LMFDB - Embedded newform 2008.2.a.d.1.9

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			1.9
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)

$f(q)$	$=$	$q-1.15579 q^{3} -4.09592 q^{5} -0.621313 q^{7} -1.66414 q^{9} +O(q^{10})$	$q-1.15579 q^{3} -4.09592 q^{5} -0.621313 q^{7} -1.66414 q^{9} +0.176674 q^{11} -6.62113 q^{13} +4.73404 q^{15} -4.87879 q^{17} -6.67634 q^{19} +0.718109 q^{21} +1.95079 q^{23} +11.7766 q^{25} +5.39078 q^{27} -4.79872 q^{29} -3.71027 q^{31} -0.204199 q^{33} +2.54485 q^{35} +6.22498 q^{37} +7.65265 q^{39} +1.84850 q^{41} -6.89738 q^{43} +6.81620 q^{45} -0.545127 q^{47} -6.61397 q^{49} +5.63887 q^{51} -4.30758 q^{53} -0.723644 q^{55} +7.71646 q^{57} +11.0491 q^{59} -3.06372 q^{61} +1.03395 q^{63} +27.1196 q^{65} -9.29195 q^{67} -2.25471 q^{69} -6.86513 q^{71} +5.09365 q^{73} -13.6113 q^{75} -0.109770 q^{77} -7.28093 q^{79} -1.23820 q^{81} -18.1951 q^{83} +19.9832 q^{85} +5.54633 q^{87} +15.4585 q^{89} +4.11379 q^{91} +4.28831 q^{93} +27.3458 q^{95} +8.30697 q^{97} -0.294011 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.15579	−0.667297	−0.333649	−	0.942697i	$-0.608280\pi$
$3$	−1.15579	−0.667297	−0.333649	+	0.942697i	$0.608280\pi$
$4$	0	0
$5$	−4.09592	−1.83175	−0.915876	−	0.401460i	$-0.868503\pi$
$5$	−4.09592	−1.83175	−0.915876	+	0.401460i	$0.868503\pi$
$6$	0	0
$7$	−0.621313	−0.234834	−0.117417	−	0.993083i	$-0.537461\pi$
$7$	−0.621313	−0.234834	−0.117417	+	0.993083i	$0.537461\pi$
$8$	0	0
$9$	−1.66414	−0.554714
$10$	0	0
$11$	0.176674	0.0532693	0.0266346	−	0.999645i	$-0.491521\pi$
$11$	0.176674	0.0532693	0.0266346	+	0.999645i	$0.491521\pi$
$12$	0	0
$13$	−6.62113	−1.83637	−0.918185	−	0.396152i	$-0.870345\pi$
$13$	−6.62113	−1.83637	−0.918185	+	0.396152i	$0.870345\pi$
$14$	0	0
$15$	4.73404	1.22232
$16$	0	0
$17$	−4.87879	−1.18328	−0.591640	−	0.806202i	$-0.701519\pi$
$17$	−4.87879	−1.18328	−0.591640	+	0.806202i	$0.701519\pi$
$18$	0	0
$19$	−6.67634	−1.53166	−0.765828	−	0.643045i	$-0.777671\pi$
$19$	−6.67634	−1.53166	−0.765828	+	0.643045i	$0.777671\pi$
$20$	0	0
$21$	0.718109	0.156704
$22$	0	0
$23$	1.95079	0.406769	0.203384	−	0.979099i	$-0.434806\pi$
$23$	1.95079	0.406769	0.203384	+	0.979099i	$0.434806\pi$
$24$	0	0
$25$	11.7766	2.35532
$26$	0	0
$27$	5.39078	1.03746
$28$	0	0
$29$	−4.79872	−0.891100	−0.445550	−	0.895257i	$-0.646992\pi$
$29$	−4.79872	−0.891100	−0.445550	+	0.895257i	$0.646992\pi$
$30$	0	0
$31$	−3.71027	−0.666385	−0.333192	−	0.942859i	$-0.608126\pi$
$31$	−3.71027	−0.666385	−0.333192	+	0.942859i	$0.608126\pi$
$32$	0	0
$33$	−0.204199	−0.0355464
$34$	0	0
$35$	2.54485	0.430158
$36$	0	0
$37$	6.22498	1.02338	0.511690	−	0.859170i	$-0.329020\pi$
$37$	6.22498	1.02338	0.511690	+	0.859170i	$0.329020\pi$
$38$	0	0
$39$	7.65265	1.22541
$40$	0	0
$41$	1.84850	0.288688	0.144344	−	0.989528i	$-0.453893\pi$
$41$	1.84850	0.288688	0.144344	+	0.989528i	$0.453893\pi$
$42$	0	0
$43$	−6.89738	−1.05184	−0.525920	−	0.850534i	$-0.676279\pi$
$43$	−6.89738	−1.05184	−0.525920	+	0.850534i	$0.676279\pi$
$44$	0	0
$45$	6.81620	1.01610
$46$	0	0
$47$	−0.545127	−0.0795150	−0.0397575	−	0.999209i	$-0.512659\pi$
$47$	−0.545127	−0.0795150	−0.0397575	+	0.999209i	$0.512659\pi$
$48$	0	0
$49$	−6.61397	−0.944853
$50$	0	0
$51$	5.63887	0.789600
$52$	0	0
$53$	−4.30758	−0.591692	−0.295846	−	0.955236i	$-0.595602\pi$
$53$	−4.30758	−0.591692	−0.295846	+	0.955236i	$0.595602\pi$
$54$	0	0
$55$	−0.723644	−0.0975761
$56$	0	0
$57$	7.71646	1.02207
$58$	0	0
$59$	11.0491	1.43847	0.719237	−	0.694765i	$-0.244492\pi$
$59$	11.0491	1.43847	0.719237	+	0.694765i	$0.244492\pi$
$60$	0	0
$61$	−3.06372	−0.392269	−0.196135	−	0.980577i	$-0.562839\pi$
$61$	−3.06372	−0.392269	−0.196135	+	0.980577i	$0.562839\pi$
$62$	0	0
$63$	1.03395	0.130266
$64$	0	0
$65$	27.1196	3.36378
$66$	0	0
$67$	−9.29195	−1.13519	−0.567596	−	0.823307i	$-0.692127\pi$
$67$	−9.29195	−1.13519	−0.567596	+	0.823307i	$0.692127\pi$
$68$	0	0
$69$	−2.25471	−0.271436
$70$	0	0
$71$	−6.86513	−0.814741	−0.407370	−	0.913263i	$-0.633554\pi$
$71$	−6.86513	−0.814741	−0.407370	+	0.913263i	$0.633554\pi$
$72$	0	0
$73$	5.09365	0.596167	0.298083	−	0.954540i	$-0.403653\pi$
$73$	5.09365	0.596167	0.298083	+	0.954540i	$0.403653\pi$
$74$	0	0
$75$	−13.6113	−1.57170
$76$	0	0
$77$	−0.109770	−0.0125095
$78$	0	0
$79$	−7.28093	−0.819169	−0.409584	−	0.912272i	$-0.634326\pi$
$79$	−7.28093	−0.819169	−0.409584	+	0.912272i	$0.634326\pi$
$80$	0	0
$81$	−1.23820	−0.137578
$82$	0	0
