LMFDB - Embedded newform 6008.2.a.e.1.32

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.32
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.13231 q^{3} -1.95898 q^{5} -2.91195 q^{7} -1.71788 q^{9} +O(q^{10})$	$q+1.13231 q^{3} -1.95898 q^{5} -2.91195 q^{7} -1.71788 q^{9} +0.469326 q^{11} -0.568218 q^{13} -2.21816 q^{15} +0.0179401 q^{17} -5.38988 q^{19} -3.29722 q^{21} +1.38567 q^{23} -1.16241 q^{25} -5.34209 q^{27} -1.74404 q^{29} +2.46772 q^{31} +0.531420 q^{33} +5.70443 q^{35} +4.93171 q^{37} -0.643397 q^{39} -0.996312 q^{41} +3.10157 q^{43} +3.36529 q^{45} -11.0080 q^{47} +1.47944 q^{49} +0.0203137 q^{51} +8.32113 q^{53} -0.919398 q^{55} -6.10299 q^{57} +8.95620 q^{59} -5.75029 q^{61} +5.00239 q^{63} +1.11313 q^{65} -4.44499 q^{67} +1.56900 q^{69} +12.6689 q^{71} +6.04979 q^{73} -1.31621 q^{75} -1.36665 q^{77} +6.75973 q^{79} -0.895225 q^{81} +6.31454 q^{83} -0.0351442 q^{85} -1.97479 q^{87} +11.8411 q^{89} +1.65462 q^{91} +2.79422 q^{93} +10.5586 q^{95} -18.4646 q^{97} -0.806247 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

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	0
$2$	0	0
$3$	1.13231	0.653737	0.326869	−	0.945070i	$-0.394007\pi$
$3$	1.13231	0.653737	0.326869	+	0.945070i	$0.394007\pi$
$4$	0	0
$5$	−1.95898	−0.876081	−0.438040	−	0.898955i	$-0.644327\pi$
$5$	−1.95898	−0.876081	−0.438040	+	0.898955i	$0.644327\pi$
$6$	0	0
$7$	−2.91195	−1.10061	−0.550306	−	0.834963i	$-0.685489\pi$
$7$	−2.91195	−1.10061	−0.550306	+	0.834963i	$0.685489\pi$
$8$	0	0
$9$	−1.71788	−0.572628
$10$	0	0
$11$	0.469326	0.141507	0.0707535	−	0.997494i	$-0.477460\pi$
$11$	0.469326	0.141507	0.0707535	+	0.997494i	$0.477460\pi$
$12$	0	0
$13$	−0.568218	−0.157595	−0.0787977	−	0.996891i	$-0.525108\pi$
$13$	−0.568218	−0.157595	−0.0787977	+	0.996891i	$0.525108\pi$
$14$	0	0
$15$	−2.21816	−0.572726
$16$	0	0
$17$	0.0179401	0.00435111	0.00217556	−	0.999998i	$-0.499307\pi$
$17$	0.0179401	0.00435111	0.00217556	+	0.999998i	$0.499307\pi$
$18$	0	0
$19$	−5.38988	−1.23652	−0.618262	−	0.785972i	$-0.712163\pi$
$19$	−5.38988	−1.23652	−0.618262	+	0.785972i	$0.712163\pi$
$20$	0	0
$21$	−3.29722	−0.719511
$22$	0	0
$23$	1.38567	0.288932	0.144466	−	0.989510i	$-0.453854\pi$
$23$	1.38567	0.288932	0.144466	+	0.989510i	$0.453854\pi$
$24$	0	0
$25$	−1.16241	−0.232483
$26$	0	0
$27$	−5.34209	−1.02809
$28$	0	0
$29$	−1.74404	−0.323861	−0.161930	−	0.986802i	$-0.551772\pi$
$29$	−1.74404	−0.323861	−0.161930	+	0.986802i	$0.551772\pi$
$30$	0	0
$31$	2.46772	0.443216	0.221608	−	0.975136i	$-0.428869\pi$
$31$	2.46772	0.443216	0.221608	+	0.975136i	$0.428869\pi$
$32$	0	0
$33$	0.531420	0.0925084
$34$	0	0
$35$	5.70443	0.964225
$36$	0	0
$37$	4.93171	0.810767	0.405384	−	0.914147i	$-0.367138\pi$
$37$	4.93171	0.810767	0.405384	+	0.914147i	$0.367138\pi$
$38$	0	0
$39$	−0.643397	−0.103026
$40$	0	0
$41$	−0.996312	−0.155598	−0.0777989	−	0.996969i	$-0.524789\pi$
$41$	−0.996312	−0.155598	−0.0777989	+	0.996969i	$0.524789\pi$
$42$	0	0
$43$	3.10157	0.472986	0.236493	−	0.971633i	$-0.424002\pi$
$43$	3.10157	0.472986	0.236493	+	0.971633i	$0.424002\pi$
$44$	0	0
$45$	3.36529	0.501668
$46$	0	0
$47$	−11.0080	−1.60569	−0.802844	−	0.596189i	$-0.796681\pi$
$47$	−11.0080	−1.60569	−0.802844	+	0.596189i	$0.796681\pi$
$48$	0	0
$49$	1.47944	0.211348
$50$	0	0
$51$	0.0203137	0.00284448
$52$	0	0
$53$	8.32113	1.14300	0.571498	−	0.820604i	$-0.306363\pi$
$53$	8.32113	1.14300	0.571498	+	0.820604i	$0.306363\pi$
$54$	0	0
$55$	−0.919398	−0.123972
$56$	0	0
$57$	−6.10299	−0.808361
$58$	0	0
$59$	8.95620	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$59$	8.95620	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$60$	0	0
$61$	−5.75029	−0.736249	−0.368124	−	0.929777i	$-0.620000\pi$
$61$	−5.75029	−0.736249	−0.368124	+	0.929777i	$0.620000\pi$
$62$	0	0
$63$	5.00239	0.630242
$64$	0	0
$65$	1.11313	0.138066
$66$	0	0
$67$	−4.44499	−0.543042	−0.271521	−	0.962433i	$-0.587527\pi$
$67$	−4.44499	−0.543042	−0.271521	+	0.962433i	$0.587527\pi$
$68$	0	0
$69$	1.56900	0.188885
$70$	0	0
$71$	12.6689	1.50352	0.751758	−	0.659439i	$-0.229206\pi$
$71$	12.6689	1.50352	0.751758	+	0.659439i	$0.229206\pi$
$72$	0	0
$73$	6.04979	0.708075	0.354037	−	0.935231i	$-0.384809\pi$
$73$	6.04979	0.708075	0.354037	+	0.935231i	$0.384809\pi$
$74$	0	0
$75$	−1.31621	−0.151983
$76$	0	0
$77$	−1.36665	−0.155744
$78$	0	0
$79$	6.75973	0.760529	0.380264	−	0.924878i	$-0.375833\pi$
$79$	6.75973	0.760529	0.380264	+	0.924878i	$0.375833\pi$
$80$	0	0
