LMFDB - Embedded newform 8001.2.a.ba.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63730 q^{2} +4.95534 q^{4} -2.98743 q^{5} +1.00000 q^{7} -7.79413 q^{8} +O(q^{10})$	\(q-2.63730 q^{2} +4.95534 q^{4} -2.98743 q^{5} +1.00000 q^{7} -7.79413 q^{8} +7.87873 q^{10} -0.978411 q^{11} +3.28196 q^{13} -2.63730 q^{14} +10.6448 q^{16} +7.98808 q^{17} +3.91836 q^{19} -14.8037 q^{20} +2.58036 q^{22} -3.09261 q^{23} +3.92471 q^{25} -8.65551 q^{26} +4.95534 q^{28} -0.459116 q^{29} -3.11913 q^{31} -12.4851 q^{32} -21.0669 q^{34} -2.98743 q^{35} +6.82365 q^{37} -10.3339 q^{38} +23.2844 q^{40} +7.22199 q^{41} -7.64450 q^{43} -4.84836 q^{44} +8.15612 q^{46} -1.97911 q^{47} +1.00000 q^{49} -10.3506 q^{50} +16.2632 q^{52} +10.4292 q^{53} +2.92293 q^{55} -7.79413 q^{56} +1.21082 q^{58} +13.5702 q^{59} +11.8190 q^{61} +8.22609 q^{62} +11.6375 q^{64} -9.80461 q^{65} +5.82880 q^{67} +39.5837 q^{68} +7.87873 q^{70} -1.58640 q^{71} +0.714793 q^{73} -17.9960 q^{74} +19.4168 q^{76} -0.978411 q^{77} -7.89812 q^{79} -31.8004 q^{80} -19.0466 q^{82} +11.1653 q^{83} -23.8638 q^{85} +20.1608 q^{86} +7.62586 q^{88} -5.97776 q^{89} +3.28196 q^{91} -15.3249 q^{92} +5.21950 q^{94} -11.7058 q^{95} -6.56685 q^{97} -2.63730 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

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 $

$2$	−2.63730	−1.86485	−0.932426	−	0.361361i	$-0.882312\pi$
$2$	−2.63730	−1.86485	−0.932426	+	0.361361i	$0.882312\pi$
$3$	0	0
$3$	0	0
$4$	4.95534	2.47767
$5$	−2.98743	−1.33602	−0.668009	−	0.744154i	$-0.732853\pi$
$5$	−2.98743	−1.33602	−0.668009	+	0.744154i	$0.732853\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−7.79413	−2.75564
$9$	0	0
$10$	7.87873	2.49147
$11$	−0.978411	−0.295002	−0.147501	−	0.989062i	$-0.547123\pi$
$11$	−0.978411	−0.295002	−0.147501	+	0.989062i	$0.547123\pi$
$12$	0	0
$13$	3.28196	0.910252	0.455126	−	0.890427i	$-0.349594\pi$
$13$	3.28196	0.910252	0.455126	+	0.890427i	$0.349594\pi$
$14$	−2.63730	−0.704848
$15$	0	0
$16$	10.6448	2.66119
$17$	7.98808	1.93739	0.968697	−	0.248248i	$-0.0798548\pi$
$17$	7.98808	1.93739	0.968697	+	0.248248i	$0.0798548\pi$
$18$	0	0
$19$	3.91836	0.898934	0.449467	−	0.893297i	$-0.351614\pi$
$19$	3.91836	0.898934	0.449467	+	0.893297i	$0.351614\pi$
$20$	−14.8037	−3.31021
$21$	0	0
$22$	2.58036	0.550135
$23$	−3.09261	−0.644853	−0.322426	−	0.946595i	$-0.604498\pi$
$23$	−3.09261	−0.644853	−0.322426	+	0.946595i	$0.604498\pi$
$24$	0	0
$25$	3.92471	0.784942
$26$	−8.65551	−1.69749
$27$	0	0
$28$	4.95534	0.936472
$29$	−0.459116	−0.0852556	−0.0426278	−	0.999091i	$-0.513573\pi$
$29$	−0.459116	−0.0852556	−0.0426278	+	0.999091i	$0.513573\pi$
$30$	0	0
$31$	−3.11913	−0.560213	−0.280106	−	0.959969i	$-0.590370\pi$
$31$	−3.11913	−0.560213	−0.280106	+	0.959969i	$0.590370\pi$
$32$	−12.4851	−2.20708
$33$	0	0
$34$	−21.0669	−3.61295
$35$	−2.98743	−0.504967
$36$	0	0
$37$	6.82365	1.12180	0.560901	−	0.827883i	$-0.310455\pi$
$37$	6.82365	1.12180	0.560901	+	0.827883i	$0.310455\pi$
$38$	−10.3339	−1.67638
$39$	0	0
$40$	23.2844	3.68158
$41$	7.22199	1.12789	0.563943	−	0.825814i	$-0.309284\pi$
$41$	7.22199	1.12789	0.563943	+	0.825814i	$0.309284\pi$
$42$	0	0
$43$	−7.64450	−1.16578	−0.582888	−	0.812553i	$-0.698077\pi$
$43$	−7.64450	−1.16578	−0.582888	+	0.812553i	$0.698077\pi$
$44$	−4.84836	−0.730918
$45$	0	0
$46$	8.15612	1.20255
$47$	−1.97911	−0.288683	−0.144341	−	0.989528i	$-0.546106\pi$
$47$	−1.97911	−0.288683	−0.144341	+	0.989528i	$0.546106\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.3506	−1.46380
$51$	0	0
$52$	16.2632	2.25531
$53$	10.4292	1.43256	0.716281	−	0.697812i	$-0.245843\pi$
$53$	10.4292	1.43256	0.716281	+	0.697812i	$0.245843\pi$
$54$	0	0
$55$	2.92293	0.394128
$56$	−7.79413	−1.04153
$57$	0	0
$58$	1.21082	0.158989
$59$	13.5702	1.76668	0.883342	−	0.468729i	$-0.155288\pi$
$59$	13.5702	1.76668	0.883342	+	0.468729i	$0.155288\pi$
$60$	0	0
$61$	11.8190	1.51327	0.756633	−	0.653840i	$-0.226843\pi$
$61$	11.8190	1.51327	0.756633	+	0.653840i	$0.226843\pi$
$62$	8.22609	1.04471
$63$	0	0
$64$	11.6375	1.45469
$65$	−9.80461	−1.21611
$66$	0	0
$67$	5.82880	0.712102	0.356051	−	0.934467i	$-0.384123\pi$
$67$	5.82880	0.712102	0.356051	+	0.934467i	$0.384123\pi$
$68$	39.5837	4.80023
$69$	0	0
$70$	7.87873	0.941689
$71$	−1.58640	−0.188270	−0.0941352	−	0.995559i	$-0.530009\pi$
$71$	−1.58640	−0.188270	−0.0941352	+	0.995559i	$0.530009\pi$
$72$	0	0
$73$	0.714793	0.0836602	0.0418301	−	0.999125i	$-0.486681\pi$
$73$	0.714793	0.0836602	0.0418301	+	0.999125i	$0.486681\pi$
$74$	−17.9960	−2.09199
$75$	0	0
$76$	19.4168	2.22726
$77$	−0.978411	−0.111500
$78$	0	0
$79$	−7.89812	−0.888608	−0.444304	−	0.895876i	$-0.646549\pi$
$79$	−7.89812	−0.888608	−0.444304	+	0.895876i	$0.646549\pi$
$80$	−31.8004	−3.55539
$81$	0	0
$82$	−19.0466	−2.10334