$83$	−18.1951	−1.99717	−0.998586	−	0.0531544i	$-0.983072\pi$
$83$	−18.1951	−1.99717	−0.998586	+	0.0531544i	$0.983072\pi$
$84$	0	0
$85$	19.9832	2.16748
$86$	0	0
$87$	5.54633	0.594629
$88$	0	0
$89$	15.4585	1.63860	0.819298	−	0.573368i	$-0.194363\pi$
$89$	15.4585	1.63860	0.819298	+	0.573368i	$0.194363\pi$
$90$	0	0
$91$	4.11379	0.431243
$92$	0	0
$93$	4.28831	0.444677
$94$	0	0
$95$	27.3458	2.80562
$96$	0	0
$97$	8.30697	0.843445	0.421722	−	0.906725i	$-0.361426\pi$
$97$	8.30697	0.843445	0.421722	+	0.906725i	$0.361426\pi$
$98$	0	0
$99$	−0.294011	−0.0295492
$100$	0	0
$101$	−9.59493	−0.954731	−0.477365	−	0.878705i	$-0.658408\pi$
$101$	−9.59493	−0.954731	−0.477365	+	0.878705i	$0.658408\pi$
$102$	0	0
$103$	5.65954	0.557651	0.278825	−	0.960342i	$-0.410055\pi$
$103$	5.65954	0.557651	0.278825	+	0.960342i	$0.410055\pi$
$104$	0	0
$105$	−2.94132	−0.287044
$106$	0	0
$107$	15.7163	1.51935	0.759674	−	0.650304i	$-0.225359\pi$
$107$	15.7163	1.51935	0.759674	+	0.650304i	$0.225359\pi$
$108$	0	0
$109$	13.9298	1.33424	0.667118	−	0.744952i	$-0.267528\pi$
$109$	13.9298	1.33424	0.667118	+	0.744952i	$0.267528\pi$
$110$	0	0
$111$	−7.19479	−0.682899
$112$	0	0
$113$	11.6659	1.09743	0.548717	−	0.836008i	$-0.315117\pi$
$113$	11.6659	1.09743	0.548717	+	0.836008i	$0.315117\pi$
$114$	0	0
$115$	−7.99030	−0.745099
$116$	0	0
$117$	11.0185	1.01866
$118$	0	0
$119$	3.03126	0.277875
$120$	0	0
$121$	−10.9688	−0.997162
$122$	0	0
$123$	−2.13649	−0.192641
$124$	0	0
$125$	−27.7564	−2.48261
$126$	0	0
$127$	17.4141	1.54525	0.772624	−	0.634864i	$-0.218944\pi$
$127$	17.4141	1.54525	0.772624	+	0.634864i	$0.218944\pi$
$128$	0	0
$129$	7.97194	0.701890
$130$	0	0
$131$	13.4754	1.17735	0.588675	−	0.808370i	$-0.299650\pi$
$131$	13.4754	1.17735	0.588675	+	0.808370i	$0.299650\pi$
$132$	0	0
$133$	4.14810	0.359686
$134$	0	0
$135$	−22.0802	−1.90036
$136$	0	0
$137$	7.07181	0.604186	0.302093	−	0.953279i	$-0.402315\pi$
$137$	7.07181	0.604186	0.302093	+	0.953279i	$0.402315\pi$
$138$	0	0
$139$	−9.12679	−0.774124	−0.387062	−	0.922054i	$-0.626510\pi$
$139$	−9.12679	−0.774124	−0.387062	+	0.922054i	$0.626510\pi$
$140$	0	0
$141$	0.630054	0.0530601
$142$	0	0
$143$	−1.16978	−0.0978221
$144$	0	0
$145$	19.6552	1.63227
$146$	0	0
$147$	7.64438	0.630498
$148$	0	0
$149$	4.03604	0.330645	0.165322	−	0.986240i	$-0.447134\pi$
$149$	4.03604	0.330645	0.165322	+	0.986240i	$0.447134\pi$
$150$	0	0
$151$	−17.1672	−1.39705	−0.698525	−	0.715585i	$-0.746160\pi$
$151$	−17.1672	−1.39705	−0.698525	+	0.715585i	$0.746160\pi$
$152$	0	0
$153$	8.11901	0.656383
$154$	0	0
$155$	15.1970	1.22065
$156$	0	0
$157$	−12.9500	−1.03352	−0.516760	−	0.856130i	$-0.672862\pi$
$157$	−12.9500	−1.03352	−0.516760	+	0.856130i	$0.672862\pi$
$158$	0	0
$159$	4.97867	0.394834
$160$	0	0
$161$	−1.21205	−0.0955232
$162$	0	0
$163$	1.48780	0.116533	0.0582666	−	0.998301i	$-0.481443\pi$
$163$	1.48780	0.116533	0.0582666	+	0.998301i	$0.481443\pi$
$164$	0	0
$165$	0.836382	0.0651123
$166$	0	0
$167$	−18.1588	−1.40517	−0.702584	−	0.711600i	$-0.747971\pi$
$167$	−18.1588	−1.40517	−0.702584	+	0.711600i	$0.747971\pi$
$168$	0	0
$169$	30.8393	2.37226
$170$	0	0
$171$	11.1104	0.849632
$172$	0	0
$173$	1.60318	0.121888	0.0609439	−	0.998141i	$-0.480589\pi$
$173$	1.60318	0.121888	0.0609439	+	0.998141i	$0.480589\pi$
$174$	0	0
$175$	−7.31695	−0.553110
$176$	0	0
$177$	−12.7705	−0.959890
$178$	0	0
$179$	−6.35905	−0.475298	−0.237649	−	0.971351i	$-0.576377\pi$
$179$	−6.35905	−0.475298	−0.237649	+	0.971351i	$0.576377\pi$
$180$	0	0
$181$	−14.2799	−1.06142	−0.530709	−	0.847554i	$-0.678074\pi$
$181$	−14.2799	−1.06142	−0.530709	+	0.847554i	$0.678074\pi$
$182$	0	0
$183$	3.54103	0.261760
$184$	0	0
$185$	−25.4970	−1.87458
$186$	0	0
$187$	−0.861956	−0.0630325
$188$	0	0
$189$	−3.34936	−0.243630
$190$	0	0
$191$	−9.85686	−0.713217	−0.356609	−	0.934254i	$-0.616067\pi$
$191$	−9.85686	−0.713217	−0.356609	+	0.934254i	$0.616067\pi$
$192$	0	0
$193$	−24.8826	−1.79109	−0.895545	−	0.444971i	$-0.853214\pi$
$193$	−24.8826	−1.79109	−0.895545	+	0.444971i	$0.853214\pi$
$194$	0	0
$195$	−31.3447	−2.24464
$196$	0	0
$197$	−2.59600	−0.184957	−0.0924787	−	0.995715i	$-0.529479\pi$
$197$	−2.59600	−0.184957	−0.0924787	+	0.995715i	$0.529479\pi$
$198$	0	0
$199$	−14.1845	−1.00551	−0.502755	−	0.864429i	$-0.667680\pi$
$199$	−14.1845	−1.00551	−0.502755	+	0.864429i	$0.667680\pi$
$200$	0	0
$201$	10.7396	0.757511
$202$	0	0
$203$	2.98151	0.209261
$204$	0	0
$205$	−7.57133	−0.528805
$206$	0	0
$207$	−3.24640	−0.225640
$208$	0	0
