LMFDB - Embedded newform 2008.2.a.d.1.21

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			1.21
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)

$f(q)$	$=$	$q+3.15136 q^{3} +4.04418 q^{5} +3.52087 q^{7} +6.93109 q^{9} +O(q^{10})$	$q+3.15136 q^{3} +4.04418 q^{5} +3.52087 q^{7} +6.93109 q^{9} -5.53748 q^{11} +0.479348 q^{13} +12.7447 q^{15} -7.91995 q^{17} -5.32818 q^{19} +11.0955 q^{21} -3.98503 q^{23} +11.3554 q^{25} +12.3883 q^{27} +4.58479 q^{29} -0.0567383 q^{31} -17.4506 q^{33} +14.2390 q^{35} -1.74446 q^{37} +1.51060 q^{39} +4.51530 q^{41} -11.5560 q^{43} +28.0306 q^{45} +11.2736 q^{47} +5.39651 q^{49} -24.9586 q^{51} +4.14986 q^{53} -22.3946 q^{55} -16.7910 q^{57} -4.02416 q^{59} -12.3346 q^{61} +24.4034 q^{63} +1.93857 q^{65} -5.97615 q^{67} -12.5583 q^{69} +5.10917 q^{71} +2.78681 q^{73} +35.7849 q^{75} -19.4967 q^{77} -2.61228 q^{79} +18.2467 q^{81} +1.48896 q^{83} -32.0297 q^{85} +14.4483 q^{87} -6.81110 q^{89} +1.68772 q^{91} -0.178803 q^{93} -21.5481 q^{95} -5.45770 q^{97} -38.3808 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.15136	1.81944	0.909720	−	0.415222i	$-0.136296\pi$
$3$	3.15136	1.81944	0.909720	+	0.415222i	$0.136296\pi$
$4$	0	0
$5$	4.04418	1.80861	0.904306	−	0.426885i	$-0.140389\pi$
$5$	4.04418	1.80861	0.904306	+	0.426885i	$0.140389\pi$
$6$	0	0
$7$	3.52087	1.33076	0.665381	−	0.746504i	$-0.268269\pi$
$7$	3.52087	1.33076	0.665381	+	0.746504i	$0.268269\pi$
$8$	0	0
$9$	6.93109	2.31036
$10$	0	0
$11$	−5.53748	−1.66961	−0.834806	−	0.550544i	$-0.814420\pi$
$11$	−5.53748	−1.66961	−0.834806	+	0.550544i	$0.814420\pi$
$12$	0	0
$13$	0.479348	0.132947	0.0664735	−	0.997788i	$-0.478825\pi$
$13$	0.479348	0.132947	0.0664735	+	0.997788i	$0.478825\pi$
$14$	0	0
$15$	12.7447	3.29066
$16$	0	0
$17$	−7.91995	−1.92087	−0.960435	−	0.278503i	$-0.910162\pi$
$17$	−7.91995	−1.92087	−0.960435	+	0.278503i	$0.910162\pi$
$18$	0	0
$19$	−5.32818	−1.22237	−0.611184	−	0.791489i	$-0.709306\pi$
$19$	−5.32818	−1.22237	−0.611184	+	0.791489i	$0.709306\pi$
$20$	0	0
$21$	11.0955	2.42124
$22$	0	0
$23$	−3.98503	−0.830936	−0.415468	−	0.909608i	$-0.636382\pi$
$23$	−3.98503	−0.830936	−0.415468	+	0.909608i	$0.636382\pi$
$24$	0	0
$25$	11.3554	2.27108
$26$	0	0
$27$	12.3883	2.38413
$28$	0	0
$29$	4.58479	0.851375	0.425687	−	0.904870i	$-0.360032\pi$
$29$	4.58479	0.851375	0.425687	+	0.904870i	$0.360032\pi$
$30$	0	0
$31$	−0.0567383	−0.0101905	−0.00509525	−	0.999987i	$-0.501622\pi$
$31$	−0.0567383	−0.0101905	−0.00509525	+	0.999987i	$0.501622\pi$
$32$	0	0
$33$	−17.4506	−3.03776
$34$	0	0
$35$	14.2390	2.40683
$36$	0	0
$37$	−1.74446	−0.286787	−0.143394	−	0.989666i	$-0.545802\pi$
$37$	−1.74446	−0.286787	−0.143394	+	0.989666i	$0.545802\pi$
$38$	0	0
$39$	1.51060	0.241889
$40$	0	0
$41$	4.51530	0.705171	0.352586	−	0.935780i	$-0.385303\pi$
$41$	4.51530	0.705171	0.352586	+	0.935780i	$0.385303\pi$
$42$	0	0
$43$	−11.5560	−1.76227	−0.881135	−	0.472864i	$-0.843220\pi$
$43$	−11.5560	−1.76227	−0.881135	+	0.472864i	$0.843220\pi$
$44$	0	0
$45$	28.0306	4.17855
$46$	0	0
$47$	11.2736	1.64442	0.822212	−	0.569181i	$-0.192740\pi$
$47$	11.2736	1.64442	0.822212	+	0.569181i	$0.192740\pi$
$48$	0	0
$49$	5.39651	0.770929
$50$	0	0
$51$	−24.9586	−3.49491
$52$	0	0
$53$	4.14986	0.570027	0.285014	−	0.958523i	$-0.408002\pi$
$53$	4.14986	0.570027	0.285014	+	0.958523i	$0.408002\pi$
$54$	0	0
$55$	−22.3946	−3.01968
$56$	0	0
$57$	−16.7910	−2.22403
$58$	0	0
$59$	−4.02416	−0.523901	−0.261951	−	0.965081i	$-0.584366\pi$
$59$	−4.02416	−0.523901	−0.261951	+	0.965081i	$0.584366\pi$
$60$	0	0
$61$	−12.3346	−1.57929	−0.789643	−	0.613566i	$-0.789734\pi$
$61$	−12.3346	−1.57929	−0.789643	+	0.613566i	$0.789734\pi$
$62$	0	0
$63$	24.4034	3.07455
$64$	0	0
$65$	1.93857	0.240450
$66$	0	0
$67$	−5.97615	−0.730103	−0.365051	−	0.930987i	$-0.618949\pi$
$67$	−5.97615	−0.730103	−0.365051	+	0.930987i	$0.618949\pi$
$68$	0	0
$69$	−12.5583	−1.51184
$70$	0	0
$71$	5.10917	0.606347	0.303173	−	0.952935i	$-0.401954\pi$
$71$	5.10917	0.606347	0.303173	+	0.952935i	$0.401954\pi$
$72$	0	0
$73$	2.78681	0.326171	0.163086	−	0.986612i	$-0.447855\pi$
$73$	2.78681	0.326171	0.163086	+	0.986612i	$0.447855\pi$
$74$	0	0
$75$	35.7849	4.13209
$76$	0	0
$77$	−19.4967	−2.22186
$78$	0	0
$79$	−2.61228	−0.293904	−0.146952	−	0.989144i	$-0.546946\pi$
$79$	−2.61228	−0.293904	−0.146952	+	0.989144i	$0.546946\pi$
$80$	0	0
$81$	18.2467	2.02742
$82$	0	0
$83$	1.48896	0.163435	0.0817173	−	0.996656i	$-0.473960\pi$
