LMFDB - Embedded newform 8044.2.a.a.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.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:	$64.2316633859$
Analytic rank:	$1$
Dimension:	$80$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85715 q^{3} +0.680485 q^{5} +2.30245 q^{7} +0.448994 q^{9} +O(q^{10})$	$q-1.85715 q^{3} +0.680485 q^{5} +2.30245 q^{7} +0.448994 q^{9} +6.56658 q^{11} -5.89757 q^{13} -1.26376 q^{15} +0.370174 q^{17} +1.59564 q^{19} -4.27598 q^{21} -0.231612 q^{23} -4.53694 q^{25} +4.73759 q^{27} -3.10755 q^{29} -2.97669 q^{31} -12.1951 q^{33} +1.56678 q^{35} +3.82043 q^{37} +10.9527 q^{39} +1.44060 q^{41} -5.08724 q^{43} +0.305534 q^{45} +6.35498 q^{47} -1.69874 q^{49} -0.687468 q^{51} -11.6276 q^{53} +4.46846 q^{55} -2.96334 q^{57} +5.42038 q^{59} -9.39493 q^{61} +1.03379 q^{63} -4.01321 q^{65} -4.61423 q^{67} +0.430137 q^{69} -0.561887 q^{71} -10.4642 q^{73} +8.42576 q^{75} +15.1192 q^{77} -13.8746 q^{79} -10.1454 q^{81} +5.72702 q^{83} +0.251898 q^{85} +5.77117 q^{87} -2.14834 q^{89} -13.5788 q^{91} +5.52815 q^{93} +1.08581 q^{95} -15.7070 q^{97} +2.94836 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * 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
$3$	−1.85715	−1.07222	−0.536112	−	0.844147i	$-0.680108\pi$
$3$	−1.85715	−1.07222	−0.536112	+	0.844147i	$0.680108\pi$
$4$	0	0
$5$	0.680485	0.304322	0.152161	−	0.988356i	$-0.451377\pi$
$5$	0.680485	0.304322	0.152161	+	0.988356i	$0.451377\pi$
$6$	0	0
$7$	2.30245	0.870243	0.435121	−	0.900372i	$-0.356705\pi$
$7$	2.30245	0.870243	0.435121	+	0.900372i	$0.356705\pi$
$8$	0	0
$9$	0.448994	0.149665
$10$	0	0
$11$	6.56658	1.97990	0.989949	−	0.141423i	$-0.0451679\pi$
$11$	6.56658	1.97990	0.989949	+	0.141423i	$0.0451679\pi$
$12$	0	0
$13$	−5.89757	−1.63569	−0.817846	−	0.575438i	$-0.804832\pi$
$13$	−5.89757	−1.63569	−0.817846	+	0.575438i	$0.804832\pi$
$14$	0	0
$15$	−1.26376	−0.326302
$16$	0	0
$17$	0.370174	0.0897804	0.0448902	−	0.998992i	$-0.485706\pi$
$17$	0.370174	0.0897804	0.0448902	+	0.998992i	$0.485706\pi$
$18$	0	0
$19$	1.59564	0.366066	0.183033	−	0.983107i	$-0.441409\pi$
$19$	1.59564	0.366066	0.183033	+	0.983107i	$0.441409\pi$
$20$	0	0
$21$	−4.27598	−0.933095
$22$	0	0
$23$	−0.231612	−0.0482944	−0.0241472	−	0.999708i	$-0.507687\pi$
$23$	−0.231612	−0.0482944	−0.0241472	+	0.999708i	$0.507687\pi$
$24$	0	0
$25$	−4.53694	−0.907388
$26$	0	0
$27$	4.73759	0.911750
$28$	0	0
$29$	−3.10755	−0.577057	−0.288528	−	0.957471i	$-0.593166\pi$
$29$	−3.10755	−0.577057	−0.288528	+	0.957471i	$0.593166\pi$
$30$	0	0
$31$	−2.97669	−0.534630	−0.267315	−	0.963609i	$-0.586136\pi$
$31$	−2.97669	−0.534630	−0.267315	+	0.963609i	$0.586136\pi$
$32$	0	0
$33$	−12.1951	−2.12290
$34$	0	0
$35$	1.56678	0.264834
$36$	0	0
$37$	3.82043	0.628075	0.314037	−	0.949411i	$-0.398318\pi$
$37$	3.82043	0.628075	0.314037	+	0.949411i	$0.398318\pi$
$38$	0	0
$39$	10.9527	1.75383
$40$	0	0
$41$	1.44060	0.224984	0.112492	−	0.993653i	$-0.464117\pi$
$41$	1.44060	0.224984	0.112492	+	0.993653i	$0.464117\pi$
$42$	0	0
$43$	−5.08724	−0.775797	−0.387899	−	0.921702i	$-0.626799\pi$
$43$	−5.08724	−0.775797	−0.387899	+	0.921702i	$0.626799\pi$
$44$	0	0
$45$	0.305534	0.0455463
$46$	0	0
$47$	6.35498	0.926969	0.463484	−	0.886105i	$-0.346599\pi$
$47$	6.35498	0.926969	0.463484	+	0.886105i	$0.346599\pi$
$48$	0	0
$49$	−1.69874	−0.242677
$50$	0	0
$51$	−0.687468	−0.0962648
$52$	0	0
$53$	−11.6276	−1.59718	−0.798590	−	0.601876i	$-0.794420\pi$
$53$	−11.6276	−1.59718	−0.798590	+	0.601876i	$0.794420\pi$
$54$	0	0
$55$	4.46846	0.602527
$56$	0	0
$57$	−2.96334	−0.392504
$58$	0	0
$59$	5.42038	0.705673	0.352837	−	0.935685i	$-0.385217\pi$
$59$	5.42038	0.705673	0.352837	+	0.935685i	$0.385217\pi$
$60$	0	0
$61$	−9.39493	−1.20290	−0.601449	−	0.798912i	$-0.705410\pi$
$61$	−9.39493	−1.20290	−0.601449	+	0.798912i	$0.705410\pi$
$62$	0	0
$63$	1.03379	0.130245
$64$	0	0
$65$	−4.01321	−0.497777
$66$	0	0
$67$	−4.61423	−0.563718	−0.281859	−	0.959456i	$-0.590951\pi$
$67$	−4.61423	−0.563718	−0.281859	+	0.959456i	$0.590951\pi$
$68$	0	0
$69$	0.430137	0.0517824
$70$	0	0
$71$	−0.561887	−0.0666837	−0.0333419	−	0.999444i	$-0.510615\pi$
$71$	−0.561887	−0.0666837	−0.0333419	+	0.999444i	$0.510615\pi$
$72$	0	0
$73$	−10.4642	−1.22474	−0.612371	−	0.790570i	$-0.709784\pi$
$73$	−10.4642	−1.22474	−0.612371	+	0.790570i	$0.709784\pi$
$74$	0	0
$75$	8.42576	0.972923
$76$	0	0
$77$	15.1192	1.72299
$78$	0	0
$79$	−13.8746	−1.56101	−0.780505	−	0.625150i	$-0.785038\pi$