$209$	−1.17954	−0.0815902
$210$	0	0
$211$	7.99464	0.550374	0.275187	−	0.961391i	$-0.411260\pi$
$211$	7.99464	0.550374	0.275187	+	0.961391i	$0.411260\pi$
$212$	0	0
$213$	7.93467	0.543674
$214$	0	0
$215$	28.2511	1.92671
$216$	0	0
$217$	2.30524	0.156490
$218$	0	0
$219$	−5.88721	−0.397821
$220$	0	0
$221$	32.3031	2.17294
$222$	0	0
$223$	−0.532491	−0.0356583	−0.0178291	−	0.999841i	$-0.505675\pi$
$223$	−0.532491	−0.0356583	−0.0178291	+	0.999841i	$0.505675\pi$
$224$	0	0
$225$	−19.5979	−1.30653
$226$	0	0
$227$	2.38209	0.158105	0.0790525	−	0.996870i	$-0.474811\pi$
$227$	2.38209	0.158105	0.0790525	+	0.996870i	$0.474811\pi$
$228$	0	0
$229$	−22.8726	−1.51146	−0.755732	−	0.654882i	$-0.772719\pi$
$229$	−22.8726	−1.51146	−0.755732	+	0.654882i	$0.772719\pi$
$230$	0	0
$231$	0.126871	0.00834752
$232$	0	0
$233$	17.8670	1.17051	0.585253	−	0.810850i	$-0.300995\pi$
$233$	17.8670	1.17051	0.585253	+	0.810850i	$0.300995\pi$
$234$	0	0
$235$	2.23280	0.145652
$236$	0	0
$237$	8.41525	0.546629
$238$	0	0
$239$	−15.6266	−1.01080	−0.505401	−	0.862884i	$-0.668656\pi$
$239$	−15.6266	−1.01080	−0.505401	+	0.862884i	$0.668656\pi$
$240$	0	0
$241$	−5.48709	−0.353455	−0.176727	−	0.984260i	$-0.556551\pi$
$241$	−5.48709	−0.353455	−0.176727	+	0.984260i	$0.556551\pi$
$242$	0	0
$243$	−14.7412	−0.945651
$244$	0	0
$245$	27.0903	1.73074
$246$	0	0
$247$	44.2049	2.81269
$248$	0	0
$249$	21.0298	1.33271
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	0.344655	0.0216683
$254$	0	0
$255$	−23.0964	−1.44635
$256$	0	0
$257$	8.10440	0.505538	0.252769	−	0.967527i	$-0.418659\pi$
$257$	8.10440	0.505538	0.252769	+	0.967527i	$0.418659\pi$
$258$	0	0
$259$	−3.86766	−0.240325
$260$	0	0
$261$	7.98575	0.494306
$262$	0	0
$263$	−9.89980	−0.610448	−0.305224	−	0.952281i	$-0.598731\pi$
$263$	−9.89980	−0.610448	−0.305224	+	0.952281i	$0.598731\pi$
$264$	0	0
$265$	17.6435	1.08383
$266$	0	0
$267$	−17.8668	−1.09343
$268$	0	0
$269$	11.7770	0.718054	0.359027	−	0.933327i	$-0.383109\pi$
$269$	11.7770	0.718054	0.359027	+	0.933327i	$0.383109\pi$
$270$	0	0
$271$	−30.1836	−1.83353	−0.916763	−	0.399432i	$-0.869207\pi$
$271$	−30.1836	−1.83353	−0.916763	+	0.399432i	$0.869207\pi$
$272$	0	0
$273$	−4.75469	−0.287767
$274$	0	0
$275$	2.08062	0.125466
$276$	0	0
$277$	−30.7074	−1.84503	−0.922516	−	0.385960i	$-0.873870\pi$
$277$	−30.7074	−1.84503	−0.922516	+	0.385960i	$0.873870\pi$
$278$	0	0
$279$	6.17442	0.369653
$280$	0	0
$281$	3.98785	0.237895	0.118948	−	0.992901i	$-0.462048\pi$
$281$	3.98785	0.237895	0.118948	+	0.992901i	$0.462048\pi$
$282$	0	0
$283$	−26.9321	−1.60095	−0.800474	−	0.599368i	$-0.795419\pi$
$283$	−26.9321	−1.60095	−0.800474	+	0.599368i	$0.795419\pi$
$284$	0	0
$285$	−31.6060	−1.87218
$286$	0	0
$287$	−1.14850	−0.0677938
$288$	0	0
$289$	6.80261	0.400153
$290$	0	0
$291$	−9.60113	−0.562828
$292$	0	0
$293$	0.717962	0.0419438	0.0209719	−	0.999780i	$-0.493324\pi$
$293$	0.717962	0.0419438	0.0209719	+	0.999780i	$0.493324\pi$
$294$	0	0
$295$	−45.2564	−2.63493
$296$	0	0
$297$	0.952412	0.0552646
$298$	0	0
$299$	−12.9165	−0.746978
$300$	0	0
$301$	4.28543	0.247008
$302$	0	0
$303$	11.0897	0.637089
$304$	0	0
$305$	12.5488	0.718540
$306$	0	0
$307$	25.8617	1.47600	0.738001	−	0.674799i	$-0.235770\pi$
$307$	25.8617	1.47600	0.738001	+	0.674799i	$0.235770\pi$
$308$	0	0
$309$	−6.54125	−0.372119
$310$	0	0
$311$	8.54054	0.484290	0.242145	−	0.970240i	$-0.422149\pi$
$311$	8.54054	0.484290	0.242145	+	0.970240i	$0.422149\pi$
$312$	0	0
$313$	−5.77877	−0.326635	−0.163318	−	0.986574i	$-0.552220\pi$
$313$	−5.77877	−0.326635	−0.163318	+	0.986574i	$0.552220\pi$
$314$	0	0
$315$	−4.23500	−0.238615
$316$	0	0
$317$	15.7025	0.881941	0.440970	−	0.897522i	$-0.354634\pi$
$317$	15.7025	0.881941	0.440970	+	0.897522i	$0.354634\pi$
$318$	0	0
$319$	−0.847810	−0.0474682
$320$	0	0
$321$	−18.1647	−1.01386
$322$	0	0
$323$	32.5724	1.81238
$324$	0	0
$325$	−77.9743	−4.32524
$326$	0	0
$327$	−16.1000	−0.890332
$328$	0	0
$329$	0.338695	0.0186728
$330$	0	0
$331$	−31.5004	−1.73142	−0.865709	−	0.500548i	$-0.833132\pi$
$331$	−31.5004	−1.73142	−0.865709	+	0.500548i	$0.833132\pi$
$332$	0	0
$333$	−10.3593	−0.567684
$334$	0	0
$335$	38.0591	2.07939
$336$	0	0
$337$	28.5754	1.55660	0.778300	−	0.627893i	$-0.216082\pi$
$337$	28.5754	1.55660	0.778300	+	0.627893i	$0.216082\pi$
$338$	0	0
$339$	−13.4833	−0.732315
$340$	0	0
$341$	−0.655509	−0.0354978
$342$	0	0
$343$	8.45854	0.456718
$344$	0	0
$345$	9.23513	0.497203
$346$	0	0
$347$	−9.66818	−0.519015	−0.259507	−	0.965741i	$-0.583560\pi$
