LMFDB - Embedded Newform 4003.2.a.a.1.2

Newspace parameters

Level:	$ N $	=	$ 4003 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	4003.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	$31.9641159291$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$
Character	$\chi$	=	4003.1

$q$-expansion

$f(q)$	$=$	$q+1.41421 q^{2} +1.41421 q^{3} +1.41421 q^{5} +2.00000 q^{6} -1.00000 q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.41421 q^{3} +1.41421 q^{5} +2.00000 q^{6} -1.00000 q^{7} -2.82843 q^{8} -1.00000 q^{9} +2.00000 q^{10} -4.82843 q^{11} +6.82843 q^{13} -1.41421 q^{14} +2.00000 q^{15} -4.00000 q^{16} -6.65685 q^{17} -1.41421 q^{18} +6.65685 q^{19} -1.41421 q^{21} -6.82843 q^{22} -3.17157 q^{23} -4.00000 q^{24} -3.00000 q^{25} +9.65685 q^{26} -5.65685 q^{27} -6.82843 q^{29} +2.82843 q^{30} -6.48528 q^{31} -6.82843 q^{33} -9.41421 q^{34} -1.41421 q^{35} -9.82843 q^{37} +9.41421 q^{38} +9.65685 q^{39} -4.00000 q^{40} +10.8284 q^{41} -2.00000 q^{42} -1.41421 q^{45} -4.48528 q^{46} +11.0711 q^{47} -5.65685 q^{48} -6.00000 q^{49} -4.24264 q^{50} -9.41421 q^{51} -13.3137 q^{53} -8.00000 q^{54} -6.82843 q^{55} +2.82843 q^{56} +9.41421 q^{57} -9.65685 q^{58} -3.17157 q^{59} +12.8284 q^{61} -9.17157 q^{62} +1.00000 q^{63} +8.00000 q^{64} +9.65685 q^{65} -9.65685 q^{66} +1.34315 q^{67} -4.48528 q^{69} -2.00000 q^{70} +6.00000 q^{71} +2.82843 q^{72} -3.00000 q^{73} -13.8995 q^{74} -4.24264 q^{75} +4.82843 q^{77} +13.6569 q^{78} -5.48528 q^{79} -5.65685 q^{80} -5.00000 q^{81} +15.3137 q^{82} -11.1421 q^{83} -9.41421 q^{85} -9.65685 q^{87} +13.6569 q^{88} -6.17157 q^{89} -2.00000 q^{90} -6.82843 q^{91} -9.17157 q^{93} +15.6569 q^{94} +9.41421 q^{95} +10.2426 q^{97} -8.48528 q^{98} +4.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q + 4q^{6} - 2q^{7} - 2q^{9} + O(q^{10}) $	$ 2q + 4q^{6} - 2q^{7} - 2q^{9} + 4q^{10} - 4q^{11} + 8q^{13} + 4q^{15} - 8q^{16} - 2q^{17} + 2q^{19} - 8q^{22} - 12q^{23} - 8q^{24} - 6q^{25} + 8q^{26} - 8q^{29} + 4q^{31} - 8q^{33} - 16q^{34} - 14q^{37} + 16q^{38} + 8q^{39} - 8q^{40} + 16q^{41} - 4q^{42} + 8q^{46} + 8q^{47} - 12q^{49} - 16q^{51} - 4q^{53} - 16q^{54} - 8q^{55} + 16q^{57} - 8q^{58} - 12q^{59} + 20q^{61} - 24q^{62} + 2q^{63} + 16q^{64} + 8q^{65} - 8q^{66} + 14q^{67} + 8q^{69} - 4q^{70} + 12q^{71} - 6q^{73} - 8q^{74} + 4q^{77} + 16q^{78} + 6q^{79} - 10q^{81} + 8q^{82} + 6q^{83} - 16q^{85} - 8q^{87} + 16q^{88} - 18q^{89} - 4q^{90} - 8q^{91} - 24q^{93} + 20q^{94} + 16q^{95} + 12q^{97} + 4q^{99} + O(q^{100}) $