$79$	−13.8746	−1.56101	−0.780505	+	0.625150i	$0.785038\pi$
$80$	0	0
$81$	−10.1454	−1.12727
$82$	0	0
$83$	5.72702	0.628622	0.314311	−	0.949320i	$-0.398227\pi$
$83$	5.72702	0.628622	0.314311	+	0.949320i	$0.398227\pi$
$84$	0	0
$85$	0.251898	0.0273222
$86$	0	0
$87$	5.77117	0.618734
$88$	0	0
$89$	−2.14834	−0.227724	−0.113862	−	0.993497i	$-0.536322\pi$
$89$	−2.14834	−0.227724	−0.113862	+	0.993497i	$0.536322\pi$
$90$	0	0
$91$	−13.5788	−1.42345
$92$	0	0
$93$	5.52815	0.573243
$94$	0	0
$95$	1.08581	0.111402
$96$	0	0
$97$	−15.7070	−1.59481	−0.797404	−	0.603446i	$-0.793794\pi$
$97$	−15.7070	−1.59481	−0.797404	+	0.603446i	$0.793794\pi$
$98$	0	0
$99$	2.94836	0.296321
$100$	0	0
$101$	−3.40968	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$101$	−3.40968	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$102$	0	0
$103$	6.13508	0.604507	0.302254	−	0.953228i	$-0.402261\pi$
$103$	6.13508	0.604507	0.302254	+	0.953228i	$0.402261\pi$
$104$	0	0
$105$	−2.90974	−0.283962
$106$	0	0
$107$	−0.996732	−0.0963577	−0.0481788	−	0.998839i	$-0.515342\pi$
$107$	−0.996732	−0.0963577	−0.0481788	+	0.998839i	$0.515342\pi$
$108$	0	0
$109$	−4.14263	−0.396792	−0.198396	−	0.980122i	$-0.563573\pi$
$109$	−4.14263	−0.396792	−0.198396	+	0.980122i	$0.563573\pi$
$110$	0	0
$111$	−7.09510	−0.673437
$112$	0	0
$113$	14.3207	1.34718	0.673590	−	0.739105i	$-0.264751\pi$
$113$	14.3207	1.34718	0.673590	+	0.739105i	$0.264751\pi$
$114$	0	0
$115$	−0.157608	−0.0146971
$116$	0	0
$117$	−2.64798	−0.244805
$118$	0	0
$119$	0.852306	0.0781308
$120$	0	0
$121$	32.1200	2.92000
$122$	0	0
$123$	−2.67541	−0.241234
$124$	0	0
$125$	−6.48975	−0.580461
$126$	0	0
$127$	−8.40137	−0.745501	−0.372751	−	0.927932i	$-0.621585\pi$
$127$	−8.40137	−0.745501	−0.372751	+	0.927932i	$0.621585\pi$
$128$	0	0
$129$	9.44776	0.831829
$130$	0	0
$131$	−7.33641	−0.640985	−0.320492	−	0.947251i	$-0.603848\pi$
$131$	−7.33641	−0.640985	−0.320492	+	0.947251i	$0.603848\pi$
$132$	0	0
$133$	3.67388	0.318566
$134$	0	0
$135$	3.22386	0.277466
$136$	0	0
$137$	20.4111	1.74384	0.871921	−	0.489647i	$-0.162874\pi$
$137$	20.4111	1.74384	0.871921	+	0.489647i	$0.162874\pi$
$138$	0	0
$139$	−18.2116	−1.54469	−0.772345	−	0.635204i	$-0.780916\pi$
$139$	−18.2116	−1.54469	−0.772345	+	0.635204i	$0.780916\pi$
$140$	0	0
$141$	−11.8021	−0.993918
$142$	0	0
$143$	−38.7269	−3.23850
$144$	0	0
$145$	−2.11464	−0.175611
$146$	0	0
$147$	3.15481	0.260205
$148$	0	0
$149$	8.17332	0.669585	0.334792	−	0.942292i	$-0.391334\pi$
$149$	8.17332	0.669585	0.334792	+	0.942292i	$0.391334\pi$
$150$	0	0
$151$	9.22683	0.750869	0.375434	−	0.926849i	$-0.377494\pi$
$151$	9.22683	0.750869	0.375434	+	0.926849i	$0.377494\pi$
$152$	0	0
$153$	0.166206	0.0134370
$154$	0	0
$155$	−2.02560	−0.162700
$156$	0	0
$157$	16.3721	1.30663	0.653316	−	0.757085i	$-0.273377\pi$
$157$	16.3721	1.30663	0.653316	+	0.757085i	$0.273377\pi$
$158$	0	0
$159$	21.5942	1.71253
$160$	0	0
$161$	−0.533273	−0.0420278
$162$	0	0
$163$	19.1020	1.49619	0.748093	−	0.663594i	$-0.230970\pi$
$163$	19.1020	1.49619	0.748093	+	0.663594i	$0.230970\pi$
$164$	0	0
$165$	−8.29859	−0.646044
$166$	0	0
$167$	22.3483	1.72936	0.864681	−	0.502321i	$-0.167521\pi$
$167$	22.3483	1.72936	0.864681	+	0.502321i	$0.167521\pi$
$168$	0	0
$169$	21.7813	1.67549
$170$	0	0
$171$	0.716435	0.0547871
$172$	0	0
$173$	4.08784	0.310793	0.155396	−	0.987852i	$-0.450334\pi$
$173$	4.08784	0.310793	0.155396	+	0.987852i	$0.450334\pi$
$174$	0	0
$175$	−10.4461	−0.789648
$176$	0	0
$177$	−10.0664	−0.756640
$178$	0	0
$179$	−3.82379	−0.285803	−0.142902	−	0.989737i	$-0.545643\pi$
$179$	−3.82379	−0.285803	−0.142902	+	0.989737i	$0.545643\pi$
$180$	0	0
$181$	−17.4569	−1.29756	−0.648782	−	0.760975i	$-0.724721\pi$
$181$	−17.4569	−1.29756	−0.648782	+	0.760975i	$0.724721\pi$
$182$	0	0
$183$	17.4478	1.28978
$184$	0	0
$185$	2.59975	0.191137
$186$	0	0
$187$	2.43078	0.177756
$188$	0	0
$189$	10.9081	0.793444
$190$	0	0
$191$	−21.9404	−1.58755	−0.793775	−	0.608212i	$-0.791887\pi$
$191$	−21.9404	−1.58755	−0.793775	+	0.608212i	$0.791887\pi$
$192$	0	0
$193$	−0.666760	−0.0479944	−0.0239972	−	0.999712i	$-0.507639\pi$
$193$	−0.666760	−0.0479944	−0.0239972	+	0.999712i	$0.507639\pi$
$194$	0	0
$195$	7.45312	0.533729
$196$	0	0
$197$	−8.54021	−0.608465	−0.304232	−	0.952598i	$-0.598400\pi$
$197$	−8.54021	−0.608465	−0.304232	+	0.952598i	$0.598400\pi$
$198$	0	0
$199$	−7.18802	−0.509545	−0.254773	−	0.967001i	$-0.582001\pi$
$199$	−7.18802	−0.509545	−0.254773	+	0.967001i	$0.582001\pi$
$200$	0	0