$83$	11.1653	1.22555	0.612773	−	0.790259i	$-0.290054\pi$
$83$	11.1653	1.22555	0.612773	+	0.790259i	$0.290054\pi$
$84$	0	0
$85$	−23.8638	−2.58839
$86$	20.1608	2.17400
$87$	0	0
$88$	7.62586	0.812919
$89$	−5.97776	−0.633642	−0.316821	−	0.948485i	$-0.602615\pi$
$89$	−5.97776	−0.633642	−0.316821	+	0.948485i	$0.602615\pi$
$90$	0	0
$91$	3.28196	0.344043
$92$	−15.3249	−1.59773
$93$	0	0
$94$	5.21950	0.538350
$95$	−11.7058	−1.20099
$96$	0	0
$97$	−6.56685	−0.666763	−0.333382	−	0.942792i	$-0.608190\pi$
$97$	−6.56685	−0.666763	−0.333382	+	0.942792i	$0.608190\pi$
$98$	−2.63730	−0.266407
$99$	0	0
$100$	19.4483	1.94483
$101$	4.00300	0.398313	0.199156	−	0.979968i	$-0.436180\pi$
$101$	4.00300	0.398313	0.199156	+	0.979968i	$0.436180\pi$
$102$	0	0
$103$	19.3768	1.90926	0.954628	−	0.297802i	$-0.0962535\pi$
$103$	19.3768	1.90926	0.954628	+	0.297802i	$0.0962535\pi$
$104$	−25.5800	−2.50833
$105$	0	0
$106$	−27.5049	−2.67152
$107$	−4.10444	−0.396791	−0.198396	−	0.980122i	$-0.563573\pi$
$107$	−4.10444	−0.396791	−0.198396	+	0.980122i	$0.563573\pi$
$108$	0	0
$109$	−2.70497	−0.259089	−0.129544	−	0.991574i	$-0.541352\pi$
$109$	−2.70497	−0.259089	−0.129544	+	0.991574i	$0.541352\pi$
$110$	−7.70864	−0.734990
$111$	0	0
$112$	10.6448	1.00583
$113$	−16.3944	−1.54225	−0.771126	−	0.636682i	$-0.780306\pi$
$113$	−16.3944	−1.54225	−0.771126	+	0.636682i	$0.780306\pi$
$114$	0	0
$115$	9.23893	0.861534
$116$	−2.27508	−0.211236
$117$	0	0
$118$	−35.7886	−3.29460
$119$	7.98808	0.732266
$120$	0	0
$121$	−10.0427	−0.912974
$122$	−31.1702	−2.82202
$123$	0	0
$124$	−15.4564	−1.38802
$125$	3.21235	0.287321
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−5.72139	−0.505704
$129$	0	0
$130$	25.8577	2.26787
$131$	14.2358	1.24379	0.621893	−	0.783102i	$-0.286364\pi$
$131$	14.2358	1.24379	0.621893	+	0.783102i	$0.286364\pi$
$132$	0	0
$133$	3.91836	0.339765
$134$	−15.3723	−1.32796
$135$	0	0
$136$	−62.2601	−5.33876
$137$	−7.98377	−0.682100	−0.341050	−	0.940045i	$-0.610782\pi$
$137$	−7.98377	−0.682100	−0.341050	+	0.940045i	$0.610782\pi$
$138$	0	0
$139$	−5.99828	−0.508767	−0.254384	−	0.967103i	$-0.581873\pi$
$139$	−5.99828	−0.508767	−0.254384	+	0.967103i	$0.581873\pi$
$140$	−14.8037	−1.25114
$141$	0	0
$142$	4.18380	0.351097
$143$	−3.21111	−0.268526
$144$	0	0
$145$	1.37157	0.113903
$146$	−1.88512	−0.156014
$147$	0	0
$148$	33.8135	2.77946
$149$	0.240352	0.0196904	0.00984522	−	0.999952i	$-0.496866\pi$
$149$	0.240352	0.0196904	0.00984522	+	0.999952i	$0.496866\pi$
$150$	0	0
$151$	4.45303	0.362383	0.181191	−	0.983448i	$-0.442005\pi$
$151$	4.45303	0.362383	0.181191	+	0.983448i	$0.442005\pi$
$152$	−30.5402	−2.47714
$153$	0	0
$154$	2.58036	0.207931
$155$	9.31818	0.748454
$156$	0	0
$157$	−6.46627	−0.516065	−0.258032	−	0.966136i	$-0.583074\pi$
$157$	−6.46627	−0.516065	−0.258032	+	0.966136i	$0.583074\pi$
$158$	20.8297	1.65712
$159$	0	0
$160$	37.2984	2.94870
$161$	−3.09261	−0.243731
$162$	0	0
$163$	−0.820706	−0.0642827	−0.0321413	−	0.999483i	$-0.510233\pi$
$163$	−0.820706	−0.0642827	−0.0321413	+	0.999483i	$0.510233\pi$
$164$	35.7875	2.79453
$165$	0	0
$166$	−29.4461	−2.28546
$167$	10.7740	0.833720	0.416860	−	0.908971i	$-0.363131\pi$
$167$	10.7740	0.833720	0.416860	+	0.908971i	$0.363131\pi$
$168$	0	0
$169$	−2.22873	−0.171441
$170$	62.9359	4.82696
$171$	0	0
$172$	−37.8811	−2.88841
$173$	−18.1333	−1.37865	−0.689323	−	0.724454i	$-0.742092\pi$
$173$	−18.1333	−1.37865	−0.689323	+	0.724454i	$0.742092\pi$
$174$	0	0
$175$	3.92471	0.296680
$176$	−10.4149	−0.785056
$177$	0	0
$178$	15.7652	1.18165
$179$	−4.32377	−0.323174	−0.161587	−	0.986858i	$-0.551661\pi$
$179$	−4.32377	−0.323174	−0.161587	+	0.986858i	$0.551661\pi$
$180$	0	0
$181$	−20.5461	−1.52718	−0.763589	−	0.645703i	$-0.776564\pi$
$181$	−20.5461	−1.52718	−0.763589	+	0.645703i	$0.776564\pi$
$182$	−8.65551	−0.641589
$183$	0	0
$184$	24.1042	1.77698
$185$	−20.3851	−1.49875
$186$	0	0
$187$	−7.81562	−0.571535
$188$	−9.80716	−0.715261
$189$	0	0
$190$	30.8717	2.23967
$191$	−0.673091	−0.0487031	−0.0243516	−	0.999703i	$-0.507752\pi$
$191$	−0.673091	−0.0487031	−0.0243516	+	0.999703i	$0.507752\pi$
$192$	0	0
$193$	15.2356	1.09668	0.548342	−	0.836254i	$-0.315259\pi$
$193$	15.2356	1.09668	0.548342	+	0.836254i	$0.315259\pi$
$194$	17.3188	1.24341
$195$	0	0
$196$	4.95534	0.353953
$197$	−3.55683	−0.253414	−0.126707	−	0.991940i	$-0.540441\pi$
$197$	−3.55683	−0.253414	−0.126707	+	0.991940i	$0.540441\pi$
$198$	0	0
$199$	18.1148	1.28413	0.642063	−	0.766652i	$-0.278079\pi$
$199$	18.1148	1.28413	0.642063	+	0.766652i	$0.278079\pi$
$200$	−30.5897	−2.16302
$201$	0	0
$202$	−10.5571	−0.742795
$203$	−0.459116	−0.0322236
$204$	0	0
$205$	−21.5752	−1.50687
$206$	−51.1025	−3.56048
$207$	0	0
$208$	34.9357	2.42235
$209$	−3.83377	−0.265187
$210$	0	0
