LMFDB - Embedded newform 6008.2.a.d.1.8

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			1.8
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)

$f(q)$	$=$	$q-2.20855 q^{3} +1.60744 q^{5} +2.65561 q^{7} +1.87768 q^{9} +O(q^{10})$	$q-2.20855 q^{3} +1.60744 q^{5} +2.65561 q^{7} +1.87768 q^{9} +4.47814 q^{11} +1.08744 q^{13} -3.55011 q^{15} +3.02699 q^{17} -5.18417 q^{19} -5.86504 q^{21} +5.23537 q^{23} -2.41613 q^{25} +2.47870 q^{27} +2.98529 q^{29} +4.02345 q^{31} -9.89019 q^{33} +4.26874 q^{35} +3.78179 q^{37} -2.40166 q^{39} +6.92897 q^{41} -0.848422 q^{43} +3.01826 q^{45} +7.09493 q^{47} +0.0522760 q^{49} -6.68525 q^{51} -10.4675 q^{53} +7.19835 q^{55} +11.4495 q^{57} +11.0083 q^{59} +7.16946 q^{61} +4.98638 q^{63} +1.74799 q^{65} -8.97542 q^{67} -11.5626 q^{69} +6.41486 q^{71} -0.477645 q^{73} +5.33614 q^{75} +11.8922 q^{77} +8.29507 q^{79} -11.1074 q^{81} -7.54030 q^{83} +4.86571 q^{85} -6.59315 q^{87} -15.6302 q^{89} +2.88781 q^{91} -8.88598 q^{93} -8.33325 q^{95} -5.09371 q^{97} +8.40851 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.20855	−1.27510	−0.637552	−	0.770407i	$-0.720053\pi$
$3$	−2.20855	−1.27510	−0.637552	+	0.770407i	$0.720053\pi$
$4$	0	0
$5$	1.60744	0.718869	0.359435	−	0.933170i	$-0.382970\pi$
$5$	1.60744	0.718869	0.359435	+	0.933170i	$0.382970\pi$
$6$	0	0
$7$	2.65561	1.00373	0.501864	−	0.864947i	$-0.332648\pi$
$7$	2.65561	1.00373	0.501864	+	0.864947i	$0.332648\pi$
$8$	0	0
$9$	1.87768	0.625892
$10$	0	0
$11$	4.47814	1.35021	0.675105	−	0.737721i	$-0.264098\pi$
$11$	4.47814	1.35021	0.675105	+	0.737721i	$0.264098\pi$
$12$	0	0
$13$	1.08744	0.301601	0.150800	−	0.988564i	$-0.451815\pi$
$13$	1.08744	0.301601	0.150800	+	0.988564i	$0.451815\pi$
$14$	0	0
$15$	−3.55011	−0.916634
$16$	0	0
$17$	3.02699	0.734154	0.367077	−	0.930191i	$-0.380359\pi$
$17$	3.02699	0.734154	0.367077	+	0.930191i	$0.380359\pi$
$18$	0	0
$19$	−5.18417	−1.18933	−0.594665	−	0.803973i	$-0.702715\pi$
$19$	−5.18417	−1.18933	−0.594665	+	0.803973i	$0.702715\pi$
$20$	0	0
$21$	−5.86504	−1.27986
$22$	0	0
$23$	5.23537	1.09165	0.545825	−	0.837899i	$-0.316216\pi$
$23$	5.23537	1.09165	0.545825	+	0.837899i	$0.316216\pi$
$24$	0	0
$25$	−2.41613	−0.483227
$26$	0	0
$27$	2.47870	0.477026
$28$	0	0
$29$	2.98529	0.554354	0.277177	−	0.960819i	$-0.410601\pi$
$29$	2.98529	0.554354	0.277177	+	0.960819i	$0.410601\pi$
$30$	0	0
$31$	4.02345	0.722633	0.361317	−	0.932443i	$-0.382327\pi$
$31$	4.02345	0.722633	0.361317	+	0.932443i	$0.382327\pi$
$32$	0	0
$33$	−9.89019	−1.72166
$34$	0	0
$35$	4.26874	0.721549
$36$	0	0
$37$	3.78179	0.621722	0.310861	−	0.950455i	$-0.399383\pi$
$37$	3.78179	0.621722	0.310861	+	0.950455i	$0.399383\pi$
$38$	0	0
$39$	−2.40166	−0.384573
$40$	0	0
$41$	6.92897	1.08212	0.541061	−	0.840983i	$-0.318023\pi$
$41$	6.92897	1.08212	0.541061	+	0.840983i	$0.318023\pi$
$42$	0	0
$43$	−0.848422	−0.129383	−0.0646916	−	0.997905i	$-0.520606\pi$
$43$	−0.848422	−0.129383	−0.0646916	+	0.997905i	$0.520606\pi$
$44$	0	0
$45$	3.01826	0.449935
$46$	0	0
$47$	7.09493	1.03490	0.517451	−	0.855713i	$-0.326881\pi$
$47$	7.09493	1.03490	0.517451	+	0.855713i	$0.326881\pi$
$48$	0	0
$49$	0.0522760	0.00746799
$50$	0	0
$51$	−6.68525	−0.936123
$52$	0	0
$53$	−10.4675	−1.43783	−0.718914	−	0.695099i	$-0.755360\pi$
$53$	−10.4675	−1.43783	−0.718914	+	0.695099i	$0.755360\pi$
$54$	0	0
$55$	7.19835	0.970625
$56$	0	0
$57$	11.4495	1.51652
$58$	0	0
$59$	11.0083	1.43315	0.716576	−	0.697509i	$-0.245708\pi$
$59$	11.0083	1.43315	0.716576	+	0.697509i	$0.245708\pi$
$60$	0	0
$61$	7.16946	0.917955	0.458978	−	0.888448i	$-0.348216\pi$
$61$	7.16946	0.917955	0.458978	+	0.888448i	$0.348216\pi$
$62$	0	0
$63$	4.98638	0.628225
$64$	0	0
$65$	1.74799	0.216812
$66$	0	0
$67$	−8.97542	−1.09652	−0.548261	−	0.836307i	$-0.684710\pi$
$67$	−8.97542	−1.09652	−0.548261	+	0.836307i	$0.684710\pi$
$68$	0	0
$69$	−11.5626	−1.39197
$70$	0	0
$71$	6.41486	0.761303	0.380652	−	0.924718i	$-0.375700\pi$
$71$	6.41486	0.761303	0.380652	+	0.924718i	$0.375700\pi$
$72$	0	0
$73$	−0.477645	−0.0559041	−0.0279520	−	0.999609i	$-0.508899\pi$
$73$	−0.477645	−0.0559041	−0.0279520	+	0.999609i	$0.508899\pi$
$74$	0	0
$75$	5.33614	0.616165
$76$	0	0
$77$	11.8922	1.35524
$78$	0	0
$79$	8.29507	0.933268	0.466634	−	0.884450i	$-0.345466\pi$
$79$	8.29507	0.933268	0.466634	+	0.884450i	$0.345466\pi$
$80$	0	0
$81$	−11.1074	−1.23415
$82$	0	0
$83$	−7.54030	−0.827655	−0.413828	−	0.910355i	$-0.635808\pi$
