LMFDB - Embedded newform 2816.2.a.p.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2816,2,Mod(1,2816)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2816.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,-4,0,0,0,5,0,5] 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:	$22.4858732092$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.380224.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 2x^{2} + 5x + 2 $ x^5 - 8x^3 - 2x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.874805$ of defining polynomial
Character	$\chi$	$=$	2816.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35300 q^{3} -4.16794 q^{5} +0.933222 q^{7} +2.53661 q^{9} +1.00000 q^{11} -2.93322 q^{13} -9.80716 q^{15} +2.44626 q^{17} -2.68283 q^{19} +2.19587 q^{21} -3.47129 q^{23} +12.3717 q^{25} -1.09035 q^{27} +4.57245 q^{29} +3.65781 q^{31} +2.35300 q^{33} -3.88961 q^{35} -4.53806 q^{37} -6.90187 q^{39} -4.12618 q^{41} -11.4650 q^{43} -10.5724 q^{45} -3.26265 q^{47} -6.12910 q^{49} +5.75606 q^{51} +0.650132 q^{53} -4.16794 q^{55} -6.31271 q^{57} -2.90965 q^{59} -10.7060 q^{61} +2.36722 q^{63} +12.2255 q^{65} -5.42623 q^{67} -8.16794 q^{69} -7.75129 q^{71} -13.0650 q^{73} +29.1106 q^{75} +0.933222 q^{77} -4.83666 q^{79} -10.1754 q^{81} +0.659664 q^{83} -10.1959 q^{85} +10.7590 q^{87} +2.74629 q^{89} -2.73735 q^{91} +8.60683 q^{93} +11.1819 q^{95} +1.82905 q^{97} +2.53661 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{5} + 5 q^{9} + 5 q^{11} - 10 q^{13} - 4 q^{15} - 2 q^{17} - 12 q^{21} - 6 q^{23} + 3 q^{25} - 12 q^{27} - 10 q^{29} + 2 q^{31} - 16 q^{37} + 12 q^{39} - 2 q^{41} + 4 q^{43} - 20 q^{45} + 2 q^{47}+ \cdots + 5 q^{99}+O(q^{100}) $ 5 * q - 4 * q^5 + 5 * q^9 + 5 * q^11 - 10 * q^13 - 4 * q^15 - 2 * q^17 - 12 * q^21 - 6 * q^23 + 3 * q^25 - 12 * q^27 - 10 * q^29 + 2 * q^31 - 16 * q^37 + 12 * q^39 - 2 * q^41 + 4 * q^43 - 20 * q^45 + 2 * q^47 - 3 * q^49 - 8 * q^53 - 4 * q^55 - 8 * q^57 - 8 * q^59 - 30 * q^61 + 20 * q^63 + 8 * q^65 - 24 * q^69 - 6 * q^71 + 2 * q^73 + 36 * q^75 - 8 * q^79 - 3 * q^81 - 28 * q^85 + 16 * q^87 + 2 * q^89 - 32 * q^91 - 16 * q^93 - 12 * q^95 - 10 * q^97 + 5 * 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
$3$	2.35300	1.35851	0.679253	−	0.733904i	$-0.262304\pi$
$3$	2.35300	1.35851	0.679253	+	0.733904i	$0.262304\pi$
$4$	0	0
$5$	−4.16794	−1.86396	−0.931979	−	0.362511i	$-0.881919\pi$
$5$	−4.16794	−1.86396	−0.931979	+	0.362511i	$0.881919\pi$
$6$	0	0
$7$	0.933222	0.352725	0.176362	−	0.984325i	$-0.443567\pi$
$7$	0.933222	0.352725	0.176362	+	0.984325i	$0.443567\pi$
$8$	0	0
$9$	2.53661	0.845538
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.93322	−0.813530	−0.406765	−	0.913533i	$-0.633343\pi$
$13$	−2.93322	−0.813530	−0.406765	+	0.913533i	$0.633343\pi$
$14$	0	0
$15$	−9.80716	−2.53220
$16$	0	0
$17$	2.44626	0.593306	0.296653	−	0.954985i	$-0.404130\pi$
$17$	2.44626	0.593306	0.296653	+	0.954985i	$0.404130\pi$
$18$	0	0
$19$	−2.68283	−0.615484	−0.307742	−	0.951470i	$-0.599573\pi$
$19$	−2.68283	−0.615484	−0.307742	+	0.951470i	$0.599573\pi$
$20$	0	0
$21$	2.19587	0.479179
$22$	0	0
$23$	−3.47129	−0.723813	−0.361907	−	0.932214i	$-0.617874\pi$
$23$	−3.47129	−0.723813	−0.361907	+	0.932214i	$0.617874\pi$
$24$	0	0
$25$	12.3717	2.47434
$26$	0	0
$27$	−1.09035	−0.209838
$28$	0	0
$29$	4.57245	0.849082	0.424541	−	0.905409i	$-0.360435\pi$
$29$	4.57245	0.849082	0.424541	+	0.905409i	$0.360435\pi$
$30$	0	0
$31$	3.65781	0.656962	0.328481	−	0.944511i	$-0.393463\pi$
$31$	3.65781	0.656962	0.328481	+	0.944511i	$0.393463\pi$
$32$	0	0
$33$	2.35300	0.409605
$34$	0	0
$35$	−3.88961	−0.657465
$36$	0	0
$37$	−4.53806	−0.746053	−0.373027	−	0.927821i	$-0.621680\pi$
$37$	−4.53806	−0.746053	−0.373027	+	0.927821i	$0.621680\pi$
$38$	0	0
$39$	−6.90187	−1.10518
$40$	0	0
$41$	−4.12618	−0.644402	−0.322201	−	0.946671i	$-0.604423\pi$
$41$	−4.12618	−0.644402	−0.322201	+	0.946671i	$0.604423\pi$
$42$	0	0
$43$	−11.4650	−1.74839	−0.874196	−	0.485573i	$-0.838611\pi$
$43$	−11.4650	−1.74839	−0.874196	+	0.485573i	$0.838611\pi$
$44$	0	0
$45$	−10.5724	−1.57605
$46$	0	0
$47$	−3.26265	−0.475907	−0.237953	−	0.971277i	$-0.576477\pi$
$47$	−3.26265	−0.475907	−0.237953	+	0.971277i	$0.576477\pi$
$48$	0	0
$49$	−6.12910	−0.875585
$50$	0	0
$51$	5.75606	0.806009
$52$	0	0
$53$	0.650132	0.0893025	0.0446512	−	0.999003i	$-0.485782\pi$
$53$	0.650132	0.0893025	0.0446512	+	0.999003i	$0.485782\pi$
$54$	0	0
$55$	−4.16794	−0.562005
$56$	0	0
$57$	−6.31271	−0.836139
$58$	0	0
$59$	−2.90965	−0.378804	−0.189402	−	0.981900i	$-0.560655\pi$
$59$	−2.90965	−0.378804	−0.189402	+	0.981900i	$0.560655\pi$
$60$	0	0
$61$	−10.7060	−1.37076	−0.685382	−	0.728184i	$-0.740365\pi$
$61$	−10.7060	−1.37076	−0.685382	+	0.728184i	$0.740365\pi$
$62$	0	0
$63$	2.36722	0.298242
$64$	0	0
$65$	12.2255	1.51639
$66$	0	0
$67$	−5.42623	−0.662919	−0.331460	−	0.943469i	$-0.607541\pi$
$67$	−5.42623	−0.662919	−0.331460	+	0.943469i	$0.607541\pi$
