LMFDB - Embedded newform 2151.4.a.f.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.28174 q^{2} -6.35714 q^{4} -12.7359 q^{5} +11.6522 q^{7} +18.4021 q^{8} +O(q^{10})$	\(q-1.28174 q^{2} -6.35714 q^{4} -12.7359 q^{5} +11.6522 q^{7} +18.4021 q^{8} +16.3241 q^{10} +3.29421 q^{11} +32.7304 q^{13} -14.9351 q^{14} +27.2704 q^{16} -32.8561 q^{17} +87.6845 q^{19} +80.9637 q^{20} -4.22232 q^{22} -124.067 q^{23} +37.2021 q^{25} -41.9519 q^{26} -74.0746 q^{28} +200.358 q^{29} +136.486 q^{31} -182.171 q^{32} +42.1130 q^{34} -148.401 q^{35} +387.819 q^{37} -112.389 q^{38} -234.367 q^{40} -498.779 q^{41} +392.687 q^{43} -20.9418 q^{44} +159.021 q^{46} +497.372 q^{47} -207.227 q^{49} -47.6835 q^{50} -208.072 q^{52} -536.333 q^{53} -41.9546 q^{55} +214.425 q^{56} -256.807 q^{58} +373.211 q^{59} -310.619 q^{61} -174.940 q^{62} +15.3320 q^{64} -416.850 q^{65} -996.205 q^{67} +208.871 q^{68} +190.211 q^{70} +533.832 q^{71} +425.470 q^{73} -497.083 q^{74} -557.423 q^{76} +38.3847 q^{77} -870.508 q^{79} -347.312 q^{80} +639.306 q^{82} -519.373 q^{83} +418.451 q^{85} -503.322 q^{86} +60.6205 q^{88} +35.1304 q^{89} +381.381 q^{91} +788.709 q^{92} -637.502 q^{94} -1116.74 q^{95} -705.471 q^{97} +265.611 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

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 $

$2$	−1.28174	−0.453163	−0.226582	−	0.973992i	$-0.572755\pi$
$2$	−1.28174	−0.453163	−0.226582	+	0.973992i	$0.572755\pi$
$3$	0	0
$3$	0	0
$4$	−6.35714	−0.794643
$5$	−12.7359	−1.13913	−0.569565	−	0.821946i	$-0.692888\pi$
$5$	−12.7359	−1.13913	−0.569565	+	0.821946i	$0.692888\pi$
$6$	0	0
$7$	11.6522	0.629159	0.314579	−	0.949231i	$-0.398137\pi$
$7$	11.6522	0.629159	0.314579	+	0.949231i	$0.398137\pi$
$8$	18.4021	0.813267
$9$	0	0
$10$	16.3241	0.516212
$11$	3.29421	0.0902947	0.0451473	−	0.998980i	$-0.485624\pi$
$11$	3.29421	0.0902947	0.0451473	+	0.998980i	$0.485624\pi$
$12$	0	0
$13$	32.7304	0.698291	0.349145	−	0.937069i	$-0.386472\pi$
$13$	32.7304	0.698291	0.349145	+	0.937069i	$0.386472\pi$
$14$	−14.9351	−0.285112
$15$	0	0
$16$	27.2704	0.426100
$17$	−32.8561	−0.468752	−0.234376	−	0.972146i	$-0.575305\pi$
$17$	−32.8561	−0.468752	−0.234376	+	0.972146i	$0.575305\pi$
$18$	0	0
$19$	87.6845	1.05875	0.529374	−	0.848389i	$-0.322427\pi$
$19$	87.6845	1.05875	0.529374	+	0.848389i	$0.322427\pi$
$20$	80.9637	0.905202
$21$	0	0
$22$	−4.22232	−0.0409183
$23$	−124.067	−1.12477	−0.562384	−	0.826876i	$-0.690116\pi$
$23$	−124.067	−1.12477	−0.562384	+	0.826876i	$0.690116\pi$
$24$	0	0
$25$	37.2021	0.297617
$26$	−41.9519	−0.316440
$27$	0	0
$28$	−74.0746	−0.499957
$29$	200.358	1.28295	0.641476	−	0.767143i	$-0.278323\pi$
$29$	200.358	1.28295	0.641476	+	0.767143i	$0.278323\pi$
$30$	0	0
$31$	136.486	0.790763	0.395381	−	0.918517i	$-0.370612\pi$
$31$	136.486	0.790763	0.395381	+	0.918517i	$0.370612\pi$
$32$	−182.171	−1.00636
$33$	0	0
$34$	42.1130	0.212421
$35$	−148.401	−0.716694
$36$	0	0
$37$	387.819	1.72316	0.861581	−	0.507620i	$-0.169475\pi$
$37$	387.819	1.72316	0.861581	+	0.507620i	$0.169475\pi$
$38$	−112.389	−0.479786
$39$	0	0
$40$	−234.367	−0.926416
$41$	−498.779	−1.89991	−0.949954	−	0.312388i	$-0.898871\pi$
$41$	−498.779	−1.89991	−0.949954	+	0.312388i	$0.898871\pi$
$42$	0	0
$43$	392.687	1.39265	0.696327	−	0.717724i	$-0.254816\pi$
$43$	392.687	1.39265	0.696327	+	0.717724i	$0.254816\pi$
$44$	−20.9418	−0.0717520
$45$	0	0
$46$	159.021	0.509704
$47$	497.372	1.54360	0.771800	−	0.635865i	$-0.219357\pi$
$47$	497.372	1.54360	0.771800	+	0.635865i	$0.219357\pi$
$48$	0	0
$49$	−207.227	−0.604159
$50$	−47.6835	−0.134869
$51$	0	0
$52$	−208.072	−0.554892
$53$	−536.333	−1.39002	−0.695010	−	0.719000i	$-0.744600\pi$
$53$	−536.333	−1.39002	−0.695010	+	0.719000i	$0.744600\pi$
$54$	0	0
$55$	−41.9546	−0.102857
$56$	214.425	0.511674
$57$	0	0
$58$	−256.807	−0.581387
$59$	373.211	0.823524	0.411762	−	0.911291i	$-0.364913\pi$
$59$	373.211	0.823524	0.411762	+	0.911291i	$0.364913\pi$
$60$	0	0
$61$	−310.619	−0.651978	−0.325989	−	0.945374i	$-0.605697\pi$
$61$	−310.619	−0.651978	−0.325989	+	0.945374i	$0.605697\pi$
$62$	−174.940	−0.358345
$63$	0	0
$64$	15.3320	0.0299453
$65$	−416.850	−0.795444
$66$	0	0
$67$	−996.205	−1.81651	−0.908253	−	0.418422i	$-0.862583\pi$
$67$	−996.205	−1.81651	−0.908253	+	0.418422i	$0.862583\pi$
$68$	208.871	0.372490
$69$	0	0
$70$	190.211	0.324779
$71$	533.832	0.892313	0.446157	−	0.894955i	$-0.352792\pi$
$71$	533.832	0.892313	0.446157	+	0.894955i	$0.352792\pi$
$72$	0	0
$73$	425.470	0.682157	0.341079	−	0.940035i	$-0.389208\pi$
$73$	425.470	0.682157	0.341079	+	0.940035i	$0.389208\pi$
$74$	−497.083	−0.780874
$75$	0	0
$76$	−557.423	−0.841326
$77$	38.3847	0.0568097
$78$	0	0
$79$	−870.508	−1.23974	−0.619872	−	0.784703i	$-0.712816\pi$
$79$	−870.508	−1.23974	−0.619872	+	0.784703i	$0.712816\pi$
