LMFDB - Embedded newform 8037.2.a.x.1.18

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			1.18
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)

$f(q)$	$=$	$q+0.229930 q^{2} -1.94713 q^{4} +1.59514 q^{5} +3.20891 q^{7} -0.907563 q^{8} +O(q^{10})$	\(q+0.229930 q^{2} -1.94713 q^{4} +1.59514 q^{5} +3.20891 q^{7} -0.907563 q^{8} +0.366771 q^{10} -3.65928 q^{11} +4.28550 q^{13} +0.737823 q^{14} +3.68559 q^{16} +2.97272 q^{17} +1.00000 q^{19} -3.10595 q^{20} -0.841378 q^{22} +8.98260 q^{23} -2.45552 q^{25} +0.985363 q^{26} -6.24817 q^{28} +2.14605 q^{29} +2.12634 q^{31} +2.66255 q^{32} +0.683516 q^{34} +5.11866 q^{35} +0.715594 q^{37} +0.229930 q^{38} -1.44769 q^{40} +5.79802 q^{41} -7.92324 q^{43} +7.12510 q^{44} +2.06537 q^{46} -1.00000 q^{47} +3.29709 q^{49} -0.564598 q^{50} -8.34443 q^{52} +0.646585 q^{53} -5.83707 q^{55} -2.91229 q^{56} +0.493440 q^{58} +0.548038 q^{59} -8.43038 q^{61} +0.488908 q^{62} -6.75898 q^{64} +6.83598 q^{65} +14.9120 q^{67} -5.78828 q^{68} +1.17693 q^{70} +0.164927 q^{71} +5.41249 q^{73} +0.164536 q^{74} -1.94713 q^{76} -11.7423 q^{77} -13.9688 q^{79} +5.87904 q^{80} +1.33314 q^{82} -2.10626 q^{83} +4.74191 q^{85} -1.82179 q^{86} +3.32103 q^{88} -8.28135 q^{89} +13.7518 q^{91} -17.4903 q^{92} -0.229930 q^{94} +1.59514 q^{95} -5.10654 q^{97} +0.758098 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8} + 18 q^{11} - 6 q^{13} + 12 q^{14} + 21 q^{16} + 36 q^{17} + 34 q^{19} + 40 q^{20} + 12 q^{22} + 38 q^{23} + 32 q^{25} + 15 q^{26} + 28 q^{28} + 14 q^{29} - 6 q^{31} + 35 q^{32} + 10 q^{34} + 46 q^{35} - 2 q^{37} + 5 q^{38} + 31 q^{40} + 18 q^{41} - 6 q^{43} + 42 q^{44} - 14 q^{46} - 34 q^{47} + 44 q^{49} + 9 q^{50} + 2 q^{52} + 32 q^{53} + 8 q^{55} - 4 q^{56} + 8 q^{58} + 62 q^{59} - 10 q^{61} + 30 q^{62} - 37 q^{64} + 8 q^{65} + 92 q^{68} - 62 q^{70} + 4 q^{71} - 8 q^{73} + 34 q^{74} + 31 q^{76} + 52 q^{77} + 40 q^{79} + 48 q^{80} - 2 q^{82} + 110 q^{83} - 12 q^{85} + 16 q^{86} - 44 q^{88} + 2 q^{89} - 28 q^{91} + 60 q^{92} - 5 q^{94} + 14 q^{95} + 2 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8 + 18 * q^11 - 6 * q^13 + 12 * q^14 + 21 * q^16 + 36 * q^17 + 34 * q^19 + 40 * q^20 + 12 * q^22 + 38 * q^23 + 32 * q^25 + 15 * q^26 + 28 * q^28 + 14 * q^29 - 6 * q^31 + 35 * q^32 + 10 * q^34 + 46 * q^35 - 2 * q^37 + 5 * q^38 + 31 * q^40 + 18 * q^41 - 6 * q^43 + 42 * q^44 - 14 * q^46 - 34 * q^47 + 44 * q^49 + 9 * q^50 + 2 * q^52 + 32 * q^53 + 8 * q^55 - 4 * q^56 + 8 * q^58 + 62 * q^59 - 10 * q^61 + 30 * q^62 - 37 * q^64 + 8 * q^65 + 92 * q^68 - 62 * q^70 + 4 * q^71 - 8 * q^73 + 34 * q^74 + 31 * q^76 + 52 * q^77 + 40 * q^79 + 48 * q^80 - 2 * q^82 + 110 * q^83 - 12 * q^85 + 16 * q^86 - 44 * q^88 + 2 * q^89 - 28 * q^91 + 60 * q^92 - 5 * q^94 + 14 * q^95 + 2 * 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 $

$2$	0.229930	0.162585	0.0812924	−	0.996690i	$-0.474095\pi$
$2$	0.229930	0.162585	0.0812924	+	0.996690i	$0.474095\pi$
$3$	0	0
$3$	0	0
$4$	−1.94713	−0.973566
$5$	1.59514	0.713369	0.356685	−	0.934225i	$-0.383907\pi$
$5$	1.59514	0.713369	0.356685	+	0.934225i	$0.383907\pi$
$6$	0	0
$7$	3.20891	1.21285	0.606426	−	0.795140i	$-0.292602\pi$
$7$	3.20891	1.21285	0.606426	+	0.795140i	$0.292602\pi$
$8$	−0.907563	−0.320872
$9$	0	0
$10$	0.366771	0.115983
$11$	−3.65928	−1.10331	−0.551657	−	0.834071i	$-0.686004\pi$
$11$	−3.65928	−1.10331	−0.551657	+	0.834071i	$0.686004\pi$
$12$	0	0
$13$	4.28550	1.18858	0.594291	−	0.804250i	$-0.297433\pi$
$13$	4.28550	1.18858	0.594291	+	0.804250i	$0.297433\pi$
$14$	0.737823	0.197192
$15$	0	0
$16$	3.68559	0.921397
$17$	2.97272	0.720990	0.360495	−	0.932761i	$-0.382608\pi$
$17$	2.97272	0.720990	0.360495	+	0.932761i	$0.382608\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.10595	−0.694512
$21$	0	0
$22$	−0.841378	−0.179382
$23$	8.98260	1.87300	0.936500	−	0.350666i	$-0.114045\pi$
$23$	8.98260	1.87300	0.936500	+	0.350666i	$0.114045\pi$
$24$	0	0
$25$	−2.45552	−0.491104
$26$	0.985363	0.193246
$27$	0	0
$28$	−6.24817	−1.18079
$29$	2.14605	0.398511	0.199255	−	0.979948i	$-0.436148\pi$
$29$	2.14605	0.398511	0.199255	+	0.979948i	$0.436148\pi$
$30$	0	0
$31$	2.12634	0.381901	0.190951	−	0.981600i	$-0.438843\pi$
$31$	2.12634	0.381901	0.190951	+	0.981600i	$0.438843\pi$
$32$	2.66255	0.470677
$33$	0	0
$34$	0.683516	0.117222
$35$	5.11866	0.865212
$36$	0	0
$37$	0.715594	0.117643	0.0588214	−	0.998269i	$-0.481266\pi$
$37$	0.715594	0.117643	0.0588214	+	0.998269i	$0.481266\pi$
$38$	0.229930	0.0372995
$39$	0	0
$40$	−1.44769	−0.228900
$41$	5.79802	0.905498	0.452749	−	0.891638i	$-0.350443\pi$
$41$	5.79802	0.905498	0.452749	+	0.891638i	$0.350443\pi$
$42$	0	0
$43$	−7.92324	−1.20828	−0.604142	−	0.796877i	$-0.706484\pi$
$43$	−7.92324	−1.20828	−0.604142	+	0.796877i	$0.706484\pi$
$44$	7.12510	1.07415
$45$	0	0
$46$	2.06537	0.304522
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	3.29709	0.471012
$50$	−0.564598	−0.0798461
$51$	0	0
$52$	−8.34443	−1.15716
$53$	0.646585	0.0888153	0.0444076	−	0.999013i	$-0.485860\pi$
$53$	0.646585	0.0888153	0.0444076	+	0.999013i	$0.485860\pi$
$54$	0	0
$55$	−5.83707	−0.787071
$56$	−2.91229	−0.389171
$57$	0	0
$58$	0.493440	0.0647918
