LMFDB - Embedded newform 8330.2.a.cn.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8330,2,Mod(1,8330)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8330, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8330.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8330.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-6,-2,6,6,2,0,-6,4,-6,-6,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.5153848837$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.12694016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 1 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.04989$ of defining polynomial
Character	$\chi$	$=$	8330.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.20207 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.20207 q^{6} -1.00000 q^{8} -1.55504 q^{9} -1.00000 q^{10} +1.04989 q^{11} +1.20207 q^{12} -1.69998 q^{13} +1.20207 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.55504 q^{18} -3.94888 q^{19} +1.00000 q^{20} -1.04989 q^{22} -2.41130 q^{23} -1.20207 q^{24} +1.00000 q^{25} +1.69998 q^{26} -5.47546 q^{27} -0.0675053 q^{29} -1.20207 q^{30} +5.15812 q^{31} -1.00000 q^{32} +1.26204 q^{33} -1.00000 q^{34} -1.55504 q^{36} -8.78536 q^{37} +3.94888 q^{38} -2.04349 q^{39} -1.00000 q^{40} +8.34869 q^{41} +11.4148 q^{43} +1.04989 q^{44} -1.55504 q^{45} +2.41130 q^{46} -4.57861 q^{47} +1.20207 q^{48} -1.00000 q^{50} +1.20207 q^{51} -1.69998 q^{52} +3.49627 q^{53} +5.47546 q^{54} +1.04989 q^{55} -4.74682 q^{57} +0.0675053 q^{58} -7.57739 q^{59} +1.20207 q^{60} +8.28619 q^{61} -5.15812 q^{62} +1.00000 q^{64} -1.69998 q^{65} -1.26204 q^{66} -9.06540 q^{67} +1.00000 q^{68} -2.89855 q^{69} +0.799200 q^{71} +1.55504 q^{72} +6.62819 q^{73} +8.78536 q^{74} +1.20207 q^{75} -3.94888 q^{76} +2.04349 q^{78} -11.1687 q^{79} +1.00000 q^{80} -1.91674 q^{81} -8.34869 q^{82} -17.5875 q^{83} +1.00000 q^{85} -11.4148 q^{86} -0.0811458 q^{87} -1.04989 q^{88} -1.91107 q^{89} +1.55504 q^{90} -2.41130 q^{92} +6.20040 q^{93} +4.57861 q^{94} -3.94888 q^{95} -1.20207 q^{96} -9.16242 q^{97} -1.63262 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 6 q^{5} + 2 q^{6} - 6 q^{8} + 4 q^{9} - 6 q^{10} - 6 q^{11} - 2 q^{12} - 2 q^{15} + 6 q^{16} + 6 q^{17} - 4 q^{18} + 6 q^{19} + 6 q^{20} + 6 q^{22} - 2 q^{23} + 2 q^{24}+ \cdots - 8 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 + 6 * q^5 + 2 * q^6 - 6 * q^8 + 4 * q^9 - 6 * q^10 - 6 * q^11 - 2 * q^12 - 2 * q^15 + 6 * q^16 + 6 * q^17 - 4 * q^18 + 6 * q^19 + 6 * q^20 + 6 * q^22 - 2 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 2 * q^29 + 2 * q^30 - 2 * q^31 - 6 * q^32 - 4 * q^33 - 6 * q^34 + 4 * q^36 - 12 * q^37 - 6 * q^38 - 6 * q^40 + 6 * q^41 - 26 * q^43 - 6 * q^44 + 4 * q^45 + 2 * q^46 - 30 * q^47 - 2 * q^48 - 6 * q^50 - 2 * q^51 + 14 * q^53 + 20 * q^54 - 6 * q^55 - 8 * q^57 + 2 * q^58 - 6 * q^59 - 2 * q^60 + 12 * q^61 + 2 * q^62 + 6 * q^64 + 4 * q^66 - 14 * q^67 + 6 * q^68 - 2 * q^69 + 12 * q^71 - 4 * q^72 - 12 * q^73 + 12 * q^74 - 2 * q^75 + 6 * q^76 - 36 * q^79 + 6 * q^80 - 2 * q^81 - 6 * q^82 - 32 * q^83 + 6 * q^85 + 26 * q^86 - 2 * q^87 + 6 * q^88 + 8 * q^89 - 4 * q^90 - 2 * q^92 - 12 * q^93 + 30 * q^94 + 6 * q^95 + 2 * q^96 + 16 * q^97 - 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.20207	0.694013	0.347007	−	0.937863i	$-0.387198\pi$
$3$	1.20207	0.694013	0.347007	+	0.937863i	$0.387198\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.20207	−0.490741
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−1.55504	−0.518346
$10$	−1.00000	−0.316228
$11$	1.04989	0.316555	0.158277	−	0.987395i	$-0.449406\pi$
$11$	1.04989	0.316555	0.158277	+	0.987395i	$0.449406\pi$
$12$	1.20207	0.347007
$13$	−1.69998	−0.471489	−0.235745	−	0.971815i	$-0.575753\pi$
$13$	−1.69998	−0.471489	−0.235745	+	0.971815i	$0.575753\pi$
$14$	0	0
$15$	1.20207	0.310372
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.55504	0.366526
$19$	−3.94888	−0.905936	−0.452968	−	0.891527i	$-0.649635\pi$
$19$	−3.94888	−0.905936	−0.452968	+	0.891527i	$0.649635\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.04989	−0.223838
$23$	−2.41130	−0.502791	−0.251396	−	0.967884i	$-0.580890\pi$
$23$	−2.41130	−0.502791	−0.251396	+	0.967884i	$0.580890\pi$
$24$	−1.20207	−0.245371
$25$	1.00000	0.200000
$26$	1.69998	0.333393
$27$	−5.47546	−1.05375
$28$	0	0
$29$	−0.0675053	−0.0125354	−0.00626771	−	0.999980i	$-0.501995\pi$
$29$	−0.0675053	−0.0125354	−0.00626771	+	0.999980i	$0.501995\pi$
$30$	−1.20207	−0.219466
$31$	5.15812	0.926426	0.463213	−	0.886247i	$-0.346697\pi$
$31$	5.15812	0.926426	0.463213	+	0.886247i	$0.346697\pi$
$32$	−1.00000	−0.176777
$33$	1.26204	0.219693
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−1.55504	−0.259173
$37$	−8.78536	−1.44430	−0.722152	−	0.691734i	$-0.756847\pi$
$37$	−8.78536	−1.44430	−0.722152	+	0.691734i	$0.756847\pi$
$38$	3.94888	0.640593
$39$	−2.04349	−0.327220
$40$	−1.00000	−0.158114
$41$	8.34869	1.30385	0.651923	−	0.758285i	$-0.273962\pi$
$41$	8.34869	1.30385	0.651923	+	0.758285i	$0.273962\pi$
$42$	0	0
$43$	11.4148	1.74074	0.870372	−	0.492395i	$-0.163878\pi$
$43$	11.4148	1.74074	0.870372	+	0.492395i	$0.163878\pi$
$44$	1.04989	0.158277
$45$	−1.55504	−0.231811
$46$	2.41130	0.355527
$47$	−4.57861	−0.667860	−0.333930	−	0.942598i	$-0.608375\pi$
$47$	−4.57861	−0.667860	−0.333930	+	0.942598i	$0.608375\pi$
$48$	1.20207	0.173503
$49$	0	0
$50$	−1.00000	−0.141421
$51$	1.20207	0.168323
$52$	−1.69998	−0.235745
$53$	3.49627	0.480249	0.240125	−	0.970742i	$-0.422812\pi$
$53$	3.49627	0.480249	0.240125	+	0.970742i	$0.422812\pi$
$54$	5.47546	0.745115
