LMFDB - Embedded newform 6008.2.a.b.1.17

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			1.17
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.09930 q^{3} -2.74603 q^{5} +2.16897 q^{7} -1.79154 q^{9} +O(q^{10})$	$q-1.09930 q^{3} -2.74603 q^{5} +2.16897 q^{7} -1.79154 q^{9} -1.58452 q^{11} -1.73180 q^{13} +3.01872 q^{15} +2.86954 q^{17} +1.23160 q^{19} -2.38435 q^{21} +1.19367 q^{23} +2.54071 q^{25} +5.26734 q^{27} -0.223779 q^{29} -1.89585 q^{31} +1.74187 q^{33} -5.95606 q^{35} +8.49852 q^{37} +1.90377 q^{39} -10.1205 q^{41} +6.87393 q^{43} +4.91962 q^{45} -2.10996 q^{47} -2.29559 q^{49} -3.15449 q^{51} +10.0047 q^{53} +4.35115 q^{55} -1.35390 q^{57} +4.74701 q^{59} -7.90097 q^{61} -3.88578 q^{63} +4.75557 q^{65} +3.43609 q^{67} -1.31221 q^{69} +11.9228 q^{71} -10.4927 q^{73} -2.79300 q^{75} -3.43677 q^{77} -4.30406 q^{79} -0.415792 q^{81} +17.4052 q^{83} -7.87986 q^{85} +0.246001 q^{87} +6.11597 q^{89} -3.75621 q^{91} +2.08411 q^{93} -3.38201 q^{95} -4.33364 q^{97} +2.83873 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.09930	−0.634682	−0.317341	−	0.948311i	$-0.602790\pi$
$3$	−1.09930	−0.634682	−0.317341	+	0.948311i	$0.602790\pi$
$4$	0	0
$5$	−2.74603	−1.22806	−0.614032	−	0.789281i	$-0.710453\pi$
$5$	−2.74603	−1.22806	−0.614032	+	0.789281i	$0.710453\pi$
$6$	0	0
$7$	2.16897	0.819792	0.409896	−	0.912132i	$-0.365565\pi$
$7$	2.16897	0.819792	0.409896	+	0.912132i	$0.365565\pi$
$8$	0	0
$9$	−1.79154	−0.597179
$10$	0	0
$11$	−1.58452	−0.477751	−0.238876	−	0.971050i	$-0.576779\pi$
$11$	−1.58452	−0.477751	−0.238876	+	0.971050i	$0.576779\pi$
$12$	0	0
$13$	−1.73180	−0.480314	−0.240157	−	0.970734i	$-0.577199\pi$
$13$	−1.73180	−0.480314	−0.240157	+	0.970734i	$0.577199\pi$
$14$	0	0
$15$	3.01872	0.779430
$16$	0	0
$17$	2.86954	0.695966	0.347983	−	0.937501i	$-0.386867\pi$
$17$	2.86954	0.695966	0.347983	+	0.937501i	$0.386867\pi$
$18$	0	0
$19$	1.23160	0.282548	0.141274	−	0.989971i	$-0.454880\pi$
$19$	1.23160	0.282548	0.141274	+	0.989971i	$0.454880\pi$
$20$	0	0
$21$	−2.38435	−0.520307
$22$	0	0
$23$	1.19367	0.248898	0.124449	−	0.992226i	$-0.460284\pi$
$23$	1.19367	0.248898	0.124449	+	0.992226i	$0.460284\pi$
$24$	0	0
$25$	2.54071	0.508142
$26$	0	0
$27$	5.26734	1.01370
$28$	0	0
$29$	−0.223779	−0.0415548	−0.0207774	−	0.999784i	$-0.506614\pi$
$29$	−0.223779	−0.0415548	−0.0207774	+	0.999784i	$0.506614\pi$
$30$	0	0
$31$	−1.89585	−0.340505	−0.170253	−	0.985400i	$-0.554458\pi$
$31$	−1.89585	−0.340505	−0.170253	+	0.985400i	$0.554458\pi$
$32$	0	0
$33$	1.74187	0.303220
$34$	0	0
$35$	−5.95606	−1.00676
$36$	0	0
$37$	8.49852	1.39715	0.698574	−	0.715538i	$-0.253818\pi$
$37$	8.49852	1.39715	0.698574	+	0.715538i	$0.253818\pi$
$38$	0	0
$39$	1.90377	0.304846
$40$	0	0
$41$	−10.1205	−1.58056	−0.790280	−	0.612745i	$-0.790065\pi$
$41$	−10.1205	−1.58056	−0.790280	+	0.612745i	$0.790065\pi$
$42$	0	0
$43$	6.87393	1.04826	0.524132	−	0.851637i	$-0.324390\pi$
$43$	6.87393	1.04826	0.524132	+	0.851637i	$0.324390\pi$
$44$	0	0
$45$	4.91962	0.733374
$46$	0	0
$47$	−2.10996	−0.307769	−0.153885	−	0.988089i	$-0.549178\pi$
$47$	−2.10996	−0.307769	−0.153885	+	0.988089i	$0.549178\pi$
$48$	0	0
$49$	−2.29559	−0.327941
$50$	0	0
$51$	−3.15449	−0.441717
$52$	0	0
$53$	10.0047	1.37425	0.687127	−	0.726537i	$-0.258872\pi$
$53$	10.0047	1.37425	0.687127	+	0.726537i	$0.258872\pi$
$54$	0	0
$55$	4.35115	0.586709
$56$	0	0
$57$	−1.35390	−0.179328
$58$	0	0
$59$	4.74701	0.618008	0.309004	−	0.951061i	$-0.400004\pi$
$59$	4.74701	0.618008	0.309004	+	0.951061i	$0.400004\pi$
$60$	0	0
$61$	−7.90097	−1.01162	−0.505808	−	0.862646i	$-0.668806\pi$
$61$	−7.90097	−1.01162	−0.505808	+	0.862646i	$0.668806\pi$
$62$	0	0
$63$	−3.88578	−0.489562
$64$	0	0
$65$	4.75557	0.589856
$66$	0	0
$67$	3.43609	0.419785	0.209893	−	0.977724i	$-0.432689\pi$
$67$	3.43609	0.419785	0.209893	+	0.977724i	$0.432689\pi$
$68$	0	0
$69$	−1.31221	−0.157971
$70$	0	0
$71$	11.9228	1.41497	0.707487	−	0.706727i	$-0.249829\pi$
$71$	11.9228	1.41497	0.707487	+	0.706727i	$0.249829\pi$
$72$	0	0
$73$	−10.4927	−1.22808	−0.614038	−	0.789276i	$-0.710456\pi$
$73$	−10.4927	−1.22808	−0.614038	+	0.789276i	$0.710456\pi$
$74$	0	0
$75$	−2.79300	−0.322508
$76$	0	0
$77$	−3.43677	−0.391657
$78$	0	0
$79$	−4.30406	−0.484244	−0.242122	−	0.970246i	$-0.577843\pi$
$79$	−4.30406	−0.484244	−0.242122	+	0.970246i	$0.577843\pi$
$80$	0	0
$81$	−0.415792	−0.0461991
$82$	0	0
$83$	17.4052	1.91047	0.955235	−	0.295849i	$-0.0956025\pi$
