LMFDB - Embedded newform 2151.4.a.b.1.11

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:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-1.36717 q^{2} -6.13085 q^{4} +15.7404 q^{5} +15.1668 q^{7} +19.3193 q^{8} +O(q^{10})$	\(q-1.36717 q^{2} -6.13085 q^{4} +15.7404 q^{5} +15.1668 q^{7} +19.3193 q^{8} -21.5198 q^{10} -22.9103 q^{11} +9.20683 q^{13} -20.7356 q^{14} +22.6340 q^{16} -51.0041 q^{17} +63.7408 q^{19} -96.5019 q^{20} +31.3223 q^{22} -104.007 q^{23} +122.760 q^{25} -12.5873 q^{26} -92.9853 q^{28} +278.692 q^{29} -111.255 q^{31} -185.499 q^{32} +69.7313 q^{34} +238.731 q^{35} -383.053 q^{37} -87.1445 q^{38} +304.093 q^{40} -45.6786 q^{41} -90.6195 q^{43} +140.460 q^{44} +142.196 q^{46} -410.472 q^{47} -112.968 q^{49} -167.834 q^{50} -56.4457 q^{52} -432.912 q^{53} -360.617 q^{55} +293.011 q^{56} -381.020 q^{58} +310.948 q^{59} -862.024 q^{61} +152.105 q^{62} +72.5362 q^{64} +144.919 q^{65} +439.435 q^{67} +312.698 q^{68} -326.386 q^{70} -786.062 q^{71} +340.108 q^{73} +523.699 q^{74} -390.785 q^{76} -347.476 q^{77} -955.097 q^{79} +356.269 q^{80} +62.4504 q^{82} +404.956 q^{83} -802.825 q^{85} +123.892 q^{86} -442.611 q^{88} -9.49422 q^{89} +139.638 q^{91} +637.652 q^{92} +561.184 q^{94} +1003.31 q^{95} -1065.05 q^{97} +154.447 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * 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 $

$2$	−1.36717	−0.483368	−0.241684	−	0.970355i	$-0.577700\pi$
$2$	−1.36717	−0.483368	−0.241684	+	0.970355i	$0.577700\pi$
$3$	0	0
$3$	0	0
$4$	−6.13085	−0.766356
$5$	15.7404	1.40786	0.703932	−	0.710268i	$-0.251426\pi$
$5$	15.7404	1.40786	0.703932	+	0.710268i	$0.251426\pi$
$6$	0	0
$7$	15.1668	0.818930	0.409465	−	0.912326i	$-0.365715\pi$
$7$	15.1668	0.818930	0.409465	+	0.912326i	$0.365715\pi$
$8$	19.3193	0.853799
$9$	0	0
$10$	−21.5198	−0.680516
$11$	−22.9103	−0.627974	−0.313987	−	0.949427i	$-0.601665\pi$
$11$	−22.9103	−0.627974	−0.313987	+	0.949427i	$0.601665\pi$
$12$	0	0
$13$	9.20683	0.196424	0.0982122	−	0.995165i	$-0.468688\pi$
$13$	9.20683	0.196424	0.0982122	+	0.995165i	$0.468688\pi$
$14$	−20.7356	−0.395844
$15$	0	0
$16$	22.6340	0.353657
$17$	−51.0041	−0.727666	−0.363833	−	0.931464i	$-0.618532\pi$
$17$	−51.0041	−0.727666	−0.363833	+	0.931464i	$0.618532\pi$
$18$	0	0
$19$	63.7408	0.769639	0.384820	−	0.922992i	$-0.374264\pi$
$19$	63.7408	0.769639	0.384820	+	0.922992i	$0.374264\pi$
$20$	−96.5019	−1.07892
$21$	0	0
$22$	31.3223	0.303543
$23$	−104.007	−0.942913	−0.471456	−	0.881889i	$-0.656272\pi$
$23$	−104.007	−0.942913	−0.471456	+	0.881889i	$0.656272\pi$
$24$	0	0
$25$	122.760	0.982080
$26$	−12.5873	−0.0949452
$27$	0	0
$28$	−92.9853	−0.627592
$29$	278.692	1.78455	0.892273	−	0.451497i	$-0.149110\pi$
$29$	278.692	1.78455	0.892273	+	0.451497i	$0.149110\pi$
$30$	0	0
$31$	−111.255	−0.644581	−0.322290	−	0.946641i	$-0.604453\pi$
$31$	−111.255	−0.644581	−0.322290	+	0.946641i	$0.604453\pi$
$32$	−185.499	−1.02475
$33$	0	0
$34$	69.7313	0.351730
$35$	238.731	1.15294
$36$	0	0
$37$	−383.053	−1.70199	−0.850994	−	0.525175i	$-0.824000\pi$
$37$	−383.053	−1.70199	−0.850994	+	0.525175i	$0.824000\pi$
$38$	−87.1445	−0.372019
$39$	0	0
$40$	304.093	1.20203
$41$	−45.6786	−0.173995	−0.0869975	−	0.996209i	$-0.527727\pi$
$41$	−45.6786	−0.173995	−0.0869975	+	0.996209i	$0.527727\pi$
$42$	0	0
$43$	−90.6195	−0.321380	−0.160690	−	0.987005i	$-0.551372\pi$
$43$	−90.6195	−0.321380	−0.160690	+	0.987005i	$0.551372\pi$
$44$	140.460	0.481252
$45$	0	0
$46$	142.196	0.455774
$47$	−410.472	−1.27390	−0.636951	−	0.770904i	$-0.719805\pi$
$47$	−410.472	−1.27390	−0.636951	+	0.770904i	$0.719805\pi$
$48$	0	0
$49$	−112.968	−0.329353
$50$	−167.834	−0.474706
$51$	0	0
$52$	−56.4457	−0.150531
$53$	−432.912	−1.12198	−0.560991	−	0.827822i	$-0.689580\pi$
$53$	−432.912	−1.12198	−0.560991	+	0.827822i	$0.689580\pi$
$54$	0	0
$55$	−360.617	−0.884103
$56$	293.011	0.699202
$57$	0	0
$58$	−381.020	−0.862592
$59$	310.948	0.686134	0.343067	−	0.939311i	$-0.388534\pi$
$59$	310.948	0.686134	0.343067	+	0.939311i	$0.388534\pi$
$60$	0	0
$61$	−862.024	−1.80936	−0.904679	−	0.426094i	$-0.859889\pi$
$61$	−862.024	−1.80936	−0.904679	+	0.426094i	$0.859889\pi$
$62$	152.105	0.311570
$63$	0	0
$64$	72.5362	0.141672
$65$	144.919	0.276539
$66$	0	0
$67$	439.435	0.801277	0.400639	−	0.916236i	$-0.368788\pi$
$67$	439.435	0.801277	0.400639	+	0.916236i	$0.368788\pi$
$68$	312.698	0.557651
$69$	0	0
$70$	−326.386	−0.557295
$71$	−786.062	−1.31392	−0.656961	−	0.753925i	$-0.728158\pi$
$71$	−786.062	−1.31392	−0.656961	+	0.753925i	$0.728158\pi$
$72$	0	0
$73$	340.108	0.545296	0.272648	−	0.962114i	$-0.412101\pi$
$73$	340.108	0.545296	0.272648	+	0.962114i	$0.412101\pi$
$74$	523.699	0.822686
$75$	0	0
$76$	−390.785	−0.589817
$77$	−347.476	−0.514267
$78$	0	0
$79$	−955.097	−1.36021	−0.680106	−	0.733114i	$-0.738066\pi$
