LMFDB - Embedded newform 2151.4.a.b.1.17

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			1.17
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)

$f(q)$	$=$	$q+1.15681 q^{2} -6.66179 q^{4} +10.4070 q^{5} +11.9554 q^{7} -16.9609 q^{8} +O(q^{10})$	\(q+1.15681 q^{2} -6.66179 q^{4} +10.4070 q^{5} +11.9554 q^{7} -16.9609 q^{8} +12.0389 q^{10} +54.2452 q^{11} -17.3276 q^{13} +13.8302 q^{14} +33.6737 q^{16} -47.9434 q^{17} -15.2259 q^{19} -69.3293 q^{20} +62.7514 q^{22} -143.707 q^{23} -16.6941 q^{25} -20.0448 q^{26} -79.6445 q^{28} -227.291 q^{29} -21.4398 q^{31} +174.641 q^{32} -55.4615 q^{34} +124.420 q^{35} +187.668 q^{37} -17.6134 q^{38} -176.512 q^{40} -195.016 q^{41} +265.167 q^{43} -361.370 q^{44} -166.242 q^{46} -571.433 q^{47} -200.068 q^{49} -19.3119 q^{50} +115.433 q^{52} +144.193 q^{53} +564.530 q^{55} -202.775 q^{56} -262.933 q^{58} +836.790 q^{59} -328.149 q^{61} -24.8018 q^{62} -67.3629 q^{64} -180.329 q^{65} -899.977 q^{67} +319.389 q^{68} +143.931 q^{70} -118.994 q^{71} +310.345 q^{73} +217.096 q^{74} +101.431 q^{76} +648.524 q^{77} -529.952 q^{79} +350.443 q^{80} -225.596 q^{82} +618.535 q^{83} -498.948 q^{85} +306.749 q^{86} -920.048 q^{88} +700.532 q^{89} -207.159 q^{91} +957.346 q^{92} -661.040 q^{94} -158.456 q^{95} +1489.08 q^{97} -231.441 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.15681	0.408994	0.204497	−	0.978867i	$-0.434444\pi$
$2$	1.15681	0.408994	0.204497	+	0.978867i	$0.434444\pi$
$3$	0	0
$3$	0	0
$4$	−6.66179	−0.832724
$5$	10.4070	0.930832	0.465416	−	0.885092i	$-0.345905\pi$
$5$	10.4070	0.930832	0.465416	+	0.885092i	$0.345905\pi$
$6$	0	0
$7$	11.9554	0.645532	0.322766	−	0.946479i	$-0.395387\pi$
$7$	11.9554	0.645532	0.322766	+	0.946479i	$0.395387\pi$
$8$	−16.9609	−0.749574
$9$	0	0
$10$	12.0389	0.380705
$11$	54.2452	1.48687	0.743434	−	0.668810i	$-0.233196\pi$
$11$	54.2452	1.48687	0.743434	+	0.668810i	$0.233196\pi$
$12$	0	0
$13$	−17.3276	−0.369679	−0.184839	−	0.982769i	$-0.559176\pi$
$13$	−17.3276	−0.369679	−0.184839	+	0.982769i	$0.559176\pi$
$14$	13.8302	0.264019
$15$	0	0
$16$	33.6737	0.526152
$17$	−47.9434	−0.683999	−0.342000	−	0.939700i	$-0.611104\pi$
$17$	−47.9434	−0.683999	−0.342000	+	0.939700i	$0.611104\pi$
$18$	0	0
$19$	−15.2259	−0.183845	−0.0919224	−	0.995766i	$-0.529301\pi$
$19$	−15.2259	−0.183845	−0.0919224	+	0.995766i	$0.529301\pi$
$20$	−69.3293	−0.775125
$21$	0	0
$22$	62.7514	0.608120
$23$	−143.707	−1.30283	−0.651413	−	0.758724i	$-0.725823\pi$
$23$	−143.707	−1.30283	−0.651413	+	0.758724i	$0.725823\pi$
$24$	0	0
$25$	−16.6941	−0.133553
$26$	−20.0448	−0.151196
$27$	0	0
$28$	−79.6445	−0.537550
$29$	−227.291	−1.45541	−0.727706	−	0.685889i	$-0.759414\pi$
$29$	−227.291	−1.45541	−0.727706	+	0.685889i	$0.759414\pi$
$30$	0	0
$31$	−21.4398	−0.124216	−0.0621080	−	0.998069i	$-0.519782\pi$
$31$	−21.4398	−0.124216	−0.0621080	+	0.998069i	$0.519782\pi$
$32$	174.641	0.964767
$33$	0	0
$34$	−55.4615	−0.279752
$35$	124.420	0.600882
$36$	0	0
$37$	187.668	0.833850	0.416925	−	0.908941i	$-0.363108\pi$
$37$	187.668	0.833850	0.416925	+	0.908941i	$0.363108\pi$
$38$	−17.6134	−0.0751915
$39$	0	0
$40$	−176.512	−0.697727
$41$	−195.016	−0.742838	−0.371419	−	0.928465i	$-0.621129\pi$
$41$	−195.016	−0.742838	−0.371419	+	0.928465i	$0.621129\pi$
$42$	0	0
$43$	265.167	0.940411	0.470205	−	0.882557i	$-0.344180\pi$
$43$	265.167	0.940411	0.470205	+	0.882557i	$0.344180\pi$
$44$	−361.370	−1.23815
$45$	0	0
$46$	−166.242	−0.532848
$47$	−571.433	−1.77345	−0.886725	−	0.462298i	$-0.847025\pi$
$47$	−571.433	−1.77345	−0.886725	+	0.462298i	$0.847025\pi$
$48$	0	0
$49$	−200.068	−0.583288
$50$	−19.3119	−0.0546222
$51$	0	0
$52$	115.433	0.307840
$53$	144.193	0.373707	0.186854	−	0.982388i	$-0.440171\pi$
$53$	144.193	0.373707	0.186854	+	0.982388i	$0.440171\pi$
$54$	0	0
$55$	564.530	1.38402
$56$	−202.775	−0.483874
$57$	0	0
$58$	−262.933	−0.595255
$59$	836.790	1.84645	0.923227	−	0.384255i	$-0.125541\pi$
$59$	836.790	1.84645	0.923227	+	0.384255i	$0.125541\pi$
$60$	0	0
$61$	−328.149	−0.688773	−0.344387	−	0.938828i	$-0.611913\pi$
$61$	−328.149	−0.688773	−0.344387	+	0.938828i	$0.611913\pi$
$62$	−24.8018	−0.0508037
$63$	0	0
$64$	−67.3629	−0.131568
$65$	−180.329	−0.344109
$66$	0	0
$67$	−899.977	−1.64104	−0.820520	−	0.571618i	$-0.806316\pi$
$67$	−899.977	−1.64104	−0.820520	+	0.571618i	$0.806316\pi$
$68$	319.389	0.569582
$69$	0	0
$70$	143.931	0.245757
$71$	−118.994	−0.198901	−0.0994507	−	0.995042i	$-0.531709\pi$
$71$	−118.994	−0.198901	−0.0994507	+	0.995042i	$0.531709\pi$
$72$	0	0
$73$	310.345	0.497577	0.248788	−	0.968558i	$-0.419968\pi$
$73$	310.345	0.497577	0.248788	+	0.968558i	$0.419968\pi$
$74$	217.096	0.341040
$75$	0	0
$76$	101.431	0.153092
$77$	648.524	0.959821
$78$	0	0
$79$	−529.952	−0.754738	−0.377369	−	0.926063i	$-0.623171\pi$
$79$	−529.952	−0.754738	−0.377369	+	0.926063i	$0.623171\pi$
