LMFDB - Embedded newform 8664.2.a.bq.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8664,2,Mod(1,8664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8664, 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("8664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8664 = 2^{3} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8664.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:	$69.1823883112$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 21x^{7} + 57x^{6} + 111x^{5} - 237x^{4} - 185x^{3} + 126x^{2} + 48x - 8 $ x^9 - 3x^8 - 21x^7 + 57x^6 + 111x^5 - 237x^4 - 185x^3 + 126x^2 + 48x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.646100$ of defining polynomial
Character	$\chi$	$=$	8664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.646100 q^{5} +3.63284 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.646100 q^{5} +3.63284 q^{7} +1.00000 q^{9} +2.72595 q^{11} +3.18758 q^{13} +0.646100 q^{15} -1.59644 q^{17} +3.63284 q^{21} +1.22471 q^{23} -4.58256 q^{25} +1.00000 q^{27} +1.41413 q^{29} -4.03604 q^{31} +2.72595 q^{33} +2.34718 q^{35} +9.34827 q^{37} +3.18758 q^{39} -0.271619 q^{41} +0.380004 q^{43} +0.646100 q^{45} -8.51378 q^{47} +6.19752 q^{49} -1.59644 q^{51} +10.7211 q^{53} +1.76124 q^{55} +13.6639 q^{59} -6.35431 q^{61} +3.63284 q^{63} +2.05949 q^{65} +7.71983 q^{67} +1.22471 q^{69} +4.31164 q^{71} -7.98582 q^{73} -4.58256 q^{75} +9.90294 q^{77} +1.75902 q^{79} +1.00000 q^{81} +11.1550 q^{83} -1.03146 q^{85} +1.41413 q^{87} -7.01939 q^{89} +11.5800 q^{91} -4.03604 q^{93} -19.1073 q^{97} +2.72595 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + 3 q^{5} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 + 3 * q^5 + 9 * q^9	$ 9 q + 9 q^{3} + 3 q^{5} + 9 q^{9} + 3 q^{11} + 3 q^{13} + 3 q^{15} + 15 q^{17} - 6 q^{23} + 6 q^{25} + 9 q^{27} + 21 q^{29} + 21 q^{31} + 3 q^{33} - 12 q^{35} + 15 q^{37} + 3 q^{39} + 15 q^{41} - 6 q^{43} + 3 q^{45} + 6 q^{47} + 21 q^{49} + 15 q^{51} + 21 q^{53} - 48 q^{55} + 9 q^{59} + 30 q^{61} - 9 q^{65} + 30 q^{67} - 6 q^{69} + 30 q^{71} - 3 q^{73} + 6 q^{75} + 15 q^{77} + 6 q^{79} + 9 q^{81} + 30 q^{83} - 21 q^{85} + 21 q^{87} - 9 q^{89} + 21 q^{91} + 21 q^{93} + 15 q^{97} + 3 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + 3 * q^5 + 9 * q^9 + 3 * q^11 + 3 * q^13 + 3 * q^15 + 15 * q^17 - 6 * q^23 + 6 * q^25 + 9 * q^27 + 21 * q^29 + 21 * q^31 + 3 * q^33 - 12 * q^35 + 15 * q^37 + 3 * q^39 + 15 * q^41 - 6 * q^43 + 3 * q^45 + 6 * q^47 + 21 * q^49 + 15 * q^51 + 21 * q^53 - 48 * q^55 + 9 * q^59 + 30 * q^61 - 9 * q^65 + 30 * q^67 - 6 * q^69 + 30 * q^71 - 3 * q^73 + 6 * q^75 + 15 * q^77 + 6 * q^79 + 9 * q^81 + 30 * q^83 - 21 * q^85 + 21 * q^87 - 9 * q^89 + 21 * q^91 + 21 * q^93 + 15 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.646100	0.288945	0.144472	−	0.989509i	$-0.453852\pi$
$5$	0.646100	0.288945	0.144472	+	0.989509i	$0.453852\pi$
$6$	0	0
$7$	3.63284	1.37308	0.686542	−	0.727090i	$-0.259128\pi$
$7$	3.63284	1.37308	0.686542	+	0.727090i	$0.259128\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.72595	0.821905	0.410952	−	0.911657i	$-0.365196\pi$
$11$	2.72595	0.821905	0.410952	+	0.911657i	$0.365196\pi$
$12$	0	0
$13$	3.18758	0.884075	0.442037	−	0.896997i	$-0.354256\pi$
$13$	3.18758	0.884075	0.442037	+	0.896997i	$0.354256\pi$
$14$	0	0
$15$	0.646100	0.166822
$16$	0	0
$17$	−1.59644	−0.387193	−0.193596	−	0.981081i	$-0.562015\pi$
$17$	−1.59644	−0.387193	−0.193596	+	0.981081i	$0.562015\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	3.63284	0.792751
$22$	0	0
$23$	1.22471	0.255369	0.127685	−	0.991815i	$-0.459245\pi$
$23$	1.22471	0.255369	0.127685	+	0.991815i	$0.459245\pi$
$24$	0	0
$25$	−4.58256	−0.916511
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.41413	0.262598	0.131299	−	0.991343i	$-0.458085\pi$
$29$	1.41413	0.262598	0.131299	+	0.991343i	$0.458085\pi$
$30$	0	0
$31$	−4.03604	−0.724894	−0.362447	−	0.932004i	$-0.618059\pi$
$31$	−4.03604	−0.724894	−0.362447	+	0.932004i	$0.618059\pi$
$32$	0	0
$33$	2.72595	0.474527
$34$	0	0
$35$	2.34718	0.396745
$36$	0	0
$37$	9.34827	1.53685	0.768423	−	0.639942i	$-0.221042\pi$
$37$	9.34827	1.53685	0.768423	+	0.639942i	$0.221042\pi$
$38$	0	0
$39$	3.18758	0.510421
$40$	0	0
$41$	−0.271619	−0.0424198	−0.0212099	−	0.999775i	$-0.506752\pi$
$41$	−0.271619	−0.0424198	−0.0212099	+	0.999775i	$0.506752\pi$
$42$	0	0
$43$	0.380004	0.0579500	0.0289750	−	0.999580i	$-0.490776\pi$
$43$	0.380004	0.0579500	0.0289750	+	0.999580i	$0.490776\pi$
$44$	0	0
$45$	0.646100	0.0963149
$46$	0	0
$47$	−8.51378	−1.24186	−0.620931	−	0.783865i	$-0.713245\pi$
$47$	−8.51378	−1.24186	−0.620931	+	0.783865i	$0.713245\pi$
$48$	0	0
$49$	6.19752	0.885361
$50$	0	0
$51$	−1.59644	−0.223546
$52$	0	0
$53$	10.7211	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$53$	10.7211	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$54$	0	0
