LMFDB - Embedded newform 6008.2.a.b.1.40

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			1.40
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)

$f(q)$	$=$	$q+2.35947 q^{3} -3.32176 q^{5} +1.00106 q^{7} +2.56708 q^{9} +O(q^{10})$	$q+2.35947 q^{3} -3.32176 q^{5} +1.00106 q^{7} +2.56708 q^{9} -0.281563 q^{11} +0.731357 q^{13} -7.83758 q^{15} +4.74830 q^{17} -5.22975 q^{19} +2.36197 q^{21} -6.35333 q^{23} +6.03409 q^{25} -1.02147 q^{27} +2.12534 q^{29} +1.17035 q^{31} -0.664339 q^{33} -3.32529 q^{35} +1.39991 q^{37} +1.72561 q^{39} -2.17976 q^{41} -11.2304 q^{43} -8.52721 q^{45} +2.59008 q^{47} -5.99787 q^{49} +11.2035 q^{51} +9.36544 q^{53} +0.935286 q^{55} -12.3394 q^{57} -12.9879 q^{59} +13.7202 q^{61} +2.56980 q^{63} -2.42939 q^{65} -2.15886 q^{67} -14.9905 q^{69} +4.52588 q^{71} +3.16162 q^{73} +14.2372 q^{75} -0.281862 q^{77} -12.5522 q^{79} -10.1113 q^{81} -2.06328 q^{83} -15.7727 q^{85} +5.01465 q^{87} -18.0090 q^{89} +0.732134 q^{91} +2.76139 q^{93} +17.3720 q^{95} -15.6753 q^{97} -0.722795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.35947	1.36224	0.681119	−	0.732173i	$-0.261494\pi$
$3$	2.35947	1.36224	0.681119	+	0.732173i	$0.261494\pi$
$4$	0	0
$5$	−3.32176	−1.48554	−0.742768	−	0.669549i	$-0.766487\pi$
$5$	−3.32176	−1.48554	−0.742768	+	0.669549i	$0.766487\pi$
$6$	0	0
$7$	1.00106	0.378366	0.189183	−	0.981942i	$-0.439416\pi$
$7$	1.00106	0.378366	0.189183	+	0.981942i	$0.439416\pi$
$8$	0	0
$9$	2.56708	0.855692
$10$	0	0
$11$	−0.281563	−0.0848945	−0.0424473	−	0.999099i	$-0.513515\pi$
$11$	−0.281563	−0.0848945	−0.0424473	+	0.999099i	$0.513515\pi$
$12$	0	0
$13$	0.731357	0.202842	0.101421	−	0.994844i	$-0.467661\pi$
$13$	0.731357	0.202842	0.101421	+	0.994844i	$0.467661\pi$
$14$	0	0
$15$	−7.83758	−2.02365
$16$	0	0
$17$	4.74830	1.15163	0.575816	−	0.817579i	$-0.304684\pi$
$17$	4.74830	1.15163	0.575816	+	0.817579i	$0.304684\pi$
$18$	0	0
$19$	−5.22975	−1.19979	−0.599894	−	0.800080i	$-0.704790\pi$
$19$	−5.22975	−1.19979	−0.599894	+	0.800080i	$0.704790\pi$
$20$	0	0
$21$	2.36197	0.515425
$22$	0	0
$23$	−6.35333	−1.32476	−0.662380	−	0.749168i	$-0.730454\pi$
$23$	−6.35333	−1.32476	−0.662380	+	0.749168i	$0.730454\pi$
$24$	0	0
$25$	6.03409	1.20682
$26$	0	0
$27$	−1.02147	−0.196581
$28$	0	0
$29$	2.12534	0.394665	0.197332	−	0.980337i	$-0.436772\pi$
$29$	2.12534	0.394665	0.197332	+	0.980337i	$0.436772\pi$
$30$	0	0
$31$	1.17035	0.210200	0.105100	−	0.994462i	$-0.466484\pi$
$31$	1.17035	0.210200	0.105100	+	0.994462i	$0.466484\pi$
$32$	0	0
$33$	−0.664339	−0.115647
$34$	0	0
$35$	−3.32529	−0.562076
$36$	0	0
$37$	1.39991	0.230143	0.115072	−	0.993357i	$-0.463290\pi$
$37$	1.39991	0.230143	0.115072	+	0.993357i	$0.463290\pi$
$38$	0	0
$39$	1.72561	0.276319
$40$	0	0
$41$	−2.17976	−0.340421	−0.170210	−	0.985408i	$-0.554445\pi$
$41$	−2.17976	−0.340421	−0.170210	+	0.985408i	$0.554445\pi$
$42$	0	0
$43$	−11.2304	−1.71261	−0.856307	−	0.516466i	$-0.827247\pi$
$43$	−11.2304	−1.71261	−0.856307	+	0.516466i	$0.827247\pi$
$44$	0	0
$45$	−8.52721	−1.27116
$46$	0	0
$47$	2.59008	0.377801	0.188901	−	0.981996i	$-0.439508\pi$
$47$	2.59008	0.377801	0.188901	+	0.981996i	$0.439508\pi$
$48$	0	0
$49$	−5.99787	−0.856839
$50$	0	0
$51$	11.2035	1.56880
$52$	0	0
$53$	9.36544	1.28644	0.643221	−	0.765680i	$-0.277598\pi$
$53$	9.36544	1.28644	0.643221	+	0.765680i	$0.277598\pi$
$54$	0	0
$55$	0.935286	0.126114
$56$	0	0
$57$	−12.3394	−1.63440
$58$	0	0
$59$	−12.9879	−1.69089	−0.845443	−	0.534066i	$-0.820664\pi$
$59$	−12.9879	−1.69089	−0.845443	+	0.534066i	$0.820664\pi$
$60$	0	0
$61$	13.7202	1.75669	0.878347	−	0.478023i	$-0.158646\pi$
$61$	13.7202	1.75669	0.878347	+	0.478023i	$0.158646\pi$
$62$	0	0
$63$	2.56980	0.323765
$64$	0	0
$65$	−2.42939	−0.301329
$66$	0	0
$67$	−2.15886	−0.263747	−0.131873	−	0.991267i	$-0.542099\pi$
$67$	−2.15886	−0.263747	−0.131873	+	0.991267i	$0.542099\pi$
$68$	0	0
$69$	−14.9905	−1.80464
$70$	0	0
$71$	4.52588	0.537123	0.268562	−	0.963262i	$-0.413452\pi$
$71$	4.52588	0.537123	0.268562	+	0.963262i	$0.413452\pi$
$72$	0	0
$73$	3.16162	0.370040	0.185020	−	0.982735i	$-0.440765\pi$
$73$	3.16162	0.370040	0.185020	+	0.982735i	$0.440765\pi$
$74$	0	0
$75$	14.2372	1.64397
$76$	0	0
$77$	−0.281862	−0.0321212
$78$	0	0
$79$	−12.5522	−1.41223	−0.706115	−	0.708097i	$-0.749554\pi$
$79$	−12.5522	−1.41223	−0.706115	+	0.708097i	$0.749554\pi$
$80$	0	0
$81$	−10.1113	−1.12348
$82$	0	0
$83$	−2.06328	−0.226474	−0.113237	−	0.993568i	$-0.536122\pi$