$80$	350.443	0.489759
$81$	0	0
$82$	−225.596	−0.303817
$83$	618.535	0.817989	0.408994	−	0.912537i	$-0.365880\pi$
$83$	618.535	0.817989	0.408994	+	0.912537i	$0.365880\pi$
$84$	0	0
$85$	−498.948	−0.636688
$86$	306.749	0.384623
$87$	0	0
$88$	−920.048	−1.11452
$89$	700.532	0.834340	0.417170	−	0.908829i	$-0.363022\pi$
$89$	700.532	0.834340	0.417170	+	0.908829i	$0.363022\pi$
$90$	0	0
$91$	−207.159	−0.238640
$92$	957.346	1.08489
$93$	0	0
$94$	−661.040	−0.725331
$95$	−158.456	−0.171129
$96$	0	0
$97$	1489.08	1.55869	0.779345	−	0.626595i	$-0.215552\pi$
$97$	1489.08	1.55869	0.779345	+	0.626595i	$0.215552\pi$
$98$	−231.441	−0.238562
$99$	0	0
$100$	111.212	0.111212
$101$	1241.69	1.22330	0.611649	−	0.791129i	$-0.290507\pi$
$101$	1241.69	1.22330	0.611649	+	0.791129i	$0.290507\pi$
$102$	0	0
$103$	−510.023	−0.487903	−0.243952	−	0.969787i	$-0.578444\pi$
$103$	−510.023	−0.487903	−0.243952	+	0.969787i	$0.578444\pi$
$104$	293.893	0.277101
$105$	0	0
$106$	166.804	0.152844
$107$	−348.494	−0.314862	−0.157431	−	0.987530i	$-0.550321\pi$
$107$	−348.494	−0.314862	−0.157431	+	0.987530i	$0.550321\pi$
$108$	0	0
$109$	−1592.83	−1.39969	−0.699843	−	0.714296i	$-0.746747\pi$
$109$	−1592.83	−1.39969	−0.699843	+	0.714296i	$0.746747\pi$
$110$	653.055	0.566058
$111$	0	0
$112$	402.584	0.339648
$113$	−1181.78	−0.983832	−0.491916	−	0.870643i	$-0.663703\pi$
$113$	−1181.78	−0.983832	−0.491916	+	0.870643i	$0.663703\pi$
$114$	0	0
$115$	−1495.56	−1.21271
$116$	1514.17	1.21196
$117$	0	0
$118$	968.008	0.755189
$119$	−573.184	−0.441544
$120$	0	0
$121$	1611.54	1.21077
$122$	−379.606	−0.281704
$123$	0	0
$124$	142.827	0.103438
$125$	−1474.61	−1.05515
$126$	0	0
$127$	−1463.50	−1.02256	−0.511279	−	0.859415i	$-0.670828\pi$
$127$	−1463.50	−1.02256	−0.511279	+	0.859415i	$0.670828\pi$
$128$	−1475.06	−1.01858
$129$	0	0
$130$	−208.607	−0.140738
$131$	−1588.20	−1.05925	−0.529624	−	0.848233i	$-0.677667\pi$
$131$	−1588.20	−1.05925	−0.529624	+	0.848233i	$0.677667\pi$
$132$	0	0
$133$	−182.032	−0.118678
$134$	−1041.10	−0.671176
$135$	0	0
$136$	813.164	0.512708
$137$	−1260.03	−0.785781	−0.392890	−	0.919585i	$-0.628525\pi$
$137$	−1260.03	−0.785781	−0.392890	+	0.919585i	$0.628525\pi$
$138$	0	0
$139$	1072.54	0.654469	0.327235	−	0.944943i	$-0.393883\pi$
$139$	1072.54	0.654469	0.327235	+	0.944943i	$0.393883\pi$
$140$	−828.862	−0.500368
$141$	0	0
$142$	−137.654	−0.0813496
$143$	−939.941	−0.549663
$144$	0	0
$145$	−2365.43	−1.35474
$146$	359.010	0.203506
$147$	0	0
$148$	−1250.21	−0.694367
$149$	2591.24	1.42471	0.712357	−	0.701817i	$-0.247628\pi$
$149$	2591.24	1.42471	0.712357	+	0.701817i	$0.247628\pi$
$150$	0	0
$151$	−893.707	−0.481648	−0.240824	−	0.970569i	$-0.577418\pi$
$151$	−893.707	−0.481648	−0.240824	+	0.970569i	$0.577418\pi$
$152$	258.245	0.137805
$153$	0	0
$154$	750.220	0.392561
$155$	−223.124	−0.115624
$156$	0	0
$157$	−103.757	−0.0527434	−0.0263717	−	0.999652i	$-0.508395\pi$
$157$	−103.757	−0.0527434	−0.0263717	+	0.999652i	$0.508395\pi$
$158$	−613.054	−0.308683
$159$	0	0
$160$	1817.50	0.898035
$161$	−1718.08	−0.841016
$162$	0	0
$163$	−1005.26	−0.483054	−0.241527	−	0.970394i	$-0.577648\pi$
$163$	−1005.26	−0.483054	−0.241527	+	0.970394i	$0.577648\pi$
$164$	1299.15	0.618579
$165$	0	0
$166$	715.528	0.334553
$167$	−2035.37	−0.943124	−0.471562	−	0.881833i	$-0.656310\pi$
$167$	−2035.37	−0.943124	−0.471562	+	0.881833i	$0.656310\pi$
$168$	0	0
$169$	−1896.75	−0.863338
$170$	−577.188	−0.260402
$171$	0	0
$172$	−1766.49	−0.783102
$173$	1880.16	0.826278	0.413139	−	0.910668i	$-0.364432\pi$
$173$	1880.16	0.826278	0.413139	+	0.910668i	$0.364432\pi$
$174$	0	0
$175$	−199.585	−0.0862125
$176$	1826.64	0.782319
$177$	0	0
$178$	810.383	0.341240
$179$	−2697.56	−1.12640	−0.563198	−	0.826322i	$-0.690429\pi$
$179$	−2697.56	−1.12640	−0.563198	+	0.826322i	$0.690429\pi$
$180$	0	0
$181$	−2103.40	−0.863780	−0.431890	−	0.901926i	$-0.642153\pi$
$181$	−2103.40	−0.863780	−0.431890	+	0.901926i	$0.642153\pi$
$182$	−239.644	−0.0976022
$183$	0	0
$184$	2437.40	0.976563
$185$	1953.06	0.776174
$186$	0	0
$187$	−2600.70	−1.01702
$188$	3806.77	1.47679
$189$	0	0
$190$	−183.303	−0.0699906
$191$	3865.44	1.46436	0.732181	−	0.681110i	$-0.238502\pi$
$191$	3865.44	1.46436	0.732181	+	0.681110i	$0.238502\pi$
$192$	0	0
$193$	−315.197	−0.117556	−0.0587781	−	0.998271i	$-0.518720\pi$
$193$	−315.197	−0.117556	−0.0587781	+	0.998271i	$0.518720\pi$
$194$	1722.58	0.637495
$195$	0	0
$196$	1332.81	0.485718
$197$	−461.364	−0.166857	−0.0834285	−	0.996514i	$-0.526587\pi$
$197$	−461.364	−0.166857	−0.0834285	+	0.996514i	$0.526587\pi$
$198$	0	0
$199$	−3914.65	−1.39448	−0.697242	−	0.716836i	$-0.745590\pi$
$199$	−3914.65	−1.39448	−0.697242	+	0.716836i	$0.745590\pi$
$200$	283.147	0.100107
$201$	0	0
$202$	1436.40	0.500322
$203$	−2717.37	−0.939516
$204$	0	0
$205$	−2029.53	−0.691457
