LMFDB - Embedded newform 6008.2.a.e.1.38

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.38
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.13238 q^{3} +3.35698 q^{5} +0.313319 q^{7} +1.54704 q^{9} +O(q^{10})$	$q+2.13238 q^{3} +3.35698 q^{5} +0.313319 q^{7} +1.54704 q^{9} +2.50247 q^{11} -6.16150 q^{13} +7.15836 q^{15} +5.24751 q^{17} +7.03909 q^{19} +0.668115 q^{21} +1.85469 q^{23} +6.26934 q^{25} -3.09826 q^{27} -7.13470 q^{29} +8.61037 q^{31} +5.33621 q^{33} +1.05181 q^{35} +11.0889 q^{37} -13.1386 q^{39} -5.42641 q^{41} -4.95676 q^{43} +5.19339 q^{45} +2.99818 q^{47} -6.90183 q^{49} +11.1897 q^{51} +6.00096 q^{53} +8.40074 q^{55} +15.0100 q^{57} +10.4199 q^{59} -12.3401 q^{61} +0.484718 q^{63} -20.6840 q^{65} -7.81203 q^{67} +3.95490 q^{69} -2.57760 q^{71} -4.30939 q^{73} +13.3686 q^{75} +0.784070 q^{77} +2.00325 q^{79} -11.2478 q^{81} -12.5725 q^{83} +17.6158 q^{85} -15.2139 q^{87} +3.54250 q^{89} -1.93052 q^{91} +18.3606 q^{93} +23.6301 q^{95} +9.22478 q^{97} +3.87142 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.13238	1.23113	0.615565	−	0.788086i	$-0.288928\pi$
$3$	2.13238	1.23113	0.615565	+	0.788086i	$0.288928\pi$
$4$	0	0
$5$	3.35698	1.50129	0.750644	−	0.660706i	$-0.229743\pi$
$5$	3.35698	1.50129	0.750644	+	0.660706i	$0.229743\pi$
$6$	0	0
$7$	0.313319	0.118424	0.0592118	−	0.998245i	$-0.481141\pi$
$7$	0.313319	0.118424	0.0592118	+	0.998245i	$0.481141\pi$
$8$	0	0
$9$	1.54704	0.515681
$10$	0	0
$11$	2.50247	0.754522	0.377261	−	0.926107i	$-0.376866\pi$
$11$	2.50247	0.754522	0.377261	+	0.926107i	$0.376866\pi$
$12$	0	0
$13$	−6.16150	−1.70889	−0.854446	−	0.519540i	$-0.826103\pi$
$13$	−6.16150	−1.70889	−0.854446	+	0.519540i	$0.826103\pi$
$14$	0	0
$15$	7.15836	1.84828
$16$	0	0
$17$	5.24751	1.27271	0.636354	−	0.771397i	$-0.280442\pi$
$17$	5.24751	1.27271	0.636354	+	0.771397i	$0.280442\pi$
$18$	0	0
$19$	7.03909	1.61488	0.807439	−	0.589951i	$-0.200853\pi$
$19$	7.03909	1.61488	0.807439	+	0.589951i	$0.200853\pi$
$20$	0	0
$21$	0.668115	0.145795
$22$	0	0
$23$	1.85469	0.386729	0.193364	−	0.981127i	$-0.438060\pi$
$23$	1.85469	0.386729	0.193364	+	0.981127i	$0.438060\pi$
$24$	0	0
$25$	6.26934	1.25387
$26$	0	0
$27$	−3.09826	−0.596260
$28$	0	0
$29$	−7.13470	−1.32488	−0.662441	−	0.749114i	$-0.730479\pi$
$29$	−7.13470	−1.32488	−0.662441	+	0.749114i	$0.730479\pi$
$30$	0	0
$31$	8.61037	1.54647	0.773234	−	0.634120i	$-0.218638\pi$
$31$	8.61037	1.54647	0.773234	+	0.634120i	$0.218638\pi$
$32$	0	0
$33$	5.33621	0.928914
$34$	0	0
$35$	1.05181	0.177788
$36$	0	0
$37$	11.0889	1.82301	0.911503	−	0.411293i	$-0.134923\pi$
$37$	11.0889	1.82301	0.911503	+	0.411293i	$0.134923\pi$
$38$	0	0
$39$	−13.1386	−2.10387
$40$	0	0
$41$	−5.42641	−0.847464	−0.423732	−	0.905788i	$-0.639280\pi$
$41$	−5.42641	−0.847464	−0.423732	+	0.905788i	$0.639280\pi$
$42$	0	0
$43$	−4.95676	−0.755899	−0.377949	−	0.925826i	$-0.623371\pi$
$43$	−4.95676	−0.755899	−0.377949	+	0.925826i	$0.623371\pi$
$44$	0	0
$45$	5.19339	0.774186
$46$	0	0
$47$	2.99818	0.437329	0.218665	−	0.975800i	$-0.429830\pi$
$47$	2.99818	0.437329	0.218665	+	0.975800i	$0.429830\pi$
$48$	0	0
$49$	−6.90183	−0.985976
$50$	0	0
$51$	11.1897	1.56687
$52$	0	0
$53$	6.00096	0.824295	0.412147	−	0.911117i	$-0.364779\pi$
$53$	6.00096	0.824295	0.412147	+	0.911117i	$0.364779\pi$
$54$	0	0
$55$	8.40074	1.13276
$56$	0	0
$57$	15.0100	1.98812
$58$	0	0
$59$	10.4199	1.35656	0.678280	−	0.734804i	$-0.262726\pi$
$59$	10.4199	1.35656	0.678280	+	0.734804i	$0.262726\pi$
$60$	0	0
$61$	−12.3401	−1.57999	−0.789994	−	0.613115i	$-0.789916\pi$
$61$	−12.3401	−1.57999	−0.789994	+	0.613115i	$0.789916\pi$
$62$	0	0
$63$	0.484718	0.0610687
$64$	0	0
$65$	−20.6840	−2.56554
$66$	0	0
$67$	−7.81203	−0.954392	−0.477196	−	0.878797i	$-0.658347\pi$
$67$	−7.81203	−0.954392	−0.477196	+	0.878797i	$0.658347\pi$
$68$	0	0
$69$	3.95490	0.476113
$70$	0	0
$71$	−2.57760	−0.305905	−0.152953	−	0.988234i	$-0.548878\pi$
$71$	−2.57760	−0.305905	−0.152953	+	0.988234i	$0.548878\pi$
$72$	0	0
$73$	−4.30939	−0.504376	−0.252188	−	0.967678i	$-0.581150\pi$
$73$	−4.30939	−0.504376	−0.252188	+	0.967678i	$0.581150\pi$
$74$	0	0
$75$	13.3686	1.54367
$76$	0	0
$77$	0.784070	0.0893531
$78$	0	0
$79$	2.00325	0.225384	0.112692	−	0.993630i	$-0.464053\pi$
$79$	2.00325	0.225384	0.112692	+	0.993630i	$0.464053\pi$
$80$	0	0
$81$	−11.2478	−1.24975
$82$	0	0
$83$	−12.5725	−1.38001	−0.690006	−	0.723803i	$-0.742392\pi$