$55$	1.76124	0.237485
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.6639	1.77889	0.889446	−	0.457041i	$-0.151091\pi$
$59$	13.6639	1.77889	0.889446	+	0.457041i	$0.151091\pi$
$60$	0	0
$61$	−6.35431	−0.813586	−0.406793	−	0.913520i	$-0.633353\pi$
$61$	−6.35431	−0.813586	−0.406793	+	0.913520i	$0.633353\pi$
$62$	0	0
$63$	3.63284	0.457695
$64$	0	0
$65$	2.05949	0.255449
$66$	0	0
$67$	7.71983	0.943127	0.471564	−	0.881832i	$-0.343690\pi$
$67$	7.71983	0.943127	0.471564	+	0.881832i	$0.343690\pi$
$68$	0	0
$69$	1.22471	0.147438
$70$	0	0
$71$	4.31164	0.511697	0.255849	−	0.966717i	$-0.417645\pi$
$71$	4.31164	0.511697	0.255849	+	0.966717i	$0.417645\pi$
$72$	0	0
$73$	−7.98582	−0.934670	−0.467335	−	0.884080i	$-0.654786\pi$
$73$	−7.98582	−0.934670	−0.467335	+	0.884080i	$0.654786\pi$
$74$	0	0
$75$	−4.58256	−0.529148
$76$	0	0
$77$	9.90294	1.12854
$78$	0	0
$79$	1.75902	0.197905	0.0989523	−	0.995092i	$-0.468451\pi$
$79$	1.75902	0.197905	0.0989523	+	0.995092i	$0.468451\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.1550	1.22442	0.612210	−	0.790695i	$-0.290281\pi$
$83$	11.1550	1.22442	0.612210	+	0.790695i	$0.290281\pi$
$84$	0	0
$85$	−1.03146	−0.111877
$86$	0	0
$87$	1.41413	0.151611
$88$	0	0
$89$	−7.01939	−0.744054	−0.372027	−	0.928222i	$-0.621337\pi$
$89$	−7.01939	−0.744054	−0.372027	+	0.928222i	$0.621337\pi$
$90$	0	0
$91$	11.5800	1.21391
$92$	0	0
$93$	−4.03604	−0.418518
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−19.1073	−1.94005	−0.970025	−	0.243005i	$-0.921867\pi$
$97$	−19.1073	−1.94005	−0.970025	+	0.243005i	$0.921867\pi$
$98$	0	0
$99$	2.72595	0.273968
$100$	0	0
$101$	2.84895	0.283481	0.141741	−	0.989904i	$-0.454730\pi$
$101$	2.84895	0.283481	0.141741	+	0.989904i	$0.454730\pi$
$102$	0	0
$103$	7.34337	0.723564	0.361782	−	0.932263i	$-0.382169\pi$
$103$	7.34337	0.723564	0.361782	+	0.932263i	$0.382169\pi$
$104$	0	0
$105$	2.34718	0.229061
$106$	0	0
$107$	12.5203	1.21038	0.605192	−	0.796079i	$-0.293096\pi$
$107$	12.5203	1.21038	0.605192	+	0.796079i	$0.293096\pi$
$108$	0	0
$109$	4.89285	0.468650	0.234325	−	0.972158i	$-0.424712\pi$
$109$	4.89285	0.468650	0.234325	+	0.972158i	$0.424712\pi$
$110$	0	0
$111$	9.34827	0.887299
$112$	0	0
$113$	8.04618	0.756920	0.378460	−	0.925618i	$-0.376454\pi$
$113$	8.04618	0.756920	0.378460	+	0.925618i	$0.376454\pi$
$114$	0	0
$115$	0.791284	0.0737876
$116$	0	0
$117$	3.18758	0.294692
$118$	0	0
$119$	−5.79960	−0.531649
$120$	0	0
$121$	−3.56920	−0.324473
$122$	0	0
$123$	−0.271619	−0.0244911
$124$	0	0
$125$	−6.19129	−0.553765
$126$	0	0
$127$	−10.9400	−0.970770	−0.485385	−	0.874300i	$-0.661321\pi$
$127$	−10.9400	−0.970770	−0.485385	+	0.874300i	$0.661321\pi$
$128$	0	0
$129$	0.380004	0.0334575
$130$	0	0
$131$	−17.9331	−1.56682	−0.783411	−	0.621504i	$-0.786522\pi$
$131$	−17.9331	−1.56682	−0.783411	+	0.621504i	$0.786522\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.646100	0.0556074
$136$	0	0
$137$	3.02269	0.258246	0.129123	−	0.991629i	$-0.458784\pi$
$137$	3.02269	0.258246	0.129123	+	0.991629i	$0.458784\pi$
$138$	0	0
$139$	−15.8097	−1.34096	−0.670482	−	0.741926i	$-0.733913\pi$
$139$	−15.8097	−1.34096	−0.670482	+	0.741926i	$0.733913\pi$
$140$	0	0
$141$	−8.51378	−0.716989
$142$	0	0
$143$	8.68918	0.726625
$144$	0	0
$145$	0.913670	0.0758762
$146$	0	0
$147$	6.19752	0.511163
$148$	0	0
$149$	2.06098	0.168842	0.0844208	−	0.996430i	$-0.473096\pi$
$149$	2.06098	0.168842	0.0844208	+	0.996430i	$0.473096\pi$
$150$	0	0
$151$	13.9120	1.13215	0.566073	−	0.824355i	$-0.308462\pi$
$151$	13.9120	1.13215	0.566073	+	0.824355i	$0.308462\pi$
$152$	0	0
$153$	−1.59644	−0.129064
$154$	0	0
$155$	−2.60768	−0.209454
$156$	0	0
$157$	9.10207	0.726424	0.363212	−	0.931706i	$-0.381680\pi$
$157$	9.10207	0.726424	0.363212	+	0.931706i	$0.381680\pi$
$158$	0	0
$159$	10.7211	0.850238
$160$	0	0
$161$	4.44917	0.350644
$162$	0	0
$163$	11.7864	0.923181	0.461590	−	0.887093i	$-0.347279\pi$
$163$	11.7864	0.923181	0.461590	+	0.887093i	$0.347279\pi$
$164$	0	0
$165$	1.76124	0.137112
$166$	0	0
$167$	7.42837	0.574825	0.287412	−	0.957807i	$-0.407205\pi$
$167$	7.42837	0.574825	0.287412	+	0.957807i	$0.407205\pi$
$168$	0	0
$169$	−2.83935	−0.218412
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−21.4052	−1.62740	−0.813702	−	0.581282i	$-0.802551\pi$
$173$	−21.4052	−1.62740	−0.813702	+	0.581282i	$0.802551\pi$
$174$	0	0
$175$	−16.6477	−1.25845
$176$	0	0
$177$	13.6639	1.02704
$178$	0	0
$179$	−20.7482	−1.55079	−0.775397	−	0.631474i	$-0.782450\pi$
$179$	−20.7482	−1.55079	−0.775397	+	0.631474i	$0.782450\pi$
$180$	0	0
$181$	14.7653	1.09750	0.548749	−	0.835987i	$-0.315104\pi$
$181$	14.7653	1.09750	0.548749	+	0.835987i	$0.315104\pi$
$182$	0	0
$183$	−6.35431	−0.469724
$184$	0	0
$185$	6.03992	0.444064
$186$	0	0
