LMFDB - Embedded newform 6008.2.a.b.1.3

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			1.3
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)

$f(q)$	$=$	$q-3.25538 q^{3} -1.89943 q^{5} -1.94419 q^{7} +7.59748 q^{9} +O(q^{10})$	$q-3.25538 q^{3} -1.89943 q^{5} -1.94419 q^{7} +7.59748 q^{9} +2.87143 q^{11} -5.69721 q^{13} +6.18336 q^{15} +0.264846 q^{17} +0.150476 q^{19} +6.32908 q^{21} -6.02516 q^{23} -1.39216 q^{25} -14.9665 q^{27} +6.65014 q^{29} -1.66567 q^{31} -9.34760 q^{33} +3.69286 q^{35} +11.9476 q^{37} +18.5466 q^{39} -4.78042 q^{41} -4.59270 q^{43} -14.4309 q^{45} +0.373316 q^{47} -3.22011 q^{49} -0.862175 q^{51} -3.24190 q^{53} -5.45409 q^{55} -0.489856 q^{57} +10.8967 q^{59} +14.2659 q^{61} -14.7710 q^{63} +10.8215 q^{65} +3.11010 q^{67} +19.6142 q^{69} -14.0569 q^{71} +5.06310 q^{73} +4.53201 q^{75} -5.58262 q^{77} +13.8592 q^{79} +25.9293 q^{81} -7.20240 q^{83} -0.503057 q^{85} -21.6487 q^{87} +17.9536 q^{89} +11.0765 q^{91} +5.42238 q^{93} -0.285819 q^{95} -4.67098 q^{97} +21.8157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.25538	−1.87949	−0.939746	−	0.341872i	$-0.888939\pi$
$3$	−3.25538	−1.87949	−0.939746	+	0.341872i	$0.888939\pi$
$4$	0	0
$5$	−1.89943	−0.849451	−0.424726	−	0.905322i	$-0.639629\pi$
$5$	−1.89943	−0.849451	−0.424726	+	0.905322i	$0.639629\pi$
$6$	0	0
$7$	−1.94419	−0.734836	−0.367418	−	0.930056i	$-0.619758\pi$
$7$	−1.94419	−0.734836	−0.367418	+	0.930056i	$0.619758\pi$
$8$	0	0
$9$	7.59748	2.53249
$10$	0	0
$11$	2.87143	0.865770	0.432885	−	0.901449i	$-0.357496\pi$
$11$	2.87143	0.865770	0.432885	+	0.901449i	$0.357496\pi$
$12$	0	0
$13$	−5.69721	−1.58012	−0.790061	−	0.613029i	$-0.789951\pi$
$13$	−5.69721	−1.58012	−0.790061	+	0.613029i	$0.789951\pi$
$14$	0	0
$15$	6.18336	1.59654
$16$	0	0
$17$	0.264846	0.0642347	0.0321173	−	0.999484i	$-0.489775\pi$
$17$	0.264846	0.0642347	0.0321173	+	0.999484i	$0.489775\pi$
$18$	0	0
$19$	0.150476	0.0345215	0.0172608	−	0.999851i	$-0.494505\pi$
$19$	0.150476	0.0345215	0.0172608	+	0.999851i	$0.494505\pi$
$20$	0	0
$21$	6.32908	1.38112
$22$	0	0
$23$	−6.02516	−1.25633	−0.628167	−	0.778079i	$-0.716194\pi$
$23$	−6.02516	−1.25633	−0.628167	+	0.778079i	$0.716194\pi$
$24$	0	0
$25$	−1.39216	−0.278432
$26$	0	0
$27$	−14.9665	−2.88031
$28$	0	0
$29$	6.65014	1.23490	0.617450	−	0.786610i	$-0.288166\pi$
$29$	6.65014	1.23490	0.617450	+	0.786610i	$0.288166\pi$
$30$	0	0
$31$	−1.66567	−0.299163	−0.149581	−	0.988749i	$-0.547793\pi$
$31$	−1.66567	−0.299163	−0.149581	+	0.988749i	$0.547793\pi$
$32$	0	0
$33$	−9.34760	−1.62721
$34$	0	0
$35$	3.69286	0.624207
$36$	0	0
$37$	11.9476	1.96417	0.982086	−	0.188432i	$-0.0603405\pi$
$37$	11.9476	1.96417	0.982086	+	0.188432i	$0.0603405\pi$
$38$	0	0
$39$	18.5466	2.96983
$40$	0	0
$41$	−4.78042	−0.746576	−0.373288	−	0.927716i	$-0.621770\pi$
$41$	−4.78042	−0.746576	−0.373288	+	0.927716i	$0.621770\pi$
$42$	0	0
$43$	−4.59270	−0.700380	−0.350190	−	0.936679i	$-0.613883\pi$
$43$	−4.59270	−0.700380	−0.350190	+	0.936679i	$0.613883\pi$
$44$	0	0
$45$	−14.4309	−2.15123
$46$	0	0
$47$	0.373316	0.0544538	0.0272269	−	0.999629i	$-0.491332\pi$
$47$	0.373316	0.0544538	0.0272269	+	0.999629i	$0.491332\pi$
$48$	0	0
$49$	−3.22011	−0.460016
$50$	0	0
$51$	−0.862175	−0.120729
$52$	0	0
$53$	−3.24190	−0.445309	−0.222654	−	0.974897i	$-0.571472\pi$
$53$	−3.24190	−0.445309	−0.222654	+	0.974897i	$0.571472\pi$
$54$	0	0
$55$	−5.45409	−0.735429
$56$	0	0
$57$	−0.489856	−0.0648830
$58$	0	0
$59$	10.8967	1.41863	0.709317	−	0.704889i	$-0.249003\pi$
$59$	10.8967	1.41863	0.709317	+	0.704889i	$0.249003\pi$
$60$	0	0
$61$	14.2659	1.82656	0.913278	−	0.407336i	$-0.133542\pi$
$61$	14.2659	1.82656	0.913278	+	0.407336i	$0.133542\pi$
$62$	0	0
$63$	−14.7710	−1.86097
$64$	0	0
$65$	10.8215	1.34224
$66$	0	0
$67$	3.11010	0.379959	0.189980	−	0.981788i	$-0.439158\pi$
$67$	3.11010	0.379959	0.189980	+	0.981788i	$0.439158\pi$
$68$	0	0
$69$	19.6142	2.36127
$70$	0	0
$71$	−14.0569	−1.66824	−0.834121	−	0.551582i	$-0.814025\pi$
$71$	−14.0569	−1.66824	−0.834121	+	0.551582i	$0.814025\pi$
$72$	0	0
$73$	5.06310	0.592591	0.296295	−	0.955096i	$-0.404249\pi$
$73$	5.06310	0.592591	0.296295	+	0.955096i	$0.404249\pi$
$74$	0	0
$75$	4.53201	0.523312
$76$	0	0
$77$	−5.58262	−0.636199
$78$	0	0
$79$	13.8592	1.55928	0.779639	−	0.626230i	$-0.215403\pi$
$79$	13.8592	1.55928	0.779639	+	0.626230i	$0.215403\pi$
$80$	0	0
$81$	25.9293	2.88103
$82$	0	0
$83$	−7.20240	−0.790566	−0.395283	−	0.918559i	$-0.629353\pi$