$211$	−4.75917	−0.327635	−0.163817	−	0.986491i	$-0.552381\pi$
$211$	−4.75917	−0.327635	−0.163817	+	0.986491i	$0.552381\pi$
$212$	51.6803	3.54942
$213$	0	0
$214$	10.8246	0.739957
$215$	22.8374	1.55750
$216$	0	0
$217$	−3.11913	−0.211741
$218$	7.13381	0.483163
$219$	0	0
$220$	14.4841	0.976519
$221$	26.2166	1.76352
$222$	0	0
$223$	8.94023	0.598682	0.299341	−	0.954146i	$-0.403233\pi$
$223$	8.94023	0.598682	0.299341	+	0.954146i	$0.403233\pi$
$224$	−12.4851	−0.834199
$225$	0	0
$226$	43.2369	2.87607
$227$	−8.51410	−0.565101	−0.282550	−	0.959252i	$-0.591180\pi$
$227$	−8.51410	−0.565101	−0.282550	+	0.959252i	$0.591180\pi$
$228$	0	0
$229$	−6.21843	−0.410925	−0.205463	−	0.978665i	$-0.565870\pi$
$229$	−6.21843	−0.410925	−0.205463	+	0.978665i	$0.565870\pi$
$230$	−24.3658	−1.60663
$231$	0	0
$232$	3.57841	0.234934
$233$	7.64112	0.500587	0.250293	−	0.968170i	$-0.419473\pi$
$233$	7.64112	0.500587	0.250293	+	0.968170i	$0.419473\pi$
$234$	0	0
$235$	5.91244	0.385685
$236$	67.2448	4.37727
$237$	0	0
$238$	−21.0669	−1.36557
$239$	0.474900	0.0307187	0.0153593	−	0.999882i	$-0.495111\pi$
$239$	0.474900	0.0307187	0.0153593	+	0.999882i	$0.495111\pi$
$240$	0	0
$241$	−25.5141	−1.64350	−0.821752	−	0.569845i	$-0.807003\pi$
$241$	−25.5141	−1.64350	−0.821752	+	0.569845i	$0.807003\pi$
$242$	26.4856	1.70256
$243$	0	0
$244$	58.5671	3.74938
$245$	−2.98743	−0.190860
$246$	0	0
$247$	12.8599	0.818256
$248$	24.3109	1.54374
$249$	0	0
$250$	−8.47192	−0.535812
$251$	9.52926	0.601482	0.300741	−	0.953706i	$-0.402766\pi$
$251$	9.52926	0.601482	0.300741	+	0.953706i	$0.402766\pi$
$252$	0	0
$253$	3.02584	0.190233
$254$	2.63730	0.165479
$255$	0	0
$256$	−8.18607	−0.511630
$257$	−16.7734	−1.04630	−0.523148	−	0.852242i	$-0.675242\pi$
$257$	−16.7734	−1.04630	−0.523148	+	0.852242i	$0.675242\pi$
$258$	0	0
$259$	6.82365	0.424001
$260$	−48.5852	−3.01313
$261$	0	0
$262$	−37.5440	−2.31948
$263$	−30.1384	−1.85841	−0.929205	−	0.369564i	$-0.879507\pi$
$263$	−30.1384	−1.85841	−0.929205	+	0.369564i	$0.879507\pi$
$264$	0	0
$265$	−31.1565	−1.91393
$266$	−10.3339	−0.633611
$267$	0	0
$268$	28.8837	1.76435
$269$	4.58834	0.279756	0.139878	−	0.990169i	$-0.455329\pi$
$269$	4.58834	0.279756	0.139878	+	0.990169i	$0.455329\pi$
$270$	0	0
$271$	20.7424	1.26001	0.630007	−	0.776590i	$-0.283052\pi$
$271$	20.7424	1.26001	0.630007	+	0.776590i	$0.283052\pi$
$272$	85.0311	5.15577
$273$	0	0
$274$	21.0556	1.27201
$275$	−3.83998	−0.231559
$276$	0	0
$277$	−18.1228	−1.08890	−0.544448	−	0.838795i	$-0.683261\pi$
$277$	−18.1228	−1.08890	−0.544448	+	0.838795i	$0.683261\pi$
$278$	15.8193	0.948776
$279$	0	0
$280$	23.2844	1.39151
$281$	1.20202	0.0717066	0.0358533	−	0.999357i	$-0.488585\pi$
$281$	1.20202	0.0717066	0.0358533	+	0.999357i	$0.488585\pi$
$282$	0	0
$283$	−13.0344	−0.774816	−0.387408	−	0.921908i	$-0.626629\pi$
$283$	−13.0344	−0.774816	−0.387408	+	0.921908i	$0.626629\pi$
$284$	−7.86114	−0.466473
$285$	0	0
$286$	8.46865	0.500761
$287$	7.22199	0.426301
$288$	0	0
$289$	46.8094	2.75349
$290$	−3.61725	−0.212412
$291$	0	0
$292$	3.54205	0.207283
$293$	−28.8206	−1.68372	−0.841858	−	0.539700i	$-0.818538\pi$
$293$	−28.8206	−1.68372	−0.841858	+	0.539700i	$0.818538\pi$
$294$	0	0
$295$	−40.5398	−2.36032
$296$	−53.1844	−3.09128
$297$	0	0
$298$	−0.633881	−0.0367197
$299$	−10.1498	−0.586979
$300$	0	0
$301$	−7.64450	−0.440622
$302$	−11.7440	−0.675790
$303$	0	0
$304$	41.7100	2.39223
$305$	−35.3083	−2.02175
$306$	0	0
$307$	−19.5065	−1.11330	−0.556649	−	0.830748i	$-0.687913\pi$
$307$	−19.5065	−1.11330	−0.556649	+	0.830748i	$0.687913\pi$
$308$	−4.84836	−0.276261
$309$	0	0
$310$	−24.5748	−1.39576
$311$	28.1883	1.59841	0.799207	−	0.601056i	$-0.205253\pi$
$311$	28.1883	1.59841	0.799207	+	0.601056i	$0.205253\pi$
$312$	0	0
$313$	−8.55468	−0.483539	−0.241770	−	0.970334i	$-0.577728\pi$
$313$	−8.55468	−0.483539	−0.241770	+	0.970334i	$0.577728\pi$
$314$	17.0535	0.962384
$315$	0	0
$316$	−39.1379	−2.20168
$317$	18.8240	1.05726	0.528630	−	0.848853i	$-0.322706\pi$
$317$	18.8240	1.05726	0.528630	+	0.848853i	$0.322706\pi$
$318$	0	0
$319$	0.449204	0.0251506
$320$	−34.7663	−1.94349
$321$	0	0
$322$	8.15612	0.454523
$323$	31.3002	1.74159
$324$	0	0
$325$	12.8807	0.714495
$326$	2.16445	0.119878
$327$	0	0
$328$	−56.2891	−3.10805
$329$	−1.97911	−0.109112
$330$	0	0
$331$	33.2533	1.82777	0.913884	−	0.405977i	$-0.133069\pi$
$331$	33.2533	1.82777	0.913884	+	0.405977i	$0.133069\pi$
$332$	55.3277	3.03650
$333$	0	0
$334$	−28.4143	−1.55476
$335$	−17.4131	−0.951380
$336$	0	0
$337$	23.4705	1.27852	0.639259	−	0.768992i	$-0.279241\pi$
$337$	23.4705	1.27852	0.639259	+	0.768992i	$0.279241\pi$
$338$	5.87784	0.319712
$339$	0	0
$340$	−118.253	−6.41318
$341$	3.05179	0.165264
$342$	0	0
$343$	1.00000	0.0539949
$344$	59.5822	3.21246
$345$	0	0
$346$	47.8228	2.57097