$83$	−2.06328	−0.226474	−0.113237	+	0.993568i	$0.536122\pi$
$84$	0	0
$85$	−15.7727	−1.71079
$86$	0	0
$87$	5.01465	0.537627
$88$	0	0
$89$	−18.0090	−1.90895	−0.954475	−	0.298291i	$-0.903583\pi$
$89$	−18.0090	−1.90895	−0.954475	+	0.298291i	$0.903583\pi$
$90$	0	0
$91$	0.732134	0.0767485
$92$	0	0
$93$	2.76139	0.286343
$94$	0	0
$95$	17.3720	1.78233
$96$	0	0
$97$	−15.6753	−1.59158	−0.795791	−	0.605571i	$-0.792945\pi$
$97$	−15.6753	−1.59158	−0.795791	+	0.605571i	$0.792945\pi$
$98$	0	0
$99$	−0.722795	−0.0726436
$100$	0	0
$101$	7.86739	0.782835	0.391417	−	0.920213i	$-0.371985\pi$
$101$	7.86739	0.782835	0.391417	+	0.920213i	$0.371985\pi$
$102$	0	0
$103$	−11.2716	−1.11063	−0.555314	−	0.831641i	$-0.687402\pi$
$103$	−11.2716	−1.11063	−0.555314	+	0.831641i	$0.687402\pi$
$104$	0	0
$105$	−7.84590	−0.765682
$106$	0	0
$107$	6.77306	0.654777	0.327388	−	0.944890i	$-0.393832\pi$
$107$	6.77306	0.654777	0.327388	+	0.944890i	$0.393832\pi$
$108$	0	0
$109$	−3.15680	−0.302367	−0.151183	−	0.988506i	$-0.548308\pi$
$109$	−3.15680	−0.302367	−0.151183	+	0.988506i	$0.548308\pi$
$110$	0	0
$111$	3.30303	0.313510
$112$	0	0
$113$	12.1854	1.14631	0.573154	−	0.819448i	$-0.305720\pi$
$113$	12.1854	1.14631	0.573154	+	0.819448i	$0.305720\pi$
$114$	0	0
$115$	21.1042	1.96798
$116$	0	0
$117$	1.87745	0.173570
$118$	0	0
$119$	4.75335	0.435739
$120$	0	0
$121$	−10.9207	−0.992793
$122$	0	0
$123$	−5.14306	−0.463734
$124$	0	0
$125$	−3.43499	−0.307235
$126$	0	0
$127$	−0.769542	−0.0682858	−0.0341429	−	0.999417i	$-0.510870\pi$
$127$	−0.769542	−0.0682858	−0.0341429	+	0.999417i	$0.510870\pi$
$128$	0	0
$129$	−26.4977	−2.33299
$130$	0	0
$131$	−20.2493	−1.76919	−0.884595	−	0.466360i	$-0.845565\pi$
$131$	−20.2493	−1.76919	−0.884595	+	0.466360i	$0.845565\pi$
$132$	0	0
$133$	−5.23531	−0.453959
$134$	0	0
$135$	3.39306	0.292028
$136$	0	0
$137$	0.378221	0.0323136	0.0161568	−	0.999869i	$-0.494857\pi$
$137$	0.378221	0.0323136	0.0161568	+	0.999869i	$0.494857\pi$
$138$	0	0
$139$	4.74189	0.402202	0.201101	−	0.979570i	$-0.435548\pi$
$139$	4.74189	0.402202	0.201101	+	0.979570i	$0.435548\pi$
$140$	0	0
$141$	6.11120	0.514656
$142$	0	0
$143$	−0.205923	−0.0172202
$144$	0	0
$145$	−7.05985	−0.586289
$146$	0	0
$147$	−14.1518	−1.16722
$148$	0	0
$149$	−6.70197	−0.549047	−0.274523	−	0.961580i	$-0.588520\pi$
$149$	−6.70197	−0.549047	−0.274523	+	0.961580i	$0.588520\pi$
$150$	0	0
$151$	−3.28695	−0.267488	−0.133744	−	0.991016i	$-0.542700\pi$
$151$	−3.28695	−0.267488	−0.133744	+	0.991016i	$0.542700\pi$
$152$	0	0
$153$	12.1893	0.985443
$154$	0	0
$155$	−3.88761	−0.312260
$156$	0	0
$157$	−12.1914	−0.972981	−0.486491	−	0.873686i	$-0.661723\pi$
$157$	−12.1914	−0.972981	−0.486491	+	0.873686i	$0.661723\pi$
$158$	0	0
$159$	22.0974	1.75244
$160$	0	0
$161$	−6.36008	−0.501244
$162$	0	0
$163$	23.1424	1.81265	0.906325	−	0.422582i	$-0.138876\pi$
$163$	23.1424	1.81265	0.906325	+	0.422582i	$0.138876\pi$
$164$	0	0
$165$	2.20677	0.171797
$166$	0	0
$167$	−15.8096	−1.22339	−0.611693	−	0.791096i	$-0.709511\pi$
$167$	−15.8096	−1.22339	−0.611693	+	0.791096i	$0.709511\pi$
$168$	0	0
$169$	−12.4651	−0.958855
$170$	0	0
$171$	−13.4252	−1.02665
$172$	0	0
$173$	11.5671	0.879432	0.439716	−	0.898137i	$-0.355079\pi$
$173$	11.5671	0.879432	0.439716	+	0.898137i	$0.355079\pi$
$174$	0	0
$175$	6.04050	0.456619
$176$	0	0
$177$	−30.6446	−2.30339
$178$	0	0
$179$	−4.49731	−0.336145	−0.168072	−	0.985775i	$-0.553754\pi$
$179$	−4.49731	−0.336145	−0.168072	+	0.985775i	$0.553754\pi$
$180$	0	0
$181$	5.94503	0.441891	0.220945	−	0.975286i	$-0.429086\pi$
$181$	5.94503	0.441891	0.220945	+	0.975286i	$0.429086\pi$
$182$	0	0
$183$	32.3724	2.39304
$184$	0	0
$185$	−4.65015	−0.341886
$186$	0	0
$187$	−1.33695	−0.0977673
$188$	0	0
$189$	−1.02255	−0.0743796
$190$	0	0
$191$	12.9688	0.938393	0.469196	−	0.883094i	$-0.344544\pi$
$191$	12.9688	0.938393	0.469196	+	0.883094i	$0.344544\pi$
$192$	0	0
$193$	−25.0382	−1.80229	−0.901146	−	0.433516i	$-0.857273\pi$
$193$	−25.0382	−1.80229	−0.901146	+	0.433516i	$0.857273\pi$
$194$	0	0
$195$	−5.73207	−0.410482
$196$	0	0
$197$	−2.83492	−0.201980	−0.100990	−	0.994887i	$-0.532201\pi$
$197$	−2.83492	−0.201980	−0.100990	+	0.994887i	$0.532201\pi$
$198$	0	0
$199$	1.83273	0.129918	0.0649592	−	0.997888i	$-0.479308\pi$
$199$	1.83273	0.129918	0.0649592	+	0.997888i	$0.479308\pi$
$200$	0	0
$201$	−5.09376	−0.359286
$202$	0	0
$203$	2.12759	0.149328
$204$	0	0
$205$	7.24063	0.505707
$206$	0	0
$207$	−16.3095	−1.13359
$208$	0	0