$83$	−7.20240	−0.790566	−0.395283	+	0.918559i	$0.629353\pi$
$84$	0	0
$85$	−0.503057	−0.0545642
$86$	0	0
$87$	−21.6487	−2.32099
$88$	0	0
$89$	17.9536	1.90308	0.951540	−	0.307526i	$-0.0995012\pi$
$89$	17.9536	1.90308	0.951540	+	0.307526i	$0.0995012\pi$
$90$	0	0
$91$	11.0765	1.16113
$92$	0	0
$93$	5.42238	0.562274
$94$	0	0
$95$	−0.285819	−0.0293244
$96$	0	0
$97$	−4.67098	−0.474267	−0.237133	−	0.971477i	$-0.576208\pi$
$97$	−4.67098	−0.474267	−0.237133	+	0.971477i	$0.576208\pi$
$98$	0	0
$99$	21.8157	2.19256
$100$	0	0
$101$	1.22719	0.122110	0.0610548	−	0.998134i	$-0.480554\pi$
$101$	1.22719	0.122110	0.0610548	+	0.998134i	$0.480554\pi$
$102$	0	0
$103$	−10.0786	−0.993072	−0.496536	−	0.868016i	$-0.665395\pi$
$103$	−10.0786	−0.993072	−0.496536	+	0.868016i	$0.665395\pi$
$104$	0	0
$105$	−12.0217	−1.17319
$106$	0	0
$107$	10.8193	1.04595	0.522973	−	0.852349i	$-0.324823\pi$
$107$	10.8193	1.04595	0.522973	+	0.852349i	$0.324823\pi$
$108$	0	0
$109$	18.0796	1.73171	0.865856	−	0.500293i	$-0.166774\pi$
$109$	18.0796	1.73171	0.865856	+	0.500293i	$0.166774\pi$
$110$	0	0
$111$	−38.8939	−3.69165
$112$	0	0
$113$	4.43265	0.416989	0.208494	−	0.978024i	$-0.433144\pi$
$113$	4.43265	0.416989	0.208494	+	0.978024i	$0.433144\pi$
$114$	0	0
$115$	11.4444	1.06719
$116$	0	0
$117$	−43.2844	−4.00165
$118$	0	0
$119$	−0.514912	−0.0472019
$120$	0	0
$121$	−2.75487	−0.250442
$122$	0	0
$123$	15.5621	1.40318
$124$	0	0
$125$	12.1415	1.08597
$126$	0	0
$127$	10.4161	0.924278	0.462139	−	0.886807i	$-0.347082\pi$
$127$	10.4161	0.924278	0.462139	+	0.886807i	$0.347082\pi$
$128$	0	0
$129$	14.9510	1.31636
$130$	0	0
$131$	−9.14024	−0.798586	−0.399293	−	0.916823i	$-0.630744\pi$
$131$	−9.14024	−0.798586	−0.399293	+	0.916823i	$0.630744\pi$
$132$	0	0
$133$	−0.292554	−0.0253677
$134$	0	0
$135$	28.4279	2.44668
$136$	0	0
$137$	−14.6319	−1.25008	−0.625042	−	0.780591i	$-0.714918\pi$
$137$	−14.6319	−1.25008	−0.625042	+	0.780591i	$0.714918\pi$
$138$	0	0
$139$	−5.80989	−0.492788	−0.246394	−	0.969170i	$-0.579246\pi$
$139$	−5.80989	−0.492788	−0.246394	+	0.969170i	$0.579246\pi$
$140$	0	0
$141$	−1.21529	−0.102345
$142$	0	0
$143$	−16.3592	−1.36802
$144$	0	0
$145$	−12.6315	−1.04899
$146$	0	0
$147$	10.4827	0.864597
$148$	0	0
$149$	−4.15689	−0.340546	−0.170273	−	0.985397i	$-0.554465\pi$
$149$	−4.15689	−0.340546	−0.170273	+	0.985397i	$0.554465\pi$
$150$	0	0
$151$	10.6480	0.866520	0.433260	−	0.901269i	$-0.357363\pi$
$151$	10.6480	0.866520	0.433260	+	0.901269i	$0.357363\pi$
$152$	0	0
$153$	2.01216	0.162674
$154$	0	0
$155$	3.16382	0.254124
$156$	0	0
$157$	20.5037	1.63637	0.818186	−	0.574953i	$-0.194980\pi$
$157$	20.5037	1.63637	0.818186	+	0.574953i	$0.194980\pi$
$158$	0	0
$159$	10.5536	0.836954
$160$	0	0
$161$	11.7141	0.923199
$162$	0	0
$163$	8.81524	0.690463	0.345231	−	0.938518i	$-0.387800\pi$
$163$	8.81524	0.690463	0.345231	+	0.938518i	$0.387800\pi$
$164$	0	0
$165$	17.7551	1.38223
$166$	0	0
$167$	6.50201	0.503141	0.251570	−	0.967839i	$-0.419053\pi$
$167$	6.50201	0.503141	0.251570	+	0.967839i	$0.419053\pi$
$168$	0	0
$169$	19.4582	1.49678
$170$	0	0
$171$	1.14324	0.0874256
$172$	0	0
$173$	10.7128	0.814479	0.407239	−	0.913321i	$-0.366491\pi$
$173$	10.7128	0.814479	0.407239	+	0.913321i	$0.366491\pi$
$174$	0	0
$175$	2.70663	0.204602
$176$	0	0
$177$	−35.4730	−2.66631
$178$	0	0
$179$	−12.3476	−0.922903	−0.461451	−	0.887165i	$-0.652671\pi$
$179$	−12.3476	−0.922903	−0.461451	+	0.887165i	$0.652671\pi$
$180$	0	0
$181$	−10.5557	−0.784599	−0.392300	−	0.919838i	$-0.628320\pi$
$181$	−10.5557	−0.784599	−0.392300	+	0.919838i	$0.628320\pi$
$182$	0	0
$183$	−46.4408	−3.43300
$184$	0	0
$185$	−22.6936	−1.66847
$186$	0	0
$187$	0.760489	0.0556124
$188$	0	0
$189$	29.0978	2.11656
$190$	0	0
$191$	13.7434	0.994440	0.497220	−	0.867624i	$-0.334354\pi$
$191$	13.7434	0.994440	0.497220	+	0.867624i	$0.334354\pi$
$192$	0	0
$193$	−20.4885	−1.47479	−0.737396	−	0.675460i	$-0.763945\pi$
$193$	−20.4885	−1.47479	−0.737396	+	0.675460i	$0.763945\pi$
$194$	0	0
$195$	−35.2279	−2.52272
$196$	0	0
$197$	17.0890	1.21754	0.608771	−	0.793346i	$-0.291663\pi$
$197$	17.0890	1.21754	0.608771	+	0.793346i	$0.291663\pi$
$198$	0	0
$199$	−25.6322	−1.81702	−0.908511	−	0.417861i	$-0.862780\pi$
$199$	−25.6322	−1.81702	−0.908511	+	0.417861i	$0.862780\pi$
$200$	0	0
$201$	−10.1245	−0.714130
$202$	0	0
$203$	−12.9292	−0.907449
$204$	0	0
$205$	9.08007	0.634180
$206$	0	0
$207$	−45.7761	−3.18166
$208$	0	0
