LMFDB - Embedded newform 2008.2.a.d.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,2,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, 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("2008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2008.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:	$16.0339607259$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.96174 q^{3} +0.807414 q^{5} +3.34299 q^{7} +0.848432 q^{9} +O(q^{10})$	$q+1.96174 q^{3} +0.807414 q^{5} +3.34299 q^{7} +0.848432 q^{9} +4.04597 q^{11} -4.44468 q^{13} +1.58394 q^{15} -5.91837 q^{17} +2.98461 q^{19} +6.55809 q^{21} +7.45495 q^{23} -4.34808 q^{25} -4.22082 q^{27} +7.98249 q^{29} +9.58166 q^{31} +7.93716 q^{33} +2.69918 q^{35} +2.67999 q^{37} -8.71932 q^{39} +3.85402 q^{41} +9.57972 q^{43} +0.685036 q^{45} -9.46625 q^{47} +4.17559 q^{49} -11.6103 q^{51} -9.89664 q^{53} +3.26677 q^{55} +5.85503 q^{57} +6.88994 q^{59} -7.56108 q^{61} +2.83630 q^{63} -3.58870 q^{65} -8.97228 q^{67} +14.6247 q^{69} -14.1969 q^{71} +3.51267 q^{73} -8.52982 q^{75} +13.5257 q^{77} -9.64808 q^{79} -10.8255 q^{81} +9.42852 q^{83} -4.77858 q^{85} +15.6596 q^{87} +0.414104 q^{89} -14.8585 q^{91} +18.7967 q^{93} +2.40981 q^{95} +8.39841 q^{97} +3.43274 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * 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
$3$	1.96174	1.13261	0.566306	−	0.824195i	$-0.308372\pi$
$3$	1.96174	1.13261	0.566306	+	0.824195i	$0.308372\pi$
$4$	0	0
$5$	0.807414	0.361086	0.180543	−	0.983567i	$-0.442214\pi$
$5$	0.807414	0.361086	0.180543	+	0.983567i	$0.442214\pi$
$6$	0	0
$7$	3.34299	1.26353	0.631766	−	0.775159i	$-0.282330\pi$
$7$	3.34299	1.26353	0.631766	+	0.775159i	$0.282330\pi$
$8$	0	0
$9$	0.848432	0.282811
$10$	0	0
$11$	4.04597	1.21991	0.609954	−	0.792437i	$-0.291188\pi$
$11$	4.04597	1.21991	0.609954	+	0.792437i	$0.291188\pi$
$12$	0	0
$13$	−4.44468	−1.23273	−0.616366	−	0.787460i	$-0.711396\pi$
$13$	−4.44468	−1.23273	−0.616366	+	0.787460i	$0.711396\pi$
$14$	0	0
$15$	1.58394	0.408971
$16$	0	0
$17$	−5.91837	−1.43542	−0.717708	−	0.696344i	$-0.754809\pi$
$17$	−5.91837	−1.43542	−0.717708	+	0.696344i	$0.754809\pi$
$18$	0	0
$19$	2.98461	0.684715	0.342358	−	0.939570i	$-0.388775\pi$
$19$	2.98461	0.684715	0.342358	+	0.939570i	$0.388775\pi$
$20$	0	0
$21$	6.55809	1.43109
$22$	0	0
$23$	7.45495	1.55447	0.777233	−	0.629213i	$-0.216623\pi$
$23$	7.45495	1.55447	0.777233	+	0.629213i	$0.216623\pi$
$24$	0	0
$25$	−4.34808	−0.869617
$26$	0	0
$27$	−4.22082	−0.812297
$28$	0	0
$29$	7.98249	1.48231	0.741155	−	0.671334i	$-0.234278\pi$
$29$	7.98249	1.48231	0.741155	+	0.671334i	$0.234278\pi$
$30$	0	0
$31$	9.58166	1.72092	0.860459	−	0.509521i	$-0.170177\pi$
$31$	9.58166	1.72092	0.860459	+	0.509521i	$0.170177\pi$
$32$	0	0
$33$	7.93716	1.38168
$34$	0	0
$35$	2.69918	0.456244
$36$	0	0
$37$	2.67999	0.440587	0.220294	−	0.975434i	$-0.429298\pi$
$37$	2.67999	0.440587	0.220294	+	0.975434i	$0.429298\pi$
$38$	0	0
$39$	−8.71932	−1.39621
$40$	0	0
$41$	3.85402	0.601896	0.300948	−	0.953641i	$-0.402697\pi$
$41$	3.85402	0.601896	0.300948	+	0.953641i	$0.402697\pi$
$42$	0	0
$43$	9.57972	1.46089	0.730447	−	0.682969i	$-0.239312\pi$
$43$	9.57972	1.46089	0.730447	+	0.682969i	$0.239312\pi$
$44$	0	0
$45$	0.685036	0.102119
$46$	0	0
$47$	−9.46625	−1.38080	−0.690398	−	0.723430i	$-0.742564\pi$
$47$	−9.46625	−1.38080	−0.690398	+	0.723430i	$0.742564\pi$
$48$	0	0
$49$	4.17559	0.596512
$50$	0	0
$51$	−11.6103	−1.62577
$52$	0	0
$53$	−9.89664	−1.35941	−0.679704	−	0.733486i	$-0.737892\pi$
$53$	−9.89664	−1.35941	−0.679704	+	0.733486i	$0.737892\pi$
$54$	0	0
$55$	3.26677	0.440492
$56$	0	0
$57$	5.85503	0.775517
$58$	0	0
$59$	6.88994	0.896994	0.448497	−	0.893784i	$-0.351959\pi$
$59$	6.88994	0.896994	0.448497	+	0.893784i	$0.351959\pi$
$60$	0	0
$61$	−7.56108	−0.968097	−0.484048	−	0.875041i	$-0.660834\pi$
$61$	−7.56108	−0.968097	−0.484048	+	0.875041i	$0.660834\pi$
$62$	0	0
$63$	2.83630	0.357340
$64$	0	0
$65$	−3.58870	−0.445123
$66$	0	0
$67$	−8.97228	−1.09614	−0.548069	−	0.836433i	$-0.684637\pi$
$67$	−8.97228	−1.09614	−0.548069	+	0.836433i	$0.684637\pi$
$68$	0	0
$69$	14.6247	1.76061
$70$	0	0
$71$	−14.1969	−1.68486	−0.842432	−	0.538802i	$-0.818877\pi$
$71$	−14.1969	−1.68486	−0.842432	+	0.538802i	$0.818877\pi$
$72$	0	0
$73$	3.51267	0.411127	0.205563	−	0.978644i	$-0.434097\pi$
$73$	3.51267	0.411127	0.205563	+	0.978644i	$0.434097\pi$
$74$	0	0
$75$	−8.52982	−0.984939
$76$	0	0
$77$	13.5257	1.54139
