LMFDB - Embedded newform 4024.2.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39336 q^{3} +0.556589 q^{5} +4.72926 q^{7} +2.72817 q^{9} +O(q^{10})$	$q-2.39336 q^{3} +0.556589 q^{5} +4.72926 q^{7} +2.72817 q^{9} -5.01195 q^{11} -3.61869 q^{13} -1.33212 q^{15} +3.83843 q^{17} -1.78519 q^{19} -11.3188 q^{21} +5.21932 q^{23} -4.69021 q^{25} +0.650590 q^{27} -1.24959 q^{29} -0.555451 q^{31} +11.9954 q^{33} +2.63225 q^{35} -3.05865 q^{37} +8.66083 q^{39} -2.20087 q^{41} +9.74301 q^{43} +1.51847 q^{45} +6.42347 q^{47} +15.3659 q^{49} -9.18674 q^{51} +9.49212 q^{53} -2.78959 q^{55} +4.27260 q^{57} -1.60506 q^{59} -11.1880 q^{61} +12.9022 q^{63} -2.01412 q^{65} +3.68589 q^{67} -12.4917 q^{69} +3.22236 q^{71} -0.536130 q^{73} +11.2254 q^{75} -23.7028 q^{77} -1.13206 q^{79} -9.74160 q^{81} -1.50141 q^{83} +2.13643 q^{85} +2.99071 q^{87} -12.0111 q^{89} -17.1137 q^{91} +1.32939 q^{93} -0.993616 q^{95} +3.66773 q^{97} -13.6734 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * 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
$3$	−2.39336	−1.38181	−0.690903	−	0.722947i	$-0.742787\pi$
$3$	−2.39336	−1.38181	−0.690903	+	0.722947i	$0.742787\pi$
$4$	0	0
$5$	0.556589	0.248914	0.124457	−	0.992225i	$-0.460281\pi$
$5$	0.556589	0.248914	0.124457	+	0.992225i	$0.460281\pi$
$6$	0	0
$7$	4.72926	1.78749	0.893746	−	0.448573i	$-0.148068\pi$
$7$	4.72926	1.78749	0.893746	+	0.448573i	$0.148068\pi$
$8$	0	0
$9$	2.72817	0.909389
$10$	0	0
$11$	−5.01195	−1.51116	−0.755580	−	0.655057i	$-0.772645\pi$
$11$	−5.01195	−1.51116	−0.755580	+	0.655057i	$0.772645\pi$
$12$	0	0
$13$	−3.61869	−1.00365	−0.501823	−	0.864971i	$-0.667337\pi$
$13$	−3.61869	−1.00365	−0.501823	+	0.864971i	$0.667337\pi$
$14$	0	0
$15$	−1.33212	−0.343951
$16$	0	0
$17$	3.83843	0.930956	0.465478	−	0.885059i	$-0.345882\pi$
$17$	3.83843	0.930956	0.465478	+	0.885059i	$0.345882\pi$
$18$	0	0
$19$	−1.78519	−0.409550	−0.204775	−	0.978809i	$-0.565646\pi$
$19$	−1.78519	−0.409550	−0.204775	+	0.978809i	$0.565646\pi$
$20$	0	0
$21$	−11.3188	−2.46997
$22$	0	0
$23$	5.21932	1.08830	0.544151	−	0.838987i	$-0.316852\pi$
$23$	5.21932	1.08830	0.544151	+	0.838987i	$0.316852\pi$
$24$	0	0
$25$	−4.69021	−0.938042
$26$	0	0
$27$	0.650590	0.125206
$28$	0	0
$29$	−1.24959	−0.232042	−0.116021	−	0.993247i	$-0.537014\pi$
$29$	−1.24959	−0.232042	−0.116021	+	0.993247i	$0.537014\pi$
$30$	0	0
$31$	−0.555451	−0.0997619	−0.0498810	−	0.998755i	$-0.515884\pi$
$31$	−0.555451	−0.0997619	−0.0498810	+	0.998755i	$0.515884\pi$
$32$	0	0
$33$	11.9954	2.08813
$34$	0	0
$35$	2.63225	0.444932
$36$	0	0
$37$	−3.05865	−0.502839	−0.251420	−	0.967878i	$-0.580897\pi$
$37$	−3.05865	−0.502839	−0.251420	+	0.967878i	$0.580897\pi$
$38$	0	0
$39$	8.66083	1.38684
$40$	0	0
$41$	−2.20087	−0.343718	−0.171859	−	0.985122i	$-0.554977\pi$
$41$	−2.20087	−0.343718	−0.171859	+	0.985122i	$0.554977\pi$
$42$	0	0
$43$	9.74301	1.48580	0.742898	−	0.669405i	$-0.233451\pi$
$43$	9.74301	1.48580	0.742898	+	0.669405i	$0.233451\pi$
$44$	0	0
$45$	1.51847	0.226360
$46$	0	0
$47$	6.42347	0.936960	0.468480	−	0.883474i	$-0.344802\pi$
$47$	6.42347	0.936960	0.468480	+	0.883474i	$0.344802\pi$
$48$	0	0
$49$	15.3659	2.19513
$50$	0	0
$51$	−9.18674	−1.28640
$52$	0	0
$53$	9.49212	1.30384	0.651921	−	0.758287i	$-0.273963\pi$
$53$	9.49212	1.30384	0.651921	+	0.758287i	$0.273963\pi$
$54$	0	0
$55$	−2.78959	−0.376149
$56$	0	0
$57$	4.27260	0.565919
$58$	0	0
$59$	−1.60506	−0.208961	−0.104480	−	0.994527i	$-0.533318\pi$
$59$	−1.60506	−0.208961	−0.104480	+	0.994527i	$0.533318\pi$
$60$	0	0
$61$	−11.1880	−1.43247	−0.716237	−	0.697857i	$-0.754137\pi$
$61$	−11.1880	−1.43247	−0.716237	+	0.697857i	$0.754137\pi$
$62$	0	0
$63$	12.9022	1.62553
$64$	0	0
$65$	−2.01412	−0.249821
$66$	0	0
$67$	3.68589	0.450303	0.225152	−	0.974324i	$-0.427712\pi$
$67$	3.68589	0.450303	0.225152	+	0.974324i	$0.427712\pi$
$68$	0	0
$69$	−12.4917	−1.50382
$70$	0	0
$71$	3.22236	0.382424	0.191212	−	0.981549i	$-0.438758\pi$
$71$	3.22236	0.382424	0.191212	+	0.981549i	$0.438758\pi$
$72$	0	0
$73$	−0.536130	−0.0627493	−0.0313746	−	0.999508i	$-0.509988\pi$
$73$	−0.536130	−0.0627493	−0.0313746	+	0.999508i	$0.509988\pi$
$74$	0	0
$75$	11.2254	1.29619
$76$	0	0
$77$	−23.7028	−2.70118
$78$	0	0
$79$	−1.13206	−0.127367	−0.0636835	−	0.997970i	$-0.520285\pi$
$79$	−1.13206	−0.127367	−0.0636835	+	0.997970i	$0.520285\pi$
$80$	0	0
$81$	−9.74160	−1.08240
$82$	0	0
$83$	−1.50141	−0.164801	−0.0824003	−	0.996599i	$-0.526259\pi$
