LMFDB - Embedded newform 8044.2.a.a.1.4

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			1.4
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)

$f(q)$	$=$	$q-3.15577 q^{3} +3.07374 q^{5} -1.51261 q^{7} +6.95887 q^{9} +O(q^{10})$	$q-3.15577 q^{3} +3.07374 q^{5} -1.51261 q^{7} +6.95887 q^{9} -6.33973 q^{11} -1.59377 q^{13} -9.70001 q^{15} -1.36121 q^{17} +0.554980 q^{19} +4.77344 q^{21} -2.03971 q^{23} +4.44788 q^{25} -12.4933 q^{27} +10.4845 q^{29} +2.47173 q^{31} +20.0067 q^{33} -4.64936 q^{35} -3.38357 q^{37} +5.02956 q^{39} +0.114170 q^{41} +12.1367 q^{43} +21.3898 q^{45} +0.356421 q^{47} -4.71202 q^{49} +4.29567 q^{51} +12.5157 q^{53} -19.4867 q^{55} -1.75139 q^{57} -5.07719 q^{59} -7.98764 q^{61} -10.5260 q^{63} -4.89882 q^{65} +7.50950 q^{67} +6.43686 q^{69} +10.9419 q^{71} -7.23541 q^{73} -14.0365 q^{75} +9.58952 q^{77} +4.09540 q^{79} +18.5493 q^{81} -7.53388 q^{83} -4.18401 q^{85} -33.0868 q^{87} -6.16492 q^{89} +2.41074 q^{91} -7.80021 q^{93} +1.70586 q^{95} +3.78596 q^{97} -44.1174 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.15577	−1.82198	−0.910992	−	0.412424i	$-0.864682\pi$
$3$	−3.15577	−1.82198	−0.910992	+	0.412424i	$0.864682\pi$
$4$	0	0
$5$	3.07374	1.37462	0.687309	−	0.726365i	$-0.258792\pi$
$5$	3.07374	1.37462	0.687309	+	0.726365i	$0.258792\pi$
$6$	0	0
$7$	−1.51261	−0.571712	−0.285856	−	0.958273i	$-0.592278\pi$
$7$	−1.51261	−0.571712	−0.285856	+	0.958273i	$0.592278\pi$
$8$	0	0
$9$	6.95887	2.31962
$10$	0	0
$11$	−6.33973	−1.91150	−0.955750	−	0.294181i	$-0.904953\pi$
$11$	−6.33973	−1.91150	−0.955750	+	0.294181i	$0.904953\pi$
$12$	0	0
$13$	−1.59377	−0.442031	−0.221016	−	0.975270i	$-0.570937\pi$
$13$	−1.59377	−0.442031	−0.221016	+	0.975270i	$0.570937\pi$
$14$	0	0
$15$	−9.70001	−2.50453
$16$	0	0
$17$	−1.36121	−0.330142	−0.165071	−	0.986282i	$-0.552785\pi$
$17$	−1.36121	−0.330142	−0.165071	+	0.986282i	$0.552785\pi$
$18$	0	0
$19$	0.554980	0.127321	0.0636606	−	0.997972i	$-0.479723\pi$
$19$	0.554980	0.127321	0.0636606	+	0.997972i	$0.479723\pi$
$20$	0	0
$21$	4.77344	1.04165
$22$	0	0
$23$	−2.03971	−0.425310	−0.212655	−	0.977127i	$-0.568211\pi$
$23$	−2.03971	−0.425310	−0.212655	+	0.977127i	$0.568211\pi$
$24$	0	0
$25$	4.44788	0.889575
$26$	0	0
$27$	−12.4933	−2.40433
$28$	0	0
$29$	10.4845	1.94693	0.973465	−	0.228834i	$-0.0734914\pi$
$29$	10.4845	1.94693	0.973465	+	0.228834i	$0.0734914\pi$
$30$	0	0
$31$	2.47173	0.443936	0.221968	−	0.975054i	$-0.428752\pi$
$31$	2.47173	0.443936	0.221968	+	0.975054i	$0.428752\pi$
$32$	0	0
$33$	20.0067	3.48272
$34$	0	0
$35$	−4.64936	−0.785886
$36$	0	0
$37$	−3.38357	−0.556255	−0.278128	−	0.960544i	$-0.589714\pi$
$37$	−3.38357	−0.556255	−0.278128	+	0.960544i	$0.589714\pi$
$38$	0	0
$39$	5.02956	0.805374
$40$	0	0
$41$	0.114170	0.0178303	0.00891516	−	0.999960i	$-0.497162\pi$
$41$	0.114170	0.0178303	0.00891516	+	0.999960i	$0.497162\pi$
$42$	0	0
$43$	12.1367	1.85083	0.925414	−	0.378959i	$-0.123718\pi$
$43$	12.1367	1.85083	0.925414	+	0.378959i	$0.123718\pi$
$44$	0	0
$45$	21.3898	3.18860
$46$	0	0
$47$	0.356421	0.0519893	0.0259947	−	0.999662i	$-0.491725\pi$
$47$	0.356421	0.0519893	0.0259947	+	0.999662i	$0.491725\pi$
$48$	0	0
$49$	−4.71202	−0.673146
$50$	0	0
$51$	4.29567	0.601514
$52$	0	0
$53$	12.5157	1.71917	0.859583	−	0.510997i	$-0.170724\pi$
$53$	12.5157	1.71917	0.859583	+	0.510997i	$0.170724\pi$
$54$	0	0
$55$	−19.4867	−2.62758
$56$	0	0
$57$	−1.75139	−0.231977
$58$	0	0
$59$	−5.07719	−0.660994	−0.330497	−	0.943807i	$-0.607216\pi$
$59$	−5.07719	−0.660994	−0.330497	+	0.943807i	$0.607216\pi$
$60$	0	0
$61$	−7.98764	−1.02271	−0.511356	−	0.859369i	$-0.670857\pi$
$61$	−7.98764	−1.02271	−0.511356	+	0.859369i	$0.670857\pi$
$62$	0	0
$63$	−10.5260	−1.32616
$64$	0	0
$65$	−4.89882	−0.607624
$66$	0	0
$67$	7.50950	0.917432	0.458716	−	0.888583i	$-0.348310\pi$
$67$	7.50950	0.917432	0.458716	+	0.888583i	$0.348310\pi$
$68$	0	0
$69$	6.43686	0.774907
$70$	0	0
$71$	10.9419	1.29856	0.649282	−	0.760548i	$-0.275070\pi$
$71$	10.9419	1.29856	0.649282	+	0.760548i	$0.275070\pi$
$72$	0	0
$73$	−7.23541	−0.846841	−0.423420	−	0.905933i	$-0.639171\pi$
$73$	−7.23541	−0.846841	−0.423420	+	0.905933i	$0.639171\pi$
$74$	0	0
$75$	−14.0365	−1.62079
$76$	0	0
$77$	9.58952	1.09283
$78$	0	0
$79$	4.09540	0.460769	0.230384	−	0.973100i	$-0.426002\pi$
$79$	4.09540	0.460769	0.230384	+	0.973100i	$0.426002\pi$
$80$	0	0
$81$	18.5493	2.06103
$82$	0	0
$83$	−7.53388	−0.826951	−0.413476	−	0.910515i	$-0.635685\pi$