$206$	−590.000	−0.199550
$207$	0	0
$208$	−583.487	−0.194507
$209$	−825.930	−0.273353
$210$	0	0
$211$	−2310.08	−0.753709	−0.376854	−	0.926272i	$-0.622994\pi$
$211$	−2310.08	−0.753709	−0.376854	+	0.926272i	$0.622994\pi$
$212$	−960.586	−0.311195
$213$	0	0
$214$	−403.142	−0.128777
$215$	2759.60	0.875364
$216$	0	0
$217$	−256.322	−0.0801855
$218$	−1842.61	−0.572464
$219$	0	0
$220$	−3760.78	−1.15251
$221$	830.747	0.252860
$222$	0	0
$223$	2314.45	0.695008	0.347504	−	0.937678i	$-0.387029\pi$
$223$	2314.45	0.695008	0.347504	+	0.937678i	$0.387029\pi$
$224$	2087.91	0.622788
$225$	0	0
$226$	−1367.10	−0.402382
$227$	−6700.60	−1.95918	−0.979591	−	0.201001i	$-0.935580\pi$
$227$	−6700.60	−1.95918	−0.979591	+	0.201001i	$0.935580\pi$
$228$	0	0
$229$	779.694	0.224994	0.112497	−	0.993652i	$-0.464115\pi$
$229$	779.694	0.224994	0.112497	+	0.993652i	$0.464115\pi$
$230$	−1730.08	−0.495992
$231$	0	0
$232$	3855.07	1.09094
$233$	−4795.01	−1.34820	−0.674102	−	0.738638i	$-0.735469\pi$
$233$	−4795.01	−1.34820	−0.674102	+	0.738638i	$0.735469\pi$
$234$	0	0
$235$	−5946.91	−1.65078
$236$	−5574.52	−1.53759
$237$	0	0
$238$	−663.065	−0.180589
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	1184.54	0.316611	0.158305	−	0.987390i	$-0.449397\pi$
$241$	1184.54	0.316611	0.158305	+	0.987390i	$0.449397\pi$
$242$	1864.25	0.495200
$243$	0	0
$244$	2186.06	0.573558
$245$	−2082.11	−0.542943
$246$	0	0
$247$	263.828	0.0679635
$248$	363.638	0.0931091
$249$	0	0
$250$	−1705.85	−0.431549
$251$	−1383.76	−0.347976	−0.173988	−	0.984748i	$-0.555665\pi$
$251$	−1383.76	−0.347976	−0.173988	+	0.984748i	$0.555665\pi$
$252$	0	0
$253$	−7795.41	−1.93713
$254$	−1693.00	−0.418221
$255$	0	0
$256$	−1167.46	−0.285024
$257$	6795.28	1.64933	0.824666	−	0.565620i	$-0.191363\pi$
$257$	6795.28	1.64933	0.824666	+	0.565620i	$0.191363\pi$
$258$	0	0
$259$	2243.65	0.538277
$260$	1201.31	0.286547
$261$	0	0
$262$	−1837.24	−0.433226
$263$	3621.49	0.849090	0.424545	−	0.905407i	$-0.360434\pi$
$263$	3621.49	0.849090	0.424545	+	0.905407i	$0.360434\pi$
$264$	0	0
$265$	1500.62	0.347858
$266$	−210.576	−0.0485385
$267$	0	0
$268$	5995.45	1.36653
$269$	−3331.24	−0.755054	−0.377527	−	0.925999i	$-0.623225\pi$
$269$	−3331.24	−0.755054	−0.377527	+	0.925999i	$0.623225\pi$
$270$	0	0
$271$	545.410	0.122256	0.0611278	−	0.998130i	$-0.480530\pi$
$271$	545.410	0.122256	0.0611278	+	0.998130i	$0.480530\pi$
$272$	−1614.44	−0.359888
$273$	0	0
$274$	−1457.62	−0.321380
$275$	−905.573	−0.198575
$276$	0	0
$277$	3300.96	0.716012	0.358006	−	0.933719i	$-0.383457\pi$
$277$	3300.96	0.716012	0.358006	+	0.933719i	$0.383457\pi$
$278$	1240.72	0.267674
$279$	0	0
$280$	−2110.28	−0.450405
$281$	3471.60	0.737005	0.368503	−	0.929627i	$-0.379871\pi$
$281$	3471.60	0.737005	0.368503	+	0.929627i	$0.379871\pi$
$282$	0	0
$283$	−3745.67	−0.786774	−0.393387	−	0.919373i	$-0.628697\pi$
$283$	−3745.67	−0.786774	−0.393387	+	0.919373i	$0.628697\pi$
$284$	792.714	0.165630
$285$	0	0
$286$	−1087.33	−0.224809
$287$	−2331.50	−0.479526
$288$	0	0
$289$	−2614.43	−0.532145
$290$	−2736.35	−0.554083
$291$	0	0
$292$	−2067.45	−0.414344
$293$	538.566	0.107383	0.0536917	−	0.998558i	$-0.482901\pi$
$293$	538.566	0.107383	0.0536917	+	0.998558i	$0.482901\pi$
$294$	0	0
$295$	8708.49	1.71874
$296$	−3183.02	−0.625032
$297$	0	0
$298$	2997.57	0.582700
$299$	2490.10	0.481627
$300$	0	0
$301$	3170.19	0.607065
$302$	−1033.85	−0.196991
$303$	0	0
$304$	−512.712	−0.0967304
$305$	−3415.05	−0.641132
$306$	0	0
$307$	8025.73	1.49203	0.746014	−	0.665931i	$-0.231965\pi$
$307$	8025.73	1.49203	0.746014	+	0.665931i	$0.231965\pi$
$308$	−4320.33	−0.799266
$309$	0	0
$310$	−258.112	−0.0472896
$311$	10345.8	1.88636	0.943182	−	0.332277i	$-0.107817\pi$
$311$	10345.8	1.88636	0.943182	+	0.332277i	$0.107817\pi$
$312$	0	0
$313$	1404.15	0.253569	0.126785	−	0.991930i	$-0.459534\pi$
$313$	1404.15	0.253569	0.126785	+	0.991930i	$0.459534\pi$
$314$	−120.027	−0.0215717
$315$	0	0
$316$	3530.43	0.628488
$317$	−627.502	−0.111180	−0.0555899	−	0.998454i	$-0.517704\pi$
$317$	−627.502	−0.111180	−0.0555899	+	0.998454i	$0.517704\pi$
$318$	0	0
$319$	−12329.5	−2.16401
$320$	−701.047	−0.122468
$321$	0	0
$322$	−1987.49	−0.343971
$323$	729.980	0.125750
$324$	0	0
$325$	289.269	0.0493715
$326$	−1162.89	−0.197566
$327$	0	0
$328$	3307.65	0.556812
$329$	−6831.73	−1.14482
$330$	0	0
$331$	−5926.24	−0.984095	−0.492048	−	0.870568i	$-0.663751\pi$
$331$	−5926.24	−0.984095	−0.492048	+	0.870568i	$0.663751\pi$
$332$	−4120.55	−0.681158
$333$	0	0
$334$	−2354.54	−0.385732
$335$	−9366.07	−1.52753
$336$	0	0
$337$	−2087.61	−0.337446	−0.168723	−	0.985664i	$-0.553964\pi$
$337$	−2087.61	−0.337446	−0.168723	+	0.985664i	$0.553964\pi$
$338$	−2194.18	−0.353100
$339$	0	0
$340$	3323.89	0.530185
$341$	−1163.00	−0.184693
$342$	0	0
$343$	−6492.61	−1.02206
$344$	−4497.48	−0.704907
$345$	0	0
