LMFDB - Embedded newform 6008.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.65403 q^{3} -0.749159 q^{5} -1.02284 q^{7} +4.04389 q^{9} +O(q^{10})$	$q-2.65403 q^{3} -0.749159 q^{5} -1.02284 q^{7} +4.04389 q^{9} +0.361430 q^{11} -5.83867 q^{13} +1.98829 q^{15} +0.216744 q^{17} -6.45487 q^{19} +2.71464 q^{21} +2.08363 q^{23} -4.43876 q^{25} -2.77053 q^{27} -5.90314 q^{29} +3.13185 q^{31} -0.959248 q^{33} +0.766266 q^{35} -7.09361 q^{37} +15.4960 q^{39} +3.00347 q^{41} -10.4797 q^{43} -3.02952 q^{45} +10.7800 q^{47} -5.95381 q^{49} -0.575245 q^{51} +9.95472 q^{53} -0.270769 q^{55} +17.1314 q^{57} -5.38839 q^{59} -11.0695 q^{61} -4.13624 q^{63} +4.37409 q^{65} -6.48381 q^{67} -5.53002 q^{69} +8.86113 q^{71} -3.64833 q^{73} +11.7806 q^{75} -0.369684 q^{77} -11.5634 q^{79} -4.77860 q^{81} +1.31671 q^{83} -0.162376 q^{85} +15.6671 q^{87} -3.00602 q^{89} +5.97200 q^{91} -8.31203 q^{93} +4.83572 q^{95} -12.2669 q^{97} +1.46159 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * 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
$3$	−2.65403	−1.53231	−0.766153	−	0.642658i	$-0.777832\pi$
$3$	−2.65403	−1.53231	−0.766153	+	0.642658i	$0.777832\pi$
$4$	0	0
$5$	−0.749159	−0.335034	−0.167517	−	0.985869i	$-0.553575\pi$
$5$	−0.749159	−0.335034	−0.167517	+	0.985869i	$0.553575\pi$
$6$	0	0
$7$	−1.02284	−0.386596	−0.193298	−	0.981140i	$-0.561918\pi$
$7$	−1.02284	−0.386596	−0.193298	+	0.981140i	$0.561918\pi$
$8$	0	0
$9$	4.04389	1.34796
$10$	0	0
$11$	0.361430	0.108975	0.0544876	−	0.998514i	$-0.482647\pi$
$11$	0.361430	0.108975	0.0544876	+	0.998514i	$0.482647\pi$
$12$	0	0
$13$	−5.83867	−1.61936	−0.809678	−	0.586875i	$-0.800358\pi$
$13$	−5.83867	−1.61936	−0.809678	+	0.586875i	$0.800358\pi$
$14$	0	0
$15$	1.98829	0.513375
$16$	0	0
$17$	0.216744	0.0525681	0.0262840	−	0.999655i	$-0.491633\pi$
$17$	0.216744	0.0525681	0.0262840	+	0.999655i	$0.491633\pi$
$18$	0	0
$19$	−6.45487	−1.48085	−0.740424	−	0.672140i	$-0.765375\pi$
$19$	−6.45487	−1.48085	−0.740424	+	0.672140i	$0.765375\pi$
$20$	0	0
$21$	2.71464	0.592383
$22$	0	0
$23$	2.08363	0.434467	0.217233	−	0.976120i	$-0.430297\pi$
$23$	2.08363	0.434467	0.217233	+	0.976120i	$0.430297\pi$
$24$	0	0
$25$	−4.43876	−0.887752
$26$	0	0
$27$	−2.77053	−0.533189
$28$	0	0
$29$	−5.90314	−1.09619	−0.548093	−	0.836417i	$-0.684646\pi$
$29$	−5.90314	−1.09619	−0.548093	+	0.836417i	$0.684646\pi$
$30$	0	0
$31$	3.13185	0.562496	0.281248	−	0.959635i	$-0.409252\pi$
$31$	3.13185	0.562496	0.281248	+	0.959635i	$0.409252\pi$
$32$	0	0
$33$	−0.959248	−0.166984
$34$	0	0
$35$	0.766266	0.129523
$36$	0	0
$37$	−7.09361	−1.16618	−0.583091	−	0.812407i	$-0.698157\pi$
$37$	−7.09361	−1.16618	−0.583091	+	0.812407i	$0.698157\pi$
$38$	0	0
$39$	15.4960	2.48135
$40$	0	0
$41$	3.00347	0.469063	0.234531	−	0.972109i	$-0.424644\pi$
$41$	3.00347	0.469063	0.234531	+	0.972109i	$0.424644\pi$
$42$	0	0
$43$	−10.4797	−1.59814	−0.799072	−	0.601235i	$-0.794676\pi$
$43$	−10.4797	−1.59814	−0.799072	+	0.601235i	$0.794676\pi$
$44$	0	0
$45$	−3.02952	−0.451614
$46$	0	0
$47$	10.7800	1.57242	0.786211	−	0.617958i	$-0.212040\pi$
$47$	10.7800	1.57242	0.786211	+	0.617958i	$0.212040\pi$
$48$	0	0
$49$	−5.95381	−0.850544
$50$	0	0
$51$	−0.575245	−0.0805505
$52$	0	0
$53$	9.95472	1.36739	0.683693	−	0.729770i	$-0.260373\pi$
$53$	9.95472	1.36739	0.683693	+	0.729770i	$0.260373\pi$
$54$	0	0
$55$	−0.270769	−0.0365104
$56$	0	0
$57$	17.1314	2.26911
$58$	0	0
$59$	−5.38839	−0.701509	−0.350754	−	0.936467i	$-0.614075\pi$
$59$	−5.38839	−0.701509	−0.350754	+	0.936467i	$0.614075\pi$
$60$	0	0
$61$	−11.0695	−1.41730	−0.708651	−	0.705559i	$-0.750696\pi$
$61$	−11.0695	−1.41730	−0.708651	+	0.705559i	$0.750696\pi$
$62$	0	0
$63$	−4.13624	−0.521117
$64$	0	0
$65$	4.37409	0.542539
$66$	0	0
$67$	−6.48381	−0.792124	−0.396062	−	0.918224i	$-0.629623\pi$
$67$	−6.48381	−0.792124	−0.396062	+	0.918224i	$0.629623\pi$
$68$	0	0
$69$	−5.53002	−0.665736
$70$	0	0
$71$	8.86113	1.05162	0.525811	−	0.850601i	$-0.323762\pi$
$71$	8.86113	1.05162	0.525811	+	0.850601i	$0.323762\pi$
$72$	0	0
$73$	−3.64833	−0.427004	−0.213502	−	0.976943i	$-0.568487\pi$
$73$	−3.64833	−0.427004	−0.213502	+	0.976943i	$0.568487\pi$
$74$	0	0
$75$	11.7806	1.36031
$76$	0	0
$77$	−0.369684	−0.0421294
$78$	0	0
$79$	−11.5634	−1.30098	−0.650491	−	0.759514i	$-0.725437\pi$
$79$	−11.5634	−1.30098	−0.650491	+	0.759514i	$0.725437\pi$
$80$	0	0
$81$	−4.77860	−0.530956
$82$	0	0
$83$	1.31671	0.144528	0.0722641	−	0.997386i	$-0.476978\pi$
