LMFDB - Embedded newform 8044.2.a.b.1.6

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:	$0$
Dimension:	$87$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.85294 q^{3} -2.90004 q^{5} -0.912476 q^{7} +5.13929 q^{9} +O(q^{10})$	$q-2.85294 q^{3} -2.90004 q^{5} -0.912476 q^{7} +5.13929 q^{9} -1.00770 q^{11} -2.16009 q^{13} +8.27366 q^{15} +4.92920 q^{17} +6.43440 q^{19} +2.60324 q^{21} -1.99322 q^{23} +3.41025 q^{25} -6.10328 q^{27} +6.10564 q^{29} +0.296995 q^{31} +2.87493 q^{33} +2.64622 q^{35} -3.90472 q^{37} +6.16261 q^{39} +5.67494 q^{41} +9.47092 q^{43} -14.9042 q^{45} -3.74669 q^{47} -6.16739 q^{49} -14.0627 q^{51} -12.7754 q^{53} +2.92239 q^{55} -18.3570 q^{57} -6.27661 q^{59} -3.33474 q^{61} -4.68948 q^{63} +6.26434 q^{65} +0.730536 q^{67} +5.68656 q^{69} +2.99023 q^{71} -7.53292 q^{73} -9.72924 q^{75} +0.919506 q^{77} +7.70866 q^{79} +1.99445 q^{81} +4.37099 q^{83} -14.2949 q^{85} -17.4191 q^{87} +9.90370 q^{89} +1.97103 q^{91} -0.847310 q^{93} -18.6600 q^{95} +6.36799 q^{97} -5.17889 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * 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
$3$	−2.85294	−1.64715	−0.823574	−	0.567209i	$-0.808023\pi$
$3$	−2.85294	−1.64715	−0.823574	+	0.567209i	$0.808023\pi$
$4$	0	0
$5$	−2.90004	−1.29694	−0.648469	−	0.761241i	$-0.724590\pi$
$5$	−2.90004	−1.29694	−0.648469	+	0.761241i	$0.724590\pi$
$6$	0	0
$7$	−0.912476	−0.344883	−0.172442	−	0.985020i	$-0.555166\pi$
$7$	−0.912476	−0.344883	−0.172442	+	0.985020i	$0.555166\pi$
$8$	0	0
$9$	5.13929	1.71310
$10$	0	0
$11$	−1.00770	−0.303834	−0.151917	−	0.988393i	$-0.548545\pi$
$11$	−1.00770	−0.303834	−0.151917	+	0.988393i	$0.548545\pi$
$12$	0	0
$13$	−2.16009	−0.599100	−0.299550	−	0.954081i	$-0.596837\pi$
$13$	−2.16009	−0.599100	−0.299550	+	0.954081i	$0.596837\pi$
$14$	0	0
$15$	8.27366	2.13625
$16$	0	0
$17$	4.92920	1.19551	0.597753	−	0.801680i	$-0.296060\pi$
$17$	4.92920	1.19551	0.597753	+	0.801680i	$0.296060\pi$
$18$	0	0
$19$	6.43440	1.47615	0.738076	−	0.674718i	$-0.235735\pi$
$19$	6.43440	1.47615	0.738076	+	0.674718i	$0.235735\pi$
$20$	0	0
$21$	2.60324	0.568074
$22$	0	0
$23$	−1.99322	−0.415616	−0.207808	−	0.978170i	$-0.566633\pi$
$23$	−1.99322	−0.415616	−0.207808	+	0.978170i	$0.566633\pi$
$24$	0	0
$25$	3.41025	0.682049
$26$	0	0
$27$	−6.10328	−1.17458
$28$	0	0
$29$	6.10564	1.13379	0.566895	−	0.823790i	$-0.308145\pi$
$29$	6.10564	1.13379	0.566895	+	0.823790i	$0.308145\pi$
$30$	0	0
$31$	0.296995	0.0533418	0.0266709	−	0.999644i	$-0.491509\pi$
$31$	0.296995	0.0533418	0.0266709	+	0.999644i	$0.491509\pi$
$32$	0	0
$33$	2.87493	0.500460
$34$	0	0
$35$	2.64622	0.447292
$36$	0	0
$37$	−3.90472	−0.641932	−0.320966	−	0.947091i	$-0.604008\pi$
$37$	−3.90472	−0.641932	−0.320966	+	0.947091i	$0.604008\pi$
$38$	0	0
$39$	6.16261	0.986807
$40$	0	0
$41$	5.67494	0.886277	0.443139	−	0.896453i	$-0.353865\pi$
$41$	5.67494	0.886277	0.443139	+	0.896453i	$0.353865\pi$
$42$	0	0
$43$	9.47092	1.44430	0.722151	−	0.691736i	$-0.243154\pi$
$43$	9.47092	1.44430	0.722151	+	0.691736i	$0.243154\pi$
$44$	0	0
$45$	−14.9042	−2.22178
$46$	0	0
$47$	−3.74669	−0.546511	−0.273255	−	0.961942i	$-0.588100\pi$
$47$	−3.74669	−0.546511	−0.273255	+	0.961942i	$0.588100\pi$
$48$	0	0
$49$	−6.16739	−0.881055
$50$	0	0
$51$	−14.0627	−1.96918
$52$	0	0
$53$	−12.7754	−1.75484	−0.877418	−	0.479726i	$-0.840736\pi$
$53$	−12.7754	−1.75484	−0.877418	+	0.479726i	$0.840736\pi$
$54$	0	0
$55$	2.92239	0.394054
$56$	0	0
$57$	−18.3570	−2.43144
$58$	0	0
$59$	−6.27661	−0.817145	−0.408573	−	0.912726i	$-0.633973\pi$
$59$	−6.27661	−0.817145	−0.408573	+	0.912726i	$0.633973\pi$
$60$	0	0
$61$	−3.33474	−0.426969	−0.213485	−	0.976946i	$-0.568481\pi$
$61$	−3.33474	−0.426969	−0.213485	+	0.976946i	$0.568481\pi$
$62$	0	0
$63$	−4.68948	−0.590819
$64$	0	0
$65$	6.26434	0.776996
$66$	0	0
$67$	0.730536	0.0892492	0.0446246	−	0.999004i	$-0.485791\pi$
$67$	0.730536	0.0892492	0.0446246	+	0.999004i	$0.485791\pi$
$68$	0	0
$69$	5.68656	0.684581
$70$	0	0
$71$	2.99023	0.354875	0.177438	−	0.984132i	$-0.443219\pi$
$71$	2.99023	0.354875	0.177438	+	0.984132i	$0.443219\pi$
$72$	0	0
$73$	−7.53292	−0.881661	−0.440831	−	0.897590i	$-0.645316\pi$
$73$	−7.53292	−0.881661	−0.440831	+	0.897590i	$0.645316\pi$
$74$	0	0
$75$	−9.72924	−1.12344
$76$	0	0
$77$	0.919506	0.104787
$78$	0	0
$79$	7.70866	0.867292	0.433646	−	0.901083i	$-0.357227\pi$
$79$	7.70866	0.867292	0.433646	+	0.901083i	$0.357227\pi$
$80$	0	0
$81$	1.99445	0.221606
