LMFDB - Embedded newform 8044.2.a.b.1.16

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			1.16
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)

$f(q)$	$=$	$q-2.09889 q^{3} +0.451798 q^{5} -0.670101 q^{7} +1.40535 q^{9} +O(q^{10})$	$q-2.09889 q^{3} +0.451798 q^{5} -0.670101 q^{7} +1.40535 q^{9} -5.79366 q^{11} -3.97032 q^{13} -0.948275 q^{15} +2.58096 q^{17} -1.43691 q^{19} +1.40647 q^{21} +8.17684 q^{23} -4.79588 q^{25} +3.34700 q^{27} -6.80994 q^{29} -6.22992 q^{31} +12.1603 q^{33} -0.302750 q^{35} +1.42916 q^{37} +8.33327 q^{39} -2.72534 q^{41} +1.72298 q^{43} +0.634934 q^{45} -6.65860 q^{47} -6.55096 q^{49} -5.41716 q^{51} -10.9880 q^{53} -2.61756 q^{55} +3.01592 q^{57} -0.223201 q^{59} -9.81346 q^{61} -0.941726 q^{63} -1.79378 q^{65} +9.79146 q^{67} -17.1623 q^{69} -10.4737 q^{71} +3.49235 q^{73} +10.0660 q^{75} +3.88234 q^{77} -14.2719 q^{79} -11.2410 q^{81} +9.99675 q^{83} +1.16607 q^{85} +14.2933 q^{87} +6.88623 q^{89} +2.66051 q^{91} +13.0759 q^{93} -0.649193 q^{95} +0.343986 q^{97} -8.14212 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.09889	−1.21180	−0.605898	−	0.795542i	$-0.707186\pi$
$3$	−2.09889	−1.21180	−0.605898	+	0.795542i	$0.707186\pi$
$4$	0	0
$5$	0.451798	0.202050	0.101025	−	0.994884i	$-0.467788\pi$
$5$	0.451798	0.202050	0.101025	+	0.994884i	$0.467788\pi$
$6$	0	0
$7$	−0.670101	−0.253275	−0.126637	−	0.991949i	$-0.540418\pi$
$7$	−0.670101	−0.253275	−0.126637	+	0.991949i	$0.540418\pi$
$8$	0	0
$9$	1.40535	0.468450
$10$	0	0
$11$	−5.79366	−1.74685	−0.873427	−	0.486955i	$-0.838108\pi$
$11$	−5.79366	−1.74685	−0.873427	+	0.486955i	$0.838108\pi$
$12$	0	0
$13$	−3.97032	−1.10117	−0.550584	−	0.834780i	$-0.685595\pi$
$13$	−3.97032	−1.10117	−0.550584	+	0.834780i	$0.685595\pi$
$14$	0	0
$15$	−0.948275	−0.244844
$16$	0	0
$17$	2.58096	0.625975	0.312987	−	0.949757i	$-0.398670\pi$
$17$	2.58096	0.625975	0.312987	+	0.949757i	$0.398670\pi$
$18$	0	0
$19$	−1.43691	−0.329650	−0.164825	−	0.986323i	$-0.552706\pi$
$19$	−1.43691	−0.329650	−0.164825	+	0.986323i	$0.552706\pi$
$20$	0	0
$21$	1.40647	0.306917
$22$	0	0
$23$	8.17684	1.70499	0.852495	−	0.522736i	$-0.175089\pi$
$23$	8.17684	1.70499	0.852495	+	0.522736i	$0.175089\pi$
$24$	0	0
$25$	−4.79588	−0.959176
$26$	0	0
$27$	3.34700	0.644131
$28$	0	0
$29$	−6.80994	−1.26457	−0.632287	−	0.774734i	$-0.717884\pi$
$29$	−6.80994	−1.26457	−0.632287	+	0.774734i	$0.717884\pi$
$30$	0	0
$31$	−6.22992	−1.11893	−0.559463	−	0.828855i	$-0.688993\pi$
$31$	−6.22992	−1.11893	−0.559463	+	0.828855i	$0.688993\pi$
$32$	0	0
$33$	12.1603	2.11683
$34$	0	0
$35$	−0.302750	−0.0511741
$36$	0	0
$37$	1.42916	0.234953	0.117477	−	0.993076i	$-0.462519\pi$
$37$	1.42916	0.234953	0.117477	+	0.993076i	$0.462519\pi$
$38$	0	0
$39$	8.33327	1.33439
$40$	0	0
$41$	−2.72534	−0.425626	−0.212813	−	0.977093i	$-0.568263\pi$
$41$	−2.72534	−0.425626	−0.212813	+	0.977093i	$0.568263\pi$
$42$	0	0
$43$	1.72298	0.262753	0.131376	−	0.991333i	$-0.458060\pi$
$43$	1.72298	0.262753	0.131376	+	0.991333i	$0.458060\pi$
$44$	0	0
$45$	0.634934	0.0946503
$46$	0	0
$47$	−6.65860	−0.971256	−0.485628	−	0.874165i	$-0.661409\pi$
$47$	−6.65860	−0.971256	−0.485628	+	0.874165i	$0.661409\pi$
$48$	0	0
$49$	−6.55096	−0.935852
$50$	0	0
$51$	−5.41716	−0.758554
$52$	0	0
$53$	−10.9880	−1.50932	−0.754660	−	0.656116i	$-0.772198\pi$
$53$	−10.9880	−1.50932	−0.754660	+	0.656116i	$0.772198\pi$
$54$	0	0
$55$	−2.61756	−0.352952
$56$	0	0
$57$	3.01592	0.399468
$58$	0	0
$59$	−0.223201	−0.0290583	−0.0145291	−	0.999894i	$-0.504625\pi$
$59$	−0.223201	−0.0290583	−0.0145291	+	0.999894i	$0.504625\pi$
$60$	0	0
$61$	−9.81346	−1.25648	−0.628242	−	0.778018i	$-0.716225\pi$
$61$	−9.81346	−1.25648	−0.628242	+	0.778018i	$0.716225\pi$
$62$	0	0
$63$	−0.941726	−0.118646
$64$	0	0
$65$	−1.79378	−0.222491
$66$	0	0
$67$	9.79146	1.19622	0.598108	−	0.801415i	$-0.295919\pi$
$67$	9.79146	1.19622	0.598108	+	0.801415i	$0.295919\pi$
$68$	0	0
$69$	−17.1623	−2.06610
$70$	0	0
$71$	−10.4737	−1.24300	−0.621502	−	0.783412i	$-0.713477\pi$
$71$	−10.4737	−1.24300	−0.621502	+	0.783412i	$0.713477\pi$
$72$	0	0
$73$	3.49235	0.408749	0.204374	−	0.978893i	$-0.434484\pi$
$73$	3.49235	0.408749	0.204374	+	0.978893i	$0.434484\pi$
$74$	0	0
$75$	10.0660	1.16233
$76$	0	0
$77$	3.88234	0.442434
$78$	0	0
$79$	−14.2719	−1.60571	−0.802855	−	0.596174i	$-0.796687\pi$
$79$	−14.2719	−1.60571	−0.802855	+	0.596174i	$0.796687\pi$
$80$	0	0
$81$	−11.2410	−1.24900
$82$	0	0