$80$	−347.312	−0.485383
$81$	0	0
$82$	639.306	0.860969
$83$	−519.373	−0.686850	−0.343425	−	0.939180i	$-0.611587\pi$
$83$	−519.373	−0.686850	−0.343425	+	0.939180i	$0.611587\pi$
$84$	0	0
$85$	418.451	0.533969
$86$	−503.322	−0.631100
$87$	0	0
$88$	60.6205	0.0734337
$89$	35.1304	0.0418406	0.0209203	−	0.999781i	$-0.493340\pi$
$89$	35.1304	0.0418406	0.0209203	+	0.999781i	$0.493340\pi$
$90$	0	0
$91$	381.381	0.439336
$92$	788.709	0.893790
$93$	0	0
$94$	−637.502	−0.699503
$95$	−1116.74	−1.20605
$96$	0	0
$97$	−705.471	−0.738451	−0.369226	−	0.929340i	$-0.620377\pi$
$97$	−705.471	−0.738451	−0.369226	+	0.929340i	$0.620377\pi$
$98$	265.611	0.273783
$99$	0	0
$100$	−236.499	−0.236499
$101$	−237.324	−0.233808	−0.116904	−	0.993143i	$-0.537297\pi$
$101$	−237.324	−0.233808	−0.116904	+	0.993143i	$0.537297\pi$
$102$	0	0
$103$	−1217.61	−1.16480	−0.582401	−	0.812902i	$-0.697887\pi$
$103$	−1217.61	−1.16480	−0.582401	+	0.812902i	$0.697887\pi$
$104$	602.309	0.567897
$105$	0	0
$106$	687.440	0.629906
$107$	−75.8195	−0.0685023	−0.0342511	−	0.999413i	$-0.510905\pi$
$107$	−75.8195	−0.0685023	−0.0342511	+	0.999413i	$0.510905\pi$
$108$	0	0
$109$	1544.88	1.35755	0.678773	−	0.734348i	$-0.262512\pi$
$109$	1544.88	1.35755	0.678773	+	0.734348i	$0.262512\pi$
$110$	53.7749	0.0466112
$111$	0	0
$112$	317.760	0.268085
$113$	−2031.58	−1.69128	−0.845642	−	0.533750i	$-0.820782\pi$
$113$	−2031.58	−1.69128	−0.845642	+	0.533750i	$0.820782\pi$
$114$	0	0
$115$	1580.10	1.28126
$116$	−1273.71	−1.01949
$117$	0	0
$118$	−478.359	−0.373191
$119$	−382.846	−0.294919
$120$	0	0
$121$	−1320.15	−0.991847
$122$	398.133	0.295453
$123$	0	0
$124$	−867.662	−0.628374
$125$	1118.18	0.800105
$126$	0	0
$127$	843.301	0.589219	0.294610	−	0.955618i	$-0.404810\pi$
$127$	843.301	0.589219	0.294610	+	0.955618i	$0.404810\pi$
$128$	1437.71	0.992790
$129$	0	0
$130$	534.293	0.360466
$131$	−135.628	−0.0904571	−0.0452285	−	0.998977i	$-0.514402\pi$
$131$	−135.628	−0.0904571	−0.0452285	+	0.998977i	$0.514402\pi$
$132$	0	0
$133$	1021.72	0.666120
$134$	1276.88	0.823174
$135$	0	0
$136$	−604.623	−0.381220
$137$	−2916.46	−1.81876	−0.909380	−	0.415967i	$-0.863443\pi$
$137$	−2916.46	−1.81876	−0.909380	+	0.415967i	$0.863443\pi$
$138$	0	0
$139$	−2677.93	−1.63409	−0.817046	−	0.576572i	$-0.804390\pi$
$139$	−2677.93	−1.63409	−0.817046	+	0.576572i	$0.804390\pi$
$140$	943.404	0.569516
$141$	0	0
$142$	−684.234	−0.404364
$143$	107.821	0.0630520
$144$	0	0
$145$	−2551.73	−1.46145
$146$	−545.342	−0.309129
$147$	0	0
$148$	−2465.42	−1.36930
$149$	1808.85	0.994542	0.497271	−	0.867595i	$-0.334335\pi$
$149$	1808.85	0.994542	0.497271	+	0.867595i	$0.334335\pi$
$150$	0	0
$151$	2793.28	1.50539	0.752696	−	0.658368i	$-0.228753\pi$
$151$	2793.28	1.50539	0.752696	+	0.658368i	$0.228753\pi$
$152$	1613.58	0.861044
$153$	0	0
$154$	−49.1993	−0.0257441
$155$	−1738.27	−0.900781
$156$	0	0
$157$	3036.96	1.54380	0.771898	−	0.635746i	$-0.219308\pi$
$157$	3036.96	1.54380	0.771898	+	0.635746i	$0.219308\pi$
$158$	1115.76	0.561807
$159$	0	0
$160$	2320.10	1.14637
$161$	−1445.65	−0.707658
$162$	0	0
$163$	3252.39	1.56287	0.781433	−	0.623989i	$-0.214489\pi$
$163$	3252.39	1.56287	0.781433	+	0.623989i	$0.214489\pi$
$164$	3170.81	1.50975
$165$	0	0
$166$	665.701	0.311255
$167$	1296.99	0.600983	0.300491	−	0.953785i	$-0.402849\pi$
$167$	1296.99	0.600983	0.300491	+	0.953785i	$0.402849\pi$
$168$	0	0
$169$	−1125.72	−0.512390
$170$	−536.345	−0.241975
$171$	0	0
$172$	−2496.37	−1.10666
$173$	147.035	0.0646179	0.0323089	−	0.999478i	$-0.489714\pi$
$173$	147.035	0.0646179	0.0323089	+	0.999478i	$0.489714\pi$
$174$	0	0
$175$	433.486	0.187249
$176$	89.8344	0.0384746
$177$	0	0
$178$	−45.0281	−0.0189607
$179$	2531.07	1.05688	0.528439	−	0.848972i	$-0.322778\pi$
$179$	2531.07	1.05688	0.528439	+	0.848972i	$0.322778\pi$
$180$	0	0
$181$	716.383	0.294190	0.147095	−	0.989122i	$-0.453008\pi$
$181$	716.383	0.294190	0.147095	+	0.989122i	$0.453008\pi$
$182$	−488.831	−0.199091
$183$	0	0
$184$	−2283.09	−0.914737
$185$	−4939.21	−1.96291
$186$	0	0
$187$	−108.235	−0.0423258
$188$	−3161.87	−1.22661
$189$	0	0
$190$	1431.37	0.546538
$191$	4308.77	1.63231	0.816155	−	0.577833i	$-0.196101\pi$
$191$	4308.77	1.63231	0.816155	+	0.577833i	$0.196101\pi$
$192$	0	0
$193$	763.306	0.284684	0.142342	−	0.989818i	$-0.454537\pi$
$193$	763.306	0.284684	0.142342	+	0.989818i	$0.454537\pi$
$194$	904.231	0.334639
$195$	0	0
$196$	1317.37	0.480091
$197$	−1672.75	−0.604967	−0.302484	−	0.953155i	$-0.597816\pi$
$197$	−1672.75	−0.604967	−0.302484	+	0.953155i	$0.597816\pi$
$198$	0	0
$199$	2953.70	1.05217	0.526085	−	0.850432i	$-0.323659\pi$
$199$	2953.70	1.05217	0.526085	+	0.850432i	$0.323659\pi$
$200$	684.598	0.242042
$201$	0	0
$202$	304.188	0.105953
$203$	2334.61	0.807180
$204$	0	0
$205$	6352.39	2.16424
$206$	1560.66	0.527846
$207$	0	0