$83$	1.48896	0.163435	0.0817173	+	0.996656i	$0.473960\pi$
$84$	0	0
$85$	−32.0297	−3.47411
$86$	0	0
$87$	14.4483	1.54903
$88$	0	0
$89$	−6.81110	−0.721976	−0.360988	−	0.932571i	$-0.617560\pi$
$89$	−6.81110	−0.721976	−0.360988	+	0.932571i	$0.617560\pi$
$90$	0	0
$91$	1.68772	0.176921
$92$	0	0
$93$	−0.178803	−0.0185410
$94$	0	0
$95$	−21.5481	−2.21079
$96$	0	0
$97$	−5.45770	−0.554145	−0.277072	−	0.960849i	$-0.589364\pi$
$97$	−5.45770	−0.554145	−0.277072	+	0.960849i	$0.589364\pi$
$98$	0	0
$99$	−38.3808	−3.85741
$100$	0	0
$101$	1.79407	0.178517	0.0892585	−	0.996008i	$-0.471550\pi$
$101$	1.79407	0.178517	0.0892585	+	0.996008i	$0.471550\pi$
$102$	0	0
$103$	7.62771	0.751581	0.375790	−	0.926705i	$-0.377371\pi$
$103$	7.62771	0.751581	0.375790	+	0.926705i	$0.377371\pi$
$104$	0	0
$105$	44.8723	4.37909
$106$	0	0
$107$	6.94989	0.671872	0.335936	−	0.941885i	$-0.390947\pi$
$107$	6.94989	0.671872	0.335936	+	0.941885i	$0.390947\pi$
$108$	0	0
$109$	−13.2310	−1.26730	−0.633650	−	0.773620i	$-0.718444\pi$
$109$	−13.2310	−1.26730	−0.633650	+	0.773620i	$0.718444\pi$
$110$	0	0
$111$	−5.49743	−0.521793
$112$	0	0
$113$	16.5091	1.55304	0.776521	−	0.630092i	$-0.216983\pi$
$113$	16.5091	1.55304	0.776521	+	0.630092i	$0.216983\pi$
$114$	0	0
$115$	−16.1162	−1.50284
$116$	0	0
$117$	3.32240	0.307156
$118$	0	0
$119$	−27.8851	−2.55622
$120$	0	0
$121$	19.6637	1.78761
$122$	0	0
$123$	14.2293	1.28302
$124$	0	0
$125$	25.7023	2.29888
$126$	0	0
$127$	1.48799	0.132037	0.0660187	−	0.997818i	$-0.478970\pi$
$127$	1.48799	0.132037	0.0660187	+	0.997818i	$0.478970\pi$
$128$	0	0
$129$	−36.4171	−3.20635
$130$	0	0
$131$	1.69420	0.148023	0.0740116	−	0.997257i	$-0.476420\pi$
$131$	1.69420	0.148023	0.0740116	+	0.997257i	$0.476420\pi$
$132$	0	0
$133$	−18.7598	−1.62668
$134$	0	0
$135$	50.1005	4.31196
$136$	0	0
$137$	9.76601	0.834367	0.417183	−	0.908822i	$-0.363017\pi$
$137$	9.76601	0.834367	0.417183	+	0.908822i	$0.363017\pi$
$138$	0	0
$139$	7.40302	0.627916	0.313958	−	0.949437i	$-0.398345\pi$
$139$	7.40302	0.627916	0.313958	+	0.949437i	$0.398345\pi$
$140$	0	0
$141$	35.5272	2.99193
$142$	0	0
$143$	−2.65438	−0.221970
$144$	0	0
$145$	18.5417	1.53981
$146$	0	0
$147$	17.0063	1.40266
$148$	0	0
$149$	13.2514	1.08560	0.542801	−	0.839862i	$-0.317364\pi$
$149$	13.2514	1.08560	0.542801	+	0.839862i	$0.317364\pi$
$150$	0	0
$151$	−6.04285	−0.491760	−0.245880	−	0.969300i	$-0.579077\pi$
$151$	−6.04285	−0.491760	−0.245880	+	0.969300i	$0.579077\pi$
$152$	0	0
$153$	−54.8939	−4.43791
$154$	0	0
$155$	−0.229460	−0.0184307
$156$	0	0
$157$	6.09159	0.486162	0.243081	−	0.970006i	$-0.421842\pi$
$157$	6.09159	0.486162	0.243081	+	0.970006i	$0.421842\pi$
$158$	0	0
$159$	13.0777	1.03713
$160$	0	0
$161$	−14.0308	−1.10578
$162$	0	0
$163$	−3.28142	−0.257021	−0.128510	−	0.991708i	$-0.541020\pi$
$163$	−3.28142	−0.257021	−0.128510	+	0.991708i	$0.541020\pi$
$164$	0	0
$165$	−70.5734	−5.49413
$166$	0	0
$167$	−7.80322	−0.603832	−0.301916	−	0.953335i	$-0.597626\pi$
$167$	−7.80322	−0.603832	−0.301916	+	0.953335i	$0.597626\pi$
$168$	0	0
$169$	−12.7702	−0.982325
$170$	0	0
$171$	−36.9301	−2.82411
$172$	0	0
$173$	−8.64285	−0.657104	−0.328552	−	0.944486i	$-0.606561\pi$
$173$	−8.64285	−0.657104	−0.328552	+	0.944486i	$0.606561\pi$
$174$	0	0
$175$	39.9808	3.02226
$176$	0	0
$177$	−12.6816	−0.953207
$178$	0	0
$179$	−15.5814	−1.16461	−0.582306	−	0.812970i	$-0.697849\pi$
$179$	−15.5814	−1.16461	−0.582306	+	0.812970i	$0.697849\pi$
$180$	0	0
$181$	11.2473	0.836008	0.418004	−	0.908445i	$-0.362730\pi$
$181$	11.2473	0.836008	0.418004	+	0.908445i	$0.362730\pi$
$182$	0	0
$183$	−38.8709	−2.87342
$184$	0	0
$185$	−7.05491	−0.518687
$186$	0	0
$187$	43.8566	3.20711
$188$	0	0
$189$	43.6175	3.17271
$190$	0	0
$191$	−1.81146	−0.131073	−0.0655364	−	0.997850i	$-0.520876\pi$
$191$	−1.81146	−0.131073	−0.0655364	+	0.997850i	$0.520876\pi$
$192$	0	0
$193$	9.19667	0.661991	0.330995	−	0.943632i	$-0.392616\pi$
$193$	9.19667	0.661991	0.330995	+	0.943632i	$0.392616\pi$
$194$	0	0
$195$	6.10913	0.437484
$196$	0	0
$197$	10.9081	0.777169	0.388585	−	0.921413i	$-0.372964\pi$
$197$	10.9081	0.777169	0.388585	+	0.921413i	$0.372964\pi$
$198$	0	0
$199$	11.1164	0.788023	0.394012	−	0.919105i	$-0.371087\pi$
$199$	11.1164	0.788023	0.394012	+	0.919105i	$0.371087\pi$
$200$	0	0
$201$	−18.8330	−1.32838
$202$	0	0
$203$	16.1424	1.13298
$204$	0	0
$205$	18.2607	1.27538
$206$	0	0
$207$	−27.6206	−1.91976
$208$	0	0
$209$	29.5047	2.04088