$81$	−0.895225	−0.0994695
$82$	0	0
$83$	6.31454	0.693111	0.346555	−	0.938030i	$-0.387351\pi$
$83$	6.31454	0.693111	0.346555	+	0.938030i	$0.387351\pi$
$84$	0	0
$85$	−0.0351442	−0.00381192
$86$	0	0
$87$	−1.97479	−0.211720
$88$	0	0
$89$	11.8411	1.25515	0.627576	−	0.778555i	$-0.284047\pi$
$89$	11.8411	1.25515	0.627576	+	0.778555i	$0.284047\pi$
$90$	0	0
$91$	1.65462	0.173452
$92$	0	0
$93$	2.79422	0.289747
$94$	0	0
$95$	10.5586	1.08329
$96$	0	0
$97$	−18.4646	−1.87480	−0.937400	−	0.348256i	$-0.886774\pi$
$97$	−18.4646	−1.87480	−0.937400	+	0.348256i	$0.886774\pi$
$98$	0	0
$99$	−0.806247	−0.0810309
$100$	0	0
$101$	−3.76171	−0.374304	−0.187152	−	0.982331i	$-0.559926\pi$
$101$	−3.76171	−0.374304	−0.187152	+	0.982331i	$0.559926\pi$
$102$	0	0
$103$	10.7020	1.05450	0.527249	−	0.849711i	$-0.323223\pi$
$103$	10.7020	1.05450	0.527249	+	0.849711i	$0.323223\pi$
$104$	0	0
$105$	6.45916	0.630350
$106$	0	0
$107$	−17.6815	−1.70934	−0.854670	−	0.519172i	$-0.826240\pi$
$107$	−17.6815	−1.70934	−0.854670	+	0.519172i	$0.826240\pi$
$108$	0	0
$109$	4.46330	0.427506	0.213753	−	0.976888i	$-0.431431\pi$
$109$	4.46330	0.427506	0.213753	+	0.976888i	$0.431431\pi$
$110$	0	0
$111$	5.58420	0.530029
$112$	0	0
$113$	15.5439	1.46225	0.731123	−	0.682246i	$-0.238997\pi$
$113$	15.5439	1.46225	0.731123	+	0.682246i	$0.238997\pi$
$114$	0	0
$115$	−2.71449	−0.253128
$116$	0	0
$117$	0.976133	0.0902435
$118$	0	0
$119$	−0.0522406	−0.00478889
$120$	0	0
$121$	−10.7797	−0.979976
$122$	0	0
$123$	−1.12813	−0.101720
$124$	0	0
$125$	12.0720	1.07975
$126$	0	0
$127$	7.76363	0.688910	0.344455	−	0.938803i	$-0.388064\pi$
$127$	7.76363	0.688910	0.344455	+	0.938803i	$0.388064\pi$
$128$	0	0
$129$	3.51193	0.309208
$130$	0	0
$131$	−16.5290	−1.44414	−0.722072	−	0.691818i	$-0.756810\pi$
$131$	−16.5290	−1.44414	−0.722072	+	0.691818i	$0.756810\pi$
$132$	0	0
$133$	15.6950	1.36093
$134$	0	0
$135$	10.4650	0.900685
$136$	0	0
$137$	15.8147	1.35114	0.675570	−	0.737296i	$-0.263897\pi$
$137$	15.8147	1.35114	0.675570	+	0.737296i	$0.263897\pi$
$138$	0	0
$139$	18.6914	1.58539	0.792694	−	0.609620i	$-0.208678\pi$
$139$	18.6914	1.58539	0.792694	+	0.609620i	$0.208678\pi$
$140$	0	0
$141$	−12.4645	−1.04970
$142$	0	0
$143$	−0.266679	−0.0223009
$144$	0	0
$145$	3.41654	0.283728
$146$	0	0
$147$	1.67518	0.138166
$148$	0	0
$149$	−20.9147	−1.71340	−0.856699	−	0.515817i	$-0.827489\pi$
$149$	−20.9147	−1.71340	−0.856699	+	0.515817i	$0.827489\pi$
$150$	0	0
$151$	−3.69325	−0.300552	−0.150276	−	0.988644i	$-0.548016\pi$
$151$	−3.69325	−0.300552	−0.150276	+	0.988644i	$0.548016\pi$
$152$	0	0
$153$	−0.0308190	−0.00249157
$154$	0	0
$155$	−4.83421	−0.388293
$156$	0	0
$157$	9.73140	0.776650	0.388325	−	0.921522i	$-0.373054\pi$
$157$	9.73140	0.776650	0.388325	+	0.921522i	$0.373054\pi$
$158$	0	0
$159$	9.42207	0.747219
$160$	0	0
$161$	−4.03499	−0.318002
$162$	0	0
$163$	−5.38709	−0.421949	−0.210975	−	0.977492i	$-0.567664\pi$
$163$	−5.38709	−0.421949	−0.210975	+	0.977492i	$0.567664\pi$
$164$	0	0
$165$	−1.04104	−0.0810448
$166$	0	0
$167$	−17.6929	−1.36912	−0.684558	−	0.728958i	$-0.740005\pi$
$167$	−17.6929	−1.36912	−0.684558	+	0.728958i	$0.740005\pi$
$168$	0	0
$169$	−12.6771	−0.975164
$170$	0	0
$171$	9.25919	0.708068
$172$	0	0
$173$	4.07975	0.310178	0.155089	−	0.987901i	$-0.450434\pi$
$173$	4.07975	0.310178	0.155089	+	0.987901i	$0.450434\pi$
$174$	0	0
$175$	3.38489	0.255874
$176$	0	0
$177$	10.1412	0.762256
$178$	0	0
$179$	6.90974	0.516458	0.258229	−	0.966084i	$-0.416861\pi$
$179$	6.90974	0.516458	0.258229	+	0.966084i	$0.416861\pi$
$180$	0	0
$181$	12.3699	0.919451	0.459726	−	0.888061i	$-0.347948\pi$
$181$	12.3699	0.919451	0.459726	+	0.888061i	$0.347948\pi$
$182$	0	0
$183$	−6.51108	−0.481313
$184$	0	0
$185$	−9.66109	−0.710298
$186$	0	0
$187$	0.00841975	0.000615713 0
$188$	0	0
$189$	15.5559	1.13152
$190$	0	0
$191$	−17.5379	−1.26900	−0.634498	−	0.772925i	$-0.718793\pi$
$191$	−17.5379	−1.26900	−0.634498	+	0.772925i	$0.718793\pi$
$192$	0	0
$193$	−2.32863	−0.167619	−0.0838094	−	0.996482i	$-0.526709\pi$
$193$	−2.32863	−0.167619	−0.0838094	+	0.996482i	$0.526709\pi$
$194$	0	0
$195$	1.26040	0.0902590
$196$	0	0
$197$	17.7413	1.26401	0.632006	−	0.774963i	$-0.282232\pi$
$197$	17.7413	1.26401	0.632006	+	0.774963i	$0.282232\pi$
$198$	0	0
$199$	21.4707	1.52202	0.761008	−	0.648742i	$-0.224705\pi$
$199$	21.4707	1.52202	0.761008	+	0.648742i	$0.224705\pi$
$200$	0	0
$201$	−5.03309	−0.355007
$202$	0	0
$203$	5.07857	0.356445
$204$	0	0
$205$	1.95175	0.136316
