LMFDB - Embedded newform 162.8.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,8,Mod(1,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	162.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:	$50.6063741284$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.69765.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 72x - 179 $ x^3 - 72*x - 179
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.52819$ of defining polynomial
Character	$\chi$	$=$	162.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.00000 q^{2} +64.0000 q^{4} -375.578 q^{5} +1057.73 q^{7} +512.000 q^{8} -3004.63 q^{10} +1699.93 q^{11} -12513.5 q^{13} +8461.86 q^{14} +4096.00 q^{16} +16879.9 q^{17} -35281.9 q^{19} -24037.0 q^{20} +13599.4 q^{22} +70965.4 q^{23} +62934.2 q^{25} -100108. q^{26} +67694.9 q^{28} -77654.8 q^{29} -126188. q^{31} +32768.0 q^{32} +135039. q^{34} -397262. q^{35} -528664. q^{37} -282255. q^{38} -192296. q^{40} -517921. q^{41} -271660. q^{43} +108795. q^{44} +567723. q^{46} -423629. q^{47} +295256. q^{49} +503474. q^{50} -800863. q^{52} +1.24314e6 q^{53} -638457. q^{55} +541559. q^{56} -621238. q^{58} -2.68173e6 q^{59} -754080. q^{61} -1.00950e6 q^{62} +262144. q^{64} +4.69980e6 q^{65} -262570. q^{67} +1.08031e6 q^{68} -3.17809e6 q^{70} +990913. q^{71} -1.04389e6 q^{73} -4.22931e6 q^{74} -2.25804e6 q^{76} +1.79807e6 q^{77} -7.33719e6 q^{79} -1.53837e6 q^{80} -4.14337e6 q^{82} -3.89221e6 q^{83} -6.33971e6 q^{85} -2.17328e6 q^{86} +870364. q^{88} +675681. q^{89} -1.32359e7 q^{91} +4.54179e6 q^{92} -3.38903e6 q^{94} +1.32511e7 q^{95} +1.62557e7 q^{97} +2.36205e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 24 q^{2} + 192 q^{4} - 54 q^{5} - 210 q^{7} + 1536 q^{8} - 432 q^{10} - 6579 q^{11} - 10092 q^{13} - 1680 q^{14} + 12288 q^{16} - 14895 q^{17} - 68745 q^{19} - 3456 q^{20} - 52632 q^{22} + 39654 q^{23}+ \cdots - 3857400 q^{98}+O(q^{100}) $ 3 * q + 24 * q^2 + 192 * q^4 - 54 * q^5 - 210 * q^7 + 1536 * q^8 - 432 * q^10 - 6579 * q^11 - 10092 * q^13 - 1680 * q^14 + 12288 * q^16 - 14895 * q^17 - 68745 * q^19 - 3456 * q^20 - 52632 * q^22 + 39654 * q^23 + 7923 * q^25 - 80736 * q^26 - 13440 * q^28 - 239832 * q^29 - 145704 * q^31 + 98304 * q^32 - 119160 * q^34 - 543888 * q^35 - 360192 * q^37 - 549960 * q^38 - 27648 * q^40 - 1086993 * q^41 + 299967 * q^43 - 421056 * q^44 + 317232 * q^46 - 131634 * q^47 - 482175 * q^49 + 63384 * q^50 - 645888 * q^52 + 954576 * q^53 - 2747412 * q^55 - 107520 * q^56 - 1918656 * q^58 - 2504853 * q^59 - 7309038 * q^61 - 1165632 * q^62 + 786432 * q^64 + 1786698 * q^65 - 3433035 * q^67 - 953280 * q^68 - 4351104 * q^70 + 1134684 * q^71 - 7950873 * q^73 - 2881536 * q^74 - 4399680 * q^76 + 6147432 * q^77 - 7076928 * q^79 - 221184 * q^80 - 8695944 * q^82 + 10914444 * q^83 - 17613396 * q^85 + 2399736 * q^86 - 3368448 * q^88 + 11900178 * q^89 - 18585276 * q^91 + 2537856 * q^92 - 1053072 * q^94 + 17342424 * q^95 - 519357 * q^97 - 3857400 * q^98

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
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−375.578	−1.34371	−0.671855	−	0.740682i	$-0.734502\pi$
$5$	−375.578	−1.34371	−0.671855	+	0.740682i	$0.734502\pi$
$6$	0	0
$7$	1057.73	1.16556	0.582778	−	0.812632i	$-0.301966\pi$
$7$	1057.73	1.16556	0.582778	+	0.812632i	$0.301966\pi$
$8$	512.000	0.353553
$9$	0	0
$10$	−3004.63	−0.950147
$11$	1699.93	0.385085	0.192542	−	0.981289i	$-0.438327\pi$
$11$	1699.93	0.385085	0.192542	+	0.981289i	$0.438327\pi$
$12$	0	0
$13$	−12513.5	−1.57971	−0.789854	−	0.613295i	$-0.789844\pi$
$13$	−12513.5	−1.57971	−0.789854	+	0.613295i	$0.789844\pi$
$14$	8461.86	0.824172
$15$	0	0
$16$	4096.00	0.250000
$17$	16879.9	0.833293	0.416646	−	0.909069i	$-0.363205\pi$
$17$	16879.9	0.833293	0.416646	+	0.909069i	$0.363205\pi$
$18$	0	0
$19$	−35281.9	−1.18009	−0.590044	−	0.807371i	$-0.700890\pi$
$19$	−35281.9	−1.18009	−0.590044	+	0.807371i	$0.700890\pi$
$20$	−24037.0	−0.671855
$21$	0	0
$22$	13599.4	0.272296
$23$	70965.4	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$23$	70965.4	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$24$	0	0
$25$	62934.2	0.805558
$26$	−100108.	−1.11702
$27$	0	0
$28$	67694.9	0.582778
$29$	−77654.8	−0.591255	−0.295628	−	0.955303i	$-0.595529\pi$
$29$	−77654.8	−0.591255	−0.295628	+	0.955303i	$0.595529\pi$
$30$	0	0
$31$	−126188.	−0.760766	−0.380383	−	0.924829i	$-0.624208\pi$
$31$	−126188.	−0.760766	−0.380383	+	0.924829i	$0.624208\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	135039.	0.589227
$35$	−397262.	−1.56617
$36$	0	0
$37$	−528664.	−1.71583	−0.857914	−	0.513794i	$-0.828240\pi$
$37$	−528664.	−1.71583	−0.857914	+	0.513794i	$0.828240\pi$
$38$	−282255.	−0.834449
$39$	0	0
$40$	−192296.	−0.475073
$41$	−517921.	−1.17360	−0.586800	−	0.809732i	$-0.699612\pi$
$41$	−517921.	−1.17360	−0.586800	+	0.809732i	$0.699612\pi$
$42$	0	0
$43$	−271660.	−0.521057	−0.260529	−	0.965466i	$-0.583897\pi$
$43$	−271660.	−0.521057	−0.260529	+	0.965466i	$0.583897\pi$
$44$	108795.	0.192542
$45$	0	0
$46$	567723.	0.859972
$47$	−423629.	−0.595173	−0.297586	−	0.954695i	$-0.596182\pi$
$47$	−423629.	−0.595173	−0.297586	+	0.954695i	$0.596182\pi$
$48$	0	0
$49$	295256.	0.358519
$50$	503474.	0.569615
$51$	0	0
$52$	−800863.	−0.789854
$53$	1.24314e6	1.14698	0.573490	−	0.819213i	$-0.305589\pi$
$53$	1.24314e6	1.14698	0.573490	+	0.819213i	$0.305589\pi$
$54$	0	0
$55$	−638457.	−0.517443
$56$	541559.	0.412086
$57$	0	0
$58$	−621238.	−0.418081
