LMFDB - Embedded newform 354.8.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,8,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, 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("354.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	354.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:	$110.584299021$
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} - 260588 x^{7} - 2627755 x^{6} + 16696953355 x^{5} + 808091684078 x^{4} + \cdots + 15\!\cdots\!00 $ x^9 - 4x^8 - 260588x^7 - 2627755x^6 + 16696953355x^5 + 808091684078x^4 - 191825159847522x^3 - 12153827093891319x^2 + 237398587398357570x + 15300107670464479800
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5}\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$115.905$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -290.312 q^{5} +216.000 q^{6} +1712.32 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -290.312 q^{5} +216.000 q^{6} +1712.32 q^{7} -512.000 q^{8} +729.000 q^{9} +2322.49 q^{10} -4936.81 q^{11} -1728.00 q^{12} +5777.55 q^{13} -13698.5 q^{14} +7838.41 q^{15} +4096.00 q^{16} +7502.10 q^{17} -5832.00 q^{18} -23867.1 q^{19} -18579.9 q^{20} -46232.5 q^{21} +39494.5 q^{22} -41148.2 q^{23} +13824.0 q^{24} +6155.85 q^{25} -46220.4 q^{26} -19683.0 q^{27} +109588. q^{28} +22509.3 q^{29} -62707.3 q^{30} -127359. q^{31} -32768.0 q^{32} +133294. q^{33} -60016.8 q^{34} -497105. q^{35} +46656.0 q^{36} +606305. q^{37} +190937. q^{38} -155994. q^{39} +148640. q^{40} +102990. q^{41} +369860. q^{42} -266332. q^{43} -315956. q^{44} -211637. q^{45} +329186. q^{46} -761237. q^{47} -110592. q^{48} +2.10848e6 q^{49} -49246.8 q^{50} -202557. q^{51} +369763. q^{52} +647979. q^{53} +157464. q^{54} +1.43321e6 q^{55} -876706. q^{56} +644411. q^{57} -180075. q^{58} +205379. q^{59} +501659. q^{60} +653167. q^{61} +1.01888e6 q^{62} +1.24828e6 q^{63} +262144. q^{64} -1.67729e6 q^{65} -1.06635e6 q^{66} +3.50363e6 q^{67} +480134. q^{68} +1.11100e6 q^{69} +3.97684e6 q^{70} +1.00412e6 q^{71} -373248. q^{72} -2.30877e6 q^{73} -4.85044e6 q^{74} -166208. q^{75} -1.52749e6 q^{76} -8.45338e6 q^{77} +1.24795e6 q^{78} -27464.1 q^{79} -1.18912e6 q^{80} +531441. q^{81} -823919. q^{82} -5.20478e6 q^{83} -2.95888e6 q^{84} -2.17795e6 q^{85} +2.13066e6 q^{86} -607752. q^{87} +2.52765e6 q^{88} +2.91991e6 q^{89} +1.69310e6 q^{90} +9.89298e6 q^{91} -2.63348e6 q^{92} +3.43871e6 q^{93} +6.08990e6 q^{94} +6.92889e6 q^{95} +884736. q^{96} -1.11246e7 q^{97} -1.68679e7 q^{98} -3.59894e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 72 q^{2} - 243 q^{3} + 576 q^{4} - 230 q^{5} + 1944 q^{6} - 340 q^{7} - 4608 q^{8} + 6561 q^{9}+O(q^{10}) $ 9 * q - 72 * q^2 - 243 * q^3 + 576 * q^4 - 230 * q^5 + 1944 * q^6 - 340 * q^7 - 4608 * q^8 + 6561 * q^9	\( 9 q - 72 q^{2} - 243 q^{3} + 576 q^{4} - 230 q^{5} + 1944 q^{6} - 340 q^{7} - 4608 q^{8} + 6561 q^{9} + 1840 q^{10} - 5472 q^{11} - 15552 q^{12} + 3144 q^{13} + 2720 q^{14} + 6210 q^{15} + 36864 q^{16} - 3662 q^{17} - 52488 q^{18} + 692 q^{19} - 14720 q^{20} + 9180 q^{21} + 43776 q^{22} - 92046 q^{23} + 124416 q^{24} + 74731 q^{25} - 25152 q^{26} - 177147 q^{27} - 21760 q^{28} - 41060 q^{29} - 49680 q^{30} - 324504 q^{31} - 294912 q^{32} + 147744 q^{33} + 29296 q^{34} - 415602 q^{35} + 419904 q^{36} + 338612 q^{37} - 5536 q^{38} - 84888 q^{39} + 117760 q^{40} - 104312 q^{41} - 73440 q^{42} + 1000602 q^{43} - 350208 q^{44} - 167670 q^{45} + 736368 q^{46} - 365148 q^{47} - 995328 q^{48} + 2307505 q^{49} - 597848 q^{50} + 98874 q^{51} + 201216 q^{52} + 2017498 q^{53} + 1417176 q^{54} + 1120520 q^{55} + 174080 q^{56} - 18684 q^{57} + 328480 q^{58} + 1848411 q^{59} + 397440 q^{60} + 5102340 q^{61} + 2596032 q^{62} - 247860 q^{63} + 2359296 q^{64} + 7512810 q^{65} - 1181952 q^{66} + 10920464 q^{67} - 234368 q^{68} + 2485242 q^{69} + 3324816 q^{70} + 3607024 q^{71} - 3359232 q^{72} + 12949418 q^{73} - 2708896 q^{74} - 2017737 q^{75} + 44288 q^{76} + 7127994 q^{77} + 679104 q^{78} + 7489472 q^{79} - 942080 q^{80} + 4782969 q^{81} + 834496 q^{82} + 2760502 q^{83} + 587520 q^{84} + 11815354 q^{85} - 8004816 q^{86} + 1108620 q^{87} + 2801664 q^{88} - 9948196 q^{89} + 1341360 q^{90} + 19400656 q^{91} - 5890944 q^{92} + 8761608 q^{93} + 2921184 q^{94} + 24045208 q^{95} + 7962624 q^{96} + 38157642 q^{97} - 18460040 q^{98} - 3989088 q^{99}+O(q^{100}) \) 9 * q - 72 * q^2 - 243 * q^3 + 576 * q^4 - 230 * q^5 + 1944 * q^6 - 340 * q^7 - 4608 * q^8 + 6561 * q^9 + 1840 * q^10 - 5472 * q^11 - 15552 * q^12 + 3144 * q^13 + 2720 * q^14 + 6210 * q^15 + 36864 * q^16 - 3662 * q^17 - 52488 * q^18 + 692 * q^19 - 14720 * q^20 + 9180 * q^21 + 43776 * q^22 - 92046 * q^23 + 124416 * q^24 + 74731 * q^25 - 25152 * q^26 - 177147 * q^27 - 21760 * q^28 - 41060 * q^29 - 49680 * q^30 - 324504 * q^31 - 294912 * q^32 + 147744 * q^33 + 29296 * q^34 - 415602 * q^35 + 419904 * q^36 + 338612 * q^37 - 5536 * q^38 - 84888 * q^39 + 117760 * q^40 - 104312 * q^41 - 73440 * q^42 + 1000602 * q^43 - 350208 * q^44 - 167670 * q^45 + 736368 * q^46 - 365148 * q^47 - 995328 * q^48 + 2307505 * q^49 - 597848 * q^50 + 98874 * q^51 + 201216 * q^52 + 2017498 * q^53 + 1417176 * q^54 + 1120520 * q^55 + 174080 * q^56 - 18684 * q^57 + 328480 * q^58 + 1848411 * q^59 + 397440 * q^60 + 5102340 * q^61 + 2596032 * q^62 - 247860 * q^63 + 2359296 * q^64 + 7512810 * q^65 - 1181952 * q^66 + 10920464 * q^67 - 234368 * q^68 + 2485242 * q^69 + 3324816 * q^70 + 3607024 * q^71 - 3359232 * q^72 + 12949418 * q^73 - 2708896 * q^74 - 2017737 * q^75 + 44288 * q^76 + 7127994 * q^77 + 679104 * q^78 + 7489472 * q^79 - 942080 * q^80 + 4782969 * q^81 + 834496 * q^82 + 2760502 * q^83 + 587520 * q^84 + 11815354 * q^85 - 8004816 * q^86 + 1108620 * q^87 + 2801664 * q^88 - 9948196 * q^89 + 1341360 * q^90 + 19400656 * q^91 - 5890944 * q^92 + 8761608 * q^93 + 2921184 * q^94 + 24045208 * q^95 + 7962624 * q^96 + 38157642 * q^97 - 18460040 * q^98 - 3989088 * 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
$5$	−290.312	−1.03865	−0.519325	−	0.854577i	$-0.673817\pi$
$5$	−290.312	−1.03865	−0.519325	+	0.854577i	$0.673817\pi$
$6$	216.000	0.408248
$7$	1712.32	1.88686	0.943432	−	0.331565i	$-0.107577\pi$
$7$	1712.32	1.88686	0.943432	+	0.331565i	$0.107577\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	2322.49	0.734437
$11$	−4936.81	−1.11834	−0.559168	−	0.829054i	$-0.688879\pi$
$11$	−4936.81	−1.11834	−0.559168	+	0.829054i	$0.688879\pi$
$12$	−1728.00	−0.288675
$13$	5777.55	0.729360	0.364680	−	0.931133i	$-0.381178\pi$
$13$	5777.55	0.729360	0.364680	+	0.931133i	$0.381178\pi$
$14$	−13698.5	−1.33421
$15$	7838.41	0.599665
$16$	4096.00	0.250000
$17$	7502.10	0.370349	0.185175	−	0.982706i	$-0.440715\pi$
$17$	7502.10	0.370349	0.185175	+	0.982706i	$0.440715\pi$
