LMFDB - Embedded newform 6008.2.a.e.1.37

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			1.37
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)

$f(q)$	$=$	$q+1.91672 q^{3} +3.07306 q^{5} -0.591659 q^{7} +0.673807 q^{9} +O(q^{10})$	$q+1.91672 q^{3} +3.07306 q^{5} -0.591659 q^{7} +0.673807 q^{9} +1.99043 q^{11} +3.60968 q^{13} +5.89019 q^{15} +2.69279 q^{17} -1.18936 q^{19} -1.13404 q^{21} -8.54373 q^{23} +4.44370 q^{25} -4.45866 q^{27} +6.20864 q^{29} -7.60191 q^{31} +3.81510 q^{33} -1.81820 q^{35} +10.3205 q^{37} +6.91875 q^{39} +8.86075 q^{41} +6.29558 q^{43} +2.07065 q^{45} +1.22888 q^{47} -6.64994 q^{49} +5.16132 q^{51} +8.63575 q^{53} +6.11672 q^{55} -2.27968 q^{57} +3.53960 q^{59} -7.16899 q^{61} -0.398664 q^{63} +11.0928 q^{65} -0.813857 q^{67} -16.3759 q^{69} +12.1257 q^{71} +10.8866 q^{73} +8.51731 q^{75} -1.17766 q^{77} +13.0985 q^{79} -10.5674 q^{81} +0.290882 q^{83} +8.27510 q^{85} +11.9002 q^{87} -9.51040 q^{89} -2.13570 q^{91} -14.5707 q^{93} -3.65499 q^{95} +8.90446 q^{97} +1.34117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.91672	1.10662	0.553309	−	0.832976i	$-0.313365\pi$
$3$	1.91672	1.10662	0.553309	+	0.832976i	$0.313365\pi$
$4$	0	0
$5$	3.07306	1.37431	0.687157	−	0.726509i	$-0.258858\pi$
$5$	3.07306	1.37431	0.687157	+	0.726509i	$0.258858\pi$
$6$	0	0
$7$	−0.591659	−0.223626	−0.111813	−	0.993729i	$-0.535666\pi$
$7$	−0.591659	−0.223626	−0.111813	+	0.993729i	$0.535666\pi$
$8$	0	0
$9$	0.673807	0.224602
$10$	0	0
$11$	1.99043	0.600138	0.300069	−	0.953918i	$-0.402990\pi$
$11$	1.99043	0.600138	0.300069	+	0.953918i	$0.402990\pi$
$12$	0	0
$13$	3.60968	1.00115	0.500573	−	0.865694i	$-0.333123\pi$
$13$	3.60968	1.00115	0.500573	+	0.865694i	$0.333123\pi$
$14$	0	0
$15$	5.89019	1.52084
$16$	0	0
$17$	2.69279	0.653097	0.326549	−	0.945180i	$-0.394114\pi$
$17$	2.69279	0.653097	0.326549	+	0.945180i	$0.394114\pi$
$18$	0	0
$19$	−1.18936	−0.272859	−0.136430	−	0.990650i	$-0.543563\pi$
$19$	−1.18936	−0.272859	−0.136430	+	0.990650i	$0.543563\pi$
$20$	0	0
$21$	−1.13404	−0.247469
$22$	0	0
$23$	−8.54373	−1.78149	−0.890745	−	0.454503i	$-0.849817\pi$
$23$	−8.54373	−1.78149	−0.890745	+	0.454503i	$0.849817\pi$
$24$	0	0
$25$	4.44370	0.888739
$26$	0	0
$27$	−4.45866	−0.858069
$28$	0	0
$29$	6.20864	1.15292	0.576458	−	0.817127i	$-0.304435\pi$
$29$	6.20864	1.15292	0.576458	+	0.817127i	$0.304435\pi$
$30$	0	0
$31$	−7.60191	−1.36534	−0.682671	−	0.730725i	$-0.739182\pi$
$31$	−7.60191	−1.36534	−0.682671	+	0.730725i	$0.739182\pi$
$32$	0	0
$33$	3.81510	0.664123
$34$	0	0
$35$	−1.81820	−0.307332
$36$	0	0
$37$	10.3205	1.69667	0.848337	−	0.529456i	$-0.177604\pi$
$37$	10.3205	1.69667	0.848337	+	0.529456i	$0.177604\pi$
$38$	0	0
$39$	6.91875	1.10789
$40$	0	0
$41$	8.86075	1.38382	0.691908	−	0.721986i	$-0.256770\pi$
$41$	8.86075	1.38382	0.691908	+	0.721986i	$0.256770\pi$
$42$	0	0
$43$	6.29558	0.960067	0.480033	−	0.877250i	$-0.340625\pi$
$43$	6.29558	0.960067	0.480033	+	0.877250i	$0.340625\pi$
$44$	0	0
$45$	2.07065	0.308674
$46$	0	0
$47$	1.22888	0.179251	0.0896254	−	0.995976i	$-0.471433\pi$
$47$	1.22888	0.179251	0.0896254	+	0.995976i	$0.471433\pi$
$48$	0	0
$49$	−6.64994	−0.949991
$50$	0	0
$51$	5.16132	0.722729
$52$	0	0
$53$	8.63575	1.18621	0.593106	−	0.805124i	$-0.297901\pi$
$53$	8.63575	1.18621	0.593106	+	0.805124i	$0.297901\pi$
$54$	0	0
$55$	6.11672	0.824778
$56$	0	0
$57$	−2.27968	−0.301951
$58$	0	0
$59$	3.53960	0.460816	0.230408	−	0.973094i	$-0.425994\pi$
$59$	3.53960	0.460816	0.230408	+	0.973094i	$0.425994\pi$
$60$	0	0
$61$	−7.16899	−0.917895	−0.458947	−	0.888463i	$-0.651773\pi$
$61$	−7.16899	−0.917895	−0.458947	+	0.888463i	$0.651773\pi$
$62$	0	0
$63$	−0.398664	−0.0502269
$64$	0	0
$65$	11.0928	1.37589
$66$	0	0
$67$	−0.813857	−0.0994285	−0.0497142	−	0.998763i	$-0.515831\pi$
$67$	−0.813857	−0.0994285	−0.0497142	+	0.998763i	$0.515831\pi$
$68$	0	0
$69$	−16.3759	−1.97143
$70$	0	0
$71$	12.1257	1.43905	0.719526	−	0.694466i	$-0.244359\pi$
$71$	12.1257	1.43905	0.719526	+	0.694466i	$0.244359\pi$
$72$	0	0
$73$	10.8866	1.27418	0.637089	−	0.770790i	$-0.280138\pi$
$73$	10.8866	1.27418	0.637089	+	0.770790i	$0.280138\pi$
$74$	0	0
$75$	8.51731	0.983494
$76$	0	0
$77$	−1.17766	−0.134206
$78$	0	0
$79$	13.0985	1.47370	0.736849	−	0.676057i	$-0.236313\pi$
$79$	13.0985	1.47370	0.736849	+	0.676057i	$0.236313\pi$
$80$	0	0
$81$	−10.5674	−1.17416
$82$	0	0
$83$	0.290882	0.0319284	0.0159642	−	0.999873i	$-0.494918\pi$
