LMFDB - Embedded newform 6008.2.a.b.1.9

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:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-2.25093 q^{3} +3.53241 q^{5} +1.19008 q^{7} +2.06668 q^{9} +O(q^{10})$	$q-2.25093 q^{3} +3.53241 q^{5} +1.19008 q^{7} +2.06668 q^{9} -4.93741 q^{11} -4.24028 q^{13} -7.95119 q^{15} -0.933955 q^{17} +5.44476 q^{19} -2.67878 q^{21} -8.71920 q^{23} +7.47789 q^{25} +2.10084 q^{27} +8.13482 q^{29} +5.52324 q^{31} +11.1138 q^{33} +4.20384 q^{35} +3.19527 q^{37} +9.54456 q^{39} +11.8689 q^{41} -9.73774 q^{43} +7.30035 q^{45} -10.8322 q^{47} -5.58371 q^{49} +2.10227 q^{51} -3.26943 q^{53} -17.4409 q^{55} -12.2558 q^{57} +1.18004 q^{59} -1.59464 q^{61} +2.45951 q^{63} -14.9784 q^{65} -2.80301 q^{67} +19.6263 q^{69} +10.4959 q^{71} -3.66483 q^{73} -16.8322 q^{75} -5.87590 q^{77} +7.27723 q^{79} -10.9289 q^{81} -10.1008 q^{83} -3.29911 q^{85} -18.3109 q^{87} +2.87868 q^{89} -5.04626 q^{91} -12.4324 q^{93} +19.2331 q^{95} -4.60723 q^{97} -10.2041 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * 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
$3$	−2.25093	−1.29957	−0.649787	−	0.760116i	$-0.725142\pi$
$3$	−2.25093	−1.29957	−0.649787	+	0.760116i	$0.725142\pi$
$4$	0	0
$5$	3.53241	1.57974	0.789870	−	0.613274i	$-0.210148\pi$
$5$	3.53241	1.57974	0.789870	+	0.613274i	$0.210148\pi$
$6$	0	0
$7$	1.19008	0.449807	0.224904	−	0.974381i	$-0.427793\pi$
$7$	1.19008	0.449807	0.224904	+	0.974381i	$0.427793\pi$
$8$	0	0
$9$	2.06668	0.688893
$10$	0	0
$11$	−4.93741	−1.48869	−0.744343	−	0.667798i	$-0.767237\pi$
$11$	−4.93741	−1.48869	−0.744343	+	0.667798i	$0.767237\pi$
$12$	0	0
$13$	−4.24028	−1.17604	−0.588020	−	0.808846i	$-0.700093\pi$
$13$	−4.24028	−1.17604	−0.588020	+	0.808846i	$0.700093\pi$
$14$	0	0
$15$	−7.95119	−2.05299
$16$	0	0
$17$	−0.933955	−0.226517	−0.113259	−	0.993566i	$-0.536129\pi$
$17$	−0.933955	−0.226517	−0.113259	+	0.993566i	$0.536129\pi$
$18$	0	0
$19$	5.44476	1.24911	0.624557	−	0.780979i	$-0.285280\pi$
$19$	5.44476	1.24911	0.624557	+	0.780979i	$0.285280\pi$
$20$	0	0
$21$	−2.67878	−0.584558
$22$	0	0
$23$	−8.71920	−1.81808	−0.909039	−	0.416711i	$-0.863183\pi$
$23$	−8.71920	−1.81808	−0.909039	+	0.416711i	$0.863183\pi$
$24$	0	0
$25$	7.47789	1.49558
$26$	0	0
$27$	2.10084	0.404306
$28$	0	0
$29$	8.13482	1.51060	0.755299	−	0.655380i	$-0.227491\pi$
$29$	8.13482	1.51060	0.755299	+	0.655380i	$0.227491\pi$
$30$	0	0
$31$	5.52324	0.992003	0.496002	−	0.868322i	$-0.334801\pi$
$31$	5.52324	0.992003	0.496002	+	0.868322i	$0.334801\pi$
$32$	0	0
$33$	11.1138	1.93466
$34$	0	0
$35$	4.20384	0.710578
$36$	0	0
$37$	3.19527	0.525299	0.262650	−	0.964891i	$-0.415404\pi$
$37$	3.19527	0.525299	0.262650	+	0.964891i	$0.415404\pi$
$38$	0	0
$39$	9.54456	1.52835
$40$	0	0
$41$	11.8689	1.85361	0.926806	−	0.375540i	$-0.122543\pi$
$41$	11.8689	1.85361	0.926806	+	0.375540i	$0.122543\pi$
$42$	0	0
$43$	−9.73774	−1.48499	−0.742495	−	0.669851i	$-0.766358\pi$
$43$	−9.73774	−1.48499	−0.742495	+	0.669851i	$0.766358\pi$
$44$	0	0
$45$	7.30035	1.08827
$46$	0	0
$47$	−10.8322	−1.58003	−0.790017	−	0.613085i	$-0.789928\pi$
$47$	−10.8322	−1.58003	−0.790017	+	0.613085i	$0.789928\pi$
$48$	0	0
$49$	−5.58371	−0.797674
$50$	0	0
$51$	2.10227	0.294376
$52$	0	0
$53$	−3.26943	−0.449091	−0.224546	−	0.974464i	$-0.572090\pi$
$53$	−3.26943	−0.449091	−0.224546	+	0.974464i	$0.572090\pi$
$54$	0	0
$55$	−17.4409	−2.35174
$56$	0	0
$57$	−12.2558	−1.62332
$58$	0	0
$59$	1.18004	0.153628	0.0768140	−	0.997045i	$-0.475525\pi$
$59$	1.18004	0.153628	0.0768140	+	0.997045i	$0.475525\pi$
$60$	0	0
$61$	−1.59464	−0.204173	−0.102087	−	0.994776i	$-0.532552\pi$
$61$	−1.59464	−0.204173	−0.102087	+	0.994776i	$0.532552\pi$
$62$	0	0
$63$	2.45951	0.309869
$64$	0	0
$65$	−14.9784	−1.85784
$66$	0	0
$67$	−2.80301	−0.342443	−0.171221	−	0.985233i	$-0.554771\pi$
$67$	−2.80301	−0.342443	−0.171221	+	0.985233i	$0.554771\pi$
$68$	0	0
$69$	19.6263	2.36273
$70$	0	0
$71$	10.4959	1.24563	0.622815	−	0.782369i	$-0.285989\pi$
$71$	10.4959	1.24563	0.622815	+	0.782369i	$0.285989\pi$
$72$	0	0
$73$	−3.66483	−0.428936	−0.214468	−	0.976731i	$-0.568802\pi$
$73$	−3.66483	−0.428936	−0.214468	+	0.976731i	$0.568802\pi$
$74$	0	0
$75$	−16.8322	−1.94361
$76$	0	0
$77$	−5.87590	−0.669622
$78$	0	0
$79$	7.27723	0.818753	0.409376	−	0.912366i	$-0.365746\pi$
$79$	7.27723	0.818753	0.409376	+	0.912366i	$0.365746\pi$
$80$	0	0
$81$	−10.9289	−1.21432
$82$	0	0