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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−2.82843	−1.00000
$9$	−1.00000	−0.333333
$10$	2.00000	0.632456
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	6.82843	1.89386	0.946932	−	0.321433i	$-0.104164\pi$
$13$	6.82843	1.89386	0.946932	+	0.321433i	$0.104164\pi$
$14$	−1.41421	−0.377964
$15$	2.00000	0.516398
$16$	−4.00000	−1.00000
$17$	−6.65685	−1.61452	−0.807262	−	0.590193i	$-0.799052\pi$
$17$	−6.65685	−1.61452	−0.807262	+	0.590193i	$0.799052\pi$
$18$	−1.41421	−0.333333
$19$	6.65685	1.52719	0.763594	−	0.645697i	$-0.223433\pi$
$19$	6.65685	1.52719	0.763594	+	0.645697i	$0.223433\pi$
$20$	0	0
$21$	−1.41421	−0.308607
$22$	−6.82843	−1.45583
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	−4.00000	−0.816497
$25$	−3.00000	−0.600000
$26$	9.65685	1.89386
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−6.82843	−1.26801	−0.634004	−	0.773330i	$-0.718590\pi$
$29$	−6.82843	−1.26801	−0.634004	+	0.773330i	$0.718590\pi$
$30$	2.82843	0.516398
$31$	−6.48528	−1.16479	−0.582395	−	0.812906i	$-0.697884\pi$
$31$	−6.48528	−1.16479	−0.582395	+	0.812906i	$0.697884\pi$
$32$	0	0
$33$	−6.82843	−1.18868
$34$	−9.41421	−1.61452
$35$	−1.41421	−0.239046
$36$	0	0
$37$	−9.82843	−1.61578	−0.807892	−	0.589331i	$-0.799391\pi$
$37$	−9.82843	−1.61578	−0.807892	+	0.589331i	$0.799391\pi$
$38$	9.41421	1.52719
$39$	9.65685	1.54633
$40$	−4.00000	−0.632456
$41$	10.8284	1.69112	0.845558	−	0.533883i	$-0.179268\pi$
$41$	10.8284	1.69112	0.845558	+	0.533883i	$0.179268\pi$
$42$	−2.00000	−0.308607
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−1.41421	−0.210819
$46$	−4.48528	−0.661319
$47$	11.0711	1.61488	0.807441	−	0.589949i	$-0.200852\pi$
$47$	11.0711	1.61488	0.807441	+	0.589949i	$0.200852\pi$
$48$	−5.65685	−0.816497
$49$	−6.00000	−0.857143
$50$	−4.24264	−0.600000
$51$	−9.41421	−1.31825
$52$	0	0
$53$	−13.3137	−1.82878	−0.914389	−	0.404836i	$-0.867329\pi$
$53$	−13.3137	−1.82878	−0.914389	+	0.404836i	$0.867329\pi$
$54$	−8.00000	−1.08866
$55$	−6.82843	−0.920745
$56$	2.82843	0.377964
$57$	9.41421	1.24694
$58$	−9.65685	−1.26801
$59$	−3.17157	−0.412904	−0.206452	−	0.978457i	$-0.566192\pi$
$59$	−3.17157	−0.412904	−0.206452	+	0.978457i	$0.566192\pi$
$60$	0	0
$61$	12.8284	1.64251	0.821256	−	0.570560i	$-0.193274\pi$
$61$	12.8284	1.64251	0.821256	+	0.570560i	$0.193274\pi$
$62$	−9.17157	−1.16479
$63$	1.00000	0.125988
$64$	8.00000	1.00000
$65$	9.65685	1.19779
$66$	−9.65685	−1.18868
$67$	1.34315	0.164091	0.0820457	−	0.996629i	$-0.473855\pi$
$67$	1.34315	0.164091	0.0820457	+	0.996629i	$0.473855\pi$
$68$	0	0
$69$	−4.48528	−0.539964
$70$	−2.00000	−0.239046
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	2.82843	0.333333
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	−13.8995	−1.61578
$75$	−4.24264	−0.489898
$76$	0	0
$77$	4.82843	0.550250
$78$	13.6569	1.54633
$79$	−5.48528	−0.617142	−0.308571	−	0.951201i	$-0.599851\pi$
$79$	−5.48528	−0.617142	−0.308571	+	0.951201i	$0.599851\pi$
$80$	−5.65685	−0.632456
$81$	−5.00000	−0.555556
$82$	15.3137	1.69112
$83$	−11.1421	−1.22301	−0.611504	−	0.791241i	$-0.709435\pi$
$83$	−11.1421	−1.22301	−0.611504	+	0.791241i	$0.709435\pi$
$84$	0	0
$85$	−9.41421	−1.02111
$86$	0	0
$87$	−9.65685	−1.03532
$88$	13.6569	1.45583
$89$	−6.17157	−0.654185	−0.327093	−	0.944992i	$-0.606069\pi$
$89$	−6.17157	−0.654185	−0.327093	+	0.944992i	$0.606069\pi$
$90$	−2.00000	−0.210819
$91$	−6.82843	−0.715814
$92$	0	0
$93$	−9.17157	−0.951048
$94$	15.6569	1.61488
$95$	9.41421	0.965878
$96$	0	0
$97$	10.2426	1.03998	0.519991	−	0.854172i	$-0.325935\pi$
$97$	10.2426	1.03998	0.519991	+	0.854172i	$0.325935\pi$
$98$	−8.48528	−0.857143
$99$	4.82843	0.485275
$100$	0	0
$101$	4.17157	0.415087	0.207544	−	0.978226i	$-0.433453\pi$
$101$	4.17157	0.415087	0.207544	+	0.978226i	$0.433453\pi$
$102$	−13.3137	−1.31825
$103$	3.48528	0.343415	0.171707	−	0.985148i	$-0.445072\pi$
$103$	3.48528	0.343415	0.171707	+	0.985148i	$0.445072\pi$
$104$	−19.3137	−1.89386
$105$	−2.00000	−0.195180
$106$	−18.8284	−1.82878
$107$	−8.48528	−0.820303	−0.410152	−	0.912017i	$-0.634524\pi$
$107$	−8.48528	−0.820303	−0.410152	+	0.912017i	$0.634524\pi$
$108$	0	0
$109$	6.48528	0.621177	0.310589	−	0.950544i	$-0.399474\pi$
$109$	6.48528	0.621177	0.310589	+	0.950544i	$0.399474\pi$
$110$	−9.65685	−0.920745
$111$	−13.8995	−1.31928
$112$	4.00000	0.377964
$113$	0.928932	0.0873866	0.0436933	−	0.999045i	$-0.486088\pi$
$113$	0.928932	0.0873866	0.0436933	+	0.999045i	$0.486088\pi$
$114$	13.3137	1.24694
$115$	−4.48528	−0.418255
$116$	0	0
$117$	−6.82843	−0.631288
$118$	−4.48528	−0.412904
$119$	6.65685	0.610233
$120$	−5.65685	−0.516398
$121$	12.3137	1.11943
$122$	18.1421	1.64251
$123$	15.3137	1.38079
$124$	0	0
$125$	−11.3137	−1.01193
$126$	1.41421	0.125988
$127$	−20.7990	−1.84561	−0.922806	−	0.385265i	$-0.874110\pi$
$127$	−20.7990	−1.84561	−0.922806	+	0.385265i	$0.874110\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	13.6569	1.19779
$131$	17.4853	1.52770	0.763848	−	0.645396i	$-0.223308\pi$
$131$	17.4853	1.52770	0.763848	+	0.645396i	$0.223308\pi$
$132$	0	0
$133$	−6.65685	−0.577222
$134$	1.89949	0.164091
$135$	−8.00000	−0.688530
$136$	18.8284	1.61452
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	−6.34315	−0.539964
$139$	−15.6569	−1.32800	−0.663999	−	0.747734i	$-0.731142\pi$
$139$	−15.6569	−1.32800	−0.663999	+	0.747734i	$0.731142\pi$
$140$	0	0
$141$	15.6569	1.31854
$142$	8.48528	0.712069
$143$	−32.9706	−2.75714
$144$	4.00000	0.333333
$145$	−9.65685	−0.801958
$146$	−4.24264	−0.351123
$147$	−8.48528	−0.699854
$148$	0	0
$149$	−21.1421	−1.73203	−0.866016	−	0.500017i	$-0.833327\pi$
$149$	−21.1421	−1.73203	−0.866016	+	0.500017i	$0.833327\pi$
$150$	−6.00000	−0.489898
$151$	−13.8284	−1.12534	−0.562671	−	0.826681i	$-0.690226\pi$
$151$	−13.8284	−1.12534	−0.562671	+	0.826681i	$0.690226\pi$
$152$	−18.8284	−1.52719
$153$	6.65685	0.538175
$154$	6.82843	0.550250
$155$	−9.17157	−0.736678
$156$	0	0
$157$	7.31371	0.583697	0.291849	−	0.956464i	$-0.405730\pi$
$157$	7.31371	0.583697	0.291849	+	0.956464i	$0.405730\pi$
$158$	−7.75736	−0.617142
$159$	−18.8284	−1.49319
