LMFDB - Embedded newform 8037.2.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30668 q^{2} +3.32079 q^{4} +0.443697 q^{5} +2.32803 q^{7} -3.04665 q^{8} +O(q^{10})$	\(q-2.30668 q^{2} +3.32079 q^{4} +0.443697 q^{5} +2.32803 q^{7} -3.04665 q^{8} -1.02347 q^{10} +2.64667 q^{11} -2.14115 q^{13} -5.37003 q^{14} +0.386074 q^{16} +3.84495 q^{17} -1.00000 q^{19} +1.47342 q^{20} -6.10504 q^{22} +2.49378 q^{23} -4.80313 q^{25} +4.93896 q^{26} +7.73090 q^{28} +9.94617 q^{29} +8.60497 q^{31} +5.20275 q^{32} -8.86909 q^{34} +1.03294 q^{35} +3.29374 q^{37} +2.30668 q^{38} -1.35179 q^{40} -1.43288 q^{41} -5.77194 q^{43} +8.78905 q^{44} -5.75236 q^{46} +1.00000 q^{47} -1.58028 q^{49} +11.0793 q^{50} -7.11032 q^{52} -3.98707 q^{53} +1.17432 q^{55} -7.09269 q^{56} -22.9427 q^{58} +6.74750 q^{59} -11.8150 q^{61} -19.8489 q^{62} -12.7732 q^{64} -0.950022 q^{65} -4.26966 q^{67} +12.7683 q^{68} -2.38266 q^{70} +9.26730 q^{71} +15.7575 q^{73} -7.59761 q^{74} -3.32079 q^{76} +6.16153 q^{77} -5.05904 q^{79} +0.171300 q^{80} +3.30520 q^{82} -3.63474 q^{83} +1.70599 q^{85} +13.3140 q^{86} -8.06348 q^{88} -9.35195 q^{89} -4.98466 q^{91} +8.28131 q^{92} -2.30668 q^{94} -0.443697 q^{95} +15.9390 q^{97} +3.64521 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * 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 $

$2$	−2.30668	−1.63107	−0.815536	−	0.578706i	$-0.803558\pi$
$2$	−2.30668	−1.63107	−0.815536	+	0.578706i	$0.803558\pi$
$3$	0	0
$3$	0	0
$4$	3.32079	1.66040
$5$	0.443697	0.198427	0.0992136	−	0.995066i	$-0.468367\pi$
$5$	0.443697	0.198427	0.0992136	+	0.995066i	$0.468367\pi$
$6$	0	0
$7$	2.32803	0.879912	0.439956	−	0.898019i	$-0.354994\pi$
$7$	2.32803	0.879912	0.439956	+	0.898019i	$0.354994\pi$
$8$	−3.04665	−1.07715
$9$	0	0
$10$	−1.02347	−0.323649
$11$	2.64667	0.798002	0.399001	−	0.916951i	$-0.369357\pi$
$11$	2.64667	0.798002	0.399001	+	0.916951i	$0.369357\pi$
$12$	0	0
$13$	−2.14115	−0.593848	−0.296924	−	0.954901i	$-0.595961\pi$
$13$	−2.14115	−0.593848	−0.296924	+	0.954901i	$0.595961\pi$
$14$	−5.37003	−1.43520
$15$	0	0
$16$	0.386074	0.0965184
$17$	3.84495	0.932538	0.466269	−	0.884643i	$-0.345598\pi$
$17$	3.84495	0.932538	0.466269	+	0.884643i	$0.345598\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.47342	0.329468
$21$	0	0
$22$	−6.10504	−1.30160
$23$	2.49378	0.519988	0.259994	−	0.965610i	$-0.416279\pi$
$23$	2.49378	0.519988	0.259994	+	0.965610i	$0.416279\pi$
$24$	0	0
$25$	−4.80313	−0.960627
$26$	4.93896	0.968609
$27$	0	0
$28$	7.73090	1.46100
$29$	9.94617	1.84696	0.923478	−	0.383650i	$-0.125333\pi$
$29$	9.94617	1.84696	0.923478	+	0.383650i	$0.125333\pi$
$30$	0	0
$31$	8.60497	1.54550	0.772749	−	0.634712i	$-0.218881\pi$
$31$	8.60497	1.54550	0.772749	+	0.634712i	$0.218881\pi$
$32$	5.20275	0.919725
$33$	0	0
$34$	−8.86909	−1.52104
$35$	1.03294	0.174599
$36$	0	0
$37$	3.29374	0.541487	0.270744	−	0.962652i	$-0.412730\pi$
$37$	3.29374	0.541487	0.270744	+	0.962652i	$0.412730\pi$
$38$	2.30668	0.374194
$39$	0	0
$40$	−1.35179	−0.213737
$41$	−1.43288	−0.223778	−0.111889	−	0.993721i	$-0.535690\pi$
$41$	−1.43288	−0.223778	−0.111889	+	0.993721i	$0.535690\pi$
$42$	0	0
$43$	−5.77194	−0.880212	−0.440106	−	0.897946i	$-0.645059\pi$
$43$	−5.77194	−0.880212	−0.440106	+	0.897946i	$0.645059\pi$
$44$	8.78905	1.32500
$45$	0	0
$46$	−5.75236	−0.848139
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−1.58028	−0.225755
$50$	11.0793	1.56685
$51$	0	0
$52$	−7.11032	−0.986023
$53$	−3.98707	−0.547667	−0.273833	−	0.961777i	$-0.588292\pi$
$53$	−3.98707	−0.547667	−0.273833	+	0.961777i	$0.588292\pi$
$54$	0	0
$55$	1.17432	0.158345
$56$	−7.09269	−0.947800
$57$	0	0
$58$	−22.9427	−3.01252
$59$	6.74750	0.878450	0.439225	−	0.898377i	$-0.355253\pi$
$59$	6.74750	0.878450	0.439225	+	0.898377i	$0.355253\pi$
$60$	0	0
$61$	−11.8150	−1.51275	−0.756375	−	0.654138i	$-0.773032\pi$
$61$	−11.8150	−1.51275	−0.756375	+	0.654138i	$0.773032\pi$
$62$	−19.8489	−2.52082
$63$	0	0
$64$	−12.7732	−1.59666
$65$	−0.950022	−0.117836
$66$	0	0
$67$	−4.26966	−0.521622	−0.260811	−	0.965390i	$-0.583990\pi$
$67$	−4.26966	−0.521622	−0.260811	+	0.965390i	$0.583990\pi$
$68$	12.7683	1.54838
$69$	0	0
$70$	−2.38266	−0.284783
$71$	9.26730	1.09983	0.549913	−	0.835222i	$-0.314661\pi$
$71$	9.26730	1.09983	0.549913	+	0.835222i	$0.314661\pi$
$72$	0	0
$73$	15.7575	1.84428	0.922141	−	0.386854i	$-0.126438\pi$
$73$	15.7575	1.84428	0.922141	+	0.386854i	$0.126438\pi$
$74$	−7.59761	−0.883204
$75$	0	0
$76$	−3.32079	−0.380921
$77$	6.16153	0.702171
$78$	0	0
$79$	−5.05904	−0.569187	−0.284593	−	0.958648i	$-0.591859\pi$
$79$	−5.05904	−0.569187	−0.284593	+	0.958648i	$0.591859\pi$
$80$	0.171300	0.0191519
$81$	0	0
$82$	3.30520	0.364999
$83$	−3.63474	−0.398964	−0.199482	−	0.979901i	$-0.563926\pi$
$83$	−3.63474	−0.398964	−0.199482	+	0.979901i	$0.563926\pi$
