LMFDB - Embedded newform 2151.4.a.b.1.19

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			1.19
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)

$f(q)$	$=$	$q+2.22027 q^{2} -3.07038 q^{4} -14.7142 q^{5} +26.0557 q^{7} -24.5793 q^{8} +O(q^{10})$	\(q+2.22027 q^{2} -3.07038 q^{4} -14.7142 q^{5} +26.0557 q^{7} -24.5793 q^{8} -32.6696 q^{10} +0.340430 q^{11} -88.6632 q^{13} +57.8507 q^{14} -30.0097 q^{16} +78.2516 q^{17} +83.2310 q^{19} +45.1783 q^{20} +0.755848 q^{22} +74.9624 q^{23} +91.5080 q^{25} -196.857 q^{26} -80.0008 q^{28} +244.634 q^{29} +101.051 q^{31} +130.005 q^{32} +173.740 q^{34} -383.388 q^{35} -408.544 q^{37} +184.796 q^{38} +361.665 q^{40} -378.718 q^{41} +377.456 q^{43} -1.04525 q^{44} +166.437 q^{46} -124.770 q^{47} +335.897 q^{49} +203.173 q^{50} +272.230 q^{52} +469.684 q^{53} -5.00916 q^{55} -640.429 q^{56} +543.154 q^{58} +198.235 q^{59} -683.617 q^{61} +224.361 q^{62} +528.723 q^{64} +1304.61 q^{65} -786.649 q^{67} -240.262 q^{68} -851.227 q^{70} -292.698 q^{71} -551.472 q^{73} -907.079 q^{74} -255.551 q^{76} +8.87012 q^{77} +623.349 q^{79} +441.569 q^{80} -840.858 q^{82} +1289.89 q^{83} -1151.41 q^{85} +838.056 q^{86} -8.36752 q^{88} -1514.36 q^{89} -2310.18 q^{91} -230.163 q^{92} -277.024 q^{94} -1224.68 q^{95} +630.790 q^{97} +745.784 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.22027	0.784985	0.392493	−	0.919755i	$-0.371613\pi$
$2$	2.22027	0.784985	0.392493	+	0.919755i	$0.371613\pi$
$3$	0	0
$3$	0	0
$4$	−3.07038	−0.383798
$5$	−14.7142	−1.31608	−0.658040	−	0.752983i	$-0.728614\pi$
$5$	−14.7142	−1.31608	−0.658040	+	0.752983i	$0.728614\pi$
$6$	0	0
$7$	26.0557	1.40687	0.703437	−	0.710758i	$-0.251648\pi$
$7$	26.0557	1.40687	0.703437	+	0.710758i	$0.251648\pi$
$8$	−24.5793	−1.08626
$9$	0	0
$10$	−32.6696	−1.03310
$11$	0.340430	0.00933122	0.00466561	−	0.999989i	$-0.498515\pi$
$11$	0.340430	0.00933122	0.00466561	+	0.999989i	$0.498515\pi$
$12$	0	0
$13$	−88.6632	−1.89160	−0.945798	−	0.324757i	$-0.894718\pi$
$13$	−88.6632	−1.89160	−0.945798	+	0.324757i	$0.894718\pi$
$14$	57.8507	1.10437
$15$	0	0
$16$	−30.0097	−0.468901
$17$	78.2516	1.11640	0.558200	−	0.829707i	$-0.311492\pi$
$17$	78.2516	1.11640	0.558200	+	0.829707i	$0.311492\pi$
$18$	0	0
$19$	83.2310	1.00497	0.502487	−	0.864585i	$-0.332419\pi$
$19$	83.2310	1.00497	0.502487	+	0.864585i	$0.332419\pi$
$20$	45.1783	0.505108
$21$	0	0
$22$	0.755848	0.00732488
$23$	74.9624	0.679597	0.339799	−	0.940498i	$-0.389641\pi$
$23$	74.9624	0.679597	0.339799	+	0.940498i	$0.389641\pi$
$24$	0	0
$25$	91.5080	0.732064
$26$	−196.857	−1.48487
$27$	0	0
$28$	−80.0008	−0.539955
$29$	244.634	1.56646	0.783230	−	0.621732i	$-0.213571\pi$
$29$	244.634	1.56646	0.783230	+	0.621732i	$0.213571\pi$
$30$	0	0
$31$	101.051	0.585460	0.292730	−	0.956195i	$-0.405436\pi$
$31$	101.051	0.585460	0.292730	+	0.956195i	$0.405436\pi$
$32$	130.005	0.718180
$33$	0	0
$34$	173.740	0.876357
$35$	−383.388	−1.85156
$36$	0	0
$37$	−408.544	−1.81525	−0.907624	−	0.419783i	$-0.862106\pi$
$37$	−408.544	−1.81525	−0.907624	+	0.419783i	$0.862106\pi$
$38$	184.796	0.788889
$39$	0	0
$40$	361.665	1.42961
$41$	−378.718	−1.44258	−0.721291	−	0.692632i	$-0.756451\pi$
$41$	−378.718	−1.44258	−0.721291	+	0.692632i	$0.756451\pi$
$42$	0	0
$43$	377.456	1.33864	0.669320	−	0.742974i	$-0.266586\pi$
$43$	377.456	1.33864	0.669320	+	0.742974i	$0.266586\pi$
$44$	−1.04525	−0.00358130
$45$	0	0
$46$	166.437	0.533474
$47$	−124.770	−0.387225	−0.193613	−	0.981078i	$-0.562020\pi$
$47$	−124.770	−0.387225	−0.193613	+	0.981078i	$0.562020\pi$
$48$	0	0
$49$	335.897	0.979292
$50$	203.173	0.574660
$51$	0	0
$52$	272.230	0.725990
$53$	469.684	1.21728	0.608642	−	0.793445i	$-0.291715\pi$
$53$	469.684	1.21728	0.608642	+	0.793445i	$0.291715\pi$
$54$	0	0
$55$	−5.00916	−0.0122806
$56$	−640.429	−1.52823
$57$	0	0
$58$	543.154	1.22965
$59$	198.235	0.437424	0.218712	−	0.975789i	$-0.429814\pi$
$59$	198.235	0.437424	0.218712	+	0.975789i	$0.429814\pi$
$60$	0	0
$61$	−683.617	−1.43489	−0.717444	−	0.696616i	$-0.754688\pi$
$61$	−683.617	−1.43489	−0.717444	+	0.696616i	$0.754688\pi$
$62$	224.361	0.459578
$63$	0	0
$64$	528.723	1.03266
$65$	1304.61	2.48949
$66$	0	0
$67$	−786.649	−1.43439	−0.717197	−	0.696870i	$-0.754575\pi$
$67$	−786.649	−1.43439	−0.717197	+	0.696870i	$0.754575\pi$
$68$	−240.262	−0.428472
$69$	0	0
$70$	−851.227	−1.45344
$71$	−292.698	−0.489251	−0.244626	−	0.969618i	$-0.578665\pi$
$71$	−292.698	−0.489251	−0.244626	+	0.969618i	$0.578665\pi$
$72$	0	0
$73$	−551.472	−0.884176	−0.442088	−	0.896972i	$-0.645762\pi$
$73$	−551.472	−0.884176	−0.442088	+	0.896972i	$0.645762\pi$
$74$	−907.079	−1.42494
$75$	0	0
$76$	−255.551	−0.385707
$77$	8.87012	0.0131278
$78$	0	0
$79$	623.349	0.887749	0.443875	−	0.896089i	$-0.353604\pi$
$79$	623.349	0.887749	0.443875	+	0.896089i	$0.353604\pi$
