LMFDB - Embedded newform 6044.2.a.b.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.07770 q^{3} +3.26525 q^{5} -1.02333 q^{7} -1.83857 q^{9} +O(q^{10})$	$q-1.07770 q^{3} +3.26525 q^{5} -1.02333 q^{7} -1.83857 q^{9} +5.58485 q^{11} +3.94997 q^{13} -3.51895 q^{15} +1.02692 q^{17} +2.70210 q^{19} +1.10284 q^{21} +4.81659 q^{23} +5.66188 q^{25} +5.21451 q^{27} +7.80636 q^{29} +0.831300 q^{31} -6.01877 q^{33} -3.34144 q^{35} -0.434442 q^{37} -4.25686 q^{39} -4.44304 q^{41} -3.33864 q^{43} -6.00341 q^{45} -1.01027 q^{47} -5.95279 q^{49} -1.10671 q^{51} +4.84576 q^{53} +18.2359 q^{55} -2.91204 q^{57} -3.03462 q^{59} -12.3907 q^{61} +1.88147 q^{63} +12.8976 q^{65} +11.7831 q^{67} -5.19082 q^{69} +6.02445 q^{71} -9.49808 q^{73} -6.10178 q^{75} -5.71516 q^{77} -0.506509 q^{79} -0.103928 q^{81} -6.88924 q^{83} +3.35315 q^{85} -8.41287 q^{87} +4.80444 q^{89} -4.04213 q^{91} -0.895888 q^{93} +8.82305 q^{95} -5.71755 q^{97} -10.2682 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.07770	−0.622208	−0.311104	−	0.950376i	$-0.600699\pi$
$3$	−1.07770	−0.622208	−0.311104	+	0.950376i	$0.600699\pi$
$4$	0	0
$5$	3.26525	1.46027	0.730133	−	0.683305i	$-0.239458\pi$
$5$	3.26525	1.46027	0.730133	+	0.683305i	$0.239458\pi$
$6$	0	0
$7$	−1.02333	−0.386783	−0.193392	−	0.981122i	$-0.561949\pi$
$7$	−1.02333	−0.386783	−0.193392	+	0.981122i	$0.561949\pi$
$8$	0	0
$9$	−1.83857	−0.612858
$10$	0	0
$11$	5.58485	1.68390	0.841948	−	0.539559i	$-0.181409\pi$
$11$	5.58485	1.68390	0.841948	+	0.539559i	$0.181409\pi$
$12$	0	0
$13$	3.94997	1.09552	0.547762	−	0.836634i	$-0.315480\pi$
$13$	3.94997	1.09552	0.547762	+	0.836634i	$0.315480\pi$
$14$	0	0
$15$	−3.51895	−0.908588
$16$	0	0
$17$	1.02692	0.249064	0.124532	−	0.992216i	$-0.460257\pi$
$17$	1.02692	0.249064	0.124532	+	0.992216i	$0.460257\pi$
$18$	0	0
$19$	2.70210	0.619905	0.309953	−	0.950752i	$-0.399687\pi$
$19$	2.70210	0.619905	0.309953	+	0.950752i	$0.399687\pi$
$20$	0	0
$21$	1.10284	0.240660
$22$	0	0
$23$	4.81659	1.00433	0.502165	−	0.864772i	$-0.332537\pi$
$23$	4.81659	1.00433	0.502165	+	0.864772i	$0.332537\pi$
$24$	0	0
$25$	5.66188	1.13238
$26$	0	0
$27$	5.21451	1.00353
$28$	0	0
$29$	7.80636	1.44960	0.724802	−	0.688957i	$-0.241931\pi$
$29$	7.80636	1.44960	0.724802	+	0.688957i	$0.241931\pi$
$30$	0	0
$31$	0.831300	0.149306	0.0746529	−	0.997210i	$-0.476215\pi$
$31$	0.831300	0.149306	0.0746529	+	0.997210i	$0.476215\pi$
$32$	0	0
$33$	−6.01877	−1.04773
$34$	0	0
$35$	−3.34144	−0.564807
$36$	0	0
$37$	−0.434442	−0.0714218	−0.0357109	−	0.999362i	$-0.511370\pi$
$37$	−0.434442	−0.0714218	−0.0357109	+	0.999362i	$0.511370\pi$
$38$	0	0
$39$	−4.25686	−0.681643
$40$	0	0
$41$	−4.44304	−0.693886	−0.346943	−	0.937886i	$-0.612780\pi$
$41$	−4.44304	−0.693886	−0.346943	+	0.937886i	$0.612780\pi$
$42$	0	0
$43$	−3.33864	−0.509138	−0.254569	−	0.967055i	$-0.581934\pi$
$43$	−3.33864	−0.509138	−0.254569	+	0.967055i	$0.581934\pi$
$44$	0	0
$45$	−6.00341	−0.894935
$46$	0	0
$47$	−1.01027	−0.147364	−0.0736818	−	0.997282i	$-0.523475\pi$
$47$	−1.01027	−0.147364	−0.0736818	+	0.997282i	$0.523475\pi$
$48$	0	0
$49$	−5.95279	−0.850399
$50$	0	0
$51$	−1.10671	−0.154970
$52$	0	0
$53$	4.84576	0.665617	0.332808	−	0.942994i	$-0.392004\pi$
$53$	4.84576	0.665617	0.332808	+	0.942994i	$0.392004\pi$
$54$	0	0
$55$	18.2359	2.45893
$56$	0	0
$57$	−2.91204	−0.385710
$58$	0	0
$59$	−3.03462	−0.395074	−0.197537	−	0.980295i	$-0.563294\pi$
$59$	−3.03462	−0.395074	−0.197537	+	0.980295i	$0.563294\pi$
$60$	0	0
$61$	−12.3907	−1.58646	−0.793230	−	0.608922i	$-0.791602\pi$
$61$	−12.3907	−1.58646	−0.793230	+	0.608922i	$0.791602\pi$
$62$	0	0
$63$	1.88147	0.237043
$64$	0	0
$65$	12.8976	1.59976
$66$	0	0
$67$	11.7831	1.43953	0.719766	−	0.694217i	$-0.244249\pi$
$67$	11.7831	1.43953	0.719766	+	0.694217i	$0.244249\pi$
$68$	0	0
$69$	−5.19082	−0.624901
$70$	0	0
$71$	6.02445	0.714971	0.357485	−	0.933919i	$-0.383634\pi$
$71$	6.02445	0.714971	0.357485	+	0.933919i	$0.383634\pi$
$72$	0	0
$73$	−9.49808	−1.11167	−0.555833	−	0.831294i	$-0.687601\pi$
$73$	−9.49808	−1.11167	−0.555833	+	0.831294i	$0.687601\pi$
$74$	0	0
$75$	−6.10178	−0.704573
$76$	0	0
$77$	−5.71516	−0.651303
$78$	0	0
$79$	−0.506509	−0.0569867	−0.0284933	−	0.999594i	$-0.509071\pi$
$79$	−0.506509	−0.0569867	−0.0284933	+	0.999594i	$0.509071\pi$
$80$	0	0
$81$	−0.103928	−0.0115476
$82$	0	0
$83$	−6.88924	−0.756192	−0.378096	−	0.925766i	$-0.623421\pi$
$83$	−6.88924	−0.756192	−0.378096	+	0.925766i	$0.623421\pi$
$84$	0	0