$209$	1.47251	0.101855
$210$	0	0
$211$	−2.22122	−0.152915	−0.0764576	−	0.997073i	$-0.524361\pi$
$211$	−2.22122	−0.152915	−0.0764576	+	0.997073i	$0.524361\pi$
$212$	0	0
$213$	10.6787	0.731690
$214$	0	0
$215$	37.3046	2.54415
$216$	0	0
$217$	1.17159	0.0795327
$218$	0	0
$219$	7.45974	0.504082
$220$	0	0
$221$	3.47271	0.233599
$222$	0	0
$223$	−23.9230	−1.60200	−0.801000	−	0.598665i	$-0.795698\pi$
$223$	−23.9230	−1.60200	−0.801000	+	0.598665i	$0.795698\pi$
$224$	0	0
$225$	15.4900	1.03266
$226$	0	0
$227$	16.5119	1.09594	0.547968	−	0.836499i	$-0.315401\pi$
$227$	16.5119	1.09594	0.547968	+	0.836499i	$0.315401\pi$
$228$	0	0
$229$	8.88608	0.587209	0.293604	−	0.955927i	$-0.405145\pi$
$229$	8.88608	0.587209	0.293604	+	0.955927i	$0.405145\pi$
$230$	0	0
$231$	−0.665045	−0.0437567
$232$	0	0
$233$	10.6385	0.696950	0.348475	−	0.937318i	$-0.386700\pi$
$233$	10.6385	0.696950	0.348475	+	0.937318i	$0.386700\pi$
$234$	0	0
$235$	−8.60361	−0.561238
$236$	0	0
$237$	−29.6164	−1.92379
$238$	0	0
$239$	8.24149	0.533098	0.266549	−	0.963821i	$-0.414117\pi$
$239$	8.24149	0.533098	0.266549	+	0.963821i	$0.414117\pi$
$240$	0	0
$241$	18.2794	1.17748	0.588738	−	0.808324i	$-0.299625\pi$
$241$	18.2794	1.17748	0.588738	+	0.808324i	$0.299625\pi$
$242$	0	0
$243$	−20.7930	−1.33387
$244$	0	0
$245$	19.9235	1.27287
$246$	0	0
$247$	−3.82481	−0.243367
$248$	0	0
$249$	−4.86823	−0.308512
$250$	0	0
$251$	−10.3594	−0.653880	−0.326940	−	0.945045i	$-0.606018\pi$
$251$	−10.3594	−0.653880	−0.326940	+	0.945045i	$0.606018\pi$
$252$	0	0
$253$	1.78886	0.112465
$254$	0	0
$255$	−37.2152	−2.33051
$256$	0	0
$257$	−0.578155	−0.0360643	−0.0180322	−	0.999837i	$-0.505740\pi$
$257$	−0.578155	−0.0360643	−0.0180322	+	0.999837i	$0.505740\pi$
$258$	0	0
$259$	1.40139	0.0870783
$260$	0	0
$261$	5.45590	0.337712
$262$	0	0
$263$	−10.1021	−0.622922	−0.311461	−	0.950259i	$-0.600818\pi$
$263$	−10.1021	−0.622922	−0.311461	+	0.950259i	$0.600818\pi$
$264$	0	0
$265$	−31.1097	−1.91106
$266$	0	0
$267$	−42.4916	−2.60044
$268$	0	0
$269$	8.98048	0.547550	0.273775	−	0.961794i	$-0.411728\pi$
$269$	8.98048	0.547550	0.273775	+	0.961794i	$0.411728\pi$
$270$	0	0
$271$	−23.3306	−1.41723	−0.708615	−	0.705595i	$-0.750680\pi$
$271$	−23.3306	−1.41723	−0.708615	+	0.705595i	$0.750680\pi$
$272$	0	0
$273$	1.72744	0.104550
$274$	0	0
$275$	−1.69898	−0.102452
$276$	0	0
$277$	−9.90856	−0.595348	−0.297674	−	0.954668i	$-0.596211\pi$
$277$	−9.90856	−0.595348	−0.297674	+	0.954668i	$0.596211\pi$
$278$	0	0
$279$	3.00437	0.179867
$280$	0	0
$281$	−11.4329	−0.682029	−0.341015	−	0.940058i	$-0.610771\pi$
$281$	−11.4329	−0.682029	−0.341015	+	0.940058i	$0.610771\pi$
$282$	0	0
$283$	0.871740	0.0518196	0.0259098	−	0.999664i	$-0.491752\pi$
$283$	0.871740	0.0518196	0.0259098	+	0.999664i	$0.491752\pi$
$284$	0	0
$285$	40.9886	2.42795
$286$	0	0
$287$	−2.18207	−0.128804
$288$	0	0
$289$	5.54639	0.326258
$290$	0	0
$291$	−36.9853	−2.16811
$292$	0	0
$293$	−6.39060	−0.373343	−0.186671	−	0.982422i	$-0.559770\pi$
$293$	−6.39060	−0.373343	−0.186671	+	0.982422i	$0.559770\pi$
$294$	0	0
$295$	43.1428	2.51187
$296$	0	0
$297$	0.287607	0.0166887
$298$	0	0
$299$	−4.64655	−0.268717
$300$	0	0
$301$	−11.2423	−0.647995
$302$	0	0
$303$	18.5628	1.06641
$304$	0	0
$305$	−45.5753	−2.60963
$306$	0	0
$307$	−31.7537	−1.81228	−0.906141	−	0.422976i	$-0.860986\pi$
$307$	−31.7537	−1.81228	−0.906141	+	0.422976i	$0.860986\pi$
$308$	0	0
$309$	−26.5951	−1.51294
$310$	0	0
$311$	−18.9859	−1.07659	−0.538296	−	0.842756i	$-0.680932\pi$
$311$	−18.9859	−1.07659	−0.538296	+	0.842756i	$0.680932\pi$
$312$	0	0
$313$	−15.9397	−0.900967	−0.450483	−	0.892785i	$-0.648748\pi$
$313$	−15.9397	−0.900967	−0.450483	+	0.892785i	$0.648748\pi$
$314$	0	0
$315$	−8.53627	−0.480964
$316$	0	0
$317$	26.8663	1.50896	0.754481	−	0.656322i	$-0.227889\pi$
$317$	26.8663	1.50896	0.754481	+	0.656322i	$0.227889\pi$
$318$	0	0
$319$	−0.598416	−0.0335049
$320$	0	0
$321$	15.9808	0.891961
$322$	0	0
$323$	−24.8324	−1.38171
$324$	0	0
$325$	4.41307	0.244793
$326$	0	0
$327$	−7.44837	−0.411896
$328$	0	0
$329$	2.59283	0.142947
$330$	0	0
$331$	23.8734	1.31220	0.656100	−	0.754674i	$-0.272205\pi$
$331$	23.8734	1.31220	0.656100	+	0.754674i	$0.272205\pi$
$332$	0	0
$333$	3.59367	0.196932
$334$	0	0
$335$	7.17121	0.391805
$336$	0	0
$337$	28.5800	1.55685	0.778426	−	0.627736i	$-0.216018\pi$
$337$	28.5800	1.55685	0.778426	+	0.627736i	$0.216018\pi$
$338$	0	0
$339$	28.7511	1.56155
$340$	0	0
$341$	−0.329527	−0.0178449