$83$	−10.1008	−1.10870	−0.554352	−	0.832282i	$-0.687034\pi$
$83$	−10.1008	−1.10870	−0.554352	+	0.832282i	$0.687034\pi$
$84$	0	0
$85$	−3.29911	−0.357838
$86$	0	0
$87$	−18.3109	−1.96314
$88$	0	0
$89$	2.87868	0.305139	0.152570	−	0.988293i	$-0.451245\pi$
$89$	2.87868	0.305139	0.152570	+	0.988293i	$0.451245\pi$
$90$	0	0
$91$	−5.04626	−0.528992
$92$	0	0
$93$	−12.4324	−1.28918
$94$	0	0
$95$	19.2331	1.97328
$96$	0	0
$97$	−4.60723	−0.467793	−0.233896	−	0.972262i	$-0.575148\pi$
$97$	−4.60723	−0.467793	−0.233896	+	0.972262i	$0.575148\pi$
$98$	0	0
$99$	−10.2041	−1.02555
$100$	0	0
$101$	−1.80928	−0.180030	−0.0900150	−	0.995940i	$-0.528691\pi$
$101$	−1.80928	−0.180030	−0.0900150	+	0.995940i	$0.528691\pi$
$102$	0	0
$103$	1.82726	0.180045	0.0900227	−	0.995940i	$-0.471306\pi$
$103$	1.82726	0.180045	0.0900227	+	0.995940i	$0.471306\pi$
$104$	0	0
$105$	−9.46254	−0.923449
$106$	0	0
$107$	1.18976	0.115018	0.0575092	−	0.998345i	$-0.481684\pi$
$107$	1.18976	0.115018	0.0575092	+	0.998345i	$0.481684\pi$
$108$	0	0
$109$	−4.85334	−0.464866	−0.232433	−	0.972612i	$-0.574669\pi$
$109$	−4.85334	−0.464866	−0.232433	+	0.972612i	$0.574669\pi$
$110$	0	0
$111$	−7.19232	−0.682665
$112$	0	0
$113$	−19.9551	−1.87721	−0.938607	−	0.344989i	$-0.887883\pi$
$113$	−19.9551	−1.87721	−0.938607	+	0.344989i	$0.887883\pi$
$114$	0	0
$115$	−30.7997	−2.87209
$116$	0	0
$117$	−8.76329	−0.810167
$118$	0	0
$119$	−1.11148	−0.101889
$120$	0	0
$121$	13.3780	1.21619
$122$	0	0
$123$	−26.7161	−2.40891
$124$	0	0
$125$	8.75291	0.782884
$126$	0	0
$127$	−17.2642	−1.53195	−0.765976	−	0.642870i	$-0.777744\pi$
$127$	−17.2642	−1.53195	−0.765976	+	0.642870i	$0.777744\pi$
$128$	0	0
$129$	21.9189	1.92986
$130$	0	0
$131$	−17.4217	−1.52214	−0.761070	−	0.648670i	$-0.775326\pi$
$131$	−17.4217	−1.52214	−0.761070	+	0.648670i	$0.775326\pi$
$132$	0	0
$133$	6.47969	0.561861
$134$	0	0
$135$	7.42100	0.638698
$136$	0	0
$137$	4.05458	0.346406	0.173203	−	0.984886i	$-0.444588\pi$
$137$	4.05458	0.346406	0.173203	+	0.984886i	$0.444588\pi$
$138$	0	0
$139$	6.87804	0.583388	0.291694	−	0.956512i	$-0.405781\pi$
$139$	6.87804	0.583388	0.291694	+	0.956512i	$0.405781\pi$
$140$	0	0
$141$	24.3824	2.05337
$142$	0	0
$143$	20.9360	1.75076
$144$	0	0
$145$	28.7355	2.38635
$146$	0	0
$147$	12.5685	1.03664
$148$	0	0
$149$	13.4939	1.10546	0.552732	−	0.833359i	$-0.313585\pi$
$149$	13.4939	1.10546	0.552732	+	0.833359i	$0.313585\pi$
$150$	0	0
$151$	−17.2737	−1.40571	−0.702856	−	0.711332i	$-0.748092\pi$
$151$	−17.2737	−1.40571	−0.702856	+	0.711332i	$0.748092\pi$
$152$	0	0
$153$	−1.93019	−0.156046
$154$	0	0
$155$	19.5103	1.56711
$156$	0	0
$157$	6.35462	0.507154	0.253577	−	0.967315i	$-0.418393\pi$
$157$	6.35462	0.507154	0.253577	+	0.967315i	$0.418393\pi$
$158$	0	0
$159$	7.35926	0.583628
$160$	0	0
$161$	−10.3765	−0.817785
$162$	0	0
$163$	−14.8835	−1.16577	−0.582884	−	0.812555i	$-0.698076\pi$
$163$	−14.8835	−1.16577	−0.582884	+	0.812555i	$0.698076\pi$
$164$	0	0
$165$	39.2583	3.05626
$166$	0	0
$167$	−2.91665	−0.225697	−0.112848	−	0.993612i	$-0.535997\pi$
$167$	−2.91665	−0.225697	−0.112848	+	0.993612i	$0.535997\pi$
$168$	0	0
$169$	4.97994	0.383072
$170$	0	0
$171$	11.2526	0.860507
$172$	0	0
$173$	11.0386	0.839252	0.419626	−	0.907697i	$-0.362161\pi$
$173$	11.0386	0.839252	0.419626	+	0.907697i	$0.362161\pi$
$174$	0	0
$175$	8.89927	0.672722
$176$	0	0
$177$	−2.65618	−0.199651
$178$	0	0
$179$	12.7589	0.953645	0.476823	−	0.878999i	$-0.341788\pi$
$179$	12.7589	0.953645	0.476823	+	0.878999i	$0.341788\pi$
$180$	0	0
$181$	−19.6760	−1.46251	−0.731254	−	0.682105i	$-0.761065\pi$
$181$	−19.6760	−1.46251	−0.731254	+	0.682105i	$0.761065\pi$
$182$	0	0
$183$	3.58943	0.265338
$184$	0	0
$185$	11.2870	0.829836
$186$	0	0
$187$	4.61132	0.337213
$188$	0	0
$189$	2.50016	0.181860
$190$	0	0
$191$	8.69296	0.629000	0.314500	−	0.949257i	$-0.398163\pi$
$191$	8.69296	0.629000	0.314500	+	0.949257i	$0.398163\pi$
$192$	0	0
$193$	−13.5177	−0.973028	−0.486514	−	0.873673i	$-0.661732\pi$
$193$	−13.5177	−0.973028	−0.486514	+	0.873673i	$0.661732\pi$
$194$	0	0
$195$	33.7153	2.41440
$196$	0	0
$197$	2.12411	0.151337	0.0756684	−	0.997133i	$-0.475891\pi$
$197$	2.12411	0.151337	0.0756684	+	0.997133i	$0.475891\pi$
$198$	0	0
$199$	−13.3621	−0.947213	−0.473606	−	0.880737i	$-0.657048\pi$
$199$	−13.3621	−0.947213	−0.473606	+	0.880737i	$0.657048\pi$
$200$	0	0
$201$	6.30938	0.445030
$202$	0	0
$203$	9.68107	0.679478
$204$	0	0
$205$	41.9258	2.92822
$206$	0	0