$347$	−9.66818	−0.519015	−0.259507	+	0.965741i	$0.583560\pi$
$348$	0	0
$349$	−20.4603	−1.09522	−0.547608	−	0.836735i	$-0.684462\pi$
$349$	−20.4603	−1.09522	−0.547608	+	0.836735i	$0.684462\pi$
$350$	0	0
$351$	−35.6931	−1.90515
$352$	0	0
$353$	−28.3991	−1.51153	−0.755767	−	0.654841i	$-0.772736\pi$
$353$	−28.3991	−1.51153	−0.755767	+	0.654841i	$0.772736\pi$
$354$	0	0
$355$	28.1190	1.49240
$356$	0	0
$357$	−3.50351	−0.185425
$358$	0	0
$359$	13.2598	0.699827	0.349914	−	0.936782i	$-0.386211\pi$
$359$	13.2598	0.699827	0.349914	+	0.936782i	$0.386211\pi$
$360$	0	0
$361$	25.5735	1.34597
$362$	0	0
$363$	12.6776	0.665404
$364$	0	0
$365$	−20.8632	−1.09203
$366$	0	0
$367$	−8.13315	−0.424547	−0.212273	−	0.977210i	$-0.568087\pi$
$367$	−8.13315	−0.424547	−0.212273	+	0.977210i	$0.568087\pi$
$368$	0	0
$369$	−3.07617	−0.160139
$370$	0	0
$371$	2.67636	0.138950
$372$	0	0
$373$	4.63673	0.240081	0.120040	−	0.992769i	$-0.461698\pi$
$373$	4.63673	0.240081	0.120040	+	0.992769i	$0.461698\pi$
$374$	0	0
$375$	32.0806	1.65664
$376$	0	0
$377$	31.7729	1.63639
$378$	0	0
$379$	−4.92306	−0.252881	−0.126440	−	0.991974i	$-0.540355\pi$
$379$	−4.92306	−0.252881	−0.126440	+	0.991974i	$0.540355\pi$
$380$	0	0
$381$	−20.1271	−1.03114
$382$	0	0
$383$	30.8916	1.57849	0.789244	−	0.614080i	$-0.210473\pi$
$383$	30.8916	1.57849	0.789244	+	0.614080i	$0.210473\pi$
$384$	0	0
$385$	0.449610	0.0229142
$386$	0	0
$387$	11.4782	0.583471
$388$	0	0
$389$	28.8261	1.46154	0.730770	−	0.682624i	$-0.239161\pi$
$389$	28.8261	1.46154	0.730770	+	0.682624i	$0.239161\pi$
$390$	0	0
$391$	−9.51751	−0.481321
$392$	0	0
$393$	−15.5748	−0.785642
$394$	0	0
$395$	29.8221	1.50051
$396$	0	0
$397$	−13.2631	−0.665654	−0.332827	−	0.942988i	$-0.608003\pi$
$397$	−13.2631	−0.665654	−0.332827	+	0.942988i	$0.608003\pi$
$398$	0	0
$399$	−4.79434	−0.240017
$400$	0	0
$401$	−10.2827	−0.513495	−0.256748	−	0.966478i	$-0.582651\pi$
$401$	−10.2827	−0.513495	−0.256748	+	0.966478i	$0.582651\pi$
$402$	0	0
$403$	24.5662	1.22373
$404$	0	0
$405$	5.07158	0.252009
$406$	0	0
$407$	1.09979	0.0545147
$408$	0	0
$409$	−2.97111	−0.146912	−0.0734559	−	0.997298i	$-0.523403\pi$
$409$	−2.97111	−0.146912	−0.0734559	+	0.997298i	$0.523403\pi$
$410$	0	0
$411$	−8.17355	−0.403172
$412$	0	0
$413$	−6.86497	−0.337803
$414$	0	0
$415$	74.5258	3.65833
$416$	0	0
$417$	10.5487	0.516571
$418$	0	0
$419$	−0.443124	−0.0216480	−0.0108240	−	0.999941i	$-0.503445\pi$
$419$	−0.443124	−0.0216480	−0.0108240	+	0.999941i	$0.503445\pi$
$420$	0	0
$421$	14.7536	0.719045	0.359522	−	0.933136i	$-0.382940\pi$
$421$	14.7536	0.719045	0.359522	+	0.933136i	$0.382940\pi$
$422$	0	0
$423$	0.907170	0.0441081
$424$	0	0
$425$	−57.4555	−2.78700
$426$	0	0
$427$	1.90353	0.0921183
$428$	0	0
$429$	1.35203	0.0652764
$430$	0	0
$431$	−15.6553	−0.754091	−0.377045	−	0.926195i	$-0.623060\pi$
$431$	−15.6553	−0.754091	−0.377045	+	0.926195i	$0.623060\pi$
$432$	0	0
$433$	35.6406	1.71278	0.856388	−	0.516332i	$-0.172703\pi$
$433$	35.6406	1.71278	0.856388	+	0.516332i	$0.172703\pi$
$434$	0	0
$435$	−22.7173	−1.08921
$436$	0	0
$437$	−13.0242	−0.623030
$438$	0	0
$439$	8.10726	0.386938	0.193469	−	0.981106i	$-0.438026\pi$
$439$	8.10726	0.386938	0.193469	+	0.981106i	$0.438026\pi$
$440$	0	0
$441$	11.0066	0.524123
$442$	0	0
$443$	22.3263	1.06076	0.530378	−	0.847761i	$-0.322050\pi$
$443$	22.3263	1.06076	0.530378	+	0.847761i	$0.322050\pi$
$444$	0	0
$445$	−63.3168	−3.00150
$446$	0	0
$447$	−4.66482	−0.220639
$448$	0	0
$449$	−10.9038	−0.514582	−0.257291	−	0.966334i	$-0.582830\pi$
$449$	−10.9038	−0.514582	−0.257291	+	0.966334i	$0.582830\pi$
$450$	0	0
$451$	0.326583	0.0153782
$452$	0	0
$453$	19.8418	0.932248
$454$	0	0
$455$	−16.8498	−0.789930
$456$	0	0
$457$	−17.4789	−0.817629	−0.408814	−	0.912618i	$-0.634058\pi$
$457$	−17.4789	−0.817629	−0.408814	+	0.912618i	$0.634058\pi$
$458$	0	0
$459$	−26.3005	−1.22760
$460$	0	0
$461$	−6.16799	−0.287272	−0.143636	−	0.989631i	$-0.545879\pi$
$461$	−6.16799	−0.287272	−0.143636	+	0.989631i	$0.545879\pi$
$462$	0	0
$463$	−12.6042	−0.585768	−0.292884	−	0.956148i	$-0.594615\pi$
$463$	−12.6042	−0.585768	−0.292884	+	0.956148i	$0.594615\pi$
$464$	0	0
$465$	−17.5646	−0.814538
$466$	0	0
$467$	33.8888	1.56819	0.784093	−	0.620643i	$-0.213129\pi$
$467$	33.8888	1.56819	0.784093	+	0.620643i	$0.213129\pi$
$468$	0	0
$469$	5.77321	0.266582
$470$	0	0
$471$	14.9675	0.689665
$472$	0	0
$473$	−1.21859	−0.0560307
$474$	0	0
$475$	−78.6245	−3.60754
$476$	0	0
