LMFDB - Embedded newform 9520.2.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9520,2,Mod(1,9520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("9520.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9520 = 2^{4} \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9520.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-3,0,-3,0,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0175827243$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.36147$ of defining polynomial
Character	$\chi$	$=$	9520.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36147 q^{3} -1.00000 q^{5} -1.00000 q^{7} +2.57653 q^{9} +3.36147 q^{11} +6.93800 q^{13} -2.36147 q^{15} -1.00000 q^{17} +6.29947 q^{19} -2.36147 q^{21} +5.29947 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.79160 q^{29} +3.14640 q^{31} +7.93800 q^{33} +1.00000 q^{35} +8.72294 q^{37} +16.3839 q^{39} -3.36147 q^{41} -3.42347 q^{43} -2.57653 q^{45} +3.51454 q^{47} +1.00000 q^{49} -2.36147 q^{51} +5.94467 q^{53} -3.36147 q^{55} +14.8760 q^{57} -10.2375 q^{59} -10.3839 q^{61} -2.57653 q^{63} -6.93800 q^{65} -12.0844 q^{67} +12.5145 q^{69} -14.8140 q^{71} -15.2441 q^{73} +2.36147 q^{75} -3.36147 q^{77} -5.56987 q^{79} -10.0911 q^{81} +13.7520 q^{83} +1.00000 q^{85} -6.59228 q^{87} +10.9380 q^{89} -6.93800 q^{91} +7.43013 q^{93} -6.29947 q^{95} +14.3839 q^{97} +8.66094 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{13} - 3 q^{17} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 12 q^{33} + 3 q^{35} + 12 q^{37} + 9 q^{39} - 3 q^{41} - 15 q^{43} - 3 q^{45} - 6 q^{47}+ \cdots + 3 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^13 - 3 * q^17 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 + 12 * q^33 + 3 * q^35 + 12 * q^37 + 9 * q^39 - 3 * q^41 - 15 * q^43 - 3 * q^45 - 6 * q^47 + 3 * q^49 + 6 * q^53 - 3 * q^55 + 21 * q^57 + 9 * q^61 - 3 * q^63 - 9 * q^65 - 15 * q^67 + 21 * q^69 - 9 * q^71 - 15 * q^73 - 3 * q^77 - 12 * q^79 - 9 * q^81 - 6 * q^83 + 3 * q^85 + 18 * q^87 + 21 * q^89 - 9 * q^91 + 27 * q^93 + 3 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.36147	1.36339	0.681697	−	0.731634i	$-0.261242\pi$
$3$	2.36147	1.36339	0.681697	+	0.731634i	$0.261242\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.57653	0.858845
$10$	0	0
$11$	3.36147	1.01352	0.506760	−	0.862087i	$-0.330843\pi$
$11$	3.36147	1.01352	0.506760	+	0.862087i	$0.330843\pi$
$12$	0	0
$13$	6.93800	1.92426	0.962128	−	0.272598i	$-0.0878830\pi$
$13$	6.93800	1.92426	0.962128	+	0.272598i	$0.0878830\pi$
$14$	0	0
$15$	−2.36147	−0.609729
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.29947	1.44520	0.722599	−	0.691267i	$-0.242947\pi$
$19$	6.29947	1.44520	0.722599	+	0.691267i	$0.242947\pi$
$20$	0	0
$21$	−2.36147	−0.515315
$22$	0	0
$23$	5.29947	1.10502	0.552508	−	0.833507i	$-0.313671\pi$
$23$	5.29947	1.10502	0.552508	+	0.833507i	$0.313671\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.79160	−0.518387	−0.259194	−	0.965825i	$-0.583457\pi$
$29$	−2.79160	−0.518387	−0.259194	+	0.965825i	$0.583457\pi$
$30$	0	0
$31$	3.14640	0.565111	0.282555	−	0.959251i	$-0.408818\pi$
$31$	3.14640	0.565111	0.282555	+	0.959251i	$0.408818\pi$
$32$	0	0
$33$	7.93800	1.38183
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	8.72294	1.43404	0.717021	−	0.697052i	$-0.245505\pi$
$37$	8.72294	1.43404	0.717021	+	0.697052i	$0.245505\pi$
$38$	0	0
$39$	16.3839	2.62352
$40$	0	0
$41$	−3.36147	−0.524973	−0.262487	−	0.964936i	$-0.584543\pi$
$41$	−3.36147	−0.524973	−0.262487	+	0.964936i	$0.584543\pi$
$42$	0	0
$43$	−3.42347	−0.522074	−0.261037	−	0.965329i	$-0.584064\pi$
$43$	−3.42347	−0.522074	−0.261037	+	0.965329i	$0.584064\pi$
$44$	0	0
$45$	−2.57653	−0.384087
$46$	0	0
$47$	3.51454	0.512648	0.256324	−	0.966591i	$-0.417489\pi$
$47$	3.51454	0.512648	0.256324	+	0.966591i	$0.417489\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.36147	−0.330672
$52$	0	0
$53$	5.94467	0.816563	0.408282	−	0.912856i	$-0.366128\pi$
$53$	5.94467	0.816563	0.408282	+	0.912856i	$0.366128\pi$
$54$	0	0
$55$	−3.36147	−0.453260
$56$	0	0
$57$	14.8760	1.97038
$58$	0	0
$59$	−10.2375	−1.33281	−0.666403	−	0.745592i	$-0.732167\pi$
$59$	−10.2375	−1.33281	−0.666403	+	0.745592i	$0.732167\pi$
$60$	0	0
$61$	−10.3839	−1.32952	−0.664760	−	0.747057i	$-0.731466\pi$
$61$	−10.3839	−1.32952	−0.664760	+	0.747057i	$0.731466\pi$
$62$	0	0
$63$	−2.57653	−0.324613
$64$	0	0
$65$	−6.93800	−0.860553
$66$	0	0
$67$	−12.0844	−1.47635	−0.738173	−	0.674612i	$-0.764311\pi$
$67$	−12.0844	−1.47635	−0.738173	+	0.674612i	$0.764311\pi$
$68$	0	0
$69$	12.5145	1.50657
$70$	0	0
$71$	−14.8140	−1.75810	−0.879050	−	0.476730i	$-0.841822\pi$
