LMFDB - Embedded newform 6008.2.a.b.1.20

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			1.20
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)

$f(q)$	$=$	$q-0.753945 q^{3} -1.16579 q^{5} +1.19034 q^{7} -2.43157 q^{9} +O(q^{10})$	$q-0.753945 q^{3} -1.16579 q^{5} +1.19034 q^{7} -2.43157 q^{9} -3.93174 q^{11} +2.90377 q^{13} +0.878938 q^{15} -1.93718 q^{17} +3.04157 q^{19} -0.897448 q^{21} +6.33561 q^{23} -3.64094 q^{25} +4.09510 q^{27} +3.20651 q^{29} +0.203587 q^{31} +2.96432 q^{33} -1.38768 q^{35} -5.22739 q^{37} -2.18928 q^{39} +10.5412 q^{41} -6.50559 q^{43} +2.83468 q^{45} -1.53219 q^{47} -5.58310 q^{49} +1.46053 q^{51} -0.0388220 q^{53} +4.58356 q^{55} -2.29318 q^{57} +4.00328 q^{59} +5.53333 q^{61} -2.89438 q^{63} -3.38517 q^{65} +12.0206 q^{67} -4.77670 q^{69} -2.94570 q^{71} +7.90151 q^{73} +2.74507 q^{75} -4.68009 q^{77} -10.4546 q^{79} +4.20722 q^{81} -4.73510 q^{83} +2.25834 q^{85} -2.41753 q^{87} -0.232387 q^{89} +3.45646 q^{91} -0.153493 q^{93} -3.54582 q^{95} +5.21513 q^{97} +9.56029 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.753945	−0.435290	−0.217645	−	0.976028i	$-0.569838\pi$
$3$	−0.753945	−0.435290	−0.217645	+	0.976028i	$0.569838\pi$
$4$	0	0
$5$	−1.16579	−0.521355	−0.260678	−	0.965426i	$-0.583946\pi$
$5$	−1.16579	−0.521355	−0.260678	+	0.965426i	$0.583946\pi$
$6$	0	0
$7$	1.19034	0.449905	0.224952	−	0.974370i	$-0.427777\pi$
$7$	1.19034	0.449905	0.224952	+	0.974370i	$0.427777\pi$
$8$	0	0
$9$	−2.43157	−0.810522
$10$	0	0
$11$	−3.93174	−1.18546	−0.592732	−	0.805400i	$-0.701951\pi$
$11$	−3.93174	−1.18546	−0.592732	+	0.805400i	$0.701951\pi$
$12$	0	0
$13$	2.90377	0.805361	0.402680	−	0.915341i	$-0.368079\pi$
$13$	2.90377	0.805361	0.402680	+	0.915341i	$0.368079\pi$
$14$	0	0
$15$	0.878938	0.226941
$16$	0	0
$17$	−1.93718	−0.469836	−0.234918	−	0.972015i	$-0.575482\pi$
$17$	−1.93718	−0.469836	−0.234918	+	0.972015i	$0.575482\pi$
$18$	0	0
$19$	3.04157	0.697784	0.348892	−	0.937163i	$-0.386558\pi$
$19$	3.04157	0.697784	0.348892	+	0.937163i	$0.386558\pi$
$20$	0	0
$21$	−0.897448	−0.195839
$22$	0	0
$23$	6.33561	1.32107	0.660533	−	0.750797i	$-0.270331\pi$
$23$	6.33561	1.32107	0.660533	+	0.750797i	$0.270331\pi$
$24$	0	0
$25$	−3.64094	−0.728189
$26$	0	0
$27$	4.09510	0.788103
$28$	0	0
$29$	3.20651	0.595434	0.297717	−	0.954654i	$-0.403775\pi$
$29$	3.20651	0.595434	0.297717	+	0.954654i	$0.403775\pi$
$30$	0	0
$31$	0.203587	0.0365652	0.0182826	−	0.999833i	$-0.494180\pi$
$31$	0.203587	0.0365652	0.0182826	+	0.999833i	$0.494180\pi$
$32$	0	0
$33$	2.96432	0.516021
$34$	0	0
$35$	−1.38768	−0.234560
$36$	0	0
$37$	−5.22739	−0.859378	−0.429689	−	0.902977i	$-0.641377\pi$
$37$	−5.22739	−0.859378	−0.429689	+	0.902977i	$0.641377\pi$
$38$	0	0
$39$	−2.18928	−0.350566
$40$	0	0
$41$	10.5412	1.64626	0.823129	−	0.567855i	$-0.192227\pi$
$41$	10.5412	1.64626	0.823129	+	0.567855i	$0.192227\pi$
$42$	0	0
$43$	−6.50559	−0.992093	−0.496046	−	0.868296i	$-0.665215\pi$
$43$	−6.50559	−0.992093	−0.496046	+	0.868296i	$0.665215\pi$
$44$	0	0
$45$	2.83468	0.422570
$46$	0	0
$47$	−1.53219	−0.223492	−0.111746	−	0.993737i	$-0.535644\pi$
$47$	−1.53219	−0.223492	−0.111746	+	0.993737i	$0.535644\pi$
$48$	0	0
$49$	−5.58310	−0.797586
$50$	0	0
$51$	1.46053	0.204515
$52$	0	0
$53$	−0.0388220	−0.00533261	−0.00266630	−	0.999996i	$-0.500849\pi$
$53$	−0.0388220	−0.00533261	−0.00266630	+	0.999996i	$0.500849\pi$
$54$	0	0
$55$	4.58356	0.618048
$56$	0	0
$57$	−2.29318	−0.303739
$58$	0	0
$59$	4.00328	0.521183	0.260591	−	0.965449i	$-0.416082\pi$
$59$	4.00328	0.521183	0.260591	+	0.965449i	$0.416082\pi$
$60$	0	0
$61$	5.53333	0.708470	0.354235	−	0.935156i	$-0.384741\pi$
$61$	5.53333	0.708470	0.354235	+	0.935156i	$0.384741\pi$
$62$	0	0
$63$	−2.89438	−0.364658
$64$	0	0
$65$	−3.38517	−0.419879
$66$	0	0
$67$	12.0206	1.46856	0.734278	−	0.678849i	$-0.237521\pi$
$67$	12.0206	1.46856	0.734278	+	0.678849i	$0.237521\pi$
$68$	0	0
$69$	−4.77670	−0.575047
$70$	0	0
$71$	−2.94570	−0.349590	−0.174795	−	0.984605i	$-0.555926\pi$
$71$	−2.94570	−0.349590	−0.174795	+	0.984605i	$0.555926\pi$
$72$	0	0
$73$	7.90151	0.924802	0.462401	−	0.886671i	$-0.346988\pi$
$73$	7.90151	0.924802	0.462401	+	0.886671i	$0.346988\pi$
$74$	0	0
$75$	2.74507	0.316974
$76$	0	0
$77$	−4.68009	−0.533346
$78$	0	0
$79$	−10.4546	−1.17624	−0.588119	−	0.808774i	$-0.700131\pi$
$79$	−10.4546	−1.17624	−0.588119	+	0.808774i	$0.700131\pi$
$80$	0	0
$81$	4.20722	0.467469
$82$	0	0
