550326	−	0.834950i	$-0.314503\pi$
$727$	1.14030e10	1.10065	0.550326	+	0.834950i	$0.314503\pi$
$728$	2.97104e9	0.285397
$729$	9.35148e9	0.893993
$730$	1.42848e10	1.35907
$731$	7.95771e9	0.753490
$732$	2.17041e10	2.04529
$733$	−1.17814e10	−1.10493	−0.552463	−	0.833538i	$-0.686312\pi$
$733$	−1.17814e10	−1.10493	−0.552463	+	0.833538i	$0.686312\pi$
$734$	3.24986e9	0.303339
$735$	−1.06770e9	−0.0991842
$736$	−1.14097e10	−1.05488
$737$	−1.32690e9	−0.122096
$738$	−2.55974e9	−0.234423
$739$	6.06138e9	0.552479	0.276240	−	0.961089i	$-0.410912\pi$
$739$	6.06138e9	0.552479	0.276240	+	0.961089i	$0.410912\pi$
$740$	3.98752e10	3.61736
$741$	−2.67470e8	−0.0241497
$742$	−1.06054e10	−0.953047
$743$	−9.09290e9	−0.813283	−0.406641	−	0.913588i	$-0.633300\pi$
$743$	−9.09290e9	−0.813283	−0.406641	+	0.913588i	$0.633300\pi$
$744$	−6.14107e9	−0.546687
$745$	−8.77043e9	−0.777094
$746$	−1.11326e10	−0.981776
$747$	9.31605e8	0.0817730
$748$	−1.62772e9	−0.142208
$749$	2.82954e9	0.246054
$750$	−1.57760e10	−1.36548
$751$	1.43886e10	1.23959	0.619797	−	0.784762i	$-0.287215\pi$
$751$	1.43886e10	1.23959	0.619797	+	0.784762i	$0.287215\pi$
$752$	−4.44829e10	−3.81444
$753$	−1.59948e10	−1.36520
$754$	3.94639e9	0.335274
$755$	1.66310e10	1.40638
$756$	−1.17722e10	−0.990908
$757$	−3.41124e9	−0.285810	−0.142905	−	0.989736i	$-0.545644\pi$
$757$	−3.41124e9	−0.285810	−0.142905	+	0.989736i	$0.545644\pi$
$758$	−2.91389e10	−2.43014
$759$	5.92180e8	0.0491595
$760$	−3.89705e9	−0.322024
$761$	−5.21143e9	−0.428657	−0.214329	−	0.976762i	$-0.568756\pi$
$761$	−5.21143e9	−0.428657	−0.214329	+	0.976762i	$0.568756\pi$
$762$	−1.48149e10	−1.21299
$763$	4.57931e9	0.373219
$764$	−5.18087e10	−4.20316
$765$	−2.45394e9	−0.198175
$766$	9.57790e9	0.769963
$767$	1.89196e9	0.151401
$768$	5.69521e9	0.453675
$769$	1.71778e10	1.36215	0.681076	−	0.732213i	$-0.261512\pi$
$769$	1.71778e10	1.36215	0.681076	+	0.732213i	$0.261512\pi$
$770$	−1.19703e9	−0.0944901
$771$	5.16027e9	0.405492
$772$	3.14496e10	2.46011
$773$	−1.62555e10	−1.26582	−0.632910	−	0.774225i	$-0.718140\pi$
$773$	−1.62555e10	−1.26582	−0.632910	+	0.774225i	$0.718140\pi$
$774$	2.11875e10	1.64242
$775$	2.06954e8	0.0159705
$776$	−5.20155e10	−3.99592
$777$	5.34339e9	0.408642
$778$	−2.61423e10	−1.99029
$779$	−4.14060e8	−0.0313821
$780$	−6.28585e9	−0.474278
$781$	6.77681e8	0.0509034
$782$	−5.16325e9	−0.386100
$783$	−9.28818e9	−0.691456
$784$	5.01813e9	0.371908
$785$	−2.22810e10	−1.64396
$786$	−2.76296e10	−2.02953
$787$	−2.34255e9	−0.171308	−0.0856538	−	0.996325i	$-0.527298\pi$
$787$	−2.34255e9	−0.171308	−0.0856538	+	0.996325i	$0.527298\pi$
$788$	−2.93992e10	−2.14039
$789$	1.38551e10	1.00425
$790$	2.80928e10	2.02721
$791$	5.44564e9	0.391229
$792$	−2.57425e9	−0.184125
$793$	−4.52400e9	−0.322156
$794$	−2.03629e10	−1.44367
$795$	1.33279e10	0.940758
