LMFDB - Embedded newform 1143.3.b.a.890.81

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,3,Mod(890,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1143.890");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1143.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$31.1444942164$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			890.81
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.66638i q^{2} -9.44232 q^{4} +0.851756i q^{5} +2.76278 q^{7} -19.9536i q^{8} +O(q^{10})$	\(q+3.66638i q^{2} -9.44232 q^{4} +0.851756i q^{5} +2.76278 q^{7} -19.9536i q^{8} -3.12286 q^{10} +14.2731i q^{11} +7.64232 q^{13} +10.1294i q^{14} +35.3881 q^{16} +9.98646i q^{17} +24.6909 q^{19} -8.04255i q^{20} -52.3304 q^{22} +26.3758i q^{23} +24.2745 q^{25} +28.0196i q^{26} -26.0870 q^{28} +32.4798i q^{29} -4.59199 q^{31} +49.9317i q^{32} -36.6141 q^{34} +2.35321i q^{35} +8.59256 q^{37} +90.5262i q^{38} +16.9956 q^{40} -60.9097i q^{41} -34.3172 q^{43} -134.771i q^{44} -96.7035 q^{46} +1.72698i q^{47} -41.3670 q^{49} +88.9995i q^{50} -72.1612 q^{52} +15.5905i q^{53} -12.1572 q^{55} -55.1274i q^{56} -119.083 q^{58} +59.5018i q^{59} +31.0892 q^{61} -16.8360i q^{62} -41.5161 q^{64} +6.50940i q^{65} -124.484 q^{67} -94.2954i q^{68} -8.62777 q^{70} -9.71062i q^{71} -98.4398 q^{73} +31.5035i q^{74} -233.140 q^{76} +39.4333i q^{77} +35.2163 q^{79} +30.1420i q^{80} +223.318 q^{82} +50.8065i q^{83} -8.50603 q^{85} -125.820i q^{86} +284.799 q^{88} -12.0062i q^{89} +21.1141 q^{91} -249.048i q^{92} -6.33175 q^{94} +21.0307i q^{95} +35.5414 q^{97} -151.667i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 160 q^{4}+O(q^{10}) $ 84 * q - 160 * q^4	$ 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) $ 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times$.

$n$	$128$	$892$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.66638i	1.83319i	0.399819	+	0.916594i	$0.369073\pi$
$2$	3.66638i	1.83319i	−0.399819	+	0.916594i	$0.630927\pi$
$3$	0	0
$3$	0	0
$4$	−9.44232	−2.36058
$5$	0.851756i	0.170351i	0.996366	+	0.0851756i	$0.0271451\pi$
$5$	0.851756i	0.170351i	−0.996366	+	0.0851756i	$0.972855\pi$
$6$	0	0
$7$	2.76278	0.394683	0.197341	−	0.980335i	$-0.436769\pi$
$7$	2.76278	0.394683	0.197341	+	0.980335i	$0.436769\pi$
$8$	− 19.9536i	− 2.49420i
$9$	0	0
$10$	−3.12286	−0.312286
$11$	14.2731i	1.29755i	0.760980	+	0.648775i	$0.224718\pi$
$11$	14.2731i	1.29755i	−0.760980	+	0.648775i	$0.775282\pi$
$12$	0	0
$13$	7.64232	0.587871	0.293936	−	0.955825i	$-0.405035\pi$
$13$	7.64232	0.587871	0.293936	+	0.955825i	$0.405035\pi$
$14$	10.1294i	0.723528i
$15$	0	0
$16$	35.3881	2.21176
$17$	9.98646i	0.587439i	0.955892	+	0.293719i	$0.0948932\pi$
$17$	9.98646i	0.587439i	−0.955892	+	0.293719i	$0.905107\pi$
$18$	0	0
$19$	24.6909	1.29952	0.649761	−	0.760138i	$-0.274869\pi$
$19$	24.6909	1.29952	0.649761	+	0.760138i	$0.274869\pi$
$20$	− 8.04255i	− 0.402128i
$21$	0	0
$22$	−52.3304	−2.37865
$23$	26.3758i	1.14677i	0.819285	+	0.573386i	$0.194371\pi$
$23$	26.3758i	1.14677i	−0.819285	+	0.573386i	$0.805629\pi$
$24$	0	0
$25$	24.2745	0.970980
$26$	28.0196i	1.07768i
$27$	0	0
$28$	−26.0870	−0.931680
$29$	32.4798i	1.11999i	0.828495	+	0.559997i	$0.189198\pi$
$29$	32.4798i	1.11999i	−0.828495	+	0.559997i	$0.810802\pi$
$30$	0	0
$31$	−4.59199	−0.148129	−0.0740644	−	0.997253i	$-0.523597\pi$
$31$	−4.59199	−0.148129	−0.0740644	+	0.997253i	$0.523597\pi$
$32$	49.9317i	1.56037i
$33$	0	0
$34$	−36.6141	−1.07689
$35$	2.35321i	0.0672347i
$36$	0	0
$37$	8.59256	0.232231	0.116116	−	0.993236i	$-0.462956\pi$
$37$	8.59256	0.232231	0.116116	+	0.993236i	$0.462956\pi$
$38$	90.5262i	2.38227i
$39$	0	0
$40$	16.9956	0.424890
$41$	− 60.9097i	− 1.48560i	−0.669512	−	0.742802i	$-0.733497\pi$
$41$	− 60.9097i	− 1.48560i	0.669512	−	0.742802i	$-0.266503\pi$
$42$	0	0
$43$	−34.3172	−0.798074	−0.399037	−	0.916935i	$-0.630656\pi$
$43$	−34.3172	−0.798074	−0.399037	+	0.916935i	$0.630656\pi$
$44$	− 134.771i	− 3.06297i
$45$	0	0
$46$	−96.7035	−2.10225
$47$	1.72698i	0.0367442i	0.999831	+	0.0183721i	$0.00584835\pi$
$47$	1.72698i	0.0367442i	−0.999831	+	0.0183721i	$0.994152\pi$
$48$	0	0
$49$	−41.3670	−0.844225
$50$	88.9995i	1.77999i
$51$	0	0
$52$	−72.1612	−1.38772
$53$	15.5905i	0.294160i	0.989125	+	0.147080i	$0.0469874\pi$
$53$	15.5905i	0.294160i	−0.989125	+	0.147080i	$0.953013\pi$
$54$	0	0
$55$	−12.1572	−0.221039
$56$	− 55.1274i	− 0.984417i
$57$	0	0
$58$	−119.083	−2.05316
$59$	59.5018i	1.00851i	0.863556	+	0.504253i	$0.168232\pi$
$59$	59.5018i	1.00851i	−0.863556	+	0.504253i	$0.831768\pi$
$60$	0	0
$61$	31.0892	0.509659	0.254830	−	0.966986i	$-0.417981\pi$
$61$	31.0892	0.509659	0.254830	+	0.966986i	$0.417981\pi$
$62$	− 16.8360i	− 0.271548i
$63$	0	0
$64$	−41.5161	−0.648690
$65$	6.50940i	0.100145i
$66$	0	0
$67$	−124.484	−1.85797	−0.928984	−	0.370119i	$-0.879317\pi$
$67$	−124.484	−1.85797	−0.928984	+	0.370119i	$0.879317\pi$
$68$	− 94.2954i	− 1.38670i
$69$	0	0
$70$	−8.62777	−0.123254
$71$	− 9.71062i	− 0.136769i	−0.997659	−	0.0683847i	$-0.978215\pi$
$71$	− 9.71062i	− 0.136769i	0.997659	−	0.0683847i	$-0.0217845\pi$