$210$	0	0
$211$	12.5194	0.861868	0.430934	−	0.902383i	$-0.358184\pi$
$211$	12.5194	0.861868	0.430934	+	0.902383i	$0.358184\pi$
$212$	0	0
$213$	16.1008	1.10321
$214$	0	0
$215$	−46.7345	−3.18726
$216$	0	0
$217$	−0.199768	−0.0135611
$218$	0	0
$219$	8.78225	0.593449
$220$	0	0
$221$	−3.79641	−0.255374
$222$	0	0
$223$	−12.9011	−0.863924	−0.431962	−	0.901892i	$-0.642179\pi$
$223$	−12.9011	−0.863924	−0.431962	+	0.901892i	$0.642179\pi$
$224$	0	0
$225$	78.7051	5.24701
$226$	0	0
$227$	−12.4519	−0.826463	−0.413231	−	0.910626i	$-0.635600\pi$
$227$	−12.4519	−0.826463	−0.413231	+	0.910626i	$0.635600\pi$
$228$	0	0
$229$	−10.6207	−0.701834	−0.350917	−	0.936407i	$-0.614130\pi$
$229$	−10.6207	−0.701834	−0.350917	+	0.936407i	$0.614130\pi$
$230$	0	0
$231$	−61.4413	−4.04254
$232$	0	0
$233$	19.6738	1.28887	0.644436	−	0.764658i	$-0.277092\pi$
$233$	19.6738	1.28887	0.644436	+	0.764658i	$0.277092\pi$
$234$	0	0
$235$	45.5925	2.97412
$236$	0	0
$237$	−8.23223	−0.534741
$238$	0	0
$239$	25.8440	1.67171	0.835855	−	0.548951i	$-0.184972\pi$
$239$	25.8440	1.67171	0.835855	+	0.548951i	$0.184972\pi$
$240$	0	0
$241$	10.3953	0.669618	0.334809	−	0.942286i	$-0.391328\pi$
$241$	10.3953	0.669618	0.334809	+	0.942286i	$0.391328\pi$
$242$	0	0
$243$	20.3372	1.30463
$244$	0	0
$245$	21.8244	1.39431
$246$	0	0
$247$	−2.55405	−0.162510
$248$	0	0
$249$	4.69225	0.297359
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	22.0670	1.38734
$254$	0	0
$255$	−100.937	−6.32093
$256$	0	0
$257$	−21.1531	−1.31949	−0.659746	−	0.751489i	$-0.729336\pi$
$257$	−21.1531	−1.31949	−0.659746	+	0.751489i	$0.729336\pi$
$258$	0	0
$259$	−6.14201	−0.381646
$260$	0	0
$261$	31.7776	1.96699
$262$	0	0
$263$	−23.8106	−1.46822	−0.734112	−	0.679028i	$-0.762401\pi$
$263$	−23.8106	−1.46822	−0.734112	+	0.679028i	$0.762401\pi$
$264$	0	0
$265$	16.7828	1.03096
$266$	0	0
$267$	−21.4643	−1.31359
$268$	0	0
$269$	17.2857	1.05392	0.526962	−	0.849889i	$-0.323331\pi$
$269$	17.2857	1.05392	0.526962	+	0.849889i	$0.323331\pi$
$270$	0	0
$271$	−28.7163	−1.74439	−0.872194	−	0.489159i	$-0.837304\pi$
$271$	−28.7163	−1.74439	−0.872194	+	0.489159i	$0.837304\pi$
$272$	0	0
$273$	5.31862	0.321897
$274$	0	0
$275$	−62.8802	−3.79182
$276$	0	0
$277$	−8.14989	−0.489679	−0.244840	−	0.969564i	$-0.578735\pi$
$277$	−8.14989	−0.489679	−0.244840	+	0.969564i	$0.578735\pi$
$278$	0	0
$279$	−0.393259	−0.0235438
$280$	0	0
$281$	29.1216	1.73725	0.868626	−	0.495469i	$-0.165004\pi$
$281$	29.1216	1.73725	0.868626	+	0.495469i	$0.165004\pi$
$282$	0	0
$283$	−0.474087	−0.0281816	−0.0140908	−	0.999901i	$-0.504485\pi$
$283$	−0.474087	−0.0281816	−0.0140908	+	0.999901i	$0.504485\pi$
$284$	0	0
$285$	−67.9059	−4.02240
$286$	0	0
$287$	15.8978	0.938416
$288$	0	0
$289$	45.7256	2.68974
$290$	0	0
$291$	−17.1992	−1.00823
$292$	0	0
$293$	28.8082	1.68299	0.841497	−	0.540261i	$-0.181675\pi$
$293$	28.8082	1.68299	0.841497	+	0.540261i	$0.181675\pi$
$294$	0	0
$295$	−16.2744	−0.947533
$296$	0	0
$297$	−68.5999	−3.98057
$298$	0	0
$299$	−1.91021	−0.110470
$300$	0	0
$301$	−40.6871	−2.34516
$302$	0	0
$303$	5.65378	0.324801
$304$	0	0
$305$	−49.8834	−2.85632
$306$	0	0
$307$	−26.9035	−1.53546	−0.767732	−	0.640771i	$-0.778615\pi$
$307$	−26.9035	−1.53546	−0.767732	+	0.640771i	$0.778615\pi$
$308$	0	0
$309$	24.0377	1.36746
$310$	0	0
$311$	16.0223	0.908540	0.454270	−	0.890864i	$-0.349900\pi$
$311$	16.0223	0.908540	0.454270	+	0.890864i	$0.349900\pi$
$312$	0	0
$313$	−16.1881	−0.915004	−0.457502	−	0.889209i	$-0.651256\pi$
$313$	−16.1881	−0.915004	−0.457502	+	0.889209i	$0.651256\pi$
$314$	0	0
$315$	98.6919	5.56066
$316$	0	0
$317$	3.76329	0.211367	0.105684	−	0.994400i	$-0.466297\pi$
$317$	3.76329	0.211367	0.105684	+	0.994400i	$0.466297\pi$
$318$	0	0
$319$	−25.3882	−1.42147
$320$	0	0
$321$	21.9016	1.22243
$322$	0	0
$323$	42.1989	2.34801
$324$	0	0
$325$	5.44317	0.301933
$326$	0	0
$327$	−41.6957	−2.30578
$328$	0	0
$329$	39.6929	2.18834
$330$	0	0
$331$	26.1622	1.43800	0.719001	−	0.695009i	$-0.244599\pi$
$331$	26.1622	1.43800	0.719001	+	0.695009i	$0.244599\pi$
$332$	0	0
$333$	−12.0910	−0.662583
$334$	0	0
$335$	−24.1686	−1.32047
$336$	0	0
$337$	15.6236	0.851070	0.425535	−	0.904942i	$-0.360086\pi$
$337$	15.6236	0.851070	0.425535	+	0.904942i	$0.360086\pi$
$338$	0	0
$339$	52.0260	2.82567
$340$	0	0
$341$	0.314187	0.0170142
$342$	0	0
$343$	−5.64569	−0.304839
$344$	0	0
$345$	−50.7879	−2.73433
$346$	0	0
$347$	27.7409	1.48921	0.744604	−	0.667506i	$-0.232638\pi$