$80$	441.569	0.617111
$81$	0	0
$82$	−840.858	−1.13241
$83$	1289.89	1.70583	0.852914	−	0.522052i	$-0.174833\pi$
$83$	1289.89	1.70583	0.852914	+	0.522052i	$0.174833\pi$
$84$	0	0
$85$	−1151.41	−1.46927
$86$	838.056	1.05081
$87$	0	0
$88$	−8.36752	−0.0101361
$89$	−1514.36	−1.80362	−0.901808	−	0.432137i	$-0.857760\pi$
$89$	−1514.36	−1.80362	−0.901808	+	0.432137i	$0.857760\pi$
$90$	0	0
$91$	−2310.18	−2.66123
$92$	−230.163	−0.260828
$93$	0	0
$94$	−277.024	−0.303966
$95$	−1224.68	−1.32262
$96$	0	0
$97$	630.790	0.660279	0.330139	−	0.943932i	$-0.392904\pi$
$97$	630.790	0.660279	0.330139	+	0.943932i	$0.392904\pi$
$98$	745.784	0.768730
$99$	0	0
$100$	−280.965	−0.280965
$101$	526.770	0.518966	0.259483	−	0.965748i	$-0.416448\pi$
$101$	526.770	0.518966	0.259483	+	0.965748i	$0.416448\pi$
$102$	0	0
$103$	−1664.35	−1.59217	−0.796086	−	0.605183i	$-0.793100\pi$
$103$	−1664.35	−1.59217	−0.796086	+	0.605183i	$0.793100\pi$
$104$	2179.28	2.05477
$105$	0	0
$106$	1042.83	0.955550
$107$	−528.185	−0.477211	−0.238606	−	0.971117i	$-0.576690\pi$
$107$	−528.185	−0.477211	−0.238606	+	0.971117i	$0.576690\pi$
$108$	0	0
$109$	69.4511	0.0610294	0.0305147	−	0.999534i	$-0.490285\pi$
$109$	69.4511	0.0610294	0.0305147	+	0.999534i	$0.490285\pi$
$110$	−11.1217	−0.00964011
$111$	0	0
$112$	−781.922	−0.659685
$113$	−1452.39	−1.20911	−0.604555	−	0.796564i	$-0.706649\pi$
$113$	−1452.39	−1.20911	−0.604555	+	0.796564i	$0.706649\pi$
$114$	0	0
$115$	−1103.01	−0.894404
$116$	−751.119	−0.601204
$117$	0	0
$118$	440.137	0.343372
$119$	2038.90	1.57063
$120$	0	0
$121$	−1330.88	−0.999913
$122$	−1517.82	−1.12637
$123$	0	0
$124$	−310.265	−0.224698
$125$	492.808	0.352625
$126$	0	0
$127$	1212.53	0.847200	0.423600	−	0.905849i	$-0.360766\pi$
$127$	1212.53	0.847200	0.423600	+	0.905849i	$0.360766\pi$
$128$	133.874	0.0924446
$129$	0	0
$130$	2896.59	1.95421
$131$	1125.58	0.750703	0.375352	−	0.926882i	$-0.377522\pi$
$131$	1125.58	0.750703	0.375352	+	0.926882i	$0.377522\pi$
$132$	0	0
$133$	2168.64	1.41387
$134$	−1746.58	−1.12598
$135$	0	0
$136$	−1923.37	−1.21270
$137$	−1465.00	−0.913601	−0.456800	−	0.889569i	$-0.651005\pi$
$137$	−1465.00	−0.913601	−0.456800	+	0.889569i	$0.651005\pi$
$138$	0	0
$139$	67.8183	0.0413833	0.0206916	−	0.999786i	$-0.493413\pi$
$139$	67.8183	0.0413833	0.0206916	+	0.999786i	$0.493413\pi$
$140$	1177.15	0.710623
$141$	0	0
$142$	−649.869	−0.384055
$143$	−30.1836	−0.0176509
$144$	0	0
$145$	−3599.59	−2.06159
$146$	−1224.42	−0.694066
$147$	0	0
$148$	1254.39	0.696689
$149$	2353.92	1.29423	0.647115	−	0.762392i	$-0.275975\pi$
$149$	2353.92	1.29423	0.647115	+	0.762392i	$0.275975\pi$
$150$	0	0
$151$	−695.079	−0.374601	−0.187300	−	0.982303i	$-0.559974\pi$
$151$	−695.079	−0.374601	−0.187300	+	0.982303i	$0.559974\pi$
$152$	−2045.76	−1.09166
$153$	0	0
$154$	19.6941	0.0103052
$155$	−1486.88	−0.770512
$156$	0	0
$157$	−2257.93	−1.14779	−0.573893	−	0.818930i	$-0.694568\pi$
$157$	−2257.93	−1.14779	−0.573893	+	0.818930i	$0.694568\pi$
$158$	1384.00	0.696870
$159$	0	0
$160$	−1912.91	−0.945182
$161$	1953.19	0.956107
$162$	0	0
$163$	−1925.92	−0.925459	−0.462729	−	0.886500i	$-0.653130\pi$
$163$	−1925.92	−0.925459	−0.462729	+	0.886500i	$0.653130\pi$
$164$	1162.81	0.553660
$165$	0	0
$166$	2863.91	1.33905
$167$	−687.821	−0.318714	−0.159357	−	0.987221i	$-0.550942\pi$
$167$	−687.821	−0.318714	−0.159357	+	0.987221i	$0.550942\pi$
$168$	0	0
$169$	5664.16	2.57813
$170$	−2556.45	−1.15336
$171$	0	0
$172$	−1158.93	−0.513767
$173$	−3624.06	−1.59267	−0.796336	−	0.604854i	$-0.793231\pi$
$173$	−3624.06	−1.59267	−0.796336	+	0.604854i	$0.793231\pi$
$174$	0	0
$175$	2384.30	1.02992
$176$	−10.2162	−0.00437542
$177$	0	0
$178$	−3362.29	−1.41581
$179$	1009.85	0.421674	0.210837	−	0.977521i	$-0.432381\pi$
$179$	1009.85	0.421674	0.210837	+	0.977521i	$0.432381\pi$
$180$	0	0
$181$	3225.92	1.32476	0.662378	−	0.749170i	$-0.269547\pi$
$181$	3225.92	1.32476	0.662378	+	0.749170i	$0.269547\pi$
$182$	−5129.23	−2.08903
$183$	0	0
$184$	−1842.52	−0.738220
$185$	6011.40	2.38901
$186$	0	0
$187$	26.6392	0.0104174
$188$	383.092	0.148616
$189$	0	0
$190$	−2719.12	−1.03824
$191$	−888.584	−0.336627	−0.168313	−	0.985734i	$-0.553832\pi$
$191$	−888.584	−0.336627	−0.168313	+	0.985734i	$0.553832\pi$
$192$	0	0
$193$	−4809.23	−1.79366	−0.896829	−	0.442377i	$-0.854135\pi$
$193$	−4809.23	−1.79366	−0.896829	+	0.442377i	$0.854135\pi$
$194$	1400.53	0.518309
$195$	0	0
$196$	−1031.33	−0.375850
$197$	−1226.98	−0.443749	−0.221874	−	0.975075i	$-0.571217\pi$
$197$	−1226.98	−0.443749	−0.221874	+	0.975075i	$0.571217\pi$
$198$	0	0
$199$	−4467.50	−1.59142	−0.795710	−	0.605678i	$-0.792902\pi$
$199$	−4467.50	−1.59142	−0.795710	+	0.605678i	$0.792902\pi$
$200$	−2249.20	−0.795213
$201$	0	0
$202$	1169.57	0.407381
$203$	6374.09	2.20381
$204$	0	0
$205$	5572.54	1.89855
$206$	−3695.32	−1.24983
