LMFDB - Embedded newform 6008.2.a.b.1.18

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			1.18
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)

$f(q)$	$=$	$q-1.03287 q^{3} -2.90339 q^{5} -0.294834 q^{7} -1.93318 q^{9} +O(q^{10})$	$q-1.03287 q^{3} -2.90339 q^{5} -0.294834 q^{7} -1.93318 q^{9} +3.78730 q^{11} -3.15992 q^{13} +2.99882 q^{15} -6.61105 q^{17} +1.57683 q^{19} +0.304526 q^{21} +5.53071 q^{23} +3.42966 q^{25} +5.09534 q^{27} +0.673445 q^{29} +6.77365 q^{31} -3.91179 q^{33} +0.856019 q^{35} +2.01059 q^{37} +3.26379 q^{39} +4.28870 q^{41} +0.897878 q^{43} +5.61277 q^{45} +6.08013 q^{47} -6.91307 q^{49} +6.82836 q^{51} +4.70499 q^{53} -10.9960 q^{55} -1.62866 q^{57} +6.44387 q^{59} -11.2916 q^{61} +0.569968 q^{63} +9.17448 q^{65} +0.579037 q^{67} -5.71250 q^{69} -14.0265 q^{71} +8.03585 q^{73} -3.54240 q^{75} -1.11663 q^{77} +13.2147 q^{79} +0.536712 q^{81} -9.03832 q^{83} +19.1945 q^{85} -0.695581 q^{87} -16.8649 q^{89} +0.931654 q^{91} -6.99631 q^{93} -4.57815 q^{95} +3.10244 q^{97} -7.32152 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.03287	−0.596328	−0.298164	−	0.954515i	$-0.596374\pi$
$3$	−1.03287	−0.596328	−0.298164	+	0.954515i	$0.596374\pi$
$4$	0	0
$5$	−2.90339	−1.29843	−0.649217	−	0.760603i	$-0.724903\pi$
$5$	−2.90339	−1.29843	−0.649217	+	0.760603i	$0.724903\pi$
$6$	0	0
$7$	−0.294834	−0.111437	−0.0557185	−	0.998447i	$-0.517745\pi$
$7$	−0.294834	−0.111437	−0.0557185	+	0.998447i	$0.517745\pi$
$8$	0	0
$9$	−1.93318	−0.644393
$10$	0	0
$11$	3.78730	1.14191	0.570957	−	0.820980i	$-0.306572\pi$
$11$	3.78730	1.14191	0.570957	+	0.820980i	$0.306572\pi$
$12$	0	0
$13$	−3.15992	−0.876405	−0.438202	−	0.898876i	$-0.644385\pi$
$13$	−3.15992	−0.876405	−0.438202	+	0.898876i	$0.644385\pi$
$14$	0	0
$15$	2.99882	0.774293
$16$	0	0
$17$	−6.61105	−1.60342	−0.801708	−	0.597716i	$-0.796075\pi$
$17$	−6.61105	−1.60342	−0.801708	+	0.597716i	$0.796075\pi$
$18$	0	0
$19$	1.57683	0.361750	0.180875	−	0.983506i	$-0.442107\pi$
$19$	1.57683	0.361750	0.180875	+	0.983506i	$0.442107\pi$
$20$	0	0
$21$	0.304526	0.0664530
$22$	0	0
$23$	5.53071	1.15323	0.576616	−	0.817015i	$-0.304373\pi$
$23$	5.53071	1.15323	0.576616	+	0.817015i	$0.304373\pi$
$24$	0	0
$25$	3.42966	0.685932
$26$	0	0
$27$	5.09534	0.980598
$28$	0	0
$29$	0.673445	0.125056	0.0625278	−	0.998043i	$-0.480084\pi$
$29$	0.673445	0.125056	0.0625278	+	0.998043i	$0.480084\pi$
$30$	0	0
$31$	6.77365	1.21658	0.608292	−	0.793713i	$-0.291855\pi$
$31$	6.77365	1.21658	0.608292	+	0.793713i	$0.291855\pi$
$32$	0	0
$33$	−3.91179	−0.680955
$34$	0	0
$35$	0.856019	0.144694
$36$	0	0
$37$	2.01059	0.330538	0.165269	−	0.986248i	$-0.447151\pi$
$37$	2.01059	0.330538	0.165269	+	0.986248i	$0.447151\pi$
$38$	0	0
$39$	3.26379	0.522625
$40$	0	0
$41$	4.28870	0.669783	0.334892	−	0.942257i	$-0.391300\pi$
$41$	4.28870	0.669783	0.334892	+	0.942257i	$0.391300\pi$
$42$	0	0
$43$	0.897878	0.136925	0.0684625	−	0.997654i	$-0.478191\pi$
$43$	0.897878	0.136925	0.0684625	+	0.997654i	$0.478191\pi$
$44$	0	0
$45$	5.61277	0.836702
$46$	0	0
$47$	6.08013	0.886878	0.443439	−	0.896304i	$-0.353758\pi$
$47$	6.08013	0.886878	0.443439	+	0.896304i	$0.353758\pi$
$48$	0	0
$49$	−6.91307	−0.987582
$50$	0	0
$51$	6.82836	0.956162
$52$	0	0
$53$	4.70499	0.646280	0.323140	−	0.946351i	$-0.395262\pi$
$53$	4.70499	0.646280	0.323140	+	0.946351i	$0.395262\pi$
$54$	0	0
$55$	−10.9960	−1.48270
$56$	0	0
$57$	−1.62866	−0.215721
$58$	0	0
$59$	6.44387	0.838920	0.419460	−	0.907774i	$-0.362219\pi$
$59$	6.44387	0.838920	0.419460	+	0.907774i	$0.362219\pi$
$60$	0	0
$61$	−11.2916	−1.44574	−0.722870	−	0.690984i	$-0.757178\pi$
$61$	−11.2916	−1.44574	−0.722870	+	0.690984i	$0.757178\pi$
$62$	0	0
$63$	0.569968	0.0718092
$64$	0	0
$65$	9.17448	1.13795
$66$	0	0
$67$	0.579037	0.0707406	0.0353703	−	0.999374i	$-0.488739\pi$
$67$	0.579037	0.0707406	0.0353703	+	0.999374i	$0.488739\pi$
$68$	0	0
$69$	−5.71250	−0.687705
$70$	0	0
$71$	−14.0265	−1.66464	−0.832320	−	0.554296i	$-0.812988\pi$
$71$	−14.0265	−1.66464	−0.832320	+	0.554296i	$0.812988\pi$
$72$	0	0
$73$	8.03585	0.940526	0.470263	−	0.882526i	$-0.344159\pi$
$73$	8.03585	0.940526	0.470263	+	0.882526i	$0.344159\pi$
$74$	0	0
$75$	−3.54240	−0.409041
$76$	0	0
$77$	−1.11663	−0.127251
$78$	0	0
$79$	13.2147	1.48677	0.743385	−	0.668864i	$-0.233219\pi$
$79$	13.2147	1.48677	0.743385	+	0.668864i	$0.233219\pi$
$80$	0	0
$81$	0.536712	0.0596347
$82$	0	0
$83$	−9.03832	−0.992084	−0.496042	−	0.868298i	$-0.665214\pi$