$83$	−7.54030	−0.827655	−0.413828	+	0.910355i	$0.635808\pi$
$84$	0	0
$85$	4.86571	0.527761
$86$	0	0
$87$	−6.59315	−0.706859
$88$	0	0
$89$	−15.6302	−1.65680	−0.828398	−	0.560139i	$-0.810748\pi$
$89$	−15.6302	−1.65680	−0.828398	+	0.560139i	$0.810748\pi$
$90$	0	0
$91$	2.88781	0.302725
$92$	0	0
$93$	−8.88598	−0.921433
$94$	0	0
$95$	−8.33325	−0.854973
$96$	0	0
$97$	−5.09371	−0.517188	−0.258594	−	0.965986i	$-0.583259\pi$
$97$	−5.09371	−0.517188	−0.258594	+	0.965986i	$0.583259\pi$
$98$	0	0
$99$	8.40851	0.845087
$100$	0	0
$101$	1.33734	0.133070	0.0665351	−	0.997784i	$-0.478806\pi$
$101$	1.33734	0.133070	0.0665351	+	0.997784i	$0.478806\pi$
$102$	0	0
$103$	−2.79026	−0.274932	−0.137466	−	0.990506i	$-0.543896\pi$
$103$	−2.79026	−0.274932	−0.137466	+	0.990506i	$0.543896\pi$
$104$	0	0
$105$	−9.42771	−0.920050
$106$	0	0
$107$	18.5180	1.79020	0.895102	−	0.445861i	$-0.147102\pi$
$107$	18.5180	1.79020	0.895102	+	0.445861i	$0.147102\pi$
$108$	0	0
$109$	−7.24626	−0.694066	−0.347033	−	0.937853i	$-0.612811\pi$
$109$	−7.24626	−0.694066	−0.347033	+	0.937853i	$0.612811\pi$
$110$	0	0
$111$	−8.35225	−0.792761
$112$	0	0
$113$	−7.63779	−0.718503	−0.359251	−	0.933241i	$-0.616968\pi$
$113$	−7.63779	−0.718503	−0.359251	+	0.933241i	$0.616968\pi$
$114$	0	0
$115$	8.41555	0.784754
$116$	0	0
$117$	2.04186	0.188770
$118$	0	0
$119$	8.03852	0.736890
$120$	0	0
$121$	9.05376	0.823069
$122$	0	0
$123$	−15.3029	−1.37982
$124$	0	0
$125$	−11.9210	−1.06625
$126$	0	0
$127$	−0.542716	−0.0481583	−0.0240791	−	0.999710i	$-0.507665\pi$
$127$	−0.542716	−0.0481583	−0.0240791	+	0.999710i	$0.507665\pi$
$128$	0	0
$129$	1.87378	0.164977
$130$	0	0
$131$	2.11006	0.184357	0.0921783	−	0.995743i	$-0.470617\pi$
$131$	2.11006	0.184357	0.0921783	+	0.995743i	$0.470617\pi$
$132$	0	0
$133$	−13.7671	−1.19376
$134$	0	0
$135$	3.98437	0.342920
$136$	0	0
$137$	10.8806	0.929595	0.464797	−	0.885417i	$-0.346127\pi$
$137$	10.8806	0.929595	0.464797	+	0.885417i	$0.346127\pi$
$138$	0	0
$139$	−16.9272	−1.43575	−0.717874	−	0.696173i	$-0.754885\pi$
$139$	−16.9272	−1.43575	−0.717874	+	0.696173i	$0.754885\pi$
$140$	0	0
$141$	−15.6695	−1.31961
$142$	0	0
$143$	4.86970	0.407225
$144$	0	0
$145$	4.79867	0.398508
$146$	0	0
$147$	−0.115454	−0.00952248
$148$	0	0
$149$	3.63520	0.297807	0.148903	−	0.988852i	$-0.452426\pi$
$149$	3.63520	0.297807	0.148903	+	0.988852i	$0.452426\pi$
$150$	0	0
$151$	−1.15497	−0.0939898	−0.0469949	−	0.998895i	$-0.514964\pi$
$151$	−1.15497	−0.0939898	−0.0469949	+	0.998895i	$0.514964\pi$
$152$	0	0
$153$	5.68372	0.459501
$154$	0	0
$155$	6.46746	0.519479
$156$	0	0
$157$	−13.6115	−1.08631	−0.543157	−	0.839631i	$-0.682771\pi$
$157$	−13.6115	−1.08631	−0.543157	+	0.839631i	$0.682771\pi$
$158$	0	0
$159$	23.1181	1.83338
$160$	0	0
$161$	13.9031	1.09572
$162$	0	0
$163$	−12.2232	−0.957394	−0.478697	−	0.877980i	$-0.658891\pi$
$163$	−12.2232	−0.957394	−0.478697	+	0.877980i	$0.658891\pi$
$164$	0	0
$165$	−15.8979	−1.23765
$166$	0	0
$167$	19.1049	1.47838	0.739192	−	0.673494i	$-0.235207\pi$
$167$	19.1049	1.47838	0.739192	+	0.673494i	$0.235207\pi$
$168$	0	0
$169$	−11.8175	−0.909037
$170$	0	0
$171$	−9.73420	−0.744393
$172$	0	0
$173$	14.6005	1.11006	0.555029	−	0.831831i	$-0.312707\pi$
$173$	14.6005	1.11006	0.555029	+	0.831831i	$0.312707\pi$
$174$	0	0
$175$	−6.41632	−0.485028
$176$	0	0
$177$	−24.3122	−1.82742
$178$	0	0
$179$	−20.3446	−1.52062	−0.760312	−	0.649558i	$-0.774954\pi$
$179$	−20.3446	−1.52062	−0.760312	+	0.649558i	$0.774954\pi$
$180$	0	0
$181$	−12.8148	−0.952517	−0.476259	−	0.879305i	$-0.658007\pi$
$181$	−12.8148	−0.952517	−0.476259	+	0.879305i	$0.658007\pi$
$182$	0	0
$183$	−15.8341	−1.17049
$184$	0	0
$185$	6.07900	0.446937
$186$	0	0
$187$	13.5553	0.991262
$188$	0	0
$189$	6.58247	0.478804
$190$	0	0
$191$	0.797137	0.0576788	0.0288394	−	0.999584i	$-0.490819\pi$
$191$	0.797137	0.0576788	0.0288394	+	0.999584i	$0.490819\pi$
$192$	0	0
$193$	−2.71354	−0.195325	−0.0976623	−	0.995220i	$-0.531137\pi$
$193$	−2.71354	−0.195325	−0.0976623	+	0.995220i	$0.531137\pi$
$194$	0	0
$195$	−3.86052	−0.276458
$196$	0	0
$197$	6.43993	0.458826	0.229413	−	0.973329i	$-0.426319\pi$
$197$	6.43993	0.458826	0.229413	+	0.973329i	$0.426319\pi$
$198$	0	0
$199$	−19.5434	−1.38540	−0.692699	−	0.721227i	$-0.743579\pi$
$199$	−19.5434	−1.38540	−0.692699	+	0.721227i	$0.743579\pi$
$200$	0	0
$201$	19.8226	1.39818
$202$	0	0
$203$	7.92776	0.556420
$204$	0	0
$205$	11.1379	0.777905
$206$	0	0
$207$	9.83034	0.683256
