LMFDB - Embedded newform 8044.2.a.b.1.3

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			1.3
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)

$f(q)$	$=$	$q-3.19446 q^{3} -0.458477 q^{5} +4.07842 q^{7} +7.20455 q^{9} +O(q^{10})$	$q-3.19446 q^{3} -0.458477 q^{5} +4.07842 q^{7} +7.20455 q^{9} -2.22869 q^{11} -6.38789 q^{13} +1.46458 q^{15} -2.59851 q^{17} -0.802188 q^{19} -13.0283 q^{21} -7.99585 q^{23} -4.78980 q^{25} -13.4313 q^{27} -0.0601791 q^{29} -7.44808 q^{31} +7.11945 q^{33} -1.86986 q^{35} -4.74273 q^{37} +20.4058 q^{39} +1.35088 q^{41} -4.32948 q^{43} -3.30312 q^{45} +9.08405 q^{47} +9.63349 q^{49} +8.30083 q^{51} +1.32097 q^{53} +1.02180 q^{55} +2.56255 q^{57} +8.33609 q^{59} -4.02739 q^{61} +29.3832 q^{63} +2.92870 q^{65} +0.719352 q^{67} +25.5424 q^{69} -6.84968 q^{71} +1.47048 q^{73} +15.3008 q^{75} -9.08952 q^{77} +4.44562 q^{79} +21.2919 q^{81} -7.83408 q^{83} +1.19136 q^{85} +0.192240 q^{87} +8.30858 q^{89} -26.0525 q^{91} +23.7926 q^{93} +0.367785 q^{95} +8.91319 q^{97} -16.0567 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.19446	−1.84432	−0.922160	−	0.386808i	$-0.873577\pi$
$3$	−3.19446	−1.84432	−0.922160	+	0.386808i	$0.873577\pi$
$4$	0	0
$5$	−0.458477	−0.205037	−0.102519	−	0.994731i	$-0.532690\pi$
$5$	−0.458477	−0.205037	−0.102519	+	0.994731i	$0.532690\pi$
$6$	0	0
$7$	4.07842	1.54150	0.770748	−	0.637140i	$-0.219883\pi$
$7$	4.07842	1.54150	0.770748	+	0.637140i	$0.219883\pi$
$8$	0	0
$9$	7.20455	2.40152
$10$	0	0
$11$	−2.22869	−0.671975	−0.335988	−	0.941866i	$-0.609070\pi$
$11$	−2.22869	−0.671975	−0.335988	+	0.941866i	$0.609070\pi$
$12$	0	0
$13$	−6.38789	−1.77168	−0.885842	−	0.463988i	$-0.846418\pi$
$13$	−6.38789	−1.77168	−0.885842	+	0.463988i	$0.846418\pi$
$14$	0	0
$15$	1.46458	0.378154
$16$	0	0
$17$	−2.59851	−0.630231	−0.315116	−	0.949053i	$-0.602043\pi$
$17$	−2.59851	−0.630231	−0.315116	+	0.949053i	$0.602043\pi$
$18$	0	0
$19$	−0.802188	−0.184034	−0.0920172	−	0.995757i	$-0.529331\pi$
$19$	−0.802188	−0.184034	−0.0920172	+	0.995757i	$0.529331\pi$
$20$	0	0
$21$	−13.0283	−2.84301
$22$	0	0
$23$	−7.99585	−1.66725	−0.833625	−	0.552331i	$-0.813738\pi$
$23$	−7.99585	−1.66725	−0.833625	+	0.552331i	$0.813738\pi$
$24$	0	0
$25$	−4.78980	−0.957960
$26$	0	0
$27$	−13.4313	−2.58485
$28$	0	0
$29$	−0.0601791	−0.0111750	−0.00558749	−	0.999984i	$-0.501779\pi$
$29$	−0.0601791	−0.0111750	−0.00558749	+	0.999984i	$0.501779\pi$
$30$	0	0
$31$	−7.44808	−1.33771	−0.668857	−	0.743391i	$-0.733216\pi$
$31$	−7.44808	−1.33771	−0.668857	+	0.743391i	$0.733216\pi$
$32$	0	0
$33$	7.11945	1.23934
$34$	0	0
$35$	−1.86986	−0.316064
$36$	0	0
$37$	−4.74273	−0.779700	−0.389850	−	0.920878i	$-0.627473\pi$
$37$	−4.74273	−0.779700	−0.389850	+	0.920878i	$0.627473\pi$
$38$	0	0
$39$	20.4058	3.26755
$40$	0	0
$41$	1.35088	0.210972	0.105486	−	0.994421i	$-0.466360\pi$
$41$	1.35088	0.210972	0.105486	+	0.994421i	$0.466360\pi$
$42$	0	0
$43$	−4.32948	−0.660240	−0.330120	−	0.943939i	$-0.607089\pi$
$43$	−4.32948	−0.660240	−0.330120	+	0.943939i	$0.607089\pi$
$44$	0	0
$45$	−3.30312	−0.492400
$46$	0	0
$47$	9.08405	1.32504	0.662522	−	0.749042i	$-0.269486\pi$
$47$	9.08405	1.32504	0.662522	+	0.749042i	$0.269486\pi$
$48$	0	0
$49$	9.63349	1.37621
$50$	0	0
$51$	8.30083	1.16235
$52$	0	0
$53$	1.32097	0.181449	0.0907243	−	0.995876i	$-0.471082\pi$
$53$	1.32097	0.181449	0.0907243	+	0.995876i	$0.471082\pi$
$54$	0	0
$55$	1.02180	0.137780
$56$	0	0
$57$	2.56255	0.339419
$58$	0	0
$59$	8.33609	1.08527	0.542633	−	0.839970i	$-0.317427\pi$
$59$	8.33609	1.08527	0.542633	+	0.839970i	$0.317427\pi$
$60$	0	0
$61$	−4.02739	−0.515655	−0.257827	−	0.966191i	$-0.583007\pi$
$61$	−4.02739	−0.515655	−0.257827	+	0.966191i	$0.583007\pi$
$62$	0	0
$63$	29.3832	3.70193
$64$	0	0
$65$	2.92870	0.363261
$66$	0	0
$67$	0.719352	0.0878828	0.0439414	−	0.999034i	$-0.486009\pi$
$67$	0.719352	0.0878828	0.0439414	+	0.999034i	$0.486009\pi$
$68$	0	0
$69$	25.5424	3.07494
$70$	0	0
$71$	−6.84968	−0.812907	−0.406453	−	0.913671i	$-0.633235\pi$
$71$	−6.84968	−0.812907	−0.406453	+	0.913671i	$0.633235\pi$
$72$	0	0
$73$	1.47048	0.172107	0.0860535	−	0.996291i	$-0.472574\pi$
$73$	1.47048	0.172107	0.0860535	+	0.996291i	$0.472574\pi$
$74$	0	0
$75$	15.3008	1.76678
$76$	0	0
$77$	−9.08952	−1.03585
$78$	0	0
$79$	4.44562	0.500172	0.250086	−	0.968224i	$-0.419541\pi$
$79$	4.44562	0.500172	0.250086	+	0.968224i	$0.419541\pi$
$80$	0	0
$81$	21.2919	2.36577
$82$	0	0
