LMFDB - Embedded newform 6008.2.a.e.1.13

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			1.13
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)

$f(q)$	$=$	$q-1.42091 q^{3} -3.43075 q^{5} +0.534513 q^{7} -0.981020 q^{9} +O(q^{10})$	$q-1.42091 q^{3} -3.43075 q^{5} +0.534513 q^{7} -0.981020 q^{9} +2.02073 q^{11} -1.33304 q^{13} +4.87478 q^{15} -2.74602 q^{17} -2.29614 q^{19} -0.759494 q^{21} -7.13304 q^{23} +6.77004 q^{25} +5.65666 q^{27} -7.32409 q^{29} +1.25667 q^{31} -2.87127 q^{33} -1.83378 q^{35} -2.73951 q^{37} +1.89413 q^{39} -10.4882 q^{41} +6.16757 q^{43} +3.36563 q^{45} +5.32098 q^{47} -6.71430 q^{49} +3.90184 q^{51} +2.73392 q^{53} -6.93262 q^{55} +3.26260 q^{57} +2.37055 q^{59} -2.88421 q^{61} -0.524368 q^{63} +4.57334 q^{65} -7.96331 q^{67} +10.1354 q^{69} -9.75154 q^{71} +2.03091 q^{73} -9.61961 q^{75} +1.08011 q^{77} +2.43818 q^{79} -5.09454 q^{81} -16.3530 q^{83} +9.42090 q^{85} +10.4069 q^{87} -1.73118 q^{89} -0.712528 q^{91} -1.78561 q^{93} +7.87748 q^{95} -10.8188 q^{97} -1.98238 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.42091	−0.820362	−0.410181	−	0.912004i	$-0.634534\pi$
$3$	−1.42091	−0.820362	−0.410181	+	0.912004i	$0.634534\pi$
$4$	0	0
$5$	−3.43075	−1.53428	−0.767139	−	0.641481i	$-0.778320\pi$
$5$	−3.43075	−1.53428	−0.767139	+	0.641481i	$0.778320\pi$
$6$	0	0
$7$	0.534513	0.202027	0.101013	−	0.994885i	$-0.467791\pi$
$7$	0.534513	0.202027	0.101013	+	0.994885i	$0.467791\pi$
$8$	0	0
$9$	−0.981020	−0.327007
$10$	0	0
$11$	2.02073	0.609273	0.304637	−	0.952469i	$-0.401465\pi$
$11$	2.02073	0.609273	0.304637	+	0.952469i	$0.401465\pi$
$12$	0	0
$13$	−1.33304	−0.369720	−0.184860	−	0.982765i	$-0.559183\pi$
$13$	−1.33304	−0.369720	−0.184860	+	0.982765i	$0.559183\pi$
$14$	0	0
$15$	4.87478	1.25866
$16$	0	0
$17$	−2.74602	−0.666007	−0.333004	−	0.942926i	$-0.608062\pi$
$17$	−2.74602	−0.666007	−0.333004	+	0.942926i	$0.608062\pi$
$18$	0	0
$19$	−2.29614	−0.526771	−0.263385	−	0.964691i	$-0.584839\pi$
$19$	−2.29614	−0.526771	−0.263385	+	0.964691i	$0.584839\pi$
$20$	0	0
$21$	−0.759494	−0.165735
$22$	0	0
$23$	−7.13304	−1.48734	−0.743671	−	0.668546i	$-0.766917\pi$
$23$	−7.13304	−1.48734	−0.743671	+	0.668546i	$0.766917\pi$
$24$	0	0
$25$	6.77004	1.35401
$26$	0	0
$27$	5.65666	1.08863
$28$	0	0
$29$	−7.32409	−1.36005	−0.680025	−	0.733189i	$-0.738031\pi$
$29$	−7.32409	−1.36005	−0.680025	+	0.733189i	$0.738031\pi$
$30$	0	0
$31$	1.25667	0.225705	0.112852	−	0.993612i	$-0.464001\pi$
$31$	1.25667	0.225705	0.112852	+	0.993612i	$0.464001\pi$
$32$	0	0
$33$	−2.87127	−0.499824
$34$	0	0
$35$	−1.83378	−0.309965
$36$	0	0
$37$	−2.73951	−0.450373	−0.225186	−	0.974316i	$-0.572299\pi$
$37$	−2.73951	−0.450373	−0.225186	+	0.974316i	$0.572299\pi$
$38$	0	0
$39$	1.89413	0.303304
$40$	0	0
$41$	−10.4882	−1.63798	−0.818990	−	0.573808i	$-0.805465\pi$
$41$	−10.4882	−1.63798	−0.818990	+	0.573808i	$0.805465\pi$
$42$	0	0
$43$	6.16757	0.940545	0.470273	−	0.882521i	$-0.344156\pi$
$43$	6.16757	0.940545	0.470273	+	0.882521i	$0.344156\pi$
$44$	0	0
$45$	3.36563	0.501719
$46$	0	0
$47$	5.32098	0.776145	0.388072	−	0.921629i	$-0.373141\pi$
$47$	5.32098	0.776145	0.388072	+	0.921629i	$0.373141\pi$
$48$	0	0
$49$	−6.71430	−0.959185
$50$	0	0
$51$	3.90184	0.546367
$52$	0	0
$53$	2.73392	0.375533	0.187766	−	0.982214i	$-0.439875\pi$
$53$	2.73392	0.375533	0.187766	+	0.982214i	$0.439875\pi$
$54$	0	0
$55$	−6.93262	−0.934794
$56$	0	0
$57$	3.26260	0.432143
$58$	0	0
$59$	2.37055	0.308620	0.154310	−	0.988022i	$-0.450685\pi$
$59$	2.37055	0.308620	0.154310	+	0.988022i	$0.450685\pi$
$60$	0	0
$61$	−2.88421	−0.369285	−0.184642	−	0.982806i	$-0.559113\pi$
$61$	−2.88421	−0.369285	−0.184642	+	0.982806i	$0.559113\pi$
$62$	0	0
$63$	−0.524368	−0.0660641
$64$	0	0
$65$	4.57334	0.567252
$66$	0	0
$67$	−7.96331	−0.972873	−0.486437	−	0.873716i	$-0.661704\pi$
$67$	−7.96331	−0.972873	−0.486437	+	0.873716i	$0.661704\pi$
$68$	0	0
$69$	10.1354	1.22016
$70$	0	0
$71$	−9.75154	−1.15729	−0.578647	−	0.815578i	$-0.696419\pi$
$71$	−9.75154	−1.15729	−0.578647	+	0.815578i	$0.696419\pi$
$72$	0	0
$73$	2.03091	0.237701	0.118850	−	0.992912i	$-0.462079\pi$
$73$	2.03091	0.237701	0.118850	+	0.992912i	$0.462079\pi$
$74$	0	0
$75$	−9.61961	−1.11078
$76$	0	0
$77$	1.08011	0.123090
$78$	0	0
$79$	2.43818	0.274317	0.137158	−	0.990549i	$-0.456203\pi$
$79$	2.43818	0.274317	0.137158	+	0.990549i	$0.456203\pi$
$80$	0	0
$81$	−5.09454	−0.566060
$82$	0	0