$79$	−955.097	−1.36021	−0.680106	+	0.733114i	$0.738066\pi$
$80$	356.269	0.497900
$81$	0	0
$82$	62.4504	0.0841036
$83$	404.956	0.535538	0.267769	−	0.963483i	$-0.413714\pi$
$83$	404.956	0.535538	0.267769	+	0.963483i	$0.413714\pi$
$84$	0	0
$85$	−802.825	−1.02445
$86$	123.892	0.155345
$87$	0	0
$88$	−442.611	−0.536164
$89$	−9.49422	−0.0113077	−0.00565385	−	0.999984i	$-0.501800\pi$
$89$	−9.49422	−0.0113077	−0.00565385	+	0.999984i	$0.501800\pi$
$90$	0	0
$91$	139.638	0.160858
$92$	637.652	0.722607
$93$	0	0
$94$	561.184	0.615763
$95$	1003.31	1.08355
$96$	0	0
$97$	−1065.05	−1.11484	−0.557421	−	0.830230i	$-0.688209\pi$
$97$	−1065.05	−1.11484	−0.557421	+	0.830230i	$0.688209\pi$
$98$	154.447	0.159199
$99$	0	0
$100$	−752.623	−0.752623
$101$	944.495	0.930502	0.465251	−	0.885179i	$-0.345964\pi$
$101$	944.495	0.930502	0.465251	+	0.885179i	$0.345964\pi$
$102$	0	0
$103$	967.919	0.925941	0.462970	−	0.886374i	$-0.346784\pi$
$103$	967.919	0.925941	0.462970	+	0.886374i	$0.346784\pi$
$104$	177.869	0.167707
$105$	0	0
$106$	591.864	0.542330
$107$	−578.075	−0.522286	−0.261143	−	0.965300i	$-0.584099\pi$
$107$	−578.075	−0.522286	−0.261143	+	0.965300i	$0.584099\pi$
$108$	0	0
$109$	358.550	0.315073	0.157536	−	0.987513i	$-0.449645\pi$
$109$	358.550	0.315073	0.157536	+	0.987513i	$0.449645\pi$
$110$	493.025	0.427347
$111$	0	0
$112$	343.286	0.289620
$113$	2133.60	1.77621	0.888107	−	0.459637i	$-0.152021\pi$
$113$	2133.60	1.77621	0.888107	+	0.459637i	$0.152021\pi$
$114$	0	0
$115$	−1637.11	−1.32749
$116$	−1708.62	−1.36760
$117$	0	0
$118$	−425.118	−0.331655
$119$	−773.569	−0.595907
$120$	0	0
$121$	−806.118	−0.605648
$122$	1178.53	0.874585
$123$	0	0
$124$	682.088	0.493978
$125$	−35.2579	−0.0252285
$126$	0	0
$127$	−57.8164	−0.0403966	−0.0201983	−	0.999796i	$-0.506430\pi$
$127$	−57.8164	−0.0403966	−0.0201983	+	0.999796i	$0.506430\pi$
$128$	1384.82	0.956266
$129$	0	0
$130$	−198.129	−0.133670
$131$	−108.174	−0.0721465	−0.0360733	−	0.999349i	$-0.511485\pi$
$131$	−108.174	−0.0721465	−0.0360733	+	0.999349i	$0.511485\pi$
$132$	0	0
$133$	966.744	0.630281
$134$	−600.783	−0.387312
$135$	0	0
$136$	−985.362	−0.621280
$137$	−97.2335	−0.0606366	−0.0303183	−	0.999540i	$-0.509652\pi$
$137$	−97.2335	−0.0606366	−0.0303183	+	0.999540i	$0.509652\pi$
$138$	0	0
$139$	−1501.47	−0.916209	−0.458105	−	0.888898i	$-0.651471\pi$
$139$	−1501.47	−0.916209	−0.458105	+	0.888898i	$0.651471\pi$
$140$	−1463.63	−0.883564
$141$	0	0
$142$	1074.68	0.635107
$143$	−210.931	−0.123349
$144$	0	0
$145$	4386.72	2.51240
$146$	−464.985	−0.263578
$147$	0	0
$148$	2348.44	1.30433
$149$	981.935	0.539888	0.269944	−	0.962876i	$-0.412995\pi$
$149$	981.935	0.539888	0.269944	+	0.962876i	$0.412995\pi$
$150$	0	0
$151$	2953.29	1.59162	0.795811	−	0.605545i	$-0.207045\pi$
$151$	2953.29	1.59162	0.795811	+	0.605545i	$0.207045\pi$
$152$	1231.43	0.657117
$153$	0	0
$154$	475.059	0.248580
$155$	−1751.20	−0.907482
$156$	0	0
$157$	74.8032	0.0380252	0.0190126	−	0.999819i	$-0.493948\pi$
$157$	74.8032	0.0380252	0.0190126	+	0.999819i	$0.493948\pi$
$158$	1305.78	0.657483
$159$	0	0
$160$	−2919.82	−1.44270
$161$	−1577.46	−0.772180
$162$	0	0
$163$	80.1812	0.0385293	0.0192647	−	0.999814i	$-0.493867\pi$
$163$	80.1812	0.0385293	0.0192647	+	0.999814i	$0.493867\pi$
$164$	280.048	0.133342
$165$	0	0
$166$	−553.643	−0.258862
$167$	1458.28	0.675719	0.337860	−	0.941197i	$-0.390297\pi$
$167$	1458.28	0.675719	0.337860	+	0.941197i	$0.390297\pi$
$168$	0	0
$169$	−2112.23	−0.961417
$170$	1097.60	0.495188
$171$	0	0
$172$	555.574	0.246292
$173$	3592.87	1.57896	0.789482	−	0.613774i	$-0.210349\pi$
$173$	3592.87	1.57896	0.789482	+	0.613774i	$0.210349\pi$
$174$	0	0
$175$	1861.88	0.804255
$176$	−518.553	−0.222087
$177$	0	0
$178$	12.9802	0.00546578
$179$	721.833	0.301410	0.150705	−	0.988579i	$-0.451846\pi$
$179$	721.833	0.301410	0.150705	+	0.988579i	$0.451846\pi$
$180$	0	0
$181$	−1373.07	−0.563865	−0.281932	−	0.959434i	$-0.590975\pi$
$181$	−1373.07	−0.563865	−0.281932	+	0.959434i	$0.590975\pi$
$182$	−190.909	−0.0777535
$183$	0	0
$184$	−2009.34	−0.805058
$185$	−6029.41	−2.39617
$186$	0	0
$187$	1168.52	0.456955
$188$	2516.54	0.976262
$189$	0	0
$190$	−1371.69	−0.523752
$191$	−956.493	−0.362353	−0.181176	−	0.983451i	$-0.557991\pi$
$191$	−956.493	−0.362353	−0.181176	+	0.983451i	$0.557991\pi$
$192$	0	0
$193$	−3027.65	−1.12920	−0.564599	−	0.825365i	$-0.690969\pi$
$193$	−3027.65	−1.12920	−0.564599	+	0.825365i	$0.690969\pi$
$194$	1456.11	0.538879
$195$	0	0
$196$	692.591	0.252402
$197$	−2574.19	−0.930984	−0.465492	−	0.885052i	$-0.654123\pi$
$197$	−2574.19	−0.930984	−0.465492	+	0.885052i	$0.654123\pi$
$198$	0	0
$199$	−490.997	−0.174904	−0.0874519	−	0.996169i	$-0.527872\pi$
$199$	−490.997	−0.174904	−0.0874519	+	0.996169i	$0.527872\pi$
$200$	2371.63	0.838499
$201$	0	0
$202$	−1291.28	−0.449775