$18$	−5832.00	−0.235702
$19$	−23867.1	−0.798292	−0.399146	−	0.916887i	$-0.630693\pi$
$19$	−23867.1	−0.798292	−0.399146	+	0.916887i	$0.630693\pi$
$20$	−18579.9	−0.519325
$21$	−46232.5	−1.08938
$22$	39494.5	0.790783
$23$	−41148.2	−0.705186	−0.352593	−	0.935777i	$-0.614700\pi$
$23$	−41148.2	−0.705186	−0.352593	+	0.935777i	$0.614700\pi$
$24$	13824.0	0.204124
$25$	6155.85	0.0787949
$26$	−46220.4	−0.515735
$27$	−19683.0	−0.192450
$28$	109588.	0.943432
$29$	22509.3	0.171384	0.0856919	−	0.996322i	$-0.472690\pi$
$29$	22509.3	0.171384	0.0856919	+	0.996322i	$0.472690\pi$
$30$	−62707.3	−0.424027
$31$	−127359.	−0.767830	−0.383915	−	0.923368i	$-0.625425\pi$
$31$	−127359.	−0.767830	−0.383915	+	0.923368i	$0.625425\pi$
$32$	−32768.0	−0.176777
$33$	133294.	0.645671
$34$	−60016.8	−0.261877
$35$	−497105.	−1.95979
$36$	46656.0	0.166667
$37$	606305.	1.96782	0.983910	−	0.178665i	$-0.0571777\pi$
$37$	606305.	1.96782	0.983910	+	0.178665i	$0.0571777\pi$
$38$	190937.	0.564478
$39$	−155994.	−0.421096
$40$	148640.	0.367218
$41$	102990.	0.233373	0.116687	−	0.993169i	$-0.462773\pi$
$41$	102990.	0.233373	0.116687	+	0.993169i	$0.462773\pi$
$42$	369860.	0.770309
$43$	−266332.	−0.510839	−0.255420	−	0.966830i	$-0.582214\pi$
$43$	−266332.	−0.510839	−0.255420	+	0.966830i	$0.582214\pi$
$44$	−315956.	−0.559168
$45$	−211637.	−0.346217
$46$	329186.	0.498642
$47$	−761237.	−1.06949	−0.534746	−	0.845013i	$-0.679593\pi$
$47$	−761237.	−1.06949	−0.534746	+	0.845013i	$0.679593\pi$
$48$	−110592.	−0.144338
$49$	2.10848e6	2.56026
$50$	−49246.8	−0.0557164
$51$	−202557.	−0.213821
$52$	369763.	0.364680
$53$	647979.	0.597855	0.298927	−	0.954276i	$-0.403371\pi$
$53$	647979.	0.597855	0.298927	+	0.954276i	$0.403371\pi$
$54$	157464.	0.136083
$55$	1.43321e6	1.16156
$56$	−876706.	−0.667107
$57$	644411.	0.460894
$58$	−180075.	−0.121187
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	501659.	0.299833
$61$	653167.	0.368443	0.184221	−	0.982885i	$-0.441024\pi$
$61$	653167.	0.368443	0.184221	+	0.982885i	$0.441024\pi$
$62$	1.01888e6	0.542938
$63$	1.24828e6	0.628955
$64$	262144.	0.125000
$65$	−1.67729e6	−0.757550
$66$	−1.06635e6	−0.456559
$67$	3.50363e6	1.42317	0.711584	−	0.702601i	$-0.247978\pi$
$67$	3.50363e6	1.42317	0.711584	+	0.702601i	$0.247978\pi$
$68$	480134.	0.185175
$69$	1.11100e6	0.407139
$70$	3.97684e6	1.38578
$71$	1.00412e6	0.332952	0.166476	−	0.986046i	$-0.446761\pi$
$71$	1.00412e6	0.332952	0.166476	+	0.986046i	$0.446761\pi$
$72$	−373248.	−0.117851
$73$	−2.30877e6	−0.694626	−0.347313	−	0.937749i	$-0.612906\pi$
$73$	−2.30877e6	−0.694626	−0.347313	+	0.937749i	$0.612906\pi$
$74$	−4.85044e6	−1.39146
$75$	−166208.	−0.0454923
$76$	−1.52749e6	−0.399146
$77$	−8.45338e6	−2.11015
$78$	1.24795e6	0.297760
$79$	−27464.1	−0.00626717	−0.00313358	−	0.999995i	$-0.500997\pi$
$79$	−27464.1	−0.00626717	−0.00313358	+	0.999995i	$0.500997\pi$
$80$	−1.18912e6	−0.259663
$81$	531441.	0.111111
$82$	−823919.	−0.165020
$83$	−5.20478e6	−0.999147	−0.499574	−	0.866272i	$-0.666510\pi$
$83$	−5.20478e6	−0.999147	−0.499574	+	0.866272i	$0.666510\pi$
$84$	−2.95888e6	−0.544691
$85$	−2.17795e6	−0.384664
$86$	2.13066e6	0.361218
$87$	−607752.	−0.0989484
$88$	2.52765e6	0.395391
$89$	2.91991e6	0.439040	0.219520	−	0.975608i	$-0.429551\pi$
$89$	2.91991e6	0.439040	0.219520	+	0.975608i	$0.429551\pi$
$90$	1.69310e6	0.244812
$91$	9.89298e6	1.37620
$92$	−2.63348e6	−0.352593
$93$	3.43871e6	0.443307
$94$	6.08990e6	0.756245
$95$	6.92889e6	0.829146
$96$	884736.	0.102062
$97$	−1.11246e7	−1.23761	−0.618805	−	0.785545i	$-0.712383\pi$
$97$	−1.11246e7	−1.23761	−0.618805	+	0.785545i	$0.712383\pi$
$98$	−1.68679e7	−1.81038
$99$	−3.59894e6	−0.372779
$100$	393975.	0.0393975
$101$	1.62194e7	1.56643	0.783215	−	0.621751i	$-0.213578\pi$
$101$	1.62194e7	1.56643	0.783215	+	0.621751i	$0.213578\pi$
$102$	1.62045e6	0.151194
$103$	−1.49629e7	−1.34923	−0.674614	−	0.738171i	$-0.735690\pi$
$103$	−1.49629e7	−1.34923	−0.674614	+	0.738171i	$0.735690\pi$
$104$	−2.95810e6	−0.257868
$105$	1.34218e7	1.13149
$106$	−5.18383e6	−0.422747
$107$	8.30685e6	0.655531	0.327765	−	0.944759i	$-0.393704\pi$
$107$	8.30685e6	0.655531	0.327765	+	0.944759i	$0.393704\pi$
$108$	−1.25971e6	−0.0962250
$109$	1.84373e7	1.36366	0.681829	−	0.731512i	$-0.261185\pi$
$109$	1.84373e7	1.36366	0.681829	+	0.731512i	$0.261185\pi$
$110$	−1.14657e7	−0.821347
$111$	−1.63702e7	−1.13612
$112$	7.01365e6	0.471716
$113$	−1.56664e7	−1.02140	−0.510698	−	0.859760i	$-0.670613\pi$
$113$	−1.56664e7	−1.02140	−0.510698	+	0.859760i	$0.670613\pi$
$114$	−5.15529e6	−0.325901
$115$	1.19458e7	0.732441
$116$	1.44060e6	0.0856919
$117$	4.21183e6	0.243120
$118$	−1.64303e6	−0.0920575
$119$	1.28460e7	0.698799
$120$	−4.01327e6	−0.212014
$121$	4.88494e6	0.250675
$122$	−5.22534e6	−0.260528
$123$	−2.78073e6	−0.134738
$124$	−8.15101e6	−0.383915
$125$	2.08935e7	0.956810
$126$	−9.98623e6	−0.444738
$127$	1.02556e7	0.444273	0.222137	−	0.975016i	$-0.428697\pi$
$127$	1.02556e7	0.444273	0.222137	+	0.975016i	$0.428697\pi$
$128$	−2.09715e6	−0.0883883
$129$	7.19097e6	0.294933
$130$	1.34183e7	0.535669
$131$	−2.23485e6	−0.0868559	−0.0434280	−	0.999057i	$-0.513828\pi$
$131$	−2.23485e6	−0.0868559	−0.0434280	+	0.999057i	$0.513828\pi$
$132$	8.53081e6	0.322836
$133$	−4.08680e7	−1.50627
$134$	−2.80290e7	−1.00633
$135$	5.71420e6	0.199888
$136$	−3.84107e6	−0.130938
$137$	−6.35937e6	−0.211297	−0.105648	−	0.994404i	$-0.533692\pi$
$137$	−6.35937e6	−0.211297	−0.105648	+	0.994404i	$0.533692\pi$
$138$	−8.88801e6	−0.287891
$139$	2.62070e7	0.827686	0.413843	−	0.910348i	$-0.364186\pi$
$139$	2.62070e7	0.827686	0.413843	+	0.910348i	$0.364186\pi$
$140$	−3.18147e7	−0.979896
$141$	2.05534e7	0.617471
$142$	−8.03296e6	−0.235433
$143$	−2.85227e7	−0.815669
$144$	2.98598e6	0.0833333
$145$	−6.53472e6	−0.178008
$146$	1.84702e7	0.491174
$147$	−5.69290e7	−1.47817
$148$	3.88035e7	0.983910
$149$	−1.62825e7	−0.403245	−0.201622	−	0.979463i	$-0.564621\pi$
$149$	−1.62825e7	−0.403245	−0.201622	+	0.979463i	$0.564621\pi$
$150$	1.32966e6	0.0321679
$151$	−5.20400e7	−1.23004	−0.615018	−	0.788513i	$-0.710851\pi$
$151$	−5.20400e7	−1.23004	−0.615018	+	0.788513i	$0.710851\pi$
$152$	1.22199e7	0.282239
$153$	5.46903e6	0.123450
$154$	6.76270e7	1.49210
$155$	3.69739e7	0.797507
$156$	−9.98360e6	−0.210548
$157$	−2.53445e7	−0.522679	−0.261339	−	0.965247i	$-0.584164\pi$
$157$	−2.53445e7	−0.522679	−0.261339	+	0.965247i	$0.584164\pi$
$158$	219713.	0.00443156
$159$	−1.74954e7	−0.345171
$160$	9.51293e6	0.183609
$161$	−7.04587e7	−1.33059
$162$	−4.25153e6	−0.0785674
$163$	−4.27187e7	−0.772613	−0.386306	−	0.922371i	$-0.626249\pi$
$163$	−4.27187e7	−0.772613	−0.386306	+	0.922371i	$0.626249\pi$
$164$	6.59135e6	0.116687
$165$	−3.86968e7	−0.670627
$166$	4.16383e7	0.706504