$477$	7.16843	0.328220
$478$	0	0
$479$	−41.2752	−1.88591	−0.942956	−	0.332917i	$-0.891967\pi$
$479$	−41.2752	−1.88591	−0.942956	+	0.332917i	$0.891967\pi$
$480$	0	0
$481$	−41.2164	−1.87931
$482$	0	0
$483$	1.40088	0.0637424
$484$	0	0
$485$	−34.0247	−1.54498
$486$	0	0
$487$	26.0800	1.18180	0.590898	−	0.806746i	$-0.298773\pi$
$487$	26.0800	1.18180	0.590898	+	0.806746i	$0.298773\pi$
$488$	0	0
$489$	−1.71958	−0.0777623
$490$	0	0
$491$	8.95179	0.403989	0.201994	−	0.979387i	$-0.435258\pi$
$491$	8.95179	0.403989	0.201994	+	0.979387i	$0.435258\pi$
$492$	0	0
$493$	23.4120	1.05442
$494$	0	0
$495$	1.20425	0.0541269
$496$	0	0
$497$	4.26539	0.191329
$498$	0	0
$499$	−2.94870	−0.132002	−0.0660010	−	0.997820i	$-0.521024\pi$
$499$	−2.94870	−0.132002	−0.0660010	+	0.997820i	$0.521024\pi$
$500$	0	0
$501$	20.9878	0.937666
$502$	0	0
$503$	4.04092	0.180176	0.0900878	−	0.995934i	$-0.471285\pi$
$503$	4.04092	0.180176	0.0900878	+	0.995934i	$0.471285\pi$
$504$	0	0
$505$	39.3001	1.74883
$506$	0	0
$507$	−35.6439	−1.58300
$508$	0	0
$509$	−10.0547	−0.445667	−0.222834	−	0.974856i	$-0.571531\pi$
$509$	−10.0547	−0.445667	−0.222834	+	0.974856i	$0.571531\pi$
$510$	0	0
$511$	−3.16475	−0.140000
$512$	0	0
$513$	−35.9907	−1.58903
$514$	0	0
$515$	−23.1810	−1.02148
$516$	0	0
$517$	−0.0963099	−0.00423570
$518$	0	0
$519$	−1.85295	−0.0813354
$520$	0	0
$521$	−14.1483	−0.619850	−0.309925	−	0.950761i	$-0.600304\pi$
$521$	−14.1483	−0.619850	−0.309925	+	0.950761i	$0.600304\pi$
$522$	0	0
$523$	−44.6455	−1.95221	−0.976106	−	0.217295i	$-0.930277\pi$
$523$	−44.6455	−1.95221	−0.976106	+	0.217295i	$0.930277\pi$
$524$	0	0
$525$	8.45688	0.369089
$526$	0	0
$527$	18.1016	0.788520
$528$	0	0
$529$	−19.1944	−0.834539
$530$	0	0
$531$	−18.3873	−0.797942
$532$	0	0
$533$	−12.2392	−0.530138
$534$	0	0
$535$	−64.3726	−2.78307
$536$	0	0
$537$	7.34975	0.317165
$538$	0	0
$539$	−1.16852	−0.0503316
$540$	0	0
$541$	42.6957	1.83563	0.917816	−	0.397005i	$-0.129951\pi$
$541$	42.6957	1.83563	0.917816	+	0.397005i	$0.129951\pi$
$542$	0	0
$543$	16.5046	0.708281
$544$	0	0
$545$	−57.0555	−2.44399
$546$	0	0
$547$	37.3069	1.59513	0.797563	−	0.603235i	$-0.206122\pi$
$547$	37.3069	1.59513	0.797563	+	0.603235i	$0.206122\pi$
$548$	0	0
$549$	5.09847	0.217597
$550$	0	0
$551$	32.0379	1.36486
$552$	0	0
$553$	4.52374	0.192369
$554$	0	0
$555$	29.4693	1.25090
$556$	0	0
$557$	−6.47875	−0.274514	−0.137257	−	0.990536i	$-0.543829\pi$
$557$	−6.47875	−0.274514	−0.137257	+	0.990536i	$0.543829\pi$
$558$	0	0
$559$	45.6684	1.93157
$560$	0	0
$561$	0.996243	0.0420614
$562$	0	0
$563$	1.47270	0.0620671	0.0310335	−	0.999518i	$-0.490120\pi$
$563$	1.47270	0.0620671	0.0310335	+	0.999518i	$0.490120\pi$
$564$	0	0
$565$	−47.7826	−2.01023
$566$	0	0
$567$	0.769311	0.0323080
$568$	0	0
$569$	−30.0094	−1.25806	−0.629030	−	0.777381i	$-0.716548\pi$
$569$	−30.0094	−1.25806	−0.629030	+	0.777381i	$0.716548\pi$
$570$	0	0
$571$	−33.9014	−1.41873	−0.709366	−	0.704841i	$-0.751019\pi$
$571$	−33.9014	−1.41873	−0.709366	+	0.704841i	$0.751019\pi$
$572$	0	0
$573$	11.3925	0.475928
$574$	0	0
$575$	22.9737	0.958069
$576$	0	0
$577$	26.4242	1.10005	0.550027	−	0.835147i	$-0.314618\pi$
$577$	26.4242	1.10005	0.550027	+	0.835147i	$0.314618\pi$
$578$	0	0
$579$	28.7592	1.19519
$580$	0	0
$581$	11.3049	0.469005
$582$	0	0
$583$	−0.761038	−0.0315190
$584$	0	0
$585$	−45.1309	−1.86593
$586$	0	0
$587$	−31.1846	−1.28713	−0.643563	−	0.765393i	$-0.722545\pi$
$587$	−31.1846	−1.28713	−0.643563	+	0.765393i	$0.722545\pi$
$588$	0	0
$589$	24.7710	1.02067
$590$	0	0
$591$	3.00044	0.123422
$592$	0	0
$593$	10.9552	0.449875	0.224938	−	0.974373i	$-0.427782\pi$
$593$	10.9552	0.449875	0.224938	+	0.974373i	$0.427782\pi$
$594$	0	0
$595$	−12.4158	−0.508998
$596$	0	0
$597$	16.3943	0.670974
$598$	0	0
$599$	34.2254	1.39841	0.699206	−	0.714920i	$-0.253537\pi$
$599$	34.2254	1.39841	0.699206	+	0.714920i	$0.253537\pi$
$600$	0	0
$601$	13.0090	0.530647	0.265323	−	0.964159i	$-0.414521\pi$
$601$	13.0090	0.530647	0.265323	+	0.964159i	$0.414521\pi$
$602$	0	0
$603$	15.4631	0.629708
$604$	0	0
$605$	44.9273	1.82655
$606$	0	0
$607$	−0.895769	−0.0363581	−0.0181791	−	0.999835i	$-0.505787\pi$
$607$	−0.895769	−0.0363581	−0.0181791	+	0.999835i	$0.505787\pi$
$608$	0	0
$609$	−3.44601	−0.139639
$610$	0	0
$611$	3.60936	0.146019
$612$	0	0
$613$	16.1728	0.653212	0.326606	−	0.945161i	$-0.394095\pi$
$613$	16.1728	0.653212	0.326606	+	0.945161i	$0.394095\pi$
$614$	0	0
$615$	8.75089	0.352870
$616$	0	0