$85$	3.35315	0.363700
$86$	0	0
$87$	−8.41287	−0.901955
$88$	0	0
$89$	4.80444	0.509270	0.254635	−	0.967037i	$-0.418045\pi$
$89$	4.80444	0.509270	0.254635	+	0.967037i	$0.418045\pi$
$90$	0	0
$91$	−4.04213	−0.423730
$92$	0	0
$93$	−0.895888	−0.0928992
$94$	0	0
$95$	8.82305	0.905226
$96$	0	0
$97$	−5.71755	−0.580529	−0.290265	−	0.956946i	$-0.593743\pi$
$97$	−5.71755	−0.580529	−0.290265	+	0.956946i	$0.593743\pi$
$98$	0	0
$99$	−10.2682	−1.03199
$100$	0	0
$101$	−7.37891	−0.734229	−0.367114	−	0.930176i	$-0.619654\pi$
$101$	−7.37891	−0.734229	−0.367114	+	0.930176i	$0.619654\pi$
$102$	0	0
$103$	−8.52909	−0.840397	−0.420198	−	0.907432i	$-0.638039\pi$
$103$	−8.52909	−0.840397	−0.420198	+	0.907432i	$0.638039\pi$
$104$	0	0
$105$	3.60105	0.351427
$106$	0	0
$107$	14.8133	1.43206	0.716030	−	0.698070i	$-0.245957\pi$
$107$	14.8133	1.43206	0.716030	+	0.698070i	$0.245957\pi$
$108$	0	0
$109$	1.36552	0.130793	0.0653966	−	0.997859i	$-0.479169\pi$
$109$	1.36552	0.130793	0.0653966	+	0.997859i	$0.479169\pi$
$110$	0	0
$111$	0.468196	0.0444392
$112$	0	0
$113$	−3.37934	−0.317901	−0.158951	−	0.987287i	$-0.550811\pi$
$113$	−3.37934	−0.317901	−0.158951	+	0.987287i	$0.550811\pi$
$114$	0	0
$115$	15.7274	1.46659
$116$	0	0
$117$	−7.26230	−0.671400
$118$	0	0
$119$	−1.05088	−0.0963340
$120$	0	0
$121$	20.1905	1.83550
$122$	0	0
$123$	4.78824	0.431741
$124$	0	0
$125$	2.16120	0.193304
$126$	0	0
$127$	−6.71680	−0.596020	−0.298010	−	0.954563i	$-0.596323\pi$
$127$	−6.71680	−0.596020	−0.298010	+	0.954563i	$0.596323\pi$
$128$	0	0
$129$	3.59804	0.316790
$130$	0	0
$131$	16.7573	1.46410	0.732048	−	0.681253i	$-0.238565\pi$
$131$	16.7573	1.46410	0.732048	+	0.681253i	$0.238565\pi$
$132$	0	0
$133$	−2.76515	−0.239769
$134$	0	0
$135$	17.0267	1.46542
$136$	0	0
$137$	6.63713	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$137$	6.63713	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$138$	0	0
$139$	−0.767276	−0.0650795	−0.0325398	−	0.999470i	$-0.510360\pi$
$139$	−0.767276	−0.0650795	−0.0325398	+	0.999470i	$0.510360\pi$
$140$	0	0
$141$	1.08877	0.0916908
$142$	0	0
$143$	22.0600	1.84475
$144$	0	0
$145$	25.4897	2.11681
$146$	0	0
$147$	6.41529	0.529124
$148$	0	0
$149$	−7.38903	−0.605333	−0.302666	−	0.953097i	$-0.597877\pi$
$149$	−7.38903	−0.605333	−0.302666	+	0.953097i	$0.597877\pi$
$150$	0	0
$151$	18.3711	1.49502	0.747509	−	0.664252i	$-0.231250\pi$
$151$	18.3711	1.49502	0.747509	+	0.664252i	$0.231250\pi$
$152$	0	0
$153$	−1.88807	−0.152641
$154$	0	0
$155$	2.71440	0.218026
$156$	0	0
$157$	−14.0085	−1.11800	−0.559001	−	0.829167i	$-0.688815\pi$
$157$	−14.0085	−1.11800	−0.559001	+	0.829167i	$0.688815\pi$
$158$	0	0
$159$	−5.22226	−0.414152
$160$	0	0
$161$	−4.92898	−0.388458
$162$	0	0
$163$	−3.38967	−0.265499	−0.132750	−	0.991150i	$-0.542381\pi$
$163$	−3.38967	−0.265499	−0.132750	+	0.991150i	$0.542381\pi$
$164$	0	0
$165$	−19.6528	−1.52997
$166$	0	0
$167$	8.99666	0.696182	0.348091	−	0.937461i	$-0.386830\pi$
$167$	8.99666	0.696182	0.348091	+	0.937461i	$0.386830\pi$
$168$	0	0
$169$	2.60224	0.200172
$170$	0	0
$171$	−4.96802	−0.379914
$172$	0	0
$173$	−24.7791	−1.88392	−0.941958	−	0.335730i	$-0.891017\pi$
$173$	−24.7791	−1.88392	−0.941958	+	0.335730i	$0.891017\pi$
$174$	0	0
$175$	−5.79399	−0.437984
$176$	0	0
$177$	3.27040	0.245818
$178$	0	0
$179$	11.9890	0.896096	0.448048	−	0.894009i	$-0.352119\pi$
$179$	11.9890	0.896096	0.448048	+	0.894009i	$0.352119\pi$
$180$	0	0
$181$	5.43127	0.403703	0.201852	−	0.979416i	$-0.435304\pi$
$181$	5.43127	0.403703	0.201852	+	0.979416i	$0.435304\pi$
$182$	0	0
$183$	13.3533	0.987108
$184$	0	0
$185$	−1.41856	−0.104295
$186$	0	0
$187$	5.73519	0.419399
$188$	0	0
$189$	−5.33618	−0.388150
$190$	0	0
$191$	−0.347185	−0.0251214	−0.0125607	−	0.999921i	$-0.503998\pi$
$191$	−0.347185	−0.0251214	−0.0125607	+	0.999921i	$0.503998\pi$
$192$	0	0
$193$	15.9039	1.14479	0.572395	−	0.819978i	$-0.306014\pi$
$193$	15.9039	1.14479	0.572395	+	0.819978i	$0.306014\pi$
$194$	0	0
$195$	−13.8997	−0.995380
$196$	0	0
$197$	16.3561	1.16532	0.582661	−	0.812715i	$-0.302012\pi$
$197$	16.3561	1.16532	0.582661	+	0.812715i	$0.302012\pi$
$198$	0	0
$199$	−24.9751	−1.77044	−0.885219	−	0.465175i	$-0.845991\pi$
$199$	−24.9751	−1.77044	−0.885219	+	0.465175i	$0.845991\pi$
$200$	0	0
$201$	−12.6986	−0.895687
$202$	0	0
$203$	−7.98850	−0.560683
$204$	0	0
$205$	−14.5076	−1.01326
$206$	0	0
$207$	−8.85566	−0.615511
$208$	0	0
$209$	15.0908	1.04386
$210$	0	0
$211$	−25.8464	−1.77934	−0.889669	−	0.456605i	$-0.849065\pi$