$208$	892.572	0.297542
$209$	288.851	0.0955993
$210$	0	0
$211$	3245.57	1.05893	0.529466	−	0.848331i	$-0.322392\pi$
$211$	3245.57	1.05893	0.529466	+	0.848331i	$0.322392\pi$
$212$	3409.55	1.10457
$213$	0	0
$214$	97.1809	0.0310427
$215$	−5001.20	−1.58641
$216$	0	0
$217$	1590.36	0.497515
$218$	−1980.13	−0.615190
$219$	0	0
$220$	266.711	0.0817349
$221$	−1075.39	−0.327325
$222$	0	0
$223$	−3926.69	−1.17915	−0.589575	−	0.807714i	$-0.700705\pi$
$223$	−3926.69	−1.17915	−0.589575	+	0.807714i	$0.700705\pi$
$224$	−2122.69	−0.633160
$225$	0	0
$226$	2603.96	0.766429
$227$	3889.26	1.13718	0.568589	−	0.822622i	$-0.307490\pi$
$227$	3889.26	1.13718	0.568589	+	0.822622i	$0.307490\pi$
$228$	0	0
$229$	4954.20	1.42962	0.714810	−	0.699319i	$-0.246513\pi$
$229$	4954.20	1.42962	0.714810	+	0.699319i	$0.246513\pi$
$230$	−2025.27	−0.580619
$231$	0	0
$232$	3687.02	1.04338
$233$	3485.08	0.979894	0.489947	−	0.871752i	$-0.337016\pi$
$233$	3485.08	0.979894	0.489947	+	0.871752i	$0.337016\pi$
$234$	0	0
$235$	−6334.47	−1.75836
$236$	−2372.56	−0.654408
$237$	0	0
$238$	490.709	0.133647
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−1613.69	−0.431314	−0.215657	−	0.976469i	$-0.569189\pi$
$241$	−1613.69	−0.431314	−0.215657	+	0.976469i	$0.569189\pi$
$242$	1692.09	0.449469
$243$	0	0
$244$	1974.65	0.518090
$245$	2639.21	0.688216
$246$	0	0
$247$	2869.95	0.739314
$248$	2511.64	0.643101
$249$	0	0
$250$	−1433.22	−0.362579
$251$	770.928	0.193867	0.0969334	−	0.995291i	$-0.469097\pi$
$251$	770.928	0.193867	0.0969334	+	0.995291i	$0.469097\pi$
$252$	0	0
$253$	−408.702	−0.101561
$254$	−1080.89	−0.267013
$255$	0	0
$256$	−1965.43	−0.479841
$257$	4733.58	1.14892	0.574461	−	0.818532i	$-0.305212\pi$
$257$	4733.58	1.14892	0.574461	+	0.818532i	$0.305212\pi$
$258$	0	0
$259$	4518.94	1.08414
$260$	2649.97	0.632094
$261$	0	0
$262$	173.840	0.0409919
$263$	−977.113	−0.229093	−0.114546	−	0.993418i	$-0.536541\pi$
$263$	−977.113	−0.229093	−0.114546	+	0.993418i	$0.536541\pi$
$264$	0	0
$265$	6830.66	1.58341
$266$	−1309.57	−0.301861
$267$	0	0
$268$	6333.02	1.44347
$269$	4108.52	0.931231	0.465615	−	0.884987i	$-0.345833\pi$
$269$	4108.52	0.931231	0.465615	+	0.884987i	$0.345833\pi$
$270$	0	0
$271$	1193.07	0.267432	0.133716	−	0.991020i	$-0.457309\pi$
$271$	1193.07	0.267432	0.133716	+	0.991020i	$0.457309\pi$
$272$	−896.000	−0.199735
$273$	0	0
$274$	3738.14	0.824195
$275$	122.552	0.0268733
$276$	0	0
$277$	3854.47	0.836075	0.418038	−	0.908430i	$-0.362718\pi$
$277$	3854.47	0.836075	0.418038	+	0.908430i	$0.362718\pi$
$278$	3432.41	0.740511
$279$	0	0
$280$	−2730.89	−0.582863
$281$	3006.79	0.638328	0.319164	−	0.947699i	$-0.396598\pi$
$281$	3006.79	0.638328	0.319164	+	0.947699i	$0.396598\pi$
$282$	0	0
$283$	5235.95	1.09980	0.549902	−	0.835229i	$-0.314665\pi$
$283$	5235.95	1.09980	0.549902	+	0.835229i	$0.314665\pi$
$284$	−3393.65	−0.709070
$285$	0	0
$286$	−138.198	−0.0285729
$287$	−5811.87	−1.19534
$288$	0	0
$289$	−3833.47	−0.780272
$290$	3270.66	0.662275
$291$	0	0
$292$	−2704.77	−0.542071
$293$	−8086.45	−1.61234	−0.806169	−	0.591685i	$-0.798463\pi$
$293$	−8086.45	−1.61234	−0.806169	+	0.591685i	$0.798463\pi$
$294$	0	0
$295$	−4753.16	−0.938101
$296$	7136.69	1.40139
$297$	0	0
$298$	−2318.48	−0.450690
$299$	−4060.75	−0.785416
$300$	0	0
$301$	4575.66	0.876201
$302$	−3580.26	−0.682189
$303$	0	0
$304$	2391.19	0.451132
$305$	3956.00	0.742688
$306$	0	0
$307$	−6692.72	−1.24421	−0.622107	−	0.782932i	$-0.713723\pi$
$307$	−6692.72	−1.24421	−0.622107	+	0.782932i	$0.713723\pi$
$308$	−244.017	−0.0451434
$309$	0	0
$310$	2228.01	0.408201
$311$	2934.07	0.534970	0.267485	−	0.963562i	$-0.413807\pi$
$311$	2934.07	0.534970	0.267485	+	0.963562i	$0.413807\pi$
$312$	0	0
$313$	416.030	0.0751290	0.0375645	−	0.999294i	$-0.488040\pi$
$313$	416.030	0.0751290	0.0375645	+	0.999294i	$0.488040\pi$
$314$	−3892.60	−0.699592
$315$	0	0
$316$	5533.94	0.985154
$317$	4100.50	0.726520	0.363260	−	0.931688i	$-0.381664\pi$
$317$	4100.50	0.726520	0.363260	+	0.931688i	$0.381664\pi$
$318$	0	0
$319$	660.022	0.115844
$320$	−195.266	−0.0341116
$321$	0	0
$322$	1852.94	0.320685
$323$	−2880.97	−0.496290
$324$	0	0
$325$	1217.64	0.207823
$326$	−4168.72	−0.708234
$327$	0	0
$328$	−9178.60	−1.54513
$329$	5795.48	0.971170
$330$	0	0
$331$	−4965.01	−0.824477	−0.412238	−	0.911076i	$-0.635253\pi$
$331$	−4965.01	−0.824477	−0.412238	+	0.911076i	$0.635253\pi$
$332$	3301.73	0.545800
$333$	0	0
$334$	−1662.40	−0.272344
$335$	12687.5	2.06924
$336$	0	0
$337$	−4108.83	−0.664161	−0.332080	−	0.943251i	$-0.607751\pi$
$337$	−4108.83	−0.664161	−0.332080	+	0.943251i	$0.607751\pi$
$338$	1442.88	0.232196
$339$	0	0
$340$	−2660.15	−0.424315
$341$	449.614	0.0714017
$342$	0	0
$343$	−6411.34	−1.00927
$344$	7226.27	1.13260
$345$	0	0
$346$	−188.461	−0.0292825
$347$	−6334.11	−0.979922	−0.489961	−	0.871744i	$-0.662989\pi$