$796$	−2.16560e10	−1.52189
$797$	1.60345e10	1.12189	0.560947	−	0.827852i	$-0.310437\pi$
$797$	1.60345e10	1.12189	0.560947	+	0.827852i	$0.310437\pi$
$798$	−8.79164e8	−0.0612434
$799$	−8.81758e9	−0.611555
$800$	1.74737e9	0.120662
$801$	1.17450e10	0.807496
$802$	1.77580e9	0.121558
$803$	1.52635e9	0.104028
$804$	−2.29030e10	−1.55416
$805$	−2.70059e9	−0.182463
$806$	2.15498e9	0.144968
$807$	6.04273e9	0.404740
$808$	−2.07977e10	−1.38700
$809$	−1.49610e10	−0.993440	−0.496720	−	0.867911i	$-0.665462\pi$
$809$	−1.49610e10	−0.993440	−0.496720	+	0.867911i	$0.665462\pi$
$810$	7.43701e9	0.491701
$811$	−1.31667e10	−0.866771	−0.433386	−	0.901209i	$-0.642681\pi$
$811$	−1.31667e10	−0.866771	−0.433386	+	0.901209i	$0.642681\pi$
$812$	9.22586e9	0.604729
$813$	1.57111e10	1.02539
$814$	5.99063e9	0.389302
$815$	2.23941e10	1.44904
$816$	−1.20571e10	−0.776829
$817$	3.42725e9	0.219871
$818$	2.76998e10	1.76946
$819$	8.05739e8	0.0512508
$820$	−9.73089e9	−0.616317
$821$	−1.91978e10	−1.21074	−0.605371	−	0.795944i	$-0.706975\pi$
$821$	−1.91978e10	−1.21074	−0.605371	+	0.795944i	$0.706975\pi$
$822$	−8.85137e9	−0.555852
$823$	8.00412e9	0.500511	0.250256	−	0.968180i	$-0.419485\pi$
$823$	8.00412e9	0.500511	0.250256	+	0.968180i	$0.419485\pi$
$824$	2.40888e10	1.49992
$825$	−9.06908e7	−0.00562308
$826$	6.21881e9	0.383952
$827$	−6.07392e9	−0.373422	−0.186711	−	0.982415i	$-0.559783\pi$
$827$	−6.07392e9	−0.373422	−0.186711	+	0.982415i	$0.559783\pi$
$828$	−9.77746e9	−0.598577
$829$	2.29024e10	1.39618	0.698088	−	0.716012i	$-0.254035\pi$
$829$	2.29024e10	1.39618	0.698088	+	0.716012i	$0.254035\pi$
$830$	4.97939e9	0.302275
$831$	−1.29046e10	−0.780082
$832$	6.20028e9	0.373233
$833$	9.94715e8	0.0596268
$834$	4.29779e9	0.256545
$835$	−1.16072e10	−0.689961
$836$	−7.01031e8	−0.0414969
$837$	−5.07194e9	−0.298975
$838$	−3.75873e9	−0.220641
$839$	8.67749e9	0.507256	0.253628	−	0.967302i	$-0.418376\pi$
$839$	8.67749e9	0.507256	0.253628	+	0.967302i	$0.418376\pi$
$840$	−1.22727e10	−0.714432
$841$	−9.97077e9	−0.578020
$842$	−1.51554e10	−0.874936
$843$	−5.44215e9	−0.312877
$844$	−1.42364e10	−0.815083
$845$	1.31022e9	0.0747042
$846$	−2.34769e10	−1.33304
$847$	6.55620e9	0.370732
$848$	−6.26407e10	−3.52754
$849$	5.14522e9	0.288554
$850$	7.90738e8	0.0441638
$851$	1.35154e10	0.751751
$852$	1.16972e10	0.647951
$853$	1.52833e10	0.843130	0.421565	−	0.906798i	$-0.361481\pi$
$853$	1.52833e10	0.843130	0.421565	+	0.906798i	$0.361481\pi$
$854$	−1.48702e10	−0.816986
$855$	−1.05687e9	−0.0578282
$856$	3.25242e10	1.77235
$857$	−3.73137e9	−0.202505	−0.101252	−	0.994861i	$-0.532285\pi$
$857$	−3.73137e9	−0.202505	−0.101252	+	0.994861i	$0.532285\pi$
$858$	−9.44350e8	−0.0510420
$859$	−1.69040e10	−0.909939	−0.454970	−	0.890507i	$-0.650350\pi$
$859$	−1.69040e10	−0.909939	−0.454970	+	0.890507i	$0.650350\pi$