$59$	0.548038	0.0713484	0.0356742	−	0.999363i	$-0.488642\pi$
$59$	0.548038	0.0713484	0.0356742	+	0.999363i	$0.488642\pi$
$60$	0	0
$61$	−8.43038	−1.07940	−0.539700	−	0.841858i	$-0.681462\pi$
$61$	−8.43038	−1.07940	−0.539700	+	0.841858i	$0.681462\pi$
$62$	0.488908	0.0620914
$63$	0	0
$64$	−6.75898	−0.844872
$65$	6.83598	0.847898
$66$	0	0
$67$	14.9120	1.82179	0.910894	−	0.412641i	$-0.135394\pi$
$67$	14.9120	1.82179	0.910894	+	0.412641i	$0.135394\pi$
$68$	−5.78828	−0.701932
$69$	0	0
$70$	1.17693	0.140670
$71$	0.164927	0.0195732	0.00978660	−	0.999952i	$-0.496885\pi$
$71$	0.164927	0.0195732	0.00978660	+	0.999952i	$0.496885\pi$
$72$	0	0
$73$	5.41249	0.633484	0.316742	−	0.948512i	$-0.397411\pi$
$73$	5.41249	0.633484	0.316742	+	0.948512i	$0.397411\pi$
$74$	0.164536	0.0191270
$75$	0	0
$76$	−1.94713	−0.223351
$77$	−11.7423	−1.33816
$78$	0	0
$79$	−13.9688	−1.57161	−0.785806	−	0.618473i	$-0.787752\pi$
$79$	−13.9688	−1.57161	−0.785806	+	0.618473i	$0.787752\pi$
$80$	5.87904	0.657296
$81$	0	0
$82$	1.33314	0.147220
$83$	−2.10626	−0.231192	−0.115596	−	0.993296i	$-0.536878\pi$
$83$	−2.10626	−0.231192	−0.115596	+	0.993296i	$0.536878\pi$
$84$	0	0
$85$	4.74191	0.514332
$86$	−1.82179	−0.196449
$87$	0	0
$88$	3.32103	0.354023
$89$	−8.28135	−0.877821	−0.438911	−	0.898531i	$-0.644636\pi$
$89$	−8.28135	−0.877821	−0.438911	+	0.898531i	$0.644636\pi$
$90$	0	0
$91$	13.7518	1.44158
$92$	−17.4903	−1.82349
$93$	0	0
$94$	−0.229930	−0.0237154
$95$	1.59514	0.163658
$96$	0	0
$97$	−5.10654	−0.518491	−0.259245	−	0.965811i	$-0.583474\pi$
$97$	−5.10654	−0.518491	−0.259245	+	0.965811i	$0.583474\pi$
$98$	0.758098	0.0765795
$99$	0	0
$100$	4.78123	0.478123
$101$	9.05962	0.901466	0.450733	−	0.892659i	$-0.351163\pi$
$101$	9.05962	0.901466	0.450733	+	0.892659i	$0.351163\pi$
$102$	0	0
$103$	2.91916	0.287634	0.143817	−	0.989604i	$-0.454062\pi$
$103$	2.91916	0.287634	0.143817	+	0.989604i	$0.454062\pi$
$104$	−3.88936	−0.381383
$105$	0	0
$106$	0.148669	0.0144400
$107$	7.71657	0.745989	0.372994	−	0.927834i	$-0.378331\pi$
$107$	7.71657	0.745989	0.372994	+	0.927834i	$0.378331\pi$
$108$	0	0
$109$	0.0666058	0.00637968	0.00318984	−	0.999995i	$-0.498985\pi$
$109$	0.0666058	0.00637968	0.00318984	+	0.999995i	$0.498985\pi$
$110$	−1.34212	−0.127966
$111$	0	0
$112$	11.8267	1.11752
$113$	17.9781	1.69123	0.845617	−	0.533790i	$-0.179233\pi$
$113$	17.9781	1.69123	0.845617	+	0.533790i	$0.179233\pi$
$114$	0	0
$115$	14.3285	1.33614
$116$	−4.17864	−0.387977
$117$	0	0
$118$	0.126010	0.0116002
$119$	9.53918	0.874455
$120$	0	0
$121$	2.39034	0.217303
$122$	−1.93839	−0.175494
$123$	0	0
$124$	−4.14026	−0.371806
$125$	−11.8926	−1.06371
$126$	0	0
$127$	−2.04201	−0.181199	−0.0905994	−	0.995887i	$-0.528878\pi$
$127$	−2.04201	−0.181199	−0.0905994	+	0.995887i	$0.528878\pi$
$128$	−6.87920	−0.608041
$129$	0	0
$130$	1.57179	0.137855
$131$	4.73909	0.414056	0.207028	−	0.978335i	$-0.433621\pi$
$131$	4.73909	0.414056	0.207028	+	0.978335i	$0.433621\pi$
$132$	0	0
$133$	3.20891	0.278248
$134$	3.42871	0.296195
$135$	0	0
$136$	−2.69793	−0.231346
$137$	−1.71922	−0.146883	−0.0734416	−	0.997300i	$-0.523398\pi$
$137$	−1.71922	−0.146883	−0.0734416	+	0.997300i	$0.523398\pi$
$138$	0	0
$139$	15.5681	1.32047	0.660235	−	0.751059i	$-0.270457\pi$
$139$	15.5681	1.32047	0.660235	+	0.751059i	$0.270457\pi$
$140$	−9.96671	−0.842341
$141$	0	0
$142$	0.0379215	0.00318231
$143$	−15.6818	−1.31138
$144$	0	0
$145$	3.42325	0.284285
$146$	1.24449	0.102995
$147$	0	0
$148$	−1.39336	−0.114533
$149$	−20.4277	−1.67350	−0.836751	−	0.547583i	$-0.815548\pi$
$149$	−20.4277	−1.67350	−0.836751	+	0.547583i	$0.815548\pi$
$150$	0	0
$151$	−3.05118	−0.248301	−0.124151	−	0.992263i	$-0.539621\pi$
$151$	−3.05118	−0.248301	−0.124151	+	0.992263i	$0.539621\pi$
$152$	−0.907563	−0.0736131
$153$	0	0
$154$	−2.69990	−0.217564
$155$	3.39181	0.272437
$156$	0	0
$157$	−9.73658	−0.777064	−0.388532	−	0.921435i	$-0.627018\pi$
$157$	−9.73658	−0.777064	−0.388532	+	0.921435i	$0.627018\pi$
$158$	−3.21184	−0.255520
$159$	0	0
$160$	4.24715	0.335767
$161$	28.8243	2.27167
$162$	0	0
$163$	−3.48275	−0.272790	−0.136395	−	0.990655i	$-0.543552\pi$
$163$	−3.48275	−0.272790	−0.136395	+	0.990655i	$0.543552\pi$
$164$	−11.2895	−0.881562
$165$	0	0
$166$	−0.484291	−0.0375883
$167$	3.23071	0.250000	0.125000	−	0.992157i	$-0.460107\pi$
$167$	3.23071	0.250000	0.125000	+	0.992157i	$0.460107\pi$
$168$	0	0
$169$	5.36548	0.412729
$170$	1.09031	0.0836226
$171$	0	0
$172$	15.4276	1.17634
$173$	19.1107	1.45296	0.726480	−	0.687187i	$-0.241155\pi$
$173$	19.1107	1.45296	0.726480	+	0.687187i	$0.241155\pi$
$174$	0	0
$175$	−7.87954	−0.595637
$176$	−13.4866	−1.01659
$177$	0	0
$178$	−1.90413	−0.142720
$179$	−2.88107	−0.215341	−0.107670	−	0.994187i	$-0.534339\pi$
$179$	−2.88107	−0.215341	−0.107670	+	0.994187i	$0.534339\pi$
$180$	0	0
$181$	0.407269	0.0302720	0.0151360	−	0.999885i	$-0.495182\pi$
$181$	0.407269	0.0302720	0.0151360	+	0.999885i	$0.495182\pi$
$182$	3.16194	0.234379
$183$	0	0
$184$	−8.15227	−0.600994
$185$	1.14147	0.0839228
$186$	0	0
$187$	−10.8780	−0.795479
$188$	1.94713	0.142009
$189$	0	0
$190$	0.366771	0.0266083
$191$	7.80195	0.564529	0.282265	−	0.959337i	$-0.408914\pi$
$191$	7.80195	0.564529	0.282265	+	0.959337i	$0.408914\pi$