$83$	−7.53388	−0.826951	−0.413476	+	0.910515i	$0.635685\pi$
$84$	0	0
$85$	−4.18401	−0.453820
$86$	0	0
$87$	−33.0868	−3.54728
$88$	0	0
$89$	−6.16492	−0.653480	−0.326740	−	0.945114i	$-0.605950\pi$
$89$	−6.16492	−0.653480	−0.326740	+	0.945114i	$0.605950\pi$
$90$	0	0
$91$	2.41074	0.252714
$92$	0	0
$93$	−7.80021	−0.808844
$94$	0	0
$95$	1.70586	0.175018
$96$	0	0
$97$	3.78596	0.384406	0.192203	−	0.981355i	$-0.438437\pi$
$97$	3.78596	0.384406	0.192203	+	0.981355i	$0.438437\pi$
$98$	0	0
$99$	−44.1174	−4.43396
$100$	0	0
$101$	−18.3508	−1.82597	−0.912987	−	0.407989i	$-0.866230\pi$
$101$	−18.3508	−1.82597	−0.912987	+	0.407989i	$0.866230\pi$
$102$	0	0
$103$	5.52613	0.544506	0.272253	−	0.962226i	$-0.412231\pi$
$103$	5.52613	0.544506	0.272253	+	0.962226i	$0.412231\pi$
$104$	0	0
$105$	14.6723	1.43187
$106$	0	0
$107$	−3.80857	−0.368189	−0.184094	−	0.982909i	$-0.558935\pi$
$107$	−3.80857	−0.368189	−0.184094	+	0.982909i	$0.558935\pi$
$108$	0	0
$109$	7.84197	0.751124	0.375562	−	0.926797i	$-0.377450\pi$
$109$	7.84197	0.751124	0.375562	+	0.926797i	$0.377450\pi$
$110$	0	0
$111$	10.6778	1.01349
$112$	0	0
$113$	13.9006	1.30766	0.653828	−	0.756643i	$-0.273162\pi$
$113$	13.9006	1.30766	0.653828	+	0.756643i	$0.273162\pi$
$114$	0	0
$115$	−6.26955	−0.584639
$116$	0	0
$117$	−11.0908	−1.02535
$118$	0	0
$119$	2.05898	0.188746
$120$	0	0
$121$	29.1921	2.65383
$122$	0	0
$123$	−0.360293	−0.0324865
$124$	0	0
$125$	−1.69708	−0.151792
$126$	0	0
$127$	−7.13896	−0.633480	−0.316740	−	0.948512i	$-0.602588\pi$
$127$	−7.13896	−0.633480	−0.316740	+	0.948512i	$0.602588\pi$
$128$	0	0
$129$	−38.3006	−3.37218
$130$	0	0
$131$	−16.6704	−1.45650	−0.728248	−	0.685314i	$-0.759665\pi$
$131$	−16.6704	−1.45650	−0.728248	+	0.685314i	$0.759665\pi$
$132$	0	0
$133$	−0.839467	−0.0727910
$134$	0	0
$135$	−38.4011	−3.30504
$136$	0	0
$137$	1.48511	0.126881	0.0634407	−	0.997986i	$-0.479793\pi$
$137$	1.48511	0.126881	0.0634407	+	0.997986i	$0.479793\pi$
$138$	0	0
$139$	−11.0744	−0.939321	−0.469661	−	0.882847i	$-0.655624\pi$
$139$	−11.0744	−0.939321	−0.469661	+	0.882847i	$0.655624\pi$
$140$	0	0
$141$	−1.12478	−0.0947237
$142$	0	0
$143$	10.1040	0.844942
$144$	0	0
$145$	32.2268	2.67629
$146$	0	0
$147$	14.8700	1.22646
$148$	0	0
$149$	−13.5938	−1.11365	−0.556825	−	0.830630i	$-0.687981\pi$
$149$	−13.5938	−1.11365	−0.556825	+	0.830630i	$0.687981\pi$
$150$	0	0
$151$	−22.2367	−1.80960	−0.904798	−	0.425841i	$-0.859978\pi$
$151$	−22.2367	−1.80960	−0.904798	+	0.425841i	$0.859978\pi$
$152$	0	0
$153$	−9.47250	−0.765806
$154$	0	0
$155$	7.59746	0.610243
$156$	0	0
$157$	−13.9080	−1.10998	−0.554990	−	0.831857i	$-0.687278\pi$
$157$	−13.9080	−1.10998	−0.554990	+	0.831857i	$0.687278\pi$
$158$	0	0
$159$	−39.4967	−3.13229
$160$	0	0
$161$	3.08529	0.243155
$162$	0	0
$163$	11.7916	0.923587	0.461793	−	0.886987i	$-0.347206\pi$
$163$	11.7916	0.923587	0.461793	+	0.886987i	$0.347206\pi$
$164$	0	0
$165$	61.4954	4.78741
$166$	0	0
$167$	18.1061	1.40109	0.700546	−	0.713608i	$-0.252940\pi$
$167$	18.1061	1.40109	0.700546	+	0.713608i	$0.252940\pi$
$168$	0	0
$169$	−10.4599	−0.804608
$170$	0	0
$171$	3.86204	0.295337
$172$	0	0
$173$	18.6519	1.41807	0.709037	−	0.705171i	$-0.249130\pi$
$173$	18.6519	1.41807	0.709037	+	0.705171i	$0.249130\pi$
$174$	0	0
$175$	−6.72789	−0.508581
$176$	0	0
$177$	16.0224	1.20432
$178$	0	0
$179$	11.8764	0.887687	0.443844	−	0.896104i	$-0.353615\pi$
$179$	11.8764	0.887687	0.443844	+	0.896104i	$0.353615\pi$
$180$	0	0
$181$	21.0721	1.56628	0.783140	−	0.621846i	$-0.213617\pi$
$181$	21.0721	1.56628	0.783140	+	0.621846i	$0.213617\pi$
$182$	0	0
$183$	25.2071	1.86337
$184$	0	0
$185$	−10.4002	−0.764638
$186$	0	0
$187$	8.62971	0.631067
$188$	0	0
$189$	18.8974	1.37459
$190$	0	0
$191$	−6.21620	−0.449788	−0.224894	−	0.974383i	$-0.572204\pi$
$191$	−6.21620	−0.449788	−0.224894	+	0.974383i	$0.572204\pi$
$192$	0	0
$193$	0.738191	0.0531362	0.0265681	−	0.999647i	$-0.491542\pi$
$193$	0.738191	0.0531362	0.0265681	+	0.999647i	$0.491542\pi$
$194$	0	0
$195$	15.4595	1.10708
$196$	0	0
$197$	8.70255	0.620031	0.310016	−	0.950731i	$-0.399666\pi$
$197$	8.70255	0.620031	0.310016	+	0.950731i	$0.399666\pi$
$198$	0	0
$199$	−27.4878	−1.94856	−0.974278	−	0.225348i	$-0.927648\pi$
$199$	−27.4878	−1.94856	−0.974278	+	0.225348i	$0.927648\pi$
$200$	0	0
$201$	−23.6983	−1.67155
$202$	0	0
$203$	−15.8590	−1.11308
$204$	0	0
$205$	0.350928	0.0245099
$206$	0	0
$207$	−14.1941	−0.986559
$208$	0	0