$167$	−6.37927e7	−1.05990	−0.529948	−	0.848030i	$-0.677789\pi$
$167$	−6.37927e7	−1.05990	−0.529948	+	0.848030i	$0.677789\pi$
$168$	2.36711e7	0.385155
$169$	−2.93685e7	−0.468035
$170$	1.74236e7	0.271998
$171$	−1.73991e7	−0.266097
$172$	−1.70453e7	−0.255420
$173$	−5.19069e7	−0.762190	−0.381095	−	0.924536i	$-0.624453\pi$
$173$	−5.19069e7	−0.762190	−0.381095	+	0.924536i	$0.624453\pi$
$174$	4.86202e6	0.0699671
$175$	1.05408e7	0.148675
$176$	−2.02212e7	−0.279584
$177$	−5.54523e6	−0.0751646
$178$	−2.33593e7	−0.310448
$179$	−1.37243e8	−1.78856	−0.894282	−	0.447504i	$-0.852313\pi$
$179$	−1.37243e8	−1.78856	−0.894282	+	0.447504i	$0.852313\pi$
$180$	−1.35448e7	−0.173108
$181$	−1.12075e8	−1.40487	−0.702434	−	0.711749i	$-0.747903\pi$
$181$	−1.12075e8	−1.40487	−0.702434	+	0.711749i	$0.747903\pi$
$182$	−7.91439e7	−0.973122
$183$	−1.76355e7	−0.212720
$184$	2.10679e7	0.249321
$185$	−1.76018e8	−2.04388
$186$	−2.75096e7	−0.313465
$187$	−3.70365e7	−0.414175
$188$	−4.87192e7	−0.534746
$189$	−3.37035e7	−0.363127
$190$	−5.54311e7	−0.586295
$191$	1.39395e8	1.44754	0.723771	−	0.690040i	$-0.242407\pi$
$191$	1.39395e8	1.44754	0.723771	+	0.690040i	$0.242407\pi$
$192$	−7.07789e6	−0.0721688
$193$	−7.72083e7	−0.773060	−0.386530	−	0.922277i	$-0.626326\pi$
$193$	−7.72083e7	−0.773060	−0.386530	+	0.922277i	$0.626326\pi$
$194$	8.89969e7	0.875122
$195$	4.52868e7	0.437372
$196$	1.34943e8	1.28013
$197$	−3.10300e7	−0.289168	−0.144584	−	0.989493i	$-0.546184\pi$
$197$	−3.10300e7	−0.289168	−0.144584	+	0.989493i	$0.546184\pi$
$198$	2.87915e7	0.263594
$199$	−1.16622e8	−1.04904	−0.524522	−	0.851397i	$-0.675756\pi$
$199$	−1.16622e8	−1.04904	−0.524522	+	0.851397i	$0.675756\pi$
$200$	−3.15180e6	−0.0278582
$201$	−9.45980e7	−0.821667
$202$	−1.29755e8	−1.10763
$203$	3.85431e7	0.323378
$204$	−1.29636e7	−0.106911
$205$	−2.98992e7	−0.242393
$206$	1.19703e8	0.954048
$207$	−2.99970e7	−0.235062
$208$	2.36648e7	0.182340
$209$	1.17827e8	0.892758
$210$	−1.07375e8	−0.800082
$211$	−1.42957e8	−1.04765	−0.523827	−	0.851825i	$-0.675496\pi$
$211$	−1.42957e8	−1.04765	−0.523827	+	0.851825i	$0.675496\pi$
$212$	4.14707e7	0.298927
$213$	−2.71112e7	−0.192230
$214$	−6.64548e7	−0.463530
$215$	7.73194e7	0.530583
$216$	1.00777e7	0.0680414
$217$	−2.18080e8	−1.44879
$218$	−1.47499e8	−0.964252
$219$	6.23368e7	0.401042
$220$	9.17257e7	0.580780
$221$	4.33437e7	0.270118
$222$	1.30962e8	0.803359
$223$	3.94168e7	0.238021	0.119010	−	0.992893i	$-0.462028\pi$
$223$	3.94168e7	0.238021	0.119010	+	0.992893i	$0.462028\pi$
$224$	−5.61092e7	−0.333554
$225$	4.48762e6	0.0262650
$226$	1.25331e8	0.722237
$227$	3.40328e7	0.193111	0.0965554	−	0.995328i	$-0.469217\pi$
$227$	3.40328e7	0.193111	0.0965554	+	0.995328i	$0.469217\pi$
$228$	4.12423e7	0.230447
$229$	1.38804e8	0.763796	0.381898	−	0.924205i	$-0.375271\pi$
$229$	1.38804e8	0.763796	0.381898	+	0.924205i	$0.375271\pi$
$230$	−9.55664e7	−0.517914
$231$	2.28241e8	1.21829
$232$	−1.15248e7	−0.0605933
$233$	−1.84744e8	−0.956806	−0.478403	−	0.878140i	$-0.658784\pi$
$233$	−1.84744e8	−0.956806	−0.478403	+	0.878140i	$0.658784\pi$
$234$	−3.36946e7	−0.171912
$235$	2.20996e8	1.11083
$236$	1.31443e7	0.0650945
$237$	741532.	0.00361835
$238$	−1.02768e8	−0.494125
$239$	1.80231e8	0.853958	0.426979	−	0.904262i	$-0.359578\pi$
$239$	1.80231e8	0.853958	0.426979	+	0.904262i	$0.359578\pi$
$240$	3.21061e7	0.149916
$241$	−1.02814e8	−0.473142	−0.236571	−	0.971614i	$-0.576024\pi$
$241$	−1.02814e8	−0.473142	−0.236571	+	0.971614i	$0.576024\pi$
$242$	−3.90795e7	−0.177254
$243$	−1.43489e7	−0.0641500
$244$	4.18027e7	0.184221
$245$	−6.12117e8	−2.65921
$246$	2.22458e7	0.0952742
$247$	−1.37893e8	−0.582242
$248$	6.52080e7	0.271469
$249$	1.40529e8	0.576858
$250$	−1.67148e8	−0.676567
$251$	−2.62299e7	−0.104698	−0.0523489	−	0.998629i	$-0.516671\pi$
$251$	−2.62299e7	−0.104698	−0.0523489	+	0.998629i	$0.516671\pi$
$252$	7.98898e7	0.314477
$253$	2.03141e8	0.788634
$254$	−8.20452e7	−0.314149
$255$	5.88046e7	0.222086
$256$	1.67772e7	0.0625000
$257$	3.01471e8	1.10785	0.553924	−	0.832568i	$-0.313130\pi$
$257$	3.01471e8	1.10785	0.553924	+	0.832568i	$0.313130\pi$
$258$	−5.75278e7	−0.208549
$259$	1.03819e9	3.71301
$260$	−1.07346e8	−0.378775
$261$	1.64093e7	0.0571279
$262$	1.78788e7	0.0614164
$263$	−1.84218e8	−0.624434	−0.312217	−	0.950011i	$-0.601072\pi$
$263$	−1.84218e8	−0.624434	−0.312217	+	0.950011i	$0.601072\pi$
$264$	−6.82465e7	−0.228279
$265$	−1.88116e8	−0.620962
$266$	3.26944e8	1.06509
$267$	−7.88376e7	−0.253480
$268$	2.24232e8	0.711584
$269$	−2.20973e8	−0.692159	−0.346079	−	0.938205i	$-0.612487\pi$
$269$	−2.20973e8	−0.692159	−0.346079	+	0.938205i	$0.612487\pi$
$270$	−4.57136e7	−0.141342
$271$	2.99895e7	0.0915328	0.0457664	−	0.998952i	$-0.485427\pi$
$271$	2.99895e7	0.0915328	0.0457664	+	0.998952i	$0.485427\pi$
$272$	3.07286e7	0.0925873
$273$	−2.67111e8	−0.794551
$274$	5.08750e7	0.149409
$275$	−3.03903e7	−0.0881192
$276$	7.11041e7	0.203570
$277$	4.62522e8	1.30754	0.653768	−	0.756695i	$-0.273187\pi$
$277$	4.62522e8	1.30754	0.653768	+	0.756695i	$0.273187\pi$
$278$	−2.09656e8	−0.585262
$279$	−9.28451e7	−0.255943
$280$	2.54518e8	0.692891
$281$	−3.53901e8	−0.951503	−0.475752	−	0.879580i	$-0.657824\pi$
$281$	−3.53901e8	−0.951503	−0.475752	+	0.879580i	$0.657824\pi$
$282$	−1.64427e8	−0.436618
$283$	−2.04042e8	−0.535141	−0.267570	−	0.963538i	$-0.586221\pi$
$283$	−2.04042e8	−0.535141	−0.267570	+	0.963538i	$0.586221\pi$
$284$	6.42637e7	0.166476
$285$	−1.87080e8	−0.478708
$286$	2.28181e8	0.576765
$287$	1.76351e8	0.440344
$288$	−2.38879e7	−0.0589256
$289$	−3.54057e8	−0.862841
$290$	5.22778e7	0.125871
$291$	3.00364e8	0.714534
$292$	−1.47761e8	−0.347313
$293$	2.21064e8	0.513431	0.256716	−	0.966487i	$-0.417360\pi$
$293$	2.21064e8	0.513431	0.256716	+	0.966487i	$0.417360\pi$
$294$	4.55432e8	1.04522
$295$	−5.96239e7	−0.135221
$296$	−3.10428e8	−0.695729
$297$	9.71713e7	0.215224
$298$	1.30260e8	0.285137
$299$	−2.37736e8	−0.514334
$300$	−1.06373e7	−0.0227461
$301$	−4.56045e8	−0.963884
$302$	4.16320e8	0.869767
$303$	−4.37925e8	−0.904379
$304$	−9.77595e7	−0.199573
$305$	−1.89622e8	−0.382683
$306$	−4.37522e7	−0.0872922
$307$	−5.17011e8	−1.01980	−0.509901	−	0.860233i	$-0.670318\pi$
$307$	−5.17011e8	−1.01980	−0.509901	+	0.860233i	$0.670318\pi$
$308$	−5.41016e8	−1.05507
$309$	4.03998e8	0.778977
$310$	−2.95792e8	−0.563923
$311$	−1.42381e8	−0.268404	−0.134202	−	0.990954i	$-0.542847\pi$
$311$	−1.42381e8	−0.268404	−0.134202	+	0.990954i	$0.542847\pi$
$312$	7.98688e7	0.148880
$313$	−1.27567e8	−0.235144	−0.117572	−	0.993064i	$-0.537511\pi$
$313$	−1.27567e8	−0.235144	−0.117572	+	0.993064i	$0.537511\pi$
$314$	2.02756e8	0.369590
$315$	−3.62390e8	−0.653264
$316$	−1.75771e6	−0.00313358
$317$	−5.44766e8	−0.960512	−0.480256	−	0.877128i	$-0.659456\pi$