$192$	0	0
$193$	8.96865	0.645577	0.322789	−	0.946471i	$-0.395380\pi$
$193$	8.96865	0.645577	0.322789	+	0.946471i	$0.395380\pi$
$194$	−1.17415	−0.0842988
$195$	0	0
$196$	−6.41986	−0.458561
$197$	−15.8153	−1.12679	−0.563397	−	0.826186i	$-0.690506\pi$
$197$	−15.8153	−1.12679	−0.563397	+	0.826186i	$0.690506\pi$
$198$	0	0
$199$	12.7779	0.905804	0.452902	−	0.891560i	$-0.350389\pi$
$199$	12.7779	0.905804	0.452902	+	0.891560i	$0.350389\pi$
$200$	2.22854	0.157582
$201$	0	0
$202$	2.08308	0.146565
$203$	6.88646	0.483335
$204$	0	0
$205$	9.24866	0.645955
$206$	0.671202	0.0467649
$207$	0	0
$208$	15.7946	1.09516
$209$	−3.65928	−0.253118
$210$	0	0
$211$	−3.02284	−0.208101	−0.104051	−	0.994572i	$-0.533180\pi$
$211$	−3.02284	−0.208101	−0.104051	+	0.994572i	$0.533180\pi$
$212$	−1.25899	−0.0864676
$213$	0	0
$214$	1.77427	0.121286
$215$	−12.6387	−0.861952
$216$	0	0
$217$	6.82322	0.463190
$218$	0.0153147	0.00103724
$219$	0	0
$220$	11.3656	0.766265
$221$	12.7396	0.856956
$222$	0	0
$223$	−17.6617	−1.18272	−0.591359	−	0.806408i	$-0.701408\pi$
$223$	−17.6617	−1.18272	−0.591359	+	0.806408i	$0.701408\pi$
$224$	8.54389	0.570862
$225$	0	0
$226$	4.13369	0.274969
$227$	21.3266	1.41549	0.707747	−	0.706466i	$-0.249712\pi$
$227$	21.3266	1.41549	0.707747	+	0.706466i	$0.249712\pi$
$228$	0	0
$229$	24.8372	1.64129	0.820645	−	0.571438i	$-0.193614\pi$
$229$	24.8372	1.64129	0.820645	+	0.571438i	$0.193614\pi$
$230$	3.29455	0.217236
$231$	0	0
$232$	−1.94767	−0.127871
$233$	9.24634	0.605748	0.302874	−	0.953031i	$-0.402054\pi$
$233$	9.24634	0.605748	0.302874	+	0.953031i	$0.402054\pi$
$234$	0	0
$235$	−1.59514	−0.104056
$236$	−1.06710	−0.0694624
$237$	0	0
$238$	2.19334	0.142173
$239$	−23.6197	−1.52783	−0.763917	−	0.645315i	$-0.776726\pi$
$239$	−23.6197	−1.52783	−0.763917	+	0.645315i	$0.776726\pi$
$240$	0	0
$241$	−28.3865	−1.82854	−0.914269	−	0.405108i	$-0.867234\pi$
$241$	−28.3865	−1.82854	−0.914269	+	0.405108i	$0.867234\pi$
$242$	0.549609	0.0353302
$243$	0	0
$244$	16.4151	1.05087
$245$	5.25932	0.336006
$246$	0	0
$247$	4.28550	0.272680
$248$	−1.92979	−0.122541
$249$	0	0
$250$	−2.73447	−0.172943
$251$	−7.30697	−0.461212	−0.230606	−	0.973047i	$-0.574071\pi$
$251$	−7.30697	−0.461212	−0.230606	+	0.973047i	$0.574071\pi$
$252$	0	0
$253$	−32.8698	−2.06651
$254$	−0.469518	−0.0294602
$255$	0	0
$256$	11.9362	0.746014
$257$	−3.11176	−0.194107	−0.0970533	−	0.995279i	$-0.530942\pi$
$257$	−3.11176	−0.194107	−0.0970533	+	0.995279i	$0.530942\pi$
$258$	0	0
$259$	2.29627	0.142683
$260$	−13.3105	−0.825485
$261$	0	0
$262$	1.08966	0.0673193
$263$	27.7971	1.71405	0.857023	−	0.515279i	$-0.172312\pi$
$263$	27.7971	1.71405	0.857023	+	0.515279i	$0.172312\pi$
$264$	0	0
$265$	1.03140	0.0633581
$266$	0.737823	0.0452388
$267$	0	0
$268$	−29.0356	−1.77363
$269$	27.0505	1.64930	0.824650	−	0.565643i	$-0.191372\pi$
$269$	27.0505	1.64930	0.824650	+	0.565643i	$0.191372\pi$
$270$	0	0
$271$	−13.7959	−0.838039	−0.419019	−	0.907977i	$-0.637626\pi$
$271$	−13.7959	−0.838039	−0.419019	+	0.907977i	$0.637626\pi$
$272$	10.9562	0.664318
$273$	0	0
$274$	−0.395300	−0.0238810
$275$	8.98544	0.541843
$276$	0	0
$277$	−6.94424	−0.417239	−0.208619	−	0.977997i	$-0.566897\pi$
$277$	−6.94424	−0.417239	−0.208619	+	0.977997i	$0.566897\pi$
$278$	3.57957	0.214688
$279$	0	0
$280$	−4.64551	−0.277622
$281$	23.1144	1.37889	0.689444	−	0.724339i	$-0.257855\pi$
$281$	23.1144	1.37889	0.689444	+	0.724339i	$0.257855\pi$
$282$	0	0
$283$	3.98060	0.236622	0.118311	−	0.992977i	$-0.462252\pi$
$283$	3.98060	0.236622	0.118311	+	0.992977i	$0.462252\pi$
$284$	−0.321134	−0.0190558
$285$	0	0
$286$	−3.60572	−0.213211
$287$	18.6053	1.09824
$288$	0	0
$289$	−8.16295	−0.480173
$290$	0.787107	0.0462205
$291$	0	0
$292$	−10.5388	−0.616738
$293$	−17.1362	−1.00111	−0.500555	−	0.865705i	$-0.666871\pi$
$293$	−17.1362	−1.00111	−0.500555	+	0.865705i	$0.666871\pi$
$294$	0	0
$295$	0.874198	0.0508978
$296$	−0.649446	−0.0377483
$297$	0	0
$298$	−4.69694	−0.272086
$299$	38.4949	2.22622
$300$	0	0
$301$	−25.4250	−1.46547
$302$	−0.701556	−0.0403700
$303$	0	0
$304$	3.68559	0.211383
$305$	−13.4476	−0.770010
$306$	0	0
$307$	34.4373	1.96544	0.982719	−	0.185102i	$-0.0592615\pi$
$307$	34.4373	1.96544	0.982719	+	0.185102i	$0.0592615\pi$
$308$	22.8638	1.30279
$309$	0	0
$310$	0.779878	0.0442941
$311$	6.34936	0.360039	0.180020	−	0.983663i	$-0.442384\pi$
$311$	6.34936	0.360039	0.180020	+	0.983663i	$0.442384\pi$
$312$	0	0
$313$	3.30734	0.186942	0.0934709	−	0.995622i	$-0.470204\pi$
$313$	3.30734	0.186942	0.0934709	+	0.995622i	$0.470204\pi$
$314$	−2.23873	−0.126339
$315$	0	0
$316$	27.1991	1.53007
$317$	−10.0432	−0.564080	−0.282040	−	0.959403i	$-0.591011\pi$
$317$	−10.0432	−0.564080	−0.282040	+	0.959403i	$0.591011\pi$
$318$	0	0
$319$	−7.85299	−0.439683
$320$	−10.7815	−0.602706
$321$	0	0
$322$	6.62757	0.369340
$323$	2.97272	0.165406
$324$	0	0
$325$	−10.5231	−0.583718
$326$	−0.800788	−0.0443516
$327$	0	0
$328$	−5.26207	−0.290549
$329$	−3.20891	−0.176913
$330$	0	0
$331$	−13.7956	−0.758273	−0.379136	−	0.925341i	$-0.623779\pi$
$331$	−13.7956	−0.758273	−0.379136	+	0.925341i	$0.623779\pi$
$332$	4.10116	0.225081
$333$	0	0
$334$	0.742835	0.0406461