$187$	−4.35181	−0.318236
$188$	0	0
$189$	3.63284	0.264250
$190$	0	0
$191$	−25.1275	−1.81816	−0.909081	−	0.416619i	$-0.863215\pi$
$191$	−25.1275	−1.81816	−0.909081	+	0.416619i	$0.863215\pi$
$192$	0	0
$193$	−8.81629	−0.634610	−0.317305	−	0.948323i	$-0.602778\pi$
$193$	−8.81629	−0.634610	−0.317305	+	0.948323i	$0.602778\pi$
$194$	0	0
$195$	2.05949	0.147483
$196$	0	0
$197$	14.1664	1.00931	0.504657	−	0.863320i	$-0.331619\pi$
$197$	14.1664	1.00931	0.504657	+	0.863320i	$0.331619\pi$
$198$	0	0
$199$	4.74461	0.336337	0.168168	−	0.985758i	$-0.446215\pi$
$199$	4.74461	0.336337	0.168168	+	0.985758i	$0.446215\pi$
$200$	0	0
$201$	7.71983	0.544515
$202$	0	0
$203$	5.13732	0.360569
$204$	0	0
$205$	−0.175493	−0.0122570
$206$	0	0
$207$	1.22471	0.0851231
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−18.4301	−1.26878	−0.634391	−	0.773012i	$-0.718749\pi$
$211$	−18.4301	−1.26878	−0.634391	+	0.773012i	$0.718749\pi$
$212$	0	0
$213$	4.31164	0.295428
$214$	0	0
$215$	0.245520	0.0167443
$216$	0	0
$217$	−14.6623	−0.995340
$218$	0	0
$219$	−7.98582	−0.539632
$220$	0	0
$221$	−5.08877	−0.342308
$222$	0	0
$223$	−11.0372	−0.739107	−0.369553	−	0.929210i	$-0.620489\pi$
$223$	−11.0372	−0.739107	−0.369553	+	0.929210i	$0.620489\pi$
$224$	0	0
$225$	−4.58256	−0.305504
$226$	0	0
$227$	−10.1730	−0.675209	−0.337604	−	0.941288i	$-0.609617\pi$
$227$	−10.1730	−0.675209	−0.337604	+	0.941288i	$0.609617\pi$
$228$	0	0
$229$	−11.9151	−0.787374	−0.393687	−	0.919245i	$-0.628801\pi$
$229$	−11.9151	−0.787374	−0.393687	+	0.919245i	$0.628801\pi$
$230$	0	0
$231$	9.90294	0.651565
$232$	0	0
$233$	20.4157	1.33748	0.668738	−	0.743498i	$-0.266835\pi$
$233$	20.4157	1.33748	0.668738	+	0.743498i	$0.266835\pi$
$234$	0	0
$235$	−5.50075	−0.358829
$236$	0	0
$237$	1.75902	0.114260
$238$	0	0
$239$	28.3921	1.83653	0.918265	−	0.395965i	$-0.129590\pi$
$239$	28.3921	1.83653	0.918265	+	0.395965i	$0.129590\pi$
$240$	0	0
$241$	−11.5920	−0.746708	−0.373354	−	0.927689i	$-0.621792\pi$
$241$	−11.5920	−0.746708	−0.373354	+	0.927689i	$0.621792\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.00422	0.255820
$246$	0	0
$247$	0	0
$248$	0	0
$249$	11.1550	0.706919
$250$	0	0
$251$	16.6863	1.05323	0.526616	−	0.850103i	$-0.323461\pi$
$251$	16.6863	1.05323	0.526616	+	0.850103i	$0.323461\pi$
$252$	0	0
$253$	3.33849	0.209889
$254$	0	0
$255$	−1.03146	−0.0645924
$256$	0	0
$257$	1.14835	0.0716323	0.0358161	−	0.999358i	$-0.488597\pi$
$257$	1.14835	0.0716323	0.0358161	+	0.999358i	$0.488597\pi$
$258$	0	0
$259$	33.9608	2.11022
$260$	0	0
$261$	1.41413	0.0875326
$262$	0	0
$263$	13.9249	0.858648	0.429324	−	0.903151i	$-0.358752\pi$
$263$	13.9249	0.858648	0.429324	+	0.903151i	$0.358752\pi$
$264$	0	0
$265$	6.92690	0.425516
$266$	0	0
$267$	−7.01939	−0.429580
$268$	0	0
$269$	31.8422	1.94145	0.970726	−	0.240190i	$-0.0772099\pi$
$269$	31.8422	1.94145	0.970726	+	0.240190i	$0.0772099\pi$
$270$	0	0
$271$	−22.8758	−1.38961	−0.694804	−	0.719199i	$-0.744509\pi$
$271$	−22.8758	−1.38961	−0.694804	+	0.719199i	$0.744509\pi$
$272$	0	0
$273$	11.5800	0.700851
$274$	0	0
$275$	−12.4918	−0.753285
$276$	0	0
$277$	6.21594	0.373480	0.186740	−	0.982409i	$-0.440208\pi$
$277$	6.21594	0.373480	0.186740	+	0.982409i	$0.440208\pi$
$278$	0	0
$279$	−4.03604	−0.241631
$280$	0	0
$281$	−1.61885	−0.0965723	−0.0482862	−	0.998834i	$-0.515376\pi$
$281$	−1.61885	−0.0965723	−0.0482862	+	0.998834i	$0.515376\pi$
$282$	0	0
$283$	−25.8756	−1.53815	−0.769073	−	0.639161i	$-0.779282\pi$
$283$	−25.8756	−1.53815	−0.769073	+	0.639161i	$0.779282\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.986749	−0.0582459
$288$	0	0
$289$	−14.4514	−0.850082
$290$	0	0
$291$	−19.1073	−1.12009
$292$	0	0
$293$	−23.2969	−1.36102	−0.680509	−	0.732740i	$-0.738241\pi$
$293$	−23.2969	−1.36102	−0.680509	+	0.732740i	$0.738241\pi$
$294$	0	0
$295$	8.82826	0.514001
$296$	0	0
$297$	2.72595	0.158176
$298$	0	0
$299$	3.90385	0.225766
$300$	0	0
$301$	1.38049	0.0795703
$302$	0	0
$303$	2.84895	0.163668
$304$	0	0
$305$	−4.10552	−0.235081
$306$	0	0
$307$	−2.22893	−0.127211	−0.0636057	−	0.997975i	$-0.520260\pi$
$307$	−2.22893	−0.127211	−0.0636057	+	0.997975i	$0.520260\pi$
$308$	0	0
$309$	7.34337	0.417750
$310$	0	0
$311$	15.2543	0.864993	0.432497	−	0.901636i	$-0.357633\pi$
$311$	15.2543	0.864993	0.432497	+	0.901636i	$0.357633\pi$
$312$	0	0
$313$	−2.08519	−0.117862	−0.0589308	−	0.998262i	$-0.518769\pi$
$313$	−2.08519	−0.117862	−0.0589308	+	0.998262i	$0.518769\pi$
$314$	0	0
$315$	2.34718	0.132248
$316$	0	0
$317$	−28.3128	−1.59021	−0.795103	−	0.606474i	$-0.792583\pi$
$317$	−28.3128	−1.59021	−0.795103	+	0.606474i	$0.792583\pi$
$318$	0	0
$319$	3.85485	0.215830
$320$	0	0
$321$	12.5203	0.698816
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−14.6072	−0.810264
$326$	0	0
$327$	4.89285	0.270575
$328$	0	0
$329$	−30.9292	−1.70518
$330$	0	0
