LMFDB - Embedded newform 352.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [352,2,Mod(1,352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 352 = 2^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	352.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:	$2.81073415115$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} -0.561553 q^{5} +3.56155 q^{9} +1.00000 q^{11} +2.00000 q^{13} -1.43845 q^{15} +7.12311 q^{17} -1.12311 q^{19} -7.68466 q^{23} -4.68466 q^{25} +1.43845 q^{27} +7.12311 q^{29} -5.43845 q^{31} +2.56155 q^{33} -5.68466 q^{37} +5.12311 q^{39} -8.24621 q^{41} -1.12311 q^{43} -2.00000 q^{45} -4.00000 q^{47} -7.00000 q^{49} +18.2462 q^{51} +8.24621 q^{53} -0.561553 q^{55} -2.87689 q^{57} -0.315342 q^{59} +9.36932 q^{61} -1.12311 q^{65} +7.68466 q^{67} -19.6847 q^{69} +15.6847 q^{71} -6.00000 q^{73} -12.0000 q^{75} -13.1231 q^{79} -7.00000 q^{81} -11.3693 q^{83} -4.00000 q^{85} +18.2462 q^{87} +0.561553 q^{89} -13.9309 q^{93} +0.630683 q^{95} +5.68466 q^{97} +3.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 7 q^{15} + 6 q^{17} + 6 q^{19} - 3 q^{23} + 3 q^{25} + 7 q^{27} + 6 q^{29} - 15 q^{31} + q^{33} + q^{37} + 2 q^{39} + 6 q^{43} - 4 q^{45} - 8 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q + q^3 + 3 * q^5 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 7 * q^15 + 6 * q^17 + 6 * q^19 - 3 * q^23 + 3 * q^25 + 7 * q^27 + 6 * q^29 - 15 * q^31 + q^33 + q^37 + 2 * q^39 + 6 * q^43 - 4 * q^45 - 8 * q^47 - 14 * q^49 + 20 * q^51 + 3 * q^55 - 14 * q^57 - 13 * q^59 - 6 * q^61 + 6 * q^65 + 3 * q^67 - 27 * q^69 + 19 * q^71 - 12 * q^73 - 24 * q^75 - 18 * q^79 - 14 * q^81 + 2 * q^83 - 8 * q^85 + 20 * q^87 - 3 * q^89 + q^93 + 26 * q^95 - q^97 + 3 * q^99

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
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	−0.561553	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	−0.561553	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.43845	−0.371405
$16$	0	0
$17$	7.12311	1.72761	0.863803	−	0.503829i	$-0.168076\pi$
$17$	7.12311	1.72761	0.863803	+	0.503829i	$0.168076\pi$
$18$	0	0
$19$	−1.12311	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.68466	−1.60236	−0.801181	−	0.598422i	$-0.795795\pi$
$23$	−7.68466	−1.60236	−0.801181	+	0.598422i	$0.795795\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	7.12311	1.32273	0.661364	−	0.750065i	$-0.269978\pi$
$29$	7.12311	1.32273	0.661364	+	0.750065i	$0.269978\pi$
$30$	0	0
$31$	−5.43845	−0.976774	−0.488387	−	0.872627i	$-0.662415\pi$
$31$	−5.43845	−0.976774	−0.488387	+	0.872627i	$0.662415\pi$
$32$	0	0
$33$	2.56155	0.445909
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.68466	−0.934552	−0.467276	−	0.884111i	$-0.654765\pi$
$37$	−5.68466	−0.934552	−0.467276	+	0.884111i	$0.654765\pi$
$38$	0	0
$39$	5.12311	0.820353
$40$	0	0
$41$	−8.24621	−1.28784	−0.643921	−	0.765092i	$-0.722693\pi$
$41$	−8.24621	−1.28784	−0.643921	+	0.765092i	$0.722693\pi$
$42$	0	0
$43$	−1.12311	−0.171272	−0.0856360	−	0.996326i	$-0.527292\pi$
$43$	−1.12311	−0.171272	−0.0856360	+	0.996326i	$0.527292\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	18.2462	2.55498
$52$	0	0
$53$	8.24621	1.13270	0.566352	−	0.824163i	$-0.308354\pi$
$53$	8.24621	1.13270	0.566352	+	0.824163i	$0.308354\pi$
$54$	0	0
$55$	−0.561553	−0.0757198
$56$	0	0
$57$	−2.87689	−0.381054
$58$	0	0
$59$	−0.315342	−0.0410540	−0.0205270	−	0.999789i	$-0.506534\pi$
$59$	−0.315342	−0.0410540	−0.0205270	+	0.999789i	$0.506534\pi$
$60$	0	0
$61$	9.36932	1.19962	0.599809	−	0.800143i	$-0.295243\pi$
$61$	9.36932	1.19962	0.599809	+	0.800143i	$0.295243\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.12311	−0.139304
$66$	0	0
$67$	7.68466	0.938830	0.469415	−	0.882978i	$-0.344465\pi$
$67$	7.68466	0.938830	0.469415	+	0.882978i	$0.344465\pi$
$68$	0	0
$69$	−19.6847	−2.36975
$70$	0	0
$71$	15.6847	1.86143	0.930713	−	0.365750i	$-0.119187\pi$
$71$	15.6847	1.86143	0.930713	+	0.365750i	$0.119187\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	−12.0000	−1.38564
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.1231	−1.47646	−0.738232	−	0.674546i	$-0.764339\pi$
$79$	−13.1231	−1.47646	−0.738232	+	0.674546i	$0.764339\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−11.3693	−1.24794	−0.623972	−	0.781446i	$-0.714482\pi$