$335$	23.7867	1.29961
$336$	0	0
$337$	−11.1739	−0.608683	−0.304341	−	0.952563i	$-0.598436\pi$
$337$	−11.1739	−0.608683	−0.304341	+	0.952563i	$0.598436\pi$
$338$	1.23368	0.0671036
$339$	0	0
$340$	−9.23312	−0.500736
$341$	−7.78086	−0.421357
$342$	0	0
$343$	−11.8823	−0.641584
$344$	7.19084	0.387704
$345$	0	0
$346$	4.39412	0.236229
$347$	−1.34032	−0.0719522	−0.0359761	−	0.999353i	$-0.511454\pi$
$347$	−1.34032	−0.0719522	−0.0359761	+	0.999353i	$0.511454\pi$
$348$	0	0
$349$	35.5001	1.90028	0.950138	−	0.311831i	$-0.100942\pi$
$349$	35.5001	1.90028	0.950138	+	0.311831i	$0.100942\pi$
$350$	−1.81174	−0.0968416
$351$	0	0
$352$	−9.74303	−0.519305
$353$	23.0804	1.22845	0.614223	−	0.789132i	$-0.289469\pi$
$353$	23.0804	1.22845	0.614223	+	0.789132i	$0.289469\pi$
$354$	0	0
$355$	0.263081	0.0139629
$356$	16.1249	0.854617
$357$	0	0
$358$	−0.662443	−0.0350112
$359$	21.1317	1.11529	0.557644	−	0.830080i	$-0.311705\pi$
$359$	21.1317	1.11529	0.557644	+	0.830080i	$0.311705\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.0936432	0.00492177
$363$	0	0
$364$	−26.7765	−1.40347
$365$	8.63369	0.451908
$366$	0	0
$367$	23.4412	1.22362	0.611811	−	0.791004i	$-0.290441\pi$
$367$	23.4412	1.22362	0.611811	+	0.791004i	$0.290441\pi$
$368$	33.1062	1.72578
$369$	0	0
$370$	0.262459	0.0136446
$371$	2.07483	0.107720
$372$	0	0
$373$	14.8473	0.768762	0.384381	−	0.923175i	$-0.374415\pi$
$373$	14.8473	0.768762	0.384381	+	0.923175i	$0.374415\pi$
$374$	−2.50118	−0.129333
$375$	0	0
$376$	0.907563	0.0468040
$377$	9.19687	0.473663
$378$	0	0
$379$	3.69621	0.189862	0.0949308	−	0.995484i	$-0.469737\pi$
$379$	3.69621	0.189862	0.0949308	+	0.995484i	$0.469737\pi$
$380$	−3.10595	−0.159332
$381$	0	0
$382$	1.79390	0.0917839
$383$	1.93804	0.0990292	0.0495146	−	0.998773i	$-0.484233\pi$
$383$	1.93804	0.0990292	0.0495146	+	0.998773i	$0.484233\pi$
$384$	0	0
$385$	−18.7306	−0.954601
$386$	2.06216	0.104961
$387$	0	0
$388$	9.94311	0.504785
$389$	−7.63124	−0.386919	−0.193459	−	0.981108i	$-0.561971\pi$
$389$	−7.63124	−0.386919	−0.193459	+	0.981108i	$0.561971\pi$
$390$	0	0
$391$	26.7027	1.35042
$392$	−2.99231	−0.151135
$393$	0	0
$394$	−3.63641	−0.183200
$395$	−22.2822	−1.12114
$396$	0	0
$397$	−20.0146	−1.00450	−0.502251	−	0.864722i	$-0.667495\pi$
$397$	−20.0146	−1.00450	−0.502251	+	0.864722i	$0.667495\pi$
$398$	2.93803	0.147270
$399$	0	0
$400$	−9.05004	−0.452502
$401$	11.9712	0.597816	0.298908	−	0.954282i	$-0.403378\pi$
$401$	11.9712	0.597816	0.298908	+	0.954282i	$0.403378\pi$
$402$	0	0
$403$	9.11241	0.453921
$404$	−17.6403	−0.877637
$405$	0	0
$406$	1.58340	0.0785830
$407$	−2.61856	−0.129797
$408$	0	0
$409$	−29.3199	−1.44978	−0.724889	−	0.688866i	$-0.758109\pi$
$409$	−29.3199	−1.44978	−0.724889	+	0.688866i	$0.758109\pi$
$410$	2.12654	0.105022
$411$	0	0
$412$	−5.68400	−0.280030
$413$	1.75860	0.0865351
$414$	0	0
$415$	−3.35978	−0.164925
$416$	11.4104	0.559439
$417$	0	0
$418$	−0.841378	−0.0411531
$419$	25.3069	1.23632	0.618161	−	0.786052i	$-0.287878\pi$
$419$	25.3069	1.23632	0.618161	+	0.786052i	$0.287878\pi$
$420$	0	0
$421$	−23.4235	−1.14159	−0.570796	−	0.821092i	$-0.693365\pi$
$421$	−23.4235	−1.14159	−0.570796	+	0.821092i	$0.693365\pi$
$422$	−0.695042	−0.0338341
$423$	0	0
$424$	−0.586817	−0.0284983
$425$	−7.29957	−0.354081
$426$	0	0
$427$	−27.0523	−1.30915
$428$	−15.0252	−0.726269
$429$	0	0
$430$	−2.90601	−0.140140
$431$	−8.44900	−0.406974	−0.203487	−	0.979078i	$-0.565227\pi$
$431$	−8.44900	−0.406974	−0.203487	+	0.979078i	$0.565227\pi$
$432$	0	0
$433$	−10.9369	−0.525595	−0.262798	−	0.964851i	$-0.584645\pi$
$433$	−10.9369	−0.525595	−0.262798	+	0.964851i	$0.584645\pi$
$434$	1.56886	0.0753077
$435$	0	0
$436$	−0.129690	−0.00621104
$437$	8.98260	0.429696
$438$	0	0
$439$	6.40093	0.305500	0.152750	−	0.988265i	$-0.451187\pi$
$439$	6.40093	0.305500	0.152750	+	0.988265i	$0.451187\pi$
$440$	5.29751	0.252549
$441$	0	0
$442$	2.92921	0.139328
$443$	−3.94803	−0.187577	−0.0937883	−	0.995592i	$-0.529898\pi$
$443$	−3.94803	−0.187577	−0.0937883	+	0.995592i	$0.529898\pi$
$444$	0	0
$445$	−13.2099	−0.626211
$446$	−4.06096	−0.192292
$447$	0	0
$448$	−21.6889	−1.02471
$449$	5.77207	0.272401	0.136200	−	0.990681i	$-0.456511\pi$
$449$	5.77207	0.272401	0.136200	+	0.990681i	$0.456511\pi$
$450$	0	0
$451$	−21.2166	−0.999049
$452$	−35.0057	−1.64653
$453$	0	0
$454$	4.90361	0.230138
$455$	21.9360	1.02838
$456$	0	0
$457$	−34.0296	−1.59184	−0.795919	−	0.605403i	$-0.793012\pi$
$457$	−34.0296	−1.59184	−0.795919	+	0.605403i	$0.793012\pi$
$458$	5.71082	0.266849
$459$	0	0
$460$	−27.8995	−1.30082
$461$	5.22953	0.243563	0.121782	−	0.992557i	$-0.461139\pi$
$461$	5.22953	0.243563	0.121782	+	0.992557i	$0.461139\pi$
$462$	0	0
$463$	8.31294	0.386335	0.193168	−	0.981166i	$-0.438124\pi$
$463$	8.31294	0.386335	0.193168	+	0.981166i	$0.438124\pi$
$464$	7.90944	0.367187
$465$	0	0
$466$	2.12601	0.0984855
$467$	−1.27848	−0.0591608	−0.0295804	−	0.999562i	$-0.509417\pi$
$467$	−1.27848	−0.0591608	−0.0295804	+	0.999562i	$0.509417\pi$
$468$	0	0
$469$	47.8511	2.20956
$470$	−0.366771	−0.0169179
$471$	0	0
$472$	−0.497379	−0.0228937
$473$	28.9934	1.33312
$474$	0	0
$475$	−2.45552	−0.112667
$476$	−18.5740	−0.851340
$477$	0	0
$478$	−5.43088	−0.248403