$346$	2174.99	0.337943
$347$	−5231.13	−0.809285	−0.404642	−	0.914475i	$-0.632604\pi$
$347$	−5231.13	−0.809285	−0.404642	+	0.914475i	$0.632604\pi$
$348$	0	0
$349$	−12233.3	−1.87632	−0.938161	−	0.346200i	$-0.887472\pi$
$349$	−12233.3	−1.87632	−0.938161	+	0.346200i	$0.887472\pi$
$350$	−230.882	−0.0352604
$351$	0	0
$352$	9473.46	1.43448
$353$	−11810.5	−1.78077	−0.890383	−	0.455213i	$-0.849563\pi$
$353$	−11810.5	−1.78077	−0.890383	+	0.455213i	$0.849563\pi$
$354$	0	0
$355$	−1238.37	−0.185144
$356$	−4666.80	−0.694774
$357$	0	0
$358$	−3120.56	−0.460690
$359$	−1193.05	−0.175395	−0.0876977	−	0.996147i	$-0.527951\pi$
$359$	−1193.05	−0.175395	−0.0876977	+	0.996147i	$0.527951\pi$
$360$	0	0
$361$	−6627.17	−0.966201
$362$	−2433.23	−0.353281
$363$	0	0
$364$	1380.05	0.198721
$365$	3229.76	0.463160
$366$	0	0
$367$	−664.808	−0.0945578	−0.0472789	−	0.998882i	$-0.515055\pi$
$367$	−664.808	−0.0945578	−0.0472789	+	0.998882i	$0.515055\pi$
$368$	−4839.15	−0.685484
$369$	0	0
$370$	2259.33	0.317451
$371$	1723.89	0.241240
$372$	0	0
$373$	2177.13	0.302218	0.151109	−	0.988517i	$-0.451716\pi$
$373$	2177.13	0.302218	0.151109	+	0.988517i	$0.451716\pi$
$374$	−3008.52	−0.415954
$375$	0	0
$376$	9692.03	1.32933
$377$	3938.43	0.538035
$378$	0	0
$379$	−10433.1	−1.41401	−0.707006	−	0.707207i	$-0.749955\pi$
$379$	−10433.1	−1.41401	−0.707006	+	0.707207i	$0.749955\pi$
$380$	1055.60	0.142503
$381$	0	0
$382$	4471.58	0.598916
$383$	11493.5	1.53340	0.766698	−	0.642008i	$-0.221899\pi$
$383$	11493.5	1.53340	0.766698	+	0.642008i	$0.221899\pi$
$384$	0	0
$385$	6749.20	0.893432
$386$	−364.623	−0.0480799
$387$	0	0
$388$	−9919.92	−1.29796
$389$	−5448.06	−0.710096	−0.355048	−	0.934848i	$-0.615536\pi$
$389$	−5448.06	−0.710096	−0.355048	+	0.934848i	$0.615536\pi$
$390$	0	0
$391$	6889.81	0.891132
$392$	3393.33	0.437217
$393$	0	0
$394$	−533.711	−0.0682435
$395$	−5515.22	−0.702534
$396$	0	0
$397$	−7773.93	−0.982777	−0.491388	−	0.870941i	$-0.663510\pi$
$397$	−7773.93	−0.982777	−0.491388	+	0.870941i	$0.663510\pi$
$398$	−4528.51	−0.570336
$399$	0	0
$400$	−562.152	−0.0702690
$401$	9569.54	1.19172	0.595860	−	0.803088i	$-0.296811\pi$
$401$	9569.54	1.19172	0.595860	+	0.803088i	$0.296811\pi$
$402$	0	0
$403$	371.501	0.0459200
$404$	−8271.90	−1.01867
$405$	0	0
$406$	−3143.48	−0.384257
$407$	10180.1	1.23982
$408$	0	0
$409$	−1261.79	−0.152547	−0.0762733	−	0.997087i	$-0.524302\pi$
$409$	−1261.79	−0.152547	−0.0762733	+	0.997087i	$0.524302\pi$
$410$	−2347.78	−0.282802
$411$	0	0
$412$	3397.67	0.406289
$413$	10004.2	1.19195
$414$	0	0
$415$	6437.10	0.761410
$416$	−3026.12	−0.356654
$417$	0	0
$418$	−955.444	−0.111800
$419$	−5477.32	−0.638627	−0.319313	−	0.947649i	$-0.603452\pi$
$419$	−5477.32	−0.638627	−0.319313	+	0.947649i	$0.603452\pi$
$420$	0	0
$421$	8096.90	0.937337	0.468668	−	0.883374i	$-0.344734\pi$
$421$	8096.90	0.937337	0.468668	+	0.883374i	$0.344734\pi$
$422$	−2672.33	−0.308263
$423$	0	0
$424$	−2445.65	−0.280121
$425$	800.371	0.0913499
$426$	0	0
$427$	−3923.16	−0.444625
$428$	2321.60	0.262193
$429$	0	0
$430$	3192.34	0.358019
$431$	−10091.3	−1.12779	−0.563897	−	0.825845i	$-0.690699\pi$
$431$	−10091.3	−1.12779	−0.563897	+	0.825845i	$0.690699\pi$
$432$	0	0
$433$	−5766.21	−0.639968	−0.319984	−	0.947423i	$-0.603678\pi$
$433$	−5766.21	−0.639968	−0.319984	+	0.947423i	$0.603678\pi$
$434$	−296.516	−0.0327954
$435$	0	0
$436$	10611.1	1.16555
$437$	2188.06	0.239518
$438$	0	0
$439$	5502.88	0.598265	0.299133	−	0.954212i	$-0.403303\pi$
$439$	5502.88	0.598265	0.299133	+	0.954212i	$0.403303\pi$
$440$	−9574.95	−1.03743
$441$	0	0
$442$	961.016	0.103418
$443$	−9398.64	−1.00800	−0.503999	−	0.863704i	$-0.668138\pi$
$443$	−9398.64	−1.00800	−0.503999	+	0.863704i	$0.668138\pi$
$444$	0	0
$445$	7290.45	0.776630
$446$	2677.38	0.284254
$447$	0	0
$448$	−805.352	−0.0849315
$449$	−6741.81	−0.708610	−0.354305	−	0.935130i	$-0.615282\pi$
$449$	−6741.81	−0.708610	−0.354305	+	0.935130i	$0.615282\pi$
$450$	0	0
$451$	−10578.7	−1.10450
$452$	7872.80	0.819260
$453$	0	0
$454$	−7751.32	−0.801294
$455$	−2155.91	−0.222133
$456$	0	0
$457$	−2853.53	−0.292084	−0.146042	−	0.989278i	$-0.546654\pi$
$457$	−2853.53	−0.292084	−0.146042	+	0.989278i	$0.546654\pi$
$458$	901.958	0.0920213
$459$	0	0
$460$	9963.11	1.00985
$461$	13101.0	1.32359	0.661793	−	0.749687i	$-0.269796\pi$
$461$	13101.0	1.32359	0.661793	+	0.749687i	$0.269796\pi$
$462$	0	0
$463$	522.432	0.0524394	0.0262197	−	0.999656i	$-0.491653\pi$
$463$	522.432	0.0524394	0.0262197	+	0.999656i	$0.491653\pi$
$464$	−7653.76	−0.765769
$465$	0	0
$466$	−5546.92	−0.551408
$467$	10688.0	1.05906	0.529529	−	0.848292i	$-0.322369\pi$
$467$	10688.0	1.05906	0.529529	+	0.848292i	$0.322369\pi$
$468$	0	0
$469$	−10759.6	−1.05934
$470$	−6879.45	−0.675161
$471$	0	0
$472$	−14192.7	−1.38405
$473$	14384.1	1.39827
$474$	0	0
$475$	254.182	0.0245530