$83$	17.4052	1.91047	0.955235	+	0.295849i	$0.0956025\pi$
$84$	0	0
$85$	−7.87986	−0.854691
$86$	0	0
$87$	0.246001	0.0263741
$88$	0	0
$89$	6.11597	0.648292	0.324146	−	0.946007i	$-0.394923\pi$
$89$	6.11597	0.648292	0.324146	+	0.946007i	$0.394923\pi$
$90$	0	0
$91$	−3.75621	−0.393757
$92$	0	0
$93$	2.08411	0.216113
$94$	0	0
$95$	−3.38201	−0.346987
$96$	0	0
$97$	−4.33364	−0.440014	−0.220007	−	0.975498i	$-0.570608\pi$
$97$	−4.33364	−0.440014	−0.220007	+	0.975498i	$0.570608\pi$
$98$	0	0
$99$	2.83873	0.285303
$100$	0	0
$101$	−8.97014	−0.892562	−0.446281	−	0.894893i	$-0.647252\pi$
$101$	−8.97014	−0.892562	−0.446281	+	0.894893i	$0.647252\pi$
$102$	0	0
$103$	−18.0865	−1.78212	−0.891059	−	0.453888i	$-0.850037\pi$
$103$	−18.0865	−1.78212	−0.891059	+	0.453888i	$0.850037\pi$
$104$	0	0
$105$	6.54750	0.638971
$106$	0	0
$107$	15.4337	1.49203	0.746017	−	0.665927i	$-0.231964\pi$
$107$	15.4337	1.49203	0.746017	+	0.665927i	$0.231964\pi$
$108$	0	0
$109$	−5.54369	−0.530989	−0.265495	−	0.964112i	$-0.585535\pi$
$109$	−5.54369	−0.530989	−0.265495	+	0.964112i	$0.585535\pi$
$110$	0	0
$111$	−9.34244	−0.886745
$112$	0	0
$113$	12.8618	1.20994	0.604968	−	0.796250i	$-0.293186\pi$
$113$	12.8618	1.20994	0.604968	+	0.796250i	$0.293186\pi$
$114$	0	0
$115$	−3.27787	−0.305663
$116$	0	0
$117$	3.10257	0.286833
$118$	0	0
$119$	6.22394	0.570547
$120$	0	0
$121$	−8.48929	−0.771754
$122$	0	0
$123$	11.1255	1.00315
$124$	0	0
$125$	6.75330	0.604034
$126$	0	0
$127$	−9.77630	−0.867506	−0.433753	−	0.901032i	$-0.642811\pi$
$127$	−9.77630	−0.867506	−0.433753	+	0.901032i	$0.642811\pi$
$128$	0	0
$129$	−7.55652	−0.665315
$130$	0	0
$131$	−14.9727	−1.30817	−0.654084	−	0.756422i	$-0.726946\pi$
$131$	−14.9727	−1.30817	−0.654084	+	0.756422i	$0.726946\pi$
$132$	0	0
$133$	2.67130	0.231631
$134$	0	0
$135$	−14.4643	−1.24489
$136$	0	0
$137$	2.50628	0.214126	0.107063	−	0.994252i	$-0.465855\pi$
$137$	2.50628	0.214126	0.107063	+	0.994252i	$0.465855\pi$
$138$	0	0
$139$	−13.1374	−1.11430	−0.557150	−	0.830412i	$-0.688105\pi$
$139$	−13.1374	−1.11430	−0.557150	+	0.830412i	$0.688105\pi$
$140$	0	0
$141$	2.31948	0.195336
$142$	0	0
$143$	2.74407	0.229470
$144$	0	0
$145$	0.614506	0.0510320
$146$	0	0
$147$	2.52354	0.208138
$148$	0	0
$149$	6.30286	0.516350	0.258175	−	0.966098i	$-0.416879\pi$
$149$	6.30286	0.516350	0.258175	+	0.966098i	$0.416879\pi$
$150$	0	0
$151$	−19.7337	−1.60591	−0.802954	−	0.596041i	$-0.796740\pi$
$151$	−19.7337	−1.60591	−0.802954	+	0.596041i	$0.796740\pi$
$152$	0	0
$153$	−5.14089	−0.415616
$154$	0	0
$155$	5.20608	0.418162
$156$	0	0
$157$	3.27204	0.261137	0.130569	−	0.991439i	$-0.458320\pi$
$157$	3.27204	0.261137	0.130569	+	0.991439i	$0.458320\pi$
$158$	0	0
$159$	−10.9982	−0.872215
$160$	0	0
$161$	2.58904	0.204045
$162$	0	0
$163$	−12.7780	−1.00085	−0.500425	−	0.865780i	$-0.666823\pi$
$163$	−12.7780	−1.00085	−0.500425	+	0.865780i	$0.666823\pi$
$164$	0	0
$165$	−4.78323	−0.372374
$166$	0	0
$167$	8.37846	0.648344	0.324172	−	0.945998i	$-0.394914\pi$
$167$	8.37846	0.648344	0.324172	+	0.945998i	$0.394914\pi$
$168$	0	0
$169$	−10.0009	−0.769299
$170$	0	0
$171$	−2.20645	−0.168732
$172$	0	0
$173$	17.0971	1.29987	0.649934	−	0.759990i	$-0.274796\pi$
$173$	17.0971	1.29987	0.649934	+	0.759990i	$0.274796\pi$
$174$	0	0
$175$	5.51071	0.416570
$176$	0	0
$177$	−5.21840	−0.392239
$178$	0	0
$179$	−6.01037	−0.449237	−0.224618	−	0.974447i	$-0.572114\pi$
$179$	−6.01037	−0.449237	−0.224618	+	0.974447i	$0.572114\pi$
$180$	0	0
$181$	2.72597	0.202620	0.101310	−	0.994855i	$-0.467697\pi$
$181$	2.72597	0.202620	0.101310	+	0.994855i	$0.467697\pi$
$182$	0	0
$183$	8.68555	0.642054
$184$	0	0
$185$	−23.3372	−1.71579
$186$	0	0
$187$	−4.54685	−0.332499
$188$	0	0
$189$	11.4247	0.831024
$190$	0	0
$191$	−11.7434	−0.849725	−0.424862	−	0.905258i	$-0.639678\pi$
$191$	−11.7434	−0.849725	−0.424862	+	0.905258i	$0.639678\pi$
$192$	0	0
$193$	−7.73685	−0.556911	−0.278455	−	0.960449i	$-0.589822\pi$
$193$	−7.73685	−0.556911	−0.278455	+	0.960449i	$0.589822\pi$
$194$	0	0
$195$	−5.22781	−0.374371
$196$	0	0
$197$	−8.37966	−0.597026	−0.298513	−	0.954406i	$-0.596491\pi$
$197$	−8.37966	−0.597026	−0.298513	+	0.954406i	$0.596491\pi$
$198$	0	0
$199$	−11.7130	−0.830310	−0.415155	−	0.909751i	$-0.636273\pi$
$199$	−11.7130	−0.830310	−0.415155	+	0.909751i	$0.636273\pi$
$200$	0	0
$201$	−3.77730	−0.266430
$202$	0	0
$203$	−0.485370	−0.0340663
$204$	0	0
$205$	27.7913	1.94103
$206$	0	0
$207$	−2.13851	−0.148637