$208$	0	0
$209$	−23.2155	−1.60585
$210$	0	0
$211$	22.9163	1.57762	0.788812	−	0.614635i	$-0.210697\pi$
$211$	22.9163	1.57762	0.788812	+	0.614635i	$0.210697\pi$
$212$	0	0
$213$	−14.1675	−0.970742
$214$	0	0
$215$	−1.36379	−0.0930096
$216$	0	0
$217$	10.6847	0.725327
$218$	0	0
$219$	1.05490	0.0712836
$220$	0	0
$221$	3.29167	0.221421
$222$	0	0
$223$	10.5578	0.707000	0.353500	−	0.935434i	$-0.384991\pi$
$223$	10.5578	0.707000	0.353500	+	0.935434i	$0.384991\pi$
$224$	0	0
$225$	−4.53672	−0.302448
$226$	0	0
$227$	−11.2185	−0.744601	−0.372301	−	0.928112i	$-0.621431\pi$
$227$	−11.2185	−0.744601	−0.372301	+	0.928112i	$0.621431\pi$
$228$	0	0
$229$	−0.718830	−0.0475016	−0.0237508	−	0.999718i	$-0.507561\pi$
$229$	−0.718830	−0.0475016	−0.0237508	+	0.999718i	$0.507561\pi$
$230$	0	0
$231$	−26.2645	−1.72808
$232$	0	0
$233$	−0.584824	−0.0383131	−0.0191566	−	0.999816i	$-0.506098\pi$
$233$	−0.584824	−0.0383131	−0.0191566	+	0.999816i	$0.506098\pi$
$234$	0	0
$235$	11.4047	0.743960
$236$	0	0
$237$	−18.3200	−1.19001
$238$	0	0
$239$	22.9018	1.48140	0.740698	−	0.671838i	$-0.234495\pi$
$239$	22.9018	1.48140	0.740698	+	0.671838i	$0.234495\pi$
$240$	0	0
$241$	12.7423	0.820806	0.410403	−	0.911904i	$-0.365388\pi$
$241$	12.7423	0.820806	0.410403	+	0.911904i	$0.365388\pi$
$242$	0	0
$243$	17.0950	1.09665
$244$	0	0
$245$	0.0840305	0.00536851
$246$	0	0
$247$	−5.63746	−0.358703
$248$	0	0
$249$	16.6531	1.05535
$250$	0	0
$251$	−2.80846	−0.177269	−0.0886343	−	0.996064i	$-0.528250\pi$
$251$	−2.80846	−0.177269	−0.0886343	+	0.996064i	$0.528250\pi$
$252$	0	0
$253$	23.4447	1.47396
$254$	0	0
$255$	−10.7461	−0.672950
$256$	0	0
$257$	−24.4565	−1.52556	−0.762778	−	0.646660i	$-0.776165\pi$
$257$	−24.4565	−1.52556	−0.762778	+	0.646660i	$0.776165\pi$
$258$	0	0
$259$	10.0430	0.624039
$260$	0	0
$261$	5.60541	0.346966
$262$	0	0
$263$	25.3538	1.56338	0.781691	−	0.623665i	$-0.214357\pi$
$263$	25.3538	1.56338	0.781691	+	0.623665i	$0.214357\pi$
$264$	0	0
$265$	−16.8260	−1.03361
$266$	0	0
$267$	34.5200	2.11259
$268$	0	0
$269$	3.93401	0.239861	0.119930	−	0.992782i	$-0.461733\pi$
$269$	3.93401	0.239861	0.119930	+	0.992782i	$0.461733\pi$
$270$	0	0
$271$	−15.8838	−0.964870	−0.482435	−	0.875932i	$-0.660248\pi$
$271$	−15.8838	−0.964870	−0.482435	+	0.875932i	$0.660248\pi$
$272$	0	0
$273$	−6.37787	−0.386006
$274$	0	0
$275$	−10.8198	−0.652458
$276$	0	0
$277$	21.5110	1.29247	0.646235	−	0.763139i	$-0.276343\pi$
$277$	21.5110	1.29247	0.646235	+	0.763139i	$0.276343\pi$
$278$	0	0
$279$	7.55475	0.452291
$280$	0	0
$281$	28.6096	1.70670	0.853352	−	0.521336i	$-0.174566\pi$
$281$	28.6096	1.70670	0.853352	+	0.521336i	$0.174566\pi$
$282$	0	0
$283$	9.41164	0.559464	0.279732	−	0.960078i	$-0.409754\pi$
$283$	9.41164	0.559464	0.279732	+	0.960078i	$0.409754\pi$
$284$	0	0
$285$	18.4044	1.09018
$286$	0	0
$287$	18.4006	1.08616
$288$	0	0
$289$	−7.83731	−0.461019
$290$	0	0
$291$	11.2497	0.659469
$292$	0	0
$293$	−10.1645	−0.593819	−0.296910	−	0.954906i	$-0.595956\pi$
$293$	−10.1645	−0.593819	−0.296910	+	0.954906i	$0.595956\pi$
$294$	0	0
$295$	17.6951	1.03025
$296$	0	0
$297$	11.1000	0.644086
$298$	0	0
$299$	5.69314	0.329243
$300$	0	0
$301$	−2.25308	−0.129865
$302$	0	0
$303$	−2.95358	−0.169679
$304$	0	0
$305$	11.5245	0.659890
$306$	0	0
$307$	15.0429	0.858546	0.429273	−	0.903175i	$-0.358770\pi$
$307$	15.0429	0.858546	0.429273	+	0.903175i	$0.358770\pi$
$308$	0	0
$309$	6.16242	0.350568
$310$	0	0
$311$	11.3959	0.646204	0.323102	−	0.946364i	$-0.395274\pi$
$311$	11.3959	0.646204	0.323102	+	0.946364i	$0.395274\pi$
$312$	0	0
$313$	4.93769	0.279095	0.139547	−	0.990215i	$-0.455435\pi$
$313$	4.93769	0.279095	0.139547	+	0.990215i	$0.455435\pi$
$314$	0	0
$315$	8.01531	0.451612
$316$	0	0
$317$	−25.4712	−1.43060	−0.715302	−	0.698816i	$-0.753711\pi$
$317$	−25.4712	−1.43060	−0.715302	+	0.698816i	$0.753711\pi$
$318$	0	0
$319$	13.3685	0.748495
$320$	0	0
$321$	−40.8979	−2.28270
$322$	0	0
$323$	−15.6924	−0.873151
$324$	0	0
$325$	−2.62740	−0.145742
$326$	0	0
$327$	16.0037	0.885006
$328$	0	0
$329$	18.8414	1.03876
$330$	0	0
$331$	26.7684	1.47132	0.735662	−	0.677349i	$-0.236871\pi$
$331$	26.7684	1.47132	0.735662	+	0.677349i	$0.236871\pi$
$332$	0	0
$333$	7.10098	0.389131
$334$	0	0
$335$	−14.4275	−0.788256
$336$	0	0
$337$	12.4559	0.678517	0.339258	−	0.940693i	$-0.389824\pi$
$337$	12.4559	0.678517	0.339258	+	0.940693i	$0.389824\pi$
$338$	0	0
$339$	16.8684	0.916166
$340$	0	0
$341$	18.0176	0.975707
