LMFDB - Embedded newform 6003.2.a.u.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35991 q^{2} -0.150655 q^{4} -4.20064 q^{5} +0.870260 q^{7} -2.92469 q^{8} +O(q^{10})$	\(q+1.35991 q^{2} -0.150655 q^{4} -4.20064 q^{5} +0.870260 q^{7} -2.92469 q^{8} -5.71248 q^{10} +3.24281 q^{11} -1.60184 q^{13} +1.18347 q^{14} -3.67599 q^{16} +7.96215 q^{17} -1.89212 q^{19} +0.632848 q^{20} +4.40992 q^{22} -1.00000 q^{23} +12.6454 q^{25} -2.17835 q^{26} -0.131109 q^{28} -1.00000 q^{29} -6.87026 q^{31} +0.850373 q^{32} +10.8278 q^{34} -3.65565 q^{35} +5.65195 q^{37} -2.57310 q^{38} +12.2856 q^{40} -5.17129 q^{41} +12.5424 q^{43} -0.488546 q^{44} -1.35991 q^{46} -4.23722 q^{47} -6.24265 q^{49} +17.1965 q^{50} +0.241325 q^{52} +6.69371 q^{53} -13.6219 q^{55} -2.54524 q^{56} -1.35991 q^{58} -1.30032 q^{59} -14.9100 q^{61} -9.34291 q^{62} +8.50841 q^{64} +6.72875 q^{65} -0.0645202 q^{67} -1.19954 q^{68} -4.97134 q^{70} +7.40240 q^{71} -4.96764 q^{73} +7.68612 q^{74} +0.285057 q^{76} +2.82209 q^{77} -3.28830 q^{79} +15.4415 q^{80} -7.03247 q^{82} +2.68133 q^{83} -33.4461 q^{85} +17.0565 q^{86} -9.48422 q^{88} +11.1742 q^{89} -1.39402 q^{91} +0.150655 q^{92} -5.76222 q^{94} +7.94809 q^{95} +1.63869 q^{97} -8.48942 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * q^98

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 $

$2$	1.35991	0.961599	0.480799	−	0.876831i	$-0.340347\pi$
$2$	1.35991	0.961599	0.480799	+	0.876831i	$0.340347\pi$
$3$	0	0
$3$	0	0
$4$	−0.150655	−0.0753275
$5$	−4.20064	−1.87858	−0.939291	−	0.343120i	$-0.888516\pi$
$5$	−4.20064	−1.87858	−0.939291	+	0.343120i	$0.888516\pi$
$6$	0	0
$7$	0.870260	0.328927	0.164464	−	0.986383i	$-0.447411\pi$
$7$	0.870260	0.328927	0.164464	+	0.986383i	$0.447411\pi$
$8$	−2.92469	−1.03403
$9$	0	0
$10$	−5.71248	−1.80644
$11$	3.24281	0.977745	0.488872	−	0.872355i	$-0.337408\pi$
$11$	3.24281	0.977745	0.488872	+	0.872355i	$0.337408\pi$
$12$	0	0
$13$	−1.60184	−0.444270	−0.222135	−	0.975016i	$-0.571303\pi$
$13$	−1.60184	−0.444270	−0.222135	+	0.975016i	$0.571303\pi$
$14$	1.18347	0.316296
$15$	0	0
$16$	−3.67599	−0.918998
$17$	7.96215	1.93110	0.965552	−	0.260210i	$-0.0837918\pi$
$17$	7.96215	1.93110	0.965552	+	0.260210i	$0.0837918\pi$
$18$	0	0
$19$	−1.89212	−0.434081	−0.217040	−	0.976163i	$-0.569640\pi$
$19$	−1.89212	−0.434081	−0.217040	+	0.976163i	$0.569640\pi$
$20$	0.632848	0.141509
$21$	0	0
$22$	4.40992	0.940198
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	12.6454	2.52907
$26$	−2.17835	−0.427210
$27$	0	0
$28$	−0.131109	−0.0247773
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.87026	−1.23394	−0.616968	−	0.786988i	$-0.711639\pi$
$31$	−6.87026	−1.23394	−0.616968	+	0.786988i	$0.711639\pi$
$32$	0.850373	0.150326
$33$	0	0
$34$	10.8278	1.85695
$35$	−3.65565	−0.617917
$36$	0	0
$37$	5.65195	0.929175	0.464587	−	0.885527i	$-0.346203\pi$
$37$	5.65195	0.929175	0.464587	+	0.885527i	$0.346203\pi$
$38$	−2.57310	−0.417412
$39$	0	0
$40$	12.2856	1.94252
$41$	−5.17129	−0.807620	−0.403810	−	0.914843i	$-0.632314\pi$
$41$	−5.17129	−0.807620	−0.403810	+	0.914843i	$0.632314\pi$
$42$	0	0
$43$	12.5424	1.91269	0.956347	−	0.292232i	$-0.0943980\pi$
$43$	12.5424	1.91269	0.956347	+	0.292232i	$0.0943980\pi$
$44$	−0.488546	−0.0736511
$45$	0	0
$46$	−1.35991	−0.200507
$47$	−4.23722	−0.618062	−0.309031	−	0.951052i	$-0.600005\pi$
$47$	−4.23722	−0.618062	−0.309031	+	0.951052i	$0.600005\pi$
$48$	0	0
$49$	−6.24265	−0.891807
$50$	17.1965	2.43195
$51$	0	0
$52$	0.241325	0.0334658
$53$	6.69371	0.919452	0.459726	−	0.888061i	$-0.347948\pi$
$53$	6.69371	0.919452	0.459726	+	0.888061i	$0.347948\pi$
$54$	0	0
$55$	−13.6219	−1.83677
$56$	−2.54524	−0.340122
$57$	0	0
$58$	−1.35991	−0.178564
$59$	−1.30032	−0.169287	−0.0846435	−	0.996411i	$-0.526975\pi$
$59$	−1.30032	−0.169287	−0.0846435	+	0.996411i	$0.526975\pi$
$60$	0	0
$61$	−14.9100	−1.90902	−0.954512	−	0.298172i	$-0.903623\pi$
$61$	−14.9100	−1.90902	−0.954512	+	0.298172i	$0.903623\pi$
$62$	−9.34291	−1.18655
$63$	0	0
$64$	8.50841	1.06355
$65$	6.72875	0.834598
$66$	0	0
$67$	−0.0645202	−0.00788240	−0.00394120	−	0.999992i	$-0.501255\pi$
$67$	−0.0645202	−0.00788240	−0.00394120	+	0.999992i	$0.501255\pi$
$68$	−1.19954	−0.145465
$69$	0	0
$70$	−4.97134	−0.594188
$71$	7.40240	0.878503	0.439251	−	0.898364i	$-0.355244\pi$
$71$	7.40240	0.878503	0.439251	+	0.898364i	$0.355244\pi$
$72$	0	0
$73$	−4.96764	−0.581419	−0.290709	−	0.956811i	$-0.593891\pi$
$73$	−4.96764	−0.581419	−0.290709	+	0.956811i	$0.593891\pi$
$74$	7.68612	0.893493
$75$	0	0
$76$	0.285057	0.0326983
$77$	2.82209	0.321607
$78$	0	0
$79$	−3.28830	−0.369962	−0.184981	−	0.982742i	$-0.559222\pi$
$79$	−3.28830	−0.369962	−0.184981	+	0.982742i	$0.559222\pi$
$80$	15.4415	1.72641
$81$	0	0
$82$	−7.03247	−0.776606
$83$	2.68133	0.294314	0.147157	−	0.989113i	$-0.452988\pi$