$83$	−9.03832	−0.992084	−0.496042	+	0.868298i	$0.665214\pi$
$84$	0	0
$85$	19.1945	2.08193
$86$	0	0
$87$	−0.695581	−0.0745741
$88$	0	0
$89$	−16.8649	−1.78768	−0.893838	−	0.448390i	$-0.851997\pi$
$89$	−16.8649	−1.78768	−0.893838	+	0.448390i	$0.851997\pi$
$90$	0	0
$91$	0.931654	0.0976639
$92$	0	0
$93$	−6.99631	−0.725483
$94$	0	0
$95$	−4.57815	−0.469708
$96$	0	0
$97$	3.10244	0.315005	0.157503	−	0.987519i	$-0.449656\pi$
$97$	3.10244	0.315005	0.157503	+	0.987519i	$0.449656\pi$
$98$	0	0
$99$	−7.32152	−0.735841
$100$	0	0
$101$	−7.78522	−0.774658	−0.387329	−	0.921942i	$-0.626602\pi$
$101$	−7.78522	−0.774658	−0.387329	+	0.921942i	$0.626602\pi$
$102$	0	0
$103$	−1.68539	−0.166066	−0.0830330	−	0.996547i	$-0.526461\pi$
$103$	−1.68539	−0.166066	−0.0830330	+	0.996547i	$0.526461\pi$
$104$	0	0
$105$	−0.884157	−0.0862849
$106$	0	0
$107$	4.87446	0.471232	0.235616	−	0.971846i	$-0.424289\pi$
$107$	4.87446	0.471232	0.235616	+	0.971846i	$0.424289\pi$
$108$	0	0
$109$	−13.3360	−1.27736	−0.638679	−	0.769473i	$-0.720519\pi$
$109$	−13.3360	−1.27736	−0.638679	+	0.769473i	$0.720519\pi$
$110$	0	0
$111$	−2.07668	−0.197109
$112$	0	0
$113$	15.0488	1.41568	0.707838	−	0.706375i	$-0.249671\pi$
$113$	15.0488	1.41568	0.707838	+	0.706375i	$0.249671\pi$
$114$	0	0
$115$	−16.0578	−1.49740
$116$	0	0
$117$	6.10869	0.564749
$118$	0	0
$119$	1.94917	0.178680
$120$	0	0
$121$	3.34364	0.303967
$122$	0	0
$123$	−4.42968	−0.399411
$124$	0	0
$125$	4.55930	0.407797
$126$	0	0
$127$	0.0329066	0.00291999	0.00146000	−	0.999999i	$-0.499535\pi$
$127$	0.0329066	0.00291999	0.00146000	+	0.999999i	$0.499535\pi$
$128$	0	0
$129$	−0.927391	−0.0816523
$130$	0	0
$131$	−11.2198	−0.980279	−0.490139	−	0.871644i	$-0.663054\pi$
$131$	−11.2198	−0.980279	−0.490139	+	0.871644i	$0.663054\pi$
$132$	0	0
$133$	−0.464904	−0.0403123
$134$	0	0
$135$	−14.7937	−1.27324
$136$	0	0
$137$	4.45518	0.380631	0.190316	−	0.981723i	$-0.439049\pi$
$137$	4.45518	0.380631	0.190316	+	0.981723i	$0.439049\pi$
$138$	0	0
$139$	11.5528	0.979895	0.489947	−	0.871752i	$-0.337016\pi$
$139$	11.5528	0.979895	0.489947	+	0.871752i	$0.337016\pi$
$140$	0	0
$141$	−6.27999	−0.528871
$142$	0	0
$143$	−11.9676	−1.00078
$144$	0	0
$145$	−1.95527	−0.162376
$146$	0	0
$147$	7.14031	0.588923
$148$	0	0
$149$	13.2627	1.08652	0.543261	−	0.839564i	$-0.317189\pi$
$149$	13.2627	1.08652	0.543261	+	0.839564i	$0.317189\pi$
$150$	0	0
$151$	5.68002	0.462233	0.231117	−	0.972926i	$-0.425762\pi$
$151$	5.68002	0.462233	0.231117	+	0.972926i	$0.425762\pi$
$152$	0	0
$153$	12.7803	1.03323
$154$	0	0
$155$	−19.6665	−1.57965
$156$	0	0
$157$	−19.6391	−1.56737	−0.783687	−	0.621156i	$-0.786663\pi$
$157$	−19.6391	−1.56737	−0.783687	+	0.621156i	$0.786663\pi$
$158$	0	0
$159$	−4.85965	−0.385395
$160$	0	0
$161$	−1.63064	−0.128513
$162$	0	0
$163$	12.9564	1.01482	0.507412	−	0.861704i	$-0.330602\pi$
$163$	12.9564	1.01482	0.507412	+	0.861704i	$0.330602\pi$
$164$	0	0
$165$	11.3574	0.884176
$166$	0	0
$167$	−15.4550	−1.19594	−0.597972	−	0.801517i	$-0.704027\pi$
$167$	−15.4550	−1.19594	−0.597972	+	0.801517i	$0.704027\pi$
$168$	0	0
$169$	−3.01489	−0.231915
$170$	0	0
$171$	−3.04829	−0.233109
$172$	0	0
$173$	15.1752	1.15375	0.576876	−	0.816832i	$-0.304272\pi$
$173$	15.1752	1.15375	0.576876	+	0.816832i	$0.304272\pi$
$174$	0	0
$175$	−1.01118	−0.0764382
$176$	0	0
$177$	−6.65568	−0.500272
$178$	0	0
$179$	9.20993	0.688383	0.344191	−	0.938900i	$-0.388153\pi$
$179$	9.20993	0.688383	0.344191	+	0.938900i	$0.388153\pi$
$180$	0	0
$181$	13.9465	1.03664	0.518319	−	0.855187i	$-0.326558\pi$
$181$	13.9465	1.03664	0.518319	+	0.855187i	$0.326558\pi$
$182$	0	0
$183$	11.6628	0.862136
$184$	0	0
$185$	−5.83751	−0.429183
$186$	0	0
$187$	−25.0380	−1.83096
$188$	0	0
$189$	−1.50228	−0.109275
$190$	0	0
$191$	5.60336	0.405445	0.202722	−	0.979236i	$-0.435021\pi$
$191$	5.60336	0.405445	0.202722	+	0.979236i	$0.435021\pi$
$192$	0	0
$193$	−22.7522	−1.63774	−0.818870	−	0.573979i	$-0.805399\pi$
$193$	−22.7522	−1.63774	−0.818870	+	0.573979i	$0.805399\pi$
$194$	0	0
$195$	−9.47605	−0.678594
$196$	0	0
$197$	1.22689	0.0874123	0.0437062	−	0.999044i	$-0.486083\pi$
$197$	1.22689	0.0874123	0.0437062	+	0.999044i	$0.486083\pi$
$198$	0	0
$199$	−11.9389	−0.846328	−0.423164	−	0.906053i	$-0.639081\pi$
$199$	−11.9389	−0.846328	−0.423164	+	0.906053i	$0.639081\pi$
$200$	0	0
$201$	−0.598070	−0.0421846
$202$	0	0
$203$	−0.198555	−0.0139358
$204$	0	0
$205$	−12.4518	−0.869669
$206$	0	0
$207$	−10.6918	−0.743134