$68$	0	0
$69$	−8.16794	−0.983304
$70$	0	0
$71$	−7.75129	−0.919909	−0.459955	−	0.887942i	$-0.652134\pi$
$71$	−7.75129	−0.919909	−0.459955	+	0.887942i	$0.652134\pi$
$72$	0	0
$73$	−13.0650	−1.52915	−0.764574	−	0.644536i	$-0.777051\pi$
$73$	−13.0650	−1.52915	−0.764574	+	0.644536i	$0.777051\pi$
$74$	0	0
$75$	29.1106	3.36141
$76$	0	0
$77$	0.933222	0.106351
$78$	0	0
$79$	−4.83666	−0.544166	−0.272083	−	0.962274i	$-0.587713\pi$
$79$	−4.83666	−0.544166	−0.272083	+	0.962274i	$0.587713\pi$
$80$	0	0
$81$	−10.1754	−1.13060
$82$	0	0
$83$	0.659664	0.0724075	0.0362038	−	0.999344i	$-0.488473\pi$
$83$	0.659664	0.0724075	0.0362038	+	0.999344i	$0.488473\pi$
$84$	0	0
$85$	−10.1959	−1.10590
$86$	0	0
$87$	10.7590	1.15348
$88$	0	0
$89$	2.74629	0.291107	0.145553	−	0.989350i	$-0.453504\pi$
$89$	2.74629	0.291107	0.145553	+	0.989350i	$0.453504\pi$
$90$	0	0
$91$	−2.73735	−0.286952
$92$	0	0
$93$	8.60683	0.892487
$94$	0	0
$95$	11.1819	1.14724
$96$	0	0
$97$	1.82905	0.185712	0.0928561	−	0.995680i	$-0.470400\pi$
$97$	1.82905	0.185712	0.0928561	+	0.995680i	$0.470400\pi$
$98$	0	0
$99$	2.53661	0.254939
$100$	0	0
$101$	12.6811	1.26182	0.630908	−	0.775857i	$-0.282682\pi$
$101$	12.6811	1.26182	0.630908	+	0.775857i	$0.282682\pi$
$102$	0	0
$103$	−3.86644	−0.380972	−0.190486	−	0.981690i	$-0.561006\pi$
$103$	−3.86644	−0.380972	−0.190486	+	0.981690i	$0.561006\pi$
$104$	0	0
$105$	−9.15226	−0.893170
$106$	0	0
$107$	12.7821	1.23570	0.617848	−	0.786298i	$-0.288005\pi$
$107$	12.7821	1.23570	0.617848	+	0.786298i	$0.288005\pi$
$108$	0	0
$109$	−1.85065	−0.177260	−0.0886299	−	0.996065i	$-0.528249\pi$
$109$	−1.85065	−0.177260	−0.0886299	+	0.996065i	$0.528249\pi$
$110$	0	0
$111$	−10.6781	−1.01352
$112$	0	0
$113$	16.6815	1.56926	0.784632	−	0.619962i	$-0.212852\pi$
$113$	16.6815	1.56926	0.784632	+	0.619962i	$0.212852\pi$
$114$	0	0
$115$	14.4681	1.34916
$116$	0	0
$117$	−7.44045	−0.687870
$118$	0	0
$119$	2.28291	0.209274
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.70891	−0.875423
$124$	0	0
$125$	−30.7248	−2.74811
$126$	0	0
$127$	6.51013	0.577680	0.288840	−	0.957377i	$-0.406730\pi$
$127$	6.51013	0.577680	0.288840	+	0.957377i	$0.406730\pi$
$128$	0	0
$129$	−26.9771	−2.37520
$130$	0	0
$131$	−6.91569	−0.604227	−0.302114	−	0.953272i	$-0.597692\pi$
$131$	−6.91569	−0.604227	−0.302114	+	0.953272i	$0.597692\pi$
$132$	0	0
$133$	−2.50368	−0.217097
$134$	0	0
$135$	4.54451	0.391129
$136$	0	0
$137$	−5.81952	−0.497195	−0.248598	−	0.968607i	$-0.579970\pi$
$137$	−5.81952	−0.497195	−0.248598	+	0.968607i	$0.579970\pi$
$138$	0	0
$139$	−9.53529	−0.808772	−0.404386	−	0.914588i	$-0.632515\pi$
$139$	−9.53529	−0.808772	−0.404386	+	0.914588i	$0.632515\pi$
$140$	0	0
$141$	−7.67702	−0.646522
$142$	0	0
$143$	−2.93322	−0.245288
$144$	0	0
$145$	−19.0577	−1.58265
$146$	0	0
$147$	−14.4218	−1.18949
$148$	0	0
$149$	−13.9221	−1.14055	−0.570273	−	0.821455i	$-0.693162\pi$
$149$	−13.9221	−1.14055	−0.570273	+	0.821455i	$0.693162\pi$
$150$	0	0
$151$	15.2226	1.23880	0.619398	−	0.785077i	$-0.287377\pi$
$151$	15.2226	1.23880	0.619398	+	0.785077i	$0.287377\pi$
$152$	0	0
$153$	6.20522	0.501662
$154$	0	0
$155$	−15.2455	−1.22455
$156$	0	0
$157$	0.873940	0.0697480	0.0348740	−	0.999392i	$-0.488897\pi$
$157$	0.873940	0.0697480	0.0348740	+	0.999392i	$0.488897\pi$
$158$	0	0
$159$	1.52976	0.121318
$160$	0	0
$161$	−3.23948	−0.255307
$162$	0	0
$163$	7.52686	0.589549	0.294775	−	0.955567i	$-0.404755\pi$
$163$	7.52686	0.589549	0.294775	+	0.955567i	$0.404755\pi$
$164$	0	0
$165$	−9.80716	−0.763487
$166$	0	0
$167$	0.142904	0.0110583	0.00552913	−	0.999985i	$-0.498240\pi$
$167$	0.142904	0.0110583	0.00552913	+	0.999985i	$0.498240\pi$
$168$	0	0
$169$	−4.39621	−0.338170
$170$	0	0
$171$	−6.80531	−0.520415
$172$	0	0
$173$	1.12910	0.0858436	0.0429218	−	0.999078i	$-0.486333\pi$
$173$	1.12910	0.0858436	0.0429218	+	0.999078i	$0.486333\pi$
$174$	0	0
$175$	11.5456	0.872762
$176$	0	0
$177$	−6.84641	−0.514608
$178$	0	0
$179$	−15.1620	−1.13326	−0.566631	−	0.823972i	$-0.691754\pi$
$179$	−15.1620	−1.13326	−0.566631	+	0.823972i	$0.691754\pi$
$180$	0	0
$181$	−16.9211	−1.25773	−0.628867	−	0.777513i	$-0.716481\pi$
$181$	−16.9211	−1.25773	−0.628867	+	0.777513i	$0.716481\pi$
$182$	0	0
$183$	−25.1912	−1.86219
$184$	0	0
$185$	18.9144	1.39061
$186$	0	0
$187$	2.44626	0.178888
$188$	0	0
$189$	−1.01754	−0.0740151
$190$	0	0
$191$	16.6720	1.20635	0.603174	−	0.797610i	$-0.293903\pi$
$191$	16.6720	1.20635	0.603174	+	0.797610i	$0.293903\pi$
$192$	0	0
$193$	0.180700	0.0130071	0.00650354	−	0.999979i	$-0.497930\pi$
$193$	0.180700	0.0130071	0.00650354	+	0.999979i	$0.497930\pi$
$194$	0	0
$195$	28.7666	2.06002
$196$	0	0
$197$	−8.38885	−0.597681	−0.298840	−	0.954303i	$-0.596600\pi$
$197$	−8.38885	−0.597681	−0.298840	+	0.954303i	$0.596600\pi$
$198$	0	0
$199$	22.0996	1.56660	0.783298	−	0.621646i	$-0.213536\pi$
$199$	22.0996	1.56660	0.783298	+	0.621646i	$0.213536\pi$
$200$	0	0
$201$	−12.7679	−0.900579
$202$	0	0