$55$	1.04989	0.141568
$56$	0	0
$57$	−4.74682	−0.628731
$58$	0.0675053	0.00886388
$59$	−7.57739	−0.986492	−0.493246	−	0.869890i	$-0.664190\pi$
$59$	−7.57739	−0.986492	−0.493246	+	0.869890i	$0.664190\pi$
$60$	1.20207	0.155186
$61$	8.28619	1.06094	0.530469	−	0.847705i	$-0.322016\pi$
$61$	8.28619	1.06094	0.530469	+	0.847705i	$0.322016\pi$
$62$	−5.15812	−0.655082
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.69998	−0.210856
$66$	−1.26204	−0.155347
$67$	−9.06540	−1.10751	−0.553757	−	0.832678i	$-0.686807\pi$
$67$	−9.06540	−1.10751	−0.553757	+	0.832678i	$0.686807\pi$
$68$	1.00000	0.121268
$69$	−2.89855	−0.348944
$70$	0	0
$71$	0.799200	0.0948476	0.0474238	−	0.998875i	$-0.484899\pi$
$71$	0.799200	0.0948476	0.0474238	+	0.998875i	$0.484899\pi$
$72$	1.55504	0.183263
$73$	6.62819	0.775771	0.387886	−	0.921707i	$-0.373206\pi$
$73$	6.62819	0.775771	0.387886	+	0.921707i	$0.373206\pi$
$74$	8.78536	1.02128
$75$	1.20207	0.138803
$76$	−3.94888	−0.452968
$77$	0	0
$78$	2.04349	0.231379
$79$	−11.1687	−1.25658	−0.628289	−	0.777980i	$-0.716244\pi$
$79$	−11.1687	−1.25658	−0.628289	+	0.777980i	$0.716244\pi$
$80$	1.00000	0.111803
$81$	−1.91674	−0.212972
$82$	−8.34869	−0.921959
$83$	−17.5875	−1.93048	−0.965242	−	0.261359i	$-0.915829\pi$
$83$	−17.5875	−1.93048	−0.965242	+	0.261359i	$0.915829\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	−11.4148	−1.23089
$87$	−0.0811458	−0.00869975
$88$	−1.04989	−0.111919
$89$	−1.91107	−0.202573	−0.101286	−	0.994857i	$-0.532296\pi$
$89$	−1.91107	−0.202573	−0.101286	+	0.994857i	$0.532296\pi$
$90$	1.55504	0.163915
$91$	0	0
$92$	−2.41130	−0.251396
$93$	6.20040	0.642951
$94$	4.57861	0.472248
$95$	−3.94888	−0.405147
$96$	−1.20207	−0.122685
$97$	−9.16242	−0.930303	−0.465151	−	0.885231i	$-0.654000\pi$
$97$	−9.16242	−0.930303	−0.465151	+	0.885231i	$0.654000\pi$
$98$	0	0
$99$	−1.63262	−0.164085
$100$	1.00000	0.100000
$101$	−13.4510	−1.33842	−0.669210	−	0.743073i	$-0.733367\pi$
$101$	−13.4510	−1.33842	−0.669210	+	0.743073i	$0.733367\pi$
$102$	−1.20207	−0.119022
$103$	1.82552	0.179874	0.0899368	−	0.995947i	$-0.471334\pi$
$103$	1.82552	0.179874	0.0899368	+	0.995947i	$0.471334\pi$
$104$	1.69998	0.166697
$105$	0	0
$106$	−3.49627	−0.339587
$107$	13.0060	1.25734	0.628671	−	0.777671i	$-0.283599\pi$
$107$	13.0060	1.25734	0.628671	+	0.777671i	$0.283599\pi$
$108$	−5.47546	−0.526876
$109$	−0.0971344	−0.00930379	−0.00465190	−	0.999989i	$-0.501481\pi$
$109$	−0.0971344	−0.00930379	−0.00465190	+	0.999989i	$0.501481\pi$
$110$	−1.04989	−0.100103
$111$	−10.5606	−1.00237
$112$	0	0
$113$	−9.08386	−0.854538	−0.427269	−	0.904125i	$-0.640524\pi$
$113$	−9.08386	−0.854538	−0.427269	+	0.904125i	$0.640524\pi$
$114$	4.74682	0.444580
$115$	−2.41130	−0.224855
$116$	−0.0675053	−0.00626771
$117$	2.64353	0.244394
$118$	7.57739	0.697555
$119$	0	0
$120$	−1.20207	−0.109733
$121$	−9.89772	−0.899793
$122$	−8.28619	−0.750196
$123$	10.0357	0.904887
$124$	5.15812	0.463213
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.08663	−0.185159	−0.0925794	−	0.995705i	$-0.529511\pi$
$127$	−2.08663	−0.185159	−0.0925794	+	0.995705i	$0.529511\pi$
$128$	−1.00000	−0.0883883
$129$	13.7214	1.20810
$130$	1.69998	0.149098
$131$	3.46614	0.302838	0.151419	−	0.988470i	$-0.451616\pi$
$131$	3.46614	0.302838	0.151419	+	0.988470i	$0.451616\pi$
$132$	1.26204	0.109847
$133$	0	0
$134$	9.06540	0.783131
$135$	−5.47546	−0.471252
$136$	−1.00000	−0.0857493
$137$	21.3247	1.82190	0.910948	−	0.412522i	$-0.135352\pi$
$137$	21.3247	1.82190	0.910948	+	0.412522i	$0.135352\pi$
$138$	2.89855	0.246741
$139$	−4.23241	−0.358989	−0.179494	−	0.983759i	$-0.557446\pi$
$139$	−4.23241	−0.358989	−0.179494	+	0.983759i	$0.557446\pi$
$140$	0	0
$141$	−5.50380	−0.463503
$142$	−0.799200	−0.0670674
$143$	−1.78480	−0.149252
$144$	−1.55504	−0.129586
$145$	−0.0675053	−0.00560601
$146$	−6.62819	−0.548553
$147$	0	0
$148$	−8.78536	−0.722152
$149$	−3.26033	−0.267097	−0.133548	−	0.991042i	$-0.542637\pi$
$149$	−3.26033	−0.267097	−0.133548	+	0.991042i	$0.542637\pi$
$150$	−1.20207	−0.0981483
$151$	18.8842	1.53677	0.768386	−	0.639987i	$-0.221060\pi$
$151$	18.8842	1.53677	0.768386	+	0.639987i	$0.221060\pi$
$152$	3.94888	0.320297
$153$	−1.55504	−0.125717
$154$	0	0
$155$	5.15812	0.414310
$156$	−2.04349	−0.163610
$157$	−5.02526	−0.401059	−0.200530	−	0.979688i	$-0.564266\pi$
$157$	−5.02526	−0.401059	−0.200530	+	0.979688i	$0.564266\pi$
$158$	11.1687	0.888534
$159$	4.20274	0.333299
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.91674	0.150594
$163$	−21.4233	−1.67800	−0.839000	−	0.544131i	$-0.816859\pi$
$163$	−21.4233	−1.67800	−0.839000	+	0.544131i	$0.816859\pi$
$164$	8.34869	0.651923
$165$	1.26204	0.0982498
$166$	17.5875	1.36506
$167$	−4.20850	−0.325663	−0.162832	−	0.986654i	$-0.552063\pi$
$167$	−4.20850	−0.325663	−0.162832	+	0.986654i	$0.552063\pi$
$168$	0	0
$169$	−10.1101	−0.777698
$170$	−1.00000	−0.0766965
$171$	6.14066	0.469588
$172$	11.4148	0.870372
$173$	−25.0195	−1.90219	−0.951097	−	0.308891i	$-0.900042\pi$
$173$	−25.0195	−1.90219	−0.951097	+	0.308891i	$0.900042\pi$
$174$	0.0811458	0.00615165
$175$	0	0
$176$	1.04989	0.0791387
$177$	−9.10852	−0.684639
$178$	1.91107	0.143240
$179$	−14.6941	−1.09829	−0.549144	−	0.835728i	$-0.685046\pi$
$179$	−14.6941	−1.09829	−0.549144	+	0.835728i	$0.685046\pi$
$180$	−1.55504	−0.115906
$181$	−15.0945	−1.12196	−0.560982	−	0.827828i	$-0.689576\pi$
$181$	−15.0945	−1.12196	−0.560982	+	0.827828i	$0.689576\pi$
$182$	0	0
$183$	9.96054	0.736304
$184$	2.41130	0.177764
$185$	−8.78536	−0.645913
$186$	−6.20040	−0.454635
$187$	1.04989	0.0767759
$188$	−4.57861	−0.333930
$189$	0	0
