LMFDB - Embedded newform 6008.2.a.e.1.14

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			1.14
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)

$f(q)$	$=$	$q-1.38456 q^{3} -1.32143 q^{5} +2.39545 q^{7} -1.08299 q^{9} +O(q^{10})$	$q-1.38456 q^{3} -1.32143 q^{5} +2.39545 q^{7} -1.08299 q^{9} +3.31333 q^{11} -3.52800 q^{13} +1.82960 q^{15} +2.72465 q^{17} +6.86584 q^{19} -3.31664 q^{21} +6.61624 q^{23} -3.25383 q^{25} +5.65315 q^{27} -7.66755 q^{29} -1.56808 q^{31} -4.58751 q^{33} -3.16541 q^{35} +1.61416 q^{37} +4.88473 q^{39} +3.10413 q^{41} -2.57942 q^{43} +1.43109 q^{45} -2.66539 q^{47} -1.26184 q^{49} -3.77244 q^{51} +3.91075 q^{53} -4.37832 q^{55} -9.50618 q^{57} +1.25988 q^{59} +11.2785 q^{61} -2.59424 q^{63} +4.66199 q^{65} +9.01948 q^{67} -9.16060 q^{69} +6.20826 q^{71} -8.76402 q^{73} +4.50513 q^{75} +7.93691 q^{77} -10.5971 q^{79} -4.57817 q^{81} -10.8953 q^{83} -3.60042 q^{85} +10.6162 q^{87} -13.0225 q^{89} -8.45112 q^{91} +2.17110 q^{93} -9.07270 q^{95} +3.05044 q^{97} -3.58830 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.38456	−0.799377	−0.399689	−	0.916651i	$-0.630882\pi$
$3$	−1.38456	−0.799377	−0.399689	+	0.916651i	$0.630882\pi$
$4$	0	0
$5$	−1.32143	−0.590960	−0.295480	−	0.955349i	$-0.595480\pi$
$5$	−1.32143	−0.590960	−0.295480	+	0.955349i	$0.595480\pi$
$6$	0	0
$7$	2.39545	0.905393	0.452697	−	0.891665i	$-0.350462\pi$
$7$	2.39545	0.905393	0.452697	+	0.891665i	$0.350462\pi$
$8$	0	0
$9$	−1.08299	−0.360996
$10$	0	0
$11$	3.31333	0.999007	0.499504	−	0.866312i	$-0.333516\pi$
$11$	3.31333	0.999007	0.499504	+	0.866312i	$0.333516\pi$
$12$	0	0
$13$	−3.52800	−0.978490	−0.489245	−	0.872146i	$-0.662728\pi$
$13$	−3.52800	−0.978490	−0.489245	+	0.872146i	$0.662728\pi$
$14$	0	0
$15$	1.82960	0.472400
$16$	0	0
$17$	2.72465	0.660824	0.330412	−	0.943837i	$-0.392812\pi$
$17$	2.72465	0.660824	0.330412	+	0.943837i	$0.392812\pi$
$18$	0	0
$19$	6.86584	1.57513	0.787566	−	0.616230i	$-0.211341\pi$
$19$	6.86584	1.57513	0.787566	+	0.616230i	$0.211341\pi$
$20$	0	0
$21$	−3.31664	−0.723751
$22$	0	0
$23$	6.61624	1.37958	0.689791	−	0.724009i	$-0.257702\pi$
$23$	6.61624	1.37958	0.689791	+	0.724009i	$0.257702\pi$
$24$	0	0
$25$	−3.25383	−0.650766
$26$	0	0
$27$	5.65315	1.08795
$28$	0	0
$29$	−7.66755	−1.42383	−0.711915	−	0.702266i	$-0.752172\pi$
$29$	−7.66755	−1.42383	−0.711915	+	0.702266i	$0.752172\pi$
$30$	0	0
$31$	−1.56808	−0.281635	−0.140818	−	0.990036i	$-0.544973\pi$
$31$	−1.56808	−0.281635	−0.140818	+	0.990036i	$0.544973\pi$
$32$	0	0
$33$	−4.58751	−0.798583
$34$	0	0
$35$	−3.16541	−0.535051
$36$	0	0
$37$	1.61416	0.265366	0.132683	−	0.991159i	$-0.457641\pi$
$37$	1.61416	0.265366	0.132683	+	0.991159i	$0.457641\pi$
$38$	0	0
$39$	4.88473	0.782182
$40$	0	0
$41$	3.10413	0.484783	0.242392	−	0.970178i	$-0.422068\pi$
$41$	3.10413	0.484783	0.242392	+	0.970178i	$0.422068\pi$
$42$	0	0
$43$	−2.57942	−0.393358	−0.196679	−	0.980468i	$-0.563016\pi$
$43$	−2.57942	−0.393358	−0.196679	+	0.980468i	$0.563016\pi$
$44$	0	0
$45$	1.43109	0.213334
$46$	0	0
$47$	−2.66539	−0.388787	−0.194394	−	0.980924i	$-0.562274\pi$
$47$	−2.66539	−0.388787	−0.194394	+	0.980924i	$0.562274\pi$
$48$	0	0
$49$	−1.26184	−0.180263
$50$	0	0
$51$	−3.77244	−0.528248
$52$	0	0
$53$	3.91075	0.537183	0.268592	−	0.963254i	$-0.413442\pi$
$53$	3.91075	0.537183	0.268592	+	0.963254i	$0.413442\pi$
$54$	0	0
$55$	−4.37832	−0.590373
$56$	0	0
$57$	−9.50618	−1.25912
$58$	0	0
$59$	1.25988	0.164022	0.0820112	−	0.996631i	$-0.473866\pi$
$59$	1.25988	0.164022	0.0820112	+	0.996631i	$0.473866\pi$
$60$	0	0
$61$	11.2785	1.44406	0.722030	−	0.691862i	$-0.243209\pi$
$61$	11.2785	1.44406	0.722030	+	0.691862i	$0.243209\pi$
$62$	0	0
$63$	−2.59424	−0.326844
$64$	0	0
$65$	4.66199	0.578248
$66$	0	0
$67$	9.01948	1.10191	0.550953	−	0.834537i	$-0.314265\pi$
$67$	9.01948	1.10191	0.550953	+	0.834537i	$0.314265\pi$
$68$	0	0
$69$	−9.16060	−1.10281
$70$	0	0
$71$	6.20826	0.736785	0.368392	−	0.929670i	$-0.379908\pi$
$71$	6.20826	0.736785	0.368392	+	0.929670i	$0.379908\pi$
$72$	0	0
$73$	−8.76402	−1.02575	−0.512875	−	0.858463i	$-0.671420\pi$
$73$	−8.76402	−1.02575	−0.512875	+	0.858463i	$0.671420\pi$
$74$	0	0
$75$	4.50513	0.520208
$76$	0	0
$77$	7.93691	0.904495
$78$	0	0
$79$	−10.5971	−1.19227	−0.596135	−	0.802884i	$-0.703298\pi$
$79$	−10.5971	−1.19227	−0.596135	+	0.802884i	$0.703298\pi$
$80$	0	0
$81$	−4.57817	−0.508685
$82$	0	0
$83$	−10.8953	−1.19591	−0.597955	−	0.801530i	$-0.704020\pi$