$347$	−10.8361	−0.581711	−0.290855	−	0.956767i	$-0.593940\pi$
$347$	−10.8361	−0.581711	−0.290855	+	0.956767i	$0.593940\pi$
$348$	0	0
$349$	11.4349	0.612097	0.306049	−	0.952016i	$-0.400993\pi$
$349$	11.4349	0.612097	0.306049	+	0.952016i	$0.400993\pi$
$350$	−10.3506	−0.553265
$351$	0	0
$352$	12.2156	0.651093
$353$	36.5040	1.94291	0.971455	−	0.237225i	$-0.0762380\pi$
$353$	36.5040	1.94291	0.971455	+	0.237225i	$0.0762380\pi$
$354$	0	0
$355$	4.73924	0.251533
$356$	−29.6219	−1.56996
$357$	0	0
$358$	11.4031	0.602672
$359$	21.3254	1.12551	0.562756	−	0.826623i	$-0.309741\pi$
$359$	21.3254	1.12551	0.562756	+	0.826623i	$0.309741\pi$
$360$	0	0
$361$	−3.64645	−0.191918
$362$	54.1861	2.84796
$363$	0	0
$364$	16.2632	0.852426
$365$	−2.13539	−0.111772
$366$	0	0
$367$	−14.6063	−0.762441	−0.381221	−	0.924484i	$-0.624496\pi$
$367$	−14.6063	−0.762441	−0.381221	+	0.924484i	$0.624496\pi$
$368$	−32.9200	−1.71607
$369$	0	0
$370$	53.7617	2.79494
$371$	10.4292	0.541457
$372$	0	0
$373$	−15.4665	−0.800827	−0.400414	−	0.916335i	$-0.631134\pi$
$373$	−15.4665	−0.800827	−0.400414	+	0.916335i	$0.631134\pi$
$374$	20.6121	1.06583
$375$	0	0
$376$	15.4254	0.795505
$377$	−1.50680	−0.0776041
$378$	0	0
$379$	23.1683	1.19007	0.595037	−	0.803698i	$-0.297137\pi$
$379$	23.1683	1.19007	0.595037	+	0.803698i	$0.297137\pi$
$380$	−58.0063	−2.97566
$381$	0	0
$382$	1.77514	0.0908241
$383$	6.29432	0.321625	0.160812	−	0.986985i	$-0.448589\pi$
$383$	6.29432	0.321625	0.160812	+	0.986985i	$0.448589\pi$
$384$	0	0
$385$	2.92293	0.148966
$386$	−40.1809	−2.04515
$387$	0	0
$388$	−32.5410	−1.65202
$389$	14.7763	0.749190	0.374595	−	0.927188i	$-0.377782\pi$
$389$	14.7763	0.749190	0.374595	+	0.927188i	$0.377782\pi$
$390$	0	0
$391$	−24.7040	−1.24933
$392$	−7.79413	−0.393663
$393$	0	0
$394$	9.38043	0.472579
$395$	23.5951	1.18720
$396$	0	0
$397$	4.63809	0.232779	0.116390	−	0.993204i	$-0.462868\pi$
$397$	4.63809	0.232779	0.116390	+	0.993204i	$0.462868\pi$
$398$	−47.7742	−2.39470
$399$	0	0
$400$	41.7776	2.08888
$401$	−19.3472	−0.966154	−0.483077	−	0.875578i	$-0.660481\pi$
$401$	−19.3472	−0.966154	−0.483077	+	0.875578i	$0.660481\pi$
$402$	0	0
$403$	−10.2369	−0.509935
$404$	19.8362	0.986889
$405$	0	0
$406$	1.21082	0.0600922
$407$	−6.67633	−0.330933
$408$	0	0
$409$	−7.60791	−0.376187	−0.188093	−	0.982151i	$-0.560231\pi$
$409$	−7.60791	−0.376187	−0.188093	+	0.982151i	$0.560231\pi$
$410$	56.9002	2.81010
$411$	0	0
$412$	96.0189	4.73051
$413$	13.5702	0.667744
$414$	0	0
$415$	−33.3554	−1.63735
$416$	−40.9757	−2.00900
$417$	0	0
$418$	10.1108	0.494535
$419$	−22.4332	−1.09593	−0.547966	−	0.836501i	$-0.684598\pi$
$419$	−22.4332	−1.09593	−0.547966	+	0.836501i	$0.684598\pi$
$420$	0	0
$421$	21.9950	1.07197	0.535985	−	0.844228i	$-0.319941\pi$
$421$	21.9950	1.07197	0.535985	+	0.844228i	$0.319941\pi$
$422$	12.5513	0.610990
$423$	0	0
$424$	−81.2866	−3.94762
$425$	31.3509	1.52074
$426$	0	0
$427$	11.8190	0.571961
$428$	−20.3389	−0.983119
$429$	0	0
$430$	−60.2290	−2.90450
$431$	5.22869	0.251857	0.125928	−	0.992039i	$-0.459809\pi$
$431$	5.22869	0.251857	0.125928	+	0.992039i	$0.459809\pi$
$432$	0	0
$433$	−11.8209	−0.568077	−0.284039	−	0.958813i	$-0.591674\pi$
$433$	−11.8209	−0.568077	−0.284039	+	0.958813i	$0.591674\pi$
$434$	8.22609	0.394865
$435$	0	0
$436$	−13.4041	−0.641938
$437$	−12.1179	−0.579680
$438$	0	0
$439$	−0.544705	−0.0259973	−0.0129987	−	0.999916i	$-0.504138\pi$
$439$	−0.544705	−0.0259973	−0.0129987	+	0.999916i	$0.504138\pi$
$440$	−22.7817	−1.08607
$441$	0	0
$442$	−69.1409	−3.28870
$443$	−5.38159	−0.255687	−0.127843	−	0.991794i	$-0.540806\pi$
$443$	−5.38159	−0.255687	−0.127843	+	0.991794i	$0.540806\pi$
$444$	0	0
$445$	17.8581	0.846556
$446$	−23.5781	−1.11645
$447$	0	0
$448$	11.6375	0.549822
$449$	−1.63526	−0.0771725	−0.0385863	−	0.999255i	$-0.512285\pi$
$449$	−1.63526	−0.0771725	−0.0385863	+	0.999255i	$0.512285\pi$
$450$	0	0
$451$	−7.06608	−0.332729
$452$	−81.2398	−3.82120
$453$	0	0
$454$	22.4542	1.05383
$455$	−9.80461	−0.459647
$456$	0	0
$457$	−23.3910	−1.09419	−0.547093	−	0.837072i	$-0.684265\pi$
$457$	−23.3910	−1.09419	−0.547093	+	0.837072i	$0.684265\pi$
$458$	16.3999	0.766315
$459$	0	0
$460$	45.7821	2.13460
$461$	−32.1352	−1.49668	−0.748342	−	0.663313i	$-0.769150\pi$
$461$	−32.1352	−1.49668	−0.748342	+	0.663313i	$0.769150\pi$
$462$	0	0
$463$	17.3884	0.808107	0.404054	−	0.914735i	$-0.367601\pi$
$463$	17.3884	0.808107	0.404054	+	0.914735i	$0.367601\pi$
$464$	−4.88717	−0.226881
$465$	0	0
$466$	−20.1519	−0.933520
$467$	−12.5611	−0.581258	−0.290629	−	0.956836i	$-0.593865\pi$
$467$	−12.5611	−0.581258	−0.290629	+	0.956836i	$0.593865\pi$
$468$	0	0
$469$	5.82880	0.269149
$470$	−15.5929	−0.719245
$471$	0	0
$472$	−105.768	−4.86835
$473$	7.47946	0.343906
$474$	0	0
$475$	15.3784	0.705611
$476$	39.5837	1.81431
$477$	0	0