$211$	−25.8464	−1.77934	−0.889669	+	0.456605i	$0.849065\pi$
$212$	0	0
$213$	−6.49252	−0.444860
$214$	0	0
$215$	−10.9015	−0.743477
$216$	0	0
$217$	−0.850696	−0.0577490
$218$	0	0
$219$	10.2360	0.691687
$220$	0	0
$221$	4.05630	0.272856
$222$	0	0
$223$	1.38828	0.0929659	0.0464830	−	0.998919i	$-0.485199\pi$
$223$	1.38828	0.0929659	0.0464830	+	0.998919i	$0.485199\pi$
$224$	0	0
$225$	−10.4098	−0.693985
$226$	0	0
$227$	−3.78819	−0.251431	−0.125715	−	0.992066i	$-0.540123\pi$
$227$	−3.78819	−0.251431	−0.125715	+	0.992066i	$0.540123\pi$
$228$	0	0
$229$	11.5246	0.761570	0.380785	−	0.924664i	$-0.375654\pi$
$229$	11.5246	0.761570	0.380785	+	0.924664i	$0.375654\pi$
$230$	0	0
$231$	6.15920	0.405246
$232$	0	0
$233$	25.5727	1.67532	0.837661	−	0.546191i	$-0.183923\pi$
$233$	25.5727	1.67532	0.837661	+	0.546191i	$0.183923\pi$
$234$	0	0
$235$	−3.29880	−0.215190
$236$	0	0
$237$	0.545862	0.0354575
$238$	0	0
$239$	5.66649	0.366535	0.183267	−	0.983063i	$-0.441333\pi$
$239$	5.66649	0.366535	0.183267	+	0.983063i	$0.441333\pi$
$240$	0	0
$241$	−11.5249	−0.742382	−0.371191	−	0.928557i	$-0.621051\pi$
$241$	−11.5249	−0.742382	−0.371191	+	0.928557i	$0.621051\pi$
$242$	0	0
$243$	−15.5315	−0.996347
$244$	0	0
$245$	−19.4374	−1.24181
$246$	0	0
$247$	10.6732	0.679121
$248$	0	0
$249$	7.42450	0.470509
$250$	0	0
$251$	−4.64362	−0.293103	−0.146551	−	0.989203i	$-0.546817\pi$
$251$	−4.64362	−0.293103	−0.146551	+	0.989203i	$0.546817\pi$
$252$	0	0
$253$	26.9000	1.69119
$254$	0	0
$255$	−3.61367	−0.226297
$256$	0	0
$257$	−22.4638	−1.40125	−0.700626	−	0.713528i	$-0.747096\pi$
$257$	−22.4638	−1.40125	−0.700626	+	0.713528i	$0.747096\pi$
$258$	0	0
$259$	0.444579	0.0276248
$260$	0	0
$261$	−14.3526	−0.888401
$262$	0	0
$263$	6.76998	0.417455	0.208727	−	0.977974i	$-0.433068\pi$
$263$	6.76998	0.417455	0.208727	+	0.977974i	$0.433068\pi$
$264$	0	0
$265$	15.8226	0.971978
$266$	0	0
$267$	−5.17772	−0.316871
$268$	0	0
$269$	−6.70515	−0.408820	−0.204410	−	0.978885i	$-0.565528\pi$
$269$	−6.70515	−0.408820	−0.204410	+	0.978885i	$0.565528\pi$
$270$	0	0
$271$	21.5962	1.31188	0.655939	−	0.754814i	$-0.272273\pi$
$271$	21.5962	1.31188	0.655939	+	0.754814i	$0.272273\pi$
$272$	0	0
$273$	4.35618	0.263648
$274$	0	0
$275$	31.6207	1.90680
$276$	0	0
$277$	−9.43952	−0.567166	−0.283583	−	0.958948i	$-0.591523\pi$
$277$	−9.43952	−0.567166	−0.283583	+	0.958948i	$0.591523\pi$
$278$	0	0
$279$	−1.52841	−0.0915032
$280$	0	0
$281$	14.2515	0.850175	0.425087	−	0.905152i	$-0.360243\pi$
$281$	14.2515	0.850175	0.425087	+	0.905152i	$0.360243\pi$
$282$	0	0
$283$	−29.4627	−1.75138	−0.875688	−	0.482877i	$-0.839592\pi$
$283$	−29.4627	−1.75138	−0.875688	+	0.482877i	$0.839592\pi$
$284$	0	0
$285$	−9.50856	−0.563239
$286$	0	0
$287$	4.54671	0.268384
$288$	0	0
$289$	−15.9454	−0.937967
$290$	0	0
$291$	6.16177	0.361210
$292$	0	0
$293$	21.4120	1.25090	0.625451	−	0.780264i	$-0.284915\pi$
$293$	21.4120	1.25090	0.625451	+	0.780264i	$0.284915\pi$
$294$	0	0
$295$	−9.90880	−0.576913
$296$	0	0
$297$	29.1222	1.68984
$298$	0	0
$299$	19.0254	1.10027
$300$	0	0
$301$	3.41654	0.196926
$302$	0	0
$303$	7.95221	0.456843
$304$	0	0
$305$	−40.4586	−2.31665
$306$	0	0
$307$	7.46147	0.425849	0.212924	−	0.977069i	$-0.431701\pi$
$307$	7.46147	0.425849	0.212924	+	0.977069i	$0.431701\pi$
$308$	0	0
$309$	9.19176	0.522901
$310$	0	0
$311$	28.2781	1.60350	0.801752	−	0.597657i	$-0.203902\pi$
$311$	28.2781	1.60350	0.801752	+	0.597657i	$0.203902\pi$
$312$	0	0
$313$	−9.37175	−0.529723	−0.264861	−	0.964287i	$-0.585326\pi$
$313$	−9.37175	−0.529723	−0.264861	+	0.964287i	$0.585326\pi$
$314$	0	0
$315$	6.14348	0.346146
$316$	0	0
$317$	17.9833	1.01004	0.505021	−	0.863107i	$-0.331485\pi$
$317$	17.9833	1.01004	0.505021	+	0.863107i	$0.331485\pi$
$318$	0	0
$319$	43.5973	2.44098
$320$	0	0
$321$	−15.9643	−0.891038
$322$	0	0
$323$	2.77484	0.154396
$324$	0	0
$325$	22.3642	1.24054
$326$	0	0
$327$	−1.47162	−0.0813805
$328$	0	0
$329$	1.03385	0.0569978
$330$	0	0
$331$	−6.14537	−0.337780	−0.168890	−	0.985635i	$-0.554018\pi$
$331$	−6.14537	−0.337780	−0.168890	+	0.985635i	$0.554018\pi$
$332$	0	0
$333$	0.798753	0.0437714
$334$	0	0
$335$	38.4747	2.10210
$336$	0	0
$337$	−5.89328	−0.321027	−0.160514	−	0.987034i	$-0.551315\pi$
$337$	−5.89328	−0.321027	−0.160514	+	0.987034i	$0.551315\pi$
$338$	0	0
$339$	3.64189	0.197800
$340$	0	0
$341$	4.64268	0.251415
$342$	0	0
$343$	13.2550	0.715703
$344$	0	0
$345$	−16.9493	−0.912522
$346$	0	0