$78$	0	0
$79$	−9.64808	−1.08549	−0.542747	−	0.839896i	$-0.682616\pi$
$79$	−9.64808	−1.08549	−0.542747	+	0.839896i	$0.682616\pi$
$80$	0	0
$81$	−10.8255	−1.20283
$82$	0	0
$83$	9.42852	1.03491	0.517457	−	0.855709i	$-0.326879\pi$
$83$	9.42852	1.03491	0.517457	+	0.855709i	$0.326879\pi$
$84$	0	0
$85$	−4.77858	−0.518309
$86$	0	0
$87$	15.6596	1.67888
$88$	0	0
$89$	0.414104	0.0438949	0.0219474	−	0.999759i	$-0.493013\pi$
$89$	0.414104	0.0438949	0.0219474	+	0.999759i	$0.493013\pi$
$90$	0	0
$91$	−14.8585	−1.55760
$92$	0	0
$93$	18.7967	1.94913
$94$	0	0
$95$	2.40981	0.247241
$96$	0	0
$97$	8.39841	0.852729	0.426365	−	0.904551i	$-0.359794\pi$
$97$	8.39841	0.852729	0.426365	+	0.904551i	$0.359794\pi$
$98$	0	0
$99$	3.43274	0.345003
$100$	0	0
$101$	−8.89308	−0.884894	−0.442447	−	0.896795i	$-0.645890\pi$
$101$	−8.89308	−0.884894	−0.442447	+	0.896795i	$0.645890\pi$
$102$	0	0
$103$	12.4778	1.22948	0.614739	−	0.788730i	$-0.289261\pi$
$103$	12.4778	1.22948	0.614739	+	0.788730i	$0.289261\pi$
$104$	0	0
$105$	5.29509	0.516748
$106$	0	0
$107$	11.1670	1.07955	0.539775	−	0.841809i	$-0.318509\pi$
$107$	11.1670	1.07955	0.539775	+	0.841809i	$0.318509\pi$
$108$	0	0
$109$	−16.2762	−1.55898	−0.779491	−	0.626413i	$-0.784522\pi$
$109$	−16.2762	−1.55898	−0.779491	+	0.626413i	$0.784522\pi$
$110$	0	0
$111$	5.25744	0.499015
$112$	0	0
$113$	−3.51526	−0.330687	−0.165344	−	0.986236i	$-0.552873\pi$
$113$	−3.51526	−0.330687	−0.165344	+	0.986236i	$0.552873\pi$
$114$	0	0
$115$	6.01923	0.561296
$116$	0	0
$117$	−3.77101	−0.348630
$118$	0	0
$119$	−19.7851	−1.81369
$120$	0	0
$121$	5.36991	0.488174
$122$	0	0
$123$	7.56059	0.681715
$124$	0	0
$125$	−7.54777	−0.675093
$126$	0	0
$127$	9.24057	0.819968	0.409984	−	0.912093i	$-0.365534\pi$
$127$	9.24057	0.819968	0.409984	+	0.912093i	$0.365534\pi$
$128$	0	0
$129$	18.7929	1.65463
$130$	0	0
$131$	−11.2973	−0.987050	−0.493525	−	0.869732i	$-0.664292\pi$
$131$	−11.2973	−0.987050	−0.493525	+	0.869732i	$0.664292\pi$
$132$	0	0
$133$	9.97751	0.865160
$134$	0	0
$135$	−3.40795	−0.293310
$136$	0	0
$137$	0.354515	0.0302882	0.0151441	−	0.999885i	$-0.495179\pi$
$137$	0.354515	0.0302882	0.0151441	+	0.999885i	$0.495179\pi$
$138$	0	0
$139$	10.6888	0.906614	0.453307	−	0.891354i	$-0.350244\pi$
$139$	10.6888	0.906614	0.453307	+	0.891354i	$0.350244\pi$
$140$	0	0
$141$	−18.5704	−1.56391
$142$	0	0
$143$	−17.9831	−1.50382
$144$	0	0
$145$	6.44517	0.535242
$146$	0	0
$147$	8.19143	0.675617
$148$	0	0
$149$	21.5692	1.76702	0.883509	−	0.468415i	$-0.155175\pi$
$149$	21.5692	1.76702	0.883509	+	0.468415i	$0.155175\pi$
$150$	0	0
$151$	−6.75789	−0.549949	−0.274975	−	0.961451i	$-0.588669\pi$
$151$	−6.75789	−0.549949	−0.274975	+	0.961451i	$0.588669\pi$
$152$	0	0
$153$	−5.02134	−0.405951
$154$	0	0
$155$	7.73636	0.621400
$156$	0	0
$157$	−4.69940	−0.375053	−0.187526	−	0.982260i	$-0.560047\pi$
$157$	−4.69940	−0.375053	−0.187526	+	0.982260i	$0.560047\pi$
$158$	0	0
$159$	−19.4147	−1.53968
$160$	0	0
$161$	24.9218	1.96412
$162$	0	0
$163$	3.02003	0.236547	0.118273	−	0.992981i	$-0.462264\pi$
$163$	3.02003	0.236547	0.118273	+	0.992981i	$0.462264\pi$
$164$	0	0
$165$	6.40857	0.498907
$166$	0	0
$167$	20.0631	1.55253	0.776263	−	0.630409i	$-0.217113\pi$
$167$	20.0631	1.55253	0.776263	+	0.630409i	$0.217113\pi$
$168$	0	0
$169$	6.75519	0.519630
$170$	0	0
$171$	2.53224	0.193645
$172$	0	0
$173$	−7.03355	−0.534751	−0.267375	−	0.963592i	$-0.586156\pi$
$173$	−7.03355	−0.534751	−0.267375	+	0.963592i	$0.586156\pi$
$174$	0	0
$175$	−14.5356	−1.09879
$176$	0	0
$177$	13.5163	1.01595
$178$	0	0
$179$	17.3752	1.29868	0.649341	−	0.760497i	$-0.275045\pi$
$179$	17.3752	1.29868	0.649341	+	0.760497i	$0.275045\pi$
$180$	0	0
$181$	−24.7797	−1.84186	−0.920929	−	0.389731i	$-0.872568\pi$
$181$	−24.7797	−1.84186	−0.920929	+	0.389731i	$0.872568\pi$
$182$	0	0
$183$	−14.8329	−1.09648
$184$	0	0
$185$	2.16386	0.159090
$186$	0	0
$187$	−23.9456	−1.75107
$188$	0	0
$189$	−14.1102	−1.02636
$190$	0	0
$191$	−1.90787	−0.138049	−0.0690245	−	0.997615i	$-0.521989\pi$
$191$	−1.90787	−0.138049	−0.0690245	+	0.997615i	$0.521989\pi$
$192$	0	0
$193$	−3.61802	−0.260431	−0.130215	−	0.991486i	$-0.541567\pi$
$193$	−3.61802	−0.260431	−0.130215	+	0.991486i	$0.541567\pi$
$194$	0	0
$195$	−7.04010	−0.504152
$196$	0	0
$197$	−2.81406	−0.200494	−0.100247	−	0.994963i	$-0.531963\pi$
$197$	−2.81406	−0.200494	−0.100247	+	0.994963i	$0.531963\pi$
$198$	0	0
$199$	−13.7072	−0.971680	−0.485840	−	0.874048i	$-0.661486\pi$