$208$	0	0
$209$	5.97193	0.413087
$210$	0	0
$211$	15.0312	1.03479	0.517394	−	0.855747i	$-0.326902\pi$
$211$	15.0312	1.03479	0.517394	+	0.855747i	$0.326902\pi$
$212$	0	0
$213$	14.4876	0.992671
$214$	0	0
$215$	−2.60689	−0.177788
$216$	0	0
$217$	−1.99711	−0.135572
$218$	0	0
$219$	−8.30000	−0.560862
$220$	0	0
$221$	20.8904	1.40524
$222$	0	0
$223$	−17.4672	−1.16969	−0.584847	−	0.811144i	$-0.698845\pi$
$223$	−17.4672	−1.16969	−0.584847	+	0.811144i	$0.698845\pi$
$224$	0	0
$225$	−6.63015	−0.442010
$226$	0	0
$227$	5.11642	0.339589	0.169794	−	0.985480i	$-0.445690\pi$
$227$	5.11642	0.339589	0.169794	+	0.985480i	$0.445690\pi$
$228$	0	0
$229$	−5.04244	−0.333214	−0.166607	−	0.986023i	$-0.553281\pi$
$229$	−5.04244	−0.333214	−0.166607	+	0.986023i	$0.553281\pi$
$230$	0	0
$231$	1.15333	0.0758836
$232$	0	0
$233$	−0.969065	−0.0634856	−0.0317428	−	0.999496i	$-0.510106\pi$
$233$	−0.969065	−0.0634856	−0.0317428	+	0.999496i	$0.510106\pi$
$234$	0	0
$235$	−17.6530	−1.15155
$236$	0	0
$237$	−13.6491	−0.886603
$238$	0	0
$239$	−20.6730	−1.33723	−0.668614	−	0.743610i	$-0.733112\pi$
$239$	−20.6730	−1.33723	−0.668614	+	0.743610i	$0.733112\pi$
$240$	0	0
$241$	−5.90726	−0.380520	−0.190260	−	0.981734i	$-0.560933\pi$
$241$	−5.90726	−0.380520	−0.190260	+	0.981734i	$0.560933\pi$
$242$	0	0
$243$	−15.8404	−1.01616
$244$	0	0
$245$	20.0713	1.28231
$246$	0	0
$247$	−4.98266	−0.317039
$248$	0	0
$249$	9.33541	0.591608
$250$	0	0
$251$	10.4728	0.661040	0.330520	−	0.943799i	$-0.392776\pi$
$251$	10.4728	0.661040	0.330520	+	0.943799i	$0.392776\pi$
$252$	0	0
$253$	20.9464	1.31689
$254$	0	0
$255$	−19.8254	−1.24151
$256$	0	0
$257$	−23.1744	−1.44558	−0.722788	−	0.691070i	$-0.757140\pi$
$257$	−23.1744	−1.44558	−0.722788	+	0.691070i	$0.757140\pi$
$258$	0	0
$259$	−0.592790	−0.0368342
$260$	0	0
$261$	−1.30189	−0.0805849
$262$	0	0
$263$	−21.2638	−1.31118	−0.655592	−	0.755115i	$-0.727581\pi$
$263$	−21.2638	−1.31118	−0.655592	+	0.755115i	$0.727581\pi$
$264$	0	0
$265$	−13.6604	−0.839152
$266$	0	0
$267$	17.4193	1.06604
$268$	0	0
$269$	6.78938	0.413956	0.206978	−	0.978346i	$-0.433637\pi$
$269$	6.78938	0.413956	0.206978	+	0.978346i	$0.433637\pi$
$270$	0	0
$271$	30.9160	1.87801	0.939006	−	0.343900i	$-0.111748\pi$
$271$	30.9160	1.87801	0.939006	+	0.343900i	$0.111748\pi$
$272$	0	0
$273$	−0.962278	−0.0582397
$274$	0	0
$275$	12.9892	0.783275
$276$	0	0
$277$	−18.8902	−1.13501	−0.567503	−	0.823372i	$-0.692090\pi$
$277$	−18.8902	−1.13501	−0.567503	+	0.823372i	$0.692090\pi$
$278$	0	0
$279$	−13.0947	−0.783958
$280$	0	0
$281$	−28.8164	−1.71904	−0.859521	−	0.511100i	$-0.829238\pi$
$281$	−28.8164	−1.71904	−0.859521	+	0.511100i	$0.829238\pi$
$282$	0	0
$283$	−29.8924	−1.77692	−0.888460	−	0.458954i	$-0.848224\pi$
$283$	−29.8924	−1.77692	−0.888460	+	0.458954i	$0.848224\pi$
$284$	0	0
$285$	4.72863	0.280100
$286$	0	0
$287$	−1.26446	−0.0746386
$288$	0	0
$289$	26.7060	1.57094
$290$	0	0
$291$	−3.20442	−0.187846
$292$	0	0
$293$	13.0526	0.762543	0.381272	−	0.924463i	$-0.375486\pi$
$293$	13.0526	0.762543	0.381272	+	0.924463i	$0.375486\pi$
$294$	0	0
$295$	−18.7090	−1.08928
$296$	0	0
$297$	19.2976	1.11976
$298$	0	0
$299$	−17.4766	−1.01070
$300$	0	0
$301$	−0.264725	−0.0152585
$302$	0	0
$303$	8.04112	0.461950
$304$	0	0
$305$	32.7839	1.87720
$306$	0	0
$307$	−33.9927	−1.94006	−0.970032	−	0.242978i	$-0.921876\pi$
$307$	−33.9927	−1.94006	−0.970032	+	0.242978i	$0.921876\pi$
$308$	0	0
$309$	1.74079	0.0990298
$310$	0	0
$311$	26.6297	1.51003	0.755016	−	0.655706i	$-0.227629\pi$
$311$	26.6297	1.51003	0.755016	+	0.655706i	$0.227629\pi$
$312$	0	0
$313$	−33.0331	−1.86714	−0.933572	−	0.358391i	$-0.883326\pi$
$313$	−33.0331	−1.86714	−0.933572	+	0.358391i	$0.883326\pi$
$314$	0	0
$315$	−1.65484	−0.0932395
$316$	0	0
$317$	−16.4227	−0.922392	−0.461196	−	0.887298i	$-0.652580\pi$
$317$	−16.4227	−0.922392	−0.461196	+	0.887298i	$0.652580\pi$
$318$	0	0
$319$	2.55054	0.142803
$320$	0	0
$321$	−5.03469	−0.281009
$322$	0	0
$323$	−10.4245	−0.580035
$324$	0	0
$325$	−10.8375	−0.601154
$326$	0	0
$327$	13.7744	0.761724
$328$	0	0
$329$	−1.79263	−0.0988310
$330$	0	0
$331$	−8.90351	−0.489381	−0.244691	−	0.969601i	$-0.578686\pi$
$331$	−8.90351	−0.489381	−0.244691	+	0.969601i	$0.578686\pi$
$332$	0	0
$333$	−3.88682	−0.212997
$334$	0	0
$335$	−1.68117	−0.0918521
$336$	0	0
$337$	−29.5849	−1.61159	−0.805797	−	0.592192i	$-0.798263\pi$
$337$	−29.5849	−1.61159	−0.805797	+	0.592192i	$0.798263\pi$
$338$	0	0
$339$	−15.5435	−0.844207