$203$	4.26711	0.299492
$204$	0	0
$205$	17.1977	1.20114
$206$	0	0
$207$	−8.80531	−0.612011
$208$	0	0
$209$	−2.68283	−0.185575
$210$	0	0
$211$	−15.0390	−1.03533	−0.517663	−	0.855585i	$-0.673198\pi$
$211$	−15.0390	−1.03533	−0.517663	+	0.855585i	$0.673198\pi$
$212$	0	0
$213$	−18.2388	−1.24970
$214$	0	0
$215$	47.7853	3.25893
$216$	0	0
$217$	3.41355	0.231727
$218$	0	0
$219$	−30.7421	−2.07736
$220$	0	0
$221$	−7.17543	−0.482672
$222$	0	0
$223$	10.2766	0.688172	0.344086	−	0.938938i	$-0.388189\pi$
$223$	10.2766	0.688172	0.344086	+	0.938938i	$0.388189\pi$
$224$	0	0
$225$	31.3822	2.09215
$226$	0	0
$227$	15.6688	1.03998	0.519989	−	0.854173i	$-0.325936\pi$
$227$	15.6688	1.03998	0.519989	+	0.854173i	$0.325936\pi$
$228$	0	0
$229$	14.0747	0.930083	0.465041	−	0.885289i	$-0.346039\pi$
$229$	14.0747	0.930083	0.465041	+	0.885289i	$0.346039\pi$
$230$	0	0
$231$	2.19587	0.144478
$232$	0	0
$233$	12.0993	0.792652	0.396326	−	0.918110i	$-0.370285\pi$
$233$	12.0993	0.792652	0.396326	+	0.918110i	$0.370285\pi$
$234$	0	0
$235$	13.5985	0.887070
$236$	0	0
$237$	−11.3807	−0.739253
$238$	0	0
$239$	21.7479	1.40675	0.703377	−	0.710817i	$-0.251675\pi$
$239$	21.7479	1.40675	0.703377	+	0.710817i	$0.251675\pi$
$240$	0	0
$241$	−6.93967	−0.447024	−0.223512	−	0.974701i	$-0.571752\pi$
$241$	−6.93967	−0.447024	−0.223512	+	0.974701i	$0.571752\pi$
$242$	0	0
$243$	−20.6718	−1.32609
$244$	0	0
$245$	25.5457	1.63205
$246$	0	0
$247$	7.86935	0.500714
$248$	0	0
$249$	1.55219	0.0983660
$250$	0	0
$251$	17.7279	1.11897	0.559486	−	0.828840i	$-0.310999\pi$
$251$	17.7279	1.11897	0.559486	+	0.828840i	$0.310999\pi$
$252$	0	0
$253$	−3.47129	−0.218238
$254$	0	0
$255$	−23.9909	−1.50237
$256$	0	0
$257$	17.7545	1.10750	0.553748	−	0.832684i	$-0.313197\pi$
$257$	17.7545	1.10750	0.553748	+	0.832684i	$0.313197\pi$
$258$	0	0
$259$	−4.23502	−0.263152
$260$	0	0
$261$	11.5985	0.717931
$262$	0	0
$263$	−27.6424	−1.70450	−0.852252	−	0.523132i	$-0.824763\pi$
$263$	−27.6424	−1.70450	−0.852252	+	0.523132i	$0.824763\pi$
$264$	0	0
$265$	−2.70971	−0.166456
$266$	0	0
$267$	6.46203	0.395470
$268$	0	0
$269$	−10.6283	−0.648020	−0.324010	−	0.946054i	$-0.605031\pi$
$269$	−10.6283	−0.648020	−0.324010	+	0.946054i	$0.605031\pi$
$270$	0	0
$271$	−19.7481	−1.19961	−0.599805	−	0.800146i	$-0.704755\pi$
$271$	−19.7481	−1.19961	−0.599805	+	0.800146i	$0.704755\pi$
$272$	0	0
$273$	−6.44098	−0.389826
$274$	0	0
$275$	12.3717	0.746042
$276$	0	0
$277$	12.5353	0.753172	0.376586	−	0.926382i	$-0.377098\pi$
$277$	12.5353	0.753172	0.376586	+	0.926382i	$0.377098\pi$
$278$	0	0
$279$	9.27845	0.555486
$280$	0	0
$281$	23.8037	1.42001	0.710006	−	0.704195i	$-0.248692\pi$
$281$	23.8037	1.42001	0.710006	+	0.704195i	$0.248692\pi$
$282$	0	0
$283$	27.7294	1.64834	0.824172	−	0.566340i	$-0.191641\pi$
$283$	27.7294	1.64834	0.824172	+	0.566340i	$0.191641\pi$
$284$	0	0
$285$	26.3110	1.55853
$286$	0	0
$287$	−3.85065	−0.227297
$288$	0	0
$289$	−11.0158	−0.647988
$290$	0	0
$291$	4.30376	0.252291
$292$	0	0
$293$	5.61389	0.327967	0.163984	−	0.986463i	$-0.447566\pi$
$293$	5.61389	0.327967	0.163984	+	0.986463i	$0.447566\pi$
$294$	0	0
$295$	12.1272	0.706075
$296$	0	0
$297$	−1.09035	−0.0632685
$298$	0	0
$299$	10.1821	0.588843
$300$	0	0
$301$	−10.6994	−0.616701
$302$	0	0
$303$	29.8386	1.71419
$304$	0	0
$305$	44.6220	2.55505
$306$	0	0
$307$	−27.8554	−1.58979	−0.794895	−	0.606747i	$-0.792474\pi$
$307$	−27.8554	−1.58979	−0.794895	+	0.606747i	$0.792474\pi$
$308$	0	0
$309$	−9.09775	−0.517553
$310$	0	0
$311$	−24.4075	−1.38402	−0.692012	−	0.721886i	$-0.743276\pi$
$311$	−24.4075	−1.38402	−0.692012	+	0.721886i	$0.743276\pi$
$312$	0	0
$313$	−9.13514	−0.516349	−0.258174	−	0.966098i	$-0.583121\pi$
$313$	−9.13514	−0.516349	−0.258174	+	0.966098i	$0.583121\pi$
$314$	0	0
$315$	−9.86644	−0.555911
$316$	0	0
$317$	17.9501	1.00818	0.504088	−	0.863652i	$-0.331829\pi$
$317$	17.9501	1.00818	0.504088	+	0.863652i	$0.331829\pi$
$318$	0	0
$319$	4.57245	0.256008
$320$	0	0
$321$	30.0764	1.67870
$322$	0	0
$323$	−6.56291	−0.365170
$324$	0	0
$325$	−36.2890	−2.01295
$326$	0	0
$327$	−4.35457	−0.240809
$328$	0	0
$329$	−3.04478	−0.167864
$330$	0	0
$331$	13.1934	0.725173	0.362586	−	0.931950i	$-0.381894\pi$
$331$	13.1934	0.725173	0.362586	+	0.931950i	$0.381894\pi$
$332$	0	0
$333$	−11.5113	−0.630816
$334$	0	0
$335$	22.6162	1.23565
$336$	0	0
$337$	23.8712	1.30035	0.650173	−	0.759787i	$-0.274697\pi$
$337$	23.8712	1.30035	0.650173	+	0.759787i	$0.274697\pi$
$338$	0	0
$339$	39.2516	2.13185
$340$	0	0
$341$	3.65781	0.198082
$342$	0	0
$343$	−12.2524	−0.661566
$344$	0	0
$345$	34.0435	1.83284
$346$	0	0
$347$	−37.2210	−1.99813	−0.999065	−	0.0432373i	$-0.986233\pi$
$347$	−37.2210	−1.99813	−0.999065	+	0.0432373i	$0.986233\pi$
$348$	0	0
$349$	−28.4140	−1.52097	−0.760483	−	0.649358i	$-0.775038\pi$
$349$	−28.4140	−1.52097	−0.760483	+	0.649358i	$0.775038\pi$
$350$	0	0
$351$	3.19824	0.170709
$352$	0	0
$353$	−6.72158	−0.357754	−0.178877	−	0.983871i	$-0.557246\pi$
