LMFDB - Embedded newform 536.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [536,2,Mod(1,536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(536, 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("536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 536 = 2^{3} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	536.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:	$4.27998154834$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.648101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 4x^{2} + 14x + 5 $ x^5 - x^4 - 8x^3 + 4x^2 + 14*x + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.05807$ of defining polynomial
Character	$\chi$	$=$	536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.05807 q^{3} +1.90008 q^{5} +4.57308 q^{7} -1.88049 q^{9} +O(q^{10})$	$q-1.05807 q^{3} +1.90008 q^{5} +4.57308 q^{7} -1.88049 q^{9} +1.38507 q^{11} -3.03582 q^{13} -2.01042 q^{15} -0.302899 q^{17} +7.61155 q^{19} -4.83864 q^{21} -1.98632 q^{23} -1.38969 q^{25} +5.16390 q^{27} +3.54897 q^{29} +1.27750 q^{31} -1.46551 q^{33} +8.68922 q^{35} +3.44776 q^{37} +3.21211 q^{39} +0.420891 q^{41} +7.34784 q^{43} -3.57308 q^{45} -3.91050 q^{47} +13.9130 q^{49} +0.320489 q^{51} +8.87260 q^{53} +2.63175 q^{55} -8.05356 q^{57} -1.39420 q^{59} -12.2727 q^{61} -8.59961 q^{63} -5.76830 q^{65} +1.00000 q^{67} +2.10166 q^{69} -11.1624 q^{71} -0.346598 q^{73} +1.47039 q^{75} +6.33405 q^{77} -3.96672 q^{79} +0.177691 q^{81} -12.8085 q^{83} -0.575533 q^{85} -3.75506 q^{87} -1.24354 q^{89} -13.8830 q^{91} -1.35169 q^{93} +14.4626 q^{95} -9.46269 q^{97} -2.60461 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{3} + q^{5} + 7 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 4 * q^3 + q^5 + 7 * q^7 + 5 * q^9	$ 5 q + 4 q^{3} + q^{5} + 7 q^{7} + 5 q^{9} + 5 q^{11} + 6 q^{13} + 9 q^{15} - 13 q^{17} + 7 q^{19} + 8 q^{21} + 5 q^{23} + 6 q^{25} + 16 q^{27} + 6 q^{29} + 20 q^{31} - 7 q^{33} + 9 q^{35} - 5 q^{37} - 2 q^{39} - 21 q^{41} + 6 q^{43} - 2 q^{45} + 8 q^{47} - 10 q^{51} + 7 q^{53} + 16 q^{55} - 41 q^{57} + 16 q^{59} + 8 q^{61} + 18 q^{63} - 41 q^{65} + 5 q^{67} + 4 q^{69} + 3 q^{71} - 15 q^{73} - 22 q^{75} - 13 q^{77} + q^{79} + 5 q^{81} + 3 q^{83} + q^{85} + q^{87} - 25 q^{89} - 13 q^{91} + 15 q^{93} - 27 q^{95} + 4 q^{97} - 33 q^{99}+O(q^{100}) $ 5 * q + 4 * q^3 + q^5 + 7 * q^7 + 5 * q^9 + 5 * q^11 + 6 * q^13 + 9 * q^15 - 13 * q^17 + 7 * q^19 + 8 * q^21 + 5 * q^23 + 6 * q^25 + 16 * q^27 + 6 * q^29 + 20 * q^31 - 7 * q^33 + 9 * q^35 - 5 * q^37 - 2 * q^39 - 21 * q^41 + 6 * q^43 - 2 * q^45 + 8 * q^47 - 10 * q^51 + 7 * q^53 + 16 * q^55 - 41 * q^57 + 16 * q^59 + 8 * q^61 + 18 * q^63 - 41 * q^65 + 5 * q^67 + 4 * q^69 + 3 * q^71 - 15 * q^73 - 22 * q^75 - 13 * q^77 + q^79 + 5 * q^81 + 3 * q^83 + q^85 + q^87 - 25 * q^89 - 13 * q^91 + 15 * q^93 - 27 * q^95 + 4 * q^97 - 33 * 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
$3$	−1.05807	−0.610877	−0.305439	−	0.952212i	$-0.598803\pi$
$3$	−1.05807	−0.610877	−0.305439	+	0.952212i	$0.598803\pi$
$4$	0	0
$5$	1.90008	0.849742	0.424871	−	0.905254i	$-0.360319\pi$
$5$	1.90008	0.849742	0.424871	+	0.905254i	$0.360319\pi$
$6$	0	0
$7$	4.57308	1.72846	0.864230	−	0.503096i	$-0.167806\pi$
$7$	4.57308	1.72846	0.864230	+	0.503096i	$0.167806\pi$
$8$	0	0
$9$	−1.88049	−0.626829
$10$	0	0
$11$	1.38507	0.417616	0.208808	−	0.977957i	$-0.433042\pi$
$11$	1.38507	0.417616	0.208808	+	0.977957i	$0.433042\pi$
$12$	0	0
$13$	−3.03582	−0.841984	−0.420992	−	0.907064i	$-0.638318\pi$
$13$	−3.03582	−0.841984	−0.420992	+	0.907064i	$0.638318\pi$
$14$	0	0
$15$	−2.01042	−0.519088
$16$	0	0
$17$	−0.302899	−0.0734638	−0.0367319	−	0.999325i	$-0.511695\pi$
$17$	−0.302899	−0.0734638	−0.0367319	+	0.999325i	$0.511695\pi$
$18$	0	0
$19$	7.61155	1.74621	0.873105	−	0.487532i	$-0.162103\pi$
$19$	7.61155	1.74621	0.873105	+	0.487532i	$0.162103\pi$
$20$	0	0
$21$	−4.83864	−1.05588
$22$	0	0
$23$	−1.98632	−0.414175	−0.207088	−	0.978322i	$-0.566399\pi$
$23$	−1.98632	−0.414175	−0.207088	+	0.978322i	$0.566399\pi$
$24$	0	0
$25$	−1.38969	−0.277938
$26$	0	0
$27$	5.16390	0.993793
$28$	0	0
$29$	3.54897	0.659028	0.329514	−	0.944151i	$-0.393115\pi$
$29$	3.54897	0.659028	0.329514	+	0.944151i	$0.393115\pi$
$30$	0	0
$31$	1.27750	0.229446	0.114723	−	0.993398i	$-0.463402\pi$
$31$	1.27750	0.229446	0.114723	+	0.993398i	$0.463402\pi$
$32$	0	0
$33$	−1.46551	−0.255112
$34$	0	0
$35$	8.68922	1.46875
$36$	0	0
$37$	3.44776	0.566809	0.283404	−	0.959001i	$-0.408536\pi$
$37$	3.44776	0.566809	0.283404	+	0.959001i	$0.408536\pi$
$38$	0	0
$39$	3.21211	0.514349
$40$	0	0
$41$	0.420891	0.0657321	0.0328660	−	0.999460i	$-0.489537\pi$
$41$	0.420891	0.0657321	0.0328660	+	0.999460i	$0.489537\pi$
$42$	0	0
$43$	7.34784	1.12054	0.560268	−	0.828312i	$-0.310698\pi$
$43$	7.34784	1.12054	0.560268	+	0.828312i	$0.310698\pi$
$44$	0	0
$45$	−3.57308	−0.532643
$46$	0	0