$207$	−18.0198	−1.25246
$208$	0	0
$209$	−26.8830	−1.85954
$210$	0	0
$211$	5.47272	0.376758	0.188379	−	0.982096i	$-0.439677\pi$
$211$	5.47272	0.376758	0.188379	+	0.982096i	$0.439677\pi$
$212$	0	0
$213$	−23.6255	−1.61879
$214$	0	0
$215$	−34.3976	−2.34590
$216$	0	0
$217$	6.57309	0.446210
$218$	0	0
$219$	8.24928	0.557435
$220$	0	0
$221$	3.96022	0.266394
$222$	0	0
$223$	−13.7689	−0.922034	−0.461017	−	0.887391i	$-0.652515\pi$
$223$	−13.7689	−0.922034	−0.461017	+	0.887391i	$0.652515\pi$
$224$	0	0
$225$	15.4544	1.03029
$226$	0	0
$227$	25.8486	1.71564	0.857818	−	0.513954i	$-0.171820\pi$
$227$	25.8486	1.71564	0.857818	+	0.513954i	$0.171820\pi$
$228$	0	0
$229$	−7.83005	−0.517424	−0.258712	−	0.965954i	$-0.583298\pi$
$229$	−7.83005	−0.517424	−0.258712	+	0.965954i	$0.583298\pi$
$230$	0	0
$231$	13.2262	0.870223
$232$	0	0
$233$	−23.8290	−1.56109	−0.780545	−	0.625099i	$-0.785059\pi$
$233$	−23.8290	−1.56109	−0.780545	+	0.625099i	$0.785059\pi$
$234$	0	0
$235$	−38.2636	−2.49604
$236$	0	0
$237$	−16.3805	−1.06403
$238$	0	0
$239$	−7.72161	−0.499469	−0.249735	−	0.968314i	$-0.580343\pi$
$239$	−7.72161	−0.499469	−0.249735	+	0.968314i	$0.580343\pi$
$240$	0	0
$241$	13.0169	0.838494	0.419247	−	0.907872i	$-0.362294\pi$
$241$	13.0169	0.838494	0.419247	+	0.907872i	$0.362294\pi$
$242$	0	0
$243$	18.2976	1.17379
$244$	0	0
$245$	−19.7239	−1.26012
$246$	0	0
$247$	−23.0873	−1.46901
$248$	0	0
$249$	22.7361	1.44084
$250$	0	0
$251$	−9.12953	−0.576251	−0.288125	−	0.957593i	$-0.593032\pi$
$251$	−9.12953	−0.576251	−0.288125	+	0.957593i	$0.593032\pi$
$252$	0	0
$253$	43.0503	2.70655
$254$	0	0
$255$	7.42605	0.465037
$256$	0	0
$257$	25.4545	1.58781	0.793903	−	0.608044i	$-0.208046\pi$
$257$	25.4545	1.58781	0.793903	+	0.608044i	$0.208046\pi$
$258$	0	0
$259$	3.80262	0.236283
$260$	0	0
$261$	16.8121	1.04064
$262$	0	0
$263$	−27.0610	−1.66866	−0.834328	−	0.551269i	$-0.814144\pi$
$263$	−27.0610	−1.66866	−0.834328	+	0.551269i	$0.814144\pi$
$264$	0	0
$265$	−11.5490	−0.709448
$266$	0	0
$267$	−6.47970	−0.396551
$268$	0	0
$269$	3.73731	0.227868	0.113934	−	0.993488i	$-0.463655\pi$
$269$	3.73731	0.227868	0.113934	+	0.993488i	$0.463655\pi$
$270$	0	0
$271$	−27.6698	−1.68082	−0.840412	−	0.541949i	$-0.817687\pi$
$271$	−27.6698	−1.68082	−0.840412	+	0.541949i	$0.817687\pi$
$272$	0	0
$273$	11.3588	0.687464
$274$	0	0
$275$	−36.9214	−2.22645
$276$	0	0
$277$	−13.1090	−0.787642	−0.393821	−	0.919187i	$-0.628847\pi$
$277$	−13.1090	−0.787642	−0.393821	+	0.919187i	$0.628847\pi$
$278$	0	0
$279$	11.4148	0.683385
$280$	0	0
$281$	−26.1373	−1.55922	−0.779612	−	0.626263i	$-0.784584\pi$
$281$	−26.1373	−1.55922	−0.779612	+	0.626263i	$0.784584\pi$
$282$	0	0
$283$	17.0584	1.01402	0.507009	−	0.861941i	$-0.330751\pi$
$283$	17.0584	1.01402	0.507009	+	0.861941i	$0.330751\pi$
$284$	0	0
$285$	−43.2924	−2.56442
$286$	0	0
$287$	14.1249	0.833768
$288$	0	0
$289$	−16.1277	−0.948690
$290$	0	0
$291$	10.3705	0.607932
$292$	0	0
$293$	17.9400	1.04806	0.524032	−	0.851699i	$-0.324427\pi$
$293$	17.9400	1.04806	0.524032	+	0.851699i	$0.324427\pi$
$294$	0	0
$295$	4.16838	0.242692
$296$	0	0
$297$	−10.3727	−0.601885
$298$	0	0
$299$	36.9718	2.13813
$300$	0	0
$301$	−11.5887	−0.667959
$302$	0	0
$303$	4.07256	0.233962
$304$	0	0
$305$	−5.63292	−0.322540
$306$	0	0
$307$	−25.5218	−1.45660	−0.728302	−	0.685256i	$-0.759690\pi$
$307$	−25.5218	−1.45660	−0.728302	+	0.685256i	$0.759690\pi$
$308$	0	0
$309$	−4.11304	−0.233982
$310$	0	0
$311$	1.05784	0.0599847	0.0299924	−	0.999550i	$-0.490452\pi$
$311$	1.05784	0.0599847	0.0299924	+	0.999550i	$0.490452\pi$
$312$	0	0
$313$	−0.516552	−0.0291972	−0.0145986	−	0.999893i	$-0.504647\pi$
$313$	−0.516552	−0.0291972	−0.0145986	+	0.999893i	$0.504647\pi$
$314$	0	0
$315$	8.68799	0.489513
$316$	0	0
$317$	2.38840	0.134146	0.0670730	−	0.997748i	$-0.478634\pi$
$317$	2.38840	0.134146	0.0670730	+	0.997748i	$0.478634\pi$
$318$	0	0
$319$	−40.1650	−2.24881
$320$	0	0
$321$	−2.67807	−0.149475
$322$	0	0
$323$	−5.08516	−0.282946
$324$	0	0
$325$	−31.7083	−1.75886
$326$	0	0
$327$	10.9245	0.604128
$328$	0	0
$329$	−12.8911	−0.710711
$330$	0	0
$331$	−15.1964	−0.835270	−0.417635	−	0.908615i	$-0.637141\pi$
$331$	−15.1964	−0.835270	−0.417635	+	0.908615i	$0.637141\pi$
$332$	0	0
$333$	6.60360	0.361875
$334$	0	0
$335$	−9.90138	−0.540970
$336$	0	0
$337$	5.99184	0.326396	0.163198	−	0.986593i	$-0.447819\pi$
$337$	5.99184	0.326396	0.163198	+	0.986593i	$0.447819\pi$
$338$	0	0
$339$	44.9174	2.43958