$353$	−6.72158	−0.357754	−0.178877	+	0.983871i	$0.557246\pi$
$354$	0	0
$355$	32.3069	1.71467
$356$	0	0
$357$	5.37168	0.284300
$358$	0	0
$359$	19.8260	1.04638	0.523188	−	0.852217i	$-0.324743\pi$
$359$	19.8260	1.04638	0.523188	+	0.852217i	$0.324743\pi$
$360$	0	0
$361$	−11.8024	−0.621179
$362$	0	0
$363$	2.35300	0.123501
$364$	0	0
$365$	54.4543	2.85027
$366$	0	0
$367$	−4.54822	−0.237415	−0.118708	−	0.992929i	$-0.537875\pi$
$367$	−4.54822	−0.237415	−0.118708	+	0.992929i	$0.537875\pi$
$368$	0	0
$369$	−10.4665	−0.544866
$370$	0	0
$371$	0.606718	0.0314992
$372$	0	0
$373$	33.3992	1.72934	0.864672	−	0.502336i	$-0.167526\pi$
$373$	33.3992	1.72934	0.864672	+	0.502336i	$0.167526\pi$
$374$	0	0
$375$	−72.2956	−3.73333
$376$	0	0
$377$	−13.4120	−0.690753
$378$	0	0
$379$	−9.24843	−0.475060	−0.237530	−	0.971380i	$-0.576338\pi$
$379$	−9.24843	−0.475060	−0.237530	+	0.971380i	$0.576338\pi$
$380$	0	0
$381$	15.3183	0.784782
$382$	0	0
$383$	−31.6080	−1.61509	−0.807547	−	0.589803i	$-0.799205\pi$
$383$	−31.6080	−1.61509	−0.807547	+	0.589803i	$0.799205\pi$
$384$	0	0
$385$	−3.88961	−0.198233
$386$	0	0
$387$	−29.0822	−1.47833
$388$	0	0
$389$	4.99543	0.253278	0.126639	−	0.991949i	$-0.459581\pi$
$389$	4.99543	0.253278	0.126639	+	0.991949i	$0.459581\pi$
$390$	0	0
$391$	−8.49168	−0.429443
$392$	0	0
$393$	−16.2726	−0.820846
$394$	0	0
$395$	20.1589	1.01430
$396$	0	0
$397$	−17.7950	−0.893107	−0.446553	−	0.894757i	$-0.647349\pi$
$397$	−17.7950	−0.893107	−0.446553	+	0.894757i	$0.647349\pi$
$398$	0	0
$399$	−5.89116	−0.294927
$400$	0	0
$401$	−12.1739	−0.607934	−0.303967	−	0.952683i	$-0.598311\pi$
$401$	−12.1739	−0.607934	−0.303967	+	0.952683i	$0.598311\pi$
$402$	0	0
$403$	−10.7292	−0.534458
$404$	0	0
$405$	42.4106	2.10740
$406$	0	0
$407$	−4.53806	−0.224943
$408$	0	0
$409$	−0.587997	−0.0290746	−0.0145373	−	0.999894i	$-0.504628\pi$
$409$	−0.587997	−0.0290746	−0.0145373	+	0.999894i	$0.504628\pi$
$410$	0	0
$411$	−13.6933	−0.675442
$412$	0	0
$413$	−2.71535	−0.133614
$414$	0	0
$415$	−2.74944	−0.134965
$416$	0	0
$417$	−22.4365	−1.09872
$418$	0	0
$419$	−11.1507	−0.544748	−0.272374	−	0.962191i	$-0.587809\pi$
$419$	−11.1507	−0.544748	−0.272374	+	0.962191i	$0.587809\pi$
$420$	0	0
$421$	−34.8398	−1.69799	−0.848995	−	0.528401i	$-0.822792\pi$
$421$	−34.8398	−1.69799	−0.848995	+	0.528401i	$0.822792\pi$
$422$	0	0
$423$	−8.27608	−0.402397
$424$	0	0
$425$	30.2645	1.46804
$426$	0	0
$427$	−9.99108	−0.483502
$428$	0	0
$429$	−6.90187	−0.333226
$430$	0	0
$431$	24.8647	1.19769	0.598846	−	0.800864i	$-0.295626\pi$
$431$	24.8647	1.19769	0.598846	+	0.800864i	$0.295626\pi$
$432$	0	0
$433$	−0.547704	−0.0263210	−0.0131605	−	0.999913i	$-0.504189\pi$
$433$	−0.547704	−0.0263210	−0.0131605	+	0.999913i	$0.504189\pi$
$434$	0	0
$435$	−44.8427	−2.15004
$436$	0	0
$437$	9.31288	0.445495
$438$	0	0
$439$	27.5308	1.31397	0.656987	−	0.753902i	$-0.271831\pi$
$439$	27.5308	1.31397	0.656987	+	0.753902i	$0.271831\pi$
$440$	0	0
$441$	−15.5471	−0.740340
$442$	0	0
$443$	8.05454	0.382683	0.191341	−	0.981524i	$-0.438716\pi$
$443$	8.05454	0.382683	0.191341	+	0.981524i	$0.438716\pi$
$444$	0	0
$445$	−11.4464	−0.542611
$446$	0	0
$447$	−32.7588	−1.54944
$448$	0	0
$449$	−33.7705	−1.59373	−0.796865	−	0.604158i	$-0.793510\pi$
$449$	−33.7705	−1.59373	−0.796865	+	0.604158i	$0.793510\pi$
$450$	0	0
$451$	−4.12618	−0.194294
$452$	0	0
$453$	35.8187	1.68291
$454$	0	0
$455$	11.4091	0.534867
$456$	0	0
$457$	3.83220	0.179263	0.0896313	−	0.995975i	$-0.471431\pi$
$457$	3.83220	0.179263	0.0896313	+	0.995975i	$0.471431\pi$
$458$	0	0
$459$	−2.66728	−0.124498
$460$	0	0
$461$	15.8380	0.737649	0.368825	−	0.929499i	$-0.379760\pi$
$461$	15.8380	0.737649	0.368825	+	0.929499i	$0.379760\pi$
$462$	0	0
$463$	−30.4131	−1.41341	−0.706707	−	0.707506i	$-0.749820\pi$
$463$	−30.4131	−1.41341	−0.706707	+	0.707506i	$0.749820\pi$
$464$	0	0
$465$	−35.8727	−1.66356
$466$	0	0
$467$	20.1815	0.933887	0.466943	−	0.884287i	$-0.345355\pi$
$467$	20.1815	0.933887	0.466943	+	0.884287i	$0.345355\pi$
$468$	0	0
$469$	−5.06388	−0.233828
$470$	0	0
$471$	2.05638	0.0947531
$472$	0	0
$473$	−11.4650	−0.527160
$474$	0	0
$475$	−33.1912	−1.52292
$476$	0	0
$477$	1.64913	0.0755086
$478$	0	0
$479$	−12.5604	−0.573901	−0.286951	−	0.957945i	$-0.592642\pi$
$479$	−12.5604	−0.573901	−0.286951	+	0.957945i	$0.592642\pi$
$480$	0	0
$481$	13.3111	0.606936
$482$	0	0
$483$	−7.62250	−0.346836
$484$	0	0
$485$	−7.62338	−0.346160
$486$	0	0
$487$	14.2519	0.645814	0.322907	−	0.946431i	$-0.395340\pi$
$487$	14.2519	0.645814	0.322907	+	0.946431i	$0.395340\pi$
$488$	0	0
$489$	17.7107	0.800906
$490$	0	0
$491$	20.4046	0.920849	0.460424	−	0.887699i	$-0.347697\pi$
$491$	20.4046	0.920849	0.460424	+	0.887699i	$0.347697\pi$
$492$	0	0
$493$	11.1854	0.503765
$494$	0	0
$495$	−10.5724	−0.475196
$496$	0	0
$497$	−7.23368	−0.324475
$498$	0	0
$499$	33.3751	1.49407	0.747037	−	0.664782i	$-0.231476\pi$
$499$	33.3751	1.49407	0.747037	+	0.664782i	$0.231476\pi$