$83$	−10.8953	−1.19591	−0.597955	+	0.801530i	$0.704020\pi$
$84$	0	0
$85$	−3.60042	−0.390521
$86$	0	0
$87$	10.6162	1.13818
$88$	0	0
$89$	−13.0225	−1.38039	−0.690194	−	0.723625i	$-0.742475\pi$
$89$	−13.0225	−1.38039	−0.690194	+	0.723625i	$0.742475\pi$
$90$	0	0
$91$	−8.45112	−0.885918
$92$	0	0
$93$	2.17110	0.225133
$94$	0	0
$95$	−9.07270	−0.930840
$96$	0	0
$97$	3.05044	0.309725	0.154863	−	0.987936i	$-0.450507\pi$
$97$	3.05044	0.309725	0.154863	+	0.987936i	$0.450507\pi$
$98$	0	0
$99$	−3.58830	−0.360638
$100$	0	0
$101$	0.567035	0.0564221	0.0282110	−	0.999602i	$-0.491019\pi$
$101$	0.567035	0.0564221	0.0282110	+	0.999602i	$0.491019\pi$
$102$	0	0
$103$	−7.62185	−0.751003	−0.375502	−	0.926822i	$-0.622530\pi$
$103$	−7.62185	−0.751003	−0.375502	+	0.926822i	$0.622530\pi$
$104$	0	0
$105$	4.38270	0.427708
$106$	0	0
$107$	−0.433918	−0.0419484	−0.0209742	−	0.999780i	$-0.506677\pi$
$107$	−0.433918	−0.0419484	−0.0209742	+	0.999780i	$0.506677\pi$
$108$	0	0
$109$	12.5223	1.19942	0.599710	−	0.800217i	$-0.295282\pi$
$109$	12.5223	1.19942	0.599710	+	0.800217i	$0.295282\pi$
$110$	0	0
$111$	−2.23490	−0.212127
$112$	0	0
$113$	11.7364	1.10407	0.552034	−	0.833821i	$-0.313852\pi$
$113$	11.7364	1.10407	0.552034	+	0.833821i	$0.313852\pi$
$114$	0	0
$115$	−8.74288	−0.815278
$116$	0	0
$117$	3.82078	0.353231
$118$	0	0
$119$	6.52675	0.598306
$120$	0	0
$121$	−0.0218313	−0.00198467
$122$	0	0
$123$	−4.29786	−0.387525
$124$	0	0
$125$	10.9068	0.975537
$126$	0	0
$127$	−5.29959	−0.470262	−0.235131	−	0.971964i	$-0.575552\pi$
$127$	−5.29959	−0.470262	−0.235131	+	0.971964i	$0.575552\pi$
$128$	0	0
$129$	3.57136	0.314441
$130$	0	0
$131$	−9.13635	−0.798246	−0.399123	−	0.916897i	$-0.630685\pi$
$131$	−9.13635	−0.798246	−0.399123	+	0.916897i	$0.630685\pi$
$132$	0	0
$133$	16.4468	1.42611
$134$	0	0
$135$	−7.47022	−0.642934
$136$	0	0
$137$	−12.2093	−1.04311	−0.521554	−	0.853218i	$-0.674648\pi$
$137$	−12.2093	−1.04311	−0.521554	+	0.853218i	$0.674648\pi$
$138$	0	0
$139$	17.2855	1.46614	0.733068	−	0.680156i	$-0.238088\pi$
$139$	17.2855	1.46614	0.733068	+	0.680156i	$0.238088\pi$
$140$	0	0
$141$	3.69040	0.310787
$142$	0	0
$143$	−11.6894	−0.977518
$144$	0	0
$145$	10.1321	0.841426
$146$	0	0
$147$	1.74709	0.144098
$148$	0	0
$149$	11.7744	0.964595	0.482298	−	0.876007i	$-0.339802\pi$
$149$	11.7744	0.964595	0.482298	+	0.876007i	$0.339802\pi$
$150$	0	0
$151$	11.3961	0.927405	0.463702	−	0.885991i	$-0.346521\pi$
$151$	11.3961	0.927405	0.463702	+	0.885991i	$0.346521\pi$
$152$	0	0
$153$	−2.95076	−0.238555
$154$	0	0
$155$	2.07210	0.166435
$156$	0	0
$157$	10.9795	0.876256	0.438128	−	0.898913i	$-0.355642\pi$
$157$	10.9795	0.876256	0.438128	+	0.898913i	$0.355642\pi$
$158$	0	0
$159$	−5.41468	−0.429412
$160$	0	0
$161$	15.8489	1.24906
$162$	0	0
$163$	−5.84552	−0.457857	−0.228928	−	0.973443i	$-0.573522\pi$
$163$	−5.84552	−0.457857	−0.228928	+	0.973443i	$0.573522\pi$
$164$	0	0
$165$	6.06206	0.471931
$166$	0	0
$167$	12.7460	0.986312	0.493156	−	0.869941i	$-0.335843\pi$
$167$	12.7460	0.986312	0.493156	+	0.869941i	$0.335843\pi$
$168$	0	0
$169$	−0.553251	−0.0425578
$170$	0	0
$171$	−7.43563	−0.568617
$172$	0	0
$173$	4.64420	0.353092	0.176546	−	0.984292i	$-0.443508\pi$
$173$	4.64420	0.353092	0.176546	+	0.984292i	$0.443508\pi$
$174$	0	0
$175$	−7.79438	−0.589200
$176$	0	0
$177$	−1.74438	−0.131116
$178$	0	0
$179$	1.76747	0.132107	0.0660536	−	0.997816i	$-0.478959\pi$
$179$	1.76747	0.132107	0.0660536	+	0.997816i	$0.478959\pi$
$180$	0	0
$181$	5.86495	0.435938	0.217969	−	0.975956i	$-0.430057\pi$
$181$	5.86495	0.435938	0.217969	+	0.975956i	$0.430057\pi$
$182$	0	0
$183$	−15.6157	−1.15435
$184$	0	0
$185$	−2.13299	−0.156820
$186$	0	0
$187$	9.02767	0.660168
$188$	0	0
$189$	13.5418	0.985022
$190$	0	0
$191$	13.5392	0.979662	0.489831	−	0.871817i	$-0.337058\pi$
$191$	13.5392	0.979662	0.489831	+	0.871817i	$0.337058\pi$
$192$	0	0
$193$	7.65319	0.550889	0.275444	−	0.961317i	$-0.411175\pi$
$193$	7.65319	0.550889	0.275444	+	0.961317i	$0.411175\pi$
$194$	0	0
$195$	−6.45481	−0.462238
$196$	0	0
$197$	−2.06922	−0.147426	−0.0737128	−	0.997280i	$-0.523485\pi$
$197$	−2.06922	−0.147426	−0.0737128	+	0.997280i	$0.523485\pi$
$198$	0	0
$199$	−4.48651	−0.318040	−0.159020	−	0.987275i	$-0.550833\pi$
$199$	−4.48651	−0.318040	−0.159020	+	0.987275i	$0.550833\pi$
$200$	0	0
$201$	−12.4880	−0.880838
$202$	0	0
$203$	−18.3672	−1.28913
$204$	0	0
$205$	−4.10188	−0.286488
$206$	0	0
$207$	−7.16532	−0.498024
$208$	0	0