$83$	−1.50141	−0.164801	−0.0824003	+	0.996599i	$0.526259\pi$
$84$	0	0
$85$	2.13643	0.231728
$86$	0	0
$87$	2.99071	0.320637
$88$	0	0
$89$	−12.0111	−1.27318	−0.636589	−	0.771203i	$-0.719655\pi$
$89$	−12.0111	−1.27318	−0.636589	+	0.771203i	$0.719655\pi$
$90$	0	0
$91$	−17.1137	−1.79401
$92$	0	0
$93$	1.32939	0.137852
$94$	0	0
$95$	−0.993616	−0.101943
$96$	0	0
$97$	3.66773	0.372402	0.186201	−	0.982512i	$-0.440382\pi$
$97$	3.66773	0.372402	0.186201	+	0.982512i	$0.440382\pi$
$98$	0	0
$99$	−13.6734	−1.37423
$100$	0	0
$101$	−1.87474	−0.186544	−0.0932720	−	0.995641i	$-0.529733\pi$
$101$	−1.87474	−0.186544	−0.0932720	+	0.995641i	$0.529733\pi$
$102$	0	0
$103$	7.90229	0.778636	0.389318	−	0.921103i	$-0.372711\pi$
$103$	7.90229	0.778636	0.389318	+	0.921103i	$0.372711\pi$
$104$	0	0
$105$	−6.29992	−0.614810
$106$	0	0
$107$	1.86428	0.180227	0.0901135	−	0.995931i	$-0.471277\pi$
$107$	1.86428	0.180227	0.0901135	+	0.995931i	$0.471277\pi$
$108$	0	0
$109$	13.2452	1.26866	0.634328	−	0.773064i	$-0.281277\pi$
$109$	13.2452	1.26866	0.634328	+	0.773064i	$0.281277\pi$
$110$	0	0
$111$	7.32045	0.694827
$112$	0	0
$113$	4.22351	0.397314	0.198657	−	0.980069i	$-0.436342\pi$
$113$	4.22351	0.397314	0.198657	+	0.980069i	$0.436342\pi$
$114$	0	0
$115$	2.90501	0.270894
$116$	0	0
$117$	−9.87241	−0.912704
$118$	0	0
$119$	18.1529	1.66408
$120$	0	0
$121$	14.1196	1.28360
$122$	0	0
$123$	5.26746	0.474951
$124$	0	0
$125$	−5.39346	−0.482406
$126$	0	0
$127$	−3.68713	−0.327179	−0.163590	−	0.986528i	$-0.552307\pi$
$127$	−3.68713	−0.327179	−0.163590	+	0.986528i	$0.552307\pi$
$128$	0	0
$129$	−23.3185	−2.05308
$130$	0	0
$131$	−15.2758	−1.33465	−0.667326	−	0.744766i	$-0.732561\pi$
$131$	−15.2758	−1.33465	−0.667326	+	0.744766i	$0.732561\pi$
$132$	0	0
$133$	−8.44262	−0.732068
$134$	0	0
$135$	0.362111	0.0311656
$136$	0	0
$137$	−1.92289	−0.164283	−0.0821416	−	0.996621i	$-0.526176\pi$
$137$	−1.92289	−0.164283	−0.0821416	+	0.996621i	$0.526176\pi$
$138$	0	0
$139$	15.3424	1.30133	0.650663	−	0.759367i	$-0.274491\pi$
$139$	15.3424	1.30133	0.650663	+	0.759367i	$0.274491\pi$
$140$	0	0
$141$	−15.3737	−1.29470
$142$	0	0
$143$	18.1367	1.51667
$144$	0	0
$145$	−0.695505	−0.0577586
$146$	0	0
$147$	−36.7761	−3.03324
$148$	0	0
$149$	−2.20176	−0.180375	−0.0901877	−	0.995925i	$-0.528747\pi$
$149$	−2.20176	−0.180375	−0.0901877	+	0.995925i	$0.528747\pi$
$150$	0	0
$151$	−5.99844	−0.488146	−0.244073	−	0.969757i	$-0.578484\pi$
$151$	−5.99844	−0.488146	−0.244073	+	0.969757i	$0.578484\pi$
$152$	0	0
$153$	10.4719	0.846602
$154$	0	0
$155$	−0.309158	−0.0248321
$156$	0	0
$157$	13.9360	1.11221	0.556105	−	0.831112i	$-0.312295\pi$
$157$	13.9360	1.11221	0.556105	+	0.831112i	$0.312295\pi$
$158$	0	0
$159$	−22.7180	−1.80166
$160$	0	0
$161$	24.6835	1.94533
$162$	0	0
$163$	9.35126	0.732447	0.366224	−	0.930527i	$-0.380651\pi$
$163$	9.35126	0.732447	0.366224	+	0.930527i	$0.380651\pi$
$164$	0	0
$165$	6.67650	0.519765
$166$	0	0
$167$	21.2765	1.64643	0.823214	−	0.567732i	$-0.192179\pi$
$167$	21.2765	1.64643	0.823214	+	0.567732i	$0.192179\pi$
$168$	0	0
$169$	0.0949466	0.00730359
$170$	0	0
$171$	−4.87029	−0.372441
$172$	0	0
$173$	25.8831	1.96785	0.983927	−	0.178573i	$-0.0571481\pi$
$173$	25.8831	1.96785	0.983927	+	0.178573i	$0.0571481\pi$
$174$	0	0
$175$	−22.1812	−1.67674
$176$	0	0
$177$	3.84148	0.288744
$178$	0	0
$179$	21.9674	1.64192	0.820958	−	0.570988i	$-0.193440\pi$
$179$	21.9674	1.64192	0.820958	+	0.570988i	$0.193440\pi$
$180$	0	0
$181$	5.67121	0.421538	0.210769	−	0.977536i	$-0.432403\pi$
$181$	5.67121	0.421538	0.210769	+	0.977536i	$0.432403\pi$
$182$	0	0
$183$	26.7768	1.97940
$184$	0	0
$185$	−1.70241	−0.125164
$186$	0	0
$187$	−19.2380	−1.40682
$188$	0	0
$189$	3.07681	0.223805
$190$	0	0
$191$	18.7019	1.35322	0.676609	−	0.736342i	$-0.263449\pi$
$191$	18.7019	1.35322	0.676609	+	0.736342i	$0.263449\pi$
$192$	0	0
$193$	−11.0679	−0.796688	−0.398344	−	0.917236i	$-0.630415\pi$
$193$	−11.0679	−0.796688	−0.398344	+	0.917236i	$0.630415\pi$
$194$	0	0
$195$	4.82052	0.345205
$196$	0	0
$197$	3.17177	0.225980	0.112990	−	0.993596i	$-0.463957\pi$
$197$	3.17177	0.225980	0.112990	+	0.993596i	$0.463957\pi$
$198$	0	0
$199$	−17.7359	−1.25727	−0.628634	−	0.777702i	$-0.716385\pi$
$199$	−17.7359	−1.25727	−0.628634	+	0.777702i	$0.716385\pi$
$200$	0	0
$201$	−8.82166	−0.622232
$202$	0	0
$203$	−5.90961	−0.414774
$204$	0	0
$205$	−1.22498	−0.0855561
$206$	0	0
$207$	14.2392	0.989691
$208$	0	0
$209$	8.94727	0.618896
