LMFDB - Embedded newform 3380.2.a.r.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 $ x^9 - x^8 - 19x^7 + 16x^6 + 106x^5 - 87x^4 - 153x^3 + 149x^2 - 26*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.65879$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.65879 q^{3} -1.00000 q^{5} +4.25414 q^{7} -0.248414 q^{9} +O(q^{10})$	$q+1.65879 q^{3} -1.00000 q^{5} +4.25414 q^{7} -0.248414 q^{9} +1.49959 q^{11} -1.65879 q^{15} +5.70395 q^{17} -0.636838 q^{19} +7.05673 q^{21} +3.01186 q^{23} +1.00000 q^{25} -5.38844 q^{27} +6.58768 q^{29} +1.96883 q^{31} +2.48750 q^{33} -4.25414 q^{35} -7.56859 q^{37} -10.0713 q^{41} +5.06615 q^{43} +0.248414 q^{45} -3.08005 q^{47} +11.0977 q^{49} +9.46166 q^{51} +2.26608 q^{53} -1.49959 q^{55} -1.05638 q^{57} +0.969910 q^{59} -11.0500 q^{61} -1.05679 q^{63} +13.1252 q^{67} +4.99604 q^{69} -7.08679 q^{71} +12.5266 q^{73} +1.65879 q^{75} +6.37945 q^{77} +13.7459 q^{79} -8.19305 q^{81} +5.57487 q^{83} -5.70395 q^{85} +10.9276 q^{87} -9.73995 q^{89} +3.26587 q^{93} +0.636838 q^{95} -10.0261 q^{97} -0.372519 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * 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
$3$	1.65879	0.957703	0.478852	−	0.877896i	$-0.341053\pi$
$3$	1.65879	0.957703	0.478852	+	0.877896i	$0.341053\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.25414	1.60791	0.803957	−	0.594687i	$-0.202724\pi$
$7$	4.25414	1.60791	0.803957	+	0.594687i	$0.202724\pi$
$8$	0	0
$9$	−0.248414	−0.0828048
$10$	0	0
$11$	1.49959	0.452142	0.226071	−	0.974111i	$-0.427412\pi$
$11$	1.49959	0.452142	0.226071	+	0.974111i	$0.427412\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.65879	−0.428298
$16$	0	0
$17$	5.70395	1.38341	0.691706	−	0.722180i	$-0.256860\pi$
$17$	5.70395	1.38341	0.691706	+	0.722180i	$0.256860\pi$
$18$	0	0
$19$	−0.636838	−0.146101	−0.0730504	−	0.997328i	$-0.523273\pi$
$19$	−0.636838	−0.146101	−0.0730504	+	0.997328i	$0.523273\pi$
$20$	0	0
$21$	7.05673	1.53990
$22$	0	0
$23$	3.01186	0.628016	0.314008	−	0.949420i	$-0.398328\pi$
$23$	3.01186	0.628016	0.314008	+	0.949420i	$0.398328\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.38844	−1.03701
$28$	0	0
$29$	6.58768	1.22330	0.611651	−	0.791128i	$-0.290506\pi$
$29$	6.58768	1.22330	0.611651	+	0.791128i	$0.290506\pi$
$30$	0	0
$31$	1.96883	0.353612	0.176806	−	0.984246i	$-0.443424\pi$
$31$	1.96883	0.353612	0.176806	+	0.984246i	$0.443424\pi$
$32$	0	0
$33$	2.48750	0.433018
$34$	0	0
$35$	−4.25414	−0.719081
$36$	0	0
$37$	−7.56859	−1.24427	−0.622134	−	0.782911i	$-0.713734\pi$
$37$	−7.56859	−1.24427	−0.622134	+	0.782911i	$0.713734\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.0713	−1.57288	−0.786440	−	0.617667i	$-0.788078\pi$
$41$	−10.0713	−1.57288	−0.786440	+	0.617667i	$0.788078\pi$
$42$	0	0
$43$	5.06615	0.772581	0.386290	−	0.922377i	$-0.373756\pi$
$43$	5.06615	0.772581	0.386290	+	0.922377i	$0.373756\pi$
$44$	0	0
$45$	0.248414	0.0370314
$46$	0	0
$47$	−3.08005	−0.449272	−0.224636	−	0.974443i	$-0.572119\pi$
$47$	−3.08005	−0.449272	−0.224636	+	0.974443i	$0.572119\pi$
$48$	0	0
$49$	11.0977	1.58539
$50$	0	0
$51$	9.46166	1.32490
$52$	0	0
$53$	2.26608	0.311270	0.155635	−	0.987815i	$-0.450258\pi$
$53$	2.26608	0.311270	0.155635	+	0.987815i	$0.450258\pi$
$54$	0	0
$55$	−1.49959	−0.202204
$56$	0	0
$57$	−1.05638	−0.139921
$58$	0	0
$59$	0.969910	0.126271	0.0631357	−	0.998005i	$-0.479890\pi$
$59$	0.969910	0.126271	0.0631357	+	0.998005i	$0.479890\pi$
$60$	0	0
$61$	−11.0500	−1.41481	−0.707404	−	0.706809i	$-0.750134\pi$
$61$	−11.0500	−1.41481	−0.707404	+	0.706809i	$0.750134\pi$
$62$	0	0
$63$	−1.05679	−0.133143
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.1252	1.60350	0.801750	−	0.597659i	$-0.203902\pi$
$67$	13.1252	1.60350	0.801750	+	0.597659i	$0.203902\pi$
$68$	0	0
$69$	4.99604	0.601453
$70$	0	0
$71$	−7.08679	−0.841047	−0.420524	−	0.907282i	$-0.638154\pi$
$71$	−7.08679	−0.841047	−0.420524	+	0.907282i	$0.638154\pi$
$72$	0	0
$73$	12.5266	1.46613	0.733064	−	0.680159i	$-0.238089\pi$
$73$	12.5266	1.46613	0.733064	+	0.680159i	$0.238089\pi$
$74$	0	0
$75$	1.65879	0.191541
$76$	0	0
$77$	6.37945	0.727006
$78$	0	0
$79$	13.7459	1.54653	0.773265	−	0.634083i	$-0.218622\pi$
$79$	13.7459	1.54653	0.773265	+	0.634083i	$0.218622\pi$
$80$	0	0
$81$	−8.19305	−0.910339
$82$	0	0
$83$	5.57487	0.611922	0.305961	−	0.952044i	$-0.401022\pi$
$83$	5.57487	0.611922	0.305961	+	0.952044i	$0.401022\pi$
$84$	0	0
$85$	−5.70395	−0.618680
$86$	0	0