$83$	−7.83408	−0.859903	−0.429951	−	0.902852i	$-0.641469\pi$
$83$	−7.83408	−0.859903	−0.429951	+	0.902852i	$0.641469\pi$
$84$	0	0
$85$	1.19136	0.129221
$86$	0	0
$87$	0.192240	0.0206102
$88$	0	0
$89$	8.30858	0.880708	0.440354	−	0.897824i	$-0.354853\pi$
$89$	8.30858	0.880708	0.440354	+	0.897824i	$0.354853\pi$
$90$	0	0
$91$	−26.0525	−2.73104
$92$	0	0
$93$	23.7926	2.46717
$94$	0	0
$95$	0.367785	0.0377339
$96$	0	0
$97$	8.91319	0.904998	0.452499	−	0.891765i	$-0.350533\pi$
$97$	8.91319	0.904998	0.452499	+	0.891765i	$0.350533\pi$
$98$	0	0
$99$	−16.0567	−1.61376
$100$	0	0
$101$	−5.10382	−0.507849	−0.253925	−	0.967224i	$-0.581722\pi$
$101$	−5.10382	−0.507849	−0.253925	+	0.967224i	$0.581722\pi$
$102$	0	0
$103$	4.22520	0.416322	0.208161	−	0.978095i	$-0.433252\pi$
$103$	4.22520	0.416322	0.208161	+	0.978095i	$0.433252\pi$
$104$	0	0
$105$	5.97319	0.582923
$106$	0	0
$107$	−14.2564	−1.37822	−0.689109	−	0.724657i	$-0.741998\pi$
$107$	−14.2564	−1.37822	−0.689109	+	0.724657i	$0.741998\pi$
$108$	0	0
$109$	4.60248	0.440838	0.220419	−	0.975405i	$-0.429258\pi$
$109$	4.60248	0.440838	0.220419	+	0.975405i	$0.429258\pi$
$110$	0	0
$111$	15.1504	1.43802
$112$	0	0
$113$	−2.86696	−0.269701	−0.134850	−	0.990866i	$-0.543055\pi$
$113$	−2.86696	−0.269701	−0.134850	+	0.990866i	$0.543055\pi$
$114$	0	0
$115$	3.66591	0.341848
$116$	0	0
$117$	−46.0219	−4.25473
$118$	0	0
$119$	−10.5978	−0.971500
$120$	0	0
$121$	−6.03294	−0.548450
$122$	0	0
$123$	−4.31532	−0.389099
$124$	0	0
$125$	4.48840	0.401454
$126$	0	0
$127$	0.352652	0.0312928	0.0156464	−	0.999878i	$-0.495019\pi$
$127$	0.352652	0.0312928	0.0156464	+	0.999878i	$0.495019\pi$
$128$	0	0
$129$	13.8303	1.21769
$130$	0	0
$131$	−7.89520	−0.689807	−0.344903	−	0.938638i	$-0.612088\pi$
$131$	−7.89520	−0.689807	−0.344903	+	0.938638i	$0.612088\pi$
$132$	0	0
$133$	−3.27166	−0.283689
$134$	0	0
$135$	6.15792	0.529989
$136$	0	0
$137$	−7.14983	−0.610851	−0.305426	−	0.952216i	$-0.598799\pi$
$137$	−7.14983	−0.610851	−0.305426	+	0.952216i	$0.598799\pi$
$138$	0	0
$139$	7.09100	0.601451	0.300726	−	0.953711i	$-0.402771\pi$
$139$	7.09100	0.601451	0.300726	+	0.953711i	$0.402771\pi$
$140$	0	0
$141$	−29.0186	−2.44381
$142$	0	0
$143$	14.2366	1.19053
$144$	0	0
$145$	0.0275907	0.00229129
$146$	0	0
$147$	−30.7738	−2.53818
$148$	0	0
$149$	−7.24954	−0.593905	−0.296953	−	0.954892i	$-0.595970\pi$
$149$	−7.24954	−0.593905	−0.296953	+	0.954892i	$0.595970\pi$
$150$	0	0
$151$	10.4811	0.852942	0.426471	−	0.904501i	$-0.359757\pi$
$151$	10.4811	0.852942	0.426471	+	0.904501i	$0.359757\pi$
$152$	0	0
$153$	−18.7211	−1.51351
$154$	0	0
$155$	3.41477	0.274281
$156$	0	0
$157$	−9.74746	−0.777932	−0.388966	−	0.921252i	$-0.627168\pi$
$157$	−9.74746	−0.777932	−0.388966	+	0.921252i	$0.627168\pi$
$158$	0	0
$159$	−4.21977	−0.334649
$160$	0	0
$161$	−32.6104	−2.57006
$162$	0	0
$163$	12.5885	0.986004	0.493002	−	0.870028i	$-0.335899\pi$
$163$	12.5885	0.986004	0.493002	+	0.870028i	$0.335899\pi$
$164$	0	0
$165$	−3.26410	−0.254110
$166$	0	0
$167$	17.2198	1.33251	0.666255	−	0.745724i	$-0.267896\pi$
$167$	17.2198	1.33251	0.666255	+	0.745724i	$0.267896\pi$
$168$	0	0
$169$	27.8052	2.13886
$170$	0	0
$171$	−5.77940	−0.441962
$172$	0	0
$173$	−17.8922	−1.36032	−0.680160	−	0.733064i	$-0.738090\pi$
$173$	−17.8922	−1.36032	−0.680160	+	0.733064i	$0.738090\pi$
$174$	0	0
$175$	−19.5348	−1.47669
$176$	0	0
$177$	−26.6293	−2.00158
$178$	0	0
$179$	19.2203	1.43660	0.718298	−	0.695736i	$-0.244922\pi$
$179$	19.2203	1.43660	0.718298	+	0.695736i	$0.244922\pi$
$180$	0	0
$181$	−5.09207	−0.378491	−0.189245	−	0.981930i	$-0.560604\pi$
$181$	−5.09207	−0.378491	−0.189245	+	0.981930i	$0.560604\pi$
$182$	0	0
$183$	12.8653	0.951033
$184$	0	0
$185$	2.17443	0.159868
$186$	0	0
$187$	5.79127	0.423500
$188$	0	0
$189$	−54.7783	−3.98453
$190$	0	0
$191$	10.8952	0.788346	0.394173	−	0.919036i	$-0.371031\pi$
$191$	10.8952	0.788346	0.394173	+	0.919036i	$0.371031\pi$
$192$	0	0
$193$	−6.34190	−0.456500	−0.228250	−	0.973603i	$-0.573300\pi$
$193$	−6.34190	−0.456500	−0.228250	+	0.973603i	$0.573300\pi$
$194$	0	0
$195$	−9.35561	−0.669969
$196$	0	0
$197$	−12.9709	−0.924135	−0.462068	−	0.886845i	$-0.652892\pi$
$197$	−12.9709	−0.924135	−0.462068	+	0.886845i	$0.652892\pi$
$198$	0	0
$199$	22.1955	1.57340	0.786699	−	0.617336i	$-0.211788\pi$
$199$	22.1955	1.57340	0.786699	+	0.617336i	$0.211788\pi$
$200$	0	0
$201$	−2.29794	−0.162084
$202$	0	0
$203$	−0.245435	−0.0172262
$204$	0	0
$205$	−0.619346	−0.0432570
$206$	0	0