$479$	−23.5256	−1.07491	−0.537456	−	0.843292i	$-0.680615\pi$
$479$	−23.5256	−1.07491	−0.537456	+	0.843292i	$0.680615\pi$
$480$	0	0
$481$	3.06667	0.139828
$482$	−6.52691	−0.297293
$483$	0	0
$484$	−4.65430	−0.211559
$485$	−8.14566	−0.369875
$486$	0	0
$487$	6.87850	0.311695	0.155847	−	0.987781i	$-0.450189\pi$
$487$	6.87850	0.311695	0.155847	+	0.987781i	$0.450189\pi$
$488$	7.65110	0.346349
$489$	0	0
$490$	1.20927	0.0546294
$491$	−7.59238	−0.342639	−0.171320	−	0.985215i	$-0.554803\pi$
$491$	−7.59238	−0.342639	−0.171320	+	0.985215i	$0.554803\pi$
$492$	0	0
$493$	6.37959	0.287322
$494$	0.985363	0.0443336
$495$	0	0
$496$	7.83680	0.351883
$497$	0.529234	0.0237394
$498$	0	0
$499$	16.8921	0.756193	0.378096	−	0.925766i	$-0.376579\pi$
$499$	16.8921	0.756193	0.378096	+	0.925766i	$0.376579\pi$
$500$	23.1565	1.03559
$501$	0	0
$502$	−1.68009	−0.0749861
$503$	15.9884	0.712886	0.356443	−	0.934317i	$-0.383989\pi$
$503$	15.9884	0.712886	0.356443	+	0.934317i	$0.383989\pi$
$504$	0	0
$505$	14.4514	0.643078
$506$	−7.55776	−0.335983
$507$	0	0
$508$	3.97606	0.176409
$509$	26.1389	1.15859	0.579293	−	0.815120i	$-0.303329\pi$
$509$	26.1389	1.15859	0.579293	+	0.815120i	$0.303329\pi$
$510$	0	0
$511$	17.3682	0.768323
$512$	16.5029	0.729331
$513$	0	0
$514$	−0.715487	−0.0315588
$515$	4.65648	0.205189
$516$	0	0
$517$	3.65928	0.160935
$518$	0.527982	0.0231982
$519$	0	0
$520$	−6.20408	−0.272067
$521$	−36.5217	−1.60004	−0.800022	−	0.599971i	$-0.795179\pi$
$521$	−36.5217	−1.60004	−0.800022	+	0.599971i	$0.795179\pi$
$522$	0	0
$523$	7.11552	0.311140	0.155570	−	0.987825i	$-0.450279\pi$
$523$	7.11552	0.311140	0.155570	+	0.987825i	$0.450279\pi$
$524$	−9.22764	−0.403111
$525$	0	0
$526$	6.39139	0.278678
$527$	6.32100	0.275347
$528$	0	0
$529$	57.6870	2.50813
$530$	0.237148	0.0103011
$531$	0	0
$532$	−6.24817	−0.270892
$533$	24.8474	1.07626
$534$	0	0
$535$	12.3090	0.532165
$536$	−13.5336	−0.584561
$537$	0	0
$538$	6.21973	0.268151
$539$	−12.0650	−0.519675
$540$	0	0
$541$	14.4900	0.622976	0.311488	−	0.950250i	$-0.399173\pi$
$541$	14.4900	0.622976	0.311488	+	0.950250i	$0.399173\pi$
$542$	−3.17208	−0.136252
$543$	0	0
$544$	7.91502	0.339354
$545$	0.106246	0.00455107
$546$	0	0
$547$	−31.4760	−1.34582	−0.672908	−	0.739727i	$-0.734955\pi$
$547$	−31.4760	−1.34582	−0.672908	+	0.739727i	$0.734955\pi$
$548$	3.34755	0.143000
$549$	0	0
$550$	2.06602	0.0880954
$551$	2.14605	0.0914246
$552$	0	0
$553$	−44.8246	−1.90613
$554$	−1.59669	−0.0678367
$555$	0	0
$556$	−30.3132	−1.28556
$557$	36.0927	1.52930	0.764649	−	0.644447i	$-0.222912\pi$
$557$	36.0927	1.52930	0.764649	+	0.644447i	$0.222912\pi$
$558$	0	0
$559$	−33.9550	−1.43614
$560$	18.8653	0.797204
$561$	0	0
$562$	5.31468	0.224186
$563$	−26.5734	−1.11994	−0.559968	−	0.828514i	$-0.689186\pi$
$563$	−26.5734	−1.11994	−0.559968	+	0.828514i	$0.689186\pi$
$564$	0	0
$565$	28.6776	1.20647
$566$	0.915259	0.0384712
$567$	0	0
$568$	−0.149681	−0.00628049
$569$	39.0951	1.63895	0.819475	−	0.573114i	$-0.194265\pi$
$569$	39.0951	1.63895	0.819475	+	0.573114i	$0.194265\pi$
$570$	0	0
$571$	29.3515	1.22832	0.614162	−	0.789180i	$-0.289494\pi$
$571$	29.3515	1.22832	0.614162	+	0.789180i	$0.289494\pi$
$572$	30.5346	1.27672
$573$	0	0
$574$	4.27791	0.178557
$575$	−22.0570	−0.919839
$576$	0	0
$577$	−34.4909	−1.43588	−0.717938	−	0.696107i	$-0.754914\pi$
$577$	−34.4909	−1.43588	−0.717938	+	0.696107i	$0.754914\pi$
$578$	−1.87690	−0.0780689
$579$	0	0
$580$	−6.66552	−0.276771
$581$	−6.75879	−0.280402
$582$	0	0
$583$	−2.36604	−0.0979912
$584$	−4.91217	−0.203267
$585$	0	0
$586$	−3.94013	−0.162765
$587$	42.8239	1.76753	0.883766	−	0.467928i	$-0.155000\pi$
$587$	42.8239	1.76753	0.883766	+	0.467928i	$0.155000\pi$
$588$	0	0
$589$	2.12634	0.0876142
$590$	0.201004	0.00827521
$591$	0	0
$592$	2.63738	0.108396
$593$	−34.8837	−1.43250	−0.716251	−	0.697843i	$-0.754143\pi$
$593$	−34.8837	−1.43250	−0.716251	+	0.697843i	$0.754143\pi$
$594$	0	0
$595$	15.2163	0.623809
$596$	39.7754	1.62927
$597$	0	0
$598$	8.85112	0.361949
$599$	3.96659	0.162071	0.0810353	−	0.996711i	$-0.474177\pi$
$599$	3.96659	0.162071	0.0810353	+	0.996711i	$0.474177\pi$
$600$	0	0
$601$	30.5532	1.24629	0.623146	−	0.782106i	$-0.285854\pi$
$601$	30.5532	1.24629	0.623146	+	0.782106i	$0.285854\pi$
$602$	−5.84595	−0.238263
$603$	0	0
$604$	5.94104	0.241738
$605$	3.81292	0.155017
$606$	0	0
$607$	−35.2006	−1.42875	−0.714374	−	0.699764i	$-0.753289\pi$
$607$	−35.2006	−1.42875	−0.714374	+	0.699764i	$0.753289\pi$
$608$	2.66255	0.107981
$609$	0	0
$610$	−3.09201	−0.125192
$611$	−4.28550	−0.173373
$612$	0	0
$613$	11.9416	0.482316	0.241158	−	0.970486i	$-0.422473\pi$
$613$	11.9416	0.482316	0.241158	+	0.970486i	$0.422473\pi$
$614$	7.91815	0.319551
$615$	0	0
$616$	10.6569	0.429378
$617$	31.7289	1.27736	0.638679	−	0.769473i	$-0.279481\pi$
$617$	31.7289	1.27736	0.638679	+	0.769473i	$0.279481\pi$
$618$	0	0
$619$	33.8311	1.35979	0.679893	−	0.733311i	$-0.262026\pi$
$619$	33.8311	1.35979	0.679893	+	0.733311i	$0.262026\pi$
$620$	−6.60430	−0.265235
$621$	0	0
$622$	1.45991	0.0585370
$623$	−26.5741	−1.06467
$624$	0	0
$625$	−6.69280	−0.267712
$626$	0.760456	0.0303939
$627$	0	0
$628$	18.9584	0.756523
$629$	2.12726	0.0848193
$630$	0	0
