LMFDB - Embedded newform 352.2.a.h.1.1

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			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.56155 q^{3} +3.56155 q^{5} -0.561553 q^{9} +1.00000 q^{11} +2.00000 q^{13} -5.56155 q^{15} -1.12311 q^{17} +7.12311 q^{19} +4.68466 q^{23} +7.68466 q^{25} +5.56155 q^{27} -1.12311 q^{29} -9.56155 q^{31} -1.56155 q^{33} +6.68466 q^{37} -3.12311 q^{39} +8.24621 q^{41} +7.12311 q^{43} -2.00000 q^{45} -4.00000 q^{47} -7.00000 q^{49} +1.75379 q^{51} -8.24621 q^{53} +3.56155 q^{55} -11.1231 q^{57} -12.6847 q^{59} -15.3693 q^{61} +7.12311 q^{65} -4.68466 q^{67} -7.31534 q^{69} +3.31534 q^{71} -6.00000 q^{73} -12.0000 q^{75} -4.87689 q^{79} -7.00000 q^{81} +13.3693 q^{83} -4.00000 q^{85} +1.75379 q^{87} -3.56155 q^{89} +14.9309 q^{93} +25.3693 q^{95} -6.68466 q^{97} -0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.56155	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$4$	0	0
$5$	3.56155	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	3.56155	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	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
$15$	−5.56155	−1.43599
$16$	0	0
$17$	−1.12311	−0.272393	−0.136197	−	0.990682i	$-0.543488\pi$
$17$	−1.12311	−0.272393	−0.136197	+	0.990682i	$0.543488\pi$
$18$	0	0
$19$	7.12311	1.63415	0.817076	−	0.576530i	$-0.195593\pi$
$19$	7.12311	1.63415	0.817076	+	0.576530i	$0.195593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.68466	0.976819	0.488409	−	0.872615i	$-0.337577\pi$
$23$	4.68466	0.976819	0.488409	+	0.872615i	$0.337577\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	5.56155	1.07032
$28$	0	0
$29$	−1.12311	−0.208555	−0.104278	−	0.994548i	$-0.533253\pi$
$29$	−1.12311	−0.208555	−0.104278	+	0.994548i	$0.533253\pi$
$30$	0	0
$31$	−9.56155	−1.71731	−0.858653	−	0.512558i	$-0.828698\pi$
$31$	−9.56155	−1.71731	−0.858653	+	0.512558i	$0.828698\pi$
$32$	0	0
$33$	−1.56155	−0.271831
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.68466	1.09895	0.549476	−	0.835510i	$-0.314828\pi$
$37$	6.68466	1.09895	0.549476	+	0.835510i	$0.314828\pi$
$38$	0	0
$39$	−3.12311	−0.500097
$40$	0	0
$41$	8.24621	1.28784	0.643921	−	0.765092i	$-0.277307\pi$
$41$	8.24621	1.28784	0.643921	+	0.765092i	$0.277307\pi$
$42$	0	0
$43$	7.12311	1.08626	0.543132	−	0.839648i	$-0.317238\pi$
$43$	7.12311	1.08626	0.543132	+	0.839648i	$0.317238\pi$
$44$	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
$51$	1.75379	0.245580
$52$	0	0
$53$	−8.24621	−1.13270	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	−8.24621	−1.13270	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	3.56155	0.480240
$56$	0	0
$57$	−11.1231	−1.47329
$58$	0	0
$59$	−12.6847	−1.65140	−0.825701	−	0.564108i	$-0.809220\pi$
$59$	−12.6847	−1.65140	−0.825701	+	0.564108i	$0.809220\pi$
$60$	0	0
$61$	−15.3693	−1.96784	−0.983920	−	0.178611i	$-0.942839\pi$
$61$	−15.3693	−1.96784	−0.983920	+	0.178611i	$0.942839\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.12311	0.883513
$66$	0	0
$67$	−4.68466	−0.572322	−0.286161	−	0.958182i	$-0.592379\pi$
$67$	−4.68466	−0.572322	−0.286161	+	0.958182i	$0.592379\pi$
$68$	0	0
$69$	−7.31534	−0.880664
$70$	0	0
$71$	3.31534	0.393459	0.196729	−	0.980458i	$-0.436968\pi$
$71$	3.31534	0.393459	0.196729	+	0.980458i	$0.436968\pi$
$72$	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
$79$	−4.87689	−0.548693	−0.274347	−	0.961631i	$-0.588462\pi$
$79$	−4.87689	−0.548693	−0.274347	+	0.961631i	$0.588462\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	13.3693	1.46747	0.733737	−	0.679434i	$-0.237775\pi$
$83$	13.3693	1.46747	0.733737	+	0.679434i	$0.237775\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	1.75379	0.188026
$88$	0	0
$89$	−3.56155	−0.377524	−0.188762	−	0.982023i	$-0.560447\pi$
$89$	−3.56155	−0.377524	−0.188762	+	0.982023i	$0.560447\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	14.9309	1.54826
$94$	0	0
$95$	25.3693	2.60284
$96$	0	0
$97$	−6.68466	−0.678724	−0.339362	−	0.940656i	$-0.610211\pi$
$97$	−6.68466	−0.678724	−0.339362	+	0.940656i	$0.610211\pi$
$98$	0	0
$99$	−0.561553	−0.0564382
$100$	0	0