$83$	1.31671	0.144528	0.0722641	+	0.997386i	$0.476978\pi$
$84$	0	0
$85$	−0.162376	−0.0176121
$86$	0	0
$87$	15.6671	1.67969
$88$	0	0
$89$	−3.00602	−0.318638	−0.159319	−	0.987227i	$-0.550930\pi$
$89$	−3.00602	−0.318638	−0.159319	+	0.987227i	$0.550930\pi$
$90$	0	0
$91$	5.97200	0.626036
$92$	0	0
$93$	−8.31203	−0.861917
$94$	0	0
$95$	4.83572	0.496134
$96$	0	0
$97$	−12.2669	−1.24552	−0.622759	−	0.782414i	$-0.713988\pi$
$97$	−12.2669	−1.24552	−0.622759	+	0.782414i	$0.713988\pi$
$98$	0	0
$99$	1.46159	0.146895
$100$	0	0
$101$	−0.793426	−0.0789489	−0.0394744	−	0.999221i	$-0.512568\pi$
$101$	−0.793426	−0.0789489	−0.0394744	+	0.999221i	$0.512568\pi$
$102$	0	0
$103$	4.08922	0.402923	0.201462	−	0.979496i	$-0.435431\pi$
$103$	4.08922	0.402923	0.201462	+	0.979496i	$0.435431\pi$
$104$	0	0
$105$	−2.03370	−0.198468
$106$	0	0
$107$	0.580414	0.0561108	0.0280554	−	0.999606i	$-0.491069\pi$
$107$	0.580414	0.0561108	0.0280554	+	0.999606i	$0.491069\pi$
$108$	0	0
$109$	−5.85282	−0.560598	−0.280299	−	0.959913i	$-0.590434\pi$
$109$	−5.85282	−0.560598	−0.280299	+	0.959913i	$0.590434\pi$
$110$	0	0
$111$	18.8267	1.78695
$112$	0	0
$113$	−15.8033	−1.48665	−0.743327	−	0.668929i	$-0.766753\pi$
$113$	−15.8033	−1.48665	−0.743327	+	0.668929i	$0.766753\pi$
$114$	0	0
$115$	−1.56097	−0.145561
$116$	0	0
$117$	−23.6110	−2.18283
$118$	0	0
$119$	−0.221693	−0.0203226
$120$	0	0
$121$	−10.8694	−0.988124
$122$	0	0
$123$	−7.97130	−0.718748
$124$	0	0
$125$	7.07113	0.632461
$126$	0	0
$127$	5.90411	0.523905	0.261952	−	0.965081i	$-0.415634\pi$
$127$	5.90411	0.523905	0.261952	+	0.965081i	$0.415634\pi$
$128$	0	0
$129$	27.8136	2.44885
$130$	0	0
$131$	−9.00928	−0.787144	−0.393572	−	0.919294i	$-0.628761\pi$
$131$	−9.00928	−0.787144	−0.393572	+	0.919294i	$0.628761\pi$
$132$	0	0
$133$	6.60227	0.572489
$134$	0	0
$135$	2.07557	0.178636
$136$	0	0
$137$	−4.49096	−0.383689	−0.191844	−	0.981425i	$-0.561447\pi$
$137$	−4.49096	−0.383689	−0.191844	+	0.981425i	$0.561447\pi$
$138$	0	0
$139$	9.91174	0.840703	0.420352	−	0.907361i	$-0.361907\pi$
$139$	9.91174	0.840703	0.420352	+	0.907361i	$0.361907\pi$
$140$	0	0
$141$	−28.6105	−2.40943
$142$	0	0
$143$	−2.11027	−0.176470
$144$	0	0
$145$	4.42239	0.367260
$146$	0	0
$147$	15.8016	1.30329
$148$	0	0
$149$	−12.9043	−1.05716	−0.528580	−	0.848883i	$-0.677275\pi$
$149$	−12.9043	−1.05716	−0.528580	+	0.848883i	$0.677275\pi$
$150$	0	0
$151$	−11.5599	−0.940729	−0.470364	−	0.882472i	$-0.655878\pi$
$151$	−11.5599	−0.940729	−0.470364	+	0.882472i	$0.655878\pi$
$152$	0	0
$153$	0.876489	0.0708599
$154$	0	0
$155$	−2.34625	−0.188455
$156$	0	0
$157$	20.1725	1.60994	0.804972	−	0.593313i	$-0.202180\pi$
$157$	20.1725	1.60994	0.804972	+	0.593313i	$0.202180\pi$
$158$	0	0
$159$	−26.4202	−2.09525
$160$	0	0
$161$	−2.13121	−0.167963
$162$	0	0
$163$	9.18997	0.719814	0.359907	−	0.932988i	$-0.382808\pi$
$163$	9.18997	0.719814	0.359907	+	0.932988i	$0.382808\pi$
$164$	0	0
$165$	0.718629	0.0559452
$166$	0	0
$167$	−6.64996	−0.514590	−0.257295	−	0.966333i	$-0.582831\pi$
$167$	−6.64996	−0.514590	−0.257295	+	0.966333i	$0.582831\pi$
$168$	0	0
$169$	21.0900	1.62231
$170$	0	0
$171$	−26.1028	−1.99613
$172$	0	0
$173$	12.2043	0.927877	0.463939	−	0.885867i	$-0.346436\pi$
$173$	12.2043	0.927877	0.463939	+	0.885867i	$0.346436\pi$
$174$	0	0
$175$	4.54012	0.343201
$176$	0	0
$177$	14.3010	1.07493
$178$	0	0
$179$	7.01493	0.524321	0.262160	−	0.965024i	$-0.415565\pi$
$179$	7.01493	0.524321	0.262160	+	0.965024i	$0.415565\pi$
$180$	0	0
$181$	−11.6342	−0.864766	−0.432383	−	0.901690i	$-0.642327\pi$
$181$	−11.6342	−0.864766	−0.432383	+	0.901690i	$0.642327\pi$
$182$	0	0
$183$	29.3788	2.17174
$184$	0	0
$185$	5.31424	0.390710
$186$	0	0
$187$	0.0783377	0.00572862
$188$	0	0
$189$	2.83380	0.206128
$190$	0	0
$191$	−6.16082	−0.445781	−0.222891	−	0.974843i	$-0.571549\pi$
$191$	−6.16082	−0.445781	−0.222891	+	0.974843i	$0.571549\pi$
$192$	0	0
$193$	8.69209	0.625671	0.312835	−	0.949807i	$-0.398721\pi$
$193$	8.69209	0.625671	0.312835	+	0.949807i	$0.398721\pi$
$194$	0	0
$195$	−11.6090	−0.831336
$196$	0	0
$197$	−11.0513	−0.787373	−0.393686	−	0.919245i	$-0.628800\pi$
$197$	−11.0513	−0.787373	−0.393686	+	0.919245i	$0.628800\pi$
$198$	0	0
$199$	13.1384	0.931357	0.465679	−	0.884954i	$-0.345810\pi$
$199$	13.1384	0.931357	0.465679	+	0.884954i	$0.345810\pi$
$200$	0	0
$201$	17.2083	1.21378
$202$	0	0
$203$	6.03795	0.423781
$204$	0	0
$205$	−2.25007	−0.157152
$206$	0	0