$83$	0.290882	0.0319284	0.0159642	+	0.999873i	$0.494918\pi$
$84$	0	0
$85$	8.27510	0.897561
$86$	0	0
$87$	11.9002	1.27584
$88$	0	0
$89$	−9.51040	−1.00810	−0.504050	−	0.863674i	$-0.668157\pi$
$89$	−9.51040	−1.00810	−0.504050	+	0.863674i	$0.668157\pi$
$90$	0	0
$91$	−2.13570	−0.223882
$92$	0	0
$93$	−14.5707	−1.51091
$94$	0	0
$95$	−3.65499	−0.374994
$96$	0	0
$97$	8.90446	0.904111	0.452055	−	0.891990i	$-0.350691\pi$
$97$	8.90446	0.904111	0.452055	+	0.891990i	$0.350691\pi$
$98$	0	0
$99$	1.34117	0.134792
$100$	0	0
$101$	−4.81662	−0.479272	−0.239636	−	0.970863i	$-0.577028\pi$
$101$	−4.81662	−0.479272	−0.239636	+	0.970863i	$0.577028\pi$
$102$	0	0
$103$	−6.11547	−0.602575	−0.301287	−	0.953533i	$-0.597416\pi$
$103$	−6.11547	−0.602575	−0.301287	+	0.953533i	$0.597416\pi$
$104$	0	0
$105$	−3.48498	−0.340099
$106$	0	0
$107$	−3.45962	−0.334454	−0.167227	−	0.985918i	$-0.553481\pi$
$107$	−3.45962	−0.334454	−0.167227	+	0.985918i	$0.553481\pi$
$108$	0	0
$109$	3.12839	0.299646	0.149823	−	0.988713i	$-0.452130\pi$
$109$	3.12839	0.299646	0.149823	+	0.988713i	$0.452130\pi$
$110$	0	0
$111$	19.7814	1.87757
$112$	0	0
$113$	1.01074	0.0950820	0.0475410	−	0.998869i	$-0.484862\pi$
$113$	1.01074	0.0950820	0.0475410	+	0.998869i	$0.484862\pi$
$114$	0	0
$115$	−26.2554	−2.44833
$116$	0	0
$117$	2.43223	0.224860
$118$	0	0
$119$	−1.59321	−0.146050
$120$	0	0
$121$	−7.03818	−0.639835
$122$	0	0
$123$	16.9835	1.53135
$124$	0	0
$125$	−1.70955	−0.152907
$126$	0	0
$127$	−7.58149	−0.672748	−0.336374	−	0.941728i	$-0.609201\pi$
$127$	−7.58149	−0.672748	−0.336374	+	0.941728i	$0.609201\pi$
$128$	0	0
$129$	12.0668	1.06243
$130$	0	0
$131$	7.81531	0.682827	0.341414	−	0.939913i	$-0.389094\pi$
$131$	7.81531	0.682827	0.341414	+	0.939913i	$0.389094\pi$
$132$	0	0
$133$	0.703698	0.0610184
$134$	0	0
$135$	−13.7017	−1.17926
$136$	0	0
$137$	−15.3633	−1.31258	−0.656289	−	0.754510i	$-0.727875\pi$
$137$	−15.3633	−1.31258	−0.656289	+	0.754510i	$0.727875\pi$
$138$	0	0
$139$	−2.53672	−0.215162	−0.107581	−	0.994196i	$-0.534310\pi$
$139$	−2.53672	−0.215162	−0.107581	+	0.994196i	$0.534310\pi$
$140$	0	0
$141$	2.35542	0.198362
$142$	0	0
$143$	7.18483	0.600826
$144$	0	0
$145$	19.0795	1.58447
$146$	0	0
$147$	−12.7461	−1.05128
$148$	0	0
$149$	11.7737	0.964538	0.482269	−	0.876023i	$-0.339813\pi$
$149$	11.7737	0.964538	0.482269	+	0.876023i	$0.339813\pi$
$150$	0	0
$151$	−1.16607	−0.0948934	−0.0474467	−	0.998874i	$-0.515108\pi$
$151$	−1.16607	−0.0948934	−0.0474467	+	0.998874i	$0.515108\pi$
$152$	0	0
$153$	1.81442	0.146687
$154$	0	0
$155$	−23.3611	−1.87641
$156$	0	0
$157$	−7.40450	−0.590943	−0.295472	−	0.955352i	$-0.595477\pi$
$157$	−7.40450	−0.590943	−0.295472	+	0.955352i	$0.595477\pi$
$158$	0	0
$159$	16.5523	1.31268
$160$	0	0
$161$	5.05497	0.398388
$162$	0	0
$163$	−12.6435	−0.990313	−0.495157	−	0.868804i	$-0.664889\pi$
$163$	−12.6435	−0.990313	−0.495157	+	0.868804i	$0.664889\pi$
$164$	0	0
$165$	11.7240	0.912714
$166$	0	0
$167$	−2.98639	−0.231094	−0.115547	−	0.993302i	$-0.536862\pi$
$167$	−2.98639	−0.231094	−0.115547	+	0.993302i	$0.536862\pi$
$168$	0	0
$169$	0.0298206	0.00229389
$170$	0	0
$171$	−0.801403	−0.0612848
$172$	0	0
$173$	−10.3330	−0.785600	−0.392800	−	0.919624i	$-0.628494\pi$
$173$	−10.3330	−0.785600	−0.392800	+	0.919624i	$0.628494\pi$
$174$	0	0
$175$	−2.62915	−0.198745
$176$	0	0
$177$	6.78441	0.509948
$178$	0	0
$179$	4.34200	0.324536	0.162268	−	0.986747i	$-0.448119\pi$
$179$	4.34200	0.324536	0.162268	+	0.986747i	$0.448119\pi$
$180$	0	0
$181$	0.818680	0.0608520	0.0304260	−	0.999537i	$-0.490314\pi$
$181$	0.818680	0.0608520	0.0304260	+	0.999537i	$0.490314\pi$
$182$	0	0
$183$	−13.7409	−1.01576
$184$	0	0
$185$	31.7154	2.33176
$186$	0	0
$187$	5.35981	0.391948
$188$	0	0
$189$	2.63800	0.191886
$190$	0	0
$191$	0.682998	0.0494200	0.0247100	−	0.999695i	$-0.492134\pi$
$191$	0.682998	0.0494200	0.0247100	+	0.999695i	$0.492134\pi$
$192$	0	0
$193$	−5.59909	−0.403031	−0.201516	−	0.979485i	$-0.564587\pi$
$193$	−5.59909	−0.403031	−0.201516	+	0.979485i	$0.564587\pi$
$194$	0	0
$195$	21.2617	1.52258
$196$	0	0
$197$	−13.9385	−0.993076	−0.496538	−	0.868015i	$-0.665396\pi$
$197$	−13.9385	−0.993076	−0.496538	+	0.868015i	$0.665396\pi$
$198$	0	0
$199$	3.68243	0.261041	0.130520	−	0.991446i	$-0.458335\pi$
$199$	3.68243	0.261041	0.130520	+	0.991446i	$0.458335\pi$
$200$	0	0
$201$	−1.55993	−0.110029
$202$	0	0
$203$	−3.67340	−0.257822
$204$	0	0
$205$	27.2296	1.90180
$206$	0	0
$207$	−5.75683	−0.400127