$347$	−8.74783	−0.469608	−0.234804	−	0.972043i	$-0.575445\pi$
$347$	−8.74783	−0.469608	−0.234804	+	0.972043i	$0.575445\pi$
$348$	0	0
$349$	−2.31251	−0.123786	−0.0618928	−	0.998083i	$-0.519714\pi$
$349$	−2.31251	−0.123786	−0.0618928	+	0.998083i	$0.519714\pi$
$350$	0	0
$351$	20.5971	1.09939
$352$	0	0
$353$	17.5710	0.935211	0.467606	−	0.883937i	$-0.345117\pi$
$353$	17.5710	0.935211	0.467606	+	0.883937i	$0.345117\pi$
$354$	0	0
$355$	19.6714	1.04405
$356$	0	0
$357$	1.13253	0.0599397
$358$	0	0
$359$	4.03256	0.212830	0.106415	−	0.994322i	$-0.466063\pi$
$359$	4.03256	0.212830	0.106415	+	0.994322i	$0.466063\pi$
$360$	0	0
$361$	−11.6986	−0.615718
$362$	0	0
$363$	−21.7593	−1.14206
$364$	0	0
$365$	−31.0136	−1.62333
$366$	0	0
$367$	14.8745	0.776444	0.388222	−	0.921566i	$-0.373089\pi$
$367$	14.8745	0.776444	0.388222	+	0.921566i	$0.373089\pi$
$368$	0	0
$369$	8.16885	0.425253
$370$	0	0
$371$	−4.95883	−0.257450
$372$	0	0
$373$	−9.64250	−0.499270	−0.249635	−	0.968340i	$-0.580311\pi$
$373$	−9.64250	−0.499270	−0.249635	+	0.968340i	$0.580311\pi$
$374$	0	0
$375$	−2.32912	−0.120275
$376$	0	0
$377$	30.8349	1.58808
$378$	0	0
$379$	9.35605	0.480588	0.240294	−	0.970700i	$-0.422756\pi$
$379$	9.35605	0.480588	0.240294	+	0.970700i	$0.422756\pi$
$380$	0	0
$381$	7.23866	0.370848
$382$	0	0
$383$	8.05517	0.411600	0.205800	−	0.978594i	$-0.434020\pi$
$383$	8.05517	0.411600	0.205800	+	0.978594i	$0.434020\pi$
$384$	0	0
$385$	−18.6614	−0.951075
$386$	0	0
$387$	6.13834	0.312029
$388$	0	0
$389$	−9.31149	−0.472111	−0.236056	−	0.971740i	$-0.575855\pi$
$389$	−9.31149	−0.472111	−0.236056	+	0.971740i	$0.575855\pi$
$390$	0	0
$391$	4.94625	0.250143
$392$	0	0
$393$	−18.0593	−0.910971
$394$	0	0
$395$	−1.65388	−0.0832157
$396$	0	0
$397$	−24.1348	−1.21129	−0.605646	−	0.795734i	$-0.707085\pi$
$397$	−24.1348	−1.21129	−0.605646	+	0.795734i	$0.707085\pi$
$398$	0	0
$399$	2.97999	0.149186
$400$	0	0
$401$	−19.8328	−0.990402	−0.495201	−	0.868778i	$-0.664906\pi$
$401$	−19.8328	−0.990402	−0.495201	+	0.868778i	$0.664906\pi$
$402$	0	0
$403$	3.28361	0.163568
$404$	0	0
$405$	−0.339353	−0.0168626
$406$	0	0
$407$	−2.42629	−0.120267
$408$	0	0
$409$	−24.5820	−1.21550	−0.607751	−	0.794127i	$-0.707928\pi$
$409$	−24.5820	−1.21550	−0.607751	+	0.794127i	$0.707928\pi$
$410$	0	0
$411$	−7.15280	−0.352822
$412$	0	0
$413$	3.10543	0.152808
$414$	0	0
$415$	−22.4951	−1.10424
$416$	0	0
$417$	0.826890	0.0404930
$418$	0	0
$419$	23.1595	1.13142	0.565709	−	0.824605i	$-0.308603\pi$
$419$	23.1595	1.13142	0.565709	+	0.824605i	$0.308603\pi$
$420$	0	0
$421$	−13.1851	−0.642605	−0.321302	−	0.946977i	$-0.604121\pi$
$421$	−13.1851	−0.642605	−0.321302	+	0.946977i	$0.604121\pi$
$422$	0	0
$423$	1.85746	0.0903130
$424$	0	0
$425$	5.81429	0.282035
$426$	0	0
$427$	12.6798	0.613617
$428$	0	0
$429$	−23.7739	−1.14782
$430$	0	0
$431$	32.3611	1.55878	0.779389	−	0.626540i	$-0.215530\pi$
$431$	32.3611	1.55878	0.779389	+	0.626540i	$0.215530\pi$
$432$	0	0
$433$	26.0876	1.25369	0.626845	−	0.779144i	$-0.284346\pi$
$433$	26.0876	1.25369	0.626845	+	0.779144i	$0.284346\pi$
$434$	0	0
$435$	−27.4702	−1.31709
$436$	0	0
$437$	13.0149	0.622589
$438$	0	0
$439$	11.1954	0.534327	0.267164	−	0.963651i	$-0.413914\pi$
$439$	11.1954	0.534327	0.267164	+	0.963651i	$0.413914\pi$
$440$	0	0
$441$	10.9446	0.521173
$442$	0	0
$443$	−16.6638	−0.791722	−0.395861	−	0.918310i	$-0.629554\pi$
$443$	−16.6638	−0.791722	−0.395861	+	0.918310i	$0.629554\pi$
$444$	0	0
$445$	15.6877	0.743669
$446$	0	0
$447$	7.96312	0.376643
$448$	0	0
$449$	33.4683	1.57947	0.789734	−	0.613450i	$-0.210219\pi$
$449$	33.4683	1.57947	0.789734	+	0.613450i	$0.210219\pi$
$450$	0	0
$451$	−24.8137	−1.16843
$452$	0	0
$453$	−19.7984	−0.930211
$454$	0	0
$455$	−13.1986	−0.618759
$456$	0	0
$457$	26.4983	1.23954	0.619769	−	0.784784i	$-0.287226\pi$
$457$	26.4983	1.23954	0.619769	+	0.784784i	$0.287226\pi$
$458$	0	0
$459$	5.35488	0.249944
$460$	0	0
$461$	−0.838141	−0.0390361	−0.0195181	−	0.999810i	$-0.506213\pi$
$461$	−0.838141	−0.0390361	−0.0195181	+	0.999810i	$0.506213\pi$
$462$	0	0
$463$	−14.2754	−0.663433	−0.331716	−	0.943379i	$-0.607628\pi$
$463$	−14.2754	−0.663433	−0.331716	+	0.943379i	$0.607628\pi$
$464$	0	0
$465$	−2.92530	−0.135658
$466$	0	0
$467$	16.2421	0.751593	0.375796	−	0.926702i	$-0.377369\pi$
$467$	16.2421	0.751593	0.375796	+	0.926702i	$0.377369\pi$
$468$	0	0
$469$	−12.0580	−0.556787
$470$	0	0
$471$	15.0969	0.695629
$472$	0	0
$473$	−18.6458	−0.857336
$474$	0	0
$475$	15.2990	0.701966