$83$	−4.73510	−0.519745	−0.259873	−	0.965643i	$-0.583681\pi$
$83$	−4.73510	−0.519745	−0.259873	+	0.965643i	$0.583681\pi$
$84$	0	0
$85$	2.25834	0.244951
$86$	0	0
$87$	−2.41753	−0.259187
$88$	0	0
$89$	−0.232387	−0.0246330	−0.0123165	−	0.999924i	$-0.503921\pi$
$89$	−0.232387	−0.0246330	−0.0123165	+	0.999924i	$0.503921\pi$
$90$	0	0
$91$	3.45646	0.362336
$92$	0	0
$93$	−0.153493	−0.0159165
$94$	0	0
$95$	−3.54582	−0.363793
$96$	0	0
$97$	5.21513	0.529516	0.264758	−	0.964315i	$-0.414708\pi$
$97$	5.21513	0.529516	0.264758	+	0.964315i	$0.414708\pi$
$98$	0	0
$99$	9.56029	0.960845
$100$	0	0
$101$	−2.18327	−0.217244	−0.108622	−	0.994083i	$-0.534644\pi$
$101$	−2.18327	−0.217244	−0.108622	+	0.994083i	$0.534644\pi$
$102$	0	0
$103$	−18.3295	−1.80606	−0.903028	−	0.429582i	$-0.858661\pi$
$103$	−18.3295	−1.80606	−0.903028	+	0.429582i	$0.858661\pi$
$104$	0	0
$105$	1.04623	0.102102
$106$	0	0
$107$	−6.21563	−0.600887	−0.300444	−	0.953800i	$-0.597135\pi$
$107$	−6.21563	−0.600887	−0.300444	+	0.953800i	$0.597135\pi$
$108$	0	0
$109$	−16.4967	−1.58010	−0.790048	−	0.613045i	$-0.789944\pi$
$109$	−16.4967	−1.58010	−0.790048	+	0.613045i	$0.789944\pi$
$110$	0	0
$111$	3.94116	0.374079
$112$	0	0
$113$	2.25911	0.212519	0.106260	−	0.994338i	$-0.466113\pi$
$113$	2.25911	0.212519	0.106260	+	0.994338i	$0.466113\pi$
$114$	0	0
$115$	−7.38596	−0.688744
$116$	0	0
$117$	−7.06071	−0.652763
$118$	0	0
$119$	−2.30590	−0.211381
$120$	0	0
$121$	4.45857	0.405325
$122$	0	0
$123$	−7.94748	−0.716600
$124$	0	0
$125$	10.0735	0.901000
$126$	0	0
$127$	−17.1911	−1.52546	−0.762731	−	0.646715i	$-0.776142\pi$
$127$	−17.1911	−1.52546	−0.762731	+	0.646715i	$0.776142\pi$
$128$	0	0
$129$	4.90486	0.431849
$130$	0	0
$131$	17.2494	1.50709	0.753546	−	0.657396i	$-0.228342\pi$
$131$	17.2494	1.50709	0.753546	+	0.657396i	$0.228342\pi$
$132$	0	0
$133$	3.62049	0.313936
$134$	0	0
$135$	−4.77401	−0.410881
$136$	0	0
$137$	−17.4442	−1.49036	−0.745178	−	0.666866i	$-0.767635\pi$
$137$	−17.4442	−1.49036	−0.745178	+	0.666866i	$0.767635\pi$
$138$	0	0
$139$	−7.39978	−0.627641	−0.313821	−	0.949482i	$-0.601609\pi$
$139$	−7.39978	−0.627641	−0.313821	+	0.949482i	$0.601609\pi$
$140$	0	0
$141$	1.15518	0.0972841
$142$	0	0
$143$	−11.4169	−0.954726
$144$	0	0
$145$	−3.73810	−0.310432
$146$	0	0
$147$	4.20935	0.347182
$148$	0	0
$149$	−18.2348	−1.49385	−0.746925	−	0.664908i	$-0.768471\pi$
$149$	−18.2348	−1.49385	−0.746925	+	0.664908i	$0.768471\pi$
$150$	0	0
$151$	−11.1641	−0.908521	−0.454261	−	0.890869i	$-0.650096\pi$
$151$	−11.1641	−0.908521	−0.454261	+	0.890869i	$0.650096\pi$
$152$	0	0
$153$	4.71039	0.380813
$154$	0	0
$155$	−0.237338	−0.0190635
$156$	0	0
$157$	7.07285	0.564475	0.282238	−	0.959345i	$-0.408923\pi$
$157$	7.07285	0.564475	0.282238	+	0.959345i	$0.408923\pi$
$158$	0	0
$159$	0.0292696	0.00232123
$160$	0	0
$161$	7.54150	0.594353
$162$	0	0
$163$	−15.3342	−1.20107	−0.600534	−	0.799599i	$-0.705045\pi$
$163$	−15.3342	−1.20107	−0.600534	+	0.799599i	$0.705045\pi$
$164$	0	0
$165$	−3.45576	−0.269030
$166$	0	0
$167$	8.75536	0.677510	0.338755	−	0.940875i	$-0.389994\pi$
$167$	8.75536	0.677510	0.338755	+	0.940875i	$0.389994\pi$
$168$	0	0
$169$	−4.56812	−0.351394
$170$	0	0
$171$	−7.39578	−0.565570
$172$	0	0
$173$	9.81476	0.746202	0.373101	−	0.927791i	$-0.378294\pi$
$173$	9.81476	0.746202	0.373101	+	0.927791i	$0.378294\pi$
$174$	0	0
$175$	−4.33395	−0.327615
$176$	0	0
$177$	−3.01826	−0.226866
$178$	0	0
$179$	−4.72461	−0.353134	−0.176567	−	0.984289i	$-0.556499\pi$
$179$	−4.72461	−0.353134	−0.176567	+	0.984289i	$0.556499\pi$
$180$	0	0
$181$	8.33219	0.619327	0.309664	−	0.950846i	$-0.399784\pi$
$181$	8.33219	0.619327	0.309664	+	0.950846i	$0.399784\pi$
$182$	0	0
$183$	−4.17182	−0.308390
$184$	0	0
$185$	6.09401	0.448041
$186$	0	0
$187$	7.61650	0.556974
$188$	0	0
$189$	4.87455	0.354571
$190$	0	0
$191$	−15.9022	−1.15064	−0.575321	−	0.817928i	$-0.695123\pi$
$191$	−15.9022	−1.15064	−0.575321	+	0.817928i	$0.695123\pi$
$192$	0	0
$193$	14.6714	1.05607	0.528036	−	0.849222i	$-0.322929\pi$
$193$	14.6714	1.05607	0.528036	+	0.849222i	$0.322929\pi$
$194$	0	0
$195$	2.55223	0.182769
$196$	0	0
$197$	−6.47639	−0.461424	−0.230712	−	0.973022i	$-0.574105\pi$
$197$	−6.47639	−0.461424	−0.230712	+	0.973022i	$0.574105\pi$
$198$	0	0
$199$	0.905920	0.0642190	0.0321095	−	0.999484i	$-0.489777\pi$
$199$	0.905920	0.0642190	0.0321095	+	0.999484i	$0.489777\pi$
$200$	0	0
$201$	−9.06291	−0.639248
$202$	0	0
$203$	3.81682	0.267888
$204$	0	0
$205$	−12.2888	−0.858284
$206$	0	0