$87$	10.9276	1.17156
$88$	0	0
$89$	−9.73995	−1.03243	−0.516217	−	0.856458i	$-0.672660\pi$
$89$	−9.73995	−1.03243	−0.516217	+	0.856458i	$0.672660\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.26587	0.338655
$94$	0	0
$95$	0.636838	0.0653382
$96$	0	0
$97$	−10.0261	−1.01800	−0.508999	−	0.860767i	$-0.669984\pi$
$97$	−10.0261	−1.01800	−0.508999	+	0.860767i	$0.669984\pi$
$98$	0	0
$99$	−0.372519	−0.0374395
$100$	0	0
$101$	13.8952	1.38263	0.691314	−	0.722555i	$-0.257032\pi$
$101$	13.8952	1.38263	0.691314	+	0.722555i	$0.257032\pi$
$102$	0	0
$103$	16.6183	1.63745	0.818724	−	0.574187i	$-0.194682\pi$
$103$	16.6183	1.63745	0.818724	+	0.574187i	$0.194682\pi$
$104$	0	0
$105$	−7.05673	−0.688666
$106$	0	0
$107$	18.2026	1.75971	0.879855	−	0.475243i	$-0.157640\pi$
$107$	18.2026	1.75971	0.879855	+	0.475243i	$0.157640\pi$
$108$	0	0
$109$	−16.3445	−1.56552	−0.782758	−	0.622326i	$-0.786188\pi$
$109$	−16.3445	−1.56552	−0.782758	+	0.622326i	$0.786188\pi$
$110$	0	0
$111$	−12.5547	−1.19164
$112$	0	0
$113$	11.6082	1.09200	0.546002	−	0.837784i	$-0.316149\pi$
$113$	11.6082	1.09200	0.546002	+	0.837784i	$0.316149\pi$
$114$	0	0
$115$	−3.01186	−0.280857
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.2654	2.22441
$120$	0	0
$121$	−8.75124	−0.795567
$122$	0	0
$123$	−16.7062	−1.50635
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.51211	0.222914	0.111457	−	0.993769i	$-0.464448\pi$
$127$	2.51211	0.222914	0.111457	+	0.993769i	$0.464448\pi$
$128$	0	0
$129$	8.40368	0.739903
$130$	0	0
$131$	−19.9291	−1.74122	−0.870608	−	0.491978i	$-0.836274\pi$
$131$	−19.9291	−1.74122	−0.870608	+	0.491978i	$0.836274\pi$
$132$	0	0
$133$	−2.70920	−0.234918
$134$	0	0
$135$	5.38844	0.463763
$136$	0	0
$137$	−11.3552	−0.970141	−0.485070	−	0.874475i	$-0.661206\pi$
$137$	−11.3552	−0.970141	−0.485070	+	0.874475i	$0.661206\pi$
$138$	0	0
$139$	−17.4904	−1.48352	−0.741759	−	0.670666i	$-0.766008\pi$
$139$	−17.4904	−1.48352	−0.741759	+	0.670666i	$0.766008\pi$
$140$	0	0
$141$	−5.10916	−0.430269
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.58768	−0.547077
$146$	0	0
$147$	18.4088	1.51833
$148$	0	0
$149$	2.27148	0.186087	0.0930434	−	0.995662i	$-0.470340\pi$
$149$	2.27148	0.186087	0.0930434	+	0.995662i	$0.470340\pi$
$150$	0	0
$151$	16.4094	1.33538	0.667689	−	0.744440i	$-0.267284\pi$
$151$	16.4094	1.33538	0.667689	+	0.744440i	$0.267284\pi$
$152$	0	0
$153$	−1.41694	−0.114553
$154$	0	0
$155$	−1.96883	−0.158140
$156$	0	0
$157$	−1.21588	−0.0970377	−0.0485188	−	0.998822i	$-0.515450\pi$
$157$	−1.21588	−0.0970377	−0.0485188	+	0.998822i	$0.515450\pi$
$158$	0	0
$159$	3.75895	0.298104
$160$	0	0
$161$	12.8129	1.00980
$162$	0	0
$163$	11.4749	0.898784	0.449392	−	0.893335i	$-0.351641\pi$
$163$	11.4749	0.898784	0.449392	+	0.893335i	$0.351641\pi$
$164$	0	0
$165$	−2.48750	−0.193652
$166$	0	0
$167$	−6.54231	−0.506259	−0.253129	−	0.967432i	$-0.581460\pi$
$167$	−6.54231	−0.506259	−0.253129	+	0.967432i	$0.581460\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.158200	0.0120978
$172$	0	0
$173$	−16.6643	−1.26696	−0.633481	−	0.773758i	$-0.718375\pi$
$173$	−16.6643	−1.26696	−0.633481	+	0.773758i	$0.718375\pi$
$174$	0	0
$175$	4.25414	0.321583
$176$	0	0
$177$	1.60888	0.120931
$178$	0	0
$179$	−0.958705	−0.0716569	−0.0358285	−	0.999358i	$-0.511407\pi$
$179$	−0.958705	−0.0716569	−0.0358285	+	0.999358i	$0.511407\pi$
$180$	0	0
$181$	5.31264	0.394885	0.197443	−	0.980314i	$-0.436736\pi$
$181$	5.31264	0.394885	0.197443	+	0.980314i	$0.436736\pi$
$182$	0	0
$183$	−18.3297	−1.35497
$184$	0	0
$185$	7.56859	0.556454
$186$	0	0
$187$	8.55357	0.625499
$188$	0	0
$189$	−22.9232	−1.66742
$190$	0	0
$191$	4.70838	0.340686	0.170343	−	0.985385i	$-0.445512\pi$
$191$	4.70838	0.340686	0.170343	+	0.985385i	$0.445512\pi$
$192$	0	0
$193$	16.3532	1.17713	0.588563	−	0.808451i	$-0.299694\pi$
$193$	16.3532	1.17713	0.588563	+	0.808451i	$0.299694\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.4002	−0.954728	−0.477364	−	0.878706i	$-0.658408\pi$
$197$	−13.4002	−0.954728	−0.477364	+	0.878706i	$0.658408\pi$
$198$	0	0
$199$	3.34363	0.237023	0.118512	−	0.992953i	$-0.462188\pi$
$199$	3.34363	0.237023	0.118512	+	0.992953i	$0.462188\pi$
$200$	0	0
$201$	21.7720	1.53568
$202$	0	0
$203$	28.0249	1.96697
$204$	0	0
$205$	10.0713	0.703413
$206$	0	0
$207$	−0.748189	−0.0520028
$208$	0	0
$209$	−0.954994	−0.0660583
$210$	0	0
$211$	5.87702	0.404591	0.202295	−	0.979325i	$-0.435160\pi$
$211$	5.87702	0.404591	0.202295	+	0.979325i	$0.435160\pi$