$207$	0	0
$208$	2660.75	0.886972
$209$	28.3343	0.00937763
$210$	0	0
$211$	−1430.04	−0.466577	−0.233289	−	0.972408i	$-0.574949\pi$
$211$	−1430.04	−0.466577	−0.233289	+	0.972408i	$0.574949\pi$
$212$	−1442.11	−0.467191
$213$	0	0
$214$	−1172.72	−0.374604
$215$	−5553.97	−1.76176
$216$	0	0
$217$	2632.95	0.823668
$218$	154.200	0.0479072
$219$	0	0
$220$	15.3800	0.00471328
$221$	−6938.03	−2.11178
$222$	0	0
$223$	−489.701	−0.147053	−0.0735264	−	0.997293i	$-0.523425\pi$
$223$	−489.701	−0.147053	−0.0735264	+	0.997293i	$0.523425\pi$
$224$	3387.35	1.01039
$225$	0	0
$226$	−3224.70	−0.949133
$227$	1889.44	0.552451	0.276225	−	0.961093i	$-0.410916\pi$
$227$	1889.44	0.552451	0.276225	+	0.961093i	$0.410916\pi$
$228$	0	0
$229$	−4118.14	−1.18836	−0.594179	−	0.804333i	$-0.702523\pi$
$229$	−4118.14	−1.18836	−0.594179	+	0.804333i	$0.702523\pi$
$230$	−2448.99	−0.702094
$231$	0	0
$232$	−6012.92	−1.70158
$233$	6505.18	1.82905	0.914525	−	0.404530i	$-0.132565\pi$
$233$	6505.18	1.82905	0.914525	+	0.404530i	$0.132565\pi$
$234$	0	0
$235$	1835.89	0.509619
$236$	−608.658	−0.167882
$237$	0	0
$238$	4526.91	1.23292
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−5854.30	−1.56477	−0.782384	−	0.622797i	$-0.785996\pi$
$241$	−5854.30	−1.56477	−0.782384	+	0.622797i	$0.785996\pi$
$242$	−2954.93	−0.784917
$243$	0	0
$244$	2098.97	0.550707
$245$	−4942.46	−1.28883
$246$	0	0
$247$	−7379.52	−1.90100
$248$	−2483.76	−0.635963
$249$	0	0
$250$	1094.17	0.276805
$251$	−2272.84	−0.571556	−0.285778	−	0.958296i	$-0.592252\pi$
$251$	−2272.84	−0.571556	−0.285778	+	0.958296i	$0.592252\pi$
$252$	0	0
$253$	25.5194	0.00634147
$254$	2692.14	0.665039
$255$	0	0
$256$	−3932.55	−0.960095
$257$	−1631.18	−0.395916	−0.197958	−	0.980211i	$-0.563431\pi$
$257$	−1631.18	−0.395916	−0.197958	+	0.980211i	$0.563431\pi$
$258$	0	0
$259$	−10644.9	−2.55382
$260$	−4005.65	−0.955460
$261$	0	0
$262$	2499.09	0.589291
$263$	−3820.45	−0.895738	−0.447869	−	0.894099i	$-0.647817\pi$
$263$	−3820.45	−0.895738	−0.447869	+	0.894099i	$0.647817\pi$
$264$	0	0
$265$	−6911.03	−1.60204
$266$	4814.97	1.10987
$267$	0	0
$268$	2415.31	0.550518
$269$	−1315.95	−0.298272	−0.149136	−	0.988817i	$-0.547649\pi$
$269$	−1315.95	−0.298272	−0.149136	+	0.988817i	$0.547649\pi$
$270$	0	0
$271$	−3093.88	−0.693506	−0.346753	−	0.937957i	$-0.612716\pi$
$271$	−3093.88	−0.693506	−0.346753	+	0.937957i	$0.612716\pi$
$272$	−2348.30	−0.523481
$273$	0	0
$274$	−3252.70	−0.717163
$275$	31.1521	0.00683105
$276$	0	0
$277$	6065.89	1.31575	0.657877	−	0.753125i	$-0.271455\pi$
$277$	6065.89	1.31575	0.657877	+	0.753125i	$0.271455\pi$
$278$	150.575	0.0324853
$279$	0	0
$280$	9423.41	2.01127
$281$	−3642.45	−0.773275	−0.386637	−	0.922232i	$-0.626364\pi$
$281$	−3642.45	−0.773275	−0.386637	+	0.922232i	$0.626364\pi$
$282$	0	0
$283$	5289.78	1.11111	0.555557	−	0.831479i	$-0.312505\pi$
$283$	5289.78	1.11111	0.555557	+	0.831479i	$0.312505\pi$
$284$	898.695	0.187774
$285$	0	0
$286$	−67.0158	−0.0138557
$287$	−9867.75	−2.02953
$288$	0	0
$289$	1210.31	0.246348
$290$	−7992.08	−1.61831
$291$	0	0
$292$	1693.23	0.339345
$293$	−4796.43	−0.956349	−0.478175	−	0.878265i	$-0.658701\pi$
$293$	−4796.43	−0.956349	−0.478175	+	0.878265i	$0.658701\pi$
$294$	0	0
$295$	−2916.88	−0.575685
$296$	10041.7	1.97183
$297$	0	0
$298$	5226.34	1.01595
$299$	−6646.40	−1.28552
$300$	0	0
$301$	9834.86	1.88330
$302$	−1543.26	−0.294056
$303$	0	0
$304$	−2497.74	−0.471233
$305$	10058.9	1.88843
$306$	0	0
$307$	−7101.16	−1.32014	−0.660072	−	0.751202i	$-0.729474\pi$
$307$	−7101.16	−1.32014	−0.660072	+	0.751202i	$0.729474\pi$
$308$	−27.2347	−0.00503844
$309$	0	0
$310$	−3301.29	−0.604841
$311$	−4374.27	−0.797564	−0.398782	−	0.917046i	$-0.630567\pi$
$311$	−4374.27	−0.797564	−0.398782	+	0.917046i	$0.630567\pi$
$312$	0	0
$313$	−10486.8	−1.89376	−0.946881	−	0.321584i	$-0.895785\pi$
$313$	−10486.8	−1.89376	−0.946881	+	0.321584i	$0.895785\pi$
$314$	−5013.23	−0.900996
$315$	0	0
$316$	−1913.92	−0.340716
$317$	894.151	0.158424	0.0792122	−	0.996858i	$-0.474760\pi$
$317$	894.151	0.158424	0.0792122	+	0.996858i	$0.474760\pi$
$318$	0	0
$319$	83.2806	0.0146170
$320$	−7779.75	−1.35907
$321$	0	0
$322$	4336.63	0.750530
$323$	6512.95	1.12195
$324$	0	0
$325$	−8113.39	−1.38477
$326$	−4276.07	−0.726471
$327$	0	0
$328$	9308.63	1.56702
$329$	−3250.96	−0.544776
$330$	0	0
$331$	−6816.78	−1.13198	−0.565988	−	0.824414i	$-0.691505\pi$
$331$	−6816.78	−1.13198	−0.565988	+	0.824414i	$0.691505\pi$
$332$	−3960.45	−0.654693
$333$	0	0
$334$	−1527.15	−0.250186
$335$	11574.9	1.88778
$336$	0	0
$337$	6905.10	1.11616	0.558078	−	0.829788i	$-0.311539\pi$
$337$	6905.10	1.11616	0.558078	+	0.829788i	$0.311539\pi$
$338$	12576.0	2.02380
$339$	0	0
$340$	3535.27	0.563903
$341$	34.4007	0.00546306
$342$	0	0
$343$	−185.072	−0.0291340
$344$	−9277.60	−1.45411
$345$	0	0
$346$	−8046.41	−1.25022