$478$	−1.25245	−0.0572858
$479$	12.2561	0.559996	0.279998	−	0.960001i	$-0.409666\pi$
$479$	12.2561	0.559996	0.279998	+	0.960001i	$0.409666\pi$
$480$	0	0
$481$	22.3949	1.02112
$482$	67.2882	3.06489
$483$	0	0
$484$	−49.7651	−2.26205
$485$	19.6180	0.890807
$486$	0	0
$487$	−4.73313	−0.214479	−0.107239	−	0.994233i	$-0.534201\pi$
$487$	−4.73313	−0.214479	−0.107239	+	0.994233i	$0.534201\pi$
$488$	−92.1186	−4.17001
$489$	0	0
$490$	7.87873	0.355925
$491$	7.71255	0.348063	0.174031	−	0.984740i	$-0.444321\pi$
$491$	7.71255	0.348063	0.174031	+	0.984740i	$0.444321\pi$
$492$	0	0
$493$	−3.66745	−0.165174
$494$	−33.9154	−1.52593
$495$	0	0
$496$	−33.2024	−1.49083
$497$	−1.58640	−0.0711595
$498$	0	0
$499$	6.42640	0.287685	0.143843	−	0.989601i	$-0.454054\pi$
$499$	6.42640	0.287685	0.143843	+	0.989601i	$0.454054\pi$
$500$	15.9183	0.711888
$501$	0	0
$502$	−25.1315	−1.12167
$503$	20.5036	0.914209	0.457105	−	0.889413i	$-0.348886\pi$
$503$	20.5036	0.914209	0.457105	+	0.889413i	$0.348886\pi$
$504$	0	0
$505$	−11.9586	−0.532153
$506$	−7.98004	−0.354756
$507$	0	0
$508$	−4.95534	−0.219858
$509$	17.4590	0.773856	0.386928	−	0.922110i	$-0.373536\pi$
$509$	17.4590	0.773856	0.386928	+	0.922110i	$0.373536\pi$
$510$	0	0
$511$	0.714793	0.0316206
$512$	33.0319	1.45982
$513$	0	0
$514$	44.2365	1.95119
$515$	−57.8868	−2.55080
$516$	0	0
$517$	1.93638	0.0851619
$518$	−17.9960	−0.790699
$519$	0	0
$520$	76.4184	3.35117
$521$	−13.3586	−0.585252	−0.292626	−	0.956227i	$-0.594529\pi$
$521$	−13.3586	−0.585252	−0.292626	+	0.956227i	$0.594529\pi$
$522$	0	0
$523$	28.0952	1.22852	0.614259	−	0.789104i	$-0.289455\pi$
$523$	28.0952	1.22852	0.614259	+	0.789104i	$0.289455\pi$
$524$	70.5432	3.08170
$525$	0	0
$526$	79.4838	3.46566
$527$	−24.9159	−1.08535
$528$	0	0
$529$	−13.4358	−0.584165
$530$	82.1689	3.56919
$531$	0	0
$532$	19.4168	0.841826
$533$	23.7023	1.02666
$534$	0	0
$535$	12.2617	0.530120
$536$	−45.4304	−1.96230
$537$	0	0
$538$	−12.1008	−0.521704
$539$	−0.978411	−0.0421431
$540$	0	0
$541$	−2.76916	−0.119055	−0.0595277	−	0.998227i	$-0.518959\pi$
$541$	−2.76916	−0.119055	−0.0595277	+	0.998227i	$0.518959\pi$
$542$	−54.7040	−2.34974
$543$	0	0
$544$	−99.7323	−4.27598
$545$	8.08089	0.346147
$546$	0	0
$547$	27.8536	1.19093	0.595466	−	0.803380i	$-0.296967\pi$
$547$	27.8536	1.19093	0.595466	+	0.803380i	$0.296967\pi$
$548$	−39.5623	−1.69002
$549$	0	0
$550$	10.1272	0.431824
$551$	−1.79898	−0.0766391
$552$	0	0
$553$	−7.89812	−0.335862
$554$	47.7953	2.03063
$555$	0	0
$556$	−29.7235	−1.26056
$557$	−46.5630	−1.97294	−0.986469	−	0.163951i	$-0.947576\pi$
$557$	−46.5630	−1.97294	−0.986469	+	0.163951i	$0.947576\pi$
$558$	0	0
$559$	−25.0889	−1.06115
$560$	−31.8004	−1.34381
$561$	0	0
$562$	−3.17009	−0.133722
$563$	−31.1430	−1.31252	−0.656260	−	0.754535i	$-0.727863\pi$
$563$	−31.1430	−1.31252	−0.656260	+	0.754535i	$0.727863\pi$
$564$	0	0
$565$	48.9770	2.06048
$566$	34.3756	1.44492
$567$	0	0
$568$	12.3646	0.518806
$569$	−34.3775	−1.44118	−0.720590	−	0.693361i	$-0.756129\pi$
$569$	−34.3775	−1.44118	−0.720590	+	0.693361i	$0.756129\pi$
$570$	0	0
$571$	32.8068	1.37292	0.686462	−	0.727166i	$-0.259163\pi$
$571$	32.8068	1.37292	0.686462	+	0.727166i	$0.259163\pi$
$572$	−15.9121	−0.665320
$573$	0	0
$574$	−19.0466	−0.794988
$575$	−12.1376	−0.506172
$576$	0	0
$577$	−23.4257	−0.975224	−0.487612	−	0.873060i	$-0.662132\pi$
$577$	−23.4257	−0.975224	−0.487612	+	0.873060i	$0.662132\pi$
$578$	−123.450	−5.13485
$579$	0	0
$580$	6.79662	0.282214
$581$	11.1653	0.463213
$582$	0	0
$583$	−10.2040	−0.422608
$584$	−5.57119	−0.230538
$585$	0	0
$586$	76.0084	3.13988
$587$	7.19164	0.296831	0.148415	−	0.988925i	$-0.452583\pi$
$587$	7.19164	0.296831	0.148415	+	0.988925i	$0.452583\pi$
$588$	0	0
$589$	−12.2219	−0.503594
$590$	106.916	4.40165
$591$	0	0
$592$	72.6361	2.98532
$593$	−26.9313	−1.10594	−0.552968	−	0.833203i	$-0.686505\pi$
$593$	−26.9313	−1.10594	−0.552968	+	0.833203i	$0.686505\pi$
$594$	0	0
$595$	−23.8638	−0.978320
$596$	1.19103	0.0487864
$597$	0	0
$598$	26.7681	1.09463
$599$	−23.4927	−0.959884	−0.479942	−	0.877300i	$-0.659342\pi$
$599$	−23.4927	−0.959884	−0.479942	+	0.877300i	$0.659342\pi$
$600$	0	0
$601$	0.294510	0.0120133	0.00600667	−	0.999982i	$-0.498088\pi$
$601$	0.294510	0.0120133	0.00600667	+	0.999982i	$0.498088\pi$
$602$	20.1608	0.821694
$603$	0	0
$604$	22.0663	0.897865
$605$	30.0019	1.21975
$606$	0	0
$607$	44.8478	1.82032	0.910159	−	0.414259i	$-0.135959\pi$
$607$	44.8478	1.82032	0.910159	+	0.414259i	$0.135959\pi$
$608$	−48.9213	−1.98402
$609$	0	0
$610$	93.1186	3.77026
$611$	−6.49535	−0.262774
$612$	0	0
$613$	38.4993	1.55497	0.777486	−	0.628900i	$-0.216495\pi$
$613$	38.4993	1.55497	0.777486	+	0.628900i	$0.216495\pi$
$614$	51.4446	2.07613
$615$	0	0
$616$	7.62586	0.307255