$347$	−6334.11	−0.979922	−0.489961	+	0.871744i	$0.662989\pi$
$348$	0	0
$349$	−7887.53	−1.20977	−0.604885	−	0.796313i	$-0.706781\pi$
$349$	−7887.53	−1.20977	−0.604885	+	0.796313i	$0.706781\pi$
$350$	−555.617	−0.0848542
$351$	0	0
$352$	−600.108	−0.0908689
$353$	−2456.48	−0.370383	−0.185192	−	0.982702i	$-0.559291\pi$
$353$	−2456.48	−0.370383	−0.185192	+	0.982702i	$0.559291\pi$
$354$	0	0
$355$	−6798.81	−1.01646
$356$	−223.329	−0.0332484
$357$	0	0
$358$	−3244.17	−0.478938
$359$	−4021.21	−0.591174	−0.295587	−	0.955316i	$-0.595515\pi$
$359$	−4021.21	−0.591174	−0.295587	+	0.955316i	$0.595515\pi$
$360$	0	0
$361$	829.571	0.120946
$362$	−918.217	−0.133316
$363$	0	0
$364$	−2424.49	−0.349115
$365$	−5418.72	−0.777066
$366$	0	0
$367$	1011.68	0.143895	0.0719473	−	0.997408i	$-0.477079\pi$
$367$	1011.68	0.143895	0.0719473	+	0.997408i	$0.477079\pi$
$368$	−3383.35	−0.479264
$369$	0	0
$370$	6330.78	0.889517
$371$	−6249.45	−0.874543
$372$	0	0
$373$	−8206.22	−1.13915	−0.569574	−	0.821940i	$-0.692892\pi$
$373$	−8206.22	−1.13915	−0.569574	+	0.821940i	$0.692892\pi$
$374$	138.729	0.0191805
$375$	0	0
$376$	9152.71	1.25536
$377$	6557.81	0.895873
$378$	0	0
$379$	2683.49	0.363699	0.181849	−	0.983326i	$-0.441792\pi$
$379$	2683.49	0.363699	0.181849	+	0.983326i	$0.441792\pi$
$380$	7099.26	0.958380
$381$	0	0
$382$	−5522.72	−0.739704
$383$	2609.88	0.348195	0.174097	−	0.984728i	$-0.444299\pi$
$383$	2609.88	0.348195	0.174097	+	0.984728i	$0.444299\pi$
$384$	0	0
$385$	−488.863	−0.0647137
$386$	−978.360	−0.129008
$387$	0	0
$388$	4484.78	0.586805
$389$	4728.69	0.616335	0.308167	−	0.951332i	$-0.400284\pi$
$389$	4728.69	0.616335	0.308167	+	0.951332i	$0.400284\pi$
$390$	0	0
$391$	4076.35	0.527238
$392$	−3813.41	−0.491342
$393$	0	0
$394$	2144.03	0.274149
$395$	11086.7	1.41223
$396$	0	0
$397$	10308.3	1.30317	0.651586	−	0.758575i	$-0.274104\pi$
$397$	10308.3	1.30317	0.651586	+	0.758575i	$0.274104\pi$
$398$	−3785.87	−0.476805
$399$	0	0
$400$	1014.52	0.126815
$401$	−1121.79	−0.139699	−0.0698496	−	0.997558i	$-0.522252\pi$
$401$	−1121.79	−0.139699	−0.0698496	+	0.997558i	$0.522252\pi$
$402$	0	0
$403$	4467.25	0.552182
$404$	1508.70	0.185794
$405$	0	0
$406$	−2992.36	−0.365785
$407$	1277.56	0.155592
$408$	0	0
$409$	−4052.98	−0.489992	−0.244996	−	0.969524i	$-0.578787\pi$
$409$	−4052.98	−0.489992	−0.244996	+	0.969524i	$0.578787\pi$
$410$	−8142.11	−0.980756
$411$	0	0
$412$	7740.52	0.925602
$413$	4348.72	0.518128
$414$	0	0
$415$	6614.66	0.782411
$416$	−5962.52	−0.702732
$417$	0	0
$418$	−370.232	−0.0433221
$419$	4379.95	0.510679	0.255339	−	0.966852i	$-0.417813\pi$
$419$	4379.95	0.510679	0.255339	+	0.966852i	$0.417813\pi$
$420$	0	0
$421$	10092.3	1.16833	0.584164	−	0.811636i	$-0.301422\pi$
$421$	10092.3	1.16833	0.584164	+	0.811636i	$0.301422\pi$
$422$	−4159.98	−0.479869
$423$	0	0
$424$	−9869.67	−1.13046
$425$	−1222.32	−0.139509
$426$	0	0
$427$	−3619.39	−0.410198
$428$	481.995	0.0544349
$429$	0	0
$430$	6410.24	0.718905
$431$	−3290.85	−0.367784	−0.183892	−	0.982946i	$-0.558870\pi$
$431$	−3290.85	−0.367784	−0.183892	+	0.982946i	$0.558870\pi$
$432$	0	0
$433$	3096.01	0.343614	0.171807	−	0.985131i	$-0.445040\pi$
$433$	3096.01	0.343614	0.171807	+	0.985131i	$0.445040\pi$
$434$	−2038.43	−0.225456
$435$	0	0
$436$	−9821.01	−1.07876
$437$	−10878.7	−1.19085
$438$	0	0
$439$	15783.4	1.71594	0.857972	−	0.513697i	$-0.171724\pi$
$439$	15783.4	1.71594	0.857972	+	0.513697i	$0.171724\pi$
$440$	−772.054	−0.0836505
$441$	0	0
$442$	1378.38	0.148332
$443$	15491.0	1.66139	0.830697	−	0.556725i	$-0.187942\pi$
$443$	15491.0	1.66139	0.830697	+	0.556725i	$0.187942\pi$
$444$	0	0
$445$	−447.416	−0.0476619
$446$	5032.99	0.534348
$447$	0	0
$448$	178.651	0.0188404
$449$	−8810.37	−0.926029	−0.463014	−	0.886351i	$-0.653232\pi$
$449$	−8810.37	−0.926029	−0.463014	+	0.886351i	$0.653232\pi$
$450$	0	0
$451$	−1643.08	−0.171552
$452$	12915.1	1.34397
$453$	0	0
$454$	−4985.02	−0.515327
$455$	−4857.21	−0.500461
$456$	0	0
$457$	4070.72	0.416675	0.208337	−	0.978057i	$-0.433195\pi$
$457$	4070.72	0.416675	0.208337	+	0.978057i	$0.433195\pi$
$458$	−6350.00	−0.647851
$459$	0	0
$460$	−10044.9	−1.01814
$461$	−14681.8	−1.48329	−0.741647	−	0.670791i	$-0.765955\pi$
$461$	−14681.8	−1.48329	−0.741647	+	0.670791i	$0.765955\pi$
$462$	0	0
$463$	2787.63	0.279811	0.139905	−	0.990165i	$-0.455320\pi$
$463$	2787.63	0.279811	0.139905	+	0.990165i	$0.455320\pi$
$464$	5463.85	0.546666
$465$	0	0
$466$	−4466.97	−0.444052
$467$	1006.31	0.0997138	0.0498569	−	0.998756i	$-0.484123\pi$
$467$	1006.31	0.0997138	0.0498569	+	0.998756i	$0.484123\pi$
$468$	0	0
$469$	−11608.0	−1.14287
$470$	8119.14	0.796825
$471$	0	0
$472$	6867.88	0.669745
$473$	1293.59	0.125749
$474$	0	0
$475$	3262.05	0.315101
$476$	2433.81	0.234356
$477$	0	0
$478$	−306.336	−0.0293127