$207$	−15.4054	−1.07075
$208$	0	0
$209$	−11.9587	−0.827198
$210$	0	0
$211$	−8.14511	−0.560732	−0.280366	−	0.959893i	$-0.590456\pi$
$211$	−8.14511	−0.560732	−0.280366	+	0.959893i	$0.590456\pi$
$212$	0	0
$213$	2.22090	0.152173
$214$	0	0
$215$	7.58412	0.517233
$216$	0	0
$217$	0.242336	0.0164509
$218$	0	0
$219$	−5.95730	−0.402557
$220$	0	0
$221$	−5.62514	−0.378388
$222$	0	0
$223$	5.25390	0.351827	0.175913	−	0.984406i	$-0.443712\pi$
$223$	5.25390	0.351827	0.175913	+	0.984406i	$0.443712\pi$
$224$	0	0
$225$	8.85320	0.590213
$226$	0	0
$227$	26.0098	1.72633	0.863166	−	0.504921i	$-0.168478\pi$
$227$	26.0098	1.72633	0.863166	+	0.504921i	$0.168478\pi$
$228$	0	0
$229$	1.40548	0.0928767	0.0464383	−	0.998921i	$-0.485213\pi$
$229$	1.40548	0.0928767	0.0464383	+	0.998921i	$0.485213\pi$
$230$	0	0
$231$	3.52853	0.232160
$232$	0	0
$233$	21.6290	1.41697	0.708483	−	0.705728i	$-0.249380\pi$
$233$	21.6290	1.41697	0.708483	+	0.705728i	$0.249380\pi$
$234$	0	0
$235$	1.78620	0.116519
$236$	0	0
$237$	7.88222	0.512005
$238$	0	0
$239$	20.7620	1.34298	0.671490	−	0.741014i	$-0.265655\pi$
$239$	20.7620	1.34298	0.671490	+	0.741014i	$0.265655\pi$
$240$	0	0
$241$	−27.9928	−1.80317	−0.901586	−	0.432600i	$-0.857596\pi$
$241$	−27.9928	−1.80317	−0.901586	+	0.432600i	$0.857596\pi$
$242$	0	0
$243$	−15.4573	−0.991588
$244$	0	0
$245$	6.50870	0.415825
$246$	0	0
$247$	8.83202	0.561968
$248$	0	0
$249$	3.57001	0.226240
$250$	0	0
$251$	−24.8832	−1.57061	−0.785306	−	0.619108i	$-0.787494\pi$
$251$	−24.8832	−1.57061	−0.785306	+	0.619108i	$0.787494\pi$
$252$	0	0
$253$	−24.9099	−1.56607
$254$	0	0
$255$	−1.70266	−0.106625
$256$	0	0
$257$	−17.0074	−1.06089	−0.530447	−	0.847718i	$-0.677976\pi$
$257$	−17.0074	−1.06089	−0.530447	+	0.847718i	$0.677976\pi$
$258$	0	0
$259$	−6.22235	−0.386638
$260$	0	0
$261$	−7.79684	−0.482612
$262$	0	0
$263$	−1.51974	−0.0937113	−0.0468557	−	0.998902i	$-0.514920\pi$
$263$	−1.51974	−0.0937113	−0.0468557	+	0.998902i	$0.514920\pi$
$264$	0	0
$265$	0.0452581	0.00278018
$266$	0	0
$267$	0.175207	0.0107225
$268$	0	0
$269$	14.8243	0.903856	0.451928	−	0.892054i	$-0.350736\pi$
$269$	14.8243	0.903856	0.451928	+	0.892054i	$0.350736\pi$
$270$	0	0
$271$	−30.9026	−1.87720	−0.938599	−	0.345010i	$-0.887875\pi$
$271$	−30.9026	−1.87720	−0.938599	+	0.345010i	$0.887875\pi$
$272$	0	0
$273$	−2.60598	−0.157721
$274$	0	0
$275$	14.3152	0.863242
$276$	0	0
$277$	2.52939	0.151976	0.0759881	−	0.997109i	$-0.475789\pi$
$277$	2.52939	0.151976	0.0759881	+	0.997109i	$0.475789\pi$
$278$	0	0
$279$	−0.495034	−0.0296369
$280$	0	0
$281$	13.6965	0.817067	0.408534	−	0.912743i	$-0.366040\pi$
$281$	13.6965	0.817067	0.408534	+	0.912743i	$0.366040\pi$
$282$	0	0
$283$	20.0621	1.19257	0.596283	−	0.802774i	$-0.296643\pi$
$283$	20.0621	1.19257	0.596283	+	0.802774i	$0.296643\pi$
$284$	0	0
$285$	2.67335	0.158356
$286$	0	0
$287$	12.5476	0.740658
$288$	0	0
$289$	−13.2473	−0.779254
$290$	0	0
$291$	−3.93192	−0.230493
$292$	0	0
$293$	−16.3008	−0.952303	−0.476151	−	0.879363i	$-0.657969\pi$
$293$	−16.3008	−0.952303	−0.476151	+	0.879363i	$0.657969\pi$
$294$	0	0
$295$	−4.66697	−0.271721
$296$	0	0
$297$	−16.1009	−0.934268
$298$	0	0
$299$	18.3971	1.06393
$300$	0	0
$301$	−7.74383	−0.446347
$302$	0	0
$303$	1.64607	0.0945642
$304$	0	0
$305$	−6.45067	−0.369364
$306$	0	0
$307$	5.53944	0.316152	0.158076	−	0.987427i	$-0.449471\pi$
$307$	5.53944	0.316152	0.158076	+	0.987427i	$0.449471\pi$
$308$	0	0
$309$	13.8194	0.786159
$310$	0	0
$311$	9.83496	0.557690	0.278845	−	0.960336i	$-0.410049\pi$
$311$	9.83496	0.557690	0.278845	+	0.960336i	$0.410049\pi$
$312$	0	0
$313$	−34.3358	−1.94077	−0.970386	−	0.241560i	$-0.922341\pi$
$313$	−34.3358	−1.94077	−0.970386	+	0.241560i	$0.922341\pi$
$314$	0	0
$315$	3.37423	0.190116
$316$	0	0
$317$	8.19253	0.460138	0.230069	−	0.973174i	$-0.426105\pi$
$317$	8.19253	0.460138	0.230069	+	0.973174i	$0.426105\pi$
$318$	0	0
$319$	−12.6072	−0.705865
$320$	0	0
$321$	4.68624	0.261561
$322$	0	0
$323$	−5.89208	−0.327844
$324$	0	0
$325$	−10.5725	−0.586455
$326$	0	0
$327$	12.4376	0.687800
$328$	0	0
$329$	−1.82382	−0.100550
$330$	0	0
$331$	−22.6716	−1.24614	−0.623072	−	0.782164i	$-0.714116\pi$
$331$	−22.6716	−1.24614	−0.623072	+	0.782164i	$0.714116\pi$
$332$	0	0
$333$	12.7107	0.696545
$334$	0	0
$335$	−14.0135	−0.765639
$336$	0	0
$337$	31.8402	1.73445	0.867224	−	0.497918i	$-0.165902\pi$
$337$	31.8402	1.73445	0.867224	+	0.497918i	$0.165902\pi$
$338$	0	0
$339$	−1.70325	−0.0925076
$340$	0	0