$631$	−17.6648	−0.703225	−0.351612	−	0.936146i	$-0.614366\pi$
$631$	−17.6648	−0.703225	−0.351612	+	0.936146i	$0.614366\pi$
$632$	12.6776	0.504287
$633$	0	0
$634$	−2.30922	−0.0917109
$635$	−3.25729	−0.129262
$636$	0	0
$637$	14.1296	0.559837
$638$	−1.80564	−0.0714858
$639$	0	0
$640$	−10.9733	−0.433758
$641$	38.9750	1.53942	0.769711	−	0.638393i	$-0.220401\pi$
$641$	38.9750	1.53942	0.769711	+	0.638393i	$0.220401\pi$
$642$	0	0
$643$	8.23280	0.324670	0.162335	−	0.986736i	$-0.448097\pi$
$643$	8.23280	0.324670	0.162335	+	0.986736i	$0.448097\pi$
$644$	−56.1248	−2.21163
$645$	0	0
$646$	0.683516	0.0268926
$647$	17.8413	0.701414	0.350707	−	0.936485i	$-0.385941\pi$
$647$	17.8413	0.701414	0.350707	+	0.936485i	$0.385941\pi$
$648$	0	0
$649$	−2.00542	−0.0787197
$650$	−2.41958	−0.0949038
$651$	0	0
$652$	6.78138	0.265579
$653$	16.5699	0.648429	0.324215	−	0.945984i	$-0.394900\pi$
$653$	16.5699	0.648429	0.324215	+	0.945984i	$0.394900\pi$
$654$	0	0
$655$	7.55953	0.295375
$656$	21.3691	0.834324
$657$	0	0
$658$	−0.737823	−0.0287633
$659$	−24.8882	−0.969507	−0.484753	−	0.874651i	$-0.661091\pi$
$659$	−24.8882	−0.969507	−0.484753	+	0.874651i	$0.661091\pi$
$660$	0	0
$661$	−21.0736	−0.819667	−0.409833	−	0.912160i	$-0.634413\pi$
$661$	−21.0736	−0.819667	−0.409833	+	0.912160i	$0.634413\pi$
$662$	−3.17201	−0.123284
$663$	0	0
$664$	1.91156	0.0741830
$665$	5.11866	0.198493
$666$	0	0
$667$	19.2771	0.746411
$668$	−6.29061	−0.243391
$669$	0	0
$670$	5.46927	0.211296
$671$	30.8491	1.19092
$672$	0	0
$673$	3.44648	0.132852	0.0664261	−	0.997791i	$-0.478840\pi$
$673$	3.44648	0.132852	0.0664261	+	0.997791i	$0.478840\pi$
$674$	−2.56922	−0.0989626
$675$	0	0
$676$	−10.4473	−0.401819
$677$	−26.9824	−1.03702	−0.518509	−	0.855072i	$-0.673513\pi$
$677$	−26.9824	−1.03702	−0.518509	+	0.855072i	$0.673513\pi$
$678$	0	0
$679$	−16.3864	−0.628853
$680$	−4.30358	−0.165035
$681$	0	0
$682$	−1.78905	−0.0685063
$683$	15.6525	0.598928	0.299464	−	0.954108i	$-0.403192\pi$
$683$	15.6525	0.598928	0.299464	+	0.954108i	$0.403192\pi$
$684$	0	0
$685$	−2.74240	−0.104782
$686$	−2.73210	−0.104312
$687$	0	0
$688$	−29.2018	−1.11331
$689$	2.77094	0.105564
$690$	0	0
$691$	11.5897	0.440892	0.220446	−	0.975399i	$-0.429249\pi$
$691$	11.5897	0.440892	0.220446	+	0.975399i	$0.429249\pi$
$692$	−37.2111	−1.41455
$693$	0	0
$694$	−0.308180	−0.0116983
$695$	24.8333	0.941982
$696$	0	0
$697$	17.2359	0.652855
$698$	8.16252	0.308956
$699$	0	0
$700$	15.3425	0.579892
$701$	41.3964	1.56352	0.781760	−	0.623579i	$-0.214322\pi$
$701$	41.3964	1.56352	0.781760	+	0.623579i	$0.214322\pi$
$702$	0	0
$703$	0.715594	0.0269891
$704$	24.7330	0.932160
$705$	0	0
$706$	5.30687	0.199727
$707$	29.0715	1.09335
$708$	0	0
$709$	−12.5148	−0.470005	−0.235002	−	0.971995i	$-0.575510\pi$
$709$	−12.5148	−0.470005	−0.235002	+	0.971995i	$0.575510\pi$
$710$	0.0604902	0.00227016
$711$	0	0
$712$	7.51585	0.281668
$713$	19.1000	0.715302
$714$	0	0
$715$	−25.0148	−0.935499
$716$	5.60982	0.209649
$717$	0	0
$718$	4.85881	0.181329
$719$	−3.10559	−0.115819	−0.0579094	−	0.998322i	$-0.518443\pi$
$719$	−3.10559	−0.115819	−0.0579094	+	0.998322i	$0.518443\pi$
$720$	0	0
$721$	9.36732	0.348857
$722$	0.229930	0.00855710
$723$	0	0
$724$	−0.793006	−0.0294718
$725$	−5.26966	−0.195710
$726$	0	0
$727$	21.6380	0.802510	0.401255	−	0.915966i	$-0.368574\pi$
$727$	21.6380	0.802510	0.401255	+	0.915966i	$0.368574\pi$
$728$	−12.4806	−0.462562
$729$	0	0
$730$	1.98514	0.0734734
$731$	−23.5536	−0.871160
$732$	0	0
$733$	−42.8168	−1.58147	−0.790737	−	0.612155i	$-0.790303\pi$
$733$	−42.8168	−1.58147	−0.790737	+	0.612155i	$0.790303\pi$
$734$	5.38984	0.198943
$735$	0	0
$736$	23.9166	0.881579
$737$	−54.5671	−2.01000
$738$	0	0
$739$	−31.5133	−1.15924	−0.579618	−	0.814889i	$-0.696798\pi$
$739$	−31.5133	−1.15924	−0.579618	+	0.814889i	$0.696798\pi$
$740$	−2.22260	−0.0817044
$741$	0	0
$742$	0.477065	0.0175136
$743$	−15.7425	−0.577535	−0.288767	−	0.957399i	$-0.593245\pi$
$743$	−15.7425	−0.577535	−0.288767	+	0.957399i	$0.593245\pi$
$744$	0	0
$745$	−32.5851	−1.19383
$746$	3.41383	0.124989
$747$	0	0
$748$	21.1809	0.774451
$749$	24.7617	0.904775
$750$	0	0
$751$	52.8004	1.92671	0.963357	−	0.268224i	$-0.0864367\pi$
$751$	52.8004	1.92671	0.963357	+	0.268224i	$0.0864367\pi$
$752$	−3.68559	−0.134400
$753$	0	0
$754$	2.11464	0.0770105
$755$	−4.86706	−0.177130
$756$	0	0
$757$	13.7588	0.500071	0.250035	−	0.968237i	$-0.419558\pi$
$757$	13.7588	0.500071	0.250035	+	0.968237i	$0.419558\pi$
$758$	0.849869	0.0308686
$759$	0	0
$760$	−1.44769	−0.0525133
$761$	−34.0473	−1.23421	−0.617107	−	0.786880i	$-0.711695\pi$
$761$	−34.0473	−1.23421	−0.617107	+	0.786880i	$0.711695\pi$
$762$	0	0
$763$	0.213732	0.00773761
$764$	−15.1914	−0.549606
$765$	0	0
$766$	0.445613	0.0161007
$767$	2.34861	0.0848035
$768$	0	0
$769$	7.40200	0.266923	0.133461	−	0.991054i	$-0.457391\pi$
$769$	7.40200	0.266923	0.133461	+	0.991054i	$0.457391\pi$
$770$	−4.30673	−0.155204
$771$	0	0
$772$	−17.4631	−0.628512
$773$	11.5780	0.416433	0.208216	−	0.978083i	$-0.433234\pi$
$773$	11.5780	0.416433	0.208216	+	0.978083i	$0.433234\pi$
$774$	0	0
$775$	−5.22127	−0.187553
$776$	4.63451	0.166369
$777$	0	0
$778$	−1.75465	−0.0629072
$779$	5.79802	0.207736
