LMFDB - Embedded newform 9464.2.a.bs.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-3,0,4,0,15,0,16,0,15,0,0,0,8,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 26 x^{13} + 78 x^{12} + 253 x^{11} - 782 x^{10} - 1087 x^{9} + 3776 x^{8} + \cdots - 344 $ x^15 - 3x^14 - 26x^13 + 78x^12 + 253x^11 - 782x^10 - 1087x^9 + 3776x^8 + 1576x^7 - 8849x^6 + 1791x^5 + 8496x^4 - 5353x^3 - 1252x^2 + 1724x - 344
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.354884$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.354884 q^{3} -1.29454 q^{5} +1.00000 q^{7} -2.87406 q^{9} +2.70106 q^{11} +0.459413 q^{15} +5.65190 q^{17} +5.06579 q^{19} -0.354884 q^{21} +5.87053 q^{23} -3.32416 q^{25} +2.08461 q^{27} +4.05535 q^{29} +1.61958 q^{31} -0.958562 q^{33} -1.29454 q^{35} -6.26172 q^{37} -0.259788 q^{41} +0.838619 q^{43} +3.72059 q^{45} +2.19448 q^{47} +1.00000 q^{49} -2.00577 q^{51} -2.42435 q^{53} -3.49664 q^{55} -1.79777 q^{57} -1.49591 q^{59} -4.14499 q^{61} -2.87406 q^{63} -8.01117 q^{67} -2.08336 q^{69} +6.10633 q^{71} +1.42716 q^{73} +1.17969 q^{75} +2.70106 q^{77} +3.64099 q^{79} +7.88238 q^{81} +16.8713 q^{83} -7.31664 q^{85} -1.43918 q^{87} -12.7106 q^{89} -0.574763 q^{93} -6.55789 q^{95} -8.46418 q^{97} -7.76300 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 3 q^{3} + 4 q^{5} + 15 q^{7} + 16 q^{9} + 15 q^{11} + 8 q^{15} + 2 q^{17} + 13 q^{19} - 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} + 19 q^{31} - 24 q^{33} + 4 q^{35} - 2 q^{37} + 30 q^{41}+ \cdots + 70 q^{99}+O(q^{100}) $ 15 * q - 3 * q^3 + 4 * q^5 + 15 * q^7 + 16 * q^9 + 15 * q^11 + 8 * q^15 + 2 * q^17 + 13 * q^19 - 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 + 19 * q^31 - 24 * q^33 + 4 * q^35 - 2 * q^37 + 30 * q^41 + 15 * q^43 - 2 * q^45 + 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 - 2 * q^57 + 22 * q^59 - q^61 + 16 * q^63 + 29 * q^67 + 32 * q^69 + 20 * q^71 + 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 + 23 * q^83 + 8 * q^85 - 22 * q^87 + 9 * q^89 + 10 * q^93 + 28 * q^95 + 11 * q^97 + 70 * 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
$3$	−0.354884	−0.204892	−0.102446	−	0.994739i	$-0.532667\pi$
$3$	−0.354884	−0.204892	−0.102446	+	0.994739i	$0.532667\pi$
$4$	0	0
$5$	−1.29454	−0.578937	−0.289469	−	0.957187i	$-0.593479\pi$
$5$	−1.29454	−0.578937	−0.289469	+	0.957187i	$0.593479\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.87406	−0.958019
$10$	0	0
$11$	2.70106	0.814400	0.407200	−	0.913339i	$-0.366505\pi$
$11$	2.70106	0.814400	0.407200	+	0.913339i	$0.366505\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.459413	0.118620
$16$	0	0
$17$	5.65190	1.37079	0.685394	−	0.728172i	$-0.259630\pi$
$17$	5.65190	1.37079	0.685394	+	0.728172i	$0.259630\pi$
$18$	0	0
$19$	5.06579	1.16217	0.581086	−	0.813842i	$-0.302628\pi$
$19$	5.06579	1.16217	0.581086	+	0.813842i	$0.302628\pi$
$20$	0	0
$21$	−0.354884	−0.0774420
$22$	0	0
$23$	5.87053	1.22409	0.612045	−	0.790823i	$-0.290347\pi$
$23$	5.87053	1.22409	0.612045	+	0.790823i	$0.290347\pi$
$24$	0	0
$25$	−3.32416	−0.664831
$26$	0	0
$27$	2.08461	0.401183
$28$	0	0
$29$	4.05535	0.753059	0.376530	−	0.926405i	$-0.377117\pi$
$29$	4.05535	0.753059	0.376530	+	0.926405i	$0.377117\pi$
$30$	0	0
$31$	1.61958	0.290885	0.145443	−	0.989367i	$-0.453539\pi$
$31$	1.61958	0.290885	0.145443	+	0.989367i	$0.453539\pi$
$32$	0	0
$33$	−0.958562	−0.166864
$34$	0	0
$35$	−1.29454	−0.218818
$36$	0	0
$37$	−6.26172	−1.02942	−0.514710	−	0.857364i	$-0.672100\pi$
$37$	−6.26172	−1.02942	−0.514710	+	0.857364i	$0.672100\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.259788	−0.0405721	−0.0202860	−	0.999794i	$-0.506458\pi$
$41$	−0.259788	−0.0405721	−0.0202860	+	0.999794i	$0.506458\pi$
$42$	0	0
$43$	0.838619	0.127888	0.0639441	−	0.997953i	$-0.479632\pi$
$43$	0.838619	0.127888	0.0639441	+	0.997953i	$0.479632\pi$
$44$	0	0
$45$	3.72059	0.554633
$46$	0	0
$47$	2.19448	0.320098	0.160049	−	0.987109i	$-0.448835\pi$
$47$	2.19448	0.320098	0.160049	+	0.987109i	$0.448835\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00577	−0.280864
$52$	0	0
$53$	−2.42435	−0.333010	−0.166505	−	0.986041i	$-0.553248\pi$
$53$	−2.42435	−0.333010	−0.166505	+	0.986041i	$0.553248\pi$
$54$	0	0
$55$	−3.49664	−0.471487
$56$	0	0
$57$	−1.79777	−0.238120
$58$	0	0
$59$	−1.49591	−0.194751	−0.0973756	−	0.995248i	$-0.531045\pi$
$59$	−1.49591	−0.194751	−0.0973756	+	0.995248i	$0.531045\pi$
$60$	0	0
$61$	−4.14499	−0.530712	−0.265356	−	0.964150i	$-0.585490\pi$
$61$	−4.14499	−0.530712	−0.265356	+	0.964150i	$0.585490\pi$
$62$	0	0
$63$	−2.87406	−0.362097
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.01117	−0.978721	−0.489360	−	0.872082i	$-0.662770\pi$