$201$	8.56930	0.604432
$202$	0	0
$203$	−7.15496	−0.502180
$204$	0	0
$205$	0.980309	0.0684678
$206$	0	0
$207$	−0.103992	−0.00722797
$208$	0	0
$209$	10.4779	0.724773
$210$	0	0
$211$	−3.66359	−0.252212	−0.126106	−	0.992017i	$-0.540248\pi$
$211$	−3.66359	−0.252212	−0.126106	+	0.992017i	$0.540248\pi$
$212$	0	0
$213$	1.04351	0.0714999
$214$	0	0
$215$	−3.46180	−0.236093
$216$	0	0
$217$	−6.85367	−0.465258
$218$	0	0
$219$	19.4336	1.31320
$220$	0	0
$221$	−2.18313	−0.146853
$222$	0	0
$223$	25.5793	1.71291	0.856457	−	0.516218i	$-0.172660\pi$
$223$	25.5793	1.71291	0.856457	+	0.516218i	$0.172660\pi$
$224$	0	0
$225$	−2.03706	−0.135804
$226$	0	0
$227$	8.62910	0.572733	0.286367	−	0.958120i	$-0.407552\pi$
$227$	8.62910	0.572733	0.286367	+	0.958120i	$0.407552\pi$
$228$	0	0
$229$	6.95696	0.459729	0.229865	−	0.973223i	$-0.426172\pi$
$229$	6.95696	0.459729	0.229865	+	0.973223i	$0.426172\pi$
$230$	0	0
$231$	−28.0786	−1.84743
$232$	0	0
$233$	−3.80201	−0.249078	−0.124539	−	0.992215i	$-0.539745\pi$
$233$	−3.80201	−0.249078	−0.124539	+	0.992215i	$0.539745\pi$
$234$	0	0
$235$	4.32447	0.282097
$236$	0	0
$237$	25.7671	1.67375
$238$	0	0
$239$	16.9650	1.09737	0.548687	−	0.836028i	$-0.315128\pi$
$239$	16.9650	1.09737	0.548687	+	0.836028i	$0.315128\pi$
$240$	0	0
$241$	−27.9314	−1.79922	−0.899608	−	0.436698i	$-0.856148\pi$
$241$	−27.9314	−1.79922	−0.899608	+	0.436698i	$0.856148\pi$
$242$	0	0
$243$	4.62870	0.296931
$244$	0	0
$245$	−1.15597	−0.0738521
$246$	0	0
$247$	−9.41041	−0.598770
$248$	0	0
$249$	−10.6359	−0.674023
$250$	0	0
$251$	−28.7637	−1.81555	−0.907775	−	0.419457i	$-0.862221\pi$
$251$	−28.7637	−1.81555	−0.907775	+	0.419457i	$0.862221\pi$
$252$	0	0
$253$	−1.52090	−0.0956180
$254$	0	0
$255$	−0.467812	−0.0292955
$256$	0	0
$257$	−18.5868	−1.15941	−0.579707	−	0.814825i	$-0.696833\pi$
$257$	−18.5868	−1.15941	−0.579707	+	0.814825i	$0.696833\pi$
$258$	0	0
$259$	8.79633	0.546578
$260$	0	0
$261$	−1.39527	−0.0863651
$262$	0	0
$263$	−25.4319	−1.56820	−0.784099	−	0.620636i	$-0.786874\pi$
$263$	−25.4319	−1.56820	−0.784099	+	0.620636i	$0.786874\pi$
$264$	0	0
$265$	−7.91244	−0.486057
$266$	0	0
$267$	3.98978	0.244171
$268$	0	0
$269$	26.2960	1.60329	0.801647	−	0.597797i	$-0.203957\pi$
$269$	26.2960	1.60329	0.801647	+	0.597797i	$0.203957\pi$
$270$	0	0
$271$	18.1101	1.10011	0.550056	−	0.835127i	$-0.314606\pi$
$271$	18.1101	1.10011	0.550056	+	0.835127i	$0.314606\pi$
$272$	0	0
$273$	25.2179	1.52626
$274$	0	0
$275$	−29.7922	−1.79654
$276$	0	0
$277$	−4.12392	−0.247782	−0.123891	−	0.992296i	$-0.539537\pi$
$277$	−4.12392	−0.247782	−0.123891	+	0.992296i	$0.539537\pi$
$278$	0	0
$279$	−1.33652	−0.0800152
$280$	0	0
$281$	−10.0638	−0.600357	−0.300178	−	0.953883i	$-0.597046\pi$
$281$	−10.0638	−0.600357	−0.300178	+	0.953883i	$0.597046\pi$
$282$	0	0
$283$	−22.8096	−1.35589	−0.677945	−	0.735113i	$-0.737129\pi$
$283$	−22.8096	−1.35589	−0.677945	+	0.735113i	$0.737129\pi$
$284$	0	0
$285$	−2.01651	−0.119448
$286$	0	0
$287$	3.31691	0.195791
$288$	0	0
$289$	−16.8630	−0.991939
$290$	0	0
$291$	29.1703	1.70999
$292$	0	0
$293$	−18.4199	−1.07610	−0.538050	−	0.842913i	$-0.680839\pi$
$293$	−18.4199	−1.07610	−0.538050	+	0.842913i	$0.680839\pi$
$294$	0	0
$295$	3.68849	0.214752
$296$	0	0
$297$	31.1098	1.80517
$298$	0	0
$299$	1.36595	0.0789947
$300$	0	0
$301$	−11.7131	−0.675132
$302$	0	0
$303$	6.33228	0.363780
$304$	0	0
$305$	−6.39311	−0.366069
$306$	0	0
$307$	−23.0224	−1.31396	−0.656979	−	0.753909i	$-0.728166\pi$
$307$	−23.0224	−1.31396	−0.656979	+	0.753909i	$0.728166\pi$
$308$	0	0
$309$	−11.3937	−0.648167
$310$	0	0
$311$	3.97607	0.225462	0.112731	−	0.993626i	$-0.464040\pi$
$311$	3.97607	0.225462	0.112731	+	0.993626i	$0.464040\pi$
$312$	0	0
$313$	6.87697	0.388710	0.194355	−	0.980931i	$-0.437739\pi$
$313$	6.87697	0.388710	0.194355	+	0.980931i	$0.437739\pi$
$314$	0	0
$315$	0.703476	0.0396364
$316$	0	0
$317$	21.9062	1.23038	0.615188	−	0.788381i	$-0.289080\pi$
$317$	21.9062	1.23038	0.615188	+	0.788381i	$0.289080\pi$
$318$	0	0
$319$	−20.4059	−1.14251
$320$	0	0
$321$	1.85108	0.103317
$322$	0	0
$323$	0.590666	0.0328655
$324$	0	0
$325$	26.7569	1.48421
$326$	0	0
$327$	7.69348	0.425450
$328$	0	0
$329$	14.6320	0.806688
$330$	0	0
$331$	−24.8035	−1.36333	−0.681663	−	0.731667i	$-0.738743\pi$
$331$	−24.8035	−1.36333	−0.681663	+	0.731667i	$0.738743\pi$
$332$	0	0
$333$	1.71535	0.0940007
$334$	0	0
$335$	−3.13992	−0.171552
$336$	0	0
$337$	−21.4053	−1.16602	−0.583010	−	0.812465i	$-0.698125\pi$
$337$	−21.4053	−1.16602	−0.583010	+	0.812465i	$0.698125\pi$