$206$	0	0
$207$	−2.38042	−0.165450
$208$	0	0
$209$	−2.52961	−0.174977
$210$	0	0
$211$	13.8591	0.954101	0.477051	−	0.878876i	$-0.341706\pi$
$211$	13.8591	0.954101	0.477051	+	0.878876i	$0.341706\pi$
$212$	0	0
$213$	14.3450	0.982904
$214$	0	0
$215$	−6.07591	−0.414374
$216$	0	0
$217$	−7.18588	−0.487810
$218$	0	0
$219$	6.85021	0.462895
$220$	0	0
$221$	−0.0101939	−0.000685715 0
$222$	0	0
$223$	12.7958	0.856869	0.428434	−	0.903573i	$-0.359065\pi$
$223$	12.7958	0.856869	0.428434	+	0.903573i	$0.359065\pi$
$224$	0	0
$225$	1.99689	0.133126
$226$	0	0
$227$	22.2017	1.47358	0.736790	−	0.676122i	$-0.236341\pi$
$227$	22.2017	1.47358	0.736790	+	0.676122i	$0.236341\pi$
$228$	0	0
$229$	8.81439	0.582471	0.291236	−	0.956651i	$-0.405934\pi$
$229$	8.81439	0.582471	0.291236	+	0.956651i	$0.405934\pi$
$230$	0	0
$231$	−1.54747	−0.101816
$232$	0	0
$233$	−9.85676	−0.645738	−0.322869	−	0.946444i	$-0.604647\pi$
$233$	−9.85676	−0.645738	−0.322869	+	0.946444i	$0.604647\pi$
$234$	0	0
$235$	21.5645	1.40671
$236$	0	0
$237$	7.65408	0.497186
$238$	0	0
$239$	−6.93235	−0.448417	−0.224208	−	0.974541i	$-0.571980\pi$
$239$	−6.93235	−0.448417	−0.224208	+	0.974541i	$0.571980\pi$
$240$	0	0
$241$	−3.67040	−0.236432	−0.118216	−	0.992988i	$-0.537717\pi$
$241$	−3.67040	−0.236432	−0.118216	+	0.992988i	$0.537717\pi$
$242$	0	0
$243$	15.0126	0.963058
$244$	0	0
$245$	−2.89819	−0.185158
$246$	0	0
$247$	3.06263	0.194870
$248$	0	0
$249$	7.14999	0.453112
$250$	0	0
$251$	14.9684	0.944797	0.472398	−	0.881385i	$-0.343388\pi$
$251$	14.9684	0.944797	0.472398	+	0.881385i	$0.343388\pi$
$252$	0	0
$253$	0.650330	0.0408859
$254$	0	0
$255$	−0.0397940	−0.00249200
$256$	0	0
$257$	25.4360	1.58665	0.793327	−	0.608795i	$-0.208347\pi$
$257$	25.4360	1.58665	0.793327	+	0.608795i	$0.208347\pi$
$258$	0	0
$259$	−14.3609	−0.892341
$260$	0	0
$261$	2.99606	0.185452
$262$	0	0
$263$	7.93251	0.489140	0.244570	−	0.969632i	$-0.421353\pi$
$263$	7.93251	0.489140	0.244570	+	0.969632i	$0.421353\pi$
$264$	0	0
$265$	−16.3009	−1.00136
$266$	0	0
$267$	13.4077	0.820539
$268$	0	0
$269$	22.5590	1.37545	0.687724	−	0.725973i	$-0.258610\pi$
$269$	22.5590	1.37545	0.687724	+	0.725973i	$0.258610\pi$
$270$	0	0
$271$	12.6823	0.770394	0.385197	−	0.922834i	$-0.374133\pi$
$271$	12.6823	0.770394	0.385197	+	0.922834i	$0.374133\pi$
$272$	0	0
$273$	1.87354	0.113392
$274$	0	0
$275$	−0.545551	−0.0328980
$276$	0	0
$277$	−17.8844	−1.07457	−0.537283	−	0.843402i	$-0.680549\pi$
$277$	−17.8844	−1.07457	−0.537283	+	0.843402i	$0.680549\pi$
$278$	0	0
$279$	−4.23926	−0.253798
$280$	0	0
$281$	17.1988	1.02600	0.512998	−	0.858390i	$-0.328535\pi$
$281$	17.1988	1.02600	0.512998	+	0.858390i	$0.328535\pi$
$282$	0	0
$283$	−29.5331	−1.75556	−0.877779	−	0.479065i	$-0.840976\pi$
$283$	−29.5331	−1.75556	−0.877779	+	0.479065i	$0.840976\pi$
$284$	0	0
$285$	11.9556	0.708189
$286$	0	0
$287$	2.90121	0.171253
$288$	0	0
$289$	−16.9997	−0.999981
$290$	0	0
$291$	−20.9076	−1.22563
$292$	0	0
$293$	−3.33697	−0.194948	−0.0974741	−	0.995238i	$-0.531076\pi$
$293$	−3.33697	−0.194948	−0.0974741	+	0.995238i	$0.531076\pi$
$294$	0	0
$295$	−17.5450	−1.02151
$296$	0	0
$297$	−2.50718	−0.145481
$298$	0	0
$299$	−0.787362	−0.0455343
$300$	0	0
$301$	−9.03162	−0.520574
$302$	0	0
$303$	−4.25940	−0.244696
$304$	0	0
$305$	11.2647	0.645013
$306$	0	0
$307$	−5.03491	−0.287357	−0.143679	−	0.989624i	$-0.545893\pi$
$307$	−5.03491	−0.287357	−0.143679	+	0.989624i	$0.545893\pi$
$308$	0	0
$309$	12.1179	0.689365
$310$	0	0
$311$	27.4509	1.55660	0.778299	−	0.627894i	$-0.216083\pi$
$311$	27.4509	1.55660	0.778299	+	0.627894i	$0.216083\pi$
$312$	0	0
$313$	9.37875	0.530118	0.265059	−	0.964232i	$-0.414609\pi$
$313$	9.37875	0.530118	0.265059	+	0.964232i	$0.414609\pi$
$314$	0	0
$315$	−9.79955	−0.552142
$316$	0	0
$317$	26.6467	1.49663	0.748314	−	0.663344i	$-0.230864\pi$
$317$	26.6467	1.49663	0.748314	+	0.663344i	$0.230864\pi$
$318$	0	0
$319$	−0.818525	−0.0458286
$320$	0	0
$321$	−20.0209	−1.11746
$322$	0	0
$323$	−0.0966949	−0.00538025
$324$	0	0
$325$	0.660505	0.0366382
$326$	0	0
$327$	5.05382	0.279477
$328$	0	0
$329$	32.0549	1.76724
$330$	0	0
$331$	−18.8732	−1.03737	−0.518683	−	0.854967i	$-0.673577\pi$
$331$	−18.8732	−1.03737	−0.518683	+	0.854967i	$0.673577\pi$
$332$	0	0
$333$	−8.47210	−0.464268
$334$	0	0
$335$	8.70763	0.475748
$336$	0	0
$337$	21.3875	1.16505	0.582525	−	0.812813i	$-0.302065\pi$
$337$	21.3875	1.16505	0.582525	+	0.812813i	$0.302065\pi$
$338$	0	0
$339$	17.6004	0.955924
$340$	0	0