$47$	−3.91050	−0.570405	−0.285203	−	0.958467i	$-0.592061\pi$
$47$	−3.91050	−0.570405	−0.285203	+	0.958467i	$0.592061\pi$
$48$	0	0
$49$	13.9130	1.98758
$50$	0	0
$51$	0.320489	0.0448774
$52$	0	0
$53$	8.87260	1.21875	0.609373	−	0.792884i	$-0.291421\pi$
$53$	8.87260	1.21875	0.609373	+	0.792884i	$0.291421\pi$
$54$	0	0
$55$	2.63175	0.354866
$56$	0	0
$57$	−8.05356	−1.06672
$58$	0	0
$59$	−1.39420	−0.181510	−0.0907548	−	0.995873i	$-0.528928\pi$
$59$	−1.39420	−0.181510	−0.0907548	+	0.995873i	$0.528928\pi$
$60$	0	0
$61$	−12.2727	−1.57136	−0.785680	−	0.618633i	$-0.787687\pi$
$61$	−12.2727	−1.57136	−0.785680	+	0.618633i	$0.787687\pi$
$62$	0	0
$63$	−8.59961	−1.08345
$64$	0	0
$65$	−5.76830	−0.715469
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	2.10166	0.253010
$70$	0	0
$71$	−11.1624	−1.32473	−0.662365	−	0.749181i	$-0.730447\pi$
$71$	−11.1624	−1.32473	−0.662365	+	0.749181i	$0.730447\pi$
$72$	0	0
$73$	−0.346598	−0.0405662	−0.0202831	−	0.999794i	$-0.506457\pi$
$73$	−0.346598	−0.0405662	−0.0202831	+	0.999794i	$0.506457\pi$
$74$	0	0
$75$	1.47039	0.169786
$76$	0	0
$77$	6.33405	0.721832
$78$	0	0
$79$	−3.96672	−0.446291	−0.223145	−	0.974785i	$-0.571633\pi$
$79$	−3.96672	−0.446291	−0.223145	+	0.974785i	$0.571633\pi$
$80$	0	0
$81$	0.177691	0.0197435
$82$	0	0
$83$	−12.8085	−1.40592	−0.702958	−	0.711231i	$-0.748138\pi$
$83$	−12.8085	−1.40592	−0.702958	+	0.711231i	$0.748138\pi$
$84$	0	0
$85$	−0.575533	−0.0624253
$86$	0	0
$87$	−3.75506	−0.402585
$88$	0	0
$89$	−1.24354	−0.131815	−0.0659073	−	0.997826i	$-0.520994\pi$
$89$	−1.24354	−0.131815	−0.0659073	+	0.997826i	$0.520994\pi$
$90$	0	0
$91$	−13.8830	−1.45534
$92$	0	0
$93$	−1.35169	−0.140163
$94$	0	0
$95$	14.4626	1.48383
$96$	0	0
$97$	−9.46269	−0.960791	−0.480395	−	0.877052i	$-0.659507\pi$
$97$	−9.46269	−0.960791	−0.480395	+	0.877052i	$0.659507\pi$
$98$	0	0
$99$	−2.60461	−0.261774
$100$	0	0
$101$	−11.8491	−1.17903	−0.589513	−	0.807759i	$-0.700680\pi$
$101$	−11.8491	−1.17903	−0.589513	+	0.807759i	$0.700680\pi$
$102$	0	0
$103$	16.1682	1.59310	0.796549	−	0.604574i	$-0.206657\pi$
$103$	16.1682	1.59310	0.796549	+	0.604574i	$0.206657\pi$
$104$	0	0
$105$	−9.19381	−0.897224
$106$	0	0
$107$	−3.28911	−0.317970	−0.158985	−	0.987281i	$-0.550822\pi$
$107$	−3.28911	−0.317970	−0.158985	+	0.987281i	$0.550822\pi$
$108$	0	0
$109$	7.15332	0.685164	0.342582	−	0.939488i	$-0.388699\pi$
$109$	7.15332	0.685164	0.342582	+	0.939488i	$0.388699\pi$
$110$	0	0
$111$	−3.64798	−0.346250
$112$	0	0
$113$	−17.0597	−1.60484	−0.802420	−	0.596759i	$-0.796455\pi$
$113$	−17.0597	−1.60484	−0.802420	+	0.596759i	$0.796455\pi$
$114$	0	0
$115$	−3.77416	−0.351942
$116$	0	0
$117$	5.70881	0.527780
$118$	0	0
$119$	−1.38518	−0.126979
$120$	0	0
$121$	−9.08157	−0.825597
$122$	0	0
$123$	−0.445332	−0.0401542
$124$	0	0
$125$	−12.1409	−1.08592
$126$	0	0
$127$	−10.7923	−0.957660	−0.478830	−	0.877908i	$-0.658939\pi$
$127$	−10.7923	−0.957660	−0.478830	+	0.877908i	$0.658939\pi$
$128$	0	0
$129$	−7.77454	−0.684510
$130$	0	0
$131$	15.9574	1.39421	0.697103	−	0.716971i	$-0.254472\pi$
$131$	15.9574	1.39421	0.697103	+	0.716971i	$0.254472\pi$
$132$	0	0
$133$	34.8082	3.01826
$134$	0	0
$135$	9.81183	0.844468
$136$	0	0
$137$	−12.1077	−1.03443	−0.517214	−	0.855856i	$-0.673031\pi$
$137$	−12.1077	−1.03443	−0.517214	+	0.855856i	$0.673031\pi$
$138$	0	0
$139$	2.63510	0.223506	0.111753	−	0.993736i	$-0.464353\pi$
$139$	2.63510	0.223506	0.111753	+	0.993736i	$0.464353\pi$
$140$	0	0
$141$	4.13759	0.348448
$142$	0	0
$143$	−4.20483	−0.351626
$144$	0	0
$145$	6.74334	0.560004
$146$	0	0
$147$	−14.7210	−1.21417
$148$	0	0
$149$	21.5995	1.76950	0.884750	−	0.466067i	$-0.154329\pi$
$149$	21.5995	1.76950	0.884750	+	0.466067i	$0.154329\pi$
$150$	0	0
$151$	23.0742	1.87775	0.938875	−	0.344259i	$-0.111870\pi$
$151$	23.0742	1.87775	0.938875	+	0.344259i	$0.111870\pi$
$152$	0	0
$153$	0.569598	0.0460493
$154$	0	0
$155$	2.42736	0.194970
$156$	0	0
$157$	−13.6355	−1.08823	−0.544117	−	0.839009i	$-0.683135\pi$
$157$	−13.6355	−1.08823	−0.544117	+	0.839009i	$0.683135\pi$
$158$	0	0
$159$	−9.38784	−0.744504
$160$	0	0
$161$	−9.08357	−0.715886
$162$	0	0
$163$	−12.2472	−0.959276	−0.479638	−	0.877467i	$-0.659232\pi$
$163$	−12.2472	−0.959276	−0.479638	+	0.877467i	$0.659232\pi$
$164$	0	0
$165$	−2.78458	−0.216779
$166$	0	0
$167$	9.14919	0.707986	0.353993	−	0.935248i	$-0.384824\pi$
$167$	9.14919	0.707986	0.353993	+	0.935248i	$0.384824\pi$
$168$	0	0
$169$	−3.78382	−0.291063
$170$	0	0
$171$	−14.3134	−1.09458
$172$	0	0
$173$	−9.47706	−0.720528	−0.360264	−	0.932850i	$-0.617313\pi$
$173$	−9.47706	−0.720528	−0.360264	+	0.932850i	$0.617313\pi$
$174$	0	0
$175$	−6.35517	−0.480405
$176$	0	0
$177$	1.47516	0.110880
$178$	0	0
$179$	−19.4553	−1.45416	−0.727078	−	0.686554i	$-0.759122\pi$
$179$	−19.4553	−1.45416	−0.727078	+	0.686554i	$0.759122\pi$
$180$	0	0
$181$	10.1587	0.755087	0.377543	−	0.925992i	$-0.376769\pi$