$347$	27.7409	1.48921	0.744604	+	0.667506i	$0.232638\pi$
$348$	0	0
$349$	−22.9806	−1.23012	−0.615062	−	0.788479i	$-0.710869\pi$
$349$	−22.9806	−1.23012	−0.615062	+	0.788479i	$0.710869\pi$
$350$	0	0
$351$	5.93830	0.316963
$352$	0	0
$353$	−6.88236	−0.366311	−0.183155	−	0.983084i	$-0.558631\pi$
$353$	−6.88236	−0.366311	−0.183155	+	0.983084i	$0.558631\pi$
$354$	0	0
$355$	20.6624	1.09665
$356$	0	0
$357$	−87.8761	−4.65089
$358$	0	0
$359$	−14.9320	−0.788083	−0.394041	−	0.919093i	$-0.628923\pi$
$359$	−14.9320	−0.788083	−0.394041	+	0.919093i	$0.628923\pi$
$360$	0	0
$361$	9.38949	0.494184
$362$	0	0
$363$	61.9674	3.25244
$364$	0	0
$365$	11.2704	0.589917
$366$	0	0
$367$	−9.32339	−0.486677	−0.243338	−	0.969941i	$-0.578243\pi$
$367$	−9.32339	−0.486677	−0.243338	+	0.969941i	$0.578243\pi$
$368$	0	0
$369$	31.2959	1.62920
$370$	0	0
$371$	14.6111	0.758571
$372$	0	0
$373$	20.1887	1.04533	0.522665	−	0.852538i	$-0.324938\pi$
$373$	20.1887	1.04533	0.522665	+	0.852538i	$0.324938\pi$
$374$	0	0
$375$	80.9972	4.18268
$376$	0	0
$377$	2.19771	0.113188
$378$	0	0
$379$	−19.1073	−0.981477	−0.490739	−	0.871307i	$-0.663273\pi$
$379$	−19.1073	−0.981477	−0.490739	+	0.871307i	$0.663273\pi$
$380$	0	0
$381$	4.68919	0.240234
$382$	0	0
$383$	−4.05445	−0.207173	−0.103586	−	0.994620i	$-0.533032\pi$
$383$	−4.05445	−0.207173	−0.103586	+	0.994620i	$0.533032\pi$
$384$	0	0
$385$	−78.8482	−4.01848
$386$	0	0
$387$	−80.0956	−4.07149
$388$	0	0
$389$	−10.3037	−0.522419	−0.261209	−	0.965282i	$-0.584121\pi$
$389$	−10.3037	−0.522419	−0.261209	+	0.965282i	$0.584121\pi$
$390$	0	0
$391$	31.5612	1.59612
$392$	0	0
$393$	5.33905	0.269319
$394$	0	0
$395$	−10.5645	−0.531558
$396$	0	0
$397$	37.7089	1.89256	0.946278	−	0.323353i	$-0.104810\pi$
$397$	37.7089	1.89256	0.946278	+	0.323353i	$0.104810\pi$
$398$	0	0
$399$	−59.1190	−2.95965
$400$	0	0
$401$	14.4754	0.722867	0.361434	−	0.932398i	$-0.382287\pi$
$401$	14.4754	0.722867	0.361434	+	0.932398i	$0.382287\pi$
$402$	0	0
$403$	−0.0271974	−0.00135480
$404$	0	0
$405$	73.7931	3.66681
$406$	0	0
$407$	9.65991	0.478824
$408$	0	0
$409$	12.5803	0.622054	0.311027	−	0.950401i	$-0.399327\pi$
$409$	12.5803	0.622054	0.311027	+	0.950401i	$0.399327\pi$
$410$	0	0
$411$	30.7763	1.51808
$412$	0	0
$413$	−14.1685	−0.697188
$414$	0	0
$415$	6.02162	0.295590
$416$	0	0
$417$	23.3296	1.14246
$418$	0	0
$419$	4.00486	0.195650	0.0978251	−	0.995204i	$-0.468811\pi$
$419$	4.00486	0.195650	0.0978251	+	0.995204i	$0.468811\pi$
$420$	0	0
$421$	6.98847	0.340597	0.170299	−	0.985392i	$-0.445527\pi$
$421$	6.98847	0.340597	0.170299	+	0.985392i	$0.445527\pi$
$422$	0	0
$423$	78.1384	3.79922
$424$	0	0
$425$	−89.9340	−4.36244
$426$	0	0
$427$	−43.4286	−2.10166
$428$	0	0
$429$	−8.36491	−0.403862
$430$	0	0
$431$	−12.3992	−0.597248	−0.298624	−	0.954371i	$-0.596528\pi$
$431$	−12.3992	−0.597248	−0.298624	+	0.954371i	$0.596528\pi$
$432$	0	0
$433$	20.9395	1.00629	0.503143	−	0.864203i	$-0.332177\pi$
$433$	20.9395	1.00629	0.503143	+	0.864203i	$0.332177\pi$
$434$	0	0
$435$	58.4317	2.80159
$436$	0	0
$437$	21.2329	1.01571
$438$	0	0
$439$	8.17498	0.390170	0.195085	−	0.980786i	$-0.437502\pi$
$439$	8.17498	0.390170	0.195085	+	0.980786i	$0.437502\pi$
$440$	0	0
$441$	37.4037	1.78113
$442$	0	0
$443$	−4.88162	−0.231933	−0.115966	−	0.993253i	$-0.536996\pi$
$443$	−4.88162	−0.231933	−0.115966	+	0.993253i	$0.536996\pi$
$444$	0	0
$445$	−27.5453	−1.30577
$446$	0	0
$447$	41.7601	1.97519
$448$	0	0
$449$	−3.88168	−0.183188	−0.0915940	−	0.995796i	$-0.529196\pi$
$449$	−3.88168	−0.183188	−0.0915940	+	0.995796i	$0.529196\pi$
$450$	0	0
$451$	−25.0034	−1.17736
$452$	0	0
$453$	−19.0432	−0.894728
$454$	0	0
$455$	6.82544	0.319981
$456$	0	0
$457$	−35.5909	−1.66487	−0.832435	−	0.554123i	$-0.813054\pi$
$457$	−35.5909	−1.66487	−0.832435	+	0.554123i	$0.813054\pi$
$458$	0	0
$459$	−98.1147	−4.57960
$460$	0	0
$461$	20.6008	0.959475	0.479738	−	0.877412i	$-0.340732\pi$
$461$	20.6008	0.959475	0.479738	+	0.877412i	$0.340732\pi$
$462$	0	0
$463$	16.5277	0.768108	0.384054	−	0.923311i	$-0.374528\pi$
$463$	16.5277	0.768108	0.384054	+	0.923311i	$0.374528\pi$
$464$	0	0
$465$	−0.723112	−0.0335335
$466$	0	0
$467$	−12.4680	−0.576950	−0.288475	−	0.957487i	$-0.593148\pi$
$467$	−12.4680	−0.576950	−0.288475	+	0.957487i	$0.593148\pi$
$468$	0	0
$469$	−21.0412	−0.971594
$470$	0	0
$471$	19.1968	0.884542
$472$	0	0
$473$	63.9910	2.94231
$474$	0	0
$475$	−60.5035	−2.77609
$476$	0	0
$477$	28.7631	1.31697