$341$	1.15817	0.0627182
$342$	0	0
$343$	16.0756	0.868000
$344$	0	0
$345$	−3.07363	−0.165479
$346$	0	0
$347$	−36.4844	−1.95859	−0.979293	−	0.202448i	$-0.935110\pi$
$347$	−36.4844	−1.95859	−0.979293	+	0.202448i	$0.935110\pi$
$348$	0	0
$349$	15.6910	0.839918	0.419959	−	0.907543i	$-0.362045\pi$
$349$	15.6910	0.839918	0.419959	+	0.907543i	$0.362045\pi$
$350$	0	0
$351$	3.03547	0.162021
$352$	0	0
$353$	−25.7321	−1.36958	−0.684791	−	0.728739i	$-0.740107\pi$
$353$	−25.7321	−1.36958	−0.684791	+	0.728739i	$0.740107\pi$
$354$	0	0
$355$	−24.8180	−1.31720
$356$	0	0
$357$	−0.0591523	−0.00313067
$358$	0	0
$359$	−10.5835	−0.558576	−0.279288	−	0.960207i	$-0.590098\pi$
$359$	−10.5835	−0.558576	−0.279288	+	0.960207i	$0.590098\pi$
$360$	0	0
$361$	10.0508	0.528990
$362$	0	0
$363$	−12.2060	−0.640646
$364$	0	0
$365$	−11.8514	−0.620330
$366$	0	0
$367$	−1.48021	−0.0772662	−0.0386331	−	0.999253i	$-0.512300\pi$
$367$	−1.48021	−0.0772662	−0.0386331	+	0.999253i	$0.512300\pi$
$368$	0	0
$369$	1.71155	0.0890997
$370$	0	0
$371$	−24.2307	−1.25800
$372$	0	0
$373$	−25.4166	−1.31602	−0.658012	−	0.753008i	$-0.728602\pi$
$373$	−25.4166	−1.31602	−0.658012	+	0.753008i	$0.728602\pi$
$374$	0	0
$375$	13.6692	0.705875
$376$	0	0
$377$	0.990998	0.0510390
$378$	0	0
$379$	1.01737	0.0522587	0.0261294	−	0.999659i	$-0.491682\pi$
$379$	1.01737	0.0522587	0.0261294	+	0.999659i	$0.491682\pi$
$380$	0	0
$381$	8.79080	0.450366
$382$	0	0
$383$	−3.49480	−0.178576	−0.0892879	−	0.996006i	$-0.528459\pi$
$383$	−3.49480	−0.178576	−0.0892879	+	0.996006i	$0.528459\pi$
$384$	0	0
$385$	2.67724	0.136445
$386$	0	0
$387$	−5.32814	−0.270845
$388$	0	0
$389$	7.20197	0.365155	0.182577	−	0.983192i	$-0.441556\pi$
$389$	7.20197	0.365155	0.182577	+	0.983192i	$0.441556\pi$
$390$	0	0
$391$	0.0248590	0.00125717
$392$	0	0
$393$	−18.7159	−0.944090
$394$	0	0
$395$	−13.2421	−0.666285
$396$	0	0
$397$	2.13497	0.107151	0.0535755	−	0.998564i	$-0.482938\pi$
$397$	2.13497	0.107151	0.0535755	+	0.998564i	$0.482938\pi$
$398$	0	0
$399$	17.7716	0.889692
$400$	0	0
$401$	−26.8772	−1.34218	−0.671092	−	0.741374i	$-0.734174\pi$
$401$	−26.8772	−1.34218	−0.671092	+	0.741374i	$0.734174\pi$
$402$	0	0
$403$	−1.40221	−0.0698489
$404$	0	0
$405$	1.75372	0.0871433
$406$	0	0
$407$	2.31458	0.114729
$408$	0	0
$409$	8.97024	0.443550	0.221775	−	0.975098i	$-0.428815\pi$
$409$	8.97024	0.443550	0.221775	+	0.975098i	$0.428815\pi$
$410$	0	0
$411$	17.9071	0.883291
$412$	0	0
$413$	−26.0800	−1.28331
$414$	0	0
$415$	−12.3700	−0.607221
$416$	0	0
$417$	21.1644	1.03643
$418$	0	0
$419$	−27.5235	−1.34461	−0.672305	−	0.740274i	$-0.734696\pi$
$419$	−27.5235	−1.34461	−0.672305	+	0.740274i	$0.734696\pi$
$420$	0	0
$421$	37.3857	1.82207	0.911034	−	0.412332i	$-0.135286\pi$
$421$	37.3857	1.82207	0.911034	+	0.412332i	$0.135286\pi$
$422$	0	0
$423$	18.9105	0.919462
$424$	0	0
$425$	−0.0208538	−0.00101156
$426$	0	0
$427$	16.7445	0.810325
$428$	0	0
$429$	−0.301963	−0.0145789
$430$	0	0
$431$	−30.9766	−1.49209	−0.746044	−	0.665897i	$-0.768049\pi$
$431$	−30.9766	−1.49209	−0.746044	+	0.665897i	$0.768049\pi$
$432$	0	0
$433$	−8.12538	−0.390481	−0.195240	−	0.980755i	$-0.562549\pi$
$433$	−8.12538	−0.390481	−0.195240	+	0.980755i	$0.562549\pi$
$434$	0	0
$435$	3.86857	0.185484
$436$	0	0
$437$	−7.46859	−0.357271
$438$	0	0
$439$	−2.18958	−0.104503	−0.0522515	−	0.998634i	$-0.516640\pi$
$439$	−2.18958	−0.104503	−0.0522515	+	0.998634i	$0.516640\pi$
$440$	0	0
$441$	−2.54150	−0.121024
$442$	0	0
$443$	−27.4515	−1.30426	−0.652130	−	0.758107i	$-0.726124\pi$
$443$	−27.4515	−1.30426	−0.652130	+	0.758107i	$0.726124\pi$
$444$	0	0
$445$	−23.1964	−1.09961
$446$	0	0
$447$	−23.6818	−1.12011
$448$	0	0
$449$	16.3971	0.773829	0.386914	−	0.922116i	$-0.373541\pi$
$449$	16.3971	0.773829	0.386914	+	0.922116i	$0.373541\pi$
$450$	0	0
$451$	−0.467595	−0.0220182
$452$	0	0
$453$	−4.18188	−0.196482
$454$	0	0
$455$	−3.24136	−0.151957
$456$	0	0
$457$	22.3609	1.04600	0.523000	−	0.852333i	$-0.324813\pi$
$457$	22.3609	1.04600	0.523000	+	0.852333i	$0.324813\pi$
$458$	0	0
$459$	−0.0958375	−0.00447331
$460$	0	0
$461$	31.6429	1.47375	0.736877	−	0.676026i	$-0.236300\pi$
$461$	31.6429	1.47375	0.736877	+	0.676026i	$0.236300\pi$
$462$	0	0
$463$	−15.2255	−0.707589	−0.353794	−	0.935323i	$-0.615109\pi$
$463$	−15.2255	−0.707589	−0.353794	+	0.935323i	$0.615109\pi$
$464$	0	0
$465$	−5.47381	−0.253842
$466$	0	0
$467$	−34.4941	−1.59619	−0.798097	−	0.602529i	$-0.794160\pi$
$467$	−34.4941	−1.59619	−0.798097	+	0.602529i	$0.794160\pi$