$84$	0	0
$85$	1.70599	0.185041
$86$	13.3140	1.43569
$87$	0	0
$88$	−8.06348	−0.859570
$89$	−9.35195	−0.991305	−0.495652	−	0.868521i	$-0.665071\pi$
$89$	−9.35195	−0.991305	−0.495652	+	0.868521i	$0.665071\pi$
$90$	0	0
$91$	−4.98466	−0.522534
$92$	8.28131	0.863387
$93$	0	0
$94$	−2.30668	−0.237916
$95$	−0.443697	−0.0455223
$96$	0	0
$97$	15.9390	1.61836	0.809181	−	0.587559i	$-0.199911\pi$
$97$	15.9390	1.61836	0.809181	+	0.587559i	$0.199911\pi$
$98$	3.64521	0.368222
$99$	0	0
$100$	−15.9502	−1.59502
$101$	12.1873	1.21268	0.606342	−	0.795204i	$-0.292636\pi$
$101$	12.1873	1.21268	0.606342	+	0.795204i	$0.292636\pi$
$102$	0	0
$103$	12.7908	1.26032	0.630159	−	0.776466i	$-0.282990\pi$
$103$	12.7908	1.26032	0.630159	+	0.776466i	$0.282990\pi$
$104$	6.52333	0.639666
$105$	0	0
$106$	9.19692	0.893284
$107$	−9.53089	−0.921385	−0.460693	−	0.887560i	$-0.652399\pi$
$107$	−9.53089	−0.921385	−0.460693	+	0.887560i	$0.652399\pi$
$108$	0	0
$109$	2.36811	0.226824	0.113412	−	0.993548i	$-0.463822\pi$
$109$	2.36811	0.226824	0.113412	+	0.993548i	$0.463822\pi$
$110$	−2.70879	−0.258273
$111$	0	0
$112$	0.898790	0.0849277
$113$	−0.368233	−0.0346405	−0.0173202	−	0.999850i	$-0.505513\pi$
$113$	−0.368233	−0.0346405	−0.0173202	+	0.999850i	$0.505513\pi$
$114$	0	0
$115$	1.10648	0.103180
$116$	33.0291	3.06668
$117$	0	0
$118$	−15.5644	−1.43281
$119$	8.95116	0.820552
$120$	0	0
$121$	−3.99513	−0.363193
$122$	27.2534	2.46741
$123$	0	0
$124$	28.5753	2.56614
$125$	−4.34962	−0.389042
$126$	0	0
$127$	7.36474	0.653515	0.326757	−	0.945108i	$-0.394044\pi$
$127$	7.36474	0.653515	0.326757	+	0.945108i	$0.394044\pi$
$128$	19.0583	1.68454
$129$	0	0
$130$	2.19140	0.192199
$131$	9.24748	0.807956	0.403978	−	0.914769i	$-0.367627\pi$
$131$	9.24748	0.807956	0.403978	+	0.914769i	$0.367627\pi$
$132$	0	0
$133$	−2.32803	−0.201866
$134$	9.84877	0.850804
$135$	0	0
$136$	−11.7142	−1.00449
$137$	11.1889	0.955934	0.477967	−	0.878378i	$-0.341374\pi$
$137$	11.1889	0.955934	0.477967	+	0.878378i	$0.341374\pi$
$138$	0	0
$139$	−18.8358	−1.59763	−0.798814	−	0.601578i	$-0.794539\pi$
$139$	−18.8358	−1.59763	−0.798814	+	0.601578i	$0.794539\pi$
$140$	3.43018	0.289903
$141$	0	0
$142$	−21.3767	−1.79390
$143$	−5.66692	−0.473892
$144$	0	0
$145$	4.41308	0.366487
$146$	−36.3477	−3.00816
$147$	0	0
$148$	10.9378	0.899083
$149$	9.91540	0.812301	0.406150	−	0.913806i	$-0.366871\pi$
$149$	9.91540	0.812301	0.406150	+	0.913806i	$0.366871\pi$
$150$	0	0
$151$	14.6445	1.19175	0.595877	−	0.803076i	$-0.296805\pi$
$151$	14.6445	1.19175	0.595877	+	0.803076i	$0.296805\pi$
$152$	3.04665	0.247116
$153$	0	0
$154$	−14.2127	−1.14529
$155$	3.81800	0.306669
$156$	0	0
$157$	7.77650	0.620632	0.310316	−	0.950633i	$-0.399565\pi$
$157$	7.77650	0.620632	0.310316	+	0.950633i	$0.399565\pi$
$158$	11.6696	0.928385
$159$	0	0
$160$	2.30844	0.182498
$161$	5.80558	0.457544
$162$	0	0
$163$	7.10841	0.556773	0.278387	−	0.960469i	$-0.410200\pi$
$163$	7.10841	0.556773	0.278387	+	0.960469i	$0.410200\pi$
$164$	−4.75830	−0.371561
$165$	0	0
$166$	8.38419	0.650740
$167$	−18.3306	−1.41847	−0.709233	−	0.704974i	$-0.750959\pi$
$167$	−18.3306	−1.41847	−0.709233	+	0.704974i	$0.750959\pi$
$168$	0	0
$169$	−8.41547	−0.647344
$170$	−3.93519	−0.301815
$171$	0	0
$172$	−19.1674	−1.46150
$173$	7.52510	0.572123	0.286061	−	0.958211i	$-0.407654\pi$
$173$	7.52510	0.572123	0.286061	+	0.958211i	$0.407654\pi$
$174$	0	0
$175$	−11.1818	−0.845267
$176$	1.02181	0.0770218
$177$	0	0
$178$	21.5720	1.61689
$179$	1.49642	0.111848	0.0559238	−	0.998435i	$-0.482190\pi$
$179$	1.49642	0.111848	0.0559238	+	0.998435i	$0.482190\pi$
$180$	0	0
$181$	−9.82481	−0.730273	−0.365136	−	0.930954i	$-0.618978\pi$
$181$	−9.82481	−0.730273	−0.365136	+	0.930954i	$0.618978\pi$
$182$	11.4980	0.852291
$183$	0	0
$184$	−7.59766	−0.560107
$185$	1.46142	0.107446
$186$	0	0
$187$	10.1763	0.744167
$188$	3.32079	0.242194
$189$	0	0
$190$	1.02347	0.0742502
$191$	−9.39102	−0.679510	−0.339755	−	0.940514i	$-0.610344\pi$
$191$	−9.39102	−0.679510	−0.339755	+	0.940514i	$0.610344\pi$
$192$	0	0
$193$	7.67620	0.552545	0.276273	−	0.961079i	$-0.410901\pi$
$193$	7.67620	0.552545	0.276273	+	0.961079i	$0.410901\pi$
$194$	−36.7663	−2.63967
$195$	0	0
$196$	−5.24779	−0.374842
$197$	12.3853	0.882414	0.441207	−	0.897405i	$-0.354550\pi$
$197$	12.3853	0.882414	0.441207	+	0.897405i	$0.354550\pi$
$198$	0	0
$199$	−9.90072	−0.701843	−0.350922	−	0.936405i	$-0.614132\pi$
$199$	−9.90072	−0.701843	−0.350922	+	0.936405i	$0.614132\pi$
$200$	14.6335	1.03474
$201$	0	0
$202$	−28.1123	−1.97797
$203$	23.1550	1.62516
$204$	0	0
$205$	−0.635765	−0.0444037
$206$	−29.5044	−2.05567
$207$	0	0
$208$	−0.826642	−0.0573173
$209$	−2.64667	−0.183074
$210$	0	0
$211$	0.345461	0.0237825	0.0118913	−	0.999929i	$-0.496215\pi$
$211$	0.345461	0.0237825	0.0118913	+	0.999929i	$0.496215\pi$