$209$	−3.51842	−0.243374
$210$	0	0
$211$	1.65556	0.113974	0.0569869	−	0.998375i	$-0.481851\pi$
$211$	1.65556	0.113974	0.0569869	+	0.998375i	$0.481851\pi$
$212$	0	0
$213$	−34.5301	−2.36596
$214$	0	0
$215$	37.3050	2.54418
$216$	0	0
$217$	−3.73876	−0.253803
$218$	0	0
$219$	22.8333	1.54293
$220$	0	0
$221$	2.16945	0.145933
$222$	0	0
$223$	18.4942	1.23846	0.619232	−	0.785208i	$-0.287444\pi$
$223$	18.4942	1.23846	0.619232	+	0.785208i	$0.287444\pi$
$224$	0	0
$225$	30.9522	2.06348
$226$	0	0
$227$	6.96075	0.462001	0.231001	−	0.972954i	$-0.425800\pi$
$227$	6.96075	0.462001	0.231001	+	0.972954i	$0.425800\pi$
$228$	0	0
$229$	17.3783	1.14839	0.574196	−	0.818718i	$-0.305315\pi$
$229$	17.3783	1.14839	0.574196	+	0.818718i	$0.305315\pi$
$230$	0	0
$231$	−30.2623	−1.99111
$232$	0	0
$233$	−24.8858	−1.63032	−0.815162	−	0.579233i	$-0.803352\pi$
$233$	−24.8858	−1.63032	−0.815162	+	0.579233i	$0.803352\pi$
$234$	0	0
$235$	1.09554	0.0714654
$236$	0	0
$237$	−12.9241	−0.839513
$238$	0	0
$239$	−17.8194	−1.15264	−0.576319	−	0.817225i	$-0.695512\pi$
$239$	−17.8194	−1.15264	−0.576319	+	0.817225i	$0.695512\pi$
$240$	0	0
$241$	−18.7179	−1.20573	−0.602864	−	0.797844i	$-0.705974\pi$
$241$	−18.7179	−1.20573	−0.602864	+	0.797844i	$0.705974\pi$
$242$	0	0
$243$	−21.0574	−1.35084
$244$	0	0
$245$	−14.4835	−0.925318
$246$	0	0
$247$	−0.884508	−0.0562799
$248$	0	0
$249$	23.7752	1.50669
$250$	0	0
$251$	15.9228	1.00504	0.502518	−	0.864567i	$-0.332407\pi$
$251$	15.9228	1.00504	0.502518	+	0.864567i	$0.332407\pi$
$252$	0	0
$253$	12.9312	0.812979
$254$	0	0
$255$	13.2038	0.826852
$256$	0	0
$257$	−5.26767	−0.328588	−0.164294	−	0.986411i	$-0.552535\pi$
$257$	−5.26767	−0.328588	−0.164294	+	0.986411i	$0.552535\pi$
$258$	0	0
$259$	5.11801	0.318018
$260$	0	0
$261$	72.9606	4.51615
$262$	0	0
$263$	22.5311	1.38933	0.694663	−	0.719335i	$-0.255553\pi$
$263$	22.5311	1.38933	0.694663	+	0.719335i	$0.255553\pi$
$264$	0	0
$265$	38.4700	2.36320
$266$	0	0
$267$	19.4550	1.19063
$268$	0	0
$269$	−18.9891	−1.15779	−0.578894	−	0.815403i	$-0.696515\pi$
$269$	−18.9891	−1.15779	−0.578894	+	0.815403i	$0.696515\pi$
$270$	0	0
$271$	−7.26849	−0.441529	−0.220765	−	0.975327i	$-0.570855\pi$
$271$	−7.26849	−0.441529	−0.220765	+	0.975327i	$0.570855\pi$
$272$	0	0
$273$	−7.60775	−0.460442
$274$	0	0
$275$	−28.1983	−1.70042
$276$	0	0
$277$	22.4936	1.35151	0.675756	−	0.737126i	$-0.263817\pi$
$277$	22.4936	1.35151	0.675756	+	0.737126i	$0.263817\pi$
$278$	0	0
$279$	17.2005	1.02976
$280$	0	0
$281$	−16.8479	−1.00506	−0.502530	−	0.864560i	$-0.667597\pi$
$281$	−16.8479	−1.00506	−0.502530	+	0.864560i	$0.667597\pi$
$282$	0	0
$283$	−5.69126	−0.338310	−0.169155	−	0.985589i	$-0.554104\pi$
$283$	−5.69126	−0.338310	−0.169155	+	0.985589i	$0.554104\pi$
$284$	0	0
$285$	−5.38331	−0.318880
$286$	0	0
$287$	−0.172694	−0.0101938
$288$	0	0
$289$	−15.1471	−0.891006
$290$	0	0
$291$	−11.9476	−0.700381
$292$	0	0
$293$	−23.9889	−1.40145	−0.700725	−	0.713432i	$-0.747140\pi$
$293$	−23.9889	−1.40145	−0.700725	+	0.713432i	$0.747140\pi$
$294$	0	0
$295$	−15.6060	−0.908615
$296$	0	0
$297$	79.2040	4.59588
$298$	0	0
$299$	3.25083	0.188000
$300$	0	0
$301$	−18.3580	−1.05814
$302$	0	0
$303$	57.9109	3.32689
$304$	0	0
$305$	−24.5519	−1.40584
$306$	0	0
$307$	−10.8734	−0.620580	−0.310290	−	0.950642i	$-0.600426\pi$
$307$	−10.8734	−0.620580	−0.310290	+	0.950642i	$0.600426\pi$
$308$	0	0
$309$	−17.4392	−0.992081
$310$	0	0
$311$	8.71606	0.494243	0.247121	−	0.968985i	$-0.420515\pi$
$311$	8.71606	0.494243	0.247121	+	0.968985i	$0.420515\pi$
$312$	0	0
$313$	2.81049	0.158858	0.0794292	−	0.996841i	$-0.474690\pi$
$313$	2.81049	0.158858	0.0794292	+	0.996841i	$0.474690\pi$
$314$	0	0
$315$	−32.3543	−1.82296
$316$	0	0
$317$	−24.7363	−1.38933	−0.694665	−	0.719333i	$-0.744448\pi$
$317$	−24.7363	−1.38933	−0.694665	+	0.719333i	$0.744448\pi$
$318$	0	0
$319$	−66.4691	−3.72156
$320$	0	0
$321$	12.0190	0.670834
$322$	0	0
$323$	−0.755445	−0.0420341
$324$	0	0
$325$	−7.08888	−0.393220
$326$	0	0
$327$	−24.7474	−1.36854
$328$	0	0
$329$	−0.539125	−0.0297229
$330$	0	0
$331$	20.0596	1.10258	0.551289	−	0.834314i	$-0.314136\pi$
$331$	20.0596	1.10258	0.551289	+	0.834314i	$0.314136\pi$
$332$	0	0
$333$	−23.5458	−1.29030
$334$	0	0
$335$	23.0823	1.26112
$336$	0	0
$337$	7.13470	0.388652	0.194326	−	0.980937i	$-0.437748\pi$
$337$	7.13470	0.388652	0.194326	+	0.980937i	$0.437748\pi$
$338$	0	0
$339$	−43.8670	−2.38253
$340$	0	0
$341$	−15.6701	−0.848583
$342$	0	0