$209$	22.7488	1.57357
$210$	0	0
$211$	22.1253	1.52317	0.761583	−	0.648067i	$-0.224422\pi$
$211$	22.1253	1.52317	0.761583	+	0.648067i	$0.224422\pi$
$212$	0	0
$213$	−8.59572	−0.588969
$214$	0	0
$215$	3.40851	0.232459
$216$	0	0
$217$	−3.75625	−0.254991
$218$	0	0
$219$	12.1343	0.819961
$220$	0	0
$221$	−9.61255	−0.646610
$222$	0	0
$223$	20.3020	1.35952	0.679760	−	0.733434i	$-0.262084\pi$
$223$	20.3020	1.35952	0.679760	+	0.733434i	$0.262084\pi$
$224$	0	0
$225$	3.52386	0.234924
$226$	0	0
$227$	14.9537	0.992509	0.496255	−	0.868177i	$-0.334708\pi$
$227$	14.9537	0.992509	0.496255	+	0.868177i	$0.334708\pi$
$228$	0	0
$229$	4.45255	0.294232	0.147116	−	0.989119i	$-0.453001\pi$
$229$	4.45255	0.294232	0.147116	+	0.989119i	$0.453001\pi$
$230$	0	0
$231$	−10.9891	−0.723032
$232$	0	0
$233$	−22.1365	−1.45021	−0.725104	−	0.688640i	$-0.758208\pi$
$233$	−22.1365	−1.45021	−0.725104	+	0.688640i	$0.758208\pi$
$234$	0	0
$235$	3.52212	0.229758
$236$	0	0
$237$	14.6724	0.953073
$238$	0	0
$239$	6.23974	0.403615	0.201808	−	0.979425i	$-0.435318\pi$
$239$	6.23974	0.403615	0.201808	+	0.979425i	$0.435318\pi$
$240$	0	0
$241$	−23.1955	−1.49415	−0.747076	−	0.664738i	$-0.768543\pi$
$241$	−23.1955	−1.49415	−0.747076	+	0.664738i	$0.768543\pi$
$242$	0	0
$243$	−10.6207	−0.681318
$244$	0	0
$245$	1.66743	0.106528
$246$	0	0
$247$	−24.2227	−1.54125
$248$	0	0
$249$	15.0852	0.955983
$250$	0	0
$251$	11.7528	0.741833	0.370916	−	0.928666i	$-0.379044\pi$
$251$	11.7528	0.741833	0.370916	+	0.928666i	$0.379044\pi$
$252$	0	0
$253$	21.9218	1.37821
$254$	0	0
$255$	4.98501	0.312173
$256$	0	0
$257$	−5.47590	−0.341577	−0.170789	−	0.985308i	$-0.554632\pi$
$257$	−5.47590	−0.341577	−0.170789	+	0.985308i	$0.554632\pi$
$258$	0	0
$259$	3.86662	0.240260
$260$	0	0
$261$	8.30388	0.513997
$262$	0	0
$263$	1.05628	0.0651333	0.0325666	−	0.999470i	$-0.489632\pi$
$263$	1.05628	0.0651333	0.0325666	+	0.999470i	$0.489632\pi$
$264$	0	0
$265$	−5.16777	−0.317454
$266$	0	0
$267$	18.0305	1.10345
$268$	0	0
$269$	−2.37386	−0.144737	−0.0723683	−	0.997378i	$-0.523056\pi$
$269$	−2.37386	−0.144737	−0.0723683	+	0.997378i	$0.523056\pi$
$270$	0	0
$271$	24.6662	1.49837	0.749184	−	0.662362i	$-0.230446\pi$
$271$	24.6662	1.49837	0.749184	+	0.662362i	$0.230446\pi$
$272$	0	0
$273$	11.7011	0.708183
$274$	0	0
$275$	−10.7810	−0.650120
$276$	0	0
$277$	−0.680049	−0.0408602	−0.0204301	−	0.999791i	$-0.506504\pi$
$277$	−0.680049	−0.0408602	−0.0204301	+	0.999791i	$0.506504\pi$
$278$	0	0
$279$	1.69821	0.101669
$280$	0	0
$281$	8.38448	0.500176	0.250088	−	0.968223i	$-0.419540\pi$
$281$	8.38448	0.500176	0.250088	+	0.968223i	$0.419540\pi$
$282$	0	0
$283$	−17.8996	−1.06402	−0.532010	−	0.846738i	$-0.678563\pi$
$283$	−17.8996	−1.06402	−0.532010	+	0.846738i	$0.678563\pi$
$284$	0	0
$285$	12.5617	0.744092
$286$	0	0
$287$	7.43577	0.438920
$288$	0	0
$289$	−9.57629	−0.563311
$290$	0	0
$291$	−4.22352	−0.247587
$292$	0	0
$293$	−9.05758	−0.529150	−0.264575	−	0.964365i	$-0.585232\pi$
$293$	−9.05758	−0.529150	−0.264575	+	0.964365i	$0.585232\pi$
$294$	0	0
$295$	−1.66484	−0.0969306
$296$	0	0
$297$	18.7308	1.08687
$298$	0	0
$299$	−23.3421	−1.34991
$300$	0	0
$301$	−6.17886	−0.356143
$302$	0	0
$303$	−0.785095	−0.0451025
$304$	0	0
$305$	−14.9037	−0.853381
$306$	0	0
$307$	28.9617	1.65293	0.826465	−	0.562988i	$-0.190348\pi$
$307$	28.9617	1.65293	0.826465	+	0.562988i	$0.190348\pi$
$308$	0	0
$309$	10.5529	0.600335
$310$	0	0
$311$	−0.161250	−0.00914363	−0.00457181	−	0.999990i	$-0.501455\pi$
$311$	−0.161250	−0.00914363	−0.00457181	+	0.999990i	$0.501455\pi$
$312$	0	0
$313$	26.0251	1.47102	0.735512	−	0.677512i	$-0.236942\pi$
$313$	26.0251	1.47102	0.735512	+	0.677512i	$0.236942\pi$
$314$	0	0
$315$	3.42810	0.193151
$316$	0	0
$317$	−6.20573	−0.348548	−0.174274	−	0.984697i	$-0.555758\pi$
$317$	−6.20573	−0.348548	−0.174274	+	0.984697i	$0.555758\pi$
$318$	0	0
$319$	−25.4052	−1.42242
$320$	0	0
$321$	0.600786	0.0335326
$322$	0	0
$323$	18.7070	1.04089
$324$	0	0
$325$	11.4795	0.636768
$326$	0	0
$327$	−17.3379	−0.958789
$328$	0	0
$329$	−6.38480	−0.352005
$330$	0	0
$331$	−19.1181	−1.05083	−0.525414	−	0.850847i	$-0.676090\pi$
$331$	−19.1181	−1.05083	−0.525414	+	0.850847i	$0.676090\pi$
$332$	0	0
$333$	−1.74811	−0.0957960
$334$	0	0
$335$	−11.9186	−0.651182
$336$	0	0
$337$	1.33950	0.0729670	0.0364835	−	0.999334i	$-0.488384\pi$
$337$	1.33950	0.0729670	0.0364835	+	0.999334i	$0.488384\pi$
$338$	0	0
$339$	−16.2498	−0.882567
$340$	0	0
$341$	−5.19557	−0.281356