$181$	10.1587	0.755087	0.377543	+	0.925992i	$0.376769\pi$
$182$	0	0
$183$	12.9854	0.959908
$184$	0	0
$185$	6.55103	0.481641
$186$	0	0
$187$	−0.419538	−0.0306796
$188$	0	0
$189$	23.6149	1.71773
$190$	0	0
$191$	−27.2745	−1.97351	−0.986755	−	0.162216i	$-0.948136\pi$
$191$	−27.2745	−1.97351	−0.986755	+	0.162216i	$0.948136\pi$
$192$	0	0
$193$	0.354480	0.0255160	0.0127580	−	0.999919i	$-0.495939\pi$
$193$	0.354480	0.0255160	0.0127580	+	0.999919i	$0.495939\pi$
$194$	0	0
$195$	6.10327	0.437064
$196$	0	0
$197$	−4.41509	−0.314562	−0.157281	−	0.987554i	$-0.550273\pi$
$197$	−4.41509	−0.314562	−0.157281	+	0.987554i	$0.550273\pi$
$198$	0	0
$199$	−4.45602	−0.315879	−0.157939	−	0.987449i	$-0.550485\pi$
$199$	−4.45602	−0.315879	−0.157939	+	0.987449i	$0.550485\pi$
$200$	0	0
$201$	−1.05807	−0.0746305
$202$	0	0
$203$	16.2297	1.13910
$204$	0	0
$205$	0.799726	0.0558553
$206$	0	0
$207$	3.73524	0.259617
$208$	0	0
$209$	10.5426	0.729245
$210$	0	0
$211$	7.90725	0.544358	0.272179	−	0.962247i	$-0.412256\pi$
$211$	7.90725	0.544358	0.272179	+	0.962247i	$0.412256\pi$
$212$	0	0
$213$	11.8106	0.809248
$214$	0	0
$215$	13.9615	0.952166
$216$	0	0
$217$	5.84212	0.396589
$218$	0	0
$219$	0.366725	0.0247810
$220$	0	0
$221$	0.919546	0.0618554
$222$	0	0
$223$	−9.59764	−0.642705	−0.321353	−	0.946960i	$-0.604138\pi$
$223$	−9.59764	−0.642705	−0.321353	+	0.946960i	$0.604138\pi$
$224$	0	0
$225$	2.61330	0.174220
$226$	0	0
$227$	23.8507	1.58303	0.791513	−	0.611153i	$-0.209294\pi$
$227$	23.8507	1.58303	0.791513	+	0.611153i	$0.209294\pi$
$228$	0	0
$229$	4.48375	0.296294	0.148147	−	0.988965i	$-0.452669\pi$
$229$	4.48375	0.296294	0.148147	+	0.988965i	$0.452669\pi$
$230$	0	0
$231$	−6.70187	−0.440951
$232$	0	0
$233$	−0.537866	−0.0352368	−0.0176184	−	0.999845i	$-0.505608\pi$
$233$	−0.537866	−0.0352368	−0.0176184	+	0.999845i	$0.505608\pi$
$234$	0	0
$235$	−7.43027	−0.484697
$236$	0	0
$237$	4.19707	0.272629
$238$	0	0
$239$	4.00535	0.259085	0.129542	−	0.991574i	$-0.458649\pi$
$239$	4.00535	0.259085	0.129542	+	0.991574i	$0.458649\pi$
$240$	0	0
$241$	13.0956	0.843564	0.421782	−	0.906697i	$-0.361405\pi$
$241$	13.0956	0.843564	0.421782	+	0.906697i	$0.361405\pi$
$242$	0	0
$243$	−15.6797	−1.00585
$244$	0	0
$245$	26.4359	1.68893
$246$	0	0
$247$	−23.1073	−1.47028
$248$	0	0
$249$	13.5523	0.858842
$250$	0	0
$251$	−16.0898	−1.01558	−0.507789	−	0.861481i	$-0.669537\pi$
$251$	−16.0898	−1.01558	−0.507789	+	0.861481i	$0.669537\pi$
$252$	0	0
$253$	−2.75119	−0.172966
$254$	0	0
$255$	0.608955	0.0381342
$256$	0	0
$257$	−30.6290	−1.91059	−0.955293	−	0.295661i	$-0.904460\pi$
$257$	−30.6290	−1.91059	−0.955293	+	0.295661i	$0.904460\pi$
$258$	0	0
$259$	15.7669	0.979706
$260$	0	0
$261$	−6.67380	−0.413098
$262$	0	0
$263$	−21.6988	−1.33801	−0.669003	−	0.743260i	$-0.733279\pi$
$263$	−21.6988	−1.33801	−0.669003	+	0.743260i	$0.733279\pi$
$264$	0	0
$265$	16.8587	1.03562
$266$	0	0
$267$	1.31575	0.0805225
$268$	0	0
$269$	32.0023	1.95122	0.975608	−	0.219522i	$-0.0704497\pi$
$269$	32.0023	1.95122	0.975608	+	0.219522i	$0.0704497\pi$
$270$	0	0
$271$	24.3840	1.48122	0.740612	−	0.671933i	$-0.234536\pi$
$271$	24.3840	1.48122	0.740612	+	0.671933i	$0.234536\pi$
$272$	0	0
$273$	14.6892	0.889032
$274$	0	0
$275$	−1.92483	−0.116071
$276$	0	0
$277$	3.74035	0.224736	0.112368	−	0.993667i	$-0.464156\pi$
$277$	3.74035	0.224736	0.112368	+	0.993667i	$0.464156\pi$
$278$	0	0
$279$	−2.40233	−0.143824
$280$	0	0
$281$	−18.0781	−1.07845	−0.539225	−	0.842162i	$-0.681283\pi$
$281$	−18.0781	−1.07845	−0.539225	+	0.842162i	$0.681283\pi$
$282$	0	0
$283$	−24.2679	−1.44258	−0.721288	−	0.692636i	$-0.756449\pi$
$283$	−24.2679	−1.44258	−0.721288	+	0.692636i	$0.756449\pi$
$284$	0	0
$285$	−15.3024	−0.906437
$286$	0	0
$287$	1.92477	0.113615
$288$	0	0
$289$	−16.9083	−0.994603
$290$	0	0
$291$	10.0122	0.586925
$292$	0	0
$293$	−5.65979	−0.330649	−0.165324	−	0.986239i	$-0.552867\pi$
$293$	−5.65979	−0.330649	−0.165324	+	0.986239i	$0.552867\pi$
$294$	0	0
$295$	−2.64910	−0.154236
$296$	0	0
$297$	7.15238	0.415023
$298$	0	0
$299$	6.03009	0.348729
$300$	0	0
$301$	33.6023	1.93680
$302$	0	0
$303$	12.5371	0.720240
$304$	0	0
$305$	−23.3192	−1.33525
$306$	0	0
$307$	11.3497	0.647761	0.323881	−	0.946098i	$-0.395012\pi$
$307$	11.3497	0.647761	0.323881	+	0.946098i	$0.395012\pi$
$308$	0	0
$309$	−17.1071	−0.973187
$310$	0	0
$311$	−1.15776	−0.0656505	−0.0328253	−	0.999461i	$-0.510450\pi$
$311$	−1.15776	−0.0656505	−0.0328253	+	0.999461i	$0.510450\pi$
$312$	0	0
$313$	25.0416	1.41543	0.707716	−	0.706497i	$-0.249726\pi$
$313$	25.0416	1.41543	0.707716	+	0.706497i	$0.249726\pi$
$314$	0	0
$315$	−16.3400	−0.920653
$316$	0	0
$317$	−23.2826	−1.30768	−0.653840	−	0.756633i	$-0.726843\pi$
$317$	−23.2826	−1.30768	−0.653840	+	0.756633i	$0.726843\pi$
$318$	0	0
$319$	4.91559	0.275220
$320$	0	0
$321$	3.48011	0.194241
$322$	0	0
$323$	−2.30553	−0.128283
$324$	0	0