$617$	−4.24126	−0.170747	−0.0853734	−	0.996349i	$-0.527208\pi$
$617$	−4.24126	−0.170747	−0.0853734	+	0.996349i	$0.527208\pi$
$618$	0	0
$619$	−18.5353	−0.744997	−0.372499	−	0.928033i	$-0.621499\pi$
$619$	−18.5353	−0.744997	−0.372499	+	0.928033i	$0.621499\pi$
$620$	46.1748	1.85442
$621$	0	0
$622$	−74.3410	−2.98080
$623$	−5.97776	−0.239494
$624$	0	0
$625$	−29.2202	−1.16881
$626$	22.5612	0.901729
$627$	0	0
$628$	−32.0426	−1.27864
$629$	54.5078	2.17337
$630$	0	0
$631$	−2.93062	−0.116666	−0.0583330	−	0.998297i	$-0.518579\pi$
$631$	−2.93062	−0.116666	−0.0583330	+	0.998297i	$0.518579\pi$
$632$	61.5590	2.44869
$633$	0	0
$634$	−49.6444	−1.97163
$635$	2.98743	0.118552
$636$	0	0
$637$	3.28196	0.130036
$638$	−1.18468	−0.0469021
$639$	0	0
$640$	17.0922	0.675629
$641$	31.1295	1.22954	0.614770	−	0.788706i	$-0.289249\pi$
$641$	31.1295	1.22954	0.614770	+	0.788706i	$0.289249\pi$
$642$	0	0
$643$	−41.4857	−1.63603	−0.818017	−	0.575194i	$-0.804927\pi$
$643$	−41.4857	−1.63603	−0.818017	+	0.575194i	$0.804927\pi$
$644$	−15.3249	−0.603887
$645$	0	0
$646$	−82.5479	−3.24780
$647$	50.2701	1.97632	0.988161	−	0.153420i	$-0.0490286\pi$
$647$	50.2701	1.97632	0.988161	+	0.153420i	$0.0490286\pi$
$648$	0	0
$649$	−13.2772	−0.521175
$650$	−33.9704	−1.33243
$651$	0	0
$652$	−4.06688	−0.159271
$653$	5.82195	0.227830	0.113915	−	0.993490i	$-0.463661\pi$
$653$	5.82195	0.227830	0.113915	+	0.993490i	$0.463661\pi$
$654$	0	0
$655$	−42.5284	−1.66172
$656$	76.8763	3.00152
$657$	0	0
$658$	5.21950	0.203477
$659$	7.59473	0.295849	0.147924	−	0.988999i	$-0.452741\pi$
$659$	7.59473	0.295849	0.147924	+	0.988999i	$0.452741\pi$
$660$	0	0
$661$	28.1340	1.09429	0.547143	−	0.837039i	$-0.315715\pi$
$661$	28.1340	1.09429	0.547143	+	0.837039i	$0.315715\pi$
$662$	−87.6989	−3.40851
$663$	0	0
$664$	−87.0235	−3.37717
$665$	−11.7058	−0.453932
$666$	0	0
$667$	1.41986	0.0549773
$668$	53.3890	2.06568
$669$	0	0
$670$	45.9236	1.77418
$671$	−11.5638	−0.446416
$672$	0	0
$673$	30.3708	1.17071	0.585355	−	0.810777i	$-0.300955\pi$
$673$	30.3708	1.17071	0.585355	+	0.810777i	$0.300955\pi$
$674$	−61.8986	−2.38425
$675$	0	0
$676$	−11.0441	−0.424775
$677$	41.1666	1.58216	0.791080	−	0.611713i	$-0.209519\pi$
$677$	41.1666	1.58216	0.791080	+	0.611713i	$0.209519\pi$
$678$	0	0
$679$	−6.56685	−0.252013
$680$	185.997	7.13267
$681$	0	0
$682$	−8.04849	−0.308193
$683$	3.02094	0.115593	0.0577966	−	0.998328i	$-0.481593\pi$
$683$	3.02094	0.115593	0.0577966	+	0.998328i	$0.481593\pi$
$684$	0	0
$685$	23.8509	0.911297
$686$	−2.63730	−0.100693
$687$	0	0
$688$	−81.3738	−3.10235
$689$	34.2282	1.30399
$690$	0	0
$691$	−18.2663	−0.694883	−0.347442	−	0.937702i	$-0.612950\pi$
$691$	−18.2663	−0.694883	−0.347442	+	0.937702i	$0.612950\pi$
$692$	−89.8566	−3.41583
$693$	0	0
$694$	28.5780	1.08480
$695$	17.9194	0.679722
$696$	0	0
$697$	57.6898	2.18516
$698$	−30.1573	−1.14147
$699$	0	0
$700$	19.4483	0.735076
$701$	−47.7121	−1.80206	−0.901031	−	0.433754i	$-0.857189\pi$
$701$	−47.7121	−1.80206	−0.901031	+	0.433754i	$0.857189\pi$
$702$	0	0
$703$	26.7375	1.00842
$704$	−11.3863	−0.429137
$705$	0	0
$706$	−96.2718	−3.62324
$707$	4.00300	0.150548
$708$	0	0
$709$	35.4011	1.32952	0.664758	−	0.747059i	$-0.268535\pi$
$709$	35.4011	1.32952	0.664758	+	0.747059i	$0.268535\pi$
$710$	−12.4988	−0.469071
$711$	0	0
$712$	46.5915	1.74609
$713$	9.64625	0.361255
$714$	0	0
$715$	9.59294	0.358756
$716$	−21.4258	−0.800719
$717$	0	0
$718$	−56.2415	−2.09891
$719$	50.4583	1.88178	0.940888	−	0.338719i	$-0.109994\pi$
$719$	50.4583	1.88178	0.940888	+	0.338719i	$0.109994\pi$
$720$	0	0
$721$	19.3768	0.721631
$722$	9.61677	0.357899
$723$	0	0
$724$	−101.813	−3.78385
$725$	−1.80190	−0.0669207
$726$	0	0
$727$	−50.3646	−1.86792	−0.933959	−	0.357379i	$-0.883670\pi$
$727$	−50.3646	−1.86792	−0.933959	+	0.357379i	$0.883670\pi$
$728$	−25.5800	−0.948059
$729$	0	0
$730$	5.63167	0.208437
$731$	−61.0648	−2.25856
$732$	0	0
$733$	3.68298	0.136034	0.0680170	−	0.997684i	$-0.478333\pi$
$733$	3.68298	0.136034	0.0680170	+	0.997684i	$0.478333\pi$
$734$	38.5211	1.42184
$735$	0	0
$736$	38.6116	1.42324
$737$	−5.70296	−0.210071
$738$	0	0
$739$	6.37616	0.234551	0.117275	−	0.993099i	$-0.462584\pi$
$739$	6.37616	0.234551	0.117275	+	0.993099i	$0.462584\pi$
$740$	−101.015	−3.71340
$741$	0	0
$742$	−27.5049	−1.00974
$743$	31.7691	1.16550	0.582748	−	0.812653i	$-0.301977\pi$
$743$	31.7691	1.16550	0.582748	+	0.812653i	$0.301977\pi$
$744$	0	0
$745$	−0.718035	−0.0263068
$746$	40.7899	1.49342
$747$	0	0
$748$	−38.7291	−1.41608
$749$	−4.10444	−0.149973
$750$	0	0
$751$	15.7023	0.572985	0.286493	−	0.958082i	$-0.407511\pi$
$751$	15.7023	0.572985	0.286493	+	0.958082i	$0.407511\pi$
$752$	−21.0671	−0.768239
$753$	0	0
$754$	3.97388	0.144720
$755$	−13.3031	−0.484149
$756$	0	0
$757$	−40.1842	−1.46052	−0.730259	−	0.683170i	$-0.760601\pi$