$83$	−12.5725	−1.38001	−0.690006	+	0.723803i	$0.742392\pi$
$84$	0	0
$85$	17.6158	1.91070
$86$	0	0
$87$	−15.2139	−1.63110
$88$	0	0
$89$	3.54250	0.375504	0.187752	−	0.982216i	$-0.439880\pi$
$89$	3.54250	0.375504	0.187752	+	0.982216i	$0.439880\pi$
$90$	0	0
$91$	−1.93052	−0.202373
$92$	0	0
$93$	18.3606	1.90390
$94$	0	0
$95$	23.6301	2.42440
$96$	0	0
$97$	9.22478	0.936634	0.468317	−	0.883560i	$-0.344861\pi$
$97$	9.22478	0.936634	0.468317	+	0.883560i	$0.344861\pi$
$98$	0	0
$99$	3.87142	0.389092
$100$	0	0
$101$	18.4902	1.83985	0.919924	−	0.392096i	$-0.128250\pi$
$101$	18.4902	1.83985	0.919924	+	0.392096i	$0.128250\pi$
$102$	0	0
$103$	2.21242	0.217997	0.108998	−	0.994042i	$-0.465236\pi$
$103$	2.21242	0.217997	0.108998	+	0.994042i	$0.465236\pi$
$104$	0	0
$105$	2.24285	0.218880
$106$	0	0
$107$	−7.86881	−0.760707	−0.380353	−	0.924841i	$-0.624198\pi$
$107$	−7.86881	−0.760707	−0.380353	+	0.924841i	$0.624198\pi$
$108$	0	0
$109$	2.53430	0.242742	0.121371	−	0.992607i	$-0.461271\pi$
$109$	2.53430	0.242742	0.121371	+	0.992607i	$0.461271\pi$
$110$	0	0
$111$	23.6458	2.24436
$112$	0	0
$113$	−12.8349	−1.20741	−0.603703	−	0.797209i	$-0.706309\pi$
$113$	−12.8349	−1.20741	−0.603703	+	0.797209i	$0.706309\pi$
$114$	0	0
$115$	6.22615	0.580592
$116$	0	0
$117$	−9.53209	−0.881242
$118$	0	0
$119$	1.64414	0.150719
$120$	0	0
$121$	−4.73767	−0.430697
$122$	0	0
$123$	−11.5712	−1.04334
$124$	0	0
$125$	4.26116	0.381130
$126$	0	0
$127$	4.30900	0.382362	0.191181	−	0.981555i	$-0.438768\pi$
$127$	4.30900	0.382362	0.191181	+	0.981555i	$0.438768\pi$
$128$	0	0
$129$	−10.5697	−0.930609
$130$	0	0
$131$	−12.3090	−1.07544	−0.537720	−	0.843123i	$-0.680714\pi$
$131$	−12.3090	−1.07544	−0.537720	+	0.843123i	$0.680714\pi$
$132$	0	0
$133$	2.20548	0.191239
$134$	0	0
$135$	−10.4008	−0.895159
$136$	0	0
$137$	6.97954	0.596302	0.298151	−	0.954519i	$-0.403630\pi$
$137$	6.97954	0.596302	0.298151	+	0.954519i	$0.403630\pi$
$138$	0	0
$139$	−10.6268	−0.901352	−0.450676	−	0.892688i	$-0.648817\pi$
$139$	−10.6268	−0.901352	−0.450676	+	0.892688i	$0.648817\pi$
$140$	0	0
$141$	6.39325	0.538409
$142$	0	0
$143$	−15.4189	−1.28940
$144$	0	0
$145$	−23.9511	−1.98903
$146$	0	0
$147$	−14.7173	−1.21386
$148$	0	0
$149$	14.8083	1.21314	0.606572	−	0.795029i	$-0.292544\pi$
$149$	14.8083	1.21314	0.606572	+	0.795029i	$0.292544\pi$
$150$	0	0
$151$	−7.22480	−0.587946	−0.293973	−	0.955814i	$-0.594978\pi$
$151$	−7.22480	−0.587946	−0.293973	+	0.955814i	$0.594978\pi$
$152$	0	0
$153$	8.11811	0.656311
$154$	0	0
$155$	28.9049	2.32170
$156$	0	0
$157$	21.9819	1.75434	0.877172	−	0.480176i	$-0.159427\pi$
$157$	21.9819	1.75434	0.877172	+	0.480176i	$0.159427\pi$
$158$	0	0
$159$	12.7963	1.01481
$160$	0	0
$161$	0.581109	0.0457978
$162$	0	0
$163$	15.0863	1.18165	0.590827	−	0.806799i	$-0.298802\pi$
$163$	15.0863	1.18165	0.590827	+	0.806799i	$0.298802\pi$
$164$	0	0
$165$	17.9136	1.39457
$166$	0	0
$167$	3.15656	0.244262	0.122131	−	0.992514i	$-0.461027\pi$
$167$	3.15656	0.244262	0.122131	+	0.992514i	$0.461027\pi$
$168$	0	0
$169$	24.9640	1.92031
$170$	0	0
$171$	10.8898	0.832761
$172$	0	0
$173$	21.8410	1.66054	0.830270	−	0.557362i	$-0.188186\pi$
$173$	21.8410	1.66054	0.830270	+	0.557362i	$0.188186\pi$
$174$	0	0
$175$	1.96431	0.148488
$176$	0	0
$177$	22.2192	1.67010
$178$	0	0
$179$	21.6640	1.61924	0.809620	−	0.586954i	$-0.199673\pi$
$179$	21.6640	1.61924	0.809620	+	0.586954i	$0.199673\pi$
$180$	0	0
$181$	5.94327	0.441760	0.220880	−	0.975301i	$-0.429107\pi$
$181$	5.94327	0.441760	0.220880	+	0.975301i	$0.429107\pi$
$182$	0	0
$183$	−26.3138	−1.94517
$184$	0	0
$185$	37.2253	2.73686
$186$	0	0
$187$	13.1317	0.960286
$188$	0	0
$189$	−0.970744	−0.0706112
$190$	0	0
$191$	−5.91783	−0.428199	−0.214100	−	0.976812i	$-0.568682\pi$
$191$	−5.91783	−0.428199	−0.214100	+	0.976812i	$0.568682\pi$
$192$	0	0
$193$	−17.8803	−1.28705	−0.643526	−	0.765424i	$-0.722529\pi$
$193$	−17.8803	−1.28705	−0.643526	+	0.765424i	$0.722529\pi$
$194$	0	0
$195$	−44.1062	−3.15851
$196$	0	0
$197$	20.1986	1.43909	0.719544	−	0.694446i	$-0.244351\pi$
$197$	20.1986	1.43909	0.719544	+	0.694446i	$0.244351\pi$
$198$	0	0
$199$	3.53340	0.250476	0.125238	−	0.992127i	$-0.460031\pi$
$199$	3.53340	0.250476	0.125238	+	0.992127i	$0.460031\pi$
$200$	0	0
$201$	−16.6582	−1.17498
$202$	0	0
$203$	−2.23544	−0.156897
$204$	0	0
$205$	−18.2164	−1.27229
$206$	0	0
$207$	2.86928	0.199429
$208$	0	0
$209$	17.6151	1.21846
$210$	0	0