$83$	2.68133	0.294314	0.147157	+	0.989113i	$0.452988\pi$
$84$	0	0
$85$	−33.4461	−3.62774
$86$	17.0565	1.83925
$87$	0	0
$88$	−9.48422	−1.01102
$89$	11.1742	1.18446	0.592232	−	0.805767i	$-0.298247\pi$
$89$	11.1742	1.18446	0.592232	+	0.805767i	$0.298247\pi$
$90$	0	0
$91$	−1.39402	−0.146133
$92$	0.150655	0.0157069
$93$	0	0
$94$	−5.76222	−0.594328
$95$	7.94809	0.815457
$96$	0	0
$97$	1.63869	0.166384	0.0831918	−	0.996534i	$-0.473489\pi$
$97$	1.63869	0.166384	0.0831918	+	0.996534i	$0.473489\pi$
$98$	−8.48942	−0.857561
$99$	0	0
$100$	−1.90509	−0.190509
$101$	−15.1598	−1.50846	−0.754230	−	0.656610i	$-0.771990\pi$
$101$	−15.1598	−1.50846	−0.754230	+	0.656610i	$0.771990\pi$
$102$	0	0
$103$	−13.6519	−1.34516	−0.672579	−	0.740026i	$-0.734813\pi$
$103$	−13.6519	−1.34516	−0.672579	+	0.740026i	$0.734813\pi$
$104$	4.68488	0.459390
$105$	0	0
$106$	9.10282	0.884144
$107$	−13.7576	−1.33000	−0.664999	−	0.746844i	$-0.731568\pi$
$107$	−13.7576	−1.33000	−0.664999	+	0.746844i	$0.731568\pi$
$108$	0	0
$109$	−6.63576	−0.635591	−0.317795	−	0.948159i	$-0.602943\pi$
$109$	−6.63576	−0.635591	−0.317795	+	0.948159i	$0.602943\pi$
$110$	−18.5245	−1.76624
$111$	0	0
$112$	−3.19907	−0.302284
$113$	−4.37941	−0.411981	−0.205990	−	0.978554i	$-0.566042\pi$
$113$	−4.37941	−0.411981	−0.205990	+	0.978554i	$0.566042\pi$
$114$	0	0
$115$	4.20064	0.391712
$116$	0.150655	0.0139880
$117$	0	0
$118$	−1.76831	−0.162786
$119$	6.92913	0.635193
$120$	0	0
$121$	−0.484172	−0.0440156
$122$	−20.2761	−1.83572
$123$	0	0
$124$	1.03504	0.0929493
$125$	−32.1154	−2.87249
$126$	0	0
$127$	−9.03976	−0.802149	−0.401075	−	0.916045i	$-0.631363\pi$
$127$	−9.03976	−0.802149	−0.401075	+	0.916045i	$0.631363\pi$
$128$	9.86990	0.872384
$129$	0	0
$130$	9.15047	0.802549
$131$	3.26008	0.284834	0.142417	−	0.989807i	$-0.454513\pi$
$131$	3.26008	0.284834	0.142417	+	0.989807i	$0.454513\pi$
$132$	0	0
$133$	−1.64663	−0.142781
$134$	−0.0877414	−0.00757970
$135$	0	0
$136$	−23.2868	−1.99683
$137$	15.5938	1.33227	0.666133	−	0.745833i	$-0.267948\pi$
$137$	15.5938	1.33227	0.666133	+	0.745833i	$0.267948\pi$
$138$	0	0
$139$	9.27779	0.786932	0.393466	−	0.919339i	$-0.371276\pi$
$139$	9.27779	0.786932	0.393466	+	0.919339i	$0.371276\pi$
$140$	0.550742	0.0465462
$141$	0	0
$142$	10.0666	0.844767
$143$	−5.19446	−0.434383
$144$	0	0
$145$	4.20064	0.348844
$146$	−6.75553	−0.559091
$147$	0	0
$148$	−0.851495	−0.0699924
$149$	−6.92426	−0.567257	−0.283629	−	0.958934i	$-0.591538\pi$
$149$	−6.92426	−0.567257	−0.283629	+	0.958934i	$0.591538\pi$
$150$	0	0
$151$	−22.0062	−1.79084	−0.895419	−	0.445224i	$-0.853124\pi$
$151$	−22.0062	−1.79084	−0.895419	+	0.445224i	$0.853124\pi$
$152$	5.53385	0.448854
$153$	0	0
$154$	3.83778	0.309257
$155$	28.8595	2.31805
$156$	0	0
$157$	−20.5974	−1.64385	−0.821926	−	0.569594i	$-0.807100\pi$
$157$	−20.5974	−1.64385	−0.821926	+	0.569594i	$0.807100\pi$
$158$	−4.47178	−0.355755
$159$	0	0
$160$	−3.57211	−0.282400
$161$	−0.870260	−0.0685861
$162$	0	0
$163$	8.65285	0.677743	0.338872	−	0.940833i	$-0.389955\pi$
$163$	8.65285	0.677743	0.338872	+	0.940833i	$0.389955\pi$
$164$	0.779081	0.0608360
$165$	0	0
$166$	3.64635	0.283012
$167$	−5.82700	−0.450907	−0.225453	−	0.974254i	$-0.572386\pi$
$167$	−5.82700	−0.450907	−0.225453	+	0.974254i	$0.572386\pi$
$168$	0	0
$169$	−10.4341	−0.802624
$170$	−45.4836	−3.48843
$171$	0	0
$172$	−1.88957	−0.144079
$173$	4.42101	0.336123	0.168061	−	0.985777i	$-0.446249\pi$
$173$	4.42101	0.336123	0.168061	+	0.985777i	$0.446249\pi$
$174$	0	0
$175$	11.0048	0.831881
$176$	−11.9206	−0.898546
$177$	0	0
$178$	15.1959	1.13898
$179$	−18.9842	−1.41895	−0.709474	−	0.704732i	$-0.751067\pi$
$179$	−18.9842	−1.41895	−0.709474	+	0.704732i	$0.751067\pi$
$180$	0	0
$181$	−9.20077	−0.683888	−0.341944	−	0.939720i	$-0.611085\pi$
$181$	−9.20077	−0.683888	−0.341944	+	0.939720i	$0.611085\pi$
$182$	−1.89573	−0.140521
$183$	0	0
$184$	2.92469	0.215611
$185$	−23.7418	−1.74553
$186$	0	0
$187$	25.8197	1.88813
$188$	0.638359	0.0465571
$189$	0	0
$190$	10.8087	0.784143
$191$	22.0536	1.59574	0.797871	−	0.602828i	$-0.205959\pi$
$191$	22.0536	1.59574	0.797871	+	0.602828i	$0.205959\pi$
$192$	0	0
$193$	−1.88602	−0.135759	−0.0678793	−	0.997694i	$-0.521623\pi$
$193$	−1.88602	−0.135759	−0.0678793	+	0.997694i	$0.521623\pi$
$194$	2.22846	0.159994
$195$	0	0
$196$	0.940487	0.0671776
$197$	−9.88749	−0.704455	−0.352227	−	0.935914i	$-0.614576\pi$
$197$	−9.88749	−0.704455	−0.352227	+	0.935914i	$0.614576\pi$
$198$	0	0
$199$	−0.824714	−0.0584624	−0.0292312	−	0.999573i	$-0.509306\pi$
$199$	−0.824714	−0.0584624	−0.0292312	+	0.999573i	$0.509306\pi$
$200$	−36.9838	−2.61515
$201$	0	0
$202$	−20.6160	−1.45053
$203$	−0.870260	−0.0610803
$204$	0	0
$205$	21.7227	1.51718
$206$	−18.5652	−1.29350
$207$	0	0
$208$	5.88835	0.408284
$209$	−6.13577	−0.424420
$210$	0	0