$83$	−16.3530	−1.79497	−0.897485	−	0.441044i	$-0.854608\pi$
$83$	−16.3530	−1.79497	−0.897485	+	0.441044i	$0.854608\pi$
$84$	0	0
$85$	9.42090	1.02184
$86$	0	0
$87$	10.4069	1.11573
$88$	0	0
$89$	−1.73118	−0.183505	−0.0917524	−	0.995782i	$-0.529247\pi$
$89$	−1.73118	−0.183505	−0.0917524	+	0.995782i	$0.529247\pi$
$90$	0	0
$91$	−0.712528	−0.0746933
$92$	0	0
$93$	−1.78561	−0.185159
$94$	0	0
$95$	7.87748	0.808213
$96$	0	0
$97$	−10.8188	−1.09848	−0.549242	−	0.835664i	$-0.685083\pi$
$97$	−10.8188	−1.09848	−0.549242	+	0.835664i	$0.685083\pi$
$98$	0	0
$99$	−1.98238	−0.199236
$100$	0	0
$101$	18.6266	1.85342	0.926709	−	0.375779i	$-0.122625\pi$
$101$	18.6266	1.85342	0.926709	+	0.375779i	$0.122625\pi$
$102$	0	0
$103$	−11.2212	−1.10566	−0.552830	−	0.833294i	$-0.686452\pi$
$103$	−11.2212	−1.10566	−0.552830	+	0.833294i	$0.686452\pi$
$104$	0	0
$105$	2.60563	0.254284
$106$	0	0
$107$	−17.3155	−1.67395	−0.836976	−	0.547240i	$-0.815679\pi$
$107$	−17.3155	−1.67395	−0.836976	+	0.547240i	$0.815679\pi$
$108$	0	0
$109$	−6.23092	−0.596814	−0.298407	−	0.954439i	$-0.596455\pi$
$109$	−6.23092	−0.596814	−0.298407	+	0.954439i	$0.596455\pi$
$110$	0	0
$111$	3.89259	0.369469
$112$	0	0
$113$	−5.26670	−0.495450	−0.247725	−	0.968830i	$-0.579683\pi$
$113$	−5.26670	−0.495450	−0.247725	+	0.968830i	$0.579683\pi$
$114$	0	0
$115$	24.4717	2.28200
$116$	0	0
$117$	1.30774	0.120901
$118$	0	0
$119$	−1.46778	−0.134551
$120$	0	0
$121$	−6.91665	−0.628786
$122$	0	0
$123$	14.9028	1.34374
$124$	0	0
$125$	−6.07258	−0.543148
$126$	0	0
$127$	3.62744	0.321884	0.160942	−	0.986964i	$-0.448547\pi$
$127$	3.62744	0.321884	0.160942	+	0.986964i	$0.448547\pi$
$128$	0	0
$129$	−8.76355	−0.771587
$130$	0	0
$131$	−5.36615	−0.468842	−0.234421	−	0.972135i	$-0.575319\pi$
$131$	−5.36615	−0.468842	−0.234421	+	0.972135i	$0.575319\pi$
$132$	0	0
$133$	−1.22732	−0.106422
$134$	0	0
$135$	−19.4066	−1.67025
$136$	0	0
$137$	−12.5089	−1.06871	−0.534355	−	0.845260i	$-0.679445\pi$
$137$	−12.5089	−1.06871	−0.534355	+	0.845260i	$0.679445\pi$
$138$	0	0
$139$	11.9616	1.01457	0.507284	−	0.861779i	$-0.330650\pi$
$139$	11.9616	1.01457	0.507284	+	0.861779i	$0.330650\pi$
$140$	0	0
$141$	−7.56062	−0.636719
$142$	0	0
$143$	−2.69372	−0.225260
$144$	0	0
$145$	25.1271	2.08669
$146$	0	0
$147$	9.54040	0.786879
$148$	0	0
$149$	6.81477	0.558288	0.279144	−	0.960249i	$-0.409949\pi$
$149$	6.81477	0.558288	0.279144	+	0.960249i	$0.409949\pi$
$150$	0	0
$151$	11.3242	0.921554	0.460777	−	0.887516i	$-0.347571\pi$
$151$	11.3242	0.921554	0.460777	+	0.887516i	$0.347571\pi$
$152$	0	0
$153$	2.69390	0.217789
$154$	0	0
$155$	−4.31132	−0.346294
$156$	0	0
$157$	4.15016	0.331219	0.165609	−	0.986191i	$-0.447041\pi$
$157$	4.15016	0.331219	0.165609	+	0.986191i	$0.447041\pi$
$158$	0	0
$159$	−3.88465	−0.308073
$160$	0	0
$161$	−3.81270	−0.300483
$162$	0	0
$163$	12.0540	0.944139	0.472070	−	0.881561i	$-0.343507\pi$
$163$	12.0540	0.944139	0.472070	+	0.881561i	$0.343507\pi$
$164$	0	0
$165$	9.85062	0.766870
$166$	0	0
$167$	22.9165	1.77334	0.886668	−	0.462407i	$-0.153014\pi$
$167$	22.9165	1.77334	0.886668	+	0.462407i	$0.153014\pi$
$168$	0	0
$169$	−11.2230	−0.863307
$170$	0	0
$171$	2.25256	0.172257
$172$	0	0
$173$	−11.2546	−0.855670	−0.427835	−	0.903857i	$-0.640724\pi$
$173$	−11.2546	−0.855670	−0.427835	+	0.903857i	$0.640724\pi$
$174$	0	0
$175$	3.61868	0.273546
$176$	0	0
$177$	−3.36834	−0.253180
$178$	0	0
$179$	−6.42395	−0.480148	−0.240074	−	0.970755i	$-0.577172\pi$
$179$	−6.42395	−0.480148	−0.240074	+	0.970755i	$0.577172\pi$
$180$	0	0
$181$	18.2203	1.35430	0.677150	−	0.735845i	$-0.263215\pi$
$181$	18.2203	1.35430	0.677150	+	0.735845i	$0.263215\pi$
$182$	0	0
$183$	4.09819	0.302947
$184$	0	0
$185$	9.39858	0.690997
$186$	0	0
$187$	−5.54896	−0.405780
$188$	0	0
$189$	3.02356	0.219932
$190$	0	0
$191$	−24.1949	−1.75068	−0.875341	−	0.483506i	$-0.839363\pi$
$191$	−24.1949	−1.75068	−0.875341	+	0.483506i	$0.839363\pi$
$192$	0	0
$193$	0.721920	0.0519649	0.0259825	−	0.999662i	$-0.491729\pi$
$193$	0.721920	0.0519649	0.0259825	+	0.999662i	$0.491729\pi$
$194$	0	0
$195$	−6.49829	−0.465352
$196$	0	0
$197$	18.5422	1.32108	0.660538	−	0.750793i	$-0.270328\pi$
$197$	18.5422	1.32108	0.660538	+	0.750793i	$0.270328\pi$
$198$	0	0
$199$	16.5856	1.17572	0.587862	−	0.808961i	$-0.299970\pi$
$199$	16.5856	1.17572	0.587862	+	0.808961i	$0.299970\pi$
$200$	0	0
$201$	11.3151	0.798108
$202$	0	0
$203$	−3.91482	−0.274767
$204$	0	0
$205$	35.9823	2.51312