$342$	0	0
$343$	−13.0117	−0.702565
$344$	0	0
$345$	49.7947	2.68086
$346$	0	0
$347$	−14.9783	−0.804075	−0.402037	−	0.915623i	$-0.631698\pi$
$347$	−14.9783	−0.804075	−0.402037	+	0.915623i	$0.631698\pi$
$348$	0	0
$349$	31.8116	1.70284	0.851419	−	0.524487i	$-0.175743\pi$
$349$	31.8116	1.70284	0.851419	+	0.524487i	$0.175743\pi$
$350$	0	0
$351$	−0.747056	−0.0398749
$352$	0	0
$353$	−29.5078	−1.57054	−0.785271	−	0.619153i	$-0.787476\pi$
$353$	−29.5078	−1.57054	−0.785271	+	0.619153i	$0.787476\pi$
$354$	0	0
$355$	−15.0339	−0.797916
$356$	0	0
$357$	11.2154	0.593580
$358$	0	0
$359$	−37.5327	−1.98090	−0.990449	−	0.137878i	$-0.955972\pi$
$359$	−37.5327	−1.98090	−0.990449	+	0.137878i	$0.955972\pi$
$360$	0	0
$361$	8.35029	0.439489
$362$	0	0
$363$	−25.7671	−1.35242
$364$	0	0
$365$	−10.5021	−0.549707
$366$	0	0
$367$	27.8906	1.45588	0.727938	−	0.685643i	$-0.240479\pi$
$367$	27.8906	1.45588	0.727938	+	0.685643i	$0.240479\pi$
$368$	0	0
$369$	−5.59560	−0.291295
$370$	0	0
$371$	9.37539	0.486746
$372$	0	0
$373$	−17.4996	−0.906096	−0.453048	−	0.891486i	$-0.649663\pi$
$373$	−17.4996	−0.906096	−0.453048	+	0.891486i	$0.649663\pi$
$374$	0	0
$375$	−8.10473	−0.418527
$376$	0	0
$377$	1.55438	0.0800546
$378$	0	0
$379$	−27.5811	−1.41675	−0.708373	−	0.705839i	$-0.750570\pi$
$379$	−27.5811	−1.41675	−0.708373	+	0.705839i	$0.750570\pi$
$380$	0	0
$381$	−1.81571	−0.0930215
$382$	0	0
$383$	2.11485	0.108064	0.0540320	−	0.998539i	$-0.482793\pi$
$383$	2.11485	0.108064	0.0540320	+	0.998539i	$0.482793\pi$
$384$	0	0
$385$	0.936279	0.0477172
$386$	0	0
$387$	−28.8292	−1.46547
$388$	0	0
$389$	−15.6467	−0.793317	−0.396659	−	0.917966i	$-0.629830\pi$
$389$	−15.6467	−0.793317	−0.396659	+	0.917966i	$0.629830\pi$
$390$	0	0
$391$	−30.1675	−1.52564
$392$	0	0
$393$	−47.7775	−2.41006
$394$	0	0
$395$	41.6953	2.09792
$396$	0	0
$397$	13.4630	0.675688	0.337844	−	0.941202i	$-0.390302\pi$
$397$	13.4630	0.675688	0.337844	+	0.941202i	$0.390302\pi$
$398$	0	0
$399$	−12.3525	−0.618400
$400$	0	0
$401$	34.8957	1.74261	0.871304	−	0.490743i	$-0.163275\pi$
$401$	34.8957	1.74261	0.871304	+	0.490743i	$0.163275\pi$
$402$	0	0
$403$	0.855941	0.0426375
$404$	0	0
$405$	33.5875	1.66897
$406$	0	0
$407$	−0.394162	−0.0195379
$408$	0	0
$409$	−21.7972	−1.07780	−0.538901	−	0.842369i	$-0.681160\pi$
$409$	−21.7972	−1.07780	−0.538901	+	0.842369i	$0.681160\pi$
$410$	0	0
$411$	0.892399	0.0440188
$412$	0	0
$413$	−13.0017	−0.639774
$414$	0	0
$415$	6.85372	0.336436
$416$	0	0
$417$	11.1883	0.547895
$418$	0	0
$419$	24.5024	1.19702	0.598511	−	0.801115i	$-0.295759\pi$
$419$	24.5024	1.19702	0.598511	+	0.801115i	$0.295759\pi$
$420$	0	0
$421$	−35.7316	−1.74145	−0.870725	−	0.491770i	$-0.836350\pi$
$421$	−35.7316	−1.74145	−0.870725	+	0.491770i	$0.836350\pi$
$422$	0	0
$423$	6.64893	0.323282
$424$	0	0
$425$	28.6517	1.38981
$426$	0	0
$427$	13.7348	0.664674
$428$	0	0
$429$	−0.485869	−0.0234580
$430$	0	0
$431$	0.467028	0.0224960	0.0112480	−	0.999937i	$-0.496420\pi$
$431$	0.467028	0.0224960	0.0112480	+	0.999937i	$0.496420\pi$
$432$	0	0
$433$	4.96811	0.238752	0.119376	−	0.992849i	$-0.461911\pi$
$433$	4.96811	0.238752	0.119376	+	0.992849i	$0.461911\pi$
$434$	0	0
$435$	−16.6575	−0.798665
$436$	0	0
$437$	33.2263	1.58943
$438$	0	0
$439$	12.2159	0.583033	0.291516	−	0.956566i	$-0.405840\pi$
$439$	12.2159	0.583033	0.291516	+	0.956566i	$0.405840\pi$
$440$	0	0
$441$	−15.3970	−0.733191
$442$	0	0
$443$	33.0371	1.56964	0.784820	−	0.619724i	$-0.212755\pi$
$443$	33.0371	1.56964	0.784820	+	0.619724i	$0.212755\pi$
$444$	0	0
$445$	59.8216	2.83581
$446$	0	0
$447$	−15.8131	−0.747933
$448$	0	0
$449$	27.8097	1.31242	0.656210	−	0.754578i	$-0.272158\pi$
$449$	27.8097	1.31242	0.656210	+	0.754578i	$0.272158\pi$
$450$	0	0
$451$	0.613739	0.0288999
$452$	0	0
$453$	−7.75545	−0.364383
$454$	0	0
$455$	−2.43197	−0.114013
$456$	0	0
$457$	−13.0900	−0.612325	−0.306163	−	0.951979i	$-0.599045\pi$
$457$	−13.0900	−0.612325	−0.306163	+	0.951979i	$0.599045\pi$
$458$	0	0
$459$	−4.85023	−0.226389
$460$	0	0
$461$	−35.6113	−1.65858	−0.829291	−	0.558817i	$-0.811256\pi$
$461$	−35.6113	−1.65858	−0.829291	+	0.558817i	$0.811256\pi$
$462$	0	0
$463$	−15.7844	−0.733562	−0.366781	−	0.930307i	$-0.619540\pi$
$463$	−15.7844	−0.733562	−0.366781	+	0.930307i	$0.619540\pi$
$464$	0	0
$465$	−9.17268	−0.425373
$466$	0	0
$467$	32.0685	1.48395	0.741976	−	0.670426i	$-0.233889\pi$
$467$	32.0685	1.48395	0.741976	+	0.670426i	$0.233889\pi$
$468$	0	0
$469$	−2.16115	−0.0997928
$470$	0	0