$211$	−19.7545	−1.35995	−0.679977	−	0.733234i	$-0.738010\pi$
$211$	−19.7545	−1.35995	−0.679977	+	0.733234i	$0.738010\pi$
$212$	−1.00844	−0.0692601
$213$	0	0
$214$	−18.7091	−1.27892
$215$	−52.6860	−3.59316
$216$	0	0
$217$	−5.97891	−0.405875
$218$	−9.02402	−0.611183
$219$	0	0
$220$	2.05221	0.138360
$221$	−12.7541	−0.857932
$222$	0	0
$223$	6.70935	0.449291	0.224646	−	0.974441i	$-0.427878\pi$
$223$	6.70935	0.449291	0.224646	+	0.974441i	$0.427878\pi$
$224$	0.740045	0.0494463
$225$	0	0
$226$	−5.95559	−0.396160
$227$	−3.30901	−0.219626	−0.109813	−	0.993952i	$-0.535025\pi$
$227$	−3.30901	−0.219626	−0.109813	+	0.993952i	$0.535025\pi$
$228$	0	0
$229$	−22.9262	−1.51501	−0.757503	−	0.652832i	$-0.773581\pi$
$229$	−22.9262	−1.51501	−0.757503	+	0.652832i	$0.773581\pi$
$230$	5.71248	0.376669
$231$	0	0
$232$	2.92469	0.192015
$233$	15.0316	0.984752	0.492376	−	0.870383i	$-0.336129\pi$
$233$	15.0316	0.984752	0.492376	+	0.870383i	$0.336129\pi$
$234$	0	0
$235$	17.7990	1.16108
$236$	0.195899	0.0127520
$237$	0	0
$238$	9.42297	0.610801
$239$	3.77404	0.244122	0.122061	−	0.992523i	$-0.461050\pi$
$239$	3.77404	0.244122	0.122061	+	0.992523i	$0.461050\pi$
$240$	0	0
$241$	15.9280	1.02602	0.513008	−	0.858384i	$-0.328531\pi$
$241$	15.9280	1.02602	0.513008	+	0.858384i	$0.328531\pi$
$242$	−0.658428	−0.0423254
$243$	0	0
$244$	2.24626	0.143802
$245$	26.2231	1.67533
$246$	0	0
$247$	3.03086	0.192849
$248$	20.0934	1.27593
$249$	0	0
$250$	−43.6740	−2.76219
$251$	−20.6886	−1.30585	−0.652925	−	0.757423i	$-0.726458\pi$
$251$	−20.6886	−1.30585	−0.652925	+	0.757423i	$0.726458\pi$
$252$	0	0
$253$	−3.24281	−0.203874
$254$	−12.2932	−0.771346
$255$	0	0
$256$	−3.59469	−0.224668
$257$	0.653474	0.0407626	0.0203813	−	0.999792i	$-0.493512\pi$
$257$	0.653474	0.0407626	0.0203813	+	0.999792i	$0.493512\pi$
$258$	0	0
$259$	4.91866	0.305631
$260$	−1.01372	−0.0628682
$261$	0	0
$262$	4.43340	0.273896
$263$	12.3265	0.760082	0.380041	−	0.924970i	$-0.375910\pi$
$263$	12.3265	0.760082	0.380041	+	0.924970i	$0.375910\pi$
$264$	0	0
$265$	−28.1179	−1.72727
$266$	−2.23926	−0.137298
$267$	0	0
$268$	0.00972029	0.000593761 0
$269$	12.2201	0.745071	0.372536	−	0.928018i	$-0.378488\pi$
$269$	12.2201	0.745071	0.372536	+	0.928018i	$0.378488\pi$
$270$	0	0
$271$	−25.7038	−1.56140	−0.780698	−	0.624909i	$-0.785136\pi$
$271$	−25.7038	−1.56140	−0.780698	+	0.624909i	$0.785136\pi$
$272$	−29.2688	−1.77468
$273$	0	0
$274$	21.2061	1.28111
$275$	41.0066	2.47279
$276$	0	0
$277$	−31.0715	−1.86691	−0.933453	−	0.358700i	$-0.883220\pi$
$277$	−31.0715	−1.86691	−0.933453	+	0.358700i	$0.883220\pi$
$278$	12.6169	0.756713
$279$	0	0
$280$	10.6916	0.638947
$281$	−15.1309	−0.902635	−0.451317	−	0.892363i	$-0.649046\pi$
$281$	−15.1309	−0.902635	−0.451317	+	0.892363i	$0.649046\pi$
$282$	0	0
$283$	11.2893	0.671078	0.335539	−	0.942026i	$-0.391082\pi$
$283$	11.2893	0.671078	0.335539	+	0.942026i	$0.391082\pi$
$284$	−1.11521	−0.0661754
$285$	0	0
$286$	−7.06398	−0.417702
$287$	−4.50036	−0.265648
$288$	0	0
$289$	46.3958	2.72916
$290$	5.71248	0.335448
$291$	0	0
$292$	0.748400	0.0437968
$293$	15.7003	0.917219	0.458609	−	0.888638i	$-0.348348\pi$
$293$	15.7003	0.917219	0.458609	+	0.888638i	$0.348348\pi$
$294$	0	0
$295$	5.46217	0.318020
$296$	−16.5302	−0.960798
$297$	0	0
$298$	−9.41634	−0.545474
$299$	1.60184	0.0926367
$300$	0	0
$301$	10.9151	0.629137
$302$	−29.9263	−1.72207
$303$	0	0
$304$	6.95540	0.398920
$305$	62.6314	3.58626
$306$	0	0
$307$	23.2734	1.32828	0.664141	−	0.747607i	$-0.268797\pi$
$307$	23.2734	1.32828	0.664141	+	0.747607i	$0.268797\pi$
$308$	−0.425162	−0.0242258
$309$	0	0
$310$	39.2462	2.22903
$311$	4.08567	0.231677	0.115839	−	0.993268i	$-0.463044\pi$
$311$	4.08567	0.231677	0.115839	+	0.993268i	$0.463044\pi$
$312$	0	0
$313$	−19.8880	−1.12414	−0.562068	−	0.827091i	$-0.689994\pi$
$313$	−19.8880	−1.12414	−0.562068	+	0.827091i	$0.689994\pi$
$314$	−28.0105	−1.58073
$315$	0	0
$316$	0.495399	0.0278683
$317$	26.3501	1.47997	0.739984	−	0.672624i	$-0.234833\pi$
$317$	26.3501	1.47997	0.739984	+	0.672624i	$0.234833\pi$
$318$	0	0
$319$	−3.24281	−0.181563
$320$	−35.7408	−1.99797
$321$	0	0
$322$	−1.18347	−0.0659523
$323$	−15.0653	−0.838256
$324$	0	0
$325$	−20.2558	−1.12359
$326$	11.7671	0.651717
$327$	0	0
$328$	15.1244	0.835106
$329$	−3.68748	−0.203297
$330$	0	0
$331$	6.24238	0.343112	0.171556	−	0.985174i	$-0.445120\pi$
$331$	6.24238	0.343112	0.171556	+	0.985174i	$0.445120\pi$
$332$	−0.403956	−0.0221699
$333$	0	0
$334$	−7.92417	−0.433591
$335$	0.271026	0.0148077
$336$	0	0
$337$	20.7024	1.12773	0.563866	−	0.825866i	$-0.309313\pi$
$337$	20.7024	1.12773	0.563866	+	0.825866i	$0.309313\pi$
$338$	−14.1894	−0.771802
$339$	0	0
$340$	5.03882	0.273269
$341$	−22.2790	−1.20647
$342$	0	0
$343$	−11.5245	−0.622267
$344$	−36.6826	−1.97779
$345$	0	0
$346$	6.01216	0.323215