$476$	3818.43	0.367684
$477$	0	0
$478$	−276.478	−0.0264556
$479$	−8318.44	−0.793485	−0.396742	−	0.917930i	$-0.629859\pi$
$479$	−8318.44	−0.793485	−0.396742	+	0.917930i	$0.629859\pi$
$480$	0	0
$481$	−3251.85	−0.308257
$482$	1370.29	0.129492
$483$	0	0
$484$	−10735.7	−1.00824
$485$	15496.8	1.45088
$486$	0	0
$487$	8839.77	0.822522	0.411261	−	0.911518i	$-0.365089\pi$
$487$	8839.77	0.822522	0.411261	+	0.911518i	$0.365089\pi$
$488$	5565.71	0.516286
$489$	0	0
$490$	−2408.61	−0.222061
$491$	14799.9	1.36031	0.680153	−	0.733070i	$-0.261913\pi$
$491$	14799.9	1.36031	0.680153	+	0.733070i	$0.261913\pi$
$492$	0	0
$493$	10897.1	0.995501
$494$	305.199	0.0277967
$495$	0	0
$496$	−721.958	−0.0653566
$497$	−1422.63	−0.128397
$498$	0	0
$499$	9306.35	0.834889	0.417444	−	0.908702i	$-0.362926\pi$
$499$	9306.35	0.834889	0.417444	+	0.908702i	$0.362926\pi$
$500$	9823.55	0.878645
$501$	0	0
$502$	−1600.75	−0.142320
$503$	−11549.2	−1.02376	−0.511881	−	0.859056i	$-0.671051\pi$
$503$	−11549.2	−1.02376	−0.511881	+	0.859056i	$0.671051\pi$
$504$	0	0
$505$	12922.3	1.13868
$506$	−9017.82	−0.792274
$507$	0	0
$508$	9749.55	0.851508
$509$	−10011.4	−0.871798	−0.435899	−	0.899996i	$-0.643570\pi$
$509$	−10011.4	−0.871798	−0.435899	+	0.899996i	$0.643570\pi$
$510$	0	0
$511$	3710.30	0.321202
$512$	10449.9	0.902004
$513$	0	0
$514$	7860.86	0.674567
$515$	−5307.82	−0.454156
$516$	0	0
$517$	−30997.5	−2.63688
$518$	2595.48	0.220152
$519$	0	0
$520$	3058.54	0.257935
$521$	18123.7	1.52402	0.762008	−	0.647568i	$-0.224214\pi$
$521$	18123.7	1.52402	0.762008	+	0.647568i	$0.224214\pi$
$522$	0	0
$523$	−10830.0	−0.905473	−0.452736	−	0.891644i	$-0.649552\pi$
$523$	−10830.0	−0.905473	−0.452736	+	0.891644i	$0.649552\pi$
$524$	10580.2	0.882061
$525$	0	0
$526$	4189.38	0.347273
$527$	1027.90	0.0849637
$528$	0	0
$529$	8484.69	0.697353
$530$	1735.94	0.142272
$531$	0	0
$532$	1212.66	0.0988258
$533$	3379.16	0.274611
$534$	0	0
$535$	−3626.79	−0.293083
$536$	15264.4	1.23008
$537$	0	0
$538$	−3853.62	−0.308813
$539$	−10852.7	−0.867272
$540$	0	0
$541$	16186.8	1.28637	0.643185	−	0.765710i	$-0.277612\pi$
$541$	16186.8	1.28637	0.643185	+	0.765710i	$0.277612\pi$
$542$	630.936	0.0500019
$543$	0	0
$544$	−8372.91	−0.659900
$545$	−16576.6	−1.30287
$546$	0	0
$547$	−2924.65	−0.228609	−0.114305	−	0.993446i	$-0.536464\pi$
$547$	−2924.65	−0.228609	−0.114305	+	0.993446i	$0.536464\pi$
$548$	8394.08	0.654338
$549$	0	0
$550$	−1047.58	−0.0812160
$551$	3460.71	0.267570
$552$	0	0
$553$	−6335.80	−0.487208
$554$	3818.58	0.292845
$555$	0	0
$556$	−7145.00	−0.544992
$557$	17292.8	1.31548	0.657738	−	0.753247i	$-0.271513\pi$
$557$	17292.8	1.31548	0.657738	+	0.753247i	$0.271513\pi$
$558$	0	0
$559$	−4594.73	−0.347650
$560$	4189.70	0.316155
$561$	0	0
$562$	4015.99	0.301431
$563$	−3369.87	−0.252261	−0.126131	−	0.992014i	$-0.540256\pi$
$563$	−3369.87	−0.252261	−0.126131	+	0.992014i	$0.540256\pi$
$564$	0	0
$565$	−12298.9	−0.915782
$566$	−4333.03	−0.321786
$567$	0	0
$568$	2018.25	0.149091
$569$	−22024.6	−1.62271	−0.811354	−	0.584555i	$-0.801269\pi$
$569$	−22024.6	−1.62271	−0.811354	+	0.584555i	$0.801269\pi$
$570$	0	0
$571$	−8768.96	−0.642679	−0.321340	−	0.946964i	$-0.604133\pi$
$571$	−8768.96	−0.642679	−0.321340	+	0.946964i	$0.604133\pi$
$572$	6261.69	0.457718
$573$	0	0
$574$	−2697.10	−0.196123
$575$	2399.05	0.173996
$576$	0	0
$577$	15885.7	1.14615	0.573075	−	0.819503i	$-0.305750\pi$
$577$	15885.7	1.14615	0.573075	+	0.819503i	$0.305750\pi$
$578$	−3024.40	−0.217644
$579$	0	0
$580$	15758.0	1.12813
$581$	7394.85	0.528038
$582$	0	0
$583$	7821.80	0.555653
$584$	−5263.73	−0.372971
$585$	0	0
$586$	623.019	0.0439192
$587$	16499.8	1.16017	0.580087	−	0.814555i	$-0.303019\pi$
$587$	16499.8	1.16017	0.580087	+	0.814555i	$0.303019\pi$
$588$	0	0
$589$	326.439	0.0228365
$590$	10074.1	0.702954
$591$	0	0
$592$	6319.49	0.438732
$593$	17008.0	1.17780	0.588900	−	0.808206i	$-0.299561\pi$
$593$	17008.0	1.17780	0.588900	+	0.808206i	$0.299561\pi$
$594$	0	0
$595$	−5965.13	−0.411003
$596$	−17262.3	−1.18639
$597$	0	0
$598$	2880.58	0.196983
$599$	−5106.71	−0.348338	−0.174169	−	0.984716i	$-0.555724\pi$
$599$	−5106.71	−0.348338	−0.174169	+	0.984716i	$0.555724\pi$
$600$	0	0
$601$	25057.8	1.70072	0.850358	−	0.526205i	$-0.176386\pi$
$601$	25057.8	1.70072	0.850358	+	0.526205i	$0.176386\pi$
$602$	3667.31	0.248286
$603$	0	0
$604$	5953.69	0.401080
$605$	16771.3	1.12703
$606$	0	0
$607$	−1034.99	−0.0692075	−0.0346037	−	0.999401i	$-0.511017\pi$
$607$	−1034.99	−0.0692075	−0.0346037	+	0.999401i	$0.511017\pi$
$608$	−2659.07	−0.177367
$609$	0	0
$610$	−3950.57	−0.262219
$611$	9901.59	0.655606
$612$	0	0
$613$	−9705.84	−0.639503	−0.319751	−	0.947501i	$-0.603599\pi$
$613$	−9705.84	−0.639503	−0.319751	+	0.947501i	$0.603599\pi$
$614$	9284.24	0.610231
$615$	0	0
$616$	−10999.6	−0.719456
$617$	−3208.40	−0.209344	−0.104672	−	0.994507i	$-0.533379\pi$