$347$	8762.09	1.35554	0.677771	−	0.735273i	$-0.262946\pi$
$347$	8762.09	1.35554	0.677771	+	0.735273i	$0.262946\pi$
$348$	0	0
$349$	−219.998	−0.0337427	−0.0168713	−	0.999858i	$-0.505371\pi$
$349$	−219.998	−0.0337427	−0.0168713	+	0.999858i	$0.505371\pi$
$350$	5293.80	0.808473
$351$	0	0
$352$	44.2574	0.00670150
$353$	10049.2	1.51519	0.757597	−	0.652723i	$-0.226373\pi$
$353$	10049.2	1.51519	0.757597	+	0.652723i	$0.226373\pi$
$354$	0	0
$355$	4306.82	0.643893
$356$	4649.66	0.692224
$357$	0	0
$358$	2242.14	0.331008
$359$	−2360.47	−0.347022	−0.173511	−	0.984832i	$-0.555511\pi$
$359$	−2360.47	−0.347022	−0.173511	+	0.984832i	$0.555511\pi$
$360$	0	0
$361$	68.3947	0.00997153
$362$	7162.43	1.03991
$363$	0	0
$364$	7093.13	1.02138
$365$	8114.47	1.16365
$366$	0	0
$367$	1676.83	0.238500	0.119250	−	0.992864i	$-0.461951\pi$
$367$	1676.83	0.238500	0.119250	+	0.992864i	$0.461951\pi$
$368$	−2249.60	−0.318664
$369$	0	0
$370$	13347.0	1.87534
$371$	12237.9	1.71256
$372$	0	0
$373$	5265.88	0.730984	0.365492	−	0.930815i	$-0.380901\pi$
$373$	5265.88	0.730984	0.365492	+	0.930815i	$0.380901\pi$
$374$	59.1463	0.00817749
$375$	0	0
$376$	3066.76	0.420627
$377$	−21690.0	−2.96311
$378$	0	0
$379$	−98.4939	−0.0133491	−0.00667453	−	0.999978i	$-0.502125\pi$
$379$	−98.4939	−0.0133491	−0.00667453	+	0.999978i	$0.502125\pi$
$380$	3760.23	0.507620
$381$	0	0
$382$	−1972.90	−0.264247
$383$	−671.343	−0.0895666	−0.0447833	−	0.998997i	$-0.514260\pi$
$383$	−671.343	−0.0895666	−0.0447833	+	0.998997i	$0.514260\pi$
$384$	0	0
$385$	−130.517	−0.0172773
$386$	−10677.8	−1.40800
$387$	0	0
$388$	−1936.77	−0.253414
$389$	−411.894	−0.0536860	−0.0268430	−	0.999640i	$-0.508545\pi$
$389$	−411.894	−0.0536860	−0.0268430	+	0.999640i	$0.508545\pi$
$390$	0	0
$391$	5865.92	0.758702
$392$	−8256.11	−1.06377
$393$	0	0
$394$	−2724.23	−0.348336
$395$	−9172.08	−1.16835
$396$	0	0
$397$	13595.3	1.71871	0.859357	−	0.511377i	$-0.170864\pi$
$397$	13595.3	1.71871	0.859357	+	0.511377i	$0.170864\pi$
$398$	−9919.07	−1.24924
$399$	0	0
$400$	−2746.13	−0.343266
$401$	3280.15	0.408487	0.204243	−	0.978920i	$-0.434527\pi$
$401$	3280.15	0.408487	0.204243	+	0.978920i	$0.434527\pi$
$402$	0	0
$403$	−8959.49	−1.10745
$404$	−1617.39	−0.199178
$405$	0	0
$406$	14152.2	1.72996
$407$	−139.081	−0.0169385
$408$	0	0
$409$	−4478.26	−0.541408	−0.270704	−	0.962663i	$-0.587256\pi$
$409$	−4478.26	−0.541408	−0.270704	+	0.962663i	$0.587256\pi$
$410$	12372.6	1.49034
$411$	0	0
$412$	5110.21	0.611072
$413$	5165.15	0.615400
$414$	0	0
$415$	−18979.7	−2.24500
$416$	−11526.6	−1.35851
$417$	0	0
$418$	62.9099	0.00736130
$419$	−12804.4	−1.49293	−0.746465	−	0.665425i	$-0.768250\pi$
$419$	−12804.4	−1.49293	−0.746465	+	0.665425i	$0.768250\pi$
$420$	0	0
$421$	−1711.02	−0.198076	−0.0990380	−	0.995084i	$-0.531577\pi$
$421$	−1711.02	−0.198076	−0.0990380	+	0.995084i	$0.531577\pi$
$422$	−3175.08	−0.366257
$423$	0	0
$424$	−11544.5	−1.32229
$425$	7160.65	0.817276
$426$	0	0
$427$	−17812.1	−2.01871
$428$	1621.73	0.183153
$429$	0	0
$430$	−12331.3	−1.38295
$431$	−9318.75	−1.04146	−0.520729	−	0.853722i	$-0.674340\pi$
$431$	−9318.75	−1.04146	−0.520729	+	0.853722i	$0.674340\pi$
$432$	0	0
$433$	1512.84	0.167905	0.0839523	−	0.996470i	$-0.473246\pi$
$433$	1512.84	0.167905	0.0839523	+	0.996470i	$0.473246\pi$
$434$	5845.86	0.646568
$435$	0	0
$436$	−213.241	−0.0234230
$437$	6239.19	0.682977
$438$	0	0
$439$	11474.1	1.24745	0.623724	−	0.781645i	$-0.285619\pi$
$439$	11474.1	1.24745	0.623724	+	0.781645i	$0.285619\pi$
$440$	123.121	0.0133400
$441$	0	0
$442$	−15404.3	−1.65771
$443$	4896.61	0.525158	0.262579	−	0.964910i	$-0.415427\pi$
$443$	4896.61	0.525158	0.262579	+	0.964910i	$0.415427\pi$
$444$	0	0
$445$	22282.6	2.37370
$446$	−1087.27	−0.115434
$447$	0	0
$448$	13776.2	1.45283
$449$	−9644.39	−1.01369	−0.506845	−	0.862037i	$-0.669188\pi$
$449$	−9644.39	−1.01369	−0.506845	+	0.862037i	$0.669188\pi$
$450$	0	0
$451$	−128.927	−0.0134611
$452$	4459.39	0.464054
$453$	0	0
$454$	4195.07	0.433666
$455$	33992.4	3.50239
$456$	0	0
$457$	7396.33	0.757081	0.378540	−	0.925585i	$-0.376426\pi$
$457$	7396.33	0.757081	0.378540	+	0.925585i	$0.376426\pi$
$458$	−9143.39	−0.932844
$459$	0	0
$460$	3386.67	0.343270
$461$	12630.3	1.27604	0.638019	−	0.770021i	$-0.279754\pi$
$461$	12630.3	1.27604	0.638019	+	0.770021i	$0.279754\pi$
$462$	0	0
$463$	2181.21	0.218940	0.109470	−	0.993990i	$-0.465085\pi$
$463$	2181.21	0.218940	0.109470	+	0.993990i	$0.465085\pi$
$464$	−7341.38	−0.734515
$465$	0	0
$466$	14443.3	1.43578
$467$	13303.6	1.31824	0.659119	−	0.752039i	$-0.270929\pi$
$467$	13303.6	1.31824	0.659119	+	0.752039i	$0.270929\pi$
$468$	0	0
$469$	−20496.7	−2.01801
$470$	4076.18	0.400043
$471$	0	0
$472$	−4872.48	−0.475157
$473$	128.497	0.0124911
$474$	0	0
$475$	7616.30	0.735705
$476$	−6260.19	−0.602805
$477$	0	0