$347$	−2.61804	−0.140544	−0.0702719	−	0.997528i	$-0.522387\pi$
$347$	−2.61804	−0.140544	−0.0702719	+	0.997528i	$0.522387\pi$
$348$	0	0
$349$	−22.4939	−1.20407	−0.602036	−	0.798469i	$-0.705644\pi$
$349$	−22.4939	−1.20407	−0.602036	+	0.798469i	$0.705644\pi$
$350$	14.9654	0.799936
$351$	0	0
$352$	2.75760	0.146981
$353$	20.1237	1.07108	0.535539	−	0.844511i	$-0.320109\pi$
$353$	20.1237	1.07108	0.535539	+	0.844511i	$0.320109\pi$
$354$	0	0
$355$	−31.0948	−1.65034
$356$	−1.68345	−0.0892228
$357$	0	0
$358$	−25.8168	−1.36446
$359$	−9.07923	−0.479183	−0.239592	−	0.970874i	$-0.577014\pi$
$359$	−9.07923	−0.479183	−0.239592	+	0.970874i	$0.577014\pi$
$360$	0	0
$361$	−15.4199	−0.811574
$362$	−12.5122	−0.657625
$363$	0	0
$364$	0.210016	0.0110078
$365$	20.8673	1.09224
$366$	0	0
$367$	14.8257	0.773896	0.386948	−	0.922102i	$-0.373529\pi$
$367$	14.8257	0.773896	0.386948	+	0.922102i	$0.373529\pi$
$368$	3.67599	0.191624
$369$	0	0
$370$	−32.2866	−1.67850
$371$	5.82527	0.302433
$372$	0	0
$373$	25.9963	1.34604	0.673018	−	0.739627i	$-0.264998\pi$
$373$	25.9963	1.34604	0.673018	+	0.739627i	$0.264998\pi$
$374$	35.1124	1.81562
$375$	0	0
$376$	12.3925	0.639097
$377$	1.60184	0.0824989
$378$	0	0
$379$	−4.07417	−0.209276	−0.104638	−	0.994510i	$-0.533368\pi$
$379$	−4.07417	−0.209276	−0.104638	+	0.994510i	$0.533368\pi$
$380$	−1.19742	−0.0614264
$381$	0	0
$382$	29.9908	1.53446
$383$	8.15209	0.416552	0.208276	−	0.978070i	$-0.433215\pi$
$383$	8.15209	0.416552	0.208276	+	0.978070i	$0.433215\pi$
$384$	0	0
$385$	−11.8546	−0.604165
$386$	−2.56481	−0.130545
$387$	0	0
$388$	−0.246877	−0.0125333
$389$	6.48252	0.328677	0.164338	−	0.986404i	$-0.447451\pi$
$389$	6.48252	0.328677	0.164338	+	0.986404i	$0.447451\pi$
$390$	0	0
$391$	−7.96215	−0.402663
$392$	18.2578	0.922158
$393$	0	0
$394$	−13.4461	−0.677403
$395$	13.8129	0.695005
$396$	0	0
$397$	23.1797	1.16335	0.581677	−	0.813420i	$-0.302397\pi$
$397$	23.1797	1.16335	0.581677	+	0.813420i	$0.302397\pi$
$398$	−1.12153	−0.0562174
$399$	0	0
$400$	−46.4843	−2.32421
$401$	−19.8688	−0.992201	−0.496100	−	0.868265i	$-0.665235\pi$
$401$	−19.8688	−0.992201	−0.496100	+	0.868265i	$0.665235\pi$
$402$	0	0
$403$	11.0051	0.548201
$404$	2.28391	0.113629
$405$	0	0
$406$	−1.18347	−0.0587347
$407$	18.3282	0.908495
$408$	0	0
$409$	−14.3957	−0.711822	−0.355911	−	0.934520i	$-0.615829\pi$
$409$	−14.3957	−0.711822	−0.355911	+	0.934520i	$0.615829\pi$
$410$	29.5409	1.45892
$411$	0	0
$412$	2.05672	0.101327
$413$	−1.13161	−0.0556831
$414$	0	0
$415$	−11.2633	−0.552893
$416$	−1.36216	−0.0667854
$417$	0	0
$418$	−8.34408	−0.408122
$419$	−37.9245	−1.85273	−0.926367	−	0.376623i	$-0.877085\pi$
$419$	−37.9245	−1.85273	−0.926367	+	0.376623i	$0.877085\pi$
$420$	0	0
$421$	−25.7114	−1.25310	−0.626548	−	0.779382i	$-0.715533\pi$
$421$	−25.7114	−1.25310	−0.626548	+	0.779382i	$0.715533\pi$
$422$	−26.8642	−1.30773
$423$	0	0
$424$	−19.5770	−0.950745
$425$	100.684	4.88390
$426$	0	0
$427$	−12.9755	−0.627930
$428$	2.07265	0.100185
$429$	0	0
$430$	−71.6480	−3.45517
$431$	4.24671	0.204557	0.102278	−	0.994756i	$-0.467387\pi$
$431$	4.24671	0.204557	0.102278	+	0.994756i	$0.467387\pi$
$432$	0	0
$433$	−14.0188	−0.673699	−0.336849	−	0.941559i	$-0.609361\pi$
$433$	−14.0188	−0.673699	−0.336849	+	0.941559i	$0.609361\pi$
$434$	−8.13076	−0.390289
$435$	0	0
$436$	0.999711	0.0478775
$437$	1.89212	0.0905121
$438$	0	0
$439$	3.32357	0.158626	0.0793128	−	0.996850i	$-0.474727\pi$
$439$	3.32357	0.158626	0.0793128	+	0.996850i	$0.474727\pi$
$440$	39.8398	1.89929
$441$	0	0
$442$	−17.3443	−0.824987
$443$	−15.7380	−0.747735	−0.373868	−	0.927482i	$-0.621969\pi$
$443$	−15.7380	−0.747735	−0.373868	+	0.927482i	$0.621969\pi$
$444$	0	0
$445$	−46.9388	−2.22511
$446$	9.12409	0.432038
$447$	0	0
$448$	7.40453	0.349831
$449$	19.6901	0.929232	0.464616	−	0.885512i	$-0.346192\pi$
$449$	19.6901	0.929232	0.464616	+	0.885512i	$0.346192\pi$
$450$	0	0
$451$	−16.7695	−0.789646
$452$	0.659781	0.0310335
$453$	0	0
$454$	−4.49994	−0.211193
$455$	5.85576	0.274522
$456$	0	0
$457$	3.06929	0.143576	0.0717878	−	0.997420i	$-0.477130\pi$
$457$	3.06929	0.143576	0.0717878	+	0.997420i	$0.477130\pi$
$458$	−31.1775	−1.45683
$459$	0	0
$460$	−0.632848	−0.0295067
$461$	−28.8928	−1.34567	−0.672836	−	0.739792i	$-0.734924\pi$
$461$	−28.8928	−1.34567	−0.672836	+	0.739792i	$0.734924\pi$
$462$	0	0
$463$	−0.754421	−0.0350609	−0.0175305	−	0.999846i	$-0.505580\pi$
$463$	−0.754421	−0.0350609	−0.0175305	+	0.999846i	$0.505580\pi$
$464$	3.67599	0.170654
$465$	0	0
$466$	20.4415	0.946936
$467$	−32.5794	−1.50759	−0.753796	−	0.657108i	$-0.771780\pi$
$467$	−32.5794	−1.50759	−0.753796	+	0.657108i	$0.771780\pi$
$468$	0	0
$469$	−0.0561493	−0.00259273
$470$	24.2050	1.11649
$471$	0	0
$472$	3.80303	0.175048
$473$	40.6726	1.87013
$474$	0	0
$475$	−23.9265	−1.09782
$476$	−1.04391	−0.0478475
$477$	0	0