$203$	4226.87	1.46142
$204$	0	0
$205$	−718.999	−0.244961
$206$	−1323.31	−0.447570
$207$	0	0
$208$	208.388	0.0694668
$209$	−1460.32	−0.483314
$210$	0	0
$211$	−1017.82	−0.332083	−0.166041	−	0.986119i	$-0.553098\pi$
$211$	−1017.82	−0.332083	−0.166041	+	0.986119i	$0.553098\pi$
$212$	2654.12	0.859837
$213$	0	0
$214$	790.327	0.252456
$215$	−1426.39	−0.452460
$216$	0	0
$217$	−1687.38	−0.527867
$218$	−490.200	−0.152296
$219$	0	0
$220$	2210.89	0.677537
$221$	−469.586	−0.142931
$222$	0	0
$223$	3290.24	0.988030	0.494015	−	0.869453i	$-0.335529\pi$
$223$	3290.24	0.988030	0.494015	+	0.869453i	$0.335529\pi$
$224$	−2813.42	−0.839195
$225$	0	0
$226$	−2916.99	−0.858564
$227$	−2533.46	−0.740756	−0.370378	−	0.928881i	$-0.620772\pi$
$227$	−2533.46	−0.740756	−0.370378	+	0.928881i	$0.620772\pi$
$228$	0	0
$229$	−2837.92	−0.818929	−0.409465	−	0.912326i	$-0.634284\pi$
$229$	−2837.92	−0.818929	−0.409465	+	0.912326i	$0.634284\pi$
$230$	2238.21	0.641667
$231$	0	0
$232$	5384.13	1.52364
$233$	5493.77	1.54467	0.772336	−	0.635214i	$-0.219088\pi$
$233$	5493.77	1.54467	0.772336	+	0.635214i	$0.219088\pi$
$234$	0	0
$235$	−6460.98	−1.79348
$236$	−1906.37	−0.525823
$237$	0	0
$238$	1057.60	0.288042
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−6457.93	−1.72611	−0.863053	−	0.505113i	$-0.831451\pi$
$241$	−6457.93	−1.72611	−0.863053	+	0.505113i	$0.831451\pi$
$242$	1102.10	0.292751
$243$	0	0
$244$	5284.93	1.38661
$245$	−1778.16	−0.463685
$246$	0	0
$247$	586.851	0.151176
$248$	−2149.37	−0.550343
$249$	0	0
$250$	48.2036	0.0121947
$251$	4354.12	1.09494	0.547469	−	0.836826i	$-0.315591\pi$
$251$	4354.12	1.09494	0.547469	+	0.836826i	$0.315591\pi$
$252$	0	0
$253$	2382.84	0.592125
$254$	79.0448	0.0195264
$255$	0	0
$256$	−2473.58	−0.603900
$257$	−2820.32	−0.684540	−0.342270	−	0.939602i	$-0.611196\pi$
$257$	−2820.32	−0.684540	−0.342270	+	0.939602i	$0.611196\pi$
$258$	0	0
$259$	−5809.69	−1.39381
$260$	−888.477	−0.211927
$261$	0	0
$262$	147.892	0.0348733
$263$	−4485.78	−1.05173	−0.525865	−	0.850568i	$-0.676258\pi$
$263$	−4485.78	−1.05173	−0.525865	+	0.850568i	$0.676258\pi$
$264$	0	0
$265$	−6814.20	−1.57960
$266$	−1321.70	−0.304657
$267$	0	0
$268$	−2694.11	−0.614064
$269$	2836.72	0.642967	0.321483	−	0.946915i	$-0.395819\pi$
$269$	2836.72	0.642967	0.321483	+	0.946915i	$0.395819\pi$
$270$	0	0
$271$	4185.39	0.938171	0.469086	−	0.883153i	$-0.344584\pi$
$271$	4185.39	0.938171	0.469086	+	0.883153i	$0.344584\pi$
$272$	−1154.43	−0.257344
$273$	0	0
$274$	132.935	0.0293098
$275$	−2812.47	−0.616721
$276$	0	0
$277$	−5049.80	−1.09535	−0.547677	−	0.836690i	$-0.684488\pi$
$277$	−5049.80	−1.09535	−0.547677	+	0.836690i	$0.684488\pi$
$278$	2052.77	0.442866
$279$	0	0
$280$	4612.12	0.984381
$281$	−4860.83	−1.03193	−0.515966	−	0.856609i	$-0.672567\pi$
$281$	−4860.83	−1.03193	−0.515966	+	0.856609i	$0.672567\pi$
$282$	0	0
$283$	−8632.99	−1.81335	−0.906674	−	0.421831i	$-0.861388\pi$
$283$	−8632.99	−1.81335	−0.906674	+	0.421831i	$0.861388\pi$
$284$	4819.23	1.00693
$285$	0	0
$286$	288.379	0.0596232
$287$	−692.798	−0.142490
$288$	0	0
$289$	−2311.58	−0.470503
$290$	−5997.40	−1.21441
$291$	0	0
$292$	−2085.15	−0.417890
$293$	−4570.88	−0.911377	−0.455689	−	0.890139i	$-0.650607\pi$
$293$	−4570.88	−0.911377	−0.455689	+	0.890139i	$0.650607\pi$
$294$	0	0
$295$	4894.44	0.965984
$296$	−7400.31	−1.45316
$297$	0	0
$298$	−1342.47	−0.260964
$299$	−957.577	−0.185211
$300$	0	0
$301$	−1374.41	−0.263188
$302$	−4037.64	−0.769339
$303$	0	0
$304$	1442.71	0.272188
$305$	−13568.6	−2.54733
$306$	0	0
$307$	315.960	0.0587388	0.0293694	−	0.999569i	$-0.490650\pi$
$307$	315.960	0.0587388	0.0293694	+	0.999569i	$0.490650\pi$
$308$	2130.32	0.394112
$309$	0	0
$310$	2394.19	0.438647
$311$	−1884.17	−0.343543	−0.171771	−	0.985137i	$-0.554949\pi$
$311$	−1884.17	−0.343543	−0.171771	+	0.985137i	$0.554949\pi$
$312$	0	0
$313$	651.041	0.117569	0.0587844	−	0.998271i	$-0.481278\pi$
$313$	651.041	0.117569	0.0587844	+	0.998271i	$0.481278\pi$
$314$	−102.269	−0.0183801
$315$	0	0
$316$	5855.55	1.04241
$317$	−5795.48	−1.02683	−0.513417	−	0.858139i	$-0.671621\pi$
$317$	−5795.48	−1.02683	−0.513417	+	0.858139i	$0.671621\pi$
$318$	0	0
$319$	−6384.92	−1.12065
$320$	1141.75	0.199455
$321$	0	0
$322$	2156.65	0.373247
$323$	−3251.04	−0.560040
$324$	0	0
$325$	1130.23	0.192904
$326$	−109.621	−0.0186238
$327$	0	0
$328$	−882.477	−0.148557
$329$	−6225.54	−1.04324
$330$	0	0
$331$	2352.23	0.390605	0.195303	−	0.980743i	$-0.437431\pi$
$331$	2352.23	0.390605	0.195303	+	0.980743i	$0.437431\pi$
$332$	−2482.72	−0.410413
$333$	0	0
$334$	−1993.72	−0.326621
$335$	6916.89	1.12809
$336$	0	0
$337$	−6137.57	−0.992091	−0.496046	−	0.868296i	$-0.665215\pi$
$337$	−6137.57	−0.992091	−0.496046	+	0.868296i	$0.665215\pi$
$338$	2887.78	0.464718
$339$	0	0
$340$	4922.00	0.785096
$341$	2548.89	0.404780
$342$	0	0