$478$	0	0
$479$	16.9698	0.775370	0.387685	−	0.921792i	$-0.373275\pi$
$479$	16.9698	0.775370	0.387685	+	0.921792i	$0.373275\pi$
$480$	0	0
$481$	−0.836203	−0.0381276
$482$	0	0
$483$	−44.2160	−2.01190
$484$	0	0
$485$	−22.0719	−1.00223
$486$	0	0
$487$	17.2328	0.780893	0.390446	−	0.920626i	$-0.372321\pi$
$487$	17.2328	0.780893	0.390446	+	0.920626i	$0.372321\pi$
$488$	0	0
$489$	−10.3409	−0.467634
$490$	0	0
$491$	−18.7874	−0.847865	−0.423933	−	0.905694i	$-0.639351\pi$
$491$	−18.7874	−0.847865	−0.423933	+	0.905694i	$0.639351\pi$
$492$	0	0
$493$	−36.3113	−1.63538
$494$	0	0
$495$	−155.219	−6.97656
$496$	0	0
$497$	17.9887	0.806904
$498$	0	0
$499$	36.7584	1.64553	0.822765	−	0.568381i	$-0.192430\pi$
$499$	36.7584	1.64553	0.822765	+	0.568381i	$0.192430\pi$
$500$	0	0
$501$	−24.5908	−1.09864
$502$	0	0
$503$	−17.0001	−0.757996	−0.378998	−	0.925397i	$-0.623731\pi$
$503$	−17.0001	−0.757996	−0.378998	+	0.925397i	$0.623731\pi$
$504$	0	0
$505$	7.25555	0.322868
$506$	0	0
$507$	−40.2436	−1.78728
$508$	0	0
$509$	11.0726	0.490783	0.245391	−	0.969424i	$-0.421084\pi$
$509$	11.0726	0.490783	0.245391	+	0.969424i	$0.421084\pi$
$510$	0	0
$511$	9.81198	0.434057
$512$	0	0
$513$	−66.0070	−2.91428
$514$	0	0
$515$	30.8478	1.35932
$516$	0	0
$517$	−62.4273	−2.74555
$518$	0	0
$519$	−27.2368	−1.19556
$520$	0	0
$521$	−0.0217845	−0.000954397 0	−0.000477198	−	1.00000i	$-0.500152\pi$
$521$	−0.0217845	−0.000954397 0	−0.000477198		1.00000i	$0.500152\pi$
$522$	0	0
$523$	−33.5812	−1.46840	−0.734201	−	0.678932i	$-0.762443\pi$
$523$	−33.5812	−1.46840	−0.734201	+	0.678932i	$0.762443\pi$
$524$	0	0
$525$	125.994	5.49883
$526$	0	0
$527$	0.449365	0.0195746
$528$	0	0
$529$	−7.11956	−0.309546
$530$	0	0
$531$	−27.8918	−1.21040
$532$	0	0
$533$	2.16440	0.0937505
$534$	0	0
$535$	28.1066	1.21515
$536$	0	0
$537$	−49.1028	−2.11894
$538$	0	0
$539$	−29.8830	−1.28715
$540$	0	0
$541$	−13.5095	−0.580817	−0.290409	−	0.956903i	$-0.593791\pi$
$541$	−13.5095	−0.580817	−0.290409	+	0.956903i	$0.593791\pi$
$542$	0	0
$543$	35.4444	1.52107
$544$	0	0
$545$	−53.5085	−2.29205
$546$	0	0
$547$	16.1666	0.691233	0.345617	−	0.938376i	$-0.387670\pi$
$547$	16.1666	0.691233	0.345617	+	0.938376i	$0.387670\pi$
$548$	0	0
$549$	−85.4924	−3.64873
$550$	0	0
$551$	−24.4286	−1.04069
$552$	0	0
$553$	−9.19748	−0.391117
$554$	0	0
$555$	−22.2326	−0.943720
$556$	0	0
$557$	−41.9289	−1.77659	−0.888293	−	0.459278i	$-0.848108\pi$
$557$	−41.9289	−1.77659	−0.888293	+	0.459278i	$0.848108\pi$
$558$	0	0
$559$	−5.53933	−0.234289
$560$	0	0
$561$	138.208	5.83515
$562$	0	0
$563$	5.70801	0.240564	0.120282	−	0.992740i	$-0.461620\pi$
$563$	5.70801	0.240564	0.120282	+	0.992740i	$0.461620\pi$
$564$	0	0
$565$	66.7656	2.80885
$566$	0	0
$567$	64.2444	2.69801
$568$	0	0
$569$	24.7184	1.03625	0.518125	−	0.855305i	$-0.326630\pi$
$569$	24.7184	1.03625	0.518125	+	0.855305i	$0.326630\pi$
$570$	0	0
$571$	−37.8063	−1.58214	−0.791072	−	0.611723i	$-0.790477\pi$
$571$	−37.8063	−1.58214	−0.791072	+	0.611723i	$0.790477\pi$
$572$	0	0
$573$	−5.70858	−0.238479
$574$	0	0
$575$	−45.2515	−1.88712
$576$	0	0
$577$	−37.7099	−1.56988	−0.784942	−	0.619569i	$-0.787307\pi$
$577$	−37.7099	−1.56988	−0.784942	+	0.619569i	$0.787307\pi$
$578$	0	0
$579$	28.9820	1.20445
$580$	0	0
$581$	5.24243	0.217493
$582$	0	0
$583$	−22.9798	−0.951725
$584$	0	0
$585$	13.4364	0.555526
$586$	0	0
$587$	1.98185	0.0817998	0.0408999	−	0.999163i	$-0.486978\pi$
$587$	1.98185	0.0817998	0.0408999	+	0.999163i	$0.486978\pi$
$588$	0	0
$589$	0.302312	0.0124566
$590$	0	0
$591$	34.3754	1.41401
$592$	0	0
$593$	−33.4131	−1.37211	−0.686055	−	0.727550i	$-0.740659\pi$
$593$	−33.4131	−1.37211	−0.686055	+	0.727550i	$0.740659\pi$
$594$	0	0
$595$	−112.772	−4.62321
$596$	0	0
$597$	35.0319	1.43376
$598$	0	0
$599$	24.4120	0.997447	0.498723	−	0.866761i	$-0.333802\pi$
$599$	24.4120	0.997447	0.498723	+	0.866761i	$0.333802\pi$
$600$	0	0
$601$	−0.535645	−0.0218494	−0.0109247	−	0.999940i	$-0.503478\pi$
$601$	−0.535645	−0.0218494	−0.0109247	+	0.999940i	$0.503478\pi$
$602$	0	0
$603$	−41.4212	−1.68680
$604$	0	0
$605$	79.5234	3.23309
$606$	0	0
$607$	−38.4095	−1.55899	−0.779497	−	0.626406i	$-0.784525\pi$
$607$	−38.4095	−1.55899	−0.779497	+	0.626406i	$0.784525\pi$
$608$	0	0
$609$	50.8707	2.06139
$610$	0	0
$611$	5.40397	0.218621
$612$	0	0
$613$	−10.0723	−0.406815	−0.203407	−	0.979094i	$-0.565202\pi$
$613$	−10.0723	−0.406815	−0.203407	+	0.979094i	$0.565202\pi$
$614$	0	0
$615$	57.5460	2.32048