$71$	−14.8140	−1.75810	−0.879050	+	0.476730i	$0.841822\pi$
$72$	0	0
$73$	−15.2441	−1.78419	−0.892096	−	0.451846i	$-0.850766\pi$
$73$	−15.2441	−1.78419	−0.892096	+	0.451846i	$0.850766\pi$
$74$	0	0
$75$	2.36147	0.272679
$76$	0	0
$77$	−3.36147	−0.383075
$78$	0	0
$79$	−5.56987	−0.626659	−0.313330	−	0.949644i	$-0.601444\pi$
$79$	−5.56987	−0.626659	−0.313330	+	0.949644i	$0.601444\pi$
$80$	0	0
$81$	−10.0911	−1.12123
$82$	0	0
$83$	13.7520	1.50948	0.754740	−	0.656024i	$-0.227763\pi$
$83$	13.7520	1.50948	0.754740	+	0.656024i	$0.227763\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−6.59228	−0.706766
$88$	0	0
$89$	10.9380	1.15943	0.579713	−	0.814821i	$-0.303165\pi$
$89$	10.9380	1.15943	0.579713	+	0.814821i	$0.303165\pi$
$90$	0	0
$91$	−6.93800	−0.727300
$92$	0	0
$93$	7.43013	0.770469
$94$	0	0
$95$	−6.29947	−0.646312
$96$	0	0
$97$	14.3839	1.46046	0.730231	−	0.683201i	$-0.239413\pi$
$97$	14.3839	1.46046	0.730231	+	0.683201i	$0.239413\pi$
$98$	0	0
$99$	8.66094	0.870457
$100$	0	0
$101$	14.6834	1.46105	0.730524	−	0.682887i	$-0.239276\pi$
$101$	14.6834	1.46105	0.730524	+	0.682887i	$0.239276\pi$
$102$	0	0
$103$	−12.5765	−1.23920	−0.619601	−	0.784917i	$-0.712706\pi$
$103$	−12.5765	−1.23920	−0.619601	+	0.784917i	$0.712706\pi$
$104$	0	0
$105$	2.36147	0.230456
$106$	0	0
$107$	−6.30614	−0.609637	−0.304819	−	0.952410i	$-0.598596\pi$
$107$	−6.30614	−0.609637	−0.304819	+	0.952410i	$0.598596\pi$
$108$	0	0
$109$	17.0224	1.63045	0.815226	−	0.579144i	$-0.196613\pi$
$109$	17.0224	1.63045	0.815226	+	0.579144i	$0.196613\pi$
$110$	0	0
$111$	20.5989	1.95517
$112$	0	0
$113$	15.4392	1.45240	0.726199	−	0.687484i	$-0.241285\pi$
$113$	15.4392	1.45240	0.726199	+	0.687484i	$0.241285\pi$
$114$	0	0
$115$	−5.29947	−0.494178
$116$	0	0
$117$	17.8760	1.65264
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	0.299472	0.0272248
$122$	0	0
$123$	−7.93800	−0.715746
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.33906	−0.207558	−0.103779	−	0.994600i	$-0.533093\pi$
$127$	−2.33906	−0.207558	−0.103779	+	0.994600i	$0.533093\pi$
$128$	0	0
$129$	−8.08441	−0.711792
$130$	0	0
$131$	−8.16881	−0.713712	−0.356856	−	0.934159i	$-0.616151\pi$
$131$	−8.16881	−0.713712	−0.356856	+	0.934159i	$0.616151\pi$
$132$	0	0
$133$	−6.29947	−0.546234
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	16.2928	1.39199	0.695994	−	0.718047i	$-0.254964\pi$
$137$	16.2928	1.39199	0.695994	+	0.718047i	$0.254964\pi$
$138$	0	0
$139$	−1.27706	−0.108319	−0.0541595	−	0.998532i	$-0.517248\pi$
$139$	−1.27706	−0.108319	−0.0541595	+	0.998532i	$0.517248\pi$
$140$	0	0
$141$	8.29947	0.698942
$142$	0	0
$143$	23.3219	1.95027
$144$	0	0
$145$	2.79160	0.231830
$146$	0	0
$147$	2.36147	0.194771
$148$	0	0
$149$	−7.02908	−0.575844	−0.287922	−	0.957654i	$-0.592964\pi$
$149$	−7.02908	−0.575844	−0.287922	+	0.957654i	$0.592964\pi$
$150$	0	0
$151$	10.9313	0.889580	0.444790	−	0.895635i	$-0.353278\pi$
$151$	10.9313	0.889580	0.444790	+	0.895635i	$0.353278\pi$
$152$	0	0
$153$	−2.57653	−0.208300
$154$	0	0
$155$	−3.14640	−0.252725
$156$	0	0
$157$	−0.430132	−0.0343283	−0.0171641	−	0.999853i	$-0.505464\pi$
$157$	−0.430132	−0.0343283	−0.0171641	+	0.999853i	$0.505464\pi$
$158$	0	0
$159$	14.0382	1.11330
$160$	0	0
$161$	−5.29947	−0.417657
$162$	0	0
$163$	−17.1531	−1.34353	−0.671766	−	0.740763i	$-0.734464\pi$
$163$	−17.1531	−1.34353	−0.671766	+	0.740763i	$0.734464\pi$
$164$	0	0
$165$	−7.93800	−0.617973
$166$	0	0
$167$	23.0157	1.78101	0.890506	−	0.454972i	$-0.150351\pi$
$167$	23.0157	1.78101	0.890506	+	0.454972i	$0.150351\pi$
$168$	0	0
$169$	35.1359	2.70276
$170$	0	0
$171$	16.2308	1.24120
$172$	0	0
$173$	2.63853	0.200604	0.100302	−	0.994957i	$-0.468019\pi$
$173$	2.63853	0.200604	0.100302	+	0.994957i	$0.468019\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−24.1755	−1.81714
$178$	0	0
$179$	−6.72294	−0.502496	−0.251248	−	0.967923i	$-0.580841\pi$
$179$	−6.72294	−0.502496	−0.251248	+	0.967923i	$0.580841\pi$
$180$	0	0
$181$	−20.5989	−1.53111	−0.765554	−	0.643372i	$-0.777535\pi$
$181$	−20.5989	−1.53111	−0.765554	+	0.643372i	$0.777535\pi$
$182$	0	0
$183$	−24.5212	−1.81266
$184$	0	0
$185$	−8.72294	−0.641323
$186$	0	0
$187$	−3.36147	−0.245815
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	25.2375	1.82612	0.913060	−	0.407826i	$-0.133713\pi$
$191$	25.2375	1.82612	0.913060	+	0.407826i	$0.133713\pi$
$192$	0	0
$193$	−26.6214	−1.91625	−0.958124	−	0.286355i	$-0.907556\pi$
$193$	−26.6214	−1.91625	−0.958124	+	0.286355i	$0.907556\pi$
$194$	0	0
$195$	−16.3839	−1.17327
$196$	0	0
$197$	16.2928	1.16081	0.580407	−	0.814326i	$-0.302893\pi$
$197$	16.2928	1.16081	0.580407	+	0.814326i	$0.302893\pi$