$59$	−2.68173e6	−1.69994	−0.849968	−	0.526834i	$-0.823379\pi$
$59$	−2.68173e6	−1.69994	−0.849968	+	0.526834i	$0.823379\pi$
$60$	0	0
$61$	−754080.	−0.425366	−0.212683	−	0.977121i	$-0.568220\pi$
$61$	−754080.	−0.425366	−0.212683	+	0.977121i	$0.568220\pi$
$62$	−1.00950e6	−0.537943
$63$	0	0
$64$	262144.	0.125000
$65$	4.69980e6	2.12267
$66$	0	0
$67$	−262570.	−0.106656	−0.0533278	−	0.998577i	$-0.516983\pi$
$67$	−262570.	−0.106656	−0.0533278	+	0.998577i	$0.516983\pi$
$68$	1.08031e6	0.416646
$69$	0	0
$70$	−3.17809e6	−1.10745
$71$	990913.	0.328573	0.164286	−	0.986413i	$-0.447468\pi$
$71$	990913.	0.328573	0.164286	+	0.986413i	$0.447468\pi$
$72$	0	0
$73$	−1.04389e6	−0.314068	−0.157034	−	0.987593i	$-0.550193\pi$
$73$	−1.04389e6	−0.314068	−0.157034	+	0.987593i	$0.550193\pi$
$74$	−4.22931e6	−1.21327
$75$	0	0
$76$	−2.25804e6	−0.590044
$77$	1.79807e6	0.448838
$78$	0	0
$79$	−7.33719e6	−1.67431	−0.837153	−	0.546968i	$-0.815782\pi$
$79$	−7.33719e6	−1.67431	−0.837153	+	0.546968i	$0.815782\pi$
$80$	−1.53837e6	−0.335928
$81$	0	0
$82$	−4.14337e6	−0.829860
$83$	−3.89221e6	−0.747177	−0.373588	−	0.927595i	$-0.621873\pi$
$83$	−3.89221e6	−0.747177	−0.373588	+	0.927595i	$0.621873\pi$
$84$	0	0
$85$	−6.33971e6	−1.11970
$86$	−2.17328e6	−0.368443
$87$	0	0
$88$	870364.	0.136148
$89$	675681.	0.101596	0.0507980	−	0.998709i	$-0.483824\pi$
$89$	675681.	0.101596	0.0507980	+	0.998709i	$0.483824\pi$
$90$	0	0
$91$	−1.32359e7	−1.84124
$92$	4.54179e6	0.608092
$93$	0	0
$94$	−3.38903e6	−0.420851
$95$	1.32511e7	1.58570
$96$	0	0
$97$	1.62557e7	1.80844	0.904221	−	0.427065i	$-0.140452\pi$
$97$	1.62557e7	1.80844	0.904221	+	0.427065i	$0.140452\pi$
$98$	2.36205e6	0.253512
$99$	0	0
$100$	4.02779e6	0.402779
$101$	−424540.	−0.0410009	−0.0205005	−	0.999790i	$-0.506526\pi$
$101$	−424540.	−0.0410009	−0.0205005	+	0.999790i	$0.506526\pi$
$102$	0	0
$103$	−4.86334e6	−0.438535	−0.219268	−	0.975665i	$-0.570367\pi$
$103$	−4.86334e6	−0.438535	−0.219268	+	0.975665i	$0.570367\pi$
$104$	−6.40691e6	−0.558511
$105$	0	0
$106$	9.94515e6	0.811037
$107$	2.83648e6	0.223839	0.111920	−	0.993717i	$-0.464300\pi$
$107$	2.83648e6	0.223839	0.111920	+	0.993717i	$0.464300\pi$
$108$	0	0
$109$	6.31389e6	0.466987	0.233493	−	0.972358i	$-0.424984\pi$
$109$	6.31389e6	0.466987	0.233493	+	0.972358i	$0.424984\pi$
$110$	−5.10766e6	−0.365887
$111$	0	0
$112$	4.33247e6	0.291389
$113$	2.14437e7	1.39806	0.699029	−	0.715094i	$-0.253616\pi$
$113$	2.14437e7	1.39806	0.699029	+	0.715094i	$0.253616\pi$
$114$	0	0
$115$	−2.66531e7	−1.63420
$116$	−4.96990e6	−0.295628
$117$	0	0
$118$	−2.14538e7	−1.20204
$119$	1.78544e7	0.971249
$120$	0	0
$121$	−1.65974e7	−0.851710
$122$	−6.03264e6	−0.300779
$123$	0	0
$124$	−8.07601e6	−0.380383
$125$	5.70534e6	0.261274
$126$	0	0
$127$	1.35681e7	0.587768	0.293884	−	0.955841i	$-0.405052\pi$
$127$	1.35681e7	0.587768	0.293884	+	0.955841i	$0.405052\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	3.75984e7	1.50095
$131$	−2.35314e7	−0.914529	−0.457265	−	0.889331i	$-0.651171\pi$
$131$	−2.35314e7	−0.914529	−0.457265	+	0.889331i	$0.651171\pi$
$132$	0	0
$133$	−3.73189e7	−1.37546
$134$	−2.10056e6	−0.0754169
$135$	0	0
$136$	8.64249e6	0.294614
$137$	−2.79905e7	−0.930012	−0.465006	−	0.885307i	$-0.653948\pi$
$137$	−2.79905e7	−0.930012	−0.465006	+	0.885307i	$0.653948\pi$
$138$	0	0
$139$	−8.00187e6	−0.252720	−0.126360	−	0.991984i	$-0.540329\pi$
$139$	−8.00187e6	−0.252720	−0.126360	+	0.991984i	$0.540329\pi$
$140$	−2.54248e7	−0.783084
$141$	0	0
$142$	7.92730e6	0.232336
$143$	−2.12721e7	−0.608322
$144$	0	0
$145$	2.91655e7	0.794476
$146$	−8.35110e6	−0.222080
$147$	0	0
$148$	−3.38345e7	−0.857914
$149$	−2.71657e7	−0.672774	−0.336387	−	0.941724i	$-0.609205\pi$
$149$	−2.71657e7	−0.672774	−0.336387	+	0.941724i	$0.609205\pi$
$150$	0	0
$151$	4.19863e7	0.992404	0.496202	−	0.868207i	$-0.334728\pi$
$151$	4.19863e7	0.992404	0.496202	+	0.868207i	$0.334728\pi$
$152$	−1.80643e7	−0.417224
$153$	0	0
$154$	1.43846e7	0.317376
$155$	4.73934e7	1.02225
$156$	0	0
$157$	8.20479e7	1.69207	0.846036	−	0.533126i	$-0.178983\pi$
$157$	8.20479e7	1.69207	0.846036	+	0.533126i	$0.178983\pi$
$158$	−5.86975e7	−1.18391
$159$	0	0
$160$	−1.23070e7	−0.237537
$161$	7.50625e7	1.41753
$162$	0	0
$163$	−2.89286e7	−0.523203	−0.261602	−	0.965176i	$-0.584251\pi$
$163$	−2.89286e7	−0.523203	−0.261602	+	0.965176i	$0.584251\pi$
$164$	−3.31469e7	−0.586800
$165$	0	0
$166$	−3.11377e7	−0.528334
$167$	2.67170e7	0.443894	0.221947	−	0.975059i	$-0.428759\pi$
$167$	2.67170e7	0.443894	0.221947	+	0.975059i	$0.428759\pi$
$168$	0	0
$169$	9.38390e7	1.49548
$170$	−5.07177e7	−0.791751
$171$	0	0
$172$	−1.73862e7	−0.260529
$173$	−2.63122e7	−0.386364	−0.193182	−	0.981163i	$-0.561881\pi$
$173$	−2.63122e7	−0.386364	−0.193182	+	0.981163i	$0.561881\pi$
$174$	0	0
$175$	6.65676e7	0.938922
$176$	6.96291e6	0.0962712
$177$	0	0
$178$	5.40545e6	0.0718392
$179$	−2.19724e7	−0.286347	−0.143174	−	0.989698i	$-0.545731\pi$
$179$	−2.19724e7	−0.286347	−0.143174	+	0.989698i	$0.545731\pi$
$180$	0	0
$181$	1.26202e8	1.58195	0.790975	−	0.611849i	$-0.209574\pi$
$181$	1.26202e8	1.58195	0.790975	+	0.611849i	$0.209574\pi$
$182$	−1.05887e8	−1.30195
$183$	0	0
$184$	3.63343e7	0.429986
$185$	1.98555e8	2.30558
$186$	0	0
$187$	2.86946e7	0.320889
$188$	−2.71122e7	−0.297586
$189$	0	0
$190$	1.06009e8	1.12126
$191$	1.79451e8	1.86350	0.931752	−	0.363096i	$-0.118280\pi$
$191$	1.79451e8	1.86350	0.931752	+	0.363096i	$0.118280\pi$
$192$	0	0
$193$	−2.98314e7	−0.298692	−0.149346	−	0.988785i	$-0.547717\pi$
