LMFDB - Embedded newform 9280.2.a.cl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9280,2,Mod(1,9280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.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:	$74.1011730757$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.580484.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - 3x^{2} + 8x + 3 $ x^5 - 7x^3 - 3x^2 + 8*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.60873$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.857770 q^{3} -1.00000 q^{5} -2.23769 q^{7} -2.26423 q^{9} -3.21746 q^{11} -3.25127 q^{13} -0.857770 q^{15} +0.870730 q^{17} +5.02654 q^{19} -1.91943 q^{21} -8.46873 q^{23} +1.00000 q^{25} -4.51550 q^{27} -1.00000 q^{29} -0.0467661 q^{31} -2.75985 q^{33} +2.23769 q^{35} -11.1129 q^{37} -2.78884 q^{39} -2.47539 q^{41} +2.87073 q^{43} +2.26423 q^{45} -2.20257 q^{47} -1.99273 q^{49} +0.746886 q^{51} +11.7668 q^{53} +3.21746 q^{55} +4.31161 q^{57} -7.68620 q^{59} +4.99273 q^{61} +5.06665 q^{63} +3.25127 q^{65} +11.6928 q^{67} -7.26423 q^{69} +3.43108 q^{71} -10.1331 q^{73} +0.857770 q^{75} +7.19970 q^{77} -6.44158 q^{79} +2.91943 q^{81} +15.4615 q^{83} -0.870730 q^{85} -0.857770 q^{87} +16.7725 q^{89} +7.27535 q^{91} -0.0401146 q^{93} -5.02654 q^{95} -7.16939 q^{97} +7.28508 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 5 q^{5} - 7 q^{7} + 8 q^{9} + 10 q^{11} - q^{13} - 3 q^{15} - q^{17} + 10 q^{19} + 8 q^{21} - q^{23} + 5 q^{25} + 12 q^{27} - 5 q^{29} - 7 q^{31} - 8 q^{33} + 7 q^{35} + 8 q^{37} - 3 q^{39}+ \cdots + 32 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 5 * q^5 - 7 * q^7 + 8 * q^9 + 10 * q^11 - q^13 - 3 * q^15 - q^17 + 10 * q^19 + 8 * q^21 - q^23 + 5 * q^25 + 12 * q^27 - 5 * q^29 - 7 * q^31 - 8 * q^33 + 7 * q^35 + 8 * q^37 - 3 * q^39 - 4 * q^41 + 9 * q^43 - 8 * q^45 - 8 * q^47 + 16 * q^49 + 2 * q^51 + 9 * q^53 - 10 * q^55 + 2 * q^57 + 29 * q^59 - q^61 - 16 * q^63 + q^65 + 24 * q^67 - 17 * q^69 + 12 * q^71 - 9 * q^73 + 3 * q^75 - 2 * q^77 - 13 * q^79 - 3 * q^81 + 10 * q^83 + q^85 - 3 * q^87 + 4 * q^89 - 4 * q^91 + 26 * q^93 - 10 * q^95 - 15 * q^97 + 32 * 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
$3$	0.857770	0.495234	0.247617	−	0.968858i	$-0.420353\pi$
$3$	0.857770	0.495234	0.247617	+	0.968858i	$0.420353\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.23769	−0.845768	−0.422884	−	0.906184i	$-0.638982\pi$
$7$	−2.23769	−0.845768	−0.422884	+	0.906184i	$0.638982\pi$
$8$	0	0
$9$	−2.26423	−0.754743
$10$	0	0
$11$	−3.21746	−0.970102	−0.485051	−	0.874486i	$-0.661199\pi$
$11$	−3.21746	−0.970102	−0.485051	+	0.874486i	$0.661199\pi$
$12$	0	0
$13$	−3.25127	−0.901740	−0.450870	−	0.892590i	$-0.648886\pi$
$13$	−3.25127	−0.901740	−0.450870	+	0.892590i	$0.648886\pi$
$14$	0	0
$15$	−0.857770	−0.221475
$16$	0	0
$17$	0.870730	0.211183	0.105591	−	0.994410i	$-0.466326\pi$
$17$	0.870730	0.211183	0.105591	+	0.994410i	$0.466326\pi$
$18$	0	0
$19$	5.02654	1.15317	0.576583	−	0.817038i	$-0.304386\pi$
$19$	5.02654	1.15317	0.576583	+	0.817038i	$0.304386\pi$
$20$	0	0
$21$	−1.91943	−0.418853
$22$	0	0
$23$	−8.46873	−1.76585	−0.882927	−	0.469511i	$-0.844430\pi$
$23$	−8.46873	−1.76585	−0.882927	+	0.469511i	$0.844430\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.51550	−0.869009
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.0467661	−0.00839945	−0.00419972	−	0.999991i	$-0.501337\pi$
$31$	−0.0467661	−0.00839945	−0.00419972	+	0.999991i	$0.501337\pi$
$32$	0	0
$33$	−2.75985	−0.480427
$34$	0	0
$35$	2.23769	0.378239
$36$	0	0
$37$	−11.1129	−1.82695	−0.913474	−	0.406898i	$-0.866611\pi$
$37$	−11.1129	−1.82695	−0.913474	+	0.406898i	$0.866611\pi$
$38$	0	0
$39$	−2.78884	−0.446572
$40$	0	0
$41$	−2.47539	−0.386590	−0.193295	−	0.981141i	$-0.561917\pi$
$41$	−2.47539	−0.386590	−0.193295	+	0.981141i	$0.561917\pi$
$42$	0	0
$43$	2.87073	0.437782	0.218891	−	0.975749i	$-0.429756\pi$
$43$	2.87073	0.437782	0.218891	+	0.975749i	$0.429756\pi$
$44$	0	0
$45$	2.26423	0.337531
$46$	0	0
$47$	−2.20257	−0.321278	−0.160639	−	0.987013i	$-0.551356\pi$
$47$	−2.20257	−0.321278	−0.160639	+	0.987013i	$0.551356\pi$
$48$	0	0
$49$	−1.99273	−0.284676
$50$	0	0
$51$	0.746886	0.104585
$52$	0	0
$53$	11.7668	1.61629	0.808145	−	0.588983i	$-0.200472\pi$
$53$	11.7668	1.61629	0.808145	+	0.588983i	$0.200472\pi$
$54$	0	0
$55$	3.21746	0.433843
$56$	0	0
$57$	4.31161	0.571087
$58$	0	0
$59$	−7.68620	−1.00066	−0.500329	−	0.865835i	$-0.666788\pi$
$59$	−7.68620	−1.00066	−0.500329	+	0.865835i	$0.666788\pi$
$60$	0	0
$61$	4.99273	0.639254	0.319627	−	0.947544i	$-0.396442\pi$
$61$	4.99273	0.639254	0.319627	+	0.947544i	$0.396442\pi$
$62$	0	0
$63$	5.06665	0.638338
$64$	0	0
$65$	3.25127	0.403271
$66$	0	0