$72$	0	0
$73$	−98.4398	−1.34849	−0.674245	−	0.738508i	$-0.735531\pi$
$73$	−98.4398	−1.34849	−0.674245	+	0.738508i	$0.735531\pi$
$74$	31.5035i	0.425724i
$75$	0	0
$76$	−233.140	−3.06763
$77$	39.4333i	0.512121i
$78$	0	0
$79$	35.2163	0.445776	0.222888	−	0.974844i	$-0.428452\pi$
$79$	35.2163	0.445776	0.222888	+	0.974844i	$0.428452\pi$
$80$	30.1420i	0.376775i
$81$	0	0
$82$	223.318	2.72339
$83$	50.8065i	0.612126i	0.952011	+	0.306063i	$0.0990119\pi$
$83$	50.8065i	0.612126i	−0.952011	+	0.306063i	$0.900988\pi$
$84$	0	0
$85$	−8.50603	−0.100071
$86$	− 125.820i	− 1.46302i
$87$	0	0
$88$	284.799	3.23635
$89$	− 12.0062i	− 0.134901i	−0.997723	−	0.0674507i	$-0.978513\pi$
$89$	− 12.0062i	− 0.134901i	0.997723	−	0.0674507i	$-0.0214865\pi$
$90$	0	0
$91$	21.1141	0.232023
$92$	− 249.048i	− 2.70705i
$93$	0	0
$94$	−6.33175	−0.0673591
$95$	21.0307i	0.221375i
$96$	0	0
$97$	35.5414	0.366406	0.183203	−	0.983075i	$-0.441353\pi$
$97$	35.5414	0.366406	0.183203	+	0.983075i	$0.441353\pi$
$98$	− 151.667i	− 1.54762i
$99$	0	0
$100$	−229.208	−2.29208
$101$	− 85.3748i	− 0.845295i	−0.906294	−	0.422647i	$-0.861101\pi$
$101$	− 85.3748i	− 0.845295i	0.906294	−	0.422647i	$-0.138899\pi$
$102$	0	0
$103$	−5.67316	−0.0550792	−0.0275396	−	0.999621i	$-0.508767\pi$
$103$	−5.67316	−0.0550792	−0.0275396	+	0.999621i	$0.508767\pi$
$104$	− 152.492i	− 1.46627i
$105$	0	0
$106$	−57.1605	−0.539250
$107$	− 163.390i	− 1.52701i	−0.645801	−	0.763506i	$-0.723476\pi$
$107$	− 163.390i	− 1.52701i	0.645801	−	0.763506i	$-0.276524\pi$
$108$	0	0
$109$	1.24644	0.0114352	0.00571762	−	0.999984i	$-0.498180\pi$
$109$	1.24644	0.0114352	0.00571762	+	0.999984i	$0.498180\pi$
$110$	− 44.5727i	− 0.405207i
$111$	0	0
$112$	97.7695	0.872942
$113$	38.9817i	0.344970i	0.985012	+	0.172485i	$0.0551797\pi$
$113$	38.9817i	0.344970i	−0.985012	+	0.172485i	$0.944820\pi$
$114$	0	0
$115$	−22.4657	−0.195354
$116$	− 306.685i	− 2.64383i
$117$	0	0
$118$	−218.156	−1.84878
$119$	27.5904i	0.231852i
$120$	0	0
$121$	−82.7200	−0.683636
$122$	113.985i	0.934302i
$123$	0	0
$124$	43.3590	0.349670
$125$	41.9699i	0.335759i
$126$	0	0
$127$	−11.2694	−0.0887357
$127$	−11.2694	−0.0887357
$128$	47.5131i	0.371196i
$129$	0	0
$130$	−23.8659	−0.183584
$131$	− 61.7794i	− 0.471599i	−0.971802	−	0.235799i	$-0.924229\pi$
$131$	− 61.7794i	− 0.471599i	0.971802	−	0.235799i	$-0.0757708\pi$
$132$	0	0
$133$	68.2156	0.512899
$134$	− 456.405i	− 3.40601i
$135$	0	0
$136$	199.266	1.46519
$137$	232.233i	1.69513i	0.530692	+	0.847565i	$0.321932\pi$
$137$	232.233i	1.69513i	−0.530692	+	0.847565i	$0.678068\pi$
$138$	0	0
$139$	84.4304	0.607413	0.303706	−	0.952766i	$-0.401776\pi$
$139$	84.4304	0.607413	0.303706	+	0.952766i	$0.401776\pi$
$140$	− 22.2198i	− 0.158713i
$141$	0	0
$142$	35.6028	0.250724
$143$	109.079i	0.762792i
$144$	0	0
$145$	−27.6649	−0.190792
$146$	− 360.917i	− 2.47204i
$147$	0	0
$148$	−81.1337	−0.548200
$149$	144.774i	0.971636i	0.874060	+	0.485818i	$0.161478\pi$
$149$	144.774i	0.971636i	−0.874060	+	0.485818i	$0.838522\pi$
$150$	0	0
$151$	−260.071	−1.72232	−0.861162	−	0.508331i	$-0.830263\pi$
$151$	−260.071	−1.72232	−0.861162	+	0.508331i	$0.830263\pi$
$152$	− 492.672i	− 3.24127i
$153$	0	0
$154$	−144.577	−0.938814
$155$	− 3.91126i	− 0.0252339i
$156$	0	0
$157$	259.318	1.65171	0.825853	−	0.563885i	$-0.190694\pi$
$157$	259.318	1.65171	0.825853	+	0.563885i	$0.190694\pi$
$158$	129.116i	0.817191i
$159$	0	0
$160$	−42.5297	−0.265810
$161$	72.8704i	0.452611i
$162$	0	0
$163$	103.310	0.633802	0.316901	−	0.948459i	$-0.397358\pi$
$163$	103.310	0.633802	0.316901	+	0.948459i	$0.397358\pi$
$164$	575.129i	3.50688i
$165$	0	0
$166$	−186.276	−1.12214
$167$	289.782i	1.73522i	0.497245	+	0.867610i	$0.334345\pi$
$167$	289.782i	1.73522i	−0.497245	+	0.867610i	$0.665655\pi$
$168$	0	0
$169$	−110.595	−0.654408
$170$	− 31.1863i	− 0.183449i
$171$	0	0
$172$	324.034	1.88392
$173$	− 128.657i	− 0.743685i	−0.928296	−	0.371842i	$-0.878726\pi$
$173$	− 128.657i	− 0.743685i	0.928296	−	0.371842i	$-0.121274\pi$
$174$	0	0
$175$	67.0651	0.383229
$176$	505.096i	2.86986i
$177$	0	0
$178$	44.0194	0.247300
$179$	258.062i	1.44169i	0.693097	+	0.720844i	$0.256246\pi$
$179$	258.062i	1.44169i	−0.693097	+	0.720844i	$0.743754\pi$
$180$	0	0
$181$	51.5156	0.284616	0.142308	−	0.989822i	$-0.454548\pi$
$181$	51.5156	0.284616	0.142308	+	0.989822i	$0.454548\pi$
$182$	77.4121i	0.425341i
$183$	0	0
$184$	526.291	2.86028
$185$	7.31876i	0.0395609i
$186$	0	0
$187$	−142.537	−0.762231
$188$	− 16.3067i	− 0.0867376i
$189$	0	0
$190$	−77.1063	−0.405823
$191$	128.808i	0.674390i	0.941435	+	0.337195i	$0.109478\pi$
$191$	128.808i	0.674390i	−0.941435	+	0.337195i	$0.890522\pi$
$192$	0	0
$193$	−60.3990	−0.312948	−0.156474	−	0.987682i	$-0.550013\pi$
$193$	−60.3990	−0.312948	−0.156474	+	0.987682i	$0.550013\pi$
$194$	130.308i	0.671692i
$195$	0	0
$196$	390.601	1.99286
$197$	− 251.511i	− 1.27671i	−0.769744	−	0.638353i	$-0.779616\pi$
$197$	− 251.511i	− 1.27671i	0.769744	−	0.638353i	$-0.220384\pi$
$198$	0	0
$199$	−272.046	−1.36707	−0.683533	−	0.729920i	$-0.739557\pi$