$67$	−8.01117	−0.978721	−0.489360	+	0.872082i	$0.662770\pi$
$68$	0	0
$69$	−2.08336	−0.250807
$70$	0	0
$71$	6.10633	0.724689	0.362344	−	0.932044i	$-0.381976\pi$
$71$	6.10633	0.724689	0.362344	+	0.932044i	$0.381976\pi$
$72$	0	0
$73$	1.42716	0.167036	0.0835181	−	0.996506i	$-0.473384\pi$
$73$	1.42716	0.167036	0.0835181	+	0.996506i	$0.473384\pi$
$74$	0	0
$75$	1.17969	0.136219
$76$	0	0
$77$	2.70106	0.307814
$78$	0	0
$79$	3.64099	0.409644	0.204822	−	0.978799i	$-0.434338\pi$
$79$	3.64099	0.409644	0.204822	+	0.978799i	$0.434338\pi$
$80$	0	0
$81$	7.88238	0.875820
$82$	0	0
$83$	16.8713	1.85186	0.925932	−	0.377690i	$-0.123282\pi$
$83$	16.8713	1.85186	0.925932	+	0.377690i	$0.123282\pi$
$84$	0	0
$85$	−7.31664	−0.793601
$86$	0	0
$87$	−1.43918	−0.154296
$88$	0	0
$89$	−12.7106	−1.34732	−0.673661	−	0.739040i	$-0.735279\pi$
$89$	−12.7106	−1.34732	−0.673661	+	0.739040i	$0.735279\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.574763	−0.0596001
$94$	0	0
$95$	−6.55789	−0.672825
$96$	0	0
$97$	−8.46418	−0.859408	−0.429704	−	0.902970i	$-0.641382\pi$
$97$	−8.46418	−0.859408	−0.429704	+	0.902970i	$0.641382\pi$
$98$	0	0
$99$	−7.76300	−0.780211
$100$	0	0
$101$	−0.505794	−0.0503284	−0.0251642	−	0.999683i	$-0.508011\pi$
$101$	−0.505794	−0.0503284	−0.0251642	+	0.999683i	$0.508011\pi$
$102$	0	0
$103$	−5.27577	−0.519837	−0.259918	−	0.965631i	$-0.583696\pi$
$103$	−5.27577	−0.519837	−0.259918	+	0.965631i	$0.583696\pi$
$104$	0	0
$105$	0.459413	0.0448341
$106$	0	0
$107$	6.83896	0.661147	0.330574	−	0.943780i	$-0.392758\pi$
$107$	6.83896	0.661147	0.330574	+	0.943780i	$0.392758\pi$
$108$	0	0
$109$	−6.77926	−0.649335	−0.324668	−	0.945828i	$-0.605252\pi$
$109$	−6.77926	−0.649335	−0.324668	+	0.945828i	$0.605252\pi$
$110$	0	0
$111$	2.22218	0.210920
$112$	0	0
$113$	−13.1698	−1.23891	−0.619456	−	0.785032i	$-0.712647\pi$
$113$	−13.1698	−1.23891	−0.619456	+	0.785032i	$0.712647\pi$
$114$	0	0
$115$	−7.59965	−0.708671
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.65190	0.518109
$120$	0	0
$121$	−3.70428	−0.336753
$122$	0	0
$123$	0.0921946	0.00831291
$124$	0	0
$125$	10.7760	0.963833
$126$	0	0
$127$	−20.5040	−1.81944	−0.909718	−	0.415227i	$-0.863702\pi$
$127$	−20.5040	−1.81944	−0.909718	+	0.415227i	$0.863702\pi$
$128$	0	0
$129$	−0.297612	−0.0262033
$130$	0	0
$131$	−9.94594	−0.868980	−0.434490	−	0.900677i	$-0.643071\pi$
$131$	−9.94594	−0.868980	−0.434490	+	0.900677i	$0.643071\pi$
$132$	0	0
$133$	5.06579	0.439260
$134$	0	0
$135$	−2.69862	−0.232260
$136$	0	0
$137$	3.64127	0.311094	0.155547	−	0.987828i	$-0.450286\pi$
$137$	3.64127	0.311094	0.155547	+	0.987828i	$0.450286\pi$
$138$	0	0
$139$	6.40454	0.543226	0.271613	−	0.962407i	$-0.412443\pi$
$139$	6.40454	0.543226	0.271613	+	0.962407i	$0.412443\pi$
$140$	0	0
$141$	−0.778786	−0.0655856
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.24983	−0.435974
$146$	0	0
$147$	−0.354884	−0.0292703
$148$	0	0
$149$	18.6698	1.52949	0.764747	−	0.644331i	$-0.222864\pi$
$149$	18.6698	1.52949	0.764747	+	0.644331i	$0.222864\pi$
$150$	0	0
$151$	16.9039	1.37562	0.687809	−	0.725892i	$-0.258573\pi$
$151$	16.9039	1.37562	0.687809	+	0.725892i	$0.258573\pi$
$152$	0	0
$153$	−16.2439	−1.31324
$154$	0	0
$155$	−2.09662	−0.168404
$156$	0	0
$157$	20.1888	1.61124	0.805622	−	0.592430i	$-0.201831\pi$
$157$	20.1888	1.61124	0.805622	+	0.592430i	$0.201831\pi$
$158$	0	0
$159$	0.860362	0.0682312
$160$	0	0
$161$	5.87053	0.462662
$162$	0	0
$163$	5.00332	0.391890	0.195945	−	0.980615i	$-0.437223\pi$
$163$	5.00332	0.391890	0.195945	+	0.980615i	$0.437223\pi$
$164$	0	0
$165$	1.24090	0.0966040
$166$	0	0
$167$	15.5426	1.20272	0.601360	−	0.798978i	$-0.294626\pi$
$167$	15.5426	1.20272	0.601360	+	0.798978i	$0.294626\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−14.5594	−1.11338
$172$	0	0
$173$	−24.4949	−1.86232	−0.931158	−	0.364616i	$-0.881200\pi$
$173$	−24.4949	−1.86232	−0.931158	+	0.364616i	$0.881200\pi$
$174$	0	0
$175$	−3.32416	−0.251283
$176$	0	0
$177$	0.530875	0.0399030
$178$	0	0
$179$	−5.75501	−0.430150	−0.215075	−	0.976598i	$-0.569000\pi$
$179$	−5.75501	−0.430150	−0.215075	+	0.976598i	$0.569000\pi$
$180$	0	0
$181$	10.9096	0.810906	0.405453	−	0.914116i	$-0.367114\pi$
$181$	10.9096	0.810906	0.405453	+	0.914116i	$0.367114\pi$
$182$	0	0
$183$	1.47099	0.108739
$184$	0	0
$185$	8.10606	0.595970
$186$	0	0
$187$	15.2661	1.11637
$188$	0	0
$189$	2.08461	0.151633
$190$	0	0
$191$	−16.1385	−1.16774	−0.583869	−	0.811848i	$-0.698462\pi$
$191$	−16.1385	−1.16774	−0.583869	+	0.811848i	$0.698462\pi$
$192$	0	0
$193$	24.2928	1.74864	0.874318	−	0.485353i	$-0.161309\pi$