$500$	0	0
$501$	0.336253	0.0150227
$502$	0	0
$503$	14.8148	0.660561	0.330281	−	0.943883i	$-0.392857\pi$
$503$	14.8148	0.660561	0.330281	+	0.943883i	$0.392857\pi$
$504$	0	0
$505$	−52.8541	−2.35197
$506$	0	0
$507$	−10.3443	−0.459405
$508$	0	0
$509$	23.9464	1.06140	0.530702	−	0.847559i	$-0.321928\pi$
$509$	23.9464	1.06140	0.530702	+	0.847559i	$0.321928\pi$
$510$	0	0
$511$	−12.1926	−0.539369
$512$	0	0
$513$	2.92523	0.129152
$514$	0	0
$515$	16.1151	0.710116
$516$	0	0
$517$	−3.26265	−0.143491
$518$	0	0
$519$	2.65676	0.116619
$520$	0	0
$521$	−20.6157	−0.903189	−0.451594	−	0.892223i	$-0.649145\pi$
$521$	−20.6157	−0.903189	−0.451594	+	0.892223i	$0.649145\pi$
$522$	0	0
$523$	22.6014	0.988292	0.494146	−	0.869379i	$-0.335481\pi$
$523$	22.6014	0.988292	0.494146	+	0.869379i	$0.335481\pi$
$524$	0	0
$525$	27.1667	1.18565
$526$	0	0
$527$	8.94797	0.389779
$528$	0	0
$529$	−10.9502	−0.476095
$530$	0	0
$531$	−7.38066	−0.320293
$532$	0	0
$533$	12.1030	0.524240
$534$	0	0
$535$	−53.2752	−2.30329
$536$	0	0
$537$	−35.6762	−1.53954
$538$	0	0
$539$	−6.12910	−0.263999
$540$	0	0
$541$	−5.25411	−0.225892	−0.112946	−	0.993601i	$-0.536029\pi$
$541$	−5.25411	−0.225892	−0.112946	+	0.993601i	$0.536029\pi$
$542$	0	0
$543$	−39.8153	−1.70864
$544$	0	0
$545$	7.71338	0.330405
$546$	0	0
$547$	31.4365	1.34413	0.672064	−	0.740493i	$-0.265408\pi$
$547$	31.4365	1.34413	0.672064	+	0.740493i	$0.265408\pi$
$548$	0	0
$549$	−27.1570	−1.15903
$550$	0	0
$551$	−12.2671	−0.522596
$552$	0	0
$553$	−4.51368	−0.191941
$554$	0	0
$555$	44.5055	1.88915
$556$	0	0
$557$	13.8905	0.588561	0.294280	−	0.955719i	$-0.404920\pi$
$557$	13.8905	0.588561	0.294280	+	0.955719i	$0.404920\pi$
$558$	0	0
$559$	33.6293	1.42237
$560$	0	0
$561$	5.75606	0.243021
$562$	0	0
$563$	20.5160	0.864647	0.432324	−	0.901719i	$-0.357694\pi$
$563$	20.5160	0.864647	0.432324	+	0.901719i	$0.357694\pi$
$564$	0	0
$565$	−69.5275	−2.92504
$566$	0	0
$567$	−9.49594	−0.398792
$568$	0	0
$569$	20.9308	0.877463	0.438731	−	0.898618i	$-0.355428\pi$
$569$	20.9308	0.877463	0.438731	+	0.898618i	$0.355428\pi$
$570$	0	0
$571$	−6.00582	−0.251336	−0.125668	−	0.992072i	$-0.540107\pi$
$571$	−6.00582	−0.251336	−0.125668	+	0.992072i	$0.540107\pi$
$572$	0	0
$573$	39.2293	1.63883
$574$	0	0
$575$	−42.9457	−1.79096
$576$	0	0
$577$	−45.4212	−1.89091	−0.945455	−	0.325753i	$-0.894382\pi$
$577$	−45.4212	−1.89091	−0.945455	+	0.325753i	$0.894382\pi$
$578$	0	0
$579$	0.425187	0.0176702
$580$	0	0
$581$	0.615613	0.0255399
$582$	0	0
$583$	0.650132	0.0269257
$584$	0	0
$585$	31.0113	1.28216
$586$	0	0
$587$	4.95230	0.204403	0.102202	−	0.994764i	$-0.467411\pi$
$587$	4.95230	0.204403	0.102202	+	0.994764i	$0.467411\pi$
$588$	0	0
$589$	−9.81329	−0.404350
$590$	0	0
$591$	−19.7390	−0.811952
$592$	0	0
$593$	−17.6332	−0.724110	−0.362055	−	0.932157i	$-0.617925\pi$
$593$	−17.6332	−0.724110	−0.362055	+	0.932157i	$0.617925\pi$
$594$	0	0
$595$	−9.51502	−0.390078
$596$	0	0
$597$	52.0003	2.12823
$598$	0	0
$599$	45.1533	1.84491	0.922456	−	0.386101i	$-0.126178\pi$
$599$	45.1533	1.84491	0.922456	+	0.386101i	$0.126178\pi$
$600$	0	0
$601$	15.6375	0.637867	0.318933	−	0.947777i	$-0.396675\pi$
$601$	15.6375	0.637867	0.318933	+	0.947777i	$0.396675\pi$
$602$	0	0
$603$	−13.7642	−0.560523
$604$	0	0
$605$	−4.16794	−0.169451
$606$	0	0
$607$	41.9318	1.70196	0.850979	−	0.525200i	$-0.176010\pi$
$607$	41.9318	1.70196	0.850979	+	0.525200i	$0.176010\pi$
$608$	0	0
$609$	10.0405	0.406862
$610$	0	0
$611$	9.57008	0.387164
$612$	0	0
$613$	27.8914	1.12652	0.563262	−	0.826278i	$-0.309546\pi$
$613$	27.8914	1.12652	0.563262	+	0.826278i	$0.309546\pi$
$614$	0	0
$615$	40.4662	1.63175
$616$	0	0
$617$	14.7471	0.593697	0.296849	−	0.954925i	$-0.404064\pi$
$617$	14.7471	0.593697	0.296849	+	0.954925i	$0.404064\pi$
$618$	0	0
$619$	5.12150	0.205851	0.102925	−	0.994689i	$-0.467180\pi$
$619$	5.12150	0.205851	0.102925	+	0.994689i	$0.467180\pi$
$620$	0	0
$621$	3.78492	0.151883
$622$	0	0
$623$	2.56290	0.102681
$624$	0	0
$625$	66.2007	2.64803
$626$	0	0
$627$	−6.31271	−0.252105
$628$	0	0
$629$	−11.1013	−0.442638
$630$	0	0
$631$	−48.5097	−1.93114	−0.965570	−	0.260142i	$-0.916231\pi$
$631$	−48.5097	−1.93114	−0.965570	+	0.260142i	$0.916231\pi$
$632$	0	0
$633$	−35.3867	−1.40650
$634$	0	0
$635$	−27.1338	−1.07677
$636$	0	0
$637$	17.9780	0.712314
$638$	0	0
$639$	−19.6620	−0.777818
$640$	0	0
$641$	−7.22155	−0.285234	−0.142617	−	0.989778i	$-0.545552\pi$
$641$	−7.22155	−0.285234	−0.142617	+	0.989778i	$0.545552\pi$
$642$	0	0
$643$	−11.9607	−0.471684	−0.235842	−	0.971791i	$-0.575785\pi$
$643$	−11.9607	−0.471684	−0.235842	+	0.971791i	$0.575785\pi$
$644$	0	0
$645$	112.439	4.42728
$646$	0	0
$647$	−26.1952	−1.02984	−0.514920	−	0.857238i	$-0.672178\pi$
$647$	−26.1952	−1.02984	−0.514920	+	0.857238i	$0.672178\pi$
$648$	0	0
$649$	−2.90965	−0.114214
$650$	0	0
$651$	8.03209	0.314802
$652$	0	0
$653$	−34.2180	−1.33906	−0.669528	−	0.742787i	$-0.733503\pi$