$342$	0	0
$343$	−19.7908	−1.06860
$344$	0	0
$345$	12.1051	0.651714
$346$	0	0
$347$	−3.36148	−0.180454	−0.0902270	−	0.995921i	$-0.528759\pi$
$347$	−3.36148	−0.180454	−0.0902270	+	0.995921i	$0.528759\pi$
$348$	0	0
$349$	7.24629	0.387885	0.193942	−	0.981013i	$-0.437872\pi$
$349$	7.24629	0.387885	0.193942	+	0.981013i	$0.437872\pi$
$350$	0	0
$351$	−19.9443	−1.06455
$352$	0	0
$353$	27.9330	1.48672	0.743361	−	0.668890i	$-0.233230\pi$
$353$	27.9330	1.48672	0.743361	+	0.668890i	$0.233230\pi$
$354$	0	0
$355$	−8.20376	−0.435410
$356$	0	0
$357$	−9.03669	−0.478272
$358$	0	0
$359$	−14.0445	−0.741242	−0.370621	−	0.928784i	$-0.620855\pi$
$359$	−14.0445	−0.741242	−0.370621	+	0.928784i	$0.620855\pi$
$360$	0	0
$361$	28.1398	1.48104
$362$	0	0
$363$	0.0302268	0.00158650
$364$	0	0
$365$	11.5810	0.606177
$366$	0	0
$367$	26.7704	1.39740	0.698701	−	0.715413i	$-0.253762\pi$
$367$	26.7704	1.39740	0.698701	+	0.715413i	$0.253762\pi$
$368$	0	0
$369$	−3.36174	−0.175005
$370$	0	0
$371$	9.36800	0.486362
$372$	0	0
$373$	9.20138	0.476429	0.238215	−	0.971213i	$-0.423438\pi$
$373$	9.20138	0.476429	0.238215	+	0.971213i	$0.423438\pi$
$374$	0	0
$375$	−15.1012	−0.779822
$376$	0	0
$377$	27.0511	1.39320
$378$	0	0
$379$	−1.97475	−0.101436	−0.0507182	−	0.998713i	$-0.516151\pi$
$379$	−1.97475	−0.101436	−0.0507182	+	0.998713i	$0.516151\pi$
$380$	0	0
$381$	7.33761	0.375917
$382$	0	0
$383$	36.7039	1.87548	0.937741	−	0.347336i	$-0.112914\pi$
$383$	36.7039	1.87548	0.937741	+	0.347336i	$0.112914\pi$
$384$	0	0
$385$	−10.4880	−0.534520
$386$	0	0
$387$	2.79348	0.142001
$388$	0	0
$389$	20.2274	1.02557	0.512784	−	0.858518i	$-0.328614\pi$
$389$	20.2274	1.02557	0.512784	+	0.858518i	$0.328614\pi$
$390$	0	0
$391$	18.0269	0.911661
$392$	0	0
$393$	12.6498	0.638100
$394$	0	0
$395$	14.0033	0.704584
$396$	0	0
$397$	−11.5935	−0.581859	−0.290929	−	0.956745i	$-0.593964\pi$
$397$	−11.5935	−0.581859	−0.290929	+	0.956745i	$0.593964\pi$
$398$	0	0
$399$	−22.7715	−1.14000
$400$	0	0
$401$	17.1601	0.856932	0.428466	−	0.903558i	$-0.359054\pi$
$401$	17.1601	0.856932	0.428466	+	0.903558i	$0.359054\pi$
$402$	0	0
$403$	5.53217	0.275577
$404$	0	0
$405$	6.04971	0.300613
$406$	0	0
$407$	5.34824	0.265102
$408$	0	0
$409$	−17.1798	−0.849486	−0.424743	−	0.905314i	$-0.639635\pi$
$409$	−17.1798	−0.849486	−0.424743	+	0.905314i	$0.639635\pi$
$410$	0	0
$411$	16.9045	0.833837
$412$	0	0
$413$	3.01797	0.148505
$414$	0	0
$415$	14.3973	0.706735
$416$	0	0
$417$	−23.9328	−1.17200
$418$	0	0
$419$	−17.5543	−0.857584	−0.428792	−	0.903403i	$-0.641061\pi$
$419$	−17.5543	−0.857584	−0.428792	+	0.903403i	$0.641061\pi$
$420$	0	0
$421$	36.9668	1.80165	0.900825	−	0.434183i	$-0.142963\pi$
$421$	36.9668	1.80165	0.900825	+	0.434183i	$0.142963\pi$
$422$	0	0
$423$	2.88659	0.140351
$424$	0	0
$425$	−8.86555	−0.430042
$426$	0	0
$427$	27.0169	1.30744
$428$	0	0
$429$	16.1847	0.781406
$430$	0	0
$431$	15.4484	0.744122	0.372061	−	0.928208i	$-0.378651\pi$
$431$	15.4484	0.744122	0.372061	+	0.928208i	$0.378651\pi$
$432$	0	0
$433$	20.1095	0.966401	0.483200	−	0.875510i	$-0.339474\pi$
$433$	20.1095	0.966401	0.483200	+	0.875510i	$0.339474\pi$
$434$	0	0
$435$	−14.0285	−0.672617
$436$	0	0
$437$	45.4261	2.17302
$438$	0	0
$439$	15.9474	0.761127	0.380564	−	0.924755i	$-0.375730\pi$
$439$	15.9474	0.761127	0.380564	+	0.924755i	$0.375730\pi$
$440$	0	0
$441$	1.36656	0.0650741
$442$	0	0
$443$	5.19141	0.246651	0.123326	−	0.992366i	$-0.460644\pi$
$443$	5.19141	0.246651	0.123326	+	0.992366i	$0.460644\pi$
$444$	0	0
$445$	17.2083	0.815754
$446$	0	0
$447$	−16.3024	−0.771075
$448$	0	0
$449$	−7.83939	−0.369964	−0.184982	−	0.982742i	$-0.559223\pi$
$449$	−7.83939	−0.369964	−0.184982	+	0.982742i	$0.559223\pi$
$450$	0	0
$451$	10.2850	0.484302
$452$	0	0
$453$	−15.7787	−0.741346
$454$	0	0
$455$	11.1675	0.523542
$456$	0	0
$457$	38.0223	1.77861	0.889303	−	0.457318i	$-0.151190\pi$
$457$	38.0223	1.77861	0.889303	+	0.457318i	$0.151190\pi$
$458$	0	0
$459$	15.4028	0.718943
$460$	0	0
$461$	−2.61241	−0.121672	−0.0608361	−	0.998148i	$-0.519377\pi$
$461$	−2.61241	−0.121672	−0.0608361	+	0.998148i	$0.519377\pi$
$462$	0	0
$463$	−1.53825	−0.0714885	−0.0357443	−	0.999361i	$-0.511380\pi$
$463$	−1.53825	−0.0714885	−0.0357443	+	0.999361i	$0.511380\pi$
$464$	0	0
$465$	−2.86895	−0.133044
$466$	0	0
$467$	−10.4328	−0.482774	−0.241387	−	0.970429i	$-0.577602\pi$
$467$	−10.4328	−0.482774	−0.241387	+	0.970429i	$0.577602\pi$
$468$	0	0
$469$	21.6057	0.997658
$470$	0	0