$338$	0	0
$339$	−26.5957	−1.44448
$340$	0	0
$341$	−19.5467	−1.05851
$342$	0	0
$343$	−20.0284	−1.08143
$344$	0	0
$345$	0.292702	0.0157585
$346$	0	0
$347$	27.2149	1.46097	0.730485	−	0.682928i	$-0.239294\pi$
$347$	27.2149	1.46097	0.730485	+	0.682928i	$0.239294\pi$
$348$	0	0
$349$	15.3301	0.820603	0.410302	−	0.911950i	$-0.365423\pi$
$349$	15.3301	0.820603	0.410302	+	0.911950i	$0.365423\pi$
$350$	0	0
$351$	−27.9403	−1.49134
$352$	0	0
$353$	−4.47905	−0.238396	−0.119198	−	0.992871i	$-0.538032\pi$
$353$	−4.47905	−0.238396	−0.119198	+	0.992871i	$0.538032\pi$
$354$	0	0
$355$	−0.382356	−0.0202933
$356$	0	0
$357$	−1.58286	−0.0837737
$358$	0	0
$359$	−14.0703	−0.742604	−0.371302	−	0.928512i	$-0.621089\pi$
$359$	−14.0703	−0.742604	−0.371302	+	0.928512i	$0.621089\pi$
$360$	0	0
$361$	−16.4539	−0.865996
$362$	0	0
$363$	−59.6515	−3.13089
$364$	0	0
$365$	−7.12074	−0.372717
$366$	0	0
$367$	6.68527	0.348968	0.174484	−	0.984660i	$-0.444174\pi$
$367$	6.68527	0.348968	0.174484	+	0.984660i	$0.444174\pi$
$368$	0	0
$369$	0.646822	0.0336722
$370$	0	0
$371$	−26.7720	−1.38993
$372$	0	0
$373$	−11.4728	−0.594037	−0.297018	−	0.954872i	$-0.595992\pi$
$373$	−11.4728	−0.594037	−0.297018	+	0.954872i	$0.595992\pi$
$374$	0	0
$375$	12.0524	0.622384
$376$	0	0
$377$	18.3270	0.943887
$378$	0	0
$379$	8.46940	0.435044	0.217522	−	0.976055i	$-0.430203\pi$
$379$	8.46940	0.435044	0.217522	+	0.976055i	$0.430203\pi$
$380$	0	0
$381$	15.6026	0.799345
$382$	0	0
$383$	24.4392	1.24879	0.624393	−	0.781110i	$-0.285346\pi$
$383$	24.4392	1.24879	0.624393	+	0.781110i	$0.285346\pi$
$384$	0	0
$385$	10.2884	0.524345
$386$	0	0
$387$	−2.28414	−0.116110
$388$	0	0
$389$	24.4081	1.23754	0.618770	−	0.785572i	$-0.287631\pi$
$389$	24.4081	1.23754	0.618770	+	0.785572i	$0.287631\pi$
$390$	0	0
$391$	−0.0857367	−0.00433589
$392$	0	0
$393$	13.6248	0.687279
$394$	0	0
$395$	−9.44143	−0.475050
$396$	0	0
$397$	−24.5542	−1.23234	−0.616171	−	0.787612i	$-0.711317\pi$
$397$	−24.5542	−1.23234	−0.616171	+	0.787612i	$0.711317\pi$
$398$	0	0
$399$	−6.82294	−0.341574
$400$	0	0
$401$	−17.6890	−0.883347	−0.441673	−	0.897176i	$-0.645615\pi$
$401$	−17.6890	−0.883347	−0.441673	+	0.897176i	$0.645615\pi$
$402$	0	0
$403$	17.5552	0.874489
$404$	0	0
$405$	−6.90379	−0.343052
$406$	0	0
$407$	25.0872	1.24352
$408$	0	0
$409$	−34.4483	−1.70336	−0.851678	−	0.524065i	$-0.824415\pi$
$409$	−34.4483	−1.70336	−0.851678	+	0.524065i	$0.824415\pi$
$410$	0	0
$411$	−37.9065	−1.86979
$412$	0	0
$413$	12.4801	0.614107
$414$	0	0
$415$	3.89715	0.191304
$416$	0	0
$417$	33.8217	1.65625
$418$	0	0
$419$	−16.2710	−0.794891	−0.397446	−	0.917626i	$-0.630103\pi$
$419$	−16.2710	−0.794891	−0.397446	+	0.917626i	$0.630103\pi$
$420$	0	0
$421$	29.8845	1.45648	0.728240	−	0.685323i	$-0.240339\pi$
$421$	29.8845	1.45648	0.728240	+	0.685323i	$0.240339\pi$
$422$	0	0
$423$	2.85335	0.138735
$424$	0	0
$425$	−1.67946	−0.0814657
$426$	0	0
$427$	−21.6313	−1.04681
$428$	0	0
$429$	71.9215	3.47240
$430$	0	0
$431$	−19.3046	−0.929871	−0.464936	−	0.885344i	$-0.653923\pi$
$431$	−19.3046	−0.929871	−0.464936	+	0.885344i	$0.653923\pi$
$432$	0	0
$433$	6.36408	0.305838	0.152919	−	0.988239i	$-0.451133\pi$
$433$	6.36408	0.305838	0.152919	+	0.988239i	$0.451133\pi$
$434$	0	0
$435$	3.92720	0.188295
$436$	0	0
$437$	−0.369570	−0.0176789
$438$	0	0
$439$	−7.21387	−0.344299	−0.172150	−	0.985071i	$-0.555071\pi$
$439$	−7.21387	−0.344299	−0.172150	+	0.985071i	$0.555071\pi$
$440$	0	0
$441$	−0.762725	−0.0363203
$442$	0	0
$443$	31.7885	1.51032	0.755160	−	0.655541i	$-0.227559\pi$
$443$	31.7885	1.51032	0.755160	+	0.655541i	$0.227559\pi$
$444$	0	0
$445$	−1.46191	−0.0693014
$446$	0	0
$447$	−15.1791	−0.717945
$448$	0	0
$449$	−5.93673	−0.280172	−0.140086	−	0.990139i	$-0.544738\pi$
$449$	−5.93673	−0.280172	−0.140086	+	0.990139i	$0.544738\pi$
$450$	0	0
$451$	9.45983	0.445446
$452$	0	0
$453$	−17.1356	−0.805100
$454$	0	0
$455$	−9.24020	−0.433187
$456$	0	0
$457$	−11.1069	−0.519558	−0.259779	−	0.965668i	$-0.583650\pi$
$457$	−11.1069	−0.519558	−0.259779	+	0.965668i	$0.583650\pi$
$458$	0	0
$459$	1.75373	0.0818573
$460$	0	0
$461$	37.2598	1.73536	0.867681	−	0.497122i	$-0.165610\pi$
$461$	37.2598	1.73536	0.867681	+	0.497122i	$0.165610\pi$
$462$	0	0
$463$	−1.96776	−0.0914493	−0.0457247	−	0.998954i	$-0.514560\pi$
$463$	−1.96776	−0.0914493	−0.0457247	+	0.998954i	$0.514560\pi$
$464$	0	0
$465$	3.76183	0.174451
$466$	0	0
$467$	3.78113	0.174970	0.0874849	−	0.996166i	$-0.472117\pi$