$160$	0	0
$161$	3.17157	0.249955
$162$	−7.07107	−0.555556
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	−9.65685	−0.751785
$166$	−15.7574	−1.22301
$167$	−2.10051	−0.162542	−0.0812710	−	0.996692i	$-0.525898\pi$
$167$	−2.10051	−0.162542	−0.0812710	+	0.996692i	$0.525898\pi$
$168$	4.00000	0.308607
$169$	33.6274	2.58672
$170$	−13.3137	−1.02111
$171$	−6.65685	−0.509062
$172$	0	0
$173$	−0.686292	−0.0521778	−0.0260889	−	0.999660i	$-0.508305\pi$
$173$	−0.686292	−0.0521778	−0.0260889	+	0.999660i	$0.508305\pi$
$174$	−13.6569	−1.03532
$175$	3.00000	0.226779
$176$	19.3137	1.45583
$177$	−4.48528	−0.337134
$178$	−8.72792	−0.654185
$179$	−1.65685	−0.123839	−0.0619196	−	0.998081i	$-0.519722\pi$
$179$	−1.65685	−0.123839	−0.0619196	+	0.998081i	$0.519722\pi$
$180$	0	0
$181$	19.3137	1.43558	0.717788	−	0.696261i	$-0.245155\pi$
$181$	19.3137	1.43558	0.717788	+	0.696261i	$0.245155\pi$
$182$	−9.65685	−0.715814
$183$	18.1421	1.34111
$184$	8.97056	0.661319
$185$	−13.8995	−1.02191
$186$	−12.9706	−0.951048
$187$	32.1421	2.35047
$188$	0	0
$189$	5.65685	0.411476
$190$	13.3137	0.965878
$191$	12.6569	0.915818	0.457909	−	0.888999i	$-0.348599\pi$
$191$	12.6569	0.915818	0.457909	+	0.888999i	$0.348599\pi$
$192$	11.3137	0.816497
$193$	−12.3137	−0.886360	−0.443180	−	0.896433i	$-0.646150\pi$
$193$	−12.3137	−0.886360	−0.443180	+	0.896433i	$0.646150\pi$
$194$	14.4853	1.03998
$195$	13.6569	0.977988
$196$	0	0
$197$	8.65685	0.616775	0.308388	−	0.951261i	$-0.400211\pi$
$197$	8.65685	0.616775	0.308388	+	0.951261i	$0.400211\pi$
$198$	6.82843	0.485275
$199$	23.8995	1.69419	0.847095	−	0.531441i	$-0.178349\pi$
$199$	23.8995	1.69419	0.847095	+	0.531441i	$0.178349\pi$
$200$	8.48528	0.600000
$201$	1.89949	0.133980
$202$	5.89949	0.415087
$203$	6.82843	0.479262
$204$	0	0
$205$	15.3137	1.06956
$206$	4.92893	0.343415
$207$	3.17157	0.220440
$208$	−27.3137	−1.89386
$209$	−32.1421	−2.22332
$210$	−2.82843	−0.195180
$211$	−9.31371	−0.641182	−0.320591	−	0.947218i	$-0.603882\pi$
$211$	−9.31371	−0.641182	−0.320591	+	0.947218i	$0.603882\pi$
$212$	0	0
$213$	8.48528	0.581402
$214$	−12.0000	−0.820303
$215$	0	0
$216$	16.0000	1.08866
$217$	6.48528	0.440250
$218$	9.17157	0.621177
$219$	−4.24264	−0.286691
$220$	0	0
$221$	−45.4558	−3.05769
$222$	−19.6569	−1.31928
$223$	1.27208	0.0851846	0.0425923	−	0.999093i	$-0.486438\pi$
$223$	1.27208	0.0851846	0.0425923	+	0.999093i	$0.486438\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	1.31371	0.0873866
$227$	−21.4142	−1.42131	−0.710656	−	0.703540i	$-0.751602\pi$
$227$	−21.4142	−1.42131	−0.710656	+	0.703540i	$0.751602\pi$
$228$	0	0
$229$	9.51472	0.628750	0.314375	−	0.949299i	$-0.398205\pi$
$229$	9.51472	0.628750	0.314375	+	0.949299i	$0.398205\pi$
$230$	−6.34315	−0.418255
$231$	6.82843	0.449278
$232$	19.3137	1.26801
$233$	−0.343146	−0.0224802	−0.0112401	−	0.999937i	$-0.503578\pi$
$233$	−0.343146	−0.0224802	−0.0112401	+	0.999937i	$0.503578\pi$
$234$	−9.65685	−0.631288
$235$	15.6569	1.02134
$236$	0	0
$237$	−7.75736	−0.503895
$238$	9.41421	0.610233
$239$	8.65685	0.559965	0.279983	−	0.960005i	$-0.409671\pi$
$239$	8.65685	0.559965	0.279983	+	0.960005i	$0.409671\pi$
$240$	−8.00000	−0.516398
$241$	11.4853	0.739832	0.369916	−	0.929065i	$-0.379386\pi$
$241$	11.4853	0.739832	0.369916	+	0.929065i	$0.379386\pi$
$242$	17.4142	1.11943
$243$	9.89949	0.635053
$244$	0	0
$245$	−8.48528	−0.542105
$246$	21.6569	1.38079
$247$	45.4558	2.89229
$248$	18.3431	1.16479
$249$	−15.7574	−0.998582
$250$	−16.0000	−1.01193
$251$	2.48528	0.156870	0.0784348	−	0.996919i	$-0.475008\pi$
$251$	2.48528	0.156870	0.0784348	+	0.996919i	$0.475008\pi$
$252$	0	0
$253$	15.3137	0.962765
$254$	−29.4142	−1.84561
$255$	−13.3137	−0.833737
$256$	0	0
$257$	−16.9706	−1.05859	−0.529297	−	0.848436i	$-0.677544\pi$
$257$	−16.9706	−1.05859	−0.529297	+	0.848436i	$0.677544\pi$
$258$	0	0
$259$	9.82843	0.610709
$260$	0	0
$261$	6.82843	0.422669
$262$	24.7279	1.52770
$263$	12.3137	0.759296	0.379648	−	0.925131i	$-0.376045\pi$
$263$	12.3137	0.759296	0.379648	+	0.925131i	$0.376045\pi$
$264$	19.3137	1.18868
$265$	−18.8284	−1.15662
$266$	−9.41421	−0.577222
$267$	−8.72792	−0.534140
$268$	0	0
$269$	−6.68629	−0.407670	−0.203835	−	0.979005i	$-0.565341\pi$
$269$	−6.68629	−0.407670	−0.203835	+	0.979005i	$0.565341\pi$
$270$	−11.3137	−0.688530
$271$	17.3431	1.05352	0.526761	−	0.850014i	$-0.323406\pi$
$271$	17.3431	1.05352	0.526761	+	0.850014i	$0.323406\pi$
$272$	26.6274	1.61452
$273$	−9.65685	−0.584459
$274$	−4.24264	−0.256307
$275$	14.4853	0.873495
$276$	0	0
$277$	−14.4853	−0.870336	−0.435168	−	0.900349i	$-0.643311\pi$
$277$	−14.4853	−0.870336	−0.435168	+	0.900349i	$0.643311\pi$
$278$	−22.1421	−1.32800
$279$	6.48528	0.388264
$280$	4.00000	0.239046
$281$	4.17157	0.248855	0.124428	−	0.992229i	$-0.460291\pi$
$281$	4.17157	0.248855	0.124428	+	0.992229i	$0.460291\pi$
$282$	22.1421	1.31854
$283$	−7.17157	−0.426306	−0.213153	−	0.977019i	$-0.568373\pi$
$283$	−7.17157	−0.426306	−0.213153	+	0.977019i	$0.568373\pi$
$284$	0	0
$285$	13.3137	0.788636
$286$	−46.6274	−2.75714
$287$	−10.8284	−0.639182
$288$	0	0
$289$	27.3137	1.60669
$290$	−13.6569	−0.801958
$291$	14.4853	0.849142
$292$	0	0
$293$	−31.3137	−1.82937	−0.914683	−	0.404172i	$-0.867560\pi$
$293$	−31.3137	−1.82937	−0.914683	+	0.404172i	$0.867560\pi$
$294$	−12.0000	−0.699854
$295$	−4.48528	−0.261143
$296$	27.7990	1.61578
$297$	27.3137	1.58490
$298$	−29.8995	−1.73203
$299$	−21.6569	−1.25245
$300$	0	0
$301$	0	0
$302$	−19.5563	−1.12534
$303$	5.89949	0.338917
$304$	−26.6274	−1.52719
$305$	18.1421	1.03882
$306$	9.41421	0.538175
$307$	−2.68629	−0.153315	−0.0766574	−	0.997057i	$-0.524425\pi$
$307$	−2.68629	−0.153315	−0.0766574	+	0.997057i	$0.524425\pi$
$308$	0	0
$309$	4.92893	0.280397
$310$	−12.9706	−0.736678
$311$	19.9706	1.13243	0.566213	−	0.824259i	$-0.308408\pi$
$311$	19.9706	1.13243	0.566213	+	0.824259i	$0.308408\pi$
$312$	−27.3137	−1.54633
$313$	−13.4853	−0.762233	−0.381117	−	0.924527i	$-0.624460\pi$
$313$	−13.4853	−0.762233	−0.381117	+	0.924527i	$0.624460\pi$
$314$	10.3431	0.583697
$315$	1.41421	0.0796819
$316$	0	0
$317$	10.2426	0.575284	0.287642	−	0.957738i	$-0.407129\pi$