$471$	−15.2017	−0.700459
$472$	0	0
$473$	−8.54647	−0.392967
$474$	0	0
$475$	−22.3403	−1.02504
$476$	0	0
$477$	−4.23530	−0.193921
$478$	0	0
$479$	−33.1317	−1.51382	−0.756912	−	0.653517i	$-0.773293\pi$
$479$	−33.1317	−1.51382	−0.756912	+	0.653517i	$0.773293\pi$
$480$	0	0
$481$	−5.69474	−0.259658
$482$	0	0
$483$	−21.9437	−0.998474
$484$	0	0
$485$	−4.03093	−0.183035
$486$	0	0
$487$	12.7899	0.579564	0.289782	−	0.957093i	$-0.406417\pi$
$487$	12.7899	0.579564	0.289782	+	0.957093i	$0.406417\pi$
$488$	0	0
$489$	8.09349	0.366000
$490$	0	0
$491$	24.9775	1.12722	0.563610	−	0.826041i	$-0.309412\pi$
$491$	24.9775	1.12722	0.563610	+	0.826041i	$0.309412\pi$
$492$	0	0
$493$	−20.8914	−0.940901
$494$	0	0
$495$	4.74168	0.213123
$496$	0	0
$497$	14.8715	0.667080
$498$	0	0
$499$	−2.92974	−0.131153	−0.0655766	−	0.997848i	$-0.520889\pi$
$499$	−2.92974	−0.131153	−0.0655766	+	0.997848i	$0.520889\pi$
$500$	0	0
$501$	−17.6476	−0.788435
$502$	0	0
$503$	15.3054	0.682433	0.341217	−	0.939985i	$-0.389161\pi$
$503$	15.3054	0.682433	0.341217	+	0.939985i	$0.389161\pi$
$504$	0	0
$505$	−0.749295	−0.0333432
$506$	0	0
$507$	0.766011	0.0340197
$508$	0	0
$509$	4.14117	0.183554	0.0917772	−	0.995780i	$-0.470745\pi$
$509$	4.14117	0.183554	0.0917772	+	0.995780i	$0.470745\pi$
$510$	0	0
$511$	−20.9937	−0.928708
$512$	0	0
$513$	38.8136	1.71366
$514$	0	0
$515$	10.0717	0.443813
$516$	0	0
$517$	−8.83132	−0.388401
$518$	0	0
$519$	−6.43018	−0.282254
$520$	0	0
$521$	25.3987	1.11274	0.556368	−	0.830936i	$-0.312195\pi$
$521$	25.3987	1.11274	0.556368	+	0.830936i	$0.312195\pi$
$522$	0	0
$523$	23.2162	1.01518	0.507588	−	0.861600i	$-0.330537\pi$
$523$	23.2162	1.01518	0.507588	+	0.861600i	$0.330537\pi$
$524$	0	0
$525$	10.7918	0.470993
$526$	0	0
$527$	−4.27246	−0.186111
$528$	0	0
$529$	20.7747	0.903246
$530$	0	0
$531$	−1.36444	−0.0592114
$532$	0	0
$533$	−10.9513	−0.474356
$534$	0	0
$535$	0.573390	0.0247898
$536$	0	0
$537$	−2.44718	−0.105603
$538$	0	0
$539$	−4.18089	−0.180084
$540$	0	0
$541$	3.82144	0.164296	0.0821482	−	0.996620i	$-0.473822\pi$
$541$	3.82144	0.164296	0.0821482	+	0.996620i	$0.473822\pi$
$542$	0	0
$543$	−8.12039	−0.348479
$544$	0	0
$545$	−16.5473	−0.708810
$546$	0	0
$547$	41.2930	1.76556	0.882781	−	0.469784i	$-0.155668\pi$
$547$	41.2930	1.76556	0.882781	+	0.469784i	$0.155668\pi$
$548$	0	0
$549$	−12.2144	−0.521300
$550$	0	0
$551$	−52.6442	−2.24272
$552$	0	0
$553$	−25.3848	−1.07947
$554$	0	0
$555$	2.95326	0.125359
$556$	0	0
$557$	−24.5868	−1.04177	−0.520887	−	0.853626i	$-0.674399\pi$
$557$	−24.5868	−1.04177	−0.520887	+	0.853626i	$0.674399\pi$
$558$	0	0
$559$	9.10018	0.384896
$560$	0	0
$561$	−12.4994	−0.527723
$562$	0	0
$563$	−36.8816	−1.55437	−0.777187	−	0.629270i	$-0.783354\pi$
$563$	−36.8816	−1.55437	−0.777187	+	0.629270i	$0.783354\pi$
$564$	0	0
$565$	−15.5088	−0.652460
$566$	0	0
$567$	−10.9668	−0.460561
$568$	0	0
$569$	−28.8053	−1.20758	−0.603790	−	0.797144i	$-0.706343\pi$
$569$	−28.8053	−1.20758	−0.603790	+	0.797144i	$0.706343\pi$
$570$	0	0
$571$	18.2124	0.762167	0.381084	−	0.924541i	$-0.375551\pi$
$571$	18.2124	0.762167	0.381084	+	0.924541i	$0.375551\pi$
$572$	0	0
$573$	−18.7459	−0.783119
$574$	0	0
$575$	−21.5281	−0.897785
$576$	0	0
$577$	31.8420	1.32560	0.662800	−	0.748796i	$-0.269368\pi$
$577$	31.8420	1.32560	0.662800	+	0.748796i	$0.269368\pi$
$578$	0	0
$579$	−10.5963	−0.440368
$580$	0	0
$581$	−26.0990	−1.08277
$582$	0	0
$583$	12.9576	0.536650
$584$	0	0
$585$	−5.04888	−0.208745
$586$	0	0
$587$	34.8832	1.43978	0.719891	−	0.694087i	$-0.244192\pi$
$587$	34.8832	1.43978	0.719891	+	0.694087i	$0.244192\pi$
$588$	0	0
$589$	−10.7662	−0.443613
$590$	0	0
$591$	2.86496	0.117849
$592$	0	0
$593$	20.4938	0.841581	0.420791	−	0.907158i	$-0.361753\pi$
$593$	20.4938	0.841581	0.420791	+	0.907158i	$0.361753\pi$
$594$	0	0
$595$	−8.62462	−0.353575
$596$	0	0
$597$	6.21185	0.254234
$598$	0	0
$599$	16.0790	0.656972	0.328486	−	0.944509i	$-0.393462\pi$
$599$	16.0790	0.656972	0.328486	+	0.944509i	$0.393462\pi$
$600$	0	0
$601$	−33.1585	−1.35256	−0.676282	−	0.736643i	$-0.736410\pi$
$601$	−33.1585	−1.35256	−0.676282	+	0.736643i	$0.736410\pi$
$602$	0	0
$603$	−9.76800	−0.397784
$604$	0	0
$605$	0.0288485	0.00117286
$606$	0	0
$607$	−20.9718	−0.851219	−0.425610	−	0.904907i	$-0.639940\pi$
$607$	−20.9718	−0.851219	−0.425610	+	0.904907i	$0.639940\pi$
$608$	0	0
$609$	25.4305	1.03050
$610$	0	0
$611$	9.40348	0.380424
$612$	0	0
$613$	31.6074	1.27661	0.638305	−	0.769784i	$-0.279636\pi$