$198$	0	0
$199$	16.3615	1.15983	0.579917	−	0.814676i	$-0.303085\pi$
$199$	16.3615	1.15983	0.579917	+	0.814676i	$0.303085\pi$
$200$	0	0
$201$	−28.5369	−2.01284
$202$	0	0
$203$	2.79160	0.195932
$204$	0	0
$205$	3.36147	0.234775
$206$	0	0
$207$	13.6543	0.949038
$208$	0	0
$209$	21.1755	1.46474
$210$	0	0
$211$	−3.13974	−0.216148	−0.108074	−	0.994143i	$-0.534468\pi$
$211$	−3.13974	−0.216148	−0.108074	+	0.994143i	$0.534468\pi$
$212$	0	0
$213$	−34.9828	−2.39698
$214$	0	0
$215$	3.42347	0.233478
$216$	0	0
$217$	−3.14640	−0.213592
$218$	0	0
$219$	−35.9986	−2.43256
$220$	0	0
$221$	−6.93800	−0.466701
$222$	0	0
$223$	−1.24414	−0.0833139	−0.0416570	−	0.999132i	$-0.513264\pi$
$223$	−1.24414	−0.0833139	−0.0416570	+	0.999132i	$0.513264\pi$
$224$	0	0
$225$	2.57653	0.171769
$226$	0	0
$227$	10.4235	0.691830	0.345915	−	0.938266i	$-0.387569\pi$
$227$	10.4235	0.691830	0.345915	+	0.938266i	$0.387569\pi$
$228$	0	0
$229$	15.4235	1.01921	0.509606	−	0.860408i	$-0.329791\pi$
$229$	15.4235	1.01921	0.509606	+	0.860408i	$0.329791\pi$
$230$	0	0
$231$	−7.93800	−0.522282
$232$	0	0
$233$	16.2599	1.06522	0.532610	−	0.846361i	$-0.321211\pi$
$233$	16.2599	1.06522	0.532610	+	0.846361i	$0.321211\pi$
$234$	0	0
$235$	−3.51454	−0.229263
$236$	0	0
$237$	−13.1531	−0.854384
$238$	0	0
$239$	−6.93134	−0.448351	−0.224175	−	0.974549i	$-0.571969\pi$
$239$	−6.93134	−0.448351	−0.224175	+	0.974549i	$0.571969\pi$
$240$	0	0
$241$	−8.57653	−0.552463	−0.276232	−	0.961091i	$-0.589086\pi$
$241$	−8.57653	−0.552463	−0.276232	+	0.961091i	$0.589086\pi$
$242$	0	0
$243$	−20.8298	−1.33623
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	43.7058	2.78093
$248$	0	0
$249$	32.4750	2.05802
$250$	0	0
$251$	−21.4063	−1.35115	−0.675576	−	0.737290i	$-0.736105\pi$
$251$	−21.4063	−1.35115	−0.675576	+	0.737290i	$0.736105\pi$
$252$	0	0
$253$	17.8140	1.11996
$254$	0	0
$255$	2.36147	0.147881
$256$	0	0
$257$	−15.5923	−0.972620	−0.486310	−	0.873786i	$-0.661657\pi$
$257$	−15.5923	−0.972620	−0.486310	+	0.873786i	$0.661657\pi$
$258$	0	0
$259$	−8.72294	−0.542017
$260$	0	0
$261$	−7.19266	−0.445214
$262$	0	0
$263$	15.1201	0.932348	0.466174	−	0.884693i	$-0.345632\pi$
$263$	15.1201	0.932348	0.466174	+	0.884693i	$0.345632\pi$
$264$	0	0
$265$	−5.94467	−0.365178
$266$	0	0
$267$	25.8298	1.58076
$268$	0	0
$269$	16.2599	0.991383	0.495691	−	0.868499i	$-0.334915\pi$
$269$	16.2599	0.991383	0.495691	+	0.868499i	$0.334915\pi$
$270$	0	0
$271$	5.70719	0.346687	0.173344	−	0.984861i	$-0.444543\pi$
$271$	5.70719	0.346687	0.173344	+	0.984861i	$0.444543\pi$
$272$	0	0
$273$	−16.3839	−0.991597
$274$	0	0
$275$	3.36147	0.202704
$276$	0	0
$277$	9.13974	0.549154	0.274577	−	0.961565i	$-0.411462\pi$
$277$	9.13974	0.549154	0.274577	+	0.961565i	$0.411462\pi$
$278$	0	0
$279$	8.10682	0.485342
$280$	0	0
$281$	19.6967	1.17501	0.587503	−	0.809222i	$-0.300111\pi$
$281$	19.6967	1.17501	0.587503	+	0.809222i	$0.300111\pi$
$282$	0	0
$283$	−1.24414	−0.0739566	−0.0369783	−	0.999316i	$-0.511773\pi$
$283$	−1.24414	−0.0739566	−0.0369783	+	0.999316i	$0.511773\pi$
$284$	0	0
$285$	−14.8760	−0.881179
$286$	0	0
$287$	3.36147	0.198421
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	33.9671	1.99119
$292$	0	0
$293$	14.0911	0.823209	0.411605	−	0.911363i	$-0.364968\pi$
$293$	14.0911	0.823209	0.411605	+	0.911363i	$0.364968\pi$
$294$	0	0
$295$	10.2375	0.596049
$296$	0	0
$297$	−3.36147	−0.195052
$298$	0	0
$299$	36.7678	2.12633
$300$	0	0
$301$	3.42347	0.197325
$302$	0	0
$303$	34.6743	1.99199
$304$	0	0
$305$	10.3839	0.594579
$306$	0	0
$307$	−22.4750	−1.28271	−0.641357	−	0.767243i	$-0.721628\pi$
$307$	−22.4750	−1.28271	−0.641357	+	0.767243i	$0.721628\pi$
$308$	0	0
$309$	−29.6991	−1.68952
$310$	0	0
$311$	9.85360	0.558746	0.279373	−	0.960183i	$-0.409873\pi$
$311$	9.85360	0.558746	0.279373	+	0.960183i	$0.409873\pi$
$312$	0	0
$313$	5.45921	0.308573	0.154286	−	0.988026i	$-0.450692\pi$
$313$	5.45921	0.308573	0.154286	+	0.988026i	$0.450692\pi$
$314$	0	0
$315$	2.57653	0.145171
$316$	0	0
$317$	−15.1531	−0.851081	−0.425541	−	0.904939i	$-0.639916\pi$
$317$	−15.1531	−0.851081	−0.425541	+	0.904939i	$0.639916\pi$
$318$	0	0
$319$	−9.38388	−0.525396
$320$	0	0
$321$	−14.8918	−0.831176
$322$	0	0
$323$	−6.29947	−0.350512
$324$	0	0
$325$	6.93800	0.384851
$326$	0	0
$327$	40.1979	2.22295
$328$	0	0
$329$	−3.51454	−0.193763
$330$	0	0
$331$	−8.38388	−0.460820	−0.230410	−	0.973094i	$-0.574007\pi$
$331$	−8.38388	−0.460820	−0.230410	+	0.973094i	$0.574007\pi$
$332$	0	0
$333$	22.4750	1.23162
$334$	0	0
$335$	12.0844	0.660242
$336$	0	0
$337$	−9.94467	−0.541721	−0.270860	−	0.962619i	$-0.587308\pi$
$337$	−9.94467	−0.541721	−0.270860	+	0.962619i	$0.587308\pi$