$208$	0	0
$209$	−2.36735	−0.163753
$210$	0	0
$211$	10.5819	0.728489	0.364244	−	0.931303i	$-0.381327\pi$
$211$	10.5819	0.728489	0.364244	+	0.931303i	$0.381327\pi$
$212$	0	0
$213$	23.2415	1.59248
$214$	0	0
$215$	19.3467	1.31943
$216$	0	0
$217$	4.49774	0.305326
$218$	0	0
$219$	20.8665	1.41003
$220$	0	0
$221$	9.72012	0.653846
$222$	0	0
$223$	−26.1852	−1.75349	−0.876744	−	0.480957i	$-0.840289\pi$
$223$	−26.1852	−1.75349	−0.876744	+	0.480957i	$0.840289\pi$
$224$	0	0
$225$	2.99419	0.199613
$226$	0	0
$227$	6.72989	0.446678	0.223339	−	0.974741i	$-0.428304\pi$
$227$	6.72989	0.446678	0.223339	+	0.974741i	$0.428304\pi$
$228$	0	0
$229$	11.2832	0.745616	0.372808	−	0.927909i	$-0.378395\pi$
$229$	11.2832	0.745616	0.372808	+	0.927909i	$0.378395\pi$
$230$	0	0
$231$	−2.25724	−0.148515
$232$	0	0
$233$	26.1165	1.71095	0.855476	−	0.517843i	$-0.173265\pi$
$233$	26.1165	1.71095	0.855476	+	0.517843i	$0.173265\pi$
$234$	0	0
$235$	3.77643	0.246347
$236$	0	0
$237$	25.1062	1.63082
$238$	0	0
$239$	−8.96766	−0.580070	−0.290035	−	0.957016i	$-0.593667\pi$
$239$	−8.96766	−0.580070	−0.290035	+	0.957016i	$0.593667\pi$
$240$	0	0
$241$	8.31270	0.535468	0.267734	−	0.963493i	$-0.413725\pi$
$241$	8.31270	0.535468	0.267734	+	0.963493i	$0.413725\pi$
$242$	0	0
$243$	−6.87877	−0.441273
$244$	0	0
$245$	−20.4357	−1.30559
$246$	0	0
$247$	−4.29323	−0.273172
$248$	0	0
$249$	0.557538	0.0353325
$250$	0	0
$251$	−0.809235	−0.0510784	−0.0255392	−	0.999674i	$-0.508130\pi$
$251$	−0.809235	−0.0510784	−0.0255392	+	0.999674i	$0.508130\pi$
$252$	0	0
$253$	−17.0057	−1.06914
$254$	0	0
$255$	15.8610	0.993256
$256$	0	0
$257$	5.50371	0.343312	0.171656	−	0.985157i	$-0.445088\pi$
$257$	5.50371	0.343312	0.171656	+	0.985157i	$0.445088\pi$
$258$	0	0
$259$	−6.10620	−0.379421
$260$	0	0
$261$	4.18343	0.258948
$262$	0	0
$263$	3.58265	0.220916	0.110458	−	0.993881i	$-0.464768\pi$
$263$	3.58265	0.220916	0.110458	+	0.993881i	$0.464768\pi$
$264$	0	0
$265$	26.5382	1.63023
$266$	0	0
$267$	−18.2288	−1.11558
$268$	0	0
$269$	−4.50112	−0.274438	−0.137219	−	0.990541i	$-0.543816\pi$
$269$	−4.50112	−0.274438	−0.137219	+	0.990541i	$0.543816\pi$
$270$	0	0
$271$	12.8762	0.782173	0.391087	−	0.920354i	$-0.372099\pi$
$271$	12.8762	0.782173	0.391087	+	0.920354i	$0.372099\pi$
$272$	0	0
$273$	−4.09354	−0.247752
$274$	0	0
$275$	8.84487	0.533366
$276$	0	0
$277$	−3.60991	−0.216899	−0.108449	−	0.994102i	$-0.534589\pi$
$277$	−3.60991	−0.216899	−0.108449	+	0.994102i	$0.534589\pi$
$278$	0	0
$279$	−5.12222	−0.306659
$280$	0	0
$281$	−8.92812	−0.532607	−0.266304	−	0.963889i	$-0.585802\pi$
$281$	−8.92812	−0.532607	−0.266304	+	0.963889i	$0.585802\pi$
$282$	0	0
$283$	6.55386	0.389587	0.194793	−	0.980844i	$-0.437596\pi$
$283$	6.55386	0.389587	0.194793	+	0.980844i	$0.437596\pi$
$284$	0	0
$285$	−7.00558	−0.414975
$286$	0	0
$287$	−5.24254	−0.309457
$288$	0	0
$289$	−9.74889	−0.573464
$290$	0	0
$291$	17.0673	1.00050
$292$	0	0
$293$	−6.70738	−0.391849	−0.195925	−	0.980619i	$-0.562771\pi$
$293$	−6.70738	−0.391849	−0.195925	+	0.980619i	$0.562771\pi$
$294$	0	0
$295$	10.8774	0.633307
$296$	0	0
$297$	−8.87465	−0.514959
$298$	0	0
$299$	−30.8402	−1.78353
$300$	0	0
$301$	−3.72483	−0.214696
$302$	0	0
$303$	−9.23211	−0.530371
$304$	0	0
$305$	−22.0307	−1.26148
$306$	0	0
$307$	−20.3998	−1.16428	−0.582140	−	0.813089i	$-0.697784\pi$
$307$	−20.3998	−1.16428	−0.582140	+	0.813089i	$0.697784\pi$
$308$	0	0
$309$	−11.7216	−0.666820
$310$	0	0
$311$	−4.96079	−0.281301	−0.140650	−	0.990059i	$-0.544919\pi$
$311$	−4.96079	−0.281301	−0.140650	+	0.990059i	$0.544919\pi$
$312$	0	0
$313$	−0.291763	−0.0164914	−0.00824572	−	0.999966i	$-0.502625\pi$
$313$	−0.291763	−0.0164914	−0.00824572	+	0.999966i	$0.502625\pi$
$314$	0	0
$315$	−1.22512	−0.0690276
$316$	0	0
$317$	25.4045	1.42686	0.713428	−	0.700728i	$-0.247142\pi$
$317$	25.4045	1.42686	0.713428	+	0.700728i	$0.247142\pi$
$318$	0	0
$319$	12.3579	0.691908
$320$	0	0
$321$	−6.63111	−0.370112
$322$	0	0
$323$	−3.20271	−0.178203
$324$	0	0
$325$	16.0403	0.889758
$326$	0	0
$327$	5.99624	0.331593
$328$	0	0
$329$	−0.727079	−0.0400852
$330$	0	0
$331$	−16.1956	−0.890192	−0.445096	−	0.895483i	$-0.646830\pi$
$331$	−16.1956	−0.890192	−0.445096	+	0.895483i	$0.646830\pi$
$332$	0	0
$333$	6.95401	0.381077
$334$	0	0
$335$	−2.50103	−0.136646
$336$	0	0
$337$	−29.3642	−1.59957	−0.799785	−	0.600287i	$-0.795053\pi$
$337$	−29.3642	−1.59957	−0.799785	+	0.600287i	$0.795053\pi$
$338$	0	0
$339$	1.93729	0.105219