$317$	−5.44766e8	−0.960512	−0.480256	+	0.877128i	$0.659456\pi$
$318$	1.39964e8	0.244073
$319$	−1.11124e8	−0.191665
$320$	−7.61035e7	−0.129831
$321$	−2.24285e8	−0.378471
$322$	5.63670e8	0.940869
$323$	−1.79053e8	−0.295647
$324$	3.40122e7	0.0555556
$325$	3.55657e7	0.0574698
$326$	3.41750e8	0.546320
$327$	−4.97808e8	−0.787308
$328$	−5.27308e7	−0.0825099
$329$	−1.30348e9	−2.01799
$330$	3.09574e8	0.474205
$331$	−5.14779e8	−0.780230	−0.390115	−	0.920766i	$-0.627565\pi$
$331$	−5.14779e8	−0.780230	−0.390115	+	0.920766i	$0.627565\pi$
$332$	−3.33106e8	−0.499574
$333$	4.41997e8	0.655940
$334$	5.10342e8	0.749460
$335$	−1.01714e9	−1.47817
$336$	−1.89368e8	−0.272345
$337$	9.25446e8	1.31718	0.658592	−	0.752501i	$-0.271152\pi$
$337$	9.25446e8	1.31718	0.658592	+	0.752501i	$0.271152\pi$
$338$	2.34948e8	0.330950
$339$	4.22993e8	0.589704
$340$	−1.39389e8	−0.192332
$341$	6.28750e8	0.858692
$342$	1.39193e8	0.188159
$343$	2.20022e9	2.94399
$344$	1.36362e8	0.180609
$345$	−3.22537e8	−0.422875
$346$	4.15255e8	0.538950
$347$	1.11363e9	1.43083	0.715414	−	0.698701i	$-0.246238\pi$
$347$	1.11363e9	1.43083	0.715414	+	0.698701i	$0.246238\pi$
$348$	−3.88961e7	−0.0494742
$349$	−9.01255e8	−1.13490	−0.567451	−	0.823407i	$-0.692071\pi$
$349$	−9.01255e8	−1.13490	−0.567451	+	0.823407i	$0.692071\pi$
$350$	−8.43261e7	−0.105129
$351$	−1.13719e8	−0.140365
$352$	1.61769e8	0.197696
$353$	−4.93795e8	−0.597496	−0.298748	−	0.954332i	$-0.596569\pi$
$353$	−4.93795e8	−0.597496	−0.298748	+	0.954332i	$0.596569\pi$
$354$	4.43619e7	0.0531494
$355$	−2.91508e8	−0.345821
$356$	1.86874e8	0.219520
$357$	−3.46841e8	−0.403452
$358$	1.09794e9	1.26471
$359$	5.35574e8	0.610927	0.305463	−	0.952204i	$-0.401189\pi$
$359$	5.35574e8	0.610927	0.305463	+	0.952204i	$0.401189\pi$
$360$	1.08358e8	0.122406
$361$	−3.24234e8	−0.362730
$362$	8.96603e8	0.993391
$363$	−1.31893e8	−0.144727
$364$	6.33151e8	0.688101
$365$	6.70263e8	0.721473
$366$	1.41084e8	0.150416
$367$	−4.67055e8	−0.493216	−0.246608	−	0.969115i	$-0.579316\pi$
$367$	−4.67055e8	−0.493216	−0.246608	+	0.969115i	$0.579316\pi$
$368$	−1.68543e8	−0.176296
$369$	7.50796e7	0.0777911
$370$	1.40814e9	1.44524
$371$	1.10954e9	1.12807
$372$	2.20077e8	0.221654
$373$	1.20310e9	1.20038	0.600191	−	0.799857i	$-0.295091\pi$
$373$	1.20310e9	1.20038	0.600191	+	0.799857i	$0.295091\pi$
$374$	2.96292e8	0.292866
$375$	−5.64124e8	−0.552415
$376$	3.89753e8	0.378123
$377$	1.30049e8	0.125000
$378$	2.69628e8	0.256770
$379$	1.12650e9	1.06291	0.531453	−	0.847088i	$-0.321646\pi$
$379$	1.12650e9	1.06291	0.531453	+	0.847088i	$0.321646\pi$
$380$	4.43449e8	0.414573
$381$	−2.76902e8	−0.256501
$382$	−1.11516e9	−1.02357
$383$	1.67937e9	1.52739	0.763695	−	0.645577i	$-0.223383\pi$
$383$	1.67937e9	1.52739	0.763695	+	0.645577i	$0.223383\pi$
$384$	5.66231e7	0.0510310
$385$	2.45411e9	2.19171
$386$	6.17666e8	0.546636
$387$	−1.94156e8	−0.170280
$388$	−7.11975e8	−0.618805
$389$	4.79543e8	0.413051	0.206526	−	0.978441i	$-0.433784\pi$
$389$	4.79543e8	0.413051	0.206526	+	0.978441i	$0.433784\pi$
$390$	−3.62294e8	−0.309268
$391$	−3.08698e8	−0.261165
$392$	−1.07954e9	−0.905188
$393$	6.03410e7	0.0501463
$394$	2.48240e8	0.204473
$395$	7.97316e6	0.00650940
$396$	−2.30332e8	−0.186389
$397$	4.12452e8	0.330832	0.165416	−	0.986224i	$-0.447103\pi$
$397$	4.12452e8	0.330832	0.165416	+	0.986224i	$0.447103\pi$
$398$	9.32973e8	0.741786
$399$	1.10344e9	0.869644
$400$	2.52144e7	0.0196987
$401$	−7.77112e8	−0.601836	−0.300918	−	0.953650i	$-0.597293\pi$
$401$	−7.77112e8	−0.601836	−0.300918	+	0.953650i	$0.597293\pi$
$402$	7.56784e8	0.581006
$403$	−7.35825e8	−0.560024
$404$	1.03804e9	0.783215
$405$	−1.54284e8	−0.115406
$406$	−3.08345e8	−0.228663
$407$	−2.99322e9	−2.20068
$408$	1.03709e8	0.0755972
$409$	1.58108e9	1.14267	0.571337	−	0.820716i	$-0.306425\pi$
$409$	1.58108e9	1.14267	0.571337	+	0.820716i	$0.306425\pi$
$410$	2.39193e8	0.171398
$411$	1.71703e8	0.121992
$412$	−9.57625e8	−0.674614
$413$	3.51674e8	0.245649
$414$	2.39976e8	0.166214
$415$	1.51101e9	1.03776
$416$	−1.89319e8	−0.128934
$417$	−7.07589e8	−0.477864
$418$	−9.42618e8	−0.631275
$419$	−1.07284e9	−0.712501	−0.356251	−	0.934390i	$-0.615945\pi$
$419$	−1.07284e9	−0.712501	−0.356251	+	0.934390i	$0.615945\pi$
$420$	8.58998e8	0.565743
$421$	2.68767e9	1.75545	0.877727	−	0.479162i	$-0.159059\pi$
$421$	2.68767e9	1.75545	0.877727	+	0.479162i	$0.159059\pi$
$422$	1.14366e9	0.740803
$423$	−5.54942e8	−0.356497
$424$	−3.31765e8	−0.211373
$425$	4.61818e7	0.0291816
$426$	2.16890e8	0.135927
$427$	1.11843e9	0.695201
$428$	5.31638e8	0.327765
$429$	7.70112e8	0.470927
$430$	−6.18555e8	−0.375179
$431$	−3.00343e9	−1.80695	−0.903476	−	0.428638i	$-0.858994\pi$
$431$	−3.00343e9	−1.80695	−0.903476	+	0.428638i	$0.858994\pi$
$432$	−8.06216e7	−0.0481125
$433$	−3.32746e9	−1.96972	−0.984862	−	0.173340i	$-0.944544\pi$
$433$	−3.32746e9	−1.96972	−0.984862	+	0.173340i	$0.944544\pi$
$434$	1.74464e9	1.02445
$435$	1.76437e8	0.102773
$436$	1.17999e9	0.681829
$437$	9.82087e8	0.562944
$438$	−4.98694e8	−0.283580
$439$	3.45144e8	0.194704	0.0973519	−	0.995250i	$-0.468963\pi$
$439$	3.45144e8	0.194704	0.0973519	+	0.995250i	$0.468963\pi$
$440$	−7.33806e8	−0.410673
$441$	1.53708e9	0.853419
$442$	−3.46750e8	−0.191002
$443$	−2.09060e9	−1.14250	−0.571252	−	0.820775i	$-0.693542\pi$
$443$	−2.09060e9	−1.14250	−0.571252	+	0.820775i	$0.693542\pi$
$444$	−1.04770e9	−0.568061
$445$	−8.47684e8	−0.456010
$446$	−3.15335e8	−0.168306
$447$	4.39627e8	0.232814
$448$	4.48873e8	0.235858
$449$	−2.57803e9	−1.34408	−0.672041	−	0.740514i	$-0.734582\pi$
$449$	−2.57803e9	−1.34408	−0.672041	+	0.740514i	$0.734582\pi$
$450$	−3.59009e7	−0.0185721
$451$	−5.08442e8	−0.260990
$452$	−1.00265e9	−0.510698
$453$	1.40508e9	0.710162
$454$	−2.72262e8	−0.136550
$455$	−2.87205e9	−1.42939
$456$	−3.29938e8	−0.162951
$457$	1.10749e9	0.542793	0.271397	−	0.962468i	$-0.412514\pi$
$457$	1.10749e9	0.542793	0.271397	+	0.962468i	$0.412514\pi$
$458$	−1.11043e9	−0.540085
$459$	−1.47664e8	−0.0712738
$460$	7.64531e8	0.366221
$461$	−2.24079e9	−1.06524	−0.532620	−	0.846354i	$-0.678793\pi$
$461$	−2.24079e9	−1.06524	−0.532620	+	0.846354i	$0.678793\pi$
$462$	−1.82593e9	−0.861464
$463$	−3.02395e9	−1.41593	−0.707965	−	0.706248i	$-0.750386\pi$
$463$	−3.02395e9	−1.41593	−0.707965	+	0.706248i	$0.750386\pi$
$464$	9.21982e7	0.0428459
$465$	−9.98296e8	−0.460441
$466$	1.47795e9	0.676564
$467$	−1.43381e9	−0.651453	−0.325726	−	0.945464i	$-0.605609\pi$
$467$	−1.43381e9	−0.651453	−0.325726	+	0.945464i	$0.605609\pi$
$468$	2.69557e8	0.121560
$469$	5.99932e9	2.68533
$470$	−1.76797e9	−0.785474
$471$	6.84301e8	0.301769
$472$	−1.05154e8	−0.0460287
$473$	1.31483e9	0.571290
$474$	−5.93226e6	−0.00255856
$475$	−1.46922e8	−0.0629013
$476$	8.22142e8	0.349399