$467$	3.78113	0.174970	0.0874849	+	0.996166i	$0.472117\pi$
$468$	0	0
$469$	−10.6240	−0.490571
$470$	0	0
$471$	−30.4053	−1.40100
$472$	0	0
$473$	−33.4058	−1.53600
$474$	0	0
$475$	−7.23934	−0.332163
$476$	0	0
$477$	−5.22075	−0.239042
$478$	0	0
$479$	−19.6119	−0.896089	−0.448045	−	0.894011i	$-0.647879\pi$
$479$	−19.6119	−0.896089	−0.448045	+	0.894011i	$0.647879\pi$
$480$	0	0
$481$	−22.5312	−1.02734
$482$	0	0
$483$	0.990367	0.0450633
$484$	0	0
$485$	−10.6884	−0.485336
$486$	0	0
$487$	−8.49717	−0.385043	−0.192522	−	0.981293i	$-0.561667\pi$
$487$	−8.49717	−0.385043	−0.192522	+	0.981293i	$0.561667\pi$
$488$	0	0
$489$	−35.4752	−1.60425
$490$	0	0
$491$	18.6851	0.843248	0.421624	−	0.906771i	$-0.361460\pi$
$491$	18.6851	0.843248	0.421624	+	0.906771i	$0.361460\pi$
$492$	0	0
$493$	−1.15033	−0.0518084
$494$	0	0
$495$	2.00631	0.0901771
$496$	0	0
$497$	−1.29371	−0.0580310
$498$	0	0
$499$	−32.4084	−1.45080	−0.725400	−	0.688328i	$-0.758345\pi$
$499$	−32.4084	−1.45080	−0.725400	+	0.688328i	$0.758345\pi$
$500$	0	0
$501$	−41.5040	−1.85426
$502$	0	0
$503$	−40.6379	−1.81195	−0.905977	−	0.423327i	$-0.860862\pi$
$503$	−40.6379	−1.81195	−0.905977	+	0.423327i	$0.860862\pi$
$504$	0	0
$505$	−2.32024	−0.103249
$506$	0	0
$507$	−40.4511	−1.79650
$508$	0	0
$509$	−6.08830	−0.269859	−0.134930	−	0.990855i	$-0.543081\pi$
$509$	−6.08830	−0.269859	−0.134930	+	0.990855i	$0.543081\pi$
$510$	0	0
$511$	−24.0933	−1.06582
$512$	0	0
$513$	7.55950	0.333760
$514$	0	0
$515$	4.17483	0.183965
$516$	0	0
$517$	41.7305	1.83530
$518$	0	0
$519$	−7.59172	−0.333240
$520$	0	0
$521$	−44.3649	−1.94366	−0.971830	−	0.235683i	$-0.924267\pi$
$521$	−44.3649	−1.94366	−0.971830	+	0.235683i	$0.924267\pi$
$522$	0	0
$523$	−4.58390	−0.200440	−0.100220	−	0.994965i	$-0.531955\pi$
$523$	−4.58390	−0.200440	−0.100220	+	0.994965i	$0.531955\pi$
$524$	0	0
$525$	19.3999	0.846680
$526$	0	0
$527$	−1.10189	−0.0479993
$528$	0	0
$529$	−22.9464	−0.997668
$530$	0	0
$531$	2.43372	0.105614
$532$	0	0
$533$	−8.49605	−0.368005
$534$	0	0
$535$	−0.678261	−0.0293238
$536$	0	0
$537$	7.10133	0.306445
$538$	0	0
$539$	−11.1549	−0.480477
$540$	0	0
$541$	−13.1671	−0.566098	−0.283049	−	0.959105i	$-0.591346\pi$
$541$	−13.1671	−0.566098	−0.283049	+	0.959105i	$0.591346\pi$
$542$	0	0
$543$	32.4201	1.39128
$544$	0	0
$545$	−2.81900	−0.120753
$546$	0	0
$547$	3.77647	0.161470	0.0807350	−	0.996736i	$-0.474273\pi$
$547$	3.77647	0.161470	0.0807350	+	0.996736i	$0.474273\pi$
$548$	0	0
$549$	−4.21827	−0.180031
$550$	0	0
$551$	−4.95853	−0.211241
$552$	0	0
$553$	−31.9454	−1.35846
$554$	0	0
$555$	−4.82811	−0.204942
$556$	0	0
$557$	34.7815	1.47374	0.736869	−	0.676036i	$-0.236304\pi$
$557$	34.7815	1.47374	0.736869	+	0.676036i	$0.236304\pi$
$558$	0	0
$559$	30.0024	1.26897
$560$	0	0
$561$	−4.51431	−0.190594
$562$	0	0
$563$	−22.6239	−0.953485	−0.476743	−	0.879043i	$-0.658183\pi$
$563$	−22.6239	−0.953485	−0.476743	+	0.879043i	$0.658183\pi$
$564$	0	0
$565$	9.74505	0.409977
$566$	0	0
$567$	−23.3592	−0.980995
$568$	0	0
$569$	−28.2911	−1.18603	−0.593013	−	0.805193i	$-0.702062\pi$
$569$	−28.2911	−1.18603	−0.593013	+	0.805193i	$0.702062\pi$
$570$	0	0
$571$	−19.9319	−0.834123	−0.417061	−	0.908878i	$-0.636940\pi$
$571$	−19.9319	−0.834123	−0.417061	+	0.908878i	$0.636940\pi$
$572$	0	0
$573$	40.7465	1.70221
$574$	0	0
$575$	1.05081	0.0438217
$576$	0	0
$577$	−33.2836	−1.38561	−0.692807	−	0.721123i	$-0.743626\pi$
$577$	−33.2836	−1.38561	−0.692807	+	0.721123i	$0.743626\pi$
$578$	0	0
$579$	1.23827	0.0514608
$580$	0	0
$581$	13.1861	0.547054
$582$	0	0
$583$	−76.3538	−3.16225
$584$	0	0
$585$	−1.80191	−0.0744998
$586$	0	0
$587$	16.7714	0.692232	0.346116	−	0.938192i	$-0.387500\pi$
$587$	16.7714	0.692232	0.346116	+	0.938192i	$0.387500\pi$
$588$	0	0
$589$	−4.74974	−0.195709
$590$	0	0
$591$	15.8604	0.652411
$592$	0	0
$593$	7.36743	0.302544	0.151272	−	0.988492i	$-0.451663\pi$
$593$	7.36743	0.302544	0.151272	+	0.988492i	$0.451663\pi$
$594$	0	0
$595$	0.579982	0.0237769
$596$	0	0
$597$	13.3492	0.546347
$598$	0	0
$599$	38.8462	1.58721	0.793607	−	0.608431i	$-0.208201\pi$
$599$	38.8462	1.58721	0.793607	+	0.608431i	$0.208201\pi$
$600$	0	0
$601$	10.3522	0.422276	0.211138	−	0.977456i	$-0.432283\pi$
$601$	10.3522	0.422276	0.211138	+	0.977456i	$0.432283\pi$
$602$	0	0
$603$	−2.07176	−0.0843687
$604$	0	0
$605$	21.8572	0.888621
$606$	0	0
$607$	6.50840	0.264168	0.132084	−	0.991239i	$-0.457833\pi$
$607$	6.50840	0.264168	0.132084	+	0.991239i	$0.457833\pi$
$608$	0	0
$609$	13.2878	0.538449
$610$	0	0