$340$	0	0
$341$	−15.1311	−0.819394
$342$	0	0
$343$	8.07611	0.436069
$344$	0	0
$345$	−50.3242	−2.70936
$346$	0	0
$347$	−11.3409	−0.608812	−0.304406	−	0.952542i	$-0.598458\pi$
$347$	−11.3409	−0.608812	−0.304406	+	0.952542i	$0.598458\pi$
$348$	0	0
$349$	−20.0166	−1.07147	−0.535733	−	0.844387i	$-0.679965\pi$
$349$	−20.0166	−1.07147	−0.535733	+	0.844387i	$0.679965\pi$
$350$	0	0
$351$	−16.0943	−0.859052
$352$	0	0
$353$	1.57600	0.0838819	0.0419409	−	0.999120i	$-0.486646\pi$
$353$	1.57600	0.0838819	0.0419409	+	0.999120i	$0.486646\pi$
$354$	0	0
$355$	37.2629	1.97771
$356$	0	0
$357$	−3.05374	−0.161621
$358$	0	0
$359$	−17.9274	−0.946172	−0.473086	−	0.881016i	$-0.656860\pi$
$359$	−17.9274	−0.946172	−0.473086	+	0.881016i	$0.656860\pi$
$360$	0	0
$361$	−17.5854	−0.925548
$362$	0	0
$363$	−13.4902	−0.708052
$364$	0	0
$365$	33.4551	1.75112
$366$	0	0
$367$	29.6454	1.54748	0.773738	−	0.633506i	$-0.218385\pi$
$367$	29.6454	1.54748	0.773738	+	0.633506i	$0.218385\pi$
$368$	0	0
$369$	5.97043	0.310808
$370$	0	0
$371$	−5.10942	−0.265268
$372$	0	0
$373$	10.6698	0.552459	0.276230	−	0.961092i	$-0.410915\pi$
$373$	10.6698	0.552459	0.276230	+	0.961092i	$0.410915\pi$
$374$	0	0
$375$	−3.27673	−0.169210
$376$	0	0
$377$	22.4112	1.15424
$378$	0	0
$379$	−20.7572	−1.06622	−0.533112	−	0.846045i	$-0.678978\pi$
$379$	−20.7572	−1.06622	−0.533112	+	0.846045i	$0.678978\pi$
$380$	0	0
$381$	−14.5316	−0.744475
$382$	0	0
$383$	10.0399	0.513013	0.256506	−	0.966543i	$-0.417429\pi$
$383$	10.0399	0.513013	0.256506	+	0.966543i	$0.417429\pi$
$384$	0	0
$385$	−3.61901	−0.184442
$386$	0	0
$387$	4.24201	0.215633
$388$	0	0
$389$	−10.7264	−0.543852	−0.271926	−	0.962318i	$-0.587661\pi$
$389$	−10.7264	−0.543852	−0.271926	+	0.962318i	$0.587661\pi$
$390$	0	0
$391$	−23.0065	−1.16349
$392$	0	0
$393$	14.9798	0.755628
$394$	0	0
$395$	40.2525	2.02532
$396$	0	0
$397$	−28.9918	−1.45506	−0.727530	−	0.686076i	$-0.759332\pi$
$397$	−28.9918	−1.45506	−0.727530	+	0.686076i	$0.759332\pi$
$398$	0	0
$399$	1.34879	0.0675240
$400$	0	0
$401$	8.66132	0.432526	0.216263	−	0.976335i	$-0.430613\pi$
$401$	8.66132	0.432526	0.216263	+	0.976335i	$0.430613\pi$
$402$	0	0
$403$	−27.4405	−1.36691
$404$	0	0
$405$	−32.4743	−1.61366
$406$	0	0
$407$	20.5422	1.01824
$408$	0	0
$409$	16.4025	0.811050	0.405525	−	0.914084i	$-0.367089\pi$
$409$	16.4025	0.811050	0.405525	+	0.914084i	$0.367089\pi$
$410$	0	0
$411$	−29.4472	−1.45252
$412$	0	0
$413$	−2.09423	−0.103051
$414$	0	0
$415$	0.893897	0.0438797
$416$	0	0
$417$	−4.86217	−0.238102
$418$	0	0
$419$	11.5441	0.563964	0.281982	−	0.959420i	$-0.409008\pi$
$419$	11.5441	0.563964	0.281982	+	0.959420i	$0.409008\pi$
$420$	0	0
$421$	−29.0480	−1.41571	−0.707855	−	0.706357i	$-0.750337\pi$
$421$	−29.0480	−1.41571	−0.707855	+	0.706357i	$0.750337\pi$
$422$	0	0
$423$	0.828029	0.0402602
$424$	0	0
$425$	11.9659	0.580433
$426$	0	0
$427$	4.24160	0.205265
$428$	0	0
$429$	13.7713	0.664884
$430$	0	0
$431$	−10.5338	−0.507394	−0.253697	−	0.967284i	$-0.581647\pi$
$431$	−10.5338	−0.507394	−0.253697	+	0.967284i	$0.581647\pi$
$432$	0	0
$433$	20.0401	0.963066	0.481533	−	0.876428i	$-0.340080\pi$
$433$	20.0401	0.963066	0.481533	+	0.876428i	$0.340080\pi$
$434$	0	0
$435$	36.5701	1.75340
$436$	0	0
$437$	10.1616	0.486096
$438$	0	0
$439$	5.26237	0.251159	0.125580	−	0.992084i	$-0.459921\pi$
$439$	5.26237	0.251159	0.125580	+	0.992084i	$0.459921\pi$
$440$	0	0
$441$	−4.48078	−0.213370
$442$	0	0
$443$	−14.7666	−0.701582	−0.350791	−	0.936454i	$-0.614087\pi$
$443$	−14.7666	−0.701582	−0.350791	+	0.936454i	$0.614087\pi$
$444$	0	0
$445$	−29.2260	−1.38545
$446$	0	0
$447$	22.5668	1.06737
$448$	0	0
$449$	−40.7260	−1.92198	−0.960990	−	0.276583i	$-0.910798\pi$
$449$	−40.7260	−1.92198	−0.960990	+	0.276583i	$0.910798\pi$
$450$	0	0
$451$	17.6367	0.830480
$452$	0	0
$453$	−2.23503	−0.105011
$454$	0	0
$455$	−6.56314	−0.307685
$456$	0	0
$457$	−11.1750	−0.522746	−0.261373	−	0.965238i	$-0.584175\pi$
$457$	−11.1750	−0.522746	−0.261373	+	0.965238i	$0.584175\pi$
$458$	0	0
$459$	−12.0062	−0.560402
$460$	0	0
$461$	−4.71844	−0.219760	−0.109880	−	0.993945i	$-0.535047\pi$
$461$	−4.71844	−0.219760	−0.109880	+	0.993945i	$0.535047\pi$
$462$	0	0
$463$	16.6542	0.773987	0.386993	−	0.922082i	$-0.373514\pi$
$463$	16.6542	0.773987	0.386993	+	0.922082i	$0.373514\pi$
$464$	0	0
$465$	−44.7767	−2.07647
$466$	0	0
$467$	−25.7421	−1.19120	−0.595600	−	0.803281i	$-0.703086\pi$
$467$	−25.7421	−1.19120	−0.595600	+	0.803281i	$0.703086\pi$