$477$	4.72377e8	0.199285
$478$	−1.44185e9	−0.603839
$479$	2.00912e9	0.835278	0.417639	−	0.908613i	$-0.362858\pi$
$479$	2.00912e9	0.835278	0.417639	+	0.908613i	$0.362858\pi$
$480$	−2.56849e8	−0.106007
$481$	3.50296e9	1.43525
$482$	8.22510e8	0.334562
$483$	1.90239e9	0.768216
$484$	3.12636e8	0.125337
$485$	3.22960e9	1.28544
$486$	1.14791e8	0.0453609
$487$	−4.20868e9	−1.65118	−0.825589	−	0.564271i	$-0.809157\pi$
$487$	−4.20868e9	−1.65118	−0.825589	+	0.564271i	$0.809157\pi$
$488$	−3.34422e8	−0.130264
$489$	1.15341e9	0.446068
$490$	4.89693e9	1.88035
$491$	4.36242e8	0.166319	0.0831595	−	0.996536i	$-0.473499\pi$
$491$	4.36242e8	0.166319	0.0831595	+	0.996536i	$0.473499\pi$
$492$	−1.77966e8	−0.0673690
$493$	1.68867e8	0.0634719
$494$	1.10315e9	0.411707
$495$	1.04481e9	0.387187
$496$	−5.21664e8	−0.191958
$497$	1.71937e9	0.628235
$498$	−1.12423e9	−0.407900
$499$	−2.56498e9	−0.924129	−0.462064	−	0.886846i	$-0.652891\pi$
$499$	−2.56498e9	−0.924129	−0.462064	+	0.886846i	$0.652891\pi$
$500$	1.33718e9	0.478405
$501$	1.72240e9	0.611931
$502$	2.09839e8	0.0740326
$503$	−1.06439e9	−0.372918	−0.186459	−	0.982463i	$-0.559701\pi$
$503$	−1.06439e9	−0.372918	−0.186459	+	0.982463i	$0.559701\pi$
$504$	−6.39118e8	−0.222369
$505$	−4.70869e9	−1.62697
$506$	−1.62513e9	−0.557649
$507$	7.92949e8	0.270220
$508$	6.56361e8	0.222137
$509$	2.02524e9	0.680712	0.340356	−	0.940297i	$-0.389452\pi$
$509$	2.02524e9	0.680712	0.340356	+	0.940297i	$0.389452\pi$
$510$	−4.70436e8	−0.157038
$511$	−3.95334e9	−1.31066
$512$	−1.34218e8	−0.0441942
$513$	4.69776e8	0.153631
$514$	−2.41177e9	−0.783366
$515$	4.34390e9	1.40138
$516$	4.60222e8	0.147467
$517$	3.75809e9	1.19605
$518$	−8.30549e9	−2.62549
$519$	1.40149e9	0.440051
$520$	8.58772e8	0.267834
$521$	−5.05258e9	−1.56524	−0.782620	−	0.622500i	$-0.786117\pi$
$521$	−5.05258e9	−1.56524	−0.782620	+	0.622500i	$0.786117\pi$
$522$	−1.31274e8	−0.0403955
$523$	−4.95225e9	−1.51372	−0.756862	−	0.653575i	$-0.773268\pi$
$523$	−4.95225e9	−1.51372	−0.756862	+	0.653575i	$0.773268\pi$
$524$	−1.43031e8	−0.0434280
$525$	−2.84601e8	−0.0858377
$526$	1.47374e9	0.441541
$527$	−9.55463e8	−0.284365
$528$	5.45972e8	0.161418
$529$	−1.71165e9	−0.502713
$530$	1.50493e9	0.439086
$531$	1.49721e8	0.0433963
$532$	−2.61555e9	−0.753134
$533$	5.95029e8	0.170213
$534$	6.30701e8	0.179237
$535$	−2.41157e9	−0.680867
$536$	−1.79386e9	−0.503166
$537$	3.70556e9	1.03263
$538$	1.76778e9	0.489430
$539$	−1.04092e10	−2.86323
$540$	3.65709e8	0.0999442
$541$	9.47260e8	0.257205	0.128602	−	0.991696i	$-0.458951\pi$
$541$	9.47260e8	0.257205	0.128602	+	0.991696i	$0.458951\pi$
$542$	−2.39916e8	−0.0647235
$543$	3.02604e9	0.811101
$544$	−2.45829e8	−0.0654691
$545$	−5.35257e9	−1.41636
$546$	2.13688e9	0.561832
$547$	−4.39957e9	−1.14936	−0.574678	−	0.818380i	$-0.694873\pi$
$547$	−4.39957e9	−1.14936	−0.574678	+	0.818380i	$0.694873\pi$
$548$	−4.07000e8	−0.105648
$549$	4.76159e8	0.122814
$550$	2.43122e8	0.0623097
$551$	−5.37232e8	−0.136814
$552$	−5.68833e8	−0.143945
$553$	−4.70273e7	−0.0118253
$554$	−3.70018e9	−0.924568
$555$	4.75247e9	1.18003
$556$	1.67725e9	0.413843
$557$	2.85632e9	0.700347	0.350173	−	0.936685i	$-0.386123\pi$
$557$	2.85632e9	0.700347	0.350173	+	0.936685i	$0.386123\pi$
$558$	7.42760e8	0.180979
$559$	−1.53875e9	−0.372585
$560$	−2.03614e9	−0.489948
$561$	9.99984e8	0.239124
$562$	2.83121e9	0.672814
$563$	8.22940e8	0.194352	0.0971760	−	0.995267i	$-0.469019\pi$
$563$	8.22940e8	0.194352	0.0971760	+	0.995267i	$0.469019\pi$
$564$	1.31542e9	0.308736
$565$	4.54814e9	1.06087
$566$	1.63234e9	0.378402
$567$	9.09995e8	0.209652
$568$	−5.14110e8	−0.117716
$569$	−6.43652e9	−1.46473	−0.732367	−	0.680911i	$-0.761584\pi$
$569$	−6.43652e9	−1.46473	−0.732367	+	0.680911i	$0.761584\pi$
$570$	1.49664e9	0.338497
$571$	−1.19244e9	−0.268047	−0.134024	−	0.990978i	$-0.542790\pi$
$571$	−1.19244e9	−0.268047	−0.134024	+	0.990978i	$0.542790\pi$
$572$	−1.82545e9	−0.407834
$573$	−3.76367e9	−0.835739
$574$	−1.41081e9	−0.311370
$575$	−2.53302e8	−0.0555650
$576$	1.91103e8	0.0416667
$577$	2.23293e9	0.483906	0.241953	−	0.970288i	$-0.422212\pi$
$577$	2.23293e9	0.483906	0.241953	+	0.970288i	$0.422212\pi$
$578$	2.83246e9	0.610121
$579$	2.08462e9	0.446326
$580$	−4.18222e8	−0.0890039
$581$	−8.91223e9	−1.88525
$582$	−2.40292e9	−0.505252
$583$	−3.19895e9	−0.668602
$584$	1.18209e9	0.245587
$585$	−1.22274e9	−0.252517
$586$	−1.76852e9	−0.363051
$587$	−5.14338e9	−1.04958	−0.524790	−	0.851232i	$-0.675856\pi$
$587$	−5.14338e9	−1.04958	−0.524790	+	0.851232i	$0.675856\pi$
$588$	−3.64346e9	−0.739083
$589$	3.03970e9	0.612953
$590$	4.76991e8	0.0956155
$591$	8.37811e8	0.166951
$592$	2.48343e9	0.491955
$593$	−4.90851e9	−0.966626	−0.483313	−	0.875448i	$-0.660567\pi$
$593$	−4.90851e9	−0.966626	−0.483313	+	0.875448i	$0.660567\pi$
$594$	−7.77370e8	−0.152186
$595$	−3.72933e9	−0.725808
$596$	−1.04208e9	−0.201622
$597$	3.14879e9	0.605666
$598$	1.90189e9	0.363689
$599$	4.66340e9	0.886561	0.443281	−	0.896383i	$-0.353815\pi$
$599$	4.66340e9	0.886561	0.443281	+	0.896383i	$0.353815\pi$
$600$	8.50985e7	0.0160839
$601$	−4.42093e9	−0.830717	−0.415359	−	0.909658i	$-0.636344\pi$
$601$	−4.42093e9	−0.830717	−0.415359	+	0.909658i	$0.636344\pi$
$602$	3.64836e9	0.681569
$603$	2.55415e9	0.474389
$604$	−3.33056e9	−0.615018
$605$	−1.41816e9	−0.260364
$606$	3.50340e9	0.639492
$607$	8.44365e9	1.53239	0.766196	−	0.642607i	$-0.222147\pi$
$607$	8.44365e9	1.53239	0.766196	+	0.642607i	$0.222147\pi$
$608$	7.82076e8	0.141119
$609$	−1.04066e9	−0.186702
$610$	1.51698e9	0.270598
$611$	−4.39808e9	−0.780044
$612$	3.50018e8	0.0617249
$613$	−7.08065e9	−1.24154	−0.620771	−	0.783992i	$-0.713180\pi$
$613$	−7.08065e9	−1.24154	−0.620771	+	0.783992i	$0.713180\pi$
$614$	4.13609e9	0.721108
$615$	8.07277e8	0.139946
$616$	4.32813e9	0.746050
$617$	2.50969e9	0.430152	0.215076	−	0.976597i	$-0.431000\pi$
$617$	2.50969e9	0.430152	0.215076	+	0.976597i	$0.431000\pi$
$618$	−3.23198e9	−0.550820
$619$	−2.07619e9	−0.351844	−0.175922	−	0.984404i	$-0.556291\pi$
$619$	−2.07619e9	−0.351844	−0.175922	+	0.984404i	$0.556291\pi$
$620$	2.36633e9	0.398754
$621$	8.09920e8	0.135713
$622$	1.13904e9	0.189790
$623$	4.99981e9	0.828410
$624$	−6.38950e8	−0.105274
$625$	−6.54655e9	−1.07259
$626$	1.02054e9	0.166272
$627$	−3.18134e9	−0.515434
$628$	−1.62205e9	−0.261339
$629$	4.54856e9	0.728781
$630$	2.89912e9	0.461928
$631$	7.52392e9	1.19218	0.596089	−	0.802918i	$-0.296720\pi$
$631$	7.52392e9	1.19218	0.596089	+	0.802918i	$0.296720\pi$
$632$	1.40616e7	0.00221578
$633$	3.85984e9	0.604863
$634$	4.35813e9	0.679185
$635$	−2.97733e9	−0.461445
$636$	−1.11971e9	−0.172586
$637$	1.21819e10	1.86735
$638$	8.88995e8	0.135527
$639$	7.32004e8	0.110984
$640$	6.08828e8	0.0918046
$641$	7.99817e9	1.19946	0.599732	−	0.800201i	$-0.295274\pi$
