LMFDB - Embedded newform 6003.2.a.t.1.7

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			1.7
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	$-0.659653\pi$
$2$	−1.35991	−0.961599	−0.480799	+	0.876831i	$0.659653\pi$
$3$	0	0
$3$	0	0
$4$	−0.150655	−0.0753275
$5$	4.20064	1.87858	0.939291	−	0.343120i	$-0.111484\pi$
$5$	4.20064	1.87858	0.939291	+	0.343120i	$0.111484\pi$
$6$	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	$-0.662592\pi$
$11$	−3.24281	−0.977745	−0.488872	+	0.872355i	$0.662592\pi$
$12$	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	$-0.916208\pi$
$17$	−7.96215	−1.93110	−0.965552	+	0.260210i	$0.916208\pi$
$18$	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	$-0.367686\pi$
$41$	5.17129	0.807620	0.403810	+	0.914843i	$0.367686\pi$
$42$	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	$-0.399995\pi$
$47$	4.23722	0.618062	0.309031	+	0.951052i	$0.399995\pi$
$48$	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	$-0.652052\pi$
$53$	−6.69371	−0.919452	−0.459726	+	0.888061i	$0.652052\pi$
$54$	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	$-0.473025\pi$
$59$	1.30032	0.169287	0.0846435	+	0.996411i	$0.473025\pi$
$60$	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	$-0.644756\pi$
$71$	−7.40240	−0.878503	−0.439251	+	0.898364i	$0.644756\pi$
$72$	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	$-0.547012\pi$
$83$	−2.68133	−0.294314	−0.147157	+	0.989113i	$0.547012\pi$
$84$	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	$-0.701753\pi$
$89$	−11.1742	−1.18446	−0.592232	+	0.805767i	$0.701753\pi$
$90$	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	$-0.228010\pi$
$101$	15.1598	1.50846	0.754230	+	0.656610i	$0.228010\pi$
$102$	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	$-0.268432\pi$
$107$	13.7576	1.33000	0.664999	+	0.746844i	$0.268432\pi$
$108$	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	$-0.433958\pi$
$113$	4.37941	0.411981	0.205990	+	0.978554i	$0.433958\pi$
$114$	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	$-0.545487\pi$
$131$	−3.26008	−0.284834	−0.142417	+	0.989807i	$0.545487\pi$
$132$	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	$-0.732052\pi$
$137$	−15.5938	−1.33227	−0.666133	+	0.745833i	$0.732052\pi$
$138$	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	$-0.408462\pi$
$149$	6.92426	0.567257	0.283629	+	0.958934i	$0.408462\pi$
$150$	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	$-0.427614\pi$
$167$	5.82700	0.450907	0.225453	+	0.974254i	$0.427614\pi$
$168$	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	$-0.553751\pi$
$173$	−4.42101	−0.336123	−0.168061	+	0.985777i	$0.553751\pi$
$174$	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	$-0.248933\pi$
$179$	18.9842	1.41895	0.709474	+	0.704732i	$0.248933\pi$
$180$	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	$-0.794041\pi$
$191$	−22.0536	−1.59574	−0.797871	+	0.602828i	$0.794041\pi$
$192$	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	$-0.385424\pi$
$197$	9.88749	0.704455	0.352227	+	0.935914i	$0.385424\pi$
$198$	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	$-0.464975\pi$
$227$	3.30901	0.219626	0.109813	+	0.993952i	$0.464975\pi$
$228$	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	$-0.663871\pi$
$233$	−15.0316	−0.984752	−0.492376	+	0.870383i	$0.663871\pi$
$234$	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	$-0.538950\pi$
$239$	−3.77404	−0.244122	−0.122061	+	0.992523i	$0.538950\pi$
$240$	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	$-0.273542\pi$
$251$	20.6886	1.30585	0.652925	+	0.757423i	$0.273542\pi$
$252$	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	$-0.506488\pi$
$257$	−0.653474	−0.0407626	−0.0203813	+	0.999792i	$0.506488\pi$
$258$	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	$-0.624090\pi$
$263$	−12.3265	−0.760082	−0.380041	+	0.924970i	$0.624090\pi$
$264$	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	$-0.621512\pi$