$617$	13.5442	0.545267	0.272633	−	0.962118i	$-0.412105\pi$
$617$	13.5442	0.545267	0.272633	+	0.962118i	$0.412105\pi$
$618$	0	0
$619$	−7.95361	−0.319683	−0.159841	−	0.987143i	$-0.551098\pi$
$619$	−7.95361	−0.319683	−0.159841	+	0.987143i	$0.551098\pi$
$620$	0	0
$621$	10.5163	0.422005
$622$	0	0
$623$	−9.60456	−0.384799
$624$	0	0
$625$	54.8051	2.19220
$626$	0	0
$627$	1.36330	0.0544449
$628$	0	0
$629$	−30.3704	−1.21095
$630$	0	0
$631$	−28.8274	−1.14760	−0.573800	−	0.818995i	$-0.694531\pi$
$631$	−28.8274	−1.14760	−0.573800	+	0.818995i	$0.694531\pi$
$632$	0	0
$633$	−9.24015	−0.367263
$634$	0	0
$635$	−71.3267	−2.83051
$636$	0	0
$637$	43.7919	1.73510
$638$	0	0
$639$	11.4246	0.451948
$640$	0	0
$641$	39.1175	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$641$	39.1175	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$642$	0	0
$643$	−6.17610	−0.243562	−0.121781	−	0.992557i	$-0.538861\pi$
$643$	−6.17610	−0.243562	−0.121781	+	0.992557i	$0.538861\pi$
$644$	0	0
$645$	−32.6525	−1.28569
$646$	0	0
$647$	−4.77971	−0.187910	−0.0939548	−	0.995576i	$-0.529951\pi$
$647$	−4.77971	−0.187910	−0.0939548	+	0.995576i	$0.529951\pi$
$648$	0	0
$649$	1.95210	0.0766265
$650$	0	0
$651$	−2.66438	−0.104425
$652$	0	0
$653$	−12.9339	−0.506144	−0.253072	−	0.967447i	$-0.581441\pi$
$653$	−12.9339	−0.506144	−0.253072	+	0.967447i	$0.581441\pi$
$654$	0	0
$655$	−55.1941	−2.15661
$656$	0	0
$657$	−8.47657	−0.330702
$658$	0	0
$659$	−29.6390	−1.15457	−0.577285	−	0.816543i	$-0.695888\pi$
$659$	−29.6390	−1.15457	−0.577285	+	0.816543i	$0.695888\pi$
$660$	0	0
$661$	36.1729	1.40696	0.703481	−	0.710714i	$-0.251628\pi$
$661$	36.1729	1.40696	0.703481	+	0.710714i	$0.251628\pi$
$662$	0	0
$663$	−37.3357	−1.45000
$664$	0	0
$665$	−16.9903	−0.658855
$666$	0	0
$667$	−9.36131	−0.362471
$668$	0	0
$669$	0.615450	0.0237947
$670$	0	0
$671$	−0.541280	−0.0208959
$672$	0	0
$673$	−7.81880	−0.301393	−0.150696	−	0.988580i	$-0.548152\pi$
$673$	−7.81880	−0.301393	−0.150696	+	0.988580i	$0.548152\pi$
$674$	0	0
$675$	63.4850	2.44354
$676$	0	0
$677$	−3.61795	−0.139049	−0.0695245	−	0.997580i	$-0.522148\pi$
$677$	−3.61795	−0.139049	−0.0695245	+	0.997580i	$0.522148\pi$
$678$	0	0
$679$	−5.16123	−0.198070
$680$	0	0
$681$	−2.75321	−0.105503
$682$	0	0
$683$	−1.68227	−0.0643704	−0.0321852	−	0.999482i	$-0.510247\pi$
$683$	−1.68227	−0.0643704	−0.0321852	+	0.999482i	$0.510247\pi$
$684$	0	0
$685$	−28.9656	−1.10672
$686$	0	0
$687$	26.4360	1.00860
$688$	0	0
$689$	28.5210	1.08657
$690$	0	0
$691$	39.8337	1.51534	0.757672	−	0.652635i	$-0.226337\pi$
$691$	39.8337	1.51534	0.757672	+	0.652635i	$0.226337\pi$
$692$	0	0
$693$	0.182673	0.00693917
$694$	0	0
$695$	37.3826	1.41800
$696$	0	0
$697$	−9.01846	−0.341599
$698$	0	0
$699$	−20.6506	−0.781076
$700$	0	0
$701$	−1.65966	−0.0626845	−0.0313423	−	0.999509i	$-0.509978\pi$
$701$	−1.65966	−0.0626845	−0.0313423	+	0.999509i	$0.509978\pi$
$702$	0	0
$703$	−41.5601	−1.56747
$704$	0	0
$705$	−2.58065	−0.0971931
$706$	0	0
$707$	5.96145	0.224204
$708$	0	0
$709$	19.3607	0.727105	0.363552	−	0.931574i	$-0.381564\pi$
$709$	19.3607	0.727105	0.363552	+	0.931574i	$0.381564\pi$
$710$	0	0
$711$	12.1165	0.454405
$712$	0	0
$713$	−7.23798	−0.271064
$714$	0	0
$715$	4.79134	0.179186
$716$	0	0
$717$	18.0612	0.674506
$718$	0	0
$719$	17.7607	0.662363	0.331181	−	0.943567i	$-0.392553\pi$
$719$	17.7607	0.662363	0.331181	+	0.943567i	$0.392553\pi$
$720$	0	0
$721$	−3.51635	−0.130956
$722$	0	0
$723$	6.34194	0.235859
$724$	0	0
$725$	−56.5126	−2.09882
$726$	0	0
$727$	29.3612	1.08895	0.544473	−	0.838779i	$-0.316730\pi$
$727$	29.3612	1.08895	0.544473	+	0.838779i	$0.316730\pi$
$728$	0	0
$729$	20.7524	0.768609
$730$	0	0
$731$	33.6509	1.24462
$732$	0	0
$733$	−27.9740	−1.03324	−0.516622	−	0.856214i	$-0.672811\pi$
$733$	−27.9740	−1.03324	−0.516622	+	0.856214i	$0.672811\pi$
$734$	0	0
$735$	−31.3108	−1.15492
$736$	0	0
$737$	−1.64165	−0.0604709
$738$	0	0
$739$	3.94807	0.145232	0.0726160	−	0.997360i	$-0.476865\pi$
$739$	3.94807	0.145232	0.0726160	+	0.997360i	$0.476865\pi$
$740$	0	0
$741$	−51.0917	−1.87690
$742$	0	0
$743$	12.8304	0.470701	0.235351	−	0.971911i	$-0.424376\pi$
$743$	12.8304	0.470701	0.235351	+	0.971911i	$0.424376\pi$
$744$	0	0
$745$	−16.5313	−0.605660
$746$	0	0
$747$	30.2793	1.10786
$748$	0	0
$749$	−9.76472	−0.356795
$750$	0	0
$751$	−10.6222	−0.387611	−0.193805	−	0.981040i	$-0.562083\pi$
$751$	−10.6222	−0.387611	−0.193805	+	0.981040i	$0.562083\pi$
$752$	0	0
$753$	1.15579	0.0421194
$754$	0	0