$207$	8.42598	0.585646
$208$	0	0
$209$	−2.33298	−0.161376
$210$	0	0
$211$	−1.50977	−0.103937	−0.0519686	−	0.998649i	$-0.516550\pi$
$211$	−1.50977	−0.103937	−0.0519686	+	0.998649i	$0.516550\pi$
$212$	0	0
$213$	−23.5177	−1.61141
$214$	0	0
$215$	7.85098	0.535433
$216$	0	0
$217$	−3.20337	−0.217459
$218$	0	0
$219$	9.68278	0.654302
$220$	0	0
$221$	−1.26550	−0.0851264
$222$	0	0
$223$	−5.75773	−0.385566	−0.192783	−	0.981241i	$-0.561751\pi$
$223$	−5.75773	−0.385566	−0.192783	+	0.981241i	$0.561751\pi$
$224$	0	0
$225$	−17.9499	−1.19666
$226$	0	0
$227$	24.8721	1.65082	0.825410	−	0.564533i	$-0.190944\pi$
$227$	24.8721	1.65082	0.825410	+	0.564533i	$0.190944\pi$
$228$	0	0
$229$	12.0540	0.796550	0.398275	−	0.917266i	$-0.369609\pi$
$229$	12.0540	0.796550	0.398275	+	0.917266i	$0.369609\pi$
$230$	0	0
$231$	0.981153	0.0645551
$232$	0	0
$233$	−4.83021	−0.316437	−0.158219	−	0.987404i	$-0.550575\pi$
$233$	−4.83021	−0.316437	−0.158219	+	0.987404i	$0.550575\pi$
$234$	0	0
$235$	−8.07592	−0.526815
$236$	0	0
$237$	30.6896	1.99350
$238$	0	0
$239$	−17.1228	−1.10758	−0.553792	−	0.832655i	$-0.686820\pi$
$239$	−17.1228	−1.10758	−0.553792	+	0.832655i	$0.686820\pi$
$240$	0	0
$241$	−1.60274	−0.103242	−0.0516209	−	0.998667i	$-0.516439\pi$
$241$	−1.60274	−0.103242	−0.0516209	+	0.998667i	$0.516439\pi$
$242$	0	0
$243$	20.9942	1.34678
$244$	0	0
$245$	4.46035	0.284961
$246$	0	0
$247$	37.6878	2.39802
$248$	0	0
$249$	−3.49460	−0.221461
$250$	0	0
$251$	20.8292	1.31473	0.657364	−	0.753573i	$-0.271671\pi$
$251$	20.8292	1.31473	0.657364	+	0.753573i	$0.271671\pi$
$252$	0	0
$253$	0.753086	0.0473461
$254$	0	0
$255$	0.430950	0.0269871
$256$	0	0
$257$	2.55308	0.159257	0.0796284	−	0.996825i	$-0.474627\pi$
$257$	2.55308	0.159257	0.0796284	+	0.996825i	$0.474627\pi$
$258$	0	0
$259$	7.25559	0.450841
$260$	0	0
$261$	−23.8717	−1.47762
$262$	0	0
$263$	−19.8717	−1.22534	−0.612671	−	0.790338i	$-0.709905\pi$
$263$	−19.8717	−1.22534	−0.612671	+	0.790338i	$0.709905\pi$
$264$	0	0
$265$	−7.45766	−0.458121
$266$	0	0
$267$	7.97808	0.488251
$268$	0	0
$269$	−10.6600	−0.649949	−0.324974	−	0.945723i	$-0.605356\pi$
$269$	−10.6600	−0.649949	−0.324974	+	0.945723i	$0.605356\pi$
$270$	0	0
$271$	−13.9838	−0.849454	−0.424727	−	0.905322i	$-0.639630\pi$
$271$	−13.9838	−0.849454	−0.424727	+	0.905322i	$0.639630\pi$
$272$	0	0
$273$	−15.8499	−0.959279
$274$	0	0
$275$	−1.60430	−0.0967431
$276$	0	0
$277$	29.9650	1.80042	0.900210	−	0.435456i	$-0.143413\pi$
$277$	29.9650	1.80042	0.900210	+	0.435456i	$0.143413\pi$
$278$	0	0
$279$	12.6649	0.758225
$280$	0	0
$281$	14.6920	0.876451	0.438226	−	0.898865i	$-0.355607\pi$
$281$	14.6920	0.876451	0.438226	+	0.898865i	$0.355607\pi$
$282$	0	0
$283$	−4.26756	−0.253680	−0.126840	−	0.991923i	$-0.540483\pi$
$283$	−4.26756	−0.253680	−0.126840	+	0.991923i	$0.540483\pi$
$284$	0	0
$285$	−12.8342	−0.760230
$286$	0	0
$287$	−3.07205	−0.181338
$288$	0	0
$289$	−16.9530	−0.997237
$290$	0	0
$291$	32.5568	1.90851
$292$	0	0
$293$	−4.20042	−0.245391	−0.122695	−	0.992444i	$-0.539154\pi$
$293$	−4.20042	−0.245391	−0.122695	+	0.992444i	$0.539154\pi$
$294$	0	0
$295$	4.03676	0.235029
$296$	0	0
$297$	−1.00135	−0.0581044
$298$	0	0
$299$	−12.1656	−0.703556
$300$	0	0
$301$	10.7190	0.617836
$302$	0	0
$303$	2.10578	0.120974
$304$	0	0
$305$	8.29280	0.474844
$306$	0	0
$307$	27.6892	1.58031	0.790154	−	0.612909i	$-0.210001\pi$
$307$	27.6892	1.58031	0.790154	+	0.612909i	$0.210001\pi$
$308$	0	0
$309$	−10.8529	−0.617402
$310$	0	0
$311$	14.6051	0.828179	0.414089	−	0.910236i	$-0.364100\pi$
$311$	14.6051	0.828179	0.414089	+	0.910236i	$0.364100\pi$
$312$	0	0
$313$	7.95356	0.449562	0.224781	−	0.974409i	$-0.427833\pi$
$313$	7.95356	0.449562	0.224781	+	0.974409i	$0.427833\pi$
$314$	0	0
$315$	3.09870	0.174592
$316$	0	0
$317$	−2.61888	−0.147091	−0.0735455	−	0.997292i	$-0.523431\pi$
$317$	−2.61888	−0.147091	−0.0735455	+	0.997292i	$0.523431\pi$
$318$	0	0
$319$	−2.13357	−0.119457
$320$	0	0
$321$	−1.54044	−0.0859789
$322$	0	0
$323$	−1.39905	−0.0778454
$324$	0	0
$325$	25.9165	1.43759
$326$	0	0
$327$	15.5336	0.859009
$328$	0	0
$329$	−11.0262	−0.607892
$330$	0	0
$331$	9.54760	0.524783	0.262392	−	0.964961i	$-0.415489\pi$
$331$	9.54760	0.524783	0.262392	+	0.964961i	$0.415489\pi$
$332$	0	0
$333$	−28.6858	−1.57197
$334$	0	0
$335$	4.85740	0.265388
$336$	0	0
$337$	2.97487	0.162052	0.0810258	−	0.996712i	$-0.474180\pi$
$337$	2.97487	0.162052	0.0810258	+	0.996712i	$0.474180\pi$
$338$	0	0
$339$	41.9426	2.27801