$325$	4.21885	0.234020
$326$	0	0
$327$	−7.56872	−0.418551
$328$	0	0
$329$	−17.8830	−0.985923
$330$	0	0
$331$	25.5689	1.40539	0.702697	−	0.711489i	$-0.251979\pi$
$331$	25.5689	1.40539	0.702697	+	0.711489i	$0.251979\pi$
$332$	0	0
$333$	−6.48347	−0.355292
$334$	0	0
$335$	1.90008	0.103813
$336$	0	0
$337$	19.9431	1.08637	0.543184	−	0.839614i	$-0.317219\pi$
$337$	19.9431	1.08637	0.543184	+	0.839614i	$0.317219\pi$
$338$	0	0
$339$	18.0504	0.980361
$340$	0	0
$341$	1.76944	0.0958203
$342$	0	0
$343$	31.6139	1.70699
$344$	0	0
$345$	3.99333	0.214994
$346$	0	0
$347$	11.6729	0.626633	0.313316	−	0.949649i	$-0.398560\pi$
$347$	11.6729	0.626633	0.313316	+	0.949649i	$0.398560\pi$
$348$	0	0
$349$	27.7463	1.48523	0.742613	−	0.669720i	$-0.233586\pi$
$349$	27.7463	1.48523	0.742613	+	0.669720i	$0.233586\pi$
$350$	0	0
$351$	−15.6767	−0.836758
$352$	0	0
$353$	−32.9130	−1.75178	−0.875892	−	0.482507i	$-0.839726\pi$
$353$	−32.9130	−1.75178	−0.875892	+	0.482507i	$0.839726\pi$
$354$	0	0
$355$	−21.2094	−1.12568
$356$	0	0
$357$	1.46562	0.0775688
$358$	0	0
$359$	−7.19436	−0.379704	−0.189852	−	0.981813i	$-0.560801\pi$
$359$	−7.19436	−0.379704	−0.189852	+	0.981813i	$0.560801\pi$
$360$	0	0
$361$	38.9358	2.04925
$362$	0	0
$363$	9.60894	0.504339
$364$	0	0
$365$	−0.658564	−0.0344708
$366$	0	0
$367$	16.4917	0.860859	0.430430	−	0.902624i	$-0.358362\pi$
$367$	16.4917	0.860859	0.430430	+	0.902624i	$0.358362\pi$
$368$	0	0
$369$	−0.791479	−0.0412028
$370$	0	0
$371$	40.5751	2.10655
$372$	0	0
$373$	−37.0141	−1.91652	−0.958258	−	0.285904i	$-0.907706\pi$
$373$	−37.0141	−1.91652	−0.958258	+	0.285904i	$0.907706\pi$
$374$	0	0
$375$	12.8460	0.663363
$376$	0	0
$377$	−10.7740	−0.554891
$378$	0	0
$379$	−3.18579	−0.163643	−0.0818216	−	0.996647i	$-0.526074\pi$
$379$	−3.18579	−0.163643	−0.0818216	+	0.996647i	$0.526074\pi$
$380$	0	0
$381$	11.4190	0.585013
$382$	0	0
$383$	12.7258	0.650259	0.325130	−	0.945669i	$-0.394592\pi$
$383$	12.7258	0.650259	0.325130	+	0.945669i	$0.394592\pi$
$384$	0	0
$385$	12.0352	0.613371
$386$	0	0
$387$	−13.8175	−0.702384
$388$	0	0
$389$	12.4683	0.632166	0.316083	−	0.948732i	$-0.397632\pi$
$389$	12.4683	0.632166	0.316083	+	0.948732i	$0.397632\pi$
$390$	0	0
$391$	0.601653	0.0304269
$392$	0	0
$393$	−16.8841	−0.851689
$394$	0	0
$395$	−7.53709	−0.379232
$396$	0	0
$397$	−6.19054	−0.310694	−0.155347	−	0.987860i	$-0.549650\pi$
$397$	−6.19054	−0.310694	−0.155347	+	0.987860i	$0.549650\pi$
$398$	0	0
$399$	−36.8296	−1.84378
$400$	0	0
$401$	27.5634	1.37645	0.688226	−	0.725496i	$-0.258390\pi$
$401$	27.5634	1.37645	0.688226	+	0.725496i	$0.258390\pi$
$402$	0	0
$403$	−3.87826	−0.193190
$404$	0	0
$405$	0.337628	0.0167768
$406$	0	0
$407$	4.77541	0.236708
$408$	0	0
$409$	32.3809	1.60113	0.800565	−	0.599245i	$-0.204533\pi$
$409$	32.3809	1.60113	0.800565	+	0.599245i	$0.204533\pi$
$410$	0	0
$411$	12.8108	0.631909
$412$	0	0
$413$	−6.37579	−0.313732
$414$	0	0
$415$	−24.3372	−1.19467
$416$	0	0
$417$	−2.78812	−0.136535
$418$	0	0
$419$	8.13191	0.397270	0.198635	−	0.980074i	$-0.436349\pi$
$419$	8.13191	0.397270	0.198635	+	0.980074i	$0.436349\pi$
$420$	0	0
$421$	0.924858	0.0450748	0.0225374	−	0.999746i	$-0.492826\pi$
$421$	0.924858	0.0450748	0.0225374	+	0.999746i	$0.492826\pi$
$422$	0	0
$423$	7.35365	0.357546
$424$	0	0
$425$	0.420936	0.0204184
$426$	0	0
$427$	−56.1241	−2.71603
$428$	0	0
$429$	4.44901	0.214800
$430$	0	0
$431$	3.27142	0.157579	0.0787895	−	0.996891i	$-0.474895\pi$
$431$	3.27142	0.157579	0.0787895	+	0.996891i	$0.474895\pi$
$432$	0	0
$433$	14.4606	0.694934	0.347467	−	0.937692i	$-0.387042\pi$
$433$	14.4606	0.694934	0.347467	+	0.937692i	$0.387042\pi$
$434$	0	0
$435$	−7.13493	−0.342094
$436$	0	0
$437$	−15.1189	−0.723237
$438$	0	0
$439$	−30.1106	−1.43710	−0.718551	−	0.695474i	$-0.755194\pi$
$439$	−30.1106	−1.43710	−0.718551	+	0.695474i	$0.755194\pi$
$440$	0	0
$441$	−26.1633	−1.24587
$442$	0	0
$443$	−25.9642	−1.23359	−0.616797	−	0.787122i	$-0.711570\pi$
$443$	−25.9642	−1.23359	−0.616797	+	0.787122i	$0.711570\pi$
$444$	0	0
$445$	−2.36282	−0.112008
$446$	0	0
$447$	−22.8538	−1.08095
$448$	0	0
$449$	−32.2654	−1.52270	−0.761348	−	0.648343i	$-0.775462\pi$
$449$	−32.2654	−1.52270	−0.761348	+	0.648343i	$0.775462\pi$
$450$	0	0
$451$	0.582965	0.0274507
$452$	0	0
$453$	−24.4141	−1.14707
$454$	0	0
$455$	−26.3789	−1.23666
$456$	0	0
$457$	−36.0500	−1.68635	−0.843175	−	0.537639i	$-0.819316\pi$
$457$	−36.0500	−1.68635	−0.843175	+	0.537639i	$0.819316\pi$
$458$	0	0
$459$	−1.56414	−0.0730078
$460$	0	0
$461$	10.4583	0.487092	0.243546	−	0.969889i	$-0.421689\pi$
$461$	10.4583	0.487092	0.243546	+	0.969889i	$0.421689\pi$
$462$	0	0
$463$	22.7754	1.05846	0.529231	−	0.848478i	$-0.322480\pi$
$463$	22.7754	1.05846	0.529231	+	0.848478i	$0.322480\pi$
$464$	0	0
$465$	−2.56832	−0.119103
$466$	0	0
$467$	−16.1114	−0.745546	−0.372773	−	0.927923i	$-0.621593\pi$
$467$	−16.1114	−0.745546	−0.372773	+	0.927923i	$0.621593\pi$