$212$	−13.2402	−0.909344
$213$	0	0
$214$	21.9847	1.50285
$215$	−2.56099	−0.174658
$216$	0	0
$217$	20.0326	1.35990
$218$	−5.46249	−0.369966
$219$	0	0
$220$	3.89967	0.262916
$221$	−8.23262	−0.553786
$222$	0	0
$223$	12.7396	0.853106	0.426553	−	0.904463i	$-0.359728\pi$
$223$	12.7396	0.853106	0.426553	+	0.904463i	$0.359728\pi$
$224$	12.1121	0.809277
$225$	0	0
$226$	0.849398	0.0565011
$227$	−4.92330	−0.326771	−0.163385	−	0.986562i	$-0.552241\pi$
$227$	−4.92330	−0.326771	−0.163385	+	0.986562i	$0.552241\pi$
$228$	0	0
$229$	2.32000	0.153310	0.0766550	−	0.997058i	$-0.475576\pi$
$229$	2.32000	0.153310	0.0766550	+	0.997058i	$0.475576\pi$
$230$	−2.55230	−0.168294
$231$	0	0
$232$	−30.3025	−1.98946
$233$	−19.9260	−1.30540	−0.652698	−	0.757618i	$-0.726363\pi$
$233$	−19.9260	−1.30540	−0.652698	+	0.757618i	$0.726363\pi$
$234$	0	0
$235$	0.443697	0.0289436
$236$	22.4070	1.45857
$237$	0	0
$238$	−20.6475	−1.33838
$239$	−21.3762	−1.38271	−0.691357	−	0.722513i	$-0.742987\pi$
$239$	−21.3762	−1.38271	−0.691357	+	0.722513i	$0.742987\pi$
$240$	0	0
$241$	−19.9728	−1.28656	−0.643279	−	0.765632i	$-0.722426\pi$
$241$	−19.9728	−1.28656	−0.643279	+	0.765632i	$0.722426\pi$
$242$	9.21549	0.592394
$243$	0	0
$244$	−39.2350	−2.51177
$245$	−0.701166	−0.0447959
$246$	0	0
$247$	2.14115	0.136238
$248$	−26.2163	−1.66474
$249$	0	0
$250$	10.0332	0.634555
$251$	21.8085	1.37654	0.688270	−	0.725454i	$-0.258370\pi$
$251$	21.8085	1.37654	0.688270	+	0.725454i	$0.258370\pi$
$252$	0	0
$253$	6.60021	0.414952
$254$	−16.9881	−1.06593
$255$	0	0
$256$	−18.4151	−1.15094
$257$	−11.2367	−0.700927	−0.350464	−	0.936576i	$-0.613976\pi$
$257$	−11.2367	−0.700927	−0.350464	+	0.936576i	$0.613976\pi$
$258$	0	0
$259$	7.66791	0.476461
$260$	−3.15482	−0.195654
$261$	0	0
$262$	−21.3310	−1.31784
$263$	−27.9499	−1.72346	−0.861731	−	0.507365i	$-0.830620\pi$
$263$	−27.9499	−1.72346	−0.861731	+	0.507365i	$0.830620\pi$
$264$	0	0
$265$	−1.76905	−0.108672
$266$	5.37003	0.329257
$267$	0	0
$268$	−14.1787	−0.866100
$269$	21.7479	1.32599	0.662997	−	0.748622i	$-0.269284\pi$
$269$	21.7479	1.32599	0.662997	+	0.748622i	$0.269284\pi$
$270$	0	0
$271$	−4.92826	−0.299370	−0.149685	−	0.988734i	$-0.547826\pi$
$271$	−4.92826	−0.299370	−0.149685	+	0.988734i	$0.547826\pi$
$272$	1.48443	0.0900071
$273$	0	0
$274$	−25.8093	−1.55920
$275$	−12.7123	−0.766582
$276$	0	0
$277$	19.2101	1.15422	0.577112	−	0.816665i	$-0.304180\pi$
$277$	19.2101	1.15422	0.577112	+	0.816665i	$0.304180\pi$
$278$	43.4481	2.60585
$279$	0	0
$280$	−3.14700	−0.188069
$281$	16.5872	0.989509	0.494754	−	0.869033i	$-0.335258\pi$
$281$	16.5872	0.989509	0.494754	+	0.869033i	$0.335258\pi$
$282$	0	0
$283$	8.90013	0.529058	0.264529	−	0.964378i	$-0.414784\pi$
$283$	8.90013	0.529058	0.264529	+	0.964378i	$0.414784\pi$
$284$	30.7748	1.82615
$285$	0	0
$286$	13.0718	0.772952
$287$	−3.33579	−0.196905
$288$	0	0
$289$	−2.21634	−0.130373
$290$	−10.1796	−0.597766
$291$	0	0
$292$	52.3275	3.06224
$293$	26.1683	1.52877	0.764385	−	0.644760i	$-0.223043\pi$
$293$	26.1683	1.52877	0.764385	+	0.644760i	$0.223043\pi$
$294$	0	0
$295$	2.99385	0.174308
$296$	−10.0349	−0.583264
$297$	0	0
$298$	−22.8717	−1.32492
$299$	−5.33955	−0.308794
$300$	0	0
$301$	−13.4372	−0.774509
$302$	−33.7803	−1.94384
$303$	0	0
$304$	−0.386074	−0.0221428
$305$	−5.24226	−0.300171
$306$	0	0
$307$	29.4634	1.68156	0.840782	−	0.541375i	$-0.182096\pi$
$307$	29.4634	1.68156	0.840782	+	0.541375i	$0.182096\pi$
$308$	20.4612	1.16588
$309$	0	0
$310$	−8.80691	−0.500199
$311$	−7.16349	−0.406204	−0.203102	−	0.979158i	$-0.565102\pi$
$311$	−7.16349	−0.406204	−0.203102	+	0.979158i	$0.565102\pi$
$312$	0	0
$313$	−27.3842	−1.54785	−0.773923	−	0.633280i	$-0.781708\pi$
$313$	−27.3842	−1.54785	−0.773923	+	0.633280i	$0.781708\pi$
$314$	−17.9379	−1.01230
$315$	0	0
$316$	−16.8000	−0.945075
$317$	8.16338	0.458501	0.229251	−	0.973367i	$-0.426372\pi$
$317$	8.16338	0.458501	0.229251	+	0.973367i	$0.426372\pi$
$318$	0	0
$319$	26.3242	1.47387
$320$	−5.66745	−0.316820
$321$	0	0
$322$	−13.3916	−0.746287
$323$	−3.84495	−0.213939
$324$	0	0
$325$	10.2842	0.570467
$326$	−16.3969	−0.908138
$327$	0	0
$328$	4.36549	0.241044
$329$	2.32803	0.128348
$330$	0	0
$331$	−7.33837	−0.403353	−0.201677	−	0.979452i	$-0.564639\pi$
$331$	−7.33837	−0.403353	−0.201677	+	0.979452i	$0.564639\pi$
$332$	−12.0702	−0.662439
$333$	0	0
$334$	42.2830	2.31362
$335$	−1.89444	−0.103504
$336$	0	0
$337$	−1.50339	−0.0818947	−0.0409473	−	0.999161i	$-0.513038\pi$
$337$	−1.50339	−0.0818947	−0.0409473	+	0.999161i	$0.513038\pi$
$338$	19.4118	1.05586
$339$	0	0
$340$	5.66525	0.307241
$341$	22.7745	1.23331
$342$	0	0
$343$	−19.9751	−1.07856
$344$	17.5851	0.948123
$345$	0	0
$346$	−17.3580	−0.933173
$347$	8.13496	0.436708	0.218354	−	0.975870i	$-0.429931\pi$