$340$	0	0
$341$	25.6539	1.38923
$342$	0	0
$343$	4.10205	0.221490
$344$	0	0
$345$	16.5856	0.892939
$346$	0	0
$347$	−9.96121	−0.534746	−0.267373	−	0.963593i	$-0.586156\pi$
$347$	−9.96121	−0.534746	−0.267373	+	0.963593i	$0.586156\pi$
$348$	0	0
$349$	−14.4869	−0.775467	−0.387733	−	0.921772i	$-0.626742\pi$
$349$	−14.4869	−0.775467	−0.387733	+	0.921772i	$0.626742\pi$
$350$	0	0
$351$	−16.1009	−0.859400
$352$	0	0
$353$	−18.6779	−0.994124	−0.497062	−	0.867715i	$-0.665588\pi$
$353$	−18.6779	−0.994124	−0.497062	+	0.867715i	$0.665588\pi$
$354$	0	0
$355$	40.7244	2.16142
$356$	0	0
$357$	−2.01324	−0.106552
$358$	0	0
$359$	3.70971	0.195791	0.0978956	−	0.995197i	$-0.468789\pi$
$359$	3.70971	0.195791	0.0978956	+	0.995197i	$0.468789\pi$
$360$	0	0
$361$	−16.5136	−0.869137
$362$	0	0
$363$	−3.45355	−0.181264
$364$	0	0
$365$	−23.3312	−1.22121
$366$	0	0
$367$	−12.3625	−0.645319	−0.322659	−	0.946515i	$-0.604577\pi$
$367$	−12.3625	−0.645319	−0.322659	+	0.946515i	$0.604577\pi$
$368$	0	0
$369$	−8.29083	−0.431603
$370$	0	0
$371$	−1.38719	−0.0720195
$372$	0	0
$373$	−28.6593	−1.48392	−0.741962	−	0.670441i	$-0.766105\pi$
$373$	−28.6593	−1.48392	−0.741962	+	0.670441i	$0.766105\pi$
$374$	0	0
$375$	−4.70917	−0.243181
$376$	0	0
$377$	−2.12803	−0.109599
$378$	0	0
$379$	−0.170698	−0.00876817	−0.00438409	−	0.999990i	$-0.501396\pi$
$379$	−0.170698	−0.00876817	−0.00438409	+	0.999990i	$0.501396\pi$
$380$	0	0
$381$	−0.0339883	−0.00174127
$382$	0	0
$383$	−25.3702	−1.29636	−0.648178	−	0.761489i	$-0.724469\pi$
$383$	−25.3702	−1.29636	−0.648178	+	0.761489i	$0.724469\pi$
$384$	0	0
$385$	3.24200	0.165228
$386$	0	0
$387$	−1.73576	−0.0882335
$388$	0	0
$389$	−8.49042	−0.430481	−0.215241	−	0.976561i	$-0.569054\pi$
$389$	−8.49042	−0.430481	−0.215241	+	0.976561i	$0.569054\pi$
$390$	0	0
$391$	−36.5638	−1.84911
$392$	0	0
$393$	11.5886	0.584568
$394$	0	0
$395$	−38.3674	−1.93047
$396$	0	0
$397$	21.9563	1.10196	0.550978	−	0.834520i	$-0.314255\pi$
$397$	21.9563	1.10196	0.550978	+	0.834520i	$0.314255\pi$
$398$	0	0
$399$	0.480185	0.0240393
$400$	0	0
$401$	26.0246	1.29961	0.649804	−	0.760102i	$-0.274851\pi$
$401$	26.0246	1.29961	0.649804	+	0.760102i	$0.274851\pi$
$402$	0	0
$403$	−21.4042	−1.06622
$404$	0	0
$405$	−1.55828	−0.0774317
$406$	0	0
$407$	7.61470	0.377446
$408$	0	0
$409$	−6.04512	−0.298912	−0.149456	−	0.988768i	$-0.547752\pi$
$409$	−6.04512	−0.298912	−0.149456	+	0.988768i	$0.547752\pi$
$410$	0	0
$411$	−4.60162	−0.226981
$412$	0	0
$413$	−1.89987	−0.0934867
$414$	0	0
$415$	26.2417	1.28816
$416$	0	0
$417$	−11.9325	−0.584339
$418$	0	0
$419$	9.29453	0.454068	0.227034	−	0.973887i	$-0.427097\pi$
$419$	9.29453	0.454068	0.227034	+	0.973887i	$0.427097\pi$
$420$	0	0
$421$	38.2347	1.86345	0.931723	−	0.363171i	$-0.118306\pi$
$421$	38.2347	1.86345	0.931723	+	0.363171i	$0.118306\pi$
$422$	0	0
$423$	−11.7540	−0.571498
$424$	0	0
$425$	−22.6737	−1.09983
$426$	0	0
$427$	3.32915	0.161109
$428$	0	0
$429$	12.3610	0.596792
$430$	0	0
$431$	36.7139	1.76845	0.884223	−	0.467066i	$-0.154689\pi$
$431$	36.7139	1.76845	0.884223	+	0.467066i	$0.154689\pi$
$432$	0	0
$433$	−27.9804	−1.34465	−0.672326	−	0.740255i	$-0.734705\pi$
$433$	−27.9804	−1.34465	−0.672326	+	0.740255i	$0.734705\pi$
$434$	0	0
$435$	2.01954	0.0968296
$436$	0	0
$437$	8.72098	0.417181
$438$	0	0
$439$	−9.47577	−0.452254	−0.226127	−	0.974098i	$-0.572606\pi$
$439$	−9.47577	−0.452254	−0.226127	+	0.974098i	$0.572606\pi$
$440$	0	0
$441$	13.3642	0.636391
$442$	0	0
$443$	13.5105	0.641902	0.320951	−	0.947096i	$-0.395997\pi$
$443$	13.5105	0.641902	0.320951	+	0.947096i	$0.395997\pi$
$444$	0	0
$445$	48.9653	2.32118
$446$	0	0
$447$	−13.6987	−0.647924
$448$	0	0
$449$	−22.9747	−1.08424	−0.542121	−	0.840300i	$-0.682379\pi$
$449$	−22.9747	−1.08424	−0.542121	+	0.840300i	$0.682379\pi$
$450$	0	0
$451$	16.2426	0.764834
$452$	0	0
$453$	−5.86672	−0.275643
$454$	0	0
$455$	−2.70495	−0.126810
$456$	0	0
$457$	−16.5105	−0.772329	−0.386164	−	0.922430i	$-0.626200\pi$
$457$	−16.5105	−0.772329	−0.386164	+	0.922430i	$0.626200\pi$
$458$	0	0
$459$	−33.6855	−1.57231
$460$	0	0
$461$	38.4598	1.79125	0.895626	−	0.444808i	$-0.146728\pi$
$461$	38.4598	1.79125	0.895626	+	0.444808i	$0.146728\pi$
$462$	0	0
$463$	15.0259	0.698315	0.349157	−	0.937064i	$-0.386468\pi$
$463$	15.0259	0.698315	0.349157	+	0.937064i	$0.386468\pi$
$464$	0	0
$465$	20.3130	0.941993
$466$	0	0
$467$	−14.1325	−0.653972	−0.326986	−	0.945029i	$-0.606033\pi$
$467$	−14.1325	−0.653972	−0.326986	+	0.945029i	$0.606033\pi$