$193$	24.2928	1.74864	0.874318	+	0.485353i	$0.161309\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.3693	1.38000	0.690001	−	0.723808i	$-0.257610\pi$
$197$	19.3693	1.38000	0.690001	+	0.723808i	$0.257610\pi$
$198$	0	0
$199$	15.5910	1.10522	0.552609	−	0.833440i	$-0.313632\pi$
$199$	15.5910	1.10522	0.552609	+	0.833440i	$0.313632\pi$
$200$	0	0
$201$	2.84304	0.200532
$202$	0	0
$203$	4.05535	0.284630
$204$	0	0
$205$	0.336307	0.0234887
$206$	0	0
$207$	−16.8722	−1.17270
$208$	0	0
$209$	13.6830	0.946473
$210$	0	0
$211$	17.9687	1.23702	0.618508	−	0.785778i	$-0.287737\pi$
$211$	17.9687	1.23702	0.618508	+	0.785778i	$0.287737\pi$
$212$	0	0
$213$	−2.16704	−0.148483
$214$	0	0
$215$	−1.08563	−0.0740392
$216$	0	0
$217$	1.61958	0.109944
$218$	0	0
$219$	−0.506475	−0.0342244
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−4.55222	−0.304839	−0.152420	−	0.988316i	$-0.548707\pi$
$223$	−4.55222	−0.304839	−0.152420	+	0.988316i	$0.548707\pi$
$224$	0	0
$225$	9.55382	0.636921
$226$	0	0
$227$	1.06050	0.0703876	0.0351938	−	0.999381i	$-0.488795\pi$
$227$	1.06050	0.0703876	0.0351938	+	0.999381i	$0.488795\pi$
$228$	0	0
$229$	6.54437	0.432464	0.216232	−	0.976342i	$-0.430623\pi$
$229$	6.54437	0.432464	0.216232	+	0.976342i	$0.430623\pi$
$230$	0	0
$231$	−0.958562	−0.0630688
$232$	0	0
$233$	−5.64509	−0.369822	−0.184911	−	0.982755i	$-0.559200\pi$
$233$	−5.64509	−0.369822	−0.184911	+	0.982755i	$0.559200\pi$
$234$	0	0
$235$	−2.84085	−0.185317
$236$	0	0
$237$	−1.29213	−0.0839329
$238$	0	0
$239$	10.8418	0.701298	0.350649	−	0.936507i	$-0.385961\pi$
$239$	10.8418	0.701298	0.350649	+	0.936507i	$0.385961\pi$
$240$	0	0
$241$	0.684200	0.0440732	0.0220366	−	0.999757i	$-0.492985\pi$
$241$	0.684200	0.0440732	0.0220366	+	0.999757i	$0.492985\pi$
$242$	0	0
$243$	−9.05116	−0.580632
$244$	0	0
$245$	−1.29454	−0.0827054
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−5.98735	−0.379433
$250$	0	0
$251$	3.11117	0.196376	0.0981878	−	0.995168i	$-0.468695\pi$
$251$	3.11117	0.196376	0.0981878	+	0.995168i	$0.468695\pi$
$252$	0	0
$253$	15.8566	0.996898
$254$	0	0
$255$	2.59656	0.162603
$256$	0	0
$257$	16.7548	1.04514	0.522568	−	0.852598i	$-0.324974\pi$
$257$	16.7548	1.04514	0.522568	+	0.852598i	$0.324974\pi$
$258$	0	0
$259$	−6.26172	−0.389084
$260$	0	0
$261$	−11.6553	−0.721445
$262$	0	0
$263$	7.22612	0.445582	0.222791	−	0.974866i	$-0.428483\pi$
$263$	7.22612	0.445582	0.222791	+	0.974866i	$0.428483\pi$
$264$	0	0
$265$	3.13842	0.192792
$266$	0	0
$267$	4.51079	0.276056
$268$	0	0
$269$	−11.3080	−0.689461	−0.344730	−	0.938702i	$-0.612030\pi$
$269$	−11.3080	−0.689461	−0.344730	+	0.938702i	$0.612030\pi$
$270$	0	0
$271$	15.3864	0.934655	0.467327	−	0.884084i	$-0.345217\pi$
$271$	15.3864	0.934655	0.467327	+	0.884084i	$0.345217\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.97874	−0.541439
$276$	0	0
$277$	1.62728	0.0977737	0.0488869	−	0.998804i	$-0.484433\pi$
$277$	1.62728	0.0977737	0.0488869	+	0.998804i	$0.484433\pi$
$278$	0	0
$279$	−4.65477	−0.278674
$280$	0	0
$281$	1.52381	0.0909029	0.0454514	−	0.998967i	$-0.485527\pi$
$281$	1.52381	0.0909029	0.0454514	+	0.998967i	$0.485527\pi$
$282$	0	0
$283$	−17.8863	−1.06323	−0.531614	−	0.846987i	$-0.678414\pi$
$283$	−17.8863	−1.06323	−0.531614	+	0.846987i	$0.678414\pi$
$284$	0	0
$285$	2.32729	0.137857
$286$	0	0
$287$	−0.259788	−0.0153348
$288$	0	0
$289$	14.9440	0.879060
$290$	0	0
$291$	3.00380	0.176086
$292$	0	0
$293$	−21.8894	−1.27879	−0.639396	−	0.768878i	$-0.720815\pi$
$293$	−21.8894	−1.27879	−0.639396	+	0.768878i	$0.720815\pi$
$294$	0	0
$295$	1.93652	0.112749
$296$	0	0
$297$	5.63065	0.326723
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0.838619	0.0483372
$302$	0	0
$303$	0.179498	0.0103119
$304$	0	0
$305$	5.36588	0.307249
$306$	0	0
$307$	31.9244	1.82202	0.911010	−	0.412383i	$-0.135304\pi$
$307$	31.9244	1.82202	0.911010	+	0.412383i	$0.135304\pi$
$308$	0	0
$309$	1.87229	0.106511
$310$	0	0
$311$	1.62986	0.0924211	0.0462105	−	0.998932i	$-0.485285\pi$
$311$	1.62986	0.0924211	0.0462105	+	0.998932i	$0.485285\pi$
$312$	0	0
$313$	32.9153	1.86048	0.930242	−	0.366947i	$-0.119597\pi$
$313$	32.9153	1.86048	0.930242	+	0.366947i	$0.119597\pi$
$314$	0	0
$315$	3.72059	0.209632
$316$	0	0
$317$	21.5434	1.21000	0.604999	−	0.796226i	$-0.293173\pi$
$317$	21.5434	1.21000	0.604999	+	0.796226i	$0.293173\pi$
$318$	0	0
$319$	10.9537	0.613291
$320$	0	0
$321$	−2.42704	−0.135464
$322$	0	0
$323$	28.6314	1.59309
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.40585	0.133044
$328$	0	0
$329$	2.19448	0.120986
$330$	0	0
$331$	−18.8986	−1.03876	−0.519379	−	0.854544i	$-0.673837\pi$
$331$	−18.8986	−1.03876	−0.519379	+	0.854544i	$0.673837\pi$
$332$	0	0