$83$	9.99675	1.09729	0.548643	−	0.836057i	$-0.315145\pi$
$83$	9.99675	1.09729	0.548643	+	0.836057i	$0.315145\pi$
$84$	0	0
$85$	1.16607	0.126478
$86$	0	0
$87$	14.2933	1.53241
$88$	0	0
$89$	6.88623	0.729939	0.364969	−	0.931020i	$-0.381079\pi$
$89$	6.88623	0.729939	0.364969	+	0.931020i	$0.381079\pi$
$90$	0	0
$91$	2.66051	0.278898
$92$	0	0
$93$	13.0759	1.35591
$94$	0	0
$95$	−0.649193	−0.0666058
$96$	0	0
$97$	0.343986	0.0349265	0.0174632	−	0.999848i	$-0.494441\pi$
$97$	0.343986	0.0349265	0.0174632	+	0.999848i	$0.494441\pi$
$98$	0	0
$99$	−8.14212	−0.818313
$100$	0	0
$101$	−14.3703	−1.42990	−0.714948	−	0.699178i	$-0.753550\pi$
$101$	−14.3703	−1.42990	−0.714948	+	0.699178i	$0.753550\pi$
$102$	0	0
$103$	0.700900	0.0690618	0.0345309	−	0.999404i	$-0.489006\pi$
$103$	0.700900	0.0690618	0.0345309	+	0.999404i	$0.489006\pi$
$104$	0	0
$105$	0.635440	0.0620126
$106$	0	0
$107$	−2.02792	−0.196047	−0.0980233	−	0.995184i	$-0.531252\pi$
$107$	−2.02792	−0.196047	−0.0980233	+	0.995184i	$0.531252\pi$
$108$	0	0
$109$	7.86105	0.752952	0.376476	−	0.926426i	$-0.377136\pi$
$109$	7.86105	0.752952	0.376476	+	0.926426i	$0.377136\pi$
$110$	0	0
$111$	−2.99966	−0.284715
$112$	0	0
$113$	10.1502	0.954849	0.477425	−	0.878673i	$-0.341570\pi$
$113$	10.1502	0.954849	0.477425	+	0.878673i	$0.341570\pi$
$114$	0	0
$115$	3.69428	0.344493
$116$	0	0
$117$	−5.57968	−0.515842
$118$	0	0
$119$	−1.72950	−0.158543
$120$	0	0
$121$	22.5665	2.05150
$122$	0	0
$123$	5.72019	0.515772
$124$	0	0
$125$	−4.42576	−0.395852
$126$	0	0
$127$	15.4899	1.37450	0.687252	−	0.726419i	$-0.258817\pi$
$127$	15.4899	1.37450	0.687252	+	0.726419i	$0.258817\pi$
$128$	0	0
$129$	−3.61636	−0.318403
$130$	0	0
$131$	−4.49929	−0.393105	−0.196553	−	0.980493i	$-0.562975\pi$
$131$	−4.49929	−0.393105	−0.196553	+	0.980493i	$0.562975\pi$
$132$	0	0
$133$	0.962875	0.0834918
$134$	0	0
$135$	1.51217	0.130147
$136$	0	0
$137$	−4.95865	−0.423646	−0.211823	−	0.977308i	$-0.567940\pi$
$137$	−4.95865	−0.423646	−0.211823	+	0.977308i	$0.567940\pi$
$138$	0	0
$139$	16.1268	1.36786	0.683930	−	0.729548i	$-0.260269\pi$
$139$	16.1268	1.36786	0.683930	+	0.729548i	$0.260269\pi$
$140$	0	0
$141$	13.9757	1.17696
$142$	0	0
$143$	23.0027	1.92358
$144$	0	0
$145$	−3.07672	−0.255507
$146$	0	0
$147$	13.7498	1.13406
$148$	0	0
$149$	−21.2745	−1.74287	−0.871436	−	0.490508i	$-0.836811\pi$
$149$	−21.2745	−1.74287	−0.871436	+	0.490508i	$0.836811\pi$
$150$	0	0
$151$	3.92308	0.319256	0.159628	−	0.987177i	$-0.448971\pi$
$151$	3.92308	0.319256	0.159628	+	0.987177i	$0.448971\pi$
$152$	0	0
$153$	3.62715	0.293238
$154$	0	0
$155$	−2.81467	−0.226079
$156$	0	0
$157$	−18.2985	−1.46038	−0.730190	−	0.683245i	$-0.760568\pi$
$157$	−18.2985	−1.46038	−0.730190	+	0.683245i	$0.760568\pi$
$158$	0	0
$159$	23.0627	1.82899
$160$	0	0
$161$	−5.47931	−0.431830
$162$	0	0
$163$	3.39824	0.266171	0.133086	−	0.991105i	$-0.457512\pi$
$163$	3.39824	0.266171	0.133086	+	0.991105i	$0.457512\pi$
$164$	0	0
$165$	5.49398	0.427706
$166$	0	0
$167$	−1.17524	−0.0909430	−0.0454715	−	0.998966i	$-0.514479\pi$
$167$	−1.17524	−0.0909430	−0.0454715	+	0.998966i	$0.514479\pi$
$168$	0	0
$169$	2.76341	0.212570
$170$	0	0
$171$	−2.01936	−0.154424
$172$	0	0
$173$	−18.6097	−1.41487	−0.707434	−	0.706780i	$-0.750147\pi$
$173$	−18.6097	−1.41487	−0.707434	+	0.706780i	$0.750147\pi$
$174$	0	0
$175$	3.21372	0.242935
$176$	0	0
$177$	0.468474	0.0352127
$178$	0	0
$179$	11.8029	0.882188	0.441094	−	0.897461i	$-0.354591\pi$
$179$	11.8029	0.882188	0.441094	+	0.897461i	$0.354591\pi$
$180$	0	0
$181$	−3.78472	−0.281316	−0.140658	−	0.990058i	$-0.544922\pi$
$181$	−3.78472	−0.281316	−0.140658	+	0.990058i	$0.544922\pi$
$182$	0	0
$183$	20.5974	1.52260
$184$	0	0
$185$	0.645693	0.0474723
$186$	0	0
$187$	−14.9532	−1.09349
$188$	0	0
$189$	−2.24283	−0.163142
$190$	0	0
$191$	−10.2371	−0.740731	−0.370366	−	0.928886i	$-0.620768\pi$
$191$	−10.2371	−0.740731	−0.370366	+	0.928886i	$0.620768\pi$
$192$	0	0
$193$	17.7882	1.28042	0.640212	−	0.768198i	$-0.278846\pi$
$193$	17.7882	1.28042	0.640212	+	0.768198i	$0.278846\pi$
$194$	0	0
$195$	3.76495	0.269614
$196$	0	0
$197$	2.28100	0.162515	0.0812573	−	0.996693i	$-0.474106\pi$
$197$	2.28100	0.162515	0.0812573	+	0.996693i	$0.474106\pi$
$198$	0	0
$199$	−6.58558	−0.466839	−0.233420	−	0.972376i	$-0.574992\pi$
$199$	−6.58558	−0.466839	−0.233420	+	0.972376i	$0.574992\pi$
$200$	0	0
$201$	−20.5512	−1.44957
$202$	0	0
$203$	4.56335	0.320284
$204$	0	0
$205$	−1.23130	−0.0859978
$206$	0	0
$207$	11.4913	0.798702