$340$	0	0
$341$	−27.2705	−1.47678
$342$	0	0
$343$	−14.9756	−0.808606
$344$	0	0
$345$	69.3280	3.73249
$346$	0	0
$347$	−23.6805	−1.27124	−0.635618	−	0.772004i	$-0.719255\pi$
$347$	−23.6805	−1.27124	−0.635618	+	0.772004i	$0.719255\pi$
$348$	0	0
$349$	−8.86011	−0.474271	−0.237135	−	0.971477i	$-0.576209\pi$
$349$	−8.86011	−0.474271	−0.237135	+	0.971477i	$0.576209\pi$
$350$	0	0
$351$	−8.90812	−0.475480
$352$	0	0
$353$	16.3451	0.869959	0.434980	−	0.900440i	$-0.356756\pi$
$353$	16.3451	0.869959	0.434980	+	0.900440i	$0.356756\pi$
$354$	0	0
$355$	37.0757	1.96777
$356$	0	0
$357$	2.50186	0.132412
$358$	0	0
$359$	4.62338	0.244013	0.122006	−	0.992529i	$-0.461067\pi$
$359$	4.62338	0.244013	0.122006	+	0.992529i	$0.461067\pi$
$360$	0	0
$361$	10.6454	0.560286
$362$	0	0
$363$	−30.1130	−1.58052
$364$	0	0
$365$	−12.9457	−0.677608
$366$	0	0
$367$	34.0996	1.77999	0.889993	−	0.455973i	$-0.150709\pi$
$367$	34.0996	1.77999	0.889993	+	0.455973i	$0.150709\pi$
$368$	0	0
$369$	24.5292	1.27694
$370$	0	0
$371$	−3.89088	−0.202005
$372$	0	0
$373$	−22.6097	−1.17068	−0.585342	−	0.810787i	$-0.699040\pi$
$373$	−22.6097	−1.17068	−0.585342	+	0.810787i	$0.699040\pi$
$374$	0	0
$375$	−19.7022	−1.01742
$376$	0	0
$377$	−34.4939	−1.77653
$378$	0	0
$379$	14.1122	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$379$	14.1122	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$380$	0	0
$381$	38.8605	1.99088
$382$	0	0
$383$	38.3368	1.95892	0.979461	−	0.201635i	$-0.0646254\pi$
$383$	38.3368	1.95892	0.979461	+	0.201635i	$0.0646254\pi$
$384$	0	0
$385$	−20.7561	−1.05783
$386$	0	0
$387$	−20.1248	−1.02300
$388$	0	0
$389$	−19.3562	−0.981398	−0.490699	−	0.871329i	$-0.663259\pi$
$389$	−19.3562	−0.981398	−0.490699	+	0.871329i	$0.663259\pi$
$390$	0	0
$391$	8.14333	0.411826
$392$	0	0
$393$	39.2150	1.97813
$394$	0	0
$395$	25.7061	1.29342
$396$	0	0
$397$	10.1580	0.509813	0.254907	−	0.966966i	$-0.417955\pi$
$397$	10.1580	0.509813	0.254907	+	0.966966i	$0.417955\pi$
$398$	0	0
$399$	−14.5853	−0.730180
$400$	0	0
$401$	−23.3331	−1.16520	−0.582599	−	0.812760i	$-0.697964\pi$
$401$	−23.3331	−1.16520	−0.582599	+	0.812760i	$0.697964\pi$
$402$	0	0
$403$	−23.4201	−1.16664
$404$	0	0
$405$	−38.6052	−1.91831
$406$	0	0
$407$	−15.7764	−0.782005
$408$	0	0
$409$	29.6766	1.46741	0.733707	−	0.679466i	$-0.237788\pi$
$409$	29.6766	1.46741	0.733707	+	0.679466i	$0.237788\pi$
$410$	0	0
$411$	−9.12656	−0.450180
$412$	0	0
$413$	1.40434	0.0691030
$414$	0	0
$415$	−35.6800	−1.75146
$416$	0	0
$417$	−15.4820	−0.758156
$418$	0	0
$419$	11.5767	0.565560	0.282780	−	0.959185i	$-0.408743\pi$
$419$	11.5767	0.565560	0.282780	+	0.959185i	$0.408743\pi$
$420$	0	0
$421$	−21.5270	−1.04916	−0.524580	−	0.851361i	$-0.675778\pi$
$421$	−21.5270	−1.04916	−0.524580	+	0.851361i	$0.675778\pi$
$422$	0	0
$423$	−22.3866	−1.08847
$424$	0	0
$425$	−6.98401	−0.338774
$426$	0	0
$427$	−1.89775	−0.0918385
$428$	0	0
$429$	−47.1254	−2.27524
$430$	0	0
$431$	−30.2392	−1.45657	−0.728285	−	0.685275i	$-0.759682\pi$
$431$	−30.2392	−1.45657	−0.728285	+	0.685275i	$0.759682\pi$
$432$	0	0
$433$	13.0660	0.627914	0.313957	−	0.949437i	$-0.398345\pi$
$433$	13.0660	0.627914	0.313957	+	0.949437i	$0.398345\pi$
$434$	0	0
$435$	−64.6816	−3.10124
$436$	0	0
$437$	−47.4740	−2.27099
$438$	0	0
$439$	30.7871	1.46939	0.734695	−	0.678398i	$-0.237325\pi$
$439$	30.7871	1.46939	0.734695	+	0.678398i	$0.237325\pi$
$440$	0	0
$441$	−11.5398	−0.549512
$442$	0	0
$443$	−3.85147	−0.182989	−0.0914944	−	0.995806i	$-0.529164\pi$
$443$	−3.85147	−0.182989	−0.0914944	+	0.995806i	$0.529164\pi$
$444$	0	0
$445$	10.1687	0.482041
$446$	0	0
$447$	−30.3738	−1.43663
$448$	0	0
$449$	−28.3363	−1.33727	−0.668636	−	0.743590i	$-0.733122\pi$
$449$	−28.3363	−1.33727	−0.668636	+	0.743590i	$0.733122\pi$
$450$	0	0
$451$	−58.6017	−2.75945
$452$	0	0
$453$	38.8818	1.82683
$454$	0	0
$455$	−17.8254	−0.835669
$456$	0	0
$457$	−24.7636	−1.15839	−0.579196	−	0.815189i	$-0.696633\pi$
$457$	−24.7636	−1.15839	−0.579196	+	0.815189i	$0.696633\pi$
$458$	0	0
$459$	−1.96209	−0.0915823
$460$	0	0
$461$	16.0171	0.745990	0.372995	−	0.927833i	$-0.378331\pi$
$461$	16.0171	0.745990	0.372995	+	0.927833i	$0.378331\pi$
$462$	0	0
$463$	−25.8891	−1.20317	−0.601585	−	0.798809i	$-0.705464\pi$
$463$	−25.8891	−1.20317	−0.601585	+	0.798809i	$0.705464\pi$
$464$	0	0
$465$	−43.9163	−2.03657
$466$	0	0
$467$	21.2722	0.984362	0.492181	−	0.870493i	$-0.336200\pi$
$467$	21.2722	0.984362	0.492181	+	0.870493i	$0.336200\pi$