$212$	0	0
$213$	−11.7555	−0.805474
$214$	0	0
$215$	−5.06615	−0.345509
$216$	0	0
$217$	8.37567	0.568577
$218$	0	0
$219$	20.7790	1.40412
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.1066	1.14554	0.572772	−	0.819715i	$-0.305868\pi$
$223$	17.1066	1.14554	0.572772	+	0.819715i	$0.305868\pi$
$224$	0	0
$225$	−0.248414	−0.0165610
$226$	0	0
$227$	−12.9169	−0.857323	−0.428661	−	0.903465i	$-0.641015\pi$
$227$	−12.9169	−0.857323	−0.428661	+	0.903465i	$0.641015\pi$
$228$	0	0
$229$	22.2246	1.46864	0.734322	−	0.678801i	$-0.237500\pi$
$229$	22.2246	1.46864	0.734322	+	0.678801i	$0.237500\pi$
$230$	0	0
$231$	10.5822	0.696256
$232$	0	0
$233$	−5.15849	−0.337944	−0.168972	−	0.985621i	$-0.554045\pi$
$233$	−5.15849	−0.337944	−0.168972	+	0.985621i	$0.554045\pi$
$234$	0	0
$235$	3.08005	0.200920
$236$	0	0
$237$	22.8015	1.48112
$238$	0	0
$239$	−30.6724	−1.98403	−0.992016	−	0.126109i	$-0.959751\pi$
$239$	−30.6724	−1.98403	−0.992016	+	0.126109i	$0.959751\pi$
$240$	0	0
$241$	17.1733	1.10623	0.553115	−	0.833105i	$-0.313439\pi$
$241$	17.1733	1.10623	0.553115	+	0.833105i	$0.313439\pi$
$242$	0	0
$243$	2.57477	0.165171
$244$	0	0
$245$	−11.0977	−0.709008
$246$	0	0
$247$	0	0
$248$	0	0
$249$	9.24755	0.586039
$250$	0	0
$251$	−4.82309	−0.304431	−0.152215	−	0.988347i	$-0.548641\pi$
$251$	−4.82309	−0.304431	−0.152215	+	0.988347i	$0.548641\pi$
$252$	0	0
$253$	4.51654	0.283953
$254$	0	0
$255$	−9.46166	−0.592512
$256$	0	0
$257$	−22.9376	−1.43081	−0.715403	−	0.698712i	$-0.753757\pi$
$257$	−22.9376	−1.43081	−0.715403	+	0.698712i	$0.753757\pi$
$258$	0	0
$259$	−32.1978	−2.00068
$260$	0	0
$261$	−1.63648	−0.101295
$262$	0	0
$263$	12.2220	0.753639	0.376820	−	0.926287i	$-0.377018\pi$
$263$	12.2220	0.753639	0.376820	+	0.926287i	$0.377018\pi$
$264$	0	0
$265$	−2.26608	−0.139204
$266$	0	0
$267$	−16.1565	−0.988764
$268$	0	0
$269$	−29.3278	−1.78815	−0.894073	−	0.447922i	$-0.852164\pi$
$269$	−29.3278	−1.78815	−0.894073	+	0.447922i	$0.852164\pi$
$270$	0	0
$271$	12.9094	0.784190	0.392095	−	0.919925i	$-0.371750\pi$
$271$	12.9094	0.784190	0.392095	+	0.919925i	$0.371750\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.49959	0.0904285
$276$	0	0
$277$	3.40741	0.204731	0.102366	−	0.994747i	$-0.467359\pi$
$277$	3.40741	0.204731	0.102366	+	0.994747i	$0.467359\pi$
$278$	0	0
$279$	−0.489085	−0.0292807
$280$	0	0
$281$	20.7553	1.23816	0.619080	−	0.785328i	$-0.287506\pi$
$281$	20.7553	1.23816	0.619080	+	0.785328i	$0.287506\pi$
$282$	0	0
$283$	−1.01218	−0.0601679	−0.0300839	−	0.999547i	$-0.509577\pi$
$283$	−1.01218	−0.0601679	−0.0300839	+	0.999547i	$0.509577\pi$
$284$	0	0
$285$	1.05638	0.0625746
$286$	0	0
$287$	−42.8449	−2.52906
$288$	0	0
$289$	15.5350	0.913826
$290$	0	0
$291$	−16.6312	−0.974940
$292$	0	0
$293$	−6.59637	−0.385364	−0.192682	−	0.981261i	$-0.561719\pi$
$293$	−6.59637	−0.385364	−0.192682	+	0.981261i	$0.561719\pi$
$294$	0	0
$295$	−0.969910	−0.0564703
$296$	0	0
$297$	−8.08043	−0.468874
$298$	0	0
$299$	0	0
$300$	0	0
$301$	21.5521	1.24224
$302$	0	0
$303$	23.0493	1.32415
$304$	0	0
$305$	11.0500	0.632722
$306$	0	0
$307$	−11.7321	−0.669587	−0.334793	−	0.942292i	$-0.608666\pi$
$307$	−11.7321	−0.669587	−0.334793	+	0.942292i	$0.608666\pi$
$308$	0	0
$309$	27.5663	1.56819
$310$	0	0
$311$	−11.1618	−0.632925	−0.316463	−	0.948605i	$-0.602495\pi$
$311$	−11.1618	−0.632925	−0.316463	+	0.948605i	$0.602495\pi$
$312$	0	0
$313$	14.5600	0.822980	0.411490	−	0.911414i	$-0.365008\pi$
$313$	14.5600	0.822980	0.411490	+	0.911414i	$0.365008\pi$
$314$	0	0
$315$	1.05679	0.0595434
$316$	0	0
$317$	8.31408	0.466965	0.233483	−	0.972361i	$-0.424988\pi$
$317$	8.31408	0.466965	0.233483	+	0.972361i	$0.424988\pi$
$318$	0	0
$319$	9.87880	0.553107
$320$	0	0
$321$	30.1943	1.68528
$322$	0	0
$323$	−3.63250	−0.202117
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−27.1120	−1.49930
$328$	0	0
$329$	−13.1030	−0.722391
$330$	0	0
$331$	−35.8927	−1.97284	−0.986421	−	0.164235i	$-0.947485\pi$
$331$	−35.8927	−1.97284	−0.986421	+	0.164235i	$0.947485\pi$
$332$	0	0
$333$	1.88015	0.103031
$334$	0	0
$335$	−13.1252	−0.717107
$336$	0	0
$337$	19.7590	1.07634	0.538171	−	0.842836i	$-0.319115\pi$
$337$	19.7590	1.07634	0.538171	+	0.842836i	$0.319115\pi$
$338$	0	0
$339$	19.2555	1.04582
$340$	0	0
$341$	2.95242	0.159883
$342$	0	0
$343$	17.4323	0.941256
$344$	0	0
$345$	−4.99604	−0.268978
$346$	0	0
$347$	10.8791	0.584020	0.292010	−	0.956415i	$-0.405676\pi$
$347$	10.8791	0.584020	0.292010	+	0.956415i	$0.405676\pi$