$343$	17.7157	0.956557
$344$	0	0
$345$	19.7852	1.06520
$346$	0	0
$347$	10.1683	0.545862	0.272931	−	0.962034i	$-0.412007\pi$
$347$	10.1683	0.545862	0.272931	+	0.962034i	$0.412007\pi$
$348$	0	0
$349$	14.4794	0.775065	0.387532	−	0.921856i	$-0.373328\pi$
$349$	14.4794	0.775065	0.387532	+	0.921856i	$0.373328\pi$
$350$	0	0
$351$	19.9114	1.06279
$352$	0	0
$353$	−2.16832	−0.115408	−0.0577039	−	0.998334i	$-0.518378\pi$
$353$	−2.16832	−0.115408	−0.0577039	+	0.998334i	$0.518378\pi$
$354$	0	0
$355$	33.6325	1.78503
$356$	0	0
$357$	−6.49766	−0.343893
$358$	0	0
$359$	−17.5743	−0.927538	−0.463769	−	0.885956i	$-0.653503\pi$
$359$	−17.5743	−0.927538	−0.463769	+	0.885956i	$0.653503\pi$
$360$	0	0
$361$	−18.6920	−0.983789
$362$	0	0
$363$	−92.1236	−4.83523
$364$	0	0
$365$	−22.2398	−1.16408
$366$	0	0
$367$	−27.5733	−1.43932	−0.719658	−	0.694329i	$-0.755701\pi$
$367$	−27.5733	−1.43932	−0.719658	+	0.694329i	$0.755701\pi$
$368$	0	0
$369$	0.794492	0.0413596
$370$	0	0
$371$	−18.9314	−0.982867
$372$	0	0
$373$	−31.7182	−1.64231	−0.821154	−	0.570707i	$-0.806669\pi$
$373$	−31.7182	−1.64231	−0.821154	+	0.570707i	$0.806669\pi$
$374$	0	0
$375$	5.35559	0.276562
$376$	0	0
$377$	−16.7099	−0.860604
$378$	0	0
$379$	0.772693	0.0396906	0.0198453	−	0.999803i	$-0.493683\pi$
$379$	0.772693	0.0396906	0.0198453	+	0.999803i	$0.493683\pi$
$380$	0	0
$381$	22.5289	1.15419
$382$	0	0
$383$	−1.26029	−0.0643979	−0.0321990	−	0.999481i	$-0.510251\pi$
$383$	−1.26029	−0.0643979	−0.0321990	+	0.999481i	$0.510251\pi$
$384$	0	0
$385$	29.4757	1.50222
$386$	0	0
$387$	84.4577	4.29322
$388$	0	0
$389$	−17.5772	−0.891198	−0.445599	−	0.895233i	$-0.647009\pi$
$389$	−17.5772	−0.891198	−0.445599	+	0.895233i	$0.647009\pi$
$390$	0	0
$391$	2.77648	0.140413
$392$	0	0
$393$	52.6078	2.65371
$394$	0	0
$395$	12.5882	0.633381
$396$	0	0
$397$	3.08790	0.154977	0.0774885	−	0.996993i	$-0.475310\pi$
$397$	3.08790	0.154977	0.0774885	+	0.996993i	$0.475310\pi$
$398$	0	0
$399$	2.64916	0.132624
$400$	0	0
$401$	13.8374	0.691007	0.345504	−	0.938417i	$-0.387708\pi$
$401$	13.8374	0.691007	0.345504	+	0.938417i	$0.387708\pi$
$402$	0	0
$403$	−3.93936	−0.196234
$404$	0	0
$405$	57.0157	2.83313
$406$	0	0
$407$	21.4509	1.06328
$408$	0	0
$409$	−4.34001	−0.214600	−0.107300	−	0.994227i	$-0.534220\pi$
$409$	−4.34001	−0.214600	−0.107300	+	0.994227i	$0.534220\pi$
$410$	0	0
$411$	−4.68666	−0.231176
$412$	0	0
$413$	7.67980	0.377898
$414$	0	0
$415$	−23.1572	−1.13674
$416$	0	0
$417$	34.9483	1.71143
$418$	0	0
$419$	8.17285	0.399270	0.199635	−	0.979870i	$-0.436024\pi$
$419$	8.17285	0.399270	0.199635	+	0.979870i	$0.436024\pi$
$420$	0	0
$421$	15.1374	0.737750	0.368875	−	0.929479i	$-0.379743\pi$
$421$	15.1374	0.737750	0.368875	+	0.929479i	$0.379743\pi$
$422$	0	0
$423$	2.48029	0.120596
$424$	0	0
$425$	−6.05450	−0.293686
$426$	0	0
$427$	12.0822	0.584697
$428$	0	0
$429$	−31.8860	−1.53947
$430$	0	0
$431$	−4.39108	−0.211511	−0.105755	−	0.994392i	$-0.533726\pi$
$431$	−4.39108	−0.211511	−0.105755	+	0.994392i	$0.533726\pi$
$432$	0	0
$433$	−6.54405	−0.314487	−0.157244	−	0.987560i	$-0.550261\pi$
$433$	−6.54405	−0.314487	−0.157244	+	0.987560i	$0.550261\pi$
$434$	0	0
$435$	−101.700	−4.87615
$436$	0	0
$437$	−1.13200	−0.0541509
$438$	0	0
$439$	−7.04653	−0.336312	−0.168156	−	0.985760i	$-0.553781\pi$
$439$	−7.04653	−0.336312	−0.168156	+	0.985760i	$0.553781\pi$
$440$	0	0
$441$	−32.7903	−1.56144
$442$	0	0
$443$	−6.59704	−0.313435	−0.156717	−	0.987643i	$-0.550091\pi$
$443$	−6.59704	−0.313435	−0.156717	+	0.987643i	$0.550091\pi$
$444$	0	0
$445$	−18.9494	−0.898285
$446$	0	0
$447$	42.8990	2.02905
$448$	0	0
$449$	−35.8404	−1.69141	−0.845707	−	0.533648i	$-0.820821\pi$
$449$	−35.8404	−1.69141	−0.845707	+	0.533648i	$0.820821\pi$
$450$	0	0
$451$	−0.723805	−0.0340826
$452$	0	0
$453$	70.1738	3.29705
$454$	0	0
$455$	7.41000	0.347386
$456$	0	0
$457$	−8.92394	−0.417444	−0.208722	−	0.977975i	$-0.566930\pi$
$457$	−8.92394	−0.417444	−0.208722	+	0.977975i	$0.566930\pi$
$458$	0	0
$459$	17.0060	0.793772
$460$	0	0
$461$	8.60872	0.400948	0.200474	−	0.979699i	$-0.435752\pi$
$461$	8.60872	0.400948	0.200474	+	0.979699i	$0.435752\pi$
$462$	0	0
$463$	19.7570	0.918185	0.459093	−	0.888388i	$-0.348175\pi$
$463$	19.7570	0.918185	0.459093	+	0.888388i	$0.348175\pi$
$464$	0	0
$465$	−23.9758	−1.11185
$466$	0	0
$467$	−23.9665	−1.10904	−0.554519	−	0.832171i	$-0.687098\pi$
$467$	−23.9665	−1.10904	−0.554519	+	0.832171i	$0.687098\pi$
$468$	0	0
$469$	−11.3589	−0.524507