$613$	31.6074	1.27661	0.638305	+	0.769784i	$0.279636\pi$
$614$	0	0
$615$	5.67930	0.229012
$616$	0	0
$617$	−13.9619	−0.562086	−0.281043	−	0.959695i	$-0.590680\pi$
$617$	−13.9619	−0.562086	−0.281043	+	0.959695i	$0.590680\pi$
$618$	0	0
$619$	21.0375	0.845567	0.422784	−	0.906231i	$-0.361053\pi$
$619$	21.0375	0.845567	0.422784	+	0.906231i	$0.361053\pi$
$620$	0	0
$621$	37.4026	1.50092
$622$	0	0
$623$	−31.1948	−1.24979
$624$	0	0
$625$	1.85658	0.0742632
$626$	0	0
$627$	−31.4971	−1.25787
$628$	0	0
$629$	4.39801	0.175360
$630$	0	0
$631$	−5.97501	−0.237861	−0.118931	−	0.992903i	$-0.537947\pi$
$631$	−5.97501	−0.237861	−0.118931	+	0.992903i	$0.537947\pi$
$632$	0	0
$633$	−30.6338	−1.21758
$634$	0	0
$635$	7.00302	0.277906
$636$	0	0
$637$	4.45176	0.176385
$638$	0	0
$639$	−6.72347	−0.265977
$640$	0	0
$641$	−31.0723	−1.22728	−0.613642	−	0.789585i	$-0.710296\pi$
$641$	−31.0723	−1.22728	−0.613642	+	0.789585i	$0.710296\pi$
$642$	0	0
$643$	2.63610	0.103958	0.0519789	−	0.998648i	$-0.483447\pi$
$643$	2.63610	0.103958	0.0519789	+	0.998648i	$0.483447\pi$
$644$	0	0
$645$	−4.71930	−0.185822
$646$	0	0
$647$	−16.7314	−0.657781	−0.328890	−	0.944368i	$-0.606675\pi$
$647$	−16.7314	−0.657781	−0.328890	+	0.944368i	$0.606675\pi$
$648$	0	0
$649$	4.17440	0.163859
$650$	0	0
$651$	5.20076	0.203834
$652$	0	0
$653$	27.4451	1.07401	0.537005	−	0.843579i	$-0.319556\pi$
$653$	27.4451	1.07401	0.537005	+	0.843579i	$0.319556\pi$
$654$	0	0
$655$	12.0730	0.471732
$656$	0	0
$657$	9.49133	0.370292
$658$	0	0
$659$	−19.9180	−0.775897	−0.387948	−	0.921681i	$-0.626816\pi$
$659$	−19.9180	−0.775897	−0.387948	+	0.921681i	$0.626816\pi$
$660$	0	0
$661$	−12.4042	−0.482466	−0.241233	−	0.970467i	$-0.577552\pi$
$661$	−12.4042	−0.482466	−0.241233	+	0.970467i	$0.577552\pi$
$662$	0	0
$663$	13.3092	0.516885
$664$	0	0
$665$	−21.7332	−0.842776
$666$	0	0
$667$	−50.7304	−1.96429
$668$	0	0
$669$	−28.1093	−1.08677
$670$	0	0
$671$	37.3693	1.44263
$672$	0	0
$673$	−21.3201	−0.821828	−0.410914	−	0.911674i	$-0.634790\pi$
$673$	−21.3201	−0.821828	−0.410914	+	0.911674i	$0.634790\pi$
$674$	0	0
$675$	−18.3944	−0.708001
$676$	0	0
$677$	−7.87306	−0.302586	−0.151293	−	0.988489i	$-0.548344\pi$
$677$	−7.87306	−0.302586	−0.151293	+	0.988489i	$0.548344\pi$
$678$	0	0
$679$	7.30716	0.280423
$680$	0	0
$681$	−20.7043	−0.793389
$682$	0	0
$683$	8.40307	0.321535	0.160767	−	0.986992i	$-0.448603\pi$
$683$	8.40307	0.321535	0.160767	+	0.986992i	$0.448603\pi$
$684$	0	0
$685$	16.1337	0.616435
$686$	0	0
$687$	−6.16482	−0.235203
$688$	0	0
$689$	−13.7971	−0.525628
$690$	0	0
$691$	36.4880	1.38807	0.694035	−	0.719941i	$-0.255831\pi$
$691$	36.4880	1.38807	0.694035	+	0.719941i	$0.255831\pi$
$692$	0	0
$693$	−8.59558	−0.326519
$694$	0	0
$695$	−22.8415	−0.866427
$696$	0	0
$697$	8.45766	0.320357
$698$	0	0
$699$	30.6493	1.15926
$700$	0	0
$701$	−25.7402	−0.972195	−0.486098	−	0.873905i	$-0.661580\pi$
$701$	−25.7402	−0.972195	−0.486098	+	0.873905i	$0.661580\pi$
$702$	0	0
$703$	11.0825	0.417986
$704$	0	0
$705$	−4.87659	−0.183663
$706$	0	0
$707$	1.35830	0.0510842
$708$	0	0
$709$	25.7008	0.965213	0.482606	−	0.875837i	$-0.339690\pi$
$709$	25.7008	0.965213	0.482606	+	0.875837i	$0.339690\pi$
$710$	0	0
$711$	11.4766	0.430405
$712$	0	0
$713$	−10.3748	−0.388539
$714$	0	0
$715$	15.4467	0.577674
$716$	0	0
$717$	−8.63931	−0.322641
$718$	0	0
$719$	16.8604	0.628787	0.314394	−	0.949293i	$-0.398199\pi$
$719$	16.8604	0.628787	0.314394	+	0.949293i	$0.398199\pi$
$720$	0	0
$721$	−18.2577	−0.679953
$722$	0	0
$723$	32.1156	1.19439
$724$	0	0
$725$	24.9489	0.926580
$726$	0	0
$727$	−43.7631	−1.62308	−0.811542	−	0.584295i	$-0.801371\pi$
$727$	−43.7631	−1.62308	−0.811542	+	0.584295i	$0.801371\pi$
$728$	0	0
$729$	28.4395	1.05332
$730$	0	0
$731$	−7.02801	−0.259940
$732$	0	0
$733$	13.5934	0.502082	0.251041	−	0.967976i	$-0.419227\pi$
$733$	13.5934	0.502082	0.251041	+	0.967976i	$0.419227\pi$
$734$	0	0
$735$	−2.30866	−0.0851560
$736$	0	0
$737$	29.8845	1.10081
$738$	0	0
$739$	−8.69122	−0.319712	−0.159856	−	0.987140i	$-0.551103\pi$
$739$	−8.69122	−0.319712	−0.159856	+	0.987140i	$0.551103\pi$
$740$	0	0
$741$	33.5378	1.23204
$742$	0	0
$743$	−33.8821	−1.24302	−0.621508	−	0.783408i	$-0.713480\pi$
$743$	−33.8821	−1.24302	−0.621508	+	0.783408i	$0.713480\pi$
$744$	0	0
$745$	−15.5590	−0.570037
$746$	0	0
$747$	11.7994	0.431719
$748$	0	0
$749$	−1.03943	−0.0379798
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−16.2725	−0.593004
$754$	0	0