$757$	−40.1842	−1.46052	−0.730259	+	0.683170i	$0.760601\pi$
$758$	−61.1016	−2.21931
$759$	0	0
$760$	91.2366	3.30950
$761$	20.3166	0.736476	0.368238	−	0.929732i	$-0.379961\pi$
$761$	20.3166	0.736476	0.368238	+	0.929732i	$0.379961\pi$
$762$	0	0
$763$	−2.70497	−0.0979264
$764$	−3.33540	−0.120670
$765$	0	0
$766$	−16.6000	−0.599783
$767$	44.5367	1.60813
$768$	0	0
$769$	32.8800	1.18568	0.592841	−	0.805319i	$-0.298006\pi$
$769$	32.8800	1.18568	0.592841	+	0.805319i	$0.298006\pi$
$770$	−7.70864	−0.277800
$771$	0	0
$772$	75.4977	2.71722
$773$	25.4568	0.915618	0.457809	−	0.889051i	$-0.348634\pi$
$773$	25.4568	0.915618	0.457809	+	0.889051i	$0.348634\pi$
$774$	0	0
$775$	−12.2417	−0.439735
$776$	51.1829	1.83736
$777$	0	0
$778$	−38.9696	−1.39713
$779$	28.2984	1.01389
$780$	0	0
$781$	1.55215	0.0555402
$782$	65.1517	2.32982
$783$	0	0
$784$	10.6448	0.380170
$785$	19.3175	0.689471
$786$	0	0
$787$	27.1030	0.966117	0.483059	−	0.875588i	$-0.339526\pi$
$787$	27.1030	0.966117	0.483059	+	0.875588i	$0.339526\pi$
$788$	−17.6253	−0.627876
$789$	0	0
$790$	−62.2272	−2.21395
$791$	−16.3944	−0.582917
$792$	0	0
$793$	38.7894	1.37745
$794$	−12.2320	−0.434099
$795$	0	0
$796$	89.7652	3.18164
$797$	−9.07537	−0.321466	−0.160733	−	0.986998i	$-0.551386\pi$
$797$	−9.07537	−0.321466	−0.160733	+	0.986998i	$0.551386\pi$
$798$	0	0
$799$	−15.8093	−0.559292
$800$	−49.0006	−1.73243
$801$	0	0
$802$	51.0244	1.80173
$803$	−0.699362	−0.0246799
$804$	0	0
$805$	9.23893	0.325629
$806$	26.9977	0.950953
$807$	0	0
$808$	−31.1999	−1.09761
$809$	13.7846	0.484640	0.242320	−	0.970196i	$-0.422092\pi$
$809$	13.7846	0.484640	0.242320	+	0.970196i	$0.422092\pi$
$810$	0	0
$811$	−2.71737	−0.0954196	−0.0477098	−	0.998861i	$-0.515192\pi$
$811$	−2.71737	−0.0954196	−0.0477098	+	0.998861i	$0.515192\pi$
$812$	−2.27508	−0.0798395
$813$	0	0
$814$	17.6075	0.617142
$815$	2.45180	0.0858828
$816$	0	0
$817$	−29.9539	−1.04795
$818$	20.0643	0.701533
$819$	0	0
$820$	−106.912	−3.73354
$821$	51.2741	1.78948	0.894740	−	0.446588i	$-0.147361\pi$
$821$	51.2741	1.78948	0.894740	+	0.446588i	$0.147361\pi$
$822$	0	0
$823$	37.7120	1.31456	0.657279	−	0.753647i	$-0.271707\pi$
$823$	37.7120	1.31456	0.657279	+	0.753647i	$0.271707\pi$
$824$	−151.025	−5.26122
$825$	0	0
$826$	−35.7886	−1.24524
$827$	18.5690	0.645706	0.322853	−	0.946449i	$-0.395358\pi$
$827$	18.5690	0.645706	0.322853	+	0.946449i	$0.395358\pi$
$828$	0	0
$829$	37.0609	1.28718	0.643589	−	0.765371i	$-0.277445\pi$
$829$	37.0609	1.28718	0.643589	+	0.765371i	$0.277445\pi$
$830$	87.9681	3.05342
$831$	0	0
$832$	38.1939	1.32414
$833$	7.98808	0.276770
$834$	0	0
$835$	−32.1866	−1.11386
$836$	−18.9976	−0.657047
$837$	0	0
$838$	59.1629	2.04375
$839$	8.75759	0.302346	0.151173	−	0.988507i	$-0.451695\pi$
$839$	8.75759	0.302346	0.151173	+	0.988507i	$0.451695\pi$
$840$	0	0
$841$	−28.7892	−0.992731
$842$	−58.0073	−1.99906
$843$	0	0
$844$	−23.5833	−0.811771
$845$	6.65818	0.229048
$846$	0	0
$847$	−10.0427	−0.345072
$848$	111.016	3.81232
$849$	0	0
$850$	−82.6816	−2.83596
$851$	−21.1029	−0.723397
$852$	0	0
$853$	−27.0921	−0.927616	−0.463808	−	0.885936i	$-0.653517\pi$
$853$	−27.0921	−0.927616	−0.463808	+	0.885936i	$0.653517\pi$
$854$	−31.1702	−1.06662
$855$	0	0
$856$	31.9905	1.09341
$857$	15.8617	0.541825	0.270912	−	0.962604i	$-0.412675\pi$
$857$	15.8617	0.541825	0.270912	+	0.962604i	$0.412675\pi$
$858$	0	0
$859$	33.3808	1.13894	0.569468	−	0.822013i	$-0.307149\pi$
$859$	33.3808	1.13894	0.569468	+	0.822013i	$0.307149\pi$
$860$	113.167	3.85896
$861$	0	0
$862$	−13.7896	−0.469676
$863$	30.6759	1.04422	0.522110	−	0.852878i	$-0.325145\pi$
$863$	30.6759	1.04422	0.522110	+	0.852878i	$0.325145\pi$
$864$	0	0
$865$	54.1718	1.84190
$866$	31.1753	1.05938
$867$	0	0
$868$	−15.4564	−0.524624
$869$	7.72761	0.262141
$870$	0	0
$871$	19.1299	0.648192
$872$	21.0829	0.713956
$873$	0	0
$874$	31.9586	1.08102
$875$	3.21235	0.108597
$876$	0	0
$877$	−3.90914	−0.132002	−0.0660011	−	0.997820i	$-0.521024\pi$
$877$	−3.90914	−0.132002	−0.0660011	+	0.997820i	$0.521024\pi$
$878$	1.43655	0.0484812
$879$	0	0
$880$	31.1139	1.04885
$881$	33.6606	1.13405	0.567027	−	0.823699i	$-0.308093\pi$
$881$	33.6606	1.13405	0.567027	+	0.823699i	$0.308093\pi$
$882$	0	0
$883$	−31.8582	−1.07211	−0.536057	−	0.844182i	$-0.680087\pi$
$883$	−31.8582	−1.07211	−0.536057	+	0.844182i	$0.680087\pi$
$884$	129.912	4.36942
$885$	0	0
$886$	14.1929	0.476818
$887$	−33.3194	−1.11876	−0.559378	−	0.828912i	$-0.688960\pi$
$887$	−33.3194	−1.11876	−0.559378	+	0.828912i	$0.688960\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−47.0972	−1.57870
$891$	0	0
$892$	44.3019	1.48334
$893$	−7.75486	−0.259506
$894$	0	0
$895$	12.9169	0.431766
$896$	−5.72139	−0.191138
$897$	0	0
$898$	4.31266	0.143915
$899$	1.43204	0.0477613
$900$	0	0
$901$	83.3093	2.77543