$317$	10.2426	0.575284	0.287642	+	0.957738i	$0.407129\pi$
$318$	−26.6274	−1.49319
$319$	32.9706	1.84600
$320$	11.3137	0.632456
$321$	−12.0000	−0.669775
$322$	4.48528	0.249955
$323$	−44.3137	−2.46568
$324$	0	0
$325$	−20.4853	−1.13632
$326$	−2.82843	−0.156652
$327$	9.17157	0.507189
$328$	−30.6274	−1.69112
$329$	−11.0711	−0.610368
$330$	−13.6569	−0.751785
$331$	14.4853	0.796183	0.398092	−	0.917346i	$-0.369673\pi$
$331$	14.4853	0.796183	0.398092	+	0.917346i	$0.369673\pi$
$332$	0	0
$333$	9.82843	0.538594
$334$	−2.97056	−0.162542
$335$	1.89949	0.103780
$336$	5.65685	0.308607
$337$	15.3137	0.834191	0.417095	−	0.908863i	$-0.363048\pi$
$337$	15.3137	0.834191	0.417095	+	0.908863i	$0.363048\pi$
$338$	47.5563	2.58672
$339$	1.31371	0.0713509
$340$	0	0
$341$	31.3137	1.69573
$342$	−9.41421	−0.509062
$343$	13.0000	0.701934
$344$	0	0
$345$	−6.34315	−0.341503
$346$	−0.970563	−0.0521778
$347$	31.2843	1.67943	0.839714	−	0.543029i	$-0.182723\pi$
$347$	31.2843	1.67943	0.839714	+	0.543029i	$0.182723\pi$
$348$	0	0
$349$	24.6569	1.31985	0.659926	−	0.751331i	$-0.270588\pi$
$349$	24.6569	1.31985	0.659926	+	0.751331i	$0.270588\pi$
$350$	4.24264	0.226779
$351$	−38.6274	−2.06178
$352$	0	0
$353$	−15.3431	−0.816633	−0.408317	−	0.912840i	$-0.633884\pi$
$353$	−15.3431	−0.816633	−0.408317	+	0.912840i	$0.633884\pi$
$354$	−6.34315	−0.337134
$355$	8.48528	0.450352
$356$	0	0
$357$	9.41421	0.498253
$358$	−2.34315	−0.123839
$359$	13.8995	0.733587	0.366794	−	0.930302i	$-0.380456\pi$
$359$	13.8995	0.733587	0.366794	+	0.930302i	$0.380456\pi$
$360$	4.00000	0.210819
$361$	25.3137	1.33230
$362$	27.3137	1.43558
$363$	17.4142	0.914009
$364$	0	0
$365$	−4.24264	−0.222070
$366$	25.6569	1.34111
$367$	−4.82843	−0.252042	−0.126021	−	0.992028i	$-0.540221\pi$
$367$	−4.82843	−0.252042	−0.126021	+	0.992028i	$0.540221\pi$
$368$	12.6863	0.661319
$369$	−10.8284	−0.563705
$370$	−19.6569	−1.02191
$371$	13.3137	0.691213
$372$	0	0
$373$	−16.4853	−0.853576	−0.426788	−	0.904352i	$-0.640355\pi$
$373$	−16.4853	−0.853576	−0.426788	+	0.904352i	$0.640355\pi$
$374$	45.4558	2.35047
$375$	−16.0000	−0.826236
$376$	−31.3137	−1.61488
$377$	−46.6274	−2.40143
$378$	8.00000	0.411476
$379$	−19.2132	−0.986916	−0.493458	−	0.869770i	$-0.664267\pi$
$379$	−19.2132	−0.986916	−0.493458	+	0.869770i	$0.664267\pi$
$380$	0	0
$381$	−29.4142	−1.50694
$382$	17.8995	0.915818
$383$	7.55635	0.386111	0.193056	−	0.981188i	$-0.438160\pi$
$383$	7.55635	0.386111	0.193056	+	0.981188i	$0.438160\pi$
$384$	16.0000	0.816497
$385$	6.82843	0.348009
$386$	−17.4142	−0.886360
$387$	0	0
$388$	0	0
$389$	10.6274	0.538831	0.269416	−	0.963024i	$-0.413169\pi$
$389$	10.6274	0.538831	0.269416	+	0.963024i	$0.413169\pi$
$390$	19.3137	0.977988
$391$	21.1127	1.06772
$392$	16.9706	0.857143
$393$	24.7279	1.24736
$394$	12.2426	0.616775
$395$	−7.75736	−0.390315
$396$	0	0
$397$	−15.0000	−0.752828	−0.376414	−	0.926451i	$-0.622843\pi$
$397$	−15.0000	−0.752828	−0.376414	+	0.926451i	$0.622843\pi$
$398$	33.7990	1.69419
$399$	−9.41421	−0.471300
$400$	12.0000	0.600000
$401$	25.4558	1.27120	0.635602	−	0.772017i	$-0.280752\pi$
$401$	25.4558	1.27120	0.635602	+	0.772017i	$0.280752\pi$
$402$	2.68629	0.133980
$403$	−44.2843	−2.20596
$404$	0	0
$405$	−7.07107	−0.351364
$406$	9.65685	0.479262
$407$	47.4558	2.35230
$408$	26.6274	1.31825
$409$	23.5563	1.16479	0.582393	−	0.812907i	$-0.302116\pi$
$409$	23.5563	1.16479	0.582393	+	0.812907i	$0.302116\pi$
$410$	21.6569	1.06956
$411$	−4.24264	−0.209274
$412$	0	0
$413$	3.17157	0.156063
$414$	4.48528	0.220440
$415$	−15.7574	−0.773498
$416$	0	0
$417$	−22.1421	−1.08431
$418$	−45.4558	−2.22332
$419$	−31.3137	−1.52977	−0.764887	−	0.644164i	$-0.777205\pi$
$419$	−31.3137	−1.52977	−0.764887	+	0.644164i	$0.777205\pi$
$420$	0	0
$421$	26.4853	1.29081	0.645407	−	0.763839i	$-0.276688\pi$
$421$	26.4853	1.29081	0.645407	+	0.763839i	$0.276688\pi$
$422$	−13.1716	−0.641182
$423$	−11.0711	−0.538294
$424$	37.6569	1.82878
$425$	19.9706	0.968715
$426$	12.0000	0.581402
$427$	−12.8284	−0.620811
$428$	0	0
$429$	−46.6274	−2.25119
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	22.6274	1.08866
$433$	−37.9706	−1.82475	−0.912374	−	0.409357i	$-0.865753\pi$
$433$	−37.9706	−1.82475	−0.912374	+	0.409357i	$0.865753\pi$
$434$	9.17157	0.440250
$435$	−13.6569	−0.654796
$436$	0	0
$437$	−21.1127	−1.00996
$438$	−6.00000	−0.286691
$439$	18.4853	0.882254	0.441127	−	0.897445i	$-0.354579\pi$
$439$	18.4853	0.882254	0.441127	+	0.897445i	$0.354579\pi$
$440$	19.3137	0.920745
$441$	6.00000	0.285714
$442$	−64.2843	−3.05769
$443$	−34.9706	−1.66150	−0.830751	−	0.556645i	$-0.812089\pi$
$443$	−34.9706	−1.66150	−0.830751	+	0.556645i	$0.812089\pi$
$444$	0	0
$445$	−8.72792	−0.413743
$446$	1.79899	0.0851846
$447$	−29.8995	−1.41420
$448$	−8.00000	−0.377964
$449$	−40.0416	−1.88968	−0.944841	−	0.327530i	$-0.893784\pi$
$449$	−40.0416	−1.88968	−0.944841	+	0.327530i	$0.893784\pi$
$450$	4.24264	0.200000
$451$	−52.2843	−2.46197
$452$	0	0
$453$	−19.5563	−0.918837
$454$	−30.2843	−1.42131
$455$	−9.65685	−0.452720
$456$	−26.6274	−1.24694
$457$	−24.6274	−1.15202	−0.576011	−	0.817442i	$-0.695391\pi$
$457$	−24.6274	−1.15202	−0.576011	+	0.817442i	$0.695391\pi$
$458$	13.4558	0.628750
$459$	37.6569	1.75767
$460$	0	0
$461$	−12.3431	−0.574878	−0.287439	−	0.957799i	$-0.592804\pi$
$461$	−12.3431	−0.574878	−0.287439	+	0.957799i	$0.592804\pi$
$462$	9.65685	0.449278
$463$	−27.2132	−1.26470	−0.632352	−	0.774681i	$-0.717911\pi$
$463$	−27.2132	−1.26470	−0.632352	+	0.774681i	$0.717911\pi$
$464$	27.3137	1.26801
$465$	−12.9706	−0.601495
$466$	−0.485281	−0.0224802
$467$	26.4853	1.22559	0.612796	−	0.790241i	$-0.290045\pi$
$467$	26.4853	1.22559	0.612796	+	0.790241i	$0.290045\pi$
$468$	0	0
$469$	−1.34315	−0.0620207
$470$	22.1421	1.02134
$471$	10.3431	0.476587
$472$	8.97056	0.412904
$473$	0	0
$474$	−10.9706	−0.503895
$475$	−19.9706	−0.916312
$476$	0	0
$477$	13.3137	0.609593
$478$	12.2426	0.559965
$479$	−16.4558	−0.751887	−0.375943	−	0.926643i	$-0.622681\pi$
$479$	−16.4558	−0.751887	−0.375943	+	0.926643i	$0.622681\pi$
$480$	0	0
$481$	−67.1127	−3.06008
$482$	16.2426	0.739832
$483$	4.48528	0.204087
$484$	0	0