$101$	13.1231	1.30580	0.652899	−	0.757445i	$-0.273553\pi$
$101$	13.1231	1.30580	0.652899	+	0.757445i	$0.273553\pi$
$102$	0	0
$103$	2.24621	0.221326	0.110663	−	0.993858i	$-0.464703\pi$
$103$	2.24621	0.221326	0.110663	+	0.993858i	$0.464703\pi$
$104$	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
$111$	−10.4384	−0.990774
$112$	0	0
$113$	15.5616	1.46391	0.731954	−	0.681354i	$-0.238609\pi$
$113$	15.5616	1.46391	0.731954	+	0.681354i	$0.238609\pi$
$114$	0	0
$115$	16.6847	1.55585
$116$	0	0
$117$	−1.12311	−0.103831
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.8769	−1.16107
$124$	0	0
$125$	9.56155	0.855211
$126$	0	0
$127$	−9.36932	−0.831392	−0.415696	−	0.909504i	$-0.636462\pi$
$127$	−9.36932	−0.831392	−0.415696	+	0.909504i	$0.636462\pi$
$128$	0	0
$129$	−11.1231	−0.979335
$130$	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
$135$	19.8078	1.70478
$136$	0	0
$137$	1.31534	0.112377	0.0561886	−	0.998420i	$-0.482105\pi$
$137$	1.31534	0.112377	0.0561886	+	0.998420i	$0.482105\pi$
$138$	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
$141$	6.24621	0.526026
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	10.9309	0.901563
$148$	0	0
$149$	13.1231	1.07509	0.537543	−	0.843236i	$-0.319352\pi$
$149$	13.1231	1.07509	0.537543	+	0.843236i	$0.319352\pi$
$150$	0	0
$151$	−9.36932	−0.762464	−0.381232	−	0.924479i	$-0.624500\pi$
$151$	−9.36932	−0.762464	−0.381232	+	0.924479i	$0.624500\pi$
$152$	0	0
$153$	0.630683	0.0509877
$154$	0	0
$155$	−34.0540	−2.73528
$156$	0	0
$157$	−10.6847	−0.852729	−0.426364	−	0.904552i	$-0.640206\pi$
$157$	−10.6847	−0.852729	−0.426364	+	0.904552i	$0.640206\pi$
$158$	0	0
$159$	12.8769	1.02120
$160$	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
$165$	−5.56155	−0.432966
$166$	0	0
$167$	−17.3693	−1.34408	−0.672039	−	0.740516i	$-0.734581\pi$
$167$	−17.3693	−1.34408	−0.672039	+	0.740516i	$0.734581\pi$
$168$	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
$177$	19.8078	1.48884
$178$	0	0
$179$	−4.68466	−0.350148	−0.175074	−	0.984555i	$-0.556016\pi$
$179$	−4.68466	−0.350148	−0.175074	+	0.984555i	$0.556016\pi$
$180$	0	0
$181$	−12.0540	−0.895965	−0.447982	−	0.894042i	$-0.647857\pi$
$181$	−12.0540	−0.895965	−0.447982	+	0.894042i	$0.647857\pi$
$182$	0	0
$183$	24.0000	1.77413
$184$	0	0
$185$	23.8078	1.75038
$186$	0	0
$187$	−1.12311	−0.0821296
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.68466	−0.338970	−0.169485	−	0.985533i	$-0.554210\pi$
$191$	−4.68466	−0.338970	−0.169485	+	0.985533i	$0.554210\pi$
$192$	0	0
$193$	11.3693	0.818381	0.409191	−	0.912449i	$-0.365811\pi$
$193$	11.3693	0.818381	0.409191	+	0.912449i	$0.365811\pi$
$194$	0	0
$195$	−11.1231	−0.796542
$196$	0	0
$197$	8.24621	0.587518	0.293759	−	0.955879i	$-0.405094\pi$
$197$	8.24621	0.587518	0.293759	+	0.955879i	$0.405094\pi$
$198$	0	0
$199$	2.24621	0.159230	0.0796148	−	0.996826i	$-0.474631\pi$
$199$	2.24621	0.159230	0.0796148	+	0.996826i	$0.474631\pi$
$200$	0	0
$201$	7.31534	0.515984
$202$	0	0
$203$	0	0
$204$	0	0
$205$	29.3693	2.05124
$206$	0	0
$207$	−2.63068	−0.182845
$208$	0	0
$209$	7.12311	0.492716
$210$	0	0
$211$	−16.4924	−1.13539	−0.567693	−	0.823241i	$-0.692164\pi$
$211$	−16.4924	−1.13539	−0.567693	+	0.823241i	$0.692164\pi$
$212$	0	0
$213$	−5.17708	−0.354728
$214$	0	0
$215$	25.3693	1.73017
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.36932	0.633120
$220$	0	0
$221$	−2.24621	−0.151097
$222$	0	0
$223$	4.68466	0.313708	0.156854	−	0.987622i	$-0.449865\pi$
$223$	4.68466	0.313708	0.156854	+	0.987622i	$0.449865\pi$
$224$	0	0
$225$	−4.31534	−0.287689
$226$	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
$229$	−9.31534	−0.615575	−0.307788	−	0.951455i	$-0.599589\pi$
$229$	−9.31534	−0.615575	−0.307788	+	0.951455i	$0.599589\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.12311	−0.0735771	−0.0367885	−	0.999323i	$-0.511713\pi$
$233$	−1.12311	−0.0735771	−0.0367885	+	0.999323i	$0.511713\pi$
$234$	0	0
$235$	−14.2462	−0.929320
$236$	0	0
$237$	7.61553	0.494682
$238$	0	0
$239$	−25.3693	−1.64100	−0.820502	−	0.571643i	$-0.806306\pi$
$239$	−25.3693	−1.64100	−0.820502	+	0.571643i	$0.806306\pi$
$240$	0	0
$241$	27.3693	1.76301	0.881506	−	0.472172i	$-0.156530\pi$