$83$	−11.3693	−1.24794	−0.623972	+	0.781446i	$0.714482\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	18.2462	1.95620
$88$	0	0
$89$	0.561553	0.0595245	0.0297622	−	0.999557i	$-0.490525\pi$
$89$	0.561553	0.0595245	0.0297622	+	0.999557i	$0.490525\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−13.9309	−1.44456
$94$	0	0
$95$	0.630683	0.0647067
$96$	0	0
$97$	5.68466	0.577190	0.288595	−	0.957451i	$-0.406812\pi$
$97$	5.68466	0.577190	0.288595	+	0.957451i	$0.406812\pi$
$98$	0	0
$99$	3.56155	0.357950
$100$	0	0
$101$	4.87689	0.485269	0.242635	−	0.970118i	$-0.421988\pi$
$101$	4.87689	0.485269	0.242635	+	0.970118i	$0.421988\pi$
$102$	0	0
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−14.5616	−1.38212
$112$	0	0
$113$	11.4384	1.07604	0.538019	−	0.842933i	$-0.319173\pi$
$113$	11.4384	1.07604	0.538019	+	0.842933i	$0.319173\pi$
$114$	0	0
$115$	4.31534	0.402408
$116$	0	0
$117$	7.12311	0.658531
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−21.1231	−1.90461
$124$	0	0
$125$	5.43845	0.486430
$126$	0	0
$127$	15.3693	1.36381	0.681903	−	0.731442i	$-0.261153\pi$
$127$	15.3693	1.36381	0.681903	+	0.731442i	$0.261153\pi$
$128$	0	0
$129$	−2.87689	−0.253296
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.807764	−0.0695213
$136$	0	0
$137$	13.6847	1.16916	0.584580	−	0.811336i	$-0.301259\pi$
$137$	13.6847	1.16916	0.584580	+	0.811336i	$0.301259\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−10.2462	−0.862887
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	−17.9309	−1.47891
$148$	0	0
$149$	4.87689	0.399531	0.199765	−	0.979844i	$-0.435982\pi$
$149$	4.87689	0.399531	0.199765	+	0.979844i	$0.435982\pi$
$150$	0	0
$151$	15.3693	1.25074	0.625369	−	0.780329i	$-0.284949\pi$
$151$	15.3693	1.25074	0.625369	+	0.780329i	$0.284949\pi$
$152$	0	0
$153$	25.3693	2.05099
$154$	0	0
$155$	3.05398	0.245301
$156$	0	0
$157$	1.68466	0.134450	0.0672252	−	0.997738i	$-0.478585\pi$
$157$	1.68466	0.134450	0.0672252	+	0.997738i	$0.478585\pi$
$158$	0	0
$159$	21.1231	1.67517
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	0	0
$165$	−1.43845	−0.111983
$166$	0	0
$167$	7.36932	0.570255	0.285127	−	0.958490i	$-0.407964\pi$
$167$	7.36932	0.570255	0.285127	+	0.958490i	$0.407964\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.807764	−0.0607153
$178$	0	0
$179$	7.68466	0.574378	0.287189	−	0.957874i	$-0.407279\pi$
$179$	7.68466	0.574378	0.287189	+	0.957874i	$0.407279\pi$
$180$	0	0
$181$	25.0540	1.86225	0.931124	−	0.364704i	$-0.118830\pi$
$181$	25.0540	1.86225	0.931124	+	0.364704i	$0.118830\pi$
$182$	0	0
$183$	24.0000	1.77413
$184$	0	0
$185$	3.19224	0.234698
$186$	0	0
$187$	7.12311	0.520893
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.68466	0.556042	0.278021	−	0.960575i	$-0.410321\pi$
$191$	7.68466	0.556042	0.278021	+	0.960575i	$0.410321\pi$
$192$	0	0
$193$	−13.3693	−0.962344	−0.481172	−	0.876626i	$-0.659789\pi$
$193$	−13.3693	−0.962344	−0.481172	+	0.876626i	$0.659789\pi$
$194$	0	0
$195$	−2.87689	−0.206019
$196$	0	0
$197$	−8.24621	−0.587518	−0.293759	−	0.955879i	$-0.594906\pi$
$197$	−8.24621	−0.587518	−0.293759	+	0.955879i	$0.594906\pi$
$198$	0	0
$199$	−14.2462	−1.00989	−0.504944	−	0.863152i	$-0.668487\pi$
$199$	−14.2462	−1.00989	−0.504944	+	0.863152i	$0.668487\pi$
$200$	0	0
$201$	19.6847	1.38845
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.63068	0.323421
$206$	0	0
$207$	−27.3693	−1.90230
$208$	0	0
$209$	−1.12311	−0.0776868
$210$	0	0
$211$	16.4924	1.13539	0.567693	−	0.823241i	$-0.307836\pi$
$211$	16.4924	1.13539	0.567693	+	0.823241i	$0.307836\pi$
$212$	0	0
$213$	40.1771	2.75289
$214$	0	0
$215$	0.630683	0.0430122
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−15.3693	−1.03856
$220$	0	0
$221$	14.2462	0.958304
$222$	0	0
$223$	−7.68466	−0.514603	−0.257301	−	0.966331i	$-0.582833\pi$
$223$	−7.68466	−0.514603	−0.257301	+	0.966331i	$0.582833\pi$
$224$	0	0
$225$	−16.6847	−1.11231
$226$	0	0
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−21.6847	−1.43296	−0.716481	−	0.697606i	$-0.754248\pi$
$229$	−21.6847	−1.43296	−0.716481	+	0.697606i	$0.754248\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.12311	0.466650	0.233325	−	0.972399i	$-0.425039\pi$