$82$	0	0
$83$	4.37099	0.479779	0.239889	−	0.970800i	$-0.422889\pi$
$83$	4.37099	0.479779	0.239889	+	0.970800i	$0.422889\pi$
$84$	0	0
$85$	−14.2949	−1.55050
$86$	0	0
$87$	−17.4191	−1.86752
$88$	0	0
$89$	9.90370	1.04979	0.524895	−	0.851167i	$-0.324105\pi$
$89$	9.90370	1.04979	0.524895	+	0.851167i	$0.324105\pi$
$90$	0	0
$91$	1.97103	0.206620
$92$	0	0
$93$	−0.847310	−0.0878619
$94$	0	0
$95$	−18.6600	−1.91448
$96$	0	0
$97$	6.36799	0.646572	0.323286	−	0.946301i	$-0.395212\pi$
$97$	6.36799	0.646572	0.323286	+	0.946301i	$0.395212\pi$
$98$	0	0
$99$	−5.17889	−0.520498
$100$	0	0
$101$	−5.51388	−0.548651	−0.274326	−	0.961637i	$-0.588455\pi$
$101$	−5.51388	−0.548651	−0.274326	+	0.961637i	$0.588455\pi$
$102$	0	0
$103$	−13.3461	−1.31503	−0.657516	−	0.753440i	$-0.728393\pi$
$103$	−13.3461	−1.31503	−0.657516	+	0.753440i	$0.728393\pi$
$104$	0	0
$105$	−7.54951	−0.736757
$106$	0	0
$107$	−12.7819	−1.23567	−0.617837	−	0.786306i	$-0.711991\pi$
$107$	−12.7819	−1.23567	−0.617837	+	0.786306i	$0.711991\pi$
$108$	0	0
$109$	−11.1608	−1.06901	−0.534505	−	0.845165i	$-0.679502\pi$
$109$	−11.1608	−1.06901	−0.534505	+	0.845165i	$0.679502\pi$
$110$	0	0
$111$	11.1400	1.05736
$112$	0	0
$113$	8.28402	0.779295	0.389647	−	0.920964i	$-0.372597\pi$
$113$	8.28402	0.779295	0.389647	+	0.920964i	$0.372597\pi$
$114$	0	0
$115$	5.78043	0.539028
$116$	0	0
$117$	−11.1013	−1.02632
$118$	0	0
$119$	−4.49777	−0.412310
$120$	0	0
$121$	−9.98453	−0.907685
$122$	0	0
$123$	−16.1903	−1.45983
$124$	0	0
$125$	4.61035	0.412363
$126$	0	0
$127$	6.14638	0.545403	0.272701	−	0.962099i	$-0.412083\pi$
$127$	6.14638	0.545403	0.272701	+	0.962099i	$0.412083\pi$
$128$	0	0
$129$	−27.0200	−2.37898
$130$	0	0
$131$	3.66598	0.320298	0.160149	−	0.987093i	$-0.448802\pi$
$131$	3.66598	0.320298	0.160149	+	0.987093i	$0.448802\pi$
$132$	0	0
$133$	−5.87123	−0.509100
$134$	0	0
$135$	17.6998	1.52335
$136$	0	0
$137$	21.4983	1.83672	0.918360	−	0.395746i	$-0.129514\pi$
$137$	21.4983	1.83672	0.918360	+	0.395746i	$0.129514\pi$
$138$	0	0
$139$	6.60040	0.559839	0.279919	−	0.960024i	$-0.409692\pi$
$139$	6.60040	0.559839	0.279919	+	0.960024i	$0.409692\pi$
$140$	0	0
$141$	10.6891	0.900184
$142$	0	0
$143$	2.17673	0.182027
$144$	0	0
$145$	−17.7066	−1.47046
$146$	0	0
$147$	17.5952	1.45123
$148$	0	0
$149$	2.26513	0.185567	0.0927834	−	0.995686i	$-0.470424\pi$
$149$	2.26513	0.185567	0.0927834	+	0.995686i	$0.470424\pi$
$150$	0	0
$151$	13.3158	1.08362	0.541811	−	0.840500i	$-0.317739\pi$
$151$	13.3158	1.08362	0.541811	+	0.840500i	$0.317739\pi$
$152$	0	0
$153$	25.3326	2.04802
$154$	0	0
$155$	−0.861298	−0.0691811
$156$	0	0
$157$	−11.5061	−0.918287	−0.459143	−	0.888362i	$-0.651844\pi$
$157$	−11.5061	−0.918287	−0.459143	+	0.888362i	$0.651844\pi$
$158$	0	0
$159$	36.4475	2.89048
$160$	0	0
$161$	1.81877	0.143339
$162$	0	0
$163$	−11.6575	−0.913085	−0.456543	−	0.889701i	$-0.650912\pi$
$163$	−11.6575	−0.913085	−0.456543	+	0.889701i	$0.650912\pi$
$164$	0	0
$165$	−8.33741	−0.649066
$166$	0	0
$167$	4.28582	0.331647	0.165824	−	0.986155i	$-0.446972\pi$
$167$	4.28582	0.331647	0.165824	+	0.986155i	$0.446972\pi$
$168$	0	0
$169$	−8.33402	−0.641079
$170$	0	0
$171$	33.0682	2.52879
$172$	0	0
$173$	−4.69109	−0.356657	−0.178329	−	0.983971i	$-0.557069\pi$
$173$	−4.69109	−0.356657	−0.178329	+	0.983971i	$0.557069\pi$
$174$	0	0
$175$	−3.11177	−0.235227
$176$	0	0
$177$	17.9068	1.34596
$178$	0	0
$179$	−0.405985	−0.0303447	−0.0151724	−	0.999885i	$-0.504830\pi$
$179$	−0.405985	−0.0303447	−0.0151724	+	0.999885i	$0.504830\pi$
$180$	0	0
$181$	−8.16158	−0.606645	−0.303323	−	0.952888i	$-0.598096\pi$
$181$	−8.16158	−0.606645	−0.303323	+	0.952888i	$0.598096\pi$
$182$	0	0
$183$	9.51382	0.703282
$184$	0	0
$185$	11.3239	0.832547
$186$	0	0
$187$	−4.96718	−0.363236
$188$	0	0
$189$	5.56910	0.405092
$190$	0	0
$191$	−15.8827	−1.14923	−0.574614	−	0.818424i	$-0.694848\pi$
$191$	−15.8827	−1.14923	−0.574614	+	0.818424i	$0.694848\pi$
$192$	0	0
$193$	−23.4922	−1.69100	−0.845502	−	0.533972i	$-0.820699\pi$
$193$	−23.4922	−1.69100	−0.845502	+	0.533972i	$0.820699\pi$
$194$	0	0
$195$	−17.8718	−1.27983
$196$	0	0
$197$	−25.2060	−1.79585	−0.897926	−	0.440146i	$-0.854927\pi$
$197$	−25.2060	−1.79585	−0.897926	+	0.440146i	$0.854927\pi$
$198$	0	0
$199$	7.28654	0.516529	0.258264	−	0.966074i	$-0.416849\pi$
$199$	7.28654	0.516529	0.258264	+	0.966074i	$0.416849\pi$
$200$	0	0
$201$	−2.08418	−0.147007
$202$	0	0
$203$	−5.57125	−0.391025
$204$	0	0
$205$	−16.4576	−1.14945