$193$	−2.98314e7	−0.298692	−0.149346	+	0.988785i	$0.547717\pi$
$194$	1.30046e8	1.27876
$195$	0	0
$196$	1.88964e7	0.179260
$197$	6.70863e7	0.625176	0.312588	−	0.949889i	$-0.398804\pi$
$197$	6.70863e7	0.625176	0.312588	+	0.949889i	$0.398804\pi$
$198$	0	0
$199$	−1.18177e8	−1.06304	−0.531519	−	0.847046i	$-0.678379\pi$
$199$	−1.18177e8	−1.06304	−0.531519	+	0.847046i	$0.678379\pi$
$200$	3.22223e7	0.284808
$201$	0	0
$202$	−3.39632e6	−0.0289920
$203$	−8.21380e7	−0.689141
$204$	0	0
$205$	1.94520e8	1.57698
$206$	−3.89067e7	−0.310091
$207$	0	0
$208$	−5.12553e7	−0.394927
$209$	−5.99768e7	−0.454434
$210$	0	0
$211$	1.03985e8	0.762050	0.381025	−	0.924565i	$-0.375571\pi$
$211$	1.03985e8	0.762050	0.381025	+	0.924565i	$0.375571\pi$
$212$	7.95612e7	0.573490
$213$	0	0
$214$	2.26919e7	0.158278
$215$	1.02030e8	0.700150
$216$	0	0
$217$	−1.33473e8	−0.886715
$218$	5.05111e7	0.330209
$219$	0	0
$220$	−4.08612e7	−0.258721
$221$	−2.11226e8	−1.31636
$222$	0	0
$223$	−1.61530e8	−0.975405	−0.487703	−	0.873010i	$-0.662165\pi$
$223$	−1.61530e8	−0.975405	−0.487703	+	0.873010i	$0.662165\pi$
$224$	3.46598e7	0.206043
$225$	0	0
$226$	1.71550e8	0.988576
$227$	3.80535e7	0.215925	0.107963	−	0.994155i	$-0.465567\pi$
$227$	3.80535e7	0.215925	0.107963	+	0.994155i	$0.465567\pi$
$228$	0	0
$229$	−2.89716e8	−1.59422	−0.797110	−	0.603834i	$-0.793639\pi$
$229$	−2.89716e8	−1.59422	−0.797110	+	0.603834i	$0.793639\pi$
$230$	−2.13225e8	−1.15555
$231$	0	0
$232$	−3.97592e7	−0.209040
$233$	−7.45094e7	−0.385892	−0.192946	−	0.981209i	$-0.561804\pi$
$233$	−7.45094e7	−0.385892	−0.192946	+	0.981209i	$0.561804\pi$
$234$	0	0
$235$	1.59106e8	0.799740
$236$	−1.71631e8	−0.849968
$237$	0	0
$238$	1.42835e8	0.686777
$239$	1.79666e8	0.851280	0.425640	−	0.904893i	$-0.360049\pi$
$239$	1.79666e8	0.851280	0.425640	+	0.904893i	$0.360049\pi$
$240$	0	0
$241$	−6.08682e7	−0.280111	−0.140056	−	0.990144i	$-0.544728\pi$
$241$	−6.08682e7	−0.280111	−0.140056	+	0.990144i	$0.544728\pi$
$242$	−1.32779e8	−0.602250
$243$	0	0
$244$	−4.82611e7	−0.212683
$245$	−1.10892e8	−0.481746
$246$	0	0
$247$	4.41500e8	1.86420
$248$	−6.46081e7	−0.268971
$249$	0	0
$250$	4.56427e7	0.184749
$251$	2.14309e8	0.855426	0.427713	−	0.903915i	$-0.359319\pi$
$251$	2.14309e8	0.855426	0.427713	+	0.903915i	$0.359319\pi$
$252$	0	0
$253$	1.20636e8	0.468334
$254$	1.08545e8	0.415614
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−5.27076e7	−0.193690	−0.0968450	−	0.995299i	$-0.530875\pi$
$257$	−5.27076e7	−0.193690	−0.0968450	+	0.995299i	$0.530875\pi$
$258$	0	0
$259$	−5.59185e8	−1.99989
$260$	3.00787e8	1.06133
$261$	0	0
$262$	−1.88251e8	−0.646670
$263$	4.88578e8	1.65611	0.828055	−	0.560647i	$-0.189448\pi$
$263$	4.88578e8	1.65611	0.828055	+	0.560647i	$0.189448\pi$
$264$	0	0
$265$	−4.66898e8	−1.54121
$266$	−2.98551e8	−0.972596
$267$	0	0
$268$	−1.68045e7	−0.0533278
$269$	−1.69083e8	−0.529622	−0.264811	−	0.964300i	$-0.585310\pi$
$269$	−1.69083e8	−0.529622	−0.264811	+	0.964300i	$0.585310\pi$
$270$	0	0
$271$	−1.99531e8	−0.609001	−0.304501	−	0.952512i	$-0.598490\pi$
$271$	−1.99531e8	−0.609001	−0.304501	+	0.952512i	$0.598490\pi$
$272$	6.91399e7	0.208323
$273$	0	0
$274$	−2.23924e8	−0.657618
$275$	1.06984e8	0.310208
$276$	0	0
$277$	−3.54876e8	−1.00322	−0.501611	−	0.865093i	$-0.667259\pi$
$277$	−3.54876e8	−1.00322	−0.501611	+	0.865093i	$0.667259\pi$
$278$	−6.40150e7	−0.178700
$279$	0	0
$280$	−2.03398e8	−0.553724
$281$	−2.80491e8	−0.754130	−0.377065	−	0.926187i	$-0.623067\pi$
$281$	−2.80491e8	−0.754130	−0.377065	+	0.926187i	$0.623067\pi$
$282$	0	0
$283$	−1.59049e8	−0.417136	−0.208568	−	0.978008i	$-0.566880\pi$
$283$	−1.59049e8	−0.417136	−0.208568	+	0.978008i	$0.566880\pi$
$284$	6.34184e7	0.164286
$285$	0	0
$286$	−1.70176e8	−0.430148
$287$	−5.47822e8	−1.36790
$288$	0	0
$289$	−1.25409e8	−0.305623
$290$	2.33324e8	0.561779
$291$	0	0
$292$	−6.68088e7	−0.157034
$293$	3.44380e8	0.799837	0.399918	−	0.916551i	$-0.369038\pi$
$293$	3.44380e8	0.799837	0.399918	+	0.916551i	$0.369038\pi$
$294$	0	0
$295$	1.00720e9	2.28422
$296$	−2.70676e8	−0.606637
$297$	0	0
$298$	−2.17326e8	−0.475723
$299$	−8.88025e8	−1.92122
$300$	0	0
$301$	−2.87343e8	−0.607321
$302$	3.35890e8	0.701735
$303$	0	0
$304$	−1.44515e8	−0.295022
$305$	2.83216e8	0.571569
$306$	0	0
$307$	5.28744e8	1.04294	0.521472	−	0.853269i	$-0.325383\pi$
$307$	5.28744e8	1.04294	0.521472	+	0.853269i	$0.325383\pi$
$308$	1.15077e8	0.224419
$309$	0	0
$310$	3.79147e8	0.722839
$311$	−2.86764e8	−0.540584	−0.270292	−	0.962778i	$-0.587120\pi$
$311$	−2.86764e8	−0.540584	−0.270292	+	0.962778i	$0.587120\pi$
$312$	0	0
$313$	2.50249e7	0.0461282	0.0230641	−	0.999734i	$-0.492658\pi$
$313$	2.50249e7	0.0461282	0.0230641	+	0.999734i	$0.492658\pi$
$314$	6.56383e8	1.19648
$315$	0	0
$316$	−4.69580e8	−0.837153
$317$	−5.20273e8	−0.917327	−0.458663	−	0.888610i	$-0.651672\pi$
$317$	−5.20273e8	−0.917327	−0.458663	+	0.888610i	$0.651672\pi$
$318$	0	0
$319$	−1.32008e8	−0.227684
$320$	−9.84556e7	−0.167964
$321$	0	0
$322$	6.00500e8	1.00234
$323$	−5.95554e8	−0.983359
$324$	0	0
$325$	−7.87527e8	−1.27255
$326$	−2.31429e8	−0.369961
$327$	0	0
$328$	−2.65176e8	−0.414930
$329$	−4.48086e8	−0.693707
$330$	0	0
$331$	−5.05039e8	−0.765468	−0.382734	−	0.923859i	$-0.625017\pi$
$331$	−5.05039e8	−0.765468	−0.382734	+	0.923859i	$0.625017\pi$
$332$	−2.49102e8	−0.373588
$333$	0	0
$334$	2.13736e8	0.313881
$335$	9.86157e7	0.143314
$336$	0	0
$337$	−1.83750e8	−0.261531	−0.130765	−	0.991413i	$-0.541743\pi$