$343$	−6915.58	−1.08865
$344$	−1750.70	−0.274394
$345$	0	0
$346$	−4912.06	−0.763220
$347$	6208.28	0.960455	0.480227	−	0.877144i	$-0.340554\pi$
$347$	6208.28	0.960455	0.480227	+	0.877144i	$0.340554\pi$
$348$	0	0
$349$	1586.55	0.243341	0.121670	−	0.992571i	$-0.461175\pi$
$349$	1586.55	0.243341	0.121670	+	0.992571i	$0.461175\pi$
$350$	−2545.50	−0.388751
$351$	0	0
$352$	4249.83	0.643514
$353$	9776.22	1.47404	0.737019	−	0.675872i	$-0.236233\pi$
$353$	9776.22	1.47404	0.737019	+	0.675872i	$0.236233\pi$
$354$	0	0
$355$	−12372.9	−1.84982
$356$	58.2076	0.00866572
$357$	0	0
$358$	−986.869	−0.145692
$359$	−8789.21	−1.29214	−0.646068	−	0.763280i	$-0.723588\pi$
$359$	−8789.21	−1.29214	−0.646068	+	0.763280i	$0.723588\pi$
$360$	0	0
$361$	−2796.11	−0.407655
$362$	1877.22	0.272554
$363$	0	0
$364$	−856.100	−0.123274
$365$	5353.43	0.767702
$366$	0	0
$367$	8535.76	1.21407	0.607034	−	0.794676i	$-0.292359\pi$
$367$	8535.76	1.21407	0.607034	+	0.794676i	$0.292359\pi$
$368$	−2354.10	−0.333467
$369$	0	0
$370$	8243.23	1.15823
$371$	−6565.89	−0.918824
$372$	0	0
$373$	−7623.20	−1.05822	−0.529108	−	0.848555i	$-0.677473\pi$
$373$	−7623.20	−1.05822	−0.529108	+	0.848555i	$0.677473\pi$
$374$	−1597.57	−0.220878
$375$	0	0
$376$	−7930.01	−1.08766
$377$	2565.87	0.350528
$378$	0	0
$379$	−4548.37	−0.616449	−0.308224	−	0.951314i	$-0.599735\pi$
$379$	−4548.37	−0.616449	−0.308224	+	0.951314i	$0.599735\pi$
$380$	−6151.11	−0.830383
$381$	0	0
$382$	1307.69	0.175150
$383$	3888.30	0.518755	0.259377	−	0.965776i	$-0.416483\pi$
$383$	3888.30	0.518755	0.259377	+	0.965776i	$0.416483\pi$
$384$	0	0
$385$	−5469.41	−0.724018
$386$	4139.32	0.545818
$387$	0	0
$388$	6529.67	0.854365
$389$	14827.7	1.93263	0.966314	−	0.257365i	$-0.0828542\pi$
$389$	14827.7	1.93263	0.966314	+	0.257365i	$0.0828542\pi$
$390$	0	0
$391$	5304.79	0.686125
$392$	−2182.46	−0.281202
$393$	0	0
$394$	3519.36	0.450007
$395$	−15033.6	−1.91499
$396$	0	0
$397$	14099.5	1.78246	0.891229	−	0.453553i	$-0.149844\pi$
$397$	14099.5	1.78246	0.891229	+	0.453553i	$0.149844\pi$
$398$	671.277	0.0845428
$399$	0	0
$400$	2778.55	0.347319
$401$	−425.743	−0.0530189	−0.0265095	−	0.999649i	$-0.508439\pi$
$401$	−425.743	−0.0530189	−0.0265095	+	0.999649i	$0.508439\pi$
$402$	0	0
$403$	−1024.31	−0.126611
$404$	−5790.55	−0.713096
$405$	0	0
$406$	−5778.85	−0.706402
$407$	8775.87	1.06881
$408$	0	0
$409$	12800.2	1.54750	0.773751	−	0.633490i	$-0.218378\pi$
$409$	12800.2	1.54750	0.773751	+	0.633490i	$0.218378\pi$
$410$	982.994	0.118406
$411$	0	0
$412$	−5934.16	−0.709600
$413$	4716.08	0.561896
$414$	0	0
$415$	6374.16	0.753965
$416$	−1707.86	−0.201285
$417$	0	0
$418$	1996.51	0.233618
$419$	−15513.5	−1.80880	−0.904398	−	0.426691i	$-0.859679\pi$
$419$	−15513.5	−1.80880	−0.904398	+	0.426691i	$0.859679\pi$
$420$	0	0
$421$	−4772.38	−0.552474	−0.276237	−	0.961090i	$-0.589088\pi$
$421$	−4772.38	−0.552474	−0.276237	+	0.961090i	$0.589088\pi$
$422$	1391.53	0.160518
$423$	0	0
$424$	−8363.54	−0.957947
$425$	−6261.27	−0.714626
$426$	0	0
$427$	−13074.1	−1.48174
$428$	3544.09	0.400257
$429$	0	0
$430$	1950.11	0.218704
$431$	12592.4	1.40732	0.703660	−	0.710536i	$-0.251548\pi$
$431$	12592.4	1.40732	0.703660	+	0.710536i	$0.251548\pi$
$432$	0	0
$433$	−875.979	−0.0972214	−0.0486107	−	0.998818i	$-0.515479\pi$
$433$	−875.979	−0.0972214	−0.0486107	+	0.998818i	$0.515479\pi$
$434$	2306.94	0.255154
$435$	0	0
$436$	−2198.22	−0.241458
$437$	−6629.50	−0.725703
$438$	0	0
$439$	−6758.78	−0.734804	−0.367402	−	0.930062i	$-0.619753\pi$
$439$	−6758.78	−0.734804	−0.367402	+	0.930062i	$0.619753\pi$
$440$	−6966.87	−0.754846
$441$	0	0
$442$	642.005	0.0690883
$443$	1522.60	0.163298	0.0816489	−	0.996661i	$-0.473981\pi$
$443$	1522.60	0.163298	0.0816489	+	0.996661i	$0.473981\pi$
$444$	0	0
$445$	−149.443	−0.0159197
$446$	−4498.32	−0.477582
$447$	0	0
$448$	1100.14	0.116020
$449$	18544.2	1.94912	0.974558	−	0.224137i	$-0.0719562\pi$
$449$	18544.2	1.94912	0.974558	+	0.224137i	$0.0719562\pi$
$450$	0	0
$451$	1046.51	0.109264
$452$	−13080.8	−1.36121
$453$	0	0
$454$	3463.67	0.358058
$455$	2197.96	0.226466
$456$	0	0
$457$	−4584.92	−0.469308	−0.234654	−	0.972079i	$-0.575396\pi$
$457$	−4584.92	−0.469308	−0.234654	+	0.972079i	$0.575396\pi$
$458$	3879.92	0.395844
$459$	0	0
$460$	10036.9	1.01733
$461$	−5752.64	−0.581187	−0.290593	−	0.956847i	$-0.593853\pi$
$461$	−5752.64	−0.581187	−0.290593	+	0.956847i	$0.593853\pi$
$462$	0	0
$463$	675.334	0.0677871	0.0338936	−	0.999425i	$-0.489209\pi$
$463$	675.334	0.0677871	0.0338936	+	0.999425i	$0.489209\pi$
$464$	6307.92	0.631116
$465$	0	0
$466$	−7510.92	−0.746645
$467$	−9388.46	−0.930291	−0.465146	−	0.885234i	$-0.653998\pi$
$467$	−9388.46	−0.930291	−0.465146	+	0.885234i	$0.653998\pi$
$468$	0	0
$469$	6664.83	0.656190
$470$	8833.27	0.866911
$471$	0	0
$472$	6007.28	0.585821
$473$	2076.12	0.201819
$474$	0	0