$478$	5.13234	0.234748
$479$	38.6620	1.76651	0.883255	−	0.468893i	$-0.155347\pi$
$479$	38.6620	1.76651	0.883255	+	0.468893i	$0.155347\pi$
$480$	0	0
$481$	−9.05351	−0.412805
$482$	21.6606	0.986615
$483$	0	0
$484$	0.0729429	0.00331559
$485$	−6.88354	−0.312565
$486$	0	0
$487$	12.2269	0.554051	0.277026	−	0.960862i	$-0.410651\pi$
$487$	12.2269	0.554051	0.277026	+	0.960862i	$0.410651\pi$
$488$	43.6070	1.97400
$489$	0	0
$490$	35.6610	1.61100
$491$	13.6997	0.618258	0.309129	−	0.951020i	$-0.399963\pi$
$491$	13.6997	0.618258	0.309129	+	0.951020i	$0.399963\pi$
$492$	0	0
$493$	−7.96215	−0.358597
$494$	4.12169	0.185444
$495$	0	0
$496$	25.2550	1.13398
$497$	6.44201	0.288963
$498$	0	0
$499$	−4.31795	−0.193298	−0.0966489	−	0.995319i	$-0.530812\pi$
$499$	−4.31795	−0.193298	−0.0966489	+	0.995319i	$0.530812\pi$
$500$	4.83835	0.216378
$501$	0	0
$502$	−28.1345	−1.25570
$503$	25.5037	1.13716	0.568578	−	0.822629i	$-0.307494\pi$
$503$	25.5037	1.13716	0.568578	+	0.822629i	$0.307494\pi$
$504$	0	0
$505$	63.6810	2.83377
$506$	−4.40992	−0.196045
$507$	0	0
$508$	1.36189	0.0604239
$509$	11.1100	0.492444	0.246222	−	0.969213i	$-0.420811\pi$
$509$	11.1100	0.492444	0.246222	+	0.969213i	$0.420811\pi$
$510$	0	0
$511$	−4.32314	−0.191244
$512$	−24.6282	−1.08842
$513$	0	0
$514$	0.888664	0.0391973
$515$	57.3465	2.52699
$516$	0	0
$517$	−13.7405	−0.604307
$518$	6.68892	0.293894
$519$	0	0
$520$	−19.6795	−0.863003
$521$	−15.0317	−0.658550	−0.329275	−	0.944234i	$-0.606804\pi$
$521$	−15.0317	−0.658550	−0.329275	+	0.944234i	$0.606804\pi$
$522$	0	0
$523$	39.8498	1.74251	0.871255	−	0.490830i	$-0.163307\pi$
$523$	39.8498	1.74251	0.871255	+	0.490830i	$0.163307\pi$
$524$	−0.491147	−0.0214559
$525$	0	0
$526$	16.7628	0.730894
$527$	−54.7020	−2.38286
$528$	0	0
$529$	1.00000	0.0434783
$530$	−38.2377	−1.66094
$531$	0	0
$532$	0.248073	0.0107553
$533$	8.28357	0.358801
$534$	0	0
$535$	57.7907	2.49851
$536$	0.188702	0.00815066
$537$	0	0
$538$	16.6182	0.716460
$539$	−20.2437	−0.871959
$540$	0	0
$541$	23.8204	1.02412	0.512059	−	0.858950i	$-0.328883\pi$
$541$	23.8204	1.02412	0.512059	+	0.858950i	$0.328883\pi$
$542$	−34.9548	−1.50144
$543$	0	0
$544$	6.77079	0.290295
$545$	27.8744	1.19401
$546$	0	0
$547$	−6.98119	−0.298494	−0.149247	−	0.988800i	$-0.547685\pi$
$547$	−6.98119	−0.298494	−0.149247	+	0.988800i	$0.547685\pi$
$548$	−2.34928	−0.100356
$549$	0	0
$550$	55.7651	2.37783
$551$	1.89212	0.0806068
$552$	0	0
$553$	−2.86167	−0.121691
$554$	−42.2543	−1.79521
$555$	0	0
$556$	−1.39775	−0.0592776
$557$	−26.5781	−1.12615	−0.563074	−	0.826406i	$-0.690382\pi$
$557$	−26.5781	−1.12615	−0.563074	+	0.826406i	$0.690382\pi$
$558$	0	0
$559$	−20.0909	−0.849753
$560$	13.4381	0.567865
$561$	0	0
$562$	−20.5766	−0.867973
$563$	−12.0275	−0.506898	−0.253449	−	0.967349i	$-0.581565\pi$
$563$	−12.0275	−0.506898	−0.253449	+	0.967349i	$0.581565\pi$
$564$	0	0
$565$	18.3963	0.773940
$566$	15.3524	0.645308
$567$	0	0
$568$	−21.6497	−0.908402
$569$	−15.0935	−0.632752	−0.316376	−	0.948634i	$-0.602466\pi$
$569$	−15.0935	−0.632752	−0.316376	+	0.948634i	$0.602466\pi$
$570$	0	0
$571$	−31.3282	−1.31105	−0.655523	−	0.755176i	$-0.727552\pi$
$571$	−31.3282	−1.31105	−0.655523	+	0.755176i	$0.727552\pi$
$572$	0.782572	0.0327210
$573$	0	0
$574$	−6.12007	−0.255447
$575$	−12.6454	−0.527348
$576$	0	0
$577$	−13.9233	−0.579637	−0.289818	−	0.957082i	$-0.593595\pi$
$577$	−13.9233	−0.579637	−0.289818	+	0.957082i	$0.593595\pi$
$578$	63.0939	2.62436
$579$	0	0
$580$	−0.632848	−0.0262776
$581$	2.33345	0.0968079
$582$	0	0
$583$	21.7065	0.898989
$584$	14.5288	0.601206
$585$	0	0
$586$	21.3509	0.881997
$587$	−33.5235	−1.38366	−0.691831	−	0.722060i	$-0.743196\pi$
$587$	−33.5235	−1.38366	−0.691831	+	0.722060i	$0.743196\pi$
$588$	0	0
$589$	12.9993	0.535628
$590$	7.42803	0.305807
$591$	0	0
$592$	−20.7765	−0.853910
$593$	5.89456	0.242061	0.121030	−	0.992649i	$-0.461380\pi$
$593$	5.89456	0.242061	0.121030	+	0.992649i	$0.461380\pi$
$594$	0	0
$595$	−29.1068	−1.19326
$596$	1.04317	0.0427301
$597$	0	0
$598$	2.17835	0.0890794
$599$	33.5549	1.37101	0.685507	−	0.728066i	$-0.259580\pi$
$599$	33.5549	1.37101	0.685507	+	0.728066i	$0.259580\pi$
$600$	0	0
$601$	−29.2001	−1.19110	−0.595549	−	0.803319i	$-0.703065\pi$
$601$	−29.2001	−1.19110	−0.595549	+	0.803319i	$0.703065\pi$
$602$	14.8435	0.604978
$603$	0	0
$604$	3.31534	0.134899
$605$	2.03383	0.0826870
$606$	0	0
$607$	−47.1693	−1.91454	−0.957272	−	0.289188i	$-0.906615\pi$
$607$	−47.1693	−1.91454	−0.957272	+	0.289188i	$0.906615\pi$
$608$	−1.60900	−0.0652537
$609$	0	0
$610$	85.1728	3.44854
$611$	6.78734	0.274587
$612$	0	0
$613$	42.5400	1.71818	0.859088	−	0.511829i	$-0.171032\pi$
$613$	42.5400	1.71818	0.859088	+	0.511829i	$0.171032\pi$
$614$	31.6496	1.27727
$615$	0	0
$616$	−8.25373	−0.332552
$617$	−44.3717	−1.78634	−0.893169	−	0.449722i	$-0.851523\pi$