$208$	0	0
$209$	−1.95149	−0.134988
$210$	0	0
$211$	3.02947	0.208557	0.104279	−	0.994548i	$-0.466747\pi$
$211$	3.02947	0.208557	0.104279	+	0.994548i	$0.466747\pi$
$212$	0	0
$213$	−13.1067	−0.898058
$214$	0	0
$215$	−18.8760	−1.28734
$216$	0	0
$217$	−4.11204	−0.279144
$218$	0	0
$219$	11.5346	0.779438
$220$	0	0
$221$	−4.96946	−0.334282
$222$	0	0
$223$	11.2040	0.750272	0.375136	−	0.926970i	$-0.377596\pi$
$223$	11.2040	0.750272	0.375136	+	0.926970i	$0.377596\pi$
$224$	0	0
$225$	−4.55177	−0.303451
$226$	0	0
$227$	−16.3342	−1.08414	−0.542069	−	0.840334i	$-0.682359\pi$
$227$	−16.3342	−1.08414	−0.542069	+	0.840334i	$0.682359\pi$
$228$	0	0
$229$	22.5748	1.49179	0.745893	−	0.666066i	$-0.232023\pi$
$229$	22.5748	1.49179	0.745893	+	0.666066i	$0.232023\pi$
$230$	0	0
$231$	3.77805	0.248577
$232$	0	0
$233$	2.64135	0.173041	0.0865204	−	0.996250i	$-0.472425\pi$
$233$	2.64135	0.173041	0.0865204	+	0.996250i	$0.472425\pi$
$234$	0	0
$235$	5.79402	0.377960
$236$	0	0
$237$	4.73146	0.307341
$238$	0	0
$239$	4.24441	0.274548	0.137274	−	0.990533i	$-0.456166\pi$
$239$	4.24441	0.274548	0.137274	+	0.990533i	$0.456166\pi$
$240$	0	0
$241$	17.2000	1.10795	0.553975	−	0.832534i	$-0.313110\pi$
$241$	17.2000	1.10795	0.553975	+	0.832534i	$0.313110\pi$
$242$	0	0
$243$	−15.3449	−0.984379
$244$	0	0
$245$	6.30376	0.402732
$246$	0	0
$247$	−2.13288	−0.135712
$248$	0	0
$249$	−19.1336	−1.21254
$250$	0	0
$251$	−11.0296	−0.696184	−0.348092	−	0.937460i	$-0.613170\pi$
$251$	−11.0296	−0.696184	−0.348092	+	0.937460i	$0.613170\pi$
$252$	0	0
$253$	−1.89140	−0.118911
$254$	0	0
$255$	8.66234	0.542457
$256$	0	0
$257$	−3.98414	−0.248524	−0.124262	−	0.992249i	$-0.539656\pi$
$257$	−3.98414	−0.248524	−0.124262	+	0.992249i	$0.539656\pi$
$258$	0	0
$259$	18.4330	1.14537
$260$	0	0
$261$	0.400909	0.0248156
$262$	0	0
$263$	−14.8103	−0.913244	−0.456622	−	0.889661i	$-0.650941\pi$
$263$	−14.8103	−0.913244	−0.456622	+	0.889661i	$0.650941\pi$
$264$	0	0
$265$	−27.4733	−1.68767
$266$	0	0
$267$	−6.72330	−0.411459
$268$	0	0
$269$	−24.3355	−1.48376	−0.741882	−	0.670531i	$-0.766066\pi$
$269$	−24.3355	−1.48376	−0.741882	+	0.670531i	$0.766066\pi$
$270$	0	0
$271$	−19.6056	−1.19096	−0.595479	−	0.803371i	$-0.703038\pi$
$271$	−19.6056	−1.19096	−0.595479	+	0.803371i	$0.703038\pi$
$272$	0	0
$273$	4.12920	0.249911
$274$	0	0
$275$	−4.02581	−0.242765
$276$	0	0
$277$	27.7508	1.66738	0.833691	−	0.552231i	$-0.186223\pi$
$277$	27.7508	1.66738	0.833691	+	0.552231i	$0.186223\pi$
$278$	0	0
$279$	3.39649	0.203343
$280$	0	0
$281$	−8.25163	−0.492251	−0.246126	−	0.969238i	$-0.579158\pi$
$281$	−8.25163	−0.492251	−0.246126	+	0.969238i	$0.579158\pi$
$282$	0	0
$283$	−15.4690	−0.919539	−0.459770	−	0.888038i	$-0.652068\pi$
$283$	−15.4690	−0.919539	−0.459770	+	0.888038i	$0.652068\pi$
$284$	0	0
$285$	3.71785	0.220227
$286$	0	0
$287$	−21.9511	−1.29573
$288$	0	0
$289$	−8.76573	−0.515631
$290$	0	0
$291$	4.76398	0.279269
$292$	0	0
$293$	12.2695	0.716790	0.358395	−	0.933570i	$-0.383324\pi$
$293$	12.2695	0.716790	0.358395	+	0.933570i	$0.383324\pi$
$294$	0	0
$295$	−13.0355	−0.758954
$296$	0	0
$297$	−8.34622	−0.484297
$298$	0	0
$299$	−2.06720	−0.119549
$300$	0	0
$301$	14.9093	0.859359
$302$	0	0
$303$	9.86089	0.566493
$304$	0	0
$305$	21.6963	1.24233
$306$	0	0
$307$	−6.68631	−0.381608	−0.190804	−	0.981628i	$-0.561109\pi$
$307$	−6.68631	−0.381608	−0.190804	+	0.981628i	$0.561109\pi$
$308$	0	0
$309$	19.8825	1.13108
$310$	0	0
$311$	1.04846	0.0594526	0.0297263	−	0.999558i	$-0.490536\pi$
$311$	1.04846	0.0594526	0.0297263	+	0.999558i	$0.490536\pi$
$312$	0	0
$313$	16.0115	0.905026	0.452513	−	0.891758i	$-0.350528\pi$
$313$	16.0115	0.905026	0.452513	+	0.891758i	$0.350528\pi$
$314$	0	0
$315$	10.6705	0.601214
$316$	0	0
$317$	18.1094	1.01712	0.508562	−	0.861025i	$-0.330177\pi$
$317$	18.1094	1.01712	0.508562	+	0.861025i	$0.330177\pi$
$318$	0	0
$319$	0.354583	0.0198529
$320$	0	0
$321$	−16.9663	−0.946967
$322$	0	0
$323$	3.53412	0.196644
$324$	0	0
$325$	−4.39999	−0.244067
$326$	0	0
$327$	6.09419	0.337009
$328$	0	0
$329$	−4.57643	−0.252307
$330$	0	0
$331$	24.7980	1.36302	0.681509	−	0.731809i	$-0.261324\pi$
$331$	24.7980	1.36302	0.681509	+	0.731809i	$0.261324\pi$
$332$	0	0
$333$	−15.2254	−0.834347
$334$	0	0
$335$	−9.43562	−0.515523
$336$	0	0
$337$	−32.2436	−1.75642	−0.878212	−	0.478272i	$-0.841263\pi$
$337$	−32.2436	−1.75642	−0.878212	+	0.478272i	$0.841263\pi$
$338$	0	0
$339$	−14.1390	−0.767924
$340$	0	0