$641$	7.99817e9	1.19946	0.599732	+	0.800201i	$0.295274\pi$
$642$	1.79428e9	0.267619
$643$	−9.47088e9	−1.40492	−0.702461	−	0.711722i	$-0.747915\pi$
$643$	−9.47088e9	−1.40492	−0.702461	+	0.711722i	$0.747915\pi$
$644$	−4.50936e9	−0.665295
$645$	−2.08762e9	−0.306332
$646$	1.43243e9	0.209054
$647$	5.95727e9	0.864734	0.432367	−	0.901698i	$-0.357679\pi$
$647$	5.95727e9	0.864734	0.432367	+	0.901698i	$0.357679\pi$
$648$	−2.72098e8	−0.0392837
$649$	−1.01392e9	−0.145595
$650$	−2.84526e8	−0.0406373
$651$	5.88815e9	0.836460
$652$	−2.73400e9	−0.386306
$653$	8.13949e9	1.14393	0.571967	−	0.820277i	$-0.306181\pi$
$653$	8.13949e9	1.14393	0.571967	+	0.820277i	$0.306181\pi$
$654$	3.98246e9	0.556711
$655$	6.48804e8	0.0902129
$656$	4.21846e8	0.0583433
$657$	−1.68309e9	−0.231542
$658$	1.04278e10	1.42693
$659$	−1.19812e10	−1.63080	−0.815402	−	0.578895i	$-0.803484\pi$
$659$	−1.19812e10	−1.63080	−0.815402	+	0.578895i	$0.803484\pi$
$660$	−2.47659e9	−0.335313
$661$	−8.32952e9	−1.12180	−0.560899	−	0.827884i	$-0.689544\pi$
$661$	−8.32952e9	−1.12180	−0.560899	+	0.827884i	$0.689544\pi$
$662$	4.11823e9	0.551706
$663$	−1.17028e9	−0.155953
$664$	2.66485e9	0.353252
$665$	1.18644e10	1.56449
$666$	−3.53597e9	−0.463820
$667$	−9.26219e8	−0.120857
$668$	−4.08273e9	−0.529948
$669$	−1.06425e9	−0.137421
$670$	8.13715e9	1.04523
$671$	−3.22456e9	−0.412043
$672$	1.51495e9	0.192577
$673$	−6.61349e9	−0.836331	−0.418165	−	0.908371i	$-0.637327\pi$
$673$	−6.61349e9	−0.836331	−0.418165	+	0.908371i	$0.637327\pi$
$674$	−7.40356e9	−0.931389
$675$	−1.21166e8	−0.0151641
$676$	−1.87958e9	−0.234017
$677$	9.70461e9	1.20204	0.601018	−	0.799235i	$-0.294762\pi$
$677$	9.70461e9	1.20204	0.601018	+	0.799235i	$0.294762\pi$
$678$	−3.38394e9	−0.416984
$679$	−1.90488e10	−2.33520
$680$	1.11511e9	0.135999
$681$	−9.18885e8	−0.111493
$682$	−5.03000e9	−0.607187
$683$	−1.25269e10	−1.50443	−0.752216	−	0.658916i	$-0.771015\pi$
$683$	−1.25269e10	−1.50443	−0.752216	+	0.658916i	$0.771015\pi$
$684$	−1.11354e9	−0.133049
$685$	1.84620e9	0.219463
$686$	−1.76018e10	−2.08172
$687$	−3.74770e9	−0.440978
$688$	−1.09090e9	−0.127710
$689$	3.74373e9	0.436051
$690$	2.58029e9	0.299018
$691$	1.50928e9	0.174019	0.0870094	−	0.996207i	$-0.472269\pi$
$691$	1.50928e9	0.174019	0.0870094	+	0.996207i	$0.472269\pi$
$692$	−3.32204e9	−0.381095
$693$	−6.16251e9	−0.703383
$694$	−8.90903e9	−1.01175
$695$	−7.60820e9	−0.859676
$696$	3.11169e8	0.0349836
$697$	7.72640e8	0.0864296
$698$	7.21004e9	0.802497
$699$	4.98808e9	0.552412
$700$	6.74609e8	0.0743376
$701$	−9.02444e9	−0.989480	−0.494740	−	0.869041i	$-0.664737\pi$
$701$	−9.02444e9	−0.989480	−0.494740	+	0.869041i	$0.664737\pi$
$702$	9.09756e8	0.0992533
$703$	−1.44707e10	−1.57089
$704$	−1.29416e9	−0.139792
$705$	−5.96689e9	−0.641337
$706$	3.95036e9	0.422494
$707$	2.77728e10	2.95564
$708$	−3.54895e8	−0.0375823
$709$	1.68505e10	1.77563	0.887814	−	0.460203i	$-0.152223\pi$
$709$	1.68505e10	1.77563	0.887814	+	0.460203i	$0.152223\pi$
$710$	2.33206e9	0.244532
$711$	−2.00214e7	−0.00208906
$712$	−1.49499e9	−0.155224
$713$	5.24061e9	0.541463
$714$	2.77473e9	0.285283
$715$	8.28046e9	0.847195
$716$	−8.78355e9	−0.894282
$717$	−4.86623e9	−0.493033
$718$	−4.28459e9	−0.431990
$719$	1.78592e10	1.79188	0.895942	−	0.444171i	$-0.146502\pi$
$719$	1.78592e10	1.79188	0.895942	+	0.444171i	$0.146502\pi$
$720$	−8.66866e8	−0.0865542
$721$	−2.56212e10	−2.54581
$722$	2.59387e9	0.256489
$723$	2.77597e9	0.273169
$724$	−7.17283e9	−0.702434
$725$	1.38564e8	0.0135042
$726$	1.05515e9	0.102338
$727$	1.03971e10	1.00356	0.501779	−	0.864996i	$-0.332679\pi$
$727$	1.03971e10	1.00356	0.501779	+	0.864996i	$0.332679\pi$
$728$	−5.06521e9	−0.486561
$729$	3.87420e8	0.0370370
$730$	−5.36210e9	−0.510159
$731$	−1.99805e9	−0.189189
$732$	−1.12867e9	−0.106360
$733$	7.65113e9	0.717566	0.358783	−	0.933421i	$-0.383192\pi$
$733$	7.65113e9	0.717566	0.358783	+	0.933421i	$0.383192\pi$
$734$	3.73644e9	0.348756
$735$	1.65272e10	1.53530
$736$	1.34834e9	0.124660
$737$	−1.72968e10	−1.59158
$738$	−6.00637e8	−0.0550066
$739$	1.14469e10	1.04335	0.521677	−	0.853143i	$-0.325306\pi$
$739$	1.14469e10	1.04335	0.521677	+	0.853143i	$0.325306\pi$
$740$	−1.12651e10	−1.02194
$741$	3.72311e9	0.336157
$742$	−8.87636e9	−0.797666
$743$	−2.80445e9	−0.250834	−0.125417	−	0.992104i	$-0.540027\pi$
$743$	−2.80445e9	−0.250834	−0.125417	+	0.992104i	$0.540027\pi$
$744$	−1.76062e9	−0.156733
$745$	4.72700e9	0.418831
$746$	−9.62477e9	−0.848798
$747$	−3.79429e9	−0.333049
$748$	−2.37033e9	−0.207087
$749$	1.42239e10	1.23690
$750$	4.51299e9	0.390616
$751$	−1.37198e10	−1.18198	−0.590988	−	0.806680i	$-0.701262\pi$
$751$	−1.37198e10	−1.18198	−0.590988	+	0.806680i	$0.701262\pi$
$752$	−3.11803e9	−0.267373
$753$	7.08206e8	0.0604473
$754$	−1.04039e9	−0.0883886
$755$	1.51078e10	1.27758
$756$	−2.15702e9	−0.181564
$757$	2.24393e10	1.88007	0.940033	−	0.341083i	$-0.110794\pi$
$757$	2.24393e10	1.88007	0.940033	+	0.341083i	$0.110794\pi$
$758$	−9.01202e9	−0.751588
$759$	−5.48481e9	−0.455318
$760$	−3.54759e9	−0.293147
$761$	−1.96261e10	−1.61431	−0.807156	−	0.590338i	$-0.798994\pi$
$761$	−1.96261e10	−1.61431	−0.807156	+	0.590338i	$0.798994\pi$
$762$	2.21522e9	0.181374
$763$	3.15705e10	2.57304
$764$	8.92130e9	0.723771
$765$	−1.58772e9	−0.128221
$766$	−1.34349e10	−1.08003
$767$	1.18659e9	0.0949545
$768$	−4.52985e8	−0.0360844
$769$	4.15726e8	0.0329659	0.0164829	−	0.999864i	$-0.494753\pi$
$769$	4.15726e8	0.0329659	0.0164829	+	0.999864i	$0.494753\pi$
$770$	−1.96329e10	−1.54977
$771$	−8.13972e9	−0.639616
$772$	−4.94133e9	−0.386530
$773$	−2.41283e10	−1.87888	−0.939440	−	0.342713i	$-0.888654\pi$
$773$	−2.41283e10	−1.87888	−0.939440	+	0.342713i	$0.888654\pi$
$774$	1.55325e9	0.120406
$775$	−7.84006e8	−0.0605011
$776$	5.69580e9	0.437561
$777$	−2.80310e10	−2.14371
$778$	−3.83634e9	−0.292071
$779$	−2.45807e9	−0.186300
$780$	2.89836e9	0.218686
$781$	−4.95715e9	−0.372352
$782$	2.46958e9	0.184672
$783$	−4.43051e8	−0.0329828
$784$	8.63634e9	0.640064
$785$	7.35780e9	0.542881
$786$	−4.82728e8	−0.0354588
$787$	−2.50819e10	−1.83421	−0.917104	−	0.398647i	$-0.869480\pi$
$787$	−2.50819e10	−1.83421	−0.917104	+	0.398647i	$0.869480\pi$
$788$	−1.98592e9	−0.144584
$789$	4.97388e9	0.360517
$790$	−6.37853e7	−0.00460284
$791$	−2.68258e10	−1.92724
$792$	1.84266e9	0.131797
$793$	3.77370e9	0.268727
$794$	−3.29962e9	−0.233933
$795$	5.07913e9	0.358513
$796$	−7.46379e9	−0.524522
$797$	−8.87544e9	−0.620991	−0.310496	−	0.950575i	$-0.600495\pi$
$797$	−8.87544e9	−0.620991	−0.310496	+	0.950575i	$0.600495\pi$
$798$	−8.82748e9	−0.614931
$799$	−5.71088e9	−0.396086
$800$	−2.01715e8	−0.0139291
$801$	2.12861e9	0.146347
$802$	6.21690e9	0.425563
$803$	1.13980e10	0.776825
$804$	−6.05427e9	−0.410833
$805$	2.04550e10	1.38202
$806$	5.88660e9	0.395997