$190$	3.94888	0.286482
$191$	−0.110856	−0.00802128	−0.00401064	−	0.999992i	$-0.501277\pi$
$191$	−0.110856	−0.00802128	−0.00401064	+	0.999992i	$0.501277\pi$
$192$	1.20207	0.0867516
$193$	−14.7351	−1.06065	−0.530327	−	0.847793i	$-0.677931\pi$
$193$	−14.7351	−1.06065	−0.530327	+	0.847793i	$0.677931\pi$
$194$	9.16242	0.657823
$195$	−2.04349	−0.146337
$196$	0	0
$197$	−2.58188	−0.183951	−0.0919755	−	0.995761i	$-0.529318\pi$
$197$	−2.58188	−0.183951	−0.0919755	+	0.995761i	$0.529318\pi$
$198$	1.63262	0.116026
$199$	−12.2402	−0.867686	−0.433843	−	0.900989i	$-0.642843\pi$
$199$	−12.2402	−0.867686	−0.433843	+	0.900989i	$0.642843\pi$
$200$	−1.00000	−0.0707107
$201$	−10.8972	−0.768630
$202$	13.4510	0.946406
$203$	0	0
$204$	1.20207	0.0841614
$205$	8.34869	0.583098
$206$	−1.82552	−0.127190
$207$	3.74967	0.260620
$208$	−1.69998	−0.117872
$209$	−4.14591	−0.286778
$210$	0	0
$211$	−25.6270	−1.76423	−0.882117	−	0.471031i	$-0.843882\pi$
$211$	−25.6270	−1.76423	−0.882117	+	0.471031i	$0.843882\pi$
$212$	3.49627	0.240125
$213$	0.960691	0.0658255
$214$	−13.0060	−0.889075
$215$	11.4148	0.778484
$216$	5.47546	0.372558
$217$	0	0
$218$	0.0971344	0.00657877
$219$	7.96752	0.538395
$220$	1.04989	0.0707838
$221$	−1.69998	−0.114353
$222$	10.5606	0.708780
$223$	15.7512	1.05478	0.527390	−	0.849623i	$-0.323171\pi$
$223$	15.7512	1.05478	0.527390	+	0.849623i	$0.323171\pi$
$224$	0	0
$225$	−1.55504	−0.103669
$226$	9.08386	0.604250
$227$	1.25487	0.0832887	0.0416443	−	0.999132i	$-0.486740\pi$
$227$	1.25487	0.0832887	0.0416443	+	0.999132i	$0.486740\pi$
$228$	−4.74682	−0.314366
$229$	13.3789	0.884102	0.442051	−	0.896990i	$-0.354251\pi$
$229$	13.3789	0.884102	0.442051	+	0.896990i	$0.354251\pi$
$230$	2.41130	0.158997
$231$	0	0
$232$	0.0675053	0.00443194
$233$	−18.7193	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$233$	−18.7193	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$234$	−2.64353	−0.172813
$235$	−4.57861	−0.298676
$236$	−7.57739	−0.493246
$237$	−13.4255	−0.872081
$238$	0	0
$239$	−14.6586	−0.948186	−0.474093	−	0.880475i	$-0.657224\pi$
$239$	−14.6586	−0.948186	−0.474093	+	0.880475i	$0.657224\pi$
$240$	1.20207	0.0775930
$241$	9.80605	0.631663	0.315832	−	0.948815i	$-0.397717\pi$
$241$	9.80605	0.631663	0.315832	+	0.948815i	$0.397717\pi$
$242$	9.89772	0.636250
$243$	14.1223	0.905947
$244$	8.28619	0.530469
$245$	0	0
$246$	−10.0357	−0.639851
$247$	6.71301	0.427139
$248$	−5.15812	−0.327541
$249$	−21.1414	−1.33978
$250$	−1.00000	−0.0632456
$251$	−13.6915	−0.864198	−0.432099	−	0.901826i	$-0.642227\pi$
$251$	−13.6915	−0.864198	−0.432099	+	0.901826i	$0.642227\pi$
$252$	0	0
$253$	−2.53161	−0.159161
$254$	2.08663	0.130927
$255$	1.20207	0.0752763
$256$	1.00000	0.0625000
$257$	−3.48500	−0.217388	−0.108694	−	0.994075i	$-0.534667\pi$
$257$	−3.48500	−0.217388	−0.108694	+	0.994075i	$0.534667\pi$
$258$	−13.7214	−0.854255
$259$	0	0
$260$	−1.69998	−0.105428
$261$	0.104973	0.00649769
$262$	−3.46614	−0.214139
$263$	−2.13558	−0.131685	−0.0658427	−	0.997830i	$-0.520974\pi$
$263$	−2.13558	−0.131685	−0.0658427	+	0.997830i	$0.520974\pi$
$264$	−1.26204	−0.0776733
$265$	3.49627	0.214774
$266$	0	0
$267$	−2.29723	−0.140588
$268$	−9.06540	−0.553757
$269$	10.3321	0.629957	0.314978	−	0.949099i	$-0.398003\pi$
$269$	10.3321	0.629957	0.314978	+	0.949099i	$0.398003\pi$
$270$	5.47546	0.333226
$271$	6.90805	0.419634	0.209817	−	0.977741i	$-0.432713\pi$
$271$	6.90805	0.419634	0.209817	+	0.977741i	$0.432713\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−21.3247	−1.28827
$275$	1.04989	0.0633110
$276$	−2.89855	−0.174472
$277$	17.9763	1.08009	0.540047	−	0.841635i	$-0.318407\pi$
$277$	17.9763	1.08009	0.540047	+	0.841635i	$0.318407\pi$
$278$	4.23241	0.253843
$279$	−8.02107	−0.480209
$280$	0	0
$281$	−12.9520	−0.772652	−0.386326	−	0.922362i	$-0.626256\pi$
$281$	−12.9520	−0.772652	−0.386326	+	0.922362i	$0.626256\pi$
$282$	5.50380	0.327746
$283$	5.35862	0.318537	0.159269	−	0.987235i	$-0.449086\pi$
$283$	5.35862	0.318537	0.159269	+	0.987235i	$0.449086\pi$
$284$	0.799200	0.0474238
$285$	−4.74682	−0.281177
$286$	1.78480	0.105537
$287$	0	0
$288$	1.55504	0.0916315
$289$	1.00000	0.0588235
$290$	0.0675053	0.00396405
$291$	−11.0138	−0.645642
$292$	6.62819	0.387886
$293$	−31.3388	−1.83083	−0.915415	−	0.402512i	$-0.868137\pi$
$293$	−31.3388	−1.83083	−0.915415	+	0.402512i	$0.868137\pi$
$294$	0	0
$295$	−7.57739	−0.441173
$296$	8.78536	0.510639
$297$	−5.74865	−0.333570
$298$	3.26033	0.188866
$299$	4.09916	0.237061
$300$	1.20207	0.0694013
$301$	0	0
$302$	−18.8842	−1.08666
$303$	−16.1689	−0.928881
$304$	−3.94888	−0.226484
$305$	8.28619	0.474466
$306$	1.55504	0.0888956
$307$	−21.5310	−1.22884	−0.614418	−	0.788980i	$-0.710609\pi$
$307$	−21.5310	−1.22884	−0.614418	+	0.788980i	$0.710609\pi$
$308$	0	0
$309$	2.19439	0.124835
$310$	−5.15812	−0.292962
$311$	5.00428	0.283767	0.141883	−	0.989883i	$-0.454684\pi$
$311$	5.00428	0.283767	0.141883	+	0.989883i	$0.454684\pi$
$312$	2.04349	0.115690
$313$	29.9140	1.69084	0.845419	−	0.534104i	$-0.179351\pi$
$313$	29.9140	1.69084	0.845419	+	0.534104i	$0.179351\pi$
$314$	5.02526	0.283592
$315$	0	0
$316$	−11.1687	−0.628289
$317$	7.04811	0.395861	0.197931	−	0.980216i	$-0.436578\pi$
$317$	7.04811	0.395861	0.197931	+	0.980216i	$0.436578\pi$
$318$	−4.20274	−0.235678
$319$	−0.0708734	−0.00396815
$320$	1.00000	0.0559017
$321$	15.6341	0.872612
$322$	0	0
$323$	−3.94888	−0.219722
$324$	−1.91674	−0.106486
$325$	−1.69998	−0.0942978
$326$	21.4233	1.18653
$327$	−0.116762	−0.00645695
$328$	−8.34869	−0.460979
$329$	0	0
$330$	−1.26204	−0.0694731
$331$	−16.6572	−0.915562	−0.457781	−	0.889065i	$-0.651356\pi$