$206$	0	0
$207$	−10.2438	−0.711991
$208$	0	0
$209$	−6.48397	−0.448506
$210$	0	0
$211$	0.190709	0.0131289	0.00656446	−	0.999978i	$-0.497910\pi$
$211$	0.190709	0.0131289	0.00656446	+	0.999978i	$0.497910\pi$
$212$	0	0
$213$	−8.53096	−0.584532
$214$	0	0
$215$	−27.4661	−1.87317
$216$	0	0
$217$	−0.271001	−0.0183967
$218$	0	0
$219$	21.4910	1.45223
$220$	0	0
$221$	−10.6475	−0.716228
$222$	0	0
$223$	13.5073	0.904513	0.452256	−	0.891888i	$-0.350619\pi$
$223$	13.5073	0.904513	0.452256	+	0.891888i	$0.350619\pi$
$224$	0	0
$225$	17.5263	1.16842
$226$	0	0
$227$	3.95747	0.262666	0.131333	−	0.991338i	$-0.458074\pi$
$227$	3.95747	0.262666	0.131333	+	0.991338i	$0.458074\pi$
$228$	0	0
$229$	2.20387	0.145636	0.0728178	−	0.997345i	$-0.476801\pi$
$229$	2.20387	0.145636	0.0728178	+	0.997345i	$0.476801\pi$
$230$	0	0
$231$	−2.62330	−0.172600
$232$	0	0
$233$	16.9971	1.11352	0.556758	−	0.830674i	$-0.312045\pi$
$233$	16.9971	1.11352	0.556758	+	0.830674i	$0.312045\pi$
$234$	0	0
$235$	10.8656	0.708791
$236$	0	0
$237$	−21.9924	−1.42856
$238$	0	0
$239$	18.1776	1.17581	0.587904	−	0.808931i	$-0.299953\pi$
$239$	18.1776	1.17581	0.587904	+	0.808931i	$0.299953\pi$
$240$	0	0
$241$	17.6820	1.13900	0.569498	−	0.821993i	$-0.307138\pi$
$241$	17.6820	1.13900	0.569498	+	0.821993i	$0.307138\pi$
$242$	0	0
$243$	12.6198	0.809560
$244$	0	0
$245$	17.8857	1.14267
$246$	0	0
$247$	−13.8989	−0.884363
$248$	0	0
$249$	−12.4702	−0.790267
$250$	0	0
$251$	−13.4541	−0.849217	−0.424608	−	0.905377i	$-0.639588\pi$
$251$	−13.4541	−0.849217	−0.424608	+	0.905377i	$0.639588\pi$
$252$	0	0
$253$	2.00858	0.126278
$254$	0	0
$255$	40.7825	2.55390
$256$	0	0
$257$	−25.3080	−1.57867	−0.789334	−	0.613965i	$-0.789574\pi$
$257$	−25.3080	−1.57867	−0.789334	+	0.613965i	$0.789574\pi$
$258$	0	0
$259$	3.56296	0.221392
$260$	0	0
$261$	31.3787	1.94229
$262$	0	0
$263$	10.6072	0.654066	0.327033	−	0.945013i	$-0.393951\pi$
$263$	10.6072	0.654066	0.327033	+	0.945013i	$0.393951\pi$
$264$	0	0
$265$	37.0492	2.27591
$266$	0	0
$267$	−28.2547	−1.72916
$268$	0	0
$269$	22.8848	1.39531	0.697655	−	0.716434i	$-0.254227\pi$
$269$	22.8848	1.39531	0.697655	+	0.716434i	$0.254227\pi$
$270$	0	0
$271$	−6.46078	−0.392464	−0.196232	−	0.980557i	$-0.562871\pi$
$271$	−6.46078	−0.392464	−0.196232	+	0.980557i	$0.562871\pi$
$272$	0	0
$273$	−5.62323	−0.340333
$274$	0	0
$275$	−3.43652	−0.207230
$276$	0	0
$277$	−12.4338	−0.747072	−0.373536	−	0.927616i	$-0.621855\pi$
$277$	−12.4338	−0.747072	−0.373536	+	0.927616i	$0.621855\pi$
$278$	0	0
$279$	1.52634	0.0913798
$280$	0	0
$281$	−8.52709	−0.508683	−0.254342	−	0.967114i	$-0.581859\pi$
$281$	−8.52709	−0.508683	−0.254342	+	0.967114i	$0.581859\pi$
$282$	0	0
$283$	30.7038	1.82515	0.912577	−	0.408905i	$-0.134089\pi$
$283$	30.7038	1.82515	0.912577	+	0.408905i	$0.134089\pi$
$284$	0	0
$285$	53.2360	3.15343
$286$	0	0
$287$	−5.17825	−0.305662
$288$	0	0
$289$	7.29699	0.429235
$290$	0	0
$291$	−18.1675	−1.06500
$292$	0	0
$293$	−12.5644	−0.734017	−0.367009	−	0.930218i	$-0.619618\pi$
$293$	−12.5644	−0.734017	−0.367009	+	0.930218i	$0.619618\pi$
$294$	0	0
$295$	18.2024	1.05979
$296$	0	0
$297$	6.15031	0.356877
$298$	0	0
$299$	4.30554	0.248996
$300$	0	0
$301$	−8.64198	−0.498116
$302$	0	0
$303$	15.7308	0.903710
$304$	0	0
$305$	9.67088	0.553753
$306$	0	0
$307$	−25.6483	−1.46382	−0.731912	−	0.681400i	$-0.761372\pi$
$307$	−25.6483	−1.46382	−0.731912	+	0.681400i	$0.761372\pi$
$308$	0	0
$309$	38.0757	2.16605
$310$	0	0
$311$	−14.9577	−0.848171	−0.424085	−	0.905622i	$-0.639404\pi$
$311$	−14.9577	−0.848171	−0.424085	+	0.905622i	$0.639404\pi$
$312$	0	0
$313$	16.1170	0.910984	0.455492	−	0.890240i	$-0.349463\pi$
$313$	16.1170	0.910984	0.455492	+	0.890240i	$0.349463\pi$
$314$	0	0
$315$	13.5997	0.766256
$316$	0	0
$317$	−2.11365	−0.118715	−0.0593573	−	0.998237i	$-0.518905\pi$
$317$	−2.11365	−0.118715	−0.0593573	+	0.998237i	$0.518905\pi$
$318$	0	0
$319$	−6.15269	−0.344484
$320$	0	0
$321$	36.4661	2.03534
$322$	0	0
$323$	31.7164	1.76475
$324$	0	0
$325$	−7.36643	−0.408616
$326$	0	0
$327$	31.8411	1.76082
$328$	0	0
$329$	3.41876	0.188482
$330$	0	0
$331$	4.89131	0.268851	0.134425	−	0.990924i	$-0.457081\pi$
$331$	4.89131	0.268851	0.134425	+	0.990924i	$0.457081\pi$
$332$	0	0
$333$	−20.0675	−1.09969
$334$	0	0
$335$	−2.11859	−0.115751
$336$	0	0
$337$	24.5994	1.34002	0.670008	−	0.742354i	$-0.266291\pi$
$337$	24.5994	1.34002	0.670008	+	0.742354i	$0.266291\pi$
$338$	0	0
$339$	−23.6338	−1.28361
$340$	0	0