$485$	14.4853	0.657743
$486$	14.0000	0.635053
$487$	39.9411	1.80991	0.904953	−	0.425512i	$-0.139906\pi$
$487$	39.9411	1.80991	0.904953	+	0.425512i	$0.139906\pi$
$488$	−36.2843	−1.64251
$489$	−2.82843	−0.127906
$490$	−12.0000	−0.542105
$491$	−4.10051	−0.185053	−0.0925266	−	0.995710i	$-0.529494\pi$
$491$	−4.10051	−0.185053	−0.0925266	+	0.995710i	$0.529494\pi$
$492$	0	0
$493$	45.4558	2.04723
$494$	64.2843	2.89229
$495$	6.82843	0.306915
$496$	25.9411	1.16479
$497$	−6.00000	−0.269137
$498$	−22.2843	−0.998582
$499$	−25.3431	−1.13452	−0.567258	−	0.823540i	$-0.691996\pi$
$499$	−25.3431	−1.13452	−0.567258	+	0.823540i	$0.691996\pi$
$500$	0	0
$501$	−2.97056	−0.132715
$502$	3.51472	0.156870
$503$	−25.9706	−1.15797	−0.578985	−	0.815338i	$-0.696551\pi$
$503$	−25.9706	−1.15797	−0.578985	+	0.815338i	$0.696551\pi$
$504$	−2.82843	−0.125988
$505$	5.89949	0.262524
$506$	21.6569	0.962765
$507$	47.5563	2.11205
$508$	0	0
$509$	−9.51472	−0.421732	−0.210866	−	0.977515i	$-0.567628\pi$
$509$	−9.51472	−0.421732	−0.210866	+	0.977515i	$0.567628\pi$
$510$	−18.8284	−0.833737
$511$	3.00000	0.132712
$512$	−22.6274	−1.00000
$513$	−37.6569	−1.66259
$514$	−24.0000	−1.05859
$515$	4.92893	0.217195
$516$	0	0
$517$	−53.4558	−2.35099
$518$	13.8995	0.610709
$519$	−0.970563	−0.0426030
$520$	−27.3137	−1.19779
$521$	25.4142	1.11342	0.556708	−	0.830708i	$-0.312064\pi$
$521$	25.4142	1.11342	0.556708	+	0.830708i	$0.312064\pi$
$522$	9.65685	0.422669
$523$	34.9706	1.52916	0.764578	−	0.644531i	$-0.222947\pi$
$523$	34.9706	1.52916	0.764578	+	0.644531i	$0.222947\pi$
$524$	0	0
$525$	4.24264	0.185164
$526$	17.4142	0.759296
$527$	43.1716	1.88058
$528$	27.3137	1.18868
$529$	−12.9411	−0.562658
$530$	−26.6274	−1.15662
$531$	3.17157	0.137635
$532$	0	0
$533$	73.9411	3.20275
$534$	−12.3431	−0.534140
$535$	−12.0000	−0.518805
$536$	−3.79899	−0.164091
$537$	−2.34315	−0.101114
$538$	−9.45584	−0.407670
$539$	28.9706	1.24785
$540$	0	0
$541$	−16.4853	−0.708758	−0.354379	−	0.935102i	$-0.615308\pi$
$541$	−16.4853	−0.708758	−0.354379	+	0.935102i	$0.615308\pi$
$542$	24.5269	1.05352
$543$	27.3137	1.17214
$544$	0	0
$545$	9.17157	0.392867
$546$	−13.6569	−0.584459
$547$	5.51472	0.235792	0.117896	−	0.993026i	$-0.462385\pi$
$547$	5.51472	0.235792	0.117896	+	0.993026i	$0.462385\pi$
$548$	0	0
$549$	−12.8284	−0.547504
$550$	20.4853	0.873495
$551$	−45.4558	−1.93648
$552$	12.6863	0.539964
$553$	5.48528	0.233258
$554$	−20.4853	−0.870336
$555$	−19.6569	−0.834387
$556$	0	0
$557$	−5.82843	−0.246958	−0.123479	−	0.992347i	$-0.539405\pi$
$557$	−5.82843	−0.246958	−0.123479	+	0.992347i	$0.539405\pi$
$558$	9.17157	0.388264
$559$	0	0
$560$	5.65685	0.239046
$561$	45.4558	1.91915
$562$	5.89949	0.248855
$563$	9.89949	0.417214	0.208607	−	0.978000i	$-0.433107\pi$
$563$	9.89949	0.417214	0.208607	+	0.978000i	$0.433107\pi$
$564$	0	0
$565$	1.31371	0.0552681
$566$	−10.1421	−0.426306
$567$	5.00000	0.209980
$568$	−16.9706	−0.712069
$569$	−46.1127	−1.93314	−0.966572	−	0.256394i	$-0.917466\pi$
$569$	−46.1127	−1.93314	−0.966572	+	0.256394i	$0.917466\pi$
$570$	18.8284	0.788636
$571$	−11.1716	−0.467516	−0.233758	−	0.972295i	$-0.575102\pi$
$571$	−11.1716	−0.467516	−0.233758	+	0.972295i	$0.575102\pi$
$572$	0	0
$573$	17.8995	0.747762
$574$	−15.3137	−0.639182
$575$	9.51472	0.396791
$576$	−8.00000	−0.333333
$577$	−13.9706	−0.581602	−0.290801	−	0.956784i	$-0.593922\pi$
$577$	−13.9706	−0.581602	−0.290801	+	0.956784i	$0.593922\pi$
$578$	38.6274	1.60669
$579$	−17.4142	−0.723710
$580$	0	0
$581$	11.1421	0.462254
$582$	20.4853	0.849142
$583$	64.2843	2.66238
$584$	8.48528	0.351123
$585$	−9.65685	−0.399262
$586$	−44.2843	−1.82937
$587$	20.8701	0.861399	0.430700	−	0.902495i	$-0.358267\pi$
$587$	20.8701	0.861399	0.430700	+	0.902495i	$0.358267\pi$
$588$	0	0
$589$	−43.1716	−1.77885
$590$	−6.34315	−0.261143
$591$	12.2426	0.503595
$592$	39.3137	1.61578
$593$	22.4558	0.922151	0.461075	−	0.887361i	$-0.347464\pi$
$593$	22.4558	0.922151	0.461075	+	0.887361i	$0.347464\pi$
$594$	38.6274	1.58490
$595$	9.41421	0.385945
$596$	0	0
$597$	33.7990	1.38330
$598$	−30.6274	−1.25245
$599$	4.34315	0.177456	0.0887281	−	0.996056i	$-0.471720\pi$
$599$	4.34315	0.177456	0.0887281	+	0.996056i	$0.471720\pi$
$600$	12.0000	0.489898
$601$	−25.3137	−1.03257	−0.516284	−	0.856418i	$-0.672685\pi$
$601$	−25.3137	−1.03257	−0.516284	+	0.856418i	$0.672685\pi$
$602$	0	0
$603$	−1.34315	−0.0546971
$604$	0	0
$605$	17.4142	0.707988
$606$	8.34315	0.338917
$607$	10.7279	0.435433	0.217716	−	0.976012i	$-0.430139\pi$
$607$	10.7279	0.435433	0.217716	+	0.976012i	$0.430139\pi$
$608$	0	0
$609$	9.65685	0.391315
$610$	25.6569	1.03882
$611$	75.5980	3.05837
$612$	0	0
$613$	11.2132	0.452897	0.226449	−	0.974023i	$-0.427288\pi$
$613$	11.2132	0.452897	0.226449	+	0.974023i	$0.427288\pi$
$614$	−3.79899	−0.153315
$615$	21.6569	0.873289
$616$	−13.6569	−0.550250
$617$	−39.3137	−1.58271	−0.791355	−	0.611357i	$-0.790624\pi$
$617$	−39.3137	−1.58271	−0.791355	+	0.611357i	$0.790624\pi$
$618$	6.97056	0.280397
$619$	12.4437	0.500153	0.250076	−	0.968226i	$-0.419544\pi$
$619$	12.4437	0.500153	0.250076	+	0.968226i	$0.419544\pi$
$620$	0	0
$621$	17.9411	0.719953
$622$	28.2426	1.13243
$623$	6.17157	0.247259
$624$	−38.6274	−1.54633
$625$	−1.00000	−0.0400000
$626$	−19.0711	−0.762233
$627$	−45.4558	−1.81533
$628$	0	0
$629$	65.4264	2.60872
$630$	2.00000	0.0796819
$631$	32.7990	1.30571	0.652854	−	0.757484i	$-0.273572\pi$
$631$	32.7990	1.30571	0.652854	+	0.757484i	$0.273572\pi$
$632$	15.5147	0.617142
$633$	−13.1716	−0.523523
$634$	14.4853	0.575284
$635$	−29.4142	−1.16727
$636$	0	0
$637$	−40.9706	−1.62331
$638$	46.6274	1.84600
$639$	−6.00000	−0.237356
$640$	16.0000	0.632456
$641$	−9.68629	−0.382586	−0.191293	−	0.981533i	$-0.561268\pi$
$641$	−9.68629	−0.382586	−0.191293	+	0.981533i	$0.561268\pi$
$642$	−16.9706	−0.669775
$643$	36.6274	1.44444	0.722222	−	0.691661i	$-0.243121\pi$
$643$	36.6274	1.44444	0.722222	+	0.691661i	$0.243121\pi$
$644$	0	0
$645$	0	0
$646$	−62.6690	−2.46568
$647$	−7.79899	−0.306610	−0.153305	−	0.988179i	$-0.548992\pi$
$647$	−7.79899	−0.306610	−0.153305	+	0.988179i	$0.548992\pi$
$648$	14.1421	0.555556
$649$	15.3137	0.601116