$341$	−0.800449	−0.0433468
$342$	0	0
$343$	−14.9781	−0.808742
$344$	0	0
$345$	5.56860	0.299804
$346$	0	0
$347$	−31.9760	−1.71656	−0.858280	−	0.513182i	$-0.828467\pi$
$347$	−31.9760	−1.71656	−0.858280	+	0.513182i	$0.828467\pi$
$348$	0	0
$349$	−0.326542	−0.0174794	−0.00873969	−	0.999962i	$-0.502782\pi$
$349$	−0.326542	−0.0174794	−0.00873969	+	0.999962i	$0.502782\pi$
$350$	0	0
$351$	11.8912	0.634707
$352$	0	0
$353$	18.3671	0.977582	0.488791	−	0.872401i	$-0.337438\pi$
$353$	18.3671	0.977582	0.488791	+	0.872401i	$0.337438\pi$
$354$	0	0
$355$	3.43405	0.182261
$356$	0	0
$357$	1.73852	0.0920123
$358$	0	0
$359$	−25.6775	−1.35520	−0.677602	−	0.735429i	$-0.736981\pi$
$359$	−25.6775	−1.35520	−0.677602	+	0.735429i	$0.736981\pi$
$360$	0	0
$361$	−9.74885	−0.513097
$362$	0	0
$363$	−3.36152	−0.176434
$364$	0	0
$365$	−9.21146	−0.482150
$366$	0	0
$367$	−9.03167	−0.471449	−0.235725	−	0.971820i	$-0.575746\pi$
$367$	−9.03167	−0.471449	−0.235725	+	0.971820i	$0.575746\pi$
$368$	0	0
$369$	−25.6316	−1.33433
$370$	0	0
$371$	−0.0462111	−0.00239916
$372$	0	0
$373$	−5.16915	−0.267649	−0.133824	−	0.991005i	$-0.542726\pi$
$373$	−5.16915	−0.267649	−0.133824	+	0.991005i	$0.542726\pi$
$374$	0	0
$375$	−7.59486	−0.392197
$376$	0	0
$377$	9.31096	0.479539
$378$	0	0
$379$	−32.3359	−1.66098	−0.830492	−	0.557030i	$-0.811941\pi$
$379$	−32.3359	−1.66098	−0.830492	+	0.557030i	$0.811941\pi$
$380$	0	0
$381$	12.9611	0.664019
$382$	0	0
$383$	−2.91126	−0.148758	−0.0743791	−	0.997230i	$-0.523697\pi$
$383$	−2.91126	−0.148758	−0.0743791	+	0.997230i	$0.523697\pi$
$384$	0	0
$385$	5.45598	0.278062
$386$	0	0
$387$	15.8188	0.804113
$388$	0	0
$389$	31.6507	1.60475	0.802376	−	0.596819i	$-0.203569\pi$
$389$	31.6507	1.60475	0.802376	+	0.596819i	$0.203569\pi$
$390$	0	0
$391$	−12.2732	−0.620684
$392$	0	0
$393$	−13.0051	−0.656022
$394$	0	0
$395$	12.1879	0.613238
$396$	0	0
$397$	17.6897	0.887818	0.443909	−	0.896072i	$-0.353591\pi$
$397$	17.6897	0.887818	0.443909	+	0.896072i	$0.353591\pi$
$398$	0	0
$399$	−2.72965	−0.136653
$400$	0	0
$401$	26.5906	1.32787	0.663936	−	0.747789i	$-0.268885\pi$
$401$	26.5906	1.32787	0.663936	+	0.747789i	$0.268885\pi$
$402$	0	0
$403$	0.591169	0.0294482
$404$	0	0
$405$	−4.90471	−0.243717
$406$	0	0
$407$	20.5527	1.01876
$408$	0	0
$409$	−2.78936	−0.137925	−0.0689624	−	0.997619i	$-0.521969\pi$
$409$	−2.78936	−0.137925	−0.0689624	+	0.997619i	$0.521969\pi$
$410$	0	0
$411$	13.1519	0.648738
$412$	0	0
$413$	4.76525	0.234483
$414$	0	0
$415$	5.52011	0.270972
$416$	0	0
$417$	5.57903	0.273206
$418$	0	0
$419$	4.60858	0.225144	0.112572	−	0.993644i	$-0.464091\pi$
$419$	4.60858	0.225144	0.112572	+	0.993644i	$0.464091\pi$
$420$	0	0
$421$	−39.8575	−1.94254	−0.971268	−	0.237989i	$-0.923512\pi$
$421$	−39.8575	−1.94254	−0.971268	+	0.237989i	$0.923512\pi$
$422$	0	0
$423$	3.72562	0.181146
$424$	0	0
$425$	7.05318	0.342129
$426$	0	0
$427$	6.58651	0.318744
$428$	0	0
$429$	8.60769	0.415583
$430$	0	0
$431$	−17.5623	−0.845947	−0.422974	−	0.906142i	$-0.639014\pi$
$431$	−17.5623	−0.845947	−0.422974	+	0.906142i	$0.639014\pi$
$432$	0	0
$433$	−18.3009	−0.879483	−0.439741	−	0.898124i	$-0.644930\pi$
$433$	−18.3009	−0.879483	−0.439741	+	0.898124i	$0.644930\pi$
$434$	0	0
$435$	2.81832	0.135128
$436$	0	0
$437$	19.2702	0.921818
$438$	0	0
$439$	−41.1332	−1.96318	−0.981590	−	0.190998i	$-0.938828\pi$
$439$	−41.1332	−1.96318	−0.981590	+	0.190998i	$0.938828\pi$
$440$	0	0
$441$	13.5757	0.646461
$442$	0	0
$443$	−29.3067	−1.39240	−0.696201	−	0.717847i	$-0.745128\pi$
$443$	−29.3067	−1.39240	−0.696201	+	0.717847i	$0.745128\pi$
$444$	0	0
$445$	0.270914	0.0128425
$446$	0	0
$447$	13.7480	0.650259
$448$	0	0
$449$	40.7352	1.92241	0.961206	−	0.275833i	$-0.0889536\pi$
$449$	40.7352	1.92241	0.961206	+	0.275833i	$0.0889536\pi$
$450$	0	0
$451$	−41.4452	−1.95158
$452$	0	0
$453$	8.41711	0.395471
$454$	0	0
$455$	−4.02949	−0.188905
$456$	0	0
$457$	−21.9525	−1.02689	−0.513446	−	0.858122i	$-0.671631\pi$
$457$	−21.9525	−1.02689	−0.513446	+	0.858122i	$0.671631\pi$
$458$	0	0
$459$	−7.93297	−0.370279
$460$	0	0
$461$	30.7593	1.43260	0.716302	−	0.697790i	$-0.245833\pi$
$461$	30.7593	1.43260	0.716302	+	0.697790i	$0.245833\pi$
$462$	0	0
$463$	−34.6219	−1.60902	−0.804508	−	0.593941i	$-0.797571\pi$
$463$	−34.6219	−1.60902	−0.804508	+	0.593941i	$0.797571\pi$
$464$	0	0
$465$	0.178940	0.00829814
$466$	0	0
$467$	−21.4310	−0.991706	−0.495853	−	0.868406i	$-0.665145\pi$
$467$	−21.4310	−0.991706	−0.495853	+	0.868406i	$0.665145\pi$