$468$	0	0
$469$	12.9436	0.597679
$470$	0	0
$471$	11.0189	0.507725
$472$	0	0
$473$	1.45565	0.0669308
$474$	0	0
$475$	6.26527	0.287470
$476$	0	0
$477$	−14.2947	−0.654511
$478$	0	0
$479$	5.80105	0.265057	0.132528	−	0.991179i	$-0.457690\pi$
$479$	5.80105	0.265057	0.132528	+	0.991179i	$0.457690\pi$
$480$	0	0
$481$	−2.80229	−0.127773
$482$	0	0
$483$	−4.56885	−0.207890
$484$	0	0
$485$	36.1718	1.64247
$486$	0	0
$487$	16.1178	0.730367	0.365183	−	0.930936i	$-0.381006\pi$
$487$	16.1178	0.730367	0.365183	+	0.930936i	$0.381006\pi$
$488$	0	0
$489$	−6.09983	−0.275844
$490$	0	0
$491$	17.9645	0.810727	0.405363	−	0.914156i	$-0.367145\pi$
$491$	17.9645	0.810727	0.405363	+	0.914156i	$0.367145\pi$
$492$	0	0
$493$	−0.0312883	−0.00140915
$494$	0	0
$495$	1.57942	0.0709896
$496$	0	0
$497$	−36.8910	−1.65479
$498$	0	0
$499$	−24.2804	−1.08694	−0.543469	−	0.839429i	$-0.682890\pi$
$499$	−24.2804	−1.08694	−0.543469	+	0.839429i	$0.682890\pi$
$500$	0	0
$501$	−20.0338	−0.895042
$502$	0	0
$503$	25.4654	1.13545	0.567724	−	0.823219i	$-0.307824\pi$
$503$	25.4654	1.13545	0.567724	+	0.823219i	$0.307824\pi$
$504$	0	0
$505$	7.36909	0.327920
$506$	0	0
$507$	−14.3544	−0.637501
$508$	0	0
$509$	39.7761	1.76304	0.881522	−	0.472143i	$-0.156519\pi$
$509$	39.7761	1.76304	0.881522	+	0.472143i	$0.156519\pi$
$510$	0	0
$511$	−17.6167	−0.779316
$512$	0	0
$513$	28.7932	1.27125
$514$	0	0
$515$	−20.9649	−0.923826
$516$	0	0
$517$	−5.16636	−0.227216
$518$	0	0
$519$	4.61952	0.202775
$520$	0	0
$521$	−15.2119	−0.666445	−0.333222	−	0.942848i	$-0.608136\pi$
$521$	−15.2119	−0.666445	−0.333222	+	0.942848i	$0.608136\pi$
$522$	0	0
$523$	0.0967773	0.00423178	0.00211589	−	0.999998i	$-0.499326\pi$
$523$	0.0967773	0.00423178	0.00211589	+	0.999998i	$0.499326\pi$
$524$	0	0
$525$	3.83273	0.167274
$526$	0	0
$527$	0.0442712	0.00192848
$528$	0	0
$529$	−21.0799	−0.916518
$530$	0	0
$531$	−15.3857	−0.667683
$532$	0	0
$533$	0.566123	0.0245215
$534$	0	0
$535$	34.6377	1.49752
$536$	0	0
$537$	7.82393	0.337628
$538$	0	0
$539$	0.694339	0.0299073
$540$	0	0
$541$	−3.15797	−0.135772	−0.0678858	−	0.997693i	$-0.521625\pi$
$541$	−3.15797	−0.135772	−0.0678858	+	0.997693i	$0.521625\pi$
$542$	0	0
$543$	14.0066	0.601079
$544$	0	0
$545$	−8.74349	−0.374530
$546$	0	0
$547$	−17.3725	−0.742795	−0.371397	−	0.928474i	$-0.621121\pi$
$547$	−17.3725	−0.742795	−0.371397	+	0.928474i	$0.621121\pi$
$548$	0	0
$549$	9.87832	0.421597
$550$	0	0
$551$	9.40019	0.400462
$552$	0	0
$553$	−19.6840	−0.837048
$554$	0	0
$555$	−10.9393	−0.464348
$556$	0	0
$557$	−7.75560	−0.328615	−0.164308	−	0.986409i	$-0.552539\pi$
$557$	−7.75560	−0.328615	−0.164308	+	0.986409i	$0.552539\pi$
$558$	0	0
$559$	−1.76237	−0.0745404
$560$	0	0
$561$	0.00953373	0.000402514 0
$562$	0	0
$563$	−12.2764	−0.517389	−0.258695	−	0.965959i	$-0.583292\pi$
$563$	−12.2764	−0.517389	−0.258695	+	0.965959i	$0.583292\pi$
$564$	0	0
$565$	−30.4501	−1.28105
$566$	0	0
$567$	2.60685	0.109477
$568$	0	0
$569$	−38.0739	−1.59614	−0.798071	−	0.602564i	$-0.794146\pi$
$569$	−38.0739	−1.59614	−0.798071	+	0.602564i	$0.794146\pi$
$570$	0	0
$571$	−1.65542	−0.0692770	−0.0346385	−	0.999400i	$-0.511028\pi$
$571$	−1.65542	−0.0692770	−0.0346385	+	0.999400i	$0.511028\pi$
$572$	0	0
$573$	−19.8582	−0.829589
$574$	0	0
$575$	−1.61072	−0.0671717
$576$	0	0
$577$	45.2771	1.88491	0.942454	−	0.334335i	$-0.108512\pi$
$577$	45.2771	1.88491	0.942454	+	0.334335i	$0.108512\pi$
$578$	0	0
$579$	−2.63673	−0.109579
$580$	0	0
$581$	−18.3876	−0.762846
$582$	0	0
$583$	3.90532	0.161742
$584$	0	0
$585$	−1.91222	−0.0790606
$586$	0	0
$587$	31.1122	1.28414	0.642070	−	0.766646i	$-0.278076\pi$
$587$	31.1122	1.28414	0.642070	+	0.766646i	$0.278076\pi$
$588$	0	0
$589$	−13.3007	−0.548047
$590$	0	0
$591$	20.0885	0.826332
$592$	0	0
$593$	6.75378	0.277345	0.138672	−	0.990338i	$-0.455717\pi$
$593$	6.75378	0.277345	0.138672	+	0.990338i	$0.455717\pi$
$594$	0	0
$595$	0.102338	0.00419545
$596$	0	0
$597$	24.3114	0.994998
$598$	0	0
$599$	−5.60991	−0.229215	−0.114607	−	0.993411i	$-0.536561\pi$
$599$	−5.60991	−0.229215	−0.114607	+	0.993411i	$0.536561\pi$
$600$	0	0
$601$	1.02510	0.0418148	0.0209074	−	0.999781i	$-0.493344\pi$
$601$	1.02510	0.0418148	0.0209074	+	0.999781i	$0.493344\pi$
$602$	0	0
$603$	7.63597	0.310961
$604$	0	0
$605$	21.1172	0.858538
$606$	0	0
$607$	10.8697	0.441189	0.220594	−	0.975366i	$-0.429200\pi$
$607$	10.8697	0.441189	0.220594	+	0.975366i	$0.429200\pi$
$608$	0	0
$609$	5.75049	0.233022
$610$	0	0
$611$	6.25497	0.253049
