LMFDB - Embedded newform 4024.2.a.f.1.18

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			1.18
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)

$f(q)$	$=$	$q+0.176667 q^{3} +2.93614 q^{5} -0.0214699 q^{7} -2.96879 q^{9} +O(q^{10})$	$q+0.176667 q^{3} +2.93614 q^{5} -0.0214699 q^{7} -2.96879 q^{9} +6.29849 q^{11} -2.27424 q^{13} +0.518719 q^{15} +4.51509 q^{17} +7.58436 q^{19} -0.00379301 q^{21} -1.87598 q^{23} +3.62093 q^{25} -1.05449 q^{27} +2.67587 q^{29} +4.43861 q^{31} +1.11273 q^{33} -0.0630386 q^{35} -1.65062 q^{37} -0.401783 q^{39} +2.00713 q^{41} -11.7382 q^{43} -8.71678 q^{45} -9.75441 q^{47} -6.99954 q^{49} +0.797667 q^{51} +4.87542 q^{53} +18.4933 q^{55} +1.33990 q^{57} -11.3354 q^{59} +10.5999 q^{61} +0.0637395 q^{63} -6.67750 q^{65} +13.0917 q^{67} -0.331423 q^{69} +15.5338 q^{71} -10.7213 q^{73} +0.639698 q^{75} -0.135228 q^{77} +3.46921 q^{79} +8.72007 q^{81} -12.3364 q^{83} +13.2570 q^{85} +0.472738 q^{87} +15.0842 q^{89} +0.0488277 q^{91} +0.784156 q^{93} +22.2688 q^{95} +11.5293 q^{97} -18.6989 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * 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
$3$	0.176667	0.101999	0.0509993	−	0.998699i	$-0.483759\pi$
$3$	0.176667	0.101999	0.0509993	+	0.998699i	$0.483759\pi$
$4$	0	0
$5$	2.93614	1.31308	0.656541	−	0.754290i	$-0.272019\pi$
$5$	2.93614	1.31308	0.656541	+	0.754290i	$0.272019\pi$
$6$	0	0
$7$	−0.0214699	−0.00811485	−0.00405742	−	0.999992i	$-0.501292\pi$
$7$	−0.0214699	−0.00811485	−0.00405742	+	0.999992i	$0.501292\pi$
$8$	0	0
$9$	−2.96879	−0.989596
$10$	0	0
$11$	6.29849	1.89907	0.949534	−	0.313665i	$-0.101557\pi$
$11$	6.29849	1.89907	0.949534	+	0.313665i	$0.101557\pi$
$12$	0	0
$13$	−2.27424	−0.630762	−0.315381	−	0.948965i	$-0.602132\pi$
$13$	−2.27424	−0.630762	−0.315381	+	0.948965i	$0.602132\pi$
$14$	0	0
$15$	0.518719	0.133933
$16$	0	0
$17$	4.51509	1.09507	0.547536	−	0.836782i	$-0.315566\pi$
$17$	4.51509	1.09507	0.547536	+	0.836782i	$0.315566\pi$
$18$	0	0
$19$	7.58436	1.73997	0.869986	−	0.493077i	$-0.164128\pi$
$19$	7.58436	1.73997	0.869986	+	0.493077i	$0.164128\pi$
$20$	0	0
$21$	−0.00379301	−0.000827703 0
$22$	0	0
$23$	−1.87598	−0.391168	−0.195584	−	0.980687i	$-0.562660\pi$
$23$	−1.87598	−0.391168	−0.195584	+	0.980687i	$0.562660\pi$
$24$	0	0
$25$	3.62093	0.724186
$26$	0	0
$27$	−1.05449	−0.202936
$28$	0	0
$29$	2.67587	0.496897	0.248449	−	0.968645i	$-0.420079\pi$
$29$	2.67587	0.496897	0.248449	+	0.968645i	$0.420079\pi$
$30$	0	0
$31$	4.43861	0.797199	0.398599	−	0.917125i	$-0.369496\pi$
$31$	4.43861	0.797199	0.398599	+	0.917125i	$0.369496\pi$
$32$	0	0
$33$	1.11273	0.193702
$34$	0	0
$35$	−0.0630386	−0.0106555
$36$	0	0
$37$	−1.65062	−0.271359	−0.135680	−	0.990753i	$-0.543322\pi$
$37$	−1.65062	−0.271359	−0.135680	+	0.990753i	$0.543322\pi$
$38$	0	0
$39$	−0.401783	−0.0643368
$40$	0	0
$41$	2.00713	0.313461	0.156730	−	0.987641i	$-0.449905\pi$
$41$	2.00713	0.313461	0.156730	+	0.987641i	$0.449905\pi$
$42$	0	0
$43$	−11.7382	−1.79006	−0.895029	−	0.446009i	$-0.852845\pi$
$43$	−11.7382	−1.79006	−0.895029	+	0.446009i	$0.852845\pi$
$44$	0	0
$45$	−8.71678	−1.29942
$46$	0	0
$47$	−9.75441	−1.42283	−0.711413	−	0.702774i	$-0.751945\pi$
$47$	−9.75441	−1.42283	−0.711413	+	0.702774i	$0.751945\pi$
$48$	0	0
$49$	−6.99954	−0.999934
$50$	0	0
$51$	0.797667	0.111696
$52$	0	0
$53$	4.87542	0.669691	0.334845	−	0.942273i	$-0.391316\pi$
$53$	4.87542	0.669691	0.334845	+	0.942273i	$0.391316\pi$
$54$	0	0
$55$	18.4933	2.49363
$56$	0	0
$57$	1.33990	0.177475
$58$	0	0
$59$	−11.3354	−1.47575	−0.737874	−	0.674938i	$-0.764170\pi$
$59$	−11.3354	−1.47575	−0.737874	+	0.674938i	$0.764170\pi$
$60$	0	0
$61$	10.5999	1.35718	0.678592	−	0.734516i	$-0.262591\pi$
$61$	10.5999	1.35718	0.678592	+	0.734516i	$0.262591\pi$
$62$	0	0
$63$	0.0637395	0.00803042
$64$	0	0
$65$	−6.67750	−0.828242
$66$	0	0
$67$	13.0917	1.59940	0.799700	−	0.600400i	$-0.204992\pi$
$67$	13.0917	1.59940	0.799700	+	0.600400i	$0.204992\pi$
$68$	0	0
$69$	−0.331423	−0.0398986
$70$	0	0
$71$	15.5338	1.84352	0.921760	−	0.387761i	$-0.126751\pi$
$71$	15.5338	1.84352	0.921760	+	0.387761i	$0.126751\pi$
$72$	0	0
$73$	−10.7213	−1.25483	−0.627417	−	0.778683i	$-0.715888\pi$
$73$	−10.7213	−1.25483	−0.627417	+	0.778683i	$0.715888\pi$
$74$	0	0
$75$	0.639698	0.0738659
$76$	0	0
$77$	−0.135228	−0.0154106
$78$	0	0
$79$	3.46921	0.390317	0.195159	−	0.980772i	$-0.437478\pi$
$79$	3.46921	0.390317	0.195159	+	0.980772i	$0.437478\pi$
$80$	0	0
$81$	8.72007	0.968897
$82$	0	0