$341$	3.00402	0.162677
$342$	0	0
$343$	−20.1618	−1.08864
$344$	0	0
$345$	3.60337	0.193999
$346$	0	0
$347$	−24.8657	−1.33486	−0.667431	−	0.744672i	$-0.732606\pi$
$347$	−24.8657	−1.33486	−0.667431	+	0.744672i	$0.732606\pi$
$348$	0	0
$349$	−20.9262	−1.12015	−0.560077	−	0.828441i	$-0.689228\pi$
$349$	−20.9262	−1.12015	−0.560077	+	0.828441i	$0.689228\pi$
$350$	0	0
$351$	−9.12196	−0.486894
$352$	0	0
$353$	0.785093	0.0417863	0.0208932	−	0.999782i	$-0.493349\pi$
$353$	0.785093	0.0417863	0.0208932	+	0.999782i	$0.493349\pi$
$354$	0	0
$355$	−32.7404	−1.73768
$356$	0	0
$357$	−6.84199	−0.362116
$358$	0	0
$359$	−20.9711	−1.10681	−0.553406	−	0.832912i	$-0.686672\pi$
$359$	−20.9711	−1.10681	−0.553406	+	0.832912i	$0.686672\pi$
$360$	0	0
$361$	−17.4832	−0.920167
$362$	0	0
$363$	9.33229	0.489818
$364$	0	0
$365$	28.8133	1.50816
$366$	0	0
$367$	−36.2502	−1.89224	−0.946122	−	0.323811i	$-0.895036\pi$
$367$	−36.2502	−1.89224	−0.946122	+	0.323811i	$0.895036\pi$
$368$	0	0
$369$	18.1313	0.943877
$370$	0	0
$371$	21.6999	1.12660
$372$	0	0
$373$	−18.7117	−0.968855	−0.484427	−	0.874831i	$-0.660972\pi$
$373$	−18.7117	−0.968855	−0.484427	+	0.874831i	$0.660972\pi$
$374$	0	0
$375$	−7.42392	−0.383369
$376$	0	0
$377$	0.387540	0.0199593
$378$	0	0
$379$	16.5080	0.847957	0.423979	−	0.905672i	$-0.360633\pi$
$379$	16.5080	0.847957	0.423979	+	0.905672i	$0.360633\pi$
$380$	0	0
$381$	10.7471	0.550591
$382$	0	0
$383$	21.7893	1.11338	0.556690	−	0.830720i	$-0.312071\pi$
$383$	21.7893	1.11338	0.556690	+	0.830720i	$0.312071\pi$
$384$	0	0
$385$	9.43750	0.480979
$386$	0	0
$387$	−12.3149	−0.626001
$388$	0	0
$389$	9.66257	0.489912	0.244956	−	0.969534i	$-0.421227\pi$
$389$	9.66257	0.489912	0.244956	+	0.969534i	$0.421227\pi$
$390$	0	0
$391$	3.42530	0.173225
$392$	0	0
$393$	16.4595	0.830271
$394$	0	0
$395$	11.8191	0.594683
$396$	0	0
$397$	−14.6358	−0.734548	−0.367274	−	0.930113i	$-0.619709\pi$
$397$	−14.6358	−0.734548	−0.367274	+	0.930113i	$0.619709\pi$
$398$	0	0
$399$	−2.93656	−0.147012
$400$	0	0
$401$	−19.3109	−0.964341	−0.482170	−	0.876077i	$-0.660151\pi$
$401$	−19.3109	−0.964341	−0.482170	+	0.876077i	$0.660151\pi$
$402$	0	0
$403$	3.28323	0.163549
$404$	0	0
$405$	1.14178	0.0567355
$406$	0	0
$407$	−13.4661	−0.667489
$408$	0	0
$409$	27.9970	1.38436	0.692182	−	0.721723i	$-0.256649\pi$
$409$	27.9970	1.38436	0.692182	+	0.721723i	$0.256649\pi$
$410$	0	0
$411$	−2.75516	−0.135902
$412$	0	0
$413$	10.2961	0.506638
$414$	0	0
$415$	−47.7953	−2.34618
$416$	0	0
$417$	14.4420	0.707226
$418$	0	0
$419$	−30.7660	−1.50302	−0.751508	−	0.659724i	$-0.770673\pi$
$419$	−30.7660	−1.50302	−0.751508	+	0.659724i	$0.770673\pi$
$420$	0	0
$421$	−31.1246	−1.51692	−0.758459	−	0.651720i	$-0.774048\pi$
$421$	−31.1246	−1.51692	−0.758459	+	0.651720i	$0.774048\pi$
$422$	0	0
$423$	3.78007	0.183793
$424$	0	0
$425$	7.29067	0.353649
$426$	0	0
$427$	−17.1369	−0.829315
$428$	0	0
$429$	−3.01656	−0.145641
$430$	0	0
$431$	−39.5001	−1.90265	−0.951327	−	0.308182i	$-0.900279\pi$
$431$	−39.5001	−1.90265	−0.951327	+	0.308182i	$0.900279\pi$
$432$	0	0
$433$	29.8505	1.43452	0.717261	−	0.696805i	$-0.245396\pi$
$433$	29.8505	1.43452	0.717261	+	0.696805i	$0.245396\pi$
$434$	0	0
$435$	−0.675528	−0.0323891
$436$	0	0
$437$	1.47013	0.0703258
$438$	0	0
$439$	25.1976	1.20262	0.601308	−	0.799017i	$-0.294646\pi$
$439$	25.1976	1.20262	0.601308	+	0.799017i	$0.294646\pi$
$440$	0	0
$441$	4.11262	0.195839
$442$	0	0
$443$	10.1704	0.483212	0.241606	−	0.970374i	$-0.422326\pi$
$443$	10.1704	0.483212	0.241606	+	0.970374i	$0.422326\pi$
$444$	0	0
$445$	−16.7947	−0.796144
$446$	0	0
$447$	−6.92874	−0.327718
$448$	0	0
$449$	−31.8335	−1.50232	−0.751158	−	0.660123i	$-0.770504\pi$
$449$	−31.8335	−1.50232	−0.751158	+	0.660123i	$0.770504\pi$
$450$	0	0
$451$	16.0362	0.755115
$452$	0	0
$453$	21.6933	1.01924
$454$	0	0
$455$	10.3147	0.483559
$456$	0	0
$457$	23.9539	1.12052	0.560258	−	0.828318i	$-0.310702\pi$
$457$	23.9539	1.12052	0.560258	+	0.828318i	$0.310702\pi$
$458$	0	0
$459$	15.1149	0.705501
$460$	0	0
$461$	−7.67836	−0.357617	−0.178808	−	0.983884i	$-0.557224\pi$
$461$	−7.67836	−0.357617	−0.178808	+	0.983884i	$0.557224\pi$
$462$	0	0
$463$	10.6720	0.495969	0.247984	−	0.968764i	$-0.420232\pi$
$463$	10.6720	0.495969	0.247984	+	0.968764i	$0.420232\pi$
$464$	0	0
$465$	−5.72305	−0.265400
$466$	0	0
$467$	−1.31574	−0.0608854	−0.0304427	−	0.999537i	$-0.509692\pi$
$467$	−1.31574	−0.0608854	−0.0304427	+	0.999537i	$0.509692\pi$