$338$	0	0
$339$	36.4592	1.98019
$340$	0	0
$341$	10.5765	0.572751
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−12.5145	−0.673760
$346$	0	0
$347$	−4.21507	−0.226277	−0.113138	−	0.993579i	$-0.536090\pi$
$347$	−4.21507	−0.226277	−0.113138	+	0.993579i	$0.536090\pi$
$348$	0	0
$349$	5.56987	0.298148	0.149074	−	0.988826i	$-0.452371\pi$
$349$	5.56987	0.298148	0.149074	+	0.988826i	$0.452371\pi$
$350$	0	0
$351$	−6.93800	−0.370323
$352$	0	0
$353$	−1.34814	−0.0717540	−0.0358770	−	0.999356i	$-0.511422\pi$
$353$	−1.34814	−0.0717540	−0.0358770	+	0.999356i	$0.511422\pi$
$354$	0	0
$355$	14.8140	0.786246
$356$	0	0
$357$	2.36147	0.124982
$358$	0	0
$359$	−9.89842	−0.522418	−0.261209	−	0.965282i	$-0.584121\pi$
$359$	−9.89842	−0.522418	−0.261209	+	0.965282i	$0.584121\pi$
$360$	0	0
$361$	20.6834	1.08860
$362$	0	0
$363$	0.707194	0.0371181
$364$	0	0
$365$	15.2441	0.797915
$366$	0	0
$367$	−30.4883	−1.59148	−0.795738	−	0.605641i	$-0.792917\pi$
$367$	−30.4883	−1.59148	−0.795738	+	0.605641i	$0.792917\pi$
$368$	0	0
$369$	−8.66094	−0.450871
$370$	0	0
$371$	−5.94467	−0.308632
$372$	0	0
$373$	10.8693	0.562793	0.281397	−	0.959592i	$-0.409202\pi$
$373$	10.8693	0.562793	0.281397	+	0.959592i	$0.409202\pi$
$374$	0	0
$375$	−2.36147	−0.121946
$376$	0	0
$377$	−19.3681	−0.997510
$378$	0	0
$379$	18.7587	0.963569	0.481784	−	0.876290i	$-0.339989\pi$
$379$	18.7587	0.963569	0.481784	+	0.876290i	$0.339989\pi$
$380$	0	0
$381$	−5.52361	−0.282983
$382$	0	0
$383$	−21.7453	−1.11114	−0.555568	−	0.831471i	$-0.687499\pi$
$383$	−21.7453	−1.11114	−0.555568	+	0.831471i	$0.687499\pi$
$384$	0	0
$385$	3.36147	0.171316
$386$	0	0
$387$	−8.82068	−0.448380
$388$	0	0
$389$	−9.45921	−0.479601	−0.239800	−	0.970822i	$-0.577082\pi$
$389$	−9.45921	−0.479601	−0.239800	+	0.970822i	$0.577082\pi$
$390$	0	0
$391$	−5.29947	−0.268006
$392$	0	0
$393$	−19.2904	−0.973072
$394$	0	0
$395$	5.56987	0.280250
$396$	0	0
$397$	25.6056	1.28511	0.642554	−	0.766240i	$-0.277875\pi$
$397$	25.6056	1.28511	0.642554	+	0.766240i	$0.277875\pi$
$398$	0	0
$399$	−14.8760	−0.744732
$400$	0	0
$401$	1.40106	0.0699654	0.0349827	−	0.999388i	$-0.488862\pi$
$401$	1.40106	0.0699654	0.0349827	+	0.999388i	$0.488862\pi$
$402$	0	0
$403$	21.8298	1.08742
$404$	0	0
$405$	10.0911	0.501429
$406$	0	0
$407$	29.3219	1.45343
$408$	0	0
$409$	8.49454	0.420028	0.210014	−	0.977698i	$-0.432649\pi$
$409$	8.49454	0.420028	0.210014	+	0.977698i	$0.432649\pi$
$410$	0	0
$411$	38.4750	1.89783
$412$	0	0
$413$	10.2375	0.503753
$414$	0	0
$415$	−13.7520	−0.675060
$416$	0	0
$417$	−3.01574	−0.147682
$418$	0	0
$419$	−11.0157	−0.538154	−0.269077	−	0.963119i	$-0.586719\pi$
$419$	−11.0157	−0.538154	−0.269077	+	0.963119i	$0.586719\pi$
$420$	0	0
$421$	−29.7968	−1.45221	−0.726104	−	0.687584i	$-0.758671\pi$
$421$	−29.7968	−1.45221	−0.726104	+	0.687584i	$0.758671\pi$
$422$	0	0
$423$	9.05533	0.440285
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	10.3839	0.502511
$428$	0	0
$429$	55.0739	2.65899
$430$	0	0
$431$	19.1397	0.921929	0.460964	−	0.887419i	$-0.347504\pi$
$431$	19.1397	0.921929	0.460964	+	0.887419i	$0.347504\pi$
$432$	0	0
$433$	−24.0582	−1.15616	−0.578081	−	0.815980i	$-0.696198\pi$
$433$	−24.0582	−1.15616	−0.578081	+	0.815980i	$0.696198\pi$
$434$	0	0
$435$	6.59228	0.316076
$436$	0	0
$437$	33.3839	1.59697
$438$	0	0
$439$	−5.89842	−0.281516	−0.140758	−	0.990044i	$-0.544954\pi$
$439$	−5.89842	−0.281516	−0.140758	+	0.990044i	$0.544954\pi$
$440$	0	0
$441$	2.57653	0.122692
$442$	0	0
$443$	13.1621	0.625352	0.312676	−	0.949860i	$-0.398775\pi$
$443$	13.1621	0.625352	0.312676	+	0.949860i	$0.398775\pi$
$444$	0	0
$445$	−10.9380	−0.518511
$446$	0	0
$447$	−16.5989	−0.785103
$448$	0	0
$449$	7.44588	0.351393	0.175696	−	0.984444i	$-0.443782\pi$
$449$	7.44588	0.351393	0.175696	+	0.984444i	$0.443782\pi$
$450$	0	0
$451$	−11.2995	−0.532071
$452$	0	0
$453$	25.8140	1.21285
$454$	0	0
$455$	6.93800	0.325259
$456$	0	0
$457$	27.0291	1.26437	0.632183	−	0.774819i	$-0.282159\pi$
$457$	27.0291	1.26437	0.632183	+	0.774819i	$0.282159\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−29.4196	−1.37021	−0.685104	−	0.728445i	$-0.740243\pi$
$461$	−29.4196	−1.37021	−0.685104	+	0.728445i	$0.740243\pi$
$462$	0	0
$463$	−12.2151	−0.567682	−0.283841	−	0.958871i	$-0.591609\pi$
$463$	−12.2151	−0.567682	−0.283841	+	0.958871i	$0.591609\pi$
$464$	0	0
$465$	−7.43013	−0.344564
$466$	0	0
$467$	−12.1555	−0.562488	−0.281244	−	0.959636i	$-0.590747\pi$
$467$	−12.1555	−0.562488	−0.281244	+	0.959636i	$0.590747\pi$
$468$	0	0
$469$	12.0844	0.558006
$470$	0	0
$471$	−1.01574	−0.0468030
$472$	0	0
$473$	−11.5079	−0.529132
$474$	0	0
$475$	6.29947	0.289040
$476$	0	0