$83$	−12.3364	−1.35409	−0.677047	−	0.735940i	$-0.736741\pi$
$83$	−12.3364	−1.35409	−0.677047	+	0.735940i	$0.736741\pi$
$84$	0	0
$85$	13.2570	1.43792
$86$	0	0
$87$	0.472738	0.0506828
$88$	0	0
$89$	15.0842	1.59892	0.799461	−	0.600718i	$-0.205119\pi$
$89$	15.0842	1.59892	0.799461	+	0.600718i	$0.205119\pi$
$90$	0	0
$91$	0.0488277	0.00511854
$92$	0	0
$93$	0.784156	0.0813132
$94$	0	0
$95$	22.2688	2.28473
$96$	0	0
$97$	11.5293	1.17063	0.585314	−	0.810807i	$-0.300971\pi$
$97$	11.5293	1.17063	0.585314	+	0.810807i	$0.300971\pi$
$98$	0	0
$99$	−18.6989	−1.87931
$100$	0	0
$101$	11.2463	1.11904	0.559522	−	0.828816i	$-0.310985\pi$
$101$	11.2463	1.11904	0.559522	+	0.828816i	$0.310985\pi$
$102$	0	0
$103$	−11.4865	−1.13180	−0.565899	−	0.824475i	$-0.691471\pi$
$103$	−11.4865	−1.13180	−0.565899	+	0.824475i	$0.691471\pi$
$104$	0	0
$105$	−0.0111368	−0.00108684
$106$	0	0
$107$	−11.2308	−1.08573	−0.542863	−	0.839821i	$-0.682660\pi$
$107$	−11.2308	−1.08573	−0.542863	+	0.839821i	$0.682660\pi$
$108$	0	0
$109$	5.85875	0.561166	0.280583	−	0.959830i	$-0.409472\pi$
$109$	5.85875	0.561166	0.280583	+	0.959830i	$0.409472\pi$
$110$	0	0
$111$	−0.291609	−0.0276783
$112$	0	0
$113$	−1.54531	−0.145370	−0.0726852	−	0.997355i	$-0.523157\pi$
$113$	−1.54531	−0.145370	−0.0726852	+	0.997355i	$0.523157\pi$
$114$	0	0
$115$	−5.50813	−0.513636
$116$	0	0
$117$	6.75175	0.624199
$118$	0	0
$119$	−0.0969385	−0.00888633
$120$	0	0
$121$	28.6710	2.60646
$122$	0	0
$123$	0.354593	0.0319726
$124$	0	0
$125$	−4.04915	−0.362167
$126$	0	0
$127$	−6.45691	−0.572958	−0.286479	−	0.958087i	$-0.592485\pi$
$127$	−6.45691	−0.572958	−0.286479	+	0.958087i	$0.592485\pi$
$128$	0	0
$129$	−2.07375	−0.182583
$130$	0	0
$131$	−13.3945	−1.17028	−0.585140	−	0.810933i	$-0.698960\pi$
$131$	−13.3945	−1.17028	−0.585140	+	0.810933i	$0.698960\pi$
$132$	0	0
$133$	−0.162835	−0.0141196
$134$	0	0
$135$	−3.09612	−0.266472
$136$	0	0
$137$	−0.00459926	−0.000392942 0	−0.000196471	−	1.00000i	$-0.500063\pi$
$137$	−0.00459926	−0.000392942 0	−0.000196471		1.00000i	$0.500063\pi$
$138$	0	0
$139$	−8.73335	−0.740753	−0.370377	−	0.928882i	$-0.620771\pi$
$139$	−8.73335	−0.740753	−0.370377	+	0.928882i	$0.620771\pi$
$140$	0	0
$141$	−1.72328	−0.145126
$142$	0	0
$143$	−14.3243	−1.19786
$144$	0	0
$145$	7.85674	0.652467
$146$	0	0
$147$	−1.23659	−0.101992
$148$	0	0
$149$	12.7870	1.04755	0.523777	−	0.851855i	$-0.324522\pi$
$149$	12.7870	1.04755	0.523777	+	0.851855i	$0.324522\pi$
$150$	0	0
$151$	18.2554	1.48560	0.742802	−	0.669511i	$-0.233496\pi$
$151$	18.2554	1.48560	0.742802	+	0.669511i	$0.233496\pi$
$152$	0	0
$153$	−13.4044	−1.08368
$154$	0	0
$155$	13.0324	1.04679
$156$	0	0
$157$	−18.8109	−1.50128	−0.750638	−	0.660714i	$-0.770254\pi$
$157$	−18.8109	−1.50128	−0.750638	+	0.660714i	$0.770254\pi$
$158$	0	0
$159$	0.861325	0.0683075
$160$	0	0
$161$	0.0402770	0.00317427
$162$	0	0
$163$	20.0510	1.57051	0.785256	−	0.619171i	$-0.212531\pi$
$163$	20.0510	1.57051	0.785256	+	0.619171i	$0.212531\pi$
$164$	0	0
$165$	3.26715	0.254347
$166$	0	0
$167$	−7.24248	−0.560440	−0.280220	−	0.959936i	$-0.590407\pi$
$167$	−7.24248	−0.560440	−0.280220	+	0.959936i	$0.590407\pi$
$168$	0	0
$169$	−7.82782	−0.602140
$170$	0	0
$171$	−22.5164	−1.72187
$172$	0	0
$173$	11.7564	0.893821	0.446911	−	0.894579i	$-0.352524\pi$
$173$	11.7564	0.893821	0.446911	+	0.894579i	$0.352524\pi$
$174$	0	0
$175$	−0.0777408	−0.00587666
$176$	0	0
$177$	−2.00260	−0.150524
$178$	0	0
$179$	−17.3442	−1.29637	−0.648184	−	0.761484i	$-0.724471\pi$
$179$	−17.3442	−1.29637	−0.648184	+	0.761484i	$0.724471\pi$
$180$	0	0
$181$	16.3375	1.21436	0.607180	−	0.794564i	$-0.292301\pi$
$181$	16.3375	1.21436	0.607180	+	0.794564i	$0.292301\pi$
$182$	0	0
$183$	1.87266	0.138431
$184$	0	0
$185$	−4.84644	−0.356317
$186$	0	0
$187$	28.4383	2.07961
$188$	0	0
$189$	0.0226397	0.00164680
$190$	0	0
$191$	−17.5343	−1.26874	−0.634368	−	0.773031i	$-0.718740\pi$
$191$	−17.5343	−1.26874	−0.634368	+	0.773031i	$0.718740\pi$
$192$	0	0
$193$	12.7833	0.920165	0.460083	−	0.887876i	$-0.347820\pi$
$193$	12.7833	0.920165	0.460083	+	0.887876i	$0.347820\pi$
$194$	0	0
$195$	−1.17969	−0.0844796
$196$	0	0
$197$	−8.54813	−0.609029	−0.304515	−	0.952508i	$-0.598494\pi$
$197$	−8.54813	−0.609029	−0.304515	+	0.952508i	$0.598494\pi$
$198$	0	0
$199$	21.3284	1.51193	0.755967	−	0.654610i	$-0.227167\pi$
$199$	21.3284	1.51193	0.755967	+	0.654610i	$0.227167\pi$
$200$	0	0
$201$	2.31286	0.163137
$202$	0	0
$203$	−0.0574506	−0.00403224
$204$	0	0
$205$	5.89321	0.411600
$206$	0	0