$612$	0	0
$613$	38.3844	1.55033	0.775167	−	0.631757i	$-0.217666\pi$
$613$	38.3844	1.55033	0.775167	+	0.631757i	$0.217666\pi$
$614$	0	0
$615$	2.20998	0.0891150
$616$	0	0
$617$	−12.9640	−0.521912	−0.260956	−	0.965351i	$-0.584038\pi$
$617$	−12.9640	−0.521912	−0.260956	+	0.965351i	$0.584038\pi$
$618$	0	0
$619$	−30.1697	−1.21262	−0.606312	−	0.795227i	$-0.707352\pi$
$619$	−30.1697	−1.21262	−0.606312	+	0.795227i	$0.707352\pi$
$620$	0	0
$621$	−7.40236	−0.297047
$622$	0	0
$623$	−34.4806	−1.38144
$624$	0	0
$625$	−17.8367	−0.713469
$626$	0	0
$627$	−2.86429	−0.114389
$628$	0	0
$629$	0.0884753	0.00352774
$630$	0	0
$631$	0.931962	0.0371008	0.0185504	−	0.999828i	$-0.494095\pi$
$631$	0.931962	0.0371008	0.0185504	+	0.999828i	$0.494095\pi$
$632$	0	0
$633$	15.6928	0.623731
$634$	0	0
$635$	−15.2088	−0.603541
$636$	0	0
$637$	−0.840644	−0.0333075
$638$	0	0
$639$	−21.7636	−0.860955
$640$	0	0
$641$	−1.62066	−0.0640121	−0.0320061	−	0.999488i	$-0.510190\pi$
$641$	−1.62066	−0.0640121	−0.0320061	+	0.999488i	$0.510190\pi$
$642$	0	0
$643$	38.8892	1.53364	0.766820	−	0.641862i	$-0.221838\pi$
$643$	38.8892	1.53364	0.766820	+	0.641862i	$0.221838\pi$
$644$	0	0
$645$	−6.87979	−0.270891
$646$	0	0
$647$	−35.2419	−1.38550	−0.692750	−	0.721178i	$-0.743601\pi$
$647$	−35.2419	−1.38550	−0.692750	+	0.721178i	$0.743601\pi$
$648$	0	0
$649$	4.20337	0.164997
$650$	0	0
$651$	−8.13662	−0.318899
$652$	0	0
$653$	−37.6976	−1.47522	−0.737611	−	0.675226i	$-0.764046\pi$
$653$	−37.6976	−1.47522	−0.737611	+	0.675226i	$0.764046\pi$
$654$	0	0
$655$	32.3799	1.26519
$656$	0	0
$657$	−10.3928	−0.405463
$658$	0	0
$659$	−27.5799	−1.07436	−0.537180	−	0.843467i	$-0.680511\pi$
$659$	−27.5799	−1.07436	−0.537180	+	0.843467i	$0.680511\pi$
$660$	0	0
$661$	8.39217	0.326418	0.163209	−	0.986592i	$-0.447816\pi$
$661$	8.39217	0.326418	0.163209	+	0.986592i	$0.447816\pi$
$662$	0	0
$663$	−0.0115426	−0.000448277 0
$664$	0	0
$665$	−30.7462	−1.19229
$666$	0	0
$667$	−2.41667	−0.0935737
$668$	0	0
$669$	14.4887	0.560167
$670$	0	0
$671$	−2.69876	−0.104184
$672$	0	0
$673$	34.5373	1.33131	0.665657	−	0.746258i	$-0.268151\pi$
$673$	34.5373	1.33131	0.665657	+	0.746258i	$0.268151\pi$
$674$	0	0
$675$	6.20972	0.239012
$676$	0	0
$677$	1.83654	0.0705838	0.0352919	−	0.999377i	$-0.488764\pi$
$677$	1.83654	0.0705838	0.0352919	+	0.999377i	$0.488764\pi$
$678$	0	0
$679$	53.7680	2.06343
$680$	0	0
$681$	25.1391	0.963333
$682$	0	0
$683$	−41.0721	−1.57158	−0.785791	−	0.618492i	$-0.787744\pi$
$683$	−41.0721	−1.57158	−0.785791	+	0.618492i	$0.787744\pi$
$684$	0	0
$685$	−30.9806	−1.18371
$686$	0	0
$687$	9.98058	0.380783
$688$	0	0
$689$	−4.72822	−0.180131
$690$	0	0
$691$	1.76631	0.0671938	0.0335969	−	0.999435i	$-0.489304\pi$
$691$	1.76631	0.0671938	0.0335969	+	0.999435i	$0.489304\pi$
$692$	0	0
$693$	2.34775	0.0891836
$694$	0	0
$695$	−36.6161	−1.38893
$696$	0	0
$697$	−0.0178739	−0.000677024 0
$698$	0	0
$699$	−11.1609	−0.422143
$700$	0	0
$701$	14.9748	0.565589	0.282794	−	0.959181i	$-0.408739\pi$
$701$	14.9748	0.565589	0.282794	+	0.959181i	$0.408739\pi$
$702$	0	0
$703$	−26.5813	−1.00253
$704$	0	0
$705$	24.4176	0.919620
$706$	0	0
$707$	10.9539	0.411964
$708$	0	0
$709$	23.0808	0.866819	0.433410	−	0.901197i	$-0.357310\pi$
$709$	23.0808	0.866819	0.433410	+	0.901197i	$0.357310\pi$
$710$	0	0
$711$	−11.6124	−0.435500
$712$	0	0
$713$	3.41945	0.128059
$714$	0	0
$715$	0.522419	0.0195373
$716$	0	0
$717$	−7.84954	−0.293147
$718$	0	0
$719$	−46.1537	−1.72124	−0.860620	−	0.509247i	$-0.829924\pi$
$719$	−46.1537	−1.72124	−0.860620	+	0.509247i	$0.829924\pi$
$720$	0	0
$721$	−31.1636	−1.16059
$722$	0	0
$723$	−4.15602	−0.154564
$724$	0	0
$725$	2.02730	0.0752921
$726$	0	0
$727$	−37.1770	−1.37882	−0.689410	−	0.724371i	$-0.742130\pi$
$727$	−37.1770	−1.37882	−0.689410	+	0.724371i	$0.742130\pi$
$728$	0	0
$729$	19.6845	0.729056
$730$	0	0
$731$	0.0556425	0.00205801
$732$	0	0
$733$	−11.1429	−0.411571	−0.205786	−	0.978597i	$-0.565975\pi$
$733$	−11.1429	−0.411571	−0.205786	+	0.978597i	$0.565975\pi$
$734$	0	0
$735$	−3.28163	−0.121045
$736$	0	0
$737$	−2.08615	−0.0768442
$738$	0	0
$739$	44.9738	1.65439	0.827194	−	0.561916i	$-0.189936\pi$
$739$	44.9738	1.65439	0.827194	+	0.561916i	$0.189936\pi$
$740$	0	0
$741$	3.46783	0.127394
$742$	0	0
$743$	33.4684	1.22784	0.613919	−	0.789369i	$-0.289592\pi$
$743$	33.4684	1.22784	0.613919	+	0.789369i	$0.289592\pi$
$744$	0	0
$745$	40.9714	1.50107
$746$	0	0
$747$	−10.8476	−0.396894
$748$	0	0
$749$	51.4877	1.88132