$471$	−28.7652	−1.32543
$472$	0	0
$473$	3.16206	0.145392
$474$	0	0
$475$	−31.5568	−1.44792
$476$	0	0
$477$	24.0418	1.10080
$478$	0	0
$479$	−7.20104	−0.329024	−0.164512	−	0.986375i	$-0.552605\pi$
$479$	−7.20104	−0.329024	−0.164512	+	0.986375i	$0.552605\pi$
$480$	0	0
$481$	1.02383	0.0466827
$482$	0	0
$483$	−15.0064	−0.682814
$484$	0	0
$485$	52.0695	2.36435
$486$	0	0
$487$	−11.7204	−0.531100	−0.265550	−	0.964097i	$-0.585554\pi$
$487$	−11.7204	−0.531100	−0.265550	+	0.964097i	$0.585554\pi$
$488$	0	0
$489$	54.6036	2.46926
$490$	0	0
$491$	16.3757	0.739024	0.369512	−	0.929226i	$-0.379525\pi$
$491$	16.3757	0.739024	0.369512	+	0.929226i	$0.379525\pi$
$492$	0	0
$493$	10.0917	0.454509
$494$	0	0
$495$	2.40095	0.107915
$496$	0	0
$497$	4.53069	0.203229
$498$	0	0
$499$	31.2214	1.39766	0.698830	−	0.715287i	$-0.253704\pi$
$499$	31.2214	1.39766	0.698830	+	0.715287i	$0.253704\pi$
$500$	0	0
$501$	−37.3023	−1.66654
$502$	0	0
$503$	28.5996	1.27519	0.637596	−	0.770371i	$-0.279929\pi$
$503$	28.5996	1.27519	0.637596	+	0.770371i	$0.279929\pi$
$504$	0	0
$505$	−26.1336	−1.16293
$506$	0	0
$507$	−29.4110	−1.30619
$508$	0	0
$509$	19.4787	0.863378	0.431689	−	0.902022i	$-0.357918\pi$
$509$	19.4787	0.863378	0.431689	+	0.902022i	$0.357918\pi$
$510$	0	0
$511$	3.16498	0.140010
$512$	0	0
$513$	5.34201	0.235856
$514$	0	0
$515$	37.4417	1.64988
$516$	0	0
$517$	−0.729270	−0.0320733
$518$	0	0
$519$	27.2922	1.19800
$520$	0	0
$521$	32.3164	1.41581	0.707904	−	0.706309i	$-0.249641\pi$
$521$	32.3164	1.41581	0.707904	+	0.706309i	$0.249641\pi$
$522$	0	0
$523$	−9.37424	−0.409907	−0.204954	−	0.978772i	$-0.565704\pi$
$523$	−9.37424	−0.409907	−0.204954	+	0.978772i	$0.565704\pi$
$524$	0	0
$525$	14.2523	0.622023
$526$	0	0
$527$	5.55716	0.242074
$528$	0	0
$529$	17.3648	0.754991
$530$	0	0
$531$	−33.3411	−1.44688
$532$	0	0
$533$	−1.59418	−0.0690516
$534$	0	0
$535$	−22.4985	−0.972694
$536$	0	0
$537$	−10.6113	−0.457909
$538$	0	0
$539$	1.68878	0.0727410
$540$	0	0
$541$	−9.81510	−0.421984	−0.210992	−	0.977488i	$-0.567669\pi$
$541$	−9.81510	−0.421984	−0.210992	+	0.977488i	$0.567669\pi$
$542$	0	0
$543$	14.0271	0.601960
$544$	0	0
$545$	10.4861	0.449177
$546$	0	0
$547$	5.69646	0.243563	0.121781	−	0.992557i	$-0.461139\pi$
$547$	5.69646	0.243563	0.121781	+	0.992557i	$0.461139\pi$
$548$	0	0
$549$	35.2209	1.50319
$550$	0	0
$551$	−11.1150	−0.473514
$552$	0	0
$553$	−12.5655	−0.534340
$554$	0	0
$555$	−10.9719	−0.465730
$556$	0	0
$557$	6.97777	0.295658	0.147829	−	0.989013i	$-0.452772\pi$
$557$	6.97777	0.295658	0.147829	+	0.989013i	$0.452772\pi$
$558$	0	0
$559$	−8.21341	−0.347390
$560$	0	0
$561$	−3.15448	−0.133182
$562$	0	0
$563$	11.4472	0.482441	0.241220	−	0.970470i	$-0.422452\pi$
$563$	11.4472	0.482441	0.241220	+	0.970470i	$0.422452\pi$
$564$	0	0
$565$	−40.4771	−1.70288
$566$	0	0
$567$	−10.1221	−0.425088
$568$	0	0
$569$	−16.5466	−0.693669	−0.346835	−	0.937926i	$-0.612744\pi$
$569$	−16.5466	−0.693669	−0.346835	+	0.937926i	$0.612744\pi$
$570$	0	0
$571$	2.28957	0.0958155	0.0479078	−	0.998852i	$-0.484745\pi$
$571$	2.28957	0.0958155	0.0479078	+	0.998852i	$0.484745\pi$
$572$	0	0
$573$	30.5995	1.27831
$574$	0	0
$575$	−38.3365	−1.59874
$576$	0	0
$577$	17.7527	0.739054	0.369527	−	0.929220i	$-0.379520\pi$
$577$	17.7527	0.739054	0.369527	+	0.929220i	$0.379520\pi$
$578$	0	0
$579$	−59.0769	−2.45515
$580$	0	0
$581$	−2.06547	−0.0856902
$582$	0	0
$583$	−2.63696	−0.109212
$584$	0	0
$585$	−6.23644	−0.257845
$586$	0	0
$587$	29.0713	1.19990	0.599950	−	0.800037i	$-0.295187\pi$
$587$	29.0713	1.19990	0.599950	+	0.800037i	$0.295187\pi$
$588$	0	0
$589$	−6.12062	−0.252196
$590$	0	0
$591$	−6.68889	−0.275144
$592$	0	0
$593$	8.72077	0.358119	0.179060	−	0.983838i	$-0.442695\pi$
$593$	8.72077	0.358119	0.179060	+	0.983838i	$0.442695\pi$
$594$	0	0
$595$	−15.7895	−0.647306
$596$	0	0
$597$	4.32425	0.176980
$598$	0	0
$599$	−13.9845	−0.571392	−0.285696	−	0.958320i	$-0.592225\pi$
$599$	−13.9845	−0.571392	−0.285696	+	0.958320i	$0.592225\pi$
$600$	0	0
$601$	26.5770	1.08410	0.542048	−	0.840347i	$-0.317649\pi$
$601$	26.5770	1.08410	0.542048	+	0.840347i	$0.317649\pi$
$602$	0	0
$603$	−5.54196	−0.225686
$604$	0	0
$605$	36.2760	1.47483
$606$	0	0
$607$	35.8461	1.45495	0.727474	−	0.686136i	$-0.240694\pi$
$607$	35.8461	1.45495	0.727474	+	0.686136i	$0.240694\pi$
$608$	0	0
$609$	5.01998	0.203420
$610$	0	0
$611$	1.89427	0.0766340
$612$	0	0
$613$	36.8956	1.49020	0.745099	−	0.666954i	$-0.232402\pi$