$241$	27.3693	1.76301	0.881506	+	0.472172i	$0.156530\pi$
$242$	0	0
$243$	−5.75379	−0.369106
$244$	0	0
$245$	−24.9309	−1.59277
$246$	0	0
$247$	14.2462	0.906465
$248$	0	0
$249$	−20.8769	−1.32302
$250$	0	0
$251$	14.0540	0.887079	0.443540	−	0.896255i	$-0.353723\pi$
$251$	14.0540	0.887079	0.443540	+	0.896255i	$0.353723\pi$
$252$	0	0
$253$	4.68466	0.294522
$254$	0	0
$255$	6.24621	0.391153
$256$	0	0
$257$	8.24621	0.514385	0.257192	−	0.966360i	$-0.417203\pi$
$257$	8.24621	0.514385	0.257192	+	0.966360i	$0.417203\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.630683	0.0390383
$262$	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
$265$	−29.3693	−1.80414
$266$	0	0
$267$	5.56155	0.340362
$268$	0	0
$269$	−22.4924	−1.37139	−0.685694	−	0.727890i	$-0.740501\pi$
$269$	−22.4924	−1.37139	−0.685694	+	0.727890i	$0.740501\pi$
$270$	0	0
$271$	−4.87689	−0.296250	−0.148125	−	0.988969i	$-0.547324\pi$
$271$	−4.87689	−0.296250	−0.148125	+	0.988969i	$0.547324\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.68466	0.463402
$276$	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
$279$	5.36932	0.321453
$280$	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
$283$	26.2462	1.56018	0.780088	−	0.625670i	$-0.215174\pi$
$283$	26.2462	1.56018	0.780088	+	0.625670i	$0.215174\pi$
$284$	0	0
$285$	−39.6155	−2.34662
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.7386	−0.925802
$290$	0	0
$291$	10.4384	0.611913
$292$	0	0
$293$	22.4924	1.31402	0.657011	−	0.753881i	$-0.271821\pi$
$293$	22.4924	1.31402	0.657011	+	0.753881i	$0.271821\pi$
$294$	0	0
$295$	−45.1771	−2.63031
$296$	0	0
$297$	5.56155	0.322714
$298$	0	0
$299$	9.36932	0.541842
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−20.4924	−1.17726
$304$	0	0
$305$	−54.7386	−3.13433
$306$	0	0
$307$	2.63068	0.150141	0.0750705	−	0.997178i	$-0.476082\pi$
$307$	2.63068	0.150141	0.0750705	+	0.997178i	$0.476082\pi$
$308$	0	0
$309$	−3.50758	−0.199539
$310$	0	0
$311$	−30.7386	−1.74303	−0.871514	−	0.490371i	$-0.836861\pi$
$311$	−30.7386	−1.74303	−0.871514	+	0.490371i	$0.836861\pi$
$312$	0	0
$313$	18.6847	1.05612	0.528060	−	0.849207i	$-0.322920\pi$
$313$	18.6847	1.05612	0.528060	+	0.849207i	$0.322920\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.05398	0.452356	0.226178	−	0.974086i	$-0.427377\pi$
$317$	8.05398	0.452356	0.226178	+	0.974086i	$0.427377\pi$
$318$	0	0
$319$	−1.12311	−0.0628818
$320$	0	0
$321$	18.7386	1.04589
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	15.3693	0.852536
$326$	0	0
$327$	21.8617	1.20896
$328$	0	0
$329$	0	0
$330$	0	0
$331$	28.6847	1.57665	0.788326	−	0.615258i	$-0.210948\pi$
$331$	28.6847	1.57665	0.788326	+	0.615258i	$0.210948\pi$
$332$	0	0
$333$	−3.75379	−0.205706
$334$	0	0
$335$	−16.6847	−0.911580
$336$	0	0
$337$	−31.3693	−1.70880	−0.854398	−	0.519619i	$-0.826074\pi$
$337$	−31.3693	−1.70880	−0.854398	+	0.519619i	$0.826074\pi$
$338$	0	0
$339$	−24.3002	−1.31980
$340$	0	0
$341$	−9.56155	−0.517787
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−26.0540	−1.40270
$346$	0	0
$347$	2.63068	0.141222	0.0706112	−	0.997504i	$-0.477505\pi$
$347$	2.63068	0.141222	0.0706112	+	0.997504i	$0.477505\pi$
$348$	0	0
$349$	11.3693	0.608586	0.304293	−	0.952579i	$-0.401580\pi$
$349$	11.3693	0.608586	0.304293	+	0.952579i	$0.401580\pi$
$350$	0	0
$351$	11.1231	0.593707
$352$	0	0
$353$	20.4384	1.08783	0.543914	−	0.839141i	$-0.316942\pi$
$353$	20.4384	1.08783	0.543914	+	0.839141i	$0.316942\pi$
$354$	0	0
$355$	11.8078	0.626691
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.36932	−0.494494	−0.247247	−	0.968953i	$-0.579526\pi$
$359$	−9.36932	−0.494494	−0.247247	+	0.968953i	$0.579526\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	0	0
$363$	−1.56155	−0.0819603
$364$	0	0
$365$	−21.3693	−1.11852
$366$	0	0
$367$	−38.0540	−1.98640	−0.993201	−	0.116415i	$-0.962860\pi$
$367$	−38.0540	−1.98640	−0.993201	+	0.116415i	$0.962860\pi$
$368$	0	0
$369$	−4.63068	−0.241064
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.36932	0.174457	0.0872283	−	0.996188i	$-0.472199\pi$
$373$	3.36932	0.174457	0.0872283	+	0.996188i	$0.472199\pi$
$374$	0	0