$347$	8.13496	0.436708	0.218354	+	0.975870i	$0.429931\pi$
$348$	0	0
$349$	−0.827674	−0.0443044	−0.0221522	−	0.999755i	$-0.507052\pi$
$349$	−0.827674	−0.0443044	−0.0221522	+	0.999755i	$0.507052\pi$
$350$	25.7930	1.37869
$351$	0	0
$352$	13.7700	0.733942
$353$	7.57671	0.403268	0.201634	−	0.979461i	$-0.435375\pi$
$353$	7.57671	0.403268	0.201634	+	0.979461i	$0.435375\pi$
$354$	0	0
$355$	4.11187	0.218236
$356$	−31.0559	−1.64596
$357$	0	0
$358$	−3.45176	−0.182431
$359$	32.1400	1.69629	0.848143	−	0.529767i	$-0.177721\pi$
$359$	32.1400	1.69629	0.848143	+	0.529767i	$0.177721\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	22.6627	1.19113
$363$	0	0
$364$	−16.5530	−0.867614
$365$	6.99157	0.365956
$366$	0	0
$367$	−26.8551	−1.40182	−0.700912	−	0.713248i	$-0.747223\pi$
$367$	−26.8551	−1.40182	−0.700912	+	0.713248i	$0.747223\pi$
$368$	0.962781	0.0501884
$369$	0	0
$370$	−3.37104	−0.175252
$371$	−9.28202	−0.481899
$372$	0	0
$373$	−28.3864	−1.46979	−0.734896	−	0.678179i	$-0.762769\pi$
$373$	−28.3864	−1.46979	−0.734896	+	0.678179i	$0.762769\pi$
$374$	−23.4736	−1.21379
$375$	0	0
$376$	−3.04665	−0.157119
$377$	−21.2962	−1.09681
$378$	0	0
$379$	−18.5816	−0.954472	−0.477236	−	0.878775i	$-0.658361\pi$
$379$	−18.5816	−0.954472	−0.477236	+	0.878775i	$0.658361\pi$
$380$	−1.47342	−0.0755851
$381$	0	0
$382$	21.6621	1.10833
$383$	−35.3971	−1.80871	−0.904354	−	0.426782i	$-0.859647\pi$
$383$	−35.3971	−1.80871	−0.904354	+	0.426782i	$0.859647\pi$
$384$	0	0
$385$	2.73385	0.139330
$386$	−17.7066	−0.901241
$387$	0	0
$388$	52.9302	2.68712
$389$	−16.6317	−0.843262	−0.421631	−	0.906768i	$-0.638542\pi$
$389$	−16.6317	−0.843262	−0.421631	+	0.906768i	$0.638542\pi$
$390$	0	0
$391$	9.58846	0.484909
$392$	4.81457	0.243172
$393$	0	0
$394$	−28.5689	−1.43928
$395$	−2.24468	−0.112942
$396$	0	0
$397$	−29.1533	−1.46316	−0.731580	−	0.681756i	$-0.761217\pi$
$397$	−29.1533	−1.46316	−0.731580	+	0.681756i	$0.761217\pi$
$398$	22.8378	1.14476
$399$	0	0
$400$	−1.85436	−0.0927181
$401$	30.4347	1.51984	0.759919	−	0.650018i	$-0.225239\pi$
$401$	30.4347	1.51984	0.759919	+	0.650018i	$0.225239\pi$
$402$	0	0
$403$	−18.4245	−0.917791
$404$	40.4716	2.01354
$405$	0	0
$406$	−53.4112	−2.65075
$407$	8.71744	0.432108
$408$	0	0
$409$	−1.88022	−0.0929709	−0.0464854	−	0.998919i	$-0.514802\pi$
$409$	−1.88022	−0.0929709	−0.0464854	+	0.998919i	$0.514802\pi$
$410$	1.46651	0.0724257
$411$	0	0
$412$	42.4757	2.09263
$413$	15.7084	0.772959
$414$	0	0
$415$	−1.61272	−0.0791654
$416$	−11.1399	−0.546177
$417$	0	0
$418$	6.10504	0.298607
$419$	1.29625	0.0633261	0.0316631	−	0.999499i	$-0.489920\pi$
$419$	1.29625	0.0633261	0.0316631	+	0.999499i	$0.489920\pi$
$420$	0	0
$421$	36.2772	1.76804	0.884022	−	0.467445i	$-0.154825\pi$
$421$	36.2772	1.76804	0.884022	+	0.467445i	$0.154825\pi$
$422$	−0.796870	−0.0387910
$423$	0	0
$424$	12.1472	0.589921
$425$	−18.4678	−0.895821
$426$	0	0
$427$	−27.5056	−1.33109
$428$	−31.6501	−1.52986
$429$	0	0
$430$	5.90739	0.284880
$431$	14.1001	0.679180	0.339590	−	0.940574i	$-0.389712\pi$
$431$	14.1001	0.679180	0.339590	+	0.940574i	$0.389712\pi$
$432$	0	0
$433$	14.9894	0.720343	0.360171	−	0.932886i	$-0.382718\pi$
$433$	14.9894	0.720343	0.360171	+	0.932886i	$0.382718\pi$
$434$	−46.2089	−2.21810
$435$	0	0
$436$	7.86401	0.376618
$437$	−2.49378	−0.119294
$438$	0	0
$439$	−14.8878	−0.710558	−0.355279	−	0.934760i	$-0.615614\pi$
$439$	−14.8878	−0.710558	−0.355279	+	0.934760i	$0.615614\pi$
$440$	−3.57774	−0.170562
$441$	0	0
$442$	18.9901	0.903265
$443$	−2.07371	−0.0985251	−0.0492626	−	0.998786i	$-0.515687\pi$
$443$	−2.07371	−0.0985251	−0.0492626	+	0.998786i	$0.515687\pi$
$444$	0	0
$445$	−4.14943	−0.196702
$446$	−29.3862	−1.39148
$447$	0	0
$448$	−29.7365	−1.40492
$449$	−36.2480	−1.71065	−0.855324	−	0.518093i	$-0.826642\pi$
$449$	−36.2480	−1.71065	−0.855324	+	0.518093i	$0.826642\pi$
$450$	0	0
$451$	−3.79237	−0.178576
$452$	−1.22283	−0.0575169
$453$	0	0
$454$	11.3565	0.532986
$455$	−2.21168	−0.103685
$456$	0	0
$457$	−31.1691	−1.45803	−0.729014	−	0.684499i	$-0.760021\pi$
$457$	−31.1691	−1.45803	−0.729014	+	0.684499i	$0.760021\pi$
$458$	−5.35151	−0.250060
$459$	0	0
$460$	3.67439	0.171319
$461$	7.99178	0.372214	0.186107	−	0.982529i	$-0.440413\pi$
$461$	7.99178	0.372214	0.186107	+	0.982529i	$0.440413\pi$
$462$	0	0
$463$	29.9640	1.39254	0.696272	−	0.717778i	$-0.254841\pi$
$463$	29.9640	1.39254	0.696272	+	0.717778i	$0.254841\pi$
$464$	3.83995	0.178265
$465$	0	0
$466$	45.9630	2.12919
$467$	−26.1168	−1.20854	−0.604272	−	0.796778i	$-0.706536\pi$
$467$	−26.1168	−1.20854	−0.604272	+	0.796778i	$0.706536\pi$
$468$	0	0
$469$	−9.93990	−0.458982
$470$	−1.02347	−0.0472091
$471$	0	0
$472$	−20.5573	−0.946225
$473$	−15.2764	−0.702411
$474$	0	0
$475$	4.80313	0.220383
$476$	29.7249	1.36244
$477$	0	0
$478$	49.3082	2.25531
$479$	25.9659	1.18641	0.593207	−	0.805050i	$-0.297862\pi$