$650$	−28.9706	−1.13632
$651$	9.17157	0.359462
$652$	0	0
$653$	22.6274	0.885479	0.442740	−	0.896650i	$-0.354007\pi$
$653$	22.6274	0.885479	0.442740	+	0.896650i	$0.354007\pi$
$654$	12.9706	0.507189
$655$	24.7279	0.966200
$656$	−43.3137	−1.69112
$657$	3.00000	0.117041
$658$	−15.6569	−0.610368
$659$	3.17157	0.123547	0.0617735	−	0.998090i	$-0.480324\pi$
$659$	3.17157	0.123547	0.0617735	+	0.998090i	$0.480324\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	20.4853	0.796183
$663$	−64.2843	−2.49659
$664$	31.5147	1.22301
$665$	−9.41421	−0.365068
$666$	13.8995	0.538594
$667$	21.6569	0.838557
$668$	0	0
$669$	1.79899	0.0695530
$670$	2.68629	0.103780
$671$	−61.9411	−2.39121
$672$	0	0
$673$	33.6985	1.29898	0.649491	−	0.760370i	$-0.274982\pi$
$673$	33.6985	1.29898	0.649491	+	0.760370i	$0.274982\pi$
$674$	21.6569	0.834191
$675$	16.9706	0.653197
$676$	0	0
$677$	3.34315	0.128488	0.0642438	−	0.997934i	$-0.479536\pi$
$677$	3.34315	0.128488	0.0642438	+	0.997934i	$0.479536\pi$
$678$	1.85786	0.0713509
$679$	−10.2426	−0.393076
$680$	26.6274	1.02111
$681$	−30.2843	−1.16050
$682$	44.2843	1.69573
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−4.24264	−0.162103
$686$	18.3848	0.701934
$687$	13.4558	0.513372
$688$	0	0
$689$	−90.9117	−3.46346
$690$	−8.97056	−0.341503
$691$	10.0294	0.381538	0.190769	−	0.981635i	$-0.438902\pi$
$691$	10.0294	0.381538	0.190769	+	0.981635i	$0.438902\pi$
$692$	0	0
$693$	−4.82843	−0.183417
$694$	44.2426	1.67943
$695$	−22.1421	−0.839899
$696$	27.3137	1.03532
$697$	−72.0833	−2.73035
$698$	34.8701	1.31985
$699$	−0.485281	−0.0183550
$700$	0	0
$701$	−33.6274	−1.27009	−0.635045	−	0.772475i	$-0.719018\pi$
$701$	−33.6274	−1.27009	−0.635045	+	0.772475i	$0.719018\pi$
$702$	−54.6274	−2.06178
$703$	−65.4264	−2.46760
$704$	−38.6274	−1.45583
$705$	22.1421	0.833921
$706$	−21.6985	−0.816633
$707$	−4.17157	−0.156888
$708$	0	0
$709$	21.9706	0.825122	0.412561	−	0.910930i	$-0.364634\pi$
$709$	21.9706	0.825122	0.412561	+	0.910930i	$0.364634\pi$
$710$	12.0000	0.450352
$711$	5.48528	0.205714
$712$	17.4558	0.654185
$713$	20.5685	0.770298
$714$	13.3137	0.498253
$715$	−46.6274	−1.74377
$716$	0	0
$717$	12.2426	0.457210
$718$	19.6569	0.733587
$719$	43.9706	1.63983	0.819913	−	0.572489i	$-0.194022\pi$
$719$	43.9706	1.63983	0.819913	+	0.572489i	$0.194022\pi$
$720$	5.65685	0.210819
$721$	−3.48528	−0.129799
$722$	35.7990	1.33230
$723$	16.2426	0.604070
$724$	0	0
$725$	20.4853	0.760804
$726$	24.6274	0.914009
$727$	16.7279	0.620404	0.310202	−	0.950671i	$-0.399603\pi$
$727$	16.7279	0.620404	0.310202	+	0.950671i	$0.399603\pi$
$728$	19.3137	0.715814
$729$	29.0000	1.07407
$730$	−6.00000	−0.222070
$731$	0	0
$732$	0	0
$733$	−43.1716	−1.59458	−0.797289	−	0.603597i	$-0.793733\pi$
$733$	−43.1716	−1.59458	−0.797289	+	0.603597i	$0.793733\pi$
$734$	−6.82843	−0.252042
$735$	−12.0000	−0.442627
$736$	0	0
$737$	−6.48528	−0.238888
$738$	−15.3137	−0.563705
$739$	7.82843	0.287973	0.143987	−	0.989580i	$-0.454008\pi$
$739$	7.82843	0.287973	0.143987	+	0.989580i	$0.454008\pi$
$740$	0	0
$741$	64.2843	2.36154
$742$	18.8284	0.691213
$743$	−36.5858	−1.34220	−0.671101	−	0.741366i	$-0.734178\pi$
$743$	−36.5858	−1.34220	−0.671101	+	0.741366i	$0.734178\pi$
$744$	25.9411	0.951048
$745$	−29.8995	−1.09543
$746$	−23.3137	−0.853576
$747$	11.1421	0.407669
$748$	0	0
$749$	8.48528	0.310045
$750$	−22.6274	−0.826236
$751$	47.4853	1.73276	0.866381	−	0.499383i	$-0.166440\pi$
$751$	47.4853	1.73276	0.866381	+	0.499383i	$0.166440\pi$
$752$	−44.2843	−1.61488
$753$	3.51472	0.128083
$754$	−65.9411	−2.40143
$755$	−19.5563	−0.711728
$756$	0	0
$757$	9.20101	0.334416	0.167208	−	0.985922i	$-0.446525\pi$
$757$	9.20101	0.334416	0.167208	+	0.985922i	$0.446525\pi$
$758$	−27.1716	−0.986916
$759$	21.6569	0.786094
$760$	−26.6274	−0.965878
$761$	−10.9706	−0.397683	−0.198841	−	0.980032i	$-0.563718\pi$
$761$	−10.9706	−0.397683	−0.198841	+	0.980032i	$0.563718\pi$
$762$	−41.5980	−1.50694
$763$	−6.48528	−0.234783
$764$	0	0
$765$	9.41421	0.340372
$766$	10.6863	0.386111
$767$	−21.6569	−0.781984
$768$	0	0
$769$	−32.1838	−1.16058	−0.580288	−	0.814411i	$-0.697060\pi$
$769$	−32.1838	−1.16058	−0.580288	+	0.814411i	$0.697060\pi$
$770$	9.65685	0.348009
$771$	−24.0000	−0.864339
$772$	0	0
$773$	−11.6152	−0.417771	−0.208885	−	0.977940i	$-0.566984\pi$
$773$	−11.6152	−0.417771	−0.208885	+	0.977940i	$0.566984\pi$
$774$	0	0
$775$	19.4558	0.698875
$776$	−28.9706	−1.03998
$777$	13.8995	0.498642
$778$	15.0294	0.538831
$779$	72.0833	2.58265
$780$	0	0
$781$	−28.9706	−1.03665
$782$	29.8579	1.06772
$783$	38.6274	1.38043
$784$	24.0000	0.857143
$785$	10.3431	0.369163
$786$	34.9706	1.24736
$787$	28.9411	1.03164	0.515820	−	0.856697i	$-0.327487\pi$
$787$	28.9411	1.03164	0.515820	+	0.856697i	$0.327487\pi$
$788$	0	0
$789$	17.4142	0.619962
$790$	−10.9706	−0.390315
$791$	−0.928932	−0.0330290
$792$	−13.6569	−0.485275
$793$	87.5980	3.11070
$794$	−21.2132	−0.752828
$795$	−26.6274	−0.944377
$796$	0	0
$797$	−30.9289	−1.09556	−0.547779	−	0.836623i	$-0.684527\pi$
$797$	−30.9289	−1.09556	−0.547779	+	0.836623i	$0.684527\pi$
$798$	−13.3137	−0.471300
$799$	−73.6985	−2.60726
$800$	0	0
$801$	6.17157	0.218062
$802$	36.0000	1.27120
$803$	14.4853	0.511174
$804$	0	0
$805$	4.48528	0.158085
$806$	−62.6274	−2.20596
$807$	−9.45584	−0.332861
$808$	−11.7990	−0.415087
$809$	1.02944	0.0361931	0.0180965	−	0.999836i	$-0.494239\pi$
$809$	1.02944	0.0361931	0.0180965	+	0.999836i	$0.494239\pi$
$810$	−10.0000	−0.351364
$811$	17.1716	0.602975	0.301488	−	0.953470i	$-0.402517\pi$
$811$	17.1716	0.602975	0.301488	+	0.953470i	$0.402517\pi$
$812$	0	0
$813$	24.5269	0.860196
$814$	67.1127	2.35230
$815$	−2.82843	−0.0990755
$816$	37.6569	1.31825
$817$	0	0
$818$	33.3137	1.16479
$819$	6.82843	0.238605
$820$	0	0
$821$	27.4142	0.956763	0.478381	−	0.878152i	$-0.341224\pi$
$821$	27.4142	0.956763	0.478381	+	0.878152i	$0.341224\pi$
$822$	−6.00000	−0.209274
$823$	13.2721	0.462636	0.231318	−	0.972878i	$-0.425696\pi$
$823$	13.2721	0.462636	0.231318	+	0.972878i	$0.425696\pi$
$824$	−9.85786	−0.343415
$825$	20.4853	0.713206
$826$	4.48528	0.156063
$827$	11.4853	0.399382	0.199691	−	0.979859i	$-0.436006\pi$