$468$	0	0
$469$	−3.33580	−0.154033
$470$	0	0
$471$	−14.3038	−0.659084
$472$	0	0
$473$	48.0792	2.21068
$474$	0	0
$475$	40.7153	1.86815
$476$	0	0
$477$	−6.75688	−0.309376
$478$	0	0
$479$	−32.4888	−1.48445	−0.742226	−	0.670150i	$-0.766230\pi$
$479$	−32.4888	−1.48445	−0.742226	+	0.670150i	$0.766230\pi$
$480$	0	0
$481$	−13.5488	−0.617773
$482$	0	0
$483$	23.3568	1.06277
$484$	0	0
$485$	−16.2746	−0.738991
$486$	0	0
$487$	13.8104	0.625807	0.312903	−	0.949785i	$-0.398698\pi$
$487$	13.8104	0.625807	0.312903	+	0.949785i	$0.398698\pi$
$488$	0	0
$489$	33.5018	1.51500
$490$	0	0
$491$	−6.46238	−0.291643	−0.145822	−	0.989311i	$-0.546583\pi$
$491$	−6.46238	−0.291643	−0.145822	+	0.989311i	$0.546583\pi$
$492$	0	0
$493$	−7.59756	−0.342177
$494$	0	0
$495$	−36.0449	−1.62010
$496$	0	0
$497$	12.4909	0.560293
$498$	0	0
$499$	−4.48661	−0.200848	−0.100424	−	0.994945i	$-0.532020\pi$
$499$	−4.48661	−0.200848	−0.100424	+	0.994945i	$0.532020\pi$
$500$	0	0
$501$	6.56516	0.293310
$502$	0	0
$503$	−22.4536	−1.00116	−0.500578	−	0.865692i	$-0.666879\pi$
$503$	−22.4536	−1.00116	−0.500578	+	0.865692i	$0.666879\pi$
$504$	0	0
$505$	−6.39111	−0.284400
$506$	0	0
$507$	−11.2095	−0.497831
$508$	0	0
$509$	−24.7392	−1.09655	−0.548274	−	0.836299i	$-0.684715\pi$
$509$	−24.7392	−1.09655	−0.548274	+	0.836299i	$0.684715\pi$
$510$	0	0
$511$	−4.36144	−0.192939
$512$	0	0
$513$	11.4386	0.505024
$514$	0	0
$515$	6.45463	0.284425
$516$	0	0
$517$	53.4829	2.35217
$518$	0	0
$519$	−24.8472	−1.09067
$520$	0	0
$521$	3.89577	0.170677	0.0853384	−	0.996352i	$-0.472803\pi$
$521$	3.89577	0.170677	0.0853384	+	0.996352i	$0.472803\pi$
$522$	0	0
$523$	19.9254	0.871278	0.435639	−	0.900122i	$-0.356522\pi$
$523$	19.9254	0.871278	0.435639	+	0.900122i	$0.356522\pi$
$524$	0	0
$525$	−20.0316	−0.874252
$526$	0	0
$527$	−5.15846	−0.224706
$528$	0	0
$529$	53.0244	2.30541
$530$	0	0
$531$	2.43876	0.105833
$532$	0	0
$533$	−50.3274	−2.17992
$534$	0	0
$535$	4.20272	0.181699
$536$	0	0
$537$	−28.7194	−1.23933
$538$	0	0
$539$	27.5691	1.18749
$540$	0	0
$541$	14.9395	0.642301	0.321150	−	0.947028i	$-0.395931\pi$
$541$	14.9395	0.642301	0.321150	+	0.947028i	$0.395931\pi$
$542$	0	0
$543$	44.2894	1.90064
$544$	0	0
$545$	−17.1440	−0.734367
$546$	0	0
$547$	33.5067	1.43264	0.716320	−	0.697771i	$-0.245825\pi$
$547$	33.5067	1.43264	0.716320	+	0.697771i	$0.245825\pi$
$548$	0	0
$549$	−3.29562	−0.140653
$550$	0	0
$551$	44.2922	1.88691
$552$	0	0
$553$	8.66048	0.368281
$554$	0	0
$555$	−25.4062	−1.07843
$556$	0	0
$557$	−1.15742	−0.0490414	−0.0245207	−	0.999699i	$-0.507806\pi$
$557$	−1.15742	−0.0490414	−0.0245207	+	0.999699i	$0.507806\pi$
$558$	0	0
$559$	41.2907	1.74641
$560$	0	0
$561$	−10.3797	−0.438233
$562$	0	0
$563$	−27.5565	−1.16137	−0.580685	−	0.814129i	$-0.697215\pi$
$563$	−27.5565	−1.16137	−0.580685	+	0.814129i	$0.697215\pi$
$564$	0	0
$565$	−70.4893	−2.96551
$566$	0	0
$567$	−13.0062	−0.546210
$568$	0	0
$569$	−35.1655	−1.47421	−0.737107	−	0.675776i	$-0.763809\pi$
$569$	−35.1655	−1.47421	−0.737107	+	0.675776i	$0.763809\pi$
$570$	0	0
$571$	3.72807	0.156015	0.0780074	−	0.996953i	$-0.475144\pi$
$571$	3.72807	0.156015	0.0780074	+	0.996953i	$0.475144\pi$
$572$	0	0
$573$	−19.5672	−0.817433
$574$	0	0
$575$	−65.2012	−2.71908
$576$	0	0
$577$	−3.79664	−0.158056	−0.0790281	−	0.996872i	$-0.525182\pi$
$577$	−3.79664	−0.158056	−0.0790281	+	0.996872i	$0.525182\pi$
$578$	0	0
$579$	30.4275	1.26452
$580$	0	0
$581$	−12.0207	−0.498703
$582$	0	0
$583$	16.1425	0.668556
$584$	0	0
$585$	−30.9555	−1.27985
$586$	0	0
$587$	16.9850	0.701048	0.350524	−	0.936554i	$-0.386004\pi$
$587$	16.9850	0.701048	0.350524	+	0.936554i	$0.386004\pi$
$588$	0	0
$589$	30.0727	1.23913
$590$	0	0
$591$	−4.78123	−0.196673
$592$	0	0
$593$	24.9726	1.02550	0.512751	−	0.858537i	$-0.328626\pi$
$593$	24.9726	1.02550	0.512751	+	0.858537i	$0.328626\pi$
$594$	0	0
$595$	−3.92619	−0.160958
$596$	0	0
$597$	30.0771	1.23097
$598$	0	0
$599$	13.6270	0.556785	0.278392	−	0.960467i	$-0.410198\pi$
$599$	13.6270	0.556785	0.278392	+	0.960467i	$0.410198\pi$
$600$	0	0
$601$	−24.3804	−0.994497	−0.497248	−	0.867608i	$-0.665656\pi$
$601$	−24.3804	−0.994497	−0.497248	+	0.867608i	$0.665656\pi$
$602$	0	0
$603$	−5.79293	−0.235906
$604$	0	0
$605$	47.2566	1.92126
$606$	0	0
$607$	11.2600	0.457028	0.228514	−	0.973541i	$-0.426613\pi$
$607$	11.2600	0.457028	0.228514	+	0.973541i	$0.426613\pi$
$608$	0	0
$609$	−21.7914	−0.883032
$610$	0	0
$611$	45.9314	1.85818