$653$	−34.2180	−1.33906	−0.669528	+	0.742787i	$0.733503\pi$
$654$	0	0
$655$	28.8242	1.12625
$656$	0	0
$657$	−33.1410	−1.29295
$658$	0	0
$659$	−2.73052	−0.106366	−0.0531831	−	0.998585i	$-0.516937\pi$
$659$	−2.73052	−0.106366	−0.0531831	+	0.998585i	$0.516937\pi$
$660$	0	0
$661$	−11.2478	−0.437488	−0.218744	−	0.975782i	$-0.570196\pi$
$661$	−11.2478	−0.437488	−0.218744	+	0.975782i	$0.570196\pi$
$662$	0	0
$663$	−16.8838	−0.655713
$664$	0	0
$665$	10.4352	0.404659
$666$	0	0
$667$	−15.8723	−0.614577
$668$	0	0
$669$	24.1808	0.934885
$670$	0	0
$671$	−10.7060	−0.413301
$672$	0	0
$673$	−20.0432	−0.772609	−0.386305	−	0.922371i	$-0.626249\pi$
$673$	−20.0432	−0.772609	−0.386305	+	0.922371i	$0.626249\pi$
$674$	0	0
$675$	−13.4895	−0.519211
$676$	0	0
$677$	27.1414	1.04313	0.521564	−	0.853212i	$-0.325349\pi$
$677$	27.1414	1.04313	0.521564	+	0.853212i	$0.325349\pi$
$678$	0	0
$679$	1.70691	0.0655053
$680$	0	0
$681$	36.8688	1.41282
$682$	0	0
$683$	−41.8019	−1.59951	−0.799753	−	0.600330i	$-0.795036\pi$
$683$	−41.8019	−1.59951	−0.799753	+	0.600330i	$0.795036\pi$
$684$	0	0
$685$	24.2554	0.926751
$686$	0	0
$687$	33.1178	1.26352
$688$	0	0
$689$	−1.90698	−0.0726502
$690$	0	0
$691$	−28.7753	−1.09466	−0.547332	−	0.836916i	$-0.684356\pi$
$691$	−28.7753	−1.09466	−0.547332	+	0.836916i	$0.684356\pi$
$692$	0	0
$693$	2.36722	0.0899234
$694$	0	0
$695$	39.7425	1.50752
$696$	0	0
$697$	−10.0937	−0.382327
$698$	0	0
$699$	28.4697	1.07682
$700$	0	0
$701$	22.5826	0.852933	0.426467	−	0.904503i	$-0.359758\pi$
$701$	22.5826	0.852933	0.426467	+	0.904503i	$0.359758\pi$
$702$	0	0
$703$	12.1749	0.459184
$704$	0	0
$705$	31.9973	1.20509
$706$	0	0
$707$	11.8343	0.445074
$708$	0	0
$709$	14.5420	0.546138	0.273069	−	0.961994i	$-0.411961\pi$
$709$	14.5420	0.546138	0.273069	+	0.961994i	$0.411961\pi$
$710$	0	0
$711$	−12.2687	−0.460113
$712$	0	0
$713$	−12.6973	−0.475518
$714$	0	0
$715$	12.2255	0.457207
$716$	0	0
$717$	51.1728	1.91108
$718$	0	0
$719$	5.98413	0.223170	0.111585	−	0.993755i	$-0.464407\pi$
$719$	5.98413	0.223170	0.111585	+	0.993755i	$0.464407\pi$
$720$	0	0
$721$	−3.60825	−0.134378
$722$	0	0
$723$	−16.3291	−0.607284
$724$	0	0
$725$	56.5690	2.10092
$726$	0	0
$727$	−13.7234	−0.508973	−0.254486	−	0.967076i	$-0.581906\pi$
$727$	−13.7234	−0.508973	−0.254486	+	0.967076i	$0.581906\pi$
$728$	0	0
$729$	−18.1144	−0.670902
$730$	0	0
$731$	−28.0463	−1.03733
$732$	0	0
$733$	−46.4631	−1.71615	−0.858076	−	0.513522i	$-0.828340\pi$
$733$	−46.4631	−1.71615	−0.858076	+	0.513522i	$0.828340\pi$
$734$	0	0
$735$	60.1090	2.21716
$736$	0	0
$737$	−5.42623	−0.199878
$738$	0	0
$739$	−12.5779	−0.462686	−0.231343	−	0.972872i	$-0.574312\pi$
$739$	−12.5779	−0.462686	−0.231343	+	0.972872i	$0.574312\pi$
$740$	0	0
$741$	18.5166	0.680223
$742$	0	0
$743$	−9.36221	−0.343466	−0.171733	−	0.985144i	$-0.554937\pi$
$743$	−9.36221	−0.343466	−0.171733	+	0.985144i	$0.554937\pi$
$744$	0	0
$745$	58.0266	2.12593
$746$	0	0
$747$	1.67331	0.0612233
$748$	0	0
$749$	11.9286	0.435861
$750$	0	0
$751$	−3.96209	−0.144579	−0.0722894	−	0.997384i	$-0.523031\pi$
$751$	−3.96209	−0.144579	−0.0722894	+	0.997384i	$0.523031\pi$
$752$	0	0
$753$	41.7137	1.52013
$754$	0	0
$755$	−63.4468	−2.30906
$756$	0	0
$757$	−3.56879	−0.129710	−0.0648550	−	0.997895i	$-0.520658\pi$
$757$	−3.56879	−0.129710	−0.0648550	+	0.997895i	$0.520658\pi$
$758$	0	0
$759$	−8.16794	−0.296477
$760$	0	0
$761$	−49.8018	−1.80532	−0.902658	−	0.430359i	$-0.858387\pi$
$761$	−49.8018	−1.80532	−0.902658	+	0.430359i	$0.858387\pi$
$762$	0	0
$763$	−1.72707	−0.0625240
$764$	0	0
$765$	−25.8630	−0.935078
$766$	0	0
$767$	8.53465	0.308168
$768$	0	0
$769$	30.5772	1.10264	0.551321	−	0.834293i	$-0.314124\pi$
$769$	30.5772	1.10264	0.551321	+	0.834293i	$0.314124\pi$
$770$	0	0
$771$	41.7764	1.50454
$772$	0	0
$773$	−23.9315	−0.860756	−0.430378	−	0.902649i	$-0.641620\pi$
$773$	−23.9315	−0.860756	−0.430378	+	0.902649i	$0.641620\pi$
$774$	0	0
$775$	45.2534	1.62555
$776$	0	0
$777$	−9.96501	−0.357493
$778$	0	0
$779$	11.0699	0.396619
$780$	0	0
$781$	−7.75129	−0.277363
$782$	0	0
$783$	−4.98557	−0.178170
$784$	0	0
$785$	−3.64253	−0.130007
$786$	0	0
$787$	13.4992	0.481196	0.240598	−	0.970625i	$-0.422656\pi$
$787$	13.4992	0.481196	0.240598	+	0.970625i	$0.422656\pi$
$788$	0	0
$789$	−65.0426	−2.31558
$790$	0	0
$791$	15.5676	0.553519
$792$	0	0
$793$	31.4031	1.11516
$794$	0	0
$795$	−6.37595	−0.226132
$796$	0	0
$797$	23.8879	0.846154	0.423077	−	0.906094i	$-0.360950\pi$
$797$	23.8879	0.846154	0.423077	+	0.906094i	$0.360950\pi$
$798$	0	0
$799$	−7.98130	−0.282358
$800$	0	0
$801$	6.96629	0.246142
$802$	0	0
$803$	−13.0650	−0.461056
$804$	0	0
$805$	13.5020	0.475882
$806$	0	0
$807$	−25.0084	−0.880339
$808$	0	0
$809$	−20.1858	−0.709694	−0.354847	−	0.934924i	$-0.615467\pi$
$809$	−20.1858	−0.709694	−0.354847	+	0.934924i	$0.615467\pi$
$810$	0	0
$811$	3.78003	0.132735	0.0663675	−	0.997795i	$-0.478859\pi$
$811$	3.78003	0.132735	0.0663675	+	0.997795i	$0.478859\pi$
$812$	0	0
