LMFDB - Embedded newform 2008.2.a.d.1.19

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			1.19
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)

$f(q)$	$=$	$q+2.53334 q^{3} +1.50187 q^{5} +0.478852 q^{7} +3.41783 q^{9} +O(q^{10})$	$q+2.53334 q^{3} +1.50187 q^{5} +0.478852 q^{7} +3.41783 q^{9} +4.20287 q^{11} +4.32908 q^{13} +3.80475 q^{15} +2.05558 q^{17} +0.535658 q^{19} +1.21310 q^{21} -6.64982 q^{23} -2.74440 q^{25} +1.05852 q^{27} +0.258982 q^{29} +4.57844 q^{31} +10.6473 q^{33} +0.719173 q^{35} -11.8796 q^{37} +10.9671 q^{39} -5.23725 q^{41} -8.69198 q^{43} +5.13313 q^{45} -11.5859 q^{47} -6.77070 q^{49} +5.20748 q^{51} +12.2460 q^{53} +6.31216 q^{55} +1.35701 q^{57} +3.96974 q^{59} +4.23862 q^{61} +1.63664 q^{63} +6.50171 q^{65} +14.8609 q^{67} -16.8463 q^{69} -16.3598 q^{71} +6.11513 q^{73} -6.95250 q^{75} +2.01256 q^{77} +10.5582 q^{79} -7.57191 q^{81} +12.1519 q^{83} +3.08720 q^{85} +0.656090 q^{87} +5.39741 q^{89} +2.07299 q^{91} +11.5988 q^{93} +0.804488 q^{95} +14.0202 q^{97} +14.3647 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.53334	1.46263	0.731314	−	0.682041i	$-0.238908\pi$
$3$	2.53334	1.46263	0.731314	+	0.682041i	$0.238908\pi$
$4$	0	0
$5$	1.50187	0.671655	0.335828	−	0.941923i	$-0.390984\pi$
$5$	1.50187	0.671655	0.335828	+	0.941923i	$0.390984\pi$
$6$	0	0
$7$	0.478852	0.180989	0.0904946	−	0.995897i	$-0.471155\pi$
$7$	0.478852	0.180989	0.0904946	+	0.995897i	$0.471155\pi$
$8$	0	0
$9$	3.41783	1.13928
$10$	0	0
$11$	4.20287	1.26721	0.633607	−	0.773655i	$-0.281574\pi$
$11$	4.20287	1.26721	0.633607	+	0.773655i	$0.281574\pi$
$12$	0	0
$13$	4.32908	1.20067	0.600336	−	0.799748i	$-0.295034\pi$
$13$	4.32908	1.20067	0.600336	+	0.799748i	$0.295034\pi$
$14$	0	0
$15$	3.80475	0.982381
$16$	0	0
$17$	2.05558	0.498551	0.249275	−	0.968433i	$-0.419808\pi$
$17$	2.05558	0.498551	0.249275	+	0.968433i	$0.419808\pi$
$18$	0	0
$19$	0.535658	0.122888	0.0614442	−	0.998111i	$-0.480429\pi$
$19$	0.535658	0.122888	0.0614442	+	0.998111i	$0.480429\pi$
$20$	0	0
$21$	1.21310	0.264720
$22$	0	0
$23$	−6.64982	−1.38658	−0.693292	−	0.720657i	$-0.743840\pi$
$23$	−6.64982	−1.38658	−0.693292	+	0.720657i	$0.743840\pi$
$24$	0	0
$25$	−2.74440	−0.548879
$26$	0	0
$27$	1.05852	0.203712
$28$	0	0
$29$	0.258982	0.0480917	0.0240459	−	0.999711i	$-0.492345\pi$
$29$	0.258982	0.0480917	0.0240459	+	0.999711i	$0.492345\pi$
$30$	0	0
$31$	4.57844	0.822311	0.411156	−	0.911565i	$-0.365125\pi$
$31$	4.57844	0.822311	0.411156	+	0.911565i	$0.365125\pi$
$32$	0	0
$33$	10.6473	1.85346
$34$	0	0
$35$	0.719173	0.121562
$36$	0	0
$37$	−11.8796	−1.95300	−0.976500	−	0.215518i	$-0.930856\pi$
$37$	−11.8796	−1.95300	−0.976500	+	0.215518i	$0.930856\pi$
$38$	0	0
$39$	10.9671	1.75614
$40$	0	0
$41$	−5.23725	−0.817921	−0.408961	−	0.912552i	$-0.634109\pi$
$41$	−5.23725	−0.817921	−0.408961	+	0.912552i	$0.634109\pi$
$42$	0	0
$43$	−8.69198	−1.32552	−0.662758	−	0.748834i	$-0.730614\pi$
$43$	−8.69198	−1.32552	−0.662758	+	0.748834i	$0.730614\pi$
$44$	0	0
$45$	5.13313	0.765202
$46$	0	0
$47$	−11.5859	−1.68998	−0.844990	−	0.534783i	$-0.820393\pi$
$47$	−11.5859	−1.68998	−0.844990	+	0.534783i	$0.820393\pi$
$48$	0	0
$49$	−6.77070	−0.967243
$50$	0	0
$51$	5.20748	0.729194
$52$	0	0
$53$	12.2460	1.68211	0.841056	−	0.540948i	$-0.181934\pi$
$53$	12.2460	1.68211	0.841056	+	0.540948i	$0.181934\pi$
$54$	0	0
$55$	6.31216	0.851131
$56$	0	0
$57$	1.35701	0.179740
$58$	0	0
$59$	3.96974	0.516816	0.258408	−	0.966036i	$-0.416802\pi$
$59$	3.96974	0.516816	0.258408	+	0.966036i	$0.416802\pi$
$60$	0	0
$61$	4.23862	0.542699	0.271350	−	0.962481i	$-0.412530\pi$
$61$	4.23862	0.542699	0.271350	+	0.962481i	$0.412530\pi$
$62$	0	0
$63$	1.63664	0.206197
$64$	0	0
$65$	6.50171	0.806438
$66$	0	0
$67$	14.8609	1.81555	0.907773	−	0.419461i	$-0.137781\pi$
$67$	14.8609	1.81555	0.907773	+	0.419461i	$0.137781\pi$
$68$	0	0
$69$	−16.8463	−2.02805
$70$	0	0
$71$	−16.3598	−1.94155	−0.970773	−	0.240001i	$-0.922852\pi$
$71$	−16.3598	−1.94155	−0.970773	+	0.240001i	$0.922852\pi$
$72$	0	0
$73$	6.11513	0.715722	0.357861	−	0.933775i	$-0.383506\pi$
$73$	6.11513	0.715722	0.357861	+	0.933775i	$0.383506\pi$
$74$	0	0
$75$	−6.95250	−0.802805
$76$	0	0
$77$	2.01256	0.229352
$78$	0	0
$79$	10.5582	1.18789	0.593944	−	0.804506i	$-0.297570\pi$
$79$	10.5582	1.18789	0.593944	+	0.804506i	$0.297570\pi$
$80$	0	0
$81$	−7.57191	−0.841324
$82$	0	0
$83$	12.1519	1.33384	0.666920	−	0.745130i	$-0.267612\pi$