$611$	−37.4789	−1.51623
$612$	0	0
$613$	−13.1002	−0.529113	−0.264557	−	0.964370i	$-0.585226\pi$
$613$	−13.1002	−0.529113	−0.264557	+	0.964370i	$0.585226\pi$
$614$	0	0
$615$	−1.82058	−0.0734128
$616$	0	0
$617$	19.2219	0.773843	0.386921	−	0.922113i	$-0.373539\pi$
$617$	19.2219	0.773843	0.386921	+	0.922113i	$0.373539\pi$
$618$	0	0
$619$	4.34390	0.174596	0.0872981	−	0.996182i	$-0.472177\pi$
$619$	4.34390	0.174596	0.0872981	+	0.996182i	$0.472177\pi$
$620$	0	0
$621$	−1.09728	−0.0440324
$622$	0	0
$623$	−4.94644	−0.198175
$624$	0	0
$625$	18.2685	0.730741
$626$	0	0
$627$	−19.4590	−0.777119
$628$	0	0
$629$	1.41422	0.0563888
$630$	0	0
$631$	28.5364	1.13601	0.568007	−	0.823023i	$-0.307714\pi$
$631$	28.5364	1.13601	0.568007	+	0.823023i	$0.307714\pi$
$632$	0	0
$633$	6.80383	0.270428
$634$	0	0
$635$	−5.71701	−0.226873
$636$	0	0
$637$	10.0184	0.396945
$638$	0	0
$639$	−0.252284	−0.00998021
$640$	0	0
$641$	−48.6773	−1.92264	−0.961320	−	0.275435i	$-0.911178\pi$
$641$	−48.6773	−1.92264	−0.961320	+	0.275435i	$0.911178\pi$
$642$	0	0
$643$	5.55293	0.218986	0.109493	−	0.993988i	$-0.465077\pi$
$643$	5.55293	0.218986	0.109493	+	0.993988i	$0.465077\pi$
$644$	0	0
$645$	6.42906	0.253144
$646$	0	0
$647$	−17.1567	−0.674500	−0.337250	−	0.941415i	$-0.609497\pi$
$647$	−17.1567	−0.674500	−0.337250	+	0.941415i	$0.609497\pi$
$648$	0	0
$649$	35.5934	1.39716
$650$	0	0
$651$	12.7283	0.498860
$652$	0	0
$653$	−27.7365	−1.08541	−0.542706	−	0.839923i	$-0.682600\pi$
$653$	−27.7365	−1.08541	−0.542706	+	0.839923i	$0.682600\pi$
$654$	0	0
$655$	−4.99232	−0.195066
$656$	0	0
$657$	−4.69837	−0.183301
$658$	0	0
$659$	−22.9429	−0.893728	−0.446864	−	0.894602i	$-0.647459\pi$
$659$	−22.9429	−0.893728	−0.446864	+	0.894602i	$0.647459\pi$
$660$	0	0
$661$	−25.5574	−0.994068	−0.497034	−	0.867731i	$-0.665578\pi$
$661$	−25.5574	−0.994068	−0.497034	+	0.867731i	$0.665578\pi$
$662$	0	0
$663$	4.05439	0.157459
$664$	0	0
$665$	2.50002	0.0969467
$666$	0	0
$667$	0.719744	0.0278686
$668$	0	0
$669$	−47.5045	−1.83663
$670$	0	0
$671$	−61.6925	−2.38161
$672$	0	0
$673$	37.3533	1.43986	0.719932	−	0.694045i	$-0.244173\pi$
$673$	37.3533	1.43986	0.719932	+	0.694045i	$0.244173\pi$
$674$	0	0
$675$	−21.4942	−0.827311
$676$	0	0
$677$	−33.1483	−1.27399	−0.636997	−	0.770866i	$-0.719824\pi$
$677$	−33.1483	−1.27399	−0.636997	+	0.770866i	$0.719824\pi$
$678$	0	0
$679$	−36.1646	−1.38787
$680$	0	0
$681$	−16.0255	−0.614099
$682$	0	0
$683$	30.7875	1.17805	0.589025	−	0.808114i	$-0.299512\pi$
$683$	30.7875	1.17805	0.589025	+	0.808114i	$0.299512\pi$
$684$	0	0
$685$	13.8895	0.530690
$686$	0	0
$687$	−12.9201	−0.492933
$688$	0	0
$689$	68.5748	2.61249
$690$	0	0
$691$	−30.7468	−1.16966	−0.584831	−	0.811155i	$-0.698839\pi$
$691$	−30.7468	−1.16966	−0.584831	+	0.811155i	$0.698839\pi$
$692$	0	0
$693$	6.78844	0.257871
$694$	0	0
$695$	−12.3927	−0.470084
$696$	0	0
$697$	0.533274	0.0201992
$698$	0	0
$699$	7.06089	0.267067
$700$	0	0
$701$	−16.1719	−0.610804	−0.305402	−	0.952223i	$-0.598791\pi$
$701$	−16.1719	−0.610804	−0.305402	+	0.952223i	$0.598791\pi$
$702$	0	0
$703$	6.09604	0.229917
$704$	0	0
$705$	−8.03118	−0.302472
$706$	0	0
$707$	−7.85061	−0.295253
$708$	0	0
$709$	23.2127	0.871770	0.435885	−	0.900002i	$-0.356436\pi$
$709$	23.2127	0.871770	0.435885	+	0.900002i	$0.356436\pi$
$710$	0	0
$711$	−6.22960	−0.233628
$712$	0	0
$713$	0.689436	0.0258196
$714$	0	0
$715$	−26.3531	−0.985549
$716$	0	0
$717$	−31.5065	−1.17663
$718$	0	0
$719$	−25.0358	−0.933678	−0.466839	−	0.884342i	$-0.654607\pi$
$719$	−25.0358	−0.933678	−0.466839	+	0.884342i	$0.654607\pi$
$720$	0	0
$721$	14.1257	0.526068
$722$	0	0
$723$	51.8726	1.92916
$724$	0	0
$725$	14.0987	0.523614
$726$	0	0
$727$	36.7280	1.36217	0.681083	−	0.732207i	$-0.261509\pi$
$727$	36.7280	1.36217	0.681083	+	0.732207i	$0.261509\pi$
$728$	0	0
$729$	21.8400	0.808889
$730$	0	0
$731$	−1.88317	−0.0696514
$732$	0	0
$733$	34.1539	1.26150	0.630751	−	0.775985i	$-0.282747\pi$
$733$	34.1539	1.26150	0.630751	+	0.775985i	$0.282747\pi$
$734$	0	0
$735$	2.14680	0.0791861
$736$	0	0
$737$	−30.2997	−1.11610
$738$	0	0
$739$	−36.4174	−1.33964	−0.669818	−	0.742525i	$-0.733628\pi$
$739$	−36.4174	−1.33964	−0.669818	+	0.742525i	$0.733628\pi$
$740$	0	0
$741$	17.4765	0.642016
$742$	0	0
$743$	14.1337	0.518517	0.259258	−	0.965808i	$-0.416522\pi$
$743$	14.1337	0.518517	0.259258	+	0.965808i	$0.416522\pi$
$744$	0	0
$745$	5.56183	0.203770
$746$	0	0
$747$	2.57140	0.0940826
$748$	0	0
$749$	−2.29492	−0.0838546
$750$	0	0