$468$	0	0
$469$	0.481526	0.0222348
$470$	0	0
$471$	−14.1923	−0.653948
$472$	0	0
$473$	12.5309	0.576172
$474$	0	0
$475$	−5.28518	−0.242501
$476$	0	0
$477$	5.81883	0.266426
$478$	0	0
$479$	4.74722	0.216906	0.108453	−	0.994102i	$-0.465410\pi$
$479$	4.74722	0.216906	0.108453	+	0.994102i	$0.465410\pi$
$480$	0	0
$481$	37.2536	1.69862
$482$	0	0
$483$	9.68896	0.440863
$484$	0	0
$485$	27.3639	1.24253
$486$	0	0
$487$	−0.0262545	−0.00118970	−0.000594852	−	1.00000i	$-0.500189\pi$
$487$	−0.0262545	−0.00118970	−0.000594852		1.00000i	$0.500189\pi$
$488$	0	0
$489$	−24.2340	−1.09590
$490$	0	0
$491$	39.3877	1.77754	0.888772	−	0.458350i	$-0.151559\pi$
$491$	39.3877	1.77754	0.888772	+	0.458350i	$0.151559\pi$
$492$	0	0
$493$	16.7186	0.752966
$494$	0	0
$495$	4.12149	0.185247
$496$	0	0
$497$	−7.17426	−0.321809
$498$	0	0
$499$	−9.81624	−0.439435	−0.219718	−	0.975563i	$-0.570514\pi$
$499$	−9.81624	−0.439435	−0.219718	+	0.975563i	$0.570514\pi$
$500$	0	0
$501$	−5.72406	−0.255732
$502$	0	0
$503$	34.9463	1.55818	0.779088	−	0.626914i	$-0.215682\pi$
$503$	34.9463	1.55818	0.779088	+	0.626914i	$0.215682\pi$
$504$	0	0
$505$	−14.8018	−0.658670
$506$	0	0
$507$	0.0571577	0.00253846
$508$	0	0
$509$	−13.0838	−0.579929	−0.289965	−	0.957037i	$-0.593644\pi$
$509$	−13.0838	−0.579929	−0.289965	+	0.957037i	$0.593644\pi$
$510$	0	0
$511$	−6.44114	−0.284939
$512$	0	0
$513$	5.30297	0.234132
$514$	0	0
$515$	−18.7932	−0.828127
$516$	0	0
$517$	2.44601	0.107575
$518$	0	0
$519$	−19.8054	−0.869359
$520$	0	0
$521$	42.4840	1.86126	0.930630	−	0.365963i	$-0.119260\pi$
$521$	42.4840	1.86126	0.930630	+	0.365963i	$0.119260\pi$
$522$	0	0
$523$	−29.4564	−1.28804	−0.644018	−	0.765010i	$-0.722734\pi$
$523$	−29.4564	−1.28804	−0.644018	+	0.765010i	$0.722734\pi$
$524$	0	0
$525$	−5.03934	−0.219935
$526$	0	0
$527$	−20.4703	−0.891702
$528$	0	0
$529$	49.9953	2.17371
$530$	0	0
$531$	2.38501	0.103500
$532$	0	0
$533$	31.9845	1.38540
$534$	0	0
$535$	−10.6316	−0.459644
$536$	0	0
$537$	8.32239	0.359138
$538$	0	0
$539$	−13.2363	−0.570126
$540$	0	0
$541$	−16.2781	−0.699849	−0.349925	−	0.936778i	$-0.613793\pi$
$541$	−16.2781	−0.699849	−0.349925	+	0.936778i	$0.613793\pi$
$542$	0	0
$543$	1.56918	0.0673399
$544$	0	0
$545$	9.61373	0.411807
$546$	0	0
$547$	−44.1323	−1.88696	−0.943480	−	0.331428i	$-0.892470\pi$
$547$	−44.1323	−1.88696	−0.943480	+	0.331428i	$0.892470\pi$
$548$	0	0
$549$	−4.83052	−0.206161
$550$	0	0
$551$	−7.38434	−0.314583
$552$	0	0
$553$	−7.74986	−0.329557
$554$	0	0
$555$	60.7895	2.58037
$556$	0	0
$557$	32.5372	1.37864	0.689322	−	0.724455i	$-0.257909\pi$
$557$	32.5372	1.37864	0.689322	+	0.724455i	$0.257909\pi$
$558$	0	0
$559$	22.7250	0.961167
$560$	0	0
$561$	10.2732	0.433737
$562$	0	0
$563$	−4.45639	−0.187814	−0.0939072	−	0.995581i	$-0.529936\pi$
$563$	−4.45639	−0.187814	−0.0939072	+	0.995581i	$0.529936\pi$
$564$	0	0
$565$	3.10605	0.130672
$566$	0	0
$567$	6.25230	0.262572
$568$	0	0
$569$	17.8170	0.746928	0.373464	−	0.927645i	$-0.378170\pi$
$569$	17.8170	0.746928	0.373464	+	0.927645i	$0.378170\pi$
$570$	0	0
$571$	28.5784	1.19597	0.597984	−	0.801508i	$-0.295968\pi$
$571$	28.5784	1.19597	0.597984	+	0.801508i	$0.295968\pi$
$572$	0	0
$573$	1.30911	0.0546891
$574$	0	0
$575$	−37.9657	−1.58328
$576$	0	0
$577$	−10.9615	−0.456331	−0.228166	−	0.973622i	$-0.573273\pi$
$577$	−10.9615	−0.456331	−0.228166	+	0.973622i	$0.573273\pi$
$578$	0	0
$579$	−10.7319	−0.446002
$580$	0	0
$581$	−0.172103	−0.00714002
$582$	0	0
$583$	17.1889	0.711891
$584$	0	0
$585$	7.47439	0.309028
$586$	0	0
$587$	−20.8966	−0.862494	−0.431247	−	0.902234i	$-0.641926\pi$
$587$	−20.8966	−0.862494	−0.431247	+	0.902234i	$0.641926\pi$
$588$	0	0
$589$	9.04144	0.372546
$590$	0	0
$591$	−26.7161	−1.09896
$592$	0	0
$593$	5.40031	0.221764	0.110882	−	0.993834i	$-0.464632\pi$
$593$	5.40031	0.221764	0.110882	+	0.993834i	$0.464632\pi$
$594$	0	0
$595$	−4.89604	−0.200718
$596$	0	0
$597$	7.05819	0.288872
$598$	0	0
$599$	12.3782	0.505758	0.252879	−	0.967498i	$-0.418623\pi$
$599$	12.3782	0.505758	0.252879	+	0.967498i	$0.418623\pi$
$600$	0	0
$601$	11.2084	0.457202	0.228601	−	0.973520i	$-0.426585\pi$
$601$	11.2084	0.457202	0.228601	+	0.973520i	$0.426585\pi$
$602$	0	0
$603$	−0.548383	−0.0223319
$604$	0	0
$605$	−21.6288	−0.879334
$606$	0	0
$607$	0.746563	0.0303020	0.0151510	−	0.999885i	$-0.495177\pi$
$607$	0.746563	0.0303020	0.0151510	+	0.999885i	$0.495177\pi$
$608$	0	0
$609$	−7.04086	−0.285310