$612$	0	0
$613$	18.7593	0.757681	0.378840	−	0.925462i	$-0.376323\pi$
$613$	18.7593	0.757681	0.378840	+	0.925462i	$0.376323\pi$
$614$	0	0
$615$	−94.3720	−3.80545
$616$	0	0
$617$	−30.0002	−1.20776	−0.603881	−	0.797074i	$-0.706380\pi$
$617$	−30.0002	−1.20776	−0.603881	+	0.797074i	$0.706380\pi$
$618$	0	0
$619$	19.2450	0.773522	0.386761	−	0.922180i	$-0.373594\pi$
$619$	19.2450	0.773522	0.386761	+	0.922180i	$0.373594\pi$
$620$	0	0
$621$	−18.3176	−0.735060
$622$	0	0
$623$	3.42585	0.137254
$624$	0	0
$625$	−6.47062	−0.258825
$626$	0	0
$627$	60.5118	2.41661
$628$	0	0
$629$	−2.98424	−0.118989
$630$	0	0
$631$	−27.6021	−1.09882	−0.549411	−	0.835552i	$-0.685148\pi$
$631$	−27.6021	−1.09882	−0.549411	+	0.835552i	$0.685148\pi$
$632$	0	0
$633$	−12.3187	−0.489625
$634$	0	0
$635$	−60.9842	−2.42008
$636$	0	0
$637$	23.6765	0.938097
$638$	0	0
$639$	21.6916	0.858107
$640$	0	0
$641$	−17.2443	−0.681110	−0.340555	−	0.940225i	$-0.610615\pi$
$641$	−17.2443	−0.681110	−0.340555	+	0.940225i	$0.610615\pi$
$642$	0	0
$643$	7.14378	0.281723	0.140862	−	0.990029i	$-0.455013\pi$
$643$	7.14378	0.281723	0.140862	+	0.990029i	$0.455013\pi$
$644$	0	0
$645$	77.4266	3.04867
$646$	0	0
$647$	−25.6846	−1.00976	−0.504882	−	0.863188i	$-0.668464\pi$
$647$	−25.6846	−1.00976	−0.504882	+	0.863188i	$0.668464\pi$
$648$	0	0
$649$	−5.82634	−0.228704
$650$	0	0
$651$	−14.7955	−0.579883
$652$	0	0
$653$	−14.3382	−0.561096	−0.280548	−	0.959840i	$-0.590516\pi$
$653$	−14.3382	−0.561096	−0.280548	+	0.959840i	$0.590516\pi$
$654$	0	0
$655$	−61.5404	−2.40458
$656$	0	0
$657$	−7.57404	−0.295492
$658$	0	0
$659$	32.3569	1.26045	0.630223	−	0.776414i	$-0.282963\pi$
$659$	32.3569	1.26045	0.630223	+	0.776414i	$0.282963\pi$
$660$	0	0
$661$	−25.9681	−1.01004	−0.505022	−	0.863107i	$-0.668515\pi$
$661$	−25.9681	−1.01004	−0.505022	+	0.863107i	$0.668515\pi$
$662$	0	0
$663$	−8.91418	−0.346198
$664$	0	0
$665$	22.8889	0.887593
$666$	0	0
$667$	−70.9291	−2.74639
$668$	0	0
$669$	30.9928	1.19825
$670$	0	0
$671$	7.87341	0.303950
$672$	0	0
$673$	46.8034	1.80414	0.902069	−	0.431592i	$-0.142048\pi$
$673$	46.8034	1.80414	0.902069	+	0.431592i	$0.142048\pi$
$674$	0	0
$675$	15.7098	0.604671
$676$	0	0
$677$	4.13878	0.159066	0.0795331	−	0.996832i	$-0.474657\pi$
$677$	4.13878	0.159066	0.0795331	+	0.996832i	$0.474657\pi$
$678$	0	0
$679$	−5.48296	−0.210417
$680$	0	0
$681$	−58.1835	−2.22960
$682$	0	0
$683$	−8.59259	−0.328786	−0.164393	−	0.986395i	$-0.552567\pi$
$683$	−8.59259	−0.328786	−0.164393	+	0.986395i	$0.552567\pi$
$684$	0	0
$685$	14.3224	0.547231
$686$	0	0
$687$	17.6249	0.672431
$688$	0	0
$689$	13.8633	0.528150
$690$	0	0
$691$	10.0833	0.383588	0.191794	−	0.981435i	$-0.438569\pi$
$691$	10.0833	0.383588	0.191794	+	0.981435i	$0.438569\pi$
$692$	0	0
$693$	−12.1436	−0.461298
$694$	0	0
$695$	24.2960	0.921601
$696$	0	0
$697$	−11.0850	−0.419875
$698$	0	0
$699$	53.6374	2.02875
$700$	0	0
$701$	−12.7822	−0.482775	−0.241388	−	0.970429i	$-0.577603\pi$
$701$	−12.7822	−0.482775	−0.241388	+	0.970429i	$0.577603\pi$
$702$	0	0
$703$	17.3975	0.656158
$704$	0	0
$705$	86.1286	3.24379
$706$	0	0
$707$	−2.15318	−0.0809788
$708$	0	0
$709$	24.8184	0.932074	0.466037	−	0.884765i	$-0.345681\pi$
$709$	24.8184	0.932074	0.466037	+	0.884765i	$0.345681\pi$
$710$	0	0
$711$	15.0397	0.564033
$712$	0	0
$713$	−48.1582	−1.80354
$714$	0	0
$715$	73.9544	2.76574
$716$	0	0
$717$	17.3808	0.649097
$718$	0	0
$719$	30.3511	1.13191	0.565953	−	0.824437i	$-0.308508\pi$
$719$	30.3511	1.13191	0.565953	+	0.824437i	$0.308508\pi$
$720$	0	0
$721$	2.17458	0.0809857
$722$	0	0
$723$	−29.3002	−1.08969
$724$	0	0
$725$	60.8313	2.25922
$726$	0	0
$727$	−18.2849	−0.678151	−0.339076	−	0.940759i	$-0.610114\pi$
$727$	−18.2849	−0.678151	−0.339076	+	0.940759i	$0.610114\pi$
$728$	0	0
$729$	−8.39999	−0.311111
$730$	0	0
$731$	9.09460	0.336376
$732$	0	0
$733$	−22.4056	−0.827568	−0.413784	−	0.910375i	$-0.635793\pi$
$733$	−22.4056	−0.827568	−0.413784	+	0.910375i	$0.635793\pi$
$734$	0	0
$735$	44.3972	1.63762
$736$	0	0
$737$	13.8396	0.509789
$738$	0	0
$739$	−1.82051	−0.0669687	−0.0334844	−	0.999439i	$-0.510660\pi$
$739$	−1.82051	−0.0669687	−0.0334844	+	0.999439i	$0.510660\pi$
$740$	0	0
$741$	51.9679	1.90909
$742$	0	0
$743$	28.0989	1.03085	0.515424	−	0.856936i	$-0.327635\pi$
$743$	28.0989	1.03085	0.515424	+	0.856936i	$0.327635\pi$
$744$	0	0
$745$	47.6659	1.74634
$746$	0	0
$747$	−20.8751	−0.763779
$748$	0	0
$749$	1.41591	0.0517361