$207$	−57.6065	−4.00393
$208$	0	0
$209$	1.78783	0.123667
$210$	0	0
$211$	12.1008	0.833055	0.416528	−	0.909123i	$-0.363247\pi$
$211$	12.1008	0.833055	0.416528	+	0.909123i	$0.363247\pi$
$212$	0	0
$213$	21.8810	1.49926
$214$	0	0
$215$	1.98497	0.135374
$216$	0	0
$217$	−30.3764	−2.06208
$218$	0	0
$219$	−4.69739	−0.317420
$220$	0	0
$221$	16.5990	1.11657
$222$	0	0
$223$	−22.9689	−1.53811	−0.769056	−	0.639182i	$-0.779273\pi$
$223$	−22.9689	−1.53811	−0.769056	+	0.639182i	$0.779273\pi$
$224$	0	0
$225$	−34.5083	−2.30056
$226$	0	0
$227$	25.8458	1.71545	0.857723	−	0.514112i	$-0.171878\pi$
$227$	25.8458	1.71545	0.857723	+	0.514112i	$0.171878\pi$
$228$	0	0
$229$	−8.40507	−0.555423	−0.277711	−	0.960665i	$-0.589576\pi$
$229$	−8.40507	−0.555423	−0.277711	+	0.960665i	$0.589576\pi$
$230$	0	0
$231$	29.0361	1.91043
$232$	0	0
$233$	23.8598	1.56310	0.781552	−	0.623840i	$-0.214428\pi$
$233$	23.8598	1.56310	0.781552	+	0.623840i	$0.214428\pi$
$234$	0	0
$235$	−4.16483	−0.271683
$236$	0	0
$237$	−14.2013	−0.922476
$238$	0	0
$239$	0.312249	0.0201977	0.0100989	−	0.999949i	$-0.496785\pi$
$239$	0.312249	0.0201977	0.0100989	+	0.999949i	$0.496785\pi$
$240$	0	0
$241$	−0.468555	−0.0301823	−0.0150911	−	0.999886i	$-0.504804\pi$
$241$	−0.468555	−0.0301823	−0.0150911	+	0.999886i	$0.504804\pi$
$242$	0	0
$243$	−27.7223	−1.77839
$244$	0	0
$245$	−4.41673	−0.282175
$246$	0	0
$247$	5.12429	0.326051
$248$	0	0
$249$	25.0256	1.58594
$250$	0	0
$251$	14.5091	0.915808	0.457904	−	0.889002i	$-0.348600\pi$
$251$	14.5091	0.915808	0.457904	+	0.889002i	$0.348600\pi$
$252$	0	0
$253$	17.8203	1.12035
$254$	0	0
$255$	−3.80574	−0.238325
$256$	0	0
$257$	13.5896	0.847696	0.423848	−	0.905733i	$-0.360679\pi$
$257$	13.5896	0.847696	0.423848	+	0.905733i	$0.360679\pi$
$258$	0	0
$259$	−19.3428	−1.20191
$260$	0	0
$261$	−0.433563	−0.0268369
$262$	0	0
$263$	2.17899	0.134362	0.0671811	−	0.997741i	$-0.478599\pi$
$263$	2.17899	0.134362	0.0671811	+	0.997741i	$0.478599\pi$
$264$	0	0
$265$	−0.605632	−0.0372037
$266$	0	0
$267$	−26.5414	−1.62431
$268$	0	0
$269$	13.8972	0.847325	0.423663	−	0.905820i	$-0.360744\pi$
$269$	13.8972	0.847325	0.423663	+	0.905820i	$0.360744\pi$
$270$	0	0
$271$	−4.03971	−0.245395	−0.122697	−	0.992444i	$-0.539154\pi$
$271$	−4.03971	−0.245395	−0.122697	+	0.992444i	$0.539154\pi$
$272$	0	0
$273$	83.2236	5.03692
$274$	0	0
$275$	10.6750	0.643725
$276$	0	0
$277$	−25.8476	−1.55303	−0.776514	−	0.630099i	$-0.783014\pi$
$277$	−25.8476	−1.55303	−0.776514	+	0.630099i	$0.783014\pi$
$278$	0	0
$279$	−53.6600	−3.21254
$280$	0	0
$281$	−4.72589	−0.281923	−0.140962	−	0.990015i	$-0.545019\pi$
$281$	−4.72589	−0.281923	−0.140962	+	0.990015i	$0.545019\pi$
$282$	0	0
$283$	8.93263	0.530990	0.265495	−	0.964112i	$-0.414465\pi$
$283$	8.93263	0.530990	0.265495	+	0.964112i	$0.414465\pi$
$284$	0	0
$285$	−1.17487	−0.0695934
$286$	0	0
$287$	5.50944	0.325212
$288$	0	0
$289$	−10.2477	−0.602808
$290$	0	0
$291$	−28.4728	−1.66911
$292$	0	0
$293$	1.41608	0.0827283	0.0413642	−	0.999144i	$-0.486830\pi$
$293$	1.41608	0.0827283	0.0413642	+	0.999144i	$0.486830\pi$
$294$	0	0
$295$	−3.82190	−0.222520
$296$	0	0
$297$	29.9341	1.73695
$298$	0	0
$299$	51.0766	2.95384
$300$	0	0
$301$	−17.6574	−1.01776
$302$	0	0
$303$	16.3039	0.936637
$304$	0	0
$305$	1.84647	0.105728
$306$	0	0
$307$	32.8282	1.87361	0.936803	−	0.349856i	$-0.113769\pi$
$307$	32.8282	1.87361	0.936803	+	0.349856i	$0.113769\pi$
$308$	0	0
$309$	−13.4972	−0.767830
$310$	0	0
$311$	8.15397	0.462369	0.231185	−	0.972910i	$-0.425740\pi$
$311$	8.15397	0.462369	0.231185	+	0.972910i	$0.425740\pi$
$312$	0	0
$313$	−22.9145	−1.29521	−0.647603	−	0.761978i	$-0.724228\pi$
$313$	−22.9145	−1.29521	−0.647603	+	0.761978i	$0.724228\pi$
$314$	0	0
$315$	−13.4715	−0.759033
$316$	0	0
$317$	24.2025	1.35935	0.679674	−	0.733514i	$-0.262121\pi$
$317$	24.2025	1.35935	0.679674	+	0.733514i	$0.262121\pi$
$318$	0	0
$319$	0.134121	0.00750931
$320$	0	0
$321$	45.5415	2.54188
$322$	0	0
$323$	2.08449	0.115984
$324$	0	0
$325$	30.5967	1.69720
$326$	0	0
$327$	−14.7024	−0.813046
$328$	0	0
$329$	37.0485	2.04255
$330$	0	0
$331$	7.27919	0.400100	0.200050	−	0.979786i	$-0.435889\pi$
$331$	7.27919	0.400100	0.200050	+	0.979786i	$0.435889\pi$
$332$	0	0
$333$	−34.1693	−1.87246
$334$	0	0
$335$	−0.329806	−0.0180192
$336$	0	0
$337$	−13.8003	−0.751753	−0.375876	−	0.926670i	$-0.622658\pi$
$337$	−13.8003	−0.751753	−0.375876	+	0.926670i	$0.622658\pi$
$338$	0	0
$339$	9.15836	0.497414