$207$	5.56938	0.387099
$208$	0	0
$209$	47.7700	3.30432
$210$	0	0
$211$	−13.1950	−0.908379	−0.454189	−	0.890905i	$-0.650071\pi$
$211$	−13.1950	−0.908379	−0.454189	+	0.890905i	$0.650071\pi$
$212$	0	0
$213$	2.74430	0.188037
$214$	0	0
$215$	−34.4650	−2.35049
$216$	0	0
$217$	−0.0952965	−0.00646915
$218$	0	0
$219$	−1.89410	−0.127991
$220$	0	0
$221$	−10.2684	−0.690729
$222$	0	0
$223$	22.3627	1.49752	0.748758	−	0.662843i	$-0.230650\pi$
$223$	22.3627	1.49752	0.748758	+	0.662843i	$0.230650\pi$
$224$	0	0
$225$	−10.7498	−0.716651
$226$	0	0
$227$	22.6226	1.50151	0.750756	−	0.660580i	$-0.229690\pi$
$227$	22.6226	1.50151	0.750756	+	0.660580i	$0.229690\pi$
$228$	0	0
$229$	−8.01132	−0.529403	−0.264701	−	0.964330i	$-0.585273\pi$
$229$	−8.01132	−0.529403	−0.264701	+	0.964330i	$0.585273\pi$
$230$	0	0
$231$	−0.0238903	−0.00157186
$232$	0	0
$233$	6.18359	0.405100	0.202550	−	0.979272i	$-0.435077\pi$
$233$	6.18359	0.405100	0.202550	+	0.979272i	$0.435077\pi$
$234$	0	0
$235$	−28.6403	−1.86829
$236$	0	0
$237$	0.612895	0.0398118
$238$	0	0
$239$	−3.27018	−0.211531	−0.105765	−	0.994391i	$-0.533729\pi$
$239$	−3.27018	−0.211531	−0.105765	+	0.994391i	$0.533729\pi$
$240$	0	0
$241$	4.79708	0.309007	0.154504	−	0.987992i	$-0.450622\pi$
$241$	4.79708	0.309007	0.154504	+	0.987992i	$0.450622\pi$
$242$	0	0
$243$	4.70401	0.301762
$244$	0	0
$245$	−20.5516	−1.31300
$246$	0	0
$247$	−17.2487	−1.09751
$248$	0	0
$249$	−2.17943	−0.138116
$250$	0	0
$251$	−10.7909	−0.681115	−0.340557	−	0.940224i	$-0.610616\pi$
$251$	−10.7909	−0.681115	−0.340557	+	0.940224i	$0.610616\pi$
$252$	0	0
$253$	−11.8158	−0.742855
$254$	0	0
$255$	2.34206	0.146666
$256$	0	0
$257$	7.89087	0.492219	0.246109	−	0.969242i	$-0.420848\pi$
$257$	7.89087	0.492219	0.246109	+	0.969242i	$0.420848\pi$
$258$	0	0
$259$	0.0354385	0.00220204
$260$	0	0
$261$	−7.94410	−0.491728
$262$	0	0
$263$	1.87596	0.115677	0.0578383	−	0.998326i	$-0.481579\pi$
$263$	1.87596	0.115677	0.0578383	+	0.998326i	$0.481579\pi$
$264$	0	0
$265$	14.3149	0.879359
$266$	0	0
$267$	2.66488	0.163088
$268$	0	0
$269$	−5.25651	−0.320495	−0.160247	−	0.987077i	$-0.551229\pi$
$269$	−5.25651	−0.320495	−0.160247	+	0.987077i	$0.551229\pi$
$270$	0	0
$271$	−3.97557	−0.241499	−0.120749	−	0.992683i	$-0.538530\pi$
$271$	−3.97557	−0.241499	−0.120749	+	0.992683i	$0.538530\pi$
$272$	0	0
$273$	0.00862624	0.000522084 0
$274$	0	0
$275$	22.8064	1.37528
$276$	0	0
$277$	8.19935	0.492651	0.246326	−	0.969187i	$-0.420777\pi$
$277$	8.19935	0.492651	0.246326	+	0.969187i	$0.420777\pi$
$278$	0	0
$279$	−13.1773	−0.788905
$280$	0	0
$281$	−14.3985	−0.858941	−0.429471	−	0.903081i	$-0.641300\pi$
$281$	−14.3985	−0.858941	−0.429471	+	0.903081i	$0.641300\pi$
$282$	0	0
$283$	−1.89084	−0.112399	−0.0561993	−	0.998420i	$-0.517898\pi$
$283$	−1.89084	−0.112399	−0.0561993	+	0.998420i	$0.517898\pi$
$284$	0	0
$285$	3.93415	0.233039
$286$	0	0
$287$	−0.0430928	−0.00254369
$288$	0	0
$289$	3.38607	0.199181
$290$	0	0
$291$	2.03685	0.119402
$292$	0	0
$293$	−30.6270	−1.78925	−0.894624	−	0.446819i	$-0.852557\pi$
$293$	−30.6270	−1.78925	−0.894624	+	0.446819i	$0.852557\pi$
$294$	0	0
$295$	−33.2824	−1.93778
$296$	0	0
$297$	−6.64168	−0.385389
$298$	0	0
$299$	4.26643	0.246734
$300$	0	0
$301$	0.252017	0.0145260
$302$	0	0
$303$	1.98684	0.114141
$304$	0	0
$305$	31.1229	1.78209
$306$	0	0
$307$	−16.4527	−0.939007	−0.469504	−	0.882931i	$-0.655567\pi$
$307$	−16.4527	−0.939007	−0.469504	+	0.882931i	$0.655567\pi$
$308$	0	0
$309$	−2.02928	−0.115442
$310$	0	0
$311$	24.1668	1.37037	0.685187	−	0.728367i	$-0.259720\pi$
$311$	24.1668	1.37037	0.685187	+	0.728367i	$0.259720\pi$
$312$	0	0
$313$	8.57106	0.484465	0.242232	−	0.970218i	$-0.422120\pi$
$313$	8.57106	0.484465	0.242232	+	0.970218i	$0.422120\pi$
$314$	0	0
$315$	0.187148	0.0105446
$316$	0	0
$317$	−6.50702	−0.365471	−0.182735	−	0.983162i	$-0.558495\pi$
$317$	−6.50702	−0.365471	−0.182735	+	0.983162i	$0.558495\pi$
$318$	0	0
$319$	16.8540	0.943641
$320$	0	0
$321$	−1.98412	−0.110743
$322$	0	0
$323$	34.2441	1.90539
$324$	0	0
$325$	−8.23487	−0.456789
$326$	0	0
$327$	1.03505	0.0572382
$328$	0	0
$329$	0.209426	0.0115460
$330$	0	0
$331$	11.8818	0.653080	0.326540	−	0.945183i	$-0.394117\pi$
$331$	11.8818	0.653080	0.326540	+	0.945183i	$0.394117\pi$
$332$	0	0
$333$	4.90033	0.268536
$334$	0	0
$335$	38.4390	2.10014
$336$	0	0
$337$	−10.2976	−0.560946	−0.280473	−	0.959862i	$-0.590491\pi$
$337$	−10.2976	−0.560946	−0.280473	+	0.959862i	$0.590491\pi$
$338$	0	0
$339$	−0.273005	−0.0148276
$340$	0	0
$341$	27.9566	1.51393