$337$	−1.83750e8	−0.261531	−0.130765	+	0.991413i	$0.541743\pi$
$338$	7.50712e8	1.05746
$339$	0	0
$340$	−4.05742e8	−0.559852
$341$	−2.14510e8	−0.292960
$342$	0	0
$343$	−5.58786e8	−0.747681
$344$	−1.39090e8	−0.184222
$345$	0	0
$346$	−2.10498e8	−0.273200
$347$	−2.86132e8	−0.367632	−0.183816	−	0.982961i	$-0.558845\pi$
$347$	−2.86132e8	−0.367632	−0.183816	+	0.982961i	$0.558845\pi$
$348$	0	0
$349$	−3.66365e7	−0.0461344	−0.0230672	−	0.999734i	$-0.507343\pi$
$349$	−3.66365e7	−0.0461344	−0.0230672	+	0.999734i	$0.507343\pi$
$350$	5.32541e8	0.663918
$351$	0	0
$352$	5.57033e7	0.0680740
$353$	4.76938e8	0.577099	0.288549	−	0.957465i	$-0.406827\pi$
$353$	4.76938e8	0.577099	0.288549	+	0.957465i	$0.406827\pi$
$354$	0	0
$355$	−3.72166e8	−0.441506
$356$	4.32436e7	0.0507980
$357$	0	0
$358$	−1.75780e8	−0.202478
$359$	−3.73983e8	−0.426600	−0.213300	−	0.976987i	$-0.568421\pi$
$359$	−3.73983e8	−0.426600	−0.213300	+	0.976987i	$0.568421\pi$
$360$	0	0
$361$	3.50942e8	0.392609
$362$	1.00962e9	1.11861
$363$	0	0
$364$	−8.47100e8	−0.920619
$365$	3.92062e8	0.422016
$366$	0	0
$367$	−5.83248e8	−0.615917	−0.307958	−	0.951400i	$-0.599646\pi$
$367$	−5.83248e8	−0.615917	−0.307958	+	0.951400i	$0.599646\pi$
$368$	2.90674e8	0.304046
$369$	0	0
$370$	1.58844e9	1.63029
$371$	1.31491e9	1.33687
$372$	0	0
$373$	1.47721e9	1.47388	0.736939	−	0.675959i	$-0.236270\pi$
$373$	1.47721e9	1.47388	0.736939	+	0.675959i	$0.236270\pi$
$374$	2.29557e8	0.226902
$375$	0	0
$376$	−2.16898e8	−0.210425
$377$	9.71732e8	0.934011
$378$	0	0
$379$	−6.08958e8	−0.574580	−0.287290	−	0.957844i	$-0.592754\pi$
$379$	−6.08958e8	−0.574580	−0.287290	+	0.957844i	$0.592754\pi$
$380$	8.48072e8	0.792849
$381$	0	0
$382$	1.43561e9	1.31770
$383$	−8.48714e8	−0.771908	−0.385954	−	0.922518i	$-0.626128\pi$
$383$	−8.48714e8	−0.771908	−0.385954	+	0.922518i	$0.626128\pi$
$384$	0	0
$385$	−6.75317e8	−0.603108
$386$	−2.38651e8	−0.211207
$387$	0	0
$388$	1.04036e9	0.904221
$389$	−2.53303e8	−0.218181	−0.109090	−	0.994032i	$-0.534794\pi$
$389$	−2.53303e8	−0.218181	−0.109090	+	0.994032i	$0.534794\pi$
$390$	0	0
$391$	1.19789e9	1.01344
$392$	1.51171e8	0.126756
$393$	0	0
$394$	5.36691e8	0.442066
$395$	2.75569e9	2.24978
$396$	0	0
$397$	9.92215e8	0.795865	0.397932	−	0.917415i	$-0.369728\pi$
$397$	9.92215e8	0.795865	0.397932	+	0.917415i	$0.369728\pi$
$398$	−9.45420e8	−0.751682
$399$	0	0
$400$	2.57778e8	0.201389
$401$	−9.48753e8	−0.734764	−0.367382	−	0.930070i	$-0.619746\pi$
$401$	−9.48753e8	−0.734764	−0.367382	+	0.930070i	$0.619746\pi$
$402$	0	0
$403$	1.57905e9	1.20179
$404$	−2.71706e7	−0.0205005
$405$	0	0
$406$	−6.57104e8	−0.487296
$407$	−8.98691e8	−0.660739
$408$	0	0
$409$	7.76766e8	0.561382	0.280691	−	0.959798i	$-0.409436\pi$
$409$	7.76766e8	0.561382	0.280691	+	0.959798i	$0.409436\pi$
$410$	1.55616e9	1.11509
$411$	0	0
$412$	−3.11254e8	−0.219268
$413$	−2.83655e9	−1.98137
$414$	0	0
$415$	1.46183e9	1.00399
$416$	−4.10042e8	−0.279256
$417$	0	0
$418$	−4.79814e8	−0.321334
$419$	−6.29887e8	−0.418325	−0.209162	−	0.977881i	$-0.567074\pi$
$419$	−6.29887e8	−0.418325	−0.209162	+	0.977881i	$0.567074\pi$
$420$	0	0
$421$	1.10485e9	0.721633	0.360816	−	0.932637i	$-0.382498\pi$
$421$	1.10485e9	0.721633	0.360816	+	0.932637i	$0.382498\pi$
$422$	8.31883e8	0.538851
$423$	0	0
$424$	6.36490e8	0.405519
$425$	1.06232e9	0.671266
$426$	0	0
$427$	−7.97615e8	−0.495788
$428$	1.81535e8	0.111920
$429$	0	0
$430$	8.16236e8	0.495081
$431$	−2.85661e8	−0.171863	−0.0859313	−	0.996301i	$-0.527387\pi$
$431$	−2.85661e8	−0.171863	−0.0859313	+	0.996301i	$0.527387\pi$
$432$	0	0
$433$	4.76788e8	0.282239	0.141120	−	0.989993i	$-0.454930\pi$
$433$	4.76788e8	0.282239	0.141120	+	0.989993i	$0.454930\pi$
$434$	−1.06778e9	−0.627002
$435$	0	0
$436$	4.04089e8	0.233493
$437$	−2.50380e9	−1.43520
$438$	0	0
$439$	2.51758e9	1.42022	0.710112	−	0.704089i	$-0.248644\pi$
$439$	2.51758e9	1.42022	0.710112	+	0.704089i	$0.248644\pi$
$440$	−3.26890e8	−0.182944
$441$	0	0
$442$	−1.68981e9	−0.930807
$443$	−4.58648e8	−0.250649	−0.125325	−	0.992116i	$-0.539997\pi$
$443$	−4.58648e8	−0.250649	−0.125325	+	0.992116i	$0.539997\pi$
$444$	0	0
$445$	−2.53771e8	−0.136516
$446$	−1.29224e9	−0.689716
$447$	0	0
$448$	2.77278e8	0.145694
$449$	−1.46433e9	−0.763444	−0.381722	−	0.924277i	$-0.624669\pi$
$449$	−1.46433e9	−0.763444	−0.381722	+	0.924277i	$0.624669\pi$
$450$	0	0
$451$	−8.80429e8	−0.451936
$452$	1.37240e9	0.699029
$453$	0	0
$454$	3.04428e8	0.152682
$455$	4.97113e9	2.47409
$456$	0	0
$457$	1.39660e9	0.684486	0.342243	−	0.939611i	$-0.388813\pi$
$457$	1.39660e9	0.684486	0.342243	+	0.939611i	$0.388813\pi$
$458$	−2.31773e9	−1.12728
$459$	0	0
$460$	−1.70580e9	−0.817100
$461$	−3.96221e9	−1.88358	−0.941789	−	0.336204i	$-0.890857\pi$
$461$	−3.96221e9	−1.88358	−0.941789	+	0.336204i	$0.890857\pi$
$462$	0	0
$463$	−1.88163e9	−0.881048	−0.440524	−	0.897741i	$-0.645207\pi$
$463$	−1.88163e9	−0.881048	−0.440524	+	0.897741i	$0.645207\pi$
$464$	−3.18074e8	−0.147814
$465$	0	0
$466$	−5.96076e8	−0.272867
$467$	9.69518e8	0.440501	0.220250	−	0.975443i	$-0.429313\pi$
$467$	9.69518e8	0.440501	0.220250	+	0.975443i	$0.429313\pi$
$468$	0	0
$469$	−2.77729e8	−0.124313
$470$	1.27285e9	0.565501
$471$	0	0
$472$	−1.37304e9	−0.601018
$473$	−4.61802e8	−0.200651
$474$	0	0
$475$	−2.22044e9	−0.950629
$476$	1.14268e9	0.485624
$477$	0	0
$478$	1.43733e9	0.601946
$479$	−2.00870e9	−0.835104	−0.417552	−	0.908653i	$-0.637112\pi$
$479$	−2.00870e9	−0.835104	−0.417552	+	0.908653i	$0.637112\pi$