$616$	0	0
$617$	−22.1667	−0.892396	−0.446198	−	0.894934i	$-0.647222\pi$
$617$	−22.1667	−0.892396	−0.446198	+	0.894934i	$0.647222\pi$
$618$	0	0
$619$	31.1859	1.25347	0.626734	−	0.779233i	$-0.284391\pi$
$619$	31.1859	1.25347	0.626734	+	0.779233i	$0.284391\pi$
$620$	0	0
$621$	−49.3677	−1.98106
$622$	0	0
$623$	−23.9810	−0.960778
$624$	0	0
$625$	47.1677	1.88671
$626$	0	0
$627$	92.9800	3.71326
$628$	0	0
$629$	13.8160	0.550882
$630$	0	0
$631$	0.787730	0.0313590	0.0156795	−	0.999877i	$-0.495009\pi$
$631$	0.787730	0.0313590	0.0156795	+	0.999877i	$0.495009\pi$
$632$	0	0
$633$	39.4531	1.56812
$634$	0	0
$635$	6.01768	0.238804
$636$	0	0
$637$	2.58680	0.102493
$638$	0	0
$639$	35.4121	1.40088
$640$	0	0
$641$	−38.1821	−1.50810	−0.754051	−	0.656816i	$-0.771903\pi$
$641$	−38.1821	−1.50810	−0.754051	+	0.656816i	$0.771903\pi$
$642$	0	0
$643$	24.7804	0.977243	0.488622	−	0.872496i	$-0.337500\pi$
$643$	24.7804	0.977243	0.488622	+	0.872496i	$0.337500\pi$
$644$	0	0
$645$	−147.277	−5.79904
$646$	0	0
$647$	−44.2350	−1.73906	−0.869528	−	0.493883i	$-0.835577\pi$
$647$	−44.2350	−1.73906	−0.869528	+	0.493883i	$0.835577\pi$
$648$	0	0
$649$	22.2837	0.874712
$650$	0	0
$651$	−0.629542	−0.0246737
$652$	0	0
$653$	−6.69132	−0.261852	−0.130926	−	0.991392i	$-0.541795\pi$
$653$	−6.69132	−0.261852	−0.130926	+	0.991392i	$0.541795\pi$
$654$	0	0
$655$	6.85166	0.267716
$656$	0	0
$657$	19.3156	0.753574
$658$	0	0
$659$	−19.5366	−0.761040	−0.380520	−	0.924773i	$-0.624255\pi$
$659$	−19.5366	−0.761040	−0.380520	+	0.924773i	$0.624255\pi$
$660$	0	0
$661$	−48.5989	−1.89028	−0.945139	−	0.326668i	$-0.894074\pi$
$661$	−48.5989	−1.89028	−0.945139	+	0.326668i	$0.894074\pi$
$662$	0	0
$663$	−11.9639	−0.464638
$664$	0	0
$665$	−75.8680	−2.94204
$666$	0	0
$667$	−18.2705	−0.707438
$668$	0	0
$669$	−40.6562	−1.57186
$670$	0	0
$671$	68.3027	2.63680
$672$	0	0
$673$	12.2954	0.473954	0.236977	−	0.971515i	$-0.423843\pi$
$673$	12.2954	0.473954	0.236977	+	0.971515i	$0.423843\pi$
$674$	0	0
$675$	140.674	5.41454
$676$	0	0
$677$	18.3173	0.703989	0.351995	−	0.936002i	$-0.385504\pi$
$677$	18.3173	0.703989	0.351995	+	0.936002i	$0.385504\pi$
$678$	0	0
$679$	−19.2158	−0.737435
$680$	0	0
$681$	−39.2405	−1.50370
$682$	0	0
$683$	3.01922	0.115527	0.0577637	−	0.998330i	$-0.481603\pi$
$683$	3.01922	0.115527	0.0577637	+	0.998330i	$0.481603\pi$
$684$	0	0
$685$	39.4955	1.50905
$686$	0	0
$687$	−33.4696	−1.27694
$688$	0	0
$689$	1.98923	0.0757834
$690$	0	0
$691$	−42.6494	−1.62246	−0.811229	−	0.584728i	$-0.801201\pi$
$691$	−42.6494	−1.62246	−0.811229	+	0.584728i	$0.801201\pi$
$692$	0	0
$693$	−135.134	−5.13330
$694$	0	0
$695$	29.9391	1.13566
$696$	0	0
$697$	−35.7610	−1.35454
$698$	0	0
$699$	61.9992	2.34503
$700$	0	0
$701$	−32.4132	−1.22423	−0.612115	−	0.790769i	$-0.709681\pi$
$701$	−32.4132	−1.22423	−0.612115	+	0.790769i	$0.709681\pi$
$702$	0	0
$703$	9.29480	0.350560
$704$	0	0
$705$	143.678	5.41124
$706$	0	0
$707$	6.31670	0.237564
$708$	0	0
$709$	−42.2051	−1.58505	−0.792524	−	0.609841i	$-0.791233\pi$
$709$	−42.2051	−1.58505	−0.792524	+	0.609841i	$0.791233\pi$
$710$	0	0
$711$	−18.1059	−0.679025
$712$	0	0
$713$	0.226104	0.00846766
$714$	0	0
$715$	−10.7348	−0.401458
$716$	0	0
$717$	81.4438	3.04158
$718$	0	0
$719$	14.1237	0.526725	0.263363	−	0.964697i	$-0.415168\pi$
$719$	14.1237	0.526725	0.263363	+	0.964697i	$0.415168\pi$
$720$	0	0
$721$	26.8562	1.00018
$722$	0	0
$723$	32.7592	1.21833
$724$	0	0
$725$	52.0621	1.93354
$726$	0	0
$727$	7.03623	0.260959	0.130480	−	0.991451i	$-0.458348\pi$
$727$	7.03623	0.260959	0.130480	+	0.991451i	$0.458348\pi$
$728$	0	0
$729$	9.34978	0.346288
$730$	0	0
$731$	91.5228	3.38509
$732$	0	0
$733$	−12.5438	−0.463315	−0.231657	−	0.972797i	$-0.574415\pi$
$733$	−12.5438	−0.463315	−0.231657	+	0.972797i	$0.574415\pi$
$734$	0	0
$735$	68.7767	2.53687
$736$	0	0
$737$	33.0928	1.21899
$738$	0	0
$739$	15.0521	0.553700	0.276850	−	0.960913i	$-0.410709\pi$
$739$	15.0521	0.553700	0.276850	+	0.960913i	$0.410709\pi$
$740$	0	0
$741$	−8.04874	−0.295678
$742$	0	0
$743$	43.8186	1.60755	0.803774	−	0.594935i	$-0.202822\pi$
$743$	43.8186	1.60755	0.803774	+	0.594935i	$0.202822\pi$
$744$	0	0
$745$	53.5912	1.96343
$746$	0	0
$747$	10.3201	0.377593
$748$	0	0
$749$	24.4697	0.894102
$750$	0	0
$751$	46.5683	1.69930	0.849650	−	0.527346i	$-0.176813\pi$
$751$	46.5683	1.69930	0.849650	+	0.527346i	$0.176813\pi$
$752$	0	0
$753$	−3.15136	−0.114842
$754$	0	0
$755$	−24.4384	−0.889403