$780$	0	0
$781$	−0.603513	−0.0215954
$782$	6.13975	0.219557
$783$	0	0
$784$	12.1517	0.433989
$785$	−15.5312	−0.554333
$786$	0	0
$787$	46.7982	1.66817	0.834087	−	0.551633i	$-0.185995\pi$
$787$	46.7982	1.66817	0.834087	+	0.551633i	$0.185995\pi$
$788$	30.7945	1.09701
$789$	0	0
$790$	−5.12334	−0.182280
$791$	57.6900	2.05122
$792$	0	0
$793$	−36.1284	−1.28296
$794$	−4.60195	−0.163317
$795$	0	0
$796$	−24.8803	−0.881860
$797$	−10.3872	−0.367933	−0.183967	−	0.982933i	$-0.558894\pi$
$797$	−10.3872	−0.367933	−0.183967	+	0.982933i	$0.558894\pi$
$798$	0	0
$799$	−2.97272	−0.105167
$800$	−6.53796	−0.231152
$801$	0	0
$802$	2.75255	0.0971958
$803$	−19.8058	−0.698932
$804$	0	0
$805$	45.9789	1.62054
$806$	2.09521	0.0738008
$807$	0	0
$808$	−8.22218	−0.289255
$809$	−45.8661	−1.61257	−0.806283	−	0.591530i	$-0.798524\pi$
$809$	−45.8661	−1.61257	−0.806283	+	0.591530i	$0.798524\pi$
$810$	0	0
$811$	5.95777	0.209206	0.104603	−	0.994514i	$-0.466643\pi$
$811$	5.95777	0.209206	0.104603	+	0.994514i	$0.466643\pi$
$812$	−13.4089	−0.470559
$813$	0	0
$814$	−0.602084	−0.0211030
$815$	−5.55548	−0.194600
$816$	0	0
$817$	−7.92324	−0.277199
$818$	−6.74153	−0.235712
$819$	0	0
$820$	−18.0084	−0.628880
$821$	54.9867	1.91905	0.959525	−	0.281624i	$-0.0908732\pi$
$821$	54.9867	1.91905	0.959525	+	0.281624i	$0.0908732\pi$
$822$	0	0
$823$	39.0551	1.36138	0.680688	−	0.732573i	$-0.261681\pi$
$823$	39.0551	1.36138	0.680688	+	0.732573i	$0.261681\pi$
$824$	−2.64932	−0.0922936
$825$	0	0
$826$	0.404355	0.0140693
$827$	−4.79264	−0.166657	−0.0833283	−	0.996522i	$-0.526555\pi$
$827$	−4.79264	−0.166657	−0.0833283	+	0.996522i	$0.526555\pi$
$828$	0	0
$829$	40.5814	1.40945	0.704726	−	0.709480i	$-0.251070\pi$
$829$	40.5814	1.40945	0.704726	+	0.709480i	$0.251070\pi$
$830$	−0.772514	−0.0268143
$831$	0	0
$832$	−28.9656	−1.00420
$833$	9.80130	0.339595
$834$	0	0
$835$	5.15343	0.178342
$836$	7.12510	0.246427
$837$	0	0
$838$	5.81880	0.201007
$839$	2.72909	0.0942185	0.0471092	−	0.998890i	$-0.484999\pi$
$839$	2.72909	0.0942185	0.0471092	+	0.998890i	$0.484999\pi$
$840$	0	0
$841$	−24.3945	−0.841189
$842$	−5.38577	−0.185606
$843$	0	0
$844$	5.88588	0.202600
$845$	8.55871	0.294428
$846$	0	0
$847$	7.67036	0.263557
$848$	2.38305	0.0818342
$849$	0	0
$850$	−1.67839	−0.0575683
$851$	6.42789	0.220345
$852$	0	0
$853$	14.9021	0.510237	0.255118	−	0.966910i	$-0.417886\pi$
$853$	14.9021	0.510237	0.255118	+	0.966910i	$0.417886\pi$
$854$	−6.22013	−0.212848
$855$	0	0
$856$	−7.00327	−0.239367
$857$	−36.5191	−1.24747	−0.623734	−	0.781636i	$-0.714385\pi$
$857$	−36.5191	−1.24747	−0.623734	+	0.781636i	$0.714385\pi$
$858$	0	0
$859$	−33.0694	−1.12831	−0.564157	−	0.825668i	$-0.690799\pi$
$859$	−33.0694	−1.12831	−0.564157	+	0.825668i	$0.690799\pi$
$860$	24.6092	0.839167
$861$	0	0
$862$	−1.94268	−0.0661678
$863$	−2.40731	−0.0819457	−0.0409728	−	0.999160i	$-0.513046\pi$
$863$	−2.40731	−0.0819457	−0.0409728	+	0.999160i	$0.513046\pi$
$864$	0	0
$865$	30.4843	1.03650
$866$	−2.51473	−0.0854539
$867$	0	0
$868$	−13.2857	−0.450946
$869$	51.1158	1.73398
$870$	0	0
$871$	63.9052	2.16535
$872$	−0.0604490	−0.00204706
$873$	0	0
$874$	2.06537	0.0698621
$875$	−38.1623	−1.29012
$876$	0	0
$877$	−4.74565	−0.160249	−0.0801246	−	0.996785i	$-0.525532\pi$
$877$	−4.74565	−0.160249	−0.0801246	+	0.996785i	$0.525532\pi$
$878$	1.47176	0.0496696
$879$	0	0
$880$	−21.5130	−0.725205
$881$	40.0714	1.35004	0.675020	−	0.737799i	$-0.264135\pi$
$881$	40.0714	1.35004	0.675020	+	0.737799i	$0.264135\pi$
$882$	0	0
$883$	−1.40149	−0.0471640	−0.0235820	−	0.999722i	$-0.507507\pi$
$883$	−1.40149	−0.0471640	−0.0235820	+	0.999722i	$0.507507\pi$
$884$	−24.8056	−0.834304
$885$	0	0
$886$	−0.907770	−0.0304971
$887$	−50.1439	−1.68367	−0.841834	−	0.539737i	$-0.818524\pi$
$887$	−50.1439	−1.68367	−0.841834	+	0.539737i	$0.818524\pi$
$888$	0	0
$889$	−6.55261	−0.219767
$890$	−3.03736	−0.101812
$891$	0	0
$892$	34.3898	1.15145
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−4.59571	−0.153618
$896$	−22.0747	−0.737464
$897$	0	0
$898$	1.32717	0.0442883
$899$	4.56322	0.152192
$900$	0	0
$901$	1.92212	0.0640349
$902$	−4.87832	−0.162430
$903$	0	0
$904$	−16.3162	−0.542670
$905$	0.649651	0.0215951
$906$	0	0
$907$	−26.2860	−0.872812	−0.436406	−	0.899750i	$-0.643749\pi$
$907$	−26.2860	−0.872812	−0.436406	+	0.899750i	$0.643749\pi$
$908$	−41.5257	−1.37808
$909$	0	0
$910$	5.04374	0.167198
$911$	49.5990	1.64329	0.821644	−	0.570001i	$-0.193057\pi$
$911$	49.5990	1.64329	0.821644	+	0.570001i	$0.193057\pi$
$912$	0	0
$913$	7.70739	0.255077
$914$	−7.82442	−0.258809
$915$	0	0
$916$	−48.3614	−1.59791
$917$	15.2073	0.502190
$918$	0	0
$919$	−6.75299	−0.222761	−0.111380	−	0.993778i	$-0.535527\pi$
$919$	−6.75299	−0.222761	−0.111380	+	0.993778i	$0.535527\pi$
$920$	−13.0040	−0.428730
$921$	0	0
$922$	1.20242	0.0395997
$923$	0.706793	0.0232644
$924$	0	0
$925$	−1.75716	−0.0577749
$926$	1.91139	0.0628123
$927$	0	0
$928$	5.71396	0.187570
$929$	−33.2500	−1.09090	−0.545449	−	0.838144i	$-0.683641\pi$
$929$	−33.2500	−1.09090	−0.545449	+	0.838144i	$0.683641\pi$
$930$	0	0
$931$	3.29709	0.108058
$932$	−18.0039	−0.589736
$933$	0	0
$934$	−0.293960	−0.00961865
$935$	−17.3520	−0.567470
$936$	0	0