$209$	0.432082	0.0298877
$210$	0	0
$211$	−17.7716	−1.22345	−0.611723	−	0.791072i	$-0.709523\pi$
$211$	−17.7716	−1.22345	−0.611723	+	0.791072i	$0.709523\pi$
$212$	0	0
$213$	45.7604	3.13545
$214$	0	0
$215$	8.72351	0.594939
$216$	0	0
$217$	3.23838	0.219836
$218$	0	0
$219$	−16.4823	−1.11377
$220$	0	0
$221$	−1.50888	−0.101499
$222$	0	0
$223$	23.8681	1.59833	0.799163	−	0.601115i	$-0.205277\pi$
$223$	23.8681	1.59833	0.799163	+	0.601115i	$0.205277\pi$
$224$	0	0
$225$	−10.5769	−0.705128
$226$	0	0
$227$	−19.2514	−1.27776	−0.638880	−	0.769306i	$-0.720602\pi$
$227$	−19.2514	−1.27776	−0.638880	+	0.769306i	$0.720602\pi$
$228$	0	0
$229$	8.36859	0.553012	0.276506	−	0.961012i	$-0.410823\pi$
$229$	8.36859	0.553012	0.276506	+	0.961012i	$0.410823\pi$
$230$	0	0
$231$	18.1735	1.19573
$232$	0	0
$233$	12.3597	0.809708	0.404854	−	0.914381i	$-0.367322\pi$
$233$	12.3597	0.809708	0.404854	+	0.914381i	$0.367322\pi$
$234$	0	0
$235$	−0.709089	−0.0462558
$236$	0	0
$237$	−45.1168	−2.93065
$238$	0	0
$239$	−17.9362	−1.16020	−0.580099	−	0.814546i	$-0.696986\pi$
$239$	−17.9362	−1.16020	−0.580099	+	0.814546i	$0.696986\pi$
$240$	0	0
$241$	−29.7508	−1.91642	−0.958208	−	0.286071i	$-0.907651\pi$
$241$	−29.7508	−1.91642	−0.958208	+	0.286071i	$0.907651\pi$
$242$	0	0
$243$	−39.5099	−2.53456
$244$	0	0
$245$	6.11638	0.390761
$246$	0	0
$247$	−0.857293	−0.0545482
$248$	0	0
$249$	23.4465	1.48586
$250$	0	0
$251$	−15.0311	−0.948756	−0.474378	−	0.880321i	$-0.657327\pi$
$251$	−15.0311	−0.948756	−0.474378	+	0.880321i	$0.657327\pi$
$252$	0	0
$253$	−17.3009	−1.08770
$254$	0	0
$255$	1.63764	0.102553
$256$	0	0
$257$	−7.01988	−0.437888	−0.218944	−	0.975737i	$-0.570261\pi$
$257$	−7.01988	−0.437888	−0.218944	+	0.975737i	$0.570261\pi$
$258$	0	0
$259$	−23.2284	−1.44334
$260$	0	0
$261$	50.5243	3.12738
$262$	0	0
$263$	−24.8230	−1.53065	−0.765326	−	0.643643i	$-0.777422\pi$
$263$	−24.8230	−1.53065	−0.765326	+	0.643643i	$0.777422\pi$
$264$	0	0
$265$	6.15776	0.378268
$266$	0	0
$267$	−58.4458	−3.57682
$268$	0	0
$269$	−4.48047	−0.273179	−0.136590	−	0.990628i	$-0.543614\pi$
$269$	−4.48047	−0.273179	−0.136590	+	0.990628i	$0.543614\pi$
$270$	0	0
$271$	7.70785	0.468218	0.234109	−	0.972210i	$-0.424783\pi$
$271$	7.70785	0.468218	0.234109	+	0.972210i	$0.424783\pi$
$272$	0	0
$273$	−36.0581	−2.18234
$274$	0	0
$275$	−3.99750	−0.241058
$276$	0	0
$277$	28.9263	1.73801	0.869007	−	0.494799i	$-0.164758\pi$
$277$	28.9263	1.73801	0.869007	+	0.494799i	$0.164758\pi$
$278$	0	0
$279$	−12.6549	−0.757628
$280$	0	0
$281$	24.8577	1.48288	0.741442	−	0.671017i	$-0.234142\pi$
$281$	24.8577	1.48288	0.741442	+	0.671017i	$0.234142\pi$
$282$	0	0
$283$	−24.7008	−1.46831	−0.734154	−	0.678983i	$-0.762421\pi$
$283$	−24.7008	−1.46831	−0.734154	+	0.678983i	$0.762421\pi$
$284$	0	0
$285$	0.930447	0.0551149
$286$	0	0
$287$	9.29405	0.548611
$288$	0	0
$289$	−16.9299	−0.995874
$290$	0	0
$291$	15.2058	0.891381
$292$	0	0
$293$	−16.3183	−0.953325	−0.476662	−	0.879086i	$-0.658154\pi$
$293$	−16.3183	−0.953325	−0.476662	+	0.879086i	$0.658154\pi$
$294$	0	0
$295$	−20.6976	−1.20506
$296$	0	0
$297$	−42.9754	−2.49369
$298$	0	0
$299$	34.3266	1.98516
$300$	0	0
$301$	8.92909	0.514664
$302$	0	0
$303$	−3.99496	−0.229504
$304$	0	0
$305$	−27.0970	−1.55157
$306$	0	0
$307$	−16.6020	−0.947525	−0.473762	−	0.880653i	$-0.657104\pi$
$307$	−16.6020	−0.947525	−0.473762	+	0.880653i	$0.657104\pi$
$308$	0	0
$309$	32.8096	1.86647
$310$	0	0
$311$	−13.8524	−0.785495	−0.392747	−	0.919646i	$-0.628475\pi$
$311$	−13.8524	−0.785495	−0.392747	+	0.919646i	$0.628475\pi$
$312$	0	0
$313$	−20.6010	−1.16444	−0.582218	−	0.813033i	$-0.697815\pi$
$313$	−20.6010	−1.16444	−0.582218	+	0.813033i	$0.697815\pi$
$314$	0	0
$315$	28.0564	1.58080
$316$	0	0
$317$	1.68101	0.0944147	0.0472073	−	0.998885i	$-0.484968\pi$
$317$	1.68101	0.0944147	0.0472073	+	0.998885i	$0.484968\pi$
$318$	0	0
$319$	19.0954	1.06914
$320$	0	0
$321$	−35.2210	−1.96585
$322$	0	0
$323$	0.0398530	0.00221748
$324$	0	0
$325$	7.93144	0.439957
$326$	0	0
$327$	−58.8560	−3.25474
$328$	0	0
$329$	−0.725799	−0.0400146
$330$	0	0
$331$	−0.0445026	−0.00244608	−0.00122304	−	0.999999i	$-0.500389\pi$
$331$	−0.0445026	−0.00244608	−0.00122304	+	0.999999i	$0.500389\pi$
$332$	0	0
$333$	90.7716	4.97425
$334$	0	0
$335$	−5.90742	−0.322757
$336$	0	0
$337$	24.2964	1.32351	0.661755	−	0.749720i	$-0.269812\pi$
$337$	24.2964	1.32351	0.661755	+	0.749720i	$0.269812\pi$
$338$	0	0
$339$	−14.4300	−0.783728
$340$	0	0