$340$	0	0
$341$	16.5994	0.898910
$342$	0	0
$343$	10.7405	0.579930
$344$	0	0
$345$	−11.7106	−0.630477
$346$	0	0
$347$	−23.1719	−1.24393	−0.621965	−	0.783045i	$-0.713666\pi$
$347$	−23.1719	−1.24393	−0.621965	+	0.783045i	$0.713666\pi$
$348$	0	0
$349$	−7.80550	−0.417819	−0.208909	−	0.977935i	$-0.566991\pi$
$349$	−7.80550	−0.417819	−0.208909	+	0.977935i	$0.566991\pi$
$350$	0	0
$351$	85.7974	4.57953
$352$	0	0
$353$	9.08002	0.483281	0.241640	−	0.970366i	$-0.422315\pi$
$353$	9.08002	0.483281	0.241640	+	0.970366i	$0.422315\pi$
$354$	0	0
$355$	3.14042	0.166676
$356$	0	0
$357$	33.8542	1.79176
$358$	0	0
$359$	−14.6006	−0.770593	−0.385296	−	0.922793i	$-0.625901\pi$
$359$	−14.6006	−0.770593	−0.385296	+	0.922793i	$0.625901\pi$
$360$	0	0
$361$	−18.3565	−0.966131
$362$	0	0
$363$	19.2720	1.01152
$364$	0	0
$365$	−0.674182	−0.0352883
$366$	0	0
$367$	29.6650	1.54850	0.774250	−	0.632880i	$-0.218127\pi$
$367$	29.6650	1.54850	0.774250	+	0.632880i	$0.218127\pi$
$368$	0	0
$369$	9.73247	0.506652
$370$	0	0
$371$	5.38745	0.279702
$372$	0	0
$373$	−2.36028	−0.122211	−0.0611054	−	0.998131i	$-0.519463\pi$
$373$	−2.36028	−0.122211	−0.0611054	+	0.998131i	$0.519463\pi$
$374$	0	0
$375$	−14.3380	−0.740411
$376$	0	0
$377$	0.384418	0.0197985
$378$	0	0
$379$	4.00835	0.205895	0.102947	−	0.994687i	$-0.467173\pi$
$379$	4.00835	0.205895	0.102947	+	0.994687i	$0.467173\pi$
$380$	0	0
$381$	−1.12653	−0.0577139
$382$	0	0
$383$	−28.0778	−1.43471	−0.717355	−	0.696708i	$-0.754647\pi$
$383$	−28.0778	−1.43471	−0.717355	+	0.696708i	$0.754647\pi$
$384$	0	0
$385$	4.16734	0.212387
$386$	0	0
$387$	−31.1920	−1.58558
$388$	0	0
$389$	−28.8612	−1.46332	−0.731659	−	0.681671i	$-0.761254\pi$
$389$	−28.8612	−1.46332	−0.731659	+	0.681671i	$0.761254\pi$
$390$	0	0
$391$	20.7773	1.05075
$392$	0	0
$393$	25.2209	1.27222
$394$	0	0
$395$	−2.03822	−0.102554
$396$	0	0
$397$	33.4505	1.67883	0.839415	−	0.543490i	$-0.182898\pi$
$397$	33.4505	1.67883	0.839415	+	0.543490i	$0.182898\pi$
$398$	0	0
$399$	10.4512	0.523213
$400$	0	0
$401$	−11.7269	−0.585613	−0.292806	−	0.956172i	$-0.594589\pi$
$401$	−11.7269	−0.585613	−0.292806	+	0.956172i	$0.594589\pi$
$402$	0	0
$403$	47.5775	2.37001
$404$	0	0
$405$	−9.76185	−0.485070
$406$	0	0
$407$	10.5701	0.523939
$408$	0	0
$409$	−25.4507	−1.25846	−0.629229	−	0.777220i	$-0.716629\pi$
$409$	−25.4507	−1.25846	−0.629229	+	0.777220i	$0.716629\pi$
$410$	0	0
$411$	22.8398	1.12661
$412$	0	0
$413$	33.9980	1.67293
$414$	0	0
$415$	3.59175	0.176312
$416$	0	0
$417$	−22.6519	−1.10927
$418$	0	0
$419$	−3.17634	−0.155174	−0.0775871	−	0.996986i	$-0.524722\pi$
$419$	−3.17634	−0.155174	−0.0775871	+	0.996986i	$0.524722\pi$
$420$	0	0
$421$	−38.5426	−1.87845	−0.939225	−	0.343301i	$-0.888455\pi$
$421$	−38.5426	−1.87845	−0.939225	+	0.343301i	$0.888455\pi$
$422$	0	0
$423$	65.4465	3.18212
$424$	0	0
$425$	12.4463	0.603736
$426$	0	0
$427$	−16.4254	−0.794880
$428$	0	0
$429$	−45.4783	−2.19571
$430$	0	0
$431$	8.19941	0.394952	0.197476	−	0.980308i	$-0.436726\pi$
$431$	8.19941	0.394952	0.197476	+	0.980308i	$0.436726\pi$
$432$	0	0
$433$	−37.2396	−1.78962	−0.894809	−	0.446448i	$-0.852689\pi$
$433$	−37.2396	−1.78962	−0.894809	+	0.446448i	$0.852689\pi$
$434$	0	0
$435$	−0.0881374	−0.00422586
$436$	0	0
$437$	6.41417	0.306831
$438$	0	0
$439$	4.94747	0.236130	0.118065	−	0.993006i	$-0.462331\pi$
$439$	4.94747	0.236130	0.118065	+	0.993006i	$0.462331\pi$
$440$	0	0
$441$	69.4049	3.30500
$442$	0	0
$443$	−38.5959	−1.83375	−0.916873	−	0.399179i	$-0.869295\pi$
$443$	−38.5959	−1.83375	−0.916873	+	0.399179i	$0.869295\pi$
$444$	0	0
$445$	−3.80929	−0.180578
$446$	0	0
$447$	23.1583	1.09535
$448$	0	0
$449$	34.5408	1.63008	0.815040	−	0.579405i	$-0.196715\pi$
$449$	34.5408	1.63008	0.815040	+	0.579405i	$0.196715\pi$
$450$	0	0
$451$	−3.01069	−0.141768
$452$	0	0
$453$	−33.4815	−1.57310
$454$	0	0
$455$	11.9445	0.559965
$456$	0	0
$457$	4.54185	0.212459	0.106229	−	0.994342i	$-0.466122\pi$
$457$	4.54185	0.212459	0.106229	+	0.994342i	$0.466122\pi$
$458$	0	0
$459$	34.9013	1.62905
$460$	0	0
$461$	−3.52079	−0.163980	−0.0819898	−	0.996633i	$-0.526127\pi$
$461$	−3.52079	−0.163980	−0.0819898	+	0.996633i	$0.526127\pi$
$462$	0	0
$463$	−18.0908	−0.840752	−0.420376	−	0.907350i	$-0.638102\pi$
$463$	−18.0908	−0.840752	−0.420376	+	0.907350i	$0.638102\pi$
$464$	0	0
$465$	−10.9083	−0.505862
$466$	0	0
$467$	−25.7915	−1.19349	−0.596743	−	0.802432i	$-0.703539\pi$
$467$	−25.7915	−1.19349	−0.596743	+	0.802432i	$0.703539\pi$