$233$	7.12311	0.466650	0.233325	+	0.972399i	$0.425039\pi$
$234$	0	0
$235$	2.24621	0.146527
$236$	0	0
$237$	−33.6155	−2.18356
$238$	0	0
$239$	−0.630683	−0.0407955	−0.0203977	−	0.999792i	$-0.506493\pi$
$239$	−0.630683	−0.0407955	−0.0203977	+	0.999792i	$0.506493\pi$
$240$	0	0
$241$	2.63068	0.169457	0.0847286	−	0.996404i	$-0.472998\pi$
$241$	2.63068	0.169457	0.0847286	+	0.996404i	$0.472998\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	3.93087	0.251134
$246$	0	0
$247$	−2.24621	−0.142923
$248$	0	0
$249$	−29.1231	−1.84560
$250$	0	0
$251$	−23.0540	−1.45515	−0.727577	−	0.686026i	$-0.759354\pi$
$251$	−23.0540	−1.45515	−0.727577	+	0.686026i	$0.759354\pi$
$252$	0	0
$253$	−7.68466	−0.483130
$254$	0	0
$255$	−10.2462	−0.641643
$256$	0	0
$257$	−8.24621	−0.514385	−0.257192	−	0.966360i	$-0.582797\pi$
$257$	−8.24621	−0.514385	−0.257192	+	0.966360i	$0.582797\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	25.3693	1.57032
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	−4.63068	−0.284461
$266$	0	0
$267$	1.43845	0.0880315
$268$	0	0
$269$	10.4924	0.639734	0.319867	−	0.947462i	$-0.396362\pi$
$269$	10.4924	0.639734	0.319867	+	0.947462i	$0.396362\pi$
$270$	0	0
$271$	−13.1231	−0.797172	−0.398586	−	0.917131i	$-0.630499\pi$
$271$	−13.1231	−0.797172	−0.398586	+	0.917131i	$0.630499\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.68466	−0.282496
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−19.3693	−1.15961
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	9.75379	0.579803	0.289901	−	0.957057i	$-0.406378\pi$
$283$	9.75379	0.579803	0.289901	+	0.957057i	$0.406378\pi$
$284$	0	0
$285$	1.61553	0.0956956
$286$	0	0
$287$	0	0
$288$	0	0
$289$	33.7386	1.98463
$290$	0	0
$291$	14.5616	0.853613
$292$	0	0
$293$	−10.4924	−0.612974	−0.306487	−	0.951875i	$-0.599154\pi$
$293$	−10.4924	−0.612974	−0.306487	+	0.951875i	$0.599154\pi$
$294$	0	0
$295$	0.177081	0.0103101
$296$	0	0
$297$	1.43845	0.0834672
$298$	0	0
$299$	−15.3693	−0.888831
$300$	0	0
$301$	0	0
$302$	0	0
$303$	12.4924	0.717671
$304$	0	0
$305$	−5.26137	−0.301265
$306$	0	0
$307$	27.3693	1.56205	0.781025	−	0.624500i	$-0.214697\pi$
$307$	27.3693	1.56205	0.781025	+	0.624500i	$0.214697\pi$
$308$	0	0
$309$	−36.4924	−2.07598
$310$	0	0
$311$	18.7386	1.06257	0.531285	−	0.847193i	$-0.321709\pi$
$311$	18.7386	1.06257	0.531285	+	0.847193i	$0.321709\pi$
$312$	0	0
$313$	6.31534	0.356964	0.178482	−	0.983943i	$-0.442881\pi$
$313$	6.31534	0.356964	0.178482	+	0.983943i	$0.442881\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.0540	−1.63183	−0.815917	−	0.578169i	$-0.803767\pi$
$317$	−29.0540	−1.63183	−0.815917	+	0.578169i	$0.803767\pi$
$318$	0	0
$319$	7.12311	0.398817
$320$	0	0
$321$	−30.7386	−1.71566
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	−9.36932	−0.519716
$326$	0	0
$327$	−35.8617	−1.98316
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.3153	0.896772	0.448386	−	0.893840i	$-0.351999\pi$
$331$	16.3153	0.896772	0.448386	+	0.893840i	$0.351999\pi$
$332$	0	0
$333$	−20.2462	−1.10949
$334$	0	0
$335$	−4.31534	−0.235772
$336$	0	0
$337$	−6.63068	−0.361196	−0.180598	−	0.983557i	$-0.557803\pi$
$337$	−6.63068	−0.361196	−0.180598	+	0.983557i	$0.557803\pi$
$338$	0	0
$339$	29.3002	1.59137
$340$	0	0
$341$	−5.43845	−0.294508
$342$	0	0
$343$	0	0
$344$	0	0
$345$	11.0540	0.595126
$346$	0	0
$347$	27.3693	1.46926	0.734631	−	0.678467i	$-0.237355\pi$
$347$	27.3693	1.46926	0.734631	+	0.678467i	$0.237355\pi$
$348$	0	0
$349$	−13.3693	−0.715643	−0.357822	−	0.933790i	$-0.616480\pi$
$349$	−13.3693	−0.715643	−0.357822	+	0.933790i	$0.616480\pi$
$350$	0	0
$351$	2.87689	0.153557
$352$	0	0
$353$	24.5616	1.30728	0.653640	−	0.756806i	$-0.273241\pi$
$353$	24.5616	1.30728	0.653640	+	0.756806i	$0.273241\pi$
$354$	0	0
$355$	−8.80776	−0.467468
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.3693	0.811162	0.405581	−	0.914059i	$-0.367069\pi$
$359$	15.3693	0.811162	0.405581	+	0.914059i	$0.367069\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	2.56155	0.134447
$364$	0	0
$365$	3.36932	0.176358
$366$	0	0
$367$	−0.946025	−0.0493821	−0.0246910	−	0.999695i	$-0.507860\pi$
$367$	−0.946025	−0.0493821	−0.0246910	+	0.999695i	$0.507860\pi$
$368$	0	0
$369$	−29.3693	−1.52891