$199$	−272.046	−1.36707	−0.683533	+	0.729920i	$0.739557\pi$
$200$	− 484.364i	− 2.42182i
$201$	0	0
$202$	313.016	1.54958
$203$	89.7346i	0.442042i
$204$	0	0
$205$	51.8802	0.253074
$206$	− 20.7999i	− 0.100971i
$207$	0	0
$208$	270.447	1.30023
$209$	352.415i	1.68620i
$210$	0	0
$211$	5.09269	0.0241360	0.0120680	−	0.999927i	$-0.496159\pi$
$211$	5.09269	0.0241360	0.0120680	+	0.999927i	$0.496159\pi$
$212$	− 147.210i	− 0.694387i
$213$	0	0
$214$	599.050	2.79930
$215$	− 29.2299i	− 0.135953i
$216$	0	0
$217$	−12.6867	−0.0584639
$218$	4.56992i	0.0209629i
$219$	0	0
$220$	114.792	0.521781
$221$	76.3198i	0.345338i
$222$	0	0
$223$	75.0319	0.336466	0.168233	−	0.985747i	$-0.446194\pi$
$223$	75.0319	0.336466	0.168233	+	0.985747i	$0.446194\pi$
$224$	137.950i	0.615850i
$225$	0	0
$226$	−142.921	−0.632396
$227$	− 221.592i	− 0.976176i	−0.872794	−	0.488088i	$-0.837694\pi$
$227$	− 221.592i	− 0.976176i	0.872794	−	0.488088i	$-0.162306\pi$
$228$	0	0
$229$	−327.428	−1.42981	−0.714907	−	0.699219i	$-0.753531\pi$
$229$	−327.428	−1.42981	−0.714907	+	0.699219i	$0.753531\pi$
$230$	− 82.3678i	− 0.358121i
$231$	0	0
$232$	648.089	2.79349
$233$	− 52.8126i	− 0.226663i	−0.993557	−	0.113332i	$-0.963848\pi$
$233$	− 52.8126i	− 0.226663i	0.993557	−	0.113332i	$-0.0361523\pi$
$234$	0	0
$235$	−1.47096	−0.00625942
$236$	− 561.835i	− 2.38066i
$237$	0	0
$238$	−101.157	−0.425028
$239$	74.9951i	0.313787i	0.987616	+	0.156894i	$0.0501480\pi$
$239$	74.9951i	0.313787i	−0.987616	+	0.156894i	$0.949852\pi$
$240$	0	0
$241$	205.835	0.854085	0.427043	−	0.904232i	$-0.359555\pi$
$241$	205.835	0.854085	0.427043	+	0.904232i	$0.359555\pi$
$242$	− 303.283i	− 1.25323i
$243$	0	0
$244$	−293.554	−1.20309
$245$	− 35.2346i	− 0.143815i
$246$	0	0
$247$	188.696	0.763952
$248$	91.6267i	0.369462i
$249$	0	0
$250$	−153.877	−0.615509
$251$	93.4507i	0.372314i	0.982520	+	0.186157i	$0.0596032\pi$
$251$	93.4507i	0.372314i	−0.982520	+	0.186157i	$0.940397\pi$
$252$	0	0
$253$	−376.463	−1.48799
$254$	− 41.3180i	− 0.162669i
$255$	0	0
$256$	−340.265	−1.32916
$257$	− 230.227i	− 0.895823i	−0.894078	−	0.447912i	$-0.852168\pi$
$257$	− 230.227i	− 0.895823i	0.894078	−	0.447912i	$-0.147832\pi$
$258$	0	0
$259$	23.7393	0.0916577
$260$	− 61.4638i	− 0.236399i
$261$	0	0
$262$	226.507	0.864529
$263$	− 428.852i	− 1.63061i	−0.579028	−	0.815307i	$-0.696568\pi$
$263$	− 428.852i	− 1.63061i	0.579028	−	0.815307i	$-0.303432\pi$
$264$	0	0
$265$	−13.2793	−0.0501105
$266$	250.104i	0.940241i
$267$	0	0
$268$	1175.42	4.38588
$269$	279.561i	1.03926i	0.854391	+	0.519630i	$0.173930\pi$
$269$	279.561i	1.03926i	−0.854391	+	0.519630i	$0.826070\pi$
$270$	0	0
$271$	145.582	0.537201	0.268601	−	0.963252i	$-0.413439\pi$
$271$	145.582	0.537201	0.268601	+	0.963252i	$0.413439\pi$
$272$	353.402i	1.29927i
$273$	0	0
$274$	−851.453	−3.10749
$275$	346.471i	1.25990i
$276$	0	0
$277$	392.296	1.41623	0.708116	−	0.706096i	$-0.249545\pi$
$277$	392.296	1.41623	0.708116	+	0.706096i	$0.249545\pi$
$278$	309.554i	1.11350i
$279$	0	0
$280$	46.9551	0.167697
$281$	− 43.1317i	− 0.153494i	−0.997051	−	0.0767468i	$-0.975547\pi$
$281$	− 43.1317i	− 0.153494i	0.997051	−	0.0767468i	$-0.0244533\pi$
$282$	0	0
$283$	87.6603	0.309754	0.154877	−	0.987934i	$-0.450502\pi$
$283$	87.6603	0.309754	0.154877	+	0.987934i	$0.450502\pi$
$284$	91.6908i	0.322855i
$285$	0	0
$286$	−399.926	−1.39834
$287$	− 168.280i	− 0.586342i
$288$	0	0
$289$	189.271	0.654915
$290$	− 101.430i	− 0.349758i
$291$	0	0
$292$	929.499	3.18322
$293$	− 158.686i	− 0.541590i	−0.962637	−	0.270795i	$-0.912713\pi$
$293$	− 158.686i	− 0.541590i	0.962637	−	0.270795i	$-0.0872865\pi$
$294$	0	0
$295$	−50.6811	−0.171800
$296$	− 171.452i	− 0.579231i
$297$	0	0
$298$	−530.795	−1.78119
$299$	201.572i	0.674154i
$300$	0	0
$301$	−94.8108	−0.314986
$302$	− 953.518i	− 3.15734i
$303$	0	0
$304$	873.765	2.87423
$305$	26.4804i	0.0868211i
$306$	0	0
$307$	120.792	0.393460	0.196730	−	0.980458i	$-0.436968\pi$
$307$	120.792	0.393460	0.196730	+	0.980458i	$0.436968\pi$
$308$	− 372.342i	− 1.20890i
$309$	0	0
$310$	14.3401	0.0462585
$311$	62.7402i	0.201737i	0.994900	+	0.100868i	$0.0321621\pi$
$311$	62.7402i	0.201737i	−0.994900	+	0.100868i	$0.967838\pi$
$312$	0	0
$313$	−177.785	−0.568003	−0.284001	−	0.958824i	$-0.591662\pi$
$313$	−177.785	−0.568003	−0.284001	+	0.958824i	$0.591662\pi$
$314$	950.757i	3.02789i
$315$	0	0
$316$	−332.523	−1.05229
$317$	202.295i	0.638155i	0.947729	+	0.319078i	$0.103373\pi$
$317$	202.295i	0.638155i	−0.947729	+	0.319078i	$0.896627\pi$
$318$	0	0
$319$	−463.586	−1.45325
$320$	− 35.3616i	− 0.110505i
$321$	0	0
$322$	−267.170	−0.829722
$323$	246.575i	0.763390i
$324$	0	0
$325$	185.514	0.570811
$326$	378.772i	1.16188i
$327$	0	0
$328$	−1215.37	−3.70539
$329$	4.77126i	0.0145023i
$330$	0	0
$331$	−147.920	−0.446888	−0.223444	−	0.974717i	$-0.571730\pi$
$331$	−147.920	−0.446888	−0.223444	+	0.974717i	$0.571730\pi$
$332$	− 479.731i	− 1.44497i
$333$	0	0
$334$	−1062.45	−3.18099
$335$	− 106.030i	− 0.316507i
$336$	0	0
$337$	−459.988	−1.36495	−0.682474	−	0.730909i	$-0.739096\pi$
$337$	−459.988	−1.36495	−0.682474	+	0.730909i	$0.739096\pi$