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	16.9488	0.617649
$754$	0	0
$755$	7.23498	0.263308
$756$	0	0
$757$	52.1947	1.89705	0.948525	−	0.316703i	$-0.102576\pi$
$757$	52.1947	1.89705	0.948525	+	0.316703i	$0.102576\pi$
$758$	0	0
$759$	0.736372	0.0267286
$760$	0	0
$761$	34.1943	1.23954	0.619772	−	0.784782i	$-0.287225\pi$
$761$	34.1943	1.23954	0.619772	+	0.784782i	$0.287225\pi$
$762$	0	0
$763$	−12.9969	−0.470519
$764$	0	0
$765$	0.0603737	0.00218281
$766$	0	0
$767$	−5.08908	−0.183756
$768$	0	0
$769$	0.0184882	0.000666700 0	0.000333350	−	1.00000i	$-0.499894\pi$
$769$	0.0184882	0.000666700 0	0.000333350		1.00000i	$0.499894\pi$
$770$	0	0
$771$	28.8013	1.03726
$772$	0	0
$773$	−27.6351	−0.993965	−0.496982	−	0.867761i	$-0.665559\pi$
$773$	−27.6351	−0.993965	−0.496982	+	0.867761i	$0.665559\pi$
$774$	0	0
$775$	−2.86852	−0.103040
$776$	0	0
$777$	−16.2609	−0.583356
$778$	0	0
$779$	5.37000	0.192400
$780$	0	0
$781$	5.94582	0.212758
$782$	0	0
$783$	9.31684	0.332957
$784$	0	0
$785$	−19.0636	−0.680408
$786$	0	0
$787$	−38.6144	−1.37645	−0.688227	−	0.725495i	$-0.741611\pi$
$787$	−38.6144	−1.37645	−0.688227	+	0.725495i	$0.741611\pi$
$788$	0	0
$789$	8.98203	0.319769
$790$	0	0
$791$	−45.2630	−1.60937
$792$	0	0
$793$	3.26742	0.116029
$794$	0	0
$795$	−18.4576	−0.654624
$796$	0	0
$797$	6.59449	0.233589	0.116794	−	0.993156i	$-0.462738\pi$
$797$	6.59449	0.233589	0.116794	+	0.993156i	$0.462738\pi$
$798$	0	0
$799$	−0.197485	−0.00698653
$800$	0	0
$801$	−20.3416	−0.718735
$802$	0	0
$803$	2.83932	0.100198
$804$	0	0
$805$	7.90445	0.278595
$806$	0	0
$807$	25.5437	0.899181
$808$	0	0
$809$	50.1555	1.76337	0.881686	−	0.471836i	$-0.156409\pi$
$809$	50.1555	1.76337	0.881686	+	0.471836i	$0.156409\pi$
$810$	0	0
$811$	−23.5456	−0.826798	−0.413399	−	0.910550i	$-0.635658\pi$
$811$	−23.5456	−0.826798	−0.413399	+	0.910550i	$0.635658\pi$
$812$	0	0
$813$	14.3602	0.503635
$814$	0	0
$815$	10.5532	0.369662
$816$	0	0
$817$	−16.7171	−0.584858
$818$	0	0
$819$	−2.84245	−0.0993232
$820$	0	0
$821$	−30.7543	−1.07333	−0.536667	−	0.843794i	$-0.680317\pi$
$821$	−30.7543	−1.07333	−0.536667	+	0.843794i	$0.680317\pi$
$822$	0	0
$823$	25.9832	0.905717	0.452859	−	0.891582i	$-0.350404\pi$
$823$	25.9832	0.905717	0.452859	+	0.891582i	$0.350404\pi$
$824$	0	0
$825$	−0.617730	−0.0215066
$826$	0	0
$827$	−0.456809	−0.0158848	−0.00794241	−	0.999968i	$-0.502528\pi$
$827$	−0.456809	−0.0158848	−0.00794241	+	0.999968i	$0.502528\pi$
$828$	0	0
$829$	45.6445	1.58530	0.792650	−	0.609677i	$-0.208701\pi$
$829$	45.6445	1.58530	0.792650	+	0.609677i	$0.208701\pi$
$830$	0	0
$831$	−20.2506	−0.702484
$832$	0	0
$833$	0.0265413	0.000919601 0
$834$	0	0
$835$	34.6599	1.19946
$836$	0	0
$837$	−13.1828	−0.455664
$838$	0	0
$839$	3.50708	0.121078	0.0605389	−	0.998166i	$-0.480718\pi$
$839$	3.50708	0.121078	0.0605389	+	0.998166i	$0.480718\pi$
$840$	0	0
$841$	−25.9583	−0.895114
$842$	0	0
$843$	19.4743	0.670731
$844$	0	0
$845$	24.8342	0.854322
$846$	0	0
$847$	31.3900	1.07857
$848$	0	0
$849$	−33.4405	−1.14767
$850$	0	0
$851$	6.83371	0.234257
$852$	0	0
$853$	51.4488	1.76157	0.880786	−	0.473515i	$-0.157015\pi$
$853$	51.4488	1.76157	0.880786	+	0.473515i	$0.157015\pi$
$854$	0	0
$855$	−18.1385	−0.620324
$856$	0	0
$857$	−5.92601	−0.202429	−0.101214	−	0.994865i	$-0.532273\pi$
$857$	−5.92601	−0.202429	−0.101214	+	0.994865i	$0.532273\pi$
$858$	0	0
$859$	54.2744	1.85182	0.925909	−	0.377746i	$-0.123301\pi$
$859$	54.2744	1.85182	0.925909	+	0.377746i	$0.123301\pi$
$860$	0	0
$861$	3.28506	0.111954
$862$	0	0
$863$	−32.4021	−1.10298	−0.551490	−	0.834182i	$-0.685940\pi$
$863$	−32.4021	−1.10298	−0.551490	+	0.834182i	$0.685940\pi$
$864$	0	0
$865$	−7.99213	−0.271741
$866$	0	0
$867$	−19.2488	−0.653725
$868$	0	0
$869$	3.17251	0.107620
$870$	0	0
$871$	2.52572	0.0855809
$872$	0	0
$873$	31.7201	1.07356
$874$	0	0
$875$	−35.1531	−1.18839
$876$	0	0
$877$	−22.2435	−0.751110	−0.375555	−	0.926800i	$-0.622548\pi$
$877$	−22.2435	−0.751110	−0.375555	+	0.926800i	$0.622548\pi$
$878$	0	0
$879$	−3.77848	−0.127445
$880$	0	0
$881$	1.03750	0.0349543	0.0174771	−	0.999847i	$-0.494437\pi$
$881$	1.03750	0.0349543	0.0174771	+	0.999847i	$0.494437\pi$
$882$	0	0
$883$	−12.5611	−0.422715	−0.211357	−	0.977409i	$-0.567788\pi$
$883$	−12.5611	−0.422715	−0.211357	+	0.977409i	$0.567788\pi$
$884$	0	0
$885$	−19.8663	−0.667798
$886$	0	0
$887$	−6.95005	−0.233360	−0.116680	−	0.993170i	$-0.537225\pi$
$887$	−6.95005	−0.233360	−0.116680	+	0.993170i	$0.537225\pi$
$888$	0	0