$480$	0	0
$481$	6.61543e9	2.71051
$482$	−4.86945e8	−0.198069
$483$	0	0
$484$	−1.06223e9	−0.425855
$485$	−6.10529e9	−2.43002
$486$	0	0
$487$	3.02884e9	1.18830	0.594148	−	0.804356i	$-0.297490\pi$
$487$	3.02884e9	1.18830	0.594148	+	0.804356i	$0.297490\pi$
$488$	−3.86089e8	−0.150390
$489$	0	0
$490$	−8.87135e8	−0.340646
$491$	4.51723e9	1.72221	0.861107	−	0.508424i	$-0.169772\pi$
$491$	4.51723e9	1.72221	0.861107	+	0.508424i	$0.169772\pi$
$492$	0	0
$493$	−1.31080e9	−0.492689
$494$	3.53200e9	1.31819
$495$	0	0
$496$	−5.16865e8	−0.190191
$497$	1.04812e9	0.382970
$498$	0	0
$499$	−4.65664e9	−1.67772	−0.838862	−	0.544343i	$-0.816779\pi$
$499$	−4.65664e9	−1.67772	−0.838862	+	0.544343i	$0.816779\pi$
$500$	3.65142e8	0.130637
$501$	0	0
$502$	1.71447e9	0.604878
$503$	2.88413e9	1.01048	0.505239	−	0.862980i	$-0.331404\pi$
$503$	2.88413e9	1.01048	0.505239	+	0.862980i	$0.331404\pi$
$504$	0	0
$505$	1.59448e8	0.0550934
$506$	9.65089e8	0.331162
$507$	0	0
$508$	8.68358e8	0.293884
$509$	−1.32072e9	−0.443915	−0.221957	−	0.975056i	$-0.571245\pi$
$509$	−1.32072e9	−0.443915	−0.221957	+	0.975056i	$0.571245\pi$
$510$	0	0
$511$	−1.10415e9	−0.366064
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	−4.21661e8	−0.136960
$515$	1.82657e9	0.589264
$516$	0	0
$517$	−7.20139e8	−0.229192
$518$	−4.47348e9	−1.41414
$519$	0	0
$520$	2.40630e9	0.750477
$521$	5.14654e9	1.59435	0.797175	−	0.603749i	$-0.206327\pi$
$521$	5.14654e9	1.59435	0.797175	+	0.603749i	$0.206327\pi$
$522$	0	0
$523$	−2.17050e9	−0.663444	−0.331722	−	0.943377i	$-0.607630\pi$
$523$	−2.17050e9	−0.663444	−0.331722	+	0.943377i	$0.607630\pi$
$524$	−1.50601e9	−0.457265
$525$	0	0
$526$	3.90863e9	1.17105
$527$	−2.13003e9	−0.633941
$528$	0	0
$529$	1.63126e9	0.479103
$530$	−3.73518e9	−1.08980
$531$	0	0
$532$	−2.38841e9	−0.687729
$533$	6.48100e9	1.85394
$534$	0	0
$535$	−1.06532e9	−0.300775
$536$	−1.34436e8	−0.0377084
$537$	0	0
$538$	−1.35266e9	−0.374499
$539$	5.01915e8	0.138060
$540$	0	0
$541$	−9.27955e7	−0.0251963	−0.0125981	−	0.999921i	$-0.504010\pi$
$541$	−9.27955e7	−0.0251963	−0.0125981	+	0.999921i	$0.504010\pi$
$542$	−1.59625e9	−0.430629
$543$	0	0
$544$	5.53119e8	0.147307
$545$	−2.37136e9	−0.627495
$546$	0	0
$547$	−5.47036e9	−1.42909	−0.714547	−	0.699588i	$-0.753367\pi$
$547$	−5.47036e9	−1.42909	−0.714547	+	0.699588i	$0.753367\pi$
$548$	−1.79139e9	−0.465006
$549$	0	0
$550$	8.55870e8	0.219350
$551$	2.73981e9	0.697734
$552$	0	0
$553$	−7.76079e9	−1.95150
$554$	−2.83901e9	−0.709385
$555$	0	0
$556$	−5.12120e8	−0.126360
$557$	4.50060e9	1.10351	0.551756	−	0.834006i	$-0.313958\pi$
$557$	4.50060e9	1.10351	0.551756	+	0.834006i	$0.313958\pi$
$558$	0	0
$559$	3.39941e9	0.823119
$560$	−1.62718e9	−0.391542
$561$	0	0
$562$	−2.24393e9	−0.533251
$563$	6.13057e9	1.44784	0.723922	−	0.689882i	$-0.242338\pi$
$563$	6.13057e9	1.44784	0.723922	+	0.689882i	$0.242338\pi$
$564$	0	0
$565$	−8.05379e9	−1.87858
$566$	−1.27239e9	−0.294960
$567$	0	0
$568$	5.07347e8	0.116168
$569$	−3.42118e9	−0.778543	−0.389271	−	0.921123i	$-0.627273\pi$
$569$	−3.42118e9	−0.778543	−0.389271	+	0.921123i	$0.627273\pi$
$570$	0	0
$571$	2.28585e9	0.513831	0.256916	−	0.966434i	$-0.417294\pi$
$571$	2.28585e9	0.513831	0.256916	+	0.966434i	$0.417294\pi$
$572$	−1.36141e9	−0.304161
$573$	0	0
$574$	−4.38258e9	−0.967248
$575$	4.46615e9	0.979706
$576$	0	0
$577$	−7.03121e9	−1.52375	−0.761877	−	0.647722i	$-0.775722\pi$
$577$	−7.03121e9	−1.52375	−0.761877	+	0.647722i	$0.775722\pi$
$578$	−1.00327e9	−0.216108
$579$	0	0
$580$	1.86659e9	0.397238
$581$	−4.11692e9	−0.870876
$582$	0	0
$583$	2.11326e9	0.441685
$584$	−5.34470e8	−0.111040
$585$	0	0
$586$	2.75504e9	0.565570
$587$	−8.94315e8	−0.182498	−0.0912488	−	0.995828i	$-0.529086\pi$
$587$	−8.94315e8	−0.182498	−0.0912488	+	0.995828i	$0.529086\pi$
$588$	0	0
$589$	4.45214e9	0.897771
$590$	8.05759e9	1.61519
$591$	0	0
$592$	−2.16541e9	−0.428957
$593$	−8.80907e9	−1.73476	−0.867379	−	0.497649i	$-0.834197\pi$
$593$	−8.80907e9	−1.73476	−0.867379	+	0.497649i	$0.834197\pi$
$594$	0	0
$595$	−6.70572e9	−1.30508
$596$	−1.73860e9	−0.336387
$597$	0	0
$598$	−7.10420e9	−1.35850
$599$	9.83190e8	0.186915	0.0934574	−	0.995623i	$-0.470208\pi$
$599$	9.83190e8	0.186915	0.0934574	+	0.995623i	$0.470208\pi$
$600$	0	0
$601$	2.04472e9	0.384214	0.192107	−	0.981374i	$-0.438468\pi$
$601$	2.04472e9	0.384214	0.192107	+	0.981374i	$0.438468\pi$
$602$	−2.29875e9	−0.429441
$603$	0	0
$604$	2.68712e9	0.496202
$605$	6.23363e9	1.14445
$606$	0	0
$607$	3.57611e9	0.649009	0.324505	−	0.945884i	$-0.394802\pi$
$607$	3.57611e9	0.649009	0.324505	+	0.945884i	$0.394802\pi$
$608$	−1.15612e9	−0.208612
$609$	0	0
$610$	2.26573e9	0.404160
$611$	5.30108e9	0.940199
$612$	0	0
$613$	1.22567e9	0.214913	0.107457	−	0.994210i	$-0.465729\pi$
$613$	1.22567e9	0.214913	0.107457	+	0.994210i	$0.465729\pi$
$614$	4.22995e9	0.737472
$615$	0	0
$616$	9.20613e8	0.158688
$617$	7.60333e9	1.30318	0.651592	−	0.758570i	$-0.274101\pi$
$617$	7.60333e9	1.30318	0.651592	+	0.758570i	$0.274101\pi$
$618$	0	0
$619$	5.75700e9	0.975616	0.487808	−	0.872951i	$-0.337797\pi$
$619$	5.75700e9	0.975616	0.487808	+	0.872951i	$0.337797\pi$
$620$	3.03318e9	0.511125
$621$	0	0
$622$	−2.29411e9	−0.382250
$623$	7.14690e8	0.118416
$624$	0	0
$625$	−7.05954e9	−1.15663
$626$	2.00199e8	0.0326176
$627$	0	0
$628$	5.25107e9	0.846036
$629$	−8.92377e9	−1.42979
$630$	0	0
$631$	−2.41912e9	−0.383313	−0.191657	−	0.981462i	$-0.561386\pi$
$631$	−2.41912e9	−0.383313	−0.191657	+	0.981462i	$0.561386\pi$