$468$	0	0
$469$	2.93382	0.135471
$470$	0	0
$471$	31.1378	1.43476
$472$	0	0
$473$	9.64907	0.443664
$474$	0	0
$475$	3.84232	0.176298
$476$	0	0
$477$	9.51696	0.435752
$478$	0	0
$479$	34.6531	1.58334	0.791669	−	0.610950i	$-0.209212\pi$
$479$	34.6531	1.58334	0.791669	+	0.610950i	$0.209212\pi$
$480$	0	0
$481$	30.2961	1.38138
$482$	0	0
$483$	104.173	4.74001
$484$	0	0
$485$	−4.08649	−0.185558
$486$	0	0
$487$	−36.0940	−1.63558	−0.817788	−	0.575519i	$-0.804800\pi$
$487$	−36.0940	−1.63558	−0.817788	+	0.575519i	$0.804800\pi$
$488$	0	0
$489$	−40.2133	−1.81851
$490$	0	0
$491$	10.3317	0.466261	0.233131	−	0.972445i	$-0.425103\pi$
$491$	10.3317	0.466261	0.233131	+	0.972445i	$0.425103\pi$
$492$	0	0
$493$	0.156376	0.00704282
$494$	0	0
$495$	7.36163	0.330881
$496$	0	0
$497$	−27.9358	−1.25309
$498$	0	0
$499$	13.5134	0.604944	0.302472	−	0.953158i	$-0.402188\pi$
$499$	13.5134	0.604944	0.302472	+	0.953158i	$0.402188\pi$
$500$	0	0
$501$	−55.0080	−2.45758
$502$	0	0
$503$	−4.63304	−0.206577	−0.103289	−	0.994651i	$-0.532937\pi$
$503$	−4.63304	−0.206577	−0.103289	+	0.994651i	$0.532937\pi$
$504$	0	0
$505$	2.33998	0.104128
$506$	0	0
$507$	−88.8225	−3.94474
$508$	0	0
$509$	4.06324	0.180100	0.0900500	−	0.995937i	$-0.471297\pi$
$509$	4.06324	0.180100	0.0900500	+	0.995937i	$0.471297\pi$
$510$	0	0
$511$	5.99724	0.265302
$512$	0	0
$513$	10.7744	0.475701
$514$	0	0
$515$	−1.93716	−0.0853614
$516$	0	0
$517$	−20.2455	−0.890397
$518$	0	0
$519$	57.1559	2.50886
$520$	0	0
$521$	13.8947	0.608737	0.304369	−	0.952554i	$-0.401555\pi$
$521$	13.8947	0.608737	0.304369	+	0.952554i	$0.401555\pi$
$522$	0	0
$523$	−35.4789	−1.55138	−0.775692	−	0.631112i	$-0.782599\pi$
$523$	−35.4789	−1.55138	−0.775692	+	0.631112i	$0.782599\pi$
$524$	0	0
$525$	62.4031	2.72349
$526$	0	0
$527$	19.3539	0.843069
$528$	0	0
$529$	40.9336	1.77972
$530$	0	0
$531$	60.0578	2.60629
$532$	0	0
$533$	−8.62926	−0.373775
$534$	0	0
$535$	6.53623	0.282586
$536$	0	0
$537$	−61.3985	−2.64954
$538$	0	0
$539$	−21.4700	−0.924780
$540$	0	0
$541$	−25.2543	−1.08577	−0.542884	−	0.839808i	$-0.682668\pi$
$541$	−25.2543	−1.08577	−0.542884	+	0.839808i	$0.682668\pi$
$542$	0	0
$543$	16.2664	0.698058
$544$	0	0
$545$	−2.11013	−0.0903881
$546$	0	0
$547$	31.9779	1.36728	0.683638	−	0.729821i	$-0.260397\pi$
$547$	31.9779	1.36728	0.683638	+	0.729821i	$0.260397\pi$
$548$	0	0
$549$	−29.0156	−1.23835
$550$	0	0
$551$	0.0482749	0.00205658
$552$	0	0
$553$	18.1311	0.771013
$554$	0	0
$555$	−6.94613	−0.294847
$556$	0	0
$557$	8.06822	0.341861	0.170931	−	0.985283i	$-0.445323\pi$
$557$	8.06822	0.341861	0.170931	+	0.985283i	$0.445323\pi$
$558$	0	0
$559$	27.6563	1.16974
$560$	0	0
$561$	−18.5000	−0.781069
$562$	0	0
$563$	33.5409	1.41358	0.706790	−	0.707424i	$-0.250143\pi$
$563$	33.5409	1.41358	0.706790	+	0.707424i	$0.250143\pi$
$564$	0	0
$565$	1.31443	0.0552986
$566$	0	0
$567$	86.8372	3.64682
$568$	0	0
$569$	32.8633	1.37770	0.688850	−	0.724904i	$-0.258116\pi$
$569$	32.8633	1.37770	0.688850	+	0.724904i	$0.258116\pi$
$570$	0	0
$571$	−0.475032	−0.0198795	−0.00993973	−	0.999951i	$-0.503164\pi$
$571$	−0.475032	−0.0198795	−0.00993973	+	0.999951i	$0.503164\pi$
$572$	0	0
$573$	−34.8041	−1.45396
$574$	0	0
$575$	38.2985	1.59716
$576$	0	0
$577$	−23.2276	−0.966978	−0.483489	−	0.875350i	$-0.660631\pi$
$577$	−23.2276	−0.966978	−0.483489	+	0.875350i	$0.660631\pi$
$578$	0	0
$579$	20.2589	0.841932
$580$	0	0
$581$	−31.9507	−1.32554
$582$	0	0
$583$	−2.94402	−0.121929
$584$	0	0
$585$	21.1000	0.872377
$586$	0	0
$587$	42.5119	1.75465	0.877327	−	0.479894i	$-0.159325\pi$
$587$	42.5119	1.75465	0.877327	+	0.479894i	$0.159325\pi$
$588$	0	0
$589$	5.97475	0.246185
$590$	0	0
$591$	41.4348	1.70440
$592$	0	0
$593$	27.6272	1.13451	0.567257	−	0.823541i	$-0.308005\pi$
$593$	27.6272	1.13451	0.567257	+	0.823541i	$0.308005\pi$
$594$	0	0
$595$	4.85885	0.199193
$596$	0	0
$597$	−70.9026	−2.90185
$598$	0	0
$599$	−12.9268	−0.528174	−0.264087	−	0.964499i	$-0.585071\pi$
$599$	−12.9268	−0.528174	−0.264087	+	0.964499i	$0.585071\pi$
$600$	0	0
$601$	14.7135	0.600176	0.300088	−	0.953911i	$-0.402984\pi$
$601$	14.7135	0.600176	0.300088	+	0.953911i	$0.402984\pi$
$602$	0	0
$603$	5.18261	0.211052
$604$	0	0
$605$	2.76597	0.112453
$606$	0	0
$607$	−31.6477	−1.28454	−0.642269	−	0.766479i	$-0.722007\pi$
$607$	−31.6477	−1.28454	−0.642269	+	0.766479i	$0.722007\pi$
$608$	0	0
$609$	0.784033	0.0317706
$610$	0	0
$611$	−58.0280	−2.34756
$612$	0	0