$468$	0	0
$469$	−0.170720	−0.00788312
$470$	0	0
$471$	20.2847	0.934669
$472$	0	0
$473$	3.40053	0.156357
$474$	0	0
$475$	5.40799	0.248136
$476$	0	0
$477$	−9.09558	−0.416458
$478$	0	0
$479$	−0.892881	−0.0407968	−0.0203984	−	0.999792i	$-0.506493\pi$
$479$	−0.892881	−0.0407968	−0.0203984	+	0.999792i	$0.506493\pi$
$480$	0	0
$481$	−6.35330	−0.289685
$482$	0	0
$483$	1.68424	0.0766357
$484$	0	0
$485$	−9.00759	−0.409013
$486$	0	0
$487$	8.98741	0.407258	0.203629	−	0.979048i	$-0.434726\pi$
$487$	8.98741	0.407258	0.203629	+	0.979048i	$0.434726\pi$
$488$	0	0
$489$	−13.3823	−0.605168
$490$	0	0
$491$	−36.7399	−1.65805	−0.829024	−	0.559212i	$-0.811104\pi$
$491$	−36.7399	−1.65805	−0.829024	+	0.559212i	$0.811104\pi$
$492$	0	0
$493$	−4.45218	−0.200516
$494$	0	0
$495$	21.2572	0.955441
$496$	0	0
$497$	4.13550	0.185502
$498$	0	0
$499$	33.3965	1.49503	0.747515	−	0.664245i	$-0.231247\pi$
$499$	33.3965	1.49503	0.747515	+	0.664245i	$0.231247\pi$
$500$	0	0
$501$	15.9630	0.713175
$502$	0	0
$503$	17.5852	0.784085	0.392042	−	0.919947i	$-0.371769\pi$
$503$	17.5852	0.784085	0.392042	+	0.919947i	$0.371769\pi$
$504$	0	0
$505$	22.6035	1.00584
$506$	0	0
$507$	3.11399	0.138297
$508$	0	0
$509$	6.57152	0.291278	0.145639	−	0.989338i	$-0.453476\pi$
$509$	6.57152	0.291278	0.145639	+	0.989338i	$0.453476\pi$
$510$	0	0
$511$	−2.36925	−0.104809
$512$	0	0
$513$	8.03448	0.354731
$514$	0	0
$515$	4.89333	0.215626
$516$	0	0
$517$	23.0273	1.01274
$518$	0	0
$519$	−15.6741	−0.688015
$520$	0	0
$521$	19.2574	0.843683	0.421842	−	0.906670i	$-0.361384\pi$
$521$	19.2574	0.843683	0.421842	+	0.906670i	$0.361384\pi$
$522$	0	0
$523$	28.6560	1.25304	0.626519	−	0.779406i	$-0.284479\pi$
$523$	28.6560	1.25304	0.626519	+	0.779406i	$0.284479\pi$
$524$	0	0
$525$	1.04442	0.0455822
$526$	0	0
$527$	−44.7810	−1.95069
$528$	0	0
$529$	7.58870	0.329944
$530$	0	0
$531$	−12.4571	−0.540594
$532$	0	0
$533$	−13.5520	−0.587001
$534$	0	0
$535$	−14.1525	−0.611864
$536$	0	0
$537$	−9.51267	−0.410502
$538$	0	0
$539$	−26.1819	−1.12773
$540$	0	0
$541$	−9.68729	−0.416489	−0.208245	−	0.978077i	$-0.566775\pi$
$541$	−9.68729	−0.416489	−0.208245	+	0.978077i	$0.566775\pi$
$542$	0	0
$543$	−14.4050	−0.618176
$544$	0	0
$545$	38.7196	1.65857
$546$	0	0
$547$	15.6364	0.668565	0.334283	−	0.942473i	$-0.391506\pi$
$547$	15.6364	0.668565	0.334283	+	0.942473i	$0.391506\pi$
$548$	0	0
$549$	21.8287	0.931625
$550$	0	0
$551$	1.06191	0.0452388
$552$	0	0
$553$	−3.89615	−0.165681
$554$	0	0
$555$	6.02940	0.255934
$556$	0	0
$557$	40.3448	1.70946	0.854731	−	0.519070i	$-0.173722\pi$
$557$	40.3448	1.70946	0.854731	+	0.519070i	$0.173722\pi$
$558$	0	0
$559$	−2.83722	−0.120002
$560$	0	0
$561$	25.8611	1.09185
$562$	0	0
$563$	23.4714	0.989201	0.494601	−	0.869120i	$-0.335314\pi$
$563$	23.4714	0.989201	0.494601	+	0.869120i	$0.335314\pi$
$564$	0	0
$565$	−43.6926	−1.83816
$566$	0	0
$567$	−0.158241	−0.00664550
$568$	0	0
$569$	−26.2503	−1.10047	−0.550235	−	0.835010i	$-0.685462\pi$
$569$	−26.2503	−1.10047	−0.550235	+	0.835010i	$0.685462\pi$
$570$	0	0
$571$	11.0641	0.463016	0.231508	−	0.972833i	$-0.425634\pi$
$571$	11.0641	0.463016	0.231508	+	0.972833i	$0.425634\pi$
$572$	0	0
$573$	−5.78754	−0.241778
$574$	0	0
$575$	18.9684	0.791039
$576$	0	0
$577$	−13.1312	−0.546659	−0.273330	−	0.961920i	$-0.588125\pi$
$577$	−13.1312	−0.546659	−0.273330	+	0.961920i	$0.588125\pi$
$578$	0	0
$579$	23.5001	0.976631
$580$	0	0
$581$	2.66481	0.110555
$582$	0	0
$583$	17.8192	0.737996
$584$	0	0
$585$	−17.7359	−0.733289
$586$	0	0
$587$	−30.1580	−1.24476	−0.622378	−	0.782717i	$-0.713833\pi$
$587$	−30.1580	−1.24476	−0.622378	+	0.782717i	$0.713833\pi$
$588$	0	0
$589$	10.6809	0.440099
$590$	0	0
$591$	−1.26722	−0.0521264
$592$	0	0
$593$	15.5670	0.639261	0.319631	−	0.947542i	$-0.396441\pi$
$593$	15.5670	0.639261	0.319631	+	0.947542i	$0.396441\pi$
$594$	0	0
$595$	−5.65919	−0.232004
$596$	0	0
$597$	12.3314	0.504689
$598$	0	0
$599$	34.7367	1.41930	0.709652	−	0.704552i	$-0.248852\pi$
$599$	34.7367	1.41930	0.709652	+	0.704552i	$0.248852\pi$
$600$	0	0
$601$	5.81123	0.237045	0.118522	−	0.992951i	$-0.462184\pi$
$601$	5.81123	0.237045	0.118522	+	0.992951i	$0.462184\pi$
$602$	0	0
$603$	−1.11938	−0.0455847
$604$	0	0
$605$	−9.70788	−0.394681
$606$	0	0
$607$	7.58877	0.308019	0.154009	−	0.988069i	$-0.450781\pi$
$607$	7.58877	0.308019	0.154009	+	0.988069i	$0.450781\pi$
$608$	0	0
$609$	0.205081	0.00831032
$610$	0	0
$611$	−19.2127	−0.777264
$612$	0	0