$348$	0	0
$349$	−4.96959	−0.266016	−0.133008	−	0.991115i	$-0.542464\pi$
$349$	−4.96959	−0.266016	−0.133008	+	0.991115i	$0.542464\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.6620	−0.833606	−0.416803	−	0.908997i	$-0.636849\pi$
$353$	−15.6620	−0.833606	−0.416803	+	0.908997i	$0.636849\pi$
$354$	0	0
$355$	7.08679	0.376128
$356$	0	0
$357$	40.2512	2.13032
$358$	0	0
$359$	−23.1319	−1.22086	−0.610428	−	0.792072i	$-0.709002\pi$
$359$	−23.1319	−1.22086	−0.610428	+	0.792072i	$0.709002\pi$
$360$	0	0
$361$	−18.5944	−0.978655
$362$	0	0
$363$	−14.5165	−0.761917
$364$	0	0
$365$	−12.5266	−0.655673
$366$	0	0
$367$	−18.9594	−0.989671	−0.494836	−	0.868987i	$-0.664772\pi$
$367$	−18.9594	−0.989671	−0.494836	+	0.868987i	$0.664772\pi$
$368$	0	0
$369$	2.50187	0.130242
$370$	0	0
$371$	9.64022	0.500495
$372$	0	0
$373$	3.11921	0.161507	0.0807534	−	0.996734i	$-0.474267\pi$
$373$	3.11921	0.161507	0.0807534	+	0.996734i	$0.474267\pi$
$374$	0	0
$375$	−1.65879	−0.0856596
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−4.68573	−0.240690	−0.120345	−	0.992732i	$-0.538400\pi$
$379$	−4.68573	−0.240690	−0.120345	+	0.992732i	$0.538400\pi$
$380$	0	0
$381$	4.16707	0.213485
$382$	0	0
$383$	−18.9467	−0.968130	−0.484065	−	0.875032i	$-0.660840\pi$
$383$	−18.9467	−0.968130	−0.484065	+	0.875032i	$0.660840\pi$
$384$	0	0
$385$	−6.37945	−0.325127
$386$	0	0
$387$	−1.25850	−0.0639734
$388$	0	0
$389$	15.7332	0.797702	0.398851	−	0.917016i	$-0.369409\pi$
$389$	15.7332	0.797702	0.398851	+	0.917016i	$0.369409\pi$
$390$	0	0
$391$	17.1795	0.868805
$392$	0	0
$393$	−33.0582	−1.66757
$394$	0	0
$395$	−13.7459	−0.691630
$396$	0	0
$397$	−13.0486	−0.654890	−0.327445	−	0.944870i	$-0.606188\pi$
$397$	−13.0486	−0.654890	−0.327445	+	0.944870i	$0.606188\pi$
$398$	0	0
$399$	−4.49400	−0.224981
$400$	0	0
$401$	2.09694	0.104716	0.0523582	−	0.998628i	$-0.483326\pi$
$401$	2.09694	0.104716	0.0523582	+	0.998628i	$0.483326\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	8.19305	0.407116
$406$	0	0
$407$	−11.3497	−0.562586
$408$	0	0
$409$	−10.0168	−0.495298	−0.247649	−	0.968850i	$-0.579658\pi$
$409$	−10.0168	−0.495298	−0.247649	+	0.968850i	$0.579658\pi$
$410$	0	0
$411$	−18.8359	−0.929107
$412$	0	0
$413$	4.12613	0.203034
$414$	0	0
$415$	−5.57487	−0.273660
$416$	0	0
$417$	−29.0129	−1.42077
$418$	0	0
$419$	−38.7042	−1.89083	−0.945413	−	0.325876i	$-0.894341\pi$
$419$	−38.7042	−1.89083	−0.945413	+	0.325876i	$0.894341\pi$
$420$	0	0
$421$	4.88277	0.237972	0.118986	−	0.992896i	$-0.462036\pi$
$421$	4.88277	0.237972	0.118986	+	0.992896i	$0.462036\pi$
$422$	0	0
$423$	0.765129	0.0372019
$424$	0	0
$425$	5.70395	0.276682
$426$	0	0
$427$	−47.0083	−2.27489
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.48596	0.408754	0.204377	−	0.978892i	$-0.434483\pi$
$431$	8.48596	0.408754	0.204377	+	0.978892i	$0.434483\pi$
$432$	0	0
$433$	−34.6384	−1.66461	−0.832307	−	0.554315i	$-0.812980\pi$
$433$	−34.6384	−1.66461	−0.832307	+	0.554315i	$0.812980\pi$
$434$	0	0
$435$	−10.9276	−0.523938
$436$	0	0
$437$	−1.91807	−0.0917537
$438$	0	0
$439$	36.2573	1.73047	0.865234	−	0.501368i	$-0.167170\pi$
$439$	36.2573	1.73047	0.865234	+	0.501368i	$0.167170\pi$
$440$	0	0
$441$	−2.75683	−0.131278
$442$	0	0
$443$	21.1718	1.00590	0.502950	−	0.864315i	$-0.332248\pi$
$443$	21.1718	1.00590	0.502950	+	0.864315i	$0.332248\pi$
$444$	0	0
$445$	9.73995	0.461718
$446$	0	0
$447$	3.76791	0.178216
$448$	0	0
$449$	−4.18848	−0.197667	−0.0988333	−	0.995104i	$-0.531511\pi$
$449$	−4.18848	−0.197667	−0.0988333	+	0.995104i	$0.531511\pi$
$450$	0	0
$451$	−15.1028	−0.711165
$452$	0	0
$453$	27.2198	1.27890
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.5088	−1.66103	−0.830516	−	0.556994i	$-0.811954\pi$
$457$	−35.5088	−1.66103	−0.830516	+	0.556994i	$0.811954\pi$
$458$	0	0
$459$	−30.7354	−1.43460
$460$	0	0
$461$	−4.40907	−0.205351	−0.102676	−	0.994715i	$-0.532740\pi$
$461$	−4.40907	−0.205351	−0.102676	+	0.994715i	$0.532740\pi$
$462$	0	0
$463$	17.5773	0.816885	0.408442	−	0.912784i	$-0.366072\pi$
$463$	17.5773	0.816885	0.408442	+	0.912784i	$0.366072\pi$
$464$	0	0
$465$	−3.26587	−0.151451
$466$	0	0
$467$	−4.21327	−0.194967	−0.0974835	−	0.995237i	$-0.531079\pi$
$467$	−4.21327	−0.194967	−0.0974835	+	0.995237i	$0.531079\pi$
$468$	0	0
$469$	55.8366	2.57829
$470$	0	0
$471$	−2.01689	−0.0929333
$472$	0	0
$473$	7.59713	0.349316
$474$	0	0
$475$	−0.636838	−0.0292202
$476$	0	0
$477$	−0.562926	−0.0257746
$478$	0	0