$341$	−0.299283	−0.0162071
$342$	0	0
$343$	12.0149	0.648745
$344$	0	0
$345$	−16.4913	−0.887859
$346$	0	0
$347$	11.9560	0.641833	0.320917	−	0.947107i	$-0.396009\pi$
$347$	11.9560	0.641833	0.320917	+	0.947107i	$0.396009\pi$
$348$	0	0
$349$	9.62391	0.515156	0.257578	−	0.966257i	$-0.417076\pi$
$349$	9.62391	0.515156	0.257578	+	0.966257i	$0.417076\pi$
$350$	0	0
$351$	13.1836	0.703690
$352$	0	0
$353$	8.97717	0.477807	0.238903	−	0.971043i	$-0.423212\pi$
$353$	8.97717	0.477807	0.238903	+	0.971043i	$0.423212\pi$
$354$	0	0
$355$	−8.67180	−0.460251
$356$	0	0
$357$	12.8319	0.679136
$358$	0	0
$359$	35.7940	1.88913	0.944566	−	0.328321i	$-0.106483\pi$
$359$	35.7940	1.88913	0.944566	+	0.328321i	$0.106483\pi$
$360$	0	0
$361$	22.4014	1.17902
$362$	0	0
$363$	28.4853	1.49509
$364$	0	0
$365$	21.8458	1.14346
$366$	0	0
$367$	−10.7659	−0.561976	−0.280988	−	0.959711i	$-0.590662\pi$
$367$	−10.7659	−0.561976	−0.280988	+	0.959711i	$0.590662\pi$
$368$	0	0
$369$	29.1652	1.51828
$370$	0	0
$371$	11.6572	0.605214
$372$	0	0
$373$	−32.6228	−1.68914	−0.844571	−	0.535443i	$-0.820145\pi$
$373$	−32.6228	−1.68914	−0.844571	+	0.535443i	$0.820145\pi$
$374$	0	0
$375$	−13.1531	−0.679222
$376$	0	0
$377$	−13.1887	−0.679254
$378$	0	0
$379$	−8.38020	−0.430462	−0.215231	−	0.976563i	$-0.569050\pi$
$379$	−8.38020	−0.430462	−0.215231	+	0.976563i	$0.569050\pi$
$380$	0	0
$381$	−17.5353	−0.898360
$382$	0	0
$383$	−2.82140	−0.144167	−0.0720835	−	0.997399i	$-0.522965\pi$
$383$	−2.82140	−0.144167	−0.0720835	+	0.997399i	$0.522965\pi$
$384$	0	0
$385$	−2.66661	−0.135903
$386$	0	0
$387$	48.6738	2.47423
$388$	0	0
$389$	−2.94656	−0.149397	−0.0746984	−	0.997206i	$-0.523799\pi$
$389$	−2.94656	−0.149397	−0.0746984	+	0.997206i	$0.523799\pi$
$390$	0	0
$391$	−9.82499	−0.496871
$392$	0	0
$393$	−10.4588	−0.527579
$394$	0	0
$395$	−22.3554	−1.12482
$396$	0	0
$397$	−13.1937	−0.662170	−0.331085	−	0.943601i	$-0.607415\pi$
$397$	−13.1937	−0.662170	−0.331085	+	0.943601i	$0.607415\pi$
$398$	0	0
$399$	16.7503	0.838563
$400$	0	0
$401$	23.0765	1.15239	0.576194	−	0.817313i	$-0.304537\pi$
$401$	23.0765	1.15239	0.576194	+	0.817313i	$0.304537\pi$
$402$	0	0
$403$	−0.641535	−0.0319571
$404$	0	0
$405$	−5.78400	−0.287409
$406$	0	0
$407$	3.93481	0.195041
$408$	0	0
$409$	5.53436	0.273656	0.136828	−	0.990595i	$-0.456309\pi$
$409$	5.53436	0.273656	0.136828	+	0.990595i	$0.456309\pi$
$410$	0	0
$411$	−61.3333	−3.02535
$412$	0	0
$413$	5.72726	0.281820
$414$	0	0
$415$	−12.6761	−0.622243
$416$	0	0
$417$	−18.8306	−0.922137
$418$	0	0
$419$	−3.30369	−0.161396	−0.0806978	−	0.996739i	$-0.525715\pi$
$419$	−3.30369	−0.161396	−0.0806978	+	0.996739i	$0.525715\pi$
$420$	0	0
$421$	−13.3384	−0.650073	−0.325037	−	0.945701i	$-0.605377\pi$
$421$	−13.3384	−0.650073	−0.325037	+	0.945701i	$0.605377\pi$
$422$	0	0
$423$	−19.2553	−0.936226
$424$	0	0
$425$	16.8098	0.815394
$426$	0	0
$427$	3.04287	0.147255
$428$	0	0
$429$	−6.21009	−0.299826
$430$	0	0
$431$	28.1665	1.35673	0.678366	−	0.734724i	$-0.262688\pi$
$431$	28.1665	1.35673	0.678366	+	0.734724i	$0.262688\pi$
$432$	0	0
$433$	17.1327	0.823343	0.411672	−	0.911332i	$-0.364945\pi$
$433$	17.1327	0.823343	0.411672	+	0.911332i	$0.364945\pi$
$434$	0	0
$435$	50.5160	2.42206
$436$	0	0
$437$	−12.8252	−0.613512
$438$	0	0
$439$	28.7004	1.36980	0.684898	−	0.728639i	$-0.259847\pi$
$439$	28.7004	1.36980	0.684898	+	0.728639i	$0.259847\pi$
$440$	0	0
$441$	−31.6960	−1.50933
$442$	0	0
$443$	−21.9867	−1.04462	−0.522310	−	0.852756i	$-0.674930\pi$
$443$	−21.9867	−1.04462	−0.522310	+	0.852756i	$0.674930\pi$
$444$	0	0
$445$	−28.7211	−1.36151
$446$	0	0
$447$	−6.46230	−0.305656
$448$	0	0
$449$	31.0457	1.46514	0.732569	−	0.680693i	$-0.238321\pi$
$449$	31.0457	1.46514	0.732569	+	0.680693i	$0.238321\pi$
$450$	0	0
$451$	−5.71867	−0.269281
$452$	0	0
$453$	−37.9892	−1.78489
$454$	0	0
$455$	−5.71606	−0.267973
$456$	0	0
$457$	0.687534	0.0321615	0.0160807	−	0.999871i	$-0.494881\pi$
$457$	0.687534	0.0321615	0.0160807	+	0.999871i	$0.494881\pi$
$458$	0	0
$459$	−30.0843	−1.40421
$460$	0	0
$461$	34.7714	1.61947	0.809733	−	0.586799i	$-0.199612\pi$
$461$	34.7714	1.61947	0.809733	+	0.586799i	$0.199612\pi$
$462$	0	0
$463$	−23.6875	−1.10085	−0.550426	−	0.834884i	$-0.685535\pi$
$463$	−23.6875	−1.10085	−0.550426	+	0.834884i	$0.685535\pi$
$464$	0	0
$465$	2.45723	0.113952
$466$	0	0
$467$	−6.47963	−0.299842	−0.149921	−	0.988698i	$-0.547902\pi$
$467$	−6.47963	−0.299842	−0.149921	+	0.988698i	$0.547902\pi$
$468$	0	0