$755$	−15.0592	−0.548059
$756$	0	0
$757$	28.7758	1.04587	0.522937	−	0.852371i	$-0.324836\pi$
$757$	28.7758	1.04587	0.522937	+	0.852371i	$0.324836\pi$
$758$	0	0
$759$	−30.3521	−1.10171
$760$	0	0
$761$	−27.6680	−1.00297	−0.501483	−	0.865168i	$-0.667212\pi$
$761$	−27.6680	−1.00297	−0.501483	+	0.865168i	$0.667212\pi$
$762$	0	0
$763$	29.9965	1.08595
$764$	0	0
$765$	3.89922	0.140977
$766$	0	0
$767$	−4.44485	−0.160494
$768$	0	0
$769$	−32.8613	−1.18501	−0.592505	−	0.805567i	$-0.701861\pi$
$769$	−32.8613	−1.18501	−0.592505	+	0.805567i	$0.701861\pi$
$770$	0	0
$771$	7.58173	0.273049
$772$	0	0
$773$	−23.7167	−0.853031	−0.426515	−	0.904480i	$-0.640259\pi$
$773$	−23.7167	−0.853031	−0.426515	+	0.904480i	$0.640259\pi$
$774$	0	0
$775$	5.10226	0.183279
$776$	0	0
$777$	−5.35358	−0.192059
$778$	0	0
$779$	21.3125	0.763598
$780$	0	0
$781$	20.5700	0.736053
$782$	0	0
$783$	−43.3458	−1.54905
$784$	0	0
$785$	−14.5085	−0.517832
$786$	0	0
$787$	42.2466	1.50593	0.752964	−	0.658062i	$-0.228623\pi$
$787$	42.2466	1.50593	0.752964	+	0.658062i	$0.228623\pi$
$788$	0	0
$789$	−1.46249	−0.0520661
$790$	0	0
$791$	28.1139	0.999616
$792$	0	0
$793$	−39.7904	−1.41300
$794$	0	0
$795$	7.15510	0.253765
$796$	0	0
$797$	−17.7775	−0.629710	−0.314855	−	0.949140i	$-0.601956\pi$
$797$	−17.7775	−0.629710	−0.314855	+	0.949140i	$0.601956\pi$
$798$	0	0
$799$	−7.26225	−0.256920
$800$	0	0
$801$	14.1033	0.498315
$802$	0	0
$803$	−29.0381	−1.02473
$804$	0	0
$805$	−20.9431	−0.738147
$806$	0	0
$807$	3.28675	0.115699
$808$	0	0
$809$	−31.0812	−1.09276	−0.546379	−	0.837538i	$-0.683994\pi$
$809$	−31.0812	−1.09276	−0.546379	+	0.837538i	$0.683994\pi$
$810$	0	0
$811$	−27.3377	−0.959956	−0.479978	−	0.877281i	$-0.659355\pi$
$811$	−27.3377	−0.959956	−0.479978	+	0.877281i	$0.659355\pi$
$812$	0	0
$813$	−34.1519	−1.19776
$814$	0	0
$815$	7.72443	0.270575
$816$	0	0
$817$	−17.7099	−0.619590
$818$	0	0
$819$	9.15247	0.319813
$820$	0	0
$821$	−15.7341	−0.549125	−0.274563	−	0.961569i	$-0.588533\pi$
$821$	−15.7341	−0.549125	−0.274563	+	0.961569i	$0.588533\pi$
$822$	0	0
$823$	−10.0741	−0.351161	−0.175581	−	0.984465i	$-0.556180\pi$
$823$	−10.0741	−0.351161	−0.175581	+	0.984465i	$0.556180\pi$
$824$	0	0
$825$	14.9270	0.519691
$826$	0	0
$827$	−33.8221	−1.17611	−0.588055	−	0.808821i	$-0.700106\pi$
$827$	−33.8221	−1.17611	−0.588055	+	0.808821i	$0.700106\pi$
$828$	0	0
$829$	10.3217	0.358487	0.179243	−	0.983805i	$-0.442635\pi$
$829$	10.3217	0.358487	0.179243	+	0.983805i	$0.442635\pi$
$830$	0	0
$831$	0.941570	0.0326627
$832$	0	0
$833$	−3.43807	−0.119122
$834$	0	0
$835$	−16.8428	−0.582871
$836$	0	0
$837$	−8.86458	−0.306405
$838$	0	0
$839$	−22.2506	−0.768175	−0.384088	−	0.923297i	$-0.625484\pi$
$839$	−22.2506	−0.768175	−0.384088	+	0.923297i	$0.625484\pi$
$840$	0	0
$841$	29.7914	1.02729
$842$	0	0
$843$	−11.6088	−0.399829
$844$	0	0
$845$	0.731081	0.0251499
$846$	0	0
$847$	−0.0522958	−0.00179690
$848$	0	0
$849$	24.7831	0.850552
$850$	0	0
$851$	10.6796	0.366094
$852$	0	0
$853$	−1.80520	−0.0618090	−0.0309045	−	0.999522i	$-0.509839\pi$
$853$	−1.80520	−0.0618090	−0.0309045	+	0.999522i	$0.509839\pi$
$854$	0	0
$855$	9.82564	0.336030
$856$	0	0
$857$	25.6244	0.875313	0.437657	−	0.899142i	$-0.355809\pi$
$857$	25.6244	0.875313	0.437657	+	0.899142i	$0.355809\pi$
$858$	0	0
$859$	25.2050	0.859982	0.429991	−	0.902833i	$-0.358517\pi$
$859$	25.2050	0.859982	0.429991	+	0.902833i	$0.358517\pi$
$860$	0	0
$861$	−10.2953	−0.350862
$862$	0	0
$863$	−30.5911	−1.04133	−0.520667	−	0.853760i	$-0.674317\pi$
$863$	−30.5911	−1.04133	−0.520667	+	0.853760i	$0.674317\pi$
$864$	0	0
$865$	−6.13697	−0.208663
$866$	0	0
$867$	13.2590	0.450298
$868$	0	0
$869$	−35.1118	−1.19109
$870$	0	0
$871$	−31.8207	−1.07820
$872$	0	0
$873$	−3.30359	−0.111810
$874$	0	0
$875$	26.1267	0.883245
$876$	0	0
$877$	2.41338	0.0814939	0.0407469	−	0.999169i	$-0.487026\pi$
$877$	2.41338	0.0814939	0.0407469	+	0.999169i	$0.487026\pi$
$878$	0	0
$879$	12.5408	0.422990
$880$	0	0
$881$	50.8491	1.71315	0.856575	−	0.516022i	$-0.172588\pi$
$881$	50.8491	1.71315	0.856575	+	0.516022i	$0.172588\pi$
$882$	0	0
$883$	−38.0832	−1.28160	−0.640801	−	0.767707i	$-0.721398\pi$
$883$	−38.0832	−1.28160	−0.640801	+	0.767707i	$0.721398\pi$
$884$	0	0
$885$	2.30507	0.0774841
$886$	0	0
$887$	16.1179	0.541188	0.270594	−	0.962694i	$-0.412780\pi$
$887$	16.1179	0.541188	0.270594	+	0.962694i	$0.412780\pi$
$888$	0	0
$889$	−12.6949	−0.425773
$890$	0	0
$891$	−15.1690	−0.508180
$892$	0	0