$340$	0	0
$341$	1.13194	0.0612982
$342$	0	0
$343$	13.2496	0.715412
$344$	0	0
$345$	4.14286	0.223044
$346$	0	0
$347$	11.9701	0.642590	0.321295	−	0.946979i	$-0.395882\pi$
$347$	11.9701	0.642590	0.321295	+	0.946979i	$0.395882\pi$
$348$	0	0
$349$	2.80250	0.150014	0.0750072	−	0.997183i	$-0.476102\pi$
$349$	2.80250	0.150014	0.0750072	+	0.997183i	$0.476102\pi$
$350$	0	0
$351$	16.1762	0.863422
$352$	0	0
$353$	9.79424	0.521295	0.260647	−	0.965434i	$-0.416064\pi$
$353$	9.79424	0.521295	0.260647	+	0.965434i	$0.416064\pi$
$354$	0	0
$355$	−6.63839	−0.352329
$356$	0	0
$357$	0.588382	0.0311405
$358$	0	0
$359$	35.1365	1.85443	0.927217	−	0.374523i	$-0.122194\pi$
$359$	35.1365	1.85443	0.927217	+	0.374523i	$0.122194\pi$
$360$	0	0
$361$	22.6653	1.19291
$362$	0	0
$363$	28.8477	1.51411
$364$	0	0
$365$	2.73318	0.143061
$366$	0	0
$367$	−14.5170	−0.757782	−0.378891	−	0.925441i	$-0.623694\pi$
$367$	−14.5170	−0.757782	−0.378891	+	0.925441i	$0.623694\pi$
$368$	0	0
$369$	12.1457	0.632280
$370$	0	0
$371$	−10.1820	−0.528625
$372$	0	0
$373$	−20.9731	−1.08594	−0.542972	−	0.839751i	$-0.682701\pi$
$373$	−20.9731	−1.08594	−0.542972	+	0.839751i	$0.682701\pi$
$374$	0	0
$375$	−18.7670	−0.969125
$376$	0	0
$377$	34.4665	1.77512
$378$	0	0
$379$	−29.3854	−1.50943	−0.754714	−	0.656054i	$-0.772224\pi$
$379$	−29.3854	−1.50943	−0.754714	+	0.656054i	$0.772224\pi$
$380$	0	0
$381$	−15.6697	−0.802783
$382$	0	0
$383$	28.9103	1.47725	0.738625	−	0.674117i	$-0.235476\pi$
$383$	28.9103	1.47725	0.738625	+	0.674117i	$0.235476\pi$
$384$	0	0
$385$	0.276952	0.0141148
$386$	0	0
$387$	−42.3789	−2.15424
$388$	0	0
$389$	−2.52146	−0.127843	−0.0639215	−	0.997955i	$-0.520361\pi$
$389$	−2.52146	−0.127843	−0.0639215	+	0.997955i	$0.520361\pi$
$390$	0	0
$391$	0.451614	0.0228391
$392$	0	0
$393$	23.9109	1.20615
$394$	0	0
$395$	8.66281	0.435873
$396$	0	0
$397$	−26.3921	−1.32458	−0.662290	−	0.749247i	$-0.730415\pi$
$397$	−26.3921	−1.32458	−0.662290	+	0.749247i	$0.730415\pi$
$398$	0	0
$399$	−17.5226	−0.877229
$400$	0	0
$401$	34.6456	1.73012	0.865058	−	0.501671i	$-0.167281\pi$
$401$	34.6456	1.73012	0.865058	+	0.501671i	$0.167281\pi$
$402$	0	0
$403$	−18.2858	−0.910881
$404$	0	0
$405$	3.57993	0.177888
$406$	0	0
$407$	−2.56384	−0.127085
$408$	0	0
$409$	21.7438	1.07516	0.537581	−	0.843212i	$-0.319338\pi$
$409$	21.7438	1.07516	0.537581	+	0.843212i	$0.319338\pi$
$410$	0	0
$411$	11.9192	0.587929
$412$	0	0
$413$	5.51144	0.271200
$414$	0	0
$415$	−0.986428	−0.0484218
$416$	0	0
$417$	−26.3061	−1.28822
$418$	0	0
$419$	−14.7755	−0.721830	−0.360915	−	0.932599i	$-0.617536\pi$
$419$	−14.7755	−0.721830	−0.360915	+	0.932599i	$0.617536\pi$
$420$	0	0
$421$	−9.60228	−0.467986	−0.233993	−	0.972238i	$-0.575179\pi$
$421$	−9.60228	−0.467986	−0.233993	+	0.972238i	$0.575179\pi$
$422$	0	0
$423$	43.5931	2.11957
$424$	0	0
$425$	−0.962074	−0.0466674
$426$	0	0
$427$	11.3223	0.547923
$428$	0	0
$429$	5.60073	0.270406
$430$	0	0
$431$	3.17211	0.152795	0.0763977	−	0.997077i	$-0.475658\pi$
$431$	3.17211	0.152795	0.0763977	+	0.997077i	$0.475658\pi$
$432$	0	0
$433$	28.0378	1.34741	0.673705	−	0.739000i	$-0.264702\pi$
$433$	28.0378	1.34741	0.673705	+	0.739000i	$0.264702\pi$
$434$	0	0
$435$	−11.7372	−0.562755
$436$	0	0
$437$	−13.4495	−0.643379
$438$	0	0
$439$	−7.15333	−0.341410	−0.170705	−	0.985322i	$-0.554605\pi$
$439$	−7.15333	−0.341410	−0.170705	+	0.985322i	$0.554605\pi$
$440$	0	0
$441$	−24.0766	−1.14650
$442$	0	0
$443$	26.0709	1.23867	0.619333	−	0.785128i	$-0.287403\pi$
$443$	26.0709	1.23867	0.619333	+	0.785128i	$0.287403\pi$
$444$	0	0
$445$	2.25199	0.106754
$446$	0	0
$447$	34.2484	1.61989
$448$	0	0
$449$	34.8401	1.64421	0.822104	−	0.569338i	$-0.192800\pi$
$449$	34.8401	1.64421	0.822104	+	0.569338i	$0.192800\pi$
$450$	0	0
$451$	1.08554	0.0511162
$452$	0	0
$453$	30.6803	1.44149
$454$	0	0
$455$	−4.47398	−0.209743
$456$	0	0
$457$	7.95860	0.372288	0.186144	−	0.982523i	$-0.440401\pi$
$457$	7.95860	0.372288	0.186144	+	0.982523i	$0.440401\pi$
$458$	0	0
$459$	−0.600495	−0.0280287
$460$	0	0
$461$	−17.0675	−0.794911	−0.397456	−	0.917621i	$-0.630107\pi$
$461$	−17.0675	−0.794911	−0.397456	+	0.917621i	$0.630107\pi$
$462$	0	0
$463$	−18.3862	−0.854480	−0.427240	−	0.904138i	$-0.640514\pi$
$463$	−18.3862	−0.854480	−0.427240	+	0.904138i	$0.640514\pi$
$464$	0	0
$465$	6.22703	0.288771
$466$	0	0
$467$	8.02122	0.371178	0.185589	−	0.982627i	$-0.440581\pi$
$467$	8.02122	0.371178	0.185589	+	0.982627i	$0.440581\pi$
$468$	0	0