$211$	−20.9948	−1.44534	−0.722670	−	0.691193i	$-0.757085\pi$
$211$	−20.9948	−1.44534	−0.722670	+	0.691193i	$0.757085\pi$
$212$	0	0
$213$	−5.49642	−0.376609
$214$	0	0
$215$	−16.6398	−1.13482
$216$	0	0
$217$	2.69780	0.183138
$218$	0	0
$219$	−9.18926	−0.620953
$220$	0	0
$221$	−32.3325	−2.17492
$222$	0	0
$223$	−12.7911	−0.856552	−0.428276	−	0.903648i	$-0.640879\pi$
$223$	−12.7911	−0.856552	−0.428276	+	0.903648i	$0.640879\pi$
$224$	0	0
$225$	9.69893	0.646596
$226$	0	0
$227$	−21.3732	−1.41859	−0.709294	−	0.704912i	$-0.750986\pi$
$227$	−21.3732	−1.41859	−0.709294	+	0.704912i	$0.750986\pi$
$228$	0	0
$229$	−12.0904	−0.798953	−0.399477	−	0.916743i	$-0.630808\pi$
$229$	−12.0904	−0.798953	−0.399477	+	0.916743i	$0.630808\pi$
$230$	0	0
$231$	1.67194	0.110005
$232$	0	0
$233$	−2.42805	−0.159067	−0.0795333	−	0.996832i	$-0.525343\pi$
$233$	−2.42805	−0.159067	−0.0795333	+	0.996832i	$0.525343\pi$
$234$	0	0
$235$	10.0648	0.656557
$236$	0	0
$237$	4.27170	0.277477
$238$	0	0
$239$	11.8069	0.763723	0.381861	−	0.924220i	$-0.375283\pi$
$239$	11.8069	0.763723	0.381861	+	0.924220i	$0.375283\pi$
$240$	0	0
$241$	6.28094	0.404591	0.202296	−	0.979325i	$-0.435160\pi$
$241$	6.28094	0.404591	0.202296	+	0.979325i	$0.435160\pi$
$242$	0	0
$243$	−14.6898	−0.942349
$244$	0	0
$245$	−23.1693	−1.48023
$246$	0	0
$247$	−43.3713	−2.75965
$248$	0	0
$249$	−26.8094	−1.69897
$250$	0	0
$251$	27.9428	1.76374	0.881868	−	0.471496i	$-0.156286\pi$
$251$	27.9428	1.76374	0.881868	+	0.471496i	$0.156286\pi$
$252$	0	0
$253$	4.64129	0.291795
$254$	0	0
$255$	37.5636	2.35232
$256$	0	0
$257$	5.42177	0.338201	0.169100	−	0.985599i	$-0.445914\pi$
$257$	5.42177	0.338201	0.169100	+	0.985599i	$0.445914\pi$
$258$	0	0
$259$	3.47437	0.215887
$260$	0	0
$261$	−11.0377	−0.683215
$262$	0	0
$263$	−14.1366	−0.871699	−0.435849	−	0.900020i	$-0.643552\pi$
$263$	−14.1366	−0.871699	−0.435849	+	0.900020i	$0.643552\pi$
$264$	0	0
$265$	20.1451	1.23750
$266$	0	0
$267$	7.55396	0.462295
$268$	0	0
$269$	−9.45961	−0.576762	−0.288381	−	0.957516i	$-0.593117\pi$
$269$	−9.45961	−0.576762	−0.288381	+	0.957516i	$0.593117\pi$
$270$	0	0
$271$	−20.9128	−1.27036	−0.635180	−	0.772364i	$-0.719074\pi$
$271$	−20.9128	−1.27036	−0.635180	+	0.772364i	$0.719074\pi$
$272$	0	0
$273$	−4.11659	−0.249147
$274$	0	0
$275$	15.6888	0.946071
$276$	0	0
$277$	−13.3043	−0.799375	−0.399688	−	0.916651i	$-0.630881\pi$
$277$	−13.3043	−0.799375	−0.399688	+	0.916651i	$0.630881\pi$
$278$	0	0
$279$	13.3206	0.797484
$280$	0	0
$281$	−23.7950	−1.41949	−0.709746	−	0.704458i	$-0.751190\pi$
$281$	−23.7950	−1.41949	−0.709746	+	0.704458i	$0.751190\pi$
$282$	0	0
$283$	−12.5028	−0.743215	−0.371607	−	0.928390i	$-0.621193\pi$
$283$	−12.5028	−0.743215	−0.371607	+	0.928390i	$0.621193\pi$
$284$	0	0
$285$	50.3884	2.98475
$286$	0	0
$287$	−1.70020	−0.100360
$288$	0	0
$289$	10.5363	0.619785
$290$	0	0
$291$	19.6707	1.15312
$292$	0	0
$293$	−19.4231	−1.13471	−0.567356	−	0.823473i	$-0.692033\pi$
$293$	−19.4231	−1.13471	−0.567356	+	0.823473i	$0.692033\pi$
$294$	0	0
$295$	34.9795	2.03659
$296$	0	0
$297$	−7.75328	−0.449891
$298$	0	0
$299$	−11.4276	−0.660878
$300$	0	0
$301$	−1.55305	−0.0895162
$302$	0	0
$303$	39.4282	2.26509
$304$	0	0
$305$	−41.4255	−2.37202
$306$	0	0
$307$	−9.75183	−0.556566	−0.278283	−	0.960499i	$-0.589765\pi$
$307$	−9.75183	−0.556566	−0.278283	+	0.960499i	$0.589765\pi$
$308$	0	0
$309$	4.71773	0.268382
$310$	0	0
$311$	8.30282	0.470810	0.235405	−	0.971897i	$-0.424358\pi$
$311$	8.30282	0.470810	0.235405	+	0.971897i	$0.424358\pi$
$312$	0	0
$313$	−9.78331	−0.552986	−0.276493	−	0.961016i	$-0.589172\pi$
$313$	−9.78331	−0.552986	−0.276493	+	0.961016i	$0.589172\pi$
$314$	0	0
$315$	1.62719	0.0916818
$316$	0	0
$317$	−7.36431	−0.413621	−0.206810	−	0.978381i	$-0.566308\pi$
$317$	−7.36431	−0.413621	−0.206810	+	0.978381i	$0.566308\pi$
$318$	0	0
$319$	−17.8543	−0.999652
$320$	0	0
$321$	−16.7793	−0.936529
$322$	0	0
$323$	36.9377	2.05527
$324$	0	0
$325$	−38.6285	−2.14273
$326$	0	0
$327$	5.40409	0.298847
$328$	0	0
$329$	0.939386	0.0517900
$330$	0	0
$331$	−14.6384	−0.804602	−0.402301	−	0.915508i	$-0.631789\pi$
$331$	−14.6384	−0.804602	−0.402301	+	0.915508i	$0.631789\pi$
$332$	0	0
$333$	17.1550	0.940089
$334$	0	0
$335$	−26.2249	−1.43282
$336$	0	0
$337$	4.36586	0.237824	0.118912	−	0.992905i	$-0.462059\pi$
$337$	4.36586	0.237824	0.118912	+	0.992905i	$0.462059\pi$
$338$	0	0
$339$	−27.3689	−1.48647
$340$	0	0
$341$	21.5472	1.16684
$342$	0	0
$343$	−4.35571	−0.235186