$206$	0	0
$207$	6.99765	0.486370
$208$	0	0
$209$	−4.63988	−0.320947
$210$	0	0
$211$	19.2846	1.32761	0.663805	−	0.747906i	$-0.268941\pi$
$211$	19.2846	1.32761	0.663805	+	0.747906i	$0.268941\pi$
$212$	0	0
$213$	13.8560	0.949400
$214$	0	0
$215$	−21.1594	−1.44306
$216$	0	0
$217$	0.671706	0.0455984
$218$	0	0
$219$	−2.88574	−0.195000
$220$	0	0
$221$	3.66056	0.246236
$222$	0	0
$223$	−6.72788	−0.450532	−0.225266	−	0.974297i	$-0.572325\pi$
$223$	−6.72788	−0.450532	−0.225266	+	0.974297i	$0.572325\pi$
$224$	0	0
$225$	−6.64155	−0.442770
$226$	0	0
$227$	−14.8744	−0.987245	−0.493623	−	0.869676i	$-0.664328\pi$
$227$	−14.8744	−0.987245	−0.493623	+	0.869676i	$0.664328\pi$
$228$	0	0
$229$	−5.46475	−0.361121	−0.180560	−	0.983564i	$-0.557791\pi$
$229$	−5.46475	−0.361121	−0.180560	+	0.983564i	$0.557791\pi$
$230$	0	0
$231$	−1.53473	−0.100978
$232$	0	0
$233$	3.81016	0.249612	0.124806	−	0.992181i	$-0.460169\pi$
$233$	3.81016	0.249612	0.124806	+	0.992181i	$0.460169\pi$
$234$	0	0
$235$	−18.2549	−1.19082
$236$	0	0
$237$	−3.46443	−0.225039
$238$	0	0
$239$	30.7876	1.99148	0.995742	−	0.0921882i	$-0.0293861\pi$
$239$	30.7876	1.99148	0.995742	+	0.0921882i	$0.0293861\pi$
$240$	0	0
$241$	17.9658	1.15728	0.578638	−	0.815585i	$-0.303584\pi$
$241$	17.9658	1.15728	0.578638	+	0.815585i	$0.303584\pi$
$242$	0	0
$243$	−9.73111	−0.624251
$244$	0	0
$245$	23.0351	1.47166
$246$	0	0
$247$	3.06085	0.194757
$248$	0	0
$249$	23.2361	1.47253
$250$	0	0
$251$	−3.04855	−0.192423	−0.0962113	−	0.995361i	$-0.530672\pi$
$251$	−3.04855	−0.192423	−0.0962113	+	0.995361i	$0.530672\pi$
$252$	0	0
$253$	−14.4140	−0.906197
$254$	0	0
$255$	−13.3862	−0.838279
$256$	0	0
$257$	17.3990	1.08532	0.542660	−	0.839953i	$-0.317417\pi$
$257$	17.3990	1.08532	0.542660	+	0.839953i	$0.317417\pi$
$258$	0	0
$259$	−1.46430	−0.0909874
$260$	0	0
$261$	7.18508	0.444745
$262$	0	0
$263$	−23.3874	−1.44213	−0.721064	−	0.692868i	$-0.756347\pi$
$263$	−23.3874	−1.44213	−0.721064	+	0.692868i	$0.756347\pi$
$264$	0	0
$265$	−9.37940	−0.576172
$266$	0	0
$267$	2.45985	0.150540
$268$	0	0
$269$	19.1181	1.16565	0.582825	−	0.812598i	$-0.301947\pi$
$269$	19.1181	1.16565	0.582825	+	0.812598i	$0.301947\pi$
$270$	0	0
$271$	−7.62607	−0.463251	−0.231625	−	0.972805i	$-0.574404\pi$
$271$	−7.62607	−0.463251	−0.231625	+	0.972805i	$0.574404\pi$
$272$	0	0
$273$	1.01244	0.0612755
$274$	0	0
$275$	13.6804	0.824961
$276$	0	0
$277$	29.6187	1.77961	0.889807	−	0.456336i	$-0.150839\pi$
$277$	29.6187	1.77961	0.889807	+	0.456336i	$0.150839\pi$
$278$	0	0
$279$	−1.23282	−0.0738069
$280$	0	0
$281$	19.0003	1.13346	0.566731	−	0.823903i	$-0.308208\pi$
$281$	19.0003	1.13346	0.566731	+	0.823903i	$0.308208\pi$
$282$	0	0
$283$	18.3732	1.09217	0.546087	−	0.837729i	$-0.316117\pi$
$283$	18.3732	1.09217	0.546087	+	0.837729i	$0.316117\pi$
$284$	0	0
$285$	−11.1932	−0.663027
$286$	0	0
$287$	−5.60607	−0.330916
$288$	0	0
$289$	−9.45938	−0.556434
$290$	0	0
$291$	15.3725	0.901154
$292$	0	0
$293$	29.4043	1.71782	0.858909	−	0.512129i	$-0.171143\pi$
$293$	29.4043	1.71782	0.858909	+	0.512129i	$0.171143\pi$
$294$	0	0
$295$	−8.13278	−0.473509
$296$	0	0
$297$	11.4306	0.663270
$298$	0	0
$299$	9.50865	0.549899
$300$	0	0
$301$	3.29664	0.190015
$302$	0	0
$303$	−26.4667	−1.52047
$304$	0	0
$305$	9.89499	0.566585
$306$	0	0
$307$	25.7169	1.46774	0.733871	−	0.679289i	$-0.237712\pi$
$307$	25.7169	1.46774	0.733871	+	0.679289i	$0.237712\pi$
$308$	0	0
$309$	15.9443	0.907041
$310$	0	0
$311$	−30.0912	−1.70631	−0.853156	−	0.521655i	$-0.825315\pi$
$311$	−30.0912	−1.70631	−0.853156	+	0.521655i	$0.825315\pi$
$312$	0	0
$313$	7.10566	0.401636	0.200818	−	0.979629i	$-0.435640\pi$
$313$	7.10566	0.401636	0.200818	+	0.979629i	$0.435640\pi$
$314$	0	0
$315$	1.79897	0.101361
$316$	0	0
$317$	8.94735	0.502533	0.251267	−	0.967918i	$-0.419153\pi$
$317$	8.94735	0.502533	0.251267	+	0.967918i	$0.419153\pi$
$318$	0	0
$319$	−14.8000	−0.828642
$320$	0	0
$321$	24.6037	1.37325
$322$	0	0
$323$	6.30524	0.350833
$324$	0	0
$325$	−9.02476	−0.500603
$326$	0	0
$327$	8.85357	0.489604
$328$	0	0
$329$	2.84413	0.156802
$330$	0	0
$331$	−4.50527	−0.247632	−0.123816	−	0.992305i	$-0.539513\pi$
$331$	−4.50527	−0.247632	−0.123816	+	0.992305i	$0.539513\pi$
$332$	0	0
$333$	2.68751	0.147275
$334$	0	0
$335$	27.3201	1.49266
$336$	0	0
$337$	30.6487	1.66954	0.834772	−	0.550596i	$-0.185600\pi$
$337$	30.6487	1.66954	0.834772	+	0.550596i	$0.185600\pi$
$338$	0	0
$339$	7.48350	0.406448