$478$	−530.646	−0.0507765
$479$	9332.30	0.890196	0.445098	−	0.895482i	$-0.353169\pi$
$479$	9332.30	0.890196	0.445098	+	0.895482i	$0.353169\pi$
$480$	0	0
$481$	36222.8	3.43372
$482$	−12998.2	−1.22832
$483$	0	0
$484$	4086.32	0.383764
$485$	−9281.58	−0.868979
$486$	0	0
$487$	−375.904	−0.0349771	−0.0174885	−	0.999847i	$-0.505567\pi$
$487$	−375.904	−0.0349771	−0.0174885	+	0.999847i	$0.505567\pi$
$488$	16802.8	1.55866
$489$	0	0
$490$	−10973.6	−1.01171
$491$	6952.36	0.639014	0.319507	−	0.947584i	$-0.396483\pi$
$491$	6952.36	0.639014	0.319507	+	0.947584i	$0.396483\pi$
$492$	0	0
$493$	19143.0	1.74880
$494$	−16384.6	−1.49226
$495$	0	0
$496$	−3032.50	−0.274523
$497$	−7626.43	−0.688315
$498$	0	0
$499$	12371.8	1.10989	0.554946	−	0.831886i	$-0.312739\pi$
$499$	12371.8	1.10989	0.554946	+	0.831886i	$0.312739\pi$
$500$	−1513.11	−0.135337
$501$	0	0
$502$	−5046.33	−0.448663
$503$	9609.65	0.851835	0.425917	−	0.904762i	$-0.359951\pi$
$503$	9609.65	0.851835	0.425917	+	0.904762i	$0.359951\pi$
$504$	0	0
$505$	−7751.01	−0.683001
$506$	56.6601	0.00497796
$507$	0	0
$508$	−3722.92	−0.325153
$509$	6067.93	0.528402	0.264201	−	0.964468i	$-0.414892\pi$
$509$	6067.93	0.528402	0.264201	+	0.964468i	$0.414892\pi$
$510$	0	0
$511$	−14369.0	−1.24392
$512$	−9802.33	−0.846105
$513$	0	0
$514$	−3621.67	−0.310788
$515$	24489.7	2.09542
$516$	0	0
$517$	−42.4754	−0.00361328
$518$	−23634.5	−2.00471
$519$	0	0
$520$	−32066.3	−2.70424
$521$	3574.38	0.300569	0.150284	−	0.988643i	$-0.451981\pi$
$521$	3574.38	0.300569	0.150284	+	0.988643i	$0.451981\pi$
$522$	0	0
$523$	−17945.9	−1.50042	−0.750210	−	0.661199i	$-0.770048\pi$
$523$	−17945.9	−1.50042	−0.750210	+	0.661199i	$0.770048\pi$
$524$	−3455.95	−0.288118
$525$	0	0
$526$	−8482.45	−0.703141
$527$	7907.39	0.653608
$528$	0	0
$529$	−6547.64	−0.538148
$530$	−15344.4	−1.25758
$531$	0	0
$532$	−6658.55	−0.542640
$533$	33578.4	2.72878
$534$	0	0
$535$	7771.83	0.628048
$536$	19335.3	1.55813
$537$	0	0
$538$	−2921.78	−0.234139
$539$	114.349	0.00913799
$540$	0	0
$541$	1120.93	0.0890808	0.0445404	−	0.999008i	$-0.485818\pi$
$541$	1120.93	0.0890808	0.0445404	+	0.999008i	$0.485818\pi$
$542$	−6869.27	−0.544392
$543$	0	0
$544$	10173.1	0.801776
$545$	−1021.92	−0.0803195
$546$	0	0
$547$	1707.21	0.133446	0.0667230	−	0.997772i	$-0.478746\pi$
$547$	1707.21	0.133446	0.0667230	+	0.997772i	$0.478746\pi$
$548$	4498.11	0.350638
$549$	0	0
$550$	69.1661	0.00536228
$551$	20361.1	1.57425
$552$	0	0
$553$	16241.8	1.24895
$554$	13467.9	1.03285
$555$	0	0
$556$	−208.228	−0.0158828
$557$	11043.7	0.840099	0.420050	−	0.907501i	$-0.362013\pi$
$557$	11043.7	0.840099	0.420050	+	0.907501i	$0.362013\pi$
$558$	0	0
$559$	−33466.4	−2.53216
$560$	11505.4	0.868197
$561$	0	0
$562$	−8087.23	−0.607010
$563$	−22213.2	−1.66283	−0.831415	−	0.555653i	$-0.812468\pi$
$563$	−22213.2	−1.66283	−0.831415	+	0.555653i	$0.812468\pi$
$564$	0	0
$565$	21370.8	1.59128
$566$	11744.8	0.872208
$567$	0	0
$568$	7194.30	0.531455
$569$	−22338.1	−1.64580	−0.822900	−	0.568186i	$-0.807645\pi$
$569$	−22338.1	−1.64580	−0.822900	+	0.568186i	$0.807645\pi$
$570$	0	0
$571$	−15993.3	−1.17215	−0.586077	−	0.810255i	$-0.699328\pi$
$571$	−15993.3	−1.17215	−0.586077	+	0.810255i	$0.699328\pi$
$572$	92.6752	0.00677438
$573$	0	0
$574$	−21909.1	−1.59315
$575$	6859.66	0.497509
$576$	0	0
$577$	−15608.3	−1.12614	−0.563069	−	0.826410i	$-0.690380\pi$
$577$	−15608.3	−1.12614	−0.563069	+	0.826410i	$0.690380\pi$
$578$	2687.21	0.193380
$579$	0	0
$580$	11052.1	0.791232
$581$	33608.9	2.39988
$582$	0	0
$583$	159.894	0.0113588
$584$	13554.8	0.960447
$585$	0	0
$586$	−10649.4	−0.750720
$587$	−22697.6	−1.59596	−0.797982	−	0.602682i	$-0.794099\pi$
$587$	−22697.6	−1.59596	−0.797982	+	0.602682i	$0.794099\pi$
$588$	0	0
$589$	8410.56	0.588372
$590$	−6476.26	−0.451904
$591$	0	0
$592$	12260.3	0.851173
$593$	8691.72	0.601899	0.300950	−	0.953640i	$-0.402696\pi$
$593$	8691.72	0.601899	0.300950	+	0.953640i	$0.402696\pi$
$594$	0	0
$595$	−30000.7	−2.06708
$596$	−7227.42	−0.496723
$597$	0	0
$598$	−14756.8	−1.00912
$599$	−17220.9	−1.17467	−0.587335	−	0.809344i	$-0.699823\pi$
$599$	−17220.9	−1.17467	−0.587335	+	0.809344i	$0.699823\pi$
$600$	0	0
$601$	5812.47	0.394502	0.197251	−	0.980353i	$-0.436799\pi$
$601$	5812.47	0.394502	0.197251	+	0.980353i	$0.436799\pi$
$602$	21836.1	1.47836
$603$	0	0
$604$	2134.16	0.143771
$605$	19582.9	1.31596
$606$	0	0
$607$	1047.92	0.0700720	0.0350360	−	0.999386i	$-0.488845\pi$
$607$	1047.92	0.0700720	0.0350360	+	0.999386i	$0.488845\pi$
$608$	10820.4	0.721752
$609$	0	0
$610$	22333.5	1.48239
$611$	11062.5	0.732473
$612$	0	0
$613$	5108.11	0.336565	0.168283	−	0.985739i	$-0.446178\pi$
$613$	5108.11	0.336565	0.168283	+	0.985739i	$0.446178\pi$
$614$	−15766.5	−1.03629
$615$	0	0
$616$	−218.021	−0.0142603
$617$	24146.1	1.57550	0.787751	−	0.615994i	$-0.211246\pi$
