LMFDB - Embedded newform 416.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(1,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, 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("416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.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:	$3.32177672409$
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	$\chi$	$=$	416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +0.561553 q^{5} -0.561553 q^{7} +3.56155 q^{9} +O(q^{10})$	$q+2.56155 q^{3} +0.561553 q^{5} -0.561553 q^{7} +3.56155 q^{9} +2.00000 q^{11} -1.00000 q^{13} +1.43845 q^{15} -0.561553 q^{17} +6.00000 q^{19} -1.43845 q^{21} -4.68466 q^{25} +1.43845 q^{27} -8.24621 q^{29} +7.12311 q^{31} +5.12311 q^{33} -0.315342 q^{35} -9.68466 q^{37} -2.56155 q^{39} +7.12311 q^{41} -8.80776 q^{43} +2.00000 q^{45} +1.68466 q^{47} -6.68466 q^{49} -1.43845 q^{51} -4.87689 q^{53} +1.12311 q^{55} +15.3693 q^{57} -6.00000 q^{59} +13.3693 q^{61} -2.00000 q^{63} -0.561553 q^{65} +6.00000 q^{67} -1.68466 q^{71} +10.0000 q^{73} -12.0000 q^{75} -1.12311 q^{77} +12.0000 q^{79} -7.00000 q^{81} -17.3693 q^{83} -0.315342 q^{85} -21.1231 q^{87} -8.24621 q^{89} +0.561553 q^{91} +18.2462 q^{93} +3.36932 q^{95} -6.00000 q^{97} +7.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 2 q + q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 4 q^{11} - 2 q^{13} + 7 q^{15} + 3 q^{17} + 12 q^{19} - 7 q^{21} + 3 q^{25} + 7 q^{27} + 6 q^{31} + 2 q^{33} - 13 q^{35} - 7 q^{37} - q^{39} + 6 q^{41} + 3 q^{43} + 4 q^{45} - 9 q^{47} - q^{49} - 7 q^{51} - 18 q^{53} - 6 q^{55} + 6 q^{57} - 12 q^{59} + 2 q^{61} - 4 q^{63} + 3 q^{65} + 12 q^{67} + 9 q^{71} + 20 q^{73} - 24 q^{75} + 6 q^{77} + 24 q^{79} - 14 q^{81} - 10 q^{83} - 13 q^{85} - 34 q^{87} - 3 q^{91} + 20 q^{93} - 18 q^{95} - 12 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 4 * q^11 - 2 * q^13 + 7 * q^15 + 3 * q^17 + 12 * q^19 - 7 * q^21 + 3 * q^25 + 7 * q^27 + 6 * q^31 + 2 * q^33 - 13 * q^35 - 7 * q^37 - q^39 + 6 * q^41 + 3 * q^43 + 4 * q^45 - 9 * q^47 - q^49 - 7 * q^51 - 18 * q^53 - 6 * q^55 + 6 * q^57 - 12 * q^59 + 2 * q^61 - 4 * q^63 + 3 * q^65 + 12 * q^67 + 9 * q^71 + 20 * q^73 - 24 * q^75 + 6 * q^77 + 24 * q^79 - 14 * q^81 - 10 * q^83 - 13 * q^85 - 34 * q^87 - 3 * q^91 + 20 * q^93 - 18 * q^95 - 12 * q^97 + 6 * 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
$3$	2.56155	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	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	0	0
$7$	−0.561553	−0.212247	−0.106124	−	0.994353i	$-0.533844\pi$
$7$	−0.561553	−0.212247	−0.106124	+	0.994353i	$0.533844\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.43845	0.371405
$16$	0	0
$17$	−0.561553	−0.136197	−0.0680983	−	0.997679i	$-0.521693\pi$
$17$	−0.561553	−0.136197	−0.0680983	+	0.997679i	$0.521693\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−1.43845	−0.313895
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	−8.24621	−1.53128	−0.765641	−	0.643268i	$-0.777578\pi$
$29$	−8.24621	−1.53128	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	7.12311	1.27935	0.639674	−	0.768647i	$-0.279069\pi$
$31$	7.12311	1.27935	0.639674	+	0.768647i	$0.279069\pi$
$32$	0	0
$33$	5.12311	0.891818
$34$	0	0
$35$	−0.315342	−0.0533025
$36$	0	0
$37$	−9.68466	−1.59215	−0.796074	−	0.605199i	$-0.793093\pi$
$37$	−9.68466	−1.59215	−0.796074	+	0.605199i	$0.793093\pi$
$38$	0	0
$39$	−2.56155	−0.410177
$40$	0	0
$41$	7.12311	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$41$	7.12311	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$42$	0	0
$43$	−8.80776	−1.34317	−0.671586	−	0.740927i	$-0.734386\pi$
$43$	−8.80776	−1.34317	−0.671586	+	0.740927i	$0.734386\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	1.68466	0.245733	0.122866	−	0.992423i	$-0.460791\pi$
$47$	1.68466	0.245733	0.122866	+	0.992423i	$0.460791\pi$
$48$	0	0
$49$	−6.68466	−0.954951
$50$	0	0
$51$	−1.43845	−0.201423
$52$	0	0
$53$	−4.87689	−0.669893	−0.334946	−	0.942237i	$-0.608718\pi$
$53$	−4.87689	−0.669893	−0.334946	+	0.942237i	$0.608718\pi$
$54$	0	0
$55$	1.12311	0.151440
$56$	0	0
$57$	15.3693	2.03572
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	13.3693	1.71177	0.855883	−	0.517170i	$-0.173014\pi$
$61$	13.3693	1.71177	0.855883	+	0.517170i	$0.173014\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−0.561553	−0.0696521
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.68466	−0.199932	−0.0999661	−	0.994991i	$-0.531873\pi$