$610$	0	0
$611$	4.43588	0.179456
$612$	0	0
$613$	4.93806	0.199447	0.0997233	−	0.995015i	$-0.468204\pi$
$613$	4.93806	0.199447	0.0997233	+	0.995015i	$0.468204\pi$
$614$	0	0
$615$	52.1915	2.10456
$616$	0	0
$617$	45.7806	1.84306	0.921530	−	0.388308i	$-0.126940\pi$
$617$	45.7806	1.84306	0.921530	+	0.388308i	$0.126940\pi$
$618$	0	0
$619$	−12.0362	−0.483776	−0.241888	−	0.970304i	$-0.577767\pi$
$619$	−12.0362	−0.483776	−0.241888	+	0.970304i	$0.577767\pi$
$620$	0	0
$621$	38.0935	1.52864
$622$	0	0
$623$	5.62691	0.225437
$624$	0	0
$625$	−27.4720	−1.09888
$626$	0	0
$627$	−4.53754	−0.181212
$628$	0	0
$629$	27.7908	1.10809
$630$	0	0
$631$	1.36801	0.0544596	0.0272298	−	0.999629i	$-0.491331\pi$
$631$	1.36801	0.0544596	0.0272298	+	0.999629i	$0.491331\pi$
$632$	0	0
$633$	20.2825	0.806159
$634$	0	0
$635$	−23.2984	−0.924568
$636$	0	0
$637$	−24.0042	−0.951080
$638$	0	0
$639$	8.17036	0.323215
$640$	0	0
$641$	−23.1653	−0.914975	−0.457487	−	0.889216i	$-0.651250\pi$
$641$	−23.1653	−0.914975	−0.457487	+	0.889216i	$0.651250\pi$
$642$	0	0
$643$	−14.6533	−0.577869	−0.288935	−	0.957349i	$-0.593301\pi$
$643$	−14.6533	−0.577869	−0.288935	+	0.957349i	$0.593301\pi$
$644$	0	0
$645$	37.0821	1.46011
$646$	0	0
$647$	39.4759	1.55196	0.775979	−	0.630758i	$-0.217256\pi$
$647$	39.4759	1.55196	0.775979	+	0.630758i	$0.217256\pi$
$648$	0	0
$649$	7.04533	0.276553
$650$	0	0
$651$	8.62089	0.337879
$652$	0	0
$653$	2.67703	0.104760	0.0523800	−	0.998627i	$-0.483319\pi$
$653$	2.67703	0.104760	0.0523800	+	0.998627i	$0.483319\pi$
$654$	0	0
$655$	24.0169	0.938419
$656$	0	0
$657$	7.33545	0.286183
$658$	0	0
$659$	−46.2181	−1.80040	−0.900202	−	0.435473i	$-0.856581\pi$
$659$	−46.2181	−1.80040	−0.900202	+	0.435473i	$0.856581\pi$
$660$	0	0
$661$	1.20188	0.0467477	0.0233739	−	0.999727i	$-0.492559\pi$
$661$	1.20188	0.0467477	0.0233739	+	0.999727i	$0.492559\pi$
$662$	0	0
$663$	18.6307	0.723557
$664$	0	0
$665$	2.16251	0.0838584
$666$	0	0
$667$	−53.0449	−2.05391
$668$	0	0
$669$	−50.1896	−1.94044
$670$	0	0
$671$	−14.2694	−0.550863
$672$	0	0
$673$	−31.5411	−1.21582	−0.607909	−	0.794007i	$-0.707992\pi$
$673$	−31.5411	−1.21582	−0.607909	+	0.794007i	$0.707992\pi$
$674$	0	0
$675$	−19.8129	−0.762599
$676$	0	0
$677$	1.50118	0.0576950	0.0288475	−	0.999584i	$-0.490816\pi$
$677$	1.50118	0.0576950	0.0288475	+	0.999584i	$0.490816\pi$
$678$	0	0
$679$	−5.26840	−0.202183
$680$	0	0
$681$	12.8993	0.494302
$682$	0	0
$683$	−38.2646	−1.46415	−0.732077	−	0.681221i	$-0.761449\pi$
$683$	−38.2646	−1.46415	−0.732077	+	0.681221i	$0.761449\pi$
$684$	0	0
$685$	−47.2124	−1.80389
$686$	0	0
$687$	21.6267	0.825112
$688$	0	0
$689$	31.1723	1.18757
$690$	0	0
$691$	−32.6983	−1.24390	−0.621952	−	0.783056i	$-0.713660\pi$
$691$	−32.6983	−1.24390	−0.621952	+	0.783056i	$0.713660\pi$
$692$	0	0
$693$	−0.793514	−0.0301431
$694$	0	0
$695$	−7.79548	−0.295700
$696$	0	0
$697$	23.8601	0.903766
$698$	0	0
$699$	50.0580	1.89337
$700$	0	0
$701$	−17.5244	−0.661886	−0.330943	−	0.943651i	$-0.607367\pi$
$701$	−17.5244	−0.661886	−0.330943	+	0.943651i	$0.607367\pi$
$702$	0	0
$703$	−12.2748	−0.462953
$704$	0	0
$705$	7.23834	0.272612
$706$	0	0
$707$	2.84980	0.107178
$708$	0	0
$709$	34.0779	1.27982	0.639912	−	0.768448i	$-0.278971\pi$
$709$	34.0779	1.27982	0.639912	+	0.768448i	$0.278971\pi$
$710$	0	0
$711$	8.82588	0.330996
$712$	0	0
$713$	64.9486	2.43235
$714$	0	0
$715$	22.0794	0.825723
$716$	0	0
$717$	−17.1885	−0.641916
$718$	0	0
$719$	29.1090	1.08558	0.542791	−	0.839868i	$-0.317368\pi$
$719$	29.1090	1.08558	0.542791	+	0.839868i	$0.317368\pi$
$720$	0	0
$721$	3.61827	0.134751
$722$	0	0
$723$	15.9331	0.592558
$724$	0	0
$725$	27.5893	1.02464
$726$	0	0
$727$	26.9888	1.00096	0.500480	−	0.865748i	$-0.333157\pi$
$727$	26.9888	1.00096	0.500480	+	0.865748i	$0.333157\pi$
$728$	0	0
$729$	18.5176	0.685835
$730$	0	0
$731$	16.9527	0.627017
$732$	0	0
$733$	6.76159	0.249745	0.124873	−	0.992173i	$-0.460148\pi$
$733$	6.76159	0.249745	0.124873	+	0.992173i	$0.460148\pi$
$734$	0	0
$735$	−39.1694	−1.44479
$736$	0	0
$737$	−1.61993	−0.0596708
$738$	0	0
$739$	14.4347	0.530989	0.265494	−	0.964112i	$-0.414465\pi$
$739$	14.4347	0.530989	0.265494	+	0.964112i	$0.414465\pi$
$740$	0	0
$741$	−8.22891	−0.302297
$742$	0	0
$743$	−0.739099	−0.0271149	−0.0135575	−	0.999908i	$-0.504316\pi$
$743$	−0.739099	−0.0271149	−0.0135575	+	0.999908i	$0.504316\pi$
$744$	0	0
$745$	36.1813	1.32558
$746$	0	0
$747$	0.195998	0.00717120
$748$	0	0
$749$	2.04691	0.0747925