$67$	11.6928	1.42851	0.714254	−	0.699886i	$-0.246766\pi$
$67$	11.6928	1.42851	0.714254	+	0.699886i	$0.246766\pi$
$68$	0	0
$69$	−7.26423	−0.874511
$70$	0	0
$71$	3.43108	0.407194	0.203597	−	0.979055i	$-0.434737\pi$
$71$	3.43108	0.407194	0.203597	+	0.979055i	$0.434737\pi$
$72$	0	0
$73$	−10.1331	−1.18599	−0.592996	−	0.805206i	$-0.702055\pi$
$73$	−10.1331	−1.18599	−0.592996	+	0.805206i	$0.702055\pi$
$74$	0	0
$75$	0.857770	0.0990468
$76$	0	0
$77$	7.19970	0.820482
$78$	0	0
$79$	−6.44158	−0.724734	−0.362367	−	0.932035i	$-0.618031\pi$
$79$	−6.44158	−0.724734	−0.362367	+	0.932035i	$0.618031\pi$
$80$	0	0
$81$	2.91943	0.324381
$82$	0	0
$83$	15.4615	1.69712	0.848558	−	0.529102i	$-0.177471\pi$
$83$	15.4615	1.69712	0.848558	+	0.529102i	$0.177471\pi$
$84$	0	0
$85$	−0.870730	−0.0944439
$86$	0	0
$87$	−0.857770	−0.0919626
$88$	0	0
$89$	16.7725	1.77788	0.888939	−	0.458026i	$-0.151443\pi$
$89$	16.7725	1.77788	0.888939	+	0.458026i	$0.151443\pi$
$90$	0	0
$91$	7.27535	0.762663
$92$	0	0
$93$	−0.0401146	−0.00415969
$94$	0	0
$95$	−5.02654	−0.515712
$96$	0	0
$97$	−7.16939	−0.727941	−0.363970	−	0.931411i	$-0.618579\pi$
$97$	−7.16939	−0.727941	−0.363970	+	0.931411i	$0.618579\pi$
$98$	0	0
$99$	7.28508	0.732178
$100$	0	0
$101$	8.87328	0.882924	0.441462	−	0.897280i	$-0.354460\pi$
$101$	8.87328	0.882924	0.441462	+	0.897280i	$0.354460\pi$
$102$	0	0
$103$	−6.74208	−0.664317	−0.332158	−	0.943224i	$-0.607777\pi$
$103$	−6.74208	−0.664317	−0.332158	+	0.943224i	$0.607777\pi$
$104$	0	0
$105$	1.91943	0.187317
$106$	0	0
$107$	−12.0304	−1.16302	−0.581511	−	0.813539i	$-0.697538\pi$
$107$	−12.0304	−1.16302	−0.581511	+	0.813539i	$0.697538\pi$
$108$	0	0
$109$	1.74270	0.166920	0.0834600	−	0.996511i	$-0.473403\pi$
$109$	1.74270	0.166920	0.0834600	+	0.996511i	$0.473403\pi$
$110$	0	0
$111$	−9.53231	−0.904766
$112$	0	0
$113$	−19.4393	−1.82870	−0.914348	−	0.404929i	$-0.867296\pi$
$113$	−19.4393	−1.82870	−0.914348	+	0.404929i	$0.867296\pi$
$114$	0	0
$115$	8.46873	0.789714
$116$	0	0
$117$	7.36162	0.680582
$118$	0	0
$119$	−1.94843	−0.178612
$120$	0	0
$121$	−0.647927	−0.0589024
$122$	0	0
$123$	−2.12331	−0.191453
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.149500	−0.0132660	−0.00663298	−	0.999978i	$-0.502111\pi$
$127$	−0.149500	−0.0132660	−0.00663298	+	0.999978i	$0.502111\pi$
$128$	0	0
$129$	2.46243	0.216805
$130$	0	0
$131$	−10.2889	−0.898947	−0.449474	−	0.893294i	$-0.648388\pi$
$131$	−10.2889	−0.898947	−0.449474	+	0.893294i	$0.648388\pi$
$132$	0	0
$133$	−11.2478	−0.975312
$134$	0	0
$135$	4.51550	0.388632
$136$	0	0
$137$	−0.898229	−0.0767409	−0.0383704	−	0.999264i	$-0.512217\pi$
$137$	−0.898229	−0.0767409	−0.0383704	+	0.999264i	$0.512217\pi$
$138$	0	0
$139$	19.8642	1.68486	0.842428	−	0.538808i	$-0.181125\pi$
$139$	19.8642	1.68486	0.842428	+	0.538808i	$0.181125\pi$
$140$	0	0
$141$	−1.88930	−0.159108
$142$	0	0
$143$	10.4608	0.874780
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	−1.70930	−0.140981
$148$	0	0
$149$	−9.45331	−0.774446	−0.387223	−	0.921986i	$-0.626566\pi$
$149$	−9.45331	−0.774446	−0.387223	+	0.921986i	$0.626566\pi$
$150$	0	0
$151$	10.5025	0.854685	0.427342	−	0.904090i	$-0.359450\pi$
$151$	10.5025	0.854685	0.427342	+	0.904090i	$0.359450\pi$
$152$	0	0
$153$	−1.97153	−0.159389
$154$	0	0
$155$	0.0467661	0.00375635
$156$	0	0
$157$	14.2324	1.13587	0.567933	−	0.823075i	$-0.307743\pi$
$157$	14.2324	1.13587	0.567933	+	0.823075i	$0.307743\pi$
$158$	0	0
$159$	10.0932	0.800442
$160$	0	0
$161$	18.9504	1.49350
$162$	0	0
$163$	−13.4245	−1.05149	−0.525744	−	0.850643i	$-0.676213\pi$
$163$	−13.4245	−1.05149	−0.525744	+	0.850643i	$0.676213\pi$
$164$	0	0
$165$	2.75985	0.214854
$166$	0	0
$167$	2.49500	0.193069	0.0965343	−	0.995330i	$-0.469224\pi$
$167$	2.49500	0.193069	0.0965343	+	0.995330i	$0.469224\pi$
$168$	0	0
$169$	−2.42924	−0.186864
$170$	0	0
$171$	−11.3812	−0.870345
$172$	0	0
$173$	−3.27350	−0.248880	−0.124440	−	0.992227i	$-0.539713\pi$
$173$	−3.27350	−0.248880	−0.124440	+	0.992227i	$0.539713\pi$
$174$	0	0
$175$	−2.23769	−0.169154
$176$	0	0
$177$	−6.59299	−0.495560
$178$	0	0
$179$	23.6862	1.77039	0.885195	−	0.465221i	$-0.154025\pi$
$179$	23.6862	1.77039	0.885195	+	0.465221i	$0.154025\pi$
$180$	0	0
$181$	19.1815	1.42575	0.712874	−	0.701292i	$-0.247393\pi$
$181$	19.1815	1.42575	0.712874	+	0.701292i	$0.247393\pi$
$182$	0	0
$183$	4.28262	0.316580
$184$	0	0
$185$	11.1129	0.817036
$186$	0	0
$187$	−2.80154	−0.204869
$188$	0	0
$189$	10.1043	0.734980
$190$	0	0
$191$	8.04677	0.582244	0.291122	−	0.956686i	$-0.405972\pi$
$191$	8.04677	0.582244	0.291122	+	0.956686i	$0.405972\pi$
$192$	0	0
$193$	−12.0525	−0.867561	−0.433781	−	0.901019i	$-0.642821\pi$
$193$	−12.0525	−0.867561	−0.433781	+	0.901019i	$0.642821\pi$