$469$	−0.666597	−0.0307806
$470$	0	0
$471$	32.8263	1.51255
$472$	0	0
$473$	−9.54389	−0.438828
$474$	0	0
$475$	21.9429	1.00681
$476$	0	0
$477$	−65.6565	−3.00621
$478$	0	0
$479$	13.9438	0.637111	0.318555	−	0.947904i	$-0.396802\pi$
$479$	13.9438	0.637111	0.318555	+	0.947904i	$0.396802\pi$
$480$	0	0
$481$	8.43454	0.384582
$482$	0	0
$483$	−5.18884	−0.236101
$484$	0	0
$485$	−18.4674	−0.838564
$486$	0	0
$487$	−27.2396	−1.23434	−0.617172	−	0.786828i	$-0.711722\pi$
$487$	−27.2396	−1.23434	−0.617172	+	0.786828i	$0.711722\pi$
$488$	0	0
$489$	33.2582	1.50399
$490$	0	0
$491$	32.2133	1.45376	0.726882	−	0.686762i	$-0.240969\pi$
$491$	32.2133	1.45376	0.726882	+	0.686762i	$0.240969\pi$
$492$	0	0
$493$	30.0959	1.35545
$494$	0	0
$495$	15.0190	0.675054
$496$	0	0
$497$	−2.72851	−0.122391
$498$	0	0
$499$	−4.40738	−0.197301	−0.0986506	−	0.995122i	$-0.531453\pi$
$499$	−4.40738	−0.197301	−0.0986506	+	0.995122i	$0.531453\pi$
$500$	0	0
$501$	−12.2272	−0.546272
$502$	0	0
$503$	34.2277	1.52614	0.763069	−	0.646317i	$-0.223692\pi$
$503$	34.2277	1.52614	0.763069	+	0.646317i	$0.223692\pi$
$504$	0	0
$505$	15.9905	0.711567
$506$	0	0
$507$	23.7765	1.05595
$508$	0	0
$509$	−7.50435	−0.332624	−0.166312	−	0.986073i	$-0.553186\pi$
$509$	−7.50435	−0.332624	−0.166312	+	0.986073i	$0.553186\pi$
$510$	0	0
$511$	6.87360	0.304070
$512$	0	0
$513$	−39.2709	−1.73385
$514$	0	0
$515$	38.7043	1.70552
$516$	0	0
$517$	3.77555	0.166049
$518$	0	0
$519$	13.3834	0.587467
$520$	0	0
$521$	−39.6001	−1.73491	−0.867456	−	0.497515i	$-0.834246\pi$
$521$	−39.6001	−1.73491	−0.867456	+	0.497515i	$0.834246\pi$
$522$	0	0
$523$	29.2929	1.28089	0.640445	−	0.768004i	$-0.278750\pi$
$523$	29.2929	1.28089	0.640445	+	0.768004i	$0.278750\pi$
$524$	0	0
$525$	8.87770	0.387454
$526$	0	0
$527$	1.46395	0.0637705
$528$	0	0
$529$	−19.0271	−0.827263
$530$	0	0
$531$	−32.2573	−1.39985
$532$	0	0
$533$	−12.2584	−0.530969
$534$	0	0
$535$	37.0681	1.60259
$536$	0	0
$537$	1.15825	0.0499823
$538$	0	0
$539$	6.21491	0.267695
$540$	0	0
$541$	−0.111380	−0.00478859	−0.00239430	−	0.999997i	$-0.500762\pi$
$541$	−0.111380	−0.00478859	−0.00239430	+	0.999997i	$0.500762\pi$
$542$	0	0
$543$	23.2845	0.999235
$544$	0	0
$545$	32.3668	1.38644
$546$	0	0
$547$	4.56616	0.195235	0.0976173	−	0.995224i	$-0.468878\pi$
$547$	4.56616	0.195235	0.0976173	+	0.995224i	$0.468878\pi$
$548$	0	0
$549$	−17.1382	−0.731440
$550$	0	0
$551$	39.2861	1.67365
$552$	0	0
$553$	−7.03396	−0.299114
$554$	0	0
$555$	−32.3063	−1.37133
$556$	0	0
$557$	−4.45481	−0.188756	−0.0943782	−	0.995536i	$-0.530086\pi$
$557$	−4.45481	−0.188756	−0.0943782	+	0.995536i	$0.530086\pi$
$558$	0	0
$559$	−20.4580	−0.865282
$560$	0	0
$561$	14.1711	0.598303
$562$	0	0
$563$	43.8217	1.84686	0.923431	−	0.383764i	$-0.125372\pi$
$563$	43.8217	1.84686	0.923431	+	0.383764i	$0.125372\pi$
$564$	0	0
$565$	−24.0240	−1.01070
$566$	0	0
$567$	−1.81989	−0.0764282
$568$	0	0
$569$	−34.9506	−1.46521	−0.732604	−	0.680655i	$-0.761695\pi$
$569$	−34.9506	−1.46521	−0.732604	+	0.680655i	$0.761695\pi$
$570$	0	0
$571$	−26.9857	−1.12932	−0.564658	−	0.825325i	$-0.690992\pi$
$571$	−26.9857	−1.12932	−0.564658	+	0.825325i	$0.690992\pi$
$572$	0	0
$573$	45.3123	1.89295
$574$	0	0
$575$	−6.79738	−0.283470
$576$	0	0
$577$	10.8490	0.451651	0.225826	−	0.974168i	$-0.427492\pi$
$577$	10.8490	0.451651	0.225826	+	0.974168i	$0.427492\pi$
$578$	0	0
$579$	67.0219	2.78533
$580$	0	0
$581$	−3.98842	−0.165468
$582$	0	0
$583$	12.8738	0.533180
$584$	0	0
$585$	32.1943	1.33107
$586$	0	0
$587$	9.50660	0.392379	0.196190	−	0.980566i	$-0.437143\pi$
$587$	9.50660	0.392379	0.196190	+	0.980566i	$0.437143\pi$
$588$	0	0
$589$	1.91098	0.0787406
$590$	0	0
$591$	71.9113	2.95804
$592$	0	0
$593$	4.32150	0.177463	0.0887314	−	0.996056i	$-0.471719\pi$
$593$	4.32150	0.177463	0.0887314	+	0.996056i	$0.471719\pi$
$594$	0	0
$595$	13.0437	0.534741
$596$	0	0
$597$	−20.7881	−0.850800
$598$	0	0
$599$	−27.5233	−1.12457	−0.562286	−	0.826943i	$-0.690078\pi$
$599$	−27.5233	−1.12457	−0.562286	+	0.826943i	$0.690078\pi$
$600$	0	0
$601$	28.9917	1.18260	0.591298	−	0.806453i	$-0.298616\pi$
$601$	28.9917	1.18260	0.591298	+	0.806453i	$0.298616\pi$
$602$	0	0
$603$	3.75444	0.152893
$604$	0	0
$605$	28.9556	1.17721
$606$	0	0
$607$	13.4950	0.547747	0.273873	−	0.961766i	$-0.411695\pi$
$607$	13.4950	0.547747	0.273873	+	0.961766i	$0.411695\pi$
$608$	0	0
$609$	15.8945	0.644077
$610$	0	0
$611$	8.09317	0.327415
$612$	0	0
$613$	−5.05805	−0.204293	−0.102146	−	0.994769i	$-0.532571\pi$