$331$	−16.6572	−0.915562	−0.457781	+	0.889065i	$0.651356\pi$
$332$	−17.5875	−0.965242
$333$	13.6616	0.748649
$334$	4.20850	0.230279
$335$	−9.06540	−0.495296
$336$	0	0
$337$	−11.5188	−0.627468	−0.313734	−	0.949511i	$-0.601580\pi$
$337$	−11.5188	−0.627468	−0.313734	+	0.949511i	$0.601580\pi$
$338$	10.1101	0.549916
$339$	−10.9194	−0.593061
$340$	1.00000	0.0542326
$341$	5.41548	0.293265
$342$	−6.14066	−0.332049
$343$	0	0
$344$	−11.4148	−0.615446
$345$	−2.89855	−0.156052
$346$	25.0195	1.34505
$347$	−4.06413	−0.218174	−0.109087	−	0.994032i	$-0.534793\pi$
$347$	−4.06413	−0.218174	−0.109087	+	0.994032i	$0.534793\pi$
$348$	−0.0811458	−0.00434987
$349$	−0.809305	−0.0433211	−0.0216606	−	0.999765i	$-0.506895\pi$
$349$	−0.809305	−0.0433211	−0.0216606	+	0.999765i	$0.506895\pi$
$350$	0	0
$351$	9.30815	0.496832
$352$	−1.04989	−0.0559595
$353$	−0.840633	−0.0447424	−0.0223712	−	0.999750i	$-0.507122\pi$
$353$	−0.840633	−0.0447424	−0.0223712	+	0.999750i	$0.507122\pi$
$354$	9.10852	0.484113
$355$	0.799200	0.0424171
$356$	−1.91107	−0.101286
$357$	0	0
$358$	14.6941	0.776607
$359$	18.8913	0.997045	0.498523	−	0.866877i	$-0.333876\pi$
$359$	18.8913	0.997045	0.498523	+	0.866877i	$0.333876\pi$
$360$	1.55504	0.0819577
$361$	−3.40633	−0.179281
$362$	15.0945	0.793348
$363$	−11.8977	−0.624468
$364$	0	0
$365$	6.62819	0.346935
$366$	−9.96054	−0.520646
$367$	−34.8805	−1.82075	−0.910374	−	0.413786i	$-0.864206\pi$
$367$	−34.8805	−1.82075	−0.910374	+	0.413786i	$0.864206\pi$
$368$	−2.41130	−0.125698
$369$	−12.9825	−0.675844
$370$	8.78536	0.456729
$371$	0	0
$372$	6.20040	0.321476
$373$	10.3388	0.535324	0.267662	−	0.963513i	$-0.413749\pi$
$373$	10.3388	0.535324	0.267662	+	0.963513i	$0.413749\pi$
$374$	−1.04989	−0.0542887
$375$	1.20207	0.0620744
$376$	4.57861	0.236124
$377$	0.114758	0.00591031
$378$	0	0
$379$	20.5012	1.05308	0.526538	−	0.850152i	$-0.323490\pi$
$379$	20.5012	1.05308	0.526538	+	0.850152i	$0.323490\pi$
$380$	−3.94888	−0.202573
$381$	−2.50827	−0.128503
$382$	0.110856	0.00567190
$383$	29.0273	1.48323	0.741614	−	0.670827i	$-0.234061\pi$
$383$	29.0273	1.48323	0.741614	+	0.670827i	$0.234061\pi$
$384$	−1.20207	−0.0613427
$385$	0	0
$386$	14.7351	0.749995
$387$	−17.7505	−0.902308
$388$	−9.16242	−0.465151
$389$	−26.8338	−1.36053	−0.680265	−	0.732966i	$-0.738135\pi$
$389$	−26.8338	−1.36053	−0.680265	+	0.732966i	$0.738135\pi$
$390$	2.04349	0.103476
$391$	−2.41130	−0.121945
$392$	0	0
$393$	4.16652	0.210173
$394$	2.58188	0.130073
$395$	−11.1687	−0.561959
$396$	−1.63262	−0.0820425
$397$	−29.1916	−1.46508	−0.732542	−	0.680722i	$-0.761666\pi$
$397$	−29.1916	−1.46508	−0.732542	+	0.680722i	$0.761666\pi$
$398$	12.2402	0.613546
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.80078	0.339615	0.169807	−	0.985477i	$-0.445685\pi$
$401$	6.80078	0.339615	0.169807	+	0.985477i	$0.445685\pi$
$402$	10.8972	0.543503
$403$	−8.76869	−0.436799
$404$	−13.4510	−0.669210
$405$	−1.91674	−0.0952438
$406$	0	0
$407$	−9.22370	−0.457202
$408$	−1.20207	−0.0595111
$409$	2.63414	0.130250	0.0651248	−	0.997877i	$-0.479255\pi$
$409$	2.63414	0.130250	0.0651248	+	0.997877i	$0.479255\pi$
$410$	−8.34869	−0.412313
$411$	25.6337	1.26442
$412$	1.82552	0.0899368
$413$	0	0
$414$	−3.74967	−0.184286
$415$	−17.5875	−0.863338
$416$	1.69998	0.0833483
$417$	−5.08764	−0.249143
$418$	4.14591	0.202783
$419$	35.2146	1.72034	0.860172	−	0.510004i	$-0.170356\pi$
$419$	35.2146	1.72034	0.860172	+	0.510004i	$0.170356\pi$
$420$	0	0
$421$	24.9152	1.21429	0.607146	−	0.794590i	$-0.292314\pi$
$421$	24.9152	1.21429	0.607146	+	0.794590i	$0.292314\pi$
$422$	25.6270	1.24750
$423$	7.11992	0.346182
$424$	−3.49627	−0.169794
$425$	1.00000	0.0485071
$426$	−0.960691	−0.0465456
$427$	0	0
$428$	13.0060	0.628671
$429$	−2.14544	−0.103583
$430$	−11.4148	−0.550472
$431$	0.314602	0.0151538	0.00757691	−	0.999971i	$-0.497588\pi$
$431$	0.314602	0.0151538	0.00757691	+	0.999971i	$0.497588\pi$
$432$	−5.47546	−0.263438
$433$	−6.16933	−0.296479	−0.148240	−	0.988951i	$-0.547361\pi$
$433$	−6.16933	−0.296479	−0.148240	+	0.988951i	$0.547361\pi$
$434$	0	0
$435$	−0.0811458	−0.00389065
$436$	−0.0971344	−0.00465190
$437$	9.52195	0.455497
$438$	−7.96752	−0.380703
$439$	29.2350	1.39531	0.697654	−	0.716435i	$-0.254227\pi$
$439$	29.2350	1.39531	0.697654	+	0.716435i	$0.254227\pi$
$440$	−1.04989	−0.0500517
$441$	0	0
$442$	1.69998	0.0808597
$443$	−7.53424	−0.357962	−0.178981	−	0.983852i	$-0.557280\pi$
$443$	−7.53424	−0.357962	−0.178981	+	0.983852i	$0.557280\pi$
$444$	−10.5606	−0.501183
$445$	−1.91107	−0.0905932
$446$	−15.7512	−0.745842
$447$	−3.91914	−0.185369
$448$	0	0
$449$	16.2256	0.765736	0.382868	−	0.923803i	$-0.374936\pi$
$449$	16.2256	0.765736	0.382868	+	0.923803i	$0.374936\pi$
$450$	1.55504	0.0733052
$451$	8.76524	0.412739
$452$	−9.08386	−0.427269
$453$	22.7000	1.06654
$454$	−1.25487	−0.0588940
$455$	0	0
$456$	4.74682	0.222290
$457$	−9.52210	−0.445425	−0.222712	−	0.974884i	$-0.571491\pi$
$457$	−9.52210	−0.445425	−0.222712	+	0.974884i	$0.571491\pi$
$458$	−13.3789	−0.625154
$459$	−5.47546	−0.255572
$460$	−2.41130	−0.112428
$461$	16.9510	0.789488	0.394744	−	0.918791i	$-0.370833\pi$
$461$	16.9510	0.789488	0.394744	+	0.918791i	$0.370833\pi$
$462$	0	0
$463$	−40.9708	−1.90407	−0.952036	−	0.305985i	$-0.901014\pi$
$463$	−40.9708	−1.90407	−0.952036	+	0.305985i	$0.901014\pi$
$464$	−0.0675053	−0.00313386
$465$	6.20040	0.287537
$466$	18.7193	0.867154
$467$	11.1039	0.513828	0.256914	−	0.966434i	$-0.417294\pi$
$467$	11.1039	0.513828	0.256914	+	0.966434i	$0.417294\pi$
$468$	2.64353	0.122197
$469$	0	0
$470$	4.57861	0.211196
$471$	−6.04069	−0.278340