$340$	0	0
$341$	2.53939	0.137516
$342$	0	0
$343$	−7.33047	−0.395808
$344$	0	0
$345$	−34.7720	−1.87206
$346$	0	0
$347$	14.6588	0.786923	0.393462	−	0.919341i	$-0.371277\pi$
$347$	14.6588	0.786923	0.393462	+	0.919341i	$0.371277\pi$
$348$	0	0
$349$	−27.0146	−1.44606	−0.723030	−	0.690817i	$-0.757251\pi$
$349$	−27.0146	−1.44606	−0.723030	+	0.690817i	$0.757251\pi$
$350$	0	0
$351$	−7.54057	−0.402486
$352$	0	0
$353$	8.73538	0.464938	0.232469	−	0.972604i	$-0.425320\pi$
$353$	8.73538	0.464938	0.232469	+	0.972604i	$0.425320\pi$
$354$	0	0
$355$	33.4551	1.77561
$356$	0	0
$357$	2.08558	0.110381
$358$	0	0
$359$	1.12674	0.0594669	0.0297334	−	0.999558i	$-0.490534\pi$
$359$	1.12674	0.0594669	0.0297334	+	0.999558i	$0.490534\pi$
$360$	0	0
$361$	−13.7277	−0.722513
$362$	0	0
$363$	9.82792	0.515832
$364$	0	0
$365$	−6.96756	−0.364699
$366$	0	0
$367$	−30.7255	−1.60386	−0.801928	−	0.597421i	$-0.796192\pi$
$367$	−30.7255	−1.60386	−0.801928	+	0.597421i	$0.796192\pi$
$368$	0	0
$369$	10.2891	0.535630
$370$	0	0
$371$	1.46132	0.0758677
$372$	0	0
$373$	4.67812	0.242224	0.121112	−	0.992639i	$-0.461354\pi$
$373$	4.67812	0.242224	0.121112	+	0.992639i	$0.461354\pi$
$374$	0	0
$375$	8.62858	0.445578
$376$	0	0
$377$	9.76332	0.502837
$378$	0	0
$379$	−9.98838	−0.513069	−0.256534	−	0.966535i	$-0.582581\pi$
$379$	−9.98838	−0.513069	−0.256534	+	0.966535i	$0.582581\pi$
$380$	0	0
$381$	−5.15426	−0.264061
$382$	0	0
$383$	−11.5330	−0.589310	−0.294655	−	0.955604i	$-0.595205\pi$
$383$	−11.5330	−0.589310	−0.294655	+	0.955604i	$0.595205\pi$
$384$	0	0
$385$	−3.70558	−0.188854
$386$	0	0
$387$	−6.05050	−0.307564
$388$	0	0
$389$	16.2244	0.822609	0.411304	−	0.911498i	$-0.365073\pi$
$389$	16.2244	0.822609	0.411304	+	0.911498i	$0.365073\pi$
$390$	0	0
$391$	19.5875	0.990581
$392$	0	0
$393$	7.62480	0.384620
$394$	0	0
$395$	−8.36479	−0.420878
$396$	0	0
$397$	28.9202	1.45147	0.725733	−	0.687977i	$-0.241501\pi$
$397$	28.9202	1.45147	0.725733	+	0.687977i	$0.241501\pi$
$398$	0	0
$399$	1.74390	0.0873044
$400$	0	0
$401$	−8.49304	−0.424122	−0.212061	−	0.977256i	$-0.568018\pi$
$401$	−8.49304	−0.424122	−0.212061	+	0.977256i	$0.568018\pi$
$402$	0	0
$403$	−1.67519	−0.0834474
$404$	0	0
$405$	17.4781	0.868494
$406$	0	0
$407$	−5.53581	−0.274400
$408$	0	0
$409$	24.0666	1.19002	0.595008	−	0.803720i	$-0.297149\pi$
$409$	24.0666	1.19002	0.595008	+	0.803720i	$0.297149\pi$
$410$	0	0
$411$	17.7740	0.876729
$412$	0	0
$413$	1.26709	0.0623495
$414$	0	0
$415$	56.1029	2.75398
$416$	0	0
$417$	−16.9963	−0.832313
$418$	0	0
$419$	14.4107	0.704009	0.352004	−	0.935998i	$-0.385500\pi$
$419$	14.4107	0.704009	0.352004	+	0.935998i	$0.385500\pi$
$420$	0	0
$421$	−25.3454	−1.23526	−0.617630	−	0.786468i	$-0.711907\pi$
$421$	−25.3454	−1.23526	−0.617630	+	0.786468i	$0.711907\pi$
$422$	0	0
$423$	−5.21998	−0.253804
$424$	0	0
$425$	−18.5907	−0.901780
$426$	0	0
$427$	−1.54165	−0.0746055
$428$	0	0
$429$	3.82753	0.184795
$430$	0	0
$431$	−24.8824	−1.19854	−0.599272	−	0.800546i	$-0.704543\pi$
$431$	−24.8824	−1.19854	−0.599272	+	0.800546i	$0.704543\pi$
$432$	0	0
$433$	−29.5652	−1.42081	−0.710406	−	0.703792i	$-0.751489\pi$
$433$	−29.5652	−1.42081	−0.710406	+	0.703792i	$0.751489\pi$
$434$	0	0
$435$	−35.7033	−1.71184
$436$	0	0
$437$	16.3785	0.783488
$438$	0	0
$439$	12.8822	0.614836	0.307418	−	0.951575i	$-0.400535\pi$
$439$	12.8822	0.614836	0.307418	+	0.951575i	$0.400535\pi$
$440$	0	0
$441$	6.58686	0.313660
$442$	0	0
$443$	7.52766	0.357650	0.178825	−	0.983881i	$-0.442770\pi$
$443$	7.52766	0.357650	0.178825	+	0.983881i	$0.442770\pi$
$444$	0	0
$445$	5.93925	0.281547
$446$	0	0
$447$	−9.68316	−0.457998
$448$	0	0
$449$	−25.3085	−1.19438	−0.597190	−	0.802100i	$-0.703716\pi$
$449$	−25.3085	−1.19438	−0.597190	+	0.802100i	$0.703716\pi$
$450$	0	0
$451$	−21.1938	−0.997977
$452$	0	0
$453$	−16.0907	−0.756007
$454$	0	0
$455$	2.44451	0.114600
$456$	0	0
$457$	15.3782	0.719363	0.359682	−	0.933075i	$-0.382885\pi$
$457$	15.3782	0.719363	0.359682	+	0.933075i	$0.382885\pi$
$458$	0	0
$459$	−15.5333	−0.725033
$460$	0	0
$461$	−17.4478	−0.812624	−0.406312	−	0.913734i	$-0.633185\pi$
$461$	−17.4478	−0.812624	−0.406312	+	0.913734i	$0.633185\pi$
$462$	0	0
$463$	−15.6244	−0.726130	−0.363065	−	0.931764i	$-0.618270\pi$
$463$	−15.6244	−0.726130	−0.363065	+	0.931764i	$0.618270\pi$
$464$	0	0
$465$	6.12599	0.284086
$466$	0	0
$467$	−9.57103	−0.442895	−0.221447	−	0.975172i	$-0.571078\pi$