$331$	8.76994	0.482039	0.241020	−	0.970520i	$-0.422518\pi$
$331$	8.76994	0.482039	0.241020	+	0.970520i	$0.422518\pi$
$332$	0	0
$333$	9.34827	0.512282
$334$	0	0
$335$	4.98778	0.272512
$336$	0	0
$337$	35.4404	1.93056	0.965282	−	0.261212i	$-0.0841221\pi$
$337$	35.4404	1.93056	0.965282	+	0.261212i	$0.0841221\pi$
$338$	0	0
$339$	8.04618	0.437008
$340$	0	0
$341$	−11.0020	−0.595794
$342$	0	0
$343$	−2.91526	−0.157409
$344$	0	0
$345$	0.791284	0.0426013
$346$	0	0
$347$	−1.79926	−0.0965893	−0.0482946	−	0.998833i	$-0.515379\pi$
$347$	−1.79926	−0.0965893	−0.0482946	+	0.998833i	$0.515379\pi$
$348$	0	0
$349$	0.514285	0.0275290	0.0137645	−	0.999905i	$-0.495618\pi$
$349$	0.514285	0.0275290	0.0137645	+	0.999905i	$0.495618\pi$
$350$	0	0
$351$	3.18758	0.170140
$352$	0	0
$353$	25.7741	1.37182	0.685909	−	0.727688i	$-0.259405\pi$
$353$	25.7741	1.37182	0.685909	+	0.727688i	$0.259405\pi$
$354$	0	0
$355$	2.78575	0.147852
$356$	0	0
$357$	−5.79960	−0.306947
$358$	0	0
$359$	5.27335	0.278317	0.139158	−	0.990270i	$-0.455560\pi$
$359$	5.27335	0.278317	0.139158	+	0.990270i	$0.455560\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−3.56920	−0.187334
$364$	0	0
$365$	−5.15964	−0.270068
$366$	0	0
$367$	−7.80668	−0.407505	−0.203753	−	0.979022i	$-0.565314\pi$
$367$	−7.80668	−0.407505	−0.203753	+	0.979022i	$0.565314\pi$
$368$	0	0
$369$	−0.271619	−0.0141399
$370$	0	0
$371$	38.9480	2.02208
$372$	0	0
$373$	22.9088	1.18617	0.593087	−	0.805138i	$-0.297909\pi$
$373$	22.9088	1.18617	0.593087	+	0.805138i	$0.297909\pi$
$374$	0	0
$375$	−6.19129	−0.319717
$376$	0	0
$377$	4.50766	0.232156
$378$	0	0
$379$	8.29656	0.426166	0.213083	−	0.977034i	$-0.431650\pi$
$379$	8.29656	0.426166	0.213083	+	0.977034i	$0.431650\pi$
$380$	0	0
$381$	−10.9400	−0.560474
$382$	0	0
$383$	−6.94366	−0.354804	−0.177402	−	0.984138i	$-0.556769\pi$
$383$	−6.94366	−0.354804	−0.177402	+	0.984138i	$0.556769\pi$
$384$	0	0
$385$	6.39829	0.326087
$386$	0	0
$387$	0.380004	0.0193167
$388$	0	0
$389$	35.1183	1.78057	0.890285	−	0.455404i	$-0.150505\pi$
$389$	35.1183	1.78057	0.890285	+	0.455404i	$0.150505\pi$
$390$	0	0
$391$	−1.95517	−0.0988772
$392$	0	0
$393$	−17.9331	−0.904605
$394$	0	0
$395$	1.13650	0.0571835
$396$	0	0
$397$	−9.10830	−0.457133	−0.228566	−	0.973528i	$-0.573404\pi$
$397$	−9.10830	−0.457133	−0.228566	+	0.973528i	$0.573404\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.5829	1.52724	0.763619	−	0.645668i	$-0.223421\pi$
$401$	30.5829	1.52724	0.763619	+	0.645668i	$0.223421\pi$
$402$	0	0
$403$	−12.8652	−0.640860
$404$	0	0
$405$	0.646100	0.0321050
$406$	0	0
$407$	25.4829	1.26314
$408$	0	0
$409$	−5.47550	−0.270746	−0.135373	−	0.990795i	$-0.543223\pi$
$409$	−5.47550	−0.270746	−0.135373	+	0.990795i	$0.543223\pi$
$410$	0	0
$411$	3.02269	0.149098
$412$	0	0
$413$	49.6389	2.44257
$414$	0	0
$415$	7.20724	0.353789
$416$	0	0
$417$	−15.8097	−0.774206
$418$	0	0
$419$	9.72227	0.474964	0.237482	−	0.971392i	$-0.423678\pi$
$419$	9.72227	0.474964	0.237482	+	0.971392i	$0.423678\pi$
$420$	0	0
$421$	7.98829	0.389325	0.194663	−	0.980870i	$-0.437639\pi$
$421$	7.98829	0.389325	0.194663	+	0.980870i	$0.437639\pi$
$422$	0	0
$423$	−8.51378	−0.413954
$424$	0	0
$425$	7.31576	0.354867
$426$	0	0
$427$	−23.0842	−1.11712
$428$	0	0
$429$	8.68918	0.419517
$430$	0	0
$431$	0.225997	0.0108859	0.00544294	−	0.999985i	$-0.498267\pi$
$431$	0.225997	0.0108859	0.00544294	+	0.999985i	$0.498267\pi$
$432$	0	0
$433$	−11.7746	−0.565853	−0.282927	−	0.959142i	$-0.591305\pi$
$433$	−11.7746	−0.565853	−0.282927	+	0.959142i	$0.591305\pi$
$434$	0	0
$435$	0.913670	0.0438071
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−19.5565	−0.933380	−0.466690	−	0.884421i	$-0.654554\pi$
$439$	−19.5565	−0.933380	−0.466690	+	0.884421i	$0.654554\pi$
$440$	0	0
$441$	6.19752	0.295120
$442$	0	0
$443$	1.57810	0.0749779	0.0374890	−	0.999297i	$-0.488064\pi$
$443$	1.57810	0.0749779	0.0374890	+	0.999297i	$0.488064\pi$
$444$	0	0
$445$	−4.53523	−0.214990
$446$	0	0
$447$	2.06098	0.0974808
$448$	0	0
$449$	−15.9340	−0.751971	−0.375985	−	0.926626i	$-0.622696\pi$
$449$	−15.9340	−0.751971	−0.375985	+	0.926626i	$0.622696\pi$
$450$	0	0
$451$	−0.740420	−0.0348650
$452$	0	0
$453$	13.9120	0.653645
$454$	0	0
$455$	7.48181	0.350753
$456$	0	0
$457$	23.4251	1.09578	0.547889	−	0.836551i	$-0.315432\pi$
$457$	23.4251	1.09578	0.547889	+	0.836551i	$0.315432\pi$
$458$	0	0
$459$	−1.59644	−0.0745153
$460$	0	0
$461$	−27.0073	−1.25786	−0.628929	−	0.777463i	$-0.716506\pi$
$461$	−27.0073	−1.25786	−0.628929	+	0.777463i	$0.716506\pi$
$462$	0	0
$463$	21.3764	0.993444	0.496722	−	0.867910i	$-0.334537\pi$
$463$	21.3764	0.993444	0.496722	+	0.867910i	$0.334537\pi$
$464$	0	0
$465$	−2.60768	−0.120928
$466$	0	0
$467$	−28.0664	−1.29876	−0.649380	−	0.760464i	$-0.724971\pi$
$467$	−28.0664	−1.29876	−0.649380	+	0.760464i	$0.724971\pi$