$613$	45.4182	1.83443	0.917213	−	0.398398i	$-0.130434\pi$
$613$	45.4182	1.83443	0.917213	+	0.398398i	$0.130434\pi$
$614$	0	0
$615$	1.97847	0.0797798
$616$	0	0
$617$	27.4125	1.10358	0.551792	−	0.833982i	$-0.313944\pi$
$617$	27.4125	1.10358	0.551792	+	0.833982i	$0.313944\pi$
$618$	0	0
$619$	46.4894	1.86857	0.934284	−	0.356530i	$-0.116040\pi$
$619$	46.4894	1.86857	0.934284	+	0.356530i	$0.116040\pi$
$620$	0	0
$621$	107.394	4.30958
$622$	0	0
$623$	33.8859	1.35761
$624$	0	0
$625$	21.8912	0.875647
$626$	0	0
$627$	−5.71114	−0.228081
$628$	0	0
$629$	12.3240	0.491392
$630$	0	0
$631$	18.6893	0.744009	0.372004	−	0.928231i	$-0.378671\pi$
$631$	18.6893	0.744009	0.372004	+	0.928231i	$0.378671\pi$
$632$	0	0
$633$	−38.6556	−1.53642
$634$	0	0
$635$	−0.161683	−0.00641618
$636$	0	0
$637$	−61.5377	−2.43821
$638$	0	0
$639$	−49.3488	−1.95221
$640$	0	0
$641$	33.8202	1.33582	0.667910	−	0.744242i	$-0.267189\pi$
$641$	33.8202	1.33582	0.667910	+	0.744242i	$0.267189\pi$
$642$	0	0
$643$	−37.9760	−1.49763	−0.748814	−	0.662780i	$-0.769376\pi$
$643$	−37.9760	−1.49763	−0.748814	+	0.662780i	$0.769376\pi$
$644$	0	0
$645$	−6.34089	−0.249672
$646$	0	0
$647$	46.1729	1.81524	0.907622	−	0.419788i	$-0.137896\pi$
$647$	46.1729	1.81524	0.907622	+	0.419788i	$0.137896\pi$
$648$	0	0
$649$	−18.5785	−0.729272
$650$	0	0
$651$	97.0360	3.80314
$652$	0	0
$653$	26.2586	1.02758	0.513790	−	0.857916i	$-0.328241\pi$
$653$	26.2586	1.02758	0.513790	+	0.857916i	$0.328241\pi$
$654$	0	0
$655$	3.61977	0.141436
$656$	0	0
$657$	10.5942	0.413318
$658$	0	0
$659$	26.6621	1.03861	0.519304	−	0.854589i	$-0.326191\pi$
$659$	26.6621	1.03861	0.519304	+	0.854589i	$0.326191\pi$
$660$	0	0
$661$	−8.20661	−0.319200	−0.159600	−	0.987182i	$-0.551020\pi$
$661$	−8.20661	−0.319200	−0.159600	+	0.987182i	$0.551020\pi$
$662$	0	0
$663$	−53.0248	−2.05931
$664$	0	0
$665$	1.49998	0.0581667
$666$	0	0
$667$	0.481183	0.0186315
$668$	0	0
$669$	73.3732	2.83677
$670$	0	0
$671$	8.97581	0.346507
$672$	0	0
$673$	26.4045	1.01782	0.508910	−	0.860820i	$-0.330049\pi$
$673$	26.4045	1.01782	0.508910	+	0.860820i	$0.330049\pi$
$674$	0	0
$675$	64.3330	2.47618
$676$	0	0
$677$	23.1298	0.888951	0.444476	−	0.895791i	$-0.353390\pi$
$677$	23.1298	0.888951	0.444476	+	0.895791i	$0.353390\pi$
$678$	0	0
$679$	36.3517	1.39505
$680$	0	0
$681$	−82.5633	−3.16383
$682$	0	0
$683$	1.82789	0.0699422	0.0349711	−	0.999388i	$-0.488866\pi$
$683$	1.82789	0.0699422	0.0349711	+	0.999388i	$0.488866\pi$
$684$	0	0
$685$	3.27803	0.125247
$686$	0	0
$687$	26.8496	1.02438
$688$	0	0
$689$	−8.43819	−0.321469
$690$	0	0
$691$	−48.5335	−1.84630	−0.923150	−	0.384439i	$-0.874395\pi$
$691$	−48.5335	−1.84630	−0.923150	+	0.384439i	$0.874395\pi$
$692$	0	0
$693$	−65.4859	−2.48761
$694$	0	0
$695$	−3.25106	−0.123320
$696$	0	0
$697$	−3.51027	−0.132961
$698$	0	0
$699$	−76.2189	−2.88287
$700$	0	0
$701$	−26.7223	−1.00929	−0.504644	−	0.863328i	$-0.668376\pi$
$701$	−26.7223	−1.00929	−0.504644	+	0.863328i	$0.668376\pi$
$702$	0	0
$703$	3.80456	0.143492
$704$	0	0
$705$	13.3044	0.501071
$706$	0	0
$707$	−20.8155	−0.782848
$708$	0	0
$709$	−27.4591	−1.03125	−0.515623	−	0.856815i	$-0.672440\pi$
$709$	−27.4591	−1.03125	−0.515623	+	0.856815i	$0.672440\pi$
$710$	0	0
$711$	32.0287	1.20117
$712$	0	0
$713$	59.5537	2.23030
$714$	0	0
$715$	−6.52717	−0.244102
$716$	0	0
$717$	−0.997466	−0.0372511
$718$	0	0
$719$	39.0130	1.45494	0.727471	−	0.686139i	$-0.240696\pi$
$719$	39.0130	1.45494	0.727471	+	0.686139i	$0.240696\pi$
$720$	0	0
$721$	17.2321	0.641758
$722$	0	0
$723$	1.49678	0.0556658
$724$	0	0
$725$	0.288246	0.0107052
$726$	0	0
$727$	13.9535	0.517506	0.258753	−	0.965943i	$-0.416688\pi$
$727$	13.9535	0.517506	0.258753	+	0.965943i	$0.416688\pi$
$728$	0	0
$729$	24.6819	0.914145
$730$	0	0
$731$	11.2502	0.416104
$732$	0	0
$733$	−6.38576	−0.235863	−0.117932	−	0.993022i	$-0.537626\pi$
$733$	−6.38576	−0.235863	−0.117932	+	0.993022i	$0.537626\pi$
$734$	0	0
$735$	14.1091	0.520420
$736$	0	0
$737$	−1.60321	−0.0590551
$738$	0	0
$739$	48.3706	1.77934	0.889671	−	0.456601i	$-0.150933\pi$
$739$	48.3706	1.77934	0.889671	+	0.456601i	$0.150933\pi$
$740$	0	0
$741$	−16.3693	−0.601342
$742$	0	0
$743$	−8.51290	−0.312308	−0.156154	−	0.987733i	$-0.549910\pi$
$743$	−8.51290	−0.312308	−0.156154	+	0.987733i	$0.549910\pi$
$744$	0	0
$745$	3.32375	0.121773
$746$	0	0
$747$	−56.4411	−2.06507
$748$	0	0
$749$	−58.1436	−2.12452
$750$	0	0
$751$	−23.9013	−0.872173	−0.436086	−	0.899905i	$-0.643636\pi$