$470$	0	0
$471$	43.8905	2.02237
$472$	0	0
$473$	−76.9433	−3.53785
$474$	0	0
$475$	2.46848	0.113262
$476$	0	0
$477$	87.0952	3.98782
$478$	0	0
$479$	−27.8199	−1.27112	−0.635562	−	0.772050i	$-0.719231\pi$
$479$	−27.8199	−1.27112	−0.635562	+	0.772050i	$0.719231\pi$
$480$	0	0
$481$	5.39262	0.245882
$482$	0	0
$483$	−9.73645	−0.443024
$484$	0	0
$485$	11.6370	0.528411
$486$	0	0
$487$	−20.9884	−0.951076	−0.475538	−	0.879695i	$-0.657747\pi$
$487$	−20.9884	−0.951076	−0.475538	+	0.879695i	$0.657747\pi$
$488$	0	0
$489$	−37.2115	−1.68276
$490$	0	0
$491$	−7.59924	−0.342949	−0.171474	−	0.985189i	$-0.554853\pi$
$491$	−7.59924	−0.342949	−0.171474	+	0.985189i	$0.554853\pi$
$492$	0	0
$493$	−14.2717	−0.642764
$494$	0	0
$495$	−135.605	−6.09500
$496$	0	0
$497$	−16.5508	−0.742404
$498$	0	0
$499$	10.9311	0.489343	0.244671	−	0.969606i	$-0.421320\pi$
$499$	10.9311	0.489343	0.244671	+	0.969606i	$0.421320\pi$
$500$	0	0
$501$	−57.1386	−2.55277
$502$	0	0
$503$	−39.0892	−1.74290	−0.871451	−	0.490483i	$-0.836820\pi$
$503$	−39.0892	−1.74290	−0.871451	+	0.490483i	$0.836820\pi$
$504$	0	0
$505$	−56.4056	−2.51002
$506$	0	0
$507$	33.0091	1.46598
$508$	0	0
$509$	−14.1037	−0.625133	−0.312567	−	0.949896i	$-0.601189\pi$
$509$	−14.1037	−0.625133	−0.312567	+	0.949896i	$0.601189\pi$
$510$	0	0
$511$	10.9443	0.484149
$512$	0	0
$513$	−6.93352	−0.306123
$514$	0	0
$515$	16.9859	0.748488
$516$	0	0
$517$	−2.25961	−0.0993775
$518$	0	0
$519$	−58.8610	−2.58371
$520$	0	0
$521$	19.7803	0.866589	0.433295	−	0.901252i	$-0.357351\pi$
$521$	19.7803	0.866589	0.433295	+	0.901252i	$0.357351\pi$
$522$	0	0
$523$	−5.74807	−0.251346	−0.125673	−	0.992072i	$-0.540109\pi$
$523$	−5.74807	−0.251346	−0.125673	+	0.992072i	$0.540109\pi$
$524$	0	0
$525$	21.2317	0.926626
$526$	0	0
$527$	−3.36455	−0.146562
$528$	0	0
$529$	−18.8396	−0.819112
$530$	0	0
$531$	−35.3315	−1.53326
$532$	0	0
$533$	−0.181960	−0.00788155
$534$	0	0
$535$	−11.7066	−0.506119
$536$	0	0
$537$	−37.4793	−1.61735
$538$	0	0
$539$	29.8729	1.28672
$540$	0	0
$541$	−35.4196	−1.52281	−0.761404	−	0.648278i	$-0.775489\pi$
$541$	−35.4196	−1.52281	−0.761404	+	0.648278i	$0.775489\pi$
$542$	0	0
$543$	−66.4988	−2.85374
$544$	0	0
$545$	24.1042	1.03251
$546$	0	0
$547$	−39.9130	−1.70656	−0.853279	−	0.521455i	$-0.825390\pi$
$547$	−39.9130	−1.70656	−0.853279	+	0.521455i	$0.825390\pi$
$548$	0	0
$549$	−55.5850	−2.37231
$550$	0	0
$551$	5.81871	0.247885
$552$	0	0
$553$	−6.19474	−0.263427
$554$	0	0
$555$	32.8206	1.39316
$556$	0	0
$557$	28.3569	1.20152	0.600760	−	0.799430i	$-0.294865\pi$
$557$	28.3569	1.20152	0.600760	+	0.799430i	$0.294865\pi$
$558$	0	0
$559$	−19.3430	−0.818123
$560$	0	0
$561$	−27.2334	−1.14979
$562$	0	0
$563$	−39.7920	−1.67703	−0.838516	−	0.544877i	$-0.816576\pi$
$563$	−39.7920	−1.67703	−0.838516	+	0.544877i	$0.816576\pi$
$564$	0	0
$565$	42.7267	1.79753
$566$	0	0
$567$	−28.0578	−1.17832
$568$	0	0
$569$	15.8735	0.665451	0.332726	−	0.943024i	$-0.392032\pi$
$569$	15.8735	0.665451	0.332726	+	0.943024i	$0.392032\pi$
$570$	0	0
$571$	−40.0531	−1.67617	−0.838086	−	0.545538i	$-0.816325\pi$
$571$	−40.0531	−1.67617	−0.838086	+	0.545538i	$0.816325\pi$
$572$	0	0
$573$	19.6169	0.819507
$574$	0	0
$575$	−9.07240	−0.378345
$576$	0	0
$577$	−24.5682	−1.02279	−0.511395	−	0.859346i	$-0.670871\pi$
$577$	−24.5682	−1.02279	−0.511395	+	0.859346i	$0.670871\pi$
$578$	0	0
$579$	−2.32956	−0.0968132
$580$	0	0
$581$	11.3958	0.472778
$582$	0	0
$583$	−79.3462	−3.28618
$584$	0	0
$585$	−34.0903	−1.40946
$586$	0	0
$587$	38.2447	1.57853	0.789263	−	0.614055i	$-0.210463\pi$
$587$	38.2447	1.57853	0.789263	+	0.614055i	$0.210463\pi$
$588$	0	0
$589$	1.37176	0.0565224
$590$	0	0
$591$	−27.4632	−1.12969
$592$	0	0
$593$	−20.1171	−0.826112	−0.413056	−	0.910706i	$-0.635539\pi$
$593$	−20.1171	−0.826112	−0.413056	+	0.910706i	$0.635539\pi$
$594$	0	0
$595$	6.32876	0.259454
$596$	0	0
$597$	86.7450	3.55024
$598$	0	0
$599$	−9.60522	−0.392459	−0.196229	−	0.980558i	$-0.562870\pi$
$599$	−9.60522	−0.392459	−0.196229	+	0.980558i	$0.562870\pi$
$600$	0	0
$601$	23.6533	0.964840	0.482420	−	0.875940i	$-0.339758\pi$
$601$	23.6533	0.964840	0.482420	+	0.875940i	$0.339758\pi$
$602$	0	0
$603$	52.2577	2.12810
$604$	0	0
$605$	89.7290	3.64800
$606$	0	0
$607$	−15.9837	−0.648757	−0.324379	−	0.945927i	$-0.605155\pi$
$607$	−15.9837	−0.648757	−0.324379	+	0.945927i	$0.605155\pi$
$608$	0	0
$609$	50.0473	2.02802
$610$	0	0
$611$	−0.568051	−0.0229809
$612$	0	0