$813$	−46.4672	−1.62968
$814$	0	0
$815$	−31.3715	−1.09890
$816$	0	0
$817$	30.7586	1.07611
$818$	0	0
$819$	−6.94360	−0.242629
$820$	0	0
$821$	−32.5119	−1.13467	−0.567336	−	0.823486i	$-0.692026\pi$
$821$	−32.5119	−1.13467	−0.567336	+	0.823486i	$0.692026\pi$
$822$	0	0
$823$	20.7651	0.723826	0.361913	−	0.932212i	$-0.382124\pi$
$823$	20.7651	0.723826	0.361913	+	0.932212i	$0.382124\pi$
$824$	0	0
$825$	29.1106	1.01350
$826$	0	0
$827$	11.2140	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$827$	11.2140	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$828$	0	0
$829$	6.82205	0.236940	0.118470	−	0.992958i	$-0.462201\pi$
$829$	6.82205	0.236940	0.118470	+	0.992958i	$0.462201\pi$
$830$	0	0
$831$	29.4955	1.02319
$832$	0	0
$833$	−14.9934	−0.519490
$834$	0	0
$835$	−0.595616	−0.0206121
$836$	0	0
$837$	−3.98829	−0.137856
$838$	0	0
$839$	−19.3720	−0.668796	−0.334398	−	0.942432i	$-0.608533\pi$
$839$	−19.3720	−0.668796	−0.334398	+	0.942432i	$0.608533\pi$
$840$	0	0
$841$	−8.09273	−0.279060
$842$	0	0
$843$	56.0102	1.92910
$844$	0	0
$845$	18.3231	0.630334
$846$	0	0
$847$	0.933222	0.0320659
$848$	0	0
$849$	65.2474	2.23928
$850$	0	0
$851$	15.7529	0.540003
$852$	0	0
$853$	−11.1509	−0.381800	−0.190900	−	0.981609i	$-0.561141\pi$
$853$	−11.1509	−0.381800	−0.190900	+	0.981609i	$0.561141\pi$
$854$	0	0
$855$	28.3641	0.970032
$856$	0	0
$857$	30.4078	1.03871	0.519355	−	0.854558i	$-0.326172\pi$
$857$	30.4078	1.03871	0.519355	+	0.854558i	$0.326172\pi$
$858$	0	0
$859$	27.7908	0.948210	0.474105	−	0.880468i	$-0.342772\pi$
$859$	27.7908	0.948210	0.474105	+	0.880468i	$0.342772\pi$
$860$	0	0
$861$	−9.06058	−0.308784
$862$	0	0
$863$	−0.221827	−0.00755109	−0.00377554	−	0.999993i	$-0.501202\pi$
$863$	−0.221827	−0.00755109	−0.00377554	+	0.999993i	$0.501202\pi$
$864$	0	0
$865$	−4.70600	−0.160009
$866$	0	0
$867$	−25.9202	−0.880295
$868$	0	0
$869$	−4.83666	−0.164072
$870$	0	0
$871$	15.9163	0.539304
$872$	0	0
$873$	4.63960	0.157027
$874$	0	0
$875$	−28.6731	−0.969328
$876$	0	0
$877$	−15.5470	−0.524985	−0.262492	−	0.964934i	$-0.584544\pi$
$877$	−15.5470	−0.524985	−0.262492	+	0.964934i	$0.584544\pi$
$878$	0	0
$879$	13.2095	0.445545
$880$	0	0
$881$	21.4338	0.722125	0.361062	−	0.932542i	$-0.382414\pi$
$881$	21.4338	0.722125	0.361062	+	0.932542i	$0.382414\pi$
$882$	0	0
$883$	−14.2700	−0.480224	−0.240112	−	0.970745i	$-0.577184\pi$
$883$	−14.2700	−0.480224	−0.240112	+	0.970745i	$0.577184\pi$
$884$	0	0
$885$	28.5354	0.959207
$886$	0	0
$887$	23.6109	0.792775	0.396388	−	0.918083i	$-0.370264\pi$
$887$	23.6109	0.792775	0.396388	+	0.918083i	$0.370264\pi$
$888$	0	0
$889$	6.07540	0.203762
$890$	0	0
$891$	−10.1754	−0.340890
$892$	0	0
$893$	8.75315	0.292913
$894$	0	0
$895$	63.1944	2.11235
$896$	0	0
$897$	23.9584	0.799947
$898$	0	0
$899$	16.7251	0.557815
$900$	0	0
$901$	1.59039	0.0529837
$902$	0	0
$903$	−25.1756	−0.837792
$904$	0	0
$905$	70.5260	2.34436
$906$	0	0
$907$	45.2882	1.50377	0.751885	−	0.659294i	$-0.229145\pi$
$907$	45.2882	1.50377	0.751885	+	0.659294i	$0.229145\pi$
$908$	0	0
$909$	32.1670	1.06691
$910$	0	0
$911$	22.4539	0.743931	0.371966	−	0.928247i	$-0.378684\pi$
$911$	22.4539	0.743931	0.371966	+	0.928247i	$0.378684\pi$
$912$	0	0
$913$	0.659664	0.0218317
$914$	0	0
$915$	104.995	3.47104
$916$	0	0
$917$	−6.45388	−0.213126
$918$	0	0
$919$	41.5329	1.37004	0.685022	−	0.728522i	$-0.259793\pi$
$919$	41.5329	1.37004	0.685022	+	0.728522i	$0.259793\pi$
$920$	0	0
$921$	−65.5437	−2.15974
$922$	0	0
$923$	22.7363	0.748373
$924$	0	0
$925$	−56.1436	−1.84599
$926$	0	0
$927$	−9.80767	−0.322126
$928$	0	0
$929$	−37.2997	−1.22376	−0.611882	−	0.790949i	$-0.709587\pi$
$929$	−37.2997	−1.22376	−0.611882	+	0.790949i	$0.709587\pi$
$930$	0	0
$931$	16.4433	0.538909
$932$	0	0
$933$	−57.4310	−1.88021
$934$	0	0
$935$	−10.1959	−0.333441
$936$	0	0
$937$	−13.1897	−0.430888	−0.215444	−	0.976516i	$-0.569120\pi$
$937$	−13.1897	−0.430888	−0.215444	+	0.976516i	$0.569120\pi$
$938$	0	0
$939$	−21.4950	−0.701463
$940$	0	0
$941$	−24.6716	−0.804270	−0.402135	−	0.915580i	$-0.631732\pi$
$941$	−24.6716	−0.804270	−0.402135	+	0.915580i	$0.631732\pi$
$942$	0	0
$943$	14.3232	0.466426
$944$	0	0
$945$	4.24104	0.137961
$946$	0	0
$947$	38.8667	1.26300	0.631500	−	0.775376i	$-0.282440\pi$
$947$	38.8667	1.26300	0.631500	+	0.775376i	$0.282440\pi$
$948$	0	0
$949$	38.3227	1.24401
$950$	0	0
$951$	42.2365	1.36961
$952$	0	0
$953$	−52.5807	−1.70326	−0.851628	−	0.524147i	$-0.824384\pi$
$953$	−52.5807	−1.70326	−0.851628	+	0.524147i	$0.824384\pi$
$954$	0	0
$955$	−69.4881	−2.24858
$956$	0	0
$957$	10.7590	0.347788
$958$	0	0
$959$	−5.43091	−0.175373
$960$	0	0
$961$	−17.6204	−0.568401
$962$	0	0
$963$	32.4233	1.04483
$964$	0	0
$965$	−0.753147	−0.0242447
$966$	0	0
$967$	−2.98370	−0.0959493	−0.0479746	−	0.998849i	$-0.515277\pi$
$967$	−2.98370	−0.0959493	−0.0479746	+	0.998849i	$0.515277\pi$
$968$	0	0
$969$	−15.4425	−0.496086
$970$	0	0
$971$	−15.0443	−0.482793	−0.241397	−	0.970427i	$-0.577605\pi$