$342$	0	0
$343$	−18.4505	−0.996231
$344$	0	0
$345$	−18.5861	−1.00064
$346$	0	0
$347$	−11.6481	−0.625300	−0.312650	−	0.949868i	$-0.601217\pi$
$347$	−11.6481	−0.625300	−0.312650	+	0.949868i	$0.601217\pi$
$348$	0	0
$349$	8.85226	0.473851	0.236925	−	0.971528i	$-0.423860\pi$
$349$	8.85226	0.473851	0.236925	+	0.971528i	$0.423860\pi$
$350$	0	0
$351$	2.69543	0.143872
$352$	0	0
$353$	−25.3446	−1.34896	−0.674478	−	0.738295i	$-0.735631\pi$
$353$	−25.3446	−1.34896	−0.674478	+	0.738295i	$0.735631\pi$
$354$	0	0
$355$	10.3115	0.547278
$356$	0	0
$357$	−17.7534	−0.939612
$358$	0	0
$359$	5.44745	0.287506	0.143753	−	0.989614i	$-0.454083\pi$
$359$	5.44745	0.287506	0.143753	+	0.989614i	$0.454083\pi$
$360$	0	0
$361$	7.87563	0.414507
$362$	0	0
$363$	−19.9957	−1.04950
$364$	0	0
$365$	−0.767785	−0.0401877
$366$	0	0
$367$	36.5424	1.90750	0.953749	−	0.300605i	$-0.0971886\pi$
$367$	36.5424	1.90750	0.953749	+	0.300605i	$0.0971886\pi$
$368$	0	0
$369$	13.0104	0.677292
$370$	0	0
$371$	−27.7977	−1.44319
$372$	0	0
$373$	14.2974	0.740290	0.370145	−	0.928974i	$-0.379308\pi$
$373$	14.2974	0.740290	0.370145	+	0.928974i	$0.379308\pi$
$374$	0	0
$375$	26.3281	1.35958
$376$	0	0
$377$	3.24631	0.167194
$378$	0	0
$379$	18.8729	0.969438	0.484719	−	0.874670i	$-0.338922\pi$
$379$	18.8729	0.969438	0.484719	+	0.874670i	$0.338922\pi$
$380$	0	0
$381$	1.19861	0.0614069
$382$	0	0
$383$	17.8179	0.910450	0.455225	−	0.890376i	$-0.349559\pi$
$383$	17.8179	0.910450	0.455225	+	0.890376i	$0.349559\pi$
$384$	0	0
$385$	19.1160	0.974243
$386$	0	0
$387$	−1.59306	−0.0809799
$388$	0	0
$389$	−3.54769	−0.179875	−0.0899375	−	0.995947i	$-0.528667\pi$
$389$	−3.54769	−0.179875	−0.0899375	+	0.995947i	$0.528667\pi$
$390$	0	0
$391$	15.8474	0.801439
$392$	0	0
$393$	−4.66016	−0.235074
$394$	0	0
$395$	13.3338	0.670898
$396$	0	0
$397$	−19.3852	−0.972917	−0.486458	−	0.873704i	$-0.661711\pi$
$397$	−19.3852	−0.972917	−0.486458	+	0.873704i	$0.661711\pi$
$398$	0	0
$399$	30.4054	1.52217
$400$	0	0
$401$	−7.71381	−0.385209	−0.192605	−	0.981276i	$-0.561693\pi$
$401$	−7.71381	−0.385209	−0.192605	+	0.981276i	$0.561693\pi$
$402$	0	0
$403$	4.37525	0.217947
$404$	0	0
$405$	−17.8544	−0.887193
$406$	0	0
$407$	16.9354	0.839456
$408$	0	0
$409$	6.06419	0.299855	0.149927	−	0.988697i	$-0.452096\pi$
$409$	6.06419	0.299855	0.149927	+	0.988697i	$0.452096\pi$
$410$	0	0
$411$	−24.0304	−1.18533
$412$	0	0
$413$	29.2337	1.43849
$414$	0	0
$415$	−12.1206	−0.594976
$416$	0	0
$417$	37.3845	1.83073
$418$	0	0
$419$	0.0131139	0.000640656 0	0.000320328	−	1.00000i	$-0.499898\pi$
$419$	0.0131139	0.000640656 0	0.000320328		1.00000i	$0.499898\pi$
$420$	0	0
$421$	−9.58321	−0.467057	−0.233529	−	0.972350i	$-0.575027\pi$
$421$	−9.58321	−0.467057	−0.233529	+	0.972350i	$0.575027\pi$
$422$	0	0
$423$	13.3220	0.647738
$424$	0	0
$425$	−7.31362	−0.354763
$426$	0	0
$427$	19.0393	0.921377
$428$	0	0
$429$	−10.7550	−0.519254
$430$	0	0
$431$	19.7659	0.952090	0.476045	−	0.879421i	$-0.342070\pi$
$431$	19.7659	0.952090	0.476045	+	0.879421i	$0.342070\pi$
$432$	0	0
$433$	23.0026	1.10544	0.552718	−	0.833368i	$-0.313591\pi$
$433$	23.0026	1.10544	0.552718	+	0.833368i	$0.313591\pi$
$434$	0	0
$435$	−10.5981	−0.508140
$436$	0	0
$437$	−27.1411	−1.29833
$438$	0	0
$439$	25.2747	1.20630	0.603148	−	0.797629i	$-0.293913\pi$
$439$	25.2747	1.20630	0.603148	+	0.797629i	$0.293913\pi$
$440$	0	0
$441$	0.0981574	0.00467416
$442$	0	0
$443$	2.85051	0.135432	0.0677159	−	0.997705i	$-0.478429\pi$
$443$	2.85051	0.135432	0.0677159	+	0.997705i	$0.478429\pi$
$444$	0	0
$445$	−25.1246	−1.19102
$446$	0	0
$447$	−8.02850	−0.379735
$448$	0	0
$449$	−18.5114	−0.873605	−0.436802	−	0.899557i	$-0.643889\pi$
$449$	−18.5114	−0.873605	−0.436802	+	0.899557i	$0.643889\pi$
$450$	0	0
$451$	31.0289	1.46109
$452$	0	0
$453$	2.55080	0.119847
$454$	0	0
$455$	4.64199	0.217620
$456$	0	0
$457$	−31.1170	−1.45559	−0.727796	−	0.685793i	$-0.759455\pi$
$457$	−31.1170	−1.45559	−0.727796	+	0.685793i	$0.759455\pi$
$458$	0	0
$459$	7.50301	0.350211
$460$	0	0
$461$	22.8861	1.06591	0.532955	−	0.846143i	$-0.321081\pi$
$461$	22.8861	1.06591	0.532955	+	0.846143i	$0.321081\pi$
$462$	0	0
$463$	−22.8132	−1.06022	−0.530110	−	0.847929i	$-0.677850\pi$
$463$	−22.8132	−1.06022	−0.530110	+	0.847929i	$0.677850\pi$
$464$	0	0
$465$	−14.2837	−0.662390
$466$	0	0
$467$	−36.2916	−1.67937	−0.839687	−	0.543071i	$-0.817261\pi$
$467$	−36.2916	−1.67937	−0.839687	+	0.543071i	$0.817261\pi$
$468$	0	0