$468$	0	0
$469$	4.57308	0.211165
$470$	0	0
$471$	14.4274	0.664778
$472$	0	0
$473$	10.1773	0.467953
$474$	0	0
$475$	−10.5777	−0.485339
$476$	0	0
$477$	−16.6848	−0.763945
$478$	0	0
$479$	23.5315	1.07518	0.537590	−	0.843206i	$-0.319335\pi$
$479$	23.5315	1.07518	0.537590	+	0.843206i	$0.319335\pi$
$480$	0	0
$481$	−10.4668	−0.477244
$482$	0	0
$483$	9.61106	0.437318
$484$	0	0
$485$	−17.9799	−0.816424
$486$	0	0
$487$	25.6428	1.16199	0.580994	−	0.813908i	$-0.302664\pi$
$487$	25.6428	1.16199	0.580994	+	0.813908i	$0.302664\pi$
$488$	0	0
$489$	12.9584	0.586000
$490$	0	0
$491$	14.9550	0.674909	0.337455	−	0.941342i	$-0.390434\pi$
$491$	14.9550	0.674909	0.337455	+	0.941342i	$0.390434\pi$
$492$	0	0
$493$	−1.07498	−0.0484147
$494$	0	0
$495$	−4.94898	−0.222440
$496$	0	0
$497$	−51.0464	−2.28974
$498$	0	0
$499$	13.6452	0.610845	0.305422	−	0.952217i	$-0.401202\pi$
$499$	13.6452	0.610845	0.305422	+	0.952217i	$0.401202\pi$
$500$	0	0
$501$	−9.68049	−0.432492
$502$	0	0
$503$	10.9054	0.486248	0.243124	−	0.969995i	$-0.421828\pi$
$503$	10.9054	0.486248	0.243124	+	0.969995i	$0.421828\pi$
$504$	0	0
$505$	−22.5142	−1.00187
$506$	0	0
$507$	4.00355	0.177804
$508$	0	0
$509$	4.91646	0.217918	0.108959	−	0.994046i	$-0.465248\pi$
$509$	4.91646	0.217918	0.108959	+	0.994046i	$0.465248\pi$
$510$	0	0
$511$	−1.58502	−0.0701171
$512$	0	0
$513$	39.3053	1.73537
$514$	0	0
$515$	30.7209	1.35372
$516$	0	0
$517$	−5.41633	−0.238210
$518$	0	0
$519$	10.0274	0.440154
$520$	0	0
$521$	0.884915	0.0387688	0.0193844	−	0.999812i	$-0.493829\pi$
$521$	0.884915	0.0387688	0.0193844	+	0.999812i	$0.493829\pi$
$522$	0	0
$523$	−24.4803	−1.07045	−0.535224	−	0.844710i	$-0.679773\pi$
$523$	−24.4803	−1.07045	−0.535224	+	0.844710i	$0.679773\pi$
$524$	0	0
$525$	6.72422	0.293469
$526$	0	0
$527$	−0.386954	−0.0168560
$528$	0	0
$529$	−19.0546	−0.828459
$530$	0	0
$531$	2.62178	0.113775
$532$	0	0
$533$	−1.27775	−0.0553453
$534$	0	0
$535$	−6.24957	−0.270192
$536$	0	0
$537$	20.5851	0.888311
$538$	0	0
$539$	19.2706	0.830043
$540$	0	0
$541$	−39.5680	−1.70116	−0.850581	−	0.525843i	$-0.823750\pi$
$541$	−39.5680	−1.70116	−0.850581	+	0.525843i	$0.823750\pi$
$542$	0	0
$543$	−10.7486	−0.461265
$544$	0	0
$545$	13.5919	0.582213
$546$	0	0
$547$	−10.7435	−0.459359	−0.229680	−	0.973266i	$-0.573768\pi$
$547$	−10.7435	−0.459359	−0.229680	+	0.973266i	$0.573768\pi$
$548$	0	0
$549$	23.0787	0.984974
$550$	0	0
$551$	27.0132	1.15080
$552$	0	0
$553$	−18.1401	−0.771397
$554$	0	0
$555$	−6.93145	−0.294224
$556$	0	0
$557$	−17.9062	−0.758711	−0.379356	−	0.925251i	$-0.623854\pi$
$557$	−17.9062	−0.758711	−0.379356	+	0.925251i	$0.623854\pi$
$558$	0	0
$559$	−22.3067	−0.943473
$560$	0	0
$561$	0.443901	0.0187415
$562$	0	0
$563$	9.42461	0.397200	0.198600	−	0.980081i	$-0.436361\pi$
$563$	9.42461	0.397200	0.198600	+	0.980081i	$0.436361\pi$
$564$	0	0
$565$	−32.4148	−1.36370
$566$	0	0
$567$	0.812595	0.0341258
$568$	0	0
$569$	19.7515	0.828028	0.414014	−	0.910271i	$-0.364126\pi$
$569$	19.7515	0.828028	0.414014	+	0.910271i	$0.364126\pi$
$570$	0	0
$571$	−13.2538	−0.554654	−0.277327	−	0.960776i	$-0.589449\pi$
$571$	−13.2538	−0.554654	−0.277327	+	0.960776i	$0.589449\pi$
$572$	0	0
$573$	28.8583	1.20557
$574$	0	0
$575$	2.76037	0.115115
$576$	0	0
$577$	−2.96805	−0.123562	−0.0617808	−	0.998090i	$-0.519678\pi$
$577$	−2.96805	−0.123562	−0.0617808	+	0.998090i	$0.519678\pi$
$578$	0	0
$579$	−0.375065	−0.0155872
$580$	0	0
$581$	−58.5743	−2.43007
$582$	0	0
$583$	12.2892	0.508967
$584$	0	0
$585$	10.8472	0.448477
$586$	0	0
$587$	−4.03785	−0.166660	−0.0833299	−	0.996522i	$-0.526556\pi$
$587$	−4.03785	−0.166660	−0.0833299	+	0.996522i	$0.526556\pi$
$588$	0	0
$589$	9.72378	0.400661
$590$	0	0
$591$	4.67147	0.192159
$592$	0	0
$593$	5.98673	0.245845	0.122923	−	0.992416i	$-0.460773\pi$
$593$	5.98673	0.245845	0.122923	+	0.992416i	$0.460773\pi$
$594$	0	0
$595$	−2.63196	−0.107900
$596$	0	0
$597$	4.71478	0.192963
$598$	0	0
$599$	46.1184	1.88435	0.942174	−	0.335125i	$-0.108779\pi$
$599$	46.1184	1.88435	0.942174	+	0.335125i	$0.108779\pi$
$600$	0	0
$601$	−44.6121	−1.81976	−0.909882	−	0.414867i	$-0.863828\pi$
$601$	−44.6121	−1.81976	−0.909882	+	0.414867i	$0.863828\pi$
$602$	0	0
$603$	−1.88049	−0.0765793
$604$	0	0
$605$	−17.2557	−0.701545
$606$	0	0
$607$	−4.02234	−0.163262	−0.0816309	−	0.996663i	$-0.526013\pi$
$607$	−4.02234	−0.163262	−0.0816309	+	0.996663i	$0.526013\pi$
$608$	0	0
$609$	−17.1722	−0.695853
$610$	0	0
$611$	11.8716	0.480272
$612$	0	0
$613$	23.7800	0.960467	0.480233	−	0.877141i	$-0.340552\pi$
$613$	23.7800	0.960467	0.480233	+	0.877141i	$0.340552\pi$
$614$	0	0
$615$	−0.846167	−0.0341207
$616$	0	0
$617$	4.37212	0.176015	0.0880075	−	0.996120i	$-0.471950\pi$
$617$	4.37212	0.176015	0.0880075	+	0.996120i	$0.471950\pi$
$618$	0	0
$619$	21.2801	0.855319	0.427660	−	0.903940i	$-0.359338\pi$
$619$	21.2801	0.855319	0.427660	+	0.903940i	$0.359338\pi$