$468$	0	0
$469$	28.0449	1.29499
$470$	0	0
$471$	9.10207	0.419401
$472$	0	0
$473$	1.03587	0.0476294
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.7211	0.490885
$478$	0	0
$479$	−42.0949	−1.92337	−0.961683	−	0.274165i	$-0.911599\pi$
$479$	−42.0949	−1.92337	−0.961683	+	0.274165i	$0.911599\pi$
$480$	0	0
$481$	29.7983	1.35869
$482$	0	0
$483$	4.44917	0.202444
$484$	0	0
$485$	−12.3452	−0.560567
$486$	0	0
$487$	10.4180	0.472083	0.236042	−	0.971743i	$-0.424150\pi$
$487$	10.4180	0.472083	0.236042	+	0.971743i	$0.424150\pi$
$488$	0	0
$489$	11.7864	0.532999
$490$	0	0
$491$	−28.2310	−1.27405	−0.637023	−	0.770845i	$-0.719834\pi$
$491$	−28.2310	−1.27405	−0.637023	+	0.770845i	$0.719834\pi$
$492$	0	0
$493$	−2.25757	−0.101676
$494$	0	0
$495$	1.76124	0.0791616
$496$	0	0
$497$	15.6635	0.702603
$498$	0	0
$499$	12.2526	0.548504	0.274252	−	0.961658i	$-0.411570\pi$
$499$	12.2526	0.548504	0.274252	+	0.961658i	$0.411570\pi$
$500$	0	0
$501$	7.42837	0.331875
$502$	0	0
$503$	−35.8937	−1.60042	−0.800211	−	0.599718i	$-0.795279\pi$
$503$	−35.8937	−1.60042	−0.800211	+	0.599718i	$0.795279\pi$
$504$	0	0
$505$	1.84071	0.0819104
$506$	0	0
$507$	−2.83935	−0.126100
$508$	0	0
$509$	7.35430	0.325974	0.162987	−	0.986628i	$-0.447887\pi$
$509$	7.35430	0.325974	0.162987	+	0.986628i	$0.447887\pi$
$510$	0	0
$511$	−29.0112	−1.28338
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.74455	0.209070
$516$	0	0
$517$	−23.2081	−1.02069
$518$	0	0
$519$	−21.4052	−0.939582
$520$	0	0
$521$	−25.2411	−1.10583	−0.552917	−	0.833236i	$-0.686485\pi$
$521$	−25.2411	−1.10583	−0.552917	+	0.833236i	$0.686485\pi$
$522$	0	0
$523$	4.01401	0.175520	0.0877602	−	0.996142i	$-0.472029\pi$
$523$	4.01401	0.175520	0.0877602	+	0.996142i	$0.472029\pi$
$524$	0	0
$525$	−16.6477	−0.726565
$526$	0	0
$527$	6.44328	0.280674
$528$	0	0
$529$	−21.5001	−0.934786
$530$	0	0
$531$	13.6639	0.592964
$532$	0	0
$533$	−0.865807	−0.0375023
$534$	0	0
$535$	8.08937	0.349734
$536$	0	0
$537$	−20.7482	−0.895351
$538$	0	0
$539$	16.8941	0.727682
$540$	0	0
$541$	−12.0392	−0.517606	−0.258803	−	0.965930i	$-0.583328\pi$
$541$	−12.0392	−0.517606	−0.258803	+	0.965930i	$0.583328\pi$
$542$	0	0
$543$	14.7653	0.633641
$544$	0	0
$545$	3.16127	0.135414
$546$	0	0
$547$	7.76140	0.331853	0.165927	−	0.986138i	$-0.446938\pi$
$547$	7.76140	0.331853	0.165927	+	0.986138i	$0.446938\pi$
$548$	0	0
$549$	−6.35431	−0.271195
$550$	0	0
$551$	0	0
$552$	0	0
$553$	6.39022	0.271740
$554$	0	0
$555$	6.03992	0.256380
$556$	0	0
$557$	−28.3946	−1.20312	−0.601559	−	0.798828i	$-0.705454\pi$
$557$	−28.3946	−1.20312	−0.601559	+	0.798828i	$0.705454\pi$
$558$	0	0
$559$	1.21129	0.0512322
$560$	0	0
$561$	−4.35181	−0.183733
$562$	0	0
$563$	37.1116	1.56407	0.782033	−	0.623237i	$-0.214183\pi$
$563$	37.1116	1.56407	0.782033	+	0.623237i	$0.214183\pi$
$564$	0	0
$565$	5.19863	0.218708
$566$	0	0
$567$	3.63284	0.152565
$568$	0	0
$569$	40.8997	1.71461	0.857303	−	0.514812i	$-0.172138\pi$
$569$	40.8997	1.71461	0.857303	+	0.514812i	$0.172138\pi$
$570$	0	0
$571$	−1.75545	−0.0734631	−0.0367315	−	0.999325i	$-0.511695\pi$
$571$	−1.75545	−0.0734631	−0.0367315	+	0.999325i	$0.511695\pi$
$572$	0	0
$573$	−25.1275	−1.04972
$574$	0	0
$575$	−5.61230	−0.234049
$576$	0	0
$577$	28.1399	1.17148	0.585740	−	0.810499i	$-0.300804\pi$
$577$	28.1399	1.17148	0.585740	+	0.810499i	$0.300804\pi$
$578$	0	0
$579$	−8.81629	−0.366392
$580$	0	0
$581$	40.5243	1.68123
$582$	0	0
$583$	29.2252	1.21038
$584$	0	0
$585$	2.05949	0.0851496
$586$	0	0
$587$	14.6703	0.605508	0.302754	−	0.953069i	$-0.402094\pi$
$587$	14.6703	0.605508	0.302754	+	0.953069i	$0.402094\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	14.1664	0.582728
$592$	0	0
$593$	10.3349	0.424405	0.212203	−	0.977226i	$-0.431936\pi$
$593$	10.3349	0.424405	0.212203	+	0.977226i	$0.431936\pi$
$594$	0	0
$595$	−3.74712	−0.153617
$596$	0	0
$597$	4.74461	0.194184
$598$	0	0
$599$	−29.6793	−1.21266	−0.606331	−	0.795212i	$-0.707359\pi$
$599$	−29.6793	−1.21266	−0.606331	+	0.795212i	$0.707359\pi$
$600$	0	0
$601$	17.1516	0.699628	0.349814	−	0.936819i	$-0.386245\pi$
$601$	17.1516	0.699628	0.349814	+	0.936819i	$0.386245\pi$
$602$	0	0
$603$	7.71983	0.314376
$604$	0	0
$605$	−2.30606	−0.0937546
$606$	0	0
$607$	46.3567	1.88156	0.940780	−	0.339016i	$-0.110094\pi$
$607$	46.3567	1.88156	0.940780	+	0.339016i	$0.110094\pi$
$608$	0	0
$609$	5.13732	0.208175
$610$	0	0
$611$	−27.1383	−1.09790
$612$	0	0
$613$	44.2860	1.78870	0.894348	−	0.447373i	$-0.147640\pi$
$613$	44.2860	1.78870	0.894348	+	0.447373i	$0.147640\pi$
$614$	0	0
$615$	−0.175493	−0.00707656
$616$	0	0
$617$	42.9658	1.72974	0.864868	−	0.501999i	$-0.167402\pi$
$617$	42.9658	1.72974	0.864868	+	0.501999i	$0.167402\pi$
$618$	0	0
$619$	−42.5071	−1.70850	−0.854251	−	0.519860i	$-0.825984\pi$
$619$	−42.5071	−1.70850	−0.854251	+	0.519860i	$0.825984\pi$