$83$	12.1519	1.33384	0.666920	+	0.745130i	$0.267612\pi$
$84$	0	0
$85$	3.08720	0.334854
$86$	0	0
$87$	0.656090	0.0703403
$88$	0	0
$89$	5.39741	0.572124	0.286062	−	0.958211i	$-0.407654\pi$
$89$	5.39741	0.572124	0.286062	+	0.958211i	$0.407654\pi$
$90$	0	0
$91$	2.07299	0.217309
$92$	0	0
$93$	11.5988	1.20274
$94$	0	0
$95$	0.804488	0.0825387
$96$	0	0
$97$	14.0202	1.42354	0.711768	−	0.702414i	$-0.247895\pi$
$97$	14.0202	1.42354	0.711768	+	0.702414i	$0.247895\pi$
$98$	0	0
$99$	14.3647	1.44371
$100$	0	0
$101$	−0.0962519	−0.00957742	−0.00478871	−	0.999989i	$-0.501524\pi$
$101$	−0.0962519	−0.00957742	−0.00478871	+	0.999989i	$0.501524\pi$
$102$	0	0
$103$	−10.7390	−1.05814	−0.529072	−	0.848577i	$-0.677460\pi$
$103$	−10.7390	−1.05814	−0.529072	+	0.848577i	$0.677460\pi$
$104$	0	0
$105$	1.82191	0.177800
$106$	0	0
$107$	19.4866	1.88384	0.941920	−	0.335836i	$-0.109019\pi$
$107$	19.4866	1.88384	0.941920	+	0.335836i	$0.109019\pi$
$108$	0	0
$109$	0.0525721	0.00503549	0.00251775	−	0.999997i	$-0.499199\pi$
$109$	0.0525721	0.00503549	0.00251775	+	0.999997i	$0.499199\pi$
$110$	0	0
$111$	−30.0952	−2.85651
$112$	0	0
$113$	2.74548	0.258273	0.129136	−	0.991627i	$-0.458780\pi$
$113$	2.74548	0.258273	0.129136	+	0.991627i	$0.458780\pi$
$114$	0	0
$115$	−9.98715	−0.931306
$116$	0	0
$117$	14.7961	1.36790
$118$	0	0
$119$	0.984318	0.0902323
$120$	0	0
$121$	6.66414	0.605831
$122$	0	0
$123$	−13.2678	−1.19631
$124$	0	0
$125$	−11.6311	−1.04031
$126$	0	0
$127$	−15.4193	−1.36824	−0.684122	−	0.729367i	$-0.739815\pi$
$127$	−15.4193	−1.36824	−0.684122	+	0.729367i	$0.739815\pi$
$128$	0	0
$129$	−22.0198	−1.93873
$130$	0	0
$131$	13.8641	1.21131	0.605656	−	0.795727i	$-0.292911\pi$
$131$	13.8641	1.21131	0.605656	+	0.795727i	$0.292911\pi$
$132$	0	0
$133$	0.256501	0.0222415
$134$	0	0
$135$	1.58975	0.136824
$136$	0	0
$137$	−14.4346	−1.23323	−0.616613	−	0.787266i	$-0.711496\pi$
$137$	−14.4346	−1.23323	−0.616613	+	0.787266i	$0.711496\pi$
$138$	0	0
$139$	−22.9011	−1.94244	−0.971221	−	0.238179i	$-0.923449\pi$
$139$	−22.9011	−1.94244	−0.971221	+	0.238179i	$0.923449\pi$
$140$	0	0
$141$	−29.3511	−2.47181
$142$	0	0
$143$	18.1946	1.52151
$144$	0	0
$145$	0.388956	0.0323011
$146$	0	0
$147$	−17.1525	−1.41472
$148$	0	0
$149$	−9.40542	−0.770522	−0.385261	−	0.922808i	$-0.625889\pi$
$149$	−9.40542	−0.770522	−0.385261	+	0.922808i	$0.625889\pi$
$150$	0	0
$151$	−7.32090	−0.595766	−0.297883	−	0.954602i	$-0.596281\pi$
$151$	−7.32090	−0.595766	−0.297883	+	0.954602i	$0.596281\pi$
$152$	0	0
$153$	7.02562	0.567988
$154$	0	0
$155$	6.87620	0.552310
$156$	0	0
$157$	7.11074	0.567499	0.283749	−	0.958898i	$-0.408422\pi$
$157$	7.11074	0.567499	0.283749	+	0.958898i	$0.408422\pi$
$158$	0	0
$159$	31.0232	2.46030
$160$	0	0
$161$	−3.18428	−0.250957
$162$	0	0
$163$	−6.35460	−0.497730	−0.248865	−	0.968538i	$-0.580058\pi$
$163$	−6.35460	−0.497730	−0.248865	+	0.968538i	$0.580058\pi$
$164$	0	0
$165$	15.9909	1.24489
$166$	0	0
$167$	−8.47308	−0.655667	−0.327833	−	0.944736i	$-0.606318\pi$
$167$	−8.47308	−0.655667	−0.327833	+	0.944736i	$0.606318\pi$
$168$	0	0
$169$	5.74097	0.441613
$170$	0	0
$171$	1.83079	0.140004
$172$	0	0
$173$	14.2270	1.08166	0.540828	−	0.841133i	$-0.318111\pi$
$173$	14.2270	1.08166	0.540828	+	0.841133i	$0.318111\pi$
$174$	0	0
$175$	−1.31416	−0.0993412
$176$	0	0
$177$	10.0567	0.755909
$178$	0	0
$179$	−8.07057	−0.603223	−0.301611	−	0.953431i	$-0.597524\pi$
$179$	−8.07057	−0.603223	−0.301611	+	0.953431i	$0.597524\pi$
$180$	0	0
$181$	17.9756	1.33612	0.668060	−	0.744108i	$-0.267125\pi$
$181$	17.9756	1.33612	0.668060	+	0.744108i	$0.267125\pi$
$182$	0	0
$183$	10.7379	0.793766
$184$	0	0
$185$	−17.8416	−1.31174
$186$	0	0
$187$	8.63933	0.631770
$188$	0	0
$189$	0.506874	0.0368696
$190$	0	0
$191$	−19.0431	−1.37791	−0.688954	−	0.724805i	$-0.741930\pi$
$191$	−19.0431	−1.37791	−0.688954	+	0.724805i	$0.741930\pi$
$192$	0	0
$193$	−2.70949	−0.195033	−0.0975166	−	0.995234i	$-0.531090\pi$
$193$	−2.70949	−0.195033	−0.0975166	+	0.995234i	$0.531090\pi$
$194$	0	0
$195$	16.4711	1.17952
$196$	0	0
$197$	−13.3153	−0.948674	−0.474337	−	0.880343i	$-0.657312\pi$
$197$	−13.3153	−0.948674	−0.474337	+	0.880343i	$0.657312\pi$
$198$	0	0
$199$	2.68736	0.190502	0.0952510	−	0.995453i	$-0.469635\pi$
$199$	2.68736	0.190502	0.0952510	+	0.995453i	$0.469635\pi$
$200$	0	0
$201$	37.6477	2.65547
$202$	0	0
$203$	0.124014	0.00870408
$204$	0	0
$205$	−7.86566	−0.549361
$206$	0	0
$207$	−22.7280	−1.57970
$208$	0	0
$209$	2.25130	0.155726