$472$	7.57739	0.348778
$473$	11.9844	0.551041
$474$	13.4255	0.616654
$475$	−3.94888	−0.181187
$476$	0	0
$477$	−5.43683	−0.248935
$478$	14.6586	0.670469
$479$	33.9667	1.55198	0.775989	−	0.630746i	$-0.217251\pi$
$479$	33.9667	1.55198	0.775989	+	0.630746i	$0.217251\pi$
$480$	−1.20207	−0.0548665
$481$	14.9349	0.680974
$482$	−9.80605	−0.446653
$483$	0	0
$484$	−9.89772	−0.449896
$485$	−9.16242	−0.416044
$486$	−14.1223	−0.640601
$487$	19.9526	0.904139	0.452070	−	0.891983i	$-0.350686\pi$
$487$	19.9526	0.904139	0.452070	+	0.891983i	$0.350686\pi$
$488$	−8.28619	−0.375098
$489$	−25.7522	−1.16455
$490$	0	0
$491$	32.6268	1.47243	0.736214	−	0.676749i	$-0.236612\pi$
$491$	32.6268	1.47243	0.736214	+	0.676749i	$0.236612\pi$
$492$	10.0357	0.452443
$493$	−0.0675053	−0.00304029
$494$	−6.71301	−0.302033
$495$	−1.63262	−0.0733810
$496$	5.15812	0.231606
$497$	0	0
$498$	21.1414	0.947368
$499$	−31.4821	−1.40933	−0.704667	−	0.709538i	$-0.748904\pi$
$499$	−31.4821	−1.40933	−0.704667	+	0.709538i	$0.748904\pi$
$500$	1.00000	0.0447214
$501$	−5.05889	−0.226015
$502$	13.6915	0.611080
$503$	0.977636	0.0435907	0.0217953	−	0.999762i	$-0.493062\pi$
$503$	0.977636	0.0435907	0.0217953	+	0.999762i	$0.493062\pi$
$504$	0	0
$505$	−13.4510	−0.598560
$506$	2.53161	0.112544
$507$	−12.1530	−0.539733
$508$	−2.08663	−0.0925794
$509$	1.26467	0.0560556	0.0280278	−	0.999607i	$-0.491077\pi$
$509$	1.26467	0.0560556	0.0280278	+	0.999607i	$0.491077\pi$
$510$	−1.20207	−0.0532284
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	21.6219	0.954631
$514$	3.48500	0.153717
$515$	1.82552	0.0804419
$516$	13.7214	0.604050
$517$	−4.80706	−0.211414
$518$	0	0
$519$	−30.0750	−1.32015
$520$	1.69998	0.0745490
$521$	28.3323	1.24126	0.620631	−	0.784103i	$-0.286877\pi$
$521$	28.3323	1.24126	0.620631	+	0.784103i	$0.286877\pi$
$522$	−0.104973	−0.00459456
$523$	3.63740	0.159052	0.0795262	−	0.996833i	$-0.474659\pi$
$523$	3.63740	0.159052	0.0795262	+	0.996833i	$0.474659\pi$
$524$	3.46614	0.151419
$525$	0	0
$526$	2.13558	0.0931157
$527$	5.15812	0.224691
$528$	1.26204	0.0549233
$529$	−17.1856	−0.747201
$530$	−3.49627	−0.151868
$531$	11.7831	0.511344
$532$	0	0
$533$	−14.1926	−0.614749
$534$	2.29723	0.0994107
$535$	13.0060	0.562301
$536$	9.06540	0.391566
$537$	−17.6633	−0.762226
$538$	−10.3321	−0.445447
$539$	0	0
$540$	−5.47546	−0.235626
$541$	−21.8687	−0.940208	−0.470104	−	0.882611i	$-0.655784\pi$
$541$	−21.8687	−0.940208	−0.470104	+	0.882611i	$0.655784\pi$
$542$	−6.90805	−0.296726
$543$	−18.1446	−0.778657
$544$	−1.00000	−0.0428746
$545$	−0.0971344	−0.00416078
$546$	0	0
$547$	−7.03126	−0.300635	−0.150318	−	0.988638i	$-0.548030\pi$
$547$	−7.03126	−0.300635	−0.150318	+	0.988638i	$0.548030\pi$
$548$	21.3247	0.910948
$549$	−12.8853	−0.549933
$550$	−1.04989	−0.0447676
$551$	0.266571	0.0113563
$552$	2.89855	0.123370
$553$	0	0
$554$	−17.9763	−0.763741
$555$	−10.5606	−0.448272
$556$	−4.23241	−0.179494
$557$	−11.8787	−0.503319	−0.251659	−	0.967816i	$-0.580976\pi$
$557$	−11.8787	−0.503319	−0.251659	+	0.967816i	$0.580976\pi$
$558$	8.02107	0.339559
$559$	−19.4049	−0.820742
$560$	0	0
$561$	1.26204	0.0532834
$562$	12.9520	0.546347
$563$	7.38299	0.311156	0.155578	−	0.987824i	$-0.450276\pi$
$563$	7.38299	0.311156	0.155578	+	0.987824i	$0.450276\pi$
$564$	−5.50380	−0.231752
$565$	−9.08386	−0.382161
$566$	−5.35862	−0.225240
$567$	0	0
$568$	−0.799200	−0.0335337
$569$	25.6898	1.07697	0.538487	−	0.842634i	$-0.318996\pi$
$569$	25.6898	1.07697	0.538487	+	0.842634i	$0.318996\pi$
$570$	4.74682	0.198822
$571$	−7.23204	−0.302652	−0.151326	−	0.988484i	$-0.548354\pi$
$571$	−7.23204	−0.302652	−0.151326	+	0.988484i	$0.548354\pi$
$572$	−1.78480	−0.0746261
$573$	−0.133257	−0.00556687
$574$	0	0
$575$	−2.41130	−0.100558
$576$	−1.55504	−0.0647932
$577$	26.3014	1.09494	0.547471	−	0.836825i	$-0.315591\pi$
$577$	26.3014	1.09494	0.547471	+	0.836825i	$0.315591\pi$
$578$	−1.00000	−0.0415945
$579$	−17.7125	−0.736107
$580$	−0.0675053	−0.00280301
$581$	0	0
$582$	11.0138	0.456538
$583$	3.67071	0.152025
$584$	−6.62819	−0.274277
$585$	2.64353	0.109296
$586$	31.3388	1.29459
$587$	33.0507	1.36415	0.682074	−	0.731283i	$-0.261078\pi$
$587$	33.0507	1.36415	0.682074	+	0.731283i	$0.261078\pi$
$588$	0	0
$589$	−20.3688	−0.839282
$590$	7.57739	0.311956
$591$	−3.10358	−0.127664
$592$	−8.78536	−0.361076
$593$	45.2312	1.85742	0.928712	−	0.370802i	$-0.120917\pi$
$593$	45.2312	1.85742	0.928712	+	0.370802i	$0.120917\pi$
$594$	5.74865	0.235870
$595$	0	0
$596$	−3.26033	−0.133548
$597$	−14.7135	−0.602185
$598$	−4.09916	−0.167627
$599$	−30.1210	−1.23071	−0.615355	−	0.788250i	$-0.710987\pi$
$599$	−30.1210	−1.23071	−0.615355	+	0.788250i	$0.710987\pi$
$600$	−1.20207	−0.0490741
$601$	18.4286	0.751720	0.375860	−	0.926676i	$-0.377347\pi$
$601$	18.4286	0.751720	0.375860	+	0.926676i	$0.377347\pi$
$602$	0	0
$603$	14.0970	0.574076
$604$	18.8842	0.768386
$605$	−9.89772	−0.402400
$606$	16.1689	0.656818
$607$	−3.89797	−0.158214	−0.0791069	−	0.996866i	$-0.525207\pi$
$607$	−3.89797	−0.158214	−0.0791069	+	0.996866i	$0.525207\pi$
$608$	3.94888	0.160148
$609$	0	0
$610$	−8.28619	−0.335498
$611$	7.78354	0.314888
$612$	−1.55504	−0.0628587
$613$	−8.37322	−0.338191	−0.169096	−	0.985600i	$-0.554085\pi$
$613$	−8.37322	−0.338191	−0.169096	+	0.985600i	$0.554085\pi$
$614$	21.5310	0.868919
$615$	10.0357	0.404678
$616$	0	0
$617$	−29.9146	−1.20432	−0.602159	−	0.798377i	$-0.705692\pi$
$617$	−29.9146	−1.20432	−0.602159	+	0.798377i	$0.705692\pi$
$618$	−2.19439	−0.0882714
$619$	−4.26919	−0.171593	−0.0857966	−	0.996313i	$-0.527344\pi$
$619$	−4.26919	−0.171593	−0.0857966	+	0.996313i	$0.527344\pi$