$199$	−13.7072	−0.971680	−0.485840	+	0.874048i	$0.661486\pi$
$200$	0	0
$201$	−17.6013	−1.24150
$202$	0	0
$203$	26.6854	1.87295
$204$	0	0
$205$	3.11179	0.217337
$206$	0	0
$207$	6.32502	0.439620
$208$	0	0
$209$	12.0756	0.835289
$210$	0	0
$211$	−23.6771	−1.63000	−0.814999	−	0.579463i	$-0.803262\pi$
$211$	−23.6771	−1.63000	−0.814999	+	0.579463i	$0.803262\pi$
$212$	0	0
$213$	−27.8507	−1.90830
$214$	0	0
$215$	7.73480	0.527509
$216$	0	0
$217$	32.0314	2.17443
$218$	0	0
$219$	6.89095	0.465647
$220$	0	0
$221$	26.3053	1.76948
$222$	0	0
$223$	−17.6488	−1.18185	−0.590927	−	0.806725i	$-0.701238\pi$
$223$	−17.6488	−1.18185	−0.590927	+	0.806725i	$0.701238\pi$
$224$	0	0
$225$	−3.68905	−0.245937
$226$	0	0
$227$	−24.1655	−1.60392	−0.801962	−	0.597375i	$-0.796210\pi$
$227$	−24.1655	−1.60392	−0.801962	+	0.597375i	$0.796210\pi$
$228$	0	0
$229$	−16.2503	−1.07385	−0.536924	−	0.843630i	$-0.680414\pi$
$229$	−16.2503	−1.07385	−0.536924	+	0.843630i	$0.680414\pi$
$230$	0	0
$231$	26.5338	1.74580
$232$	0	0
$233$	−13.6794	−0.896169	−0.448085	−	0.893991i	$-0.647894\pi$
$233$	−13.6794	−0.896169	−0.448085	+	0.893991i	$0.647894\pi$
$234$	0	0
$235$	−7.64318	−0.498586
$236$	0	0
$237$	−18.9271	−1.22944
$238$	0	0
$239$	−17.1632	−1.11020	−0.555099	−	0.831784i	$-0.687320\pi$
$239$	−17.1632	−1.11020	−0.555099	+	0.831784i	$0.687320\pi$
$240$	0	0
$241$	−12.0334	−0.775139	−0.387570	−	0.921840i	$-0.626685\pi$
$241$	−12.0334	−0.775139	−0.387570	+	0.921840i	$0.626685\pi$
$242$	0	0
$243$	−8.57430	−0.550041
$244$	0	0
$245$	3.37143	0.215393
$246$	0	0
$247$	−13.2656	−0.844071
$248$	0	0
$249$	18.4963	1.17216
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	30.1626	1.89630
$254$	0	0
$255$	−9.37433	−0.587043
$256$	0	0
$257$	22.2631	1.38873	0.694367	−	0.719621i	$-0.255685\pi$
$257$	22.2631	1.38873	0.694367	+	0.719621i	$0.255685\pi$
$258$	0	0
$259$	8.95917	0.556696
$260$	0	0
$261$	6.77260	0.419213
$262$	0	0
$263$	30.0510	1.85302	0.926511	−	0.376268i	$-0.122793\pi$
$263$	30.0510	1.85302	0.926511	+	0.376268i	$0.122793\pi$
$264$	0	0
$265$	−7.99069	−0.490864
$266$	0	0
$267$	0.812364	0.0497159
$268$	0	0
$269$	13.9225	0.848871	0.424435	−	0.905458i	$-0.360473\pi$
$269$	13.9225	0.848871	0.424435	+	0.905458i	$0.360473\pi$
$270$	0	0
$271$	−32.0375	−1.94614	−0.973071	−	0.230505i	$-0.925962\pi$
$271$	−32.0375	−1.94614	−0.973071	+	0.230505i	$0.925962\pi$
$272$	0	0
$273$	−29.1486	−1.76415
$274$	0	0
$275$	−17.5922	−1.06085
$276$	0	0
$277$	27.4178	1.64737	0.823687	−	0.567044i	$-0.191913\pi$
$277$	27.4178	1.64737	0.823687	+	0.567044i	$0.191913\pi$
$278$	0	0
$279$	8.12939	0.486694
$280$	0	0
$281$	−2.36183	−0.140895	−0.0704474	−	0.997515i	$-0.522443\pi$
$281$	−2.36183	−0.140895	−0.0704474	+	0.997515i	$0.522443\pi$
$282$	0	0
$283$	−6.40656	−0.380831	−0.190415	−	0.981704i	$-0.560983\pi$
$283$	−6.40656	−0.380831	−0.190415	+	0.981704i	$0.560983\pi$
$284$	0	0
$285$	4.72743	0.280029
$286$	0	0
$287$	12.8839	0.760515
$288$	0	0
$289$	18.0271	1.06042
$290$	0	0
$291$	16.4755	0.965812
$292$	0	0
$293$	−2.38006	−0.139045	−0.0695223	−	0.997580i	$-0.522147\pi$
$293$	−2.38006	−0.139045	−0.0695223	+	0.997580i	$0.522147\pi$
$294$	0	0
$295$	5.56303	0.323892
$296$	0	0
$297$	−17.0773	−0.990927
$298$	0	0
$299$	−33.1349	−1.91624
$300$	0	0
$301$	32.0249	1.84589
$302$	0	0
$303$	−17.4459	−1.00224
$304$	0	0
$305$	−6.10492	−0.349566
$306$	0	0
$307$	11.5792	0.660861	0.330430	−	0.943830i	$-0.392806\pi$
$307$	11.5792	0.660861	0.330430	+	0.943830i	$0.392806\pi$
$308$	0	0
$309$	24.4783	1.39252
$310$	0	0
$311$	18.5945	1.05440	0.527198	−	0.849743i	$-0.323243\pi$
$311$	18.5945	1.05440	0.527198	+	0.849743i	$0.323243\pi$
$312$	0	0
$313$	−13.4775	−0.761791	−0.380895	−	0.924618i	$-0.624384\pi$
$313$	−13.4775	−0.761791	−0.380895	+	0.924618i	$0.624384\pi$
$314$	0	0
$315$	2.29007	0.129031
$316$	0	0
$317$	−24.2903	−1.36428	−0.682141	−	0.731221i	$-0.738951\pi$
$317$	−24.2903	−1.36428	−0.682141	+	0.731221i	$0.738951\pi$
$318$	0	0
$319$	32.2969	1.80828
$320$	0	0
$321$	21.9067	1.22271
$322$	0	0
$323$	−17.6640	−0.982852
$324$	0	0
$325$	19.3258	1.07200
$326$	0	0
$327$	−31.9298	−1.76572
$328$	0	0
$329$	−31.6456	−1.74468
$330$	0	0
$331$	27.5827	1.51608	0.758041	−	0.652207i	$-0.226157\pi$
$331$	27.5827	1.51608	0.758041	+	0.652207i	$0.226157\pi$
$332$	0	0
$333$	2.27379	0.124603
$334$	0	0
$335$	−7.24434	−0.395800
$336$	0	0
$337$	−13.0865	−0.712868	−0.356434	−	0.934320i	$-0.616008\pi$
$337$	−13.0865	−0.712868	−0.356434	+	0.934320i	$0.616008\pi$
$338$	0	0