$479$	−17050.6	−1.62644	−0.813219	−	0.581958i	$-0.802287\pi$
$479$	−17050.6	−1.62644	−0.813219	+	0.581958i	$0.802287\pi$
$480$	0	0
$481$	12693.5	1.20327
$482$	2068.33	0.195456
$483$	0	0
$484$	8392.37	0.788164
$485$	8984.79	0.841192
$486$	0	0
$487$	−1671.85	−0.155562	−0.0777810	−	0.996970i	$-0.524784\pi$
$487$	−1671.85	−0.155562	−0.0777810	+	0.996970i	$0.524784\pi$
$488$	−5716.05	−0.530232
$489$	0	0
$490$	−3382.78	−0.311874
$491$	5750.51	0.528548	0.264274	−	0.964448i	$-0.414868\pi$
$491$	5750.51	0.528548	0.264274	+	0.964448i	$0.414868\pi$
$492$	0	0
$493$	−6583.00	−0.601386
$494$	−3678.53	−0.335030
$495$	0	0
$496$	3722.03	0.336944
$497$	6220.31	0.561407
$498$	0	0
$499$	7773.48	0.697372	0.348686	−	0.937240i	$-0.386628\pi$
$499$	7773.48	0.697372	0.348686	+	0.937240i	$0.386628\pi$
$500$	−7108.44	−0.635798
$501$	0	0
$502$	−988.130	−0.0878534
$503$	9395.33	0.832837	0.416419	−	0.909173i	$-0.363285\pi$
$503$	9395.33	0.832837	0.416419	+	0.909173i	$0.363285\pi$
$504$	0	0
$505$	3022.53	0.266338
$506$	523.849	0.0460236
$507$	0	0
$508$	−5360.98	−0.468219
$509$	11810.2	1.02844	0.514220	−	0.857658i	$-0.328081\pi$
$509$	11810.2	1.02844	0.514220	+	0.857658i	$0.328081\pi$
$510$	0	0
$511$	4957.65	0.429185
$512$	−8982.53	−0.775343
$513$	0	0
$514$	−6067.22	−0.520649
$515$	15507.3	1.32686
$516$	0	0
$517$	1638.45	0.139379
$518$	−5792.10	−0.491294
$519$	0	0
$520$	−7670.92	−0.646908
$521$	3038.32	0.255492	0.127746	−	0.991807i	$-0.459226\pi$
$521$	3038.32	0.255492	0.127746	+	0.991807i	$0.459226\pi$
$522$	0	0
$523$	−2346.37	−0.196175	−0.0980876	−	0.995178i	$-0.531273\pi$
$523$	−2346.37	−0.196175	−0.0980876	+	0.995178i	$0.531273\pi$
$524$	862.207	0.0718811
$525$	0	0
$526$	1252.40	0.103816
$527$	−4484.41	−0.370671
$528$	0	0
$529$	3225.53	0.265105
$530$	−8755.14	−0.717545
$531$	0	0
$532$	−6495.20	−0.529328
$533$	−16325.3	−1.32669
$534$	0	0
$535$	965.626	0.0780330
$536$	−18332.3	−1.47730
$537$	0	0
$538$	−5266.06	−0.422000
$539$	−682.648	−0.0545524
$540$	0	0
$541$	−21671.3	−1.72222	−0.861110	−	0.508419i	$-0.830230\pi$
$541$	−21671.3	−1.72222	−0.861110	+	0.508419i	$0.830230\pi$
$542$	−1529.21	−0.121190
$543$	0	0
$544$	5985.42	0.471733
$545$	−19675.4	−1.54642
$546$	0	0
$547$	−2439.49	−0.190686	−0.0953428	−	0.995444i	$-0.530395\pi$
$547$	−2439.49	−0.190686	−0.0953428	+	0.995444i	$0.530395\pi$
$548$	18540.4	1.44526
$549$	0	0
$550$	−157.079	−0.0121780
$551$	17568.3	1.35832
$552$	0	0
$553$	−10143.3	−0.779996
$554$	−4940.43	−0.378879
$555$	0	0
$556$	17024.0	1.29852
$557$	−6513.85	−0.495513	−0.247756	−	0.968822i	$-0.579693\pi$
$557$	−6513.85	−0.495513	−0.247756	+	0.968822i	$0.579693\pi$
$558$	0	0
$559$	12852.8	0.972478
$560$	−4046.95	−0.305383
$561$	0	0
$562$	−3853.93	−0.289267
$563$	26513.8	1.98476	0.992382	−	0.123198i	$-0.0393151\pi$
$563$	26513.8	1.98476	0.992382	+	0.123198i	$0.0393151\pi$
$564$	0	0
$565$	25873.9	1.92659
$566$	−6711.12	−0.498391
$567$	0	0
$568$	9823.65	0.725688
$569$	−14543.3	−1.07151	−0.535754	−	0.844374i	$-0.679973\pi$
$569$	−14543.3	−1.07151	−0.535754	+	0.844374i	$0.679973\pi$
$570$	0	0
$571$	3354.57	0.245857	0.122928	−	0.992416i	$-0.460771\pi$
$571$	3354.57	0.245857	0.122928	+	0.992416i	$0.460771\pi$
$572$	−685.433	−0.0501038
$573$	0	0
$574$	7449.31	0.541687
$575$	−4615.55	−0.334751
$576$	0	0
$577$	−14887.7	−1.07415	−0.537075	−	0.843535i	$-0.680471\pi$
$577$	−14887.7	−1.07415	−0.537075	+	0.843535i	$0.680471\pi$
$578$	4913.52	0.353591
$579$	0	0
$580$	16221.7	1.16133
$581$	−6051.83	−0.432138
$582$	0	0
$583$	−1766.79	−0.125511
$584$	7829.55	0.554776
$585$	0	0
$586$	10364.7	0.730653
$587$	22565.7	1.58669	0.793345	−	0.608772i	$-0.208338\pi$
$587$	22565.7	1.58669	0.793345	+	0.608772i	$0.208338\pi$
$588$	0	0
$589$	11967.7	0.837218
$590$	6092.32	0.425113
$591$	0	0
$592$	10576.0	0.734240
$593$	12359.6	0.855899	0.427949	−	0.903803i	$-0.359236\pi$
$593$	12359.6	0.855899	0.427949	+	0.903803i	$0.359236\pi$
$594$	0	0
$595$	4875.87	0.335952
$596$	−11499.1	−0.790306
$597$	0	0
$598$	5204.83	0.355922
$599$	18280.2	1.24692	0.623461	−	0.781854i	$-0.285726\pi$
$599$	18280.2	1.24692	0.623461	+	0.781854i	$0.285726\pi$
$600$	0	0
$601$	16554.7	1.12360	0.561799	−	0.827274i	$-0.310110\pi$
$601$	16554.7	1.12360	0.561799	+	0.827274i	$0.310110\pi$
$602$	−5864.80	−0.397062
$603$	0	0
$604$	−17757.3	−1.19625
$605$	16813.2	1.12984
$606$	0	0
$607$	−9063.55	−0.606059	−0.303030	−	0.952981i	$-0.597998\pi$
$607$	−9063.55	−0.606059	−0.303030	+	0.952981i	$0.597998\pi$
$608$	−15973.5	−1.06548
$609$	0	0
$610$	−5070.56	−0.336559
$611$	16279.2	1.07788
$612$	0	0
$613$	27605.4	1.81888	0.909439	−	0.415837i	$-0.136511\pi$
$613$	27605.4	1.81888	0.909439	+	0.415837i	$0.136511\pi$
$614$	8578.32	0.563832
$615$	0	0
$616$	706.361	0.0462014
$617$	−14668.8	−0.957120	−0.478560	−	0.878055i	$-0.658841\pi$