$477$	15.3166	0.701301
$478$	0	0
$479$	−12.3839	−0.565834	−0.282917	−	0.959144i	$-0.591302\pi$
$479$	−12.3839	−0.565834	−0.282917	+	0.959144i	$0.591302\pi$
$480$	0	0
$481$	60.5198	2.75946
$482$	0	0
$483$	−12.5145	−0.569431
$484$	0	0
$485$	−14.3839	−0.653138
$486$	0	0
$487$	−20.5145	−0.929602	−0.464801	−	0.885415i	$-0.653874\pi$
$487$	−20.5145	−0.929602	−0.464801	+	0.885415i	$0.653874\pi$
$488$	0	0
$489$	−40.5064	−1.83176
$490$	0	0
$491$	8.20173	0.370139	0.185070	−	0.982725i	$-0.440749\pi$
$491$	8.20173	0.370139	0.185070	+	0.982725i	$0.440749\pi$
$492$	0	0
$493$	2.79160	0.125727
$494$	0	0
$495$	−8.66094	−0.389280
$496$	0	0
$497$	14.8140	0.664499
$498$	0	0
$499$	−22.9447	−1.02714	−0.513572	−	0.858046i	$-0.671678\pi$
$499$	−22.9447	−1.02714	−0.513572	+	0.858046i	$0.671678\pi$
$500$	0	0
$501$	54.3510	2.42822
$502$	0	0
$503$	7.33522	0.327061	0.163531	−	0.986538i	$-0.447712\pi$
$503$	7.33522	0.327061	0.163531	+	0.986538i	$0.447712\pi$
$504$	0	0
$505$	−14.6834	−0.653401
$506$	0	0
$507$	82.9723	3.68493
$508$	0	0
$509$	−3.89842	−0.172794	−0.0863971	−	0.996261i	$-0.527535\pi$
$509$	−3.89842	−0.172794	−0.0863971	+	0.996261i	$0.527535\pi$
$510$	0	0
$511$	15.2441	0.674361
$512$	0	0
$513$	−6.29947	−0.278128
$514$	0	0
$515$	12.5765	0.554188
$516$	0	0
$517$	11.8140	0.519580
$518$	0	0
$519$	6.23081	0.273502
$520$	0	0
$521$	−7.46828	−0.327192	−0.163596	−	0.986527i	$-0.552309\pi$
$521$	−7.46828	−0.327192	−0.163596	+	0.986527i	$0.552309\pi$
$522$	0	0
$523$	31.7653	1.38900	0.694501	−	0.719492i	$-0.255625\pi$
$523$	31.7653	1.38900	0.694501	+	0.719492i	$0.255625\pi$
$524$	0	0
$525$	−2.36147	−0.103063
$526$	0	0
$527$	−3.14640	−0.137059
$528$	0	0
$529$	5.08441	0.221061
$530$	0	0
$531$	−26.3772	−1.14467
$532$	0	0
$533$	−23.3219	−1.01018
$534$	0	0
$535$	6.30614	0.272638
$536$	0	0
$537$	−15.8760	−0.685100
$538$	0	0
$539$	3.36147	0.144789
$540$	0	0
$541$	−29.3930	−1.26370	−0.631851	−	0.775090i	$-0.717705\pi$
$541$	−29.3930	−1.26370	−0.631851	+	0.775090i	$0.717705\pi$
$542$	0	0
$543$	−48.6438	−2.08750
$544$	0	0
$545$	−17.0224	−0.729160
$546$	0	0
$547$	−25.6609	−1.09718	−0.548591	−	0.836091i	$-0.684836\pi$
$547$	−25.6609	−1.09718	−0.548591	+	0.836091i	$0.684836\pi$
$548$	0	0
$549$	−26.7544	−1.14185
$550$	0	0
$551$	−17.5856	−0.749172
$552$	0	0
$553$	5.56987	0.236855
$554$	0	0
$555$	−20.5989	−0.874376
$556$	0	0
$557$	−4.46828	−0.189327	−0.0946637	−	0.995509i	$-0.530178\pi$
$557$	−4.46828	−0.189327	−0.0946637	+	0.995509i	$0.530178\pi$
$558$	0	0
$559$	−23.7520	−1.00460
$560$	0	0
$561$	−7.93800	−0.335143
$562$	0	0
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	−15.4392	−0.649532
$566$	0	0
$567$	10.0911	0.423785
$568$	0	0
$569$	7.99333	0.335098	0.167549	−	0.985864i	$-0.446415\pi$
$569$	7.99333	0.335098	0.167549	+	0.985864i	$0.446415\pi$
$570$	0	0
$571$	−13.4855	−0.564349	−0.282175	−	0.959363i	$-0.591056\pi$
$571$	−13.4855	−0.564349	−0.282175	+	0.959363i	$0.591056\pi$
$572$	0	0
$573$	59.5975	2.48972
$574$	0	0
$575$	5.29947	0.221003
$576$	0	0
$577$	−13.7783	−0.573597	−0.286798	−	0.957991i	$-0.592591\pi$
$577$	−13.7783	−0.573597	−0.286798	+	0.957991i	$0.592591\pi$
$578$	0	0
$579$	−62.8655	−2.61260
$580$	0	0
$581$	−13.7520	−0.570530
$582$	0	0
$583$	19.9828	0.827604
$584$	0	0
$585$	−17.8760	−0.739082
$586$	0	0
$587$	−31.1846	−1.28712	−0.643562	−	0.765394i	$-0.722544\pi$
$587$	−31.1846	−1.28712	−0.643562	+	0.765394i	$0.722544\pi$
$588$	0	0
$589$	19.8207	0.816697
$590$	0	0
$591$	38.4750	1.58265
$592$	0	0
$593$	8.56320	0.351649	0.175824	−	0.984422i	$-0.443741\pi$
$593$	8.56320	0.351649	0.175824	+	0.984422i	$0.443741\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	0	0
$597$	38.6371	1.58131
$598$	0	0
$599$	33.2203	1.35734	0.678672	−	0.734441i	$-0.262556\pi$
$599$	33.2203	1.35734	0.678672	+	0.734441i	$0.262556\pi$
$600$	0	0
$601$	5.25081	0.214185	0.107092	−	0.994249i	$-0.465846\pi$
$601$	5.25081	0.214185	0.107092	+	0.994249i	$0.465846\pi$
$602$	0	0
$603$	−31.1359	−1.26795
$604$	0	0
$605$	−0.299472	−0.0121753
$606$	0	0
$607$	−16.0582	−0.651780	−0.325890	−	0.945408i	$-0.605664\pi$
$607$	−16.0582	−0.651780	−0.325890	+	0.945408i	$0.605664\pi$
$608$	0	0
$609$	6.59228	0.267133
$610$	0	0
$611$	24.3839	0.986466
$612$	0	0
$613$	−0.507872	−0.0205127	−0.0102564	−	0.999947i	$-0.503265\pi$
$613$	−0.507872	−0.0205127	−0.0102564	+	0.999947i	$0.503265\pi$
$614$	0	0
$615$	7.93800	0.320091
$616$	0	0
$617$	14.5303	0.584967	0.292484	−	0.956271i	$-0.405518\pi$
$617$	14.5303	0.584967	0.292484	+	0.956271i	$0.405518\pi$
$618$	0	0
$619$	11.7072	0.470552	0.235276	−	0.971929i	$-0.424401\pi$
$619$	11.7072	0.470552	0.235276	+	0.971929i	$0.424401\pi$