$827$	11.4853	0.399382	0.199691	+	0.979859i	$0.436006\pi$
$828$	0	0
$829$	−22.7279	−0.789373	−0.394687	−	0.918816i	$-0.629147\pi$
$829$	−22.7279	−0.789373	−0.394687	+	0.918816i	$0.629147\pi$
$830$	−22.2843	−0.773498
$831$	−20.4853	−0.710627
$832$	54.6274	1.89386
$833$	39.9411	1.38388
$834$	−31.3137	−1.08431
$835$	−2.97056	−0.102801
$836$	0	0
$837$	36.6863	1.26806
$838$	−44.2843	−1.52977
$839$	−6.20101	−0.214083	−0.107041	−	0.994255i	$-0.534138\pi$
$839$	−6.20101	−0.214083	−0.107041	+	0.994255i	$0.534138\pi$
$840$	5.65685	0.195180
$841$	17.6274	0.607842
$842$	37.4558	1.29081
$843$	5.89949	0.203189
$844$	0	0
$845$	47.5563	1.63599
$846$	−15.6569	−0.538294
$847$	−12.3137	−0.423104
$848$	53.2548	1.82878
$849$	−10.1421	−0.348077
$850$	28.2426	0.968715
$851$	31.1716	1.06855
$852$	0	0
$853$	−8.68629	−0.297413	−0.148706	−	0.988881i	$-0.547511\pi$
$853$	−8.68629	−0.297413	−0.148706	+	0.988881i	$0.547511\pi$
$854$	−18.1421	−0.620811
$855$	−9.41421	−0.321959
$856$	24.0000	0.820303
$857$	−15.3431	−0.524112	−0.262056	−	0.965053i	$-0.584400\pi$
$857$	−15.3431	−0.524112	−0.262056	+	0.965053i	$0.584400\pi$
$858$	−65.9411	−2.25119
$859$	−34.7696	−1.18632	−0.593161	−	0.805084i	$-0.702120\pi$
$859$	−34.7696	−1.18632	−0.593161	+	0.805084i	$0.702120\pi$
$860$	0	0
$861$	−15.3137	−0.521890
$862$	−25.4558	−0.867029
$863$	57.1127	1.94414	0.972069	−	0.234693i	$-0.0754086\pi$
$863$	57.1127	1.94414	0.972069	+	0.234693i	$0.0754086\pi$
$864$	0	0
$865$	−0.970563	−0.0330001
$866$	−53.6985	−1.82475
$867$	38.6274	1.31186
$868$	0	0
$869$	26.4853	0.898452
$870$	−19.3137	−0.654796
$871$	9.17157	0.310767
$872$	−18.3431	−0.621177
$873$	−10.2426	−0.346661
$874$	−29.8579	−1.00996
$875$	11.3137	0.382473
$876$	0	0
$877$	5.34315	0.180425	0.0902126	−	0.995923i	$-0.471245\pi$
$877$	5.34315	0.180425	0.0902126	+	0.995923i	$0.471245\pi$
$878$	26.1421	0.882254
$879$	−44.2843	−1.49367
$880$	27.3137	0.920745
$881$	13.0711	0.440375	0.220188	−	0.975458i	$-0.429333\pi$
$881$	13.0711	0.440375	0.220188	+	0.975458i	$0.429333\pi$
$882$	8.48528	0.285714
$883$	−41.4558	−1.39510	−0.697550	−	0.716536i	$-0.745727\pi$
$883$	−41.4558	−1.39510	−0.697550	+	0.716536i	$0.745727\pi$
$884$	0	0
$885$	−6.34315	−0.213223
$886$	−49.4558	−1.66150
$887$	−28.6569	−0.962203	−0.481101	−	0.876665i	$-0.659763\pi$
$887$	−28.6569	−0.962203	−0.481101	+	0.876665i	$0.659763\pi$
$888$	39.3137	1.31928
$889$	20.7990	0.697576
$890$	−12.3431	−0.413743
$891$	24.1421	0.808792
$892$	0	0
$893$	73.6985	2.46623
$894$	−42.2843	−1.41420
$895$	−2.34315	−0.0783227
$896$	−11.3137	−0.377964
$897$	−30.6274	−1.02262
$898$	−56.6274	−1.88968
$899$	44.2843	1.47696
$900$	0	0
$901$	88.6274	2.95261
$902$	−73.9411	−2.46197
$903$	0	0
$904$	−2.62742	−0.0873866
$905$	27.3137	0.907938
$906$	−27.6569	−0.918837
$907$	−28.5269	−0.947221	−0.473610	−	0.880735i	$-0.657049\pi$
$907$	−28.5269	−0.947221	−0.473610	+	0.880735i	$0.657049\pi$
$908$	0	0
$909$	−4.17157	−0.138362
$910$	−13.6569	−0.452720
$911$	−52.3137	−1.73323	−0.866615	−	0.498977i	$-0.833709\pi$
$911$	−52.3137	−1.73323	−0.866615	+	0.498977i	$0.833709\pi$
$912$	−37.6569	−1.24694
$913$	53.7990	1.78049
$914$	−34.8284	−1.15202
$915$	25.6569	0.848189
$916$	0	0
$917$	−17.4853	−0.577415
$918$	53.2548	1.75767
$919$	−49.2132	−1.62339	−0.811697	−	0.584079i	$-0.801456\pi$
$919$	−49.2132	−1.62339	−0.811697	+	0.584079i	$0.801456\pi$
$920$	12.6863	0.418255
$921$	−3.79899	−0.125181
$922$	−17.4558	−0.574878
$923$	40.9706	1.34856
$924$	0	0
$925$	29.4853	0.969470
$926$	−38.4853	−1.26470
$927$	−3.48528	−0.114472
$928$	0	0
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	−18.3431	−0.601495
$931$	−39.9411	−1.30902
$932$	0	0
$933$	28.2426	0.924623
$934$	37.4558	1.22559
$935$	45.4558	1.48657
$936$	19.3137	0.631288
$937$	−31.5147	−1.02954	−0.514770	−	0.857328i	$-0.672123\pi$
$937$	−31.5147	−1.02954	−0.514770	+	0.857328i	$0.672123\pi$
$938$	−1.89949	−0.0620207
$939$	−19.0711	−0.622361
$940$	0	0
$941$	−20.2843	−0.661248	−0.330624	−	0.943763i	$-0.607259\pi$
$941$	−20.2843	−0.661248	−0.330624	+	0.943763i	$0.607259\pi$
$942$	14.6274	0.476587
$943$	−34.3431	−1.11837
$944$	12.6863	0.412904
$945$	8.00000	0.260240
$946$	0	0
$947$	−56.1838	−1.82573	−0.912864	−	0.408265i	$-0.866134\pi$
$947$	−56.1838	−1.82573	−0.912864	+	0.408265i	$0.866134\pi$
$948$	0	0
$949$	−20.4853	−0.664980
$950$	−28.2426	−0.916312
$951$	14.4853	0.469717
$952$	−18.8284	−0.610233
$953$	−24.0416	−0.778785	−0.389392	−	0.921072i	$-0.627315\pi$
$953$	−24.0416	−0.778785	−0.389392	+	0.921072i	$0.627315\pi$
$954$	18.8284	0.609593
$955$	17.8995	0.579214
$956$	0	0
$957$	46.6274	1.50725
$958$	−23.2721	−0.751887
$959$	3.00000	0.0968751
$960$	16.0000	0.516398
$961$	11.0589	0.356738
$962$	−94.9117	−3.06008
$963$	8.48528	0.273434
$964$	0	0
$965$	−17.4142	−0.560583
$966$	6.34315	0.204087
$967$	−20.7279	−0.666565	−0.333283	−	0.942827i	$-0.608156\pi$
$967$	−20.7279	−0.666565	−0.333283	+	0.942827i	$0.608156\pi$
$968$	−34.8284	−1.11943
$969$	−62.6690	−2.01322
$970$	20.4853	0.657743
$971$	−24.6274	−0.790331	−0.395166	−	0.918610i	$-0.629313\pi$
$971$	−24.6274	−0.790331	−0.395166	+	0.918610i	$0.629313\pi$
$972$	0	0
$973$	15.6569	0.501936
$974$	56.4853	1.80991
$975$	−28.9706	−0.927801
$976$	−51.3137	−1.64251
$977$	−52.9706	−1.69468	−0.847339	−	0.531052i	$-0.821797\pi$
$977$	−52.9706	−1.69468	−0.847339	+	0.531052i	$0.821797\pi$
$978$	−4.00000	−0.127906
$979$	29.7990	0.952380
$980$	0	0
$981$	−6.48528	−0.207059
$982$	−5.79899	−0.185053
$983$	−29.4853	−0.940434	−0.470217	−	0.882551i	$-0.655824\pi$
$983$	−29.4853	−0.940434	−0.470217	+	0.882551i	$0.655824\pi$
$984$	−43.3137	−1.38079
$985$	12.2426	0.390083
$986$	64.2843	2.04723
$987$	−15.6569	−0.498363
$988$	0	0
$989$	0	0
$990$	9.65685	0.306915
$991$	30.4558	0.967462	0.483731	−	0.875217i	$-0.339281\pi$
$991$	30.4558	0.967462	0.483731	+	0.875217i	$0.339281\pi$
$992$	0	0
$993$	20.4853	0.650081
$994$	−8.48528	−0.269137
$995$	33.7990	1.07150
$996$	0	0
$997$	32.3848	1.02564	0.512818	−	0.858497i	$-0.328602\pi$
$997$	32.3848	1.02564	0.512818	+	0.858497i	$0.328602\pi$
$998$	−35.8406	−1.13452
$999$	55.5980	1.75904