$479$	25.9659	1.18641	0.593207	+	0.805050i	$0.297862\pi$
$480$	0	0
$481$	−7.05239	−0.321561
$482$	46.0708	2.09847
$483$	0	0
$484$	−13.2670	−0.603045
$485$	7.07209	0.321127
$486$	0	0
$487$	0.448610	0.0203285	0.0101642	−	0.999948i	$-0.496765\pi$
$487$	0.448610	0.0203285	0.0101642	+	0.999948i	$0.496765\pi$
$488$	35.9960	1.62946
$489$	0	0
$490$	1.61737	0.0730653
$491$	11.2975	0.509851	0.254925	−	0.966961i	$-0.417949\pi$
$491$	11.2975	0.509851	0.254925	+	0.966961i	$0.417949\pi$
$492$	0	0
$493$	38.2425	1.72236
$494$	−4.93896	−0.222214
$495$	0	0
$496$	3.32215	0.149169
$497$	21.5745	0.967750
$498$	0	0
$499$	7.96923	0.356752	0.178376	−	0.983962i	$-0.442916\pi$
$499$	7.96923	0.356752	0.178376	+	0.983962i	$0.442916\pi$
$500$	−14.4442	−0.645963
$501$	0	0
$502$	−50.3053	−2.24524
$503$	−17.2733	−0.770179	−0.385090	−	0.922879i	$-0.625829\pi$
$503$	−17.2733	−0.770179	−0.385090	+	0.922879i	$0.625829\pi$
$504$	0	0
$505$	5.40748	0.240630
$506$	−15.2246	−0.676816
$507$	0	0
$508$	24.4568	1.08509
$509$	14.4497	0.640473	0.320237	−	0.947338i	$-0.396238\pi$
$509$	14.4497	0.640473	0.320237	+	0.947338i	$0.396238\pi$
$510$	0	0
$511$	36.6840	1.62281
$512$	4.36110	0.192735
$513$	0	0
$514$	25.9196	1.14326
$515$	5.67525	0.250081
$516$	0	0
$517$	2.64667	0.116401
$518$	−17.6875	−0.777142
$519$	0	0
$520$	2.89438	0.126927
$521$	−16.0593	−0.703572	−0.351786	−	0.936080i	$-0.614426\pi$
$521$	−16.0593	−0.703572	−0.351786	+	0.936080i	$0.614426\pi$
$522$	0	0
$523$	15.1792	0.663740	0.331870	−	0.943325i	$-0.392321\pi$
$523$	15.1792	0.663740	0.331870	+	0.943325i	$0.392321\pi$
$524$	30.7090	1.34153
$525$	0	0
$526$	64.4715	2.81109
$527$	33.0857	1.44124
$528$	0	0
$529$	−16.7811	−0.729612
$530$	4.08064	0.177252
$531$	0	0
$532$	−7.73090	−0.335177
$533$	3.06801	0.132890
$534$	0	0
$535$	−4.22882	−0.182828
$536$	13.0082	0.561867
$537$	0	0
$538$	−50.1656	−2.16279
$539$	−4.18249	−0.180153
$540$	0	0
$541$	17.6418	0.758482	0.379241	−	0.925298i	$-0.376185\pi$
$541$	17.6418	0.758482	0.379241	+	0.925298i	$0.376185\pi$
$542$	11.3679	0.488295
$543$	0	0
$544$	20.0043	0.857678
$545$	1.05072	0.0450081
$546$	0	0
$547$	22.6232	0.967300	0.483650	−	0.875261i	$-0.339311\pi$
$547$	22.6232	0.967300	0.483650	+	0.875261i	$0.339311\pi$
$548$	37.1561	1.58723
$549$	0	0
$550$	29.3233	1.25035
$551$	−9.94617	−0.423721
$552$	0	0
$553$	−11.7776	−0.500834
$554$	−44.3116	−1.88262
$555$	0	0
$556$	−62.5496	−2.65270
$557$	18.7297	0.793602	0.396801	−	0.917905i	$-0.370120\pi$
$557$	18.7297	0.793602	0.396801	+	0.917905i	$0.370120\pi$
$558$	0	0
$559$	12.3586	0.522712
$560$	0.398790	0.0168520
$561$	0	0
$562$	−38.2614	−1.61396
$563$	14.7946	0.623518	0.311759	−	0.950161i	$-0.399082\pi$
$563$	14.7946	0.623518	0.311759	+	0.950161i	$0.399082\pi$
$564$	0	0
$565$	−0.163384	−0.00687362
$566$	−20.5298	−0.862931
$567$	0	0
$568$	−28.2342	−1.18468
$569$	−12.2688	−0.514334	−0.257167	−	0.966367i	$-0.582789\pi$
$569$	−12.2688	−0.514334	−0.257167	+	0.966367i	$0.582789\pi$
$570$	0	0
$571$	4.46202	0.186730	0.0933648	−	0.995632i	$-0.470238\pi$
$571$	4.46202	0.186730	0.0933648	+	0.995632i	$0.470238\pi$
$572$	−18.8187	−0.786848
$573$	0	0
$574$	7.69461	0.321167
$575$	−11.9779	−0.499515
$576$	0	0
$577$	−25.7647	−1.07260	−0.536299	−	0.844028i	$-0.680178\pi$
$577$	−25.7647	−1.07260	−0.536299	+	0.844028i	$0.680178\pi$
$578$	5.11239	0.212647
$579$	0	0
$580$	14.6549	0.608513
$581$	−8.46177	−0.351054
$582$	0	0
$583$	−10.5525	−0.437039
$584$	−48.0077	−1.98657
$585$	0	0
$586$	−60.3620	−2.49353
$587$	−17.0342	−0.703076	−0.351538	−	0.936174i	$-0.614341\pi$
$587$	−17.0342	−0.703076	−0.351538	+	0.936174i	$0.614341\pi$
$588$	0	0
$589$	−8.60497	−0.354562
$590$	−6.90585	−0.284310
$591$	0	0
$592$	1.27162	0.0522635
$593$	−21.0934	−0.866204	−0.433102	−	0.901345i	$-0.642581\pi$
$593$	−21.0934	−0.866204	−0.433102	+	0.901345i	$0.642581\pi$
$594$	0	0
$595$	3.97160	0.162820
$596$	32.9270	1.34874
$597$	0	0
$598$	12.3167	0.503666
$599$	−12.5100	−0.511144	−0.255572	−	0.966790i	$-0.582264\pi$
$599$	−12.5100	−0.511144	−0.255572	+	0.966790i	$0.582264\pi$
$600$	0	0
$601$	−30.1536	−1.22999	−0.614995	−	0.788531i	$-0.710842\pi$
$601$	−30.1536	−1.22999	−0.614995	+	0.788531i	$0.710842\pi$
$602$	30.9954	1.26328
$603$	0	0
$604$	48.6314	1.97878
$605$	−1.77262	−0.0720674
$606$	0	0
$607$	−13.9699	−0.567021	−0.283510	−	0.958969i	$-0.591499\pi$
$607$	−13.9699	−0.567021	−0.283510	+	0.958969i	$0.591499\pi$
$608$	−5.20275	−0.210999
$609$	0	0
$610$	12.0922	0.489601
$611$	−2.14115	−0.0866217
$612$	0	0
$613$	7.02445	0.283715	0.141857	−	0.989887i	$-0.454693\pi$
$613$	7.02445	0.283715	0.141857	+	0.989887i	$0.454693\pi$
$614$	−67.9627	−2.74275
$615$	0	0
$616$	−18.7720	−0.756346
$617$	−4.36482	−0.175721	−0.0878604	−	0.996133i	$-0.528003\pi$