$807$	5.96626e9	0.399618
$808$	−8.30435e9	−0.553817
$809$	2.79821e10	1.85806	0.929030	−	0.370004i	$-0.120644\pi$
$809$	2.79821e10	1.85806	0.929030	+	0.370004i	$0.120644\pi$
$810$	1.23427e9	0.0816041
$811$	1.50662e10	0.991815	0.495908	−	0.868375i	$-0.334835\pi$
$811$	1.50662e10	0.991815	0.495908	+	0.868375i	$0.334835\pi$
$812$	2.46676e9	0.161689
$813$	−8.09716e8	−0.0528465
$814$	2.39457e10	1.55612
$815$	1.24017e10	0.802475
$816$	−8.29672e8	−0.0534553
$817$	6.35657e9	0.407799
$818$	−1.26486e10	−0.807992
$819$	7.21198e9	0.458734
$820$	−1.91355e9	−0.121197
$821$	−1.12791e10	−0.711336	−0.355668	−	0.934612i	$-0.615747\pi$
$821$	−1.12791e10	−0.711336	−0.355668	+	0.934612i	$0.615747\pi$
$822$	−1.37362e9	−0.0862614
$823$	−2.25258e10	−1.40858	−0.704289	−	0.709913i	$-0.748734\pi$
$823$	−2.25258e10	−1.40858	−0.704289	+	0.709913i	$0.748734\pi$
$824$	7.66100e9	0.477024
$825$	8.20538e8	0.0508756
$826$	−2.81339e9	−0.173700
$827$	1.04048e10	0.639683	0.319841	−	0.947471i	$-0.396370\pi$
$827$	1.04048e10	0.639683	0.319841	+	0.947471i	$0.396370\pi$
$828$	−1.91981e9	−0.117531
$829$	2.48413e9	0.151437	0.0757187	−	0.997129i	$-0.475875\pi$
$829$	2.48413e9	0.151437	0.0757187	+	0.997129i	$0.475875\pi$
$830$	−1.20881e10	−0.733810
$831$	−1.24881e10	−0.754906
$832$	1.51455e9	0.0911699
$833$	1.58180e10	0.948190
$834$	5.66071e9	0.337901
$835$	1.85198e10	1.10086
$836$	7.54095e9	0.446379
$837$	2.50682e9	0.147769
$838$	8.58271e9	0.503814
$839$	−5.23659e9	−0.306113	−0.153057	−	0.988217i	$-0.548912\pi$
$839$	−5.23659e9	−0.306113	−0.153057	+	0.988217i	$0.548912\pi$
$840$	−6.87198e9	−0.400041
$841$	−1.67432e10	−0.970628
$842$	−2.15014e10	−1.24129
$843$	9.55534e9	0.549351
$844$	−9.14926e9	−0.523827
$845$	8.52601e9	0.486124
$846$	4.43954e9	0.252082
$847$	8.36456e9	0.472989
$848$	2.65412e9	0.149464
$849$	5.50915e9	0.308964
$850$	−3.69454e8	−0.0206345
$851$	−2.49484e10	−1.38768
$852$	−1.73512e9	−0.0961149
$853$	−3.59219e9	−0.198170	−0.0990851	−	0.995079i	$-0.531592\pi$
$853$	−3.59219e9	−0.198170	−0.0990851	+	0.995079i	$0.531592\pi$
$854$	−8.94742e9	−0.491581
$855$	5.05116e9	0.276382
$856$	−4.25311e9	−0.231765
$857$	−1.04795e10	−0.568733	−0.284366	−	0.958716i	$-0.591783\pi$
$857$	−1.04795e10	−0.568733	−0.284366	+	0.958716i	$0.591783\pi$
$858$	−6.16089e9	−0.332995
$859$	1.96445e10	1.05746	0.528731	−	0.848790i	$-0.322668\pi$
$859$	1.96445e10	1.05746	0.528731	+	0.848790i	$0.322668\pi$
$860$	4.94844e9	0.265292
$861$	−4.76148e9	−0.254232
$862$	2.40274e10	1.27771
$863$	−3.21335e10	−1.70185	−0.850923	−	0.525291i	$-0.823957\pi$
$863$	−3.21335e10	−1.70185	−0.850923	+	0.525291i	$0.823957\pi$
$864$	6.44973e8	0.0340207
$865$	1.50692e10	0.791649
$866$	2.66197e10	1.39281
$867$	9.55954e9	0.498162
$868$	−1.39571e10	−0.724396
$869$	1.35585e8	0.00700880
$870$	−1.41150e9	−0.0726714
$871$	2.02424e10	1.03800
$872$	−9.43991e9	−0.482126
$873$	−8.10984e9	−0.412537
$874$	−7.85670e9	−0.398061
$875$	3.57762e10	1.80537
$876$	3.98956e9	0.200521
$877$	−1.48899e10	−0.745408	−0.372704	−	0.927950i	$-0.621569\pi$
$877$	−1.48899e10	−0.745408	−0.372704	+	0.927950i	$0.621569\pi$
$878$	−2.76115e9	−0.137676
$879$	−5.96874e9	−0.296430
$880$	5.87044e9	0.290390
$881$	1.38787e10	0.683808	0.341904	−	0.939735i	$-0.388928\pi$
$881$	1.38787e10	0.683808	0.341904	+	0.939735i	$0.388928\pi$
$882$	−1.22967e10	−0.603458
$883$	−2.95156e9	−0.144274	−0.0721372	−	0.997395i	$-0.522982\pi$
$883$	−2.95156e9	−0.144274	−0.0721372	+	0.997395i	$0.522982\pi$
$884$	2.77400e9	0.135059
$885$	1.60985e9	0.0780698
$886$	1.67248e10	0.807873
$887$	−3.03155e10	−1.45859	−0.729293	−	0.684201i	$-0.760151\pi$
$887$	−3.03155e10	−1.45859	−0.729293	+	0.684201i	$0.760151\pi$
$888$	8.38157e9	0.401680
$889$	1.75609e10	0.838283
$890$	6.78147e9	0.322447
$891$	−2.62362e9	−0.124260
$892$	2.52268e9	0.119010
$893$	1.81685e10	0.853767
$894$	−3.51702e9	−0.164624
$895$	3.98432e10	1.85769
$896$	−3.59099e9	−0.166777
$897$	6.41886e9	0.296951
$898$	2.06242e10	0.950410
$899$	−2.86678e9	−0.131594
$900$	2.87207e8	0.0131325
$901$	4.86120e9	0.221415
$902$	4.06753e9	0.184547
$903$	1.23132e10	0.556499
$904$	8.02119e9	0.361118
$905$	3.25368e10	1.45917
$906$	−1.12406e10	−0.502160
$907$	3.08660e10	1.37358	0.686791	−	0.726855i	$-0.259019\pi$
$907$	3.08660e10	1.37358	0.686791	+	0.726855i	$0.259019\pi$
$908$	2.17810e9	0.0965554
$909$	1.18240e10	0.522143
$910$	2.29764e10	1.01073
$911$	−4.26998e10	−1.87116	−0.935582	−	0.353111i	$-0.885124\pi$
$911$	−4.26998e10	−1.87116	−0.935582	+	0.353111i	$0.885124\pi$
$912$	2.63951e9	0.115223
$913$	2.56950e10	1.11738
$914$	−8.85995e9	−0.383813
$915$	5.11979e9	0.220942
$916$	8.88344e9	0.381898
$917$	−3.82677e9	−0.163885
$918$	1.18131e9	0.0503982
$919$	−3.63542e10	−1.54508	−0.772540	−	0.634967i	$-0.781014\pi$
$919$	−3.63542e10	−1.54508	−0.772540	+	0.634967i	$0.781014\pi$
$920$	−6.11625e9	−0.258957
$921$	1.39593e10	0.588783
$922$	1.79263e10	0.753239
$923$	5.80135e9	0.242842
$924$	1.46074e10	0.609147
$925$	3.73233e9	0.155054
$926$	2.41916e10	1.00121
$927$	−1.09079e10	−0.449742
$928$	−7.37586e8	−0.0302967
$929$	1.56640e10	0.640983	0.320491	−	0.947251i	$-0.396152\pi$
$929$	1.56640e10	0.640983	0.320491	+	0.947251i	$0.396152\pi$
$930$	7.98637e9	0.325581
$931$	−5.03233e10	−2.04383
$932$	−1.18236e10	−0.478403
$933$	3.84427e9	0.154963
$934$	1.14705e10	0.460647
$935$	1.07521e10	0.430183
$936$	−2.15646e9	−0.0859558
$937$	−1.06025e10	−0.421035	−0.210518	−	0.977590i	$-0.567515\pi$
$937$	−1.06025e10	−0.421035	−0.210518	+	0.977590i	$0.567515\pi$
$938$	−4.79945e10	−1.89881
$939$	3.44432e9	0.135761
$940$	1.41437e10	0.555414
$941$	3.52493e10	1.37907	0.689537	−	0.724251i	$-0.257814\pi$
$941$	3.52493e10	1.37907	0.689537	+	0.724251i	$0.257814\pi$
$942$	−5.47441e9	−0.213383
$943$	−4.23785e9	−0.164571
$944$	8.41232e8	0.0325472
$945$	9.78452e9	0.377162
$946$	−1.05187e10	−0.403963
$947$	−2.40472e10	−0.920108	−0.460054	−	0.887891i	$-0.652170\pi$
$947$	−2.40472e10	−0.920108	−0.460054	+	0.887891i	$0.652170\pi$
$948$	4.74580e7	0.00180918
$949$	−1.33390e10	−0.506632
$950$	1.17538e9	0.0444779
$951$	1.47087e10	0.554552
$952$	−6.57713e9	−0.247063
$953$	−2.66516e10	−0.997468	−0.498734	−	0.866755i	$-0.666202\pi$
$953$	−2.66516e10	−0.997468	−0.498734	+	0.866755i	$0.666202\pi$
$954$	−3.77901e9	−0.140916
$955$	−4.04681e10	−1.50349
$956$	1.15348e10	0.426979
$957$	3.00036e9	0.110658
$958$	−1.60729e10	−0.590631
$959$	−1.08893e10	−0.398688
$960$	2.05479e9	0.0749581
$961$	−1.12922e10	−0.410436
$962$	−2.80237e10	−1.01487
$963$	6.05569e9	0.218510
$964$	−6.58008e9	−0.236571
$965$	2.24145e10	0.802939
$966$	−1.52191e10	−0.543211
$967$	4.06439e10	1.44545	0.722725	−	0.691136i	$-0.242889\pi$
$967$	4.06439e10	1.44545	0.722725	+	0.691136i	$0.242889\pi$
$968$	−2.50109e9	−0.0886269