$469$	−23.8352	−1.10061
$470$	0	0
$471$	30.0615	1.38516
$472$	0	0
$473$	−3.79935	−0.174694
$474$	0	0
$475$	12.5257	0.574716
$476$	0	0
$477$	−19.6547	−0.899925
$478$	0	0
$479$	19.2731	0.880612	0.440306	−	0.897848i	$-0.354870\pi$
$479$	19.2731	0.880612	0.440306	+	0.897848i	$0.354870\pi$
$480$	0	0
$481$	4.11246	0.187512
$482$	0	0
$483$	−30.7057	−1.39716
$484$	0	0
$485$	−8.18784	−0.371791
$486$	0	0
$487$	4.66094	0.211208	0.105604	−	0.994408i	$-0.466322\pi$
$487$	4.66094	0.211208	0.105604	+	0.994408i	$0.466322\pi$
$488$	0	0
$489$	26.9955	1.22078
$490$	0	0
$491$	−31.3204	−1.41347	−0.706735	−	0.707478i	$-0.749833\pi$
$491$	−31.3204	−1.41347	−0.706735	+	0.707478i	$0.749833\pi$
$492$	0	0
$493$	9.03644	0.406981
$494$	0	0
$495$	13.5162	0.607507
$496$	0	0
$497$	17.0354	0.764141
$498$	0	0
$499$	−23.6492	−1.05868	−0.529342	−	0.848409i	$-0.677561\pi$
$499$	−23.6492	−1.05868	−0.529342	+	0.848409i	$0.677561\pi$
$500$	0	0
$501$	−42.1942	−1.88510
$502$	0	0
$503$	32.5488	1.45128	0.725639	−	0.688076i	$-0.241544\pi$
$503$	32.5488	1.45128	0.725639	+	0.688076i	$0.241544\pi$
$504$	0	0
$505$	2.14969	0.0956602
$506$	0	0
$507$	26.0995	1.15912
$508$	0	0
$509$	−6.74314	−0.298884	−0.149442	−	0.988770i	$-0.547748\pi$
$509$	−6.74314	−0.298884	−0.149442	+	0.988770i	$0.547748\pi$
$510$	0	0
$511$	−1.26844	−0.0561124
$512$	0	0
$513$	−12.8500	−0.567342
$514$	0	0
$515$	−4.48518	−0.197640
$516$	0	0
$517$	31.7721	1.39734
$518$	0	0
$519$	−32.2460	−1.41544
$520$	0	0
$521$	30.4860	1.33562	0.667808	−	0.744334i	$-0.267233\pi$
$521$	30.4860	1.33562	0.667808	+	0.744334i	$0.267233\pi$
$522$	0	0
$523$	−35.3799	−1.54705	−0.773527	−	0.633764i	$-0.781509\pi$
$523$	−35.3799	−1.54705	−0.773527	+	0.633764i	$0.781509\pi$
$524$	0	0
$525$	14.1707	0.618461
$526$	0	0
$527$	12.1790	0.530524
$528$	0	0
$529$	4.40911	0.191701
$530$	0	0
$531$	20.6700	0.896999
$532$	0	0
$533$	7.53482	0.326369
$534$	0	0
$535$	29.7666	1.28692
$536$	0	0
$537$	44.9319	1.93895
$538$	0	0
$539$	0.234099	0.0100834
$540$	0	0
$541$	1.92707	0.0828512	0.0414256	−	0.999142i	$-0.486810\pi$
$541$	1.92707	0.0828512	0.0414256	+	0.999142i	$0.486810\pi$
$542$	0	0
$543$	28.3021	1.21456
$544$	0	0
$545$	−11.6479	−0.498943
$546$	0	0
$547$	−42.8308	−1.83131	−0.915656	−	0.401963i	$-0.868328\pi$
$547$	−42.8308	−1.83131	−0.915656	+	0.401963i	$0.868328\pi$
$548$	0	0
$549$	13.4619	0.574541
$550$	0	0
$551$	−15.4762	−0.659310
$552$	0	0
$553$	22.0285	0.936747
$554$	0	0
$555$	−13.4258	−0.569891
$556$	0	0
$557$	−6.70629	−0.284155	−0.142077	−	0.989856i	$-0.545378\pi$
$557$	−6.70629	−0.284155	−0.142077	+	0.989856i	$0.545378\pi$
$558$	0	0
$559$	−0.922606	−0.0390221
$560$	0	0
$561$	−29.9375	−1.26396
$562$	0	0
$563$	34.5296	1.45525	0.727624	−	0.685976i	$-0.240624\pi$
$563$	34.5296	1.45525	0.727624	+	0.685976i	$0.240624\pi$
$564$	0	0
$565$	−12.2773	−0.516510
$566$	0	0
$567$	−29.4968	−1.23875
$568$	0	0
$569$	24.7919	1.03933	0.519665	−	0.854370i	$-0.326057\pi$
$569$	24.7919	1.03933	0.519665	+	0.854370i	$0.326057\pi$
$570$	0	0
$571$	0.782690	0.0327545	0.0163773	−	0.999866i	$-0.494787\pi$
$571$	0.782690	0.0327545	0.0163773	+	0.999866i	$0.494787\pi$
$572$	0	0
$573$	−1.76051	−0.0735466
$574$	0	0
$575$	−12.6494	−0.527515
$576$	0	0
$577$	42.1602	1.75515	0.877576	−	0.479437i	$-0.159159\pi$
$577$	42.1602	1.75515	0.877576	+	0.479437i	$0.159159\pi$
$578$	0	0
$579$	5.99297	0.249059
$580$	0	0
$581$	−20.0241	−0.830740
$582$	0	0
$583$	−46.8752	−1.94137
$584$	0	0
$585$	3.28216	0.135701
$586$	0	0
$587$	−8.15934	−0.336772	−0.168386	−	0.985721i	$-0.553855\pi$
$587$	−8.15934	−0.336772	−0.168386	+	0.985721i	$0.553855\pi$
$588$	0	0
$589$	−20.8583	−0.859450
$590$	0	0
$591$	−14.2229	−0.585051
$592$	0	0
$593$	29.2316	1.20040	0.600199	−	0.799851i	$-0.295088\pi$
$593$	29.2316	1.20040	0.600199	+	0.799851i	$0.295088\pi$
$594$	0	0
$595$	12.9214	0.529728
$596$	0	0
$597$	43.1626	1.76653
$598$	0	0
$599$	−8.45872	−0.345614	−0.172807	−	0.984956i	$-0.555284\pi$
$599$	−8.45872	−0.345614	−0.172807	+	0.984956i	$0.555284\pi$
$600$	0	0
$601$	−11.7306	−0.478499	−0.239250	−	0.970958i	$-0.576901\pi$
$601$	−11.7306	−0.478499	−0.239250	+	0.970958i	$0.576901\pi$
$602$	0	0
$603$	−16.8529	−0.686305
$604$	0	0
$605$	14.5534	0.591679
$606$	0	0
$607$	3.54902	0.144050	0.0720252	−	0.997403i	$-0.477054\pi$
$607$	3.54902	0.144050	0.0720252	+	0.997403i	$0.477054\pi$
$608$	0	0
$609$	−17.5088	−0.709494
$610$	0	0
$611$	7.71530	0.312128
$612$	0	0