$617$	−4.36482	−0.175721	−0.0878604	+	0.996133i	$0.528003\pi$
$618$	0	0
$619$	−3.67693	−0.147788	−0.0738941	−	0.997266i	$-0.523543\pi$
$619$	−3.67693	−0.147788	−0.0738941	+	0.997266i	$0.523543\pi$
$620$	12.6788	0.509192
$621$	0	0
$622$	16.5239	0.662548
$623$	−21.7716	−0.872261
$624$	0	0
$625$	22.0858	0.883430
$626$	63.1667	2.52465
$627$	0	0
$628$	25.8241	1.03050
$629$	12.6643	0.504957
$630$	0	0
$631$	25.6402	1.02072	0.510360	−	0.859961i	$-0.329512\pi$
$631$	25.6402	1.02072	0.510360	+	0.859961i	$0.329512\pi$
$632$	15.4131	0.613101
$633$	0	0
$634$	−18.8303	−0.747849
$635$	3.26771	0.129675
$636$	0	0
$637$	3.38362	0.134064
$638$	−60.7217	−2.40400
$639$	0	0
$640$	8.45613	0.334258
$641$	30.9418	1.22213	0.611063	−	0.791582i	$-0.290742\pi$
$641$	30.9418	1.22213	0.611063	+	0.791582i	$0.290742\pi$
$642$	0	0
$643$	−19.0804	−0.752457	−0.376229	−	0.926527i	$-0.622779\pi$
$643$	−19.0804	−0.752457	−0.376229	+	0.926527i	$0.622779\pi$
$644$	19.2791	0.759704
$645$	0	0
$646$	8.86909	0.348950
$647$	46.2218	1.81717	0.908584	−	0.417703i	$-0.137165\pi$
$647$	46.2218	1.81717	0.908584	+	0.417703i	$0.137165\pi$
$648$	0	0
$649$	17.8584	0.701004
$650$	−23.7225	−0.930472
$651$	0	0
$652$	23.6055	0.924464
$653$	26.1224	1.02225	0.511125	−	0.859506i	$-0.329229\pi$
$653$	26.1224	1.02225	0.511125	+	0.859506i	$0.329229\pi$
$654$	0	0
$655$	4.10308	0.160321
$656$	−0.553197	−0.0215987
$657$	0	0
$658$	−5.37003	−0.209345
$659$	−13.7492	−0.535591	−0.267796	−	0.963476i	$-0.586295\pi$
$659$	−13.7492	−0.535591	−0.267796	+	0.963476i	$0.586295\pi$
$660$	0	0
$661$	38.8759	1.51210	0.756048	−	0.654516i	$-0.227128\pi$
$661$	38.8759	1.51210	0.756048	+	0.654516i	$0.227128\pi$
$662$	16.9273	0.657899
$663$	0	0
$664$	11.0738	0.429746
$665$	−1.03294	−0.0400557
$666$	0	0
$667$	24.8035	0.960396
$668$	−60.8722	−2.35522
$669$	0	0
$670$	4.36987	0.168823
$671$	−31.2703	−1.20718
$672$	0	0
$673$	1.97389	0.0760878	0.0380439	−	0.999276i	$-0.487887\pi$
$673$	1.97389	0.0760878	0.0380439	+	0.999276i	$0.487887\pi$
$674$	3.46784	0.133576
$675$	0	0
$676$	−27.9460	−1.07485
$677$	10.9005	0.418942	0.209471	−	0.977815i	$-0.432826\pi$
$677$	10.9005	0.418942	0.209471	+	0.977815i	$0.432826\pi$
$678$	0	0
$679$	37.1065	1.42402
$680$	−5.19756	−0.199317
$681$	0	0
$682$	−52.5336	−2.01162
$683$	11.9705	0.458037	0.229019	−	0.973422i	$-0.426448\pi$
$683$	11.9705	0.458037	0.229019	+	0.973422i	$0.426448\pi$
$684$	0	0
$685$	4.96449	0.189683
$686$	46.0763	1.75920
$687$	0	0
$688$	−2.22839	−0.0849566
$689$	8.53693	0.325231
$690$	0	0
$691$	−10.2376	−0.389455	−0.194728	−	0.980857i	$-0.562382\pi$
$691$	−10.2376	−0.389455	−0.194728	+	0.980857i	$0.562382\pi$
$692$	24.9893	0.949950
$693$	0	0
$694$	−18.7648	−0.712302
$695$	−8.35737	−0.317013
$696$	0	0
$697$	−5.50936	−0.208682
$698$	1.90918	0.0722636
$699$	0	0
$700$	−37.1325	−1.40348
$701$	−17.5250	−0.661910	−0.330955	−	0.943647i	$-0.607371\pi$
$701$	−17.5250	−0.661910	−0.330955	+	0.943647i	$0.607371\pi$
$702$	0	0
$703$	−3.29374	−0.124226
$704$	−33.8066	−1.27413
$705$	0	0
$706$	−17.4771	−0.657759
$707$	28.3724	1.06706
$708$	0	0
$709$	−37.9457	−1.42508	−0.712541	−	0.701630i	$-0.752456\pi$
$709$	−37.9457	−1.42508	−0.712541	+	0.701630i	$0.752456\pi$
$710$	−9.48479	−0.355958
$711$	0	0
$712$	28.4921	1.06779
$713$	21.4589	0.803641
$714$	0	0
$715$	−2.51440	−0.0940331
$716$	4.96929	0.185711
$717$	0	0
$718$	−74.1369	−2.76677
$719$	32.7848	1.22267	0.611334	−	0.791373i	$-0.290633\pi$
$719$	32.7848	1.22267	0.611334	+	0.791373i	$0.290633\pi$
$720$	0	0
$721$	29.7774	1.10897
$722$	−2.30668	−0.0858459
$723$	0	0
$724$	−32.6262	−1.21254
$725$	−47.7728	−1.77424
$726$	0	0
$727$	−36.3288	−1.34736	−0.673680	−	0.739023i	$-0.735288\pi$
$727$	−36.3288	−1.34736	−0.673680	+	0.739023i	$0.735288\pi$
$728$	15.1865	0.562850
$729$	0	0
$730$	−16.1274	−0.596900
$731$	−22.1928	−0.820831
$732$	0	0
$733$	36.5549	1.35018	0.675092	−	0.737733i	$-0.264104\pi$
$733$	36.5549	1.35018	0.675092	+	0.737733i	$0.264104\pi$
$734$	61.9462	2.28648
$735$	0	0
$736$	12.9745	0.478246
$737$	−11.3004	−0.416256
$738$	0	0
$739$	28.0523	1.03192	0.515960	−	0.856613i	$-0.327435\pi$
$739$	28.0523	1.03192	0.515960	+	0.856613i	$0.327435\pi$
$740$	4.85307	0.178403
$741$	0	0
$742$	21.4107	0.786011
$743$	20.8116	0.763505	0.381753	−	0.924265i	$-0.375321\pi$
$743$	20.8116	0.763505	0.381753	+	0.924265i	$0.375321\pi$
$744$	0	0
$745$	4.39943	0.161183
$746$	65.4785	2.39734
$747$	0	0
$748$	33.7935	1.23561
$749$	−22.1882	−0.810738
$750$	0	0
$751$	38.6479	1.41028	0.705141	−	0.709067i	$-0.250884\pi$
$751$	38.6479	1.41028	0.705141	+	0.709067i	$0.250884\pi$
$752$	0.386074	0.0140787
$753$	0	0
$754$	49.1237	1.78898
$755$	6.49773	0.236477
$756$	0	0
$757$	41.6802	1.51489	0.757446	−	0.652897i	$-0.226447\pi$
$757$	41.6802	1.51489	0.757446	+	0.652897i	$0.226447\pi$
$758$	42.8618	1.55681
$759$	0	0