$210$	0	0
$211$	−21.6787	−1.49242	−0.746212	−	0.665708i	$-0.768130\pi$
$211$	−21.6787	−1.49242	−0.746212	+	0.665708i	$0.768130\pi$
$212$	0	0
$213$	−7.71226	−0.528436
$214$	0	0
$215$	5.42285	0.369835
$216$	0	0
$217$	−2.62687	−0.178324
$218$	0	0
$219$	1.28315	0.0867074
$220$	0	0
$221$	−13.8901	−0.934349
$222$	0	0
$223$	22.2772	1.49179	0.745894	−	0.666064i	$-0.232022\pi$
$223$	22.2772	1.49179	0.745894	+	0.666064i	$0.232022\pi$
$224$	0	0
$225$	−12.7957	−0.853045
$226$	0	0
$227$	1.67817	0.111384	0.0556919	−	0.998448i	$-0.482264\pi$
$227$	1.67817	0.111384	0.0556919	+	0.998448i	$0.482264\pi$
$228$	0	0
$229$	−22.3005	−1.47366	−0.736831	−	0.676077i	$-0.763678\pi$
$229$	−22.3005	−1.47366	−0.736831	+	0.676077i	$0.763678\pi$
$230$	0	0
$231$	56.7293	3.73251
$232$	0	0
$233$	−11.3494	−0.743526	−0.371763	−	0.928328i	$-0.621247\pi$
$233$	−11.3494	−0.743526	−0.371763	+	0.928328i	$0.621247\pi$
$234$	0	0
$235$	3.57523	0.233222
$236$	0	0
$237$	2.70943	0.175997
$238$	0	0
$239$	16.6634	1.07786	0.538932	−	0.842349i	$-0.318828\pi$
$239$	16.6634	1.07786	0.538932	+	0.842349i	$0.318828\pi$
$240$	0	0
$241$	28.2809	1.82173	0.910866	−	0.412703i	$-0.135415\pi$
$241$	28.2809	1.82173	0.910866	+	0.412703i	$0.135415\pi$
$242$	0	0
$243$	21.3634	1.37046
$244$	0	0
$245$	8.55248	0.546398
$246$	0	0
$247$	6.46005	0.411043
$248$	0	0
$249$	3.59340	0.227723
$250$	0	0
$251$	19.3219	1.21959	0.609794	−	0.792560i	$-0.291252\pi$
$251$	19.3219	1.21959	0.609794	+	0.792560i	$0.291252\pi$
$252$	0	0
$253$	−26.1589	−1.64460
$254$	0	0
$255$	−5.11324	−0.320203
$256$	0	0
$257$	−17.9007	−1.11661	−0.558306	−	0.829635i	$-0.688549\pi$
$257$	−17.9007	−1.11661	−0.558306	+	0.829635i	$0.688549\pi$
$258$	0	0
$259$	−14.4652	−0.898821
$260$	0	0
$261$	−3.40908	−0.211017
$262$	0	0
$263$	12.4085	0.765139	0.382570	−	0.923927i	$-0.375039\pi$
$263$	12.4085	0.765139	0.382570	+	0.923927i	$0.375039\pi$
$264$	0	0
$265$	5.28320	0.324545
$266$	0	0
$267$	28.7470	1.75929
$268$	0	0
$269$	14.5355	0.886244	0.443122	−	0.896461i	$-0.353871\pi$
$269$	14.5355	0.886244	0.443122	+	0.896461i	$0.353871\pi$
$270$	0	0
$271$	5.20678	0.316289	0.158145	−	0.987416i	$-0.449449\pi$
$271$	5.20678	0.316289	0.158145	+	0.987416i	$0.449449\pi$
$272$	0	0
$273$	40.9593	2.47897
$274$	0	0
$275$	23.5071	1.41753
$276$	0	0
$277$	12.8779	0.773757	0.386879	−	0.922131i	$-0.373553\pi$
$277$	12.8779	0.773757	0.386879	+	0.922131i	$0.373553\pi$
$278$	0	0
$279$	−1.51536	−0.0907224
$280$	0	0
$281$	−26.7897	−1.59814	−0.799071	−	0.601237i	$-0.794675\pi$
$281$	−26.7897	−1.59814	−0.799071	+	0.601237i	$0.794675\pi$
$282$	0	0
$283$	30.9869	1.84198	0.920989	−	0.389588i	$-0.127383\pi$
$283$	30.9869	1.84198	0.920989	+	0.389588i	$0.127383\pi$
$284$	0	0
$285$	2.37808	0.140865
$286$	0	0
$287$	−10.4085	−0.614392
$288$	0	0
$289$	−2.26646	−0.133321
$290$	0	0
$291$	−8.77820	−0.514587
$292$	0	0
$293$	−3.47574	−0.203055	−0.101528	−	0.994833i	$-0.532373\pi$
$293$	−3.47574	−0.203055	−0.101528	+	0.994833i	$0.532373\pi$
$294$	0	0
$295$	−0.893358	−0.0520133
$296$	0	0
$297$	−3.26073	−0.189206
$298$	0	0
$299$	−18.8871	−1.09227
$300$	0	0
$301$	46.0772	2.65585
$302$	0	0
$303$	4.48694	0.257768
$304$	0	0
$305$	−6.22710	−0.356563
$306$	0	0
$307$	−17.2465	−0.984310	−0.492155	−	0.870508i	$-0.663791\pi$
$307$	−17.2465	−0.984310	−0.492155	+	0.870508i	$0.663791\pi$
$308$	0	0
$309$	−18.9130	−1.07592
$310$	0	0
$311$	31.1026	1.76366	0.881832	−	0.471563i	$-0.156310\pi$
$311$	31.1026	1.76366	0.881832	+	0.471563i	$0.156310\pi$
$312$	0	0
$313$	−2.74917	−0.155392	−0.0776961	−	0.996977i	$-0.524756\pi$
$313$	−2.74917	−0.155392	−0.0776961	+	0.996977i	$0.524756\pi$
$314$	0	0
$315$	7.18123	0.404616
$316$	0	0
$317$	34.1867	1.92012	0.960058	−	0.279802i	$-0.0902688\pi$
$317$	34.1867	1.92012	0.960058	+	0.279802i	$0.0902688\pi$
$318$	0	0
$319$	6.26286	0.350653
$320$	0	0
$321$	−4.46190	−0.249039
$322$	0	0
$323$	−6.85232	−0.381273
$324$	0	0
$325$	16.9724	0.941461
$326$	0	0
$327$	−31.7004	−1.75304
$328$	0	0
$329$	30.3783	1.67481
$330$	0	0
$331$	8.09376	0.444873	0.222437	−	0.974947i	$-0.428599\pi$
$331$	8.09376	0.444873	0.222437	+	0.974947i	$0.428599\pi$
$332$	0	0
$333$	−8.34452	−0.457277
$334$	0	0
$335$	2.05152	0.112087
$336$	0	0
$337$	17.8836	0.974181	0.487091	−	0.873351i	$-0.338058\pi$
$337$	17.8836	0.974181	0.487091	+	0.873351i	$0.338058\pi$
$338$	0	0
$339$	−10.1084	−0.549012
$340$	0	0
$341$	2.78389	0.150756
$342$	0	0
$343$	39.5645	2.13628
$344$	0	0