$613$	−18.7773	−0.758410	−0.379205	−	0.925313i	$-0.623802\pi$
$613$	−18.7773	−0.758410	−0.379205	+	0.925313i	$0.623802\pi$
$614$	0	0
$615$	−1.10745	−0.0446566
$616$	0	0
$617$	−17.5542	−0.706706	−0.353353	−	0.935490i	$-0.614959\pi$
$617$	−17.5542	−0.706706	−0.353353	+	0.935490i	$0.614959\pi$
$618$	0	0
$619$	−18.6163	−0.748254	−0.374127	−	0.927378i	$-0.622058\pi$
$619$	−18.6163	−0.748254	−0.374127	+	0.927378i	$0.622058\pi$
$620$	0	0
$621$	25.4827	1.02259
$622$	0	0
$623$	9.32510	0.373602
$624$	0	0
$625$	−27.4558	−1.09823
$626$	0	0
$627$	11.1033	0.443424
$628$	0	0
$629$	4.60575	0.183643
$630$	0	0
$631$	16.4381	0.654389	0.327194	−	0.944957i	$-0.393897\pi$
$631$	16.4381	0.654389	0.327194	+	0.944957i	$0.393897\pi$
$632$	0	0
$633$	−5.22458	−0.207658
$634$	0	0
$635$	−21.9433	−0.870794
$636$	0	0
$637$	7.50986	0.297551
$638$	0	0
$639$	76.1433	3.01218
$640$	0	0
$641$	−16.4006	−0.647785	−0.323892	−	0.946094i	$-0.604992\pi$
$641$	−16.4006	−0.647785	−0.323892	+	0.946094i	$0.604992\pi$
$642$	0	0
$643$	−11.8571	−0.467597	−0.233799	−	0.972285i	$-0.575116\pi$
$643$	−11.8571	−0.467597	−0.233799	+	0.972285i	$0.575116\pi$
$644$	0	0
$645$	−117.726	−4.63546
$646$	0	0
$647$	31.2338	1.22793	0.613964	−	0.789334i	$-0.289574\pi$
$647$	31.2338	1.22793	0.613964	+	0.789334i	$0.289574\pi$
$648$	0	0
$649$	32.1880	1.26349
$650$	0	0
$651$	11.7987	0.462426
$652$	0	0
$653$	38.2358	1.49628	0.748142	−	0.663539i	$-0.230946\pi$
$653$	38.2358	1.49628	0.748142	+	0.663539i	$0.230946\pi$
$654$	0	0
$655$	−51.2403	−2.00213
$656$	0	0
$657$	−50.3503	−1.96435
$658$	0	0
$659$	13.5015	0.525945	0.262973	−	0.964803i	$-0.415297\pi$
$659$	13.5015	0.525945	0.262973	+	0.964803i	$0.415297\pi$
$660$	0	0
$661$	20.8372	0.810474	0.405237	−	0.914212i	$-0.367189\pi$
$661$	20.8372	0.810474	0.405237	+	0.914212i	$0.367189\pi$
$662$	0	0
$663$	−6.84629	−0.265888
$664$	0	0
$665$	−2.58030	−0.100060
$666$	0	0
$667$	−21.3855	−0.828049
$668$	0	0
$669$	−58.3635	−2.25646
$670$	0	0
$671$	50.6394	1.95491
$672$	0	0
$673$	5.83795	0.225036	0.112518	−	0.993650i	$-0.464108\pi$
$673$	5.83795	0.225036	0.112518	+	0.993650i	$0.464108\pi$
$674$	0	0
$675$	−55.5686	−2.13884
$676$	0	0
$677$	−14.3207	−0.550390	−0.275195	−	0.961388i	$-0.588742\pi$
$677$	−14.3207	−0.550390	−0.275195	+	0.961388i	$0.588742\pi$
$678$	0	0
$679$	−5.72666	−0.219769
$680$	0	0
$681$	−21.9665	−0.841759
$682$	0	0
$683$	2.59082	0.0991351	0.0495676	−	0.998771i	$-0.484216\pi$
$683$	2.59082	0.0991351	0.0495676	+	0.998771i	$0.484216\pi$
$684$	0	0
$685$	4.56484	0.174414
$686$	0	0
$687$	−54.8420	−2.09235
$688$	0	0
$689$	−19.9471	−0.759925
$690$	0	0
$691$	−12.4598	−0.473994	−0.236997	−	0.971510i	$-0.576163\pi$
$691$	−12.4598	−0.473994	−0.236997	+	0.971510i	$0.576163\pi$
$692$	0	0
$693$	66.7322	2.53495
$694$	0	0
$695$	−34.0399	−1.29121
$696$	0	0
$697$	−0.155409	−0.00588654
$698$	0	0
$699$	78.5338	2.97042
$700$	0	0
$701$	−5.26778	−0.198961	−0.0994807	−	0.995039i	$-0.531718\pi$
$701$	−5.26778	−0.198961	−0.0994807	+	0.995039i	$0.531718\pi$
$702$	0	0
$703$	−1.87781	−0.0708230
$704$	0	0
$705$	−3.45728	−0.130209
$706$	0	0
$707$	27.7576	1.04393
$708$	0	0
$709$	30.4770	1.14459	0.572293	−	0.820049i	$-0.306054\pi$
$709$	30.4770	1.14459	0.572293	+	0.820049i	$0.306054\pi$
$710$	0	0
$711$	28.4994	1.06881
$712$	0	0
$713$	−5.04162	−0.188810
$714$	0	0
$715$	31.0572	1.16147
$716$	0	0
$717$	56.2338	2.10009
$718$	0	0
$719$	18.9377	0.706258	0.353129	−	0.935575i	$-0.385118\pi$
$719$	18.9377	0.706258	0.353129	+	0.935575i	$0.385118\pi$
$720$	0	0
$721$	−8.35887	−0.311301
$722$	0	0
$723$	59.0694	2.19682
$724$	0	0
$725$	46.6340	1.73194
$726$	0	0
$727$	−30.6129	−1.13537	−0.567684	−	0.823246i	$-0.692160\pi$
$727$	−30.6129	−1.13537	−0.567684	+	0.823246i	$0.692160\pi$
$728$	0	0
$729$	10.8045	0.400166
$730$	0	0
$731$	−16.5206	−0.611036
$732$	0	0
$733$	−35.1036	−1.29658	−0.648290	−	0.761394i	$-0.724516\pi$
$733$	−35.1036	−1.29658	−0.648290	+	0.761394i	$0.724516\pi$
$734$	0	0
$735$	45.7066	1.68591
$736$	0	0
$737$	−47.6082	−1.75367
$738$	0	0
$739$	51.1520	1.88166	0.940829	−	0.338881i	$-0.110048\pi$
$739$	51.1520	1.88166	0.940829	+	0.338881i	$0.110048\pi$
$740$	0	0
$741$	2.79130	0.102541
$742$	0	0
$743$	−11.1185	−0.407899	−0.203950	−	0.978981i	$-0.565378\pi$
$743$	−11.1185	−0.407899	−0.203950	+	0.978981i	$0.565378\pi$
$744$	0	0
$745$	−41.7839	−1.53084
$746$	0	0
$747$	−52.4274	−1.91822
$748$	0	0
$749$	5.76088	0.210498
$750$	0	0
$751$	48.5238	1.77066	0.885330	−	0.464964i	$-0.153933\pi$