$860$	8.05443e10	4.31808
$861$	−1.30397e9	−0.0696234
$862$	3.79480e10	2.01797
$863$	2.72267e10	1.44197	0.720987	−	0.692948i	$-0.243689\pi$
$863$	2.72267e10	1.44197	0.720987	+	0.692948i	$0.243689\pi$
$864$	−4.28238e10	−2.25885
$865$	−1.77212e10	−0.930970
$866$	−3.89340e9	−0.203712
$867$	1.13289e10	0.590365
$868$	5.03792e9	0.261476
$869$	3.00176e9	0.155169
$870$	−1.63016e10	−0.839291
$871$	4.77388e9	0.244798
$872$	5.26369e10	2.68833
$873$	−1.41065e10	−0.717577
$874$	−2.22372e9	−0.112665
$875$	7.68749e9	0.387933
$876$	2.63457e10	1.32417
$877$	−1.97100e10	−0.986705	−0.493352	−	0.869830i	$-0.664229\pi$
$877$	−1.97100e10	−0.986705	−0.493352	+	0.869830i	$0.664229\pi$
$878$	8.81530e9	0.439548
$879$	−2.04130e10	−1.01378
$880$	−7.07021e9	−0.349738
$881$	2.54470e9	0.125378	0.0626889	−	0.998033i	$-0.480032\pi$
$881$	2.54470e9	0.125378	0.0626889	+	0.998033i	$0.480032\pi$
$882$	2.64844e9	0.129972
$883$	−1.86843e10	−0.913303	−0.456652	−	0.889646i	$-0.650951\pi$
$883$	−1.86843e10	−0.913303	−0.456652	+	0.889646i	$0.650951\pi$
$884$	5.85618e9	0.285123
$885$	−7.81523e9	−0.379001
$886$	−4.65667e10	−2.24935
$887$	−8.96021e9	−0.431108	−0.215554	−	0.976492i	$-0.569156\pi$
$887$	−8.96021e9	−0.431108	−0.215554	+	0.976492i	$0.569156\pi$
$888$	6.14196e10	2.94348
$889$	7.21913e9	0.344611
$890$	6.27767e10	2.98493
$891$	7.94657e8	0.0376364
$892$	7.33020e10	3.45811
$893$	−3.79758e9	−0.178454
$894$	−2.27429e10	−1.06454
$895$	3.24309e10	1.51209
$896$	3.10989e9	0.144433
$897$	−2.13053e9	−0.0985632
$898$	1.50260e10	0.692432
$899$	3.97486e9	0.182458
$900$	1.49739e9	0.0684678
$901$	−1.24169e10	−0.565557
$902$	−1.46191e9	−0.0663283
$903$	1.07932e10	0.487799
$904$	6.25949e10	2.81806
$905$	2.00728e10	0.900198
$906$	4.31263e10	1.92661
$907$	2.36513e9	0.105252	0.0526259	−	0.998614i	$-0.483241\pi$
$907$	2.36513e9	0.105252	0.0526259	+	0.998614i	$0.483241\pi$
$908$	7.26646e10	3.22123
$909$	−5.64028e9	−0.249073
$910$	4.30664e9	0.189450
$911$	4.27646e10	1.87400	0.937000	−	0.349329i	$-0.113590\pi$
$911$	4.27646e10	1.87400	0.937000	+	0.349329i	$0.113590\pi$
$912$	−5.19276e9	−0.226682
$913$	5.32056e8	0.0231371
$914$	4.48430e8	0.0194260
$915$	1.86875e10	0.806452
$916$	2.07561e10	0.892304
$917$	1.34636e10	0.576591
$918$	−1.93791e10	−0.826768
$919$	4.05262e10	1.72239	0.861195	−	0.508275i	$-0.169717\pi$
$919$	4.05262e10	1.72239	0.861195	+	0.508275i	$0.169717\pi$
$920$	−3.10420e10	−1.31429
$921$	1.16062e10	0.489534
$922$	−7.01000e10	−2.94551
$923$	−2.43815e9	−0.102060
$924$	−2.20770e9	−0.0920637
$925$	−2.06984e9	−0.0859886
$926$	8.74856e10	3.62075
$927$	6.53281e9	0.269352
$928$	3.35608e10	1.37852
$929$	−2.08054e10	−0.851375	−0.425687	−	0.904870i	$-0.639968\pi$
$929$	−2.08054e10	−0.851375	−0.425687	+	0.904870i	$0.639968\pi$
$930$	−8.90172e9	−0.362897