$467$	−9.57103	−0.442895	−0.221447	+	0.975172i	$0.571078\pi$
$468$	0	0
$469$	−4.25649	−0.196547
$470$	0	0
$471$	−5.89699	−0.271719
$472$	0	0
$473$	12.4630	0.573049
$474$	0	0
$475$	−15.5450	−0.713252
$476$	0	0
$477$	−2.68203	−0.122802
$478$	0	0
$479$	28.0103	1.27982	0.639912	−	0.768448i	$-0.278971\pi$
$479$	28.0103	1.27982	0.639912	+	0.768448i	$0.278971\pi$
$480$	0	0
$481$	3.65189	0.166512
$482$	0	0
$483$	5.41750	0.246505
$484$	0	0
$485$	37.1166	1.68538
$486$	0	0
$487$	−2.40682	−0.109063	−0.0545317	−	0.998512i	$-0.517367\pi$
$487$	−2.40682	−0.109063	−0.0545317	+	0.998512i	$0.517367\pi$
$488$	0	0
$489$	−17.1276	−0.774536
$490$	0	0
$491$	−19.7804	−0.892677	−0.446339	−	0.894864i	$-0.647272\pi$
$491$	−19.7804	−0.892677	−0.446339	+	0.894864i	$0.647272\pi$
$492$	0	0
$493$	20.1121	0.905803
$494$	0	0
$495$	6.80104	0.305684
$496$	0	0
$497$	−5.21232	−0.233805
$498$	0	0
$499$	10.4300	0.466911	0.233455	−	0.972368i	$-0.424997\pi$
$499$	10.4300	0.466911	0.233455	+	0.972368i	$0.424997\pi$
$500$	0	0
$501$	−32.5623	−1.45478
$502$	0	0
$503$	−4.51291	−0.201221	−0.100610	−	0.994926i	$-0.532080\pi$
$503$	−4.51291	−0.201221	−0.100610	+	0.994926i	$0.532080\pi$
$504$	0	0
$505$	−63.9033	−2.84366
$506$	0	0
$507$	15.9469	0.708224
$508$	0	0
$509$	−5.16980	−0.229147	−0.114574	−	0.993415i	$-0.536550\pi$
$509$	−5.16980	−0.229147	−0.114574	+	0.993415i	$0.536550\pi$
$510$	0	0
$511$	1.08555	0.0480219
$512$	0	0
$513$	−12.9885	−0.573456
$514$	0	0
$515$	38.4972	1.69639
$516$	0	0
$517$	10.7523	0.472884
$518$	0	0
$519$	15.9917	0.701959
$520$	0	0
$521$	1.32701	0.0581373	0.0290687	−	0.999577i	$-0.490746\pi$
$521$	1.32701	0.0581373	0.0290687	+	0.999577i	$0.490746\pi$
$522$	0	0
$523$	−37.7140	−1.64912	−0.824560	−	0.565774i	$-0.808577\pi$
$523$	−37.7140	−1.64912	−0.824560	+	0.565774i	$0.808577\pi$
$524$	0	0
$525$	−5.14181	−0.224407
$526$	0	0
$527$	−3.45084	−0.150321
$528$	0	0
$529$	27.8803	1.21219
$530$	0	0
$531$	−2.32556	−0.100921
$532$	0	0
$533$	13.9812	0.605593
$534$	0	0
$535$	59.4051	2.56831
$536$	0	0
$537$	9.12784	0.393895
$538$	0	0
$539$	−13.5678	−0.584406
$540$	0	0
$541$	17.5239	0.753410	0.376705	−	0.926333i	$-0.377057\pi$
$541$	17.5239	0.753410	0.376705	+	0.926333i	$0.377057\pi$
$542$	0	0
$543$	−25.8893	−1.11102
$544$	0	0
$545$	21.3767	0.915679
$546$	0	0
$547$	−43.4808	−1.85911	−0.929553	−	0.368689i	$-0.879807\pi$
$547$	−43.4808	−1.85911	−0.929553	+	0.368689i	$0.879807\pi$
$548$	0	0
$549$	2.82946	0.120759
$550$	0	0
$551$	16.8171	0.716434
$552$	0	0
$553$	1.30324	0.0554194
$554$	0	0
$555$	−13.3545	−0.566868
$556$	0	0
$557$	−16.8982	−0.716000	−0.358000	−	0.933722i	$-0.616541\pi$
$557$	−16.8982	−0.716000	−0.358000	+	0.933722i	$0.616541\pi$
$558$	0	0
$559$	−8.22163	−0.347738
$560$	0	0
$561$	7.88457	0.332887
$562$	0	0
$563$	10.5645	0.445241	0.222621	−	0.974905i	$-0.428539\pi$
$563$	10.5645	0.445241	0.222621	+	0.974905i	$0.428539\pi$
$564$	0	0
$565$	18.0687	0.760158
$566$	0	0
$567$	−2.72310	−0.114359
$568$	0	0
$569$	−38.9609	−1.63333	−0.816663	−	0.577115i	$-0.804179\pi$
$569$	−38.9609	−1.63333	−0.816663	+	0.577115i	$0.804179\pi$
$570$	0	0
$571$	2.89631	0.121207	0.0606034	−	0.998162i	$-0.480698\pi$
$571$	2.89631	0.121207	0.0606034	+	0.998162i	$0.480698\pi$
$572$	0	0
$573$	34.3788	1.43619
$574$	0	0
$575$	−48.2910	−2.01387
$576$	0	0
$577$	10.7421	0.447198	0.223599	−	0.974681i	$-0.428219\pi$
$577$	10.7421	0.447198	0.223599	+	0.974681i	$0.428219\pi$
$578$	0	0
$579$	−1.02578	−0.0426300
$580$	0	0
$581$	−8.74087	−0.362632
$582$	0	0
$583$	5.52452	0.228802
$584$	0	0
$585$	−4.48653	−0.185495
$586$	0	0
$587$	9.34105	0.385546	0.192773	−	0.981243i	$-0.438252\pi$
$587$	9.34105	0.385546	0.192773	+	0.981243i	$0.438252\pi$
$588$	0	0
$589$	−2.88549	−0.118895
$590$	0	0
$591$	−26.3467	−1.08376
$592$	0	0
$593$	11.7326	0.481802	0.240901	−	0.970550i	$-0.422557\pi$
$593$	11.7326	0.481802	0.240901	+	0.970550i	$0.422557\pi$
$594$	0	0
$595$	5.03559	0.206439
$596$	0	0
$597$	−23.5667	−0.964520
$598$	0	0
$599$	5.93707	0.242582	0.121291	−	0.992617i	$-0.461297\pi$
$599$	5.93707	0.242582	0.121291	+	0.992617i	$0.461297\pi$
$600$	0	0
$601$	29.3099	1.19558	0.597788	−	0.801654i	$-0.296046\pi$
$601$	29.3099	1.19558	0.597788	+	0.801654i	$0.296046\pi$
$602$	0	0
$603$	7.81216	0.318136
$604$	0	0
$605$	23.7293	0.964733
$606$	0	0
$607$	36.4339	1.47881	0.739403	−	0.673263i	$-0.235108\pi$
$607$	36.4339	1.47881	0.739403	+	0.673263i	$0.235108\pi$
$608$	0	0
$609$	5.56260	0.225408
$610$	0	0