$902$	18.6354	0.620489
$903$	0	0
$904$	127.780	4.24989
$905$	61.3798	2.04034
$906$	0	0
$907$	18.4293	0.611933	0.305967	−	0.952042i	$-0.401020\pi$
$907$	18.4293	0.611933	0.305967	+	0.952042i	$0.401020\pi$
$908$	−42.1903	−1.40013
$909$	0	0
$910$	25.8577	0.857174
$911$	48.7976	1.61674	0.808368	−	0.588678i	$-0.200351\pi$
$911$	48.7976	1.61674	0.808368	+	0.588678i	$0.200351\pi$
$912$	0	0
$913$	−10.9242	−0.361539
$914$	61.6891	2.04049
$915$	0	0
$916$	−30.8145	−1.01814
$917$	14.2358	0.470107
$918$	0	0
$919$	14.0377	0.463061	0.231530	−	0.972828i	$-0.425627\pi$
$919$	14.0377	0.463061	0.231530	+	0.972828i	$0.425627\pi$
$920$	−72.0094	−2.37408
$921$	0	0
$922$	84.7500	2.79109
$923$	−5.20649	−0.171374
$924$	0	0
$925$	26.7808	0.880549
$926$	−45.8584	−1.50700
$927$	0	0
$928$	5.73212	0.188166
$929$	44.8202	1.47050	0.735252	−	0.677793i	$-0.237064\pi$
$929$	44.8202	1.47050	0.735252	+	0.677793i	$0.237064\pi$
$930$	0	0
$931$	3.91836	0.128419
$932$	37.8644	1.24029
$933$	0	0
$934$	33.1273	1.08396
$935$	23.3486	0.763580
$936$	0	0
$937$	−10.7243	−0.350346	−0.175173	−	0.984538i	$-0.556049\pi$
$937$	−10.7243	−0.350346	−0.175173	+	0.984538i	$0.556049\pi$
$938$	−15.3723	−0.501923
$939$	0	0
$940$	29.2982	0.955601
$941$	11.3539	0.370127	0.185064	−	0.982727i	$-0.440751\pi$
$941$	11.3539	0.370127	0.185064	+	0.982727i	$0.440751\pi$
$942$	0	0
$943$	−22.3348	−0.727320
$944$	144.451	4.70148
$945$	0	0
$946$	−19.7256	−0.641334
$947$	−42.5428	−1.38246	−0.691228	−	0.722636i	$-0.742930\pi$
$947$	−42.5428	−1.38246	−0.691228	+	0.722636i	$0.742930\pi$
$948$	0	0
$949$	2.34592	0.0761519
$950$	−40.5575	−1.31586
$951$	0	0
$952$	−62.2601	−2.01786
$953$	−32.1697	−1.04208	−0.521039	−	0.853533i	$-0.674456\pi$
$953$	−32.1697	−1.04208	−0.521039	+	0.853533i	$0.674456\pi$
$954$	0	0
$955$	2.01081	0.0650682
$956$	2.35329	0.0761109
$957$	0	0
$958$	−32.3230	−1.04431
$959$	−7.98377	−0.257809
$960$	0	0
$961$	−21.2710	−0.686162
$962$	−59.0622	−1.90424
$963$	0	0
$964$	−126.431	−4.07207
$965$	−45.5153	−1.46519
$966$	0	0
$967$	49.6638	1.59708	0.798540	−	0.601942i	$-0.205606\pi$
$967$	49.6638	1.59708	0.798540	+	0.601942i	$0.205606\pi$
$968$	78.2742	2.51583
$969$	0	0
$970$	−51.7385	−1.66122
$971$	−1.76184	−0.0565403	−0.0282701	−	0.999600i	$-0.509000\pi$
$971$	−1.76184	−0.0565403	−0.0282701	+	0.999600i	$0.509000\pi$
$972$	0	0
$973$	−5.99828	−0.192296
$974$	12.4827	0.399971
$975$	0	0
$976$	125.810	4.02708
$977$	−10.0805	−0.322505	−0.161252	−	0.986913i	$-0.551553\pi$
$977$	−10.0805	−0.322505	−0.161252	+	0.986913i	$0.551553\pi$
$978$	0	0
$979$	5.84871	0.186926
$980$	−14.8037	−0.472888
$981$	0	0
$982$	−20.3403	−0.649085
$983$	39.2502	1.25189	0.625943	−	0.779868i	$-0.284714\pi$
$983$	39.2502	1.25189	0.625943	+	0.779868i	$0.284714\pi$
$984$	0	0
$985$	10.6258	0.338565
$986$	9.67216	0.308024
$987$	0	0
$988$	63.7253	2.02737
$989$	23.6414	0.751753
$990$	0	0
$991$	−30.9846	−0.984259	−0.492129	−	0.870522i	$-0.663781\pi$
$991$	−30.9846	−0.984259	−0.492129	+	0.870522i	$0.663781\pi$
$992$	38.9428	1.23644
$993$	0	0
$994$	4.18380	0.132702
$995$	−54.1167	−1.71561
$996$	0	0
$997$	−40.4655	−1.28155	−0.640777	−	0.767727i	$-0.721388\pi$
$997$	−40.4655	−1.28155	−0.640777	+	0.767727i	$0.721388\pi$
$998$	−16.9483	−0.536490
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.4	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.37	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.4	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.37	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.63730 q^{2} +4.95534 q^{4} -2.98743 q^{5} +1.00000 q^{7} -7.79413 q^{8} +O(q^{10})\)	\(q-2.63730 q^{2} +4.95534 q^{4} -2.98743 q^{5} +1.00000 q^{7} -7.79413 q^{8} +7.87873 q^{10} -0.978411 q^{11} +3.28196 q^{13} -2.63730 q^{14} +10.6448 q^{16} +7.98808 q^{17} +3.91836 q^{19} -14.8037 q^{20} +2.58036 q^{22} -3.09261 q^{23} +3.92471 q^{25} -8.65551 q^{26} +4.95534 q^{28} -0.459116 q^{29} -3.11913 q^{31} -12.4851 q^{32} -21.0669 q^{34} -2.98743 q^{35} +6.82365 q^{37} -10.3339 q^{38} +23.2844 q^{40} +7.22199 q^{41} -7.64450 q^{43} -4.84836 q^{44} +8.15612 q^{46} -1.97911 q^{47} +1.00000 q^{49} -10.3506 q^{50} +16.2632 q^{52} +10.4292 q^{53} +2.92293 q^{55} -7.79413 q^{56} +1.21082 q^{58} +13.5702 q^{59} +11.8190 q^{61} +8.22609 q^{62} +11.6375 q^{64} -9.80461 q^{65} +5.82880 q^{67} +39.5837 q^{68} +7.87873 q^{70} -1.58640 q^{71} +0.714793 q^{73} -17.9960 q^{74} +19.4168 q^{76} -0.978411 q^{77} -7.89812 q^{79} -31.8004 q^{80} -19.0466 q^{82} +11.1653 q^{83} -23.8638 q^{85} +20.1608 q^{86} +7.62586 q^{88} -5.97776 q^{89} +3.28196 q^{91} -15.3249 q^{92} +5.21950 q^{94} -11.7058 q^{95} -6.56685 q^{97} -2.63730 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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