$751$	48.5238	1.77066	0.885330	+	0.464964i	$0.153933\pi$
$752$	0	0
$753$	−50.2485	−1.83116
$754$	0	0
$755$	−68.3498	−2.48750
$756$	0	0
$757$	−13.0860	−0.475618	−0.237809	−	0.971312i	$-0.576429\pi$
$757$	−13.0860	−0.475618	−0.237809	+	0.971312i	$0.576429\pi$
$758$	0	0
$759$	−40.8080	−1.48124
$760$	0	0
$761$	52.9952	1.92107	0.960536	−	0.278154i	$-0.0897227\pi$
$761$	52.9952	1.92107	0.960536	+	0.278154i	$0.0897227\pi$
$762$	0	0
$763$	−11.8618	−0.429427
$764$	0	0
$765$	−29.1160	−1.05269
$766$	0	0
$767$	8.09186	0.292180
$768$	0	0
$769$	30.2741	1.09171	0.545856	−	0.837879i	$-0.316205\pi$
$769$	30.2741	1.09171	0.545856	+	0.837879i	$0.316205\pi$
$770$	0	0
$771$	16.6235	0.598682
$772$	0	0
$773$	29.2667	1.05265	0.526325	−	0.850283i	$-0.323569\pi$
$773$	29.2667	1.05265	0.526325	+	0.850283i	$0.323569\pi$
$774$	0	0
$775$	10.9940	0.394915
$776$	0	0
$777$	−16.1513	−0.579423
$778$	0	0
$779$	0.0633619	0.00227018
$780$	0	0
$781$	−69.3686	−2.48220
$782$	0	0
$783$	−130.986	−4.68107
$784$	0	0
$785$	−42.7496	−1.52580
$786$	0	0
$787$	46.8364	1.66954	0.834769	−	0.550600i	$-0.185601\pi$
$787$	46.8364	1.66954	0.834769	+	0.550600i	$0.185601\pi$
$788$	0	0
$789$	−71.1029	−2.53133
$790$	0	0
$791$	−21.0261	−0.747602
$792$	0	0
$793$	12.7304	0.452071
$794$	0	0
$795$	−121.403	−4.30570
$796$	0	0
$797$	5.47039	0.193771	0.0968856	−	0.995296i	$-0.469112\pi$
$797$	5.47039	0.193771	0.0968856	+	0.995296i	$0.469112\pi$
$798$	0	0
$799$	−0.485164	−0.0171639
$800$	0	0
$801$	−42.9009	−1.51583
$802$	0	0
$803$	45.8705	1.61874
$804$	0	0
$805$	9.48337	0.334245
$806$	0	0
$807$	59.9253	2.10947
$808$	0	0
$809$	−16.4437	−0.578129	−0.289065	−	0.957310i	$-0.593344\pi$
$809$	−16.4437	−0.578129	−0.289065	+	0.957310i	$0.593344\pi$
$810$	0	0
$811$	−20.0747	−0.704917	−0.352459	−	0.935827i	$-0.614654\pi$
$811$	−20.0747	−0.704917	−0.352459	+	0.935827i	$0.614654\pi$
$812$	0	0
$813$	22.9377	0.804459
$814$	0	0
$815$	36.2442	1.26958
$816$	0	0
$817$	6.73562	0.235649
$818$	0	0
$819$	16.7761	0.586203
$820$	0	0
$821$	19.3219	0.674338	0.337169	−	0.941444i	$-0.390531\pi$
$821$	19.3219	0.674338	0.337169	+	0.941444i	$0.390531\pi$
$822$	0	0
$823$	17.0093	0.592907	0.296453	−	0.955047i	$-0.404196\pi$
$823$	17.0093	0.592907	0.296453	+	0.955047i	$0.404196\pi$
$824$	0	0
$825$	88.9874	3.09814
$826$	0	0
$827$	7.87975	0.274006	0.137003	−	0.990571i	$-0.456253\pi$
$827$	7.87975	0.274006	0.137003	+	0.990571i	$0.456253\pi$
$828$	0	0
$829$	−52.3711	−1.81892	−0.909462	−	0.415786i	$-0.863506\pi$
$829$	−52.3711	−1.81892	−0.909462	+	0.415786i	$0.863506\pi$
$830$	0	0
$831$	−70.9847	−2.46243
$832$	0	0
$833$	6.41405	0.222234
$834$	0	0
$835$	55.6534	1.92597
$836$	0	0
$837$	−30.8800	−1.06737
$838$	0	0
$839$	5.22792	0.180488	0.0902439	−	0.995920i	$-0.471235\pi$
$839$	5.22792	0.180488	0.0902439	+	0.995920i	$0.471235\pi$
$840$	0	0
$841$	80.9257	2.79054
$842$	0	0
$843$	53.1679	1.83120
$844$	0	0
$845$	−32.1510	−1.10603
$846$	0	0
$847$	−44.1562	−1.51723
$848$	0	0
$849$	17.9603	0.616396
$850$	0	0
$851$	6.90151	0.236581
$852$	0	0
$853$	−0.0429338	−0.00147003	−0.000735013	−	1.00000i	$-0.500234\pi$
$853$	−0.0429338	−0.00147003	−0.000735013		1.00000i	$0.500234\pi$
$854$	0	0
$855$	11.8709	0.405976
$856$	0	0
$857$	−11.6249	−0.397099	−0.198550	−	0.980091i	$-0.563623\pi$
$857$	−11.6249	−0.397099	−0.198550	+	0.980091i	$0.563623\pi$
$858$	0	0
$859$	6.22576	0.212420	0.106210	−	0.994344i	$-0.466128\pi$
$859$	6.22576	0.212420	0.106210	+	0.994344i	$0.466128\pi$
$860$	0	0
$861$	0.544982	0.0185729
$862$	0	0
$863$	8.33264	0.283646	0.141823	−	0.989892i	$-0.454704\pi$
$863$	8.33264	0.283646	0.141823	+	0.989892i	$0.454704\pi$
$864$	0	0
$865$	57.3310	1.94931
$866$	0	0
$867$	47.8008	1.62340
$868$	0	0
$869$	−25.9637	−0.880759
$870$	0	0
$871$	−11.9684	−0.405534
$872$	0	0
$873$	26.3460	0.891677
$874$	0	0
$875$	2.56702	0.0867810
$876$	0	0
$877$	5.72491	0.193316	0.0966582	−	0.995318i	$-0.469185\pi$
$877$	5.72491	0.193316	0.0966582	+	0.995318i	$0.469185\pi$
$878$	0	0
$879$	75.7036	2.55342
$880$	0	0
$881$	−10.9131	−0.367672	−0.183836	−	0.982957i	$-0.558852\pi$
$881$	−10.9131	−0.367672	−0.183836	+	0.982957i	$0.558852\pi$
$882$	0	0
$883$	24.5053	0.824670	0.412335	−	0.911032i	$-0.364713\pi$
$883$	24.5053	0.824670	0.412335	+	0.911032i	$0.364713\pi$
$884$	0	0
$885$	49.2488	1.65548
$886$	0	0
$887$	−18.8988	−0.634561	−0.317280	−	0.948332i	$-0.602770\pi$
$887$	−18.8988	−0.634561	−0.317280	+	0.948332i	$0.602770\pi$