$469$	6.63187	0.306231
$470$	0	0
$471$	−53.5386	−2.46693
$472$	0	0
$473$	−3.78769	−0.174158
$474$	0	0
$475$	28.6516	1.31463
$476$	0	0
$477$	40.2558	1.84319
$478$	0	0
$479$	−12.1758	−0.556325	−0.278163	−	0.960534i	$-0.589725\pi$
$479$	−12.1758	−0.556325	−0.278163	+	0.960534i	$0.589725\pi$
$480$	0	0
$481$	41.4172	1.88846
$482$	0	0
$483$	5.65631	0.257371
$484$	0	0
$485$	9.18987	0.417291
$486$	0	0
$487$	19.6492	0.890388	0.445194	−	0.895434i	$-0.353135\pi$
$487$	19.6492	0.890388	0.445194	+	0.895434i	$0.353135\pi$
$488$	0	0
$489$	−24.3905	−1.10298
$490$	0	0
$491$	17.7757	0.802207	0.401103	−	0.916033i	$-0.368627\pi$
$491$	17.7757	0.802207	0.401103	+	0.916033i	$0.368627\pi$
$492$	0	0
$493$	−1.27947	−0.0576244
$494$	0	0
$495$	−1.09496	−0.0492148
$496$	0	0
$497$	−9.06348	−0.406553
$498$	0	0
$499$	20.5753	0.921076	0.460538	−	0.887640i	$-0.347656\pi$
$499$	20.5753	0.921076	0.460538	+	0.887640i	$0.347656\pi$
$500$	0	0
$501$	17.6492	0.788509
$502$	0	0
$503$	−29.5848	−1.31912	−0.659560	−	0.751652i	$-0.729257\pi$
$503$	−29.5848	−1.31912	−0.659560	+	0.751652i	$0.729257\pi$
$504$	0	0
$505$	0.594402	0.0264506
$506$	0	0
$507$	−55.9737	−2.48588
$508$	0	0
$509$	−10.4920	−0.465049	−0.232524	−	0.972591i	$-0.574699\pi$
$509$	−10.4920	−0.465049	−0.232524	+	0.972591i	$0.574699\pi$
$510$	0	0
$511$	3.73164	0.165078
$512$	0	0
$513$	17.8834	0.789571
$514$	0	0
$515$	−3.06348	−0.134993
$516$	0	0
$517$	3.89621	0.171355
$518$	0	0
$519$	−32.3907	−1.42179
$520$	0	0
$521$	1.27316	0.0557783	0.0278892	−	0.999611i	$-0.491121\pi$
$521$	1.27316	0.0557783	0.0278892	+	0.999611i	$0.491121\pi$
$522$	0	0
$523$	16.2273	0.709572	0.354786	−	0.934948i	$-0.384554\pi$
$523$	16.2273	0.709572	0.354786	+	0.934948i	$0.384554\pi$
$524$	0	0
$525$	−12.0496	−0.525889
$526$	0	0
$527$	0.678808	0.0295694
$528$	0	0
$529$	−18.6585	−0.811239
$530$	0	0
$531$	−21.7901	−0.945609
$532$	0	0
$533$	−17.5362	−0.759579
$534$	0	0
$535$	−0.434822	−0.0187990
$536$	0	0
$537$	−18.6179	−0.803420
$538$	0	0
$539$	−2.15189	−0.0926883
$540$	0	0
$541$	18.5729	0.798512	0.399256	−	0.916839i	$-0.369268\pi$
$541$	18.5729	0.798512	0.399256	+	0.916839i	$0.369268\pi$
$542$	0	0
$543$	30.8777	1.32509
$544$	0	0
$545$	4.38469	0.187820
$546$	0	0
$547$	29.3125	1.25331	0.626656	−	0.779296i	$-0.284423\pi$
$547$	29.3125	1.25331	0.626656	+	0.779296i	$0.284423\pi$
$548$	0	0
$549$	−44.7638	−1.91047
$550$	0	0
$551$	38.1040	1.62329
$552$	0	0
$553$	11.8274	0.502954
$554$	0	0
$555$	−14.1042	−0.598688
$556$	0	0
$557$	12.9498	0.548702	0.274351	−	0.961630i	$-0.411537\pi$
$557$	12.9498	0.548702	0.274351	+	0.961630i	$0.411537\pi$
$558$	0	0
$559$	61.1877	2.58796
$560$	0	0
$561$	−0.207911	−0.00877801
$562$	0	0
$563$	26.0703	1.09873	0.549366	−	0.835582i	$-0.314869\pi$
$563$	26.0703	1.09873	0.549366	+	0.835582i	$0.314869\pi$
$564$	0	0
$565$	11.8392	0.498079
$566$	0	0
$567$	4.88773	0.205265
$568$	0	0
$569$	−16.3225	−0.684273	−0.342137	−	0.939650i	$-0.611151\pi$
$569$	−16.3225	−0.684273	−0.342137	+	0.939650i	$0.611151\pi$
$570$	0	0
$571$	−35.0774	−1.46795	−0.733973	−	0.679179i	$-0.762336\pi$
$571$	−35.0774	−1.46795	−0.733973	+	0.679179i	$0.762336\pi$
$572$	0	0
$573$	16.3510	0.683074
$574$	0	0
$575$	−9.24873	−0.385699
$576$	0	0
$577$	−15.8374	−0.659319	−0.329659	−	0.944100i	$-0.606934\pi$
$577$	−15.8374	−0.659319	−0.329659	+	0.944100i	$0.606934\pi$
$578$	0	0
$579$	−23.0691	−0.958719
$580$	0	0
$581$	−1.34678	−0.0558739
$582$	0	0
$583$	3.59794	0.149011
$584$	0	0
$585$	17.6884	0.731323
$586$	0	0
$587$	−25.8863	−1.06844	−0.534221	−	0.845345i	$-0.679395\pi$
$587$	−25.8863	−1.06844	−0.534221	+	0.845345i	$0.679395\pi$
$588$	0	0
$589$	−20.2157	−0.832971
$590$	0	0
$591$	29.3305	1.20650
$592$	0	0
$593$	12.7643	0.524167	0.262084	−	0.965045i	$-0.415590\pi$
$593$	12.7643	0.524167	0.262084	+	0.965045i	$0.415590\pi$
$594$	0	0
$595$	0.166084	0.00680876
$596$	0	0
$597$	−34.8698	−1.42713
$598$	0	0
$599$	−38.1404	−1.55837	−0.779187	−	0.626791i	$-0.784368\pi$
$599$	−38.1404	−1.55837	−0.779187	+	0.626791i	$0.784368\pi$
$600$	0	0
$601$	26.4900	1.08055	0.540275	−	0.841489i	$-0.318320\pi$
$601$	26.4900	1.08055	0.540275	+	0.841489i	$0.318320\pi$
$602$	0	0
$603$	−26.2198	−1.06775
$604$	0	0
$605$	8.14288	0.331055
$606$	0	0
$607$	−32.3601	−1.31346	−0.656728	−	0.754127i	$-0.728060\pi$
$607$	−32.3601	−1.31346	−0.656728	+	0.754127i	$0.728060\pi$
$608$	0	0
$609$	−16.0249	−0.649362
$610$	0	0
$611$	−62.9408	−2.54631
$612$	0	0