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−21.3693	−1.10646	−0.553231	−	0.833028i	$-0.686605\pi$
$373$	−21.3693	−1.10646	−0.553231	+	0.833028i	$0.686605\pi$
$374$	0	0
$375$	13.9309	0.719387
$376$	0	0
$377$	14.2462	0.733717
$378$	0	0
$379$	18.5616	0.953443	0.476721	−	0.879054i	$-0.341825\pi$
$379$	18.5616	0.953443	0.476721	+	0.879054i	$0.341825\pi$
$380$	0	0
$381$	39.3693	2.01695
$382$	0	0
$383$	8.31534	0.424894	0.212447	−	0.977173i	$-0.431857\pi$
$383$	8.31534	0.424894	0.212447	+	0.977173i	$0.431857\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−26.8078	−1.35921	−0.679604	−	0.733579i	$-0.737848\pi$
$389$	−26.8078	−1.35921	−0.679604	+	0.733579i	$0.737848\pi$
$390$	0	0
$391$	−54.7386	−2.76825
$392$	0	0
$393$	10.2462	0.516853
$394$	0	0
$395$	7.36932	0.370791
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.4924	1.12322	0.561609	−	0.827403i	$-0.310183\pi$
$401$	22.4924	1.12322	0.561609	+	0.827403i	$0.310183\pi$
$402$	0	0
$403$	−10.8769	−0.541817
$404$	0	0
$405$	3.93087	0.195326
$406$	0	0
$407$	−5.68466	−0.281778
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	35.0540	1.72909
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6.38447	0.313401
$416$	0	0
$417$	30.7386	1.50528
$418$	0	0
$419$	−18.7386	−0.915442	−0.457721	−	0.889096i	$-0.651334\pi$
$419$	−18.7386	−0.915442	−0.457721	+	0.889096i	$0.651334\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	−14.2462	−0.692674
$424$	0	0
$425$	−33.3693	−1.61865
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5.12311	0.247346
$430$	0	0
$431$	−38.7386	−1.86597	−0.932987	−	0.359910i	$-0.882808\pi$
$431$	−38.7386	−1.86597	−0.932987	+	0.359910i	$0.882808\pi$
$432$	0	0
$433$	5.68466	0.273187	0.136594	−	0.990627i	$-0.456385\pi$
$433$	5.68466	0.273187	0.136594	+	0.990627i	$0.456385\pi$
$434$	0	0
$435$	−10.2462	−0.491268
$436$	0	0
$437$	8.63068	0.412862
$438$	0	0
$439$	−26.2462	−1.25266	−0.626332	−	0.779557i	$-0.715444\pi$
$439$	−26.2462	−1.25266	−0.626332	+	0.779557i	$0.715444\pi$
$440$	0	0
$441$	−24.9309	−1.18718
$442$	0	0
$443$	−8.31534	−0.395074	−0.197537	−	0.980295i	$-0.563294\pi$
$443$	−8.31534	−0.395074	−0.197537	+	0.980295i	$0.563294\pi$
$444$	0	0
$445$	−0.315342	−0.0149486
$446$	0	0
$447$	12.4924	0.590871
$448$	0	0
$449$	2.80776	0.132507	0.0662533	−	0.997803i	$-0.478895\pi$
$449$	2.80776	0.132507	0.0662533	+	0.997803i	$0.478895\pi$
$450$	0	0
$451$	−8.24621	−0.388299
$452$	0	0
$453$	39.3693	1.84973
$454$	0	0
$455$	0	0
$456$	0	0
$457$	41.3693	1.93518	0.967588	−	0.252536i	$-0.0812646\pi$
$457$	41.3693	1.93518	0.967588	+	0.252536i	$0.0812646\pi$
$458$	0	0
$459$	10.2462	0.478252
$460$	0	0
$461$	−19.1231	−0.890652	−0.445326	−	0.895369i	$-0.646912\pi$
$461$	−19.1231	−0.890652	−0.445326	+	0.895369i	$0.646912\pi$
$462$	0	0
$463$	33.9309	1.57690	0.788451	−	0.615098i	$-0.210884\pi$
$463$	33.9309	1.57690	0.788451	+	0.615098i	$0.210884\pi$
$464$	0	0
$465$	7.82292	0.362779
$466$	0	0
$467$	−8.31534	−0.384788	−0.192394	−	0.981318i	$-0.561625\pi$
$467$	−8.31534	−0.384788	−0.192394	+	0.981318i	$0.561625\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.31534	0.198840
$472$	0	0
$473$	−1.12311	−0.0516405
$474$	0	0
$475$	5.26137	0.241408
$476$	0	0
$477$	29.3693	1.34473
$478$	0	0
$479$	0.630683	0.0288166	0.0144083	−	0.999896i	$-0.495414\pi$
$479$	0.630683	0.0288166	0.0144083	+	0.999896i	$0.495414\pi$
$480$	0	0
$481$	−11.3693	−0.518396
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.19224	−0.144952
$486$	0	0
$487$	−31.6847	−1.43577	−0.717884	−	0.696162i	$-0.754889\pi$
$487$	−31.6847	−1.43577	−0.717884	+	0.696162i	$0.754889\pi$
$488$	0	0
$489$	−30.7386	−1.39005
$490$	0	0
$491$	−26.7386	−1.20670	−0.603349	−	0.797477i	$-0.706167\pi$
$491$	−26.7386	−1.20670	−0.603349	+	0.797477i	$0.706167\pi$
$492$	0	0
$493$	50.7386	2.28515
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−40.4924	−1.81269	−0.906345	−	0.422539i	$-0.861139\pi$
$499$	−40.4924	−1.81269	−0.906345	+	0.422539i	$0.861139\pi$
$500$	0	0
$501$	18.8769	0.843357
$502$	0	0
$503$	15.3693	0.685284	0.342642	−	0.939466i	$-0.388678\pi$
$503$	15.3693	0.685284	0.342642	+	0.939466i	$0.388678\pi$
$504$	0	0
$505$	−2.73863	−0.121868
$506$	0	0
$507$	−23.0540	−1.02386
$508$	0	0
$509$	−13.6847	−0.606562	−0.303281	−	0.952901i	$-0.598082\pi$