$617$	−14668.8	−0.957120	−0.478560	+	0.878055i	$0.658841\pi$
$618$	0	0
$619$	−2551.38	−0.165668	−0.0828341	−	0.996563i	$-0.526397\pi$
$619$	−2551.38	−0.165668	−0.0828341	+	0.996563i	$0.526397\pi$
$620$	11050.4	0.715799
$621$	0	0
$622$	−3760.71	−0.242429
$623$	409.346	0.0263244
$624$	0	0
$625$	−18891.3	−1.20904
$626$	−533.242	−0.0340457
$627$	0	0
$628$	−19306.4	−1.22677
$629$	−12742.2	−0.807736
$630$	0	0
$631$	8523.52	0.537743	0.268872	−	0.963176i	$-0.413349\pi$
$631$	8523.52	0.537743	0.268872	+	0.963176i	$0.413349\pi$
$632$	−16019.2	−1.00824
$633$	0	0
$634$	−5255.77	−0.329232
$635$	−10740.2	−0.671197
$636$	0	0
$637$	−6782.61	−0.421879
$638$	−845.977	−0.0524961
$639$	0	0
$640$	−18310.5	−1.13092
$641$	11465.1	0.706465	0.353232	−	0.935536i	$-0.385082\pi$
$641$	11465.1	0.706465	0.353232	+	0.935536i	$0.385082\pi$
$642$	0	0
$643$	20283.1	1.24399	0.621997	−	0.783019i	$-0.286321\pi$
$643$	20283.1	1.24399	0.621997	+	0.783019i	$0.286321\pi$
$644$	9190.19	0.562336
$645$	0	0
$646$	3692.66	0.224900
$647$	8928.01	0.542498	0.271249	−	0.962509i	$-0.412563\pi$
$647$	8928.01	0.542498	0.271249	+	0.962509i	$0.412563\pi$
$648$	0	0
$649$	1229.44	0.0743599
$650$	−1560.70	−0.0941780
$651$	0	0
$652$	−20675.9	−1.24192
$653$	20197.4	1.21039	0.605196	−	0.796077i	$-0.293095\pi$
$653$	20197.4	1.21039	0.605196	+	0.796077i	$0.293095\pi$
$654$	0	0
$655$	1727.34	0.103042
$656$	−13601.9	−0.809551
$657$	0	0
$658$	−7428.29	−0.440099
$659$	27932.1	1.65111	0.825554	−	0.564323i	$-0.190863\pi$
$659$	27932.1	1.65111	0.825554	+	0.564323i	$0.190863\pi$
$660$	0	0
$661$	−11314.0	−0.665757	−0.332878	−	0.942970i	$-0.608020\pi$
$661$	−11314.0	−0.665757	−0.332878	+	0.942970i	$0.608020\pi$
$662$	6363.86	0.373623
$663$	0	0
$664$	−9557.56	−0.558592
$665$	−13012.4	−0.758798
$666$	0	0
$667$	−24857.8	−1.44302
$668$	−8245.15	−0.477567
$669$	0	0
$670$	−16262.1	−0.937702
$671$	−1023.24	−0.0588702
$672$	0	0
$673$	3137.30	0.179694	0.0898470	−	0.995956i	$-0.471362\pi$
$673$	3137.30	0.179694	0.0898470	+	0.995956i	$0.471362\pi$
$674$	5266.45	0.300973
$675$	0	0
$676$	7156.36	0.407167
$677$	−16691.7	−0.947582	−0.473791	−	0.880637i	$-0.657115\pi$
$677$	−16691.7	−0.947582	−0.473791	+	0.880637i	$0.657115\pi$
$678$	0	0
$679$	−8220.29	−0.464603
$680$	7700.39	0.434259
$681$	0	0
$682$	−576.288	−0.0323566
$683$	25651.5	1.43708	0.718542	−	0.695484i	$-0.244810\pi$
$683$	25651.5	1.43708	0.718542	+	0.695484i	$0.244810\pi$
$684$	0	0
$685$	37143.6	2.07180
$686$	8217.67	0.457365
$687$	0	0
$688$	10708.7	0.593410
$689$	−17554.4	−0.970638
$690$	0	0
$691$	4481.89	0.246743	0.123371	−	0.992361i	$-0.460629\pi$
$691$	4481.89	0.246743	0.123371	+	0.992361i	$0.460629\pi$
$692$	−934.725	−0.0513481
$693$	0	0
$694$	8118.68	0.444065
$695$	34105.7	1.86144
$696$	0	0
$697$	16388.0	0.890586
$698$	10109.8	0.548223
$699$	0	0
$700$	−2755.73	−0.148796
$701$	21668.2	1.16747	0.583734	−	0.811945i	$-0.301591\pi$
$701$	21668.2	1.16747	0.583734	+	0.811945i	$0.301591\pi$
$702$	0	0
$703$	34005.7	1.82439
$704$	50.5069	0.00270390
$705$	0	0
$706$	3148.57	0.167844
$707$	−2765.35	−0.147103
$708$	0	0
$709$	−27233.1	−1.44254	−0.721269	−	0.692655i	$-0.756441\pi$
$709$	−27233.1	−1.44254	−0.721269	+	0.692655i	$0.756441\pi$
$710$	8714.31	0.460623
$711$	0	0
$712$	646.474	0.0340276
$713$	−16933.4	−0.889425
$714$	0	0
$715$	−1373.19	−0.0718244
$716$	−16090.4	−0.839840
$717$	0	0
$718$	5154.15	0.267898
$719$	15035.8	0.779888	0.389944	−	0.920839i	$-0.372494\pi$
$719$	15035.8	0.779888	0.389944	+	0.920839i	$0.372494\pi$
$720$	0	0
$721$	−14187.8	−0.732846
$722$	−1063.29	−0.0548085
$723$	0	0
$724$	−4554.15	−0.233776
$725$	7453.76	0.381828
$726$	0	0
$727$	−9861.28	−0.503074	−0.251537	−	0.967848i	$-0.580936\pi$
$727$	−9861.28	−0.503074	−0.251537	+	0.967848i	$0.580936\pi$
$728$	7018.22	0.357297
$729$	0	0
$730$	6945.39	0.352138
$731$	−12902.2	−0.652810
$732$	0	0
$733$	−24706.5	−1.24496	−0.622481	−	0.782635i	$-0.713875\pi$
$733$	−24706.5	−1.24496	−0.622481	+	0.782635i	$0.713875\pi$
$734$	−1296.71	−0.0652078
$735$	0	0
$736$	22601.3	1.13192
$737$	−3281.71	−0.164021
$738$	0	0
$739$	−5280.19	−0.262835	−0.131418	−	0.991327i	$-0.541953\pi$
$739$	−5280.19	−0.262835	−0.131418	+	0.991327i	$0.541953\pi$
$740$	31399.2	1.55981
$741$	0	0
$742$	8010.18	0.396311
$743$	28765.9	1.42035	0.710174	−	0.704026i	$-0.248616\pi$
$743$	28765.9	1.42035	0.710174	+	0.704026i	$0.248616\pi$
$744$	0	0
$745$	−23037.3	−1.13291
$746$	10518.2	0.516220
$747$	0	0
$748$	688.065	0.0336339
$749$	−883.463	−0.0430988
$750$	0	0
$751$	36656.6	1.78111	0.890557	−	0.454871i	$-0.150315\pi$
$751$	36656.6	1.78111	0.890557	+	0.454871i	$0.150315\pi$
$752$	13563.5	0.657728
$753$	0	0
$754$	−8405.40	−0.405977
$755$	−35574.9	−1.71484
$756$	0	0
$757$	−133.645	−0.00641665	−0.00320833	−	0.999995i	$-0.501021\pi$
$757$	−133.645	−0.00641665	−0.00320833	+	0.999995i	$0.501021\pi$