$468$	0	0
$469$	14.3086	0.660710
$470$	0	0
$471$	−5.33254	−0.245711
$472$	0	0
$473$	25.5783	1.17609
$474$	0	0
$475$	−11.0742	−0.508119
$476$	0	0
$477$	0.0943982	0.00432220
$478$	0	0
$479$	−29.9036	−1.36633	−0.683164	−	0.730265i	$-0.739397\pi$
$479$	−29.9036	−1.36633	−0.683164	+	0.730265i	$0.739397\pi$
$480$	0	0
$481$	−15.1791	−0.692109
$482$	0	0
$483$	−5.68587	−0.258716
$484$	0	0
$485$	−6.07972	−0.276066
$486$	0	0
$487$	−15.2037	−0.688947	−0.344474	−	0.938796i	$-0.611943\pi$
$487$	−15.2037	−0.688947	−0.344474	+	0.938796i	$0.611943\pi$
$488$	0	0
$489$	11.5612	0.522814
$490$	0	0
$491$	−39.4964	−1.78245	−0.891223	−	0.453566i	$-0.850152\pi$
$491$	−39.4964	−1.78245	−0.891223	+	0.453566i	$0.850152\pi$
$492$	0	0
$493$	−6.21159	−0.279756
$494$	0	0
$495$	−11.1452	−0.500941
$496$	0	0
$497$	−3.50637	−0.157282
$498$	0	0
$499$	−18.0985	−0.810202	−0.405101	−	0.914272i	$-0.632764\pi$
$499$	−18.0985	−0.810202	−0.405101	+	0.914272i	$0.632764\pi$
$500$	0	0
$501$	−6.60106	−0.294914
$502$	0	0
$503$	9.34057	0.416475	0.208238	−	0.978078i	$-0.433227\pi$
$503$	9.34057	0.416475	0.208238	+	0.978078i	$0.433227\pi$
$504$	0	0
$505$	2.54523	0.113261
$506$	0	0
$507$	3.44411	0.152958
$508$	0	0
$509$	−7.79856	−0.345665	−0.172833	−	0.984951i	$-0.555292\pi$
$509$	−7.79856	−0.345665	−0.172833	+	0.984951i	$0.555292\pi$
$510$	0	0
$511$	9.40545	0.416072
$512$	0	0
$513$	12.4555	0.549926
$514$	0	0
$515$	21.3682	0.941596
$516$	0	0
$517$	6.02416	0.264942
$518$	0	0
$519$	−7.39979	−0.324815
$520$	0	0
$521$	−42.1313	−1.84581	−0.922903	−	0.385031i	$-0.874191\pi$
$521$	−42.1313	−1.84581	−0.922903	+	0.385031i	$0.874191\pi$
$522$	0	0
$523$	40.1060	1.75371	0.876856	−	0.480753i	$-0.159637\pi$
$523$	40.1060	1.75371	0.876856	+	0.480753i	$0.159637\pi$
$524$	0	0
$525$	3.26756	0.142608
$526$	0	0
$527$	−0.394385	−0.0171797
$528$	0	0
$529$	17.1399	0.745213
$530$	0	0
$531$	−9.73425	−0.422430
$532$	0	0
$533$	30.6092	1.32583
$534$	0	0
$535$	7.24609	0.313276
$536$	0	0
$537$	3.56209	0.153716
$538$	0	0
$539$	21.9513	0.945509
$540$	0	0
$541$	29.7312	1.27824	0.639122	−	0.769105i	$-0.279298\pi$
$541$	29.7312	1.27824	0.639122	+	0.769105i	$0.279298\pi$
$542$	0	0
$543$	−6.28202	−0.269587
$544$	0	0
$545$	19.2316	0.823791
$546$	0	0
$547$	23.9669	1.02475	0.512376	−	0.858761i	$-0.328765\pi$
$547$	23.9669	1.02475	0.512376	+	0.858761i	$0.328765\pi$
$548$	0	0
$549$	−13.4547	−0.574231
$550$	0	0
$551$	9.75282	0.415484
$552$	0	0
$553$	−12.4445	−0.529195
$554$	0	0
$555$	−4.59455	−0.195028
$556$	0	0
$557$	0.749692	0.0317655	0.0158827	−	0.999874i	$-0.494944\pi$
$557$	0.749692	0.0317655	0.0158827	+	0.999874i	$0.494944\pi$
$558$	0	0
$559$	−18.8907	−0.798993
$560$	0	0
$561$	−5.74242	−0.242445
$562$	0	0
$563$	−2.73187	−0.115135	−0.0575673	−	0.998342i	$-0.518334\pi$
$563$	−2.73187	−0.115135	−0.0575673	+	0.998342i	$0.518334\pi$
$564$	0	0
$565$	−2.63364	−0.110798
$566$	0	0
$567$	5.00800	0.210316
$568$	0	0
$569$	−23.8731	−1.00081	−0.500406	−	0.865791i	$-0.666816\pi$
$569$	−23.8731	−1.00081	−0.500406	+	0.865791i	$0.666816\pi$
$570$	0	0
$571$	3.68911	0.154384	0.0771922	−	0.997016i	$-0.475404\pi$
$571$	3.68911	0.154384	0.0771922	+	0.997016i	$0.475404\pi$
$572$	0	0
$573$	11.9894	0.500863
$574$	0	0
$575$	−23.0676	−0.961985
$576$	0	0
$577$	−25.8555	−1.07638	−0.538189	−	0.842824i	$-0.680891\pi$
$577$	−25.8555	−1.07638	−0.538189	+	0.842824i	$0.680891\pi$
$578$	0	0
$579$	−11.0614	−0.459698
$580$	0	0
$581$	−5.63636	−0.233836
$582$	0	0
$583$	0.152638	0.00632161
$584$	0	0
$585$	8.23127	0.340321
$586$	0	0
$587$	−18.0297	−0.744163	−0.372082	−	0.928200i	$-0.621356\pi$
$587$	−18.0297	−0.744163	−0.372082	+	0.928200i	$0.621356\pi$
$588$	0	0
$589$	0.619223	0.0255146
$590$	0	0
$591$	4.88284	0.200853
$592$	0	0
$593$	−42.2040	−1.73311	−0.866555	−	0.499082i	$-0.833671\pi$
$593$	−42.2040	−1.73311	−0.866555	+	0.499082i	$0.833671\pi$
$594$	0	0
$595$	2.68818	0.110205
$596$	0	0
$597$	−0.683014	−0.0279539
$598$	0	0
$599$	−8.44815	−0.345182	−0.172591	−	0.984994i	$-0.555214\pi$
$599$	−8.44815	−0.345182	−0.172591	+	0.984994i	$0.555214\pi$
$600$	0	0
$601$	23.6713	0.965574	0.482787	−	0.875738i	$-0.339625\pi$
$601$	23.6713	0.965574	0.482787	+	0.875738i	$0.339625\pi$
$602$	0	0
$603$	−29.2290	−1.19030
$604$	0	0
$605$	−5.19774	−0.211318
$606$	0	0
$607$	−12.3845	−0.502670	−0.251335	−	0.967900i	$-0.580870\pi$
$607$	−12.3845	−0.502670	−0.251335	+	0.967900i	$0.580870\pi$
$608$	0	0
$609$	−2.87767	−0.116609
$610$	0	0
$611$	−4.44912	−0.179992