$931$	4.28407e8	0.0173993
$932$	−6.37610e10	−2.57988
$933$	2.41818e10	0.974770
$934$	−3.65172e10	−1.46650
$935$	−1.40149e9	−0.0560723
$936$	9.26157e9	0.369164
$937$	−8.09118e9	−0.321309	−0.160655	−	0.987011i	$-0.551361\pi$
$937$	−8.09118e9	−0.321309	−0.160655	+	0.987011i	$0.551361\pi$
$938$	1.56916e10	0.620807
$939$	−2.34233e10	−0.923247
$940$	−8.92475e10	−3.50468
$941$	−4.10268e10	−1.60511	−0.802554	−	0.596580i	$-0.796526\pi$
$941$	−4.10268e10	−1.60511	−0.802554	+	0.596580i	$0.796526\pi$
$942$	−5.77775e10	−2.25206
$943$	−3.29820e9	−0.128081
$944$	3.67313e10	1.42113
$945$	−1.01361e10	−0.390713
$946$	1.21005e10	0.464713
$947$	1.78600e9	0.0683370	0.0341685	−	0.999416i	$-0.489122\pi$
$947$	1.78600e9	0.0683370	0.0341685	+	0.999416i	$0.489122\pi$
$948$	5.18120e10	1.97516
$949$	−5.49147e9	−0.208572
$950$	3.40557e8	0.0128872
$951$	1.36972e10	0.516418
$952$	1.14338e10	0.429497
$953$	−2.42777e10	−0.908620	−0.454310	−	0.890844i	$-0.650114\pi$
$953$	−2.42777e10	−0.908620	−0.454310	+	0.890844i	$0.650114\pi$
$954$	−3.30601e10	−1.23278
$955$	−4.46079e10	−1.65730
$956$	−1.17755e8	−0.00435889
$957$	−1.74185e9	−0.0642420
$958$	−8.59263e10	−3.15753
$959$	4.31317e9	0.157918
$960$	−2.56119e10	−0.934312
$961$	−2.53421e10	−0.921108
$962$	−2.15530e10	−0.780538
$963$	8.82048e9	0.318273
$964$	−8.88528e10	−3.19449
$965$	2.70785e10	0.970016
$966$	−7.00299e9	−0.249956
$967$	1.84102e10	0.654736	0.327368	−	0.944897i	$-0.393838\pi$
$967$	1.84102e10	0.654736	0.327368	+	0.944897i	$0.393838\pi$
$968$	7.53602e10	2.67041
$969$	−1.02933e9	−0.0363431
$970$	−7.53985e10	−2.65254
$971$	−7.05610e9	−0.247342	−0.123671	−	0.992323i	$-0.539467\pi$
$971$	−7.05610e9	−0.247342	−0.123671	+	0.992323i	$0.539467\pi$
$972$	−6.13447e10	−2.14262
$973$	−2.09426e9	−0.0728847
$974$	−7.03345e10	−2.43900
$975$	3.26285e8	0.0112741
$976$	−8.78306e10	−3.02393
$977$	5.59885e10	1.92074	0.960368	−	0.278736i	$-0.0899156\pi$
$977$	5.59885e10	1.92074	0.960368	+	0.278736i	$0.0899156\pi$
$978$	5.80708e10	1.98505
$979$	6.70779e9	0.228476
$980$	1.00681e10	0.341707
$981$	1.42750e10	0.482763
$982$	−4.37914e10	−1.47570
$983$	2.80226e10	0.940959	0.470480	−	0.882411i	$-0.344081\pi$
$983$	2.80226e10	0.940959	0.470480	+	0.882411i	$0.344081\pi$
$984$	−1.49884e10	−0.501503
$985$	−2.53130e10	−0.843951
$986$	1.51873e10	0.504558
$987$	−1.19594e10	−0.395913
$988$	2.52216e9	0.0831998
$989$	2.72998e10	0.897372
$990$	−3.73147e9	−0.122224
$991$	1.16663e10	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$991$	1.16663e10	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$992$	1.83264e10	0.596054
$993$	1.31077e10	0.424821
$994$	−8.01411e9	−0.258823
$995$	−1.86461e10	−0.600077
$996$	9.18359e9	0.294513
$997$	4.53066e10	1.44786	0.723932	−	0.689871i	$-0.242333\pi$