$194$	0	0
$195$	2.78884	0.199713
$196$	0	0
$197$	17.2509	1.22908	0.614539	−	0.788887i	$-0.289342\pi$
$197$	17.2509	1.22908	0.614539	+	0.788887i	$0.289342\pi$
$198$	0	0
$199$	3.09738	0.219567	0.109784	−	0.993956i	$-0.464984\pi$
$199$	3.09738	0.219567	0.109784	+	0.993956i	$0.464984\pi$
$200$	0	0
$201$	10.0298	0.707446
$202$	0	0
$203$	2.23769	0.157055
$204$	0	0
$205$	2.47539	0.172888
$206$	0	0
$207$	19.1752	1.33277
$208$	0	0
$209$	−16.1727	−1.11869
$210$	0	0
$211$	−9.65239	−0.664498	−0.332249	−	0.943192i	$-0.607807\pi$
$211$	−9.65239	−0.664498	−0.332249	+	0.943192i	$0.607807\pi$
$212$	0	0
$213$	2.94308	0.201657
$214$	0	0
$215$	−2.87073	−0.195782
$216$	0	0
$217$	0.104648	0.00710399
$218$	0	0
$219$	−8.69189	−0.587343
$220$	0	0
$221$	−2.83098	−0.190432
$222$	0	0
$223$	−3.89982	−0.261151	−0.130576	−	0.991438i	$-0.541682\pi$
$223$	−3.89982	−0.261151	−0.130576	+	0.991438i	$0.541682\pi$
$224$	0	0
$225$	−2.26423	−0.150949
$226$	0	0
$227$	27.9458	1.85483	0.927413	−	0.374038i	$-0.122027\pi$
$227$	27.9458	1.85483	0.927413	+	0.374038i	$0.122027\pi$
$228$	0	0
$229$	−19.5354	−1.29093	−0.645467	−	0.763788i	$-0.723337\pi$
$229$	−19.5354	−1.29093	−0.645467	+	0.763788i	$0.723337\pi$
$230$	0	0
$231$	6.17569	0.406330
$232$	0	0
$233$	−9.61815	−0.630106	−0.315053	−	0.949074i	$-0.602022\pi$
$233$	−9.61815	−0.630106	−0.315053	+	0.949074i	$0.602022\pi$
$234$	0	0
$235$	2.20257	0.143680
$236$	0	0
$237$	−5.52540	−0.358913
$238$	0	0
$239$	15.8932	1.02804	0.514022	−	0.857777i	$-0.328155\pi$
$239$	15.8932	1.02804	0.514022	+	0.857777i	$0.328155\pi$
$240$	0	0
$241$	16.1085	1.03764	0.518820	−	0.854883i	$-0.326371\pi$
$241$	16.1085	1.03764	0.518820	+	0.854883i	$0.326371\pi$
$242$	0	0
$243$	16.0507	1.02965
$244$	0	0
$245$	1.99273	0.127311
$246$	0	0
$247$	−16.3426	−1.03986
$248$	0	0
$249$	13.2624	0.840470
$250$	0	0
$251$	13.6783	0.863367	0.431684	−	0.902025i	$-0.357920\pi$
$251$	13.6783	0.863367	0.431684	+	0.902025i	$0.357920\pi$
$252$	0	0
$253$	27.2478	1.71306
$254$	0	0
$255$	−0.746886	−0.0467718
$256$	0	0
$257$	−10.4754	−0.653437	−0.326718	−	0.945122i	$-0.605943\pi$
$257$	−10.4754	−0.653437	−0.326718	+	0.945122i	$0.605943\pi$
$258$	0	0
$259$	24.8672	1.54517
$260$	0	0
$261$	2.26423	0.140152
$262$	0	0
$263$	11.0414	0.680843	0.340421	−	0.940273i	$-0.389430\pi$
$263$	11.0414	0.680843	0.340421	+	0.940273i	$0.389430\pi$
$264$	0	0
$265$	−11.7668	−0.722827
$266$	0	0
$267$	14.3869	0.880465
$268$	0	0
$269$	11.5228	0.702556	0.351278	−	0.936271i	$-0.385747\pi$
$269$	11.5228	0.702556	0.351278	+	0.936271i	$0.385747\pi$
$270$	0	0
$271$	23.7757	1.44427	0.722136	−	0.691752i	$-0.243161\pi$
$271$	23.7757	1.44427	0.722136	+	0.691752i	$0.243161\pi$
$272$	0	0
$273$	6.24058	0.377697
$274$	0	0
$275$	−3.21746	−0.194020
$276$	0	0
$277$	−10.6972	−0.642730	−0.321365	−	0.946955i	$-0.604142\pi$
$277$	−10.6972	−0.642730	−0.321365	+	0.946955i	$0.604142\pi$
$278$	0	0
$279$	0.105889	0.00633943
$280$	0	0
$281$	−12.7435	−0.760211	−0.380106	−	0.924943i	$-0.624112\pi$
$281$	−12.7435	−0.760211	−0.380106	+	0.924943i	$0.624112\pi$
$282$	0	0
$283$	−0.187700	−0.0111576	−0.00557879	−	0.999984i	$-0.501776\pi$
$283$	−0.187700	−0.0111576	−0.00557879	+	0.999984i	$0.501776\pi$
$284$	0	0
$285$	−4.31161	−0.255398
$286$	0	0
$287$	5.53915	0.326966
$288$	0	0
$289$	−16.2418	−0.955402
$290$	0	0
$291$	−6.14969	−0.360501
$292$	0	0
$293$	21.8468	1.27630	0.638152	−	0.769910i	$-0.279699\pi$
$293$	21.8468	1.27630	0.638152	+	0.769910i	$0.279699\pi$
$294$	0	0
$295$	7.68620	0.447508
$296$	0	0
$297$	14.5285	0.843027
$298$	0	0
$299$	27.5341	1.59234
$300$	0	0
$301$	−6.42381	−0.370262
$302$	0	0
$303$	7.61124	0.437254
$304$	0	0
$305$	−4.99273	−0.285883
$306$	0	0
$307$	−17.8064	−1.01626	−0.508131	−	0.861280i	$-0.669663\pi$
$307$	−17.8064	−1.01626	−0.508131	+	0.861280i	$0.669663\pi$
$308$	0	0
$309$	−5.78315	−0.328992
$310$	0	0
$311$	−1.73112	−0.0981629	−0.0490814	−	0.998795i	$-0.515629\pi$
$311$	−1.73112	−0.0981629	−0.0490814	+	0.998795i	$0.515629\pi$
$312$	0	0
$313$	26.0089	1.47011	0.735056	−	0.678007i	$-0.237156\pi$
$313$	26.0089	1.47011	0.735056	+	0.678007i	$0.237156\pi$
$314$	0	0
$315$	−5.06665	−0.285473
$316$	0	0
$317$	27.5517	1.54746	0.773728	−	0.633518i	$-0.218390\pi$
$317$	27.5517	1.54746	0.773728	+	0.633518i	$0.218390\pi$
$318$	0	0
$319$	3.21746	0.180143
$320$	0	0
$321$	−10.3193	−0.575968
$322$	0	0
$323$	4.37675	0.243529
$324$	0	0
$325$	−3.25127	−0.180348
$326$	0	0
$327$	1.49483	0.0826645
$328$	0	0
$329$	4.92868	0.271727
$330$	0	0
$331$	−24.2744	−1.33424	−0.667120	−	0.744950i	$-0.732473\pi$
$331$	−24.2744	−1.33424	−0.667120	+	0.744950i	$0.732473\pi$
$332$	0	0
$333$	25.1621	1.37888
$334$	0	0
$335$	−11.6928	−0.638849
$336$	0	0