$339$	−6.89603	−0.374541
$340$	0	0
$341$	38.7672	2.09936
$342$	0	0
$343$	−9.44199	−0.509819
$344$	0	0
$345$	11.8082	0.635731
$346$	0	0
$347$	14.0578	0.754661	0.377330	−	0.926079i	$-0.376842\pi$
$347$	14.0578	0.754661	0.377330	+	0.926079i	$0.376842\pi$
$348$	0	0
$349$	−22.6500	−1.21243	−0.606213	−	0.795302i	$-0.707312\pi$
$349$	−22.6500	−1.21243	−0.606213	+	0.795302i	$0.707312\pi$
$350$	0	0
$351$	18.7602	1.00135
$352$	0	0
$353$	14.0640	0.748552	0.374276	−	0.927317i	$-0.377891\pi$
$353$	14.0640	0.748552	0.374276	+	0.927317i	$0.377891\pi$
$354$	0	0
$355$	−11.4628	−0.608382
$356$	0	0
$357$	−38.8132	−2.05421
$358$	0	0
$359$	−5.72191	−0.301991	−0.150995	−	0.988534i	$-0.548248\pi$
$359$	−5.72191	−0.301991	−0.150995	+	0.988534i	$0.548248\pi$
$360$	0	0
$361$	−10.0921	−0.531165
$362$	0	0
$363$	10.5344	0.552911
$364$	0	0
$365$	2.83618	0.148452
$366$	0	0
$367$	35.9154	1.87477	0.937385	−	0.348294i	$-0.113239\pi$
$367$	35.9154	1.87477	0.937385	+	0.348294i	$0.113239\pi$
$368$	0	0
$369$	3.26987	0.170223
$370$	0	0
$371$	−33.0844	−1.71766
$372$	0	0
$373$	−21.9660	−1.13735	−0.568677	−	0.822561i	$-0.692545\pi$
$373$	−21.9660	−1.13735	−0.568677	+	0.822561i	$0.692545\pi$
$374$	0	0
$375$	−14.8068	−0.764619
$376$	0	0
$377$	−35.4796	−1.82729
$378$	0	0
$379$	−1.92693	−0.0989795	−0.0494898	−	0.998775i	$-0.515760\pi$
$379$	−1.92693	−0.0989795	−0.0494898	+	0.998775i	$0.515760\pi$
$380$	0	0
$381$	18.1276	0.928706
$382$	0	0
$383$	−24.2000	−1.23656	−0.618280	−	0.785958i	$-0.712170\pi$
$383$	−24.2000	−1.23656	−0.618280	+	0.785958i	$0.712170\pi$
$384$	0	0
$385$	10.9208	0.556575
$386$	0	0
$387$	8.12775	0.413157
$388$	0	0
$389$	−8.04246	−0.407769	−0.203885	−	0.978995i	$-0.565357\pi$
$389$	−8.04246	−0.407769	−0.203885	+	0.978995i	$0.565357\pi$
$390$	0	0
$391$	−44.1212	−2.23130
$392$	0	0
$393$	−22.1624	−1.11795
$394$	0	0
$395$	−7.78999	−0.391957
$396$	0	0
$397$	−26.3765	−1.32380	−0.661900	−	0.749592i	$-0.730250\pi$
$397$	−26.3765	−1.32380	−0.661900	+	0.749592i	$0.730250\pi$
$398$	0	0
$399$	19.5733	0.979890
$400$	0	0
$401$	24.2236	1.20967	0.604835	−	0.796351i	$-0.293239\pi$
$401$	24.2236	1.20967	0.604835	+	0.796351i	$0.293239\pi$
$402$	0	0
$403$	−42.5874	−2.12143
$404$	0	0
$405$	−8.74062	−0.434325
$406$	0	0
$407$	10.8432	0.537475
$408$	0	0
$409$	−23.4802	−1.16102	−0.580512	−	0.814252i	$-0.697147\pi$
$409$	−23.4802	−1.16102	−0.580512	+	0.814252i	$0.697147\pi$
$410$	0	0
$411$	0.695466	0.0343048
$412$	0	0
$413$	23.0330	1.13338
$414$	0	0
$415$	7.61271	0.373693
$416$	0	0
$417$	20.9687	1.02684
$418$	0	0
$419$	4.91738	0.240229	0.120115	−	0.992760i	$-0.461674\pi$
$419$	4.91738	0.240229	0.120115	+	0.992760i	$0.461674\pi$
$420$	0	0
$421$	9.28392	0.452471	0.226235	−	0.974073i	$-0.427358\pi$
$421$	9.28392	0.452471	0.226235	+	0.974073i	$0.427358\pi$
$422$	0	0
$423$	−8.03148	−0.390504
$424$	0	0
$425$	25.7336	1.24826
$426$	0	0
$427$	−25.2766	−1.22322
$428$	0	0
$429$	−35.2781	−1.70324
$430$	0	0
$431$	−34.4793	−1.66081	−0.830404	−	0.557162i	$-0.811890\pi$
$431$	−34.4793	−1.66081	−0.830404	+	0.557162i	$0.811890\pi$
$432$	0	0
$433$	17.7460	0.852817	0.426408	−	0.904531i	$-0.359779\pi$
$433$	17.7460	0.852817	0.426408	+	0.904531i	$0.359779\pi$
$434$	0	0
$435$	12.6438	0.606222
$436$	0	0
$437$	22.2501	1.06437
$438$	0	0
$439$	−36.2208	−1.72872	−0.864362	−	0.502871i	$-0.832277\pi$
$439$	−36.2208	−1.72872	−0.864362	+	0.502871i	$0.832277\pi$
$440$	0	0
$441$	3.54270	0.168700
$442$	0	0
$443$	1.32623	0.0630109	0.0315055	−	0.999504i	$-0.489970\pi$
$443$	1.32623	0.0630109	0.0315055	+	0.999504i	$0.489970\pi$
$444$	0	0
$445$	0.334353	0.0158498
$446$	0	0
$447$	42.3132	2.00135
$448$	0	0
$449$	−9.76016	−0.460611	−0.230305	−	0.973118i	$-0.573972\pi$
$449$	−9.76016	−0.460611	−0.230305	+	0.973118i	$0.573972\pi$
$450$	0	0
$451$	15.5933	0.734257
$452$	0	0
$453$	−13.2572	−0.622880
$454$	0	0
$455$	−11.9970	−0.562427
$456$	0	0
$457$	34.0108	1.59096	0.795479	−	0.605981i	$-0.207219\pi$
$457$	34.0108	1.59096	0.795479	+	0.605981i	$0.207219\pi$
$458$	0	0
$459$	24.9804	1.16598
$460$	0	0
$461$	−1.06453	−0.0495800	−0.0247900	−	0.999693i	$-0.507892\pi$
$461$	−1.06453	−0.0495800	−0.0247900	+	0.999693i	$0.507892\pi$
$462$	0	0
$463$	23.9463	1.11288	0.556440	−	0.830888i	$-0.312167\pi$
$463$	23.9463	1.11288	0.556440	+	0.830888i	$0.312167\pi$
$464$	0	0
$465$	15.1768	0.703805
$466$	0	0
$467$	38.9525	1.80251	0.901253	−	0.433293i	$-0.142649\pi$
$467$	38.9525	1.80251	0.901253	+	0.433293i	$0.142649\pi$
$468$	0	0