$613$	8.42995	0.340483	0.170241	−	0.985402i	$-0.445545\pi$
$613$	8.42995	0.340483	0.170241	+	0.985402i	$0.445545\pi$
$614$	0	0
$615$	−24.5986	−0.991910
$616$	0	0
$617$	−28.6871	−1.15490	−0.577449	−	0.816427i	$-0.695952\pi$
$617$	−28.6871	−1.15490	−0.577449	+	0.816427i	$0.695952\pi$
$618$	0	0
$619$	−25.2215	−1.01374	−0.506868	−	0.862024i	$-0.669197\pi$
$619$	−25.2215	−1.01374	−0.506868	+	0.862024i	$0.669197\pi$
$620$	0	0
$621$	12.9769	0.520746
$622$	0	0
$623$	−41.5077	−1.66297
$624$	0	0
$625$	−7.08162	−0.283265
$626$	0	0
$627$	51.2724	2.04762
$628$	0	0
$629$	11.4474	0.456439
$630$	0	0
$631$	30.6730	1.22107	0.610536	−	0.791989i	$-0.290954\pi$
$631$	30.6730	1.22107	0.610536	+	0.791989i	$0.290954\pi$
$632$	0	0
$633$	−50.6117	−2.01164
$634$	0	0
$635$	−0.872384	−0.0346195
$636$	0	0
$637$	0.0568469	0.00225235
$638$	0	0
$639$	12.0450	0.476494
$640$	0	0
$641$	−33.3878	−1.31874	−0.659370	−	0.751819i	$-0.729177\pi$
$641$	−33.3878	−1.31874	−0.659370	+	0.751819i	$0.729177\pi$
$642$	0	0
$643$	−6.72479	−0.265200	−0.132600	−	0.991170i	$-0.542333\pi$
$643$	−6.72479	−0.265200	−0.132600	+	0.991170i	$0.542333\pi$
$644$	0	0
$645$	3.01199	0.118597
$646$	0	0
$647$	−10.2355	−0.402399	−0.201200	−	0.979550i	$-0.564484\pi$
$647$	−10.2355	−0.402399	−0.201200	+	0.979550i	$0.564484\pi$
$648$	0	0
$649$	49.2965	1.93506
$650$	0	0
$651$	−23.5977	−0.924868
$652$	0	0
$653$	22.4659	0.879158	0.439579	−	0.898204i	$-0.355128\pi$
$653$	22.4659	0.879158	0.439579	+	0.898204i	$0.355128\pi$
$654$	0	0
$655$	3.39179	0.132528
$656$	0	0
$657$	−0.896862	−0.0349899
$658$	0	0
$659$	−10.6052	−0.413120	−0.206560	−	0.978434i	$-0.566227\pi$
$659$	−10.6052	−0.413120	−0.206560	+	0.978434i	$0.566227\pi$
$660$	0	0
$661$	0.705434	0.0274382	0.0137191	−	0.999906i	$-0.495633\pi$
$661$	0.705434	0.0274382	0.0137191	+	0.999906i	$0.495633\pi$
$662$	0	0
$663$	−7.26980	−0.282336
$664$	0	0
$665$	−22.1299	−0.858160
$666$	0	0
$667$	15.6291	0.605161
$668$	0	0
$669$	−23.3173	−0.901499
$670$	0	0
$671$	32.1059	1.23943
$672$	0	0
$673$	31.8397	1.22733	0.613665	−	0.789567i	$-0.289695\pi$
$673$	31.8397	1.22733	0.613665	+	0.789567i	$0.289695\pi$
$674$	0	0
$675$	−5.98888	−0.230512
$676$	0	0
$677$	41.0631	1.57818	0.789091	−	0.614277i	$-0.210552\pi$
$677$	41.0631	1.57818	0.789091	+	0.614277i	$0.210552\pi$
$678$	0	0
$679$	−13.5269	−0.519116
$680$	0	0
$681$	24.7767	0.949444
$682$	0	0
$683$	−23.7471	−0.908657	−0.454328	−	0.890834i	$-0.650121\pi$
$683$	−23.7471	−0.908657	−0.454328	+	0.890834i	$0.650121\pi$
$684$	0	0
$685$	17.4900	0.668257
$686$	0	0
$687$	1.58757	0.0605696
$688$	0	0
$689$	−11.3828	−0.433650
$690$	0	0
$691$	45.5998	1.73470	0.867349	−	0.497700i	$-0.165822\pi$
$691$	45.5998	1.73470	0.867349	+	0.497700i	$0.165822\pi$
$692$	0	0
$693$	22.3297	0.848236
$694$	0	0
$695$	−27.2095	−1.03211
$696$	0	0
$697$	20.9739	0.794444
$698$	0	0
$699$	1.29161	0.0488532
$700$	0	0
$701$	−15.9376	−0.601956	−0.300978	−	0.953631i	$-0.597313\pi$
$701$	−15.9376	−0.601956	−0.300978	+	0.953631i	$0.597313\pi$
$702$	0	0
$703$	−19.6054	−0.739433
$704$	0	0
$705$	−25.1878	−0.948627
$706$	0	0
$707$	3.55146	0.133566
$708$	0	0
$709$	34.0498	1.27877	0.639384	−	0.768888i	$-0.279190\pi$
$709$	34.0498	1.27877	0.639384	+	0.768888i	$0.279190\pi$
$710$	0	0
$711$	15.5755	0.584126
$712$	0	0
$713$	21.0643	0.788863
$714$	0	0
$715$	7.82776	0.292742
$716$	0	0
$717$	−50.5798	−1.88894
$718$	0	0
$719$	−21.0650	−0.785591	−0.392795	−	0.919626i	$-0.628492\pi$
$719$	−21.0650	−0.785591	−0.392795	+	0.919626i	$0.628492\pi$
$720$	0	0
$721$	−7.40984	−0.275957
$722$	0	0
$723$	−28.1420	−1.04661
$724$	0	0
$725$	−7.21285	−0.267879
$726$	0	0
$727$	42.8528	1.58932	0.794661	−	0.607053i	$-0.207649\pi$
$727$	42.8528	1.58932	0.794661	+	0.607053i	$0.207649\pi$
$728$	0	0
$729$	−4.43305	−0.164187
$730$	0	0
$731$	−2.56817	−0.0949871
$732$	0	0
$733$	28.0540	1.03620	0.518098	−	0.855321i	$-0.326640\pi$
$733$	28.0540	1.03620	0.518098	+	0.855321i	$0.326640\pi$
$734$	0	0
$735$	−0.185585	−0.00684542
$736$	0	0
$737$	−40.1932	−1.48054
$738$	0	0
$739$	−25.0444	−0.921275	−0.460637	−	0.887588i	$-0.652379\pi$
$739$	−25.0444	−0.921275	−0.460637	+	0.887588i	$0.652379\pi$
$740$	0	0
$741$	12.4506	0.457384
$742$	0	0
$743$	−0.425752	−0.0156193	−0.00780966	−	0.999970i	$-0.502486\pi$
$743$	−0.425752	−0.0156193	−0.00780966	+	0.999970i	$0.502486\pi$
$744$	0	0
$745$	5.84336	0.214084
$746$	0	0
$747$	−14.1583	−0.518023
$748$	0	0
$749$	49.1767	1.79688