$969$	4.83444e9	0.170692
$970$	−2.58368e10	−0.908946
$971$	9.70862e9	0.340322	0.170161	−	0.985416i	$-0.445571\pi$
$971$	9.70862e9	0.340322	0.170161	+	0.985416i	$0.445571\pi$
$972$	−9.18330e8	−0.0320750
$973$	4.48746e10	1.56173
$974$	3.36694e10	1.16756
$975$	−9.60274e8	−0.0331802
$976$	2.67537e9	0.0921106
$977$	1.77961e10	0.610511	0.305255	−	0.952271i	$-0.401258\pi$
$977$	1.77961e10	0.610511	0.305255	+	0.952271i	$0.401258\pi$
$978$	−9.22725e9	−0.315418
$979$	−1.44150e10	−0.490995
$980$	−3.91755e10	−1.32961
$981$	1.34408e10	0.454553
$982$	−3.48993e9	−0.117605
$983$	5.89750e10	1.98030	0.990149	−	0.140015i	$-0.0447151\pi$
$983$	5.89750e10	1.98030	0.990149	+	0.140015i	$0.0447151\pi$
$984$	1.42373e9	0.0476371
$985$	9.00838e9	0.300345
$986$	−1.35094e9	−0.0448814
$987$	3.51939e10	1.16508
$988$	−8.82516e9	−0.291121
$989$	1.09591e10	0.360236
$990$	−8.35850e9	−0.273782
$991$	−4.50007e10	−1.46880	−0.734398	−	0.678719i	$-0.762536\pi$
$991$	−4.50007e10	−1.46880	−0.734398	+	0.678719i	$0.762536\pi$
$992$	4.17332e9	0.135735
$993$	1.38990e10	0.450466
$994$	−1.37550e10	−0.444229
$995$	3.38566e10	1.08959
$996$	8.99387e9	0.288429
$997$	−1.71722e10	−0.548772	−0.274386	−	0.961620i	$-0.588475\pi$
$997$	−1.71722e10	−0.548772	−0.274386	+	0.961620i	$0.588475\pi$
$998$	2.05199e10	0.653458
$999$	−1.19339e10	−0.378707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.8.a.e.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.8.a.e.1.3	✓	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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -290.312 q^{5} +216.000 q^{6} +1712.32 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -290.312 q^{5} +216.000 q^{6} +1712.32 q^{7} -512.000 q^{8} +729.000 q^{9} +2322.49 q^{10} -4936.81 q^{11} -1728.00 q^{12} +5777.55 q^{13} -13698.5 q^{14} +7838.41 q^{15} +4096.00 q^{16} +7502.10 q^{17} -5832.00 q^{18} -23867.1 q^{19} -18579.9 q^{20} -46232.5 q^{21} +39494.5 q^{22} -41148.2 q^{23} +13824.0 q^{24} +6155.85 q^{25} -46220.4 q^{26} -19683.0 q^{27} +109588. q^{28} +22509.3 q^{29} -62707.3 q^{30} -127359. q^{31} -32768.0 q^{32} +133294. q^{33} -60016.8 q^{34} -497105. q^{35} +46656.0 q^{36} +606305. q^{37} +190937. q^{38} -155994. q^{39} +148640. q^{40} +102990. q^{41} +369860. q^{42} -266332. q^{43} -315956. q^{44} -211637. q^{45} +329186. q^{46} -761237. q^{47} -110592. q^{48} +2.10848e6 q^{49} -49246.8 q^{50} -202557. q^{51} +369763. q^{52} +647979. q^{53} +157464. q^{54} +1.43321e6 q^{55} -876706. q^{56} +644411. q^{57} -180075. q^{58} +205379. q^{59} +501659. q^{60} +653167. q^{61} +1.01888e6 q^{62} +1.24828e6 q^{63} +262144. q^{64} -1.67729e6 q^{65} -1.06635e6 q^{66} +3.50363e6 q^{67} +480134. q^{68} +1.11100e6 q^{69} +3.97684e6 q^{70} +1.00412e6 q^{71} -373248. q^{72} -2.30877e6 q^{73} -4.85044e6 q^{74} -166208. q^{75} -1.52749e6 q^{76} -8.45338e6 q^{77} +1.24795e6 q^{78} -27464.1 q^{79} -1.18912e6 q^{80} +531441. q^{81} -823919. q^{82} -5.20478e6 q^{83} -2.95888e6 q^{84} -2.17795e6 q^{85} +2.13066e6 q^{86} -607752. q^{87} +2.52765e6 q^{88} +2.91991e6 q^{89} +1.69310e6 q^{90} +9.89298e6 q^{91} -2.63348e6 q^{92} +3.43871e6 q^{93} +6.08990e6 q^{94} +6.92889e6 q^{95} +884736. q^{96} -1.11246e7 q^{97} -1.68679e7 q^{98} -3.59894e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 72 q^{2} - 243 q^{3} + 576 q^{4} - 230 q^{5} + 1944 q^{6} - 340 q^{7} - 4608 q^{8} + 6561 q^{9}+O(q^{10}) \) 9 * q - 72 * q^2 - 243 * q^3 + 576 * q^4 - 230 * q^5 + 1944 * q^6 - 340 * q^7 - 4608 * q^8 + 6561 * q^9	\( 9 q - 72 q^{2} - 243 q^{3} + 576 q^{4} - 230 q^{5} + 1944 q^{6} - 340 q^{7} - 4608 q^{8} + 6561 q^{9} + 1840 q^{10} - 5472 q^{11} - 15552 q^{12} + 3144 q^{13} + 2720 q^{14} + 6210 q^{15} + 36864 q^{16} - 3662 q^{17} - 52488 q^{18} + 692 q^{19} - 14720 q^{20} + 9180 q^{21} + 43776 q^{22} - 92046 q^{23} + 124416 q^{24} + 74731 q^{25} - 25152 q^{26} - 177147 q^{27} - 21760 q^{28} - 41060 q^{29} - 49680 q^{30} - 324504 q^{31} - 294912 q^{32} + 147744 q^{33} + 29296 q^{34} - 415602 q^{35} + 419904 q^{36} + 338612 q^{37} - 5536 q^{38} - 84888 q^{39} + 117760 q^{40} - 104312 q^{41} - 73440 q^{42} + 1000602 q^{43} - 350208 q^{44} - 167670 q^{45} + 736368 q^{46} - 365148 q^{47} - 995328 q^{48} + 2307505 q^{49} - 597848 q^{50} + 98874 q^{51} + 201216 q^{52} + 2017498 q^{53} + 1417176 q^{54} + 1120520 q^{55} + 174080 q^{56} - 18684 q^{57} + 328480 q^{58} + 1848411 q^{59} + 397440 q^{60} + 5102340 q^{61} + 2596032 q^{62} - 247860 q^{63} + 2359296 q^{64} + 7512810 q^{65} - 1181952 q^{66} + 10920464 q^{67} - 234368 q^{68} + 2485242 q^{69} + 3324816 q^{70} + 3607024 q^{71} - 3359232 q^{72} + 12949418 q^{73} - 2708896 q^{74} - 2017737 q^{75} + 44288 q^{76} + 7127994 q^{77} + 679104 q^{78} + 7489472 q^{79} - 942080 q^{80} + 4782969 q^{81} + 834496 q^{82} + 2760502 q^{83} + 587520 q^{84} + 11815354 q^{85} - 8004816 q^{86} + 1108620 q^{87} + 2801664 q^{88} - 9948196 q^{89} + 1341360 q^{90} + 19400656 q^{91} - 5890944 q^{92} + 8761608 q^{93} + 2921184 q^{94} + 24045208 q^{95} + 7962624 q^{96} + 38157642 q^{97} - 18460040 q^{98} - 3989088 q^{99}+O(q^{100}) \) 9 * q - 72 * q^2 - 243 * q^3 + 576 * q^4 - 230 * q^5 + 1944 * q^6 - 340 * q^7 - 4608 * q^8 + 6561 * q^9 + 1840 * q^10 - 5472 * q^11 - 15552 * q^12 + 3144 * q^13 + 2720 * q^14 + 6210 * q^15 + 36864 * q^16 - 3662 * q^17 - 52488 * q^18 + 692 * q^19 - 14720 * q^20 + 9180 * q^21 + 43776 * q^22 - 92046 * q^23 + 124416 * q^24 + 74731 * q^25 - 25152 * q^26 - 177147 * q^27 - 21760 * q^28 - 41060 * q^29 - 49680 * q^30 - 324504 * q^31 - 294912 * q^32 + 147744 * q^33 + 29296 * q^34 - 415602 * q^35 + 419904 * q^36 + 338612 * q^37 - 5536 * q^38 - 84888 * q^39 + 117760 * q^40 - 104312 * q^41 - 73440 * q^42 + 1000602 * q^43 - 350208 * q^44 - 167670 * q^45 + 736368 * q^46 - 365148 * q^47 - 995328 * q^48 + 2307505 * q^49 - 597848 * q^50 + 98874 * q^51 + 201216 * q^52 + 2017498 * q^53 + 1417176 * q^54 + 1120520 * q^55 + 174080 * q^56 - 18684 * q^57 + 328480 * q^58 + 1848411 * q^59 + 397440 * q^60 + 5102340 * q^61 + 2596032 * q^62 - 247860 * q^63 + 2359296 * q^64 + 7512810 * q^65 - 1181952 * q^66 + 10920464 * q^67 - 234368 * q^68 + 2485242 * q^69 + 3324816 * q^70 + 3607024 * q^71 - 3359232 * q^72 + 12949418 * q^73 - 2708896 * q^74 - 2017737 * q^75 + 44288 * q^76 + 7127994 * q^77 + 679104 * q^78 + 7489472 * q^79 - 942080 * q^80 + 4782969 * q^81 + 834496 * q^82 + 2760502 * q^83 + 587520 * q^84 + 11815354 * q^85 - 8004816 * q^86 + 1108620 * q^87 + 2801664 * q^88 - 9948196 * q^89 + 1341360 * q^90 + 19400656 * q^91 - 5890944 * q^92 + 8761608 * q^93 + 2921184 * q^94 + 24045208 * q^95 + 7962624 * q^96 + 38157642 * q^97 - 18460040 * q^98 - 3989088 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more