$617$	24146.1	1.57550	0.787751	+	0.615994i	$0.211246\pi$
$618$	0	0
$619$	−2044.51	−0.132756	−0.0663778	−	0.997795i	$-0.521144\pi$
$619$	−2044.51	−0.132756	−0.0663778	+	0.997795i	$0.521144\pi$
$620$	4565.30	0.295721
$621$	0	0
$622$	−9712.08	−0.626076
$623$	−39457.6	−2.53746
$624$	0	0
$625$	−18689.8	−1.19615
$626$	−23283.5	−1.48658
$627$	0	0
$628$	6932.71	0.440518
$629$	−31969.2	−2.02654
$630$	0	0
$631$	−7840.35	−0.494642	−0.247321	−	0.968934i	$-0.579550\pi$
$631$	−7840.35	−0.494642	−0.247321	+	0.968934i	$0.579550\pi$
$632$	−15321.5	−0.964328
$633$	0	0
$634$	1985.26	0.124361
$635$	−17841.4	−1.11498
$636$	0	0
$637$	−29781.7	−1.85242
$638$	184.906	0.0114741
$639$	0	0
$640$	−1969.85	−0.121664
$641$	2419.72	0.149100	0.0745500	−	0.997217i	$-0.476248\pi$
$641$	2419.72	0.149100	0.0745500	+	0.997217i	$0.476248\pi$
$642$	0	0
$643$	12466.3	0.764574	0.382287	−	0.924044i	$-0.375137\pi$
$643$	12466.3	0.764574	0.382287	+	0.924044i	$0.375137\pi$
$644$	−5997.05	−0.366952
$645$	0	0
$646$	14460.5	0.880716
$647$	−5047.48	−0.306703	−0.153352	−	0.988172i	$-0.549007\pi$
$647$	−5047.48	−0.306703	−0.153352	+	0.988172i	$0.549007\pi$
$648$	0	0
$649$	67.4852	0.00408170
$650$	−18014.0	−1.08702
$651$	0	0
$652$	5913.31	0.355189
$653$	14985.3	0.898042	0.449021	−	0.893521i	$-0.351773\pi$
$653$	14985.3	0.898042	0.449021	+	0.893521i	$0.351773\pi$
$654$	0	0
$655$	−16562.0	−0.987985
$656$	11365.2	0.676429
$657$	0	0
$658$	−7218.03	−0.427642
$659$	−28076.3	−1.65963	−0.829817	−	0.558036i	$-0.811555\pi$
$659$	−28076.3	−1.65963	−0.829817	+	0.558036i	$0.811555\pi$
$660$	0	0
$661$	−2369.40	−0.139424	−0.0697118	−	0.997567i	$-0.522208\pi$
$661$	−2369.40	−0.139424	−0.0697118	+	0.997567i	$0.522208\pi$
$662$	−15135.1	−0.888584
$663$	0	0
$664$	−31704.5	−1.85297
$665$	−31909.8	−1.86076
$666$	0	0
$667$	18338.3	1.06456
$668$	2111.87	0.122322
$669$	0	0
$670$	25699.5	1.48188
$671$	−232.724	−0.0133893
$672$	0	0
$673$	18820.2	1.07796	0.538978	−	0.842320i	$-0.318811\pi$
$673$	18820.2	1.07796	0.538978	+	0.842320i	$0.318811\pi$
$674$	15331.2	0.876166
$675$	0	0
$676$	−17391.1	−0.989482
$677$	−570.476	−0.0323858	−0.0161929	−	0.999869i	$-0.505155\pi$
$677$	−570.476	−0.0323858	−0.0161929	+	0.999869i	$0.505155\pi$
$678$	0	0
$679$	16435.6	0.928928
$680$	28300.8	1.59601
$681$	0	0
$682$	76.3790	0.00428842
$683$	−4958.43	−0.277788	−0.138894	−	0.990307i	$-0.544355\pi$
$683$	−4958.43	−0.277788	−0.138894	+	0.990307i	$0.544355\pi$
$684$	0	0
$685$	21556.3	1.20237
$686$	−410.911	−0.0228697
$687$	0	0
$688$	−11327.3	−0.627690
$689$	−41643.7	−2.30261
$690$	0	0
$691$	23201.8	1.27734	0.638668	−	0.769483i	$-0.279486\pi$
$691$	23201.8	1.27734	0.638668	+	0.769483i	$0.279486\pi$
$692$	11127.3	0.611264
$693$	0	0
$694$	19454.2	1.06408
$695$	−997.893	−0.0544636
$696$	0	0
$697$	−29635.3	−1.61050
$698$	−488.455	−0.0264875
$699$	0	0
$700$	−7320.72	−0.395282
$701$	3653.17	0.196831	0.0984155	−	0.995145i	$-0.468623\pi$
$701$	3653.17	0.196831	0.0984155	+	0.995145i	$0.468623\pi$
$702$	0	0
$703$	−34003.5	−1.82428
$704$	179.993	0.00963601
$705$	0	0
$706$	22311.9	1.18941
$707$	13725.3	0.730120
$708$	0	0
$709$	−15973.9	−0.846139	−0.423070	−	0.906097i	$-0.639047\pi$
$709$	−15973.9	−0.846139	−0.423070	+	0.906097i	$0.639047\pi$
$710$	9562.32	0.505447
$711$	0	0
$712$	37221.9	1.95920
$713$	7575.01	0.397877
$714$	0	0
$715$	444.128	0.0232300
$716$	−3100.62	−0.161837
$717$	0	0
$718$	−5240.90	−0.272407
$719$	26074.2	1.35244	0.676218	−	0.736701i	$-0.263618\pi$
$719$	26074.2	1.35244	0.676218	+	0.736701i	$0.263618\pi$
$720$	0	0
$721$	−43365.8	−2.23998
$722$	151.855	0.00782751
$723$	0	0
$724$	−9904.82	−0.508439
$725$	22385.9	1.14675
$726$	0	0
$727$	−10573.4	−0.539405	−0.269702	−	0.962944i	$-0.586925\pi$
$727$	−10573.4	−0.539405	−0.269702	+	0.962944i	$0.586925\pi$
$728$	56782.5	2.89080
$729$	0	0
$730$	18016.4	0.913445
$731$	29536.5	1.49446
$732$	0	0
$733$	13075.1	0.658856	0.329428	−	0.944181i	$-0.393144\pi$
$733$	13075.1	0.658856	0.329428	+	0.944181i	$0.393144\pi$
$734$	3723.01	0.187219
$735$	0	0
$736$	9745.45	0.488073
$737$	−267.799	−0.0133847
$738$	0	0
$739$	20558.6	1.02336	0.511679	−	0.859177i	$-0.329024\pi$
$739$	20558.6	1.02336	0.511679	+	0.859177i	$0.329024\pi$
$740$	−18457.3	−0.916897
$741$	0	0
$742$	27171.5	1.34434
$743$	−31508.1	−1.55575	−0.777874	−	0.628420i	$-0.783702\pi$
$743$	−31508.1	−1.55575	−0.777874	+	0.628420i	$0.783702\pi$
$744$	0	0
$745$	−34636.0	−1.70331
$746$	11691.7	0.573812
$747$	0	0
$748$	−81.7924	−0.00399817
$749$	−13762.2	−0.671376
$750$	0	0
$751$	29028.2	1.41046	0.705229	−	0.708979i	$-0.250844\pi$
$751$	29028.2	1.41046	0.705229	+	0.708979i	$0.250844\pi$
$752$	3744.31	0.181570
$753$	0	0
$754$	−48157.7	−2.32600
$755$	10227.5	0.493004
$756$	0	0
$757$	13319.1	0.639485	0.319742	−	0.947505i	$-0.396404\pi$