$333$	17.9965	0.986204
$334$	0	0
$335$	10.3708	0.566618
$336$	0	0
$337$	−17.0379	−0.928114	−0.464057	−	0.885805i	$-0.653607\pi$
$337$	−17.0379	−0.928114	−0.464057	+	0.885805i	$0.653607\pi$
$338$	0	0
$339$	4.67375	0.253843
$340$	0	0
$341$	4.37458	0.236897
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.69700	0.145201
$346$	0	0
$347$	6.10071	0.327504	0.163752	−	0.986502i	$-0.447640\pi$
$347$	6.10071	0.327504	0.163752	+	0.986502i	$0.447640\pi$
$348$	0	0
$349$	−16.1111	−0.862407	−0.431204	−	0.902255i	$-0.641911\pi$
$349$	−16.1111	−0.862407	−0.431204	+	0.902255i	$0.641911\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.38709	0.286726	0.143363	−	0.989670i	$-0.454208\pi$
$353$	5.38709	0.286726	0.143363	+	0.989670i	$0.454208\pi$
$354$	0	0
$355$	−7.90492	−0.419549
$356$	0	0
$357$	−2.00577	−0.106157
$358$	0	0
$359$	26.2145	1.38355	0.691774	−	0.722114i	$-0.256829\pi$
$359$	26.2145	1.38355	0.691774	+	0.722114i	$0.256829\pi$
$360$	0	0
$361$	6.66224	0.350644
$362$	0	0
$363$	1.31459	0.0689981
$364$	0	0
$365$	−1.84752	−0.0967035
$366$	0	0
$367$	−17.7366	−0.925844	−0.462922	−	0.886399i	$-0.653199\pi$
$367$	−17.7366	−0.925844	−0.462922	+	0.886399i	$0.653199\pi$
$368$	0	0
$369$	0.746646	0.0388688
$370$	0	0
$371$	−2.42435	−0.125866
$372$	0	0
$373$	13.0990	0.678241	0.339120	−	0.940743i	$-0.389871\pi$
$373$	13.0990	0.678241	0.339120	+	0.940743i	$0.389871\pi$
$374$	0	0
$375$	−3.82422	−0.197482
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−16.3094	−0.837759	−0.418879	−	0.908042i	$-0.637577\pi$
$379$	−16.3094	−0.837759	−0.418879	+	0.908042i	$0.637577\pi$
$380$	0	0
$381$	7.27654	0.372788
$382$	0	0
$383$	8.74491	0.446844	0.223422	−	0.974722i	$-0.428277\pi$
$383$	8.74491	0.446844	0.223422	+	0.974722i	$0.428277\pi$
$384$	0	0
$385$	−3.49664	−0.178205
$386$	0	0
$387$	−2.41024	−0.122519
$388$	0	0
$389$	29.3389	1.48754	0.743770	−	0.668435i	$-0.233036\pi$
$389$	29.3389	1.48754	0.743770	+	0.668435i	$0.233036\pi$
$390$	0	0
$391$	33.1797	1.67797
$392$	0	0
$393$	3.52965	0.178047
$394$	0	0
$395$	−4.71342	−0.237158
$396$	0	0
$397$	−9.83870	−0.493790	−0.246895	−	0.969042i	$-0.579410\pi$
$397$	−9.83870	−0.493790	−0.246895	+	0.969042i	$0.579410\pi$
$398$	0	0
$399$	−1.79777	−0.0900010
$400$	0	0
$401$	−28.2640	−1.41144	−0.705718	−	0.708493i	$-0.749375\pi$
$401$	−28.2640	−1.41144	−0.705718	+	0.708493i	$0.749375\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−10.2041	−0.507045
$406$	0	0
$407$	−16.9133	−0.838359
$408$	0	0
$409$	34.2555	1.69383	0.846913	−	0.531731i	$-0.178458\pi$
$409$	34.2555	1.69383	0.846913	+	0.531731i	$0.178458\pi$
$410$	0	0
$411$	−1.29223	−0.0637409
$412$	0	0
$413$	−1.49591	−0.0736090
$414$	0	0
$415$	−21.8406	−1.07211
$416$	0	0
$417$	−2.27287	−0.111303
$418$	0	0
$419$	−6.33712	−0.309589	−0.154794	−	0.987947i	$-0.549471\pi$
$419$	−6.33712	−0.309589	−0.154794	+	0.987947i	$0.549471\pi$
$420$	0	0
$421$	26.4362	1.28842	0.644210	−	0.764849i	$-0.277186\pi$
$421$	26.4362	1.28842	0.644210	+	0.764849i	$0.277186\pi$
$422$	0	0
$423$	−6.30706	−0.306660
$424$	0	0
$425$	−18.7878	−0.911343
$426$	0	0
$427$	−4.14499	−0.200590
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.1652	−1.06766	−0.533830	−	0.845592i	$-0.679248\pi$
$431$	−22.1652	−1.06766	−0.533830	+	0.845592i	$0.679248\pi$
$432$	0	0
$433$	−24.0234	−1.15449	−0.577244	−	0.816571i	$-0.695872\pi$
$433$	−24.0234	−1.15449	−0.577244	+	0.816571i	$0.695872\pi$
$434$	0	0
$435$	1.86308	0.0893278
$436$	0	0
$437$	29.7389	1.42260
$438$	0	0
$439$	−26.1253	−1.24689	−0.623445	−	0.781867i	$-0.714268\pi$
$439$	−26.1253	−1.24689	−0.623445	+	0.781867i	$0.714268\pi$
$440$	0	0
$441$	−2.87406	−0.136860
$442$	0	0
$443$	19.6003	0.931240	0.465620	−	0.884985i	$-0.345831\pi$
$443$	19.6003	0.931240	0.465620	+	0.884985i	$0.345831\pi$
$444$	0	0
$445$	16.4544	0.780016
$446$	0	0
$447$	−6.62563	−0.313381
$448$	0	0
$449$	3.95912	0.186842	0.0934212	−	0.995627i	$-0.470220\pi$
$449$	3.95912	0.186842	0.0934212	+	0.995627i	$0.470220\pi$
$450$	0	0
$451$	−0.701703	−0.0330419
$452$	0	0
$453$	−5.99892	−0.281854
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.40988	0.299842	0.149921	−	0.988698i	$-0.452098\pi$
$457$	6.40988	0.299842	0.149921	+	0.988698i	$0.452098\pi$
$458$	0	0
$459$	11.7820	0.549937
$460$	0	0
$461$	19.8252	0.923351	0.461676	−	0.887049i	$-0.347248\pi$
$461$	19.8252	0.923351	0.461676	+	0.887049i	$0.347248\pi$
$462$	0	0
$463$	−9.06580	−0.421324	−0.210662	−	0.977559i	$-0.567562\pi$
$463$	−9.06580	−0.421324	−0.210662	+	0.977559i	$0.567562\pi$
$464$	0	0
$465$	0.744056	0.0345048
$466$	0	0
$467$	−34.2617	−1.58544	−0.792721	−	0.609584i	$-0.791336\pi$