$755$	70.3157	2.55905
$756$	0	0
$757$	2.07752	0.0755086	0.0377543	−	0.999287i	$-0.487980\pi$
$757$	2.07752	0.0755086	0.0377543	+	0.999287i	$0.487980\pi$
$758$	0	0
$759$	−0.398350	−0.0144592
$760$	0	0
$761$	0.610314	0.0221239	0.0110619	−	0.999939i	$-0.496479\pi$
$761$	0.610314	0.0221239	0.0110619	+	0.999939i	$0.496479\pi$
$762$	0	0
$763$	−8.65479	−0.313324
$764$	0	0
$765$	−33.2548	−1.20233
$766$	0	0
$767$	−73.1577	−2.64157
$768$	0	0
$769$	42.9945	1.55042	0.775210	−	0.631704i	$-0.217644\pi$
$769$	42.9945	1.55042	0.775210	+	0.631704i	$0.217644\pi$
$770$	0	0
$771$	−9.36701	−0.337344
$772$	0	0
$773$	28.3483	1.01962	0.509809	−	0.860288i	$-0.329716\pi$
$773$	28.3483	1.01962	0.509809	+	0.860288i	$0.329716\pi$
$774$	0	0
$775$	−43.6944	−1.56955
$776$	0	0
$777$	4.47022	0.160368
$778$	0	0
$779$	−12.3412	−0.442170
$780$	0	0
$781$	−1.21289	−0.0434006
$782$	0	0
$783$	−25.8689	−0.924478
$784$	0	0
$785$	53.0421	1.89315
$786$	0	0
$787$	38.3435	1.36680	0.683399	−	0.730045i	$-0.260501\pi$
$787$	38.3435	1.36680	0.683399	+	0.730045i	$0.260501\pi$
$788$	0	0
$789$	11.4421	0.407350
$790$	0	0
$791$	−7.24817	−0.257715
$792$	0	0
$793$	20.2853	0.720351
$794$	0	0
$795$	−20.3923	−0.723239
$796$	0	0
$797$	7.48153	0.265010	0.132505	−	0.991182i	$-0.457698\pi$
$797$	7.48153	0.265010	0.132505	+	0.991182i	$0.457698\pi$
$798$	0	0
$799$	2.65956	0.0940886
$800$	0	0
$801$	−25.7251	−0.908953
$802$	0	0
$803$	0.899917	0.0317574
$804$	0	0
$805$	4.96448	0.174975
$806$	0	0
$807$	−13.6117	−0.479156
$808$	0	0
$809$	34.6803	1.21929	0.609646	−	0.792674i	$-0.291311\pi$
$809$	34.6803	1.21929	0.609646	+	0.792674i	$0.291311\pi$
$810$	0	0
$811$	39.0793	1.37226	0.686130	−	0.727479i	$-0.259308\pi$
$811$	39.0793	1.37226	0.686130	+	0.727479i	$0.259308\pi$
$812$	0	0
$813$	34.8860	1.22351
$814$	0	0
$815$	−6.09390	−0.213460
$816$	0	0
$817$	46.0492	1.61106
$818$	0	0
$819$	−6.84594	−0.239217
$820$	0	0
$821$	−22.7662	−0.794545	−0.397273	−	0.917701i	$-0.630043\pi$
$821$	−22.7662	−0.794545	−0.397273	+	0.917701i	$0.630043\pi$
$822$	0	0
$823$	20.9505	0.730289	0.365145	−	0.930951i	$-0.381020\pi$
$823$	20.9505	0.730289	0.365145	+	0.930951i	$0.381020\pi$
$824$	0	0
$825$	−2.40476	−0.0837232
$826$	0	0
$827$	7.80711	0.271480	0.135740	−	0.990744i	$-0.456659\pi$
$827$	7.80711	0.271480	0.135740	+	0.990744i	$0.456659\pi$
$828$	0	0
$829$	−35.4966	−1.23285	−0.616423	−	0.787415i	$-0.711419\pi$
$829$	−35.4966	−1.23285	−0.616423	+	0.787415i	$0.711419\pi$
$830$	0	0
$831$	35.4914	1.23118
$832$	0	0
$833$	32.2682	1.11803
$834$	0	0
$835$	74.3770	2.57392
$836$	0	0
$837$	−20.0013	−0.691345
$838$	0	0
$839$	−28.3734	−0.979559	−0.489780	−	0.871846i	$-0.662923\pi$
$839$	−28.3734	−0.979559	−0.489780	+	0.871846i	$0.662923\pi$
$840$	0	0
$841$	−5.97228	−0.205941
$842$	0	0
$843$	−4.60913	−0.158747
$844$	0	0
$845$	−126.316	−4.34539
$846$	0	0
$847$	6.81505	0.234168
$848$	0	0
$849$	31.1279	1.06831
$850$	0	0
$851$	12.1437	0.416279
$852$	0	0
$853$	49.0270	1.67865	0.839326	−	0.543628i	$-0.182950\pi$
$853$	49.0270	1.67865	0.839326	+	0.543628i	$0.182950\pi$
$854$	0	0
$855$	−45.5072	−1.55632
$856$	0	0
$857$	−32.1040	−1.09665	−0.548325	−	0.836265i	$-0.684734\pi$
$857$	−32.1040	−1.09665	−0.548325	+	0.836265i	$0.684734\pi$
$858$	0	0
$859$	8.31632	0.283749	0.141875	−	0.989885i	$-0.454687\pi$
$859$	8.31632	0.283749	0.141875	+	0.989885i	$0.454687\pi$
$860$	0	0
$861$	1.32743	0.0452386
$862$	0	0
$863$	46.7903	1.59276	0.796380	−	0.604796i	$-0.206745\pi$
$863$	46.7903	1.59276	0.796380	+	0.604796i	$0.206745\pi$
$864$	0	0
$865$	−6.56652	−0.223268
$866$	0	0
$867$	−7.86240	−0.267021
$868$	0	0
$869$	−1.28635	−0.0436365
$870$	0	0
$871$	61.5232	2.08463
$872$	0	0
$873$	−13.8240	−0.467871
$874$	0	0
$875$	17.2454	0.583001
$876$	0	0
$877$	17.5910	0.594007	0.297004	−	0.954876i	$-0.404013\pi$
$877$	17.5910	0.594007	0.297004	+	0.954876i	$0.404013\pi$
$878$	0	0
$879$	−0.829815	−0.0279890
$880$	0	0
$881$	−42.9312	−1.44639	−0.723195	−	0.690644i	$-0.757327\pi$
$881$	−42.9312	−1.44639	−0.723195	+	0.690644i	$0.757327\pi$
$882$	0	0
$883$	−16.2432	−0.546628	−0.273314	−	0.961925i	$-0.588120\pi$
$883$	−16.2432	−0.546628	−0.273314	+	0.961925i	$0.588120\pi$
$884$	0	0
$885$	52.3070	1.75828
$886$	0	0
$887$	−27.1735	−0.912398	−0.456199	−	0.889878i	$-0.650790\pi$
$887$	−27.1735	−0.912398	−0.456199	+	0.889878i	$0.650790\pi$
$888$	0	0
$889$	−10.8196	−0.362877
$890$	0	0
$891$	−0.218758	−0.00732867
$892$	0	0
$893$	3.63945	0.121790
$894$	0	0
$895$	26.0462	0.870628