$342$	0	0
$343$	0.300568	0.0162292
$344$	0	0
$345$	−0.973105	−0.0523902
$346$	0	0
$347$	−17.8632	−0.958947	−0.479474	−	0.877556i	$-0.659172\pi$
$347$	−17.8632	−0.958947	−0.479474	+	0.877556i	$0.659172\pi$
$348$	0	0
$349$	29.5910	1.58397	0.791985	−	0.610541i	$-0.209048\pi$
$349$	29.5910	1.58397	0.791985	+	0.610541i	$0.209048\pi$
$350$	0	0
$351$	2.39816	0.128004
$352$	0	0
$353$	−5.02308	−0.267352	−0.133676	−	0.991025i	$-0.542678\pi$
$353$	−5.02308	−0.267352	−0.133676	+	0.991025i	$0.542678\pi$
$354$	0	0
$355$	45.6094	2.42069
$356$	0	0
$357$	−0.0171258	−0.000906394 0
$358$	0	0
$359$	22.9349	1.21046	0.605228	−	0.796052i	$-0.293082\pi$
$359$	22.9349	1.21046	0.605228	+	0.796052i	$0.293082\pi$
$360$	0	0
$361$	38.5225	2.02750
$362$	0	0
$363$	5.06522	0.265855
$364$	0	0
$365$	−31.4793	−1.64770
$366$	0	0
$367$	18.6345	0.972714	0.486357	−	0.873760i	$-0.338326\pi$
$367$	18.6345	0.972714	0.486357	+	0.873760i	$0.338326\pi$
$368$	0	0
$369$	−5.95874	−0.310200
$370$	0	0
$371$	−0.104675	−0.00543444
$372$	0	0
$373$	−23.8626	−1.23556	−0.617779	−	0.786352i	$-0.711967\pi$
$373$	−23.8626	−1.23556	−0.617779	+	0.786352i	$0.711967\pi$
$374$	0	0
$375$	−0.715351	−0.0369406
$376$	0	0
$377$	−6.08559	−0.313424
$378$	0	0
$379$	−22.5585	−1.15875	−0.579376	−	0.815061i	$-0.696704\pi$
$379$	−22.5585	−1.15875	−0.579376	+	0.815061i	$0.696704\pi$
$380$	0	0
$381$	−1.14072	−0.0584409
$382$	0	0
$383$	−7.65981	−0.391398	−0.195699	−	0.980664i	$-0.562698\pi$
$383$	−7.65981	−0.391398	−0.195699	+	0.980664i	$0.562698\pi$
$384$	0	0
$385$	−0.397048	−0.0202354
$386$	0	0
$387$	34.8482	1.77143
$388$	0	0
$389$	−7.20357	−0.365235	−0.182618	−	0.983184i	$-0.558457\pi$
$389$	−7.20357	−0.365235	−0.182618	+	0.983184i	$0.558457\pi$
$390$	0	0
$391$	−8.47021	−0.428357
$392$	0	0
$393$	−2.36636	−0.119367
$394$	0	0
$395$	10.1861	0.512519
$396$	0	0
$397$	17.4797	0.877282	0.438641	−	0.898662i	$-0.355460\pi$
$397$	17.4797	0.877282	0.438641	+	0.898662i	$0.355460\pi$
$398$	0	0
$399$	−0.0287676	−0.00144018
$400$	0	0
$401$	−34.0067	−1.69821	−0.849106	−	0.528223i	$-0.822859\pi$
$401$	−34.0067	−1.69821	−0.849106	+	0.528223i	$0.822859\pi$
$402$	0	0
$403$	−10.0945	−0.502842
$404$	0	0
$405$	25.6034	1.27224
$406$	0	0
$407$	−10.3964	−0.515330
$408$	0	0
$409$	−10.0657	−0.497717	−0.248859	−	0.968540i	$-0.580055\pi$
$409$	−10.0657	−0.497717	−0.248859	+	0.968540i	$0.580055\pi$
$410$	0	0
$411$	−0.000812537 0	−4.00795e−5 0
$412$	0	0
$413$	0.243370	0.0119755
$414$	0	0
$415$	−36.2214	−1.77804
$416$	0	0
$417$	−1.54289	−0.0755558
$418$	0	0
$419$	−2.82828	−0.138170	−0.0690852	−	0.997611i	$-0.522008\pi$
$419$	−2.82828	−0.138170	−0.0690852	+	0.997611i	$0.522008\pi$
$420$	0	0
$421$	−9.00033	−0.438649	−0.219325	−	0.975652i	$-0.570385\pi$
$421$	−9.00033	−0.438649	−0.219325	+	0.975652i	$0.570385\pi$
$422$	0	0
$423$	28.9588	1.40802
$424$	0	0
$425$	16.3488	0.793035
$426$	0	0
$427$	−0.227579	−0.0110133
$428$	0	0
$429$	−2.53063	−0.122180
$430$	0	0
$431$	−31.2794	−1.50668	−0.753338	−	0.657633i	$-0.771558\pi$
$431$	−31.2794	−1.50668	−0.753338	+	0.657633i	$0.771558\pi$
$432$	0	0
$433$	−34.3162	−1.64913	−0.824567	−	0.565765i	$-0.808581\pi$
$433$	−34.3162	−1.64913	−0.824567	+	0.565765i	$0.808581\pi$
$434$	0	0
$435$	1.38803	0.0665507
$436$	0	0
$437$	−14.2281	−0.680622
$438$	0	0
$439$	−36.9964	−1.76574	−0.882872	−	0.469614i	$-0.844393\pi$
$439$	−36.9964	−1.76574	−0.882872	+	0.469614i	$0.844393\pi$
$440$	0	0
$441$	20.7802	0.989531
$442$	0	0
$443$	5.66135	0.268979	0.134489	−	0.990915i	$-0.457061\pi$
$443$	5.66135	0.268979	0.134489	+	0.990915i	$0.457061\pi$
$444$	0	0
$445$	44.2893	2.09952
$446$	0	0
$447$	2.25905	0.106849
$448$	0	0
$449$	−0.628289	−0.0296508	−0.0148254	−	0.999890i	$-0.504719\pi$
$449$	−0.628289	−0.0296508	−0.0148254	+	0.999890i	$0.504719\pi$
$450$	0	0
$451$	12.6419	0.595283
$452$	0	0
$453$	3.22513	0.151530
$454$	0	0
$455$	0.143365	0.00672106
$456$	0	0
$457$	7.95443	0.372093	0.186046	−	0.982541i	$-0.440433\pi$
$457$	7.95443	0.372093	0.186046	+	0.982541i	$0.440433\pi$
$458$	0	0
$459$	−4.76111	−0.222229
$460$	0	0
$461$	40.3617	1.87983	0.939915	−	0.341408i	$-0.110904\pi$
$461$	40.3617	1.87983	0.939915	+	0.341408i	$0.110904\pi$
$462$	0	0
$463$	−27.8717	−1.29531	−0.647653	−	0.761935i	$-0.724249\pi$
$463$	−27.8717	−1.29531	−0.647653	+	0.761935i	$0.724249\pi$
$464$	0	0
$465$	2.30239	0.106771
$466$	0	0
$467$	11.1488	0.515906	0.257953	−	0.966157i	$-0.416952\pi$
$467$	11.1488	0.515906	0.257953	+	0.966157i	$0.416952\pi$
$468$	0	0
$469$	−0.281076	−0.0129789