$612$	0	0
$613$	21.1281	0.853355	0.426677	−	0.904404i	$-0.359684\pi$
$613$	21.1281	0.853355	0.426677	+	0.904404i	$0.359684\pi$
$614$	0	0
$615$	9.26505	0.373603
$616$	0	0
$617$	−8.60266	−0.346330	−0.173165	−	0.984893i	$-0.555399\pi$
$617$	−8.60266	−0.346330	−0.173165	+	0.984893i	$0.555399\pi$
$618$	0	0
$619$	44.1235	1.77347	0.886736	−	0.462276i	$-0.152967\pi$
$619$	44.1235	1.77347	0.886736	+	0.462276i	$0.152967\pi$
$620$	0	0
$621$	25.9450	1.04114
$622$	0	0
$623$	−0.276619	−0.0110825
$624$	0	0
$625$	6.46120	0.258448
$626$	0	0
$627$	9.01617	0.360071
$628$	0	0
$629$	10.1264	0.403767
$630$	0	0
$631$	44.8292	1.78462	0.892312	−	0.451420i	$-0.149082\pi$
$631$	44.8292	1.78462	0.892312	+	0.451420i	$0.149082\pi$
$632$	0	0
$633$	6.14096	0.244081
$634$	0	0
$635$	20.0411	0.795308
$636$	0	0
$637$	−16.2120	−0.642345
$638$	0	0
$639$	7.16267	0.283351
$640$	0	0
$641$	1.38935	0.0548760	0.0274380	−	0.999624i	$-0.491265\pi$
$641$	1.38935	0.0548760	0.0274380	+	0.999624i	$0.491265\pi$
$642$	0	0
$643$	13.5612	0.534802	0.267401	−	0.963585i	$-0.413835\pi$
$643$	13.5612	0.534802	0.267401	+	0.963585i	$0.413835\pi$
$644$	0	0
$645$	−5.71801	−0.225146
$646$	0	0
$647$	−18.7636	−0.737674	−0.368837	−	0.929494i	$-0.620244\pi$
$647$	−18.7636	−0.737674	−0.368837	+	0.929494i	$0.620244\pi$
$648$	0	0
$649$	−15.7399	−0.617844
$650$	0	0
$651$	−0.182708	−0.00716090
$652$	0	0
$653$	−19.6968	−0.770797	−0.385399	−	0.922750i	$-0.625936\pi$
$653$	−19.6968	−0.770797	−0.385399	+	0.922750i	$0.625936\pi$
$654$	0	0
$655$	−20.1092	−0.785730
$656$	0	0
$657$	−19.2130	−0.749572
$658$	0	0
$659$	−36.3069	−1.41432	−0.707159	−	0.707055i	$-0.750023\pi$
$659$	−36.3069	−1.41432	−0.707159	+	0.707055i	$0.750023\pi$
$660$	0	0
$661$	46.1595	1.79540	0.897698	−	0.440610i	$-0.145238\pi$
$661$	46.1595	1.79540	0.897698	+	0.440610i	$0.145238\pi$
$662$	0	0
$663$	4.24104	0.164708
$664$	0	0
$665$	−4.22071	−0.163672
$666$	0	0
$667$	20.3152	0.786607
$668$	0	0
$669$	−3.96115	−0.153147
$670$	0	0
$671$	−21.7556	−0.839866
$672$	0	0
$673$	−37.8370	−1.45851	−0.729254	−	0.684243i	$-0.760133\pi$
$673$	−37.8370	−1.45851	−0.729254	+	0.684243i	$0.760133\pi$
$674$	0	0
$675$	−14.9100	−0.573888
$676$	0	0
$677$	−17.1305	−0.658380	−0.329190	−	0.944264i	$-0.606776\pi$
$677$	−17.1305	−0.658380	−0.329190	+	0.944264i	$0.606776\pi$
$678$	0	0
$679$	6.20775	0.238232
$680$	0	0
$681$	−19.6100	−0.751455
$682$	0	0
$683$	−11.6402	−0.445401	−0.222701	−	0.974887i	$-0.571487\pi$
$683$	−11.6402	−0.445401	−0.222701	+	0.974887i	$0.571487\pi$
$684$	0	0
$685$	20.3362	0.777004
$686$	0	0
$687$	−1.05965	−0.0404283
$688$	0	0
$689$	−0.112730	−0.00429467
$690$	0	0
$691$	−23.2923	−0.886079	−0.443040	−	0.896502i	$-0.646100\pi$
$691$	−23.2923	−0.886079	−0.443040	+	0.896502i	$0.646100\pi$
$692$	0	0
$693$	11.3799	0.432288
$694$	0	0
$695$	8.62655	0.327224
$696$	0	0
$697$	−20.4202	−0.773471
$698$	0	0
$699$	−16.3071	−0.616792
$700$	0	0
$701$	46.3984	1.75244	0.876222	−	0.481909i	$-0.160056\pi$
$701$	46.3984	1.75244	0.876222	+	0.481909i	$0.160056\pi$
$702$	0	0
$703$	−15.8995	−0.599660
$704$	0	0
$705$	−1.34670	−0.0507196
$706$	0	0
$707$	−2.59883	−0.0977390
$708$	0	0
$709$	22.0094	0.826581	0.413290	−	0.910599i	$-0.364379\pi$
$709$	22.0094	0.826581	0.413290	+	0.910599i	$0.364379\pi$
$710$	0	0
$711$	25.4211	0.953367
$712$	0	0
$713$	1.28984	0.0483050
$714$	0	0
$715$	13.3096	0.497751
$716$	0	0
$717$	−15.6534	−0.584586
$718$	0	0
$719$	−42.0701	−1.56895	−0.784475	−	0.620160i	$-0.787068\pi$
$719$	−42.0701	−1.56895	−0.784475	+	0.620160i	$0.787068\pi$
$720$	0	0
$721$	−21.8182	−0.812552
$722$	0	0
$723$	21.1050	0.784903
$724$	0	0
$725$	−11.6747	−0.433588
$726$	0	0
$727$	19.3608	0.718053	0.359027	−	0.933327i	$-0.383109\pi$
$727$	19.3608	0.718053	0.359027	+	0.933327i	$0.383109\pi$
$728$	0	0
$729$	−0.967681	−0.0358400
$730$	0	0
$731$	12.6025	0.466121
$732$	0	0
$733$	25.3443	0.936113	0.468057	−	0.883698i	$-0.344954\pi$
$733$	25.3443	0.936113	0.468057	+	0.883698i	$0.344954\pi$
$734$	0	0
$735$	−4.90720	−0.181005
$736$	0	0
$737$	−47.2620	−1.74092
$738$	0	0
$739$	13.4490	0.494731	0.247366	−	0.968922i	$-0.420435\pi$
$739$	13.4490	0.494731	0.247366	+	0.968922i	$0.420435\pi$
$740$	0	0
$741$	−6.65886	−0.244619
$742$	0	0
$743$	36.2620	1.33032	0.665162	−	0.746699i	$-0.268363\pi$
$743$	36.2620	1.33032	0.665162	+	0.746699i	$0.268363\pi$
$744$	0	0
$745$	21.2578	0.778826
$746$	0	0
$747$	11.5137	0.421265
$748$	0	0
$749$	−7.39868	−0.270342