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	20.5499	0.748881
$754$	0	0
$755$	−61.0177	−2.22066
$756$	0	0
$757$	−8.03911	−0.292187	−0.146093	−	0.989271i	$-0.546670\pi$
$757$	−8.03911	−0.292187	−0.146093	+	0.989271i	$0.546670\pi$
$758$	0	0
$759$	−96.9031	−3.51736
$760$	0	0
$761$	−3.57818	−0.129709	−0.0648544	−	0.997895i	$-0.520658\pi$
$761$	−3.57818	−0.129709	−0.0648544	+	0.997895i	$0.520658\pi$
$762$	0	0
$763$	−5.77585	−0.209100
$764$	0	0
$765$	−6.81820	−0.246512
$766$	0	0
$767$	−5.00369	−0.180673
$768$	0	0
$769$	−2.99846	−0.108127	−0.0540636	−	0.998537i	$-0.517217\pi$
$769$	−2.99846	−0.108127	−0.0540636	+	0.998537i	$0.517217\pi$
$770$	0	0
$771$	−57.2962	−2.06347
$772$	0	0
$773$	18.8092	0.676519	0.338259	−	0.941053i	$-0.390162\pi$
$773$	18.8092	0.676519	0.338259	+	0.941053i	$0.390162\pi$
$774$	0	0
$775$	41.3022	1.48362
$776$	0	0
$777$	−8.55943	−0.307068
$778$	0	0
$779$	64.6234	2.31537
$780$	0	0
$781$	−51.8224	−1.85435
$782$	0	0
$783$	17.0899	0.610744
$784$	0	0
$785$	22.4471	0.801171
$786$	0	0
$787$	25.4340	0.906625	0.453313	−	0.891352i	$-0.350242\pi$
$787$	25.4340	0.906625	0.453313	+	0.891352i	$0.350242\pi$
$788$	0	0
$789$	60.9125	2.16854
$790$	0	0
$791$	−23.7481	−0.844384
$792$	0	0
$793$	6.76172	0.240116
$794$	0	0
$795$	25.9959	0.921980
$796$	0	0
$797$	−30.8058	−1.09120	−0.545599	−	0.838046i	$-0.683698\pi$
$797$	−30.8058	−1.09120	−0.545599	+	0.838046i	$0.683698\pi$
$798$	0	0
$799$	10.1168	0.357905
$800$	0	0
$801$	5.94931	0.210209
$802$	0	0
$803$	18.0948	0.638552
$804$	0	0
$805$	−36.6541	−1.29189
$806$	0	0
$807$	−8.41241	−0.296131
$808$	0	0
$809$	−0.831446	−0.0292321	−0.0146160	−	0.999893i	$-0.504653\pi$
$809$	−0.831446	−0.0292321	−0.0146160	+	0.999893i	$0.504653\pi$
$810$	0	0
$811$	20.8900	0.733547	0.366773	−	0.930310i	$-0.380462\pi$
$811$	20.8900	0.733547	0.366773	+	0.930310i	$0.380462\pi$
$812$	0	0
$813$	62.2828	2.18435
$814$	0	0
$815$	−52.5747	−1.84161
$816$	0	0
$817$	−53.0197	−1.85492
$818$	0	0
$819$	−10.4290	−0.364419
$820$	0	0
$821$	24.0976	0.841011	0.420506	−	0.907290i	$-0.361853\pi$
$821$	24.0976	0.841011	0.420506	+	0.907290i	$0.361853\pi$
$822$	0	0
$823$	24.2839	0.846482	0.423241	−	0.906017i	$-0.360892\pi$
$823$	24.2839	0.846482	0.423241	+	0.906017i	$0.360892\pi$
$824$	0	0
$825$	83.1075	2.89343
$826$	0	0
$827$	9.51253	0.330783	0.165391	−	0.986228i	$-0.447111\pi$
$827$	9.51253	0.330783	0.165391	+	0.986228i	$0.447111\pi$
$828$	0	0
$829$	−15.8158	−0.549306	−0.274653	−	0.961543i	$-0.588563\pi$
$829$	−15.8158	−0.549306	−0.274653	+	0.961543i	$0.588563\pi$
$830$	0	0
$831$	29.5074	1.02360
$832$	0	0
$833$	5.21494	0.180687
$834$	0	0
$835$	−10.3028	−0.356542
$836$	0	0
$837$	11.6034	0.401073
$838$	0	0
$839$	−50.1793	−1.73238	−0.866191	−	0.499712i	$-0.833439\pi$
$839$	−50.1793	−1.73238	−0.866191	+	0.499712i	$0.833439\pi$
$840$	0	0
$841$	37.1754	1.28191
$842$	0	0
$843$	58.8333	2.02633
$844$	0	0
$845$	17.5912	0.605154
$846$	0	0
$847$	15.9209	0.547049
$848$	0	0
$849$	−38.3973	−1.31779
$850$	0	0
$851$	−27.8602	−0.955035
$852$	0	0
$853$	−24.7996	−0.849123	−0.424562	−	0.905399i	$-0.639572\pi$
$853$	−24.7996	−0.849123	−0.424562	+	0.905399i	$0.639572\pi$
$854$	0	0
$855$	39.7487	1.35938
$856$	0	0
$857$	29.2281	0.998413	0.499206	−	0.866483i	$-0.333625\pi$
$857$	29.2281	0.998413	0.499206	+	0.866483i	$0.333625\pi$
$858$	0	0
$859$	1.04234	0.0355640	0.0177820	−	0.999842i	$-0.494340\pi$
$859$	1.04234	0.0355640	0.0177820	+	0.999842i	$0.494340\pi$
$860$	0	0
$861$	−31.7942	−1.08354
$862$	0	0
$863$	4.84799	0.165027	0.0825137	−	0.996590i	$-0.473705\pi$
$863$	4.84799	0.165027	0.0825137	+	0.996590i	$0.473705\pi$
$864$	0	0
$865$	38.9929	1.32580
$866$	0	0
$867$	36.3024	1.23289
$868$	0	0
$869$	−35.9307	−1.21887
$870$	0	0
$871$	11.8855	0.402726
$872$	0	0
$873$	−9.52166	−0.322260
$874$	0	0
$875$	10.4166	0.352147
$876$	0	0
$877$	−23.7251	−0.801140	−0.400570	−	0.916266i	$-0.631188\pi$
$877$	−23.7251	−0.801140	−0.400570	+	0.916266i	$0.631188\pi$
$878$	0	0
$879$	−40.3816	−1.36204
$880$	0	0
$881$	−12.2267	−0.411929	−0.205965	−	0.978559i	$-0.566033\pi$
$881$	−12.2267	−0.411929	−0.205965	+	0.978559i	$0.566033\pi$
$882$	0	0
$883$	−25.4653	−0.856977	−0.428489	−	0.903547i	$-0.640954\pi$
$883$	−25.4653	−0.856977	−0.428489	+	0.903547i	$0.640954\pi$
$884$	0	0
$885$	−9.38272	−0.315397
$886$	0	0
$887$	−0.757954	−0.0254496	−0.0127248	−	0.999919i	$-0.504051\pi$
$887$	−0.757954	−0.0254496	−0.0127248	+	0.999919i	$0.504051\pi$