$479$	20.7016	0.945880	0.472940	−	0.881095i	$-0.343193\pi$
$479$	20.7016	0.945880	0.472940	+	0.881095i	$0.343193\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	21.2539	0.967085
$484$	0	0
$485$	10.0261	0.455263
$486$	0	0
$487$	15.4795	0.701442	0.350721	−	0.936480i	$-0.385937\pi$
$487$	15.4795	0.701442	0.350721	+	0.936480i	$0.385937\pi$
$488$	0	0
$489$	19.0345	0.860768
$490$	0	0
$491$	−34.3346	−1.54950	−0.774750	−	0.632268i	$-0.782124\pi$
$491$	−34.3346	−1.54950	−0.774750	+	0.632268i	$0.782124\pi$
$492$	0	0
$493$	37.5758	1.69233
$494$	0	0
$495$	0.372519	0.0167435
$496$	0	0
$497$	−30.1482	−1.35233
$498$	0	0
$499$	21.1450	0.946579	0.473290	−	0.880907i	$-0.343066\pi$
$499$	21.1450	0.946579	0.473290	+	0.880907i	$0.343066\pi$
$500$	0	0
$501$	−10.8523	−0.484846
$502$	0	0
$503$	3.46540	0.154515	0.0772573	−	0.997011i	$-0.475384\pi$
$503$	3.46540	0.154515	0.0772573	+	0.997011i	$0.475384\pi$
$504$	0	0
$505$	−13.8952	−0.618330
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.5573	0.778213	0.389107	−	0.921193i	$-0.372784\pi$
$509$	17.5573	0.778213	0.389107	+	0.921193i	$0.372784\pi$
$510$	0	0
$511$	53.2900	2.35741
$512$	0	0
$513$	3.43157	0.151507
$514$	0	0
$515$	−16.6183	−0.732289
$516$	0	0
$517$	−4.61880	−0.203135
$518$	0	0
$519$	−27.6426	−1.21337
$520$	0	0
$521$	39.7429	1.74117	0.870585	−	0.492018i	$-0.163741\pi$
$521$	39.7429	1.74117	0.870585	+	0.492018i	$0.163741\pi$
$522$	0	0
$523$	−27.7468	−1.21328	−0.606641	−	0.794976i	$-0.707483\pi$
$523$	−27.7468	−1.21328	−0.606641	+	0.794976i	$0.707483\pi$
$524$	0	0
$525$	7.05673	0.307981
$526$	0	0
$527$	11.2301	0.489190
$528$	0	0
$529$	−13.9287	−0.605596
$530$	0	0
$531$	−0.240939	−0.0104559
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−18.2026	−0.786966
$536$	0	0
$537$	−1.59029	−0.0686261
$538$	0	0
$539$	16.6420	0.716822
$540$	0	0
$541$	−43.1354	−1.85453	−0.927267	−	0.374400i	$-0.877849\pi$
$541$	−43.1354	−1.85453	−0.927267	+	0.374400i	$0.877849\pi$
$542$	0	0
$543$	8.81255	0.378183
$544$	0	0
$545$	16.3445	0.700120
$546$	0	0
$547$	−22.4698	−0.960740	−0.480370	−	0.877066i	$-0.659498\pi$
$547$	−22.4698	−0.960740	−0.480370	+	0.877066i	$0.659498\pi$
$548$	0	0
$549$	2.74498	0.117153
$550$	0	0
$551$	−4.19529	−0.178725
$552$	0	0
$553$	58.4769	2.48669
$554$	0	0
$555$	12.5547	0.532917
$556$	0	0
$557$	−24.2107	−1.02584	−0.512920	−	0.858436i	$-0.671436\pi$
$557$	−24.2107	−1.02584	−0.512920	+	0.858436i	$0.671436\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	14.1886	0.599042
$562$	0	0
$563$	−8.87480	−0.374028	−0.187014	−	0.982357i	$-0.559881\pi$
$563$	−8.87480	−0.374028	−0.187014	+	0.982357i	$0.559881\pi$
$564$	0	0
$565$	−11.6082	−0.488359
$566$	0	0
$567$	−34.8544	−1.46375
$568$	0	0
$569$	−27.1897	−1.13985	−0.569927	−	0.821695i	$-0.693028\pi$
$569$	−27.1897	−1.13985	−0.569927	+	0.821695i	$0.693028\pi$
$570$	0	0
$571$	13.5393	0.566604	0.283302	−	0.959031i	$-0.408570\pi$
$571$	13.5393	0.566604	0.283302	+	0.959031i	$0.408570\pi$
$572$	0	0
$573$	7.81021	0.326276
$574$	0	0
$575$	3.01186	0.125603
$576$	0	0
$577$	40.3084	1.67806	0.839031	−	0.544084i	$-0.183123\pi$
$577$	40.3084	1.67806	0.839031	+	0.544084i	$0.183123\pi$
$578$	0	0
$579$	27.1265	1.12734
$580$	0	0
$581$	23.7163	0.983918
$582$	0	0
$583$	3.39818	0.140738
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.9302	−0.740060	−0.370030	−	0.929020i	$-0.620653\pi$
$587$	−17.9302	−0.740060	−0.370030	+	0.929020i	$0.620653\pi$
$588$	0	0
$589$	−1.25382	−0.0516629
$590$	0	0
$591$	−22.2282	−0.914346
$592$	0	0
$593$	−44.4851	−1.82678	−0.913392	−	0.407081i	$-0.866547\pi$
$593$	−44.4851	−1.82678	−0.913392	+	0.407081i	$0.866547\pi$
$594$	0	0
$595$	−24.2654	−0.994785
$596$	0	0
$597$	5.54637	0.226998
$598$	0	0
$599$	0.145172	0.00593155	0.00296577	−	0.999996i	$-0.499056\pi$
$599$	0.145172	0.00593155	0.00296577	+	0.999996i	$0.499056\pi$
$600$	0	0
$601$	19.1209	0.779958	0.389979	−	0.920824i	$-0.372482\pi$
$601$	19.1209	0.779958	0.389979	+	0.920824i	$0.372482\pi$
$602$	0	0
$603$	−3.26049	−0.132778
$604$	0	0
$605$	8.75124	0.355789
$606$	0	0
$607$	−33.6276	−1.36490	−0.682451	−	0.730931i	$-0.739086\pi$
$607$	−33.6276	−1.36490	−0.682451	+	0.730931i	$0.739086\pi$
$608$	0	0
$609$	46.4875	1.88377
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−31.2683	−1.26292	−0.631458	−	0.775410i	$-0.717543\pi$
$613$	−31.2683	−1.26292	−0.631458	+	0.775410i	$0.717543\pi$
$614$	0	0
$615$	16.7062	0.673661
$616$	0	0
$617$	8.22626	0.331177	0.165588	−	0.986195i	$-0.447048\pi$