$611$	−7.09309	−0.286956
$612$	0	0
$613$	−17.1098	−0.691059	−0.345529	−	0.938408i	$-0.612301\pi$
$613$	−17.1098	−0.691059	−0.345529	+	0.938408i	$0.612301\pi$
$614$	0	0
$615$	−51.1276	−2.06166
$616$	0	0
$617$	−23.4037	−0.942197	−0.471098	−	0.882081i	$-0.656142\pi$
$617$	−23.4037	−0.942197	−0.471098	+	0.882081i	$0.656142\pi$
$618$	0	0
$619$	−46.2329	−1.85826	−0.929129	−	0.369755i	$-0.879442\pi$
$619$	−46.2329	−1.85826	−0.929129	+	0.369755i	$0.879442\pi$
$620$	0	0
$621$	−40.3492	−1.61916
$622$	0	0
$623$	−0.925339	−0.0370729
$624$	0	0
$625$	−13.0167	−0.520669
$626$	0	0
$627$	6.59285	0.263293
$628$	0	0
$629$	7.52275	0.299952
$630$	0	0
$631$	−31.1563	−1.24031	−0.620157	−	0.784478i	$-0.712931\pi$
$631$	−31.1563	−1.24031	−0.620157	+	0.784478i	$0.712931\pi$
$632$	0	0
$633$	−27.4017	−1.08912
$634$	0	0
$635$	−12.4449	−0.493859
$636$	0	0
$637$	8.95044	0.354629
$638$	0	0
$639$	9.56645	0.378443
$640$	0	0
$641$	22.7425	0.898276	0.449138	−	0.893463i	$-0.351731\pi$
$641$	22.7425	0.898276	0.449138	+	0.893463i	$0.351731\pi$
$642$	0	0
$643$	20.9350	0.825596	0.412798	−	0.910823i	$-0.364551\pi$
$643$	20.9350	0.825596	0.412798	+	0.910823i	$0.364551\pi$
$644$	0	0
$645$	30.0655	1.18383
$646$	0	0
$647$	27.4638	1.07971	0.539856	−	0.841757i	$-0.318479\pi$
$647$	27.4638	1.07971	0.539856	+	0.841757i	$0.318479\pi$
$648$	0	0
$649$	4.79025	0.188034
$650$	0	0
$651$	−0.954433	−0.0374072
$652$	0	0
$653$	21.7438	0.850900	0.425450	−	0.904982i	$-0.360116\pi$
$653$	21.7438	0.850900	0.425450	+	0.904982i	$0.360116\pi$
$654$	0	0
$655$	18.4099	0.719334
$656$	0	0
$657$	−1.99237	−0.0777296
$658$	0	0
$659$	−18.5840	−0.723931	−0.361966	−	0.932191i	$-0.617894\pi$
$659$	−18.5840	−0.723931	−0.361966	+	0.932191i	$0.617894\pi$
$660$	0	0
$661$	27.0551	1.05232	0.526161	−	0.850385i	$-0.323631\pi$
$661$	27.0551	1.05232	0.526161	+	0.850385i	$0.323631\pi$
$662$	0	0
$663$	−5.20132	−0.202003
$664$	0	0
$665$	4.21062	0.163281
$666$	0	0
$667$	52.2430	2.02286
$668$	0	0
$669$	9.55971	0.369600
$670$	0	0
$671$	−5.82820	−0.224995
$672$	0	0
$673$	−40.4382	−1.55878	−0.779389	−	0.626541i	$-0.784470\pi$
$673$	−40.4382	−1.55878	−0.779389	+	0.626541i	$0.784470\pi$
$674$	0	0
$675$	38.2959	1.47401
$676$	0	0
$677$	39.4535	1.51632	0.758161	−	0.652068i	$-0.226098\pi$
$677$	39.4535	1.51632	0.758161	+	0.652068i	$0.226098\pi$
$678$	0	0
$679$	−5.78279	−0.221923
$680$	0	0
$681$	21.1351	0.809898
$682$	0	0
$683$	3.44550	0.131838	0.0659192	−	0.997825i	$-0.479002\pi$
$683$	3.44550	0.131838	0.0659192	+	0.997825i	$0.479002\pi$
$684$	0	0
$685$	42.9150	1.63970
$686$	0	0
$687$	7.76491	0.296250
$688$	0	0
$689$	−3.64443	−0.138842
$690$	0	0
$691$	−19.3908	−0.737659	−0.368830	−	0.929497i	$-0.620241\pi$
$691$	−19.3908	−0.737659	−0.368830	+	0.929497i	$0.620241\pi$
$692$	0	0
$693$	−1.05961	−0.0402511
$694$	0	0
$695$	−41.0372	−1.55663
$696$	0	0
$697$	28.8008	1.09091
$698$	0	0
$699$	−5.41388	−0.204772
$700$	0	0
$701$	−18.6975	−0.706194	−0.353097	−	0.935587i	$-0.614871\pi$
$701$	−18.6975	−0.706194	−0.353097	+	0.935587i	$0.614871\pi$
$702$	0	0
$703$	6.29030	0.237243
$704$	0	0
$705$	25.9386	0.976904
$706$	0	0
$707$	9.95617	0.374440
$708$	0	0
$709$	1.27090	0.0477298	0.0238649	−	0.999715i	$-0.492403\pi$
$709$	1.27090	0.0477298	0.0238649	+	0.999715i	$0.492403\pi$
$710$	0	0
$711$	−2.39190	−0.0897034
$712$	0	0
$713$	−8.96388	−0.335700
$714$	0	0
$715$	9.24148	0.345612
$716$	0	0
$717$	−43.7463	−1.63374
$718$	0	0
$719$	9.16395	0.341758	0.170879	−	0.985292i	$-0.445339\pi$
$719$	9.16395	0.341758	0.170879	+	0.985292i	$0.445339\pi$
$720$	0	0
$721$	−5.99788	−0.223373
$722$	0	0
$723$	−25.5277	−0.949385
$724$	0	0
$725$	−49.5844	−1.84152
$726$	0	0
$727$	12.0913	0.448440	0.224220	−	0.974539i	$-0.428017\pi$
$727$	12.0913	0.448440	0.224220	+	0.974539i	$0.428017\pi$
$728$	0	0
$729$	29.1106	1.07817
$730$	0	0
$731$	−16.9363	−0.626410
$732$	0	0
$733$	12.0391	0.444673	0.222337	−	0.974970i	$-0.428632\pi$
$733$	12.0391	0.444673	0.222337	+	0.974970i	$0.428632\pi$
$734$	0	0
$735$	−32.7307	−1.20729
$736$	0	0
$737$	−16.0917	−0.592745
$738$	0	0
$739$	32.4300	1.19296	0.596478	−	0.802630i	$-0.296566\pi$
$739$	32.4300	1.19296	0.596478	+	0.802630i	$0.296566\pi$
$740$	0	0
$741$	−4.34919	−0.159772
$742$	0	0
$743$	21.2839	0.780830	0.390415	−	0.920639i	$-0.372332\pi$
$743$	21.2839	0.780830	0.390415	+	0.920639i	$0.372332\pi$
$744$	0	0
$745$	−23.3798	−0.856568
$746$	0	0
$747$	16.0426	0.586967
$748$	0	0