$509$	−13.6847	−0.606562	−0.303281	+	0.952901i	$0.598082\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.61553	−0.0713273
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	46.1080	2.02391
$520$	0	0
$521$	−17.0540	−0.747148	−0.373574	−	0.927600i	$-0.621868\pi$
$521$	−17.0540	−0.747148	−0.373574	+	0.927600i	$0.621868\pi$
$522$	0	0
$523$	34.1080	1.49144	0.745718	−	0.666261i	$-0.232106\pi$
$523$	34.1080	1.49144	0.745718	+	0.666261i	$0.232106\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−38.7386	−1.68748
$528$	0	0
$529$	36.0540	1.56756
$530$	0	0
$531$	−1.12311	−0.0487386
$532$	0	0
$533$	−16.4924	−0.714366
$534$	0	0
$535$	6.73863	0.291337
$536$	0	0
$537$	19.6847	0.849456
$538$	0	0
$539$	−7.00000	−0.301511
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	64.1771	2.75410
$544$	0	0
$545$	7.86174	0.336760
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	33.3693	1.42417
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.17708	0.347098
$556$	0	0
$557$	33.3693	1.41390	0.706952	−	0.707262i	$-0.250070\pi$
$557$	33.3693	1.41390	0.706952	+	0.707262i	$0.250070\pi$
$558$	0	0
$559$	−2.24621	−0.0950046
$560$	0	0
$561$	18.2462	0.770356
$562$	0	0
$563$	18.7386	0.789739	0.394870	−	0.918737i	$-0.370790\pi$
$563$	18.7386	0.789739	0.394870	+	0.918737i	$0.370790\pi$
$564$	0	0
$565$	−6.42329	−0.270230
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.3693	−0.895848	−0.447924	−	0.894072i	$-0.647837\pi$
$569$	−21.3693	−0.895848	−0.447924	+	0.894072i	$0.647837\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	19.6847	0.822338
$574$	0	0
$575$	36.0000	1.50130
$576$	0	0
$577$	5.68466	0.236655	0.118328	−	0.992975i	$-0.462247\pi$
$577$	5.68466	0.236655	0.118328	+	0.992975i	$0.462247\pi$
$578$	0	0
$579$	−34.2462	−1.42322
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.24621	0.341523
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	6.10795	0.251674
$590$	0	0
$591$	−21.1231	−0.868888
$592$	0	0
$593$	−25.8617	−1.06201	−0.531007	−	0.847367i	$-0.678186\pi$
$593$	−25.8617	−1.06201	−0.531007	+	0.847367i	$0.678186\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−36.4924	−1.49354
$598$	0	0
$599$	−34.7386	−1.41938	−0.709691	−	0.704513i	$-0.751165\pi$
$599$	−34.7386	−1.41938	−0.709691	+	0.704513i	$0.751165\pi$
$600$	0	0
$601$	32.1080	1.30971	0.654855	−	0.755754i	$-0.272730\pi$
$601$	32.1080	1.30971	0.654855	+	0.755754i	$0.272730\pi$
$602$	0	0
$603$	27.3693	1.11456
$604$	0	0
$605$	−0.561553	−0.0228304
$606$	0	0
$607$	37.1231	1.50678	0.753390	−	0.657574i	$-0.228417\pi$
$607$	37.1231	1.50678	0.753390	+	0.657574i	$0.228417\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	17.3693	0.701540	0.350770	−	0.936462i	$-0.385920\pi$
$613$	17.3693	0.701540	0.350770	+	0.936462i	$0.385920\pi$
$614$	0	0
$615$	11.8617	0.478311
$616$	0	0
$617$	−32.2462	−1.29818	−0.649092	−	0.760710i	$-0.724851\pi$
$617$	−32.2462	−1.29818	−0.649092	+	0.760710i	$0.724851\pi$
$618$	0	0
$619$	−7.68466	−0.308873	−0.154436	−	0.988003i	$-0.549356\pi$
$619$	−7.68466	−0.308873	−0.154436	+	0.988003i	$0.549356\pi$
$620$	0	0
$621$	−11.0540	−0.443581
$622$	0	0
$623$	0	0
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	−2.87689	−0.114892
$628$	0	0
$629$	−40.4924	−1.61454
$630$	0	0
$631$	−38.4233	−1.52961	−0.764804	−	0.644264i	$-0.777164\pi$
$631$	−38.4233	−1.52961	−0.764804	+	0.644264i	$0.777164\pi$
$632$	0	0
$633$	42.2462	1.67914
$634$	0	0
$635$	−8.63068	−0.342498
$636$	0	0
$637$	−14.0000	−0.554700
$638$	0	0
$639$	55.8617	2.20986
$640$	0	0
$641$	−27.9309	−1.10320	−0.551602	−	0.834108i	$-0.685983\pi$
$641$	−27.9309	−1.10320	−0.551602	+	0.834108i	$0.685983\pi$
$642$	0	0
$643$	−42.5616	−1.67846	−0.839232	−	0.543774i	$-0.816995\pi$
$643$	−42.5616	−1.67846	−0.839232	+	0.543774i	$0.816995\pi$
$644$	0	0
$645$	1.61553	0.0636114
$646$	0	0
$647$	31.6847	1.24565	0.622826	−	0.782360i	$-0.285984\pi$
$647$	31.6847	1.24565	0.622826	+	0.782360i	$0.285984\pi$
$648$	0	0
$649$	−0.315342	−0.0123782
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13.6847	−0.535522	−0.267761	−	0.963485i	$-0.586284\pi$
$653$	−13.6847	−0.535522	−0.267761	+	0.963485i	$0.586284\pi$
$654$	0	0
$655$	−2.24621	−0.0877667
$656$	0	0
$657$	−21.3693	−0.833696
$658$	0	0
$659$	−12.6307	−0.492022	−0.246011	−	0.969267i	$-0.579120\pi$