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	18.7605	0.683672
$754$	0	0
$755$	13.0149	0.473662
$756$	0	0
$757$	19.3753	0.704206	0.352103	−	0.935961i	$-0.385467\pi$
$757$	19.3753	0.704206	0.352103	+	0.935961i	$0.385467\pi$
$758$	0	0
$759$	18.7807	0.681697
$760$	0	0
$761$	24.4960	0.887980	0.443990	−	0.896032i	$-0.353563\pi$
$761$	24.4960	0.887980	0.443990	+	0.896032i	$0.353563\pi$
$762$	0	0
$763$	−19.6366	−0.710892
$764$	0	0
$765$	−5.49130	−0.198539
$766$	0	0
$767$	11.6246	0.419740
$768$	0	0
$769$	25.4597	0.918101	0.459050	−	0.888410i	$-0.348190\pi$
$769$	25.4597	0.918101	0.459050	+	0.888410i	$0.348190\pi$
$770$	0	0
$771$	12.8227	0.461797
$772$	0	0
$773$	29.0973	1.04656	0.523279	−	0.852161i	$-0.324709\pi$
$773$	29.0973	1.04656	0.523279	+	0.852161i	$0.324709\pi$
$774$	0	0
$775$	−0.741247	−0.0266264
$776$	0	0
$777$	4.69131	0.168300
$778$	0	0
$779$	32.0618	1.14873
$780$	0	0
$781$	11.5817	0.414427
$782$	0	0
$783$	13.1310	0.469263
$784$	0	0
$785$	−8.24543	−0.294292
$786$	0	0
$787$	20.9705	0.747517	0.373758	−	0.927526i	$-0.378069\pi$
$787$	20.9705	0.747517	0.373758	+	0.927526i	$0.378069\pi$
$788$	0	0
$789$	1.14580	0.0407916
$790$	0	0
$791$	2.68910	0.0956134
$792$	0	0
$793$	16.0675	0.570574
$794$	0	0
$795$	−0.0341221	−0.00121019
$796$	0	0
$797$	32.6754	1.15742	0.578712	−	0.815532i	$-0.303556\pi$
$797$	32.6754	1.15742	0.578712	+	0.815532i	$0.303556\pi$
$798$	0	0
$799$	2.96813	0.105005
$800$	0	0
$801$	0.565065	0.0199656
$802$	0	0
$803$	−31.0667	−1.09632
$804$	0	0
$805$	−8.79176	−0.309869
$806$	0	0
$807$	−11.1767	−0.393440
$808$	0	0
$809$	−19.7965	−0.696006	−0.348003	−	0.937493i	$-0.613140\pi$
$809$	−19.7965	−0.696006	−0.348003	+	0.937493i	$0.613140\pi$
$810$	0	0
$811$	−8.18101	−0.287274	−0.143637	−	0.989630i	$-0.545880\pi$
$811$	−8.18101	−0.287274	−0.143637	+	0.989630i	$0.545880\pi$
$812$	0	0
$813$	23.2988	0.817126
$814$	0	0
$815$	17.8764	0.626183
$816$	0	0
$817$	−19.7872	−0.692267
$818$	0	0
$819$	−8.40462	−0.293681
$820$	0	0
$821$	−54.0434	−1.88613	−0.943065	−	0.332610i	$-0.892071\pi$
$821$	−54.0434	−1.88613	−0.943065	+	0.332610i	$0.892071\pi$
$822$	0	0
$823$	4.14058	0.144331	0.0721657	−	0.997393i	$-0.477009\pi$
$823$	4.14058	0.144331	0.0721657	+	0.997393i	$0.477009\pi$
$824$	0	0
$825$	−10.7929	−0.375761
$826$	0	0
$827$	37.5430	1.30550	0.652750	−	0.757574i	$-0.273615\pi$
$827$	37.5430	1.30550	0.652750	+	0.757574i	$0.273615\pi$
$828$	0	0
$829$	39.1228	1.35879	0.679395	−	0.733773i	$-0.262242\pi$
$829$	39.1228	1.35879	0.679395	+	0.733773i	$0.262242\pi$
$830$	0	0
$831$	−1.90702	−0.0661537
$832$	0	0
$833$	10.8155	0.374735
$834$	0	0
$835$	−10.2069	−0.353223
$836$	0	0
$837$	0.833708	0.0288172
$838$	0	0
$839$	48.6630	1.68003	0.840017	−	0.542560i	$-0.182545\pi$
$839$	48.6630	1.68003	0.840017	+	0.542560i	$0.182545\pi$
$840$	0	0
$841$	−18.7183	−0.645459
$842$	0	0
$843$	−10.3264	−0.355661
$844$	0	0
$845$	5.32545	0.183201
$846$	0	0
$847$	5.30720	0.182357
$848$	0	0
$849$	−15.1257	−0.519113
$850$	0	0
$851$	−33.1187	−1.13529
$852$	0	0
$853$	−17.8323	−0.610567	−0.305284	−	0.952261i	$-0.598751\pi$
$853$	−17.8323	−0.610567	−0.305284	+	0.952261i	$0.598751\pi$
$854$	0	0
$855$	8.62189	0.294863
$856$	0	0
$857$	25.2337	0.861967	0.430984	−	0.902360i	$-0.358167\pi$
$857$	25.2337	0.861967	0.430984	+	0.902360i	$0.358167\pi$
$858$	0	0
$859$	8.85134	0.302004	0.151002	−	0.988533i	$-0.451750\pi$
$859$	8.85134	0.302004	0.151002	+	0.988533i	$0.451750\pi$
$860$	0	0
$861$	−9.46016	−0.322402
$862$	0	0
$863$	−20.4808	−0.697175	−0.348587	−	0.937276i	$-0.613339\pi$
$863$	−20.4808	−0.697175	−0.348587	+	0.937276i	$0.613339\pi$
$864$	0	0
$865$	−11.4419	−0.389036
$866$	0	0
$867$	9.98775	0.339202
$868$	0	0
$869$	41.1049	1.39439
$870$	0	0
$871$	34.9052	1.18272
$872$	0	0
$873$	−12.6809	−0.429185
$874$	0	0
$875$	11.9908	0.405364
$876$	0	0
$877$	−5.50921	−0.186033	−0.0930164	−	0.995665i	$-0.529651\pi$
$877$	−5.50921	−0.186033	−0.0930164	+	0.995665i	$0.529651\pi$
$878$	0	0
$879$	12.2899	0.414528
$880$	0	0
$881$	−32.6893	−1.10133	−0.550665	−	0.834726i	$-0.685626\pi$
$881$	−32.6893	−1.10133	−0.550665	+	0.834726i	$0.685626\pi$
$882$	0	0
$883$	4.87717	0.164130	0.0820650	−	0.996627i	$-0.473849\pi$
$883$	4.87717	0.164130	0.0820650	+	0.996627i	$0.473849\pi$
$884$	0	0
$885$	3.51864	0.118278
$886$	0	0
$887$	−11.9276	−0.400491	−0.200245	−	0.979746i	$-0.564174\pi$
$887$	−11.9276	−0.400491	−0.200245	+	0.979746i	$0.564174\pi$
$888$	0	0
$889$	−20.4632	−0.686313
$890$	0	0