$617$	−44.3717	−1.78634	−0.893169	+	0.449722i	$0.851523\pi$
$618$	0	0
$619$	−18.8365	−0.757102	−0.378551	−	0.925580i	$-0.623578\pi$
$619$	−18.8365	−0.757102	−0.378551	+	0.925580i	$0.623578\pi$
$620$	−4.34783	−0.174613
$621$	0	0
$622$	5.55613	0.222780
$623$	9.72447	0.389603
$624$	0	0
$625$	71.6785	2.86714
$626$	−27.0458	−1.08097
$627$	0	0
$628$	3.10310	0.123827
$629$	45.0016	1.79433
$630$	0	0
$631$	−37.2650	−1.48350	−0.741749	−	0.670678i	$-0.766003\pi$
$631$	−37.2650	−1.48350	−0.741749	+	0.670678i	$0.766003\pi$
$632$	9.61725	0.382554
$633$	0	0
$634$	35.8337	1.42314
$635$	37.9728	1.50690
$636$	0	0
$637$	9.99972	0.396203
$638$	−4.40992	−0.174590
$639$	0	0
$640$	−41.4599	−1.63885
$641$	8.03569	0.317391	0.158695	−	0.987328i	$-0.449271\pi$
$641$	8.03569	0.317391	0.158695	+	0.987328i	$0.449271\pi$
$642$	0	0
$643$	−39.4899	−1.55733	−0.778664	−	0.627441i	$-0.784103\pi$
$643$	−39.4899	−1.55733	−0.778664	+	0.627441i	$0.784103\pi$
$644$	0.131109	0.00516642
$645$	0	0
$646$	−20.4874	−0.806066
$647$	3.52753	0.138681	0.0693407	−	0.997593i	$-0.477910\pi$
$647$	3.52753	0.138681	0.0693407	+	0.997593i	$0.477910\pi$
$648$	0	0
$649$	−4.21669	−0.165519
$650$	−27.5461	−1.08045
$651$	0	0
$652$	−1.30360	−0.0510527
$653$	30.9353	1.21059	0.605295	−	0.796001i	$-0.293055\pi$
$653$	30.9353	1.21059	0.605295	+	0.796001i	$0.293055\pi$
$654$	0	0
$655$	−13.6944	−0.535085
$656$	19.0096	0.742201
$657$	0	0
$658$	−5.01463	−0.195491
$659$	−35.1637	−1.36978	−0.684891	−	0.728646i	$-0.740150\pi$
$659$	−35.1637	−1.36978	−0.684891	+	0.728646i	$0.740150\pi$
$660$	0	0
$661$	18.4718	0.718468	0.359234	−	0.933248i	$-0.383038\pi$
$661$	18.4718	0.718468	0.359234	+	0.933248i	$0.383038\pi$
$662$	8.48905	0.329936
$663$	0	0
$664$	−7.84205	−0.304331
$665$	6.91690	0.268226
$666$	0	0
$667$	1.00000	0.0387202
$668$	0.877867	0.0339657
$669$	0	0
$670$	0.368570	0.0142391
$671$	−48.3502	−1.86654
$672$	0	0
$673$	37.6630	1.45180	0.725902	−	0.687798i	$-0.241423\pi$
$673$	37.6630	1.45180	0.725902	+	0.687798i	$0.241423\pi$
$674$	28.1534	1.08443
$675$	0	0
$676$	1.57195	0.0604597
$677$	49.9213	1.91863	0.959315	−	0.282338i	$-0.0911101\pi$
$677$	49.9213	1.91863	0.959315	+	0.282338i	$0.0911101\pi$
$678$	0	0
$679$	1.42608	0.0547281
$680$	97.8195	3.75120
$681$	0	0
$682$	−30.2973	−1.16014
$683$	14.1168	0.540165	0.270083	−	0.962837i	$-0.412949\pi$
$683$	14.1168	0.540165	0.270083	+	0.962837i	$0.412949\pi$
$684$	0	0
$685$	−65.5038	−2.50277
$686$	−15.6723	−0.598371
$687$	0	0
$688$	−46.1057	−1.75776
$689$	−10.7223	−0.408485
$690$	0	0
$691$	42.4830	1.61613	0.808064	−	0.589095i	$-0.200516\pi$
$691$	42.4830	1.61613	0.808064	+	0.589095i	$0.200516\pi$
$692$	−0.666047	−0.0253193
$693$	0	0
$694$	−3.56029	−0.135147
$695$	−38.9726	−1.47832
$696$	0	0
$697$	−41.1746	−1.55960
$698$	−30.5896	−1.15783
$699$	0	0
$700$	−1.65792	−0.0626636
$701$	−4.16302	−0.157235	−0.0786175	−	0.996905i	$-0.525051\pi$
$701$	−4.16302	−0.157235	−0.0786175	+	0.996905i	$0.525051\pi$
$702$	0	0
$703$	−10.6941	−0.403337
$704$	27.5912	1.03988
$705$	0	0
$706$	27.3664	1.02995
$707$	−13.1930	−0.496174
$708$	0	0
$709$	22.4150	0.841812	0.420906	−	0.907104i	$-0.361712\pi$
$709$	22.4150	0.841812	0.420906	+	0.907104i	$0.361712\pi$
$710$	−42.2860	−1.58697
$711$	0	0
$712$	−32.6811	−1.22478
$713$	6.87026	0.257293
$714$	0	0
$715$	21.8201	0.816024
$716$	2.86007	0.106886
$717$	0	0
$718$	−12.3469	−0.460782
$719$	−6.23578	−0.232555	−0.116278	−	0.993217i	$-0.537096\pi$
$719$	−6.23578	−0.232555	−0.116278	+	0.993217i	$0.537096\pi$
$720$	0	0
$721$	−11.8807	−0.442459
$722$	−20.9696	−0.780408
$723$	0	0
$724$	1.38614	0.0515156
$725$	−12.6454	−0.469637
$726$	0	0
$727$	2.68791	0.0996891	0.0498446	−	0.998757i	$-0.484127\pi$
$727$	2.68791	0.0996891	0.0498446	+	0.998757i	$0.484127\pi$
$728$	4.07706	0.151106
$729$	0	0
$730$	28.3775	1.05030
$731$	99.8642	3.69361
$732$	0	0
$733$	−10.3347	−0.381720	−0.190860	−	0.981617i	$-0.561128\pi$
$733$	−10.3347	−0.381720	−0.190860	+	0.981617i	$0.561128\pi$
$734$	20.1616	0.744178
$735$	0	0
$736$	−0.850373	−0.0313452
$737$	−0.209227	−0.00770697
$738$	0	0
$739$	−7.45273	−0.274153	−0.137077	−	0.990560i	$-0.543771\pi$
$739$	−7.45273	−0.274153	−0.137077	+	0.990560i	$0.543771\pi$
$740$	3.57682	0.131487
$741$	0	0
$742$	7.92182	0.290819
$743$	−17.7690	−0.651882	−0.325941	−	0.945390i	$-0.605681\pi$
$743$	−17.7690	−0.651882	−0.325941	+	0.945390i	$0.605681\pi$
$744$	0	0
$745$	29.0863	1.06564
$746$	35.3525	1.29435
$747$	0	0
$748$	−3.88987	−0.142228
$749$	−11.9727	−0.437473
$750$	0	0
$751$	20.0716	0.732424	0.366212	−	0.930531i	$-0.380654\pi$
$751$	20.0716	0.732424	0.366212	+	0.930531i	$0.380654\pi$
$752$	15.5760	0.567998
$753$	0	0
$754$	2.17835	0.0793309
$755$	92.4400	3.36424
$756$	0	0
$757$	−37.4069	−1.35958	−0.679788	−	0.733409i	$-0.737928\pi$