$341$	−4.78285	−0.259006
$342$	0	0
$343$	19.8699	1.07287
$344$	0	0
$345$	−37.2558	−2.00578
$346$	0	0
$347$	9.96838	0.535131	0.267565	−	0.963540i	$-0.413781\pi$
$347$	9.96838	0.535131	0.267565	+	0.963540i	$0.413781\pi$
$348$	0	0
$349$	−2.59904	−0.139123	−0.0695617	−	0.997578i	$-0.522160\pi$
$349$	−2.59904	−0.139123	−0.0695617	+	0.997578i	$0.522160\pi$
$350$	0	0
$351$	85.2675	4.55124
$352$	0	0
$353$	9.86509	0.525066	0.262533	−	0.964923i	$-0.415442\pi$
$353$	9.86509	0.525066	0.262533	+	0.964923i	$0.415442\pi$
$354$	0	0
$355$	26.7000	1.41709
$356$	0	0
$357$	1.67623	0.0887157
$358$	0	0
$359$	−16.8267	−0.888077	−0.444038	−	0.896008i	$-0.646455\pi$
$359$	−16.8267	−0.888077	−0.444038	+	0.896008i	$0.646455\pi$
$360$	0	0
$361$	−18.9774	−0.998808
$362$	0	0
$363$	8.96813	0.470705
$364$	0	0
$365$	−9.61700	−0.503377
$366$	0	0
$367$	3.82197	0.199505	0.0997525	−	0.995012i	$-0.468195\pi$
$367$	3.82197	0.199505	0.0997525	+	0.995012i	$0.468195\pi$
$368$	0	0
$369$	−36.3191	−1.89070
$370$	0	0
$371$	6.30287	0.327229
$372$	0	0
$373$	−27.4302	−1.42028	−0.710140	−	0.704060i	$-0.751368\pi$
$373$	−27.4302	−1.42028	−0.710140	+	0.704060i	$0.751368\pi$
$374$	0	0
$375$	−39.5251	−2.04107
$376$	0	0
$377$	−37.8872	−1.95129
$378$	0	0
$379$	−4.89613	−0.251497	−0.125749	−	0.992062i	$-0.540133\pi$
$379$	−4.89613	−0.251497	−0.125749	+	0.992062i	$0.540133\pi$
$380$	0	0
$381$	−33.9083	−1.73717
$382$	0	0
$383$	28.4284	1.45262	0.726312	−	0.687365i	$-0.241233\pi$
$383$	28.4284	1.45262	0.726312	+	0.687365i	$0.241233\pi$
$384$	0	0
$385$	10.6038	0.540420
$386$	0	0
$387$	−34.8929	−1.77371
$388$	0	0
$389$	−14.6206	−0.741293	−0.370646	−	0.928774i	$-0.620864\pi$
$389$	−14.6206	−0.741293	−0.370646	+	0.928774i	$0.620864\pi$
$390$	0	0
$391$	−1.59574	−0.0807002
$392$	0	0
$393$	29.7549	1.50094
$394$	0	0
$395$	−26.3245	−1.32453
$396$	0	0
$397$	−15.0317	−0.754420	−0.377210	−	0.926128i	$-0.623116\pi$
$397$	−15.0317	−0.754420	−0.377210	+	0.926128i	$0.623116\pi$
$398$	0	0
$399$	0.952374	0.0476784
$400$	0	0
$401$	−8.69181	−0.434048	−0.217024	−	0.976166i	$-0.569635\pi$
$401$	−8.69181	−0.434048	−0.217024	+	0.976166i	$0.569635\pi$
$402$	0	0
$403$	9.48966	0.472713
$404$	0	0
$405$	−49.2509	−2.44729
$406$	0	0
$407$	34.3067	1.70052
$408$	0	0
$409$	−8.05233	−0.398162	−0.199081	−	0.979983i	$-0.563796\pi$
$409$	−8.05233	−0.398162	−0.199081	+	0.979983i	$0.563796\pi$
$410$	0	0
$411$	47.6322	2.34952
$412$	0	0
$413$	−21.1854	−1.04246
$414$	0	0
$415$	13.6805	0.671547
$416$	0	0
$417$	18.9134	0.926192
$418$	0	0
$419$	11.7729	0.575145	0.287572	−	0.957759i	$-0.407152\pi$
$419$	11.7729	0.575145	0.287572	+	0.957759i	$0.407152\pi$
$420$	0	0
$421$	−10.2287	−0.498519	−0.249259	−	0.968437i	$-0.580187\pi$
$421$	−10.2287	−0.498519	−0.249259	+	0.968437i	$0.580187\pi$
$422$	0	0
$423$	2.83626	0.137904
$424$	0	0
$425$	−0.368709	−0.0178850
$426$	0	0
$427$	−27.7356	−1.34222
$428$	0	0
$429$	53.2552	2.57119
$430$	0	0
$431$	7.93027	0.381988	0.190994	−	0.981591i	$-0.438829\pi$
$431$	7.93027	0.381988	0.190994	+	0.981591i	$0.438829\pi$
$432$	0	0
$433$	−4.88646	−0.234828	−0.117414	−	0.993083i	$-0.537460\pi$
$433$	−4.88646	−0.234828	−0.117414	+	0.993083i	$0.537460\pi$
$434$	0	0
$435$	41.1202	1.97156
$436$	0	0
$437$	−0.906642	−0.0433706
$438$	0	0
$439$	−29.3402	−1.40033	−0.700167	−	0.713980i	$-0.746891\pi$
$439$	−29.3402	−1.40033	−0.700167	+	0.713980i	$0.746891\pi$
$440$	0	0
$441$	−24.4648	−1.16499
$442$	0	0
$443$	−6.18870	−0.294034	−0.147017	−	0.989134i	$-0.546967\pi$
$443$	−6.18870	−0.294034	−0.147017	+	0.989134i	$0.546967\pi$
$444$	0	0
$445$	−34.1016	−1.61657
$446$	0	0
$447$	13.5322	0.640053
$448$	0	0
$449$	3.70276	0.174744	0.0873720	−	0.996176i	$-0.472153\pi$
$449$	3.70276	0.174744	0.0873720	+	0.996176i	$0.472153\pi$
$450$	0	0
$451$	−13.7267	−0.646363
$452$	0	0
$453$	−34.6632	−1.62862
$454$	0	0
$455$	−21.0390	−0.986323
$456$	0	0
$457$	6.34604	0.296855	0.148428	−	0.988923i	$-0.452579\pi$
$457$	6.34604	0.296855	0.148428	+	0.988923i	$0.452579\pi$
$458$	0	0
$459$	−3.96383	−0.185016
$460$	0	0
$461$	10.7694	0.501581	0.250790	−	0.968041i	$-0.419310\pi$
$461$	10.7694	0.501581	0.250790	+	0.968041i	$0.419310\pi$
$462$	0	0
$463$	−10.3467	−0.480853	−0.240427	−	0.970667i	$-0.577287\pi$
$463$	−10.3467	−0.480853	−0.240427	+	0.970667i	$0.577287\pi$
$464$	0	0
$465$	−10.2994	−0.477625
$466$	0	0
$467$	−9.90174	−0.458198	−0.229099	−	0.973403i	$-0.573578\pi$
$467$	−9.90174	−0.458198	−0.229099	+	0.973403i	$0.573578\pi$