$613$	−6.71730	−0.271309	−0.135655	−	0.990756i	$-0.543314\pi$
$613$	−6.71730	−0.271309	−0.135655	+	0.990756i	$0.543314\pi$
$614$	0	0
$615$	5.97177	0.240805
$616$	0	0
$617$	8.91622	0.358954	0.179477	−	0.983762i	$-0.442560\pi$
$617$	8.91622	0.358954	0.179477	+	0.983762i	$0.442560\pi$
$618$	0	0
$619$	−41.7440	−1.67783	−0.838917	−	0.544260i	$-0.816811\pi$
$619$	−41.7440	−1.67783	−0.838917	+	0.544260i	$0.816811\pi$
$620$	0	0
$621$	−5.77276	−0.231653
$622$	0	0
$623$	3.07467	0.123184
$624$	0	0
$625$	16.8964	0.675856
$626$	0	0
$627$	6.19182	0.247277
$628$	0	0
$629$	−1.53750	−0.0613039
$630$	0	0
$631$	7.64725	0.304432	0.152216	−	0.988347i	$-0.451359\pi$
$631$	7.64725	0.304432	0.152216	+	0.988347i	$0.451359\pi$
$632$	0	0
$633$	4.00699	0.159264
$634$	0	0
$635$	−4.42311	−0.175526
$636$	0	0
$637$	34.7623	1.37733
$638$	0	0
$639$	35.8334	1.41755
$640$	0	0
$641$	−8.51056	−0.336147	−0.168073	−	0.985774i	$-0.553755\pi$
$641$	−8.51056	−0.336147	−0.168073	+	0.985774i	$0.553755\pi$
$642$	0	0
$643$	16.5731	0.653578	0.326789	−	0.945097i	$-0.394033\pi$
$643$	16.5731	0.653578	0.326789	+	0.945097i	$0.394033\pi$
$644$	0	0
$645$	−20.8368	−0.820447
$646$	0	0
$647$	46.9691	1.84655	0.923274	−	0.384143i	$-0.125503\pi$
$647$	46.9691	1.84655	0.923274	+	0.384143i	$0.125503\pi$
$648$	0	0
$649$	−1.94753	−0.0764471
$650$	0	0
$651$	8.50184	0.333213
$652$	0	0
$653$	−27.0187	−1.05732	−0.528662	−	0.848833i	$-0.677306\pi$
$653$	−27.0187	−1.05732	−0.528662	+	0.848833i	$0.677306\pi$
$654$	0	0
$655$	6.74938	0.263720
$656$	0	0
$657$	−14.7534	−0.575587
$658$	0	0
$659$	24.5294	0.955529	0.477764	−	0.878488i	$-0.341447\pi$
$659$	24.5294	0.955529	0.477764	+	0.878488i	$0.341447\pi$
$660$	0	0
$661$	−36.3608	−1.41427	−0.707136	−	0.707077i	$-0.750013\pi$
$661$	−36.3608	−1.41427	−0.707136	+	0.707077i	$0.750013\pi$
$662$	0	0
$663$	3.35867	0.130440
$664$	0	0
$665$	−4.94615	−0.191803
$666$	0	0
$667$	−12.3000	−0.476257
$668$	0	0
$669$	15.2812	0.590805
$670$	0	0
$671$	−4.00084	−0.154451
$672$	0	0
$673$	−23.7466	−0.915364	−0.457682	−	0.889116i	$-0.651320\pi$
$673$	−23.7466	−0.915364	−0.457682	+	0.889116i	$0.651320\pi$
$674$	0	0
$675$	12.2977	0.473339
$676$	0	0
$677$	−16.2456	−0.624367	−0.312184	−	0.950022i	$-0.601060\pi$
$677$	−16.2456	−0.624367	−0.312184	+	0.950022i	$0.601060\pi$
$678$	0	0
$679$	12.5470	0.481511
$680$	0	0
$681$	−66.0114	−2.52956
$682$	0	0
$683$	−36.8814	−1.41123	−0.705613	−	0.708597i	$-0.749328\pi$
$683$	−36.8814	−1.41123	−0.705613	+	0.708597i	$0.749328\pi$
$684$	0	0
$685$	3.36444	0.128549
$686$	0	0
$687$	−31.9917	−1.22056
$688$	0	0
$689$	−58.1223	−2.21428
$690$	0	0
$691$	−45.1039	−1.71583	−0.857917	−	0.513789i	$-0.828241\pi$
$691$	−45.1039	−1.71583	−0.857917	+	0.513789i	$0.828241\pi$
$692$	0	0
$693$	−1.49496	−0.0567889
$694$	0	0
$695$	−7.42547	−0.281664
$696$	0	0
$697$	0.650983	0.0246577
$698$	0	0
$699$	12.8195	0.484879
$700$	0	0
$701$	17.0841	0.645257	0.322628	−	0.946526i	$-0.395434\pi$
$701$	17.0841	0.645257	0.322628	+	0.946526i	$0.395434\pi$
$702$	0	0
$703$	45.7883	1.72694
$704$	0	0
$705$	21.4338	0.807242
$706$	0	0
$707$	0.811545	0.0305213
$708$	0	0
$709$	−36.4780	−1.36996	−0.684981	−	0.728561i	$-0.740189\pi$
$709$	−36.4780	−1.36996	−0.684981	+	0.728561i	$0.740189\pi$
$710$	0	0
$711$	−46.7611	−1.75368
$712$	0	0
$713$	6.52561	0.244386
$714$	0	0
$715$	1.58093	0.0591233
$716$	0	0
$717$	45.4445	1.69716
$718$	0	0
$719$	24.7768	0.924020	0.462010	−	0.886875i	$-0.347128\pi$
$719$	24.7768	0.924020	0.462010	+	0.886875i	$0.347128\pi$
$720$	0	0
$721$	−4.18261	−0.155768
$722$	0	0
$723$	4.25373	0.158198
$724$	0	0
$725$	26.2027	0.973142
$726$	0	0
$727$	29.4861	1.09358	0.546790	−	0.837270i	$-0.315850\pi$
$727$	29.4861	1.09358	0.546790	+	0.837270i	$0.315850\pi$
$728$	0	0
$729$	−41.3834	−1.53272
$730$	0	0
$731$	−2.27142	−0.0840114
$732$	0	0
$733$	−4.79139	−0.176974	−0.0884871	−	0.996077i	$-0.528203\pi$
$733$	−4.79139	−0.176974	−0.0884871	+	0.996077i	$0.528203\pi$
$734$	0	0
$735$	−11.8379	−0.436648
$736$	0	0
$737$	−2.34344	−0.0863219
$738$	0	0
$739$	−42.3443	−1.55766	−0.778830	−	0.627235i	$-0.784187\pi$
$739$	−42.3443	−1.55766	−0.778830	+	0.627235i	$0.784187\pi$
$740$	0	0
$741$	−100.025	−3.67450
$742$	0	0
$743$	4.77021	0.175002	0.0875010	−	0.996164i	$-0.472112\pi$
$743$	4.77021	0.175002	0.0875010	+	0.996164i	$0.472112\pi$
$744$	0	0
$745$	9.66736	0.354185
$746$	0	0
$747$	5.32465	0.194819
$748$	0	0
$749$	−0.593669	−0.0216922