$751$	−23.9013	−0.872173	−0.436086	+	0.899905i	$0.643636\pi$
$752$	0	0
$753$	−46.3488	−1.68904
$754$	0	0
$755$	−4.80536	−0.174885
$756$	0	0
$757$	33.6566	1.22327	0.611634	−	0.791141i	$-0.290512\pi$
$757$	33.6566	1.22327	0.611634	+	0.791141i	$0.290512\pi$
$758$	0	0
$759$	−56.9261	−2.06628
$760$	0	0
$761$	26.0811	0.945438	0.472719	−	0.881213i	$-0.343273\pi$
$761$	26.0811	0.945438	0.472719	+	0.881213i	$0.343273\pi$
$762$	0	0
$763$	18.7708	0.679550
$764$	0	0
$765$	8.58319	0.310326
$766$	0	0
$767$	−53.2501	−1.92275
$768$	0	0
$769$	6.37456	0.229872	0.114936	−	0.993373i	$-0.463334\pi$
$769$	6.37456	0.229872	0.114936	+	0.993373i	$0.463334\pi$
$770$	0	0
$771$	−43.4114	−1.56342
$772$	0	0
$773$	17.4469	0.627521	0.313760	−	0.949502i	$-0.398411\pi$
$773$	17.4469	0.627521	0.313760	+	0.949502i	$0.398411\pi$
$774$	0	0
$775$	35.6748	1.28148
$776$	0	0
$777$	61.7899	2.21670
$778$	0	0
$779$	−1.08366	−0.0388261
$780$	0	0
$781$	15.2658	0.546253
$782$	0	0
$783$	0.808281	0.0288856
$784$	0	0
$785$	4.46899	0.159505
$786$	0	0
$787$	5.76592	0.205533	0.102766	−	0.994706i	$-0.467231\pi$
$787$	5.76592	0.205533	0.102766	+	0.994706i	$0.467231\pi$
$788$	0	0
$789$	−6.96068	−0.247807
$790$	0	0
$791$	−11.6926	−0.415742
$792$	0	0
$793$	25.7266	0.913577
$794$	0	0
$795$	1.93467	0.0686155
$796$	0	0
$797$	−36.1384	−1.28009	−0.640043	−	0.768339i	$-0.721084\pi$
$797$	−36.1384	−1.28009	−0.640043	+	0.768339i	$0.721084\pi$
$798$	0	0
$799$	−23.6050	−0.835085
$800$	0	0
$801$	59.8596	2.11503
$802$	0	0
$803$	−3.27725	−0.115652
$804$	0	0
$805$	14.9511	0.526958
$806$	0	0
$807$	−44.3939	−1.56274
$808$	0	0
$809$	−6.10783	−0.214740	−0.107370	−	0.994219i	$-0.534243\pi$
$809$	−6.10783	−0.214740	−0.107370	+	0.994219i	$0.534243\pi$
$810$	0	0
$811$	45.6275	1.60220	0.801098	−	0.598533i	$-0.204249\pi$
$811$	45.6275	1.60220	0.801098	+	0.598533i	$0.204249\pi$
$812$	0	0
$813$	12.9047	0.452586
$814$	0	0
$815$	−5.77152	−0.202168
$816$	0	0
$817$	3.47306	0.121507
$818$	0	0
$819$	−187.697	−6.55865
$820$	0	0
$821$	24.7270	0.862978	0.431489	−	0.902118i	$-0.357988\pi$
$821$	24.7270	0.862978	0.431489	+	0.902118i	$0.357988\pi$
$822$	0	0
$823$	−8.15204	−0.284162	−0.142081	−	0.989855i	$-0.545379\pi$
$823$	−8.15204	−0.284162	−0.142081	+	0.989855i	$0.545379\pi$
$824$	0	0
$825$	−34.1007	−1.18724
$826$	0	0
$827$	−12.2508	−0.426003	−0.213001	−	0.977052i	$-0.568324\pi$
$827$	−12.2508	−0.426003	−0.213001	+	0.977052i	$0.568324\pi$
$828$	0	0
$829$	−33.6429	−1.16847	−0.584234	−	0.811585i	$-0.698605\pi$
$829$	−33.6429	−1.16847	−0.584234	+	0.811585i	$0.698605\pi$
$830$	0	0
$831$	82.5689	2.86428
$832$	0	0
$833$	−25.0327	−0.867332
$834$	0	0
$835$	−7.89489	−0.273214
$836$	0	0
$837$	100.037	3.45778
$838$	0	0
$839$	−12.9394	−0.446719	−0.223359	−	0.974736i	$-0.571702\pi$
$839$	−12.9394	−0.446719	−0.223359	+	0.974736i	$0.571702\pi$
$840$	0	0
$841$	−28.9964	−0.999875
$842$	0	0
$843$	15.0967	0.519957
$844$	0	0
$845$	−12.7480	−0.438546
$846$	0	0
$847$	−24.6049	−0.845433
$848$	0	0
$849$	−28.5349	−0.979316
$850$	0	0
$851$	37.9222	1.29996
$852$	0	0
$853$	7.37820	0.252625	0.126312	−	0.991991i	$-0.459686\pi$
$853$	7.37820	0.252625	0.126312	+	0.991991i	$0.459686\pi$
$854$	0	0
$855$	2.64972	0.0906186
$856$	0	0
$857$	26.4466	0.903397	0.451699	−	0.892171i	$-0.350818\pi$
$857$	26.4466	0.903397	0.451699	+	0.892171i	$0.350818\pi$
$858$	0	0
$859$	−10.3594	−0.353460	−0.176730	−	0.984259i	$-0.556552\pi$
$859$	−10.3594	−0.353460	−0.176730	+	0.984259i	$0.556552\pi$
$860$	0	0
$861$	−17.5997	−0.599795
$862$	0	0
$863$	−28.6654	−0.975780	−0.487890	−	0.872905i	$-0.662233\pi$
$863$	−28.6654	−0.975780	−0.487890	+	0.872905i	$0.662233\pi$
$864$	0	0
$865$	8.20316	0.278916
$866$	0	0
$867$	32.7360	1.11177
$868$	0	0
$869$	−9.90791	−0.336103
$870$	0	0
$871$	−4.59515	−0.155701
$872$	0	0
$873$	64.2155	2.17337
$874$	0	0
$875$	18.3056	0.618841
$876$	0	0
$877$	−18.2265	−0.615467	−0.307733	−	0.951473i	$-0.599570\pi$
$877$	−18.2265	−0.615467	−0.307733	+	0.951473i	$0.599570\pi$
$878$	0	0
$879$	−4.52361	−0.152578
$880$	0	0
$881$	21.5095	0.724673	0.362337	−	0.932047i	$-0.381979\pi$
$881$	21.5095	0.724673	0.362337	+	0.932047i	$0.381979\pi$
$882$	0	0
$883$	4.71146	0.158553	0.0792767	−	0.996853i	$-0.474739\pi$
$883$	4.71146	0.158553	0.0792767	+	0.996853i	$0.474739\pi$
$884$	0	0
$885$	12.2089	0.410398
$886$	0	0
$887$	9.95539	0.334269	0.167135	−	0.985934i	$-0.446549\pi$
$887$	9.95539	0.334269	0.167135	+	0.985934i	$0.446549\pi$
$888$	0	0
$889$	1.43826	0.0482377