$468$	0	0
$469$	−6.04663	−0.279208
$470$	0	0
$471$	−66.7472	−3.07555
$472$	0	0
$473$	−13.1876	−0.606368
$474$	0	0
$475$	−0.209487	−0.00961192
$476$	0	0
$477$	−24.6302	−1.12774
$478$	0	0
$479$	34.5995	1.58089	0.790445	−	0.612533i	$-0.209849\pi$
$479$	34.5995	1.58089	0.790445	+	0.612533i	$0.209849\pi$
$480$	0	0
$481$	−68.0679	−3.10363
$482$	0	0
$483$	−38.1337	−1.73515
$484$	0	0
$485$	8.87221	0.402866
$486$	0	0
$487$	16.1438	0.731544	0.365772	−	0.930704i	$-0.380805\pi$
$487$	16.1438	0.731544	0.365772	+	0.930704i	$0.380805\pi$
$488$	0	0
$489$	−28.6969	−1.29772
$490$	0	0
$491$	6.43857	0.290569	0.145284	−	0.989390i	$-0.453590\pi$
$491$	6.43857	0.290569	0.145284	+	0.989390i	$0.453590\pi$
$492$	0	0
$493$	1.76126	0.0793234
$494$	0	0
$495$	−41.4373	−1.86247
$496$	0	0
$497$	27.3292	1.22588
$498$	0	0
$499$	11.1805	0.500508	0.250254	−	0.968180i	$-0.419486\pi$
$499$	11.1805	0.500508	0.250254	+	0.968180i	$0.419486\pi$
$500$	0	0
$501$	−21.1665	−0.945649
$502$	0	0
$503$	−15.2723	−0.680957	−0.340478	−	0.940252i	$-0.610589\pi$
$503$	−15.2723	−0.680957	−0.340478	+	0.940252i	$0.610589\pi$
$504$	0	0
$505$	−2.33096	−0.103726
$506$	0	0
$507$	−63.3438	−2.81319
$508$	0	0
$509$	−25.1006	−1.11256	−0.556282	−	0.830994i	$-0.687772\pi$
$509$	−25.1006	−1.11256	−0.556282	+	0.830994i	$0.687772\pi$
$510$	0	0
$511$	−9.84364	−0.435457
$512$	0	0
$513$	−2.25210	−0.0994328
$514$	0	0
$515$	19.1436	0.843566
$516$	0	0
$517$	1.07195	0.0471444
$518$	0	0
$519$	−34.8742	−1.53081
$520$	0	0
$521$	−12.8602	−0.563417	−0.281708	−	0.959500i	$-0.590901\pi$
$521$	−12.8602	−0.563417	−0.281708	+	0.959500i	$0.590901\pi$
$522$	0	0
$523$	18.1990	0.795785	0.397892	−	0.917432i	$-0.369742\pi$
$523$	18.1990	0.795785	0.397892	+	0.917432i	$0.369742\pi$
$524$	0	0
$525$	−8.81111	−0.384548
$526$	0	0
$527$	−0.441146	−0.0192166
$528$	0	0
$529$	13.3026	0.578374
$530$	0	0
$531$	82.7878	3.59268
$532$	0	0
$533$	27.2350	1.17968
$534$	0	0
$535$	−20.5506	−0.888480
$536$	0	0
$537$	40.1961	1.73459
$538$	0	0
$539$	−9.24634	−0.398268
$540$	0	0
$541$	21.6770	0.931966	0.465983	−	0.884794i	$-0.345701\pi$
$541$	21.6770	0.931966	0.465983	+	0.884794i	$0.345701\pi$
$542$	0	0
$543$	34.3628	1.47465
$544$	0	0
$545$	−34.3410	−1.47101
$546$	0	0
$547$	−30.1348	−1.28847	−0.644235	−	0.764827i	$-0.722824\pi$
$547$	−30.1348	−1.28847	−0.644235	+	0.764827i	$0.722824\pi$
$548$	0	0
$549$	108.385	4.62574
$550$	0	0
$551$	1.00069	0.0426307
$552$	0	0
$553$	−26.9449	−1.14581
$554$	0	0
$555$	73.8763	3.13588
$556$	0	0
$557$	7.90863	0.335099	0.167550	−	0.985864i	$-0.446415\pi$
$557$	7.90863	0.335099	0.167550	+	0.985864i	$0.446415\pi$
$558$	0	0
$559$	26.1656	1.10669
$560$	0	0
$561$	−2.47568	−0.104523
$562$	0	0
$563$	−10.1074	−0.425976	−0.212988	−	0.977055i	$-0.568319\pi$
$563$	−10.1074	−0.425976	−0.212988	+	0.977055i	$0.568319\pi$
$564$	0	0
$565$	−8.41952	−0.354212
$566$	0	0
$567$	−50.4115	−2.11708
$568$	0	0
$569$	−32.7454	−1.37276	−0.686380	−	0.727243i	$-0.740801\pi$
$569$	−32.7454	−1.37276	−0.686380	+	0.727243i	$0.740801\pi$
$570$	0	0
$571$	10.4423	0.436997	0.218499	−	0.975837i	$-0.429884\pi$
$571$	10.4423	0.436997	0.218499	+	0.975837i	$0.429884\pi$
$572$	0	0
$573$	−44.7401	−1.86904
$574$	0	0
$575$	8.38800	0.349804
$576$	0	0
$577$	16.9248	0.704591	0.352295	−	0.935889i	$-0.385401\pi$
$577$	16.9248	0.704591	0.352295	+	0.935889i	$0.385401\pi$
$578$	0	0
$579$	66.6977	2.77186
$580$	0	0
$581$	14.0028	0.580936
$582$	0	0
$583$	−9.30889	−0.385535
$584$	0	0
$585$	82.2158	3.39920
$586$	0	0
$587$	30.4346	1.25617	0.628086	−	0.778144i	$-0.283839\pi$
$587$	30.4346	1.25617	0.628086	+	0.778144i	$0.283839\pi$
$588$	0	0
$589$	−0.250643	−0.0103276
$590$	0	0
$591$	−55.6312	−2.28836
$592$	0	0
$593$	15.7745	0.647783	0.323891	−	0.946094i	$-0.395009\pi$
$593$	15.7745	0.647783	0.323891	+	0.946094i	$0.395009\pi$
$594$	0	0
$595$	0.978040	0.0400957
$596$	0	0
$597$	83.4426	3.41508
$598$	0	0
$599$	27.8825	1.13925	0.569623	−	0.821906i	$-0.307089\pi$
$599$	27.8825	1.13925	0.569623	+	0.821906i	$0.307089\pi$
$600$	0	0
$601$	−14.9743	−0.610813	−0.305406	−	0.952222i	$-0.598792\pi$
$601$	−14.9743	−0.610813	−0.305406	+	0.952222i	$0.598792\pi$
$602$	0	0
$603$	23.6289	0.962244
$604$	0	0
$605$	5.23268	0.212739
$606$	0	0
$607$	33.5976	1.36369	0.681843	−	0.731498i	$-0.261179\pi$
$607$	33.5976	1.36369	0.681843	+	0.731498i	$0.261179\pi$
$608$	0	0
$609$	42.0893	1.70554
$610$	0	0
$611$	−2.12686	−0.0860436
$612$	0	0