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−1.55107	−0.0565243
$754$	0	0
$755$	−3.58340	−0.130413
$756$	0	0
$757$	3.11668	0.113278	0.0566388	−	0.998395i	$-0.481962\pi$
$757$	3.11668	0.113278	0.0566388	+	0.998395i	$0.481962\pi$
$758$	0	0
$759$	−32.5951	−1.18313
$760$	0	0
$761$	−24.4341	−0.885736	−0.442868	−	0.896587i	$-0.646039\pi$
$761$	−24.4341	−0.885736	−0.442868	+	0.896587i	$0.646039\pi$
$762$	0	0
$763$	−1.85094	−0.0670086
$764$	0	0
$765$	5.57582	0.201594
$766$	0	0
$767$	12.7768	0.461345
$768$	0	0
$769$	0.429693	0.0154951	0.00774757	−	0.999970i	$-0.497534\pi$
$769$	0.429693	0.0154951	0.00774757	+	0.999970i	$0.497534\pi$
$770$	0	0
$771$	10.5491	0.379915
$772$	0	0
$773$	−38.5249	−1.38564	−0.692821	−	0.721109i	$-0.743633\pi$
$773$	−38.5249	−1.38564	−0.692821	+	0.721109i	$0.743633\pi$
$774$	0	0
$775$	−33.7806	−1.21343
$776$	0	0
$777$	−11.7039	−0.419874
$778$	0	0
$779$	−10.5387	−0.377587
$780$	0	0
$781$	24.1353	0.863629
$782$	0	0
$783$	−27.6822	−0.989280
$784$	0	0
$785$	−22.7545	−0.812142
$786$	0	0
$787$	7.54525	0.268959	0.134480	−	0.990916i	$-0.457064\pi$
$787$	7.54525	0.268959	0.134480	+	0.990916i	$0.457064\pi$
$788$	0	0
$789$	6.86693	0.244469
$790$	0	0
$791$	−0.598011	−0.0212628
$792$	0	0
$793$	−25.8778	−0.918947
$794$	0	0
$795$	50.8662	1.80404
$796$	0	0
$797$	−34.8787	−1.23547	−0.617733	−	0.786388i	$-0.711949\pi$
$797$	−34.8787	−1.23547	−0.617733	+	0.786388i	$0.711949\pi$
$798$	0	0
$799$	3.30912	0.117068
$800$	0	0
$801$	−6.40818	−0.226422
$802$	0	0
$803$	21.6690	0.764682
$804$	0	0
$805$	15.5342	0.547510
$806$	0	0
$807$	−8.62738	−0.303698
$808$	0	0
$809$	18.3170	0.643990	0.321995	−	0.946741i	$-0.395647\pi$
$809$	18.3170	0.643990	0.321995	+	0.946741i	$0.395647\pi$
$810$	0	0
$811$	−17.2941	−0.607279	−0.303639	−	0.952787i	$-0.598202\pi$
$811$	−17.2941	−0.607279	−0.303639	+	0.952787i	$0.598202\pi$
$812$	0	0
$813$	24.6800	0.865566
$814$	0	0
$815$	−38.8541	−1.36100
$816$	0	0
$817$	−7.48774	−0.261963
$818$	0	0
$819$	−1.43905	−0.0502845
$820$	0	0
$821$	9.34588	0.326173	0.163087	−	0.986612i	$-0.447855\pi$
$821$	9.34588	0.326173	0.163087	+	0.986612i	$0.447855\pi$
$822$	0	0
$823$	17.5716	0.612508	0.306254	−	0.951950i	$-0.400924\pi$
$823$	17.5716	0.612508	0.306254	+	0.951950i	$0.400924\pi$
$824$	0	0
$825$	16.9531	0.590232
$826$	0	0
$827$	3.37473	0.117351	0.0586755	−	0.998277i	$-0.481312\pi$
$827$	3.37473	0.117351	0.0586755	+	0.998277i	$0.481312\pi$
$828$	0	0
$829$	47.9279	1.66460	0.832302	−	0.554323i	$-0.187023\pi$
$829$	47.9279	1.66460	0.832302	+	0.554323i	$0.187023\pi$
$830$	0	0
$831$	−6.91919	−0.240024
$832$	0	0
$833$	−17.9069	−0.620437
$834$	0	0
$835$	−9.17735	−0.317595
$836$	0	0
$837$	33.8943	1.17156
$838$	0	0
$839$	40.5911	1.40136	0.700680	−	0.713476i	$-0.252880\pi$
$839$	40.5911	1.40136	0.700680	+	0.713476i	$0.252880\pi$
$840$	0	0
$841$	9.54719	0.329213
$842$	0	0
$843$	−17.1127	−0.589392
$844$	0	0
$845$	0.0916405	0.00315253
$846$	0	0
$847$	4.16420	0.143084
$848$	0	0
$849$	12.5619	0.431124
$850$	0	0
$851$	−88.1753	−3.02261
$852$	0	0
$853$	−14.3673	−0.491928	−0.245964	−	0.969279i	$-0.579104\pi$
$853$	−14.3673	−0.491928	−0.245964	+	0.969279i	$0.579104\pi$
$854$	0	0
$855$	−2.46276	−0.0842246
$856$	0	0
$857$	−10.3293	−0.352843	−0.176422	−	0.984315i	$-0.556452\pi$
$857$	−10.3293	−0.352843	−0.176422	+	0.984315i	$0.556452\pi$
$858$	0	0
$859$	22.5762	0.770289	0.385144	−	0.922856i	$-0.374152\pi$
$859$	22.5762	0.770289	0.385144	+	0.922856i	$0.374152\pi$
$860$	0	0
$861$	−10.0485	−0.342451
$862$	0	0
$863$	22.6420	0.770742	0.385371	−	0.922762i	$-0.374073\pi$
$863$	22.6420	0.770742	0.385371	+	0.922762i	$0.374073\pi$
$864$	0	0
$865$	−31.7538	−1.07966
$866$	0	0
$867$	−18.6859	−0.634605
$868$	0	0
$869$	26.0717	0.884422
$870$	0	0
$871$	−2.93777	−0.0995424
$872$	0	0
$873$	5.99989	0.203065
$874$	0	0
$875$	1.01147	0.0341940
$876$	0	0
$877$	56.2365	1.89897	0.949485	−	0.313812i	$-0.101606\pi$
$877$	56.2365	1.89897	0.949485	+	0.313812i	$0.101606\pi$
$878$	0	0
$879$	−12.8562	−0.433627
$880$	0	0
$881$	−53.9520	−1.81769	−0.908844	−	0.417136i	$-0.863034\pi$
$881$	−53.9520	−1.81769	−0.908844	+	0.417136i	$0.863034\pi$
$882$	0	0
$883$	−9.13695	−0.307483	−0.153741	−	0.988111i	$-0.549132\pi$
$883$	−9.13695	−0.307483	−0.153741	+	0.988111i	$0.549132\pi$
$884$	0	0
$885$	20.8489	0.700828
$886$	0	0
$887$	−42.2248	−1.41777	−0.708884	−	0.705325i	$-0.750801\pi$
$887$	−42.2248	−1.41777	−0.708884	+	0.705325i	$0.750801\pi$