$896$	0	0
$897$	14.9287	0.498456
$898$	0	0
$899$	17.8046	0.593815
$900$	0	0
$901$	21.0158	0.700138
$902$	0	0
$903$	−4.95307	−0.164828
$904$	0	0
$905$	58.4894	1.94425
$906$	0	0
$907$	−32.4060	−1.07602	−0.538011	−	0.842938i	$-0.680824\pi$
$907$	−32.4060	−1.07602	−0.538011	+	0.842938i	$0.680824\pi$
$908$	0	0
$909$	15.9673	0.529603
$910$	0	0
$911$	22.2655	0.737688	0.368844	−	0.929491i	$-0.379754\pi$
$911$	22.2655	0.737688	0.368844	+	0.929491i	$0.379754\pi$
$912$	0	0
$913$	−3.21461	−0.106388
$914$	0	0
$915$	−14.5038	−0.479480
$916$	0	0
$917$	−8.37244	−0.276482
$918$	0	0
$919$	−53.9357	−1.77918	−0.889588	−	0.456764i	$-0.849008\pi$
$919$	−53.9357	−1.77918	−0.889588	+	0.456764i	$0.849008\pi$
$920$	0	0
$921$	−29.8907	−0.984933
$922$	0	0
$923$	45.4549	1.49617
$924$	0	0
$925$	73.3090	2.41039
$926$	0	0
$927$	−9.41828	−0.309337
$928$	0	0
$929$	30.0362	0.985456	0.492728	−	0.870183i	$-0.336000\pi$
$929$	30.0362	0.985456	0.492728	+	0.870183i	$0.336000\pi$
$930$	0	0
$931$	44.1571	1.44719
$932$	0	0
$933$	−9.87110	−0.323165
$934$	0	0
$935$	3.53051	0.115460
$936$	0	0
$937$	4.90313	0.160178	0.0800892	−	0.996788i	$-0.474479\pi$
$937$	4.90313	0.160178	0.0800892	+	0.996788i	$0.474479\pi$
$938$	0	0
$939$	6.67906	0.217963
$940$	0	0
$941$	−34.8436	−1.13587	−0.567934	−	0.823074i	$-0.692257\pi$
$941$	−34.8436	−1.13587	−0.567934	+	0.823074i	$0.692257\pi$
$942$	0	0
$943$	3.60605	0.117429
$944$	0	0
$945$	13.7187	0.446271
$946$	0	0
$947$	53.9666	1.75368	0.876840	−	0.480782i	$-0.159647\pi$
$947$	53.9666	1.75368	0.876840	+	0.480782i	$0.159647\pi$
$948$	0	0
$949$	−33.7257	−1.09478
$950$	0	0
$951$	−18.1489	−0.588517
$952$	0	0
$953$	17.8743	0.579005	0.289502	−	0.957177i	$-0.406510\pi$
$953$	17.8743	0.579005	0.289502	+	0.957177i	$0.406510\pi$
$954$	0	0
$955$	40.3729	1.30644
$956$	0	0
$957$	0.979893	0.0316754
$958$	0	0
$959$	−4.39381	−0.141884
$960$	0	0
$961$	−17.2339	−0.555931
$962$	0	0
$963$	−26.1541	−0.842804
$964$	0	0
$965$	101.917	3.28083
$966$	0	0
$967$	−28.2898	−0.909739	−0.454869	−	0.890558i	$-0.650314\pi$
$967$	−28.2898	−0.909739	−0.454869	+	0.890558i	$0.650314\pi$
$968$	0	0
$969$	−37.6470	−1.20940
$970$	0	0
$971$	−11.5535	−0.370770	−0.185385	−	0.982666i	$-0.559353\pi$
$971$	−11.5535	−0.370770	−0.185385	+	0.982666i	$0.559353\pi$
$972$	0	0
$973$	5.67060	0.181791
$974$	0	0
$975$	90.1221	2.88622
$976$	0	0
$977$	−27.3097	−0.873714	−0.436857	−	0.899531i	$-0.643908\pi$
$977$	−27.3097	−0.873714	−0.436857	+	0.899531i	$0.643908\pi$
$978$	0	0
$979$	2.73112	0.0872868
$980$	0	0
$981$	−23.1812	−0.740119
$982$	0	0
$983$	5.50331	0.175528	0.0877641	−	0.996141i	$-0.472028\pi$
$983$	5.50331	0.175528	0.0877641	+	0.996141i	$0.472028\pi$
$984$	0	0
$985$	10.6330	0.338796
$986$	0	0
$987$	−0.391461	−0.0124603
$988$	0	0
$989$	−13.4554	−0.427855
$990$	0	0
$991$	−22.3080	−0.708636	−0.354318	−	0.935125i	$-0.615287\pi$
$991$	−22.3080	−0.708636	−0.354318	+	0.935125i	$0.615287\pi$
$992$	0	0
$993$	36.4079	1.15537
$994$	0	0
$995$	58.0985	1.84184
$996$	0	0
$997$	−1.01117	−0.0320240	−0.0160120	−	0.999872i	$-0.505097\pi$
$997$	−1.01117	−0.0320240	−0.0160120	+	0.999872i	$0.505097\pi$
$998$	0	0
$999$	33.5575	1.06171

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.9	✓	23
4.3	odd	2		4016.2.a.m.1.15		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.9	✓	23	1.1	even	1	trivial
4016.2.a.m.1.15		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)

\(f(q)\)	\(=\)	\(q-1.15579 q^{3} -4.09592 q^{5} -0.621313 q^{7} -1.66414 q^{9} +O(q^{10})\)	\(q-1.15579 q^{3} -4.09592 q^{5} -0.621313 q^{7} -1.66414 q^{9} +0.176674 q^{11} -6.62113 q^{13} +4.73404 q^{15} -4.87879 q^{17} -6.67634 q^{19} +0.718109 q^{21} +1.95079 q^{23} +11.7766 q^{25} +5.39078 q^{27} -4.79872 q^{29} -3.71027 q^{31} -0.204199 q^{33} +2.54485 q^{35} +6.22498 q^{37} +7.65265 q^{39} +1.84850 q^{41} -6.89738 q^{43} +6.81620 q^{45} -0.545127 q^{47} -6.61397 q^{49} +5.63887 q^{51} -4.30758 q^{53} -0.723644 q^{55} +7.71646 q^{57} +11.0491 q^{59} -3.06372 q^{61} +1.03395 q^{63} +27.1196 q^{65} -9.29195 q^{67} -2.25471 q^{69} -6.86513 q^{71} +5.09365 q^{73} -13.6113 q^{75} -0.109770 q^{77} -7.28093 q^{79} -1.23820 q^{81} -18.1951 q^{83} +19.9832 q^{85} +5.54633 q^{87} +15.4585 q^{89} +4.11379 q^{91} +4.28831 q^{93} +27.3458 q^{95} +8.30697 q^{97} -0.294011 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 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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