$632$	−3.75664e9	−0.591957
$633$	0	0
$634$	−4.16219e9	−0.648648
$635$	−5.09588e9	−0.789789
$636$	0	0
$637$	−3.69469e9	−0.566356
$638$	−1.05606e9	−0.160997
$639$	0	0
$640$	−7.87645e8	−0.118768
$641$	−7.91782e9	−1.18741	−0.593707	−	0.804681i	$-0.702336\pi$
$641$	−7.91782e9	−1.18741	−0.593707	+	0.804681i	$0.702336\pi$
$642$	0	0
$643$	−1.07171e10	−1.58978	−0.794891	−	0.606752i	$-0.792472\pi$
$643$	−1.07171e10	−1.58978	−0.794891	+	0.606752i	$0.792472\pi$
$644$	4.80400e9	0.708765
$645$	0	0
$646$	−4.76443e9	−0.695340
$647$	8.06643e9	1.17089	0.585446	−	0.810712i	$-0.300920\pi$
$647$	8.06643e9	1.17089	0.585446	+	0.810712i	$0.300920\pi$
$648$	0	0
$649$	−4.55875e9	−0.654620
$650$	−6.30021e9	−0.899826
$651$	0	0
$652$	−1.85143e9	−0.261602
$653$	1.00136e10	1.40732	0.703661	−	0.710535i	$-0.251547\pi$
$653$	1.00136e10	1.40732	0.703661	+	0.710535i	$0.251547\pi$
$654$	0	0
$655$	8.83787e9	1.22886
$656$	−2.12140e9	−0.293400
$657$	0	0
$658$	−3.58469e9	−0.490525
$659$	1.41114e10	1.92075	0.960376	−	0.278707i	$-0.0899057\pi$
$659$	1.41114e10	1.92075	0.960376	+	0.278707i	$0.0899057\pi$
$660$	0	0
$661$	9.62149e9	1.29580	0.647899	−	0.761726i	$-0.275648\pi$
$661$	9.62149e9	1.29580	0.647899	+	0.761726i	$0.275648\pi$
$662$	−4.04031e9	−0.541268
$663$	0	0
$664$	−1.99281e9	−0.264167
$665$	1.40162e10	1.84822
$666$	0	0
$667$	−5.51080e9	−0.719075
$668$	1.70989e9	0.221947
$669$	0	0
$670$	7.88926e8	0.101338
$671$	−1.28188e9	−0.163802
$672$	0	0
$673$	1.16858e10	1.47777	0.738884	−	0.673833i	$-0.235353\pi$
$673$	1.16858e10	1.47777	0.738884	+	0.673833i	$0.235353\pi$
$674$	−1.47000e9	−0.184930
$675$	0	0
$676$	6.00569e9	0.747738
$677$	−1.11144e10	−1.37665	−0.688327	−	0.725400i	$-0.741655\pi$
$677$	−1.11144e10	−1.37665	−0.688327	+	0.725400i	$0.741655\pi$
$678$	0	0
$679$	1.71942e10	2.10784
$680$	−3.24593e9	−0.395875
$681$	0	0
$682$	−1.71608e9	−0.207154
$683$	−1.61450e10	−1.93894	−0.969472	−	0.245201i	$-0.921146\pi$
$683$	−1.61450e10	−1.93894	−0.969472	+	0.245201i	$0.921146\pi$
$684$	0	0
$685$	1.05126e10	1.24967
$686$	−4.47029e9	−0.528690
$687$	0	0
$688$	−1.11272e9	−0.130264
$689$	−1.55561e10	−1.81189
$690$	0	0
$691$	−1.39780e9	−0.161166	−0.0805829	−	0.996748i	$-0.525678\pi$
$691$	−1.39780e9	−0.161166	−0.0805829	+	0.996748i	$0.525678\pi$
$692$	−1.68398e9	−0.193182
$693$	0	0
$694$	−2.28905e9	−0.259955
$695$	3.00533e9	0.339583
$696$	0	0
$697$	−8.74243e9	−0.977952
$698$	−2.93092e8	−0.0326219
$699$	0	0
$700$	4.26033e9	0.469461
$701$	−1.34244e9	−0.147191	−0.0735956	−	0.997288i	$-0.523447\pi$
$701$	−1.34244e9	−0.147191	−0.0735956	+	0.997288i	$0.523447\pi$
$702$	0	0
$703$	1.86523e10	2.02483
$704$	4.45626e8	0.0481356
$705$	0	0
$706$	3.81550e9	0.408071
$707$	−4.49050e8	−0.0477889
$708$	0	0
$709$	−1.58631e10	−1.67157	−0.835786	−	0.549055i	$-0.814988\pi$
$709$	−1.58631e10	−1.67157	−0.835786	+	0.549055i	$0.814988\pi$
$710$	−2.97732e9	−0.312192
$711$	0	0
$712$	3.45949e8	0.0359196
$713$	−8.95496e9	−0.925231
$714$	0	0
$715$	7.98933e9	0.817408
$716$	−1.40624e9	−0.143174
$717$	0	0
$718$	−2.99186e9	−0.301652
$719$	2.14766e9	0.215483	0.107742	−	0.994179i	$-0.465638\pi$
$719$	2.14766e9	0.215483	0.107742	+	0.994179i	$0.465638\pi$
$720$	0	0
$721$	−5.14412e9	−0.511137
$722$	2.80754e9	0.277616
$723$	0	0
$724$	8.07695e9	0.790975
$725$	−4.88714e9	−0.476290
$726$	0	0
$727$	−1.02867e10	−0.992903	−0.496451	−	0.868065i	$-0.665364\pi$
$727$	−1.02867e10	−0.992903	−0.496451	+	0.868065i	$0.665364\pi$
$728$	−6.77680e9	−0.650976
$729$	0	0
$730$	3.13649e9	0.298411
$731$	−4.58558e9	−0.434193
$732$	0	0
$733$	−3.20421e9	−0.300509	−0.150254	−	0.988647i	$-0.548009\pi$
$733$	−3.20421e9	−0.300509	−0.150254	+	0.988647i	$0.548009\pi$
$734$	−4.66598e9	−0.435519
$735$	0	0
$736$	2.32539e9	0.214993
$737$	−4.46351e8	−0.0410715
$738$	0	0
$739$	1.29766e10	1.18279	0.591394	−	0.806383i	$-0.298578\pi$
$739$	1.29766e10	1.18279	0.591394	+	0.806383i	$0.298578\pi$
$740$	1.27075e10	1.15279
$741$	0	0
$742$	1.05193e10	0.945309
$743$	7.24706e9	0.648188	0.324094	−	0.946025i	$-0.394941\pi$
$743$	7.24706e9	0.648188	0.324094	+	0.946025i	$0.394941\pi$
$744$	0	0
$745$	1.02029e10	0.904013
$746$	1.18177e10	1.04219
$747$	0	0
$748$	1.83645e9	0.160444
$749$	3.00024e9	0.260897
$750$	0	0
$751$	−8.94246e9	−0.770402	−0.385201	−	0.922833i	$-0.625868\pi$
$751$	−8.94246e9	−0.770402	−0.385201	+	0.922833i	$0.625868\pi$
$752$	−1.73518e9	−0.148793
$753$	0	0
$754$	7.77386e9	0.660445
$755$	−1.57692e10	−1.33350
$756$	0	0
$757$	1.36108e10	1.14037	0.570187	−	0.821515i	$-0.306871\pi$
$757$	1.36108e10	1.14037	0.570187	+	0.821515i	$0.306871\pi$
$758$	−4.87167e9	−0.406289
$759$	0	0
$760$	6.78458e9	0.560629
$761$	−2.20245e10	−1.81159	−0.905794	−	0.423718i	$-0.860725\pi$
$761$	−2.20245e10	−1.81159	−0.905794	+	0.423718i	$0.860725\pi$
$762$	0	0
$763$	6.67841e9	0.544299
$764$	1.14849e10	0.931752
$765$	0	0
$766$	−6.78971e9	−0.545821
$767$	3.35578e10	2.68540
$768$	0	0
$769$	1.03422e10	0.820110	0.410055	−	0.912061i	$-0.365510\pi$
$769$	1.03422e10	0.820110	0.410055	+	0.912061i	$0.365510\pi$
$770$	−5.40254e9	−0.426462
$771$	0	0
$772$	−1.90921e9	−0.149346
$773$	3.65169e9	0.284358	0.142179	−	0.989841i	$-0.454589\pi$
$773$	3.65169e9	0.284358	0.142179	+	0.989841i	$0.454589\pi$
$774$	0	0
$775$	−7.94152e9	−0.612841
$776$	8.32292e9	0.639381
$777$	0	0
$778$	−2.02642e9	−0.154277
$779$	1.82732e10	1.38495
$780$	0	0
$781$	1.68448e9	0.126528
$782$	9.58309e9	0.716608
$783$	0	0
$784$	1.20937e9	0.0896299
$785$	−3.08154e10	−2.27365