$617$	8.22626	0.331177	0.165588	+	0.986195i	$0.447048\pi$
$618$	0	0
$619$	−17.6835	−0.710759	−0.355379	−	0.934722i	$-0.615648\pi$
$619$	−17.6835	−0.710759	−0.355379	+	0.934722i	$0.615648\pi$
$620$	0	0
$621$	−16.2292	−0.651256
$622$	0	0
$623$	−41.4351	−1.66006
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.58414	−0.0632643
$628$	0	0
$629$	−43.1708	−1.72133
$630$	0	0
$631$	32.8689	1.30849	0.654245	−	0.756283i	$-0.272987\pi$
$631$	32.8689	1.30849	0.654245	+	0.756283i	$0.272987\pi$
$632$	0	0
$633$	9.74874	0.387478
$634$	0	0
$635$	−2.51211	−0.0996901
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.76046	0.0696428
$640$	0	0
$641$	−14.9749	−0.591471	−0.295736	−	0.955270i	$-0.595565\pi$
$641$	−14.9749	−0.591471	−0.295736	+	0.955270i	$0.595565\pi$
$642$	0	0
$643$	22.0696	0.870341	0.435170	−	0.900348i	$-0.356688\pi$
$643$	22.0696	0.870341	0.435170	+	0.900348i	$0.356688\pi$
$644$	0	0
$645$	−8.40368	−0.330895
$646$	0	0
$647$	30.2132	1.18780	0.593902	−	0.804538i	$-0.297587\pi$
$647$	30.2132	1.18780	0.593902	+	0.804538i	$0.297587\pi$
$648$	0	0
$649$	1.45446	0.0570927
$650$	0	0
$651$	13.8935	0.544528
$652$	0	0
$653$	30.5523	1.19561	0.597803	−	0.801643i	$-0.296041\pi$
$653$	30.5523	1.19561	0.597803	+	0.801643i	$0.296041\pi$
$654$	0	0
$655$	19.9291	0.778695
$656$	0	0
$657$	−3.11179	−0.121402
$658$	0	0
$659$	−33.9179	−1.32125	−0.660626	−	0.750715i	$-0.729709\pi$
$659$	−33.9179	−1.32125	−0.660626	+	0.750715i	$0.729709\pi$
$660$	0	0
$661$	−17.9952	−0.699931	−0.349965	−	0.936763i	$-0.613807\pi$
$661$	−17.9952	−0.699931	−0.349965	+	0.936763i	$0.613807\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.70920	0.105058
$666$	0	0
$667$	19.8412	0.768254
$668$	0	0
$669$	28.3763	1.09709
$670$	0	0
$671$	−16.5704	−0.639695
$672$	0	0
$673$	−29.6026	−1.14110	−0.570548	−	0.821265i	$-0.693269\pi$
$673$	−29.6026	−1.14110	−0.570548	+	0.821265i	$0.693269\pi$
$674$	0	0
$675$	−5.38844	−0.207401
$676$	0	0
$677$	3.06371	0.117748	0.0588740	−	0.998265i	$-0.481249\pi$
$677$	3.06371	0.117748	0.0588740	+	0.998265i	$0.481249\pi$
$678$	0	0
$679$	−42.6525	−1.63685
$680$	0	0
$681$	−21.4264	−0.821060
$682$	0	0
$683$	−24.2517	−0.927964	−0.463982	−	0.885845i	$-0.653580\pi$
$683$	−24.2517	−0.927964	−0.463982	+	0.885845i	$0.653580\pi$
$684$	0	0
$685$	11.3552	0.433860
$686$	0	0
$687$	36.8660	1.40653
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−26.4961	−1.00796	−0.503979	−	0.863716i	$-0.668131\pi$
$691$	−26.4961	−1.00796	−0.503979	+	0.863716i	$0.668131\pi$
$692$	0	0
$693$	−1.58475	−0.0601996
$694$	0	0
$695$	17.4904	0.663449
$696$	0	0
$697$	−57.4464	−2.17594
$698$	0	0
$699$	−8.55685	−0.323650
$700$	0	0
$701$	−31.4444	−1.18764	−0.593819	−	0.804599i	$-0.702380\pi$
$701$	−31.4444	−1.18764	−0.593819	+	0.804599i	$0.702380\pi$
$702$	0	0
$703$	4.81997	0.181788
$704$	0	0
$705$	5.10916	0.192422
$706$	0	0
$707$	59.1123	2.22315
$708$	0	0
$709$	34.6756	1.30227	0.651134	−	0.758963i	$-0.274294\pi$
$709$	34.6756	1.30227	0.651134	+	0.758963i	$0.274294\pi$
$710$	0	0
$711$	−3.41467	−0.128060
$712$	0	0
$713$	5.92983	0.222074
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−50.8791	−1.90011
$718$	0	0
$719$	−25.4213	−0.948053	−0.474027	−	0.880511i	$-0.657200\pi$
$719$	−25.4213	−0.948053	−0.474027	+	0.880511i	$0.657200\pi$
$720$	0	0
$721$	70.6966	2.63288
$722$	0	0
$723$	28.4869	1.05944
$724$	0	0
$725$	6.58768	0.244660
$726$	0	0
$727$	−5.13306	−0.190375	−0.0951873	−	0.995459i	$-0.530345\pi$
$727$	−5.13306	−0.190375	−0.0951873	+	0.995459i	$0.530345\pi$
$728$	0	0
$729$	28.8501	1.06852
$730$	0	0
$731$	28.8971	1.06880
$732$	0	0
$733$	−22.2813	−0.822979	−0.411490	−	0.911414i	$-0.634991\pi$
$733$	−22.2813	−0.822979	−0.411490	+	0.911414i	$0.634991\pi$
$734$	0	0
$735$	−18.4088	−0.679019
$736$	0	0
$737$	19.6824	0.725011
$738$	0	0
$739$	−1.27189	−0.0467871	−0.0233935	−	0.999726i	$-0.507447\pi$
$739$	−1.27189	−0.0467871	−0.0233935	+	0.999726i	$0.507447\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.3848	0.857906	0.428953	−	0.903327i	$-0.358883\pi$
$743$	23.3848	0.857906	0.428953	+	0.903327i	$0.358883\pi$
$744$	0	0
$745$	−2.27148	−0.0832205
$746$	0	0
$747$	−1.38488	−0.0506701
$748$	0	0
$749$	77.4364	2.82946
$750$	0	0
$751$	−2.31524	−0.0844843	−0.0422422	−	0.999107i	$-0.513450\pi$
$751$	−2.31524	−0.0844843	−0.0422422	+	0.999107i	$0.513450\pi$
$752$	0	0
$753$	−8.00050	−0.291554
$754$	0	0
$755$	−16.4094	−0.597199
$756$	0	0
$757$	41.4080	1.50500	0.752500	−	0.658592i	$-0.228848\pi$