$997$	4.53066e10	1.44786	0.723932	+	0.689871i	$0.242333\pi$
$998$	−5.50614e9	−0.175344
$999$	5.07268e10	1.60975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.a.b.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.a.b.1.1	✓	9	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:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.a (trivial)

\(f(q)\)	\(=\)	\(q-21.0538 q^{2} -33.4331 q^{3} +315.264 q^{4} +271.446 q^{5} +703.895 q^{6} -343.000 q^{7} -3942.62 q^{8} -1069.23 q^{9} +O(q^{10})\)	\(q-21.0538 q^{2} -33.4331 q^{3} +315.264 q^{4} +271.446 q^{5} +703.895 q^{6} -343.000 q^{7} -3942.62 q^{8} -1069.23 q^{9} -5714.98 q^{10} -610.654 q^{11} -10540.2 q^{12} +2197.00 q^{13} +7221.46 q^{14} -9075.28 q^{15} +42653.4 q^{16} +8454.94 q^{17} +22511.3 q^{18} +3641.40 q^{19} +85577.0 q^{20} +11467.6 q^{21} +12856.6 q^{22} +29005.6 q^{23} +131814. q^{24} -4442.13 q^{25} -46255.3 q^{26} +108866. q^{27} -108135. q^{28} -85317.7 q^{29} +191069. q^{30} -46589.0 q^{31} -393363. q^{32} +20416.1 q^{33} -178009. q^{34} -93105.9 q^{35} -337089. q^{36} +465957. q^{37} -76665.3 q^{38} -73452.5 q^{39} -1.07021e6 q^{40} -113709. q^{41} -241436. q^{42} +941191. q^{43} -192517. q^{44} -290237. q^{45} -610679. q^{46} -1.04289e6 q^{47} -1.42604e6 q^{48} +117649. q^{49} +93523.8 q^{50} -282675. q^{51} +692634. q^{52} -1.46860e6 q^{53} -2.29204e6 q^{54} -165760. q^{55} +1.35232e6 q^{56} -121743. q^{57} +1.79626e6 q^{58} +861156. q^{59} -2.86111e6 q^{60} -2.05917e6 q^{61} +980876. q^{62} +366745. q^{63} +2.82216e6 q^{64} +596367. q^{65} -429836. q^{66} +2.17291e6 q^{67} +2.66554e6 q^{68} -969747. q^{69} +1.96024e6 q^{70} -1.10976e6 q^{71} +4.21556e6 q^{72} -2.49953e6 q^{73} -9.81018e6 q^{74} +148514. q^{75} +1.14800e6 q^{76} +209454. q^{77} +1.54646e6 q^{78} -4.91564e6 q^{79} +1.15781e7 q^{80} -1.30132e6 q^{81} +2.39401e6 q^{82} -871288. q^{83} +3.61530e6 q^{84} +2.29506e6 q^{85} -1.98157e7 q^{86} +2.85243e6 q^{87} +2.40758e6 q^{88} -1.09846e7 q^{89} +6.11061e6 q^{90} -753571. q^{91} +9.14441e6 q^{92} +1.55761e6 q^{93} +2.19568e7 q^{94} +988442. q^{95} +1.31513e7 q^{96} +1.31931e7 q^{97} -2.47696e6 q^{98} +652928. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9}+O(q^{10}) \) 9 * q - 5 * q^2 - 26 * q^3 + 393 * q^4 - 181 * q^5 + 703 * q^6 - 3087 * q^7 + 1197 * q^8 + 3211 * q^9	\( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9} - 5124 q^{10} - 9826 q^{11} - 20919 q^{12} + 19773 q^{13} + 1715 q^{14} - 20346 q^{15} + 31113 q^{16} - 22766 q^{17} - 12978 q^{18} - 17769 q^{19} - 44204 q^{20} + 8918 q^{21} - 203553 q^{22} - 49103 q^{23} + 52737 q^{24} + 227466 q^{25} - 10985 q^{26} + 103624 q^{27} - 134799 q^{28} - 487455 q^{29} - 287992 q^{30} - 63843 q^{31} - 587099 q^{32} - 314392 q^{33} - 576240 q^{34} + 62083 q^{35} - 1514926 q^{36} - 796926 