$620$	0	0
$621$	−5.29947	−0.212660
$622$	0	0
$623$	−10.9380	−0.438222
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	50.0052	1.99702
$628$	0	0
$629$	−8.72294	−0.347806
$630$	0	0
$631$	16.9762	0.675810	0.337905	−	0.941180i	$-0.390282\pi$
$631$	16.9762	0.675810	0.337905	+	0.941180i	$0.390282\pi$
$632$	0	0
$633$	−7.41439	−0.294695
$634$	0	0
$635$	2.33906	0.0928227
$636$	0	0
$637$	6.93800	0.274894
$638$	0	0
$639$	−38.1688	−1.50993
$640$	0	0
$641$	−22.7544	−0.898746	−0.449373	−	0.893344i	$-0.648353\pi$
$641$	−22.7544	−0.898746	−0.449373	+	0.893344i	$0.648353\pi$
$642$	0	0
$643$	3.38772	0.133599	0.0667994	−	0.997766i	$-0.478721\pi$
$643$	3.38772	0.133599	0.0667994	+	0.997766i	$0.478721\pi$
$644$	0	0
$645$	8.08441	0.318323
$646$	0	0
$647$	−3.18599	−0.125254	−0.0626271	−	0.998037i	$-0.519948\pi$
$647$	−3.18599	−0.125254	−0.0626271	+	0.998037i	$0.519948\pi$
$648$	0	0
$649$	−34.4130	−1.35083
$650$	0	0
$651$	−7.43013	−0.291210
$652$	0	0
$653$	−32.6304	−1.27693	−0.638464	−	0.769652i	$-0.720430\pi$
$653$	−32.6304	−1.27693	−0.638464	+	0.769652i	$0.720430\pi$
$654$	0	0
$655$	8.16881	0.319182
$656$	0	0
$657$	−39.2771	−1.53234
$658$	0	0
$659$	9.23081	0.359581	0.179791	−	0.983705i	$-0.442458\pi$
$659$	9.23081	0.359581	0.179791	+	0.983705i	$0.442458\pi$
$660$	0	0
$661$	49.9919	1.94446	0.972230	−	0.234028i	$-0.0751909\pi$
$661$	49.9919	1.94446	0.972230	+	0.234028i	$0.0751909\pi$
$662$	0	0
$663$	−16.3839	−0.636297
$664$	0	0
$665$	6.29947	0.244283
$666$	0	0
$667$	−14.7940	−0.572826
$668$	0	0
$669$	−2.93800	−0.113590
$670$	0	0
$671$	−34.9051	−1.34750
$672$	0	0
$673$	−25.3285	−0.976344	−0.488172	−	0.872747i	$-0.662336\pi$
$673$	−25.3285	−0.976344	−0.488172	+	0.872747i	$0.662336\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−30.4392	−1.16987	−0.584937	−	0.811079i	$-0.698881\pi$
$677$	−30.4392	−1.16987	−0.584937	+	0.811079i	$0.698881\pi$
$678$	0	0
$679$	−14.3839	−0.552003
$680$	0	0
$681$	24.6147	0.943237
$682$	0	0
$683$	−33.1201	−1.26731	−0.633654	−	0.773617i	$-0.718446\pi$
$683$	−33.1201	−1.26731	−0.633654	+	0.773617i	$0.718446\pi$
$684$	0	0
$685$	−16.2928	−0.622516
$686$	0	0
$687$	36.4220	1.38959
$688$	0	0
$689$	41.2441	1.57128
$690$	0	0
$691$	−30.7544	−1.16995	−0.584977	−	0.811050i	$-0.698896\pi$
$691$	−30.7544	−1.16995	−0.584977	+	0.811050i	$0.698896\pi$
$692$	0	0
$693$	−8.66094	−0.329002
$694$	0	0
$695$	1.27706	0.0484417
$696$	0	0
$697$	3.36147	0.127325
$698$	0	0
$699$	38.3972	1.45232
$700$	0	0
$701$	−22.4883	−0.849371	−0.424685	−	0.905341i	$-0.639615\pi$
$701$	−22.4883	−0.849371	−0.424685	+	0.905341i	$0.639615\pi$
$702$	0	0
$703$	54.9499	2.07247
$704$	0	0
$705$	−8.29947	−0.312576
$706$	0	0
$707$	−14.6834	−0.552224
$708$	0	0
$709$	−23.1292	−0.868636	−0.434318	−	0.900760i	$-0.643011\pi$
$709$	−23.1292	−0.868636	−0.434318	+	0.900760i	$0.643011\pi$
$710$	0	0
$711$	−14.3510	−0.538203
$712$	0	0
$713$	16.6743	0.624456
$714$	0	0
$715$	−23.3219	−0.872189
$716$	0	0
$717$	−16.3681	−0.611279
$718$	0	0
$719$	1.00908	0.0376322	0.0188161	−	0.999823i	$-0.494010\pi$
$719$	1.00908	0.0376322	0.0188161	+	0.999823i	$0.494010\pi$
$720$	0	0
$721$	12.5765	0.468375
$722$	0	0
$723$	−20.2532	−0.753225
$724$	0	0
$725$	−2.79160	−0.103677
$726$	0	0
$727$	24.3748	0.904011	0.452006	−	0.892015i	$-0.350709\pi$
$727$	24.3748	0.904011	0.452006	+	0.892015i	$0.350709\pi$
$728$	0	0
$729$	−18.9156	−0.700578
$730$	0	0
$731$	3.42347	0.126621
$732$	0	0
$733$	−7.15307	−0.264205	−0.132102	−	0.991236i	$-0.542173\pi$
$733$	−7.15307	−0.264205	−0.132102	+	0.991236i	$0.542173\pi$
$734$	0	0
$735$	−2.36147	−0.0871041
$736$	0	0
$737$	−40.6214	−1.49631
$738$	0	0
$739$	−7.92226	−0.291425	−0.145713	−	0.989327i	$-0.546547\pi$
$739$	−7.92226	−0.291425	−0.145713	+	0.989327i	$0.546547\pi$
$740$	0	0
$741$	103.210	3.79151
$742$	0	0
$743$	7.59653	0.278690	0.139345	−	0.990244i	$-0.455500\pi$
$743$	7.59653	0.278690	0.139345	+	0.990244i	$0.455500\pi$
$744$	0	0
$745$	7.02908	0.257525
$746$	0	0
$747$	35.4325	1.29641
$748$	0	0
$749$	6.30614	0.230421
$750$	0	0
$751$	−45.1516	−1.64761	−0.823803	−	0.566876i	$-0.808152\pi$
$751$	−45.1516	−1.64761	−0.823803	+	0.566876i	$0.808152\pi$
$752$	0	0
$753$	−50.5503	−1.84215
$754$	0	0
$755$	−10.9313	−0.397832
$756$	0	0
$757$	38.3839	1.39509	0.697543	−	0.716543i	$-0.254277\pi$
$757$	38.3839	1.39509	0.697543	+	0.716543i	$0.254277\pi$
$758$	0	0
$759$	42.0672	1.52694
$760$	0	0
$761$	20.0582	0.727107	0.363554	−	0.931573i	$-0.381563\pi$
$761$	20.0582	0.727107	0.363554	+	0.931573i	$0.381563\pi$
$762$	0	0
$763$	−17.0224	−0.616253
$764$	0	0
$765$	2.57653	0.0931548
$766$	0	0
$767$	−71.0276	−2.56466
$768$	0	0