$613$	−5.05805	−0.204293	−0.102146	+	0.994769i	$0.532571\pi$
$614$	0	0
$615$	46.9526	1.89331
$616$	0	0
$617$	−23.4345	−0.943436	−0.471718	−	0.881749i	$-0.656366\pi$
$617$	−23.4345	−0.943436	−0.471718	+	0.881749i	$0.656366\pi$
$618$	0	0
$619$	25.1968	1.01274	0.506372	−	0.862315i	$-0.330986\pi$
$619$	25.1968	1.01274	0.506372	+	0.862315i	$0.330986\pi$
$620$	0	0
$621$	12.1652	0.488173
$622$	0	0
$623$	−9.03688	−0.362055
$624$	0	0
$625$	−30.4215	−1.21686
$626$	0	0
$627$	18.4984	0.738755
$628$	0	0
$629$	−19.2471	−0.767434
$630$	0	0
$631$	−5.15715	−0.205303	−0.102651	−	0.994717i	$-0.532733\pi$
$631$	−5.15715	−0.205303	−0.102651	+	0.994717i	$0.532733\pi$
$632$	0	0
$633$	−0.544081	−0.0216253
$634$	0	0
$635$	−17.8248	−0.707354
$636$	0	0
$637$	13.3221	0.527841
$638$	0	0
$639$	15.3677	0.607936
$640$	0	0
$641$	31.8391	1.25757	0.628784	−	0.777580i	$-0.283553\pi$
$641$	31.8391	1.25757	0.628784	+	0.777580i	$0.283553\pi$
$642$	0	0
$643$	7.81343	0.308132	0.154066	−	0.988061i	$-0.450763\pi$
$643$	7.81343	0.308132	0.154066	+	0.988061i	$0.450763\pi$
$644$	0	0
$645$	78.3592	3.08539
$646$	0	0
$647$	3.95855	0.155627	0.0778133	−	0.996968i	$-0.475206\pi$
$647$	3.95855	0.155627	0.0778133	+	0.996968i	$0.475206\pi$
$648$	0	0
$649$	6.32497	0.248277
$650$	0	0
$651$	0.773150	0.0303021
$652$	0	0
$653$	8.73084	0.341664	0.170832	−	0.985300i	$-0.445354\pi$
$653$	8.73084	0.341664	0.170832	+	0.985300i	$0.445354\pi$
$654$	0	0
$655$	−10.6315	−0.415407
$656$	0	0
$657$	−38.7139	−1.51037
$658$	0	0
$659$	−31.6664	−1.23355	−0.616773	−	0.787141i	$-0.711561\pi$
$659$	−31.6664	−1.23355	−0.616773	+	0.787141i	$0.711561\pi$
$660$	0	0
$661$	3.23626	0.125876	0.0629379	−	0.998017i	$-0.479953\pi$
$661$	3.23626	0.125876	0.0629379	+	0.998017i	$0.479953\pi$
$662$	0	0
$663$	30.3767	1.17973
$664$	0	0
$665$	17.0268	0.660271
$666$	0	0
$667$	−12.1699	−0.471221
$668$	0	0
$669$	−38.5355	−1.48987
$670$	0	0
$671$	3.36043	0.129728
$672$	0	0
$673$	−9.74375	−0.375594	−0.187797	−	0.982208i	$-0.560135\pi$
$673$	−9.74375	−0.375594	−0.187797	+	0.982208i	$0.560135\pi$
$674$	0	0
$675$	−20.8137	−0.801120
$676$	0	0
$677$	12.9090	0.496131	0.248066	−	0.968743i	$-0.420205\pi$
$677$	12.9090	0.496131	0.248066	+	0.968743i	$0.420205\pi$
$678$	0	0
$679$	−5.81064	−0.222992
$680$	0	0
$681$	−11.2904	−0.432650
$682$	0	0
$683$	2.41094	0.0922521	0.0461261	−	0.998936i	$-0.485312\pi$
$683$	2.41094	0.0922521	0.0461261	+	0.998936i	$0.485312\pi$
$684$	0	0
$685$	−62.3458	−2.38211
$686$	0	0
$687$	−6.28751	−0.239884
$688$	0	0
$689$	27.5960	1.05132
$690$	0	0
$691$	47.9969	1.82589	0.912945	−	0.408083i	$-0.133803\pi$
$691$	47.9969	1.82589	0.912945	+	0.408083i	$0.133803\pi$
$692$	0	0
$693$	4.72561	0.179511
$694$	0	0
$695$	−19.1414	−0.726076
$696$	0	0
$697$	27.9729	1.05955
$698$	0	0
$699$	−48.4918	−1.83413
$700$	0	0
$701$	2.58201	0.0975213	0.0487606	−	0.998810i	$-0.484473\pi$
$701$	2.58201	0.0975213	0.0487606	+	0.998810i	$0.484473\pi$
$702$	0	0
$703$	−25.1245	−0.947589
$704$	0	0
$705$	−30.9988	−1.16748
$706$	0	0
$707$	5.03128	0.189221
$708$	0	0
$709$	16.5647	0.622100	0.311050	−	0.950394i	$-0.399319\pi$
$709$	16.5647	0.622100	0.311050	+	0.950394i	$0.399319\pi$
$710$	0	0
$711$	39.6170	1.48576
$712$	0	0
$713$	−0.591977	−0.0221697
$714$	0	0
$715$	−6.31261	−0.236078
$716$	0	0
$717$	−51.8595	−1.93673
$718$	0	0
$719$	28.0509	1.04612	0.523061	−	0.852295i	$-0.324790\pi$
$719$	28.0509	1.04612	0.523061	+	0.852295i	$0.324790\pi$
$720$	0	0
$721$	12.1780	0.453533
$722$	0	0
$723$	−50.4457	−1.87610
$724$	0	0
$725$	20.8217	0.773300
$726$	0	0
$727$	5.46525	0.202695	0.101347	−	0.994851i	$-0.467685\pi$
$727$	5.46525	0.202695	0.101347	+	0.994851i	$0.467685\pi$
$728$	0	0
$729$	−41.9869	−1.55507
$730$	0	0
$731$	46.6840	1.72667
$732$	0	0
$733$	36.4900	1.34779	0.673895	−	0.738827i	$-0.264620\pi$
$733$	36.4900	1.34779	0.673895	+	0.738827i	$0.264620\pi$
$734$	0	0
$735$	−51.0269	−1.88215
$736$	0	0
$737$	−0.736165	−0.0271170
$738$	0	0
$739$	−4.98096	−0.183227	−0.0916137	−	0.995795i	$-0.529203\pi$
$739$	−4.98096	−0.183227	−0.0916137	+	0.995795i	$0.529203\pi$
$740$	0	0
$741$	39.6527	1.45668
$742$	0	0
$743$	36.7935	1.34982	0.674911	−	0.737899i	$-0.264182\pi$
$743$	36.7935	1.34982	0.674911	+	0.737899i	$0.264182\pi$
$744$	0	0
$745$	−6.56898	−0.240669
$746$	0	0
$747$	22.4638	0.821908
$748$	0	0
$749$	11.6632	0.426164
$750$	0	0
$751$	6.12287	0.223427	0.111713	−	0.993740i	$-0.464366\pi$