$617$	−3208.40	−0.209344	−0.104672	+	0.994507i	$0.533379\pi$
$618$	0	0
$619$	−8779.96	−0.570107	−0.285054	−	0.958512i	$-0.592011\pi$
$619$	−8779.96	−0.570107	−0.285054	+	0.958512i	$0.592011\pi$
$620$	1486.41	0.0962830
$621$	0	0
$622$	11968.2	0.771512
$623$	8375.16	0.538593
$624$	0	0
$625$	−13259.5	−0.848611
$626$	1624.33	0.103708
$627$	0	0
$628$	691.207	0.0439207
$629$	−8997.45	−0.570353
$630$	0	0
$631$	−29775.2	−1.87849	−0.939247	−	0.343241i	$-0.888475\pi$
$631$	−29775.2	−1.87849	−0.939247	+	0.343241i	$0.888475\pi$
$632$	8988.47	0.565732
$633$	0	0
$634$	−725.901	−0.0454719
$635$	−15230.7	−0.951830
$636$	0	0
$637$	3466.70	0.215629
$638$	−14262.9	−0.885066
$639$	0	0
$640$	−15350.9	−0.948124
$641$	5550.85	0.342037	0.171018	−	0.985268i	$-0.445294\pi$
$641$	5550.85	0.342037	0.171018	+	0.985268i	$0.445294\pi$
$642$	0	0
$643$	−9355.36	−0.573778	−0.286889	−	0.957964i	$-0.592621\pi$
$643$	−9355.36	−0.573778	−0.286889	+	0.957964i	$0.592621\pi$
$644$	11445.5	0.700334
$645$	0	0
$646$	844.449	0.0514309
$647$	30963.5	1.88145	0.940727	−	0.339165i	$-0.110144\pi$
$647$	30963.5	1.88145	0.940727	+	0.339165i	$0.110144\pi$
$648$	0	0
$649$	45391.8	2.74543
$650$	334.629	0.0201927
$651$	0	0
$652$	6696.81	0.402251
$653$	12389.6	0.742486	0.371243	−	0.928536i	$-0.378932\pi$
$653$	12389.6	0.742486	0.371243	+	0.928536i	$0.378932\pi$
$654$	0	0
$655$	−16528.4	−0.985981
$656$	−6566.91	−0.390846
$657$	0	0
$658$	−7903.01	−0.468224
$659$	18072.8	1.06831	0.534154	−	0.845387i	$-0.320630\pi$
$659$	18072.8	1.06831	0.534154	+	0.845387i	$0.320630\pi$
$660$	0	0
$661$	−23625.0	−1.39018	−0.695088	−	0.718925i	$-0.744634\pi$
$661$	−23625.0	−1.39018	−0.695088	+	0.718925i	$0.744634\pi$
$662$	−6855.54	−0.402489
$663$	0	0
$664$	−10490.9	−0.613143
$665$	−1894.41	−0.110469
$666$	0	0
$667$	32663.4	1.89615
$668$	13559.2	0.785362
$669$	0	0
$670$	−10834.8	−0.624752
$671$	−17800.5	−1.02411
$672$	0	0
$673$	13935.3	0.798164	0.399082	−	0.916915i	$-0.369329\pi$
$673$	13935.3	0.798164	0.399082	+	0.916915i	$0.369329\pi$
$674$	−2414.97	−0.138013
$675$	0	0
$676$	12635.8	0.718922
$677$	−20965.2	−1.19019	−0.595095	−	0.803655i	$-0.702886\pi$
$677$	−20965.2	−1.19019	−0.595095	+	0.803655i	$0.702886\pi$
$678$	0	0
$679$	17802.5	1.00618
$680$	8462.61	0.477245
$681$	0	0
$682$	−1345.38	−0.0755383
$683$	35311.9	1.97829	0.989145	−	0.146945i	$-0.0469440\pi$
$683$	35311.9	1.97829	0.989145	+	0.146945i	$0.0469440\pi$
$684$	0	0
$685$	−13113.2	−0.731429
$686$	−7510.72	−0.418018
$687$	0	0
$688$	8929.18	0.494799
$689$	−2498.53	−0.138152
$690$	0	0
$691$	−5960.98	−0.328171	−0.164086	−	0.986446i	$-0.552467\pi$
$691$	−5960.98	−0.328171	−0.164086	+	0.986446i	$0.552467\pi$
$692$	−12525.2	−0.688061
$693$	0	0
$694$	−6051.43	−0.330993
$695$	11161.9	0.609201
$696$	0	0
$697$	9349.73	0.508101
$698$	−14151.7	−0.767405
$699$	0	0
$700$	1329.59	0.0717912
$701$	−1876.16	−0.101086	−0.0505431	−	0.998722i	$-0.516095\pi$
$701$	−1876.16	−0.101086	−0.0505431	+	0.998722i	$0.516095\pi$
$702$	0	0
$703$	−2857.41	−0.153299
$704$	−3654.11	−0.195624
$705$	0	0
$706$	−13662.5	−0.728323
$707$	14845.0	0.789678
$708$	0	0
$709$	2912.38	0.154269	0.0771344	−	0.997021i	$-0.475423\pi$
$709$	2912.38	0.154269	0.0771344	+	0.997021i	$0.475423\pi$
$710$	−1432.56	−0.0757227
$711$	0	0
$712$	−11881.7	−0.625399
$713$	3081.04	0.161832
$714$	0	0
$715$	−9781.98	−0.511644
$716$	17970.6	0.937977
$717$	0	0
$718$	−1380.14	−0.0717357
$719$	23312.6	1.20920	0.604598	−	0.796531i	$-0.293334\pi$
$719$	23312.6	1.20920	0.604598	+	0.796531i	$0.293334\pi$
$720$	0	0
$721$	−6097.54	−0.314957
$722$	−7666.38	−0.395171
$723$	0	0
$724$	14012.4	0.719290
$725$	3794.42	0.194374
$726$	0	0
$727$	−1478.33	−0.0754172	−0.0377086	−	0.999289i	$-0.512006\pi$
$727$	−1478.33	−0.0754172	−0.0377086	+	0.999289i	$0.512006\pi$
$728$	3513.61	0.178878
$729$	0	0
$730$	3736.22	0.189430
$731$	−12713.0	−0.643240
$732$	0	0
$733$	−13746.1	−0.692663	−0.346332	−	0.938112i	$-0.612573\pi$
$733$	−13746.1	−0.692663	−0.346332	+	0.938112i	$0.612573\pi$
$734$	−769.057	−0.0386736
$735$	0	0
$736$	−25097.2	−1.25692
$737$	−48819.4	−2.44001
$738$	0	0
$739$	−34696.2	−1.72709	−0.863545	−	0.504273i	$-0.831761\pi$
$739$	−34696.2	−1.72709	−0.863545	+	0.504273i	$0.831761\pi$
$740$	−13010.9	−0.646338
$741$	0	0
$742$	1994.22	0.0986658
$743$	30866.0	1.52404	0.762020	−	0.647553i	$-0.224208\pi$
$743$	30866.0	1.52404	0.762020	+	0.647553i	$0.224208\pi$
$744$	0	0
$745$	26967.0	1.32617
$746$	2518.52	0.123606
$747$	0	0
$748$	17325.3	0.846894
$749$	−4166.40	−0.203254
$750$	0	0
$751$	−32713.8	−1.58954	−0.794770	−	0.606910i	$-0.792409\pi$
$751$	−32713.8	−1.58954	−0.794770	+	0.606910i	$0.792409\pi$
$752$	−19242.3	−0.933104
$753$	0	0
$754$	4556.01	0.220053
$755$	−9300.82	−0.448333
$756$	0	0
$757$	30136.8	1.44695	0.723476	−	0.690350i	$-0.242543\pi$