$344$	0	0
$345$	13.2765	0.714784
$346$	0	0
$347$	23.8023	1.27778	0.638888	−	0.769300i	$-0.279395\pi$
$347$	23.8023	1.27778	0.638888	+	0.769300i	$0.279395\pi$
$348$	0	0
$349$	5.40629	0.289392	0.144696	−	0.989476i	$-0.453780\pi$
$349$	5.40629	0.289392	0.144696	+	0.989476i	$0.453780\pi$
$350$	0	0
$351$	19.0899	1.01894
$352$	0	0
$353$	−27.1508	−1.44509	−0.722545	−	0.691324i	$-0.757028\pi$
$353$	−27.1508	−1.44509	−0.722545	+	0.691324i	$0.757028\pi$
$354$	0	0
$355$	−8.65297	−0.459252
$356$	0	0
$357$	3.50594	0.185554
$358$	0	0
$359$	28.1469	1.48553	0.742767	−	0.669550i	$-0.233513\pi$
$359$	28.1469	1.48553	0.742767	+	0.669550i	$0.233513\pi$
$360$	0	0
$361$	30.5488	1.60783
$362$	0	0
$363$	−10.1025	−0.530244
$364$	0	0
$365$	−14.4666	−0.757215
$366$	0	0
$367$	−23.6066	−1.23225	−0.616126	−	0.787648i	$-0.711299\pi$
$367$	−23.6066	−1.23225	−0.616126	+	0.787648i	$0.711299\pi$
$368$	0	0
$369$	−8.39489	−0.437021
$370$	0	0
$371$	1.88021	0.0976159
$372$	0	0
$373$	3.46416	0.179367	0.0896837	−	0.995970i	$-0.471414\pi$
$373$	3.46416	0.179367	0.0896837	+	0.995970i	$0.471414\pi$
$374$	0	0
$375$	9.08642	0.469220
$376$	0	0
$377$	43.9605	2.26408
$378$	0	0
$379$	−24.9895	−1.28362	−0.641812	−	0.766862i	$-0.721817\pi$
$379$	−24.9895	−1.28362	−0.641812	+	0.766862i	$0.721817\pi$
$380$	0	0
$381$	9.18843	0.470737
$382$	0	0
$383$	11.5945	0.592450	0.296225	−	0.955118i	$-0.404272\pi$
$383$	11.5945	0.592450	0.296225	+	0.955118i	$0.404272\pi$
$384$	0	0
$385$	2.63211	0.134145
$386$	0	0
$387$	−7.66831	−0.389802
$388$	0	0
$389$	−13.1452	−0.666490	−0.333245	−	0.942840i	$-0.608144\pi$
$389$	−13.1452	−0.666490	−0.333245	+	0.942840i	$0.608144\pi$
$390$	0	0
$391$	9.73248	0.492193
$392$	0	0
$393$	−26.2474	−1.32401
$394$	0	0
$395$	6.72489	0.338366
$396$	0	0
$397$	−17.7720	−0.891953	−0.445977	−	0.895045i	$-0.647144\pi$
$397$	−17.7720	−0.891953	−0.445977	+	0.895045i	$0.647144\pi$
$398$	0	0
$399$	4.70292	0.235441
$400$	0	0
$401$	11.4734	0.572953	0.286477	−	0.958087i	$-0.407516\pi$
$401$	11.4734	0.572953	0.286477	+	0.958087i	$0.407516\pi$
$402$	0	0
$403$	−53.0528	−2.64275
$404$	0	0
$405$	−37.7586	−1.87624
$406$	0	0
$407$	27.7496	1.37550
$408$	0	0
$409$	9.21292	0.455549	0.227775	−	0.973714i	$-0.426855\pi$
$409$	9.21292	0.455549	0.227775	+	0.973714i	$0.426855\pi$
$410$	0	0
$411$	14.8830	0.734126
$412$	0	0
$413$	3.26476	0.160649
$414$	0	0
$415$	−42.2057	−2.07180
$416$	0	0
$417$	−22.6603	−1.10968
$418$	0	0
$419$	−25.7212	−1.25656	−0.628281	−	0.777987i	$-0.716241\pi$
$419$	−25.7212	−1.25656	−0.628281	+	0.777987i	$0.716241\pi$
$420$	0	0
$421$	1.98970	0.0969722	0.0484861	−	0.998824i	$-0.484560\pi$
$421$	1.98970	0.0969722	0.0484861	+	0.998824i	$0.484560\pi$
$422$	0	0
$423$	4.63831	0.225522
$424$	0	0
$425$	32.8984	1.59581
$426$	0	0
$427$	−3.86639	−0.187108
$428$	0	0
$429$	−32.8790	−1.58741
$430$	0	0
$431$	3.69245	0.177859	0.0889295	−	0.996038i	$-0.471655\pi$
$431$	3.69245	0.177859	0.0889295	+	0.996038i	$0.471655\pi$
$432$	0	0
$433$	26.8669	1.29114	0.645571	−	0.763700i	$-0.276620\pi$
$433$	26.8669	1.29114	0.645571	+	0.763700i	$0.276620\pi$
$434$	0	0
$435$	−51.0728	−2.44875
$436$	0	0
$437$	13.0553	0.624520
$438$	0	0
$439$	−0.131440	−0.00627329	−0.00313664	−	0.999995i	$-0.500998\pi$
$439$	−0.131440	−0.00627329	−0.00313664	+	0.999995i	$0.500998\pi$
$440$	0	0
$441$	−10.6774	−0.508449
$442$	0	0
$443$	17.6536	0.838748	0.419374	−	0.907814i	$-0.362250\pi$
$443$	17.6536	0.838748	0.419374	+	0.907814i	$0.362250\pi$
$444$	0	0
$445$	11.8921	0.563741
$446$	0	0
$447$	31.5769	1.49354
$448$	0	0
$449$	19.3359	0.912518	0.456259	−	0.889847i	$-0.349189\pi$
$449$	19.3359	0.912518	0.456259	+	0.889847i	$0.349189\pi$
$450$	0	0
$451$	−13.5794	−0.639430
$452$	0	0
$453$	−15.4060	−0.723837
$454$	0	0
$455$	−6.48071	−0.303820
$456$	0	0
$457$	1.94628	0.0910430	0.0455215	−	0.998963i	$-0.485505\pi$
$457$	1.94628	0.0910430	0.0455215	+	0.998963i	$0.485505\pi$
$458$	0	0
$459$	−16.2581	−0.758865
$460$	0	0
$461$	14.3829	0.669877	0.334939	−	0.942240i	$-0.391284\pi$
$461$	14.3829	0.669877	0.334939	+	0.942240i	$0.391284\pi$
$462$	0	0
$463$	17.2676	0.802492	0.401246	−	0.915970i	$-0.368577\pi$
$463$	17.2676	0.802492	0.401246	+	0.915970i	$0.368577\pi$
$464$	0	0
$465$	61.6362	2.85831
$466$	0	0
$467$	−27.1021	−1.25414	−0.627069	−	0.778964i	$-0.715745\pi$
$467$	−27.1021	−1.25414	−0.627069	+	0.778964i	$0.715745\pi$
$468$	0	0
$469$	−2.44766	−0.113022
$470$	0	0