$476$	0	0
$477$	−8.90929	−0.407929
$478$	0	0
$479$	−31.9587	−1.46023	−0.730115	−	0.683324i	$-0.760534\pi$
$479$	−31.9587	−1.46023	−0.730115	+	0.683324i	$0.760534\pi$
$480$	0	0
$481$	−1.71603	−0.0782443
$482$	0	0
$483$	5.31194	0.241701
$484$	0	0
$485$	−18.6692	−0.847727
$486$	0	0
$487$	−3.28288	−0.148761	−0.0743806	−	0.997230i	$-0.523698\pi$
$487$	−3.28288	−0.148761	−0.0743806	+	0.997230i	$0.523698\pi$
$488$	0	0
$489$	3.65303	0.165196
$490$	0	0
$491$	39.0942	1.76430	0.882149	−	0.470970i	$-0.156096\pi$
$491$	39.0942	1.76430	0.882149	+	0.470970i	$0.156096\pi$
$492$	0	0
$493$	8.01650	0.361045
$494$	0	0
$495$	−33.5281	−1.50698
$496$	0	0
$497$	−6.16502	−0.276539
$498$	0	0
$499$	7.85640	0.351701	0.175850	−	0.984417i	$-0.443733\pi$
$499$	7.85640	0.351701	0.175850	+	0.984417i	$0.443733\pi$
$500$	0	0
$501$	−9.69566	−0.433170
$502$	0	0
$503$	−21.6376	−0.964773	−0.482387	−	0.875958i	$-0.660230\pi$
$503$	−21.6376	−0.964773	−0.482387	+	0.875958i	$0.660230\pi$
$504$	0	0
$505$	−24.0940	−1.07217
$506$	0	0
$507$	−2.80442	−0.124549
$508$	0	0
$509$	−14.6709	−0.650277	−0.325139	−	0.945666i	$-0.605411\pi$
$509$	−14.6709	−0.650277	−0.325139	+	0.945666i	$0.605411\pi$
$510$	0	0
$511$	9.71970	0.429974
$512$	0	0
$513$	14.0901	0.622095
$514$	0	0
$515$	−27.8497	−1.22720
$516$	0	0
$517$	−5.64223	−0.248145
$518$	0	0
$519$	26.7043	1.17219
$520$	0	0
$521$	18.6579	0.817419	0.408709	−	0.912665i	$-0.365979\pi$
$521$	18.6579	0.817419	0.408709	+	0.912665i	$0.365979\pi$
$522$	0	0
$523$	−29.7376	−1.30033	−0.650166	−	0.759792i	$-0.725301\pi$
$523$	−29.7376	−1.30033	−0.650166	+	0.759792i	$0.725301\pi$
$524$	0	0
$525$	6.24415	0.272517
$526$	0	0
$527$	0.853678	0.0371868
$528$	0	0
$529$	0.199579	0.00867735
$530$	0	0
$531$	5.57937	0.242124
$532$	0	0
$533$	−17.5499	−0.760169
$534$	0	0
$535$	48.3693	2.09119
$536$	0	0
$537$	−12.9204	−0.557558
$538$	0	0
$539$	−33.2454	−1.43198
$540$	0	0
$541$	−21.9996	−0.945835	−0.472918	−	0.881107i	$-0.656799\pi$
$541$	−21.9996	−0.945835	−0.472918	+	0.881107i	$0.656799\pi$
$542$	0	0
$543$	−5.85326	−0.251187
$544$	0	0
$545$	4.45877	0.190993
$546$	0	0
$547$	10.3020	0.440482	0.220241	−	0.975445i	$-0.429316\pi$
$547$	10.3020	0.440482	0.220241	+	0.975445i	$0.429316\pi$
$548$	0	0
$549$	22.7811	0.972275
$550$	0	0
$551$	21.0936	0.898617
$552$	0	0
$553$	0.518327	0.0220415
$554$	0	0
$555$	1.52878	0.0648930
$556$	0	0
$557$	−9.19434	−0.389577	−0.194788	−	0.980845i	$-0.562402\pi$
$557$	−9.19434	−0.389577	−0.194788	+	0.980845i	$0.562402\pi$
$558$	0	0
$559$	−13.1875	−0.557773
$560$	0	0
$561$	−6.18078	−0.260953
$562$	0	0
$563$	6.24021	0.262993	0.131497	−	0.991317i	$-0.458022\pi$
$563$	6.24021	0.262993	0.131497	+	0.991317i	$0.458022\pi$
$564$	0	0
$565$	−11.0344	−0.464220
$566$	0	0
$567$	0.106353	0.00446642
$568$	0	0
$569$	33.4044	1.40039	0.700194	−	0.713953i	$-0.253097\pi$
$569$	33.4044	1.40039	0.700194	+	0.713953i	$0.253097\pi$
$570$	0	0
$571$	33.6605	1.40865	0.704323	−	0.709879i	$-0.251250\pi$
$571$	33.6605	1.40865	0.704323	+	0.709879i	$0.251250\pi$
$572$	0	0
$573$	0.374160	0.0156307
$574$	0	0
$575$	27.2710	1.13728
$576$	0	0
$577$	−4.84503	−0.201701	−0.100851	−	0.994902i	$-0.532156\pi$
$577$	−4.84503	−0.201701	−0.100851	+	0.994902i	$0.532156\pi$
$578$	0	0
$579$	−17.1396	−0.712297
$580$	0	0
$581$	7.04998	0.292483
$582$	0	0
$583$	27.0629	1.12083
$584$	0	0
$585$	−23.7133	−0.980423
$586$	0	0
$587$	31.6615	1.30681	0.653404	−	0.757009i	$-0.273340\pi$
$587$	31.6615	1.30681	0.653404	+	0.757009i	$0.273340\pi$
$588$	0	0
$589$	2.24626	0.0925555
$590$	0	0
$591$	−17.6269	−0.725072
$592$	0	0
$593$	−33.1295	−1.36046	−0.680232	−	0.732996i	$-0.738121\pi$
$593$	−33.1295	−1.36046	−0.680232	+	0.732996i	$0.738121\pi$
$594$	0	0
$595$	−3.43139	−0.140673
$596$	0	0
$597$	26.9155	1.10158
$598$	0	0
$599$	5.77773	0.236072	0.118036	−	0.993009i	$-0.462340\pi$
$599$	5.77773	0.236072	0.118036	+	0.993009i	$0.462340\pi$
$600$	0	0
$601$	28.9118	1.17934	0.589669	−	0.807645i	$-0.299258\pi$
$601$	28.9118	1.17934	0.589669	+	0.807645i	$0.299258\pi$
$602$	0	0
$603$	−21.6640	−0.882228
$604$	0	0
$605$	65.9273	2.68032
$606$	0	0
$607$	25.4911	1.03465	0.517326	−	0.855789i	$-0.326928\pi$
$607$	25.4911	1.03465	0.517326	+	0.855789i	$0.326928\pi$
$608$	0	0
$609$	8.60917	0.348861
$610$	0	0
$611$	−3.99055	−0.161440
$612$	0	0
$613$	−13.3730	−0.540129	−0.270065	−	0.962842i	$-0.587045\pi$
$613$	−13.3730	−0.540129	−0.270065	+	0.962842i	$0.587045\pi$
$614$	0	0
$615$	15.6348	0.630457