$210$	0	0
$211$	−9.55720	−0.657945	−0.328973	−	0.944339i	$-0.606702\pi$
$211$	−9.55720	−0.657945	−0.328973	+	0.944339i	$0.606702\pi$
$212$	0	0
$213$	−41.4449	−2.83976
$214$	0	0
$215$	−13.0542	−0.890289
$216$	0	0
$217$	2.19240	0.148829
$218$	0	0
$219$	15.4917	1.04683
$220$	0	0
$221$	8.89876	0.598596
$222$	0	0
$223$	4.23789	0.283790	0.141895	−	0.989882i	$-0.454680\pi$
$223$	4.23789	0.283790	0.141895	+	0.989882i	$0.454680\pi$
$224$	0	0
$225$	−9.37989	−0.625326
$226$	0	0
$227$	−2.23005	−0.148013	−0.0740067	−	0.997258i	$-0.523579\pi$
$227$	−2.23005	−0.148013	−0.0740067	+	0.997258i	$0.523579\pi$
$228$	0	0
$229$	27.2320	1.79954	0.899769	−	0.436366i	$-0.143735\pi$
$229$	27.2320	1.79954	0.899769	+	0.436366i	$0.143735\pi$
$230$	0	0
$231$	5.09850	0.335456
$232$	0	0
$233$	17.4212	1.14130	0.570649	−	0.821194i	$-0.306692\pi$
$233$	17.4212	1.14130	0.570649	+	0.821194i	$0.306692\pi$
$234$	0	0
$235$	−17.4005	−1.13508
$236$	0	0
$237$	26.7475	1.73744
$238$	0	0
$239$	−29.2187	−1.89000	−0.945002	−	0.327066i	$-0.893940\pi$
$239$	−29.2187	−1.89000	−0.945002	+	0.327066i	$0.893940\pi$
$240$	0	0
$241$	−0.793095	−0.0510877	−0.0255439	−	0.999674i	$-0.508132\pi$
$241$	−0.793095	−0.0510877	−0.0255439	+	0.999674i	$0.508132\pi$
$242$	0	0
$243$	−22.3578	−1.43425
$244$	0	0
$245$	−10.1687	−0.649654
$246$	0	0
$247$	2.31891	0.147549
$248$	0	0
$249$	30.7848	1.95091
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−27.9484	−1.75710
$254$	0	0
$255$	7.82095	0.489767
$256$	0	0
$257$	−5.86990	−0.366154	−0.183077	−	0.983099i	$-0.558606\pi$
$257$	−5.86990	−0.366154	−0.183077	+	0.983099i	$0.558606\pi$
$258$	0	0
$259$	−5.68859	−0.353472
$260$	0	0
$261$	0.885157	0.0547898
$262$	0	0
$263$	−16.2437	−1.00163	−0.500815	−	0.865554i	$-0.666966\pi$
$263$	−16.2437	−1.00163	−0.500815	+	0.865554i	$0.666966\pi$
$264$	0	0
$265$	18.3918	1.12980
$266$	0	0
$267$	13.6735	0.836804
$268$	0	0
$269$	20.6368	1.25825	0.629125	−	0.777304i	$-0.283413\pi$
$269$	20.6368	1.25825	0.629125	+	0.777304i	$0.283413\pi$
$270$	0	0
$271$	−31.0148	−1.88401	−0.942007	−	0.335593i	$-0.891063\pi$
$271$	−31.0148	−1.88401	−0.942007	+	0.335593i	$0.891063\pi$
$272$	0	0
$273$	5.25160	0.317841
$274$	0	0
$275$	−11.5343	−0.695547
$276$	0	0
$277$	4.90285	0.294584	0.147292	−	0.989093i	$-0.452944\pi$
$277$	4.90285	0.294584	0.147292	+	0.989093i	$0.452944\pi$
$278$	0	0
$279$	15.6483	0.936841
$280$	0	0
$281$	15.7041	0.936827	0.468413	−	0.883509i	$-0.344826\pi$
$281$	15.7041	0.936827	0.468413	+	0.883509i	$0.344826\pi$
$282$	0	0
$283$	18.2286	1.08358	0.541790	−	0.840514i	$-0.317747\pi$
$283$	18.2286	1.08358	0.541790	+	0.840514i	$0.317747\pi$
$284$	0	0
$285$	2.03804	0.120723
$286$	0	0
$287$	−2.50787	−0.148035
$288$	0	0
$289$	−12.7746	−0.751447
$290$	0	0
$291$	35.5180	2.08210
$292$	0	0
$293$	−1.47789	−0.0863394	−0.0431697	−	0.999068i	$-0.513746\pi$
$293$	−1.47789	−0.0863394	−0.0431697	+	0.999068i	$0.513746\pi$
$294$	0	0
$295$	5.96202	0.347122
$296$	0	0
$297$	4.44882	0.258147
$298$	0	0
$299$	−28.7876	−1.66483
$300$	0	0
$301$	−4.16218	−0.239904
$302$	0	0
$303$	−0.243839	−0.0140082
$304$	0	0
$305$	6.36584	0.364507
$306$	0	0
$307$	−28.7263	−1.63950	−0.819748	−	0.572725i	$-0.805886\pi$
$307$	−28.7263	−1.63950	−0.819748	+	0.572725i	$0.805886\pi$
$308$	0	0
$309$	−27.2056	−1.54767
$310$	0	0
$311$	17.8515	1.01227	0.506133	−	0.862455i	$-0.331074\pi$
$311$	17.8515	1.01227	0.506133	+	0.862455i	$0.331074\pi$
$312$	0	0
$313$	−13.9352	−0.787662	−0.393831	−	0.919183i	$-0.628850\pi$
$313$	−13.9352	−0.787662	−0.393831	+	0.919183i	$0.628850\pi$
$314$	0	0
$315$	2.45801	0.138493
$316$	0	0
$317$	17.6130	0.989247	0.494623	−	0.869107i	$-0.335306\pi$
$317$	17.6130	0.989247	0.494623	+	0.869107i	$0.335306\pi$
$318$	0	0
$319$	1.08847	0.0609425
$320$	0	0
$321$	49.3663	2.75536
$322$	0	0
$323$	1.10109	0.0612661
$324$	0	0
$325$	−11.8807	−0.659024
$326$	0	0
$327$	0.133183	0.00736505
$328$	0	0
$329$	−5.54794	−0.305868
$330$	0	0
$331$	−12.0310	−0.661284	−0.330642	−	0.943756i	$-0.607265\pi$
$331$	−12.0310	−0.661284	−0.330642	+	0.943756i	$0.607265\pi$
$332$	0	0
$333$	−40.6026	−2.22501
$334$	0	0
$335$	22.3191	1.21942
$336$	0	0
$337$	17.3931	0.947463	0.473732	−	0.880669i	$-0.342907\pi$
$337$	17.3931	0.947463	0.473732	+	0.880669i	$0.342907\pi$
$338$	0	0
$339$	6.95524	0.377757
$340$	0	0
$341$	19.2426	1.04204
$342$	0	0
$343$	−6.59413	−0.356050
$344$	0	0
$345$	−25.3009	−1.36215
$346$	0	0
$347$	15.6671	0.841054	0.420527	−	0.907280i	$-0.361845\pi$