$613$	36.8956	1.49020	0.745099	+	0.666954i	$0.232402\pi$
$614$	0	0
$615$	17.0840	0.688894
$616$	0	0
$617$	−16.8681	−0.679083	−0.339542	−	0.940591i	$-0.610272\pi$
$617$	−16.8681	−0.679083	−0.339542	+	0.940591i	$0.610272\pi$
$618$	0	0
$619$	−8.20858	−0.329931	−0.164965	−	0.986299i	$-0.552751\pi$
$619$	−8.20858	−0.329931	−0.164965	+	0.986299i	$0.552751\pi$
$620$	0	0
$621$	6.48971	0.260423
$622$	0	0
$623$	−18.0281	−0.722282
$624$	0	0
$625$	−18.7602	−0.750409
$626$	0	0
$627$	3.47433	0.138751
$628$	0	0
$629$	6.64718	0.265040
$630$	0	0
$631$	22.4666	0.894383	0.447192	−	0.894438i	$-0.352424\pi$
$631$	22.4666	0.894383	0.447192	+	0.894438i	$0.352424\pi$
$632$	0	0
$633$	−5.24090	−0.208307
$634$	0	0
$635$	2.55623	0.101441
$636$	0	0
$637$	−4.38659	−0.173803
$638$	0	0
$639$	11.6183	0.459612
$640$	0	0
$641$	17.0927	0.675123	0.337561	−	0.941303i	$-0.390398\pi$
$641$	17.0927	0.675123	0.337561	+	0.941303i	$0.390398\pi$
$642$	0	0
$643$	12.4259	0.490028	0.245014	−	0.969519i	$-0.421207\pi$
$643$	12.4259	0.490028	0.245014	+	0.969519i	$0.421207\pi$
$644$	0	0
$645$	88.0189	3.46574
$646$	0	0
$647$	−12.1415	−0.477330	−0.238665	−	0.971102i	$-0.576710\pi$
$647$	−12.1415	−0.477330	−0.238665	+	0.971102i	$0.576710\pi$
$648$	0	0
$649$	3.65693	0.143547
$650$	0	0
$651$	2.76432	0.108342
$652$	0	0
$653$	42.3458	1.65712	0.828561	−	0.559900i	$-0.189160\pi$
$653$	42.3458	1.65712	0.828561	+	0.559900i	$0.189160\pi$
$654$	0	0
$655$	67.2633	2.62820
$656$	0	0
$657$	8.11613	0.316640
$658$	0	0
$659$	−30.1150	−1.17311	−0.586556	−	0.809908i	$-0.699517\pi$
$659$	−30.1150	−1.17311	−0.586556	+	0.809908i	$0.699517\pi$
$660$	0	0
$661$	−32.2119	−1.25290	−0.626449	−	0.779462i	$-0.715493\pi$
$661$	−32.2119	−1.25290	−0.626449	+	0.779462i	$0.715493\pi$
$662$	0	0
$663$	8.19373	0.318218
$664$	0	0
$665$	17.3904	0.674372
$666$	0	0
$667$	−13.5030	−0.522836
$668$	0	0
$669$	−56.4454	−2.18230
$670$	0	0
$671$	−3.86311	−0.149134
$672$	0	0
$673$	−26.9776	−1.03991	−0.519954	−	0.854194i	$-0.674051\pi$
$673$	−26.9776	−1.03991	−0.519954	+	0.854194i	$0.674051\pi$
$674$	0	0
$675$	−6.16361	−0.237238
$676$	0	0
$677$	−5.26957	−0.202526	−0.101263	−	0.994860i	$-0.532288\pi$
$677$	−5.26957	−0.202526	−0.101263	+	0.994860i	$0.532288\pi$
$678$	0	0
$679$	−15.6919	−0.602201
$680$	0	0
$681$	38.9594	1.49293
$682$	0	0
$683$	29.3817	1.12426	0.562130	−	0.827049i	$-0.309982\pi$
$683$	29.3817	1.12426	0.562130	+	0.827049i	$0.309982\pi$
$684$	0	0
$685$	−1.25636	−0.0480030
$686$	0	0
$687$	20.9664	0.799918
$688$	0	0
$689$	6.84948	0.260944
$690$	0	0
$691$	19.5268	0.742835	0.371417	−	0.928466i	$-0.378872\pi$
$691$	19.5268	0.742835	0.371417	+	0.928466i	$0.378872\pi$
$692$	0	0
$693$	−0.723563	−0.0274859
$694$	0	0
$695$	−15.7514	−0.597486
$696$	0	0
$697$	−10.3501	−0.392040
$698$	0	0
$699$	25.1011	0.949412
$700$	0	0
$701$	−1.64757	−0.0622279	−0.0311139	−	0.999516i	$-0.509905\pi$
$701$	−1.64757	−0.0622279	−0.0311139	+	0.999516i	$0.509905\pi$
$702$	0	0
$703$	−7.32116	−0.276123
$704$	0	0
$705$	−20.2999	−0.764539
$706$	0	0
$707$	7.87575	0.296198
$708$	0	0
$709$	1.40550	0.0527847	0.0263923	−	0.999652i	$-0.491598\pi$
$709$	1.40550	0.0527847	0.0263923	+	0.999652i	$0.491598\pi$
$710$	0	0
$711$	−32.2224	−1.20843
$712$	0	0
$713$	−7.43559	−0.278465
$714$	0	0
$715$	0.684028	0.0255812
$716$	0	0
$717$	19.4455	0.726206
$718$	0	0
$719$	−48.1292	−1.79492	−0.897458	−	0.441100i	$-0.854588\pi$
$719$	−48.1292	−1.79492	−0.897458	+	0.441100i	$0.854588\pi$
$720$	0	0
$721$	−11.2836	−0.420224
$722$	0	0
$723$	43.1295	1.60400
$724$	0	0
$725$	12.8245	0.476288
$726$	0	0
$727$	16.6350	0.616959	0.308480	−	0.951231i	$-0.400180\pi$
$727$	16.6350	0.616959	0.308480	+	0.951231i	$0.400180\pi$
$728$	0	0
$729$	−18.7263	−0.693565
$730$	0	0
$731$	−53.3252	−1.97230
$732$	0	0
$733$	37.3466	1.37943	0.689714	−	0.724082i	$-0.257736\pi$
$733$	37.3466	1.37943	0.689714	+	0.724082i	$0.257736\pi$
$734$	0	0
$735$	47.0088	1.73395
$736$	0	0
$737$	0.607856	0.0223907
$738$	0	0
$739$	49.4985	1.82083	0.910416	−	0.413693i	$-0.135761\pi$
$739$	49.4985	1.82083	0.910416	+	0.413693i	$0.135761\pi$
$740$	0	0
$741$	−9.02452	−0.331524
$742$	0	0
$743$	26.6742	0.978581	0.489290	−	0.872121i	$-0.337256\pi$
$743$	26.6742	0.978581	0.489290	+	0.872121i	$0.337256\pi$
$744$	0	0
$745$	22.2623	0.815629
$746$	0	0
$747$	−5.29660	−0.193792
$748$	0	0
$749$	6.78026	0.247745
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−24.4427	−0.890741
$754$	0	0
$755$	10.9185	0.397364
$756$	0	0