$756$	0	0
$757$	30.6548	1.11417	0.557083	−	0.830457i	$-0.311920\pi$
$757$	30.6548	1.11417	0.557083	+	0.830457i	$0.311920\pi$
$758$	0	0
$759$	69.5412	2.52418
$760$	0	0
$761$	25.2379	0.914873	0.457437	−	0.889242i	$-0.348768\pi$
$761$	25.2379	0.914873	0.457437	+	0.889242i	$0.348768\pi$
$762$	0	0
$763$	−46.5846	−1.68648
$764$	0	0
$765$	−222.001	−8.02645
$766$	0	0
$767$	−1.92897	−0.0696511
$768$	0	0
$769$	7.00969	0.252776	0.126388	−	0.991981i	$-0.459662\pi$
$769$	7.00969	0.252776	0.126388	+	0.991981i	$0.459662\pi$
$770$	0	0
$771$	−66.6610	−2.40074
$772$	0	0
$773$	20.9877	0.754873	0.377437	−	0.926035i	$-0.376806\pi$
$773$	20.9877	0.754873	0.377437	+	0.926035i	$0.376806\pi$
$774$	0	0
$775$	−0.644285	−0.0231434
$776$	0	0
$777$	−19.3557	−0.694382
$778$	0	0
$779$	−24.0583	−0.861979
$780$	0	0
$781$	−28.2919	−1.01236
$782$	0	0
$783$	56.7978	2.02979
$784$	0	0
$785$	24.6355	0.879278
$786$	0	0
$787$	34.8551	1.24245	0.621225	−	0.783632i	$-0.286635\pi$
$787$	34.8551	1.24245	0.621225	+	0.783632i	$0.286635\pi$
$788$	0	0
$789$	−75.0359	−2.67135
$790$	0	0
$791$	58.1262	2.06673
$792$	0	0
$793$	−5.91257	−0.209962
$794$	0	0
$795$	52.8886	1.87577
$796$	0	0
$797$	−37.7830	−1.33834	−0.669171	−	0.743109i	$-0.733351\pi$
$797$	−37.7830	−1.33834	−0.669171	+	0.743109i	$0.733351\pi$
$798$	0	0
$799$	−89.2864	−3.15873
$800$	0	0
$801$	−47.2084	−1.66803
$802$	0	0
$803$	−15.4319	−0.544580
$804$	0	0
$805$	−56.7429	−1.99992
$806$	0	0
$807$	54.4734	1.91755
$808$	0	0
$809$	5.97233	0.209976	0.104988	−	0.994473i	$-0.466520\pi$
$809$	5.97233	0.209976	0.104988	+	0.994473i	$0.466520\pi$
$810$	0	0
$811$	15.0374	0.528035	0.264017	−	0.964518i	$-0.414952\pi$
$811$	15.0374	0.528035	0.264017	+	0.964518i	$0.414952\pi$
$812$	0	0
$813$	−90.4954	−3.17381
$814$	0	0
$815$	−13.2706	−0.464850
$816$	0	0
$817$	61.5723	2.15414
$818$	0	0
$819$	11.6977	0.408752
$820$	0	0
$821$	7.73076	0.269805	0.134903	−	0.990859i	$-0.456928\pi$
$821$	7.73076	0.269805	0.134903	+	0.990859i	$0.456928\pi$
$822$	0	0
$823$	−40.1506	−1.39956	−0.699782	−	0.714357i	$-0.746719\pi$
$823$	−40.1506	−1.39956	−0.699782	+	0.714357i	$0.746719\pi$
$824$	0	0
$825$	−198.158	−6.89898
$826$	0	0
$827$	−51.3430	−1.78537	−0.892686	−	0.450680i	$-0.851182\pi$
$827$	−51.3430	−1.78537	−0.892686	+	0.450680i	$0.851182\pi$
$828$	0	0
$829$	16.3083	0.566411	0.283205	−	0.959059i	$-0.408602\pi$
$829$	16.3083	0.566411	0.283205	+	0.959059i	$0.408602\pi$
$830$	0	0
$831$	−25.6833	−0.890942
$832$	0	0
$833$	−42.7401	−1.48086
$834$	0	0
$835$	−31.5576	−1.09210
$836$	0	0
$837$	−0.702891	−0.0242955
$838$	0	0
$839$	22.5986	0.780191	0.390096	−	0.920774i	$-0.372442\pi$
$839$	22.5986	0.780191	0.390096	+	0.920774i	$0.372442\pi$
$840$	0	0
$841$	−7.97967	−0.275161
$842$	0	0
$843$	91.7729	3.16083
$844$	0	0
$845$	−51.6451	−1.77664
$846$	0	0
$847$	69.2332	2.37888
$848$	0	0
$849$	−1.49402	−0.0512747
$850$	0	0
$851$	6.95172	0.238302
$852$	0	0
$853$	44.4187	1.52087	0.760434	−	0.649416i	$-0.224986\pi$
$853$	44.4187	1.52087	0.760434	+	0.649416i	$0.224986\pi$
$854$	0	0
$855$	−149.352	−5.10773
$856$	0	0
$857$	−34.4871	−1.17806	−0.589028	−	0.808113i	$-0.700489\pi$
$857$	−34.4871	−1.17806	−0.589028	+	0.808113i	$0.700489\pi$
$858$	0	0
$859$	−10.7049	−0.365247	−0.182624	−	0.983183i	$-0.558459\pi$
$859$	−10.7049	−0.365247	−0.182624	+	0.983183i	$0.558459\pi$
$860$	0	0
$861$	50.0996	1.70739
$862$	0	0
$863$	23.1619	0.788439	0.394220	−	0.919016i	$-0.371015\pi$
$863$	23.1619	0.788439	0.394220	+	0.919016i	$0.371015\pi$
$864$	0	0
$865$	−34.9532	−1.18844
$866$	0	0
$867$	144.098	4.89383
$868$	0	0
$869$	14.4654	0.490706
$870$	0	0
$871$	−2.86465	−0.0970650
$872$	0	0
$873$	−37.8278	−1.28028
$874$	0	0
$875$	90.4943	3.05927
$876$	0	0
$877$	43.4215	1.46624	0.733119	−	0.680100i	$-0.238064\pi$
$877$	43.4215	1.46624	0.733119	+	0.680100i	$0.238064\pi$
$878$	0	0
$879$	90.7852	3.06211
$880$	0	0
$881$	5.13169	0.172891	0.0864454	−	0.996257i	$-0.472449\pi$
$881$	5.13169	0.172891	0.0864454	+	0.996257i	$0.472449\pi$
$882$	0	0
$883$	23.4895	0.790486	0.395243	−	0.918577i	$-0.370660\pi$
$883$	23.4895	0.790486	0.395243	+	0.918577i	$0.370660\pi$
$884$	0	0
$885$	−51.2866	−1.72398
$886$	0	0
$887$	−14.5715	−0.489263	−0.244632	−	0.969616i	$-0.578667\pi$
$887$	−14.5715	−0.489263	−0.244632	+	0.969616i	$0.578667\pi$
$888$	0	0
$889$	5.23900	0.175711
$890$	0	0
$891$	−101.041	−3.38500
$892$	0	0
$893$	−60.0678	−2.01009
$894$	0	0
$895$	−63.0141	−2.10633
$896$	0	0