$269$	−12.2201	−0.745071	−0.372536	+	0.928018i	$0.621512\pi$
$270$	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	$-0.350954\pi$
$281$	15.1309	0.902635	0.451317	+	0.892363i	$0.350954\pi$
$282$	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	$-0.651652\pi$
$293$	−15.7003	−0.917219	−0.458609	+	0.888638i	$0.651652\pi$
$294$	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	$-0.536956\pi$
$311$	−4.08567	−0.231677	−0.115839	+	0.993268i	$0.536956\pi$
$312$	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	$-0.765167\pi$
$317$	−26.3501	−1.47997	−0.739984	+	0.672624i	$0.765167\pi$
$318$	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	$-0.477613\pi$
$347$	2.61804	0.140544	0.0702719	+	0.997528i	$0.477613\pi$
$348$	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	$-0.679891\pi$
$353$	−20.1237	−1.07108	−0.535539	+	0.844511i	$0.679891\pi$
$354$	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	$-0.422986\pi$
$359$	9.07923	0.479183	0.239592	+	0.970874i	$0.422986\pi$
$360$	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	$-0.566785\pi$
$383$	−8.15209	−0.416552	−0.208276	+	0.978070i	$0.566785\pi$
$384$	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	$-0.552549\pi$
$389$	−6.48252	−0.328677	−0.164338	+	0.986404i	$0.552549\pi$
$390$	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	$-0.334765\pi$
$401$	19.8688	0.992201	0.496100	+	0.868265i	$0.334765\pi$
$402$	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	$-0.122915\pi$
$419$	37.9245	1.85273	0.926367	+	0.376623i	$0.122915\pi$
$420$	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	$-0.532613\pi$
$431$	−4.24671	−0.204557	−0.102278	+	0.994756i	$0.532613\pi$
$432$	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	$-0.378031\pi$
$443$	15.7380	0.747735	0.373868	+	0.927482i	$0.378031\pi$
$444$	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	$-0.653808\pi$
$449$	−19.6901	−0.929232	−0.464616	+	0.885512i	$0.653808\pi$
$450$	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	$-0.265076\pi$
$461$	28.8928	1.34567	0.672836	+	0.739792i	$0.265076\pi$
$462$	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	$-0.228220\pi$
$467$	32.5794	1.50759	0.753796	+	0.657108i	$0.228220\pi$
$468$	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	$-0.844653\pi$
$479$	−38.6620	−1.76651	−0.883255	+	0.468893i	$0.844653\pi$
$480$	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	$-0.600037\pi$
$491$	−13.6997	−0.618258	−0.309129	+	0.951020i	$0.600037\pi$
$492$	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	$-0.692506\pi$
$503$	−25.5037	−1.13716	−0.568578	+	0.822629i	$0.692506\pi$
$504$	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	$-0.579189\pi$
$509$	−11.1100	−0.492444	−0.246222	+	0.969213i	$0.579189\pi$
$510$	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	$-0.393196\pi$
$521$	15.0317	0.658550	0.329275	+	0.944234i	$0.393196\pi$
$522$	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	$-0.309618\pi$
$557$	26.5781	1.12615	0.563074	+	0.826406i	$0.309618\pi$
$558$	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	$-0.418435\pi$
$563$	12.0275	0.506898	0.253449	+	0.967349i	$0.418435\pi$
$564$	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	$-0.397534\pi$
$569$	15.0935	0.632752	0.316376	+	0.948634i	$0.397534\pi$
$570$	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	$-0.256804\pi$
$587$	33.5235	1.38366	0.691831	+	0.722060i	$0.256804\pi$
$588$	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	$-0.538620\pi$
$593$	−5.89456	−0.242061	−0.121030	+	0.992649i	$0.538620\pi$
$594$	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	$-0.740420\pi$
$599$	−33.5549	−1.37101	−0.685507	+	0.728066i	$0.740420\pi$
$600$	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	$-0.148477\pi$
$617$	44.3717	1.78634	0.893169	+	0.449722i	$0.148477\pi$
$618$	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	$-0.550729\pi$