$338$	− 405.483i	− 1.19965i
$339$	0	0
$340$	80.3167	0.236225
$341$	− 65.5417i	− 0.192204i
$342$	0	0
$343$	−249.664	−0.727884
$344$	684.751i	1.99056i
$345$	0	0
$346$	471.707	1.36331
$347$	547.855i	1.57883i	0.613858	+	0.789416i	$0.289617\pi$
$347$	547.855i	1.57883i	−0.613858	+	0.789416i	$0.710383\pi$
$348$	0	0
$349$	−30.6558	−0.0878389	−0.0439195	−	0.999035i	$-0.513985\pi$
$349$	−30.6558	−0.0878389	−0.0439195	+	0.999035i	$0.513985\pi$
$350$	245.886i	0.702531i
$351$	0	0
$352$	−712.678	−2.02465
$353$	− 162.085i	− 0.459166i	−0.973289	−	0.229583i	$-0.926264\pi$
$353$	− 162.085i	− 0.459166i	0.973289	−	0.229583i	$-0.0737362\pi$
$354$	0	0
$355$	8.27108	0.0232988
$356$	113.367i	0.318446i
$357$	0	0
$358$	−946.153	−2.64289
$359$	− 119.205i	− 0.332046i	−0.986122	−	0.166023i	$-0.946907\pi$
$359$	− 119.205i	− 0.332046i	0.986122	−	0.166023i	$-0.0530926\pi$
$360$	0	0
$361$	248.642	0.688758
$362$	188.875i	0.521755i
$363$	0	0
$364$	−199.366	−0.547708
$365$	− 83.8467i	− 0.229717i
$366$	0	0
$367$	507.592	1.38309	0.691543	−	0.722336i	$-0.256931\pi$
$367$	507.592	1.38309	0.691543	+	0.722336i	$0.256931\pi$
$368$	933.388i	2.53638i
$369$	0	0
$370$	−26.8333	−0.0725226
$371$	43.0730i	0.116100i
$372$	0	0
$373$	−499.234	−1.33843	−0.669215	−	0.743069i	$-0.733369\pi$
$373$	−499.234	−1.33843	−0.669215	+	0.743069i	$0.733369\pi$
$374$	− 522.595i	− 1.39731i
$375$	0	0
$376$	34.4594	0.0916473
$377$	248.221i	0.658412i
$378$	0	0
$379$	26.0902	0.0688395	0.0344198	−	0.999407i	$-0.489042\pi$
$379$	26.0902	0.0688395	0.0344198	+	0.999407i	$0.489042\pi$
$380$	− 198.578i	− 0.522574i
$381$	0	0
$382$	−472.260	−1.23628
$383$	327.847i	0.855998i	0.903779	+	0.427999i	$0.140781\pi$
$383$	327.847i	0.855998i	−0.903779	+	0.427999i	$0.859219\pi$
$384$	0	0
$385$	−33.5876	−0.0872404
$386$	− 221.446i	− 0.573693i
$387$	0	0
$388$	−335.593	−0.864931
$389$	80.5907i	0.207174i	0.994620	+	0.103587i	$0.0330320\pi$
$389$	80.5907i	0.207174i	−0.994620	+	0.103587i	$0.966968\pi$
$390$	0	0
$391$	−263.401	−0.673659
$392$	825.421i	2.10567i
$393$	0	0
$394$	922.134	2.34044
$395$	29.9957i	0.0759384i
$396$	0	0
$397$	−71.5458	−0.180216	−0.0901080	−	0.995932i	$-0.528721\pi$
$397$	−71.5458	−0.180216	−0.0901080	+	0.995932i	$0.528721\pi$
$398$	− 997.423i	− 2.50609i
$399$	0	0
$400$	859.029	2.14757
$401$	− 139.812i	− 0.348658i	−0.984687	−	0.174329i	$-0.944224\pi$
$401$	− 139.812i	− 0.348658i	0.984687	−	0.174329i	$-0.0557756\pi$
$402$	0	0
$403$	−35.0935	−0.0870806
$404$	806.136i	1.99539i
$405$	0	0
$406$	−329.001	−0.810347
$407$	122.642i	0.301332i
$408$	0	0
$409$	658.477	1.60997	0.804984	−	0.593297i	$-0.202174\pi$
$409$	658.477	1.60997	0.804984	+	0.593297i	$0.202174\pi$
$410$	190.213i	0.463933i
$411$	0	0
$412$	53.5678	0.130019
$413$	164.390i	0.398040i
$414$	0	0
$415$	−43.2747	−0.104276
$416$	381.594i	0.917294i
$417$	0	0
$418$	−1292.09	−3.09111
$419$	− 83.3121i	− 0.198835i	−0.995046	−	0.0994177i	$-0.968302\pi$
$419$	− 83.3121i	− 0.198835i	0.995046	−	0.0994177i	$-0.0316980\pi$
$420$	0	0
$421$	−44.9743	−0.106827	−0.0534137	−	0.998572i	$-0.517010\pi$
$421$	−44.9743	−0.106827	−0.0534137	+	0.998572i	$0.517010\pi$
$422$	18.6717i	0.0442458i
$423$	0	0
$424$	311.086	0.733693
$425$	242.417i	0.570392i
$426$	0	0
$427$	85.8927	0.201154
$428$	1542.78i	3.60463i
$429$	0	0
$430$	107.168	0.249227
$431$	− 488.409i	− 1.13320i	−0.823993	−	0.566599i	$-0.808259\pi$
$431$	− 488.409i	− 1.13320i	0.823993	−	0.566599i	$-0.191741\pi$
$432$	0	0
$433$	−659.033	−1.52202	−0.761009	−	0.648742i	$-0.775296\pi$
$433$	−659.033	−1.52202	−0.761009	+	0.648742i	$0.775296\pi$
$434$	− 46.5141i	− 0.107175i
$435$	0	0
$436$	−11.7693	−0.0269938
$437$	651.242i	1.49026i
$438$	0	0
$439$	86.1838	0.196319	0.0981593	−	0.995171i	$-0.468705\pi$
$439$	86.1838	0.196319	0.0981593	+	0.995171i	$0.468705\pi$
$440$	242.579i	0.551316i
$441$	0	0
$442$	−279.817	−0.633070
$443$	521.457i	1.17710i	0.808459	+	0.588552i	$0.200302\pi$
$443$	521.457i	1.17710i	−0.808459	+	0.588552i	$0.799698\pi$
$444$	0	0
$445$	10.2264	0.0229806
$446$	275.095i	0.616805i
$447$	0	0
$448$	−114.700	−0.256027
$449$	543.281i	1.20998i	0.796233	+	0.604990i	$0.206823\pi$
$449$	543.281i	1.20998i	−0.796233	+	0.604990i	$0.793177\pi$
$450$	0	0
$451$	869.368	1.92764
$452$	− 368.077i	− 0.814330i
$453$	0	0
$454$	812.440	1.78951
$455$	17.9840i	0.0395253i
$456$	0	0
$457$	179.592	0.392981	0.196491	−	0.980506i	$-0.437045\pi$
$457$	179.592	0.392981	0.196491	+	0.980506i	$0.437045\pi$
$458$	− 1200.47i	− 2.62112i
$459$	0	0
$460$	212.128	0.461149
$461$	− 156.830i	− 0.340196i	−0.985427	−	0.170098i	$-0.945592\pi$
$461$	− 156.830i	− 0.340196i	0.985427	−	0.170098i	$-0.0544084\pi$
$462$	0	0
$463$	488.330	1.05471	0.527354	−	0.849646i	$-0.323184\pi$
$463$	488.330	1.05471	0.527354	+	0.849646i	$0.323184\pi$
$464$	1149.40i	2.47715i
$465$	0	0
$466$	193.631	0.415517
$467$	− 314.991i	− 0.674498i	−0.941415	−	0.337249i	$-0.890504\pi$
$467$	− 314.991i	− 0.674498i	0.941415	−	0.337249i	$-0.109496\pi$
$468$	0	0
$469$	−343.922	−0.733308
$470$	− 5.39311i	− 0.0114747i
$471$	0	0