$888$	0	0
$889$	10.7984	0.362168
$890$	0	0
$891$	−117.597	−3.93966
$892$	0	0
$893$	0.197806	0.00661934
$894$	0	0
$895$	36.5051	1.22023
$896$	0	0
$897$	−10.2589	−0.342533
$898$	0	0
$899$	25.9150	0.864313
$900$	0	0
$901$	−17.0365	−0.567569
$902$	0	0
$903$	57.9337	1.92791
$904$	0	0
$905$	64.7703	2.15304
$906$	0	0
$907$	−2.05830	−0.0683446	−0.0341723	−	0.999416i	$-0.510880\pi$
$907$	−2.05830	−0.0683446	−0.0341723	+	0.999416i	$0.510880\pi$
$908$	0	0
$909$	−127.701	−4.23557
$910$	0	0
$911$	−53.0316	−1.75702	−0.878508	−	0.477728i	$-0.841460\pi$
$911$	−53.0316	−1.75702	−0.878508	+	0.477728i	$0.841460\pi$
$912$	0	0
$913$	47.7628	1.58072
$914$	0	0
$915$	77.4802	2.56142
$916$	0	0
$917$	25.2157	0.832696
$918$	0	0
$919$	−32.9602	−1.08726	−0.543628	−	0.839326i	$-0.682950\pi$
$919$	−32.9602	−1.08726	−0.543628	+	0.839326i	$0.682950\pi$
$920$	0	0
$921$	34.3141	1.13069
$922$	0	0
$923$	−17.4388	−0.574006
$924$	0	0
$925$	−15.0497	−0.494831
$926$	0	0
$927$	38.4557	1.26305
$928$	0	0
$929$	−35.8255	−1.17539	−0.587697	−	0.809081i	$-0.699965\pi$
$929$	−35.8255	−1.17539	−0.587697	+	0.809081i	$0.699965\pi$
$930$	0	0
$931$	−2.61508	−0.0857056
$932$	0	0
$933$	−27.5059	−0.900502
$934$	0	0
$935$	26.5255	0.867476
$936$	0	0
$937$	−12.4523	−0.406800	−0.203400	−	0.979096i	$-0.565199\pi$
$937$	−12.4523	−0.406800	−0.203400	+	0.979096i	$0.565199\pi$
$938$	0	0
$939$	−8.86927	−0.289438
$940$	0	0
$941$	−36.9370	−1.20411	−0.602057	−	0.798453i	$-0.705652\pi$
$941$	−36.9370	−1.20411	−0.602057	+	0.798453i	$0.705652\pi$
$942$	0	0
$943$	−0.232874	−0.00758341
$944$	0	0
$945$	58.0858	1.88953
$946$	0	0
$947$	21.3470	0.693684	0.346842	−	0.937924i	$-0.387254\pi$
$947$	21.3470	0.693684	0.346842	+	0.937924i	$0.387254\pi$
$948$	0	0
$949$	11.5316	0.374330
$950$	0	0
$951$	78.0621	2.53134
$952$	0	0
$953$	18.6712	0.604821	0.302410	−	0.953178i	$-0.402209\pi$
$953$	18.6712	0.604821	0.302410	+	0.953178i	$0.402209\pi$
$954$	0	0
$955$	−19.1070	−0.618287
$956$	0	0
$957$	209.761	6.78062
$958$	0	0
$959$	−2.24639	−0.0725396
$960$	0	0
$961$	−24.8905	−0.802921
$962$	0	0
$963$	−26.5034	−0.854060
$964$	0	0
$965$	2.26901	0.0730420
$966$	0	0
$967$	−41.1480	−1.32323	−0.661615	−	0.749844i	$-0.730129\pi$
$967$	−41.1480	−1.32323	−0.661615	+	0.749844i	$0.730129\pi$
$968$	0	0
$969$	2.38401	0.0765854
$970$	0	0
$971$	2.52662	0.0810830	0.0405415	−	0.999178i	$-0.487092\pi$
$971$	2.52662	0.0810830	0.0405415	+	0.999178i	$0.487092\pi$
$972$	0	0
$973$	16.7513	0.537021
$974$	0	0
$975$	22.3709	0.716441
$976$	0	0
$977$	−40.2952	−1.28916	−0.644579	−	0.764538i	$-0.722967\pi$
$977$	−40.2952	−1.28916	−0.644579	+	0.764538i	$0.722967\pi$
$978$	0	0
$979$	39.0839	1.24913
$980$	0	0
$981$	54.5713	1.74233
$982$	0	0
$983$	16.7401	0.533926	0.266963	−	0.963707i	$-0.413980\pi$
$983$	16.7401	0.533926	0.266963	+	0.963707i	$0.413980\pi$
$984$	0	0
$985$	26.7494	0.852306
$986$	0	0
$987$	1.70135	0.0541546
$988$	0	0
$989$	−24.7554	−0.787175
$990$	0	0
$991$	−57.7676	−1.83505	−0.917524	−	0.397681i	$-0.869815\pi$
$991$	−57.7676	−1.83505	−0.917524	+	0.397681i	$0.869815\pi$
$992$	0	0
$993$	−63.3036	−2.00888
$994$	0	0
$995$	−84.4903	−2.67852
$996$	0	0
$997$	−8.44775	−0.267543	−0.133771	−	0.991012i	$-0.542709\pi$
$997$	−8.44775	−0.267543	−0.133771	+	0.991012i	$0.542709\pi$
$998$	0	0
$999$	42.2719	1.33742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.4	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-3.15577 q^{3} +3.07374 q^{5} -1.51261 q^{7} +6.95887 q^{9} +O(q^{10})\)	\(q-3.15577 q^{3} +3.07374 q^{5} -1.51261 q^{7} +6.95887 q^{9} -6.33973 q^{11} -1.59377 q^{13} -9.70001 q^{15} -1.36121 q^{17} +0.554980 q^{19} +4.77344 q^{21} -2.03971 q^{23} +4.44788 q^{25} -12.4933 q^{27} +10.4845 q^{29} +2.47173 q^{31} +20.0067 q^{33} -4.64936 q^{35} -3.38357 q^{37} +5.02956 q^{39} +0.114170 q^{41} +12.1367 q^{43} +21.3898 q^{45} +0.356421 q^{47} -4.71202 q^{49} +4.29567 q^{51} +12.5157 q^{53} -19.4867 q^{55} -1.75139 q^{57} -5.07719 q^{59} -7.98764 q^{61} -10.5260 q^{63} -4.89882 q^{65} +7.50950 q^{67} +6.43686 q^{69} +10.9419 q^{71} -7.23541 q^{73} -14.0365 q^{75} +9.58952 q^{77} +4.09540 q^{79} +18.5493 q^{81} -7.53388 q^{83} -4.18401 q^{85} -33.0868 q^{87} -6.16492 q^{89} +2.41074 q^{91} -7.80021 q^{93} +1.70586 q^{95} +3.78596 q^{97} -44.1174 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 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

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