$208$	0	0
$209$	8.32496	0.575850
$210$	0	0
$211$	−12.6671	−0.872036	−0.436018	−	0.899938i	$-0.643612\pi$
$211$	−12.6671	−0.872036	−0.436018	+	0.899938i	$0.643612\pi$
$212$	0	0
$213$	21.9833	1.50627
$214$	0	0
$215$	0.778441	0.0530892
$216$	0	0
$217$	4.17468	0.283396
$218$	0	0
$219$	−7.33007	−0.495320
$220$	0	0
$221$	−10.2472	−0.689303
$222$	0	0
$223$	−11.1204	−0.744679	−0.372340	−	0.928097i	$-0.621444\pi$
$223$	−11.1204	−0.744679	−0.372340	+	0.928097i	$0.621444\pi$
$224$	0	0
$225$	−6.73988	−0.449326
$226$	0	0
$227$	−11.3084	−0.750564	−0.375282	−	0.926911i	$-0.622454\pi$
$227$	−11.3084	−0.750564	−0.375282	+	0.926911i	$0.622454\pi$
$228$	0	0
$229$	−1.78101	−0.117692	−0.0588461	−	0.998267i	$-0.518742\pi$
$229$	−1.78101	−0.117692	−0.0588461	+	0.998267i	$0.518742\pi$
$230$	0	0
$231$	−8.14861	−0.536139
$232$	0	0
$233$	−29.5470	−1.93569	−0.967844	−	0.251551i	$-0.919059\pi$
$233$	−29.5470	−1.93569	−0.967844	+	0.251551i	$0.919059\pi$
$234$	0	0
$235$	−3.00834	−0.196242
$236$	0	0
$237$	29.9551	1.94579
$238$	0	0
$239$	17.7983	1.15128	0.575639	−	0.817704i	$-0.304754\pi$
$239$	17.7983	1.15128	0.575639	+	0.817704i	$0.304754\pi$
$240$	0	0
$241$	4.61554	0.297313	0.148656	−	0.988889i	$-0.452505\pi$
$241$	4.61554	0.297313	0.148656	+	0.988889i	$0.452505\pi$
$242$	0	0
$243$	13.5527	0.869408
$244$	0	0
$245$	−2.95971	−0.189089
$246$	0	0
$247$	5.70498	0.362999
$248$	0	0
$249$	−20.9821	−1.32969
$250$	0	0
$251$	21.3672	1.34869	0.674343	−	0.738418i	$-0.264427\pi$
$251$	21.3672	1.34869	0.674343	+	0.738418i	$0.264427\pi$
$252$	0	0
$253$	−47.3738	−2.97837
$254$	0	0
$255$	−2.44746	−0.153266
$256$	0	0
$257$	23.2925	1.45295	0.726474	−	0.687194i	$-0.241158\pi$
$257$	23.2925	1.45295	0.726474	+	0.687194i	$0.241158\pi$
$258$	0	0
$259$	−0.957685	−0.0595076
$260$	0	0
$261$	−9.57035	−0.592390
$262$	0	0
$263$	1.61367	0.0995030	0.0497515	−	0.998762i	$-0.484157\pi$
$263$	1.61367	0.0995030	0.0497515	+	0.998762i	$0.484157\pi$
$264$	0	0
$265$	−4.96436	−0.304958
$266$	0	0
$267$	−14.4535	−0.884537
$268$	0	0
$269$	−2.56830	−0.156592	−0.0782959	−	0.996930i	$-0.524948\pi$
$269$	−2.56830	−0.156592	−0.0782959	+	0.996930i	$0.524948\pi$
$270$	0	0
$271$	30.2225	1.83589	0.917943	−	0.396712i	$-0.129849\pi$
$271$	30.2225	1.83589	0.917943	+	0.396712i	$0.129849\pi$
$272$	0	0
$273$	−5.58413	−0.337967
$274$	0	0
$275$	27.7857	1.67554
$276$	0	0
$277$	27.4255	1.64784	0.823920	−	0.566706i	$-0.191782\pi$
$277$	27.4255	1.64784	0.823920	+	0.566706i	$0.191782\pi$
$278$	0	0
$279$	−8.75522	−0.524161
$280$	0	0
$281$	11.3108	0.674747	0.337374	−	0.941371i	$-0.390461\pi$
$281$	11.3108	0.674747	0.337374	+	0.941371i	$0.390461\pi$
$282$	0	0
$283$	15.6017	0.927424	0.463712	−	0.885986i	$-0.346517\pi$
$283$	15.6017	0.927424	0.463712	+	0.885986i	$0.346517\pi$
$284$	0	0
$285$	1.36259	0.0807126
$286$	0	0
$287$	1.82625	0.107800
$288$	0	0
$289$	−10.3386	−0.608156
$290$	0	0
$291$	−0.721990	−0.0423238
$292$	0	0
$293$	29.5110	1.72405	0.862024	−	0.506867i	$-0.169197\pi$
$293$	29.5110	1.72405	0.862024	+	0.506867i	$0.169197\pi$
$294$	0	0
$295$	−0.100842	−0.00587122
$296$	0	0
$297$	−19.3914	−1.12520
$298$	0	0
$299$	−32.4646	−1.87748
$300$	0	0
$301$	−1.15457	−0.0665485
$302$	0	0
$303$	30.1617	1.73274
$304$	0	0
$305$	−4.43370	−0.253873
$306$	0	0
$307$	−20.2681	−1.15676	−0.578380	−	0.815767i	$-0.696315\pi$
$307$	−20.2681	−1.15676	−0.578380	+	0.815767i	$0.696315\pi$
$308$	0	0
$309$	−1.47111	−0.0836888
$310$	0	0
$311$	24.1815	1.37121	0.685603	−	0.727976i	$-0.259539\pi$
$311$	24.1815	1.37121	0.685603	+	0.727976i	$0.259539\pi$
$312$	0	0
$313$	15.6838	0.886502	0.443251	−	0.896398i	$-0.353825\pi$
$313$	15.6838	0.886502	0.443251	+	0.896398i	$0.353825\pi$
$314$	0	0
$315$	−0.425470	−0.0239725
$316$	0	0
$317$	−20.5677	−1.15520	−0.577599	−	0.816321i	$-0.696010\pi$
$317$	−20.5677	−1.15520	−0.577599	+	0.816321i	$0.696010\pi$
$318$	0	0
$319$	39.4545	2.20903
$320$	0	0
$321$	4.25639	0.237569
$322$	0	0
$323$	−3.70861	−0.206352
$324$	0	0
$325$	19.0412	1.05621
$326$	0	0
$327$	−16.4995	−0.912425
$328$	0	0
$329$	4.46194	0.245994
$330$	0	0
$331$	−20.9059	−1.14909	−0.574547	−	0.818472i	$-0.694822\pi$
$331$	−20.9059	−1.14909	−0.574547	+	0.818472i	$0.694822\pi$
$332$	0	0
$333$	2.00847	0.110064
$334$	0	0
$335$	4.42376	0.241696
$336$	0	0
$337$	18.6834	1.01775	0.508876	−	0.860840i	$-0.330061\pi$
$337$	18.6834	1.01775	0.508876	+	0.860840i	$0.330061\pi$
$338$	0	0
$339$	−21.3041	−1.15708
$340$	0	0
$341$	36.0940	1.95460
$342$	0	0