$758$	−3439.54	−0.164815
$759$	0	0
$760$	−20550.3	−0.980841
$761$	−6803.10	−0.324064	−0.162032	−	0.986786i	$-0.551805\pi$
$761$	−6803.10	−0.324064	−0.162032	+	0.986786i	$0.551805\pi$
$762$	0	0
$763$	18001.2	0.854112
$764$	−27391.4	−1.29710
$765$	0	0
$766$	−3345.19	−0.157789
$767$	12215.4	0.575060
$768$	0	0
$769$	−26354.9	−1.23587	−0.617935	−	0.786229i	$-0.712030\pi$
$769$	−26354.9	−1.23587	−0.617935	+	0.786229i	$0.712030\pi$
$770$	626.595	0.0293259
$771$	0	0
$772$	−4852.44	−0.226222
$773$	24334.7	1.13229	0.566143	−	0.824307i	$-0.308435\pi$
$773$	24334.7	1.13229	0.566143	+	0.824307i	$0.308435\pi$
$774$	0	0
$775$	5077.58	0.235345
$776$	−12982.2	−0.600558
$777$	0	0
$778$	−6060.95	−0.279300
$779$	−43735.2	−2.01152
$780$	0	0
$781$	1758.56	0.0805711
$782$	−5224.82	−0.238925
$783$	0	0
$784$	−5651.15	−0.257432
$785$	−38678.3	−1.75858
$786$	0	0
$787$	2038.24	0.0923194	0.0461597	−	0.998934i	$-0.485302\pi$
$787$	2038.24	0.0923194	0.0461597	+	0.998934i	$0.485302\pi$
$788$	10633.9	0.480733
$789$	0	0
$790$	−14210.2	−0.639971
$791$	−23672.4	−1.06409
$792$	0	0
$793$	−10166.7	−0.455271
$794$	−13212.6	−0.590550
$795$	0	0
$796$	−18777.1	−0.836100
$797$	−16414.1	−0.729509	−0.364755	−	0.931104i	$-0.618847\pi$
$797$	−16414.1	−0.729509	−0.364755	+	0.931104i	$0.618847\pi$
$798$	0	0
$799$	−16341.7	−0.723566
$800$	−6777.14	−0.299510
$801$	0	0
$802$	1437.84	0.0633066
$803$	1401.59	0.0615952
$804$	0	0
$805$	18411.6	0.806115
$806$	−5725.85	−0.250229
$807$	0	0
$808$	−4367.27	−0.190148
$809$	−42254.2	−1.83632	−0.918158	−	0.396215i	$-0.870323\pi$
$809$	−42254.2	−1.83632	−0.918158	+	0.396215i	$0.870323\pi$
$810$	0	0
$811$	9168.05	0.396959	0.198479	−	0.980105i	$-0.436400\pi$
$811$	9168.05	0.396959	0.198479	+	0.980105i	$0.436400\pi$
$812$	−14841.5	−0.641420
$813$	0	0
$814$	−1637.49	−0.0705088
$815$	−41422.0	−1.78031
$816$	0	0
$817$	34432.5	1.47447
$818$	5194.86	0.222047
$819$	0	0
$820$	−40383.0	−1.71980
$821$	−2631.57	−0.111867	−0.0559333	−	0.998435i	$-0.517813\pi$
$821$	−2631.57	−0.111867	−0.0559333	+	0.998435i	$0.517813\pi$
$822$	0	0
$823$	−27765.9	−1.17601	−0.588006	−	0.808856i	$-0.700087\pi$
$823$	−27765.9	−1.17601	−0.588006	+	0.808856i	$0.700087\pi$
$824$	−22406.6	−0.947295
$825$	0	0
$826$	−5573.93	−0.234797
$827$	−25470.4	−1.07097	−0.535486	−	0.844544i	$-0.679872\pi$
$827$	−25470.4	−1.07097	−0.535486	+	0.844544i	$0.679872\pi$
$828$	0	0
$829$	31737.9	1.32968	0.664840	−	0.746986i	$-0.268500\pi$
$829$	31737.9	1.32968	0.664840	+	0.746986i	$0.268500\pi$
$830$	−8478.27	−0.354560
$831$	0	0
$832$	501.823	0.0209106
$833$	6808.66	0.283201
$834$	0	0
$835$	−16518.3	−0.684598
$836$	−1836.27	−0.0759673
$837$	0	0
$838$	−5613.95	−0.231421
$839$	3300.01	0.135792	0.0678958	−	0.997692i	$-0.478371\pi$
$839$	3300.01	0.135792	0.0678958	+	0.997692i	$0.478371\pi$
$840$	0	0
$841$	15754.4	0.645964
$842$	−12935.6	−0.529444
$843$	0	0
$844$	−20632.6	−0.841473
$845$	14337.0	0.583679
$846$	0	0
$847$	−15382.6	−0.624029
$848$	−14626.0	−0.592287
$849$	0	0
$850$	1566.69	0.0632202
$851$	−48115.4	−1.93816
$852$	0	0
$853$	8444.72	0.338970	0.169485	−	0.985533i	$-0.445790\pi$
$853$	8444.72	0.338970	0.169485	+	0.985533i	$0.445790\pi$
$854$	4639.12	0.185887
$855$	0	0
$856$	−1395.24	−0.0557106
$857$	161.549	0.00643921	0.00321961	−	0.999995i	$-0.498975\pi$
$857$	161.549	0.00643921	0.00321961	+	0.999995i	$0.498975\pi$
$858$	0	0
$859$	−13536.6	−0.537677	−0.268838	−	0.963185i	$-0.586640\pi$
$859$	−13536.6	−0.537677	−0.268838	+	0.963185i	$0.586640\pi$
$860$	31793.4	1.26063
$861$	0	0
$862$	4218.02	0.166666
$863$	−1113.95	−0.0439388	−0.0219694	−	0.999759i	$-0.506994\pi$
$863$	−1113.95	−0.0439388	−0.0219694	+	0.999759i	$0.506994\pi$
$864$	0	0
$865$	−1872.62	−0.0736082
$866$	−3968.28	−0.155713
$867$	0	0
$868$	−10110.2	−0.395347
$869$	−2867.64	−0.111942
$870$	0	0
$871$	−32606.2	−1.26845
$872$	28429.0	1.10405
$873$	0	0
$874$	13943.7	0.539648
$875$	13029.3	0.503393
$876$	0	0
$877$	17292.9	0.665839	0.332920	−	0.942955i	$-0.391966\pi$
$877$	17292.9	0.665839	0.332920	+	0.942955i	$0.391966\pi$
$878$	−20230.2	−0.777603
$879$	0	0
$880$	−1144.12	−0.0438275
$881$	−50831.3	−1.94387	−0.971936	−	0.235245i	$-0.924411\pi$
$881$	−50831.3	−1.94387	−0.971936	+	0.235245i	$0.924411\pi$
$882$	0	0
$883$	−8641.30	−0.329335	−0.164668	−	0.986349i	$-0.552655\pi$
$883$	−8641.30	−0.329335	−0.164668	+	0.986349i	$0.552655\pi$
$884$	6836.44	0.260107
$885$	0	0
$886$	−19855.4	−0.752883
$887$	−28352.5	−1.07326	−0.536631	−	0.843817i	$-0.680303\pi$
$887$	−28352.5	−1.07326	−0.536631	+	0.843817i	$0.680303\pi$
$888$	0	0
$889$	9826.30	0.370713
$890$	573.471	0.0215986
$891$	0	0
$892$	24962.5	0.937003
$893$	43611.8	1.63428
$894$	0	0
$895$	−32235.3	−1.20392
$896$	16752.5	0.624622
$897$	0	0
$898$	11292.6	0.419643
$899$	27346.1	1.01451
$900$	0	0