$471$	46.8737	2.15983
$472$	0	0
$473$	−12.4041	−0.570342
$474$	0	0
$475$	44.1305	2.02484
$476$	0	0
$477$	9.28373	0.425073
$478$	0	0
$479$	−13.3602	−0.610443	−0.305222	−	0.952281i	$-0.598731\pi$
$479$	−13.3602	−0.610443	−0.305222	+	0.952281i	$0.598731\pi$
$480$	0	0
$481$	−68.3243	−3.11532
$482$	0	0
$483$	1.23914	0.0563830
$484$	0	0
$485$	30.9674	1.40616
$486$	0	0
$487$	−19.3904	−0.878665	−0.439332	−	0.898325i	$-0.644785\pi$
$487$	−19.3904	−0.878665	−0.439332	+	0.898325i	$0.644785\pi$
$488$	0	0
$489$	32.1698	1.45477
$490$	0	0
$491$	−22.3790	−1.00995	−0.504974	−	0.863135i	$-0.668498\pi$
$491$	−22.3790	−1.00995	−0.504974	+	0.863135i	$0.668498\pi$
$492$	0	0
$493$	−37.4394	−1.68619
$494$	0	0
$495$	12.9963	0.584140
$496$	0	0
$497$	−0.807612	−0.0362264
$498$	0	0
$499$	−3.54098	−0.158516	−0.0792581	−	0.996854i	$-0.525255\pi$
$499$	−3.54098	−0.158516	−0.0792581	+	0.996854i	$0.525255\pi$
$500$	0	0
$501$	6.73099	0.300718
$502$	0	0
$503$	21.5119	0.959167	0.479583	−	0.877496i	$-0.340788\pi$
$503$	21.5119	0.959167	0.479583	+	0.877496i	$0.340788\pi$
$504$	0	0
$505$	62.0715	2.76214
$506$	0	0
$507$	53.2328	2.36415
$508$	0	0
$509$	−22.6017	−1.00180	−0.500902	−	0.865504i	$-0.666998\pi$
$509$	−22.6017	−1.00180	−0.500902	+	0.865504i	$0.666998\pi$
$510$	0	0
$511$	−1.35022	−0.0597300
$512$	0	0
$513$	−21.8089	−0.962887
$514$	0	0
$515$	7.42707	0.327276
$516$	0	0
$517$	7.50283	0.329974
$518$	0	0
$519$	46.5733	2.04434
$520$	0	0
$521$	−17.2507	−0.755767	−0.377883	−	0.925853i	$-0.623348\pi$
$521$	−17.2507	−0.755767	−0.377883	+	0.925853i	$0.623348\pi$
$522$	0	0
$523$	9.86567	0.431395	0.215698	−	0.976460i	$-0.430797\pi$
$523$	9.86567	0.431395	0.215698	+	0.976460i	$0.430797\pi$
$524$	0	0
$525$	4.18864	0.182807
$526$	0	0
$527$	45.1830	1.96820
$528$	0	0
$529$	−19.5601	−0.850441
$530$	0	0
$531$	16.1201	0.699551
$532$	0	0
$533$	33.4348	1.44822
$534$	0	0
$535$	−26.4155	−1.14204
$536$	0	0
$537$	46.1958	1.99350
$538$	0	0
$539$	−17.2716	−0.743940
$540$	0	0
$541$	−43.0278	−1.84991	−0.924954	−	0.380079i	$-0.875897\pi$
$541$	−43.0278	−1.84991	−0.924954	+	0.380079i	$0.875897\pi$
$542$	0	0
$543$	12.6733	0.543863
$544$	0	0
$545$	8.50761	0.364426
$546$	0	0
$547$	−1.38431	−0.0591887	−0.0295944	−	0.999562i	$-0.509422\pi$
$547$	−1.38431	−0.0591887	−0.0295944	+	0.999562i	$0.509422\pi$
$548$	0	0
$549$	−19.0906	−0.814769
$550$	0	0
$551$	−50.2218	−2.13952
$552$	0	0
$553$	0.627658	0.0266907
$554$	0	0
$555$	79.3785	3.36943
$556$	0	0
$557$	−40.7280	−1.72570	−0.862850	−	0.505460i	$-0.831323\pi$
$557$	−40.7280	−1.72570	−0.862850	+	0.505460i	$0.831323\pi$
$558$	0	0
$559$	30.5411	1.29175
$560$	0	0
$561$	28.0018	1.18224
$562$	0	0
$563$	16.8633	0.710701	0.355351	−	0.934733i	$-0.384361\pi$
$563$	16.8633	0.710701	0.355351	+	0.934733i	$0.384361\pi$
$564$	0	0
$565$	−43.0865	−1.81266
$566$	0	0
$567$	−3.52415	−0.148000
$568$	0	0
$569$	−5.38083	−0.225576	−0.112788	−	0.993619i	$-0.535978\pi$
$569$	−5.38083	−0.225576	−0.112788	+	0.993619i	$0.535978\pi$
$570$	0	0
$571$	8.16422	0.341662	0.170831	−	0.985300i	$-0.445355\pi$
$571$	8.16422	0.341662	0.170831	+	0.985300i	$0.445355\pi$
$572$	0	0
$573$	−12.6191	−0.527169
$574$	0	0
$575$	11.6277	0.484907
$576$	0	0
$577$	38.3390	1.59607	0.798036	−	0.602610i	$-0.205872\pi$
$577$	38.3390	1.59607	0.798036	+	0.602610i	$0.205872\pi$
$578$	0	0
$579$	−38.1276	−1.58453
$580$	0	0
$581$	−3.93921	−0.163426
$582$	0	0
$583$	15.0172	0.621948
$584$	0	0
$585$	−31.9991	−1.32300
$586$	0	0
$587$	−35.8650	−1.48031	−0.740153	−	0.672439i	$-0.765247\pi$
$587$	−35.8650	−1.48031	−0.740153	+	0.672439i	$0.765247\pi$
$588$	0	0
$589$	60.6092	2.49736
$590$	0	0
$591$	43.0710	1.77171
$592$	0	0
$593$	43.6135	1.79099	0.895495	−	0.445071i	$-0.146822\pi$
$593$	43.6135	1.79099	0.895495	+	0.445071i	$0.146822\pi$
$594$	0	0
$595$	5.51937	0.226272
$596$	0	0
$597$	7.53454	0.308368
$598$	0	0
$599$	−17.2171	−0.703474	−0.351737	−	0.936099i	$-0.614409\pi$
$599$	−17.2171	−0.703474	−0.351737	+	0.936099i	$0.614409\pi$
$600$	0	0
$601$	16.2288	0.661988	0.330994	−	0.943633i	$-0.392616\pi$
$601$	16.2288	0.661988	0.330994	+	0.943633i	$0.392616\pi$
$602$	0	0
$603$	−12.0855	−0.492161
$604$	0	0
$605$	−15.9043	−0.646601
$606$	0	0
$607$	−32.3908	−1.31470	−0.657351	−	0.753584i	$-0.728323\pi$
$607$	−32.3908	−1.31470	−0.657351	+	0.753584i	$0.728323\pi$
$608$	0	0
$609$	−4.76680	−0.193161
$610$	0	0
$611$	−18.4733	−0.747348
$612$	0	0
$613$	34.0607	1.37570	0.687849	−	0.725853i	$-0.258555\pi$