$893$	−18.3001	−0.612391
$894$	0	0
$895$	−2.33559	−0.0780701
$896$	0	0
$897$	32.3185	1.07908
$898$	0	0
$899$	12.0233	0.401001
$900$	0	0
$901$	10.6554	0.354984
$902$	0	0
$903$	8.55501	0.284693
$904$	0	0
$905$	−7.75010	−0.257622
$906$	0	0
$907$	38.9490	1.29328	0.646640	−	0.762796i	$-0.276174\pi$
$907$	38.9490	1.29328	0.646640	+	0.762796i	$0.276174\pi$
$908$	0	0
$909$	−0.614092	−0.0203682
$910$	0	0
$911$	−41.5336	−1.37607	−0.688034	−	0.725678i	$-0.741526\pi$
$911$	−41.5336	−1.37607	−0.688034	+	0.725678i	$0.741526\pi$
$912$	0	0
$913$	−36.0996	−1.19472
$914$	0	0
$915$	20.6350	0.682173
$916$	0	0
$917$	−21.8856	−0.722727
$918$	0	0
$919$	11.7811	0.388621	0.194311	−	0.980940i	$-0.437753\pi$
$919$	11.7811	0.388621	0.194311	+	0.980940i	$0.437753\pi$
$920$	0	0
$921$	−40.0992	−1.32131
$922$	0	0
$923$	−21.9027	−0.720936
$924$	0	0
$925$	−5.25219	−0.172691
$926$	0	0
$927$	8.25438	0.271109
$928$	0	0
$929$	−45.5816	−1.49549	−0.747743	−	0.663989i	$-0.768862\pi$
$929$	−45.5816	−1.49549	−0.747743	+	0.663989i	$0.768862\pi$
$930$	0	0
$931$	−8.66358	−0.283937
$932$	0	0
$933$	0.223260	0.00730921
$934$	0	0
$935$	−11.9294	−0.390133
$936$	0	0
$937$	17.0635	0.557441	0.278720	−	0.960372i	$-0.410090\pi$
$937$	17.0635	0.557441	0.278720	+	0.960372i	$0.410090\pi$
$938$	0	0
$939$	−36.0333	−1.17590
$940$	0	0
$941$	9.03747	0.294613	0.147306	−	0.989091i	$-0.452940\pi$
$941$	9.03747	0.294613	0.147306	+	0.989091i	$0.452940\pi$
$942$	0	0
$943$	20.5377	0.668798
$944$	0	0
$945$	−17.8945	−0.582109
$946$	0	0
$947$	−20.2044	−0.656556	−0.328278	−	0.944581i	$-0.606468\pi$
$947$	−20.2044	−0.656556	−0.328278	+	0.944581i	$0.606468\pi$
$948$	0	0
$949$	30.9194	1.00369
$950$	0	0
$951$	8.59221	0.278622
$952$	0	0
$953$	7.77078	0.251720	0.125860	−	0.992048i	$-0.459831\pi$
$953$	7.77078	0.251720	0.125860	+	0.992048i	$0.459831\pi$
$954$	0	0
$955$	−17.8911	−0.578941
$956$	0	0
$957$	35.1750	1.13705
$958$	0	0
$959$	−29.2467	−0.944424
$960$	0	0
$961$	−28.5411	−0.920682
$962$	0	0
$963$	0.469928	0.0151432
$964$	0	0
$965$	−10.1131	−0.325553
$966$	0	0
$967$	37.8630	1.21759	0.608795	−	0.793327i	$-0.291653\pi$
$967$	37.8630	1.21759	0.608795	+	0.793327i	$0.291653\pi$
$968$	0	0
$969$	−25.9010	−0.832060
$970$	0	0
$971$	−5.69990	−0.182919	−0.0914593	−	0.995809i	$-0.529153\pi$
$971$	−5.69990	−0.182919	−0.0914593	+	0.995809i	$0.529153\pi$
$972$	0	0
$973$	41.4064	1.32743
$974$	0	0
$975$	−15.8941	−0.509018
$976$	0	0
$977$	0.107457	0.00343785	0.00171893	−	0.999999i	$-0.499453\pi$
$977$	0.107457	0.00343785	0.00171893	+	0.999999i	$0.499453\pi$
$978$	0	0
$979$	−43.1480	−1.37902
$980$	0	0
$981$	−13.5615	−0.432986
$982$	0	0
$983$	0.277299	0.00884447	0.00442224	−	0.999990i	$-0.498592\pi$
$983$	0.277299	0.00884447	0.00442224	+	0.999990i	$0.498592\pi$
$984$	0	0
$985$	2.73432	0.0871226
$986$	0	0
$987$	8.84015	0.281385
$988$	0	0
$989$	−17.0661	−0.542669
$990$	0	0
$991$	−16.8235	−0.534418	−0.267209	−	0.963639i	$-0.586101\pi$
$991$	−16.8235	−0.534418	−0.267209	+	0.963639i	$0.586101\pi$
$992$	0	0
$993$	26.4702	0.840008
$994$	0	0
$995$	5.92859	0.187949
$996$	0	0
$997$	1.20514	0.0381671	0.0190835	−	0.999818i	$-0.493925\pi$
$997$	1.20514	0.0381671	0.0190835	+	0.999818i	$0.493925\pi$
$998$	0	0
$999$	9.12507	0.288704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.14	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.14	✓	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)

\(f(q)\)	\(=\)	\(q-1.38456 q^{3} -1.32143 q^{5} +2.39545 q^{7} -1.08299 q^{9} +O(q^{10})\)	\(q-1.38456 q^{3} -1.32143 q^{5} +2.39545 q^{7} -1.08299 q^{9} +3.31333 q^{11} -3.52800 q^{13} +1.82960 q^{15} +2.72465 q^{17} +6.86584 q^{19} -3.31664 q^{21} +6.61624 q^{23} -3.25383 q^{25} +5.65315 q^{27} -7.66755 q^{29} -1.56808 q^{31} -4.58751 q^{33} -3.16541 q^{35} +1.61416 q^{37} +4.88473 q^{39} +3.10413 q^{41} -2.57942 q^{43} +1.43109 q^{45} -2.66539 q^{47} -1.26184 q^{49} -3.77244 q^{51} +3.91075 q^{53} -4.37832 q^{55} -9.50618 q^{57} +1.25988 q^{59} +11.2785 q^{61} -2.59424 q^{63} +4.66199 q^{65} +9.01948 q^{67} -9.16060 q^{69} +6.20826 q^{71} -8.76402 q^{73} +4.50513 q^{75} +7.93691 q^{77} -10.5971 q^{79} -4.57817 q^{81} -10.8953 q^{83} -3.60042 q^{85} +10.6162 q^{87} -13.0225 q^{89} -8.45112 q^{91} +2.17110 q^{93} -9.07270 q^{95} +3.05044 q^{97} -3.58830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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

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