$769$	−14.9828	−0.540294	−0.270147	−	0.962819i	$-0.587072\pi$
$769$	−14.9828	−0.540294	−0.270147	+	0.962819i	$0.587072\pi$
$770$	0	0
$771$	−36.8207	−1.32606
$772$	0	0
$773$	33.8760	1.21844	0.609218	−	0.793003i	$-0.291484\pi$
$773$	33.8760	1.21844	0.609218	+	0.793003i	$0.291484\pi$
$774$	0	0
$775$	3.14640	0.113022
$776$	0	0
$777$	−20.5989	−0.738983
$778$	0	0
$779$	−21.1755	−0.758690
$780$	0	0
$781$	−49.7968	−1.78187
$782$	0	0
$783$	2.79160	0.0997637
$784$	0	0
$785$	0.430132	0.0153521
$786$	0	0
$787$	6.60561	0.235465	0.117732	−	0.993045i	$-0.462438\pi$
$787$	6.60561	0.235465	0.117732	+	0.993045i	$0.462438\pi$
$788$	0	0
$789$	35.7058	1.27116
$790$	0	0
$791$	−15.4392	−0.548955
$792$	0	0
$793$	−72.0434	−2.55834
$794$	0	0
$795$	−14.0382	−0.497882
$796$	0	0
$797$	−18.4616	−0.653944	−0.326972	−	0.945034i	$-0.606028\pi$
$797$	−18.4616	−0.653944	−0.326972	+	0.945034i	$0.606028\pi$
$798$	0	0
$799$	−3.51454	−0.124335
$800$	0	0
$801$	28.1821	0.995767
$802$	0	0
$803$	−51.2427	−1.80832
$804$	0	0
$805$	5.29947	0.186782
$806$	0	0
$807$	38.3972	1.35165
$808$	0	0
$809$	28.9499	1.01782	0.508912	−	0.860818i	$-0.330048\pi$
$809$	28.9499	1.01782	0.508912	+	0.860818i	$0.330048\pi$
$810$	0	0
$811$	51.6280	1.81291	0.906453	−	0.422308i	$-0.138780\pi$
$811$	51.6280	1.81291	0.906453	+	0.422308i	$0.138780\pi$
$812$	0	0
$813$	13.4774	0.472672
$814$	0	0
$815$	17.1531	0.600846
$816$	0	0
$817$	−21.5660	−0.754500
$818$	0	0
$819$	−17.8760	−0.624638
$820$	0	0
$821$	−10.5660	−0.368757	−0.184378	−	0.982855i	$-0.559027\pi$
$821$	−10.5660	−0.368757	−0.184378	+	0.982855i	$0.559027\pi$
$822$	0	0
$823$	28.0977	0.979426	0.489713	−	0.871884i	$-0.337102\pi$
$823$	28.0977	0.979426	0.489713	+	0.871884i	$0.337102\pi$
$824$	0	0
$825$	7.93800	0.276366
$826$	0	0
$827$	8.54079	0.296992	0.148496	−	0.988913i	$-0.452557\pi$
$827$	8.54079	0.296992	0.148496	+	0.988913i	$0.452557\pi$
$828$	0	0
$829$	15.2823	0.530776	0.265388	−	0.964142i	$-0.414500\pi$
$829$	15.2823	0.530776	0.265388	+	0.964142i	$0.414500\pi$
$830$	0	0
$831$	21.5832	0.748713
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−23.0157	−0.796493
$836$	0	0
$837$	−3.14640	−0.108756
$838$	0	0
$839$	10.3481	0.357257	0.178629	−	0.983917i	$-0.442834\pi$
$839$	10.3481	0.357257	0.178629	+	0.983917i	$0.442834\pi$
$840$	0	0
$841$	−21.2070	−0.731275
$842$	0	0
$843$	46.5131	1.60200
$844$	0	0
$845$	−35.1359	−1.20871
$846$	0	0
$847$	−0.299472	−0.0102900
$848$	0	0
$849$	−2.93800	−0.100832
$850$	0	0
$851$	46.2270	1.58464
$852$	0	0
$853$	−26.0091	−0.890534	−0.445267	−	0.895398i	$-0.646891\pi$
$853$	−26.0091	−0.890534	−0.445267	+	0.895398i	$0.646891\pi$
$854$	0	0
$855$	−16.2308	−0.555082
$856$	0	0
$857$	8.24655	0.281697	0.140848	−	0.990031i	$-0.455017\pi$
$857$	8.24655	0.281697	0.140848	+	0.990031i	$0.455017\pi$
$858$	0	0
$859$	42.5975	1.45341	0.726704	−	0.686951i	$-0.241051\pi$
$859$	42.5975	1.45341	0.726704	+	0.686951i	$0.241051\pi$
$860$	0	0
$861$	7.93800	0.270526
$862$	0	0
$863$	−25.4011	−0.864662	−0.432331	−	0.901715i	$-0.642309\pi$
$863$	−25.4011	−0.864662	−0.432331	+	0.901715i	$0.642309\pi$
$864$	0	0
$865$	−2.63853	−0.0897128
$866$	0	0
$867$	2.36147	0.0801997
$868$	0	0
$869$	−18.7229	−0.635132
$870$	0	0
$871$	−83.8417	−2.84087
$872$	0	0
$873$	37.0606	1.25431
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	25.2771	0.853546	0.426773	−	0.904359i	$-0.359650\pi$
$877$	25.2771	0.853546	0.426773	+	0.904359i	$0.359650\pi$
$878$	0	0
$879$	33.2756	1.12236
$880$	0	0
$881$	−28.5107	−0.960550	−0.480275	−	0.877118i	$-0.659463\pi$
$881$	−28.5107	−0.960550	−0.480275	+	0.877118i	$0.659463\pi$
$882$	0	0
$883$	−17.1979	−0.578755	−0.289378	−	0.957215i	$-0.593448\pi$
$883$	−17.1979	−0.578755	−0.289378	+	0.957215i	$0.593448\pi$
$884$	0	0
$885$	24.1755	0.812650
$886$	0	0
$887$	−53.4115	−1.79338	−0.896692	−	0.442656i	$-0.854036\pi$
$887$	−53.4115	−1.79338	−0.896692	+	0.442656i	$0.854036\pi$
$888$	0	0
$889$	2.33906	0.0784495
$890$	0	0
$891$	−33.9208	−1.13639
$892$	0	0
$893$	22.1397	0.740878
$894$	0	0
$895$	6.72294	0.224723
$896$	0	0
$897$	86.8259	2.89903
$898$	0	0
$899$	−8.78350	−0.292946
$900$	0	0
$901$	−5.94467	−0.198046
$902$	0	0
$903$	8.08441	0.269032
$904$	0	0
$905$	20.5989	0.684732
$906$	0	0
$907$	−2.80068	−0.0929950	−0.0464975	−	0.998918i	$-0.514806\pi$
$907$	−2.80068	−0.0929950	−0.0464975	+	0.998918i	$0.514806\pi$
$908$	0	0
$909$	37.8322	1.25481
$910$	0	0
$911$	−21.5699	−0.714642	−0.357321	−	0.933982i	$-0.616310\pi$
$911$	−21.5699	−0.714642	−0.357321	+	0.933982i	$0.616310\pi$
$912$	0	0
$913$	46.2270	1.52989
$914$	0	0
$915$	24.5212	0.810646
$916$	0	0
$917$	8.16881	0.269758
$918$	0	0
$919$	−18.5765	−0.612783	−0.306392	−	0.951906i	$-0.599122\pi$