$901$	17621.8	0.651574
$902$	2106.01	0.0777410
$903$	0	0
$904$	−37385.4	−1.37547
$905$	−9123.75	−0.335120
$906$	0	0
$907$	24595.6	0.900425	0.450212	−	0.892922i	$-0.351348\pi$
$907$	24595.6	0.900425	0.450212	+	0.892922i	$0.351348\pi$
$908$	−24724.6	−0.903650
$909$	0	0
$910$	6225.68	0.226791
$911$	11289.5	0.410579	0.205290	−	0.978701i	$-0.434186\pi$
$911$	11289.5	0.410579	0.205290	+	0.978701i	$0.434186\pi$
$912$	0	0
$913$	−1710.92	−0.0620189
$914$	−5217.61	−0.188822
$915$	0	0
$916$	−31494.6	−1.13604
$917$	−1580.36	−0.0569119
$918$	0	0
$919$	36395.2	1.30638	0.653192	−	0.757193i	$-0.273430\pi$
$919$	36395.2	1.30638	0.653192	+	0.757193i	$0.273430\pi$
$920$	29077.1	1.04200
$921$	0	0
$922$	18818.2	0.672174
$923$	17472.5	0.623094
$924$	0	0
$925$	14427.7	0.512843
$926$	−3573.02	−0.126800
$927$	0	0
$928$	−36499.4	−1.29111
$929$	12241.9	0.432341	0.216171	−	0.976356i	$-0.430643\pi$
$929$	12241.9	0.432341	0.216171	+	0.976356i	$0.430643\pi$
$930$	0	0
$931$	−18170.6	−0.639652
$932$	−22155.2	−0.778666
$933$	0	0
$934$	−1289.82	−0.0451866
$935$	1378.47	0.0482146
$936$	0	0
$937$	−6795.65	−0.236931	−0.118465	−	0.992958i	$-0.537797\pi$
$937$	−6795.65	−0.236931	−0.118465	+	0.992958i	$0.537797\pi$
$938$	14878.4	0.517907
$939$	0	0
$940$	40269.1	1.39727
$941$	39544.4	1.36994	0.684969	−	0.728573i	$-0.259816\pi$
$941$	39544.4	1.36994	0.684969	+	0.728573i	$0.259816\pi$
$942$	0	0
$943$	61881.9	2.13696
$944$	10177.6	0.350904
$945$	0	0
$946$	−1658.05	−0.0569850
$947$	36454.9	1.25092	0.625462	−	0.780255i	$-0.284911\pi$
$947$	36454.9	1.25092	0.625462	+	0.780255i	$0.284911\pi$
$948$	0	0
$949$	13925.8	0.476344
$950$	−4181.10	−0.142792
$951$	0	0
$952$	−7045.18	−0.239848
$953$	−13131.5	−0.446350	−0.223175	−	0.974778i	$-0.571642\pi$
$953$	−13131.5	−0.446350	−0.223175	+	0.974778i	$0.571642\pi$
$954$	0	0
$955$	−54875.8	−1.85941
$956$	−1519.36	−0.0514012
$957$	0	0
$958$	21854.5	0.737042
$959$	−33983.1	−1.14429
$960$	0	0
$961$	−11162.5	−0.374695
$962$	−16269.7	−0.545277
$963$	0	0
$964$	10258.4	0.342740
$965$	−9721.36	−0.324292
$966$	0	0
$967$	−42537.3	−1.41459	−0.707294	−	0.706920i	$-0.750084\pi$
$967$	−42537.3	−1.41459	−0.707294	+	0.706920i	$0.750084\pi$
$968$	−24293.5	−0.806636
$969$	0	0
$970$	−11516.2	−0.381198
$971$	−1119.68	−0.0370054	−0.0185027	−	0.999829i	$-0.505890\pi$
$971$	−1119.68	−0.0370054	−0.0185027	+	0.999829i	$0.505890\pi$
$972$	0	0
$973$	−31203.7	−1.02810
$974$	2142.88	0.0704950
$975$	0	0
$976$	−8470.70	−0.277808
$977$	48678.5	1.59403	0.797014	−	0.603961i	$-0.206412\pi$
$977$	48678.5	1.59403	0.797014	+	0.603961i	$0.206412\pi$
$978$	0	0
$979$	115.727	0.00377799
$980$	−16777.8	−0.546886
$981$	0	0
$982$	−7370.66	−0.239519
$983$	−11480.4	−0.372499	−0.186250	−	0.982502i	$-0.559633\pi$
$983$	−11480.4	−0.372499	−0.186250	+	0.982502i	$0.559633\pi$
$984$	0	0
$985$	21303.9	0.689136
$986$	8437.69	0.272526
$987$	0	0
$988$	−18244.7	−0.587490
$989$	−48719.3	−1.56641
$990$	0	0
$991$	−8389.19	−0.268912	−0.134456	−	0.990920i	$-0.542929\pi$
$991$	−8389.19	−0.268912	−0.134456	+	0.990920i	$0.542929\pi$
$992$	−24863.8	−0.795791
$993$	0	0
$994$	−7972.82	−0.254409
$995$	−37617.9	−1.19856
$996$	0	0
$997$	41853.0	1.32949	0.664743	−	0.747072i	$-0.268541\pi$
$997$	41853.0	1.32949	0.664743	+	0.747072i	$0.268541\pi$
$998$	−9963.58	−0.316024
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.f.1.15		37
3.2	odd	2		239.4.a.b.1.23	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.b.1.23	✓	37	3.2	odd	2
2151.4.a.f.1.15		37	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-1.28174 q^{2} -6.35714 q^{4} -12.7359 q^{5} +11.6522 q^{7} +18.4021 q^{8} +O(q^{10})\)	\(q-1.28174 q^{2} -6.35714 q^{4} -12.7359 q^{5} +11.6522 q^{7} +18.4021 q^{8} +16.3241 q^{10} +3.29421 q^{11} +32.7304 q^{13} -14.9351 q^{14} +27.2704 q^{16} -32.8561 q^{17} +87.6845 q^{19} +80.9637 q^{20} -4.22232 q^{22} -124.067 q^{23} +37.2021 q^{25} -41.9519 q^{26} -74.0746 q^{28} +200.358 q^{29} +136.486 q^{31} -182.171 q^{32} +42.1130 q^{34} -148.401 q^{35} +387.819 q^{37} -112.389 q^{38} -234.367 q^{40} -498.779 q^{41} +392.687 q^{43} -20.9418 q^{44} +159.021 q^{46} +497.372 q^{47} -207.227 q^{49} -47.6835 q^{50} -208.072 q^{52} -536.333 q^{53} -41.9546 q^{55} +214.425 q^{56} -256.807 q^{58} +373.211 q^{59} -310.619 q^{61} -174.940 q^{62} +15.3320 q^{64} -416.850 q^{65} -996.205 q^{67} +208.871 q^{68} +190.211 q^{70} +533.832 q^{71} +425.470 q^{73} -497.083 q^{74} -557.423 q^{76} +38.3847 q^{77} -870.508 q^{79} -347.312 q^{80} +639.306 q^{82} -519.373 q^{83} +418.451 q^{85} -503.322 q^{86} +60.6205 q^{88} +35.1304 q^{89} +381.381 q^{91} +788.709 q^{92} -637.502 q^{94} -1116.74 q^{95} -705.471 q^{97} +265.611 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Introduction

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