$375$	−14.9309	−0.771027
$376$	0	0
$377$	−2.24621	−0.115686
$378$	0	0
$379$	14.4384	0.741653	0.370827	−	0.928702i	$-0.379074\pi$
$379$	14.4384	0.741653	0.370827	+	0.928702i	$0.379074\pi$
$380$	0	0
$381$	14.6307	0.749553
$382$	0	0
$383$	20.6847	1.05694	0.528468	−	0.848953i	$-0.322767\pi$
$383$	20.6847	1.05694	0.528468	+	0.848953i	$0.322767\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−6.19224	−0.313959	−0.156979	−	0.987602i	$-0.550176\pi$
$389$	−6.19224	−0.313959	−0.156979	+	0.987602i	$0.550176\pi$
$390$	0	0
$391$	−5.26137	−0.266079
$392$	0	0
$393$	−6.24621	−0.315080
$394$	0	0
$395$	−17.3693	−0.873945
$396$	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
$401$	−10.4924	−0.523967	−0.261983	−	0.965072i	$-0.584377\pi$
$401$	−10.4924	−0.523967	−0.261983	+	0.965072i	$0.584377\pi$
$402$	0	0
$403$	−19.1231	−0.952590
$404$	0	0
$405$	−24.9309	−1.23882
$406$	0	0
$407$	6.68466	0.331346
$408$	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
$411$	−2.05398	−0.101315
$412$	0	0
$413$	0	0
$414$	0	0
$415$	47.6155	2.33735
$416$	0	0
$417$	−18.7386	−0.917635
$418$	0	0
$419$	30.7386	1.50168	0.750840	−	0.660484i	$-0.229649\pi$
$419$	30.7386	1.50168	0.750840	+	0.660484i	$0.229649\pi$
$420$	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
$423$	2.24621	0.109215
$424$	0	0
$425$	−8.63068	−0.418650
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.12311	−0.150785
$430$	0	0
$431$	10.7386	0.517262	0.258631	−	0.965976i	$-0.416729\pi$
$431$	10.7386	0.517262	0.258631	+	0.965976i	$0.416729\pi$
$432$	0	0
$433$	−6.68466	−0.321244	−0.160622	−	0.987016i	$-0.551350\pi$
$433$	−6.68466	−0.321244	−0.160622	+	0.987016i	$0.551350\pi$
$434$	0	0
$435$	6.24621	0.299483
$436$	0	0
$437$	33.3693	1.59627
$438$	0	0
$439$	−9.75379	−0.465523	−0.232761	−	0.972534i	$-0.574776\pi$
$439$	−9.75379	−0.465523	−0.232761	+	0.972534i	$0.574776\pi$
$440$	0	0
$441$	3.93087	0.187184
$442$	0	0
$443$	−20.6847	−0.982758	−0.491379	−	0.870946i	$-0.663507\pi$
$443$	−20.6847	−0.982758	−0.491379	+	0.870946i	$0.663507\pi$
$444$	0	0
$445$	−12.6847	−0.601310
$446$	0	0
$447$	−20.4924	−0.969258
$448$	0	0
$449$	−17.8078	−0.840400	−0.420200	−	0.907431i	$-0.638040\pi$
$449$	−17.8078	−0.840400	−0.420200	+	0.907431i	$0.638040\pi$
$450$	0	0
$451$	8.24621	0.388299
$452$	0	0
$453$	14.6307	0.687409
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.6307	0.777951	0.388975	−	0.921248i	$-0.372829\pi$
$457$	16.6307	0.777951	0.388975	+	0.921248i	$0.372829\pi$
$458$	0	0
$459$	−6.24621	−0.291548
$460$	0	0
$461$	−10.8769	−0.506587	−0.253294	−	0.967389i	$-0.581514\pi$
$461$	−10.8769	−0.506587	−0.253294	+	0.967389i	$0.581514\pi$
$462$	0	0
$463$	5.06913	0.235582	0.117791	−	0.993038i	$-0.462419\pi$
$463$	5.06913	0.235582	0.117791	+	0.993038i	$0.462419\pi$
$464$	0	0
$465$	53.1771	2.46603
$466$	0	0
$467$	−20.6847	−0.957172	−0.478586	−	0.878041i	$-0.658850\pi$
$467$	−20.6847	−0.957172	−0.478586	+	0.878041i	$0.658850\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.6847	0.768788
$472$	0	0
$473$	7.12311	0.327521
$474$	0	0
$475$	54.7386	2.51158
$476$	0	0
$477$	4.63068	0.212024
$478$	0	0
$479$	25.3693	1.15915	0.579577	−	0.814918i	$-0.303218\pi$
$479$	25.3693	1.15915	0.579577	+	0.814918i	$0.303218\pi$
$480$	0	0
$481$	13.3693	0.609588
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−23.8078	−1.08105
$486$	0	0
$487$	−19.3153	−0.875262	−0.437631	−	0.899155i	$-0.644182\pi$
$487$	−19.3153	−0.875262	−0.437631	+	0.899155i	$0.644182\pi$
$488$	0	0
$489$	18.7386	0.847390
$490$	0	0
$491$	22.7386	1.02618	0.513090	−	0.858335i	$-0.328501\pi$
$491$	22.7386	1.02618	0.513090	+	0.858335i	$0.328501\pi$
$492$	0	0
$493$	1.26137	0.0568091
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.50758	−0.336085	−0.168043	−	0.985780i	$-0.553745\pi$
$499$	−7.50758	−0.336085	−0.168043	+	0.985780i	$0.553745\pi$
$500$	0	0
$501$	27.1231	1.21177
$502$	0	0
$503$	−9.36932	−0.417757	−0.208879	−	0.977942i	$-0.566981\pi$
$503$	−9.36932	−0.417757	−0.208879	+	0.977942i	$0.566981\pi$
$504$	0	0
$505$	46.7386	2.07984
$506$	0	0
$507$	14.0540	0.624159
$508$	0	0
$509$	−1.31534	−0.0583015	−0.0291507	−	0.999575i	$-0.509280\pi$