$345$	−6.95274	−0.374323
$346$	0	0
$347$	31.4688	1.68933	0.844667	−	0.535291i	$-0.179798\pi$
$347$	31.4688	1.68933	0.844667	+	0.535291i	$0.179798\pi$
$348$	0	0
$349$	4.62098	0.247355	0.123678	−	0.992322i	$-0.460531\pi$
$349$	4.62098	0.247355	0.123678	+	0.992322i	$0.460531\pi$
$350$	0	0
$351$	−2.35429	−0.125663
$352$	0	0
$353$	30.0672	1.60031	0.800157	−	0.599791i	$-0.204750\pi$
$353$	30.0672	1.60031	0.800157	+	0.599791i	$0.204750\pi$
$354$	0	0
$355$	1.79353	0.0951906
$356$	0	0
$357$	−43.4465	−2.29943
$358$	0	0
$359$	18.4955	0.976153	0.488077	−	0.872801i	$-0.337699\pi$
$359$	18.4955	0.976153	0.488077	+	0.872801i	$0.337699\pi$
$360$	0	0
$361$	−15.8131	−0.832269
$362$	0	0
$363$	−33.7933	−1.77369
$364$	0	0
$365$	−0.298404	−0.0156192
$366$	0	0
$367$	−22.4529	−1.17203	−0.586017	−	0.810299i	$-0.699305\pi$
$367$	−22.4529	−1.17203	−0.586017	+	0.810299i	$0.699305\pi$
$368$	0	0
$369$	−6.00433	−0.312573
$370$	0	0
$371$	44.8907	2.33061
$372$	0	0
$373$	−23.5087	−1.21723	−0.608616	−	0.793465i	$-0.708275\pi$
$373$	−23.5087	−1.21723	−0.608616	+	0.793465i	$0.708275\pi$
$374$	0	0
$375$	12.9085	0.666591
$376$	0	0
$377$	4.52187	0.232888
$378$	0	0
$379$	−10.0252	−0.514958	−0.257479	−	0.966284i	$-0.582892\pi$
$379$	−10.0252	−0.514958	−0.257479	+	0.966284i	$0.582892\pi$
$380$	0	0
$381$	8.82462	0.452099
$382$	0	0
$383$	15.7927	0.806969	0.403485	−	0.914986i	$-0.367799\pi$
$383$	15.7927	0.806969	0.403485	+	0.914986i	$0.367799\pi$
$384$	0	0
$385$	−13.1927	−0.672363
$386$	0	0
$387$	26.5806	1.35117
$388$	0	0
$389$	30.4099	1.54184	0.770922	−	0.636929i	$-0.219796\pi$
$389$	30.4099	1.54184	0.770922	+	0.636929i	$0.219796\pi$
$390$	0	0
$391$	20.0340	1.01316
$392$	0	0
$393$	36.5605	1.84423
$394$	0	0
$395$	−0.630093	−0.0317034
$396$	0	0
$397$	1.55746	0.0781666	0.0390833	−	0.999236i	$-0.487556\pi$
$397$	1.55746	0.0781666	0.0390833	+	0.999236i	$0.487556\pi$
$398$	0	0
$399$	20.2062	1.01158
$400$	0	0
$401$	−19.3973	−0.968657	−0.484329	−	0.874886i	$-0.660936\pi$
$401$	−19.3973	−0.968657	−0.484329	+	0.874886i	$0.660936\pi$
$402$	0	0
$403$	2.01001	0.100126
$404$	0	0
$405$	−5.42207	−0.269425
$406$	0	0
$407$	15.3298	0.759870
$408$	0	0
$409$	−22.5868	−1.11685	−0.558423	−	0.829556i	$-0.688593\pi$
$409$	−22.5868	−1.11685	−0.558423	+	0.829556i	$0.688593\pi$
$410$	0	0
$411$	4.60216	0.227008
$412$	0	0
$413$	−7.59074	−0.373516
$414$	0	0
$415$	−0.835665	−0.0410212
$416$	0	0
$417$	−36.7199	−1.79818
$418$	0	0
$419$	20.4919	1.00110	0.500548	−	0.865709i	$-0.333132\pi$
$419$	20.4919	1.00110	0.500548	+	0.865709i	$0.333132\pi$
$420$	0	0
$421$	−13.0068	−0.633911	−0.316956	−	0.948440i	$-0.602661\pi$
$421$	−13.0068	−0.633911	−0.316956	+	0.948440i	$0.602661\pi$
$422$	0	0
$423$	17.5243	0.852061
$424$	0	0
$425$	−18.0030	−0.873276
$426$	0	0
$427$	−52.9108	−2.56053
$428$	0	0
$429$	−43.4077	−2.09574
$430$	0	0
$431$	−3.96578	−0.191025	−0.0955125	−	0.995428i	$-0.530449\pi$
$431$	−3.96578	−0.191025	−0.0955125	+	0.995428i	$0.530449\pi$
$432$	0	0
$433$	−8.14654	−0.391498	−0.195749	−	0.980654i	$-0.562714\pi$
$433$	−8.14654	−0.391498	−0.195749	+	0.980654i	$0.562714\pi$
$434$	0	0
$435$	1.66459	0.0798112
$436$	0	0
$437$	−9.31746	−0.445715
$438$	0	0
$439$	−20.3322	−0.970404	−0.485202	−	0.874402i	$-0.661254\pi$
$439$	−20.3322	−0.970404	−0.485202	+	0.874402i	$0.661254\pi$
$440$	0	0
$441$	41.9207	1.99623
$442$	0	0
$443$	−34.8372	−1.65517	−0.827583	−	0.561343i	$-0.810285\pi$
$443$	−34.8372	−1.65517	−0.827583	+	0.561343i	$0.810285\pi$
$444$	0	0
$445$	−6.68526	−0.316912
$446$	0	0
$447$	5.26961	0.249244
$448$	0	0
$449$	6.99147	0.329948	0.164974	−	0.986298i	$-0.447246\pi$
$449$	6.99147	0.329948	0.164974	+	0.986298i	$0.447246\pi$
$450$	0	0
$451$	11.0306	0.519412
$452$	0	0
$453$	14.3564	0.674523
$454$	0	0
$455$	−9.52531	−0.446554
$456$	0	0
$457$	−25.4179	−1.18900	−0.594500	−	0.804096i	$-0.702650\pi$
$457$	−25.4179	−1.18900	−0.594500	+	0.804096i	$0.702650\pi$
$458$	0	0
$459$	2.49725	0.116561
$460$	0	0
$461$	25.6449	1.19440	0.597201	−	0.802092i	$-0.296280\pi$
$461$	25.6449	1.19440	0.597201	+	0.802092i	$0.296280\pi$
$462$	0	0
$463$	36.2239	1.68347	0.841733	−	0.539893i	$-0.181535\pi$
$463$	36.2239	1.68347	0.841733	+	0.539893i	$0.181535\pi$
$464$	0	0
$465$	0.739925	0.0343132
$466$	0	0
$467$	−4.84581	−0.224237	−0.112119	−	0.993695i	$-0.535764\pi$
$467$	−4.84581	−0.224237	−0.112119	+	0.993695i	$0.535764\pi$
$468$	0	0
$469$	17.4315	0.804913
$470$	0	0
$471$	−33.3538	−1.53686