$347$	15.6671	0.841054	0.420527	+	0.907280i	$0.361845\pi$
$348$	0	0
$349$	21.9367	1.17425	0.587123	−	0.809498i	$-0.300260\pi$
$349$	21.9367	1.17425	0.587123	+	0.809498i	$0.300260\pi$
$350$	0	0
$351$	4.58241	0.244591
$352$	0	0
$353$	32.9804	1.75537	0.877686	−	0.479237i	$-0.159087\pi$
$353$	32.9804	1.75537	0.877686	+	0.479237i	$0.159087\pi$
$354$	0	0
$355$	−24.5702	−1.30405
$356$	0	0
$357$	2.49362	0.131976
$358$	0	0
$359$	−17.6493	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$359$	−17.6493	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$360$	0	0
$361$	−18.7131	−0.984898
$362$	0	0
$363$	16.8826	0.886105
$364$	0	0
$365$	9.18411	0.480718
$366$	0	0
$367$	−31.5782	−1.64837	−0.824183	−	0.566323i	$-0.808366\pi$
$367$	−31.5782	−1.64837	−0.824183	+	0.566323i	$0.808366\pi$
$368$	0	0
$369$	−17.9001	−0.931840
$370$	0	0
$371$	5.86401	0.304444
$372$	0	0
$373$	14.7157	0.761949	0.380975	−	0.924585i	$-0.375589\pi$
$373$	14.7157	0.761949	0.380975	+	0.924585i	$0.375589\pi$
$374$	0	0
$375$	−29.4655	−1.52159
$376$	0	0
$377$	1.12115	0.0577424
$378$	0	0
$379$	25.6456	1.31732	0.658662	−	0.752439i	$-0.271123\pi$
$379$	25.6456	1.31732	0.658662	+	0.752439i	$0.271123\pi$
$380$	0	0
$381$	−39.0625	−2.00123
$382$	0	0
$383$	14.6100	0.746537	0.373268	−	0.927723i	$-0.378237\pi$
$383$	14.6100	0.746537	0.373268	+	0.927723i	$0.378237\pi$
$384$	0	0
$385$	3.02259	0.154046
$386$	0	0
$387$	−29.7078	−1.51013
$388$	0	0
$389$	−7.94533	−0.402844	−0.201422	−	0.979505i	$-0.564556\pi$
$389$	−7.94533	−0.402844	−0.201422	+	0.979505i	$0.564556\pi$
$390$	0	0
$391$	−13.6692	−0.691282
$392$	0	0
$393$	35.1225	1.77170
$394$	0	0
$395$	15.8570	0.797852
$396$	0	0
$397$	−6.55835	−0.329154	−0.164577	−	0.986364i	$-0.552626\pi$
$397$	−6.55835	−0.329154	−0.164577	+	0.986364i	$0.552626\pi$
$398$	0	0
$399$	0.649806	0.0325310
$400$	0	0
$401$	−5.20130	−0.259741	−0.129870	−	0.991531i	$-0.541456\pi$
$401$	−5.20130	−0.259741	−0.129870	+	0.991531i	$0.541456\pi$
$402$	0	0
$403$	19.8204	0.987326
$404$	0	0
$405$	−11.3720	−0.565079
$406$	0	0
$407$	−49.9286	−2.47487
$408$	0	0
$409$	14.7419	0.728939	0.364470	−	0.931215i	$-0.381250\pi$
$409$	14.7419	0.728939	0.364470	+	0.931215i	$0.381250\pi$
$410$	0	0
$411$	−36.5677	−1.80375
$412$	0	0
$413$	1.90092	0.0935382
$414$	0	0
$415$	18.2505	0.895880
$416$	0	0
$417$	−58.0163	−2.84107
$418$	0	0
$419$	29.9425	1.46279	0.731394	−	0.681955i	$-0.238870\pi$
$419$	29.9425	1.46279	0.731394	+	0.681955i	$0.238870\pi$
$420$	0	0
$421$	−0.851309	−0.0414903	−0.0207451	−	0.999785i	$-0.506604\pi$
$421$	−0.851309	−0.0414903	−0.0207451	+	0.999785i	$0.506604\pi$
$422$	0	0
$423$	−39.5987	−1.92536
$424$	0	0
$425$	−5.64132	−0.273644
$426$	0	0
$427$	2.02967	0.0982227
$428$	0	0
$429$	46.0932	2.22540
$430$	0	0
$431$	12.5827	0.606086	0.303043	−	0.952977i	$-0.401997\pi$
$431$	12.5827	0.606086	0.303043	+	0.952977i	$0.401997\pi$
$432$	0	0
$433$	−10.9864	−0.527974	−0.263987	−	0.964526i	$-0.585038\pi$
$433$	−10.9864	−0.527974	−0.263987	+	0.964526i	$0.585038\pi$
$434$	0	0
$435$	0.985360	0.0472444
$436$	0	0
$437$	−3.56203	−0.170395
$438$	0	0
$439$	−21.3714	−1.02000	−0.510000	−	0.860175i	$-0.670355\pi$
$439$	−21.3714	−1.02000	−0.510000	+	0.860175i	$0.670355\pi$
$440$	0	0
$441$	−23.1411	−1.10196
$442$	0	0
$443$	37.4243	1.77808	0.889040	−	0.457829i	$-0.151373\pi$
$443$	37.4243	1.77808	0.889040	+	0.457829i	$0.151373\pi$
$444$	0	0
$445$	8.10618	0.384270
$446$	0	0
$447$	−23.8272	−1.12699
$448$	0	0
$449$	30.9688	1.46151	0.730755	−	0.682640i	$-0.239168\pi$
$449$	30.9688	1.46151	0.730755	+	0.682640i	$0.239168\pi$
$450$	0	0
$451$	−22.0115	−1.03648
$452$	0	0
$453$	−18.5464	−0.871384
$454$	0	0
$455$	3.11336	0.145957
$456$	0	0
$457$	−9.39062	−0.439275	−0.219637	−	0.975582i	$-0.570487\pi$
$457$	−9.39062	−0.439275	−0.219637	+	0.975582i	$0.570487\pi$
$458$	0	0
$459$	2.17587	0.101561
$460$	0	0
$461$	−5.75111	−0.267856	−0.133928	−	0.990991i	$-0.542759\pi$
$461$	−5.75111	−0.267856	−0.133928	+	0.990991i	$0.542759\pi$
$462$	0	0
$463$	−25.7631	−1.19731	−0.598656	−	0.801006i	$-0.704299\pi$
$463$	−25.7631	−1.19731	−0.598656	+	0.801006i	$0.704299\pi$
$464$	0	0
$465$	17.4198	0.807823
$466$	0	0
$467$	29.5070	1.36542	0.682710	−	0.730689i	$-0.260801\pi$
$467$	29.5070	1.36542	0.682710	+	0.730689i	$0.260801\pi$
$468$	0	0
$469$	7.11617	0.328594
$470$	0	0
$471$	18.0139	0.830039
$472$	0	0
$473$	−36.5313	−1.67971
$474$	0	0
$475$	−1.47006	−0.0674509