$470$	0	0
$471$	−3.32327	−0.153128
$472$	0	0
$473$	−73.9329	−3.39944
$474$	0	0
$475$	27.4624	1.26006
$476$	0	0
$477$	−14.4741	−0.662723
$478$	0	0
$479$	21.0275	0.960770	0.480385	−	0.877058i	$-0.340497\pi$
$479$	21.0275	0.960770	0.480385	+	0.877058i	$0.340497\pi$
$480$	0	0
$481$	3.75390	0.171163
$482$	0	0
$483$	0.00711561	0.000323771 0
$484$	0	0
$485$	33.8518	1.53713
$486$	0	0
$487$	17.5834	0.796780	0.398390	−	0.917216i	$-0.369569\pi$
$487$	17.5834	0.796780	0.398390	+	0.917216i	$0.369569\pi$
$488$	0	0
$489$	3.54234	0.160190
$490$	0	0
$491$	2.91648	0.131619	0.0658094	−	0.997832i	$-0.479037\pi$
$491$	2.91648	0.131619	0.0658094	+	0.997832i	$0.479037\pi$
$492$	0	0
$493$	12.0818	0.544138
$494$	0	0
$495$	−54.9026	−2.46769
$496$	0	0
$497$	−0.333508	−0.0149599
$498$	0	0
$499$	−1.89433	−0.0848018	−0.0424009	−	0.999101i	$-0.513501\pi$
$499$	−1.89433	−0.0848018	−0.0424009	+	0.999101i	$0.513501\pi$
$500$	0	0
$501$	−1.27951	−0.0571641
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	33.0206	1.46940
$506$	0	0
$507$	−1.38292	−0.0614174
$508$	0	0
$509$	14.2568	0.631922	0.315961	−	0.948772i	$-0.397673\pi$
$509$	14.2568	0.631922	0.315961	+	0.948772i	$0.397673\pi$
$510$	0	0
$511$	0.230185	0.0101828
$512$	0	0
$513$	−7.99761	−0.353103
$514$	0	0
$515$	−33.7260	−1.48614
$516$	0	0
$517$	−61.4381	−2.70204
$518$	0	0
$519$	2.07696	0.0911686
$520$	0	0
$521$	25.3197	1.10928	0.554639	−	0.832091i	$-0.312856\pi$
$521$	25.3197	1.10928	0.554639	+	0.832091i	$0.312856\pi$
$522$	0	0
$523$	−14.8465	−0.649192	−0.324596	−	0.945853i	$-0.605228\pi$
$523$	−14.8465	−0.649192	−0.324596	+	0.945853i	$0.605228\pi$
$524$	0	0
$525$	−0.0137342	−0.000599411 0
$526$	0	0
$527$	20.0408	0.872989
$528$	0	0
$529$	−19.4807	−0.846987
$530$	0	0
$531$	33.6525	1.46039
$532$	0	0
$533$	−4.56470	−0.197719
$534$	0	0
$535$	−32.9753	−1.42565
$536$	0	0
$537$	−3.06415	−0.132228
$538$	0	0
$539$	−44.0866	−1.89894
$540$	0	0
$541$	−9.98450	−0.429267	−0.214634	−	0.976695i	$-0.568856\pi$
$541$	−9.98450	−0.429267	−0.214634	+	0.976695i	$0.568856\pi$
$542$	0	0
$543$	2.88630	0.123863
$544$	0	0
$545$	17.2021	0.736858
$546$	0	0
$547$	32.4471	1.38734	0.693669	−	0.720294i	$-0.255993\pi$
$547$	32.4471	1.38734	0.693669	+	0.720294i	$0.255993\pi$
$548$	0	0
$549$	−31.4690	−1.34306
$550$	0	0
$551$	20.2948	0.864587
$552$	0	0
$553$	−0.0744836	−0.00316736
$554$	0	0
$555$	−0.856205	−0.0363439
$556$	0	0
$557$	−9.26551	−0.392592	−0.196296	−	0.980545i	$-0.562891\pi$
$557$	−9.26551	−0.392592	−0.196296	+	0.980545i	$0.562891\pi$
$558$	0	0
$559$	26.6955	1.12910
$560$	0	0
$561$	5.02410	0.212118
$562$	0	0
$563$	7.65196	0.322492	0.161246	−	0.986914i	$-0.448449\pi$
$563$	7.65196	0.322492	0.161246	+	0.986914i	$0.448449\pi$
$564$	0	0
$565$	−4.53725	−0.190883
$566$	0	0
$567$	−0.187219	−0.00786245
$568$	0	0
$569$	25.5417	1.07077	0.535383	−	0.844610i	$-0.320167\pi$
$569$	25.5417	1.07077	0.535383	+	0.844610i	$0.320167\pi$
$570$	0	0
$571$	36.2498	1.51701	0.758504	−	0.651669i	$-0.225931\pi$
$571$	36.2498	1.51701	0.758504	+	0.651669i	$0.225931\pi$
$572$	0	0
$573$	−3.09773	−0.129409
$574$	0	0
$575$	−6.79278	−0.283278
$576$	0	0
$577$	23.7393	0.988280	0.494140	−	0.869382i	$-0.335483\pi$
$577$	23.7393	0.988280	0.494140	+	0.869382i	$0.335483\pi$
$578$	0	0
$579$	2.25839	0.0938556
$580$	0	0
$581$	0.264860	0.0109883
$582$	0	0
$583$	30.7078	1.27179
$584$	0	0
$585$	19.8241	0.819625
$586$	0	0
$587$	−36.6343	−1.51206	−0.756029	−	0.654538i	$-0.772863\pi$
$587$	−36.6343	−1.51206	−0.756029	+	0.654538i	$0.772863\pi$
$588$	0	0
$589$	33.6641	1.38710
$590$	0	0
$591$	−1.51017	−0.0621201
$592$	0	0
$593$	−5.37378	−0.220675	−0.110337	−	0.993894i	$-0.535193\pi$
$593$	−5.37378	−0.220675	−0.110337	+	0.993894i	$0.535193\pi$
$594$	0	0
$595$	−0.284625	−0.0116685
$596$	0	0
$597$	3.76803	0.154215
$598$	0	0
$599$	−6.75259	−0.275903	−0.137952	−	0.990439i	$-0.544052\pi$
$599$	−6.75259	−0.275903	−0.137952	+	0.990439i	$0.544052\pi$
$600$	0	0
$601$	−42.2711	−1.72427	−0.862137	−	0.506675i	$-0.830874\pi$
$601$	−42.2711	−1.72427	−0.862137	+	0.506675i	$0.830874\pi$
$602$	0	0
$603$	−38.8664	−1.58276
$604$	0	0
$605$	84.1822	3.42249
$606$	0	0
$607$	−15.0929	−0.612600	−0.306300	−	0.951935i	$-0.599091\pi$
$607$	−15.0929	−0.612600	−0.306300	+	0.951935i	$0.599091\pi$
$608$	0	0
$609$	−0.0101496	−0.000411283 0
$610$	0	0
$611$	22.1839	0.897465
$612$	0	0
$613$	22.0223	0.889471	0.444735	−	0.895662i	$-0.353298\pi$
$613$	22.0223	0.889471	0.444735	+	0.895662i	$0.353298\pi$
$614$	0	0
$615$	1.04113	0.0419826