$751$	4.48411	0.163627	0.0818137	−	0.996648i	$-0.473929\pi$
$751$	4.48411	0.163627	0.0818137	+	0.996648i	$0.473929\pi$
$752$	0	0
$753$	53.4185	1.94668
$754$	0	0
$755$	6.27873	0.228506
$756$	0	0
$757$	53.9372	1.96038	0.980191	−	0.198053i	$-0.0634618\pi$
$757$	53.9372	1.96038	0.980191	+	0.198053i	$0.0634618\pi$
$758$	0	0
$759$	2.82453	0.102524
$760$	0	0
$761$	32.7104	1.18575	0.592876	−	0.805294i	$-0.297992\pi$
$761$	32.7104	1.18575	0.592876	+	0.805294i	$0.297992\pi$
$762$	0	0
$763$	−9.53819	−0.345306
$764$	0	0
$765$	0.113101	0.00408917
$766$	0	0
$767$	−31.9671	−1.15426
$768$	0	0
$769$	3.04603	0.109843	0.0549213	−	0.998491i	$-0.482509\pi$
$769$	3.04603	0.109843	0.0549213	+	0.998491i	$0.482509\pi$
$770$	0	0
$771$	34.5185	1.24315
$772$	0	0
$773$	10.6985	0.384799	0.192400	−	0.981317i	$-0.438373\pi$
$773$	10.6985	0.384799	0.192400	+	0.981317i	$0.438373\pi$
$774$	0	0
$775$	13.5051	0.485116
$776$	0	0
$777$	−16.3361	−0.586054
$778$	0	0
$779$	2.29869	0.0823590
$780$	0	0
$781$	−3.68968	−0.132027
$782$	0	0
$783$	−14.7223	−0.526132
$784$	0	0
$785$	11.1409	0.397637
$786$	0	0
$787$	−33.3092	−1.18734	−0.593672	−	0.804707i	$-0.702322\pi$
$787$	−33.3092	−1.18734	−0.593672	+	0.804707i	$0.702322\pi$
$788$	0	0
$789$	47.2308	1.68146
$790$	0	0
$791$	32.9727	1.17237
$792$	0	0
$793$	55.4072	1.96757
$794$	0	0
$795$	14.6946	0.521162
$796$	0	0
$797$	−15.5569	−0.551054	−0.275527	−	0.961293i	$-0.588852\pi$
$797$	−15.5569	−0.551054	−0.275527	+	0.961293i	$0.588852\pi$
$798$	0	0
$799$	2.35245	0.0832237
$800$	0	0
$801$	−0.964593	−0.0340822
$802$	0	0
$803$	−68.7141	−2.42487
$804$	0	0
$805$	−0.362885	−0.0127900
$806$	0	0
$807$	−48.8355	−1.71909
$808$	0	0
$809$	−27.7581	−0.975923	−0.487961	−	0.872865i	$-0.662259\pi$
$809$	−27.7581	−0.975923	−0.487961	+	0.872865i	$0.662259\pi$
$810$	0	0
$811$	8.38201	0.294332	0.147166	−	0.989112i	$-0.452985\pi$
$811$	8.38201	0.294332	0.147166	+	0.989112i	$0.452985\pi$
$812$	0	0
$813$	−33.6332	−1.17957
$814$	0	0
$815$	12.9986	0.455323
$816$	0	0
$817$	−8.11742	−0.283993
$818$	0	0
$819$	−6.09682	−0.213040
$820$	0	0
$821$	−25.2785	−0.882224	−0.441112	−	0.897452i	$-0.645416\pi$
$821$	−25.2785	−0.882224	−0.441112	+	0.897452i	$0.645416\pi$
$822$	0	0
$823$	27.5385	0.959933	0.479966	−	0.877287i	$-0.340649\pi$
$823$	27.5385	0.959933	0.479966	+	0.877287i	$0.340649\pi$
$824$	0	0
$825$	55.3284	1.92629
$826$	0	0
$827$	1.85259	0.0644209	0.0322104	−	0.999481i	$-0.489745\pi$
$827$	1.85259	0.0644209	0.0322104	+	0.999481i	$0.489745\pi$
$828$	0	0
$829$	37.4656	1.30123	0.650617	−	0.759406i	$-0.274510\pi$
$829$	37.4656	1.30123	0.650617	+	0.759406i	$0.274510\pi$
$830$	0	0
$831$	7.65872	0.265678
$832$	0	0
$833$	−0.628830	−0.0217877
$834$	0	0
$835$	15.2077	0.526284
$836$	0	0
$837$	−14.1023	−0.487448
$838$	0	0
$839$	−22.1728	−0.765490	−0.382745	−	0.923854i	$-0.625021\pi$
$839$	−22.1728	−0.765490	−0.382745	+	0.923854i	$0.625021\pi$
$840$	0	0
$841$	−19.3432	−0.667005
$842$	0	0
$843$	18.6900	0.643717
$844$	0	0
$845$	14.8219	0.509888
$846$	0	0
$847$	73.9545	2.54111
$848$	0	0
$849$	42.3608	1.45382
$850$	0	0
$851$	−0.884856	−0.0303325
$852$	0	0
$853$	−25.4261	−0.870573	−0.435287	−	0.900292i	$-0.643353\pi$
$853$	−25.4261	−0.870573	−0.435287	+	0.900292i	$0.643353\pi$
$854$	0	0
$855$	0.487523	0.0166730
$856$	0	0
$857$	−28.2101	−0.963639	−0.481820	−	0.876270i	$-0.660024\pi$
$857$	−28.2101	−0.963639	−0.481820	+	0.876270i	$0.660024\pi$
$858$	0	0
$859$	−5.20655	−0.177645	−0.0888226	−	0.996047i	$-0.528310\pi$
$859$	−5.20655	−0.177645	−0.0888226	+	0.996047i	$0.528310\pi$
$860$	0	0
$861$	−6.15999	−0.209932
$862$	0	0
$863$	41.1135	1.39952	0.699761	−	0.714377i	$-0.253290\pi$
$863$	41.1135	1.39952	0.699761	+	0.714377i	$0.253290\pi$
$864$	0	0
$865$	2.78172	0.0945812
$866$	0	0
$867$	31.3170	1.06358
$868$	0	0
$869$	−91.1084	−3.09064
$870$	0	0
$871$	27.2127	0.922068
$872$	0	0
$873$	−7.05237	−0.238687
$874$	0	0
$875$	−14.9423	−0.505142
$876$	0	0
$877$	33.2642	1.12325	0.561626	−	0.827391i	$-0.310176\pi$
$877$	33.2642	1.12325	0.561626	+	0.827391i	$0.310176\pi$
$878$	0	0
$879$	34.2084	1.15382
$880$	0	0
$881$	9.99610	0.336777	0.168389	−	0.985721i	$-0.446144\pi$
$881$	9.99610	0.336777	0.168389	+	0.985721i	$0.446144\pi$
$882$	0	0
$883$	33.1715	1.11631	0.558156	−	0.829736i	$-0.311509\pi$
$883$	33.1715	1.11631	0.558156	+	0.829736i	$0.311509\pi$
$884$	0	0
$885$	−6.85007	−0.230262
$886$	0	0
$887$	−49.2150	−1.65248	−0.826239	−	0.563320i	$-0.809524\pi$
$887$	−49.2150	−1.65248	−0.826239	+	0.563320i	$0.809524\pi$