$476$	0	0
$477$	41.8547	1.91639
$478$	0	0
$479$	19.6959	0.899927	0.449963	−	0.893047i	$-0.351437\pi$
$479$	19.6959	0.899927	0.449963	+	0.893047i	$0.351437\pi$
$480$	0	0
$481$	−51.4279	−2.34491
$482$	0	0
$483$	−8.06688	−0.367056
$484$	0	0
$485$	21.0565	0.956126
$486$	0	0
$487$	10.1758	0.461108	0.230554	−	0.973060i	$-0.425946\pi$
$487$	10.1758	0.461108	0.230554	+	0.973060i	$0.425946\pi$
$488$	0	0
$489$	−16.0984	−0.727994
$490$	0	0
$491$	−30.0822	−1.35759	−0.678796	−	0.734327i	$-0.737498\pi$
$491$	−30.0822	−1.35759	−0.678796	+	0.734327i	$0.737498\pi$
$492$	0	0
$493$	0.532357	0.0239762
$494$	0	0
$495$	21.5739	0.969675
$496$	0	0
$497$	−7.83391	−0.351399
$498$	0	0
$499$	−28.6262	−1.28149	−0.640743	−	0.767756i	$-0.721373\pi$
$499$	−28.6262	−1.28149	−0.640743	+	0.767756i	$0.721373\pi$
$500$	0	0
$501$	−21.4652	−0.958996
$502$	0	0
$503$	−5.95326	−0.265443	−0.132721	−	0.991153i	$-0.542372\pi$
$503$	−5.95326	−0.265443	−0.132721	+	0.991153i	$0.542372\pi$
$504$	0	0
$505$	−0.144558	−0.00643273
$506$	0	0
$507$	14.5438	0.645915
$508$	0	0
$509$	−9.90797	−0.439163	−0.219582	−	0.975594i	$-0.570469\pi$
$509$	−9.90797	−0.439163	−0.219582	+	0.975594i	$0.570469\pi$
$510$	0	0
$511$	2.92824	0.129538
$512$	0	0
$513$	0.567004	0.0250338
$514$	0	0
$515$	−16.1285	−0.710709
$516$	0	0
$517$	−48.6941	−2.14157
$518$	0	0
$519$	36.0418	1.58206
$520$	0	0
$521$	2.43628	0.106735	0.0533677	−	0.998575i	$-0.483004\pi$
$521$	2.43628	0.106735	0.0533677	+	0.998575i	$0.483004\pi$
$522$	0	0
$523$	31.7452	1.38812	0.694061	−	0.719916i	$-0.255820\pi$
$523$	31.7452	1.38812	0.694061	+	0.719916i	$0.255820\pi$
$524$	0	0
$525$	−3.32922	−0.145299
$526$	0	0
$527$	9.41133	0.409964
$528$	0	0
$529$	21.2201	0.922614
$530$	0	0
$531$	13.5679	0.588797
$532$	0	0
$533$	−22.6725	−0.982055
$534$	0	0
$535$	29.2663	1.26529
$536$	0	0
$537$	−20.4455	−0.882290
$538$	0	0
$539$	−28.4564	−1.22570
$540$	0	0
$541$	0.211389	0.00908834	0.00454417	−	0.999990i	$-0.498554\pi$
$541$	0.211389	0.00908834	0.00454417	+	0.999990i	$0.498554\pi$
$542$	0	0
$543$	45.5385	1.95424
$544$	0	0
$545$	0.0789563	0.00338212
$546$	0	0
$547$	−8.20570	−0.350850	−0.175425	−	0.984493i	$-0.556130\pi$
$547$	−8.20570	−0.350850	−0.175425	+	0.984493i	$0.556130\pi$
$548$	0	0
$549$	14.4869	0.618285
$550$	0	0
$551$	0.138726	0.00590992
$552$	0	0
$553$	5.05581	0.214995
$554$	0	0
$555$	−45.1990	−1.91859
$556$	0	0
$557$	−39.8194	−1.68720	−0.843602	−	0.536970i	$-0.819569\pi$
$557$	−39.8194	−1.68720	−0.843602	+	0.536970i	$0.819569\pi$
$558$	0	0
$559$	−37.6283	−1.59151
$560$	0	0
$561$	21.8864	0.924044
$562$	0	0
$563$	5.24909	0.221223	0.110611	−	0.993864i	$-0.464719\pi$
$563$	5.24909	0.221223	0.110611	+	0.993864i	$0.464719\pi$
$564$	0	0
$565$	4.12334	0.173470
$566$	0	0
$567$	−3.62583	−0.152270
$568$	0	0
$569$	−8.91025	−0.373537	−0.186769	−	0.982404i	$-0.559801\pi$
$569$	−8.91025	−0.373537	−0.186769	+	0.982404i	$0.559801\pi$
$570$	0	0
$571$	−14.0518	−0.588051	−0.294025	−	0.955798i	$-0.594995\pi$
$571$	−14.0518	−0.588051	−0.294025	+	0.955798i	$0.594995\pi$
$572$	0	0
$573$	−48.2426	−2.01536
$574$	0	0
$575$	18.2497	0.761067
$576$	0	0
$577$	2.31977	0.0965735	0.0482867	−	0.998834i	$-0.484624\pi$
$577$	2.31977	0.0965735	0.0482867	+	0.998834i	$0.484624\pi$
$578$	0	0
$579$	−6.86407	−0.285261
$580$	0	0
$581$	5.81894	0.241410
$582$	0	0
$583$	51.4682	2.13160
$584$	0	0
$585$	22.2218	0.918757
$586$	0	0
$587$	−7.53569	−0.311031	−0.155516	−	0.987833i	$-0.549704\pi$
$587$	−7.53569	−0.311031	−0.155516	+	0.987833i	$0.549704\pi$
$588$	0	0
$589$	2.45248	0.101053
$590$	0	0
$591$	−33.7322	−1.38756
$592$	0	0
$593$	20.1818	0.828768	0.414384	−	0.910102i	$-0.363997\pi$
$593$	20.1818	0.828768	0.414384	+	0.910102i	$0.363997\pi$
$594$	0	0
$595$	1.47831	0.0606050
$596$	0	0
$597$	6.80802	0.278634
$598$	0	0
$599$	−11.2813	−0.460943	−0.230471	−	0.973079i	$-0.574027\pi$
$599$	−11.2813	−0.460943	−0.230471	+	0.973079i	$0.574027\pi$
$600$	0	0
$601$	−37.6425	−1.53547	−0.767734	−	0.640769i	$-0.778616\pi$
$601$	−37.6425	−1.53547	−0.767734	+	0.640769i	$0.778616\pi$
$602$	0	0
$603$	50.7921	2.06841
$604$	0	0
$605$	10.0087	0.406910
$606$	0	0
$607$	−38.9414	−1.58058	−0.790291	−	0.612732i	$-0.790071\pi$
$607$	−38.9414	−1.58058	−0.790291	+	0.612732i	$0.790071\pi$
$608$	0	0
$609$	0.314170	0.0127308
$610$	0	0
$611$	−50.1564	−2.02911
$612$	0	0
$613$	−33.6687	−1.35987	−0.679934	−	0.733274i	$-0.737991\pi$
$613$	−33.6687	−1.35987	−0.679934	+	0.733274i	$0.737991\pi$