$71$	−1.68466	−0.199932	−0.0999661	+	0.994991i	$0.531873\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	−12.0000	−1.38564
$76$	0	0
$77$	−1.12311	−0.127990
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−17.3693	−1.90653	−0.953265	−	0.302135i	$-0.902301\pi$
$83$	−17.3693	−1.90653	−0.953265	+	0.302135i	$0.902301\pi$
$84$	0	0
$85$	−0.315342	−0.0342036
$86$	0	0
$87$	−21.1231	−2.26463
$88$	0	0
$89$	−8.24621	−0.874097	−0.437048	−	0.899438i	$-0.643976\pi$
$89$	−8.24621	−0.874097	−0.437048	+	0.899438i	$0.643976\pi$
$90$	0	0
$91$	0.561553	0.0588667
$92$	0	0
$93$	18.2462	1.89204
$94$	0	0
$95$	3.36932	0.345685
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	7.12311	0.715899
$100$	0	0
$101$	−7.12311	−0.708776	−0.354388	−	0.935099i	$-0.615311\pi$
$101$	−7.12311	−0.708776	−0.354388	+	0.935099i	$0.615311\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−0.807764	−0.0788297
$106$	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
$109$	−1.68466	−0.161361	−0.0806805	−	0.996740i	$-0.525709\pi$
$109$	−1.68466	−0.161361	−0.0806805	+	0.996740i	$0.525709\pi$
$110$	0	0
$111$	−24.8078	−2.35465
$112$	0	0
$113$	8.24621	0.775738	0.387869	−	0.921714i	$-0.373211\pi$
$113$	8.24621	0.775738	0.387869	+	0.921714i	$0.373211\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.56155	−0.329266
$118$	0	0
$119$	0.315342	0.0289073
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	18.2462	1.64521
$124$	0	0
$125$	−5.43845	−0.486430
$126$	0	0
$127$	2.24621	0.199319	0.0996595	−	0.995022i	$-0.468225\pi$
$127$	2.24621	0.199319	0.0996595	+	0.995022i	$0.468225\pi$
$128$	0	0
$129$	−22.5616	−1.98643
$130$	0	0
$131$	15.6847	1.37037	0.685187	−	0.728367i	$-0.259720\pi$
$131$	15.6847	1.37037	0.685187	+	0.728367i	$0.259720\pi$
$132$	0	0
$133$	−3.36932	−0.292157
$134$	0	0
$135$	0.807764	0.0695213
$136$	0	0
$137$	4.87689	0.416661	0.208331	−	0.978058i	$-0.433197\pi$
$137$	4.87689	0.416661	0.208331	+	0.978058i	$0.433197\pi$
$138$	0	0
$139$	7.68466	0.651804	0.325902	−	0.945404i	$-0.394332\pi$
$139$	7.68466	0.651804	0.325902	+	0.945404i	$0.394332\pi$
$140$	0	0
$141$	4.31534	0.363417
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−4.63068	−0.384557
$146$	0	0
$147$	−17.1231	−1.41229
$148$	0	0
$149$	−8.24621	−0.675556	−0.337778	−	0.941226i	$-0.609675\pi$
$149$	−8.24621	−0.675556	−0.337778	+	0.941226i	$0.609675\pi$
$150$	0	0
$151$	10.3153	0.839451	0.419725	−	0.907651i	$-0.362126\pi$
$151$	10.3153	0.839451	0.419725	+	0.907651i	$0.362126\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−12.4924	−0.990714
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.36932	0.733862	0.366931	−	0.930248i	$-0.380409\pi$
$163$	9.36932	0.733862	0.366931	+	0.930248i	$0.380409\pi$
$164$	0	0
$165$	2.87689	0.223966
$166$	0	0
$167$	−9.36932	−0.725020	−0.362510	−	0.931980i	$-0.618080\pi$
$167$	−9.36932	−0.725020	−0.362510	+	0.931980i	$0.618080\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	21.3693	1.63415
$172$	0	0
$173$	21.3693	1.62468	0.812340	−	0.583185i	$-0.198194\pi$
$173$	21.3693	1.62468	0.812340	+	0.583185i	$0.198194\pi$
$174$	0	0
$175$	2.63068	0.198861
$176$	0	0
$177$	−15.3693	−1.15523
$178$	0	0
$179$	−19.0540	−1.42416	−0.712080	−	0.702098i	$-0.752247\pi$
$179$	−19.0540	−1.42416	−0.712080	+	0.702098i	$0.752247\pi$
$180$	0	0
$181$	17.3693	1.29105	0.645526	−	0.763739i	$-0.276638\pi$
$181$	17.3693	1.29105	0.645526	+	0.763739i	$0.276638\pi$
$182$	0	0
$183$	34.2462	2.53155
$184$	0	0
$185$	−5.43845	−0.399843
$186$	0	0
$187$	−1.12311	−0.0821296
$188$	0	0
$189$	−0.807764	−0.0587562
$190$	0	0
$191$	−19.3693	−1.40151	−0.700757	−	0.713400i	$-0.747154\pi$
$191$	−19.3693	−1.40151	−0.700757	+	0.713400i	$0.747154\pi$
$192$	0	0
$193$	1.36932	0.0985656	0.0492828	−	0.998785i	$-0.484306\pi$
$193$	1.36932	0.0985656	0.0492828	+	0.998785i	$0.484306\pi$
$194$	0	0
$195$	−1.43845	−0.103009
$196$	0	0
$197$	13.6847	0.974992	0.487496	−	0.873125i	$-0.337910\pi$
$197$	13.6847	0.974992	0.487496	+	0.873125i	$0.337910\pi$
$198$	0	0
$199$	15.3693	1.08950	0.544751	−	0.838598i	$-0.316624\pi$
$199$	15.3693	1.08950	0.544751	+	0.838598i	$0.316624\pi$
$200$	0	0
$201$	15.3693	1.08407
$202$	0	0
$203$	4.63068	0.325010
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	3.19224	0.219763	0.109881	−	0.993945i	$-0.464953\pi$
$211$	3.19224	0.219763	0.109881	+	0.993945i	$0.464953\pi$
$212$	0	0
$213$	−4.31534	−0.295682
$214$	0	0
$215$	−4.94602	−0.337316
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	25.6155	1.73094
$220$	0	0
$221$	0.561553	0.0377741