$757$	−0.115052	−0.00418163	−0.00209082	−	0.999998i	$-0.500666\pi$
$757$	−0.115052	−0.00418163	−0.00209082	+	0.999998i	$0.500666\pi$
$758$	0	0
$759$	4.22076	0.153204
$760$	0	0
$761$	34.5931	1.25400	0.626999	−	0.779020i	$-0.284283\pi$
$761$	34.5931	1.25400	0.626999	+	0.779020i	$0.284283\pi$
$762$	0	0
$763$	−3.16016	−0.114405
$764$	0	0
$765$	−40.4898	−1.46391
$766$	0	0
$767$	−9.49882	−0.342983
$768$	0	0
$769$	12.0756	0.435458	0.217729	−	0.976009i	$-0.430135\pi$
$769$	12.0756	0.435458	0.217729	+	0.976009i	$0.430135\pi$
$770$	0	0
$771$	−1.36414	−0.0491282
$772$	0	0
$773$	40.5322	1.45784	0.728922	−	0.684597i	$-0.240022\pi$
$773$	40.5322	1.45784	0.728922	+	0.684597i	$0.240022\pi$
$774$	0	0
$775$	7.06197	0.253673
$776$	0	0
$777$	3.30654	0.118621
$778$	0	0
$779$	11.3996	0.408432
$780$	0	0
$781$	−1.27432	−0.0455988
$782$	0	0
$783$	−2.17096	−0.0775837
$784$	0	0
$785$	40.4970	1.44540
$786$	0	0
$787$	−54.8747	−1.95607	−0.978036	−	0.208434i	$-0.933163\pi$
$787$	−54.8747	−1.95607	−0.978036	+	0.208434i	$0.933163\pi$
$788$	0	0
$789$	−23.8356	−0.848568
$790$	0	0
$791$	12.1984	0.433724
$792$	0	0
$793$	10.0344	0.356331
$794$	0	0
$795$	−73.4024	−2.60331
$796$	0	0
$797$	−26.6234	−0.943051	−0.471525	−	0.881852i	$-0.656296\pi$
$797$	−26.6234	−0.943051	−0.471525	+	0.881852i	$0.656296\pi$
$798$	0	0
$799$	12.2985	0.435089
$800$	0	0
$801$	−46.2305	−1.63347
$802$	0	0
$803$	−0.890196	−0.0314144
$804$	0	0
$805$	21.1267	0.744617
$806$	0	0
$807$	21.1891	0.745893
$808$	0	0
$809$	47.1837	1.65889	0.829445	−	0.558588i	$-0.188657\pi$
$809$	47.1837	1.65889	0.829445	+	0.558588i	$0.188657\pi$
$810$	0	0
$811$	−11.9553	−0.419807	−0.209904	−	0.977722i	$-0.567315\pi$
$811$	−11.9553	−0.419807	−0.209904	+	0.977722i	$0.567315\pi$
$812$	0	0
$813$	−55.0477	−1.93061
$814$	0	0
$815$	−76.8733	−2.69276
$816$	0	0
$817$	58.7320	2.05477
$818$	0	0
$819$	1.87944	0.0656731
$820$	0	0
$821$	42.5235	1.48408	0.742041	−	0.670355i	$-0.233858\pi$
$821$	42.5235	1.48408	0.742041	+	0.670355i	$0.233858\pi$
$822$	0	0
$823$	−32.4576	−1.13140	−0.565701	−	0.824611i	$-0.691394\pi$
$823$	−32.4576	−1.13140	−0.565701	+	0.824611i	$0.691394\pi$
$824$	0	0
$825$	−4.00868	−0.139564
$826$	0	0
$827$	43.0834	1.49816	0.749078	−	0.662482i	$-0.230497\pi$
$827$	43.0834	1.49816	0.749078	+	0.662482i	$0.230497\pi$
$828$	0	0
$829$	0.335496	0.0116523	0.00582613	−	0.999983i	$-0.498145\pi$
$829$	0.335496	0.0116523	0.00582613	+	0.999983i	$0.498145\pi$
$830$	0	0
$831$	−23.3789	−0.811006
$832$	0	0
$833$	−28.4797	−0.986764
$834$	0	0
$835$	52.5158	1.81738
$836$	0	0
$837$	−1.19547	−0.0413214
$838$	0	0
$839$	−52.7061	−1.81962	−0.909808	−	0.415029i	$-0.863771\pi$
$839$	−52.7061	−1.81962	−0.909808	+	0.415029i	$0.863771\pi$
$840$	0	0
$841$	−24.4830	−0.844240
$842$	0	0
$843$	−26.9755	−0.929086
$844$	0	0
$845$	41.4061	1.42441
$846$	0	0
$847$	−10.9323	−0.375639
$848$	0	0
$849$	2.05684	0.0705906
$850$	0	0
$851$	−8.89407	−0.304885
$852$	0	0
$853$	−20.1373	−0.689490	−0.344745	−	0.938696i	$-0.612035\pi$
$853$	−20.1373	−0.689490	−0.344745	+	0.938696i	$0.612035\pi$
$854$	0	0
$855$	44.5952	1.52512
$856$	0	0
$857$	−2.09060	−0.0714137	−0.0357068	−	0.999362i	$-0.511368\pi$
$857$	−2.09060	−0.0714137	−0.0357068	+	0.999362i	$0.511368\pi$
$858$	0	0
$859$	−32.8173	−1.11971	−0.559856	−	0.828590i	$-0.689144\pi$
$859$	−32.8173	−1.11971	−0.559856	+	0.828590i	$0.689144\pi$
$860$	0	0
$861$	−5.14852	−0.175461
$862$	0	0
$863$	9.60960	0.327114	0.163557	−	0.986534i	$-0.447703\pi$
$863$	9.60960	0.327114	0.163557	+	0.986534i	$0.447703\pi$
$864$	0	0
$865$	−38.4232	−1.30643
$866$	0	0
$867$	13.0865	0.444441
$868$	0	0
$869$	3.53423	0.119891
$870$	0	0
$871$	−1.57890	−0.0534989
$872$	0	0
$873$	−40.2396	−1.36191
$874$	0	0
$875$	−3.43864	−0.116247
$876$	0	0
$877$	−55.1799	−1.86329	−0.931647	−	0.363364i	$-0.881628\pi$
$877$	−55.1799	−1.86329	−0.931647	+	0.363364i	$0.881628\pi$
$878$	0	0
$879$	−15.0784	−0.508582
$880$	0	0
$881$	35.9126	1.20993	0.604963	−	0.796254i	$-0.293188\pi$
$881$	35.9126	1.20993	0.604963	+	0.796254i	$0.293188\pi$
$882$	0	0
$883$	1.62196	0.0545834	0.0272917	−	0.999628i	$-0.491312\pi$
$883$	1.62196	0.0545834	0.0272917	+	0.999628i	$0.491312\pi$
$884$	0	0
$885$	101.794	3.42177
$886$	0	0
$887$	41.9062	1.40707	0.703537	−	0.710659i	$-0.251603\pi$
$887$	41.9062	1.40707	0.703537	+	0.710659i	$0.251603\pi$
$888$	0	0
$889$	−0.770359	−0.0258370
$890$	0	0
$891$	2.84698	0.0953775
$892$	0	0
$893$	−13.5455	−0.453281
$894$	0	0
$895$	14.9390	0.499355