$613$	−23.6306	−0.954432	−0.477216	−	0.878786i	$-0.658354\pi$
$613$	−23.6306	−0.954432	−0.477216	+	0.878786i	$0.658354\pi$
$614$	0	0
$615$	12.8611	0.518608
$616$	0	0
$617$	−14.8838	−0.599198	−0.299599	−	0.954065i	$-0.596853\pi$
$617$	−14.8838	−0.599198	−0.299599	+	0.954065i	$0.596853\pi$
$618$	0	0
$619$	3.66844	0.147447	0.0737236	−	0.997279i	$-0.476512\pi$
$619$	3.66844	0.147447	0.0737236	+	0.997279i	$0.476512\pi$
$620$	0	0
$621$	28.1808	1.13086
$622$	0	0
$623$	4.97235	0.199213
$624$	0	0
$625$	−30.3857	−1.21543
$626$	0	0
$627$	−6.16823	−0.246335
$628$	0	0
$629$	−13.2921	−0.529991
$630$	0	0
$631$	−1.47434	−0.0586924	−0.0293462	−	0.999569i	$-0.509343\pi$
$631$	−1.47434	−0.0586924	−0.0293462	+	0.999569i	$0.509343\pi$
$632$	0	0
$633$	−15.5253	−0.617074
$634$	0	0
$635$	−0.0955407	−0.00379142
$636$	0	0
$637$	21.8448	0.865521
$638$	0	0
$639$	27.1157	1.07268
$640$	0	0
$641$	−39.2066	−1.54857	−0.774283	−	0.632839i	$-0.781889\pi$
$641$	−39.2066	−1.54857	−0.774283	+	0.632839i	$0.781889\pi$
$642$	0	0
$643$	−1.62711	−0.0641669	−0.0320835	−	0.999485i	$-0.510214\pi$
$643$	−1.62711	−0.0641669	−0.0320835	+	0.999485i	$0.510214\pi$
$644$	0	0
$645$	2.69258	0.106020
$646$	0	0
$647$	−24.0334	−0.944849	−0.472424	−	0.881371i	$-0.656621\pi$
$647$	−24.0334	−0.944849	−0.472424	+	0.881371i	$0.656621\pi$
$648$	0	0
$649$	24.4049	0.957974
$650$	0	0
$651$	2.06275	0.0808457
$652$	0	0
$653$	−2.99602	−0.117243	−0.0586216	−	0.998280i	$-0.518671\pi$
$653$	−2.99602	−0.117243	−0.0586216	+	0.998280i	$0.518671\pi$
$654$	0	0
$655$	32.5754	1.27283
$656$	0	0
$657$	−15.5347	−0.606068
$658$	0	0
$659$	21.4521	0.835654	0.417827	−	0.908527i	$-0.362792\pi$
$659$	21.4521	0.835654	0.417827	+	0.908527i	$0.362792\pi$
$660$	0	0
$661$	−2.81111	−0.109339	−0.0546697	−	0.998504i	$-0.517411\pi$
$661$	−2.81111	−0.109339	−0.0546697	+	0.998504i	$0.517411\pi$
$662$	0	0
$663$	−21.5771	−0.837985
$664$	0	0
$665$	1.34980	0.0523428
$666$	0	0
$667$	3.72462	0.144218
$668$	0	0
$669$	18.0414	0.697521
$670$	0	0
$671$	−42.7646	−1.65091
$672$	0	0
$673$	23.2536	0.896361	0.448181	−	0.893943i	$-0.352072\pi$
$673$	23.2536	0.896361	0.448181	+	0.893943i	$0.352072\pi$
$674$	0	0
$675$	17.4753	0.672624
$676$	0	0
$677$	36.9377	1.41963	0.709815	−	0.704389i	$-0.248779\pi$
$677$	36.9377	1.41963	0.709815	+	0.704389i	$0.248779\pi$
$678$	0	0
$679$	−0.914706	−0.0351032
$680$	0	0
$681$	−5.28460	−0.202506
$682$	0	0
$683$	25.6082	0.979870	0.489935	−	0.871759i	$-0.337021\pi$
$683$	25.6082	0.979870	0.489935	+	0.871759i	$0.337021\pi$
$684$	0	0
$685$	−12.9351	−0.494225
$686$	0	0
$687$	5.20819	0.198705
$688$	0	0
$689$	−14.8674	−0.566403
$690$	0	0
$691$	−37.8360	−1.43935	−0.719675	−	0.694311i	$-0.755709\pi$
$691$	−37.8360	−1.43935	−0.719675	+	0.694311i	$0.755709\pi$
$692$	0	0
$693$	2.15864	0.0819999
$694$	0	0
$695$	−33.5422	−1.27233
$696$	0	0
$697$	−28.3529	−1.07394
$698$	0	0
$699$	1.00092	0.0378582
$700$	0	0
$701$	43.5010	1.64301	0.821504	−	0.570203i	$-0.193135\pi$
$701$	43.5010	1.64301	0.821504	+	0.570203i	$0.193135\pi$
$702$	0	0
$703$	3.17035	0.119572
$704$	0	0
$705$	18.2332	0.686704
$706$	0	0
$707$	2.29535	0.0863255
$708$	0	0
$709$	−6.12453	−0.230012	−0.115006	−	0.993365i	$-0.536689\pi$
$709$	−6.12453	−0.230012	−0.115006	+	0.993365i	$0.536689\pi$
$710$	0	0
$711$	−25.5464	−0.958064
$712$	0	0
$713$	37.4631	1.40300
$714$	0	0
$715$	34.7465	1.29945
$716$	0	0
$717$	21.3526	0.797426
$718$	0	0
$719$	−41.4399	−1.54545	−0.772725	−	0.634741i	$-0.781107\pi$
$719$	−41.4399	−1.54545	−0.772725	+	0.634741i	$0.781107\pi$
$720$	0	0
$721$	0.496910	0.0185059
$722$	0	0
$723$	6.10144	0.226915
$724$	0	0
$725$	2.30969	0.0857796
$726$	0	0
$727$	−40.3836	−1.49774	−0.748872	−	0.662715i	$-0.769404\pi$
$727$	−40.3836	−1.49774	−0.748872	+	0.662715i	$0.769404\pi$
$728$	0	0
$729$	14.7509	0.546330
$730$	0	0
$731$	−5.93592	−0.219548
$732$	0	0
$733$	−18.5995	−0.686987	−0.343494	−	0.939155i	$-0.611610\pi$
$733$	−18.5995	−0.686987	−0.343494	+	0.939155i	$0.611610\pi$
$734$	0	0
$735$	−20.7311	−0.764678
$736$	0	0
$737$	2.19299	0.0807797
$738$	0	0
$739$	−14.7396	−0.542206	−0.271103	−	0.962550i	$-0.587388\pi$
$739$	−14.7396	−0.542206	−0.271103	+	0.962550i	$0.587388\pi$
$740$	0	0
$741$	5.14644	0.189059
$742$	0	0
$743$	−24.0762	−0.883269	−0.441635	−	0.897195i	$-0.645601\pi$
$743$	−24.0762	−0.883269	−0.441635	+	0.897195i	$0.645601\pi$
$744$	0	0
$745$	−38.5068	−1.41078
$746$	0	0
$747$	17.4727	0.639292
$748$	0	0
$749$	−1.43716	−0.0525127