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	−55.2814	−2.01457
$754$	0	0
$755$	8.66018	0.315176
$756$	0	0
$757$	33.1774	1.20585	0.602926	−	0.797797i	$-0.294001\pi$
$757$	33.1774	1.20585	0.602926	+	0.797797i	$0.294001\pi$
$758$	0	0
$759$	−1.99872	−0.0725488
$760$	0	0
$761$	45.7465	1.65831	0.829155	−	0.559019i	$-0.188822\pi$
$761$	45.7465	1.65831	0.829155	+	0.559019i	$0.188822\pi$
$762$	0	0
$763$	5.98647	0.216725
$764$	0	0
$765$	−0.656629	−0.0237405
$766$	0	0
$767$	31.4610	1.13599
$768$	0	0
$769$	−13.6843	−0.493469	−0.246735	−	0.969083i	$-0.579358\pi$
$769$	−13.6843	−0.493469	−0.246735	+	0.969083i	$0.579358\pi$
$770$	0	0
$771$	−6.77596	−0.244030
$772$	0	0
$773$	−39.3867	−1.41664	−0.708321	−	0.705890i	$-0.750547\pi$
$773$	−39.3867	−1.41664	−0.708321	+	0.705890i	$0.750547\pi$
$774$	0	0
$775$	−13.9015	−0.499357
$776$	0	0
$777$	−19.2566	−0.690826
$778$	0	0
$779$	−19.3870	−0.694610
$780$	0	0
$781$	3.20268	0.114601
$782$	0	0
$783$	16.3548	0.584474
$784$	0	0
$785$	−15.1124	−0.539386
$786$	0	0
$787$	20.0518	0.714768	0.357384	−	0.933958i	$-0.383669\pi$
$787$	20.0518	0.714768	0.357384	+	0.933958i	$0.383669\pi$
$788$	0	0
$789$	52.7402	1.87760
$790$	0	0
$791$	16.1642	0.574734
$792$	0	0
$793$	64.6310	2.29512
$794$	0	0
$795$	19.7929	0.701982
$796$	0	0
$797$	53.0732	1.87995	0.939974	−	0.341247i	$-0.110849\pi$
$797$	53.0732	1.87995	0.939974	+	0.341247i	$0.110849\pi$
$798$	0	0
$799$	2.33650	0.0826593
$800$	0	0
$801$	−12.1560	−0.429512
$802$	0	0
$803$	−1.31862	−0.0465329
$804$	0	0
$805$	1.59662	0.0562733
$806$	0	0
$807$	28.2919	0.995921
$808$	0	0
$809$	−38.7084	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$809$	−38.7084	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$810$	0	0
$811$	40.6245	1.42652	0.713260	−	0.700900i	$-0.247218\pi$
$811$	40.6245	1.42652	0.713260	+	0.700900i	$0.247218\pi$
$812$	0	0
$813$	37.1134	1.30162
$814$	0	0
$815$	−6.88475	−0.241162
$816$	0	0
$817$	67.6453	2.36661
$818$	0	0
$819$	24.1501	0.843874
$820$	0	0
$821$	−11.1692	−0.389808	−0.194904	−	0.980822i	$-0.562439\pi$
$821$	−11.1692	−0.389808	−0.194904	+	0.980822i	$0.562439\pi$
$822$	0	0
$823$	−51.7537	−1.80402	−0.902011	−	0.431713i	$-0.857909\pi$
$823$	−51.7537	−1.80402	−0.902011	+	0.431713i	$0.857909\pi$
$824$	0	0
$825$	4.25787	0.148240
$826$	0	0
$827$	−51.5032	−1.79094	−0.895471	−	0.445120i	$-0.853161\pi$
$827$	−51.5032	−1.79094	−0.895471	+	0.445120i	$0.853161\pi$
$828$	0	0
$829$	−28.4980	−0.989776	−0.494888	−	0.868957i	$-0.664791\pi$
$829$	−28.4980	−0.989776	−0.494888	+	0.868957i	$0.664791\pi$
$830$	0	0
$831$	−79.5280	−2.75880
$832$	0	0
$833$	−1.29045	−0.0447115
$834$	0	0
$835$	4.98188	0.172405
$836$	0	0
$837$	−8.67687	−0.299917
$838$	0	0
$839$	22.4196	0.774010	0.387005	−	0.922078i	$-0.373510\pi$
$839$	22.4196	0.774010	0.387005	+	0.922078i	$0.373510\pi$
$840$	0	0
$841$	5.84712	0.201625
$842$	0	0
$843$	−38.9931	−1.34299
$844$	0	0
$845$	−15.7998	−0.543529
$846$	0	0
$847$	11.1176	0.382005
$848$	0	0
$849$	11.3262	0.388716
$850$	0	0
$851$	−14.7804	−0.506667
$852$	0	0
$853$	24.0778	0.824409	0.412205	−	0.911091i	$-0.364759\pi$
$853$	24.0778	0.824409	0.412205	+	0.911091i	$0.364759\pi$
$854$	0	0
$855$	19.5551	0.668772
$856$	0	0
$857$	8.44243	0.288388	0.144194	−	0.989549i	$-0.453941\pi$
$857$	8.44243	0.288388	0.144194	+	0.989549i	$0.453941\pi$
$858$	0	0
$859$	46.0324	1.57060	0.785302	−	0.619112i	$-0.212507\pi$
$859$	46.0324	1.57060	0.785302	+	0.619112i	$0.212507\pi$
$860$	0	0
$861$	8.15333	0.277865
$862$	0	0
$863$	−35.8375	−1.21992	−0.609962	−	0.792431i	$-0.708815\pi$
$863$	−35.8375	−1.21992	−0.609962	+	0.792431i	$0.708815\pi$
$864$	0	0
$865$	−9.14297	−0.310870
$866$	0	0
$867$	44.9939	1.52807
$868$	0	0
$869$	−4.17935	−0.141775
$870$	0	0
$871$	37.8568	1.28273
$872$	0	0
$873$	−49.6061	−1.67891
$874$	0	0
$875$	−7.23261	−0.244507
$876$	0	0
$877$	34.0034	1.14821	0.574107	−	0.818781i	$-0.305349\pi$
$877$	34.0034	1.14821	0.574107	+	0.818781i	$0.305349\pi$
$878$	0	0
$879$	11.1480	0.376014
$880$	0	0
$881$	−22.5460	−0.759593	−0.379796	−	0.925070i	$-0.624006\pi$
$881$	−22.5460	−0.759593	−0.379796	+	0.925070i	$0.624006\pi$
$882$	0	0
$883$	−21.9935	−0.740141	−0.370070	−	0.929004i	$-0.620666\pi$
$883$	−21.9935	−0.740141	−0.370070	+	0.929004i	$0.620666\pi$
$884$	0	0
$885$	−10.7137	−0.360137
$886$	0	0
$887$	6.08194	0.204211	0.102106	−	0.994774i	$-0.467442\pi$
$887$	6.08194	0.204211	0.102106	+	0.994774i	$0.467442\pi$
$888$	0	0
$889$	−6.03893	−0.202539
$890$	0	0