$888$	0	0
$889$	−20.5458	−0.689083
$890$	0	0
$891$	53.9603	1.80774
$892$	0	0
$893$	−58.9786	−1.97364
$894$	0	0
$895$	45.0696	1.50651
$896$	0	0
$897$	−83.2209	−2.77866
$898$	0	0
$899$	44.9306	1.49852
$900$	0	0
$901$	3.05350	0.101727
$902$	0	0
$903$	26.0853	0.868063
$904$	0	0
$905$	−69.5037	−2.31038
$906$	0	0
$907$	54.9506	1.82461	0.912303	−	0.409516i	$-0.134303\pi$
$907$	54.9506	1.82461	0.912303	+	0.409516i	$0.134303\pi$
$908$	0	0
$909$	−3.73920	−0.124021
$910$	0	0
$911$	8.67904	0.287549	0.143775	−	0.989610i	$-0.454076\pi$
$911$	8.67904	0.287549	0.143775	+	0.989610i	$0.454076\pi$
$912$	0	0
$913$	49.8717	1.65051
$914$	0	0
$915$	12.6793	0.419165
$916$	0	0
$917$	−20.7332	−0.684669
$918$	0	0
$919$	−23.1237	−0.762781	−0.381390	−	0.924414i	$-0.624555\pi$
$919$	−23.1237	−0.762781	−0.381390	+	0.924414i	$0.624555\pi$
$920$	0	0
$921$	57.4477	1.89297
$922$	0	0
$923$	−44.5054	−1.46491
$924$	0	0
$925$	23.8939	0.785626
$926$	0	0
$927$	3.77637	0.124032
$928$	0	0
$929$	−17.8344	−0.585129	−0.292564	−	0.956246i	$-0.594509\pi$
$929$	−17.8344	−0.585129	−0.292564	+	0.956246i	$0.594509\pi$
$930$	0	0
$931$	−30.4020	−0.996385
$932$	0	0
$933$	−2.38113	−0.0779546
$934$	0	0
$935$	16.2890	0.532709
$936$	0	0
$937$	−54.6764	−1.78620	−0.893101	−	0.449857i	$-0.851475\pi$
$937$	−54.6764	−1.78620	−0.893101	+	0.449857i	$0.851475\pi$
$938$	0	0
$939$	1.16272	0.0379440
$940$	0	0
$941$	8.54393	0.278524	0.139262	−	0.990256i	$-0.455527\pi$
$941$	8.54393	0.278524	0.139262	+	0.990256i	$0.455527\pi$
$942$	0	0
$943$	−103.487	−3.37001
$944$	0	0
$945$	8.83157	0.287291
$946$	0	0
$947$	34.9613	1.13609	0.568044	−	0.822998i	$-0.307700\pi$
$947$	34.9613	1.13609	0.568044	+	0.822998i	$0.307700\pi$
$948$	0	0
$949$	15.5399	0.504447
$950$	0	0
$951$	−5.37612	−0.174333
$952$	0	0
$953$	−58.0334	−1.87989	−0.939944	−	0.341330i	$-0.889123\pi$
$953$	−58.0334	−1.87989	−0.939944	+	0.341330i	$0.889123\pi$
$954$	0	0
$955$	30.7071	0.993657
$956$	0	0
$957$	90.4085	2.92249
$958$	0	0
$959$	4.82526	0.155816
$960$	0	0
$961$	−0.493821	−0.0159297
$962$	0	0
$963$	2.45885	0.0792355
$964$	0	0
$965$	−47.7501	−1.53713
$966$	0	0
$967$	−7.15038	−0.229941	−0.114970	−	0.993369i	$-0.536677\pi$
$967$	−7.15038	−0.229941	−0.114970	+	0.993369i	$0.536677\pi$
$968$	0	0
$969$	11.4463	0.367709
$970$	0	0
$971$	1.83517	0.0588933	0.0294466	−	0.999566i	$-0.490625\pi$
$971$	1.83517	0.0588933	0.0294466	+	0.999566i	$0.490625\pi$
$972$	0	0
$973$	8.18541	0.262412
$974$	0	0
$975$	71.3731	2.28577
$976$	0	0
$977$	−35.6590	−1.14083	−0.570416	−	0.821356i	$-0.693218\pi$
$977$	−35.6590	−1.14083	−0.570416	+	0.821356i	$0.693218\pi$
$978$	0	0
$979$	−14.2132	−0.454257
$980$	0	0
$981$	−10.0303	−0.320243
$982$	0	0
$983$	46.9647	1.49794	0.748970	−	0.662603i	$-0.230548\pi$
$983$	46.9647	1.49794	0.748970	+	0.662603i	$0.230548\pi$
$984$	0	0
$985$	7.50323	0.239073
$986$	0	0
$987$	29.0170	0.923621
$988$	0	0
$989$	84.9052	2.69983
$990$	0	0
$991$	6.27640	0.199376	0.0996882	−	0.995019i	$-0.468215\pi$
$991$	6.27640	0.199376	0.0996882	+	0.995019i	$0.468215\pi$
$992$	0	0
$993$	34.2060	1.08550
$994$	0	0
$995$	−47.2003	−1.49635
$996$	0	0
$997$	3.78831	0.119977	0.0599885	−	0.998199i	$-0.480894\pi$
$997$	3.78831	0.119977	0.0599885	+	0.998199i	$0.480894\pi$
$998$	0	0
$999$	6.71274	0.212382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.9	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.9	✓	44	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)

\(f(q)\)	\(=\)	\(q-2.25093 q^{3} +3.53241 q^{5} +1.19008 q^{7} +2.06668 q^{9} +O(q^{10})\)	\(q-2.25093 q^{3} +3.53241 q^{5} +1.19008 q^{7} +2.06668 q^{9} -4.93741 q^{11} -4.24028 q^{13} -7.95119 q^{15} -0.933955 q^{17} +5.44476 q^{19} -2.67878 q^{21} -8.71920 q^{23} +7.47789 q^{25} +2.10084 q^{27} +8.13482 q^{29} +5.52324 q^{31} +11.1138 q^{33} +4.20384 q^{35} +3.19527 q^{37} +9.54456 q^{39} +11.8689 q^{41} -9.73774 q^{43} +7.30035 q^{45} -10.8322 q^{47} -5.58371 q^{49} +2.10227 q^{51} -3.26943 q^{53} -17.4409 q^{55} -12.2558 q^{57} +1.18004 q^{59} -1.59464 q^{61} +2.45951 q^{63} -14.9784 q^{65} -2.80301 q^{67} +19.6263 q^{69} +10.4959 q^{71} -3.66483 q^{73} -16.8322 q^{75} -5.87590 q^{77} +7.27723 q^{79} -10.9289 q^{81} -10.1008 q^{83} -3.29911 q^{85} -18.3109 q^{87} +2.87868 q^{89} -5.04626 q^{91} -12.4324 q^{93} +19.2331 q^{95} -4.60723 q^{97} -10.2041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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