$472$	1187.28	2.51541
$473$	− 489.811i	− 1.03554i
$474$	0	0
$475$	599.360	1.26181
$476$	− 260.517i	− 0.547305i
$477$	0	0
$478$	−274.960	−0.575231
$479$	− 683.816i	− 1.42759i	−0.700354	−	0.713796i	$-0.746974\pi$
$479$	− 683.816i	− 1.42759i	0.700354	−	0.713796i	$-0.253026\pi$
$480$	0	0
$481$	65.6671	0.136522
$482$	754.667i	1.56570i
$483$	0	0
$484$	781.068	1.61378
$485$	30.2726i	0.0624178i
$486$	0	0
$487$	690.839	1.41856	0.709280	−	0.704927i	$-0.249020\pi$
$487$	690.839	1.41856	0.709280	+	0.704927i	$0.249020\pi$
$488$	− 620.341i	− 1.27119i
$489$	0	0
$490$	129.183	0.263640
$491$	525.752i	1.07078i	0.844606	+	0.535389i	$0.179835\pi$
$491$	525.752i	1.07078i	−0.844606	+	0.535389i	$0.820165\pi$
$492$	0	0
$493$	−324.358	−0.657928
$494$	691.831i	1.40047i
$495$	0	0
$496$	−162.502	−0.327624
$497$	− 26.8283i	− 0.0539805i
$498$	0	0
$499$	−99.2375	−0.198873	−0.0994364	−	0.995044i	$-0.531704\pi$
$499$	−99.2375	−0.198873	−0.0994364	+	0.995044i	$0.531704\pi$
$500$	− 396.293i	− 0.792586i
$501$	0	0
$502$	−342.626	−0.682521
$503$	776.756i	1.54425i	0.635472	+	0.772124i	$0.280805\pi$
$503$	776.756i	1.54425i	−0.635472	+	0.772124i	$0.719195\pi$
$504$	0	0
$505$	72.7185	0.143997
$506$	− 1380.25i	− 2.72777i
$507$	0	0
$508$	106.410	0.209468
$509$	− 644.281i	− 1.26578i	−0.774242	−	0.632889i	$-0.781869\pi$
$509$	− 644.281i	− 1.26578i	0.774242	−	0.632889i	$-0.218131\pi$
$510$	0	0
$511$	−271.967	−0.532226
$512$	− 1057.49i	− 2.06541i
$513$	0	0
$514$	844.097	1.64221
$515$	− 4.83215i	− 0.00938282i
$516$	0	0
$517$	−24.6492	−0.0476774
$518$	87.0374i	0.168026i
$519$	0	0
$520$	129.886	0.249780
$521$	498.797i	0.957384i	0.877983	+	0.478692i	$0.158889\pi$
$521$	498.797i	0.957384i	−0.877983	+	0.478692i	$0.841111\pi$
$522$	0	0
$523$	552.330	1.05608	0.528040	−	0.849219i	$-0.322927\pi$
$523$	552.330	1.05608	0.528040	+	0.849219i	$0.322927\pi$
$524$	583.341i	1.11325i
$525$	0	0
$526$	1572.33	2.98922
$527$	− 45.8577i	− 0.0870166i
$528$	0	0
$529$	−166.681	−0.315087
$530$	− 48.6868i	− 0.0918620i
$531$	0	0
$532$	−644.113	−1.21074
$533$	− 465.492i	− 0.873343i
$534$	0	0
$535$	139.169	0.260128
$536$	2483.90i	4.63414i
$537$	0	0
$538$	−1024.98	−1.90516
$539$	− 590.434i	− 1.09542i
$540$	0	0
$541$	307.368	0.568149	0.284074	−	0.958802i	$-0.408314\pi$
$541$	307.368	0.568149	0.284074	+	0.958802i	$0.408314\pi$
$542$	533.757i	0.984791i
$543$	0	0
$544$	−498.641	−0.916620
$545$	1.06166i	0.00194801i
$546$	0	0
$547$	−1037.81	−1.89727	−0.948637	−	0.316367i	$-0.897537\pi$
$547$	−1037.81	−1.89727	−0.948637	+	0.316367i	$0.897537\pi$
$548$	− 2192.82i	− 4.00149i
$549$	0	0
$550$	−1270.29	−2.30963
$551$	801.957i	1.45546i
$552$	0	0
$553$	97.2948	0.175940
$554$	1438.31i	2.59622i
$555$	0	0
$556$	−797.218	−1.43385
$557$	94.2544i	0.169218i	0.996414	+	0.0846090i	$0.0269641\pi$
$557$	94.2544i	0.169218i	−0.996414	+	0.0846090i	$0.973036\pi$
$558$	0	0
$559$	−262.263	−0.469165
$560$	83.2758i	0.148707i
$561$	0	0
$562$	158.137	0.281383
$563$	− 917.628i	− 1.62989i	−0.579538	−	0.814945i	$-0.696767\pi$
$563$	− 917.628i	− 1.62989i	0.579538	−	0.814945i	$-0.303233\pi$
$564$	0	0
$565$	−33.2029	−0.0587662
$566$	321.396i	0.567837i
$567$	0	0
$568$	−193.762	−0.341130
$569$	− 97.4460i	− 0.171258i	−0.996327	−	0.0856292i	$-0.972710\pi$
$569$	− 97.4460i	− 0.171258i	0.996327	−	0.0856292i	$-0.0272900\pi$
$570$	0	0
$571$	299.153	0.523911	0.261955	−	0.965080i	$-0.415633\pi$
$571$	299.153	0.523911	0.261955	+	0.965080i	$0.415633\pi$
$572$	− 1029.96i	− 1.80063i
$573$	0	0
$574$	616.978	1.07488
$575$	640.259i	1.11349i
$576$	0	0
$577$	532.221	0.922393	0.461196	−	0.887298i	$-0.347420\pi$
$577$	532.221	0.922393	0.461196	+	0.887298i	$0.347420\pi$
$578$	693.937i	1.20058i
$579$	0	0
$580$	261.221	0.450380
$581$	140.367i	0.241596i
$582$	0	0
$583$	−222.524	−0.381687
$584$	1964.23i	3.36340i
$585$	0	0
$586$	581.802	0.992836
$587$	− 475.281i	− 0.809677i	−0.914388	−	0.404839i	$-0.867328\pi$
$587$	− 475.281i	− 0.809677i	0.914388	−	0.404839i	$-0.132672\pi$
$588$	0	0
$589$	−113.380	−0.192497
$590$	− 185.816i	− 0.314942i
$591$	0	0
$592$	304.074	0.513639
$593$	321.262i	0.541758i	0.962613	+	0.270879i	$0.0873143\pi$
$593$	321.262i	0.541758i	−0.962613	+	0.270879i	$0.912686\pi$
$594$	0	0
$595$	−23.5003	−0.0394963
$596$	− 1367.00i	− 2.29362i
$597$	0	0
$598$	−739.039	−1.23585
$599$	− 231.313i	− 0.386166i	−0.981182	−	0.193083i	$-0.938151\pi$
$599$	− 231.313i	− 0.386166i	0.981182	−	0.193083i	$-0.0618486\pi$
$600$	0	0
$601$	980.113	1.63080	0.815402	−	0.578895i	$-0.196516\pi$
$601$	980.113	1.63080	0.815402	+	0.578895i	$0.196516\pi$
$602$	− 347.612i	− 0.577429i
$603$	0	0
$604$	2455.67	4.06568
$605$	− 70.4573i	− 0.116458i
$606$	0	0
$607$	533.312	0.878602	0.439301	−	0.898340i	$-0.355226\pi$
$607$	533.312	0.878602	0.439301	+	0.898340i	$0.355226\pi$
$608$	1232.86i	2.02773i
$609$	0	0
$610$	−97.0873	−0.159159
$611$	13.1981i	0.0216009i
$612$	0	0
$613$	−826.976	−1.34906	−0.674532	−	0.738246i	$-0.735655\pi$
$613$	−826.976	−1.34906	−0.674532	+	0.738246i	$0.735655\pi$
$614$	442.870i	0.721287i
$615$	0	0
$616$	786.836	1.27733