$757$	13319.1	0.639485	0.319742	+	0.947505i	$0.396404\pi$
$758$	−218.683	−0.0104788
$759$	0	0
$760$	30101.7	1.43672
$761$	3668.86	0.174765	0.0873823	−	0.996175i	$-0.472150\pi$
$761$	3668.86	0.174765	0.0873823	+	0.996175i	$0.472150\pi$
$762$	0	0
$763$	1809.59	0.0858606
$764$	2728.29	0.129197
$765$	0	0
$766$	−1490.57	−0.0703085
$767$	−17576.2	−0.827430
$768$	0	0
$769$	700.614	0.0328541	0.0164270	−	0.999865i	$-0.494771\pi$
$769$	700.614	0.0328541	0.0164270	+	0.999865i	$0.494771\pi$
$770$	−289.783	−0.0135624
$771$	0	0
$772$	14766.2	0.688402
$773$	−15789.8	−0.734694	−0.367347	−	0.930084i	$-0.619734\pi$
$773$	−15789.8	−0.734694	−0.367347	+	0.930084i	$0.619734\pi$
$774$	0	0
$775$	9246.96	0.428594
$776$	−15504.4	−0.717235
$777$	0	0
$778$	−914.518	−0.0421428
$779$	−31521.1	−1.44976
$780$	0	0
$781$	−99.6431	−0.00456531
$782$	13024.0	0.595570
$783$	0	0
$784$	−10080.2	−0.459191
$785$	33223.7	1.51058
$786$	0	0
$787$	−38813.7	−1.75802	−0.879009	−	0.476805i	$-0.841795\pi$
$787$	−38813.7	−1.75802	−0.879009	+	0.476805i	$0.841795\pi$
$788$	3767.29	0.170310
$789$	0	0
$790$	−20364.5	−0.917136
$791$	−37843.0	−1.70106
$792$	0	0
$793$	60611.7	2.71423
$794$	30185.3	1.34916
$795$	0	0
$796$	13716.9	0.610783
$797$	−20453.0	−0.909012	−0.454506	−	0.890744i	$-0.650184\pi$
$797$	−20453.0	−0.909012	−0.454506	+	0.890744i	$0.650184\pi$
$798$	0	0
$799$	−9763.45	−0.432298
$800$	11896.5	0.525754
$801$	0	0
$802$	7282.84	0.320656
$803$	−187.737	−0.00825045
$804$	0	0
$805$	−28739.7	−1.25831
$806$	−19892.5	−0.869335
$807$	0	0
$808$	−12947.6	−0.563733
$809$	−16290.3	−0.707955	−0.353978	−	0.935254i	$-0.615171\pi$
$809$	−16290.3	−0.707955	−0.353978	+	0.935254i	$0.615171\pi$
$810$	0	0
$811$	−30790.8	−1.33318	−0.666591	−	0.745424i	$-0.732247\pi$
$811$	−30790.8	−1.33318	−0.666591	+	0.745424i	$0.732247\pi$
$812$	−19570.9	−0.845818
$813$	0	0
$814$	−308.797	−0.0132965
$815$	28338.4	1.21798
$816$	0	0
$817$	31416.0	1.34530
$818$	−9942.97	−0.424997
$819$	0	0
$820$	−17109.8	−0.728660
$821$	3949.81	0.167904	0.0839521	−	0.996470i	$-0.473246\pi$
$821$	3949.81	0.167904	0.0839521	+	0.996470i	$0.473246\pi$
$822$	0	0
$823$	−28712.6	−1.21611	−0.608055	−	0.793895i	$-0.708050\pi$
$823$	−28712.6	−1.21611	−0.608055	+	0.793895i	$0.708050\pi$
$824$	40908.6	1.72951
$825$	0	0
$826$	11468.0	0.483080
$827$	−15218.1	−0.639885	−0.319943	−	0.947437i	$-0.603664\pi$
$827$	−15218.1	−0.639885	−0.319943	+	0.947437i	$0.603664\pi$
$828$	0	0
$829$	34435.8	1.44271	0.721354	−	0.692567i	$-0.243520\pi$
$829$	34435.8	1.44271	0.721354	+	0.692567i	$0.243520\pi$
$830$	−42140.1	−1.76230
$831$	0	0
$832$	−46878.3	−1.95338
$833$	26284.5	1.09328
$834$	0	0
$835$	10120.7	0.419453
$836$	−86.9972	−0.00359912
$837$	0	0
$838$	−28429.3	−1.17193
$839$	−16733.4	−0.688561	−0.344280	−	0.938867i	$-0.611877\pi$
$839$	−16733.4	−0.688561	−0.344280	+	0.938867i	$0.611877\pi$
$840$	0	0
$841$	35456.7	1.45380
$842$	−3798.93	−0.155487
$843$	0	0
$844$	4390.76	0.179071
$845$	−83343.6	−3.39303
$846$	0	0
$847$	−34677.1	−1.40675
$848$	−14095.1	−0.570786
$849$	0	0
$850$	15898.6	0.641550
$851$	−30625.4	−1.23364
$852$	0	0
$853$	29427.8	1.18123	0.590615	−	0.806953i	$-0.298885\pi$
$853$	29427.8	1.18123	0.590615	+	0.806953i	$0.298885\pi$
$854$	−39547.7	−1.58465
$855$	0	0
$856$	12982.4	0.518376
$857$	30020.8	1.19661	0.598304	−	0.801269i	$-0.295842\pi$
$857$	30020.8	1.19661	0.598304	+	0.801269i	$0.295842\pi$
$858$	0	0
$859$	31857.6	1.26539	0.632693	−	0.774403i	$-0.281949\pi$
$859$	31857.6	1.26539	0.632693	+	0.774403i	$0.281949\pi$
$860$	17052.8	0.676158
$861$	0	0
$862$	−20690.2	−0.817529
$863$	39561.9	1.56049	0.780246	−	0.625473i	$-0.215094\pi$
$863$	39561.9	1.56049	0.780246	+	0.625473i	$0.215094\pi$
$864$	0	0
$865$	53325.2	2.09608
$866$	3358.93	0.131803
$867$	0	0
$868$	−8084.15	−0.316122
$869$	212.206	0.00828379
$870$	0	0
$871$	69746.8	2.71329
$872$	−1707.06	−0.0662939
$873$	0	0
$874$	13852.7	0.536127
$875$	12840.4	0.496098
$876$	0	0
$877$	−19557.8	−0.753044	−0.376522	−	0.926408i	$-0.622880\pi$
$877$	−19557.8	−0.753044	−0.376522	+	0.926408i	$0.622880\pi$
$878$	25475.7	0.979229
$879$	0	0
$880$	150.323	0.00575840
$881$	34019.0	1.30094	0.650470	−	0.759532i	$-0.274572\pi$
$881$	34019.0	1.30094	0.650470	+	0.759532i	$0.274572\pi$
$882$	0	0
$883$	−15049.5	−0.573565	−0.286782	−	0.957996i	$-0.592586\pi$
$883$	−15049.5	−0.573565	−0.286782	+	0.957996i	$0.592586\pi$
$884$	21302.4	0.810495
$885$	0	0
$886$	10871.8	0.412242
$887$	−17648.9	−0.668086	−0.334043	−	0.942558i	$-0.608413\pi$
$887$	−17648.9	−0.668086	−0.334043	+	0.942558i	$0.608413\pi$
$888$	0	0
$889$	31593.2	1.19190
$890$	49473.5	1.86332
$891$	0	0
$892$	1503.57	0.0564386
$893$	−10384.7	−0.389151
$894$	0	0
$895$	−14859.1	−0.554956
$896$	3488.18	0.130058
$897$	0	0
$898$	−21413.2	−0.795732
$899$	24720.4	0.917100
$900$	0	0
$901$	36753.5	1.35898