$891$	−16.5417	−0.554167
$892$	0	0
$893$	−4.66025	−0.155949
$894$	0	0
$895$	5.50788	0.184108
$896$	0	0
$897$	−13.8704	−0.463120
$898$	0	0
$899$	0.652802	0.0217722
$900$	0	0
$901$	0.0752052	0.00250545
$902$	0	0
$903$	5.83842	0.194291
$904$	0	0
$905$	−9.71355	−0.322889
$906$	0	0
$907$	51.0492	1.69506	0.847531	−	0.530746i	$-0.178088\pi$
$907$	51.0492	1.69506	0.847531	+	0.530746i	$0.178088\pi$
$908$	0	0
$909$	5.30878	0.176081
$910$	0	0
$911$	−1.23533	−0.0409283	−0.0204641	−	0.999791i	$-0.506514\pi$
$911$	−1.23533	−0.0409283	−0.0204641	+	0.999791i	$0.506514\pi$
$912$	0	0
$913$	18.6172	0.616139
$914$	0	0
$915$	4.86345	0.160781
$916$	0	0
$917$	20.5326	0.678047
$918$	0	0
$919$	44.3490	1.46294	0.731470	−	0.681874i	$-0.238835\pi$
$919$	44.3490	1.46294	0.731470	+	0.681874i	$0.238835\pi$
$920$	0	0
$921$	−4.17643	−0.137618
$922$	0	0
$923$	−8.55364	−0.281546
$924$	0	0
$925$	19.0326	0.625789
$926$	0	0
$927$	44.5693	1.46385
$928$	0	0
$929$	28.1291	0.922885	0.461443	−	0.887170i	$-0.347332\pi$
$929$	28.1291	0.922885	0.461443	+	0.887170i	$0.347332\pi$
$930$	0	0
$931$	−16.9814	−0.556543
$932$	0	0
$933$	−7.41502	−0.242757
$934$	0	0
$935$	−8.87920	−0.290381
$936$	0	0
$937$	43.0468	1.40628	0.703139	−	0.711053i	$-0.251781\pi$
$937$	43.0468	1.40628	0.703139	+	0.711053i	$0.251781\pi$
$938$	0	0
$939$	25.8873	0.844799
$940$	0	0
$941$	−20.0151	−0.652473	−0.326236	−	0.945288i	$-0.605780\pi$
$941$	−20.0151	−0.652473	−0.326236	+	0.945288i	$0.605780\pi$
$942$	0	0
$943$	66.7848	2.17481
$944$	0	0
$945$	−5.68267	−0.184857
$946$	0	0
$947$	1.31980	0.0428877	0.0214439	−	0.999770i	$-0.493174\pi$
$947$	1.31980	0.0428877	0.0214439	+	0.999770i	$0.493174\pi$
$948$	0	0
$949$	22.9442	0.744799
$950$	0	0
$951$	−6.17672	−0.200294
$952$	0	0
$953$	31.9537	1.03508	0.517541	−	0.855658i	$-0.326847\pi$
$953$	31.9537	1.03508	0.517541	+	0.855658i	$0.326847\pi$
$954$	0	0
$955$	18.5385	0.599893
$956$	0	0
$957$	9.50510	0.307256
$958$	0	0
$959$	−20.7644	−0.670518
$960$	0	0
$961$	−30.9586	−0.998663
$962$	0	0
$963$	15.1137	0.487033
$964$	0	0
$965$	−17.1037	−0.550588
$966$	0	0
$967$	−47.3793	−1.52362	−0.761808	−	0.647803i	$-0.775688\pi$
$967$	−47.3793	−1.52362	−0.761808	+	0.647803i	$0.775688\pi$
$968$	0	0
$969$	4.44230	0.142707
$970$	0	0
$971$	16.3405	0.524390	0.262195	−	0.965015i	$-0.415554\pi$
$971$	16.3405	0.524390	0.262195	+	0.965015i	$0.415554\pi$
$972$	0	0
$973$	−8.80822	−0.282379
$974$	0	0
$975$	7.97106	0.255278
$976$	0	0
$977$	55.3239	1.76997	0.884984	−	0.465621i	$-0.154169\pi$
$977$	55.3239	1.76997	0.884984	+	0.465621i	$0.154169\pi$
$978$	0	0
$979$	0.913687	0.0292015
$980$	0	0
$981$	40.1128	1.28070
$982$	0	0
$983$	41.4214	1.32114	0.660569	−	0.750766i	$-0.270315\pi$
$983$	41.4214	1.32114	0.660569	+	0.750766i	$0.270315\pi$
$984$	0	0
$985$	7.55008	0.240565
$986$	0	0
$987$	1.37506	0.0437686
$988$	0	0
$989$	−41.2168	−1.31062
$990$	0	0
$991$	−42.8024	−1.35966	−0.679831	−	0.733369i	$-0.737947\pi$
$991$	−42.8024	−1.35966	−0.679831	+	0.733369i	$0.737947\pi$
$992$	0	0
$993$	17.0931	0.542435
$994$	0	0
$995$	−1.05611	−0.0334809
$996$	0	0
$997$	−39.8522	−1.26213	−0.631067	−	0.775729i	$-0.717383\pi$
$997$	−39.8522	−1.26213	−0.631067	+	0.775729i	$0.717383\pi$
$998$	0	0
$999$	−21.4067	−0.677278

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.20	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.20	✓	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)

\(f(q)\)	\(=\)	\(q-0.753945 q^{3} -1.16579 q^{5} +1.19034 q^{7} -2.43157 q^{9} +O(q^{10})\)	\(q-0.753945 q^{3} -1.16579 q^{5} +1.19034 q^{7} -2.43157 q^{9} -3.93174 q^{11} +2.90377 q^{13} +0.878938 q^{15} -1.93718 q^{17} +3.04157 q^{19} -0.897448 q^{21} +6.33561 q^{23} -3.64094 q^{25} +4.09510 q^{27} +3.20651 q^{29} +0.203587 q^{31} +2.96432 q^{33} -1.38768 q^{35} -5.22739 q^{37} -2.18928 q^{39} +10.5412 q^{41} -6.50559 q^{43} +2.83468 q^{45} -1.53219 q^{47} -5.58310 q^{49} +1.46053 q^{51} -0.0388220 q^{53} +4.58356 q^{55} -2.29318 q^{57} +4.00328 q^{59} +5.53333 q^{61} -2.89438 q^{63} -3.38517 q^{65} +12.0206 q^{67} -4.77670 q^{69} -2.94570 q^{71} +7.90151 q^{73} +2.74507 q^{75} -4.68009 q^{77} -10.4546 q^{79} +4.20722 q^{81} -4.73510 q^{83} +2.25834 q^{85} -2.41753 q^{87} -0.232387 q^{89} +3.45646 q^{91} -0.153493 q^{93} -3.54582 q^{95} +5.21513 q^{97} +9.56029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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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