$467$	−34.2617	−1.58544	−0.792721	+	0.609584i	$0.791336\pi$
$468$	0	0
$469$	−8.01117	−0.369922
$470$	0	0
$471$	−7.16469	−0.330132
$472$	0	0
$473$	2.26516	0.104152
$474$	0	0
$475$	−16.8395	−0.772649
$476$	0	0
$477$	6.96772	0.319030
$478$	0	0
$479$	−20.5465	−0.938794	−0.469397	−	0.882987i	$-0.655529\pi$
$479$	−20.5465	−0.938794	−0.469397	+	0.882987i	$0.655529\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−2.08336	−0.0947960
$484$	0	0
$485$	10.9573	0.497543
$486$	0	0
$487$	24.7311	1.12067	0.560337	−	0.828264i	$-0.310672\pi$
$487$	24.7311	1.12067	0.560337	+	0.828264i	$0.310672\pi$
$488$	0	0
$489$	−1.77560	−0.0802952
$490$	0	0
$491$	−28.0991	−1.26810	−0.634048	−	0.773293i	$-0.718608\pi$
$491$	−28.0991	−1.26810	−0.634048	+	0.773293i	$0.718608\pi$
$492$	0	0
$493$	22.9204	1.03229
$494$	0	0
$495$	10.0495	0.451693
$496$	0	0
$497$	6.10633	0.273907
$498$	0	0
$499$	−20.7117	−0.927185	−0.463592	−	0.886049i	$-0.653440\pi$
$499$	−20.7117	−0.927185	−0.463592	+	0.886049i	$0.653440\pi$
$500$	0	0
$501$	−5.51581	−0.246428
$502$	0	0
$503$	18.6729	0.832584	0.416292	−	0.909231i	$-0.363329\pi$
$503$	18.6729	0.832584	0.416292	+	0.909231i	$0.363329\pi$
$504$	0	0
$505$	0.654772	0.0291370
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−28.2436	−1.25187	−0.625937	−	0.779874i	$-0.715283\pi$
$509$	−28.2436	−1.25187	−0.625937	+	0.779874i	$0.715283\pi$
$510$	0	0
$511$	1.42716	0.0631337
$512$	0	0
$513$	10.5602	0.466244
$514$	0	0
$515$	6.82971	0.300953
$516$	0	0
$517$	5.92742	0.260687
$518$	0	0
$519$	8.69286	0.381574
$520$	0	0
$521$	0.617556	0.0270556	0.0135278	−	0.999908i	$-0.495694\pi$
$521$	0.617556	0.0270556	0.0135278	+	0.999908i	$0.495694\pi$
$522$	0	0
$523$	−12.6813	−0.554514	−0.277257	−	0.960796i	$-0.589425\pi$
$523$	−12.6813	−0.554514	−0.277257	+	0.960796i	$0.589425\pi$
$524$	0	0
$525$	1.17969	0.0514859
$526$	0	0
$527$	9.15371	0.398742
$528$	0	0
$529$	11.4631	0.498396
$530$	0	0
$531$	4.29934	0.186575
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−8.85333	−0.382763
$536$	0	0
$537$	2.04236	0.0881344
$538$	0	0
$539$	2.70106	0.116343
$540$	0	0
$541$	16.2059	0.696745	0.348372	−	0.937356i	$-0.386734\pi$
$541$	16.2059	0.696745	0.348372	+	0.937356i	$0.386734\pi$
$542$	0	0
$543$	−3.87165	−0.166148
$544$	0	0
$545$	8.77605	0.375925
$546$	0	0
$547$	12.1627	0.520040	0.260020	−	0.965603i	$-0.416271\pi$
$547$	12.1627	0.520040	0.260020	+	0.965603i	$0.416271\pi$
$548$	0	0
$549$	11.9130	0.508432
$550$	0	0
$551$	20.5436	0.875185
$552$	0	0
$553$	3.64099	0.154831
$554$	0	0
$555$	−2.87671	−0.122110
$556$	0	0
$557$	−6.19015	−0.262285	−0.131143	−	0.991364i	$-0.541865\pi$
$557$	−6.19015	−0.262285	−0.131143	+	0.991364i	$0.541865\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−5.41770	−0.228736
$562$	0	0
$563$	−8.76617	−0.369450	−0.184725	−	0.982790i	$-0.559139\pi$
$563$	−8.76617	−0.369450	−0.184725	+	0.982790i	$0.559139\pi$
$564$	0	0
$565$	17.0489	0.717252
$566$	0	0
$567$	7.88238	0.331029
$568$	0	0
$569$	−19.2367	−0.806446	−0.403223	−	0.915102i	$-0.632110\pi$
$569$	−19.2367	−0.806446	−0.403223	+	0.915102i	$0.632110\pi$
$570$	0	0
$571$	16.4143	0.686919	0.343459	−	0.939168i	$-0.388401\pi$
$571$	16.4143	0.686919	0.343459	+	0.939168i	$0.388401\pi$
$572$	0	0
$573$	5.72728	0.239261
$574$	0	0
$575$	−19.5146	−0.813813
$576$	0	0
$577$	−10.8255	−0.450670	−0.225335	−	0.974281i	$-0.572348\pi$
$577$	−10.8255	−0.450670	−0.225335	+	0.974281i	$0.572348\pi$
$578$	0	0
$579$	−8.62114	−0.358282
$580$	0	0
$581$	16.8713	0.699939
$582$	0	0
$583$	−6.54831	−0.271203
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.5679	−0.725106	−0.362553	−	0.931963i	$-0.618095\pi$
$587$	−17.5679	−0.725106	−0.362553	+	0.931963i	$0.618095\pi$
$588$	0	0
$589$	8.20445	0.338059
$590$	0	0
$591$	−6.87384	−0.282752
$592$	0	0
$593$	−33.5792	−1.37893	−0.689465	−	0.724319i	$-0.742154\pi$
$593$	−33.5792	−1.37893	−0.689465	+	0.724319i	$0.742154\pi$
$594$	0	0
$595$	−7.31664	−0.299953
$596$	0	0
$597$	−5.53301	−0.226451
$598$	0	0
$599$	42.8842	1.75220	0.876100	−	0.482130i	$-0.160137\pi$
$599$	42.8842	1.75220	0.876100	+	0.482130i	$0.160137\pi$
$600$	0	0
$601$	41.1591	1.67892	0.839458	−	0.543425i	$-0.182873\pi$
$601$	41.1591	1.67892	0.839458	+	0.543425i	$0.182873\pi$
$602$	0	0
$603$	23.0246	0.937633
$604$	0	0
$605$	4.79536	0.194959
$606$	0	0
$607$	−27.4675	−1.11487	−0.557435	−	0.830221i	$-0.688214\pi$
$607$	−27.4675	−1.11487	−0.557435	+	0.830221i	$0.688214\pi$
$608$	0	0
$609$	−1.43918	−0.0583185
$610$	0	0
$611$	0	0
$612$	0	0
$613$	28.4353	1.14849	0.574245	−	0.818684i	$-0.305296\pi$
$613$	28.4353	1.14849	0.574245	+	0.818684i	$0.305296\pi$
$614$	0	0
$615$	−0.119350	−0.00481265
$616$	0	0