$897$	−6.01978	−0.200994
$898$	0	0
$899$	−0.260134	−0.00867594
$900$	0	0
$901$	−32.8667	−1.09495
$902$	0	0
$903$	−128.220	−4.26689
$904$	0	0
$905$	45.4862	1.51201
$906$	0	0
$907$	−13.5526	−0.450006	−0.225003	−	0.974358i	$-0.572239\pi$
$907$	−13.5526	−0.450006	−0.225003	+	0.974358i	$0.572239\pi$
$908$	0	0
$909$	12.4349	0.412439
$910$	0	0
$911$	−23.1056	−0.765522	−0.382761	−	0.923847i	$-0.625027\pi$
$911$	−23.1056	−0.765522	−0.382761	+	0.923847i	$0.625027\pi$
$912$	0	0
$913$	−8.24508	−0.272872
$914$	0	0
$915$	−157.201	−5.19690
$916$	0	0
$917$	5.96506	0.196984
$918$	0	0
$919$	−28.0667	−0.925834	−0.462917	−	0.886402i	$-0.653197\pi$
$919$	−28.0667	−0.925834	−0.462917	+	0.886402i	$0.653197\pi$
$920$	0	0
$921$	−84.7827	−2.79369
$922$	0	0
$923$	2.44907	0.0806121
$924$	0	0
$925$	−19.8090	−0.651316
$926$	0	0
$927$	52.8684	1.73642
$928$	0	0
$929$	11.0125	0.361309	0.180654	−	0.983547i	$-0.442178\pi$
$929$	11.0125	0.361309	0.180654	+	0.983547i	$0.442178\pi$
$930$	0	0
$931$	−28.7535	−0.942359
$932$	0	0
$933$	50.4920	1.65303
$934$	0	0
$935$	177.364	5.80042
$936$	0	0
$937$	37.1633	1.21407	0.607037	−	0.794674i	$-0.292358\pi$
$937$	37.1633	1.21407	0.607037	+	0.794674i	$0.292358\pi$
$938$	0	0
$939$	−51.0145	−1.66480
$940$	0	0
$941$	15.3274	0.499660	0.249830	−	0.968290i	$-0.419625\pi$
$941$	15.3274	0.499660	0.249830	+	0.968290i	$0.419625\pi$
$942$	0	0
$943$	−17.9936	−0.585952
$944$	0	0
$945$	176.397	5.73820
$946$	0	0
$947$	0.476399	0.0154809	0.00774045	−	0.999970i	$-0.497536\pi$
$947$	0.476399	0.0154809	0.00774045	+	0.999970i	$0.497536\pi$
$948$	0	0
$949$	1.33585	0.0433635
$950$	0	0
$951$	11.8595	0.384570
$952$	0	0
$953$	48.8116	1.58116	0.790581	−	0.612357i	$-0.209778\pi$
$953$	48.8116	1.58116	0.790581	+	0.612357i	$0.209778\pi$
$954$	0	0
$955$	−7.32588	−0.237060
$956$	0	0
$957$	−80.0074	−2.58627
$958$	0	0
$959$	34.3848	1.11034
$960$	0	0
$961$	−30.9968	−0.999896
$962$	0	0
$963$	48.1703	1.55227
$964$	0	0
$965$	37.1930	1.19728
$966$	0	0
$967$	10.1506	0.326421	0.163211	−	0.986591i	$-0.447815\pi$
$967$	10.1506	0.326421	0.163211	+	0.986591i	$0.447815\pi$
$968$	0	0
$969$	132.984	4.27207
$970$	0	0
$971$	−23.7954	−0.763631	−0.381815	−	0.924239i	$-0.624701\pi$
$971$	−23.7954	−0.763631	−0.381815	+	0.924239i	$0.624701\pi$
$972$	0	0
$973$	26.0651	0.835607
$974$	0	0
$975$	17.1534	0.549349
$976$	0	0
$977$	−3.16766	−0.101342	−0.0506712	−	0.998715i	$-0.516136\pi$
$977$	−3.16766	−0.101342	−0.0506712	+	0.998715i	$0.516136\pi$
$978$	0	0
$979$	37.7163	1.20542
$980$	0	0
$981$	−91.7053	−2.92792
$982$	0	0
$983$	55.7595	1.77845	0.889227	−	0.457467i	$-0.151243\pi$
$983$	55.7595	1.77845	0.889227	+	0.457467i	$0.151243\pi$
$984$	0	0
$985$	44.1143	1.40560
$986$	0	0
$987$	125.087	3.98155
$988$	0	0
$989$	46.0509	1.46433
$990$	0	0
$991$	45.3174	1.43956	0.719778	−	0.694204i	$-0.244244\pi$
$991$	45.3174	1.43956	0.719778	+	0.694204i	$0.244244\pi$
$992$	0	0
$993$	82.4465	2.61636
$994$	0	0
$995$	44.9569	1.42523
$996$	0	0
$997$	36.1146	1.14376	0.571880	−	0.820337i	$-0.306214\pi$
$997$	36.1146	1.14376	0.571880	+	0.820337i	$0.306214\pi$
$998$	0	0
$999$	−21.6109	−0.683738

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.21	✓	23
4.3	odd	2		4016.2.a.m.1.3		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.21	✓	23	1.1	even	1	trivial
4016.2.a.m.1.3		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)

\(f(q)\)	\(=\)	\(q+3.15136 q^{3} +4.04418 q^{5} +3.52087 q^{7} +6.93109 q^{9} +O(q^{10})\)	\(q+3.15136 q^{3} +4.04418 q^{5} +3.52087 q^{7} +6.93109 q^{9} -5.53748 q^{11} +0.479348 q^{13} +12.7447 q^{15} -7.91995 q^{17} -5.32818 q^{19} +11.0955 q^{21} -3.98503 q^{23} +11.3554 q^{25} +12.3883 q^{27} +4.58479 q^{29} -0.0567383 q^{31} -17.4506 q^{33} +14.2390 q^{35} -1.74446 q^{37} +1.51060 q^{39} +4.51530 q^{41} -11.5560 q^{43} +28.0306 q^{45} +11.2736 q^{47} +5.39651 q^{49} -24.9586 q^{51} +4.14986 q^{53} -22.3946 q^{55} -16.7910 q^{57} -4.02416 q^{59} -12.3346 q^{61} +24.4034 q^{63} +1.93857 q^{65} -5.97615 q^{67} -12.5583 q^{69} +5.10917 q^{71} +2.78681 q^{73} +35.7849 q^{75} -19.4967 q^{77} -2.61228 q^{79} +18.2467 q^{81} +1.48896 q^{83} -32.0297 q^{85} +14.4483 q^{87} -6.81110 q^{89} +1.68772 q^{91} -0.178803 q^{93} -21.5481 q^{95} -5.45770 q^{97} -38.3808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 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

Introduction

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