$337$	9.56420	0.520995	0.260498	−	0.965475i	$-0.416113\pi$
$337$	9.56420	0.520995	0.260498	+	0.965475i	$0.416113\pi$
$338$	0	0
$339$	−16.6745	−0.905632
$340$	0	0
$341$	0.150468	0.00814832
$342$	0	0
$343$	20.1230	1.08654
$344$	0	0
$345$	7.26423	0.391093
$346$	0	0
$347$	−26.6291	−1.42952	−0.714762	−	0.699368i	$-0.753465\pi$
$347$	−26.6291	−1.42952	−0.714762	+	0.699368i	$0.753465\pi$
$348$	0	0
$349$	−17.3242	−0.927346	−0.463673	−	0.886006i	$-0.653469\pi$
$349$	−17.3242	−0.927346	−0.463673	+	0.886006i	$0.653469\pi$
$350$	0	0
$351$	14.6811	0.783620
$352$	0	0
$353$	5.24785	0.279315	0.139657	−	0.990200i	$-0.455400\pi$
$353$	5.24785	0.279315	0.139657	+	0.990200i	$0.455400\pi$
$354$	0	0
$355$	−3.43108	−0.182103
$356$	0	0
$357$	−1.67130	−0.0884547
$358$	0	0
$359$	11.6337	0.614005	0.307002	−	0.951709i	$-0.400674\pi$
$359$	11.6337	0.614005	0.307002	+	0.951709i	$0.400674\pi$
$360$	0	0
$361$	6.26607	0.329793
$362$	0	0
$363$	−0.555772	−0.0291705
$364$	0	0
$365$	10.1331	0.530392
$366$	0	0
$367$	−15.1090	−0.788685	−0.394342	−	0.918964i	$-0.629028\pi$
$367$	−15.1090	−0.788685	−0.394342	+	0.918964i	$0.629028\pi$
$368$	0	0
$369$	5.60484	0.291776
$370$	0	0
$371$	−26.3304	−1.36701
$372$	0	0
$373$	0.762926	0.0395028	0.0197514	−	0.999805i	$-0.493713\pi$
$373$	0.762926	0.0395028	0.0197514	+	0.999805i	$0.493713\pi$
$374$	0	0
$375$	−0.857770	−0.0442951
$376$	0	0
$377$	3.25127	0.167449
$378$	0	0
$379$	23.3003	1.19686	0.598428	−	0.801176i	$-0.295792\pi$
$379$	23.3003	1.19686	0.598428	+	0.801176i	$0.295792\pi$
$380$	0	0
$381$	−0.128236	−0.00656975
$382$	0	0
$383$	13.4115	0.685294	0.342647	−	0.939464i	$-0.388676\pi$
$383$	13.4115	0.685294	0.342647	+	0.939464i	$0.388676\pi$
$384$	0	0
$385$	−7.19970	−0.366930
$386$	0	0
$387$	−6.49999	−0.330413
$388$	0	0
$389$	−24.3540	−1.23480	−0.617399	−	0.786650i	$-0.711813\pi$
$389$	−24.3540	−1.23480	−0.617399	+	0.786650i	$0.711813\pi$
$390$	0	0
$391$	−7.37398	−0.372918
$392$	0	0
$393$	−8.82553	−0.445189
$394$	0	0
$395$	6.44158	0.324111
$396$	0	0
$397$	−22.0665	−1.10748	−0.553742	−	0.832688i	$-0.686801\pi$
$397$	−22.0665	−1.10748	−0.553742	+	0.832688i	$0.686801\pi$
$398$	0	0
$399$	−9.64807	−0.483008
$400$	0	0
$401$	1.50981	0.0753964	0.0376982	−	0.999289i	$-0.487997\pi$
$401$	1.50981	0.0753964	0.0376982	+	0.999289i	$0.487997\pi$
$402$	0	0
$403$	0.152049	0.00757412
$404$	0	0
$405$	−2.91943	−0.145067
$406$	0	0
$407$	35.7553	1.77232
$408$	0	0
$409$	−5.81676	−0.287621	−0.143810	−	0.989605i	$-0.545935\pi$
$409$	−5.81676	−0.287621	−0.143810	+	0.989605i	$0.545935\pi$
$410$	0	0
$411$	−0.770474	−0.0380047
$412$	0	0
$413$	17.1994	0.846325
$414$	0	0
$415$	−15.4615	−0.758973
$416$	0	0
$417$	17.0389	0.834398
$418$	0	0
$419$	−33.1082	−1.61744	−0.808719	−	0.588195i	$-0.799839\pi$
$419$	−33.1082	−1.61744	−0.808719	+	0.588195i	$0.799839\pi$
$420$	0	0
$421$	−37.7099	−1.83787	−0.918935	−	0.394410i	$-0.870949\pi$
$421$	−37.7099	−1.83787	−0.918935	+	0.394410i	$0.870949\pi$
$422$	0	0
$423$	4.98713	0.242483
$424$	0	0
$425$	0.870730	0.0422366
$426$	0	0
$427$	−11.1722	−0.540661
$428$	0	0
$429$	8.97300	0.433221
$430$	0	0
$431$	−9.55177	−0.460093	−0.230046	−	0.973180i	$-0.573888\pi$
$431$	−9.55177	−0.460093	−0.230046	+	0.973180i	$0.573888\pi$
$432$	0	0
$433$	8.42066	0.404671	0.202335	−	0.979316i	$-0.435147\pi$
$433$	8.42066	0.404671	0.202335	+	0.979316i	$0.435147\pi$
$434$	0	0
$435$	0.857770	0.0411269
$436$	0	0
$437$	−42.5684	−2.03632
$438$	0	0
$439$	−2.59607	−0.123904	−0.0619519	−	0.998079i	$-0.519733\pi$
$439$	−2.59607	−0.123904	−0.0619519	+	0.998079i	$0.519733\pi$
$440$	0	0
$441$	4.51200	0.214857
$442$	0	0
$443$	18.3331	0.871034	0.435517	−	0.900181i	$-0.356566\pi$
$443$	18.3331	0.871034	0.435517	+	0.900181i	$0.356566\pi$
$444$	0	0
$445$	−16.7725	−0.795091
$446$	0	0
$447$	−8.10877	−0.383532
$448$	0	0
$449$	−20.6567	−0.974850	−0.487425	−	0.873165i	$-0.662064\pi$
$449$	−20.6567	−0.974850	−0.487425	+	0.873165i	$0.662064\pi$
$450$	0	0
$451$	7.96446	0.375032
$452$	0	0
$453$	9.00877	0.423269
$454$	0	0
$455$	−7.27535	−0.341073
$456$	0	0
$457$	21.6038	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$457$	21.6038	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$458$	0	0
$459$	−3.93178	−0.183520
$460$	0	0
$461$	10.7447	0.500431	0.250215	−	0.968190i	$-0.419499\pi$
$461$	10.7447	0.500431	0.250215	+	0.968190i	$0.419499\pi$
$462$	0	0
$463$	23.9652	1.11376	0.556880	−	0.830593i	$-0.311998\pi$
$463$	23.9652	1.11376	0.556880	+	0.830593i	$0.311998\pi$
$464$	0	0
$465$	0.0401146	0.00186027
$466$	0	0
$467$	6.17357	0.285679	0.142839	−	0.989746i	$-0.454377\pi$
$467$	6.17357	0.285679	0.142839	+	0.989746i	$0.454377\pi$
$468$	0	0
$469$	−26.1650	−1.20819
$470$	0	0
$471$	12.2081	0.562519