$889$	−22.6073	−0.758224
$890$	0	0
$891$	−0.420152	−0.0140756
$892$	0	0
$893$	59.3320	1.98547
$894$	0	0
$895$	−13.5360	−0.452459
$896$	0	0
$897$	−0.891535	−0.0297675
$898$	0	0
$899$	−4.30382	−0.143540
$900$	0	0
$901$	0.149282	0.00497330
$902$	0	0
$903$	−10.2266	−0.340319
$904$	0	0
$905$	−24.2324	−0.805513
$906$	0	0
$907$	−29.4503	−0.977882	−0.488941	−	0.872317i	$-0.662617\pi$
$907$	−29.4503	−0.977882	−0.488941	+	0.872317i	$0.662617\pi$
$908$	0	0
$909$	6.46217	0.214337
$910$	0	0
$911$	2.88626	0.0956260	0.0478130	−	0.998856i	$-0.484775\pi$
$911$	2.88626	0.0956260	0.0478130	+	0.998856i	$0.484775\pi$
$912$	0	0
$913$	2.96358	0.0980800
$914$	0	0
$915$	12.7551	0.421669
$916$	0	0
$917$	48.1315	1.58944
$918$	0	0
$919$	19.5777	0.645809	0.322905	−	0.946431i	$-0.395341\pi$
$919$	19.5777	0.645809	0.322905	+	0.946431i	$0.395341\pi$
$920$	0	0
$921$	−5.70106	−0.187856
$922$	0	0
$923$	−7.19868	−0.236947
$924$	0	0
$925$	−5.73268	−0.188490
$926$	0	0
$927$	−18.3848	−0.603835
$928$	0	0
$929$	−60.5576	−1.98683	−0.993414	−	0.114577i	$-0.963449\pi$
$929$	−60.5576	−1.98683	−0.993414	+	0.114577i	$0.963449\pi$
$930$	0	0
$931$	−7.97400	−0.261337
$932$	0	0
$933$	31.0828	1.01761
$934$	0	0
$935$	−0.0164941	−0.000539414 0
$936$	0	0
$937$	−15.3177	−0.500407	−0.250203	−	0.968193i	$-0.580497\pi$
$937$	−15.3177	−0.500407	−0.250203	+	0.968193i	$0.580497\pi$
$938$	0	0
$939$	10.6196	0.346558
$940$	0	0
$941$	−16.6993	−0.544383	−0.272192	−	0.962243i	$-0.587748\pi$
$941$	−16.6993	−0.544383	−0.272192	+	0.962243i	$0.587748\pi$
$942$	0	0
$943$	−1.38056	−0.0449572
$944$	0	0
$945$	−30.4736	−0.991306
$946$	0	0
$947$	46.3409	1.50588	0.752938	−	0.658091i	$-0.228636\pi$
$947$	46.3409	1.50588	0.752938	+	0.658091i	$0.228636\pi$
$948$	0	0
$949$	−3.43760	−0.111589
$950$	0	0
$951$	30.1722	0.978401
$952$	0	0
$953$	49.0506	1.58891	0.794453	−	0.607326i	$-0.207758\pi$
$953$	49.0506	1.58891	0.794453	+	0.607326i	$0.207758\pi$
$954$	0	0
$955$	34.3563	1.11174
$956$	0	0
$957$	−0.926820	−0.0299598
$958$	0	0
$959$	−46.0516	−1.48708
$960$	0	0
$961$	−24.9103	−0.803559
$962$	0	0
$963$	30.3748	0.978815
$964$	0	0
$965$	4.56174	0.146848
$966$	0	0
$967$	−10.8296	−0.348255	−0.174128	−	0.984723i	$-0.555711\pi$
$967$	−10.8296	−0.348255	−0.174128	+	0.984723i	$0.555711\pi$
$968$	0	0
$969$	−0.109488	−0.00351727
$970$	0	0
$971$	6.42742	0.206266	0.103133	−	0.994668i	$-0.467113\pi$
$971$	6.42742	0.206266	0.103133	+	0.994668i	$0.467113\pi$
$972$	0	0
$973$	−54.4285	−1.74490
$974$	0	0
$975$	0.747894	0.0239518
$976$	0	0
$977$	22.7256	0.727055	0.363527	−	0.931584i	$-0.381572\pi$
$977$	22.7256	0.727055	0.363527	+	0.931584i	$0.381572\pi$
$978$	0	0
$979$	5.55732	0.177613
$980$	0	0
$981$	−7.66742	−0.244802
$982$	0	0
$983$	−0.861312	−0.0274716	−0.0137358	−	0.999906i	$-0.504372\pi$
$983$	−0.861312	−0.0274716	−0.0137358	+	0.999906i	$0.504372\pi$
$984$	0	0
$985$	−34.7547	−1.10738
$986$	0	0
$987$	36.2959	1.15531
$988$	0	0
$989$	4.29775	0.136661
$990$	0	0
$991$	0.928715	0.0295016	0.0147508	−	0.999891i	$-0.495305\pi$
$991$	0.928715	0.0295016	0.0147508	+	0.999891i	$0.495305\pi$
$992$	0	0
$993$	−21.3703	−0.678164
$994$	0	0
$995$	−42.0605	−1.33341
$996$	0	0
$997$	27.5025	0.871013	0.435507	−	0.900186i	$-0.356569\pi$
$997$	27.5025	0.871013	0.435507	+	0.900186i	$0.356569\pi$
$998$	0	0
$999$	−26.3456	−0.833538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.32	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.32	✓	50	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.13231 q^{3} -1.95898 q^{5} -2.91195 q^{7} -1.71788 q^{9} +O(q^{10})\)	\(q+1.13231 q^{3} -1.95898 q^{5} -2.91195 q^{7} -1.71788 q^{9} +0.469326 q^{11} -0.568218 q^{13} -2.21816 q^{15} +0.0179401 q^{17} -5.38988 q^{19} -3.29722 q^{21} +1.38567 q^{23} -1.16241 q^{25} -5.34209 q^{27} -1.74404 q^{29} +2.46772 q^{31} +0.531420 q^{33} +5.70443 q^{35} +4.93171 q^{37} -0.643397 q^{39} -0.996312 q^{41} +3.10157 q^{43} +3.36529 q^{45} -11.0080 q^{47} +1.47944 q^{49} +0.0203137 q^{51} +8.32113 q^{53} -0.919398 q^{55} -6.10299 q^{57} +8.95620 q^{59} -5.75029 q^{61} +5.00239 q^{63} +1.11313 q^{65} -4.44499 q^{67} +1.56900 q^{69} +12.6689 q^{71} +6.04979 q^{73} -1.31621 q^{75} -1.36665 q^{77} +6.75973 q^{79} -0.895225 q^{81} +6.31454 q^{83} -0.0351442 q^{85} -1.97479 q^{87} +11.8411 q^{89} +1.65462 q^{91} +2.79422 q^{93} +10.5586 q^{95} -18.4646 q^{97} -0.806247 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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