$472$	0	0
$473$	−48.8315	−2.24527
$474$	0	0
$475$	8.37291	0.384175
$476$	0	0
$477$	25.8961	1.18570
$478$	0	0
$479$	−11.9663	−0.546756	−0.273378	−	0.961907i	$-0.588141\pi$
$479$	−11.9663	−0.546756	−0.273378	+	0.961907i	$0.588141\pi$
$480$	0	0
$481$	11.0683	0.504672
$482$	0	0
$483$	−59.0765	−2.68807
$484$	0	0
$485$	2.04142	0.0926960
$486$	0	0
$487$	−15.3211	−0.694266	−0.347133	−	0.937816i	$-0.612845\pi$
$487$	−15.3211	−0.694266	−0.347133	+	0.937816i	$0.612845\pi$
$488$	0	0
$489$	−22.3809	−1.01210
$490$	0	0
$491$	9.42468	0.425330	0.212665	−	0.977125i	$-0.431786\pi$
$491$	9.42468	0.425330	0.212665	+	0.977125i	$0.431786\pi$
$492$	0	0
$493$	−4.79645	−0.216021
$494$	0	0
$495$	−7.61048	−0.342066
$496$	0	0
$497$	15.2394	0.683579
$498$	0	0
$499$	−40.8400	−1.82825	−0.914124	−	0.405435i	$-0.867120\pi$
$499$	−40.8400	−1.82825	−0.914124	+	0.405435i	$0.867120\pi$
$500$	0	0
$501$	−50.9224	−2.27504
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−1.04346	−0.0464334
$506$	0	0
$507$	−0.227241	−0.0100921
$508$	0	0
$509$	4.03493	0.178845	0.0894226	−	0.995994i	$-0.471498\pi$
$509$	4.03493	0.178845	0.0894226	+	0.995994i	$0.471498\pi$
$510$	0	0
$511$	−2.53550	−0.112164
$512$	0	0
$513$	−1.16143	−0.0512782
$514$	0	0
$515$	4.39832	0.193813
$516$	0	0
$517$	−32.1941	−1.41590
$518$	0	0
$519$	−61.9475	−2.71919
$520$	0	0
$521$	−11.0220	−0.482882	−0.241441	−	0.970416i	$-0.577620\pi$
$521$	−11.0220	−0.482882	−0.241441	+	0.970416i	$0.577620\pi$
$522$	0	0
$523$	−19.7235	−0.862449	−0.431225	−	0.902245i	$-0.641918\pi$
$523$	−19.7235	−0.862449	−0.431225	+	0.902245i	$0.641918\pi$
$524$	0	0
$525$	53.0876	2.31693
$526$	0	0
$527$	−2.13206	−0.0928740
$528$	0	0
$529$	4.24126	0.184403
$530$	0	0
$531$	−4.37887	−0.190027
$532$	0	0
$533$	7.96426	0.344971
$534$	0	0
$535$	1.03764	0.0448610
$536$	0	0
$537$	−52.5758	−2.26881
$538$	0	0
$539$	−77.0130	−3.31719
$540$	0	0
$541$	41.4964	1.78407	0.892035	−	0.451966i	$-0.149277\pi$
$541$	41.4964	1.78407	0.892035	+	0.451966i	$0.149277\pi$
$542$	0	0
$543$	−13.5732	−0.582484
$544$	0	0
$545$	7.37210	0.315786
$546$	0	0
$547$	1.34144	0.0573558	0.0286779	−	0.999589i	$-0.490870\pi$
$547$	1.34144	0.0573558	0.0286779	+	0.999589i	$0.490870\pi$
$548$	0	0
$549$	−30.5227	−1.30268
$550$	0	0
$551$	2.23075	0.0950330
$552$	0	0
$553$	−5.35382	−0.227667
$554$	0	0
$555$	4.07448	0.172952
$556$	0	0
$557$	31.6239	1.33995	0.669973	−	0.742385i	$-0.266306\pi$
$557$	31.6239	1.33995	0.669973	+	0.742385i	$0.266306\pi$
$558$	0	0
$559$	−35.2570	−1.49121
$560$	0	0
$561$	46.0435	1.94396
$562$	0	0
$563$	25.5575	1.07712	0.538560	−	0.842587i	$-0.318968\pi$
$563$	25.5575	1.07712	0.538560	+	0.842587i	$0.318968\pi$
$564$	0	0
$565$	2.35076	0.0988971
$566$	0	0
$567$	−46.0706	−1.93478
$568$	0	0
$569$	−15.4475	−0.647592	−0.323796	−	0.946127i	$-0.604959\pi$
$569$	−15.4475	−0.647592	−0.323796	+	0.946127i	$0.604959\pi$
$570$	0	0
$571$	−14.0266	−0.586996	−0.293498	−	0.955960i	$-0.594819\pi$
$571$	−14.0266	−0.586996	−0.293498	+	0.955960i	$0.594819\pi$
$572$	0	0
$573$	−44.7603	−1.86989
$574$	0	0
$575$	−24.4797	−1.02087
$576$	0	0
$577$	−17.0740	−0.710798	−0.355399	−	0.934715i	$-0.615655\pi$
$577$	−17.0740	−0.710798	−0.355399	+	0.934715i	$0.615655\pi$
$578$	0	0
$579$	26.4896	1.10087
$580$	0	0
$581$	−7.10054	−0.294580
$582$	0	0
$583$	−47.5740	−1.97031
$584$	0	0
$585$	−5.49487	−0.227185
$586$	0	0
$587$	13.7032	0.565590	0.282795	−	0.959180i	$-0.408738\pi$
$587$	13.7032	0.565590	0.282795	+	0.959180i	$0.408738\pi$
$588$	0	0
$589$	0.991584	0.0408575
$590$	0	0
$591$	−7.59120	−0.312260
$592$	0	0
$593$	−14.3472	−0.589171	−0.294585	−	0.955625i	$-0.595181\pi$
$593$	−14.3472	−0.589171	−0.294585	+	0.955625i	$0.595181\pi$
$594$	0	0
$595$	10.1037	0.414212
$596$	0	0
$597$	42.4485	1.73730
$598$	0	0
$599$	−23.8707	−0.975329	−0.487664	−	0.873031i	$-0.662151\pi$
$599$	−23.8707	−0.975329	−0.487664	+	0.873031i	$0.662151\pi$
$600$	0	0
$601$	30.9334	1.26180	0.630900	−	0.775864i	$-0.282686\pi$
$601$	30.9334	1.26180	0.630900	+	0.775864i	$0.282686\pi$
$602$	0	0
$603$	10.0557	0.409501
$604$	0	0
$605$	7.85882	0.319506
$606$	0	0
$607$	−8.53021	−0.346230	−0.173115	−	0.984902i	$-0.555383\pi$
$607$	−8.53021	−0.346230	−0.173115	+	0.984902i	$0.555383\pi$
$608$	0	0
$609$	14.1438	0.573137
$610$	0	0
$611$	−23.2446	−0.940375
$612$	0	0
$613$	47.7746	1.92960	0.964799	−	0.262988i	$-0.0847079\pi$
$613$	47.7746	1.92960	0.964799	+	0.262988i	$0.0847079\pi$
$614$	0	0
$615$	2.93181	0.118222