$613$	34.0607	1.37570	0.687849	+	0.725853i	$0.258555\pi$
$614$	0	0
$615$	−38.8443	−1.56635
$616$	0	0
$617$	−17.0616	−0.686873	−0.343437	−	0.939176i	$-0.611591\pi$
$617$	−17.0616	−0.686873	−0.343437	+	0.939176i	$0.611591\pi$
$618$	0	0
$619$	−3.65747	−0.147006	−0.0735031	−	0.997295i	$-0.523418\pi$
$619$	−3.65747	−0.147006	−0.0735031	+	0.997295i	$0.523418\pi$
$620$	0	0
$621$	−5.74630	−0.230591
$622$	0	0
$623$	1.10993	0.0444685
$624$	0	0
$625$	−17.0421	−0.681682
$626$	0	0
$627$	37.5620	1.50008
$628$	0	0
$629$	58.1892	2.32015
$630$	0	0
$631$	2.18425	0.0869537	0.0434769	−	0.999054i	$-0.486157\pi$
$631$	2.18425	0.0869537	0.0434769	+	0.999054i	$0.486157\pi$
$632$	0	0
$633$	−44.7688	−1.77940
$634$	0	0
$635$	14.4653	0.574036
$636$	0	0
$637$	42.5256	1.68493
$638$	0	0
$639$	−3.98766	−0.157749
$640$	0	0
$641$	−33.4019	−1.31930	−0.659648	−	0.751575i	$-0.729295\pi$
$641$	−33.4019	−1.31930	−0.659648	+	0.751575i	$0.729295\pi$
$642$	0	0
$643$	−40.4816	−1.59644	−0.798219	−	0.602368i	$-0.794224\pi$
$643$	−40.4816	−1.59644	−0.798219	+	0.602368i	$0.794224\pi$
$644$	0	0
$645$	−35.4823	−1.39711
$646$	0	0
$647$	10.0519	0.395181	0.197591	−	0.980285i	$-0.436688\pi$
$647$	10.0519	0.395181	0.197591	+	0.980285i	$0.436688\pi$
$648$	0	0
$649$	26.0755	1.02355
$650$	0	0
$651$	5.75272	0.225467
$652$	0	0
$653$	24.4580	0.957114	0.478557	−	0.878056i	$-0.341160\pi$
$653$	24.4580	0.957114	0.478557	+	0.878056i	$0.341160\pi$
$654$	0	0
$655$	−41.3210	−1.61455
$656$	0	0
$657$	−6.66681	−0.260097
$658$	0	0
$659$	41.2636	1.60740	0.803702	−	0.595032i	$-0.202861\pi$
$659$	41.2636	1.60740	0.803702	+	0.595032i	$0.202861\pi$
$660$	0	0
$661$	20.5331	0.798645	0.399323	−	0.916811i	$-0.369245\pi$
$661$	20.5331	0.798645	0.399323	+	0.916811i	$0.369245\pi$
$662$	0	0
$663$	−68.9452	−2.67761
$664$	0	0
$665$	7.40377	0.287106
$666$	0	0
$667$	−13.2326	−0.512370
$668$	0	0
$669$	−27.2754	−1.05453
$670$	0	0
$671$	−30.8807	−1.19214
$672$	0	0
$673$	23.7108	0.913986	0.456993	−	0.889470i	$-0.348927\pi$
$673$	23.7108	0.913986	0.456993	+	0.889470i	$0.348927\pi$
$674$	0	0
$675$	−19.4240	−0.747632
$676$	0	0
$677$	17.6801	0.679501	0.339750	−	0.940516i	$-0.389657\pi$
$677$	17.6801	0.679501	0.339750	+	0.940516i	$0.389657\pi$
$678$	0	0
$679$	2.89030	0.110920
$680$	0	0
$681$	−45.5758	−1.74647
$682$	0	0
$683$	4.14755	0.158702	0.0793508	−	0.996847i	$-0.474715\pi$
$683$	4.14755	0.158702	0.0793508	+	0.996847i	$0.474715\pi$
$684$	0	0
$685$	23.4302	0.895222
$686$	0	0
$687$	−25.7812	−0.983615
$688$	0	0
$689$	−36.9749	−1.40863
$690$	0	0
$691$	−16.5345	−0.629002	−0.314501	−	0.949257i	$-0.601837\pi$
$691$	−16.5345	−0.629002	−0.314501	+	0.949257i	$0.601837\pi$
$692$	0	0
$693$	1.21299	0.0460777
$694$	0	0
$695$	−35.6740	−1.35319
$696$	0	0
$697$	−28.4752	−1.07857
$698$	0	0
$699$	−5.17751	−0.195832
$700$	0	0
$701$	−37.2653	−1.40749	−0.703745	−	0.710453i	$-0.748490\pi$
$701$	−37.2653	−1.40749	−0.703745	+	0.710453i	$0.748490\pi$
$702$	0	0
$703$	78.0559	2.94393
$704$	0	0
$705$	21.4620	0.808307
$706$	0	0
$707$	5.79335	0.217881
$708$	0	0
$709$	−43.0375	−1.61631	−0.808154	−	0.588972i	$-0.799533\pi$
$709$	−43.0375	−1.61631	−0.808154	+	0.588972i	$0.799533\pi$
$710$	0	0
$711$	3.09912	0.116226
$712$	0	0
$713$	15.9695	0.598064
$714$	0	0
$715$	−51.7611	−1.93576
$716$	0	0
$717$	25.1767	0.940242
$718$	0	0
$719$	−27.8865	−1.03999	−0.519995	−	0.854169i	$-0.674066\pi$
$719$	−27.8865	−1.03999	−0.519995	+	0.854169i	$0.674066\pi$
$720$	0	0
$721$	0.693195	0.0258159
$722$	0	0
$723$	13.3934	0.498104
$724$	0	0
$725$	−44.7299	−1.66123
$726$	0	0
$727$	−35.4483	−1.31471	−0.657353	−	0.753583i	$-0.728324\pi$
$727$	−35.4483	−1.31471	−0.657353	+	0.753583i	$0.728324\pi$
$728$	0	0
$729$	2.41919	0.0895996
$730$	0	0
$731$	−26.0106	−0.962038
$732$	0	0
$733$	6.68560	0.246938	0.123469	−	0.992348i	$-0.460598\pi$
$733$	6.68560	0.246938	0.123469	+	0.992348i	$0.460598\pi$
$734$	0	0
$735$	−49.4058	−1.82236
$736$	0	0
$737$	−19.5493	−0.720109
$738$	0	0
$739$	−40.3367	−1.48381	−0.741905	−	0.670505i	$-0.766077\pi$
$739$	−40.3367	−1.48381	−0.741905	+	0.670505i	$0.766077\pi$
$740$	0	0
$741$	−92.4841	−3.39749
$742$	0	0
$743$	44.6906	1.63954	0.819769	−	0.572694i	$-0.194102\pi$
$743$	44.6906	1.63954	0.819769	+	0.572694i	$0.194102\pi$
$744$	0	0
$745$	49.7112	1.82128
$746$	0	0
$747$	−19.4502	−0.711646
$748$	0	0
$749$	−2.46545	−0.0900856
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	59.5847	2.17139