$616$	0	0
$617$	−18.7272	−0.753930	−0.376965	−	0.926227i	$-0.623032\pi$
$617$	−18.7272	−0.753930	−0.376965	+	0.926227i	$0.623032\pi$
$618$	0	0
$619$	23.5446	0.946337	0.473168	−	0.880972i	$-0.343110\pi$
$619$	23.5446	0.946337	0.473168	+	0.880972i	$0.343110\pi$
$620$	0	0
$621$	25.1162	1.00788
$622$	0	0
$623$	−4.91654	−0.196977
$624$	0	0
$625$	−21.2525	−0.850101
$626$	0	0
$627$	−16.2633	−0.649495
$628$	0	0
$629$	−0.446137	−0.0177886
$630$	0	0
$631$	9.79006	0.389736	0.194868	−	0.980829i	$-0.437572\pi$
$631$	9.79006	0.389736	0.194868	+	0.980829i	$0.437572\pi$
$632$	0	0
$633$	27.8545	1.10712
$634$	0	0
$635$	−21.9321	−0.870347
$636$	0	0
$637$	−23.5133	−0.931632
$638$	0	0
$639$	−11.0764	−0.438175
$640$	0	0
$641$	18.7333	0.739921	0.369961	−	0.929047i	$-0.379371\pi$
$641$	18.7333	0.739921	0.369961	+	0.929047i	$0.379371\pi$
$642$	0	0
$643$	−37.9147	−1.49521	−0.747605	−	0.664144i	$-0.768796\pi$
$643$	−37.9147	−1.49521	−0.747605	+	0.664144i	$0.768796\pi$
$644$	0	0
$645$	11.7485	0.462597
$646$	0	0
$647$	−16.2575	−0.639148	−0.319574	−	0.947561i	$-0.603540\pi$
$647$	−16.2575	−0.639148	−0.319574	+	0.947561i	$0.603540\pi$
$648$	0	0
$649$	−16.9479	−0.665263
$650$	0	0
$651$	0.916791	0.0359319
$652$	0	0
$653$	−24.3861	−0.954303	−0.477151	−	0.878821i	$-0.658331\pi$
$653$	−24.3861	−0.954303	−0.477151	+	0.878821i	$0.658331\pi$
$654$	0	0
$655$	54.7170	2.13797
$656$	0	0
$657$	17.4629	0.681293
$658$	0	0
$659$	−3.85316	−0.150098	−0.0750490	−	0.997180i	$-0.523911\pi$
$659$	−3.85316	−0.150098	−0.0750490	+	0.997180i	$0.523911\pi$
$660$	0	0
$661$	−12.6972	−0.493866	−0.246933	−	0.969033i	$-0.579423\pi$
$661$	−12.6972	−0.493866	−0.246933	+	0.969033i	$0.579423\pi$
$662$	0	0
$663$	−4.37145	−0.169773
$664$	0	0
$665$	−9.02892	−0.350126
$666$	0	0
$667$	37.6001	1.45588
$668$	0	0
$669$	−1.49614	−0.0578441
$670$	0	0
$671$	−69.1999	−2.67143
$672$	0	0
$673$	−33.8980	−1.30667	−0.653336	−	0.757068i	$-0.726631\pi$
$673$	−33.8980	−1.30667	−0.653336	+	0.757068i	$0.726631\pi$
$674$	0	0
$675$	29.5239	1.13638
$676$	0	0
$677$	4.64918	0.178682	0.0893412	−	0.996001i	$-0.471524\pi$
$677$	4.64918	0.178682	0.0893412	+	0.996001i	$0.471524\pi$
$678$	0	0
$679$	5.85095	0.224539
$680$	0	0
$681$	4.08251	0.156442
$682$	0	0
$683$	3.40110	0.130139	0.0650697	−	0.997881i	$-0.479273\pi$
$683$	3.40110	0.130139	0.0650697	+	0.997881i	$0.479273\pi$
$684$	0	0
$685$	21.6719	0.828041
$686$	0	0
$687$	−12.4200	−0.473854
$688$	0	0
$689$	19.1406	0.729199
$690$	0	0
$691$	14.2080	0.540499	0.270250	−	0.962790i	$-0.412894\pi$
$691$	14.2080	0.540499	0.270250	+	0.962790i	$0.412894\pi$
$692$	0	0
$693$	10.5077	0.399156
$694$	0	0
$695$	−2.50535	−0.0950334
$696$	0	0
$697$	−4.56264	−0.172822
$698$	0	0
$699$	−27.5595	−1.04240
$700$	0	0
$701$	28.1574	1.06349	0.531746	−	0.846904i	$-0.321536\pi$
$701$	28.1574	1.06349	0.531746	+	0.846904i	$0.321536\pi$
$702$	0	0
$703$	−1.17391	−0.0442747
$704$	0	0
$705$	3.55510	0.133893
$706$	0	0
$707$	7.55108	0.283988
$708$	0	0
$709$	23.5793	0.885540	0.442770	−	0.896635i	$-0.353996\pi$
$709$	23.5793	0.885540	0.442770	+	0.896635i	$0.353996\pi$
$710$	0	0
$711$	0.931253	0.0349247
$712$	0	0
$713$	4.00403	0.149952
$714$	0	0
$715$	72.0314	2.69382
$716$	0	0
$717$	−6.10675	−0.228061
$718$	0	0
$719$	25.1580	0.938233	0.469117	−	0.883136i	$-0.344572\pi$
$719$	25.1580	0.938233	0.469117	+	0.883136i	$0.344572\pi$
$720$	0	0
$721$	8.72810	0.325051
$722$	0	0
$723$	12.4203	0.461916
$724$	0	0
$725$	44.1986	1.64150
$726$	0	0
$727$	−1.67960	−0.0622927	−0.0311464	−	0.999515i	$-0.509916\pi$
$727$	−1.67960	−0.0622927	−0.0311464	+	0.999515i	$0.509916\pi$
$728$	0	0
$729$	17.0500	0.631482
$730$	0	0
$731$	−3.42852	−0.126808
$732$	0	0
$733$	0.315863	0.0116667	0.00583334	−	0.999983i	$-0.498143\pi$
$733$	0.315863	0.0116667	0.00583334	+	0.999983i	$0.498143\pi$
$734$	0	0
$735$	20.9476	0.772662
$736$	0	0
$737$	65.8067	2.42402
$738$	0	0
$739$	−7.07248	−0.260165	−0.130083	−	0.991503i	$-0.541524\pi$
$739$	−7.07248	−0.260165	−0.130083	+	0.991503i	$0.541524\pi$
$740$	0	0
$741$	−11.5025	−0.422554
$742$	0	0
$743$	−0.815443	−0.0299157	−0.0149579	−	0.999888i	$-0.504761\pi$
$743$	−0.815443	−0.0299157	−0.0149579	+	0.999888i	$0.504761\pi$
$744$	0	0
$745$	−24.1270	−0.883946
$746$	0	0
$747$	12.6664	0.463438
$748$	0	0
$749$	−15.1590	−0.553897
$750$	0	0
$751$	−26.3878	−0.962905	−0.481453	−	0.876472i	$-0.659891\pi$
$751$	−26.3878	−0.962905	−0.481453	+	0.876472i	$0.659891\pi$
$752$	0	0
$753$	5.00441	0.182371