$760$	1.35179	0.0490345
$761$	−5.98577	−0.216984	−0.108492	−	0.994097i	$-0.534602\pi$
$761$	−5.98577	−0.216984	−0.108492	+	0.994097i	$0.534602\pi$
$762$	0	0
$763$	5.51304	0.199585
$764$	−31.1856	−1.12826
$765$	0	0
$766$	81.6500	2.95013
$767$	−14.4474	−0.521666
$768$	0	0
$769$	−21.6388	−0.780313	−0.390157	−	0.920748i	$-0.627579\pi$
$769$	−21.6388	−0.780313	−0.390157	+	0.920748i	$0.627579\pi$
$770$	−6.30613	−0.227257
$771$	0	0
$772$	25.4911	0.917444
$773$	40.0533	1.44062	0.720308	−	0.693654i	$-0.244000\pi$
$773$	40.0533	1.44062	0.720308	+	0.693654i	$0.244000\pi$
$774$	0	0
$775$	−41.3308	−1.48465
$776$	−48.5606	−1.74322
$777$	0	0
$778$	38.3641	1.37542
$779$	1.43288	0.0513383
$780$	0	0
$781$	24.5275	0.877663
$782$	−22.1175	−0.790922
$783$	0	0
$784$	−0.610105	−0.0217895
$785$	3.45041	0.123150
$786$	0	0
$787$	44.7751	1.59606	0.798029	−	0.602619i	$-0.205876\pi$
$787$	44.7751	1.59606	0.798029	+	0.602619i	$0.205876\pi$
$788$	41.1289	1.46516
$789$	0	0
$790$	5.17777	0.184217
$791$	−0.857258	−0.0304806
$792$	0	0
$793$	25.2976	0.898345
$794$	67.2474	2.38652
$795$	0	0
$796$	−32.8782	−1.16534
$797$	37.6088	1.33217	0.666085	−	0.745876i	$-0.267969\pi$
$797$	37.6088	1.33217	0.666085	+	0.745876i	$0.267969\pi$
$798$	0	0
$799$	3.84495	0.136025
$800$	−24.9895	−0.883512
$801$	0	0
$802$	−70.2033	−2.47897
$803$	41.7051	1.47174
$804$	0	0
$805$	2.57592	0.0907892
$806$	42.4996	1.49698
$807$	0	0
$808$	−37.1305	−1.30625
$809$	31.6712	1.11350	0.556751	−	0.830680i	$-0.312048\pi$
$809$	31.6712	1.11350	0.556751	+	0.830680i	$0.312048\pi$
$810$	0	0
$811$	−50.2797	−1.76556	−0.882779	−	0.469789i	$-0.844330\pi$
$811$	−50.2797	−1.76556	−0.882779	+	0.469789i	$0.844330\pi$
$812$	76.8928	2.69841
$813$	0	0
$814$	−20.1084	−0.704799
$815$	3.15398	0.110479
$816$	0	0
$817$	5.77194	0.201934
$818$	4.33707	0.151642
$819$	0	0
$820$	−2.11124	−0.0737278
$821$	−7.17944	−0.250564	−0.125282	−	0.992121i	$-0.539984\pi$
$821$	−7.17944	−0.250564	−0.125282	+	0.992121i	$0.539984\pi$
$822$	0	0
$823$	54.6853	1.90621	0.953105	−	0.302639i	$-0.0978677\pi$
$823$	54.6853	1.90621	0.953105	+	0.302639i	$0.0978677\pi$
$824$	−38.9692	−1.35756
$825$	0	0
$826$	−36.2343	−1.26075
$827$	44.9897	1.56444	0.782222	−	0.622999i	$-0.214086\pi$
$827$	44.9897	1.56444	0.782222	+	0.622999i	$0.214086\pi$
$828$	0	0
$829$	−15.6005	−0.541829	−0.270915	−	0.962603i	$-0.587326\pi$
$829$	−15.6005	−0.541829	−0.270915	+	0.962603i	$0.587326\pi$
$830$	3.72004	0.129124
$831$	0	0
$832$	27.3494	0.948171
$833$	−6.07611	−0.210525
$834$	0	0
$835$	−8.13324	−0.281463
$836$	−8.78905	−0.303976
$837$	0	0
$838$	−2.99005	−0.103289
$839$	28.2839	0.976467	0.488233	−	0.872713i	$-0.337641\pi$
$839$	28.2839	0.976467	0.488233	+	0.872713i	$0.337641\pi$
$840$	0	0
$841$	69.9262	2.41125
$842$	−83.6801	−2.88381
$843$	0	0
$844$	1.14720	0.0394884
$845$	−3.73392	−0.128451
$846$	0	0
$847$	−9.30077	−0.319578
$848$	−1.53930	−0.0528599
$849$	0	0
$850$	42.5994	1.46115
$851$	8.21385	0.281567
$852$	0	0
$853$	35.0647	1.20059	0.600297	−	0.799778i	$-0.295049\pi$
$853$	35.0647	1.20059	0.600297	+	0.799778i	$0.295049\pi$
$854$	63.4467	2.17110
$855$	0	0
$856$	29.0373	0.992473
$857$	−50.5747	−1.72760	−0.863800	−	0.503836i	$-0.831922\pi$
$857$	−50.5747	−1.72760	−0.863800	+	0.503836i	$0.831922\pi$
$858$	0	0
$859$	4.56934	0.155904	0.0779520	−	0.996957i	$-0.475162\pi$
$859$	4.56934	0.155904	0.0779520	+	0.996957i	$0.475162\pi$
$860$	−8.50451	−0.290001
$861$	0	0
$862$	−32.5246	−1.10779
$863$	0.173763	0.00591496	0.00295748	−	0.999996i	$-0.499059\pi$
$863$	0.173763	0.00591496	0.00295748	+	0.999996i	$0.499059\pi$
$864$	0	0
$865$	3.33886	0.113525
$866$	−34.5757	−1.17493
$867$	0	0
$868$	66.5241	2.25798
$869$	−13.3896	−0.454212
$870$	0	0
$871$	9.14199	0.309765
$872$	−7.21481	−0.244324
$873$	0	0
$874$	5.75236	0.194576
$875$	−10.1260	−0.342323
$876$	0	0
$877$	42.4446	1.43325	0.716627	−	0.697457i	$-0.245685\pi$
$877$	42.4446	1.43325	0.716627	+	0.697457i	$0.245685\pi$
$878$	34.3416	1.15897
$879$	0	0
$880$	0.453374	0.0152832
$881$	6.19141	0.208594	0.104297	−	0.994546i	$-0.466741\pi$
$881$	6.19141	0.208594	0.104297	+	0.994546i	$0.466741\pi$
$882$	0	0
$883$	−8.14477	−0.274093	−0.137047	−	0.990565i	$-0.543761\pi$
$883$	−8.14477	−0.274093	−0.137047	+	0.990565i	$0.543761\pi$
$884$	−27.3388	−0.919504
$885$	0	0
$886$	4.78340	0.160702
$887$	17.6640	0.593099	0.296550	−	0.955017i	$-0.404164\pi$
$887$	17.6640	0.593099	0.296550	+	0.955017i	$0.404164\pi$
$888$	0	0
$889$	17.1453	0.575036
$890$	9.57142	0.320835
$891$	0	0
$892$	42.3055	1.41649
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	0.663956	0.0221936
$896$	44.3684	1.48224
$897$	0	0
$898$	83.6127	2.79019
$899$	85.5864	2.85447
$900$	0	0
$901$	−15.3301	−0.510720
$902$	8.74779	0.291270
$903$	0	0
$904$	1.12188	0.0373131
$905$	−4.35924	−0.144906