$472$	0	0
$473$	−9.23647	−0.424693
$474$	0	0
$475$	5.02654	0.230633
$476$	0	0
$477$	−26.6427	−1.21988
$478$	0	0
$479$	22.2182	1.01518	0.507588	−	0.861600i	$-0.330537\pi$
$479$	22.2182	1.01518	0.507588	+	0.861600i	$0.330537\pi$
$480$	0	0
$481$	36.1310	1.64743
$482$	0	0
$483$	16.2551	0.739633
$484$	0	0
$485$	7.16939	0.325545
$486$	0	0
$487$	−1.58434	−0.0717933	−0.0358966	−	0.999356i	$-0.511429\pi$
$487$	−1.58434	−0.0717933	−0.0358966	+	0.999356i	$0.511429\pi$
$488$	0	0
$489$	−11.5151	−0.520733
$490$	0	0
$491$	19.6119	0.885074	0.442537	−	0.896750i	$-0.354079\pi$
$491$	19.6119	0.885074	0.442537	+	0.896750i	$0.354079\pi$
$492$	0	0
$493$	−0.870730	−0.0392157
$494$	0	0
$495$	−7.28508	−0.327440
$496$	0	0
$497$	−7.67771	−0.344392
$498$	0	0
$499$	16.1490	0.722927	0.361464	−	0.932386i	$-0.382277\pi$
$499$	16.1490	0.722927	0.361464	+	0.932386i	$0.382277\pi$
$500$	0	0
$501$	2.14013	0.0956142
$502$	0	0
$503$	38.1824	1.70247	0.851235	−	0.524785i	$-0.175854\pi$
$503$	38.1824	1.70247	0.851235	+	0.524785i	$0.175854\pi$
$504$	0	0
$505$	−8.87328	−0.394856
$506$	0	0
$507$	−2.08373	−0.0925416
$508$	0	0
$509$	−42.4255	−1.88048	−0.940238	−	0.340517i	$-0.889398\pi$
$509$	−42.4255	−1.88048	−0.940238	+	0.340517i	$0.889398\pi$
$510$	0	0
$511$	22.6748	1.00307
$512$	0	0
$513$	−22.6973	−1.00211
$514$	0	0
$515$	6.74208	0.297091
$516$	0	0
$517$	7.08670	0.311673
$518$	0	0
$519$	−2.80791	−0.123254
$520$	0	0
$521$	−8.30977	−0.364058	−0.182029	−	0.983293i	$-0.558266\pi$
$521$	−8.30977	−0.364058	−0.182029	+	0.983293i	$0.558266\pi$
$522$	0	0
$523$	−15.7890	−0.690405	−0.345203	−	0.938528i	$-0.612190\pi$
$523$	−15.7890	−0.690405	−0.345203	+	0.938528i	$0.612190\pi$
$524$	0	0
$525$	−1.91943	−0.0837707
$526$	0	0
$527$	−0.0407207	−0.00177382
$528$	0	0
$529$	48.7195	2.11824
$530$	0	0
$531$	17.4033	0.755240
$532$	0	0
$533$	8.04815	0.348604
$534$	0	0
$535$	12.0304	0.520119
$536$	0	0
$537$	20.3173	0.876757
$538$	0	0
$539$	6.41154	0.276164
$540$	0	0
$541$	−37.6007	−1.61658	−0.808290	−	0.588784i	$-0.799607\pi$
$541$	−37.6007	−1.61658	−0.808290	+	0.588784i	$0.799607\pi$
$542$	0	0
$543$	16.4533	0.706079
$544$	0	0
$545$	−1.74270	−0.0746489
$546$	0	0
$547$	−7.46146	−0.319029	−0.159515	−	0.987196i	$-0.550993\pi$
$547$	−7.46146	−0.319029	−0.159515	+	0.987196i	$0.550993\pi$
$548$	0	0
$549$	−11.3047	−0.482472
$550$	0	0
$551$	−5.02654	−0.214138
$552$	0	0
$553$	14.4143	0.612957
$554$	0	0
$555$	9.53231	0.404624
$556$	0	0
$557$	−36.7286	−1.55624	−0.778120	−	0.628116i	$-0.783826\pi$
$557$	−36.7286	−1.55624	−0.778120	+	0.628116i	$0.783826\pi$
$558$	0	0
$559$	−9.33352	−0.394766
$560$	0	0
$561$	−2.40308	−0.101458
$562$	0	0
$563$	27.6746	1.16634	0.583172	−	0.812349i	$-0.301811\pi$
$563$	27.6746	1.16634	0.583172	+	0.812349i	$0.301811\pi$
$564$	0	0
$565$	19.4393	0.817818
$566$	0	0
$567$	−6.53278	−0.274351
$568$	0	0
$569$	−42.4477	−1.77950	−0.889750	−	0.456447i	$-0.849122\pi$
$569$	−42.4477	−1.77950	−0.889750	+	0.456447i	$0.849122\pi$
$570$	0	0
$571$	39.9871	1.67341	0.836705	−	0.547654i	$-0.184479\pi$
$571$	39.9871	1.67341	0.836705	+	0.547654i	$0.184479\pi$
$572$	0	0
$573$	6.90228	0.288347
$574$	0	0
$575$	−8.46873	−0.353171
$576$	0	0
$577$	−38.4120	−1.59911	−0.799555	−	0.600592i	$-0.794932\pi$
$577$	−38.4120	−1.59911	−0.799555	+	0.600592i	$0.794932\pi$
$578$	0	0
$579$	−10.3383	−0.429646
$580$	0	0
$581$	−34.5980	−1.43537
$582$	0	0
$583$	−37.8592	−1.56797
$584$	0	0
$585$	−7.36162	−0.304366
$586$	0	0
$587$	−12.2591	−0.505989	−0.252994	−	0.967468i	$-0.581415\pi$
$587$	−12.2591	−0.505989	−0.252994	+	0.967468i	$0.581415\pi$
$588$	0	0
$589$	−0.235072	−0.00968596
$590$	0	0
$591$	14.7973	0.608681
$592$	0	0
$593$	1.19970	0.0492656	0.0246328	−	0.999697i	$-0.492158\pi$
$593$	1.19970	0.0492656	0.0246328	+	0.999697i	$0.492158\pi$
$594$	0	0
$595$	1.94843	0.0798777
$596$	0	0
$597$	2.65684	0.108737
$598$	0	0
$599$	−3.58978	−0.146675	−0.0733373	−	0.997307i	$-0.523365\pi$
$599$	−3.58978	−0.146675	−0.0733373	+	0.997307i	$0.523365\pi$
$600$	0	0
$601$	−20.0462	−0.817703	−0.408851	−	0.912601i	$-0.634071\pi$
$601$	−20.0462	−0.817703	−0.408851	+	0.912601i	$0.634071\pi$
$602$	0	0
$603$	−26.4753	−1.07816
$604$	0	0
$605$	0.647927	0.0263420
$606$	0	0
$607$	10.3397	0.419677	0.209839	−	0.977736i	$-0.432706\pi$
$607$	10.3397	0.419677	0.209839	+	0.977736i	$0.432706\pi$
$608$	0	0
$609$	1.91943	0.0777791
$610$	0	0
$611$	7.16116	0.289710
$612$	0	0
$613$	−14.8344	−0.599156	−0.299578	−	0.954072i	$-0.596846\pi$
$613$	−14.8344	−0.599156	−0.299578	+	0.954072i	$0.596846\pi$
$614$	0	0
$615$	2.12331	0.0856202
$616$	0	0
$617$	−28.2046	−1.13547	−0.567737	−	0.823210i	$-0.692181\pi$
$617$	−28.2046	−1.13547	−0.567737	+	0.823210i	$0.692181\pi$
$618$	0	0