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−10.8171	−0.394197
$754$	0	0
$755$	−16.4913	−0.600179
$756$	0	0
$757$	9.92844	0.360855	0.180428	−	0.983588i	$-0.442252\pi$
$757$	9.92844	0.360855	0.180428	+	0.983588i	$0.442252\pi$
$758$	0	0
$759$	−21.6350	−0.785299
$760$	0	0
$761$	−0.917360	−0.0332543	−0.0166271	−	0.999862i	$-0.505293\pi$
$761$	−0.917360	−0.0332543	−0.0166271	+	0.999862i	$0.505293\pi$
$762$	0	0
$763$	3.93191	0.142345
$764$	0	0
$765$	−37.1063	−1.34158
$766$	0	0
$767$	−20.3621	−0.735234
$768$	0	0
$769$	30.2720	1.09164	0.545818	−	0.837904i	$-0.316219\pi$
$769$	30.2720	1.09164	0.545818	+	0.837904i	$0.316219\pi$
$770$	0	0
$771$	23.9361	0.862038
$772$	0	0
$773$	−30.0130	−1.07949	−0.539747	−	0.841827i	$-0.681480\pi$
$773$	−30.0130	−1.07949	−0.539747	+	0.841827i	$0.681480\pi$
$774$	0	0
$775$	23.2313	0.834494
$776$	0	0
$777$	0.612276	0.0219653
$778$	0	0
$779$	6.76256	0.242294
$780$	0	0
$781$	−53.1226	−1.90087
$782$	0	0
$783$	3.43143	0.122629
$784$	0	0
$785$	57.0200	2.03513
$786$	0	0
$787$	−23.0844	−0.822872	−0.411436	−	0.911439i	$-0.634973\pi$
$787$	−23.0844	−0.822872	−0.411436	+	0.911439i	$0.634973\pi$
$788$	0	0
$789$	21.9628	0.781896
$790$	0	0
$791$	−4.43692	−0.157759
$792$	0	0
$793$	35.6806	1.26705
$794$	0	0
$795$	14.1094	0.500410
$796$	0	0
$797$	−31.4740	−1.11487	−0.557433	−	0.830222i	$-0.688214\pi$
$797$	−31.4740	−1.11487	−0.557433	+	0.830222i	$0.688214\pi$
$798$	0	0
$799$	−40.1961	−1.42204
$800$	0	0
$801$	32.6029	1.15197
$802$	0	0
$803$	30.4342	1.07400
$804$	0	0
$805$	4.73439	0.166865
$806$	0	0
$807$	−7.01255	−0.246854
$808$	0	0
$809$	−45.2043	−1.58930	−0.794650	−	0.607068i	$-0.792346\pi$
$809$	−45.2043	−1.58930	−0.794650	+	0.607068i	$0.792346\pi$
$810$	0	0
$811$	18.6776	0.655859	0.327930	−	0.944702i	$-0.393649\pi$
$811$	18.6776	0.655859	0.327930	+	0.944702i	$0.393649\pi$
$812$	0	0
$813$	−31.9322	−1.11991
$814$	0	0
$815$	−37.6175	−1.31768
$816$	0	0
$817$	1.41580	0.0495326
$818$	0	0
$819$	−1.80105	−0.0629339
$820$	0	0
$821$	48.9989	1.71007	0.855036	−	0.518568i	$-0.173535\pi$
$821$	48.9989	1.71007	0.855036	+	0.518568i	$0.173535\pi$
$822$	0	0
$823$	−2.49958	−0.0871297	−0.0435649	−	0.999051i	$-0.513872\pi$
$823$	−2.49958	−0.0871297	−0.0435649	+	0.999051i	$0.513872\pi$
$824$	0	0
$825$	−13.4161	−0.467089
$826$	0	0
$827$	46.8695	1.62981	0.814905	−	0.579594i	$-0.196789\pi$
$827$	46.8695	1.62981	0.814905	+	0.579594i	$0.196789\pi$
$828$	0	0
$829$	38.4392	1.33505	0.667525	−	0.744588i	$-0.267354\pi$
$829$	38.4392	1.33505	0.667525	+	0.744588i	$0.267354\pi$
$830$	0	0
$831$	19.5112	0.676835
$832$	0	0
$833$	45.7027	1.58350
$834$	0	0
$835$	44.8719	1.55286
$836$	0	0
$837$	34.5140	1.19298
$838$	0	0
$839$	34.0332	1.17496	0.587478	−	0.809240i	$-0.300121\pi$
$839$	34.0332	1.17496	0.587478	+	0.809240i	$0.300121\pi$
$840$	0	0
$841$	−28.5465	−0.984361
$842$	0	0
$843$	29.7636	1.02511
$844$	0	0
$845$	8.75340	0.301126
$846$	0	0
$847$	−0.985820	−0.0338732
$848$	0	0
$849$	30.8750	1.05963
$850$	0	0
$851$	11.1200	0.381187
$852$	0	0
$853$	35.8524	1.22756	0.613781	−	0.789477i	$-0.289648\pi$
$853$	35.8524	1.22756	0.613781	+	0.789477i	$0.289648\pi$
$854$	0	0
$855$	8.85038	0.302676
$856$	0	0
$857$	−6.32951	−0.216212	−0.108106	−	0.994139i	$-0.534479\pi$
$857$	−6.32951	−0.216212	−0.108106	+	0.994139i	$0.534479\pi$
$858$	0	0
$859$	−31.8565	−1.08693	−0.543464	−	0.839432i	$-0.682888\pi$
$859$	−31.8565	−1.08693	−0.543464	+	0.839432i	$0.682888\pi$
$860$	0	0
$861$	1.30602	0.0445091
$862$	0	0
$863$	−24.0747	−0.819513	−0.409756	−	0.912195i	$-0.634386\pi$
$863$	−24.0747	−0.819513	−0.409756	+	0.912195i	$0.634386\pi$
$864$	0	0
$865$	−44.0596	−1.49807
$866$	0	0
$867$	−27.5839	−0.936798
$868$	0	0
$869$	50.0480	1.69776
$870$	0	0
$871$	−1.82971	−0.0619974
$872$	0	0
$873$	−5.99757	−0.202987
$874$	0	0
$875$	−1.34424	−0.0454436
$876$	0	0
$877$	−5.73142	−0.193536	−0.0967682	−	0.995307i	$-0.530851\pi$
$877$	−5.73142	−0.193536	−0.0967682	+	0.995307i	$0.530851\pi$
$878$	0	0
$879$	−13.4817	−0.454726
$880$	0	0
$881$	19.6830	0.663136	0.331568	−	0.943431i	$-0.392422\pi$
$881$	19.6830	0.663136	0.331568	+	0.943431i	$0.392422\pi$
$882$	0	0
$883$	−4.27683	−0.143927	−0.0719634	−	0.997407i	$-0.522926\pi$
$883$	−4.27683	−0.143927	−0.0719634	+	0.997407i	$0.522926\pi$
$884$	0	0
$885$	19.3240	0.649570
$886$	0	0
$887$	29.7672	0.999486	0.499743	−	0.866174i	$-0.333428\pi$
$887$	29.7672	0.999486	0.499743	+	0.866174i	$0.333428\pi$
$888$	0	0