$641$	−8.03569	−0.317391	−0.158695	+	0.987328i	$0.550729\pi$
$642$	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	$-0.522090\pi$
$647$	−3.52753	−0.138681	−0.0693407	+	0.997593i	$0.522090\pi$
$648$	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	$-0.706945\pi$
$653$	−30.9353	−1.21059	−0.605295	+	0.796001i	$0.706945\pi$
$654$	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	$-0.259850\pi$
$659$	35.1637	1.36978	0.684891	+	0.728646i	$0.259850\pi$
$660$	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	$-0.908890\pi$
$677$	−49.9213	−1.91863	−0.959315	+	0.282338i	$0.908890\pi$
$678$	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	$-0.587051\pi$
$683$	−14.1168	−0.540165	−0.270083	+	0.962837i	$0.587051\pi$
$684$	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	$-0.474949\pi$
$701$	4.16302	0.157235	0.0786175	+	0.996905i	$0.474949\pi$
$702$	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	$-0.462904\pi$
$719$	6.23578	0.232555	0.116278	+	0.993217i	$0.462904\pi$
$720$	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	$-0.394319\pi$
$743$	17.7690	0.651882	0.325941	+	0.945390i	$0.394319\pi$
$744$	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	$-0.340449\pi$
$761$	26.5113	0.961033	0.480516	+	0.876986i	$0.340449\pi$
$762$	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	$-0.499766\pi$
$773$	0.0409421	0.00147259	0.000736293		1.00000i	$0.499766\pi$
$774$	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	$-0.817833\pi$
$797$	−47.4657	−1.68132	−0.840661	+	0.541562i	$0.817833\pi$
$798$	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	$-0.203225\pi$
$809$	45.6804	1.60604	0.803019	+	0.595953i	$0.203225\pi$
$810$	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	$-0.414838\pi$
$821$	15.1497	0.528727	0.264363	+	0.964423i	$0.414838\pi$
$822$	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	$-0.285343\pi$
$827$	35.9126	1.24880	0.624402	+	0.781103i	$0.285343\pi$
$828$	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	$-0.260977\pi$
$839$	39.5267	1.36461	0.682307	+	0.731066i	$0.260977\pi$
$840$	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	$-0.300936\pi$
$857$	34.2748	1.17081	0.585403	+	0.810743i	$0.300936\pi$
$858$	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	$-0.515057\pi$
$863$	−2.77819	−0.0945708	−0.0472854	+	0.998881i	$0.515057\pi$
$864$	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	$-0.685329\pi$
$881$	−32.6430	−1.09977	−0.549886	+	0.835240i	$0.685329\pi$
$882$	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	$-0.615650\pi$
$887$	−21.1685	−0.710769	−0.355384	+	0.934720i	$0.615650\pi$
$888$	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	$-0.426314\pi$
$911$	13.8496	0.458857	0.229429	+	0.973325i	$0.426314\pi$
$912$	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	$-0.696746\pi$
$929$	−35.3248	−1.15897	−0.579485	+	0.814983i	$0.696746\pi$
$930$	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	$-0.700825\pi$
$941$	−36.1900	−1.17976	−0.589880	+	0.807491i	$0.700825\pi$
$942$	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	$-0.608930\pi$
$947$	−20.6534	−0.671146	−0.335573	+	0.942014i	$0.608930\pi$
$948$	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	$-0.552232\pi$
$953$	−10.0858	−0.326710	−0.163355	+	0.986567i	$0.552232\pi$
$954$	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	$-0.327967\pi$
$971$	32.0663	1.02906	0.514528	+	0.857473i	$0.327967\pi$
$972$	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	$-0.444050\pi$
$977$	10.9317	0.349737	0.174868	+	0.984592i	$0.444050\pi$
$978$	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	$-0.233550\pi$
$983$	46.5709	1.48538	0.742690	+	0.669635i	$0.233550\pi$
$984$	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.t.1.7	✓	22
3.2	odd	2		6003.2.a.u.1.16	yes	22

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

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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