$616$	0	0
$617$	8.22910	0.331291	0.165646	−	0.986185i	$-0.447029\pi$
$617$	8.22910	0.331291	0.165646	+	0.986185i	$0.447029\pi$
$618$	0	0
$619$	−46.4395	−1.86656	−0.933280	−	0.359150i	$-0.883066\pi$
$619$	−46.4395	−1.86656	−0.933280	+	0.359150i	$0.883066\pi$
$620$	0	0
$621$	3.39564	0.136262
$622$	0	0
$623$	−56.8038	−2.27579
$624$	0	0
$625$	20.4491	0.817964
$626$	0	0
$627$	−21.4140	−0.855194
$628$	0	0
$629$	−11.7404	−0.468121
$630$	0	0
$631$	−14.5185	−0.577971	−0.288986	−	0.957333i	$-0.593318\pi$
$631$	−14.5185	−0.577971	−0.288986	+	0.957333i	$0.593318\pi$
$632$	0	0
$633$	51.8850	2.06224
$634$	0	0
$635$	−2.05221	−0.0814396
$636$	0	0
$637$	−55.6045	−2.20313
$638$	0	0
$639$	8.79114	0.347772
$640$	0	0
$641$	48.7941	1.92725	0.963626	−	0.267256i	$-0.0861168\pi$
$641$	48.7941	1.92725	0.963626	+	0.267256i	$0.0861168\pi$
$642$	0	0
$643$	37.5296	1.48002	0.740011	−	0.672595i	$-0.234820\pi$
$643$	37.5296	1.48002	0.740011	+	0.672595i	$0.234820\pi$
$644$	0	0
$645$	−12.9788	−0.511041
$646$	0	0
$647$	−4.11112	−0.161625	−0.0808125	−	0.996729i	$-0.525751\pi$
$647$	−4.11112	−0.161625	−0.0808125	+	0.996729i	$0.525751\pi$
$648$	0	0
$649$	8.04447	0.315773
$650$	0	0
$651$	6.28705	0.246409
$652$	0	0
$653$	−2.93868	−0.114999	−0.0574997	−	0.998346i	$-0.518313\pi$
$653$	−2.93868	−0.114999	−0.0574997	+	0.998346i	$0.518313\pi$
$654$	0	0
$655$	−8.50233	−0.332214
$656$	0	0
$657$	−1.46265	−0.0570635
$658$	0	0
$659$	18.0107	0.701595	0.350798	−	0.936451i	$-0.385910\pi$
$659$	18.0107	0.701595	0.350798	+	0.936451i	$0.385910\pi$
$660$	0	0
$661$	−1.40775	−0.0547549	−0.0273775	−	0.999625i	$-0.508716\pi$
$661$	−1.40775	−0.0547549	−0.0273775	+	0.999625i	$0.508716\pi$
$662$	0	0
$663$	33.2440	1.29109
$664$	0	0
$665$	−4.69907	−0.182222
$666$	0	0
$667$	−6.52198	−0.252532
$668$	0	0
$669$	−53.3172	−2.06136
$670$	0	0
$671$	56.0735	2.16470
$672$	0	0
$673$	32.1948	1.24102	0.620508	−	0.784200i	$-0.286926\pi$
$673$	32.1948	1.24102	0.620508	+	0.784200i	$0.286926\pi$
$674$	0	0
$675$	−3.05141	−0.117449
$676$	0	0
$677$	−24.7016	−0.949358	−0.474679	−	0.880159i	$-0.657436\pi$
$677$	−24.7016	−0.949358	−0.474679	+	0.880159i	$0.657436\pi$
$678$	0	0
$679$	17.3457	0.665665
$680$	0	0
$681$	−4.01645	−0.153911
$682$	0	0
$683$	−35.6349	−1.36353	−0.681767	−	0.731570i	$-0.738788\pi$
$683$	−35.6349	−1.36353	−0.681767	+	0.731570i	$0.738788\pi$
$684$	0	0
$685$	−1.07026	−0.0408924
$686$	0	0
$687$	53.3732	2.03631
$688$	0	0
$689$	−34.3491	−1.30860
$690$	0	0
$691$	1.83431	0.0697804	0.0348902	−	0.999391i	$-0.488892\pi$
$691$	1.83431	0.0697804	0.0348902	+	0.999391i	$0.488892\pi$
$692$	0	0
$693$	−64.6652	−2.45643
$694$	0	0
$695$	8.53941	0.323918
$696$	0	0
$697$	−8.44787	−0.319986
$698$	0	0
$699$	27.1633	1.02741
$700$	0	0
$701$	2.06223	0.0778893	0.0389446	−	0.999241i	$-0.487600\pi$
$701$	2.06223	0.0778893	0.0389446	+	0.999241i	$0.487600\pi$
$702$	0	0
$703$	5.46027	0.205938
$704$	0	0
$705$	−8.55682	−0.322268
$706$	0	0
$707$	−8.86615	−0.333446
$708$	0	0
$709$	−5.06927	−0.190380	−0.0951901	−	0.995459i	$-0.530346\pi$
$709$	−5.06927	−0.190380	−0.0951901	+	0.995459i	$0.530346\pi$
$710$	0	0
$711$	−3.08846	−0.115826
$712$	0	0
$713$	−2.89907	−0.108571
$714$	0	0
$715$	10.0947	0.377520
$716$	0	0
$717$	−39.8815	−1.48940
$718$	0	0
$719$	−16.0455	−0.598396	−0.299198	−	0.954191i	$-0.596719\pi$
$719$	−16.0455	−0.598396	−0.299198	+	0.954191i	$0.596719\pi$
$720$	0	0
$721$	37.3720	1.39180
$722$	0	0
$723$	−67.6863	−2.51728
$724$	0	0
$725$	5.86082	0.217665
$726$	0	0
$727$	−16.9366	−0.628144	−0.314072	−	0.949399i	$-0.601693\pi$
$727$	−16.9366	−0.628144	−0.314072	+	0.949399i	$0.601693\pi$
$728$	0	0
$729$	−21.9054	−0.811313
$730$	0	0
$731$	37.3979	1.38321
$732$	0	0
$733$	−5.74218	−0.212092	−0.106046	−	0.994361i	$-0.533819\pi$
$733$	−5.74218	−0.212092	−0.106046	+	0.994361i	$0.533819\pi$
$734$	0	0
$735$	−20.4692	−0.755016
$736$	0	0
$737$	−18.4735	−0.680479
$738$	0	0
$739$	21.4798	0.790149	0.395075	−	0.918649i	$-0.370719\pi$
$739$	21.4798	0.790149	0.395075	+	0.918649i	$0.370719\pi$
$740$	0	0
$741$	−15.4612	−0.567982
$742$	0	0
$743$	−17.0005	−0.623687	−0.311843	−	0.950134i	$-0.600946\pi$
$743$	−17.0005	−0.623687	−0.311843	+	0.950134i	$0.600946\pi$
$744$	0	0
$745$	−1.22548	−0.0448980
$746$	0	0
$747$	−4.09609	−0.149868
$748$	0	0
$749$	8.81668	0.322154
$750$	0	0
$751$	−22.3236	−0.814601	−0.407301	−	0.913294i	$-0.633530\pi$
$751$	−22.3236	−0.814601	−0.407301	+	0.913294i	$0.633530\pi$