$343$	9.08052	0.490302
$344$	0	0
$345$	−7.75390	−0.417456
$346$	0	0
$347$	−5.24159	−0.281383	−0.140692	−	0.990053i	$-0.544933\pi$
$347$	−5.24159	−0.281383	−0.140692	+	0.990053i	$0.544933\pi$
$348$	0	0
$349$	−35.4487	−1.89753	−0.948763	−	0.315988i	$-0.897664\pi$
$349$	−35.4487	−1.89753	−0.948763	+	0.315988i	$0.897664\pi$
$350$	0	0
$351$	−13.2886	−0.709296
$352$	0	0
$353$	13.8846	0.739003	0.369502	−	0.929230i	$-0.379528\pi$
$353$	13.8846	0.739003	0.369502	+	0.929230i	$0.379528\pi$
$354$	0	0
$355$	−4.73201	−0.251149
$356$	0	0
$357$	3.63004	0.192122
$358$	0	0
$359$	−29.9207	−1.57916	−0.789578	−	0.613651i	$-0.789700\pi$
$359$	−29.9207	−1.57916	−0.789578	+	0.613651i	$0.789700\pi$
$360$	0	0
$361$	−16.9353	−0.891331
$362$	0	0
$363$	−47.3646	−2.48600
$364$	0	0
$365$	1.57784	0.0825878
$366$	0	0
$367$	27.3770	1.42907	0.714533	−	0.699602i	$-0.246639\pi$
$367$	27.3770	1.42907	0.714533	+	0.699602i	$0.246639\pi$
$368$	0	0
$369$	−3.83005	−0.199384
$370$	0	0
$371$	7.36308	0.382272
$372$	0	0
$373$	−15.4177	−0.798297	−0.399148	−	0.916886i	$-0.630694\pi$
$373$	−15.4177	−0.798297	−0.399148	+	0.916886i	$0.630694\pi$
$374$	0	0
$375$	9.28919	0.479692
$376$	0	0
$377$	27.0376	1.39251
$378$	0	0
$379$	20.3007	1.04278	0.521389	−	0.853319i	$-0.325414\pi$
$379$	20.3007	1.04278	0.521389	+	0.853319i	$0.325414\pi$
$380$	0	0
$381$	−32.5116	−1.66562
$382$	0	0
$383$	−7.31509	−0.373784	−0.186892	−	0.982381i	$-0.559841\pi$
$383$	−7.31509	−0.373784	−0.186892	+	0.982381i	$0.559841\pi$
$384$	0	0
$385$	1.75403	0.0893938
$386$	0	0
$387$	2.42139	0.123086
$388$	0	0
$389$	24.1013	1.22198	0.610991	−	0.791638i	$-0.290771\pi$
$389$	24.1013	1.22198	0.610991	+	0.791638i	$0.290771\pi$
$390$	0	0
$391$	21.1041	1.06728
$392$	0	0
$393$	9.44353	0.476363
$394$	0	0
$395$	−6.44800	−0.324434
$396$	0	0
$397$	−34.6918	−1.74113	−0.870565	−	0.492054i	$-0.836246\pi$
$397$	−34.6918	−1.74113	−0.870565	+	0.492054i	$0.836246\pi$
$398$	0	0
$399$	−2.02097	−0.101175
$400$	0	0
$401$	−1.12904	−0.0563818	−0.0281909	−	0.999603i	$-0.508975\pi$
$401$	−1.12904	−0.0563818	−0.0281909	+	0.999603i	$0.508975\pi$
$402$	0	0
$403$	24.7348	1.23213
$404$	0	0
$405$	−5.07868	−0.252362
$406$	0	0
$407$	−8.28009	−0.410429
$408$	0	0
$409$	13.2759	0.656450	0.328225	−	0.944599i	$-0.393550\pi$
$409$	13.2759	0.656450	0.328225	+	0.944599i	$0.393550\pi$
$410$	0	0
$411$	10.4077	0.513372
$412$	0	0
$413$	0.149567	0.00735972
$414$	0	0
$415$	4.51651	0.221707
$416$	0	0
$417$	−33.8485	−1.65757
$418$	0	0
$419$	34.4102	1.68105	0.840524	−	0.541775i	$-0.182247\pi$
$419$	34.4102	1.68105	0.840524	+	0.541775i	$0.182247\pi$
$420$	0	0
$421$	−2.88192	−0.140456	−0.0702280	−	0.997531i	$-0.522373\pi$
$421$	−2.88192	−0.140456	−0.0702280	+	0.997531i	$0.522373\pi$
$422$	0	0
$423$	−9.35766	−0.454985
$424$	0	0
$425$	−12.3780	−0.600420
$426$	0	0
$427$	6.57601	0.318236
$428$	0	0
$429$	−48.2801	−2.33099
$430$	0	0
$431$	8.16017	0.393062	0.196531	−	0.980498i	$-0.437032\pi$
$431$	8.16017	0.393062	0.196531	+	0.980498i	$0.437032\pi$
$432$	0	0
$433$	−10.7318	−0.515736	−0.257868	−	0.966180i	$-0.583020\pi$
$433$	−10.7318	−0.515736	−0.257868	+	0.966180i	$0.583020\pi$
$434$	0	0
$435$	6.45770	0.309623
$436$	0	0
$437$	−11.7494	−0.562049
$438$	0	0
$439$	2.53413	0.120947	0.0604737	−	0.998170i	$-0.480739\pi$
$439$	2.53413	0.120947	0.0604737	+	0.998170i	$0.480739\pi$
$440$	0	0
$441$	−9.20639	−0.438400
$442$	0	0
$443$	−11.2533	−0.534659	−0.267330	−	0.963605i	$-0.586141\pi$
$443$	−11.2533	−0.534659	−0.267330	+	0.963605i	$0.586141\pi$
$444$	0	0
$445$	3.11118	0.147484
$446$	0	0
$447$	44.6528	2.11201
$448$	0	0
$449$	28.4304	1.34171	0.670857	−	0.741587i	$-0.265927\pi$
$449$	28.4304	1.34171	0.670857	+	0.741587i	$0.265927\pi$
$450$	0	0
$451$	15.7897	0.743507
$452$	0	0
$453$	−8.23413	−0.386873
$454$	0	0
$455$	1.20201	0.0563513
$456$	0	0
$457$	4.01804	0.187956	0.0939780	−	0.995574i	$-0.470042\pi$
$457$	4.01804	0.187956	0.0939780	+	0.995574i	$0.470042\pi$
$458$	0	0
$459$	8.63847	0.403209
$460$	0	0
$461$	1.50223	0.0699660	0.0349830	−	0.999388i	$-0.488862\pi$
$461$	1.50223	0.0699660	0.0349830	+	0.999388i	$0.488862\pi$
$462$	0	0
$463$	36.7961	1.71006	0.855031	−	0.518577i	$-0.173538\pi$
$463$	36.7961	1.71006	0.855031	+	0.518577i	$0.173538\pi$
$464$	0	0
$465$	5.90768	0.273962
$466$	0	0
$467$	2.08106	0.0963001	0.0481500	−	0.998840i	$-0.484667\pi$
$467$	2.08106	0.0963001	0.0481500	+	0.998840i	$0.484667\pi$
$468$	0	0
$469$	−6.56127	−0.302971
$470$	0	0
$471$	38.4066	1.76968