$469$	−29.9942	−1.38500
$470$	0	0
$471$	−9.21901	−0.424790
$472$	0	0
$473$	38.7593	1.78216
$474$	0	0
$475$	−12.9773	−0.595440
$476$	0	0
$477$	−8.39663	−0.384455
$478$	0	0
$479$	−0.611331	−0.0279324	−0.0139662	−	0.999902i	$-0.504446\pi$
$479$	−0.611331	−0.0279324	−0.0139662	+	0.999902i	$0.504446\pi$
$480$	0	0
$481$	−11.9117	−0.543126
$482$	0	0
$483$	48.8902	2.22458
$484$	0	0
$485$	6.78099	0.307909
$486$	0	0
$487$	19.7281	0.893965	0.446983	−	0.894543i	$-0.352499\pi$
$487$	19.7281	0.893965	0.446983	+	0.894543i	$0.352499\pi$
$488$	0	0
$489$	5.92451	0.267916
$490$	0	0
$491$	−19.5698	−0.883173	−0.441587	−	0.897219i	$-0.645584\pi$
$491$	−19.5698	−0.883173	−0.441587	+	0.897219i	$0.645584\pi$
$492$	0	0
$493$	−47.2433	−2.12773
$494$	0	0
$495$	2.77164	0.124576
$496$	0	0
$497$	−47.4602	−2.12888
$498$	0	0
$499$	9.20005	0.411851	0.205925	−	0.978568i	$-0.433980\pi$
$499$	9.20005	0.411851	0.205925	+	0.978568i	$0.433980\pi$
$500$	0	0
$501$	39.3586	1.75841
$502$	0	0
$503$	32.5699	1.45222	0.726110	−	0.687578i	$-0.241326\pi$
$503$	32.5699	1.45222	0.726110	+	0.687578i	$0.241326\pi$
$504$	0	0
$505$	−7.18039	−0.319523
$506$	0	0
$507$	13.2519	0.588539
$508$	0	0
$509$	2.60435	0.115436	0.0577178	−	0.998333i	$-0.481618\pi$
$509$	2.60435	0.115436	0.0577178	+	0.998333i	$0.481618\pi$
$510$	0	0
$511$	11.7428	0.519472
$512$	0	0
$513$	−12.5975	−0.556193
$514$	0	0
$515$	10.0748	0.443948
$516$	0	0
$517$	−38.3002	−1.68444
$518$	0	0
$519$	−13.7980	−0.605665
$520$	0	0
$521$	−20.2071	−0.885288	−0.442644	−	0.896697i	$-0.645959\pi$
$521$	−20.2071	−0.885288	−0.442644	+	0.896697i	$0.645959\pi$
$522$	0	0
$523$	17.8234	0.779362	0.389681	−	0.920950i	$-0.372585\pi$
$523$	17.8234	0.779362	0.389681	+	0.920950i	$0.372585\pi$
$524$	0	0
$525$	−28.5151	−1.24450
$526$	0	0
$527$	−56.7078	−2.47023
$528$	0	0
$529$	32.5763	1.41636
$530$	0	0
$531$	5.84565	0.253680
$532$	0	0
$533$	−17.1299	−0.741977
$534$	0	0
$535$	9.01635	0.389811
$536$	0	0
$537$	34.0856	1.47090
$538$	0	0
$539$	16.8943	0.727690
$540$	0	0
$541$	−3.19983	−0.137571	−0.0687857	−	0.997631i	$-0.521912\pi$
$541$	−3.19983	−0.137571	−0.0687857	+	0.997631i	$0.521912\pi$
$542$	0	0
$543$	−48.6113	−2.08611
$544$	0	0
$545$	−13.1417	−0.562927
$546$	0	0
$547$	−40.4498	−1.72951	−0.864753	−	0.502197i	$-0.832525\pi$
$547$	−40.4498	−1.72951	−0.864753	+	0.502197i	$0.832525\pi$
$548$	0	0
$549$	−6.41506	−0.273788
$550$	0	0
$551$	23.8246	1.01496
$552$	0	0
$553$	−32.2535	−1.37156
$554$	0	0
$555$	4.24493	0.180187
$556$	0	0
$557$	−21.0135	−0.890369	−0.445184	−	0.895439i	$-0.646862\pi$
$557$	−21.0135	−0.890369	−0.445184	+	0.895439i	$0.646862\pi$
$558$	0	0
$559$	−42.5788	−1.80089
$560$	0	0
$561$	−46.9751	−1.98329
$562$	0	0
$563$	−26.9398	−1.13538	−0.567688	−	0.823244i	$-0.692162\pi$
$563$	−26.9398	−1.13538	−0.567688	+	0.823244i	$0.692162\pi$
$564$	0	0
$565$	−2.83827	−0.119407
$566$	0	0
$567$	−36.1894	−1.51981
$568$	0	0
$569$	−10.5424	−0.441959	−0.220979	−	0.975279i	$-0.570925\pi$
$569$	−10.5424	−0.441959	−0.220979	+	0.975279i	$0.570925\pi$
$570$	0	0
$571$	−12.2056	−0.510788	−0.255394	−	0.966837i	$-0.582205\pi$
$571$	−12.2056	−0.510788	−0.255394	+	0.966837i	$0.582205\pi$
$572$	0	0
$573$	−3.74276	−0.156356
$574$	0	0
$575$	−32.4148	−1.35179
$576$	0	0
$577$	4.42277	0.184122	0.0920612	−	0.995753i	$-0.470654\pi$
$577$	4.42277	0.184122	0.0920612	+	0.995753i	$0.470654\pi$
$578$	0	0
$579$	−7.09762	−0.294967
$580$	0	0
$581$	31.5194	1.30765
$582$	0	0
$583$	−40.0416	−1.65835
$584$	0	0
$585$	−3.04477	−0.125886
$586$	0	0
$587$	0.600308	0.0247774	0.0123887	−	0.999923i	$-0.496056\pi$
$587$	0.600308	0.0247774	0.0123887	+	0.999923i	$0.496056\pi$
$588$	0	0
$589$	28.5975	1.17834
$590$	0	0
$591$	−5.52047	−0.227082
$592$	0	0
$593$	14.4881	0.594955	0.297478	−	0.954729i	$-0.403855\pi$
$593$	14.4881	0.594955	0.297478	+	0.954729i	$0.403855\pi$
$594$	0	0
$595$	−15.9747	−0.654900
$596$	0	0
$597$	−26.8900	−1.10054
$598$	0	0
$599$	29.3814	1.20049	0.600245	−	0.799816i	$-0.295070\pi$
$599$	29.3814	1.20049	0.600245	+	0.799816i	$0.295070\pi$
$600$	0	0
$601$	−0.578613	−0.0236021	−0.0118011	−	0.999930i	$-0.503756\pi$
$601$	−0.578613	−0.0236021	−0.0118011	+	0.999930i	$0.503756\pi$
$602$	0	0
$603$	−7.61237	−0.310000
$604$	0	0
$605$	4.33574	0.176273
$606$	0	0
$607$	5.51793	0.223966	0.111983	−	0.993710i	$-0.464280\pi$
$607$	5.51793	0.223966	0.111983	+	0.993710i	$0.464280\pi$
$608$	0	0
$609$	52.3498	2.12132
$610$	0	0
$611$	42.0745	1.70215
$612$	0	0