$616$	0	0
$617$	0.859169	0.0345888	0.0172944	−	0.999850i	$-0.494495\pi$
$617$	0.859169	0.0345888	0.0172944	+	0.999850i	$0.494495\pi$
$618$	0	0
$619$	14.2478	0.572667	0.286333	−	0.958130i	$-0.407563\pi$
$619$	14.2478	0.572667	0.286333	+	0.958130i	$0.407563\pi$
$620$	0	0
$621$	1.97819	0.0793822
$622$	0	0
$623$	−0.323856	−0.0129750
$624$	0	0
$625$	−29.9935	−1.19974
$626$	0	0
$627$	8.43938	0.337036
$628$	0	0
$629$	−7.45268	−0.297158
$630$	0	0
$631$	−22.1829	−0.883086	−0.441543	−	0.897240i	$-0.645569\pi$
$631$	−22.1829	−0.883086	−0.441543	+	0.897240i	$0.645569\pi$
$632$	0	0
$633$	−2.33111	−0.0926534
$634$	0	0
$635$	−18.9584	−0.752341
$636$	0	0
$637$	15.9187	0.630720
$638$	0	0
$639$	−46.1165	−1.82434
$640$	0	0
$641$	−7.41272	−0.292785	−0.146393	−	0.989227i	$-0.546766\pi$
$641$	−7.41272	−0.292785	−0.146393	+	0.989227i	$0.546766\pi$
$642$	0	0
$643$	−8.48312	−0.334542	−0.167271	−	0.985911i	$-0.553495\pi$
$643$	−8.48312	−0.334542	−0.167271	+	0.985911i	$0.553495\pi$
$644$	0	0
$645$	−6.08882	−0.239747
$646$	0	0
$647$	−41.0743	−1.61480	−0.807399	−	0.590006i	$-0.799125\pi$
$647$	−41.0743	−1.61480	−0.807399	+	0.590006i	$0.799125\pi$
$648$	0	0
$649$	−71.3962	−2.80254
$650$	0	0
$651$	−0.0168357	−0.000659844 0
$652$	0	0
$653$	5.74274	0.224731	0.112365	−	0.993667i	$-0.464157\pi$
$653$	5.74274	0.224731	0.112365	+	0.993667i	$0.464157\pi$
$654$	0	0
$655$	−39.3280	−1.53667
$656$	0	0
$657$	31.8293	1.24178
$658$	0	0
$659$	−43.0135	−1.67557	−0.837785	−	0.546001i	$-0.816150\pi$
$659$	−43.0135	−1.67557	−0.837785	+	0.546001i	$0.816150\pi$
$660$	0	0
$661$	−11.8931	−0.462588	−0.231294	−	0.972884i	$-0.574296\pi$
$661$	−11.8931	−0.462588	−0.231294	+	0.972884i	$0.574296\pi$
$662$	0	0
$663$	−1.81409	−0.0704534
$664$	0	0
$665$	−0.478107	−0.0185402
$666$	0	0
$667$	−5.01988	−0.194370
$668$	0	0
$669$	3.95075	0.152745
$670$	0	0
$671$	66.7637	2.57738
$672$	0	0
$673$	14.5940	0.562558	0.281279	−	0.959626i	$-0.409241\pi$
$673$	14.5940	0.562558	0.281279	+	0.959626i	$0.409241\pi$
$674$	0	0
$675$	−3.81822	−0.146963
$676$	0	0
$677$	15.8167	0.607886	0.303943	−	0.952690i	$-0.401697\pi$
$677$	15.8167	0.607886	0.303943	+	0.952690i	$0.401697\pi$
$678$	0	0
$679$	−0.247534	−0.00949947
$680$	0	0
$681$	3.99665	0.153152
$682$	0	0
$683$	−24.7270	−0.946151	−0.473076	−	0.881022i	$-0.656856\pi$
$683$	−24.7270	−0.946151	−0.473076	+	0.881022i	$0.656856\pi$
$684$	0	0
$685$	−0.0135041	−0.000515965 0
$686$	0	0
$687$	−1.41533	−0.0539984
$688$	0	0
$689$	−11.0879	−0.422415
$690$	0	0
$691$	−10.9475	−0.416463	−0.208232	−	0.978080i	$-0.566771\pi$
$691$	−10.9475	−0.416463	−0.208232	+	0.978080i	$0.566771\pi$
$692$	0	0
$693$	0.401463	0.0152503
$694$	0	0
$695$	−25.6424	−0.972670
$696$	0	0
$697$	9.06237	0.343262
$698$	0	0
$699$	1.09243	0.0413197
$700$	0	0
$701$	−34.1212	−1.28874	−0.644370	−	0.764714i	$-0.722880\pi$
$701$	−34.1212	−1.28874	−0.644370	+	0.764714i	$0.722880\pi$
$702$	0	0
$703$	−12.5189	−0.472158
$704$	0	0
$705$	−5.05980	−0.190563
$706$	0	0
$707$	−0.241456	−0.00908087
$708$	0	0
$709$	−42.7035	−1.60376	−0.801881	−	0.597483i	$-0.796167\pi$
$709$	−42.7035	−1.60376	−0.801881	+	0.597483i	$0.796167\pi$
$710$	0	0
$711$	−10.2994	−0.386256
$712$	0	0
$713$	−8.32674	−0.311839
$714$	0	0
$715$	−42.0582	−1.57289
$716$	0	0
$717$	−0.577733	−0.0215758
$718$	0	0
$719$	−1.97249	−0.0735615	−0.0367808	−	0.999323i	$-0.511710\pi$
$719$	−1.97249	−0.0735615	−0.0367808	+	0.999323i	$0.511710\pi$
$720$	0	0
$721$	0.246613	0.00918436
$722$	0	0
$723$	0.847485	0.0315183
$724$	0	0
$725$	9.68914	0.359846
$726$	0	0
$727$	−48.1145	−1.78447	−0.892235	−	0.451572i	$-0.850863\pi$
$727$	−48.1145	−1.78447	−0.892235	+	0.451572i	$0.850863\pi$
$728$	0	0
$729$	−25.3292	−0.938118
$730$	0	0
$731$	−52.9990	−1.96024
$732$	0	0
$733$	10.3605	0.382674	0.191337	−	0.981524i	$-0.438718\pi$
$733$	10.3605	0.382674	0.191337	+	0.981524i	$0.438718\pi$
$734$	0	0
$735$	−3.63079	−0.133924
$736$	0	0
$737$	82.4577	3.03737
$738$	0	0
$739$	10.8592	0.399463	0.199731	−	0.979851i	$-0.435993\pi$
$739$	10.8592	0.399463	0.199731	+	0.979851i	$0.435993\pi$
$740$	0	0
$741$	−3.04727	−0.111944
$742$	0	0
$743$	44.6501	1.63805	0.819026	−	0.573756i	$-0.194514\pi$
$743$	44.6501	1.63805	0.819026	+	0.573756i	$0.194514\pi$
$744$	0	0
$745$	37.5445	1.37553
$746$	0	0
$747$	36.6241	1.34001
$748$	0	0
$749$	0.241125	0.00881050
$750$	0	0
$751$	−36.3333	−1.32582	−0.662911	−	0.748698i	$-0.730679\pi$
$751$	−36.3333	−1.32582	−0.662911	+	0.748698i	$0.730679\pi$
$752$	0	0
$753$	−1.90639	−0.0694728