$620$	0	0
$621$	−10.2571	−0.411605
$622$	0	0
$623$	−5.68679	−0.227836
$624$	0	0
$625$	−16.1203	−0.644812
$626$	0	0
$627$	−11.1548	−0.445479
$628$	0	0
$629$	−1.04432	−0.0416399
$630$	0	0
$631$	10.2531	0.408169	0.204085	−	0.978953i	$-0.434578\pi$
$631$	10.2531	0.408169	0.204085	+	0.978953i	$0.434578\pi$
$632$	0	0
$633$	−8.36643	−0.332536
$634$	0	0
$635$	−20.5062	−0.813764
$636$	0	0
$637$	−42.2374	−1.67351
$638$	0	0
$639$	20.9907	0.830379
$640$	0	0
$641$	18.4000	0.726756	0.363378	−	0.931642i	$-0.381623\pi$
$641$	18.4000	0.726756	0.363378	+	0.931642i	$0.381623\pi$
$642$	0	0
$643$	17.0712	0.673222	0.336611	−	0.941644i	$-0.390719\pi$
$643$	17.0712	0.673222	0.336611	+	0.941644i	$0.390719\pi$
$644$	0	0
$645$	−14.7722	−0.581657
$646$	0	0
$647$	5.27806	0.207502	0.103751	−	0.994603i	$-0.466915\pi$
$647$	5.27806	0.207502	0.103751	+	0.994603i	$0.466915\pi$
$648$	0	0
$649$	−1.93107	−0.0758012
$650$	0	0
$651$	−6.18137	−0.242267
$652$	0	0
$653$	33.1565	1.29752	0.648758	−	0.760995i	$-0.275289\pi$
$653$	33.1565	1.29752	0.648758	+	0.760995i	$0.275289\pi$
$654$	0	0
$655$	30.3204	1.18472
$656$	0	0
$657$	0.651773	0.0254281
$658$	0	0
$659$	21.1843	0.825224	0.412612	−	0.910907i	$-0.364617\pi$
$659$	21.1843	0.825224	0.412612	+	0.910907i	$0.364617\pi$
$660$	0	0
$661$	−19.2421	−0.748431	−0.374216	−	0.927342i	$-0.622088\pi$
$661$	−19.2421	−0.748431	−0.374216	+	0.927342i	$0.622088\pi$
$662$	0	0
$663$	−0.972945	−0.0377860
$664$	0	0
$665$	66.1385	2.56474
$666$	0	0
$667$	−7.04938	−0.272953
$668$	0	0
$669$	10.1550	0.392614
$670$	0	0
$671$	−16.9986	−0.656224
$672$	0	0
$673$	−29.6424	−1.14263	−0.571315	−	0.820731i	$-0.693566\pi$
$673$	−29.6424	−1.14263	−0.571315	+	0.820731i	$0.693566\pi$
$674$	0	0
$675$	−7.17623	−0.276213
$676$	0	0
$677$	8.27015	0.317848	0.158924	−	0.987291i	$-0.449198\pi$
$677$	8.27015	0.317848	0.158924	+	0.987291i	$0.449198\pi$
$678$	0	0
$679$	−43.2736	−1.66069
$680$	0	0
$681$	−25.2357	−0.967034
$682$	0	0
$683$	16.5298	0.632496	0.316248	−	0.948676i	$-0.397577\pi$
$683$	16.5298	0.632496	0.316248	+	0.948676i	$0.397577\pi$
$684$	0	0
$685$	−23.0056	−0.878998
$686$	0	0
$687$	−4.74412	−0.181000
$688$	0	0
$689$	−26.9356	−1.02616
$690$	0	0
$691$	28.3837	1.07977	0.539883	−	0.841740i	$-0.318469\pi$
$691$	28.3837	1.07977	0.539883	+	0.841740i	$0.318469\pi$
$692$	0	0
$693$	−11.9111	−0.452465
$694$	0	0
$695$	5.00690	0.189923
$696$	0	0
$697$	−0.127487	−0.00482893
$698$	0	0
$699$	0.569100	0.0215254
$700$	0	0
$701$	30.1787	1.13983	0.569916	−	0.821703i	$-0.306976\pi$
$701$	30.1787	1.13983	0.569916	+	0.821703i	$0.306976\pi$
$702$	0	0
$703$	26.2428	0.989767
$704$	0	0
$705$	7.86175	0.296091
$706$	0	0
$707$	−54.1867	−2.03790
$708$	0	0
$709$	−7.04420	−0.264551	−0.132275	−	0.991213i	$-0.542228\pi$
$709$	−7.04420	−0.264551	−0.132275	+	0.991213i	$0.542228\pi$
$710$	0	0
$711$	7.45937	0.279748
$712$	0	0
$713$	−2.53752	−0.0950310
$714$	0	0
$715$	−7.98952	−0.298791
$716$	0	0
$717$	−4.23794	−0.158269
$718$	0	0
$719$	4.35882	0.162557	0.0812783	−	0.996691i	$-0.474100\pi$
$719$	4.35882	0.162557	0.0812783	+	0.996691i	$0.474100\pi$
$720$	0	0
$721$	73.9383	2.75361
$722$	0	0
$723$	−13.8561	−0.515314
$724$	0	0
$725$	−4.93198	−0.183169
$726$	0	0
$727$	3.29869	0.122341	0.0611707	−	0.998127i	$-0.480517\pi$
$727$	3.29869	0.122341	0.0611707	+	0.998127i	$0.480517\pi$
$728$	0	0
$729$	16.0572	0.594710
$730$	0	0
$731$	−2.22566	−0.0823189
$732$	0	0
$733$	13.6906	0.505673	0.252836	−	0.967509i	$-0.418637\pi$
$733$	13.6906	0.505673	0.252836	+	0.967509i	$0.418637\pi$
$734$	0	0
$735$	−27.9710	−1.03173
$736$	0	0
$737$	1.38507	0.0510199
$738$	0	0
$739$	−29.6985	−1.09248	−0.546238	−	0.837630i	$-0.683940\pi$
$739$	−29.6985	−1.09248	−0.546238	+	0.837630i	$0.683940\pi$
$740$	0	0
$741$	24.4491	0.898161
$742$	0	0
$743$	46.6323	1.71077	0.855386	−	0.517991i	$-0.173320\pi$
$743$	46.6323	1.71077	0.855386	+	0.517991i	$0.173320\pi$
$744$	0	0
$745$	41.0408	1.50362
$746$	0	0
$747$	24.0862	0.881269
$748$	0	0
$749$	−15.0413	−0.549599
$750$	0	0
$751$	−35.0094	−1.27751	−0.638755	−	0.769410i	$-0.720550\pi$
$751$	−35.0094	−1.27751	−0.638755	+	0.769410i	$0.720550\pi$
$752$	0	0
$753$	17.0241	0.620394
$754$	0	0
$755$	43.8428	1.59560
$756$	0	0
$757$	3.97408	0.144441	0.0722203	−	0.997389i	$-0.476992\pi$
$757$	3.97408	0.144441	0.0722203	+	0.997389i	$0.476992\pi$
$758$	0	0
$759$	2.91096	0.105661
$760$	0	0
$761$	10.3461	0.375044	0.187522	−	0.982260i	$-0.439954\pi$
$761$	10.3461	0.375044	0.187522	+	0.982260i	$0.439954\pi$
$762$	0	0
$763$	32.7127	1.18428
$764$	0	0
$765$	1.08228	0.0391300
$766$	0	0
$767$	4.23254	0.152828
$768$	0	0
$769$	53.5524	1.93115	0.965574	−	0.260128i	$-0.0837648\pi$
$769$	53.5524	1.93115	0.965574	+	0.260128i	$0.0837648\pi$
$770$	0	0
$771$	32.4077	1.16713
$772$	0	0
$773$	36.6192	1.31710	0.658551	−	0.752536i	$-0.271170\pi$
$773$	36.6192	1.31710	0.658551	+	0.752536i	$0.271170\pi$