$509$	−1.31534	−0.0583015	−0.0291507	+	0.999575i	$0.509280\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	39.6155	1.74907
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	−28.1080	−1.23380
$520$	0	0
$521$	20.0540	0.878581	0.439290	−	0.898345i	$-0.355230\pi$
$521$	20.0540	0.878581	0.439290	+	0.898345i	$0.355230\pi$
$522$	0	0
$523$	−40.1080	−1.75380	−0.876899	−	0.480674i	$-0.840392\pi$
$523$	−40.1080	−1.75380	−0.876899	+	0.480674i	$0.840392\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.7386	0.467782
$528$	0	0
$529$	−1.05398	−0.0458250
$530$	0	0
$531$	7.12311	0.309116
$532$	0	0
$533$	16.4924	0.714366
$534$	0	0
$535$	−42.7386	−1.84775
$536$	0	0
$537$	7.31534	0.315680
$538$	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
$543$	18.8229	0.807769
$544$	0	0
$545$	−49.8617	−2.13584
$546$	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
$549$	8.63068	0.368349
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−37.1771	−1.57808
$556$	0	0
$557$	8.63068	0.365694	0.182847	−	0.983141i	$-0.441469\pi$
$557$	8.63068	0.365694	0.182847	+	0.983141i	$0.441469\pi$
$558$	0	0
$559$	14.2462	0.602551
$560$	0	0
$561$	1.75379	0.0740450
$562$	0	0
$563$	−30.7386	−1.29548	−0.647739	−	0.761862i	$-0.724285\pi$
$563$	−30.7386	−1.29548	−0.647739	+	0.761862i	$0.724285\pi$
$564$	0	0
$565$	55.4233	2.33168
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.36932	0.141249	0.0706246	−	0.997503i	$-0.477501\pi$
$569$	3.36932	0.141249	0.0706246	+	0.997503i	$0.477501\pi$
$570$	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
$573$	7.31534	0.305603
$574$	0	0
$575$	36.0000	1.50130
$576$	0	0
$577$	−6.68466	−0.278286	−0.139143	−	0.990272i	$-0.544435\pi$
$577$	−6.68466	−0.278286	−0.139143	+	0.990272i	$0.544435\pi$
$578$	0	0
$579$	−17.7538	−0.737822
$580$	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
$589$	−68.1080	−2.80634
$590$	0	0
$591$	−12.8769	−0.529685
$592$	0	0
$593$	31.8617	1.30840	0.654202	−	0.756320i	$-0.273004\pi$
$593$	31.8617	1.30840	0.654202	+	0.756320i	$0.273004\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.50758	−0.143556
$598$	0	0
$599$	14.7386	0.602204	0.301102	−	0.953592i	$-0.402645\pi$
$599$	14.7386	0.602204	0.301102	+	0.953592i	$0.402645\pi$
$600$	0	0
$601$	−42.1080	−1.71762	−0.858810	−	0.512295i	$-0.828795\pi$
$601$	−42.1080	−1.71762	−0.858810	+	0.512295i	$0.828795\pi$
$602$	0	0
$603$	2.63068	0.107130
$604$	0	0
$605$	3.56155	0.144798
$606$	0	0
$607$	28.8769	1.17208	0.586038	−	0.810283i	$-0.300687\pi$
$607$	28.8769	1.17208	0.586038	+	0.810283i	$0.300687\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−7.36932	−0.297644	−0.148822	−	0.988864i	$-0.547548\pi$
$613$	−7.36932	−0.297644	−0.148822	+	0.988864i	$0.547548\pi$
$614$	0	0
$615$	−45.8617	−1.84932
$616$	0	0
$617$	−15.7538	−0.634224	−0.317112	−	0.948388i	$-0.602713\pi$
$617$	−15.7538	−0.634224	−0.317112	+	0.948388i	$0.602713\pi$
$618$	0	0
$619$	4.68466	0.188292	0.0941462	−	0.995558i	$-0.469988\pi$
$619$	4.68466	0.188292	0.0941462	+	0.995558i	$0.469988\pi$
$620$	0	0
$621$	26.0540	1.04551
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	−11.1231	−0.444214
$628$	0	0
$629$	−7.50758	−0.299347
$630$	0	0
$631$	23.4233	0.932467	0.466233	−	0.884662i	$-0.345611\pi$
$631$	23.4233	0.932467	0.466233	+	0.884662i	$0.345611\pi$
$632$	0	0
$633$	25.7538	1.02362
$634$	0	0
$635$	−33.3693	−1.32422
$636$	0	0
$637$	−14.0000	−0.554700
$638$	0	0
$639$	−1.86174	−0.0736493
$640$	0	0
$641$	0.930870	0.0367671	0.0183836	−	0.999831i	$-0.494148\pi$
$641$	0.930870	0.0367671	0.0183836	+	0.999831i	$0.494148\pi$
$642$	0	0
$643$	−38.4384	−1.51586	−0.757932	−	0.652333i	$-0.773790\pi$
$643$	−38.4384	−1.51586	−0.757932	+	0.652333i	$0.773790\pi$
$644$	0	0
$645$	−39.6155	−1.55986
$646$	0	0
$647$	19.3153	0.759364	0.379682	−	0.925117i	$-0.376033\pi$
$647$	19.3153	0.759364	0.379682	+	0.925117i	$0.376033\pi$
$648$	0	0
$649$	−12.6847	−0.497916
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.31534	−0.0514733	−0.0257366	−	0.999669i	$-0.508193\pi$
$653$	−1.31534	−0.0514733	−0.0257366	+	0.999669i	$0.508193\pi$
$654$	0	0
$655$	14.2462	0.556646
$656$	0	0