$472$	0	0
$473$	−9.98238	−0.458991
$474$	0	0
$475$	6.89124	0.316192
$476$	0	0
$477$	−15.4420	−0.707041
$478$	0	0
$479$	38.7510	1.77058	0.885290	−	0.465039i	$-0.153960\pi$
$479$	38.7510	1.77058	0.885290	+	0.465039i	$0.153960\pi$
$480$	0	0
$481$	−5.67423	−0.258723
$482$	0	0
$483$	11.5005	0.523290
$484$	0	0
$485$	0.155412	0.00705690
$486$	0	0
$487$	−31.1208	−1.41022	−0.705110	−	0.709098i	$-0.749102\pi$
$487$	−31.1208	−1.41022	−0.705110	+	0.709098i	$0.749102\pi$
$488$	0	0
$489$	−7.13255	−0.322545
$490$	0	0
$491$	−34.2835	−1.54719	−0.773596	−	0.633679i	$-0.781544\pi$
$491$	−34.2835	−1.54719	−0.773596	+	0.633679i	$0.781544\pi$
$492$	0	0
$493$	−17.5762	−0.791592
$494$	0	0
$495$	−3.67859	−0.165340
$496$	0	0
$497$	7.01847	0.314821
$498$	0	0
$499$	−14.0422	−0.628616	−0.314308	−	0.949321i	$-0.601773\pi$
$499$	−14.0422	−0.628616	−0.314308	+	0.949321i	$0.601773\pi$
$500$	0	0
$501$	2.46671	0.110204
$502$	0	0
$503$	19.5734	0.872733	0.436366	−	0.899769i	$-0.356265\pi$
$503$	19.5734	0.872733	0.436366	+	0.899769i	$0.356265\pi$
$504$	0	0
$505$	−6.49246	−0.288911
$506$	0	0
$507$	−5.80010	−0.257591
$508$	0	0
$509$	3.13231	0.138837	0.0694186	−	0.997588i	$-0.477886\pi$
$509$	3.13231	0.138837	0.0694186	+	0.997588i	$0.477886\pi$
$510$	0	0
$511$	−2.34023	−0.103526
$512$	0	0
$513$	−4.80934	−0.212337
$514$	0	0
$515$	0.316665	0.0139539
$516$	0	0
$517$	38.5777	1.69664
$518$	0	0
$519$	39.0597	1.71453
$520$	0	0
$521$	−23.7235	−1.03935	−0.519673	−	0.854365i	$-0.673946\pi$
$521$	−23.7235	−1.03935	−0.519673	+	0.854365i	$0.673946\pi$
$522$	0	0
$523$	−10.6942	−0.467627	−0.233813	−	0.972282i	$-0.575120\pi$
$523$	−10.6942	−0.467627	−0.233813	+	0.972282i	$0.575120\pi$
$524$	0	0
$525$	−6.74526	−0.294387
$526$	0	0
$527$	−16.0792	−0.700420
$528$	0	0
$529$	43.8607	1.90699
$530$	0	0
$531$	−0.313675	−0.0136123
$532$	0	0
$533$	10.8204	0.468686
$534$	0	0
$535$	−0.916211	−0.0396113
$536$	0	0
$537$	−24.7730	−1.06903
$538$	0	0
$539$	37.9541	1.63480
$540$	0	0
$541$	24.3570	1.04719	0.523595	−	0.851967i	$-0.324591\pi$
$541$	24.3570	1.04719	0.523595	+	0.851967i	$0.324591\pi$
$542$	0	0
$543$	7.94371	0.340897
$544$	0	0
$545$	3.55161	0.152134
$546$	0	0
$547$	−11.5072	−0.492013	−0.246007	−	0.969268i	$-0.579118\pi$
$547$	−11.5072	−0.492013	−0.246007	+	0.969268i	$0.579118\pi$
$548$	0	0
$549$	−13.7913	−0.588600
$550$	0	0
$551$	9.78527	0.416866
$552$	0	0
$553$	9.56360	0.406686
$554$	0	0
$555$	−1.35524	−0.0575268
$556$	0	0
$557$	20.0212	0.848325	0.424162	−	0.905586i	$-0.360569\pi$
$557$	20.0212	0.848325	0.424162	+	0.905586i	$0.360569\pi$
$558$	0	0
$559$	−6.84079	−0.289335
$560$	0	0
$561$	31.3852	1.32508
$562$	0	0
$563$	−8.94930	−0.377168	−0.188584	−	0.982057i	$-0.560390\pi$
$563$	−8.94930	−0.377168	−0.188584	+	0.982057i	$0.560390\pi$
$564$	0	0
$565$	4.58583	0.192927
$566$	0	0
$567$	7.53264	0.316341
$568$	0	0
$569$	32.2315	1.35122	0.675608	−	0.737261i	$-0.263881\pi$
$569$	32.2315	1.35122	0.675608	+	0.737261i	$0.263881\pi$
$570$	0	0
$571$	28.4846	1.19204	0.596022	−	0.802968i	$-0.296747\pi$
$571$	28.4846	1.19204	0.596022	+	0.802968i	$0.296747\pi$
$572$	0	0
$573$	21.4866	0.897615
$574$	0	0
$575$	−39.2151	−1.63538
$576$	0	0
$577$	27.0355	1.12550	0.562752	−	0.826626i	$-0.309743\pi$
$577$	27.0355	1.12550	0.562752	+	0.826626i	$0.309743\pi$
$578$	0	0
$579$	−37.3356	−1.55161
$580$	0	0
$581$	−6.69884	−0.277915
$582$	0	0
$583$	63.6608	2.63656
$584$	0	0
$585$	−2.52089	−0.104226
$586$	0	0
$587$	−25.0229	−1.03281	−0.516404	−	0.856345i	$-0.672730\pi$
$587$	−25.0229	−1.03281	−0.516404	+	0.856345i	$0.672730\pi$
$588$	0	0
$589$	8.95183	0.368854
$590$	0	0
$591$	−4.78758	−0.196935
$592$	0	0
$593$	20.3534	0.835815	0.417908	−	0.908490i	$-0.362764\pi$
$593$	20.3534	0.835815	0.417908	+	0.908490i	$0.362764\pi$
$594$	0	0
$595$	−0.781387	−0.0320337
$596$	0	0
$597$	13.8224	0.565714
$598$	0	0
$599$	24.0326	0.981945	0.490972	−	0.871175i	$-0.336642\pi$
$599$	24.0326	0.981945	0.490972	+	0.871175i	$0.336642\pi$
$600$	0	0
$601$	−22.1521	−0.903601	−0.451801	−	0.892119i	$-0.649218\pi$
$601$	−22.1521	−0.903601	−0.451801	+	0.892119i	$0.649218\pi$
$602$	0	0
$603$	13.7604	0.560367
$604$	0	0
$605$	10.1955	0.414506
$606$	0	0
$607$	−26.1060	−1.05961	−0.529805	−	0.848119i	$-0.677735\pi$
$607$	−26.1060	−1.05961	−0.529805	+	0.848119i	$0.677735\pi$
$608$	0	0
$609$	−9.57798	−0.388119
$610$	0	0
$611$	26.4367	1.06952
$612$	0	0
$613$	11.2999	0.456398	0.228199	−	0.973615i	$-0.426716\pi$