$620$	5.15812	0.207155
$621$	13.2030	0.529817
$622$	−5.00428	−0.200653
$623$	0	0
$624$	−2.04349	−0.0818049
$625$	1.00000	0.0400000
$626$	−29.9140	−1.19560
$627$	−4.98365	−0.199028
$628$	−5.02526	−0.200530
$629$	−8.78536	−0.350295
$630$	0	0
$631$	3.27476	0.130366	0.0651830	−	0.997873i	$-0.479237\pi$
$631$	3.27476	0.130366	0.0651830	+	0.997873i	$0.479237\pi$
$632$	11.1687	0.444267
$633$	−30.8053	−1.22440
$634$	−7.04811	−0.279916
$635$	−2.08663	−0.0828056
$636$	4.20274	0.166650
$637$	0	0
$638$	0.0708734	0.00280591
$639$	−1.24279	−0.0491639
$640$	−1.00000	−0.0395285
$641$	9.73319	0.384438	0.192219	−	0.981352i	$-0.438432\pi$
$641$	9.73319	0.384438	0.192219	+	0.981352i	$0.438432\pi$
$642$	−15.6341	−0.617030
$643$	−37.7945	−1.49047	−0.745236	−	0.666801i	$-0.767663\pi$
$643$	−37.7945	−1.49047	−0.745236	+	0.666801i	$0.767663\pi$
$644$	0	0
$645$	13.7214	0.540278
$646$	3.94888	0.155367
$647$	1.37695	0.0541335	0.0270667	−	0.999634i	$-0.491383\pi$
$647$	1.37695	0.0541335	0.0270667	+	0.999634i	$0.491383\pi$
$648$	1.91674	0.0752968
$649$	−7.95546	−0.312279
$650$	1.69998	0.0666786
$651$	0	0
$652$	−21.4233	−0.839000
$653$	−17.3424	−0.678661	−0.339331	−	0.940667i	$-0.610201\pi$
$653$	−17.3424	−0.678661	−0.339331	+	0.940667i	$0.610201\pi$
$654$	0.116762	0.00456575
$655$	3.46614	0.135433
$656$	8.34869	0.325962
$657$	−10.3071	−0.402118
$658$	0	0
$659$	−4.25802	−0.165869	−0.0829345	−	0.996555i	$-0.526429\pi$
$659$	−4.25802	−0.165869	−0.0829345	+	0.996555i	$0.526429\pi$
$660$	1.26204	0.0491249
$661$	28.9277	1.12516	0.562578	−	0.826744i	$-0.309809\pi$
$661$	28.9277	1.12516	0.562578	+	0.826744i	$0.309809\pi$
$662$	16.6572	0.647400
$663$	−2.04349	−0.0793624
$664$	17.5875	0.682529
$665$	0	0
$666$	−13.6616	−0.529375
$667$	0.162776	0.00630270
$668$	−4.20850	−0.162832
$669$	18.9340	0.732031
$670$	9.06540	0.350227
$671$	8.69962	0.335845
$672$	0	0
$673$	2.75557	0.106220	0.0531098	−	0.998589i	$-0.483087\pi$
$673$	2.75557	0.106220	0.0531098	+	0.998589i	$0.483087\pi$
$674$	11.5188	0.443687
$675$	−5.47546	−0.210750
$676$	−10.1101	−0.388849
$677$	−26.5363	−1.01987	−0.509937	−	0.860212i	$-0.670331\pi$
$677$	−26.5363	−1.01987	−0.509937	+	0.860212i	$0.670331\pi$
$678$	10.9194	0.419357
$679$	0	0
$680$	−1.00000	−0.0383482
$681$	1.50844	0.0578034
$682$	−5.41548	−0.207369
$683$	−12.2708	−0.469528	−0.234764	−	0.972052i	$-0.575432\pi$
$683$	−12.2708	−0.469528	−0.234764	+	0.972052i	$0.575432\pi$
$684$	6.14066	0.234794
$685$	21.3247	0.814776
$686$	0	0
$687$	16.0823	0.613578
$688$	11.4148	0.435186
$689$	−5.94358	−0.226432
$690$	2.89855	0.110346
$691$	−31.2784	−1.18989	−0.594943	−	0.803768i	$-0.702825\pi$
$691$	−31.2784	−1.18989	−0.594943	+	0.803768i	$0.702825\pi$
$692$	−25.0195	−0.951097
$693$	0	0
$694$	4.06413	0.154272
$695$	−4.23241	−0.160545
$696$	0.0811458	0.00307583
$697$	8.34869	0.316229
$698$	0.809305	0.0306327
$699$	−22.5018	−0.851097
$700$	0	0
$701$	19.3741	0.731750	0.365875	−	0.930664i	$-0.380770\pi$
$701$	19.3741	0.731750	0.365875	+	0.930664i	$0.380770\pi$
$702$	−9.30815	−0.351314
$703$	34.6924	1.30845
$704$	1.04989	0.0395694
$705$	−5.50380	−0.207285
$706$	0.840633	0.0316376
$707$	0	0
$708$	−9.10852	−0.342319
$709$	25.4406	0.955443	0.477722	−	0.878511i	$-0.341463\pi$
$709$	25.4406	0.955443	0.477722	+	0.878511i	$0.341463\pi$
$710$	−0.799200	−0.0299934
$711$	17.3678	0.651342
$712$	1.91107	0.0716202
$713$	−12.4378	−0.465799
$714$	0	0
$715$	−1.78480	−0.0667476
$716$	−14.6941	−0.549144
$717$	−17.6206	−0.658054
$718$	−18.8913	−0.705018
$719$	5.02463	0.187387	0.0936935	−	0.995601i	$-0.470133\pi$
$719$	5.02463	0.187387	0.0936935	+	0.995601i	$0.470133\pi$
$720$	−1.55504	−0.0579528
$721$	0	0
$722$	3.40633	0.126770
$723$	11.7875	0.438383
$724$	−15.0945	−0.560982
$725$	−0.0675053	−0.00250708
$726$	11.8977	0.441566
$727$	−28.1289	−1.04324	−0.521621	−	0.853177i	$-0.674672\pi$
$727$	−28.1289	−1.04324	−0.521621	+	0.853177i	$0.674672\pi$
$728$	0	0
$729$	22.7262	0.841710
$730$	−6.62819	−0.245320
$731$	11.4148	0.422192
$732$	9.96054	0.368152
$733$	−24.8910	−0.919371	−0.459686	−	0.888082i	$-0.652038\pi$
$733$	−24.8910	−0.919371	−0.459686	+	0.888082i	$0.652038\pi$
$734$	34.8805	1.28746
$735$	0	0
$736$	2.41130	0.0888818
$737$	−9.51771	−0.350589
$738$	12.9825	0.477894
$739$	−36.0732	−1.32698	−0.663488	−	0.748187i	$-0.730925\pi$
$739$	−36.0732	−1.32698	−0.663488	+	0.748187i	$0.730925\pi$
$740$	−8.78536	−0.322956
$741$	8.06948	0.296440
$742$	0	0
$743$	35.2945	1.29483	0.647414	−	0.762138i	$-0.275850\pi$
$743$	35.2945	1.29483	0.647414	+	0.762138i	$0.275850\pi$
$744$	−6.20040	−0.227318
$745$	−3.26033	−0.119449
$746$	−10.3388	−0.378531
$747$	27.3493	1.00066
$748$	1.04989	0.0383879
$749$	0	0
$750$	−1.20207	−0.0438932
$751$	−0.368766	−0.0134565	−0.00672823	−	0.999977i	$-0.502142\pi$
$751$	−0.368766	−0.0134565	−0.00672823	+	0.999977i	$0.502142\pi$
$752$	−4.57861	−0.166965
$753$	−16.4581	−0.599765
$754$	−0.114758	−0.00417922
$755$	18.8842	0.687265
$756$	0	0
$757$	−54.7646	−1.99046	−0.995228	−	0.0975816i	$-0.968889\pi$
$757$	−54.7646	−1.99046	−0.995228	+	0.0975816i	$0.968889\pi$
$758$	−20.5012	−0.744637
$759$	−3.04317	−0.110460
$760$	3.94888	0.143241
$761$	3.68087	0.133431	0.0667157	−	0.997772i	$-0.478748\pi$
$761$	3.68087	0.133431	0.0667157	+	0.997772i	$0.478748\pi$
$762$	2.50827	0.0908651
$763$	0	0
$764$	−0.110856	−0.00401064
$765$	−1.55504	−0.0562225
$766$	−29.0273	−1.04880
$767$	12.8814	0.465120
$768$	1.20207	0.0433758
$769$	43.9351	1.58434	0.792170	−	0.610301i	$-0.208951\pi$
$769$	43.9351	1.58434	0.792170	+	0.610301i	$0.208951\pi$
$770$	0	0
$771$	−4.18920	−0.150870