$657$	3.36932	0.131450
$658$	0	0
$659$	−37.3693	−1.45570	−0.727851	−	0.685735i	$-0.759481\pi$
$659$	−37.3693	−1.45570	−0.727851	+	0.685735i	$0.759481\pi$
$660$	0	0
$661$	16.0540	0.624427	0.312214	−	0.950012i	$-0.398930\pi$
$661$	16.0540	0.624427	0.312214	+	0.950012i	$0.398930\pi$
$662$	0	0
$663$	3.50758	0.136223
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.26137	−0.203721
$668$	0	0
$669$	−7.31534	−0.282827
$670$	0	0
$671$	−15.3693	−0.593326
$672$	0	0
$673$	−24.7386	−0.953604	−0.476802	−	0.879011i	$-0.658204\pi$
$673$	−24.7386	−0.953604	−0.476802	+	0.879011i	$0.658204\pi$
$674$	0	0
$675$	42.7386	1.64501
$676$	0	0
$677$	−19.8617	−0.763349	−0.381674	−	0.924297i	$-0.624652\pi$
$677$	−19.8617	−0.763349	−0.381674	+	0.924297i	$0.624652\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−31.2311	−1.19678
$682$	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
$685$	4.68466	0.178992
$686$	0	0
$687$	14.5464	0.554980
$688$	0	0
$689$	−16.4924	−0.628311
$690$	0	0
$691$	28.3002	1.07659	0.538295	−	0.842757i	$-0.319069\pi$
$691$	28.3002	1.07659	0.538295	+	0.842757i	$0.319069\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	42.7386	1.62117
$696$	0	0
$697$	−9.26137	−0.350799
$698$	0	0
$699$	1.75379	0.0663344
$700$	0	0
$701$	−6.38447	−0.241138	−0.120569	−	0.992705i	$-0.538472\pi$
$701$	−6.38447	−0.241138	−0.120569	+	0.992705i	$0.538472\pi$
$702$	0	0
$703$	47.6155	1.79585
$704$	0	0
$705$	22.2462	0.837841
$706$	0	0
$707$	0	0
$708$	0	0
$709$	22.6847	0.851940	0.425970	−	0.904737i	$-0.359933\pi$
$709$	22.6847	0.851940	0.425970	+	0.904737i	$0.359933\pi$
$710$	0	0
$711$	2.73863	0.102707
$712$	0	0
$713$	−44.7926	−1.67750
$714$	0	0
$715$	7.12311	0.266389
$716$	0	0
$717$	39.6155	1.47947
$718$	0	0
$719$	28.6847	1.06976	0.534879	−	0.844929i	$-0.320357\pi$
$719$	28.6847	1.06976	0.534879	+	0.844929i	$0.320357\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−42.7386	−1.58947
$724$	0	0
$725$	−8.63068	−0.320536
$726$	0	0
$727$	28.6847	1.06386	0.531928	−	0.846790i	$-0.321468\pi$
$727$	28.6847	1.06386	0.531928	+	0.846790i	$0.321468\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	38.1080	1.40755	0.703775	−	0.710423i	$-0.251496\pi$
$733$	38.1080	1.40755	0.703775	+	0.710423i	$0.251496\pi$
$734$	0	0
$735$	38.9309	1.43599
$736$	0	0
$737$	−4.68466	−0.172562
$738$	0	0
$739$	−11.6155	−0.427284	−0.213642	−	0.976912i	$-0.568533\pi$
$739$	−11.6155	−0.427284	−0.213642	+	0.976912i	$0.568533\pi$
$740$	0	0
$741$	−22.2462	−0.817235
$742$	0	0
$743$	18.7386	0.687454	0.343727	−	0.939070i	$-0.388311\pi$
$743$	18.7386	0.687454	0.343727	+	0.939070i	$0.388311\pi$
$744$	0	0
$745$	46.7386	1.71237
$746$	0	0
$747$	−7.50758	−0.274688
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−33.1771	−1.21065	−0.605324	−	0.795979i	$-0.706957\pi$
$751$	−33.1771	−1.21065	−0.605324	+	0.795979i	$0.706957\pi$
$752$	0	0
$753$	−21.9460	−0.799758
$754$	0	0
$755$	−33.3693	−1.21443
$756$	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
$759$	−7.31534	−0.265530
$760$	0	0
$761$	−20.2462	−0.733925	−0.366962	−	0.930236i	$-0.619602\pi$
$761$	−20.2462	−0.733925	−0.366962	+	0.930236i	$0.619602\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.24621	0.0812119
$766$	0	0
$767$	−25.3693	−0.916033
$768$	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
$771$	−12.8769	−0.463750
$772$	0	0
$773$	−8.24621	−0.296596	−0.148298	−	0.988943i	$-0.547379\pi$
$773$	−8.24621	−0.296596	−0.148298	+	0.988943i	$0.547379\pi$
$774$	0	0
$775$	−73.4773	−2.63938
$776$	0	0
$777$	0	0
$778$	0	0
$779$	58.7386	2.10453
$780$	0	0
$781$	3.31534	0.118632
$782$	0	0
$783$	−6.24621	−0.223221
$784$	0	0
$785$	−38.0540	−1.35820
$786$	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
$789$	−12.4924	−0.444742
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−30.7386	−1.09156
$794$	0	0
$795$	45.8617	1.62655
$796$	0	0
$797$	−20.4384	−0.723967	−0.361983	−	0.932185i	$-0.617900\pi$
$797$	−20.4384	−0.723967	−0.361983	+	0.932185i	$0.617900\pi$
$798$	0	0
$799$	4.49242	0.158930
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	35.1231	1.23639
$808$	0	0
$809$	8.24621	0.289921	0.144961	−	0.989437i	$-0.453694\pi$