$749$	−9.25535	−0.338183
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	4.33170	0.157856
$754$	0	0
$755$	−38.8506	−1.41392
$756$	0	0
$757$	−31.7941	−1.15558	−0.577788	−	0.816187i	$-0.696084\pi$
$757$	−31.7941	−1.15558	−0.577788	+	0.816187i	$0.696084\pi$
$758$	0	0
$759$	20.4809	0.743410
$760$	0	0
$761$	−44.1947	−1.60206	−0.801029	−	0.598625i	$-0.795714\pi$
$761$	−44.1947	−1.60206	−0.801029	+	0.598625i	$0.795714\pi$
$762$	0	0
$763$	−3.33051	−0.120573
$764$	0	0
$765$	−9.24209	−0.334148
$766$	0	0
$767$	−3.16005	−0.114103
$768$	0	0
$769$	5.10314	0.184024	0.0920120	−	0.995758i	$-0.470670\pi$
$769$	5.10314	0.184024	0.0920120	+	0.995758i	$0.470670\pi$
$770$	0	0
$771$	−24.7224	−0.890354
$772$	0	0
$773$	−19.8193	−0.712850	−0.356425	−	0.934324i	$-0.616005\pi$
$773$	−19.8193	−0.712850	−0.356425	+	0.934324i	$0.616005\pi$
$774$	0	0
$775$	8.50771	0.305606
$776$	0	0
$777$	2.08064	0.0746426
$778$	0	0
$779$	24.0823	0.862840
$780$	0	0
$781$	−19.7052	−0.705108
$782$	0	0
$783$	−41.4299	−1.48058
$784$	0	0
$785$	−14.2382	−0.508181
$786$	0	0
$787$	26.7823	0.954686	0.477343	−	0.878717i	$-0.341600\pi$
$787$	26.7823	0.954686	0.477343	+	0.878717i	$0.341600\pi$
$788$	0	0
$789$	33.2313	1.18307
$790$	0	0
$791$	−2.81512	−0.100094
$792$	0	0
$793$	3.84477	0.136532
$794$	0	0
$795$	13.3273	0.472669
$796$	0	0
$797$	31.3741	1.11133	0.555664	−	0.831407i	$-0.312464\pi$
$797$	31.3741	1.11133	0.555664	+	0.831407i	$0.312464\pi$
$798$	0	0
$799$	−14.6115	−0.516918
$800$	0	0
$801$	1.69832	0.0600073
$802$	0	0
$803$	4.10393	0.144825
$804$	0	0
$805$	13.0804	0.461024
$806$	0	0
$807$	−27.1650	−0.956255
$808$	0	0
$809$	−36.2157	−1.27328	−0.636638	−	0.771163i	$-0.719675\pi$
$809$	−36.2157	−1.27328	−0.636638	+	0.771163i	$0.719675\pi$
$810$	0	0
$811$	−15.2165	−0.534324	−0.267162	−	0.963652i	$-0.586086\pi$
$811$	−15.2165	−0.534324	−0.267162	+	0.963652i	$0.586086\pi$
$812$	0	0
$813$	10.8360	0.380033
$814$	0	0
$815$	−41.3541	−1.44857
$816$	0	0
$817$	−14.1616	−0.495452
$818$	0	0
$819$	0.699004	0.0244252
$820$	0	0
$821$	2.00689	0.0700409	0.0350204	−	0.999387i	$-0.488850\pi$
$821$	2.00689	0.0700409	0.0350204	+	0.999387i	$0.488850\pi$
$822$	0	0
$823$	22.3096	0.777664	0.388832	−	0.921309i	$-0.372879\pi$
$823$	22.3096	0.777664	0.388832	+	0.921309i	$0.372879\pi$
$824$	0	0
$825$	−19.4386	−0.676767
$826$	0	0
$827$	−41.4512	−1.44140	−0.720700	−	0.693247i	$-0.756180\pi$
$827$	−41.4512	−1.44140	−0.720700	+	0.693247i	$0.756180\pi$
$828$	0	0
$829$	−35.4228	−1.23028	−0.615142	−	0.788417i	$-0.710901\pi$
$829$	−35.4228	−1.23028	−0.615142	+	0.788417i	$0.710901\pi$
$830$	0	0
$831$	−42.0854	−1.45993
$832$	0	0
$833$	18.4376	0.638824
$834$	0	0
$835$	−78.6209	−2.72079
$836$	0	0
$837$	7.10856	0.245708
$838$	0	0
$839$	−45.1507	−1.55878	−0.779388	−	0.626542i	$-0.784470\pi$
$839$	−45.1507	−1.55878	−0.779388	+	0.626542i	$0.784470\pi$
$840$	0	0
$841$	24.6423	0.849735
$842$	0	0
$843$	−26.9977	−0.929850
$844$	0	0
$845$	38.5033	1.32455
$846$	0	0
$847$	−3.69704	−0.127032
$848$	0	0
$849$	−26.1066	−0.895977
$850$	0	0
$851$	19.5410	0.669858
$852$	0	0
$853$	39.6427	1.35734	0.678670	−	0.734443i	$-0.262557\pi$
$853$	39.6427	1.35734	0.678670	+	0.734443i	$0.262557\pi$
$854$	0	0
$855$	−7.72796	−0.264291
$856$	0	0
$857$	53.2397	1.81863	0.909317	−	0.416105i	$-0.136605\pi$
$857$	53.2397	1.81863	0.909317	+	0.416105i	$0.136605\pi$
$858$	0	0
$859$	−32.1333	−1.09637	−0.548186	−	0.836356i	$-0.684682\pi$
$859$	−32.1333	−1.09637	−0.548186	+	0.836356i	$0.684682\pi$
$860$	0	0
$861$	7.96571	0.271471
$862$	0	0
$863$	−27.3278	−0.930248	−0.465124	−	0.885246i	$-0.653990\pi$
$863$	−27.3278	−0.930248	−0.465124	+	0.885246i	$0.653990\pi$
$864$	0	0
$865$	38.6117	1.31284
$866$	0	0
$867$	13.4409	0.456477
$868$	0	0
$869$	4.92691	0.167134
$870$	0	0
$871$	10.6154	0.359690
$872$	0	0
$873$	10.6135	0.359211
$874$	0	0
$875$	−3.24587	−0.109730
$876$	0	0
$877$	21.6237	0.730180	0.365090	−	0.930972i	$-0.381038\pi$
$877$	21.6237	0.730180	0.365090	+	0.930972i	$0.381038\pi$
$878$	0	0
$879$	−41.7808	−1.40923
$880$	0	0
$881$	−51.1335	−1.72273	−0.861366	−	0.507985i	$-0.830391\pi$
$881$	−51.1335	−1.72273	−0.861366	+	0.507985i	$0.830391\pi$
$882$	0	0
$883$	16.9290	0.569708	0.284854	−	0.958571i	$-0.408055\pi$
$883$	16.9290	0.569708	0.284854	+	0.958571i	$0.408055\pi$
$884$	0	0
$885$	11.5559	0.388449
$886$	0	0
$887$	−43.8163	−1.47121	−0.735604	−	0.677412i	$-0.763102\pi$
$887$	−43.8163	−1.47121	−0.735604	+	0.677412i	$0.763102\pi$