$613$	9.18481	0.370971	0.185486	−	0.982647i	$-0.440614\pi$
$613$	9.18481	0.370971	0.185486	+	0.982647i	$0.440614\pi$
$614$	0	0
$615$	6.10452	0.246158
$616$	0	0
$617$	47.0623	1.89466	0.947329	−	0.320261i	$-0.103771\pi$
$617$	47.0623	1.89466	0.947329	+	0.320261i	$0.103771\pi$
$618$	0	0
$619$	−9.03710	−0.363232	−0.181616	−	0.983370i	$-0.558133\pi$
$619$	−9.03710	−0.363232	−0.181616	+	0.983370i	$0.558133\pi$
$620$	0	0
$621$	−31.4660	−1.26269
$622$	0	0
$623$	1.38434	0.0554626
$624$	0	0
$625$	15.6462	0.625850
$626$	0	0
$627$	23.6893	0.946059
$628$	0	0
$629$	−15.8612	−0.632426
$630$	0	0
$631$	−35.3376	−1.40677	−0.703384	−	0.710810i	$-0.748328\pi$
$631$	−35.3376	−1.40677	−0.703384	+	0.710810i	$0.748328\pi$
$632$	0	0
$633$	−46.4483	−1.84615
$634$	0	0
$635$	7.46096	0.296079
$636$	0	0
$637$	−18.5592	−0.735340
$638$	0	0
$639$	−12.0451	−0.476498
$640$	0	0
$641$	−38.4145	−1.51728	−0.758640	−	0.651510i	$-0.774136\pi$
$641$	−38.4145	−1.51728	−0.758640	+	0.651510i	$0.774136\pi$
$642$	0	0
$643$	−0.214027	−0.00844038	−0.00422019	−	0.999991i	$-0.501343\pi$
$643$	−0.214027	−0.00844038	−0.00422019	+	0.999991i	$0.501343\pi$
$644$	0	0
$645$	15.1737	0.597463
$646$	0	0
$647$	13.2869	0.522362	0.261181	−	0.965290i	$-0.415888\pi$
$647$	13.2869	0.522362	0.261181	+	0.965290i	$0.415888\pi$
$648$	0	0
$649$	27.8765	1.09425
$650$	0	0
$651$	62.8374	2.46279
$652$	0	0
$653$	−16.6369	−0.651051	−0.325526	−	0.945533i	$-0.605541\pi$
$653$	−16.6369	−0.651051	−0.325526	+	0.945533i	$0.605541\pi$
$654$	0	0
$655$	−9.12160	−0.356410
$656$	0	0
$657$	2.98026	0.116271
$658$	0	0
$659$	34.2857	1.33558	0.667790	−	0.744350i	$-0.267240\pi$
$659$	34.2857	1.33558	0.667790	+	0.744350i	$0.267240\pi$
$660$	0	0
$661$	−30.9807	−1.20501	−0.602506	−	0.798115i	$-0.705831\pi$
$661$	−30.9807	−1.20501	−0.602506	+	0.798115i	$0.705831\pi$
$662$	0	0
$663$	51.6042	2.00414
$664$	0	0
$665$	8.05598	0.312397
$666$	0	0
$667$	59.5091	2.30420
$668$	0	0
$669$	−34.6225	−1.33858
$670$	0	0
$671$	−30.5919	−1.18099
$672$	0	0
$673$	−15.9279	−0.613977	−0.306989	−	0.951713i	$-0.599321\pi$
$673$	−15.9279	−0.613977	−0.306989	+	0.951713i	$0.599321\pi$
$674$	0	0
$675$	18.3525	0.706387
$676$	0	0
$677$	−36.9190	−1.41891	−0.709457	−	0.704749i	$-0.751060\pi$
$677$	−36.9190	−1.41891	−0.709457	+	0.704749i	$0.751060\pi$
$678$	0	0
$679$	28.0758	1.07745
$680$	0	0
$681$	−47.4066	−1.81662
$682$	0	0
$683$	31.4896	1.20492	0.602459	−	0.798150i	$-0.294188\pi$
$683$	31.4896	1.20492	0.602459	+	0.798150i	$0.294188\pi$
$684$	0	0
$685$	0.286240	0.0109367
$686$	0	0
$687$	−31.8789	−1.21625
$688$	0	0
$689$	43.9874	1.67579
$690$	0	0
$691$	−5.15383	−0.196061	−0.0980305	−	0.995183i	$-0.531254\pi$
$691$	−5.15383	−0.196061	−0.0980305	+	0.995183i	$0.531254\pi$
$692$	0	0
$693$	11.4756	0.435922
$694$	0	0
$695$	8.63030	0.327366
$696$	0	0
$697$	−22.8095	−0.863972
$698$	0	0
$699$	−26.8355	−1.01501
$700$	0	0
$701$	−6.46137	−0.244043	−0.122021	−	0.992527i	$-0.538938\pi$
$701$	−6.46137	−0.244043	−0.122021	+	0.992527i	$0.538938\pi$
$702$	0	0
$703$	7.99870	0.301677
$704$	0	0
$705$	−14.9940	−0.564705
$706$	0	0
$707$	−29.7295	−1.11809
$708$	0	0
$709$	5.47356	0.205564	0.102782	−	0.994704i	$-0.467226\pi$
$709$	5.47356	0.205564	0.102782	+	0.994704i	$0.467226\pi$
$710$	0	0
$711$	−8.18575	−0.306989
$712$	0	0
$713$	71.4308	2.67511
$714$	0	0
$715$	−14.5198	−0.543009
$716$	0	0
$717$	−33.6699	−1.25742
$718$	0	0
$719$	42.2216	1.57460	0.787300	−	0.616570i	$-0.211478\pi$
$719$	42.2216	1.57460	0.787300	+	0.616570i	$0.211478\pi$
$720$	0	0
$721$	41.7133	1.55349
$722$	0	0
$723$	−23.6064	−0.877932
$724$	0	0
$725$	−34.7085	−1.28904
$726$	0	0
$727$	3.78101	0.140230	0.0701149	−	0.997539i	$-0.477663\pi$
$727$	3.78101	0.140230	0.0701149	+	0.997539i	$0.477663\pi$
$728$	0	0
$729$	15.6558	0.579845
$730$	0	0
$731$	−56.6964	−2.09699
$732$	0	0
$733$	−3.93004	−0.145159	−0.0725797	−	0.997363i	$-0.523123\pi$
$733$	−3.93004	−0.145159	−0.0725797	+	0.997363i	$0.523123\pi$
$734$	0	0
$735$	6.61387	0.243956
$736$	0	0
$737$	−36.3016	−1.33719
$738$	0	0
$739$	13.9677	0.513810	0.256905	−	0.966437i	$-0.417297\pi$
$739$	13.9677	0.513810	0.256905	+	0.966437i	$0.417297\pi$
$740$	0	0
$741$	−26.0237	−0.956005
$742$	0	0
$743$	14.9788	0.549520	0.274760	−	0.961513i	$-0.411402\pi$
$743$	14.9788	0.549520	0.274760	+	0.961513i	$0.411402\pi$
$744$	0	0
$745$	17.4153	0.638046
$746$	0	0
$747$	7.99946	0.292685
$748$	0	0
$749$	37.3310	1.36405
$750$	0	0
$751$	38.1760	1.39306	0.696530	−	0.717528i	$-0.254726\pi$