$889$	−0.00970201	−0.000325395 0
$890$	0	0
$891$	2.03269	0.0680976
$892$	0	0
$893$	9.58733	0.320828
$894$	0	0
$895$	−26.7400	−0.893820
$896$	0	0
$897$	18.0511	0.602708
$898$	0	0
$899$	4.56168	0.152141
$900$	0	0
$901$	−31.1049	−1.03626
$902$	0	0
$903$	0.273427	0.00909908
$904$	0	0
$905$	−40.4922	−1.34601
$906$	0	0
$907$	−39.2603	−1.30362	−0.651808	−	0.758384i	$-0.725989\pi$
$907$	−39.2603	−1.30362	−0.651808	+	0.758384i	$0.725989\pi$
$908$	0	0
$909$	15.0502	0.499184
$910$	0	0
$911$	26.6583	0.883227	0.441614	−	0.897205i	$-0.354406\pi$
$911$	26.6583	0.883227	0.441614	+	0.897205i	$0.354406\pi$
$912$	0	0
$913$	−34.2308	−1.13287
$914$	0	0
$915$	−33.8615	−1.11943
$916$	0	0
$917$	3.30798	0.109239
$918$	0	0
$919$	−56.0219	−1.84799	−0.923995	−	0.382404i	$-0.875096\pi$
$919$	−56.0219	−1.84799	−0.923995	+	0.382404i	$0.875096\pi$
$920$	0	0
$921$	35.1100	1.15691
$922$	0	0
$923$	44.3226	1.45890
$924$	0	0
$925$	6.89563	0.226727
$926$	0	0
$927$	3.25815	0.107012
$928$	0	0
$929$	−25.0444	−0.821681	−0.410840	−	0.911707i	$-0.634765\pi$
$929$	−25.0444	−0.821681	−0.410840	+	0.911707i	$0.634765\pi$
$930$	0	0
$931$	−10.9007	−0.357257
$932$	0	0
$933$	−27.5050	−0.900475
$934$	0	0
$935$	72.6951	2.37739
$936$	0	0
$937$	−41.2844	−1.34870	−0.674351	−	0.738411i	$-0.735576\pi$
$937$	−41.2844	−1.34870	−0.674351	+	0.738411i	$0.735576\pi$
$938$	0	0
$939$	34.1190	1.11343
$940$	0	0
$941$	−3.64571	−0.118847	−0.0594234	−	0.998233i	$-0.518926\pi$
$941$	−3.64571	−0.118847	−0.0594234	+	0.998233i	$0.518926\pi$
$942$	0	0
$943$	23.7196	0.772415
$944$	0	0
$945$	4.36170	0.141886
$946$	0	0
$947$	17.7881	0.578036	0.289018	−	0.957324i	$-0.406671\pi$
$947$	17.7881	0.578036	0.289018	+	0.957324i	$0.406671\pi$
$948$	0	0
$949$	−25.3927	−0.824281
$950$	0	0
$951$	16.9626	0.550049
$952$	0	0
$953$	28.0504	0.908642	0.454321	−	0.890838i	$-0.349882\pi$
$953$	28.0504	0.908642	0.454321	+	0.890838i	$0.349882\pi$
$954$	0	0
$955$	−16.2687	−0.526443
$956$	0	0
$957$	−2.63437	−0.0851572
$958$	0	0
$959$	−1.31354	−0.0424164
$960$	0	0
$961$	14.8824	0.480077
$962$	0	0
$963$	−9.42321	−0.303659
$964$	0	0
$965$	66.0585	2.12650
$966$	0	0
$967$	40.4338	1.30026	0.650131	−	0.759822i	$-0.274714\pi$
$967$	40.4338	1.30026	0.650131	+	0.759822i	$0.274714\pi$
$968$	0	0
$969$	10.7672	0.345891
$970$	0	0
$971$	−46.8956	−1.50495	−0.752475	−	0.658621i	$-0.771140\pi$
$971$	−46.8956	−1.50495	−0.752475	+	0.658621i	$0.771140\pi$
$972$	0	0
$973$	−3.40616	−0.109196
$974$	0	0
$975$	11.1937	0.358485
$976$	0	0
$977$	−40.3543	−1.29105	−0.645525	−	0.763739i	$-0.723361\pi$
$977$	−40.3543	−1.29105	−0.645525	+	0.763739i	$0.723361\pi$
$978$	0	0
$979$	−63.8724	−2.04137
$980$	0	0
$981$	25.7809	0.823120
$982$	0	0
$983$	58.8995	1.87860	0.939301	−	0.343095i	$-0.111475\pi$
$983$	58.8995	1.87860	0.939301	+	0.343095i	$0.111475\pi$
$984$	0	0
$985$	−3.56214	−0.113499
$986$	0	0
$987$	1.85156	0.0589357
$988$	0	0
$989$	4.96590	0.157906
$990$	0	0
$991$	2.93776	0.0933210	0.0466605	−	0.998911i	$-0.485142\pi$
$991$	2.93776	0.0933210	0.0466605	+	0.998911i	$0.485142\pi$
$992$	0	0
$993$	9.19617	0.291832
$994$	0	0
$995$	34.6633	1.09890
$996$	0	0
$997$	−3.53314	−0.111896	−0.0559478	−	0.998434i	$-0.517818\pi$
$997$	−3.53314	−0.111896	−0.0559478	+	0.998434i	$0.517818\pi$
$998$	0	0
$999$	10.2446	0.324125

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.18	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.18	✓	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)

\(f(q)\)	\(=\)	\(q-1.03287 q^{3} -2.90339 q^{5} -0.294834 q^{7} -1.93318 q^{9} +O(q^{10})\)	\(q-1.03287 q^{3} -2.90339 q^{5} -0.294834 q^{7} -1.93318 q^{9} +3.78730 q^{11} -3.15992 q^{13} +2.99882 q^{15} -6.61105 q^{17} +1.57683 q^{19} +0.304526 q^{21} +5.53071 q^{23} +3.42966 q^{25} +5.09534 q^{27} +0.673445 q^{29} +6.77365 q^{31} -3.91179 q^{33} +0.856019 q^{35} +2.01059 q^{37} +3.26379 q^{39} +4.28870 q^{41} +0.897878 q^{43} +5.61277 q^{45} +6.08013 q^{47} -6.91307 q^{49} +6.82836 q^{51} +4.70499 q^{53} -10.9960 q^{55} -1.62866 q^{57} +6.44387 q^{59} -11.2916 q^{61} +0.569968 q^{63} +9.17448 q^{65} +0.579037 q^{67} -5.71250 q^{69} -14.0265 q^{71} +8.03585 q^{73} -3.54240 q^{75} -1.11663 q^{77} +13.2147 q^{79} +0.536712 q^{81} -9.03832 q^{83} +19.1945 q^{85} -0.695581 q^{87} -16.8649 q^{89} +0.931654 q^{91} -6.99631 q^{93} -4.57815 q^{95} +3.10244 q^{97} -7.32152 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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