$613$	11.2999	0.456398	0.228199	+	0.973615i	$0.426716\pi$
$614$	0	0
$615$	2.58437	0.104212
$616$	0	0
$617$	22.9849	0.925338	0.462669	−	0.886531i	$-0.346892\pi$
$617$	22.9849	0.925338	0.462669	+	0.886531i	$0.346892\pi$
$618$	0	0
$619$	−7.82283	−0.314426	−0.157213	−	0.987565i	$-0.550251\pi$
$619$	−7.82283	−0.314426	−0.157213	+	0.987565i	$0.550251\pi$
$620$	0	0
$621$	27.3679	1.09824
$622$	0	0
$623$	−4.61447	−0.184875
$624$	0	0
$625$	21.9798	0.879194
$626$	0	0
$627$	−17.4732	−0.697813
$628$	0	0
$629$	3.68862	0.147075
$630$	0	0
$631$	9.92915	0.395273	0.197637	−	0.980275i	$-0.436673\pi$
$631$	9.92915	0.395273	0.197637	+	0.980275i	$0.436673\pi$
$632$	0	0
$633$	26.5868	1.05673
$634$	0	0
$635$	6.99829	0.277719
$636$	0	0
$637$	26.0094	1.03053
$638$	0	0
$639$	−14.7193	−0.582285
$640$	0	0
$641$	39.2782	1.55139	0.775697	−	0.631105i	$-0.217398\pi$
$641$	39.2782	1.55139	0.775697	+	0.631105i	$0.217398\pi$
$642$	0	0
$643$	−38.1556	−1.50471	−0.752354	−	0.658759i	$-0.771082\pi$
$643$	−38.1556	−1.50471	−0.752354	+	0.658759i	$0.771082\pi$
$644$	0	0
$645$	−1.63386	−0.0643333
$646$	0	0
$647$	40.4979	1.59214	0.796068	−	0.605208i	$-0.206910\pi$
$647$	40.4979	1.59214	0.796068	+	0.605208i	$0.206910\pi$
$648$	0	0
$649$	1.29315	0.0507605
$650$	0	0
$651$	−8.76220	−0.343418
$652$	0	0
$653$	−17.3653	−0.679558	−0.339779	−	0.940505i	$-0.610352\pi$
$653$	−17.3653	−0.679558	−0.339779	+	0.940505i	$0.610352\pi$
$654$	0	0
$655$	−2.03277	−0.0794269
$656$	0	0
$657$	4.90797	0.191478
$658$	0	0
$659$	47.9589	1.86821	0.934106	−	0.356996i	$-0.116199\pi$
$659$	47.9589	1.86821	0.934106	+	0.356996i	$0.116199\pi$
$660$	0	0
$661$	−22.6216	−0.879879	−0.439939	−	0.898027i	$-0.645000\pi$
$661$	−22.6216	−0.879879	−0.439939	+	0.898027i	$0.645000\pi$
$662$	0	0
$663$	21.5078	0.835295
$664$	0	0
$665$	0.435025	0.0168695
$666$	0	0
$667$	−55.6838	−2.15609
$668$	0	0
$669$	23.3406	0.902400
$670$	0	0
$671$	56.8558	2.19490
$672$	0	0
$673$	6.02761	0.232347	0.116174	−	0.993229i	$-0.462937\pi$
$673$	6.02761	0.232347	0.116174	+	0.993229i	$0.462937\pi$
$674$	0	0
$675$	−16.0518	−0.617834
$676$	0	0
$677$	19.6428	0.754935	0.377467	−	0.926023i	$-0.376795\pi$
$677$	19.6428	0.754935	0.377467	+	0.926023i	$0.376795\pi$
$678$	0	0
$679$	−0.230506	−0.00884599
$680$	0	0
$681$	23.7351	0.909530
$682$	0	0
$683$	50.2221	1.92169	0.960847	−	0.277078i	$-0.0893660\pi$
$683$	50.2221	1.92169	0.960847	+	0.277078i	$0.0893660\pi$
$684$	0	0
$685$	−2.24031	−0.0855977
$686$	0	0
$687$	3.73814	0.142619
$688$	0	0
$689$	43.6259	1.66201
$690$	0	0
$691$	9.75166	0.370971	0.185485	−	0.982647i	$-0.440614\pi$
$691$	9.75166	0.370971	0.185485	+	0.982647i	$0.440614\pi$
$692$	0	0
$693$	5.45604	0.207258
$694$	0	0
$695$	7.28607	0.276376
$696$	0	0
$697$	−7.03399	−0.266431
$698$	0	0
$699$	62.0160	2.34566
$700$	0	0
$701$	25.8899	0.977849	0.488925	−	0.872326i	$-0.337389\pi$
$701$	25.8899	0.977849	0.488925	+	0.872326i	$0.337389\pi$
$702$	0	0
$703$	−2.05358	−0.0774522
$704$	0	0
$705$	6.31418	0.237806
$706$	0	0
$707$	9.62954	0.362156
$708$	0	0
$709$	−4.84690	−0.182029	−0.0910146	−	0.995850i	$-0.529011\pi$
$709$	−4.84690	−0.182029	−0.0910146	+	0.995850i	$0.529011\pi$
$710$	0	0
$711$	−20.0570	−0.752195
$712$	0	0
$713$	−50.9411	−1.90776
$714$	0	0
$715$	10.3926	0.388659
$716$	0	0
$717$	−37.3567	−1.39511
$718$	0	0
$719$	−38.6906	−1.44291	−0.721457	−	0.692459i	$-0.756527\pi$
$719$	−38.6906	−1.44291	−0.721457	+	0.692459i	$0.756527\pi$
$720$	0	0
$721$	−0.469674	−0.0174916
$722$	0	0
$723$	−9.68752	−0.360283
$724$	0	0
$725$	32.6597	1.21295
$726$	0	0
$727$	−45.3940	−1.68357	−0.841785	−	0.539813i	$-0.818495\pi$
$727$	−45.3940	−1.68357	−0.841785	+	0.539813i	$0.818495\pi$
$728$	0	0
$729$	5.27739	0.195459
$730$	0	0
$731$	4.44695	0.164477
$732$	0	0
$733$	−3.33164	−0.123057	−0.0615286	−	0.998105i	$-0.519598\pi$
$733$	−3.33164	−0.123057	−0.0615286	+	0.998105i	$0.519598\pi$
$734$	0	0
$735$	6.21212	0.229137
$736$	0	0
$737$	−56.7284	−2.08962
$738$	0	0
$739$	−29.3101	−1.07819	−0.539095	−	0.842245i	$-0.681233\pi$
$739$	−29.3101	−1.07819	−0.539095	+	0.842245i	$0.681233\pi$
$740$	0	0
$741$	−11.9741	−0.439881
$742$	0	0
$743$	−18.5826	−0.681728	−0.340864	−	0.940113i	$-0.610720\pi$
$743$	−18.5826	−0.681728	−0.340864	+	0.940113i	$0.610720\pi$
$744$	0	0
$745$	−9.61176	−0.352148
$746$	0	0
$747$	14.0489	0.514024
$748$	0	0
$749$	1.35891	0.0496536
$750$	0	0
$751$	−50.9586	−1.85951	−0.929754	−	0.368182i	$-0.879980\pi$
$751$	−50.9586	−1.85951	−0.929754	+	0.368182i	$0.879980\pi$