$888$	0	0
$889$	−19.3437	−0.648767
$890$	0	0
$891$	−66.6205	−2.23187
$892$	0	0
$893$	10.1403	0.339331
$894$	0	0
$895$	−2.60203	−0.0869763
$896$	0	0
$897$	−2.53676	−0.0847000
$898$	0	0
$899$	9.25020	0.308512
$900$	0	0
$901$	−4.30425	−0.143395
$902$	0	0
$903$	21.7530	0.723893
$904$	0	0
$905$	−11.8792	−0.394877
$906$	0	0
$907$	−21.2825	−0.706673	−0.353336	−	0.935496i	$-0.614953\pi$
$907$	−21.2825	−0.706673	−0.353336	+	0.935496i	$0.614953\pi$
$908$	0	0
$909$	−1.53093	−0.0507777
$910$	0	0
$911$	−24.1420	−0.799861	−0.399931	−	0.916545i	$-0.630966\pi$
$911$	−24.1420	−0.799861	−0.399931	+	0.916545i	$0.630966\pi$
$912$	0	0
$913$	37.6069	1.24461
$914$	0	0
$915$	11.8729	0.392508
$916$	0	0
$917$	−16.8917	−0.557812
$918$	0	0
$919$	−34.0218	−1.12228	−0.561138	−	0.827722i	$-0.689636\pi$
$919$	−34.0218	−1.12228	−0.561138	+	0.827722i	$0.689636\pi$
$920$	0	0
$921$	42.7560	1.40886
$922$	0	0
$923$	3.31377	0.109074
$924$	0	0
$925$	−17.3331	−0.569907
$926$	0	0
$927$	2.75462	0.0904735
$928$	0	0
$929$	13.4502	0.441288	0.220644	−	0.975354i	$-0.429184\pi$
$929$	13.4502	0.441288	0.220644	+	0.975354i	$0.429184\pi$
$930$	0	0
$931$	−2.71058	−0.0888358
$932$	0	0
$933$	−7.38415	−0.241746
$934$	0	0
$935$	1.65411	0.0540952
$936$	0	0
$937$	−33.6409	−1.09900	−0.549500	−	0.835494i	$-0.685182\pi$
$937$	−33.6409	−1.09900	−0.549500	+	0.835494i	$0.685182\pi$
$938$	0	0
$939$	−12.7716	−0.416784
$940$	0	0
$941$	−15.2118	−0.495889	−0.247945	−	0.968774i	$-0.579755\pi$
$941$	−15.2118	−0.495889	−0.247945	+	0.968774i	$0.579755\pi$
$942$	0	0
$943$	−0.333660	−0.0108655
$944$	0	0
$945$	7.42277	0.241463
$946$	0	0
$947$	42.8411	1.39215	0.696074	−	0.717970i	$-0.254929\pi$
$947$	42.8411	1.39215	0.696074	+	0.717970i	$0.254929\pi$
$948$	0	0
$949$	61.7134	2.00330
$950$	0	0
$951$	−40.6830	−1.31924
$952$	0	0
$953$	−41.7959	−1.35390	−0.676951	−	0.736028i	$-0.736699\pi$
$953$	−41.7959	−1.35390	−0.676951	+	0.736028i	$0.736699\pi$
$954$	0	0
$955$	−14.9301	−0.483127
$956$	0	0
$957$	37.8968	1.22503
$958$	0	0
$959$	46.9955	1.51757
$960$	0	0
$961$	−22.1393	−0.714171
$962$	0	0
$963$	−0.447527	−0.0144214
$964$	0	0
$965$	−0.453720	−0.0146058
$966$	0	0
$967$	16.9647	0.545547	0.272774	−	0.962078i	$-0.412059\pi$
$967$	16.9647	0.545547	0.272774	+	0.962078i	$0.412059\pi$
$968$	0	0
$969$	−1.09695	−0.0352392
$970$	0	0
$971$	26.4192	0.847831	0.423915	−	0.905702i	$-0.360655\pi$
$971$	26.4192	0.847831	0.423915	+	0.905702i	$0.360655\pi$
$972$	0	0
$973$	−41.9313	−1.34425
$974$	0	0
$975$	−49.6915	−1.59140
$976$	0	0
$977$	38.2390	1.22337	0.611687	−	0.791100i	$-0.290491\pi$
$977$	38.2390	1.22337	0.611687	+	0.791100i	$0.290491\pi$
$978$	0	0
$979$	−14.1073	−0.450870
$980$	0	0
$981$	−1.86002	−0.0593859
$982$	0	0
$983$	−38.6524	−1.23282	−0.616410	−	0.787426i	$-0.711413\pi$
$983$	−38.6524	−1.23282	−0.616410	+	0.787426i	$0.711413\pi$
$984$	0	0
$985$	−5.81149	−0.185169
$986$	0	0
$987$	−27.1738	−0.864950
$988$	0	0
$989$	1.17827	0.0374667
$990$	0	0
$991$	−4.64624	−0.147593	−0.0737963	−	0.997273i	$-0.523511\pi$
$991$	−4.64624	−0.147593	−0.0737963	+	0.997273i	$0.523511\pi$
$992$	0	0
$993$	46.0638	1.46179
$994$	0	0
$995$	−4.89134	−0.155066
$996$	0	0
$997$	−22.3137	−0.706682	−0.353341	−	0.935495i	$-0.614955\pi$
$997$	−22.3137	−0.706682	−0.353341	+	0.935495i	$0.614955\pi$
$998$	0	0
$999$	18.0996	0.572647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.20	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.20	✓	80	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-1.85715 q^{3} +0.680485 q^{5} +2.30245 q^{7} +0.448994 q^{9} +O(q^{10})\)	\(q-1.85715 q^{3} +0.680485 q^{5} +2.30245 q^{7} +0.448994 q^{9} +6.56658 q^{11} -5.89757 q^{13} -1.26376 q^{15} +0.370174 q^{17} +1.59564 q^{19} -4.27598 q^{21} -0.231612 q^{23} -4.53694 q^{25} +4.73759 q^{27} -3.10755 q^{29} -2.97669 q^{31} -12.1951 q^{33} +1.56678 q^{35} +3.82043 q^{37} +10.9527 q^{39} +1.44060 q^{41} -5.08724 q^{43} +0.305534 q^{45} +6.35498 q^{47} -1.69874 q^{49} -0.687468 q^{51} -11.6276 q^{53} +4.46846 q^{55} -2.96334 q^{57} +5.42038 q^{59} -9.39493 q^{61} +1.03379 q^{63} -4.01321 q^{65} -4.61423 q^{67} +0.430137 q^{69} -0.561887 q^{71} -10.4642 q^{73} +8.42576 q^{75} +15.1192 q^{77} -13.8746 q^{79} -10.1454 q^{81} +5.72702 q^{83} +0.251898 q^{85} +5.77117 q^{87} -2.14834 q^{89} -13.5788 q^{91} +5.52815 q^{93} +1.08581 q^{95} -15.7070 q^{97} +2.94836 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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