$222$	0	0
$223$	26.8078	1.79518	0.897590	−	0.440831i	$-0.145316\pi$
$223$	26.8078	1.79518	0.897590	+	0.440831i	$0.145316\pi$
$224$	0	0
$225$	−16.6847	−1.11231
$226$	0	0
$227$	21.3693	1.41833	0.709166	−	0.705042i	$-0.249072\pi$
$227$	21.3693	1.41833	0.709166	+	0.705042i	$0.249072\pi$
$228$	0	0
$229$	13.6847	0.904308	0.452154	−	0.891940i	$-0.350656\pi$
$229$	13.6847	0.904308	0.452154	+	0.891940i	$0.350656\pi$
$230$	0	0
$231$	−2.87689	−0.189286
$232$	0	0
$233$	13.6847	0.896512	0.448256	−	0.893905i	$-0.352045\pi$
$233$	13.6847	0.896512	0.448256	+	0.893905i	$0.352045\pi$
$234$	0	0
$235$	0.946025	0.0617118
$236$	0	0
$237$	30.7386	1.99669
$238$	0	0
$239$	21.6847	1.40266	0.701332	−	0.712835i	$-0.252589\pi$
$239$	21.6847	1.40266	0.701332	+	0.712835i	$0.252589\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	−3.75379	−0.239821
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−44.4924	−2.81959
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.807764	−0.0505841
$256$	0	0
$257$	15.9309	0.993740	0.496870	−	0.867825i	$-0.334483\pi$
$257$	15.9309	0.993740	0.496870	+	0.867825i	$0.334483\pi$
$258$	0	0
$259$	5.43845	0.337929
$260$	0	0
$261$	−29.3693	−1.81792
$262$	0	0
$263$	26.7386	1.64877	0.824387	−	0.566026i	$-0.191520\pi$
$263$	26.7386	1.64877	0.824387	+	0.566026i	$0.191520\pi$
$264$	0	0
$265$	−2.73863	−0.168233
$266$	0	0
$267$	−21.1231	−1.29271
$268$	0	0
$269$	−21.3693	−1.30291	−0.651455	−	0.758687i	$-0.725841\pi$
$269$	−21.3693	−1.30291	−0.651455	+	0.758687i	$0.725841\pi$
$270$	0	0
$271$	1.68466	0.102336	0.0511679	−	0.998690i	$-0.483706\pi$
$271$	1.68466	0.102336	0.0511679	+	0.998690i	$0.483706\pi$
$272$	0	0
$273$	1.43845	0.0870588
$274$	0	0
$275$	−9.36932	−0.564991
$276$	0	0
$277$	−1.36932	−0.0822743	−0.0411371	−	0.999154i	$-0.513098\pi$
$277$	−1.36932	−0.0822743	−0.0411371	+	0.999154i	$0.513098\pi$
$278$	0	0
$279$	25.3693	1.51882
$280$	0	0
$281$	20.2462	1.20779	0.603894	−	0.797065i	$-0.293615\pi$
$281$	20.2462	1.20779	0.603894	+	0.797065i	$0.293615\pi$
$282$	0	0
$283$	2.24621	0.133523	0.0667617	−	0.997769i	$-0.478733\pi$
$283$	2.24621	0.133523	0.0667617	+	0.997769i	$0.478733\pi$
$284$	0	0
$285$	8.63068	0.511238
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−16.6847	−0.981450
$290$	0	0
$291$	−15.3693	−0.900965
$292$	0	0
$293$	−23.4384	−1.36929	−0.684644	−	0.728877i	$-0.740042\pi$
$293$	−23.4384	−1.36929	−0.684644	+	0.728877i	$0.740042\pi$
$294$	0	0
$295$	−3.36932	−0.196169
$296$	0	0
$297$	2.87689	0.166934
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.94602	0.285084
$302$	0	0
$303$	−18.2462	−1.04822
$304$	0	0
$305$	7.50758	0.429883
$306$	0	0
$307$	−6.00000	−0.342438	−0.171219	−	0.985233i	$-0.554771\pi$
$307$	−6.00000	−0.342438	−0.171219	+	0.985233i	$0.554771\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.3693	−0.871514	−0.435757	−	0.900064i	$-0.643519\pi$
$311$	−15.3693	−0.871514	−0.435757	+	0.900064i	$0.643519\pi$
$312$	0	0
$313$	17.0540	0.963948	0.481974	−	0.876186i	$-0.339920\pi$
$313$	17.0540	0.963948	0.481974	+	0.876186i	$0.339920\pi$
$314$	0	0
$315$	−1.12311	−0.0632798
$316$	0	0
$317$	−32.2462	−1.81113	−0.905564	−	0.424210i	$-0.860552\pi$
$317$	−32.2462	−1.81113	−0.905564	+	0.424210i	$0.860552\pi$
$318$	0	0
$319$	−16.4924	−0.923398
$320$	0	0
$321$	−30.7386	−1.71566
$322$	0	0
$323$	−3.36932	−0.187474
$324$	0	0
$325$	4.68466	0.259858
$326$	0	0
$327$	−4.31534	−0.238639
$328$	0	0
$329$	−0.946025	−0.0521560
$330$	0	0
$331$	21.3693	1.17456	0.587282	−	0.809382i	$-0.300198\pi$
$331$	21.3693	1.17456	0.587282	+	0.809382i	$0.300198\pi$
$332$	0	0
$333$	−34.4924	−1.89017
$334$	0	0
$335$	3.36932	0.184085
$336$	0	0
$337$	1.05398	0.0574137	0.0287068	−	0.999588i	$-0.490861\pi$
$337$	1.05398	0.0574137	0.0287068	+	0.999588i	$0.490861\pi$
$338$	0	0
$339$	21.1231	1.14725
$340$	0	0
$341$	14.2462	0.771476
$342$	0	0
$343$	7.68466	0.414933
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.3153	−0.875853	−0.437927	−	0.899011i	$-0.644287\pi$
$347$	−16.3153	−0.875853	−0.437927	+	0.899011i	$0.644287\pi$
$348$	0	0
$349$	29.0540	1.55522	0.777612	−	0.628745i	$-0.216431\pi$
$349$	29.0540	1.55522	0.777612	+	0.628745i	$0.216431\pi$
$350$	0	0
$351$	−1.43845	−0.0767786
$352$	0	0
$353$	−25.8617	−1.37648	−0.688241	−	0.725482i	$-0.741617\pi$
$353$	−25.8617	−1.37648	−0.688241	+	0.725482i	$0.741617\pi$
$354$	0	0
$355$	−0.946025	−0.0502098
$356$	0	0
$357$	0.807764	0.0427514
$358$	0	0
$359$	−17.3693	−0.916717	−0.458359	−	0.888767i	$-0.651563\pi$