$919$	−18.5765	−0.612783	−0.306392	+	0.951906i	$0.599122\pi$
$920$	0	0
$921$	−53.0739	−1.74884
$922$	0	0
$923$	−102.780	−3.38303
$924$	0	0
$925$	8.72294	0.286808
$926$	0	0
$927$	−32.4039	−1.06428
$928$	0	0
$929$	−13.7520	−0.451189	−0.225594	−	0.974221i	$-0.572432\pi$
$929$	−13.7520	−0.451189	−0.225594	+	0.974221i	$0.572432\pi$
$930$	0	0
$931$	6.29947	0.206457
$932$	0	0
$933$	23.2690	0.761792
$934$	0	0
$935$	3.36147	0.109932
$936$	0	0
$937$	−37.8536	−1.23662	−0.618312	−	0.785933i	$-0.712183\pi$
$937$	−37.8536	−1.23662	−0.618312	+	0.785933i	$0.712183\pi$
$938$	0	0
$939$	12.8918	0.420706
$940$	0	0
$941$	31.2175	1.01766	0.508830	−	0.860867i	$-0.330078\pi$
$941$	31.2175	1.01766	0.508830	+	0.860867i	$0.330078\pi$
$942$	0	0
$943$	−17.8140	−0.580104
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	20.9695	0.681417	0.340708	−	0.940169i	$-0.389333\pi$
$947$	20.9695	0.681417	0.340708	+	0.940169i	$0.389333\pi$
$948$	0	0
$949$	−105.764	−3.43324
$950$	0	0
$951$	−35.7835	−1.16036
$952$	0	0
$953$	−10.7096	−0.346918	−0.173459	−	0.984841i	$-0.555494\pi$
$953$	−10.7096	−0.346918	−0.173459	+	0.984841i	$0.555494\pi$
$954$	0	0
$955$	−25.2375	−0.816666
$956$	0	0
$957$	−22.1597	−0.716323
$958$	0	0
$959$	−16.2928	−0.526122
$960$	0	0
$961$	−21.1001	−0.680650
$962$	0	0
$963$	−16.2480	−0.523584
$964$	0	0
$965$	26.6214	0.856972
$966$	0	0
$967$	7.45921	0.239872	0.119936	−	0.992782i	$-0.461731\pi$
$967$	7.45921	0.239872	0.119936	+	0.992782i	$0.461731\pi$
$968$	0	0
$969$	−14.8760	−0.477886
$970$	0	0
$971$	−43.5107	−1.39632	−0.698162	−	0.715940i	$-0.745999\pi$
$971$	−43.5107	−1.39632	−0.698162	+	0.715940i	$0.745999\pi$
$972$	0	0
$973$	1.27706	0.0409407
$974$	0	0
$975$	16.3839	0.524704
$976$	0	0
$977$	43.6414	1.39621	0.698105	−	0.715995i	$-0.254027\pi$
$977$	43.6414	1.39621	0.698105	+	0.715995i	$0.254027\pi$
$978$	0	0
$979$	36.7678	1.17510
$980$	0	0
$981$	43.8588	1.40030
$982$	0	0
$983$	22.7544	0.725753	0.362877	−	0.931837i	$-0.381795\pi$
$983$	22.7544	0.725753	0.362877	+	0.931837i	$0.381795\pi$
$984$	0	0
$985$	−16.2928	−0.519132
$986$	0	0
$987$	−8.29947	−0.264175
$988$	0	0
$989$	−18.1426	−0.576900
$990$	0	0
$991$	−4.65853	−0.147983	−0.0739916	−	0.997259i	$-0.523574\pi$
$991$	−4.65853	−0.147983	−0.0739916	+	0.997259i	$0.523574\pi$
$992$	0	0
$993$	−19.7983	−0.628279
$994$	0	0
$995$	−16.3615	−0.518693
$996$	0	0
$997$	−41.0739	−1.30082	−0.650412	−	0.759582i	$-0.725404\pi$
$997$	−41.0739	−1.30082	−0.650412	+	0.759582i	$0.725404\pi$
$998$	0	0
$999$	−8.72294	−0.275982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9520.2.a.u.1.3		3
4.3	odd	2		1190.2.a.l.1.1	✓	3
20.19	odd	2		5950.2.a.bk.1.3		3
28.27	even	2		8330.2.a.bs.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.a.l.1.1	✓	3	4.3	odd	2
5950.2.a.bk.1.3		3	20.19	odd	2
8330.2.a.bs.1.3		3	28.27	even	2
9520.2.a.u.1.3		3	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:	\( N \)	\(=\)	\( 9520 = 2^{4} \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9520.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36147 q^{3} -1.00000 q^{5} -1.00000 q^{7} +2.57653 q^{9} +3.36147 q^{11} +6.93800 q^{13} -2.36147 q^{15} -1.00000 q^{17} +6.29947 q^{19} -2.36147 q^{21} +5.29947 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.79160 q^{29} +3.14640 q^{31} +7.93800 q^{33} +1.00000 q^{35} +8.72294 q^{37} +16.3839 q^{39} -3.36147 q^{41} -3.42347 q^{43} -2.57653 q^{45} +3.51454 q^{47} +1.00000 q^{49} -2.36147 q^{51} +5.94467 q^{53} -3.36147 q^{55} +14.8760 q^{57} -10.2375 q^{59} -10.3839 q^{61} -2.57653 q^{63} -6.93800 q^{65} -12.0844 q^{67} +12.5145 q^{69} -14.8140 q^{71} -15.2441 q^{73} +2.36147 q^{75} -3.36147 q^{77} -5.56987 q^{79} -10.0911 q^{81} +13.7520 q^{83} +1.00000 q^{85} -6.59228 q^{87} +10.9380 q^{89} -6.93800 q^{91} +7.43013 q^{93} -6.29947 q^{95} +14.3839 q^{97} +8.66094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{13} - 3 q^{17} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 12 q^{33} + 3 q^{35} + 12 q^{37} + 9 q^{39} - 3 q^{41} - 15 q^{43} - 3 q^{45} - 6 q^{47}+ \cdots + 3 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^13 - 3 * q^17 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 + 12 * q^33 + 3 * q^35 + 12 * q^37 + 9 * q^39 - 3 * q^41 - 15 * q^43 - 3 * q^45 - 6 * q^47 + 3 * q^49 + 6 * q^53 - 3 * q^55 + 21 * q^57 + 9 * q^61 - 3 * q^63 - 9 * q^65 - 15 * q^67 + 21 * q^69 - 9 * q^71 - 15 * q^73 - 3 * q^77 - 12 * q^79 - 9 * q^81 - 6 * q^83 + 3 * q^85 + 18 * q^87 + 21 * q^89 - 9 * q^91 + 27 * q^93 + 3 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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