$468$	0	0
$469$	7.45276	0.344137
$470$	0	0
$471$	−3.59696	−0.165739
$472$	0	0
$473$	−10.8919	−0.500809
$474$	0	0
$475$	3.12913	0.143574
$476$	0	0
$477$	−17.9238	−0.820675
$478$	0	0
$479$	−26.0250	−1.18911	−0.594557	−	0.804053i	$-0.702673\pi$
$479$	−26.0250	−1.18911	−0.594557	+	0.804053i	$0.702673\pi$
$480$	0	0
$481$	−14.7177	−0.671069
$482$	0	0
$483$	−2.84614	−0.129504
$484$	0	0
$485$	11.9003	0.540366
$486$	0	0
$487$	36.4637	1.65233	0.826164	−	0.563430i	$-0.190519\pi$
$487$	36.4637	1.65233	0.826164	+	0.563430i	$0.190519\pi$
$488$	0	0
$489$	14.0469	0.635222
$490$	0	0
$491$	−26.9479	−1.21614	−0.608070	−	0.793883i	$-0.708056\pi$
$491$	−26.9479	−1.21614	−0.608070	+	0.793883i	$0.708056\pi$
$492$	0	0
$493$	−0.642144	−0.0289207
$494$	0	0
$495$	−7.79524	−0.350370
$496$	0	0
$497$	25.8601	1.15998
$498$	0	0
$499$	−22.4133	−1.00336	−0.501678	−	0.865055i	$-0.667284\pi$
$499$	−22.4133	−1.00336	−0.501678	+	0.865055i	$0.667284\pi$
$500$	0	0
$501$	−9.21045	−0.411493
$502$	0	0
$503$	8.78473	0.391692	0.195846	−	0.980635i	$-0.437255\pi$
$503$	8.78473	0.391692	0.195846	+	0.980635i	$0.437255\pi$
$504$	0	0
$505$	24.6323	1.09612
$506$	0	0
$507$	10.9940	0.488260
$508$	0	0
$509$	−40.6673	−1.80255	−0.901273	−	0.433252i	$-0.857366\pi$
$509$	−40.6673	−1.80255	−0.901273	+	0.433252i	$0.857366\pi$
$510$	0	0
$511$	−22.7583	−1.00677
$512$	0	0
$513$	6.48725	0.286419
$514$	0	0
$515$	49.6662	2.18855
$516$	0	0
$517$	3.34328	0.147037
$518$	0	0
$519$	−18.7949	−0.825003
$520$	0	0
$521$	−40.6019	−1.77880	−0.889400	−	0.457130i	$-0.848877\pi$
$521$	−40.6019	−1.77880	−0.889400	+	0.457130i	$0.848877\pi$
$522$	0	0
$523$	−35.3124	−1.54410	−0.772052	−	0.635559i	$-0.780770\pi$
$523$	−35.3124	−1.54410	−0.772052	+	0.635559i	$0.780770\pi$
$524$	0	0
$525$	−6.05793	−0.264390
$526$	0	0
$527$	−5.44023	−0.236980
$528$	0	0
$529$	−21.5751	−0.938050
$530$	0	0
$531$	−8.50444	−0.369061
$532$	0	0
$533$	17.5267	0.759165
$534$	0	0
$535$	−42.3815	−1.83231
$536$	0	0
$537$	6.60722	0.285122
$538$	0	0
$539$	3.63740	0.156674
$540$	0	0
$541$	−5.07096	−0.218018	−0.109009	−	0.994041i	$-0.534768\pi$
$541$	−5.07096	−0.218018	−0.109009	+	0.994041i	$0.534768\pi$
$542$	0	0
$543$	−2.99666	−0.128599
$544$	0	0
$545$	15.2232	0.652089
$546$	0	0
$547$	−1.61212	−0.0689291	−0.0344646	−	0.999406i	$-0.510973\pi$
$547$	−1.61212	−0.0689291	−0.0344646	+	0.999406i	$0.510973\pi$
$548$	0	0
$549$	14.1549	0.604115
$550$	0	0
$551$	−0.275606	−0.0117412
$552$	0	0
$553$	−9.33535	−0.396980
$554$	0	0
$555$	25.6547	1.08898
$556$	0	0
$557$	12.4610	0.527988	0.263994	−	0.964524i	$-0.414960\pi$
$557$	12.4610	0.527988	0.263994	+	0.964524i	$0.414960\pi$
$558$	0	0
$559$	−11.9042	−0.503496
$560$	0	0
$561$	4.99836	0.211031
$562$	0	0
$563$	−10.7354	−0.452442	−0.226221	−	0.974076i	$-0.572637\pi$
$563$	−10.7354	−0.452442	−0.226221	+	0.974076i	$0.572637\pi$
$564$	0	0
$565$	−35.3189	−1.48588
$566$	0	0
$567$	−0.901839	−0.0378737
$568$	0	0
$569$	43.9395	1.84204	0.921020	−	0.389515i	$-0.127357\pi$
$569$	43.9395	1.84204	0.921020	+	0.389515i	$0.127357\pi$
$570$	0	0
$571$	6.99777	0.292848	0.146424	−	0.989222i	$-0.453224\pi$
$571$	6.99777	0.292848	0.146424	+	0.989222i	$0.453224\pi$
$572$	0	0
$573$	12.9096	0.539305
$574$	0	0
$575$	3.03278	0.126476
$576$	0	0
$577$	−8.57486	−0.356976	−0.178488	−	0.983942i	$-0.557121\pi$
$577$	−8.57486	−0.356976	−0.178488	+	0.983942i	$0.557121\pi$
$578$	0	0
$579$	8.50514	0.353461
$580$	0	0
$581$	37.7513	1.56619
$582$	0	0
$583$	−15.8527	−0.656551
$584$	0	0
$585$	−8.51977	−0.352249
$586$	0	0
$587$	−43.1314	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$587$	−43.1314	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$588$	0	0
$589$	−2.33493	−0.0962091
$590$	0	0
$591$	9.21177	0.378922
$592$	0	0
$593$	29.5291	1.21261	0.606306	−	0.795231i	$-0.292650\pi$
$593$	29.5291	1.21261	0.606306	+	0.795231i	$0.292650\pi$
$594$	0	0
$595$	−17.0912	−0.700669
$596$	0	0
$597$	12.8761	0.526983
$598$	0	0
$599$	37.9585	1.55094	0.775471	−	0.631383i	$-0.217512\pi$
$599$	37.9585	1.55094	0.775471	+	0.631383i	$0.217512\pi$
$600$	0	0
$601$	0.0309671	0.00126318	0.000631588	−	1.00000i	$-0.499799\pi$
$601$	0.0309671	0.00126318	0.000631588		1.00000i	$0.499799\pi$
$602$	0	0
$603$	−6.15588	−0.250687
$604$	0	0
$605$	23.3119	0.947763
$606$	0	0
$607$	12.6469	0.513323	0.256662	−	0.966501i	$-0.417377\pi$
$607$	12.6469	0.513323	0.256662	+	0.966501i	$0.417377\pi$
$608$	0	0
$609$	0.533568	0.0216213
$610$	0	0