$619$	−1.45009	−0.0582839	−0.0291419	−	0.999575i	$-0.509277\pi$
$619$	−1.45009	−0.0582839	−0.0291419	+	0.999575i	$0.509277\pi$
$620$	0	0
$621$	38.2406	1.53454
$622$	0	0
$623$	−37.5316	−1.50367
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−13.8725	−0.554013
$628$	0	0
$629$	−9.67632	−0.385820
$630$	0	0
$631$	20.9338	0.833361	0.416680	−	0.909053i	$-0.363193\pi$
$631$	20.9338	0.833361	0.416680	+	0.909053i	$0.363193\pi$
$632$	0	0
$633$	−8.27954	−0.329082
$634$	0	0
$635$	0.149500	0.00593271
$636$	0	0
$637$	6.47890	0.256704
$638$	0	0
$639$	−7.76876	−0.307327
$640$	0	0
$641$	16.4929	0.651431	0.325716	−	0.945468i	$-0.394395\pi$
$641$	16.4929	0.651431	0.325716	+	0.945468i	$0.394395\pi$
$642$	0	0
$643$	31.8407	1.25567	0.627837	−	0.778345i	$-0.283940\pi$
$643$	31.8407	1.25567	0.627837	+	0.778345i	$0.283940\pi$
$644$	0	0
$645$	−2.46243	−0.0969580
$646$	0	0
$647$	−20.8262	−0.818761	−0.409381	−	0.912364i	$-0.634255\pi$
$647$	−20.8262	−0.818761	−0.409381	+	0.912364i	$0.634255\pi$
$648$	0	0
$649$	24.7301	0.970740
$650$	0	0
$651$	0.0897642	0.00351814
$652$	0	0
$653$	6.56096	0.256750	0.128375	−	0.991726i	$-0.459024\pi$
$653$	6.56096	0.256750	0.128375	+	0.991726i	$0.459024\pi$
$654$	0	0
$655$	10.2889	0.402022
$656$	0	0
$657$	22.9437	0.895119
$658$	0	0
$659$	29.3559	1.14354	0.571771	−	0.820413i	$-0.306257\pi$
$659$	29.3559	1.14354	0.571771	+	0.820413i	$0.306257\pi$
$660$	0	0
$661$	−22.8255	−0.887810	−0.443905	−	0.896074i	$-0.646407\pi$
$661$	−22.8255	−0.887810	−0.443905	+	0.896074i	$0.646407\pi$
$662$	0	0
$663$	−2.42833	−0.0943085
$664$	0	0
$665$	11.2478	0.436173
$666$	0	0
$667$	8.46873	0.327911
$668$	0	0
$669$	−3.34515	−0.129331
$670$	0	0
$671$	−16.0639	−0.620141
$672$	0	0
$673$	24.3647	0.939190	0.469595	−	0.882882i	$-0.344400\pi$
$673$	24.3647	0.939190	0.469595	+	0.882882i	$0.344400\pi$
$674$	0	0
$675$	−4.51550	−0.173802
$676$	0	0
$677$	−4.46866	−0.171745	−0.0858723	−	0.996306i	$-0.527368\pi$
$677$	−4.46866	−0.171745	−0.0858723	+	0.996306i	$0.527368\pi$
$678$	0	0
$679$	16.0429	0.615669
$680$	0	0
$681$	23.9711	0.918573
$682$	0	0
$683$	46.9824	1.79773	0.898866	−	0.438224i	$-0.144392\pi$
$683$	46.9824	1.79773	0.898866	+	0.438224i	$0.144392\pi$
$684$	0	0
$685$	0.898229	0.0343196
$686$	0	0
$687$	−16.7569	−0.639315
$688$	0	0
$689$	−38.2570	−1.45747
$690$	0	0
$691$	−27.4786	−1.04534	−0.522668	−	0.852536i	$-0.675063\pi$
$691$	−27.4786	−1.04534	−0.522668	+	0.852536i	$0.675063\pi$
$692$	0	0
$693$	−16.3018	−0.619253
$694$	0	0
$695$	−19.8642	−0.753491
$696$	0	0
$697$	−2.15539	−0.0816413
$698$	0	0
$699$	−8.25016	−0.312050
$700$	0	0
$701$	39.9711	1.50969	0.754844	−	0.655905i	$-0.227713\pi$
$701$	39.9711	1.50969	0.754844	+	0.655905i	$0.227713\pi$
$702$	0	0
$703$	−55.8593	−2.10677
$704$	0	0
$705$	1.88930	0.0711552
$706$	0	0
$707$	−19.8557	−0.746750
$708$	0	0
$709$	−5.66247	−0.212658	−0.106329	−	0.994331i	$-0.533910\pi$
$709$	−5.66247	−0.212658	−0.106329	+	0.994331i	$0.533910\pi$
$710$	0	0
$711$	14.5852	0.546988
$712$	0	0
$713$	0.396050	0.0148322
$714$	0	0
$715$	−10.4608	−0.391213
$716$	0	0
$717$	13.6327	0.509122
$718$	0	0
$719$	17.5518	0.654571	0.327285	−	0.944926i	$-0.393866\pi$
$719$	17.5518	0.654571	0.327285	+	0.944926i	$0.393866\pi$
$720$	0	0
$721$	15.0867	0.561858
$722$	0	0
$723$	13.8174	0.513875
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−20.7531	−0.769691	−0.384845	−	0.922981i	$-0.625745\pi$
$727$	−20.7531	−0.769691	−0.384845	+	0.922981i	$0.625745\pi$
$728$	0	0
$729$	5.00953	0.185538
$730$	0	0
$731$	2.49963	0.0924521
$732$	0	0
$733$	30.6723	1.13291	0.566455	−	0.824093i	$-0.308315\pi$
$733$	30.6723	1.13291	0.566455	+	0.824093i	$0.308315\pi$
$734$	0	0
$735$	1.70930	0.0630487
$736$	0	0
$737$	−37.6213	−1.38580
$738$	0	0
$739$	12.8691	0.473397	0.236698	−	0.971583i	$-0.423935\pi$
$739$	12.8691	0.473397	0.236698	+	0.971583i	$0.423935\pi$
$740$	0	0
$741$	−14.0182	−0.514972
$742$	0	0
$743$	−17.4763	−0.641145	−0.320572	−	0.947224i	$-0.603875\pi$
$743$	−17.4763	−0.641145	−0.320572	+	0.947224i	$0.603875\pi$
$744$	0	0
$745$	9.45331	0.346343
$746$	0	0
$747$	−35.0083	−1.28089
$748$	0	0
$749$	26.9203	0.983646
$750$	0	0
$751$	46.9691	1.71393	0.856963	−	0.515378i	$-0.172348\pi$
$751$	46.9691	1.71393	0.856963	+	0.515378i	$0.172348\pi$
$752$	0	0
$753$	11.7328	0.427569
$754$	0	0
$755$	−10.5025	−0.382227
$756$	0	0
$757$	−21.5788	−0.784296	−0.392148	−	0.919902i	$-0.628268\pi$
$757$	−21.5788	−0.784296	−0.392148	+	0.919902i	$0.628268\pi$
$758$	0	0
$759$	23.3724	0.848364
$760$	0	0
$761$	27.5835	0.999902	0.499951	−	0.866054i	$-0.333351\pi$
$761$	27.5835	0.999902	0.499951	+	0.866054i	$0.333351\pi$
$762$	0	0
$763$	−3.89962	−0.141176
$764$	0	0
$765$	1.97153	0.0712809
$766$	0	0