$754$	0	0
$755$	53.6005	1.95072
$756$	0	0
$757$	−18.1570	−0.659929	−0.329964	−	0.943993i	$-0.607037\pi$
$757$	−18.1570	−0.659929	−0.329964	+	0.943993i	$0.607037\pi$
$758$	0	0
$759$	−2.08747	−0.0757702
$760$	0	0
$761$	−20.7030	−0.750483	−0.375242	−	0.926927i	$-0.622440\pi$
$761$	−20.7030	−0.750483	−0.375242	+	0.926927i	$0.622440\pi$
$762$	0	0
$763$	−0.125787	−0.00455378
$764$	0	0
$765$	−39.3571	−1.42296
$766$	0	0
$767$	25.7795	0.930845
$768$	0	0
$769$	−34.7933	−1.25468	−0.627340	−	0.778745i	$-0.715856\pi$
$769$	−34.7933	−1.25468	−0.627340	+	0.778745i	$0.715856\pi$
$770$	0	0
$771$	1.39405	0.0502056
$772$	0	0
$773$	12.2523	0.440684	0.220342	−	0.975423i	$-0.429283\pi$
$773$	12.2523	0.440684	0.220342	+	0.975423i	$0.429283\pi$
$774$	0	0
$775$	16.0719	0.577320
$776$	0	0
$777$	0.00626081	0.000224605 0
$778$	0	0
$779$	15.2228	0.545413
$780$	0	0
$781$	97.8394	3.50097
$782$	0	0
$783$	−2.82167	−0.100838
$784$	0	0
$785$	−55.2315	−1.97130
$786$	0	0
$787$	−43.5760	−1.55332	−0.776658	−	0.629922i	$-0.783087\pi$
$787$	−43.5760	−1.55332	−0.776658	+	0.629922i	$0.783087\pi$
$788$	0	0
$789$	0.331420	0.0117989
$790$	0	0
$791$	0.0331776	0.00117966
$792$	0	0
$793$	−24.1068	−0.856059
$794$	0	0
$795$	2.52897	0.0896934
$796$	0	0
$797$	51.7166	1.83189	0.915947	−	0.401298i	$-0.131441\pi$
$797$	51.7166	1.83189	0.915947	+	0.401298i	$0.131441\pi$
$798$	0	0
$799$	−44.0421	−1.55810
$800$	0	0
$801$	−44.7818	−1.58229
$802$	0	0
$803$	−67.5281	−2.38301
$804$	0	0
$805$	0.118259	0.00416808
$806$	0	0
$807$	−0.928651	−0.0326900
$808$	0	0
$809$	41.5511	1.46086	0.730430	−	0.682988i	$-0.239320\pi$
$809$	41.5511	1.46086	0.730430	+	0.682988i	$0.239320\pi$
$810$	0	0
$811$	−50.1638	−1.76149	−0.880744	−	0.473593i	$-0.842957\pi$
$811$	−50.1638	−1.76149	−0.880744	+	0.473593i	$0.842957\pi$
$812$	0	0
$813$	−0.702352	−0.0246326
$814$	0	0
$815$	58.8725	2.06221
$816$	0	0
$817$	−89.0267	−3.11465
$818$	0	0
$819$	−0.144959	−0.00506528
$820$	0	0
$821$	−8.26804	−0.288557	−0.144278	−	0.989537i	$-0.546086\pi$
$821$	−8.26804	−0.288557	−0.144278	+	0.989537i	$0.546086\pi$
$822$	0	0
$823$	−0.407938	−0.0142198	−0.00710992	−	0.999975i	$-0.502263\pi$
$823$	−0.407938	−0.0142198	−0.00710992	+	0.999975i	$0.502263\pi$
$824$	0	0
$825$	4.02913	0.140276
$826$	0	0
$827$	−3.18904	−0.110894	−0.0554469	−	0.998462i	$-0.517658\pi$
$827$	−3.18904	−0.110894	−0.0554469	+	0.998462i	$0.517658\pi$
$828$	0	0
$829$	41.5278	1.44232	0.721160	−	0.692768i	$-0.243609\pi$
$829$	41.5278	1.44232	0.721160	+	0.692768i	$0.243609\pi$
$830$	0	0
$831$	1.44855	0.0502497
$832$	0	0
$833$	−31.6036	−1.09500
$834$	0	0
$835$	−21.2650	−0.735904
$836$	0	0
$837$	−4.68046	−0.161780
$838$	0	0
$839$	−48.6757	−1.68047	−0.840236	−	0.542221i	$-0.817583\pi$
$839$	−48.6757	−1.68047	−0.840236	+	0.542221i	$0.817583\pi$
$840$	0	0
$841$	−21.8397	−0.753093
$842$	0	0
$843$	−2.54373	−0.0876108
$844$	0	0
$845$	−22.9836	−0.790659
$846$	0	0
$847$	−0.615563	−0.0211510
$848$	0	0
$849$	−0.334048	−0.0114645
$850$	0	0
$851$	3.09652	0.106147
$852$	0	0
$853$	−42.0583	−1.44005	−0.720025	−	0.693948i	$-0.755870\pi$
$853$	−42.0583	−1.44005	−0.720025	+	0.693948i	$0.755870\pi$
$854$	0	0
$855$	−66.1112	−2.26096
$856$	0	0
$857$	14.6823	0.501539	0.250769	−	0.968047i	$-0.419316\pi$
$857$	14.6823	0.501539	0.250769	+	0.968047i	$0.419316\pi$
$858$	0	0
$859$	−42.2691	−1.44220	−0.721101	−	0.692830i	$-0.756364\pi$
$859$	−42.2691	−1.44220	−0.721101	+	0.692830i	$0.756364\pi$
$860$	0	0
$861$	−0.00761306	−0.000259452 0
$862$	0	0
$863$	26.9427	0.917142	0.458571	−	0.888658i	$-0.348362\pi$
$863$	26.9427	0.917142	0.458571	+	0.888658i	$0.348362\pi$
$864$	0	0
$865$	34.5184	1.17366
$866$	0	0
$867$	0.598207	0.0203162
$868$	0	0
$869$	21.8508	0.741238
$870$	0	0
$871$	−29.7736	−1.00884
$872$	0	0
$873$	−34.2282	−1.15845
$874$	0	0
$875$	0.0869348	0.00293893
$876$	0	0
$877$	−14.1250	−0.476967	−0.238484	−	0.971147i	$-0.576650\pi$
$877$	−14.1250	−0.476967	−0.238484	+	0.971147i	$0.576650\pi$
$878$	0	0
$879$	−5.41077	−0.182501
$880$	0	0
$881$	7.32706	0.246855	0.123427	−	0.992354i	$-0.460611\pi$
$881$	7.32706	0.246855	0.123427	+	0.992354i	$0.460611\pi$
$882$	0	0
$883$	17.0776	0.574705	0.287353	−	0.957825i	$-0.407225\pi$
$883$	17.0776	0.574705	0.287353	+	0.957825i	$0.407225\pi$
$884$	0	0
$885$	−5.87990	−0.197651
$886$	0	0
$887$	33.0467	1.10960	0.554800	−	0.831983i	$-0.312795\pi$
$887$	33.0467	1.10960	0.554800	+	0.831983i	$0.312795\pi$
$888$	0	0
$889$	0.138629	0.00464947
$890$	0	0
$891$	54.9233	1.84000
$892$	0	0