$757$	30136.8	1.44695	0.723476	+	0.690350i	$0.242543\pi$
$758$	−12069.1	−0.578323
$759$	0	0
$760$	2687.55	0.128273
$761$	11739.0	0.559185	0.279593	−	0.960119i	$-0.409801\pi$
$761$	11739.0	0.559185	0.279593	+	0.960119i	$0.409801\pi$
$762$	0	0
$763$	−19043.0	−0.903543
$764$	−25750.7	−1.21941
$765$	0	0
$766$	13295.8	0.627150
$767$	−14499.6	−0.682595
$768$	0	0
$769$	−8154.63	−0.382397	−0.191199	−	0.981551i	$-0.561237\pi$
$769$	−8154.63	−0.382397	−0.191199	+	0.981551i	$0.561237\pi$
$770$	7807.55	0.365408
$771$	0	0
$772$	2099.77	0.0978919
$773$	−8529.13	−0.396858	−0.198429	−	0.980115i	$-0.563584\pi$
$773$	−8529.13	−0.396858	−0.198429	+	0.980115i	$0.563584\pi$
$774$	0	0
$775$	357.917	0.0165894
$776$	−25256.1	−1.16835
$777$	0	0
$778$	−6302.37	−0.290425
$779$	2969.28	0.136567
$780$	0	0
$781$	−6454.86	−0.295740
$782$	7970.20	0.364468
$783$	0	0
$784$	−6737.03	−0.306898
$785$	−1079.80	−0.0490952
$786$	0	0
$787$	−26540.5	−1.20212	−0.601060	−	0.799204i	$-0.705255\pi$
$787$	−26540.5	−1.20212	−0.601060	+	0.799204i	$0.705255\pi$
$788$	3073.51	0.138946
$789$	0	0
$790$	−6380.07	−0.287332
$791$	−14128.7	−0.635095
$792$	0	0
$793$	5686.05	0.254625
$794$	−8992.97	−0.401950
$795$	0	0
$796$	26078.6	1.16122
$797$	−22075.0	−0.981100	−0.490550	−	0.871413i	$-0.663204\pi$
$797$	−22075.0	−0.981100	−0.490550	+	0.871413i	$0.663204\pi$
$798$	0	0
$799$	27396.5	1.21304
$800$	−2915.48	−0.128847
$801$	0	0
$802$	11070.1	0.487407
$803$	16834.7	0.739831
$804$	0	0
$805$	−17880.1	−0.782844
$806$	429.756	0.0187810
$807$	0	0
$808$	−21060.2	−0.916952
$809$	−36247.8	−1.57528	−0.787642	−	0.616134i	$-0.788698\pi$
$809$	−36247.8	−1.57528	−0.787642	+	0.616134i	$0.788698\pi$
$810$	0	0
$811$	−32218.6	−1.39500	−0.697501	−	0.716584i	$-0.745705\pi$
$811$	−32218.6	−1.39500	−0.697501	+	0.716584i	$0.745705\pi$
$812$	18102.5	0.782357
$813$	0	0
$814$	11776.4	0.507081
$815$	−10461.7	−0.449642
$816$	0	0
$817$	−4037.40	−0.172890
$818$	−1459.65	−0.0623907
$819$	0	0
$820$	13520.3	0.575793
$821$	−11347.5	−0.482377	−0.241188	−	0.970478i	$-0.577537\pi$
$821$	−11347.5	−0.482377	−0.241188	+	0.970478i	$0.577537\pi$
$822$	0	0
$823$	37987.4	1.60894	0.804470	−	0.593993i	$-0.202449\pi$
$823$	37987.4	1.60894	0.804470	+	0.593993i	$0.202449\pi$
$824$	8650.46	0.365720
$825$	0	0
$826$	11572.9	0.487499
$827$	22954.6	0.965185	0.482593	−	0.875845i	$-0.339695\pi$
$827$	22954.6	0.965185	0.482593	+	0.875845i	$0.339695\pi$
$828$	0	0
$829$	4524.92	0.189574	0.0947871	−	0.995498i	$-0.469783\pi$
$829$	4524.92	0.189574	0.0947871	+	0.995498i	$0.469783\pi$
$830$	7446.51	0.311412
$831$	0	0
$832$	1167.24	0.0486379
$833$	9591.94	0.398969
$834$	0	0
$835$	−21182.1	−0.877890
$836$	5502.17	0.227627
$837$	0	0
$838$	−6336.22	−0.261195
$839$	−11574.3	−0.476267	−0.238134	−	0.971232i	$-0.576536\pi$
$839$	−11574.3	−0.476267	−0.238134	+	0.971232i	$0.576536\pi$
$840$	0	0
$841$	27272.4	1.11823
$842$	9366.58	0.383366
$843$	0	0
$844$	15389.3	0.627631
$845$	−19739.5	−0.803622
$846$	0	0
$847$	19266.7	0.781594
$848$	4855.53	0.196627
$849$	0	0
$850$	925.878	0.0373616
$851$	−26969.2	−1.08636
$852$	0	0
$853$	−8582.93	−0.344518	−0.172259	−	0.985052i	$-0.555107\pi$
$853$	−8582.93	−0.344518	−0.172259	+	0.985052i	$0.555107\pi$
$854$	−4538.35	−0.181849
$855$	0	0
$856$	5910.78	0.236012
$857$	−4380.93	−0.174621	−0.0873103	−	0.996181i	$-0.527827\pi$
$857$	−4380.93	−0.174621	−0.0873103	+	0.996181i	$0.527827\pi$
$858$	0	0
$859$	−5431.28	−0.215731	−0.107866	−	0.994165i	$-0.534402\pi$
$859$	−5431.28	−0.215731	−0.107866	+	0.994165i	$0.534402\pi$
$860$	−18383.9	−0.728936
$861$	0	0
$862$	−11673.7	−0.461262
$863$	17433.6	0.687655	0.343828	−	0.939033i	$-0.388276\pi$
$863$	17433.6	0.687655	0.343828	+	0.939033i	$0.388276\pi$
$864$	0	0
$865$	19566.9	0.769126
$866$	−6670.41	−0.261743
$867$	0	0
$868$	1707.56	0.0667723
$869$	−28747.4	−1.12220
$870$	0	0
$871$	15594.5	0.606657
$872$	27015.9	1.04917
$873$	0	0
$874$	2531.17	0.0979614
$875$	−17629.6	−0.681131
$876$	0	0
$877$	6100.21	0.234880	0.117440	−	0.993080i	$-0.462531\pi$
$877$	6100.21	0.234880	0.117440	+	0.993080i	$0.462531\pi$
$878$	6365.80	0.244687
$879$	0	0
$880$	19009.9	0.728207
$881$	−23328.3	−0.892112	−0.446056	−	0.895005i	$-0.647172\pi$
$881$	−23328.3	−0.892112	−0.446056	+	0.895005i	$0.647172\pi$
$882$	0	0
$883$	−14303.7	−0.545139	−0.272569	−	0.962136i	$-0.587873\pi$
$883$	−14303.7	−0.545139	−0.272569	+	0.962136i	$0.587873\pi$
$884$	−5534.26	−0.210563
$885$	0	0
$886$	−10872.4	−0.412265
$887$	31165.7	1.17975	0.589877	−	0.807493i	$-0.299176\pi$
$887$	31165.7	1.17975	0.589877	+	0.807493i	$0.299176\pi$
$888$	0	0
$889$	−17496.8	−0.660094
$890$	8433.66	0.317637
$891$	0	0
$892$	−15418.4	−0.578750
$893$	8700.56	0.326040
$894$	0	0
$895$	−28073.5	−1.04848
$896$	−17634.9	−0.657525
$897$	0	0
$898$	−7799.00	−0.289817
$899$	4873.08	0.180786
$900$	0	0