$751$	38.1760	1.39306	0.696530	+	0.717528i	$0.254726\pi$
$752$	0	0
$753$	−1.96174	−0.0714899
$754$	0	0
$755$	−5.45641	−0.198579
$756$	0	0
$757$	−43.6988	−1.58826	−0.794130	−	0.607748i	$-0.792073\pi$
$757$	−43.6988	−1.58826	−0.794130	+	0.607748i	$0.792073\pi$
$758$	0	0
$759$	59.1711	2.14778
$760$	0	0
$761$	−39.9011	−1.44641	−0.723206	−	0.690632i	$-0.757332\pi$
$761$	−39.9011	−1.44641	−0.723206	+	0.690632i	$0.757332\pi$
$762$	0	0
$763$	−54.4114	−1.96982
$764$	0	0
$765$	−4.05430	−0.146583
$766$	0	0
$767$	−30.6236	−1.10575
$768$	0	0
$769$	−10.3619	−0.373659	−0.186830	−	0.982392i	$-0.559821\pi$
$769$	−10.3619	−0.373659	−0.186830	+	0.982392i	$0.559821\pi$
$770$	0	0
$771$	43.6744	1.57290
$772$	0	0
$773$	−12.5439	−0.451174	−0.225587	−	0.974223i	$-0.572430\pi$
$773$	−12.5439	−0.451174	−0.225587	+	0.974223i	$0.572430\pi$
$774$	0	0
$775$	−41.6619	−1.49654
$776$	0	0
$777$	17.5756	0.630521
$778$	0	0
$779$	11.5027	0.412128
$780$	0	0
$781$	−57.4404	−2.05538
$782$	0	0
$783$	−33.6926	−1.20408
$784$	0	0
$785$	−3.79436	−0.135426
$786$	0	0
$787$	11.0440	0.393677	0.196838	−	0.980436i	$-0.436933\pi$
$787$	11.0440	0.393677	0.196838	+	0.980436i	$0.436933\pi$
$788$	0	0
$789$	58.9522	2.09876
$790$	0	0
$791$	−11.7515	−0.417834
$792$	0	0
$793$	33.6066	1.19340
$794$	0	0
$795$	−15.6757	−0.555959
$796$	0	0
$797$	18.6839	0.661818	0.330909	−	0.943663i	$-0.392645\pi$
$797$	18.6839	0.661818	0.330909	+	0.943663i	$0.392645\pi$
$798$	0	0
$799$	56.0248	1.98202
$800$	0	0
$801$	0.351339	0.0124139
$802$	0	0
$803$	14.2122	0.501537
$804$	0	0
$805$	20.1222	0.709216
$806$	0	0
$807$	27.3124	0.961441
$808$	0	0
$809$	47.3125	1.66342	0.831709	−	0.555211i	$-0.187363\pi$
$809$	47.3125	1.66342	0.831709	+	0.555211i	$0.187363\pi$
$810$	0	0
$811$	−5.37107	−0.188604	−0.0943019	−	0.995544i	$-0.530062\pi$
$811$	−5.37107	−0.188604	−0.0943019	+	0.995544i	$0.530062\pi$
$812$	0	0
$813$	−62.8494	−2.20422
$814$	0	0
$815$	2.43841	0.0854138
$816$	0	0
$817$	28.5917	1.00030
$818$	0	0
$819$	−12.6065	−0.440505
$820$	0	0
$821$	3.31427	0.115669	0.0578344	−	0.998326i	$-0.481580\pi$
$821$	3.31427	0.115669	0.0578344	+	0.998326i	$0.481580\pi$
$822$	0	0
$823$	3.62186	0.126250	0.0631250	−	0.998006i	$-0.479893\pi$
$823$	3.62186	0.126250	0.0631250	+	0.998006i	$0.479893\pi$
$824$	0	0
$825$	−34.5114	−1.20153
$826$	0	0
$827$	−32.6087	−1.13392	−0.566959	−	0.823746i	$-0.691880\pi$
$827$	−32.6087	−1.13392	−0.566959	+	0.823746i	$0.691880\pi$
$828$	0	0
$829$	−3.90979	−0.135793	−0.0678963	−	0.997692i	$-0.521629\pi$
$829$	−3.90979	−0.135793	−0.0678963	+	0.997692i	$0.521629\pi$
$830$	0	0
$831$	53.7866	1.86584
$832$	0	0
$833$	−24.7127	−0.856244
$834$	0	0
$835$	16.1992	0.560596
$836$	0	0
$837$	−40.4425	−1.39790
$838$	0	0
$839$	16.9656	0.585718	0.292859	−	0.956156i	$-0.405393\pi$
$839$	16.9656	0.585718	0.292859	+	0.956156i	$0.405393\pi$
$840$	0	0
$841$	34.7201	1.19724
$842$	0	0
$843$	−4.63330	−0.159579
$844$	0	0
$845$	5.45423	0.187631
$846$	0	0
$847$	17.9516	0.616823
$848$	0	0
$849$	−12.5680	−0.431334
$850$	0	0
$851$	19.9792	0.684877
$852$	0	0
$853$	−46.0843	−1.57790	−0.788948	−	0.614459i	$-0.789374\pi$
$853$	−46.0843	−1.57790	−0.788948	+	0.614459i	$0.789374\pi$
$854$	0	0
$855$	2.04456	0.0699225
$856$	0	0
$857$	−12.5346	−0.428173	−0.214087	−	0.976815i	$-0.568678\pi$
$857$	−12.5346	−0.428173	−0.214087	+	0.976815i	$0.568678\pi$
$858$	0	0
$859$	−42.6511	−1.45524	−0.727618	−	0.685982i	$-0.759373\pi$
$859$	−42.6511	−1.45524	−0.727618	+	0.685982i	$0.759373\pi$
$860$	0	0
$861$	25.2750	0.861369
$862$	0	0
$863$	57.3229	1.95129	0.975647	−	0.219346i	$-0.0703922\pi$
$863$	57.3229	1.95129	0.975647	+	0.219346i	$0.0703922\pi$
$864$	0	0
$865$	−5.67898	−0.193091
$866$	0	0
$867$	35.3646	1.20104
$868$	0	0
$869$	−39.0359	−1.32420
$870$	0	0
$871$	39.8789	1.35125
$872$	0	0
$873$	7.12548	0.241161
$874$	0	0
$875$	−25.2321	−0.853002
$876$	0	0
$877$	32.6917	1.10392	0.551960	−	0.833870i	$-0.313880\pi$
$877$	32.6917	1.10392	0.551960	+	0.833870i	$0.313880\pi$
$878$	0	0
$879$	−4.66906	−0.157484
$880$	0	0
$881$	−13.4415	−0.452854	−0.226427	−	0.974028i	$-0.572704\pi$
$881$	−13.4415	−0.452854	−0.226427	+	0.974028i	$0.572704\pi$
$882$	0	0
$883$	6.02293	0.202688	0.101344	−	0.994851i	$-0.467686\pi$
$883$	6.02293	0.202688	0.101344	+	0.994851i	$0.467686\pi$
$884$	0	0
$885$	10.9132	0.366844
$886$	0	0
$887$	40.8986	1.37324	0.686620	−	0.727016i	$-0.259094\pi$
$887$	40.8986	1.37324	0.686620	+	0.727016i	$0.259094\pi$
$888$	0	0
$889$	30.8911	1.03606