$617$	35.4166	1.42582	0.712910	−	0.701255i	$-0.247377\pi$
$617$	35.4166	1.42582	0.712910	+	0.701255i	$0.247377\pi$
$618$	0	0
$619$	19.5809	0.787021	0.393511	−	0.919320i	$-0.371260\pi$
$619$	19.5809	0.787021	0.393511	+	0.919320i	$0.371260\pi$
$620$	0	0
$621$	12.2378	0.491084
$622$	0	0
$623$	−12.7106	−0.509240
$624$	0	0
$625$	2.67081	0.106832
$626$	0	0
$627$	−4.85588	−0.193925
$628$	0	0
$629$	−35.3906	−1.41112
$630$	0	0
$631$	27.9767	1.11373	0.556867	−	0.830602i	$-0.312003\pi$
$631$	27.9767	1.11373	0.556867	+	0.830602i	$0.312003\pi$
$632$	0	0
$633$	−6.37681	−0.253455
$634$	0	0
$635$	26.5433	1.05334
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−17.5500	−0.694266
$640$	0	0
$641$	13.0690	0.516194	0.258097	−	0.966119i	$-0.416905\pi$
$641$	13.0690	0.516194	0.258097	+	0.966119i	$0.416905\pi$
$642$	0	0
$643$	29.5312	1.16460	0.582298	−	0.812975i	$-0.302154\pi$
$643$	29.5312	1.16460	0.582298	+	0.812975i	$0.302154\pi$
$644$	0	0
$645$	0.385272	0.0151701
$646$	0	0
$647$	5.22069	0.205247	0.102623	−	0.994720i	$-0.467276\pi$
$647$	5.22069	0.205247	0.102623	+	0.994720i	$0.467276\pi$
$648$	0	0
$649$	−4.04055	−0.158605
$650$	0	0
$651$	−0.574763	−0.0225267
$652$	0	0
$653$	45.0760	1.76396	0.881980	−	0.471287i	$-0.156211\pi$
$653$	45.0760	1.76396	0.881980	+	0.471287i	$0.156211\pi$
$654$	0	0
$655$	12.8754	0.503085
$656$	0	0
$657$	−4.10173	−0.160024
$658$	0	0
$659$	−48.9003	−1.90489	−0.952443	−	0.304718i	$-0.901438\pi$
$659$	−48.9003	−1.90489	−0.952443	+	0.304718i	$0.901438\pi$
$660$	0	0
$661$	12.3715	0.481194	0.240597	−	0.970625i	$-0.422657\pi$
$661$	12.3715	0.481194	0.240597	+	0.970625i	$0.422657\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.55789	−0.254304
$666$	0	0
$667$	23.8070	0.921812
$668$	0	0
$669$	1.61551	0.0624593
$670$	0	0
$671$	−11.1959	−0.432212
$672$	0	0
$673$	32.2469	1.24303	0.621513	−	0.783403i	$-0.286518\pi$
$673$	32.2469	1.24303	0.621513	+	0.783403i	$0.286518\pi$
$674$	0	0
$675$	−6.92957	−0.266719
$676$	0	0
$677$	28.5883	1.09874	0.549369	−	0.835580i	$-0.314868\pi$
$677$	28.5883	1.09874	0.549369	+	0.835580i	$0.314868\pi$
$678$	0	0
$679$	−8.46418	−0.324826
$680$	0	0
$681$	−0.376353	−0.0144219
$682$	0	0
$683$	4.18894	0.160285	0.0801426	−	0.996783i	$-0.474462\pi$
$683$	4.18894	0.160285	0.0801426	+	0.996783i	$0.474462\pi$
$684$	0	0
$685$	−4.71378	−0.180104
$686$	0	0
$687$	−2.32249	−0.0886086
$688$	0	0
$689$	0	0
$690$	0	0
$691$	39.6200	1.50722	0.753609	−	0.657323i	$-0.228311\pi$
$691$	39.6200	1.50722	0.753609	+	0.657323i	$0.228311\pi$
$692$	0	0
$693$	−7.76300	−0.294892
$694$	0	0
$695$	−8.29096	−0.314494
$696$	0	0
$697$	−1.46830	−0.0556157
$698$	0	0
$699$	2.00335	0.0757737
$700$	0	0
$701$	33.5649	1.26773	0.633865	−	0.773444i	$-0.281467\pi$
$701$	33.5649	1.26773	0.633865	+	0.773444i	$0.281467\pi$
$702$	0	0
$703$	−31.7205	−1.19636
$704$	0	0
$705$	1.00817	0.0379699
$706$	0	0
$707$	−0.505794	−0.0190223
$708$	0	0
$709$	15.9669	0.599649	0.299825	−	0.953994i	$-0.403072\pi$
$709$	15.9669	0.599649	0.299825	+	0.953994i	$0.403072\pi$
$710$	0	0
$711$	−10.4644	−0.392447
$712$	0	0
$713$	9.50779	0.356070
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.84758	−0.143691
$718$	0	0
$719$	−45.1792	−1.68490	−0.842450	−	0.538774i	$-0.818887\pi$
$719$	−45.1792	−1.68490	−0.842450	+	0.538774i	$0.818887\pi$
$720$	0	0
$721$	−5.27577	−0.196480
$722$	0	0
$723$	−0.242812	−0.00903026
$724$	0	0
$725$	−13.4806	−0.500658
$726$	0	0
$727$	−24.7244	−0.916975	−0.458488	−	0.888701i	$-0.651609\pi$
$727$	−24.7244	−0.916975	−0.458488	+	0.888701i	$0.651609\pi$
$728$	0	0
$729$	−20.4350	−0.756853
$730$	0	0
$731$	4.73979	0.175308
$732$	0	0
$733$	0.277679	0.0102563	0.00512815	−	0.999987i	$-0.498368\pi$
$733$	0.277679	0.0102563	0.00512815	+	0.999987i	$0.498368\pi$
$734$	0	0
$735$	0.459413	0.0169457
$736$	0	0
$737$	−21.6386	−0.797070
$738$	0	0
$739$	−27.0377	−0.994600	−0.497300	−	0.867579i	$-0.665675\pi$
$739$	−27.0377	−0.994600	−0.497300	+	0.867579i	$0.665675\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.9240	−0.804314	−0.402157	−	0.915571i	$-0.631739\pi$
$743$	−21.9240	−0.804314	−0.402157	+	0.915571i	$0.631739\pi$
$744$	0	0
$745$	−24.1689	−0.885481
$746$	0	0
$747$	−48.4890	−1.77412
$748$	0	0
$749$	6.83896	0.249890
$750$	0	0
$751$	−13.4452	−0.490623	−0.245311	−	0.969444i	$-0.578890\pi$
$751$	−13.4452	−0.490623	−0.245311	+	0.969444i	$0.578890\pi$
$752$	0	0
$753$	−1.10411	−0.0402358
$754$	0	0
$755$	−21.8828	−0.796397
$756$	0	0
$757$	−37.1671	−1.35086	−0.675430	−	0.737424i	$-0.736042\pi$
$757$	−37.1671	−1.35086	−0.675430	+	0.737424i	$0.736042\pi$
$758$	0	0
$759$	−5.62727	−0.204257
$760$	0	0
$761$	−45.7346	−1.65788	−0.828939	−	0.559339i	$-0.811055\pi$