q^{37} - 766702 q^{38} - 57122 q^{39} - 2887296 q^{40} - 1567546 q^{41} - 241129 q^{42} - 277899 q^{43} - 1281195 q^{44} - 1650593 q^{45} - 1907445 q^{46} + 1077367 q^{47} - 1110835 q^{48} + 1058841 q^{49} - 267459 q^{50} - 3054368 q^{51} + 863421 q^{52} - 7322659 q^{53} - 3355387 q^{54} - 2613324 q^{55} - 410571 q^{56} - 3751946 q^{57} - 2992332 q^{58} - 169804 q^{59} - 2754416 q^{60} - 6352284 q^{61} + 6001087 q^{62} - 1101373 q^{63} + 1657017 q^{64} - 397657 q^{65} - 5962713 q^{66} + 921120 q^{67} + 5615224 q^{68} - 5202780 q^{69} + 1757532 q^{70} + 3786654 q^{71} + 2229758 q^{72} + 5792889 q^{73} - 1991961 q^{74} + 145628 q^{75} - 2806026 q^{76} + 3370318 q^{77} + 1544491 q^{78} + 3464037 q^{79} + 15422512 q^{80} - 5010363 q^{81} - 12539943 q^{82} + 6834945 q^{83} + 7175217 q^{84} + 3880662 q^{85} - 7977524 q^{86} + 3727078 q^{87} + 7013709 q^{88} - 20408371 q^{89} + 34910060 q^{90} - 6782139 q^{91} - 3544371 q^{92} + 3121742 q^{93} + 61343967 q^{94} + 3360807 q^{95} + 23547905 q^{96} + 41644125 q^{97} - 588245 q^{98} + 50754068 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 - 26 * q^3 + 393 * q^4 - 181 * q^5 + 703 * q^6 - 3087 * q^7 + 1197 * q^8 + 3211 * q^9 - 5124 * q^10 - 9826 * q^11 - 20919 * q^12 + 19773 * q^13 + 1715 * q^14 - 20346 * q^15 + 31113 * q^16 - 22766 * q^17 - 12978 * q^18 - 17769 * q^19 - 44204 * q^20 + 8918 * q^21 - 203553 * q^22 - 49103 * q^23 + 52737 * q^24 + 227466 * q^25 - 10985 * q^26 + 103624 * q^27 - 134799 * q^28 - 487455 * q^29 - 287992 * q^30 - 63843 * q^31 - 587099 * q^32 - 314392 * q^33 - 576240 * q^34 + 62083 * q^35 - 1514926 * q^36 - 796926 * q^37 - 766702 * q^38 - 57122 * q^39 - 2887296 * q^40 - 1567546 * q^41 - 241129 * q^42 - 277899 * q^43 - 1281195 * q^44 - 1650593 * q^45 - 1907445 * q^46 + 1077367 * q^47 - 1110835 * q^48 + 1058841 * q^49 - 267459 * q^50 - 3054368 * q^51 + 863421 * q^52 - 7322659 * q^53 - 3355387 * q^54 - 2613324 * q^55 - 410571 * q^56 - 3751946 * q^57 - 2992332 * q^58 - 169804 * q^59 - 2754416 * q^60 - 6352284 * q^61 + 6001087 * q^62 - 1101373 * q^63 + 1657017 * q^64 - 397657 * q^65 - 5962713 * q^66 + 921120 * q^67 + 5615224 * q^68 - 5202780 * q^69 + 1757532 * q^70 + 3786654 * q^71 + 2229758 * q^72 + 5792889 * q^73 - 1991961 * q^74 + 145628 * q^75 - 2806026 * q^76 + 3370318 * q^77 + 1544491 * q^78 + 3464037 * q^79 + 15422512 * q^80 - 5010363 * q^81 - 12539943 * q^82 + 6834945 * q^83 + 7175217 * q^84 + 3880662 * q^85 - 7977524 * q^86 + 3727078 * q^87 + 7013709 * q^88 - 20408371 * q^89 + 34910060 * q^90 - 6782139 * q^91 - 3544371 * q^92 + 3121742 * q^93 + 61343967 * q^94 + 3360807 * q^95 + 23547905 * q^96 + 41644125 * q^97 - 588245 * q^98 + 50754068 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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