$901$	−6913.13	−0.255616
$902$	−12237.5	−0.451735
$903$	0	0
$904$	20044.2	0.737454
$905$	−21890.1	−0.804034
$906$	0	0
$907$	10824.0	0.396258	0.198129	−	0.980176i	$-0.436514\pi$
$907$	10824.0	0.396258	0.198129	+	0.980176i	$0.436514\pi$
$908$	44638.0	1.63146
$909$	0	0
$910$	−2493.98	−0.0908512
$911$	39254.9	1.42763	0.713815	−	0.700334i	$-0.246966\pi$
$911$	39254.9	1.42763	0.713815	+	0.700334i	$0.246966\pi$
$912$	0	0
$913$	33552.6	1.21624
$914$	−3301.00	−0.119461
$915$	0	0
$916$	−5194.16	−0.187358
$917$	−18987.6	−0.683779
$918$	0	0
$919$	44401.8	1.59377	0.796887	−	0.604128i	$-0.206478\pi$
$919$	44401.8	1.59377	0.796887	+	0.604128i	$0.206478\pi$
$920$	25366.1	0.909016
$921$	0	0
$922$	15155.3	0.541339
$923$	2061.89	0.0735296
$924$	0	0
$925$	−3132.95	−0.111363
$926$	604.355	0.0214474
$927$	0	0
$928$	−39694.5	−1.40413
$929$	−5971.45	−0.210890	−0.105445	−	0.994425i	$-0.533627\pi$
$929$	−5971.45	−0.210890	−0.105445	+	0.994425i	$0.533627\pi$
$930$	0	0
$931$	3046.21	0.107235
$932$	31943.3	1.12268
$933$	0	0
$934$	12364.0	0.433149
$935$	−27065.5	−0.946671
$936$	0	0
$937$	1518.43	0.0529401	0.0264700	−	0.999650i	$-0.491573\pi$
$937$	1518.43	0.0529401	0.0264700	+	0.999650i	$0.491573\pi$
$938$	−12446.8	−0.433266
$939$	0	0
$940$	39617.1	1.37465
$941$	52025.7	1.80233	0.901164	−	0.433479i	$-0.142714\pi$
$941$	52025.7	1.80233	0.901164	+	0.433479i	$0.142714\pi$
$942$	0	0
$943$	28025.1	0.967788
$944$	28177.9	0.971516
$945$	0	0
$946$	16639.6	0.571883
$947$	684.767	0.0234973	0.0117487	−	0.999931i	$-0.496260\pi$
$947$	684.767	0.0234973	0.0117487	+	0.999931i	$0.496260\pi$
$948$	0	0
$949$	−5377.54	−0.183944
$950$	294.040	0.0100420
$951$	0	0
$952$	9721.73	0.330969
$953$	30046.2	1.02129	0.510646	−	0.859791i	$-0.329406\pi$
$953$	30046.2	1.02129	0.510646	+	0.859791i	$0.329406\pi$
$954$	0	0
$955$	40227.7	1.36308
$956$	1592.17	0.0538644
$957$	0	0
$958$	−9622.86	−0.324531
$959$	−15064.2	−0.507247
$960$	0	0
$961$	−29331.3	−0.984570
$962$	−3761.77	−0.126075
$963$	0	0
$964$	−7891.18	−0.263649
$965$	−3280.26	−0.109425
$966$	0	0
$967$	−10648.3	−0.354111	−0.177055	−	0.984201i	$-0.556657\pi$
$967$	−10648.3	−0.354111	−0.177055	+	0.984201i	$0.556657\pi$
$968$	−27333.2	−0.907565
$969$	0	0
$970$	17926.9	0.593401
$971$	−41951.3	−1.38649	−0.693245	−	0.720702i	$-0.743820\pi$
$971$	−41951.3	−1.38649	−0.693245	+	0.720702i	$0.743820\pi$
$972$	0	0
$973$	12822.6	0.422481
$974$	10225.9	0.336407
$975$	0	0
$976$	−11050.0	−0.362400
$977$	−7227.71	−0.236678	−0.118339	−	0.992973i	$-0.537757\pi$
$977$	−7227.71	−0.236678	−0.118339	+	0.992973i	$0.537757\pi$
$978$	0	0
$979$	38000.5	1.24055
$980$	13870.6	0.452121
$981$	0	0
$982$	17120.7	0.556357
$983$	−11233.8	−0.364498	−0.182249	−	0.983252i	$-0.558338\pi$
$983$	−11233.8	−0.364498	−0.182249	+	0.983252i	$0.558338\pi$
$984$	0	0
$985$	−4801.42	−0.155316
$986$	12605.9	0.407154
$987$	0	0
$988$	−1757.57	−0.0565948
$989$	−38106.4	−1.22519
$990$	0	0
$991$	37046.0	1.18749	0.593746	−	0.804652i	$-0.297648\pi$
$991$	37046.0	1.18749	0.593746	+	0.804652i	$0.297648\pi$
$992$	−3744.27	−0.119840
$993$	0	0
$994$	−1645.71	−0.0525138
$995$	−40739.8	−1.29803
$996$	0	0
$997$	44043.4	1.39907	0.699533	−	0.714601i	$-0.253392\pi$
$997$	44043.4	1.39907	0.699533	+	0.714601i	$0.253392\pi$
$998$	10765.7	0.341465
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.17		28
3.2	odd	2		717.4.a.b.1.12	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.12	✓	28	3.2	odd	2
2151.4.a.b.1.17		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)

\(f(q)\)	\(=\)	\(q+1.15681 q^{2} -6.66179 q^{4} +10.4070 q^{5} +11.9554 q^{7} -16.9609 q^{8} +O(q^{10})\)	\(q+1.15681 q^{2} -6.66179 q^{4} +10.4070 q^{5} +11.9554 q^{7} -16.9609 q^{8} +12.0389 q^{10} +54.2452 q^{11} -17.3276 q^{13} +13.8302 q^{14} +33.6737 q^{16} -47.9434 q^{17} -15.2259 q^{19} -69.3293 q^{20} +62.7514 q^{22} -143.707 q^{23} -16.6941 q^{25} -20.0448 q^{26} -79.6445 q^{28} -227.291 q^{29} -21.4398 q^{31} +174.641 q^{32} -55.4615 q^{34} +124.420 q^{35} +187.668 q^{37} -17.6134 q^{38} -176.512 q^{40} -195.016 q^{41} +265.167 q^{43} -361.370 q^{44} -166.242 q^{46} -571.433 q^{47} -200.068 q^{49} -19.3119 q^{50} +115.433 q^{52} +144.193 q^{53} +564.530 q^{55} -202.775 q^{56} -262.933 q^{58} +836.790 q^{59} -328.149 q^{61} -24.8018 q^{62} -67.3629 q^{64} -180.329 q^{65} -899.977 q^{67} +319.389 q^{68} +143.931 q^{70} -118.994 q^{71} +310.345 q^{73} +217.096 q^{74} +101.431 q^{76} +648.524 q^{77} -529.952 q^{79} +350.443 q^{80} -225.596 q^{82} +618.535 q^{83} -498.948 q^{85} +306.749 q^{86} -920.048 q^{88} +700.532 q^{89} -207.159 q^{91} +957.346 q^{92} -661.040 q^{94} -158.456 q^{95} +1489.08 q^{97} -231.441 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 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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