$613$	42.7245	1.72563	0.862814	−	0.505522i	$-0.168700\pi$
$613$	42.7245	1.72563	0.862814	+	0.505522i	$0.168700\pi$
$614$	0	0
$615$	−29.5591	−1.19194
$616$	0	0
$617$	−39.1632	−1.57665	−0.788326	−	0.615258i	$-0.789052\pi$
$617$	−39.1632	−1.57665	−0.788326	+	0.615258i	$0.789052\pi$
$618$	0	0
$619$	−32.8180	−1.31907	−0.659534	−	0.751675i	$-0.729246\pi$
$619$	−32.8180	−1.31907	−0.659534	+	0.751675i	$0.729246\pi$
$620$	0	0
$621$	90.1758	3.61863
$622$	0	0
$623$	−34.9053	−1.39845
$624$	0	0
$625$	−16.1011	−0.644043
$626$	0	0
$627$	−1.40659	−0.0561738
$628$	0	0
$629$	3.16428	0.126168
$630$	0	0
$631$	23.3532	0.929676	0.464838	−	0.885396i	$-0.346113\pi$
$631$	23.3532	0.929676	0.464838	+	0.885396i	$0.346113\pi$
$632$	0	0
$633$	57.8532	2.29946
$634$	0	0
$635$	−19.7846	−0.785129
$636$	0	0
$637$	18.3457	0.726882
$638$	0	0
$639$	−106.797	−4.22481
$640$	0	0
$641$	25.4903	1.00680	0.503402	−	0.864052i	$-0.332081\pi$
$641$	25.4903	1.00680	0.503402	+	0.864052i	$0.332081\pi$
$642$	0	0
$643$	−26.8236	−1.05782	−0.528909	−	0.848678i	$-0.677399\pi$
$643$	−26.8236	−1.05782	−0.528909	+	0.848678i	$0.677399\pi$
$644$	0	0
$645$	−28.3983	−1.11818
$646$	0	0
$647$	23.2640	0.914604	0.457302	−	0.889311i	$-0.348816\pi$
$647$	23.2640	0.914604	0.457302	+	0.889311i	$0.348816\pi$
$648$	0	0
$649$	31.2893	1.22821
$650$	0	0
$651$	−10.5421	−0.413179
$652$	0	0
$653$	15.4286	0.603767	0.301884	−	0.953345i	$-0.402385\pi$
$653$	15.4286	0.603767	0.301884	+	0.953345i	$0.402385\pi$
$654$	0	0
$655$	17.3612	0.678360
$656$	0	0
$657$	38.4668	1.50073
$658$	0	0
$659$	18.5481	0.722531	0.361265	−	0.932463i	$-0.382345\pi$
$659$	18.5481	0.722531	0.361265	+	0.932463i	$0.382345\pi$
$660$	0	0
$661$	−32.2144	−1.25300	−0.626498	−	0.779423i	$-0.715512\pi$
$661$	−32.2144	−1.25300	−0.626498	+	0.779423i	$0.715512\pi$
$662$	0	0
$663$	4.91199	0.190766
$664$	0	0
$665$	0.555687	0.0215486
$666$	0	0
$667$	−40.0682	−1.55145
$668$	0	0
$669$	−77.6996	−3.00404
$670$	0	0
$671$	40.9635	1.58138
$672$	0	0
$673$	−34.1127	−1.31495	−0.657475	−	0.753476i	$-0.728375\pi$
$673$	−34.1127	−1.31495	−0.657475	+	0.753476i	$0.728375\pi$
$674$	0	0
$675$	20.8358	0.801972
$676$	0	0
$677$	3.62537	0.139334	0.0696671	−	0.997570i	$-0.477806\pi$
$677$	3.62537	0.139334	0.0696671	+	0.997570i	$0.477806\pi$
$678$	0	0
$679$	9.08130	0.348508
$680$	0	0
$681$	62.6706	2.40154
$682$	0	0
$683$	43.5808	1.66757	0.833787	−	0.552087i	$-0.186168\pi$
$683$	43.5808	1.66757	0.833787	+	0.552087i	$0.186168\pi$
$684$	0	0
$685$	27.7922	1.06189
$686$	0	0
$687$	−27.2429	−1.03938
$688$	0	0
$689$	18.4698	0.703642
$690$	0	0
$691$	−2.79869	−0.106467	−0.0532337	−	0.998582i	$-0.516953\pi$
$691$	−2.79869	−0.106467	−0.0532337	+	0.998582i	$0.516953\pi$
$692$	0	0
$693$	−42.4139	−1.61117
$694$	0	0
$695$	11.0355	0.418600
$696$	0	0
$697$	−1.26608	−0.0479560
$698$	0	0
$699$	−40.2353	−1.52184
$700$	0	0
$701$	2.55077	0.0963412	0.0481706	−	0.998839i	$-0.484661\pi$
$701$	2.55077	0.0963412	0.0481706	+	0.998839i	$0.484661\pi$
$702$	0	0
$703$	1.79783	0.0678063
$704$	0	0
$705$	2.30835	0.0869375
$706$	0	0
$707$	−2.38589	−0.0897306
$708$	0	0
$709$	10.9622	0.411693	0.205846	−	0.978584i	$-0.434005\pi$
$709$	10.9622	0.411693	0.205846	+	0.978584i	$0.434005\pi$
$710$	0	0
$711$	105.295	3.94886
$712$	0	0
$713$	10.0359	0.375848
$714$	0	0
$715$	31.0731	1.16207
$716$	0	0
$717$	58.3891	2.18058
$718$	0	0
$719$	−45.2556	−1.68775	−0.843875	−	0.536539i	$-0.819731\pi$
$719$	−45.2556	−1.68775	−0.843875	+	0.536539i	$0.819731\pi$
$720$	0	0
$721$	19.5947	0.729745
$722$	0	0
$723$	96.8500	3.60189
$724$	0	0
$725$	−9.25807	−0.343836
$726$	0	0
$727$	−2.98672	−0.110771	−0.0553856	−	0.998465i	$-0.517639\pi$
$727$	−2.98672	−0.110771	−0.0553856	+	0.998465i	$0.517639\pi$
$728$	0	0
$729$	50.8320	1.88267
$730$	0	0
$731$	−1.21636	−0.0449887
$732$	0	0
$733$	−42.9041	−1.58470	−0.792350	−	0.610067i	$-0.791142\pi$
$733$	−42.9041	−1.58470	−0.792350	+	0.610067i	$0.791142\pi$
$734$	0	0
$735$	−19.9111	−0.734433
$736$	0	0
$737$	8.93044	0.328957
$738$	0	0
$739$	−4.92719	−0.181249	−0.0906247	−	0.995885i	$-0.528886\pi$
$739$	−4.92719	−0.181249	−0.0906247	+	0.995885i	$0.528886\pi$
$740$	0	0
$741$	2.79081	0.102523
$742$	0	0
$743$	21.8293	0.800841	0.400420	−	0.916332i	$-0.368864\pi$
$743$	21.8293	0.800841	0.400420	+	0.916332i	$0.368864\pi$
$744$	0	0
$745$	7.89572	0.289277
$746$	0	0
$747$	−54.7201	−2.00210
$748$	0	0
$749$	−21.0349	−0.768598
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	48.9319	1.78318