$757$	−37.4069	−1.35958	−0.679788	+	0.733409i	$0.737928\pi$
$758$	−5.54049	−0.201240
$759$	0	0
$760$	−23.2457	−0.843210
$761$	−26.5113	−0.961033	−0.480516	−	0.876986i	$-0.659551\pi$
$761$	−26.5113	−0.961033	−0.480516	+	0.876986i	$0.659551\pi$
$762$	0	0
$763$	−5.77484	−0.209063
$764$	−3.32249	−0.120203
$765$	0	0
$766$	11.0861	0.400556
$767$	2.08290	0.0752092
$768$	0	0
$769$	25.5111	0.919955	0.459978	−	0.887931i	$-0.347857\pi$
$769$	25.5111	0.919955	0.459978	+	0.887931i	$0.347857\pi$
$770$	−16.1211	−0.580964
$771$	0	0
$772$	0.284138	0.0102264
$773$	−0.0409421	−0.00147259	−0.000736293	−	1.00000i	$-0.500234\pi$
$773$	−0.0409421	−0.00147259	−0.000736293		1.00000i	$0.500234\pi$
$774$	0	0
$775$	−86.8770	−3.12071
$776$	−4.79265	−0.172046
$777$	0	0
$778$	8.81562	0.316055
$779$	9.78467	0.350572
$780$	0	0
$781$	24.0046	0.858951
$782$	−10.8278	−0.387200
$783$	0	0
$784$	22.9479	0.819569
$785$	86.5223	3.08811
$786$	0	0
$787$	33.4211	1.19133	0.595666	−	0.803232i	$-0.296888\pi$
$787$	33.4211	1.19133	0.595666	+	0.803232i	$0.296888\pi$
$788$	1.48960	0.0530648
$789$	0	0
$790$	18.7843	0.668316
$791$	−3.81123	−0.135512
$792$	0	0
$793$	23.8834	0.848123
$794$	31.5222	1.11868
$795$	0	0
$796$	0.124247	0.00440383
$797$	47.4657	1.68132	0.840661	−	0.541562i	$-0.182167\pi$
$797$	47.4657	1.68132	0.840661	+	0.541562i	$0.182167\pi$
$798$	0	0
$799$	−33.7374	−1.19354
$800$	10.7533	0.380186
$801$	0	0
$802$	−27.0197	−0.954099
$803$	−16.1091	−0.568479
$804$	0	0
$805$	3.65565	0.128845
$806$	14.9658	0.527149
$807$	0	0
$808$	44.3378	1.55980
$809$	−45.6804	−1.60604	−0.803019	−	0.595953i	$-0.796775\pi$
$809$	−45.6804	−1.60604	−0.803019	+	0.595953i	$0.796775\pi$
$810$	0	0
$811$	12.6623	0.444633	0.222316	−	0.974975i	$-0.428638\pi$
$811$	12.6623	0.444633	0.222316	+	0.974975i	$0.428638\pi$
$812$	0.131109	0.00460103
$813$	0	0
$814$	24.9246	0.873608
$815$	−36.3475	−1.27320
$816$	0	0
$817$	−23.7316	−0.830264
$818$	−19.5768	−0.684488
$819$	0	0
$820$	−3.27264	−0.114285
$821$	−15.1497	−0.528727	−0.264363	−	0.964423i	$-0.585162\pi$
$821$	−15.1497	−0.528727	−0.264363	+	0.964423i	$0.585162\pi$
$822$	0	0
$823$	−33.7831	−1.17761	−0.588803	−	0.808276i	$-0.700401\pi$
$823$	−33.7831	−1.17761	−0.588803	+	0.808276i	$0.700401\pi$
$824$	39.9274	1.39094
$825$	0	0
$826$	−1.53889	−0.0535448
$827$	−35.9126	−1.24880	−0.624402	−	0.781103i	$-0.714657\pi$
$827$	−35.9126	−1.24880	−0.624402	+	0.781103i	$0.714657\pi$
$828$	0	0
$829$	−23.8358	−0.827853	−0.413927	−	0.910310i	$-0.635843\pi$
$829$	−23.8358	−0.827853	−0.413927	+	0.910310i	$0.635843\pi$
$830$	−15.3170	−0.531661
$831$	0	0
$832$	−13.6291	−0.472504
$833$	−49.7049	−1.72217
$834$	0	0
$835$	24.4771	0.847065
$836$	0.924385	0.0319705
$837$	0	0
$838$	−51.5738	−1.78159
$839$	−39.5267	−1.36461	−0.682307	−	0.731066i	$-0.739023\pi$
$839$	−39.5267	−1.36461	−0.682307	+	0.731066i	$0.739023\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−34.9651	−1.20498
$843$	0	0
$844$	2.97611	0.102442
$845$	43.8299	1.50780
$846$	0	0
$847$	−0.421355	−0.0144779
$848$	−24.6060	−0.844975
$849$	0	0
$850$	136.921	4.69636
$851$	−5.65195	−0.193746
$852$	0	0
$853$	−10.7801	−0.369105	−0.184552	−	0.982823i	$-0.559084\pi$
$853$	−10.7801	−0.369105	−0.184552	+	0.982823i	$0.559084\pi$
$854$	−17.6455	−0.603817
$855$	0	0
$856$	40.2367	1.37526
$857$	−34.2748	−1.17081	−0.585403	−	0.810743i	$-0.699064\pi$
$857$	−34.2748	−1.17081	−0.585403	+	0.810743i	$0.699064\pi$
$858$	0	0
$859$	−5.67545	−0.193644	−0.0968220	−	0.995302i	$-0.530868\pi$
$859$	−5.67545	−0.193644	−0.0968220	+	0.995302i	$0.530868\pi$
$860$	7.93741	0.270664
$861$	0	0
$862$	5.77513	0.196702
$863$	2.77819	0.0945708	0.0472854	−	0.998881i	$-0.484943\pi$
$863$	2.77819	0.0945708	0.0472854	+	0.998881i	$0.484943\pi$
$864$	0	0
$865$	−18.5711	−0.631435
$866$	−19.0642	−0.647828
$867$	0	0
$868$	0.900753	0.0305736
$869$	−10.6633	−0.361729
$870$	0	0
$871$	0.103351	0.00350191
$872$	19.4075	0.657222
$873$	0	0
$874$	2.57310	0.0870364
$875$	−27.9488	−0.944841
$876$	0	0
$877$	28.3972	0.958906	0.479453	−	0.877567i	$-0.340835\pi$
$877$	28.3972	0.958906	0.479453	+	0.877567i	$0.340835\pi$
$878$	4.51975	0.152534
$879$	0	0
$880$	50.0739	1.68799
$881$	32.6430	1.09977	0.549886	−	0.835240i	$-0.314671\pi$
$881$	32.6430	1.09977	0.549886	+	0.835240i	$0.314671\pi$
$882$	0	0
$883$	−32.7949	−1.10363	−0.551817	−	0.833965i	$-0.686065\pi$
$883$	−32.7949	−1.10363	−0.551817	+	0.833965i	$0.686065\pi$
$884$	1.92147	0.0646259
$885$	0	0
$886$	−21.4022	−0.719022
$887$	21.1685	0.710769	0.355384	−	0.934720i	$-0.384350\pi$
$887$	21.1685	0.710769	0.355384	+	0.934720i	$0.384350\pi$
$888$	0	0
$889$	−7.86694	−0.263849
$890$	−63.8324	−2.13967
$891$	0	0
$892$	−1.01080	−0.0338440
$893$	8.01731	0.268289
$894$	0	0
$895$	79.7459	2.66561
$896$	8.58937	0.286951
$897$	0	0
$898$	26.7766	0.893548
$899$	6.87026	0.229136