$751$	6.12287	0.223427	0.111713	+	0.993740i	$0.464366\pi$
$752$	0	0
$753$	38.3839	1.39879
$754$	0	0
$755$	−38.6163	−1.40539
$756$	0	0
$757$	−11.7003	−0.425255	−0.212627	−	0.977133i	$-0.568202\pi$
$757$	−11.7003	−0.425255	−0.212627	+	0.977133i	$0.568202\pi$
$758$	0	0
$759$	−5.73037	−0.207999
$760$	0	0
$761$	−50.5056	−1.83083	−0.915413	−	0.402515i	$-0.868136\pi$
$761$	−50.5056	−1.83083	−0.915413	+	0.402515i	$0.868136\pi$
$762$	0	0
$763$	10.1840	0.368684
$764$	0	0
$765$	−73.4656	−2.65615
$766$	0	0
$767$	13.5580	0.489552
$768$	0	0
$769$	−44.6678	−1.61076	−0.805381	−	0.592757i	$-0.798039\pi$
$769$	−44.6678	−1.61076	−0.805381	+	0.592757i	$0.798039\pi$
$770$	0	0
$771$	72.2022	2.60030
$772$	0	0
$773$	−39.2915	−1.41322	−0.706608	−	0.707605i	$-0.749776\pi$
$773$	−39.2915	−1.41322	−0.706608	+	0.707605i	$0.749776\pi$
$774$	0	0
$775$	1.01283	0.0363818
$776$	0	0
$777$	−10.1649	−0.364665
$778$	0	0
$779$	36.5148	1.30828
$780$	0	0
$781$	−3.01327	−0.107823
$782$	0	0
$783$	−37.2645	−1.33172
$784$	0	0
$785$	33.3682	1.19096
$786$	0	0
$787$	−1.10481	−0.0393823	−0.0196911	−	0.999806i	$-0.506268\pi$
$787$	−1.10481	−0.0393823	−0.0196911	+	0.999806i	$0.506268\pi$
$788$	0	0
$789$	−30.2617	−1.07734
$790$	0	0
$791$	−7.55896	−0.268766
$792$	0	0
$793$	7.20333	0.255798
$794$	0	0
$795$	−105.699	−3.74877
$796$	0	0
$797$	−18.0646	−0.639881	−0.319941	−	0.947438i	$-0.603663\pi$
$797$	−18.0646	−0.639881	−0.319941	+	0.947438i	$0.603663\pi$
$798$	0	0
$799$	−18.4682	−0.653357
$800$	0	0
$801$	50.8980	1.79839
$802$	0	0
$803$	7.59095	0.267879
$804$	0	0
$805$	−5.27450	−0.185902
$806$	0	0
$807$	−65.2890	−2.29828
$808$	0	0
$809$	48.5352	1.70641	0.853203	−	0.521578i	$-0.174657\pi$
$809$	48.5352	1.70641	0.853203	+	0.521578i	$0.174657\pi$
$810$	0	0
$811$	9.47896	0.332851	0.166426	−	0.986054i	$-0.446777\pi$
$811$	9.47896	0.332851	0.166426	+	0.986054i	$0.446777\pi$
$812$	0	0
$813$	18.4322	0.646447
$814$	0	0
$815$	33.8072	1.18422
$816$	0	0
$817$	60.9396	2.13201
$818$	0	0
$819$	10.1297	0.353960
$820$	0	0
$821$	−1.89558	−0.0661563	−0.0330781	−	0.999453i	$-0.510531\pi$
$821$	−1.89558	−0.0661563	−0.0330781	+	0.999453i	$0.510531\pi$
$822$	0	0
$823$	23.1549	0.807130	0.403565	−	0.914951i	$-0.367771\pi$
$823$	23.1549	0.807130	0.403565	+	0.914951i	$0.367771\pi$
$824$	0	0
$825$	9.80420	0.341339
$826$	0	0
$827$	9.24230	0.321386	0.160693	−	0.987004i	$-0.448627\pi$
$827$	9.24230	0.321386	0.160693	+	0.987004i	$0.448627\pi$
$828$	0	0
$829$	−27.5292	−0.956128	−0.478064	−	0.878325i	$-0.658661\pi$
$829$	−27.5292	−0.956128	−0.478064	+	0.878325i	$0.658661\pi$
$830$	0	0
$831$	35.4728	1.23054
$832$	0	0
$833$	−30.4003	−1.05331
$834$	0	0
$835$	−12.4291	−0.430126
$836$	0	0
$837$	−1.81264	−0.0626541
$838$	0	0
$839$	−20.2998	−0.700827	−0.350413	−	0.936595i	$-0.613959\pi$
$839$	−20.2998	−0.700827	−0.350413	+	0.936595i	$0.613959\pi$
$840$	0	0
$841$	8.27889	0.285479
$842$	0	0
$843$	24.3273	0.837877
$844$	0	0
$845$	24.1690	0.831440
$846$	0	0
$847$	9.11064	0.313045
$848$	0	0
$849$	−87.5964	−3.00630
$850$	0	0
$851$	7.78298	0.266797
$852$	0	0
$853$	−23.8255	−0.815770	−0.407885	−	0.913033i	$-0.633734\pi$
$853$	−23.8255	−0.815770	−0.407885	+	0.913033i	$0.633734\pi$
$854$	0	0
$855$	−95.8993	−3.27969
$856$	0	0
$857$	−4.67993	−0.159863	−0.0799317	−	0.996800i	$-0.525470\pi$
$857$	−4.67993	−0.159863	−0.0799317	+	0.996800i	$0.525470\pi$
$858$	0	0
$859$	2.11829	0.0722751	0.0361375	−	0.999347i	$-0.488495\pi$
$859$	2.11829	0.0722751	0.0361375	+	0.999347i	$0.488495\pi$
$860$	0	0
$861$	14.7733	0.503471
$862$	0	0
$863$	−20.1642	−0.686398	−0.343199	−	0.939263i	$-0.611510\pi$
$863$	−20.1642	−0.686398	−0.343199	+	0.939263i	$0.611510\pi$
$864$	0	0
$865$	13.6044	0.462562
$866$	0	0
$867$	−20.8179	−0.707014
$868$	0	0
$869$	−7.76805	−0.263513
$870$	0	0
$871$	−1.57802	−0.0534693
$872$	0	0
$873$	32.7270	1.10764
$874$	0	0
$875$	−4.20684	−0.142217
$876$	0	0
$877$	−38.3640	−1.29546	−0.647730	−	0.761870i	$-0.724281\pi$
$877$	−38.3640	−1.29546	−0.647730	+	0.761870i	$0.724281\pi$
$878$	0	0
$879$	35.8454	1.20904
$880$	0	0
$881$	−55.2325	−1.86083	−0.930415	−	0.366508i	$-0.880553\pi$
$881$	−55.2325	−1.86083	−0.930415	+	0.366508i	$0.880553\pi$
$882$	0	0
$883$	6.49383	0.218535	0.109267	−	0.994012i	$-0.465150\pi$
$883$	6.49383	0.218535	0.109267	+	0.994012i	$0.465150\pi$
$884$	0	0
$885$	−51.9306	−1.74563
$886$	0	0
$887$	42.4044	1.42380	0.711899	−	0.702281i	$-0.247835\pi$
$887$	42.4044	1.42380	0.711899	+	0.702281i	$0.247835\pi$
$888$	0	0