$757$	41.4080	1.50500	0.752500	+	0.658592i	$0.228848\pi$
$758$	0	0
$759$	7.49200	0.271942
$760$	0	0
$761$	23.5369	0.853212	0.426606	−	0.904438i	$-0.359709\pi$
$761$	23.5369	0.853212	0.426606	+	0.904438i	$0.359709\pi$
$762$	0	0
$763$	−69.5317	−2.51722
$764$	0	0
$765$	1.41694	0.0512297
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−8.84837	−0.319080	−0.159540	−	0.987191i	$-0.551001\pi$
$769$	−8.84837	−0.319080	−0.159540	+	0.987191i	$0.551001\pi$
$770$	0	0
$771$	−38.0486	−1.37029
$772$	0	0
$773$	6.15591	0.221413	0.110706	−	0.993853i	$-0.464689\pi$
$773$	6.15591	0.221413	0.110706	+	0.993853i	$0.464689\pi$
$774$	0	0
$775$	1.96883	0.0707223
$776$	0	0
$777$	−53.4095	−1.91605
$778$	0	0
$779$	6.41382	0.229799
$780$	0	0
$781$	−10.6273	−0.380273
$782$	0	0
$783$	−35.4973	−1.26857
$784$	0	0
$785$	1.21588	0.0433966
$786$	0	0
$787$	7.12007	0.253803	0.126901	−	0.991915i	$-0.459497\pi$
$787$	7.12007	0.253803	0.126901	+	0.991915i	$0.459497\pi$
$788$	0	0
$789$	20.2737	0.721763
$790$	0	0
$791$	49.3828	1.75585
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−3.75895	−0.133316
$796$	0	0
$797$	4.93234	0.174712	0.0873562	−	0.996177i	$-0.472158\pi$
$797$	4.93234	0.174712	0.0873562	+	0.996177i	$0.472158\pi$
$798$	0	0
$799$	−17.5685	−0.621528
$800$	0	0
$801$	2.41954	0.0854904
$802$	0	0
$803$	18.7847	0.662899
$804$	0	0
$805$	−12.8129	−0.451595
$806$	0	0
$807$	−48.6486	−1.71251
$808$	0	0
$809$	−47.6258	−1.67443	−0.837217	−	0.546871i	$-0.815819\pi$
$809$	−47.6258	−1.67443	−0.837217	+	0.546871i	$0.815819\pi$
$810$	0	0
$811$	−15.2644	−0.536005	−0.268002	−	0.963418i	$-0.586363\pi$
$811$	−15.2644	−0.536005	−0.268002	+	0.963418i	$0.586363\pi$
$812$	0	0
$813$	21.4140	0.751021
$814$	0	0
$815$	−11.4749	−0.401948
$816$	0	0
$817$	−3.22632	−0.112875
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.3914	0.851267	0.425634	−	0.904896i	$-0.360051\pi$
$821$	24.3914	0.851267	0.425634	+	0.904896i	$0.360051\pi$
$822$	0	0
$823$	7.63554	0.266158	0.133079	−	0.991105i	$-0.457514\pi$
$823$	7.63554	0.266158	0.133079	+	0.991105i	$0.457514\pi$
$824$	0	0
$825$	2.48750	0.0866036
$826$	0	0
$827$	−32.8801	−1.14335	−0.571677	−	0.820479i	$-0.693707\pi$
$827$	−32.8801	−1.14335	−0.571677	+	0.820479i	$0.693707\pi$
$828$	0	0
$829$	2.73117	0.0948576	0.0474288	−	0.998875i	$-0.484897\pi$
$829$	2.73117	0.0948576	0.0474288	+	0.998875i	$0.484897\pi$
$830$	0	0
$831$	5.65217	0.196072
$832$	0	0
$833$	63.3009	2.19325
$834$	0	0
$835$	6.54231	0.226406
$836$	0	0
$837$	−10.6089	−0.366697
$838$	0	0
$839$	32.0624	1.10692	0.553459	−	0.832877i	$-0.313308\pi$
$839$	32.0624	1.10692	0.553459	+	0.832877i	$0.313308\pi$
$840$	0	0
$841$	14.3976	0.496468
$842$	0	0
$843$	34.4287	1.18579
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−37.2290	−1.27920
$848$	0	0
$849$	−1.67900	−0.0576230
$850$	0	0
$851$	−22.7955	−0.781420
$852$	0	0
$853$	−39.7908	−1.36241	−0.681206	−	0.732092i	$-0.738544\pi$
$853$	−39.7908	−1.36241	−0.681206	+	0.732092i	$0.738544\pi$
$854$	0	0
$855$	−0.158200	−0.00541032
$856$	0	0
$857$	−42.3226	−1.44571	−0.722856	−	0.690999i	$-0.757171\pi$
$857$	−42.3226	−1.44571	−0.722856	+	0.690999i	$0.757171\pi$
$858$	0	0
$859$	18.1451	0.619104	0.309552	−	0.950883i	$-0.399821\pi$
$859$	18.1451	0.619104	0.309552	+	0.950883i	$0.399821\pi$
$860$	0	0
$861$	−71.0707	−2.42208
$862$	0	0
$863$	−19.3054	−0.657165	−0.328582	−	0.944475i	$-0.606571\pi$
$863$	−19.3054	−0.657165	−0.328582	+	0.944475i	$0.606571\pi$
$864$	0	0
$865$	16.6643	0.566603
$866$	0	0
$867$	25.7694	0.875174
$868$	0	0
$869$	20.6131	0.699252
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.49063	0.0842951
$874$	0	0
$875$	−4.25414	−0.143816
$876$	0	0
$877$	−14.1636	−0.478270	−0.239135	−	0.970986i	$-0.576864\pi$
$877$	−14.1636	−0.478270	−0.239135	+	0.970986i	$0.576864\pi$
$878$	0	0
$879$	−10.9420	−0.369064
$880$	0	0
$881$	17.0142	0.573222	0.286611	−	0.958047i	$-0.407471\pi$
$881$	17.0142	0.573222	0.286611	+	0.958047i	$0.407471\pi$
$882$	0	0
$883$	0.539474	0.0181548	0.00907738	−	0.999959i	$-0.497111\pi$
$883$	0.539474	0.0181548	0.00907738	+	0.999959i	$0.497111\pi$
$884$	0	0
$885$	−1.60888	−0.0540818
$886$	0	0
$887$	7.96905	0.267574	0.133787	−	0.991010i	$-0.457286\pi$
$887$	7.96905	0.267574	0.133787	+	0.991010i	$0.457286\pi$
$888$	0	0
$889$	10.6869	0.358427
$890$	0	0
$891$	−12.2862	−0.411603
$892$	0	0
$893$	1.96150	0.0656390
$894$	0	0
$895$	0.958705	0.0320460
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.9700	0.432574