$359$	−17.3693	−0.916717	−0.458359	+	0.888767i	$0.651563\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−17.9309	−0.941127
$364$	0	0
$365$	5.61553	0.293930
$366$	0	0
$367$	−27.3693	−1.42867	−0.714333	−	0.699806i	$-0.753270\pi$
$367$	−27.3693	−1.42867	−0.714333	+	0.699806i	$0.753270\pi$
$368$	0	0
$369$	25.3693	1.32067
$370$	0	0
$371$	2.73863	0.142183
$372$	0	0
$373$	−13.3693	−0.692237	−0.346118	−	0.938191i	$-0.612500\pi$
$373$	−13.3693	−0.692237	−0.346118	+	0.938191i	$0.612500\pi$
$374$	0	0
$375$	−13.9309	−0.719387
$376$	0	0
$377$	8.24621	0.424701
$378$	0	0
$379$	16.8769	0.866908	0.433454	−	0.901176i	$-0.357295\pi$
$379$	16.8769	0.866908	0.433454	+	0.901176i	$0.357295\pi$
$380$	0	0
$381$	5.75379	0.294776
$382$	0	0
$383$	−1.68466	−0.0860820	−0.0430410	−	0.999073i	$-0.513705\pi$
$383$	−1.68466	−0.0860820	−0.0430410	+	0.999073i	$0.513705\pi$
$384$	0	0
$385$	−0.630683	−0.0321426
$386$	0	0
$387$	−31.3693	−1.59459
$388$	0	0
$389$	20.2462	1.02652	0.513262	−	0.858232i	$-0.328437\pi$
$389$	20.2462	1.02652	0.513262	+	0.858232i	$0.328437\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	40.1771	2.02667
$394$	0	0
$395$	6.73863	0.339057
$396$	0	0
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	0	0
$399$	−8.63068	−0.432075
$400$	0	0
$401$	31.1231	1.55421	0.777107	−	0.629369i	$-0.216686\pi$
$401$	31.1231	1.55421	0.777107	+	0.629369i	$0.216686\pi$
$402$	0	0
$403$	−7.12311	−0.354827
$404$	0	0
$405$	−3.93087	−0.195326
$406$	0	0
$407$	−19.3693	−0.960101
$408$	0	0
$409$	−6.63068	−0.327866	−0.163933	−	0.986471i	$-0.552418\pi$
$409$	−6.63068	−0.327866	−0.163933	+	0.986471i	$0.552418\pi$
$410$	0	0
$411$	12.4924	0.616206
$412$	0	0
$413$	3.36932	0.165793
$414$	0	0
$415$	−9.75379	−0.478795
$416$	0	0
$417$	19.6847	0.963962
$418$	0	0
$419$	15.0540	0.735435	0.367717	−	0.929938i	$-0.380139\pi$
$419$	15.0540	0.735435	0.367717	+	0.929938i	$0.380139\pi$
$420$	0	0
$421$	−33.6847	−1.64169	−0.820845	−	0.571151i	$-0.806497\pi$
$421$	−33.6847	−1.64169	−0.820845	+	0.571151i	$0.806497\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	2.63068	0.127607
$426$	0	0
$427$	−7.50758	−0.363317
$428$	0	0
$429$	−5.12311	−0.247346
$430$	0	0
$431$	5.68466	0.273820	0.136910	−	0.990583i	$-0.456283\pi$
$431$	5.68466	0.273820	0.136910	+	0.990583i	$0.456283\pi$
$432$	0	0
$433$	14.3153	0.687951	0.343976	−	0.938979i	$-0.388226\pi$
$433$	14.3153	0.687951	0.343976	+	0.938979i	$0.388226\pi$
$434$	0	0
$435$	−11.8617	−0.568727
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−41.6155	−1.98620	−0.993100	−	0.117267i	$-0.962587\pi$
$439$	−41.6155	−1.98620	−0.993100	+	0.117267i	$0.962587\pi$
$440$	0	0
$441$	−23.8078	−1.13370
$442$	0	0
$443$	−7.68466	−0.365109	−0.182555	−	0.983196i	$-0.558437\pi$
$443$	−7.68466	−0.365109	−0.182555	+	0.983196i	$0.558437\pi$
$444$	0	0
$445$	−4.63068	−0.219515
$446$	0	0
$447$	−21.1231	−0.999089
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	14.2462	0.670828
$452$	0	0
$453$	26.4233	1.24147
$454$	0	0
$455$	0.315342	0.0147834
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	−0.807764	−0.0377032
$460$	0	0
$461$	−21.1922	−0.987021	−0.493510	−	0.869740i	$-0.664287\pi$
$461$	−21.1922	−0.987021	−0.493510	+	0.869740i	$0.664287\pi$
$462$	0	0
$463$	23.6155	1.09751	0.548753	−	0.835984i	$-0.315103\pi$
$463$	23.6155	1.09751	0.548753	+	0.835984i	$0.315103\pi$
$464$	0	0
$465$	10.2462	0.475157
$466$	0	0
$467$	14.7386	0.682023	0.341011	−	0.940059i	$-0.389231\pi$
$467$	14.7386	0.682023	0.341011	+	0.940059i	$0.389231\pi$
$468$	0	0
$469$	−3.36932	−0.155581
$470$	0	0
$471$	5.12311	0.236060
$472$	0	0
$473$	−17.6155	−0.809963
$474$	0	0
$475$	−28.1080	−1.28968
$476$	0	0
$477$	−17.3693	−0.795286
$478$	0	0
$479$	33.6847	1.53909	0.769546	−	0.638592i	$-0.220483\pi$
$479$	33.6847	1.53909	0.769546	+	0.638592i	$0.220483\pi$
$480$	0	0
$481$	9.68466	0.441582
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.36932	−0.152993
$486$	0	0
$487$	−19.1231	−0.866551	−0.433275	−	0.901262i	$-0.642642\pi$
$487$	−19.1231	−0.866551	−0.433275	+	0.901262i	$0.642642\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	35.6847	1.61043	0.805213	−	0.592986i	$-0.202051\pi$
$491$	35.6847	1.61043	0.805213	+	0.592986i	$0.202051\pi$
$492$	0	0
$493$	4.63068	0.208555
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	0.946025	0.0424350
$498$	0	0
$499$	16.8769	0.755514	0.377757	−	0.925905i	$-0.376696\pi$
$499$	16.8769	0.755514	0.377757	+	0.925905i	$0.376696\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	−4.63068	−0.206472	−0.103236	−	0.994657i	$-0.532920\pi$