$617$	− 1225.99i	− 1.98701i	−0.113785	−	0.993505i	$-0.536297\pi$
$617$	− 1225.99i	− 1.98701i	0.113785	−	0.993505i	$-0.463703\pi$
$618$	0	0
$619$	804.256	1.29928	0.649641	−	0.760241i	$-0.274919\pi$
$619$	804.256	1.29928	0.649641	+	0.760241i	$0.274919\pi$
$620$	36.9313i	0.0595667i
$621$	0	0
$622$	−230.029	−0.369822
$623$	− 33.1706i	− 0.0532433i
$624$	0	0
$625$	571.115	0.913783
$626$	− 651.826i	− 1.04126i
$627$	0	0
$628$	−2448.56	−3.89898
$629$	85.8092i	0.136422i
$630$	0	0
$631$	−541.991	−0.858939	−0.429470	−	0.903081i	$-0.641299\pi$
$631$	−541.991	−0.858939	−0.429470	+	0.903081i	$0.641299\pi$
$632$	− 702.691i	− 1.11185i
$633$	0	0
$634$	−741.690	−1.16986
$635$	− 9.59881i	− 0.0151162i
$636$	0	0
$637$	−316.140	−0.496296
$638$	− 1699.68i	− 2.66408i
$639$	0	0
$640$	−40.4696	−0.0632337
$641$	427.172i	0.666416i	0.942853	+	0.333208i	$0.108131\pi$
$641$	427.172i	0.666416i	−0.942853	+	0.333208i	$0.891869\pi$
$642$	0	0
$643$	1273.41	1.98042	0.990210	−	0.139589i	$-0.0445781\pi$
$643$	1273.41	1.98042	0.990210	+	0.139589i	$0.0445781\pi$
$644$	− 688.066i	− 1.06842i
$645$	0	0
$646$	−904.037	−1.39944
$647$	344.452i	0.532383i	0.963920	+	0.266191i	$0.0857653\pi$
$647$	344.452i	0.532383i	−0.963920	+	0.266191i	$0.914235\pi$
$648$	0	0
$649$	−849.273	−1.30859
$650$	680.163i	1.04640i
$651$	0	0
$652$	−975.483	−1.49614
$653$	132.502i	0.202913i	0.994840	+	0.101456i	$0.0323502\pi$
$653$	132.502i	0.202913i	−0.994840	+	0.101456i	$0.967650\pi$
$654$	0	0
$655$	52.6210	0.0803374
$656$	− 2155.48i	− 3.28579i
$657$	0	0
$658$	−17.4932	−0.0265855
$659$	731.028i	1.10930i	0.832084	+	0.554650i	$0.187148\pi$
$659$	731.028i	1.10930i	−0.832084	+	0.554650i	$0.812852\pi$
$660$	0	0
$661$	513.775	0.777270	0.388635	−	0.921392i	$-0.372947\pi$
$661$	513.775	0.777270	0.388635	+	0.921392i	$0.372947\pi$
$662$	− 542.330i	− 0.819230i
$663$	0	0
$664$	1013.77	1.52676
$665$	58.1031i	0.0873730i
$666$	0	0
$667$	−856.680	−1.28438
$668$	− 2736.21i	− 4.09613i
$669$	0	0
$670$	388.746	0.580218
$671$	443.738i	0.661309i
$672$	0	0
$673$	−216.699	−0.321990	−0.160995	−	0.986955i	$-0.551470\pi$
$673$	−216.699	−0.321990	−0.160995	+	0.986955i	$0.551470\pi$
$674$	− 1686.49i	− 2.50221i
$675$	0	0
$676$	1044.27	1.54478
$677$	676.400i	0.999114i	0.866281	+	0.499557i	$0.166504\pi$
$677$	676.400i	0.999114i	−0.866281	+	0.499557i	$0.833496\pi$
$678$	0	0
$679$	98.1931	0.144614
$680$	169.726i	0.249597i
$681$	0	0
$682$	240.301	0.352347
$683$	− 96.7435i	− 0.141645i	−0.997489	−	0.0708225i	$-0.977438\pi$
$683$	− 96.7435i	− 0.141645i	0.997489	−	0.0708225i	$-0.0225624\pi$
$684$	0	0
$685$	−197.806	−0.288767
$686$	− 915.363i	− 1.33435i
$687$	0	0
$688$	−1214.42	−1.76515
$689$	119.147i	0.172928i
$690$	0	0
$691$	−152.026	−0.220008	−0.110004	−	0.993931i	$-0.535086\pi$
$691$	−152.026	−0.220008	−0.110004	+	0.993931i	$0.535086\pi$
$692$	1214.82i	1.75553i
$693$	0	0
$694$	−2008.64	−2.89430
$695$	71.9141i	0.103474i
$696$	0	0
$697$	608.273	0.872701
$698$	− 112.396i	− 0.161025i
$699$	0	0
$700$	−633.250	−0.904643
$701$	− 516.572i	− 0.736907i	−0.929646	−	0.368454i	$-0.879887\pi$
$701$	− 516.572i	− 0.736907i	0.929646	−	0.368454i	$-0.120113\pi$
$702$	0	0
$703$	212.158	0.301790
$704$	− 592.562i	− 0.841707i
$705$	0	0
$706$	594.266	0.841737
$707$	− 235.872i	− 0.333623i
$708$	0	0
$709$	652.207	0.919897	0.459949	−	0.887946i	$-0.347868\pi$
$709$	652.207	0.919897	0.459949	+	0.887946i	$0.347868\pi$
$710$	30.3249i	0.0427111i
$711$	0	0
$712$	−239.567	−0.336471
$713$	− 121.117i	− 0.169870i
$714$	0	0
$715$	−92.9090	−0.129943
$716$	− 2436.71i	− 3.40322i
$717$	0	0
$718$	437.049	0.608703
$719$	− 1380.67i	− 1.92026i	−0.279552	−	0.960131i	$-0.590186\pi$
$719$	− 1380.67i	− 1.92026i	0.279552	−	0.960131i	$-0.409814\pi$
$720$	0	0
$721$	−15.6737	−0.0217388
$722$	911.614i	1.26262i
$723$	0	0
$724$	−486.426	−0.671859
$725$	788.432i	1.08749i
$726$	0	0
$727$	733.171	1.00849	0.504244	−	0.863561i	$-0.331771\pi$
$727$	733.171	1.00849	0.504244	+	0.863561i	$0.331771\pi$
$728$	− 421.301i	− 0.578710i
$729$	0	0
$730$	307.414	0.421114
$731$	− 342.707i	− 0.468820i
$732$	0	0
$733$	8.24724	0.0112513	0.00562567	−	0.999984i	$-0.498209\pi$
$733$	8.24724	0.0112513	0.00562567	+	0.999984i	$0.498209\pi$
$734$	1861.02i	2.53546i
$735$	0	0
$736$	−1316.99	−1.78938
$737$	− 1776.77i	− 2.41081i
$738$	0	0
$739$	944.003	1.27741	0.638703	−	0.769453i	$-0.279471\pi$
$739$	944.003	1.27741	0.638703	+	0.769453i	$0.279471\pi$
$740$	− 69.1061i	− 0.0933866i
$741$	0	0
$742$	−157.922	−0.212833
$743$	467.178i	0.628773i	0.949295	+	0.314386i	$0.101799\pi$
$743$	467.178i	0.628773i	−0.949295	+	0.314386i	$0.898201\pi$
$744$	0	0
$745$	−123.312	−0.165519
$746$	− 1830.38i	− 2.45359i
$747$	0	0
$748$	1345.88	1.79931
$749$	− 451.411i	− 0.602686i
$750$	0	0
$751$	721.896	0.961247	0.480623	−	0.876927i	$-0.340410\pi$
$751$	721.896	0.961247	0.480623	+	0.876927i	$0.340410\pi$
$752$	61.1144i	0.0812692i
$753$	0	0
$754$	−910.073	−1.20699
$755$	− 221.517i	− 0.293400i
$756$	0	0
$757$	1140.81	1.50701	0.753507	−	0.657440i	$-0.228361\pi$
$757$	1140.81	1.50701	0.753507	+	0.657440i	$0.228361\pi$