$900$	0	0
$901$	12.9256	0.430614
$902$	0	0
$903$	35.7505	1.18970
$904$	0	0
$905$	−5.31264	−0.176598
$906$	0	0
$907$	−43.8197	−1.45501	−0.727505	−	0.686103i	$-0.759320\pi$
$907$	−43.8197	−1.45501	−0.727505	+	0.686103i	$0.759320\pi$
$908$	0	0
$909$	−3.45178	−0.114488
$910$	0	0
$911$	−8.95127	−0.296569	−0.148284	−	0.988945i	$-0.547375\pi$
$911$	−8.95127	−0.296569	−0.148284	+	0.988945i	$0.547375\pi$
$912$	0	0
$913$	8.36000	0.276676
$914$	0	0
$915$	18.3297	0.605960
$916$	0	0
$917$	−84.7813	−2.79973
$918$	0	0
$919$	41.1943	1.35887	0.679437	−	0.733734i	$-0.262224\pi$
$919$	41.1943	1.35887	0.679437	+	0.733734i	$0.262224\pi$
$920$	0	0
$921$	−19.4611	−0.641265
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−7.56859	−0.248854
$926$	0	0
$927$	−4.12822	−0.135589
$928$	0	0
$929$	−7.32126	−0.240203	−0.120101	−	0.992762i	$-0.538322\pi$
$929$	−7.32126	−0.240203	−0.120101	+	0.992762i	$0.538322\pi$
$930$	0	0
$931$	−7.06746	−0.231627
$932$	0	0
$933$	−18.5150	−0.606154
$934$	0	0
$935$	−8.55357	−0.279732
$936$	0	0
$937$	−6.88696	−0.224987	−0.112494	−	0.993652i	$-0.535884\pi$
$937$	−6.88696	−0.224987	−0.112494	+	0.993652i	$0.535884\pi$
$938$	0	0
$939$	24.1520	0.788171
$940$	0	0
$941$	51.9727	1.69426	0.847131	−	0.531384i	$-0.178328\pi$
$941$	51.9727	1.69426	0.847131	+	0.531384i	$0.178328\pi$
$942$	0	0
$943$	−30.3335	−0.987794
$944$	0	0
$945$	22.9232	0.745691
$946$	0	0
$947$	−44.0866	−1.43262	−0.716310	−	0.697782i	$-0.754170\pi$
$947$	−44.0866	−1.43262	−0.716310	+	0.697782i	$0.754170\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	13.7913	0.447214
$952$	0	0
$953$	−16.9713	−0.549755	−0.274877	−	0.961479i	$-0.588637\pi$
$953$	−16.9713	−0.549755	−0.274877	+	0.961479i	$0.588637\pi$
$954$	0	0
$955$	−4.70838	−0.152359
$956$	0	0
$957$	16.3869	0.529712
$958$	0	0
$959$	−48.3067	−1.55990
$960$	0	0
$961$	−27.1237	−0.874959
$962$	0	0
$963$	−4.52178	−0.145712
$964$	0	0
$965$	−16.3532	−0.526427
$966$	0	0
$967$	−8.23698	−0.264883	−0.132442	−	0.991191i	$-0.542282\pi$
$967$	−8.23698	−0.264883	−0.132442	+	0.991191i	$0.542282\pi$
$968$	0	0
$969$	−6.02555	−0.193568
$970$	0	0
$971$	−26.1164	−0.838114	−0.419057	−	0.907960i	$-0.637639\pi$
$971$	−26.1164	−0.838114	−0.419057	+	0.907960i	$0.637639\pi$
$972$	0	0
$973$	−74.4067	−2.38537
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.16938	0.229369	0.114684	−	0.993402i	$-0.463414\pi$
$977$	7.16938	0.229369	0.114684	+	0.993402i	$0.463414\pi$
$978$	0	0
$979$	−14.6059	−0.466807
$980$	0	0
$981$	4.06020	0.129632
$982$	0	0
$983$	−28.4268	−0.906675	−0.453338	−	0.891339i	$-0.649767\pi$
$983$	−28.4268	−0.906675	−0.453338	+	0.891339i	$0.649767\pi$
$984$	0	0
$985$	13.4002	0.426967
$986$	0	0
$987$	−21.7351	−0.691836
$988$	0	0
$989$	15.2585	0.485193
$990$	0	0
$991$	−40.0219	−1.27134	−0.635669	−	0.771962i	$-0.719276\pi$
$991$	−40.0219	−1.27134	−0.635669	+	0.771962i	$0.719276\pi$
$992$	0	0
$993$	−59.5385	−1.88940
$994$	0	0
$995$	−3.34363	−0.106000
$996$	0	0
$997$	53.5360	1.69550	0.847751	−	0.530395i	$-0.177956\pi$
$997$	53.5360	1.69550	0.847751	+	0.530395i	$0.177956\pi$
$998$	0	0
$999$	40.7829	1.29031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.r.1.7	✓	9
13.5	odd	4		3380.2.f.j.3041.14		18
13.8	odd	4		3380.2.f.j.3041.13		18
13.12	even	2		3380.2.a.s.1.7	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.7	✓	9	1.1	even	1	trivial
3380.2.a.s.1.7	yes	9	13.12	even	2
3380.2.f.j.3041.13		18	13.8	odd	4
3380.2.f.j.3041.14		18	13.5	odd	4

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q+1.65879 q^{3} -1.00000 q^{5} +4.25414 q^{7} -0.248414 q^{9} +O(q^{10})\)	\(q+1.65879 q^{3} -1.00000 q^{5} +4.25414 q^{7} -0.248414 q^{9} +1.49959 q^{11} -1.65879 q^{15} +5.70395 q^{17} -0.636838 q^{19} +7.05673 q^{21} +3.01186 q^{23} +1.00000 q^{25} -5.38844 q^{27} +6.58768 q^{29} +1.96883 q^{31} +2.48750 q^{33} -4.25414 q^{35} -7.56859 q^{37} -10.0713 q^{41} +5.06615 q^{43} +0.248414 q^{45} -3.08005 q^{47} +11.0977 q^{49} +9.46166 q^{51} +2.26608 q^{53} -1.49959 q^{55} -1.05638 q^{57} +0.969910 q^{59} -11.0500 q^{61} -1.05679 q^{63} +13.1252 q^{67} +4.99604 q^{69} -7.08679 q^{71} +12.5266 q^{73} +1.65879 q^{75} +6.37945 q^{77} +13.7459 q^{79} -8.19305 q^{81} +5.57487 q^{83} -5.70395 q^{85} +10.9276 q^{87} -9.73995 q^{89} +3.26587 q^{93} +0.636838 q^{95} -10.0261 q^{97} -0.372519 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

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