$614$	0	0
$615$	−19.9264	−0.803511
$616$	0	0
$617$	6.86536	0.276389	0.138194	−	0.990405i	$-0.455870\pi$
$617$	6.86536	0.276389	0.138194	+	0.990405i	$0.455870\pi$
$618$	0	0
$619$	−14.3773	−0.577873	−0.288936	−	0.957348i	$-0.593302\pi$
$619$	−14.3773	−0.577873	−0.288936	+	0.957348i	$0.593302\pi$
$620$	0	0
$621$	−7.03895	−0.282464
$622$	0	0
$623$	2.58456	0.103548
$624$	0	0
$625$	−3.74632	−0.149853
$626$	0	0
$627$	5.70333	0.227769
$628$	0	0
$629$	−24.4195	−0.973669
$630$	0	0
$631$	23.1422	0.921278	0.460639	−	0.887588i	$-0.347620\pi$
$631$	23.1422	0.921278	0.460639	+	0.887588i	$0.347620\pi$
$632$	0	0
$633$	−24.2117	−0.962328
$634$	0	0
$635$	−23.1578	−0.918989
$636$	0	0
$637$	−29.3109	−1.16134
$638$	0	0
$639$	−55.9149	−2.21196
$640$	0	0
$641$	25.5776	1.01026	0.505128	−	0.863045i	$-0.331445\pi$
$641$	25.5776	1.01026	0.505128	+	0.863045i	$0.331445\pi$
$642$	0	0
$643$	2.45113	0.0966633	0.0483316	−	0.998831i	$-0.484610\pi$
$643$	2.45113	0.0966633	0.0483316	+	0.998831i	$0.484610\pi$
$644$	0	0
$645$	−33.0708	−1.30216
$646$	0	0
$647$	43.9315	1.72712	0.863562	−	0.504243i	$-0.168228\pi$
$647$	43.9315	1.72712	0.863562	+	0.504243i	$0.168228\pi$
$648$	0	0
$649$	16.6843	0.654917
$650$	0	0
$651$	5.55409	0.217682
$652$	0	0
$653$	35.4190	1.38605	0.693025	−	0.720913i	$-0.256277\pi$
$653$	35.4190	1.38605	0.693025	+	0.720913i	$0.256277\pi$
$654$	0	0
$655$	20.8220	0.813584
$656$	0	0
$657$	20.9005	0.815406
$658$	0	0
$659$	40.6085	1.58188	0.790940	−	0.611893i	$-0.209592\pi$
$659$	40.6085	1.58188	0.790940	+	0.611893i	$0.209592\pi$
$660$	0	0
$661$	8.84140	0.343891	0.171945	−	0.985106i	$-0.444995\pi$
$661$	8.84140	0.343891	0.171945	+	0.985106i	$0.444995\pi$
$662$	0	0
$663$	22.5436	0.875522
$664$	0	0
$665$	0.385231	0.0149386
$666$	0	0
$667$	−1.72218	−0.0666832
$668$	0	0
$669$	10.7360	0.415079
$670$	0	0
$671$	17.8144	0.687716
$672$	0	0
$673$	43.8126	1.68885	0.844426	−	0.535672i	$-0.179942\pi$
$673$	43.8126	1.68885	0.844426	+	0.535672i	$0.179942\pi$
$674$	0	0
$675$	−2.90499	−0.111813
$676$	0	0
$677$	−32.7378	−1.25822	−0.629109	−	0.777317i	$-0.716580\pi$
$677$	−32.7378	−1.25822	−0.629109	+	0.777317i	$0.716580\pi$
$678$	0	0
$679$	6.71361	0.257645
$680$	0	0
$681$	−5.64948	−0.216488
$682$	0	0
$683$	−18.2997	−0.700218	−0.350109	−	0.936709i	$-0.613855\pi$
$683$	−18.2997	−0.700218	−0.350109	+	0.936709i	$0.613855\pi$
$684$	0	0
$685$	−21.6788	−0.828303
$686$	0	0
$687$	68.9879	2.63205
$688$	0	0
$689$	53.0138	2.01966
$690$	0	0
$691$	19.0415	0.724372	0.362186	−	0.932106i	$-0.382031\pi$
$691$	19.0415	0.724372	0.362186	+	0.932106i	$0.382031\pi$
$692$	0	0
$693$	6.87858	0.261296
$694$	0	0
$695$	−34.3943	−1.30465
$696$	0	0
$697$	−10.7656	−0.407775
$698$	0	0
$699$	44.1338	1.66929
$700$	0	0
$701$	−42.8488	−1.61838	−0.809189	−	0.587549i	$-0.800093\pi$
$701$	−42.8488	−1.61838	−0.809189	+	0.587549i	$0.800093\pi$
$702$	0	0
$703$	−6.36343	−0.240001
$704$	0	0
$705$	−44.0815	−1.66020
$706$	0	0
$707$	−0.0460905	−0.00173341
$708$	0	0
$709$	−33.0324	−1.24056	−0.620279	−	0.784382i	$-0.712980\pi$
$709$	−33.0324	−1.24056	−0.620279	+	0.784382i	$0.712980\pi$
$710$	0	0
$711$	36.0861	1.35334
$712$	0	0
$713$	−30.4458	−1.14020
$714$	0	0
$715$	27.3259	1.02193
$716$	0	0
$717$	−74.0211	−2.76437
$718$	0	0
$719$	44.0194	1.64165	0.820824	−	0.571182i	$-0.193515\pi$
$719$	44.0194	1.64165	0.820824	+	0.571182i	$0.193515\pi$
$720$	0	0
$721$	−5.14239	−0.191513
$722$	0	0
$723$	−2.00918	−0.0747223
$724$	0	0
$725$	−0.710749	−0.0263965
$726$	0	0
$727$	−11.1168	−0.412300	−0.206150	−	0.978520i	$-0.566094\pi$
$727$	−11.1168	−0.412300	−0.206150	+	0.978520i	$0.566094\pi$
$728$	0	0
$729$	−33.9243	−1.25646
$730$	0	0
$731$	−17.8670	−0.660836
$732$	0	0
$733$	−11.6867	−0.431659	−0.215829	−	0.976431i	$-0.569246\pi$
$733$	−11.6867	−0.431659	−0.215829	+	0.976431i	$0.569246\pi$
$734$	0	0
$735$	−25.7608	−0.950201
$736$	0	0
$737$	62.4584	2.30069
$738$	0	0
$739$	−44.9919	−1.65505	−0.827527	−	0.561426i	$-0.810253\pi$
$739$	−44.9919	−1.65505	−0.827527	+	0.561426i	$0.810253\pi$
$740$	0	0
$741$	5.87460	0.215809
$742$	0	0
$743$	26.7461	0.981220	0.490610	−	0.871379i	$-0.336774\pi$
$743$	26.7461	0.981220	0.490610	+	0.871379i	$0.336774\pi$
$744$	0	0
$745$	−14.1257	−0.517525
$746$	0	0
$747$	41.5330	1.51961
$748$	0	0
$749$	9.33121	0.340955
$750$	0	0
$751$	−12.7444	−0.465049	−0.232525	−	0.972591i	$-0.574699\pi$
$751$	−12.7444	−0.465049	−0.232525	+	0.972591i	$0.574699\pi$
$752$	0	0