$659$	−12.6307	−0.492022	−0.246011	+	0.969267i	$0.579120\pi$
$660$	0	0
$661$	−21.0540	−0.818905	−0.409452	−	0.912331i	$-0.634280\pi$
$661$	−21.0540	−0.818905	−0.409452	+	0.912331i	$0.634280\pi$
$662$	0	0
$663$	36.4924	1.41725
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−54.7386	−2.11949
$668$	0	0
$669$	−19.6847	−0.761053
$670$	0	0
$671$	9.36932	0.361698
$672$	0	0
$673$	24.7386	0.953604	0.476802	−	0.879011i	$-0.341796\pi$
$673$	24.7386	0.953604	0.476802	+	0.879011i	$0.341796\pi$
$674$	0	0
$675$	−6.73863	−0.259370
$676$	0	0
$677$	37.8617	1.45514	0.727572	−	0.686031i	$-0.240649\pi$
$677$	37.8617	1.45514	0.727572	+	0.686031i	$0.240649\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	51.2311	1.96318
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−7.68466	−0.293616
$686$	0	0
$687$	−55.5464	−2.11923
$688$	0	0
$689$	16.4924	0.628311
$690$	0	0
$691$	−25.3002	−0.962464	−0.481232	−	0.876593i	$-0.659811\pi$
$691$	−25.3002	−0.962464	−0.481232	+	0.876593i	$0.659811\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.73863	−0.255611
$696$	0	0
$697$	−58.7386	−2.22488
$698$	0	0
$699$	18.2462	0.690135
$700$	0	0
$701$	−47.6155	−1.79841	−0.899207	−	0.437524i	$-0.855856\pi$
$701$	−47.6155	−1.79841	−0.899207	+	0.437524i	$0.855856\pi$
$702$	0	0
$703$	6.38447	0.240795
$704$	0	0
$705$	5.75379	0.216700
$706$	0	0
$707$	0	0
$708$	0	0
$709$	10.3153	0.387401	0.193700	−	0.981061i	$-0.437951\pi$
$709$	10.3153	0.387401	0.193700	+	0.981061i	$0.437951\pi$
$710$	0	0
$711$	−46.7386	−1.75284
$712$	0	0
$713$	41.7926	1.56515
$714$	0	0
$715$	−1.12311	−0.0420018
$716$	0	0
$717$	−1.61553	−0.0603330
$718$	0	0
$719$	16.3153	0.608460	0.304230	−	0.952599i	$-0.401601\pi$
$719$	16.3153	0.608460	0.304230	+	0.952599i	$0.401601\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	6.73863	0.250612
$724$	0	0
$725$	−33.3693	−1.23931
$726$	0	0
$727$	16.3153	0.605103	0.302551	−	0.953133i	$-0.402162\pi$
$727$	16.3153	0.605103	0.302551	+	0.953133i	$0.402162\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−36.1080	−1.33368	−0.666839	−	0.745202i	$-0.732353\pi$
$733$	−36.1080	−1.33368	−0.666839	+	0.745202i	$0.732353\pi$
$734$	0	0
$735$	10.0691	0.371405
$736$	0	0
$737$	7.68466	0.283068
$738$	0	0
$739$	29.6155	1.08942	0.544712	−	0.838623i	$-0.316639\pi$
$739$	29.6155	1.08942	0.544712	+	0.838623i	$0.316639\pi$
$740$	0	0
$741$	−5.75379	−0.211371
$742$	0	0
$743$	−30.7386	−1.12769	−0.563846	−	0.825880i	$-0.690679\pi$
$743$	−30.7386	−1.12769	−0.563846	+	0.825880i	$0.690679\pi$
$744$	0	0
$745$	−2.73863	−0.100336
$746$	0	0
$747$	−40.4924	−1.48154
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.1771	0.444348	0.222174	−	0.975007i	$-0.428685\pi$
$751$	12.1771	0.444348	0.222174	+	0.975007i	$0.428685\pi$
$752$	0	0
$753$	−59.0540	−2.15205
$754$	0	0
$755$	−8.63068	−0.314103
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	−19.6847	−0.714508
$760$	0	0
$761$	−3.75379	−0.136075	−0.0680374	−	0.997683i	$-0.521674\pi$
$761$	−3.75379	−0.136075	−0.0680374	+	0.997683i	$0.521674\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−14.2462	−0.515073
$766$	0	0
$767$	−0.630683	−0.0227726
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	−21.1231	−0.760730
$772$	0	0
$773$	8.24621	0.296596	0.148298	−	0.988943i	$-0.452621\pi$
$773$	8.24621	0.296596	0.148298	+	0.988943i	$0.452621\pi$
$774$	0	0
$775$	25.4773	0.915170
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.26137	0.331823
$780$	0	0
$781$	15.6847	0.561241
$782$	0	0
$783$	10.2462	0.366170
$784$	0	0
$785$	−0.946025	−0.0337651
$786$	0	0
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	0	0
$789$	20.4924	0.729550
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18.7386	0.665428
$794$	0	0
$795$	−11.8617	−0.420693
$796$	0	0
$797$	−24.5616	−0.870015	−0.435007	−	0.900427i	$-0.643254\pi$
$797$	−24.5616	−0.870015	−0.435007	+	0.900427i	$0.643254\pi$
$798$	0	0
$799$	−28.4924	−1.00799
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.8769	0.946111
$808$	0	0
$809$	−8.24621	−0.289921	−0.144961	−	0.989437i	$-0.546306\pi$
$809$	−8.24621	−0.289921	−0.144961	+	0.989437i	$0.546306\pi$
$810$	0	0
$811$	51.3693	1.80382	0.901910	−	0.431923i	$-0.142165\pi$
$811$	51.3693	1.80382	0.901910	+	0.431923i	$0.142165\pi$