$786$	0	0
$787$	2.39073e10	1.74832	0.874158	−	0.485642i	$-0.161414\pi$
$787$	2.39073e10	1.74832	0.874158	+	0.485642i	$0.161414\pi$
$788$	4.29353e9	0.312588
$789$	0	0
$790$	2.20455e10	1.59084
$791$	2.26817e10	1.62951
$792$	0	0
$793$	9.43617e9	0.671954
$794$	7.93772e9	0.562761
$795$	0	0
$796$	−7.56336e9	−0.531519
$797$	2.72163e10	1.90426	0.952128	−	0.305699i	$-0.0988901\pi$
$797$	2.72163e10	1.90426	0.952128	+	0.305699i	$0.0988901\pi$
$798$	0	0
$799$	−7.15080e9	−0.495953
$800$	2.06223e9	0.142404
$801$	0	0
$802$	−7.59002e9	−0.519556
$803$	−1.77453e9	−0.120943
$804$	0	0
$805$	−2.81918e10	−1.90475
$806$	1.26324e10	0.849792
$807$	0	0
$808$	−2.17365e8	−0.0144960
$809$	−2.09999e10	−1.39443	−0.697216	−	0.716861i	$-0.745578\pi$
$809$	−2.09999e10	−1.39443	−0.697216	+	0.716861i	$0.745578\pi$
$810$	0	0
$811$	−1.21414e10	−0.799276	−0.399638	−	0.916673i	$-0.630864\pi$
$811$	−1.21414e10	−0.799276	−0.399638	+	0.916673i	$0.630864\pi$
$812$	−5.25683e9	−0.344570
$813$	0	0
$814$	−7.18953e9	−0.467213
$815$	1.08650e10	0.703034
$816$	0	0
$817$	9.58468e9	0.614894
$818$	6.21413e9	0.396957
$819$	0	0
$820$	1.24493e10	0.788489
$821$	−1.13220e10	−0.714041	−0.357020	−	0.934097i	$-0.616207\pi$
$821$	−1.13220e10	−0.714041	−0.357020	+	0.934097i	$0.616207\pi$
$822$	0	0
$823$	−5.82799e9	−0.364434	−0.182217	−	0.983258i	$-0.558327\pi$
$823$	−5.82799e9	−0.364434	−0.182217	+	0.983258i	$0.558327\pi$
$824$	−2.49003e9	−0.155046
$825$	0	0
$826$	−2.26924e10	−1.40104
$827$	2.11794e10	1.30210	0.651050	−	0.759034i	$-0.274329\pi$
$827$	2.11794e10	1.30210	0.651050	+	0.759034i	$0.274329\pi$
$828$	0	0
$829$	−1.77492e10	−1.08203	−0.541013	−	0.841014i	$-0.681959\pi$
$829$	−1.77492e10	−1.08203	−0.541013	+	0.841014i	$0.681959\pi$
$830$	1.16947e10	0.709927
$831$	0	0
$832$	−3.28034e9	−0.197463
$833$	4.98388e9	0.298752
$834$	0	0
$835$	−1.00343e10	−0.596465
$836$	−3.83851e9	−0.227217
$837$	0	0
$838$	−5.03909e9	−0.295800
$839$	−2.63821e10	−1.54221	−0.771104	−	0.636710i	$-0.780295\pi$
$839$	−2.63821e10	−1.54221	−0.771104	+	0.636710i	$0.780295\pi$
$840$	0	0
$841$	−1.12196e10	−0.650417
$842$	8.83881e9	0.510271
$843$	0	0
$844$	6.65506e9	0.381025
$845$	−3.52439e10	−2.00949
$846$	0	0
$847$	−1.75556e10	−0.992715
$848$	5.09192e9	0.286745
$849$	0	0
$850$	8.49856e9	0.474656
$851$	−3.75168e10	−2.08676
$852$	0	0
$853$	2.67806e10	1.47740	0.738701	−	0.674033i	$-0.235440\pi$
$853$	2.67806e10	1.47740	0.738701	+	0.674033i	$0.235440\pi$
$854$	−6.38092e9	−0.350575
$855$	0	0
$856$	1.45228e9	0.0791392
$857$	1.28880e10	0.699442	0.349721	−	0.936854i	$-0.386276\pi$
$857$	1.28880e10	0.699442	0.349721	+	0.936854i	$0.386276\pi$
$858$	0	0
$859$	4.27588e9	0.230170	0.115085	−	0.993356i	$-0.463286\pi$
$859$	4.27588e9	0.230170	0.115085	+	0.993356i	$0.463286\pi$
$860$	6.52989e9	0.350075
$861$	0	0
$862$	−2.28529e9	−0.121525
$863$	−2.63998e10	−1.39818	−0.699089	−	0.715035i	$-0.746411\pi$
$863$	−2.63998e10	−1.39818	−0.699089	+	0.715035i	$0.746411\pi$
$864$	0	0
$865$	9.88231e9	0.519161
$866$	3.81430e9	0.199573
$867$	0	0
$868$	−8.54227e9	−0.443357
$869$	−1.24727e10	−0.644750
$870$	0	0
$871$	3.28567e9	0.168485
$872$	3.23271e9	0.165105
$873$	0	0
$874$	−2.00304e10	−1.01484
$875$	6.03472e9	0.304529
$876$	0	0
$877$	1.43925e10	0.720505	0.360252	−	0.932855i	$-0.382691\pi$
$877$	1.43925e10	0.720505	0.360252	+	0.932855i	$0.382691\pi$
$878$	2.01406e10	1.00425
$879$	0	0
$880$	−2.61512e9	−0.129361
$881$	2.86878e10	1.41345	0.706727	−	0.707487i	$-0.250171\pi$
$881$	2.86878e10	1.41345	0.706727	+	0.707487i	$0.250171\pi$
$882$	0	0
$883$	1.10176e10	0.538550	0.269275	−	0.963063i	$-0.413216\pi$
$883$	1.10176e10	0.538550	0.269275	+	0.963063i	$0.413216\pi$
$884$	−1.35185e10	−0.658180
$885$	0	0
$886$	−3.66919e9	−0.177236
$887$	1.75507e10	0.844426	0.422213	−	0.906497i	$-0.361254\pi$
$887$	1.75507e10	0.844426	0.422213	+	0.906497i	$0.361254\pi$
$888$	0	0
$889$	1.43514e10	0.685076
$890$	−2.03017e9	−0.0965311
$891$	0	0
$892$	−1.03379e10	−0.487703
$893$	1.49464e10	0.702357
$894$	0	0
$895$	8.25238e9	0.384768
$896$	2.21823e9	0.103022
$897$	0	0
$898$	−1.17147e10	−0.539837
$899$	9.79907e9	0.449807
$900$	0	0
$901$	2.09841e10	0.955770
$902$	−7.04343e9	−0.319567
$903$	0	0
$904$	1.09792e10	0.494288
$905$	−4.73989e10	−2.12568
$906$	0	0
$907$	−1.68445e10	−0.749605	−0.374803	−	0.927105i	$-0.622290\pi$
$907$	−1.68445e10	−0.749605	−0.374803	+	0.927105i	$0.622290\pi$
$908$	2.43542e9	0.107963
$909$	0	0
$910$	3.97691e10	1.74945
$911$	1.47271e10	0.645363	0.322682	−	0.946508i	$-0.395416\pi$
$911$	1.47271e10	0.645363	0.322682	+	0.946508i	$0.395416\pi$
$912$	0	0
$913$	−6.61649e9	−0.287726
$914$	1.11728e10	0.484005
$915$	0	0
$916$	−1.85418e10	−0.797110
$917$	−2.48899e10	−1.06593
$918$	0	0
$919$	7.11506e9	0.302395	0.151197	−	0.988504i	$-0.451687\pi$
$919$	7.11506e9	0.302395	0.151197	+	0.988504i	$0.451687\pi$
$920$	−1.36464e10	−0.577777
$921$	0	0
$922$	−3.16976e10	−1.33189
$923$	−1.23998e10	−0.519049
$924$	0	0
$925$	−3.32710e10	−1.38220
$926$	−1.50530e10	−0.622995
$927$	0	0
$928$	−2.54459e9	−0.104520
$929$	−3.39405e10	−1.38887	−0.694437	−	0.719553i	$-0.744347\pi$
$929$	−3.39405e10	−1.38887	−0.694437	+	0.719553i	$0.744347\pi$
$930$	0	0
$931$	−1.04172e10	−0.423085
$932$	−4.76860e9	−0.192946
$933$	0	0
$934$	7.75614e9	0.311481
$935$	−1.07771e10	−0.431181
$936$	0	0
$937$	−3.78364e10	−1.50252	−0.751261	−	0.660005i	$-0.770554\pi$
$937$	−3.78364e10	−1.50252	−0.751261	+	0.660005i	$0.770554\pi$
$938$	−2.22183e9	−0.0879025