$774$	0	0
$775$	−1.77533	−0.0637719
$776$	0	0
$777$	−16.6825	−0.598480
$778$	0	0
$779$	3.20363	0.114782
$780$	0	0
$781$	−15.4607	−0.553228
$782$	0	0
$783$	18.3265	0.654937
$784$	0	0
$785$	−25.9086	−0.924718
$786$	0	0
$787$	24.9907	0.890823	0.445411	−	0.895326i	$-0.353057\pi$
$787$	24.9907	0.890823	0.445411	+	0.895326i	$0.353057\pi$
$788$	0	0
$789$	22.9589	0.817357
$790$	0	0
$791$	−78.0153	−2.77390
$792$	0	0
$793$	37.2577	1.32306
$794$	0	0
$795$	−17.8377	−0.632637
$796$	0	0
$797$	44.6569	1.58183	0.790914	−	0.611928i	$-0.209606\pi$
$797$	44.6569	1.58183	0.790914	+	0.611928i	$0.209606\pi$
$798$	0	0
$799$	1.18449	0.0419042
$800$	0	0
$801$	2.33845	0.0826252
$802$	0	0
$803$	−0.480064	−0.0169411
$804$	0	0
$805$	−17.2595	−0.608318
$806$	0	0
$807$	−33.8607	−1.19195
$808$	0	0
$809$	39.2956	1.38156	0.690779	−	0.723066i	$-0.257268\pi$
$809$	39.2956	1.38156	0.690779	+	0.723066i	$0.257268\pi$
$810$	0	0
$811$	10.3816	0.364546	0.182273	−	0.983248i	$-0.441654\pi$
$811$	10.3816	0.364546	0.182273	+	0.983248i	$0.441654\pi$
$812$	0	0
$813$	−25.8000	−0.904846
$814$	0	0
$815$	−23.2707	−0.815137
$816$	0	0
$817$	55.9285	1.95669
$818$	0	0
$819$	26.1068	0.912247
$820$	0	0
$821$	−26.8595	−0.937402	−0.468701	−	0.883357i	$-0.655278\pi$
$821$	−26.8595	−0.937402	−0.468701	+	0.883357i	$0.655278\pi$
$822$	0	0
$823$	31.2147	1.08807	0.544037	−	0.839061i	$-0.316895\pi$
$823$	31.2147	1.08807	0.544037	+	0.839061i	$0.316895\pi$
$824$	0	0
$825$	2.03660	0.0709054
$826$	0	0
$827$	−17.3446	−0.603130	−0.301565	−	0.953446i	$-0.597509\pi$
$827$	−17.3446	−0.603130	−0.301565	+	0.953446i	$0.597509\pi$
$828$	0	0
$829$	9.92619	0.344751	0.172375	−	0.985031i	$-0.444856\pi$
$829$	9.92619	0.344751	0.172375	+	0.985031i	$0.444856\pi$
$830$	0	0
$831$	−3.95755	−0.137286
$832$	0	0
$833$	−4.21425	−0.146015
$834$	0	0
$835$	17.3842	0.601605
$836$	0	0
$837$	6.59689	0.228022
$838$	0	0
$839$	15.1775	0.523987	0.261993	−	0.965070i	$-0.415620\pi$
$839$	15.1775	0.523987	0.261993	+	0.965070i	$0.415620\pi$
$840$	0	0
$841$	−16.4048	−0.565682
$842$	0	0
$843$	19.1279	0.658800
$844$	0	0
$845$	−7.18956	−0.247328
$846$	0	0
$847$	−41.5307	−1.42701
$848$	0	0
$849$	25.6771	0.881236
$850$	0	0
$851$	−6.84834	−0.234758
$852$	0	0
$853$	−11.9096	−0.407778	−0.203889	−	0.978994i	$-0.565358\pi$
$853$	−11.9096	−0.407778	−0.203889	+	0.978994i	$0.565358\pi$
$854$	0	0
$855$	−27.1967	−0.930107
$856$	0	0
$857$	44.0014	1.50306	0.751529	−	0.659700i	$-0.229317\pi$
$857$	44.0014	1.50306	0.751529	+	0.659700i	$0.229317\pi$
$858$	0	0
$859$	−19.2705	−0.657500	−0.328750	−	0.944417i	$-0.606627\pi$
$859$	−19.2705	−0.657500	−0.328750	+	0.944417i	$0.606627\pi$
$860$	0	0
$861$	−2.03654	−0.0694050
$862$	0	0
$863$	2.70007	0.0919115	0.0459558	−	0.998943i	$-0.485367\pi$
$863$	2.70007	0.0919115	0.0459558	+	0.998943i	$0.485367\pi$
$864$	0	0
$865$	−18.0072	−0.612263
$866$	0	0
$867$	17.8901	0.607580
$868$	0	0
$869$	−5.49420	−0.186378
$870$	0	0
$871$	−3.03582	−0.102865
$872$	0	0
$873$	17.7945	0.602251
$874$	0	0
$875$	−55.5214	−1.87697
$876$	0	0
$877$	−3.50197	−0.118253	−0.0591266	−	0.998250i	$-0.518832\pi$
$877$	−3.50197	−0.118253	−0.0591266	+	0.998250i	$0.518832\pi$
$878$	0	0
$879$	5.98846	0.201986
$880$	0	0
$881$	9.98124	0.336277	0.168138	−	0.985763i	$-0.446224\pi$
$881$	9.98124	0.336277	0.168138	+	0.985763i	$0.446224\pi$
$882$	0	0
$883$	−37.4747	−1.26112	−0.630562	−	0.776139i	$-0.717176\pi$
$883$	−37.4747	−1.26112	−0.630562	+	0.776139i	$0.717176\pi$
$884$	0	0
$885$	2.80293	0.0942195
$886$	0	0
$887$	53.4674	1.79526	0.897629	−	0.440752i	$-0.145288\pi$
$887$	53.4674	1.79526	0.897629	+	0.440752i	$0.145288\pi$
$888$	0	0
$889$	−49.3539	−1.65528
$890$	0	0
$891$	0.246115	0.00824518
$892$	0	0
$893$	−29.7650	−0.996047
$894$	0	0
$895$	−36.9666	−1.23566
$896$	0	0
$897$	−6.38026	−0.213031
$898$	0	0
$899$	4.53382	0.151211
$900$	0	0
$901$	−2.68750	−0.0895338
$902$	0	0
$903$	−35.5536	−1.18315
$904$	0	0
$905$	19.3023	0.641629
$906$	0	0
$907$	22.3149	0.740955	0.370478	−	0.928841i	$-0.379194\pi$
$907$	22.3149	0.740955	0.370478	+	0.928841i	$0.379194\pi$
$908$	0	0
$909$	22.2820	0.739047
$910$	0	0
$911$	−25.3958	−0.841401	−0.420700	−	0.907200i	$-0.638216\pi$
$911$	−25.3958	−0.841401	−0.420700	+	0.907200i	$0.638216\pi$
$912$	0	0
$913$	−17.7407	−0.587132
$914$	0	0
$915$	24.6733	0.815675
$916$	0	0
$917$	72.9745	2.40983
$918$	0	0
$919$	10.6898	0.352625	0.176313	−	0.984334i	$-0.443583\pi$
$919$	10.6898	0.352625	0.176313	+	0.984334i	$0.443583\pi$
$920$	0	0
$921$	−12.0088	−0.395703
$922$	0	0
$923$	33.8869	1.11540
$924$	0	0
$925$	−4.79133	−0.157538
$926$	0	0
$927$	−30.4040	−0.998600
$928$	0	0
$929$	−50.6856	−1.66294	−0.831471	−	0.555568i	$-0.812501\pi$
$929$	−50.6856	−1.66294	−0.831471	+	0.555568i	$0.812501\pi$
$930$	0	0
$931$	105.900	3.47073
$932$	0	0
$933$	1.22499	0.0401044
$934$	0	0
$935$	−0.797156	−0.0260698
$936$	0	0