$620$	0	0
$621$	1.22471	0.0491459
$622$	0	0
$623$	−25.5003	−1.02165
$624$	0	0
$625$	18.9126	0.756504
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14.9239	−0.595056
$630$	0	0
$631$	−14.8469	−0.591044	−0.295522	−	0.955336i	$-0.595494\pi$
$631$	−14.8469	−0.591044	−0.295522	+	0.955336i	$0.595494\pi$
$632$	0	0
$633$	−18.4301	−0.732532
$634$	0	0
$635$	−7.06835	−0.280499
$636$	0	0
$637$	19.7551	0.782725
$638$	0	0
$639$	4.31164	0.170566
$640$	0	0
$641$	40.4858	1.59909	0.799547	−	0.600603i	$-0.205073\pi$
$641$	40.4858	1.59909	0.799547	+	0.600603i	$0.205073\pi$
$642$	0	0
$643$	−45.0711	−1.77743	−0.888716	−	0.458458i	$-0.848402\pi$
$643$	−45.0711	−1.77743	−0.888716	+	0.458458i	$0.848402\pi$
$644$	0	0
$645$	0.245520	0.00966735
$646$	0	0
$647$	−4.25397	−0.167241	−0.0836205	−	0.996498i	$-0.526648\pi$
$647$	−4.25397	−0.167241	−0.0836205	+	0.996498i	$0.526648\pi$
$648$	0	0
$649$	37.2472	1.46208
$650$	0	0
$651$	−14.6623	−0.574660
$652$	0	0
$653$	−6.03091	−0.236008	−0.118004	−	0.993013i	$-0.537650\pi$
$653$	−6.03091	−0.236008	−0.118004	+	0.993013i	$0.537650\pi$
$654$	0	0
$655$	−11.5866	−0.452725
$656$	0	0
$657$	−7.98582	−0.311557
$658$	0	0
$659$	−27.4532	−1.06943	−0.534713	−	0.845034i	$-0.679580\pi$
$659$	−27.4532	−1.06943	−0.534713	+	0.845034i	$0.679580\pi$
$660$	0	0
$661$	11.3643	0.442018	0.221009	−	0.975272i	$-0.429065\pi$
$661$	11.3643	0.442018	0.221009	+	0.975272i	$0.429065\pi$
$662$	0	0
$663$	−5.08877	−0.197631
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.73190	0.0670594
$668$	0	0
$669$	−11.0372	−0.426723
$670$	0	0
$671$	−17.3215	−0.668690
$672$	0	0
$673$	−2.58273	−0.0995571	−0.0497786	−	0.998760i	$-0.515852\pi$
$673$	−2.58273	−0.0995571	−0.0497786	+	0.998760i	$0.515852\pi$
$674$	0	0
$675$	−4.58256	−0.176383
$676$	0	0
$677$	−44.4198	−1.70719	−0.853596	−	0.520935i	$-0.825583\pi$
$677$	−44.4198	−1.70719	−0.853596	+	0.520935i	$0.825583\pi$
$678$	0	0
$679$	−69.4137	−2.66385
$680$	0	0
$681$	−10.1730	−0.389832
$682$	0	0
$683$	−8.91746	−0.341217	−0.170608	−	0.985339i	$-0.554573\pi$
$683$	−8.91746	−0.341217	−0.170608	+	0.985339i	$0.554573\pi$
$684$	0	0
$685$	1.95296	0.0746188
$686$	0	0
$687$	−11.9151	−0.454590
$688$	0	0
$689$	34.1743	1.30194
$690$	0	0
$691$	31.2643	1.18935	0.594674	−	0.803967i	$-0.297281\pi$
$691$	31.2643	1.18935	0.594674	+	0.803967i	$0.297281\pi$
$692$	0	0
$693$	9.90294	0.376182
$694$	0	0
$695$	−10.2147	−0.387464
$696$	0	0
$697$	0.433623	0.0164246
$698$	0	0
$699$	20.4157	0.772193
$700$	0	0
$701$	−5.48950	−0.207335	−0.103668	−	0.994612i	$-0.533058\pi$
$701$	−5.48950	−0.207335	−0.103668	+	0.994612i	$0.533058\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−5.50075	−0.207170
$706$	0	0
$707$	10.3498	0.389244
$708$	0	0
$709$	−28.2687	−1.06165	−0.530827	−	0.847480i	$-0.678119\pi$
$709$	−28.2687	−1.06165	−0.530827	+	0.847480i	$0.678119\pi$
$710$	0	0
$711$	1.75902	0.0659682
$712$	0	0
$713$	−4.94297	−0.185116
$714$	0	0
$715$	5.61407	0.209954
$716$	0	0
$717$	28.3921	1.06032
$718$	0	0
$719$	22.0636	0.822834	0.411417	−	0.911447i	$-0.365034\pi$
$719$	22.0636	0.822834	0.411417	+	0.911447i	$0.365034\pi$
$720$	0	0
$721$	26.6773	0.993514
$722$	0	0
$723$	−11.5920	−0.431112
$724$	0	0
$725$	−6.48034	−0.240674
$726$	0	0
$727$	39.9189	1.48051	0.740254	−	0.672327i	$-0.234705\pi$
$727$	39.9189	1.48051	0.740254	+	0.672327i	$0.234705\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.606652	−0.0224378
$732$	0	0
$733$	−40.4529	−1.49416	−0.747081	−	0.664733i	$-0.768545\pi$
$733$	−40.4529	−1.49416	−0.747081	+	0.664733i	$0.768545\pi$
$734$	0	0
$735$	4.00422	0.147698
$736$	0	0
$737$	21.0439	0.775161
$738$	0	0
$739$	−41.3245	−1.52015	−0.760074	−	0.649837i	$-0.774837\pi$
$739$	−41.3245	−1.52015	−0.760074	+	0.649837i	$0.774837\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.9255	0.804368	0.402184	−	0.915559i	$-0.368251\pi$
$743$	21.9255	0.804368	0.402184	+	0.915559i	$0.368251\pi$
$744$	0	0
$745$	1.33160	0.0487859
$746$	0	0
$747$	11.1550	0.408140
$748$	0	0
$749$	45.4843	1.66196
$750$	0	0
$751$	−34.5296	−1.26000	−0.630002	−	0.776593i	$-0.716946\pi$
$751$	−34.5296	−1.26000	−0.630002	+	0.776593i	$0.716946\pi$
$752$	0	0
$753$	16.6863	0.608084
$754$	0	0
$755$	8.98857	0.327128
$756$	0	0
$757$	−31.1501	−1.13217	−0.566085	−	0.824347i	$-0.691543\pi$
$757$	−31.1501	−1.13217	−0.566085	+	0.824347i	$0.691543\pi$
$758$	0	0
$759$	3.33849	0.121180
$760$	0	0
$761$	5.72463	0.207518	0.103759	−	0.994602i	$-0.466913\pi$
$761$	5.72463	0.207518	0.103759	+	0.994602i	$0.466913\pi$
$762$	0	0
$763$	17.7749	0.643496
$764$	0	0
$765$	−1.03146	−0.0372924
$766$	0	0
$767$	43.5548	1.57267
$768$	0	0
$769$	4.72383	0.170346	0.0851728	−	0.996366i	$-0.472856\pi$
$769$	4.72383	0.170346	0.0851728	+	0.996366i	$0.472856\pi$
$770$	0	0
$771$	1.14835	0.0413569
$772$	0	0
$773$	27.8029	1.00000	0.500000	−	0.866026i	$-0.333333\pi$