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	6.20262	0.226036
$754$	0	0
$755$	−1.85654	−0.0675664
$756$	0	0
$757$	30.3086	1.10159	0.550793	−	0.834642i	$-0.314325\pi$
$757$	30.3086	1.10159	0.550793	+	0.834642i	$0.314325\pi$
$758$	0	0
$759$	−51.7788	−1.87945
$760$	0	0
$761$	0.338351	0.0122652	0.00613260	−	0.999981i	$-0.498048\pi$
$761$	0.338351	0.0122652	0.00613260	+	0.999981i	$0.498048\pi$
$762$	0	0
$763$	−19.2433	−0.696652
$764$	0	0
$765$	9.13624	0.330321
$766$	0	0
$767$	11.9708	0.432240
$768$	0	0
$769$	−36.2516	−1.30727	−0.653633	−	0.756812i	$-0.726756\pi$
$769$	−36.2516	−1.30727	−0.653633	+	0.756812i	$0.726756\pi$
$770$	0	0
$771$	54.0134	1.94524
$772$	0	0
$773$	20.6655	0.743288	0.371644	−	0.928375i	$-0.378794\pi$
$773$	20.6655	0.743288	0.371644	+	0.928375i	$0.378794\pi$
$774$	0	0
$775$	−9.72120	−0.349196
$776$	0	0
$777$	−22.1803	−0.795715
$778$	0	0
$779$	−35.9209	−1.28700
$780$	0	0
$781$	28.7266	1.02792
$782$	0	0
$783$	7.39964	0.264441
$784$	0	0
$785$	−21.8796	−0.780917
$786$	0	0
$787$	14.2909	0.509415	0.254707	−	0.967018i	$-0.418021\pi$
$787$	14.2909	0.509415	0.254707	+	0.967018i	$0.418021\pi$
$788$	0	0
$789$	−55.9951	−1.99348
$790$	0	0
$791$	−20.2830	−0.721181
$792$	0	0
$793$	7.79634	0.276856
$794$	0	0
$795$	37.1609	1.31796
$796$	0	0
$797$	1.90991	0.0676523	0.0338262	−	0.999428i	$-0.489231\pi$
$797$	1.90991	0.0676523	0.0338262	+	0.999428i	$0.489231\pi$
$798$	0	0
$799$	21.4763	0.759777
$800$	0	0
$801$	−29.3485	−1.03698
$802$	0	0
$803$	−2.13896	−0.0754823
$804$	0	0
$805$	22.3484	0.787679
$806$	0	0
$807$	−8.68844	−0.305848
$808$	0	0
$809$	−12.0175	−0.422511	−0.211256	−	0.977431i	$-0.567755\pi$
$809$	−12.0175	−0.422511	−0.211256	+	0.977431i	$0.567755\pi$
$810$	0	0
$811$	0.607304	0.0213253	0.0106627	−	0.999943i	$-0.496606\pi$
$811$	0.607304	0.0213253	0.0106627	+	0.999943i	$0.496606\pi$
$812$	0	0
$813$	35.0800	1.23031
$814$	0	0
$815$	−19.6480	−0.688241
$816$	0	0
$817$	4.39836	0.153879
$818$	0	0
$819$	5.42238	0.189473
$820$	0	0
$821$	44.1372	1.54040	0.770200	−	0.637802i	$-0.220156\pi$
$821$	44.1372	1.54040	0.770200	+	0.637802i	$0.220156\pi$
$822$	0	0
$823$	45.9532	1.60183	0.800913	−	0.598780i	$-0.204348\pi$
$823$	45.9532	1.60183	0.800913	+	0.598780i	$0.204348\pi$
$824$	0	0
$825$	23.8960	0.831953
$826$	0	0
$827$	−44.6020	−1.55096	−0.775482	−	0.631370i	$-0.782493\pi$
$827$	−44.6020	−1.55096	−0.775482	+	0.631370i	$0.782493\pi$
$828$	0	0
$829$	−9.83199	−0.341479	−0.170739	−	0.985316i	$-0.554616\pi$
$829$	−9.83199	−0.341479	−0.170739	+	0.985316i	$0.554616\pi$
$830$	0	0
$831$	−47.5080	−1.64803
$832$	0	0
$833$	0.158239	0.00548266
$834$	0	0
$835$	30.7101	1.06277
$836$	0	0
$837$	9.97294	0.344715
$838$	0	0
$839$	26.6718	0.920814	0.460407	−	0.887708i	$-0.347704\pi$
$839$	26.6718	0.920814	0.460407	+	0.887708i	$0.347704\pi$
$840$	0	0
$841$	−20.0881	−0.692692
$842$	0	0
$843$	−63.1856	−2.17623
$844$	0	0
$845$	−18.9959	−0.653479
$846$	0	0
$847$	24.0433	0.826137
$848$	0	0
$849$	−20.7861	−0.713376
$850$	0	0
$851$	19.7991	0.678703
$852$	0	0
$853$	−2.61905	−0.0896745	−0.0448373	−	0.998994i	$-0.514277\pi$
$853$	−2.61905	−0.0896745	−0.0448373	+	0.998994i	$0.514277\pi$
$854$	0	0
$855$	−15.6472	−0.535121
$856$	0	0
$857$	−11.6206	−0.396952	−0.198476	−	0.980106i	$-0.563599\pi$
$857$	−11.6206	−0.396952	−0.198476	+	0.980106i	$0.563599\pi$
$858$	0	0
$859$	−33.3272	−1.13711	−0.568554	−	0.822646i	$-0.692497\pi$
$859$	−33.3272	−1.13711	−0.568554	+	0.822646i	$0.692497\pi$
$860$	0	0
$861$	−40.6387	−1.38496
$862$	0	0
$863$	−53.1382	−1.80885	−0.904423	−	0.426636i	$-0.859698\pi$
$863$	−53.1382	−1.80885	−0.904423	+	0.426636i	$0.859698\pi$
$864$	0	0
$865$	23.4695	0.797987
$866$	0	0
$867$	17.3091	0.587847
$868$	0	0
$869$	37.1465	1.26011
$870$	0	0
$871$	−9.76021	−0.330712
$872$	0	0
$873$	−9.56435	−0.323704
$874$	0	0
$875$	−31.6575	−1.07022
$876$	0	0
$877$	−17.1446	−0.578931	−0.289466	−	0.957188i	$-0.593478\pi$
$877$	−17.1446	−0.578931	−0.289466	+	0.957188i	$0.593478\pi$
$878$	0	0
$879$	22.4489	0.757182
$880$	0	0
$881$	−54.7788	−1.84554	−0.922772	−	0.385347i	$-0.874082\pi$
$881$	−54.7788	−1.84554	−0.922772	+	0.385347i	$0.874082\pi$
$882$	0	0
$883$	−35.7119	−1.20180	−0.600900	−	0.799324i	$-0.705191\pi$
$883$	−35.7119	−1.20180	−0.600900	+	0.799324i	$0.705191\pi$
$884$	0	0
$885$	−39.0805	−1.31368
$886$	0	0
$887$	36.5614	1.22761	0.613805	−	0.789457i	$-0.289638\pi$
$887$	36.5614	1.22761	0.613805	+	0.789457i	$0.289638\pi$
$888$	0	0