$772$	−14.7351	−0.530327
$773$	7.75284	0.278850	0.139425	−	0.990233i	$-0.455475\pi$
$773$	7.75284	0.278850	0.139425	+	0.990233i	$0.455475\pi$
$774$	17.7505	0.638028
$775$	5.15812	0.185285
$776$	9.16242	0.328912
$777$	0	0
$778$	26.8338	0.962040
$779$	−32.9680	−1.18120
$780$	−2.04349	−0.0731685
$781$	0.839075	0.0300245
$782$	2.41130	0.0862280
$783$	0.369622	0.0132092
$784$	0	0
$785$	−5.02526	−0.179359
$786$	−4.16652	−0.148615
$787$	7.10864	0.253396	0.126698	−	0.991941i	$-0.459562\pi$
$787$	7.10864	0.253396	0.126698	+	0.991941i	$0.459562\pi$
$788$	−2.58188	−0.0919755
$789$	−2.56711	−0.0913914
$790$	11.1687	0.397365
$791$	0	0
$792$	1.63262	0.0580128
$793$	−14.0863	−0.500220
$794$	29.1916	1.03597
$795$	4.20274	0.149056
$796$	−12.2402	−0.433843
$797$	44.9458	1.59206	0.796031	−	0.605256i	$-0.206929\pi$
$797$	44.9458	1.59206	0.796031	+	0.605256i	$0.206929\pi$
$798$	0	0
$799$	−4.57861	−0.161980
$800$	−1.00000	−0.0353553
$801$	2.97178	0.105003
$802$	−6.80078	−0.240144
$803$	6.95890	0.245574
$804$	−10.8972	−0.384315
$805$	0	0
$806$	8.76869	0.308864
$807$	12.4198	0.437198
$808$	13.4510	0.473203
$809$	−41.0294	−1.44252	−0.721258	−	0.692666i	$-0.756436\pi$
$809$	−41.0294	−1.44252	−0.721258	+	0.692666i	$0.756436\pi$
$810$	1.91674	0.0673475
$811$	14.0749	0.494237	0.247118	−	0.968985i	$-0.420516\pi$
$811$	14.0749	0.494237	0.247118	+	0.968985i	$0.420516\pi$
$812$	0	0
$813$	8.30393	0.291232
$814$	9.22370	0.323291
$815$	−21.4233	−0.750425
$816$	1.20207	0.0420807
$817$	−45.0758	−1.57700
$818$	−2.63414	−0.0921004
$819$	0	0
$820$	8.34869	0.291549
$821$	−33.2171	−1.15928	−0.579642	−	0.814871i	$-0.696808\pi$
$821$	−33.2171	−1.15928	−0.579642	+	0.814871i	$0.696808\pi$
$822$	−25.6337	−0.894079
$823$	−2.72911	−0.0951309	−0.0475655	−	0.998868i	$-0.515146\pi$
$823$	−2.72911	−0.0951309	−0.0475655	+	0.998868i	$0.515146\pi$
$824$	−1.82552	−0.0635949
$825$	1.26204	0.0439387
$826$	0	0
$827$	−45.5343	−1.58338	−0.791692	−	0.610921i	$-0.790799\pi$
$827$	−45.5343	−1.58338	−0.791692	+	0.610921i	$0.790799\pi$
$828$	3.74967	0.130310
$829$	8.26239	0.286965	0.143482	−	0.989653i	$-0.454170\pi$
$829$	8.26239	0.286965	0.143482	+	0.989653i	$0.454170\pi$
$830$	17.5875	0.610472
$831$	21.6087	0.749599
$832$	−1.69998	−0.0589361
$833$	0	0
$834$	5.08764	0.176171
$835$	−4.20850	−0.145641
$836$	−4.14591	−0.143389
$837$	−28.2431	−0.976223
$838$	−35.2146	−1.21647
$839$	−5.27258	−0.182030	−0.0910148	−	0.995850i	$-0.529011\pi$
$839$	−5.27258	−0.182030	−0.0910148	+	0.995850i	$0.529011\pi$
$840$	0	0
$841$	−28.9954	−0.999843
$842$	−24.9152	−0.858635
$843$	−15.5692	−0.536230
$844$	−25.6270	−0.882117
$845$	−10.1101	−0.347797
$846$	−7.11992	−0.244788
$847$	0	0
$848$	3.49627	0.120062
$849$	6.44142	0.221069
$850$	−1.00000	−0.0342997
$851$	21.1842	0.726184
$852$	0.960691	0.0329127
$853$	19.9929	0.684546	0.342273	−	0.939601i	$-0.388803\pi$
$853$	19.9929	0.684546	0.342273	+	0.939601i	$0.388803\pi$
$854$	0	0
$855$	6.14066	0.210006
$856$	−13.0060	−0.444538
$857$	−37.2286	−1.27171	−0.635853	−	0.771810i	$-0.719352\pi$
$857$	−37.2286	−1.27171	−0.635853	+	0.771810i	$0.719352\pi$
$858$	2.14544	0.0732442
$859$	27.5317	0.939371	0.469685	−	0.882834i	$-0.344367\pi$
$859$	27.5317	0.939371	0.469685	+	0.882834i	$0.344367\pi$
$860$	11.4148	0.389242
$861$	0	0
$862$	−0.314602	−0.0107154
$863$	15.9663	0.543501	0.271750	−	0.962368i	$-0.412398\pi$
$863$	15.9663	0.543501	0.271750	+	0.962368i	$0.412398\pi$
$864$	5.47546	0.186279
$865$	−25.0195	−0.850687
$866$	6.16933	0.209642
$867$	1.20207	0.0408243
$868$	0	0
$869$	−11.7260	−0.397776
$870$	0.0811458	0.00275110
$871$	15.4110	0.522181
$872$	0.0971344	0.00328939
$873$	14.2479	0.482219
$874$	−9.52195	−0.322085
$875$	0	0
$876$	7.96752	0.269198
$877$	−50.2999	−1.69851	−0.849254	−	0.527985i	$-0.822948\pi$
$877$	−50.2999	−1.69851	−0.849254	+	0.527985i	$0.822948\pi$
$878$	−29.2350	−0.986632
$879$	−37.6712	−1.27062
$880$	1.04989	0.0353919
$881$	6.63857	0.223659	0.111830	−	0.993727i	$-0.464329\pi$
$881$	6.63857	0.223659	0.111830	+	0.993727i	$0.464329\pi$
$882$	0	0
$883$	−44.7937	−1.50743	−0.753713	−	0.657203i	$-0.771739\pi$
$883$	−44.7937	−1.50743	−0.753713	+	0.657203i	$0.771739\pi$
$884$	−1.69998	−0.0571764
$885$	−9.10852	−0.306180
$886$	7.53424	0.253118
$887$	−19.6376	−0.659366	−0.329683	−	0.944092i	$-0.606942\pi$
$887$	−19.6376	−0.659366	−0.329683	+	0.944092i	$0.606942\pi$
$888$	10.5606	0.354390
$889$	0	0
$890$	1.91107	0.0640591
$891$	−2.01238	−0.0674172
$892$	15.7512	0.527390
$893$	18.0804	0.605038
$894$	3.91914	0.131075
$895$	−14.6941	−0.491169
$896$	0	0
$897$	4.92746	0.164523
$898$	−16.2256	−0.541457
$899$	−0.348201	−0.0116131
$900$	−1.55504	−0.0518346
$901$	3.49627	0.116478
$902$	−8.76524	−0.291851
$903$	0	0
$904$	9.08386	0.302125
$905$	−15.0945	−0.501757
$906$	−22.7000	−0.754157
$907$	−11.6571	−0.387067	−0.193534	−	0.981094i	$-0.561995\pi$
$907$	−11.6571	−0.387067	−0.193534	+	0.981094i	$0.561995\pi$
$908$	1.25487	0.0416443
$909$	20.9167	0.693764
$910$	0	0
$911$	−9.27858	−0.307413	−0.153707	−	0.988117i	$-0.549121\pi$
$911$	−9.27858	−0.307413	−0.153707	+	0.988117i	$0.549121\pi$
$912$	−4.74682	−0.157183
$913$	−18.4651	−0.611104
$914$	9.52210	0.314963
$915$	9.96054	0.329285
$916$	13.3789	0.442051
$917$	0	0
$918$	5.47546	0.180717
$919$	−35.7191	−1.17827	−0.589133	−	0.808036i	$-0.700531\pi$
$919$	−35.7191	−1.17827	−0.589133	+	0.808036i	$0.700531\pi$
$920$	2.41130	0.0794983
$921$	−25.8816	−0.852829
$922$	−16.9510	−0.558252
$923$	−1.35862	−0.0447196
$924$	0	0
$925$	−8.78536	−0.288861
$926$	40.9708	1.34638
$927$	−2.83875	−0.0932367
$928$	0.0675053	0.00221597