$773$	27.8029	1.00000	0.500000	+	0.866026i	$0.333333\pi$
$774$	0	0
$775$	18.4954	0.664373
$776$	0	0
$777$	33.9608	1.21834
$778$	0	0
$779$	0	0
$780$	0	0
$781$	11.7533	0.420566
$782$	0	0
$783$	1.41413	0.0505370
$784$	0	0
$785$	5.88085	0.209896
$786$	0	0
$787$	−15.5857	−0.555572	−0.277786	−	0.960643i	$-0.589601\pi$
$787$	−15.5857	−0.555572	−0.277786	+	0.960643i	$0.589601\pi$
$788$	0	0
$789$	13.9249	0.495741
$790$	0	0
$791$	29.2305	1.03932
$792$	0	0
$793$	−20.2548	−0.719271
$794$	0	0
$795$	6.92690	0.245672
$796$	0	0
$797$	−21.6702	−0.767597	−0.383798	−	0.923417i	$-0.625384\pi$
$797$	−21.6702	−0.767597	−0.383798	+	0.923417i	$0.625384\pi$
$798$	0	0
$799$	13.5917	0.480840
$800$	0	0
$801$	−7.01939	−0.248018
$802$	0	0
$803$	−21.7689	−0.768209
$804$	0	0
$805$	2.87461	0.101317
$806$	0	0
$807$	31.8422	1.12090
$808$	0	0
$809$	14.5206	0.510517	0.255259	−	0.966873i	$-0.417839\pi$
$809$	14.5206	0.510517	0.255259	+	0.966873i	$0.417839\pi$
$810$	0	0
$811$	15.4923	0.544007	0.272003	−	0.962296i	$-0.412314\pi$
$811$	15.4923	0.544007	0.272003	+	0.962296i	$0.412314\pi$
$812$	0	0
$813$	−22.8758	−0.802290
$814$	0	0
$815$	7.61518	0.266748
$816$	0	0
$817$	0	0
$818$	0	0
$819$	11.5800	0.404636
$820$	0	0
$821$	15.5937	0.544223	0.272111	−	0.962266i	$-0.412278\pi$
$821$	15.5937	0.544223	0.272111	+	0.962266i	$0.412278\pi$
$822$	0	0
$823$	−1.02581	−0.0357574	−0.0178787	−	0.999840i	$-0.505691\pi$
$823$	−1.02581	−0.0357574	−0.0178787	+	0.999840i	$0.505691\pi$
$824$	0	0
$825$	−12.4918	−0.434909
$826$	0	0
$827$	24.5373	0.853246	0.426623	−	0.904430i	$-0.359703\pi$
$827$	24.5373	0.853246	0.426623	+	0.904430i	$0.359703\pi$
$828$	0	0
$829$	−46.3855	−1.61103	−0.805517	−	0.592573i	$-0.798112\pi$
$829$	−46.3855	−1.61103	−0.805517	+	0.592573i	$0.798112\pi$
$830$	0	0
$831$	6.21594	0.215629
$832$	0	0
$833$	−9.89396	−0.342805
$834$	0	0
$835$	4.79947	0.166092
$836$	0	0
$837$	−4.03604	−0.139506
$838$	0	0
$839$	−14.8378	−0.512257	−0.256128	−	0.966643i	$-0.582447\pi$
$839$	−14.8378	−0.512257	−0.256128	+	0.966643i	$0.582447\pi$
$840$	0	0
$841$	−27.0002	−0.931042
$842$	0	0
$843$	−1.61885	−0.0557561
$844$	0	0
$845$	−1.83450	−0.0631088
$846$	0	0
$847$	−12.9663	−0.445528
$848$	0	0
$849$	−25.8756	−0.888049
$850$	0	0
$851$	11.4489	0.392464
$852$	0	0
$853$	−13.6977	−0.469000	−0.234500	−	0.972116i	$-0.575345\pi$
$853$	−13.6977	−0.469000	−0.234500	+	0.972116i	$0.575345\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.32069	0.0792732	0.0396366	−	0.999214i	$-0.487380\pi$
$857$	2.32069	0.0792732	0.0396366	+	0.999214i	$0.487380\pi$
$858$	0	0
$859$	−0.771028	−0.0263071	−0.0131536	−	0.999913i	$-0.504187\pi$
$859$	−0.771028	−0.0263071	−0.0131536	+	0.999913i	$0.504187\pi$
$860$	0	0
$861$	−0.986749	−0.0336283
$862$	0	0
$863$	−39.0649	−1.32979	−0.664893	−	0.746939i	$-0.731523\pi$
$863$	−39.0649	−1.32979	−0.664893	+	0.746939i	$0.731523\pi$
$864$	0	0
$865$	−13.8299	−0.470230
$866$	0	0
$867$	−14.4514	−0.490795
$868$	0	0
$869$	4.79499	0.162659
$870$	0	0
$871$	24.6076	0.833795
$872$	0	0
$873$	−19.1073	−0.646683
$874$	0	0
$875$	−22.4920	−0.760367
$876$	0	0
$877$	57.8572	1.95370	0.976849	−	0.213931i	$-0.0686269\pi$
$877$	57.8572	1.95370	0.976849	+	0.213931i	$0.0686269\pi$
$878$	0	0
$879$	−23.2969	−0.785784
$880$	0	0
$881$	−45.3205	−1.52689	−0.763443	−	0.645875i	$-0.776493\pi$
$881$	−45.3205	−1.52689	−0.763443	+	0.645875i	$0.776493\pi$
$882$	0	0
$883$	−52.3958	−1.76326	−0.881629	−	0.471943i	$-0.843553\pi$
$883$	−52.3958	−1.76326	−0.881629	+	0.471943i	$0.843553\pi$
$884$	0	0
$885$	8.82826	0.296759
$886$	0	0
$887$	−51.3048	−1.72265	−0.861324	−	0.508057i	$-0.830364\pi$
$887$	−51.3048	−1.72265	−0.861324	+	0.508057i	$0.830364\pi$
$888$	0	0
$889$	−39.7434	−1.33295
$890$	0	0
$891$	2.72595	0.0913227
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−13.4054	−0.448094
$896$	0	0
$897$	3.90385	0.130346
$898$	0	0
$899$	−5.70749	−0.190355
$900$	0	0
$901$	−17.1156	−0.570202
$902$	0	0
$903$	1.38049	0.0459399
$904$	0	0
$905$	9.53987	0.317116
$906$	0	0
$907$	6.42523	0.213346	0.106673	−	0.994294i	$-0.465980\pi$
$907$	6.42523	0.213346	0.106673	+	0.994294i	$0.465980\pi$
$908$	0	0
$909$	2.84895	0.0944938
$910$	0	0
$911$	19.1355	0.633989	0.316994	−	0.948427i	$-0.397326\pi$
$911$	19.1355	0.633989	0.316994	+	0.948427i	$0.397326\pi$
$912$	0	0
$913$	30.4079	1.00636
$914$	0	0
$915$	−4.10552	−0.135724
$916$	0	0
$917$	−65.1481	−2.15138
$918$	0	0
$919$	−38.5266	−1.27087	−0.635437	−	0.772153i	$-0.719180\pi$
$919$	−38.5266	−1.27087	−0.635437	+	0.772153i	$0.719180\pi$
$920$	0	0
$921$	−2.22893	−0.0734456
$922$	0	0
$923$	13.7437	0.452379
$924$	0	0
$925$	−42.8390	−1.40854
$926$	0	0
$927$	7.34337	0.241188
$928$	0	0
$929$	47.2779	1.55114	0.775569	−	0.631263i	$-0.217463\pi$