$890$	0	0
$891$	−47.4530	−1.58974
$892$	0	0
$893$	−7.28711	−0.243854
$894$	0	0
$895$	−8.81208	−0.294555
$896$	0	0
$897$	−163.162	−5.44782
$898$	0	0
$899$	0.448218	0.0149489
$900$	0	0
$901$	−3.43254	−0.114355
$902$	0	0
$903$	56.4059	1.87707
$904$	0	0
$905$	2.33460	0.0776047
$906$	0	0
$907$	−40.0131	−1.32861	−0.664306	−	0.747461i	$-0.731273\pi$
$907$	−40.0131	−1.32861	−0.664306	+	0.747461i	$0.731273\pi$
$908$	0	0
$909$	−36.7707	−1.21961
$910$	0	0
$911$	1.46991	0.0487002	0.0243501	−	0.999703i	$-0.492248\pi$
$911$	1.46991	0.0487002	0.0243501	+	0.999703i	$0.492248\pi$
$912$	0	0
$913$	17.4597	0.577833
$914$	0	0
$915$	−5.89846	−0.194997
$916$	0	0
$917$	−32.1999	−1.06333
$918$	0	0
$919$	−9.38670	−0.309639	−0.154819	−	0.987943i	$-0.549480\pi$
$919$	−9.38670	−0.309639	−0.154819	+	0.987943i	$0.549480\pi$
$920$	0	0
$921$	−104.868	−3.45553
$922$	0	0
$923$	43.7550	1.44021
$924$	0	0
$925$	22.7167	0.746922
$926$	0	0
$927$	30.4407	0.999803
$928$	0	0
$929$	−55.7583	−1.82937	−0.914686	−	0.404166i	$-0.867562\pi$
$929$	−55.7583	−1.82937	−0.914686	+	0.404166i	$0.867562\pi$
$930$	0	0
$931$	−7.72787	−0.253271
$932$	0	0
$933$	−26.0475	−0.852757
$934$	0	0
$935$	−2.65516	−0.0868332
$936$	0	0
$937$	−56.5998	−1.84903	−0.924517	−	0.381141i	$-0.875531\pi$
$937$	−56.5998	−1.84903	−0.924517	+	0.381141i	$0.875531\pi$
$938$	0	0
$939$	73.1995	2.38877
$940$	0	0
$941$	−34.4111	−1.12177	−0.560886	−	0.827893i	$-0.689539\pi$
$941$	−34.4111	−1.12177	−0.560886	+	0.827893i	$0.689539\pi$
$942$	0	0
$943$	−10.8014	−0.351742
$944$	0	0
$945$	25.1146	0.816977
$946$	0	0
$947$	5.68968	0.184890	0.0924449	−	0.995718i	$-0.470532\pi$
$947$	5.68968	0.184890	0.0924449	+	0.995718i	$0.470532\pi$
$948$	0	0
$949$	−9.39328	−0.304919
$950$	0	0
$951$	−77.3138	−2.50707
$952$	0	0
$953$	−30.2338	−0.979370	−0.489685	−	0.871899i	$-0.662888\pi$
$953$	−30.2338	−0.979370	−0.489685	+	0.871899i	$0.662888\pi$
$954$	0	0
$955$	−4.99518	−0.161640
$956$	0	0
$957$	−0.428442	−0.0138496
$958$	0	0
$959$	−29.1600	−0.941626
$960$	0	0
$961$	24.4738	0.789478
$962$	0	0
$963$	−102.711	−3.30982
$964$	0	0
$965$	2.90761	0.0935994
$966$	0	0
$967$	−29.6360	−0.953028	−0.476514	−	0.879167i	$-0.658100\pi$
$967$	−29.6360	−0.953028	−0.476514	+	0.879167i	$0.658100\pi$
$968$	0	0
$969$	−6.65882	−0.213912
$970$	0	0
$971$	−10.2103	−0.327665	−0.163833	−	0.986488i	$-0.552386\pi$
$971$	−10.2103	−0.327665	−0.163833	+	0.986488i	$0.552386\pi$
$972$	0	0
$973$	28.9201	0.927135
$974$	0	0
$975$	−97.7399	−3.13018
$976$	0	0
$977$	33.9678	1.08673	0.543363	−	0.839498i	$-0.317151\pi$
$977$	33.9678	1.08673	0.543363	+	0.839498i	$0.317151\pi$
$978$	0	0
$979$	−18.5172	−0.591814
$980$	0	0
$981$	33.1588	1.05868
$982$	0	0
$983$	17.1044	0.545547	0.272774	−	0.962078i	$-0.412059\pi$
$983$	17.1044	0.545547	0.272774	+	0.962078i	$0.412059\pi$
$984$	0	0
$985$	5.94684	0.189482
$986$	0	0
$987$	−118.350	−3.76712
$988$	0	0
$989$	34.6179	1.10078
$990$	0	0
$991$	−27.9936	−0.889247	−0.444623	−	0.895718i	$-0.646662\pi$
$991$	−27.9936	−0.889247	−0.444623	+	0.895718i	$0.646662\pi$
$992$	0	0
$993$	−23.2530	−0.737913
$994$	0	0
$995$	−10.1761	−0.322605
$996$	0	0
$997$	−19.0959	−0.604774	−0.302387	−	0.953185i	$-0.597784\pi$
$997$	−19.0959	−0.604774	−0.302387	+	0.953185i	$0.597784\pi$
$998$	0	0
$999$	63.7008	2.01541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.3	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-3.19446 q^{3} -0.458477 q^{5} +4.07842 q^{7} +7.20455 q^{9} +O(q^{10})\)	\(q-3.19446 q^{3} -0.458477 q^{5} +4.07842 q^{7} +7.20455 q^{9} -2.22869 q^{11} -6.38789 q^{13} +1.46458 q^{15} -2.59851 q^{17} -0.802188 q^{19} -13.0283 q^{21} -7.99585 q^{23} -4.78980 q^{25} -13.4313 q^{27} -0.0601791 q^{29} -7.44808 q^{31} +7.11945 q^{33} -1.86986 q^{35} -4.74273 q^{37} +20.4058 q^{39} +1.35088 q^{41} -4.32948 q^{43} -3.30312 q^{45} +9.08405 q^{47} +9.63349 q^{49} +8.30083 q^{51} +1.32097 q^{53} +1.02180 q^{55} +2.56255 q^{57} +8.33609 q^{59} -4.02739 q^{61} +29.3832 q^{63} +2.92870 q^{65} +0.719352 q^{67} +25.5424 q^{69} -6.84968 q^{71} +1.47048 q^{73} +15.3008 q^{75} -9.08952 q^{77} +4.44562 q^{79} +21.2919 q^{81} -7.83408 q^{83} +1.19136 q^{85} +0.192240 q^{87} +8.30858 q^{89} -26.0525 q^{91} +23.7926 q^{93} +0.367785 q^{95} +8.91319 q^{97} -16.0567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 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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