$758$	95.6564i	0.126196i
$759$	0	0
$760$	419.637	0.552154
$761$	− 646.929i	− 0.850103i	−0.905169	−	0.425052i	$-0.860256\pi$
$761$	− 646.929i	− 0.850103i	0.905169	−	0.425052i	$-0.139744\pi$
$762$	0	0
$763$	3.44364	0.00451329
$764$	− 1216.25i	− 1.59195i
$765$	0	0
$766$	−1202.01	−1.56921
$767$	454.732i	0.592871i
$768$	0	0
$769$	868.292	1.12912	0.564559	−	0.825393i	$-0.309046\pi$
$769$	868.292	1.12912	0.564559	+	0.825393i	$0.309046\pi$
$770$	− 123.145i	− 0.159928i
$771$	0	0
$772$	570.307	0.738739
$773$	273.329i	0.353596i	0.984247	+	0.176798i	$0.0565739\pi$
$773$	273.329i	0.353596i	−0.984247	+	0.176798i	$0.943426\pi$
$774$	0	0
$775$	−111.468	−0.143830
$776$	− 709.178i	− 0.913890i
$777$	0	0
$778$	−295.476	−0.379789
$779$	− 1503.92i	− 1.93057i
$780$	0	0
$781$	138.600	0.177465
$782$	− 965.726i	− 1.23494i
$783$	0	0
$784$	−1463.90	−1.86722
$785$	220.876i	0.281370i
$786$	0	0
$787$	1042.66	1.32485	0.662425	−	0.749128i	$-0.269527\pi$
$787$	1042.66	1.32485	0.662425	+	0.749128i	$0.269527\pi$
$788$	2374.85i	3.01376i
$789$	0	0
$790$	−109.975	−0.139209
$791$	107.698i	0.136154i
$792$	0	0
$793$	237.594	0.299614
$794$	− 262.314i	− 0.330370i
$795$	0	0
$796$	2568.75	3.22707
$797$	− 67.4321i	− 0.0846074i	−0.999105	−	0.0423037i	$-0.986530\pi$
$797$	− 67.4321i	− 0.0846074i	0.999105	−	0.0423037i	$-0.0134697\pi$
$798$	0	0
$799$	−17.2464	−0.0215850
$800$	1212.07i	1.51509i
$801$	0	0
$802$	512.602	0.639155
$803$	− 1405.04i	− 1.74973i
$804$	0	0
$805$	−62.0678	−0.0771029
$806$	− 128.666i	− 0.159635i
$807$	0	0
$808$	−1703.53	−2.10833
$809$	− 1496.12i	− 1.84935i	−0.380758	−	0.924675i	$-0.624337\pi$
$809$	− 1496.12i	− 1.84935i	0.380758	−	0.924675i	$-0.375663\pi$
$810$	0	0
$811$	217.664	0.268389	0.134195	−	0.990955i	$-0.457155\pi$
$811$	217.664	0.268389	0.134195	+	0.990955i	$0.457155\pi$
$812$	− 847.302i	− 1.04348i
$813$	0	0
$814$	−449.652	−0.552398
$815$	87.9947i	0.107969i
$816$	0	0
$817$	−847.323	−1.03712
$818$	2414.22i	2.95137i
$819$	0	0
$820$	−489.870	−0.597402
$821$	352.519i	0.429378i	0.976682	+	0.214689i	$0.0688738\pi$
$821$	352.519i	0.429378i	−0.976682	+	0.214689i	$0.931126\pi$
$822$	0	0
$823$	−1222.88	−1.48588	−0.742941	−	0.669357i	$-0.766569\pi$
$823$	−1222.88	−1.48588	−0.742941	+	0.669357i	$0.766569\pi$
$824$	113.200i	0.137379i
$825$	0	0
$826$	−602.717	−0.729682
$827$	454.985i	0.550163i	0.961421	+	0.275082i	$0.0887049\pi$
$827$	454.985i	0.550163i	−0.961421	+	0.275082i	$0.911295\pi$
$828$	0	0
$829$	1307.17	1.57681	0.788404	−	0.615157i	$-0.210908\pi$
$829$	1307.17	1.57681	0.788404	+	0.615157i	$0.210908\pi$
$830$	− 158.662i	− 0.191158i
$831$	0	0
$832$	−317.280	−0.381346
$833$	− 413.110i	− 0.495931i
$834$	0	0
$835$	−246.824	−0.295597
$836$	− 3327.61i	− 3.98040i
$837$	0	0
$838$	305.453	0.364503
$839$	1460.75i	1.74106i	0.492116	+	0.870529i	$0.336223\pi$
$839$	1460.75i	1.74106i	−0.492116	+	0.870529i	$0.663777\pi$
$840$	0	0
$841$	−213.938	−0.254386
$842$	− 164.893i	− 0.195835i
$843$	0	0
$844$	−48.0868	−0.0569749
$845$	− 94.1999i	− 0.111479i
$846$	0	0
$847$	−228.537	−0.269819
$848$	551.717i	0.650609i
$849$	0	0
$850$	−888.790	−1.04564
$851$	226.635i	0.266316i
$852$	0	0
$853$	815.030	0.955487	0.477744	−	0.878499i	$-0.341455\pi$
$853$	815.030	0.955487	0.477744	+	0.878499i	$0.341455\pi$
$854$	314.915i	0.368753i
$855$	0	0
$856$	−3260.22	−3.80867
$857$	− 409.071i	− 0.477329i	−0.971102	−	0.238665i	$-0.923290\pi$
$857$	− 409.071i	− 0.477329i	0.971102	−	0.238665i	$-0.0767096\pi$
$858$	0	0
$859$	1179.16	1.37271	0.686357	−	0.727265i	$-0.259209\pi$
$859$	1179.16	1.37271	0.686357	+	0.727265i	$0.259209\pi$
$860$	275.998i	0.320928i
$861$	0	0
$862$	1790.69	2.07737
$863$	− 1131.09i	− 1.31065i	−0.755347	−	0.655325i	$-0.772532\pi$
$863$	− 1131.09i	− 1.31065i	0.755347	−	0.655325i	$-0.227468\pi$
$864$	0	0
$865$	109.585	0.126688
$866$	− 2416.26i	− 2.79014i
$867$	0	0
$868$	119.791	0.138009
$869$	502.644i	0.578416i
$870$	0	0
$871$	−951.346	−1.09225
$872$	− 24.8710i	− 0.0285217i
$873$	0	0
$874$	−2387.70	−2.73192
$875$	115.954i	0.132518i
$876$	0	0
$877$	−647.836	−0.738695	−0.369348	−	0.929291i	$-0.620419\pi$
$877$	−647.836	−0.738695	−0.369348	+	0.929291i	$0.620419\pi$
$878$	315.982i	0.359889i
$879$	0	0
$880$	−430.219	−0.488885
$881$	− 1224.05i	− 1.38939i	−0.719304	−	0.694696i	$-0.755539\pi$
$881$	− 1224.05i	− 1.38939i	0.719304	−	0.694696i	$-0.244461\pi$
$882$	0	0
$883$	1114.32	1.26197	0.630986	−	0.775794i	$-0.282650\pi$
$883$	1114.32	1.26197	0.630986	+	0.775794i	$0.282650\pi$
$884$	− 720.636i	− 0.815199i
$885$	0	0
$886$	−1911.86	−2.15785
$887$	1180.73i	1.33115i	0.746333	+	0.665573i	$0.231813\pi$
$887$	1180.73i	1.33115i	−0.746333	+	0.665573i	$0.768187\pi$
$888$	0	0
$889$	−31.1349	−0.0350224
$890$	37.4938i	0.0421278i
$891$	0	0
$892$	−708.475	−0.794255
$893$	42.6407i	0.0477499i
$894$	0	0
$895$	−219.806	−0.245593
$896$	131.268i	0.146505i
$897$	0	0
$898$	−1991.87	−2.21812
$899$	− 149.147i	− 0.165903i
$900$	0	0
$901$	−155.694	−0.172801
$902$	3187.43i	3.53374i
$903$	0	0
$904$	777.824	0.860425
$905$	43.8787i	0.0484847i
$906$	0	0