$900$	0	0
$901$	53.2963	1.77556
$902$	−22.8050	−0.759322
$903$	0	0
$904$	12.8084	0.426002
$905$	38.6491	1.28474
$906$	0	0
$907$	−46.1813	−1.53343	−0.766713	−	0.641990i	$-0.778109\pi$
$907$	−46.1813	−1.53343	−0.766713	+	0.641990i	$0.778109\pi$
$908$	0.498518	0.0165439
$909$	0	0
$910$	7.96328	0.263980
$911$	−13.8496	−0.458857	−0.229429	−	0.973325i	$-0.573686\pi$
$911$	−13.8496	−0.458857	−0.229429	+	0.973325i	$0.573686\pi$
$912$	0	0
$913$	8.69504	0.287764
$914$	4.17395	0.138062
$915$	0	0
$916$	3.45395	0.114122
$917$	2.83711	0.0936897
$918$	0	0
$919$	31.9148	1.05277	0.526386	−	0.850246i	$-0.323547\pi$
$919$	31.9148	1.05277	0.526386	+	0.850246i	$0.323547\pi$
$920$	−12.2856	−0.405043
$921$	0	0
$922$	−39.2915	−1.29400
$923$	−11.8574	−0.390293
$924$	0	0
$925$	71.4710	2.34995
$926$	−1.02594	−0.0337146
$927$	0	0
$928$	−0.850373	−0.0279149
$929$	35.3248	1.15897	0.579485	−	0.814983i	$-0.303254\pi$
$929$	35.3248	1.15897	0.579485	+	0.814983i	$0.303254\pi$
$930$	0	0
$931$	11.8118	0.387116
$932$	−2.26458	−0.0741789
$933$	0	0
$934$	−44.3049	−1.44970
$935$	−108.459	−3.54700
$936$	0	0
$937$	9.47258	0.309456	0.154728	−	0.987957i	$-0.450550\pi$
$937$	9.47258	0.309456	0.154728	+	0.987957i	$0.450550\pi$
$938$	−0.0763578	−0.00249317
$939$	0	0
$940$	−2.68151	−0.0874613
$941$	36.1900	1.17976	0.589880	−	0.807491i	$-0.299175\pi$
$941$	36.1900	1.17976	0.589880	+	0.807491i	$0.299175\pi$
$942$	0	0
$943$	5.17129	0.168400
$944$	4.77996	0.155574
$945$	0	0
$946$	55.3109	1.79831
$947$	20.6534	0.671146	0.335573	−	0.942014i	$-0.391070\pi$
$947$	20.6534	0.671146	0.335573	+	0.942014i	$0.391070\pi$
$948$	0	0
$949$	7.95736	0.258307
$950$	−32.5378	−1.05567
$951$	0	0
$952$	−20.2656	−0.656811
$953$	10.0858	0.326710	0.163355	−	0.986567i	$-0.447768\pi$
$953$	10.0858	0.326710	0.163355	+	0.986567i	$0.447768\pi$
$954$	0	0
$955$	−92.6392	−2.99774
$956$	−0.568578	−0.0183891
$957$	0	0
$958$	52.5766	1.69867
$959$	13.5706	0.438219
$960$	0	0
$961$	16.2005	0.522597
$962$	−12.3119	−0.396952
$963$	0	0
$964$	−2.39964	−0.0772872
$965$	7.92248	0.255034
$966$	0	0
$967$	38.0671	1.22416	0.612078	−	0.790798i	$-0.290334\pi$
$967$	38.0671	1.22416	0.612078	+	0.790798i	$0.290334\pi$
$968$	1.41605	0.0455136
$969$	0	0
$970$	−9.36096	−0.300562
$971$	−32.0663	−1.02906	−0.514528	−	0.857473i	$-0.672033\pi$
$971$	−32.0663	−1.02906	−0.514528	+	0.857473i	$0.672033\pi$
$972$	0	0
$973$	8.07408	0.258843
$974$	16.6274	0.532775
$975$	0	0
$976$	54.8089	1.75439
$977$	−10.9317	−0.349737	−0.174868	−	0.984592i	$-0.555950\pi$
$977$	−10.9317	−0.349737	−0.174868	+	0.984592i	$0.555950\pi$
$978$	0	0
$979$	36.2359	1.15810
$980$	−3.95064	−0.126199
$981$	0	0
$982$	18.6303	0.594516
$983$	−46.5709	−1.48538	−0.742690	−	0.669635i	$-0.766450\pi$
$983$	−46.5709	−1.48538	−0.742690	+	0.669635i	$0.766450\pi$
$984$	0	0
$985$	41.5338	1.32338
$986$	−10.8278	−0.344827
$987$	0	0
$988$	−0.456615	−0.0145269
$989$	−12.5424	−0.398824
$990$	0	0
$991$	−4.69862	−0.149257	−0.0746283	−	0.997211i	$-0.523777\pi$
$991$	−4.69862	−0.149257	−0.0746283	+	0.997211i	$0.523777\pi$
$992$	−5.84228	−0.185493
$993$	0	0
$994$	8.76053	0.277867
$995$	3.46433	0.109827
$996$	0	0
$997$	−12.7393	−0.403458	−0.201729	−	0.979441i	$-0.564656\pi$
$997$	−12.7393	−0.403458	−0.201729	+	0.979441i	$0.564656\pi$
$998$	−5.87200	−0.185875
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.u.1.16	yes	22
3.2	odd	2		6003.2.a.t.1.7	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.7	✓	22	3.2	odd	2
6003.2.a.u.1.16	yes	22	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35991 q^{2} -0.150655 q^{4} -4.20064 q^{5} +0.870260 q^{7} -2.92469 q^{8} +O(q^{10})\)	\(q+1.35991 q^{2} -0.150655 q^{4} -4.20064 q^{5} +0.870260 q^{7} -2.92469 q^{8} -5.71248 q^{10} +3.24281 q^{11} -1.60184 q^{13} +1.18347 q^{14} -3.67599 q^{16} +7.96215 q^{17} -1.89212 q^{19} +0.632848 q^{20} +4.40992 q^{22} -1.00000 q^{23} +12.6454 q^{25} -2.17835 q^{26} -0.131109 q^{28} -1.00000 q^{29} -6.87026 q^{31} +0.850373 q^{32} +10.8278 q^{34} -3.65565 q^{35} +5.65195 q^{37} -2.57310 q^{38} +12.2856 q^{40} -5.17129 q^{41} +12.5424 q^{43} -0.488546 q^{44} -1.35991 q^{46} -4.23722 q^{47} -6.24265 q^{49} +17.1965 q^{50} +0.241325 q^{52} +6.69371 q^{53} -13.6219 q^{55} -2.54524 q^{56} -1.35991 q^{58} -1.30032 q^{59} -14.9100 q^{61} -9.34291 q^{62} +8.50841 q^{64} +6.72875 q^{65} -0.0645202 q^{67} -1.19954 q^{68} -4.97134 q^{70} +7.40240 q^{71} -4.96764 q^{73} +7.68612 q^{74} +0.285057 q^{76} +2.82209 q^{77} -3.28830 q^{79} +15.4415 q^{80} -7.03247 q^{82} +2.68133 q^{83} -33.4461 q^{85} +17.0565 q^{86} -9.48422 q^{88} +11.1742 q^{89} -1.39402 q^{91} +0.150655 q^{92} -5.76222 q^{94} +7.94809 q^{95} +1.63869 q^{97} -8.48942 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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