$754$	0	0
$755$	−24.2535	−0.882676
$756$	0	0
$757$	8.74113	0.317702	0.158851	−	0.987303i	$-0.449221\pi$
$757$	8.74113	0.317702	0.158851	+	0.987303i	$0.449221\pi$
$758$	0	0
$759$	9.89699	0.359238
$760$	0	0
$761$	−7.14722	−0.259087	−0.129543	−	0.991574i	$-0.541351\pi$
$761$	−7.14722	−0.259087	−0.129543	+	0.991574i	$0.541351\pi$
$762$	0	0
$763$	0.794045	0.0287464
$764$	0	0
$765$	27.2524	0.985312
$766$	0	0
$767$	−64.2024	−2.31821
$768$	0	0
$769$	−9.23824	−0.333140	−0.166570	−	0.986030i	$-0.553269\pi$
$769$	−9.23824	−0.333140	−0.166570	+	0.986030i	$0.553269\pi$
$770$	0	0
$771$	11.5613	0.416369
$772$	0	0
$773$	−40.7693	−1.46637	−0.733185	−	0.680029i	$-0.761967\pi$
$773$	−40.7693	−1.46637	−0.733185	+	0.680029i	$0.761967\pi$
$774$	0	0
$775$	53.9814	1.93907
$776$	0	0
$777$	7.40867	0.265785
$778$	0	0
$779$	−38.1970	−1.36855
$780$	0	0
$781$	−6.45036	−0.230812
$782$	0	0
$783$	22.1052	0.789974
$784$	0	0
$785$	73.7928	2.63378
$786$	0	0
$787$	45.4381	1.61969	0.809847	−	0.586642i	$-0.199550\pi$
$787$	45.4381	1.61969	0.809847	+	0.586642i	$0.199550\pi$
$788$	0	0
$789$	−30.1445	−1.07317
$790$	0	0
$791$	−4.02142	−0.142985
$792$	0	0
$793$	76.0335	2.70003
$794$	0	0
$795$	42.9570	1.52353
$796$	0	0
$797$	39.5721	1.40172	0.700858	−	0.713301i	$-0.252801\pi$
$797$	39.5721	1.40172	0.700858	+	0.713301i	$0.252801\pi$
$798$	0	0
$799$	15.7330	0.556592
$800$	0	0
$801$	5.48040	0.193640
$802$	0	0
$803$	−10.7841	−0.380563
$804$	0	0
$805$	1.95077	0.0687557
$806$	0	0
$807$	−20.1715	−0.710069
$808$	0	0
$809$	21.7708	0.765419	0.382710	−	0.923869i	$-0.374991\pi$
$809$	21.7708	0.765419	0.382710	+	0.923869i	$0.374991\pi$
$810$	0	0
$811$	−2.13128	−0.0748393	−0.0374197	−	0.999300i	$-0.511914\pi$
$811$	−2.13128	−0.0748393	−0.0374197	+	0.999300i	$0.511914\pi$
$812$	0	0
$813$	−44.5939	−1.56398
$814$	0	0
$815$	50.6446	1.77400
$816$	0	0
$817$	−34.8911	−1.22068
$818$	0	0
$819$	−2.98659	−0.104360
$820$	0	0
$821$	51.9223	1.81210	0.906051	−	0.423169i	$-0.139082\pi$
$821$	51.9223	1.81210	0.906051	+	0.423169i	$0.139082\pi$
$822$	0	0
$823$	13.3988	0.467054	0.233527	−	0.972350i	$-0.424973\pi$
$823$	13.3988	0.467054	0.233527	+	0.972350i	$0.424973\pi$
$824$	0	0
$825$	33.4545	1.16474
$826$	0	0
$827$	34.3066	1.19296	0.596478	−	0.802629i	$-0.296566\pi$
$827$	34.3066	1.19296	0.596478	+	0.802629i	$0.296566\pi$
$828$	0	0
$829$	27.8188	0.966186	0.483093	−	0.875569i	$-0.339513\pi$
$829$	27.8188	0.966186	0.483093	+	0.875569i	$0.339513\pi$
$830$	0	0
$831$	−28.3697	−0.984135
$832$	0	0
$833$	−36.2174	−1.25486
$834$	0	0
$835$	10.5965	0.366708
$836$	0	0
$837$	−26.6772	−0.922098
$838$	0	0
$839$	−24.7978	−0.856116	−0.428058	−	0.903751i	$-0.640802\pi$
$839$	−24.7978	−0.856116	−0.428058	+	0.903751i	$0.640802\pi$
$840$	0	0
$841$	21.9040	0.755310
$842$	0	0
$843$	−50.7400	−1.74758
$844$	0	0
$845$	83.8039	2.88294
$846$	0	0
$847$	−1.48440	−0.0510046
$848$	0	0
$849$	−26.6607	−0.914994
$850$	0	0
$851$	20.5665	0.705009
$852$	0	0
$853$	30.7719	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$853$	30.7719	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$854$	0	0
$855$	36.5568	1.25022
$856$	0	0
$857$	−47.1993	−1.61230	−0.806149	−	0.591713i	$-0.798452\pi$
$857$	−47.1993	−1.61230	−0.806149	+	0.591713i	$0.798452\pi$
$858$	0	0
$859$	−13.7918	−0.470570	−0.235285	−	0.971926i	$-0.575602\pi$
$859$	−13.7918	−0.470570	−0.235285	+	0.971926i	$0.575602\pi$
$860$	0	0
$861$	−3.62547	−0.123556
$862$	0	0
$863$	−17.9383	−0.610628	−0.305314	−	0.952252i	$-0.598761\pi$
$863$	−17.9383	−0.610628	−0.305314	+	0.952252i	$0.598761\pi$
$864$	0	0
$865$	73.3199	2.49295
$866$	0	0
$867$	22.4675	0.763036
$868$	0	0
$869$	5.01307	0.170057
$870$	0	0
$871$	48.1338	1.63095
$872$	0	0
$873$	14.2711	0.483004
$874$	0	0
$875$	1.33510	0.0451348
$876$	0	0
$877$	23.6917	0.800011	0.400005	−	0.916513i	$-0.369008\pi$
$877$	23.6917	0.800011	0.400005	+	0.916513i	$0.369008\pi$
$878$	0	0
$879$	−41.4175	−1.39698
$880$	0	0
$881$	−5.25534	−0.177057	−0.0885284	−	0.996074i	$-0.528216\pi$
$881$	−5.25534	−0.177057	−0.0885284	+	0.996074i	$0.528216\pi$
$882$	0	0
$883$	22.1745	0.746232	0.373116	−	0.927785i	$-0.378289\pi$
$883$	22.1745	0.746232	0.373116	+	0.927785i	$0.378289\pi$
$884$	0	0
$885$	74.5897	2.50730
$886$	0	0
$887$	−8.20195	−0.275394	−0.137697	−	0.990474i	$-0.543970\pi$
$887$	−8.20195	−0.275394	−0.137697	+	0.990474i	$0.543970\pi$
$888$	0	0
$889$	1.35009	0.0452807
$890$	0	0