$611$	3.65402	0.147826
$612$	0	0
$613$	−30.5519	−1.23398	−0.616989	−	0.786971i	$-0.711648\pi$
$613$	−30.5519	−1.23398	−0.616989	+	0.786971i	$0.711648\pi$
$614$	0	0
$615$	−30.5510	−1.23194
$616$	0	0
$617$	−26.5044	−1.06703	−0.533513	−	0.845792i	$-0.679128\pi$
$617$	−26.5044	−1.06703	−0.533513	+	0.845792i	$0.679128\pi$
$618$	0	0
$619$	−47.5115	−1.90965	−0.954824	−	0.297172i	$-0.903957\pi$
$619$	−47.5115	−1.90965	−0.954824	+	0.297172i	$0.903957\pi$
$620$	0	0
$621$	6.28750	0.252309
$622$	0	0
$623$	13.2653	0.531465
$624$	0	0
$625$	−31.2483	−1.24993
$626$	0	0
$627$	2.14528	0.0856743
$628$	0	0
$629$	24.3869	0.972368
$630$	0	0
$631$	−0.276788	−0.0110187	−0.00550937	−	0.999985i	$-0.501754\pi$
$631$	−0.276788	−0.0110187	−0.00550937	+	0.999985i	$0.501754\pi$
$632$	0	0
$633$	−3.33030	−0.132368
$634$	0	0
$635$	26.8461	1.06535
$636$	0	0
$637$	3.97548	0.157514
$638$	0	0
$639$	−21.3601	−0.844992
$640$	0	0
$641$	43.7215	1.72690	0.863448	−	0.504438i	$-0.168300\pi$
$641$	43.7215	1.72690	0.863448	+	0.504438i	$0.168300\pi$
$642$	0	0
$643$	−12.1504	−0.479166	−0.239583	−	0.970876i	$-0.577011\pi$
$643$	−12.1504	−0.479166	−0.239583	+	0.970876i	$0.577011\pi$
$644$	0	0
$645$	20.7505	0.817049
$646$	0	0
$647$	20.8906	0.821295	0.410647	−	0.911794i	$-0.365303\pi$
$647$	20.8906	0.821295	0.410647	+	0.911794i	$0.365303\pi$
$648$	0	0
$649$	−7.52174	−0.295254
$650$	0	0
$651$	4.52037	0.177167
$652$	0	0
$653$	−0.341625	−0.0133688	−0.00668441	−	0.999978i	$-0.502128\pi$
$653$	−0.341625	−0.0133688	−0.00668441	+	0.999978i	$0.502128\pi$
$654$	0	0
$655$	41.1155	1.60651
$656$	0	0
$657$	18.7980	0.733381
$658$	0	0
$659$	15.9077	0.619675	0.309837	−	0.950790i	$-0.399725\pi$
$659$	15.9077	0.619675	0.309837	+	0.950790i	$0.399725\pi$
$660$	0	0
$661$	29.8124	1.15957	0.579784	−	0.814770i	$-0.303137\pi$
$661$	29.8124	1.15957	0.579784	+	0.814770i	$0.303137\pi$
$662$	0	0
$663$	5.46293	0.212163
$664$	0	0
$665$	−7.33547	−0.284457
$666$	0	0
$667$	−0.267120	−0.0103429
$668$	0	0
$669$	−12.3165	−0.476185
$670$	0	0
$671$	12.5193	0.483301
$672$	0	0
$673$	33.7290	1.30016	0.650079	−	0.759867i	$-0.274736\pi$
$673$	33.7290	1.30016	0.650079	+	0.759867i	$0.274736\pi$
$674$	0	0
$675$	13.3828	0.515103
$676$	0	0
$677$	−42.5081	−1.63372	−0.816859	−	0.576837i	$-0.804287\pi$
$677$	−42.5081	−1.63372	−0.816859	+	0.576837i	$0.804287\pi$
$678$	0	0
$679$	−9.39952	−0.360720
$680$	0	0
$681$	17.9562	0.688083
$682$	0	0
$683$	−43.2500	−1.65492	−0.827459	−	0.561527i	$-0.810214\pi$
$683$	−43.2500	−1.65492	−0.827459	+	0.561527i	$0.810214\pi$
$684$	0	0
$685$	−6.88233	−0.262960
$686$	0	0
$687$	−24.8165	−0.946809
$688$	0	0
$689$	−17.3261	−0.660073
$690$	0	0
$691$	27.9817	1.06447	0.532236	−	0.846596i	$-0.321352\pi$
$691$	27.9817	1.06447	0.532236	+	0.846596i	$0.321352\pi$
$692$	0	0
$693$	6.15710	0.233889
$694$	0	0
$695$	36.0758	1.36843
$696$	0	0
$697$	−29.0413	−1.10002
$698$	0	0
$699$	−2.90364	−0.109826
$700$	0	0
$701$	13.9590	0.527226	0.263613	−	0.964629i	$-0.415086\pi$
$701$	13.9590	0.527226	0.263613	+	0.964629i	$0.415086\pi$
$702$	0	0
$703$	10.4668	0.394762
$704$	0	0
$705$	−6.36938	−0.239885
$706$	0	0
$707$	−19.4559	−0.731716
$708$	0	0
$709$	−2.14351	−0.0805013	−0.0402506	−	0.999190i	$-0.512816\pi$
$709$	−2.14351	−0.0805013	−0.0402506	+	0.999190i	$0.512816\pi$
$710$	0	0
$711$	7.71087	0.289180
$712$	0	0
$713$	−2.26303	−0.0847512
$714$	0	0
$715$	−7.53530	−0.281804
$716$	0	0
$717$	−4.66588	−0.174251
$718$	0	0
$719$	−12.2714	−0.457646	−0.228823	−	0.973468i	$-0.573488\pi$
$719$	−12.2714	−0.457646	−0.228823	+	0.973468i	$0.573488\pi$
$720$	0	0
$721$	−39.2290	−1.46097
$722$	0	0
$723$	−18.9080	−0.703195
$724$	0	0
$725$	−0.568558	−0.0211157
$726$	0	0
$727$	−3.20764	−0.118965	−0.0594824	−	0.998229i	$-0.518945\pi$
$727$	−3.20764	−0.118965	−0.0594824	+	0.998229i	$0.518945\pi$
$728$	0	0
$729$	18.1161	0.670967
$730$	0	0
$731$	19.7250	0.729556
$732$	0	0
$733$	10.1991	0.376711	0.188356	−	0.982101i	$-0.439684\pi$
$733$	10.1991	0.376711	0.188356	+	0.982101i	$0.439684\pi$
$734$	0	0
$735$	−6.92973	−0.255607
$736$	0	0
$737$	−5.44456	−0.200553
$738$	0	0
$739$	39.4814	1.45235	0.726174	−	0.687511i	$-0.241297\pi$
$739$	39.4814	1.45235	0.726174	+	0.687511i	$0.241297\pi$
$740$	0	0
$741$	2.34467	0.0861338
$742$	0	0
$743$	42.7144	1.56704	0.783520	−	0.621367i	$-0.213422\pi$
$743$	42.7144	1.56704	0.783520	+	0.621367i	$0.213422\pi$
$744$	0	0
$745$	−17.3079	−0.634111
$746$	0	0
$747$	−31.1820	−1.14089
$748$	0	0