$503$	−4.63068	−0.206472	−0.103236	+	0.994657i	$0.532920\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	2.56155	0.113763
$508$	0	0
$509$	−10.4924	−0.465068	−0.232534	−	0.972588i	$-0.574702\pi$
$509$	−10.4924	−0.465068	−0.232534	+	0.972588i	$0.574702\pi$
$510$	0	0
$511$	−5.61553	−0.248416
$512$	0	0
$513$	8.63068	0.381054
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.36932	0.148182
$518$	0	0
$519$	54.7386	2.40276
$520$	0	0
$521$	12.5616	0.550332	0.275166	−	0.961397i	$-0.411267\pi$
$521$	12.5616	0.550332	0.275166	+	0.961397i	$0.411267\pi$
$522$	0	0
$523$	−40.4924	−1.77061	−0.885305	−	0.465011i	$-0.846050\pi$
$523$	−40.4924	−1.77061	−0.885305	+	0.465011i	$0.846050\pi$
$524$	0	0
$525$	6.73863	0.294098
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−21.3693	−0.927349
$532$	0	0
$533$	−7.12311	−0.308536
$534$	0	0
$535$	−6.73863	−0.291337
$536$	0	0
$537$	−48.8078	−2.10621
$538$	0	0
$539$	−13.3693	−0.575857
$540$	0	0
$541$	−18.3153	−0.787438	−0.393719	−	0.919231i	$-0.628812\pi$
$541$	−18.3153	−0.787438	−0.393719	+	0.919231i	$0.628812\pi$
$542$	0	0
$543$	44.4924	1.90935
$544$	0	0
$545$	−0.946025	−0.0405232
$546$	0	0
$547$	23.0540	0.985717	0.492858	−	0.870110i	$-0.335952\pi$
$547$	23.0540	0.985717	0.492858	+	0.870110i	$0.335952\pi$
$548$	0	0
$549$	47.6155	2.03218
$550$	0	0
$551$	−49.4773	−2.10780
$552$	0	0
$553$	−6.73863	−0.286556
$554$	0	0
$555$	−13.9309	−0.591332
$556$	0	0
$557$	−43.3002	−1.83469	−0.917344	−	0.398096i	$-0.869671\pi$
$557$	−43.3002	−1.83469	−0.917344	+	0.398096i	$0.869671\pi$
$558$	0	0
$559$	8.80776	0.372529
$560$	0	0
$561$	−2.87689	−0.121463
$562$	0	0
$563$	43.6847	1.84109	0.920544	−	0.390638i	$-0.127746\pi$
$563$	43.6847	1.84109	0.920544	+	0.390638i	$0.127746\pi$
$564$	0	0
$565$	4.63068	0.194814
$566$	0	0
$567$	3.93087	0.165081
$568$	0	0
$569$	−25.6847	−1.07676	−0.538378	−	0.842703i	$-0.680963\pi$
$569$	−25.6847	−1.07676	−0.538378	+	0.842703i	$0.680963\pi$
$570$	0	0
$571$	−17.4384	−0.729776	−0.364888	−	0.931051i	$-0.618893\pi$
$571$	−17.4384	−0.729776	−0.364888	+	0.931051i	$0.618893\pi$
$572$	0	0
$573$	−49.6155	−2.07272
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.73863	−0.197272	−0.0986360	−	0.995124i	$-0.531448\pi$
$577$	−4.73863	−0.197272	−0.0986360	+	0.995124i	$0.531448\pi$
$578$	0	0
$579$	3.50758	0.145770
$580$	0	0
$581$	9.75379	0.404655
$582$	0	0
$583$	−9.75379	−0.403961
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−9.36932	−0.386713	−0.193357	−	0.981129i	$-0.561937\pi$
$587$	−9.36932	−0.386713	−0.193357	+	0.981129i	$0.561937\pi$
$588$	0	0
$589$	42.7386	1.76101
$590$	0	0
$591$	35.0540	1.44193
$592$	0	0
$593$	20.2462	0.831412	0.415706	−	0.909499i	$-0.363534\pi$
$593$	20.2462	0.831412	0.415706	+	0.909499i	$0.363534\pi$
$594$	0	0
$595$	0.177081	0.00725961
$596$	0	0
$597$	39.3693	1.61128
$598$	0	0
$599$	35.3693	1.44515	0.722576	−	0.691292i	$-0.242958\pi$
$599$	35.3693	1.44515	0.722576	+	0.691292i	$0.242958\pi$
$600$	0	0
$601$	−21.0540	−0.858810	−0.429405	−	0.903112i	$-0.641277\pi$
$601$	−21.0540	−0.858810	−0.429405	+	0.903112i	$0.641277\pi$
$602$	0	0
$603$	21.3693	0.870226
$604$	0	0
$605$	−3.93087	−0.159813
$606$	0	0
$607$	31.8617	1.29323	0.646614	−	0.762817i	$-0.276184\pi$
$607$	31.8617	1.29323	0.646614	+	0.762817i	$0.276184\pi$
$608$	0	0
$609$	11.8617	0.480662
$610$	0	0
$611$	−1.68466	−0.0681540
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	0	0
$615$	10.2462	0.413167
$616$	0	0
$617$	−23.6155	−0.950725	−0.475363	−	0.879790i	$-0.657683\pi$
$617$	−23.6155	−0.950725	−0.475363	+	0.879790i	$0.657683\pi$
$618$	0	0
$619$	9.36932	0.376585	0.188292	−	0.982113i	$-0.439705\pi$
$619$	9.36932	0.376585	0.188292	+	0.982113i	$0.439705\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.63068	0.185524
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	30.7386	1.22758
$628$	0	0
$629$	5.43845	0.216845
$630$	0	0
$631$	41.0540	1.63433	0.817166	−	0.576402i	$-0.195544\pi$
$631$	41.0540	1.63433	0.817166	+	0.576402i	$0.195544\pi$
$632$	0	0
$633$	8.17708	0.325010
$634$	0	0
$635$	1.26137	0.0500558
$636$	0	0
$637$	6.68466	0.264856
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	10.4924	0.414426	0.207213	−	0.978296i	$-0.433561\pi$
$641$	10.4924	0.414426	0.207213	+	0.978296i	$0.433561\pi$
$642$	0	0
$643$	−18.0000	−0.709851	−0.354925	−	0.934895i	$-0.615494\pi$
$643$	−18.0000	−0.709851	−0.354925	+	0.934895i	$0.615494\pi$
$644$	0	0
$645$	−12.6695	−0.498861
$646$	0	0