$809$	8.24621	0.289921	0.144961	+	0.989437i	$0.453694\pi$
$810$	0	0
$811$	26.6307	0.935130	0.467565	−	0.883959i	$-0.345131\pi$
$811$	26.6307	0.935130	0.467565	+	0.883959i	$0.345131\pi$
$812$	0	0
$813$	7.61553	0.267088
$814$	0	0
$815$	−42.7386	−1.49707
$816$	0	0
$817$	50.7386	1.77512
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.7386	−1.70099	−0.850495	−	0.525983i	$-0.823698\pi$
$821$	−48.7386	−1.70099	−0.850495	+	0.525983i	$0.823698\pi$
$822$	0	0
$823$	14.4384	0.503293	0.251646	−	0.967819i	$-0.419028\pi$
$823$	14.4384	0.503293	0.251646	+	0.967819i	$0.419028\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−21.3693	−0.743084	−0.371542	−	0.928416i	$-0.621171\pi$
$827$	−21.3693	−0.743084	−0.371542	+	0.928416i	$0.621171\pi$
$828$	0	0
$829$	5.31534	0.184609	0.0923047	−	0.995731i	$-0.470577\pi$
$829$	5.31534	0.184609	0.0923047	+	0.995731i	$0.470577\pi$
$830$	0	0
$831$	−3.12311	−0.108339
$832$	0	0
$833$	7.86174	0.272393
$834$	0	0
$835$	−61.8617	−2.14081
$836$	0	0
$837$	−53.1771	−1.83807
$838$	0	0
$839$	31.4233	1.08485	0.542426	−	0.840103i	$-0.317506\pi$
$839$	31.4233	1.08485	0.542426	+	0.840103i	$0.317506\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	0	0
$843$	9.36932	0.322696
$844$	0	0
$845$	−32.0540	−1.10269
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−40.9848	−1.40660
$850$	0	0
$851$	31.3153	1.07348
$852$	0	0
$853$	−8.73863	−0.299205	−0.149603	−	0.988746i	$-0.547799\pi$
$853$	−8.73863	−0.299205	−0.149603	+	0.988746i	$0.547799\pi$
$854$	0	0
$855$	−14.2462	−0.487210
$856$	0	0
$857$	13.1231	0.448277	0.224138	−	0.974557i	$-0.428043\pi$
$857$	13.1231	0.448277	0.224138	+	0.974557i	$0.428043\pi$
$858$	0	0
$859$	29.0691	0.991826	0.495913	−	0.868372i	$-0.334833\pi$
$859$	29.0691	0.991826	0.495913	+	0.868372i	$0.334833\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.7386	−1.04636	−0.523178	−	0.852224i	$-0.675254\pi$
$863$	−30.7386	−1.04636	−0.523178	+	0.852224i	$0.675254\pi$
$864$	0	0
$865$	64.1080	2.17974
$866$	0	0
$867$	24.5767	0.834669
$868$	0	0
$869$	−4.87689	−0.165437
$870$	0	0
$871$	−9.36932	−0.317467
$872$	0	0
$873$	3.75379	0.127047
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−56.7386	−1.91593	−0.957964	−	0.286889i	$-0.907379\pi$
$877$	−56.7386	−1.91593	−0.957964	+	0.286889i	$0.907379\pi$
$878$	0	0
$879$	−35.1231	−1.18467
$880$	0	0
$881$	34.3002	1.15560	0.577801	−	0.816177i	$-0.303911\pi$
$881$	34.3002	1.15560	0.577801	+	0.816177i	$0.303911\pi$
$882$	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
$885$	70.5464	2.37139
$886$	0	0
$887$	−52.1080	−1.74961	−0.874807	−	0.484472i	$-0.839012\pi$
$887$	−52.1080	−1.74961	−0.874807	+	0.484472i	$0.839012\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	−28.4924	−0.953463
$894$	0	0
$895$	−16.6847	−0.557707
$896$	0	0
$897$	−14.6307	−0.488504
$898$	0	0
$899$	10.7386	0.358153
$900$	0	0
$901$	9.26137	0.308541
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−42.9309	−1.42707
$906$	0	0
$907$	16.4924	0.547622	0.273811	−	0.961784i	$-0.411716\pi$
$907$	16.4924	0.547622	0.273811	+	0.961784i	$0.411716\pi$
$908$	0	0
$909$	−7.36932	−0.244425
$910$	0	0
$911$	−22.7386	−0.753365	−0.376682	−	0.926343i	$-0.622935\pi$
$911$	−22.7386	−0.753365	−0.376682	+	0.926343i	$0.622935\pi$
$912$	0	0
$913$	13.3693	0.442460
$914$	0	0
$915$	85.4773	2.82579
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−23.6155	−0.779004	−0.389502	−	0.921026i	$-0.627353\pi$
$919$	−23.6155	−0.779004	−0.389502	+	0.921026i	$0.627353\pi$
$920$	0	0
$921$	−4.10795	−0.135362
$922$	0	0
$923$	6.63068	0.218252
$924$	0	0
$925$	51.3693	1.68901
$926$	0	0
$927$	−1.26137	−0.0414287
$928$	0	0
$929$	−10.4924	−0.344245	−0.172123	−	0.985076i	$-0.555063\pi$
$929$	−10.4924	−0.344245	−0.172123	+	0.985076i	$0.555063\pi$
$930$	0	0
$931$	−49.8617	−1.63415
$932$	0	0
$933$	48.0000	1.57145
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	28.7386	0.938850	0.469425	−	0.882972i	$-0.344461\pi$
$937$	28.7386	0.938850	0.469425	+	0.882972i	$0.344461\pi$
$938$	0	0
$939$	−29.1771	−0.952158
$940$	0	0
$941$	8.24621	0.268819	0.134409	−	0.990926i	$-0.457086\pi$