$752$	0	0
$753$	−46.2443	−1.68524
$754$	0	0
$755$	−3.33866	−0.121506
$756$	0	0
$757$	32.8959	1.19562	0.597811	−	0.801637i	$-0.296037\pi$
$757$	32.8959	1.19562	0.597811	+	0.801637i	$0.296037\pi$
$758$	0	0
$759$	62.6077	2.27252
$760$	0	0
$761$	9.47572	0.343495	0.171747	−	0.985141i	$-0.445059\pi$
$761$	9.47572	0.343495	0.171747	+	0.985141i	$0.445059\pi$
$762$	0	0
$763$	62.6397	2.26771
$764$	0	0
$765$	5.82853	0.210731
$766$	0	0
$767$	5.80822	0.209723
$768$	0	0
$769$	26.4708	0.954563	0.477282	−	0.878750i	$-0.341622\pi$
$769$	26.4708	0.954563	0.477282	+	0.878750i	$0.341622\pi$
$770$	0	0
$771$	42.8427	1.54294
$772$	0	0
$773$	12.3606	0.444580	0.222290	−	0.974981i	$-0.428647\pi$
$773$	12.3606	0.444580	0.222290	+	0.974981i	$0.428647\pi$
$774$	0	0
$775$	2.60518	0.0935809
$776$	0	0
$777$	34.6203	1.24200
$778$	0	0
$779$	3.92896	0.140770
$780$	0	0
$781$	−16.1503	−0.577903
$782$	0	0
$783$	−0.812968	−0.0290531
$784$	0	0
$785$	7.75660	0.276845
$786$	0	0
$787$	−37.2543	−1.32797	−0.663986	−	0.747745i	$-0.731136\pi$
$787$	−37.2543	−1.32797	−0.663986	+	0.747745i	$0.731136\pi$
$788$	0	0
$789$	−29.6979	−1.05727
$790$	0	0
$791$	19.9741	0.710196
$792$	0	0
$793$	40.4859	1.43769
$794$	0	0
$795$	−12.6446	−0.448458
$796$	0	0
$797$	−30.6111	−1.08430	−0.542150	−	0.840282i	$-0.682390\pi$
$797$	−30.6111	−1.08430	−0.542150	+	0.840282i	$0.682390\pi$
$798$	0	0
$799$	24.6560	0.872268
$800$	0	0
$801$	−32.7684	−1.15781
$802$	0	0
$803$	2.68706	0.0948241
$804$	0	0
$805$	13.7386	0.484220
$806$	0	0
$807$	−34.7886	−1.22462
$808$	0	0
$809$	2.14826	0.0755287	0.0377643	−	0.999287i	$-0.487976\pi$
$809$	2.14826	0.0755287	0.0377643	+	0.999287i	$0.487976\pi$
$810$	0	0
$811$	−19.1301	−0.671750	−0.335875	−	0.941907i	$-0.609032\pi$
$811$	−19.1301	−0.671750	−0.335875	+	0.941907i	$0.609032\pi$
$812$	0	0
$813$	−12.4617	−0.437051
$814$	0	0
$815$	5.20481	0.182316
$816$	0	0
$817$	−17.3931	−0.608508
$818$	0	0
$819$	−46.6892	−1.63145
$820$	0	0
$821$	50.2707	1.75446	0.877229	−	0.480071i	$-0.159389\pi$
$821$	50.2707	1.75446	0.877229	+	0.480071i	$0.159389\pi$
$822$	0	0
$823$	−7.78833	−0.271484	−0.135742	−	0.990744i	$-0.543342\pi$
$823$	−7.78833	−0.271484	−0.135742	+	0.990744i	$0.543342\pi$
$824$	0	0
$825$	−56.2609	−1.95875
$826$	0	0
$827$	−3.76052	−0.130766	−0.0653831	−	0.997860i	$-0.520827\pi$
$827$	−3.76052	−0.130766	−0.0653831	+	0.997860i	$0.520827\pi$
$828$	0	0
$829$	12.6728	0.440145	0.220072	−	0.975484i	$-0.429371\pi$
$829$	12.6728	0.440145	0.220072	+	0.975484i	$0.429371\pi$
$830$	0	0
$831$	−30.8214	−1.06918
$832$	0	0
$833$	58.9809	2.04357
$834$	0	0
$835$	11.8423	0.409819
$836$	0	0
$837$	−0.361371	−0.0124908
$838$	0	0
$839$	−16.3273	−0.563681	−0.281840	−	0.959461i	$-0.590945\pi$
$839$	−16.3273	−0.563681	−0.281840	+	0.959461i	$0.590945\pi$
$840$	0	0
$841$	−27.4385	−0.946156
$842$	0	0
$843$	64.1175	2.20832
$844$	0	0
$845$	0.0528462	0.00181797
$846$	0	0
$847$	66.7753	2.29443
$848$	0	0
$849$	−74.1627	−2.54526
$850$	0	0
$851$	−15.9641	−0.547241
$852$	0	0
$853$	−36.6000	−1.25316	−0.626580	−	0.779357i	$-0.715546\pi$
$853$	−36.6000	−1.25316	−0.626580	+	0.779357i	$0.715546\pi$
$854$	0	0
$855$	−2.71075	−0.0927057
$856$	0	0
$857$	17.6791	0.603905	0.301953	−	0.953323i	$-0.402362\pi$
$857$	17.6791	0.603905	0.301953	+	0.953323i	$0.402362\pi$
$858$	0	0
$859$	−29.3642	−1.00189	−0.500947	−	0.865478i	$-0.667015\pi$
$859$	−29.3642	−1.00189	−0.500947	+	0.865478i	$0.667015\pi$
$860$	0	0
$861$	24.9112	0.848972
$862$	0	0
$863$	36.3829	1.23849	0.619245	−	0.785198i	$-0.287439\pi$
$863$	36.3829	1.23849	0.619245	+	0.785198i	$0.287439\pi$
$864$	0	0
$865$	14.4062	0.489826
$866$	0	0
$867$	5.42445	0.184224
$868$	0	0
$869$	5.67384	0.192472
$870$	0	0
$871$	−13.3381	−0.451944
$872$	0	0
$873$	10.0062	0.338658
$874$	0	0
$875$	−25.5071	−0.862296
$876$	0	0
$877$	10.4787	0.353841	0.176921	−	0.984225i	$-0.443386\pi$
$877$	10.4787	0.353841	0.176921	+	0.984225i	$0.443386\pi$
$878$	0	0
$879$	8.31871	0.280583
$880$	0	0
$881$	−23.6147	−0.795599	−0.397799	−	0.917472i	$-0.630226\pi$
$881$	−23.6147	−0.795599	−0.397799	+	0.917472i	$0.630226\pi$
$882$	0	0
$883$	5.54626	0.186647	0.0933233	−	0.995636i	$-0.470251\pi$
$883$	5.54626	0.186647	0.0933233	+	0.995636i	$0.470251\pi$
$884$	0	0
$885$	2.13813	0.0718723
$886$	0	0
$887$	−28.0476	−0.941747	−0.470874	−	0.882201i	$-0.656061\pi$
$887$	−28.0476	−0.941747	−0.470874	+	0.882201i	$0.656061\pi$
$888$	0	0
$889$	−17.4374	−0.584831
$890$	0	0
$891$	48.8244	1.63568