$889$	−1.44124	−0.0483378
$890$	0	0
$891$	−49.7403	−1.66636
$892$	0	0
$893$	−36.7814	−1.23084
$894$	0	0
$895$	−32.7027	−1.09313
$896$	0	0
$897$	−12.5736	−0.419819
$898$	0	0
$899$	12.0112	0.400595
$900$	0	0
$901$	−31.6852	−1.05559
$902$	0	0
$903$	4.97603	0.165592
$904$	0	0
$905$	−20.5990	−0.684735
$906$	0	0
$907$	−32.4497	−1.07747	−0.538737	−	0.842474i	$-0.681098\pi$
$907$	−32.4497	−1.07747	−0.538737	+	0.842474i	$0.681098\pi$
$908$	0	0
$909$	2.51109	0.0832877
$910$	0	0
$911$	12.6143	0.417929	0.208964	−	0.977923i	$-0.432991\pi$
$911$	12.6143	0.417929	0.208964	+	0.977923i	$0.432991\pi$
$912$	0	0
$913$	−33.7665	−1.11751
$914$	0	0
$915$	−25.4524	−0.841429
$916$	0	0
$917$	5.60349	0.185044
$918$	0	0
$919$	11.5072	0.379588	0.189794	−	0.981824i	$-0.439218\pi$
$919$	11.5072	0.379588	0.189794	+	0.981824i	$0.439218\pi$
$920$	0	0
$921$	−33.2230	−1.09474
$922$	0	0
$923$	6.97576	0.229610
$924$	0	0
$925$	−9.13731	−0.300433
$926$	0	0
$927$	−5.23921	−0.172078
$928$	0	0
$929$	14.9189	0.489473	0.244737	−	0.969590i	$-0.421299\pi$
$929$	14.9189	0.489473	0.244737	+	0.969590i	$0.421299\pi$
$930$	0	0
$931$	−0.271008	−0.00888191
$932$	0	0
$933$	−25.1684	−0.823978
$934$	0	0
$935$	21.7894	0.712588
$936$	0	0
$937$	−20.3883	−0.666058	−0.333029	−	0.942917i	$-0.608071\pi$
$937$	−20.3883	−0.666058	−0.333029	+	0.942917i	$0.608071\pi$
$938$	0	0
$939$	−10.9051	−0.355875
$940$	0	0
$941$	−60.0180	−1.95653	−0.978265	−	0.207359i	$-0.933513\pi$
$941$	−60.0180	−1.95653	−0.978265	+	0.207359i	$0.933513\pi$
$942$	0	0
$943$	36.2757	1.18130
$944$	0	0
$945$	10.5809	0.344198
$946$	0	0
$947$	−45.7314	−1.48607	−0.743036	−	0.669252i	$-0.766615\pi$
$947$	−45.7314	−1.48607	−0.743036	+	0.669252i	$0.766615\pi$
$948$	0	0
$949$	−0.519409	−0.0168607
$950$	0	0
$951$	56.2542	1.82417
$952$	0	0
$953$	47.0813	1.52511	0.762556	−	0.646922i	$-0.223944\pi$
$953$	47.0813	1.52511	0.762556	+	0.646922i	$0.223944\pi$
$954$	0	0
$955$	1.28135	0.0414635
$956$	0	0
$957$	−29.5250	−0.954409
$958$	0	0
$959$	28.8947	0.933059
$960$	0	0
$961$	−14.8118	−0.477801
$962$	0	0
$963$	34.7709	1.12048
$964$	0	0
$965$	−4.36185	−0.140413
$966$	0	0
$967$	−16.7960	−0.540124	−0.270062	−	0.962843i	$-0.587044\pi$
$967$	−16.7960	−0.540124	−0.270062	+	0.962843i	$0.587044\pi$
$968$	0	0
$969$	34.6575	1.11336
$970$	0	0
$971$	−29.9338	−0.960621	−0.480310	−	0.877099i	$-0.659476\pi$
$971$	−29.9338	−0.960621	−0.480310	+	0.877099i	$0.659476\pi$
$972$	0	0
$973$	−44.9521	−1.44110
$974$	0	0
$975$	5.80273	0.185836
$976$	0	0
$977$	38.6717	1.23722	0.618608	−	0.785700i	$-0.287697\pi$
$977$	38.6717	1.23722	0.618608	+	0.785700i	$0.287697\pi$
$978$	0	0
$979$	−69.9942	−2.23703
$980$	0	0
$981$	−13.6061	−0.434410
$982$	0	0
$983$	7.61706	0.242946	0.121473	−	0.992595i	$-0.461238\pi$
$983$	7.61706	0.242946	0.121473	+	0.992595i	$0.461238\pi$
$984$	0	0
$985$	10.3518	0.329836
$986$	0	0
$987$	−41.6121	−1.32453
$988$	0	0
$989$	−4.44180	−0.141241
$990$	0	0
$991$	−29.5352	−0.938218	−0.469109	−	0.883140i	$-0.655425\pi$
$991$	−29.5352	−0.938218	−0.469109	+	0.883140i	$0.655425\pi$
$992$	0	0
$993$	−59.1192	−1.87609
$994$	0	0
$995$	−31.4149	−0.995920
$996$	0	0
$997$	−33.4567	−1.05958	−0.529792	−	0.848127i	$-0.677730\pi$
$997$	−33.4567	−1.05958	−0.529792	+	0.848127i	$0.677730\pi$
$998$	0	0
$999$	9.37392	0.296578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.8	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-2.20855 q^{3} +1.60744 q^{5} +2.65561 q^{7} +1.87768 q^{9} +O(q^{10})\)	\(q-2.20855 q^{3} +1.60744 q^{5} +2.65561 q^{7} +1.87768 q^{9} +4.47814 q^{11} +1.08744 q^{13} -3.55011 q^{15} +3.02699 q^{17} -5.18417 q^{19} -5.86504 q^{21} +5.23537 q^{23} -2.41613 q^{25} +2.47870 q^{27} +2.98529 q^{29} +4.02345 q^{31} -9.89019 q^{33} +4.26874 q^{35} +3.78179 q^{37} -2.40166 q^{39} +6.92897 q^{41} -0.848422 q^{43} +3.01826 q^{45} +7.09493 q^{47} +0.0522760 q^{49} -6.68525 q^{51} -10.4675 q^{53} +7.19835 q^{55} +11.4495 q^{57} +11.0083 q^{59} +7.16946 q^{61} +4.98638 q^{63} +1.74799 q^{65} -8.97542 q^{67} -11.5626 q^{69} +6.41486 q^{71} -0.477645 q^{73} +5.33614 q^{75} +11.8922 q^{77} +8.29507 q^{79} -11.1074 q^{81} -7.54030 q^{83} +4.86571 q^{85} -6.59315 q^{87} -15.6302 q^{89} +2.88781 q^{91} -8.88598 q^{93} -8.33325 q^{95} -5.09371 q^{97} +8.40851 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 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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