$937$	−41.1124	−1.34308	−0.671541	−	0.740967i	$-0.734367\pi$
$937$	−41.1124	−1.34308	−0.671541	+	0.740967i	$0.734367\pi$
$938$	0	0
$939$	−26.4957	−0.864656
$940$	0	0
$941$	8.20556	0.267494	0.133747	−	0.991016i	$-0.457299\pi$
$941$	8.20556	0.267494	0.133747	+	0.991016i	$0.457299\pi$
$942$	0	0
$943$	−0.836021	−0.0272246
$944$	0	0
$945$	44.8702	1.45963
$946$	0	0
$947$	−50.7742	−1.64994	−0.824970	−	0.565177i	$-0.808808\pi$
$947$	−50.7742	−1.64994	−0.824970	+	0.565177i	$0.808808\pi$
$948$	0	0
$949$	1.05221	0.0341561
$950$	0	0
$951$	24.6346	0.798832
$952$	0	0
$953$	−14.3967	−0.466356	−0.233178	−	0.972434i	$-0.574913\pi$
$953$	−14.3967	−0.466356	−0.233178	+	0.972434i	$0.574913\pi$
$954$	0	0
$955$	−51.8237	−1.67698
$956$	0	0
$957$	−5.20104	−0.168126
$958$	0	0
$959$	−55.3694	−1.78797
$960$	0	0
$961$	−29.3680	−0.947354
$962$	0	0
$963$	6.18512	0.199313
$964$	0	0
$965$	0.673541	0.0216821
$966$	0	0
$967$	−45.1153	−1.45081	−0.725405	−	0.688322i	$-0.758347\pi$
$967$	−45.1153	−1.45081	−0.725405	+	0.688322i	$0.758347\pi$
$968$	0	0
$969$	2.43942	0.0783654
$970$	0	0
$971$	−14.8841	−0.477654	−0.238827	−	0.971062i	$-0.576763\pi$
$971$	−14.8841	−0.477654	−0.238827	+	0.971062i	$0.576763\pi$
$972$	0	0
$973$	12.0505	0.386322
$974$	0	0
$975$	−4.46384	−0.142957
$976$	0	0
$977$	−39.6136	−1.26735	−0.633676	−	0.773599i	$-0.718455\pi$
$977$	−39.6136	−1.26735	−0.633676	+	0.773599i	$0.718455\pi$
$978$	0	0
$979$	−1.72239	−0.0550478
$980$	0	0
$981$	−13.4517	−0.429481
$982$	0	0
$983$	29.9192	0.954274	0.477137	−	0.878829i	$-0.341674\pi$
$983$	29.9192	0.954274	0.477137	+	0.878829i	$0.341674\pi$
$984$	0	0
$985$	−8.38903	−0.267297
$986$	0	0
$987$	18.9215	0.602278
$988$	0	0
$989$	−14.5951	−0.464098
$990$	0	0
$991$	−9.09477	−0.288905	−0.144452	−	0.989512i	$-0.546142\pi$
$991$	−9.09477	−0.288905	−0.144452	+	0.989512i	$0.546142\pi$
$992$	0	0
$993$	−27.0537	−0.858523
$994$	0	0
$995$	−8.46680	−0.268416
$996$	0	0
$997$	19.2682	0.610229	0.305115	−	0.952316i	$-0.401305\pi$
$997$	19.2682	0.610229	0.305115	+	0.952316i	$0.401305\pi$
$998$	0	0
$999$	17.8039	0.563290

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	536.2.a.e.1.2	✓	5
3.2	odd	2		4824.2.a.t.1.2		5
4.3	odd	2		1072.2.a.m.1.4		5
8.3	odd	2		4288.2.a.bd.1.2		5
8.5	even	2		4288.2.a.ba.1.4		5
12.11	even	2		9648.2.a.cc.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.2.a.e.1.2	✓	5	1.1	even	1	trivial
1072.2.a.m.1.4		5	4.3	odd	2
4288.2.a.ba.1.4		5	8.5	even	2
4288.2.a.bd.1.2		5	8.3	odd	2
4824.2.a.t.1.2		5	3.2	odd	2
9648.2.a.cc.1.2		5	12.11	even	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:	\( N \)	\(=\)	\( 536 = 2^{3} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	536.a (trivial)

\(f(q)\)	\(=\)	\(q-1.05807 q^{3} +1.90008 q^{5} +4.57308 q^{7} -1.88049 q^{9} +O(q^{10})\)	\(q-1.05807 q^{3} +1.90008 q^{5} +4.57308 q^{7} -1.88049 q^{9} +1.38507 q^{11} -3.03582 q^{13} -2.01042 q^{15} -0.302899 q^{17} +7.61155 q^{19} -4.83864 q^{21} -1.98632 q^{23} -1.38969 q^{25} +5.16390 q^{27} +3.54897 q^{29} +1.27750 q^{31} -1.46551 q^{33} +8.68922 q^{35} +3.44776 q^{37} +3.21211 q^{39} +0.420891 q^{41} +7.34784 q^{43} -3.57308 q^{45} -3.91050 q^{47} +13.9130 q^{49} +0.320489 q^{51} +8.87260 q^{53} +2.63175 q^{55} -8.05356 q^{57} -1.39420 q^{59} -12.2727 q^{61} -8.59961 q^{63} -5.76830 q^{65} +1.00000 q^{67} +2.10166 q^{69} -11.1624 q^{71} -0.346598 q^{73} +1.47039 q^{75} +6.33405 q^{77} -3.96672 q^{79} +0.177691 q^{81} -12.8085 q^{83} -0.575533 q^{85} -3.75506 q^{87} -1.24354 q^{89} -13.8830 q^{91} -1.35169 q^{93} +14.4626 q^{95} -9.46269 q^{97} -2.60461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{3} + q^{5} + 7 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 4 * q^3 + q^5 + 7 * q^7 + 5 * q^9	\( 5 q + 4 q^{3} + q^{5} + 7 q^{7} + 5 q^{9} + 5 q^{11} + 6 q^{13} + 9 q^{15} - 13 q^{17} + 7 q^{19} + 8 q^{21} + 5 q^{23} + 6 q^{25} + 16 q^{27} + 6 q^{29} + 20 q^{31} - 7 q^{33} + 9 q^{35} - 5 q^{37} - 2 q^{39} - 21 q^{41} + 6 q^{43} - 2 q^{45} + 8 q^{47} - 10 q^{51} + 7 q^{53} + 16 q^{55} - 41 q^{57} + 16 q^{59} + 8 q^{61} + 18 q^{63} - 41 q^{65} + 5 q^{67} + 4 q^{69} + 3 q^{71} - 15 q^{73} - 22 q^{75} - 13 q^{77} + q^{79} + 5 q^{81} + 3 q^{83} + q^{85} + q^{87} - 25 q^{89} - 13 q^{91} + 15 q^{93} - 27 q^{95} + 4 q^{97} - 33 q^{99}+O(q^{100}) \) 5 * q + 4 * q^3 + q^5 + 7 * q^7 + 5 * q^9 + 5 * q^11 + 6 * q^13 + 9 * q^15 - 13 * q^17 + 7 * q^19 + 8 * q^21 + 5 * q^23 + 6 * q^25 + 16 * q^27 + 6 * q^29 + 20 * q^31 - 7 * q^33 + 9 * q^35 - 5 * q^37 - 2 * q^39 - 21 * q^41 + 6 * q^43 - 2 * q^45 + 8 * q^47 - 10 * q^51 + 7 * q^53 + 16 * q^55 - 41 * q^57 + 16 * q^59 + 8 * q^61 + 18 * q^63 - 41 * q^65 + 5 * q^67 + 4 * q^69 + 3 * q^71 - 15 * q^73 - 22 * q^75 - 13 * q^77 + q^79 + 5 * q^81 + 3 * q^83 + q^85 + q^87 - 25 * q^89 - 13 * q^91 + 15 * q^93 - 27 * q^95 + 4 * q^97 - 33 * q^99

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