$475$	7824.82	0.755848
$476$	4742.63	0.456677
$477$	0	0
$478$	326.754	0.0312665
$479$	7795.49	0.743601	0.371800	−	0.928313i	$-0.378741\pi$
$479$	7795.49	0.743601	0.371800	+	0.928313i	$0.378741\pi$
$480$	0	0
$481$	−3526.71	−0.334312
$482$	8829.09	0.834344
$483$	0	0
$484$	4942.18	0.464142
$485$	−16764.3	−1.56955
$486$	0	0
$487$	8972.87	0.834907	0.417453	−	0.908698i	$-0.362923\pi$
$487$	8972.87	0.834907	0.417453	+	0.908698i	$0.362923\pi$
$488$	−16653.7	−1.54483
$489$	0	0
$490$	2431.05	0.224130
$491$	3758.98	0.345500	0.172750	−	0.984966i	$-0.444735\pi$
$491$	3758.98	0.345500	0.172750	+	0.984966i	$0.444735\pi$
$492$	0	0
$493$	−14214.4	−1.29855
$494$	−802.325	−0.0730735
$495$	0	0
$496$	−2518.15	−0.227960
$497$	−11922.0	−1.07601
$498$	0	0
$499$	−5070.51	−0.454884	−0.227442	−	0.973792i	$-0.573036\pi$
$499$	−5070.51	−0.454884	−0.227442	+	0.973792i	$0.573036\pi$
$500$	216.161	0.0193340
$501$	0	0
$502$	−5952.82	−0.529258
$503$	−22359.4	−1.98202	−0.991009	−	0.133798i	$-0.957283\pi$
$503$	−22359.4	−1.98202	−0.991009	+	0.133798i	$0.957283\pi$
$504$	0	0
$505$	14866.7	1.31002
$506$	−3257.74	−0.286214
$507$	0	0
$508$	354.463	0.0309582
$509$	−15824.9	−1.37805	−0.689026	−	0.724737i	$-0.741961\pi$
$509$	−15824.9	−1.37805	−0.689026	+	0.724737i	$0.741961\pi$
$510$	0	0
$511$	5158.34	0.446559
$512$	−7696.77	−0.664360
$513$	0	0
$514$	3855.86	0.330884
$515$	15235.4	1.30360
$516$	0	0
$517$	9404.03	0.799978
$518$	7942.84	0.673723
$519$	0	0
$520$	2799.73	0.236109
$521$	−14358.5	−1.20741	−0.603704	−	0.797209i	$-0.706309\pi$
$521$	−14358.5	−1.20741	−0.603704	+	0.797209i	$0.706309\pi$
$522$	0	0
$523$	−8713.68	−0.728533	−0.364267	−	0.931295i	$-0.618680\pi$
$523$	−8713.68	−0.728533	−0.364267	+	0.931295i	$0.618680\pi$
$524$	663.197	0.0552899
$525$	0	0
$526$	6132.82	0.508372
$527$	5674.47	0.469039
$528$	0	0
$529$	−1349.50	−0.110915
$530$	9316.18	0.763526
$531$	0	0
$532$	−5926.96	−0.483019
$533$	−420.555	−0.0341768
$534$	0	0
$535$	−9099.13	−0.735308
$536$	8489.57	0.684130
$537$	0	0
$538$	−3878.28	−0.310789
$539$	2588.14	0.206826
$540$	0	0
$541$	−9602.09	−0.763080	−0.381540	−	0.924352i	$-0.624606\pi$
$541$	−9602.09	−0.763080	−0.381540	+	0.924352i	$0.624606\pi$
$542$	−5722.14	−0.453482
$543$	0	0
$544$	9461.20	0.745672
$545$	5643.73	0.443579
$546$	0	0
$547$	−12181.3	−0.952166	−0.476083	−	0.879400i	$-0.657944\pi$
$547$	−12181.3	−0.952166	−0.476083	+	0.879400i	$0.657944\pi$
$548$	596.124	0.0464692
$549$	0	0
$550$	3845.13	0.298103
$551$	17764.1	1.37346
$552$	0	0
$553$	−14485.8	−1.11392
$554$	6903.94	0.529459
$555$	0	0
$556$	9205.28	0.702142
$557$	−8012.40	−0.609509	−0.304754	−	0.952431i	$-0.598574\pi$
$557$	−8012.40	−0.609509	−0.304754	+	0.952431i	$0.598574\pi$
$558$	0	0
$559$	−834.319	−0.0631269
$560$	5403.45	0.407746
$561$	0	0
$562$	6645.58	0.498802
$563$	16423.1	1.22940	0.614698	−	0.788762i	$-0.289278\pi$
$563$	16423.1	1.22940	0.614698	+	0.788762i	$0.289278\pi$
$564$	0	0
$565$	33583.7	2.50067
$566$	11802.8	0.876514
$567$	0	0
$568$	−15186.2	−1.12183
$569$	−6489.32	−0.478113	−0.239057	−	0.971006i	$-0.576838\pi$
$569$	−6489.32	−0.478113	−0.239057	+	0.971006i	$0.576838\pi$
$570$	0	0
$571$	14856.8	1.08886	0.544429	−	0.838807i	$-0.316746\pi$
$571$	14856.8	1.08886	0.544429	+	0.838807i	$0.316746\pi$
$572$	1293.19	0.0945296
$573$	0	0
$574$	947.172	0.0688749
$575$	−12767.9	−0.926016
$576$	0	0
$577$	−6519.01	−0.470347	−0.235173	−	0.971953i	$-0.575566\pi$
$577$	−6519.01	−0.470347	−0.235173	+	0.971953i	$0.575566\pi$
$578$	3160.32	0.227426
$579$	0	0
$580$	−26894.3	−1.92539
$581$	6141.88	0.438568
$582$	0	0
$583$	9918.15	0.704576
$584$	6570.63	0.465573
$585$	0	0
$586$	6249.17	0.440530
$587$	−19352.2	−1.36073	−0.680367	−	0.732872i	$-0.738180\pi$
$587$	−19352.2	−1.36073	−0.680367	+	0.732872i	$0.738180\pi$
$588$	0	0
$589$	−7091.49	−0.496095
$590$	−6691.53	−0.466925
$591$	0	0
$592$	−8670.04	−0.601920
$593$	−16139.8	−1.11768	−0.558839	−	0.829276i	$-0.688753\pi$
$593$	−16139.8	−1.11768	−0.558839	+	0.829276i	$0.688753\pi$
$594$	0	0
$595$	−12176.3	−0.838956
$596$	−6020.09	−0.413746
$597$	0	0
$598$	1309.17	0.0895250
$599$	14523.1	0.990649	0.495324	−	0.868708i	$-0.335049\pi$
$599$	14523.1	0.990649	0.495324	+	0.868708i	$0.335049\pi$
$600$	0	0
$601$	−14174.1	−0.962016	−0.481008	−	0.876716i	$-0.659729\pi$
$601$	−14174.1	−0.962016	−0.481008	+	0.876716i	$0.659729\pi$
$602$	1879.05	0.127217
$603$	0	0
$604$	−18106.1	−1.21975
$605$	−12688.6	−0.852670
$606$	0	0
$607$	15889.5	1.06249	0.531247	−	0.847217i	$-0.321724\pi$
$607$	15889.5	1.06249	0.531247	+	0.847217i	$0.321724\pi$
$608$	−11823.8	−0.788684
$609$	0	0
$610$	18550.6	1.23130
$611$	−3779.14	−0.250225
$612$	0	0
$613$	−20686.0	−1.36297	−0.681483	−	0.731834i	$-0.738665\pi$
$613$	−20686.0	−1.36297	−0.681483	+	0.731834i	$0.738665\pi$
$614$	−431.972	−0.0283924
$615$	0	0