$906$	0	0
$907$	−51.8418	−1.72138	−0.860689	−	0.509131i	$-0.829967\pi$
$907$	−51.8418	−1.72138	−0.860689	+	0.509131i	$0.829967\pi$
$908$	−16.3492	−0.542569
$909$	0	0
$910$	5.10164	0.169118
$911$	−31.6870	−1.04984	−0.524919	−	0.851152i	$-0.675904\pi$
$911$	−31.6870	−1.04984	−0.524919	+	0.851152i	$0.675904\pi$
$912$	0	0
$913$	−9.61996	−0.318374
$914$	71.8972	2.37815
$915$	0	0
$916$	7.70424	0.254555
$917$	21.5284	0.710931
$918$	0	0
$919$	16.2868	0.537251	0.268625	−	0.963245i	$-0.413431\pi$
$919$	16.2868	0.537251	0.268625	+	0.963245i	$0.413431\pi$
$920$	−3.37106	−0.111141
$921$	0	0
$922$	−18.4345	−0.607109
$923$	−19.8427	−0.653130
$924$	0	0
$925$	−15.8203	−0.520167
$926$	−69.1175	−2.27134
$927$	0	0
$928$	51.7474	1.69869
$929$	−33.6298	−1.10336	−0.551679	−	0.834056i	$-0.686013\pi$
$929$	−33.6298	−1.10336	−0.551679	+	0.834056i	$0.686013\pi$
$930$	0	0
$931$	1.58028	0.0517917
$932$	−66.1701	−2.16747
$933$	0	0
$934$	60.2433	1.97122
$935$	4.51521	0.147663
$936$	0	0
$937$	43.1079	1.40827	0.704137	−	0.710064i	$-0.251334\pi$
$937$	43.1079	1.40827	0.704137	+	0.710064i	$0.251334\pi$
$938$	22.9282	0.748633
$939$	0	0
$940$	1.47342	0.0480578
$941$	−31.9752	−1.04236	−0.521180	−	0.853447i	$-0.674508\pi$
$941$	−31.9752	−1.04236	−0.521180	+	0.853447i	$0.674508\pi$
$942$	0	0
$943$	−3.57329	−0.116362
$944$	2.60503	0.0847866
$945$	0	0
$946$	35.2379	1.14568
$947$	−31.4929	−1.02338	−0.511690	−	0.859170i	$-0.670980\pi$
$947$	−31.4929	−1.02338	−0.511690	+	0.859170i	$0.670980\pi$
$948$	0	0
$949$	−33.7393	−1.09522
$950$	−11.0793	−0.359460
$951$	0	0
$952$	−27.2710	−0.883860
$953$	−20.5806	−0.666669	−0.333335	−	0.942809i	$-0.608174\pi$
$953$	−20.5806	−0.666669	−0.333335	+	0.942809i	$0.608174\pi$
$954$	0	0
$955$	−4.16676	−0.134833
$956$	−70.9860	−2.29585
$957$	0	0
$958$	−59.8952	−1.93513
$959$	26.0481	0.841138
$960$	0	0
$961$	43.0455	1.38856
$962$	16.2676	0.524489
$963$	0	0
$964$	−66.3253	−2.13620
$965$	3.40591	0.109640
$966$	0	0
$967$	−11.8183	−0.380052	−0.190026	−	0.981779i	$-0.560857\pi$
$967$	−11.8183	−0.380052	−0.190026	+	0.981779i	$0.560857\pi$
$968$	12.1717	0.391215
$969$	0	0
$970$	−16.3131	−0.523782
$971$	−49.0549	−1.57425	−0.787123	−	0.616796i	$-0.788430\pi$
$971$	−49.0549	−1.57425	−0.787123	+	0.616796i	$0.788430\pi$
$972$	0	0
$973$	−43.8502	−1.40577
$974$	−1.03480	−0.0331572
$975$	0	0
$976$	−4.56144	−0.146008
$977$	−0.324513	−0.0103821	−0.00519104	−	0.999987i	$-0.501652\pi$
$977$	−0.324513	−0.0103821	−0.00519104	+	0.999987i	$0.501652\pi$
$978$	0	0
$979$	−24.7515	−0.791063
$980$	−2.32843	−0.0743789
$981$	0	0
$982$	−26.0599	−0.831603
$983$	54.4676	1.73725	0.868623	−	0.495473i	$-0.165005\pi$
$983$	54.4676	1.73725	0.868623	+	0.495473i	$0.165005\pi$
$984$	0	0
$985$	5.49531	0.175095
$986$	−88.2135	−2.80929
$987$	0	0
$988$	7.11032	0.226209
$989$	−14.3939	−0.457700
$990$	0	0
$991$	−11.2265	−0.356621	−0.178311	−	0.983974i	$-0.557063\pi$
$991$	−11.2265	−0.356621	−0.178311	+	0.983974i	$0.557063\pi$
$992$	44.7695	1.42143
$993$	0	0
$994$	−49.7657	−1.57847
$995$	−4.39292	−0.139265
$996$	0	0
$997$	−20.1722	−0.638861	−0.319430	−	0.947610i	$-0.603492\pi$
$997$	−20.1722	−0.638861	−0.319430	+	0.947610i	$0.603492\pi$
$998$	−18.3825	−0.581888
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.4	yes	34
3.2	odd	2		8037.2.a.v.1.31	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.31	✓	34	3.2	odd	2
8037.2.a.w.1.4	yes	34	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30668 q^{2} +3.32079 q^{4} +0.443697 q^{5} +2.32803 q^{7} -3.04665 q^{8} +O(q^{10})\)	\(q-2.30668 q^{2} +3.32079 q^{4} +0.443697 q^{5} +2.32803 q^{7} -3.04665 q^{8} -1.02347 q^{10} +2.64667 q^{11} -2.14115 q^{13} -5.37003 q^{14} +0.386074 q^{16} +3.84495 q^{17} -1.00000 q^{19} +1.47342 q^{20} -6.10504 q^{22} +2.49378 q^{23} -4.80313 q^{25} +4.93896 q^{26} +7.73090 q^{28} +9.94617 q^{29} +8.60497 q^{31} +5.20275 q^{32} -8.86909 q^{34} +1.03294 q^{35} +3.29374 q^{37} +2.30668 q^{38} -1.35179 q^{40} -1.43288 q^{41} -5.77194 q^{43} +8.78905 q^{44} -5.75236 q^{46} +1.00000 q^{47} -1.58028 q^{49} +11.0793 q^{50} -7.11032 q^{52} -3.98707 q^{53} +1.17432 q^{55} -7.09269 q^{56} -22.9427 q^{58} +6.74750 q^{59} -11.8150 q^{61} -19.8489 q^{62} -12.7732 q^{64} -0.950022 q^{65} -4.26966 q^{67} +12.7683 q^{68} -2.38266 q^{70} +9.26730 q^{71} +15.7575 q^{73} -7.59761 q^{74} -3.32079 q^{76} +6.16153 q^{77} -5.05904 q^{79} +0.171300 q^{80} +3.30520 q^{82} -3.63474 q^{83} +1.70599 q^{85} +13.3140 q^{86} -8.06348 q^{88} -9.35195 q^{89} -4.98466 q^{91} +8.28131 q^{92} -2.30668 q^{94} -0.443697 q^{95} +15.9390 q^{97} +3.64521 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Introduction

L-functions

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Database

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