$647$	−30.1080	−1.18367	−0.591833	−	0.806061i	$-0.701595\pi$
$647$	−30.1080	−1.18367	−0.591833	+	0.806061i	$0.701595\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−10.2462	−0.401581
$652$	0	0
$653$	−47.6155	−1.86334	−0.931670	−	0.363306i	$-0.881648\pi$
$653$	−47.6155	−1.86334	−0.931670	+	0.363306i	$0.881648\pi$
$654$	0	0
$655$	8.80776	0.344148
$656$	0	0
$657$	35.6155	1.38949
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−20.7386	−0.806639	−0.403320	−	0.915059i	$-0.632144\pi$
$661$	−20.7386	−0.806639	−0.403320	+	0.915059i	$0.632144\pi$
$662$	0	0
$663$	1.43845	0.0558647
$664$	0	0
$665$	−1.89205	−0.0733705
$666$	0	0
$667$	0	0
$668$	0	0
$669$	68.6695	2.65492
$670$	0	0
$671$	26.7386	1.03223
$672$	0	0
$673$	−21.0540	−0.811571	−0.405786	−	0.913968i	$-0.633002\pi$
$673$	−21.0540	−0.811571	−0.405786	+	0.913968i	$0.633002\pi$
$674$	0	0
$675$	−6.73863	−0.259370
$676$	0	0
$677$	21.3693	0.821290	0.410645	−	0.911795i	$-0.365304\pi$
$677$	21.3693	0.821290	0.410645	+	0.911795i	$0.365304\pi$
$678$	0	0
$679$	3.36932	0.129303
$680$	0	0
$681$	54.7386	2.09759
$682$	0	0
$683$	−5.36932	−0.205451	−0.102726	−	0.994710i	$-0.532756\pi$
$683$	−5.36932	−0.205451	−0.102726	+	0.994710i	$0.532756\pi$
$684$	0	0
$685$	2.73863	0.104638
$686$	0	0
$687$	35.0540	1.33739
$688$	0	0
$689$	4.87689	0.185795
$690$	0	0
$691$	−4.87689	−0.185526	−0.0927629	−	0.995688i	$-0.529570\pi$
$691$	−4.87689	−0.185526	−0.0927629	+	0.995688i	$0.529570\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	4.31534	0.163690
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	35.0540	1.32586
$700$	0	0
$701$	9.36932	0.353874	0.176937	−	0.984222i	$-0.443381\pi$
$701$	9.36932	0.353874	0.176937	+	0.984222i	$0.443381\pi$
$702$	0	0
$703$	−58.1080	−2.19158
$704$	0	0
$705$	2.42329	0.0912665
$706$	0	0
$707$	4.00000	0.150435
$708$	0	0
$709$	16.7386	0.628633	0.314316	−	0.949318i	$-0.398225\pi$
$709$	16.7386	0.628633	0.314316	+	0.949318i	$0.398225\pi$
$710$	0	0
$711$	42.7386	1.60282
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−1.12311	−0.0420018
$716$	0	0
$717$	55.5464	2.07442
$718$	0	0
$719$	−15.3693	−0.573179	−0.286589	−	0.958054i	$-0.592522\pi$
$719$	−15.3693	−0.573179	−0.286589	+	0.958054i	$0.592522\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	5.12311	0.190530
$724$	0	0
$725$	38.6307	1.43471
$726$	0	0
$727$	−41.6155	−1.54343	−0.771717	−	0.635966i	$-0.780602\pi$
$727$	−41.6155	−1.54343	−0.771717	+	0.635966i	$0.780602\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	4.94602	0.182935
$732$	0	0
$733$	−25.0540	−0.925390	−0.462695	−	0.886518i	$-0.653117\pi$
$733$	−25.0540	−0.925390	−0.462695	+	0.886518i	$0.653117\pi$
$734$	0	0
$735$	−9.61553	−0.354674
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	−52.1080	−1.91682	−0.958411	−	0.285392i	$-0.907876\pi$
$739$	−52.1080	−1.91682	−0.958411	+	0.285392i	$0.907876\pi$
$740$	0	0
$741$	−15.3693	−0.564606
$742$	0	0
$743$	13.6847	0.502041	0.251021	−	0.967982i	$-0.419234\pi$
$743$	13.6847	0.502041	0.251021	+	0.967982i	$0.419234\pi$
$744$	0	0
$745$	−4.63068	−0.169655
$746$	0	0
$747$	−61.8617	−2.26340
$748$	0	0
$749$	6.73863	0.246224
$750$	0	0
$751$	−26.2462	−0.957738	−0.478869	−	0.877886i	$-0.658953\pi$
$751$	−26.2462	−0.957738	−0.478869	+	0.877886i	$0.658953\pi$
$752$	0	0
$753$	−40.9848	−1.49357
$754$	0	0
$755$	5.79261	0.210815
$756$	0	0
$757$	−9.36932	−0.340534	−0.170267	−	0.985398i	$-0.554463\pi$
$757$	−9.36932	−0.340534	−0.170267	+	0.985398i	$0.554463\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	0.946025	0.0342484
$764$	0	0
$765$	−1.12311	−0.0406060
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	9.36932	0.337866	0.168933	−	0.985628i	$-0.445968\pi$
$769$	9.36932	0.337866	0.168933	+	0.985628i	$0.445968\pi$
$770$	0	0
$771$	40.8078	1.46966
$772$	0	0
$773$	−25.6847	−0.923813	−0.461906	−	0.886929i	$-0.652834\pi$
$773$	−25.6847	−0.923813	−0.461906	+	0.886929i	$0.652834\pi$
$774$	0	0
$775$	−33.3693	−1.19866
$776$	0	0
$777$	13.9309	0.499767
$778$	0	0
$779$	42.7386	1.53127
$780$	0	0
$781$	−3.36932	−0.120564
$782$	0	0
$783$	−11.8617	−0.423904
$784$	0	0
$785$	1.12311	0.0400854
$786$	0	0
$787$	35.6155	1.26956	0.634778	−	0.772694i	$-0.281091\pi$
$787$	35.6155	1.26956	0.634778	+	0.772694i	$0.281091\pi$
$788$	0	0
$789$	68.4924	2.43839
$790$	0	0
$791$	−4.63068	−0.164648
$792$	0	0
$793$	−13.3693	−0.474758
$794$	0	0
$795$	−7.01515	−0.248802
$796$	0	0
$797$	−12.7386	−0.451226	−0.225613	−	0.974217i	$-0.572438\pi$
$797$	−12.7386	−0.451226	−0.225613	+	0.974217i	$0.572438\pi$