$971$	−15.0443	−0.482793	−0.241397	+	0.970427i	$0.577605\pi$
$972$	0	0
$973$	−8.89854	−0.285274
$974$	0	0
$975$	−85.3880	−2.73460
$976$	0	0
$977$	−11.4313	−0.365720	−0.182860	−	0.983139i	$-0.558536\pi$
$977$	−11.4313	−0.365720	−0.182860	+	0.983139i	$0.558536\pi$
$978$	0	0
$979$	2.74629	0.0877720
$980$	0	0
$981$	−4.69438	−0.149880
$982$	0	0
$983$	58.1032	1.85321	0.926603	−	0.376042i	$-0.122715\pi$
$983$	58.1032	1.85321	0.926603	+	0.376042i	$0.122715\pi$
$984$	0	0
$985$	34.9642	1.11405
$986$	0	0
$987$	−7.16437	−0.228044
$988$	0	0
$989$	39.7982	1.26551
$990$	0	0
$991$	−30.2307	−0.960311	−0.480156	−	0.877183i	$-0.659420\pi$
$991$	−30.2307	−0.960311	−0.480156	+	0.877183i	$0.659420\pi$
$992$	0	0
$993$	31.0440	0.985152
$994$	0	0
$995$	−92.1096	−2.92007
$996$	0	0
$997$	13.3996	0.424368	0.212184	−	0.977230i	$-0.431942\pi$
$997$	13.3996	0.424368	0.212184	+	0.977230i	$0.431942\pi$
$998$	0	0
$999$	4.94808	0.156550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.a.p.1.5		5
4.3	odd	2		2816.2.a.o.1.1		5
8.3	odd	2		2816.2.a.r.1.5		5
8.5	even	2		2816.2.a.q.1.1		5
16.3	odd	4		88.2.c.a.45.7	✓	10
16.5	even	4		352.2.c.a.177.2		10
16.11	odd	4		88.2.c.a.45.8	yes	10
16.13	even	4		352.2.c.a.177.9		10
48.5	odd	4		3168.2.f.g.1585.10		10
48.11	even	4		792.2.f.g.397.3		10
48.29	odd	4		3168.2.f.g.1585.1		10
48.35	even	4		792.2.f.g.397.4		10
176.3	odd	20		968.2.o.g.493.4		40
176.19	even	20		968.2.o.h.493.7		40
176.21	odd	4		3872.2.c.f.1937.2		10
176.27	odd	20		968.2.o.g.245.2		40
176.35	even	20		968.2.o.h.565.9		40
176.43	even	4		968.2.c.d.485.3		10
176.51	even	20		968.2.o.h.269.1		40
176.59	odd	20		968.2.o.g.269.4		40
176.75	odd	20		968.2.o.g.565.6		40
176.83	even	20		968.2.o.h.245.5		40
176.91	odd	20		968.2.o.g.493.10		40
176.107	even	20		968.2.o.h.493.1		40
176.109	odd	4		3872.2.c.f.1937.9		10
176.115	odd	20		968.2.o.g.245.6		40
176.123	even	20		968.2.o.h.565.5		40
176.131	even	4		968.2.c.d.485.4		10
176.139	even	20		968.2.o.h.269.7		40
176.147	odd	20		968.2.o.g.269.10		40
176.163	odd	20		968.2.o.g.565.2		40
176.171	even	20		968.2.o.h.245.9		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.c.a.45.7	✓	10	16.3	odd	4
88.2.c.a.45.8	yes	10	16.11	odd	4
352.2.c.a.177.2		10	16.5	even	4
352.2.c.a.177.9		10	16.13	even	4
792.2.f.g.397.3		10	48.11	even	4
792.2.f.g.397.4		10	48.35	even	4
968.2.c.d.485.3		10	176.43	even	4
968.2.c.d.485.4		10	176.131	even	4
968.2.o.g.245.2		40	176.27	odd	20
968.2.o.g.245.6		40	176.115	odd	20
968.2.o.g.269.4		40	176.59	odd	20
968.2.o.g.269.10		40	176.147	odd	20
968.2.o.g.493.4		40	176.3	odd	20
968.2.o.g.493.10		40	176.91	odd	20
968.2.o.g.565.2		40	176.163	odd	20
968.2.o.g.565.6		40	176.75	odd	20
968.2.o.h.245.5		40	176.83	even	20
968.2.o.h.245.9		40	176.171	even	20
968.2.o.h.269.1		40	176.51	even	20
968.2.o.h.269.7		40	176.139	even	20
968.2.o.h.493.1		40	176.107	even	20
968.2.o.h.493.7		40	176.19	even	20
968.2.o.h.565.5		40	176.123	even	20
968.2.o.h.565.9		40	176.35	even	20
2816.2.a.o.1.1		5	4.3	odd	2
2816.2.a.p.1.5		5	1.1	even	1	trivial
2816.2.a.q.1.1		5	8.5	even	2
2816.2.a.r.1.5		5	8.3	odd	2
3168.2.f.g.1585.1		10	48.29	odd	4
3168.2.f.g.1585.10		10	48.5	odd	4
3872.2.c.f.1937.2		10	176.21	odd	4
3872.2.c.f.1937.9		10	176.109	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35300 q^{3} -4.16794 q^{5} +0.933222 q^{7} +2.53661 q^{9} +1.00000 q^{11} -2.93322 q^{13} -9.80716 q^{15} +2.44626 q^{17} -2.68283 q^{19} +2.19587 q^{21} -3.47129 q^{23} +12.3717 q^{25} -1.09035 q^{27} +4.57245 q^{29} +3.65781 q^{31} +2.35300 q^{33} -3.88961 q^{35} -4.53806 q^{37} -6.90187 q^{39} -4.12618 q^{41} -11.4650 q^{43} -10.5724 q^{45} -3.26265 q^{47} -6.12910 q^{49} +5.75606 q^{51} +0.650132 q^{53} -4.16794 q^{55} -6.31271 q^{57} -2.90965 q^{59} -10.7060 q^{61} +2.36722 q^{63} +12.2255 q^{65} -5.42623 q^{67} -8.16794 q^{69} -7.75129 q^{71} -13.0650 q^{73} +29.1106 q^{75} +0.933222 q^{77} -4.83666 q^{79} -10.1754 q^{81} +0.659664 q^{83} -10.1959 q^{85} +10.7590 q^{87} +2.74629 q^{89} -2.73735 q^{91} +8.60683 q^{93} +11.1819 q^{95} +1.82905 q^{97} +2.53661 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{5} + 5 q^{9} + 5 q^{11} - 10 q^{13} - 4 q^{15} - 2 q^{17} - 12 q^{21} - 6 q^{23} + 3 q^{25} - 12 q^{27} - 10 q^{29} + 2 q^{31} - 16 q^{37} + 12 q^{39} - 2 q^{41} + 4 q^{43} - 20 q^{45} + 2 q^{47}+ \cdots + 5 q^{99}+O(q^{100}) \) 5 * q - 4 * q^5 + 5 * q^9 + 5 * q^11 - 10 * q^13 - 4 * q^15 - 2 * q^17 - 12 * q^21 - 6 * q^23 + 3 * q^25 - 12 * q^27 - 10 * q^29 + 2 * q^31 - 16 * q^37 + 12 * q^39 - 2 * q^41 + 4 * q^43 - 20 * q^45 + 2 * q^47 - 3 * q^49 - 8 * q^53 - 4 * q^55 - 8 * q^57 - 8 * q^59 - 30 * q^61 + 20 * q^63 + 8 * q^65 - 24 * q^69 - 6 * q^71 + 2 * q^73 + 36 * q^75 - 8 * q^79 - 3 * q^81 - 28 * q^85 + 16 * q^87 + 2 * q^89 - 32 * q^91 - 16 * q^93 - 12 * q^95 - 10 * q^97 + 5 * q^99

Introduction

L-functions

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