$767$	24.9899	0.902333
$768$	0	0
$769$	27.3491	0.986233	0.493116	−	0.869963i	$-0.335858\pi$
$769$	27.3491	0.986233	0.493116	+	0.869963i	$0.335858\pi$
$770$	0	0
$771$	−8.98548	−0.323604
$772$	0	0
$773$	−29.7475	−1.06994	−0.534972	−	0.844870i	$-0.679678\pi$
$773$	−29.7475	−1.06994	−0.534972	+	0.844870i	$0.679678\pi$
$774$	0	0
$775$	−0.0467661	−0.00167989
$776$	0	0
$777$	21.3304	0.765223
$778$	0	0
$779$	−12.4426	−0.445803
$780$	0	0
$781$	−11.0394	−0.395020
$782$	0	0
$783$	4.51550	0.161371
$784$	0	0
$785$	−14.2324	−0.507975
$786$	0	0
$787$	9.90091	0.352929	0.176465	−	0.984307i	$-0.443534\pi$
$787$	9.90091	0.352929	0.176465	+	0.984307i	$0.443534\pi$
$788$	0	0
$789$	9.47100	0.337176
$790$	0	0
$791$	43.4992	1.54665
$792$	0	0
$793$	−16.2327	−0.576441
$794$	0	0
$795$	−10.0932	−0.357969
$796$	0	0
$797$	0.167726	0.00594117	0.00297058	−	0.999996i	$-0.499054\pi$
$797$	0.167726	0.00594117	0.00297058	+	0.999996i	$0.499054\pi$
$798$	0	0
$799$	−1.91785	−0.0678485
$800$	0	0
$801$	−37.9767	−1.34184
$802$	0	0
$803$	32.6029	1.15053
$804$	0	0
$805$	−18.9504	−0.667915
$806$	0	0
$807$	9.88389	0.347929
$808$	0	0
$809$	2.24400	0.0788949	0.0394474	−	0.999222i	$-0.487440\pi$
$809$	2.24400	0.0788949	0.0394474	+	0.999222i	$0.487440\pi$
$810$	0	0
$811$	11.9666	0.420205	0.210103	−	0.977679i	$-0.432620\pi$
$811$	11.9666	0.420205	0.210103	+	0.977679i	$0.432620\pi$
$812$	0	0
$813$	20.3941	0.715252
$814$	0	0
$815$	13.4245	0.470240
$816$	0	0
$817$	14.4298	0.504836
$818$	0	0
$819$	−16.4731	−0.575615
$820$	0	0
$821$	8.42162	0.293917	0.146958	−	0.989143i	$-0.453052\pi$
$821$	8.42162	0.293917	0.146958	+	0.989143i	$0.453052\pi$
$822$	0	0
$823$	−12.6027	−0.439301	−0.219650	−	0.975579i	$-0.570492\pi$
$823$	−12.6027	−0.439301	−0.219650	+	0.975579i	$0.570492\pi$
$824$	0	0
$825$	−2.75985	−0.0960855
$826$	0	0
$827$	22.4328	0.780065	0.390033	−	0.920801i	$-0.372464\pi$
$827$	22.4328	0.780065	0.390033	+	0.920801i	$0.372464\pi$
$828$	0	0
$829$	50.5065	1.75416	0.877081	−	0.480342i	$-0.159488\pi$
$829$	50.5065	1.75416	0.877081	+	0.480342i	$0.159488\pi$
$830$	0	0
$831$	−9.17570	−0.318302
$832$	0	0
$833$	−1.73513	−0.0601187
$834$	0	0
$835$	−2.49500	−0.0863429
$836$	0	0
$837$	0.211173	0.00729919
$838$	0	0
$839$	−4.18085	−0.144339	−0.0721695	−	0.997392i	$-0.522992\pi$
$839$	−4.18085	−0.144339	−0.0721695	+	0.997392i	$0.522992\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−10.9310	−0.376482
$844$	0	0
$845$	2.42924	0.0835683
$846$	0	0
$847$	1.44986	0.0498178
$848$	0	0
$849$	−0.161003	−0.00552561
$850$	0	0
$851$	94.1121	3.22612
$852$	0	0
$853$	28.3569	0.970921	0.485460	−	0.874259i	$-0.338652\pi$
$853$	28.3569	0.970921	0.485460	+	0.874259i	$0.338652\pi$
$854$	0	0
$855$	11.3812	0.389230
$856$	0	0
$857$	−39.8616	−1.36165	−0.680824	−	0.732447i	$-0.738378\pi$
$857$	−39.8616	−1.36165	−0.680824	+	0.732447i	$0.738378\pi$
$858$	0	0
$859$	20.8839	0.712551	0.356275	−	0.934381i	$-0.384047\pi$
$859$	20.8839	0.712551	0.356275	+	0.934381i	$0.384047\pi$
$860$	0	0
$861$	4.75132	0.161925
$862$	0	0
$863$	2.01051	0.0684387	0.0342193	−	0.999414i	$-0.489106\pi$
$863$	2.01051	0.0684387	0.0342193	+	0.999414i	$0.489106\pi$
$864$	0	0
$865$	3.27350	0.111302
$866$	0	0
$867$	−13.9318	−0.473147
$868$	0	0
$869$	20.7255	0.703066
$870$	0	0
$871$	−38.0166	−1.28814
$872$	0	0
$873$	16.2331	0.549408
$874$	0	0
$875$	2.23769	0.0756478
$876$	0	0
$877$	−33.7480	−1.13959	−0.569795	−	0.821787i	$-0.692977\pi$
$877$	−33.7480	−1.13959	−0.569795	+	0.821787i	$0.692977\pi$
$878$	0	0
$879$	18.7396	0.632069
$880$	0	0
$881$	33.8799	1.14144	0.570721	−	0.821144i	$-0.306664\pi$
$881$	33.8799	1.14144	0.570721	+	0.821144i	$0.306664\pi$
$882$	0	0
$883$	−15.5808	−0.524335	−0.262167	−	0.965022i	$-0.584437\pi$
$883$	−15.5808	−0.524335	−0.262167	+	0.965022i	$0.584437\pi$
$884$	0	0
$885$	6.59299	0.221621
$886$	0	0
$887$	−25.4530	−0.854630	−0.427315	−	0.904103i	$-0.640540\pi$
$887$	−25.4530	−0.854630	−0.427315	+	0.904103i	$0.640540\pi$
$888$	0	0
$889$	0.334534	0.0112199
$890$	0	0
$891$	−9.39315	−0.314682
$892$	0	0
$893$	−11.0713	−0.370487
$894$	0	0
$895$	−23.6862	−0.791742
$896$	0	0
$897$	23.6180	0.788581
$898$	0	0
$899$	0.0467661	0.00155974
$900$	0	0
$901$	10.2457	0.341333
$902$	0	0
$903$	−5.51016	−0.183366
$904$	0	0
$905$	−19.1815	−0.637614
$906$	0	0
$907$	−20.0086	−0.664374	−0.332187	−	0.943214i	$-0.607786\pi$
$907$	−20.0086	−0.664374	−0.332187	+	0.943214i	$0.607786\pi$
$908$	0	0
$909$	−20.0911	−0.666381
$910$	0	0
$911$	−36.7350	−1.21708	−0.608542	−	0.793522i	$-0.708245\pi$
$911$	−36.7350	−1.21708	−0.608542	+	0.793522i	$0.708245\pi$
$912$	0	0
$913$	−49.7467	−1.64638
$914$	0	0
$915$	−4.28262	−0.141579
$916$	0	0
$917$	23.0235	0.760301
$918$	0	0