$798$	0	0
$799$	−0.946025	−0.0334679
$800$	0	0
$801$	−29.3693	−1.03771
$802$	0	0
$803$	20.0000	0.705785
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−54.7386	−1.92689
$808$	0	0
$809$	−39.9309	−1.40389	−0.701947	−	0.712229i	$-0.747686\pi$
$809$	−39.9309	−1.40389	−0.701947	+	0.712229i	$0.747686\pi$
$810$	0	0
$811$	25.8617	0.908128	0.454064	−	0.890969i	$-0.349974\pi$
$811$	25.8617	0.908128	0.454064	+	0.890969i	$0.349974\pi$
$812$	0	0
$813$	4.31534	0.151346
$814$	0	0
$815$	5.26137	0.184298
$816$	0	0
$817$	−52.8466	−1.84887
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	5.05398	0.176385	0.0881925	−	0.996103i	$-0.471891\pi$
$821$	5.05398	0.176385	0.0881925	+	0.996103i	$0.471891\pi$
$822$	0	0
$823$	−39.3693	−1.37233	−0.686164	−	0.727447i	$-0.740707\pi$
$823$	−39.3693	−1.37233	−0.686164	+	0.727447i	$0.740707\pi$
$824$	0	0
$825$	−24.0000	−0.835573
$826$	0	0
$827$	−36.7386	−1.27753	−0.638764	−	0.769403i	$-0.720554\pi$
$827$	−36.7386	−1.27753	−0.638764	+	0.769403i	$0.720554\pi$
$828$	0	0
$829$	−32.7386	−1.13706	−0.568530	−	0.822663i	$-0.692488\pi$
$829$	−32.7386	−1.13706	−0.568530	+	0.822663i	$0.692488\pi$
$830$	0	0
$831$	−3.50758	−0.121677
$832$	0	0
$833$	3.75379	0.130061
$834$	0	0
$835$	−5.26137	−0.182077
$836$	0	0
$837$	10.2462	0.354161
$838$	0	0
$839$	36.1080	1.24658	0.623292	−	0.781989i	$-0.285795\pi$
$839$	36.1080	1.24658	0.623292	+	0.781989i	$0.285795\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	0	0
$843$	51.8617	1.78621
$844$	0	0
$845$	0.561553	0.0193180
$846$	0	0
$847$	3.93087	0.135066
$848$	0	0
$849$	5.75379	0.197470
$850$	0	0
$851$	0	0
$852$	0	0
$853$	21.6847	0.742469	0.371234	−	0.928539i	$-0.378935\pi$
$853$	21.6847	0.742469	0.371234	+	0.928539i	$0.378935\pi$
$854$	0	0
$855$	12.0000	0.410391
$856$	0	0
$857$	−32.2462	−1.10151	−0.550755	−	0.834667i	$-0.685660\pi$
$857$	−32.2462	−1.10151	−0.550755	+	0.834667i	$0.685660\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	−10.2462	−0.349190
$862$	0	0
$863$	−25.0540	−0.852847	−0.426424	−	0.904524i	$-0.640227\pi$
$863$	−25.0540	−0.852847	−0.426424	+	0.904524i	$0.640227\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	−42.7386	−1.45148
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	−6.00000	−0.203302
$872$	0	0
$873$	−21.3693	−0.723242
$874$	0	0
$875$	3.05398	0.103243
$876$	0	0
$877$	−31.7926	−1.07356	−0.536780	−	0.843722i	$-0.680360\pi$
$877$	−31.7926	−1.07356	−0.536780	+	0.843722i	$0.680360\pi$
$878$	0	0
$879$	−60.0388	−2.02506
$880$	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
$883$	−17.4384	−0.586850	−0.293425	−	0.955982i	$-0.594795\pi$
$883$	−17.4384	−0.586850	−0.293425	+	0.955982i	$0.594795\pi$
$884$	0	0
$885$	−8.63068	−0.290117
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	−1.26137	−0.0423049
$890$	0	0
$891$	−14.0000	−0.469018
$892$	0	0
$893$	10.1080	0.338250
$894$	0	0
$895$	−10.6998	−0.357655
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−58.7386	−1.95904
$900$	0	0
$901$	2.73863	0.0912371
$902$	0	0
$903$	12.6695	0.421615
$904$	0	0
$905$	9.75379	0.324227
$906$	0	0
$907$	26.4233	0.877371	0.438686	−	0.898641i	$-0.355444\pi$
$907$	26.4233	0.877371	0.438686	+	0.898641i	$0.355444\pi$
$908$	0	0
$909$	−25.3693	−0.841447
$910$	0	0
$911$	−27.3693	−0.906786	−0.453393	−	0.891311i	$-0.649787\pi$
$911$	−27.3693	−0.906786	−0.453393	+	0.891311i	$0.649787\pi$
$912$	0	0
$913$	−34.7386	−1.14968
$914$	0	0
$915$	19.2311	0.635759
$916$	0	0
$917$	−8.80776	−0.290858
$918$	0	0
$919$	33.7538	1.11343	0.556717	−	0.830702i	$-0.312061\pi$
$919$	33.7538	1.11343	0.556717	+	0.830702i	$0.312061\pi$
$920$	0	0
$921$	−15.3693	−0.506436
$922$	0	0
$923$	1.68466	0.0554512
$924$	0	0
$925$	45.3693	1.49173
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.6155	1.16851	0.584254	−	0.811571i	$-0.301387\pi$
$929$	35.6155	1.16851	0.584254	+	0.811571i	$0.301387\pi$
$930$	0	0
$931$	−40.1080	−1.31448
$932$	0	0
$933$	−39.3693	−1.28889
$934$	0	0
$935$	−0.630683	−0.0206255
$936$	0	0
$937$	40.7386	1.33087	0.665437	−	0.746454i	$-0.268245\pi$
$937$	40.7386	1.33087	0.665437	+	0.746454i	$0.268245\pi$
$938$	0	0
$939$	43.6847	1.42559
$940$	0	0
$941$	37.6847	1.22848	0.614242	−	0.789117i	$-0.289462\pi$
$941$	37.6847	1.22848	0.614242	+	0.789117i	$0.289462\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−0.453602	−0.0147557
$946$	0	0
$947$	2.00000	0.0649913	0.0324956	−	0.999472i	$-0.489654\pi$
$947$	2.00000	0.0649913	0.0324956	+	0.999472i	$0.489654\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	−82.6004	−2.67850