$929$	47.2779	1.55114	0.775569	+	0.631263i	$0.217463\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15.2543	0.499404
$934$	0	0
$935$	−2.81170	−0.0919525
$936$	0	0
$937$	41.0738	1.34182	0.670911	−	0.741538i	$-0.265903\pi$
$937$	41.0738	1.34182	0.670911	+	0.741538i	$0.265903\pi$
$938$	0	0
$939$	−2.08519	−0.0680475
$940$	0	0
$941$	3.23253	0.105377	0.0526887	−	0.998611i	$-0.483221\pi$
$941$	3.23253	0.105377	0.0526887	+	0.998611i	$0.483221\pi$
$942$	0	0
$943$	−0.332654	−0.0108327
$944$	0	0
$945$	2.34718	0.0763537
$946$	0	0
$947$	30.2542	0.983130	0.491565	−	0.870841i	$-0.336425\pi$
$947$	30.2542	0.983130	0.491565	+	0.870841i	$0.336425\pi$
$948$	0	0
$949$	−25.4554	−0.826318
$950$	0	0
$951$	−28.3128	−0.918106
$952$	0	0
$953$	−4.86253	−0.157513	−0.0787565	−	0.996894i	$-0.525095\pi$
$953$	−4.86253	−0.157513	−0.0787565	+	0.996894i	$0.525095\pi$
$954$	0	0
$955$	−16.2349	−0.525348
$956$	0	0
$957$	3.85485	0.124610
$958$	0	0
$959$	10.9810	0.354594
$960$	0	0
$961$	−14.7104	−0.474529
$962$	0	0
$963$	12.5203	0.403461
$964$	0	0
$965$	−5.69620	−0.183367
$966$	0	0
$967$	−29.1640	−0.937851	−0.468925	−	0.883238i	$-0.655359\pi$
$967$	−29.1640	−0.937851	−0.468925	+	0.883238i	$0.655359\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.2360	1.03450	0.517251	−	0.855833i	$-0.326955\pi$
$971$	32.2360	1.03450	0.517251	+	0.855833i	$0.326955\pi$
$972$	0	0
$973$	−57.4342	−1.84126
$974$	0	0
$975$	−14.6072	−0.467806
$976$	0	0
$977$	23.7107	0.758573	0.379287	−	0.925279i	$-0.376169\pi$
$977$	23.7107	0.758573	0.379287	+	0.925279i	$0.376169\pi$
$978$	0	0
$979$	−19.1345	−0.611541
$980$	0	0
$981$	4.89285	0.156217
$982$	0	0
$983$	−23.1634	−0.738797	−0.369398	−	0.929271i	$-0.620436\pi$
$983$	−23.1634	−0.738797	−0.369398	+	0.929271i	$0.620436\pi$
$984$	0	0
$985$	9.15291	0.291636
$986$	0	0
$987$	−30.9292	−0.984487
$988$	0	0
$989$	0.465394	0.0147987
$990$	0	0
$991$	−10.4383	−0.331583	−0.165792	−	0.986161i	$-0.553018\pi$
$991$	−10.4383	−0.331583	−0.165792	+	0.986161i	$0.553018\pi$
$992$	0	0
$993$	8.76994	0.278306
$994$	0	0
$995$	3.06549	0.0971827
$996$	0	0
$997$	47.4368	1.50234	0.751169	−	0.660110i	$-0.229490\pi$
$997$	47.4368	1.50234	0.751169	+	0.660110i	$0.229490\pi$
$998$	0	0
$999$	9.34827	0.295766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8664.2.a.bq.1.6		9
19.3	odd	18		456.2.bg.c.313.2	yes	18
19.13	odd	18		456.2.bg.c.169.2	✓	18
19.18	odd	2		8664.2.a.bo.1.6		9
76.3	even	18		912.2.bo.k.769.2		18
76.51	even	18		912.2.bo.k.625.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bg.c.169.2	✓	18	19.13	odd	18
456.2.bg.c.313.2	yes	18	19.3	odd	18
912.2.bo.k.625.2		18	76.51	even	18
912.2.bo.k.769.2		18	76.3	even	18
8664.2.a.bo.1.6		9	19.18	odd	2
8664.2.a.bq.1.6		9	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8664.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.646100 q^{5} +3.63284 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.646100 q^{5} +3.63284 q^{7} +1.00000 q^{9} +2.72595 q^{11} +3.18758 q^{13} +0.646100 q^{15} -1.59644 q^{17} +3.63284 q^{21} +1.22471 q^{23} -4.58256 q^{25} +1.00000 q^{27} +1.41413 q^{29} -4.03604 q^{31} +2.72595 q^{33} +2.34718 q^{35} +9.34827 q^{37} +3.18758 q^{39} -0.271619 q^{41} +0.380004 q^{43} +0.646100 q^{45} -8.51378 q^{47} +6.19752 q^{49} -1.59644 q^{51} +10.7211 q^{53} +1.76124 q^{55} +13.6639 q^{59} -6.35431 q^{61} +3.63284 q^{63} +2.05949 q^{65} +7.71983 q^{67} +1.22471 q^{69} +4.31164 q^{71} -7.98582 q^{73} -4.58256 q^{75} +9.90294 q^{77} +1.75902 q^{79} +1.00000 q^{81} +11.1550 q^{83} -1.03146 q^{85} +1.41413 q^{87} -7.01939 q^{89} +11.5800 q^{91} -4.03604 q^{93} -19.1073 q^{97} +2.72595 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 3 q^{5} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 + 3 * q^5 + 9 * q^9	\( 9 q + 9 q^{3} + 3 q^{5} + 9 q^{9} + 3 q^{11} + 3 q^{13} + 3 q^{15} + 15 q^{17} - 6 q^{23} + 6 q^{25} + 9 q^{27} + 21 q^{29} + 21 q^{31} + 3 q^{33} - 12 q^{35} + 15 q^{37} + 3 q^{39} + 15 q^{41} - 6 q^{43} + 3 q^{45} + 6 q^{47} + 21 q^{49} + 15 q^{51} + 21 q^{53} - 48 q^{55} + 9 q^{59} + 30 q^{61} - 9 q^{65} + 30 q^{67} - 6 q^{69} + 30 q^{71} - 3 q^{73} + 6 q^{75} + 15 q^{77} + 6 q^{79} + 9 q^{81} + 30 q^{83} - 21 q^{85} + 21 q^{87} - 9 q^{89} + 21 q^{91} + 21 q^{93} + 15 q^{97} + 3 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + 3 * q^5 + 9 * q^9 + 3 * q^11 + 3 * q^13 + 3 * q^15 + 15 * q^17 - 6 * q^23 + 6 * q^25 + 9 * q^27 + 21 * q^29 + 21 * q^31 + 3 * q^33 - 12 * q^35 + 15 * q^37 + 3 * q^39 + 15 * q^41 - 6 * q^43 + 3 * q^45 + 6 * q^47 + 21 * q^49 + 15 * q^51 + 21 * q^53 - 48 * q^55 + 9 * q^59 + 30 * q^61 - 9 * q^65 + 30 * q^67 - 6 * q^69 + 30 * q^71 - 3 * q^73 + 6 * q^75 + 15 * q^77 + 6 * q^79 + 9 * q^81 + 30 * q^83 - 21 * q^85 + 21 * q^87 - 9 * q^89 + 21 * q^91 + 21 * q^93 + 15 * q^97 + 3 * q^99

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