$889$	−5.60842	−0.188100
$890$	0	0
$891$	−2.00982	−0.0673315
$892$	0	0
$893$	−24.1077	−0.806732
$894$	0	0
$895$	1.17737	0.0393553
$896$	0	0
$897$	−12.2835	−0.410133
$898$	0	0
$899$	1.81334	0.0604784
$900$	0	0
$901$	−62.9725	−2.09792
$902$	0	0
$903$	24.6551	0.820470
$904$	0	0
$905$	23.6689	0.786782
$906$	0	0
$907$	−23.8104	−0.790611	−0.395305	−	0.918550i	$-0.629361\pi$
$907$	−23.8104	−0.790611	−0.395305	+	0.918550i	$0.629361\pi$
$908$	0	0
$909$	−28.3374	−0.939893
$910$	0	0
$911$	−32.8132	−1.08715	−0.543574	−	0.839361i	$-0.682929\pi$
$911$	−32.8132	−1.08715	−0.543574	+	0.839361i	$0.682929\pi$
$912$	0	0
$913$	−4.40467	−0.145773
$914$	0	0
$915$	−27.5905	−0.912113
$916$	0	0
$917$	−3.34512	−0.110466
$918$	0	0
$919$	−27.1972	−0.897153	−0.448576	−	0.893745i	$-0.648069\pi$
$919$	−27.1972	−0.897153	−0.448576	+	0.893745i	$0.648069\pi$
$920$	0	0
$921$	73.1730	2.41113
$922$	0	0
$923$	−6.45916	−0.212606
$924$	0	0
$925$	−13.3161	−0.437829
$926$	0	0
$927$	−68.5896	−2.25278
$928$	0	0
$929$	22.4298	0.735899	0.367949	−	0.929846i	$-0.380060\pi$
$929$	22.4298	0.735899	0.367949	+	0.929846i	$0.380060\pi$
$930$	0	0
$931$	−39.6834	−1.30057
$932$	0	0
$933$	42.6734	1.39706
$934$	0	0
$935$	14.4050	0.471095
$936$	0	0
$937$	15.4495	0.504713	0.252357	−	0.967634i	$-0.418794\pi$
$937$	15.4495	0.504713	0.252357	+	0.967634i	$0.418794\pi$
$938$	0	0
$939$	−45.9808	−1.50053
$940$	0	0
$941$	49.7328	1.62124	0.810621	−	0.585571i	$-0.199130\pi$
$941$	49.7328	1.62124	0.810621	+	0.585571i	$0.199130\pi$
$942$	0	0
$943$	−11.3114	−0.368351
$944$	0	0
$945$	−16.1506	−0.525380
$946$	0	0
$947$	32.2236	1.04713	0.523563	−	0.851987i	$-0.324603\pi$
$947$	32.2236	1.04713	0.523563	+	0.851987i	$0.324603\pi$
$948$	0	0
$949$	16.2718	0.528204
$950$	0	0
$951$	6.03014	0.195541
$952$	0	0
$953$	13.3762	0.433299	0.216649	−	0.976249i	$-0.430487\pi$
$953$	13.3762	0.433299	0.216649	+	0.976249i	$0.430487\pi$
$954$	0	0
$955$	46.0604	1.49048
$956$	0	0
$957$	17.5533	0.567417
$958$	0	0
$959$	−19.6166	−0.633454
$960$	0	0
$961$	−30.9118	−0.997155
$962$	0	0
$963$	−65.6900	−2.11683
$964$	0	0
$965$	68.1283	2.19313
$966$	0	0
$967$	36.7388	1.18144	0.590721	−	0.806876i	$-0.298844\pi$
$967$	36.7388	1.18144	0.590721	+	0.806876i	$0.298844\pi$
$968$	0	0
$969$	−90.4852	−2.90680
$970$	0	0
$971$	48.6218	1.56035	0.780174	−	0.625562i	$-0.215130\pi$
$971$	48.6218	1.56035	0.780174	+	0.625562i	$0.215130\pi$
$972$	0	0
$973$	−6.02271	−0.193079
$974$	0	0
$975$	21.0160	0.673051
$976$	0	0
$977$	23.6105	0.755367	0.377684	−	0.925935i	$-0.376721\pi$
$977$	23.6105	0.755367	0.377684	+	0.925935i	$0.376721\pi$
$978$	0	0
$979$	−9.98000	−0.318962
$980$	0	0
$981$	−57.3586	−1.83132
$982$	0	0
$983$	14.3065	0.456305	0.228153	−	0.973625i	$-0.426732\pi$
$983$	14.3065	0.456305	0.228153	+	0.973625i	$0.426732\pi$
$984$	0	0
$985$	73.0985	2.32911
$986$	0	0
$987$	−9.75354	−0.310459
$988$	0	0
$989$	−18.8777	−0.600275
$990$	0	0
$991$	38.9215	1.23638	0.618192	−	0.786027i	$-0.287866\pi$
$991$	38.9215	1.23638	0.618192	+	0.786027i	$0.287866\pi$
$992$	0	0
$993$	−13.9546	−0.442837
$994$	0	0
$995$	−21.1313	−0.669906
$996$	0	0
$997$	1.46612	0.0464324	0.0232162	−	0.999730i	$-0.492609\pi$
$997$	1.46612	0.0464324	0.0232162	+	0.999730i	$0.492609\pi$
$998$	0	0
$999$	23.8316	0.753999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.6	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.6	✓	87	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)

\(f(q)\)	\(=\)	\(q-2.85294 q^{3} -2.90004 q^{5} -0.912476 q^{7} +5.13929 q^{9} +O(q^{10})\)	\(q-2.85294 q^{3} -2.90004 q^{5} -0.912476 q^{7} +5.13929 q^{9} -1.00770 q^{11} -2.16009 q^{13} +8.27366 q^{15} +4.92920 q^{17} +6.43440 q^{19} +2.60324 q^{21} -1.99322 q^{23} +3.41025 q^{25} -6.10328 q^{27} +6.10564 q^{29} +0.296995 q^{31} +2.87493 q^{33} +2.64622 q^{35} -3.90472 q^{37} +6.16261 q^{39} +5.67494 q^{41} +9.47092 q^{43} -14.9042 q^{45} -3.74669 q^{47} -6.16739 q^{49} -14.0627 q^{51} -12.7754 q^{53} +2.92239 q^{55} -18.3570 q^{57} -6.27661 q^{59} -3.33474 q^{61} -4.68948 q^{63} +6.26434 q^{65} +0.730536 q^{67} +5.68656 q^{69} +2.99023 q^{71} -7.53292 q^{73} -9.72924 q^{75} +0.919506 q^{77} +7.70866 q^{79} +1.99445 q^{81} +4.37099 q^{83} -14.2949 q^{85} -17.4191 q^{87} +9.90370 q^{89} +1.97103 q^{91} -0.847310 q^{93} -18.6600 q^{95} +6.36799 q^{97} -5.17889 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * q^99

Introduction

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