$812$	0	0
$813$	−33.6155	−1.17895
$814$	0	0
$815$	6.73863	0.236044
$816$	0	0
$817$	1.26137	0.0441296
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.738634	0.0257785	0.0128892	−	0.999917i	$-0.495897\pi$
$821$	0.738634	0.0257785	0.0128892	+	0.999917i	$0.495897\pi$
$822$	0	0
$823$	18.5616	0.647015	0.323508	−	0.946226i	$-0.395138\pi$
$823$	18.5616	0.647015	0.323508	+	0.946226i	$0.395138\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	3.36932	0.117163	0.0585813	−	0.998283i	$-0.481342\pi$
$827$	3.36932	0.117163	0.0585813	+	0.998283i	$0.481342\pi$
$828$	0	0
$829$	17.6847	0.614214	0.307107	−	0.951675i	$-0.400639\pi$
$829$	17.6847	0.614214	0.307107	+	0.951675i	$0.400639\pi$
$830$	0	0
$831$	5.12311	0.177719
$832$	0	0
$833$	−49.8617	−1.72761
$834$	0	0
$835$	−4.13826	−0.143210
$836$	0	0
$837$	−7.82292	−0.270400
$838$	0	0
$839$	−30.4233	−1.05033	−0.525164	−	0.851001i	$-0.675996\pi$
$839$	−30.4233	−1.05033	−0.525164	+	0.851001i	$0.675996\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	0	0
$843$	−15.3693	−0.529347
$844$	0	0
$845$	5.05398	0.173862
$846$	0	0
$847$	0	0
$848$	0	0
$849$	24.9848	0.857478
$850$	0	0
$851$	43.6847	1.49749
$852$	0	0
$853$	40.7386	1.39486	0.697432	−	0.716651i	$-0.254326\pi$
$853$	40.7386	1.39486	0.697432	+	0.716651i	$0.254326\pi$
$854$	0	0
$855$	2.24621	0.0768188
$856$	0	0
$857$	4.87689	0.166592	0.0832958	−	0.996525i	$-0.473455\pi$
$857$	4.87689	0.166592	0.0832958	+	0.996525i	$0.473455\pi$
$858$	0	0
$859$	57.9309	1.97658	0.988288	−	0.152601i	$-0.0487649\pi$
$859$	57.9309	1.97658	0.988288	+	0.152601i	$0.0487649\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.7386	0.637871	0.318935	−	0.947777i	$-0.396675\pi$
$863$	18.7386	0.637871	0.318935	+	0.947777i	$0.396675\pi$
$864$	0	0
$865$	−10.1080	−0.343681
$866$	0	0
$867$	86.4233	2.93509
$868$	0	0
$869$	−13.1231	−0.445171
$870$	0	0
$871$	15.3693	0.520769
$872$	0	0
$873$	20.2462	0.685230
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.26137	−0.245199	−0.122599	−	0.992456i	$-0.539123\pi$
$877$	−7.26137	−0.245199	−0.122599	+	0.992456i	$0.539123\pi$
$878$	0	0
$879$	−26.8769	−0.906535
$880$	0	0
$881$	−19.3002	−0.650240	−0.325120	−	0.945673i	$-0.605405\pi$
$881$	−19.3002	−0.650240	−0.325120	+	0.945673i	$0.605405\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	0.453602	0.0152477
$886$	0	0
$887$	22.1080	0.742312	0.371156	−	0.928570i	$-0.378961\pi$
$887$	22.1080	0.742312	0.371156	+	0.928570i	$0.378961\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	4.49242	0.150333
$894$	0	0
$895$	−4.31534	−0.144246
$896$	0	0
$897$	−39.3693	−1.31450
$898$	0	0
$899$	−38.7386	−1.29201
$900$	0	0
$901$	58.7386	1.95687
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.0691	−0.467674
$906$	0	0
$907$	−16.4924	−0.547622	−0.273811	−	0.961784i	$-0.588284\pi$
$907$	−16.4924	−0.547622	−0.273811	+	0.961784i	$0.588284\pi$
$908$	0	0
$909$	17.3693	0.576104
$910$	0	0
$911$	26.7386	0.885890	0.442945	−	0.896549i	$-0.353934\pi$
$911$	26.7386	0.885890	0.442945	+	0.896549i	$0.353934\pi$
$912$	0	0
$913$	−11.3693	−0.376269
$914$	0	0
$915$	−13.4773	−0.445545
$916$	0	0
$917$	0	0
$918$	0	0
$919$	17.6155	0.581083	0.290541	−	0.956862i	$-0.406165\pi$
$919$	17.6155	0.581083	0.290541	+	0.956862i	$0.406165\pi$
$920$	0	0
$921$	70.1080	2.31014
$922$	0	0
$923$	31.3693	1.03253
$924$	0	0
$925$	26.6307	0.875611
$926$	0	0
$927$	−50.7386	−1.66648
$928$	0	0
$929$	22.4924	0.737952	0.368976	−	0.929439i	$-0.379708\pi$
$929$	22.4924	0.737952	0.368976	+	0.929439i	$0.379708\pi$
$930$	0	0
$931$	7.86174	0.257658
$932$	0	0
$933$	48.0000	1.57145
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−20.7386	−0.677502	−0.338751	−	0.940876i	$-0.610004\pi$
$937$	−20.7386	−0.677502	−0.338751	+	0.940876i	$0.610004\pi$
$938$	0	0
$939$	16.1771	0.527919
$940$	0	0
$941$	−8.24621	−0.268819	−0.134409	−	0.990926i	$-0.542914\pi$
$941$	−8.24621	−0.268819	−0.134409	+	0.990926i	$0.542914\pi$
$942$	0	0
$943$	63.3693	2.06359
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.6847	1.02961	0.514807	−	0.857306i	$-0.327864\pi$
$947$	31.6847	1.02961	0.514807	+	0.857306i	$0.327864\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−74.4233	−2.41334
$952$	0	0
$953$	−47.6155	−1.54242	−0.771209	−	0.636582i	$-0.780348\pi$