$891$	−1.72713	−0.0578611
$892$	0	0
$893$	−69.5834	−2.32852
$894$	0	0
$895$	−5.25530	−0.175665
$896$	0	0
$897$	32.2880	1.07806
$898$	0	0
$899$	−18.4877	−0.616601
$900$	0	0
$901$	2.15762	0.0718809
$902$	0	0
$903$	−28.4487	−0.946714
$904$	0	0
$905$	8.71589	0.289726
$906$	0	0
$907$	21.4858	0.713424	0.356712	−	0.934214i	$-0.383898\pi$
$907$	21.4858	0.713424	0.356712	+	0.934214i	$0.383898\pi$
$908$	0	0
$909$	−3.20853	−0.106420
$910$	0	0
$911$	30.2438	1.00202	0.501011	−	0.865441i	$-0.332961\pi$
$911$	30.2438	1.00202	0.501011	+	0.865441i	$0.332961\pi$
$912$	0	0
$913$	0.475900	0.0157500
$914$	0	0
$915$	−22.0094	−0.727607
$916$	0	0
$917$	9.21501	0.304307
$918$	0	0
$919$	−40.1761	−1.32529	−0.662644	−	0.748935i	$-0.730566\pi$
$919$	−40.1761	−1.32529	−0.662644	+	0.748935i	$0.730566\pi$
$920$	0	0
$921$	−73.4881	−2.42152
$922$	0	0
$923$	−51.7372	−1.70295
$924$	0	0
$925$	31.4868	1.03528
$926$	0	0
$927$	16.5364	0.543126
$928$	0	0
$929$	−12.9827	−0.425947	−0.212974	−	0.977058i	$-0.568315\pi$
$929$	−12.9827	−0.425947	−0.212974	+	0.977058i	$0.568315\pi$
$930$	0	0
$931$	38.4310	1.25953
$932$	0	0
$933$	−38.7624	−1.26902
$934$	0	0
$935$	−0.0586874	−0.00191928
$936$	0	0
$937$	13.0214	0.425390	0.212695	−	0.977119i	$-0.431776\pi$
$937$	13.0214	0.425390	0.212695	+	0.977119i	$0.431776\pi$
$938$	0	0
$939$	−21.1090	−0.688867
$940$	0	0
$941$	−23.9808	−0.781752	−0.390876	−	0.920443i	$-0.627828\pi$
$941$	−23.9808	−0.781752	−0.390876	+	0.920443i	$0.627828\pi$
$942$	0	0
$943$	6.25811	0.203792
$944$	0	0
$945$	−2.12296	−0.0690600
$946$	0	0
$947$	1.21685	0.0395424	0.0197712	−	0.999805i	$-0.493706\pi$
$947$	1.21685	0.0395424	0.0197712	+	0.999805i	$0.493706\pi$
$948$	0	0
$949$	21.3014	0.691472
$950$	0	0
$951$	6.95060	0.225388
$952$	0	0
$953$	37.8403	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$953$	37.8403	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$954$	0	0
$955$	4.61543	0.149352
$956$	0	0
$957$	5.66258	0.183045
$958$	0	0
$959$	4.59352	0.148332
$960$	0	0
$961$	−21.1915	−0.683598
$962$	0	0
$963$	2.34713	0.0756353
$964$	0	0
$965$	−6.51176	−0.209621
$966$	0	0
$967$	−57.0960	−1.83608	−0.918041	−	0.396485i	$-0.870230\pi$
$967$	−57.0960	−1.83608	−0.918041	+	0.396485i	$0.870230\pi$
$968$	0	0
$969$	3.71313	0.119283
$970$	0	0
$971$	43.4246	1.39356	0.696780	−	0.717285i	$-0.254615\pi$
$971$	43.4246	1.39356	0.696780	+	0.717285i	$0.254615\pi$
$972$	0	0
$973$	−10.1381	−0.325012
$974$	0	0
$975$	−68.7831	−2.20282
$976$	0	0
$977$	−16.2035	−0.518396	−0.259198	−	0.965824i	$-0.583458\pi$
$977$	−16.2035	−0.518396	−0.259198	+	0.965824i	$0.583458\pi$
$978$	0	0
$979$	−1.08647	−0.0347236
$980$	0	0
$981$	−23.6682	−0.755667
$982$	0	0
$983$	30.2002	0.963236	0.481618	−	0.876381i	$-0.340049\pi$
$983$	30.2002	0.963236	0.481618	+	0.876381i	$0.340049\pi$
$984$	0	0
$985$	8.27918	0.263797
$986$	0	0
$987$	29.2638	0.931477
$988$	0	0
$989$	−21.8359	−0.694341
$990$	0	0
$991$	43.1432	1.37049	0.685244	−	0.728314i	$-0.259696\pi$
$991$	43.1432	1.37049	0.685244	+	0.728314i	$0.259696\pi$
$992$	0	0
$993$	−25.3396	−0.804129
$994$	0	0
$995$	−9.84276	−0.312036
$996$	0	0
$997$	−44.5040	−1.40946	−0.704728	−	0.709478i	$-0.748931\pi$
$997$	−44.5040	−1.40946	−0.704728	+	0.709478i	$0.748931\pi$
$998$	0	0
$999$	19.6530	0.621795

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.4	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.4	✓	49	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)

\(f(q)\)	\(=\)	\(q-2.65403 q^{3} -0.749159 q^{5} -1.02284 q^{7} +4.04389 q^{9} +O(q^{10})\)	\(q-2.65403 q^{3} -0.749159 q^{5} -1.02284 q^{7} +4.04389 q^{9} +0.361430 q^{11} -5.83867 q^{13} +1.98829 q^{15} +0.216744 q^{17} -6.45487 q^{19} +2.71464 q^{21} +2.08363 q^{23} -4.43876 q^{25} -2.77053 q^{27} -5.90314 q^{29} +3.13185 q^{31} -0.959248 q^{33} +0.766266 q^{35} -7.09361 q^{37} +15.4960 q^{39} +3.00347 q^{41} -10.4797 q^{43} -3.02952 q^{45} +10.7800 q^{47} -5.95381 q^{49} -0.575245 q^{51} +9.95472 q^{53} -0.270769 q^{55} +17.1314 q^{57} -5.38839 q^{59} -11.0695 q^{61} -4.13624 q^{63} +4.37409 q^{65} -6.48381 q^{67} -5.53002 q^{69} +8.86113 q^{71} -3.64833 q^{73} +11.7806 q^{75} -0.369684 q^{77} -11.5634 q^{79} -4.77860 q^{81} +1.31671 q^{83} -0.162376 q^{85} +15.6671 q^{87} -3.00602 q^{89} +5.97200 q^{91} -8.31203 q^{93} +4.83572 q^{95} -12.2669 q^{97} +1.46159 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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