$752$	0	0
$753$	−44.8475	−1.63433
$754$	0	0
$755$	1.77244	0.0645057
$756$	0	0
$757$	−7.20747	−0.261960	−0.130980	−	0.991385i	$-0.541812\pi$
$757$	−7.20747	−0.261960	−0.130980	+	0.991385i	$0.541812\pi$
$758$	0	0
$759$	99.4326	3.60917
$760$	0	0
$761$	−10.6482	−0.385998	−0.192999	−	0.981199i	$-0.561821\pi$
$761$	−10.6482	−0.385998	−0.192999	+	0.981199i	$0.561821\pi$
$762$	0	0
$763$	−5.26770	−0.190704
$764$	0	0
$765$	1.63874	0.0592487
$766$	0	0
$767$	0.886177	0.0319980
$768$	0	0
$769$	−17.9979	−0.649021	−0.324511	−	0.945882i	$-0.605200\pi$
$769$	−17.9979	−0.649021	−0.324511	+	0.945882i	$0.605200\pi$
$770$	0	0
$771$	−48.8885	−1.76068
$772$	0	0
$773$	21.5986	0.776849	0.388424	−	0.921481i	$-0.373019\pi$
$773$	21.5986	0.776849	0.388424	+	0.921481i	$0.373019\pi$
$774$	0	0
$775$	29.8779	1.07325
$776$	0	0
$777$	2.01008	0.0721111
$778$	0	0
$779$	3.91606	0.140307
$780$	0	0
$781$	60.6813	2.17135
$782$	0	0
$783$	−22.7929	−0.814551
$784$	0	0
$785$	−8.26722	−0.295070
$786$	0	0
$787$	6.56664	0.234075	0.117038	−	0.993127i	$-0.462660\pi$
$787$	6.56664	0.234075	0.117038	+	0.993127i	$0.462660\pi$
$788$	0	0
$789$	−3.38691	−0.120577
$790$	0	0
$791$	−6.80165	−0.241839
$792$	0	0
$793$	38.9625	1.38360
$794$	0	0
$795$	10.4197	0.369547
$796$	0	0
$797$	34.8389	1.23406	0.617029	−	0.786940i	$-0.288336\pi$
$797$	34.8389	1.23406	0.617029	+	0.786940i	$0.288336\pi$
$798$	0	0
$799$	−17.1856	−0.607982
$800$	0	0
$801$	9.67756	0.341940
$802$	0	0
$803$	−20.2335	−0.714025
$804$	0	0
$805$	−2.47554	−0.0872514
$806$	0	0
$807$	5.39057	0.189757
$808$	0	0
$809$	−35.8751	−1.26130	−0.630651	−	0.776066i	$-0.717212\pi$
$809$	−35.8751	−1.26130	−0.630651	+	0.776066i	$0.717212\pi$
$810$	0	0
$811$	19.4910	0.684421	0.342211	−	0.939623i	$-0.388824\pi$
$811$	19.4910	0.684421	0.342211	+	0.939623i	$0.388824\pi$
$812$	0	0
$813$	−63.4338	−2.22472
$814$	0	0
$815$	1.53532	0.0537799
$816$	0	0
$817$	−2.47577	−0.0866163
$818$	0	0
$819$	3.73895	0.130650
$820$	0	0
$821$	−47.1210	−1.64453	−0.822267	−	0.569102i	$-0.807291\pi$
$821$	−47.1210	−1.64453	−0.822267	+	0.569102i	$0.807291\pi$
$822$	0	0
$823$	15.0547	0.524772	0.262386	−	0.964963i	$-0.415491\pi$
$823$	15.0547	0.524772	0.262386	+	0.964963i	$0.415491\pi$
$824$	0	0
$825$	−58.3192	−2.03041
$826$	0	0
$827$	1.88277	0.0654704	0.0327352	−	0.999464i	$-0.489578\pi$
$827$	1.88277	0.0654704	0.0327352	+	0.999464i	$0.489578\pi$
$828$	0	0
$829$	−44.6607	−1.55113	−0.775565	−	0.631267i	$-0.782535\pi$
$829$	−44.6607	−1.55113	−0.775565	+	0.631267i	$0.782535\pi$
$830$	0	0
$831$	−57.5632	−1.99685
$832$	0	0
$833$	−16.9078	−0.585820
$834$	0	0
$835$	−0.530972	−0.0183750
$836$	0	0
$837$	−20.8515	−0.720735
$838$	0	0
$839$	2.15704	0.0744693	0.0372346	−	0.999307i	$-0.488145\pi$
$839$	2.15704	0.0744693	0.0372346	+	0.999307i	$0.488145\pi$
$840$	0	0
$841$	17.3753	0.599148
$842$	0	0
$843$	−23.7402	−0.817656
$844$	0	0
$845$	1.24850	0.0429498
$846$	0	0
$847$	−15.1218	−0.519593
$848$	0	0
$849$	−32.7463	−1.12385
$850$	0	0
$851$	11.6860	0.400593
$852$	0	0
$853$	3.31295	0.113433	0.0567166	−	0.998390i	$-0.481937\pi$
$853$	3.31295	0.113433	0.0567166	+	0.998390i	$0.481937\pi$
$854$	0	0
$855$	−0.912342	−0.0312014
$856$	0	0
$857$	−20.6347	−0.704867	−0.352434	−	0.935837i	$-0.614646\pi$
$857$	−20.6347	−0.704867	−0.352434	+	0.935837i	$0.614646\pi$
$858$	0	0
$859$	−20.3082	−0.692906	−0.346453	−	0.938067i	$-0.612614\pi$
$859$	−20.3082	−0.692906	−0.346453	+	0.938067i	$0.612614\pi$
$860$	0	0
$861$	−3.83311	−0.130632
$862$	0	0
$863$	17.5222	0.596461	0.298231	−	0.954494i	$-0.403604\pi$
$863$	17.5222	0.596461	0.298231	+	0.954494i	$0.403604\pi$
$864$	0	0
$865$	−8.40781	−0.285874
$866$	0	0
$867$	21.6997	0.736961
$868$	0	0
$869$	82.6864	2.80494
$870$	0	0
$871$	−38.8752	−1.31724
$872$	0	0
$873$	0.483421	0.0163613
$874$	0	0
$875$	2.96571	0.100259
$876$	0	0
$877$	−22.5295	−0.760768	−0.380384	−	0.924829i	$-0.624208\pi$
$877$	−22.5295	−0.760768	−0.380384	+	0.924829i	$0.624208\pi$
$878$	0	0
$879$	−61.9403	−2.08920
$880$	0	0
$881$	−15.3569	−0.517387	−0.258693	−	0.965960i	$-0.583292\pi$
$881$	−15.3569	−0.517387	−0.258693	+	0.965960i	$0.583292\pi$
$882$	0	0
$883$	−40.4207	−1.36027	−0.680133	−	0.733089i	$-0.738078\pi$
$883$	−40.4207	−1.36027	−0.680133	+	0.733089i	$0.738078\pi$
$884$	0	0
$885$	0.211656	0.00711473
$886$	0	0
$887$	−57.2605	−1.92262	−0.961309	−	0.275472i	$-0.911166\pi$
$887$	−57.2605	−1.92262	−0.961309	+	0.275472i	$0.911166\pi$
$888$	0	0
$889$	−10.3798	−0.348127