$893$	−73.9809	−2.47568
$894$	0	0
$895$	−50.9251	−1.70224
$896$	0	0
$897$	0.753736	0.0251665
$898$	0	0
$899$	11.8772	0.396126
$900$	0	0
$901$	22.0130	0.733359
$902$	0	0
$903$	0.0445231	0.00148164
$904$	0	0
$905$	47.9694	1.59456
$906$	0	0
$907$	−42.5284	−1.41213	−0.706066	−	0.708146i	$-0.749532\pi$
$907$	−42.5284	−1.41213	−0.706066	+	0.708146i	$0.749532\pi$
$908$	0	0
$909$	−33.3877	−1.10740
$910$	0	0
$911$	−44.2899	−1.46739	−0.733696	−	0.679478i	$-0.762206\pi$
$911$	−44.2899	−1.46739	−0.733696	+	0.679478i	$0.762206\pi$
$912$	0	0
$913$	−77.7006	−2.57151
$914$	0	0
$915$	5.49839	0.181771
$916$	0	0
$917$	0.287577	0.00949664
$918$	0	0
$919$	−22.5512	−0.743895	−0.371948	−	0.928254i	$-0.621310\pi$
$919$	−22.5512	−0.743895	−0.371948	+	0.928254i	$0.621310\pi$
$920$	0	0
$921$	−2.90665	−0.0957775
$922$	0	0
$923$	−35.3276	−1.16282
$924$	0	0
$925$	−5.97676	−0.196515
$926$	0	0
$927$	34.1010	1.12002
$928$	0	0
$929$	39.9003	1.30909	0.654544	−	0.756024i	$-0.272861\pi$
$929$	39.9003	1.30909	0.654544	+	0.756024i	$0.272861\pi$
$930$	0	0
$931$	−53.0870	−1.73986
$932$	0	0
$933$	4.26947	0.139776
$934$	0	0
$935$	83.4988	2.73070
$936$	0	0
$937$	−0.767078	−0.0250593	−0.0125297	−	0.999922i	$-0.503988\pi$
$937$	−0.767078	−0.0250593	−0.0125297	+	0.999922i	$0.503988\pi$
$938$	0	0
$939$	1.51422	0.0494148
$940$	0	0
$941$	14.1553	0.461448	0.230724	−	0.973019i	$-0.425890\pi$
$941$	14.1553	0.461448	0.230724	+	0.973019i	$0.425890\pi$
$942$	0	0
$943$	−3.76533	−0.122616
$944$	0	0
$945$	0.0664734	0.00216238
$946$	0	0
$947$	9.75532	0.317005	0.158503	−	0.987359i	$-0.449333\pi$
$947$	9.75532	0.317005	0.158503	+	0.987359i	$0.449333\pi$
$948$	0	0
$949$	24.3829	0.791501
$950$	0	0
$951$	−1.14957	−0.0372775
$952$	0	0
$953$	−50.4779	−1.63514	−0.817570	−	0.575829i	$-0.804679\pi$
$953$	−50.4779	−1.63514	−0.817570	+	0.575829i	$0.804679\pi$
$954$	0	0
$955$	−51.4832	−1.66596
$956$	0	0
$957$	2.97754	0.0962501
$958$	0	0
$959$	9.87456e−5 0	3.18866e−6 0
$960$	0	0
$961$	−11.2987	−0.364474
$962$	0	0
$963$	33.3420	1.07443
$964$	0	0
$965$	37.5337	1.20825
$966$	0	0
$967$	−20.2951	−0.652645	−0.326322	−	0.945259i	$-0.605810\pi$
$967$	−20.2951	−0.652645	−0.326322	+	0.945259i	$0.605810\pi$
$968$	0	0
$969$	6.04980	0.194347
$970$	0	0
$971$	−24.5285	−0.787157	−0.393578	−	0.919291i	$-0.628763\pi$
$971$	−24.5285	−0.787157	−0.393578	+	0.919291i	$0.628763\pi$
$972$	0	0
$973$	0.187504	0.00601110
$974$	0	0
$975$	−1.45483	−0.0465918
$976$	0	0
$977$	52.8834	1.69189	0.845945	−	0.533270i	$-0.179037\pi$
$977$	52.8834	1.69189	0.845945	+	0.533270i	$0.179037\pi$
$978$	0	0
$979$	95.0077	3.03646
$980$	0	0
$981$	−17.3934	−0.555328
$982$	0	0
$983$	−29.4827	−0.940352	−0.470176	−	0.882573i	$-0.655810\pi$
$983$	−29.4827	−0.940352	−0.470176	+	0.882573i	$0.655810\pi$
$984$	0	0
$985$	−25.0985	−0.799705
$986$	0	0
$987$	0.0369986	0.00117768
$988$	0	0
$989$	22.0206	0.700214
$990$	0	0
$991$	−2.50115	−0.0794517	−0.0397259	−	0.999211i	$-0.512648\pi$
$991$	−2.50115	−0.0794517	−0.0397259	+	0.999211i	$0.512648\pi$
$992$	0	0
$993$	2.09911	0.0666133
$994$	0	0
$995$	62.6233	1.98529
$996$	0	0
$997$	24.7918	0.785165	0.392582	−	0.919717i	$-0.371582\pi$
$997$	24.7918	0.785165	0.392582	+	0.919717i	$0.371582\pi$
$998$	0	0
$999$	1.74055	0.0550686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.18	✓	33
4.3	odd	2		8048.2.a.y.1.16		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.18	✓	33	1.1	even	1	trivial
8048.2.a.y.1.16		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)

\(f(q)\)	\(=\)	\(q+0.176667 q^{3} +2.93614 q^{5} -0.0214699 q^{7} -2.96879 q^{9} +O(q^{10})\)	\(q+0.176667 q^{3} +2.93614 q^{5} -0.0214699 q^{7} -2.96879 q^{9} +6.29849 q^{11} -2.27424 q^{13} +0.518719 q^{15} +4.51509 q^{17} +7.58436 q^{19} -0.00379301 q^{21} -1.87598 q^{23} +3.62093 q^{25} -1.05449 q^{27} +2.67587 q^{29} +4.43861 q^{31} +1.11273 q^{33} -0.0630386 q^{35} -1.65062 q^{37} -0.401783 q^{39} +2.00713 q^{41} -11.7382 q^{43} -8.71678 q^{45} -9.75441 q^{47} -6.99954 q^{49} +0.797667 q^{51} +4.87542 q^{53} +18.4933 q^{55} +1.33990 q^{57} -11.3354 q^{59} +10.5999 q^{61} +0.0637395 q^{63} -6.67750 q^{65} +13.0917 q^{67} -0.331423 q^{69} +15.5338 q^{71} -10.7213 q^{73} +0.639698 q^{75} -0.135228 q^{77} +3.46921 q^{79} +8.72007 q^{81} -12.3364 q^{83} +13.2570 q^{85} +0.472738 q^{87} +15.0842 q^{89} +0.0488277 q^{91} +0.784156 q^{93} +22.2688 q^{95} +11.5293 q^{97} -18.6989 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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