$754$	0	0
$755$	59.9862	2.18312
$756$	0	0
$757$	−35.2082	−1.27966	−0.639831	−	0.768516i	$-0.720996\pi$
$757$	−35.2082	−1.27966	−0.639831	+	0.768516i	$0.720996\pi$
$758$	0	0
$759$	−28.9900	−1.05227
$760$	0	0
$761$	19.7712	0.716705	0.358352	−	0.933586i	$-0.383339\pi$
$761$	19.7712	0.716705	0.358352	+	0.933586i	$0.383339\pi$
$762$	0	0
$763$	−1.39738	−0.0505886
$764$	0	0
$765$	−6.16501	−0.222897
$766$	0	0
$767$	−11.9867	−0.432813
$768$	0	0
$769$	37.0240	1.33512	0.667560	−	0.744556i	$-0.267339\pi$
$769$	37.0240	1.33512	0.667560	+	0.744556i	$0.267339\pi$
$770$	0	0
$771$	24.2091	0.871870
$772$	0	0
$773$	−30.8502	−1.10960	−0.554802	−	0.831983i	$-0.687206\pi$
$773$	−30.8502	−1.10960	−0.554802	+	0.831983i	$0.687206\pi$
$774$	0	0
$775$	4.70672	0.169070
$776$	0	0
$777$	−0.479120	−0.0171883
$778$	0	0
$779$	−12.0056	−0.430144
$780$	0	0
$781$	33.6457	1.20394
$782$	0	0
$783$	40.7063	1.45472
$784$	0	0
$785$	−45.7414	−1.63258
$786$	0	0
$787$	−40.6557	−1.44922	−0.724610	−	0.689159i	$-0.757980\pi$
$787$	−40.6557	−1.44922	−0.724610	+	0.689159i	$0.757980\pi$
$788$	0	0
$789$	−7.29597	−0.259743
$790$	0	0
$791$	3.45818	0.122959
$792$	0	0
$793$	−48.9427	−1.73801
$794$	0	0
$795$	−17.0520	−0.604772
$796$	0	0
$797$	2.13809	0.0757351	0.0378676	−	0.999283i	$-0.487944\pi$
$797$	2.13809	0.0757351	0.0378676	+	0.999283i	$0.487944\pi$
$798$	0	0
$799$	−1.03747	−0.0367031
$800$	0	0
$801$	−8.83332	−0.312110
$802$	0	0
$803$	−53.0454	−1.87193
$804$	0	0
$805$	−16.0944	−0.567252
$806$	0	0
$807$	7.22611	0.254371
$808$	0	0
$809$	9.31987	0.327669	0.163835	−	0.986488i	$-0.447614\pi$
$809$	9.31987	0.327669	0.163835	+	0.986488i	$0.447614\pi$
$810$	0	0
$811$	12.1012	0.424931	0.212466	−	0.977169i	$-0.431851\pi$
$811$	12.1012	0.424931	0.212466	+	0.977169i	$0.431851\pi$
$812$	0	0
$813$	−23.2742	−0.816261
$814$	0	0
$815$	−11.0681	−0.387699
$816$	0	0
$817$	−9.02136	−0.315617
$818$	0	0
$819$	7.43175	0.259686
$820$	0	0
$821$	−33.5145	−1.16966	−0.584832	−	0.811154i	$-0.698840\pi$
$821$	−33.5145	−1.16966	−0.584832	+	0.811154i	$0.698840\pi$
$822$	0	0
$823$	−15.0591	−0.524928	−0.262464	−	0.964942i	$-0.584535\pi$
$823$	−15.0591	−0.524928	−0.262464	+	0.964942i	$0.584535\pi$
$824$	0	0
$825$	−34.0775	−1.18643
$826$	0	0
$827$	40.2745	1.40048	0.700241	−	0.713906i	$-0.253076\pi$
$827$	40.2745	1.40048	0.700241	+	0.713906i	$0.253076\pi$
$828$	0	0
$829$	38.9890	1.35414	0.677072	−	0.735917i	$-0.263249\pi$
$829$	38.9890	1.35414	0.677072	+	0.735917i	$0.263249\pi$
$830$	0	0
$831$	10.1729	0.352895
$832$	0	0
$833$	−6.11303	−0.211804
$834$	0	0
$835$	29.3764	1.01661
$836$	0	0
$837$	4.33482	0.149833
$838$	0	0
$839$	−45.7849	−1.58067	−0.790336	−	0.612674i	$-0.790094\pi$
$839$	−45.7849	−1.58067	−0.790336	+	0.612674i	$0.790094\pi$
$840$	0	0
$841$	31.9392	1.10135
$842$	0	0
$843$	−15.3588	−0.528985
$844$	0	0
$845$	8.49697	0.292305
$846$	0	0
$847$	−20.6616	−0.709943
$848$	0	0
$849$	31.7518	1.08972
$850$	0	0
$851$	−2.09253	−0.0717310
$852$	0	0
$853$	−55.3210	−1.89416	−0.947078	−	0.321003i	$-0.895980\pi$
$853$	−55.3210	−1.89416	−0.947078	+	0.321003i	$0.895980\pi$
$854$	0	0
$855$	−16.2218	−0.554775
$856$	0	0
$857$	22.7870	0.778390	0.389195	−	0.921155i	$-0.372753\pi$
$857$	22.7870	0.778390	0.389195	+	0.921155i	$0.372753\pi$
$858$	0	0
$859$	41.9020	1.42968	0.714838	−	0.699290i	$-0.246500\pi$
$859$	41.9020	1.42968	0.714838	+	0.699290i	$0.246500\pi$
$860$	0	0
$861$	−4.89996	−0.166990
$862$	0	0
$863$	2.28371	0.0777383	0.0388692	−	0.999244i	$-0.487624\pi$
$863$	2.28371	0.0777383	0.0388692	+	0.999244i	$0.487624\pi$
$864$	0	0
$865$	−80.9099	−2.75102
$866$	0	0
$867$	17.1843	0.583610
$868$	0	0
$869$	−2.82877	−0.0959596
$870$	0	0
$871$	46.5427	1.57704
$872$	0	0
$873$	10.5121	0.355782
$874$	0	0
$875$	−2.21163	−0.0747667
$876$	0	0
$877$	6.09011	0.205648	0.102824	−	0.994700i	$-0.467212\pi$
$877$	6.09011	0.205648	0.102824	+	0.994700i	$0.467212\pi$
$878$	0	0
$879$	−23.0756	−0.778320
$880$	0	0
$881$	−15.2840	−0.514930	−0.257465	−	0.966288i	$-0.582887\pi$
$881$	−15.2840	−0.514930	−0.257465	+	0.966288i	$0.582887\pi$
$882$	0	0
$883$	24.2776	0.817008	0.408504	−	0.912757i	$-0.366051\pi$
$883$	24.2776	0.817008	0.408504	+	0.912757i	$0.366051\pi$
$884$	0	0
$885$	10.6787	0.358960
$886$	0	0
$887$	−50.7597	−1.70434	−0.852172	−	0.523262i	$-0.824715\pi$
$887$	−50.7597	−1.70434	−0.852172	+	0.523262i	$0.824715\pi$
$888$	0	0
$889$	6.87352	0.230530
$890$	0	0
$891$	−0.580425	−0.0194450
$892$	0	0
$893$	−2.72987	−0.0913515