$753$	−2.53334	−0.0923202
$754$	0	0
$755$	−10.9950	−0.400150
$756$	0	0
$757$	10.2279	0.371739	0.185870	−	0.982574i	$-0.440490\pi$
$757$	10.2279	0.371739	0.185870	+	0.982574i	$0.440490\pi$
$758$	0	0
$759$	−70.8028	−2.56998
$760$	0	0
$761$	22.1670	0.803555	0.401777	−	0.915737i	$-0.368393\pi$
$761$	22.1670	0.803555	0.401777	+	0.915737i	$0.368393\pi$
$762$	0	0
$763$	0.0251743	0.000911370 0
$764$	0	0
$765$	10.5515	0.381492
$766$	0	0
$767$	17.1853	0.620527
$768$	0	0
$769$	51.3208	1.85067	0.925337	−	0.379146i	$-0.123782\pi$
$769$	51.3208	1.85067	0.925337	+	0.379146i	$0.123782\pi$
$770$	0	0
$771$	−14.8705	−0.535547
$772$	0	0
$773$	6.17332	0.222039	0.111019	−	0.993818i	$-0.464588\pi$
$773$	6.17332	0.222039	0.111019	+	0.993818i	$0.464588\pi$
$774$	0	0
$775$	−12.5650	−0.451350
$776$	0	0
$777$	−14.4112	−0.516997
$778$	0	0
$779$	−2.80538	−0.100513
$780$	0	0
$781$	−68.7580	−2.46035
$782$	0	0
$783$	0.274137	0.00979686
$784$	0	0
$785$	10.6794	0.381164
$786$	0	0
$787$	19.7681	0.704658	0.352329	−	0.935876i	$-0.385390\pi$
$787$	19.7681	0.704658	0.352329	+	0.935876i	$0.385390\pi$
$788$	0	0
$789$	−41.1509	−1.46501
$790$	0	0
$791$	1.31468	0.0467446
$792$	0	0
$793$	18.3493	0.651604
$794$	0	0
$795$	46.5928	1.65248
$796$	0	0
$797$	−0.193962	−0.00687048	−0.00343524	−	0.999994i	$-0.501093\pi$
$797$	−0.193962	−0.00687048	−0.00343524	+	0.999994i	$0.501093\pi$
$798$	0	0
$799$	−23.8157	−0.842540
$800$	0	0
$801$	18.4474	0.651808
$802$	0	0
$803$	25.7011	0.906973
$804$	0	0
$805$	−4.78237	−0.168556
$806$	0	0
$807$	52.2802	1.84035
$808$	0	0
$809$	−25.7244	−0.904422	−0.452211	−	0.891911i	$-0.649365\pi$
$809$	−25.7244	−0.904422	−0.452211	+	0.891911i	$0.649365\pi$
$810$	0	0
$811$	27.3805	0.961459	0.480730	−	0.876869i	$-0.340372\pi$
$811$	27.3805	0.961459	0.480730	+	0.876869i	$0.340372\pi$
$812$	0	0
$813$	−78.5711	−2.75561
$814$	0	0
$815$	−9.54376	−0.334303
$816$	0	0
$817$	−4.65593	−0.162891
$818$	0	0
$819$	7.08514	0.247575
$820$	0	0
$821$	−22.8188	−0.796380	−0.398190	−	0.917303i	$-0.630362\pi$
$821$	−22.8188	−0.796380	−0.398190	+	0.917303i	$0.630362\pi$
$822$	0	0
$823$	34.9298	1.21758	0.608789	−	0.793332i	$-0.291656\pi$
$823$	34.9298	1.21758	0.608789	+	0.793332i	$0.291656\pi$
$824$	0	0
$825$	−29.2205	−1.01733
$826$	0	0
$827$	8.95134	0.311269	0.155634	−	0.987815i	$-0.450258\pi$
$827$	8.95134	0.311269	0.155634	+	0.987815i	$0.450258\pi$
$828$	0	0
$829$	40.6186	1.41074	0.705372	−	0.708838i	$-0.250780\pi$
$829$	40.6186	1.41074	0.705372	+	0.708838i	$0.250780\pi$
$830$	0	0
$831$	12.4206	0.430866
$832$	0	0
$833$	−13.9177	−0.482220
$834$	0	0
$835$	−12.7254	−0.440382
$836$	0	0
$837$	4.84636	0.167515
$838$	0	0
$839$	23.2914	0.804108	0.402054	−	0.915616i	$-0.368296\pi$
$839$	23.2914	0.804108	0.402054	+	0.915616i	$0.368296\pi$
$840$	0	0
$841$	−28.9329	−0.997687
$842$	0	0
$843$	39.7838	1.37023
$844$	0	0
$845$	8.62217	0.296612
$846$	0	0
$847$	3.19114	0.109649
$848$	0	0
$849$	46.1794	1.58487
$850$	0	0
$851$	78.9974	2.70800
$852$	0	0
$853$	5.70196	0.195231	0.0976157	−	0.995224i	$-0.468878\pi$
$853$	5.70196	0.195231	0.0976157	+	0.995224i	$0.468878\pi$
$854$	0	0
$855$	2.74961	0.0940345
$856$	0	0
$857$	3.35878	0.114734	0.0573669	−	0.998353i	$-0.481730\pi$
$857$	3.35878	0.114734	0.0573669	+	0.998353i	$0.481730\pi$
$858$	0	0
$859$	−9.73347	−0.332102	−0.166051	−	0.986117i	$-0.553102\pi$
$859$	−9.73347	−0.332102	−0.166051	+	0.986117i	$0.553102\pi$
$860$	0	0
$861$	−6.35330	−0.216520
$862$	0	0
$863$	−9.75375	−0.332022	−0.166011	−	0.986124i	$-0.553089\pi$
$863$	−9.75375	−0.332022	−0.166011	+	0.986124i	$0.553089\pi$
$864$	0	0
$865$	21.3670	0.726500
$866$	0	0
$867$	−32.3625	−1.09909
$868$	0	0
$869$	44.3747	1.50531
$870$	0	0
$871$	64.3340	2.17988
$872$	0	0
$873$	47.9187	1.62180
$874$	0	0
$875$	−5.56956	−0.188285
$876$	0	0
$877$	−11.7419	−0.396495	−0.198247	−	0.980152i	$-0.563525\pi$
$877$	−11.7419	−0.396495	−0.198247	+	0.980152i	$0.563525\pi$
$878$	0	0
$879$	−3.74401	−0.126282
$880$	0	0
$881$	37.1291	1.25091	0.625456	−	0.780259i	$-0.284913\pi$
$881$	37.1291	1.25091	0.625456	+	0.780259i	$0.284913\pi$
$882$	0	0
$883$	−18.9646	−0.638210	−0.319105	−	0.947719i	$-0.603382\pi$
$883$	−18.9646	−0.638210	−0.319105	+	0.947719i	$0.603382\pi$
$884$	0	0
$885$	15.1039	0.507711
$886$	0	0
$887$	−23.0150	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$887$	−23.0150	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$888$	0	0
$889$	−7.38359	−0.247638
$890$	0	0