$952$	0	0
$953$	−21.1922	−0.686484	−0.343242	−	0.939247i	$-0.611525\pi$
$953$	−21.1922	−0.686484	−0.343242	+	0.939247i	$0.611525\pi$
$954$	0	0
$955$	−10.8769	−0.351968
$956$	0	0
$957$	−42.2462	−1.36563
$958$	0	0
$959$	−2.73863	−0.0884351
$960$	0	0
$961$	19.7386	0.636730
$962$	0	0
$963$	−42.7386	−1.37723
$964$	0	0
$965$	0.768944	0.0247532
$966$	0	0
$967$	−8.42329	−0.270875	−0.135437	−	0.990786i	$-0.543244\pi$
$967$	−8.42329	−0.270875	−0.135437	+	0.990786i	$0.543244\pi$
$968$	0	0
$969$	−8.63068	−0.277257
$970$	0	0
$971$	−15.6847	−0.503345	−0.251672	−	0.967813i	$-0.580981\pi$
$971$	−15.6847	−0.503345	−0.251672	+	0.967813i	$0.580981\pi$
$972$	0	0
$973$	−4.31534	−0.138343
$974$	0	0
$975$	12.0000	0.384308
$976$	0	0
$977$	50.9848	1.63115	0.815575	−	0.578652i	$-0.196421\pi$
$977$	50.9848	1.63115	0.815575	+	0.578652i	$0.196421\pi$
$978$	0	0
$979$	−16.4924	−0.527100
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	40.4233	1.28930	0.644651	−	0.764477i	$-0.277003\pi$
$983$	40.4233	1.28930	0.644651	+	0.764477i	$0.277003\pi$
$984$	0	0
$985$	7.68466	0.244854
$986$	0	0
$987$	−2.42329	−0.0771342
$988$	0	0
$989$	0	0
$990$	0	0
$991$	3.36932	0.107030	0.0535149	−	0.998567i	$-0.482958\pi$
$991$	3.36932	0.107030	0.0535149	+	0.998567i	$0.482958\pi$
$992$	0	0
$993$	54.7386	1.73708
$994$	0	0
$995$	8.63068	0.273611
$996$	0	0
$997$	44.7386	1.41689	0.708443	−	0.705768i	$-0.249398\pi$
$997$	44.7386	1.41689	0.708443	+	0.705768i	$0.249398\pi$
$998$	0	0
$999$	−13.9309	−0.440753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.a.e.1.2	yes	2
3.2	odd	2		3744.2.a.y.1.1		2
4.3	odd	2		416.2.a.c.1.1	✓	2
8.3	odd	2		832.2.a.o.1.2		2
8.5	even	2		832.2.a.l.1.1		2
12.11	even	2		3744.2.a.x.1.1		2
13.12	even	2		5408.2.a.bd.1.2		2
16.3	odd	4		3328.2.b.ba.1665.1		4
16.5	even	4		3328.2.b.u.1665.1		4
16.11	odd	4		3328.2.b.ba.1665.4		4
16.13	even	4		3328.2.b.u.1665.4		4
24.5	odd	2		7488.2.a.cf.1.2		2
24.11	even	2		7488.2.a.ce.1.2		2
52.51	odd	2		5408.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.c.1.1	✓	2	4.3	odd	2
416.2.a.e.1.2	yes	2	1.1	even	1	trivial
832.2.a.l.1.1		2	8.5	even	2
832.2.a.o.1.2		2	8.3	odd	2
3328.2.b.u.1665.1		4	16.5	even	4
3328.2.b.u.1665.4		4	16.13	even	4
3328.2.b.ba.1665.1		4	16.3	odd	4
3328.2.b.ba.1665.4		4	16.11	odd	4
3744.2.a.x.1.1		2	12.11	even	2
3744.2.a.y.1.1		2	3.2	odd	2
5408.2.a.p.1.1		2	52.51	odd	2
5408.2.a.bd.1.2		2	13.12	even	2
7488.2.a.ce.1.2		2	24.11	even	2
7488.2.a.cf.1.2		2	24.5	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +0.561553 q^{5} -0.561553 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q+2.56155 q^{3} +0.561553 q^{5} -0.561553 q^{7} +3.56155 q^{9} +2.00000 q^{11} -1.00000 q^{13} +1.43845 q^{15} -0.561553 q^{17} +6.00000 q^{19} -1.43845 q^{21} -4.68466 q^{25} +1.43845 q^{27} -8.24621 q^{29} +7.12311 q^{31} +5.12311 q^{33} -0.315342 q^{35} -9.68466 q^{37} -2.56155 q^{39} +7.12311 q^{41} -8.80776 q^{43} +2.00000 q^{45} +1.68466 q^{47} -6.68466 q^{49} -1.43845 q^{51} -4.87689 q^{53} +1.12311 q^{55} +15.3693 q^{57} -6.00000 q^{59} +13.3693 q^{61} -2.00000 q^{63} -0.561553 q^{65} +6.00000 q^{67} -1.68466 q^{71} +10.0000 q^{73} -12.0000 q^{75} -1.12311 q^{77} +12.0000 q^{79} -7.00000 q^{81} -17.3693 q^{83} -0.315342 q^{85} -21.1231 q^{87} -8.24621 q^{89} +0.561553 q^{91} +18.2462 q^{93} +3.36932 q^{95} -6.00000 q^{97} +7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 2 q + q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 4 q^{11} - 2 q^{13} + 7 q^{15} + 3 q^{17} + 12 q^{19} - 7 q^{21} + 3 q^{25} + 7 q^{27} + 6 q^{31} + 2 q^{33} - 13 q^{35} - 7 q^{37} - q^{39} + 6 q^{41} + 3 q^{43} + 4 q^{45} - 9 q^{47} - q^{49} - 7 q^{51} - 18 q^{53} - 6 q^{55} + 6 q^{57} - 12 q^{59} + 2 q^{61} - 4 q^{63} + 3 q^{65} + 12 q^{67} + 9 q^{71} + 20 q^{73} - 24 q^{75} + 6 q^{77} + 24 q^{79} - 14 q^{81} - 10 q^{83} - 13 q^{85} - 34 q^{87} - 3 q^{91} + 20 q^{93} - 18 q^{95} - 12 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 4 * q^11 - 2 * q^13 + 7 * q^15 + 3 * q^17 + 12 * q^19 - 7 * q^21 + 3 * q^25 + 7 * q^27 + 6 * q^31 + 2 * q^33 - 13 * q^35 - 7 * q^37 - q^39 + 6 * q^41 + 3 * q^43 + 4 * q^45 - 9 * q^47 - q^49 - 7 * q^51 - 18 * q^53 - 6 * q^55 + 6 * q^57 - 12 * q^59 + 2 * q^61 - 4 * q^63 + 3 * q^65 + 12 * q^67 + 9 * q^71 + 20 * q^73 - 24 * q^75 + 6 * q^77 + 24 * q^79 - 14 * q^81 - 10 * q^83 - 13 * q^85 - 34 * q^87 - 3 * q^91 + 20 * q^93 - 18 * q^95 - 12 * q^97 + 6 * q^99

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