$937$	−39.1307	−1.27834	−0.639171	−	0.769064i	$-0.720723\pi$
$937$	−39.1307	−1.27834	−0.639171	+	0.769064i	$0.720723\pi$
$938$	11.0024	0.359241
$939$	0	0
$940$	3.10595	0.101305
$941$	−57.1163	−1.86194	−0.930969	−	0.365099i	$-0.881035\pi$
$941$	−57.1163	−1.86194	−0.930969	+	0.365099i	$0.881035\pi$
$942$	0	0
$943$	52.0813	1.69600
$944$	2.01984	0.0657402
$945$	0	0
$946$	6.66644	0.216745
$947$	−30.3772	−0.987128	−0.493564	−	0.869709i	$-0.664306\pi$
$947$	−30.3772	−0.987128	−0.493564	+	0.869709i	$0.664306\pi$
$948$	0	0
$949$	23.1952	0.752948
$950$	−0.564598	−0.0183180
$951$	0	0
$952$	−8.65741	−0.280588
$953$	−26.7274	−0.865785	−0.432893	−	0.901446i	$-0.642507\pi$
$953$	−26.7274	−0.865785	−0.432893	+	0.901446i	$0.642507\pi$
$954$	0	0
$955$	12.4452	0.402718
$956$	45.9908	1.48745
$957$	0	0
$958$	−5.40923	−0.174764
$959$	−5.51683	−0.178148
$960$	0	0
$961$	−26.4787	−0.854151
$962$	0.705120	0.0227340
$963$	0	0
$964$	55.2723	1.78020
$965$	14.3063	0.460535
$966$	0	0
$967$	23.3685	0.751480	0.375740	−	0.926725i	$-0.377389\pi$
$967$	23.3685	0.751480	0.375740	+	0.926725i	$0.377389\pi$
$968$	−2.16938	−0.0697265
$969$	0	0
$970$	−1.87293	−0.0601361
$971$	4.93768	0.158458	0.0792289	−	0.996856i	$-0.474754\pi$
$971$	4.93768	0.158458	0.0792289	+	0.996856i	$0.474754\pi$
$972$	0	0
$973$	49.9566	1.60154
$974$	1.58157	0.0506768
$975$	0	0
$976$	−31.0709	−0.994555
$977$	−40.1025	−1.28299	−0.641496	−	0.767126i	$-0.721686\pi$
$977$	−40.1025	−1.28299	−0.641496	+	0.767126i	$0.721686\pi$
$978$	0	0
$979$	30.3038	0.968513
$980$	−10.2406	−0.327124
$981$	0	0
$982$	−1.74571	−0.0557080
$983$	−15.9799	−0.509680	−0.254840	−	0.966983i	$-0.582023\pi$
$983$	−15.9799	−0.509680	−0.254840	+	0.966983i	$0.582023\pi$
$984$	0	0
$985$	−25.2277	−0.803821
$986$	1.46686	0.0467143
$987$	0	0
$988$	−8.34443	−0.265472
$989$	−71.1713	−2.26312
$990$	0	0
$991$	3.22904	0.102574	0.0512869	−	0.998684i	$-0.483668\pi$
$991$	3.22904	0.102574	0.0512869	+	0.998684i	$0.483668\pi$
$992$	5.66148	0.179752
$993$	0	0
$994$	0.121687	0.00385967
$995$	20.3826	0.646173
$996$	0	0
$997$	−50.1355	−1.58781	−0.793904	−	0.608043i	$-0.791955\pi$
$997$	−50.1355	−1.58781	−0.793904	+	0.608043i	$0.791955\pi$
$998$	3.88399	0.122946
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.x.1.18	yes	34
3.2	odd	2		8037.2.a.u.1.17	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.u.1.17	✓	34	3.2	odd	2
8037.2.a.x.1.18	yes	34	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.229930 q^{2} -1.94713 q^{4} +1.59514 q^{5} +3.20891 q^{7} -0.907563 q^{8} +O(q^{10})\)	\(q+0.229930 q^{2} -1.94713 q^{4} +1.59514 q^{5} +3.20891 q^{7} -0.907563 q^{8} +0.366771 q^{10} -3.65928 q^{11} +4.28550 q^{13} +0.737823 q^{14} +3.68559 q^{16} +2.97272 q^{17} +1.00000 q^{19} -3.10595 q^{20} -0.841378 q^{22} +8.98260 q^{23} -2.45552 q^{25} +0.985363 q^{26} -6.24817 q^{28} +2.14605 q^{29} +2.12634 q^{31} +2.66255 q^{32} +0.683516 q^{34} +5.11866 q^{35} +0.715594 q^{37} +0.229930 q^{38} -1.44769 q^{40} +5.79802 q^{41} -7.92324 q^{43} +7.12510 q^{44} +2.06537 q^{46} -1.00000 q^{47} +3.29709 q^{49} -0.564598 q^{50} -8.34443 q^{52} +0.646585 q^{53} -5.83707 q^{55} -2.91229 q^{56} +0.493440 q^{58} +0.548038 q^{59} -8.43038 q^{61} +0.488908 q^{62} -6.75898 q^{64} +6.83598 q^{65} +14.9120 q^{67} -5.78828 q^{68} +1.17693 q^{70} +0.164927 q^{71} +5.41249 q^{73} +0.164536 q^{74} -1.94713 q^{76} -11.7423 q^{77} -13.9688 q^{79} +5.87904 q^{80} +1.33314 q^{82} -2.10626 q^{83} +4.74191 q^{85} -1.82179 q^{86} +3.32103 q^{88} -8.28135 q^{89} +13.7518 q^{91} -17.4903 q^{92} -0.229930 q^{94} +1.59514 q^{95} -5.10654 q^{97} +0.758098 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8} + 18 q^{11} - 6 q^{13} + 12 q^{14} + 21 q^{16} + 36 q^{17} + 34 q^{19} + 40 q^{20} + 12 q^{22} + 38 q^{23} + 32 q^{25} + 15 q^{26} + 28 q^{28} + 14 q^{29} - 6 q^{31} + 35 q^{32} + 10 q^{34} + 46 q^{35} - 2 q^{37} + 5 q^{38} + 31 q^{40} + 18 q^{41} - 6 q^{43} + 42 q^{44} - 14 q^{46} - 34 q^{47} + 44 q^{49} + 9 q^{50} + 2 q^{52} + 32 q^{53} + 8 q^{55} - 4 q^{56} + 8 q^{58} + 62 q^{59} - 10 q^{61} + 30 q^{62} - 37 q^{64} + 8 q^{65} + 92 q^{68} - 62 q^{70} + 4 q^{71} - 8 q^{73} + 34 q^{74} + 31 q^{76} + 52 q^{77} + 40 q^{79} + 48 q^{80} - 2 q^{82} + 110 q^{83} - 12 q^{85} + 16 q^{86} - 44 q^{88} + 2 q^{89} - 28 q^{91} + 60 q^{92} - 5 q^{94} + 14 q^{95} + 2 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8 + 18 * q^11 - 6 * q^13 + 12 * q^14 + 21 * q^16 + 36 * q^17 + 34 * q^19 + 40 * q^20 + 12 * q^22 + 38 * q^23 + 32 * q^25 + 15 * q^26 + 28 * q^28 + 14 * q^29 - 6 * q^31 + 35 * q^32 + 10 * q^34 + 46 * q^35 - 2 * q^37 + 5 * q^38 + 31 * q^40 + 18 * q^41 - 6 * q^43 + 42 * q^44 - 14 * q^46 - 34 * q^47 + 44 * q^49 + 9 * q^50 + 2 * q^52 + 32 * q^53 + 8 * q^55 - 4 * q^56 + 8 * q^58 + 62 * q^59 - 10 * q^61 + 30 * q^62 - 37 * q^64 + 8 * q^65 + 92 * q^68 - 62 * q^70 + 4 * q^71 - 8 * q^73 + 34 * q^74 + 31 * q^76 + 52 * q^77 + 40 * q^79 + 48 * q^80 - 2 * q^82 + 110 * q^83 - 12 * q^85 + 16 * q^86 - 44 * q^88 + 2 * q^89 - 28 * q^91 + 60 * q^92 - 5 * q^94 + 14 * q^95 + 2 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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