$616$	−6712.99	−0.439081
$617$	17979.1	1.17312	0.586558	−	0.809908i	$-0.300483\pi$
$617$	17979.1	1.17312	0.586558	+	0.809908i	$0.300483\pi$
$618$	0	0
$619$	−20936.0	−1.35943	−0.679716	−	0.733475i	$-0.737897\pi$
$619$	−20936.0	−1.35943	−0.679716	+	0.733475i	$0.737897\pi$
$620$	10736.3	0.695454
$621$	0	0
$622$	2575.99	0.166057
$623$	−143.997	−0.00926022
$624$	0	0
$625$	−15900.0	−1.01760
$626$	−890.085	−0.0568290
$627$	0	0
$628$	−458.607	−0.0291408
$629$	19537.3	1.23848
$630$	0	0
$631$	1825.85	0.115192	0.0575959	−	0.998340i	$-0.481657\pi$
$631$	1825.85	0.115192	0.0575959	+	0.998340i	$0.481657\pi$
$632$	−18451.8	−1.16135
$633$	0	0
$634$	7923.40	0.496338
$635$	−910.052	−0.0568730
$636$	0	0
$637$	−1040.08	−0.0646930
$638$	8729.28	0.541686
$639$	0	0
$640$	21797.6	1.34629
$641$	22387.4	1.37948	0.689742	−	0.724055i	$-0.257724\pi$
$641$	22387.4	1.37948	0.689742	+	0.724055i	$0.257724\pi$
$642$	0	0
$643$	−17727.1	−1.08723	−0.543615	−	0.839335i	$-0.682945\pi$
$643$	−17727.1	−1.08723	−0.543615	+	0.839335i	$0.682945\pi$
$644$	9671.14	0.591764
$645$	0	0
$646$	4444.73	0.270705
$647$	21754.8	1.32190	0.660950	−	0.750430i	$-0.270154\pi$
$647$	21754.8	1.32190	0.660950	+	0.750430i	$0.270154\pi$
$648$	0	0
$649$	−7123.91	−0.430875
$650$	−1545.22	−0.0932438
$651$	0	0
$652$	−491.579	−0.0295272
$653$	10154.8	0.608559	0.304280	−	0.952583i	$-0.401584\pi$
$653$	10154.8	0.608559	0.304280	+	0.952583i	$0.401584\pi$
$654$	0	0
$655$	−1702.70	−0.101572
$656$	−1033.89	−0.0615345
$657$	0	0
$658$	8511.37	0.504267
$659$	6661.61	0.393778	0.196889	−	0.980426i	$-0.436916\pi$
$659$	6661.61	0.393778	0.196889	+	0.980426i	$0.436916\pi$
$660$	0	0
$661$	13823.1	0.813397	0.406698	−	0.913562i	$-0.366680\pi$
$661$	13823.1	0.813397	0.406698	+	0.913562i	$0.366680\pi$
$662$	−3215.90	−0.188806
$663$	0	0
$664$	7823.45	0.457242
$665$	15216.9	0.887349
$666$	0	0
$667$	−28986.0	−1.68267
$668$	−8940.49	−0.517841
$669$	0	0
$670$	−9456.56	−0.545282
$671$	19749.2	1.13623
$672$	0	0
$673$	32126.9	1.84012	0.920061	−	0.391775i	$-0.128139\pi$
$673$	32126.9	1.84012	0.920061	+	0.391775i	$0.128139\pi$
$674$	8391.11	0.479545
$675$	0	0
$676$	12949.8	0.736788
$677$	4701.02	0.266876	0.133438	−	0.991057i	$-0.457398\pi$
$677$	4701.02	0.266876	0.133438	+	0.991057i	$0.457398\pi$
$678$	0	0
$679$	−16153.4	−0.912978
$680$	−15510.0	−0.874678
$681$	0	0
$682$	−3484.77	−0.195658
$683$	−11752.4	−0.658409	−0.329204	−	0.944259i	$-0.606781\pi$
$683$	−11752.4	−0.658409	−0.329204	+	0.944259i	$0.606781\pi$
$684$	0	0
$685$	−1530.49	−0.0853681
$686$	9454.77	0.526217
$687$	0	0
$688$	−2051.09	−0.113658
$689$	−3985.75	−0.220384
$690$	0	0
$691$	−15325.2	−0.843702	−0.421851	−	0.906665i	$-0.638619\pi$
$691$	−15325.2	−0.843702	−0.421851	+	0.906665i	$0.638619\pi$
$692$	−22027.3	−1.21005
$693$	0	0
$694$	−8487.77	−0.464253
$695$	−23633.7	−1.28990
$696$	0	0
$697$	2329.80	0.126610
$698$	−2169.08	−0.117623
$699$	0	0
$700$	−11414.9	−0.616345
$701$	−9247.66	−0.498258	−0.249129	−	0.968470i	$-0.580144\pi$
$701$	−9247.66	−0.498258	−0.249129	+	0.968470i	$0.580144\pi$
$702$	0	0
$703$	−24416.1	−1.30992
$704$	−1661.83	−0.0889665
$705$	0	0
$706$	−13365.8	−0.712503
$707$	14325.0	0.762016
$708$	0	0
$709$	−16405.3	−0.868989	−0.434494	−	0.900675i	$-0.643073\pi$
$709$	−16405.3	−0.868989	−0.434494	+	0.900675i	$0.643073\pi$
$710$	16915.9	0.894145
$711$	0	0
$712$	−183.422	−0.00965451
$713$	11571.3	0.607784
$714$	0	0
$715$	−3320.14	−0.173659
$716$	−4425.45	−0.230987
$717$	0	0
$718$	12016.3	0.624576
$719$	16768.1	0.869744	0.434872	−	0.900492i	$-0.356794\pi$
$719$	16768.1	0.869744	0.434872	+	0.900492i	$0.356794\pi$
$720$	0	0
$721$	14680.2	0.758281
$722$	3822.76	0.197047
$723$	0	0
$724$	8418.08	0.432121
$725$	34212.2	1.75257
$726$	0	0
$727$	−11761.6	−0.600019	−0.300009	−	0.953936i	$-0.596990\pi$
$727$	−11761.6	−0.600019	−0.300009	+	0.953936i	$0.596990\pi$
$728$	2697.71	0.137340
$729$	0	0
$730$	−7319.05	−0.371082
$731$	4621.97	0.233857
$732$	0	0
$733$	−10179.6	−0.512949	−0.256475	−	0.966551i	$-0.582561\pi$
$733$	−10179.6	−0.512949	−0.256475	+	0.966551i	$0.582561\pi$
$734$	−11669.8	−0.586841
$735$	0	0
$736$	19293.2	0.966246
$737$	−10067.6	−0.503182
$738$	0	0
$739$	26465.2	1.31737	0.658687	−	0.752417i	$-0.271112\pi$
$739$	26465.2	1.31737	0.658687	+	0.752417i	$0.271112\pi$
$740$	36965.4	1.83632
$741$	0	0
$742$	8976.68	0.444130
$743$	−5214.81	−0.257487	−0.128743	−	0.991678i	$-0.541094\pi$
$743$	−5214.81	−0.257487	−0.128743	+	0.991678i	$0.541094\pi$
$744$	0	0
$745$	15456.0	0.760088
$746$	10422.2	0.511507
$747$	0	0
$748$	−7164.02	−0.350190
$749$	−8767.55	−0.427716
$750$	0	0
$751$	−36463.1	−1.77171	−0.885857	−	0.463959i	$-0.846429\pi$
$751$	−36463.1	−1.77171	−0.885857	+	0.463959i	$0.846429\pi$
$752$	−9290.62	−0.450524
$753$	0	0
$754$	−3507.98	−0.169434
$755$	46485.9	2.24079
$756$	0	0