$890$	0	0
$891$	−43.7995	−1.46734
$892$	0	0
$893$	−28.2530	−0.945452
$894$	0	0
$895$	14.0290	0.468937
$896$	0	0
$897$	−65.0021	−2.17036
$898$	0	0
$899$	76.4855	2.55093
$900$	0	0
$901$	58.5720	1.95132
$902$	0	0
$903$	62.8247	2.09067
$904$	0	0
$905$	−20.0074	−0.665070
$906$	0	0
$907$	−15.0074	−0.498313	−0.249157	−	0.968463i	$-0.580153\pi$
$907$	−15.0074	−0.498313	−0.249157	+	0.968463i	$0.580153\pi$
$908$	0	0
$909$	−7.54517	−0.250258
$910$	0	0
$911$	42.5509	1.40978	0.704888	−	0.709319i	$-0.250997\pi$
$911$	42.5509	1.40978	0.704888	+	0.709319i	$0.250997\pi$
$912$	0	0
$913$	38.1475	1.26250
$914$	0	0
$915$	−11.9763	−0.395923
$916$	0	0
$917$	−37.7668	−1.24717
$918$	0	0
$919$	3.59673	0.118645	0.0593226	−	0.998239i	$-0.481106\pi$
$919$	3.59673	0.118645	0.0593226	+	0.998239i	$0.481106\pi$
$920$	0	0
$921$	22.7154	0.748499
$922$	0	0
$923$	63.1008	2.07699
$924$	0	0
$925$	−11.6528	−0.383142
$926$	0	0
$927$	10.5866	0.347710
$928$	0	0
$929$	28.1899	0.924882	0.462441	−	0.886650i	$-0.346974\pi$
$929$	28.1899	0.924882	0.462441	+	0.886650i	$0.346974\pi$
$930$	0	0
$931$	12.4625	0.408441
$932$	0	0
$933$	36.4776	1.19422
$934$	0	0
$935$	−19.3340	−0.632289
$936$	0	0
$937$	47.0171	1.53598	0.767990	−	0.640462i	$-0.221257\pi$
$937$	47.0171	1.53598	0.767990	+	0.640462i	$0.221257\pi$
$938$	0	0
$939$	−26.4393	−0.862813
$940$	0	0
$941$	−3.16805	−0.103275	−0.0516377	−	0.998666i	$-0.516444\pi$
$941$	−3.16805	−0.103275	−0.0516377	+	0.998666i	$0.516444\pi$
$942$	0	0
$943$	28.7315	0.935627
$944$	0	0
$945$	−11.3927	−0.370606
$946$	0	0
$947$	−6.73552	−0.218875	−0.109438	−	0.993994i	$-0.534905\pi$
$947$	−6.73552	−0.218875	−0.109438	+	0.993994i	$0.534905\pi$
$948$	0	0
$949$	−15.6127	−0.506810
$950$	0	0
$951$	−47.6514	−1.54520
$952$	0	0
$953$	11.4333	0.370362	0.185181	−	0.982704i	$-0.440713\pi$
$953$	11.4333	0.370362	0.185181	+	0.982704i	$0.440713\pi$
$954$	0	0
$955$	−1.54044	−0.0498476
$956$	0	0
$957$	63.3583	2.04808
$958$	0	0
$959$	1.18514	0.0382701
$960$	0	0
$961$	60.8082	1.96156
$962$	0	0
$963$	9.47441	0.305309
$964$	0	0
$965$	−2.92124	−0.0940380
$966$	0	0
$967$	47.7263	1.53477	0.767387	−	0.641185i	$-0.221557\pi$
$967$	47.7263	1.53477	0.767387	+	0.641185i	$0.221557\pi$
$968$	0	0
$969$	−34.6522	−1.11319
$970$	0	0
$971$	25.5475	0.819860	0.409930	−	0.912117i	$-0.365553\pi$
$971$	25.5475	0.819860	0.409930	+	0.912117i	$0.365553\pi$
$972$	0	0
$973$	35.7326	1.14554
$974$	0	0
$975$	37.9123	1.21417
$976$	0	0
$977$	18.3454	0.586922	0.293461	−	0.955971i	$-0.405193\pi$
$977$	18.3454	0.586922	0.293461	+	0.955971i	$0.405193\pi$
$978$	0	0
$979$	1.67545	0.0535477
$980$	0	0
$981$	−13.8093	−0.440897
$982$	0	0
$983$	3.50579	0.111817	0.0559087	−	0.998436i	$-0.482194\pi$
$983$	3.50579	0.111817	0.0559087	+	0.998436i	$0.482194\pi$
$984$	0	0
$985$	−2.27211	−0.0723955
$986$	0	0
$987$	−62.0805	−1.97604
$988$	0	0
$989$	71.4164	2.27091
$990$	0	0
$991$	12.4935	0.396869	0.198434	−	0.980114i	$-0.436414\pi$
$991$	12.4935	0.396869	0.198434	+	0.980114i	$0.436414\pi$
$992$	0	0
$993$	54.1101	1.71713
$994$	0	0
$995$	−11.0674	−0.350860
$996$	0	0
$997$	−62.1074	−1.96696	−0.983480	−	0.181016i	$-0.942061\pi$
$997$	−62.1074	−1.96696	−0.983480	+	0.181016i	$0.942061\pi$
$998$	0	0
$999$	−11.3117	−0.357888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.17	✓	23
4.3	odd	2		4016.2.a.m.1.7		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.17	✓	23	1.1	even	1	trivial
4016.2.a.m.1.7		23	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.96174 q^{3} +0.807414 q^{5} +3.34299 q^{7} +0.848432 q^{9} +O(q^{10})\)	\(q+1.96174 q^{3} +0.807414 q^{5} +3.34299 q^{7} +0.848432 q^{9} +4.04597 q^{11} -4.44468 q^{13} +1.58394 q^{15} -5.91837 q^{17} +2.98461 q^{19} +6.55809 q^{21} +7.45495 q^{23} -4.34808 q^{25} -4.22082 q^{27} +7.98249 q^{29} +9.58166 q^{31} +7.93716 q^{33} +2.69918 q^{35} +2.67999 q^{37} -8.71932 q^{39} +3.85402 q^{41} +9.57972 q^{43} +0.685036 q^{45} -9.46625 q^{47} +4.17559 q^{49} -11.6103 q^{51} -9.89664 q^{53} +3.26677 q^{55} +5.85503 q^{57} +6.88994 q^{59} -7.56108 q^{61} +2.83630 q^{63} -3.58870 q^{65} -8.97228 q^{67} +14.6247 q^{69} -14.1969 q^{71} +3.51267 q^{73} -8.52982 q^{75} +13.5257 q^{77} -9.64808 q^{79} -10.8255 q^{81} +9.42852 q^{83} -4.77858 q^{85} +15.6596 q^{87} +0.414104 q^{89} -14.8585 q^{91} +18.7967 q^{93} +2.40981 q^{95} +8.39841 q^{97} +3.43274 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * q^99

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