$919$	21.5715	0.711579	0.355790	−	0.934566i	$-0.384212\pi$
$919$	21.5715	0.711579	0.355790	+	0.934566i	$0.384212\pi$
$920$	0	0
$921$	−15.2738	−0.503288
$922$	0	0
$923$	−11.1554	−0.367184
$924$	0	0
$925$	−11.1129	−0.365389
$926$	0	0
$927$	15.2656	0.501389
$928$	0	0
$929$	−11.4215	−0.374728	−0.187364	−	0.982291i	$-0.559994\pi$
$929$	−11.4215	−0.374728	−0.187364	+	0.982291i	$0.559994\pi$
$930$	0	0
$931$	−10.0165	−0.328279
$932$	0	0
$933$	−1.48490	−0.0486136
$934$	0	0
$935$	2.80154	0.0916202
$936$	0	0
$937$	13.7970	0.450729	0.225364	−	0.974275i	$-0.427643\pi$
$937$	13.7970	0.450729	0.225364	+	0.974275i	$0.427643\pi$
$938$	0	0
$939$	22.3097	0.728049
$940$	0	0
$941$	40.0879	1.30683	0.653414	−	0.757000i	$-0.273336\pi$
$941$	40.0879	1.30683	0.653414	+	0.757000i	$0.273336\pi$
$942$	0	0
$943$	20.9634	0.682662
$944$	0	0
$945$	−10.1043	−0.328693
$946$	0	0
$947$	−48.5899	−1.57896	−0.789480	−	0.613776i	$-0.789650\pi$
$947$	−48.5899	−1.57896	−0.789480	+	0.613776i	$0.789650\pi$
$948$	0	0
$949$	32.9455	1.06946
$950$	0	0
$951$	23.6330	0.766353
$952$	0	0
$953$	46.2200	1.49721	0.748607	−	0.663014i	$-0.230723\pi$
$953$	46.2200	1.49721	0.748607	+	0.663014i	$0.230723\pi$
$954$	0	0
$955$	−8.04677	−0.260387
$956$	0	0
$957$	2.75985	0.0892131
$958$	0	0
$959$	2.00996	0.0649050
$960$	0	0
$961$	−30.9978	−0.999929
$962$	0	0
$963$	27.2396	0.877782
$964$	0	0
$965$	12.0525	0.387985
$966$	0	0
$967$	−36.9491	−1.18820	−0.594102	−	0.804390i	$-0.702492\pi$
$967$	−36.9491	−1.18820	−0.594102	+	0.804390i	$0.702492\pi$
$968$	0	0
$969$	3.75425	0.120604
$970$	0	0
$971$	−50.6568	−1.62565	−0.812827	−	0.582505i	$-0.802072\pi$
$971$	−50.6568	−1.62565	−0.812827	+	0.582505i	$0.802072\pi$
$972$	0	0
$973$	−44.4499	−1.42500
$974$	0	0
$975$	−2.78884	−0.0893145
$976$	0	0
$977$	−53.3369	−1.70640	−0.853199	−	0.521586i	$-0.825341\pi$
$977$	−53.3369	−1.70640	−0.853199	+	0.521586i	$0.825341\pi$
$978$	0	0
$979$	−53.9648	−1.72472
$980$	0	0
$981$	−3.94587	−0.125982
$982$	0	0
$983$	15.7670	0.502888	0.251444	−	0.967872i	$-0.419095\pi$
$983$	15.7670	0.502888	0.251444	+	0.967872i	$0.419095\pi$
$984$	0	0
$985$	−17.2509	−0.549660
$986$	0	0
$987$	4.22768	0.134568
$988$	0	0
$989$	−24.3114	−0.773059
$990$	0	0
$991$	−7.87093	−0.250028	−0.125014	−	0.992155i	$-0.539898\pi$
$991$	−7.87093	−0.250028	−0.125014	+	0.992155i	$0.539898\pi$
$992$	0	0
$993$	−20.8218	−0.660761
$994$	0	0
$995$	−3.09738	−0.0981935
$996$	0	0
$997$	18.5245	0.586677	0.293338	−	0.956009i	$-0.405234\pi$
$997$	18.5245	0.586677	0.293338	+	0.956009i	$0.405234\pi$
$998$	0	0
$999$	50.1802	1.58763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.cl.1.3		5
4.3	odd	2		9280.2.a.cg.1.3		5
8.3	odd	2		1160.2.a.j.1.3	✓	5
8.5	even	2		2320.2.a.u.1.3		5
40.19	odd	2		5800.2.a.t.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.j.1.3	✓	5	8.3	odd	2
2320.2.a.u.1.3		5	8.5	even	2
5800.2.a.t.1.3		5	40.19	odd	2
9280.2.a.cg.1.3		5	4.3	odd	2
9280.2.a.cl.1.3		5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q+0.857770 q^{3} -1.00000 q^{5} -2.23769 q^{7} -2.26423 q^{9} -3.21746 q^{11} -3.25127 q^{13} -0.857770 q^{15} +0.870730 q^{17} +5.02654 q^{19} -1.91943 q^{21} -8.46873 q^{23} +1.00000 q^{25} -4.51550 q^{27} -1.00000 q^{29} -0.0467661 q^{31} -2.75985 q^{33} +2.23769 q^{35} -11.1129 q^{37} -2.78884 q^{39} -2.47539 q^{41} +2.87073 q^{43} +2.26423 q^{45} -2.20257 q^{47} -1.99273 q^{49} +0.746886 q^{51} +11.7668 q^{53} +3.21746 q^{55} +4.31161 q^{57} -7.68620 q^{59} +4.99273 q^{61} +5.06665 q^{63} +3.25127 q^{65} +11.6928 q^{67} -7.26423 q^{69} +3.43108 q^{71} -10.1331 q^{73} +0.857770 q^{75} +7.19970 q^{77} -6.44158 q^{79} +2.91943 q^{81} +15.4615 q^{83} -0.870730 q^{85} -0.857770 q^{87} +16.7725 q^{89} +7.27535 q^{91} -0.0401146 q^{93} -5.02654 q^{95} -7.16939 q^{97} +7.28508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 5 q^{5} - 7 q^{7} + 8 q^{9} + 10 q^{11} - q^{13} - 3 q^{15} - q^{17} + 10 q^{19} + 8 q^{21} - q^{23} + 5 q^{25} + 12 q^{27} - 5 q^{29} - 7 q^{31} - 8 q^{33} + 7 q^{35} + 8 q^{37} - 3 q^{39}+ \cdots + 32 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 5 * q^5 - 7 * q^7 + 8 * q^9 + 10 * q^11 - q^13 - 3 * q^15 - q^17 + 10 * q^19 + 8 * q^21 - q^23 + 5 * q^25 + 12 * q^27 - 5 * q^29 - 7 * q^31 - 8 * q^33 + 7 * q^35 + 8 * q^37 - 3 * q^39 - 4 * q^41 + 9 * q^43 - 8 * q^45 - 8 * q^47 + 16 * q^49 + 2 * q^51 + 9 * q^53 - 10 * q^55 + 2 * q^57 + 29 * q^59 - q^61 - 16 * q^63 + q^65 + 24 * q^67 - 17 * q^69 + 12 * q^71 - 9 * q^73 + 3 * q^75 - 2 * q^77 - 13 * q^79 - 3 * q^81 + 10 * q^83 + q^85 - 3 * q^87 + 4 * q^89 - 4 * q^91 + 26 * q^93 - 10 * q^95 - 15 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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