$892$	0	0
$893$	−11.4671	−0.383732
$894$	0	0
$895$	12.2268	0.408696
$896$	0	0
$897$	45.2036	1.50931
$898$	0	0
$899$	0.694083	0.0231490
$900$	0	0
$901$	36.4348	1.21382
$902$	0	0
$903$	−110.279	−3.66987
$904$	0	0
$905$	3.15653	0.104927
$906$	0	0
$907$	11.4064	0.378744	0.189372	−	0.981905i	$-0.439355\pi$
$907$	11.4064	0.378744	0.189372	+	0.981905i	$0.439355\pi$
$908$	0	0
$909$	−5.11462	−0.169641
$910$	0	0
$911$	17.7377	0.587675	0.293838	−	0.955855i	$-0.405068\pi$
$911$	17.7377	0.587675	0.293838	+	0.955855i	$0.405068\pi$
$912$	0	0
$913$	7.52497	0.249040
$914$	0	0
$915$	14.9037	0.492701
$916$	0	0
$917$	−72.2432	−2.38568
$918$	0	0
$919$	−33.1398	−1.09318	−0.546591	−	0.837400i	$-0.684075\pi$
$919$	−33.1398	−1.09318	−0.546591	+	0.837400i	$0.684075\pi$
$920$	0	0
$921$	41.2771	1.36013
$922$	0	0
$923$	−11.6607	−0.383818
$924$	0	0
$925$	14.3457	0.471684
$926$	0	0
$927$	21.5588	0.708083
$928$	0	0
$929$	41.0369	1.34638	0.673188	−	0.739472i	$-0.264925\pi$
$929$	41.0369	1.34638	0.673188	+	0.739472i	$0.264925\pi$
$930$	0	0
$931$	−27.4310	−0.899015
$932$	0	0
$933$	−74.4396	−2.43704
$934$	0	0
$935$	−10.7077	−0.350178
$936$	0	0
$937$	8.60118	0.280988	0.140494	−	0.990082i	$-0.455131\pi$
$937$	8.60118	0.280988	0.140494	+	0.990082i	$0.455131\pi$
$938$	0	0
$939$	6.57975	0.214722
$940$	0	0
$941$	−24.5480	−0.800243	−0.400121	−	0.916462i	$-0.631032\pi$
$941$	−24.5480	−0.800243	−0.400121	+	0.916462i	$0.631032\pi$
$942$	0	0
$943$	−11.4870	−0.374069
$944$	0	0
$945$	1.71252	0.0557082
$946$	0	0
$947$	−39.8951	−1.29642	−0.648209	−	0.761463i	$-0.724482\pi$
$947$	−39.8951	−1.29642	−0.648209	+	0.761463i	$0.724482\pi$
$948$	0	0
$949$	1.94009	0.0629780
$950$	0	0
$951$	−81.8210	−2.65323
$952$	0	0
$953$	−18.4346	−0.597155	−0.298577	−	0.954385i	$-0.596512\pi$
$953$	−18.4346	−0.597155	−0.298577	+	0.954385i	$0.596512\pi$
$954$	0	0
$955$	10.4092	0.336835
$956$	0	0
$957$	−14.9893	−0.484534
$958$	0	0
$959$	−9.09383	−0.293655
$960$	0	0
$961$	−30.6915	−0.990048
$962$	0	0
$963$	5.08608	0.163897
$964$	0	0
$965$	−6.16029	−0.198307
$966$	0	0
$967$	36.6547	1.17874	0.589368	−	0.807865i	$-0.299377\pi$
$967$	36.6547	1.17874	0.589368	+	0.807865i	$0.299377\pi$
$968$	0	0
$969$	16.4001	0.526846
$970$	0	0
$971$	−31.4494	−1.00926	−0.504629	−	0.863336i	$-0.668371\pi$
$971$	−31.4494	−1.00926	−0.504629	+	0.863336i	$0.668371\pi$
$972$	0	0
$973$	72.5582	2.32611
$974$	0	0
$975$	−40.6211	−1.30092
$976$	0	0
$977$	−19.5937	−0.626858	−0.313429	−	0.949612i	$-0.601478\pi$
$977$	−19.5937	−0.626858	−0.313429	+	0.949612i	$0.601478\pi$
$978$	0	0
$979$	60.1992	1.92397
$980$	0	0
$981$	36.1350	1.15370
$982$	0	0
$983$	−18.6754	−0.595652	−0.297826	−	0.954620i	$-0.596261\pi$
$983$	−18.6754	−0.595652	−0.297826	+	0.954620i	$0.596261\pi$
$984$	0	0
$985$	1.76537	0.0562495
$986$	0	0
$987$	−72.7061	−2.31426
$988$	0	0
$989$	50.8519	1.61700
$990$	0	0
$991$	36.5565	1.16125	0.580627	−	0.814170i	$-0.302807\pi$
$991$	36.5565	1.16125	0.580627	+	0.814170i	$0.302807\pi$
$992$	0	0
$993$	−19.3713	−0.614729
$994$	0	0
$995$	−9.87162	−0.312951
$996$	0	0
$997$	40.0432	1.26818	0.634090	−	0.773259i	$-0.281375\pi$
$997$	40.0432	1.26818	0.634090	+	0.773259i	$0.281375\pi$
$998$	0	0
$999$	−1.98993	−0.0629586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.5	✓	33
4.3	odd	2		8048.2.a.x.1.29		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.5	✓	33	1.1	even	1	trivial
8048.2.a.x.1.29		33	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39336 q^{3} +0.556589 q^{5} +4.72926 q^{7} +2.72817 q^{9} +O(q^{10})\)	\(q-2.39336 q^{3} +0.556589 q^{5} +4.72926 q^{7} +2.72817 q^{9} -5.01195 q^{11} -3.61869 q^{13} -1.33212 q^{15} +3.83843 q^{17} -1.78519 q^{19} -11.3188 q^{21} +5.21932 q^{23} -4.69021 q^{25} +0.650590 q^{27} -1.24959 q^{29} -0.555451 q^{31} +11.9954 q^{33} +2.63225 q^{35} -3.05865 q^{37} +8.66083 q^{39} -2.20087 q^{41} +9.74301 q^{43} +1.51847 q^{45} +6.42347 q^{47} +15.3659 q^{49} -9.18674 q^{51} +9.49212 q^{53} -2.78959 q^{55} +4.27260 q^{57} -1.60506 q^{59} -11.1880 q^{61} +12.9022 q^{63} -2.01412 q^{65} +3.68589 q^{67} -12.4917 q^{69} +3.22236 q^{71} -0.536130 q^{73} +11.2254 q^{75} -23.7028 q^{77} -1.13206 q^{79} -9.74160 q^{81} -1.50141 q^{83} +2.13643 q^{85} +2.99071 q^{87} -12.0111 q^{89} -17.1137 q^{91} +1.32939 q^{93} -0.993616 q^{95} +3.66773 q^{97} -13.6734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

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