$902$	−286.253	−0.0105667
$903$	0	0
$904$	35698.7	1.31341
$905$	−47466.9	−1.74348
$906$	0	0
$907$	−32871.7	−1.20340	−0.601701	−	0.798721i	$-0.705510\pi$
$907$	−32871.7	−1.20340	−0.601701	+	0.798721i	$0.705510\pi$
$908$	−5801.30	−0.212029
$909$	0	0
$910$	75472.5	2.74933
$911$	−44747.8	−1.62740	−0.813699	−	0.581287i	$-0.802550\pi$
$911$	−44747.8	−1.62740	−0.813699	+	0.581287i	$0.802550\pi$
$912$	0	0
$913$	439.117	0.0159175
$914$	16421.9	0.594297
$915$	0	0
$916$	12644.3	0.456089
$917$	29327.6	1.05614
$918$	0	0
$919$	−22345.7	−0.802086	−0.401043	−	0.916059i	$-0.631352\pi$
$919$	−22345.7	−0.802086	−0.401043	+	0.916059i	$0.631352\pi$
$920$	27111.3	0.971556
$921$	0	0
$922$	28042.8	1.00167
$923$	25951.5	0.925465
$924$	0	0
$925$	−37385.0	−1.32888
$926$	4842.88	0.171865
$927$	0	0
$928$	31803.5	1.12500
$929$	−28504.2	−1.00667	−0.503333	−	0.864093i	$-0.667893\pi$
$929$	−28504.2	−1.00667	−0.503333	+	0.864093i	$0.667893\pi$
$930$	0	0
$931$	27957.0	0.984162
$932$	−19973.4	−0.701985
$933$	0	0
$934$	29537.6	1.03480
$935$	−391.974	−0.0137101
$936$	0	0
$937$	−3736.11	−0.130260	−0.0651298	−	0.997877i	$-0.520746\pi$
$937$	−3736.11	−0.130260	−0.0651298	+	0.997877i	$0.520746\pi$
$938$	−45508.2	−1.58411
$939$	0	0
$940$	−5636.89	−0.195591
$941$	6143.66	0.212835	0.106417	−	0.994322i	$-0.466062\pi$
$941$	6143.66	0.212835	0.106417	+	0.994322i	$0.466062\pi$
$942$	0	0
$943$	−28389.6	−0.980375
$944$	−5948.98	−0.205109
$945$	0	0
$946$	285.299	0.00980537
$947$	−7558.34	−0.259359	−0.129680	−	0.991556i	$-0.541395\pi$
$947$	−7558.34	−0.259359	−0.129680	+	0.991556i	$0.541395\pi$
$948$	0	0
$949$	48895.2	1.67250
$950$	16910.3	0.577518
$951$	0	0
$952$	−50114.6	−1.70612
$953$	2318.18	0.0787967	0.0393984	−	0.999224i	$-0.487456\pi$
$953$	2318.18	0.0787967	0.0393984	+	0.999224i	$0.487456\pi$
$954$	0	0
$955$	13074.8	0.443027
$956$	733.821	0.0248258
$957$	0	0
$958$	20720.3	0.698791
$959$	−38171.5	−1.28532
$960$	0	0
$961$	−19579.7	−0.657236
$962$	80424.5	2.69542
$963$	0	0
$964$	17975.0	0.600554
$965$	70764.0	2.36060
$966$	0	0
$967$	24054.9	0.799952	0.399976	−	0.916526i	$-0.369018\pi$
$967$	24054.9	0.799952	0.399976	+	0.916526i	$0.369018\pi$
$968$	32712.2	1.08617
$969$	0	0
$970$	−20607.7	−0.682136
$971$	14369.3	0.474905	0.237453	−	0.971399i	$-0.423688\pi$
$971$	14369.3	0.474905	0.237453	+	0.971399i	$0.423688\pi$
$972$	0	0
$973$	1767.05	0.0582210
$974$	−834.610	−0.0274565
$975$	0	0
$976$	20515.1	0.672821
$977$	4622.39	0.151365	0.0756824	−	0.997132i	$-0.475886\pi$
$977$	4622.39	0.151365	0.0756824	+	0.997132i	$0.475886\pi$
$978$	0	0
$979$	−515.533	−0.0168299
$980$	15175.2	0.494648
$981$	0	0
$982$	15436.2	0.501616
$983$	−18957.8	−0.615116	−0.307558	−	0.951529i	$-0.599512\pi$
$983$	−18957.8	−0.615116	−0.307558	+	0.951529i	$0.599512\pi$
$984$	0	0
$985$	18054.0	0.584008
$986$	42502.6	1.37278
$987$	0	0
$988$	22658.0	0.729601
$989$	28295.0	0.909736
$990$	0	0
$991$	−19224.0	−0.616218	−0.308109	−	0.951351i	$-0.599696\pi$
$991$	−19224.0	−0.616218	−0.308109	+	0.951351i	$0.599696\pi$
$992$	13137.1	0.420466
$993$	0	0
$994$	−16932.8	−0.540317
$995$	65735.7	2.09443
$996$	0	0
$997$	−34507.6	−1.09616	−0.548078	−	0.836427i	$-0.684640\pi$
$997$	−34507.6	−1.09616	−0.548078	+	0.836427i	$0.684640\pi$
$998$	27468.7	0.871249
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.19		28
3.2	odd	2		717.4.a.b.1.10	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.10	✓	28	3.2	odd	2
2151.4.a.b.1.19		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)

\(f(q)\)	\(=\)	\(q+2.22027 q^{2} -3.07038 q^{4} -14.7142 q^{5} +26.0557 q^{7} -24.5793 q^{8} +O(q^{10})\)	\(q+2.22027 q^{2} -3.07038 q^{4} -14.7142 q^{5} +26.0557 q^{7} -24.5793 q^{8} -32.6696 q^{10} +0.340430 q^{11} -88.6632 q^{13} +57.8507 q^{14} -30.0097 q^{16} +78.2516 q^{17} +83.2310 q^{19} +45.1783 q^{20} +0.755848 q^{22} +74.9624 q^{23} +91.5080 q^{25} -196.857 q^{26} -80.0008 q^{28} +244.634 q^{29} +101.051 q^{31} +130.005 q^{32} +173.740 q^{34} -383.388 q^{35} -408.544 q^{37} +184.796 q^{38} +361.665 q^{40} -378.718 q^{41} +377.456 q^{43} -1.04525 q^{44} +166.437 q^{46} -124.770 q^{47} +335.897 q^{49} +203.173 q^{50} +272.230 q^{52} +469.684 q^{53} -5.00916 q^{55} -640.429 q^{56} +543.154 q^{58} +198.235 q^{59} -683.617 q^{61} +224.361 q^{62} +528.723 q^{64} +1304.61 q^{65} -786.649 q^{67} -240.262 q^{68} -851.227 q^{70} -292.698 q^{71} -551.472 q^{73} -907.079 q^{74} -255.551 q^{76} +8.87012 q^{77} +623.349 q^{79} +441.569 q^{80} -840.858 q^{82} +1289.89 q^{83} -1151.41 q^{85} +838.056 q^{86} -8.36752 q^{88} -1514.36 q^{89} -2310.18 q^{91} -230.163 q^{92} -277.024 q^{94} -1224.68 q^{95} +630.790 q^{97} +745.784 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 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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