$888$	0	0
$889$	1.93892	0.0650291
$890$	0	0
$891$	−10.2947	−0.344885
$892$	0	0
$893$	−12.2177	−0.408850
$894$	0	0
$895$	22.0389	0.736681
$896$	0	0
$897$	−13.5109	−0.451116
$898$	0	0
$899$	−9.20397	−0.306969
$900$	0	0
$901$	−7.50739	−0.250108
$902$	0	0
$903$	−4.68423	−0.155881
$904$	0	0
$905$	−62.5091	−2.07787
$906$	0	0
$907$	52.2717	1.73565	0.867827	−	0.496867i	$-0.165516\pi$
$907$	52.2717	1.73565	0.867827	+	0.496867i	$0.165516\pi$
$908$	0	0
$909$	−18.2731	−0.606080
$910$	0	0
$911$	−18.4449	−0.611106	−0.305553	−	0.952175i	$-0.598841\pi$
$911$	−18.4449	−0.611106	−0.305553	+	0.952175i	$0.598841\pi$
$912$	0	0
$913$	−33.0449	−1.09363
$914$	0	0
$915$	−14.0599	−0.464805
$916$	0	0
$917$	−2.86827	−0.0947188
$918$	0	0
$919$	−0.355688	−0.0117331	−0.00586653	−	0.999983i	$-0.501867\pi$
$919$	−0.355688	−0.0117331	−0.00586653	+	0.999983i	$0.501867\pi$
$920$	0	0
$921$	−36.5414	−1.20408
$922$	0	0
$923$	12.9992	0.427874
$924$	0	0
$925$	−18.5466	−0.609809
$926$	0	0
$927$	11.0082	0.361558
$928$	0	0
$929$	−8.90560	−0.292183	−0.146092	−	0.989271i	$-0.546669\pi$
$929$	−8.90560	−0.292183	−0.146092	+	0.989271i	$0.546669\pi$
$930$	0	0
$931$	15.4170	0.505271
$932$	0	0
$933$	42.7568	1.39979
$934$	0	0
$935$	19.0371	0.622580
$936$	0	0
$937$	13.8456	0.452316	0.226158	−	0.974091i	$-0.427384\pi$
$937$	13.8456	0.452316	0.226158	+	0.974091i	$0.427384\pi$
$938$	0	0
$939$	−10.0965	−0.329486
$940$	0	0
$941$	43.9045	1.43125	0.715623	−	0.698486i	$-0.246143\pi$
$941$	43.9045	1.43125	0.715623	+	0.698486i	$0.246143\pi$
$942$	0	0
$943$	74.8127	2.43624
$944$	0	0
$945$	−10.3731	−0.337436
$946$	0	0
$947$	16.4787	0.535485	0.267742	−	0.963491i	$-0.413722\pi$
$947$	16.4787	0.535485	0.267742	+	0.963491i	$0.413722\pi$
$948$	0	0
$949$	−2.70730	−0.0878825
$950$	0	0
$951$	−12.7134	−0.412259
$952$	0	0
$953$	25.5958	0.829129	0.414564	−	0.910020i	$-0.363934\pi$
$953$	25.5958	0.829129	0.414564	+	0.910020i	$0.363934\pi$
$954$	0	0
$955$	83.0067	2.68603
$956$	0	0
$957$	21.0295	0.679786
$958$	0	0
$959$	−6.68618	−0.215908
$960$	0	0
$961$	−29.4208	−0.949057
$962$	0	0
$963$	16.9868	0.547393
$964$	0	0
$965$	−2.47673	−0.0797286
$966$	0	0
$967$	13.7793	0.443111	0.221556	−	0.975148i	$-0.428887\pi$
$967$	13.7793	0.443111	0.221556	+	0.975148i	$0.428887\pi$
$968$	0	0
$969$	−8.95917	−0.287810
$970$	0	0
$971$	35.7294	1.14661	0.573306	−	0.819341i	$-0.305661\pi$
$971$	35.7294	1.14661	0.573306	+	0.819341i	$0.305661\pi$
$972$	0	0
$973$	6.39362	0.204970
$974$	0	0
$975$	12.8234	0.410676
$976$	0	0
$977$	−40.7899	−1.30498	−0.652492	−	0.757795i	$-0.726276\pi$
$977$	−40.7899	−1.30498	−0.652492	+	0.757795i	$0.726276\pi$
$978$	0	0
$979$	−3.49825	−0.111805
$980$	0	0
$981$	6.11266	0.195162
$982$	0	0
$983$	−27.7071	−0.883719	−0.441860	−	0.897084i	$-0.645681\pi$
$983$	−27.7071	−0.883719	−0.441860	+	0.897084i	$0.645681\pi$
$984$	0	0
$985$	−63.6136	−2.02690
$986$	0	0
$987$	−4.04125	−0.128634
$988$	0	0
$989$	−43.9935	−1.39891
$990$	0	0
$991$	−5.84738	−0.185748	−0.0928740	−	0.995678i	$-0.529605\pi$
$991$	−5.84738	−0.185748	−0.0928740	+	0.995678i	$0.529605\pi$
$992$	0	0
$993$	6.40157	0.203148
$994$	0	0
$995$	−56.9012	−1.80389
$996$	0	0
$997$	12.2585	0.388231	0.194115	−	0.980979i	$-0.437816\pi$
$997$	12.2585	0.388231	0.194115	+	0.980979i	$0.437816\pi$
$998$	0	0
$999$	−15.4965	−0.490287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.13	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-1.42091 q^{3} -3.43075 q^{5} +0.534513 q^{7} -0.981020 q^{9} +O(q^{10})\)	\(q-1.42091 q^{3} -3.43075 q^{5} +0.534513 q^{7} -0.981020 q^{9} +2.02073 q^{11} -1.33304 q^{13} +4.87478 q^{15} -2.74602 q^{17} -2.29614 q^{19} -0.759494 q^{21} -7.13304 q^{23} +6.77004 q^{25} +5.65666 q^{27} -7.32409 q^{29} +1.25667 q^{31} -2.87127 q^{33} -1.83378 q^{35} -2.73951 q^{37} +1.89413 q^{39} -10.4882 q^{41} +6.16757 q^{43} +3.36563 q^{45} +5.32098 q^{47} -6.71430 q^{49} +3.90184 q^{51} +2.73392 q^{53} -6.93262 q^{55} +3.26260 q^{57} +2.37055 q^{59} -2.88421 q^{61} -0.524368 q^{63} +4.57334 q^{65} -7.96331 q^{67} +10.1354 q^{69} -9.75154 q^{71} +2.03091 q^{73} -9.61961 q^{75} +1.08011 q^{77} +2.43818 q^{79} -5.09454 q^{81} -16.3530 q^{83} +9.42090 q^{85} +10.4069 q^{87} -1.73118 q^{89} -0.712528 q^{91} -1.78561 q^{93} +7.87748 q^{95} -10.8188 q^{97} -1.98238 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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