$754$	0	0
$755$	−20.2251	−0.736067
$756$	0	0
$757$	−37.4801	−1.36224	−0.681118	−	0.732173i	$-0.738506\pi$
$757$	−37.4801	−1.36224	−0.681118	+	0.732173i	$0.738506\pi$
$758$	0	0
$759$	56.3208	2.04432
$760$	0	0
$761$	7.99404	0.289784	0.144892	−	0.989448i	$-0.453717\pi$
$761$	7.99404	0.289784	0.144892	+	0.989448i	$0.453717\pi$
$762$	0	0
$763$	−35.1503	−1.27252
$764$	0	0
$765$	−3.82197	−0.138184
$766$	0	0
$767$	−62.0810	−2.24162
$768$	0	0
$769$	−4.56017	−0.164444	−0.0822220	−	0.996614i	$-0.526202\pi$
$769$	−4.56017	−0.164444	−0.0822220	+	0.996614i	$0.526202\pi$
$770$	0	0
$771$	22.8523	0.823007
$772$	0	0
$773$	−21.0761	−0.758053	−0.379027	−	0.925386i	$-0.623741\pi$
$773$	−21.0761	−0.758053	−0.379027	+	0.925386i	$0.623741\pi$
$774$	0	0
$775$	2.31888	0.0832966
$776$	0	0
$777$	75.6173	2.71276
$778$	0	0
$779$	−0.719338	−0.0257730
$780$	0	0
$781$	−40.3633	−1.44431
$782$	0	0
$783$	−99.5295	−3.55690
$784$	0	0
$785$	−38.9453	−1.39002
$786$	0	0
$787$	−25.3383	−0.903211	−0.451606	−	0.892218i	$-0.649149\pi$
$787$	−25.3383	−0.903211	−0.451606	+	0.892218i	$0.649149\pi$
$788$	0	0
$789$	80.8082	2.87685
$790$	0	0
$791$	−8.61793	−0.306418
$792$	0	0
$793$	−81.2756	−2.88618
$794$	0	0
$795$	−20.0458	−0.710952
$796$	0	0
$797$	41.5673	1.47239	0.736194	−	0.676771i	$-0.236621\pi$
$797$	41.5673	1.47239	0.736194	+	0.676771i	$0.236621\pi$
$798$	0	0
$799$	0.0988714	0.00349782
$800$	0	0
$801$	136.402	4.81954
$802$	0	0
$803$	14.5383	0.513047
$804$	0	0
$805$	−22.2501	−0.784212
$806$	0	0
$807$	14.5856	0.513438
$808$	0	0
$809$	−39.4663	−1.38756	−0.693781	−	0.720186i	$-0.744056\pi$
$809$	−39.4663	−1.38756	−0.693781	+	0.720186i	$0.744056\pi$
$810$	0	0
$811$	−26.1658	−0.918806	−0.459403	−	0.888228i	$-0.651937\pi$
$811$	−26.1658	−0.918806	−0.459403	+	0.888228i	$0.651937\pi$
$812$	0	0
$813$	−25.0920	−0.880013
$814$	0	0
$815$	−16.7439	−0.586515
$816$	0	0
$817$	−0.691091	−0.0241782
$818$	0	0
$819$	84.1533	2.94055
$820$	0	0
$821$	−44.3337	−1.54726	−0.773629	−	0.633639i	$-0.781560\pi$
$821$	−44.3337	−1.54726	−0.773629	+	0.633639i	$0.781560\pi$
$822$	0	0
$823$	−17.1910	−0.599242	−0.299621	−	0.954058i	$-0.596860\pi$
$823$	−17.1910	−0.599242	−0.299621	+	0.954058i	$0.596860\pi$
$824$	0	0
$825$	13.0134	0.453068
$826$	0	0
$827$	−34.8054	−1.21030	−0.605152	−	0.796110i	$-0.706887\pi$
$827$	−34.8054	−1.21030	−0.605152	+	0.796110i	$0.706887\pi$
$828$	0	0
$829$	11.4690	0.398335	0.199168	−	0.979965i	$-0.436176\pi$
$829$	11.4690	0.398335	0.199168	+	0.979965i	$0.436176\pi$
$830$	0	0
$831$	−94.1661	−3.26659
$832$	0	0
$833$	−0.852835	−0.0295490
$834$	0	0
$835$	−12.3501	−0.427393
$836$	0	0
$837$	24.9293	0.861682
$838$	0	0
$839$	40.4858	1.39773	0.698863	−	0.715255i	$-0.253690\pi$
$839$	40.4858	1.39773	0.698863	+	0.715255i	$0.253690\pi$
$840$	0	0
$841$	15.2244	0.524978
$842$	0	0
$843$	−80.9210	−2.78707
$844$	0	0
$845$	−36.9595	−1.27145
$846$	0	0
$847$	5.35599	0.184034
$848$	0	0
$849$	80.4103	2.75967
$850$	0	0
$851$	−71.9862	−2.46766
$852$	0	0
$853$	−42.9963	−1.47216	−0.736082	−	0.676893i	$-0.763326\pi$
$853$	−42.9963	−1.47216	−0.736082	+	0.676893i	$0.763326\pi$
$854$	0	0
$855$	−2.17150	−0.0742638
$856$	0	0
$857$	−37.6398	−1.28575	−0.642875	−	0.765971i	$-0.722259\pi$
$857$	−37.6398	−1.28575	−0.642875	+	0.765971i	$0.722259\pi$
$858$	0	0
$859$	10.1473	0.346220	0.173110	−	0.984902i	$-0.444618\pi$
$859$	10.1473	0.346220	0.173110	+	0.984902i	$0.444618\pi$
$860$	0	0
$861$	−30.2557	−1.03111
$862$	0	0
$863$	−8.98553	−0.305871	−0.152935	−	0.988236i	$-0.548873\pi$
$863$	−8.98553	−0.305871	−0.152935	+	0.988236i	$0.548873\pi$
$864$	0	0
$865$	−20.3482	−0.691860
$866$	0	0
$867$	55.1131	1.87174
$868$	0	0
$869$	39.7957	1.34998
$870$	0	0
$871$	−17.7189	−0.600381
$872$	0	0
$873$	−35.4877	−1.20108
$874$	0	0
$875$	−23.6054	−0.798007
$876$	0	0
$877$	−40.0623	−1.35281	−0.676404	−	0.736531i	$-0.736463\pi$
$877$	−40.0623	−1.35281	−0.676404	+	0.736531i	$0.736463\pi$
$878$	0	0
$879$	53.1222	1.79177
$880$	0	0
$881$	16.1564	0.544322	0.272161	−	0.962252i	$-0.412262\pi$
$881$	16.1564	0.544322	0.272161	+	0.962252i	$0.412262\pi$
$882$	0	0
$883$	8.24155	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$883$	8.24155	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$884$	0	0
$885$	67.3785	2.26490
$886$	0	0
$887$	−13.0485	−0.438127	−0.219063	−	0.975711i	$-0.570300\pi$
$887$	−13.0485	−0.438127	−0.219063	+	0.975711i	$0.570300\pi$
$888$	0	0
$889$	−20.2509	−0.679193
$890$	0	0