Introduction	Features
Universe	News

Level:	\( N \)	=	\( 4003 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	4003.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.41421 q^{3} +1.41421 q^{5} +2.00000 q^{6} -1.00000 q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.41421 q^{3} +1.41421 q^{5} +2.00000 q^{6} -1.00000 q^{7} -2.82843 q^{8} -1.00000 q^{9} +2.00000 q^{10} -4.82843 q^{11} +6.82843 q^{13} -1.41421 q^{14} +2.00000 q^{15} -4.00000 q^{16} -6.65685 q^{17} -1.41421 q^{18} +6.65685 q^{19} -1.41421 q^{21} -6.82843 q^{22} -3.17157 q^{23} -4.00000 q^{24} -3.00000 q^{25} +9.65685 q^{26} -5.65685 q^{27} -6.82843 q^{29} +2.82843 q^{30} -6.48528 q^{31} -6.82843 q^{33} -9.41421 q^{34} -1.41421 q^{35} -9.82843 q^{37} +9.41421 q^{38} +9.65685 q^{39} -4.00000 q^{40} +10.8284 q^{41} -2.00000 q^{42} -1.41421 q^{45} -4.48528 q^{46} +11.0711 q^{47} -5.65685 q^{48} -6.00000 q^{49} -4.24264 q^{50} -9.41421 q^{51} -13.3137 q^{53} -8.00000 q^{54} -6.82843 q^{55} +2.82843 q^{56} +9.41421 q^{57} -9.65685 q^{58} -3.17157 q^{59} +12.8284 q^{61} -9.17157 q^{62} +1.00000 q^{63} +8.00000 q^{64} +9.65685 q^{65} -9.65685 q^{66} +1.34315 q^{67} -4.48528 q^{69} -2.00000 q^{70} +6.00000 q^{71} +2.82843 q^{72} -3.00000 q^{73} -13.8995 q^{74} -4.24264 q^{75} +4.82843 q^{77} +13.6569 q^{78} -5.48528 q^{79} -5.65685 q^{80} -5.00000 q^{81} +15.3137 q^{82} -11.1421 q^{83} -9.41421 q^{85} -9.65685 q^{87} +13.6569 q^{88} -6.17157 q^{89} -2.00000 q^{90} -6.82843 q^{91} -9.17157 q^{93} +15.6569 q^{94} +9.41421 q^{95} +10.2426 q^{97} -8.48528 q^{98} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q + 4q^{6} - 2q^{7} - 2q^{9} + O(q^{10}) \)	\( 2q + 4q^{6} - 2q^{7} - 2q^{9} + 4q^{10} - 4q^{11} + 8q^{13} + 4q^{15} - 8q^{16} - 2q^{17} + 2q^{19} - 8q^{22} - 12q^{23} - 8q^{24} - 6q^{25} + 8q^{26} - 8q^{29} + 4q^{31} - 8q^{33} - 16q^{34} - 14q^{37} + 16q^{38} + 8q^{39} - 8q^{40} + 16q^{41} - 4q^{42} + 8q^{46} + 8q^{47} - 12q^{49} - 16q^{51} - 4q^{53} - 16q^{54} - 8q^{55} + 16q^{57} - 8q^{58} - 12q^{59} + 20q^{61} - 24q^{62} + 2q^{63} + 16q^{64} + 8q^{65} - 8q^{66} + 14q^{67} + 8q^{69} - 4q^{70} + 12q^{71} - 6q^{73} - 8q^{74} + 4q^{77} + 16q^{78} + 6q^{79} - 10q^{81} + 8q^{82} + 6q^{83} - 16q^{85} - 8q^{87} + 16q^{88} - 18q^{89} - 4q^{90} - 8q^{91} - 24q^{93} + 20q^{94} + 16q^{95} + 12q^{97} + 4q^{99} + O(q^{100}) \)

Introduction and more

L-functions

Modular Forms

Varieties

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Representations

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Knowledge

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$-expansion

Coefficient data

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4003.2.a.a.1.2

Level	4003
Weight	2
Character	4003.1
Self dual	Yes
Analytic conductor	31.964
Analytic rank	1
Dimension	2
CM	No
Inner twists	1

Self dual:	Yes
Analytic conductor:	\(31.9641159291\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{8})^+\)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$