$941$	8.24621	0.268819	0.134409	+	0.990926i	$0.457086\pi$
$942$	0	0
$943$	38.6307	1.25799
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.3153	0.627664	0.313832	−	0.949478i	$-0.398387\pi$
$947$	19.3153	0.627664	0.313832	+	0.949478i	$0.398387\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−12.5767	−0.407828
$952$	0	0
$953$	−6.38447	−0.206813	−0.103407	−	0.994639i	$-0.532974\pi$
$953$	−6.38447	−0.206813	−0.103407	+	0.994639i	$0.532974\pi$
$954$	0	0
$955$	−16.6847	−0.539903
$956$	0	0
$957$	1.75379	0.0566919
$958$	0	0
$959$	0	0
$960$	0	0
$961$	60.4233	1.94914
$962$	0	0
$963$	6.73863	0.217149
$964$	0	0
$965$	40.4924	1.30350
$966$	0	0
$967$	18.7386	0.602594	0.301297	−	0.953530i	$-0.402580\pi$
$967$	18.7386	0.602594	0.301297	+	0.953530i	$0.402580\pi$
$968$	0	0
$969$	12.4924	0.401314
$970$	0	0
$971$	28.6847	0.920534	0.460267	−	0.887780i	$-0.347754\pi$
$971$	28.6847	0.920534	0.460267	+	0.887780i	$0.347754\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−24.0000	−0.768615
$976$	0	0
$977$	53.0388	1.69686	0.848431	−	0.529306i	$-0.177548\pi$
$977$	53.0388	1.69686	0.848431	+	0.529306i	$0.177548\pi$
$978$	0	0
$979$	−3.56155	−0.113828
$980$	0	0
$981$	7.86174	0.251006
$982$	0	0
$983$	28.6847	0.914899	0.457449	−	0.889236i	$-0.348763\pi$
$983$	28.6847	0.914899	0.457449	+	0.889236i	$0.348763\pi$
$984$	0	0
$985$	29.3693	0.935784
$986$	0	0
$987$	0	0
$988$	0	0
$989$	33.3693	1.06108
$990$	0	0
$991$	54.7386	1.73883	0.869415	−	0.494083i	$-0.164496\pi$
$991$	54.7386	1.73883	0.869415	+	0.494083i	$0.164496\pi$
$992$	0	0
$993$	−44.7926	−1.42145
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	0.630683	0.0199739	0.00998697	−	0.999950i	$-0.496821\pi$
$997$	0.630683	0.0199739	0.00998697	+	0.999950i	$0.496821\pi$
$998$	0	0
$999$	37.1771	1.17623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	352.2.a.h.1.1	yes	2
3.2	odd	2		3168.2.a.bc.1.1		2
4.3	odd	2		352.2.a.g.1.2	✓	2
5.4	even	2		8800.2.a.bd.1.2		2
8.3	odd	2		704.2.a.o.1.1		2
8.5	even	2		704.2.a.n.1.2		2
11.10	odd	2		3872.2.a.ba.1.1		2
12.11	even	2		3168.2.a.bd.1.1		2
16.3	odd	4		2816.2.c.t.1409.3		4
16.5	even	4		2816.2.c.s.1409.3		4
16.11	odd	4		2816.2.c.t.1409.2		4
16.13	even	4		2816.2.c.s.1409.2		4
20.19	odd	2		8800.2.a.be.1.1		2
24.5	odd	2		6336.2.a.cw.1.2		2
24.11	even	2		6336.2.a.cv.1.2		2
44.43	even	2		3872.2.a.p.1.2		2
88.21	odd	2		7744.2.a.bw.1.2		2
88.43	even	2		7744.2.a.cm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.a.g.1.2	✓	2	4.3	odd	2
352.2.a.h.1.1	yes	2	1.1	even	1	trivial
704.2.a.n.1.2		2	8.5	even	2
704.2.a.o.1.1		2	8.3	odd	2
2816.2.c.s.1409.2		4	16.13	even	4
2816.2.c.s.1409.3		4	16.5	even	4
2816.2.c.t.1409.2		4	16.11	odd	4
2816.2.c.t.1409.3		4	16.3	odd	4
3168.2.a.bc.1.1		2	3.2	odd	2
3168.2.a.bd.1.1		2	12.11	even	2
3872.2.a.p.1.2		2	44.43	even	2
3872.2.a.ba.1.1		2	11.10	odd	2
6336.2.a.cv.1.2		2	24.11	even	2
6336.2.a.cw.1.2		2	24.5	odd	2
7744.2.a.bw.1.2		2	88.21	odd	2
7744.2.a.cm.1.1		2	88.43	even	2
8800.2.a.bd.1.2		2	5.4	even	2
8800.2.a.be.1.1		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)

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} +3.56155 q^{5} -0.561553 q^{9} +1.00000 q^{11} +2.00000 q^{13} -5.56155 q^{15} -1.12311 q^{17} +7.12311 q^{19} +4.68466 q^{23} +7.68466 q^{25} +5.56155 q^{27} -1.12311 q^{29} -9.56155 q^{31} -1.56155 q^{33} +6.68466 q^{37} -3.12311 q^{39} +8.24621 q^{41} +7.12311 q^{43} -2.00000 q^{45} -4.00000 q^{47} -7.00000 q^{49} +1.75379 q^{51} -8.24621 q^{53} +3.56155 q^{55} -11.1231 q^{57} -12.6847 q^{59} -15.3693 q^{61} +7.12311 q^{65} -4.68466 q^{67} -7.31534 q^{69} +3.31534 q^{71} -6.00000 q^{73} -12.0000 q^{75} -4.87689 q^{79} -7.00000 q^{81} +13.3693 q^{83} -4.00000 q^{85} +1.75379 q^{87} -3.56155 q^{89} +14.9309 q^{93} +25.3693 q^{95} -6.68466 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 7 q^{15} + 6 q^{17} + 6 q^{19} - 3 q^{23} + 3 q^{25} + 7 q^{27} + 6 q^{29} - 15 q^{31} + q^{33} + q^{37} + 2 q^{39} + 6 q^{43} - 4 q^{45} - 8 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) \) 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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