$891$	−28.1472	−0.942967
$892$	0	0
$893$	21.1044	0.706233
$894$	0	0
$895$	72.7256	2.43095
$896$	0	0
$897$	−24.3681	−0.813626
$898$	0	0
$899$	−61.4325	−2.04889
$900$	0	0
$901$	31.4901	1.04909
$902$	0	0
$903$	−3.31169	−0.110206
$904$	0	0
$905$	19.9515	0.663209
$906$	0	0
$907$	0.0575697	0.00191157	0.000955785	−	1.00000i	$-0.499696\pi$
$907$	0.0575697	0.00191157	0.000955785		1.00000i	$0.499696\pi$
$908$	0	0
$909$	28.6052	0.948774
$910$	0	0
$911$	−40.9087	−1.35537	−0.677683	−	0.735354i	$-0.737016\pi$
$911$	−40.9087	−1.35537	−0.677683	+	0.735354i	$0.737016\pi$
$912$	0	0
$913$	−31.4623	−1.04125
$914$	0	0
$915$	−88.3349	−2.92026
$916$	0	0
$917$	−3.85664	−0.127357
$918$	0	0
$919$	−23.1149	−0.762490	−0.381245	−	0.924474i	$-0.624504\pi$
$919$	−23.1149	−0.762490	−0.381245	+	0.924474i	$0.624504\pi$
$920$	0	0
$921$	−20.7946	−0.685205
$922$	0	0
$923$	15.8819	0.522759
$924$	0	0
$925$	69.5202	2.28581
$926$	0	0
$927$	3.42271	0.112417
$928$	0	0
$929$	−10.8963	−0.357497	−0.178748	−	0.983895i	$-0.557205\pi$
$929$	−10.8963	−0.357497	−0.178748	+	0.983895i	$0.557205\pi$
$930$	0	0
$931$	−48.5826	−1.59223
$932$	0	0
$933$	17.7048	0.579628
$934$	0	0
$935$	44.0829	1.44167
$936$	0	0
$937$	3.23087	0.105548	0.0527739	−	0.998606i	$-0.483194\pi$
$937$	3.23087	0.105548	0.0527739	+	0.998606i	$0.483194\pi$
$938$	0	0
$939$	−20.8617	−0.680797
$940$	0	0
$941$	−2.70781	−0.0882721	−0.0441360	−	0.999026i	$-0.514054\pi$
$941$	−2.70781	−0.0882721	−0.0441360	+	0.999026i	$0.514054\pi$
$942$	0	0
$943$	−10.0643	−0.327739
$944$	0	0
$945$	−3.25877	−0.106008
$946$	0	0
$947$	−24.6709	−0.801698	−0.400849	−	0.916144i	$-0.631285\pi$
$947$	−24.6709	−0.801698	−0.400849	+	0.916144i	$0.631285\pi$
$948$	0	0
$949$	26.5523	0.861925
$950$	0	0
$951$	−15.7035	−0.509221
$952$	0	0
$953$	−6.38885	−0.206955	−0.103478	−	0.994632i	$-0.532997\pi$
$953$	−6.38885	−0.206955	−0.103478	+	0.994632i	$0.532997\pi$
$954$	0	0
$955$	−19.8661	−0.642851
$956$	0	0
$957$	−38.0722	−1.23070
$958$	0	0
$959$	2.18682	0.0706162
$960$	0	0
$961$	43.1385	1.39157
$962$	0	0
$963$	−12.1734	−0.392282
$964$	0	0
$965$	−60.0239	−1.93224
$966$	0	0
$967$	−39.8299	−1.28084	−0.640422	−	0.768023i	$-0.721240\pi$
$967$	−39.8299	−1.28084	−0.640422	+	0.768023i	$0.721240\pi$
$968$	0	0
$969$	78.7651	2.53030
$970$	0	0
$971$	−53.3171	−1.71103	−0.855513	−	0.517781i	$-0.826758\pi$
$971$	−53.3171	−1.71103	−0.855513	+	0.517781i	$0.826758\pi$
$972$	0	0
$973$	−3.32958	−0.106741
$974$	0	0
$975$	−82.3707	−2.63797
$976$	0	0
$977$	5.88120	0.188156	0.0940781	−	0.995565i	$-0.470010\pi$
$977$	5.88120	0.188156	0.0940781	+	0.995565i	$0.470010\pi$
$978$	0	0
$979$	8.86499	0.283326
$980$	0	0
$981$	3.92067	0.125177
$982$	0	0
$983$	−37.7762	−1.20487	−0.602436	−	0.798167i	$-0.705803\pi$
$983$	−37.7762	−1.20487	−0.602436	+	0.798167i	$0.705803\pi$
$984$	0	0
$985$	67.8063	2.16049
$986$	0	0
$987$	2.00313	0.0637603
$988$	0	0
$989$	−9.19323	−0.292328
$990$	0	0
$991$	25.2590	0.802380	0.401190	−	0.915995i	$-0.368597\pi$
$991$	25.2590	0.802380	0.401190	+	0.915995i	$0.368597\pi$
$992$	0	0
$993$	−31.2147	−0.990569
$994$	0	0
$995$	11.8616	0.376036
$996$	0	0
$997$	−19.3072	−0.611467	−0.305733	−	0.952117i	$-0.598902\pi$
$997$	−19.3072	−0.611467	−0.305733	+	0.952117i	$0.598902\pi$
$998$	0	0
$999$	−34.3563	−1.08699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.38	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.38	✓	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+2.13238 q^{3} +3.35698 q^{5} +0.313319 q^{7} +1.54704 q^{9} +O(q^{10})\)	\(q+2.13238 q^{3} +3.35698 q^{5} +0.313319 q^{7} +1.54704 q^{9} +2.50247 q^{11} -6.16150 q^{13} +7.15836 q^{15} +5.24751 q^{17} +7.03909 q^{19} +0.668115 q^{21} +1.85469 q^{23} +6.26934 q^{25} -3.09826 q^{27} -7.13470 q^{29} +8.61037 q^{31} +5.33621 q^{33} +1.05181 q^{35} +11.0889 q^{37} -13.1386 q^{39} -5.42641 q^{41} -4.95676 q^{43} +5.19339 q^{45} +2.99818 q^{47} -6.90183 q^{49} +11.1897 q^{51} +6.00096 q^{53} +8.40074 q^{55} +15.0100 q^{57} +10.4199 q^{59} -12.3401 q^{61} +0.484718 q^{63} -20.6840 q^{65} -7.81203 q^{67} +3.95490 q^{69} -2.57760 q^{71} -4.30939 q^{73} +13.3686 q^{75} +0.784070 q^{77} +2.00325 q^{79} -11.2478 q^{81} -12.5725 q^{83} +17.6158 q^{85} -15.2139 q^{87} +3.54250 q^{89} -1.93052 q^{91} +18.3606 q^{93} +23.6301 q^{95} +9.22478 q^{97} +3.87142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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