$953$	−47.6155	−1.54242	−0.771209	+	0.636582i	$0.780348\pi$
$954$	0	0
$955$	−4.31534	−0.139641
$956$	0	0
$957$	18.2462	0.589816
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.42329	−0.0459127
$962$	0	0
$963$	−42.7386	−1.37723
$964$	0	0
$965$	7.50758	0.241677
$966$	0	0
$967$	−30.7386	−0.988488	−0.494244	−	0.869323i	$-0.664555\pi$
$967$	−30.7386	−0.988488	−0.494244	+	0.869323i	$0.664555\pi$
$968$	0	0
$969$	−20.4924	−0.658311
$970$	0	0
$971$	16.3153	0.523584	0.261792	−	0.965124i	$-0.415687\pi$
$971$	16.3153	0.523584	0.261792	+	0.965124i	$0.415687\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−24.0000	−0.768615
$976$	0	0
$977$	−50.0388	−1.60088	−0.800442	−	0.599410i	$-0.795402\pi$
$977$	−50.0388	−1.60088	−0.800442	+	0.599410i	$0.795402\pi$
$978$	0	0
$979$	0.561553	0.0179473
$980$	0	0
$981$	−49.8617	−1.59196
$982$	0	0
$983$	16.3153	0.520379	0.260189	−	0.965558i	$-0.416215\pi$
$983$	16.3153	0.520379	0.260189	+	0.965558i	$0.416215\pi$
$984$	0	0
$985$	4.63068	0.147546
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.63068	0.274440
$990$	0	0
$991$	5.26137	0.167133	0.0835664	−	0.996502i	$-0.473369\pi$
$991$	5.26137	0.167133	0.0835664	+	0.996502i	$0.473369\pi$
$992$	0	0
$993$	41.7926	1.32625
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	25.3693	0.803454	0.401727	−	0.915759i	$-0.368410\pi$
$997$	25.3693	0.803454	0.401727	+	0.915759i	$0.368410\pi$
$998$	0	0
$999$	−8.17708	−0.258711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	352.2.a.h.1.2	yes	2
3.2	odd	2		3168.2.a.bc.1.2		2
4.3	odd	2		352.2.a.g.1.1	✓	2
5.4	even	2		8800.2.a.bd.1.1		2
8.3	odd	2		704.2.a.o.1.2		2
8.5	even	2		704.2.a.n.1.1		2
11.10	odd	2		3872.2.a.ba.1.2		2
12.11	even	2		3168.2.a.bd.1.2		2
16.3	odd	4		2816.2.c.t.1409.1		4
16.5	even	4		2816.2.c.s.1409.1		4
16.11	odd	4		2816.2.c.t.1409.4		4
16.13	even	4		2816.2.c.s.1409.4		4
20.19	odd	2		8800.2.a.be.1.2		2
24.5	odd	2		6336.2.a.cw.1.1		2
24.11	even	2		6336.2.a.cv.1.1		2
44.43	even	2		3872.2.a.p.1.1		2
88.21	odd	2		7744.2.a.bw.1.1		2
88.43	even	2		7744.2.a.cm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.a.g.1.1	✓	2	4.3	odd	2
352.2.a.h.1.2	yes	2	1.1	even	1	trivial
704.2.a.n.1.1		2	8.5	even	2
704.2.a.o.1.2		2	8.3	odd	2
2816.2.c.s.1409.1		4	16.5	even	4
2816.2.c.s.1409.4		4	16.13	even	4
2816.2.c.t.1409.1		4	16.3	odd	4
2816.2.c.t.1409.4		4	16.11	odd	4
3168.2.a.bc.1.2		2	3.2	odd	2
3168.2.a.bd.1.2		2	12.11	even	2
3872.2.a.p.1.1		2	44.43	even	2
3872.2.a.ba.1.2		2	11.10	odd	2
6336.2.a.cv.1.1		2	24.11	even	2
6336.2.a.cw.1.1		2	24.5	odd	2
7744.2.a.bw.1.1		2	88.21	odd	2
7744.2.a.cm.1.2		2	88.43	even	2
8800.2.a.bd.1.1		2	5.4	even	2
8800.2.a.be.1.2		2	20.19	odd	2

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:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} -0.561553 q^{5} +3.56155 q^{9} +1.00000 q^{11} +2.00000 q^{13} -1.43845 q^{15} +7.12311 q^{17} -1.12311 q^{19} -7.68466 q^{23} -4.68466 q^{25} +1.43845 q^{27} +7.12311 q^{29} -5.43845 q^{31} +2.56155 q^{33} -5.68466 q^{37} +5.12311 q^{39} -8.24621 q^{41} -1.12311 q^{43} -2.00000 q^{45} -4.00000 q^{47} -7.00000 q^{49} +18.2462 q^{51} +8.24621 q^{53} -0.561553 q^{55} -2.87689 q^{57} -0.315342 q^{59} +9.36932 q^{61} -1.12311 q^{65} +7.68466 q^{67} -19.6847 q^{69} +15.6847 q^{71} -6.00000 q^{73} -12.0000 q^{75} -13.1231 q^{79} -7.00000 q^{81} -11.3693 q^{83} -4.00000 q^{85} +18.2462 q^{87} +0.561553 q^{89} -13.9309 q^{93} +0.630683 q^{95} +5.68466 q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 7 q^{15} + 6 q^{17} + 6 q^{19} - 3 q^{23} + 3 q^{25} + 7 q^{27} + 6 q^{29} - 15 q^{31} + q^{33} + q^{37} + 2 q^{39} + 6 q^{43} - 4 q^{45} - 8 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q + q^3 + 3 * q^5 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 7 * q^15 + 6 * q^17 + 6 * q^19 - 3 * q^23 + 3 * q^25 + 7 * q^27 + 6 * q^29 - 15 * q^31 + q^33 + q^37 + 2 * q^39 + 6 * q^43 - 4 * q^45 - 8 * q^47 - 14 * q^49 + 20 * q^51 + 3 * q^55 - 14 * q^57 - 13 * q^59 - 6 * q^61 + 6 * q^65 + 3 * q^67 - 27 * q^69 + 19 * q^71 - 12 * q^73 - 24 * q^75 - 18 * q^79 - 14 * q^81 + 2 * q^83 - 8 * q^85 + 20 * q^87 - 3 * q^89 + q^93 + 26 * q^95 - q^97 + 3 * q^99

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