$888$	0	0
$889$	4.48566	0.150444
$890$	0	0
$891$	−21.0337	−0.704655
$892$	0	0
$893$	−1.46159	−0.0489102
$894$	0	0
$895$	13.3432	0.446015
$896$	0	0
$897$	−59.1119	−1.97369
$898$	0	0
$899$	−47.1975	−1.57412
$900$	0	0
$901$	23.2543	0.774712
$902$	0	0
$903$	−7.13946	−0.237586
$904$	0	0
$905$	2.51585	0.0836298
$906$	0	0
$907$	−26.6789	−0.885857	−0.442929	−	0.896557i	$-0.646061\pi$
$907$	−26.6789	−0.885857	−0.442929	+	0.896557i	$0.646061\pi$
$908$	0	0
$909$	−3.24548	−0.107646
$910$	0	0
$911$	−48.4608	−1.60558	−0.802789	−	0.596263i	$-0.796652\pi$
$911$	−48.4608	−1.60558	−0.802789	+	0.596263i	$0.796652\pi$
$912$	0	0
$913$	0.578980	0.0191614
$914$	0	0
$915$	−42.2267	−1.39597
$916$	0	0
$917$	−4.62400	−0.152698
$918$	0	0
$919$	8.67151	0.286047	0.143023	−	0.989719i	$-0.454318\pi$
$919$	8.67151	0.286047	0.143023	+	0.989719i	$0.454318\pi$
$920$	0	0
$921$	−39.1007	−1.28841
$922$	0	0
$923$	43.7698	1.44070
$924$	0	0
$925$	45.8610	1.50790
$926$	0	0
$927$	−4.12065	−0.135340
$928$	0	0
$929$	−52.4420	−1.72057	−0.860283	−	0.509817i	$-0.829713\pi$
$929$	−52.4420	−1.72057	−0.860283	+	0.509817i	$0.829713\pi$
$930$	0	0
$931$	7.90921	0.259214
$932$	0	0
$933$	−9.50844	−0.311292
$934$	0	0
$935$	16.4710	0.538660
$936$	0	0
$937$	−6.01007	−0.196340	−0.0981702	−	0.995170i	$-0.531299\pi$
$937$	−6.01007	−0.196340	−0.0981702	+	0.995170i	$0.531299\pi$
$938$	0	0
$939$	−0.559228	−0.0182497
$940$	0	0
$941$	−19.8695	−0.647728	−0.323864	−	0.946104i	$-0.604982\pi$
$941$	−19.8695	−0.647728	−0.323864	+	0.946104i	$0.604982\pi$
$942$	0	0
$943$	−75.7038	−2.46526
$944$	0	0
$945$	8.10674	0.263712
$946$	0	0
$947$	10.6701	0.346731	0.173365	−	0.984858i	$-0.444536\pi$
$947$	10.6701	0.346731	0.173365	+	0.984858i	$0.444536\pi$
$948$	0	0
$949$	39.2971	1.27564
$950$	0	0
$951$	48.6932	1.57898
$952$	0	0
$953$	−52.7635	−1.70918	−0.854589	−	0.519305i	$-0.826191\pi$
$953$	−52.7635	−1.70918	−0.854589	+	0.519305i	$0.826191\pi$
$954$	0	0
$955$	2.09889	0.0679186
$956$	0	0
$957$	23.6866	0.765677
$958$	0	0
$959$	9.08985	0.293527
$960$	0	0
$961$	26.7890	0.864161
$962$	0	0
$963$	−2.33111	−0.0751191
$964$	0	0
$965$	−17.2063	−0.553892
$966$	0	0
$967$	−22.6426	−0.728138	−0.364069	−	0.931372i	$-0.618613\pi$
$967$	−22.6426	−0.728138	−0.364069	+	0.931372i	$0.618613\pi$
$968$	0	0
$969$	−6.13869	−0.197203
$970$	0	0
$971$	28.7801	0.923599	0.461799	−	0.886984i	$-0.347204\pi$
$971$	28.7801	0.923599	0.461799	+	0.886984i	$0.347204\pi$
$972$	0	0
$973$	1.50087	0.0481157
$974$	0	0
$975$	30.7448	0.984622
$976$	0	0
$977$	13.3853	0.428235	0.214118	−	0.976808i	$-0.431312\pi$
$977$	13.3853	0.428235	0.214118	+	0.976808i	$0.431312\pi$
$978$	0	0
$979$	−18.9298	−0.604999
$980$	0	0
$981$	2.10793	0.0673011
$982$	0	0
$983$	−46.5305	−1.48409	−0.742047	−	0.670348i	$-0.766145\pi$
$983$	−46.5305	−1.48409	−0.742047	+	0.670348i	$0.766145\pi$
$984$	0	0
$985$	−42.8338	−1.36480
$986$	0	0
$987$	−1.39360	−0.0443589
$988$	0	0
$989$	−53.7877	−1.71035
$990$	0	0
$991$	−22.6594	−0.719798	−0.359899	−	0.932991i	$-0.617189\pi$
$991$	−22.6594	−0.719798	−0.359899	+	0.932991i	$0.617189\pi$
$992$	0	0
$993$	−31.0424	−0.985102
$994$	0	0
$995$	11.3163	0.358752
$996$	0	0
$997$	43.4733	1.37681	0.688406	−	0.725325i	$-0.258311\pi$
$997$	43.4733	1.37681	0.688406	+	0.725325i	$0.258311\pi$
$998$	0	0
$999$	−46.0154	−1.45586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.37	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.37	✓	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)

\(f(q)\)	\(=\)	\(q+1.91672 q^{3} +3.07306 q^{5} -0.591659 q^{7} +0.673807 q^{9} +O(q^{10})\)	\(q+1.91672 q^{3} +3.07306 q^{5} -0.591659 q^{7} +0.673807 q^{9} +1.99043 q^{11} +3.60968 q^{13} +5.89019 q^{15} +2.69279 q^{17} -1.18936 q^{19} -1.13404 q^{21} -8.54373 q^{23} +4.44370 q^{25} -4.45866 q^{27} +6.20864 q^{29} -7.60191 q^{31} +3.81510 q^{33} -1.81820 q^{35} +10.3205 q^{37} +6.91875 q^{39} +8.86075 q^{41} +6.29558 q^{43} +2.07065 q^{45} +1.22888 q^{47} -6.64994 q^{49} +5.16132 q^{51} +8.63575 q^{53} +6.11672 q^{55} -2.27968 q^{57} +3.53960 q^{59} -7.16899 q^{61} -0.398664 q^{63} +11.0928 q^{65} -0.813857 q^{67} -16.3759 q^{69} +12.1257 q^{71} +10.8866 q^{73} +8.51731 q^{75} -1.17766 q^{77} +13.0985 q^{79} -10.5674 q^{81} +0.290882 q^{83} +8.27510 q^{85} +11.9002 q^{87} -9.51040 q^{89} -2.13570 q^{91} -14.5707 q^{93} -3.65499 q^{95} +8.90446 q^{97} +1.34117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more