$761$	−45.7346	−1.65788	−0.828939	+	0.559339i	$0.811055\pi$
$762$	0	0
$763$	−6.77926	−0.245426
$764$	0	0
$765$	21.0284	0.760285
$766$	0	0
$767$	0	0
$768$	0	0
$769$	49.1871	1.77373	0.886866	−	0.462026i	$-0.152877\pi$
$769$	49.1871	1.77373	0.886866	+	0.462026i	$0.152877\pi$
$770$	0	0
$771$	−5.94601	−0.214140
$772$	0	0
$773$	−6.65141	−0.239235	−0.119617	−	0.992820i	$-0.538167\pi$
$773$	−6.65141	−0.239235	−0.119617	+	0.992820i	$0.538167\pi$
$774$	0	0
$775$	−5.38374	−0.193390
$776$	0	0
$777$	2.22218	0.0797204
$778$	0	0
$779$	−1.31603	−0.0471517
$780$	0	0
$781$	16.4936	0.590186
$782$	0	0
$783$	8.45382	0.302115
$784$	0	0
$785$	−26.1353	−0.932810
$786$	0	0
$787$	49.5475	1.76618	0.883089	−	0.469206i	$-0.155460\pi$
$787$	49.5475	1.76618	0.883089	+	0.469206i	$0.155460\pi$
$788$	0	0
$789$	−2.56443	−0.0912963
$790$	0	0
$791$	−13.1698	−0.468265
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1.11378	−0.0395016
$796$	0	0
$797$	−19.6344	−0.695487	−0.347744	−	0.937590i	$-0.613052\pi$
$797$	−19.6344	−0.695487	−0.347744	+	0.937590i	$0.613052\pi$
$798$	0	0
$799$	12.4030	0.438786
$800$	0	0
$801$	36.5310	1.29076
$802$	0	0
$803$	3.85483	0.136034
$804$	0	0
$805$	−7.59965	−0.267853
$806$	0	0
$807$	4.01303	0.141265
$808$	0	0
$809$	49.8888	1.75400	0.876998	−	0.480494i	$-0.159543\pi$
$809$	49.8888	1.75400	0.876998	+	0.480494i	$0.159543\pi$
$810$	0	0
$811$	23.4443	0.823242	0.411621	−	0.911355i	$-0.364963\pi$
$811$	23.4443	0.823242	0.411621	+	0.911355i	$0.364963\pi$
$812$	0	0
$813$	−5.46037	−0.191504
$814$	0	0
$815$	−6.47701	−0.226880
$816$	0	0
$817$	4.24827	0.148628
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.2650	0.428050	0.214025	−	0.976828i	$-0.431343\pi$
$821$	12.2650	0.428050	0.214025	+	0.976828i	$0.431343\pi$
$822$	0	0
$823$	48.2863	1.68315	0.841577	−	0.540138i	$-0.181628\pi$
$823$	48.2863	1.68315	0.841577	+	0.540138i	$0.181628\pi$
$824$	0	0
$825$	3.18641	0.110937
$826$	0	0
$827$	44.8076	1.55811	0.779056	−	0.626955i	$-0.215699\pi$
$827$	44.8076	1.55811	0.779056	+	0.626955i	$0.215699\pi$
$828$	0	0
$829$	−5.68706	−0.197520	−0.0987599	−	0.995111i	$-0.531488\pi$
$829$	−5.68706	−0.197520	−0.0987599	+	0.995111i	$0.531488\pi$
$830$	0	0
$831$	−0.577495	−0.0200331
$832$	0	0
$833$	5.65190	0.195827
$834$	0	0
$835$	−20.1205	−0.696300
$836$	0	0
$837$	3.37619	0.116698
$838$	0	0
$839$	−25.5046	−0.880516	−0.440258	−	0.897871i	$-0.645113\pi$
$839$	−25.5046	−0.880516	−0.440258	+	0.897871i	$0.645113\pi$
$840$	0	0
$841$	−12.5541	−0.432901
$842$	0	0
$843$	−0.540776	−0.0186253
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.70428	−0.127281
$848$	0	0
$849$	6.34755	0.217847
$850$	0	0
$851$	−36.7596	−1.26010
$852$	0	0
$853$	10.3542	0.354523	0.177261	−	0.984164i	$-0.443276\pi$
$853$	10.3542	0.354523	0.177261	+	0.984164i	$0.443276\pi$
$854$	0	0
$855$	18.8477	0.644579
$856$	0	0
$857$	−3.47023	−0.118541	−0.0592704	−	0.998242i	$-0.518877\pi$
$857$	−3.47023	−0.118541	−0.0592704	+	0.998242i	$0.518877\pi$
$858$	0	0
$859$	37.6507	1.28463	0.642313	−	0.766443i	$-0.277975\pi$
$859$	37.6507	1.28463	0.642313	+	0.766443i	$0.277975\pi$
$860$	0	0
$861$	0.0921946	0.00314198
$862$	0	0
$863$	4.51649	0.153743	0.0768716	−	0.997041i	$-0.475507\pi$
$863$	4.51649	0.153743	0.0768716	+	0.997041i	$0.475507\pi$
$864$	0	0
$865$	31.7098	1.07816
$866$	0	0
$867$	−5.30339	−0.180113
$868$	0	0
$869$	9.83454	0.333614
$870$	0	0
$871$	0	0
$872$	0	0
$873$	24.3266	0.823329
$874$	0	0
$875$	10.7760	0.364295
$876$	0	0
$877$	−27.5324	−0.929703	−0.464851	−	0.885389i	$-0.653892\pi$
$877$	−27.5324	−0.929703	−0.464851	+	0.885389i	$0.653892\pi$
$878$	0	0
$879$	7.76819	0.262015
$880$	0	0
$881$	12.3395	0.415727	0.207863	−	0.978158i	$-0.433349\pi$
$881$	12.3395	0.415727	0.207863	+	0.978158i	$0.433349\pi$
$882$	0	0
$883$	−17.6551	−0.594140	−0.297070	−	0.954856i	$-0.596009\pi$
$883$	−17.6551	−0.594140	−0.297070	+	0.954856i	$0.596009\pi$
$884$	0	0
$885$	−0.687241	−0.0231014
$886$	0	0
$887$	−21.0907	−0.708157	−0.354078	−	0.935216i	$-0.615205\pi$
$887$	−21.0907	−0.708157	−0.354078	+	0.935216i	$0.615205\pi$
$888$	0	0
$889$	−20.5040	−0.687682
$890$	0	0
$891$	21.2908	0.713267
$892$	0	0
$893$	11.1168	0.372009
$894$	0	0
$895$	7.45011	0.249030
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.56796	0.219054
$900$	0	0
$901$	−13.7022	−0.456486
$902$	0	0
$903$	−0.297612	−0.00990392
$904$	0	0
$905$	−14.1230	−0.469464
$906$	0	0
$907$	6.47035	0.214845	0.107422	−	0.994213i	$-0.465740\pi$
$907$	6.47035	0.214845	0.107422	+	0.994213i	$0.465740\pi$
$908$	0	0
$909$	1.45368	0.0482156
$910$	0	0
$911$	−23.2455	−0.770158	−0.385079	−	0.922884i	$-0.625826\pi$