$757$	1286.49	0.0617678	0.0308839	−	0.999523i	$-0.490168\pi$
$757$	1286.49	0.0617678	0.0308839	+	0.999523i	$0.490168\pi$
$758$	6218.39	0.297971
$759$	0	0
$760$	19383.1	0.925132
$761$	4673.76	0.222633	0.111317	−	0.993785i	$-0.464493\pi$
$761$	4673.76	0.222633	0.111317	+	0.993785i	$0.464493\pi$
$762$	0	0
$763$	5438.06	0.258022
$764$	5864.11	0.277691
$765$	0	0
$766$	−5315.97	−0.250749
$767$	2862.84	0.134773
$768$	0	0
$769$	3110.88	0.145879	0.0729397	−	0.997336i	$-0.476762\pi$
$769$	3110.88	0.145879	0.0729397	+	0.997336i	$0.476762\pi$
$770$	7477.62	0.349967
$771$	0	0
$772$	18562.1	0.865368
$773$	−16677.9	−0.776020	−0.388010	−	0.921655i	$-0.626837\pi$
$773$	−16677.9	−0.776020	−0.388010	+	0.921655i	$0.626837\pi$
$774$	0	0
$775$	−13657.7	−0.633030
$776$	−20576.0	−0.951851
$777$	0	0
$778$	−20271.9	−0.934170
$779$	−2911.59	−0.133913
$780$	0	0
$781$	18008.9	0.825109
$782$	−7252.56	−0.331651
$783$	0	0
$784$	−2556.93	−0.116478
$785$	1177.43	0.0535342
$786$	0	0
$787$	−11660.2	−0.528132	−0.264066	−	0.964505i	$-0.585064\pi$
$787$	−11660.2	−0.528132	−0.264066	+	0.964505i	$0.585064\pi$
$788$	15782.0	0.713465
$789$	0	0
$790$	20553.5	0.925646
$791$	32359.9	1.45459
$792$	0	0
$793$	−7936.51	−0.355402
$794$	−19276.5	−0.861583
$795$	0	0
$796$	3010.23	0.134039
$797$	−6729.57	−0.299089	−0.149544	−	0.988755i	$-0.547781\pi$
$797$	−6729.57	−0.299089	−0.149544	+	0.988755i	$0.547781\pi$
$798$	0	0
$799$	20935.7	0.926975
$800$	−22771.8	−1.00638
$801$	0	0
$802$	582.063	0.0256276
$803$	−7791.97	−0.342432
$804$	0	0
$805$	−24829.8	−1.08712
$806$	1400.40	0.0611998
$807$	0	0
$808$	18246.9	0.794462
$809$	−1479.41	−0.0642935	−0.0321468	−	0.999483i	$-0.510234\pi$
$809$	−1479.41	−0.0642935	−0.0321468	+	0.999483i	$0.510234\pi$
$810$	0	0
$811$	38845.5	1.68193	0.840967	−	0.541086i	$-0.181987\pi$
$811$	38845.5	1.68193	0.840967	+	0.541086i	$0.181987\pi$
$812$	−25914.3	−1.11997
$813$	0	0
$814$	−11998.1	−0.516626
$815$	1262.08	0.0542440
$816$	0	0
$817$	−5776.16	−0.247347
$818$	−17500.0	−0.748012
$819$	0	0
$820$	4408.07	0.187727
$821$	−39212.3	−1.66689	−0.833446	−	0.552601i	$-0.813635\pi$
$821$	−39212.3	−1.66689	−0.833446	+	0.552601i	$0.813635\pi$
$822$	0	0
$823$	−34862.8	−1.47660	−0.738299	−	0.674474i	$-0.764371\pi$
$823$	−34862.8	−1.47660	−0.738299	+	0.674474i	$0.764371\pi$
$824$	18699.5	0.790567
$825$	0	0
$826$	−6447.68	−0.271602
$827$	29815.6	1.25368	0.626839	−	0.779149i	$-0.284348\pi$
$827$	29815.6	1.25368	0.626839	+	0.779149i	$0.284348\pi$
$828$	0	0
$829$	27837.6	1.16627	0.583136	−	0.812374i	$-0.301825\pi$
$829$	27837.6	1.16627	0.583136	+	0.812374i	$0.301825\pi$
$830$	−8714.57	−0.364442
$831$	0	0
$832$	667.828	0.0278279
$833$	5761.85	0.239659
$834$	0	0
$835$	22953.9	0.951320
$836$	8953.01	0.370390
$837$	0	0
$838$	21209.6	0.874313
$839$	16228.4	0.667778	0.333889	−	0.942612i	$-0.391639\pi$
$839$	16228.4	0.667778	0.333889	+	0.942612i	$0.391639\pi$
$840$	0	0
$841$	53280.3	2.18460
$842$	6524.66	0.267048
$843$	0	0
$844$	6240.08	0.254493
$845$	−33247.4	−1.35354
$846$	0	0
$847$	−12226.2	−0.495983
$848$	−9798.54	−0.396796
$849$	0	0
$850$	8560.22	0.345427
$851$	39840.3	1.60483
$852$	0	0
$853$	−6284.42	−0.252256	−0.126128	−	0.992014i	$-0.540255\pi$
$853$	−6284.42	−0.252256	−0.126128	+	0.992014i	$0.540255\pi$
$854$	17874.6	0.716224
$855$	0	0
$856$	−11168.0	−0.445928
$857$	−39626.6	−1.57948	−0.789742	−	0.613440i	$-0.789785\pi$
$857$	−39626.6	−1.57948	−0.789742	+	0.613440i	$0.789785\pi$
$858$	0	0
$859$	−21271.4	−0.844900	−0.422450	−	0.906386i	$-0.638830\pi$
$859$	−21271.4	−0.844900	−0.422450	+	0.906386i	$0.638830\pi$
$860$	8744.96	0.346745
$861$	0	0
$862$	−17216.0	−0.680253
$863$	−12359.4	−0.487508	−0.243754	−	0.969837i	$-0.578379\pi$
$863$	−12359.4	−0.487508	−0.243754	+	0.969837i	$0.578379\pi$
$864$	0	0
$865$	56553.2	2.22297
$866$	1197.61	0.0469937
$867$	0	0
$868$	10345.1	0.404534
$869$	21881.6	0.854179
$870$	0	0
$871$	4045.81	0.157390
$872$	6926.93	0.269009
$873$	0	0
$874$	9063.66	0.350781
$875$	−534.750	−0.0206604
$876$	0	0
$877$	22665.2	0.872692	0.436346	−	0.899779i	$-0.356272\pi$
$877$	22665.2	0.872692	0.436346	+	0.899779i	$0.356272\pi$
$878$	9240.40	0.355181
$879$	0	0
$880$	−8162.22	−0.312669
$881$	43070.8	1.64710	0.823549	−	0.567246i	$-0.191991\pi$
$881$	43070.8	1.64710	0.823549	+	0.567246i	$0.191991\pi$
$882$	0	0
$883$	46120.4	1.75773	0.878865	−	0.477071i	$-0.158301\pi$
$883$	46120.4	1.75773	0.878865	+	0.477071i	$0.158301\pi$
$884$	2878.96	0.109536
$885$	0	0
$886$	−2081.65	−0.0789329
$887$	−2846.88	−0.107767	−0.0538833	−	0.998547i	$-0.517160\pi$
$887$	−2846.88	−0.107767	−0.0538833	+	0.998547i	$0.517160\pi$
$888$	0	0
$889$	−876.889	−0.0330820
$890$	204.314	0.00769507
$891$	0	0
$892$	−20171.9	−0.757183
$893$	−26163.8	−0.980445
$894$	0	0
$895$	11361.9	0.424344
$896$	21003.3	0.783115
$897$	0	0
$898$	−25353.0	−0.942139