$896$	0	0
$897$	−10.9634	−0.366057
$898$	0	0
$899$	2.48738	0.0829587
$900$	0	0
$901$	44.4700	1.48151
$902$	0	0
$903$	−26.5258	−0.882724
$904$	0	0
$905$	−19.7480	−0.656444
$906$	0	0
$907$	−55.8895	−1.85578	−0.927889	−	0.372856i	$-0.878379\pi$
$907$	−55.8895	−1.85578	−0.927889	+	0.372856i	$0.878379\pi$
$908$	0	0
$909$	20.1962	0.669866
$910$	0	0
$911$	−46.6680	−1.54618	−0.773089	−	0.634297i	$-0.781290\pi$
$911$	−46.6680	−1.54618	−0.773089	+	0.634297i	$0.781290\pi$
$912$	0	0
$913$	0.580944	0.0192264
$914$	0	0
$915$	−107.533	−3.55494
$916$	0	0
$917$	−20.2708	−0.669401
$918$	0	0
$919$	−8.89643	−0.293466	−0.146733	−	0.989176i	$-0.546876\pi$
$919$	−8.89643	−0.293466	−0.146733	+	0.989176i	$0.546876\pi$
$920$	0	0
$921$	−74.9219	−2.46876
$922$	0	0
$923$	3.31004	0.108951
$924$	0	0
$925$	8.44715	0.277741
$926$	0	0
$927$	−28.9352	−0.950356
$928$	0	0
$929$	−17.2744	−0.566754	−0.283377	−	0.959009i	$-0.591455\pi$
$929$	−17.2744	−0.566754	−0.283377	+	0.959009i	$0.591455\pi$
$930$	0	0
$931$	31.3674	1.02802
$932$	0	0
$933$	−44.7966	−1.46658
$934$	0	0
$935$	4.44102	0.145237
$936$	0	0
$937$	25.8783	0.845408	0.422704	−	0.906268i	$-0.361081\pi$
$937$	25.8783	0.845408	0.422704	+	0.906268i	$0.361081\pi$
$938$	0	0
$939$	−37.6092	−1.22733
$940$	0	0
$941$	−50.4155	−1.64350	−0.821749	−	0.569850i	$-0.807001\pi$
$941$	−50.4155	−1.64350	−0.821749	+	0.569850i	$0.807001\pi$
$942$	0	0
$943$	13.8487	0.450976
$944$	0	0
$945$	3.39667	0.110494
$946$	0	0
$947$	−2.34953	−0.0763495	−0.0381747	−	0.999271i	$-0.512154\pi$
$947$	−2.34953	−0.0763495	−0.0381747	+	0.999271i	$0.512154\pi$
$948$	0	0
$949$	2.31227	0.0750596
$950$	0	0
$951$	63.3901	2.05556
$952$	0	0
$953$	−11.6309	−0.376761	−0.188380	−	0.982096i	$-0.560324\pi$
$953$	−11.6309	−0.376761	−0.188380	+	0.982096i	$0.560324\pi$
$954$	0	0
$955$	−43.0794	−1.39402
$956$	0	0
$957$	−1.41194	−0.0456416
$958$	0	0
$959$	0.378623	0.0122264
$960$	0	0
$961$	−29.6303	−0.955816
$962$	0	0
$963$	17.3870	0.560287
$964$	0	0
$965$	83.1710	2.67737
$966$	0	0
$967$	7.09441	0.228141	0.114070	−	0.993473i	$-0.463611\pi$
$967$	7.09441	0.228141	0.114070	+	0.993473i	$0.463611\pi$
$968$	0	0
$969$	−58.5913	−1.88222
$970$	0	0
$971$	16.0048	0.513617	0.256809	−	0.966462i	$-0.417329\pi$
$971$	16.0048	0.513617	0.256809	+	0.966462i	$0.417329\pi$
$972$	0	0
$973$	4.74693	0.152180
$974$	0	0
$975$	10.4125	0.333467
$976$	0	0
$977$	54.2809	1.73660	0.868300	−	0.496040i	$-0.165213\pi$
$977$	54.2809	1.73660	0.868300	+	0.496040i	$0.165213\pi$
$978$	0	0
$979$	5.07067	0.162059
$980$	0	0
$981$	−8.10376	−0.258733
$982$	0	0
$983$	43.6941	1.39363	0.696813	−	0.717253i	$-0.254601\pi$
$983$	43.6941	1.39363	0.696813	+	0.717253i	$0.254601\pi$
$984$	0	0
$985$	9.41692	0.300048
$986$	0	0
$987$	6.11769	0.194728
$988$	0	0
$989$	71.3502	2.26881
$990$	0	0
$991$	−31.2775	−0.993562	−0.496781	−	0.867876i	$-0.665485\pi$
$991$	−31.2775	−0.993562	−0.496781	+	0.867876i	$0.665485\pi$
$992$	0	0
$993$	56.3285	1.78753
$994$	0	0
$995$	−6.08787	−0.192999
$996$	0	0
$997$	23.9404	0.758200	0.379100	−	0.925356i	$-0.376234\pi$
$997$	23.9404	0.758200	0.379100	+	0.925356i	$0.376234\pi$
$998$	0	0
$999$	−1.42996	−0.0452418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.40	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.40	✓	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)

\(f(q)\)	\(=\)	\(q+2.35947 q^{3} -3.32176 q^{5} +1.00106 q^{7} +2.56708 q^{9} +O(q^{10})\)	\(q+2.35947 q^{3} -3.32176 q^{5} +1.00106 q^{7} +2.56708 q^{9} -0.281563 q^{11} +0.731357 q^{13} -7.83758 q^{15} +4.74830 q^{17} -5.22975 q^{19} +2.36197 q^{21} -6.35333 q^{23} +6.03409 q^{25} -1.02147 q^{27} +2.12534 q^{29} +1.17035 q^{31} -0.664339 q^{33} -3.32529 q^{35} +1.39991 q^{37} +1.72561 q^{39} -2.17976 q^{41} -11.2304 q^{43} -8.52721 q^{45} +2.59008 q^{47} -5.99787 q^{49} +11.2035 q^{51} +9.36544 q^{53} +0.935286 q^{55} -12.3394 q^{57} -12.9879 q^{59} +13.7202 q^{61} +2.56980 q^{63} -2.42939 q^{65} -2.15886 q^{67} -14.9905 q^{69} +4.52588 q^{71} +3.16162 q^{73} +14.2372 q^{75} -0.281862 q^{77} -12.5522 q^{79} -10.1113 q^{81} -2.06328 q^{83} -15.7727 q^{85} +5.01465 q^{87} -18.0090 q^{89} +0.732134 q^{91} +2.76139 q^{93} +17.3720 q^{95} -15.6753 q^{97} -0.722795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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

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Groups

Database

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