$907$	688.839	0.759470	0.379735	−	0.925095i	$-0.376015\pi$
$907$	688.839	0.759470	0.379735	+	0.925095i	$0.376015\pi$
$908$	2092.34i	2.30434i
$909$	0	0
$910$	−65.9362	−0.0724574
$911$	904.227i	0.992565i	0.868161	+	0.496283i	$0.165302\pi$
$911$	904.227i	0.992565i	−0.868161	+	0.496283i	$0.834698\pi$
$912$	0	0
$913$	−725.164	−0.794265
$914$	658.453i	0.720408i
$915$	0	0
$916$	3091.67	3.37519
$917$	− 170.683i	− 0.186132i
$918$	0	0
$919$	591.385	0.643509	0.321754	−	0.946823i	$-0.395727\pi$
$919$	591.385	0.643509	0.321754	+	0.946823i	$0.395727\pi$
$920$	448.272i	0.487252i
$921$	0	0
$922$	574.999	0.623643
$923$	− 74.2117i	− 0.0804027i
$924$	0	0
$925$	208.580	0.225492
$926$	1790.40i	1.93348i
$927$	0	0
$928$	−1621.77	−1.74760
$929$	1652.28i	1.77856i	0.457364	+	0.889280i	$0.348794\pi$
$929$	1652.28i	1.77856i	−0.457364	+	0.889280i	$0.651206\pi$
$930$	0	0
$931$	−1021.39	−1.09709
$932$	498.673i	0.535057i
$933$	0	0
$934$	1154.87	1.23648
$935$	− 121.407i	− 0.129847i
$936$	0	0
$937$	−1538.88	−1.64235	−0.821174	−	0.570678i	$-0.806680\pi$
$937$	−1538.88	−1.64235	−0.821174	+	0.570678i	$0.806680\pi$
$938$	− 1260.95i	− 1.34429i
$939$	0	0
$940$	13.8893	0.0147759
$941$	− 1733.41i	− 1.84210i	−0.389450	−	0.921048i	$-0.627335\pi$
$941$	− 1733.41i	− 1.84210i	0.389450	−	0.921048i	$-0.372665\pi$
$942$	0	0
$943$	1606.54	1.70365
$944$	2105.66i	2.23057i
$945$	0	0
$946$	1795.83	1.89834
$947$	− 160.332i	− 0.169305i	−0.996411	−	0.0846527i	$-0.973022\pi$
$947$	− 160.332i	− 0.169305i	0.996411	−	0.0846527i	$-0.0269781\pi$
$948$	0	0
$949$	−752.309	−0.792738
$950$	2197.48i	2.31314i
$951$	0	0
$952$	550.527	0.578285
$953$	102.479i	0.107533i	0.998554	+	0.0537667i	$0.0171227\pi$
$953$	102.479i	0.107533i	−0.998554	+	0.0537667i	$0.982877\pi$
$954$	0	0
$955$	−109.713	−0.114883
$956$	− 708.128i	− 0.740719i
$957$	0	0
$958$	2507.13	2.61704
$959$	641.608i	0.669039i
$960$	0	0
$961$	−939.914	−0.978058
$962$	240.760i	0.250271i
$963$	0	0
$964$	−1943.55	−2.01614
$965$	− 51.4453i	− 0.0533111i
$966$	0	0
$967$	−483.412	−0.499908	−0.249954	−	0.968258i	$-0.580416\pi$
$967$	−483.412	−0.499908	−0.249954	+	0.968258i	$0.580416\pi$
$968$	1650.56i	1.70512i
$969$	0	0
$970$	−110.991	−0.114424
$971$	1891.66i	1.94816i	0.226206	+	0.974079i	$0.427368\pi$
$971$	1891.66i	1.94816i	−0.226206	+	0.974079i	$0.572632\pi$
$972$	0	0
$973$	233.263	0.239735
$974$	2532.88i	2.60049i
$975$	0	0
$976$	1100.19	1.12724
$977$	1540.72i	1.57699i	0.615043	+	0.788493i	$0.289139\pi$
$977$	1540.72i	1.57699i	−0.615043	+	0.788493i	$0.710861\pi$
$978$	0	0
$979$	171.366	0.175041
$980$	332.697i	0.339486i
$981$	0	0
$982$	−1927.60	−1.96294
$983$	1251.38i	1.27302i	0.771266	+	0.636512i	$0.219624\pi$
$983$	1251.38i	1.27302i	−0.771266	+	0.636512i	$0.780376\pi$
$984$	0	0
$985$	214.226	0.217488
$986$	− 1189.22i	− 1.20611i
$987$	0	0
$988$	−1781.73	−1.80337
$989$	− 905.142i	− 0.915210i
$990$	0	0
$991$	−857.100	−0.864884	−0.432442	−	0.901662i	$-0.642348\pi$
$991$	−857.100	−0.864884	−0.432442	+	0.901662i	$0.642348\pi$
$992$	− 229.286i	− 0.231135i
$993$	0	0
$994$	98.3627	0.0989564
$995$	− 231.717i	− 0.232881i
$996$	0	0
$997$	−1671.70	−1.67673	−0.838364	−	0.545111i	$-0.816487\pi$
$997$	−1671.70	−1.67673	−0.838364	+	0.545111i	$0.816487\pi$
$998$	− 363.842i	− 0.364571i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.81	yes	84
3.2	odd	2	inner	1143.3.b.a.890.4	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.4	✓	84	3.2	odd	2	inner
1143.3.b.a.890.81	yes	84	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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1143.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.66638i q^{2} -9.44232 q^{4} +0.851756i q^{5} +2.76278 q^{7} -19.9536i q^{8} +O(q^{10})\)	\(q+3.66638i q^{2} -9.44232 q^{4} +0.851756i q^{5} +2.76278 q^{7} -19.9536i q^{8} -3.12286 q^{10} +14.2731i q^{11} +7.64232 q^{13} +10.1294i q^{14} +35.3881 q^{16} +9.98646i q^{17} +24.6909 q^{19} -8.04255i q^{20} -52.3304 q^{22} +26.3758i q^{23} +24.2745 q^{25} +28.0196i q^{26} -26.0870 q^{28} +32.4798i q^{29} -4.59199 q^{31} +49.9317i q^{32} -36.6141 q^{34} +2.35321i q^{35} +8.59256 q^{37} +90.5262i q^{38} +16.9956 q^{40} -60.9097i q^{41} -34.3172 q^{43} -134.771i q^{44} -96.7035 q^{46} +1.72698i q^{47} -41.3670 q^{49} +88.9995i q^{50} -72.1612 q^{52} +15.5905i q^{53} -12.1572 q^{55} -55.1274i q^{56} -119.083 q^{58} +59.5018i q^{59} +31.0892 q^{61} -16.8360i q^{62} -41.5161 q^{64} +6.50940i q^{65} -124.484 q^{67} -94.2954i q^{68} -8.62777 q^{70} -9.71062i q^{71} -98.4398 q^{73} +31.5035i q^{74} -233.140 q^{76} +39.4333i q^{77} +35.2163 q^{79} +30.1420i q^{80} +223.318 q^{82} +50.8065i q^{83} -8.50603 q^{85} -125.820i q^{86} +284.799 q^{88} -12.0062i q^{89} +21.1141 q^{91} -249.048i q^{92} -6.33175 q^{94} +21.0307i q^{95} +35.5414 q^{97} -151.667i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 84 * q - 160 * q^4	\( 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) \) 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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