$911$	−23.2455	−0.770158	−0.385079	+	0.922884i	$0.625826\pi$
$912$	0	0
$913$	45.5703	1.50816
$914$	0	0
$915$	−1.90426	−0.0629530
$916$	0	0
$917$	−9.94594	−0.328444
$918$	0	0
$919$	36.4849	1.20353	0.601763	−	0.798675i	$-0.294465\pi$
$919$	36.4849	1.20353	0.601763	+	0.798675i	$0.294465\pi$
$920$	0	0
$921$	−11.3295	−0.373318
$922$	0	0
$923$	0	0
$924$	0	0
$925$	20.8149	0.684391
$926$	0	0
$927$	15.1629	0.498014
$928$	0	0
$929$	39.7444	1.30397	0.651986	−	0.758231i	$-0.273936\pi$
$929$	39.7444	1.30397	0.651986	+	0.758231i	$0.273936\pi$
$930$	0	0
$931$	5.06579	0.166025
$932$	0	0
$933$	−0.578412	−0.0189364
$934$	0	0
$935$	−19.7627	−0.646308
$936$	0	0
$937$	−25.2520	−0.824946	−0.412473	−	0.910970i	$-0.635335\pi$
$937$	−25.2520	−0.824946	−0.412473	+	0.910970i	$0.635335\pi$
$938$	0	0
$939$	−11.6811	−0.381199
$940$	0	0
$941$	27.9746	0.911945	0.455973	−	0.889994i	$-0.349292\pi$
$941$	27.9746	0.911945	0.455973	+	0.889994i	$0.349292\pi$
$942$	0	0
$943$	−1.52509	−0.0496639
$944$	0	0
$945$	−2.69862	−0.0877860
$946$	0	0
$947$	5.19982	0.168971	0.0844857	−	0.996425i	$-0.473075\pi$
$947$	5.19982	0.168971	0.0844857	+	0.996425i	$0.473075\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−7.64541	−0.247919
$952$	0	0
$953$	−39.0886	−1.26620	−0.633102	−	0.774069i	$-0.718218\pi$
$953$	−39.0886	−1.26620	−0.633102	+	0.774069i	$0.718218\pi$
$954$	0	0
$955$	20.8920	0.676048
$956$	0	0
$957$	−3.88731	−0.125659
$958$	0	0
$959$	3.64127	0.117583
$960$	0	0
$961$	−28.3770	−0.915386
$962$	0	0
$963$	−19.6556	−0.633392
$964$	0	0
$965$	−31.4481	−1.01235
$966$	0	0
$967$	43.4306	1.39663	0.698317	−	0.715789i	$-0.253933\pi$
$967$	43.4306	1.39663	0.698317	+	0.715789i	$0.253933\pi$
$968$	0	0
$969$	−10.1608	−0.326412
$970$	0	0
$971$	18.1168	0.581395	0.290698	−	0.956815i	$-0.406113\pi$
$971$	18.1168	0.581395	0.290698	+	0.956815i	$0.406113\pi$
$972$	0	0
$973$	6.40454	0.205320
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.6622	1.39688	0.698439	−	0.715669i	$-0.253878\pi$
$977$	43.6622	1.39688	0.698439	+	0.715669i	$0.253878\pi$
$978$	0	0
$979$	−34.3321	−1.09726
$980$	0	0
$981$	19.4840	0.622076
$982$	0	0
$983$	−58.7257	−1.87306	−0.936530	−	0.350589i	$-0.885982\pi$
$983$	−58.7257	−1.87306	−0.936530	+	0.350589i	$0.885982\pi$
$984$	0	0
$985$	−25.0743	−0.798935
$986$	0	0
$987$	−0.778786	−0.0247890
$988$	0	0
$989$	4.92313	0.156547
$990$	0	0
$991$	28.4382	0.903369	0.451684	−	0.892178i	$-0.350823\pi$
$991$	28.4382	0.903369	0.451684	+	0.892178i	$0.350823\pi$
$992$	0	0
$993$	6.70680	0.212834
$994$	0	0
$995$	−20.1833	−0.639853
$996$	0	0
$997$	36.8258	1.16629	0.583143	−	0.812370i	$-0.301823\pi$
$997$	36.8258	1.16629	0.583143	+	0.812370i	$0.301823\pi$
$998$	0	0
$999$	−13.0532	−0.412986

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.bs.1.9	yes	15
13.12	even	2		9464.2.a.br.1.9	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.9	✓	15	13.12	even	2
9464.2.a.bs.1.9	yes	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q-0.354884 q^{3} -1.29454 q^{5} +1.00000 q^{7} -2.87406 q^{9} +2.70106 q^{11} +0.459413 q^{15} +5.65190 q^{17} +5.06579 q^{19} -0.354884 q^{21} +5.87053 q^{23} -3.32416 q^{25} +2.08461 q^{27} +4.05535 q^{29} +1.61958 q^{31} -0.958562 q^{33} -1.29454 q^{35} -6.26172 q^{37} -0.259788 q^{41} +0.838619 q^{43} +3.72059 q^{45} +2.19448 q^{47} +1.00000 q^{49} -2.00577 q^{51} -2.42435 q^{53} -3.49664 q^{55} -1.79777 q^{57} -1.49591 q^{59} -4.14499 q^{61} -2.87406 q^{63} -8.01117 q^{67} -2.08336 q^{69} +6.10633 q^{71} +1.42716 q^{73} +1.17969 q^{75} +2.70106 q^{77} +3.64099 q^{79} +7.88238 q^{81} +16.8713 q^{83} -7.31664 q^{85} -1.43918 q^{87} -12.7106 q^{89} -0.574763 q^{93} -6.55789 q^{95} -8.46418 q^{97} -7.76300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} + 4 q^{5} + 15 q^{7} + 16 q^{9} + 15 q^{11} + 8 q^{15} + 2 q^{17} + 13 q^{19} - 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} + 19 q^{31} - 24 q^{33} + 4 q^{35} - 2 q^{37} + 30 q^{41}+ \cdots + 70 q^{99}+O(q^{100}) \) 15 * q - 3 * q^3 + 4 * q^5 + 15 * q^7 + 16 * q^9 + 15 * q^11 + 8 * q^15 + 2 * q^17 + 13 * q^19 - 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 + 19 * q^31 - 24 * q^33 + 4 * q^35 - 2 * q^37 + 30 * q^41 + 15 * q^43 - 2 * q^45 + 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 - 2 * q^57 + 22 * q^59 - q^61 + 16 * q^63 + 29 * q^67 + 32 * q^69 + 20 * q^71 + 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 + 23 * q^83 + 8 * q^85 - 22 * q^87 + 9 * q^89 + 10 * q^93 + 28 * q^95 + 11 * q^97 + 70 * q^99

Introduction

L-functions

Modular forms

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Database

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