$891$	−31.8238	−1.06614
$892$	0	0
$893$	−6.20609	−0.207679
$894$	0	0
$895$	−12.1209	−0.405158
$896$	0	0
$897$	−72.9290	−2.43503
$898$	0	0
$899$	1.18573	0.0395464
$900$	0	0
$901$	25.1725	0.838618
$902$	0	0
$903$	−10.5442	−0.350890
$904$	0	0
$905$	26.9970	0.897411
$906$	0	0
$907$	−23.5773	−0.782873	−0.391436	−	0.920205i	$-0.628022\pi$
$907$	−23.5773	−0.782873	−0.391436	+	0.920205i	$0.628022\pi$
$908$	0	0
$909$	−0.328973	−0.0109114
$910$	0	0
$911$	−20.6900	−0.685492	−0.342746	−	0.939428i	$-0.611357\pi$
$911$	−20.6900	−0.685492	−0.342746	+	0.939428i	$0.611357\pi$
$912$	0	0
$913$	51.0727	1.69026
$914$	0	0
$915$	16.1269	0.533137
$916$	0	0
$917$	6.63886	0.219234
$918$	0	0
$919$	39.6254	1.30712	0.653560	−	0.756875i	$-0.273275\pi$
$919$	39.6254	1.30712	0.653560	+	0.756875i	$0.273275\pi$
$920$	0	0
$921$	−72.7736	−2.39797
$922$	0	0
$923$	−70.8227	−2.33116
$924$	0	0
$925$	32.6024	1.07196
$926$	0	0
$927$	−36.7041	−1.20552
$928$	0	0
$929$	−8.16495	−0.267883	−0.133942	−	0.990989i	$-0.542763\pi$
$929$	−8.16495	−0.267883	−0.133942	+	0.990989i	$0.542763\pi$
$930$	0	0
$931$	−3.62678	−0.118863
$932$	0	0
$933$	45.2240	1.48057
$934$	0	0
$935$	12.9751	0.424332
$936$	0	0
$937$	−15.2089	−0.496853	−0.248426	−	0.968651i	$-0.579913\pi$
$937$	−15.2089	−0.496853	−0.248426	+	0.968651i	$0.579913\pi$
$938$	0	0
$939$	−35.3026	−1.15206
$940$	0	0
$941$	28.5538	0.930827	0.465414	−	0.885093i	$-0.345906\pi$
$941$	28.5538	0.930827	0.465414	+	0.885093i	$0.345906\pi$
$942$	0	0
$943$	34.8268	1.13412
$944$	0	0
$945$	0.761257	0.0247637
$946$	0	0
$947$	15.8580	0.515317	0.257658	−	0.966236i	$-0.417049\pi$
$947$	15.8580	0.515317	0.257658	+	0.966236i	$0.417049\pi$
$948$	0	0
$949$	26.4729	0.859347
$950$	0	0
$951$	44.6199	1.44690
$952$	0	0
$953$	−15.4793	−0.501423	−0.250711	−	0.968062i	$-0.580665\pi$
$953$	−15.4793	−0.501423	−0.250711	+	0.968062i	$0.580665\pi$
$954$	0	0
$955$	−28.6001	−0.925479
$956$	0	0
$957$	2.75746	0.0891361
$958$	0	0
$959$	−6.91202	−0.223201
$960$	0	0
$961$	−10.0379	−0.323804
$962$	0	0
$963$	66.6020	2.14622
$964$	0	0
$965$	−4.06929	−0.130995
$966$	0	0
$967$	−46.7818	−1.50440	−0.752201	−	0.658933i	$-0.771008\pi$
$967$	−46.7818	−1.50440	−0.752201	+	0.658933i	$0.771008\pi$
$968$	0	0
$969$	2.78943	0.0896095
$970$	0	0
$971$	4.16808	0.133760	0.0668801	−	0.997761i	$-0.478696\pi$
$971$	4.16808	0.133760	0.0668801	+	0.997761i	$0.478696\pi$
$972$	0	0
$973$	−10.9662	−0.351561
$974$	0	0
$975$	−30.0980	−0.963906
$976$	0	0
$977$	10.8271	0.346389	0.173195	−	0.984888i	$-0.444591\pi$
$977$	10.8271	0.346389	0.173195	+	0.984888i	$0.444591\pi$
$978$	0	0
$979$	22.6846	0.725003
$980$	0	0
$981$	0.179683	0.00573683
$982$	0	0
$983$	30.8392	0.983616	0.491808	−	0.870704i	$-0.336336\pi$
$983$	30.8392	0.983616	0.491808	+	0.870704i	$0.336336\pi$
$984$	0	0
$985$	−19.9978	−0.637182
$986$	0	0
$987$	−14.0548	−0.447371
$988$	0	0
$989$	57.8001	1.83794
$990$	0	0
$991$	−22.0982	−0.701973	−0.350987	−	0.936380i	$-0.614154\pi$
$991$	−22.0982	−0.701973	−0.350987	+	0.936380i	$0.614154\pi$
$992$	0	0
$993$	−30.4787	−0.967212
$994$	0	0
$995$	4.03606	0.127952
$996$	0	0
$997$	5.31981	0.168480	0.0842400	−	0.996445i	$-0.473154\pi$
$997$	5.31981	0.168480	0.0842400	+	0.996445i	$0.473154\pi$
$998$	0	0
$999$	−12.5748	−0.397849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.19	✓	23
4.3	odd	2		4016.2.a.m.1.5		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.19	✓	23	1.1	even	1	trivial
4016.2.a.m.1.5		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)

\(f(q)\)	\(=\)	\(q+2.53334 q^{3} +1.50187 q^{5} +0.478852 q^{7} +3.41783 q^{9} +O(q^{10})\)	\(q+2.53334 q^{3} +1.50187 q^{5} +0.478852 q^{7} +3.41783 q^{9} +4.20287 q^{11} +4.32908 q^{13} +3.80475 q^{15} +2.05558 q^{17} +0.535658 q^{19} +1.21310 q^{21} -6.64982 q^{23} -2.74440 q^{25} +1.05852 q^{27} +0.258982 q^{29} +4.57844 q^{31} +10.6473 q^{33} +0.719173 q^{35} -11.8796 q^{37} +10.9671 q^{39} -5.23725 q^{41} -8.69198 q^{43} +5.13313 q^{45} -11.5859 q^{47} -6.77070 q^{49} +5.20748 q^{51} +12.2460 q^{53} +6.31216 q^{55} +1.35701 q^{57} +3.96974 q^{59} +4.23862 q^{61} +1.63664 q^{63} +6.50171 q^{65} +14.8609 q^{67} -16.8463 q^{69} -16.3598 q^{71} +6.11513 q^{73} -6.95250 q^{75} +2.01256 q^{77} +10.5582 q^{79} -7.57191 q^{81} +12.1519 q^{83} +3.08720 q^{85} +0.656090 q^{87} +5.39741 q^{89} +2.07299 q^{91} +11.5988 q^{93} +0.804488 q^{95} +14.0202 q^{97} +14.3647 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more