$749$	33.4752	1.22316
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	12.1249	0.441855
$754$	0	0
$755$	54.1895	1.97216
$756$	0	0
$757$	−21.6557	−0.787091	−0.393546	−	0.919305i	$-0.628752\pi$
$757$	−21.6557	−0.787091	−0.393546	+	0.919305i	$0.628752\pi$
$758$	0	0
$759$	2.07922	0.0754710
$760$	0	0
$761$	29.7939	1.08003	0.540013	−	0.841657i	$-0.318419\pi$
$761$	29.7939	1.08003	0.540013	+	0.841657i	$0.318419\pi$
$762$	0	0
$763$	−12.0241	−0.435301
$764$	0	0
$765$	14.1171	0.510403
$766$	0	0
$767$	−8.22085	−0.296838
$768$	0	0
$769$	10.4364	0.376348	0.188174	−	0.982136i	$-0.439743\pi$
$769$	10.4364	0.376348	0.188174	+	0.982136i	$0.439743\pi$
$770$	0	0
$771$	4.37977	0.157734
$772$	0	0
$773$	25.8927	0.931295	0.465648	−	0.884970i	$-0.345821\pi$
$773$	25.8927	0.931295	0.465648	+	0.884970i	$0.345821\pi$
$774$	0	0
$775$	−4.81681	−0.173025
$776$	0	0
$777$	−20.2634	−0.726947
$778$	0	0
$779$	−12.4644	−0.446584
$780$	0	0
$781$	−18.8919	−0.676005
$782$	0	0
$783$	−1.17872	−0.0421241
$784$	0	0
$785$	−8.98514	−0.320693
$786$	0	0
$787$	31.3764	1.11845	0.559224	−	0.829017i	$-0.311099\pi$
$787$	31.3764	1.11845	0.559224	+	0.829017i	$0.311099\pi$
$788$	0	0
$789$	16.2810	0.579619
$790$	0	0
$791$	27.8968	0.991896
$792$	0	0
$793$	13.6829	0.485893
$794$	0	0
$795$	30.2015	1.07114
$796$	0	0
$797$	−30.5511	−1.08217	−0.541087	−	0.840966i	$-0.681987\pi$
$797$	−30.5511	−1.08217	−0.541087	+	0.840966i	$0.681987\pi$
$798$	0	0
$799$	−6.05462	−0.214197
$800$	0	0
$801$	−10.9570	−0.387146
$802$	0	0
$803$	16.6259	0.586715
$804$	0	0
$805$	−7.10960	−0.250580
$806$	0	0
$807$	26.7521	0.941718
$808$	0	0
$809$	34.0041	1.19552	0.597760	−	0.801675i	$-0.296058\pi$
$809$	34.0041	1.19552	0.597760	+	0.801675i	$0.296058\pi$
$810$	0	0
$811$	−16.6465	−0.584538	−0.292269	−	0.956336i	$-0.594410\pi$
$811$	−16.6465	−0.584538	−0.292269	+	0.956336i	$0.594410\pi$
$812$	0	0
$813$	21.5525	0.755879
$814$	0	0
$815$	35.0889	1.22911
$816$	0	0
$817$	8.46592	0.296185
$818$	0	0
$819$	6.72938	0.235143
$820$	0	0
$821$	−3.44212	−0.120131	−0.0600655	−	0.998194i	$-0.519131\pi$
$821$	−3.44212	−0.120131	−0.0600655	+	0.998194i	$0.519131\pi$
$822$	0	0
$823$	37.5459	1.30877	0.654384	−	0.756163i	$-0.272928\pi$
$823$	37.5459	1.30877	0.654384	+	0.756163i	$0.272928\pi$
$824$	0	0
$825$	4.42557	0.154079
$826$	0	0
$827$	35.6422	1.23940	0.619700	−	0.784839i	$-0.287254\pi$
$827$	35.6422	1.23940	0.619700	+	0.784839i	$0.287254\pi$
$828$	0	0
$829$	−11.8902	−0.412963	−0.206482	−	0.978450i	$-0.566201\pi$
$829$	−11.8902	−0.412963	−0.206482	+	0.978450i	$0.566201\pi$
$830$	0	0
$831$	−30.5065	−1.05826
$832$	0	0
$833$	−6.58728	−0.228236
$834$	0	0
$835$	−23.0075	−0.796209
$836$	0	0
$837$	−9.98611	−0.345171
$838$	0	0
$839$	47.2189	1.63018	0.815089	−	0.579335i	$-0.196688\pi$
$839$	47.2189	1.63018	0.815089	+	0.579335i	$0.196688\pi$
$840$	0	0
$841$	−28.9499	−0.998273
$842$	0	0
$843$	9.07103	0.312423
$844$	0	0
$845$	27.4628	0.944748
$846$	0	0
$847$	−18.4130	−0.632678
$848$	0	0
$849$	17.0051	0.583615
$850$	0	0
$851$	10.1445	0.347748
$852$	0	0
$853$	42.9014	1.46892	0.734458	−	0.678654i	$-0.237436\pi$
$853$	42.9014	1.46892	0.734458	+	0.678654i	$0.237436\pi$
$854$	0	0
$855$	6.05900	0.207213
$856$	0	0
$857$	−21.4026	−0.731099	−0.365549	−	0.930792i	$-0.619119\pi$
$857$	−21.4026	−0.731099	−0.365549	+	0.930792i	$0.619119\pi$
$858$	0	0
$859$	6.70785	0.228869	0.114434	−	0.993431i	$-0.463494\pi$
$859$	6.70785	0.228869	0.114434	+	0.993431i	$0.463494\pi$
$860$	0	0
$861$	24.1309	0.822377
$862$	0	0
$863$	46.3084	1.57636	0.788178	−	0.615448i	$-0.211025\pi$
$863$	46.3084	1.57636	0.788178	+	0.615448i	$0.211025\pi$
$864$	0	0
$865$	−46.9492	−1.59632
$866$	0	0
$867$	9.63618	0.327262
$868$	0	0
$869$	6.81987	0.231348
$870$	0	0
$871$	−5.95060	−0.201629
$872$	0	0
$873$	7.76387	0.262767
$874$	0	0
$875$	14.6477	0.495182
$876$	0	0
$877$	21.2141	0.716351	0.358175	−	0.933654i	$-0.383399\pi$
$877$	21.2141	0.716351	0.358175	+	0.933654i	$0.383399\pi$
$878$	0	0
$879$	−13.4878	−0.454934
$880$	0	0
$881$	25.0157	0.842800	0.421400	−	0.906875i	$-0.361539\pi$
$881$	25.0157	0.842800	0.421400	+	0.906875i	$0.361539\pi$
$882$	0	0
$883$	14.1355	0.475699	0.237849	−	0.971302i	$-0.423557\pi$
$883$	14.1355	0.475699	0.237849	+	0.971302i	$0.423557\pi$
$884$	0	0
$885$	14.3299	0.481694
$886$	0	0
$887$	−18.9050	−0.634769	−0.317384	−	0.948297i	$-0.602805\pi$
$887$	−18.9050	−0.634769	−0.317384	+	0.948297i	$0.602805\pi$
$888$	0	0