$929$	−15.8688	−0.520640	−0.260320	−	0.965522i	$-0.583828\pi$
$929$	−15.8688	−0.520640	−0.260320	+	0.965522i	$0.583828\pi$
$930$	−6.20040	−0.203319
$931$	0	0
$932$	−18.7193	−0.613171
$933$	6.01547	0.196938
$934$	−11.1039	−0.363331
$935$	1.04989	0.0343352
$936$	−2.64353	−0.0864065
$937$	10.6576	0.348170	0.174085	−	0.984731i	$-0.444303\pi$
$937$	10.6576	0.348170	0.174085	+	0.984731i	$0.444303\pi$
$938$	0	0
$939$	35.9586	1.17346
$940$	−4.57861	−0.149338
$941$	12.9896	0.423450	0.211725	−	0.977329i	$-0.432092\pi$
$941$	12.9896	0.423450	0.211725	+	0.977329i	$0.432092\pi$
$942$	6.04069	0.196816
$943$	−20.1312	−0.655563
$944$	−7.57739	−0.246623
$945$	0	0
$946$	−11.9844	−0.389645
$947$	46.4450	1.50926	0.754630	−	0.656150i	$-0.227816\pi$
$947$	46.4450	1.50926	0.754630	+	0.656150i	$0.227816\pi$
$948$	−13.4255	−0.436041
$949$	−11.2678	−0.365768
$950$	3.94888	0.128119
$951$	8.47229	0.274733
$952$	0	0
$953$	29.3425	0.950496	0.475248	−	0.879852i	$-0.342358\pi$
$953$	29.3425	0.950496	0.475248	+	0.879852i	$0.342358\pi$
$954$	5.43683	0.176024
$955$	−0.110856	−0.00358723
$956$	−14.6586	−0.474093
$957$	−0.0851945	−0.00275395
$958$	−33.9667	−1.09741
$959$	0	0
$960$	1.20207	0.0387965
$961$	−4.39380	−0.141736
$962$	−14.9349	−0.481521
$963$	−20.2249	−0.651738
$964$	9.80605	0.315832
$965$	−14.7351	−0.474339
$966$	0	0
$967$	26.4863	0.851744	0.425872	−	0.904784i	$-0.359967\pi$
$967$	26.4863	0.851744	0.425872	+	0.904784i	$0.359967\pi$
$968$	9.89772	0.318125
$969$	−4.74682	−0.152490
$970$	9.16242	0.294188
$971$	29.6774	0.952394	0.476197	−	0.879339i	$-0.342015\pi$
$971$	29.6774	0.952394	0.476197	+	0.879339i	$0.342015\pi$
$972$	14.1223	0.452973
$973$	0	0
$974$	−19.9526	−0.639323
$975$	−2.04349	−0.0654439
$976$	8.28619	0.265234
$977$	37.2644	1.19219	0.596097	−	0.802912i	$-0.296717\pi$
$977$	37.2644	1.19219	0.596097	+	0.802912i	$0.296717\pi$
$978$	25.7522	0.823464
$979$	−2.00642	−0.0641253
$980$	0	0
$981$	0.151048	0.00482258
$982$	−32.6268	−1.04116
$983$	10.2325	0.326365	0.163182	−	0.986596i	$-0.447824\pi$
$983$	10.2325	0.326365	0.163182	+	0.986596i	$0.447824\pi$
$984$	−10.0357	−0.319926
$985$	−2.58188	−0.0822654
$986$	0.0675053	0.00214981
$987$	0	0
$988$	6.71301	0.213569
$989$	−27.5246	−0.875231
$990$	1.63262	0.0518882
$991$	−43.7261	−1.38901	−0.694503	−	0.719489i	$-0.744376\pi$
$991$	−43.7261	−1.38901	−0.694503	+	0.719489i	$0.744376\pi$
$992$	−5.15812	−0.163770
$993$	−20.0230	−0.635412
$994$	0	0
$995$	−12.2402	−0.388041
$996$	−21.1414	−0.669890
$997$	14.0207	0.444039	0.222019	−	0.975042i	$-0.428735\pi$
$997$	14.0207	0.444039	0.222019	+	0.975042i	$0.428735\pi$
$998$	31.4821	0.996550
$999$	48.1039	1.52194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8330.2.a.cn.1.5	✓	6
7.6	odd	2		8330.2.a.cq.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8330.2.a.cn.1.5	✓	6	1.1	even	1	trivial
8330.2.a.cq.1.2	yes	6	7.6	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8330.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.20207 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.20207 q^{6} -1.00000 q^{8} -1.55504 q^{9} -1.00000 q^{10} +1.04989 q^{11} +1.20207 q^{12} -1.69998 q^{13} +1.20207 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.55504 q^{18} -3.94888 q^{19} +1.00000 q^{20} -1.04989 q^{22} -2.41130 q^{23} -1.20207 q^{24} +1.00000 q^{25} +1.69998 q^{26} -5.47546 q^{27} -0.0675053 q^{29} -1.20207 q^{30} +5.15812 q^{31} -1.00000 q^{32} +1.26204 q^{33} -1.00000 q^{34} -1.55504 q^{36} -8.78536 q^{37} +3.94888 q^{38} -2.04349 q^{39} -1.00000 q^{40} +8.34869 q^{41} +11.4148 q^{43} +1.04989 q^{44} -1.55504 q^{45} +2.41130 q^{46} -4.57861 q^{47} +1.20207 q^{48} -1.00000 q^{50} +1.20207 q^{51} -1.69998 q^{52} +3.49627 q^{53} +5.47546 q^{54} +1.04989 q^{55} -4.74682 q^{57} +0.0675053 q^{58} -7.57739 q^{59} +1.20207 q^{60} +8.28619 q^{61} -5.15812 q^{62} +1.00000 q^{64} -1.69998 q^{65} -1.26204 q^{66} -9.06540 q^{67} +1.00000 q^{68} -2.89855 q^{69} +0.799200 q^{71} +1.55504 q^{72} +6.62819 q^{73} +8.78536 q^{74} +1.20207 q^{75} -3.94888 q^{76} +2.04349 q^{78} -11.1687 q^{79} +1.00000 q^{80} -1.91674 q^{81} -8.34869 q^{82} -17.5875 q^{83} +1.00000 q^{85} -11.4148 q^{86} -0.0811458 q^{87} -1.04989 q^{88} -1.91107 q^{89} +1.55504 q^{90} -2.41130 q^{92} +6.20040 q^{93} +4.57861 q^{94} -3.94888 q^{95} -1.20207 q^{96} -9.16242 q^{97} -1.63262 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 6 q^{5} + 2 q^{6} - 6 q^{8} + 4 q^{9} - 6 q^{10} - 6 q^{11} - 2 q^{12} - 2 q^{15} + 6 q^{16} + 6 q^{17} - 4 q^{18} + 6 q^{19} + 6 q^{20} + 6 q^{22} - 2 q^{23} + 2 q^{24}+ \cdots - 8 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 + 6 * q^5 + 2 * q^6 - 6 * q^8 + 4 * q^9 - 6 * q^10 - 6 * q^11 - 2 * q^12 - 2 * q^15 + 6 * q^16 + 6 * q^17 - 4 * q^18 + 6 * q^19 + 6 * q^20 + 6 * q^22 - 2 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 2 * q^29 + 2 * q^30 - 2 * q^31 - 6 * q^32 - 4 * q^33 - 6 * q^34 + 4 * q^36 - 12 * q^37 - 6 * q^38 - 6 * q^40 + 6 * q^41 - 26 * q^43 - 6 * q^44 + 4 * q^45 + 2 * q^46 - 30 * q^47 - 2 * q^48 - 6 * q^50 - 2 * q^51 + 14 * q^53 + 20 * q^54 - 6 * q^55 - 8 * q^57 + 2 * q^58 - 6 * q^59 - 2 * q^60 + 12 * q^61 + 2 * q^62 + 6 * q^64 + 4 * q^66 - 14 * q^67 + 6 * q^68 - 2 * q^69 + 12 * q^71 - 4 * q^72 - 12 * q^73 + 12 * q^74 - 2 * q^75 + 6 * q^76 - 36 * q^79 + 6 * q^80 - 2 * q^81 - 6 * q^82 - 32 * q^83 + 6 * q^85 + 26 * q^86 - 2 * q^87 + 6 * q^88 + 8 * q^89 - 4 * q^90 - 2 * q^92 - 12 * q^93 + 30 * q^94 + 6 * q^95 + 2 * q^96 + 16 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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