$899$	−31005.9	−1.15028
$900$	0	0
$901$	22080.3	0.816427
$902$	−1430.76	−0.0528149
$903$	0	0
$904$	41219.6	1.51653
$905$	−21612.7	−0.793845
$906$	0	0
$907$	32042.0	1.17303	0.586515	−	0.809939i	$-0.300500\pi$
$907$	32042.0	1.17303	0.586515	+	0.809939i	$0.300500\pi$
$908$	15532.3	0.567683
$909$	0	0
$910$	−3004.99	−0.109466
$911$	9197.00	0.334479	0.167239	−	0.985916i	$-0.446515\pi$
$911$	9197.00	0.334479	0.167239	+	0.985916i	$0.446515\pi$
$912$	0	0
$913$	−9277.66	−0.336304
$914$	6268.37	0.226848
$915$	0	0
$916$	17398.8	0.627591
$917$	−1640.65	−0.0590830
$918$	0	0
$919$	2726.95	0.0978823	0.0489412	−	0.998802i	$-0.484415\pi$
$919$	2726.95	0.0978823	0.0489412	+	0.998802i	$0.484415\pi$
$920$	−31627.9	−1.13341
$921$	0	0
$922$	7864.84	0.280927
$923$	−7237.15	−0.258086
$924$	0	0
$925$	−47023.6	−1.67149
$926$	−923.297	−0.0327661
$927$	0	0
$928$	−51697.0	−1.82870
$929$	−40137.1	−1.41750	−0.708749	−	0.705461i	$-0.750740\pi$
$929$	−40137.1	−1.41750	−0.708749	+	0.705461i	$0.750740\pi$
$930$	0	0
$931$	−7200.69	−0.253483
$932$	−33681.4	−1.18377
$933$	0	0
$934$	12835.6	0.449673
$935$	18393.0	0.643331
$936$	0	0
$937$	41074.1	1.43205	0.716026	−	0.698073i	$-0.245959\pi$
$937$	41074.1	1.43205	0.716026	+	0.698073i	$0.245959\pi$
$938$	−9111.96	−0.317181
$939$	0	0
$940$	39611.3	1.37444
$941$	−50939.1	−1.76468	−0.882342	−	0.470609i	$-0.844034\pi$
$941$	−50939.1	−1.76468	−0.882342	+	0.470609i	$0.844034\pi$
$942$	0	0
$943$	4750.90	0.164062
$944$	7038.00	0.242656
$945$	0	0
$946$	−2838.41	−0.0975526
$947$	43191.4	1.48208	0.741042	−	0.671459i	$-0.234332\pi$
$947$	43191.4	1.48208	0.741042	+	0.671459i	$0.234332\pi$
$948$	0	0
$949$	3131.31	0.107109
$950$	−10697.9	−0.365352
$951$	0	0
$952$	−14944.8	−0.508785
$953$	6762.27	0.229855	0.114927	−	0.993374i	$-0.463336\pi$
$953$	6762.27	0.229855	0.114927	+	0.993374i	$0.463336\pi$
$954$	0	0
$955$	−15055.6	−0.510144
$956$	1465.27	0.0495714
$957$	0	0
$958$	−10657.8	−0.359433
$959$	−1474.72	−0.0496572
$960$	0	0
$961$	−17413.3	−0.584516
$962$	4821.61	0.161596
$963$	0	0
$964$	39592.6	1.32281
$965$	−47656.5	−1.58976
$966$	0	0
$967$	−14831.2	−0.493216	−0.246608	−	0.969115i	$-0.579316\pi$
$967$	−14831.2	−0.493216	−0.246608	+	0.969115i	$0.579316\pi$
$968$	−15573.6	−0.517102
$969$	0	0
$970$	22919.7	0.758668
$971$	14700.8	0.485861	0.242930	−	0.970044i	$-0.421891\pi$
$971$	14700.8	0.485861	0.242930	+	0.970044i	$0.421891\pi$
$972$	0	0
$973$	−22772.5	−0.750311
$974$	−12267.4	−0.403567
$975$	0	0
$976$	−19511.1	−0.639892
$977$	9818.77	0.321525	0.160763	−	0.986993i	$-0.448605\pi$
$977$	9818.77	0.321525	0.160763	+	0.986993i	$0.448605\pi$
$978$	0	0
$979$	217.516	0.00710095
$980$	10901.7	0.355348
$981$	0	0
$982$	−5139.17	−0.167003
$983$	14628.3	0.474639	0.237319	−	0.971432i	$-0.423731\pi$
$983$	14628.3	0.474639	0.237319	+	0.971432i	$0.423731\pi$
$984$	0	0
$985$	−40518.8	−1.31070
$986$	19433.6	0.627678
$987$	0	0
$988$	−3597.89	−0.115854
$989$	9425.08	0.303034
$990$	0	0
$991$	477.422	0.0153036	0.00765178	−	0.999971i	$-0.497564\pi$
$991$	477.422	0.0153036	0.00765178	+	0.999971i	$0.497564\pi$
$992$	20637.7	0.660531
$993$	0	0
$994$	16299.5	0.520109
$995$	−7728.49	−0.246241
$996$	0	0
$997$	34084.4	1.08271	0.541356	−	0.840793i	$-0.317911\pi$
$997$	34084.4	1.08271	0.541356	+	0.840793i	$0.317911\pi$
$998$	6932.26	0.219876
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.11		28
3.2	odd	2		717.4.a.b.1.18	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.18	✓	28	3.2	odd	2
2151.4.a.b.1.11		28	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-1.36717 q^{2} -6.13085 q^{4} +15.7404 q^{5} +15.1668 q^{7} +19.3193 q^{8} +O(q^{10})\)	\(q-1.36717 q^{2} -6.13085 q^{4} +15.7404 q^{5} +15.1668 q^{7} +19.3193 q^{8} -21.5198 q^{10} -22.9103 q^{11} +9.20683 q^{13} -20.7356 q^{14} +22.6340 q^{16} -51.0041 q^{17} +63.7408 q^{19} -96.5019 q^{20} +31.3223 q^{22} -104.007 q^{23} +122.760 q^{25} -12.5873 q^{26} -92.9853 q^{28} +278.692 q^{29} -111.255 q^{31} -185.499 q^{32} +69.7313 q^{34} +238.731 q^{35} -383.053 q^{37} -87.1445 q^{38} +304.093 q^{40} -45.6786 q^{41} -90.6195 q^{43} +140.460 q^{44} +142.196 q^{46} -410.472 q^{47} -112.968 q^{49} -167.834 q^{50} -56.4457 q^{52} -432.912 q^{53} -360.617 q^{55} +293.011 q^{56} -381.020 q^{58} +310.948 q^{59} -862.024 q^{61} +152.105 q^{62} +72.5362 q^{64} +144.919 q^{65} +439.435 q^{67} +312.698 q^{68} -326.386 q^{70} -786.062 q^{71} +340.108 q^{73} +523.699 q^{74} -390.785 q^{76} -347.476 q^{77} -955.097 q^{79} +356.269 q^{80} +62.4504 q^{82} +404.956 q^{83} -802.825 q^{85} +123.892 q^{86} -442.611 q^{88} -9.49422 q^{89} +139.638 q^{91} +637.652 q^{92} +561.184 q^{94} +1003.31 q^{95} -1065.05 q^{97} +154.447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * q^98

Introduction

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