$894$	0	0
$895$	39.1470	1.30854
$896$	0	0
$897$	−20.5036	−0.684594
$898$	0	0
$899$	6.48942	0.216434
$900$	0	0
$901$	4.97621	0.165782
$902$	0	0
$903$	−3.68199	−0.122529
$904$	0	0
$905$	17.7345	0.589514
$906$	0	0
$907$	23.8807	0.792945	0.396473	−	0.918047i	$-0.370234\pi$
$907$	23.8807	0.792945	0.396473	+	0.918047i	$0.370234\pi$
$908$	0	0
$909$	13.5667	0.449978
$910$	0	0
$911$	−21.5208	−0.713017	−0.356508	−	0.934292i	$-0.616033\pi$
$911$	−21.5208	−0.713017	−0.356508	+	0.934292i	$0.616033\pi$
$912$	0	0
$913$	−38.4754	−1.27335
$914$	0	0
$915$	43.6021	1.44144
$916$	0	0
$917$	−17.1483	−0.566288
$918$	0	0
$919$	−48.4260	−1.59743	−0.798713	−	0.601712i	$-0.794486\pi$
$919$	−48.4260	−1.59743	−0.798713	+	0.601712i	$0.794486\pi$
$920$	0	0
$921$	−8.04119	−0.264966
$922$	0	0
$923$	23.7964	0.783268
$924$	0	0
$925$	−2.45976	−0.0808763
$926$	0	0
$927$	15.6814	0.515044
$928$	0	0
$929$	−9.19380	−0.301639	−0.150819	−	0.988561i	$-0.548191\pi$
$929$	−9.19380	−0.301639	−0.150819	+	0.988561i	$0.548191\pi$
$930$	0	0
$931$	−16.0851	−0.527166
$932$	0	0
$933$	−30.4752	−0.997712
$934$	0	0
$935$	18.7268	0.612433
$936$	0	0
$937$	−4.82338	−0.157573	−0.0787865	−	0.996892i	$-0.525105\pi$
$937$	−4.82338	−0.157573	−0.0787865	+	0.996892i	$0.525105\pi$
$938$	0	0
$939$	10.0999	0.329597
$940$	0	0
$941$	35.7267	1.16466	0.582329	−	0.812953i	$-0.302142\pi$
$941$	35.7267	1.16466	0.582329	+	0.812953i	$0.302142\pi$
$942$	0	0
$943$	−21.4003	−0.696890
$944$	0	0
$945$	−17.4240	−0.566802
$946$	0	0
$947$	21.8224	0.709133	0.354567	−	0.935031i	$-0.384628\pi$
$947$	21.8224	0.709133	0.354567	+	0.935031i	$0.384628\pi$
$948$	0	0
$949$	−37.5171	−1.21786
$950$	0	0
$951$	−19.3805	−0.628456
$952$	0	0
$953$	32.6185	1.05662	0.528308	−	0.849053i	$-0.322827\pi$
$953$	32.6185	1.05662	0.528308	+	0.849053i	$0.322827\pi$
$954$	0	0
$955$	−1.13365	−0.0366840
$956$	0	0
$957$	−46.9846	−1.51880
$958$	0	0
$959$	−6.79199	−0.219325
$960$	0	0
$961$	−30.3089	−0.977708
$962$	0	0
$963$	−27.2354	−0.877649
$964$	0	0
$965$	51.9304	1.67170
$966$	0	0
$967$	−18.6116	−0.598508	−0.299254	−	0.954173i	$-0.596738\pi$
$967$	−18.6116	−0.598508	−0.299254	+	0.954173i	$0.596738\pi$
$968$	0	0
$969$	−2.99043	−0.0960666
$970$	0	0
$971$	−50.5621	−1.62262	−0.811308	−	0.584620i	$-0.801244\pi$
$971$	−50.5621	−1.62262	−0.811308	+	0.584620i	$0.801244\pi$
$972$	0	0
$973$	0.785179	0.0251717
$974$	0	0
$975$	−24.1018	−0.771876
$976$	0	0
$977$	−43.5538	−1.39341	−0.696704	−	0.717358i	$-0.745351\pi$
$977$	−43.5538	−1.39341	−0.696704	+	0.717358i	$0.745351\pi$
$978$	0	0
$979$	26.8321	0.857557
$980$	0	0
$981$	−2.51061	−0.0801576
$982$	0	0
$983$	25.5619	0.815299	0.407649	−	0.913139i	$-0.366349\pi$
$983$	25.5619	0.815299	0.407649	+	0.913139i	$0.366349\pi$
$984$	0	0
$985$	53.4067	1.70168
$986$	0	0
$987$	−1.11417	−0.0354645
$988$	0	0
$989$	−16.0809	−0.511342
$990$	0	0
$991$	−31.8426	−1.01151	−0.505756	−	0.862676i	$-0.668787\pi$
$991$	−31.8426	−1.01151	−0.505756	+	0.862676i	$0.668787\pi$
$992$	0	0
$993$	6.62283	0.210169
$994$	0	0
$995$	−81.5500	−2.58531
$996$	0	0
$997$	27.2984	0.864550	0.432275	−	0.901742i	$-0.357711\pi$
$997$	27.2984	0.864550	0.432275	+	0.901742i	$0.357711\pi$
$998$	0	0
$999$	−2.26540	−0.0716741

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.20	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.20	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-1.07770 q^{3} +3.26525 q^{5} -1.02333 q^{7} -1.83857 q^{9} +O(q^{10})\)	\(q-1.07770 q^{3} +3.26525 q^{5} -1.02333 q^{7} -1.83857 q^{9} +5.58485 q^{11} +3.94997 q^{13} -3.51895 q^{15} +1.02692 q^{17} +2.70210 q^{19} +1.10284 q^{21} +4.81659 q^{23} +5.66188 q^{25} +5.21451 q^{27} +7.80636 q^{29} +0.831300 q^{31} -6.01877 q^{33} -3.34144 q^{35} -0.434442 q^{37} -4.25686 q^{39} -4.44304 q^{41} -3.33864 q^{43} -6.00341 q^{45} -1.01027 q^{47} -5.95279 q^{49} -1.10671 q^{51} +4.84576 q^{53} +18.2359 q^{55} -2.91204 q^{57} -3.03462 q^{59} -12.3907 q^{61} +1.88147 q^{63} +12.8976 q^{65} +11.7831 q^{67} -5.19082 q^{69} +6.02445 q^{71} -9.49808 q^{73} -6.10178 q^{75} -5.71516 q^{77} -0.506509 q^{79} -0.103928 q^{81} -6.88924 q^{83} +3.35315 q^{85} -8.41287 q^{87} +4.80444 q^{89} -4.04213 q^{91} -0.895888 q^{93} +8.82305 q^{95} -5.71755 q^{97} -10.2682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

Introduction

L-functions

Modular forms

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Database

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