$889$	−21.2045	−0.711175
$890$	0	0
$891$	0.658831	0.0220717
$892$	0	0
$893$	−2.59862	−0.0869596
$894$	0	0
$895$	16.5047	0.551691
$896$	0	0
$897$	2.27248	0.0758758
$898$	0	0
$899$	0.424253	0.0141496
$900$	0	0
$901$	28.7090	0.956434
$902$	0	0
$903$	−16.3898	−0.545420
$904$	0	0
$905$	−7.48560	−0.248830
$906$	0	0
$907$	23.6291	0.784591	0.392296	−	0.919839i	$-0.371681\pi$
$907$	23.6291	0.784591	0.392296	+	0.919839i	$0.371681\pi$
$908$	0	0
$909$	16.0703	0.533019
$910$	0	0
$911$	−11.8946	−0.394087	−0.197044	−	0.980395i	$-0.563134\pi$
$911$	−11.8946	−0.394087	−0.197044	+	0.980395i	$0.563134\pi$
$912$	0	0
$913$	−27.5789	−0.912729
$914$	0	0
$915$	−23.8508	−0.788484
$916$	0	0
$917$	−32.4752	−1.07243
$918$	0	0
$919$	−29.4710	−0.972160	−0.486080	−	0.873914i	$-0.661574\pi$
$919$	−29.4710	−0.972160	−0.486080	+	0.873914i	$0.661574\pi$
$920$	0	0
$921$	7.35027	0.242200
$922$	0	0
$923$	−20.6478	−0.679631
$924$	0	0
$925$	21.5923	0.709949
$926$	0	0
$927$	32.4026	1.06424
$928$	0	0
$929$	12.9547	0.425029	0.212515	−	0.977158i	$-0.431835\pi$
$929$	12.9547	0.425029	0.212515	+	0.977158i	$0.431835\pi$
$930$	0	0
$931$	−2.82724	−0.0926591
$932$	0	0
$933$	−1.15257	−0.0377335
$934$	0	0
$935$	12.4858	0.408330
$936$	0	0
$937$	43.2695	1.41355	0.706776	−	0.707438i	$-0.250149\pi$
$937$	43.2695	1.41355	0.706776	+	0.707438i	$0.250149\pi$
$938$	0	0
$939$	−17.6015	−0.574404
$940$	0	0
$941$	13.0194	0.424420	0.212210	−	0.977224i	$-0.431934\pi$
$941$	13.0194	0.424420	0.212210	+	0.977224i	$0.431934\pi$
$942$	0	0
$943$	−12.0806	−0.393399
$944$	0	0
$945$	−31.3726	−1.02055
$946$	0	0
$947$	−25.5145	−0.829111	−0.414555	−	0.910024i	$-0.636063\pi$
$947$	−25.5145	−0.829111	−0.414555	+	0.910024i	$0.636063\pi$
$948$	0	0
$949$	18.1712	0.589862
$950$	0	0
$951$	−19.9077	−0.645551
$952$	0	0
$953$	19.0101	0.615796	0.307898	−	0.951419i	$-0.400374\pi$
$953$	19.0101	0.615796	0.307898	+	0.951419i	$0.400374\pi$
$954$	0	0
$955$	32.2479	1.04352
$956$	0	0
$957$	−0.389794	−0.0126002
$958$	0	0
$959$	5.43603	0.175539
$960$	0	0
$961$	−27.4057	−0.884056
$962$	0	0
$963$	−27.6501	−0.891010
$964$	0	0
$965$	21.2457	0.683922
$966$	0	0
$967$	12.2436	0.393728	0.196864	−	0.980431i	$-0.436924\pi$
$967$	12.2436	0.393728	0.196864	+	0.980431i	$0.436924\pi$
$968$	0	0
$969$	−3.88507	−0.124806
$970$	0	0
$971$	−0.946139	−0.0303630	−0.0151815	−	0.999885i	$-0.504833\pi$
$971$	−0.946139	−0.0303630	−0.0151815	+	0.999885i	$0.504833\pi$
$972$	0	0
$973$	−28.4946	−0.913495
$974$	0	0
$975$	4.83691	0.154905
$976$	0	0
$977$	−51.0007	−1.63166	−0.815829	−	0.578294i	$-0.803719\pi$
$977$	−51.0007	−1.63166	−0.815829	+	0.578294i	$0.803719\pi$
$978$	0	0
$979$	−9.69089	−0.309722
$980$	0	0
$981$	9.93172	0.317095
$982$	0	0
$983$	−17.2201	−0.549236	−0.274618	−	0.961553i	$-0.588551\pi$
$983$	−17.2201	−0.549236	−0.274618	+	0.961553i	$0.588551\pi$
$984$	0	0
$985$	23.0108	0.733186
$986$	0	0
$987$	5.03088	0.160135
$988$	0	0
$989$	8.20524	0.260911
$990$	0	0
$991$	43.8871	1.39412	0.697060	−	0.717013i	$-0.254491\pi$
$991$	43.8871	1.39412	0.697060	+	0.717013i	$0.254491\pi$
$992$	0	0
$993$	−27.2604	−0.865084
$994$	0	0
$995$	32.1642	1.01967
$996$	0	0
$997$	16.0745	0.509086	0.254543	−	0.967061i	$-0.418075\pi$
$997$	16.0745	0.509086	0.254543	+	0.967061i	$0.418075\pi$
$998$	0	0
$999$	44.7646	1.41629

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.17	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.17	✓	44	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.09930 q^{3} -2.74603 q^{5} +2.16897 q^{7} -1.79154 q^{9} +O(q^{10})\)	\(q-1.09930 q^{3} -2.74603 q^{5} +2.16897 q^{7} -1.79154 q^{9} -1.58452 q^{11} -1.73180 q^{13} +3.01872 q^{15} +2.86954 q^{17} +1.23160 q^{19} -2.38435 q^{21} +1.19367 q^{23} +2.54071 q^{25} +5.26734 q^{27} -0.223779 q^{29} -1.89585 q^{31} +1.74187 q^{33} -5.95606 q^{35} +8.49852 q^{37} +1.90377 q^{39} -10.1205 q^{41} +6.87393 q^{43} +4.91962 q^{45} -2.10996 q^{47} -2.29559 q^{49} -3.15449 q^{51} +10.0047 q^{53} +4.35115 q^{55} -1.35390 q^{57} +4.74701 q^{59} -7.90097 q^{61} -3.88578 q^{63} +4.75557 q^{65} +3.43609 q^{67} -1.31221 q^{69} +11.9228 q^{71} -10.4927 q^{73} -2.79300 q^{75} -3.43677 q^{77} -4.30406 q^{79} -0.415792 q^{81} +17.4052 q^{83} -7.87986 q^{85} +0.246001 q^{87} +6.11597 q^{89} -3.75621 q^{91} +2.08411 q^{93} -3.38201 q^{95} -4.33364 q^{97} +2.83873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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