$939$	0	0
$940$	1.01828e10	0.399870
$941$	1.68892e10	0.660763	0.330382	−	0.943847i	$-0.392823\pi$
$941$	1.68892e10	0.660763	0.330382	+	0.943847i	$0.392823\pi$
$942$	0	0
$943$	−3.67545e10	−1.42731
$944$	−1.09844e10	−0.424984
$945$	0	0
$946$	−3.69442e9	−0.141882
$947$	1.16912e10	0.447337	0.223668	−	0.974665i	$-0.428197\pi$
$947$	1.16912e10	0.447337	0.223668	+	0.974665i	$0.428197\pi$
$948$	0	0
$949$	1.30627e10	0.496136
$950$	−1.77635e10	−0.672197
$951$	0	0
$952$	9.14145e9	0.343388
$953$	5.10551e10	1.91079	0.955397	−	0.295323i	$-0.0954274\pi$
$953$	5.10551e10	1.91079	0.955397	+	0.295323i	$0.0954274\pi$
$954$	0	0
$955$	−6.73981e10	−2.50401
$956$	1.14986e10	0.425640
$957$	0	0
$958$	−1.60696e10	−0.590508
$959$	−2.96065e10	−1.08398
$960$	0	0
$961$	−1.15893e10	−0.421235
$962$	5.29235e10	1.91662
$963$	0	0
$964$	−3.89556e9	−0.140056
$965$	1.12040e10	0.401355
$966$	0	0
$967$	−8.68730e9	−0.308953	−0.154476	−	0.987996i	$-0.549369\pi$
$967$	−8.68730e9	−0.308953	−0.154476	+	0.987996i	$0.549369\pi$
$968$	−8.49787e9	−0.301125
$969$	0	0
$970$	−4.88423e10	−1.71829
$971$	3.91347e10	1.37181	0.685906	−	0.727690i	$-0.259406\pi$
$971$	3.91347e10	1.37181	0.685906	+	0.727690i	$0.259406\pi$
$972$	0	0
$973$	−8.46384e9	−0.294559
$974$	2.42307e10	0.840252
$975$	0	0
$976$	−3.08871e9	−0.106342
$977$	−2.72511e10	−0.934874	−0.467437	−	0.884026i	$-0.654822\pi$
$977$	−2.72511e10	−0.934874	−0.467437	+	0.884026i	$0.654822\pi$
$978$	0	0
$979$	1.14861e9	0.0391231
$980$	−7.09708e9	−0.240873
$981$	0	0
$982$	3.61379e10	1.21779
$983$	7.13353e9	0.239534	0.119767	−	0.992802i	$-0.461785\pi$
$983$	7.13353e9	0.239534	0.119767	+	0.992802i	$0.461785\pi$
$984$	0	0
$985$	−2.51962e10	−0.840055
$986$	−1.04864e10	−0.348384
$987$	0	0
$988$	2.82560e10	0.932098
$989$	−1.92784e10	−0.633702
$990$	0	0
$991$	−1.99378e10	−0.650760	−0.325380	−	0.945583i	$-0.605492\pi$
$991$	−1.99378e10	−0.650760	−0.325380	+	0.945583i	$0.605492\pi$
$992$	−4.13492e9	−0.134486
$993$	0	0
$994$	8.38497e9	0.270800
$995$	4.43849e10	1.42842
$996$	0	0
$997$	−1.27404e10	−0.407146	−0.203573	−	0.979060i	$-0.565255\pi$
$997$	−1.27404e10	−0.407146	−0.203573	+	0.979060i	$0.565255\pi$
$998$	−3.72531e10	−1.18633
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.a.f.1.1		3
3.2	odd	2		162.8.a.e.1.3		3
9.2	odd	6		54.8.c.a.37.1		6
9.4	even	3		18.8.c.a.7.2	✓	6
9.5	odd	6		54.8.c.a.19.1		6
9.7	even	3		18.8.c.a.13.2	yes	6
36.7	odd	6		144.8.i.a.49.2		6
36.11	even	6		432.8.i.a.145.1		6
36.23	even	6		432.8.i.a.289.1		6
36.31	odd	6		144.8.i.a.97.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.8.c.a.7.2	✓	6	9.4	even	3
18.8.c.a.13.2	yes	6	9.7	even	3
54.8.c.a.19.1		6	9.5	odd	6
54.8.c.a.37.1		6	9.2	odd	6
144.8.i.a.49.2		6	36.7	odd	6
144.8.i.a.97.2		6	36.31	odd	6
162.8.a.e.1.3		3	3.2	odd	2
162.8.a.f.1.1		3	1.1	even	1	trivial
432.8.i.a.145.1		6	36.11	even	6
432.8.i.a.289.1		6	36.23	even	6

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +64.0000 q^{4} -375.578 q^{5} +1057.73 q^{7} +512.000 q^{8} -3004.63 q^{10} +1699.93 q^{11} -12513.5 q^{13} +8461.86 q^{14} +4096.00 q^{16} +16879.9 q^{17} -35281.9 q^{19} -24037.0 q^{20} +13599.4 q^{22} +70965.4 q^{23} +62934.2 q^{25} -100108. q^{26} +67694.9 q^{28} -77654.8 q^{29} -126188. q^{31} +32768.0 q^{32} +135039. q^{34} -397262. q^{35} -528664. q^{37} -282255. q^{38} -192296. q^{40} -517921. q^{41} -271660. q^{43} +108795. q^{44} +567723. q^{46} -423629. q^{47} +295256. q^{49} +503474. q^{50} -800863. q^{52} +1.24314e6 q^{53} -638457. q^{55} +541559. q^{56} -621238. q^{58} -2.68173e6 q^{59} -754080. q^{61} -1.00950e6 q^{62} +262144. q^{64} +4.69980e6 q^{65} -262570. q^{67} +1.08031e6 q^{68} -3.17809e6 q^{70} +990913. q^{71} -1.04389e6 q^{73} -4.22931e6 q^{74} -2.25804e6 q^{76} +1.79807e6 q^{77} -7.33719e6 q^{79} -1.53837e6 q^{80} -4.14337e6 q^{82} -3.89221e6 q^{83} -6.33971e6 q^{85} -2.17328e6 q^{86} +870364. q^{88} +675681. q^{89} -1.32359e7 q^{91} +4.54179e6 q^{92} -3.38903e6 q^{94} +1.32511e7 q^{95} +1.62557e7 q^{97} +2.36205e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 24 q^{2} + 192 q^{4} - 54 q^{5} - 210 q^{7} + 1536 q^{8} - 432 q^{10} - 6579 q^{11} - 10092 q^{13} - 1680 q^{14} + 12288 q^{16} - 14895 q^{17} - 68745 q^{19} - 3456 q^{20} - 52632 q^{22} + 39654 q^{23}+ \cdots - 3857400 q^{98}+O(q^{100}) \) 3 * q + 24 * q^2 + 192 * q^4 - 54 * q^5 - 210 * q^7 + 1536 * q^8 - 432 * q^10 - 6579 * q^11 - 10092 * q^13 - 1680 * q^14 + 12288 * q^16 - 14895 * q^17 - 68745 * q^19 - 3456 * q^20 - 52632 * q^22 + 39654 * q^23 + 7923 * q^25 - 80736 * q^26 - 13440 * q^28 - 239832 * q^29 - 145704 * q^31 + 98304 * q^32 - 119160 * q^34 - 543888 * q^35 - 360192 * q^37 - 549960 * q^38 - 27648 * q^40 - 1086993 * q^41 + 299967 * q^43 - 421056 * q^44 + 317232 * q^46 - 131634 * q^47 - 482175 * q^49 + 63384 * q^50 - 645888 * q^52 + 954576 * q^53 - 2747412 * q^55 - 107520 * q^56 - 1918656 * q^58 - 2504853 * q^59 - 7309038 * q^61 - 1165632 * q^62 + 786432 * q^64 + 1786698 * q^65 - 3433035 * q^67 - 953280 * q^68 - 4351104 * q^70 + 1134684 * q^71 - 7950873 * q^73 - 2881536 * q^74 - 4399680 * q^76 + 6147432 * q^77 - 7076928 * q^79 - 221184 * q^80 - 8695944 * q^82 + 10914444 * q^83 - 17613396 * q^85 + 2399736 * q^86 - 3368448 * q^88 + 11900178 * q^89 - 18585276 * q^91 + 2537856 * q^92 - 1053072 * q^94 + 17342424 * q^95 - 519357 * q^97 - 3857400 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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