$890$	0	0
$891$	65.1268	2.18183
$892$	0	0
$893$	9.56780	0.320174
$894$	0	0
$895$	5.33251	0.178246
$896$	0	0
$897$	68.1398	2.27512
$898$	0	0
$899$	42.4254	1.41497
$900$	0	0
$901$	−28.3596	−0.944796
$902$	0	0
$903$	2.42333	0.0806433
$904$	0	0
$905$	−1.70993	−0.0568399
$906$	0	0
$907$	13.0610	0.433682	0.216841	−	0.976207i	$-0.430425\pi$
$907$	13.0610	0.433682	0.216841	+	0.976207i	$0.430425\pi$
$908$	0	0
$909$	−20.1952	−0.669834
$910$	0	0
$911$	−13.3982	−0.443901	−0.221951	−	0.975058i	$-0.571242\pi$
$911$	−13.3982	−0.443901	−0.221951	+	0.975058i	$0.571242\pi$
$912$	0	0
$913$	−57.9178	−1.91680
$914$	0	0
$915$	9.30586	0.307642
$916$	0	0
$917$	3.01498	0.0995635
$918$	0	0
$919$	28.1052	0.927106	0.463553	−	0.886069i	$-0.346574\pi$
$919$	28.1052	0.927106	0.463553	+	0.886069i	$0.346574\pi$
$920$	0	0
$921$	42.5405	1.40176
$922$	0	0
$923$	41.5841	1.36876
$924$	0	0
$925$	−6.85410	−0.225361
$926$	0	0
$927$	0.985010	0.0323520
$928$	0	0
$929$	25.7158	0.843708	0.421854	−	0.906664i	$-0.361379\pi$
$929$	25.7158	0.843708	0.421854	+	0.906664i	$0.361379\pi$
$930$	0	0
$931$	9.41314	0.308503
$932$	0	0
$933$	−50.7543	−1.66162
$934$	0	0
$935$	−6.75583	−0.220939
$936$	0	0
$937$	21.6758	0.708119	0.354059	−	0.935223i	$-0.384801\pi$
$937$	21.6758	0.708119	0.354059	+	0.935223i	$0.384801\pi$
$938$	0	0
$939$	−32.9186	−1.07426
$940$	0	0
$941$	46.5491	1.51746	0.758729	−	0.651407i	$-0.225821\pi$
$941$	46.5491	1.51746	0.758729	+	0.651407i	$0.225821\pi$
$942$	0	0
$943$	−22.2846	−0.725688
$944$	0	0
$945$	−1.01331	−0.0329628
$946$	0	0
$947$	8.24300	0.267862	0.133931	−	0.990991i	$-0.457240\pi$
$947$	8.24300	0.267862	0.133931	+	0.990991i	$0.457240\pi$
$948$	0	0
$949$	−13.8657	−0.450101
$950$	0	0
$951$	43.1694	1.39986
$952$	0	0
$953$	39.7322	1.28705	0.643527	−	0.765424i	$-0.277471\pi$
$953$	39.7322	1.28705	0.643527	+	0.765424i	$0.277471\pi$
$954$	0	0
$955$	−4.62510	−0.149665
$956$	0	0
$957$	−82.8107	−2.67689
$958$	0	0
$959$	3.32280	0.107299
$960$	0	0
$961$	7.81192	0.251997
$962$	0	0
$963$	−2.84994	−0.0918380
$964$	0	0
$965$	8.03668	0.258710
$966$	0	0
$967$	44.6828	1.43690	0.718451	−	0.695577i	$-0.244851\pi$
$967$	44.6828	1.43690	0.718451	+	0.695577i	$0.244851\pi$
$968$	0	0
$969$	7.78396	0.250057
$970$	0	0
$971$	22.9784	0.737412	0.368706	−	0.929546i	$-0.379801\pi$
$971$	22.9784	0.737412	0.368706	+	0.929546i	$0.379801\pi$
$972$	0	0
$973$	−10.8066	−0.346444
$974$	0	0
$975$	−39.9653	−1.27992
$976$	0	0
$977$	7.85417	0.251277	0.125639	−	0.992076i	$-0.459902\pi$
$977$	7.85417	0.251277	0.125639	+	0.992076i	$0.459902\pi$
$978$	0	0
$979$	−39.8965	−1.27510
$980$	0	0
$981$	11.0475	0.352720
$982$	0	0
$983$	27.9404	0.891159	0.445580	−	0.895242i	$-0.352998\pi$
$983$	27.9404	0.891159	0.445580	+	0.895242i	$0.352998\pi$
$984$	0	0
$985$	1.03055	0.0328361
$986$	0	0
$987$	−9.36512	−0.298095
$988$	0	0
$989$	14.0886	0.447990
$990$	0	0
$991$	38.0870	1.20987	0.604936	−	0.796274i	$-0.293199\pi$
$991$	38.0870	1.20987	0.604936	+	0.796274i	$0.293199\pi$
$992$	0	0
$993$	43.8793	1.39247
$994$	0	0
$995$	−2.97535	−0.0943249
$996$	0	0
$997$	−11.4607	−0.362963	−0.181482	−	0.983394i	$-0.558089\pi$
$997$	−11.4607	−0.362963	−0.181482	+	0.983394i	$0.558089\pi$
$998$	0	0
$999$	4.78341	0.151340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.16	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.16	✓	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)

\(f(q)\)	\(=\)	\(q-2.09889 q^{3} +0.451798 q^{5} -0.670101 q^{7} +1.40535 q^{9} +O(q^{10})\)	\(q-2.09889 q^{3} +0.451798 q^{5} -0.670101 q^{7} +1.40535 q^{9} -5.79366 q^{11} -3.97032 q^{13} -0.948275 q^{15} +2.58096 q^{17} -1.43691 q^{19} +1.40647 q^{21} +8.17684 q^{23} -4.79588 q^{25} +3.34700 q^{27} -6.80994 q^{29} -6.22992 q^{31} +12.1603 q^{33} -0.302750 q^{35} +1.42916 q^{37} +8.33327 q^{39} -2.72534 q^{41} +1.72298 q^{43} +0.634934 q^{45} -6.65860 q^{47} -6.55096 q^{49} -5.41716 q^{51} -10.9880 q^{53} -2.61756 q^{55} +3.01592 q^{57} -0.223201 q^{59} -9.81346 q^{61} -0.941726 q^{63} -1.79378 q^{65} +9.79146 q^{67} -17.1623 q^{69} -10.4737 q^{71} +3.49235 q^{73} +10.0660 q^{75} +3.88234 q^{77} -14.2719 q^{79} -11.2410 q^{81} +9.99675 q^{83} +1.16607 q^{85} +14.2933 q^{87} +6.88623 q^{89} +2.66051 q^{91} +13.0759 q^{93} -0.649193 q^{95} +0.343986 q^{97} -8.14212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 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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