$891$	74.4542	2.49431
$892$	0	0
$893$	0.0561751	0.00187983
$894$	0	0
$895$	23.4534	0.783961
$896$	0	0
$897$	−111.746	−3.73109
$898$	0	0
$899$	−11.0769	−0.369436
$900$	0	0
$901$	−0.858604	−0.0286042
$902$	0	0
$903$	−29.0676	−0.967308
$904$	0	0
$905$	20.0498	0.666479
$906$	0	0
$907$	43.5156	1.44491	0.722456	−	0.691417i	$-0.243013\pi$
$907$	43.5156	1.44491	0.722456	+	0.691417i	$0.243013\pi$
$908$	0	0
$909$	9.32353	0.309242
$910$	0	0
$911$	40.1001	1.32858	0.664288	−	0.747477i	$-0.268735\pi$
$911$	40.1001	1.32858	0.664288	+	0.747477i	$0.268735\pi$
$912$	0	0
$913$	−20.6812	−0.684448
$914$	0	0
$915$	88.2110	2.91617
$916$	0	0
$917$	17.7704	0.586830
$918$	0	0
$919$	25.2636	0.833368	0.416684	−	0.909051i	$-0.363192\pi$
$919$	25.2636	0.833368	0.416684	+	0.909051i	$0.363192\pi$
$920$	0	0
$921$	54.0457	1.78087
$922$	0	0
$923$	80.0849	2.63602
$924$	0	0
$925$	−16.6330	−0.546889
$926$	0	0
$927$	−76.5718	−2.51495
$928$	0	0
$929$	−2.98452	−0.0979189	−0.0489595	−	0.998801i	$-0.515591\pi$
$929$	−2.98452	−0.0979189	−0.0489595	+	0.998801i	$0.515591\pi$
$930$	0	0
$931$	−0.484550	−0.0158805
$932$	0	0
$933$	45.0946	1.47633
$934$	0	0
$935$	−1.44450	−0.0472401
$936$	0	0
$937$	30.6853	1.00244	0.501222	−	0.865319i	$-0.332884\pi$
$937$	30.6853	1.00244	0.501222	+	0.865319i	$0.332884\pi$
$938$	0	0
$939$	67.0639	2.18855
$940$	0	0
$941$	−17.8260	−0.581112	−0.290556	−	0.956858i	$-0.593840\pi$
$941$	−17.8260	−0.581112	−0.290556	+	0.956858i	$0.593840\pi$
$942$	0	0
$943$	28.8028	0.937948
$944$	0	0
$945$	−55.2693	−1.79791
$946$	0	0
$947$	−50.5741	−1.64344	−0.821719	−	0.569892i	$-0.806985\pi$
$947$	−50.5741	−1.64344	−0.821719	+	0.569892i	$0.806985\pi$
$948$	0	0
$949$	−28.8455	−0.936365
$950$	0	0
$951$	−5.47231	−0.177452
$952$	0	0
$953$	3.94969	0.127943	0.0639715	−	0.997952i	$-0.479623\pi$
$953$	3.94969	0.127943	0.0639715	+	0.997952i	$0.479623\pi$
$954$	0	0
$955$	−26.1047	−0.844729
$956$	0	0
$957$	−62.1629	−2.00944
$958$	0	0
$959$	28.4471	0.918606
$960$	0	0
$961$	−28.2256	−0.910502
$962$	0	0
$963$	82.1997	2.64885
$964$	0	0
$965$	38.9164	1.25276
$966$	0	0
$967$	20.5721	0.661553	0.330777	−	0.943709i	$-0.392689\pi$
$967$	20.5721	0.661553	0.330777	+	0.943709i	$0.392689\pi$
$968$	0	0
$969$	−0.129737	−0.00416774
$970$	0	0
$971$	31.1573	0.999885	0.499942	−	0.866059i	$-0.333355\pi$
$971$	31.1573	0.999885	0.499942	+	0.866059i	$0.333355\pi$
$972$	0	0
$973$	11.2955	0.362118
$974$	0	0
$975$	−25.8198	−0.826896
$976$	0	0
$977$	33.8762	1.08380	0.541898	−	0.840444i	$-0.317706\pi$
$977$	33.8762	1.08380	0.541898	+	0.840444i	$0.317706\pi$
$978$	0	0
$979$	51.5526	1.64763
$980$	0	0
$981$	137.359	4.38555
$982$	0	0
$983$	−50.3897	−1.60718	−0.803592	−	0.595181i	$-0.797080\pi$
$983$	−50.3897	−1.60718	−0.803592	+	0.595181i	$0.797080\pi$
$984$	0	0
$985$	−32.4594	−1.03424
$986$	0	0
$987$	2.36275	0.0752071
$988$	0	0
$989$	27.6718	0.879911
$990$	0	0
$991$	10.6068	0.336937	0.168469	−	0.985707i	$-0.446118\pi$
$991$	10.6068	0.336937	0.168469	+	0.985707i	$0.446118\pi$
$992$	0	0
$993$	0.144873	0.00459740
$994$	0	0
$995$	48.6867	1.54347
$996$	0	0
$997$	−49.8630	−1.57918	−0.789588	−	0.613637i	$-0.789706\pi$
$997$	−49.8630	−1.57918	−0.789588	+	0.613637i	$0.789706\pi$
$998$	0	0
$999$	−178.814	−5.65743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.3	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-3.25538 q^{3} -1.89943 q^{5} -1.94419 q^{7} +7.59748 q^{9} +O(q^{10})\)	\(q-3.25538 q^{3} -1.89943 q^{5} -1.94419 q^{7} +7.59748 q^{9} +2.87143 q^{11} -5.69721 q^{13} +6.18336 q^{15} +0.264846 q^{17} +0.150476 q^{19} +6.32908 q^{21} -6.02516 q^{23} -1.39216 q^{25} -14.9665 q^{27} +6.65014 q^{29} -1.66567 q^{31} -9.34760 q^{33} +3.69286 q^{35} +11.9476 q^{37} +18.5466 q^{39} -4.78042 q^{41} -4.59270 q^{43} -14.4309 q^{45} +0.373316 q^{47} -3.22011 q^{49} -0.862175 q^{51} -3.24190 q^{53} -5.45409 q^{55} -0.489856 q^{57} +10.8967 q^{59} +14.2659 q^{61} -14.7710 q^{63} +10.8215 q^{65} +3.11010 q^{67} +19.6142 q^{69} -14.0569 q^{71} +5.06310 q^{73} +4.53201 q^{75} -5.58262 q^{77} +13.8592 q^{79} +25.9293 q^{81} -7.20240 q^{83} -0.503057 q^{85} -21.6487 q^{87} +17.9536 q^{89} +11.0765 q^{91} +5.42238 q^{93} -0.285819 q^{95} -4.67098 q^{97} +21.8157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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

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Database

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